LMFDB - Embedded newform 2800.2.g.t.449.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2800,2,Mod(449,2800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2800.449"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-6,0,-2,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	2800.449
Dual form			2800.2.g.t.449.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{3} -1.00000i q^{7} -3.56155 q^{9} -2.56155 q^{11} +4.56155i q^{13} +4.56155i q^{17} +1.12311 q^{19} +2.56155 q^{21} +5.12311i q^{23} -1.43845i q^{27} +5.68466 q^{29} -6.56155i q^{33} -6.00000i q^{37} -11.6847 q^{39} -3.12311 q^{41} -9.12311i q^{43} +3.68466i q^{47} -1.00000 q^{49} -11.6847 q^{51} +3.12311i q^{53} +2.87689i q^{57} -4.00000 q^{59} -9.36932 q^{61} +3.56155i q^{63} -6.24621i q^{67} -13.1231 q^{69} -8.00000 q^{71} +4.24621i q^{73} +2.56155i q^{77} -6.56155 q^{79} -7.00000 q^{81} -4.00000i q^{83} +14.5616i q^{87} -7.12311 q^{89} +4.56155 q^{91} +14.8078i q^{97} +9.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9} - 2 q^{11} - 12 q^{19} + 2 q^{21} - 2 q^{29} - 22 q^{39} + 4 q^{41} - 4 q^{49} - 22 q^{51} - 16 q^{59} + 12 q^{61} - 36 q^{69} - 32 q^{71} - 18 q^{79} - 28 q^{81} - 12 q^{89} + 10 q^{91} + 20 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 2 * q^11 - 12 * q^19 + 2 * q^21 - 2 * q^29 - 22 * q^39 + 4 * q^41 - 4 * q^49 - 22 * q^51 - 16 * q^59 + 12 * q^61 - 36 * q^69 - 32 * q^71 - 18 * q^79 - 28 * q^81 - 12 * q^89 + 10 * q^91 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	4.56155i	1.26515i	0.774500	+	0.632574i	$0.218001\pi$
$13$	4.56155i	1.26515i	−0.774500	+	0.632574i	$0.781999\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.56155i	1.10634i	0.833069	+	0.553170i	$0.186582\pi$
$17$	4.56155i	1.10634i	−0.833069	+	0.553170i	$0.813418\pi$
$18$	0	0
$19$	1.12311	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$19$	1.12311	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	0	0
$23$	5.12311i	1.06824i	0.845408	+	0.534121i	$0.179357\pi$
$23$	5.12311i	1.06824i	−0.845408	+	0.534121i	$0.820643\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.43845i	− 0.276829i
$28$	0	0
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	− 6.56155i	− 1.14222i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	−11.6847	−1.87104
$40$	0	0
$41$	−3.12311	−0.487747	−0.243874	−	0.969807i	$-0.578418\pi$
$41$	−3.12311	−0.487747	−0.243874	+	0.969807i	$0.578418\pi$
$42$	0	0
$43$	− 9.12311i	− 1.39126i	−0.718400	−	0.695630i	$-0.755125\pi$
$43$	− 9.12311i	− 1.39126i	0.718400	−	0.695630i	$-0.244875\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.68466i	0.537463i	0.963215	+	0.268731i	$0.0866044\pi$
$47$	3.68466i	0.537463i	−0.963215	+	0.268731i	$0.913396\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−11.6847	−1.63618
$52$	0	0
$53$	3.12311i	0.428992i	0.976725	+	0.214496i	$0.0688108\pi$
$53$	3.12311i	0.428992i	−0.976725	+	0.214496i	$0.931189\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.87689i	0.381054i
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−9.36932	−1.19962	−0.599809	−	0.800143i	$-0.704757\pi$
$61$	−9.36932	−1.19962	−0.599809	+	0.800143i	$0.704757\pi$
$62$	0	0
$63$	3.56155i	0.448713i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.24621i	− 0.763096i	−0.924349	−	0.381548i	$-0.875391\pi$
$67$	− 6.24621i	− 0.763096i	0.924349	−	0.381548i	$-0.124609\pi$
$68$	0	0
$69$	−13.1231	−1.57984
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	4.24621i	0.496981i	0.968634	+	0.248491i	$0.0799345\pi$
$73$	4.24621i	0.496981i	−0.968634	+	0.248491i	$0.920065\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.56155i	0.291916i
$78$	0	0
$79$	−6.56155	−0.738232	−0.369116	−	0.929383i	$-0.620340\pi$
$79$	−6.56155	−0.738232	−0.369116	+	0.929383i	$0.620340\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	14.5616i	1.56116i
$88$	0	0
$89$	−7.12311	−0.755048	−0.377524	−	0.926000i	$-0.623224\pi$
$89$	−7.12311	−0.755048	−0.377524	+	0.926000i	$0.623224\pi$
$90$	0	0
$91$	4.56155	0.478181
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.8078i	1.50350i	0.659448	+	0.751750i	$0.270790\pi$
$97$	14.8078i	1.50350i	−0.659448	+	0.751750i	$0.729210\pi$
$98$	0	0
$99$	9.12311	0.916907
$100$	0	0
$101$	0.246211	0.0244989	0.0122495	−	0.999925i	$-0.496101\pi$
$101$	0.246211	0.0244989	0.0122495	+	0.999925i	$0.496101\pi$
$102$	0	0
$103$	− 1.43845i	− 0.141734i	−0.997486	−	0.0708672i	$-0.977423\pi$
$103$	− 1.43845i	− 0.141734i	0.997486	−	0.0708672i	$-0.0225767\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.3693i	− 1.09911i	−0.835456	−	0.549557i	$-0.814797\pi$
