LMFDB - Embedded newform 4100.2.d.g.1149.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4100,2,Mod(1149,4100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4100.1149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4100 = 2^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4100.d (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:	$32.7386648287$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 261x^{10} + 1141x^{8} + 2289x^{6} + 1896x^{4} + 549x^{2} + 9 $ x^14 + 27x^12 + 261x^10 + 1141x^8 + 2289x^6 + 1896x^4 + 549x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.12
Root			$-2.20421i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.g.1149.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.20421i q^{3} -0.688745i q^{7} -1.85852 q^{9} +O(q^{10})$	$q+2.20421i q^{3} -0.688745i q^{7} -1.85852 q^{9} +3.47071 q^{11} -4.02968i q^{13} -6.46988i q^{17} -1.54726 q^{19} +1.51814 q^{21} +8.11903i q^{23} +2.51606i q^{27} -2.03360 q^{29} -4.74546 q^{31} +7.65016i q^{33} -11.4472i q^{37} +8.88223 q^{39} +1.00000 q^{41} -12.8840i q^{43} +1.89811i q^{47} +6.52563 q^{49} +14.2609 q^{51} -11.4513i q^{53} -3.41049i q^{57} -11.6881 q^{59} -1.25034 q^{61} +1.28005i q^{63} -5.80935i q^{67} -17.8960 q^{69} +6.85018 q^{71} -9.91064i q^{73} -2.39043i q^{77} -12.9458 q^{79} -11.1215 q^{81} +0.577760i q^{83} -4.48246i q^{87} +3.94821 q^{89} -2.77542 q^{91} -10.4600i q^{93} -18.5176i q^{97} -6.45038 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 12 q^{9}+O(q^{10}) $ 14 * q - 12 * q^9	$ 14 q - 12 q^{9} - 2 q^{11} + 2 q^{19} - 2 q^{21} + 10 q^{29} + 8 q^{31} + 22 q^{39} + 14 q^{41} + 6 q^{49} + 54 q^{51} + 32 q^{59} + 20 q^{61} + 38 q^{69} + 54 q^{71} + 2 q^{79} + 54 q^{81} + 32 q^{89} + 18 q^{91} + 80 q^{99}+O(q^{100}) $ 14 * q - 12 * q^9 - 2 * q^11 + 2 * q^19 - 2 * q^21 + 10 * q^29 + 8 * q^31 + 22 * q^39 + 14 * q^41 + 6 * q^49 + 54 * q^51 + 32 * q^59 + 20 * q^61 + 38 * q^69 + 54 * q^71 + 2 * q^79 + 54 * q^81 + 32 * q^89 + 18 * q^91 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times$.

$n$	$1477$	$2051$	$3901$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.20421i	1.27260i	0.771442	+	0.636299i	$0.219536\pi$
$3$	2.20421i	1.27260i	−0.771442	+	0.636299i	$0.780464\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.688745i	− 0.260321i	−0.991493	−	0.130161i	$-0.958451\pi$
$7$	− 0.688745i	− 0.260321i	0.991493	−	0.130161i	$-0.0415493\pi$
$8$	0	0
$9$	−1.85852	−0.619507
$10$	0	0
$11$	3.47071	1.04646	0.523229	−	0.852192i	$-0.324727\pi$
$11$	3.47071	1.04646	0.523229	+	0.852192i	$0.324727\pi$
$12$	0	0
$13$	− 4.02968i	− 1.11763i	−0.829292	−	0.558815i	$-0.811256\pi$
$13$	− 4.02968i	− 1.11763i	0.829292	−	0.558815i	$-0.188744\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.46988i	− 1.56918i	−0.620018	−	0.784588i	$-0.712875\pi$
$17$	− 6.46988i	− 1.56918i	0.620018	−	0.784588i	$-0.287125\pi$
$18$	0	0
$19$	−1.54726	−0.354967	−0.177483	−	0.984124i	$-0.556796\pi$
$19$	−1.54726	−0.354967	−0.177483	+	0.984124i	$0.556796\pi$
$20$	0	0
$21$	1.51814	0.331284
$22$	0	0
$23$	8.11903i	1.69293i	0.532441	+	0.846467i	$0.321275\pi$
$23$	8.11903i	1.69293i	−0.532441	+	0.846467i	$0.678725\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.51606i	0.484215i
$28$	0	0
$29$	−2.03360	−0.377629	−0.188815	−	0.982013i	$-0.560465\pi$
$29$	−2.03360	−0.377629	−0.188815	+	0.982013i	$0.560465\pi$
$30$	0	0
$31$	−4.74546	−0.852309	−0.426154	−	0.904650i	$-0.640132\pi$
$31$	−4.74546	−0.852309	−0.426154	+	0.904650i	$0.640132\pi$
$32$	0	0
$33$	7.65016i	1.33172i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 11.4472i	− 1.88190i	−0.338543	−	0.940951i	$-0.609934\pi$
$37$	− 11.4472i	− 1.88190i	0.338543	−	0.940951i	$-0.390066\pi$
$38$	0	0
$39$	8.88223	1.42230
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	− 12.8840i	− 1.96480i	−0.186796	−	0.982399i	$-0.559810\pi$
$43$	− 12.8840i	− 1.96480i	0.186796	−	0.982399i	$-0.440190\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.89811i	0.276867i	0.990372	+	0.138434i	$0.0442067\pi$
$47$	1.89811i	0.276867i	−0.990372	+	0.138434i	$0.955793\pi$
$48$	0	0
$49$	6.52563	0.932233
$50$	0	0
$51$	14.2609	1.99693
$52$	0	0
$53$	− 11.4513i	− 1.57296i	−0.617614	−	0.786481i	$-0.711901\pi$
$53$	− 11.4513i	− 1.57296i	0.617614	−	0.786481i	$-0.288099\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.41049i	− 0.451730i
$58$	0	0
$59$	−11.6881	−1.52167	−0.760833	−	0.648948i	$-0.775209\pi$
$59$	−11.6881	−1.52167	−0.760833	+	0.648948i	$0.775209\pi$
$60$	0	0
$61$	−1.25034	−0.160089	−0.0800446	−	0.996791i	$-0.525506\pi$
$61$	−1.25034	−0.160089	−0.0800446	+	0.996791i	$0.525506\pi$
$62$	0	0
$63$	1.28005i	0.161271i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.80935i	− 0.709725i	−0.934918	−	0.354863i	$-0.884528\pi$
