LMFDB - Embedded newform 4100.2.d.g.1149.1

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			1149.1
Root			$3.41812i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.g.1149.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41812i q^{3} -2.44400i q^{7} -8.68355 q^{9} +O(q^{10})$	$q-3.41812i q^{3} -2.44400i q^{7} -8.68355 q^{9} -3.18330 q^{11} -2.56659i q^{13} +3.03436i q^{17} -5.23955 q^{19} -8.35389 q^{21} +6.06586i q^{23} +19.4271i q^{27} +3.49177 q^{29} +2.10304 q^{31} +10.8809i q^{33} +11.7337i q^{37} -8.77290 q^{39} +1.00000 q^{41} -11.0294i q^{43} -3.36333i q^{47} +1.02686 q^{49} +10.3718 q^{51} +1.72687i q^{53} +17.9094i q^{57} -5.43391 q^{59} +9.91414 q^{61} +21.2226i q^{63} -11.4266i q^{67} +20.7338 q^{69} -0.455977 q^{71} -11.9120i q^{73} +7.77999i q^{77} -4.85299 q^{79} +40.3534 q^{81} +4.15742i q^{83} -11.9353i q^{87} -12.0233 q^{89} -6.27274 q^{91} -7.18844i q^{93} -0.605138i q^{97} +27.6424 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 3.41812i	− 1.97345i	−0.162391	−	0.986727i	$-0.551921\pi$
$3$	− 3.41812i	− 1.97345i	0.162391	−	0.986727i	$-0.448079\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.44400i	− 0.923746i	−0.886946	−	0.461873i	$-0.847178\pi$
$7$	− 2.44400i	− 0.923746i	0.886946	−	0.461873i	$-0.152822\pi$
$8$	0	0
$9$	−8.68355	−2.89452
$10$	0	0
$11$	−3.18330	−0.959802	−0.479901	−	0.877323i	$-0.659327\pi$
$11$	−3.18330	−0.959802	−0.479901	+	0.877323i	$0.659327\pi$
$12$	0	0
$13$	− 2.56659i	− 0.711843i	−0.934516	−	0.355921i	$-0.884167\pi$
$13$	− 2.56659i	− 0.711843i	0.934516	−	0.355921i	$-0.115833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.03436i	0.735940i	0.929838	+	0.367970i	$0.119947\pi$
$17$	3.03436i	0.735940i	−0.929838	+	0.367970i	$0.880053\pi$
$18$	0	0
$19$	−5.23955	−1.20203	−0.601017	−	0.799236i	$-0.705238\pi$
$19$	−5.23955	−1.20203	−0.601017	+	0.799236i	$0.705238\pi$
$20$	0	0
$21$	−8.35389	−1.82297
$22$	0	0
$23$	6.06586i	1.26482i	0.774634	+	0.632410i	$0.217934\pi$
$23$	6.06586i	1.26482i	−0.774634	+	0.632410i	$0.782066\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	19.4271i	3.73874i
$28$	0	0
$29$	3.49177	0.648405	0.324203	−	0.945988i	$-0.394904\pi$
$29$	3.49177	0.648405	0.324203	+	0.945988i	$0.394904\pi$
$30$	0	0
$31$	2.10304	0.377717	0.188859	−	0.982004i	$-0.439521\pi$
$31$	2.10304	0.377717	0.188859	+	0.982004i	$0.439521\pi$
$32$	0	0
$33$	10.8809i	1.89412i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.7337i	1.92901i	0.264074	+	0.964503i	$0.414934\pi$
$37$	11.7337i	1.92901i	−0.264074	+	0.964503i	$0.585066\pi$
$38$	0	0
$39$	−8.77290	−1.40479
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	− 11.0294i	− 1.68196i	−0.541063	−	0.840982i	$-0.681978\pi$
$43$	− 11.0294i	− 1.68196i	0.541063	−	0.840982i	$-0.318022\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.36333i	− 0.490592i	−0.969448	−	0.245296i	$-0.921115\pi$
$47$	− 3.36333i	− 0.490592i	0.969448	−	0.245296i	$-0.0788851\pi$
$48$	0	0
$49$	1.02686	0.146694
$50$	0	0
$51$	10.3718	1.45234
$52$	0	0
$53$	1.72687i	0.237203i	0.992942	+	0.118602i	$0.0378412\pi$
$53$	1.72687i	0.237203i	−0.992942	+	0.118602i	$0.962159\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	17.9094i	2.37216i
$58$	0	0
$59$	−5.43391	−0.707435	−0.353717	−	0.935352i	$-0.615083\pi$
$59$	−5.43391	−0.707435	−0.353717	+	0.935352i	$0.615083\pi$
$60$	0	0
$61$	9.91414	1.26938	0.634688	−	0.772768i	$-0.281129\pi$
$61$	9.91414	1.26938	0.634688	+	0.772768i	$0.281129\pi$
$62$	0	0
$63$	21.2226i	2.67380i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 11.4266i	− 1.39598i	−0.716108	−	0.697990i	$-0.754078\pi$
$67$	− 11.4266i	− 1.39598i	0.716108	−	0.697990i	$-0.245922\pi$
$68$	0	0
$69$	20.7338	2.49606
$70$	0	0
$71$	−0.455977	−0.0541145	−0.0270573	−	0.999634i	$-0.508614\pi$
$71$	−0.455977	−0.0541145	−0.0270573	+	0.999634i	$0.508614\pi$
$72$	0	0
$73$	− 11.9120i	− 1.39419i	−0.716977	−	0.697097i	$-0.754475\pi$
$73$	− 11.9120i	− 1.39419i	0.716977	−	0.697097i	$-0.245525\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.77999i	0.886613i
$78$	0	0
$79$	−4.85299	−0.546004	−0.273002	−	0.962013i	$-0.588017\pi$
$79$	−4.85299	−0.546004	−0.273002	+	0.962013i	$0.588017\pi$
$80$	0	0
$81$	40.3534	4.48371
$82$	0	0
$83$	4.15742i	0.456336i	0.973622	+	0.228168i	$0.0732736\pi$
$83$	4.15742i	0.456336i	−0.973622	+	0.228168i	$0.926726\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 11.9353i	− 1.27960i
$88$	0	0
$89$	−12.0233	−1.27446	−0.637232	−	0.770672i	$-0.719921\pi$
$89$	−12.0233	−1.27446	−0.637232	+	0.770672i	$0.719921\pi$
