LMFDB - Embedded newform 6100.2.c.f.4149.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6100,2,Mod(4149,6100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6100, 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("6100.4149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6100 = 2^{2} \cdot 5^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6100.c (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:	$48.7087452330$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4149.5
Root			$1.24698i$ of defining polynomial
Character	$\chi$	$=$	6100.4149
Dual form			6100.2.c.f.4149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.801938i q^{3} +1.69202i q^{7} +2.35690 q^{9} +O(q^{10})$	$q+0.801938i q^{3} +1.69202i q^{7} +2.35690 q^{9} +0.335126 q^{11} -5.24698i q^{13} +3.44504i q^{17} +0.417895 q^{19} -1.35690 q^{21} -9.34481i q^{23} +4.29590i q^{27} -4.54288 q^{29} +0.198062 q^{31} +0.268750i q^{33} -5.67994i q^{37} +4.20775 q^{39} -5.07606 q^{41} -4.29590i q^{43} -8.45473i q^{47} +4.13706 q^{49} -2.76271 q^{51} -3.08815i q^{53} +0.335126i q^{57} -1.51573 q^{59} -1.00000 q^{61} +3.98792i q^{63} -9.25667i q^{67} +7.49396 q^{69} -2.38404 q^{71} -6.46681i q^{73} +0.567040i q^{77} -7.83877 q^{79} +3.62565 q^{81} +12.1075i q^{83} -3.64310i q^{87} +1.71917 q^{89} +8.87800 q^{91} +0.158834i q^{93} +4.23729i q^{97} +0.789856 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^9	$ 6 q + 6 q^{9} + 14 q^{19} + 10 q^{29} + 10 q^{31} - 10 q^{39} + 14 q^{49} + 18 q^{51} + 16 q^{59} - 6 q^{61} + 26 q^{69} + 6 q^{71} + 18 q^{79} - 2 q^{81} - 12 q^{89} + 14 q^{91} - 42 q^{99}+O(q^{100}) $ 6 * q + 6 * q^9 + 14 * q^19 + 10 * q^29 + 10 * q^31 - 10 * q^39 + 14 * q^49 + 18 * q^51 + 16 * q^59 - 6 * q^61 + 26 * q^69 + 6 * q^71 + 18 * q^79 - 2 * q^81 - 12 * q^89 + 14 * q^91 - 42 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6100\mathbb{Z}\right)^\times$.

$n$	$977$	$3051$	$3601$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0.801938i	0.462999i	0.972835	+	0.231499i	$0.0743632\pi$
$3$	0.801938i	0.462999i	−0.972835	+	0.231499i	$0.925637\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.69202i	0.639524i	0.947498	+	0.319762i	$0.103603\pi$
$7$	1.69202i	0.639524i	−0.947498	+	0.319762i	$0.896397\pi$
$8$	0	0
$9$	2.35690	0.785632
$10$	0	0
$11$	0.335126	0.101044	0.0505221	−	0.998723i	$-0.483911\pi$
$11$	0.335126	0.101044	0.0505221	+	0.998723i	$0.483911\pi$
$12$	0	0
$13$	− 5.24698i	− 1.45525i	−0.685975	−	0.727625i	$-0.740624\pi$
$13$	− 5.24698i	− 1.45525i	0.685975	−	0.727625i	$-0.259376\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.44504i	0.835545i	0.908552	+	0.417773i	$0.137189\pi$
$17$	3.44504i	0.835545i	−0.908552	+	0.417773i	$0.862811\pi$
$18$	0	0
$19$	0.417895	0.0958716	0.0479358	−	0.998850i	$-0.484736\pi$
$19$	0.417895	0.0958716	0.0479358	+	0.998850i	$0.484736\pi$
$20$	0	0
$21$	−1.35690	−0.296099
$22$	0	0
$23$	− 9.34481i	− 1.94853i	−0.225409	−	0.974264i	$-0.572372\pi$
$23$	− 9.34481i	− 1.94853i	0.225409	−	0.974264i	$-0.427628\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.29590i	0.826746i
$28$	0	0
$29$	−4.54288	−0.843591	−0.421795	−	0.906691i	$-0.638600\pi$
$29$	−4.54288	−0.843591	−0.421795	+	0.906691i	$0.638600\pi$
$30$	0	0
$31$	0.198062	0.0355730	0.0177865	−	0.999842i	$-0.494338\pi$
$31$	0.198062	0.0355730	0.0177865	+	0.999842i	$0.494338\pi$
$32$	0	0
$33$	0.268750i	0.0467833i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.67994i	− 0.933776i	−0.884316	−	0.466888i	$-0.845375\pi$
$37$	− 5.67994i	− 0.933776i	0.884316	−	0.466888i	$-0.154625\pi$
$38$	0	0
$39$	4.20775	0.673779
$40$	0	0
$41$	−5.07606	−0.792748	−0.396374	−	0.918089i	$-0.629732\pi$
$41$	−5.07606	−0.792748	−0.396374	+	0.918089i	$0.629732\pi$
$42$	0	0
$43$	− 4.29590i	− 0.655118i	−0.944831	−	0.327559i	$-0.893774\pi$
$43$	− 4.29590i	− 0.655118i	0.944831	−	0.327559i	$-0.106226\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.45473i	− 1.23325i	−0.787257	−	0.616625i	$-0.788500\pi$
$47$	− 8.45473i	− 1.23325i	0.787257	−	0.616625i	$-0.211500\pi$
$48$	0	0
$49$	4.13706	0.591009
$50$	0	0
$51$	−2.76271	−0.386857
$52$	0	0
$53$	− 3.08815i	− 0.424189i	−0.977249	−	0.212095i	$-0.931971\pi$
$53$	− 3.08815i	− 0.424189i	0.977249	−	0.212095i	$-0.0680285\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.335126i	0.0443885i
$58$	0	0
$59$	−1.51573	−0.197331	−0.0986656	−	0.995121i	$-0.531457\pi$
$59$	−1.51573	−0.197331	−0.0986656	+	0.995121i	$0.531457\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	3.98792i	0.502430i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.25667i	− 1.13088i	−0.824789	−	0.565441i	$-0.808706\pi$
$67$	− 9.25667i	− 1.13088i	0.824789	−	0.565441i	$-0.191294\pi$
$68$	0	0
$69$	7.49396	0.902167
