LMFDB - Embedded newform 3800.2.d.l.3649.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3800,2,Mod(3649,3800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3800.3649"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3356224.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.3
Root			$-2.11491i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.l.3649.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.642074i q^{3} -3.58774i q^{7} +2.58774 q^{9} +1.35793i q^{13} -5.58774i q^{17} -1.00000 q^{19} -2.30359 q^{21} -4.87189i q^{23} -3.58774i q^{27} -9.58774 q^{29} -7.17548 q^{31} +0.945668i q^{37} +0.871889 q^{39} +10.4596 q^{41} -2.71585i q^{43} +5.89134i q^{47} -5.87189 q^{49} -3.58774 q^{51} +9.81756i q^{53} +0.642074i q^{57} -10.1560 q^{59} +3.28415 q^{61} -9.28415i q^{63} -10.3859i q^{67} -3.12811 q^{69} +14.3510 q^{71} -4.15604i q^{73} -1.28415 q^{79} +5.45963 q^{81} -11.1755i q^{83} +6.15604i q^{87} +6.45963 q^{89} +4.87189 q^{91} +4.60719i q^{93} +13.4053i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100}) $ 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 0.642074i	− 0.370701i	−0.982672	−	0.185351i	$-0.940658\pi$
$3$	− 0.642074i	− 0.370701i	0.982672	−	0.185351i	$-0.0593421\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.58774i	− 1.35604i	−0.735044	−	0.678019i	$-0.762839\pi$
$7$	− 3.58774i	− 1.35604i	0.735044	−	0.678019i	$-0.237161\pi$
$8$	0	0
$9$	2.58774	0.862580
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.35793i	0.376621i	0.982110	+	0.188311i	$0.0603011\pi$
$13$	1.35793i	0.376621i	−0.982110	+	0.188311i	$0.939699\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.58774i	− 1.35523i	−0.735419	−	0.677613i	$-0.763014\pi$
$17$	− 5.58774i	− 1.35523i	0.735419	−	0.677613i	$-0.236986\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.30359	−0.502685
$22$	0	0
$23$	− 4.87189i	− 1.01586i	−0.861399	−	0.507930i	$-0.830411\pi$
$23$	− 4.87189i	− 1.01586i	0.861399	−	0.507930i	$-0.169589\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3.58774i	− 0.690461i
$28$	0	0
$29$	−9.58774	−1.78040	−0.890199	−	0.455571i	$-0.849435\pi$
$29$	−9.58774	−1.78040	−0.890199	+	0.455571i	$0.849435\pi$
$30$	0	0
$31$	−7.17548	−1.28875	−0.644377	−	0.764708i	$-0.722883\pi$
$31$	−7.17548	−1.28875	−0.644377	+	0.764708i	$0.722883\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.945668i	0.155467i	0.996974	+	0.0777334i	$0.0247683\pi$
$37$	0.945668i	0.155467i	−0.996974	+	0.0777334i	$0.975232\pi$
$38$	0	0
$39$	0.871889	0.139614
$40$	0	0
$41$	10.4596	1.63352	0.816760	−	0.576978i	$-0.195768\pi$
$41$	10.4596	1.63352	0.816760	+	0.576978i	$0.195768\pi$
$42$	0	0
$43$	− 2.71585i	− 0.414164i	−0.978324	−	0.207082i	$-0.933603\pi$
$43$	− 2.71585i	− 0.414164i	0.978324	−	0.207082i	$-0.0663966\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.89134i	0.859340i	0.902986	+	0.429670i	$0.141370\pi$
$47$	5.89134i	0.859340i	−0.902986	+	0.429670i	$0.858630\pi$
$48$	0	0
$49$	−5.87189	−0.838841
$50$	0	0
$51$	−3.58774	−0.502384
$52$	0	0
$53$	9.81756i	1.34855i	0.738483	+	0.674273i	$0.235543\pi$
$53$	9.81756i	1.34855i	−0.738483	+	0.674273i	$0.764457\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.642074i	0.0850447i
$58$	0	0
$59$	−10.1560	−1.32220	−0.661102	−	0.750296i	$-0.729911\pi$
$59$	−10.1560	−1.32220	−0.661102	+	0.750296i	$0.729911\pi$
$60$	0	0
$61$	3.28415	0.420492	0.210246	−	0.977649i	$-0.432574\pi$
$61$	3.28415	0.420492	0.210246	+	0.977649i	$0.432574\pi$
$62$	0	0
$63$	− 9.28415i	− 1.16969i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.3859i	− 1.26883i	−0.772991	−	0.634417i	$-0.781240\pi$
$67$	− 10.3859i	− 1.26883i	0.772991	−	0.634417i	$-0.218760\pi$
$68$	0	0
$69$	−3.12811	−0.376580
$70$	0	0
$71$	14.3510	1.70315	0.851573	−	0.524236i	$-0.175649\pi$
$71$	14.3510	1.70315	0.851573	+	0.524236i	$0.175649\pi$
$72$	0	0
$73$	− 4.15604i	− 0.486427i	−0.969973	−	0.243214i	$-0.921798\pi$
$73$	− 4.15604i	− 0.486427i	0.969973	−	0.243214i	$-0.0782016\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.28415	−0.144478	−0.0722389	−	0.997387i	$-0.523014\pi$
$79$	−1.28415	−0.144478	−0.0722389	+	0.997387i	$0.523014\pi$
$80$	0	0
$81$	5.45963	0.606626
$82$	0	0
$83$	− 11.1755i	− 1.22667i	−0.789823	−	0.613334i	$-0.789828\pi$
$83$	− 11.1755i	− 1.22667i	0.789823	−	0.613334i	$-0.210172\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.15604i	0.659996i
$88$	0	0
$89$	6.45963	0.684719	0.342360	−	0.939569i	$-0.388774\pi$
