LMFDB - Embedded newform 2800.2.g.r.449.4

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:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$3.37228i$ of defining polynomial
Character	$\chi$	$=$	2800.449
Dual form			2800.2.g.r.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37228i q^{3} -1.00000i q^{7} -8.37228 q^{9} +O(q^{10})$	$q+3.37228i q^{3} -1.00000i q^{7} -8.37228 q^{9} -0.627719 q^{11} -1.37228i q^{13} -5.37228i q^{17} +6.74456 q^{19} +3.37228 q^{21} -6.74456i q^{23} -18.1168i q^{27} -1.37228 q^{29} +8.00000 q^{31} -2.11684i q^{33} +2.00000i q^{37} +4.62772 q^{39} -4.74456 q^{41} -2.74456i q^{43} +10.1168i q^{47} -1.00000 q^{49} +18.1168 q^{51} -0.744563i q^{53} +22.7446i q^{57} +8.00000 q^{59} +8.74456 q^{61} +8.37228i q^{63} -4.00000i q^{67} +22.7446 q^{69} -8.00000 q^{71} -6.00000i q^{73} +0.627719i q^{77} -2.11684 q^{79} +35.9783 q^{81} -13.4891i q^{83} -4.62772i q^{87} -3.25544 q^{89} -1.37228 q^{91} +26.9783i q^{93} -18.8614i q^{97} +5.25544 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 22 q^{9}+O(q^{10}) $ 4 * q - 22 * q^9	$ 4 q - 22 q^{9} - 14 q^{11} + 4 q^{19} + 2 q^{21} + 6 q^{29} + 32 q^{31} + 30 q^{39} + 4 q^{41} - 4 q^{49} + 38 q^{51} + 32 q^{59} + 12 q^{61} + 68 q^{69} - 32 q^{71} + 26 q^{79} + 52 q^{81} - 36 q^{89} + 6 q^{91} + 44 q^{99}+O(q^{100}) $ 4 * q - 22 * q^9 - 14 * q^11 + 4 * q^19 + 2 * q^21 + 6 * q^29 + 32 * q^31 + 30 * q^39 + 4 * q^41 - 4 * q^49 + 38 * q^51 + 32 * q^59 + 12 * q^61 + 68 * q^69 - 32 * q^71 + 26 * q^79 + 52 * q^81 - 36 * q^89 + 6 * q^91 + 44 * q^99

Display 100 coefficients 10 coefficients

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
$3$	3.37228i	1.94699i	0.228714	+	0.973494i	$0.426548\pi$
$3$	3.37228i	1.94699i	−0.228714	+	0.973494i	$0.573452\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−8.37228	−2.79076
$10$	0	0
$11$	−0.627719	−0.189264	−0.0946322	−	0.995512i	$-0.530167\pi$
$11$	−0.627719	−0.189264	−0.0946322	+	0.995512i	$0.530167\pi$
$12$	0	0
$13$	− 1.37228i	− 0.380602i	−0.981726	−	0.190301i	$-0.939054\pi$
$13$	− 1.37228i	− 0.380602i	0.981726	−	0.190301i	$-0.0609465\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.37228i	− 1.30297i	−0.758662	−	0.651485i	$-0.774146\pi$
$17$	− 5.37228i	− 1.30297i	0.758662	−	0.651485i	$-0.225854\pi$
$18$	0	0
$19$	6.74456	1.54731	0.773654	−	0.633608i	$-0.218427\pi$
$19$	6.74456	1.54731	0.773654	+	0.633608i	$0.218427\pi$
$20$	0	0
$21$	3.37228	0.735892
$22$	0	0
$23$	− 6.74456i	− 1.40634i	−0.711022	−	0.703169i	$-0.751768\pi$
$23$	− 6.74456i	− 1.40634i	0.711022	−	0.703169i	$-0.248232\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 18.1168i	− 3.48659i
$28$	0	0
$29$	−1.37228	−0.254826	−0.127413	−	0.991850i	$-0.540667\pi$
$29$	−1.37228	−0.254826	−0.127413	+	0.991850i	$0.540667\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	− 2.11684i	− 0.368495i
$34$	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
$39$	4.62772	0.741028
$40$	0	0
$41$	−4.74456	−0.740976	−0.370488	−	0.928837i	$-0.620810\pi$
$41$	−4.74456	−0.740976	−0.370488	+	0.928837i	$0.620810\pi$
$42$	0	0
$43$	− 2.74456i	− 0.418542i	−0.977858	−	0.209271i	$-0.932891\pi$
$43$	− 2.74456i	− 0.418542i	0.977858	−	0.209271i	$-0.0671091\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.1168i	1.47569i	0.674968	+	0.737847i	$0.264157\pi$
$47$	10.1168i	1.47569i	−0.674968	+	0.737847i	$0.735843\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	18.1168	2.53687
$52$	0	0
$53$	− 0.744563i	− 0.102274i	−0.998692	−	0.0511368i	$-0.983716\pi$
$53$	− 0.744563i	− 0.102274i	0.998692	−	0.0511368i	$-0.0162844\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	22.7446i	3.01259i
$58$	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
$61$	8.74456	1.11963	0.559813	−	0.828619i	$-0.310873\pi$
$61$	8.74456	1.11963	0.559813	+	0.828619i	$0.310873\pi$
$62$	0	0
$63$	8.37228i	1.05481i
$64$	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
$69$	22.7446	2.73812
$70$	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
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.627719i	0.0715352i
$78$	0	0
$79$	−2.11684	−0.238164	−0.119082	−	0.992884i	$-0.537995\pi$
$79$	−2.11684	−0.238164	−0.119082	+	0.992884i	$0.537995\pi$
$80$	0	0
$81$	35.9783	3.99758
$82$	0	0
$83$	− 13.4891i	− 1.48062i	−0.672264	−	0.740312i	$-0.734678\pi$
$83$	− 13.4891i	− 1.48062i	0.672264	−	0.740312i	$-0.265322\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.62772i	− 0.496144i
$88$	0	0
