LMFDB - Embedded newform 108.4.b.a.107.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(107,108)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("108.107");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.37220628062$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{10} + 112x^{8} - 368x^{6} + 928x^{4} - 256x^{2} + 64 $ x^12 - 12x^10 + 112x^8 - 368x^6 + 928x^4 - 256*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			107.8
Root			$-2.48442 - 1.43438i$ of defining polynomial
Character	$\chi$	$=$	108.107
Dual form			108.4.b.a.107.7

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.889241 + 2.68500i) q^{2} +(-6.41850 + 4.77523i) q^{4} -14.9230i q^{5} -30.0528i q^{7} +(-18.5291 - 12.9874i) q^{8} +O(q^{10})$	\(q+(0.889241 + 2.68500i) q^{2} +(-6.41850 + 4.77523i) q^{4} -14.9230i q^{5} -30.0528i q^{7} +(-18.5291 - 12.9874i) q^{8} +(40.0683 - 13.2701i) q^{10} -55.9380 q^{11} +57.4627 q^{13} +(80.6918 - 26.7241i) q^{14} +(18.3943 - 61.2997i) q^{16} +29.2840i q^{17} +0.709738i q^{19} +(71.2608 + 95.7833i) q^{20} +(-49.7423 - 150.194i) q^{22} -48.0368 q^{23} -97.6960 q^{25} +(51.0981 + 154.288i) q^{26} +(143.509 + 192.894i) q^{28} -172.964i q^{29} -45.2268i q^{31} +(180.947 - 5.12130i) q^{32} +(-78.6277 + 26.0405i) q^{34} -448.477 q^{35} +248.625 q^{37} +(-1.90565 + 0.631128i) q^{38} +(-193.811 + 276.510i) q^{40} +51.3323i q^{41} -19.9660i q^{43} +(359.038 - 267.117i) q^{44} +(-42.7163 - 128.979i) q^{46} +10.8215 q^{47} -560.168 q^{49} +(-86.8753 - 262.314i) q^{50} +(-368.824 + 274.398i) q^{52} -37.0817i q^{53} +834.763i q^{55} +(-390.306 + 556.851i) q^{56} +(464.410 - 153.807i) q^{58} -411.262 q^{59} -308.855 q^{61} +(121.434 - 40.2175i) q^{62} +(174.656 + 481.289i) q^{64} -857.516i q^{65} -113.616i q^{67} +(-139.838 - 187.959i) q^{68} +(-398.804 - 1204.16i) q^{70} +1134.56 q^{71} +728.560 q^{73} +(221.088 + 667.560i) q^{74} +(-3.38916 - 4.55545i) q^{76} +1681.09i q^{77} -487.025i q^{79} +(-914.775 - 274.499i) q^{80} +(-137.827 + 45.6468i) q^{82} +1165.71 q^{83} +437.005 q^{85} +(53.6088 - 17.7546i) q^{86} +(1036.48 + 726.488i) q^{88} +1198.68i q^{89} -1726.91i q^{91} +(308.324 - 229.387i) q^{92} +(9.62289 + 29.0557i) q^{94} +10.5914 q^{95} +624.472 q^{97} +(-498.124 - 1504.05i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4}+O(q^{10}) $ 12 * q - 12 * q^4	$ 12 q - 12 q^{4} + 24 q^{10} + 36 q^{13} + 24 q^{16} + 120 q^{22} - 132 q^{25} + 420 q^{28} - 360 q^{34} + 516 q^{37} - 1152 q^{40} - 696 q^{46} - 720 q^{49} + 204 q^{52} + 2832 q^{58} - 972 q^{61} + 2496 q^{64} - 1848 q^{70} + 660 q^{73} - 5004 q^{76} - 3888 q^{82} + 1056 q^{85} + 3168 q^{88} + 7608 q^{94} + 2532 q^{97}+O(q^{100}) $ 12 * q - 12 * q^4 + 24 * q^10 + 36 * q^13 + 24 * q^16 + 120 * q^22 - 132 * q^25 + 420 * q^28 - 360 * q^34 + 516 * q^37 - 1152 * q^40 - 696 * q^46 - 720 * q^49 + 204 * q^52 + 2832 * q^58 - 972 * q^61 + 2496 * q^64 - 1848 * q^70 + 660 * q^73 - 5004 * q^76 - 3888 * q^82 + 1056 * q^85 + 3168 * q^88 + 7608 * q^94 + 2532 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.889241	+	2.68500i	0.314394	+	0.949293i
$2$	0.889241	+	2.68500i	0.314394	+	0.949293i
$3$		0			0
$3$		0			0
$4$	−6.41850	+	4.77523i	−0.802313	+	0.596904i
$5$		−	14.9230i		−	1.33475i	−0.744720	−	0.667377i	$-0.767417\pi$
$5$		−	14.9230i		−	1.33475i	0.744720	−	0.667377i	$-0.232583\pi$
$6$		0			0
$7$		−	30.0528i		−	1.62270i	−0.584563	−	0.811348i	$-0.698734\pi$
$7$		−	30.0528i		−	1.62270i	0.584563	−	0.811348i	$-0.301266\pi$
$8$	−18.5291	−	12.9874i	−0.818879	−	0.573966i
$9$		0			0
$10$	40.0683	−	13.2701i	1.26707	−	0.419639i
$11$	−55.9380			−1.53327			−0.766634	−	0.642085i	$-0.778070\pi$
$11$	−55.9380			−1.53327			−0.766634	+	0.642085i	$0.778070\pi$
$12$		0			0
$13$	57.4627			1.22594			0.612972	−	0.790105i	$-0.289974\pi$
$13$	57.4627			1.22594			0.612972	+	0.790105i	$0.289974\pi$
$14$	80.6918	−	26.7241i	1.54041	−	0.510166i
$15$		0			0
$16$	18.3943	−	61.2997i	0.287411	−	0.957807i
$17$			29.2840i			0.417789i	0.977938	+	0.208895i	$0.0669865\pi$
$17$			29.2840i			0.417789i	−0.977938	+	0.208895i	$0.933013\pi$
$18$		0			0
$19$			0.709738i			0.00856974i	0.999991	+	0.00428487i	$0.00136392\pi$
$19$			0.709738i			0.00856974i	−0.999991	+	0.00428487i	$0.998636\pi$
$20$	71.2608	+	95.7833i	0.796720	+	1.07089i
$21$		0			0
$22$	−49.7423	−	150.194i	−0.482050	−	1.45552i
$23$	−48.0368			−0.435494			−0.217747	−	0.976005i	$-0.569871\pi$
$23$	−48.0368			−0.435494			−0.217747	+	0.976005i	$0.569871\pi$
$24$		0			0
$25$	−97.6960			−0.781568
$26$	51.0981	+	154.288i	0.385430	+	1.16378i
$27$		0			0
$28$	143.509	+	192.894i	0.968594	+	1.30191i
$29$		−	172.964i		−	1.10754i	−0.832670	−	0.553770i	$-0.813189\pi$
$29$		−	172.964i		−	1.10754i	0.832670	−	0.553770i	$-0.186811\pi$
$30$		0			0
$31$		−	45.2268i		−	0.262031i	−0.991380	−	0.131016i	$-0.958176\pi$
$31$		−	45.2268i		−	0.262031i	0.991380	−	0.131016i	$-0.0418238\pi$
$32$	180.947	−	5.12130i	0.999600	−	0.0282914i
$33$		0			0
$34$	−78.6277	+	26.0405i	−0.396604	+	0.131350i
$35$	−448.477			−2.16590
$36$		0			0
$37$	248.625			1.10470			0.552348	−	0.833613i	$-0.313732\pi$
$37$	248.625			1.10470			0.552348	+	0.833613i	$0.313732\pi$
$38$	−1.90565	+	0.631128i	−0.00813519	+	0.00269427i
$39$		0			0
$40$	−193.811	+	276.510i	−0.766104	+	1.09300i
$41$			51.3323i			0.195531i	0.995209	+	0.0977653i	$0.0311695\pi$
$41$			51.3323i			0.195531i	−0.995209	+	0.0977653i	$0.968831\pi$
$42$		0			0
$43$		−	19.9660i		−	0.0708090i	−0.999373	−	0.0354045i	$-0.988728\pi$
$43$		−	19.9660i		−	0.0708090i	0.999373	−	0.0354045i	$-0.0112720\pi$
$44$	359.038	−	267.117i	1.23016	−	0.915213i
$45$		0			0
$46$	−42.7163	−	128.979i	−0.136917	−	0.413411i
$47$	10.8215			0.0335845			0.0167923	−	0.999859i	$-0.494655\pi$
$47$	10.8215			0.0335845			0.0167923	+	0.999859i	$0.494655\pi$
$48$		0			0
$49$	−560.168			−1.63314
$50$	−86.8753	−	262.314i	−0.245720	−	0.741937i
$51$		0			0
$52$	−368.824	+	274.398i	−0.983591	+	0.731771i
$53$		−	37.0817i		−	0.0961051i	−0.998845	−	0.0480525i	$-0.984699\pi$
$53$		−	37.0817i		−	0.0961051i	0.998845	−	0.0480525i	$-0.0153015\pi$
$54$		0			0
$55$			834.763i			2.04653i
$56$	−390.306	+	556.851i	−0.931373	+	1.32879i
$57$		0			0
$58$	464.410	−	153.807i	1.05138	−	0.348204i
$59$	−411.262			−0.907487			−0.453744	−	0.891132i	$-0.649912\pi$
$59$	−411.262			−0.907487			−0.453744	+	0.891132i	$0.649912\pi$
$60$		0			0
$61$	−308.855			−0.648275			−0.324138	−	0.946010i	$-0.605074\pi$
$61$	−308.855			−0.648275			−0.324138	+	0.946010i	$0.605074\pi$
$62$	121.434	−	40.2175i	0.248744	−	0.0823811i
$63$		0			0
$64$	174.656	+	481.289i	0.341125	+	0.940018i
$65$		−	857.516i		−	1.63633i
