LMFDB - Embedded newform 2450.2.c.f.99.1

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2450.99
Dual form			2450.2.c.f.99.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} +5.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -3.00000i q^{22} +3.00000i q^{23} +2.00000 q^{24} +5.00000 q^{26} -4.00000i q^{27} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -11.0000i q^{37} -1.00000i q^{38} +10.0000 q^{39} +3.00000 q^{41} -10.0000i q^{43} -3.00000 q^{44} +3.00000 q^{46} -3.00000i q^{47} -2.00000i q^{48} -12.0000 q^{51} -5.00000i q^{52} +3.00000i q^{53} -4.00000 q^{54} -2.00000i q^{57} -6.00000i q^{58} -4.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -6.00000 q^{66} +4.00000i q^{67} +6.00000i q^{68} +6.00000 q^{69} +12.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -11.0000 q^{74} -1.00000 q^{76} -10.0000i q^{78} +10.0000 q^{79} -11.0000 q^{81} -3.00000i q^{82} -12.0000i q^{83} -10.0000 q^{86} -12.0000i q^{87} +3.00000i q^{88} -6.00000 q^{89} -3.00000i q^{92} +8.00000i q^{93} -3.00000 q^{94} -2.00000 q^{96} -14.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 6 q^{11} + 2 q^{16} + 2 q^{19} + 4 q^{24} + 10 q^{26} + 12 q^{29} - 8 q^{31} - 12 q^{34} + 2 q^{36} + 20 q^{39} + 6 q^{41} - 6 q^{44} + 6 q^{46} - 24 q^{51} - 8 q^{54} - 8 q^{61} - 2 q^{64} - 12 q^{66} + 12 q^{69} + 24 q^{71} - 22 q^{74} - 2 q^{76} + 20 q^{79} - 22 q^{81} - 20 q^{86} - 12 q^{89} - 6 q^{94} - 4 q^{96} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 6 * q^11 + 2 * q^16 + 2 * q^19 + 4 * q^24 + 10 * q^26 + 12 * q^29 - 8 * q^31 - 12 * q^34 + 2 * q^36 + 20 * q^39 + 6 * q^41 - 6 * q^44 + 6 * q^46 - 24 * q^51 - 8 * q^54 - 8 * q^61 - 2 * q^64 - 12 * q^66 + 12 * q^69 + 24 * q^71 - 22 * q^74 - 2 * q^76 + 20 * q^79 - 22 * q^81 - 20 * q^86 - 12 * q^89 - 6 * q^94 - 4 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

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
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	2.00000i	0.577350i
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	1.00000i	0.235702i
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	− 3.00000i	− 0.639602i
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	5.00000	0.980581
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 6.00000i	− 1.04447i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	− 11.0000i	− 1.80839i	−0.427121	−	0.904194i	$-0.640472\pi$
$37$	− 11.0000i	− 1.80839i	0.427121	−	0.904194i	$-0.359528\pi$
$38$	− 1.00000i	− 0.162221i
$39$	10.0000	1.60128
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	3.00000	0.442326
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	− 2.00000i	− 0.288675i
$49$	0	0
$50$	0	0
$51$	−12.0000	−1.68034
$52$	− 5.00000i	− 0.693375i
$53$	3.00000i	0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$	3.00000i	0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	0	0
$57$	− 2.00000i	− 0.264906i
$58$	− 6.00000i	− 0.787839i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	6.00000i	0.727607i
$69$	6.00000	0.722315
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−11.0000	−1.27872
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	− 10.0000i	− 1.13228i
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 3.00000i	− 0.331295i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	−10.0000	−1.07833
$87$	− 12.0000i	− 1.28654i
$88$	3.00000i	0.319801i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	− 3.00000i	− 0.312772i
$93$	8.00000i	0.829561i
$94$	−3.00000	−0.309426
$95$	0	0
$96$	−2.00000	−0.204124
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	12.0000i	1.18818i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−5.00000	−0.490290
