LMFDB - Embedded newform 2450.2.c.v.99.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(99,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.c (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:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 98)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.4
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2450.99
Dual form			2450.2.c.v.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.41421i q^{3} -1.00000 q^{4} -1.41421 q^{6} -1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.41421i q^{12} +1.00000 q^{16} -1.41421i q^{17} +1.00000i q^{18} +7.07107 q^{19} -2.00000i q^{22} +4.00000i q^{23} +1.41421 q^{24} +5.65685i q^{27} -2.00000 q^{29} +8.48528 q^{31} +1.00000i q^{32} -2.82843i q^{33} +1.41421 q^{34} -1.00000 q^{36} +10.0000i q^{37} +7.07107i q^{38} -9.89949 q^{41} -2.00000i q^{43} +2.00000 q^{44} -4.00000 q^{46} +2.82843i q^{47} +1.41421i q^{48} +2.00000 q^{51} +2.00000i q^{53} -5.65685 q^{54} +10.0000i q^{57} -2.00000i q^{58} +1.41421 q^{59} +2.82843 q^{61} +8.48528i q^{62} -1.00000 q^{64} +2.82843 q^{66} +12.0000i q^{67} +1.41421i q^{68} -5.65685 q^{69} -12.0000 q^{71} -1.00000i q^{72} +1.41421i q^{73} -10.0000 q^{74} -7.07107 q^{76} +4.00000 q^{79} -5.00000 q^{81} -9.89949i q^{82} -9.89949i q^{83} +2.00000 q^{86} -2.82843i q^{87} +2.00000i q^{88} +7.07107 q^{89} -4.00000i q^{92} +12.0000i q^{93} -2.82843 q^{94} -1.41421 q^{96} +9.89949i q^{97} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{9} - 8 q^{11} + 4 q^{16} - 8 q^{29} - 4 q^{36} + 8 q^{44} - 16 q^{46} + 8 q^{51} - 4 q^{64} - 48 q^{71} - 40 q^{74} + 16 q^{79} - 20 q^{81} + 8 q^{86} - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^9 - 8 * q^11 + 4 * q^16 - 8 * q^29 - 4 * q^36 + 8 * q^44 - 16 * q^46 + 8 * q^51 - 4 * q^64 - 48 * q^71 - 40 * q^74 + 16 * q^79 - 20 * q^81 + 8 * q^86 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.41421i	0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$	1.41421i	0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	− 1.41421i	− 0.408248i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 1.41421i	− 0.342997i	−0.985184	−	0.171499i	$-0.945139\pi$
$17$	− 1.41421i	− 0.342997i	0.985184	−	0.171499i	$-0.0548609\pi$
$18$	1.00000i	0.235702i
$19$	7.07107	1.62221	0.811107	−	0.584898i	$-0.198865\pi$
$19$	7.07107	1.62221	0.811107	+	0.584898i	$0.198865\pi$
$20$	0	0
$21$	0	0
$22$	− 2.00000i	− 0.426401i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	1.41421	0.288675
$25$	0	0
$26$	0	0
$27$	5.65685i	1.08866i
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	1.00000i	0.176777i
$33$	− 2.82843i	− 0.492366i
$34$	1.41421	0.242536
$35$	0	0
$36$	−1.00000	−0.166667
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	7.07107i	1.14708i
$39$	0	0
$40$	0	0
$41$	−9.89949	−1.54604	−0.773021	−	0.634381i	$-0.781255\pi$
$41$	−9.89949	−1.54604	−0.773021	+	0.634381i	$0.781255\pi$
$42$	0	0
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−4.00000	−0.589768
$47$	2.82843i	0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$	2.82843i	0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$	1.41421i	0.204124i
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	−5.65685	−0.769800
$55$	0	0
$56$	0	0
$57$	10.0000i	1.32453i
$58$	− 2.00000i	− 0.262613i
$59$	1.41421	0.184115	0.0920575	−	0.995754i	$-0.470656\pi$
$59$	1.41421	0.184115	0.0920575	+	0.995754i	$0.470656\pi$
$60$	0	0
$61$	2.82843	0.362143	0.181071	−	0.983470i	$-0.442043\pi$
$61$	2.82843	0.362143	0.181071	+	0.983470i	$0.442043\pi$
$62$	8.48528i	1.07763i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.82843	0.348155
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	1.41421i	0.171499i
$69$	−5.65685	−0.681005
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	− 1.00000i	− 0.117851i
$73$	1.41421i	0.165521i	0.996569	+	0.0827606i	$0.0263737\pi$
$73$	1.41421i	0.165521i	−0.996569	+	0.0827606i	$0.973626\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−7.07107	−0.811107
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	− 9.89949i	− 1.09322i
$83$	− 9.89949i	− 1.08661i	−0.839535	−	0.543305i	$-0.817173\pi$
$83$	− 9.89949i	− 1.08661i	0.839535	−	0.543305i	$-0.182827\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	− 2.82843i	− 0.303239i
