LMFDB - Embedded newform 2312.2.b.g.577.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,2,Mod(577,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2312.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.b (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:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.g.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607i q^{3} +2.00000i q^{5} +1.23607i q^{7} +1.47214 q^{9} +O(q^{10})$	$q+1.23607i q^{3} +2.00000i q^{5} +1.23607i q^{7} +1.47214 q^{9} +1.23607i q^{11} +4.47214 q^{13} -2.47214 q^{15} +6.47214 q^{19} -1.52786 q^{21} +1.23607i q^{23} +1.00000 q^{25} +5.52786i q^{27} +2.00000i q^{29} +1.23607i q^{31} -1.52786 q^{33} -2.47214 q^{35} -10.9443i q^{37} +5.52786i q^{39} -2.00000i q^{41} +1.52786 q^{43} +2.94427i q^{45} +12.9443 q^{47} +5.47214 q^{49} +2.00000 q^{53} -2.47214 q^{55} +8.00000i q^{57} -14.4721 q^{59} -6.94427i q^{61} +1.81966i q^{63} +8.94427i q^{65} -12.0000 q^{67} -1.52786 q^{69} +9.23607i q^{71} +14.9443i q^{73} +1.23607i q^{75} -1.52786 q^{77} -11.7082i q^{79} -2.41641 q^{81} -1.52786 q^{83} -2.47214 q^{87} -7.52786 q^{89} +5.52786i q^{91} -1.52786 q^{93} +12.9443i q^{95} +2.00000i q^{97} +1.81966i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 8 q^{15} + 8 q^{19} - 24 q^{21} + 4 q^{25} - 24 q^{33} + 8 q^{35} + 24 q^{43} + 16 q^{47} + 4 q^{49} + 8 q^{53} + 8 q^{55} - 40 q^{59} - 48 q^{67} - 24 q^{69} - 24 q^{77} + 44 q^{81} - 24 q^{83} + 8 q^{87} - 48 q^{89} - 24 q^{93}+O(q^{100}) $ 4 * q - 12 * q^9 + 8 * q^15 + 8 * q^19 - 24 * q^21 + 4 * q^25 - 24 * q^33 + 8 * q^35 + 24 * q^43 + 16 * q^47 + 4 * q^49 + 8 * q^53 + 8 * q^55 - 40 * q^59 - 48 * q^67 - 24 * q^69 - 24 * q^77 + 44 * q^81 - 24 * q^83 + 8 * q^87 - 48 * q^89 - 24 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.23607i	0.713644i	0.934172	+	0.356822i	$0.116140\pi$
$3$	1.23607i	0.713644i	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	1.23607i	0.467190i	0.972334	+	0.233595i	$0.0750489\pi$
$7$	1.23607i	0.467190i	−0.972334	+	0.233595i	$0.924951\pi$
$8$	0	0
$9$	1.47214	0.490712
$10$	0	0
$11$	1.23607i	0.372689i	0.982485	+	0.186344i	$0.0596640\pi$
$11$	1.23607i	0.372689i	−0.982485	+	0.186344i	$0.940336\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	−2.47214	−0.638303
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	0	0
$21$	−1.52786	−0.333407
$22$	0	0
$23$	1.23607i	0.257738i	0.991662	+	0.128869i	$0.0411347\pi$
$23$	1.23607i	0.257738i	−0.991662	+	0.128869i	$0.958865\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.52786i	1.06384i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	1.23607i	0.222004i	0.993820	+	0.111002i	$0.0354061\pi$
$31$	1.23607i	0.222004i	−0.993820	+	0.111002i	$0.964594\pi$
$32$	0	0
$33$	−1.52786	−0.265967
$34$	0	0
$35$	−2.47214	−0.417867
$36$	0	0
$37$	− 10.9443i	− 1.79923i	−0.436687	−	0.899614i	$-0.643848\pi$
$37$	− 10.9443i	− 1.79923i	0.436687	−	0.899614i	$-0.356152\pi$
$38$	0	0
$39$	5.52786i	0.885167i
$40$	0	0
$41$	− 2.00000i	− 0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$	− 2.00000i	− 0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$	0	0
$43$	1.52786	0.232997	0.116499	−	0.993191i	$-0.462833\pi$
$43$	1.52786	0.232997	0.116499	+	0.993191i	$0.462833\pi$
$44$	0	0
$45$	2.94427i	0.438906i
$46$	0	0
$47$	12.9443	1.88812	0.944058	−	0.329779i	$-0.106974\pi$
$47$	12.9443	1.88812	0.944058	+	0.329779i	$0.106974\pi$
$48$	0	0
$49$	5.47214	0.781734
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−2.47214	−0.333343
$56$	0	0
$57$	8.00000i	1.05963i
$58$	0	0
$59$	−14.4721	−1.88411	−0.942056	−	0.335456i	$-0.891110\pi$
$59$	−14.4721	−1.88411	−0.942056	+	0.335456i	$0.891110\pi$
$60$	0	0
$61$	− 6.94427i	− 0.889123i	−0.895748	−	0.444561i	$-0.853360\pi$
$61$	− 6.94427i	− 0.889123i	0.895748	−	0.444561i	$-0.146640\pi$
$62$	0	0
$63$	1.81966i	0.229256i
$64$	0	0
$65$	8.94427i	1.10940i
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−1.52786	−0.183933
$70$	0	0
$71$	9.23607i	1.09612i	0.836439	+	0.548060i	$0.184633\pi$
$71$	9.23607i	1.09612i	−0.836439	+	0.548060i	$0.815367\pi$
$72$	0	0
$73$	14.9443i	1.74909i	0.484940	+	0.874547i	$0.338841\pi$
$73$	14.9443i	1.74909i	−0.484940	+	0.874547i	$0.661159\pi$
$74$	0	0
$75$	1.23607i	0.142729i
$76$	0	0
$77$	−1.52786	−0.174116
$78$	0	0
$79$	− 11.7082i	− 1.31728i	−0.752460	−	0.658638i	$-0.771133\pi$
$79$	− 11.7082i	− 1.31728i	0.752460	−	0.658638i	$-0.228867\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−1.52786	−0.167705	−0.0838524	−	0.996478i	$-0.526722\pi$
$83$	−1.52786	−0.167705	−0.0838524	+	0.996478i	$0.526722\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.47214	−0.265041