$107$	− 11.3693i	− 1.09911i	0.835456	−	0.549557i	$-0.185203\pi$
$108$	0	0
$109$	−17.6847	−1.69388	−0.846942	−	0.531686i	$-0.821559\pi$
$109$	−17.6847	−1.69388	−0.846942	+	0.531686i	$0.821559\pi$
$110$	0	0
$111$	15.3693	1.45879
$112$	0	0
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 16.2462i	− 1.50196i
$118$	0	0
$119$	4.56155	0.418157
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	− 8.00000i	− 0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.2462i	0.909204i	0.890695	+	0.454602i	$0.150219\pi$
$127$	10.2462i	0.909204i	−0.890695	+	0.454602i	$0.849781\pi$
$128$	0	0
$129$	23.3693	2.05755
$130$	0	0
$131$	9.12311	0.797089	0.398545	−	0.917149i	$-0.369515\pi$
$131$	9.12311	0.797089	0.398545	+	0.917149i	$0.369515\pi$
$132$	0	0
$133$	− 1.12311i	− 0.0973856i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.87689i	0.758404i	0.925314	+	0.379202i	$0.123801\pi$
$137$	8.87689i	0.758404i	−0.925314	+	0.379202i	$0.876199\pi$
$138$	0	0
$139$	−6.87689	−0.583291	−0.291645	−	0.956527i	$-0.594203\pi$
$139$	−6.87689	−0.583291	−0.291645	+	0.956527i	$0.594203\pi$
$140$	0	0
$141$	−9.43845	−0.794861
$142$	0	0
$143$	− 11.6847i	− 0.977120i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.56155i	− 0.211273i
$148$	0	0
$149$	4.24621	0.347863	0.173932	−	0.984758i	$-0.444353\pi$
$149$	4.24621	0.347863	0.173932	+	0.984758i	$0.444353\pi$
$150$	0	0
$151$	−21.9309	−1.78471	−0.892354	−	0.451335i	$-0.850948\pi$
$151$	−21.9309	−1.78471	−0.892354	+	0.451335i	$0.850948\pi$
$152$	0	0
$153$	− 16.2462i	− 1.31343i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3.75379i	− 0.299585i	−0.988717	−	0.149792i	$-0.952139\pi$
$157$	− 3.75379i	− 0.299585i	0.988717	−	0.149792i	$-0.0478606\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	5.12311	0.403757
$162$	0	0
$163$	− 1.12311i	− 0.0879684i	−0.999032	−	0.0439842i	$-0.985995\pi$
$163$	− 1.12311i	− 0.0879684i	0.999032	−	0.0439842i	$-0.0140051\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.9309i	1.69706i	0.529146	+	0.848531i	$0.322512\pi$
$167$	21.9309i	1.69706i	−0.529146	+	0.848531i	$0.677488\pi$
$168$	0	0
$169$	−7.80776	−0.600597
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	− 8.56155i	− 0.650923i	−0.945555	−	0.325461i	$-0.894480\pi$
$173$	− 8.56155i	− 0.650923i	0.945555	−	0.325461i	$-0.105520\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 10.2462i	− 0.770152i
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	23.6155	1.75533	0.877664	−	0.479276i	$-0.159101\pi$
$181$	23.6155	1.75533	0.877664	+	0.479276i	$0.159101\pi$
$182$	0	0
$183$	− 24.0000i	− 1.77413i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 11.6847i	− 0.854467i
$188$	0	0
$189$	−1.43845	−0.104632
$190$	0	0
$191$	9.43845	0.682942	0.341471	−	0.939892i	$-0.389075\pi$
$191$	9.43845	0.682942	0.341471	+	0.939892i	$0.389075\pi$
$192$	0	0
$193$	− 5.36932i	− 0.386492i	−0.981150	−	0.193246i	$-0.938098\pi$
$193$	− 5.36932i	− 0.386492i	0.981150	−	0.193246i	$-0.0619015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.12311i	0.507500i	0.967270	+	0.253750i	$0.0816641\pi$
$197$	7.12311i	0.507500i	−0.967270	+	0.253750i	$0.918336\pi$
$198$	0	0
$199$	−18.2462	−1.29344	−0.646720	−	0.762728i	$-0.723860\pi$
$199$	−18.2462	−1.29344	−0.646720	+	0.762728i	$0.723860\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	− 5.68466i	− 0.398985i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 18.2462i	− 1.26820i
$208$	0	0
$209$	−2.87689	−0.198999
$210$	0	0
$211$	23.0540	1.58710	0.793551	−	0.608504i	$-0.208230\pi$
$211$	23.0540	1.58710	0.793551	+	0.608504i	$0.208230\pi$
$212$	0	0
$213$	− 20.4924i	− 1.40412i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−10.8769	−0.734992
$220$	0	0
$221$	−20.8078	−1.39968
$222$	0	0
$223$	6.56155i	0.439394i	0.975568	+	0.219697i	$0.0705069\pi$
$223$	6.56155i	0.439394i	−0.975568	+	0.219697i	$0.929493\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.6847i	1.57201i	0.618223	+	0.786003i	$0.287853\pi$
$227$	23.6847i	1.57201i	−0.618223	+	0.786003i	$0.712147\pi$
$228$	0	0
$229$	−19.1231	−1.26369	−0.631845	−	0.775095i	$-0.717702\pi$
$229$	−19.1231	−1.26369	−0.631845	+	0.775095i	$0.717702\pi$
$230$	0	0
$231$	−6.56155	−0.431718
$232$	0	0
$233$	− 3.12311i	− 0.204601i	−0.994754	−	0.102301i	$-0.967380\pi$
$233$	− 3.12311i	− 0.204601i	0.994754	−	0.102301i	$-0.0326204\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 16.8078i	− 1.09178i
$238$	0	0
$239$	−0.807764	−0.0522499	−0.0261250	−	0.999659i	$-0.508317\pi$
