LMFDB - Embedded newform 1840.2.m.g.1839.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1839,1840)]

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([1, 0, 1, 1]))

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

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

chi := DirichletCharacter("1840.1839");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.m (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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1839.15
Character	$\chi$	$=$	1840.1839
Dual form			1840.2.m.g.1839.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.49850 q^{3} +(-0.689327 + 2.12716i) q^{5} -4.18735i q^{7} -0.754490 q^{9} +O(q^{10})$	$q-1.49850 q^{3} +(-0.689327 + 2.12716i) q^{5} -4.18735i q^{7} -0.754490 q^{9} +6.19450 q^{11} -0.773792i q^{13} +(1.03296 - 3.18756i) q^{15} -0.0856933 q^{17} -4.78278 q^{19} +6.27475i q^{21} +(-3.29689 + 3.48288i) q^{23} +(-4.04966 - 2.93262i) q^{25} +5.62611 q^{27} +6.23922 q^{29} +1.76282i q^{31} -9.28247 q^{33} +(8.90717 + 2.88645i) q^{35} -6.81613 q^{37} +1.15953i q^{39} -6.59345 q^{41} -1.54990i q^{43} +(0.520090 - 1.60492i) q^{45} +9.34948 q^{47} -10.5339 q^{49} +0.128412 q^{51} -8.12198 q^{53} +(-4.27003 + 13.1767i) q^{55} +7.16700 q^{57} -12.7626i q^{59} +0.753523i q^{61} +3.15931i q^{63} +(1.64598 + 0.533395i) q^{65} -3.43119i q^{67} +(4.94040 - 5.21911i) q^{69} -5.70649i q^{71} -11.8761i q^{73} +(6.06842 + 4.39454i) q^{75} -25.9385i q^{77} -12.2799 q^{79} -6.16727 q^{81} -14.0112i q^{83} +(0.0590707 - 0.182284i) q^{85} -9.34948 q^{87} -0.211260i q^{89} -3.24013 q^{91} -2.64160i q^{93} +(3.29689 - 10.1738i) q^{95} +5.35178 q^{97} -4.67369 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 80 q^{9}+O(q^{10}) $ 40 * q + 80 * q^9	$ 40 q + 80 q^{9} - 24 q^{25} + 24 q^{41} - 16 q^{49} + 80 q^{69} + 40 q^{81} - 8 q^{85}+O(q^{100}) $ 40 * q + 80 * q^9 - 24 * q^25 + 24 * q^41 - 16 * q^49 + 80 * q^69 + 40 * q^81 - 8 * q^85

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
$3$	−1.49850			−0.865161			−0.432580	−	0.901595i	$-0.642397\pi$
$3$	−1.49850			−0.865161			−0.432580	+	0.901595i	$0.642397\pi$
$4$		0			0
$5$	−0.689327	+	2.12716i	−0.308276	+	0.951297i
$5$	−0.689327	+	2.12716i	−0.308276	+	0.951297i
$6$		0			0
$7$		−	4.18735i		−	1.58267i	−0.611384	−	0.791334i	$-0.709387\pi$
$7$		−	4.18735i		−	1.58267i	0.611384	−	0.791334i	$-0.290613\pi$
$8$		0			0
$9$	−0.754490			−0.251497
$10$		0			0
$11$	6.19450			1.86771			0.933855	−	0.357651i	$-0.116422\pi$
$11$	6.19450			1.86771			0.933855	+	0.357651i	$0.116422\pi$
$12$		0			0
$13$		−	0.773792i		−	0.214611i	−0.994226	−	0.107306i	$-0.965778\pi$
$13$		−	0.773792i		−	0.214611i	0.994226	−	0.107306i	$-0.0342223\pi$
$14$		0			0
$15$	1.03296	−	3.18756i	0.266709	−	0.823025i
$16$		0			0
$17$	−0.0856933			−0.0207837			−0.0103918	−	0.999946i	$-0.503308\pi$
$17$	−0.0856933			−0.0207837			−0.0103918	+	0.999946i	$0.503308\pi$
$18$		0			0
$19$	−4.78278			−1.09724			−0.548622	−	0.836071i	$-0.684847\pi$
$19$	−4.78278			−1.09724			−0.548622	+	0.836071i	$0.684847\pi$
$20$		0			0
$21$			6.27475i			1.36926i
$22$		0			0
$23$	−3.29689	+	3.48288i	−0.687450	+	0.726232i
$23$	−3.29689	+	3.48288i	−0.687450	+	0.726232i
$24$		0			0
$25$	−4.04966	−	2.93262i	−0.809932	−	0.586524i
$26$		0			0
$27$	5.62611			1.08275
$28$		0			0
$29$	6.23922			1.15859			0.579297	−	0.815117i	$-0.303327\pi$
$29$	6.23922			1.15859			0.579297	+	0.815117i	$0.303327\pi$
$30$		0			0
$31$			1.76282i			0.316613i	0.987390	+	0.158306i	$0.0506033\pi$
$31$			1.76282i			0.316613i	−0.987390	+	0.158306i	$0.949397\pi$
$32$		0			0
$33$	−9.28247			−1.61587
$34$		0			0
$35$	8.90717	+	2.88645i	1.50559	+	0.487899i
$36$		0			0
$37$	−6.81613			−1.12056			−0.560282	−	0.828302i	$-0.689307\pi$
$37$	−6.81613			−1.12056			−0.560282	+	0.828302i	$0.689307\pi$
$38$		0			0
$39$			1.15953i			0.185673i
$40$		0			0
$41$	−6.59345			−1.02972			−0.514862	−	0.857273i	$-0.672157\pi$
$41$	−6.59345			−1.02972			−0.514862	+	0.857273i	$0.672157\pi$
$42$		0			0
$43$		−	1.54990i		−	0.236358i	−0.992992	−	0.118179i	$-0.962294\pi$
$43$		−	1.54990i		−	0.236358i	0.992992	−	0.118179i	$-0.0377056\pi$
$44$		0			0
$45$	0.520090	−	1.60492i	0.0775305	−	0.239248i
$46$		0			0
$47$	9.34948			1.36376			0.681881	−	0.731463i	$-0.261162\pi$
$47$	9.34948			1.36376			0.681881	+	0.731463i	$0.261162\pi$
$48$		0			0
$49$	−10.5339			−1.50484
$50$		0			0
$51$	0.128412			0.0179812
$52$		0			0
$53$	−8.12198			−1.11564			−0.557820	−	0.829962i	$-0.688362\pi$
$53$	−8.12198			−1.11564			−0.557820	+	0.829962i	$0.688362\pi$
$54$		0			0
$55$	−4.27003	+	13.1767i	−0.575771	+	1.77675i
$56$		0			0
$57$	7.16700			0.949292
$58$		0			0
$59$		−	12.7626i		−	1.66156i	−0.556605	−	0.830778i	$-0.687896\pi$
$59$		−	12.7626i		−	1.66156i	0.556605	−	0.830778i	$-0.312104\pi$
$60$		0			0
$61$			0.753523i			0.0964787i	0.998836	+	0.0482394i	$0.0153610\pi$
$61$			0.753523i			0.0964787i	−0.998836	+	0.0482394i	$0.984639\pi$
$62$		0			0
$63$			3.15931i			0.398036i
$64$		0			0
$65$	1.64598	+	0.533395i	0.204159	+	0.0661596i
$66$		0			0
$67$		−	3.43119i		−	0.419187i	−0.977789	−	0.209593i	$-0.932786\pi$
$67$		−	3.43119i		−	0.419187i	0.977789	−	0.209593i	$-0.0672140\pi$
$68$		0			0
$69$	4.94040	−	5.21911i	0.594755	−	0.628307i
$70$		0			0
$71$		−	5.70649i		−	0.677236i	−0.940924	−	0.338618i	$-0.890041\pi$
$71$		−	5.70649i		−	0.677236i	0.940924	−	0.338618i	$-0.109959\pi$
$72$		0			0
$73$		−	11.8761i		−	1.38999i	−0.719013	−	0.694997i	$-0.755406\pi$
$73$		−	11.8761i		−	1.38999i	0.719013	−	0.694997i	$-0.244594\pi$