$90$	0	0
$91$	−6.27274	−0.657562
$92$	0	0
$93$	− 7.18844i	− 0.745407i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 0.605138i	− 0.0614424i	−0.999528	−	0.0307212i	$-0.990220\pi$
$97$	− 0.605138i	− 0.0614424i	0.999528	−	0.0307212i	$-0.00978040\pi$
$98$	0	0
$99$	27.6424	2.77816
$100$	0	0
$101$	−15.2932	−1.52173	−0.760864	−	0.648912i	$-0.775225\pi$
$101$	−15.2932	−1.52173	−0.760864	+	0.648912i	$0.775225\pi$
$102$	0	0
$103$	10.9287i	1.07683i	0.842678	+	0.538417i	$0.180978\pi$
$103$	10.9287i	1.07683i	−0.842678	+	0.538417i	$0.819022\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.2317i	− 1.47251i	−0.676705	−	0.736254i	$-0.736593\pi$
$107$	− 15.2317i	− 1.47251i	0.676705	−	0.736254i	$-0.263407\pi$
$108$	0	0
$109$	1.17299	0.112352	0.0561761	−	0.998421i	$-0.482109\pi$
$109$	1.17299	0.112352	0.0561761	+	0.998421i	$0.482109\pi$
$110$	0	0
$111$	40.1071	3.80680
$112$	0	0
$113$	9.12161i	0.858089i	0.903283	+	0.429044i	$0.141150\pi$
$113$	9.12161i	0.858089i	−0.903283	+	0.429044i	$0.858850\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	22.2871i	2.06044i
$118$	0	0
$119$	7.41598	0.679822
$120$	0	0
$121$	−0.866589	−0.0787808
$122$	0	0
$123$	− 3.41812i	− 0.308202i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.3183i	1.35928i	0.733546	+	0.679640i	$0.237864\pi$
$127$	15.3183i	1.35928i	−0.733546	+	0.679640i	$0.762136\pi$
$128$	0	0
$129$	−37.6997	−3.31928
$130$	0	0
$131$	−8.16919	−0.713745	−0.356873	−	0.934153i	$-0.616157\pi$
$131$	−8.16919	−0.713745	−0.356873	+	0.934153i	$0.616157\pi$
$132$	0	0
$133$	12.8055i	1.11037i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.6993i	1.51215i	0.654485	+	0.756075i	$0.272885\pi$
$137$	17.6993i	1.51215i	−0.654485	+	0.756075i	$0.727115\pi$
$138$	0	0
$139$	4.00217	0.339460	0.169730	−	0.985491i	$-0.445711\pi$
$139$	4.00217	0.339460	0.169730	+	0.985491i	$0.445711\pi$
$140$	0	0
$141$	−11.4963	−0.968160
$142$	0	0
$143$	8.17022i	0.683228i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.50992i	− 0.289493i
$148$	0	0
$149$	−6.20281	−0.508154	−0.254077	−	0.967184i	$-0.581772\pi$
$149$	−6.20281	−0.508154	−0.254077	+	0.967184i	$0.581772\pi$
$150$	0	0
$151$	−16.4161	−1.33592	−0.667960	−	0.744197i	$-0.732832\pi$
$151$	−16.4161	−1.33592	−0.667960	+	0.744197i	$0.732832\pi$
$152$	0	0
$153$	− 26.3490i	− 2.13019i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.2371i	1.21606i	0.793915	+	0.608028i	$0.208039\pi$
$157$	15.2371i	1.21606i	−0.793915	+	0.608028i	$0.791961\pi$
$158$	0	0
$159$	5.90264	0.468110
$160$	0	0
$161$	14.8250	1.16837
$162$	0	0
$163$	− 11.6298i	− 0.910912i	−0.890258	−	0.455456i	$-0.849476\pi$
$163$	− 11.6298i	− 0.910912i	0.890258	−	0.455456i	$-0.150524\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2.39272i	− 0.185154i	−0.995706	−	0.0925770i	$-0.970490\pi$
$167$	− 2.39272i	− 0.185154i	0.995706	−	0.0925770i	$-0.0295104\pi$
$168$	0	0
$169$	6.41263	0.493280
$170$	0	0
$171$	45.4979	3.47931
$172$	0	0
$173$	24.8633i	1.89032i	0.326608	+	0.945160i	$0.394094\pi$
$173$	24.8633i	1.89032i	−0.326608	+	0.945160i	$0.605906\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.5738i	1.39609i
$178$	0	0
$179$	−3.86509	−0.288890	−0.144445	−	0.989513i	$-0.546140\pi$
$179$	−3.86509	−0.288890	−0.144445	+	0.989513i	$0.546140\pi$
$180$	0	0
$181$	−13.2681	−0.986212	−0.493106	−	0.869969i	$-0.664138\pi$
$181$	−13.2681	−0.986212	−0.493106	+	0.869969i	$0.664138\pi$
$182$	0	0
$183$	− 33.8877i	− 2.50505i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 9.65928i	− 0.706357i
$188$	0	0
$189$	47.4798	3.45365
$190$	0	0
$191$	10.4202	0.753980	0.376990	−	0.926217i	$-0.376959\pi$
$191$	10.4202	0.753980	0.376990	+	0.926217i	$0.376959\pi$
$192$	0	0
$193$	− 14.5924i	− 1.05039i	−0.850983	−	0.525193i	$-0.823993\pi$
$193$	− 14.5924i	− 1.05039i	0.850983	−	0.525193i	$-0.176007\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.20040i	0.228019i	0.993480	+	0.114010i	$0.0363694\pi$
$197$	3.20040i	0.228019i	−0.993480	+	0.114010i	$0.963631\pi$
$198$	0	0
$199$	15.8036	1.12029	0.560143	−	0.828396i	$-0.310746\pi$
$199$	15.8036	1.12029	0.560143	+	0.828396i	$0.310746\pi$
$200$	0	0
$201$	−39.0575	−2.75490
$202$	0	0
$203$	− 8.53389i	− 0.598962i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 52.6732i	− 3.66104i
$208$	0	0
$209$	16.6791	1.15372
$210$	0	0
$211$	−12.2144	−0.840876	−0.420438	−	0.907321i	$-0.638124\pi$
$211$	−12.2144	−0.840876	−0.420438	+	0.907321i	$0.638124\pi$
$212$	0	0
$213$	1.55859i	0.106793i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.13983i	− 0.348915i