$89$	−3.25544	−0.345076	−0.172538	−	0.985003i	$-0.555197\pi$
$89$	−3.25544	−0.345076	−0.172538	+	0.985003i	$0.555197\pi$
$90$	0	0
$91$	−1.37228	−0.143854
$92$	0	0
$93$	26.9783i	2.79751i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 18.8614i	− 1.91509i	−0.288291	−	0.957543i	$-0.593087\pi$
$97$	− 18.8614i	− 1.91509i	0.288291	−	0.957543i	$-0.406913\pi$
$98$	0	0
$99$	5.25544	0.528191
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	11.3723i	1.12054i	0.828309	+	0.560272i	$0.189303\pi$
$103$	11.3723i	1.12054i	−0.828309	+	0.560272i	$0.810697\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.74456i	− 0.265327i	−0.991161	−	0.132663i	$-0.957647\pi$
$107$	− 2.74456i	− 0.265327i	0.991161	−	0.132663i	$-0.0423529\pi$
$108$	0	0
$109$	5.37228	0.514571	0.257286	−	0.966335i	$-0.417172\pi$
$109$	5.37228	0.514571	0.257286	+	0.966335i	$0.417172\pi$
$110$	0	0
$111$	−6.74456	−0.640166
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.4891i	1.06217i
$118$	0	0
$119$	−5.37228	−0.492476
$120$	0	0
$121$	−10.6060	−0.964179
$122$	0	0
$123$	− 16.0000i	− 1.44267i
$124$	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
$129$	9.25544	0.814896
$130$	0	0
$131$	−6.74456	−0.589275	−0.294638	−	0.955609i	$-0.595199\pi$
$131$	−6.74456	−0.589275	−0.294638	+	0.955609i	$0.595199\pi$
$132$	0	0
$133$	− 6.74456i	− 0.584828i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.25544i	− 0.278131i	−0.990283	−	0.139065i	$-0.955590\pi$
$137$	− 3.25544i	− 0.278131i	0.990283	−	0.139065i	$-0.0444098\pi$
$138$	0	0
$139$	6.74456	0.572066	0.286033	−	0.958220i	$-0.407663\pi$
$139$	6.74456	0.572066	0.286033	+	0.958220i	$0.407663\pi$
$140$	0	0
$141$	−34.1168	−2.87316
$142$	0	0
$143$	0.861407i	0.0720344i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.37228i	− 0.278141i
$148$	0	0
$149$	7.48913	0.613533	0.306767	−	0.951785i	$-0.400753\pi$
$149$	7.48913	0.613533	0.306767	+	0.951785i	$0.400753\pi$
$150$	0	0
$151$	2.11684	0.172266	0.0861332	−	0.996284i	$-0.472549\pi$
$151$	2.11684	0.172266	0.0861332	+	0.996284i	$0.472549\pi$
$152$	0	0
$153$	44.9783i	3.63628i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 7.48913i	− 0.597697i	−0.954301	−	0.298849i	$-0.903397\pi$
$157$	− 7.48913i	− 0.597697i	0.954301	−	0.298849i	$-0.0966026\pi$
$158$	0	0
$159$	2.51087	0.199125
$160$	0	0
$161$	−6.74456	−0.531546
$162$	0	0
$163$	5.25544i	0.411638i	0.978590	+	0.205819i	$0.0659858\pi$
$163$	5.25544i	0.411638i	−0.978590	+	0.205819i	$0.934014\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 11.3723i	− 0.880014i	−0.897994	−	0.440007i	$-0.854976\pi$
$167$	− 11.3723i	− 0.880014i	0.897994	−	0.440007i	$-0.145024\pi$
$168$	0	0
$169$	11.1168	0.855142
$170$	0	0
$171$	−56.4674	−4.31817
$172$	0	0
$173$	5.37228i	0.408447i	0.978924	+	0.204223i	$0.0654669\pi$
$173$	5.37228i	0.408447i	−0.978924	+	0.204223i	$0.934533\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	26.9783i	2.02781i
$178$	0	0
$179$	22.9783	1.71748	0.858738	−	0.512416i	$-0.171249\pi$
$179$	22.9783	1.71748	0.858738	+	0.512416i	$0.171249\pi$
$180$	0	0
$181$	−18.2337	−1.35530	−0.677650	−	0.735385i	$-0.737001\pi$
$181$	−18.2337	−1.35530	−0.677650	+	0.735385i	$0.737001\pi$
$182$	0	0
$183$	29.4891i	2.17990i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.37228i	0.246606i
$188$	0	0
$189$	−18.1168	−1.31781
$190$	0	0
$191$	24.8614	1.79891	0.899454	−	0.437015i	$-0.143964\pi$
$191$	24.8614	1.79891	0.899454	+	0.437015i	$0.143964\pi$
$192$	0	0
$193$	− 4.74456i	− 0.341521i	−0.985313	−	0.170761i	$-0.945378\pi$
$193$	− 4.74456i	− 0.341521i	0.985313	−	0.170761i	$-0.0546225\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 26.2337i	− 1.86907i	−0.355867	−	0.934536i	$-0.615815\pi$
$197$	− 26.2337i	− 1.86907i	0.355867	−	0.934536i	$-0.384185\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	13.4891	0.951450
$202$	0	0
$203$	1.37228i	0.0963153i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	56.4674i	3.92475i
$208$	0	0
$209$	−4.23369	−0.292850
$210$	0	0
$211$	8.62772	0.593957	0.296978	−	0.954884i	$-0.404021\pi$
$211$	8.62772	0.593957	0.296978	+	0.954884i	$0.404021\pi$
$212$	0	0
$213$	− 26.9783i	− 1.84852i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 8.00000i	− 0.543075i
$218$	0	0
$219$	20.2337	1.36727
$220$	0	0
$221$	−7.37228	−0.495913
$222$	0	0
$223$	11.3723i	0.761544i	0.924669	+	0.380772i	$0.124342\pi$
$223$	11.3723i	0.761544i	−0.924669	+	0.380772i	$0.875658\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.86141i	− 0.588152i	−0.955782	−	0.294076i	$-0.904988\pi$
