LMFDB - Embedded newform 1600.2.n.v.1407.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1343,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("1600.1343");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.n (of order $4$, degree $2$, 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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.3317760000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{4} + 16 $ x^8 - 7*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1407.3
Root			$0.178197 + 1.40294i$ of defining polynomial
Character	$\chi$	$=$	1600.1407
Dual form			1600.2.n.v.1343.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58114 + 1.58114i) q^{3} +2.00000i q^{9} +O(q^{10})$	$q+(1.58114 + 1.58114i) q^{3} +2.00000i q^{9} -3.87298i q^{11} +(2.44949 - 2.44949i) q^{13} +(1.22474 + 1.22474i) q^{17} +3.87298 q^{19} +(3.16228 + 3.16228i) q^{23} +(1.58114 - 1.58114i) q^{27} +6.00000i q^{29} -7.74597i q^{31} +(6.12372 - 6.12372i) q^{33} +(4.89898 + 4.89898i) q^{37} +7.74597 q^{39} -3.00000 q^{41} +(3.16228 - 3.16228i) q^{47} +7.00000i q^{49} +3.87298i q^{51} +(2.44949 - 2.44949i) q^{53} +(6.12372 + 6.12372i) q^{57} -7.74597 q^{59} +8.00000 q^{61} +(-4.74342 + 4.74342i) q^{67} +10.0000i q^{69} +7.74597i q^{71} +(3.67423 - 3.67423i) q^{73} -15.4919 q^{79} +11.0000 q^{81} +(-1.58114 - 1.58114i) q^{83} +(-9.48683 + 9.48683i) q^{87} +9.00000i q^{89} +(12.2474 - 12.2474i) q^{93} +(-4.89898 - 4.89898i) q^{97} +7.74597 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{41} + 64 q^{61} + 88 q^{81}+O(q^{100}) $ 8 * q - 24 * q^41 + 64 * q^61 + 88 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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
$3$	1.58114	+	1.58114i	0.912871	+	0.912871i	0.996497	−	0.0836263i	$-0.0266502\pi$
$3$	1.58114	+	1.58114i	0.912871	+	0.912871i	−0.0836263	+	0.996497i	$0.526650\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$			2.00000i			0.666667i
$10$		0			0
$11$		−	3.87298i		−	1.16775i	−0.811844	−	0.583874i	$-0.801536\pi$
$11$		−	3.87298i		−	1.16775i	0.811844	−	0.583874i	$-0.198464\pi$
$12$		0			0
$13$	2.44949	−	2.44949i	0.679366	−	0.679366i	−0.280491	−	0.959857i	$-0.590497\pi$
$13$	2.44949	−	2.44949i	0.679366	−	0.679366i	0.959857	+	0.280491i	$0.0904971\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.22474	+	1.22474i	0.297044	+	0.297044i	0.839855	−	0.542811i	$-0.182640\pi$
$17$	1.22474	+	1.22474i	0.297044	+	0.297044i	−0.542811	+	0.839855i	$0.682640\pi$
$18$		0			0
$19$	3.87298			0.888523			0.444262	−	0.895897i	$-0.353466\pi$
$19$	3.87298			0.888523			0.444262	+	0.895897i	$0.353466\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.16228	+	3.16228i	0.659380	+	0.659380i	0.955233	−	0.295853i	$-0.0956039\pi$
$23$	3.16228	+	3.16228i	0.659380	+	0.659380i	−0.295853	+	0.955233i	$0.595604\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.58114	−	1.58114i	0.304290	−	0.304290i
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$		−	7.74597i		−	1.39122i	−0.718421	−	0.695608i	$-0.755135\pi$
$31$		−	7.74597i		−	1.39122i	0.718421	−	0.695608i	$-0.244865\pi$
$32$		0			0
$33$	6.12372	−	6.12372i	1.06600	−	1.06600i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.89898	+	4.89898i	0.805387	+	0.805387i	0.983932	−	0.178545i	$-0.0571389\pi$
$37$	4.89898	+	4.89898i	0.805387	+	0.805387i	−0.178545	+	0.983932i	$0.557139\pi$
$38$		0			0
$39$	7.74597			1.24035
$40$		0			0
$41$	−3.00000			−0.468521			−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000			−0.468521			−0.234261	+	0.972174i	$0.575267\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.16228	−	3.16228i	0.461266	−	0.461266i	−0.437805	−	0.899070i	$-0.644244\pi$
$47$	3.16228	−	3.16228i	0.461266	−	0.461266i	0.899070	+	0.437805i	$0.144244\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$			3.87298i			0.542326i
$52$		0			0
$53$	2.44949	−	2.44949i	0.336463	−	0.336463i	−0.518571	−	0.855034i	$-0.673536\pi$
$53$	2.44949	−	2.44949i	0.336463	−	0.336463i	0.855034	+	0.518571i	$0.173536\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.12372	+	6.12372i	0.811107	+	0.811107i
$58$		0			0
$59$	−7.74597			−1.00844			−0.504219	−	0.863576i	$-0.668220\pi$
$59$	−7.74597			−1.00844			−0.504219	+	0.863576i	$0.668220\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.74342	+	4.74342i	−0.579501	+	0.579501i	−0.934766	−	0.355265i	$-0.884391\pi$
$67$	−4.74342	+	4.74342i	−0.579501	+	0.579501i	0.355265	+	0.934766i	$0.384391\pi$
$68$		0			0
$69$			10.0000i			1.20386i
$70$		0			0
$71$			7.74597i			0.919277i	0.888106	+	0.459639i	$0.152021\pi$
$71$			7.74597i			0.919277i	−0.888106	+	0.459639i	$0.847979\pi$
$72$		0			0
$73$	3.67423	−	3.67423i	0.430037	−	0.430037i	−0.458604	−	0.888641i	$-0.651650\pi$
$73$	3.67423	−	3.67423i	0.430037	−	0.430037i	0.888641	+	0.458604i	$0.151650\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−15.4919			−1.74298			−0.871489	−	0.490414i	$-0.836845\pi$
$79$	−15.4919			−1.74298			−0.871489	+	0.490414i	$0.836845\pi$
$80$		0			0
$81$	11.0000			1.22222
$82$		0			0
$83$	−1.58114	−	1.58114i	−0.173553	−	0.173553i	0.614986	−	0.788538i	$-0.289162\pi$