$67$	− 5.80935i	− 0.709725i	0.934918	−	0.354863i	$-0.115472\pi$
$68$	0	0
$69$	−17.8960	−2.15442
$70$	0	0
$71$	6.85018	0.812967	0.406484	−	0.913658i	$-0.366755\pi$
$71$	6.85018	0.812967	0.406484	+	0.913658i	$0.366755\pi$
$72$	0	0
$73$	− 9.91064i	− 1.15995i	−0.814633	−	0.579976i	$-0.803062\pi$
$73$	− 9.91064i	− 1.15995i	0.814633	−	0.579976i	$-0.196938\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.39043i	− 0.272415i
$78$	0	0
$79$	−12.9458	−1.45651	−0.728256	−	0.685306i	$-0.759669\pi$
$79$	−12.9458	−1.45651	−0.728256	+	0.685306i	$0.759669\pi$
$80$	0	0
$81$	−11.1215	−1.23572
$82$	0	0
$83$	0.577760i	0.0634174i	0.999497	+	0.0317087i	$0.0100949\pi$
$83$	0.577760i	0.0634174i	−0.999497	+	0.0317087i	$0.989905\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.48246i	− 0.480570i
$88$	0	0
$89$	3.94821	0.418509	0.209255	−	0.977861i	$-0.432896\pi$
$89$	3.94821	0.418509	0.209255	+	0.977861i	$0.432896\pi$
$90$	0	0
$91$	−2.77542	−0.290943
$92$	0	0
$93$	− 10.4600i	− 1.08465i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 18.5176i	− 1.88017i	−0.340933	−	0.940087i	$-0.610743\pi$
$97$	− 18.5176i	− 1.88017i	0.340933	−	0.940087i	$-0.389257\pi$
$98$	0	0
$99$	−6.45038	−0.648288
$100$	0	0
$101$	3.85518	0.383605	0.191802	−	0.981434i	$-0.438567\pi$
$101$	3.85518	0.383605	0.191802	+	0.981434i	$0.438567\pi$
$102$	0	0
$103$	9.99501i	0.984837i	0.870358	+	0.492419i	$0.163887\pi$
$103$	9.99501i	0.984837i	−0.870358	+	0.492419i	$0.836113\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 19.7103i	− 1.90547i	−0.303801	−	0.952735i	$-0.598256\pi$
$107$	− 19.7103i	− 1.90547i	0.303801	−	0.952735i	$-0.401744\pi$
$108$	0	0
$109$	14.0577	1.34649	0.673243	−	0.739421i	$-0.264901\pi$
$109$	14.0577	1.34649	0.673243	+	0.739421i	$0.264901\pi$
$110$	0	0
$111$	25.2319	2.39490
$112$	0	0
$113$	14.2263i	1.33830i	0.743127	+	0.669151i	$0.233342\pi$
$113$	14.2263i	1.33830i	−0.743127	+	0.669151i	$0.766658\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.48923i	0.692380i
$118$	0	0
$119$	−4.45610	−0.408490
$120$	0	0
$121$	1.04583	0.0950754
$122$	0	0
$123$	2.20421i	0.198746i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.8514i	0.962910i	0.876471	+	0.481455i	$0.159892\pi$
$127$	10.8514i	0.962910i	−0.876471	+	0.481455i	$0.840108\pi$
$128$	0	0
$129$	28.3991	2.50040
$130$	0	0
$131$	13.3943	1.17027	0.585134	−	0.810937i	$-0.301042\pi$
$131$	13.3943	1.17027	0.585134	+	0.810937i	$0.301042\pi$
$132$	0	0
$133$	1.06567i	0.0924054i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.35557i	− 0.372121i	−0.982538	−	0.186061i	$-0.940428\pi$
$137$	− 4.35557i	− 0.372121i	0.982538	−	0.186061i	$-0.0595721\pi$
$138$	0	0
$139$	17.1104	1.45128	0.725641	−	0.688074i	$-0.241544\pi$
$139$	17.1104	1.45128	0.725641	+	0.688074i	$0.241544\pi$
$140$	0	0
$141$	−4.18381	−0.352341
$142$	0	0
$143$	− 13.9858i	− 1.16955i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.3838i	1.18636i
$148$	0	0
$149$	4.23785	0.347178	0.173589	−	0.984818i	$-0.444464\pi$
$149$	4.23785	0.347178	0.173589	+	0.984818i	$0.444464\pi$
$150$	0	0
$151$	−6.06396	−0.493478	−0.246739	−	0.969082i	$-0.579359\pi$
$151$	−6.06396	−0.493478	−0.246739	+	0.969082i	$0.579359\pi$
$152$	0	0
$153$	12.0244i	0.972114i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.55789i	0.363759i	0.983321	+	0.181880i	$0.0582181\pi$
$157$	4.55789i	0.363759i	−0.983321	+	0.181880i	$0.941782\pi$
$158$	0	0
$159$	25.2411	2.00175
$160$	0	0
$161$	5.59194	0.440707
$162$	0	0
$163$	24.7609i	1.93942i	0.244250	+	0.969712i	$0.421458\pi$
$163$	24.7609i	1.93942i	−0.244250	+	0.969712i	$0.578542\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.60955i	− 0.124551i	−0.998059	−	0.0622753i	$-0.980164\pi$
$167$	− 1.60955i	− 0.124551i	0.998059	−	0.0622753i	$-0.0198357\pi$
$168$	0	0
$169$	−3.23829	−0.249099
$170$	0	0
$171$	2.87562	0.219904
$172$	0	0
$173$	10.0860i	0.766826i	0.923577	+	0.383413i	$0.125251\pi$
$173$	10.0860i	0.766826i	−0.923577	+	0.383413i	$0.874749\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 25.7630i	− 1.93647i
$178$	0	0
$179$	−24.9008	−1.86117	−0.930586	−	0.366072i	$-0.880702\pi$
$179$	−24.9008	−1.86117	−0.930586	+	0.366072i	$0.880702\pi$
$180$	0	0
$181$	16.2719	1.20948	0.604740	−	0.796423i	$-0.293277\pi$
$181$	16.2719	1.20948	0.604740	+	0.796423i	$0.293277\pi$
$182$	0	0
$183$	− 2.75600i	− 0.203729i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 22.4551i	− 1.64208i
$188$	0	0
$189$	1.73292	0.126052
$190$	0	0
$191$	−18.6670	−1.35069	−0.675347	−	0.737500i	$-0.736006\pi$
$191$	−18.6670	−1.35069	−0.675347	+	0.737500i	$0.736006\pi$
$192$	0	0
$193$	− 1.09429i	− 0.0787688i	−0.999224	−	0.0393844i	$-0.987460\pi$