$66$		0			0
$67$		−	113.616i		−	0.207170i	−0.994621	−	0.103585i	$-0.966969\pi$
$67$		−	113.616i		−	0.207170i	0.994621	−	0.103585i	$-0.0330314\pi$
$68$	−139.838	−	187.959i	−0.249380	−	0.335197i
$69$		0			0
$70$	−398.804	−	1204.16i	−0.680946	−	2.05607i
$71$	1134.56			1.89645			0.948224	−	0.317603i	$-0.102878\pi$
$71$	1134.56			1.89645			0.948224	+	0.317603i	$0.102878\pi$
$72$		0			0
$73$	728.560			1.16810			0.584051	−	0.811717i	$-0.301467\pi$
$73$	728.560			1.16810			0.584051	+	0.811717i	$0.301467\pi$
$74$	221.088	+	667.560i	0.347310	+	1.04868i
$75$		0			0
$76$	−3.38916	−	4.55545i	−0.00511531	−	0.00687561i
$77$			1681.09i			2.48803i
$78$		0			0
$79$		−	487.025i		−	0.693602i	−0.937939	−	0.346801i	$-0.887268\pi$
$79$		−	487.025i		−	0.693602i	0.937939	−	0.346801i	$-0.112732\pi$
$80$	−914.775	−	274.499i	−1.27844	−	0.383623i
$81$		0			0
$82$	−137.827	+	45.6468i	−0.185616	+	0.0614737i
$83$	1165.71			1.54161			0.770803	−	0.637074i	$-0.219855\pi$
$83$	1165.71			1.54161			0.770803	+	0.637074i	$0.219855\pi$
$84$		0			0
$85$	437.005			0.557646
$86$	53.6088	−	17.7546i	0.0672184	−	0.0222619i
$87$		0			0
$88$	1036.48	+	726.488i	1.25556	+	0.880044i
$89$			1198.68i			1.42764i	0.700332	+	0.713818i	$0.253035\pi$
$89$			1198.68i			1.42764i	−0.700332	+	0.713818i	$0.746965\pi$
$90$		0			0
$91$		−	1726.91i		−	1.98934i
$92$	308.324	−	229.387i	0.349402	−	0.259948i
$93$		0			0
$94$	9.62289	+	29.0557i	0.0105588	+	0.0318815i
$95$	10.5914			0.0114385
$96$		0			0
$97$	624.472			0.653665			0.326833	−	0.945082i	$-0.394019\pi$
$97$	624.472			0.653665			0.326833	+	0.945082i	$0.394019\pi$
$98$	−498.124	−	1504.05i	−0.513450	−	1.55033i
$99$		0			0
$100$	627.062	−	466.521i	0.627062	−	0.466521i
$101$		−	1757.10i		−	1.73107i	−0.500849	−	0.865535i	$-0.666979\pi$
$101$		−	1757.10i		−	1.73107i	0.500849	−	0.865535i	$-0.333021\pi$
$102$		0			0
$103$		−	3.90175i		−	0.00373254i	−0.999998	−	0.00186627i	$-0.999406\pi$
$103$		−	3.90175i		−	0.00373254i	0.999998	−	0.00186627i	$-0.000594052\pi$
$104$	−1064.73	−	746.289i	−1.00390	−	0.703651i
$105$		0			0
$106$	99.5646	−	32.9746i	0.0912318	−	0.0302149i
$107$	77.3509			0.0698859			0.0349429	−	0.999389i	$-0.488875\pi$
$107$	77.3509			0.0698859			0.0349429	+	0.999389i	$0.488875\pi$
$108$		0			0
$109$	−1660.75			−1.45937			−0.729683	−	0.683785i	$-0.760333\pi$
$109$	−1660.75			−1.45937			−0.729683	+	0.683785i	$0.760333\pi$
$110$	−2241.34	+	742.305i	−1.94276	+	0.643418i
$111$		0			0
$112$	−1842.22	−	552.800i	−1.55423	−	0.466381i
$113$		−	253.390i		−	0.210946i	−0.994422	−	0.105473i	$-0.966364\pi$
$113$		−	253.390i		−	0.210946i	0.994422	−	0.105473i	$-0.0336357\pi$
$114$		0			0
$115$			716.853i			0.581277i
$116$	825.945	+	1110.17i	0.661095	+	0.888594i
$117$		0			0
$118$	−365.711	−	1104.24i	−0.285309	−	0.861471i
$119$	880.065			0.677945
$120$		0			0
$121$	1798.06			1.35091
$122$	−274.646	−	829.276i	−0.203814	−	0.615403i
$123$		0			0
$124$	215.968	+	290.288i	0.156408	+	0.210231i
$125$		−	407.457i		−	0.291553i
$126$		0			0
$127$			1429.82i			0.999022i	0.866307	+	0.499511i	$0.166487\pi$
$127$			1429.82i			0.999022i	−0.866307	+	0.499511i	$0.833513\pi$
$128$	−1136.95	+	896.934i	−0.785104	+	0.619364i
$129$		0			0
$130$	2302.43	−	762.538i	1.55336	−	0.514454i
$131$	2037.82			1.35912			0.679560	−	0.733620i	$-0.262171\pi$
$131$	2037.82			1.35912			0.679560	+	0.733620i	$0.262171\pi$
$132$		0			0
$133$	21.3296			0.0139061
$134$	305.059	−	101.032i	0.196665	−	0.0651330i
$135$		0			0
$136$	380.322	−	542.607i	0.239797	−	0.342119i
$137$			804.869i			0.501931i	0.967996	+	0.250966i	$0.0807481\pi$
$137$			804.869i			0.501931i	−0.967996	+	0.250966i	$0.919252\pi$
$138$		0			0
$139$			413.813i			0.252512i	0.991998	+	0.126256i	$0.0402961\pi$
$139$			413.813i			0.252512i	−0.991998	+	0.126256i	$0.959704\pi$
$140$	2878.55	−	2141.58i	1.73773	−	1.29283i
$141$		0			0
$142$	1008.90	+	3046.31i	0.596232	+	1.80028i
$143$	−3214.35			−1.87970
$144$		0			0
$145$	−2581.15			−1.47829
$146$	647.865	+	1956.19i	0.367245	+	1.10887i
$147$		0			0
$148$	−1595.80	+	1187.24i	−0.886312	+	0.659398i
$149$			1459.08i			0.802233i	0.916027	+	0.401116i	$0.131378\pi$
$149$			1459.08i			0.802233i	−0.916027	+	0.401116i	$0.868622\pi$
$150$		0			0
$151$			1668.38i			0.899144i	0.893244	+	0.449572i	$0.148424\pi$
$151$			1668.38i			0.899144i	−0.893244	+	0.449572i	$0.851576\pi$
$152$	9.21763	−	13.1508i	0.00491874	−	0.00701757i
$153$		0			0
$154$	−4513.74	+	1494.89i	−2.36187	+	0.782221i
$155$	−674.920			−0.349747
$156$		0			0
$157$	−1773.81			−0.901691			−0.450846	−	0.892602i	$-0.648877\pi$
$157$	−1773.81			−0.901691			−0.450846	+	0.892602i	$0.648877\pi$
$158$	1307.66	−	433.082i	0.658431	−	0.218064i
$159$		0			0
$160$	−76.4251	−	2700.27i	−0.0377621	−	1.33422i
$161$			1443.64i			0.706674i
$162$		0			0
$163$		−	3549.13i		−	1.70546i	−0.522356	−	0.852728i	$-0.674947\pi$
$163$		−	3549.13i		−	1.70546i	0.522356	−	0.852728i	$-0.325053\pi$
$164$	−245.124	−	329.476i	−0.116713	−	0.156877i
$165$		0			0
$166$	1036.60	+	3129.94i	0.484672	+	1.46343i
$167$	−1888.67			−0.875146			−0.437573	−	0.899183i	$-0.644162\pi$
$167$	−1888.67			−0.875146			−0.437573	+	0.899183i	$0.644162\pi$
$168$		0			0
$169$	1104.96			0.502939
$170$	388.603	+	1173.36i	0.175320	+	0.529369i
$171$		0			0
$172$	95.3422	+	128.152i	0.0422661	+	0.0568109i
$173$		−	2104.10i		−	0.924694i	−0.886699	−	0.462347i	$-0.847008\pi$
$173$		−	2104.10i		−	0.924694i	0.886699	−	0.462347i	$-0.152992\pi$
$174$		0			0
$175$			2936.03i			1.26825i
$176$	−1028.94	+	3428.98i	−0.440678	+	1.46857i
$177$		0			0
$178$	−3218.45	+	1065.91i	−1.35524	+	0.448840i
$179$	1830.10			0.764178			0.382089	−	0.924126i	$-0.375205\pi$
$179$	1830.10			0.764178			0.382089	+	0.924126i	$0.375205\pi$
$180$		0			0
$181$	3333.54			1.36895			0.684475	−	0.729036i	$-0.260032\pi$
$181$	3333.54			1.36895			0.684475	+	0.729036i	$0.260032\pi$
$182$	4636.77	−	1535.64i	1.88846	−	0.625435i
$183$		0			0
$184$	890.079	+	623.872i	0.356617	+	0.249959i
$185$		−	3710.24i		−	1.47450i
$186$		0			0
$187$		−	1638.09i		−	0.640582i
$188$	−69.4576	+	51.6750i	−0.0269453	+	0.0200467i
$189$		0			0
$190$	9.41832	+	28.4380i	0.00359619	+	0.0108585i
$191$	−3622.23			−1.37223			−0.686114	−	0.727494i	$-0.740685\pi$
$191$	−3622.23			−1.37223			−0.686114	+	0.727494i	$0.740685\pi$
$192$		0			0
$193$	−2588.68			−0.965479			−0.482740	−	0.875764i	$-0.660358\pi$
$193$	−2588.68			−0.965479			−0.482740	+	0.875764i	$0.660358\pi$
$194$	555.306	+	1676.71i	0.205508	+	0.620520i
$195$		0			0
$196$	3595.44	−	2674.93i	1.31029	−	0.974829i
$197$		−	1752.24i		−	0.633717i	−0.948473	−	0.316858i	$-0.897372\pi$