$227$	− 8.86141i	− 0.588152i	0.955782	−	0.294076i	$-0.0950119\pi$
$228$	0	0
$229$	−22.2337	−1.46924	−0.734622	−	0.678477i	$-0.762640\pi$
$229$	−22.2337	−1.46924	−0.734622	+	0.678477i	$0.762640\pi$
$230$	0	0
$231$	−2.11684	−0.139278
$232$	0	0
$233$	− 12.7446i	− 0.834924i	−0.908694	−	0.417462i	$-0.862920\pi$
$233$	− 12.7446i	− 0.834924i	0.908694	−	0.417462i	$-0.137080\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 7.13859i	− 0.463701i
$238$	0	0
$239$	19.3723	1.25309	0.626544	−	0.779386i	$-0.284469\pi$
$239$	19.3723	1.25309	0.626544	+	0.779386i	$0.284469\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	66.9783i	4.29666i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 9.25544i	− 0.588909i
$248$	0	0
$249$	45.4891	2.88276
$250$	0	0
$251$	6.74456	0.425713	0.212857	−	0.977083i	$-0.431723\pi$
$251$	6.74456	0.425713	0.212857	+	0.977083i	$0.431723\pi$
$252$	0	0
$253$	4.23369i	0.266170i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.510875i	0.0318675i	0.999873	+	0.0159337i	$0.00507208\pi$
$257$	0.510875i	0.0318675i	−0.999873	+	0.0159337i	$0.994928\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	11.4891	0.711159
$262$	0	0
$263$	12.2337i	0.754362i	0.926140	+	0.377181i	$0.123106\pi$
$263$	12.2337i	0.754362i	−0.926140	+	0.377181i	$0.876894\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 10.9783i	− 0.671858i
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	13.4891	0.819406	0.409703	−	0.912219i	$-0.365632\pi$
$271$	13.4891	0.819406	0.409703	+	0.912219i	$0.365632\pi$
$272$	0	0
$273$	− 4.62772i	− 0.280082i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.9783i	1.26046i	0.776408	+	0.630230i	$0.217040\pi$
$277$	20.9783i	1.26046i	−0.776408	+	0.630230i	$0.782960\pi$
$278$	0	0
$279$	−66.9783	−4.00988
$280$	0	0
$281$	−21.6060	−1.28890	−0.644452	−	0.764645i	$-0.722914\pi$
$281$	−21.6060	−1.28890	−0.644452	+	0.764645i	$0.722914\pi$
$282$	0	0
$283$	26.1168i	1.55249i	0.630434	+	0.776243i	$0.282877\pi$
$283$	26.1168i	1.55249i	−0.630434	+	0.776243i	$0.717123\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.74456i	0.280063i
$288$	0	0
$289$	−11.8614	−0.697730
$290$	0	0
$291$	63.6060	3.72865
$292$	0	0
$293$	7.88316i	0.460539i	0.973127	+	0.230269i	$0.0739608\pi$
$293$	7.88316i	0.460539i	−0.973127	+	0.230269i	$0.926039\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	11.3723i	0.659887i
$298$	0	0
$299$	−9.25544	−0.535256
$300$	0	0
$301$	−2.74456	−0.158194
$302$	0	0
$303$	− 20.2337i	− 1.16240i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.8832i	0.792354i	0.918174	+	0.396177i	$0.129663\pi$
$307$	13.8832i	0.792354i	−0.918174	+	0.396177i	$0.870337\pi$
$308$	0	0
$309$	−38.3505	−2.18169
$310$	0	0
$311$	1.25544	0.0711893	0.0355947	−	0.999366i	$-0.488667\pi$
$311$	1.25544	0.0711893	0.0355947	+	0.999366i	$0.488667\pi$
$312$	0	0
$313$	20.1168i	1.13707i	0.822659	+	0.568536i	$0.192490\pi$
$313$	20.1168i	1.13707i	−0.822659	+	0.568536i	$0.807510\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	0	0
$319$	0.861407	0.0482295
$320$	0	0
$321$	9.25544	0.516588
$322$	0	0
$323$	− 36.2337i	− 2.01610i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.1168i	1.00186i
$328$	0	0
$329$	10.1168	0.557760
$330$	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
$333$	− 16.7446i	− 0.917596i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 15.4891i	− 0.843746i	−0.906655	−	0.421873i	$-0.861373\pi$
$337$	− 15.4891i	− 0.843746i	0.906655	−	0.421873i	$-0.138627\pi$
$338$	0	0
$339$	−6.74456	−0.366314
$340$	0	0
$341$	−5.02175	−0.271943
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 13.2554i	− 0.711589i	−0.934564	−	0.355795i	$-0.884210\pi$
$347$	− 13.2554i	− 0.711589i	0.934564	−	0.355795i	$-0.115790\pi$
$348$	0	0
$349$	3.48913	0.186769	0.0933843	−	0.995630i	$-0.470231\pi$
$349$	3.48913	0.186769	0.0933843	+	0.995630i	$0.470231\pi$
$350$	0	0
$351$	−24.8614	−1.32700
$352$	0	0
$353$	26.8614i	1.42969i	0.699284	+	0.714844i	$0.253502\pi$
$353$	26.8614i	1.42969i	−0.699284	+	0.714844i	$0.746498\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 18.1168i	− 0.958845i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	0	0
$363$	− 35.7663i	− 1.87724i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8.86141i	− 0.462562i	−0.972887	−	0.231281i	$-0.925708\pi$
$367$	− 8.86141i	− 0.462562i	0.972887	−	0.231281i	$-0.0742916\pi$
$368$	0	0
$369$	39.7228	2.06789
$370$	0	0
$371$	−0.744563	−0.0386558
$372$	0	0