$74$		0			0
$75$	6.06842	+	4.39454i	0.700721	+	0.507438i
$76$		0			0
$77$		−	25.9385i		−	2.95597i
$78$		0			0
$79$	−12.2799			−1.38160			−0.690798	−	0.723047i	$-0.742741\pi$
$79$	−12.2799			−1.38160			−0.690798	+	0.723047i	$0.742741\pi$
$80$		0			0
$81$	−6.16727			−0.685253
$82$		0			0
$83$		−	14.0112i		−	1.53793i	−0.639293	−	0.768964i	$-0.720773\pi$
$83$		−	14.0112i		−	1.53793i	0.639293	−	0.768964i	$-0.279227\pi$
$84$		0			0
$85$	0.0590707	−	0.182284i	0.00640712	−	0.0197715i
$86$		0			0
$87$	−9.34948			−1.00237
$88$		0			0
$89$		−	0.211260i		−	0.0223935i	−0.999937	−	0.0111968i	$-0.996436\pi$
$89$		−	0.211260i		−	0.0223935i	0.999937	−	0.0111968i	$-0.00356411\pi$
$90$		0			0
$91$	−3.24013			−0.339658
$92$		0			0
$93$		−	2.64160i		−	0.273921i
$94$		0			0
$95$	3.29689	−	10.1738i	0.338254	−	1.04380i
$96$		0			0
$97$	5.35178			0.543391			0.271696	−	0.962383i	$-0.412416\pi$
$97$	5.35178			0.543391			0.271696	+	0.962383i	$0.412416\pi$
$98$		0			0
$99$	−4.67369			−0.469723
$100$		0			0
$101$	1.08031			0.107494			0.0537472	−	0.998555i	$-0.482883\pi$
$101$	1.08031			0.107494			0.0537472	+	0.998555i	$0.482883\pi$
$102$		0			0
$103$		−	8.77530i		−	0.864656i	−0.901716	−	0.432328i	$-0.857692\pi$
$103$		−	8.77530i		−	0.864656i	0.901716	−	0.432328i	$-0.142308\pi$
$104$		0			0
$105$	−13.3474	−	4.32535i	−1.30257	−	0.422111i
$106$		0			0
$107$			13.5086i			1.30593i	0.757390	+	0.652963i	$0.226474\pi$
$107$			13.5086i			1.30593i	−0.757390	+	0.652963i	$0.773526\pi$
$108$		0			0
$109$		−	6.77696i		−	0.649115i	−0.945866	−	0.324558i	$-0.894785\pi$
$109$		−	6.77696i		−	0.649115i	0.945866	−	0.324558i	$-0.105215\pi$
$110$		0			0
$111$	10.2140			0.969469
$112$		0			0
$113$	−17.8410			−1.67834			−0.839171	−	0.543868i	$-0.816959\pi$
$113$	−17.8410			−1.67834			−0.839171	+	0.543868i	$0.816959\pi$
$114$		0			0
$115$	−5.13603	−	9.41388i	−0.478937	−	0.877849i
$116$		0			0
$117$			0.583818i			0.0539740i
$118$		0			0
$119$			0.358827i			0.0328937i
$120$		0			0
$121$	27.3718			2.48834
$122$		0			0
$123$	9.88031			0.890877
$124$		0			0
$125$	9.02971	−	6.59275i	0.807642	−	0.589674i
$126$		0			0
$127$	−2.78736			−0.247339			−0.123669	−	0.992323i	$-0.539466\pi$
$127$	−2.78736			−0.247339			−0.123669	+	0.992323i	$0.539466\pi$
$128$		0			0
$129$			2.32253i			0.204487i
$130$		0			0
$131$		−	13.3587i		−	1.16716i	−0.812057	−	0.583578i	$-0.801652\pi$
$131$		−	13.3587i		−	1.16716i	0.812057	−	0.583578i	$-0.198348\pi$
$132$		0			0
$133$			20.0271i			1.73657i
$134$		0			0
$135$	−3.87823	+	11.9677i	−0.333785	+	1.03001i
$136$		0			0
$137$	−11.9841			−1.02387			−0.511936	−	0.859023i	$-0.671072\pi$
$137$	−11.9841			−1.02387			−0.511936	+	0.859023i	$0.671072\pi$
$138$		0			0
$139$		−	1.57990i		−	0.134006i	−0.997753	−	0.0670029i	$-0.978656\pi$
$139$		−	1.57990i		−	0.134006i	0.997753	−	0.0670029i	$-0.0213437\pi$
$140$		0			0
$141$	−14.0102			−1.17987
$142$		0			0
$143$		−	4.79325i		−	0.400832i
$144$		0			0
$145$	−4.30086	+	13.2718i	−0.357167	+	1.10217i
$146$		0			0
$147$	15.7850			1.30193
$148$		0			0
$149$			14.4701i			1.18544i	0.805409	+	0.592720i	$0.201946\pi$
$149$			14.4701i			1.18544i	−0.805409	+	0.592720i	$0.798054\pi$
$150$		0			0
$151$		−	16.2708i		−	1.32410i	−0.749460	−	0.662050i	$-0.769687\pi$
$151$		−	16.2708i		−	1.32410i	0.749460	−	0.662050i	$-0.230313\pi$
$152$		0			0
$153$	0.0646548			0.00522703
$154$		0			0
$155$	−3.74982	−	1.21516i	−0.301193	−	0.0976041i
$156$		0			0
$157$	−18.7313			−1.49492			−0.747460	−	0.664307i	$-0.768727\pi$
$157$	−18.7313			−1.49492			−0.747460	+	0.664307i	$0.768727\pi$
$158$		0			0
$159$	12.1708			0.965208
$160$		0			0
$161$	14.5840	+	13.8052i	1.14938	+	1.08800i
$162$		0			0
$163$	7.90236			0.618961			0.309480	−	0.950906i	$-0.399845\pi$
$163$	7.90236			0.618961			0.309480	+	0.950906i	$0.399845\pi$
$164$		0			0
$165$	6.39865	−	19.7453i	0.498134	−	1.53717i
$166$		0			0
$167$	−17.8143			−1.37852			−0.689258	−	0.724516i	$-0.742063\pi$
$167$	−17.8143			−1.37852			−0.689258	+	0.724516i	$0.742063\pi$
$168$		0			0
$169$	12.4012			0.953942
$170$		0			0
$171$	3.60856			0.275953
$172$		0			0
$173$		−	4.96297i		−	0.377328i	−0.982042	−	0.188664i	$-0.939584\pi$
$173$		−	4.96297i		−	0.377328i	0.982042	−	0.188664i	$-0.0604157\pi$
$174$		0			0
$175$	−12.2799	+	16.9573i	−0.928273	+	1.28185i
$176$		0			0
$177$			19.1249i			1.43751i
$178$		0			0
$179$		−	5.03374i		−	0.376240i	−0.982146	−	0.188120i	$-0.939761\pi$
$179$		−	5.03374i		−	0.376240i	0.982146	−	0.188120i	$-0.0602393\pi$
$180$		0			0
$181$			3.19869i			0.237757i	0.992909	+	0.118879i	$0.0379299\pi$
$181$			3.19869i			0.237757i	−0.992909	+	0.118879i	$0.962070\pi$
$182$		0			0
$183$		−	1.12916i		−	0.0834696i
$184$		0			0
$185$	4.69854	−	14.4990i	0.345443	−	1.06599i
$186$		0			0
$187$	−0.530827			−0.0388179
$188$		0			0
$189$		−	23.5585i		−	1.71363i
$190$		0			0
$191$	7.49964			0.542655			0.271327	−	0.962487i	$-0.412537\pi$
$191$	7.49964			0.542655			0.271327	+	0.962487i	$0.412537\pi$
$192$		0			0
$193$			16.5583i			1.19190i	0.803023	+	0.595948i	$0.203224\pi$
$193$			16.5583i			1.19190i	−0.803023	+	0.595948i	$0.796776\pi$
$194$		0			0
$195$	−2.46651	−	0.799294i	−0.176630	−	0.0572387i
$196$		0			0
$197$		−	15.8736i		−	1.13094i	−0.824768	−	0.565472i	$-0.808694\pi$
$197$		−	15.8736i		−	1.13094i	0.824768	−	0.565472i	$-0.191306\pi$
$198$		0			0
$199$	−13.0374			−0.924199			−0.462099	−	0.886828i	$-0.652904\pi$
$199$	−13.0374			−0.924199			−0.462099	+	0.886828i	$0.652904\pi$
$200$		0			0
$201$			5.14165i			0.362664i
$202$		0			0
