LMFDB - Embedded newform 6000.2.f.a.1249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6000,2,Mod(1249,6000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6000 = 2^{4} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6000.f (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:	$47.9102412128$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 750)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	6000.1249
Dual form			6000.2.f.a.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.23607i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.23607i q^{7} -1.00000 q^{9} -1.38197 q^{11} +2.85410i q^{13} +4.85410i q^{17} +6.00000 q^{19} -1.23607 q^{21} -3.09017i q^{23} +1.00000i q^{27} -9.32624 q^{29} -2.14590 q^{31} +1.38197i q^{33} -7.85410i q^{37} +2.85410 q^{39} -3.23607 q^{41} -0.145898i q^{43} -10.8541i q^{47} +5.47214 q^{49} +4.85410 q^{51} -3.23607i q^{53} -6.00000i q^{57} +6.38197 q^{59} +13.4164 q^{61} +1.23607i q^{63} +2.38197i q^{67} -3.09017 q^{69} +8.94427 q^{71} +12.0000i q^{73} +1.70820i q^{77} -1.14590 q^{79} +1.00000 q^{81} -8.18034i q^{83} +9.32624i q^{87} +9.23607 q^{89} +3.52786 q^{91} +2.14590i q^{93} -18.1803i q^{97} +1.38197 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 10 q^{11} + 24 q^{19} + 4 q^{21} - 6 q^{29} - 22 q^{31} - 2 q^{39} - 4 q^{41} + 4 q^{49} + 6 q^{51} + 30 q^{59} + 10 q^{69} - 18 q^{79} + 4 q^{81} + 28 q^{89} + 32 q^{91} + 10 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 10 * q^11 + 24 * q^19 + 4 * q^21 - 6 * q^29 - 22 * q^31 - 2 * q^39 - 4 * q^41 + 4 * q^49 + 6 * q^51 + 30 * q^59 + 10 * q^69 - 18 * q^79 + 4 * q^81 + 28 * q^89 + 32 * q^91 + 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6000\mathbb{Z}\right)^\times$.

$n$	$751$	$4001$	$4501$	$5377$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.23607i	− 0.467190i	−0.972334	−	0.233595i	$-0.924951\pi$
$7$	− 1.23607i	− 0.467190i	0.972334	−	0.233595i	$-0.0750489\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.38197	−0.416678	−0.208339	−	0.978057i	$-0.566806\pi$
$11$	−1.38197	−0.416678	−0.208339	+	0.978057i	$0.566806\pi$
$12$	0	0
$13$	2.85410i	0.791585i	0.918340	+	0.395793i	$0.129530\pi$
$13$	2.85410i	0.791585i	−0.918340	+	0.395793i	$0.870470\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.85410i	1.17729i	0.808391	+	0.588646i	$0.200339\pi$
$17$	4.85410i	1.17729i	−0.808391	+	0.588646i	$0.799661\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−1.23607	−0.269732
$22$	0	0
$23$	− 3.09017i	− 0.644345i	−0.946681	−	0.322172i	$-0.895587\pi$
$23$	− 3.09017i	− 0.644345i	0.946681	−	0.322172i	$-0.104413\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−9.32624	−1.73184	−0.865919	−	0.500183i	$-0.833266\pi$
$29$	−9.32624	−1.73184	−0.865919	+	0.500183i	$0.833266\pi$
$30$	0	0
$31$	−2.14590	−0.385415	−0.192707	−	0.981256i	$-0.561727\pi$
$31$	−2.14590	−0.385415	−0.192707	+	0.981256i	$0.561727\pi$
$32$	0	0
$33$	1.38197i	0.240569i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.85410i	− 1.29121i	−0.763673	−	0.645603i	$-0.776606\pi$
$37$	− 7.85410i	− 1.29121i	0.763673	−	0.645603i	$-0.223394\pi$
$38$	0	0
$39$	2.85410	0.457022
$40$	0	0
$41$	−3.23607	−0.505389	−0.252694	−	0.967546i	$-0.581317\pi$
$41$	−3.23607	−0.505389	−0.252694	+	0.967546i	$0.581317\pi$
$42$	0	0
$43$	− 0.145898i	− 0.0222492i	−0.999938	−	0.0111246i	$-0.996459\pi$
$43$	− 0.145898i	− 0.0222492i	0.999938	−	0.0111246i	$-0.00354115\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.8541i	− 1.58323i	−0.611018	−	0.791617i	$-0.709240\pi$
$47$	− 10.8541i	− 1.58323i	0.611018	−	0.791617i	$-0.290760\pi$
$48$	0	0
$49$	5.47214	0.781734
$50$	0	0
$51$	4.85410	0.679710
$52$	0	0
$53$	− 3.23607i	− 0.444508i	−0.974989	−	0.222254i	$-0.928659\pi$
$53$	− 3.23607i	− 0.444508i	0.974989	−	0.222254i	$-0.0713414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.00000i	− 0.794719i
$58$	0	0
$59$	6.38197	0.830861	0.415431	−	0.909625i	$-0.363631\pi$
$59$	6.38197	0.830861	0.415431	+	0.909625i	$0.363631\pi$
$60$	0	0
$61$	13.4164	1.71780	0.858898	−	0.512148i	$-0.171150\pi$
$61$	13.4164	1.71780	0.858898	+	0.512148i	$0.171150\pi$
$62$	0	0
$63$	1.23607i	0.155730i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.38197i	0.291003i	0.989358	+	0.145502i	$0.0464796\pi$
$67$	2.38197i	0.291003i	−0.989358	+	0.145502i	$0.953520\pi$
$68$	0	0
$69$	−3.09017	−0.372013
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	0	0
$73$	12.0000i	1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$	12.0000i	1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.70820i	0.194668i
$78$	0	0
$79$	−1.14590	−0.128924	−0.0644618	−	0.997920i	$-0.520533\pi$