$373$	− 19.2554i	− 0.997009i	−0.866887	−	0.498504i	$-0.833883\pi$
$373$	− 19.2554i	− 0.997009i	0.866887	−	0.498504i	$-0.166117\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.88316i	0.0969875i
$378$	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
$381$	−26.9783	−1.38214
$382$	0	0
$383$	− 5.48913i	− 0.280481i	−0.990117	−	0.140241i	$-0.955212\pi$
$383$	− 5.48913i	− 0.280481i	0.990117	−	0.140241i	$-0.0447876\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	22.9783i	1.16805i
$388$	0	0
$389$	10.8614	0.550695	0.275348	−	0.961345i	$-0.411207\pi$
$389$	10.8614	0.550695	0.275348	+	0.961345i	$0.411207\pi$
$390$	0	0
$391$	−36.2337	−1.83242
$392$	0	0
$393$	− 22.7446i	− 1.14731i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 37.3723i	− 1.87566i	−0.347094	−	0.937831i	$-0.612831\pi$
$397$	− 37.3723i	− 1.87566i	0.347094	−	0.937831i	$-0.387169\pi$
$398$	0	0
$399$	22.7446	1.13865
$400$	0	0
$401$	1.60597	0.0801983	0.0400991	−	0.999196i	$-0.487233\pi$
$401$	1.60597	0.0801983	0.0400991	+	0.999196i	$0.487233\pi$
$402$	0	0
$403$	− 10.9783i	− 0.546866i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.25544i	− 0.0622297i
$408$	0	0
$409$	11.4891	0.568101	0.284050	−	0.958809i	$-0.408322\pi$
$409$	11.4891	0.568101	0.284050	+	0.958809i	$0.408322\pi$
$410$	0	0
$411$	10.9783	0.541517
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	22.7446i	1.11381i
$418$	0	0
$419$	−37.4891	−1.83146	−0.915732	−	0.401790i	$-0.868388\pi$
$419$	−37.4891	−1.83146	−0.915732	+	0.401790i	$0.868388\pi$
$420$	0	0
$421$	21.6060	1.05301	0.526505	−	0.850172i	$-0.323502\pi$
$421$	21.6060	1.05301	0.526505	+	0.850172i	$0.323502\pi$
$422$	0	0
$423$	− 84.7011i	− 4.11831i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.74456i	− 0.423179i
$428$	0	0
$429$	−2.90491	−0.140250
$430$	0	0
$431$	−12.6277	−0.608256	−0.304128	−	0.952631i	$-0.598365\pi$
$431$	−12.6277	−0.608256	−0.304128	+	0.952631i	$0.598365\pi$
$432$	0	0
$433$	− 16.9783i	− 0.815923i	−0.912999	−	0.407961i	$-0.866240\pi$
$433$	− 16.9783i	− 0.815923i	0.912999	−	0.407961i	$-0.133760\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 45.4891i	− 2.17604i
$438$	0	0
$439$	28.2337	1.34752	0.673760	−	0.738950i	$-0.264678\pi$
$439$	28.2337	1.34752	0.673760	+	0.738950i	$0.264678\pi$
$440$	0	0
$441$	8.37228	0.398680
$442$	0	0
$443$	5.25544i	0.249693i	0.992176	+	0.124847i	$0.0398439\pi$
$443$	5.25544i	0.249693i	−0.992176	+	0.124847i	$0.960156\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	25.2554i	1.19454i
$448$	0	0
$449$	0.116844	0.00551421	0.00275710	−	0.999996i	$-0.499122\pi$
$449$	0.116844	0.00551421	0.00275710	+	0.999996i	$0.499122\pi$
$450$	0	0
$451$	2.97825	0.140240
$452$	0	0
$453$	7.13859i	0.335400i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 16.7446i	− 0.783278i	−0.920119	−	0.391639i	$-0.871908\pi$
$457$	− 16.7446i	− 0.783278i	0.920119	−	0.391639i	$-0.128092\pi$
$458$	0	0
$459$	−97.3288	−4.54292
$460$	0	0
$461$	−12.7446	−0.593573	−0.296787	−	0.954944i	$-0.595915\pi$
$461$	−12.7446	−0.593573	−0.296787	+	0.954944i	$0.595915\pi$
$462$	0	0
$463$	29.4891i	1.37048i	0.728319	+	0.685238i	$0.240302\pi$
$463$	29.4891i	1.37048i	−0.728319	+	0.685238i	$0.759698\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 31.6060i	− 1.46255i	−0.682083	−	0.731275i	$-0.738926\pi$
$467$	− 31.6060i	− 1.46255i	0.682083	−	0.731275i	$-0.261074\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	25.2554	1.16371
$472$	0	0
$473$	1.72281i	0.0792150i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.23369i	0.285421i
$478$	0	0
$479$	−12.2337	−0.558971	−0.279486	−	0.960150i	$-0.590164\pi$
$479$	−12.2337	−0.558971	−0.279486	+	0.960150i	$0.590164\pi$
$480$	0	0
$481$	2.74456	0.125141
$482$	0	0
$483$	− 22.7446i	− 1.03491i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.23369i	− 0.191847i	−0.995389	−	0.0959234i	$-0.969420\pi$
$487$	− 4.23369i	− 0.191847i	0.995389	−	0.0959234i	$-0.0305804\pi$
$488$	0	0
$489$	−17.7228	−0.801453
$490$	0	0
$491$	−17.8832	−0.807056	−0.403528	−	0.914967i	$-0.632216\pi$
$491$	−17.8832	−0.807056	−0.403528	+	0.914967i	$0.632216\pi$
$492$	0	0
$493$	7.37228i	0.332031i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	3.13859	0.140503	0.0702514	−	0.997529i	$-0.477620\pi$
$499$	3.13859	0.140503	0.0702514	+	0.997529i	$0.477620\pi$
$500$	0	0
$501$	38.3505	1.71338
$502$	0	0
$503$	12.6277i	0.563042i	0.959555	+	0.281521i	$0.0908389\pi$
$503$	12.6277i	0.563042i	−0.959555	+	0.281521i	$0.909161\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	37.4891i	1.66495i
