LMFDB - Embedded newform 2450.2.c.s.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.s.99.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{6} +1.00000i q^{8} -6.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{6} +1.00000i q^{8} -6.00000 q^{9} -2.00000 q^{11} -3.00000i q^{12} +1.00000 q^{16} +4.00000i q^{17} +6.00000i q^{18} +6.00000 q^{19} +2.00000i q^{22} +3.00000i q^{23} -3.00000 q^{24} -9.00000i q^{27} -9.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +4.00000 q^{34} +6.00000 q^{36} +4.00000i q^{37} -6.00000i q^{38} -7.00000 q^{41} -5.00000i q^{43} +2.00000 q^{44} +3.00000 q^{46} -8.00000i q^{47} +3.00000i q^{48} -12.0000 q^{51} -2.00000i q^{53} -9.00000 q^{54} +18.0000i q^{57} +9.00000i q^{58} -10.0000 q^{59} +1.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -6.00000 q^{66} +9.00000i q^{67} -4.00000i q^{68} -9.00000 q^{69} +2.00000 q^{71} -6.00000i q^{72} -4.00000i q^{73} +4.00000 q^{74} -6.00000 q^{76} -10.0000 q^{79} +9.00000 q^{81} +7.00000i q^{82} -7.00000i q^{83} -5.00000 q^{86} -27.0000i q^{87} -2.00000i q^{88} -1.00000 q^{89} -3.00000i q^{92} -12.0000i q^{93} -8.00000 q^{94} +3.00000 q^{96} -14.0000i q^{97} +12.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9	$ 2 q - 2 q^{4} + 6 q^{6} - 12 q^{9} - 4 q^{11} + 2 q^{16} + 12 q^{19} - 6 q^{24} - 18 q^{29} - 8 q^{31} + 8 q^{34} + 12 q^{36} - 14 q^{41} + 4 q^{44} + 6 q^{46} - 24 q^{51} - 18 q^{54} - 20 q^{59} + 2 q^{61} - 2 q^{64} - 12 q^{66} - 18 q^{69} + 4 q^{71} + 8 q^{74} - 12 q^{76} - 20 q^{79} + 18 q^{81} - 10 q^{86} - 2 q^{89} - 16 q^{94} + 6 q^{96} + 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9 - 4 * q^11 + 2 * q^16 + 12 * q^19 - 6 * q^24 - 18 * q^29 - 8 * q^31 + 8 * q^34 + 12 * q^36 - 14 * q^41 + 4 * q^44 + 6 * q^46 - 24 * q^51 - 18 * q^54 - 20 * q^59 + 2 * q^61 - 2 * q^64 - 12 * q^66 - 18 * q^69 + 4 * q^71 + 8 * q^74 - 12 * q^76 - 20 * q^79 + 18 * q^81 - 10 * q^86 - 2 * q^89 - 16 * q^94 + 6 * q^96 + 24 * 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
$3$	3.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	3.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	3.00000	1.22474
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	−6.00000	−2.00000
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	− 3.00000i	− 0.866025i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	6.00000i	1.41421i
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	2.00000i	0.426401i
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	0	0
$27$	− 9.00000i	− 1.73205i
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	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
$34$	4.00000	0.685994
$35$	0	0
$36$	6.00000	1.00000
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	− 6.00000i	− 0.973329i
$39$	0	0
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	− 5.00000i	− 0.762493i	−0.924473	−	0.381246i	$-0.875495\pi$
$43$	− 5.00000i	− 0.762493i	0.924473	−	0.381246i	$-0.124505\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	3.00000	0.442326
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	3.00000i	0.433013i
$49$	0	0
$50$	0	0
$51$	−12.0000	−1.68034
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	−9.00000	−1.22474
$55$	0	0
$56$	0	0
$57$	18.0000i	2.38416i
$58$	9.00000i	1.18176i
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	9.00000i	1.09952i	0.835321	+	0.549762i	$0.185282\pi$
$67$	9.00000i	1.09952i	−0.835321	+	0.549762i	$0.814718\pi$
$68$	− 4.00000i	− 0.485071i
$69$	−9.00000	−1.08347