$218$	0	0
$219$	−40.7167	−2.75138
$220$	0	0
$221$	7.78794	0.523874
$222$	0	0
$223$	− 9.23181i	− 0.618208i	−0.951028	−	0.309104i	$-0.899971\pi$
$223$	− 9.23181i	− 0.618208i	0.951028	−	0.309104i	$-0.100029\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.7515i	0.779977i	0.920820	+	0.389989i	$0.127521\pi$
$227$	11.7515i	0.779977i	−0.920820	+	0.389989i	$0.872479\pi$
$228$	0	0
$229$	4.54517	0.300353	0.150177	−	0.988659i	$-0.452016\pi$
$229$	4.54517	0.300353	0.150177	+	0.988659i	$0.452016\pi$
$230$	0	0
$231$	26.5930	1.74969
$232$	0	0
$233$	− 4.93013i	− 0.322983i	−0.986874	−	0.161492i	$-0.948370\pi$
$233$	− 4.93013i	− 0.322983i	0.986874	−	0.161492i	$-0.0516305\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.5881i	1.07751i
$238$	0	0
$239$	9.29201	0.601050	0.300525	−	0.953774i	$-0.402838\pi$
$239$	9.29201	0.601050	0.300525	+	0.953774i	$0.402838\pi$
$240$	0	0
$241$	−14.7633	−0.950991	−0.475495	−	0.879718i	$-0.657731\pi$
$241$	−14.7633	−0.950991	−0.475495	+	0.879718i	$0.657731\pi$
$242$	0	0
$243$	− 79.6516i	− 5.10965i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.4478i	0.855660i
$248$	0	0
$249$	14.2106	0.900558
$250$	0	0
$251$	6.43055	0.405893	0.202946	−	0.979190i	$-0.434948\pi$
$251$	6.43055	0.405893	0.202946	+	0.979190i	$0.434948\pi$
$252$	0	0
$253$	− 19.3095i	− 1.21398i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.6485i	0.913748i	0.889531	+	0.456874i	$0.151031\pi$
$257$	14.6485i	0.913748i	−0.889531	+	0.456874i	$0.848969\pi$
$258$	0	0
$259$	28.6771	1.78191
$260$	0	0
$261$	−30.3210	−1.87682
$262$	0	0
$263$	21.0714i	1.29932i	0.760225	+	0.649659i	$0.225089\pi$
$263$	21.0714i	1.29932i	−0.760225	+	0.649659i	$0.774911\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	41.0970i	2.51510i
$268$	0	0
$269$	−8.99810	−0.548624	−0.274312	−	0.961641i	$-0.588450\pi$
$269$	−8.99810	−0.548624	−0.274312	+	0.961641i	$0.588450\pi$
$270$	0	0
$271$	−2.12399	−0.129023	−0.0645116	−	0.997917i	$-0.520549\pi$
$271$	−2.12399	−0.129023	−0.0645116	+	0.997917i	$0.520549\pi$
$272$	0	0
$273$	21.4410i	1.29767i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 16.1807i	− 0.972204i	−0.873902	−	0.486102i	$-0.838418\pi$
$277$	− 16.1807i	− 0.972204i	0.873902	−	0.486102i	$-0.161582\pi$
$278$	0	0
$279$	−18.2619	−1.09331
$280$	0	0
$281$	−30.0551	−1.79293	−0.896467	−	0.443110i	$-0.853875\pi$
$281$	−30.0551	−1.79293	−0.896467	+	0.443110i	$0.853875\pi$
$282$	0	0
$283$	13.7698i	0.818531i	0.912415	+	0.409266i	$0.134215\pi$
$283$	13.7698i	0.818531i	−0.912415	+	0.409266i	$0.865785\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.44400i	− 0.144265i
$288$	0	0
$289$	7.79267	0.458392
$290$	0	0
$291$	−2.06843	−0.121254
$292$	0	0
$293$	− 9.70105i	− 0.566741i	−0.959010	−	0.283371i	$-0.908547\pi$
$293$	− 9.70105i	− 0.566741i	0.959010	−	0.283371i	$-0.0914527\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 61.8422i	− 3.58845i
$298$	0	0
$299$	15.5686	0.900353
$300$	0	0
$301$	−26.9558	−1.55371
$302$	0	0
$303$	52.2739i	3.00306i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 8.96214i	− 0.511497i	−0.966743	−	0.255748i	$-0.917678\pi$
$307$	− 8.96214i	− 0.511497i	0.966743	−	0.255748i	$-0.0823218\pi$
$308$	0	0
$309$	37.3555	2.12508
$310$	0	0
$311$	−0.800475	−0.0453908	−0.0226954	−	0.999742i	$-0.507225\pi$
$311$	−0.800475	−0.0453908	−0.0226954	+	0.999742i	$0.507225\pi$
$312$	0	0
$313$	− 16.9504i	− 0.958091i	−0.877790	−	0.479045i	$-0.840983\pi$
$313$	− 16.9504i	− 0.958091i	0.877790	−	0.479045i	$-0.159017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.7050i	1.78073i	0.455249	+	0.890364i	$0.349550\pi$
$317$	31.7050i	1.78073i	−0.455249	+	0.890364i	$0.650450\pi$
$318$	0	0
$319$	−11.1154	−0.622341
$320$	0	0
$321$	−52.0639	−2.90593
$322$	0	0
$323$	− 15.8987i	− 0.884626i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4.00943i	− 0.221722i
$328$	0	0
$329$	−8.21998	−0.453182
$330$	0	0
$331$	−10.6087	−0.583105	−0.291553	−	0.956555i	$-0.594172\pi$
$331$	−10.6087	−0.583105	−0.291553	+	0.956555i	$0.594172\pi$
$332$	0	0
$333$	− 101.890i	− 5.58354i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7.41865i	− 0.404120i	−0.979373	−	0.202060i	$-0.935236\pi$
$337$	− 7.41865i	− 0.404120i	0.979373	−	0.202060i	$-0.0647636\pi$
$338$	0	0
$339$	31.1788	1.69340
$340$	0	0
$341$	−6.69461	−0.362533
$342$	0	0
$343$	− 19.6176i	− 1.05925i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.47347i	0.347514i	0.984789	+	0.173757i	$0.0555907\pi$
$347$	6.47347i	0.347514i	−0.984789	+	0.173757i	$0.944409\pi$
$348$	0	0
