LMFDB - Embedded newform 1764.3.bk.g.557.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(557,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1764.557");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.bk (of order $6$, degree $2$, not 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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.329365073333488765586374656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 199x^{12} + 24960x^{8} - 2913559x^{4} + 214358881 $ x^16 - 199x^12 + 24960x^8 - 2913559*x^4 + 214358881
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			557.8
Root			$3.03759 + 1.33156i$ of defining polynomial
Character	$\chi$	$=$	1764.557
Dual form			1764.3.bk.g.1745.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.27698 - 4.77872i) q^{5} +O(q^{10})$	$q+(8.27698 - 4.77872i) q^{5} +(8.03119 + 4.63681i) q^{11} +4.24264 q^{13} +(-13.4731 - 7.77872i) q^{17} +(-17.4460 - 30.2173i) q^{19} +(15.3797 - 8.87945i) q^{23} +(33.1723 - 57.4561i) q^{25} +26.2442i q^{29} +(-16.0317 + 27.7678i) q^{31} +(-27.6723 - 47.9299i) q^{37} -38.4426i q^{41} +29.3446 q^{43} +(59.0887 - 34.1149i) q^{47} +(-0.943355 - 0.544646i) q^{53} +88.6320 q^{55} +(-59.0887 - 34.1149i) q^{59} +(23.8099 + 41.2400i) q^{61} +(35.1163 - 20.2744i) q^{65} +(4.32768 - 7.49577i) q^{67} +95.7031i q^{71} +(45.0231 - 77.9823i) q^{73} +(74.3446 + 128.769i) q^{79} -88.8851i q^{83} -148.689 q^{85} +(-66.0316 + 38.1234i) q^{89} +(-288.800 - 166.739i) q^{95} -12.7519 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 216 q^{25} - 128 q^{37} - 160 q^{43} + 384 q^{67} + 560 q^{79} - 1120 q^{85}+O(q^{100}) $ 16 * q + 216 * q^25 - 128 * q^37 - 160 * q^43 + 384 * q^67 + 560 * q^79 - 1120 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	8.27698	−	4.77872i	1.65540	−	0.955744i	0.680598	−	0.732657i	$-0.261720\pi$
$5$	8.27698	−	4.77872i	1.65540	−	0.955744i	0.974798	−	0.223087i	$-0.0716135\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.03119	+	4.63681i	0.730108	+	0.421528i	0.818462	−	0.574561i	$-0.194827\pi$
$11$	8.03119	+	4.63681i	0.730108	+	0.421528i	−0.0883536	+	0.996089i	$0.528161\pi$
$12$		0			0
$13$	4.24264			0.326357			0.163178	−	0.986597i	$-0.447825\pi$
$13$	4.24264			0.326357			0.163178	+	0.986597i	$0.447825\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−13.4731	−	7.77872i	−0.792537	−	0.457572i	0.0483176	−	0.998832i	$-0.484614\pi$
$17$	−13.4731	−	7.77872i	−0.792537	−	0.457572i	−0.840855	+	0.541260i	$0.817947\pi$
$18$		0			0
$19$	−17.4460	−	30.2173i	−0.918209	−	1.59038i	−0.802135	−	0.597143i	$-0.796302\pi$
$19$	−17.4460	−	30.2173i	−0.918209	−	1.59038i	−0.116074	−	0.993241i	$-0.537031\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	15.3797	−	8.87945i	0.668681	−	0.386063i	−0.126896	−	0.991916i	$-0.540501\pi$
$23$	15.3797	−	8.87945i	0.668681	−	0.386063i	0.795577	+	0.605853i	$0.207168\pi$
$24$		0			0
$25$	33.1723	−	57.4561i	1.32689	−	2.29825i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			26.2442i			0.904972i	0.891772	+	0.452486i	$0.149463\pi$
$29$			26.2442i			0.904972i	−0.891772	+	0.452486i	$0.850537\pi$
$30$		0			0
$31$	−16.0317	+	27.7678i	−0.517153	+	0.895736i	0.482648	+	0.875814i	$0.339675\pi$
$31$	−16.0317	+	27.7678i	−0.517153	+	0.895736i	−0.999802	+	0.0199213i	$0.993658\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−27.6723	−	47.9299i	−0.747900	−	1.29540i	−0.948827	−	0.315795i	$-0.897729\pi$
$37$	−27.6723	−	47.9299i	−0.747900	−	1.29540i	0.200927	−	0.979606i	$-0.435605\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	38.4426i		−	0.937623i	−0.883298	−	0.468812i	$-0.844682\pi$
$41$		−	38.4426i		−	0.937623i	0.883298	−	0.468812i	$-0.155318\pi$
$42$		0			0
$43$	29.3446			0.682433			0.341217	−	0.939985i	$-0.389161\pi$
$43$	29.3446			0.682433			0.341217	+	0.939985i	$0.389161\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	59.0887	−	34.1149i	1.25721	−	0.725848i	0.284676	−	0.958624i	$-0.408114\pi$
$47$	59.0887	−	34.1149i	1.25721	−	0.725848i	0.972531	+	0.232775i	$0.0747807\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.943355	−	0.544646i	−0.0177991	−	0.0102763i	0.491074	−	0.871118i	$-0.336604\pi$
$53$	−0.943355	−	0.544646i	−0.0177991	−	0.0102763i	−0.508873	+	0.860842i	$0.669938\pi$
$54$		0			0
$55$	88.6320			1.61149
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−59.0887	−	34.1149i	−1.00150	−	0.578218i	−0.0928106	−	0.995684i	$-0.529585\pi$
$59$	−59.0887	−	34.1149i	−1.00150	−	0.578218i	−0.908693	+	0.417466i	$0.862918\pi$
$60$		0			0
$61$	23.8099	+	41.2400i	0.390327	+	0.676066i	0.992493	−	0.122305i	$-0.0390288\pi$
$61$	23.8099	+	41.2400i	0.390327	+	0.676066i	−0.602166	+	0.798371i	$0.705695\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	35.1163	−	20.2744i	0.540250	−	0.311914i
$66$		0			0
$67$	4.32768	−	7.49577i	0.0645923	−	0.111877i	−0.831921	−	0.554894i	$-0.812759\pi$
$67$	4.32768	−	7.49577i	0.0645923	−	0.111877i	0.896513	+	0.443017i	$0.146092\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			95.7031i			1.34793i	0.738763	+	0.673966i	$0.235410\pi$
$71$			95.7031i			1.34793i	−0.738763	+	0.673966i	$0.764590\pi$
$72$		0			0
$73$	45.0231	−	77.9823i	0.616755	−	1.06825i	−0.373319	−	0.927703i	$-0.621780\pi$
$73$	45.0231	−	77.9823i	0.616755	−	1.06825i	0.990074	−	0.140548i	$-0.0448865\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	74.3446	+	128.769i	0.941071	+	1.62998i	0.763433	+	0.645887i	$0.223512\pi$
