LMFDB - Embedded newform 4100.2.d.g.1149.13

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			1149.13
Root			$-2.47795i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.g.1149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.47795i q^{3} -0.154217i q^{7} -3.14025 q^{9} +O(q^{10})$	$q+2.47795i q^{3} -0.154217i q^{7} -3.14025 q^{9} -4.66070 q^{11} -6.51290i q^{13} +0.170128i q^{17} -1.98604 q^{19} +0.382142 q^{21} +0.648409i q^{23} -0.347539i q^{27} +2.94159 q^{29} -3.23134 q^{31} -11.5490i q^{33} -0.224856i q^{37} +16.1387 q^{39} +1.00000 q^{41} +1.55814i q^{43} +0.959526i q^{47} +6.97622 q^{49} -0.421568 q^{51} +3.23328i q^{53} -4.92130i q^{57} +12.5538 q^{59} +13.0886 q^{61} +0.484280i q^{63} +2.84725i q^{67} -1.60673 q^{69} +2.61291 q^{71} -13.5928i q^{73} +0.718759i q^{77} +8.14710 q^{79} -8.55957 q^{81} +2.02853i q^{83} +7.28913i q^{87} +10.7892 q^{89} -1.00440 q^{91} -8.00710i q^{93} +7.77063i q^{97} +14.6358 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.47795i	1.43065i	0.698793	+	0.715323i	$0.253721\pi$
$3$	2.47795i	1.43065i	−0.698793	+	0.715323i	$0.746279\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.154217i	− 0.0582885i	−0.999575	−	0.0291443i	$-0.990722\pi$
$7$	− 0.154217i	− 0.0582885i	0.999575	−	0.0291443i	$-0.00927822\pi$
$8$	0	0
$9$	−3.14025	−1.04675
$10$	0	0
$11$	−4.66070	−1.40525	−0.702627	−	0.711558i	$-0.747990\pi$
$11$	−4.66070	−1.40525	−0.702627	+	0.711558i	$0.747990\pi$
$12$	0	0
$13$	− 6.51290i	− 1.80635i	−0.429268	−	0.903177i	$-0.641228\pi$
$13$	− 6.51290i	− 1.80635i	0.429268	−	0.903177i	$-0.358772\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.170128i	0.0412620i	0.999787	+	0.0206310i	$0.00656752\pi$
$17$	0.170128i	0.0412620i	−0.999787	+	0.0206310i	$0.993432\pi$
$18$	0	0
$19$	−1.98604	−0.455628	−0.227814	−	0.973705i	$-0.573158\pi$
$19$	−1.98604	−0.455628	−0.227814	+	0.973705i	$0.573158\pi$
$20$	0	0
$21$	0.382142	0.0833903
$22$	0	0
$23$	0.648409i	0.135203i	0.997712	+	0.0676013i	$0.0215346\pi$
$23$	0.648409i	0.135203i	−0.997712	+	0.0676013i	$0.978465\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 0.347539i	− 0.0668838i
$28$	0	0
$29$	2.94159	0.546240	0.273120	−	0.961980i	$-0.411944\pi$
$29$	2.94159	0.546240	0.273120	+	0.961980i	$0.411944\pi$
$30$	0	0
$31$	−3.23134	−0.580365	−0.290182	−	0.956971i	$-0.593716\pi$
$31$	−3.23134	−0.580365	−0.290182	+	0.956971i	$0.593716\pi$
$32$	0	0
$33$	− 11.5490i	− 2.01042i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.224856i	− 0.0369662i	−0.999829	−	0.0184831i	$-0.994116\pi$
$37$	− 0.224856i	− 0.0369662i	0.999829	−	0.0184831i	$-0.00588368\pi$
$38$	0	0
$39$	16.1387	2.58426
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	1.55814i	0.237614i	0.992917	+	0.118807i	$0.0379070\pi$
$43$	1.55814i	0.237614i	−0.992917	+	0.118807i	$0.962093\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.959526i	0.139961i	0.997548	+	0.0699806i	$0.0222937\pi$
$47$	0.959526i	0.139961i	−0.997548	+	0.0699806i	$0.977706\pi$
$48$	0	0
$49$	6.97622	0.996602
$50$	0	0
$51$	−0.421568	−0.0590314
$52$	0	0
$53$	3.23328i	0.444125i	0.975032	+	0.222063i	$0.0712790\pi$
$53$	3.23328i	0.444125i	−0.975032	+	0.222063i	$0.928721\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.92130i	− 0.651842i
$58$	0	0
$59$	12.5538	1.63437	0.817183	−	0.576378i	$-0.195535\pi$
$59$	12.5538	1.63437	0.817183	+	0.576378i	$0.195535\pi$
$60$	0	0
$61$	13.0886	1.67583	0.837913	−	0.545804i	$-0.183776\pi$
$61$	13.0886	1.67583	0.837913	+	0.545804i	$0.183776\pi$
$62$	0	0
$63$	0.484280i	0.0610135i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.84725i	0.347847i	0.984759	+	0.173924i	$0.0556446\pi$
$67$	2.84725i	0.347847i	−0.984759	+	0.173924i	$0.944355\pi$
$68$	0	0
$69$	−1.60673	−0.193427
$70$	0	0
$71$	2.61291	0.310095	0.155048	−	0.987907i	$-0.450447\pi$
$71$	2.61291	0.310095	0.155048	+	0.987907i	$0.450447\pi$
$72$	0	0
$73$	− 13.5928i	− 1.59092i	−0.606008	−	0.795459i	$-0.707230\pi$
$73$	− 13.5928i	− 1.59092i	0.606008	−	0.795459i	$-0.292770\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.718759i	0.0819102i
$78$	0	0
$79$	8.14710	0.916620	0.458310	−	0.888792i	$-0.348455\pi$
$79$	8.14710	0.916620	0.458310	+	0.888792i	$0.348455\pi$
$80$	0	0
$81$	−8.55957	−0.951064
$82$	0	0
$83$	2.02853i	0.222660i	0.993783	+	0.111330i	$0.0355111\pi$
$83$	2.02853i	0.222660i	−0.993783	+	0.111330i	$0.964489\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.28913i	0.781477i
$88$	0	0
$89$	10.7892	1.14365	0.571826	−	0.820375i	$-0.306235\pi$
$89$	10.7892	1.14365	0.571826	+	0.820375i	$0.306235\pi$