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	− 6.00000i	− 0.707107i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	7.00000i	0.773021i
$83$	− 7.00000i	− 0.768350i	−0.923260	−	0.384175i	$-0.874486\pi$
$83$	− 7.00000i	− 0.768350i	0.923260	−	0.384175i	$-0.125514\pi$
$84$	0	0
$85$	0	0
$86$	−5.00000	−0.539164
$87$	− 27.0000i	− 2.89470i
$88$	− 2.00000i	− 0.213201i
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	0	0
$92$	− 3.00000i	− 0.312772i
$93$	− 12.0000i	− 1.24434i
$94$	−8.00000	−0.825137
$95$	0	0
$96$	3.00000	0.306186
$97$	− 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
$99$	12.0000	1.20605
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	12.0000i	1.18818i
$103$	1.00000i	0.0985329i	0.998786	+	0.0492665i	$0.0156884\pi$
$103$	1.00000i	0.0985329i	−0.998786	+	0.0492665i	$0.984312\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	9.00000i	0.866025i
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	18.0000	1.68585
$115$	0	0
$116$	9.00000	0.835629
$117$	0	0
$118$	10.0000i	0.920575i
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 1.00000i	− 0.0905357i
$123$	− 21.0000i	− 1.89351i
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	1.00000i	0.0883883i
$129$	15.0000	1.32068
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	6.00000i	0.522233i
$133$	0	0
$134$	9.00000	0.777482
$135$	0	0
$136$	−4.00000	−0.342997
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	9.00000i	0.766131i
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	− 2.00000i	− 0.167836i
$143$	0	0
$144$	−6.00000	−0.500000
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	− 4.00000i	− 0.328798i
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	6.00000i	0.486664i
$153$	− 24.0000i	− 1.94029i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	10.0000i	0.795557i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	− 9.00000i	− 0.707107i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	−7.00000	−0.543305
$167$	21.0000i	1.62503i	0.582941	+	0.812514i	$0.301902\pi$
$167$	21.0000i	1.62503i	−0.582941	+	0.812514i	$0.698098\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	−36.0000	−2.75299
$172$	5.00000i	0.381246i
$173$	8.00000i	0.608229i	0.952636	+	0.304114i	$0.0983605\pi$
$173$	8.00000i	0.608229i	−0.952636	+	0.304114i	$0.901639\pi$
$174$	−27.0000	−2.04686
$175$	0	0
$176$	−2.00000	−0.150756
$177$	− 30.0000i	− 2.25494i
$178$	1.00000i	0.0749532i
$179$	−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
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	3.00000i	0.221766i
$184$	−3.00000	−0.221163
$185$	0	0
$186$	−12.0000	−0.879883
$187$	− 8.00000i	− 0.585018i
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	− 3.00000i	− 0.216506i
$193$	26.0000i	1.87152i	0.352636	+	0.935760i	$0.385285\pi$
$193$	26.0000i	1.87152i	−0.352636	+	0.935760i	$0.614715\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	0	0
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	− 12.0000i	− 0.852803i
$199$	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
$201$	−27.0000	−1.90443
$202$	− 3.00000i	− 0.211079i
$203$	0	0
$204$	12.0000	0.840168
$205$	0	0
$206$	1.00000	0.0696733
$207$	− 18.0000i	− 1.25109i
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	2.00000i	0.137361i
$213$	6.00000i	0.411113i
$214$	−3.00000	−0.205076
$215$	0	0
$216$	9.00000	0.612372
$217$	0	0
$218$	− 9.00000i	− 0.609557i
$219$	12.0000	0.810885
$220$	0	0
$221$	0	0
$222$	12.0000i	0.805387i
$223$	28.0000i	1.87502i	0.347960	+	0.937509i	$0.386874\pi$