$349$	10.4941	0.561738	0.280869	−	0.959746i	$-0.409377\pi$
$349$	10.4941	0.561738	0.280869	+	0.959746i	$0.409377\pi$
$350$	0	0
$351$	49.8612	2.66140
$352$	0	0
$353$	19.7117i	1.04915i	0.851364	+	0.524575i	$0.175776\pi$
$353$	19.7117i	1.04915i	−0.851364	+	0.524575i	$0.824224\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 25.3487i	− 1.34160i
$358$	0	0
$359$	35.2995	1.86304	0.931519	−	0.363691i	$-0.118484\pi$
$359$	35.2995	1.86304	0.931519	+	0.363691i	$0.118484\pi$
$360$	0	0
$361$	8.45287	0.444888
$362$	0	0
$363$	2.96210i	0.155470i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.8136i	1.50406i	0.659129	+	0.752030i	$0.270925\pi$
$367$	28.8136i	1.50406i	−0.659129	+	0.752030i	$0.729075\pi$
$368$	0	0
$369$	−8.68355	−0.452048
$370$	0	0
$371$	4.22047	0.219116
$372$	0	0
$373$	29.7765i	1.54177i	0.636974	+	0.770885i	$0.280186\pi$
$373$	29.7765i	1.54177i	−0.636974	+	0.770885i	$0.719814\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.96193i	− 0.461563i
$378$	0	0
$379$	16.8562	0.865847	0.432923	−	0.901431i	$-0.357482\pi$
$379$	16.8562	0.865847	0.432923	+	0.901431i	$0.357482\pi$
$380$	0	0
$381$	52.3598	2.68247
$382$	0	0
$383$	− 7.64286i	− 0.390532i	−0.980750	−	0.195266i	$-0.937443\pi$
$383$	− 7.64286i	− 0.390532i	0.980750	−	0.195266i	$-0.0625570\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	95.7741i	4.86847i
$388$	0	0
$389$	31.6008	1.60223	0.801113	−	0.598514i	$-0.204242\pi$
$389$	31.6008	1.60223	0.801113	+	0.598514i	$0.204242\pi$
$390$	0	0
$391$	−18.4060	−0.930831
$392$	0	0
$393$	27.9233i	1.40854i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 30.4558i	− 1.52853i	−0.644902	−	0.764265i	$-0.723102\pi$
$397$	− 30.4558i	− 1.52853i	0.644902	−	0.764265i	$-0.276898\pi$
$398$	0	0
$399$	43.7706	2.19127
$400$	0	0
$401$	38.6884	1.93201	0.966004	−	0.258526i	$-0.0832367\pi$
$401$	38.6884	1.93201	0.966004	+	0.258526i	$0.0832367\pi$
$402$	0	0
$403$	− 5.39763i	− 0.268875i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 37.3518i	− 1.85146i
$408$	0	0
$409$	−29.4757	−1.45748	−0.728740	−	0.684791i	$-0.759894\pi$
$409$	−29.4757	−1.45748	−0.728740	+	0.684791i	$0.759894\pi$
$410$	0	0
$411$	60.4982	2.98416
$412$	0	0
$413$	13.2805i	0.653490i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 13.6799i	− 0.669908i
$418$	0	0
$419$	−10.3797	−0.507081	−0.253540	−	0.967325i	$-0.581595\pi$
$419$	−10.3797	−0.507081	−0.253540	+	0.967325i	$0.581595\pi$
$420$	0	0
$421$	−7.97020	−0.388444	−0.194222	−	0.980958i	$-0.562218\pi$
$421$	−7.97020	−0.388444	−0.194222	+	0.980958i	$0.562218\pi$
$422$	0	0
$423$	29.2056i	1.42003i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 24.2302i	− 1.17258i
$428$	0	0
$429$	27.9268	1.34832
$430$	0	0
$431$	−23.1019	−1.11278	−0.556389	−	0.830922i	$-0.687814\pi$
$431$	−23.1019	−1.11278	−0.556389	+	0.830922i	$0.687814\pi$
$432$	0	0
$433$	39.1216i	1.88006i	0.341089	+	0.940031i	$0.389204\pi$
$433$	39.1216i	1.88006i	−0.341089	+	0.940031i	$0.610796\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 31.7824i	− 1.52036i
$438$	0	0
$439$	−7.17197	−0.342300	−0.171150	−	0.985245i	$-0.554748\pi$
$439$	−7.17197	−0.342300	−0.171150	+	0.985245i	$0.554748\pi$
$440$	0	0
$441$	−8.91676	−0.424608
$442$	0	0
$443$	10.9242i	0.519024i	0.965740	+	0.259512i	$0.0835617\pi$
$443$	10.9242i	0.519024i	−0.965740	+	0.259512i	$0.916438\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21.2019i	1.00282i
$448$	0	0
$449$	38.7384	1.82818	0.914089	−	0.405514i	$-0.132907\pi$
$449$	38.7384	1.82818	0.914089	+	0.405514i	$0.132907\pi$
$450$	0	0
$451$	−3.18330	−0.149896
$452$	0	0
$453$	56.1121i	2.63638i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.0202i	1.26395i	0.774988	+	0.631976i	$0.217756\pi$
$457$	27.0202i	1.26395i	−0.774988	+	0.631976i	$0.782244\pi$
$458$	0	0
$459$	−58.9487	−2.75149
$460$	0	0
$461$	17.2737	0.804515	0.402258	−	0.915526i	$-0.368226\pi$
$461$	17.2737	0.804515	0.402258	+	0.915526i	$0.368226\pi$
$462$	0	0
$463$	36.6217i	1.70196i	0.525201	+	0.850978i	$0.323990\pi$
$463$	36.6217i	1.70196i	−0.525201	+	0.850978i	$0.676010\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 31.7871i	− 1.47093i	−0.677561	−	0.735467i	$-0.736963\pi$
$467$	− 31.7871i	− 1.47093i	0.677561	−	0.735467i	$-0.263037\pi$
$468$	0	0
$469$	−27.9266	−1.28953
$470$	0	0
$471$	52.0824	2.39983
$472$	0	0
$473$	35.1098i	1.61435i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 14.9953i	− 0.686589i
$478$	0	0
$479$	−21.4479	−0.979982	−0.489991	−	0.871728i	$-0.663000\pi$
$479$	−21.4479	−0.979982	−0.489991	+	0.871728i	$0.663000\pi$
$480$	0	0