$105$	0	0
$106$	3.00000	0.291386
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	4.00000i	0.384900i
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	−22.0000	−2.08815
$112$	0	0
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−6.00000	−0.557086
$117$	− 5.00000i	− 0.462250i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	4.00000i	0.362143i
$123$	− 6.00000i	− 0.541002i
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	19.0000i	1.68598i	0.537931	+	0.842989i	$0.319206\pi$
$127$	19.0000i	1.68598i	−0.537931	+	0.842989i	$0.680794\pi$
$128$	1.00000i	0.0883883i
$129$	−20.0000	−1.76090
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	6.00000i	0.522233i
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	− 6.00000i	− 0.510754i
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	− 12.0000i	− 1.00702i
$143$	15.0000i	1.25436i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	11.0000i	0.904194i
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	1.00000i	0.0811107i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	−10.0000	−0.800641
$157$	− 5.00000i	− 0.399043i	−0.979893	−	0.199522i	$-0.936061\pi$
$157$	− 5.00000i	− 0.399043i	0.979893	−	0.199522i	$-0.0639388\pi$
$158$	− 10.0000i	− 0.795557i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	11.0000i	0.864242i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 9.00000i	− 0.696441i	−0.937413	−	0.348220i	$-0.886786\pi$
$167$	− 9.00000i	− 0.696441i	0.937413	−	0.348220i	$-0.113214\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	10.0000i	0.762493i
$173$	3.00000i	0.228086i	0.993476	+	0.114043i	$0.0363801\pi$
$173$	3.00000i	0.228086i	−0.993476	+	0.114043i	$0.963620\pi$
$174$	−12.0000	−0.909718
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	6.00000i	0.449719i
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	8.00000i	0.591377i
$184$	−3.00000	−0.221163
$185$	0	0
$186$	8.00000	0.586588
$187$	− 18.0000i	− 1.31629i
$188$	3.00000i	0.218797i
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	2.00000i	0.144338i
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	0	0
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	3.00000i	0.213201i
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	12.0000i	0.844317i
$203$	0	0
$204$	12.0000	0.840168
$205$	0	0
$206$	−4.00000	−0.278693
$207$	− 3.00000i	− 0.208514i
$208$	5.00000i	0.346688i
$209$	3.00000	0.207514
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	− 3.00000i	− 0.206041i
$213$	− 24.0000i	− 1.64445i
$214$	12.0000	0.820303
$215$	0	0
$216$	4.00000	0.272166
$217$	0	0
$218$	− 4.00000i	− 0.270914i
$219$	−8.00000	−0.540590
$220$	0	0
$221$	30.0000	2.01802
$222$	22.0000i	1.47654i
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	0	0
$226$	12.0000	0.798228
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	2.00000i	0.132453i
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−5.00000	−0.326860
$235$	0	0
$236$	0	0
$237$	− 20.0000i	− 1.29914i
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	2.00000i	0.128565i
$243$	10.0000i	0.641500i
$244$	4.00000	0.256074
$245$	0	0
$246$	−6.00000	−0.382546
$247$	5.00000i	0.318142i
$248$	− 4.00000i	− 0.254000i
$249$	−24.0000	−1.52094
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	9.00000i	0.565825i
$254$	19.0000	1.19217
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	20.0000i	1.24515i
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	− 3.00000i	− 0.185341i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	0	0
$267$	12.0000i	0.734388i
$268$	− 4.00000i	− 0.244339i
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 6.00000i	− 0.363803i
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	−6.00000	−0.361158