$88$	2.00000i	0.213201i
$89$	7.07107	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$90$	0	0
$91$	0	0
$92$	− 4.00000i	− 0.417029i
$93$	12.0000i	1.24434i
$94$	−2.82843	−0.291730
$95$	0	0
$96$	−1.41421	−0.144338
$97$	9.89949i	1.00514i	0.864536	+	0.502571i	$0.167612\pi$
$97$	9.89949i	1.00514i	−0.864536	+	0.502571i	$0.832388\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	8.48528	0.844317	0.422159	−	0.906522i	$-0.361273\pi$
$101$	8.48528	0.844317	0.422159	+	0.906522i	$0.361273\pi$
$102$	2.00000i	0.198030i
$103$	− 2.82843i	− 0.278693i	−0.990244	−	0.139347i	$-0.955500\pi$
$103$	− 2.82843i	− 0.278693i	0.990244	−	0.139347i	$-0.0445002\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	− 5.65685i	− 0.544331i
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−14.1421	−1.34231
$112$	0	0
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	−10.0000	−0.936586
$115$	0	0
$116$	2.00000	0.185695
$117$	0	0
$118$	1.41421i	0.130189i
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.82843i	0.256074i
$123$	− 14.0000i	− 1.26234i
$124$	−8.48528	−0.762001
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	2.82843	0.249029
$130$	0	0
$131$	12.7279	1.11204	0.556022	−	0.831168i	$-0.312327\pi$
$131$	12.7279	1.11204	0.556022	+	0.831168i	$0.312327\pi$
$132$	2.82843i	0.246183i
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	−1.41421	−0.121268
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	− 5.65685i	− 0.481543i
$139$	−9.89949	−0.839664	−0.419832	−	0.907602i	$-0.637911\pi$
$139$	−9.89949	−0.839664	−0.419832	+	0.907602i	$0.637911\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	− 12.0000i	− 1.00702i
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−1.41421	−0.117041
$147$	0	0
$148$	− 10.0000i	− 0.821995i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	− 7.07107i	− 0.573539i
$153$	− 1.41421i	− 0.114332i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 11.3137i	− 0.902932i	−0.892288	−	0.451466i	$-0.850901\pi$
$157$	− 11.3137i	− 0.902932i	0.892288	−	0.451466i	$-0.149099\pi$
$158$	4.00000i	0.318223i
$159$	−2.82843	−0.224309
$160$	0	0
$161$	0	0
$162$	− 5.00000i	− 0.392837i
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	9.89949	0.773021
$165$	0	0
$166$	9.89949	0.768350
$167$	− 19.7990i	− 1.53209i	−0.642786	−	0.766046i	$-0.722221\pi$
$167$	− 19.7990i	− 1.53209i	0.642786	−	0.766046i	$-0.277779\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	7.07107	0.540738
$172$	2.00000i	0.152499i
$173$	16.9706i	1.29025i	0.764078	+	0.645124i	$0.223194\pi$
$173$	16.9706i	1.29025i	−0.764078	+	0.645124i	$0.776806\pi$
$174$	2.82843	0.214423
$175$	0	0
$176$	−2.00000	−0.150756
$177$	2.00000i	0.150329i
$178$	7.07107i	0.529999i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	4.00000i	0.295689i
$184$	4.00000	0.294884
$185$	0	0
$186$	−12.0000	−0.879883
$187$	2.82843i	0.206835i
$188$	− 2.82843i	− 0.206284i
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	− 1.41421i	− 0.102062i
$193$	16.0000i	1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$	16.0000i	1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	−9.89949	−0.710742
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	− 2.00000i	− 0.142134i
$199$	−8.48528	−0.601506	−0.300753	−	0.953702i	$-0.597238\pi$
$199$	−8.48528	−0.601506	−0.300753	+	0.953702i	$0.597238\pi$
$200$	0	0
$201$	−16.9706	−1.19701
$202$	8.48528i	0.597022i
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	2.82843	0.197066
$207$	4.00000i	0.278019i
$208$	0	0
$209$	−14.1421	−0.978232
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	− 2.00000i	− 0.137361i
$213$	− 16.9706i	− 1.16280i
$214$	4.00000	0.273434
$215$	0	0
$216$	5.65685	0.384900
$217$	0	0
$218$	2.00000i	0.135457i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0	0
$222$	− 14.1421i	− 0.949158i
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	−12.0000	−0.798228
$227$	− 21.2132i	− 1.40797i	−0.710215	−	0.703985i	$-0.751402\pi$
$227$	− 21.2132i	− 1.40797i	0.710215	−	0.703985i	$-0.248598\pi$
$228$	− 10.0000i	− 0.662266i
$229$	16.9706	1.12145	0.560723	−	0.828003i	$-0.310523\pi$
$229$	16.9706	1.12145	0.560723	+	0.828003i	$0.310523\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	0	0
$236$	−1.41421	−0.0920575