$88$	0	0
$89$	−7.52786	−0.797952	−0.398976	−	0.916961i	$-0.630634\pi$
$89$	−7.52786	−0.797952	−0.398976	+	0.916961i	$0.630634\pi$
$90$	0	0
$91$	5.52786i	0.579478i
$92$	0	0
$93$	−1.52786	−0.158432
$94$	0	0
$95$	12.9443i	1.32805i
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	1.81966i	0.182883i
$100$	0	0
$101$	−13.4164	−1.33498	−0.667491	−	0.744618i	$-0.732632\pi$
$101$	−13.4164	−1.33498	−0.667491	+	0.744618i	$0.732632\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	− 3.05573i	− 0.298209i
$106$	0	0
$107$	− 3.70820i	− 0.358486i	−0.983805	−	0.179243i	$-0.942635\pi$
$107$	− 3.70820i	− 0.358486i	0.983805	−	0.179243i	$-0.0573648\pi$
$108$	0	0
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	0	0
$111$	13.5279	1.28401
$112$	0	0
$113$	− 14.9443i	− 1.40584i	−0.711269	−	0.702919i	$-0.751879\pi$
$113$	− 14.9443i	− 1.40584i	0.711269	−	0.702919i	$-0.248121\pi$
$114$	0	0
$115$	−2.47214	−0.230528
$116$	0	0
$117$	6.58359	0.608653
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.47214	0.861103
$122$	0	0
$123$	2.47214	0.222905
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	−2.47214	−0.219367	−0.109683	−	0.993967i	$-0.534984\pi$
$127$	−2.47214	−0.219367	−0.109683	+	0.993967i	$0.534984\pi$
$128$	0	0
$129$	1.88854i	0.166277i
$130$	0	0
$131$	19.1246i	1.67093i	0.549547	+	0.835463i	$0.314800\pi$
$131$	19.1246i	1.67093i	−0.549547	+	0.835463i	$0.685200\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	−11.0557	−0.951526
$136$	0	0
$137$	−7.52786	−0.643149	−0.321574	−	0.946884i	$-0.604212\pi$
$137$	−7.52786	−0.643149	−0.321574	+	0.946884i	$0.604212\pi$
$138$	0	0
$139$	− 6.76393i	− 0.573709i	−0.957974	−	0.286855i	$-0.907390\pi$
$139$	− 6.76393i	− 0.573709i	0.957974	−	0.286855i	$-0.0926097\pi$
$140$	0	0
$141$	16.0000i	1.34744i
$142$	0	0
$143$	5.52786i	0.462263i
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	6.76393i	0.557880i
$148$	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
$151$	−5.52786	−0.449851	−0.224926	−	0.974376i	$-0.572214\pi$
$151$	−5.52786	−0.449851	−0.224926	+	0.974376i	$0.572214\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.47214	−0.198567
$156$	0	0
$157$	−11.8885	−0.948809	−0.474405	−	0.880307i	$-0.657337\pi$
$157$	−11.8885	−0.948809	−0.474405	+	0.880307i	$0.657337\pi$
$158$	0	0
$159$	2.47214i	0.196053i
$160$	0	0
$161$	−1.52786	−0.120413
$162$	0	0
$163$	22.1803i	1.73730i	0.495428	+	0.868649i	$0.335011\pi$
$163$	22.1803i	1.73730i	−0.495428	+	0.868649i	$0.664989\pi$
$164$	0	0
$165$	− 3.05573i	− 0.237888i
$166$	0	0
$167$	− 19.7082i	− 1.52507i	−0.646949	−	0.762533i	$-0.723955\pi$
$167$	− 19.7082i	− 1.52507i	0.646949	−	0.762533i	$-0.276045\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	9.52786	0.728614
$172$	0	0
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	0	0
$175$	1.23607i	0.0934380i
$176$	0	0
$177$	− 17.8885i	− 1.34459i
$178$	0	0
$179$	−14.4721	−1.08170	−0.540849	−	0.841120i	$-0.681897\pi$
$179$	−14.4721	−1.08170	−0.540849	+	0.841120i	$0.681897\pi$
$180$	0	0
$181$	− 10.0000i	− 0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$	− 10.0000i	− 0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$	0	0
$183$	8.58359	0.634517
$184$	0	0
$185$	21.8885	1.60928
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−6.83282	−0.497014
$190$	0	0
$191$	−12.9443	−0.936615	−0.468307	−	0.883566i	$-0.655136\pi$
$191$	−12.9443	−0.936615	−0.468307	+	0.883566i	$0.655136\pi$
$192$	0	0
$193$	− 14.9443i	− 1.07571i	−0.843037	−	0.537856i	$-0.819234\pi$
$193$	− 14.9443i	− 1.07571i	0.843037	−	0.537856i	$-0.180766\pi$
$194$	0	0
$195$	−11.0557	−0.791717
$196$	0	0
$197$	− 6.94427i	− 0.494759i	−0.968919	−	0.247379i	$-0.920431\pi$
$197$	− 6.94427i	− 0.494759i	0.968919	−	0.247379i	$-0.0795694\pi$
$198$	0	0
$199$	22.1803i	1.57232i	0.618021	+	0.786161i	$0.287935\pi$
$199$	22.1803i	1.57232i	−0.618021	+	0.786161i	$0.712065\pi$
$200$	0	0
$201$	− 14.8328i	− 1.04623i
$202$	0	0
$203$	−2.47214	−0.173510
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	1.81966i	0.126475i
$208$	0	0
$209$	8.00000i	0.553372i
$210$	0	0
$211$	1.23607i	0.0850944i	0.999094	+	0.0425472i	$0.0135473\pi$
$211$	1.23607i	0.0850944i	−0.999094	+	0.0425472i	$0.986453\pi$
$212$	0	0
$213$	−11.4164	−0.782239
$214$	0	0
$215$	3.05573i	0.208399i
$216$	0	0
$217$	−1.52786	−0.103718
$218$	0	0
$219$	−18.4721	−1.24823
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5.52786	−0.370173	−0.185087	−	0.982722i	$-0.559257\pi$
$223$	−5.52786	−0.370173	−0.185087	+	0.982722i	$0.559257\pi$
$224$	0	0
$225$	1.47214	0.0981424
$226$	0	0