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 4.00000i	− 0.239904i
$279$	4.00000	0.239474
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	6.00000i	0.357295i
$283$	26.0000i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$	26.0000i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	15.0000	0.886969
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−28.0000	−1.64139
$292$	4.00000i	0.234082i
$293$	− 27.0000i	− 1.57736i	−0.614806	−	0.788678i	$-0.710766\pi$
$293$	− 27.0000i	− 1.57736i	0.614806	−	0.788678i	$-0.289234\pi$
$294$	0	0
$295$	0	0
$296$	11.0000	0.639362
$297$	− 12.0000i	− 0.696311i
$298$	18.0000i	1.04271i
$299$	−15.0000	−0.867472
$300$	0	0
$301$	0	0
$302$	− 14.0000i	− 0.805609i
$303$	24.0000i	1.37876i
$304$	1.00000	0.0573539
$305$	0	0
$306$	6.00000	0.342997
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	10.0000i	0.566139i
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	−5.00000	−0.282166
$315$	0	0
$316$	−10.0000	−0.562544
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	− 6.00000i	− 0.336463i
$319$	18.0000	1.00781
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	11.0000	0.611111
$325$	0	0
$326$	−4.00000	−0.221540
$327$	− 8.00000i	− 0.442401i
$328$	3.00000i	0.165647i
$329$	0	0
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	12.0000i	0.658586i
$333$	11.0000i	0.602796i
$334$	−9.00000	−0.492458
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	12.0000i	0.652714i
$339$	24.0000	1.30350
$340$	0	0
$341$	−12.0000	−0.649836
$342$	1.00000i	0.0540738i
$343$	0	0
$344$	10.0000	0.539164
$345$	0	0
$346$	3.00000	0.161281
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	12.0000i	0.643268i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	− 3.00000i	− 0.159901i
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	− 3.00000i	− 0.158555i
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	− 2.00000i	− 0.105118i
$363$	4.00000i	0.209946i
$364$	0	0
$365$	0	0
$366$	8.00000	0.418167
$367$	1.00000i	0.0521996i	0.999659	+	0.0260998i	$0.00830876\pi$
$367$	1.00000i	0.0521996i	−0.999659	+	0.0260998i	$0.991691\pi$
$368$	3.00000i	0.156386i
$369$	−3.00000	−0.156174
$370$	0	0
$371$	0	0
$372$	− 8.00000i	− 0.414781i
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	3.00000	0.154713
$377$	30.0000i	1.54508i
$378$	0	0
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	0	0
$381$	38.0000	1.94680
$382$	− 12.0000i	− 0.613973i
$383$	15.0000i	0.766464i	0.923652	+	0.383232i	$0.125189\pi$
$383$	15.0000i	0.766464i	−0.923652	+	0.383232i	$0.874811\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	−4.00000	−0.203595
$387$	10.0000i	0.508329i
$388$	14.0000i	0.710742i
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	− 6.00000i	− 0.302660i
$394$	3.00000	0.151138
$395$	0	0
$396$	3.00000	0.150756
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	− 4.00000i	− 0.200502i
$399$	0	0
$400$	0	0
$401$	21.0000	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$402$	− 8.00000i	− 0.399004i
$403$	− 20.0000i	− 0.996271i
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	− 33.0000i	− 1.63575i
$408$	− 12.0000i	− 0.594089i
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−24.0000	−1.18383
$412$	4.00000i	0.197066i
$413$	0	0
$414$	−3.00000	−0.147442
$415$	0	0
$416$	5.00000	0.245145
$417$	− 8.00000i	− 0.391762i
$418$	− 3.00000i	− 0.146735i
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	1.00000i	0.0486792i
$423$	3.00000i	0.145865i
$424$	−3.00000	−0.145693
$425$	0	0
$426$	−24.0000	−1.16280
$427$	0	0
$428$	− 12.0000i	− 0.580042i
$429$	30.0000	1.44841
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 4.00000i	− 0.192450i