$481$	30.1155	1.37315
$482$	0	0
$483$	− 50.6735i	− 2.30573i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.62993i	0.255117i	0.991831	+	0.127558i	$0.0407140\pi$
$487$	5.62993i	0.255117i	−0.991831	+	0.127558i	$0.959286\pi$
$488$	0	0
$489$	−39.7519	−1.79764
$490$	0	0
$491$	−20.0891	−0.906607	−0.453303	−	0.891356i	$-0.649755\pi$
$491$	−20.0891	−0.906607	−0.453303	+	0.891356i	$0.649755\pi$
$492$	0	0
$493$	10.5953i	0.477188i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.11441i	0.0499881i
$498$	0	0
$499$	−3.36607	−0.150686	−0.0753429	−	0.997158i	$-0.524005\pi$
$499$	−3.36607	−0.150686	−0.0753429	+	0.997158i	$0.524005\pi$
$500$	0	0
$501$	−8.17860	−0.365393
$502$	0	0
$503$	− 32.4855i	− 1.44846i	−0.689559	−	0.724229i	$-0.742196\pi$
$503$	− 32.4855i	− 1.44846i	0.689559	−	0.724229i	$-0.257804\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 21.9192i	− 0.973464i
$508$	0	0
$509$	4.25867	0.188762	0.0943812	−	0.995536i	$-0.469913\pi$
$509$	4.25867	0.188762	0.0943812	+	0.995536i	$0.469913\pi$
$510$	0	0
$511$	−29.1129	−1.28788
$512$	0	0
$513$	− 101.789i	− 4.49410i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.7065i	0.470871i
$518$	0	0
$519$	84.9857	3.73046
$520$	0	0
$521$	−6.96895	−0.305315	−0.152658	−	0.988279i	$-0.548783\pi$
$521$	−6.96895	−0.305315	−0.152658	+	0.988279i	$0.548783\pi$
$522$	0	0
$523$	− 0.445743i	− 0.0194910i	−0.999953	−	0.00974548i	$-0.996898\pi$
$523$	− 0.445743i	− 0.0194910i	0.999953	−	0.00974548i	$-0.00310213\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.38138i	0.277977i
$528$	0	0
$529$	−13.7946	−0.599767
$530$	0	0
$531$	47.1856	2.04768
$532$	0	0
$533$	− 2.56659i	− 0.111171i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	13.2113i	0.570111i
$538$	0	0
$539$	−3.26879	−0.140797
$540$	0	0
$541$	−14.0618	−0.604565	−0.302283	−	0.953218i	$-0.597749\pi$
$541$	−14.0618	−0.604565	−0.302283	+	0.953218i	$0.597749\pi$
$542$	0	0
$543$	45.3520i	1.94624i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.27542i	0.182804i	0.995814	+	0.0914018i	$0.0291348\pi$
$547$	4.27542i	0.182804i	−0.995814	+	0.0914018i	$0.970865\pi$
$548$	0	0
$549$	−86.0900	−3.67423
$550$	0	0
$551$	−18.2953	−0.779406
$552$	0	0
$553$	11.8607i	0.504369i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.71264i	− 0.411538i	−0.978601	−	0.205769i	$-0.934030\pi$
$557$	− 9.71264i	− 0.411538i	0.978601	−	0.205769i	$-0.0659695\pi$
$558$	0	0
$559$	−28.3078	−1.19729
$560$	0	0
$561$	−33.0166	−1.39396
$562$	0	0
$563$	23.6336i	0.996039i	0.867166	+	0.498019i	$0.165939\pi$
$563$	23.6336i	0.996039i	−0.867166	+	0.498019i	$0.834061\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 98.6238i	− 4.14181i
$568$	0	0
$569$	−33.0697	−1.38636	−0.693178	−	0.720766i	$-0.743790\pi$
$569$	−33.0697	−1.38636	−0.693178	+	0.720766i	$0.743790\pi$
$570$	0	0
$571$	−19.9951	−0.836771	−0.418385	−	0.908270i	$-0.637404\pi$
$571$	−19.9951	−0.836771	−0.418385	+	0.908270i	$0.637404\pi$
$572$	0	0
$573$	− 35.6175i	− 1.48794i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.7293i	1.11275i	0.830930	+	0.556377i	$0.187809\pi$
$577$	26.7293i	1.11275i	−0.830930	+	0.556377i	$0.812191\pi$
$578$	0	0
$579$	−49.8787	−2.07289
$580$	0	0
$581$	10.1607	0.421539
$582$	0	0
$583$	− 5.49714i	− 0.227668i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.88153i	0.407855i	0.978986	+	0.203927i	$0.0653706\pi$
$587$	9.88153i	0.407855i	−0.978986	+	0.203927i	$0.934629\pi$
$588$	0	0
$589$	−11.0190	−0.454029
$590$	0	0
$591$	10.9394	0.449985
$592$	0	0
$593$	− 15.7122i	− 0.645223i	−0.946532	−	0.322611i	$-0.895439\pi$
$593$	− 15.7122i	− 0.645223i	0.946532	−	0.322611i	$-0.104561\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 54.0185i	− 2.21083i
$598$	0	0
$599$	36.4941	1.49111	0.745554	−	0.666445i	$-0.232185\pi$
$599$	36.4941	1.49111	0.745554	+	0.666445i	$0.232185\pi$
$600$	0	0
$601$	−23.2879	−0.949934	−0.474967	−	0.880004i	$-0.657540\pi$
$601$	−23.2879	−0.949934	−0.474967	+	0.880004i	$0.657540\pi$
$602$	0	0
$603$	99.2233i	4.04069i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.1677i	1.14329i	0.820501	+	0.571645i	$0.193695\pi$
$607$	28.1677i	1.14329i	−0.820501	+	0.571645i	$0.806305\pi$
$608$	0	0
$609$	−29.1699	−1.18202
$610$	0	0
$611$	−8.63227	−0.349224
$612$	0	0
$613$	− 6.09963i	− 0.246362i	−0.992384	−	0.123181i	$-0.960690\pi$
$613$	− 6.09963i	− 0.246362i	0.992384	−	0.123181i	$-0.0393095\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.4533i	1.38704i	0.720439	+	0.693519i	$0.243941\pi$
$617$	34.4533i	1.38704i	−0.720439	+	0.693519i	$0.756059\pi$