$239$	−0.807764	−0.0522499	−0.0261250	+	0.999659i	$0.508317\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	0	0
$243$	− 22.2462i	− 1.42710i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.12311i	0.325975i
$248$	0	0
$249$	10.2462	0.649327
$250$	0	0
$251$	17.1231	1.08080	0.540400	−	0.841408i	$-0.318273\pi$
$251$	17.1231	1.08080	0.540400	+	0.841408i	$0.318273\pi$
$252$	0	0
$253$	− 13.1231i	− 0.825043i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 22.4924i	− 1.40304i	−0.712650	−	0.701519i	$-0.752505\pi$
$257$	− 22.4924i	− 1.40304i	0.712650	−	0.701519i	$-0.247495\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−20.2462	−1.25321
$262$	0	0
$263$	21.1231i	1.30251i	0.758860	+	0.651253i	$0.225756\pi$
$263$	21.1231i	1.30251i	−0.758860	+	0.651253i	$0.774244\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 18.2462i	− 1.11665i
$268$	0	0
$269$	−28.7386	−1.75223	−0.876113	−	0.482106i	$-0.839872\pi$
$269$	−28.7386	−1.75223	−0.876113	+	0.482106i	$0.839872\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	11.6847i	0.707188i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 16.2462i	− 0.976140i	−0.872804	−	0.488070i	$-0.837701\pi$
$277$	− 16.2462i	− 0.976140i	0.872804	−	0.488070i	$-0.162299\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.5616	0.987979	0.493990	−	0.869468i	$-0.335538\pi$
$281$	16.5616	0.987979	0.493990	+	0.869468i	$0.335538\pi$
$282$	0	0
$283$	23.6847i	1.40791i	0.710246	+	0.703953i	$0.248584\pi$
$283$	23.6847i	1.40791i	−0.710246	+	0.703953i	$0.751416\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.12311i	0.184351i
$288$	0	0
$289$	−3.80776	−0.223986
$290$	0	0
$291$	−37.9309	−2.22355
$292$	0	0
$293$	9.68466i	0.565784i	0.959152	+	0.282892i	$0.0912938\pi$
$293$	9.68466i	0.565784i	−0.959152	+	0.282892i	$0.908706\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.68466i	0.213806i
$298$	0	0
$299$	−23.3693	−1.35148
$300$	0	0
$301$	−9.12311	−0.525847
$302$	0	0
$303$	0.630683i	0.0362318i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 31.6847i	− 1.80834i	−0.427174	−	0.904169i	$-0.640491\pi$
$307$	− 31.6847i	− 1.80834i	0.427174	−	0.904169i	$-0.359509\pi$
$308$	0	0
$309$	3.68466	0.209613
$310$	0	0
$311$	9.61553	0.545247	0.272623	−	0.962121i	$-0.412109\pi$
$311$	9.61553	0.545247	0.272623	+	0.962121i	$0.412109\pi$
$312$	0	0
$313$	31.3002i	1.76919i	0.466359	+	0.884596i	$0.345566\pi$
$313$	31.3002i	1.76919i	−0.466359	+	0.884596i	$0.654434\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.4924i	1.26330i	0.775254	+	0.631650i	$0.217622\pi$
$317$	22.4924i	1.26330i	−0.775254	+	0.631650i	$0.782378\pi$
$318$	0	0
$319$	−14.5616	−0.815290
$320$	0	0
$321$	29.1231	1.62549
$322$	0	0
$323$	5.12311i	0.285057i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 45.3002i	− 2.50511i
$328$	0	0
$329$	3.68466	0.203142
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	21.3693i	1.17103i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	34.4924i	1.87892i	0.342656	+	0.939461i	$0.388674\pi$
$337$	34.4924i	1.87892i	−0.342656	+	0.939461i	$0.611326\pi$
$338$	0	0
$339$	35.8617	1.94774
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.12311i	− 0.0602915i	−0.999546	−	0.0301457i	$-0.990403\pi$
$347$	− 1.12311i	− 0.0602915i	0.999546	−	0.0301457i	$-0.00959714\pi$
$348$	0	0
$349$	22.4924	1.20399	0.601996	−	0.798499i	$-0.294372\pi$
$349$	22.4924	1.20399	0.601996	+	0.798499i	$0.294372\pi$
$350$	0	0
$351$	6.56155	0.350230
$352$	0	0
$353$	− 14.8078i	− 0.788138i	−0.919081	−	0.394069i	$-0.871067\pi$
$353$	− 14.8078i	− 0.788138i	0.919081	−	0.394069i	$-0.128933\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11.6847i	0.618418i
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	− 11.3693i	− 0.596734i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.68466i	0.192338i	0.995365	+	0.0961688i	$0.0306589\pi$
$367$	3.68466i	0.192338i	−0.995365	+	0.0961688i	$0.969341\pi$
$368$	0	0
$369$	11.1231	0.579046
$370$	0	0
$371$	3.12311	0.162144
$372$	0	0
$373$	29.3693i	1.52069i	0.649522	+	0.760343i	$0.274969\pi$
$373$	29.3693i	1.52069i	−0.649522	+	0.760343i	$0.725031\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	25.9309i	1.33551i
$378$	0	0
$379$	16.4924	0.847159	0.423579	−	0.905859i	$-0.360773\pi$
$379$	16.4924	0.847159	0.423579	+	0.905859i	$0.360773\pi$
$380$	0	0
$381$	−26.2462	−1.34463
$382$	0	0
$383$	10.2462i	0.523557i	0.965128	+	0.261778i	$0.0843090\pi$
$383$	10.2462i	0.523557i	−0.965128	+	0.261778i	$0.915691\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	32.4924i	1.65168i