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	3.00000i	0.143509i
$438$	8.00000i	0.382255i
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	− 30.0000i	− 1.42695i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	22.0000	1.04407
$445$	0	0
$446$	8.00000	0.378811
$447$	36.0000i	1.70274i
$448$	0	0
$449$	3.00000	0.141579	0.0707894	−	0.997491i	$-0.477448\pi$
$449$	3.00000	0.141579	0.0707894	+	0.997491i	$0.477448\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	− 12.0000i	− 0.564433i
$453$	− 28.0000i	− 1.31555i
$454$	24.0000	1.12638
$455$	0	0
$456$	2.00000	0.0936586
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	− 28.0000i	− 1.30835i
$459$	−24.0000	−1.12022
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	− 19.0000i	− 0.883005i	−0.897260	−	0.441502i	$-0.854446\pi$
$463$	− 19.0000i	− 0.883005i	0.897260	−	0.441502i	$-0.145554\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	18.0000i	0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$	18.0000i	0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$	5.00000i	0.231125i
$469$	0	0
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	− 30.0000i	− 1.37940i
$474$	−20.0000	−0.918630
$475$	0	0
$476$	0	0
$477$	− 3.00000i	− 0.137361i
$478$	6.00000i	0.274434i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	55.0000	2.50778
$482$	25.0000i	1.13872i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	10.0000	0.453609
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	− 4.00000i	− 0.181071i
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.00000i	0.270501i
$493$	− 36.0000i	− 1.62136i
$494$	5.00000	0.224961
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	24.0000i	1.07547i
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	15.0000i	0.669483i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	9.00000	0.400099
$507$	24.0000i	1.06588i
$508$	− 19.0000i	− 0.842989i
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	12.0000	0.529297
$515$	0	0
$516$	20.0000	0.880451
$517$	− 9.00000i	− 0.395820i
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	6.00000i	0.262613i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	0	0
$527$	24.0000i	1.04546i
$528$	− 6.00000i	− 0.261116i
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15.0000i	0.649722i
$534$	12.0000	0.519291
$535$	0	0
$536$	−4.00000	−0.172774
$537$	− 6.00000i	− 0.258919i
$538$	− 12.0000i	− 0.517357i
$539$	0	0
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	16.0000i	0.687259i
$543$	− 4.00000i	− 0.171656i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	12.0000i	0.512615i
$549$	4.00000	0.170716
$550$	0	0
$551$	6.00000	0.255609
$552$	6.00000i	0.255377i
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−4.00000	−0.169638
$557$	27.0000i	1.14403i	0.820244	+	0.572013i	$0.193837\pi$
$557$	27.0000i	1.14403i	−0.820244	+	0.572013i	$0.806163\pi$
$558$	− 4.00000i	− 0.169334i
$559$	50.0000	2.11477
$560$	0	0
$561$	−36.0000	−1.51992
$562$	3.00000i	0.126547i
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	26.0000	1.09286
$567$	0	0
$568$	12.0000i	0.503509i
$569$	3.00000	0.125767	0.0628833	−	0.998021i	$-0.479970\pi$
$569$	3.00000	0.125767	0.0628833	+	0.998021i	$0.479970\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	− 15.0000i	− 0.627182i
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 20.0000i	− 0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$	− 20.0000i	− 0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$	19.0000i	0.790296i
$579$	−8.00000	−0.332469
$580$	0	0
$581$	0	0
$582$	28.0000i	1.16064i
$583$	9.00000i	0.372742i
$584$	4.00000	0.165521
$585$	0	0
$586$	−27.0000	−1.11536