$90$	0	0
$91$	−1.00440	−0.105290
$92$	0	0
$93$	− 8.00710i	− 0.830297i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.77063i	0.788988i	0.918899	+	0.394494i	$0.129080\pi$
$97$	7.77063i	0.788988i	−0.918899	+	0.394494i	$0.870920\pi$
$98$	0	0
$99$	14.6358	1.47095
$100$	0	0
$101$	10.5694	1.05169	0.525846	−	0.850579i	$-0.323749\pi$
$101$	10.5694	1.05169	0.525846	+	0.850579i	$0.323749\pi$
$102$	0	0
$103$	− 9.76162i	− 0.961841i	−0.876764	−	0.480921i	$-0.840303\pi$
$103$	− 9.76162i	− 0.961841i	0.876764	−	0.480921i	$-0.159697\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.692361i	0.0669330i	0.999440	+	0.0334665i	$0.0106547\pi$
$107$	0.692361i	0.0669330i	−0.999440	+	0.0334665i	$0.989345\pi$
$108$	0	0
$109$	10.4239	0.998430	0.499215	−	0.866478i	$-0.333622\pi$
$109$	10.4239	0.998430	0.499215	+	0.866478i	$0.333622\pi$
$110$	0	0
$111$	0.557184	0.0528855
$112$	0	0
$113$	− 8.87059i	− 0.834475i	−0.908798	−	0.417237i	$-0.862998\pi$
$113$	− 8.87059i	− 0.834475i	0.908798	−	0.417237i	$-0.137002\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	20.4522i	1.89080i
$118$	0	0
$119$	0.0262366	0.00240510
$120$	0	0
$121$	10.7221	0.974739
$122$	0	0
$123$	2.47795i	0.223430i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10.8673i	− 0.964314i	−0.876085	−	0.482157i	$-0.839853\pi$
$127$	− 10.8673i	− 0.964314i	0.876085	−	0.482157i	$-0.160147\pi$
$128$	0	0
$129$	−3.86100	−0.339942
$130$	0	0
$131$	4.89093	0.427322	0.213661	−	0.976908i	$-0.431461\pi$
$131$	4.89093	0.427322	0.213661	+	0.976908i	$0.431461\pi$
$132$	0	0
$133$	0.306280i	0.0265579i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.5582i	− 1.07292i	−0.843925	−	0.536461i	$-0.819761\pi$
$137$	− 12.5582i	− 1.07292i	0.843925	−	0.536461i	$-0.180239\pi$
$138$	0	0
$139$	−0.960202	−0.0814433	−0.0407216	−	0.999171i	$-0.512966\pi$
$139$	−0.960202	−0.0814433	−0.0407216	+	0.999171i	$0.512966\pi$
$140$	0	0
$141$	−2.37766	−0.200235
$142$	0	0
$143$	30.3547i	2.53839i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.2867i	1.42579i
$148$	0	0
$149$	8.36853	0.685576	0.342788	−	0.939413i	$-0.388629\pi$
$149$	8.36853	0.685576	0.342788	+	0.939413i	$0.388629\pi$
$150$	0	0
$151$	0.236367	0.0192353	0.00961765	−	0.999954i	$-0.496939\pi$
$151$	0.236367	0.0192353	0.00961765	+	0.999954i	$0.496939\pi$
$152$	0	0
$153$	− 0.534244i	− 0.0431911i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 7.48073i	− 0.597027i	−0.954405	−	0.298514i	$-0.903509\pi$
$157$	− 7.48073i	− 0.597027i	0.954405	−	0.298514i	$-0.0964908\pi$
$158$	0	0
$159$	−8.01192	−0.635387
$160$	0	0
$161$	0.0999957	0.00788076
$162$	0	0
$163$	− 15.2851i	− 1.19722i	−0.801041	−	0.598609i	$-0.795720\pi$
$163$	− 15.2851i	− 1.19722i	0.801041	−	0.598609i	$-0.204280\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.6104i	0.975823i	0.872893	+	0.487911i	$0.162241\pi$
$167$	12.6104i	0.975823i	−0.872893	+	0.487911i	$0.837759\pi$
$168$	0	0
$169$	−29.4179	−2.26292
$170$	0	0
$171$	6.23665	0.476929
$172$	0	0
$173$	− 7.96446i	− 0.605527i	−0.953066	−	0.302763i	$-0.902091\pi$
$173$	− 7.96446i	− 0.605527i	0.953066	−	0.302763i	$-0.0979092\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	31.1077i	2.33820i
$178$	0	0
$179$	−22.2654	−1.66419	−0.832096	−	0.554632i	$-0.812859\pi$
$179$	−22.2654	−1.66419	−0.832096	+	0.554632i	$0.812859\pi$
$180$	0	0
$181$	22.3831	1.66372	0.831860	−	0.554985i	$-0.187276\pi$
$181$	22.3831	1.66372	0.831860	+	0.554985i	$0.187276\pi$
$182$	0	0
$183$	32.4330i	2.39751i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.792914i	− 0.0579836i
$188$	0	0
$189$	−0.0535963	−0.00389856
$190$	0	0
$191$	7.36888	0.533194	0.266597	−	0.963808i	$-0.414101\pi$
$191$	7.36888	0.533194	0.266597	+	0.963808i	$0.414101\pi$
$192$	0	0
$193$	5.47417i	0.394040i	0.980400	+	0.197020i	$0.0631263\pi$
$193$	5.47417i	0.394040i	−0.980400	+	0.197020i	$0.936874\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.7806i	0.839332i	0.907679	+	0.419666i	$0.137853\pi$
$197$	11.7806i	0.839332i	−0.907679	+	0.419666i	$0.862147\pi$
$198$	0	0
$199$	−14.4712	−1.02583	−0.512917	−	0.858438i	$-0.671435\pi$
$199$	−14.4712	−1.02583	−0.512917	+	0.858438i	$0.671435\pi$
$200$	0	0
$201$	−7.05536	−0.497647
$202$	0	0
$203$	− 0.453643i	− 0.0318395i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.03617i	− 0.141524i
$208$	0	0
$209$	9.25632	0.640273
$210$	0	0
$211$	−12.6590	−0.871481	−0.435741	−	0.900072i	$-0.643513\pi$
$211$	−12.6590	−0.871481	−0.435741	+	0.900072i	$0.643513\pi$
$212$	0	0
$213$	6.47466i	0.443637i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.498326i	0.0338286i