$223$	28.0000i	1.87502i	−0.347960	+	0.937509i	$0.613126\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	0.133038
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	− 18.0000i	− 1.19208i
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	− 9.00000i	− 0.590879i
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	0	0
$235$	0	0
$236$	10.0000	0.650945
$237$	− 30.0000i	− 1.94871i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	7.00000i	0.449977i
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	−21.0000	−1.33891
$247$	0	0
$248$	− 4.00000i	− 0.254000i
$249$	21.0000	1.33082
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 6.00000i	− 0.377217i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 8.00000i	− 0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$	− 8.00000i	− 0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$	− 15.0000i	− 0.933859i
$259$	0	0
$260$	0	0
$261$	54.0000	3.34252
$262$	− 8.00000i	− 0.494242i
$263$	5.00000i	0.308313i	0.988046	+	0.154157i	$0.0492660\pi$
$263$	5.00000i	0.308313i	−0.988046	+	0.154157i	$0.950734\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	0	0
$267$	− 3.00000i	− 0.183597i
$268$	− 9.00000i	− 0.549762i
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	4.00000i	0.242536i
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	9.00000	0.541736
$277$	− 12.0000i	− 0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$	− 12.0000i	− 0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$	− 14.0000i	− 0.839664i
$279$	24.0000	1.43684
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	− 24.0000i	− 1.42918i
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	0	0
$287$	0	0
$288$	6.00000i	0.353553i
$289$	1.00000	0.0588235
$290$	0	0
$291$	42.0000	2.46208
$292$	4.00000i	0.234082i
$293$	28.0000i	1.63578i	0.575376	+	0.817889i	$0.304856\pi$
$293$	28.0000i	1.63578i	−0.575376	+	0.817889i	$0.695144\pi$
$294$	0	0
$295$	0	0
$296$	−4.00000	−0.232495
$297$	18.0000i	1.04447i
$298$	3.00000i	0.173785i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	16.0000i	0.920697i
$303$	9.00000i	0.517036i
$304$	6.00000	0.344124
$305$	0	0
$306$	−24.0000	−1.37199
$307$	− 7.00000i	− 0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$	− 7.00000i	− 0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	0	0
$309$	−3.00000	−0.170664
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	10.0000	0.562544
$317$	32.0000i	1.79730i	0.438667	+	0.898650i	$0.355451\pi$
$317$	32.0000i	1.79730i	−0.438667	+	0.898650i	$0.644549\pi$
$318$	− 6.00000i	− 0.336463i
$319$	18.0000	1.00781
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	24.0000i	1.33540i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−4.00000	−0.221540
$327$	27.0000i	1.49310i
$328$	− 7.00000i	− 0.386510i
$329$	0	0
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	7.00000i	0.384175i
$333$	− 24.0000i	− 1.31519i
$334$	21.0000	1.14907
$335$	0	0
$336$	0	0
$337$	26.0000i	1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$	26.0000i	1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$	− 13.0000i	− 0.707107i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	8.00000	0.433224
$342$	36.0000i	1.94666i
$343$	0	0
$344$	5.00000	0.269582
$345$	0	0
$346$	8.00000	0.430083
$347$	− 19.0000i	− 1.01997i	−0.860182	−	0.509987i	$-0.829650\pi$
$347$	− 19.0000i	− 1.01997i	0.860182	−	0.509987i	$-0.170350\pi$
$348$	27.0000i	1.44735i
$349$	35.0000	1.87351	0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000	1.87351	0.936754	+	0.349990i	$0.113815\pi$
$350$	0	0
$351$	0	0
$352$	2.00000i	0.106600i