$203$		−	26.1258i		−	1.83367i
$204$		0			0
$205$	4.54504	−	14.0254i	0.317440	−	0.979574i
$206$		0			0
$207$	2.48747	−	2.62780i	0.172891	−	0.182645i
$208$		0			0
$209$	−29.6269			−2.04933
$210$		0			0
$211$		−	12.6908i		−	0.873673i	−0.899541	−	0.436837i	$-0.856099\pi$
$211$		−	12.6908i		−	0.873673i	0.899541	−	0.436837i	$-0.143901\pi$
$212$		0			0
$213$			8.55119i			0.585918i
$214$		0			0
$215$	3.29689	+	1.06839i	0.224846	+	0.0728635i
$216$		0			0
$217$	7.38155			0.501093
$218$		0			0
$219$			17.7964i			1.20257i
$220$		0			0
$221$			0.0663088i			0.00446041i
$222$		0			0
$223$	−24.9762			−1.67253			−0.836266	−	0.548323i	$-0.815266\pi$
$223$	−24.9762			−1.67253			−0.836266	+	0.548323i	$0.815266\pi$
$224$		0			0
$225$	3.05543	+	2.21263i	0.203695	+	0.147509i
$226$		0			0
$227$		−	18.2726i		−	1.21279i	−0.795163	−	0.606396i	$-0.792615\pi$
$227$		−	18.2726i		−	1.21279i	0.795163	−	0.606396i	$-0.207385\pi$
$228$		0			0
$229$			12.0847i			0.798579i	0.916825	+	0.399290i	$0.130743\pi$
$229$			12.0847i			0.798579i	−0.916825	+	0.399290i	$0.869257\pi$
$230$		0			0
$231$			38.8689i			2.55739i
$232$		0			0
$233$			28.4879i			1.86630i	0.359485	+	0.933151i	$0.382952\pi$
$233$			28.4879i			1.86630i	−0.359485	+	0.933151i	$0.617048\pi$
$234$		0			0
$235$	−6.44485	+	19.8879i	−0.420415	+	1.29734i
$236$		0			0
$237$	18.4015			1.19530
$238$		0			0
$239$			17.8453i			1.15432i	0.816632	+	0.577159i	$0.195839\pi$
$239$			17.8453i			1.15432i	−0.816632	+	0.577159i	$0.804161\pi$
$240$		0			0
$241$		−	20.1463i		−	1.29774i	−0.760901	−	0.648868i	$-0.775243\pi$
$241$		−	20.1463i		−	1.29774i	0.760901	−	0.648868i	$-0.224757\pi$
$242$		0			0
$243$	−7.63666			−0.489892
$244$		0			0
$245$	7.26127	−	22.4073i	0.463905	−	1.43155i
$246$		0			0
$247$			3.70087i			0.235481i
$248$		0			0
$249$			20.9958i			1.33055i
$250$		0			0
$251$	−4.13434			−0.260957			−0.130479	−	0.991451i	$-0.541651\pi$
$251$	−4.13434			−0.260957			−0.130479	+	0.991451i	$0.541651\pi$
$252$		0			0
$253$	−20.4226	+	21.5747i	−1.28396	+	1.35639i
$254$		0			0
$255$	−0.0885176	+	0.273153i	−0.00554319	+	0.0171055i
$256$		0			0
$257$		−	18.9792i		−	1.18389i	−0.805978	−	0.591946i	$-0.798360\pi$
$257$		−	18.9792i		−	1.18389i	0.805978	−	0.591946i	$-0.201640\pi$
$258$		0			0
$259$			28.5415i			1.77348i
$260$		0			0
$261$	−4.70743			−0.291382
$262$		0			0
$263$			12.5948i			0.776630i	0.921527	+	0.388315i	$0.126943\pi$
$263$			12.5948i			0.776630i	−0.921527	+	0.388315i	$0.873057\pi$
$264$		0			0
$265$	5.59870	−	17.2768i	0.343925	−	1.06130i
$266$		0			0
$267$			0.316574i			0.0193740i
$268$		0			0
$269$	3.75973			0.229235			0.114618	−	0.993410i	$-0.463436\pi$
$269$	3.75973			0.229235			0.114618	+	0.993410i	$0.463436\pi$
$270$		0			0
$271$		−	4.98461i		−	0.302794i	−0.988473	−	0.151397i	$-0.951623\pi$
$271$		−	4.98461i		−	0.302794i	0.988473	−	0.151397i	$-0.0483771\pi$
$272$		0			0
$273$	4.85535			0.293859
$274$		0			0
$275$	−25.0856	−	18.1661i	−1.51272	−	1.09546i
$276$		0			0
$277$			13.7883i			0.828457i	0.910173	+	0.414228i	$0.135949\pi$
$277$			13.7883i			0.828457i	−0.910173	+	0.414228i	$0.864051\pi$
$278$		0			0
$279$		−	1.33003i		−	0.0796270i
$280$		0			0
$281$			13.2010i			0.787506i	0.919216	+	0.393753i	$0.128824\pi$
$281$			13.2010i			0.787506i	−0.919216	+	0.393753i	$0.871176\pi$
$282$		0			0
$283$		−	9.33878i		−	0.555133i	−0.960706	−	0.277566i	$-0.910472\pi$
$283$		−	9.33878i		−	0.555133i	0.960706	−	0.277566i	$-0.0895278\pi$
$284$		0			0
$285$	−4.94040	+	15.2454i	−0.292644	+	0.903059i
$286$		0			0
$287$			27.6091i			1.62971i
$288$		0			0
$289$	−16.9927			−0.999568
$290$		0			0
$291$	−8.01966			−0.470121
$292$		0			0
$293$	9.20080			0.537516			0.268758	−	0.963208i	$-0.413387\pi$
$293$	9.20080			0.537516			0.268758	+	0.963208i	$0.413387\pi$
$294$		0			0
$295$	27.1483	+	8.79763i	1.58063	+	0.512218i
$296$		0			0
$297$	34.8509			2.02226
$298$		0			0
$299$	2.69503	+	2.55111i	0.155858	+	0.147535i
$300$		0			0
$301$	−6.48997			−0.374076
$302$		0			0
$303$	−1.61884			−0.0929999
$304$		0			0
$305$	−1.60287	−	0.519423i	−0.0917799	−	0.0297421i
$306$		0			0
$307$	22.1002			1.26133			0.630663	−	0.776057i	$-0.282783\pi$
$307$	22.1002			1.26133			0.630663	+	0.776057i	$0.282783\pi$
$308$		0			0
$309$			13.1498i			0.748066i
$310$		0			0
$311$		−	15.3138i		−	0.868364i	−0.900825	−	0.434182i	$-0.857038\pi$
$311$		−	15.3138i		−	0.868364i	0.900825	−	0.434182i	$-0.142962\pi$
$312$		0			0
$313$	−8.21578			−0.464383			−0.232192	−	0.972670i	$-0.574590\pi$
$313$	−8.21578			−0.464383			−0.232192	+	0.972670i	$0.574590\pi$
$314$		0			0
$315$	−6.72037	−	2.17780i	−0.378650	−	0.122705i
$316$		0			0
$317$			11.4120i			0.640964i	0.947255	+	0.320482i	$0.103845\pi$
$317$			11.4120i			0.640964i	−0.947255	+	0.320482i	$0.896155\pi$
$318$		0			0
$319$	38.6488			2.16392
$320$		0			0
$321$		−	20.2427i		−	1.12984i
$322$		0			0
$323$	0.409852			0.0228048
$324$		0			0
$325$	−2.26924	+	3.13359i	−0.125875	+	0.173820i
$326$		0			0
$327$			10.1553i			0.561589i
$328$		0			0
$329$		−	39.1495i		−	2.15838i
$330$		0			0
$331$		−	19.6675i		−	1.08103i	−0.841336	−	0.540513i	$-0.818230\pi$
$331$		−	19.6675i		−	1.08103i	0.841336	−	0.540513i	$-0.181770\pi$
$332$		0			0
$333$	5.14270			0.281818
$334$		0			0
$335$	7.29871	+	2.36521i	0.398771	+	0.129225i
$336$		0			0
$337$	22.7906			1.24148			0.620742	−	0.784015i	$-0.286831\pi$
$337$	22.7906			1.24148			0.620742	+	0.784015i	$0.286831\pi$
$338$		0			0
$339$	26.7348			1.45204
$340$		0			0
$341$			10.9198i			0.591341i
$342$		0			0
$343$			14.7975i			0.798989i
$344$		0			0