$237$	5.65685i	0.367452i
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−21.2132	−1.36646	−0.683231	−	0.730202i	$-0.739426\pi$
$241$	−21.2132	−1.36646	−0.683231	+	0.730202i	$0.739426\pi$
$242$	− 7.00000i	− 0.449977i
$243$	9.89949i	0.635053i
$244$	−2.82843	−0.181071
$245$	0	0
$246$	14.0000	0.892607
$247$	0	0
$248$	− 8.48528i	− 0.538816i
$249$	14.0000	0.887214
$250$	0	0
$251$	9.89949	0.624851	0.312425	−	0.949942i	$-0.398859\pi$
$251$	9.89949	0.624851	0.312425	+	0.949942i	$0.398859\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.7279i	0.793946i	0.917830	+	0.396973i	$0.129939\pi$
$257$	12.7279i	0.793946i	−0.917830	+	0.396973i	$0.870061\pi$
$258$	2.82843i	0.176090i
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	12.7279i	0.786334i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	−2.82843	−0.174078
$265$	0	0
$266$	0	0
$267$	10.0000i	0.611990i
$268$	− 12.0000i	− 0.733017i
$269$	11.3137	0.689809	0.344904	−	0.938638i	$-0.387911\pi$
$269$	11.3137	0.689809	0.344904	+	0.938638i	$0.387911\pi$
$270$	0	0
$271$	22.6274	1.37452	0.687259	−	0.726413i	$-0.258814\pi$
$271$	22.6274	1.37452	0.687259	+	0.726413i	$0.258814\pi$
$272$	− 1.41421i	− 0.0857493i
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	5.65685	0.340503
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 9.89949i	− 0.593732i
$279$	8.48528	0.508001
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	− 4.00000i	− 0.238197i
$283$	1.41421i	0.0840663i	0.999116	+	0.0420331i	$0.0133835\pi$
$283$	1.41421i	0.0840663i	−0.999116	+	0.0420331i	$0.986616\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	15.0000	0.882353
$290$	0	0
$291$	−14.0000	−0.820695
$292$	− 1.41421i	− 0.0827606i
$293$	19.7990i	1.15667i	0.815800	+	0.578335i	$0.196297\pi$
$293$	19.7990i	1.15667i	−0.815800	+	0.578335i	$0.803703\pi$
$294$	0	0
$295$	0	0
$296$	10.0000	0.581238
$297$	− 11.3137i	− 0.656488i
$298$	− 10.0000i	− 0.579284i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 16.0000i	− 0.920697i
$303$	12.0000i	0.689382i
$304$	7.07107	0.405554
$305$	0	0
$306$	1.41421	0.0808452
$307$	− 9.89949i	− 0.564994i	−0.959268	−	0.282497i	$-0.908837\pi$
$307$	− 9.89949i	− 0.564994i	0.959268	−	0.282497i	$-0.0911627\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	− 12.7279i	− 0.719425i	−0.933063	−	0.359712i	$-0.882875\pi$
$313$	− 12.7279i	− 0.719425i	0.933063	−	0.359712i	$-0.117125\pi$
$314$	11.3137	0.638470
$315$	0	0
$316$	−4.00000	−0.225018
$317$	10.0000i	0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$	10.0000i	0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$	− 2.82843i	− 0.158610i
$319$	4.00000	0.223957
$320$	0	0
$321$	5.65685	0.315735
$322$	0	0
$323$	− 10.0000i	− 0.556415i
$324$	5.00000	0.277778
$325$	0	0
$326$	10.0000	0.553849
$327$	2.82843i	0.156412i
$328$	9.89949i	0.546608i
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	9.89949i	0.543305i
$333$	10.0000i	0.547997i
$334$	19.7990	1.08335
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	13.0000i	0.707107i
$339$	−16.9706	−0.921714
$340$	0	0
$341$	−16.9706	−0.919007
$342$	7.07107i	0.382360i
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−16.9706	−0.912343
$347$	− 30.0000i	− 1.61048i	−0.592946	−	0.805242i	$-0.702035\pi$
$347$	− 30.0000i	− 1.61048i	0.592946	−	0.805242i	$-0.297965\pi$
$348$	2.82843i	0.151620i
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	− 2.00000i	− 0.106600i
$353$	1.41421i	0.0752710i	0.999292	+	0.0376355i	$0.0119826\pi$
$353$	1.41421i	0.0752710i	−0.999292	+	0.0376355i	$0.988017\pi$
$354$	−2.00000	−0.106299
$355$	0	0
$356$	−7.07107	−0.374766
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	31.0000	1.63158
$362$	0	0
$363$	− 9.89949i	− 0.519589i
$364$	0	0
$365$	0	0
$366$	−4.00000	−0.209083
$367$	28.2843i	1.47643i	0.674567	+	0.738213i	$0.264330\pi$
$367$	28.2843i	1.47643i	−0.674567	+	0.738213i	$0.735670\pi$
$368$	4.00000i	0.208514i
$369$	−9.89949	−0.515347
$370$	0	0
$371$	0	0
$372$	− 12.0000i	− 0.622171i
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	−2.82843	−0.146254
$375$	0	0
$376$	2.82843	0.145865
$377$	0	0
$378$	0	0
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	−22.6274	−1.15924
$382$	− 4.00000i	− 0.204658i
$383$	36.7696i	1.87884i	0.342773	+	0.939418i	$0.388634\pi$