$197$		−	1752.24i		−	0.633717i	0.948473	−	0.316858i	$-0.102628\pi$
$198$		0			0
$199$			3316.58i			1.18144i	0.806877	+	0.590719i	$0.201156\pi$
$199$			3316.58i			1.18144i	−0.806877	+	0.590719i	$0.798844\pi$
$200$	1810.22	+	1268.82i	0.640010	+	0.448594i
$201$		0			0
$202$	4717.82	−	1562.49i	1.64329	−	0.544238i
$203$	−5198.05			−1.79720
$204$		0			0
$205$	766.032			0.260985
$206$	10.4762	−	3.46960i	0.00354327	−	0.00117349i
$207$		0			0
$208$	1056.99	−	3522.44i	0.352350	−	1.17422i
$209$		−	39.7013i		−	0.0131397i
$210$		0			0
$211$		−	5960.10i		−	1.94460i	−0.233738	−	0.972300i	$-0.575096\pi$
$211$		−	5960.10i		−	1.94460i	0.233738	−	0.972300i	$-0.424904\pi$
$212$	177.074	+	238.009i	0.0573655	+	0.0771063i
$213$		0			0
$214$	68.7836	+	207.687i	0.0219717	+	0.0663422i
$215$	−297.953			−0.0945125
$216$		0			0
$217$	−1359.19			−0.425197
$218$	−1476.81	−	4459.12i	−0.458816	−	1.38537i
$219$		0			0
$220$	−3986.19	−	5357.93i	−1.22158	−	1.64196i
$221$			1682.74i			0.512186i
$222$		0			0
$223$			1995.23i			0.599150i	0.954073	+	0.299575i	$0.0968449\pi$
$223$			1995.23i			0.599150i	−0.954073	+	0.299575i	$0.903155\pi$
$224$	−153.909	−	5437.95i	−0.0459084	−	1.62205i
$225$		0			0
$226$	680.354	−	225.325i	0.200250	−	0.0663203i
$227$	−2280.12			−0.666682			−0.333341	−	0.942806i	$-0.608176\pi$
$227$	−2280.12			−0.666682			−0.333341	+	0.942806i	$0.608176\pi$
$228$		0			0
$229$	4647.94			1.34124			0.670621	−	0.741800i	$-0.266028\pi$
$229$	4647.94			1.34124			0.670621	+	0.741800i	$0.266028\pi$
$230$	−1924.75	+	637.455i	−0.551802	+	0.182750i
$231$		0			0
$232$	−2246.35	+	3204.88i	−0.635691	+	0.906941i
$233$		−	2824.81i		−	0.794247i	−0.917765	−	0.397124i	$-0.870008\pi$
$233$		−	2824.81i		−	0.794247i	0.917765	−	0.397124i	$-0.129992\pi$
$234$		0			0
$235$		−	161.489i		−	0.0448271i
$236$	2639.69	−	1963.87i	0.728089	−	0.541683i
$237$		0			0
$238$	782.590	+	2362.98i	0.213142	+	0.643568i
$239$	2405.85			0.651136			0.325568	−	0.945519i	$-0.394445\pi$
$239$	2405.85			0.651136			0.325568	+	0.945519i	$0.394445\pi$
$240$		0			0
$241$	−226.108			−0.0604352			−0.0302176	−	0.999543i	$-0.509620\pi$
$241$	−226.108			−0.0604352			−0.0302176	+	0.999543i	$0.509620\pi$
$242$	1598.91	+	4827.80i	0.424718	+	1.28241i
$243$		0			0
$244$	1982.38	−	1474.85i	0.520120	−	0.386958i
$245$			8359.39i			2.17984i
$246$		0			0
$247$			40.7834i			0.0105060i
$248$	−587.377	+	838.012i	−0.150397	+	0.214572i
$249$		0			0
$250$	1094.02	−	362.328i	0.276769	−	0.0916624i
$251$	1071.12			0.269357			0.134679	−	0.990889i	$-0.457000\pi$
$251$	1071.12			0.269357			0.134679	+	0.990889i	$0.457000\pi$
$252$		0			0
$253$	2687.08			0.667729
$254$	−3839.07	+	1271.45i	−0.948365	+	0.314087i
$255$		0			0
$256$	−3419.30	−	2255.13i	−0.834789	−	0.550569i
$257$		−	1844.28i		−	0.447638i	−0.974631	−	0.223819i	$-0.928148\pi$
$257$		−	1844.28i		−	0.447638i	0.974631	−	0.223819i	$-0.0718525\pi$
$258$		0			0
$259$		−	7471.88i		−	1.79259i
$260$	4094.84	+	5503.96i	0.976734	+	1.31285i
$261$		0			0
$262$	1812.11	+	5471.54i	0.427299	+	1.29020i
$263$	7502.45			1.75901			0.879507	−	0.475886i	$-0.157872\pi$
$263$	7502.45			1.75901			0.879507	+	0.475886i	$0.157872\pi$
$264$		0			0
$265$	−553.371			−0.128277
$266$	18.9671	+	57.2700i	0.00437199	+	0.0132009i
$267$		0			0
$268$	542.542	+	729.243i	0.123661	+	0.166215i
$269$		−	3465.21i		−	0.785418i	−0.919663	−	0.392709i	$-0.871538\pi$
$269$		−	3465.21i		−	0.785418i	0.919663	−	0.392709i	$-0.128462\pi$
$270$		0			0
$271$		−	2922.27i		−	0.655038i	−0.944845	−	0.327519i	$-0.893788\pi$
$271$		−	2922.27i		−	0.655038i	0.944845	−	0.327519i	$-0.106212\pi$
$272$	1795.10	+	538.660i	0.400161	+	0.120077i
$273$		0			0
$274$	−2161.08	+	715.722i	−0.476480	+	0.157804i
$275$	5464.92			1.19835
$276$		0			0
$277$	−6644.62			−1.44129			−0.720643	−	0.693306i	$-0.756153\pi$
$277$	−6644.62			−1.44129			−0.720643	+	0.693306i	$0.756153\pi$
$278$	−1111.09	+	367.980i	−0.239708	+	0.0793883i
$279$		0			0
$280$	8309.89	+	5824.54i	1.77361	+	1.24315i
$281$			6612.13i			1.40373i	0.712312	+	0.701863i	$0.247648\pi$
$281$			6612.13i			1.40373i	−0.712312	+	0.701863i	$0.752352\pi$
$282$		0			0
$283$			2658.33i			0.558379i	0.960236	+	0.279190i	$0.0900658\pi$
$283$			2658.33i			0.558379i	−0.960236	+	0.279190i	$0.909934\pi$
$284$	−7282.19	+	5417.80i	−1.52154	+	1.13200i
$285$		0			0
$286$	−2858.33	−	8630.53i	−0.590967	−	1.78439i
$287$	1542.68			0.317287
$288$		0			0
$289$	4055.45			0.825452
$290$	−2295.26	−	6930.39i	−0.464767	−	1.40333i
$291$		0			0
$292$	−4676.26	+	3479.04i	−0.937184	+	0.697245i
$293$		−	5553.60i		−	1.10732i	−0.832743	−	0.553660i	$-0.813231\pi$
$293$		−	5553.60i		−	1.10732i	0.832743	−	0.553660i	$-0.186769\pi$
$294$		0			0
$295$			6137.26i			1.21127i
$296$	−4606.81	−	3228.99i	−0.904613	−	0.634059i
$297$		0			0
$298$	−3917.64	+	1297.48i	−0.761554	+	0.252217i
$299$	−2760.32			−0.533891
$300$		0			0
$301$	−600.033			−0.114901
$302$	−4479.61	+	1483.59i	−0.853551	+	0.282686i
$303$		0			0
$304$	43.5067	+	13.0551i	0.00820815	+	0.00246304i
$305$			4609.04i			0.865288i
$306$		0			0
$307$			8694.95i			1.61644i	0.588880	+	0.808220i	$0.299569\pi$
$307$			8694.95i			1.61644i	−0.588880	+	0.808220i	$0.700431\pi$
$308$	−8027.60	−	10790.1i	−1.48511	−	1.99618i
$309$		0			0
$310$	−600.166	−	1812.16i	−0.109959	−	0.332013i
$311$	6717.09			1.22473			0.612365	−	0.790575i	$-0.290218\pi$
$311$	6717.09			1.22473			0.612365	+	0.790575i	$0.290218\pi$
$312$		0			0
$313$	−1389.06			−0.250844			−0.125422	−	0.992103i	$-0.540028\pi$
$313$	−1389.06			−0.250844			−0.125422	+	0.992103i	$0.540028\pi$
$314$	−1577.34	−	4762.69i	−0.283486	−	0.855969i
$315$		0			0
$316$	2325.66	+	3125.97i	0.414014	+	0.556486i
$317$			1457.63i			0.258260i	0.991628	+	0.129130i	$0.0412185\pi$
$317$			1457.63i			0.258260i	−0.991628	+	0.129130i	$0.958782\pi$
$318$		0			0
$319$			9675.28i			1.69816i
$320$	7182.28	−	2606.39i	1.25469	−	0.455318i
$321$		0			0
$322$	−3876.17	+	1283.74i	−0.670841	+	0.222174i
$323$	−20.7840			−0.00358034
$324$		0			0
$325$	−5613.87			−0.958159
$326$	9529.43	−	3156.03i	1.61898	−	0.536185i
$327$		0			0
$328$	666.672	−	951.142i	0.112228	−	0.160116i
$329$		−	325.215i		−	0.0544975i
$330$		0			0
$331$		−	4966.82i		−	0.824776i	−0.911008	−	0.412388i	$-0.864695\pi$
$331$		−	4966.82i		−	0.824776i	0.911008	−	0.412388i	$-0.135305\pi$
$332$	−7482.11	+	5566.53i	−1.23685	+	0.920190i
$333$		0			0
$334$	−1679.48	−	5071.08i	−0.275141	−	0.830769i
$335$	−1695.49			−0.276521
$336$		0			0
$337$	−4892.90			−0.790899			−0.395450	−	0.918488i	$-0.629411\pi$
$337$	−4892.90			−0.790899			−0.395450	+	0.918488i	$0.629411\pi$
$338$	982.574	+	2966.82i	0.158121	+	0.477437i
$339$		0			0