$79$	74.3446	+	128.769i	0.941071	+	1.62998i	0.177639	+	0.984096i	$0.443154\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	88.8851i		−	1.07091i	−0.844565	−	0.535453i	$-0.820141\pi$
$83$		−	88.8851i		−	1.07091i	0.844565	−	0.535453i	$-0.179859\pi$
$84$		0			0
$85$	−148.689			−1.74929
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−66.0316	+	38.1234i	−0.741928	+	0.428352i	−0.822770	−	0.568375i	$-0.807572\pi$
$89$	−66.0316	+	38.1234i	−0.741928	+	0.428352i	0.0808420	+	0.996727i	$0.474239\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−288.800	−	166.739i	−3.04000	−	1.75514i
$96$		0			0
$97$	−12.7519			−0.131463			−0.0657314	−	0.997837i	$-0.520938\pi$
$97$	−12.7519			−0.131463			−0.0657314	+	0.997837i	$0.520938\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.64326	+	2.68079i	0.0459729	+	0.0265425i	0.522810	−	0.852449i	$-0.324884\pi$
$101$	4.64326	+	2.68079i	0.0459729	+	0.0265425i	−0.476837	+	0.878992i	$0.658217\pi$
$102$		0			0
$103$	68.3576	+	118.399i	0.663667	+	1.14950i	0.979645	+	0.200738i	$0.0643339\pi$
$103$	68.3576	+	118.399i	0.663667	+	1.14950i	−0.315979	+	0.948766i	$0.602333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−129.182	+	74.5831i	−1.20731	+	0.697038i	−0.962170	−	0.272450i	$-0.912166\pi$
$107$	−129.182	+	74.5831i	−1.20731	+	0.697038i	−0.245136	+	0.969489i	$0.578833\pi$
$108$		0			0
$109$	−14.3446	+	24.8456i	−0.131602	+	0.227942i	−0.924294	−	0.381681i	$-0.875345\pi$
$109$	−14.3446	+	24.8456i	−0.131602	+	0.227942i	0.792692	+	0.609622i	$0.208679\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	145.712i		−	1.28949i	−0.764399	−	0.644743i	$-0.776964\pi$
$113$		−	145.712i		−	1.28949i	0.764399	−	0.644743i	$-0.223036\pi$
$114$		0			0
$115$	84.8648	−	146.990i	0.737955	−	1.27818i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−17.5000	−	30.3109i	−0.144628	−	0.250503i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	395.149i		−	3.16119i
$126$		0			0
$127$	−119.345			−0.939722			−0.469861	−	0.882741i	$-0.655696\pi$
$127$	−119.345			−0.939722			−0.469861	+	0.882741i	$0.655696\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	83.1384	−	48.0000i	0.634645	−	0.366412i	−0.147904	−	0.989002i	$-0.547253\pi$
$131$	83.1384	−	48.0000i	0.634645	−	0.366412i	0.782549	+	0.622590i	$0.213919\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−149.018	−	86.0353i	−1.08772	−	0.627995i	−0.154751	−	0.987954i	$-0.549458\pi$
$137$	−149.018	−	86.0353i	−1.08772	−	0.627995i	−0.932968	+	0.359958i	$0.882791\pi$
$138$		0			0
$139$	211.205			1.51946			0.759731	−	0.650238i	$-0.225331\pi$
$139$	211.205			1.51946			0.759731	+	0.650238i	$0.225331\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	34.0735	+	19.6723i	0.238276	+	0.137569i
$144$		0			0
$145$	125.414	+	217.223i	0.864921	+	1.49809i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	130.808	−	75.5219i	0.877905	−	0.506859i	0.00793782	−	0.999968i	$-0.497473\pi$
$149$	130.808	−	75.5219i	0.877905	−	0.506859i	0.869967	+	0.493110i	$0.164140\pi$
$150$		0			0
$151$	−6.01695	+	10.4217i	−0.0398473	+	0.0690176i	−0.885261	−	0.465094i	$-0.846020\pi$
$151$	−6.01695	+	10.4217i	−0.0398473	+	0.0690176i	0.845414	+	0.534112i	$0.179354\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			306.445i			1.97706i
$156$		0			0
$157$	80.3785	−	139.220i	0.511965	−	0.886749i	−0.487939	−	0.872878i	$-0.662251\pi$
$157$	80.3785	−	139.220i	0.511965	−	0.886749i	0.999904	−	0.0138712i	$-0.00441550\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	129.017	+	223.464i	0.791515	+	1.37094i	0.925029	+	0.379897i	$0.124041\pi$
$163$	129.017	+	223.464i	0.791515	+	1.37094i	−0.133514	+	0.991047i	$0.542626\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			107.838i			0.645737i	0.946444	+	0.322868i	$0.104647\pi$
$167$			107.838i			0.645737i	−0.946444	+	0.322868i	$0.895353\pi$
$168$		0			0
$169$	−151.000			−0.893491
$170$		0			0
$171$		0			0
$172$		0			0
$173$	21.9344	−	12.6638i	0.126789	−	0.0732014i	−0.435264	−	0.900303i	$-0.643345\pi$
$173$	21.9344	−	12.6638i	0.126789	−	0.0732014i	0.562053	+	0.827101i	$0.310012\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−48.5470	−	28.0286i	−0.271212	−	0.156584i	0.358226	−	0.933635i	$-0.383382\pi$
$179$	−48.5470	−	28.0286i	−0.271212	−	0.156584i	−0.629438	+	0.777050i	$0.716715\pi$
$180$		0			0
$181$	118.307			0.653627			0.326814	−	0.945089i	$-0.394025\pi$
$181$	118.307			0.653627			0.326814	+	0.945089i	$0.394025\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−458.087	−	264.476i	−2.47614	−	1.42960i
$186$		0			0
$187$	−72.1369	−	124.945i	−0.385759	−	0.668154i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	81.5159	−	47.0632i	0.426785	−	0.246404i	−0.271191	−	0.962525i	$-0.587418\pi$
$191$	81.5159	−	47.0632i	0.426785	−	0.246404i	0.697976	+	0.716121i	$0.254084\pi$
$192$		0			0
$193$	152.689	−	264.466i	0.791136	−	1.37029i	−0.134128	−	0.990964i	$-0.542823\pi$
$193$	152.689	−	264.466i	0.791136	−	1.37029i	0.925264	−	0.379324i	$-0.123843\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			369.483i			1.87555i	0.347249	+	0.937773i	$0.387116\pi$
$197$			369.483i			1.87555i	−0.347249	+	0.937773i	$0.612884\pi$
$198$		0			0
$199$	−134.374	+	232.743i	−0.675248	+	1.16956i	0.301149	+	0.953577i	$0.402630\pi$
$199$	−134.374	+	232.743i	−0.675248	+	1.16956i	−0.976396	+	0.215986i	$0.930703\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−183.706	−	318.188i	−0.896128	−	1.55214i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	323.574i		−	1.54820i