$383$	36.7696i	1.87884i	−0.342773	+	0.939418i	$0.611366\pi$
$384$	1.41421	0.0721688
$385$	0	0
$386$	−16.0000	−0.814379
$387$	− 2.00000i	− 0.101666i
$388$	− 9.89949i	− 0.502571i
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	18.0000i	0.907980i
$394$	−2.00000	−0.100759
$395$	0	0
$396$	2.00000	0.100504
$397$	22.6274i	1.13564i	0.823154	+	0.567819i	$0.192213\pi$
$397$	22.6274i	1.13564i	−0.823154	+	0.567819i	$0.807787\pi$
$398$	− 8.48528i	− 0.425329i
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	− 16.9706i	− 0.846415i
$403$	0	0
$404$	−8.48528	−0.422159
$405$	0	0
$406$	0	0
$407$	− 20.0000i	− 0.991363i
$408$	− 2.00000i	− 0.0990148i
$409$	−38.1838	−1.88807	−0.944033	−	0.329851i	$-0.893001\pi$
$409$	−38.1838	−1.88807	−0.944033	+	0.329851i	$0.893001\pi$
$410$	0	0
$411$	−16.9706	−0.837096
$412$	2.82843i	0.139347i
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	0	0
$417$	− 14.0000i	− 0.685583i
$418$	− 14.1421i	− 0.691714i
$419$	9.89949	0.483622	0.241811	−	0.970323i	$-0.422259\pi$
$419$	9.89949	0.483622	0.241811	+	0.970323i	$0.422259\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	− 12.0000i	− 0.584151i
$423$	2.82843i	0.137523i
$424$	2.00000	0.0971286
$425$	0	0
$426$	16.9706	0.822226
$427$	0	0
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	5.65685i	0.272166i
$433$	29.6985i	1.42722i	0.700544	+	0.713609i	$0.252941\pi$
$433$	29.6985i	1.42722i	−0.700544	+	0.713609i	$0.747059\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	28.2843i	1.35302i
$438$	− 2.00000i	− 0.0955637i
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	14.1421	0.671156
$445$	0	0
$446$	0	0
$447$	− 14.1421i	− 0.668900i
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	19.7990	0.932298
$452$	− 12.0000i	− 0.564433i
$453$	− 22.6274i	− 1.06313i
$454$	21.2132	0.995585
$455$	0	0
$456$	10.0000	0.468293
$457$	24.0000i	1.12267i	0.827588	+	0.561336i	$0.189713\pi$
$457$	24.0000i	1.12267i	−0.827588	+	0.561336i	$0.810287\pi$
$458$	16.9706i	0.792982i
$459$	8.00000	0.373408
$460$	0	0
$461$	39.5980	1.84426	0.922131	−	0.386878i	$-0.126447\pi$
$461$	39.5980	1.84426	0.922131	+	0.386878i	$0.126447\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	24.0000	1.11178
$467$	32.5269i	1.50517i	0.658497	+	0.752583i	$0.271192\pi$
$467$	32.5269i	1.50517i	−0.658497	+	0.752583i	$0.728808\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.0000	0.737241
$472$	− 1.41421i	− 0.0650945i
$473$	4.00000i	0.183920i
$474$	−5.65685	−0.259828
$475$	0	0
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	12.0000i	0.548867i
$479$	31.1127	1.42158	0.710788	−	0.703407i	$-0.248339\pi$
$479$	31.1127	1.42158	0.710788	+	0.703407i	$0.248339\pi$
$480$	0	0
$481$	0	0
$482$	− 21.2132i	− 0.966235i
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	−9.89949	−0.449050
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	− 2.82843i	− 0.128037i
$489$	14.1421	0.639529
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	14.0000i	0.631169i
$493$	2.82843i	0.127386i
$494$	0	0
$495$	0	0
$496$	8.48528	0.381000
$497$	0	0
$498$	14.0000i	0.627355i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	28.0000	1.25095
$502$	9.89949i	0.441836i
$503$	− 39.5980i	− 1.76559i	−0.469762	−	0.882793i	$-0.655660\pi$
$503$	− 39.5980i	− 1.76559i	0.469762	−	0.882793i	$-0.344340\pi$
$504$	0	0
$505$	0	0
$506$	8.00000	0.355643
$507$	18.3848i	0.816497i
$508$	− 16.0000i	− 0.709885i
$509$	−22.6274	−1.00294	−0.501471	−	0.865174i	$-0.667208\pi$
$509$	−22.6274	−1.00294	−0.501471	+	0.865174i	$0.667208\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	40.0000i	1.76604i
$514$	−12.7279	−0.561405
$515$	0	0
$516$	−2.82843	−0.124515
$517$	− 5.65685i	− 0.248788i
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	− 12.7279i	− 0.556553i	−0.960501	−	0.278277i	$-0.910237\pi$
$523$	− 12.7279i	− 0.556553i	0.960501	−	0.278277i	$-0.0897632\pi$
$524$	−12.7279	−0.556022
$525$	0	0
$526$	12.0000	0.523225
$527$	− 12.0000i	− 0.522728i
$528$	− 2.82843i	− 0.123091i
$529$	7.00000	0.304348
$530$	0	0
$531$	1.41421	0.0613716
$532$	0	0
$533$	0	0
$534$	−10.0000	−0.432742
$535$	0	0
$536$	12.0000	0.518321
$537$	− 16.9706i	− 0.732334i
$538$	11.3137i	0.487769i
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	22.6274i	0.971931i
$543$	0	0