$340$	−2804.92	+	2086.80i	−0.447406	+	0.332861i
$341$			2529.90i			0.401764i
$342$		0			0
$343$			6526.50i			1.02740i
$344$	−259.306	+	369.952i	−0.0406420	+	0.0579840i
$345$		0			0
$346$	5649.53	−	1871.05i	0.877805	−	0.290718i
$347$	−253.981			−0.0392923			−0.0196461	−	0.999807i	$-0.506254\pi$
$347$	−253.981			−0.0392923			−0.0196461	+	0.999807i	$0.506254\pi$
$348$		0			0
$349$	−6464.21			−0.991465			−0.495732	−	0.868475i	$-0.665100\pi$
$349$	−6464.21			−0.991465			−0.495732	+	0.868475i	$0.665100\pi$
$350$	−7883.27	+	2610.84i	−1.20394	+	0.398730i
$351$		0			0
$352$	−10121.8	+	286.475i	−1.53265	+	0.0433783i
$353$			4033.67i			0.608188i	0.952642	+	0.304094i	$0.0983537\pi$
$353$			4033.67i			0.608188i	−0.952642	+	0.304094i	$0.901646\pi$
$354$		0			0
$355$		−	16931.1i		−	2.53129i
$356$	−5723.96	−	7693.71i	−0.852161	−	1.14541i
$357$		0			0
$358$	1627.40	+	4913.82i	0.240253	+	0.725428i
$359$	10670.4			1.56870			0.784350	−	0.620319i	$-0.212997\pi$
$359$	10670.4			1.56870			0.784350	+	0.620319i	$0.212997\pi$
$360$		0			0
$361$	6858.50			0.999927
$362$	2964.32	+	8950.56i	0.430390	+	1.29953i
$363$		0			0
$364$	8246.40	+	11084.2i	1.18744	+	1.59607i
$365$		−	10872.3i		−	1.55913i
$366$		0			0
$367$			2559.84i			0.364094i	0.983290	+	0.182047i	$0.0582724\pi$
$367$			2559.84i			0.364094i	−0.983290	+	0.182047i	$0.941728\pi$
$368$	−883.604	+	2944.64i	−0.125166	+	0.417119i
$369$		0			0
$370$	9962.01	−	3299.29i	1.39973	−	0.463573i
$371$	−1114.41			−0.155949
$372$		0			0
$373$	1935.92			0.268735			0.134368	−	0.990932i	$-0.457100\pi$
$373$	1935.92			0.268735			0.134368	+	0.990932i	$0.457100\pi$
$374$	4398.28	−	1456.65i	0.608100	−	0.201395i
$375$		0			0
$376$	−200.512	−	140.542i	−0.0275017	−	0.0192764i
$377$		−	9938.99i		−	1.35778i
$378$		0			0
$379$		−	5944.90i		−	0.805722i	−0.915261	−	0.402861i	$-0.868016\pi$
$379$		−	5944.90i		−	0.805722i	0.915261	−	0.402861i	$-0.131984\pi$
$380$	−67.9810	+	50.5765i	−0.00917724	+	0.00682768i
$381$		0			0
$382$	−3221.03	−	9725.71i	−0.431420	−	1.30264i
$383$	1696.80			0.226378			0.113189	−	0.993573i	$-0.463893\pi$
$383$	1696.80			0.226378			0.113189	+	0.993573i	$0.463893\pi$
$384$		0			0
$385$	25086.9			3.32090
$386$	−2301.96	−	6950.63i	−0.303541	−	0.916522i
$387$		0			0
$388$	−4008.17	+	2982.00i	−0.524444	+	0.390175i
$389$			3913.71i			0.510111i	0.966926	+	0.255055i	$0.0820937\pi$
$389$			3913.71i			0.510111i	−0.966926	+	0.255055i	$0.917906\pi$
$390$		0			0
$391$		−	1406.71i		−	0.181945i
$392$	10379.4	+	7275.11i	1.33735	+	0.937369i
$393$		0			0
$394$	4704.78	−	1558.17i	0.601583	−	0.199237i
$395$	−7267.87			−0.925788
$396$		0			0
$397$	5079.98			0.642208			0.321104	−	0.947044i	$-0.395946\pi$
$397$	5079.98			0.642208			0.321104	+	0.947044i	$0.395946\pi$
$398$	−8905.04	+	2949.24i	−1.12153	+	0.371437i
$399$		0			0
$400$	−1797.05	+	5988.73i	−0.224632	+	0.748592i
$401$			9374.44i			1.16742i	0.811961	+	0.583712i	$0.198400\pi$
$401$			9374.44i			1.16742i	−0.811961	+	0.583712i	$0.801600\pi$
$402$		0			0
$403$		−	2598.85i		−	0.321236i
$404$	8390.56	+	11278.0i	1.03328	+	1.38886i
$405$		0			0
$406$	−4622.32	−	13956.8i	−0.565029	−	1.70607i
$407$	−13907.6			−1.69379
$408$		0			0
$409$	−12392.7			−1.49824			−0.749120	−	0.662434i	$-0.769523\pi$
$409$	−12392.7			−1.49824			−0.749120	+	0.662434i	$0.769523\pi$
$410$	681.187	+	2056.80i	0.0820522	+	0.247751i
$411$		0			0
$412$	18.6318	+	25.0434i	0.00222797	+	0.00299466i
$413$			12359.6i			1.47258i
$414$		0			0
$415$		−	17395.9i		−	2.05766i
$416$	10397.7	−	294.283i	1.22545	−	0.0346837i
$417$		0			0
$418$	106.598	−	35.3040i	0.0124734	−	0.00413104i
$419$	266.866			0.0311151			0.0155576	−	0.999879i	$-0.495048\pi$
$419$	266.866			0.0311151			0.0155576	+	0.999879i	$0.495048\pi$
$420$		0			0
$421$	−9952.62			−1.15216			−0.576082	−	0.817392i	$-0.695419\pi$
$421$	−9952.62			−1.15216			−0.576082	+	0.817392i	$0.695419\pi$
$422$	16002.9	−	5299.97i	1.84599	−	0.611371i
$423$		0			0
$424$	−481.595	+	687.092i	−0.0551611	+	0.0786984i
$425$		−	2860.93i		−	0.326531i
$426$		0			0
$427$			9281.94i			1.05195i
$428$	−496.477	+	369.368i	−0.0560703	+	0.0417152i
$429$		0			0
$430$	−264.952	−	800.004i	−0.0297142	−	0.0897200i
$431$	−6583.33			−0.735749			−0.367875	−	0.929875i	$-0.619914\pi$
$431$	−6583.33			−0.735749			−0.367875	+	0.929875i	$0.619914\pi$
$432$		0			0
$433$	−13747.0			−1.52572			−0.762860	−	0.646564i	$-0.776205\pi$
$433$	−13747.0			−1.52572			−0.762860	+	0.646564i	$0.776205\pi$
$434$	−1208.65	−	3649.43i	−0.133679	−	0.403637i
$435$		0			0
$436$	10659.5	−	7930.46i	1.17087	−	0.871102i
$437$		−	34.0935i		−	0.00373207i
$438$		0			0
$439$			13380.0i			1.45466i	0.686290	+	0.727328i	$0.259238\pi$
$439$			13380.0i			1.45466i	−0.686290	+	0.727328i	$0.740762\pi$
$440$	10841.4	−	15467.4i	1.17464	−	1.67586i
$441$		0			0
$442$	−4518.16	+	1496.36i	−0.486214	+	0.161028i
$443$	16108.1			1.72758			0.863791	−	0.503851i	$-0.168084\pi$
$443$	16108.1			1.72758			0.863791	+	0.503851i	$0.168084\pi$
$444$		0			0
$445$	17887.9			1.90554
$446$	−5357.20	+	1774.24i	−0.568769	+	0.188369i
$447$		0			0
$448$	14464.1	−	5248.90i	1.52536	−	0.553542i
$449$		−	11283.5i		−	1.18597i	−0.805214	−	0.592984i	$-0.797950\pi$
$449$		−	11283.5i		−	1.18597i	0.805214	−	0.592984i	$-0.202050\pi$
$450$		0			0
$451$		−	2871.42i		−	0.299801i
$452$	1210.00	+	1626.39i	0.125915	+	0.169245i
$453$		0			0
$454$	−2027.57	−	6122.13i	−0.209601	−	0.632876i
$455$	−25770.7			−2.65527
$456$		0			0
$457$	1984.72			0.203154			0.101577	−	0.994828i	$-0.467611\pi$
$457$	1984.72			0.203154			0.101577	+	0.994828i	$0.467611\pi$
$458$	4133.14	+	12479.7i	0.421679	+	1.27323i
$459$		0			0
$460$	−3423.14	−	4601.12i	−0.346967	−	0.466366i
$461$		−	9634.41i		−	0.973360i	−0.873580	−	0.486680i	$-0.838208\pi$
$461$		−	9634.41i		−	0.973360i	0.873580	−	0.486680i	$-0.161792\pi$
$462$		0			0
$463$			1392.38i			0.139762i	0.997555	+	0.0698808i	$0.0222619\pi$
$463$			1392.38i			0.139762i	−0.997555	+	0.0698808i	$0.977738\pi$
$464$	−10602.7	−	3181.56i	−1.06081	−	0.318320i
$465$		0			0
$466$	7584.64	−	2511.94i	0.753973	−	0.249707i
$467$	−15161.4			−1.50232			−0.751161	−	0.660119i	$-0.770506\pi$
$467$	−15161.4			−1.50232			−0.751161	+	0.660119i	$0.770506\pi$
$468$		0			0
$469$	−3414.47			−0.336174
$470$	433.598	−	143.602i	0.0425540	−	0.0140934i
$471$		0			0
$472$	7620.32	+	5341.22i	0.743122	+	0.520867i
$473$			1116.86i			0.108569i
$474$		0			0
$475$		−	69.3385i		−	0.00669783i
$476$	−5648.70	+	4202.51i	−0.543924	+	0.404668i
$477$		0			0
$478$	2139.38	+	6459.72i	0.204713	+	0.618118i
$479$	273.224			0.0260625			0.0130312	−	0.999915i	$-0.495852\pi$
$479$	273.224			0.0260625			0.0130312	+	0.999915i	$0.495852\pi$