$388$	0	0
$389$	−3.93087	−0.199303	−0.0996515	−	0.995022i	$-0.531773\pi$
$389$	−3.93087	−0.199303	−0.0996515	+	0.995022i	$0.531773\pi$
$390$	0	0
$391$	−23.3693	−1.18184
$392$	0	0
$393$	23.3693i	1.17883i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 23.4384i	− 1.17634i	−0.808737	−	0.588171i	$-0.799848\pi$
$397$	− 23.4384i	− 1.17634i	0.808737	−	0.588171i	$-0.200152\pi$
$398$	0	0
$399$	2.87689	0.144025
$400$	0	0
$401$	27.4384	1.37021	0.685105	−	0.728444i	$-0.259756\pi$
$401$	27.4384	1.37021	0.685105	+	0.728444i	$0.259756\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.3693i	0.761829i
$408$	0	0
$409$	26.4924	1.30997	0.654983	−	0.755644i	$-0.272676\pi$
$409$	26.4924	1.30997	0.654983	+	0.755644i	$0.272676\pi$
$410$	0	0
$411$	−22.7386	−1.12161
$412$	0	0
$413$	4.00000i	0.196827i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 17.6155i	− 0.862636i
$418$	0	0
$419$	9.75379	0.476504	0.238252	−	0.971203i	$-0.423426\pi$
$419$	9.75379	0.476504	0.238252	+	0.971203i	$0.423426\pi$
$420$	0	0
$421$	9.68466	0.472001	0.236001	−	0.971753i	$-0.424163\pi$
$421$	9.68466	0.472001	0.236001	+	0.971753i	$0.424163\pi$
$422$	0	0
$423$	− 13.1231i	− 0.638067i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.36932i	0.453413i
$428$	0	0
$429$	29.9309	1.44508
$430$	0	0
$431$	−0.807764	−0.0389086	−0.0194543	−	0.999811i	$-0.506193\pi$
$431$	−0.807764	−0.0389086	−0.0194543	+	0.999811i	$0.506193\pi$
$432$	0	0
$433$	− 8.24621i	− 0.396288i	−0.980173	−	0.198144i	$-0.936509\pi$
$433$	− 8.24621i	− 0.396288i	0.980173	−	0.198144i	$-0.0634913\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.75379i	0.275241i
$438$	0	0
$439$	−15.3693	−0.733537	−0.366769	−	0.930312i	$-0.619536\pi$
$439$	−15.3693	−0.733537	−0.366769	+	0.930312i	$0.619536\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	0	0
$443$	27.3693i	1.30036i	0.759782	+	0.650178i	$0.225306\pi$
$443$	27.3693i	1.30036i	−0.759782	+	0.650178i	$0.774694\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.8769i	0.514459i
$448$	0	0
$449$	−18.8078	−0.887593	−0.443797	−	0.896128i	$-0.646369\pi$
$449$	−18.8078	−0.887593	−0.443797	+	0.896128i	$0.646369\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	− 56.1771i	− 2.63943i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.87689i	0.415244i	0.978209	+	0.207622i	$0.0665723\pi$
$457$	8.87689i	0.415244i	−0.978209	+	0.207622i	$0.933428\pi$
$458$	0	0
$459$	6.56155	0.306267
$460$	0	0
$461$	−4.87689	−0.227140	−0.113570	−	0.993530i	$-0.536229\pi$
$461$	−4.87689	−0.227140	−0.113570	+	0.993530i	$0.536229\pi$
$462$	0	0
$463$	20.4924i	0.952364i	0.879347	+	0.476182i	$0.157980\pi$
$463$	20.4924i	0.952364i	−0.879347	+	0.476182i	$0.842020\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.5616i	1.22912i	0.788869	+	0.614561i	$0.210667\pi$
$467$	26.5616i	1.22912i	−0.788869	+	0.614561i	$0.789333\pi$
$468$	0	0
$469$	−6.24621	−0.288423
$470$	0	0
$471$	9.61553	0.443060
$472$	0	0
$473$	23.3693i	1.07452i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 11.1231i	− 0.509292i
$478$	0	0
$479$	13.1231	0.599610	0.299805	−	0.954001i	$-0.403078\pi$
$479$	13.1231	0.599610	0.299805	+	0.954001i	$0.403078\pi$
$480$	0	0
$481$	27.3693	1.24793
$482$	0	0
$483$	13.1231i	0.597122i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.12311i	0.232150i	0.993240	+	0.116075i	$0.0370313\pi$
$487$	5.12311i	0.232150i	−0.993240	+	0.116075i	$0.962969\pi$
$488$	0	0
$489$	2.87689	0.130098
$490$	0	0
$491$	−4.17708	−0.188509	−0.0942545	−	0.995548i	$-0.530047\pi$
$491$	−4.17708	−0.188509	−0.0942545	+	0.995548i	$0.530047\pi$
$492$	0	0
$493$	25.9309i	1.16787i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	−4.17708	−0.186992	−0.0934959	−	0.995620i	$-0.529804\pi$
$499$	−4.17708	−0.186992	−0.0934959	+	0.995620i	$0.529804\pi$
$500$	0	0
$501$	−56.1771	−2.50981
$502$	0	0
$503$	− 10.0691i	− 0.448960i	−0.974479	−	0.224480i	$-0.927932\pi$
$503$	− 10.0691i	− 0.448960i	0.974479	−	0.224480i	$-0.0720684\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 20.0000i	− 0.888231i
$508$	0	0
$509$	28.2462	1.25199	0.625996	−	0.779827i	$-0.284693\pi$
$509$	28.2462	1.25199	0.625996	+	0.779827i	$0.284693\pi$
$510$	0	0
$511$	4.24621	0.187841
$512$	0	0
$513$	− 1.61553i	− 0.0713273i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9.43845i	− 0.415102i
$518$	0	0
$519$	21.9309	0.962658
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	− 7.50758i	− 0.328283i	−0.986437	−	0.164142i	$-0.947515\pi$