$508$	0	0
$509$	−4.97825	−0.220657	−0.110329	−	0.993895i	$-0.535190\pi$
$509$	−4.97825	−0.220657	−0.110329	+	0.993895i	$0.535190\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	− 122.190i	− 5.39483i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.35053i	− 0.279296i
$518$	0	0
$519$	−18.1168	−0.795241
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	− 32.4674i	− 1.41970i	−0.704353	−	0.709850i	$-0.748763\pi$
$523$	− 32.4674i	− 1.41970i	0.704353	−	0.709850i	$-0.251237\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 42.9783i	− 1.87216i
$528$	0	0
$529$	−22.4891	−0.977788
$530$	0	0
$531$	−66.9783	−2.90661
$532$	0	0
$533$	6.51087i	0.282017i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	77.4891i	3.34390i
$538$	0	0
$539$	0.627719	0.0270378
$540$	0	0
$541$	20.3505	0.874938	0.437469	−	0.899234i	$-0.355875\pi$
$541$	20.3505	0.874938	0.437469	+	0.899234i	$0.355875\pi$
$542$	0	0
$543$	− 61.4891i	− 2.63875i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.9783i	0.640424i	0.947346	+	0.320212i	$0.103754\pi$
$547$	14.9783i	0.640424i	−0.947346	+	0.320212i	$0.896246\pi$
$548$	0	0
$549$	−73.2119	−3.12461
$550$	0	0
$551$	−9.25544	−0.394295
$552$	0	0
$553$	2.11684i	0.0900174i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.2337i	1.62001i	0.586421	+	0.810007i	$0.300537\pi$
$557$	38.2337i	1.62001i	−0.586421	+	0.810007i	$0.699463\pi$
$558$	0	0
$559$	−3.76631	−0.159298
$560$	0	0
$561$	−11.3723	−0.480138
$562$	0	0
$563$	5.48913i	0.231339i	0.993288	+	0.115670i	$0.0369014\pi$
$563$	5.48913i	0.231339i	−0.993288	+	0.115670i	$0.963099\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 35.9783i	− 1.51094i
$568$	0	0
$569$	−20.9783	−0.879454	−0.439727	−	0.898131i	$-0.644925\pi$
$569$	−20.9783	−0.879454	−0.439727	+	0.898131i	$0.644925\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	83.8397i	3.50245i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 17.6060i	− 0.732946i	−0.930429	−	0.366473i	$-0.880565\pi$
$577$	− 17.6060i	− 0.732946i	0.930429	−	0.366473i	$-0.119435\pi$
$578$	0	0
$579$	16.0000	0.664937
$580$	0	0
$581$	−13.4891	−0.559623
$582$	0	0
$583$	0.467376i	0.0193567i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.9783i	0.453121i	0.973997	+	0.226560i	$0.0727481\pi$
$587$	10.9783i	0.453121i	−0.973997	+	0.226560i	$0.927252\pi$
$588$	0	0
$589$	53.9565	2.22324
$590$	0	0
$591$	88.4674	3.63906
$592$	0	0
$593$	− 25.3723i	− 1.04191i	−0.853583	−	0.520957i	$-0.825575\pi$
$593$	− 25.3723i	− 1.04191i	0.853583	−	0.520957i	$-0.174425\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 53.9565i	− 2.20829i
$598$	0	0
$599$	7.60597	0.310771	0.155386	−	0.987854i	$-0.450338\pi$
$599$	7.60597	0.310771	0.155386	+	0.987854i	$0.450338\pi$
$600$	0	0
$601$	39.4891	1.61080	0.805398	−	0.592735i	$-0.201952\pi$
$601$	39.4891	1.61080	0.805398	+	0.592735i	$0.201952\pi$
$602$	0	0
$603$	33.4891i	1.36378i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 15.6060i	− 0.633427i	−0.948521	−	0.316713i	$-0.897421\pi$
$607$	− 15.6060i	− 0.633427i	0.948521	−	0.316713i	$-0.102579\pi$
$608$	0	0
$609$	−4.62772	−0.187525
$610$	0	0
$611$	13.8832	0.561652
$612$	0	0
$613$	− 31.4891i	− 1.27183i	−0.771758	−	0.635917i	$-0.780622\pi$
$613$	− 31.4891i	− 1.27183i	0.771758	−	0.635917i	$-0.219378\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	−1.25544	−0.0504603	−0.0252301	−	0.999682i	$-0.508032\pi$
$619$	−1.25544	−0.0504603	−0.0252301	+	0.999682i	$0.508032\pi$
$620$	0	0
$621$	−122.190	−4.90332
$622$	0	0
$623$	3.25544i	0.130426i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 14.2772i	− 0.570176i
$628$	0	0
$629$	10.7446	0.428414
$630$	0	0
$631$	3.37228	0.134248	0.0671242	−	0.997745i	$-0.478618\pi$
$631$	3.37228	0.134248	0.0671242	+	0.997745i	$0.478618\pi$
$632$	0	0
$633$	29.0951i	1.15643i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.37228i	0.0543718i
$638$	0	0
$639$	66.9783	2.64962
$640$	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
$643$	− 12.6277i	− 0.497989i	−0.968505	−	0.248994i	$-0.919900\pi$
$643$	− 12.6277i	− 0.497989i	0.968505	−	0.248994i	$-0.0801000\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 16.0000i	− 0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$	− 16.0000i	− 0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$	0	0
$649$	−5.02175	−0.197121
$650$	0	0
$651$	26.9783	1.05736
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	50.2337i	1.95980i