$70$	0	0
$71$	−2.38404	−0.282934	−0.141467	−	0.989943i	$-0.545182\pi$
$71$	−2.38404	−0.282934	−0.141467	+	0.989943i	$0.545182\pi$
$72$	0	0
$73$	− 6.46681i	− 0.756883i	−0.925625	−	0.378442i	$-0.876460\pi$
$73$	− 6.46681i	− 0.756883i	0.925625	−	0.378442i	$-0.123540\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.567040i	0.0646202i
$78$	0	0
$79$	−7.83877	−0.881931	−0.440965	−	0.897524i	$-0.645364\pi$
$79$	−7.83877	−0.881931	−0.440965	+	0.897524i	$0.645364\pi$
$80$	0	0
$81$	3.62565	0.402850
$82$	0	0
$83$	12.1075i	1.32897i	0.747300	+	0.664487i	$0.231350\pi$
$83$	12.1075i	1.32897i	−0.747300	+	0.664487i	$0.768650\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.64310i	− 0.390582i
$88$	0	0
$89$	1.71917	0.182232	0.0911158	−	0.995840i	$-0.470957\pi$
$89$	1.71917	0.182232	0.0911158	+	0.995840i	$0.470957\pi$
$90$	0	0
$91$	8.87800	0.930668
$92$	0	0
$93$	0.158834i	0.0164703i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.23729i	0.430232i	0.976589	+	0.215116i	$0.0690129\pi$
$97$	4.23729i	0.430232i	−0.976589	+	0.215116i	$0.930987\pi$
$98$	0	0
$99$	0.789856	0.0793835
$100$	0	0
$101$	−16.9922	−1.69079	−0.845395	−	0.534142i	$-0.820635\pi$
$101$	−16.9922	−1.69079	−0.845395	+	0.534142i	$0.820635\pi$
$102$	0	0
$103$	− 7.53750i	− 0.742692i	−0.928495	−	0.371346i	$-0.878896\pi$
$103$	− 7.53750i	− 0.742692i	0.928495	−	0.371346i	$-0.121104\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.5797i	1.11945i	0.828677	+	0.559727i	$0.189094\pi$
$107$	11.5797i	1.11945i	−0.828677	+	0.559727i	$0.810906\pi$
$108$	0	0
$109$	−18.3424	−1.75689	−0.878443	−	0.477848i	$-0.841417\pi$
$109$	−18.3424	−1.75689	−0.878443	+	0.477848i	$0.841417\pi$
$110$	0	0
$111$	4.55496	0.432337
$112$	0	0
$113$	− 18.7235i	− 1.76136i	−0.473715	−	0.880678i	$-0.657087\pi$
$113$	− 18.7235i	− 1.76136i	0.473715	−	0.880678i	$-0.342913\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 12.3666i	− 1.14329i
$118$	0	0
$119$	−5.82908	−0.534351
$120$	0	0
$121$	−10.8877	−0.989790
$122$	0	0
$123$	− 4.07069i	− 0.367042i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 9.69202i	− 0.860028i	−0.902822	−	0.430014i	$-0.858509\pi$
$127$	− 9.69202i	− 0.860028i	0.902822	−	0.430014i	$-0.141491\pi$
$128$	0	0
$129$	3.44504	0.303319
$130$	0	0
$131$	16.5743	1.44811	0.724053	−	0.689744i	$-0.242277\pi$
$131$	16.5743	1.44811	0.724053	+	0.689744i	$0.242277\pi$
$132$	0	0
$133$	0.707087i	0.0613122i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.04354i	0.430899i	0.976515	+	0.215449i	$0.0691216\pi$
$137$	5.04354i	0.430899i	−0.976515	+	0.215449i	$0.930878\pi$
$138$	0	0
$139$	14.8562	1.26009	0.630045	−	0.776559i	$-0.283037\pi$
$139$	14.8562	1.26009	0.630045	+	0.776559i	$0.283037\pi$
$140$	0	0
$141$	6.78017	0.570993
$142$	0	0
$143$	− 1.75840i	− 0.147045i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.31767i	0.273637i
$148$	0	0
$149$	−11.3448	−0.929403	−0.464702	−	0.885467i	$-0.653838\pi$
$149$	−11.3448	−0.929403	−0.464702	+	0.885467i	$0.653838\pi$
$150$	0	0
$151$	5.35690	0.435938	0.217969	−	0.975956i	$-0.430057\pi$
$151$	5.35690	0.435938	0.217969	+	0.975956i	$0.430057\pi$
$152$	0	0
$153$	8.11960i	0.656431i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	23.2325i	1.85416i	0.374869	+	0.927078i	$0.377688\pi$
$157$	23.2325i	1.85416i	−0.374869	+	0.927078i	$0.622312\pi$
$158$	0	0
$159$	2.47650	0.196399
$160$	0	0
$161$	15.8116	1.24613
$162$	0	0
$163$	− 9.11529i	− 0.713965i	−0.934111	−	0.356982i	$-0.883806\pi$
$163$	− 9.11529i	− 0.713965i	0.934111	−	0.356982i	$-0.116194\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 10.3123i	− 0.797989i	−0.916953	−	0.398994i	$-0.869359\pi$
$167$	− 10.3123i	− 0.797989i	0.916953	−	0.398994i	$-0.130641\pi$
$168$	0	0
$169$	−14.5308	−1.11775
$170$	0	0
$171$	0.984935	0.0753198
$172$	0	0
$173$	6.87263i	0.522516i	0.965269	+	0.261258i	$0.0841373\pi$
$173$	6.87263i	0.522516i	−0.965269	+	0.261258i	$0.915863\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.21552i	− 0.0913641i
$178$	0	0
$179$	8.13467	0.608014	0.304007	−	0.952670i	$-0.401675\pi$
$179$	8.13467	0.608014	0.304007	+	0.952670i	$0.401675\pi$
$180$	0	0
$181$	−5.18598	−0.385471	−0.192735	−	0.981251i	$-0.561736\pi$
$181$	−5.18598	−0.385471	−0.192735	+	0.981251i	$0.561736\pi$
$182$	0	0
$183$	− 0.801938i	− 0.0592809i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.15452i	0.0844270i
$188$	0	0
$189$	−7.26875	−0.528724
$190$	0	0
$191$	−4.89440	−0.354146	−0.177073	−	0.984198i	$-0.556663\pi$
$191$	−4.89440	−0.354146	−0.177073	+	0.984198i	$0.556663\pi$
$192$	0	0
$193$	− 14.2717i	− 1.02730i	−0.857999	−	0.513651i	$-0.828293\pi$
$193$	− 14.2717i	− 1.02730i	0.857999	−	0.513651i	$-0.171707\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.37867i	0.0982259i	0.998793	+	0.0491129i	$0.0156394\pi$