$345$	7.69636	+	14.1067i	0.414358	+	0.759481i
$346$		0			0
$347$	17.1569			0.921032			0.460516	−	0.887651i	$-0.347664\pi$
$347$	17.1569			0.921032			0.460516	+	0.887651i	$0.347664\pi$
$348$		0			0
$349$	5.84127			0.312676			0.156338	−	0.987704i	$-0.450031\pi$
$349$	5.84127			0.312676			0.156338	+	0.987704i	$0.450031\pi$
$350$		0			0
$351$		−	4.35344i		−	0.232369i
$352$		0			0
$353$		−	5.64630i		−	0.300522i	−0.988646	−	0.150261i	$-0.951989\pi$
$353$		−	5.64630i		−	0.300522i	0.988646	−	0.150261i	$-0.0480115\pi$
$354$		0			0
$355$	12.1386	+	3.93363i	0.644252	+	0.208776i
$356$		0			0
$357$		−	0.537704i		−	0.0284583i
$358$		0			0
$359$	19.0220			1.00394			0.501972	−	0.864884i	$-0.332608\pi$
$359$	19.0220			1.00394			0.501972	+	0.864884i	$0.332608\pi$
$360$		0			0
$361$	3.87494			0.203944
$362$		0			0
$363$	−41.0167			−2.15282
$364$		0			0
$365$	25.2625	+	8.18652i	1.32230	+	0.428502i
$366$		0			0
$367$			6.49340i			0.338953i	0.985534	+	0.169476i	$0.0542076\pi$
$367$			6.49340i			0.338953i	−0.985534	+	0.169476i	$0.945792\pi$
$368$		0			0
$369$	4.97470			0.258972
$370$		0			0
$371$			34.0095i			1.76569i
$372$		0			0
$373$	6.37142			0.329900			0.164950	−	0.986302i	$-0.447254\pi$
$373$	6.37142			0.329900			0.164950	+	0.986302i	$0.447254\pi$
$374$		0			0
$375$	−13.5310	+	9.87926i	−0.698740	+	0.510163i
$376$		0			0
$377$		−	4.82786i		−	0.248647i
$378$		0			0
$379$	10.4998			0.539336			0.269668	−	0.962953i	$-0.413086\pi$
$379$	10.4998			0.539336			0.269668	+	0.962953i	$0.413086\pi$
$380$		0			0
$381$	4.17687			0.213988
$382$		0			0
$383$			7.22801i			0.369334i	0.982801	+	0.184667i	$0.0591207\pi$
$383$			7.22801i			0.369334i	−0.982801	+	0.184667i	$0.940879\pi$
$384$		0			0
$385$	55.1754	+	17.8801i	2.81200	+	0.911254i
$386$		0			0
$387$			1.16938i			0.0594432i
$388$		0			0
$389$			29.0245i			1.47160i	0.677197	+	0.735801i	$0.263194\pi$
$389$			29.0245i			1.47160i	−0.677197	+	0.735801i	$0.736806\pi$
$390$		0			0
$391$	0.282522	−	0.298460i	0.0142877	−	0.0150938i
$392$		0			0
$393$			20.0181i			1.00978i
$394$		0			0
$395$	8.46486	−	26.1214i	0.425913	−	1.31431i
$396$		0			0
$397$			33.4508i			1.67885i	0.543476	+	0.839425i	$0.317108\pi$
$397$			33.4508i			1.67885i	−0.543476	+	0.839425i	$0.682892\pi$
$398$		0			0
$399$		−	30.0107i		−	1.50241i
$400$		0			0
$401$		−	35.0523i		−	1.75043i	−0.483734	−	0.875215i	$-0.660720\pi$
$401$		−	35.0523i		−	1.75043i	0.483734	−	0.875215i	$-0.339280\pi$
$402$		0			0
$403$	1.36406			0.0679486
$404$		0			0
$405$	4.25127	−	13.1188i	0.211247	−	0.651879i
$406$		0			0
$407$	−42.2225			−2.09289
$408$		0			0
$409$	28.3385			1.40125			0.700624	−	0.713531i	$-0.252905\pi$
$409$	28.3385			1.40125			0.700624	+	0.713531i	$0.252905\pi$
$410$		0			0
$411$	17.9582			0.885815
$412$		0			0
$413$	−53.4416			−2.62969
$414$		0			0
$415$	29.8041	+	9.65828i	1.46303	+	0.474106i
$416$		0			0
$417$			2.36749i			0.115937i
$418$		0			0
$419$	−7.54215			−0.368458			−0.184229	−	0.982883i	$-0.558979\pi$
$419$	−7.54215			−0.368458			−0.184229	+	0.982883i	$0.558979\pi$
$420$		0			0
$421$			0.184602i			0.00899695i	0.999990	+	0.00449847i	$0.00143191\pi$
$421$			0.184602i			0.00899695i	−0.999990	+	0.00449847i	$0.998568\pi$
$422$		0			0
$423$	−7.05409			−0.342982
$424$		0			0
$425$	0.347029	+	0.251306i	0.0168334	+	0.0121901i
$426$		0			0
$427$	3.15526			0.152694
$428$		0			0
$429$			7.18270i			0.346784i
$430$		0			0
$431$	16.4117			0.790525			0.395263	−	0.918568i	$-0.370654\pi$
$431$	16.4117			0.790525			0.395263	+	0.918568i	$0.370654\pi$
$432$		0			0
$433$	0.590803			0.0283922			0.0141961	−	0.999899i	$-0.495481\pi$
$433$	0.590803			0.0283922			0.0141961	+	0.999899i	$0.495481\pi$
$434$		0			0
$435$	6.44485	−	19.8879i	0.309007	−	0.953551i
$436$		0			0
$437$	15.7683	−	16.6579i	0.754300	−	0.796853i
$438$		0			0
$439$			32.2613i			1.53975i	0.638196	+	0.769874i	$0.279681\pi$
$439$			32.2613i			1.53975i	−0.638196	+	0.769874i	$0.720319\pi$
$440$		0			0
$441$	7.94769			0.378462
$442$		0			0
$443$	−0.175821			−0.00835352			−0.00417676	−	0.999991i	$-0.501330\pi$
$443$	−0.175821			−0.00835352			−0.00417676	+	0.999991i	$0.501330\pi$
$444$		0			0
$445$	0.449385	+	0.145627i	0.0213029	+	0.00690339i
$446$		0			0
$447$		−	21.6835i		−	1.02560i
$448$		0			0
$449$	−8.47744			−0.400075			−0.200038	−	0.979788i	$-0.564106\pi$
$449$	−8.47744			−0.400075			−0.200038	+	0.979788i	$0.564106\pi$
$450$		0			0
$451$	−40.8431			−1.92323
$452$		0			0
$453$			24.3819i			1.14556i
$454$		0			0
$455$	2.23351	−	6.89230i	0.104709	−	0.323116i
$456$		0			0
$457$	−10.0816			−0.471597			−0.235798	−	0.971802i	$-0.575770\pi$
$457$	−10.0816			−0.471597			−0.235798	+	0.971802i	$0.575770\pi$
$458$		0			0
$459$	−0.482120			−0.0225034
$460$		0			0
$461$	−16.0577			−0.747881			−0.373940	−	0.927453i	$-0.621994\pi$
$461$	−16.0577			−0.747881			−0.373940	+	0.927453i	$0.621994\pi$
$462$		0			0
$463$	−14.5657			−0.676927			−0.338464	−	0.940979i	$-0.609907\pi$
$463$	−14.5657			−0.676927			−0.338464	+	0.940979i	$0.609907\pi$
$464$		0			0
$465$	5.61911	+	1.82092i	0.260580	+	0.0844433i
$466$		0			0
$467$			19.4162i			0.898474i	0.893413	+	0.449237i	$0.148304\pi$
$467$			19.4162i			0.898474i	−0.893413	+	0.449237i	$0.851696\pi$
$468$		0			0
$469$	−14.3676			−0.663433
$470$		0			0
$471$	28.0689			1.29335
$472$		0			0
$473$		−	9.60085i		−	0.441448i
$474$		0			0
$475$	19.3686	+	14.0261i	0.888692	+	0.643560i
$476$		0			0
$477$	6.12796			0.280580
$478$		0			0
$479$	−37.9747			−1.73511			−0.867553	−	0.497344i	$-0.834309\pi$
$479$	−37.9747			−1.73511			−0.867553	+	0.497344i	$0.834309\pi$