$83$	−1.58114	−	1.58114i	−0.173553	−	0.173553i	−0.788538	+	0.614986i	$0.789162\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−9.48683	+	9.48683i	−1.01710	+	1.01710i
$88$		0			0
$89$			9.00000i			0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$			9.00000i			0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	12.2474	−	12.2474i	1.27000	−	1.27000i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.89898	−	4.89898i	−0.497416	−	0.497416i	0.413217	−	0.910633i	$-0.364405\pi$
$97$	−4.89898	−	4.89898i	−0.497416	−	0.497416i	−0.910633	+	0.413217i	$0.864405\pi$
$98$		0			0
$99$	7.74597			0.778499
$100$		0			0
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$	−9.48683	−	9.48683i	−0.934765	−	0.934765i	0.0632333	−	0.997999i	$-0.479859\pi$
$103$	−9.48683	−	9.48683i	−0.934765	−	0.934765i	−0.997999	+	0.0632333i	$0.979859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.90569	−	7.90569i	0.764272	−	0.764272i	−0.212819	−	0.977092i	$-0.568265\pi$
$107$	7.90569	−	7.90569i	0.764272	−	0.764272i	0.977092	+	0.212819i	$0.0682646\pi$
$108$		0			0
$109$		−	4.00000i		−	0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$		−	4.00000i		−	0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$		0			0
$111$			15.4919i			1.47043i
$112$		0			0
$113$	3.67423	−	3.67423i	0.345643	−	0.345643i	−0.512841	−	0.858484i	$-0.671407\pi$
$113$	3.67423	−	3.67423i	0.345643	−	0.345643i	0.858484	+	0.512841i	$0.171407\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	4.89898	+	4.89898i	0.452911	+	0.452911i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−4.00000			−0.363636
$122$		0			0
$123$	−4.74342	−	4.74342i	−0.427699	−	0.427699i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	9.48683	−	9.48683i	0.841820	−	0.841820i	−0.147275	−	0.989096i	$-0.547050\pi$
$127$	9.48683	−	9.48683i	0.841820	−	0.841820i	0.989096	+	0.147275i	$0.0470503\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			7.74597i			0.676768i	0.941008	+	0.338384i	$0.109880\pi$
$131$			7.74597i			0.676768i	−0.941008	+	0.338384i	$0.890120\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.4722	+	13.4722i	1.15101	+	1.15101i	0.986351	+	0.164656i	$0.0526514\pi$
$137$	13.4722	+	13.4722i	1.15101	+	1.15101i	0.164656	+	0.986351i	$0.447349\pi$
$138$		0			0
$139$	−11.6190			−0.985506			−0.492753	−	0.870169i	$-0.664009\pi$
$139$	−11.6190			−0.985506			−0.492753	+	0.870169i	$0.664009\pi$
$140$		0			0
$141$	10.0000			0.842152
$142$		0			0
$143$	−9.48683	−	9.48683i	−0.793329	−	0.793329i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−11.0680	+	11.0680i	−0.912871	+	0.912871i
$148$		0			0
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−2.44949	+	2.44949i	−0.198030	+	0.198030i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−7.34847	−	7.34847i	−0.586472	−	0.586472i	0.350202	−	0.936674i	$-0.386113\pi$
$157$	−7.34847	−	7.34847i	−0.586472	−	0.586472i	−0.936674	+	0.350202i	$0.886113\pi$
$158$		0			0
$159$	7.74597			0.614295
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.74342	−	4.74342i	−0.371533	−	0.371533i	0.496502	−	0.868035i	$-0.334617\pi$
$163$	−4.74342	−	4.74342i	−0.371533	−	0.371533i	−0.868035	+	0.496502i	$0.834617\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.16228	+	3.16228i	−0.244704	+	0.244704i	−0.818793	−	0.574089i	$-0.805356\pi$
$167$	−3.16228	+	3.16228i	−0.244704	+	0.244704i	0.574089	+	0.818793i	$0.305356\pi$
$168$		0			0
$169$			1.00000i			0.0769231i
$170$		0			0
$171$			7.74597i			0.592349i
$172$		0			0
$173$	−9.79796	+	9.79796i	−0.744925	+	0.744925i	−0.973521	−	0.228596i	$-0.926586\pi$
$173$	−9.79796	+	9.79796i	−0.744925	+	0.744925i	0.228596	+	0.973521i	$0.426586\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12.2474	−	12.2474i	−0.920575	−	0.920575i
$178$		0			0
$179$	−3.87298			−0.289480			−0.144740	−	0.989470i	$-0.546235\pi$
$179$	−3.87298			−0.289480			−0.144740	+	0.989470i	$0.546235\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	12.6491	+	12.6491i	0.935049	+	0.935049i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.74342	−	4.74342i	0.346873	−	0.346873i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	23.2379i		−	1.68144i	−0.541474	−	0.840718i	$-0.682133\pi$
$191$		−	23.2379i		−	1.68144i	0.541474	−	0.840718i	$-0.317867\pi$
$192$		0			0
$193$	3.67423	−	3.67423i	0.264477	−	0.264477i	−0.562393	−	0.826870i	$-0.690119\pi$
$193$	3.67423	−	3.67423i	0.264477	−	0.264477i	0.826870	+	0.562393i	$0.190119\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.1464	+	17.1464i	1.22163	+	1.22163i	0.967051	+	0.254581i	$0.0819375\pi$
$197$	17.1464	+	17.1464i	1.22163	+	1.22163i	0.254581	+	0.967051i	$0.418062\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−15.0000			−1.05802
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−6.32456	+	6.32456i	−0.439587	+	0.439587i
$208$		0			0
$209$		−	15.0000i		−	1.03757i
$210$		0			0
$211$		−	3.87298i		−	0.266627i	−0.991074	−	0.133314i	$-0.957438\pi$
$211$		−	3.87298i		−	0.266627i	0.991074	−	0.133314i	$-0.0425617\pi$
$212$		0			0
$213$	−12.2474	+	12.2474i	−0.839181	+	0.839181i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	11.6190			0.785136
$220$		0			0
$221$	6.00000			0.403604
$222$		0			0
$223$	18.9737	+	18.9737i	1.27057	+	1.27057i	0.945789	+	0.324782i	$0.105291\pi$