$544$	1.41421	0.0606339
$545$	0	0
$546$	0	0
$547$	− 26.0000i	− 1.11168i	−0.831289	−	0.555840i	$-0.812397\pi$
$547$	− 26.0000i	− 1.11168i	0.831289	−	0.555840i	$-0.187603\pi$
$548$	− 12.0000i	− 0.512615i
$549$	2.82843	0.120714
$550$	0	0
$551$	−14.1421	−0.602475
$552$	5.65685i	0.240772i
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	9.89949	0.419832
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	8.48528i	0.359211i
$559$	0	0
$560$	0	0
$561$	−4.00000	−0.168880
$562$	16.0000i	0.674919i
$563$	1.41421i	0.0596020i	0.999556	+	0.0298010i	$0.00948736\pi$
$563$	1.41421i	0.0596020i	−0.999556	+	0.0298010i	$0.990513\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−1.41421	−0.0594438
$567$	0	0
$568$	12.0000i	0.503509i
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	0	0
$573$	− 5.65685i	− 0.236318i
$574$	0	0
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	− 21.2132i	− 0.883117i	−0.897232	−	0.441559i	$-0.854426\pi$
$577$	− 21.2132i	− 0.883117i	0.897232	−	0.441559i	$-0.145574\pi$
$578$	15.0000i	0.623918i
$579$	−22.6274	−0.940363
$580$	0	0
$581$	0	0
$582$	− 14.0000i	− 0.580319i
$583$	− 4.00000i	− 0.165663i
$584$	1.41421	0.0585206
$585$	0	0
$586$	−19.7990	−0.817889
$587$	− 29.6985i	− 1.22579i	−0.790165	−	0.612894i	$-0.790005\pi$
$587$	− 29.6985i	− 1.22579i	0.790165	−	0.612894i	$-0.209995\pi$
$588$	0	0
$589$	60.0000	2.47226
$590$	0	0
$591$	−2.82843	−0.116346
$592$	10.0000i	0.410997i
$593$	7.07107i	0.290374i	0.989404	+	0.145187i	$0.0463784\pi$
$593$	7.07107i	0.290374i	−0.989404	+	0.145187i	$0.953622\pi$
$594$	11.3137	0.464207
$595$	0	0
$596$	10.0000	0.409616
$597$	− 12.0000i	− 0.491127i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	29.6985	1.21143	0.605713	−	0.795683i	$-0.292888\pi$
$601$	29.6985	1.21143	0.605713	+	0.795683i	$0.292888\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	16.0000	0.651031
$605$	0	0
$606$	−12.0000	−0.487467
$607$	− 16.9706i	− 0.688814i	−0.938820	−	0.344407i	$-0.888080\pi$
$607$	− 16.9706i	− 0.688814i	0.938820	−	0.344407i	$-0.111920\pi$
$608$	7.07107i	0.286770i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.41421i	0.0571662i
$613$	30.0000i	1.21169i	0.795583	+	0.605844i	$0.207165\pi$
$613$	30.0000i	1.21169i	−0.795583	+	0.605844i	$0.792835\pi$
$614$	9.89949	0.399511
$615$	0	0
$616$	0	0
$617$	− 26.0000i	− 1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$	− 26.0000i	− 1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$	4.00000i	0.160904i
$619$	−18.3848	−0.738947	−0.369473	−	0.929241i	$-0.620462\pi$
$619$	−18.3848	−0.738947	−0.369473	+	0.929241i	$0.620462\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	− 11.3137i	− 0.453638i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	12.7279	0.508710
$627$	− 20.0000i	− 0.798723i
$628$	11.3137i	0.451466i
$629$	14.1421	0.563884
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	− 4.00000i	− 0.159111i
$633$	− 16.9706i	− 0.674519i
$634$	−10.0000	−0.397151
$635$	0	0
$636$	2.82843	0.112154
$637$	0	0
$638$	4.00000i	0.158362i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	5.65685i	0.223258i
$643$	− 9.89949i	− 0.390398i	−0.980764	−	0.195199i	$-0.937465\pi$
$643$	− 9.89949i	− 0.390398i	0.980764	−	0.195199i	$-0.0625353\pi$
$644$	0	0
$645$	0	0
$646$	10.0000	0.393445
$647$	8.48528i	0.333591i	0.985992	+	0.166795i	$0.0533419\pi$
$647$	8.48528i	0.333591i	−0.985992	+	0.166795i	$0.946658\pi$
$648$	5.00000i	0.196419i
$649$	−2.82843	−0.111025
$650$	0	0
$651$	0	0
$652$	10.0000i	0.391630i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	−2.82843	−0.110600
$655$	0	0
$656$	−9.89949	−0.386510
$657$	1.41421i	0.0551737i
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	8.48528	0.330039	0.165020	−	0.986290i	$-0.447231\pi$
$661$	8.48528	0.330039	0.165020	+	0.986290i	$0.447231\pi$
$662$	10.0000i	0.388661i
$663$	0	0
$664$	−9.89949	−0.384175
$665$	0	0
$666$	−10.0000	−0.387492
$667$	− 8.00000i	− 0.309761i
$668$	19.7990i	0.766046i
$669$	0	0
$670$	0	0
$671$	−5.65685	−0.218380
$672$	0	0
$673$	12.0000i	0.462566i	0.972887	+	0.231283i	$0.0742923\pi$
$673$	12.0000i	0.462566i	−0.972887	+	0.231283i	$0.925708\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−13.0000	−0.500000
$677$	− 16.9706i	− 0.652232i	−0.945330	−	0.326116i	$-0.894260\pi$