$523$	− 7.50758i	− 0.328283i	0.986437	−	0.164142i	$-0.0524854\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.24621	−0.141140
$530$	0	0
$531$	14.2462	0.618233
$532$	0	0
$533$	− 14.2462i	− 0.617072i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	51.2311i	2.21078i
$538$	0	0
$539$	2.56155	0.110334
$540$	0	0
$541$	−17.1922	−0.739152	−0.369576	−	0.929201i	$-0.620497\pi$
$541$	−17.1922	−0.739152	−0.369576	+	0.929201i	$0.620497\pi$
$542$	0	0
$543$	60.4924i	2.59598i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.2462i	0.609124i	0.952493	+	0.304562i	$0.0985101\pi$
$547$	14.2462i	0.609124i	−0.952493	+	0.304562i	$0.901490\pi$
$548$	0	0
$549$	33.3693	1.42417
$550$	0	0
$551$	6.38447	0.271988
$552$	0	0
$553$	6.56155i	0.279026i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.87689i	0.206641i	0.994648	+	0.103320i	$0.0329467\pi$
$557$	4.87689i	0.206641i	−0.994648	+	0.103320i	$0.967053\pi$
$558$	0	0
$559$	41.6155	1.76015
$560$	0	0
$561$	29.9309	1.26368
$562$	0	0
$563$	28.0000i	1.18006i	0.807382	+	0.590030i	$0.200884\pi$
$563$	28.0000i	1.18006i	−0.807382	+	0.590030i	$0.799116\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.00000i	0.293972i
$568$	0	0
$569$	−34.9848	−1.46664	−0.733320	−	0.679883i	$-0.762031\pi$
$569$	−34.9848	−1.46664	−0.733320	+	0.679883i	$0.762031\pi$
$570$	0	0
$571$	−7.50758	−0.314182	−0.157091	−	0.987584i	$-0.550212\pi$
$571$	−7.50758	−0.314182	−0.157091	+	0.987584i	$0.550212\pi$
$572$	0	0
$573$	24.1771i	1.01001i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 13.0540i	− 0.543444i	−0.962376	−	0.271722i	$-0.912407\pi$
$577$	− 13.0540i	− 0.543444i	0.962376	−	0.271722i	$-0.0875931\pi$
$578$	0	0
$579$	13.7538	0.571588
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	− 8.00000i	− 0.331326i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.75379i	− 0.402582i	−0.979531	−	0.201291i	$-0.935486\pi$
$587$	− 9.75379i	− 0.402582i	0.979531	−	0.201291i	$-0.0645137\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−18.2462	−0.750549
$592$	0	0
$593$	− 23.4384i	− 0.962502i	−0.876583	−	0.481251i	$-0.840183\pi$
$593$	− 23.4384i	− 0.962502i	0.876583	−	0.481251i	$-0.159817\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 46.7386i	− 1.91288i
$598$	0	0
$599$	8.80776	0.359875	0.179938	−	0.983678i	$-0.442410\pi$
$599$	8.80776	0.359875	0.179938	+	0.983678i	$0.442410\pi$
$600$	0	0
$601$	−26.4924	−1.08065	−0.540324	−	0.841457i	$-0.681698\pi$
$601$	−26.4924	−1.08065	−0.540324	+	0.841457i	$0.681698\pi$
$602$	0	0
$603$	22.2462i	0.905936i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.94602i	− 0.200753i	−0.994950	−	0.100376i	$-0.967995\pi$
$607$	− 4.94602i	− 0.200753i	0.994950	−	0.100376i	$-0.0320047\pi$
$608$	0	0
$609$	14.5616	0.590064
$610$	0	0
$611$	−16.8078	−0.679969
$612$	0	0
$613$	− 8.73863i	− 0.352950i	−0.984305	−	0.176475i	$-0.943531\pi$
$613$	− 8.73863i	− 0.352950i	0.984305	−	0.176475i	$-0.0564695\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 15.7538i	− 0.634224i	−0.948388	−	0.317112i	$-0.897287\pi$
$617$	− 15.7538i	− 0.634224i	0.948388	−	0.317112i	$-0.102713\pi$
$618$	0	0
$619$	−42.1080	−1.69246	−0.846231	−	0.532817i	$-0.821134\pi$
$619$	−42.1080	−1.69246	−0.846231	+	0.532817i	$0.821134\pi$
$620$	0	0
$621$	7.36932	0.295720
$622$	0	0
$623$	7.12311i	0.285381i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 7.36932i	− 0.294302i
$628$	0	0
$629$	27.3693	1.09129
$630$	0	0
$631$	−8.80776	−0.350632	−0.175316	−	0.984512i	$-0.556095\pi$
$631$	−8.80776	−0.350632	−0.175316	+	0.984512i	$0.556095\pi$
$632$	0	0
$633$	59.0540i	2.34718i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.56155i	− 0.180735i
$638$	0	0
$639$	28.4924	1.12714
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	2.56155i	0.101018i	0.998724	+	0.0505089i	$0.0160843\pi$
$643$	2.56155i	0.101018i	−0.998724	+	0.0505089i	$0.983916\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.50758i	0.137897i	0.997620	+	0.0689486i	$0.0219644\pi$
$647$	3.50758i	0.137897i	−0.997620	+	0.0689486i	$0.978036\pi$
$648$	0	0
$649$	10.2462	0.402199
$650$	0	0
$651$	0	0
$652$	0	0
$653$	49.2311i	1.92656i	0.268499	+	0.963280i	$0.413473\pi$
$653$	49.2311i	1.92656i	−0.268499	+	0.963280i	$0.586527\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 15.1231i	− 0.590009i
$658$	0	0
$659$	−36.1771	−1.40926	−0.704629	−	0.709575i	$-0.748887\pi$
$659$	−36.1771	−1.40926	−0.704629	+	0.709575i	$0.748887\pi$
$660$	0	0
$661$	3.12311	0.121475	0.0607374	−	0.998154i	$-0.480655\pi$
$661$	3.12311	0.121475	0.0607374	+	0.998154i	$0.480655\pi$