$210$		0			0
$211$	296.068			1.40316			0.701582	−	0.712588i	$-0.252477\pi$
$211$	296.068			1.40316			0.701582	+	0.712588i	$0.252477\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	242.885	−	140.230i	1.12970	−	0.652231i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−57.1617	−	33.0023i	−0.258650	−	0.149332i
$222$		0			0
$223$	298.934			1.34051			0.670256	−	0.742130i	$-0.266184\pi$
$223$	298.934			1.34051			0.670256	+	0.742130i	$0.266184\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	114.912	+	66.3446i	0.506221	+	0.292267i	0.731279	−	0.682078i	$-0.238924\pi$
$227$	114.912	+	66.3446i	0.506221	+	0.292267i	−0.225058	+	0.974345i	$0.572257\pi$
$228$		0			0
$229$	−117.647	−	203.771i	−0.513744	−	0.889831i	−0.999873	−	0.0159436i	$-0.994925\pi$
$229$	−117.647	−	203.771i	−0.513744	−	0.889831i	0.486129	−	0.873887i	$-0.338409\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−82.8813	+	47.8516i	−0.355714	+	0.205371i	−0.667199	−	0.744880i	$-0.732507\pi$
$233$	−82.8813	+	47.8516i	−0.355714	+	0.205371i	0.311485	+	0.950251i	$0.399174\pi$
$234$		0			0
$235$	326.051	−	564.737i	1.38745	−	2.40313i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	382.397i		−	1.59999i	−0.600009	−	0.799994i	$-0.704836\pi$
$239$		−	382.397i		−	1.59999i	0.600009	−	0.799994i	$-0.295164\pi$
$240$		0			0
$241$	−4.03490	+	6.98865i	−0.0167423	+	0.0289985i	−0.874275	−	0.485431i	$-0.838663\pi$
$241$	−4.03490	+	6.98865i	−0.0167423	+	0.0289985i	0.857533	+	0.514429i	$0.171996\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−74.0169	−	128.201i	−0.299664	−	0.519033i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	216.264i		−	0.861608i	−0.902445	−	0.430804i	$-0.858230\pi$
$251$		−	216.264i		−	0.861608i	0.902445	−	0.430804i	$-0.141770\pi$
$252$		0			0
$253$	164.689			0.650946
$254$		0			0
$255$		0			0
$256$		0			0
$257$	250.056	−	144.370i	0.972982	−	0.561751i	0.0728377	−	0.997344i	$-0.476795\pi$
$257$	250.056	−	144.370i	0.972982	−	0.561751i	0.900144	+	0.435593i	$0.143461\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	58.4279	+	33.7334i	0.222159	+	0.128264i	0.606950	−	0.794740i	$-0.292393\pi$
$263$	58.4279	+	33.7334i	0.222159	+	0.128264i	−0.384790	+	0.923004i	$0.625726\pi$
$264$		0			0
$265$	−10.4108			−0.0392862
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−50.5834	−	29.2043i	−0.188042	−	0.108566i	0.403023	−	0.915190i	$-0.367959\pi$
$269$	−50.5834	−	29.2043i	−0.188042	−	0.108566i	−0.591066	+	0.806623i	$0.701293\pi$
$270$		0			0
$271$	−69.3324	−	120.087i	−0.255839	−	0.443126i	0.709284	−	0.704923i	$-0.249018\pi$
$271$	−69.3324	−	120.087i	−0.255839	−	0.443126i	−0.965123	+	0.261796i	$0.915685\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	532.826	−	307.627i	1.93755	−	1.11865i
$276$		0			0
$277$	64.3446	−	111.448i	0.232291	−	0.402340i	−0.726191	−	0.687493i	$-0.758711\pi$
$277$	64.3446	−	111.448i	0.232291	−	0.402340i	0.958482	+	0.285153i	$0.0920445\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			225.534i			0.802611i	0.915944	+	0.401306i	$0.131443\pi$
$281$			225.534i			0.802611i	−0.915944	+	0.401306i	$0.868557\pi$
$282$		0			0
$283$	−11.7651	+	20.3778i	−0.0415729	+	0.0720064i	−0.886063	−	0.463565i	$-0.846570\pi$
$283$	−11.7651	+	20.3778i	−0.0415729	+	0.0720064i	0.844490	+	0.535571i	$0.179904\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−23.4831	−	40.6738i	−0.0812562	−	0.140740i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			301.855i			1.03022i	0.857124	+	0.515111i	$0.172249\pi$
$293$			301.855i			1.03022i	−0.857124	+	0.515111i	$0.827751\pi$
$294$		0			0
$295$	−652.102			−2.21051
$296$		0			0
$297$		0			0
$298$		0			0
$299$	65.2504	−	37.6723i	0.218229	−	0.125994i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	394.149	+	227.562i	1.29229	+	0.746104i
$306$		0			0
$307$	−320.611			−1.04434			−0.522168	−	0.852843i	$-0.674877\pi$
$307$	−320.611			−1.04434			−0.522168	+	0.852843i	$0.674877\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−339.400	−	195.953i	−1.09132	−	0.630074i	−0.157392	−	0.987536i	$-0.550309\pi$
$311$	−339.400	−	195.953i	−1.09132	−	0.630074i	−0.933927	+	0.357463i	$0.883642\pi$
$312$		0			0
$313$	−199.648	−	345.800i	−0.637852	−	1.10479i	−0.985903	−	0.167317i	$-0.946490\pi$
$313$	−199.648	−	345.800i	−0.637852	−	1.10479i	0.348051	−	0.937476i	$-0.386844\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−496.345	+	286.565i	−1.56576	+	0.903990i	−0.569101	+	0.822268i	$0.692708\pi$
$317$	−496.345	+	286.565i	−1.56576	+	0.903990i	−0.996655	+	0.0817216i	$0.973958\pi$
$318$		0			0
$319$	−121.689	+	210.772i	−0.381471	+	0.660727i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			542.829i			1.68058i
$324$		0			0
$325$	140.738	−	243.766i	0.433041	−	0.750048i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	211.345	+	366.060i	0.638503	+	1.10592i	0.985761	+	0.168151i	$0.0537795\pi$
$331$	211.345	+	366.060i	0.638503	+	1.10592i	−0.347258	+	0.937770i	$0.612887\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	82.7232i		−	0.246935i
$336$		0			0
$337$	299.345			0.888263			0.444132	−	0.895962i	$-0.353512\pi$
$337$	299.345			0.888263			0.444132	+	0.895962i	$0.353512\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−257.508	+	148.672i	−0.755155	+	0.435989i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−367.908	−	212.412i	−1.06025	−	0.612137i	−0.134751	−	0.990880i	$-0.543023\pi$