$218$	0	0
$219$	33.6823	2.27604
$220$	0	0
$221$	1.10803	0.0745338
$222$	0	0
$223$	− 11.0958i	− 0.743030i	−0.928427	−	0.371515i	$-0.878838\pi$
$223$	− 11.0958i	− 0.743030i	0.928427	−	0.371515i	$-0.121162\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 9.90556i	− 0.657455i	−0.944425	−	0.328728i	$-0.893380\pi$
$227$	− 9.90556i	− 0.657455i	0.944425	−	0.328728i	$-0.106620\pi$
$228$	0	0
$229$	−29.3952	−1.94249	−0.971244	−	0.238085i	$-0.923480\pi$
$229$	−29.3952	−1.94249	−0.971244	+	0.238085i	$0.923480\pi$
$230$	0	0
$231$	−1.78105	−0.117185
$232$	0	0
$233$	− 14.7057i	− 0.963403i	−0.876335	−	0.481702i	$-0.840019\pi$
$233$	− 14.7057i	− 0.963403i	0.876335	−	0.481702i	$-0.159981\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	20.1881i	1.31136i
$238$	0	0
$239$	−2.07555	−0.134256	−0.0671280	−	0.997744i	$-0.521384\pi$
$239$	−2.07555	−0.134256	−0.0671280	+	0.997744i	$0.521384\pi$
$240$	0	0
$241$	−1.55460	−0.100141	−0.0500704	−	0.998746i	$-0.515945\pi$
$241$	−1.55460	−0.100141	−0.0500704	+	0.998746i	$0.515945\pi$
$242$	0	0
$243$	− 22.2528i	− 1.42752i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.9349i	0.823025i
$248$	0	0
$249$	−5.02660	−0.318548
$250$	0	0
$251$	−6.19705	−0.391155	−0.195577	−	0.980688i	$-0.562658\pi$
$251$	−6.19705	−0.391155	−0.195577	+	0.980688i	$0.562658\pi$
$252$	0	0
$253$	− 3.02204i	− 0.189994i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 26.9345i	− 1.68013i	−0.542486	−	0.840065i	$-0.682517\pi$
$257$	− 26.9345i	− 1.68013i	0.542486	−	0.840065i	$-0.317483\pi$
$258$	0	0
$259$	−0.0346767	−0.00215470
$260$	0	0
$261$	−9.23735	−0.571777
$262$	0	0
$263$	22.0511i	1.35973i	0.733336	+	0.679866i	$0.237962\pi$
$263$	22.0511i	1.35973i	−0.733336	+	0.679866i	$0.762038\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	26.7351i	1.63616i
$268$	0	0
$269$	−9.74494	−0.594160	−0.297080	−	0.954853i	$-0.596013\pi$
$269$	−9.74494	−0.594160	−0.297080	+	0.954853i	$0.596013\pi$
$270$	0	0
$271$	10.6785	0.648670	0.324335	−	0.945942i	$-0.394859\pi$
$271$	10.6785	0.648670	0.324335	+	0.945942i	$0.394859\pi$
$272$	0	0
$273$	− 2.48886i	− 0.150632i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 0.231637i	− 0.0139177i	−0.999976	−	0.00695886i	$-0.997785\pi$
$277$	− 0.231637i	− 0.0139177i	0.999976	−	0.00695886i	$-0.00221509\pi$
$278$	0	0
$279$	10.1472	0.607497
$280$	0	0
$281$	31.8715	1.90130	0.950648	−	0.310272i	$-0.100420\pi$
$281$	31.8715	1.90130	0.950648	+	0.310272i	$0.100420\pi$
$282$	0	0
$283$	− 29.9752i	− 1.78184i	−0.454162	−	0.890919i	$-0.650061\pi$
$283$	− 29.9752i	− 1.78184i	0.454162	−	0.890919i	$-0.349939\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 0.154217i	− 0.00910314i
$288$	0	0
$289$	16.9711	0.998297
$290$	0	0
$291$	−19.2553	−1.12876
$292$	0	0
$293$	− 3.50405i	− 0.204709i	−0.994748	−	0.102354i	$-0.967362\pi$
$293$	− 3.50405i	− 0.204709i	0.994748	−	0.102354i	$-0.0326375\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.61977i	0.0939888i
$298$	0	0
$299$	4.22303	0.244224
$300$	0	0
$301$	0.240292	0.0138502
$302$	0	0
$303$	26.1904i	1.50460i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.6803i	0.951994i	0.879447	+	0.475997i	$0.157913\pi$
$307$	16.6803i	0.951994i	−0.879447	+	0.475997i	$0.842087\pi$
$308$	0	0
$309$	24.1888	1.37606
$310$	0	0
$311$	19.9664	1.13219	0.566096	−	0.824339i	$-0.308453\pi$
$311$	19.9664	1.13219	0.566096	+	0.824339i	$0.308453\pi$
$312$	0	0
$313$	− 21.2399i	− 1.20055i	−0.799794	−	0.600275i	$-0.795058\pi$
$313$	− 21.2399i	− 1.20055i	0.799794	−	0.600275i	$-0.204942\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.6530i	1.32848i	0.747518	+	0.664241i	$0.231245\pi$
$317$	23.6530i	1.32848i	−0.747518	+	0.664241i	$0.768755\pi$
$318$	0	0
$319$	−13.7099	−0.767606
$320$	0	0
$321$	−1.71564	−0.0957575
$322$	0	0
$323$	− 0.337880i	− 0.0188001i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	25.8300i	1.42840i
$328$	0	0
$329$	0.147975	0.00815813
$330$	0	0
$331$	20.5943	1.13196	0.565982	−	0.824417i	$-0.308497\pi$
$331$	20.5943	1.13196	0.565982	+	0.824417i	$0.308497\pi$
$332$	0	0
$333$	0.706106i	0.0386944i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 15.2777i	− 0.832232i	−0.909312	−	0.416116i	$-0.863391\pi$
$337$	− 15.2777i	− 0.832232i	0.909312	−	0.416116i	$-0.136609\pi$
$338$	0	0
$339$	21.9809	1.19384
$340$	0	0
$341$	15.0603	0.815560
$342$	0	0
$343$	− 2.15537i	− 0.116379i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.1039i	1.40133i	0.713489	+	0.700666i	$0.247114\pi$
$347$	26.1039i	1.40133i	−0.713489	+	0.700666i	$0.752886\pi$
$348$	0	0