$197$	1.37867i	0.0982259i	−0.998793	+	0.0491129i	$0.984361\pi$
$198$	0	0
$199$	12.3817	0.877712	0.438856	−	0.898557i	$-0.355384\pi$
$199$	12.3817	0.877712	0.438856	+	0.898557i	$0.355384\pi$
$200$	0	0
$201$	7.42327	0.523597
$202$	0	0
$203$	− 7.68664i	− 0.539497i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 22.0248i	− 1.53083i
$208$	0	0
$209$	0.140047	0.00968727
$210$	0	0
$211$	19.5429	1.34539	0.672694	−	0.739921i	$-0.265137\pi$
$211$	19.5429	1.34539	0.672694	+	0.739921i	$0.265137\pi$
$212$	0	0
$213$	− 1.91185i	− 0.130998i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.335126i	0.0227498i
$218$	0	0
$219$	5.18598	0.350436
$220$	0	0
$221$	18.0761	1.21593
$222$	0	0
$223$	− 4.27114i	− 0.286017i	−0.989721	−	0.143008i	$-0.954322\pi$
$223$	− 4.27114i	− 0.286017i	0.989721	−	0.143008i	$-0.0456776\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 25.2717i	− 1.67734i	−0.544637	−	0.838672i	$-0.683333\pi$
$227$	− 25.2717i	− 1.67734i	0.544637	−	0.838672i	$-0.316667\pi$
$228$	0	0
$229$	10.1588	0.671315	0.335657	−	0.941984i	$-0.391042\pi$
$229$	10.1588	0.671315	0.335657	+	0.941984i	$0.391042\pi$
$230$	0	0
$231$	−0.454731	−0.0299191
$232$	0	0
$233$	10.4383i	0.683838i	0.939729	+	0.341919i	$0.111077\pi$
$233$	10.4383i	0.683838i	−0.939729	+	0.341919i	$0.888923\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 6.28621i	− 0.408333i
$238$	0	0
$239$	−6.25906	−0.404865	−0.202433	−	0.979296i	$-0.564885\pi$
$239$	−6.25906	−0.404865	−0.202433	+	0.979296i	$0.564885\pi$
$240$	0	0
$241$	−2.66487	−0.171660	−0.0858298	−	0.996310i	$-0.527354\pi$
$241$	−2.66487	−0.171660	−0.0858298	+	0.996310i	$0.527354\pi$
$242$	0	0
$243$	15.7952i	1.01326i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.19269i	− 0.139517i
$248$	0	0
$249$	−9.70948	−0.615313
$250$	0	0
$251$	30.5013	1.92522	0.962611	−	0.270887i	$-0.0873171\pi$
$251$	30.5013	1.92522	0.962611	+	0.270887i	$0.0873171\pi$
$252$	0	0
$253$	− 3.13169i	− 0.196887i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.95539i	0.496244i	0.968729	+	0.248122i	$0.0798134\pi$
$257$	7.95539i	0.496244i	−0.968729	+	0.248122i	$0.920187\pi$
$258$	0	0
$259$	9.61058	0.597172
$260$	0	0
$261$	−10.7071	−0.662752
$262$	0	0
$263$	− 0.652793i	− 0.0402529i	−0.999797	−	0.0201265i	$-0.993593\pi$
$263$	− 0.652793i	− 0.0402529i	0.999797	−	0.0201265i	$-0.00640689\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.37867i	0.0843730i
$268$	0	0
$269$	−9.71379	−0.592260	−0.296130	−	0.955148i	$-0.595696\pi$
$269$	−9.71379	−0.592260	−0.296130	+	0.955148i	$0.595696\pi$
$270$	0	0
$271$	11.0610	0.671908	0.335954	−	0.941878i	$-0.390941\pi$
$271$	11.0610	0.671908	0.335954	+	0.941878i	$0.390941\pi$
$272$	0	0
$273$	7.11960i	0.430898i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.0271471i	0.00163111i	1.00000		0.000815555i	$0.000259599\pi$
$277$	0.0271471i	0.00163111i	−1.00000		0.000815555i	$0.999740\pi$
$278$	0	0
$279$	0.466812	0.0279473
$280$	0	0
$281$	20.7735	1.23924	0.619620	−	0.784902i	$-0.287287\pi$
$281$	20.7735	1.23924	0.619620	+	0.784902i	$0.287287\pi$
$282$	0	0
$283$	1.92154i	0.114224i	0.998368	+	0.0571119i	$0.0181892\pi$
$283$	1.92154i	0.114224i	−0.998368	+	0.0571119i	$0.981811\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.58881i	− 0.506981i
$288$	0	0
$289$	5.13169	0.301864
$290$	0	0
$291$	−3.39804	−0.199197
$292$	0	0
$293$	− 16.6950i	− 0.975333i	−0.873030	−	0.487666i	$-0.837848\pi$
$293$	− 16.6950i	− 0.975333i	0.873030	−	0.487666i	$-0.162152\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.43967i	0.0835378i
$298$	0	0
$299$	−49.0320	−2.83560
$300$	0	0
$301$	7.26875	0.418964
$302$	0	0
$303$	− 13.6267i	− 0.782834i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.1414i	0.750018i	0.927021	+	0.375009i	$0.122360\pi$
$307$	13.1414i	0.750018i	−0.927021	+	0.375009i	$0.877640\pi$
$308$	0	0
$309$	6.04461	0.343866
$310$	0	0
$311$	7.87800	0.446721	0.223360	−	0.974736i	$-0.428297\pi$
$311$	7.87800	0.446721	0.223360	+	0.974736i	$0.428297\pi$
$312$	0	0
$313$	− 25.1008i	− 1.41878i	−0.704815	−	0.709391i	$-0.748970\pi$
$313$	− 25.1008i	− 1.41878i	0.704815	−	0.709391i	$-0.251030\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 24.9071i	− 1.39892i	−0.714672	−	0.699460i	$-0.753424\pi$
$317$	− 24.9071i	− 1.39892i	0.714672	−	0.699460i	$-0.246576\pi$
$318$	0	0
$319$	−1.52243	−0.0852400
$320$	0	0
$321$	−9.28621	−0.518306
$322$	0	0
$323$	1.43967i	0.0801051i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 14.7095i	− 0.813436i
$328$	0	0
$329$	14.3056	0.788692
$330$	0	0
$331$	1.42758	0.0784671	0.0392335	−	0.999230i	$-0.487508\pi$
$331$	1.42758	0.0784671	0.0392335	+	0.999230i	$0.487508\pi$
$332$	0	0
