LMFDB - Embedded newform 1840.2.e.e.369.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (of order $2$, degree $1$, 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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.11574317056.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 45x^{4} + 4 $ x^8 + 45*x^4 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.3
Root			$-0.386289 - 0.386289i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.e.369.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.44270i q^{3} +(-0.386289 - 2.20245i) q^{5} +3.25886i q^{7} +0.918614 q^{9} +O(q^{10})$	$q-1.44270i q^{3} +(-0.386289 - 2.20245i) q^{5} +3.25886i q^{7} +0.918614 q^{9} -1.32988 q^{11} -1.25886i q^{13} +(-3.17748 + 0.557299i) q^{15} -4.62018i q^{17} -3.37169 q^{19} +4.70156 q^{21} -1.00000i q^{23} +(-4.70156 + 1.70156i) q^{25} -5.65339i q^{27} -4.29207 q^{29} -2.37346 q^{31} +1.91861i q^{33} +(7.17748 - 1.25886i) q^{35} +5.74514i q^{37} -1.81616 q^{39} -5.13790 q^{41} -10.4678i q^{43} +(-0.354850 - 2.02320i) q^{45} -6.06288i q^{47} -3.62018 q^{49} -6.66553 q^{51} -11.1275i q^{53} +(0.513716 + 2.92898i) q^{55} +4.86433i q^{57} -2.72263 q^{59} +12.3653 q^{61} +2.99364i q^{63} +(-2.77258 + 0.486284i) q^{65} -6.60981i q^{67} -1.44270 q^{69} -0.265226 q^{71} +5.26464i q^{73} +(2.45485 + 6.78295i) q^{75} -4.33388i q^{77} -15.3275 q^{79} -5.40031 q^{81} +2.02566i q^{83} +(-10.1757 + 1.78472i) q^{85} +6.19218i q^{87} -16.1500 q^{89} +4.10245 q^{91} +3.42419i q^{93} +(1.30244 + 7.42597i) q^{95} +8.62018i q^{97} -1.22164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{9}+O(q^{10}) $ 8 * q - 14 * q^9	$ 8 q - 14 q^{9} - 10 q^{11} + 16 q^{15} - 2 q^{19} + 12 q^{21} - 12 q^{25} - 4 q^{29} - 10 q^{31} + 16 q^{35} + 46 q^{41} - 26 q^{45} + 18 q^{49} - 14 q^{51} + 18 q^{55} + 32 q^{59} + 18 q^{61} - 16 q^{65} - 6 q^{69} - 38 q^{71} + 32 q^{75} - 12 q^{79} + 32 q^{81} - 24 q^{85} - 60 q^{89} + 26 q^{91} - 18 q^{95} - 54 q^{99}+O(q^{100}) $ 8 * q - 14 * q^9 - 10 * q^11 + 16 * q^15 - 2 * q^19 + 12 * q^21 - 12 * q^25 - 4 * q^29 - 10 * q^31 + 16 * q^35 + 46 * q^41 - 26 * q^45 + 18 * q^49 - 14 * q^51 + 18 * q^55 + 32 * q^59 + 18 * q^61 - 16 * q^65 - 6 * q^69 - 38 * q^71 + 32 * q^75 - 12 * q^79 + 32 * q^81 - 24 * q^85 - 60 * q^89 + 26 * q^91 - 18 * q^95 - 54 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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			0
$2$		0			0
$3$		−	1.44270i		−	0.832944i	−0.909149	−	0.416472i	$-0.863266\pi$
$3$		−	1.44270i		−	0.832944i	0.909149	−	0.416472i	$-0.136734\pi$
$4$		0			0
$5$	−0.386289	−	2.20245i	−0.172754	−	0.984965i
$5$	−0.386289	−	2.20245i	−0.172754	−	0.984965i
$6$		0			0
$7$			3.25886i			1.23173i	0.787850	+	0.615867i	$0.211194\pi$
$7$			3.25886i			1.23173i	−0.787850	+	0.615867i	$0.788806\pi$
$8$		0			0
$9$	0.918614			0.306205
$10$		0			0
$11$	−1.32988			−0.400973			−0.200486	−	0.979696i	$-0.564252\pi$
$11$	−1.32988			−0.400973			−0.200486	+	0.979696i	$0.564252\pi$
$12$		0			0
$13$		−	1.25886i		−	0.349145i	−0.984644	−	0.174573i	$-0.944146\pi$
$13$		−	1.25886i		−	0.349145i	0.984644	−	0.174573i	$-0.0558544\pi$
$14$		0			0
$15$	−3.17748	+	0.557299i	−0.820421	+	0.143894i
$16$		0			0
$17$		−	4.62018i		−	1.12056i	−0.828304	−	0.560279i	$-0.810694\pi$
$17$		−	4.62018i		−	1.12056i	0.828304	−	0.560279i	$-0.189306\pi$
$18$		0			0
$19$	−3.37169			−0.773518			−0.386759	−	0.922181i	$-0.626405\pi$
$19$	−3.37169			−0.773518			−0.386759	+	0.922181i	$0.626405\pi$
$20$		0			0
$21$	4.70156			1.02596
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.70156	+	1.70156i	−0.940312	+	0.340312i
$26$		0			0
$27$		−	5.65339i		−	1.08800i
$28$		0			0
$29$	−4.29207			−0.797018			−0.398509	−	0.917164i	$-0.630472\pi$
$29$	−4.29207			−0.797018			−0.398509	+	0.917164i	$0.630472\pi$
$30$		0			0
$31$	−2.37346			−0.426286			−0.213143	−	0.977021i	$-0.568370\pi$
$31$	−2.37346			−0.426286			−0.213143	+	0.977021i	$0.568370\pi$
$32$		0			0
$33$			1.91861i			0.333988i
$34$		0			0
$35$	7.17748	−	1.25886i	1.21321	−	0.212786i
$36$		0			0
$37$			5.74514i			0.944496i	0.881466	+	0.472248i	$0.156557\pi$
$37$			5.74514i			0.944496i	−0.881466	+	0.472248i	$0.843443\pi$
$38$		0			0
$39$	−1.81616			−0.290818
$40$		0			0
$41$	−5.13790			−0.802405			−0.401202	−	0.915989i	$-0.631408\pi$
$41$	−5.13790			−0.802405			−0.401202	+	0.915989i	$0.631408\pi$
$42$		0			0
$43$		−	10.4678i		−	1.59632i	−0.602445	−	0.798160i	$-0.705807\pi$
$43$		−	10.4678i		−	1.59632i	0.602445	−	0.798160i	$-0.294193\pi$
$44$		0			0
$45$	−0.354850	−	2.02320i	−0.0528979	−	0.301601i
$46$		0			0
$47$		−	6.06288i		−	0.884362i	−0.896926	−	0.442181i	$-0.854205\pi$
$47$		−	6.06288i		−	0.884362i	0.896926	−	0.442181i	$-0.145795\pi$
$48$		0			0
$49$	−3.62018			−0.517168
$50$		0			0
$51$	−6.66553			−0.933361
$52$		0			0
$53$		−	11.1275i		−	1.52848i	−0.644930	−	0.764242i	$-0.723113\pi$
$53$		−	11.1275i		−	1.52848i	0.644930	−	0.764242i	$-0.276887\pi$
$54$		0			0
$55$	0.513716	+	2.92898i	0.0692695	+	0.394944i
$56$		0			0
$57$			4.86433i			0.644297i
$58$		0			0
$59$	−2.72263			−0.354456			−0.177228	−	0.984170i	$-0.556713\pi$
$59$	−2.72263			−0.354456			−0.177228	+	0.984170i	$0.556713\pi$
$60$		0			0
$61$	12.3653			1.58322			0.791609	−	0.611029i	$-0.209244\pi$
$61$	12.3653			1.58322			0.791609	+	0.611029i	$0.209244\pi$
$62$		0			0
$63$			2.99364i			0.377163i
$64$		0			0
$65$	−2.77258	+	0.486284i	−0.343896	+	0.0603161i
$66$		0			0
$67$		−	6.60981i		−	0.807516i	−0.914866	−	0.403758i	$-0.867704\pi$
$67$		−	6.60981i		−	0.807516i	0.914866	−	0.403758i	$-0.132296\pi$
$68$		0			0