$89$	6.45963	0.684719	0.342360	+	0.939569i	$0.388774\pi$
$90$	0	0
$91$	4.87189	0.510713
$92$	0	0
$93$	4.60719i	0.477743i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.4053i	1.36110i	0.732701	+	0.680551i	$0.238259\pi$
$97$	13.4053i	1.36110i	−0.732701	+	0.680551i	$0.761741\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.0668	−1.89722	−0.948610	−	0.316449i	$-0.897510\pi$
$101$	−19.0668	−1.89722	−0.948610	+	0.316449i	$0.897510\pi$
$102$	0	0
$103$	− 9.05433i	− 0.892150i	−0.894996	−	0.446075i	$-0.852821\pi$
$103$	− 9.05433i	− 0.892150i	0.894996	−	0.446075i	$-0.147179\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.24926i	0.507465i	0.967274	+	0.253733i	$0.0816583\pi$
$107$	5.24926i	0.507465i	−0.967274	+	0.253733i	$0.918342\pi$
$108$	0	0
$109$	6.04737	0.579233	0.289617	−	0.957143i	$-0.406472\pi$
$109$	6.04737	0.579233	0.289617	+	0.957143i	$0.406472\pi$
$110$	0	0
$111$	0.607188	0.0576318
$112$	0	0
$113$	− 6.22982i	− 0.586052i	−0.956105	−	0.293026i	$-0.905338\pi$
$113$	− 6.22982i	− 0.586052i	0.956105	−	0.293026i	$-0.0946622\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.51396i	0.324866i
$118$	0	0
$119$	−20.0474	−1.83774
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 6.71585i	− 0.605548i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.1212i	1.25305i	0.779402	+	0.626525i	$0.215523\pi$
$127$	14.1212i	1.25305i	−0.779402	+	0.626525i	$0.784477\pi$
$128$	0	0
$129$	−1.74378	−0.153531
$130$	0	0
$131$	−18.3510	−1.60333	−0.801666	−	0.597773i	$-0.796053\pi$
$131$	−18.3510	−1.60333	−0.801666	+	0.597773i	$0.796053\pi$
$132$	0	0
$133$	3.58774i	0.311097i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.01945i	− 0.257969i	−0.991647	−	0.128984i	$-0.958828\pi$
$137$	− 3.01945i	− 0.257969i	0.991647	−	0.128984i	$-0.0411717\pi$
$138$	0	0
$139$	−11.7827	−0.999393	−0.499697	−	0.866201i	$-0.666555\pi$
$139$	−11.7827	−0.999393	−0.499697	+	0.866201i	$0.666555\pi$
$140$	0	0
$141$	3.78267	0.318558
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.77018i	0.310960i
$148$	0	0
$149$	−20.4985	−1.67930	−0.839652	−	0.543124i	$-0.817241\pi$
$149$	−20.4985	−1.67930	−0.839652	+	0.543124i	$0.817241\pi$
$150$	0	0
$151$	−3.85244	−0.313507	−0.156754	−	0.987638i	$-0.550103\pi$
$151$	−3.85244	−0.313507	−0.156754	+	0.987638i	$0.550103\pi$
$152$	0	0
$153$	− 14.4596i	− 1.16899i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3.89134i	− 0.310562i	−0.987870	−	0.155281i	$-0.950372\pi$
$157$	− 3.89134i	− 0.310562i	0.987870	−	0.155281i	$-0.0496284\pi$
$158$	0	0
$159$	6.30359	0.499908
$160$	0	0
$161$	−17.4791	−1.37754
$162$	0	0
$163$	− 18.7159i	− 1.46594i	−0.680262	−	0.732969i	$-0.738134\pi$
$163$	− 18.7159i	− 1.46594i	0.680262	−	0.732969i	$-0.261866\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.09323i	0.239361i	0.992812	+	0.119681i	$0.0381870\pi$
$167$	3.09323i	0.239361i	−0.992812	+	0.119681i	$0.961813\pi$
$168$	0	0
$169$	11.1560	0.858157
$170$	0	0
$171$	−2.58774	−0.197890
$172$	0	0
$173$	7.51396i	0.571276i	0.958338	+	0.285638i	$0.0922055\pi$
$173$	7.51396i	0.571276i	−0.958338	+	0.285638i	$0.907795\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.52092i	0.490143i
$178$	0	0
$179$	4.45963	0.333328	0.166664	−	0.986014i	$-0.446700\pi$
$179$	4.45963	0.333328	0.166664	+	0.986014i	$0.446700\pi$
$180$	0	0
$181$	8.56829	0.636876	0.318438	−	0.947944i	$-0.396842\pi$
$181$	8.56829	0.636876	0.318438	+	0.947944i	$0.396842\pi$
$182$	0	0
$183$	− 2.10866i	− 0.155877i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−12.8719	−0.936292
$190$	0	0
$191$	7.73530	0.559707	0.279853	−	0.960043i	$-0.409714\pi$
$191$	7.73530	0.559707	0.279853	+	0.960043i	$0.409714\pi$
$192$	0	0
$193$	6.08226i	0.437810i	0.975746	+	0.218905i	$0.0702486\pi$
$193$	6.08226i	0.437810i	−0.975746	+	0.218905i	$0.929751\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	6.91078	0.489892	0.244946	−	0.969537i	$-0.421230\pi$
$199$	6.91078	0.489892	0.244946	+	0.969537i	$0.421230\pi$
$200$	0	0
$201$	−6.66848	−0.470358
$202$	0	0
$203$	34.3983i	2.41429i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.6072i	− 0.876260i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.1949	−0.839534	−0.419767	−	0.907632i	$-0.637888\pi$
$211$	−12.1949	−0.839534	−0.419767	+	0.907632i	$0.637888\pi$
$212$	0	0
$213$	− 9.21438i	− 0.631359i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	25.7438i	1.74760i
$218$	0	0
$219$	−2.66848	−0.180319
$220$	0	0
$221$	7.58774	0.510407
$222$	0	0