$79$	−1.14590	−0.128924	−0.0644618	+	0.997920i	$0.520533\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 8.18034i	− 0.897909i	−0.893555	−	0.448954i	$-0.851797\pi$
$83$	− 8.18034i	− 0.897909i	0.893555	−	0.448954i	$-0.148203\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.32624i	0.999878i
$88$	0	0
$89$	9.23607	0.979021	0.489511	−	0.871997i	$-0.337175\pi$
$89$	9.23607	0.979021	0.489511	+	0.871997i	$0.337175\pi$
$90$	0	0
$91$	3.52786	0.369821
$92$	0	0
$93$	2.14590i	0.222519i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 18.1803i	− 1.84593i	−0.384879	−	0.922967i	$-0.625757\pi$
$97$	− 18.1803i	− 1.84593i	0.384879	−	0.922967i	$-0.374243\pi$
$98$	0	0
$99$	1.38197	0.138893
$100$	0	0
$101$	−2.32624	−0.231469	−0.115735	−	0.993280i	$-0.536922\pi$
$101$	−2.32624	−0.231469	−0.115735	+	0.993280i	$0.536922\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.18034i	− 0.790823i	−0.918504	−	0.395412i	$-0.870602\pi$
$107$	− 8.18034i	− 0.790823i	0.918504	−	0.395412i	$-0.129398\pi$
$108$	0	0
$109$	13.4164	1.28506	0.642529	−	0.766261i	$-0.277885\pi$
$109$	13.4164	1.28506	0.642529	+	0.766261i	$0.277885\pi$
$110$	0	0
$111$	−7.85410	−0.745478
$112$	0	0
$113$	12.3262i	1.15955i	0.814775	+	0.579777i	$0.196861\pi$
$113$	12.3262i	1.15955i	−0.814775	+	0.579777i	$0.803139\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.85410i	− 0.263862i
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	−9.09017	−0.826379
$122$	0	0
$123$	3.23607i	0.291786i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.4164i	− 1.19051i	−0.803535	−	0.595257i	$-0.797050\pi$
$127$	− 13.4164i	− 1.19051i	0.803535	−	0.595257i	$-0.202950\pi$
$128$	0	0
$129$	−0.145898	−0.0128456
$130$	0	0
$131$	−17.8885	−1.56293	−0.781465	−	0.623949i	$-0.785527\pi$
$131$	−17.8885	−1.56293	−0.781465	+	0.623949i	$0.785527\pi$
$132$	0	0
$133$	− 7.41641i	− 0.643084i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.61803i	− 0.394545i	−0.980349	−	0.197273i	$-0.936792\pi$
$137$	− 4.61803i	− 0.394545i	0.980349	−	0.197273i	$-0.0632084\pi$
$138$	0	0
$139$	−21.2361	−1.80122	−0.900610	−	0.434628i	$-0.856880\pi$
$139$	−21.2361	−1.80122	−0.900610	+	0.434628i	$0.856880\pi$
$140$	0	0
$141$	−10.8541	−0.914080
$142$	0	0
$143$	− 3.94427i	− 0.329837i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 5.47214i	− 0.451334i
$148$	0	0
$149$	20.4721	1.67714	0.838571	−	0.544792i	$-0.183391\pi$
$149$	20.4721	1.67714	0.838571	+	0.544792i	$0.183391\pi$
$150$	0	0
$151$	17.8541	1.45295	0.726473	−	0.687195i	$-0.241158\pi$
$151$	17.8541	1.45295	0.726473	+	0.687195i	$0.241158\pi$
$152$	0	0
$153$	− 4.85410i	− 0.392431i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.437694i	0.0349318i	0.999847	+	0.0174659i	$0.00555985\pi$
$157$	0.437694i	0.0349318i	−0.999847	+	0.0174659i	$0.994440\pi$
$158$	0	0
$159$	−3.23607	−0.256637
$160$	0	0
$161$	−3.81966	−0.301031
$162$	0	0
$163$	− 20.2705i	− 1.58771i	−0.608108	−	0.793854i	$-0.708071\pi$
$163$	− 20.2705i	− 1.58771i	0.608108	−	0.793854i	$-0.291929\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 23.0902i	− 1.78677i	−0.449291	−	0.893385i	$-0.648323\pi$
$167$	− 23.0902i	− 1.78677i	0.449291	−	0.893385i	$-0.351677\pi$
$168$	0	0
$169$	4.85410	0.373392
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	− 9.23607i	− 0.702205i	−0.936337	−	0.351103i	$-0.885807\pi$
$173$	− 9.23607i	− 0.702205i	0.936337	−	0.351103i	$-0.114193\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.38197i	− 0.479698i
$178$	0	0
$179$	7.41641	0.554328	0.277164	−	0.960823i	$-0.410605\pi$
$179$	7.41641	0.554328	0.277164	+	0.960823i	$0.410605\pi$
$180$	0	0
$181$	7.41641	0.551257	0.275629	−	0.961264i	$-0.411114\pi$
$181$	7.41641	0.551257	0.275629	+	0.961264i	$0.411114\pi$
$182$	0	0
$183$	− 13.4164i	− 0.991769i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 6.70820i	− 0.490552i
$188$	0	0
$189$	1.23607	0.0899107
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	− 17.1246i	− 1.23266i	−0.787489	−	0.616328i	$-0.788619\pi$
$193$	− 17.1246i	− 1.23266i	0.787489	−	0.616328i	$-0.211381\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.52786i	0.108856i	0.998518	+	0.0544279i	$0.0173335\pi$
$197$	1.52786i	0.108856i	−0.998518	+	0.0544279i	$0.982666\pi$
$198$	0	0
$199$	−15.6180	−1.10713	−0.553567	−	0.832805i	$-0.686734\pi$
$199$	−15.6180	−1.10713	−0.553567	+	0.832805i	$0.686734\pi$
$200$	0	0
$201$	2.38197	0.168011
$202$	0	0
$203$	11.5279i	0.809097i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.09017i	0.214782i
$208$	0	0
$209$	−8.29180	−0.573556
$210$	0	0
$211$	−3.41641	−0.235195	−0.117598	−	0.993061i	$-0.537519\pi$
$211$	−3.41641	−0.235195	−0.117598	+	0.993061i	$0.537519\pi$
$212$	0	0