$227$	1.23607i	0.0820407i	0.999158	+	0.0410204i	$0.0130608\pi$
$227$	1.23607i	0.0820407i	−0.999158	+	0.0410204i	$0.986939\pi$
$228$	0	0
$229$	24.4721	1.61716	0.808582	−	0.588383i	$-0.200235\pi$
$229$	24.4721	1.61716	0.808582	+	0.588383i	$0.200235\pi$
$230$	0	0
$231$	− 1.88854i	− 0.124257i
$232$	0	0
$233$	− 23.8885i	− 1.56499i	−0.622657	−	0.782495i	$-0.713947\pi$
$233$	− 23.8885i	− 1.56499i	0.622657	−	0.782495i	$-0.286053\pi$
$234$	0	0
$235$	25.8885i	1.68878i
$236$	0	0
$237$	14.4721	0.940066
$238$	0	0
$239$	17.8885	1.15711	0.578557	−	0.815642i	$-0.303616\pi$
$239$	17.8885	1.15711	0.578557	+	0.815642i	$0.303616\pi$
$240$	0	0
$241$	− 1.05573i	− 0.0680054i	−0.999422	−	0.0340027i	$-0.989175\pi$
$241$	− 1.05573i	− 0.0680054i	0.999422	−	0.0340027i	$-0.0108255\pi$
$242$	0	0
$243$	13.5967i	0.872232i
$244$	0	0
$245$	10.9443i	0.699204i
$246$	0	0
$247$	28.9443	1.84168
$248$	0	0
$249$	− 1.88854i	− 0.119682i
$250$	0	0
$251$	−8.94427	−0.564557	−0.282279	−	0.959332i	$-0.591090\pi$
$251$	−8.94427	−0.564557	−0.282279	+	0.959332i	$0.591090\pi$
$252$	0	0
$253$	−1.52786	−0.0960560
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.3607	1.39482	0.697410	−	0.716672i	$-0.254335\pi$
$257$	22.3607	1.39482	0.697410	+	0.716672i	$0.254335\pi$
$258$	0	0
$259$	13.5279	0.840581
$260$	0	0
$261$	2.94427i	0.182246i
$262$	0	0
$263$	15.4164	0.950616	0.475308	−	0.879819i	$-0.342337\pi$
$263$	15.4164	0.950616	0.475308	+	0.879819i	$0.342337\pi$
$264$	0	0
$265$	4.00000i	0.245718i
$266$	0	0
$267$	− 9.30495i	− 0.569454i
$268$	0	0
$269$	− 1.05573i	− 0.0643689i	−0.999482	−	0.0321844i	$-0.989754\pi$
$269$	− 1.05573i	− 0.0643689i	0.999482	−	0.0321844i	$-0.0102464\pi$
$270$	0	0
$271$	−17.8885	−1.08665	−0.543326	−	0.839522i	$-0.682835\pi$
$271$	−17.8885	−1.08665	−0.543326	+	0.839522i	$0.682835\pi$
$272$	0	0
$273$	−6.83282	−0.413541
$274$	0	0
$275$	1.23607i	0.0745377i
$276$	0	0
$277$	− 20.8328i	− 1.25172i	−0.779934	−	0.625861i	$-0.784748\pi$
$277$	− 20.8328i	− 1.25172i	0.779934	−	0.625861i	$-0.215252\pi$
$278$	0	0
$279$	1.81966i	0.108940i
$280$	0	0
$281$	−19.8885	−1.18645	−0.593226	−	0.805036i	$-0.702146\pi$
$281$	−19.8885	−1.18645	−0.593226	+	0.805036i	$0.702146\pi$
$282$	0	0
$283$	1.23607i	0.0734766i	0.999325	+	0.0367383i	$0.0116968\pi$
$283$	1.23607i	0.0734766i	−0.999325	+	0.0367383i	$0.988303\pi$
$284$	0	0
$285$	−16.0000	−0.947758
$286$	0	0
$287$	2.47214	0.145926
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−2.47214	−0.144919
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	− 28.9443i	− 1.68520i
$296$	0	0
$297$	−6.83282	−0.396480
$298$	0	0
$299$	5.52786i	0.319685i
$300$	0	0
$301$	1.88854i	0.108854i
$302$	0	0
$303$	− 16.5836i	− 0.952702i
$304$	0	0
$305$	13.8885	0.795256
$306$	0	0
$307$	5.88854	0.336077	0.168038	−	0.985780i	$-0.446257\pi$
$307$	5.88854	0.336077	0.168038	+	0.985780i	$0.446257\pi$
$308$	0	0
$309$	− 9.88854i	− 0.562540i
$310$	0	0
$311$	− 14.7639i	− 0.837186i	−0.908174	−	0.418593i	$-0.862523\pi$
$311$	− 14.7639i	− 0.837186i	0.908174	−	0.418593i	$-0.137477\pi$
$312$	0	0
$313$	7.88854i	0.445887i	0.974831	+	0.222943i	$0.0715665\pi$
$313$	7.88854i	0.445887i	−0.974831	+	0.222943i	$0.928433\pi$
$314$	0	0
$315$	−3.63932	−0.205052
$316$	0	0
$317$	15.8885i	0.892390i	0.894936	+	0.446195i	$0.147221\pi$
$317$	15.8885i	0.892390i	−0.894936	+	0.446195i	$0.852779\pi$
$318$	0	0
$319$	−2.47214	−0.138413
$320$	0	0
$321$	4.58359	0.255831
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.47214	0.248069
$326$	0	0
$327$	12.3607	0.683547
$328$	0	0
$329$	16.0000i	0.882109i
$330$	0	0
$331$	1.52786	0.0839790	0.0419895	−	0.999118i	$-0.486630\pi$
$331$	1.52786	0.0839790	0.0419895	+	0.999118i	$0.486630\pi$
$332$	0	0
$333$	− 16.1115i	− 0.882902i
$334$	0	0
$335$	− 24.0000i	− 1.31126i
$336$	0	0
$337$	11.8885i	0.647610i	0.946124	+	0.323805i	$0.104962\pi$
$337$	11.8885i	0.647610i	−0.946124	+	0.323805i	$0.895038\pi$
$338$	0	0
$339$	18.4721	1.00327
$340$	0	0
$341$	−1.52786	−0.0827385
$342$	0	0
$343$	15.4164i	0.832408i
$344$	0	0
$345$	− 3.05573i	− 0.164515i
$346$	0	0
$347$	9.23607i	0.495818i	0.968783	+	0.247909i	$0.0797434\pi$
$347$	9.23607i	0.495818i	−0.968783	+	0.247909i	$0.920257\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	24.7214i	1.31953i
$352$	0	0
$353$	11.8885	0.632763	0.316382	−	0.948632i	$-0.397532\pi$
$353$	11.8885	0.632763	0.316382	+	0.948632i	$0.397532\pi$
$354$	0	0
$355$	−18.4721	−0.980399
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.5279	−0.713973	−0.356987	−	0.934109i	$-0.616196\pi$
$359$	−13.5279	−0.713973	−0.356987	+	0.934109i	$0.616196\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	11.7082i	0.614521i