$223$	− 1.87885i	− 0.125817i	−0.998019	−	0.0629085i	$-0.979962\pi$
$223$	− 1.87885i	− 0.125817i	0.998019	−	0.0629085i	$-0.0200376\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 7.35793i	− 0.488363i	−0.969730	−	0.244181i	$-0.921481\pi$
$227$	− 7.35793i	− 0.488363i	0.969730	−	0.244181i	$-0.0785192\pi$
$228$	0	0
$229$	−13.0279	−0.860909	−0.430455	−	0.902612i	$-0.641647\pi$
$229$	−13.0279	−0.860909	−0.430455	+	0.902612i	$0.641647\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 26.7019i	− 1.74930i	−0.484753	−	0.874651i	$-0.661091\pi$
$233$	− 26.7019i	− 1.74930i	0.484753	−	0.874651i	$-0.338909\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.824517i	0.0535581i
$238$	0	0
$239$	9.47908	0.613151	0.306575	−	0.951846i	$-0.400817\pi$
$239$	9.47908	0.613151	0.306575	+	0.951846i	$0.400817\pi$
$240$	0	0
$241$	−24.2034	−1.55908	−0.779539	−	0.626353i	$-0.784547\pi$
$241$	−24.2034	−1.55908	−0.779539	+	0.626353i	$0.784547\pi$
$242$	0	0
$243$	− 14.2687i	− 0.915338i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.35793i	− 0.0864028i
$248$	0	0
$249$	−7.17548	−0.454728
$250$	0	0
$251$	9.43171	0.595324	0.297662	−	0.954671i	$-0.403793\pi$
$251$	9.43171	0.595324	0.297662	+	0.954671i	$0.403793\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 25.1102i	− 1.56633i	−0.621814	−	0.783165i	$-0.713604\pi$
$257$	− 25.1102i	− 1.56633i	0.621814	−	0.783165i	$-0.286396\pi$
$258$	0	0
$259$	3.39281	0.210819
$260$	0	0
$261$	−24.8106	−1.53574
$262$	0	0
$263$	− 28.7019i	− 1.76984i	−0.465746	−	0.884918i	$-0.654214\pi$
$263$	− 28.7019i	− 1.76984i	0.465746	−	0.884918i	$-0.345786\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.14756i	− 0.253826i
$268$	0	0
$269$	0.350966	0.0213988	0.0106994	−	0.999943i	$-0.496594\pi$
$269$	0.350966	0.0213988	0.0106994	+	0.999943i	$0.496594\pi$
$270$	0	0
$271$	16.8719	1.02489	0.512447	−	0.858719i	$-0.328739\pi$
$271$	16.8719	1.02489	0.512447	+	0.858719i	$0.328739\pi$
$272$	0	0
$273$	− 3.12811i	− 0.189322i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.14756i	0.369371i	0.982798	+	0.184685i	$0.0591266\pi$
$277$	6.14756i	0.369371i	−0.982798	+	0.184685i	$0.940873\pi$
$278$	0	0
$279$	−18.5683	−1.11165
$280$	0	0
$281$	−17.0279	−1.01580	−0.507900	−	0.861416i	$-0.669578\pi$
$281$	−17.0279	−1.01580	−0.507900	+	0.861416i	$0.669578\pi$
$282$	0	0
$283$	5.74378i	0.341432i	0.985320	+	0.170716i	$0.0546081\pi$
$283$	5.74378i	0.341432i	−0.985320	+	0.170716i	$0.945392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 37.5264i	− 2.21512i
$288$	0	0
$289$	−14.2229	−0.836639
$290$	0	0
$291$	8.60719	0.504562
$292$	0	0
$293$	− 14.2772i	− 0.834082i	−0.908888	−	0.417041i	$-0.863067\pi$
$293$	− 14.2772i	− 0.834082i	0.908888	−	0.417041i	$-0.136933\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.61567	0.382594
$300$	0	0
$301$	−9.74378	−0.561622
$302$	0	0
$303$	12.2423i	0.703302i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 8.83700i	− 0.504354i	−0.967681	−	0.252177i	$-0.918853\pi$
$307$	− 8.83700i	− 0.504354i	0.967681	−	0.252177i	$-0.0811466\pi$
$308$	0	0
$309$	−5.81355	−0.330721
$310$	0	0
$311$	23.2229	1.31685	0.658424	−	0.752648i	$-0.271224\pi$
$311$	23.2229	1.31685	0.658424	+	0.752648i	$0.271224\pi$
$312$	0	0
$313$	30.1421i	1.70373i	0.523759	+	0.851867i	$0.324529\pi$
$313$	30.1421i	1.70373i	−0.523759	+	0.851867i	$0.675471\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5.96511i	− 0.335034i	−0.985869	−	0.167517i	$-0.946425\pi$
$317$	− 5.96511i	− 0.335034i	0.985869	−	0.167517i	$-0.0535749\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	3.37041	0.188118
$322$	0	0
$323$	5.58774i	0.310910i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.88286i	− 0.214723i
$328$	0	0
$329$	21.1366	1.16530
$330$	0	0
$331$	−2.15604	−0.118506	−0.0592532	−	0.998243i	$-0.518872\pi$
$331$	−2.15604	−0.118506	−0.0592532	+	0.998243i	$0.518872\pi$
$332$	0	0
$333$	2.44714i	0.134103i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 0.945668i	− 0.0515138i	−0.999668	−	0.0257569i	$-0.991800\pi$
$337$	− 0.945668i	− 0.0515138i	0.999668	−	0.0257569i	$-0.00819958\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 4.04737i	− 0.218538i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10.2562i	− 0.550583i	−0.961361	−	0.275291i	$-0.911226\pi$
$347$	− 10.2562i	− 0.550583i	0.961361	−	0.275291i	$-0.0887743\pi$
$348$	0	0
$349$	24.0558	1.28768	0.643840	−	0.765160i	$-0.277340\pi$
$349$	24.0558	1.28768	0.643840	+	0.765160i	$0.277340\pi$