$223$	18.9737	+	18.9737i	1.27057	+	1.27057i	0.324782	+	0.945789i	$0.394709\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.32456	−	6.32456i	0.419775	−	0.419775i	−0.465351	−	0.885126i	$-0.654072\pi$
$227$	6.32456	−	6.32456i	0.419775	−	0.419775i	0.885126	+	0.465351i	$0.154072\pi$
$228$		0			0
$229$		−	4.00000i		−	0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$		−	4.00000i		−	0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.6969	+	14.6969i	−0.962828	+	0.962828i	−0.999333	−	0.0365050i	$-0.988378\pi$
$233$	−14.6969	+	14.6969i	−0.962828	+	0.962828i	0.0365050	+	0.999333i	$0.488378\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−24.4949	−	24.4949i	−1.59111	−	1.59111i
$238$		0			0
$239$	−7.74597			−0.501045			−0.250522	−	0.968111i	$-0.580602\pi$
$239$	−7.74597			−0.501045			−0.250522	+	0.968111i	$0.580602\pi$
$240$		0			0
$241$	17.0000			1.09507			0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000			1.09507			0.547533	+	0.836784i	$0.315567\pi$
$242$		0			0
$243$	12.6491	+	12.6491i	0.811441	+	0.811441i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	9.48683	−	9.48683i	0.603633	−	0.603633i
$248$		0			0
$249$		−	5.00000i		−	0.316862i
$250$		0			0
$251$		−	19.3649i		−	1.22230i	−0.791514	−	0.611151i	$-0.790707\pi$
$251$		−	19.3649i		−	1.22230i	0.791514	−	0.611151i	$-0.209293\pi$
$252$		0			0
$253$	12.2474	−	12.2474i	0.769991	−	0.769991i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.89898	−	4.89898i	−0.305590	−	0.305590i	0.537606	−	0.843196i	$-0.319329\pi$
$257$	−4.89898	−	4.89898i	−0.305590	−	0.305590i	−0.843196	+	0.537606i	$0.819329\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−12.0000			−0.742781
$262$		0			0
$263$	−3.16228	−	3.16228i	−0.194994	−	0.194994i	0.602856	−	0.797850i	$-0.294029\pi$
$263$	−3.16228	−	3.16228i	−0.194994	−	0.194994i	−0.797850	+	0.602856i	$0.794029\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−14.2302	+	14.2302i	−0.870877	+	0.870877i
$268$		0			0
$269$		−	24.0000i		−	1.46331i	−0.681677	−	0.731653i	$-0.738749\pi$
$269$		−	24.0000i		−	1.46331i	0.681677	−	0.731653i	$-0.261251\pi$
$270$		0			0
$271$			7.74597i			0.470534i	0.971931	+	0.235267i	$0.0755965\pi$
$271$			7.74597i			0.470534i	−0.971931	+	0.235267i	$0.924404\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−7.34847	−	7.34847i	−0.441527	−	0.441527i	0.450998	−	0.892525i	$-0.351068\pi$
$277$	−7.34847	−	7.34847i	−0.441527	−	0.441527i	−0.892525	+	0.450998i	$0.851068\pi$
$278$		0			0
$279$	15.4919			0.927478
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	14.2302	+	14.2302i	0.845901	+	0.845901i	0.989619	−	0.143718i	$-0.0459059\pi$
$283$	14.2302	+	14.2302i	0.845901	+	0.845901i	−0.143718	+	0.989619i	$0.545906\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	14.0000i		−	0.823529i
$290$		0			0
$291$		−	15.4919i		−	0.908153i
$292$		0			0
$293$	−9.79796	+	9.79796i	−0.572403	+	0.572403i	−0.932799	−	0.360396i	$-0.882641\pi$
$293$	−9.79796	+	9.79796i	−0.572403	+	0.572403i	0.360396	+	0.932799i	$0.382641\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−6.12372	−	6.12372i	−0.355335	−	0.355335i
$298$		0			0
$299$	15.4919			0.895922
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−18.9737	−	18.9737i	−1.09001	−	1.09001i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	23.7171	−	23.7171i	1.35361	−	1.35361i	0.472015	−	0.881591i	$-0.343527\pi$
$307$	23.7171	−	23.7171i	1.35361	−	1.35361i	0.881591	−	0.472015i	$-0.156473\pi$
$308$		0			0
$309$		−	30.0000i		−	1.70664i
$310$		0			0
$311$		−	15.4919i		−	0.878467i	−0.898373	−	0.439233i	$-0.855250\pi$
$311$		−	15.4919i		−	0.878467i	0.898373	−	0.439233i	$-0.144750\pi$
$312$		0			0
$313$	−14.6969	+	14.6969i	−0.830720	+	0.830720i	−0.987615	−	0.156895i	$-0.949852\pi$
$313$	−14.6969	+	14.6969i	−0.830720	+	0.830720i	0.156895	+	0.987615i	$0.449852\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.89898	+	4.89898i	0.275154	+	0.275154i	0.831171	−	0.556017i	$-0.187671\pi$
$317$	4.89898	+	4.89898i	0.275154	+	0.275154i	−0.556017	+	0.831171i	$0.687671\pi$
$318$		0			0
$319$	23.2379			1.30107
$320$		0			0
$321$	25.0000			1.39536
$322$		0			0
$323$	4.74342	+	4.74342i	0.263931	+	0.263931i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	6.32456	−	6.32456i	0.349749	−	0.349749i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	11.6190i		−	0.638635i	−0.947648	−	0.319318i	$-0.896546\pi$
$331$		−	11.6190i		−	0.638635i	0.947648	−	0.319318i	$-0.103454\pi$
$332$		0			0
$333$	−9.79796	+	9.79796i	−0.536925	+	0.536925i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−11.0227	−	11.0227i	−0.600445	−	0.600445i	0.339986	−	0.940431i	$-0.389578\pi$
$337$	−11.0227	−	11.0227i	−0.600445	−	0.600445i	−0.940431	+	0.339986i	$0.889578\pi$
$338$		0			0
$339$	11.6190			0.631055
$340$		0			0
$341$	−30.0000			−1.62459
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.0680	+	11.0680i	−0.594160	+	0.594160i	−0.938752	−	0.344593i	$-0.888017\pi$
$347$	−11.0680	+	11.0680i	−0.594160	+	0.594160i	0.344593	+	0.938752i	$0.388017\pi$
$348$		0			0
$349$			16.0000i			0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$			16.0000i			0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$		0			0
$351$		−	7.74597i		−	0.413449i