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	6.00000	0.246807
$592$	− 11.0000i	− 0.452097i
$593$	− 36.0000i	− 1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$	− 36.0000i	− 1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$	−12.0000	−0.492366
$595$	0	0
$596$	18.0000	0.737309
$597$	− 8.00000i	− 0.327418i
$598$	15.0000i	0.613396i
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	−14.0000	−0.569652
$605$	0	0
$606$	24.0000	0.974933
$607$	19.0000i	0.771186i	0.922669	+	0.385593i	$0.126003\pi$
$607$	19.0000i	0.771186i	−0.922669	+	0.385593i	$0.873997\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	0	0
$610$	0	0
$611$	15.0000	0.606835
$612$	− 6.00000i	− 0.242536i
$613$	47.0000i	1.89831i	0.314806	+	0.949156i	$0.398061\pi$
$613$	47.0000i	1.89831i	−0.314806	+	0.949156i	$0.601939\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	8.00000i	0.321807i
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	0	0
$621$	12.0000	0.481543
$622$	− 12.0000i	− 0.481156i
$623$	0	0
$624$	10.0000	0.400320
$625$	0	0
$626$	8.00000	0.319744
$627$	− 6.00000i	− 0.239617i
$628$	5.00000i	0.199522i
$629$	−66.0000	−2.63159
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	10.0000i	0.397779i
$633$	2.00000i	0.0794929i
$634$	−18.0000	−0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	0	0
$638$	− 18.0000i	− 0.712627i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−45.0000	−1.77739	−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000	−1.77739	−0.888697	+	0.458496i	$0.848388\pi$
$642$	− 24.0000i	− 0.947204i
$643$	38.0000i	1.49857i	0.662246	+	0.749287i	$0.269604\pi$
$643$	38.0000i	1.49857i	−0.662246	+	0.749287i	$0.730396\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	21.0000i	0.825595i	0.910823	+	0.412798i	$0.135448\pi$
$647$	21.0000i	0.825595i	−0.910823	+	0.412798i	$0.864552\pi$
$648$	− 11.0000i	− 0.432121i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	21.0000i	0.821794i	0.911682	+	0.410897i	$0.134784\pi$
$653$	21.0000i	0.821794i	−0.911682	+	0.410897i	$0.865216\pi$
$654$	−8.00000	−0.312825
$655$	0	0
$656$	3.00000	0.117130
$657$	4.00000i	0.156055i
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	44.0000	1.71140	0.855701	−	0.517471i	$-0.173126\pi$
$661$	44.0000	1.71140	0.855701	+	0.517471i	$0.173126\pi$
$662$	7.00000i	0.272063i
$663$	− 60.0000i	− 2.33021i
$664$	12.0000	0.465690
$665$	0	0
$666$	11.0000	0.426241
$667$	18.0000i	0.696963i
$668$	9.00000i	0.348220i
$669$	16.0000	0.618596
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	12.0000	0.461538
$677$	3.00000i	0.115299i	0.998337	+	0.0576497i	$0.0183606\pi$
$677$	3.00000i	0.115299i	−0.998337	+	0.0576497i	$0.981639\pi$
$678$	− 24.0000i	− 0.921714i
$679$	0	0
$680$	0	0
$681$	48.0000	1.83936
$682$	12.0000i	0.459504i
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	0	0
$687$	− 56.0000i	− 2.13653i
$688$	− 10.0000i	− 0.381246i
$689$	−15.0000	−0.571454
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	− 3.00000i	− 0.114043i
$693$	0	0
$694$	−24.0000	−0.911028
$695$	0	0
$696$	12.0000	0.454859
$697$	− 18.0000i	− 0.681799i
$698$	− 10.0000i	− 0.378506i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	− 20.0000i	− 0.754851i
$703$	− 11.0000i	− 0.414873i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	12.0000	0.451626
$707$	0	0
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	− 6.00000i	− 0.224860i
$713$	− 12.0000i	− 0.449404i
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	12.0000i	0.448148i
$718$	6.00000i	0.223918i
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	18.0000i	0.669891i
$723$	50.0000i	1.85952i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	4.00000	0.148454