$213$	− 8.94427i	− 0.612851i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.65248i	0.180062i
$218$	0	0
$219$	12.0000	0.810885
$220$	0	0
$221$	−13.8541	−0.931928
$222$	0	0
$223$	21.7082i	1.45369i	0.686802	+	0.726844i	$0.259014\pi$
$223$	21.7082i	1.45369i	−0.686802	+	0.726844i	$0.740986\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.4721i	− 1.09329i	−0.837363	−	0.546647i	$-0.815904\pi$
$227$	− 16.4721i	− 1.09329i	0.837363	−	0.546647i	$-0.184096\pi$
$228$	0	0
$229$	1.70820	0.112881	0.0564406	−	0.998406i	$-0.482025\pi$
$229$	1.70820	0.112881	0.0564406	+	0.998406i	$0.482025\pi$
$230$	0	0
$231$	1.70820	0.112392
$232$	0	0
$233$	7.32624i	0.479958i	0.970778	+	0.239979i	$0.0771405\pi$
$233$	7.32624i	0.479958i	−0.970778	+	0.239979i	$0.922859\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.14590i	0.0744341i
$238$	0	0
$239$	−26.6525	−1.72401	−0.862003	−	0.506904i	$-0.830790\pi$
$239$	−26.6525	−1.72401	−0.862003	+	0.506904i	$0.830790\pi$
$240$	0	0
$241$	0.0901699	0.00580836	0.00290418	−	0.999996i	$-0.499076\pi$
$241$	0.0901699	0.00580836	0.00290418	+	0.999996i	$0.499076\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	17.1246i	1.08961i
$248$	0	0
$249$	−8.18034	−0.518408
$250$	0	0
$251$	−7.90983	−0.499264	−0.249632	−	0.968341i	$-0.580310\pi$
$251$	−7.90983	−0.499264	−0.249632	+	0.968341i	$0.580310\pi$
$252$	0	0
$253$	4.27051i	0.268485i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 23.5066i	− 1.46630i	−0.680067	−	0.733150i	$-0.738049\pi$
$257$	− 23.5066i	− 1.46630i	0.680067	−	0.733150i	$-0.261951\pi$
$258$	0	0
$259$	−9.70820	−0.603238
$260$	0	0
$261$	9.32624	0.577280
$262$	0	0
$263$	15.9787i	0.985290i	0.870230	+	0.492645i	$0.163970\pi$
$263$	15.9787i	0.985290i	−0.870230	+	0.492645i	$0.836030\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 9.23607i	− 0.565238i
$268$	0	0
$269$	−22.6180	−1.37905	−0.689523	−	0.724264i	$-0.742180\pi$
$269$	−22.6180	−1.37905	−0.689523	+	0.724264i	$0.742180\pi$
$270$	0	0
$271$	8.56231	0.520123	0.260062	−	0.965592i	$-0.416257\pi$
$271$	8.56231	0.520123	0.260062	+	0.965592i	$0.416257\pi$
$272$	0	0
$273$	− 3.52786i	− 0.213516i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 14.3607i	− 0.862850i	−0.902149	−	0.431425i	$-0.858011\pi$
$277$	− 14.3607i	− 0.862850i	0.902149	−	0.431425i	$-0.141989\pi$
$278$	0	0
$279$	2.14590	0.128472
$280$	0	0
$281$	22.4721	1.34058	0.670288	−	0.742101i	$-0.266171\pi$
$281$	22.4721	1.34058	0.670288	+	0.742101i	$0.266171\pi$
$282$	0	0
$283$	21.0902i	1.25368i	0.779148	+	0.626840i	$0.215652\pi$
$283$	21.0902i	1.25368i	−0.779148	+	0.626840i	$0.784348\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00000i	0.236113i
$288$	0	0
$289$	−6.56231	−0.386018
$290$	0	0
$291$	−18.1803	−1.06575
$292$	0	0
$293$	− 15.5967i	− 0.911172i	−0.890192	−	0.455586i	$-0.849430\pi$
$293$	− 15.5967i	− 0.911172i	0.890192	−	0.455586i	$-0.150570\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.38197i	− 0.0801898i
$298$	0	0
$299$	8.81966	0.510054
$300$	0	0
$301$	−0.180340	−0.0103946
$302$	0	0
$303$	2.32624i	0.133639i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 33.0902i	− 1.88856i	−0.329149	−	0.944278i	$-0.606762\pi$
$307$	− 33.0902i	− 1.88856i	0.329149	−	0.944278i	$-0.393238\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	4.47214	0.253592	0.126796	−	0.991929i	$-0.459531\pi$
$311$	4.47214	0.253592	0.126796	+	0.991929i	$0.459531\pi$
$312$	0	0
$313$	30.9443i	1.74907i	0.484959	+	0.874537i	$0.338834\pi$
$313$	30.9443i	1.74907i	−0.484959	+	0.874537i	$0.661166\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.70820i	− 0.545267i	−0.962118	−	0.272634i	$-0.912105\pi$
$317$	− 9.70820i	− 0.545267i	0.962118	−	0.272634i	$-0.0878947\pi$
$318$	0	0
$319$	12.8885	0.721620
$320$	0	0
$321$	−8.18034	−0.456582
$322$	0	0
$323$	29.1246i	1.62054i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 13.4164i	− 0.741929i
$328$	0	0
$329$	−13.4164	−0.739671
$330$	0	0
$331$	21.4164	1.17715	0.588576	−	0.808442i	$-0.299689\pi$
$331$	21.4164	1.17715	0.588576	+	0.808442i	$0.299689\pi$
$332$	0	0
$333$	7.85410i	0.430402i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.9443i	0.705119i	0.935789	+	0.352560i	$0.114689\pi$
$337$	12.9443i	0.705119i	−0.935789	+	0.352560i	$0.885311\pi$
$338$	0	0
$339$	12.3262	0.669469
$340$	0	0
$341$	2.96556	0.160594
$342$	0	0
$343$	− 15.4164i	− 0.832408i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.3607i	1.30775i	0.756603	+	0.653875i	$0.226858\pi$
$347$	24.3607i	1.30775i	−0.756603	+	0.653875i	$0.773142\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	−2.85410	−0.152341
$352$	0	0
$353$	1.20163i	0.0639561i	0.999489	+	0.0319781i	$0.0101807\pi$
