LMFDB - Embedded newform 2800.2.g.m.449.2

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 = [2,0,0,0,0,0,0,0,4,0,10,0,0,0,0,0,0,0,-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:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2800.449
Dual form			2800.2.g.m.449.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} +5.00000 q^{11} +1.00000i q^{13} -3.00000i q^{17} -6.00000 q^{19} +1.00000 q^{21} +6.00000i q^{23} +5.00000i q^{27} +9.00000 q^{29} +5.00000i q^{33} -6.00000i q^{37} -1.00000 q^{39} +8.00000 q^{41} -6.00000i q^{43} +3.00000i q^{47} -1.00000 q^{49} +3.00000 q^{51} -12.0000i q^{53} -6.00000i q^{57} +8.00000 q^{59} -4.00000 q^{61} -2.00000i q^{63} -4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} +10.0000i q^{73} -5.00000i q^{77} -3.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +9.00000i q^{87} +16.0000 q^{89} +1.00000 q^{91} -7.00000i q^{97} +10.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} + 10 q^{11} - 12 q^{19} + 2 q^{21} + 18 q^{29} - 2 q^{39} + 16 q^{41} - 2 q^{49} + 6 q^{51} + 16 q^{59} - 8 q^{61} - 12 q^{69} - 16 q^{71} - 6 q^{79} + 2 q^{81} + 32 q^{89} + 2 q^{91} + 20 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 + 10 * q^11 - 12 * q^19 + 2 * q^21 + 18 * q^29 - 2 * q^39 + 16 * q^41 - 2 * q^49 + 6 * q^51 + 16 * q^59 - 8 * q^61 - 12 * q^69 - 16 * q^71 - 6 * q^79 + 2 * q^81 + 32 * q^89 + 2 * 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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	5.00000i	0.870388i
$34$	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
$39$	−1.00000	−0.160128
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.00000i	− 0.794719i
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	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
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.00000i	− 0.569803i
$78$	0	0
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.00000i	0.964901i
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 7.00000i	− 0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$	− 7.00000i	− 0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$	0	0
$99$	10.0000	1.00504
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	9.00000i	0.886796i	0.896325	+	0.443398i	$0.146227\pi$
$103$	9.00000i	0.886796i	−0.896325	+	0.443398i	$0.853773\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	8.00000i	0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	6.00000i	0.520266i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.0000i	1.36697i	0.729964	+	0.683486i	$0.239537\pi$
$137$	16.0000i	1.36697i	−0.729964	+	0.683486i	$0.760463\pi$
$138$	0	0
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	5.00000i	0.418121i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 9.00000i	− 0.696441i	−0.937413	−	0.348220i	$-0.886786\pi$
$167$	− 9.00000i	− 0.696441i	0.937413	−	0.348220i	$-0.113214\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−12.0000	−0.917663
$172$	0	0
$173$	19.0000i	1.44454i	0.691609	+	0.722272i	$0.256902\pi$
$173$	19.0000i	1.44454i	−0.691609	+	0.722272i	$0.743098\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.00000i	0.601317i
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	− 4.00000i	− 0.295689i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 15.0000i	− 1.09691i
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	−11.0000	−0.795932	−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000	−0.795932	−0.397966	+	0.917400i	$0.630284\pi$
$192$	0	0
$193$	8.00000i	0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$	8.00000i	0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 9.00000i	− 0.631676i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.0000i	0.834058i
$208$	0	0
$209$	−30.0000	−2.07514
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	25.0000i	1.67412i	0.547108	+	0.837062i	$0.315729\pi$