$193$	− 1.09429i	− 0.0787688i	0.999224	−	0.0393844i	$-0.0125397\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.9866i	1.63772i	0.573990	+	0.818862i	$0.305395\pi$
$197$	22.9866i	1.63772i	−0.573990	+	0.818862i	$0.694605\pi$
$198$	0	0
$199$	15.2888	1.08379	0.541896	−	0.840446i	$-0.317707\pi$
$199$	15.2888	1.08379	0.541896	+	0.840446i	$0.317707\pi$
$200$	0	0
$201$	12.8050	0.903195
$202$	0	0
$203$	1.40063i	0.0983049i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 15.0894i	− 1.04878i
$208$	0	0
$209$	−5.37011	−0.371458
$210$	0	0
$211$	−6.95707	−0.478944	−0.239472	−	0.970903i	$-0.576974\pi$
$211$	−6.95707	−0.478944	−0.239472	+	0.970903i	$0.576974\pi$
$212$	0	0
$213$	15.0992i	1.03458i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.26841i	0.221874i
$218$	0	0
$219$	21.8451	1.47615
$220$	0	0
$221$	−26.0715	−1.75376
$222$	0	0
$223$	− 7.50216i	− 0.502382i	−0.967938	−	0.251191i	$-0.919178\pi$
$223$	− 7.50216i	− 0.502382i	0.967938	−	0.251191i	$-0.0808222\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 9.14500i	− 0.606975i	−0.952835	−	0.303488i	$-0.901849\pi$
$227$	− 9.14500i	− 0.606975i	0.952835	−	0.303488i	$-0.0981511\pi$
$228$	0	0
$229$	12.2854	0.811842	0.405921	−	0.913908i	$-0.366951\pi$
$229$	12.2854	0.811842	0.405921	+	0.913908i	$0.366951\pi$
$230$	0	0
$231$	5.26901	0.346675
$232$	0	0
$233$	9.52356i	0.623909i	0.950097	+	0.311955i	$0.100984\pi$
$233$	9.52356i	0.623909i	−0.950097	+	0.311955i	$0.899016\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 28.5351i	− 1.85355i
$238$	0	0
$239$	18.6193	1.20439	0.602193	−	0.798351i	$-0.294294\pi$
$239$	18.6193	1.20439	0.602193	+	0.798351i	$0.294294\pi$
$240$	0	0
$241$	−21.7495	−1.40101	−0.700505	−	0.713647i	$-0.747042\pi$
$241$	−21.7495	−1.40101	−0.700505	+	0.713647i	$0.747042\pi$
$242$	0	0
$243$	− 16.9658i	− 1.08836i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.23498i	0.396722i
$248$	0	0
$249$	−1.27350	−0.0807049
$250$	0	0
$251$	14.4211	0.910254	0.455127	−	0.890426i	$-0.349594\pi$
$251$	14.4211	0.910254	0.455127	+	0.890426i	$0.349594\pi$
$252$	0	0
$253$	28.1788i	1.77159i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 15.0101i	− 0.936304i	−0.883648	−	0.468152i	$-0.844920\pi$
$257$	− 15.0101i	− 0.936304i	0.883648	−	0.468152i	$-0.155080\pi$
$258$	0	0
$259$	−7.88418	−0.489899
$260$	0	0
$261$	3.77948	0.233944
$262$	0	0
$263$	− 26.5038i	− 1.63430i	−0.576427	−	0.817149i	$-0.695554\pi$
$263$	− 26.5038i	− 1.63430i	0.576427	−	0.817149i	$-0.304446\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.70266i	0.532594i
$268$	0	0
$269$	26.0166	1.58626	0.793129	−	0.609053i	$-0.208450\pi$
$269$	26.0166	1.58626	0.793129	+	0.609053i	$0.208450\pi$
$270$	0	0
$271$	−4.53914	−0.275733	−0.137867	−	0.990451i	$-0.544025\pi$
$271$	−4.53914	−0.275733	−0.137867	+	0.990451i	$0.544025\pi$
$272$	0	0
$273$	− 6.11759i	− 0.370254i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 26.9762i	− 1.62084i	−0.585847	−	0.810422i	$-0.699238\pi$
$277$	− 26.9762i	− 1.62084i	0.585847	−	0.810422i	$-0.300762\pi$
$278$	0	0
$279$	8.81952	0.528011
$280$	0	0
$281$	−14.5029	−0.865172	−0.432586	−	0.901593i	$-0.642399\pi$
$281$	−14.5029	−0.865172	−0.432586	+	0.901593i	$0.642399\pi$
$282$	0	0
$283$	15.1459i	0.900330i	0.892945	+	0.450165i	$0.148635\pi$
$283$	15.1459i	0.900330i	−0.892945	+	0.450165i	$0.851365\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 0.688745i	− 0.0406553i
$288$	0	0
$289$	−24.8593	−1.46231
$290$	0	0
$291$	40.8165	2.39271
$292$	0	0
$293$	27.0856i	1.58236i	0.611583	+	0.791180i	$0.290533\pi$
$293$	27.0856i	1.58236i	−0.611583	+	0.791180i	$0.709467\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.73250i	0.506711i
$298$	0	0
$299$	32.7170	1.89208
$300$	0	0
$301$	−8.87382	−0.511478
$302$	0	0
$303$	8.49761i	0.488175i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.0799i	1.65968i	0.558004	+	0.829838i	$0.311567\pi$
$307$	29.0799i	1.65968i	−0.558004	+	0.829838i	$0.688433\pi$
$308$	0	0
$309$	−22.0310	−1.25330
$310$	0	0
$311$	0.958125	0.0543303	0.0271652	−	0.999631i	$-0.491352\pi$
$311$	0.958125	0.0543303	0.0271652	+	0.999631i	$0.491352\pi$
$312$	0	0
$313$	− 2.88181i	− 0.162890i	−0.996678	−	0.0814448i	$-0.974047\pi$
$313$	− 2.88181i	− 0.162890i	0.996678	−	0.0814448i	$-0.0259534\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.9767i	1.29050i	0.763972	+	0.645249i	$0.223246\pi$
$317$	22.9767i	1.29050i	−0.763972	+	0.645249i	$0.776754\pi$
$318$	0	0
$319$	−7.05802	−0.395173
$320$	0	0
$321$	43.4456	2.42490
$322$	0	0
$323$	10.0106i	0.557005i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	30.9861i	1.71354i
$328$	0	0
$329$	1.30731	0.0720744
$330$	0	0
$331$	4.76712	0.262024	0.131012	−	0.991381i	$-0.458177\pi$
$331$	4.76712	0.262024	0.131012	+	0.991381i	$0.458177\pi$
$332$	0	0