$353$	1.20163i	0.0639561i	−0.999489	+	0.0319781i	$0.989819\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.00000i	− 0.317554i
$358$	0	0
$359$	23.8885	1.26079	0.630395	−	0.776275i	$-0.282893\pi$
$359$	23.8885	1.26079	0.630395	+	0.776275i	$0.282893\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	9.09017i	0.477110i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.2918i	0.537227i	0.963248	+	0.268614i	$0.0865655\pi$
$367$	10.2918i	0.537227i	−0.963248	+	0.268614i	$0.913434\pi$
$368$	0	0
$369$	3.23607	0.168463
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	− 32.7426i	− 1.69535i	−0.530516	−	0.847675i	$-0.678002\pi$
$373$	− 32.7426i	− 1.69535i	0.530516	−	0.847675i	$-0.321998\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 26.6180i	− 1.37090i
$378$	0	0
$379$	−1.23607	−0.0634925	−0.0317463	−	0.999496i	$-0.510107\pi$
$379$	−1.23607	−0.0634925	−0.0317463	+	0.999496i	$0.510107\pi$
$380$	0	0
$381$	−13.4164	−0.687343
$382$	0	0
$383$	− 17.8885i	− 0.914062i	−0.889451	−	0.457031i	$-0.848913\pi$
$383$	− 17.8885i	− 0.914062i	0.889451	−	0.457031i	$-0.151087\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.145898i	0.00741641i
$388$	0	0
$389$	−18.2705	−0.926352	−0.463176	−	0.886266i	$-0.653290\pi$
$389$	−18.2705	−0.926352	−0.463176	+	0.886266i	$0.653290\pi$
$390$	0	0
$391$	15.0000	0.758583
$392$	0	0
$393$	17.8885i	0.902358i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 32.4721i	− 1.62973i	−0.579651	−	0.814865i	$-0.696811\pi$
$397$	− 32.4721i	− 1.62973i	0.579651	−	0.814865i	$-0.303189\pi$
$398$	0	0
$399$	−7.41641	−0.371285
$400$	0	0
$401$	−10.4721	−0.522954	−0.261477	−	0.965210i	$-0.584209\pi$
$401$	−10.4721	−0.522954	−0.261477	+	0.965210i	$0.584209\pi$
$402$	0	0
$403$	− 6.12461i	− 0.305089i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.8541i	0.538018i
$408$	0	0
$409$	−13.5623	−0.670613	−0.335306	−	0.942109i	$-0.608840\pi$
$409$	−13.5623	−0.670613	−0.335306	+	0.942109i	$0.608840\pi$
$410$	0	0
$411$	−4.61803	−0.227791
$412$	0	0
$413$	− 7.88854i	− 0.388170i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	21.2361i	1.03993i
$418$	0	0
$419$	−8.94427	−0.436956	−0.218478	−	0.975842i	$-0.570109\pi$
$419$	−8.94427	−0.436956	−0.218478	+	0.975842i	$0.570109\pi$
$420$	0	0
$421$	14.4721	0.705329	0.352664	−	0.935750i	$-0.385276\pi$
$421$	14.4721	0.705329	0.352664	+	0.935750i	$0.385276\pi$
$422$	0	0
$423$	10.8541i	0.527744i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 16.5836i	− 0.802536i
$428$	0	0
$429$	−3.94427	−0.190431
$430$	0	0
$431$	21.7082	1.04565	0.522824	−	0.852441i	$-0.324879\pi$
$431$	21.7082	1.04565	0.522824	+	0.852441i	$0.324879\pi$
$432$	0	0
$433$	− 0.180340i	− 0.00866658i	−0.999991	−	0.00433329i	$-0.998621\pi$
$433$	− 0.180340i	− 0.00866658i	0.999991	−	0.00433329i	$-0.00137933\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 18.5410i	− 0.886937i
$438$	0	0
$439$	28.3262	1.35194	0.675969	−	0.736930i	$-0.263725\pi$
$439$	28.3262	1.35194	0.675969	+	0.736930i	$0.263725\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	0	0
$443$	− 32.6525i	− 1.55137i	−0.631123	−	0.775683i	$-0.717406\pi$
$443$	− 32.6525i	− 1.55137i	0.631123	−	0.775683i	$-0.282594\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 20.4721i	− 0.968299i
$448$	0	0
$449$	−2.94427	−0.138949	−0.0694744	−	0.997584i	$-0.522132\pi$
$449$	−2.94427	−0.138949	−0.0694744	+	0.997584i	$0.522132\pi$
$450$	0	0
$451$	4.47214	0.210585
$452$	0	0
$453$	− 17.8541i	− 0.838859i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.9443i	1.07329i	0.843809	+	0.536644i	$0.180308\pi$
$457$	22.9443i	1.07329i	−0.843809	+	0.536644i	$0.819692\pi$
$458$	0	0
$459$	−4.85410	−0.226570
$460$	0	0
$461$	−15.9787	−0.744203	−0.372101	−	0.928192i	$-0.621363\pi$
$461$	−15.9787	−0.744203	−0.372101	+	0.928192i	$0.621363\pi$
$462$	0	0
$463$	2.29180i	0.106509i	0.998581	+	0.0532544i	$0.0169594\pi$
$463$	2.29180i	0.106509i	−0.998581	+	0.0532544i	$0.983041\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.29180i	0.106052i	0.998593	+	0.0530258i	$0.0168866\pi$
$467$	2.29180i	0.106052i	−0.998593	+	0.0530258i	$0.983113\pi$
$468$	0	0
$469$	2.94427	0.135954
$470$	0	0
$471$	0.437694	0.0201679
$472$	0	0
$473$	0.201626i	0.00927078i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.23607i	0.148169i
$478$	0	0
$479$	6.65248	0.303959	0.151980	−	0.988384i	$-0.451435\pi$
$479$	6.65248	0.303959	0.151980	+	0.988384i	$0.451435\pi$
$480$	0	0
$481$	22.4164	1.02210
$482$	0	0
$483$	3.81966i	0.173801i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 17.5967i	− 0.797385i	−0.917085	−	0.398692i	$-0.869464\pi$
$487$	− 17.5967i	− 0.797385i	0.917085	−	0.398692i	$-0.130536\pi$