$662$	0	0
$663$	− 53.3002i	− 2.07001i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	29.1231i	1.12765i
$668$	0	0
$669$	−16.8078	−0.649826
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	− 25.8617i	− 0.996897i	−0.866919	−	0.498448i	$-0.833903\pi$
$673$	− 25.8617i	− 0.996897i	0.866919	−	0.498448i	$-0.166097\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.9309i	0.919738i	0.887987	+	0.459869i	$0.152104\pi$
$677$	23.9309i	0.919738i	−0.887987	+	0.459869i	$0.847896\pi$
$678$	0	0
$679$	14.8078	0.568270
$680$	0	0
$681$	−60.6695	−2.32486
$682$	0	0
$683$	− 42.7386i	− 1.63535i	−0.575681	−	0.817674i	$-0.695263\pi$
$683$	− 42.7386i	− 1.63535i	0.575681	−	0.817674i	$-0.304737\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 48.9848i	− 1.86889i
$688$	0	0
$689$	−14.2462	−0.542737
$690$	0	0
$691$	−8.49242	−0.323067	−0.161533	−	0.986867i	$-0.551644\pi$
$691$	−8.49242	−0.323067	−0.161533	+	0.986867i	$0.551644\pi$
$692$	0	0
$693$	− 9.12311i	− 0.346558i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 14.2462i	− 0.539614i
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	0.0691303	0.00261102	0.00130551	−	0.999999i	$-0.499584\pi$
$701$	0.0691303	0.00261102	0.00130551	+	0.999999i	$0.499584\pi$
$702$	0	0
$703$	− 6.73863i	− 0.254152i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 0.246211i	− 0.00925973i
$708$	0	0
$709$	18.1771	0.682655	0.341327	−	0.939945i	$-0.389124\pi$
$709$	18.1771	0.682655	0.341327	+	0.939945i	$0.389124\pi$
$710$	0	0
$711$	23.3693	0.876418
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.06913i	− 0.0772731i
$718$	0	0
$719$	49.6155	1.85035	0.925173	−	0.379544i	$-0.123919\pi$
$719$	49.6155	1.85035	0.925173	+	0.379544i	$0.123919\pi$
$720$	0	0
$721$	−1.43845	−0.0535706
$722$	0	0
$723$	31.3693i	1.16664i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19.5076i	0.723496i	0.932276	+	0.361748i	$0.117820\pi$
$727$	19.5076i	0.723496i	−0.932276	+	0.361748i	$0.882180\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	41.6155	1.53921
$732$	0	0
$733$	− 5.68466i	− 0.209968i	−0.994474	−	0.104984i	$-0.966521\pi$
$733$	− 5.68466i	− 0.209968i	0.994474	−	0.104984i	$-0.0334791\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.0000i	0.589368i
$738$	0	0
$739$	6.06913	0.223257	0.111628	−	0.993750i	$-0.464393\pi$
$739$	6.06913	0.223257	0.111628	+	0.993750i	$0.464393\pi$
$740$	0	0
$741$	−13.1231	−0.482089
$742$	0	0
$743$	− 32.9848i	− 1.21010i	−0.796189	−	0.605048i	$-0.793154\pi$
$743$	− 32.9848i	− 1.21010i	0.796189	−	0.605048i	$-0.206846\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14.2462i	0.521242i
$748$	0	0
$749$	−11.3693	−0.415426
$750$	0	0
$751$	−45.9309	−1.67604	−0.838021	−	0.545639i	$-0.816287\pi$
$751$	−45.9309	−1.67604	−0.838021	+	0.545639i	$0.816287\pi$
$752$	0	0
$753$	43.8617i	1.59841i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 14.6307i	− 0.531761i	−0.964006	−	0.265881i	$-0.914337\pi$
$757$	− 14.6307i	− 0.531761i	0.964006	−	0.265881i	$-0.0856627\pi$
$758$	0	0
$759$	33.6155	1.22017
$760$	0	0
$761$	31.7538	1.15107	0.575537	−	0.817776i	$-0.304793\pi$
$761$	31.7538	1.15107	0.575537	+	0.817776i	$0.304793\pi$
$762$	0	0
$763$	17.6847i	0.640228i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 18.2462i	− 0.658833i
$768$	0	0
$769$	9.50758	0.342852	0.171426	−	0.985197i	$-0.445163\pi$
$769$	9.50758	0.342852	0.171426	+	0.985197i	$0.445163\pi$
$770$	0	0
$771$	57.6155	2.07497
$772$	0	0
$773$	8.06913i	0.290226i	0.989415	+	0.145113i	$0.0463546\pi$
$773$	8.06913i	0.290226i	−0.989415	+	0.145113i	$0.953645\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 15.3693i	− 0.551371i
$778$	0	0
$779$	−3.50758	−0.125672
$780$	0	0
$781$	20.4924	0.733277
$782$	0	0
$783$	− 8.17708i	− 0.292225i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 3.82292i	− 0.136272i	−0.997676	−	0.0681362i	$-0.978295\pi$
$787$	− 3.82292i	− 0.136272i	0.997676	−	0.0681362i	$-0.0217052\pi$
$788$	0	0
$789$	−54.1080	−1.92629
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	− 42.7386i	− 1.51769i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.0540i	0.462396i	0.972907	+	0.231198i	$0.0742644\pi$
$797$	13.0540i	0.462396i	−0.972907	+	0.231198i	$0.925736\pi$
$798$	0	0
$799$	−16.8078	−0.594616
$800$	0	0
$801$	25.3693	0.896381
$802$	0	0
$803$	− 10.8769i	− 0.383837i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 73.6155i	− 2.59139i
$808$	0	0
$809$	53.5464	1.88259	0.941296	−	0.337584i	$-0.109610\pi$
$809$	53.5464	1.88259	0.941296	+	0.337584i	$0.109610\pi$
$810$	0	0