$69$	−1.44270			−0.173681
$70$		0			0
$71$	−0.265226			−0.0314765			−0.0157383	−	0.999876i	$-0.505010\pi$
$71$	−0.265226			−0.0314765			−0.0157383	+	0.999876i	$0.505010\pi$
$72$		0			0
$73$			5.26464i			0.616180i	0.951357	+	0.308090i	$0.0996897\pi$
$73$			5.26464i			0.616180i	−0.951357	+	0.308090i	$0.900310\pi$
$74$		0			0
$75$	2.45485	+	6.78295i	0.283461	+	0.783227i
$76$		0			0
$77$		−	4.33388i		−	0.493892i
$78$		0			0
$79$	−15.3275			−1.72448			−0.862240	−	0.506500i	$-0.830939\pi$
$79$	−15.3275			−1.72448			−0.862240	+	0.506500i	$0.830939\pi$
$80$		0			0
$81$	−5.40031			−0.600034
$82$		0			0
$83$			2.02566i			0.222345i	0.993801	+	0.111172i	$0.0354606\pi$
$83$			2.02566i			0.222345i	−0.993801	+	0.111172i	$0.964539\pi$
$84$		0			0
$85$	−10.1757	+	1.78472i	−1.10371	+	0.193580i
$86$		0			0
$87$			6.19218i			0.663871i
$88$		0			0
$89$	−16.1500			−1.71190			−0.855951	−	0.517058i	$-0.827027\pi$
$89$	−16.1500			−1.71190			−0.855951	+	0.517058i	$0.827027\pi$
$90$		0			0
$91$	4.10245			0.430054
$92$		0			0
$93$			3.42419i			0.355072i
$94$		0			0
$95$	1.30244	+	7.42597i	0.133628	+	0.761888i
$96$		0			0
$97$			8.62018i			0.875246i	0.899158	+	0.437623i	$0.144180\pi$
$97$			8.62018i			0.875246i	−0.899158	+	0.437623i	$0.855820\pi$
$98$		0			0
$99$	−1.22164			−0.122780
$100$		0			0
$101$	14.4934			1.44215			0.721075	−	0.692857i	$-0.243648\pi$
$101$	14.4934			1.44215			0.721075	+	0.692857i	$0.243648\pi$
$102$		0			0
$103$			18.5410i			1.82690i	0.406950	+	0.913451i	$0.366592\pi$
$103$			18.5410i			1.82690i	−0.406950	+	0.913451i	$0.633408\pi$
$104$		0			0
$105$	−1.81616	−	10.3550i	−0.177239	−	1.01054i
$106$		0			0
$107$		−	4.48050i		−	0.433147i	−0.976266	−	0.216573i	$-0.930512\pi$
$107$		−	4.48050i		−	0.433147i	0.976266	−	0.216573i	$-0.0694880\pi$
$108$		0			0
$109$	−17.1298			−1.64073			−0.820367	−	0.571838i	$-0.806231\pi$
$109$	−17.1298			−1.64073			−0.820367	+	0.571838i	$0.806231\pi$
$110$		0			0
$111$	8.28853			0.786712
$112$		0			0
$113$		−	9.77435i		−	0.919494i	−0.888050	−	0.459747i	$-0.847940\pi$
$113$		−	9.77435i		−	0.919494i	0.888050	−	0.459747i	$-0.152060\pi$
$114$		0			0
$115$	−2.20245	+	0.386289i	−0.205379	+	0.0360216i
$116$		0			0
$117$		−	1.15641i		−	0.106910i
$118$		0			0
$119$	15.0565			1.38023
$120$		0			0
$121$	−9.23143			−0.839221
$122$		0			0
$123$			7.41245i			0.668358i
$124$		0			0
$125$	5.56376	+	9.69766i	0.497638	+	0.867385i
$126$		0			0
$127$			1.02743i			0.0911699i	0.998960	+	0.0455849i	$0.0145152\pi$
$127$			1.02743i			0.0911699i	−0.998960	+	0.0455849i	$0.985485\pi$
$128$		0			0
$129$	−15.1019			−1.32965
$130$		0			0
$131$	−12.2885			−1.07365			−0.536827	−	0.843692i	$-0.680377\pi$
$131$	−12.2885			−1.07365			−0.536827	+	0.843692i	$0.680377\pi$
$132$		0			0
$133$		−	10.9879i		−	0.952768i
$134$		0			0
$135$	−12.4513	+	2.18384i	−1.07164	+	0.187955i
$136$		0			0
$137$			8.82830i			0.754253i	0.926162	+	0.377126i	$0.123088\pi$
$137$			8.82830i			0.754253i	−0.926162	+	0.377126i	$0.876912\pi$
$138$		0			0
$139$	−11.5532			−0.979927			−0.489963	−	0.871743i	$-0.662990\pi$
$139$	−11.5532			−0.979927			−0.489963	+	0.871743i	$0.662990\pi$
$140$		0			0
$141$	−8.74692			−0.736623
$142$		0			0
$143$			1.67413i			0.139998i
$144$		0			0
$145$	1.65798	+	9.45307i	0.137688	+	0.785035i
$146$		0			0
$147$			5.22283i			0.430772i
$148$		0			0
$149$	11.1333			0.912076			0.456038	−	0.889960i	$-0.349268\pi$
$149$	11.1333			0.912076			0.456038	+	0.889960i	$0.349268\pi$
$150$		0			0
$151$	3.23779			0.263488			0.131744	−	0.991284i	$-0.457942\pi$
$151$	3.23779			0.263488			0.131744	+	0.991284i	$0.457942\pi$
$152$		0			0
$153$		−	4.24416i		−	0.343120i
$154$		0			0
$155$	0.916840	+	5.22742i	0.0736424	+	0.419877i
$156$		0			0
$157$			17.6452i			1.40824i	0.710079	+	0.704122i	$0.248659\pi$
$157$			17.6452i			1.40824i	−0.710079	+	0.704122i	$0.751341\pi$
$158$		0			0
$159$	−16.0537			−1.27314
$160$		0			0
$161$	3.25886			0.256834
$162$		0			0
$163$		−	12.2107i		−	0.956415i	−0.878247	−	0.478207i	$-0.841287\pi$
$163$		−	12.2107i		−	0.956415i	0.878247	−	0.478207i	$-0.158713\pi$
$164$		0			0
$165$	4.22565	−	0.741139i	0.328966	−	0.0576976i
$166$		0			0
$167$			2.06288i			0.159630i	0.996810	+	0.0798151i	$0.0254330\pi$
$167$			2.06288i			0.159630i	−0.996810	+	0.0798151i	$0.974567\pi$
$168$		0			0
$169$	11.4153			0.878098
$170$		0			0
$171$	−3.09728			−0.236855
$172$		0			0
$173$		−	2.10245i		−	0.159847i	−0.996801	−	0.0799233i	$-0.974532\pi$
$173$		−	2.10245i		−	0.159847i	0.996801	−	0.0799233i	$-0.0254675\pi$
$174$		0			0
$175$	−5.54515	−	15.3217i	−0.419174	−	1.15821i
$176$		0			0
$177$			3.92794i			0.295242i
$178$		0			0
$179$	20.4421			1.52792			0.763958	−	0.645266i	$-0.223254\pi$
$179$	20.4421			1.52792			0.763958	+	0.645266i	$0.223254\pi$
$180$		0			0
$181$	−7.12175			−0.529355			−0.264678	−	0.964337i	$-0.585266\pi$
$181$	−7.12175			−0.529355			−0.264678	+	0.964337i	$0.585266\pi$
$182$		0			0
$183$		−	17.8395i		−	1.31873i
$184$		0			0
$185$	12.6534	−	2.21928i	0.930296	−	0.163165i
$186$		0			0
$187$			6.14426i			0.449313i
$188$		0			0
$189$	18.4236			1.34012
$190$		0			0
$191$	16.7191			1.20975			0.604875	−	0.796320i	$-0.293223\pi$
$191$	16.7191			1.20975			0.604875	+	0.796320i	$0.293223\pi$
$192$		0			0
$193$		−	6.51772i		−	0.469156i	−0.972097	−	0.234578i	$-0.924629\pi$
$193$		−	6.51772i		−	0.469156i	0.972097	−	0.234578i	$-0.0753708\pi$
$194$		0			0
$195$	0.701562	+	4.00000i	0.0502399	+	0.286446i
$196$		0			0
$197$		−	17.8184i		−	1.26951i	−0.772714	−	0.634754i	$-0.781101\pi$
$197$		−	17.8184i		−	1.26951i	0.772714	−	0.634754i	$-0.218899\pi$
$198$		0			0