$364$	0	0
$365$	−29.8885	−1.56444
$366$	0	0
$367$	− 14.7639i	− 0.770671i	−0.922777	−	0.385335i	$-0.874086\pi$
$367$	− 14.7639i	− 0.770671i	0.922777	−	0.385335i	$-0.125914\pi$
$368$	0	0
$369$	− 2.94427i	− 0.153273i
$370$	0	0
$371$	2.47214i	0.128347i
$372$	0	0
$373$	12.4721	0.645783	0.322891	−	0.946436i	$-0.395345\pi$
$373$	12.4721	0.645783	0.322891	+	0.946436i	$0.395345\pi$
$374$	0	0
$375$	−14.8328	−0.765963
$376$	0	0
$377$	8.94427i	0.460653i
$378$	0	0
$379$	− 32.6525i	− 1.67725i	−0.544713	−	0.838623i	$-0.683361\pi$
$379$	− 32.6525i	− 1.67725i	0.544713	−	0.838623i	$-0.316639\pi$
$380$	0	0
$381$	− 3.05573i	− 0.156550i
$382$	0	0
$383$	23.4164	1.19652	0.598261	−	0.801301i	$-0.295858\pi$
$383$	23.4164	1.19652	0.598261	+	0.801301i	$0.295858\pi$
$384$	0	0
$385$	− 3.05573i	− 0.155734i
$386$	0	0
$387$	2.24922	0.114334
$388$	0	0
$389$	−1.41641	−0.0718147	−0.0359074	−	0.999355i	$-0.511432\pi$
$389$	−1.41641	−0.0718147	−0.0359074	+	0.999355i	$0.511432\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−23.6393	−1.19245
$394$	0	0
$395$	23.4164	1.17821
$396$	0	0
$397$	− 13.0557i	− 0.655248i	−0.944808	−	0.327624i	$-0.893752\pi$
$397$	− 13.0557i	− 0.655248i	0.944808	−	0.327624i	$-0.106248\pi$
$398$	0	0
$399$	−9.88854	−0.495046
$400$	0	0
$401$	− 30.9443i	− 1.54528i	−0.634842	−	0.772642i	$-0.718935\pi$
$401$	− 30.9443i	− 1.54528i	0.634842	−	0.772642i	$-0.281065\pi$
$402$	0	0
$403$	5.52786i	0.275363i
$404$	0	0
$405$	− 4.83282i	− 0.240145i
$406$	0	0
$407$	13.5279	0.670551
$408$	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
$411$	− 9.30495i	− 0.458979i
$412$	0	0
$413$	− 17.8885i	− 0.880238i
$414$	0	0
$415$	− 3.05573i	− 0.150000i
$416$	0	0
$417$	8.36068	0.409424
$418$	0	0
$419$	25.2361i	1.23286i	0.787409	+	0.616431i	$0.211422\pi$
$419$	25.2361i	1.23286i	−0.787409	+	0.616431i	$0.788578\pi$
$420$	0	0
$421$	6.36068	0.310001	0.155000	−	0.987914i	$-0.450462\pi$
$421$	6.36068	0.310001	0.155000	+	0.987914i	$0.450462\pi$
$422$	0	0
$423$	19.0557	0.926521
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.58359	0.415389
$428$	0	0
$429$	−6.83282	−0.329891
$430$	0	0
$431$	− 37.5967i	− 1.81097i	−0.424377	−	0.905486i	$-0.639507\pi$
$431$	− 37.5967i	− 1.81097i	0.424377	−	0.905486i	$-0.360493\pi$
$432$	0	0
$433$	28.4721	1.36828	0.684142	−	0.729349i	$-0.260177\pi$
$433$	28.4721	1.36828	0.684142	+	0.729349i	$0.260177\pi$
$434$	0	0
$435$	− 4.94427i	− 0.237060i
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	− 0.652476i	− 0.0311410i	−0.999879	−	0.0155705i	$-0.995044\pi$
$439$	− 0.652476i	− 0.0311410i	0.999879	−	0.0155705i	$-0.00495644\pi$
$440$	0	0
$441$	8.05573	0.383606
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	− 15.0557i	− 0.713710i
$446$	0	0
$447$	7.41641i	0.350784i
$448$	0	0
$449$	7.88854i	0.372283i	0.982523	+	0.186142i	$0.0595984\pi$
$449$	7.88854i	0.372283i	−0.982523	+	0.186142i	$0.940402\pi$
$450$	0	0
$451$	2.47214	0.116408
$452$	0	0
$453$	− 6.83282i	− 0.321034i
$454$	0	0
$455$	−11.0557	−0.518301
$456$	0	0
$457$	−37.4164	−1.75027	−0.875133	−	0.483883i	$-0.839226\pi$
$457$	−37.4164	−1.75027	−0.875133	+	0.483883i	$0.839226\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−31.8885	−1.48520	−0.742599	−	0.669737i	$-0.766407\pi$
$461$	−31.8885	−1.48520	−0.742599	+	0.669737i	$0.766407\pi$
$462$	0	0
$463$	30.8328	1.43292	0.716461	−	0.697627i	$-0.245761\pi$
$463$	30.8328	1.43292	0.716461	+	0.697627i	$0.245761\pi$
$464$	0	0
$465$	− 3.05573i	− 0.141706i
$466$	0	0
$467$	21.3050	0.985876	0.492938	−	0.870065i	$-0.335923\pi$
$467$	21.3050	0.985876	0.492938	+	0.870065i	$0.335923\pi$
$468$	0	0
$469$	− 14.8328i	− 0.684916i
$470$	0	0
$471$	− 14.6950i	− 0.677112i
$472$	0	0
$473$	1.88854i	0.0868353i
$474$	0	0
$475$	6.47214	0.296962
$476$	0	0
$477$	2.94427	0.134809
$478$	0	0
$479$	20.2918i	0.927156i	0.886056	+	0.463578i	$0.153435\pi$
$479$	20.2918i	0.927156i	−0.886056	+	0.463578i	$0.846565\pi$
$480$	0	0
$481$	− 48.9443i	− 2.23167i
$482$	0	0
$483$	− 1.88854i	− 0.0859317i
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	− 38.7639i	− 1.75656i	−0.478145	−	0.878281i	$-0.658691\pi$
$487$	− 38.7639i	− 1.75656i	0.478145	−	0.878281i	$-0.341309\pi$
$488$	0	0
$489$	−27.4164	−1.23981
$490$	0	0
$491$	−1.52786	−0.0689515	−0.0344758	−	0.999406i	$-0.510976\pi$
$491$	−1.52786	−0.0689515	−0.0344758	+	0.999406i	$0.510976\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−3.63932	−0.163575
$496$	0	0
$497$	−11.4164	−0.512096
$498$	0	0
$499$	38.1803i	1.70919i	0.519298	+	0.854593i	$0.326194\pi$
$499$	38.1803i	1.70919i	−0.519298	+	0.854593i	$0.673806\pi$