$349$	−20.0961	−1.07572	−0.537860	−	0.843034i	$-0.680767\pi$
$349$	−20.0961	−1.07572	−0.537860	+	0.843034i	$0.680767\pi$
$350$	0	0
$351$	−2.26349	−0.120816
$352$	0	0
$353$	31.1689i	1.65895i	0.558543	+	0.829476i	$0.311361\pi$
$353$	31.1689i	1.65895i	−0.558543	+	0.829476i	$0.688639\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.0650130i	0.00344085i
$358$	0	0
$359$	−15.6085	−0.823783	−0.411892	−	0.911233i	$-0.635132\pi$
$359$	−15.6085	−0.823783	−0.411892	+	0.911233i	$0.635132\pi$
$360$	0	0
$361$	−15.0557	−0.792403
$362$	0	0
$363$	26.5689i	1.39451i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 9.55565i	− 0.498801i	−0.968400	−	0.249400i	$-0.919766\pi$
$367$	− 9.55565i	− 0.498801i	0.968400	−	0.249400i	$-0.0802335\pi$
$368$	0	0
$369$	−3.14025	−0.163475
$370$	0	0
$371$	0.498627	0.0258874
$372$	0	0
$373$	− 9.57524i	− 0.495787i	−0.968787	−	0.247894i	$-0.920262\pi$
$373$	− 9.57524i	− 0.495787i	0.968787	−	0.247894i	$-0.0797383\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 19.1583i	− 0.986704i
$378$	0	0
$379$	−4.29539	−0.220639	−0.110320	−	0.993896i	$-0.535187\pi$
$379$	−4.29539	−0.220639	−0.110320	+	0.993896i	$0.535187\pi$
$380$	0	0
$381$	26.9286	1.37959
$382$	0	0
$383$	− 16.3030i	− 0.833047i	−0.909125	−	0.416523i	$-0.863248\pi$
$383$	− 16.3030i	− 0.833047i	0.909125	−	0.416523i	$-0.136752\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.89295i	− 0.248723i
$388$	0	0
$389$	−8.61713	−0.436906	−0.218453	−	0.975847i	$-0.570101\pi$
$389$	−8.61713	−0.436906	−0.218453	+	0.975847i	$0.570101\pi$
$390$	0	0
$391$	−0.110312	−0.00557874
$392$	0	0
$393$	12.1195i	0.611347i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	37.6126i	1.88772i	0.330343	+	0.943861i	$0.392836\pi$
$397$	37.6126i	1.88772i	−0.330343	+	0.943861i	$0.607164\pi$
$398$	0	0
$399$	−0.758948	−0.0379949
$400$	0	0
$401$	−15.7328	−0.785659	−0.392829	−	0.919611i	$-0.628504\pi$
$401$	−15.7328	−0.785659	−0.392829	+	0.919611i	$0.628504\pi$
$402$	0	0
$403$	21.0454i	1.04834i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.04799i	0.0519469i
$408$	0	0
$409$	−10.9923	−0.543536	−0.271768	−	0.962363i	$-0.587608\pi$
$409$	−10.9923	−0.543536	−0.271768	+	0.962363i	$0.587608\pi$
$410$	0	0
$411$	31.1187	1.53497
$412$	0	0
$413$	− 1.93601i	− 0.0952647i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.37934i	− 0.116517i
$418$	0	0
$419$	29.2578	1.42934	0.714669	−	0.699463i	$-0.246577\pi$
$419$	29.2578	1.42934	0.714669	+	0.699463i	$0.246577\pi$
$420$	0	0
$421$	−23.2518	−1.13323	−0.566613	−	0.823984i	$-0.691746\pi$
$421$	−23.2518	−1.13323	−0.566613	+	0.823984i	$0.691746\pi$
$422$	0	0
$423$	− 3.01315i	− 0.146505i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.01849i	− 0.0976814i
$428$	0	0
$429$	−75.2175	−3.63154
$430$	0	0
$431$	11.8089	0.568817	0.284408	−	0.958703i	$-0.408203\pi$
$431$	11.8089	0.568817	0.284408	+	0.958703i	$0.408203\pi$
$432$	0	0
$433$	12.2525i	0.588818i	0.955680	+	0.294409i	$0.0951227\pi$
$433$	12.2525i	0.588818i	−0.955680	+	0.294409i	$0.904877\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.28776i	− 0.0616021i
$438$	0	0
$439$	13.5394	0.646199	0.323100	−	0.946365i	$-0.395275\pi$
$439$	13.5394	0.646199	0.323100	+	0.946365i	$0.395275\pi$
$440$	0	0
$441$	−21.9071	−1.04319
$442$	0	0
$443$	− 1.19769i	− 0.0569038i	−0.999595	−	0.0284519i	$-0.990942\pi$
$443$	− 1.19769i	− 0.0569038i	0.999595	−	0.0284519i	$-0.00905775\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.7368i	0.980818i
$448$	0	0
$449$	7.10735	0.335416	0.167708	−	0.985837i	$-0.446363\pi$
$449$	7.10735	0.335416	0.167708	+	0.985837i	$0.446363\pi$
$450$	0	0
$451$	−4.66070	−0.219464
$452$	0	0
$453$	0.585707i	0.0275189i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 14.8030i	− 0.692456i	−0.938150	−	0.346228i	$-0.887462\pi$
$457$	− 14.8030i	− 0.692456i	0.938150	−	0.346228i	$-0.112538\pi$
$458$	0	0
$459$	0.0591259	0.00275976
$460$	0	0
$461$	18.8972	0.880130	0.440065	−	0.897966i	$-0.354955\pi$
$461$	18.8972	0.880130	0.440065	+	0.897966i	$0.354955\pi$
$462$	0	0
$463$	26.2693i	1.22084i	0.792079	+	0.610419i	$0.208999\pi$
$463$	26.2693i	1.22084i	−0.792079	+	0.610419i	$0.791001\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 18.0617i	− 0.835796i	−0.908494	−	0.417898i	$-0.862767\pi$
$467$	− 18.0617i	− 0.835796i	0.908494	−	0.417898i	$-0.137233\pi$
$468$	0	0
$469$	0.439095	0.0202755
$470$	0	0
$471$	18.5369	0.854135
$472$	0	0
$473$	− 7.26203i	− 0.333908i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.1533i	− 0.464889i
$478$	0	0
$479$	−15.5343	−0.709782	−0.354891	−	0.934908i	$-0.615482\pi$