$199$	−3.77435			−0.267557			−0.133778	−	0.991011i	$-0.542711\pi$
$199$	−3.77435			−0.267557			−0.133778	+	0.991011i	$0.542711\pi$
$200$		0			0
$201$	−9.53597			−0.672616
$202$		0			0
$203$		−	13.9873i		−	0.981714i
$204$		0			0
$205$	1.98471	+	11.3160i	0.138618	+	0.790341i
$206$		0			0
$207$		−	0.918614i		−	0.0638481i
$208$		0			0
$209$	4.48393			0.310160
$210$		0			0
$211$	23.9630			1.64968			0.824840	−	0.565366i	$-0.191265\pi$
$211$	23.9630			1.64968			0.824840	+	0.565366i	$0.191265\pi$
$212$		0			0
$213$			0.382642i			0.0262182i
$214$		0			0
$215$	−23.0547	+	4.04358i	−1.57232	+	0.275770i
$216$		0			0
$217$		−	7.73477i		−	0.525071i
$218$		0			0
$219$	7.59530			0.513243
$220$		0			0
$221$	−5.81616			−0.391237
$222$		0			0
$223$		−	20.2094i		−	1.35332i	−0.736296	−	0.676660i	$-0.763427\pi$
$223$		−	20.2094i		−	1.35332i	0.736296	−	0.676660i	$-0.236573\pi$
$224$		0			0
$225$	−4.31892	+	1.56308i	−0.287928	+	0.104205i
$226$		0			0
$227$		−	1.60666i		−	0.106638i	−0.998578	−	0.0533189i	$-0.983020\pi$
$227$		−	1.60666i		−	0.106638i	0.998578	−	0.0533189i	$-0.0169800\pi$
$228$		0			0
$229$	18.9019			1.24907			0.624536	−	0.780996i	$-0.285288\pi$
$229$	18.9019			1.24907			0.624536	+	0.780996i	$0.285288\pi$
$230$		0			0
$231$	−6.25250			−0.411384
$232$		0			0
$233$		−	19.0112i		−	1.24546i	−0.782436	−	0.622731i	$-0.786023\pi$
$233$		−	19.0112i		−	1.24546i	0.782436	−	0.622731i	$-0.213977\pi$
$234$		0			0
$235$	−13.3532	+	2.34202i	−0.871065	+	0.152777i
$236$		0			0
$237$			22.1130i			1.43640i
$238$		0			0
$239$	28.3387			1.83308			0.916538	−	0.399947i	$-0.130972\pi$
$239$	28.3387			1.83308			0.916538	+	0.399947i	$0.130972\pi$
$240$		0			0
$241$	6.35140			0.409130			0.204565	−	0.978853i	$-0.434422\pi$
$241$	6.35140			0.409130			0.204565	+	0.978853i	$0.434422\pi$
$242$		0			0
$243$		−	9.16914i		−	0.588200i
$244$		0			0
$245$	1.39843	+	7.97325i	0.0893426	+	0.509392i
$246$		0			0
$247$			4.24448i			0.270070i
$248$		0			0
$249$	2.92242			0.185201
$250$		0			0
$251$	−19.7231			−1.24491			−0.622455	−	0.782655i	$-0.713865\pi$
$251$	−19.7231			−1.24491			−0.622455	+	0.782655i	$0.713865\pi$
$252$		0			0
$253$			1.32988i			0.0836086i
$254$		0			0
$255$	2.57482	+	14.6805i	0.161241	+	0.919328i
$256$		0			0
$257$		−	1.18102i		−	0.0736701i	−0.999321	−	0.0368351i	$-0.988272\pi$
$257$		−	1.18102i		−	0.0736701i	0.999321	−	0.0368351i	$-0.0117276\pi$
$258$		0			0
$259$	−18.7226			−1.16337
$260$		0			0
$261$	−3.94276			−0.244051
$262$		0			0
$263$		−	24.3189i		−	1.49957i	−0.661682	−	0.749784i	$-0.730157\pi$
$263$		−	24.3189i		−	1.49957i	0.661682	−	0.749784i	$-0.269843\pi$
$264$		0			0
$265$	−24.5078	+	4.29844i	−1.50550	+	0.264051i
$266$		0			0
$267$			23.2997i			1.42592i
$268$		0			0
$269$	−7.91283			−0.482454			−0.241227	−	0.970469i	$-0.577550\pi$
$269$	−7.91283			−0.482454			−0.241227	+	0.970469i	$0.577550\pi$
$270$		0			0
$271$	18.2979			1.11152			0.555758	−	0.831344i	$-0.312428\pi$
$271$	18.2979			1.11152			0.555758	+	0.831344i	$0.312428\pi$
$272$		0			0
$273$		−	5.91861i		−	0.358211i
$274$		0			0
$275$	6.25250	−	2.26287i	0.377040	−	0.136456i
$276$		0			0
$277$		−	28.6550i		−	1.72171i	−0.508847	−	0.860857i	$-0.669928\pi$
$277$		−	28.6550i		−	1.72171i	0.508847	−	0.860857i	$-0.330072\pi$
$278$		0			0
$279$	−2.18029			−0.130531
$280$		0			0
$281$	2.74692			0.163867			0.0819337	−	0.996638i	$-0.473890\pi$
$281$	2.74692			0.163867			0.0819337	+	0.996638i	$0.473890\pi$
$282$		0			0
$283$			5.45307i			0.324151i	0.986778	+	0.162076i	$0.0518189\pi$
$283$			5.45307i			0.324151i	−0.986778	+	0.162076i	$0.948181\pi$
$284$		0			0
$285$	10.7134	−	1.87904i	0.634610	−	0.111305i
$286$		0			0
$287$		−	16.7437i		−	0.988349i
$288$		0			0
$289$	−4.34603			−0.255649
$290$		0			0
$291$	12.4363			0.729031
$292$		0			0
$293$			18.3806i			1.07381i	0.843644	+	0.536903i	$0.180406\pi$
$293$			18.3806i			1.07381i	−0.843644	+	0.536903i	$0.819594\pi$
$294$		0			0
$295$	1.05172	+	5.99645i	0.0612336	+	0.349127i
$296$		0			0
$297$			7.51831i			0.436256i
$298$		0			0
$299$	−1.25886			−0.0728018
$300$		0			0
$301$	34.1130			1.96624
$302$		0			0
$303$		−	20.9097i		−	1.20123i
$304$		0			0
$305$	−4.77658	−	27.2340i	−0.273506	−	1.55941i
$306$		0			0
$307$		−	15.9234i		−	0.908797i	−0.890799	−	0.454398i	$-0.849854\pi$
$307$		−	15.9234i		−	0.908797i	0.890799	−	0.454398i	$-0.150146\pi$
$308$		0			0
$309$	26.7492			1.52171
$310$		0			0
$311$	11.3518			0.643702			0.321851	−	0.946790i	$-0.395695\pi$
$311$	11.3518			0.643702			0.321851	+	0.946790i	$0.395695\pi$
$312$		0			0
$313$		−	28.7913i		−	1.62738i	−0.581299	−	0.813690i	$-0.697455\pi$
$313$		−	28.7913i		−	1.62738i	0.581299	−	0.813690i	$-0.302545\pi$
$314$		0			0
$315$	6.59333	−	1.15641i	0.371492	−	0.0651562i
$316$		0			0
$317$			31.2299i			1.75404i	0.480451	+	0.877022i	$0.340473\pi$
$317$			31.2299i			1.75404i	−0.480451	+	0.877022i	$0.659527\pi$
$318$		0			0
$319$	5.70793			0.319583
$320$		0			0
$321$	−6.46403			−0.360787
$322$		0			0
$323$			15.5778i			0.866771i
$324$		0			0
$325$	2.14203	+	5.91861i	0.118818	+	0.328306i
$326$		0			0
$327$			24.7131i			1.36664i
$328$		0			0
$329$	19.7581			1.08930
$330$		0			0
$331$	−17.6917			−0.972421			−0.486211	−	0.873842i	$-0.661621\pi$
$331$	−17.6917			−0.972421			−0.486211	+	0.873842i	$0.661621\pi$
$332$		0			0
$333$			5.27757i			0.289209i
$334$		0			0
$335$	−14.5578	+	2.55329i	−0.795375	+	0.139501i
$336$		0			0
$337$			1.33801i			0.0728863i	0.999336	+	0.0364431i	$0.0116028\pi$
$337$			1.33801i			0.0728863i	−0.999336	+	0.0364431i	$0.988397\pi$
$338$		0			0
$339$	−14.1015			−0.765886
$340$		0			0