$500$	0	0
$501$	24.3607	1.08835
$502$	0	0
$503$	− 27.7082i	− 1.23545i	−0.786395	−	0.617724i	$-0.788055\pi$
$503$	− 27.7082i	− 1.23545i	0.786395	−	0.617724i	$-0.211945\pi$
$504$	0	0
$505$	− 26.8328i	− 1.19404i
$506$	0	0
$507$	8.65248i	0.384270i
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	−18.4721	−0.817159
$512$	0	0
$513$	35.7771i	1.57960i
$514$	0	0
$515$	− 16.0000i	− 0.705044i
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	−2.47214	−0.108515
$520$	0	0
$521$	23.8885i	1.04658i	0.852156	+	0.523288i	$0.175295\pi$
$521$	23.8885i	1.04658i	−0.852156	+	0.523288i	$0.824705\pi$
$522$	0	0
$523$	−24.9443	−1.09074	−0.545368	−	0.838196i	$-0.683610\pi$
$523$	−24.9443	−1.09074	−0.545368	+	0.838196i	$0.683610\pi$
$524$	0	0
$525$	−1.52786	−0.0666815
$526$	0	0
$527$	0	0
$528$	0	0
$529$	21.4721	0.933571
$530$	0	0
$531$	−21.3050	−0.924556
$532$	0	0
$533$	− 8.94427i	− 0.387419i
$534$	0	0
$535$	7.41641	0.320639
$536$	0	0
$537$	− 17.8885i	− 0.771948i
$538$	0	0
$539$	6.76393i	0.291343i
$540$	0	0
$541$	− 30.0000i	− 1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$541$	− 30.0000i	− 1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$542$	0	0
$543$	12.3607	0.530448
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	− 11.7082i	− 0.500607i	−0.968167	−	0.250303i	$-0.919470\pi$
$547$	− 11.7082i	− 0.500607i	0.968167	−	0.250303i	$-0.0805303\pi$
$548$	0	0
$549$	− 10.2229i	− 0.436303i
$550$	0	0
$551$	12.9443i	0.551445i
$552$	0	0
$553$	14.4721	0.615418
$554$	0	0
$555$	27.0557i	1.14845i
$556$	0	0
$557$	−5.41641	−0.229501	−0.114750	−	0.993394i	$-0.536607\pi$
$557$	−5.41641	−0.229501	−0.114750	+	0.993394i	$0.536607\pi$
$558$	0	0
$559$	6.83282	0.288997
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.3607	1.02668	0.513340	−	0.858185i	$-0.328408\pi$
$563$	24.3607	1.02668	0.513340	+	0.858185i	$0.328408\pi$
$564$	0	0
$565$	29.8885	1.25742
$566$	0	0
$567$	− 2.98684i	− 0.125436i
$568$	0	0
$569$	−0.111456	−0.00467249	−0.00233624	−	0.999997i	$-0.500744\pi$
$569$	−0.111456	−0.00467249	−0.00233624	+	0.999997i	$0.500744\pi$
$570$	0	0
$571$	1.23607i	0.0517278i	0.999665	+	0.0258639i	$0.00823366\pi$
$571$	1.23607i	0.0517278i	−0.999665	+	0.0258639i	$0.991766\pi$
$572$	0	0
$573$	− 16.0000i	− 0.668410i
$574$	0	0
$575$	1.23607i	0.0515476i
$576$	0	0
$577$	0.472136	0.0196553	0.00982764	−	0.999952i	$-0.496872\pi$
$577$	0.472136	0.0196553	0.00982764	+	0.999952i	$0.496872\pi$
$578$	0	0
$579$	18.4721	0.767676
$580$	0	0
$581$	− 1.88854i	− 0.0783500i
$582$	0	0
$583$	2.47214i	0.102385i
$584$	0	0
$585$	13.1672i	0.544396i
$586$	0	0
$587$	−32.3607	−1.33567	−0.667834	−	0.744310i	$-0.732778\pi$
$587$	−32.3607	−1.33567	−0.667834	+	0.744310i	$0.732778\pi$
$588$	0	0
$589$	8.00000i	0.329634i
$590$	0	0
$591$	8.58359	0.353082
$592$	0	0
$593$	23.8885	0.980985	0.490492	−	0.871445i	$-0.336817\pi$
$593$	23.8885	0.980985	0.490492	+	0.871445i	$0.336817\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−27.4164	−1.12208
$598$	0	0
$599$	−6.83282	−0.279181	−0.139591	−	0.990209i	$-0.544579\pi$
$599$	−6.83282	−0.279181	−0.139591	+	0.990209i	$0.544579\pi$
$600$	0	0
$601$	− 21.0557i	− 0.858881i	−0.903095	−	0.429441i	$-0.858711\pi$
$601$	− 21.0557i	− 0.858881i	0.903095	−	0.429441i	$-0.141289\pi$
$602$	0	0
$603$	−17.6656	−0.719400
$604$	0	0
$605$	18.9443i	0.770194i
$606$	0	0
$607$	− 19.7082i	− 0.799931i	−0.916530	−	0.399966i	$-0.869022\pi$
$607$	− 19.7082i	− 0.799931i	0.916530	−	0.399966i	$-0.130978\pi$
$608$	0	0
$609$	− 3.05573i	− 0.123824i
$610$	0	0
$611$	57.8885	2.34192
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	0	0
$615$	4.94427i	0.199372i
$616$	0	0
$617$	34.0000i	1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$	34.0000i	1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$	0	0
$619$	− 32.6525i	− 1.31241i	−0.754581	−	0.656207i	$-0.772160\pi$
$619$	− 32.6525i	− 1.31241i	0.754581	−	0.656207i	$-0.227840\pi$
$620$	0	0
$621$	−6.83282	−0.274191
$622$	0	0
$623$	− 9.30495i	− 0.372795i
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−9.88854	−0.394910
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−33.3050	−1.32585	−0.662925	−	0.748686i	$-0.730685\pi$
$631$	−33.3050	−1.32585	−0.662925	+	0.748686i	$0.730685\pi$
$632$	0	0
$633$	−1.52786	−0.0607271
$634$	0	0
$635$	− 4.94427i	− 0.196207i
$636$	0	0
$637$	24.4721	0.969621
$638$	0	0
$639$	13.5967i	0.537879i
$640$	0	0
$641$	27.8885i	1.10153i	0.834660	+	0.550766i	$0.185664\pi$
$641$	27.8885i	1.10153i	−0.834660	+	0.550766i	$0.814336\pi$
$642$	0	0
$643$	− 18.5410i	− 0.731186i	−0.930775	−	0.365593i	$-0.880866\pi$
$643$	− 18.5410i	− 0.731186i	0.930775	−	0.365593i	$-0.119134\pi$