$480$		0			0
$481$			5.27427i			0.240486i
$482$		0			0
$483$	−21.8542	−	20.6872i	−0.994402	−	0.941299i
$484$		0			0
$485$	−3.68913	+	11.3841i	−0.167515	+	0.516926i
$486$		0			0
$487$	−16.5430			−0.749637			−0.374819	−	0.927098i	$-0.622295\pi$
$487$	−16.5430			−0.749637			−0.374819	+	0.927098i	$0.622295\pi$
$488$		0			0
$489$	−11.8417			−0.535501
$490$		0			0
$491$			25.9783i			1.17238i	0.810172	+	0.586192i	$0.199374\pi$
$491$			25.9783i			1.17238i	−0.810172	+	0.586192i	$0.800626\pi$
$492$		0			0
$493$	−0.534659			−0.0240798
$494$		0			0
$495$	3.22170	−	9.94170i	0.144804	−	0.446846i
$496$		0			0
$497$	−23.8950			−1.07184
$498$		0			0
$499$		−	42.1048i		−	1.88487i	−0.334388	−	0.942436i	$-0.608530\pi$
$499$		−	42.1048i		−	1.88487i	0.334388	−	0.942436i	$-0.391470\pi$
$500$		0			0
$501$	26.6948			1.19264
$502$		0			0
$503$			11.6206i			0.518139i	0.965859	+	0.259070i	$0.0834159\pi$
$503$			11.6206i			0.518139i	−0.965859	+	0.259070i	$0.916584\pi$
$504$		0			0
$505$	−0.744683	+	2.29799i	−0.0331380	+	0.102259i
$506$		0			0
$507$	−18.5833			−0.825313
$508$		0			0
$509$	32.8783			1.45730			0.728652	−	0.684885i	$-0.240147\pi$
$509$	32.8783			1.45730			0.728652	+	0.684885i	$0.240147\pi$
$510$		0			0
$511$	−49.7294			−2.19990
$512$		0			0
$513$	−26.9084			−1.18804
$514$		0			0
$515$	18.6665	+	6.04905i	0.822544	+	0.266553i
$516$		0			0
$517$	57.9153			2.54711
$518$		0			0
$519$			7.43703i			0.326449i
$520$		0			0
$521$		−	18.0441i		−	0.790525i	−0.918568	−	0.395262i	$-0.870654\pi$
$521$		−	18.0441i		−	0.790525i	0.918568	−	0.395262i	$-0.129346\pi$
$522$		0			0
$523$			17.5755i			0.768522i	0.923225	+	0.384261i	$0.125544\pi$
$523$			17.5755i			0.768522i	−0.923225	+	0.384261i	$0.874456\pi$
$524$		0			0
$525$	18.4015	−	25.4106i	0.803106	−	1.10901i
$526$		0			0
$527$		−	0.151062i		−	0.00658038i
$528$		0			0
$529$	−1.26098	−	22.9654i	−0.0548250	−	0.998496i
$530$		0			0
$531$			9.62929i			0.417876i
$532$		0			0
$533$			5.10196i			0.220990i
$534$		0			0
$535$	−28.7350	−	9.31184i	−1.24232	−	0.402586i
$536$		0			0
$537$			7.54308i			0.325508i
$538$		0			0
$539$	−65.2519			−2.81060
$540$		0			0
$541$	16.4640			0.707841			0.353920	−	0.935276i	$-0.384848\pi$
$541$	16.4640			0.707841			0.353920	+	0.935276i	$0.384848\pi$
$542$		0			0
$543$		−	4.79325i		−	0.205698i
$544$		0			0
$545$	14.4157	+	4.67154i	0.617501	+	0.200107i
$546$		0			0
$547$	15.9060			0.680091			0.340045	−	0.940409i	$-0.389558\pi$
$547$	15.9060			0.680091			0.340045	+	0.940409i	$0.389558\pi$
$548$		0			0
$549$		−	0.568526i		−	0.0242641i
$550$		0			0
$551$	−29.8408			−1.27126
$552$		0			0
$553$			51.4202i			2.18661i
$554$		0			0
$555$	−7.04077	+	21.7268i	−0.298864	+	0.922252i
$556$		0			0
$557$	−1.28870			−0.0546042			−0.0273021	−	0.999627i	$-0.508692\pi$
$557$	−1.28870			−0.0546042			−0.0273021	+	0.999627i	$0.508692\pi$
$558$		0			0
$559$	−1.19930			−0.0507250
$560$		0			0
$561$	0.795445			0.0335837
$562$		0			0
$563$		−	17.9985i		−	0.758546i	−0.925285	−	0.379273i	$-0.876174\pi$
$563$		−	17.9985i		−	0.758546i	0.925285	−	0.379273i	$-0.123826\pi$
$564$		0			0
$565$	12.2983	−	37.9508i	0.517393	−	1.59660i
$566$		0			0
$567$			25.8245i			1.08453i
$568$		0			0
$569$		−	31.5889i		−	1.32428i	−0.749381	−	0.662139i	$-0.769649\pi$
$569$		−	31.5889i		−	1.32428i	0.749381	−	0.662139i	$-0.230351\pi$
$570$		0			0
$571$	23.3895			0.978818			0.489409	−	0.872054i	$-0.337213\pi$
$571$	23.3895			0.978818			0.489409	+	0.872054i	$0.337213\pi$
$572$		0			0
$573$	−11.2382			−0.469483
$574$		0			0
$575$	23.5653	−	4.43595i	0.982740	−	0.184992i
$576$		0			0
$577$		−	22.7148i		−	0.945631i	−0.881161	−	0.472816i	$-0.843238\pi$
$577$		−	22.7148i		−	0.945631i	0.881161	−	0.472816i	$-0.156762\pi$
$578$		0			0
$579$		−	24.8127i		−	1.03118i
$580$		0			0
$581$	−58.6697			−2.43403
$582$		0			0
$583$	−50.3116			−2.08369
$584$		0			0
$585$	−1.24188	−	0.402442i	−0.0513453	−	0.0166389i
$586$		0			0
$587$	27.3910			1.13055			0.565274	−	0.824903i	$-0.308770\pi$
$587$	27.3910			1.13055			0.565274	+	0.824903i	$0.308770\pi$
$588$		0			0
$589$		−	8.43119i		−	0.347401i
$590$		0			0
$591$			23.7866i			0.978448i
$592$		0			0
$593$			13.7156i			0.563232i	0.959527	+	0.281616i	$0.0908705\pi$
$593$			13.7156i			0.563232i	−0.959527	+	0.281616i	$0.909130\pi$
$594$		0			0
$595$	−0.763285	−	0.247349i	−0.0312916	−	0.0101403i
$596$		0			0
$597$	19.5366			0.799580
$598$		0			0
$599$			18.6396i			0.761595i	0.924658	+	0.380798i	$0.124351\pi$
$599$			18.6396i			0.761595i	−0.924658	+	0.380798i	$0.875649\pi$
$600$		0			0
$601$	18.0029			0.734353			0.367176	−	0.930151i	$-0.380325\pi$
$601$	18.0029			0.734353			0.367176	+	0.930151i	$0.380325\pi$
$602$		0			0
$603$			2.58880i			0.105424i
$604$		0			0
$605$	−18.8681	+	58.2243i	−0.767097	+	2.36715i
$606$		0			0
$607$	−24.7246			−1.00354			−0.501771	−	0.865000i	$-0.667318\pi$
$607$	−24.7246			−1.00354			−0.501771	+	0.865000i	$0.667318\pi$
$608$		0			0
$609$			39.1495i			1.58642i
$610$		0			0
$611$		−	7.23455i		−	0.292679i
$612$		0			0
$613$	6.97815			0.281845			0.140922	−	0.990021i	$-0.454993\pi$
$613$	6.97815			0.281845			0.140922	+	0.990021i	$0.454993\pi$
$614$		0			0
$615$	−6.81076	+	21.0170i	−0.274636	+	0.847489i
$616$		0			0
$617$	−43.4869			−1.75072			−0.875359	−	0.483473i	$-0.839375\pi$
$617$	−43.4869			−1.75072			−0.875359	+	0.483473i	$0.839375\pi$
$618$		0			0
$619$	22.2396			0.893884			0.446942	−	0.894563i	$-0.352513\pi$
$619$	22.2396			0.893884			0.446942	+	0.894563i	$0.352513\pi$
$620$		0			0