$347$	−367.908	−	212.412i	−1.06025	−	0.612137i	−0.925502	+	0.378742i	$0.876357\pi$
$348$		0			0
$349$	182.505			0.522938			0.261469	−	0.965212i	$-0.415793\pi$
$349$	182.505			0.522938			0.261469	+	0.965212i	$0.415793\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	182.875	+	105.583i	0.518059	+	0.299102i	0.736140	−	0.676829i	$-0.236646\pi$
$353$	182.875	+	105.583i	0.518059	+	0.299102i	−0.218081	+	0.975931i	$0.569980\pi$
$354$		0			0
$355$	457.338	+	792.133i	1.28828	+	2.23136i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	80.1504	−	46.2749i	0.223260	−	0.128899i	−0.384199	−	0.923250i	$-0.625522\pi$
$359$	80.1504	−	46.2749i	0.223260	−	0.128899i	0.607459	+	0.794351i	$0.292189\pi$
$360$		0			0
$361$	−428.223	+	741.704i	−1.18621	+	2.05458i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	860.612i		−	2.35784i
$366$		0			0
$367$	−151.808	+	262.940i	−0.413646	+	0.716457i	−0.995285	−	0.0969905i	$-0.969078\pi$
$367$	−151.808	+	262.940i	−0.413646	+	0.716457i	0.581639	+	0.813447i	$0.302412\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	173.723	+	300.897i	0.465746	+	0.806695i	0.999235	−	0.0391117i	$-0.0124528\pi$
$373$	173.723	+	300.897i	0.465746	+	0.806695i	−0.533489	+	0.845807i	$0.679119\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			111.345i			0.295344i
$378$		0			0
$379$	233.379			0.615774			0.307887	−	0.951423i	$-0.400378\pi$
$379$	233.379			0.615774			0.307887	+	0.951423i	$0.400378\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−191.380	+	110.493i	−0.499687	+	0.288495i	−0.728584	−	0.684956i	$-0.759821\pi$
$383$	−191.380	+	110.493i	−0.499687	+	0.288495i	0.228897	+	0.973451i	$0.426488\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−113.442	−	65.4959i	−0.291625	−	0.168370i	0.347049	−	0.937847i	$-0.387184\pi$
$389$	−113.442	−	65.4959i	−0.291625	−	0.168370i	−0.638675	+	0.769477i	$0.720517\pi$
$390$		0			0
$391$	−276.283			−0.706606
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1230.70	+	710.544i	3.11569	+	1.79885i
$396$		0			0
$397$	98.3238	+	170.302i	0.247667	+	0.428972i	0.962878	−	0.269937i	$-0.0870028\pi$
$397$	98.3238	+	170.302i	0.247667	+	0.428972i	−0.715211	+	0.698909i	$0.753669\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−550.775	+	317.990i	−1.37350	+	0.792993i	−0.991368	−	0.131112i	$-0.958145\pi$
$401$	−550.775	+	317.990i	−1.37350	+	0.792993i	−0.382137	+	0.924106i	$0.624812\pi$
$402$		0			0
$403$	−68.0169	+	117.809i	−0.168777	+	0.292330i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	513.245i		−	1.26104i
$408$		0			0
$409$	35.1236	−	60.8359i	0.0858768	−	0.148743i	−0.819888	−	0.572524i	$-0.805964\pi$
$409$	35.1236	−	60.8359i	0.0858768	−	0.148743i	0.905765	+	0.423781i	$0.139298\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−424.757	−	735.701i	−1.02351	−	1.77277i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			436.196i			1.04104i	0.853849	+	0.520520i	$0.174262\pi$
$419$			436.196i			1.04104i	−0.853849	+	0.520520i	$0.825738\pi$
$420$		0			0
$421$	7.37852			0.0175262			0.00876309	−	0.999962i	$-0.497211\pi$
$421$	7.37852			0.0175262			0.00876309	+	0.999962i	$0.497211\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−893.870	+	516.076i	−2.10322	+	1.21430i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−173.595	−	100.225i	−0.402774	−	0.232542i	0.284906	−	0.958555i	$-0.408037\pi$
$431$	−173.595	−	100.225i	−0.402774	−	0.232542i	−0.687680	+	0.726014i	$0.741371\pi$
$432$		0			0
$433$	454.106			1.04874			0.524372	−	0.851489i	$-0.324300\pi$
$433$	454.106			1.04874			0.524372	+	0.851489i	$0.324300\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−536.626	−	309.821i	−1.22798	−	0.708973i
$438$		0			0
$439$	131.106	+	227.083i	0.298648	+	0.517273i	0.975827	−	0.218545i	$-0.0701311\pi$
$439$	131.106	+	227.083i	0.298648	+	0.517273i	−0.677179	+	0.735818i	$0.736798\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	201.499	−	116.336i	0.454852	−	0.262609i	−0.255025	−	0.966934i	$-0.582084\pi$
$443$	201.499	−	116.336i	0.454852	−	0.262609i	0.709877	+	0.704326i	$0.248750\pi$
$444$		0			0
$445$	−364.362	+	631.093i	−0.818790	+	1.41819i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			92.0197i			0.204944i	0.994736	+	0.102472i	$0.0326752\pi$
$449$			92.0197i			0.204944i	−0.994736	+	0.102472i	$0.967325\pi$
$450$		0			0
$451$	178.251	−	308.739i	0.395235	−	0.684566i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	347.412	+	601.736i	0.760202	+	1.31671i	0.942746	+	0.333511i	$0.108233\pi$
$457$	347.412	+	601.736i	0.760202	+	1.31671i	−0.182544	+	0.983198i	$0.558433\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			432.476i			0.938127i	0.883164	+	0.469063i	$0.155408\pi$
$461$			432.476i			0.938127i	−0.883164	+	0.469063i	$0.844592\pi$
$462$		0			0
$463$	−310.169			−0.669912			−0.334956	−	0.942234i	$-0.608722\pi$
$463$	−310.169			−0.669912			−0.334956	+	0.942234i	$0.608722\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−360.097	+	207.902i	−0.771086	+	0.445186i	−0.833262	−	0.552879i	$-0.813529\pi$
$467$	−360.097	+	207.902i	−0.771086	+	0.445186i	0.0621762	+	0.998065i	$0.480196\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	235.672	+	136.065i	0.498250	+	0.287665i
$474$		0			0
$475$	−2314.89			−4.87346
$476$		0			0
$477$		0			0
$478$		0			0
$479$	247.993	+	143.179i	0.517731	+	0.298912i	0.736006	−	0.676975i	$-0.236710\pi$