$644$	0	0
$645$	−3.77709	−0.148723
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	− 17.8885i	− 0.702187i
$650$	0	0
$651$	− 1.88854i	− 0.0740179i
$652$	0	0
$653$	− 32.8328i	− 1.28485i	−0.766350	−	0.642424i	$-0.777929\pi$
$653$	− 32.8328i	− 1.28485i	0.766350	−	0.642424i	$-0.222071\pi$
$654$	0	0
$655$	−38.2492	−1.49452
$656$	0	0
$657$	22.0000i	0.858302i
$658$	0	0
$659$	−34.8328	−1.35689	−0.678447	−	0.734649i	$-0.737347\pi$
$659$	−34.8328	−1.35689	−0.678447	+	0.734649i	$0.737347\pi$
$660$	0	0
$661$	−4.11146	−0.159917	−0.0799586	−	0.996798i	$-0.525479\pi$
$661$	−4.11146	−0.159917	−0.0799586	+	0.996798i	$0.525479\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	−2.47214	−0.0957215
$668$	0	0
$669$	− 6.83282i	− 0.264172i
$670$	0	0
$671$	8.58359	0.331366
$672$	0	0
$673$	33.7771i	1.30201i	0.759073	+	0.651006i	$0.225653\pi$
$673$	33.7771i	1.30201i	−0.759073	+	0.651006i	$0.774347\pi$
$674$	0	0
$675$	5.52786i	0.212768i
$676$	0	0
$677$	8.11146i	0.311749i	0.987777	+	0.155874i	$0.0498195\pi$
$677$	8.11146i	0.311749i	−0.987777	+	0.155874i	$0.950181\pi$
$678$	0	0
$679$	−2.47214	−0.0948719
$680$	0	0
$681$	−1.52786	−0.0585479
$682$	0	0
$683$	8.06888i	0.308747i	0.988013	+	0.154374i	$0.0493359\pi$
$683$	8.06888i	0.308747i	−0.988013	+	0.154374i	$0.950664\pi$
$684$	0	0
$685$	− 15.0557i	− 0.575250i
$686$	0	0
$687$	30.2492i	1.15408i
$688$	0	0
$689$	8.94427	0.340750
$690$	0	0
$691$	− 37.5967i	− 1.43025i	−0.698998	−	0.715124i	$-0.746370\pi$
$691$	− 37.5967i	− 1.43025i	0.698998	−	0.715124i	$-0.253630\pi$
$692$	0	0
$693$	−2.24922	−0.0854409
$694$	0	0
$695$	13.5279	0.513141
$696$	0	0
$697$	0	0
$698$	0	0
$699$	29.5279	1.11685
$700$	0	0
$701$	14.3607	0.542395	0.271198	−	0.962524i	$-0.412580\pi$
$701$	14.3607	0.542395	0.271198	+	0.962524i	$0.412580\pi$
$702$	0	0
$703$	− 70.8328i	− 2.67151i
$704$	0	0
$705$	−32.0000	−1.20519
$706$	0	0
$707$	− 16.5836i	− 0.623690i
$708$	0	0
$709$	14.9443i	0.561244i	0.959818	+	0.280622i	$0.0905407\pi$
$709$	14.9443i	0.561244i	−0.959818	+	0.280622i	$0.909459\pi$
$710$	0	0
$711$	− 17.2361i	− 0.646403i
$712$	0	0
$713$	−1.52786	−0.0572190
$714$	0	0
$715$	−11.0557	−0.413461
$716$	0	0
$717$	22.1115i	0.825767i
$718$	0	0
$719$	− 8.65248i	− 0.322683i	−0.986899	−	0.161341i	$-0.948418\pi$
$719$	− 8.65248i	− 0.322683i	0.986899	−	0.161341i	$-0.0515820\pi$
$720$	0	0
$721$	− 9.88854i	− 0.368269i
$722$	0	0
$723$	1.30495	0.0485317
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	20.9443	0.776780	0.388390	−	0.921495i	$-0.373031\pi$
$727$	20.9443	0.776780	0.388390	+	0.921495i	$0.373031\pi$
$728$	0	0
$729$	−24.0557	−0.890953
$730$	0	0
$731$	0	0
$732$	0	0
$733$	35.8885	1.32557	0.662787	−	0.748808i	$-0.269374\pi$
$733$	35.8885	1.32557	0.662787	+	0.748808i	$0.269374\pi$
$734$	0	0
$735$	−13.5279	−0.498983
$736$	0	0
$737$	− 14.8328i	− 0.546374i
$738$	0	0
$739$	43.4164	1.59710	0.798549	−	0.601930i	$-0.205601\pi$
$739$	43.4164	1.59710	0.798549	+	0.601930i	$0.205601\pi$
$740$	0	0
$741$	35.7771i	1.31430i
$742$	0	0
$743$	− 48.6525i	− 1.78489i	−0.451160	−	0.892443i	$-0.648990\pi$
$743$	− 48.6525i	− 1.78489i	0.451160	−	0.892443i	$-0.351010\pi$
$744$	0	0
$745$	12.0000i	0.439646i
$746$	0	0
$747$	−2.24922	−0.0822948
$748$	0	0
$749$	4.58359	0.167481
$750$	0	0
$751$	− 35.7082i	− 1.30301i	−0.758644	−	0.651505i	$-0.774138\pi$
$751$	− 35.7082i	− 1.30301i	0.758644	−	0.651505i	$-0.225862\pi$
$752$	0	0
$753$	− 11.0557i	− 0.402893i
$754$	0	0
$755$	− 11.0557i	− 0.402359i
$756$	0	0
$757$	−7.52786	−0.273605	−0.136802	−	0.990598i	$-0.543683\pi$
$757$	−7.52786	−0.273605	−0.136802	+	0.990598i	$0.543683\pi$
$758$	0	0
$759$	− 1.88854i	− 0.0685498i
$760$	0	0
$761$	34.3607	1.24557	0.622787	−	0.782392i	$-0.286000\pi$
$761$	34.3607	1.24557	0.622787	+	0.782392i	$0.286000\pi$
$762$	0	0
$763$	12.3607	0.447487
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−64.7214	−2.33695
$768$	0	0
$769$	16.4721	0.594000	0.297000	−	0.954877i	$-0.404014\pi$
$769$	16.4721	0.594000	0.297000	+	0.954877i	$0.404014\pi$
$770$	0	0
$771$	27.6393i	0.995406i
$772$	0	0
$773$	−27.3050	−0.982091	−0.491045	−	0.871134i	$-0.663385\pi$
$773$	−27.3050	−0.982091	−0.491045	+	0.871134i	$0.663385\pi$
$774$	0	0
$775$	1.23607i	0.0444009i
$776$	0	0
$777$	16.7214i	0.599875i
$778$	0	0
$779$	− 12.9443i	− 0.463777i
$780$	0	0
$781$	−11.4164	−0.408511
$782$	0	0
$783$	−11.0557	−0.395099
$784$	0	0
$785$	− 23.7771i	− 0.848641i
$786$	0	0
$787$	− 24.6525i	− 0.878766i	−0.898300	−	0.439383i	$-0.855197\pi$
$787$	− 24.6525i	− 0.878766i	0.898300	−	0.439383i	$-0.144803\pi$
$788$	0	0
$789$	19.0557i	0.678402i
$790$	0	0