$621$	−18.5487	+	19.5951i	−0.744334	+	0.786324i
$622$		0			0
$623$	−0.884619			−0.0354415
$624$		0			0
$625$	7.79945	+	23.7522i	0.311978	+	0.950089i
$626$		0			0
$627$	44.3960			1.77300
$628$		0			0
$629$	0.584097			0.0232895
$630$		0			0
$631$	−5.28097			−0.210232			−0.105116	−	0.994460i	$-0.533521\pi$
$631$	−5.28097			−0.210232			−0.105116	+	0.994460i	$0.533521\pi$
$632$		0			0
$633$			19.0173i			0.755868i
$634$		0			0
$635$	1.92140	−	5.92918i	0.0762486	−	0.235292i
$636$		0			0
$637$			8.15102i			0.322955i
$638$		0			0
$639$			4.30549i			0.170323i
$640$		0			0
$641$			25.3147i			0.999871i	0.866063	+	0.499936i	$0.166643\pi$
$641$			25.3147i			0.999871i	−0.866063	+	0.499936i	$0.833357\pi$
$642$		0			0
$643$		−	13.2107i		−	0.520979i	−0.965477	−	0.260490i	$-0.916116\pi$
$643$		−	13.2107i		−	0.520979i	0.965477	−	0.260490i	$-0.0838840\pi$
$644$		0			0
$645$	−4.94040	−	1.60098i	−0.194528	−	0.0630386i
$646$		0			0
$647$	29.5611			1.16216			0.581082	−	0.813845i	$-0.302629\pi$
$647$	29.5611			1.16216			0.581082	+	0.813845i	$0.302629\pi$
$648$		0			0
$649$		−	79.0582i		−	3.10330i
$650$		0			0
$651$	−11.0613			−0.433526
$652$		0			0
$653$		−	23.1775i		−	0.907004i	−0.891255	−	0.453502i	$-0.850174\pi$
$653$		−	23.1775i		−	0.907004i	0.891255	−	0.453502i	$-0.149826\pi$
$654$		0			0
$655$	28.4162	+	9.20852i	1.11031	+	0.359807i
$656$		0			0
$657$			8.96041i			0.349579i
$658$		0			0
$659$	−37.1054			−1.44542			−0.722711	−	0.691151i	$-0.757104\pi$
$659$	−37.1054			−1.44542			−0.722711	+	0.691151i	$0.757104\pi$
$660$		0			0
$661$		−	29.1795i		−	1.13495i	−0.823391	−	0.567475i	$-0.807920\pi$
$661$		−	29.1795i		−	1.13495i	0.823391	−	0.567475i	$-0.192080\pi$
$662$		0			0
$663$		−	0.0993639i		−	0.00385897i
$664$		0			0
$665$	−42.6010	−	13.8052i	−1.65200	−	0.535344i
$666$		0			0
$667$	−20.5700	+	21.7305i	−0.796475	+	0.841407i
$668$		0			0
$669$	37.4270			1.44701
$670$		0			0
$671$			4.66769i			0.180194i
$672$		0			0
$673$		−	2.14920i		−	0.0828454i	−0.999142	−	0.0414227i	$-0.986811\pi$
$673$		−	2.14920i		−	0.0828454i	0.999142	−	0.0414227i	$-0.0131890\pi$
$674$		0			0
$675$	−22.7838	−	16.4993i	−0.876950	−	0.635057i
$676$		0			0
$677$	−29.9326			−1.15040			−0.575202	−	0.818012i	$-0.695076\pi$
$677$	−29.9326			−1.15040			−0.575202	+	0.818012i	$0.695076\pi$
$678$		0			0
$679$		−	22.4098i		−	0.860008i
$680$		0			0
$681$			27.3815i			1.04926i
$682$		0			0
$683$	45.2257			1.73051			0.865257	−	0.501328i	$-0.167155\pi$
$683$	45.2257			1.73051			0.865257	+	0.501328i	$0.167155\pi$
$684$		0			0
$685$	8.26098	−	25.4922i	0.315636	−	0.974007i
$686$		0			0
$687$		−	18.1089i		−	0.690899i
$688$		0			0
$689$			6.28472i			0.239429i
$690$		0			0
$691$		−	24.6029i		−	0.935938i	−0.883745	−	0.467969i	$-0.844986\pi$
$691$		−	24.6029i		−	0.935938i	0.883745	−	0.467969i	$-0.155014\pi$
$692$		0			0
$693$			19.5703i			0.743416i
$694$		0			0
$695$	3.36072	+	1.08907i	0.127479	+	0.0413108i
$696$		0			0
$697$	0.565015			0.0214015
$698$		0			0
$699$		−	42.6891i		−	1.61465i
$700$		0			0
$701$			0.713990i			0.0269670i	0.999909	+	0.0134835i	$0.00429207\pi$
$701$			0.713990i			0.0269670i	−0.999909	+	0.0134835i	$0.995708\pi$
$702$		0			0
$703$	32.6000			1.22953
$704$		0			0
$705$	9.65762	−	29.8020i	0.363727	−	1.12241i
$706$		0			0
$707$		−	4.52361i		−	0.170128i
$708$		0			0
$709$			45.7415i			1.71786i	0.512096	+	0.858928i	$0.328869\pi$
$709$			45.7415i			1.71786i	−0.512096	+	0.858928i	$0.671131\pi$
$710$		0			0
$711$	9.26506			0.347467
$712$		0			0
$713$	−6.13971	−	5.81185i	−0.229934	−	0.217655i
$714$		0			0
$715$	10.1960	+	3.30412i	0.381310	+	0.123567i
$716$		0			0
$717$		−	26.7412i		−	0.998670i
$718$		0			0
$719$		−	51.6632i		−	1.92671i	−0.268226	−	0.963356i	$-0.586438\pi$
$719$		−	51.6632i		−	1.92671i	0.268226	−	0.963356i	$-0.413562\pi$
$720$		0			0
$721$	−36.7452			−1.36846
$722$		0			0
$723$			30.1893i			1.12275i
$724$		0			0
$725$	−25.2667	−	18.2973i	−0.938381	−	0.679543i
$726$		0			0
$727$		−	21.0843i		−	0.781973i	−0.920396	−	0.390987i	$-0.872134\pi$
$727$		−	21.0843i		−	0.781973i	0.920396	−	0.390987i	$-0.127866\pi$
$728$		0			0
$729$	29.9454			1.10909
$730$		0			0
$731$			0.132816i			0.00491238i
$732$		0			0
$733$	22.4345			0.828637			0.414319	−	0.910132i	$-0.364020\pi$
$733$	22.4345			0.828637			0.414319	+	0.910132i	$0.364020\pi$
$734$		0			0
$735$	−10.8810	+	33.5773i	−0.401353	+	1.23852i
$736$		0			0
$737$		−	21.2545i		−	0.782919i
$738$		0			0
$739$			17.5899i			0.647057i	0.946218	+	0.323528i	$0.104869\pi$
$739$			17.5899i			0.647057i	−0.946218	+	0.323528i	$0.895131\pi$
$740$		0			0
$741$		−	5.54577i		−	0.203729i
$742$		0			0
$743$		−	39.1058i		−	1.43465i	−0.696737	−	0.717327i	$-0.745366\pi$
$743$		−	39.1058i		−	1.43465i	0.696737	−	0.717327i	$-0.254634\pi$
$744$		0			0
$745$	−30.7804	−	9.97465i	−1.12771	−	0.365443i
$746$		0			0
$747$			10.5713i			0.386784i
$748$		0			0
$749$	56.5652			2.06685
$750$		0			0
$751$	−11.8049			−0.430767			−0.215383	−	0.976530i	$-0.569100\pi$
$751$	−11.8049			−0.430767			−0.215383	+	0.976530i	$0.569100\pi$
$752$		0			0
$753$	6.19532			0.225770
$754$		0			0
$755$	34.6107	+	11.2159i	1.25961	+	0.408189i
$756$		0			0
$757$	6.53262			0.237432			0.118716	−	0.992928i	$-0.462122\pi$
$757$	6.53262			0.237432			0.118716	+	0.992928i	$0.462122\pi$
$758$		0			0
$759$	30.6033	−	32.3298i	1.11083	−	1.17350i
$760$		0			0
$761$	−3.21601			−0.116580			−0.0582902	−	0.998300i	$-0.518565\pi$
$761$	−3.21601			−0.116580			−0.0582902	+	0.998300i	$0.518565\pi$
$762$		0			0
$763$	−28.3775			−1.02733
$764$		0			0