$479$	−15.5343	−0.709782	−0.354891	+	0.934908i	$0.615482\pi$
$480$	0	0
$481$	−1.46447	−0.0667740
$482$	0	0
$483$	0.247785i	0.0112746i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.2478i	0.555001i	0.960726	+	0.277500i	$0.0895060\pi$
$487$	12.2478i	0.555001i	−0.960726	+	0.277500i	$0.910494\pi$
$488$	0	0
$489$	37.8757	1.71280
$490$	0	0
$491$	32.6791	1.47479	0.737393	−	0.675463i	$-0.236056\pi$
$491$	32.6791	1.47479	0.737393	+	0.675463i	$0.236056\pi$
$492$	0	0
$493$	0.500447i	0.0225390i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 0.402954i	− 0.0180750i
$498$	0	0
$499$	7.25695	0.324866	0.162433	−	0.986720i	$-0.448066\pi$
$499$	7.25695	0.324866	0.162433	+	0.986720i	$0.448066\pi$
$500$	0	0
$501$	−31.2480	−1.39606
$502$	0	0
$503$	− 12.5784i	− 0.560842i	−0.959877	−	0.280421i	$-0.909526\pi$
$503$	− 12.5784i	− 0.560842i	0.959877	−	0.280421i	$-0.0904741\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 72.8962i	− 3.23744i
$508$	0	0
$509$	29.0927	1.28951	0.644755	−	0.764389i	$-0.276959\pi$
$509$	29.0927	1.28951	0.644755	+	0.764389i	$0.276959\pi$
$510$	0	0
$511$	−2.09624	−0.0927322
$512$	0	0
$513$	0.690224i	0.0304741i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.47206i	− 0.196681i
$518$	0	0
$519$	19.7356	0.866295
$520$	0	0
$521$	3.16815	0.138799	0.0693997	−	0.997589i	$-0.477892\pi$
$521$	3.16815	0.138799	0.0693997	+	0.997589i	$0.477892\pi$
$522$	0	0
$523$	19.1447i	0.837137i	0.908185	+	0.418569i	$0.137468\pi$
$523$	19.1447i	0.837137i	−0.908185	+	0.418569i	$0.862532\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 0.549740i	− 0.0239470i
$528$	0	0
$529$	22.5796	0.981720
$530$	0	0
$531$	−39.4221	−1.71077
$532$	0	0
$533$	− 6.51290i	− 0.282105i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 55.1725i	− 2.38087i
$538$	0	0
$539$	−32.5141	−1.40048
$540$	0	0
$541$	−6.42973	−0.276436	−0.138218	−	0.990402i	$-0.544137\pi$
$541$	−6.42973	−0.276436	−0.138218	+	0.990402i	$0.544137\pi$
$542$	0	0
$543$	55.4642i	2.38020i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.5264i	1.04867i	0.851511	+	0.524337i	$0.175687\pi$
$547$	24.5264i	1.04867i	−0.851511	+	0.524337i	$0.824313\pi$
$548$	0	0
$549$	−41.1016	−1.75417
$550$	0	0
$551$	−5.84211	−0.248882
$552$	0	0
$553$	− 1.25642i	− 0.0534284i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.2156i	1.28028i	0.768260	+	0.640138i	$0.221123\pi$
$557$	30.2156i	1.28028i	−0.768260	+	0.640138i	$0.778877\pi$
$558$	0	0
$559$	10.1480	0.429215
$560$	0	0
$561$	1.96480	0.0829541
$562$	0	0
$563$	− 19.3955i	− 0.817424i	−0.912663	−	0.408712i	$-0.865978\pi$
$563$	− 19.3955i	− 0.817424i	0.912663	−	0.408712i	$-0.134022\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.32003i	0.0554361i
$568$	0	0
$569$	−36.0474	−1.51118	−0.755592	−	0.655043i	$-0.772651\pi$
$569$	−36.0474	−1.51118	−0.755592	+	0.655043i	$0.772651\pi$
$570$	0	0
$571$	−25.2533	−1.05682	−0.528409	−	0.848990i	$-0.677211\pi$
$571$	−25.2533	−1.05682	−0.528409	+	0.848990i	$0.677211\pi$
$572$	0	0
$573$	18.2597i	0.762812i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.0765i	0.586011i	0.956111	+	0.293006i	$0.0946555\pi$
$577$	14.0765i	0.586011i	−0.956111	+	0.293006i	$0.905345\pi$
$578$	0	0
$579$	−13.5647	−0.563732
$580$	0	0
$581$	0.312834	0.0129785
$582$	0	0
$583$	− 15.0694i	− 0.624109i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.24494i	− 0.257756i	−0.991660	−	0.128878i	$-0.958862\pi$
$587$	− 6.24494i	− 0.257756i	0.991660	−	0.128878i	$-0.0411376\pi$
$588$	0	0
$589$	6.41755	0.264430
$590$	0	0
$591$	−29.1917	−1.20079
$592$	0	0
$593$	− 33.4069i	− 1.37186i	−0.727669	−	0.685929i	$-0.759396\pi$
$593$	− 33.4069i	− 1.37186i	0.727669	−	0.685929i	$-0.240604\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 35.8589i	− 1.46761i
$598$	0	0
$599$	1.91202	0.0781231	0.0390616	−	0.999237i	$-0.487563\pi$
$599$	1.91202	0.0781231	0.0390616	+	0.999237i	$0.487563\pi$
$600$	0	0
$601$	−15.2706	−0.622903	−0.311451	−	0.950262i	$-0.600815\pi$
$601$	−15.2706	−0.622903	−0.311451	+	0.950262i	$0.600815\pi$
$602$	0	0
$603$	− 8.94109i	− 0.364110i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 36.3016i	− 1.47343i	−0.676201	−	0.736717i	$-0.736375\pi$
$607$	− 36.3016i	− 1.47343i	0.676201	−	0.736717i	$-0.263625\pi$
$608$	0	0
$609$	1.12411	0.0455511
$610$	0	0
$611$	6.24930	0.252820
$612$	0	0
$613$	− 29.5816i	− 1.19479i	−0.801947	−	0.597395i	$-0.796203\pi$
$613$	− 29.5816i	− 1.19479i	0.801947	−	0.597395i	$-0.203797\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 37.6855i	− 1.51716i	−0.651580	−	0.758580i	$-0.725894\pi$