$480$		0			0
$481$	14286.7			1.35430
$482$	−201.064	−	607.100i	−0.0190005	−	0.0573707i
$483$		0			0
$484$	−11540.8	+	8586.15i	−1.08385	+	0.806362i
$485$		−	9319.00i		−	0.872482i
$486$		0			0
$487$			8706.60i			0.810131i	0.914288	+	0.405065i	$0.132751\pi$
$487$			8706.60i			0.810131i	−0.914288	+	0.405065i	$0.867249\pi$
$488$	5722.80	+	4011.21i	0.530859	+	0.372088i
$489$		0			0
$490$	−22445.0	+	7433.51i	−2.06931	+	0.685330i
$491$	−4442.47			−0.408321			−0.204161	−	0.978937i	$-0.565446\pi$
$491$	−4442.47			−0.408321			−0.204161	+	0.978937i	$0.565446\pi$
$492$		0			0
$493$	5065.09			0.462718
$494$	−109.504	+	36.2663i	−0.00997328	+	0.00330303i
$495$		0			0
$496$	−2772.39	−	831.916i	−0.250975	−	0.0753108i
$497$		−	34096.7i		−	3.07736i
$498$		0			0
$499$			4920.29i			0.441408i	0.975341	+	0.220704i	$0.0708355\pi$
$499$			4920.29i			0.441408i	−0.975341	+	0.220704i	$0.929165\pi$
$500$	1945.70	+	2615.27i	0.174029	+	0.233916i
$501$		0			0
$502$	952.486	+	2875.97i	0.0846843	+	0.255699i
$503$	−5575.54			−0.494237			−0.247118	−	0.968985i	$-0.579484\pi$
$503$	−5575.54			−0.494237			−0.247118	+	0.968985i	$0.579484\pi$
$504$		0			0
$505$	−26221.2			−2.31055
$506$	2389.46	+	7214.83i	0.209930	+	0.633870i
$507$		0			0
$508$	−6827.71	−	9177.29i	−0.596320	−	0.801528i
$509$			17508.5i			1.52466i	0.647190	+	0.762328i	$0.275944\pi$
$509$			17508.5i			1.52466i	−0.647190	+	0.762328i	$0.724056\pi$
$510$		0			0
$511$		−	21895.2i		−	1.89548i
$512$	3014.46	−	11186.2i	0.260199	−	0.965555i
$513$		0			0
$514$	4951.90	−	1640.01i	0.424940	−	0.140735i
$515$	−58.2259			−0.00498202
$516$		0			0
$517$	−605.331			−0.0514941
$518$	20062.0	−	6644.30i	1.70169	−	0.563579i
$519$		0			0
$520$	−11136.9	+	15889.0i	−0.939201	+	1.33996i
$521$			12662.1i			1.06475i	0.846507	+	0.532377i	$0.178701\pi$
$521$			12662.1i			1.06475i	−0.846507	+	0.532377i	$0.821299\pi$
$522$		0			0
$523$			2988.40i			0.249854i	0.992166	+	0.124927i	$0.0398697\pi$
$523$			2988.40i			0.249854i	−0.992166	+	0.124927i	$0.960130\pi$
$524$	−13079.7	+	9731.04i	−1.09044	+	0.811264i
$525$		0			0
$526$	6671.48	+	20144.1i	0.553024	+	1.66982i
$527$	1324.42			0.109474
$528$		0			0
$529$	−9859.47			−0.810345
$530$	−492.080	−	1485.80i	−0.0403294	−	0.121772i
$531$		0			0
$532$	−136.904	+	101.854i	−0.0111570	+	0.00830059i
$533$			2949.69i			0.239710i
$534$		0			0
$535$		−	1154.31i		−	0.0932805i
$536$	−1475.57	+	2105.20i	−0.118909	+	0.169647i
$537$		0			0
$538$	9304.10	−	3081.40i	0.745591	−	0.246931i
$539$	31334.7			2.50404
$540$		0			0
$541$	9079.39			0.721541			0.360770	−	0.932655i	$-0.382514\pi$
$541$	9079.39			0.721541			0.360770	+	0.932655i	$0.382514\pi$
$542$	7846.31	−	2598.60i	0.621822	−	0.205940i
$543$		0			0
$544$	149.972	+	5298.85i	0.0118199	+	0.417622i
$545$			24783.4i			1.94790i
$546$		0			0
$547$			21690.2i			1.69544i	0.530444	+	0.847720i	$0.322025\pi$
$547$			21690.2i			1.69544i	−0.530444	+	0.847720i	$0.677975\pi$
$548$	−3843.43	−	5166.05i	−0.299605	−	0.402706i
$549$		0			0
$550$	4859.63	+	14673.3i	0.376755	+	1.13759i
$551$	122.759			0.00949133
$552$		0			0
$553$	−14636.4			−1.12551
$554$	−5908.66	−	17840.8i	−0.453132	−	1.36820i
$555$		0			0
$556$	−1976.05	−	2656.06i	−0.150725	−	0.202594i
$557$			18129.0i			1.37908i	0.724247	+	0.689541i	$0.242188\pi$
$557$			18129.0i			1.37908i	−0.724247	+	0.689541i	$0.757812\pi$
$558$		0			0
$559$		−	1147.30i		−	0.0868078i
$560$	−8249.44	+	27491.5i	−0.622504	+	2.07451i
$561$		0			0
$562$	−17753.6	+	5879.78i	−1.33255	+	0.441323i
$563$	4922.54			0.368491			0.184245	−	0.982880i	$-0.441016\pi$
$563$	4922.54			0.368491			0.184245	+	0.982880i	$0.441016\pi$
$564$		0			0
$565$	−3781.34			−0.281562
$566$	−7137.63	+	2363.90i	−0.530065	+	0.175551i
$567$		0			0
$568$	−21022.4	−	14735.0i	−1.55296	−	1.08850i
$569$			17731.7i			1.30641i	0.757179	+	0.653207i	$0.226577\pi$
$569$			17731.7i			1.30641i	−0.757179	+	0.653207i	$0.773423\pi$
$570$		0			0
$571$			17237.0i			1.26330i	0.775254	+	0.631650i	$0.217622\pi$
$571$			17237.0i			1.26330i	−0.775254	+	0.631650i	$0.782378\pi$
$572$	20631.3	−	15349.2i	1.50811	−	1.12200i
$573$		0			0
$574$	1371.81	+	4142.09i	0.0997531	+	0.301198i
$575$	4693.00			0.340368
$576$		0			0
$577$	6166.73			0.444929			0.222465	−	0.974941i	$-0.428590\pi$
$577$	6166.73			0.444929			0.222465	+	0.974941i	$0.428590\pi$
$578$	3606.27	+	10888.9i	0.259517	+	0.783596i
$579$		0			0
$580$	16567.1	−	12325.6i	1.18605	−	0.882399i
$581$		−	35032.8i		−	2.50156i
$582$		0			0
$583$			2074.28i			0.147355i
$584$	−13499.6	−	9462.09i	−0.956535	−	0.670452i
$585$		0			0
$586$	14911.5	−	4938.49i	1.05117	−	0.348135i
$587$	−11268.2			−0.792311			−0.396155	−	0.918183i	$-0.629656\pi$
$587$	−11268.2			−0.792311			−0.396155	+	0.918183i	$0.629656\pi$
$588$		0			0
$589$	32.0992			0.00224554
$590$	−16478.6	+	5457.51i	−1.14985	+	0.380817i
$591$		0			0
$592$	4573.30	−	15240.7i	0.317502	−	1.05809i
$593$			4766.20i			0.330058i	0.986289	+	0.165029i	$0.0527718\pi$
$593$			4766.20i			0.330058i	−0.986289	+	0.165029i	$0.947228\pi$
$594$		0			0
$595$		−	13133.2i		−	0.904889i
$596$	−6967.46	−	9365.12i	−0.478856	−	0.643642i
$597$		0			0
$598$	−2454.59	−	7411.48i	−0.167852	−	0.506819i
$599$	11994.7			0.818177			0.409089	−	0.912495i	$-0.365847\pi$
$599$	11994.7			0.818177			0.409089	+	0.912495i	$0.365847\pi$
$600$		0			0
$601$	23975.3			1.62724			0.813622	−	0.581394i	$-0.197493\pi$
$601$	23975.3			1.62724			0.813622	+	0.581394i	$0.197493\pi$
$602$	−533.574	−	1611.09i	−0.0361243	−	0.109075i
$603$		0			0
$604$	−7966.90	−	10708.5i	−0.536703	−	0.721395i
$605$		−	26832.4i		−	1.80313i
$606$		0			0
$607$		−	18629.0i		−	1.24568i	−0.782349	−	0.622840i	$-0.785979\pi$
$607$		−	18629.0i		−	1.24568i	0.782349	−	0.622840i	$-0.214021\pi$
$608$	3.63478	+	128.425i	0.000242450	+	0.00856630i
$609$		0			0
$610$	−12375.3	+	4098.55i	−0.821412	+	0.272041i
$611$	621.830			0.0411728
$612$		0			0
$613$	18552.5			1.22240			0.611198	−	0.791478i	$-0.290688\pi$
$613$	18552.5			1.22240			0.611198	+	0.791478i	$0.290688\pi$
$614$	−23346.0	+	7731.91i	−1.53447	+	0.508199i
$615$		0			0
$616$	21833.0	−	31149.1i	1.42804	−	2.03739i
$617$			7836.95i			0.511351i	0.966763	+	0.255676i	$0.0822979\pi$
$617$			7836.95i			0.511351i	−0.966763	+	0.255676i	$0.917702\pi$
$618$		0			0
$619$			16023.6i			1.04045i	0.854028	+	0.520227i	$0.174153\pi$
$619$			16023.6i			1.04045i	−0.854028	+	0.520227i	$0.825847\pi$
$620$	4331.97	−	3222.90i	0.280607	−	0.208766i
$621$		0			0
$622$	5973.11	+	18035.4i	0.385048	+	1.16263i
$623$	36023.6			2.31662
$624$		0			0
$625$	−18292.5			−1.17072
$626$	−1235.21	−	3729.62i	−0.0788638	−	0.238124i
$627$		0			0