$727$	− 29.0000i	− 1.07555i	−0.843088	−	0.537775i	$-0.819265\pi$
$727$	− 29.0000i	− 1.07555i	0.843088	−	0.537775i	$-0.180735\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−60.0000	−2.21918
$732$	− 8.00000i	− 0.295689i
$733$	47.0000i	1.73598i	0.496578	+	0.867992i	$0.334590\pi$
$733$	47.0000i	1.73598i	−0.496578	+	0.867992i	$0.665410\pi$
$734$	1.00000	0.0369107
$735$	0	0
$736$	3.00000	0.110581
$737$	12.0000i	0.442026i
$738$	3.00000i	0.110432i
$739$	37.0000	1.36107	0.680534	−	0.732717i	$-0.261748\pi$
$739$	37.0000	1.36107	0.680534	+	0.732717i	$0.261748\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	0	0
$743$	9.00000i	0.330178i	0.986279	+	0.165089i	$0.0527911\pi$
$743$	9.00000i	0.330178i	−0.986279	+	0.165089i	$0.947209\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−34.0000	−1.24483
$747$	12.0000i	0.439057i
$748$	18.0000i	0.658145i
$749$	0	0
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	− 3.00000i	− 0.109399i
$753$	30.0000i	1.09326i
$754$	30.0000	1.09254
$755$	0	0
$756$	0	0
$757$	− 26.0000i	− 0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	− 26.0000i	− 0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$	− 25.0000i	− 0.908041i
$759$	18.0000	0.653359
$760$	0	0
$761$	−51.0000	−1.84875	−0.924374	−	0.381487i	$-0.875412\pi$
$761$	−51.0000	−1.84875	−0.924374	+	0.381487i	$0.875412\pi$
$762$	− 38.0000i	− 1.37659i
$763$	0	0
$764$	−12.0000	−0.434145
$765$	0	0
$766$	15.0000	0.541972
$767$	0	0
$768$	− 2.00000i	− 0.0721688i
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	4.00000i	0.143963i
$773$	39.0000i	1.40273i	0.712801	+	0.701366i	$0.247426\pi$
$773$	39.0000i	1.40273i	−0.712801	+	0.701366i	$0.752574\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	− 24.0000i	− 0.860442i
$779$	3.00000	0.107486
$780$	0	0
$781$	36.0000	1.28818
$782$	− 18.0000i	− 0.643679i
$783$	− 24.0000i	− 0.857690i
$784$	0	0
$785$	0	0
$786$	−6.00000	−0.214013
$787$	34.0000i	1.21197i	0.795476	+	0.605985i	$0.207221\pi$
$787$	34.0000i	1.21197i	−0.795476	+	0.605985i	$0.792779\pi$
$788$	− 3.00000i	− 0.106871i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	− 3.00000i	− 0.106600i
$793$	− 20.0000i	− 0.710221i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−4.00000	−0.141776
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	0	0
$801$	6.00000	0.212000
$802$	− 21.0000i	− 0.741536i
$803$	− 12.0000i	− 0.423471i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−20.0000	−0.704470
$807$	− 24.0000i	− 0.844840i
$808$	− 12.0000i	− 0.422159i
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	47.0000	1.65039	0.825197	−	0.564846i	$-0.191064\pi$
$811$	47.0000	1.65039	0.825197	+	0.564846i	$0.191064\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	−33.0000	−1.15665
$815$	0	0
$816$	−12.0000	−0.420084
$817$	− 10.0000i	− 0.349856i
$818$	− 22.0000i	− 0.769212i
$819$	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	$0.601703\pi$
$822$	24.0000i	0.837096i
$823$	44.0000i	1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$	44.0000i	1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	54.0000i	1.87776i	0.344239	+	0.938882i	$0.388137\pi$
$827$	54.0000i	1.87776i	−0.344239	+	0.938882i	$0.611863\pi$
$828$	3.00000i	0.104257i
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	− 5.00000i	− 0.173344i
$833$	0	0
$834$	−8.00000	−0.277017
$835$	0	0
$836$	−3.00000	−0.103757
$837$	16.0000i	0.553041i
$838$	15.0000i	0.518166i
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	34.0000i	1.17172i
$843$	6.00000i	0.206651i
$844$	1.00000	0.0344214
$845$	0	0
$846$	3.00000	0.103142
$847$	0	0
$848$	3.00000i	0.103020i
$849$	52.0000	1.78464
$850$	0	0
$851$	33.0000	1.13123
$852$	24.0000i	0.822226i