$350$	0	0
$351$	4.87189	0.260042
$352$	0	0
$353$	15.3315i	0.816014i	0.912979	+	0.408007i	$0.133776\pi$
$353$	15.3315i	0.816014i	−0.912979	+	0.408007i	$0.866224\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	12.8719i	0.681253i
$358$	0	0
$359$	−15.1281	−0.798431	−0.399216	−	0.916857i	$-0.630718\pi$
$359$	−15.1281	−0.798431	−0.399216	+	0.916857i	$0.630718\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.06281i	0.370701i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	33.6740i	1.75777i	0.477035	+	0.878884i	$0.341712\pi$
$367$	33.6740i	1.75777i	−0.477035	+	0.878884i	$0.658288\pi$
$368$	0	0
$369$	27.0668	1.40904
$370$	0	0
$371$	35.2229	1.82868
$372$	0	0
$373$	16.0738i	0.832269i	0.909303	+	0.416134i	$0.136615\pi$
$373$	16.0738i	0.832269i	−0.909303	+	0.416134i	$0.863385\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 13.0194i	− 0.670536i
$378$	0	0
$379$	6.00848	0.308635	0.154317	−	0.988021i	$-0.450682\pi$
$379$	6.00848	0.308635	0.154317	+	0.988021i	$0.450682\pi$
$380$	0	0
$381$	9.06682	0.464507
$382$	0	0
$383$	− 15.8649i	− 0.810660i	−0.914170	−	0.405330i	$-0.867157\pi$
$383$	− 15.8649i	− 0.810660i	0.914170	−	0.405330i	$-0.132843\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.02792i	− 0.357249i
$388$	0	0
$389$	−15.7438	−0.798241	−0.399121	−	0.916898i	$-0.630685\pi$
$389$	−15.7438	−0.798241	−0.399121	+	0.916898i	$0.630685\pi$
$390$	0	0
$391$	−27.2229	−1.37672
$392$	0	0
$393$	11.7827i	0.594357i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.9861i	1.20383i	0.798561	+	0.601913i	$0.205595\pi$
$397$	23.9861i	1.20383i	−0.798561	+	0.601913i	$0.794405\pi$
$398$	0	0
$399$	2.30359	0.115324
$400$	0	0
$401$	−24.5683	−1.22688	−0.613441	−	0.789741i	$-0.710215\pi$
$401$	−24.5683	−1.22688	−0.613441	+	0.789741i	$0.710215\pi$
$402$	0	0
$403$	− 9.74378i	− 0.485372i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.9193	0.737710	0.368855	−	0.929487i	$-0.379750\pi$
$409$	14.9193	0.737710	0.368855	+	0.929487i	$0.379750\pi$
$410$	0	0
$411$	−1.93871	−0.0956294
$412$	0	0
$413$	36.4372i	1.79296i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.56534i	0.370476i
$418$	0	0
$419$	−14.0389	−0.685845	−0.342922	−	0.939364i	$-0.611417\pi$
$419$	−14.0389	−0.685845	−0.342922	+	0.939364i	$0.611417\pi$
$420$	0	0
$421$	39.9387	1.94649	0.973247	−	0.229762i	$-0.0737949\pi$
$421$	39.9387	1.94649	0.973247	+	0.229762i	$0.0737949\pi$
$422$	0	0
$423$	15.2453i	0.741250i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 11.7827i	− 0.570203i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	27.0279	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$431$	27.0279	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$432$	0	0
$433$	− 40.0683i	− 1.92556i	−0.270284	−	0.962781i	$-0.587118\pi$
$433$	− 40.0683i	− 1.92556i	0.270284	−	0.962781i	$-0.412882\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.87189i	0.233054i
$438$	0	0
$439$	−6.42074	−0.306445	−0.153223	−	0.988192i	$-0.548965\pi$
$439$	−6.42074	−0.306445	−0.153223	+	0.988192i	$0.548965\pi$
$440$	0	0
$441$	−15.1949	−0.723568
$442$	0	0
$443$	− 35.2702i	− 1.67574i	−0.545871	−	0.837870i	$-0.683801\pi$
$443$	− 35.2702i	− 1.67574i	0.545871	−	0.837870i	$-0.316199\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.1616i	0.622521i
$448$	0	0
$449$	−5.78267	−0.272901	−0.136451	−	0.990647i	$-0.543569\pi$
$449$	−5.78267	−0.272901	−0.136451	+	0.990647i	$0.543569\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.47355i	0.116218i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.12811i	0.426995i	0.976944	+	0.213498i	$0.0684855\pi$
$457$	9.12811i	0.426995i	−0.976944	+	0.213498i	$0.931514\pi$
$458$	0	0
$459$	−20.0474	−0.935731
$460$	0	0
$461$	8.86341	0.412810	0.206405	−	0.978467i	$-0.433824\pi$
$461$	8.86341	0.412810	0.206405	+	0.978467i	$0.433824\pi$
$462$	0	0
$463$	− 21.1366i	− 0.982301i	−0.871075	−	0.491150i	$-0.836577\pi$
$463$	− 21.1366i	− 0.982301i	0.871075	−	0.491150i	$-0.163423\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.78267i	− 0.360139i	−0.983654	−	0.180070i	$-0.942368\pi$
$467$	− 7.78267i	− 0.360139i	0.983654	−	0.180070i	$-0.0576323\pi$
$468$	0	0
$469$	−37.2617	−1.72059
$470$	0	0
$471$	−2.49852	−0.115126
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	25.4053i	1.16323i
$478$	0	0
$479$	−40.4068	−1.84623	−0.923117	−	0.384519i	$-0.874367\pi$
$479$	−40.4068	−1.84623	−0.923117	+	0.384519i	$0.874367\pi$
$480$	0	0
$481$	−1.28415	−0.0585521