$333$	21.2748i	1.16585i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10.2572i	− 0.558743i	−0.960183	−	0.279372i	$-0.909874\pi$
$337$	− 10.2572i	− 0.558743i	0.960183	−	0.279372i	$-0.0901262\pi$
$338$	0	0
$339$	−31.3578	−1.70312
$340$	0	0
$341$	−16.4701	−0.891906
$342$	0	0
$343$	− 9.31571i	− 0.503001i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5.88092i	− 0.315705i	−0.987463	−	0.157852i	$-0.949543\pi$
$347$	− 5.88092i	− 0.315705i	0.987463	−	0.157852i	$-0.0504570\pi$
$348$	0	0
$349$	1.43307	0.0767105	0.0383553	−	0.999264i	$-0.487788\pi$
$349$	1.43307	0.0767105	0.0383553	+	0.999264i	$0.487788\pi$
$350$	0	0
$351$	10.1389	0.541174
$352$	0	0
$353$	− 32.7676i	− 1.74405i	−0.489465	−	0.872023i	$-0.662808\pi$
$353$	− 32.7676i	− 1.74405i	0.489465	−	0.872023i	$-0.337192\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 9.82215i	− 0.519843i
$358$	0	0
$359$	19.5777	1.03327	0.516635	−	0.856206i	$-0.327184\pi$
$359$	19.5777	1.03327	0.516635	+	0.856206i	$0.327184\pi$
$360$	0	0
$361$	−16.6060	−0.873998
$362$	0	0
$363$	2.30522i	0.120993i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.9313i	0.779409i	0.920940	+	0.389704i	$0.127423\pi$
$367$	14.9313i	0.779409i	−0.920940	+	0.389704i	$0.872577\pi$
$368$	0	0
$369$	−1.85852	−0.0967507
$370$	0	0
$371$	−7.88706	−0.409476
$372$	0	0
$373$	0.858506i	0.0444517i	0.999753	+	0.0222259i	$0.00707530\pi$
$373$	0.858506i	0.0444517i	−0.999753	+	0.0222259i	$0.992925\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.19473i	0.422050i
$378$	0	0
$379$	−1.81065	−0.0930066	−0.0465033	−	0.998918i	$-0.514808\pi$
$379$	−1.81065	−0.0930066	−0.0465033	+	0.998918i	$0.514808\pi$
$380$	0	0
$381$	−23.9188	−1.22540
$382$	0	0
$383$	− 5.90035i	− 0.301494i	−0.988572	−	0.150747i	$-0.951832\pi$
$383$	− 5.90035i	− 0.301494i	0.988572	−	0.150747i	$-0.0481678\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	23.9452i	1.21720i
$388$	0	0
$389$	4.83830	0.245312	0.122656	−	0.992449i	$-0.460859\pi$
$389$	4.83830	0.245312	0.122656	+	0.992449i	$0.460859\pi$
$390$	0	0
$391$	52.5291	2.65651
$392$	0	0
$393$	29.5238i	1.48928i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.92639i	0.498191i	0.968479	+	0.249096i	$0.0801334\pi$
$397$	9.92639i	0.498191i	−0.968479	+	0.249096i	$0.919867\pi$
$398$	0	0
$399$	−2.34896	−0.117595
$400$	0	0
$401$	−15.9687	−0.797441	−0.398720	−	0.917073i	$-0.630546\pi$
$401$	−15.9687	−0.797441	−0.398720	+	0.917073i	$0.630546\pi$
$402$	0	0
$403$	19.1226i	0.952567i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 39.7298i	− 1.96933i
$408$	0	0
$409$	12.4657	0.616390	0.308195	−	0.951323i	$-0.400275\pi$
$409$	12.4657	0.616390	0.308195	+	0.951323i	$0.400275\pi$
$410$	0	0
$411$	9.60057	0.473561
$412$	0	0
$413$	8.05015i	0.396122i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	37.7147i	1.84690i
$418$	0	0
$419$	27.8648	1.36128	0.680642	−	0.732616i	$-0.261701\pi$
$419$	27.8648	1.36128	0.680642	+	0.732616i	$0.261701\pi$
$420$	0	0
$421$	−35.6607	−1.73800	−0.868999	−	0.494813i	$-0.835236\pi$
$421$	−35.6607	−1.73800	−0.868999	+	0.494813i	$0.835236\pi$
$422$	0	0
$423$	− 3.52767i	− 0.171521i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.861163i	0.0416746i
$428$	0	0
$429$	30.8277	1.48837
$430$	0	0
$431$	−4.91116	−0.236562	−0.118281	−	0.992980i	$-0.537738\pi$
$431$	−4.91116	−0.236562	−0.118281	+	0.992980i	$0.537738\pi$
$432$	0	0
$433$	− 20.5406i	− 0.987118i	−0.869712	−	0.493559i	$-0.835696\pi$
$433$	− 20.5406i	− 0.987118i	0.869712	−	0.493559i	$-0.164304\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.5623i	− 0.600936i
$438$	0	0
$439$	−5.46749	−0.260949	−0.130474	−	0.991452i	$-0.541650\pi$
$439$	−5.46749	−0.260949	−0.130474	+	0.991452i	$0.541650\pi$
$440$	0	0
$441$	−12.1280	−0.577524
$442$	0	0
$443$	8.21242i	0.390184i	0.980785	+	0.195092i	$0.0625005\pi$
$443$	8.21242i	0.390184i	−0.980785	+	0.195092i	$0.937500\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.34109i	0.441818i
$448$	0	0
$449$	−4.52370	−0.213486	−0.106743	−	0.994287i	$-0.534042\pi$
$449$	−4.52370	−0.213486	−0.106743	+	0.994287i	$0.534042\pi$
$450$	0	0
$451$	3.47071	0.163429
$452$	0	0
$453$	− 13.3662i	− 0.627999i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 23.0796i	− 1.07962i	−0.841787	−	0.539810i	$-0.818496\pi$
$457$	− 23.0796i	− 1.07962i	0.841787	−	0.539810i	$-0.181504\pi$
$458$	0	0
$459$	16.2786	0.759819
$460$	0	0
$461$	−2.63243	−0.122605	−0.0613023	−	0.998119i	$-0.519525\pi$
$461$	−2.63243	−0.122605	−0.0613023	+	0.998119i	$0.519525\pi$
$462$	0	0
$463$	− 6.68944i	− 0.310884i	−0.987845	−	0.155442i	$-0.950320\pi$
$463$	− 6.68944i	− 0.310884i	0.987845	−	0.155442i	$-0.0496802\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.97479i	0.0913822i	0.998956	+	0.0456911i	$0.0145490\pi$