$479$	247.993	+	143.179i	0.517731	+	0.298912i	−0.218275	+	0.975887i	$0.570043\pi$
$480$		0			0
$481$	−117.404	−	203.349i	−0.244083	−	0.422763i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−105.547	+	60.9377i	−0.217623	+	0.125645i
$486$		0			0
$487$	−242.706	+	420.379i	−0.498370	+	0.863202i	−0.999998	−	0.00188110i	$-0.999401\pi$
$487$	−242.706	+	420.379i	−0.498370	+	0.863202i	0.501628	+	0.865083i	$0.332735\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	138.130i		−	0.281323i	−0.990058	−	0.140661i	$-0.955077\pi$
$491$		−	138.130i		−	0.281323i	0.990058	−	0.140661i	$-0.0449229\pi$
$492$		0			0
$493$	204.146	−	353.591i	0.414089	−	0.717224i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	72.6723	+	125.872i	0.145636	+	0.252249i	0.929610	−	0.368545i	$-0.120144\pi$
$499$	72.6723	+	125.872i	0.145636	+	0.252249i	−0.783974	+	0.620794i	$0.786811\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			619.115i			1.23084i	0.788198	+	0.615422i	$0.211015\pi$
$503$			619.115i			1.23084i	−0.788198	+	0.615422i	$0.788985\pi$
$504$		0			0
$505$	51.2430			0.101471
$506$		0			0
$507$		0			0
$508$		0			0
$509$	200.851	−	115.961i	0.394599	−	0.227822i	−0.289552	−	0.957162i	$-0.593506\pi$
$509$	200.851	−	115.961i	0.394599	−	0.227822i	0.684151	+	0.729340i	$0.260173\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1131.59	+	653.324i	2.19726	+	1.26859i
$516$		0			0
$517$	632.737			1.22386
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−45.7558	−	26.4171i	−0.0878231	−	0.0507047i	0.455445	−	0.890264i	$-0.349480\pi$
$521$	−45.7558	−	26.4171i	−0.0878231	−	0.0507047i	−0.543268	+	0.839559i	$0.682813\pi$
$522$		0			0
$523$	−28.3082	−	49.0313i	−0.0541266	−	0.0937501i	0.837693	−	0.546142i	$-0.183904\pi$
$523$	−28.3082	−	49.0313i	−0.0541266	−	0.0937501i	−0.891819	+	0.452392i	$0.850571\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	431.996	−	249.413i	0.819727	−	0.473269i
$528$		0			0
$529$	−106.811	+	185.002i	−0.201911	+	0.349720i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	163.098i		−	0.306000i
$534$		0			0
$535$	−712.824	+	1234.65i	−1.33238	+	2.30775i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−45.6215	−	79.0187i	−0.0843281	−	0.146060i	0.820777	−	0.571249i	$-0.193541\pi$
$541$	−45.6215	−	79.0187i	−0.0843281	−	0.146060i	−0.905105	+	0.425189i	$0.860208\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			274.196i			0.503112i
$546$		0			0
$547$	356.825			0.652331			0.326165	−	0.945313i	$-0.394243\pi$
$547$	356.825			0.652331			0.326165	+	0.945313i	$0.394243\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	793.028	−	457.855i	1.43925	−	0.830953i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	812.850	+	469.299i	1.45934	+	0.842548i	0.998979	−	0.0451859i	$-0.0143880\pi$
$557$	812.850	+	469.299i	1.45934	+	0.842548i	0.460357	+	0.887734i	$0.347721\pi$
$558$		0			0
$559$	124.499			0.222717
$560$		0			0
$561$		0			0
$562$		0			0
$563$	877.413	+	506.574i	1.55846	+	0.899777i	0.997405	+	0.0719980i	$0.0229375\pi$
$563$	877.413	+	506.574i	1.55846	+	0.899777i	0.561055	+	0.827779i	$0.310396\pi$
$564$		0			0
$565$	−696.316	−	1206.06i	−1.23242	−	2.13461i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−837.689	+	483.640i	−1.47221	+	0.849982i	−0.999512	−	0.0312415i	$-0.990054\pi$
$569$	−837.689	+	483.640i	−1.47221	+	0.849982i	−0.472700	+	0.881223i	$0.656721\pi$
$570$		0			0
$571$	315.757	−	546.907i	0.552990	−	0.957806i	−0.445067	−	0.895497i	$-0.646820\pi$
$571$	315.757	−	546.907i	0.552990	−	0.957806i	0.998057	−	0.0623089i	$-0.0198464\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1178.21i		−	2.04906i
$576$		0			0
$577$	−307.176	+	532.044i	−0.532367	+	0.922087i	0.466918	+	0.884300i	$0.345364\pi$
$577$	−307.176	+	532.044i	−0.532367	+	0.922087i	−0.999286	+	0.0377869i	$0.987969\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−5.05084	−	8.74831i	−0.00866353	−	0.0150057i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			411.243i			0.700584i	0.936641	+	0.350292i	$0.113918\pi$
$587$			411.243i			0.700584i	−0.936641	+	0.350292i	$0.886082\pi$
$588$		0			0
$589$	1118.76			1.89942
$590$		0			0
$591$		0			0
$592$		0			0
$593$	330.755	−	190.961i	0.557765	−	0.322026i	−0.194483	−	0.980906i	$-0.562303\pi$
$593$	330.755	−	190.961i	0.557765	−	0.322026i	0.752248	+	0.658880i	$0.228970\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	212.423	+	122.642i	0.354629	+	0.204745i	0.666722	−	0.745306i	$-0.267697\pi$
$599$	212.423	+	122.642i	0.354629	+	0.204745i	−0.312093	+	0.950052i	$0.601030\pi$
$600$		0			0
$601$	−168.243			−0.279939			−0.139970	−	0.990156i	$-0.544700\pi$
$601$	−168.243			−0.279939			−0.139970	+	0.990156i	$0.544700\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−289.694	−	167.255i	−0.478834	−	0.276455i
$606$		0			0
$607$	−279.183	−	483.560i	−0.459940	−	0.796639i	0.539018	−	0.842294i	$-0.318796\pi$
$607$	−279.183	−	483.560i	−0.459940	−	0.796639i	−0.998957	+	0.0456558i	$0.985462\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	250.692	−	144.737i	0.410298	−	0.236886i
$612$		0			0
$613$	130.085	−	225.313i	0.212210	−	0.367559i	−0.740196	−	0.672391i	$-0.765267\pi$
$613$	130.085	−	225.313i	0.212210	−	0.367559i	0.952406	+	0.304833i	$0.0986006\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			351.007i			0.568893i	0.958692	+	0.284447i	$0.0918099\pi$
$617$			351.007i			0.568893i	−0.958692	+	0.284447i	$0.908190\pi$
$618$		0			0
$619$	158.903	−	275.229i	0.256710	−	0.444634i	−0.708649	−	0.705561i	$-0.750695\pi$