$341$	3.15641			0.170929
$342$		0			0
$343$			11.0144i			0.594720i
$344$		0			0
$345$	0.557299	+	3.17748i	0.0300040	+	0.171070i
$346$		0			0
$347$		−	5.36355i		−	0.287930i	−0.989583	−	0.143965i	$-0.954015\pi$
$347$		−	5.36355i		−	0.287930i	0.989583	−	0.143965i	$-0.0459853\pi$
$348$		0			0
$349$	18.4306			0.986565			0.493283	−	0.869869i	$-0.335797\pi$
$349$	18.4306			0.986565			0.493283	+	0.869869i	$0.335797\pi$
$350$		0			0
$351$	−7.11683			−0.379868
$352$		0			0
$353$			6.76477i			0.360052i	0.983662	+	0.180026i	$0.0576182\pi$
$353$			6.76477i			0.360052i	−0.983662	+	0.180026i	$0.942382\pi$
$354$		0			0
$355$	0.102454	+	0.584146i	0.00543768	+	0.0310033i
$356$		0			0
$357$		−	21.7220i		−	1.14965i
$358$		0			0
$359$	27.5324			1.45311			0.726553	−	0.687111i	$-0.241121\pi$
$359$	27.5324			1.45311			0.726553	+	0.687111i	$0.241121\pi$
$360$		0			0
$361$	−7.63174			−0.401670
$362$		0			0
$363$			13.3182i			0.699024i
$364$		0			0
$365$	11.5951	−	2.03367i	0.606915	−	0.106447i
$366$		0			0
$367$			24.0744i			1.25668i	0.777941	+	0.628338i	$0.216264\pi$
$367$			24.0744i			1.25668i	−0.777941	+	0.628338i	$0.783736\pi$
$368$		0			0
$369$	−4.71975			−0.245700
$370$		0			0
$371$	36.2631			1.88268
$372$		0			0
$373$		−	1.31459i		−	0.0680668i	−0.999421	−	0.0340334i	$-0.989165\pi$
$373$		−	1.31459i		−	0.0680668i	0.999421	−	0.0340334i	$-0.0108353\pi$
$374$		0			0
$375$	13.9908	−	8.02685i	0.722483	−	0.414505i
$376$		0			0
$377$			5.40312i			0.278275i
$378$		0			0
$379$	8.38961			0.430945			0.215473	−	0.976510i	$-0.430871\pi$
$379$	8.38961			0.430945			0.215473	+	0.976510i	$0.430871\pi$
$380$		0			0
$381$	1.48228			0.0759394
$382$		0			0
$383$			24.1258i			1.23277i	0.787446	+	0.616384i	$0.211403\pi$
$383$			24.1258i			1.23277i	−0.787446	+	0.616384i	$0.788597\pi$
$384$		0			0
$385$	−9.54515	+	1.67413i	−0.486466	+	0.0853216i
$386$		0			0
$387$		−	9.61584i		−	0.488801i
$388$		0			0
$389$	−12.5702			−0.637336			−0.318668	−	0.947866i	$-0.603235\pi$
$389$	−12.5702			−0.637336			−0.318668	+	0.947866i	$0.603235\pi$
$390$		0			0
$391$	−4.62018			−0.233652
$392$		0			0
$393$			17.7287i			0.894293i
$394$		0			0
$395$	5.92085	+	33.7581i	0.297910	+	1.69855i
$396$		0			0
$397$		−	28.5318i		−	1.43197i	−0.698115	−	0.715986i	$-0.745977\pi$
$397$		−	28.5318i		−	1.43197i	0.698115	−	0.715986i	$-0.254023\pi$
$398$		0			0
$399$	−15.8522			−0.793602
$400$		0			0
$401$	−19.7130			−0.984422			−0.492211	−	0.870476i	$-0.663811\pi$
$401$	−19.7130			−0.984422			−0.492211	+	0.870476i	$0.663811\pi$
$402$		0			0
$403$			2.98786i			0.148836i
$404$		0			0
$405$	2.08608	+	11.8939i	0.103658	+	0.591013i
$406$		0			0
$407$		−	7.64033i		−	0.378717i
$408$		0			0
$409$	−20.3400			−1.00575			−0.502874	−	0.864360i	$-0.667724\pi$
$409$	−20.3400			−1.00575			−0.502874	+	0.864360i	$0.667724\pi$
$410$		0			0
$411$	12.7366			0.628250
$412$		0			0
$413$		−	8.87267i		−	0.436596i
$414$		0			0
$415$	4.46141	−	0.782489i	0.219002	−	0.0384109i
$416$		0			0
$417$			16.6678i			0.816224i
$418$		0			0
$419$	9.91146			0.484207			0.242103	−	0.970250i	$-0.422163\pi$
$419$	9.91146			0.484207			0.242103	+	0.970250i	$0.422163\pi$
$420$		0			0
$421$	−28.7927			−1.40327			−0.701634	−	0.712537i	$-0.747546\pi$
$421$	−28.7927			−1.40327			−0.701634	+	0.712537i	$0.747546\pi$
$422$		0			0
$423$		−	5.56944i		−	0.270796i
$424$		0			0
$425$	7.86152	+	21.7220i	0.381340	+	1.05367i
$426$		0			0
$427$			40.2969i			1.95010i
$428$		0			0
$429$	2.41527			0.116610
$430$		0			0
$431$	−36.6821			−1.76691			−0.883456	−	0.468513i	$-0.844790\pi$
$431$	−36.6821			−1.76691			−0.883456	+	0.468513i	$0.844790\pi$
$432$		0			0
$433$		−	29.6894i		−	1.42678i	−0.700766	−	0.713391i	$-0.747158\pi$
$433$		−	29.6894i		−	1.42678i	0.700766	−	0.713391i	$-0.252842\pi$
$434$		0			0
$435$	13.6380	−	2.39197i	0.653890	−	0.114686i
$436$		0			0
$437$			3.37169i			0.161290i
$438$		0			0
$439$	34.5378			1.64840			0.824200	−	0.566299i	$-0.191625\pi$
$439$	34.5378			1.64840			0.824200	+	0.566299i	$0.191625\pi$
$440$		0			0
$441$	−3.32554			−0.158359
$442$		0			0
$443$		−	36.6735i		−	1.74241i	−0.490917	−	0.871206i	$-0.663338\pi$
$443$		−	36.6735i		−	1.74241i	0.490917	−	0.871206i	$-0.336662\pi$
$444$		0			0
$445$	6.23858	+	35.5696i	0.295737	+	1.68616i
$446$		0			0
$447$		−	16.0620i		−	0.759708i
$448$		0			0
$449$	13.2138			0.623599			0.311800	−	0.950148i	$-0.399068\pi$
$449$	13.2138			0.623599			0.311800	+	0.950148i	$0.399068\pi$
$450$		0			0
$451$	6.83277			0.321743
$452$		0			0
$453$		−	4.67117i		−	0.219471i
$454$		0			0
$455$	−1.58473	−	9.03544i	−0.0742934	−	0.423588i
$456$		0			0
$457$			12.2292i			0.572058i	0.958221	+	0.286029i	$0.0923353\pi$
$457$			12.2292i			0.572058i	−0.958221	+	0.286029i	$0.907665\pi$
$458$		0			0
$459$	−26.1196			−1.21916
$460$		0			0
$461$	24.0872			1.12185			0.560926	−	0.827866i	$-0.310445\pi$
$461$	24.0872			1.12185			0.560926	+	0.827866i	$0.310445\pi$
$462$		0			0
$463$		−	15.9173i		−	0.739740i	−0.929084	−	0.369870i	$-0.879402\pi$
$463$		−	15.9173i		−	0.739740i	0.929084	−	0.369870i	$-0.120598\pi$
$464$		0			0
$465$	7.54161	−	1.32273i	0.349734	−	0.0613400i
$466$		0			0
$467$			9.70011i			0.448868i	0.974489	+	0.224434i	$0.0720533\pi$
$467$			9.70011i			0.448868i	−0.974489	+	0.224434i	$0.927947\pi$
$468$		0			0
$469$	21.5404			0.994645
$470$		0			0
$471$	25.4568			1.17299
$472$		0			0
$473$			13.9208i			0.640081i
$474$		0			0
$475$	15.8522	−	5.73713i	0.727348	−	0.263238i
$476$		0			0
$477$		−	10.2219i		−	0.468029i
$478$		0			0
$479$	14.1385			0.646004			0.323002	−	0.946398i	$-0.395308\pi$
$479$	14.1385			0.646004			0.323002	+	0.946398i	$0.395308\pi$