$467$	1.97479i	0.0913822i	−0.998956	+	0.0456911i	$0.985451\pi$
$468$	0	0
$469$	−4.00116	−0.184757
$470$	0	0
$471$	−10.0465	−0.462920
$472$	0	0
$473$	− 44.7168i	− 2.05608i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	21.2825i	0.974461i
$478$	0	0
$479$	38.6725	1.76699	0.883495	−	0.468440i	$-0.155184\pi$
$479$	38.6725	1.76699	0.883495	+	0.468440i	$0.155184\pi$
$480$	0	0
$481$	−46.1283	−2.10327
$482$	0	0
$483$	12.3258i	0.560843i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.75025i	0.124626i	0.998057	+	0.0623129i	$0.0198477\pi$
$487$	2.75025i	0.124626i	−0.998057	+	0.0623129i	$0.980152\pi$
$488$	0	0
$489$	−54.5781	−2.46811
$490$	0	0
$491$	11.1025	0.501048	0.250524	−	0.968110i	$-0.419397\pi$
$491$	11.1025	0.501048	0.250524	+	0.968110i	$0.419397\pi$
$492$	0	0
$493$	13.1571i	0.592566i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.71803i	− 0.211633i
$498$	0	0
$499$	16.7715	0.750797	0.375399	−	0.926863i	$-0.377506\pi$
$499$	16.7715	0.750797	0.375399	+	0.926863i	$0.377506\pi$
$500$	0	0
$501$	3.54778	0.158503
$502$	0	0
$503$	− 5.35443i	− 0.238742i	−0.992850	−	0.119371i	$-0.961912\pi$
$503$	− 5.35443i	− 0.238742i	0.992850	−	0.119371i	$-0.0380878\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7.13785i	− 0.317003i
$508$	0	0
$509$	16.1533	0.715980	0.357990	−	0.933725i	$-0.383462\pi$
$509$	16.1533	0.715980	0.357990	+	0.933725i	$0.383462\pi$
$510$	0	0
$511$	−6.82590	−0.301960
$512$	0	0
$513$	− 3.89301i	− 0.171880i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.58777i	0.289730i
$518$	0	0
$519$	−22.2317	−0.975862
$520$	0	0
$521$	14.2841	0.625798	0.312899	−	0.949786i	$-0.398700\pi$
$521$	14.2841	0.625798	0.312899	+	0.949786i	$0.398700\pi$
$522$	0	0
$523$	− 32.9804i	− 1.44213i	−0.692866	−	0.721067i	$-0.743652\pi$
$523$	− 32.9804i	− 1.44213i	0.692866	−	0.721067i	$-0.256348\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	30.7025i	1.33742i
$528$	0	0
$529$	−42.9186	−1.86603
$530$	0	0
$531$	21.7226	0.942682
$532$	0	0
$533$	− 4.02968i	− 0.174545i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 54.8865i	− 2.36853i
$538$	0	0
$539$	22.6486	0.975543
$540$	0	0
$541$	15.8331	0.680716	0.340358	−	0.940296i	$-0.389452\pi$
$541$	15.8331	0.680716	0.340358	+	0.940296i	$0.389452\pi$
$542$	0	0
$543$	35.8666i	1.53918i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 35.2771i	− 1.50834i	−0.656680	−	0.754169i	$-0.728040\pi$
$547$	− 35.2771i	− 1.50834i	0.656680	−	0.754169i	$-0.271960\pi$
$548$	0	0
$549$	2.32377	0.0991763
$550$	0	0
$551$	3.14651	0.134046
$552$	0	0
$553$	8.91633i	0.379161i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.201168i	0.00852374i	0.999991	+	0.00426187i	$0.00135660\pi$
$557$	0.201168i	0.00852374i	−0.999991	+	0.00426187i	$0.998643\pi$
$558$	0	0
$559$	−51.9185	−2.19592
$560$	0	0
$561$	49.4956	2.08970
$562$	0	0
$563$	− 27.1130i	− 1.14267i	−0.820715	−	0.571337i	$-0.806425\pi$
$563$	− 27.1130i	− 1.14267i	0.820715	−	0.571337i	$-0.193575\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.65985i	0.321684i
$568$	0	0
$569$	−0.686600	−0.0287838	−0.0143919	−	0.999896i	$-0.504581\pi$
$569$	−0.686600	−0.0287838	−0.0143919	+	0.999896i	$0.504581\pi$
$570$	0	0
$571$	−9.92428	−0.415318	−0.207659	−	0.978201i	$-0.566584\pi$
$571$	−9.92428	−0.415318	−0.207659	+	0.978201i	$0.566584\pi$
$572$	0	0
$573$	− 41.1458i	− 1.71889i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 29.7498i	− 1.23850i	−0.785194	−	0.619250i	$-0.787437\pi$
$577$	− 29.7498i	− 1.23850i	0.785194	−	0.619250i	$-0.212563\pi$
$578$	0	0
$579$	2.41204	0.100241
$580$	0	0
$581$	0.397929	0.0165089
$582$	0	0
$583$	− 39.7443i	− 1.64604i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.3489i	1.00499i	0.864581	+	0.502494i	$0.167584\pi$
$587$	24.3489i	1.00499i	−0.864581	+	0.502494i	$0.832416\pi$
$588$	0	0
$589$	7.34248	0.302541
$590$	0	0
$591$	−50.6671	−2.08417
$592$	0	0
$593$	34.4008i	1.41267i	0.707877	+	0.706336i	$0.249653\pi$
$593$	34.4008i	1.41267i	−0.707877	+	0.706336i	$0.750347\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	33.6996i	1.37923i
$598$	0	0
$599$	7.97180	0.325719	0.162859	−	0.986649i	$-0.447928\pi$
$599$	7.97180	0.325719	0.162859	+	0.986649i	$0.447928\pi$
$600$	0	0
$601$	18.7989	0.766823	0.383411	−	0.923578i	$-0.374749\pi$
$601$	18.7989	0.766823	0.383411	+	0.923578i	$0.374749\pi$
$602$	0	0
$603$	10.7968i	0.439679i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 31.7750i	− 1.28971i	−0.764306	−	0.644854i	$-0.776918\pi$
$607$	− 31.7750i	− 1.28971i	0.764306	−	0.644854i	$-0.223082\pi$
$608$	0	0
$609$	−3.08727	−0.125103
$610$	0	0
$611$	7.64875	0.309435
$612$	0	0
$613$	35.2325i	1.42303i	0.702672	+	0.711514i	$0.251990\pi$