$628$	11385.2	−	8470.36i	0.723438	−	0.538223i
$629$			7280.75i			0.461530i
$630$		0			0
$631$			18451.1i			1.16407i	0.813164	+	0.582035i	$0.197743\pi$
$631$			18451.1i			1.16407i	−0.813164	+	0.582035i	$0.802257\pi$
$632$	−6325.18	+	9024.14i	−0.398104	+	0.567976i
$633$		0			0
$634$	−3913.73	+	1296.18i	−0.245164	+	0.0811955i
$635$	21337.2			1.33345
$636$		0			0
$637$	−32188.7			−2.00214
$638$	−25978.2	+	8603.65i	−1.61205	+	0.533890i
$639$		0			0
$640$	13385.0	+	16966.7i	0.826698	+	1.04792i
$641$		−	20544.4i		−	1.26592i	−0.774185	−	0.632960i	$-0.781840\pi$
$641$		−	20544.4i		−	1.26592i	0.774185	−	0.632960i	$-0.218160\pi$
$642$		0			0
$643$		−	27413.8i		−	1.68133i	−0.541555	−	0.840665i	$-0.682164\pi$
$643$		−	27413.8i		−	1.68133i	0.541555	−	0.840665i	$-0.317836\pi$
$644$	−6893.70	−	9265.99i	−0.421817	−	0.566974i
$645$		0			0
$646$	−18.4819	−	55.8050i	−0.00112564	−	0.00339879i
$647$	−23051.0			−1.40066			−0.700332	−	0.713817i	$-0.746965\pi$
$647$	−23051.0			−1.40066			−0.700332	+	0.713817i	$0.746965\pi$
$648$		0			0
$649$	23005.2			1.39142
$650$	−4992.09	−	15073.3i	−0.301240	−	0.909573i
$651$		0			0
$652$	16947.9	+	22780.1i	1.01799	+	1.36831i
$653$			27762.7i			1.66376i	0.554953	+	0.831882i	$0.312736\pi$
$653$			27762.7i			1.66376i	−0.554953	+	0.831882i	$0.687264\pi$
$654$		0			0
$655$		−	30410.3i		−	1.81409i
$656$	3146.65	+	944.223i	0.187281	+	0.0561977i
$657$		0			0
$658$	873.203	−	289.194i	0.0517341	−	0.0171337i
$659$	9857.22			0.582675			0.291338	−	0.956620i	$-0.405900\pi$
$659$	9857.22			0.582675			0.291338	+	0.956620i	$0.405900\pi$
$660$		0			0
$661$	3542.53			0.208454			0.104227	−	0.994554i	$-0.466763\pi$
$661$	3542.53			0.208454			0.104227	+	0.994554i	$0.466763\pi$
$662$	13335.9	−	4416.70i	0.782954	−	0.259305i
$663$		0			0
$664$	−21599.6	−	15139.5i	−1.26239	−	0.884830i
$665$		−	318.301i		−	0.0185612i
$666$		0			0
$667$			8308.65i			0.482327i
$668$	12122.4	−	9018.82i	0.702141	−	0.522378i
$669$		0			0
$670$	−1507.70	−	4552.40i	−0.0869365	−	0.262499i
$671$	17276.7			0.993979
$672$		0			0
$673$	19216.6			1.10066			0.550330	−	0.834947i	$-0.314502\pi$
$673$	19216.6			1.10066			0.550330	+	0.834947i	$0.314502\pi$
$674$	−4350.96	−	13137.5i	−0.248654	−	0.750795i
$675$		0			0
$676$	−7092.17	+	5276.43i	−0.403515	+	0.300206i
$677$			15731.9i			0.893093i	0.894760	+	0.446547i	$0.147346\pi$
$677$			15731.9i			0.893093i	−0.894760	+	0.446547i	$0.852654\pi$
$678$		0			0
$679$		−	18767.1i		−	1.06070i
$680$	−8097.32	−	5675.55i	−0.456644	−	0.320070i
$681$		0			0
$682$	−6792.78	+	2249.69i	−0.381392	+	0.126312i
$683$	−16870.5			−0.945142			−0.472571	−	0.881293i	$-0.656674\pi$
$683$	−16870.5			−0.945142			−0.472571	+	0.881293i	$0.656674\pi$
$684$		0			0
$685$	12011.1			0.669955
$686$	−17523.7	+	5803.63i	−0.975302	+	0.323008i
$687$		0			0
$688$	−1223.91	−	367.261i	−0.0678213	−	0.0203513i
$689$		−	2130.82i		−	0.117819i
$690$		0			0
$691$		−	29234.5i		−	1.60945i	−0.593645	−	0.804727i	$-0.702312\pi$
$691$		−	29234.5i		−	1.60945i	0.593645	−	0.804727i	$-0.297688\pi$
$692$	10047.6	+	13505.2i	0.551953	+	0.741894i
$693$		0			0
$694$	−225.850	−	681.941i	−0.0123533	−	0.0372999i
$695$	6175.33			0.337041
$696$		0			0
$697$	−1503.21			−0.0816905
$698$	−5748.24	−	17356.4i	−0.311711	−	0.941190i
$699$		0			0
$700$	−14020.2	−	18844.9i	−0.757022	−	1.01753i
$701$		−	14904.4i		−	0.803041i	−0.915850	−	0.401520i	$-0.868482\pi$
$701$		−	14904.4i		−	0.803041i	0.915850	−	0.401520i	$-0.131518\pi$
$702$		0			0
$703$			176.459i			0.00946696i
$704$	−9769.91	−	26922.3i	−0.523036	−	1.44130i
$705$		0			0
$706$	−10830.4	+	3586.90i	−0.577348	+	0.191211i
$707$	−52805.7			−2.80900
$708$		0			0
$709$	−17930.1			−0.949758			−0.474879	−	0.880051i	$-0.657508\pi$
$709$	−17930.1			−0.949758			−0.474879	+	0.880051i	$0.657508\pi$
$710$	45460.0	−	15055.8i	2.40294	−	0.795823i
$711$		0			0
$712$	15567.7	−	22210.4i	0.819415	−	1.16906i
$713$			2172.55i			0.114113i
$714$		0			0
$715$			47967.7i			2.50894i
$716$	−11746.5	+	8739.13i	−0.613109	+	0.456141i
$717$		0			0
$718$	9488.57	+	28650.1i	0.493190	+	1.48915i
$719$	19033.4			0.987238			0.493619	−	0.869678i	$-0.335674\pi$
$719$	19033.4			0.987238			0.493619	+	0.869678i	$0.335674\pi$
$720$		0			0
$721$	−117.258			−0.00605677
$722$	6098.85	+	18415.1i	0.314371	+	0.949223i
$723$		0			0
$724$	−21396.3	+	15918.4i	−1.09833	+	0.817131i
$725$			16897.9i			0.865618i
$726$		0			0
$727$		−	19643.0i		−	1.00209i	−0.865422	−	0.501043i	$-0.832950\pi$
$727$		−	19643.0i		−	1.00209i	0.865422	−	0.501043i	$-0.167050\pi$
$728$	−22428.0	+	31998.1i	−1.14181	+	1.62902i
$729$		0			0
$730$	29192.2	−	9668.10i	1.48007	−	0.490181i
$731$	584.684			0.0295832
$732$		0			0
$733$	8335.12			0.420006			0.210003	−	0.977701i	$-0.432653\pi$
$733$	8335.12			0.420006			0.210003	+	0.977701i	$0.432653\pi$
$734$	−6873.19	+	2276.32i	−0.345632	+	0.114469i
$735$		0			0
$736$	−8692.10	+	246.011i	−0.435320	+	0.0123208i
$737$			6355.44i			0.317647i
$738$		0			0
$739$			12308.0i			0.612661i	0.951925	+	0.306331i	$0.0991013\pi$
$739$			12308.0i			0.612661i	−0.951925	+	0.306331i	$0.900899\pi$
$740$	17717.2	+	23814.2i	0.880134	+	1.18301i
$741$		0			0
$742$	−990.977	−	2992.19i	−0.0490295	−	0.148042i
$743$	26227.7			1.29502			0.647511	−	0.762056i	$-0.275810\pi$
$743$	26227.7			1.29502			0.647511	+	0.762056i	$0.275810\pi$
$744$		0			0
$745$	21773.9			1.07078
$746$	1721.50	+	5197.96i	0.0844887	+	0.255108i
$747$		0			0
$748$	7822.25	+	10514.1i	0.382366	+	0.513947i
$749$		−	2324.61i		−	0.113404i
$750$		0			0
$751$		−	2741.63i		−	0.133214i	−0.997779	−	0.0666068i	$-0.978783\pi$
$751$		−	2741.63i		−	0.133214i	0.997779	−	0.0666068i	$-0.0212173\pi$
$752$	199.054	−	663.352i	0.00965258	−	0.0321675i
$753$		0			0
$754$	26686.2	−	8838.16i	1.28893	−	0.426879i
$755$	24897.2			1.20014
$756$		0			0
$757$	−17816.8			−0.855434			−0.427717	−	0.903913i	$-0.640682\pi$
$757$	−17816.8			−0.855434			−0.427717	+	0.903913i	$0.640682\pi$
$758$	15962.1	−	5286.44i	0.764866	−	0.253314i
$759$		0			0
$760$	−196.250	−	137.555i	−0.00936674	−	0.00656531i
$761$		−	32228.2i		−	1.53518i	−0.640940	−	0.767591i	$-0.721455\pi$
$761$		−	32228.2i		−	1.53518i	0.640940	−	0.767591i	$-0.278545\pi$
$762$		0			0
$763$			49910.1i			2.36811i
$764$	23249.3	−	17297.0i	1.10096	−	0.819088i
$765$		0			0
$766$	1508.87	+	4555.93i	0.0711718	+	0.214899i
$767$	−23632.2			−1.11253
$768$		0			0
$769$	−4236.60			−0.198668			−0.0993340	−	0.995054i	$-0.531671\pi$
$769$	−4236.60			−0.198668			−0.0993340	+	0.995054i	$0.531671\pi$
$770$	22308.3	+	67358.5i	1.04407	+	3.15251i
$771$		0			0
$772$	16615.5	−	12361.6i	0.774616	−	0.576298i
$773$			1754.64i			0.0816431i	0.999166	+	0.0408215i	$0.0129975\pi$