$619$	158.903	−	275.229i	0.256710	−	0.444634i	0.965358	+	0.260927i	$0.0840283\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1059.00	−	1834.24i	−1.69440	−	2.93478i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			861.021i			1.36887i
$630$		0			0
$631$	297.966			0.472213			0.236106	−	0.971727i	$-0.424129\pi$
$631$	297.966			0.472213			0.236106	+	0.971727i	$0.424129\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−987.814	+	570.314i	−1.55561	+	0.898133i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	659.835	+	380.956i	1.02938	+	0.594315i	0.916808	−	0.399328i	$-0.130756\pi$
$641$	659.835	+	380.956i	1.02938	+	0.594315i	0.112576	+	0.993643i	$0.464090\pi$
$642$		0			0
$643$	−21.5808			−0.0335626			−0.0167813	−	0.999859i	$-0.505342\pi$
$643$	−21.5808			−0.0335626			−0.0167813	+	0.999859i	$0.505342\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	90.5462	+	52.2768i	0.139948	+	0.0807988i	0.568339	−	0.822794i	$-0.307586\pi$
$647$	90.5462	+	52.2768i	0.139948	+	0.0807988i	−0.428391	+	0.903593i	$0.640920\pi$
$648$		0			0
$649$	−316.368	−	547.966i	−0.487471	−	0.844324i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−311.528	+	179.861i	−0.477072	+	0.275438i	−0.719195	−	0.694808i	$-0.755489\pi$
$653$	−311.528	+	179.861i	−0.477072	+	0.275438i	0.242123	+	0.970245i	$0.422156\pi$
$654$		0			0
$655$	458.757	−	794.591i	0.700392	−	1.21312i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			983.046i			1.49172i	0.666100	+	0.745862i	$0.267962\pi$
$659$			983.046i			1.49172i	−0.666100	+	0.745862i	$0.732038\pi$
$660$		0			0
$661$	−255.813	+	443.081i	−0.387009	+	0.670319i	−0.992046	−	0.125879i	$-0.959825\pi$
$661$	−255.813	+	443.081i	−0.387009	+	0.670319i	0.605037	+	0.796197i	$0.293158\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	233.034	+	403.627i	0.349376	+	0.605137i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			441.608i			0.658135i
$672$		0			0
$673$	−49.8644			−0.0740928			−0.0370464	−	0.999314i	$-0.511795\pi$
$673$	−49.8644			−0.0740928			−0.0370464	+	0.999314i	$0.511795\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−488.482	+	282.025i	−0.721540	+	0.416581i	−0.815319	−	0.579012i	$-0.803438\pi$
$677$	−488.482	+	282.025i	−0.721540	+	0.416581i	0.0937794	+	0.995593i	$0.470105\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−114.683	−	66.2124i	−0.167911	−	0.0969434i	0.413689	−	0.910418i	$-0.364240\pi$
$683$	−114.683	−	66.2124i	−0.167911	−	0.0969434i	−0.581600	+	0.813475i	$0.697573\pi$
$684$		0			0
$685$	−1644.55			−2.40081
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.00232	−	2.31074i	−0.00580888	−	0.00335376i
$690$		0			0
$691$	−551.056	−	954.457i	−0.797476	−	1.38127i	−0.921255	−	0.388959i	$-0.872835\pi$
$691$	−551.056	−	954.457i	−0.797476	−	1.38127i	0.123779	−	0.992310i	$-0.460499\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1748.14	−	1009.29i	2.51531	−	1.45222i
$696$		0			0
$697$	−299.034	+	517.942i	−0.429030	+	0.743102i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	302.130i		−	0.430999i	−0.976504	−	0.215500i	$-0.930862\pi$
$701$		−	302.130i		−	0.430999i	0.976504	−	0.215500i	$-0.0691380\pi$
$702$		0			0
$703$	−965.540	+	1672.36i	−1.37346	+	2.37890i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6.31074	+	10.9305i	0.00890090	+	0.0154168i	0.870442	−	0.492272i	$-0.163833\pi$
$709$	6.31074	+	10.9305i	0.00890090	+	0.0154168i	−0.861541	+	0.507689i	$0.830500\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			569.412i			0.798615i
$714$		0			0
$715$	376.034			0.525922
$716$		0			0
$717$		0			0
$718$		0			0
$719$	657.840	−	379.804i	0.914938	−	0.528239i	0.0329210	−	0.999458i	$-0.489519\pi$
$719$	657.840	−	379.804i	0.914938	−	0.528239i	0.882016	+	0.471219i	$0.156186\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1507.89	+	870.580i	2.07985	+	1.20080i
$726$		0			0
$727$	829.041			1.14036			0.570179	−	0.821520i	$-0.306874\pi$
$727$	829.041			1.14036			0.570179	+	0.821520i	$0.306874\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−395.364	−	228.264i	−0.540854	−	0.312262i
$732$		0			0
$733$	677.260	+	1173.05i	0.923957	+	1.60034i	0.793230	+	0.608922i	$0.208398\pi$
$733$	677.260	+	1173.05i	0.923957	+	1.60034i	0.130727	+	0.991418i	$0.458269\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	69.5129	−	40.1333i	0.0943187	−	0.0544549i
$738$		0			0
$739$	−17.0169	+	29.4742i	−0.0230270	+	0.0398839i	−0.877309	−	0.479925i	$-0.840664\pi$
$739$	−17.0169	+	29.4742i	−0.0230270	+	0.0398839i	0.854282	+	0.519809i	$0.173997\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	61.0163i		−	0.0821215i	−0.999157	−	0.0410607i	$-0.986926\pi$
$743$		−	61.0163i		−	0.0821215i	0.999157	−	0.0410607i	$-0.0130737\pi$
$744$		0			0
$745$	721.796	−	1250.19i	0.968854	−	1.67810i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	70.0678	+	121.361i	0.0932993	+	0.161599i	0.908898	−	0.417019i	$-0.136925\pi$
$751$	70.0678	+	121.361i	0.0932993	+	0.161599i	−0.815598	+	0.578619i	$0.803592\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			115.013i			0.152335i
$756$		0			0
$757$	518.000			0.684280			0.342140	−	0.939649i	$-0.388848\pi$
$757$	518.000			0.684280			0.342140	+	0.939649i	$0.388848\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−117.905	+	68.0725i	−0.154934	+	0.0894514i	−0.575463	−	0.817828i	$-0.695178\pi$