$482$	0	0
$483$	11.2229i	0.510658i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5.82603i	− 0.264003i	−0.991250	−	0.132001i	$-0.957860\pi$
$487$	− 5.82603i	− 0.264003i	0.991250	−	0.132001i	$-0.0421403\pi$
$488$	0	0
$489$	−12.0170	−0.543426
$490$	0	0
$491$	22.6630	1.02277	0.511384	−	0.859352i	$-0.329133\pi$
$491$	22.6630	1.02277	0.511384	+	0.859352i	$0.329133\pi$
$492$	0	0
$493$	53.5738i	2.41284i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 51.4876i	− 2.30953i
$498$	0	0
$499$	25.7438	1.15245	0.576225	−	0.817291i	$-0.304525\pi$
$499$	25.7438	1.15245	0.576225	+	0.817291i	$0.304525\pi$
$500$	0	0
$501$	1.98608	0.0887315
$502$	0	0
$503$	25.4791i	1.13606i	0.823009	+	0.568028i	$0.192293\pi$
$503$	25.4791i	1.13606i	−0.823009	+	0.568028i	$0.807707\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7.16300i	− 0.318120i
$508$	0	0
$509$	9.54037	0.422869	0.211435	−	0.977392i	$-0.432186\pi$
$509$	9.54037	0.422869	0.211435	+	0.977392i	$0.432186\pi$
$510$	0	0
$511$	−14.9108	−0.659614
$512$	0	0
$513$	3.58774i	0.158403i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	4.82452	0.211773
$520$	0	0
$521$	6.31207	0.276537	0.138268	−	0.990395i	$-0.455846\pi$
$521$	6.31207	0.276537	0.138268	+	0.990395i	$0.455846\pi$
$522$	0	0
$523$	8.93719i	0.390796i	0.980724	+	0.195398i	$0.0625999\pi$
$523$	8.93719i	0.390796i	−0.980724	+	0.195398i	$0.937400\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	40.0947i	1.74655i
$528$	0	0
$529$	−0.735300	−0.0319696
$530$	0	0
$531$	−26.2812	−1.14051
$532$	0	0
$533$	14.2034i	0.615218i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.86341i	− 0.123565i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.71585	0.374724	0.187362	−	0.982291i	$-0.440006\pi$
$541$	8.71585	0.374724	0.187362	+	0.982291i	$0.440006\pi$
$542$	0	0
$543$	− 5.50148i	− 0.236091i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	26.9457i	1.15211i	0.817410	+	0.576057i	$0.195409\pi$
$547$	26.9457i	1.15211i	−0.817410	+	0.576057i	$0.804591\pi$
$548$	0	0
$549$	8.49852	0.362708
$550$	0	0
$551$	9.58774	0.408452
$552$	0	0
$553$	4.60719i	0.195918i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.3619i	0.989877i	0.868928	+	0.494938i	$0.164809\pi$
$557$	23.3619i	0.989877i	−0.868928	+	0.494938i	$0.835191\pi$
$558$	0	0
$559$	3.68793	0.155983
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.6476i	1.33379i	0.745153	+	0.666894i	$0.232376\pi$
$563$	31.6476i	1.33379i	−0.745153	+	0.666894i	$0.767624\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 19.5877i	− 0.822608i
$568$	0	0
$569$	20.8804	0.875350	0.437675	−	0.899133i	$-0.355802\pi$
$569$	20.8804	0.875350	0.437675	+	0.899133i	$0.355802\pi$
$570$	0	0
$571$	42.6630	1.78539	0.892696	−	0.450659i	$-0.148811\pi$
$571$	42.6630	1.78539	0.892696	+	0.450659i	$0.148811\pi$
$572$	0	0
$573$	− 4.96663i	− 0.207484i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.59871i	0.357969i	0.983852	+	0.178985i	$0.0572812\pi$
$577$	8.59871i	0.357969i	−0.983852	+	0.178985i	$0.942719\pi$
$578$	0	0
$579$	3.90526	0.162297
$580$	0	0
$581$	−40.0947	−1.66341
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 29.7438i	− 1.22766i	−0.789439	−	0.613829i	$-0.789629\pi$
$587$	− 29.7438i	− 1.22766i	0.789439	−	0.613829i	$-0.210371\pi$
$588$	0	0
$589$	7.17548	0.295661
$590$	0	0
$591$	1.28415	0.0528228
$592$	0	0
$593$	4.56829i	0.187597i	0.995591	+	0.0937987i	$0.0299010\pi$
$593$	4.56829i	0.187597i	−0.995591	+	0.0937987i	$0.970099\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.43723i	− 0.181604i
$598$	0	0
$599$	−14.7159	−0.601273	−0.300637	−	0.953739i	$-0.597199\pi$
$599$	−14.7159	−0.601273	−0.300637	+	0.953739i	$0.597199\pi$
$600$	0	0
$601$	37.3230	1.52244	0.761219	−	0.648495i	$-0.224601\pi$
$601$	37.3230	1.52244	0.761219	+	0.648495i	$0.224601\pi$
$602$	0	0
$603$	− 26.8759i	− 1.09447i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 23.0404i	− 0.935181i	−0.883945	−	0.467591i	$-0.845122\pi$
$607$	− 23.0404i	− 0.935181i	0.883945	−	0.467591i	$-0.154878\pi$
$608$	0	0
$609$	22.0863	0.894981
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	32.4985i	1.31260i	0.754499	+	0.656302i	$0.227880\pi$
$613$	32.4985i	1.31260i	−0.754499	+	0.656302i	$0.772120\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.2982i	1.30027i	0.759817	+	0.650137i	$0.225289\pi$
$617$	32.2982i	1.30027i	−0.759817	+	0.650137i	$0.774711\pi$
$618$	0	0
$619$	34.5683	1.38942	0.694709	−	0.719291i	$-0.255533\pi$