$613$	35.2325i	1.42303i	−0.702672	+	0.711514i	$0.748010\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 28.0241i	− 1.12821i	−0.825703	−	0.564104i	$-0.809222\pi$
$617$	− 28.0241i	− 1.12821i	0.825703	−	0.564104i	$-0.190778\pi$
$618$	0	0
$619$	−22.5596	−0.906747	−0.453374	−	0.891321i	$-0.649780\pi$
$619$	−22.5596	−0.906747	−0.453374	+	0.891321i	$0.649780\pi$
$620$	0	0
$621$	−20.4279	−0.819745
$622$	0	0
$623$	− 2.71931i	− 0.108947i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 11.8368i	− 0.472717i
$628$	0	0
$629$	−74.0617	−2.95303
$630$	0	0
$631$	8.24013	0.328034	0.164017	−	0.986457i	$-0.447555\pi$
$631$	8.24013	0.328034	0.164017	+	0.986457i	$0.447555\pi$
$632$	0	0
$633$	− 15.3348i	− 0.609504i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 26.2962i	− 1.04189i
$638$	0	0
$639$	−12.7312	−0.503638
$640$	0	0
$641$	−9.49524	−0.375039	−0.187520	−	0.982261i	$-0.560045\pi$
$641$	−9.49524	−0.375039	−0.187520	+	0.982261i	$0.560045\pi$
$642$	0	0
$643$	17.1605i	0.676745i	0.941012	+	0.338372i	$0.109876\pi$
$643$	17.1605i	0.676745i	−0.941012	+	0.338372i	$0.890124\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.9852i	0.471185i	0.971852	+	0.235593i	$0.0757031\pi$
$647$	11.9852i	0.471185i	−0.971852	+	0.235593i	$0.924297\pi$
$648$	0	0
$649$	−40.5661	−1.59236
$650$	0	0
$651$	−7.20424	−0.282357
$652$	0	0
$653$	− 11.7743i	− 0.460763i	−0.973100	−	0.230382i	$-0.926003\pi$
$653$	− 11.7743i	− 0.460763i	0.973100	−	0.230382i	$-0.0739975\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	18.4191i	0.718598i
$658$	0	0
$659$	22.7545	0.886390	0.443195	−	0.896425i	$-0.353845\pi$
$659$	22.7545	0.886390	0.443195	+	0.896425i	$0.353845\pi$
$660$	0	0
$661$	−1.81641	−0.0706501	−0.0353251	−	0.999376i	$-0.511247\pi$
$661$	−1.81641	−0.0706501	−0.0353251	+	0.999376i	$0.511247\pi$
$662$	0	0
$663$	− 57.4669i	− 2.23183i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.5108i	− 0.639301i
$668$	0	0
$669$	16.5363	0.639330
$670$	0	0
$671$	−4.33955	−0.167527
$672$	0	0
$673$	19.6691i	0.758188i	0.925358	+	0.379094i	$0.123764\pi$
$673$	19.6691i	0.758188i	−0.925358	+	0.379094i	$0.876236\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	46.1292i	1.77289i	0.462835	+	0.886445i	$0.346832\pi$
$677$	46.1292i	1.77289i	−0.462835	+	0.886445i	$0.653168\pi$
$678$	0	0
$679$	−12.7539	−0.489449
$680$	0	0
$681$	20.1575	0.772436
$682$	0	0
$683$	− 32.5263i	− 1.24459i	−0.782785	−	0.622293i	$-0.786201\pi$
$683$	− 32.5263i	− 1.24459i	0.782785	−	0.622293i	$-0.213799\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	27.0795i	1.03315i
$688$	0	0
$689$	−46.1452	−1.75799
$690$	0	0
$691$	−33.1552	−1.26128	−0.630642	−	0.776074i	$-0.717208\pi$
$691$	−33.1552	−1.26128	−0.630642	+	0.776074i	$0.717208\pi$
$692$	0	0
$693$	4.44267i	0.168763i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.46988i	− 0.245064i
$698$	0	0
$699$	−20.9919	−0.793986
$700$	0	0
$701$	−8.56333	−0.323433	−0.161716	−	0.986837i	$-0.551703\pi$
$701$	−8.56333	−0.323433	−0.161716	+	0.986837i	$0.551703\pi$
$702$	0	0
$703$	17.7118i	0.668013i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.65524i	− 0.0998605i
$708$	0	0
$709$	−6.32957	−0.237712	−0.118856	−	0.992912i	$-0.537923\pi$
$709$	−6.32957	−0.237712	−0.118856	+	0.992912i	$0.537923\pi$
$710$	0	0
$711$	24.0599	0.902318
$712$	0	0
$713$	− 38.5285i	− 1.44290i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	41.0409i	1.53270i
$718$	0	0
$719$	−32.7306	−1.22065	−0.610323	−	0.792153i	$-0.708960\pi$
$719$	−32.7306	−1.22065	−0.610323	+	0.792153i	$0.708960\pi$
$720$	0	0
$721$	6.88401	0.256374
$722$	0	0
$723$	− 47.9404i	− 1.78292i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	43.3028i	1.60601i	0.595970	+	0.803006i	$0.296768\pi$
$727$	43.3028i	1.60601i	−0.595970	+	0.803006i	$0.703232\pi$
$728$	0	0
$729$	4.03175	0.149324
$730$	0	0
$731$	−83.3581	−3.08311
$732$	0	0
$733$	50.9757i	1.88283i	0.337252	+	0.941414i	$0.390502\pi$
$733$	50.9757i	1.88283i	−0.337252	+	0.941414i	$0.609498\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 20.1626i	− 0.742698i
$738$	0	0
$739$	−11.1501	−0.410161	−0.205081	−	0.978745i	$-0.565746\pi$
$739$	−11.1501	−0.410161	−0.205081	+	0.978745i	$0.565746\pi$
$740$	0	0
$741$	−13.7432	−0.504868
$742$	0	0
$743$	45.5924i	1.67262i	0.548255	+	0.836311i	$0.315292\pi$
$743$	45.5924i	1.67262i	−0.548255	+	0.836311i	$0.684708\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.07378i	− 0.0392875i
$748$	0	0
$749$	−13.5754	−0.496034
$750$	0	0
$751$	−18.7367	−0.683711	−0.341856	−	0.939753i	$-0.611055\pi$
$751$	−18.7367	−0.683711	−0.341856	+	0.939753i	$0.611055\pi$
$752$	0	0
$753$	31.7871i	1.15839i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23.2987i	0.846805i	0.905942	+	0.423403i	$0.139164\pi$
$757$	23.2987i	0.846805i	−0.905942	+	0.423403i	$0.860836\pi$
$758$	0	0