$677$	− 16.9706i	− 0.652232i	0.945330	−	0.326116i	$-0.105740\pi$
$678$	− 16.9706i	− 0.651751i
$679$	0	0
$680$	0	0
$681$	30.0000	1.14960
$682$	− 16.9706i	− 0.649836i
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	−7.07107	−0.270369
$685$	0	0
$686$	0	0
$687$	24.0000i	0.915657i
$688$	− 2.00000i	− 0.0762493i
$689$	0	0
$690$	0	0
$691$	12.7279	0.484193	0.242096	−	0.970252i	$-0.422165\pi$
$691$	12.7279	0.484193	0.242096	+	0.970252i	$0.422165\pi$
$692$	− 16.9706i	− 0.645124i
$693$	0	0
$694$	30.0000	1.13878
$695$	0	0
$696$	−2.82843	−0.107211
$697$	14.0000i	0.530288i
$698$	0	0
$699$	33.9411	1.28377
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	70.7107i	2.66690i
$704$	2.00000	0.0753778
$705$	0	0
$706$	−1.41421	−0.0532246
$707$	0	0
$708$	− 2.00000i	− 0.0751646i
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	− 7.07107i	− 0.264999i
$713$	33.9411i	1.27111i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	16.9706i	0.633777i
$718$	32.0000i	1.19423i
$719$	−2.82843	−0.105483	−0.0527413	−	0.998608i	$-0.516796\pi$
$719$	−2.82843	−0.105483	−0.0527413	+	0.998608i	$0.516796\pi$
$720$	0	0
$721$	0	0
$722$	31.0000i	1.15370i
$723$	− 30.0000i	− 1.11571i
$724$	0	0
$725$	0	0
$726$	9.89949	0.367405
$727$	− 19.7990i	− 0.734304i	−0.930161	−	0.367152i	$-0.880333\pi$
$727$	− 19.7990i	− 0.734304i	0.930161	−	0.367152i	$-0.119667\pi$
$728$	0	0
$729$	−29.0000	−1.07407
$730$	0	0
$731$	−2.82843	−0.104613
$732$	− 4.00000i	− 0.147844i
$733$	− 42.4264i	− 1.56706i	−0.621357	−	0.783528i	$-0.713418\pi$
$733$	− 42.4264i	− 1.56706i	0.621357	−	0.783528i	$-0.286582\pi$
$734$	−28.2843	−1.04399
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 24.0000i	− 0.884051i
$738$	− 9.89949i	− 0.364405i
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	10.0000	0.366126
$747$	− 9.89949i	− 0.362204i
$748$	− 2.82843i	− 0.103418i
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	2.82843i	0.103142i
$753$	14.0000i	0.510188i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	26.0000i	0.944363i
$759$	11.3137	0.410662
$760$	0	0
$761$	−7.07107	−0.256326	−0.128163	−	0.991753i	$-0.540908\pi$
$761$	−7.07107	−0.256326	−0.128163	+	0.991753i	$0.540908\pi$
$762$	− 22.6274i	− 0.819705i
$763$	0	0
$764$	4.00000	0.144715
$765$	0	0
$766$	−36.7696	−1.32854
$767$	0	0
$768$	1.41421i	0.0510310i
$769$	−29.6985	−1.07095	−0.535477	−	0.844550i	$-0.679868\pi$
$769$	−29.6985	−1.07095	−0.535477	+	0.844550i	$0.679868\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	− 16.0000i	− 0.575853i
$773$	− 48.0833i	− 1.72943i	−0.502259	−	0.864717i	$-0.667498\pi$
$773$	− 48.0833i	− 1.72943i	0.502259	−	0.864717i	$-0.332502\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	9.89949	0.355371
$777$	0	0
$778$	− 26.0000i	− 0.932145i
$779$	−70.0000	−2.50801
$780$	0	0
$781$	24.0000	0.858788
$782$	5.65685i	0.202289i
$783$	− 11.3137i	− 0.404319i
$784$	0	0
$785$	0	0
$786$	−18.0000	−0.642039
$787$	− 1.41421i	− 0.0504113i	−0.999682	−	0.0252056i	$-0.991976\pi$
$787$	− 1.41421i	− 0.0504113i	0.999682	−	0.0252056i	$-0.00802405\pi$
$788$	− 2.00000i	− 0.0712470i
$789$	16.9706	0.604168
$790$	0	0
$791$	0	0
$792$	2.00000i	0.0710669i
$793$	0	0
$794$	−22.6274	−0.803017
$795$	0	0
$796$	8.48528	0.300753
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	7.07107	0.249844
$802$	− 18.0000i	− 0.635602i
$803$	− 2.82843i	− 0.0998130i
$804$	16.9706	0.598506
$805$	0	0
$806$	0	0
$807$	16.0000i	0.563227i
$808$	− 8.48528i	− 0.298511i
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	−29.6985	−1.04285	−0.521427	−	0.853296i	$-0.674600\pi$
$811$	−29.6985	−1.04285	−0.521427	+	0.853296i	$0.674600\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	20.0000	0.701000
$815$	0	0
$816$	2.00000	0.0700140
$817$	− 14.1421i	− 0.494771i
$818$	− 38.1838i	− 1.33506i
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	− 16.9706i	− 0.591916i
$823$	− 40.0000i	− 1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$	− 40.0000i	− 1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	−2.82843	−0.0985329
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	− 4.00000i	− 0.139010i
$829$	31.1127	1.08059	0.540294	−	0.841476i	$-0.318313\pi$
$829$	31.1127	1.08059	0.540294	+	0.841476i	$0.318313\pi$
$830$	0	0
$831$	2.82843	0.0981170
$832$	0	0
$833$	0	0
$834$	14.0000	0.484780