$352$		0			0
$353$	−14.6969	+	14.6969i	−0.782239	+	0.782239i	−0.980208	−	0.197969i	$-0.936565\pi$
$353$	−14.6969	+	14.6969i	−0.782239	+	0.782239i	0.197969	+	0.980208i	$0.436565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.9839			−1.63527			−0.817633	−	0.575740i	$-0.804714\pi$
$359$	−30.9839			−1.63527			−0.817633	+	0.575740i	$0.804714\pi$
$360$		0			0
$361$	−4.00000			−0.210526
$362$		0			0
$363$	−6.32456	−	6.32456i	−0.331953	−	0.331953i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−18.9737	+	18.9737i	−0.990417	+	0.990417i	−0.999955	−	0.00953722i	$-0.996964\pi$
$367$	−18.9737	+	18.9737i	−0.990417	+	0.990417i	0.00953722	+	0.999955i	$0.496964\pi$
$368$		0			0
$369$		−	6.00000i		−	0.312348i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−22.0454	+	22.0454i	−1.14147	+	1.14147i	−0.153286	+	0.988182i	$0.548986\pi$
$373$	−22.0454	+	22.0454i	−1.14147	+	1.14147i	−0.988182	+	0.153286i	$0.951014\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.6969	+	14.6969i	0.756931	+	0.756931i
$378$		0			0
$379$	−3.87298			−0.198942			−0.0994709	−	0.995040i	$-0.531715\pi$
$379$	−3.87298			−0.198942			−0.0994709	+	0.995040i	$0.531715\pi$
$380$		0			0
$381$	30.0000			1.53695
$382$		0			0
$383$	25.2982	+	25.2982i	1.29268	+	1.29268i	0.933124	+	0.359555i	$0.117071\pi$
$383$	25.2982	+	25.2982i	1.29268	+	1.29268i	0.359555	+	0.933124i	$0.382929\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$		0			0
$391$			7.74597i			0.391730i
$392$		0			0
$393$	−12.2474	+	12.2474i	−0.617802	+	0.617802i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−19.5959	−	19.5959i	−0.983491	−	0.983491i	0.0163750	−	0.999866i	$-0.494787\pi$
$397$	−19.5959	−	19.5959i	−0.983491	−	0.983491i	−0.999866	+	0.0163750i	$0.994787\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−33.0000			−1.64794			−0.823971	−	0.566632i	$-0.808246\pi$
$401$	−33.0000			−1.64794			−0.823971	+	0.566632i	$0.808246\pi$
$402$		0			0
$403$	−18.9737	−	18.9737i	−0.945146	−	0.945146i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.9737	−	18.9737i	0.940490	−	0.940490i
$408$		0			0
$409$		−	31.0000i		−	1.53285i	−0.642333	−	0.766426i	$-0.722033\pi$
$409$		−	31.0000i		−	1.53285i	0.642333	−	0.766426i	$-0.277967\pi$
$410$		0			0
$411$			42.6028i			2.10144i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−18.3712	−	18.3712i	−0.899640	−	0.899640i
$418$		0			0
$419$	−34.8569			−1.70287			−0.851434	−	0.524461i	$-0.824267\pi$
$419$	−34.8569			−1.70287			−0.851434	+	0.524461i	$0.824267\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$		0			0
$423$	6.32456	+	6.32456i	0.307510	+	0.307510i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		−	30.0000i		−	1.44841i
$430$		0			0
$431$			30.9839i			1.49244i	0.665699	+	0.746220i	$0.268133\pi$
$431$			30.9839i			1.49244i	−0.665699	+	0.746220i	$0.731867\pi$
$432$		0			0
$433$	15.9217	−	15.9217i	0.765147	−	0.765147i	−0.212101	−	0.977248i	$-0.568030\pi$
$433$	15.9217	−	15.9217i	0.765147	−	0.765147i	0.977248	+	0.212101i	$0.0680304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.2474	+	12.2474i	0.585875	+	0.585875i
$438$		0			0
$439$	−7.74597			−0.369695			−0.184847	−	0.982767i	$-0.559179\pi$
$439$	−7.74597			−0.369695			−0.184847	+	0.982767i	$0.559179\pi$
$440$		0			0
$441$	−14.0000			−0.666667
$442$		0			0
$443$	−7.90569	−	7.90569i	−0.375611	−	0.375611i	0.493905	−	0.869516i	$-0.335569\pi$
$443$	−7.90569	−	7.90569i	−0.375611	−	0.375611i	−0.869516	+	0.493905i	$0.835569\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−9.48683	+	9.48683i	−0.448712	+	0.448712i
$448$		0			0
$449$			9.00000i			0.424736i	0.977190	+	0.212368i	$0.0681176\pi$
$449$			9.00000i			0.424736i	−0.977190	+	0.212368i	$0.931882\pi$
$450$		0			0
$451$			11.6190i			0.547115i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.22474	+	1.22474i	0.0572911	+	0.0572911i	0.735172	−	0.677881i	$-0.237101\pi$
$457$	1.22474	+	1.22474i	0.0572911	+	0.0572911i	−0.677881	+	0.735172i	$0.737101\pi$
$458$		0			0
$459$	3.87298			0.180775
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$	−18.9737	−	18.9737i	−0.881781	−	0.881781i	0.111935	−	0.993716i	$-0.464295\pi$
$463$	−18.9737	−	18.9737i	−0.881781	−	0.881781i	−0.993716	+	0.111935i	$0.964295\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.6491	+	12.6491i	−0.585331	+	0.585331i	−0.936363	−	0.351032i	$-0.885831\pi$
$467$	−12.6491	+	12.6491i	−0.585331	+	0.585331i	0.351032	+	0.936363i	$0.385831\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		−	23.2379i		−	1.07075i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.89898	+	4.89898i	0.224309	+	0.224309i
$478$		0			0
$479$	23.2379			1.06177			0.530883	−	0.847445i	$-0.321860\pi$
$479$	23.2379			1.06177			0.530883	+	0.847445i	$0.321860\pi$
$480$		0			0
$481$	24.0000			1.09431
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−9.48683	+	9.48683i	−0.429889	+	0.429889i	−0.888591	−	0.458701i	$-0.848315\pi$
$487$	−9.48683	+	9.48683i	−0.429889	+	0.429889i	0.458701	+	0.888591i	$0.348315\pi$
$488$		0			0
$489$		−	15.0000i		−	0.678323i
$490$		0			0
$491$			23.2379i			1.04871i	0.851499	+	0.524356i	$0.175694\pi$
$491$			23.2379i			1.04871i	−0.851499	+	0.524356i	$0.824306\pi$
$492$		0			0