$223$	25.0000i	1.67412i	−0.547108	+	0.837062i	$0.684271\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 13.0000i	− 0.862840i	−0.902151	−	0.431420i	$-0.858013\pi$
$227$	− 13.0000i	− 0.862840i	0.902151	−	0.431420i	$-0.141987\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	0	0
$233$	8.00000i	0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$	8.00000i	0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.00000i	− 0.194871i
$238$	0	0
$239$	−7.00000	−0.452792	−0.226396	−	0.974035i	$-0.572694\pi$
$239$	−7.00000	−0.452792	−0.226396	+	0.974035i	$0.572694\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.00000i	− 0.381771i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	0	0
$253$	30.0000i	1.88608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	18.0000	1.11417
$262$	0	0
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.0000i	0.979184i
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	1.00000i	0.0605228i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	0	0
$283$	− 29.0000i	− 1.72387i	−0.507018	−	0.861936i	$-0.669252\pi$
$283$	− 29.0000i	− 1.72387i	0.507018	−	0.861936i	$-0.330748\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.00000i	− 0.472225i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	7.00000	0.410347
$292$	0	0
$293$	− 1.00000i	− 0.0584206i	−0.999573	−	0.0292103i	$-0.990701\pi$
$293$	− 1.00000i	− 0.0584206i	0.999573	−	0.0292103i	$-0.00929925\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	25.0000i	1.45065i
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−6.00000	−0.345834
$302$	0	0
$303$	− 14.0000i	− 0.804279i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 27.0000i	− 1.54097i	−0.637457	−	0.770486i	$-0.720014\pi$
$307$	− 27.0000i	− 1.54097i	0.637457	−	0.770486i	$-0.279986\pi$
$308$	0	0
$309$	−9.00000	−0.511992
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	29.0000i	1.63918i	0.572953	+	0.819588i	$0.305798\pi$
$313$	29.0000i	1.63918i	−0.572953	+	0.819588i	$0.694202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	0	0
$319$	45.0000	2.51952
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	18.0000i	1.00155i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.0000i	0.608301i
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	− 12.0000i	− 0.657596i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.0000i	1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$	26.0000i	1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10.0000i	− 0.536828i	−0.963304	−	0.268414i	$-0.913500\pi$
$347$	− 10.0000i	− 0.536828i	0.963304	−	0.268414i	$-0.0864995\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	− 33.0000i	− 1.75641i	−0.478282	−	0.878206i	$-0.658740\pi$
$353$	− 33.0000i	− 1.75641i	0.478282	−	0.878206i	$-0.341260\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.00000i	− 0.158777i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	14.0000i	0.734809i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 29.0000i	− 1.51379i	−0.653538	−	0.756894i	$-0.726716\pi$
$367$	− 29.0000i	− 1.51379i	0.653538	−	0.756894i	$-0.273284\pi$
$368$	0	0
$369$	16.0000	0.832927
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.00000i	0.463524i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 12.0000i	− 0.609994i
$388$	0	0
$389$	−25.0000	−1.26755	−0.633775	−	0.773517i	$-0.718496\pi$
$389$	−25.0000	−1.26755	−0.633775	+	0.773517i	$0.718496\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	6.00000i	0.302660i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	29.0000i	1.45547i	0.685859	+	0.727734i	$0.259427\pi$
$397$	29.0000i	1.45547i	−0.685859	+	0.727734i	$0.740573\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	9.00000	0.449439	0.224719	−	0.974424i	$-0.427853\pi$
$401$	9.00000	0.449439	0.224719	+	0.974424i	$0.427853\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 30.0000i	− 1.48704i