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	−30.0000	−1.59448
$355$	0	0
$356$	1.00000	0.0529999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 7.00000i	− 0.367912i
$363$	− 21.0000i	− 1.10221i
$364$	0	0
$365$	0	0
$366$	3.00000	0.156813
$367$	11.0000i	0.574195i	0.957901	+	0.287098i	$0.0926904\pi$
$367$	11.0000i	0.574195i	−0.957901	+	0.287098i	$0.907310\pi$
$368$	3.00000i	0.156386i
$369$	42.0000	2.18643
$370$	0	0
$371$	0	0
$372$	12.0000i	0.622171i
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	0	0
$381$	48.0000	2.45911
$382$	18.0000i	0.920960i
$383$	15.0000i	0.766464i	0.923652	+	0.383232i	$0.125189\pi$
$383$	15.0000i	0.766464i	−0.923652	+	0.383232i	$0.874811\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	26.0000	1.32337
$387$	30.0000i	1.52499i
$388$	14.0000i	0.710742i
$389$	−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
$391$	−12.0000	−0.606866
$392$	0	0
$393$	24.0000i	1.21064i
$394$	−2.00000	−0.100759
$395$	0	0
$396$	−12.0000	−0.603023
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	− 4.00000i	− 0.200502i
$399$	0	0
$400$	0	0
$401$	31.0000	1.54807	0.774033	−	0.633145i	$-0.218236\pi$
$401$	31.0000	1.54807	0.774033	+	0.633145i	$0.218236\pi$
$402$	27.0000i	1.34664i
$403$	0	0
$404$	−3.00000	−0.149256
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	− 12.0000i	− 0.594089i
$409$	−3.00000	−0.148340	−0.0741702	−	0.997246i	$-0.523631\pi$
$409$	−3.00000	−0.148340	−0.0741702	+	0.997246i	$0.523631\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	− 1.00000i	− 0.0492665i
$413$	0	0
$414$	−18.0000	−0.884652
$415$	0	0
$416$	0	0
$417$	42.0000i	2.05675i
$418$	12.0000i	0.586939i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	26.0000i	1.26566i
$423$	48.0000i	2.33384i
$424$	2.00000	0.0971286
$425$	0	0
$426$	6.00000	0.290701
$427$	0	0
$428$	3.00000i	0.145010i
$429$	0	0
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	− 9.00000i	− 0.433013i
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	−9.00000	−0.431022
$437$	18.0000i	0.861057i
$438$	− 12.0000i	− 0.573382i
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.0000i	1.47285i	0.676517	+	0.736427i	$0.263489\pi$
$443$	31.0000i	1.47285i	−0.676517	+	0.736427i	$0.736511\pi$
$444$	12.0000	0.569495
$445$	0	0
$446$	28.0000	1.32584
$447$	− 9.00000i	− 0.425685i
$448$	0	0
$449$	33.0000	1.55737	0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000	1.55737	0.778683	+	0.627417i	$0.215888\pi$
$450$	0	0
$451$	14.0000	0.659234
$452$	− 2.00000i	− 0.0940721i
$453$	− 48.0000i	− 2.25524i
$454$	4.00000	0.187729
$455$	0	0
$456$	−18.0000	−0.842927
$457$	32.0000i	1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$	32.0000i	1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$	22.0000i	1.02799i
$459$	36.0000	1.68034
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	− 19.0000i	− 0.883005i	−0.897260	−	0.441502i	$-0.854446\pi$
$463$	− 19.0000i	− 0.883005i	0.897260	−	0.441502i	$-0.145554\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	24.0000	1.11178
$467$	13.0000i	0.601568i	0.953692	+	0.300784i	$0.0972484\pi$
$467$	13.0000i	0.601568i	−0.953692	+	0.300784i	$0.902752\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	30.0000	1.38233
$472$	− 10.0000i	− 0.460287i
$473$	10.0000i	0.459800i
$474$	−30.0000	−1.37795
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	16.0000i	0.731823i
$479$	−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
$481$	0	0
$482$	− 10.0000i	− 0.455488i
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	1.00000i	0.0452679i