$811$	21.6155	0.759024	0.379512	−	0.925187i	$-0.376092\pi$
$811$	21.6155	0.759024	0.379512	+	0.925187i	$0.376092\pi$
$812$	0	0
$813$	40.9848i	1.43740i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10.2462i	− 0.358470i
$818$	0	0
$819$	−16.2462	−0.567689
$820$	0	0
$821$	40.4233	1.41078	0.705391	−	0.708818i	$-0.250771\pi$
$821$	40.4233	1.41078	0.705391	+	0.708818i	$0.250771\pi$
$822$	0	0
$823$	3.50758i	0.122266i	0.998130	+	0.0611332i	$0.0194715\pi$
$823$	3.50758i	0.122266i	−0.998130	+	0.0611332i	$0.980529\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.3693i	0.673537i	0.941587	+	0.336769i	$0.109334\pi$
$827$	19.3693i	0.673537i	−0.941587	+	0.336769i	$0.890666\pi$
$828$	0	0
$829$	−43.1231	−1.49773	−0.748864	−	0.662724i	$-0.769400\pi$
$829$	−43.1231	−1.49773	−0.748864	+	0.662724i	$0.769400\pi$
$830$	0	0
$831$	41.6155	1.44363
$832$	0	0
$833$	− 4.56155i	− 0.158048i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37.1231	−1.28163	−0.640816	−	0.767695i	$-0.721404\pi$
$839$	−37.1231	−1.28163	−0.640816	+	0.767695i	$0.721404\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	0	0
$843$	42.4233i	1.46114i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.43845i	0.152507i
$848$	0	0
$849$	−60.6695	−2.08217
$850$	0	0
$851$	30.7386	1.05371
$852$	0	0
$853$	− 56.7386i	− 1.94269i	−0.237666	−	0.971347i	$-0.576382\pi$
$853$	− 56.7386i	− 1.94269i	0.237666	−	0.971347i	$-0.423618\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.2462i	1.10151i	0.834667	+	0.550755i	$0.185660\pi$
$857$	32.2462i	1.10151i	−0.834667	+	0.550755i	$0.814340\pi$
$858$	0	0
$859$	16.4924	0.562714	0.281357	−	0.959603i	$-0.409215\pi$
$859$	16.4924	0.562714	0.281357	+	0.959603i	$0.409215\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	42.2462i	1.43808i	0.694970	+	0.719039i	$0.255418\pi$
$863$	42.2462i	1.43808i	−0.694970	+	0.719039i	$0.744582\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 9.75379i	− 0.331256i
$868$	0	0
$869$	16.8078	0.570164
$870$	0	0
$871$	28.4924	0.965429
$872$	0	0
$873$	− 52.7386i	− 1.78493i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.7538i	0.802108i	0.916054	+	0.401054i	$0.131356\pi$
$877$	23.7538i	0.802108i	−0.916054	+	0.401054i	$0.868644\pi$
$878$	0	0
$879$	−24.8078	−0.836745
$880$	0	0
$881$	45.8617	1.54512	0.772561	−	0.634941i	$-0.218976\pi$
$881$	45.8617	1.54512	0.772561	+	0.634941i	$0.218976\pi$
$882$	0	0
$883$	− 24.4924i	− 0.824236i	−0.911131	−	0.412118i	$-0.864789\pi$
$883$	− 24.4924i	− 0.824236i	0.911131	−	0.412118i	$-0.135211\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 12.4924i	− 0.419454i	−0.977760	−	0.209727i	$-0.932742\pi$
$887$	− 12.4924i	− 0.419454i	0.977760	−	0.209727i	$-0.0672576\pi$
$888$	0	0
$889$	10.2462	0.343647
$890$	0	0
$891$	17.9309	0.600707
$892$	0	0
$893$	4.13826i	0.138482i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 59.8617i	− 1.99873i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−14.2462	−0.474610
$902$	0	0
$903$	− 23.3693i	− 0.777682i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	50.1080i	1.66381i	0.554920	+	0.831904i	$0.312749\pi$
$907$	50.1080i	1.66381i	−0.554920	+	0.831904i	$0.687251\pi$
$908$	0	0
$909$	−0.876894	−0.0290848
$910$	0	0
$911$	−4.49242	−0.148841	−0.0744203	−	0.997227i	$-0.523711\pi$
$911$	−4.49242	−0.148841	−0.0744203	+	0.997227i	$0.523711\pi$
$912$	0	0
$913$	10.2462i	0.339100i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9.12311i	− 0.301271i
$918$	0	0
$919$	−13.3002	−0.438733	−0.219366	−	0.975643i	$-0.570399\pi$
$919$	−13.3002	−0.438733	−0.219366	+	0.975643i	$0.570399\pi$
$920$	0	0
$921$	81.1619	2.67438
$922$	0	0
$923$	− 36.4924i	− 1.20116i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.12311i	0.168265i
$928$	0	0
$929$	52.1080	1.70961	0.854803	−	0.518952i	$-0.173678\pi$
$929$	52.1080	1.70961	0.854803	+	0.518952i	$0.173678\pi$
$930$	0	0
$931$	−1.12311	−0.0368083
$932$	0	0
$933$	24.6307i	0.806372i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 22.6695i	− 0.740580i	−0.928916	−	0.370290i	$-0.879258\pi$
$937$	− 22.6695i	− 0.740580i	0.928916	−	0.370290i	$-0.120742\pi$
$938$	0	0
$939$	−80.1771	−2.61648
$940$	0	0
$941$	−13.8617	−0.451880	−0.225940	−	0.974141i	$-0.572545\pi$
$941$	−13.8617	−0.451880	−0.225940	+	0.974141i	$0.572545\pi$
$942$	0	0
$943$	− 16.0000i	− 0.521032i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	−19.3693	−0.628755
$950$	0	0
$951$	−57.6155	−1.86831
$952$	0	0
$953$	− 24.8769i	− 0.805842i	−0.915235	−	0.402921i	$-0.867995\pi$