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−16.0000	−0.789222
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.0000i	0.881464i
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	−5.00000	−0.241402
$430$	0	0
$431$	−23.0000	−1.10787	−0.553936	−	0.832560i	$-0.686875\pi$
$431$	−23.0000	−1.10787	−0.553936	+	0.832560i	$0.686875\pi$
$432$	0	0
$433$	− 30.0000i	− 1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$	− 30.0000i	− 1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 36.0000i	− 1.72211i
$438$	0	0
$439$	−34.0000	−1.62273	−0.811366	−	0.584539i	$-0.801275\pi$
$439$	−34.0000	−1.62273	−0.811366	+	0.584539i	$0.801275\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	− 30.0000i	− 1.42534i	−0.701498	−	0.712672i	$-0.747485\pi$
$443$	− 30.0000i	− 1.42534i	0.701498	−	0.712672i	$-0.252515\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.0000i	0.662177i
$448$	0	0
$449$	33.0000	1.55737	0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000	1.55737	0.778683	+	0.627417i	$0.215888\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	0	0
$453$	19.0000i	0.892698i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.0000i	1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$	28.0000i	1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	− 20.0000i	− 0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	− 20.0000i	− 0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.0000i	1.52706i	0.645774	+	0.763529i	$0.276535\pi$
$467$	33.0000i	1.52706i	−0.645774	+	0.763529i	$0.723465\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	− 30.0000i	− 1.37940i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 24.0000i	− 1.09888i
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.0000i	1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$	26.0000i	1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$	0	0
$489$	6.00000	0.271329
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	0	0
$493$	− 27.0000i	− 1.21602i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	0	0
$503$	31.0000i	1.38222i	0.722749	+	0.691111i	$0.242878\pi$
$503$	31.0000i	1.38222i	−0.722749	+	0.691111i	$0.757122\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.0000i	0.532939i
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	− 30.0000i	− 1.32453i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15.0000i	0.659699i
$518$	0	0
$519$	−19.0000	−0.834007
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	16.0000	0.694341
$532$	0	0
$533$	8.00000i	0.346518i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.00000i	0.172613i
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−9.00000	−0.386940	−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000	−0.386940	−0.193470	+	0.981106i	$0.561974\pi$
$542$	0	0
$543$	− 20.0000i	− 0.858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−54.0000	−2.30048
$552$	0	0
$553$	3.00000i	0.127573i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.0000i	− 1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$	− 24.0000i	− 1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$	0	0
$559$	6.00000	0.253773
$560$	0	0
$561$	15.0000	0.633300
$562$	0	0
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	− 11.0000i	− 0.459532i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 17.0000i	− 0.707719i	−0.935299	−	0.353860i	$-0.884869\pi$
$577$	− 17.0000i	− 0.707719i	0.935299	−	0.353860i	$-0.115131\pi$
$578$	0	0
$579$	−8.00000	−0.332469
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 60.0000i	− 2.48495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.00000i	0.330195i	0.986277	+	0.165098i	$0.0527939\pi$
$587$	8.00000i	0.330195i	−0.986277	+	0.165098i	$0.947206\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	− 7.00000i	− 0.287456i	−0.989617	−	0.143728i	$-0.954091\pi$