$489$	12.0000	0.542659
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	21.0000i	0.946753i
$493$	− 36.0000i	− 1.62136i
$494$	0	0
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	− 21.0000i	− 0.941033i
$499$	18.0000	0.805791	0.402895	−	0.915246i	$-0.368004\pi$
$499$	18.0000	0.805791	0.402895	+	0.915246i	$0.368004\pi$
$500$	0	0
$501$	−63.0000	−2.81463
$502$	0	0
$503$	− 21.0000i	− 0.936344i	−0.883637	−	0.468172i	$-0.844913\pi$
$503$	− 21.0000i	− 0.936344i	0.883637	−	0.468172i	$-0.155087\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	39.0000i	1.73205i
$508$	16.0000i	0.709885i
$509$	−1.00000	−0.0443242	−0.0221621	−	0.999754i	$-0.507055\pi$
$509$	−1.00000	−0.0443242	−0.0221621	+	0.999754i	$0.507055\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 54.0000i	− 2.38416i
$514$	−8.00000	−0.352865
$515$	0	0
$516$	−15.0000	−0.660338
$517$	16.0000i	0.703679i
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	− 54.0000i	− 2.36352i
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	5.00000	0.218010
$527$	− 16.0000i	− 0.696971i
$528$	− 6.00000i	− 0.261116i
$529$	14.0000	0.608696
$530$	0	0
$531$	60.0000	2.60378
$532$	0	0
$533$	0	0
$534$	−3.00000	−0.129823
$535$	0	0
$536$	−9.00000	−0.388741
$537$	− 36.0000i	− 1.55351i
$538$	3.00000i	0.129339i
$539$	0	0
$540$	0	0
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	6.00000i	0.257722i
$543$	21.0000i	0.901196i
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	33.0000i	1.41098i	0.708721	+	0.705489i	$0.249273\pi$
$547$	33.0000i	1.41098i	−0.708721	+	0.705489i	$0.750727\pi$
$548$	12.0000i	0.512615i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−54.0000	−2.30048
$552$	− 9.00000i	− 0.383065i
$553$	0	0
$554$	−12.0000	−0.509831
$555$	0	0
$556$	−14.0000	−0.593732
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	− 24.0000i	− 1.01600i
$559$	0	0
$560$	0	0
$561$	24.0000	1.01328
$562$	− 2.00000i	− 0.0843649i
$563$	17.0000i	0.716465i	0.933632	+	0.358232i	$0.116620\pi$
$563$	17.0000i	0.716465i	−0.933632	+	0.358232i	$0.883380\pi$
$564$	−24.0000	−1.01058
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	2.00000i	0.0839181i
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−30.0000	−1.25546	−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$572$	0	0
$573$	− 54.0000i	− 2.25588i
$574$	0	0
$575$	0	0
$576$	6.00000	0.250000
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−78.0000	−3.24157
$580$	0	0
$581$	0	0
$582$	− 42.0000i	− 1.74096i
$583$	4.00000i	0.165663i
$584$	4.00000	0.165521
$585$	0	0
$586$	28.0000	1.15667
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	6.00000	0.246807
$592$	4.00000i	0.164399i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	18.0000	0.738549
$595$	0	0
$596$	3.00000	0.122885
$597$	12.0000i	0.491127i
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	− 54.0000i	− 2.19905i
$604$	16.0000	0.651031
$605$	0	0
$606$	9.00000	0.365600
$607$	− 1.00000i	− 0.0405887i	−0.999794	−	0.0202944i	$-0.993540\pi$
$607$	− 1.00000i	− 0.0405887i	0.999794	−	0.0202944i	$-0.00646034\pi$
$608$	− 6.00000i	− 0.243332i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	24.0000i	0.970143i
$613$	12.0000i	0.484675i	0.970192	+	0.242338i	$0.0779142\pi$
$613$	12.0000i	0.484675i	−0.970192	+	0.242338i	$0.922086\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	0	0
$617$	− 44.0000i	− 1.77137i	−0.464283	−	0.885687i	$-0.653688\pi$
$617$	− 44.0000i	− 1.77137i	0.464283	−	0.885687i	$-0.346312\pi$
$618$	3.00000i	0.120678i
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	0	0