$488$	0	0
$489$	−20.2705	−0.916664
$490$	0	0
$491$	−2.61803	−0.118150	−0.0590751	−	0.998254i	$-0.518815\pi$
$491$	−2.61803	−0.118150	−0.0590751	+	0.998254i	$0.518815\pi$
$492$	0	0
$493$	− 45.2705i	− 2.03888i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 11.0557i	− 0.495917i
$498$	0	0
$499$	13.2361	0.592528	0.296264	−	0.955106i	$-0.404259\pi$
$499$	13.2361	0.592528	0.296264	+	0.955106i	$0.404259\pi$
$500$	0	0
$501$	−23.0902	−1.03159
$502$	0	0
$503$	10.4721i	0.466929i	0.972365	+	0.233465i	$0.0750063\pi$
$503$	10.4721i	0.466929i	−0.972365	+	0.233465i	$0.924994\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.85410i	− 0.215578i
$508$	0	0
$509$	−2.94427	−0.130503	−0.0652513	−	0.997869i	$-0.520785\pi$
$509$	−2.94427	−0.130503	−0.0652513	+	0.997869i	$0.520785\pi$
$510$	0	0
$511$	14.8328	0.656165
$512$	0	0
$513$	6.00000i	0.264906i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15.0000i	0.659699i
$518$	0	0
$519$	−9.23607	−0.405418
$520$	0	0
$521$	−8.18034	−0.358387	−0.179194	−	0.983814i	$-0.557349\pi$
$521$	−8.18034	−0.358387	−0.179194	+	0.983814i	$0.557349\pi$
$522$	0	0
$523$	− 11.9098i	− 0.520781i	−0.965503	−	0.260390i	$-0.916149\pi$
$523$	− 11.9098i	− 0.520781i	0.965503	−	0.260390i	$-0.0838512\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10.4164i	− 0.453746i
$528$	0	0
$529$	13.4508	0.584820
$530$	0	0
$531$	−6.38197	−0.276954
$532$	0	0
$533$	− 9.23607i	− 0.400059i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 7.41641i	− 0.320042i
$538$	0	0
$539$	−7.56231	−0.325732
$540$	0	0
$541$	11.7082	0.503375	0.251688	−	0.967809i	$-0.419014\pi$
$541$	11.7082	0.503375	0.251688	+	0.967809i	$0.419014\pi$
$542$	0	0
$543$	− 7.41641i	− 0.318269i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5.43769i	− 0.232499i	−0.993220	−	0.116250i	$-0.962913\pi$
$547$	− 5.43769i	− 0.232499i	0.993220	−	0.116250i	$-0.0370872\pi$
$548$	0	0
$549$	−13.4164	−0.572598
$550$	0	0
$551$	−55.9574	−2.38387
$552$	0	0
$553$	1.41641i	0.0602318i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10.3607i	− 0.438996i	−0.975613	−	0.219498i	$-0.929558\pi$
$557$	− 10.3607i	− 0.438996i	0.975613	−	0.219498i	$-0.0704420\pi$
$558$	0	0
$559$	0.416408	0.0176122
$560$	0	0
$561$	−6.70820	−0.283221
$562$	0	0
$563$	12.6525i	0.533238i	0.963802	+	0.266619i	$0.0859066\pi$
$563$	12.6525i	0.533238i	−0.963802	+	0.266619i	$0.914093\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.23607i	− 0.0519100i
$568$	0	0
$569$	13.0557	0.547325	0.273662	−	0.961826i	$-0.411765\pi$
$569$	13.0557	0.547325	0.273662	+	0.961826i	$0.411765\pi$
$570$	0	0
$571$	−12.3607	−0.517278	−0.258639	−	0.965974i	$-0.583274\pi$
$571$	−12.3607	−0.517278	−0.258639	+	0.965974i	$0.583274\pi$
$572$	0	0
$573$	6.00000i	0.250654i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 4.11146i	− 0.171162i	−0.996331	−	0.0855811i	$-0.972725\pi$
$577$	− 4.11146i	− 0.171162i	0.996331	−	0.0855811i	$-0.0272747\pi$
$578$	0	0
$579$	−17.1246	−0.711675
$580$	0	0
$581$	−10.1115	−0.419494
$582$	0	0
$583$	4.47214i	0.185217i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.6525i	0.769870i	0.922944	+	0.384935i	$0.125776\pi$
$587$	18.6525i	0.769870i	−0.922944	+	0.384935i	$0.874224\pi$
$588$	0	0
$589$	−12.8754	−0.530521
$590$	0	0
$591$	1.52786	0.0628479
$592$	0	0
$593$	34.7984i	1.42900i	0.699636	+	0.714499i	$0.253345\pi$
$593$	34.7984i	1.42900i	−0.699636	+	0.714499i	$0.746655\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.6180i	0.639204i
$598$	0	0
$599$	−31.8885	−1.30293	−0.651465	−	0.758678i	$-0.725845\pi$
$599$	−31.8885	−1.30293	−0.651465	+	0.758678i	$0.725845\pi$
$600$	0	0
$601$	40.6869	1.65965	0.829827	−	0.558021i	$-0.188439\pi$
$601$	40.6869	1.65965	0.829827	+	0.558021i	$0.188439\pi$
$602$	0	0
$603$	− 2.38197i	− 0.0970012i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 47.4164i	− 1.92457i	−0.272039	−	0.962286i	$-0.587698\pi$
$607$	− 47.4164i	− 1.92457i	0.272039	−	0.962286i	$-0.412302\pi$
$608$	0	0
$609$	11.5279	0.467133
$610$	0	0
$611$	30.9787	1.25326
$612$	0	0
$613$	37.7771i	1.52580i	0.646515	+	0.762901i	$0.276226\pi$
$613$	37.7771i	1.52580i	−0.646515	+	0.762901i	$0.723774\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.3050i	1.26029i	0.776478	+	0.630145i	$0.217005\pi$
$617$	31.3050i	1.26029i	−0.776478	+	0.630145i	$0.782995\pi$
$618$	0	0
$619$	−47.7082	−1.91755	−0.958777	−	0.284159i	$-0.908286\pi$
$619$	−47.7082	−1.91755	−0.958777	+	0.284159i	$0.908286\pi$
$620$	0	0
$621$	3.09017	0.124004
$622$	0	0
$623$	− 11.4164i	− 0.457389i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.29180i	0.331142i
$628$	0	0
$629$	38.1246	1.52013
$630$	0	0
$631$	15.0344	0.598512	0.299256	−	0.954173i	$-0.403262\pi$