$773$			1754.64i			0.0816431i	−0.999166	+	0.0408215i	$0.987002\pi$
$774$		0			0
$775$			4418.48i			0.204795i
$776$	−11570.9	−	8110.25i	−0.535273	−	0.375182i
$777$		0			0
$778$	−10508.3	+	3480.23i	−0.484244	+	0.160376i
$779$	−36.4324			−0.00167565
$780$		0			0
$781$	−63465.1			−2.90776
$782$	3777.02	−	1250.90i	0.172719	−	0.0572023i
$783$		0			0
$784$	−10303.9	+	34338.1i	−0.469384	+	1.56424i
$785$			26470.6i			1.20354i
$786$		0			0
$787$			11896.9i			0.538857i	0.963020	+	0.269428i	$0.0868347\pi$
$787$			11896.9i			0.538857i	−0.963020	+	0.269428i	$0.913165\pi$
$788$	8367.37	+	11246.8i	0.378268	+	0.508439i
$789$		0			0
$790$	−6462.89	−	19514.3i	−0.291062	−	0.878844i
$791$	−7615.07			−0.342302
$792$		0			0
$793$	−17747.6			−0.794749
$794$	4517.32	+	13639.8i	0.201906	+	0.609644i
$795$		0			0
$796$	−15837.4	−	21287.5i	−0.705205	−	0.947883i
$797$		−	16762.4i		−	0.744986i	−0.928035	−	0.372493i	$-0.878503\pi$
$797$		−	16762.4i		−	0.744986i	0.928035	−	0.372493i	$-0.121497\pi$
$798$		0			0
$799$			316.896i			0.0140313i
$800$	−17677.8	+	500.330i	−0.781255	+	0.0221117i
$801$		0			0
$802$	−25170.4	+	8336.13i	−1.10823	+	0.367031i
$803$	−40754.2			−1.79101
$804$		0			0
$805$	21543.4			0.943236
$806$	6977.93	−	2311.00i	0.304947	−	0.100995i
$807$		0			0
$808$	−22820.1	+	32557.5i	−0.993576	+	1.41754i
$809$		−	28039.3i		−	1.21855i	−0.792957	−	0.609277i	$-0.791460\pi$
$809$		−	28039.3i		−	1.21855i	0.792957	−	0.609277i	$-0.208540\pi$
$810$		0			0
$811$			14131.8i			0.611882i	0.952051	+	0.305941i	$0.0989710\pi$
$811$			14131.8i			0.611882i	−0.952051	+	0.305941i	$0.901029\pi$
$812$	33363.7	−	24821.9i	1.44192	−	1.07276i
$813$		0			0
$814$	−12367.2	−	37342.0i	−0.532519	−	1.60791i
$815$	−52963.7			−2.27636
$816$		0			0
$817$	14.1706			0.000606814
$818$	−11020.1	−	33274.5i	−0.471038	−	1.42227i
$819$		0			0
$820$	−4916.78	+	3657.98i	−0.209392	+	0.155783i
$821$			8233.70i			0.350010i	0.984568	+	0.175005i	$0.0559942\pi$
$821$			8233.70i			0.350010i	−0.984568	+	0.175005i	$0.944006\pi$
$822$		0			0
$823$		−	5824.72i		−	0.246704i	−0.992363	−	0.123352i	$-0.960636\pi$
$823$		−	5824.72i		−	0.246704i	0.992363	−	0.123352i	$-0.0393644\pi$
$824$	−50.6736	+	72.2961i	−0.00214235	+	0.00305650i
$825$		0			0
$826$	−33185.5	+	10990.6i	−1.39791	+	0.462969i
$827$	−20581.6			−0.865406			−0.432703	−	0.901536i	$-0.642440\pi$
$827$	−20581.6			−0.865406			−0.432703	+	0.901536i	$0.642440\pi$
$828$		0			0
$829$	2844.67			0.119179			0.0595896	−	0.998223i	$-0.481021\pi$
$829$	2844.67			0.119179			0.0595896	+	0.998223i	$0.481021\pi$
$830$	46708.0	−	15469.1i	1.95332	−	0.646917i
$831$		0			0
$832$	10036.2	+	27656.2i	0.418200	+	1.15241i
$833$		−	16404.0i		−	0.682309i
$834$		0			0
$835$			28184.6i			1.16810i
$836$	189.583	+	254.823i	0.00784313	+	0.0105421i
$837$		0			0
$838$	237.308	+	716.535i	0.00978241	+	0.0295374i
$839$	−12881.0			−0.530037			−0.265019	−	0.964243i	$-0.585378\pi$
$839$	−12881.0			−0.530037			−0.265019	+	0.964243i	$0.585378\pi$
$840$		0			0
$841$	−5527.65			−0.226645
$842$	−8850.28	−	26722.8i	−0.362234	−	1.09374i
$843$		0			0
$844$	28460.9	+	38254.9i	1.16074	+	1.56018i
$845$		−	16489.3i		−	0.671300i
$846$		0			0
$847$		−	54036.6i		−	2.19211i
$848$	−2273.10	−	682.094i	−0.0920501	−	0.0276217i
$849$		0			0
$850$	7681.61	−	2544.06i	0.309973	−	0.102659i
$851$	−11943.2			−0.481089
$852$		0			0
$853$	43159.3			1.73241			0.866205	−	0.499689i	$-0.166552\pi$
$853$	43159.3			1.73241			0.866205	+	0.499689i	$0.166552\pi$
$854$	−24922.0	+	8253.87i	−0.998612	+	0.330728i
$855$		0			0
$856$	−1433.24	−	1004.59i	−0.0572281	−	0.0401122i
$857$			236.518i			0.00942741i	0.999989	+	0.00471371i	$0.00150042\pi$
$857$			236.518i			0.00942741i	−0.999989	+	0.00471371i	$0.998500\pi$
$858$		0			0
$859$		−	1103.24i		−	0.0438210i	−0.999760	−	0.0219105i	$-0.993025\pi$
$859$		−	1103.24i		−	0.0438210i	0.999760	−	0.0219105i	$-0.00697488\pi$
$860$	1912.41	−	1422.79i	0.0758286	−	0.0564149i
$861$		0			0
$862$	−5854.17	−	17676.3i	−0.231315	−	0.698441i
$863$	22297.9			0.879522			0.439761	−	0.898115i	$-0.355063\pi$
$863$	22297.9			0.879522			0.439761	+	0.898115i	$0.355063\pi$
$864$		0			0
$865$	−31399.5			−1.23424
$866$	−12224.4	−	36910.7i	−0.479677	−	1.44835i
$867$		0			0
$868$	8723.96	−	6490.44i	0.341141	−	0.253802i
$869$			27243.2i			1.06348i
$870$		0			0
$871$		−	6528.67i		−	0.253979i
$872$	30772.2	+	21568.8i	1.19504	+	0.837627i
$873$		0			0
$874$	91.5412	−	30.3173i	0.00354282	−	0.00117334i
$875$	−12245.2			−0.473101
$876$		0			0
$877$	29308.0			1.12846			0.564231	−	0.825617i	$-0.309173\pi$
$877$	29308.0			1.12846			0.564231	+	0.825617i	$0.309173\pi$
$878$	−35925.4	+	11898.1i	−1.38089	+	0.457335i
$879$		0			0
$880$	51170.7	+	15354.9i	1.96019	+	0.588197i
$881$			25857.5i			0.988831i	0.869226	+	0.494415i	$0.164618\pi$
$881$			25857.5i			0.988831i	−0.869226	+	0.494415i	$0.835382\pi$
$882$		0			0
$883$		−	15147.3i		−	0.577290i	−0.957436	−	0.288645i	$-0.906795\pi$
$883$		−	15147.3i		−	0.577290i	0.957436	−	0.288645i	$-0.0932047\pi$
$884$	−8035.46	−	10800.6i	−0.305726	−	0.410933i
$885$		0			0
$886$	14324.0	+	43250.3i	0.543141	+	1.63998i
$887$	−18719.8			−0.708624			−0.354312	−	0.935127i	$-0.615285\pi$
$887$	−18719.8			−0.708624			−0.354312	+	0.935127i	$0.615285\pi$
$888$		0			0
$889$	42970.0			1.62111
$890$	15906.6	+	48029.0i	0.599091	+	1.80892i
$891$		0			0
$892$	−9527.68	−	12806.4i	−0.357635	−	0.480706i
$893$			7.68040i			0.000287811i
$894$		0			0
$895$		−	27310.5i		−	1.01999i
$896$	26955.3	+	34168.6i	1.00504	+	1.27399i
$897$		0			0
$898$	30296.2	−	10033.7i	1.12583	−	0.372861i
$899$	−7822.62			−0.290210
$900$		0			0
$901$	1085.90			0.0401516
$902$	7709.79	−	2553.39i	0.284599	−	0.0942556i
$903$		0			0
$904$	−3290.87	+	4695.09i	−0.121076	+	0.172740i
$905$		−	49746.4i		−	1.82721i
$906$		0			0
$907$		−	10374.7i		−	0.379807i	−0.981803	−	0.189904i	$-0.939182\pi$
$907$		−	10374.7i		−	0.379807i	0.981803	−	0.189904i	$-0.0608175\pi$
$908$	14634.9	−	10888.1i	0.534887	−	0.397945i
$909$		0			0
$910$	−22916.4	−	69194.5i	−0.834802	−	2.52063i
$911$	10860.1			0.394962			0.197481	−	0.980307i	$-0.436724\pi$
$911$	10860.1			0.394962			0.197481	+	0.980307i	$0.436724\pi$
$912$		0			0
$913$	−65207.4			−2.36369
$914$	1764.90	+	5328.99i	0.0638704	+	0.192853i
$915$		0			0
$916$	−29832.8	+	22195.0i	−1.07610	+	0.800593i
$917$		−	61242.0i		−	2.20544i
$918$		0			0
$919$		−	5743.96i		−	0.206176i	−0.994672	−	0.103088i	$-0.967128\pi$
$919$		−	5743.96i		−	0.206176i	0.994672	−	0.103088i	$-0.0328723\pi$
$920$	9310.04	−	13282.6i	0.333634	−	0.475996i
$921$		0			0
$922$	25868.4	−	8567.31i	0.924004	−	0.306019i