$761$	−117.905	+	68.0725i	−0.154934	+	0.0894514i	0.420528	+	0.907279i	$0.361845\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−250.692	−	144.737i	−0.326848	−	0.188706i
$768$		0			0
$769$	−237.244			−0.308510			−0.154255	−	0.988031i	$-0.549298\pi$
$769$	−237.244			−0.308510			−0.154255	+	0.988031i	$0.549298\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−747.641	−	431.651i	−0.967194	−	0.558410i	−0.0688142	−	0.997629i	$-0.521922\pi$
$773$	−747.641	−	431.651i	−0.967194	−	0.558410i	−0.898380	+	0.439220i	$0.855255\pi$
$774$		0			0
$775$	1063.62	+	1842.24i	1.37241	+	2.37709i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1161.63	+	670.667i	−1.49118	+	0.860934i
$780$		0			0
$781$	−443.757	+	768.610i	−0.568191	+	0.984135i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	1536.42i		−	1.95723i
$786$		0			0
$787$	382.421	−	662.372i	0.485922	−	0.841642i	−0.513947	−	0.857822i	$-0.671817\pi$
$787$	382.421	−	662.372i	0.485922	−	0.841642i	0.999869	+	0.0161799i	$0.00515044\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	101.017	+	174.966i	0.127386	+	0.220639i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1413.60i		−	1.77365i	−0.462106	−	0.886825i	$-0.652906\pi$
$797$		−	1413.60i		−	1.77365i	0.462106	−	0.886825i	$-0.347094\pi$
$798$		0			0
$799$	−1061.48			−1.32851
$800$		0			0
$801$		0			0
$802$		0			0
$803$	723.178	−	417.527i	0.900596	−	0.519959i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−624.645	−	360.639i	−0.772120	−	0.445784i	0.0615102	−	0.998106i	$-0.480408\pi$
$809$	−624.645	−	360.639i	−0.772120	−	0.445784i	−0.833630	+	0.552323i	$0.813742\pi$
$810$		0			0
$811$	928.795			1.14525			0.572623	−	0.819819i	$-0.305926\pi$
$811$	928.795			1.14525			0.572623	+	0.819819i	$0.305926\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2135.74	+	1233.07i	2.62054	+	1.51297i
$816$		0			0
$817$	−511.945	−	886.715i	−0.626616	−	1.08533i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−876.578	+	506.093i	−1.06770	+	0.616434i	−0.927551	−	0.373697i	$-0.878090\pi$
$821$	−876.578	+	506.093i	−1.06770	+	0.616434i	−0.140145	+	0.990131i	$0.544757\pi$
$822$		0			0
$823$	−665.791	+	1153.18i	−0.808980	+	1.40120i	0.104590	+	0.994515i	$0.466647\pi$
$823$	−665.791	+	1153.18i	−0.808980	+	1.40120i	−0.913571	+	0.406680i	$0.866686\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	96.8217i		−	0.117076i	−0.998285	−	0.0585379i	$-0.981356\pi$
$827$		−	96.8217i		−	0.117076i	0.998285	−	0.0585379i	$-0.0186438\pi$
$828$		0			0
$829$	137.422	−	238.023i	0.165769	−	0.287120i	−0.771159	−	0.636642i	$-0.780323\pi$
$829$	137.422	−	238.023i	0.165769	−	0.287120i	0.936928	+	0.349522i	$0.113656\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	515.328	+	892.574i	0.617159	+	1.06895i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1196.53i			1.42614i	0.701091	+	0.713072i	$0.252697\pi$
$839$			1196.53i			1.42614i	−0.701091	+	0.713072i	$0.747303\pi$
$840$		0			0
$841$	152.243			0.181026
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1249.82	+	721.587i	−1.47908	+	0.853949i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−851.182	−	491.430i	−1.00021	−	0.577473i
$852$		0			0
$853$	−761.454			−0.892678			−0.446339	−	0.894864i	$-0.647272\pi$
$853$	−761.454			−0.892678			−0.446339	+	0.894864i	$0.647272\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	177.030	+	102.208i	0.206569	+	0.119263i	0.599716	−	0.800213i	$-0.295280\pi$
$857$	177.030	+	102.208i	0.206569	+	0.119263i	−0.393147	+	0.919476i	$0.628614\pi$
$858$		0			0
$859$	28.2244	+	48.8860i	0.0328572	+	0.0569104i	0.881986	−	0.471275i	$-0.156206\pi$
$859$	28.2244	+	48.8860i	0.0328572	+	0.0569104i	−0.849129	+	0.528185i	$0.822873\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−768.748	+	443.837i	−0.890785	+	0.514295i	−0.874199	−	0.485567i	$-0.838613\pi$
$863$	−768.748	+	443.837i	−0.890785	+	0.514295i	−0.0165860	+	0.999862i	$0.505280\pi$
$864$		0			0
$865$	121.034	−	209.637i	0.139924	−	0.242355i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1378.89i			1.58675i
$870$		0			0
$871$	18.3608	−	31.8019i	0.0210801	−	0.0365119i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−38.5706	−	66.8063i	−0.0439802	−	0.0761759i	0.843197	−	0.537604i	$-0.180671\pi$
$877$	−38.5706	−	66.8063i	−0.0439802	−	0.0761759i	−0.887178	+	0.461428i	$0.847337\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	918.740i		−	1.04284i	−0.853301	−	0.521419i	$-0.825403\pi$
$881$		−	918.740i		−	1.04284i	0.853301	−	0.521419i	$-0.174597\pi$
$882$		0			0
$883$	6.72316			0.00761399			0.00380700	−	0.999993i	$-0.498788\pi$
$883$	6.72316			0.00761399			0.00380700	+	0.999993i	$0.498788\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−136.610	+	78.8719i	−0.154014	+	0.0889199i	−0.575026	−	0.818135i	$-0.695008\pi$
$887$	−136.610	+	78.8719i	−0.154014	+	0.0889199i	0.421013	+	0.907055i	$0.361675\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2061.72	−	1190.33i	−2.30876	−	1.33296i
$894$		0			0
$895$	−535.763			−0.598618
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−728.743	−	420.740i	−0.810615	−	0.468009i
$900$		0			0
$901$	8.47330	+	14.6762i	0.00940433	+	0.0162888i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	979.222	−	565.354i	1.08201	−	0.624700i
$906$		0			0
$907$	−306.085	+	530.154i	−0.337469	+	0.584514i	−0.983956	−	0.178411i	$-0.942904\pi$
$907$	−306.085	+	530.154i	−0.337469	+	0.584514i	0.646487	+	0.762925i	$0.276238\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	570.650i		−	0.626399i	−0.949687	−	0.313200i	$-0.898599\pi$
