LMFDB - Embedded newform 2800.2.g.c.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
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.c.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +5.00000 q^{11} +3.00000i q^{13} -1.00000i q^{17} +6.00000 q^{19} -3.00000 q^{21} +6.00000i q^{23} -9.00000i q^{27} +9.00000 q^{29} +4.00000 q^{31} +15.0000i q^{33} +2.00000i q^{37} -9.00000 q^{39} -4.00000 q^{41} +10.0000i q^{43} +1.00000i q^{47} -1.00000 q^{49} +3.00000 q^{51} -4.00000i q^{53} +18.0000i q^{57} -8.00000 q^{59} -8.00000 q^{61} -6.00000i q^{63} -12.0000i q^{67} -18.0000 q^{69} -8.00000 q^{71} -2.00000i q^{73} +5.00000i q^{77} +13.0000 q^{79} +9.00000 q^{81} -4.00000i q^{83} +27.0000i q^{87} -4.00000 q^{89} -3.00000 q^{91} +12.0000i q^{93} -13.0000i q^{97} -30.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9} + 10 q^{11} + 12 q^{19} - 6 q^{21} + 18 q^{29} + 8 q^{31} - 18 q^{39} - 8 q^{41} - 2 q^{49} + 6 q^{51} - 16 q^{59} - 16 q^{61} - 36 q^{69} - 16 q^{71} + 26 q^{79} + 18 q^{81} - 8 q^{89} - 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100}) $ 2 * q - 12 * q^9 + 10 * q^11 + 12 * q^19 - 6 * q^21 + 18 * q^29 + 8 * q^31 - 18 * q^39 - 8 * q^41 - 2 * q^49 + 6 * q^51 - 16 * q^59 - 16 * q^61 - 36 * q^69 - 16 * q^71 + 26 * q^79 + 18 * q^81 - 8 * q^89 - 6 * q^91 - 60 * 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
$3$	3.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	3.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	−6.00000	−2.00000
$10$	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
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.00000i	− 0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$	− 1.00000i	− 0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$	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
$21$	−3.00000	−0.654654
$22$	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
$27$	− 9.00000i	− 1.73205i
$28$	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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	15.0000i	2.61116i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−9.00000	−1.44115
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.00000i	0.145865i	0.997337	+	0.0729325i	$0.0232358\pi$
$47$	1.00000i	0.145865i	−0.997337	+	0.0729325i	$0.976764\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	− 4.00000i	− 0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	− 4.00000i	− 0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	18.0000i	2.38416i
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	− 6.00000i	− 0.755929i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	−18.0000	−2.16695
$70$	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
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.00000i	0.569803i
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	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
$87$	27.0000i	2.89470i
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	12.0000i	1.24434i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.0000i	− 1.31995i	−0.751288	−	0.659975i	$-0.770567\pi$
$97$	− 13.0000i	− 1.31995i	0.751288	−	0.659975i	$-0.229433\pi$
$98$	0	0
$99$	−30.0000	−3.01511
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	19.0000i	1.87213i	0.351833	+	0.936063i	$0.385559\pi$
$103$	19.0000i	1.87213i	−0.351833	+	0.936063i	$0.614441\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	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
$117$	− 18.0000i	− 1.66410i
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	− 12.0000i	− 1.08200i
$124$	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
$129$	−30.0000	−2.64135
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	0	0
$133$	6.00000i	0.520266i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	15.0000i	1.25436i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	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	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	−36.0000	−2.75299
$172$	0	0
$173$	1.00000i	0.0760286i	0.999277	+	0.0380143i	$0.0121032\pi$
$173$	1.00000i	0.0760286i	−0.999277	+	0.0380143i	$0.987897\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 24.0000i	− 1.80395i