$618$	0	0
$619$	−16.9633	−0.681815	−0.340907	−	0.940097i	$-0.610734\pi$
$619$	−16.9633	−0.681815	−0.340907	+	0.940097i	$0.610734\pi$
$620$	0	0
$621$	−117.842	−4.72883
$622$	0	0
$623$	29.3849i	1.17728i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 57.0111i	− 2.27680i
$628$	0	0
$629$	−35.6042	−1.41963
$630$	0	0
$631$	30.7888	1.22568	0.612842	−	0.790206i	$-0.290026\pi$
$631$	30.7888	1.22568	0.612842	+	0.790206i	$0.290026\pi$
$632$	0	0
$633$	41.7504i	1.65943i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.63552i	− 0.104423i
$638$	0	0
$639$	3.95950	0.156635
$640$	0	0
$641$	−35.3165	−1.39492	−0.697459	−	0.716625i	$-0.745686\pi$
$641$	−35.3165	−1.39492	−0.697459	+	0.716625i	$0.745686\pi$
$642$	0	0
$643$	− 37.6680i	− 1.48548i	−0.669578	−	0.742741i	$-0.733525\pi$
$643$	− 37.6680i	− 1.48548i	0.669578	−	0.742741i	$-0.266475\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.6352i	1.28302i	0.767113	+	0.641512i	$0.221693\pi$
$647$	32.6352i	1.28302i	−0.767113	+	0.641512i	$0.778307\pi$
$648$	0	0
$649$	17.2978	0.678997
$650$	0	0
$651$	−17.5686	−0.688566
$652$	0	0
$653$	− 46.9891i	− 1.83882i	−0.393296	−	0.919412i	$-0.628665\pi$
$653$	− 46.9891i	− 1.83882i	0.393296	−	0.919412i	$-0.371335\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	103.438i	4.03552i
$658$	0	0
$659$	23.7156	0.923828	0.461914	−	0.886925i	$-0.347163\pi$
$659$	23.7156	0.923828	0.461914	+	0.886925i	$0.347163\pi$
$660$	0	0
$661$	−2.21371	−0.0861034	−0.0430517	−	0.999073i	$-0.513708\pi$
$661$	−2.21371	−0.0861034	−0.0430517	+	0.999073i	$0.513708\pi$
$662$	0	0
$663$	− 26.6201i	− 1.03384i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.1806i	0.820116i
$668$	0	0
$669$	−31.5554	−1.22000
$670$	0	0
$671$	−31.5597	−1.21835
$672$	0	0
$673$	49.0687i	1.89146i	0.324956	+	0.945729i	$0.394651\pi$
$673$	49.0687i	1.89146i	−0.324956	+	0.945729i	$0.605349\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10.0165i	− 0.384965i	−0.981300	−	0.192482i	$-0.938346\pi$
$677$	− 10.0165i	− 0.384965i	0.981300	−	0.192482i	$-0.0616538\pi$
$678$	0	0
$679$	−1.47896	−0.0567572
$680$	0	0
$681$	40.1682	1.53925
$682$	0	0
$683$	− 27.1075i	− 1.03724i	−0.855005	−	0.518619i	$-0.826446\pi$
$683$	− 27.1075i	− 1.03724i	0.855005	−	0.518619i	$-0.173554\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 15.5359i	− 0.592733i
$688$	0	0
$689$	4.43215	0.168852
$690$	0	0
$691$	−15.3309	−0.583214	−0.291607	−	0.956538i	$-0.594190\pi$
$691$	−15.3309	−0.583214	−0.291607	+	0.956538i	$0.594190\pi$
$692$	0	0
$693$	− 67.5580i	− 2.56632i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.03436i	0.114935i
$698$	0	0
$699$	−16.8518	−0.637392
$700$	0	0
$701$	16.9263	0.639299	0.319649	−	0.947536i	$-0.396435\pi$
$701$	16.9263	0.639299	0.319649	+	0.947536i	$0.396435\pi$
$702$	0	0
$703$	− 61.4792i	− 2.31873i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	37.3765i	1.40569i
$708$	0	0
$709$	19.2986	0.724775	0.362388	−	0.932027i	$-0.381962\pi$
$709$	19.2986	0.724775	0.362388	+	0.932027i	$0.381962\pi$
$710$	0	0
$711$	42.1412	1.58042
$712$	0	0
$713$	12.7567i	0.477744i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 31.7612i	− 1.18614i
$718$	0	0
$719$	−46.1504	−1.72112	−0.860560	−	0.509349i	$-0.829886\pi$
$719$	−46.1504	−1.72112	−0.860560	+	0.509349i	$0.829886\pi$
$720$	0	0
$721$	26.7097	0.994722
$722$	0	0
$723$	50.4629i	1.87674i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.52076i	0.0934897i	0.998907	+	0.0467449i	$0.0148848\pi$
$727$	2.52076i	0.0934897i	−0.998907	+	0.0467449i	$0.985115\pi$
$728$	0	0
$729$	−151.199	−5.59995
$730$	0	0
$731$	33.4671	1.23782
$732$	0	0
$733$	− 13.5118i	− 0.499070i	−0.968366	−	0.249535i	$-0.919722\pi$
$733$	− 13.5118i	− 0.499070i	0.968366	−	0.249535i	$-0.0802778\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.3743i	1.33986i
$738$	0	0
$739$	−40.0364	−1.47276	−0.736382	−	0.676566i	$-0.763467\pi$
$739$	−40.0364	−1.47276	−0.736382	+	0.676566i	$0.763467\pi$
$740$	0	0
$741$	45.9661	1.68861
$742$	0	0
$743$	35.4611i	1.30094i	0.759531	+	0.650471i	$0.225429\pi$
$743$	35.4611i	1.30094i	−0.759531	+	0.650471i	$0.774571\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 36.1012i	− 1.32087i
$748$	0	0
$749$	−37.2264	−1.36022
$750$	0	0
$751$	−25.2693	−0.922092	−0.461046	−	0.887376i	$-0.652526\pi$
$751$	−25.2693	−0.922092	−0.461046	+	0.887376i	$0.652526\pi$
$752$	0	0
$753$	− 21.9804i	− 0.801011i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35.1240i	1.27660i	0.769787	+	0.638301i	$0.220363\pi$
$757$	35.1240i	1.27660i	−0.769787	+	0.638301i	$0.779637\pi$