$631$	15.0344	0.598512	0.299256	+	0.954173i	$0.403262\pi$
$632$	0	0
$633$	3.41641i	0.135790i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15.6180i	0.618809i
$638$	0	0
$639$	−8.94427	−0.353830
$640$	0	0
$641$	−30.6525	−1.21070	−0.605350	−	0.795959i	$-0.706967\pi$
$641$	−30.6525	−1.21070	−0.605350	+	0.795959i	$0.706967\pi$
$642$	0	0
$643$	31.5623i	1.24470i	0.782741	+	0.622348i	$0.213821\pi$
$643$	31.5623i	1.24470i	−0.782741	+	0.622348i	$0.786179\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.6180i	0.496066i	0.968752	+	0.248033i	$0.0797842\pi$
$647$	12.6180i	0.496066i	−0.968752	+	0.248033i	$0.920216\pi$
$648$	0	0
$649$	−8.81966	−0.346202
$650$	0	0
$651$	2.65248	0.103959
$652$	0	0
$653$	− 9.88854i	− 0.386969i	−0.981103	−	0.193484i	$-0.938021\pi$
$653$	− 9.88854i	− 0.386969i	0.981103	−	0.193484i	$-0.0619789\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 12.0000i	− 0.468165i
$658$	0	0
$659$	2.02129	0.0787381	0.0393691	−	0.999225i	$-0.487465\pi$
$659$	2.02129	0.0787381	0.0393691	+	0.999225i	$0.487465\pi$
$660$	0	0
$661$	5.70820	0.222023	0.111012	−	0.993819i	$-0.464591\pi$
$661$	5.70820	0.222023	0.111012	+	0.993819i	$0.464591\pi$
$662$	0	0
$663$	13.8541i	0.538049i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.8197i	1.11590i
$668$	0	0
$669$	21.7082	0.839288
$670$	0	0
$671$	−18.5410	−0.715768
$672$	0	0
$673$	47.3050i	1.82347i	0.410777	+	0.911736i	$0.365258\pi$
$673$	47.3050i	1.82347i	−0.410777	+	0.911736i	$0.634742\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 31.2361i	− 1.20050i	−0.799813	−	0.600250i	$-0.795068\pi$
$677$	− 31.2361i	− 1.20050i	0.799813	−	0.600250i	$-0.204932\pi$
$678$	0	0
$679$	−22.4721	−0.862401
$680$	0	0
$681$	−16.4721	−0.631214
$682$	0	0
$683$	− 10.7639i	− 0.411870i	−0.978566	−	0.205935i	$-0.933976\pi$
$683$	− 10.7639i	− 0.411870i	0.978566	−	0.205935i	$-0.0660236\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.70820i	− 0.0651720i
$688$	0	0
$689$	9.23607	0.351866
$690$	0	0
$691$	−26.7639	−1.01815	−0.509074	−	0.860723i	$-0.670012\pi$
$691$	−26.7639	−1.01815	−0.509074	+	0.860723i	$0.670012\pi$
$692$	0	0
$693$	− 1.70820i	− 0.0648893i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 15.7082i	− 0.594991i
$698$	0	0
$699$	7.32624	0.277104
$700$	0	0
$701$	12.2148	0.461346	0.230673	−	0.973031i	$-0.425907\pi$
$701$	12.2148	0.461346	0.230673	+	0.973031i	$0.425907\pi$
$702$	0	0
$703$	− 47.1246i	− 1.77734i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.87539i	0.108140i
$708$	0	0
$709$	6.87539	0.258211	0.129105	−	0.991631i	$-0.458789\pi$
$709$	6.87539	0.258211	0.129105	+	0.991631i	$0.458789\pi$
$710$	0	0
$711$	1.14590	0.0429745
$712$	0	0
$713$	6.63119i	0.248340i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	26.6525i	0.995355i
$718$	0	0
$719$	−7.23607	−0.269860	−0.134930	−	0.990855i	$-0.543081\pi$
$719$	−7.23607	−0.269860	−0.134930	+	0.990855i	$0.543081\pi$
$720$	0	0
$721$	4.94427	0.184134
$722$	0	0
$723$	− 0.0901699i	− 0.00335346i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 7.34752i	− 0.272505i	−0.990674	−	0.136252i	$-0.956494\pi$
$727$	− 7.34752i	− 0.272505i	0.990674	−	0.136252i	$-0.0435058\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0.708204	0.0261939
$732$	0	0
$733$	− 34.6869i	− 1.28119i	−0.767879	−	0.640595i	$-0.778688\pi$
$733$	− 34.6869i	− 1.28119i	0.767879	−	0.640595i	$-0.221312\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.29180i	− 0.121255i
$738$	0	0
$739$	8.65248	0.318286	0.159143	−	0.987256i	$-0.449127\pi$
$739$	8.65248	0.318286	0.159143	+	0.987256i	$0.449127\pi$
$740$	0	0
$741$	17.1246	0.629088
$742$	0	0
$743$	− 19.6180i	− 0.719716i	−0.933007	−	0.359858i	$-0.882825\pi$
$743$	− 19.6180i	− 0.719716i	0.933007	−	0.359858i	$-0.117175\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.18034i	0.299303i
$748$	0	0
$749$	−10.1115	−0.369465
$750$	0	0
$751$	43.4164	1.58429	0.792144	−	0.610335i	$-0.208965\pi$
$751$	43.4164	1.58429	0.792144	+	0.610335i	$0.208965\pi$
$752$	0	0
$753$	7.90983i	0.288250i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.83282i	0.175652i	0.996136	+	0.0878258i	$0.0279919\pi$
$757$	4.83282i	0.175652i	−0.996136	+	0.0878258i	$0.972008\pi$
$758$	0	0
$759$	4.27051	0.155010
$760$	0	0
$761$	−32.6525	−1.18365	−0.591826	−	0.806066i	$-0.701593\pi$
$761$	−32.6525	−1.18365	−0.591826	+	0.806066i	$0.701593\pi$
$762$	0	0
$763$	− 16.5836i	− 0.600366i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.2148i	0.657698i
$768$	0	0
$769$	22.9230	0.826624	0.413312	−	0.910589i	$-0.364372\pi$
$769$	22.9230	0.826624	0.413312	+	0.910589i	$0.364372\pi$
$770$	0	0
$771$	−23.5066	−0.846569
$772$	0	0