$765$	−0.0445682	+	0.137531i	−0.00161137	+	0.00497246i
$766$		0			0
$767$	−9.87563			−0.356588
$768$		0			0
$769$			32.9411i			1.18789i	0.804507	+	0.593943i	$0.202429\pi$
$769$			32.9411i			1.18789i	−0.804507	+	0.593943i	$0.797571\pi$
$770$		0			0
$771$			28.4404i			1.02426i
$772$		0			0
$773$	−14.7419			−0.530229			−0.265114	−	0.964217i	$-0.585410\pi$
$773$	−14.7419			−0.530229			−0.265114	+	0.964217i	$0.585410\pi$
$774$		0			0
$775$	5.16970	−	7.13884i	0.185701	−	0.256435i
$776$		0			0
$777$		−	42.7695i		−	1.53435i
$778$		0			0
$779$	31.5350			1.12986
$780$		0			0
$781$		−	35.3488i		−	1.26488i
$782$		0			0
$783$	35.1025			1.25446
$784$		0			0
$785$	12.9120	−	39.8446i	0.460848	−	1.42211i
$786$		0			0
$787$			27.5536i			0.982179i	0.871109	+	0.491090i	$0.163401\pi$
$787$			27.5536i			0.982179i	−0.871109	+	0.491090i	$0.836599\pi$
$788$		0			0
$789$		−	18.8734i		−	0.671910i
$790$		0			0
$791$			74.7065i			2.65626i
$792$		0			0
$793$	0.583070			0.0207054
$794$		0			0
$795$	−8.38966	+	25.8893i	−0.297551	+	0.918199i
$796$		0			0
$797$	49.3861			1.74935			0.874673	−	0.484714i	$-0.161076\pi$
$797$	49.3861			1.74935			0.874673	+	0.484714i	$0.161076\pi$
$798$		0			0
$799$	−0.801188			−0.0283440
$800$		0			0
$801$			0.159394i			0.00563190i
$802$		0			0
$803$		−	73.5665i		−	2.59611i
$804$		0			0
$805$	−39.4192	+	21.5063i	−1.38934	+	0.757999i
$806$		0			0
$807$	−5.63397			−0.198325
$808$		0			0
$809$	−49.3397			−1.73469			−0.867346	−	0.497705i	$-0.834176\pi$
$809$	−49.3397			−1.73469			−0.867346	+	0.497705i	$0.834176\pi$
$810$		0			0
$811$			8.90107i			0.312559i	0.987713	+	0.156279i	$0.0499500\pi$
$811$			8.90107i			0.312559i	−0.987713	+	0.156279i	$0.950050\pi$
$812$		0			0
$813$			7.46945i			0.261965i
$814$		0			0
$815$	−5.44731	+	16.8096i	−0.190811	+	0.588816i
$816$		0			0
$817$			7.41283i			0.259342i
$818$		0			0
$819$	2.44465			0.0854230
$820$		0			0
$821$	14.9757			0.522655			0.261328	−	0.965250i	$-0.415840\pi$
$821$	14.9757			0.522655			0.261328	+	0.965250i	$0.415840\pi$
$822$		0			0
$823$	14.6394			0.510296			0.255148	−	0.966902i	$-0.417876\pi$
$823$	14.6394			0.510296			0.255148	+	0.966902i	$0.417876\pi$
$824$		0			0
$825$	37.5908	+	27.2220i	1.30874	+	0.947747i
$826$		0			0
$827$			2.22205i			0.0772682i	0.999253	+	0.0386341i	$0.0123007\pi$
$827$			2.22205i			0.0772682i	−0.999253	+	0.0386341i	$0.987699\pi$
$828$		0			0
$829$	−32.6474			−1.13389			−0.566945	−	0.823756i	$-0.691875\pi$
$829$	−32.6474			−1.13389			−0.566945	+	0.823756i	$0.691875\pi$
$830$		0			0
$831$		−	20.6618i		−	0.716748i
$832$		0			0
$833$	0.902681			0.0312761
$834$		0			0
$835$	12.2799	−	37.8940i	0.424963	−	1.31138i
$836$		0			0
$837$			9.91785i			0.342811i
$838$		0			0
$839$	25.9980			0.897549			0.448775	−	0.893645i	$-0.351861\pi$
$839$	25.9980			0.897549			0.448775	+	0.893645i	$0.351861\pi$
$840$		0			0
$841$	9.92782			0.342339
$842$		0			0
$843$		−	19.7817i		−	0.681320i
$844$		0			0
$845$	−8.54851	+	26.3795i	−0.294078	+	0.907482i
$846$		0			0
$847$		−	114.615i		−	3.93822i
$848$		0			0
$849$			13.9942i			0.480279i
$850$		0			0
$851$	22.4721	−	23.7398i	0.770332	−	0.813790i
$852$		0			0
$853$		−	41.8316i		−	1.43229i	−0.697953	−	0.716143i	$-0.745906\pi$
$853$		−	41.8316i		−	1.43229i	0.697953	−	0.716143i	$-0.254094\pi$
$854$		0			0
$855$	−2.48747	+	7.67599i	−0.0850698	+	0.262513i
$856$		0			0
$857$		−	30.8358i		−	1.05333i	−0.850073	−	0.526665i	$-0.823442\pi$
$857$		−	30.8358i		−	1.05333i	0.850073	−	0.526665i	$-0.176558\pi$
$858$		0			0
$859$			9.41831i			0.321349i	0.987007	+	0.160674i	$0.0513669\pi$
$859$			9.41831i			0.321349i	−0.987007	+	0.160674i	$0.948633\pi$
$860$		0			0
$861$		−	41.3723i		−	1.40996i
$862$		0			0
$863$	−10.8621			−0.369750			−0.184875	−	0.982762i	$-0.559188\pi$
$863$	−10.8621			−0.369750			−0.184875	+	0.982762i	$0.559188\pi$
$864$		0			0
$865$	10.5571	+	3.42111i	0.358951	+	0.116321i
$866$		0			0
$867$	25.4635			0.864787
$868$		0			0
$869$	−76.0678			−2.58042
$870$		0			0
$871$	−2.65503			−0.0899622
$872$		0			0
$873$	−4.03787			−0.136661
$874$		0			0
$875$	−27.6061	−	37.8105i	−0.933258	−	1.27823i
$876$		0			0
$877$		−	6.70958i		−	0.226566i	−0.993563	−	0.113283i	$-0.963863\pi$
$877$		−	6.70958i		−	0.226566i	0.993563	−	0.113283i	$-0.0361368\pi$
$878$		0			0
$879$	−13.7874			−0.465038
$880$		0			0
$881$			20.1686i			0.679497i	0.940516	+	0.339749i	$0.110342\pi$
$881$			20.1686i			0.679497i	−0.940516	+	0.339749i	$0.889658\pi$
$882$		0			0
$883$	35.9396			1.20946			0.604732	−	0.796429i	$-0.293280\pi$
$883$	35.9396			1.20946			0.604732	+	0.796429i	$0.293280\pi$
$884$		0			0
$885$	−40.6817	−	13.1833i	−1.36750	−	0.443151i
$886$		0			0
$887$	−22.6398			−0.760171			−0.380086	−	0.924951i	$-0.624105\pi$
$887$	−22.6398			−0.760171			−0.380086	+	0.924951i	$0.624105\pi$
$888$		0			0
$889$			11.6717i			0.391455i
$890$		0			0
$891$	−38.2032			−1.27985
$892$		0			0
$893$	−44.7165			−1.49638
$894$		0			0
$895$	10.7076	+	3.46989i	0.357916	+	0.115986i
$896$		0			0
$897$	−4.03851	−	3.82285i	−0.134842	−	0.127641i
$898$		0			0
$899$			10.9986i			0.366825i
$900$		0			0
$901$	0.696000			0.0231871
$902$		0			0
$903$	9.72524			0.323636
$904$		0			0
$905$	−6.80415	−	2.20494i	−0.226178	−	0.0732948i
$906$		0			0
$907$			28.4304i			0.944017i	0.881594	+	0.472008i	$0.156471\pi$
$907$			28.4304i			0.944017i	−0.881594	+	0.472008i	$0.843529\pi$
$908$		0			0
$909$	−0.815080			−0.0270345
$910$		0			0
$911$	−2.24437			−0.0743592			−0.0371796	−	0.999309i	$-0.511837\pi$
$911$	−2.24437			−0.0743592			−0.0371796	+	0.999309i	$0.511837\pi$
$912$		0			0