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	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
$183$	− 24.0000i	− 1.77413i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5.00000i	− 0.365636i
$188$	0	0
$189$	9.00000	0.654654
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	36.0000	2.53924
$202$	0	0
$203$	9.00000i	0.631676i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 36.0000i	− 2.50217i
$208$	0	0
$209$	30.0000	2.07514
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	0	0
$213$	− 24.0000i	− 1.64445i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.00000i	0.271538i
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	− 5.00000i	− 0.334825i	−0.985887	−	0.167412i	$-0.946459\pi$
$223$	− 5.00000i	− 0.334825i	0.985887	−	0.167412i	$-0.0535411\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.00000i	0.0663723i	0.999449	+	0.0331862i	$0.0105654\pi$
$227$	1.00000i	0.0663723i	−0.999449	+	0.0331862i	$0.989435\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	0	0
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	39.0000i	2.53332i
$238$	0	0
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.0000i	1.14531i
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	30.0000i	1.88608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−54.0000	−3.34252
$262$	0	0
$263$	2.00000i	0.123325i	0.998097	+	0.0616626i	$0.0196403\pi$
$263$	2.00000i	0.123325i	−0.998097	+	0.0616626i	$0.980360\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.0000i	− 0.734388i
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	− 9.00000i	− 0.544705i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.00000i	0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$	6.00000i	0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$	0	0
$279$	−24.0000	−1.43684
$280$	0	0
$281$	11.0000	0.656205	0.328102	−	0.944642i	$-0.393591\pi$
$281$	11.0000	0.656205	0.328102	+	0.944642i	$0.393591\pi$
$282$	0	0
$283$	− 31.0000i	− 1.84276i	−0.388664	−	0.921379i	$-0.627063\pi$
$283$	− 31.0000i	− 1.84276i	0.388664	−	0.921379i	$-0.372937\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.00000i	− 0.236113i
$288$	0	0
$289$	16.0000	0.941176
$290$	0	0
$291$	39.0000	2.28622
$292$	0	0
$293$	5.00000i	0.292103i	0.989277	+	0.146052i	$0.0466565\pi$
$293$	5.00000i	0.292103i	−0.989277	+	0.146052i	$0.953343\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 45.0000i	− 2.61116i
$298$	0	0
$299$	−18.0000	−1.04097
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	18.0000i	1.03407i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.0000i	1.31268i	0.754466	+	0.656340i	$0.227896\pi$
$307$	23.0000i	1.31268i	−0.754466	+	0.656340i	$0.772104\pi$
$308$	0	0
$309$	−57.0000	−3.24262
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	7.00000i	0.395663i	0.980236	+	0.197832i	$0.0633900\pi$
$313$	7.00000i	0.395663i	−0.980236	+	0.197832i	$0.936610\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 26.0000i	− 1.46031i	−0.683284	−	0.730153i	$-0.739449\pi$
$317$	− 26.0000i	− 1.46031i	0.683284	−	0.730153i	$-0.260551\pi$
$318$	0	0
$319$	45.0000	2.51952
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.00000i	0.497701i
$328$	0	0
$329$	−1.00000	−0.0551318
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	− 12.0000i	− 0.657596i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	0	0
$339$	42.0000	2.28113
$340$	0	0
$341$	20.0000	1.08306
$342$	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
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	27.0000	1.44115
$352$	0	0
$353$	− 3.00000i	− 0.159674i	−0.996808	−	0.0798369i	$-0.974560\pi$
$353$	− 3.00000i	− 0.159674i	0.996808	−	0.0798369i	$-0.0254400\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.00000i	0.158777i
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	42.0000i	2.20443i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.0000i	0.887393i	0.896177	+	0.443696i	$0.146333\pi$
$367$	17.0000i	0.887393i	−0.896177	+	0.443696i	$0.853667\pi$
$368$	0	0
$369$	24.0000	1.24939
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	27.0000i	1.39057i
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	−24.0000	−1.22956
$382$	0	0
$383$	36.0000i	1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$	36.0000i	1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 60.0000i	− 3.04997i