$493$	−7.34847	+	7.34847i	−0.330958	+	0.330958i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−38.7298			−1.73379			−0.866893	−	0.498495i	$-0.833886\pi$
$499$	−38.7298			−1.73379			−0.866893	+	0.498495i	$0.833886\pi$
$500$		0			0
$501$	−10.0000			−0.446767
$502$		0			0
$503$	−25.2982	−	25.2982i	−1.12799	−	1.12799i	−0.990503	−	0.137489i	$-0.956097\pi$
$503$	−25.2982	−	25.2982i	−1.12799	−	1.12799i	−0.137489	−	0.990503i	$-0.543903\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.58114	+	1.58114i	−0.0702208	+	0.0702208i
$508$		0			0
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	6.12372	−	6.12372i	0.270369	−	0.270369i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−12.2474	−	12.2474i	−0.538642	−	0.538642i
$518$		0			0
$519$	−30.9839			−1.36004
$520$		0			0
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$		0			0
$523$	4.74342	+	4.74342i	0.207415	+	0.207415i	0.803168	−	0.595753i	$-0.203146\pi$
$523$	4.74342	+	4.74342i	0.207415	+	0.207415i	−0.595753	+	0.803168i	$0.703146\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.48683	−	9.48683i	0.413253	−	0.413253i
$528$		0			0
$529$		−	3.00000i		−	0.130435i
$530$		0			0
$531$		−	15.4919i		−	0.672293i
$532$		0			0
$533$	−7.34847	+	7.34847i	−0.318298	+	0.318298i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−6.12372	−	6.12372i	−0.264258	−	0.264258i
$538$		0			0
$539$	27.1109			1.16775
$540$		0			0
$541$	8.00000			0.343947			0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000			0.343947			0.171973	+	0.985102i	$0.444986\pi$
$542$		0			0
$543$	−3.16228	−	3.16228i	−0.135706	−	0.135706i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4.74342	−	4.74342i	0.202814	−	0.202814i	−0.598391	−	0.801204i	$-0.704193\pi$
$547$	4.74342	−	4.74342i	0.202814	−	0.202814i	0.801204	+	0.598391i	$0.204193\pi$
$548$		0			0
$549$			16.0000i			0.682863i
$550$		0			0
$551$			23.2379i			0.989968i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.34847	−	7.34847i	−0.311365	−	0.311365i	0.534073	−	0.845438i	$-0.320661\pi$
$557$	−7.34847	−	7.34847i	−0.311365	−	0.311365i	−0.845438	+	0.534073i	$0.820661\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	15.0000			0.633300
$562$		0			0
$563$	−12.6491	−	12.6491i	−0.533096	−	0.533096i	0.388396	−	0.921493i	$-0.373029\pi$
$563$	−12.6491	−	12.6491i	−0.533096	−	0.533096i	−0.921493	+	0.388396i	$0.873029\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			39.0000i			1.63497i	0.575953	+	0.817483i	$0.304631\pi$
$569$			39.0000i			1.63497i	−0.575953	+	0.817483i	$0.695369\pi$
$570$		0			0
$571$		−	7.74597i		−	0.324159i	−0.986778	−	0.162079i	$-0.948180\pi$
$571$		−	7.74597i		−	0.324159i	0.986778	−	0.162079i	$-0.0518200\pi$
$572$		0			0
$573$	36.7423	−	36.7423i	1.53493	−	1.53493i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.22474	+	1.22474i	0.0509868	+	0.0509868i	0.732140	−	0.681154i	$-0.238521\pi$
$577$	1.22474	+	1.22474i	0.0509868	+	0.0509868i	−0.681154	+	0.732140i	$0.738521\pi$
$578$		0			0
$579$	11.6190			0.482867
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−9.48683	−	9.48683i	−0.392904	−	0.392904i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.58114	−	1.58114i	0.0652606	−	0.0652606i	−0.673723	−	0.738984i	$-0.735306\pi$
$587$	1.58114	−	1.58114i	0.0652606	−	0.0652606i	0.738984	+	0.673723i	$0.235306\pi$
$588$		0			0
$589$		−	30.0000i		−	1.23613i
$590$		0			0
$591$			54.2218i			2.23039i
$592$		0			0
$593$	−20.8207	+	20.8207i	−0.855002	+	0.855002i	−0.990744	−	0.135742i	$-0.956658\pi$
$593$	−20.8207	+	20.8207i	−0.855002	+	0.855002i	0.135742	+	0.990744i	$0.456658\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	17.0000			0.693444			0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000			0.693444			0.346722	+	0.937968i	$0.387295\pi$
$602$		0			0
$603$	−9.48683	−	9.48683i	−0.386334	−	0.386334i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	15.4919i		−	0.626737i
$612$		0			0
$613$	26.9444	−	26.9444i	1.08827	−	1.08827i	0.0925671	−	0.995706i	$-0.470493\pi$
$613$	26.9444	−	26.9444i	1.08827	−	1.08827i	0.995706	−	0.0925671i	$-0.0295073\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	19.5959	+	19.5959i	0.788902	+	0.788902i	0.981314	−	0.192412i	$-0.0616311\pi$
$617$	19.5959	+	19.5959i	0.788902	+	0.788902i	−0.192412	+	0.981314i	$0.561631\pi$
$618$		0			0
$619$	23.2379			0.934010			0.467005	−	0.884255i	$-0.345333\pi$
$619$	23.2379			0.934010			0.467005	+	0.884255i	$0.345333\pi$
$620$		0			0
$621$	10.0000			0.401286
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	23.7171	−	23.7171i	0.947169	−	0.947169i
$628$		0			0
$629$			12.0000i			0.478471i
$630$		0			0
$631$		−	7.74597i		−	0.308362i	−0.988043	−	0.154181i	$-0.950726\pi$
$631$		−	7.74597i		−	0.308362i	0.988043	−	0.154181i	$-0.0492739\pi$
$632$		0			0
$633$	6.12372	−	6.12372i	0.243396	−	0.243396i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	17.1464	+	17.1464i	0.679366	+	0.679366i
$638$		0			0
$639$	−15.4919			−0.612851
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−12.6491	+	12.6491i	−0.497288	+	0.497288i	−0.910593	−	0.413305i	$-0.864374\pi$
$647$	−12.6491	+	12.6491i	−0.497288	+	0.497288i	0.413305	+	0.910593i	$0.364374\pi$
$648$		0			0