$593$	− 7.00000i	− 0.287456i	0.989617	−	0.143728i	$-0.0459090\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−33.0000	−1.34834	−0.674172	−	0.738575i	$-0.735499\pi$
$599$	−33.0000	−1.34834	−0.674172	+	0.738575i	$0.735499\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	− 8.00000i	− 0.325785i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.0000i	0.690009i	0.938601	+	0.345004i	$0.112123\pi$
$607$	17.0000i	0.690009i	−0.938601	+	0.345004i	$0.887877\pi$
$608$	0	0
$609$	9.00000	0.364698
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.0000i	− 0.402585i	−0.979531	−	0.201292i	$-0.935486\pi$
$617$	− 10.0000i	− 0.402585i	0.979531	−	0.201292i	$-0.0645141\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	0	0
$623$	− 16.0000i	− 0.641026i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 30.0000i	− 1.19808i
$628$	0	0
$629$	−18.0000	−0.717707
$630$	0	0
$631$	9.00000	0.358284	0.179142	−	0.983823i	$-0.442668\pi$
$631$	9.00000	0.358284	0.179142	+	0.983823i	$0.442668\pi$
$632$	0	0
$633$	− 13.0000i	− 0.516704i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.00000i	− 0.0396214i
$638$	0	0
$639$	−16.0000	−0.632950
$640$	0	0
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	0	0
$643$	− 47.0000i	− 1.85350i	−0.375680	−	0.926750i	$-0.622591\pi$
$643$	− 47.0000i	− 1.85350i	0.375680	−	0.926750i	$-0.377409\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	40.0000	1.57014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	20.0000i	0.780274i
$658$	0	0
$659$	−25.0000	−0.973862	−0.486931	−	0.873441i	$-0.661884\pi$
$659$	−25.0000	−0.973862	−0.486931	+	0.873441i	$0.661884\pi$
$660$	0	0
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	0	0
$663$	3.00000i	0.116510i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	54.0000i	2.09089i
$668$	0	0
$669$	−25.0000	−0.966556
$670$	0	0
$671$	−20.0000	−0.772091
$672$	0	0
$673$	32.0000i	1.23351i	0.787155	+	0.616755i	$0.211553\pi$
$673$	32.0000i	1.23351i	−0.787155	+	0.616755i	$0.788447\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 33.0000i	− 1.26829i	−0.773213	−	0.634147i	$-0.781352\pi$
$677$	− 33.0000i	− 1.26829i	0.773213	−	0.634147i	$-0.218648\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	0	0
$681$	13.0000	0.498161
$682$	0	0
$683$	20.0000i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$	20.0000i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 16.0000i	− 0.610438i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	− 10.0000i	− 0.379869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 24.0000i	− 0.909065i
$698$	0	0
$699$	−8.00000	−0.302588
$700$	0	0
$701$	−21.0000	−0.793159	−0.396580	−	0.918000i	$-0.629803\pi$
$701$	−21.0000	−0.793159	−0.396580	+	0.918000i	$0.629803\pi$
$702$	0	0
$703$	36.0000i	1.35777i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.0000i	0.526524i
$708$	0	0
$709$	41.0000	1.53979	0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000	1.53979	0.769894	+	0.638172i	$0.220309\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 7.00000i	− 0.261420i
$718$	0	0
$719$	50.0000	1.86469	0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000	1.86469	0.932343	+	0.361576i	$0.117761\pi$
$720$	0	0
$721$	9.00000	0.335178
$722$	0	0
$723$	18.0000i	0.669427i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−18.0000	−0.665754
$732$	0	0
$733$	5.00000i	0.184679i	0.995728	+	0.0923396i	$0.0294345\pi$
$733$	5.00000i	0.184679i	−0.995728	+	0.0923396i	$0.970565\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 20.0000i	− 0.736709i
$738$	0	0
$739$	−37.0000	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	0	0
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24.0000i	0.878114i
$748$	0	0
$749$	2.00000	0.0730784
$750$	0	0
$751$	35.0000	1.27717	0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000	1.27717	0.638584	+	0.769552i	$0.279520\pi$