$619$	34.5683	1.38942	0.694709	+	0.719291i	$0.255533\pi$
$620$	0	0
$621$	−17.4791	−0.701411
$622$	0	0
$623$	− 23.1755i	− 0.928506i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.28415	0.210693
$630$	0	0
$631$	−19.2702	−0.767136	−0.383568	−	0.923513i	$-0.625305\pi$
$631$	−19.2702	−0.767136	−0.383568	+	0.923513i	$0.625305\pi$
$632$	0	0
$633$	7.83004i	0.311216i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 7.97359i	− 0.315925i
$638$	0	0
$639$	37.1366	1.46910
$640$	0	0
$641$	−3.06682	−0.121132	−0.0605660	−	0.998164i	$-0.519291\pi$
$641$	−3.06682	−0.121132	−0.0605660	+	0.998164i	$0.519291\pi$
$642$	0	0
$643$	− 25.5264i	− 1.00666i	−0.864093	−	0.503332i	$-0.832107\pi$
$643$	− 25.5264i	− 1.00666i	0.864093	−	0.503332i	$-0.167893\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 25.4791i	− 1.00169i	−0.865538	−	0.500843i	$-0.833023\pi$
$647$	− 25.4791i	− 1.00169i	0.865538	−	0.500843i	$-0.166977\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	16.5294	0.647838
$652$	0	0
$653$	− 36.2034i	− 1.41675i	−0.705837	−	0.708374i	$-0.749429\pi$
$653$	− 36.2034i	− 1.41675i	0.705837	−	0.708374i	$-0.250571\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.7547i	− 0.419583i
$658$	0	0
$659$	42.6406	1.66104	0.830522	−	0.556986i	$-0.188042\pi$
$659$	42.6406	1.66104	0.830522	+	0.556986i	$0.188042\pi$
$660$	0	0
$661$	24.4681	0.951699	0.475850	−	0.879527i	$-0.342141\pi$
$661$	24.4681	0.951699	0.475850	+	0.879527i	$0.342141\pi$
$662$	0	0
$663$	− 4.87189i	− 0.189208i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	46.7104i	1.80863i
$668$	0	0
$669$	−1.20636	−0.0466406
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 26.7672i	− 1.03180i	−0.856649	−	0.515901i	$-0.827457\pi$
$673$	− 26.7672i	− 1.03180i	0.856649	−	0.515901i	$-0.172543\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.65304i	0.217264i	0.994082	+	0.108632i	$0.0346470\pi$
$677$	5.65304i	0.217264i	−0.994082	+	0.108632i	$0.965353\pi$
$678$	0	0
$679$	48.0947	1.84571
$680$	0	0
$681$	−4.72433	−0.181037
$682$	0	0
$683$	− 11.1102i	− 0.425119i	−0.977148	−	0.212560i	$-0.931820\pi$
$683$	− 11.1102i	− 0.425119i	0.977148	−	0.212560i	$-0.0681800\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.36489i	0.319140i
$688$	0	0
$689$	−13.3315	−0.507890
$690$	0	0
$691$	−5.96111	−0.226771	−0.113386	−	0.993551i	$-0.536170\pi$
$691$	−5.96111	−0.226771	−0.113386	+	0.993551i	$0.536170\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 58.4457i	− 2.21379i
$698$	0	0
$699$	−17.1446	−0.648469
$700$	0	0
$701$	−25.8524	−0.976433	−0.488217	−	0.872722i	$-0.662352\pi$
$701$	−25.8524	−0.976433	−0.488217	+	0.872722i	$0.662352\pi$
$702$	0	0
$703$	− 0.945668i	− 0.0356665i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	68.4068i	2.57270i
$708$	0	0
$709$	23.4317	0.879996	0.439998	−	0.897999i	$-0.354979\pi$
$709$	23.4317	0.879996	0.439998	+	0.897999i	$0.354979\pi$
$710$	0	0
$711$	−3.32304	−0.124624
$712$	0	0
$713$	34.9582i	1.30919i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.08627i	− 0.227296i
$718$	0	0
$719$	6.20885	0.231551	0.115776	−	0.993275i	$-0.463065\pi$
$719$	6.20885	0.231551	0.115776	+	0.993275i	$0.463065\pi$
$720$	0	0
$721$	−32.4846	−1.20979
$722$	0	0
$723$	15.5404i	0.577953i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 29.7214i	− 1.10230i	−0.834405	−	0.551152i	$-0.814188\pi$
$727$	− 29.7214i	− 1.10230i	0.834405	−	0.551152i	$-0.185812\pi$
$728$	0	0
$729$	7.21733	0.267308
$730$	0	0
$731$	−15.1755	−0.561286
$732$	0	0
$733$	− 7.52645i	− 0.277996i	−0.990293	−	0.138998i	$-0.955612\pi$
$733$	− 7.52645i	− 0.277996i	0.990293	−	0.138998i	$-0.0443881\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−22.9582	−0.844529	−0.422265	−	0.906473i	$-0.638765\pi$
$739$	−22.9582	−0.844529	−0.422265	+	0.906473i	$0.638765\pi$
$740$	0	0
$741$	−0.871889	−0.0320296
$742$	0	0
$743$	0.524931i	0.0192579i	0.999954	+	0.00962893i	$0.00306503\pi$
$743$	0.524931i	0.0192579i	−0.999954	+	0.00962893i	$0.996935\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 28.9193i	− 1.05810i
$748$	0	0
$749$	18.8330	0.688143
$750$	0	0
$751$	21.3619	0.779508	0.389754	−	0.920919i	$-0.372560\pi$
$751$	21.3619	0.779508	0.389754	+	0.920919i	$0.372560\pi$
$752$	0	0
$753$	− 6.05585i	− 0.220687i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 50.4068i	− 1.83207i	−0.401102	−	0.916033i	$-0.631373\pi$
$757$	− 50.4068i	− 1.83207i	0.401102	−	0.916033i	$-0.368627\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.2338	−0.733476	−0.366738	−	0.930324i	$-0.619525\pi$