$773$	6.47214i	0.232787i	0.993203	+	0.116393i	$0.0371333\pi$
$773$	6.47214i	0.232787i	−0.993203	+	0.116393i	$0.962867\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.70820i	0.348280i
$778$	0	0
$779$	−19.4164	−0.695665
$780$	0	0
$781$	−12.3607	−0.442300
$782$	0	0
$783$	− 9.32624i	− 0.333293i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.5623i	1.37460i	0.726375	+	0.687299i	$0.241204\pi$
$787$	38.5623i	1.37460i	−0.726375	+	0.687299i	$0.758796\pi$
$788$	0	0
$789$	15.9787	0.568857
$790$	0	0
$791$	15.2361	0.541732
$792$	0	0
$793$	38.2918i	1.35978i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.4721i	1.22107i	0.791991	+	0.610533i	$0.209045\pi$
$797$	34.4721i	1.22107i	−0.791991	+	0.610533i	$0.790955\pi$
$798$	0	0
$799$	52.6869	1.86393
$800$	0	0
$801$	−9.23607	−0.326340
$802$	0	0
$803$	− 16.5836i	− 0.585222i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.6180i	0.796193i
$808$	0	0
$809$	29.5279	1.03814	0.519072	−	0.854730i	$-0.326278\pi$
$809$	29.5279	1.03814	0.519072	+	0.854730i	$0.326278\pi$
$810$	0	0
$811$	34.0000	1.19390	0.596951	−	0.802278i	$-0.296379\pi$
$811$	34.0000	1.19390	0.596951	+	0.802278i	$0.296379\pi$
$812$	0	0
$813$	− 8.56231i	− 0.300293i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 0.875388i	− 0.0306260i
$818$	0	0
$819$	−3.52786	−0.123274
$820$	0	0
$821$	36.2148	1.26390	0.631952	−	0.775007i	$-0.282254\pi$
$821$	36.2148	1.26390	0.631952	+	0.775007i	$0.282254\pi$
$822$	0	0
$823$	15.1246i	0.527211i	0.964631	+	0.263605i	$0.0849117\pi$
$823$	15.1246i	0.527211i	−0.964631	+	0.263605i	$0.915088\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	0	0
$829$	−34.1803	−1.18713	−0.593566	−	0.804785i	$-0.702280\pi$
$829$	−34.1803	−1.18713	−0.593566	+	0.804785i	$0.702280\pi$
$830$	0	0
$831$	−14.3607	−0.498166
$832$	0	0
$833$	26.5623i	0.920329i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.14590i	− 0.0741731i
$838$	0	0
$839$	−11.1246	−0.384064	−0.192032	−	0.981389i	$-0.561508\pi$
$839$	−11.1246	−0.384064	−0.192032	+	0.981389i	$0.561508\pi$
$840$	0	0
$841$	57.9787	1.99927
$842$	0	0
$843$	− 22.4721i	− 0.773981i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.2361i	0.386076i
$848$	0	0
$849$	21.0902	0.723813
$850$	0	0
$851$	−24.2705	−0.831982
$852$	0	0
$853$	− 29.0902i	− 0.996028i	−0.867169	−	0.498014i	$-0.834063\pi$
$853$	− 29.0902i	− 0.996028i	0.867169	−	0.498014i	$-0.165937\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 12.3262i	− 0.421056i	−0.977588	−	0.210528i	$-0.932482\pi$
$857$	− 12.3262i	− 0.421056i	0.977588	−	0.210528i	$-0.0675184\pi$
$858$	0	0
$859$	−15.7082	−0.535957	−0.267979	−	0.963425i	$-0.586356\pi$
$859$	−15.7082	−0.535957	−0.267979	+	0.963425i	$0.586356\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	8.56231i	0.291464i	0.989324	+	0.145732i	$0.0465538\pi$
$863$	8.56231i	0.291464i	−0.989324	+	0.145732i	$0.953446\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.56231i	0.222868i
$868$	0	0
$869$	1.58359	0.0537197
$870$	0	0
$871$	−6.79837	−0.230354
$872$	0	0
$873$	18.1803i	0.615311i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 26.3951i	− 0.891300i	−0.895207	−	0.445650i	$-0.852973\pi$
$877$	− 26.3951i	− 0.891300i	0.895207	−	0.445650i	$-0.147027\pi$
$878$	0	0
$879$	−15.5967	−0.526065
$880$	0	0
$881$	−5.05573	−0.170332	−0.0851659	−	0.996367i	$-0.527142\pi$
$881$	−5.05573	−0.170332	−0.0851659	+	0.996367i	$0.527142\pi$
$882$	0	0
$883$	− 46.9787i	− 1.58096i	−0.612488	−	0.790480i	$-0.709831\pi$
$883$	− 46.9787i	− 1.58096i	0.612488	−	0.790480i	$-0.290169\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	54.4721i	1.82900i	0.404591	+	0.914498i	$0.367414\pi$
$887$	54.4721i	1.82900i	−0.404591	+	0.914498i	$0.632586\pi$
$888$	0	0
$889$	−16.5836	−0.556196
$890$	0	0
$891$	−1.38197	−0.0462976
$892$	0	0
$893$	− 65.1246i	− 2.17931i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8.81966i	− 0.294480i
$898$	0	0
$899$	20.0132	0.667476
$900$	0	0
$901$	15.7082	0.523316
$902$	0	0
$903$	0.180340i	0.00600134i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.1459i	0.768547i	0.923219	+	0.384273i	$0.125548\pi$
$907$	23.1459i	0.768547i	−0.923219	+	0.384273i	$0.874452\pi$
$908$	0	0
$909$	2.32624	0.0771564
$910$	0	0
$911$	−37.5279	−1.24335	−0.621677	−	0.783274i	$-0.713548\pi$
$911$	−37.5279	−1.24335	−0.621677	+	0.783274i	$0.713548\pi$
$912$	0	0
$913$	11.3050i	0.374139i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	22.1115i	0.730185i
$918$	0	0
$919$	21.8885	0.722036	0.361018	−	0.932559i	$-0.382429\pi$
$919$	21.8885	0.722036	0.361018	+	0.932559i	$0.382429\pi$
$920$	0	0
$921$	−33.0902	−1.09036
$922$	0	0
$923$	25.5279i	0.840260i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4.00000i	− 0.131377i