$617$	− 37.6855i	− 1.51716i	0.651580	−	0.758580i	$-0.274106\pi$
$618$	0	0
$619$	25.6797	1.03216	0.516078	−	0.856542i	$-0.327392\pi$
$619$	25.6797	1.03216	0.516078	+	0.856542i	$0.327392\pi$
$620$	0	0
$621$	0.225347	0.00904287
$622$	0	0
$623$	− 1.66388i	− 0.0666618i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	22.9367i	0.916004i
$628$	0	0
$629$	0.0382543	0.00152530
$630$	0	0
$631$	−6.36803	−0.253507	−0.126754	−	0.991934i	$-0.540456\pi$
$631$	−6.36803	−0.253507	−0.126754	+	0.991934i	$0.540456\pi$
$632$	0	0
$633$	− 31.3684i	− 1.24678i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 45.4354i	− 1.80022i
$638$	0	0
$639$	−8.20519	−0.324592
$640$	0	0
$641$	−24.4076	−0.964041	−0.482020	−	0.876160i	$-0.660097\pi$
$641$	−24.4076	−0.964041	−0.482020	+	0.876160i	$0.660097\pi$
$642$	0	0
$643$	10.1020i	0.398383i	0.979961	+	0.199192i	$0.0638317\pi$
$643$	10.1020i	0.398383i	−0.979961	+	0.199192i	$0.936168\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 45.8189i	− 1.80133i	−0.434518	−	0.900663i	$-0.643081\pi$
$647$	− 45.8189i	− 1.80133i	0.434518	−	0.900663i	$-0.356919\pi$
$648$	0	0
$649$	−58.5095	−2.29670
$650$	0	0
$651$	−1.23483	−0.0483968
$652$	0	0
$653$	− 47.3168i	− 1.85165i	−0.377952	−	0.925825i	$-0.623372\pi$
$653$	− 47.3168i	− 1.85165i	0.377952	−	0.925825i	$-0.376628\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	42.6848i	1.66529i
$658$	0	0
$659$	−2.97112	−0.115738	−0.0578691	−	0.998324i	$-0.518431\pi$
$659$	−2.97112	−0.115738	−0.0578691	+	0.998324i	$0.518431\pi$
$660$	0	0
$661$	0.202587	0.00787972	0.00393986	−	0.999992i	$-0.498746\pi$
$661$	0.202587	0.00787972	0.00393986	+	0.999992i	$0.498746\pi$
$662$	0	0
$663$	2.74563i	0.106632i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.90736i	0.0738532i
$668$	0	0
$669$	27.4949	1.06301
$670$	0	0
$671$	−61.0021	−2.35496
$672$	0	0
$673$	− 37.5673i	− 1.44811i	−0.689741	−	0.724057i	$-0.742275\pi$
$673$	− 37.5673i	− 1.44811i	0.689741	−	0.724057i	$-0.257725\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.5295i	0.827447i	0.910403	+	0.413723i	$0.135772\pi$
$677$	21.5295i	0.827447i	−0.910403	+	0.413723i	$0.864228\pi$
$678$	0	0
$679$	1.19836	0.0459889
$680$	0	0
$681$	24.5455	0.940586
$682$	0	0
$683$	7.32947i	0.280454i	0.990119	+	0.140227i	$0.0447833\pi$
$683$	7.32947i	0.280454i	−0.990119	+	0.140227i	$0.955217\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 72.8399i	− 2.77902i
$688$	0	0
$689$	21.0581	0.802248
$690$	0	0
$691$	−36.7528	−1.39814	−0.699071	−	0.715052i	$-0.746403\pi$
$691$	−36.7528	−1.39814	−0.699071	+	0.715052i	$0.746403\pi$
$692$	0	0
$693$	− 2.25708i	− 0.0857395i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.170128i	0.00644405i
$698$	0	0
$699$	36.4401	1.37829
$700$	0	0
$701$	−4.14975	−0.156734	−0.0783669	−	0.996925i	$-0.524971\pi$
$701$	−4.14975	−0.156734	−0.0783669	+	0.996925i	$0.524971\pi$
$702$	0	0
$703$	0.446573i	0.0168428i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.62998i	− 0.0613016i
$708$	0	0
$709$	7.94137	0.298245	0.149122	−	0.988819i	$-0.452355\pi$
$709$	7.94137	0.298245	0.149122	+	0.988819i	$0.452355\pi$
$710$	0	0
$711$	−25.5839	−0.959473
$712$	0	0
$713$	− 2.09523i	− 0.0784669i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 5.14311i	− 0.192073i
$718$	0	0
$719$	47.2916	1.76368	0.881840	−	0.471549i	$-0.156305\pi$
$719$	47.2916	1.76368	0.881840	+	0.471549i	$0.156305\pi$
$720$	0	0
$721$	−1.50541	−0.0560643
$722$	0	0
$723$	− 3.85223i	− 0.143266i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11.8115i	0.438062i	0.975718	+	0.219031i	$0.0702897\pi$
$727$	11.8115i	0.438062i	−0.975718	+	0.219031i	$0.929710\pi$
$728$	0	0
$729$	29.4628	1.09121
$730$	0	0
$731$	−0.265083	−0.00980444
$732$	0	0
$733$	21.8546i	0.807219i	0.914931	+	0.403609i	$0.132245\pi$
$733$	21.8546i	0.807219i	−0.914931	+	0.403609i	$0.867755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 13.2702i	− 0.488814i
$738$	0	0
$739$	40.0907	1.47476	0.737380	−	0.675478i	$-0.236063\pi$
$739$	40.0907	1.47476	0.737380	+	0.675478i	$0.236063\pi$
$740$	0	0
$741$	−32.0520	−1.17746
$742$	0	0
$743$	30.4553i	1.11730i	0.829405	+	0.558648i	$0.188680\pi$
$743$	30.4553i	1.11730i	−0.829405	+	0.558648i	$0.811320\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 6.37010i	− 0.233070i
$748$	0	0
$749$	0.106774	0.00390143
$750$	0	0
$751$	38.1436	1.39188	0.695940	−	0.718100i	$-0.254988\pi$
$751$	38.1436	1.39188	0.695940	+	0.718100i	$0.254988\pi$
$752$	0	0
$753$	− 15.3560i	− 0.559604i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.9587i	0.689066i	0.938774	+	0.344533i	$0.111963\pi$