$953$	− 24.8769i	− 0.805842i	0.915235	−	0.402921i	$-0.132005\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 37.3002i	− 1.20574i
$958$	0	0
$959$	8.87689	0.286650
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	40.4924i	1.30485i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 26.8769i	− 0.864303i	−0.901801	−	0.432151i	$-0.857755\pi$
$967$	− 26.8769i	− 0.864303i	0.901801	−	0.432151i	$-0.142245\pi$
$968$	0	0
$969$	−13.1231	−0.421575
$970$	0	0
$971$	49.4773	1.58780	0.793901	−	0.608048i	$-0.208047\pi$
$971$	49.4773	1.58780	0.793901	+	0.608048i	$0.208047\pi$
$972$	0	0
$973$	6.87689i	0.220463i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.2311i	1.57504i	0.616288	+	0.787521i	$0.288636\pi$
$977$	49.2311i	1.57504i	−0.616288	+	0.787521i	$0.711364\pi$
$978$	0	0
$979$	18.2462	0.583151
$980$	0	0
$981$	62.9848	2.01095
$982$	0	0
$983$	10.4233i	0.332451i	0.986088	+	0.166226i	$0.0531580\pi$
$983$	10.4233i	0.332451i	−0.986088	+	0.166226i	$0.946842\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	9.43845i	0.300429i
$988$	0	0
$989$	46.7386	1.48620
$990$	0	0
$991$	20.4924	0.650963	0.325482	−	0.945548i	$-0.394474\pi$
$991$	20.4924	0.650963	0.325482	+	0.945548i	$0.394474\pi$
$992$	0	0
$993$	− 30.7386i	− 0.975461i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 9.68466i	− 0.306716i	−0.988171	−	0.153358i	$-0.950991\pi$
$997$	− 9.68466i	− 0.306716i	0.988171	−	0.153358i	$-0.0490088\pi$
$998$	0	0
$999$	−8.63068	−0.273063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.g.t.449.4		4
4.3	odd	2		175.2.b.b.99.2		4
5.2	odd	4		560.2.a.i.1.2		2
5.3	odd	4		2800.2.a.bi.1.1		2
5.4	even	2	inner	2800.2.g.t.449.1		4
12.11	even	2		1575.2.d.e.1324.3		4
15.2	even	4		5040.2.a.bt.1.2		2
20.3	even	4		175.2.a.f.1.1		2
20.7	even	4		35.2.a.b.1.2	✓	2
20.19	odd	2		175.2.b.b.99.3		4
28.27	even	2		1225.2.b.f.99.2		4
35.27	even	4		3920.2.a.bs.1.1		2
40.27	even	4		2240.2.a.bh.1.2		2
40.37	odd	4		2240.2.a.bd.1.1		2
60.23	odd	4		1575.2.a.p.1.2		2
60.47	odd	4		315.2.a.e.1.1		2
60.59	even	2		1575.2.d.e.1324.2		4
140.27	odd	4		245.2.a.d.1.2		2
140.47	odd	12		245.2.e.h.116.1		4
140.67	even	12		245.2.e.i.226.1		4
140.83	odd	4		1225.2.a.s.1.1		2
140.87	odd	12		245.2.e.h.226.1		4
140.107	even	12		245.2.e.i.116.1		4
140.139	even	2		1225.2.b.f.99.3		4
220.87	odd	4		4235.2.a.m.1.1		2
260.207	even	4		5915.2.a.l.1.1		2
420.167	even	4		2205.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.b.1.2	✓	2	20.7	even	4
175.2.a.f.1.1		2	20.3	even	4
175.2.b.b.99.2		4	4.3	odd	2
175.2.b.b.99.3		4	20.19	odd	2
245.2.a.d.1.2		2	140.27	odd	4
245.2.e.h.116.1		4	140.47	odd	12
245.2.e.h.226.1		4	140.87	odd	12
245.2.e.i.116.1		4	140.107	even	12
245.2.e.i.226.1		4	140.67	even	12
315.2.a.e.1.1		2	60.47	odd	4
560.2.a.i.1.2		2	5.2	odd	4
1225.2.a.s.1.1		2	140.83	odd	4
1225.2.b.f.99.2		4	28.27	even	2
1225.2.b.f.99.3		4	140.139	even	2
1575.2.a.p.1.2		2	60.23	odd	4
1575.2.d.e.1324.2		4	60.59	even	2
1575.2.d.e.1324.3		4	12.11	even	2
2205.2.a.x.1.1		2	420.167	even	4
2240.2.a.bd.1.1		2	40.37	odd	4
2240.2.a.bh.1.2		2	40.27	even	4
2800.2.a.bi.1.1		2	5.3	odd	4
2800.2.g.t.449.1		4	5.4	even	2	inner
2800.2.g.t.449.4		4	1.1	even	1	trivial
3920.2.a.bs.1.1		2	35.27	even	4
4235.2.a.m.1.1		2	220.87	odd	4
5040.2.a.bt.1.2		2	15.2	even	4
5915.2.a.l.1.1		2	260.207	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155i q^{3} -1.00000i q^{7} -3.56155 q^{9} -2.56155 q^{11} +4.56155i q^{13} +4.56155i q^{17} +1.12311 q^{19} +2.56155 q^{21} +5.12311i q^{23} -1.43845i q^{27} +5.68466 q^{29} -6.56155i q^{33} -6.00000i q^{37} -11.6847 q^{39} -3.12311 q^{41} -9.12311i q^{43} +3.68466i q^{47} -1.00000 q^{49} -11.6847 q^{51} +3.12311i q^{53} +2.87689i q^{57} -4.00000 q^{59} -9.36932 q^{61} +3.56155i q^{63} -6.24621i q^{67} -13.1231 q^{69} -8.00000 q^{71} +4.24621i q^{73} +2.56155i q^{77} -6.56155 q^{79} -7.00000 q^{81} -4.00000i q^{83} +14.5616i q^{87} -7.12311 q^{89} +4.56155 q^{91} +14.8078i q^{97} +9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} - 2 q^{11} - 12 q^{19} + 2 q^{21} - 2 q^{29} - 22 q^{39} + 4 q^{41} - 4 q^{49} - 22 q^{51} - 16 q^{59} + 12 q^{61} - 36 q^{69} - 32 q^{71} - 18 q^{79} - 28 q^{81} - 12 q^{89} + 10 q^{91} + 20 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 2 * q^11 - 12 * q^19 + 2 * q^21 - 2 * q^29 - 22 * q^39 + 4 * q^41 - 4 * q^49 - 22 * q^51 - 16 * q^59 + 12 * q^61 - 36 * q^69 - 32 * q^71 - 18 * q^79 - 28 * q^81 - 12 * q^89 + 10 * q^91 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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