$752$	0	0
$753$	− 14.0000i	− 0.510188i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	0	0
$759$	−30.0000	−1.08893
$760$	0	0
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	0	0
$763$	− 11.0000i	− 0.398227i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000i	0.288863i
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	1.00000i	0.0359675i	0.999838	+	0.0179838i	$0.00572471\pi$
$773$	1.00000i	0.0359675i	−0.999838	+	0.0179838i	$0.994275\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6.00000i	− 0.215249i
$778$	0	0
$779$	−48.0000	−1.71978
$780$	0	0
$781$	−40.0000	−1.43131
$782$	0	0
$783$	45.0000i	1.60817i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	11.0000i	0.392108i	0.980593	+	0.196054i	$0.0628127\pi$
$787$	11.0000i	0.392108i	−0.980593	+	0.196054i	$0.937187\pi$
$788$	0	0
$789$	18.0000	0.640817
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 35.0000i	− 1.23976i	−0.784695	−	0.619882i	$-0.787181\pi$
$797$	− 35.0000i	− 1.23976i	0.784695	−	0.619882i	$-0.212819\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	0	0
$801$	32.0000	1.13066
$802$	0	0
$803$	50.0000i	1.76446i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10.0000i	− 0.352017i
$808$	0	0
$809$	23.0000	0.808637	0.404318	−	0.914618i	$-0.367509\pi$
$809$	23.0000	0.808637	0.404318	+	0.914618i	$0.367509\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	0	0
$813$	− 4.00000i	− 0.140286i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36.0000i	1.25948i
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−7.00000	−0.244302	−0.122151	−	0.992512i	$-0.538979\pi$
$821$	−7.00000	−0.244302	−0.122151	+	0.992512i	$0.538979\pi$
$822$	0	0
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.00000i	0.0695468i	0.999395	+	0.0347734i	$0.0110710\pi$
$827$	2.00000i	0.0695468i	−0.999395	+	0.0347734i	$0.988929\pi$
$828$	0	0
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	3.00000i	0.103944i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	− 13.0000i	− 0.447744i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 14.0000i	− 0.481046i
$848$	0	0
$849$	29.0000	0.995277
$850$	0	0
$851$	36.0000	1.23406
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	0	0
$863$	− 20.0000i	− 0.680808i	−0.940279	−	0.340404i	$-0.889436\pi$
$863$	− 20.0000i	− 0.680808i	0.940279	−	0.340404i	$-0.110564\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000i	0.271694i
$868$	0	0
$869$	−15.0000	−0.508840
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	− 14.0000i	− 0.473828i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	0	0
$879$	1.00000	0.0337292
$880$	0	0
$881$	−16.0000	−0.539054	−0.269527	−	0.962993i	$-0.586867\pi$
$881$	−16.0000	−0.539054	−0.269527	+	0.962993i	$0.586867\pi$
$882$	0	0
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	− 18.0000i	− 0.602347i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 6.00000i	− 0.200334i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	− 6.00000i	− 0.199667i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 30.0000i	− 0.996134i	−0.867139	−	0.498067i	$-0.834043\pi$
$907$	− 30.0000i	− 0.996134i	0.867139	−	0.498067i	$-0.165957\pi$
$908$	0	0
$909$	−28.0000	−0.928701
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	60.0000i	1.98571i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.00000i	− 0.198137i
$918$	0	0
$919$	25.0000	0.824674	0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000	0.824674	0.412337	+	0.911031i	$0.364713\pi$
$920$	0	0
$921$	27.0000	0.889680
$922$	0	0
$923$	− 8.00000i	− 0.263323i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	18.0000i	0.591198i
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	14.0000i	0.458339i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 7.00000i	− 0.228680i	−0.993442	−	0.114340i	$-0.963525\pi$
$937$	− 7.00000i	− 0.228680i	0.993442	−	0.114340i	$-0.0364753\pi$