$913$		−	86.7922i		−	2.87240i
$914$		0			0
$915$	2.40190	+	0.778357i	0.0794044	+	0.0257317i
$916$		0			0
$917$	−55.9376			−1.84722
$918$		0			0
$919$	−42.2555			−1.39388			−0.696939	−	0.717130i	$-0.745455\pi$
$919$	−42.2555			−1.39388			−0.696939	+	0.717130i	$0.745455\pi$
$920$		0			0
$921$	−33.1172			−1.09125
$922$		0			0
$923$	−4.41564			−0.145342
$924$		0			0
$925$	27.6030	+	19.9891i	0.907581	+	0.657239i
$926$		0			0
$927$			6.62088i			0.217458i
$928$		0			0
$929$	29.5878			0.970744			0.485372	−	0.874308i	$-0.338684\pi$
$929$	29.5878			0.970744			0.485372	+	0.874308i	$0.338684\pi$
$930$		0			0
$931$	50.3811			1.65117
$932$		0			0
$933$			22.9477i			0.751274i
$934$		0			0
$935$	0.365913	−	1.12916i	0.0119666	−	0.0369274i
$936$		0			0
$937$	−39.2442			−1.28205			−0.641026	−	0.767519i	$-0.721491\pi$
$937$	−39.2442			−1.28205			−0.641026	+	0.767519i	$0.721491\pi$
$938$		0			0
$939$	12.3114			0.401766
$940$		0			0
$941$			24.7562i			0.807028i	0.914973	+	0.403514i	$0.132211\pi$
$941$			24.7562i			0.807028i	−0.914973	+	0.403514i	$0.867789\pi$
$942$		0			0
$943$	21.7379	−	22.9642i	0.707884	−	0.747819i
$944$		0			0
$945$	50.1128	+	16.2395i	1.63017	+	0.528270i
$946$		0			0
$947$	3.59054			0.116677			0.0583384	−	0.998297i	$-0.481420\pi$
$947$	3.59054			0.116677			0.0583384	+	0.998297i	$0.481420\pi$
$948$		0			0
$949$	−9.18964			−0.298308
$950$		0			0
$951$		−	17.1010i		−	0.554537i
$952$		0			0
$953$	−2.68459			−0.0869624			−0.0434812	−	0.999054i	$-0.513845\pi$
$953$	−2.68459			−0.0869624			−0.0434812	+	0.999054i	$0.513845\pi$
$954$		0			0
$955$	−5.16970	+	15.9530i	−0.167288	+	0.516226i
$956$		0			0
$957$	−57.9153			−1.87214
$958$		0			0
$959$			50.1817i			1.62045i
$960$		0			0
$961$	27.8924			0.899756
$962$		0			0
$963$		−	10.1921i		−	0.328436i
$964$		0			0
$965$	−35.2223	−	11.4141i	−1.13385	−	0.367433i
$966$		0			0
$967$	1.70496			0.0548278			0.0274139	−	0.999624i	$-0.491273\pi$
$967$	1.70496			0.0548278			0.0274139	+	0.999624i	$0.491273\pi$
$968$		0			0
$969$	−0.614164			−0.0197298
$970$		0			0
$971$	−29.1443			−0.935286			−0.467643	−	0.883917i	$-0.654897\pi$
$971$	−29.1443			−0.935286			−0.467643	+	0.883917i	$0.654897\pi$
$972$		0			0
$973$	−6.61560			−0.212087
$974$		0			0
$975$	3.40046	−	4.69570i	0.108902	−	0.150383i
$976$		0			0
$977$	61.2752			1.96037			0.980184	−	0.198087i	$-0.0634728\pi$
$977$	61.2752			1.96037			0.980184	+	0.198087i	$0.0634728\pi$
$978$		0			0
$979$		−	1.30865i		−	0.0418246i
$980$		0			0
$981$			5.11315i			0.163250i
$982$		0			0
$983$			14.9815i			0.477834i	0.971040	+	0.238917i	$0.0767925\pi$
$983$			14.9815i			0.477834i	−0.971040	+	0.238917i	$0.923208\pi$
$984$		0			0
$985$	33.7657	+	10.9421i	1.07586	+	0.348643i
$986$		0			0
$987$			58.6656i			1.86735i
$988$		0			0
$989$	5.39813	+	5.10986i	0.171650	+	0.162484i
$990$		0			0
$991$			11.1736i			0.354940i	0.984126	+	0.177470i	$0.0567913\pi$
$991$			11.1736i			0.354940i	−0.984126	+	0.177470i	$0.943209\pi$
$992$		0			0
$993$			29.4718i			0.935261i
$994$		0			0
$995$	8.98704	−	27.7327i	0.284908	−	0.879187i
$996$		0			0
$997$			59.4888i			1.88403i	0.335570	+	0.942015i	$0.391071\pi$
$997$			59.4888i			1.88403i	−0.335570	+	0.942015i	$0.608929\pi$
$998$		0			0
$999$	−38.3483			−1.21329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.m.g.1839.15	yes	40
4.3	odd	2	inner	1840.2.m.g.1839.26	yes	40
5.4	even	2	inner	1840.2.m.g.1839.28	yes	40
20.19	odd	2	inner	1840.2.m.g.1839.13	✓	40
23.22	odd	2	inner	1840.2.m.g.1839.16	yes	40
92.91	even	2	inner	1840.2.m.g.1839.25	yes	40
115.114	odd	2	inner	1840.2.m.g.1839.27	yes	40
460.459	even	2	inner	1840.2.m.g.1839.14	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.m.g.1839.13	✓	40	20.19	odd	2	inner
1840.2.m.g.1839.14	yes	40	460.459	even	2	inner
1840.2.m.g.1839.15	yes	40	1.1	even	1	trivial
1840.2.m.g.1839.16	yes	40	23.22	odd	2	inner
1840.2.m.g.1839.25	yes	40	92.91	even	2	inner
1840.2.m.g.1839.26	yes	40	4.3	odd	2	inner
1840.2.m.g.1839.27	yes	40	115.114	odd	2	inner
1840.2.m.g.1839.28	yes	40	5.4	even	2	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.49850 q^{3} +(-0.689327 + 2.12716i) q^{5} -4.18735i q^{7} -0.754490 q^{9} +O(q^{10})\)	\(q-1.49850 q^{3} +(-0.689327 + 2.12716i) q^{5} -4.18735i q^{7} -0.754490 q^{9} +6.19450 q^{11} -0.773792i q^{13} +(1.03296 - 3.18756i) q^{15} -0.0856933 q^{17} -4.78278 q^{19} +6.27475i q^{21} +(-3.29689 + 3.48288i) q^{23} +(-4.04966 - 2.93262i) q^{25} +5.62611 q^{27} +6.23922 q^{29} +1.76282i q^{31} -9.28247 q^{33} +(8.90717 + 2.88645i) q^{35} -6.81613 q^{37} +1.15953i q^{39} -6.59345 q^{41} -1.54990i q^{43} +(0.520090 - 1.60492i) q^{45} +9.34948 q^{47} -10.5339 q^{49} +0.128412 q^{51} -8.12198 q^{53} +(-4.27003 + 13.1767i) q^{55} +7.16700 q^{57} -12.7626i q^{59} +0.753523i q^{61} +3.15931i q^{63} +(1.64598 + 0.533395i) q^{65} -3.43119i q^{67} +(4.94040 - 5.21911i) q^{69} -5.70649i q^{71} -11.8761i q^{73} +(6.06842 + 4.39454i) q^{75} -25.9385i q^{77} -12.2799 q^{79} -6.16727 q^{81} -14.0112i q^{83} +(0.0590707 - 0.182284i) q^{85} -9.34948 q^{87} -0.211260i q^{89} -3.24013 q^{91} -2.64160i q^{93} +(3.29689 - 10.1738i) q^{95} +5.35178 q^{97} -4.67369 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 80 q^{9}+O(q^{10}) \) 40 * q + 80 * q^9	\( 40 q + 80 q^{9} - 24 q^{25} + 24 q^{41} - 16 q^{49} + 80 q^{69} + 40 q^{81} - 8 q^{85}+O(q^{100}) \) 40 * q + 80 * q^9 - 24 * q^25 + 24 * q^41 - 16 * q^49 + 80 * q^69 + 40 * q^81 - 8 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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