$658$	0	0
$659$	6.11684	0.238278	0.119139	−	0.992878i	$-0.461987\pi$
$659$	6.11684	0.238278	0.119139	+	0.992878i	$0.461987\pi$
$660$	0	0
$661$	3.25544	0.126622	0.0633109	−	0.997994i	$-0.479834\pi$
$661$	3.25544	0.126622	0.0633109	+	0.997994i	$0.479834\pi$
$662$	0	0
$663$	− 24.8614i	− 0.965537i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.25544i	0.358372i
$668$	0	0
$669$	−38.3505	−1.48272
$670$	0	0
$671$	−5.48913	−0.211905
$672$	0	0
$673$	− 31.7228i	− 1.22282i	−0.791312	−	0.611412i	$-0.790602\pi$
$673$	− 31.7228i	− 1.22282i	0.791312	−	0.611412i	$-0.209398\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.3505i	1.39706i	0.715579	+	0.698532i	$0.246163\pi$
$677$	36.3505i	1.39706i	−0.715579	+	0.698532i	$0.753837\pi$
$678$	0	0
$679$	−18.8614	−0.723834
$680$	0	0
$681$	29.8832	1.14513
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 74.9783i	− 2.86060i
$688$	0	0
$689$	−1.02175	−0.0389256
$690$	0	0
$691$	−13.4891	−0.513151	−0.256575	−	0.966524i	$-0.582594\pi$
$691$	−13.4891	−0.513151	−0.256575	+	0.966524i	$0.582594\pi$
$692$	0	0
$693$	− 5.25544i	− 0.199638i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	25.4891i	0.965469i
$698$	0	0
$699$	42.9783	1.62559
$700$	0	0
$701$	30.8614	1.16562	0.582810	−	0.812609i	$-0.301953\pi$
$701$	30.8614	1.16562	0.582810	+	0.812609i	$0.301953\pi$
$702$	0	0
$703$	13.4891i	0.508752i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	33.6060	1.26210	0.631049	−	0.775743i	$-0.282625\pi$
$709$	33.6060	1.26210	0.631049	+	0.775743i	$0.282625\pi$
$710$	0	0
$711$	17.7228	0.664657
$712$	0	0
$713$	− 53.9565i	− 2.02069i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	65.3288i	2.43975i
$718$	0	0
$719$	14.7446	0.549879	0.274940	−	0.961461i	$-0.411342\pi$
$719$	14.7446	0.549879	0.274940	+	0.961461i	$0.411342\pi$
$720$	0	0
$721$	11.3723	0.423526
$722$	0	0
$723$	87.6793i	3.26083i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32.0000i	1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$	32.0000i	1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$	0	0
$729$	−117.935	−4.36795
$730$	0	0
$731$	−14.7446	−0.545347
$732$	0	0
$733$	− 38.8614i	− 1.43538i	−0.696363	−	0.717689i	$-0.745200\pi$
$733$	− 38.8614i	− 1.43538i	0.696363	−	0.717689i	$-0.254800\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.51087i	0.0924893i
$738$	0	0
$739$	19.6060	0.721217	0.360609	−	0.932717i	$-0.382569\pi$
$739$	19.6060	0.721217	0.360609	+	0.932717i	$0.382569\pi$
$740$	0	0
$741$	31.2119	1.14660
$742$	0	0
$743$	29.4891i	1.08185i	0.841070	+	0.540926i	$0.181926\pi$
$743$	29.4891i	1.08185i	−0.841070	+	0.540926i	$0.818074\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	112.935i	4.13207i
$748$	0	0
$749$	−2.74456	−0.100284
$750$	0	0
$751$	−48.8614	−1.78298	−0.891489	−	0.453042i	$-0.850339\pi$
$751$	−48.8614	−1.78298	−0.891489	+	0.453042i	$0.850339\pi$
$752$	0	0
$753$	22.7446i	0.828858i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.2337i	1.38963i	0.719190	+	0.694814i	$0.244513\pi$
$757$	38.2337i	1.38963i	−0.719190	+	0.694814i	$0.755487\pi$
$758$	0	0
$759$	−14.2772	−0.518229
$760$	0	0
$761$	−40.9783	−1.48546	−0.742730	−	0.669591i	$-0.766469\pi$
$761$	−40.9783	−1.48546	−0.742730	+	0.669591i	$0.766469\pi$
$762$	0	0
$763$	− 5.37228i	− 0.194490i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 10.9783i	− 0.396402i
$768$	0	0
$769$	19.4891	0.702796	0.351398	−	0.936226i	$-0.385706\pi$
$769$	19.4891	0.702796	0.351398	+	0.936226i	$0.385706\pi$
$770$	0	0
$771$	−1.72281	−0.0620456
$772$	0	0
$773$	− 1.37228i	− 0.0493575i	−0.999695	−	0.0246788i	$-0.992144\pi$
$773$	− 1.37228i	− 0.0493575i	0.999695	−	0.0246788i	$-0.00785629\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.74456i	0.241960i
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	5.02175	0.179692
$782$	0	0
$783$	24.8614i	0.888474i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 13.8832i	− 0.494881i	−0.968903	−	0.247441i	$-0.920411\pi$
$787$	− 13.8832i	− 0.494881i	0.968903	−	0.247441i	$-0.0795895\pi$
$788$	0	0
$789$	−41.2554	−1.46873
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	− 12.0000i	− 0.426132i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 21.3723i	− 0.757045i	−0.925592	−	0.378523i	$-0.876432\pi$
$797$	− 21.3723i	− 0.757045i	0.925592	−	0.378523i	$-0.123568\pi$
$798$	0	0
$799$	54.3505	1.92278
$800$	0	0
$801$	27.2554	0.963024
$802$	0	0
$803$	3.76631i	0.132910i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.2337i	0.712260i