$761$	−20.2338	−0.733476	−0.366738	+	0.930324i	$0.619525\pi$
$762$	0	0
$763$	− 21.6964i	− 0.785463i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 13.7911i	− 0.497970i
$768$	0	0
$769$	27.1142	0.977763	0.488881	−	0.872350i	$-0.337405\pi$
$769$	27.1142	0.977763	0.488881	+	0.872350i	$0.337405\pi$
$770$	0	0
$771$	−16.1226	−0.580641
$772$	0	0
$773$	− 12.1685i	− 0.437671i	−0.975762	−	0.218836i	$-0.929774\pi$
$773$	− 12.1685i	− 0.437671i	0.975762	−	0.218836i	$-0.0702259\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.17843i	− 0.0781509i
$778$	0	0
$779$	−10.4596	−0.374755
$780$	0	0
$781$	0	0
$782$	0	0
$783$	34.3983i	1.22930i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.1296i	1.28788i	0.765075	+	0.643941i	$0.222702\pi$
$787$	36.1296i	1.28788i	−0.765075	+	0.643941i	$0.777298\pi$
$788$	0	0
$789$	−18.4288	−0.656081
$790$	0	0
$791$	−22.3510	−0.794709
$792$	0	0
$793$	4.45963i	0.158366i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 42.4417i	− 1.50336i	−0.659527	−	0.751681i	$-0.729243\pi$
$797$	− 42.4417i	− 1.50336i	0.659527	−	0.751681i	$-0.270757\pi$
$798$	0	0
$799$	32.9193	1.16460
$800$	0	0
$801$	16.7159	0.590626
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 0.225346i	− 0.00793255i
$808$	0	0
$809$	37.3006	1.31142	0.655710	−	0.755013i	$-0.272369\pi$
$809$	37.3006	1.31142	0.655710	+	0.755013i	$0.272369\pi$
$810$	0	0
$811$	27.4402	0.963555	0.481778	−	0.876294i	$-0.339991\pi$
$811$	27.4402	0.963555	0.481778	+	0.876294i	$0.339991\pi$
$812$	0	0
$813$	− 10.8330i	− 0.379930i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.71585i	0.0950157i
$818$	0	0
$819$	12.6072	0.440531
$820$	0	0
$821$	47.0140	1.64080	0.820400	−	0.571790i	$-0.193751\pi$
$821$	47.0140	1.64080	0.820400	+	0.571790i	$0.193751\pi$
$822$	0	0
$823$	35.9776i	1.25410i	0.778979	+	0.627050i	$0.215738\pi$
$823$	35.9776i	1.25410i	−0.778979	+	0.627050i	$0.784262\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 14.6032i	− 0.507802i	−0.967230	−	0.253901i	$-0.918286\pi$
$827$	− 14.6032i	− 0.507802i	0.967230	−	0.253901i	$-0.0817138\pi$
$828$	0	0
$829$	6.96663	0.241961	0.120981	−	0.992655i	$-0.461396\pi$
$829$	6.96663	0.241961	0.120981	+	0.992655i	$0.461396\pi$
$830$	0	0
$831$	3.94719	0.136926
$832$	0	0
$833$	32.8106i	1.13682i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	25.7438i	0.889835i
$838$	0	0
$839$	0.459630	0.0158682	0.00793410	−	0.999969i	$-0.497474\pi$
$839$	0.459630	0.0158682	0.00793410	+	0.999969i	$0.497474\pi$
$840$	0	0
$841$	62.9248	2.16982
$842$	0	0
$843$	10.9332i	0.376559i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	39.4652i	1.35604i
$848$	0	0
$849$	3.68793	0.126569
$850$	0	0
$851$	4.60719	0.157932
$852$	0	0
$853$	− 5.48755i	− 0.187890i	−0.995577	−	0.0939451i	$-0.970052\pi$
$853$	− 5.48755i	− 0.187890i	0.995577	−	0.0939451i	$-0.0299478\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 52.6586i	− 1.79878i	−0.437145	−	0.899391i	$-0.644010\pi$
$857$	− 52.6586i	− 1.79878i	0.437145	−	0.899391i	$-0.355990\pi$
$858$	0	0
$859$	4.29512	0.146547	0.0732737	−	0.997312i	$-0.476655\pi$
$859$	4.29512	0.146547	0.0732737	+	0.997312i	$0.476655\pi$
$860$	0	0
$861$	−24.0947	−0.821147
$862$	0	0
$863$	− 14.6506i	− 0.498711i	−0.968412	−	0.249355i	$-0.919781\pi$
$863$	− 14.6506i	− 0.498711i	0.968412	−	0.249355i	$-0.0802187\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.13212i	0.310143i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	14.1032	0.477869
$872$	0	0
$873$	34.6894i	1.17406i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.7089i	0.665522i	0.943011	+	0.332761i	$0.107980\pi$
$877$	19.7089i	0.665522i	−0.943011	+	0.332761i	$0.892020\pi$
$878$	0	0
$879$	−9.16701	−0.309195
$880$	0	0
$881$	1.32304	0.0445744	0.0222872	−	0.999752i	$-0.492905\pi$
$881$	1.32304	0.0445744	0.0222872	+	0.999752i	$0.492905\pi$
$882$	0	0
$883$	− 25.0668i	− 0.843566i	−0.906697	−	0.421783i	$-0.861404\pi$
$883$	− 25.0668i	− 0.843566i	0.906697	−	0.421783i	$-0.138596\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.5669i	1.22779i	0.789386	+	0.613897i	$0.210399\pi$
$887$	36.5669i	1.22779i	−0.789386	+	0.613897i	$0.789601\pi$
$888$	0	0
$889$	50.6630	1.69918
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.89134i	− 0.197146i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.24774i	− 0.141828i
$898$	0	0
$899$	68.7967	2.29450
$900$	0	0
$901$	54.8580	1.82758
$902$	0	0