$853$	− 1.00000i	− 0.0342393i	−0.999853	−	0.0171197i	$-0.994550\pi$
$853$	− 1.00000i	− 0.0342393i	0.999853	−	0.0171197i	$-0.00544963\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	− 30.0000i	− 1.02418i
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.0000i	1.32758i	0.747921	+	0.663788i	$0.231052\pi$
$863$	39.0000i	1.32758i	−0.747921	+	0.663788i	$0.768948\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	−16.0000	−0.543702
$867$	38.0000i	1.29055i
$868$	0	0
$869$	30.0000	1.01768
$870$	0	0
$871$	−20.0000	−0.677674
$872$	4.00000i	0.135457i
$873$	14.0000i	0.473828i
$874$	3.00000	0.101477
$875$	0	0
$876$	8.00000	0.270295
$877$	7.00000i	0.236373i	0.992991	+	0.118187i	$0.0377081\pi$
$877$	7.00000i	0.236373i	−0.992991	+	0.118187i	$0.962292\pi$
$878$	− 10.0000i	− 0.337484i
$879$	−54.0000	−1.82137
$880$	0	0
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	0	0
$883$	8.00000i	0.269221i	0.990899	+	0.134611i	$0.0429784\pi$
$883$	8.00000i	0.269221i	−0.990899	+	0.134611i	$0.957022\pi$
$884$	−30.0000	−1.00901
$885$	0	0
$886$	−24.0000	−0.806296
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	− 22.0000i	− 0.738272i
$889$	0	0
$890$	0	0
$891$	−33.0000	−1.10554
$892$	− 8.00000i	− 0.267860i
$893$	− 3.00000i	− 0.100391i
$894$	36.0000	1.20402
$895$	0	0
$896$	0	0
$897$	30.0000i	1.00167i
$898$	− 3.00000i	− 0.100111i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	18.0000	0.599667
$902$	− 9.00000i	− 0.299667i
$903$	0	0
$904$	−12.0000	−0.399114
$905$	0	0
$906$	−28.0000	−0.930238
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	− 24.0000i	− 0.796468i
$909$	12.0000	0.398015
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	− 2.00000i	− 0.0662266i
$913$	− 36.0000i	− 1.19143i
$914$	22.0000	0.727695
$915$	0	0
$916$	−28.0000	−0.925146
$917$	0	0
$918$	24.0000i	0.792118i
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	− 6.00000i	− 0.197599i
$923$	60.0000i	1.97492i
$924$	0	0
$925$	0	0
$926$	−19.0000	−0.624379
$927$	4.00000i	0.131377i
$928$	− 6.00000i	− 0.196960i
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	0	0
$932$	6.00000i	0.196537i
$933$	− 24.0000i	− 0.785725i
$934$	18.0000	0.588978
$935$	0	0
$936$	5.00000	0.163430
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	10.0000i	0.325818i
$943$	9.00000i	0.293080i
$944$	0	0
$945$	0	0
$946$	−30.0000	−0.975384
$947$	− 30.0000i	− 0.974869i	−0.873160	−	0.487435i	$-0.837933\pi$
$947$	− 30.0000i	− 0.974869i	0.873160	−	0.487435i	$-0.162067\pi$
$948$	20.0000i	0.649570i
$949$	20.0000	0.649227
$950$	0	0
$951$	−36.0000	−1.16738
$952$	0	0
$953$	− 12.0000i	− 0.388718i	−0.980930	−	0.194359i	$-0.937737\pi$
$953$	− 12.0000i	− 0.388718i	0.980930	−	0.194359i	$-0.0622627\pi$
$954$	−3.00000	−0.0971286
$955$	0	0
$956$	6.00000	0.194054
$957$	− 36.0000i	− 1.16371i
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 55.0000i	− 1.77327i
$963$	− 12.0000i	− 0.386695i
$964$	25.0000	0.805196
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	−12.0000	−0.385496
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	− 10.0000i	− 0.320750i
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	−4.00000	−0.128037
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	8.00000i	0.255812i
$979$	−18.0000	−0.575282
$980$	0	0
$981$	−4.00000	−0.127710
$982$	12.0000i	0.382935i
$983$	57.0000i	1.81802i	0.416777	+	0.909009i	$0.363160\pi$
$983$	57.0000i	1.81802i	−0.416777	+	0.909009i	$0.636840\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	− 5.00000i	− 0.159071i
$989$	30.0000	0.953945
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	4.00000i	0.127000i
$993$	14.0000i	0.444277i
$994$	0	0
$995$	0	0
$996$	24.0000	0.760469