$757$	18.9587i	0.689066i	−0.938774	+	0.344533i	$0.888037\pi$
$758$	0	0
$759$	7.48848	0.271815
$760$	0	0
$761$	0.687053	0.0249057	0.0124528	−	0.999922i	$-0.496036\pi$
$761$	0.687053	0.0249057	0.0124528	+	0.999922i	$0.496036\pi$
$762$	0	0
$763$	− 1.60754i	− 0.0581970i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 81.7617i	− 2.95224i
$768$	0	0
$769$	7.94514	0.286509	0.143255	−	0.989686i	$-0.454243\pi$
$769$	7.94514	0.286509	0.143255	+	0.989686i	$0.454243\pi$
$770$	0	0
$771$	66.7425	2.40367
$772$	0	0
$773$	− 40.0866i	− 1.44182i	−0.693031	−	0.720908i	$-0.743725\pi$
$773$	− 40.0866i	− 1.44182i	0.693031	−	0.720908i	$-0.256275\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 0.0859271i	− 0.00308262i
$778$	0	0
$779$	−1.98604	−0.0711571
$780$	0	0
$781$	−12.1780	−0.435762
$782$	0	0
$783$	− 1.02232i	− 0.0365346i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 16.2733i	− 0.580080i	−0.957015	−	0.290040i	$-0.906331\pi$
$787$	− 16.2733i	− 0.580080i	0.957015	−	0.290040i	$-0.0936685\pi$
$788$	0	0
$789$	−54.6417	−1.94530
$790$	0	0
$791$	−1.36799	−0.0486403
$792$	0	0
$793$	− 85.2449i	− 3.02714i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.6565i	0.660848i	0.943833	+	0.330424i	$0.107192\pi$
$797$	18.6565i	0.660848i	−0.943833	+	0.330424i	$0.892808\pi$
$798$	0	0
$799$	−0.163242	−0.00577508
$800$	0	0
$801$	−33.8808	−1.19712
$802$	0	0
$803$	63.3520i	2.23564i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 24.1475i	− 0.850033i
$808$	0	0
$809$	15.4673	0.543801	0.271900	−	0.962325i	$-0.412348\pi$
$809$	15.4673	0.543801	0.271900	+	0.962325i	$0.412348\pi$
$810$	0	0
$811$	−11.0327	−0.387412	−0.193706	−	0.981060i	$-0.562051\pi$
$811$	−11.0327	−0.387412	−0.193706	+	0.981060i	$0.562051\pi$
$812$	0	0
$813$	26.4607i	0.928018i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.09452i	− 0.108264i
$818$	0	0
$819$	3.15407	0.110212
$820$	0	0
$821$	−23.1031	−0.806303	−0.403152	−	0.915133i	$-0.632085\pi$
$821$	−23.1031	−0.806303	−0.403152	+	0.915133i	$0.632085\pi$
$822$	0	0
$823$	− 21.0523i	− 0.733836i	−0.930253	−	0.366918i	$-0.880413\pi$
$823$	− 21.0523i	− 0.733836i	0.930253	−	0.366918i	$-0.119587\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.5524i	1.30582i	0.757434	+	0.652912i	$0.226453\pi$
$827$	37.5524i	1.30582i	−0.757434	+	0.652912i	$0.773547\pi$
$828$	0	0
$829$	3.99462	0.138739	0.0693694	−	0.997591i	$-0.477901\pi$
$829$	3.99462	0.138739	0.0693694	+	0.997591i	$0.477901\pi$
$830$	0	0
$831$	0.573986	0.0199113
$832$	0	0
$833$	1.18685i	0.0411218i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.12301i	0.0388170i
$838$	0	0
$839$	−36.6872	−1.26658	−0.633292	−	0.773913i	$-0.718297\pi$
$839$	−36.6872	−1.26658	−0.633292	+	0.773913i	$0.718297\pi$
$840$	0	0
$841$	−20.3470	−0.701622
$842$	0	0
$843$	78.9761i	2.72008i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.65353i	− 0.0568161i
$848$	0	0
$849$	74.2770	2.54918
$850$	0	0
$851$	0.145799	0.00499793
$852$	0	0
$853$	49.1310i	1.68221i	0.540869	+	0.841107i	$0.318096\pi$
$853$	49.1310i	1.68221i	−0.540869	+	0.841107i	$0.681904\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 37.8776i	− 1.29387i	−0.762544	−	0.646937i	$-0.776050\pi$
$857$	− 37.8776i	− 1.29387i	0.762544	−	0.646937i	$-0.223950\pi$
$858$	0	0
$859$	−38.5601	−1.31565	−0.657826	−	0.753170i	$-0.728524\pi$
$859$	−38.5601	−1.31565	−0.657826	+	0.753170i	$0.728524\pi$
$860$	0	0
$861$	0.382142	0.0130234
$862$	0	0
$863$	− 25.8156i	− 0.878773i	−0.898298	−	0.439386i	$-0.855196\pi$
$863$	− 25.8156i	− 0.878773i	0.898298	−	0.439386i	$-0.144804\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	42.0535i	1.42821i
$868$	0	0
$869$	−37.9712	−1.28808
$870$	0	0
$871$	18.5439	0.628336
$872$	0	0
$873$	− 24.4017i	− 0.825874i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.47018i	− 0.0496443i	−0.999692	−	0.0248222i	$-0.992098\pi$
$877$	− 1.47018i	− 0.0496443i	0.999692	−	0.0248222i	$-0.00790195\pi$
$878$	0	0
$879$	8.68286	0.292866
$880$	0	0
$881$	−9.11691	−0.307156	−0.153578	−	0.988136i	$-0.549080\pi$
$881$	−9.11691	−0.307156	−0.153578	+	0.988136i	$0.549080\pi$
$882$	0	0
$883$	− 14.1889i	− 0.477494i	−0.971082	−	0.238747i	$-0.923263\pi$
$883$	− 14.1889i	− 0.477494i	0.971082	−	0.238747i	$-0.0767367\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 37.5575i	− 1.26106i	−0.776166	−	0.630528i	$-0.782838\pi$
$887$	− 37.5575i	− 1.26106i	0.776166	−	0.630528i	$-0.217162\pi$
$888$	0	0
$889$	−1.67592	−0.0562084
$890$	0	0
$891$	39.8936	1.33649
$892$	0	0
$893$	− 1.90565i	− 0.0637702i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.4645i	0.349398i