$903$	6.25622i	0.208194i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.7368i	1.35264i	0.736606	+	0.676322i	$0.236427\pi$
$907$	40.7368i	1.35264i	−0.736606	+	0.676322i	$0.763573\pi$
$908$	0	0
$909$	−49.3400	−1.63650
$910$	0	0
$911$	−58.2982	−1.93150	−0.965752	−	0.259467i	$-0.916453\pi$
$911$	−58.2982	−1.93150	−0.965752	+	0.259467i	$0.916453\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	65.8385i	2.17418i
$918$	0	0
$919$	14.5209	0.479001	0.239501	−	0.970896i	$-0.423016\pi$
$919$	14.5209	0.479001	0.239501	+	0.970896i	$0.423016\pi$
$920$	0	0
$921$	−5.67401	−0.186965
$922$	0	0
$923$	19.4876i	0.641441i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 23.4303i	− 0.769551i
$928$	0	0
$929$	−13.7523	−0.451197	−0.225598	−	0.974220i	$-0.572434\pi$
$929$	−13.7523	−0.451197	−0.225598	+	0.974220i	$0.572434\pi$
$930$	0	0
$931$	5.87189	0.192443
$932$	0	0
$933$	− 14.9108i	− 0.488157i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.4512i	0.406761i	0.979100	+	0.203381i	$0.0651929\pi$
$937$	12.4512i	0.406761i	−0.979100	+	0.203381i	$0.934807\pi$
$938$	0	0
$939$	19.3535	0.631576
$940$	0	0
$941$	−34.3594	−1.12009	−0.560043	−	0.828464i	$-0.689215\pi$
$941$	−34.3594	−1.12009	−0.560043	+	0.828464i	$0.689215\pi$
$942$	0	0
$943$	− 50.9582i	− 1.65943i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 23.4876i	− 0.763243i	−0.924319	−	0.381621i	$-0.875366\pi$
$947$	− 23.4876i	− 0.763243i	0.924319	−	0.381621i	$-0.124634\pi$
$948$	0	0
$949$	5.64359	0.183199
$950$	0	0
$951$	−3.83004	−0.124198
$952$	0	0
$953$	48.5808i	1.57369i	0.617153	+	0.786843i	$0.288286\pi$
$953$	48.5808i	1.57369i	−0.617153	+	0.786843i	$0.711714\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.8330	−0.349816
$960$	0	0
$961$	20.4876	0.660889
$962$	0	0
$963$	13.5837i	0.437730i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 53.3261i	− 1.71485i	−0.514608	−	0.857425i	$-0.672063\pi$
$967$	− 53.3261i	− 1.71485i	0.514608	−	0.857425i	$-0.327937\pi$
$968$	0	0
$969$	3.58774	0.115255
$970$	0	0
$971$	−33.1366	−1.06340	−0.531702	−	0.846932i	$-0.678447\pi$
$971$	−33.1366	−1.06340	−0.531702	+	0.846932i	$0.678447\pi$
$972$	0	0
$973$	42.2732i	1.35522i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 5.62263i	− 0.179884i	−0.995947	−	0.0899419i	$-0.971332\pi$
$977$	− 5.62263i	− 0.179884i	0.995947	−	0.0899419i	$-0.0286681\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	15.6490	0.499635
$982$	0	0
$983$	21.3664i	0.681482i	0.940157	+	0.340741i	$0.110678\pi$
$983$	21.3664i	0.681482i	−0.940157	+	0.340741i	$0.889322\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 13.5712i	− 0.431978i
$988$	0	0
$989$	−13.2313	−0.420732
$990$	0	0
$991$	41.1446	1.30700	0.653501	−	0.756926i	$-0.273300\pi$
$991$	41.1446	1.30700	0.653501	+	0.756926i	$0.273300\pi$
$992$	0	0
$993$	1.38433i	0.0439305i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	48.5155i	1.53650i	0.640150	+	0.768250i	$0.278872\pi$
$997$	48.5155i	1.53650i	−0.640150	+	0.768250i	$0.721128\pi$
$998$	0	0
$999$	3.39281	0.107344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.l.3649.3		6
5.2	odd	4		760.2.a.j.1.2	✓	3
5.3	odd	4		3800.2.a.x.1.2		3
5.4	even	2	inner	3800.2.d.l.3649.4		6
15.2	even	4		6840.2.a.bg.1.3		3
20.3	even	4		7600.2.a.bq.1.2		3
20.7	even	4		1520.2.a.s.1.2		3
40.27	even	4		6080.2.a.bq.1.2		3
40.37	odd	4		6080.2.a.bv.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.j.1.2	✓	3	5.2	odd	4
1520.2.a.s.1.2		3	20.7	even	4
3800.2.a.x.1.2		3	5.3	odd	4
3800.2.d.l.3649.3		6	1.1	even	1	trivial
3800.2.d.l.3649.4		6	5.4	even	2	inner
6080.2.a.bq.1.2		3	40.27	even	4
6080.2.a.bv.1.2		3	40.37	odd	4
6840.2.a.bg.1.3		3	15.2	even	4
7600.2.a.bq.1.2		3	20.3	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.642074i q^{3} -3.58774i q^{7} +2.58774 q^{9} +1.35793i q^{13} -5.58774i q^{17} -1.00000 q^{19} -2.30359 q^{21} -4.87189i q^{23} -3.58774i q^{27} -9.58774 q^{29} -7.17548 q^{31} +0.945668i q^{37} +0.871889 q^{39} +10.4596 q^{41} -2.71585i q^{43} +5.89134i q^{47} -5.87189 q^{49} -3.58774 q^{51} +9.81756i q^{53} +0.642074i q^{57} -10.1560 q^{59} +3.28415 q^{61} -9.28415i q^{63} -10.3859i q^{67} -3.12811 q^{69} +14.3510 q^{71} -4.15604i q^{73} -1.28415 q^{79} +5.45963 q^{81} -11.1755i q^{83} +6.15604i q^{87} +6.45963 q^{89} +4.87189 q^{91} +4.60719i q^{93} +13.4053i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100}) \) 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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