$621$	27.0000	1.08347
$622$	18.0000i	0.721734i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	8.00000	0.319744
$627$	− 36.0000i	− 1.43770i
$628$	10.0000i	0.399043i
$629$	−16.0000	−0.637962
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	− 10.0000i	− 0.397779i
$633$	− 78.0000i	− 3.10022i
$634$	32.0000	1.27088
$635$	0	0
$636$	−6.00000	−0.237915
$637$	0	0
$638$	− 18.0000i	− 0.712627i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	− 9.00000i	− 0.355202i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	11.0000i	0.432455i	0.976343	+	0.216227i	$0.0693752\pi$
$647$	11.0000i	0.432455i	−0.976343	+	0.216227i	$0.930625\pi$
$648$	9.00000i	0.353553i
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	− 4.00000i	− 0.156532i	−0.996933	−	0.0782660i	$-0.975062\pi$
$653$	− 4.00000i	− 0.156532i	0.996933	−	0.0782660i	$-0.0249384\pi$
$654$	27.0000	1.05578
$655$	0	0
$656$	−7.00000	−0.273304
$657$	24.0000i	0.936329i
$658$	0	0
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	32.0000i	1.24372i
$663$	0	0
$664$	7.00000	0.271653
$665$	0	0
$666$	−24.0000	−0.929981
$667$	− 27.0000i	− 1.04544i
$668$	− 21.0000i	− 0.812514i
$669$	−84.0000	−3.24763
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	16.0000i	0.616755i	0.951264	+	0.308377i	$0.0997859\pi$
$673$	16.0000i	0.616755i	−0.951264	+	0.308377i	$0.900214\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	−13.0000	−0.500000
$677$	48.0000i	1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$677$	48.0000i	1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$678$	6.00000i	0.230429i
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	− 8.00000i	− 0.306336i
$683$	− 37.0000i	− 1.41577i	−0.706330	−	0.707883i	$-0.749650\pi$
$683$	− 37.0000i	− 1.41577i	0.706330	−	0.707883i	$-0.250350\pi$
$684$	36.0000	1.37649
$685$	0	0
$686$	0	0
$687$	− 66.0000i	− 2.51806i
$688$	− 5.00000i	− 0.190623i
$689$	0	0
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	− 8.00000i	− 0.304114i
$693$	0	0
$694$	−19.0000	−0.721230
$695$	0	0
$696$	27.0000	1.02343
$697$	− 28.0000i	− 1.06058i
$698$	− 35.0000i	− 1.32477i
$699$	−72.0000	−2.72329
$700$	0	0
$701$	−47.0000	−1.77517	−0.887583	−	0.460648i	$-0.847617\pi$
$701$	−47.0000	−1.77517	−0.887583	+	0.460648i	$0.847617\pi$
$702$	0	0
$703$	24.0000i	0.905177i
$704$	2.00000	0.0753778
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	30.0000i	1.12747i
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	0	0
$711$	60.0000	2.25018
$712$	− 1.00000i	− 0.0374766i
$713$	− 12.0000i	− 0.449404i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	− 48.0000i	− 1.79259i
$718$	− 4.00000i	− 0.149279i
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	0	0
$722$	− 17.0000i	− 0.632674i
$723$	30.0000i	1.11571i
$724$	−7.00000	−0.260153
$725$	0	0
$726$	−21.0000	−0.779383
$727$	21.0000i	0.778847i	0.921059	+	0.389423i	$0.127326\pi$
$727$	21.0000i	0.778847i	−0.921059	+	0.389423i	$0.872674\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	20.0000	0.739727
$732$	− 3.00000i	− 0.110883i
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	11.0000	0.406017
$735$	0	0
$736$	3.00000	0.110581
$737$	− 18.0000i	− 0.663039i
$738$	− 42.0000i	− 1.54604i
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.00000i	0.330178i	0.986279	+	0.165089i	$0.0527911\pi$
$743$	9.00000i	0.330178i	−0.986279	+	0.165089i	$0.947209\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	−4.00000	−0.146450
$747$	42.0000i	1.53670i
$748$	8.00000i	0.292509i