$388$	0	0
$389$	31.0000	1.57176	0.785881	−	0.618378i	$-0.212210\pi$
$389$	31.0000	1.57176	0.785881	+	0.618378i	$0.212210\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	30.0000i	1.51330i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.00000i	0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$	7.00000i	0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$	0	0
$399$	−18.0000	−0.901127
$400$	0	0
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	0	0
$403$	12.0000i	0.597763i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.0000i	0.495682i
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 42.0000i	− 2.05675i
$418$	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
$427$	− 8.00000i	− 0.387147i
$428$	0	0
$429$	−45.0000	−2.17262
$430$	0	0
$431$	9.00000	0.433515	0.216757	−	0.976226i	$-0.430452\pi$
$431$	9.00000	0.433515	0.216757	+	0.976226i	$0.430452\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	36.0000i	1.72211i
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	− 14.0000i	− 0.665160i	−0.943075	−	0.332580i	$-0.892081\pi$
$443$	− 14.0000i	− 0.665160i	0.943075	−	0.332580i	$-0.107919\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	18.0000i	0.851371i
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	0	0
$453$	− 15.0000i	− 0.704761i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.0000i	0.748448i	0.927338	+	0.374224i	$0.122091\pi$
$457$	16.0000i	0.748448i	−0.927338	+	0.374224i	$0.877909\pi$
$458$	0	0
$459$	−9.00000	−0.420084
$460$	0	0
$461$	40.0000	1.86299	0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000	1.86299	0.931493	+	0.363760i	$0.118507\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.00000i	− 0.231372i	−0.993286	−	0.115686i	$-0.963093\pi$
$467$	− 5.00000i	− 0.231372i	0.993286	−	0.115686i	$-0.0369067\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	6.00000	0.276465
$472$	0	0
$473$	50.0000i	2.29900i
$474$	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	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	− 18.0000i	− 0.819028i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	0	0
$489$	30.0000	1.35665
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	− 9.00000i	− 0.405340i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	−9.00000	−0.402895	−0.201448	−	0.979499i	$-0.564565\pi$
$499$	−9.00000	−0.402895	−0.201448	+	0.979499i	$0.564565\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	0	0
$503$	− 3.00000i	− 0.133763i	−0.997761	−	0.0668817i	$-0.978695\pi$
$503$	− 3.00000i	− 0.133763i	0.997761	−	0.0668817i	$-0.0213050\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.0000i	0.532939i
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	− 54.0000i	− 2.38416i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.00000i	0.219900i
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.00000i	− 0.174243i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	48.0000	2.08302
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 36.0000i	− 1.55351i
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	39.0000	1.67674	0.838370	−	0.545101i	$-0.183509\pi$
$541$	39.0000	1.67674	0.838370	+	0.545101i	$0.183509\pi$
$542$	0	0
$543$	− 60.0000i	− 2.57485i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.0000i	1.02617i	0.858339	+	0.513083i	$0.171497\pi$
$547$	24.0000i	1.02617i	−0.858339	+	0.513083i	$0.828503\pi$
$548$	0	0
$549$	48.0000	2.04859
$550$	0	0
$551$	54.0000	2.30048
$552$	0	0
$553$	13.0000i	0.552816i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.0000i	1.69485i	0.530912	+	0.847427i	$0.321850\pi$
$557$	40.0000i	1.69485i	−0.530912	+	0.847427i	$0.678150\pi$
$558$	0	0
$559$	−30.0000	−1.26886
$560$	0	0
$561$	15.0000	0.633300
$562$	0	0
$563$	− 28.0000i	− 1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$	− 28.0000i	− 1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.00000i	0.377964i
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	− 9.00000i	− 0.375980i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.0000i	0.541197i	0.962692	+	0.270599i	$0.0872216\pi$
$577$	13.0000i	0.541197i	−0.962692	+	0.270599i	$0.912778\pi$
$578$	0	0
$579$	−12.0000	−0.498703