$928$	0	0
$929$	24.5410	0.805165	0.402582	−	0.915384i	$-0.368113\pi$
$929$	24.5410	0.805165	0.402582	+	0.915384i	$0.368113\pi$
$930$	0	0
$931$	32.8328	1.07605
$932$	0	0
$933$	− 4.47214i	− 0.146411i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.4164i	0.895655i	0.894120	+	0.447828i	$0.147802\pi$
$937$	27.4164i	0.895655i	−0.894120	+	0.447828i	$0.852198\pi$
$938$	0	0
$939$	30.9443	1.00983
$940$	0	0
$941$	−22.4508	−0.731877	−0.365938	−	0.930639i	$-0.619252\pi$
$941$	−22.4508	−0.731877	−0.365938	+	0.930639i	$0.619252\pi$
$942$	0	0
$943$	10.0000i	0.325645i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 55.3050i	− 1.79717i	−0.438800	−	0.898585i	$-0.644596\pi$
$947$	− 55.3050i	− 1.79717i	0.438800	−	0.898585i	$-0.355404\pi$
$948$	0	0
$949$	−34.2492	−1.11178
$950$	0	0
$951$	−9.70820	−0.314810
$952$	0	0
$953$	− 1.63932i	− 0.0531028i	−0.999647	−	0.0265514i	$-0.991547\pi$
$953$	− 1.63932i	− 0.0531028i	0.999647	−	0.0265514i	$-0.00845257\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 12.8885i	− 0.416627i
$958$	0	0
$959$	−5.70820	−0.184328
$960$	0	0
$961$	−26.3951	−0.851456
$962$	0	0
$963$	8.18034i	0.263608i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	39.0132i	1.25458i	0.778786	+	0.627289i	$0.215836\pi$
$967$	39.0132i	1.25458i	−0.778786	+	0.627289i	$0.784164\pi$
$968$	0	0
$969$	29.1246	0.935617
$970$	0	0
$971$	−52.0902	−1.67165	−0.835827	−	0.548994i	$-0.815011\pi$
$971$	−52.0902	−1.67165	−0.835827	+	0.548994i	$0.815011\pi$
$972$	0	0
$973$	26.2492i	0.841511i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.2837i	1.73669i	0.495962	+	0.868344i	$0.334815\pi$
$977$	54.2837i	1.73669i	−0.495962	+	0.868344i	$0.665185\pi$
$978$	0	0
$979$	−12.7639	−0.407937
$980$	0	0
$981$	−13.4164	−0.428353
$982$	0	0
$983$	24.9787i	0.796697i	0.917234	+	0.398349i	$0.130417\pi$
$983$	24.9787i	0.796697i	−0.917234	+	0.398349i	$0.869583\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.4164i	0.427049i
$988$	0	0
$989$	−0.450850	−0.0143362
$990$	0	0
$991$	8.68692	0.275949	0.137975	−	0.990436i	$-0.455941\pi$
$991$	8.68692	0.275949	0.137975	+	0.990436i	$0.455941\pi$
$992$	0	0
$993$	− 21.4164i	− 0.679629i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	38.2705i	1.21204i	0.795450	+	0.606020i	$0.207235\pi$
$997$	38.2705i	1.21204i	−0.795450	+	0.606020i	$0.792765\pi$
$998$	0	0
$999$	7.85410	0.248493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.f.a.1249.1		4
4.3	odd	2		750.2.c.b.499.4		4
5.2	odd	4		6000.2.a.d.1.2		2
5.3	odd	4		6000.2.a.y.1.1		2
5.4	even	2	inner	6000.2.f.a.1249.4		4
12.11	even	2		2250.2.c.b.1999.2		4
20.3	even	4		750.2.a.f.1.2	yes	2
20.7	even	4		750.2.a.c.1.1	✓	2
20.19	odd	2		750.2.c.b.499.1		4
60.23	odd	4		2250.2.a.c.1.2		2
60.47	odd	4		2250.2.a.n.1.1		2
60.59	even	2		2250.2.c.b.1999.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.2.a.c.1.1	✓	2	20.7	even	4
750.2.a.f.1.2	yes	2	20.3	even	4
750.2.c.b.499.1		4	20.19	odd	2
750.2.c.b.499.4		4	4.3	odd	2
2250.2.a.c.1.2		2	60.23	odd	4
2250.2.a.n.1.1		2	60.47	odd	4
2250.2.c.b.1999.2		4	12.11	even	2
2250.2.c.b.1999.3		4	60.59	even	2
6000.2.a.d.1.2		2	5.2	odd	4
6000.2.a.y.1.1		2	5.3	odd	4
6000.2.f.a.1249.1		4	1.1	even	1	trivial
6000.2.f.a.1249.4		4	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:	\( N \)	\(=\)	\( 6000 = 2^{4} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6000.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.23607i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.23607i q^{7} -1.00000 q^{9} -1.38197 q^{11} +2.85410i q^{13} +4.85410i q^{17} +6.00000 q^{19} -1.23607 q^{21} -3.09017i q^{23} +1.00000i q^{27} -9.32624 q^{29} -2.14590 q^{31} +1.38197i q^{33} -7.85410i q^{37} +2.85410 q^{39} -3.23607 q^{41} -0.145898i q^{43} -10.8541i q^{47} +5.47214 q^{49} +4.85410 q^{51} -3.23607i q^{53} -6.00000i q^{57} +6.38197 q^{59} +13.4164 q^{61} +1.23607i q^{63} +2.38197i q^{67} -3.09017 q^{69} +8.94427 q^{71} +12.0000i q^{73} +1.70820i q^{77} -1.14590 q^{79} +1.00000 q^{81} -8.18034i q^{83} +9.32624i q^{87} +9.23607 q^{89} +3.52786 q^{91} +2.14590i q^{93} -18.1803i q^{97} +1.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 10 q^{11} + 24 q^{19} + 4 q^{21} - 6 q^{29} - 22 q^{31} - 2 q^{39} - 4 q^{41} + 4 q^{49} + 6 q^{51} + 30 q^{59} + 10 q^{69} - 18 q^{79} + 4 q^{81} + 28 q^{89} + 32 q^{91} + 10 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 10 * q^11 + 24 * q^19 + 4 * q^21 - 6 * q^29 - 22 * q^31 - 2 * q^39 - 4 * q^41 + 4 * q^49 + 6 * q^51 + 30 * q^59 + 10 * q^69 - 18 * q^79 + 4 * q^81 + 28 * q^89 + 32 * q^91 + 10 * q^99

Introduction

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