$333$	− 13.3870i	− 0.733605i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10.0435i	− 0.547107i	−0.961857	−	0.273553i	$-0.911801\pi$
$337$	− 10.0435i	− 0.547107i	0.961857	−	0.273553i	$-0.0881990\pi$
$338$	0	0
$339$	15.0151	0.815506
$340$	0	0
$341$	0.0663757	0.00359445
$342$	0	0
$343$	18.8442i	1.01749i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.1812i	1.67389i	0.547284	+	0.836947i	$0.315662\pi$
$347$	31.1812i	1.67389i	−0.547284	+	0.836947i	$0.684338\pi$
$348$	0	0
$349$	1.28621	0.0688491	0.0344246	−	0.999407i	$-0.489040\pi$
$349$	1.28621	0.0688491	0.0344246	+	0.999407i	$0.489040\pi$
$350$	0	0
$351$	22.5405	1.20312
$352$	0	0
$353$	− 25.7536i	− 1.37073i	−0.728201	−	0.685363i	$-0.759643\pi$
$353$	− 25.7536i	− 1.37073i	0.728201	−	0.685363i	$-0.240357\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.67456i	− 0.247404i
$358$	0	0
$359$	−11.3991	−0.601622	−0.300811	−	0.953684i	$-0.597257\pi$
$359$	−11.3991	−0.601622	−0.300811	+	0.953684i	$0.597257\pi$
$360$	0	0
$361$	−18.8254	−0.990809
$362$	0	0
$363$	− 8.73125i	− 0.458272i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 12.6431i	− 0.659965i	−0.943987	−	0.329982i	$-0.892957\pi$
$367$	− 12.6431i	− 0.659965i	0.943987	−	0.329982i	$-0.107043\pi$
$368$	0	0
$369$	−11.9638	−0.622808
$370$	0	0
$371$	5.22521	0.271279
$372$	0	0
$373$	13.1715i	0.681995i	0.940064	+	0.340997i	$0.110765\pi$
$373$	13.1715i	0.681995i	−0.940064	+	0.340997i	$0.889235\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.8364i	1.22764i
$378$	0	0
$379$	27.6179	1.41864	0.709318	−	0.704889i	$-0.249003\pi$
$379$	27.6179	1.41864	0.709318	+	0.704889i	$0.249003\pi$
$380$	0	0
$381$	7.77240	0.398192
$382$	0	0
$383$	− 7.27114i	− 0.371538i	−0.982593	−	0.185769i	$-0.940522\pi$
$383$	− 7.27114i	− 0.371538i	0.982593	−	0.185769i	$-0.0594776\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.1250i	− 0.514682i
$388$	0	0
$389$	−10.0248	−0.508275	−0.254138	−	0.967168i	$-0.581792\pi$
$389$	−10.0248	−0.508275	−0.254138	+	0.967168i	$0.581792\pi$
$390$	0	0
$391$	32.1933	1.62808
$392$	0	0
$393$	13.2916i	0.670472i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 0.943313i	− 0.0473435i	−0.999720	−	0.0236718i	$-0.992464\pi$
$397$	− 0.943313i	− 0.0473435i	0.999720	−	0.0236718i	$-0.00753566\pi$
$398$	0	0
$399$	−0.567040	−0.0283875
$400$	0	0
$401$	30.6262	1.52940	0.764701	−	0.644386i	$-0.222887\pi$
$401$	30.6262	1.52940	0.764701	+	0.644386i	$0.222887\pi$
$402$	0	0
$403$	− 1.03923i	− 0.0517677i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.90349i	− 0.0943527i
$408$	0	0
$409$	8.53212	0.421886	0.210943	−	0.977498i	$-0.432347\pi$
$409$	8.53212	0.421886	0.210943	+	0.977498i	$0.432347\pi$
$410$	0	0
$411$	−4.04461	−0.199506
$412$	0	0
$413$	− 2.56465i	− 0.126198i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.9138i	0.583420i
$418$	0	0
$419$	−14.3569	−0.701380	−0.350690	−	0.936492i	$-0.614053\pi$
$419$	−14.3569	−0.701380	−0.350690	+	0.936492i	$0.614053\pi$
$420$	0	0
$421$	−18.3894	−0.896245	−0.448123	−	0.893972i	$-0.647907\pi$
$421$	−18.3894	−0.896245	−0.448123	+	0.893972i	$0.647907\pi$
$422$	0	0
$423$	− 19.9269i	− 0.968880i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.69202i	− 0.0818827i
$428$	0	0
$429$	1.41013	0.0680815
$430$	0	0
$431$	−25.5362	−1.23003	−0.615017	−	0.788514i	$-0.710851\pi$
$431$	−25.5362	−1.23003	−0.615017	+	0.788514i	$0.710851\pi$
$432$	0	0
$433$	− 6.25368i	− 0.300533i	−0.988646	−	0.150266i	$-0.951987\pi$
$433$	− 6.25368i	− 0.300533i	0.988646	−	0.150266i	$-0.0480132\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.90515i	− 0.186809i
$438$	0	0
$439$	13.2107	0.630514	0.315257	−	0.949006i	$-0.397909\pi$
$439$	13.2107	0.630514	0.315257	+	0.949006i	$0.397909\pi$
$440$	0	0
$441$	9.75063	0.464316
$442$	0	0
$443$	1.35152i	0.0642126i	0.999484	+	0.0321063i	$0.0102215\pi$
$443$	1.35152i	0.0642126i	−0.999484	+	0.0321063i	$0.989778\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 9.09783i	− 0.430313i
$448$	0	0
$449$	9.15751	0.432169	0.216085	−	0.976375i	$-0.430671\pi$
$449$	9.15751	0.432169	0.216085	+	0.976375i	$0.430671\pi$
$450$	0	0
$451$	−1.70112	−0.0801026
$452$	0	0
$453$	4.29590i	0.201839i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.93230i	0.277501i	0.990327	+	0.138751i	$0.0443086\pi$
$457$	5.93230i	0.277501i	−0.990327	+	0.138751i	$0.955691\pi$
$458$	0	0
$459$	−14.7995	−0.690784
$460$	0	0
$461$	23.7627	1.10674	0.553370	−	0.832936i	$-0.313342\pi$
$461$	23.7627	1.10674	0.553370	+	0.832936i	$0.313342\pi$
$462$	0	0
$463$	3.60388i	0.167486i	0.996487	+	0.0837431i	$0.0266875\pi$
$463$	3.60388i	0.167486i	−0.996487	+	0.0837431i	$0.973312\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.50604i	0.115966i	0.998318	+	0.0579829i	$0.0184669\pi$
$467$	2.50604i	0.115966i	−0.998318	+	0.0579829i	$0.981533\pi$
$468$	0	0