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	− 20.0000i	− 0.828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.0000i	0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$	16.0000i	0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	−24.0000	−0.987228
$592$	0	0
$593$	27.0000i	1.10876i	0.832265	+	0.554379i	$0.187044\pi$
$593$	27.0000i	1.10876i	−0.832265	+	0.554379i	$0.812956\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	24.0000i	0.982255i
$598$	0	0
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	72.0000i	2.93207i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 13.0000i	− 0.527654i	−0.964570	−	0.263827i	$-0.915015\pi$
$607$	− 13.0000i	− 0.527654i	0.964570	−	0.263827i	$-0.0849848\pi$
$608$	0	0
$609$	−27.0000	−1.09410
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	− 42.0000i	− 1.69636i	−0.529705	−	0.848182i	$-0.677697\pi$
$613$	− 42.0000i	− 1.69636i	0.529705	−	0.848182i	$-0.322303\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 46.0000i	− 1.85189i	−0.377658	−	0.925945i	$-0.623271\pi$
$617$	− 46.0000i	− 1.85189i	0.377658	−	0.925945i	$-0.376729\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	54.0000	2.16695
$622$	0	0
$623$	− 4.00000i	− 0.160257i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	90.0000i	3.59425i
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	−47.0000	−1.87104	−0.935520	−	0.353273i	$-0.885069\pi$
$631$	−47.0000	−1.87104	−0.935520	+	0.353273i	$0.885069\pi$
$632$	0	0
$633$	33.0000i	1.31163i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.00000i	− 0.118864i
$638$	0	0
$639$	48.0000	1.89885
$640$	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
$643$	19.0000i	0.749287i	0.927169	+	0.374643i	$0.122235\pi$
$643$	19.0000i	0.749287i	−0.927169	+	0.374643i	$0.877765\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	0	0
$649$	−40.0000	−1.57014
$650$	0	0
$651$	−12.0000	−0.470317
$652$	0	0
$653$	− 2.00000i	− 0.0782660i	−0.999234	−	0.0391330i	$-0.987540\pi$
$653$	− 2.00000i	− 0.0782660i	0.999234	−	0.0391330i	$-0.0124596\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.0000i	0.468165i
$658$	0	0
$659$	31.0000	1.20759	0.603794	−	0.797140i	$-0.293655\pi$
$659$	31.0000	1.20759	0.603794	+	0.797140i	$0.293655\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	9.00000i	0.349531i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	54.0000i	2.09089i
$668$	0	0
$669$	15.0000	0.579934
$670$	0	0
$671$	−40.0000	−1.54418
$672$	0	0
$673$	− 4.00000i	− 0.154189i	−0.997024	−	0.0770943i	$-0.975436\pi$
$673$	− 4.00000i	− 0.154189i	0.997024	−	0.0770943i	$-0.0245643\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 27.0000i	− 1.03769i	−0.854867	−	0.518847i	$-0.826361\pi$
$677$	− 27.0000i	− 1.03769i	0.854867	−	0.518847i	$-0.173639\pi$
$678$	0	0
$679$	13.0000	0.498894
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	8.00000i	0.306111i	0.988218	+	0.153056i	$0.0489114\pi$
$683$	8.00000i	0.306111i	−0.988218	+	0.153056i	$0.951089\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 12.0000i	− 0.457829i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	− 30.0000i	− 1.13961i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.00000i	0.151511i
$698$	0	0
$699$	72.0000	2.72329
$700$	0	0
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	0	0
$703$	12.0000i	0.452589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	−78.0000	−2.92523
$712$	0	0
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.00000i	0.112037i
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−19.0000	−0.707597
$722$	0	0
$723$	− 78.0000i	− 2.90085i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 24.0000i	− 0.890111i	−0.895503	−	0.445055i	$-0.853184\pi$
$727$	− 24.0000i	− 0.890111i	0.895503	−	0.445055i	$-0.146816\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	10.0000	0.369863
$732$	0	0
$733$	47.0000i	1.73598i	0.496578	+	0.867992i	$0.334590\pi$
$733$	47.0000i	1.73598i	−0.496578	+	0.867992i	$0.665410\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 60.0000i	− 2.21013i
$738$	0	0
$739$	−5.00000	−0.183928	−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000	−0.183928	−0.0919640	+	0.995762i	$0.529314\pi$
$740$	0	0
$741$	−54.0000	−1.98374
$742$	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
$749$	−6.00000	−0.219235