$759$	−62.1118	−2.25452
$760$	0	0
$761$	1.44791	0.0524865	0.0262433	−	0.999656i	$-0.491646\pi$
$761$	1.44791	0.0524865	0.0262433	+	0.999656i	$0.491646\pi$
$762$	0	0
$763$	− 9.68220i	− 0.350519i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	47.0994i	1.70066i
$768$	0	0
$769$	37.1153	1.33841	0.669206	−	0.743077i	$-0.266634\pi$
$769$	37.1153	1.33841	0.669206	+	0.743077i	$0.266634\pi$
$770$	0	0
$771$	33.0853	1.19154
$772$	0	0
$773$	14.3889i	0.517531i	0.965940	+	0.258766i	$0.0833157\pi$
$773$	14.3889i	0.517531i	−0.965940	+	0.258766i	$0.916684\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 17.3783i	− 0.623445i
$778$	0	0
$779$	−1.54726	−0.0554365
$780$	0	0
$781$	23.7750	0.850736
$782$	0	0
$783$	− 5.11664i	− 0.182854i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.4763i	0.480376i	0.970726	+	0.240188i	$0.0772092\pi$
$787$	13.4763i	0.480376i	−0.970726	+	0.240188i	$0.922791\pi$
$788$	0	0
$789$	58.4199	2.07980
$790$	0	0
$791$	9.79832	0.348388
$792$	0	0
$793$	5.03845i	0.178921i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.5092i	− 0.868161i	−0.900874	−	0.434080i	$-0.857073\pi$
$797$	− 24.5092i	− 0.868161i	0.900874	−	0.434080i	$-0.142927\pi$
$798$	0	0
$799$	12.2805	0.434453
$800$	0	0
$801$	−7.33782	−0.259269
$802$	0	0
$803$	− 34.3969i	− 1.21384i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	57.3459i	2.01867i
$808$	0	0
$809$	−40.1448	−1.41142	−0.705708	−	0.708503i	$-0.749371\pi$
$809$	−40.1448	−1.41142	−0.705708	+	0.708503i	$0.749371\pi$
$810$	0	0
$811$	−26.2644	−0.922268	−0.461134	−	0.887330i	$-0.652557\pi$
$811$	−26.2644	−0.922268	−0.461134	+	0.887330i	$0.652557\pi$
$812$	0	0
$813$	− 10.0052i	− 0.350898i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	19.9350i	0.697438i
$818$	0	0
$819$	5.15817	0.180241
$820$	0	0
$821$	22.4784	0.784501	0.392251	−	0.919858i	$-0.371697\pi$
$821$	22.4784	0.784501	0.392251	+	0.919858i	$0.371697\pi$
$822$	0	0
$823$	− 14.0543i	− 0.489902i	−0.969535	−	0.244951i	$-0.921228\pi$
$823$	− 14.0543i	− 0.489902i	0.969535	−	0.244951i	$-0.0787719\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 45.7734i	− 1.59170i	−0.605496	−	0.795848i	$-0.707025\pi$
$827$	− 45.7734i	− 1.59170i	0.605496	−	0.795848i	$-0.292975\pi$
$828$	0	0
$829$	−16.5229	−0.573863	−0.286931	−	0.957951i	$-0.592635\pi$
$829$	−16.5229	−0.573863	−0.286931	+	0.957951i	$0.592635\pi$
$830$	0	0
$831$	59.4611	2.06268
$832$	0	0
$833$	− 42.2200i	− 1.46284i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 11.9398i	− 0.412701i
$838$	0	0
$839$	50.8270	1.75474	0.877371	−	0.479812i	$-0.159295\pi$
$839$	50.8270	1.75474	0.877371	+	0.479812i	$0.159295\pi$
$840$	0	0
$841$	−24.8645	−0.857396
$842$	0	0
$843$	− 31.9674i	− 1.10102i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.720310i	− 0.0247501i
$848$	0	0
$849$	−33.3847	−1.14576
$850$	0	0
$851$	92.9398	3.18594
$852$	0	0
$853$	− 34.8928i	− 1.19471i	−0.801979	−	0.597353i	$-0.796219\pi$
$853$	− 34.8928i	− 1.19471i	0.801979	−	0.597353i	$-0.203781\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.5599i	1.28302i	0.767114	+	0.641511i	$0.221692\pi$
$857$	37.5599i	1.28302i	−0.767114	+	0.641511i	$0.778308\pi$
$858$	0	0
$859$	−36.4423	−1.24340	−0.621698	−	0.783257i	$-0.713557\pi$
$859$	−36.4423	−1.24340	−0.621698	+	0.783257i	$0.713557\pi$
$860$	0	0
$861$	1.51814	0.0517379
$862$	0	0
$863$	− 41.9871i	− 1.42926i	−0.699504	−	0.714629i	$-0.746596\pi$
$863$	− 41.9871i	− 1.42926i	0.699504	−	0.714629i	$-0.253404\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 54.7950i	− 1.86093i
$868$	0	0
$869$	−44.9310	−1.52418
$870$	0	0
$871$	−23.4098	−0.793211
$872$	0	0
$873$	34.4153i	1.16478i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 20.7398i	− 0.700334i	−0.936687	−	0.350167i	$-0.886125\pi$
$877$	− 20.7398i	− 0.700334i	0.936687	−	0.350167i	$-0.113875\pi$
$878$	0	0
$879$	−59.7023	−2.01371
$880$	0	0
$881$	31.2867	1.05408	0.527039	−	0.849841i	$-0.323302\pi$
$881$	31.2867	1.05408	0.527039	+	0.849841i	$0.323302\pi$
$882$	0	0
$883$	7.20600i	0.242501i	0.992622	+	0.121251i	$0.0386905\pi$
$883$	7.20600i	0.242501i	−0.992622	+	0.121251i	$0.961310\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.4163i	0.651937i	0.945381	+	0.325969i	$0.105690\pi$
$887$	19.4163i	0.651937i	−0.945381	+	0.325969i	$0.894310\pi$
$888$	0	0
$889$	7.47388	0.250666
$890$	0	0
$891$	−38.5994	−1.29313
$892$	0	0
$893$	− 2.93687i	− 0.0982787i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	72.1151i	2.40785i
$898$	0	0
$899$	9.65034	0.321857
$900$	0	0
$901$	−74.0887	−2.46825
$902$	0	0
$903$	− 19.5597i	− 0.650907i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.2946i	0.640668i	0.947305	+	0.320334i	$0.103795\pi$
$907$	19.2946i	0.640668i	−0.947305	+	0.320334i	$0.896205\pi$