$835$	0	0
$836$	14.1421	0.489116
$837$	48.0000i	1.65912i
$838$	9.89949i	0.341972i
$839$	−19.7990	−0.683537	−0.341769	−	0.939784i	$-0.611026\pi$
$839$	−19.7990	−0.683537	−0.341769	+	0.939784i	$0.611026\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	30.0000i	1.03387i
$843$	22.6274i	0.779330i
$844$	12.0000	0.413057
$845$	0	0
$846$	−2.82843	−0.0972433
$847$	0	0
$848$	2.00000i	0.0686803i
$849$	−2.00000	−0.0686398
$850$	0	0
$851$	−40.0000	−1.37118
$852$	16.9706i	0.581402i
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	18.3848i	0.628012i	0.949421	+	0.314006i	$0.101671\pi$
$857$	18.3848i	0.628012i	−0.949421	+	0.314006i	$0.898329\pi$
$858$	0	0
$859$	26.8701	0.916795	0.458397	−	0.888747i	$-0.348424\pi$
$859$	26.8701	0.916795	0.458397	+	0.888747i	$0.348424\pi$
$860$	0	0
$861$	0	0
$862$	12.0000i	0.408722i
$863$	4.00000i	0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$	4.00000i	0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$	−5.65685	−0.192450
$865$	0	0
$866$	−29.6985	−1.00920
$867$	21.2132i	0.720438i
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	0	0
$872$	− 2.00000i	− 0.0677285i
$873$	9.89949i	0.335047i
$874$	−28.2843	−0.956730
$875$	0	0
$876$	2.00000	0.0675737
$877$	− 46.0000i	− 1.55331i	−0.629926	−	0.776655i	$-0.716915\pi$
$877$	− 46.0000i	− 1.55331i	0.629926	−	0.776655i	$-0.283085\pi$
$878$	16.9706i	0.572729i
$879$	−28.0000	−0.944417
$880$	0	0
$881$	29.6985	1.00057	0.500284	−	0.865862i	$-0.333229\pi$
$881$	29.6985	1.00057	0.500284	+	0.865862i	$0.333229\pi$
$882$	0	0
$883$	− 44.0000i	− 1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$	− 44.0000i	− 1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$	0	0
$885$	0	0
$886$	−4.00000	−0.134383
$887$	− 36.7696i	− 1.23460i	−0.786728	−	0.617300i	$-0.788226\pi$
$887$	− 36.7696i	− 1.23460i	0.786728	−	0.617300i	$-0.211774\pi$
$888$	14.1421i	0.474579i
$889$	0	0
$890$	0	0
$891$	10.0000	0.335013
$892$	0	0
$893$	20.0000i	0.669274i
$894$	14.1421	0.472984
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 30.0000i	− 1.00111i
$899$	−16.9706	−0.566000
$900$	0	0
$901$	2.82843	0.0942286
$902$	19.7990i	0.659234i
$903$	0	0
$904$	12.0000	0.399114
$905$	0	0
$906$	22.6274	0.751746
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	21.2132i	0.703985i
$909$	8.48528	0.281439
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	10.0000i	0.331133i
$913$	19.7990i	0.655251i
$914$	−24.0000	−0.793849
$915$	0	0
$916$	−16.9706	−0.560723
$917$	0	0
$918$	8.00000i	0.264039i
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	39.5980i	1.30409i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	− 2.82843i	− 0.0928977i
$928$	− 2.00000i	− 0.0656532i
$929$	−32.5269	−1.06717	−0.533587	−	0.845745i	$-0.679156\pi$
$929$	−32.5269	−1.06717	−0.533587	+	0.845745i	$0.679156\pi$
$930$	0	0
$931$	0	0
$932$	24.0000i	0.786146i
$933$	− 16.0000i	− 0.523816i
$934$	−32.5269	−1.06431
$935$	0	0
$936$	0	0
$937$	9.89949i	0.323402i	0.986840	+	0.161701i	$0.0516981\pi$
$937$	9.89949i	0.323402i	−0.986840	+	0.161701i	$0.948302\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	−31.1127	−1.01424	−0.507122	−	0.861874i	$-0.669291\pi$
$941$	−31.1127	−1.01424	−0.507122	+	0.861874i	$0.669291\pi$
$942$	16.0000i	0.521308i
$943$	− 39.5980i	− 1.28949i
$944$	1.41421	0.0460287
$945$	0	0
$946$	−4.00000	−0.130051
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	− 5.65685i	− 0.183726i
$949$	0	0
$950$	0	0
$951$	−14.1421	−0.458590
$952$	0	0
$953$	26.0000i	0.842223i	0.907009	+	0.421111i	$0.138360\pi$
$953$	26.0000i	0.842223i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−12.0000	−0.388108
$957$	5.65685i	0.182860i
$958$	31.1127i	1.00521i
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	21.2132	0.683231
$965$	0	0
$966$	0	0
$967$	− 12.0000i	− 0.385894i	−0.981209	−	0.192947i	$-0.938195\pi$
$967$	− 12.0000i	− 0.385894i	0.981209	−	0.192947i	$-0.0618045\pi$
$968$	7.00000i	0.224989i
$969$	14.1421	0.454311
$970$	0	0
$971$	32.5269	1.04384	0.521919	−	0.852995i	$-0.325216\pi$
$971$	32.5269	1.04384	0.521919	+	0.852995i	$0.325216\pi$
$972$	− 9.89949i	− 0.317526i
$973$	0	0
$974$	−12.0000	−0.384505
$975$	0	0
$976$	2.82843	0.0905357
$977$	12.0000i	0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$	12.0000i	0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$	14.1421i	0.452216i