$791$	18.4721	0.656794
$792$	0	0
$793$	− 31.0557i	− 1.10282i
$794$	0	0
$795$	−4.94427	−0.175355
$796$	0	0
$797$	−15.8885	−0.562801	−0.281401	−	0.959590i	$-0.590799\pi$
$797$	−15.8885	−0.562801	−0.281401	+	0.959590i	$0.590799\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−11.0820	−0.391565
$802$	0	0
$803$	−18.4721	−0.651868
$804$	0	0
$805$	− 3.05573i	− 0.107700i
$806$	0	0
$807$	1.30495	0.0459365
$808$	0	0
$809$	42.9443i	1.50984i	0.655817	+	0.754920i	$0.272324\pi$
$809$	42.9443i	1.50984i	−0.655817	+	0.754920i	$0.727676\pi$
$810$	0	0
$811$	27.1246i	0.952474i	0.879317	+	0.476237i	$0.158000\pi$
$811$	27.1246i	0.952474i	−0.879317	+	0.476237i	$0.842000\pi$
$812$	0	0
$813$	− 22.1115i	− 0.775483i
$814$	0	0
$815$	−44.3607	−1.55389
$816$	0	0
$817$	9.88854	0.345956
$818$	0	0
$819$	8.13777i	0.284357i
$820$	0	0
$821$	50.0000i	1.74501i	0.488603	+	0.872506i	$0.337507\pi$
$821$	50.0000i	1.74501i	−0.488603	+	0.872506i	$0.662493\pi$
$822$	0	0
$823$	2.40325i	0.0837721i	0.999122	+	0.0418861i	$0.0133366\pi$
$823$	2.40325i	0.0837721i	−0.999122	+	0.0418861i	$0.986663\pi$
$824$	0	0
$825$	−1.52786	−0.0531934
$826$	0	0
$827$	25.2361i	0.877544i	0.898598	+	0.438772i	$0.144586\pi$
$827$	25.2361i	0.877544i	−0.898598	+	0.438772i	$0.855414\pi$
$828$	0	0
$829$	−43.8885	−1.52431	−0.762156	−	0.647393i	$-0.775859\pi$
$829$	−43.8885	−1.52431	−0.762156	+	0.647393i	$0.775859\pi$
$830$	0	0
$831$	25.7508	0.893285
$832$	0	0
$833$	0	0
$834$	0	0
$835$	39.4164	1.36406
$836$	0	0
$837$	−6.83282	−0.236177
$838$	0	0
$839$	32.0689i	1.10714i	0.832802	+	0.553570i	$0.186735\pi$
$839$	32.0689i	1.10714i	−0.832802	+	0.553570i	$0.813265\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 24.5836i	− 0.846704i
$844$	0	0
$845$	14.0000i	0.481615i
$846$	0	0
$847$	11.7082i	0.402299i
$848$	0	0
$849$	−1.52786	−0.0524362
$850$	0	0
$851$	13.5279	0.463729
$852$	0	0
$853$	37.7771i	1.29346i	0.762718	+	0.646731i	$0.223865\pi$
$853$	37.7771i	1.29346i	−0.762718	+	0.646731i	$0.776135\pi$
$854$	0	0
$855$	19.0557i	0.651692i
$856$	0	0
$857$	7.88854i	0.269468i	0.990882	+	0.134734i	$0.0430179\pi$
$857$	7.88854i	0.269468i	−0.990882	+	0.134734i	$0.956982\pi$
$858$	0	0
$859$	−50.2492	−1.71448	−0.857241	−	0.514916i	$-0.827823\pi$
$859$	−50.2492	−1.71448	−0.857241	+	0.514916i	$0.827823\pi$
$860$	0	0
$861$	3.05573i	0.104139i
$862$	0	0
$863$	−30.8328	−1.04956	−0.524781	−	0.851238i	$-0.675853\pi$
$863$	−30.8328	−1.04956	−0.524781	+	0.851238i	$0.675853\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.4721	0.490934
$870$	0	0
$871$	−53.6656	−1.81839
$872$	0	0
$873$	2.94427i	0.0996485i
$874$	0	0
$875$	−14.8328	−0.501441
$876$	0	0
$877$	− 3.88854i	− 0.131307i	−0.997842	−	0.0656534i	$-0.979087\pi$
$877$	− 3.88854i	− 0.131307i	0.997842	−	0.0656534i	$-0.0209132\pi$
$878$	0	0
$879$	− 12.3607i	− 0.416915i
$880$	0	0
$881$	− 7.88854i	− 0.265772i	−0.991131	−	0.132886i	$-0.957576\pi$
$881$	− 7.88854i	− 0.265772i	0.991131	−	0.132886i	$-0.0424244\pi$
$882$	0	0
$883$	−2.11146	−0.0710562	−0.0355281	−	0.999369i	$-0.511311\pi$
$883$	−2.11146	−0.0710562	−0.0355281	+	0.999369i	$0.511311\pi$
$884$	0	0
$885$	35.7771	1.20263
$886$	0	0
$887$	11.1246i	0.373528i	0.982405	+	0.186764i	$0.0598000\pi$
$887$	11.1246i	0.373528i	−0.982405	+	0.186764i	$0.940200\pi$
$888$	0	0
$889$	− 3.05573i	− 0.102486i
$890$	0	0
$891$	− 2.98684i	− 0.100063i
$892$	0	0
$893$	83.7771	2.80349
$894$	0	0
$895$	− 28.9443i	− 0.967500i
$896$	0	0
$897$	−6.83282	−0.228141
$898$	0	0
$899$	−2.47214	−0.0824504
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−2.33437	−0.0776829
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	59.1246i	1.96320i	0.190947	+	0.981600i	$0.438844\pi$
$907$	59.1246i	1.96320i	−0.190947	+	0.981600i	$0.561156\pi$
$908$	0	0
$909$	−19.7508	−0.655092
$910$	0	0
$911$	− 27.7082i	− 0.918014i	−0.888433	−	0.459007i	$-0.848205\pi$
$911$	− 27.7082i	− 0.918014i	0.888433	−	0.459007i	$-0.151795\pi$
$912$	0	0
$913$	− 1.88854i	− 0.0625017i
$914$	0	0
$915$	17.1672i	0.567530i
$916$	0	0
$917$	−23.6393	−0.780639
$918$	0	0
$919$	−41.8885	−1.38178	−0.690888	−	0.722962i	$-0.742780\pi$
$919$	−41.8885	−1.38178	−0.690888	+	0.722962i	$0.742780\pi$
$920$	0	0
$921$	7.27864i	0.239839i
$922$	0	0
$923$	41.3050i	1.35957i
$924$	0	0
$925$	− 10.9443i	− 0.359845i
$926$	0	0
$927$	−11.7771	−0.386810
$928$	0	0
$929$	26.9443i	0.884013i	0.897012	+	0.442006i	$0.145733\pi$
$929$	26.9443i	0.884013i	−0.897012	+	0.442006i	$0.854267\pi$
$930$	0	0
$931$	35.4164	1.16073
$932$	0	0
$933$	18.2492	0.597453
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	−9.75078	−0.318205
$940$	0	0
$941$	25.7771i	0.840309i	0.907453	+	0.420155i	$0.138024\pi$