$908$	0	0
$909$	−7.16493	−0.237646
$910$	0	0
$911$	−3.06573	−0.101572	−0.0507861	−	0.998710i	$-0.516173\pi$
$911$	−3.06573	−0.101572	−0.0507861	+	0.998710i	$0.516173\pi$
$912$	0	0
$913$	2.00524i	0.0663637i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9.22528i	− 0.304646i
$918$	0	0
$919$	5.87166	0.193688	0.0968441	−	0.995300i	$-0.469125\pi$
$919$	5.87166	0.193688	0.0968441	+	0.995300i	$0.469125\pi$
$920$	0	0
$921$	−64.0980	−2.11210
$922$	0	0
$923$	− 27.6040i	− 0.908597i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 18.5759i	− 0.610113i
$928$	0	0
$929$	−23.1716	−0.760237	−0.380119	−	0.924938i	$-0.624117\pi$
$929$	−23.1716	−0.760237	−0.380119	+	0.924938i	$0.624117\pi$
$930$	0	0
$931$	−10.0969	−0.330912
$932$	0	0
$933$	2.11190i	0.0691407i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.2287i	0.726181i	0.931754	+	0.363091i	$0.118278\pi$
$937$	22.2287i	0.726181i	−0.931754	+	0.363091i	$0.881722\pi$
$938$	0	0
$939$	6.35210	0.207293
$940$	0	0
$941$	13.0223	0.424513	0.212257	−	0.977214i	$-0.431919\pi$
$941$	13.0223	0.424513	0.212257	+	0.977214i	$0.431919\pi$
$942$	0	0
$943$	8.11903i	0.264392i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.7492i	− 0.414293i	−0.978310	−	0.207147i	$-0.933582\pi$
$947$	− 12.7492i	− 0.414293i	0.978310	−	0.207147i	$-0.0664177\pi$
$948$	0	0
$949$	−39.9367	−1.29640
$950$	0	0
$951$	−50.6453	−1.64229
$952$	0	0
$953$	− 34.4952i	− 1.11741i	−0.829367	−	0.558704i	$-0.811299\pi$
$953$	− 34.4952i	− 1.11741i	0.829367	−	0.558704i	$-0.188701\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 15.5573i	− 0.502897i
$958$	0	0
$959$	−2.99988	−0.0968711
$960$	0	0
$961$	−8.48065	−0.273569
$962$	0	0
$963$	36.6321i	1.18045i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.14314i	− 0.0689186i	−0.999406	−	0.0344593i	$-0.989029\pi$
$967$	− 2.14314i	− 0.0689186i	0.999406	−	0.0344593i	$-0.0109709\pi$
$968$	0	0
$969$	−22.0654	−0.708844
$970$	0	0
$971$	−27.8804	−0.894725	−0.447363	−	0.894353i	$-0.647637\pi$
$971$	−27.8804	−0.894725	−0.447363	+	0.894353i	$0.647637\pi$
$972$	0	0
$973$	− 11.7847i	− 0.377799i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	36.8029i	1.17743i	0.808340	+	0.588715i	$0.200366\pi$
$977$	36.8029i	1.17743i	−0.808340	+	0.588715i	$0.799634\pi$
$978$	0	0
$979$	13.7031	0.437952
$980$	0	0
$981$	−26.1266	−0.834157
$982$	0	0
$983$	− 1.39487i	− 0.0444896i	−0.999753	−	0.0222448i	$-0.992919\pi$
$983$	− 1.39487i	− 0.0444896i	0.999753	−	0.0222448i	$-0.00708132\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.88158i	0.0917218i
$988$	0	0
$989$	104.606	3.32627
$990$	0	0
$991$	0.0100125	0.000318057 0	0.000159029	−	1.00000i	$-0.499949\pi$
$991$	0.0100125	0.000318057 0	0.000159029		1.00000i	$0.499949\pi$
$992$	0	0
$993$	10.5077i	0.333452i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	40.3618i	1.27827i	0.769094	+	0.639136i	$0.220708\pi$
$997$	40.3618i	1.27827i	−0.769094	+	0.639136i	$0.779292\pi$
$998$	0	0
$999$	28.8017	0.911246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.g.1149.12		14
5.2	odd	4		4100.2.a.j.1.6	yes	7
5.3	odd	4		4100.2.a.g.1.2	✓	7
5.4	even	2	inner	4100.2.d.g.1149.3		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4100.2.a.g.1.2	✓	7	5.3	odd	4
4100.2.a.j.1.6	yes	7	5.2	odd	4
4100.2.d.g.1149.3		14	5.4	even	2	inner
4100.2.d.g.1149.12		14	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.20421i q^{3} -0.688745i q^{7} -1.85852 q^{9} +O(q^{10})\)	\(q+2.20421i q^{3} -0.688745i q^{7} -1.85852 q^{9} +3.47071 q^{11} -4.02968i q^{13} -6.46988i q^{17} -1.54726 q^{19} +1.51814 q^{21} +8.11903i q^{23} +2.51606i q^{27} -2.03360 q^{29} -4.74546 q^{31} +7.65016i q^{33} -11.4472i q^{37} +8.88223 q^{39} +1.00000 q^{41} -12.8840i q^{43} +1.89811i q^{47} +6.52563 q^{49} +14.2609 q^{51} -11.4513i q^{53} -3.41049i q^{57} -11.6881 q^{59} -1.25034 q^{61} +1.28005i q^{63} -5.80935i q^{67} -17.8960 q^{69} +6.85018 q^{71} -9.91064i q^{73} -2.39043i q^{77} -12.9458 q^{79} -11.1215 q^{81} +0.577760i q^{83} -4.48246i q^{87} +3.94821 q^{89} -2.77542 q^{91} -10.4600i q^{93} -18.5176i q^{97} -6.45038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 12 q^{9}+O(q^{10}) \) 14 * q - 12 * q^9	\( 14 q - 12 q^{9} - 2 q^{11} + 2 q^{19} - 2 q^{21} + 10 q^{29} + 8 q^{31} + 22 q^{39} + 14 q^{41} + 6 q^{49} + 54 q^{51} + 32 q^{59} + 20 q^{61} + 38 q^{69} + 54 q^{71} + 2 q^{79} + 54 q^{81} + 32 q^{89} + 18 q^{91} + 80 q^{99}+O(q^{100}) \) 14 * q - 12 * q^9 - 2 * q^11 + 2 * q^19 - 2 * q^21 + 10 * q^29 + 8 * q^31 + 22 * q^39 + 14 * q^41 + 6 * q^49 + 54 * q^51 + 32 * q^59 + 20 * q^61 + 38 * q^69 + 54 * q^71 + 2 * q^79 + 54 * q^81 + 32 * q^89 + 18 * q^91 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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