$750$	0	0
$751$	19.0000	0.693320	0.346660	−	0.937991i	$-0.387316\pi$
$751$	19.0000	0.693320	0.346660	+	0.937991i	$0.387316\pi$
$752$	0	0
$753$	− 90.0000i	− 3.27978i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 32.0000i	− 1.16306i	−0.813525	−	0.581530i	$-0.802454\pi$
$757$	− 32.0000i	− 1.16306i	0.813525	−	0.581530i	$-0.197546\pi$
$758$	0	0
$759$	−90.0000	−3.26679
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	3.00000i	0.108607i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	−42.0000	−1.51259
$772$	0	0
$773$	− 45.0000i	− 1.61854i	−0.587439	−	0.809269i	$-0.699864\pi$
$773$	− 45.0000i	− 1.61854i	0.587439	−	0.809269i	$-0.300136\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6.00000i	− 0.215249i
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−40.0000	−1.43131
$782$	0	0
$783$	− 81.0000i	− 2.89470i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 31.0000i	− 1.10503i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	− 31.0000i	− 1.10503i	0.833503	−	0.552515i	$-0.186332\pi$
$788$	0	0
$789$	−6.00000	−0.213606
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	− 24.0000i	− 0.852265i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 33.0000i	− 1.16892i	−0.811423	−	0.584460i	$-0.801306\pi$
$797$	− 33.0000i	− 1.16892i	0.811423	−	0.584460i	$-0.198694\pi$
$798$	0	0
$799$	1.00000	0.0353775
$800$	0	0
$801$	24.0000	0.847998
$802$	0	0
$803$	− 10.0000i	− 0.352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 54.0000i	− 1.90089i
$808$	0	0
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	60.0000i	2.09913i
$818$	0	0
$819$	18.0000	0.628971
$820$	0	0
$821$	−39.0000	−1.36111	−0.680555	−	0.732697i	$-0.738261\pi$
$821$	−39.0000	−1.36111	−0.680555	+	0.732697i	$0.738261\pi$
$822$	0	0
$823$	28.0000i	0.976019i	0.872838	+	0.488009i	$0.162277\pi$
$823$	28.0000i	0.976019i	−0.872838	+	0.488009i	$0.837723\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 50.0000i	− 1.73867i	−0.494223	−	0.869335i	$-0.664547\pi$
$827$	− 50.0000i	− 1.73867i	0.494223	−	0.869335i	$-0.335453\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	1.00000i	0.0346479i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 36.0000i	− 1.24434i
$838$	0	0
$839$	2.00000	0.0690477	0.0345238	−	0.999404i	$-0.489009\pi$
$839$	2.00000	0.0690477	0.0345238	+	0.999404i	$0.489009\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	33.0000i	1.13658i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0000i	0.481046i
$848$	0	0
$849$	93.0000	3.19175
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	48.0000i	1.63017i
$868$	0	0
$869$	65.0000	2.20497
$870$	0	0
$871$	36.0000	1.21981
$872$	0	0
$873$	78.0000i	2.63990i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26.0000i	0.877958i	0.898497	+	0.438979i	$0.144660\pi$
$877$	26.0000i	0.877958i	−0.898497	+	0.438979i	$0.855340\pi$
$878$	0	0
$879$	−15.0000	−0.505937
$880$	0	0
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	0	0
$883$	− 36.0000i	− 1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$	− 36.0000i	− 1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	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
$891$	45.0000	1.50756
$892$	0	0
$893$	6.00000i	0.200782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 54.0000i	− 1.80301i
$898$	0	0
$899$	36.0000	1.20067
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	− 30.0000i	− 0.998337i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 14.0000i	− 0.464862i	−0.972613	−	0.232431i	$-0.925332\pi$
$907$	− 14.0000i	− 0.464862i	0.972613	−	0.232431i	$-0.0746680\pi$
$908$	0	0
$909$	−36.0000	−1.19404
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	− 20.0000i	− 0.661903i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.0000i	0.330229i
$918$	0	0
$919$	−31.0000	−1.02260	−0.511298	−	0.859404i	$-0.670835\pi$
$919$	−31.0000	−1.02260	−0.511298	+	0.859404i	$0.670835\pi$
$920$	0	0
$921$	−69.0000	−2.27363
$922$	0	0
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 114.000i	− 3.74425i
$928$	0	0
$929$	−28.0000	−0.918650	−0.459325	−	0.888268i	$-0.651909\pi$
$929$	−28.0000	−0.918650	−0.459325	+	0.888268i	$0.651909\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	54.0000i	1.76788i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 37.0000i	− 1.20874i	−0.796705	−	0.604369i	$-0.793425\pi$