$911$		−	570.650i		−	0.626399i	0.949687	−	0.313200i	$-0.101401\pi$
$912$		0			0
$913$	412.143	−	713.853i	0.451417	−	0.781876i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	633.463	+	1097.19i	0.689296	+	1.19390i	0.972066	+	0.234708i	$0.0754135\pi$
$919$	633.463	+	1097.19i	0.689296	+	1.19390i	−0.282770	+	0.959188i	$0.591253\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			406.034i			0.439907i
$924$		0			0
$925$	−3671.82			−3.96953
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1192.81	+	688.668i	−1.28397	+	0.741300i	−0.977572	−	0.210604i	$-0.932457\pi$
$929$	−1192.81	+	688.668i	−1.28397	+	0.741300i	−0.306398	+	0.951904i	$0.599124\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1194.15	−	689.444i	−1.27717	−	0.737373i
$936$		0			0
$937$	754.751			0.805497			0.402748	−	0.915311i	$-0.368055\pi$
$937$	754.751			0.805497			0.402748	+	0.915311i	$0.368055\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	658.289	+	380.063i	0.699563	+	0.403893i	0.807185	−	0.590299i	$-0.200990\pi$
$941$	658.289	+	380.063i	0.699563	+	0.403893i	−0.107622	+	0.994192i	$0.534324\pi$
$942$		0			0
$943$	−341.349	−	591.233i	−0.361982	−	0.626971i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.0058	−	19.0559i	0.0348531	−	0.0201224i	−0.482472	−	0.875911i	$-0.660261\pi$
$947$	33.0058	−	19.0559i	0.0348531	−	0.0201224i	0.517325	+	0.855789i	$0.326928\pi$
$948$		0			0
$949$	191.017	−	330.851i	0.201282	−	0.348631i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1140.96i		−	1.19723i	−0.801039	−	0.598613i	$-0.795719\pi$
$953$		−	1140.96i		−	1.19723i	0.801039	−	0.598613i	$-0.204281\pi$
$954$		0			0
$955$	449.804	−	779.083i	0.470999	−	0.815794i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−33.5339	−	58.0824i	−0.0348948	−	0.0604395i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	2918.64i		−	3.02449i
$966$		0			0
$967$	1212.79			1.25418			0.627089	−	0.778947i	$-0.284246\pi$
$967$	1212.79			1.25418			0.627089	+	0.778947i	$0.284246\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−411.462	+	237.557i	−0.423750	+	0.244652i	−0.696681	−	0.717381i	$-0.745341\pi$
$971$	−411.462	+	237.557i	−0.423750	+	0.244652i	0.272930	+	0.962034i	$0.412007\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1156.85	+	667.908i	1.18409	+	0.683632i	0.956956	−	0.290233i	$-0.0937328\pi$
$977$	1156.85	+	667.908i	1.18409	+	0.683632i	0.227129	+	0.973865i	$0.427066\pi$
$978$		0			0
$979$	−707.083			−0.722250
$980$		0			0
$981$		0			0
$982$		0			0
$983$	94.1799	+	54.3748i	0.0958086	+	0.0553151i	0.547139	−	0.837042i	$-0.315717\pi$
$983$	94.1799	+	54.3748i	0.0958086	+	0.0553151i	−0.451330	+	0.892357i	$0.649050\pi$
$984$		0			0
$985$	1765.65	+	3058.20i	1.79254	+	3.10477i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	451.310	−	260.564i	0.456330	−	0.263462i
$990$		0			0
$991$	−275.429	+	477.058i	−0.277931	+	0.481390i	−0.970870	−	0.239605i	$-0.922982\pi$
$991$	−275.429	+	477.058i	−0.277931	+	0.481390i	0.692940	+	0.720996i	$0.256315\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			2568.55i			2.58145i
$996$		0			0
$997$	−35.7308	+	61.8876i	−0.0358384	+	0.0620739i	−0.883388	−	0.468642i	$-0.844743\pi$
$997$	−35.7308	+	61.8876i	−0.0358384	+	0.0620739i	0.847550	+	0.530716i	$0.178077\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.bk.g.557.8		16
3.2	odd	2	inner	1764.3.bk.g.557.1		16
7.2	even	3	inner	1764.3.bk.g.1745.1		16
7.3	odd	6		1764.3.c.h.197.1	✓	8
7.4	even	3		1764.3.c.h.197.7	yes	8
7.5	odd	6	inner	1764.3.bk.g.1745.7		16
7.6	odd	2	inner	1764.3.bk.g.557.2		16
21.2	odd	6	inner	1764.3.bk.g.1745.8		16
21.5	even	6	inner	1764.3.bk.g.1745.2		16
21.11	odd	6		1764.3.c.h.197.2	yes	8
21.17	even	6		1764.3.c.h.197.8	yes	8
21.20	even	2	inner	1764.3.bk.g.557.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.c.h.197.1	✓	8	7.3	odd	6
1764.3.c.h.197.2	yes	8	21.11	odd	6
1764.3.c.h.197.7	yes	8	7.4	even	3
1764.3.c.h.197.8	yes	8	21.17	even	6
1764.3.bk.g.557.1		16	3.2	odd	2	inner
1764.3.bk.g.557.2		16	7.6	odd	2	inner
1764.3.bk.g.557.7		16	21.20	even	2	inner
1764.3.bk.g.557.8		16	1.1	even	1	trivial
1764.3.bk.g.1745.1		16	7.2	even	3	inner
1764.3.bk.g.1745.2		16	21.5	even	6	inner
1764.3.bk.g.1745.7		16	7.5	odd	6	inner
1764.3.bk.g.1745.8		16	21.2	odd	6	inner

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.bk (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(8.27698 - 4.77872i) q^{5} +O(q^{10})\)	\(q+(8.27698 - 4.77872i) q^{5} +(8.03119 + 4.63681i) q^{11} +4.24264 q^{13} +(-13.4731 - 7.77872i) q^{17} +(-17.4460 - 30.2173i) q^{19} +(15.3797 - 8.87945i) q^{23} +(33.1723 - 57.4561i) q^{25} +26.2442i q^{29} +(-16.0317 + 27.7678i) q^{31} +(-27.6723 - 47.9299i) q^{37} -38.4426i q^{41} +29.3446 q^{43} +(59.0887 - 34.1149i) q^{47} +(-0.943355 - 0.544646i) q^{53} +88.6320 q^{55} +(-59.0887 - 34.1149i) q^{59} +(23.8099 + 41.2400i) q^{61} +(35.1163 - 20.2744i) q^{65} +(4.32768 - 7.49577i) q^{67} +95.7031i q^{71} +(45.0231 - 77.9823i) q^{73} +(74.3446 + 128.769i) q^{79} -88.8851i q^{83} -148.689 q^{85} +(-66.0316 + 38.1234i) q^{89} +(-288.800 - 166.739i) q^{95} -12.7519 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 216 q^{25} - 128 q^{37} - 160 q^{43} + 384 q^{67} + 560 q^{79} - 1120 q^{85}+O(q^{100}) \) 16 * q + 216 * q^25 - 128 * q^37 - 160 * q^43 + 384 * q^67 + 560 * q^79 - 1120 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more