$898$	0	0
$899$	−9.50528	−0.317019
$900$	0	0
$901$	−0.550071	−0.0183255
$902$	0	0
$903$	0.595431i	0.0198147i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 14.0139i	− 0.465324i	−0.972558	−	0.232662i	$-0.925256\pi$
$907$	− 14.0139i	− 0.465324i	0.972558	−	0.232662i	$-0.0747436\pi$
$908$	0	0
$909$	−33.1905	−1.10086
$910$	0	0
$911$	46.2606	1.53268	0.766340	−	0.642435i	$-0.222076\pi$
$911$	46.2606	1.53268	0.766340	+	0.642435i	$0.222076\pi$
$912$	0	0
$913$	− 9.45437i	− 0.312894i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.754264i	− 0.0249080i
$918$	0	0
$919$	−14.0106	−0.462168	−0.231084	−	0.972934i	$-0.574227\pi$
$919$	−14.0106	−0.462168	−0.231084	+	0.972934i	$0.574227\pi$
$920$	0	0
$921$	−41.3330	−1.36197
$922$	0	0
$923$	− 17.0176i	− 0.560142i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	30.6540i	1.00681i
$928$	0	0
$929$	−32.0911	−1.05287	−0.526437	−	0.850214i	$-0.676472\pi$
$929$	−32.0911	−1.05287	−0.526437	+	0.850214i	$0.676472\pi$
$930$	0	0
$931$	−13.8550	−0.454080
$932$	0	0
$933$	49.4759i	1.61977i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.3982i	0.862390i	0.902259	+	0.431195i	$0.141908\pi$
$937$	26.3982i	0.862390i	−0.902259	+	0.431195i	$0.858092\pi$
$938$	0	0
$939$	52.6315	1.71756
$940$	0	0
$941$	48.1936	1.57107	0.785533	−	0.618819i	$-0.212389\pi$
$941$	48.1936	1.57107	0.785533	+	0.618819i	$0.212389\pi$
$942$	0	0
$943$	0.648409i	0.0211151i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.35993i	− 0.239166i	−0.992824	−	0.119583i	$-0.961844\pi$
$947$	− 7.35993i	− 0.239166i	0.992824	−	0.119583i	$-0.0381557\pi$
$948$	0	0
$949$	−88.5286	−2.87376
$950$	0	0
$951$	−58.6109	−1.90059
$952$	0	0
$953$	47.6666i	1.54407i	0.635578	+	0.772036i	$0.280762\pi$
$953$	47.6666i	1.54407i	−0.635578	+	0.772036i	$0.719238\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 33.9725i	− 1.09817i
$958$	0	0
$959$	−1.93669	−0.0625390
$960$	0	0
$961$	−20.5585	−0.663177
$962$	0	0
$963$	− 2.17419i	− 0.0700622i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 33.8414i	− 1.08827i	−0.838999	−	0.544133i	$-0.816859\pi$
$967$	− 33.8414i	− 1.08827i	0.838999	−	0.544133i	$-0.183141\pi$
$968$	0	0
$969$	0.837250	0.0268963
$970$	0	0
$971$	49.6036	1.59185	0.795927	−	0.605393i	$-0.206984\pi$
$971$	49.6036	1.59185	0.795927	+	0.605393i	$0.206984\pi$
$972$	0	0
$973$	0.148079i	0.00474721i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.48799i	0.0795980i	0.999208	+	0.0397990i	$0.0126718\pi$
$977$	2.48799i	0.0795980i	−0.999208	+	0.0397990i	$0.987328\pi$
$978$	0	0
$979$	−50.2852	−1.60712
$980$	0	0
$981$	−32.7337	−1.04511
$982$	0	0
$983$	41.4324i	1.32149i	0.750611	+	0.660744i	$0.229759\pi$
$983$	41.4324i	1.32149i	−0.750611	+	0.660744i	$0.770241\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.366675i	0.0116714i
$988$	0	0
$989$	−1.01031	−0.0321261
$990$	0	0
$991$	15.1066	0.479876	0.239938	−	0.970788i	$-0.422873\pi$
$991$	15.1066	0.479876	0.239938	+	0.970788i	$0.422873\pi$
$992$	0	0
$993$	51.0317i	1.61944i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	49.0108i	1.55219i	0.630618	+	0.776093i	$0.282802\pi$
$997$	49.0108i	1.55219i	−0.630618	+	0.776093i	$0.717198\pi$
$998$	0	0
$999$	−0.0781463	−0.00247244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.g.1149.13		14
5.2	odd	4		4100.2.a.g.1.7	✓	7
5.3	odd	4		4100.2.a.j.1.1	yes	7
5.4	even	2	inner	4100.2.d.g.1149.2		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4100.2.a.g.1.7	✓	7	5.2	odd	4
4100.2.a.j.1.1	yes	7	5.3	odd	4
4100.2.d.g.1149.2		14	5.4	even	2	inner
4100.2.d.g.1149.13		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)

\(f(q)\)	\(=\)	\(q+2.47795i q^{3} -0.154217i q^{7} -3.14025 q^{9} +O(q^{10})\)	\(q+2.47795i q^{3} -0.154217i q^{7} -3.14025 q^{9} -4.66070 q^{11} -6.51290i q^{13} +0.170128i q^{17} -1.98604 q^{19} +0.382142 q^{21} +0.648409i q^{23} -0.347539i q^{27} +2.94159 q^{29} -3.23134 q^{31} -11.5490i q^{33} -0.224856i q^{37} +16.1387 q^{39} +1.00000 q^{41} +1.55814i q^{43} +0.959526i q^{47} +6.97622 q^{49} -0.421568 q^{51} +3.23328i q^{53} -4.92130i q^{57} +12.5538 q^{59} +13.0886 q^{61} +0.484280i q^{63} +2.84725i q^{67} -1.60673 q^{69} +2.61291 q^{71} -13.5928i q^{73} +0.718759i q^{77} +8.14710 q^{79} -8.55957 q^{81} +2.02853i q^{83} +7.28913i q^{87} +10.7892 q^{89} -1.00440 q^{91} -8.00710i q^{93} +7.77063i q^{97} +14.6358 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 12 q^{9}+O(q^{10}) \) 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

Groups

Database

Properties

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