$758$	0	0
$759$	−66.0021	−2.39572
$760$	0	0
$761$	−24.2268	−0.878222	−0.439111	−	0.898433i	$-0.644706\pi$
$761$	−24.2268	−0.878222	−0.439111	+	0.898433i	$0.644706\pi$
$762$	0	0
$763$	− 2.86679i	− 0.103785i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.9466i	0.503583i
$768$	0	0
$769$	−32.6079	−1.17587	−0.587935	−	0.808908i	$-0.700059\pi$
$769$	−32.6079	−1.17587	−0.587935	+	0.808908i	$0.700059\pi$
$770$	0	0
$771$	50.0703	1.80324
$772$	0	0
$773$	− 39.8279i	− 1.43251i	−0.697838	−	0.716256i	$-0.745854\pi$
$773$	− 39.8279i	− 1.43251i	0.697838	−	0.716256i	$-0.254146\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 98.0219i	− 3.51652i
$778$	0	0
$779$	−5.23955	−0.187726
$780$	0	0
$781$	1.45151	0.0519392
$782$	0	0
$783$	67.8348i	2.42422i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 17.1252i	− 0.610449i	−0.952280	−	0.305224i	$-0.901269\pi$
$787$	− 17.1252i	− 0.610449i	0.952280	−	0.305224i	$-0.0987315\pi$
$788$	0	0
$789$	72.0246	2.56414
$790$	0	0
$791$	22.2932	0.792656
$792$	0	0
$793$	− 25.4455i	− 0.903596i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 27.2288i	− 0.964493i	−0.876036	−	0.482246i	$-0.839821\pi$
$797$	− 27.2288i	− 0.964493i	0.876036	−	0.482246i	$-0.160179\pi$
$798$	0	0
$799$	10.2055	0.361046
$800$	0	0
$801$	104.405	3.68896
$802$	0	0
$803$	37.9195i	1.33815i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.7566i	1.08268i
$808$	0	0
$809$	−2.85574	−0.100403	−0.0502013	−	0.998739i	$-0.515986\pi$
$809$	−2.85574	−0.100403	−0.0502013	+	0.998739i	$0.515986\pi$
$810$	0	0
$811$	−31.1909	−1.09526	−0.547631	−	0.836720i	$-0.684470\pi$
$811$	−31.1909	−1.09526	−0.547631	+	0.836720i	$0.684470\pi$
$812$	0	0
$813$	7.26005i	0.254621i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	57.7890i	2.02178i
$818$	0	0
$819$	54.4697	1.90332
$820$	0	0
$821$	−9.65646	−0.337013	−0.168506	−	0.985701i	$-0.553894\pi$
$821$	−9.65646	−0.337013	−0.168506	+	0.985701i	$0.553894\pi$
$822$	0	0
$823$	− 13.9327i	− 0.485663i	−0.970068	−	0.242832i	$-0.921924\pi$
$823$	− 13.9327i	− 0.485663i	0.970068	−	0.242832i	$-0.0780762\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 14.8217i	− 0.515402i	−0.966225	−	0.257701i	$-0.917035\pi$
$827$	− 14.8217i	− 0.515402i	0.966225	−	0.257701i	$-0.0829650\pi$
$828$	0	0
$829$	−9.24514	−0.321097	−0.160549	−	0.987028i	$-0.551326\pi$
$829$	−9.24514	−0.321097	−0.160549	+	0.987028i	$0.551326\pi$
$830$	0	0
$831$	−55.3076	−1.91860
$832$	0	0
$833$	3.11585i	0.107958i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	40.8559i	1.41219i
$838$	0	0
$839$	−40.3934	−1.39453	−0.697267	−	0.716811i	$-0.745601\pi$
$839$	−40.3934	−1.39453	−0.697267	+	0.716811i	$0.745601\pi$
$840$	0	0
$841$	−16.8075	−0.579570
$842$	0	0
$843$	102.732i	3.53827i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.11794i	0.0727734i
$848$	0	0
$849$	47.0669	1.61533
$850$	0	0
$851$	−71.1748	−2.43984
$852$	0	0
$853$	− 50.2511i	− 1.72056i	−0.509818	−	0.860282i	$-0.670287\pi$
$853$	− 50.2511i	− 1.72056i	0.509818	−	0.860282i	$-0.329713\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.87991i	− 0.0642166i	−0.999484	−	0.0321083i	$-0.989778\pi$
$857$	− 1.87991i	− 0.0642166i	0.999484	−	0.0321083i	$-0.0102221\pi$
$858$	0	0
$859$	32.4292	1.10647	0.553236	−	0.833025i	$-0.313393\pi$
$859$	32.4292	1.10647	0.553236	+	0.833025i	$0.313393\pi$
$860$	0	0
$861$	−8.35389	−0.284700
$862$	0	0
$863$	24.9471i	0.849208i	0.905379	+	0.424604i	$0.139587\pi$
$863$	24.9471i	0.849208i	−0.905379	+	0.424604i	$0.860413\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 26.6363i	− 0.904615i
$868$	0	0
$869$	15.4485	0.524055
$870$	0	0
$871$	−29.3273	−0.993718
$872$	0	0
$873$	5.25474i	0.177846i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.5853i	1.13409i	0.823685	+	0.567047i	$0.191914\pi$
$877$	33.5853i	1.13409i	−0.823685	+	0.567047i	$0.808086\pi$
$878$	0	0
$879$	−33.1594	−1.11844
$880$	0	0
$881$	34.6265	1.16660	0.583298	−	0.812258i	$-0.301762\pi$
$881$	34.6265	1.16660	0.583298	+	0.812258i	$0.301762\pi$
$882$	0	0
$883$	25.7586i	0.866845i	0.901191	+	0.433422i	$0.142694\pi$
$883$	25.7586i	0.866845i	−0.901191	+	0.433422i	$0.857306\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.13623i	− 0.0381510i	−0.999818	−	0.0190755i	$-0.993928\pi$
$887$	− 1.13623i	− 0.0381510i	0.999818	−	0.0190755i	$-0.00607229\pi$
$888$	0	0
$889$	37.4379	1.25563
$890$	0	0
$891$	−128.457	−4.30347
$892$	0	0
$893$	17.6223i	0.589708i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 53.2152i	− 1.77680i
$898$	0	0
$899$	7.34333	0.244914