$979$	−14.1421	−0.451985
$980$	0	0
$981$	2.00000	0.0638551
$982$	− 12.0000i	− 0.382935i
$983$	− 48.0833i	− 1.53362i	−0.641875	−	0.766809i	$-0.721843\pi$
$983$	− 48.0833i	− 1.53362i	0.641875	−	0.766809i	$-0.278157\pi$
$984$	−14.0000	−0.446304
$985$	0	0
$986$	−2.82843	−0.0900755
$987$	0	0
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.48528i	0.269408i
$993$	14.1421i	0.448787i
$994$	0	0
$995$	0	0
$996$	−14.0000	−0.443607
$997$	− 31.1127i	− 0.985349i	−0.870214	−	0.492675i	$-0.836019\pi$
$997$	− 31.1127i	− 0.985349i	0.870214	−	0.492675i	$-0.163981\pi$
$998$	4.00000i	0.126618i
$999$	−56.5685	−1.78975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.c.v.99.4		4
5.2	odd	4		2450.2.a.bj.1.2		2
5.3	odd	4		98.2.a.b.1.1	✓	2
5.4	even	2	inner	2450.2.c.v.99.1		4
7.6	odd	2	inner	2450.2.c.v.99.3		4
15.8	even	4		882.2.a.n.1.1		2
20.3	even	4		784.2.a.l.1.2		2
35.3	even	12		98.2.c.c.79.1		4
35.13	even	4		98.2.a.b.1.2	yes	2
35.18	odd	12		98.2.c.c.79.2		4
35.23	odd	12		98.2.c.c.67.2		4
35.27	even	4		2450.2.a.bj.1.1		2
35.33	even	12		98.2.c.c.67.1		4
35.34	odd	2	inner	2450.2.c.v.99.2		4
40.3	even	4		3136.2.a.bm.1.1		2
40.13	odd	4		3136.2.a.bn.1.2		2
60.23	odd	4		7056.2.a.cl.1.1		2
105.23	even	12		882.2.g.l.361.2		4
105.38	odd	12		882.2.g.l.667.1		4
105.53	even	12		882.2.g.l.667.2		4
105.68	odd	12		882.2.g.l.361.1		4
105.83	odd	4		882.2.a.n.1.2		2
140.3	odd	12		784.2.i.m.177.2		4
140.23	even	12		784.2.i.m.753.1		4
140.83	odd	4		784.2.a.l.1.1		2
140.103	odd	12		784.2.i.m.753.2		4
140.123	even	12		784.2.i.m.177.1		4
280.13	even	4		3136.2.a.bn.1.1		2
280.83	odd	4		3136.2.a.bm.1.2		2
420.83	even	4		7056.2.a.cl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.2.a.b.1.1	✓	2	5.3	odd	4
98.2.a.b.1.2	yes	2	35.13	even	4
98.2.c.c.67.1		4	35.33	even	12
98.2.c.c.67.2		4	35.23	odd	12
98.2.c.c.79.1		4	35.3	even	12
98.2.c.c.79.2		4	35.18	odd	12
784.2.a.l.1.1		2	140.83	odd	4
784.2.a.l.1.2		2	20.3	even	4
784.2.i.m.177.1		4	140.123	even	12
784.2.i.m.177.2		4	140.3	odd	12
784.2.i.m.753.1		4	140.23	even	12
784.2.i.m.753.2		4	140.103	odd	12
882.2.a.n.1.1		2	15.8	even	4
882.2.a.n.1.2		2	105.83	odd	4
882.2.g.l.361.1		4	105.68	odd	12
882.2.g.l.361.2		4	105.23	even	12
882.2.g.l.667.1		4	105.38	odd	12
882.2.g.l.667.2		4	105.53	even	12
2450.2.a.bj.1.1		2	35.27	even	4
2450.2.a.bj.1.2		2	5.2	odd	4
2450.2.c.v.99.1		4	5.4	even	2	inner
2450.2.c.v.99.2		4	35.34	odd	2	inner
2450.2.c.v.99.3		4	7.6	odd	2	inner
2450.2.c.v.99.4		4	1.1	even	1	trivial
3136.2.a.bm.1.1		2	40.3	even	4
3136.2.a.bm.1.2		2	280.83	odd	4
3136.2.a.bn.1.1		2	280.13	even	4
3136.2.a.bn.1.2		2	40.13	odd	4
7056.2.a.cl.1.1		2	60.23	odd	4
7056.2.a.cl.1.2		2	420.83	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.41421i q^{3} -1.00000 q^{4} -1.41421 q^{6} -1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.41421i q^{12} +1.00000 q^{16} -1.41421i q^{17} +1.00000i q^{18} +7.07107 q^{19} -2.00000i q^{22} +4.00000i q^{23} +1.41421 q^{24} +5.65685i q^{27} -2.00000 q^{29} +8.48528 q^{31} +1.00000i q^{32} -2.82843i q^{33} +1.41421 q^{34} -1.00000 q^{36} +10.0000i q^{37} +7.07107i q^{38} -9.89949 q^{41} -2.00000i q^{43} +2.00000 q^{44} -4.00000 q^{46} +2.82843i q^{47} +1.41421i q^{48} +2.00000 q^{51} +2.00000i q^{53} -5.65685 q^{54} +10.0000i q^{57} -2.00000i q^{58} +1.41421 q^{59} +2.82843 q^{61} +8.48528i q^{62} -1.00000 q^{64} +2.82843 q^{66} +12.0000i q^{67} +1.41421i q^{68} -5.65685 q^{69} -12.0000 q^{71} -1.00000i q^{72} +1.41421i q^{73} -10.0000 q^{74} -7.07107 q^{76} +4.00000 q^{79} -5.00000 q^{81} -9.89949i q^{82} -9.89949i q^{83} +2.00000 q^{86} -2.82843i q^{87} +2.00000i q^{88} +7.07107 q^{89} -4.00000i q^{92} +12.0000i q^{93} -2.82843 q^{94} -1.41421 q^{96} +9.89949i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{9} - 8 q^{11} + 4 q^{16} - 8 q^{29} - 4 q^{36} + 8 q^{44} - 16 q^{46} + 8 q^{51} - 4 q^{64} - 48 q^{71} - 40 q^{74} + 16 q^{79} - 20 q^{81} + 8 q^{86} - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^9 - 8 * q^11 + 4 * q^16 - 8 * q^29 - 4 * q^36 + 8 * q^44 - 16 * q^46 + 8 * q^51 - 4 * q^64 - 48 * q^71 - 40 * q^74 + 16 * q^79 - 20 * q^81 + 8 * q^86 - 8 * q^99

Introduction

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