$469$	15.6625	0.723226
$470$	0	0
$471$	−18.6310	−0.858472
$472$	0	0
$473$	− 1.43967i	− 0.0661959i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 7.27844i	− 0.333257i
$478$	0	0
$479$	−8.09890	−0.370048	−0.185024	−	0.982734i	$-0.559236\pi$
$479$	−8.09890	−0.370048	−0.185024	+	0.982734i	$0.559236\pi$
$480$	0	0
$481$	−29.8025	−1.35888
$482$	0	0
$483$	12.6799i	0.576957i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 43.3193i	− 1.96299i	−0.191498	−	0.981493i	$-0.561335\pi$
$487$	− 43.3193i	− 1.96299i	0.191498	−	0.981493i	$-0.438665\pi$
$488$	0	0
$489$	7.30990	0.330565
$490$	0	0
$491$	−32.0277	−1.44539	−0.722696	−	0.691166i	$-0.757097\pi$
$491$	−32.0277	−1.44539	−0.722696	+	0.691166i	$0.757097\pi$
$492$	0	0
$493$	− 15.6504i	− 0.704859i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.03385i	− 0.180943i
$498$	0	0
$499$	28.5429	1.27775	0.638877	−	0.769309i	$-0.279399\pi$
$499$	28.5429	1.27775	0.638877	+	0.769309i	$0.279399\pi$
$500$	0	0
$501$	8.26981	0.369468
$502$	0	0
$503$	− 30.8552i	− 1.37576i	−0.725823	−	0.687882i	$-0.758541\pi$
$503$	− 30.8552i	− 1.37576i	0.725823	−	0.687882i	$-0.241459\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 11.6528i	− 0.517519i
$508$	0	0
$509$	−3.36360	−0.149089	−0.0745445	−	0.997218i	$-0.523750\pi$
$509$	−3.36360	−0.149089	−0.0745445	+	0.997218i	$0.523750\pi$
$510$	0	0
$511$	10.9420	0.484045
$512$	0	0
$513$	1.79523i	0.0792615i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.83340i	− 0.124613i
$518$	0	0
$519$	−5.51142	−0.241924
$520$	0	0
$521$	44.3933	1.94490	0.972452	−	0.233103i	$-0.0748880\pi$
$521$	44.3933	1.94490	0.972452	+	0.233103i	$0.0748880\pi$
$522$	0	0
$523$	− 8.56571i	− 0.374552i	−0.982307	−	0.187276i	$-0.940034\pi$
$523$	− 8.56571i	− 0.374552i	0.982307	−	0.187276i	$-0.0599659\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.682333i	0.0297229i
$528$	0	0
$529$	−64.3256	−2.79676
$530$	0	0
$531$	−3.57242	−0.155030
$532$	0	0
$533$	26.6340i	1.15365i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.52350i	0.281510i
$538$	0	0
$539$	1.38644	0.0597180
$540$	0	0
$541$	−39.9845	−1.71907	−0.859533	−	0.511080i	$-0.829246\pi$
$541$	−39.9845	−1.71907	−0.859533	+	0.511080i	$0.829246\pi$
$542$	0	0
$543$	− 4.15883i	− 0.178473i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.1118i	1.50127i	0.660715	+	0.750637i	$0.270253\pi$
$547$	35.1118i	1.50127i	−0.660715	+	0.750637i	$0.729747\pi$
$548$	0	0
$549$	−2.35690	−0.100590
$550$	0	0
$551$	−1.89844	−0.0808765
$552$	0	0
$553$	− 13.2634i	− 0.564016i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.5120i	1.08098i	0.841351	+	0.540489i	$0.181761\pi$
$557$	25.5120i	1.08098i	−0.841351	+	0.540489i	$0.818239\pi$
$558$	0	0
$559$	−22.5405	−0.953361
$560$	0	0
$561$	−0.925855	−0.0390896
$562$	0	0
$563$	− 24.9681i	− 1.05228i	−0.850398	−	0.526139i	$-0.823639\pi$
$563$	− 24.9681i	− 1.05228i	0.850398	−	0.526139i	$-0.176361\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.13467i	0.257632i
$568$	0	0
$569$	−1.24027	−0.0519950	−0.0259975	−	0.999662i	$-0.508276\pi$
$569$	−1.24027	−0.0519950	−0.0259975	+	0.999662i	$0.508276\pi$
$570$	0	0
$571$	−9.66487	−0.404462	−0.202231	−	0.979338i	$-0.564819\pi$
$571$	−9.66487	−0.404462	−0.202231	+	0.979338i	$0.564819\pi$
$572$	0	0
$573$	− 3.92500i	− 0.163969i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	40.7525i	1.69655i	0.529555	+	0.848275i	$0.322359\pi$
$577$	40.7525i	1.69655i	−0.529555	+	0.848275i	$0.677641\pi$
$578$	0	0
$579$	11.4450	0.475640
$580$	0	0
$581$	−20.4862	−0.849910
$582$	0	0
$583$	− 1.03492i	− 0.0428619i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.49875i	− 0.0618598i	−0.999522	−	0.0309299i	$-0.990153\pi$
$587$	− 1.49875i	− 0.0618598i	0.999522	−	0.0309299i	$-0.00984687\pi$
$588$	0	0
$589$	0.0827692	0.00341045
$590$	0	0
$591$	−1.10560	−0.0454785
$592$	0	0
$593$	8.66487i	0.355824i	0.984046	+	0.177912i	$0.0569342\pi$
$593$	8.66487i	0.355824i	−0.984046	+	0.177912i	$0.943066\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.92931i	0.406380i
$598$	0	0
$599$	−8.34614	−0.341014	−0.170507	−	0.985356i	$-0.554541\pi$
$599$	−8.34614	−0.341014	−0.170507	+	0.985356i	$0.554541\pi$
$600$	0	0
$601$	−22.3284	−0.910795	−0.455398	−	0.890288i	$-0.650503\pi$
$601$	−22.3284	−0.910795	−0.455398	+	0.890288i	$0.650503\pi$
$602$	0	0
$603$	− 21.8170i	− 0.888457i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 22.4805i	− 0.912457i	−0.889863	−	0.456229i	$-0.849200\pi$
$607$	− 22.4805i	− 0.912457i	0.889863	−	0.456229i	$-0.150800\pi$
$608$	0	0
$609$	6.16421	0.249786
$610$	0	0
$611$	−44.3618	−1.79469
$612$	0	0
$613$	− 40.6601i	− 1.64225i	−0.570752	−	0.821123i	$-0.693348\pi$
$613$	− 40.6601i	− 1.64225i	0.570752	−	0.821123i	$-0.306652\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	47.8146i	1.92494i	0.271381	+	0.962472i	$0.412520\pi$