$941$	25.7771i	0.840309i	−0.907453	+	0.420155i	$0.861976\pi$
$942$	0	0
$943$	2.47214	0.0805038
$944$	0	0
$945$	− 13.6656i	− 0.444543i
$946$	0	0
$947$	37.0132i	1.20277i	0.798961	+	0.601383i	$0.205383\pi$
$947$	37.0132i	1.20277i	−0.798961	+	0.601383i	$0.794617\pi$
$948$	0	0
$949$	66.8328i	2.16949i
$950$	0	0
$951$	−19.6393	−0.636849
$952$	0	0
$953$	−23.5279	−0.762142	−0.381071	−	0.924546i	$-0.624445\pi$
$953$	−23.5279	−0.762142	−0.381071	+	0.924546i	$0.624445\pi$
$954$	0	0
$955$	− 25.8885i	− 0.837734i
$956$	0	0
$957$	− 3.05573i	− 0.0987777i
$958$	0	0
$959$	− 9.30495i	− 0.300473i
$960$	0	0
$961$	29.4721	0.950714
$962$	0	0
$963$	− 5.45898i	− 0.175913i
$964$	0	0
$965$	29.8885	0.962146
$966$	0	0
$967$	2.47214	0.0794985	0.0397493	−	0.999210i	$-0.487344\pi$
$967$	2.47214	0.0794985	0.0397493	+	0.999210i	$0.487344\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.52786	−0.305764	−0.152882	−	0.988244i	$-0.548855\pi$
$971$	−9.52786	−0.305764	−0.152882	+	0.988244i	$0.548855\pi$
$972$	0	0
$973$	8.36068	0.268031
$974$	0	0
$975$	5.52786i	0.177033i
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	0	0
$979$	− 9.30495i	− 0.297388i
$980$	0	0
$981$	− 14.7214i	− 0.470017i
$982$	0	0
$983$	− 6.76393i	− 0.215736i	−0.994165	−	0.107868i	$-0.965598\pi$
$983$	− 6.76393i	− 0.215736i	0.994165	−	0.107868i	$-0.0344024\pi$
$984$	0	0
$985$	13.8885	0.442526
$986$	0	0
$987$	−19.7771	−0.629512
$988$	0	0
$989$	1.88854i	0.0600522i
$990$	0	0
$991$	17.2361i	0.547522i	0.961798	+	0.273761i	$0.0882677\pi$
$991$	17.2361i	0.547522i	−0.961798	+	0.273761i	$0.911732\pi$
$992$	0	0
$993$	1.88854i	0.0599311i
$994$	0	0
$995$	−44.3607	−1.40633
$996$	0	0
$997$	− 29.0557i	− 0.920204i	−0.887866	−	0.460102i	$-0.847813\pi$
$997$	− 29.0557i	− 0.920204i	0.887866	−	0.460102i	$-0.152187\pi$
$998$	0	0
$999$	60.4984	1.91409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.b.g.577.3		4
17.4	even	4		136.2.a.c.1.2	✓	2
17.13	even	4		2312.2.a.m.1.1		2
17.16	even	2	inner	2312.2.b.g.577.2		4
51.38	odd	4		1224.2.a.i.1.1		2
68.47	odd	4		4624.2.a.h.1.2		2
68.55	odd	4		272.2.a.f.1.1		2
85.4	even	4		3400.2.a.i.1.1		2
85.38	odd	4		3400.2.e.f.2449.3		4
85.72	odd	4		3400.2.e.f.2449.2		4
119.55	odd	4		6664.2.a.i.1.1		2
136.21	even	4		1088.2.a.s.1.1		2
136.123	odd	4		1088.2.a.o.1.2		2
204.191	even	4		2448.2.a.u.1.2		2
340.259	odd	4		6800.2.a.bd.1.2		2
408.293	odd	4		9792.2.a.db.1.1		2
408.395	even	4		9792.2.a.da.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.a.c.1.2	✓	2	17.4	even	4
272.2.a.f.1.1		2	68.55	odd	4
1088.2.a.o.1.2		2	136.123	odd	4
1088.2.a.s.1.1		2	136.21	even	4
1224.2.a.i.1.1		2	51.38	odd	4
2312.2.a.m.1.1		2	17.13	even	4
2312.2.b.g.577.2		4	17.16	even	2	inner
2312.2.b.g.577.3		4	1.1	even	1	trivial
2448.2.a.u.1.2		2	204.191	even	4
3400.2.a.i.1.1		2	85.4	even	4
3400.2.e.f.2449.2		4	85.72	odd	4
3400.2.e.f.2449.3		4	85.38	odd	4
4624.2.a.h.1.2		2	68.47	odd	4
6664.2.a.i.1.1		2	119.55	odd	4
6800.2.a.bd.1.2		2	340.259	odd	4
9792.2.a.da.1.2		2	408.395	even	4
9792.2.a.db.1.1		2	408.293	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.23607i q^{3} +2.00000i q^{5} +1.23607i q^{7} +1.47214 q^{9} +O(q^{10})\)	\(q+1.23607i q^{3} +2.00000i q^{5} +1.23607i q^{7} +1.47214 q^{9} +1.23607i q^{11} +4.47214 q^{13} -2.47214 q^{15} +6.47214 q^{19} -1.52786 q^{21} +1.23607i q^{23} +1.00000 q^{25} +5.52786i q^{27} +2.00000i q^{29} +1.23607i q^{31} -1.52786 q^{33} -2.47214 q^{35} -10.9443i q^{37} +5.52786i q^{39} -2.00000i q^{41} +1.52786 q^{43} +2.94427i q^{45} +12.9443 q^{47} +5.47214 q^{49} +2.00000 q^{53} -2.47214 q^{55} +8.00000i q^{57} -14.4721 q^{59} -6.94427i q^{61} +1.81966i q^{63} +8.94427i q^{65} -12.0000 q^{67} -1.52786 q^{69} +9.23607i q^{71} +14.9443i q^{73} +1.23607i q^{75} -1.52786 q^{77} -11.7082i q^{79} -2.41641 q^{81} -1.52786 q^{83} -2.47214 q^{87} -7.52786 q^{89} +5.52786i q^{91} -1.52786 q^{93} +12.9443i q^{95} +2.00000i q^{97} +1.81966i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 8 q^{15} + 8 q^{19} - 24 q^{21} + 4 q^{25} - 24 q^{33} + 8 q^{35} + 24 q^{43} + 16 q^{47} + 4 q^{49} + 8 q^{53} + 8 q^{55} - 40 q^{59} - 48 q^{67} - 24 q^{69} - 24 q^{77} + 44 q^{81} - 24 q^{83} + 8 q^{87} - 48 q^{89} - 24 q^{93}+O(q^{100}) \) 4 * q - 12 * q^9 + 8 * q^15 + 8 * q^19 - 24 * q^21 + 4 * q^25 - 24 * q^33 + 8 * q^35 + 24 * q^43 + 16 * q^47 + 4 * q^49 + 8 * q^53 + 8 * q^55 - 40 * q^59 - 48 * q^67 - 24 * q^69 - 24 * q^77 + 44 * q^81 - 24 * q^83 + 8 * q^87 - 48 * q^89 - 24 * q^93

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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