$649$			30.0000i			1.17760i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−34.2929	+	34.2929i	−1.34198	+	1.34198i	−0.447899	+	0.894084i	$0.647828\pi$
$653$	−34.2929	+	34.2929i	−1.34198	+	1.34198i	−0.894084	+	0.447899i	$0.852172\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	7.34847	+	7.34847i	0.286691	+	0.286691i
$658$		0			0
$659$	11.6190			0.452610			0.226305	−	0.974056i	$-0.427335\pi$
$659$	11.6190			0.452610			0.226305	+	0.974056i	$0.427335\pi$
$660$		0			0
$661$	−2.00000			−0.0777910			−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000			−0.0777910			−0.0388955	+	0.999243i	$0.512384\pi$
$662$		0			0
$663$	9.48683	+	9.48683i	0.368438	+	0.368438i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−18.9737	+	18.9737i	−0.734663	+	0.734663i
$668$		0			0
$669$			60.0000i			2.31973i
$670$		0			0
$671$		−	30.9839i		−	1.19612i
$672$		0			0
$673$	9.79796	−	9.79796i	0.377684	−	0.377684i	−0.492582	−	0.870266i	$-0.663947\pi$
$673$	9.79796	−	9.79796i	0.377684	−	0.377684i	0.870266	+	0.492582i	$0.163947\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.3939	+	29.3939i	1.12970	+	1.12970i	0.990226	+	0.139473i	$0.0445407\pi$
$677$	29.3939	+	29.3939i	1.12970	+	1.12970i	0.139473	+	0.990226i	$0.455459\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	20.0000			0.766402
$682$		0			0
$683$	30.0416	+	30.0416i	1.14951	+	1.14951i	0.986649	+	0.162863i	$0.0520727\pi$
$683$	30.0416	+	30.0416i	1.14951	+	1.14951i	0.162863	+	0.986649i	$0.447927\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	6.32456	−	6.32456i	0.241297	−	0.241297i
$688$		0			0
$689$		−	12.0000i		−	0.457164i
$690$		0			0
$691$			3.87298i			0.147335i	0.997283	+	0.0736676i	$0.0234704\pi$
$691$			3.87298i			0.147335i	−0.997283	+	0.0736676i	$0.976530\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.67423	−	3.67423i	−0.139172	−	0.139172i
$698$		0			0
$699$	−46.4758			−1.75788
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$		0			0
$703$	18.9737	+	18.9737i	0.715605	+	0.715605i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	4.00000i		−	0.150223i	−0.997175	−	0.0751116i	$-0.976069\pi$
$709$		−	4.00000i		−	0.150223i	0.997175	−	0.0751116i	$-0.0239313\pi$
$710$		0			0
$711$		−	30.9839i		−	1.16199i
$712$		0			0
$713$	24.4949	−	24.4949i	0.917341	−	0.917341i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−12.2474	−	12.2474i	−0.457389	−	0.457389i
$718$		0			0
$719$	15.4919			0.577752			0.288876	−	0.957367i	$-0.406719\pi$
$719$	15.4919			0.577752			0.288876	+	0.957367i	$0.406719\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	26.8794	+	26.8794i	0.999654	+	0.999654i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	9.48683	−	9.48683i	0.351847	−	0.351847i	−0.508949	−	0.860796i	$-0.669966\pi$
$727$	9.48683	−	9.48683i	0.351847	−	0.351847i	0.860796	+	0.508949i	$0.169966\pi$
$728$		0			0
$729$			7.00000i			0.259259i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	26.9444	−	26.9444i	0.995214	−	0.995214i	−0.00477495	−	0.999989i	$-0.501520\pi$
$733$	26.9444	−	26.9444i	0.995214	−	0.995214i	0.999989	+	0.00477495i	$0.00151992\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18.3712	+	18.3712i	0.676711	+	0.676711i
$738$		0			0
$739$	7.74597			0.284940			0.142470	−	0.989799i	$-0.454496\pi$
$739$	7.74597			0.284940			0.142470	+	0.989799i	$0.454496\pi$
$740$		0			0
$741$	30.0000			1.10208
$742$		0			0
$743$	15.8114	+	15.8114i	0.580064	+	0.580064i	0.934921	−	0.354857i	$-0.115470\pi$
$743$	15.8114	+	15.8114i	0.580064	+	0.580064i	−0.354857	+	0.934921i	$0.615470\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	3.16228	−	3.16228i	0.115702	−	0.115702i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	38.7298i		−	1.41327i	−0.707577	−	0.706636i	$-0.750212\pi$
$751$		−	38.7298i		−	1.41327i	0.707577	−	0.706636i	$-0.249788\pi$
$752$		0			0
$753$	30.6186	−	30.6186i	1.11580	−	1.11580i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	4.89898	+	4.89898i	0.178056	+	0.178056i	0.790508	−	0.612452i	$-0.209817\pi$
$757$	4.89898	+	4.89898i	0.178056	+	0.178056i	−0.612452	+	0.790508i	$0.709817\pi$
$758$		0			0
$759$	38.7298			1.40580
$760$		0			0
$761$	−33.0000			−1.19625			−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000			−1.19625			−0.598125	+	0.801403i	$0.704087\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.9737	+	18.9737i	−0.685099	+	0.685099i
$768$		0			0
$769$			19.0000i			0.685158i	0.939489	+	0.342579i	$0.111300\pi$
$769$			19.0000i			0.685158i	−0.939489	+	0.342579i	$0.888700\pi$
$770$		0			0
$771$		−	15.4919i		−	0.557928i
$772$		0			0
$773$	26.9444	−	26.9444i	0.969122	−	0.969122i	−0.0304151	−	0.999537i	$-0.509683\pi$
$773$	26.9444	−	26.9444i	0.969122	−	0.969122i	0.999537	+	0.0304151i	$0.00968292\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.6190			−0.416292
$780$		0			0
$781$	30.0000			1.07348
$782$		0			0
$783$	9.48683	+	9.48683i	0.339032	+	0.339032i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−37.9473	+	37.9473i	−1.35268	+	1.35268i	−0.470021	+	0.882655i	$0.655754\pi$
$787$	−37.9473	+	37.9473i	−1.35268	+	1.35268i	−0.882655	+	0.470021i	$0.844246\pi$
$788$		0			0