$749$	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	$0.523251\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	30.0000i	1.08965i
$759$	18.0000	0.653359
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	− 48.0000i	− 1.73886i
$763$	0	0
$764$	18.0000	0.651217
$765$	0	0
$766$	15.0000	0.541972
$767$	0	0
$768$	3.00000i	0.108253i
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	− 26.0000i	− 0.935760i
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	30.0000	1.07833
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	26.0000i	0.932145i
$779$	−42.0000	−1.50481
$780$	0	0
$781$	−4.00000	−0.143131
$782$	12.0000i	0.429119i
$783$	81.0000i	2.89470i
$784$	0	0
$785$	0	0
$786$	24.0000	0.856052
$787$	− 31.0000i	− 1.10503i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	− 31.0000i	− 1.10503i	0.833503	−	0.552515i	$-0.186332\pi$
$788$	2.00000i	0.0712470i
$789$	−15.0000	−0.534014
$790$	0	0
$791$	0	0
$792$	12.0000i	0.426401i
$793$	0	0
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−4.00000	−0.141776
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	0	0
$801$	6.00000	0.212000
$802$	− 31.0000i	− 1.09465i
$803$	8.00000i	0.282314i
$804$	27.0000	0.952217
$805$	0	0
$806$	0	0
$807$	− 9.00000i	− 0.316815i
$808$	3.00000i	0.105540i
$809$	51.0000	1.79306	0.896532	−	0.442978i	$-0.146078\pi$
$809$	51.0000	1.79306	0.896532	+	0.442978i	$0.146078\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	− 18.0000i	− 0.631288i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	−12.0000	−0.420084
$817$	− 30.0000i	− 1.04957i
$818$	3.00000i	0.104893i
$819$	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	$0.601703\pi$
$822$	− 36.0000i	− 1.25564i
$823$	19.0000i	0.662298i	0.943578	+	0.331149i	$0.107436\pi$
$823$	19.0000i	0.662298i	−0.943578	+	0.331149i	$0.892564\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	0	0
$827$	19.0000i	0.660695i	0.943859	+	0.330347i	$0.107166\pi$
$827$	19.0000i	0.660695i	−0.943859	+	0.330347i	$0.892834\pi$
$828$	18.0000i	0.625543i
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	36.0000	1.24883
$832$	0	0
$833$	0	0
$834$	42.0000	1.45434
$835$	0	0
$836$	12.0000	0.415029
$837$	36.0000i	1.24434i
$838$	0	0
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	19.0000i	0.654783i
$843$	6.00000i	0.206651i
$844$	26.0000	0.894957
$845$	0	0
$846$	48.0000	1.65027
$847$	0	0
$848$	− 2.00000i	− 0.0686803i
$849$	12.0000	0.411839
$850$	0	0
$851$	−12.0000	−0.411355
$852$	− 6.00000i	− 0.205557i
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	0	0
$855$	0	0
$856$	3.00000	0.102538
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	48.0000	1.63774	0.818869	−	0.573980i	$-0.194601\pi$
$859$	48.0000	1.63774	0.818869	+	0.573980i	$0.194601\pi$
$860$	0	0
$861$	0	0
$862$	30.0000i	1.02180i
$863$	− 11.0000i	− 0.374444i	−0.982318	−	0.187222i	$-0.940052\pi$
$863$	− 11.0000i	− 0.374444i	0.982318	−	0.187222i	$-0.0599484\pi$
$864$	−9.00000	−0.306186
$865$	0	0
$866$	14.0000	0.475739
$867$	3.00000i	0.101885i
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	0	0
$872$	9.00000i	0.304778i
$873$	84.0000i	2.84297i
$874$	18.0000	0.608859
$875$	0	0
$876$	−12.0000	−0.405442
$877$	− 38.0000i	− 1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$	− 38.0000i	− 1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$	− 20.0000i	− 0.674967i
$879$	−84.0000	−2.83325
$880$	0	0
$881$	−7.00000	−0.235836	−0.117918	−	0.993023i	$-0.537622\pi$
$881$	−7.00000	−0.235836	−0.117918	+	0.993023i	$0.537622\pi$
$882$	0	0
$883$	− 12.0000i	− 0.403832i	−0.979403	−	0.201916i	$-0.935283\pi$