$900$	0	0
$901$	−5.23993	−0.174568
$902$	0	0
$903$	92.1382i	3.06617i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 22.1054i	− 0.733997i	−0.930222	−	0.366998i	$-0.880385\pi$
$907$	− 22.1054i	− 0.733997i	0.930222	−	0.366998i	$-0.119615\pi$
$908$	0	0
$909$	132.799	4.40467
$910$	0	0
$911$	25.6265	0.849044	0.424522	−	0.905418i	$-0.360442\pi$
$911$	25.6265	0.849044	0.424522	+	0.905418i	$0.360442\pi$
$912$	0	0
$913$	− 13.2343i	− 0.437992i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.9655i	0.659319i
$918$	0	0
$919$	49.7798	1.64209	0.821043	−	0.570867i	$-0.193393\pi$
$919$	49.7798	1.64209	0.821043	+	0.570867i	$0.193393\pi$
$920$	0	0
$921$	−30.6337	−1.00941
$922$	0	0
$923$	1.17031i	0.0385211i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 94.8997i	− 3.11692i
$928$	0	0
$929$	9.95284	0.326542	0.163271	−	0.986581i	$-0.447796\pi$
$929$	9.95284	0.326542	0.163271	+	0.986581i	$0.447796\pi$
$930$	0	0
$931$	−5.38027	−0.176331
$932$	0	0
$933$	2.73612i	0.0895766i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.97467i	0.0645097i	0.999480	+	0.0322548i	$0.0102688\pi$
$937$	1.97467i	0.0645097i	−0.999480	+	0.0322548i	$0.989731\pi$
$938$	0	0
$939$	−57.9384	−1.89075
$940$	0	0
$941$	−56.9458	−1.85638	−0.928190	−	0.372106i	$-0.878636\pi$
$941$	−56.9458	−1.85638	−0.928190	+	0.372106i	$0.878636\pi$
$942$	0	0
$943$	6.06586i	0.197532i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.90002i	− 0.159229i	−0.996826	−	0.0796147i	$-0.974631\pi$
$947$	− 4.90002i	− 0.159229i	0.996826	−	0.0796147i	$-0.0253690\pi$
$948$	0	0
$949$	−30.5732	−0.992447
$950$	0	0
$951$	108.371	3.51418
$952$	0	0
$953$	− 0.233767i	− 0.00757245i	−0.999993	−	0.00378623i	$-0.998795\pi$
$953$	− 0.233767i	− 0.00757245i	0.999993	−	0.00378623i	$-0.00120520\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	37.9936i	1.22816i
$958$	0	0
$959$	43.2570	1.39684
$960$	0	0
$961$	−26.5772	−0.857330
$962$	0	0
$963$	132.266i	4.26220i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 12.2741i	− 0.394709i	−0.980332	−	0.197354i	$-0.936765\pi$
$967$	− 12.2741i	− 0.394709i	0.980332	−	0.197354i	$-0.0632349\pi$
$968$	0	0
$969$	−54.3436	−1.74577
$970$	0	0
$971$	41.8268	1.34229	0.671143	−	0.741328i	$-0.265804\pi$
$971$	41.8268	1.34229	0.671143	+	0.741328i	$0.265804\pi$
$972$	0	0
$973$	− 9.78131i	− 0.313574i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 57.1375i	− 1.82799i	−0.405724	−	0.913995i	$-0.632981\pi$
$977$	− 57.1375i	− 1.82799i	0.405724	−	0.913995i	$-0.367019\pi$
$978$	0	0
$979$	38.2737	1.22323
$980$	0	0
$981$	−10.1857	−0.325205
$982$	0	0
$983$	− 11.7659i	− 0.375275i	−0.982238	−	0.187638i	$-0.939917\pi$
$983$	− 11.7659i	− 0.375275i	0.982238	−	0.187638i	$-0.0600831\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	28.0969i	0.894333i
$988$	0	0
$989$	66.9026	2.12738
$990$	0	0
$991$	−41.5426	−1.31964	−0.659821	−	0.751423i	$-0.729368\pi$
$991$	−41.5426	−1.31964	−0.659821	+	0.751423i	$0.729368\pi$
$992$	0	0
$993$	36.2617i	1.15073i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.59405i	0.145495i	0.997350	+	0.0727475i	$0.0231767\pi$
$997$	4.59405i	0.145495i	−0.997350	+	0.0727475i	$0.976823\pi$
$998$	0	0
$999$	−227.951	−7.21205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.g.1149.1		14
5.2	odd	4		4100.2.a.g.1.1	✓	7
5.3	odd	4		4100.2.a.j.1.7	yes	7
5.4	even	2	inner	4100.2.d.g.1149.14		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4100.2.a.g.1.1	✓	7	5.2	odd	4
4100.2.a.j.1.7	yes	7	5.3	odd	4
4100.2.d.g.1149.1		14	1.1	even	1	trivial
4100.2.d.g.1149.14		14	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:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.41812i q^{3} -2.44400i q^{7} -8.68355 q^{9} +O(q^{10})\)	\(q-3.41812i q^{3} -2.44400i q^{7} -8.68355 q^{9} -3.18330 q^{11} -2.56659i q^{13} +3.03436i q^{17} -5.23955 q^{19} -8.35389 q^{21} +6.06586i q^{23} +19.4271i q^{27} +3.49177 q^{29} +2.10304 q^{31} +10.8809i q^{33} +11.7337i q^{37} -8.77290 q^{39} +1.00000 q^{41} -11.0294i q^{43} -3.36333i q^{47} +1.02686 q^{49} +10.3718 q^{51} +1.72687i q^{53} +17.9094i q^{57} -5.43391 q^{59} +9.91414 q^{61} +21.2226i q^{63} -11.4266i q^{67} +20.7338 q^{69} -0.455977 q^{71} -11.9120i q^{73} +7.77999i q^{77} -4.85299 q^{79} +40.3534 q^{81} +4.15742i q^{83} -11.9353i q^{87} -12.0233 q^{89} -6.27274 q^{91} -7.18844i q^{93} -0.605138i q^{97} +27.6424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 12 q^{9}+O(q^{10}) \) 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

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