$789$		−	10.0000i		−	0.356009i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	19.5959	−	19.5959i	0.695871	−	0.695871i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.1464	+	17.1464i	0.607358	+	0.607358i	0.942255	−	0.334897i	$-0.108702\pi$
$797$	17.1464	+	17.1464i	0.607358	+	0.607358i	−0.334897	+	0.942255i	$0.608702\pi$
$798$		0			0
$799$	7.74597			0.274033
$800$		0			0
$801$	−18.0000			−0.635999
$802$		0			0
$803$	−14.2302	−	14.2302i	−0.502175	−	0.502175i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	37.9473	−	37.9473i	1.33581	−	1.33581i
$808$		0			0
$809$		−	6.00000i		−	0.210949i	−0.994422	−	0.105474i	$-0.966364\pi$
$809$		−	6.00000i		−	0.210949i	0.994422	−	0.105474i	$-0.0336361\pi$
$810$		0			0
$811$		−	23.2379i		−	0.815993i	−0.912983	−	0.407997i	$-0.866228\pi$
$811$		−	23.2379i		−	0.815993i	0.912983	−	0.407997i	$-0.133772\pi$
$812$		0			0
$813$	−12.2474	+	12.2474i	−0.429537	+	0.429537i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.0000			0.628204			0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000			0.628204			0.314102	+	0.949389i	$0.398297\pi$
$822$		0			0
$823$	18.9737	+	18.9737i	0.661380	+	0.661380i	0.955705	−	0.294325i	$-0.0950948\pi$
$823$	18.9737	+	18.9737i	0.661380	+	0.661380i	−0.294325	+	0.955705i	$0.595095\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	30.0416	−	30.0416i	1.04465	−	1.04465i	0.0456946	−	0.998955i	$-0.485450\pi$
$827$	30.0416	−	30.0416i	1.04465	−	1.04465i	0.998955	−	0.0456946i	$-0.0145501\pi$
$828$		0			0
$829$			46.0000i			1.59765i	0.601566	+	0.798823i	$0.294544\pi$
$829$			46.0000i			1.59765i	−0.601566	+	0.798823i	$0.705456\pi$
$830$		0			0
$831$		−	23.2379i		−	0.806114i
$832$		0			0
$833$	−8.57321	+	8.57321i	−0.297044	+	0.297044i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−12.2474	−	12.2474i	−0.423334	−	0.423334i
$838$		0			0
$839$	30.9839			1.06968			0.534841	−	0.844953i	$-0.320372\pi$
$839$	30.9839			1.06968			0.534841	+	0.844953i	$0.320372\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	−28.4605	−	28.4605i	−0.980232	−	0.980232i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$			45.0000i			1.54440i
$850$		0			0
$851$			30.9839i			1.06211i
$852$		0			0
$853$	2.44949	−	2.44949i	0.0838689	−	0.0838689i	−0.663928	−	0.747797i	$-0.731112\pi$
$853$	2.44949	−	2.44949i	0.0838689	−	0.0838689i	0.747797	+	0.663928i	$0.231112\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.22474	+	1.22474i	0.0418365	+	0.0418365i	0.727716	−	0.685879i	$-0.240582\pi$
$857$	1.22474	+	1.22474i	0.0418365	+	0.0418365i	−0.685879	+	0.727716i	$0.740582\pi$
$858$		0			0
$859$	50.3488			1.71788			0.858939	−	0.512078i	$-0.171124\pi$
$859$	50.3488			1.71788			0.858939	+	0.512078i	$0.171124\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.16228	−	3.16228i	−0.107645	−	0.107645i	0.651233	−	0.758878i	$-0.274252\pi$
$863$	−3.16228	−	3.16228i	−0.107645	−	0.107645i	−0.758878	+	0.651233i	$0.774252\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	22.1359	−	22.1359i	0.751776	−	0.751776i
$868$		0			0
$869$			60.0000i			2.03536i
$870$		0			0
$871$			23.2379i			0.787386i
$872$		0			0
$873$	9.79796	−	9.79796i	0.331611	−	0.331611i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	4.89898	+	4.89898i	0.165427	+	0.165427i	0.784966	−	0.619539i	$-0.212680\pi$
$877$	4.89898	+	4.89898i	0.165427	+	0.165427i	−0.619539	+	0.784966i	$0.712680\pi$
$878$		0			0
$879$	−30.9839			−1.04506
$880$		0			0
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$		0			0
$883$	14.2302	+	14.2302i	0.478886	+	0.478886i	0.904775	−	0.425889i	$-0.140039\pi$
$883$	14.2302	+	14.2302i	0.478886	+	0.478886i	−0.425889	+	0.904775i	$0.640039\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−25.2982	+	25.2982i	−0.849431	+	0.849431i	−0.990062	−	0.140631i	$-0.955087\pi$
$887$	−25.2982	+	25.2982i	−0.849431	+	0.849431i	0.140631	+	0.990062i	$0.455087\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	42.6028i		−	1.42725i
$892$		0			0
$893$	12.2474	−	12.2474i	0.409845	−	0.409845i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	24.4949	+	24.4949i	0.817861	+	0.817861i
$898$		0			0
$899$	46.4758			1.55005
$900$		0			0
$901$	6.00000			0.199889
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		−	24.0000i		−	0.796030i
$910$		0			0
$911$		−	15.4919i		−	0.513271i	−0.966508	−	0.256635i	$-0.917386\pi$
$911$		−	15.4919i		−	0.513271i	0.966508	−	0.256635i	$-0.0826139\pi$
$912$		0			0
$913$	−6.12372	+	6.12372i	−0.202666	+	0.202666i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−23.2379			−0.766548			−0.383274	−	0.923635i	$-0.625203\pi$
$919$	−23.2379			−0.766548			−0.383274	+	0.923635i	$0.625203\pi$
$920$		0			0
$921$	75.0000			2.47133
$922$		0			0
$923$	18.9737	+	18.9737i	0.624526	+	0.624526i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	18.9737	−	18.9737i	0.623177	−	0.623177i
$928$		0			0
$929$		−	6.00000i		−	0.196854i	−0.995144	−	0.0984268i	$-0.968619\pi$
$929$		−	6.00000i		−	0.196854i	0.995144	−	0.0984268i	$-0.0313810\pi$
$930$		0			0
$931$			27.1109i			0.888523i
$932$		0			0
$933$	24.4949	−	24.4949i	0.801927	−	0.801927i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−35.5176	−	35.5176i	−1.16031	−	1.16031i	−0.984408	−	0.175902i	$-0.943716\pi$
$937$	−35.5176	−	35.5176i	−1.16031	−	1.16031i	−0.175902	−	0.984408i	$-0.556284\pi$