$938$	0	0
$939$	−29.0000	−0.946379
$940$	0	0
$941$	4.00000	0.130396	0.0651981	−	0.997872i	$-0.479232\pi$
$941$	4.00000	0.130396	0.0651981	+	0.997872i	$0.479232\pi$
$942$	0	0
$943$	48.0000i	1.56310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	20.0000i	0.647864i	0.946080	+	0.323932i	$0.105005\pi$
$953$	20.0000i	0.647864i	−0.946080	+	0.323932i	$0.894995\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	45.0000i	1.45464i
$958$	0	0
$959$	16.0000	0.516667
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	22.0000i	0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$	22.0000i	0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$	0	0
$969$	−18.0000	−0.578243
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	− 18.0000i	− 0.577054i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	80.0000	2.55681
$980$	0	0
$981$	22.0000	0.702406
$982$	0	0
$983$	− 9.00000i	− 0.287055i	−0.989646	−	0.143528i	$-0.954155\pi$
$983$	− 9.00000i	− 0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.00000i	0.0954911i
$988$	0	0
$989$	36.0000	1.14473
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	20.0000i	0.634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.00000i	0.0316703i	0.999875	+	0.0158352i	$0.00504070\pi$
$997$	1.00000i	0.0316703i	−0.999875	+	0.0158352i	$0.994959\pi$
$998$	0	0
$999$	30.0000	0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.g.m.449.2		2
4.3	odd	2		1400.2.g.e.449.1		2
5.2	odd	4		560.2.a.e.1.1		1
5.3	odd	4		2800.2.a.i.1.1		1
5.4	even	2	inner	2800.2.g.m.449.1		2
15.2	even	4		5040.2.a.be.1.1		1
20.3	even	4		1400.2.a.k.1.1		1
20.7	even	4		280.2.a.b.1.1	✓	1
20.19	odd	2		1400.2.g.e.449.2		2
35.27	even	4		3920.2.a.r.1.1		1
40.27	even	4		2240.2.a.v.1.1		1
40.37	odd	4		2240.2.a.j.1.1		1
60.47	odd	4		2520.2.a.p.1.1		1
140.27	odd	4		1960.2.a.k.1.1		1
140.47	odd	12		1960.2.q.e.361.1		2
140.67	even	12		1960.2.q.m.961.1		2
140.83	odd	4		9800.2.a.n.1.1		1
140.87	odd	12		1960.2.q.e.961.1		2
140.107	even	12		1960.2.q.m.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.b.1.1	✓	1	20.7	even	4
560.2.a.e.1.1		1	5.2	odd	4
1400.2.a.k.1.1		1	20.3	even	4
1400.2.g.e.449.1		2	4.3	odd	2
1400.2.g.e.449.2		2	20.19	odd	2
1960.2.a.k.1.1		1	140.27	odd	4
1960.2.q.e.361.1		2	140.47	odd	12
1960.2.q.e.961.1		2	140.87	odd	12
1960.2.q.m.361.1		2	140.107	even	12
1960.2.q.m.961.1		2	140.67	even	12
2240.2.a.j.1.1		1	40.37	odd	4
2240.2.a.v.1.1		1	40.27	even	4
2520.2.a.p.1.1		1	60.47	odd	4
2800.2.a.i.1.1		1	5.3	odd	4
2800.2.g.m.449.1		2	5.4	even	2	inner
2800.2.g.m.449.2		2	1.1	even	1	trivial
3920.2.a.r.1.1		1	35.27	even	4
5040.2.a.be.1.1		1	15.2	even	4
9800.2.a.n.1.1		1	140.83	odd	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} +5.00000 q^{11} +1.00000i q^{13} -3.00000i q^{17} -6.00000 q^{19} +1.00000 q^{21} +6.00000i q^{23} +5.00000i q^{27} +9.00000 q^{29} +5.00000i q^{33} -6.00000i q^{37} -1.00000 q^{39} +8.00000 q^{41} -6.00000i q^{43} +3.00000i q^{47} -1.00000 q^{49} +3.00000 q^{51} -12.0000i q^{53} -6.00000i q^{57} +8.00000 q^{59} -4.00000 q^{61} -2.00000i q^{63} -4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} +10.0000i q^{73} -5.00000i q^{77} -3.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +9.00000i q^{87} +16.0000 q^{89} +1.00000 q^{91} -7.00000i q^{97} +10.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} + 10 q^{11} - 12 q^{19} + 2 q^{21} + 18 q^{29} - 2 q^{39} + 16 q^{41} - 2 q^{49} + 6 q^{51} + 16 q^{59} - 8 q^{61} - 12 q^{69} - 16 q^{71} - 6 q^{79} + 2 q^{81} + 32 q^{89} + 2 q^{91} + 20 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 + 10 * q^11 - 12 * q^19 + 2 * q^21 + 18 * q^29 - 2 * q^39 + 16 * q^41 - 2 * q^49 + 6 * q^51 + 16 * q^59 - 8 * q^61 - 12 * q^69 - 16 * q^71 - 6 * q^79 + 2 * q^81 + 32 * q^89 + 2 * q^91 + 20 * q^99

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