$937$	− 37.0000i	− 1.20874i	0.796705	−	0.604369i	$-0.206575\pi$
$938$	0	0
$939$	−21.0000	−0.685309
$940$	0	0
$941$	32.0000	1.04317	0.521585	−	0.853199i	$-0.325341\pi$
$941$	32.0000	1.04317	0.521585	+	0.853199i	$0.325341\pi$
$942$	0	0
$943$	− 24.0000i	− 0.781548i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	78.0000	2.52932
$952$	0	0
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	135.000i	4.36393i
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 36.0000i	− 1.16008i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	0	0
$969$	18.0000	0.578243
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	− 14.0000i	− 0.448819i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	−20.0000	−0.639203
$980$	0	0
$981$	−18.0000	−0.574696
$982$	0	0
$983$	− 43.0000i	− 1.37149i	−0.727843	−	0.685744i	$-0.759477\pi$
$983$	− 43.0000i	− 1.37149i	0.727843	−	0.685744i	$-0.240523\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 3.00000i	− 0.0954911i
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	− 84.0000i	− 2.66566i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11.0000i	0.348373i	0.984713	+	0.174187i	$0.0557296\pi$
$997$	11.0000i	0.348373i	−0.984713	+	0.174187i	$0.944270\pi$
$998$	0	0
$999$	18.0000	0.569495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.g.c.449.2		2
4.3	odd	2		700.2.e.a.449.1		2
5.2	odd	4		2800.2.a.be.1.1		1
5.3	odd	4		560.2.a.a.1.1		1
5.4	even	2	inner	2800.2.g.c.449.1		2
12.11	even	2		6300.2.k.p.6049.1		2
15.8	even	4		5040.2.a.bd.1.1		1
20.3	even	4		140.2.a.b.1.1	✓	1
20.7	even	4		700.2.a.b.1.1		1
20.19	odd	2		700.2.e.a.449.2		2
28.27	even	2		4900.2.e.a.2549.2		2
35.13	even	4		3920.2.a.bl.1.1		1
40.3	even	4		2240.2.a.c.1.1		1
40.13	odd	4		2240.2.a.bb.1.1		1
60.23	odd	4		1260.2.a.h.1.1		1
60.47	odd	4		6300.2.a.bf.1.1		1
60.59	even	2		6300.2.k.p.6049.2		2
140.3	odd	12		980.2.i.j.961.1		2
140.23	even	12		980.2.i.b.361.1		2
140.27	odd	4		4900.2.a.u.1.1		1
140.83	odd	4		980.2.a.b.1.1		1
140.103	odd	12		980.2.i.j.361.1		2
140.123	even	12		980.2.i.b.961.1		2
140.139	even	2		4900.2.e.a.2549.1		2
420.83	even	4		8820.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.a.b.1.1	✓	1	20.3	even	4
560.2.a.a.1.1		1	5.3	odd	4
700.2.a.b.1.1		1	20.7	even	4
700.2.e.a.449.1		2	4.3	odd	2
700.2.e.a.449.2		2	20.19	odd	2
980.2.a.b.1.1		1	140.83	odd	4
980.2.i.b.361.1		2	140.23	even	12
980.2.i.b.961.1		2	140.123	even	12
980.2.i.j.361.1		2	140.103	odd	12
980.2.i.j.961.1		2	140.3	odd	12
1260.2.a.h.1.1		1	60.23	odd	4
2240.2.a.c.1.1		1	40.3	even	4
2240.2.a.bb.1.1		1	40.13	odd	4
2800.2.a.be.1.1		1	5.2	odd	4
2800.2.g.c.449.1		2	5.4	even	2	inner
2800.2.g.c.449.2		2	1.1	even	1	trivial
3920.2.a.bl.1.1		1	35.13	even	4
4900.2.a.u.1.1		1	140.27	odd	4
4900.2.e.a.2549.1		2	140.139	even	2
4900.2.e.a.2549.2		2	28.27	even	2
5040.2.a.bd.1.1		1	15.8	even	4
6300.2.a.bf.1.1		1	60.47	odd	4
6300.2.k.p.6049.1		2	12.11	even	2
6300.2.k.p.6049.2		2	60.59	even	2
8820.2.a.n.1.1		1	420.83	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +5.00000 q^{11} +3.00000i q^{13} -1.00000i q^{17} +6.00000 q^{19} -3.00000 q^{21} +6.00000i q^{23} -9.00000i q^{27} +9.00000 q^{29} +4.00000 q^{31} +15.0000i q^{33} +2.00000i q^{37} -9.00000 q^{39} -4.00000 q^{41} +10.0000i q^{43} +1.00000i q^{47} -1.00000 q^{49} +3.00000 q^{51} -4.00000i q^{53} +18.0000i q^{57} -8.00000 q^{59} -8.00000 q^{61} -6.00000i q^{63} -12.0000i q^{67} -18.0000 q^{69} -8.00000 q^{71} -2.00000i q^{73} +5.00000i q^{77} +13.0000 q^{79} +9.00000 q^{81} -4.00000i q^{83} +27.0000i q^{87} -4.00000 q^{89} -3.00000 q^{91} +12.0000i q^{93} -13.0000i q^{97} -30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9} + 10 q^{11} + 12 q^{19} - 6 q^{21} + 18 q^{29} + 8 q^{31} - 18 q^{39} - 8 q^{41} - 2 q^{49} + 6 q^{51} - 16 q^{59} - 16 q^{61} - 36 q^{69} - 16 q^{71} + 26 q^{79} + 18 q^{81} - 8 q^{89} - 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100}) \) 2 * q - 12 * q^9 + 10 * q^11 + 12 * q^19 - 6 * q^21 + 18 * q^29 + 8 * q^31 - 18 * q^39 - 8 * q^41 - 2 * q^49 + 6 * q^51 - 16 * q^59 - 16 * q^61 - 36 * q^69 - 16 * q^71 + 26 * q^79 + 18 * q^81 - 8 * q^89 - 6 * q^91 - 60 * q^99

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