$938$		0			0
$939$	−46.4758			−1.51668
$940$		0			0
$941$	−42.0000			−1.36916			−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000			−1.36916			−0.684580	+	0.728937i	$0.740015\pi$
$942$		0			0
$943$	−9.48683	−	9.48683i	−0.308934	−	0.308934i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.6491	−	12.6491i	0.411041	−	0.411041i	−0.471060	−	0.882101i	$-0.656129\pi$
$947$	12.6491	−	12.6491i	0.411041	−	0.411041i	0.882101	+	0.471060i	$0.156129\pi$
$948$		0			0
$949$		−	18.0000i		−	0.584305i
$950$		0			0
$951$			15.4919i			0.502360i
$952$		0			0
$953$	3.67423	−	3.67423i	0.119020	−	0.119020i	−0.645088	−	0.764108i	$-0.723179\pi$
$953$	3.67423	−	3.67423i	0.119020	−	0.119020i	0.764108	+	0.645088i	$0.223179\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	36.7423	+	36.7423i	1.18771	+	1.18771i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−29.0000			−0.935484
$962$		0			0
$963$	15.8114	+	15.8114i	0.509515	+	0.509515i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−18.9737	+	18.9737i	−0.610152	+	0.610152i	−0.942986	−	0.332834i	$-0.891995\pi$
$967$	−18.9737	+	18.9737i	−0.610152	+	0.610152i	0.332834	+	0.942986i	$0.391995\pi$
$968$		0			0
$969$			15.0000i			0.481869i
$970$		0			0
$971$		−	27.1109i		−	0.870030i	−0.900423	−	0.435015i	$-0.856743\pi$
$971$		−	27.1109i		−	0.870030i	0.900423	−	0.435015i	$-0.143257\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.7196	+	25.7196i	0.822844	+	0.822844i	0.986515	−	0.163671i	$-0.0523335\pi$
$977$	25.7196	+	25.7196i	0.822844	+	0.822844i	−0.163671	+	0.986515i	$0.552333\pi$
$978$		0			0
$979$	34.8569			1.11403
$980$		0			0
$981$	8.00000			0.255420
$982$		0			0
$983$	−6.32456	−	6.32456i	−0.201722	−	0.201722i	0.599015	−	0.800737i	$-0.295559\pi$
$983$	−6.32456	−	6.32456i	−0.201722	−	0.201722i	−0.800737	+	0.599015i	$0.795559\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			15.4919i			0.492117i	0.969255	+	0.246059i	$0.0791356\pi$
$991$			15.4919i			0.492117i	−0.969255	+	0.246059i	$0.920864\pi$
$992$		0			0
$993$	18.3712	−	18.3712i	0.582992	−	0.582992i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−19.5959	−	19.5959i	−0.620609	−	0.620609i	0.325078	−	0.945687i	$-0.394609\pi$
$997$	−19.5959	−	19.5959i	−0.620609	−	0.620609i	−0.945687	+	0.325078i	$0.894609\pi$
$998$		0			0
$999$	15.4919			0.490143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.n.v.1407.3		8
4.3	odd	2	inner	1600.2.n.v.1407.2		8
5.2	odd	4	inner	1600.2.n.v.1343.3		8
5.3	odd	4	inner	1600.2.n.v.1343.1		8
5.4	even	2	inner	1600.2.n.v.1407.1		8
8.3	odd	2		100.2.e.d.7.2	yes	8
8.5	even	2		100.2.e.d.7.4	yes	8
20.3	even	4	inner	1600.2.n.v.1343.4		8
20.7	even	4	inner	1600.2.n.v.1343.2		8
20.19	odd	2	inner	1600.2.n.v.1407.4		8
24.5	odd	2		900.2.k.j.307.1		8
24.11	even	2		900.2.k.j.307.3		8
40.3	even	4		100.2.e.d.43.4	yes	8
40.13	odd	4		100.2.e.d.43.2	yes	8
40.19	odd	2		100.2.e.d.7.3	yes	8
40.27	even	4		100.2.e.d.43.1	yes	8
40.29	even	2		100.2.e.d.7.1	✓	8
40.37	odd	4		100.2.e.d.43.3	yes	8
120.29	odd	2		900.2.k.j.307.4		8
120.53	even	4		900.2.k.j.343.3		8
120.59	even	2		900.2.k.j.307.2		8
120.77	even	4		900.2.k.j.343.2		8
120.83	odd	4		900.2.k.j.343.1		8
120.107	odd	4		900.2.k.j.343.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.e.d.7.1	✓	8	40.29	even	2
100.2.e.d.7.2	yes	8	8.3	odd	2
100.2.e.d.7.3	yes	8	40.19	odd	2
100.2.e.d.7.4	yes	8	8.5	even	2
100.2.e.d.43.1	yes	8	40.27	even	4
100.2.e.d.43.2	yes	8	40.13	odd	4
100.2.e.d.43.3	yes	8	40.37	odd	4
100.2.e.d.43.4	yes	8	40.3	even	4
900.2.k.j.307.1		8	24.5	odd	2
900.2.k.j.307.2		8	120.59	even	2
900.2.k.j.307.3		8	24.11	even	2
900.2.k.j.307.4		8	120.29	odd	2
900.2.k.j.343.1		8	120.83	odd	4
900.2.k.j.343.2		8	120.77	even	4
900.2.k.j.343.3		8	120.53	even	4
900.2.k.j.343.4		8	120.107	odd	4
1600.2.n.v.1343.1		8	5.3	odd	4	inner
1600.2.n.v.1343.2		8	20.7	even	4	inner
1600.2.n.v.1343.3		8	5.2	odd	4	inner
1600.2.n.v.1343.4		8	20.3	even	4	inner
1600.2.n.v.1407.1		8	5.4	even	2	inner
1600.2.n.v.1407.2		8	4.3	odd	2	inner
1600.2.n.v.1407.3		8	1.1	even	1	trivial
1600.2.n.v.1407.4		8	20.19	odd	2	inner

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.58114 + 1.58114i) q^{3} +2.00000i q^{9} +O(q^{10})\)	\(q+(1.58114 + 1.58114i) q^{3} +2.00000i q^{9} -3.87298i q^{11} +(2.44949 - 2.44949i) q^{13} +(1.22474 + 1.22474i) q^{17} +3.87298 q^{19} +(3.16228 + 3.16228i) q^{23} +(1.58114 - 1.58114i) q^{27} +6.00000i q^{29} -7.74597i q^{31} +(6.12372 - 6.12372i) q^{33} +(4.89898 + 4.89898i) q^{37} +7.74597 q^{39} -3.00000 q^{41} +(3.16228 - 3.16228i) q^{47} +7.00000i q^{49} +3.87298i q^{51} +(2.44949 - 2.44949i) q^{53} +(6.12372 + 6.12372i) q^{57} -7.74597 q^{59} +8.00000 q^{61} +(-4.74342 + 4.74342i) q^{67} +10.0000i q^{69} +7.74597i q^{71} +(3.67423 - 3.67423i) q^{73} -15.4919 q^{79} +11.0000 q^{81} +(-1.58114 - 1.58114i) q^{83} +(-9.48683 + 9.48683i) q^{87} +9.00000i q^{89} +(12.2474 - 12.2474i) q^{93} +(-4.89898 - 4.89898i) q^{97} +7.74597 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{41} + 64 q^{61} + 88 q^{81}+O(q^{100}) \) 8 * q - 24 * q^41 + 64 * q^61 + 88 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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