$617$	47.8146i	1.92494i	−0.271381	+	0.962472i	$0.587480\pi$
$618$	0	0
$619$	24.6853	0.992187	0.496093	−	0.868269i	$-0.334767\pi$
$619$	24.6853	0.992187	0.496093	+	0.868269i	$0.334767\pi$
$620$	0	0
$621$	40.1444	1.61094
$622$	0	0
$623$	2.90887i	0.116541i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.112309i	0.00448520i
$628$	0	0
$629$	19.5676	0.780213
$630$	0	0
$631$	39.5743	1.57543	0.787715	−	0.616040i	$-0.211264\pi$
$631$	39.5743	1.57543	0.787715	+	0.616040i	$0.211264\pi$
$632$	0	0
$633$	15.6722i	0.622913i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 21.7071i	− 0.860066i
$638$	0	0
$639$	−5.61894	−0.222282
$640$	0	0
$641$	−17.1793	−0.678541	−0.339270	−	0.940689i	$-0.610180\pi$
$641$	−17.1793	−0.678541	−0.339270	+	0.940689i	$0.610180\pi$
$642$	0	0
$643$	9.46309i	0.373188i	0.982437	+	0.186594i	$0.0597449\pi$
$643$	9.46309i	0.373188i	−0.982437	+	0.186594i	$0.940255\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.0067i	0.393404i	0.980463	+	0.196702i	$0.0630232\pi$
$647$	10.0067i	0.393404i	−0.980463	+	0.196702i	$0.936977\pi$
$648$	0	0
$649$	−0.507960	−0.0199392
$650$	0	0
$651$	−0.268750	−0.0105331
$652$	0	0
$653$	24.8006i	0.970523i	0.874369	+	0.485261i	$0.161276\pi$
$653$	24.8006i	0.970523i	−0.874369	+	0.485261i	$0.838724\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 15.2416i	− 0.594632i
$658$	0	0
$659$	24.2150	0.943284	0.471642	−	0.881790i	$-0.343661\pi$
$659$	24.2150	0.943284	0.471642	+	0.881790i	$0.343661\pi$
$660$	0	0
$661$	22.4771	0.874258	0.437129	−	0.899399i	$-0.355995\pi$
$661$	22.4771	0.874258	0.437129	+	0.899399i	$0.355995\pi$
$662$	0	0
$663$	14.4959i	0.562973i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	42.4523i	1.64376i
$668$	0	0
$669$	3.42519	0.132426
$670$	0	0
$671$	−0.335126	−0.0129374
$672$	0	0
$673$	− 35.7284i	− 1.37723i	−0.725128	−	0.688614i	$-0.758219\pi$
$673$	− 35.7284i	− 1.37723i	0.725128	−	0.688614i	$-0.241781\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.1903i	1.54464i	0.635235	+	0.772319i	$0.280903\pi$
$677$	40.1903i	1.54464i	−0.635235	+	0.772319i	$0.719097\pi$
$678$	0	0
$679$	−7.16959	−0.275144
$680$	0	0
$681$	20.2664	0.776608
$682$	0	0
$683$	2.97823i	0.113959i	0.998375	+	0.0569794i	$0.0181469\pi$
$683$	2.97823i	0.113959i	−0.998375	+	0.0569794i	$0.981853\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.14675i	0.310818i
$688$	0	0
$689$	−16.2034	−0.617302
$690$	0	0
$691$	−45.0532	−1.71391	−0.856953	−	0.515395i	$-0.827645\pi$
$691$	−45.0532	−1.71391	−0.856953	+	0.515395i	$0.827645\pi$
$692$	0	0
$693$	1.33645i	0.0507677i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 17.4873i	− 0.662377i
$698$	0	0
$699$	−8.37090	−0.316616
$700$	0	0
$701$	34.0881	1.28749	0.643746	−	0.765239i	$-0.277379\pi$
$701$	34.0881	1.28749	0.643746	+	0.765239i	$0.277379\pi$
$702$	0	0
$703$	− 2.37362i	− 0.0895227i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 28.7512i	− 1.08130i
$708$	0	0
$709$	43.6383	1.63887	0.819436	−	0.573171i	$-0.194287\pi$
$709$	43.6383	1.63887	0.819436	+	0.573171i	$0.194287\pi$
$710$	0	0
$711$	−18.4752	−0.692873
$712$	0	0
$713$	− 1.85086i	− 0.0693151i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 5.01938i	− 0.187452i
$718$	0	0
$719$	43.6515	1.62792	0.813962	−	0.580917i	$-0.197306\pi$
$719$	43.6515	1.62792	0.813962	+	0.580917i	$0.197306\pi$
$720$	0	0
$721$	12.7536	0.474969
$722$	0	0
$723$	− 2.13706i	− 0.0794782i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.83399i	0.327634i	0.986491	+	0.163817i	$0.0523807\pi$
$727$	8.83399i	0.327634i	−0.986491	+	0.163817i	$0.947619\pi$
$728$	0	0
$729$	−1.78986	−0.0662910
$730$	0	0
$731$	14.7995	0.547381
$732$	0	0
$733$	31.0592i	1.14720i	0.819136	+	0.573599i	$0.194453\pi$
$733$	31.0592i	1.14720i	−0.819136	+	0.573599i	$0.805547\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.10215i	− 0.114269i
$738$	0	0
$739$	12.0597	0.443622	0.221811	−	0.975090i	$-0.428803\pi$
$739$	12.0597	0.443622	0.221811	+	0.975090i	$0.428803\pi$
$740$	0	0
$741$	1.75840	0.0645963
$742$	0	0
$743$	41.6883i	1.52940i	0.644389	+	0.764698i	$0.277112\pi$
$743$	41.6883i	1.52940i	−0.644389	+	0.764698i	$0.722888\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	28.5362i	1.04408i
$748$	0	0
$749$	−19.5931	−0.715917
$750$	0	0
$751$	−26.1269	−0.953384	−0.476692	−	0.879070i	$-0.658164\pi$
$751$	−26.1269	−0.953384	−0.476692	+	0.879070i	$0.658164\pi$
$752$	0	0
$753$	24.4601i	0.891376i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	46.3763i	1.68557i	0.538247	+	0.842787i	$0.319087\pi$
$757$	46.3763i	1.68557i	−0.538247	+	0.842787i	$0.680913\pi$
$758$	0	0
$759$	2.51142	0.0911587
$760$	0	0
$761$	21.6039	0.783140	0.391570	−	0.920148i	$-0.371932\pi$