$480$		0			0
$481$	7.23234			0.329766
$482$		0			0
$483$		−	4.70156i		−	0.213928i
$484$		0			0
$485$	18.9855	−	3.32988i	0.862087	−	0.151202i
$486$		0			0
$487$			3.75006i			0.169932i	0.996384	+	0.0849658i	$0.0270781\pi$
$487$			3.75006i			0.169932i	−0.996384	+	0.0849658i	$0.972922\pi$
$488$		0			0
$489$	−17.6164			−0.796640
$490$		0			0
$491$	3.19020			0.143972			0.0719860	−	0.997406i	$-0.477066\pi$
$491$	3.19020			0.143972			0.0719860	+	0.997406i	$0.477066\pi$
$492$		0			0
$493$			19.8301i			0.893104i
$494$		0			0
$495$	0.471907	+	2.69061i	0.0212106	+	0.120934i
$496$		0			0
$497$		−	0.864334i		−	0.0387707i
$498$		0			0
$499$	12.9355			0.579075			0.289537	−	0.957167i	$-0.406499\pi$
$499$	12.9355			0.579075			0.289537	+	0.957167i	$0.406499\pi$
$500$		0			0
$501$	2.97611			0.132963
$502$		0			0
$503$		−	15.8845i		−	0.708254i	−0.935197	−	0.354127i	$-0.884778\pi$
$503$		−	15.8845i		−	0.708254i	0.935197	−	0.354127i	$-0.115222\pi$
$504$		0			0
$505$	−5.59865	−	31.9210i	−0.249137	−	1.42047i
$506$		0			0
$507$		−	16.4688i		−	0.731406i
$508$		0			0
$509$	−22.6470			−1.00381			−0.501906	−	0.864922i	$-0.667368\pi$
$509$	−22.6470			−1.00381			−0.501906	+	0.864922i	$0.667368\pi$
$510$		0			0
$511$	−17.1567			−0.758969
$512$		0			0
$513$			19.0614i			0.841583i
$514$		0			0
$515$	40.8357	−	7.16219i	1.79943	−	0.315604i
$516$		0			0
$517$			8.06288i			0.354605i
$518$		0			0
$519$	−3.03321			−0.133143
$520$		0			0
$521$	22.7728			0.997693			0.498847	−	0.866690i	$-0.333757\pi$
$521$	22.7728			0.997693			0.498847	+	0.866690i	$0.333757\pi$
$522$		0			0
$523$		−	16.1920i		−	0.708026i	−0.935241	−	0.354013i	$-0.884817\pi$
$523$		−	16.1920i		−	0.708026i	0.935241	−	0.354013i	$-0.115183\pi$
$524$		0			0
$525$	−22.1047	+	8.00000i	−0.964728	+	0.349149i
$526$		0			0
$527$			10.9658i			0.477678i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−2.50105			−0.108536
$532$		0			0
$533$			6.46790i			0.280156i
$534$		0			0
$535$	−9.86808	+	1.73077i	−0.426634	+	0.0748276i
$536$		0			0
$537$		−	29.4919i		−	1.27267i
$538$		0			0
$539$	4.81439			0.207370
$540$		0			0
$541$	14.4350			0.620610			0.310305	−	0.950637i	$-0.399569\pi$
$541$	14.4350			0.620610			0.310305	+	0.950637i	$0.399569\pi$
$542$		0			0
$543$			10.2746i			0.440923i
$544$		0			0
$545$	6.61703	+	37.7274i	0.283443	+	1.61607i
$546$		0			0
$547$		−	0.846151i		−	0.0361788i	−0.999836	−	0.0180894i	$-0.994242\pi$
$547$		−	0.846151i		−	0.0361788i	0.999836	−	0.0180894i	$-0.00575835\pi$
$548$		0			0
$549$	11.3590			0.484788
$550$		0			0
$551$	14.4715			0.616508
$552$		0			0
$553$		−	49.9503i		−	2.12410i
$554$		0			0
$555$	−3.20176	−	18.2551i	−0.135907	−	0.774884i
$556$		0			0
$557$		−	19.3532i		−	0.820020i	−0.912081	−	0.410010i	$-0.865525\pi$
$557$		−	19.3532i		−	0.820020i	0.912081	−	0.410010i	$-0.134475\pi$
$558$		0			0
$559$	−13.1775			−0.557348
$560$		0			0
$561$	8.86433			0.374252
$562$		0			0
$563$			29.3647i			1.23758i	0.785558	+	0.618788i	$0.212376\pi$
$563$			29.3647i			1.23758i	−0.785558	+	0.618788i	$0.787624\pi$
$564$		0			0
$565$	−21.5275	+	3.77572i	−0.905669	+	0.158846i
$566$		0			0
$567$		−	17.5988i		−	0.739082i
$568$		0			0
$569$	−6.41900			−0.269098			−0.134549	−	0.990907i	$-0.542959\pi$
$569$	−6.41900			−0.269098			−0.134549	+	0.990907i	$0.542959\pi$
$570$		0			0
$571$	−35.3009			−1.47730			−0.738648	−	0.674092i	$-0.764535\pi$
$571$	−35.3009			−1.47730			−0.738648	+	0.674092i	$0.764535\pi$
$572$		0			0
$573$		−	24.1206i		−	1.00765i
$574$		0			0
$575$	1.70156	+	4.70156i	0.0709600	+	0.196069i
$576$		0			0
$577$			6.96339i			0.289890i	0.989440	+	0.144945i	$0.0463004\pi$
$577$			6.96339i			0.289890i	−0.989440	+	0.144945i	$0.953700\pi$
$578$		0			0
$579$	−9.40312			−0.390781
$580$		0			0
$581$	−6.60134			−0.273870
$582$		0			0
$583$			14.7982i			0.612880i
$584$		0			0
$585$	−2.54693	+	0.446707i	−0.105303	+	0.0184691i
$586$		0			0
$587$			23.7392i			0.979823i	0.871772	+	0.489912i	$0.162971\pi$
$587$			23.7392i			0.979823i	−0.871772	+	0.489912i	$0.837029\pi$
$588$		0			0
$589$	8.00256			0.329740
$590$		0			0
$591$	−25.7066			−1.05743
$592$		0			0
$593$			23.1612i			0.951116i	0.879684	+	0.475558i	$0.157754\pi$
$593$			23.1612i			0.951116i	−0.879684	+	0.475558i	$0.842246\pi$
$594$		0			0
$595$	−5.81616	−	33.1612i	−0.238439	−	1.35948i
$596$		0			0
$597$			5.44526i			0.222860i
$598$		0			0
$599$	33.1344			1.35383			0.676916	−	0.736060i	$-0.263316\pi$
$599$	33.1344			1.35383			0.676916	+	0.736060i	$0.263316\pi$
$600$		0			0
$601$	40.0432			1.63340			0.816698	−	0.577065i	$-0.195802\pi$
$601$	40.0432			1.63340			0.816698	+	0.577065i	$0.195802\pi$
$602$		0			0
$603$		−	6.07186i		−	0.247265i
$604$		0			0
$605$	3.56600	+	20.3317i	0.144978	+	0.826603i
$606$		0			0
$607$		−	1.81294i		−	0.0735849i	−0.999323	−	0.0367925i	$-0.988286\pi$
$607$		−	1.81294i		−	0.0735849i	0.999323	−	0.0367925i	$-0.0117140\pi$
$608$		0			0
$609$	−20.1794			−0.817713
$610$		0			0
$611$	−7.63232			−0.308771
$612$		0			0
$613$		−	11.9329i		−	0.481964i	−0.970530	−	0.240982i	$-0.922531\pi$
$613$		−	11.9329i		−	0.481964i	0.970530	−	0.240982i	$-0.0774694\pi$
$614$		0			0
$615$	16.3255	−	2.86335i	0.658309	−	0.115461i
$616$		0			0
$617$			16.5923i			0.667982i	0.942576	+	0.333991i	$0.108396\pi$
$617$			16.5923i			0.667982i	−0.942576	+	0.333991i	$0.891604\pi$
$618$		0			0
$619$	−1.43811			−0.0578025			−0.0289013	−	0.999582i	$-0.509201\pi$
$619$	−1.43811			−0.0578025			−0.0289013	+	0.999582i	$0.509201\pi$
$620$		0			0
$621$	−5.65339			−0.226863
$622$		0			0
$623$		−	52.6307i		−	2.10861i
$624$		0			0