$808$	0	0
$809$	−14.6277	−0.514283	−0.257142	−	0.966374i	$-0.582781\pi$
$809$	−14.6277	−0.514283	−0.257142	+	0.966374i	$0.582781\pi$
$810$	0	0
$811$	−1.25544	−0.0440844	−0.0220422	−	0.999757i	$-0.507017\pi$
$811$	−1.25544	−0.0440844	−0.0220422	+	0.999757i	$0.507017\pi$
$812$	0	0
$813$	45.4891i	1.59537i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 18.5109i	− 0.647614i
$818$	0	0
$819$	11.4891	0.401463
$820$	0	0
$821$	−7.88316	−0.275124	−0.137562	−	0.990493i	$-0.543927\pi$
$821$	−7.88316	−0.275124	−0.137562	+	0.990493i	$0.543927\pi$
$822$	0	0
$823$	8.00000i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$	8.00000i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.2554i	0.460937i	0.973080	+	0.230468i	$0.0740258\pi$
$827$	13.2554i	0.460937i	−0.973080	+	0.230468i	$0.925974\pi$
$828$	0	0
$829$	−22.2337	−0.772208	−0.386104	−	0.922455i	$-0.626179\pi$
$829$	−22.2337	−0.772208	−0.386104	+	0.922455i	$0.626179\pi$
$830$	0	0
$831$	−70.7446	−2.45410
$832$	0	0
$833$	5.37228i	0.186139i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 144.935i	− 5.00968i
$838$	0	0
$839$	−22.7446	−0.785230	−0.392615	−	0.919703i	$-0.628429\pi$
$839$	−22.7446	−0.785230	−0.392615	+	0.919703i	$0.628429\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	0	0
$843$	− 72.8614i	− 2.50948i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.6060i	0.364425i
$848$	0	0
$849$	−88.0733	−3.02267
$850$	0	0
$851$	13.4891	0.462401
$852$	0	0
$853$	− 16.5109i	− 0.565322i	−0.959220	−	0.282661i	$-0.908783\pi$
$853$	− 16.5109i	− 0.565322i	0.959220	−	0.282661i	$-0.0912171\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	−24.4674	−0.834816	−0.417408	−	0.908719i	$-0.637061\pi$
$859$	−24.4674	−0.834816	−0.417408	+	0.908719i	$0.637061\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	0	0
$863$	− 2.51087i	− 0.0854712i	−0.999086	−	0.0427356i	$-0.986393\pi$
$863$	− 2.51087i	− 0.0854712i	0.999086	−	0.0427356i	$-0.0136073\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 40.0000i	− 1.35847i
$868$	0	0
$869$	1.32878	0.0450759
$870$	0	0
$871$	−5.48913	−0.185992
$872$	0	0
$873$	157.913i	5.34455i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	52.9783i	1.78895i	0.447120	+	0.894474i	$0.352450\pi$
$877$	52.9783i	1.78895i	−0.447120	+	0.894474i	$0.647550\pi$
$878$	0	0
$879$	−26.5842	−0.896663
$880$	0	0
$881$	−36.7446	−1.23796	−0.618978	−	0.785408i	$-0.712453\pi$
$881$	−36.7446	−1.23796	−0.618978	+	0.785408i	$0.712453\pi$
$882$	0	0
$883$	33.4891i	1.12700i	0.826116	+	0.563499i	$0.190545\pi$
$883$	33.4891i	1.12700i	−0.826116	+	0.563499i	$0.809455\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 58.9783i	− 1.98030i	−0.140025	−	0.990148i	$-0.544718\pi$
$887$	− 58.9783i	− 1.98030i	0.140025	−	0.990148i	$-0.455282\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−22.5842	−0.756600
$892$	0	0
$893$	68.2337i	2.28335i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 31.2119i	− 1.04214i
$898$	0	0
$899$	−10.9783	−0.366145
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	− 9.25544i	− 0.308002i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 45.7228i	− 1.51820i	−0.650973	−	0.759101i	$-0.725639\pi$
$907$	− 45.7228i	− 1.51820i	0.650973	−	0.759101i	$-0.274361\pi$
$908$	0	0
$909$	50.2337	1.66615
$910$	0	0
$911$	−45.9565	−1.52261	−0.761303	−	0.648396i	$-0.775440\pi$
$911$	−45.9565	−1.52261	−0.761303	+	0.648396i	$0.775440\pi$
$912$	0	0
$913$	8.46738i	0.280229i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.74456i	0.222725i
$918$	0	0
$919$	19.3723	0.639033	0.319516	−	0.947581i	$-0.396480\pi$
$919$	19.3723	0.639033	0.319516	+	0.947581i	$0.396480\pi$
$920$	0	0
$921$	−46.8179	−1.54270
$922$	0	0
$923$	10.9783i	0.361354i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 95.2119i	− 3.12717i
$928$	0	0
$929$	42.2337	1.38564	0.692821	−	0.721109i	$-0.256368\pi$
$929$	42.2337	1.38564	0.692821	+	0.721109i	$0.256368\pi$
$930$	0	0
$931$	−6.74456	−0.221044
$932$	0	0
$933$	4.23369i	0.138605i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.0951i	1.14651i	0.819378	+	0.573253i	$0.194319\pi$
$937$	35.0951i	1.14651i	−0.819378	+	0.573253i	$0.805681\pi$
$938$	0	0
$939$	−67.8397	−2.21386
$940$	0	0
$941$	−2.23369	−0.0728161	−0.0364081	−	0.999337i	$-0.511592\pi$
$941$	−2.23369	−0.0728161	−0.0364081	+	0.999337i	$0.511592\pi$
$942$	0	0
$943$	32.0000i	1.04206i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 28.0000i	− 0.909878i	−0.890523	−	0.454939i	$-0.849661\pi$