$923$	65195.0			2.32494
$924$		0			0
$925$	−24289.7			−0.863396
$926$	−3738.56	+	1238.17i	−0.132675	+	0.0439402i
$927$		0			0
$928$	−885.802	−	31297.4i	−0.0313339	−	1.10710i
$929$			28379.7i			1.00227i	0.865369	+	0.501135i	$0.167084\pi$
$929$			28379.7i			1.00227i	−0.865369	+	0.501135i	$0.832916\pi$
$930$		0			0
$931$		−	397.572i		−	0.0139956i
$932$	13489.1	+	18131.1i	0.474089	+	0.637235i
$933$		0			0
$934$	−13482.1	−	40708.4i	−0.472321	−	1.42614i
$935$	−24445.2			−0.855020
$936$		0			0
$937$	24896.7			0.868025			0.434013	−	0.900907i	$-0.357097\pi$
$937$	24896.7			0.868025			0.434013	+	0.900907i	$0.357097\pi$
$938$	−3036.29	−	9167.87i	−0.105691	−	0.319127i
$939$		0			0
$940$	771.146	+	1036.52i	0.0267575	+	0.0359653i
$941$		−	601.120i		−	0.0208246i	−0.999946	−	0.0104123i	$-0.996686\pi$
$941$		−	601.120i		−	0.0208246i	0.999946	−	0.0104123i	$-0.00331440\pi$
$942$		0			0
$943$		−	2465.84i		−	0.0851524i
$944$	−7564.89	+	25210.2i	−0.260822	+	0.869198i
$945$		0			0
$946$	−2998.77	+	993.155i	−0.103064	+	0.0341335i
$947$	25774.5			0.884432			0.442216	−	0.896908i	$-0.354192\pi$
$947$	25774.5			0.884432			0.442216	+	0.896908i	$0.354192\pi$
$948$		0			0
$949$	41865.0			1.43203
$950$	186.174	−	61.6587i	0.00635820	−	0.00210576i
$951$		0			0
$952$	−16306.8	−	11429.7i	−0.555155	−	0.389118i
$953$			40615.9i			1.38056i	0.723541	+	0.690282i	$0.242513\pi$
$953$			40615.9i			1.38056i	−0.723541	+	0.690282i	$0.757487\pi$
$954$		0			0
$955$			54054.6i			1.83159i
$956$	−15441.9	+	11488.5i	−0.522415	+	0.388666i
$957$		0			0
$958$	242.962	+	733.608i	0.00819389	+	0.0247409i
$959$	24188.5			0.814482
$960$		0			0
$961$	27745.5			0.931340
$962$	12704.3	+	38359.8i	0.425783	+	1.28562i
$963$		0			0
$964$	1451.27	−	1079.72i	0.0484879	−	0.0360740i
$965$			38630.9i			1.28868i
$966$		0			0
$967$			52148.0i			1.73420i	0.498138	+	0.867098i	$0.334017\pi$
$967$			52148.0i			1.73420i	−0.498138	+	0.867098i	$0.665983\pi$
$968$	−33316.4	−	23352.1i	−1.10623	−	0.775376i
$969$		0			0
$970$	25021.6	−	8286.83i	0.828241	−	0.274303i
$971$	−29877.8			−0.987461			−0.493731	−	0.869615i	$-0.664367\pi$
$971$	−29877.8			−0.987461			−0.493731	+	0.869615i	$0.664367\pi$
$972$		0			0
$973$	12436.2			0.409750
$974$	−23377.3	+	7742.26i	−0.769051	+	0.254700i
$975$		0			0
$976$	−5681.18	+	18932.7i	−0.186322	+	0.620923i
$977$		−	10385.8i		−	0.340094i	−0.985436	−	0.170047i	$-0.945608\pi$
$977$		−	10385.8i		−	0.340094i	0.985436	−	0.170047i	$-0.0543920\pi$
$978$		0			0
$979$		−	67051.6i		−	2.18895i
$980$	−39918.0	−	53654.8i	−1.30116	−	1.74892i
$981$		0			0
$982$	−3950.42	−	11928.0i	−0.128374	−	0.387616i
$983$	−14316.7			−0.464530			−0.232265	−	0.972652i	$-0.574614\pi$
$983$	−14316.7			−0.464530			−0.232265	+	0.972652i	$0.574614\pi$
$984$		0			0
$985$	−26148.7			−0.845856
$986$	4504.08	+	13599.8i	0.145476	+	0.439255i
$987$		0			0
$988$	−194.750	−	261.768i	−0.00627108	−	0.00842911i
$989$			959.102i			0.0308369i
$990$		0			0
$991$		−	30199.0i		−	0.968013i	−0.875064	−	0.484007i	$-0.839181\pi$
$991$		−	30199.0i		−	0.968013i	0.875064	−	0.484007i	$-0.160819\pi$
$992$	−231.620	−	8183.65i	−0.00741324	−	0.261926i
$993$		0			0
$994$	91549.9	−	30320.2i	2.92131	−	0.967503i
$995$	49493.3			1.57693
$996$		0			0
$997$	17508.8			0.556178			0.278089	−	0.960555i	$-0.410299\pi$
$997$	17508.8			0.556178			0.278089	+	0.960555i	$0.410299\pi$
$998$	−13211.0	+	4375.32i	−0.419025	+	0.138776i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.b.a.107.8	yes	12
3.2	odd	2	inner	108.4.b.a.107.5	✓	12
4.3	odd	2	inner	108.4.b.a.107.6	yes	12
8.3	odd	2		1728.4.c.i.1727.12		12
8.5	even	2		1728.4.c.i.1727.11		12
12.11	even	2	inner	108.4.b.a.107.7	yes	12
24.5	odd	2		1728.4.c.i.1727.1		12
24.11	even	2		1728.4.c.i.1727.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.b.a.107.5	✓	12	3.2	odd	2	inner
108.4.b.a.107.6	yes	12	4.3	odd	2	inner
108.4.b.a.107.7	yes	12	12.11	even	2	inner
108.4.b.a.107.8	yes	12	1.1	even	1	trivial
1728.4.c.i.1727.1		12	24.5	odd	2
1728.4.c.i.1727.2		12	24.11	even	2
1728.4.c.i.1727.11		12	8.5	even	2
1728.4.c.i.1727.12		12	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.889241 + 2.68500i) q^{2} +(-6.41850 + 4.77523i) q^{4} -14.9230i q^{5} -30.0528i q^{7} +(-18.5291 - 12.9874i) q^{8} +O(q^{10})\)	\(q+(0.889241 + 2.68500i) q^{2} +(-6.41850 + 4.77523i) q^{4} -14.9230i q^{5} -30.0528i q^{7} +(-18.5291 - 12.9874i) q^{8} +(40.0683 - 13.2701i) q^{10} -55.9380 q^{11} +57.4627 q^{13} +(80.6918 - 26.7241i) q^{14} +(18.3943 - 61.2997i) q^{16} +29.2840i q^{17} +0.709738i q^{19} +(71.2608 + 95.7833i) q^{20} +(-49.7423 - 150.194i) q^{22} -48.0368 q^{23} -97.6960 q^{25} +(51.0981 + 154.288i) q^{26} +(143.509 + 192.894i) q^{28} -172.964i q^{29} -45.2268i q^{31} +(180.947 - 5.12130i) q^{32} +(-78.6277 + 26.0405i) q^{34} -448.477 q^{35} +248.625 q^{37} +(-1.90565 + 0.631128i) q^{38} +(-193.811 + 276.510i) q^{40} +51.3323i q^{41} -19.9660i q^{43} +(359.038 - 267.117i) q^{44} +(-42.7163 - 128.979i) q^{46} +10.8215 q^{47} -560.168 q^{49} +(-86.8753 - 262.314i) q^{50} +(-368.824 + 274.398i) q^{52} -37.0817i q^{53} +834.763i q^{55} +(-390.306 + 556.851i) q^{56} +(464.410 - 153.807i) q^{58} -411.262 q^{59} -308.855 q^{61} +(121.434 - 40.2175i) q^{62} +(174.656 + 481.289i) q^{64} -857.516i q^{65} -113.616i q^{67} +(-139.838 - 187.959i) q^{68} +(-398.804 - 1204.16i) q^{70} +1134.56 q^{71} +728.560 q^{73} +(221.088 + 667.560i) q^{74} +(-3.38916 - 4.55545i) q^{76} +1681.09i q^{77} -487.025i q^{79} +(-914.775 - 274.499i) q^{80} +(-137.827 + 45.6468i) q^{82} +1165.71 q^{83} +437.005 q^{85} +(53.6088 - 17.7546i) q^{86} +(1036.48 + 726.488i) q^{88} +1198.68i q^{89} -1726.91i q^{91} +(308.324 - 229.387i) q^{92} +(9.62289 + 29.0557i) q^{94} +10.5914 q^{95} +624.472 q^{97} +(-498.124 - 1504.05i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4}+O(q^{10}) \) 12 * q - 12 * q^4	\( 12 q - 12 q^{4} + 24 q^{10} + 36 q^{13} + 24 q^{16} + 120 q^{22} - 132 q^{25} + 420 q^{28} - 360 q^{34} + 516 q^{37} - 1152 q^{40} - 696 q^{46} - 720 q^{49} + 204 q^{52} + 2832 q^{58} - 972 q^{61} + 2496 q^{64} - 1848 q^{70} + 660 q^{73} - 5004 q^{76} - 3888 q^{82} + 1056 q^{85} + 3168 q^{88} + 7608 q^{94} + 2532 q^{97}+O(q^{100}) \) 12 * q - 12 * q^4 + 24 * q^10 + 36 * q^13 + 24 * q^16 + 120 * q^22 - 132 * q^25 + 420 * q^28 - 360 * q^34 + 516 * q^37 - 1152 * q^40 - 696 * q^46 - 720 * q^49 + 204 * q^52 + 2832 * q^58 - 972 * q^61 + 2496 * q^64 - 1848 * q^70 + 660 * q^73 - 5004 * q^76 - 3888 * q^82 + 1056 * q^85 + 3168 * q^88 + 7608 * q^94 + 2532 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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