$625$	19.2094	−	16.0000i	0.768375	−	0.640000i
$626$		0			0
$627$		−	6.46896i		−	0.258346i
$628$		0			0
$629$	26.5436			1.05836
$630$		0			0
$631$	−7.67446			−0.305515			−0.152758	−	0.988264i	$-0.548815\pi$
$631$	−7.67446			−0.305515			−0.152758	+	0.988264i	$0.548815\pi$
$632$		0			0
$633$		−	34.5714i		−	1.37409i
$634$		0			0
$635$	2.26287	−	0.396885i	0.0897991	−	0.0157499i
$636$		0			0
$637$			4.55730i			0.180567i
$638$		0			0
$639$	−0.243640			−0.00963826
$640$		0			0
$641$	−23.0562			−0.910665			−0.455332	−	0.890322i	$-0.650480\pi$
$641$	−23.0562			−0.910665			−0.455332	+	0.890322i	$0.650480\pi$
$642$		0			0
$643$			9.51595i			0.375272i	0.982239	+	0.187636i	$0.0600826\pi$
$643$			9.51595i			0.375272i	−0.982239	+	0.187636i	$0.939917\pi$
$644$		0			0
$645$	5.83368	+	33.2611i	0.229701	+	1.30965i
$646$		0			0
$647$			38.9857i			1.53269i	0.642432	+	0.766343i	$0.277926\pi$
$647$			38.9857i			1.53269i	−0.642432	+	0.766343i	$0.722074\pi$
$648$		0			0
$649$	3.62076			0.142127
$650$		0			0
$651$	−11.1590			−0.437354
$652$		0			0
$653$			7.32529i			0.286661i	0.989675	+	0.143330i	$0.0457811\pi$
$653$			7.32529i			0.286661i	−0.989675	+	0.143330i	$0.954219\pi$
$654$		0			0
$655$	4.74692	+	27.0649i	0.185477	+	1.05751i
$656$		0			0
$657$			4.83617i			0.188677i
$658$		0			0
$659$	−20.7177			−0.807048			−0.403524	−	0.914969i	$-0.632215\pi$
$659$	−20.7177			−0.807048			−0.403524	+	0.914969i	$0.632215\pi$
$660$		0			0
$661$	12.9749			0.504666			0.252333	−	0.967640i	$-0.418802\pi$
$661$	12.9749			0.504666			0.252333	+	0.967640i	$0.418802\pi$
$662$		0			0
$663$			8.39098i			0.325879i
$664$		0			0
$665$	−24.2002	+	4.24448i	−0.938443	+	0.164594i
$666$		0			0
$667$			4.29207i			0.166190i
$668$		0			0
$669$	−29.1561			−1.12724
$670$		0			0
$671$	−16.4443			−0.634827
$672$		0			0
$673$		−	15.8660i		−	0.611589i	−0.952098	−	0.305794i	$-0.901078\pi$
$673$		−	15.8660i		−	0.611589i	0.952098	−	0.305794i	$-0.0989220\pi$
$674$		0			0
$675$	9.61959	+	26.5798i	0.370258	+	1.02306i
$676$		0			0
$677$		−	23.2787i		−	0.894675i	−0.894365	−	0.447337i	$-0.852372\pi$
$677$		−	23.2787i		−	0.894675i	0.894365	−	0.447337i	$-0.147628\pi$
$678$		0			0
$679$	−28.0920			−1.07807
$680$		0			0
$681$	−2.31793			−0.0888234
$682$		0			0
$683$			27.5797i			1.05531i	0.849460	+	0.527654i	$0.176928\pi$
$683$			27.5797i			1.05531i	−0.849460	+	0.527654i	$0.823072\pi$
$684$		0			0
$685$	19.4439	−	3.41027i	0.742913	−	0.130300i
$686$		0			0
$687$		−	27.2698i		−	1.04041i
$688$		0			0
$689$	−14.0080			−0.533663
$690$		0			0
$691$	−8.61472			−0.327719			−0.163860	−	0.986484i	$-0.552394\pi$
$691$	−8.61472			−0.327719			−0.163860	+	0.986484i	$0.552394\pi$
$692$		0			0
$693$		−	3.98116i		−	0.151232i
$694$		0			0
$695$	4.46286	+	25.4453i	0.169286	+	0.965194i
$696$		0			0
$697$			23.7380i			0.899141i
$698$		0			0
$699$	−27.4274			−1.03740
$700$		0			0
$701$	17.4254			0.658148			0.329074	−	0.944304i	$-0.393264\pi$
$701$	17.4254			0.658148			0.329074	+	0.944304i	$0.393264\pi$
$702$		0			0
$703$		−	19.3708i		−	0.730584i
$704$		0			0
$705$	3.37884	+	19.2646i	0.127254	+	0.725548i
$706$		0			0
$707$			47.2321i			1.77635i
$708$		0			0
$709$	6.42019			0.241115			0.120558	−	0.992706i	$-0.461532\pi$
$709$	6.42019			0.241115			0.120558	+	0.992706i	$0.461532\pi$
$710$		0			0
$711$	−14.0801			−0.528044
$712$		0			0
$713$			2.37346i			0.0888867i
$714$		0			0
$715$	3.68719	−	0.646697i	0.137893	−	0.0241851i
$716$		0			0
$717$		−	40.8842i		−	1.52685i
$718$		0			0
$719$	1.50079			0.0559699			0.0279849	−	0.999608i	$-0.491091\pi$
$719$	1.50079			0.0559699			0.0279849	+	0.999608i	$0.491091\pi$
$720$		0			0
$721$	−60.4226			−2.25026
$722$		0			0
$723$		−	9.16318i		−	0.340782i
$724$		0			0
$725$	20.1794	−	7.30323i	0.749446	−	0.271235i
$726$		0			0
$727$			6.97867i			0.258825i	0.991591	+	0.129412i	$0.0413091\pi$
$727$			6.97867i			0.258825i	−0.991591	+	0.129412i	$0.958691\pi$
$728$		0			0
$729$	−29.4292			−1.08997
$730$		0			0
$731$	−48.3630			−1.78877
$732$		0			0
$733$		−	3.61138i		−	0.133389i	−0.997773	−	0.0666947i	$-0.978755\pi$
$733$		−	3.61138i		−	0.133389i	0.997773	−	0.0666947i	$-0.0212453\pi$
$734$		0			0
$735$	11.5030	−	2.01752i	0.424295	−	0.0744174i
$736$		0			0
$737$			8.79022i			0.323792i
$738$		0			0
$739$	5.24953			0.193107			0.0965536	−	0.995328i	$-0.469218\pi$
$739$	5.24953			0.193107			0.0965536	+	0.995328i	$0.469218\pi$
$740$		0			0
$741$	6.12352			0.224953
$742$		0			0
$743$		−	2.72603i		−	0.100008i	−0.998749	−	0.0500042i	$-0.984077\pi$
$743$		−	2.72603i		−	0.100008i	0.998749	−	0.0500042i	$-0.0159235\pi$
$744$		0			0
$745$	−4.30067	−	24.5205i	−0.157564	−	0.898363i
$746$		0			0
$747$			1.86080i			0.0680831i
$748$		0			0
$749$	14.6013			0.533521
$750$		0			0
$751$	20.6729			0.754364			0.377182	−	0.926139i	$-0.376893\pi$
$751$	20.6729			0.754364			0.377182	+	0.926139i	$0.376893\pi$
$752$		0			0
$753$			28.4545i			1.03694i
$754$		0			0
$755$	−1.25072	−	7.13107i	−0.0455185	−	0.259526i
$756$		0			0
$757$		−	8.87759i		−	0.322661i	−0.986900	−	0.161331i	$-0.948421\pi$
$757$		−	8.87759i		−	0.322661i	0.986900	−	0.161331i	$-0.0515786\pi$
$758$		0			0
$759$	1.91861			0.0696413
$760$		0			0
$761$	31.8816			1.15571			0.577853	−	0.816141i	$-0.303891\pi$
$761$	31.8816			1.15571			0.577853	+	0.816141i	$0.303891\pi$
$762$		0			0
$763$		−	55.8235i		−	2.02095i
$764$		0			0
$765$	−9.34754	+	1.63947i	−0.337961	+	0.0592752i
$766$		0			0
$767$			3.42741i			0.123757i
$768$		0			0
$769$	−37.6697			−1.35841			−0.679203	−	0.733951i	$-0.737674\pi$
$769$	−37.6697			−1.35841			−0.679203	+	0.733951i	$0.737674\pi$
$770$		0			0