$947$	− 28.0000i	− 0.909878i	0.890523	−	0.454939i	$-0.150339\pi$
$948$	0	0
$949$	−8.23369	−0.267277
$950$	0	0
$951$	47.2119	1.53095
$952$	0	0
$953$	48.7446i	1.57899i	0.613756	+	0.789496i	$0.289658\pi$
$953$	48.7446i	1.57899i	−0.613756	+	0.789496i	$0.710342\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.90491i	0.0939023i
$958$	0	0
$959$	−3.25544	−0.105124
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	22.9783i	0.740464i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	36.2337i	1.16520i	0.812760	+	0.582598i	$0.197964\pi$
$967$	36.2337i	1.16520i	−0.812760	+	0.582598i	$0.802036\pi$
$968$	0	0
$969$	122.190	3.92531
$970$	0	0
$971$	−10.5109	−0.337310	−0.168655	−	0.985675i	$-0.553942\pi$
$971$	−10.5109	−0.337310	−0.168655	+	0.985675i	$0.553942\pi$
$972$	0	0
$973$	− 6.74456i	− 0.216221i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 7.02175i	− 0.224646i	−0.993672	−	0.112323i	$-0.964171\pi$
$977$	− 7.02175i	− 0.224646i	0.993672	−	0.112323i	$-0.0358291\pi$
$978$	0	0
$979$	2.04350	0.0653105
$980$	0	0
$981$	−44.9783	−1.43605
$982$	0	0
$983$	4.62772i	0.147601i	0.997273	+	0.0738007i	$0.0235129\pi$
$983$	4.62772i	0.147601i	−0.997273	+	0.0738007i	$0.976487\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	34.1168i	1.08595i
$988$	0	0
$989$	−18.5109	−0.588612
$990$	0	0
$991$	−53.9565	−1.71398	−0.856992	−	0.515329i	$-0.827670\pi$
$991$	−53.9565	−1.71398	−0.856992	+	0.515329i	$0.827670\pi$
$992$	0	0
$993$	− 40.4674i	− 1.28419i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24.1168i	0.763788i	0.924206	+	0.381894i	$0.124728\pi$
$997$	24.1168i	0.763788i	−0.924206	+	0.381894i	$0.875272\pi$
$998$	0	0
$999$	36.2337	1.14638

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.g.r.449.4		4
4.3	odd	2		1400.2.g.i.449.1		4
5.2	odd	4		560.2.a.h.1.2		2
5.3	odd	4		2800.2.a.bk.1.1		2
5.4	even	2	inner	2800.2.g.r.449.1		4
15.2	even	4		5040.2.a.by.1.1		2
20.3	even	4		1400.2.a.r.1.2		2
20.7	even	4		280.2.a.c.1.1	✓	2
20.19	odd	2		1400.2.g.i.449.4		4
35.27	even	4		3920.2.a.bt.1.1		2
40.27	even	4		2240.2.a.bk.1.2		2
40.37	odd	4		2240.2.a.bg.1.1		2
60.47	odd	4		2520.2.a.x.1.2		2
140.27	odd	4		1960.2.a.s.1.2		2
140.47	odd	12		1960.2.q.r.361.1		4
140.67	even	12		1960.2.q.t.961.2		4
140.83	odd	4		9800.2.a.bu.1.1		2
140.87	odd	12		1960.2.q.r.961.1		4
140.107	even	12		1960.2.q.t.361.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.c.1.1	✓	2	20.7	even	4
560.2.a.h.1.2		2	5.2	odd	4
1400.2.a.r.1.2		2	20.3	even	4
1400.2.g.i.449.1		4	4.3	odd	2
1400.2.g.i.449.4		4	20.19	odd	2
1960.2.a.s.1.2		2	140.27	odd	4
1960.2.q.r.361.1		4	140.47	odd	12
1960.2.q.r.961.1		4	140.87	odd	12
1960.2.q.t.361.2		4	140.107	even	12
1960.2.q.t.961.2		4	140.67	even	12
2240.2.a.bg.1.1		2	40.37	odd	4
2240.2.a.bk.1.2		2	40.27	even	4
2520.2.a.x.1.2		2	60.47	odd	4
2800.2.a.bk.1.1		2	5.3	odd	4
2800.2.g.r.449.1		4	5.4	even	2	inner
2800.2.g.r.449.4		4	1.1	even	1	trivial
3920.2.a.bt.1.1		2	35.27	even	4
5040.2.a.by.1.1		2	15.2	even	4
9800.2.a.bu.1.1		2	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+3.37228i q^{3} -1.00000i q^{7} -8.37228 q^{9} +O(q^{10})\)	\(q+3.37228i q^{3} -1.00000i q^{7} -8.37228 q^{9} -0.627719 q^{11} -1.37228i q^{13} -5.37228i q^{17} +6.74456 q^{19} +3.37228 q^{21} -6.74456i q^{23} -18.1168i q^{27} -1.37228 q^{29} +8.00000 q^{31} -2.11684i q^{33} +2.00000i q^{37} +4.62772 q^{39} -4.74456 q^{41} -2.74456i q^{43} +10.1168i q^{47} -1.00000 q^{49} +18.1168 q^{51} -0.744563i q^{53} +22.7446i q^{57} +8.00000 q^{59} +8.74456 q^{61} +8.37228i q^{63} -4.00000i q^{67} +22.7446 q^{69} -8.00000 q^{71} -6.00000i q^{73} +0.627719i q^{77} -2.11684 q^{79} +35.9783 q^{81} -13.4891i q^{83} -4.62772i q^{87} -3.25544 q^{89} -1.37228 q^{91} +26.9783i q^{93} -18.8614i q^{97} +5.25544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 22 q^{9}+O(q^{10}) \) 4 * q - 22 * q^9	\( 4 q - 22 q^{9} - 14 q^{11} + 4 q^{19} + 2 q^{21} + 6 q^{29} + 32 q^{31} + 30 q^{39} + 4 q^{41} - 4 q^{49} + 38 q^{51} + 32 q^{59} + 12 q^{61} + 68 q^{69} - 32 q^{71} + 26 q^{79} + 52 q^{81} - 36 q^{89} + 6 q^{91} + 44 q^{99}+O(q^{100}) \) 4 * q - 22 * q^9 - 14 * q^11 + 4 * q^19 + 2 * q^21 + 6 * q^29 + 32 * q^31 + 30 * q^39 + 4 * q^41 - 4 * q^49 + 38 * q^51 + 32 * q^59 + 12 * q^61 + 68 * q^69 - 32 * q^71 + 26 * q^79 + 52 * q^81 - 36 * q^89 + 6 * q^91 + 44 * q^99

Introduction

L-functions

Modular forms

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Database

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