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	− 28.0000i	− 0.886325i
$999$	−44.0000	−1.39210

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.c.f.99.1		2
5.2	odd	4		490.2.a.g.1.1		1
5.3	odd	4		2450.2.a.p.1.1		1
5.4	even	2	inner	2450.2.c.f.99.2		2
7.2	even	3		350.2.j.a.249.1		4
7.4	even	3		350.2.j.a.149.2		4
7.6	odd	2		2450.2.c.p.99.1		2
15.2	even	4		4410.2.a.m.1.1		1
20.7	even	4		3920.2.a.be.1.1		1
35.2	odd	12		70.2.e.b.11.1	✓	2
35.4	even	6		350.2.j.a.149.1		4
35.9	even	6		350.2.j.a.249.2		4
35.12	even	12		490.2.e.a.361.1		2
35.13	even	4		2450.2.a.f.1.1		1
35.17	even	12		490.2.e.a.471.1		2
35.18	odd	12		350.2.e.h.51.1		2
35.23	odd	12		350.2.e.h.151.1		2
35.27	even	4		490.2.a.j.1.1		1
35.32	odd	12		70.2.e.b.51.1	yes	2
35.34	odd	2		2450.2.c.p.99.2		2
105.2	even	12		630.2.k.e.361.1		2
105.32	even	12		630.2.k.e.541.1		2
105.62	odd	4		4410.2.a.c.1.1		1
140.27	odd	4		3920.2.a.g.1.1		1
140.67	even	12		560.2.q.d.401.1		2
140.107	even	12		560.2.q.d.81.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.e.b.11.1	✓	2	35.2	odd	12
70.2.e.b.51.1	yes	2	35.32	odd	12
350.2.e.h.51.1		2	35.18	odd	12
350.2.e.h.151.1		2	35.23	odd	12
350.2.j.a.149.1		4	35.4	even	6
350.2.j.a.149.2		4	7.4	even	3
350.2.j.a.249.1		4	7.2	even	3
350.2.j.a.249.2		4	35.9	even	6
490.2.a.g.1.1		1	5.2	odd	4
490.2.a.j.1.1		1	35.27	even	4
490.2.e.a.361.1		2	35.12	even	12
490.2.e.a.471.1		2	35.17	even	12
560.2.q.d.81.1		2	140.107	even	12
560.2.q.d.401.1		2	140.67	even	12
630.2.k.e.361.1		2	105.2	even	12
630.2.k.e.541.1		2	105.32	even	12
2450.2.a.f.1.1		1	35.13	even	4
2450.2.a.p.1.1		1	5.3	odd	4
2450.2.c.f.99.1		2	1.1	even	1	trivial
2450.2.c.f.99.2		2	5.4	even	2	inner
2450.2.c.p.99.1		2	7.6	odd	2
2450.2.c.p.99.2		2	35.34	odd	2
3920.2.a.g.1.1		1	140.27	odd	4
3920.2.a.be.1.1		1	20.7	even	4
4410.2.a.c.1.1		1	105.62	odd	4
4410.2.a.m.1.1		1	15.2	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} +5.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -3.00000i q^{22} +3.00000i q^{23} +2.00000 q^{24} +5.00000 q^{26} -4.00000i q^{27} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -11.0000i q^{37} -1.00000i q^{38} +10.0000 q^{39} +3.00000 q^{41} -10.0000i q^{43} -3.00000 q^{44} +3.00000 q^{46} -3.00000i q^{47} -2.00000i q^{48} -12.0000 q^{51} -5.00000i q^{52} +3.00000i q^{53} -4.00000 q^{54} -2.00000i q^{57} -6.00000i q^{58} -4.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -6.00000 q^{66} +4.00000i q^{67} +6.00000i q^{68} +6.00000 q^{69} +12.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -11.0000 q^{74} -1.00000 q^{76} -10.0000i q^{78} +10.0000 q^{79} -11.0000 q^{81} -3.00000i q^{82} -12.0000i q^{83} -10.0000 q^{86} -12.0000i q^{87} +3.00000i q^{88} -6.00000 q^{89} -3.00000i q^{92} +8.00000i q^{93} -3.00000 q^{94} -2.00000 q^{96} -14.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 6 q^{11} + 2 q^{16} + 2 q^{19} + 4 q^{24} + 10 q^{26} + 12 q^{29} - 8 q^{31} - 12 q^{34} + 2 q^{36} + 20 q^{39} + 6 q^{41} - 6 q^{44} + 6 q^{46} - 24 q^{51} - 8 q^{54} - 8 q^{61} - 2 q^{64} - 12 q^{66} + 12 q^{69} + 24 q^{71} - 22 q^{74} - 2 q^{76} + 20 q^{79} - 22 q^{81} - 20 q^{86} - 12 q^{89} - 6 q^{94} - 4 q^{96} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 6 * q^11 + 2 * q^16 + 2 * q^19 + 4 * q^24 + 10 * q^26 + 12 * q^29 - 8 * q^31 - 12 * q^34 + 2 * q^36 + 20 * q^39 + 6 * q^41 - 6 * q^44 + 6 * q^46 - 24 * q^51 - 8 * q^54 - 8 * q^61 - 2 * q^64 - 12 * q^66 + 12 * q^69 + 24 * q^71 - 22 * q^74 - 2 * q^76 + 20 * q^79 - 22 * q^81 - 20 * q^86 - 12 * q^89 - 6 * q^94 - 4 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

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Database

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