$771$	−1.70386			−0.0613631
$772$		0			0
$773$		−	6.14066i		−	0.220864i	−0.993884	−	0.110432i	$-0.964777\pi$
$773$		−	6.14066i		−	0.220864i	0.993884	−	0.110432i	$-0.0352235\pi$
$774$		0			0
$775$	11.1590	−	4.03859i	0.400842	−	0.145070i
$776$		0			0
$777$			27.0112i			0.969020i
$778$		0			0
$779$	17.3234			0.620674
$780$		0			0
$781$	0.352718			0.0126212
$782$		0			0
$783$			24.2648i			0.867152i
$784$		0			0
$785$	38.8628	−	6.81616i	1.38707	−	0.243279i
$786$		0			0
$787$			17.9373i			0.639397i	0.947519	+	0.319698i	$0.103582\pi$
$787$			17.9373i			0.639397i	−0.947519	+	0.319698i	$0.896418\pi$
$788$		0			0
$789$	−35.0849			−1.24906
$790$		0			0
$791$	31.8533			1.13257
$792$		0			0
$793$		−	15.5662i		−	0.552773i
$794$		0			0
$795$	6.20136	+	35.3574i	0.219940	+	1.25400i
$796$		0			0
$797$			6.03367i			0.213724i	0.994274	+	0.106862i	$0.0340802\pi$
$797$			6.03367i			0.213724i	−0.994274	+	0.106862i	$0.965920\pi$
$798$		0			0
$799$	−28.0116			−0.990978
$800$		0			0
$801$	−14.8357			−0.524192
$802$		0			0
$803$		−	7.00132i		−	0.247071i
$804$		0			0
$805$	−1.25886	−	7.17748i	−0.0443690	−	0.252973i
$806$		0			0
$807$			11.4159i			0.401857i
$808$		0			0
$809$	51.6850			1.81715			0.908574	−	0.417723i	$-0.137172\pi$
$809$	51.6850			1.81715			0.908574	+	0.417723i	$0.137172\pi$
$810$		0			0
$811$	−11.6368			−0.408623			−0.204311	−	0.978906i	$-0.565496\pi$
$811$	−11.6368			−0.408623			−0.204311	+	0.978906i	$0.565496\pi$
$812$		0			0
$813$		−	26.3983i		−	0.925830i
$814$		0			0
$815$	−26.8934	+	4.71685i	−0.942035	+	0.165224i
$816$		0			0
$817$			35.2940i			1.23478i
$818$		0			0
$819$	3.76857			0.131685
$820$		0			0
$821$	6.02312			0.210208			0.105104	−	0.994461i	$-0.466482\pi$
$821$	6.02312			0.210208			0.105104	+	0.994461i	$0.466482\pi$
$822$		0			0
$823$			40.1839i			1.40072i	0.713788	+	0.700362i	$0.246978\pi$
$823$			40.1839i			1.40072i	−0.713788	+	0.700362i	$0.753022\pi$
$824$		0			0
$825$	−3.26464	−	9.02048i	−0.113660	−	0.314053i
$826$		0			0
$827$		−	30.5980i		−	1.06400i	−0.846745	−	0.531998i	$-0.821441\pi$
$827$		−	30.5980i		−	1.06400i	0.846745	−	0.531998i	$-0.178559\pi$
$828$		0			0
$829$	−54.9892			−1.90986			−0.954928	−	0.296838i	$-0.904068\pi$
$829$	−54.9892			−1.90986			−0.954928	+	0.296838i	$0.904068\pi$
$830$		0			0
$831$	−41.3406			−1.43409
$832$		0			0
$833$			16.7259i			0.579516i
$834$		0			0
$835$	4.54338	−	0.796866i	0.157230	−	0.0275767i
$836$		0			0
$837$			13.4181i			0.463797i
$838$		0			0
$839$	−12.5121			−0.431965			−0.215983	−	0.976397i	$-0.569295\pi$
$839$	−12.5121			−0.431965			−0.215983	+	0.976397i	$0.569295\pi$
$840$		0			0
$841$	−10.5781			−0.364762
$842$		0			0
$843$		−	3.96298i		−	0.136492i
$844$		0			0
$845$	−4.40959	−	25.1415i	−0.151694	−	0.864895i
$846$		0			0
$847$		−	30.0839i		−	1.03370i
$848$		0			0
$849$	7.86715			0.270000
$850$		0			0
$851$	5.74514			0.196941
$852$		0			0
$853$			52.3308i			1.79177i	0.444283	+	0.895886i	$0.353458\pi$
$853$			52.3308i			1.79177i	−0.444283	+	0.895886i	$0.646542\pi$
$854$		0			0
$855$	1.19644	+	6.82160i	0.0409175	+	0.233294i
$856$		0			0
$857$			4.53557i			0.154932i	0.996995	+	0.0774660i	$0.0246829\pi$
$857$			4.53557i			0.154932i	−0.996995	+	0.0774660i	$0.975317\pi$
$858$		0			0
$859$	10.1708			0.347023			0.173511	−	0.984832i	$-0.444489\pi$
$859$	10.1708			0.347023			0.173511	+	0.984832i	$0.444489\pi$
$860$		0			0
$861$	−24.1561			−0.823239
$862$		0			0
$863$		−	0.861113i		−	0.0293126i	−0.999893	−	0.0146563i	$-0.995335\pi$
$863$		−	0.861113i		−	0.0293126i	0.999893	−	0.0146563i	$-0.00466542\pi$
$864$		0			0
$865$	−4.63055	+	0.812154i	−0.157443	+	0.0276141i
$866$		0			0
$867$			6.27002i			0.212941i
$868$		0			0
$869$	20.3837			0.691470
$870$		0			0
$871$	−8.32083			−0.281940
$872$		0			0
$873$			7.91861i			0.268004i
$874$		0			0
$875$	−31.6033	+	18.1315i	−1.06839	+	0.612958i
$876$		0			0
$877$			26.4746i			0.893984i	0.894538	+	0.446992i	$0.147505\pi$
$877$			26.4746i			0.893984i	−0.894538	+	0.446992i	$0.852495\pi$
$878$		0			0
$879$	26.5177			0.894421
$880$		0			0
$881$	32.8670			1.10732			0.553659	−	0.832743i	$-0.313231\pi$
$881$	32.8670			1.10732			0.553659	+	0.832743i	$0.313231\pi$
$882$		0			0
$883$		−	29.1507i		−	0.980999i	−0.871441	−	0.490500i	$-0.836814\pi$
$883$		−	29.1507i		−	0.980999i	0.871441	−	0.490500i	$-0.163186\pi$
$884$		0			0
$885$	8.65109	−	1.51732i	0.290803	−	0.0510041i
$886$		0			0
$887$		−	29.6697i		−	0.996212i	−0.867116	−	0.498106i	$-0.834029\pi$
$887$		−	29.6697i		−	0.996212i	0.867116	−	0.498106i	$-0.165971\pi$
$888$		0			0
$889$	−3.34826			−0.112297
$890$		0			0
$891$	7.18174			0.240597
$892$		0			0
$893$			20.4421i			0.684069i
$894$		0			0
$895$	−7.89656	−	45.0227i	−0.263953	−	1.50494i
$896$		0			0
$897$			1.81616i			0.0606398i
$898$		0			0
$899$	10.1871			0.339757
$900$		0			0
$901$	−51.4111			−1.71275
$902$		0			0
$903$		−	49.2149i		−	1.63777i
$904$		0			0
$905$	2.75105	+	15.6853i	0.0914480	+	0.521397i
$906$		0			0
$907$		−	7.66153i		−	0.254397i	−0.991877	−	0.127198i	$-0.959402\pi$
$907$		−	7.66153i		−	0.254397i	0.991877	−	0.127198i	$-0.0405985\pi$
$908$		0			0
$909$	13.3139			0.441593
$910$		0			0
$911$	−53.4450			−1.77071			−0.885356	−	0.464914i	$-0.846085\pi$
$911$	−53.4450			−1.77071			−0.885356	+	0.464914i	$0.846085\pi$
$912$		0			0
$913$		−	2.69388i		−	0.0891543i
$914$		0			0
$915$	−39.2905	+	6.89118i	−1.29890	+	0.227815i
$916$		0			0
$917$		−	40.0466i		−	1.32246i
$918$		0			0
$919$	22.7485			0.750403			0.375202	−	0.926943i	$-0.377573\pi$
$919$	22.7485			0.750403			0.375202	+	0.926943i	$0.377573\pi$
$920$		0			0
$921$	−22.9727			−0.756977