$761$	21.6039	0.783140	0.391570	+	0.920148i	$0.371932\pi$
$762$	0	0
$763$	− 31.0358i	− 1.12357i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.95300i	0.287166i
$768$	0	0
$769$	−29.7399	−1.07245	−0.536224	−	0.844076i	$-0.680150\pi$
$769$	−29.7399	−1.07245	−0.536224	+	0.844076i	$0.680150\pi$
$770$	0	0
$771$	−6.37973	−0.229760
$772$	0	0
$773$	5.28621i	0.190132i	0.995471	+	0.0950658i	$0.0303062\pi$
$773$	5.28621i	0.190132i	−0.995471	+	0.0950658i	$0.969694\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.70709i	0.276490i
$778$	0	0
$779$	−2.12126	−0.0760021
$780$	0	0
$781$	−0.798954	−0.0285888
$782$	0	0
$783$	− 19.5157i	− 0.697435i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 17.8418i	− 0.635990i	−0.948092	−	0.317995i	$-0.896990\pi$
$787$	− 17.8418i	− 0.635990i	0.948092	−	0.317995i	$-0.103010\pi$
$788$	0	0
$789$	0.523499	0.0186371
$790$	0	0
$791$	31.6805	1.12643
$792$	0	0
$793$	5.24698i	0.186326i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.26145i	0.150948i	0.997148	+	0.0754742i	$0.0240471\pi$
$797$	4.26145i	0.150948i	−0.997148	+	0.0754742i	$0.975953\pi$
$798$	0	0
$799$	29.1269	1.03044
$800$	0	0
$801$	4.05190	0.143167
$802$	0	0
$803$	− 2.16719i	− 0.0764786i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 7.78986i	− 0.274216i
$808$	0	0
$809$	−32.3159	−1.13617	−0.568083	−	0.822972i	$-0.692315\pi$
$809$	−32.3159	−1.13617	−0.568083	+	0.822972i	$0.692315\pi$
$810$	0	0
$811$	−17.2459	−0.605586	−0.302793	−	0.953056i	$-0.597919\pi$
$811$	−17.2459	−0.605586	−0.302793	+	0.953056i	$0.597919\pi$
$812$	0	0
$813$	8.87023i	0.311093i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.79523i	− 0.0628073i
$818$	0	0
$819$	20.9245	0.731162
$820$	0	0
$821$	18.6116	0.649551	0.324775	−	0.945791i	$-0.394711\pi$
$821$	18.6116	0.649551	0.324775	+	0.945791i	$0.394711\pi$
$822$	0	0
$823$	− 30.5133i	− 1.06363i	−0.846861	−	0.531814i	$-0.821511\pi$
$823$	− 30.5133i	− 1.06363i	0.846861	−	0.531814i	$-0.178489\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.35749i	− 0.290618i	−0.989386	−	0.145309i	$-0.953582\pi$
$827$	− 8.35749i	− 0.290618i	0.989386	−	0.145309i	$-0.0464177\pi$
$828$	0	0
$829$	53.3169	1.85177	0.925887	−	0.377801i	$-0.123320\pi$
$829$	53.3169	1.85177	0.925887	+	0.377801i	$0.123320\pi$
$830$	0	0
$831$	−0.0217703	−0.000755202 0
$832$	0	0
$833$	14.2524i	0.493815i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.850855i	0.0294099i
$838$	0	0
$839$	5.61489	0.193848	0.0969238	−	0.995292i	$-0.469100\pi$
$839$	5.61489	0.193848	0.0969238	+	0.995292i	$0.469100\pi$
$840$	0	0
$841$	−8.36227	−0.288354
$842$	0	0
$843$	16.6590i	0.573767i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 18.4222i	− 0.632995i
$848$	0	0
$849$	−1.54096	−0.0528855
$850$	0	0
$851$	−53.0780	−1.81949
$852$	0	0
$853$	25.9148i	0.887307i	0.896198	+	0.443654i	$0.146318\pi$
$853$	25.9148i	0.887307i	−0.896198	+	0.443654i	$0.853682\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.9685i	1.12618i	0.826394	+	0.563092i	$0.190388\pi$
$857$	32.9685i	1.12618i	−0.826394	+	0.563092i	$0.809612\pi$
$858$	0	0
$859$	40.6752	1.38782	0.693909	−	0.720063i	$-0.255887\pi$
$859$	40.6752	1.38782	0.693909	+	0.720063i	$0.255887\pi$
$860$	0	0
$861$	6.88769	0.234732
$862$	0	0
$863$	40.1769i	1.36764i	0.729652	+	0.683819i	$0.239682\pi$
$863$	40.1769i	1.36764i	−0.729652	+	0.683819i	$0.760318\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.11529i	0.139763i
$868$	0	0
$869$	−2.62697	−0.0891140
$870$	0	0
$871$	−48.5695	−1.64572
$872$	0	0
$873$	9.98685i	0.338004i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.3435i	0.484345i	0.970233	+	0.242173i	$0.0778600\pi$
$877$	14.3435i	0.484345i	−0.970233	+	0.242173i	$0.922140\pi$
$878$	0	0
$879$	13.3884	0.451578
$880$	0	0
$881$	−2.44696	−0.0824402	−0.0412201	−	0.999150i	$-0.513124\pi$
$881$	−2.44696	−0.0824402	−0.0412201	+	0.999150i	$0.513124\pi$
$882$	0	0
$883$	− 7.85922i	− 0.264484i	−0.991217	−	0.132242i	$-0.957782\pi$
$883$	− 7.85922i	− 0.264484i	0.991217	−	0.132242i	$-0.0422175\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 10.6256i	− 0.356774i	−0.983960	−	0.178387i	$-0.942912\pi$
$887$	− 10.6256i	− 0.356774i	0.983960	−	0.178387i	$-0.0570879\pi$
$888$	0	0
$889$	16.3991	0.550008
$890$	0	0
$891$	1.21505	0.0407056
$892$	0	0
$893$	− 3.53319i	− 0.118234i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 39.3207i	− 1.31288i
$898$	0	0
$899$	−0.899772	−0.0300091
$900$	0	0
$901$	10.6388	0.354430
$902$	0	0
$903$	5.82908i	0.193980i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 49.9332i	− 1.65800i	−0.559246	−	0.829002i	$-0.688909\pi$
$907$	− 49.9332i	− 1.65800i	0.559246	−	0.829002i	$-0.311091\pi$
$908$	0	0
$909$	−40.0489	−1.32834
$910$	0	0
$911$	43.6424	1.44594	0.722968	−	0.690881i	$-0.242777\pi$
$911$	43.6424	1.44594	0.722968	+	0.690881i	$0.242777\pi$
$912$	0	0
$913$	4.05754i	0.134285i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	28.0441i	0.926099i
$918$	0	0
$919$	9.36168	0.308813	0.154407	−	0.988007i	$-0.450653\pi$
$919$	9.36168	0.308813	0.154407	+	0.988007i	$0.450653\pi$
$920$	0	0
$921$	−10.5386	−0.347258
$922$	0	0
$923$	12.5090i	0.411740i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 17.7651i	− 0.583483i
$928$	0	0
$929$	−7.23968	−0.237526	−0.118763	−	0.992923i	$-0.537893\pi$
$929$	−7.23968	−0.237526	−0.118763	+	0.992923i	$0.537893\pi$
$930$	0	0
$931$	1.72886	0.0566610
$932$	0	0
$933$	6.31767i	0.206831i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 55.1861i	− 1.80285i	−0.432934	−	0.901426i	$-0.642522\pi$
$937$	− 55.1861i	− 1.80285i	0.432934	−	0.901426i	$-0.357478\pi$
$938$	0	0
$939$	20.1293	0.656895
$940$	0	0
$941$	−8.44994	−0.275460	−0.137730	−	0.990470i	$-0.543981\pi$
$941$	−8.44994	−0.275460	−0.137730	+	0.990470i	$0.543981\pi$
$942$	0	0
$943$	47.4349i	1.54469i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 14.6907i	− 0.477383i	−0.971095	−	0.238692i	$-0.923281\pi$
$947$	− 14.6907i	− 0.477383i	0.971095	−	0.238692i	$-0.0767185\pi$
$948$	0	0
$949$	−33.9312	−1.10145
$950$	0	0
$951$	19.9739	0.647699
$952$	0	0
$953$	3.29099i	0.106606i	0.998578	+	0.0533029i	$0.0169749\pi$
$953$	3.29099i	0.106606i	−0.998578	+	0.0533029i	$0.983025\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.22090i	− 0.0394660i
$958$	0	0
$959$	−8.53378	−0.275570
$960$	0	0
$961$	−30.9608	−0.998735
$962$	0	0
$963$	27.2922i	0.879478i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 56.7985i	− 1.82652i	−0.407382	−	0.913258i	$-0.633558\pi$
$967$	− 56.7985i	− 1.82652i	0.407382	−	0.913258i	$-0.366442\pi$
$968$	0	0
$969$	−1.15452	−0.0370886
$970$	0	0
$971$	−20.0140	−0.642280	−0.321140	−	0.947032i	$-0.604066\pi$
$971$	−20.0140	−0.642280	−0.321140	+	0.947032i	$0.604066\pi$
$972$	0	0
$973$	25.1371i	0.805857i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 31.0847i	− 0.994487i	−0.867611	−	0.497244i	$-0.834346\pi$
$977$	− 31.0847i	− 0.994487i	0.867611	−	0.497244i	$-0.165654\pi$
$978$	0	0
$979$	0.576137	0.0184134
$980$	0	0
$981$	−43.2312	−1.38027
$982$	0	0
$983$	38.4752i	1.22717i	0.789630	+	0.613584i	$0.210273\pi$
$983$	38.4752i	1.22717i	−0.789630	+	0.613584i	$0.789727\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11.4722i	0.365164i
$988$	0	0
$989$	−40.1444	−1.27652
$990$	0	0
$991$	18.7192	0.594634	0.297317	−	0.954779i	$-0.403908\pi$
$991$	18.7192	0.594634	0.297317	+	0.954779i	$0.403908\pi$
$992$	0	0
$993$	1.14483i	0.0363302i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 30.8495i	− 0.977014i	−0.872560	−	0.488507i	$-0.837542\pi$
$997$	− 30.8495i	− 0.977014i	0.872560	−	0.488507i	$-0.162458\pi$
$998$	0	0
$999$	24.4004	0.771996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6100.2.c.f.4149.5		6
5.2	odd	4		1220.2.a.c.1.3	✓	3
5.3	odd	4		6100.2.a.h.1.1		3
5.4	even	2	inner	6100.2.c.f.4149.2		6
20.7	even	4		4880.2.a.q.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1220.2.a.c.1.3	✓	3	5.2	odd	4
4880.2.a.q.1.1		3	20.7	even	4
6100.2.a.h.1.1		3	5.3	odd	4
6100.2.c.f.4149.2		6	5.4	even	2	inner
6100.2.c.f.4149.5		6	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:	\( N \)	\(=\)	\( 6100 = 2^{2} \cdot 5^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6100.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.801938i q^{3} +1.69202i q^{7} +2.35690 q^{9} +O(q^{10})\)	\(q+0.801938i q^{3} +1.69202i q^{7} +2.35690 q^{9} +0.335126 q^{11} -5.24698i q^{13} +3.44504i q^{17} +0.417895 q^{19} -1.35690 q^{21} -9.34481i q^{23} +4.29590i q^{27} -4.54288 q^{29} +0.198062 q^{31} +0.268750i q^{33} -5.67994i q^{37} +4.20775 q^{39} -5.07606 q^{41} -4.29590i q^{43} -8.45473i q^{47} +4.13706 q^{49} -2.76271 q^{51} -3.08815i q^{53} +0.335126i q^{57} -1.51573 q^{59} -1.00000 q^{61} +3.98792i q^{63} -9.25667i q^{67} +7.49396 q^{69} -2.38404 q^{71} -6.46681i q^{73} +0.567040i q^{77} -7.83877 q^{79} +3.62565 q^{81} +12.1075i q^{83} -3.64310i q^{87} +1.71917 q^{89} +8.87800 q^{91} +0.158834i q^{93} +4.23729i q^{97} +0.789856 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^9	\( 6 q + 6 q^{9} + 14 q^{19} + 10 q^{29} + 10 q^{31} - 10 q^{39} + 14 q^{49} + 18 q^{51} + 16 q^{59} - 6 q^{61} + 26 q^{69} + 6 q^{71} + 18 q^{79} - 2 q^{81} - 12 q^{89} + 14 q^{91} - 42 q^{99}+O(q^{100}) \) 6 * q + 6 * q^9 + 14 * q^19 + 10 * q^29 + 10 * q^31 - 10 * q^39 + 14 * q^49 + 18 * q^51 + 16 * q^59 - 6 * q^61 + 26 * q^69 + 6 * q^71 + 18 * q^79 - 2 * q^81 - 12 * q^89 + 14 * q^91 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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