$922$		0			0
$923$			0.333883i			0.0109899i
$924$		0			0
$925$	−9.77572	−	27.0112i	−0.321424	−	0.888121i
$926$		0			0
$927$			17.0320i			0.559406i
$928$		0			0
$929$	−23.1810			−0.760545			−0.380272	−	0.924875i	$-0.624170\pi$
$929$	−23.1810			−0.760545			−0.380272	+	0.924875i	$0.624170\pi$
$930$		0			0
$931$	12.2061			0.400039
$932$		0			0
$933$		−	16.3773i		−	0.536167i
$934$		0			0
$935$	13.5324	−	2.37346i	0.442558	−	0.0776204i
$936$		0			0
$937$			12.6027i			0.411711i	0.978582	+	0.205855i	$0.0659977\pi$
$937$			12.6027i			0.411711i	−0.978582	+	0.205855i	$0.934002\pi$
$938$		0			0
$939$	−41.5372			−1.35552
$940$		0			0
$941$	−47.9093			−1.56180			−0.780899	−	0.624657i	$-0.785239\pi$
$941$	−47.9093			−1.56180			−0.780899	+	0.624657i	$0.785239\pi$
$942$		0			0
$943$			5.13790i			0.167313i
$944$		0			0
$945$	−7.11683	−	40.5771i	−0.231511	−	1.31997i
$946$		0			0
$947$		−	30.4073i		−	0.988105i	−0.869432	−	0.494053i	$-0.835515\pi$
$947$		−	30.4073i		−	0.988105i	0.869432	−	0.494053i	$-0.164485\pi$
$948$		0			0
$949$	6.62745			0.215136
$950$		0			0
$951$	45.0553			1.46102
$952$		0			0
$953$			45.8534i			1.48534i	0.669658	+	0.742669i	$0.266441\pi$
$953$			45.8534i			1.48534i	−0.669658	+	0.742669i	$0.733559\pi$
$954$		0			0
$955$	−6.45839	−	36.8229i	−0.208989	−	1.19156i
$956$		0			0
$957$		−	8.23483i		−	0.266194i
$958$		0			0
$959$	−28.7702			−0.929039
$960$		0			0
$961$	−25.3667			−0.818280
$962$		0			0
$963$		−	4.11585i		−	0.132632i
$964$		0			0
$965$	−14.3550	+	2.51772i	−0.462102	+	0.0810483i
$966$		0			0
$967$		−	57.3471i		−	1.84416i	−0.387002	−	0.922079i	$-0.626489\pi$
$967$		−	57.3471i		−	1.84416i	0.387002	−	0.922079i	$-0.373511\pi$
$968$		0			0
$969$	22.4741			0.721971
$970$		0			0
$971$	44.6948			1.43432			0.717162	−	0.696906i	$-0.245441\pi$
$971$	44.6948			1.43432			0.717162	+	0.696906i	$0.245441\pi$
$972$		0			0
$973$		−	37.6502i		−	1.20701i
$974$		0			0
$975$	8.53879	−	3.09031i	0.273460	−	0.0989691i
$976$		0			0
$977$		−	10.7663i		−	0.344446i	−0.985058	−	0.172223i	$-0.944905\pi$
$977$		−	10.7663i		−	0.344446i	0.985058	−	0.172223i	$-0.0550950\pi$
$978$		0			0
$979$	21.4776			0.686426
$980$		0			0
$981$	−15.7356			−0.502400
$982$		0			0
$983$			4.38626i			0.139900i	0.997550	+	0.0699500i	$0.0222840\pi$
$983$			4.38626i			0.139900i	−0.997550	+	0.0699500i	$0.977716\pi$
$984$		0			0
$985$	−39.2441	+	6.88304i	−1.25042	+	0.219312i
$986$		0			0
$987$		−	28.5050i		−	0.907324i
$988$		0			0
$989$	−10.4678			−0.332856
$990$		0			0
$991$	1.49757			0.0475717			0.0237859	−	0.999717i	$-0.492428\pi$
$991$	1.49757			0.0475717			0.0237859	+	0.999717i	$0.492428\pi$
$992$		0			0
$993$			25.5238i			0.809972i
$994$		0			0
$995$	1.45799	+	8.31281i	0.0462214	+	0.263534i
$996$		0			0
$997$			45.4135i			1.43826i	0.694875	+	0.719130i	$0.255460\pi$
$997$			45.4135i			1.43826i	−0.694875	+	0.719130i	$0.744540\pi$
$998$		0			0
$999$	32.4795			1.02761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.e.369.3		8
4.3	odd	2		230.2.b.b.139.3	✓	8
5.2	odd	4		9200.2.a.cj.1.2		4
5.3	odd	4		9200.2.a.cr.1.3		4
5.4	even	2	inner	1840.2.e.e.369.6		8
12.11	even	2		2070.2.d.f.829.7		8
20.3	even	4		1150.2.a.r.1.2		4
20.7	even	4		1150.2.a.s.1.3		4
20.19	odd	2		230.2.b.b.139.6	yes	8
60.59	even	2		2070.2.d.f.829.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.b.b.139.3	✓	8	4.3	odd	2
230.2.b.b.139.6	yes	8	20.19	odd	2
1150.2.a.r.1.2		4	20.3	even	4
1150.2.a.s.1.3		4	20.7	even	4
1840.2.e.e.369.3		8	1.1	even	1	trivial
1840.2.e.e.369.6		8	5.4	even	2	inner
2070.2.d.f.829.3		8	60.59	even	2
2070.2.d.f.829.7		8	12.11	even	2
9200.2.a.cj.1.2		4	5.2	odd	4
9200.2.a.cr.1.3		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.44270i q^{3} +(-0.386289 - 2.20245i) q^{5} +3.25886i q^{7} +0.918614 q^{9} +O(q^{10})\)	\(q-1.44270i q^{3} +(-0.386289 - 2.20245i) q^{5} +3.25886i q^{7} +0.918614 q^{9} -1.32988 q^{11} -1.25886i q^{13} +(-3.17748 + 0.557299i) q^{15} -4.62018i q^{17} -3.37169 q^{19} +4.70156 q^{21} -1.00000i q^{23} +(-4.70156 + 1.70156i) q^{25} -5.65339i q^{27} -4.29207 q^{29} -2.37346 q^{31} +1.91861i q^{33} +(7.17748 - 1.25886i) q^{35} +5.74514i q^{37} -1.81616 q^{39} -5.13790 q^{41} -10.4678i q^{43} +(-0.354850 - 2.02320i) q^{45} -6.06288i q^{47} -3.62018 q^{49} -6.66553 q^{51} -11.1275i q^{53} +(0.513716 + 2.92898i) q^{55} +4.86433i q^{57} -2.72263 q^{59} +12.3653 q^{61} +2.99364i q^{63} +(-2.77258 + 0.486284i) q^{65} -6.60981i q^{67} -1.44270 q^{69} -0.265226 q^{71} +5.26464i q^{73} +(2.45485 + 6.78295i) q^{75} -4.33388i q^{77} -15.3275 q^{79} -5.40031 q^{81} +2.02566i q^{83} +(-10.1757 + 1.78472i) q^{85} +6.19218i q^{87} -16.1500 q^{89} +4.10245 q^{91} +3.42419i q^{93} +(1.30244 + 7.42597i) q^{95} +8.62018i q^{97} -1.22164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{9}+O(q^{10}) \) 8 * q - 14 * q^9	\( 8 q - 14 q^{9} - 10 q^{11} + 16 q^{15} - 2 q^{19} + 12 q^{21} - 12 q^{25} - 4 q^{29} - 10 q^{31} + 16 q^{35} + 46 q^{41} - 26 q^{45} + 18 q^{49} - 14 q^{51} + 18 q^{55} + 32 q^{59} + 18 q^{61} - 16 q^{65} - 6 q^{69} - 38 q^{71} + 32 q^{75} - 12 q^{79} + 32 q^{81} - 24 q^{85} - 60 q^{89} + 26 q^{91} - 18 q^{95} - 54 q^{99}+O(q^{100}) \) 8 * q - 14 * q^9 - 10 * q^11 + 16 * q^15 - 2 * q^19 + 12 * q^21 - 12 * q^25 - 4 * q^29 - 10 * q^31 + 16 * q^35 + 46 * q^41 - 26 * q^45 + 18 * q^49 - 14 * q^51 + 18 * q^55 + 32 * q^59 + 18 * q^61 - 16 * q^65 - 6 * q^69 - 38 * q^71 + 32 * q^75 - 12 * q^79 + 32 * q^81 - 24 * q^85 - 60 * q^89 + 26 * q^91 - 18 * q^95 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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