$883$	− 12.0000i	− 0.403832i	0.979403	−	0.201916i	$-0.0647168\pi$
$884$	0	0
$885$	0	0
$886$	31.0000	1.04147
$887$	− 29.0000i	− 0.973725i	−0.873479	−	0.486862i	$-0.838141\pi$
$887$	− 29.0000i	− 0.973725i	0.873479	−	0.486862i	$-0.161859\pi$
$888$	− 12.0000i	− 0.402694i
$889$	0	0
$890$	0	0
$891$	−18.0000	−0.603023
$892$	− 28.0000i	− 0.937509i
$893$	− 48.0000i	− 1.60626i
$894$	−9.00000	−0.301005
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 33.0000i	− 1.10122i
$899$	36.0000	1.20067
$900$	0	0
$901$	8.00000	0.266519
$902$	− 14.0000i	− 0.466149i
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	−48.0000	−1.59469
$907$	− 5.00000i	− 0.166022i	−0.996549	−	0.0830111i	$-0.973546\pi$
$907$	− 5.00000i	− 0.166022i	0.996549	−	0.0830111i	$-0.0264537\pi$
$908$	− 4.00000i	− 0.132745i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	18.0000i	0.596040i
$913$	14.0000i	0.463332i
$914$	32.0000	1.05847
$915$	0	0
$916$	22.0000	0.726900
$917$	0	0
$918$	− 36.0000i	− 1.18818i
$919$	−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
$921$	21.0000	0.691974
$922$	14.0000i	0.461065i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−19.0000	−0.624379
$927$	− 6.00000i	− 0.197066i
$928$	9.00000i	0.295439i
$929$	−43.0000	−1.41078	−0.705392	−	0.708817i	$-0.749229\pi$
$929$	−43.0000	−1.41078	−0.705392	+	0.708817i	$0.749229\pi$
$930$	0	0
$931$	0	0
$932$	− 24.0000i	− 0.786146i
$933$	− 54.0000i	− 1.76788i
$934$	13.0000	0.425373
$935$	0	0
$936$	0	0
$937$	28.0000i	0.914720i	0.889282	+	0.457360i	$0.151205\pi$
$937$	28.0000i	0.914720i	−0.889282	+	0.457360i	$0.848795\pi$
$938$	0	0
$939$	−24.0000	−0.783210
$940$	0	0
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	− 30.0000i	− 0.977453i
$943$	− 21.0000i	− 0.683854i
$944$	−10.0000	−0.325472
$945$	0	0
$946$	10.0000	0.325128
$947$	25.0000i	0.812391i	0.913786	+	0.406195i	$0.133145\pi$
$947$	25.0000i	0.812391i	−0.913786	+	0.406195i	$0.866855\pi$
$948$	30.0000i	0.974355i
$949$	0	0
$950$	0	0
$951$	−96.0000	−3.11301
$952$	0	0
$953$	− 12.0000i	− 0.388718i	−0.980930	−	0.194359i	$-0.937737\pi$
$953$	− 12.0000i	− 0.388718i	0.980930	−	0.194359i	$-0.0622627\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	16.0000	0.517477
$957$	54.0000i	1.74557i
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	18.0000i	0.580042i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	− 37.0000i	− 1.18984i	−0.803785	−	0.594920i	$-0.797184\pi$
$967$	− 37.0000i	− 1.18984i	0.803785	−	0.594920i	$-0.202816\pi$
$968$	− 7.00000i	− 0.224989i
$969$	−72.0000	−2.31297
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	1.00000	0.0320092
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	− 12.0000i	− 0.383718i
$979$	2.00000	0.0639203
$980$	0	0
$981$	−54.0000	−1.72409
$982$	12.0000i	0.382935i
$983$	17.0000i	0.542216i	0.962549	+	0.271108i	$0.0873900\pi$
$983$	17.0000i	0.542216i	−0.962549	+	0.271108i	$0.912610\pi$
$984$	21.0000	0.669456
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	0	0
$989$	15.0000	0.476972
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	4.00000i	0.127000i
$993$	− 96.0000i	− 3.04647i
$994$	0	0
$995$	0	0
$996$	−21.0000	−0.665410
$997$	46.0000i	1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$	46.0000i	1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$	− 18.0000i	− 0.569780i
$999$	36.0000	1.13899

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{6} +1.00000i q^{8} -6.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{6} +1.00000i q^{8} -6.00000 q^{9} -2.00000 q^{11} -3.00000i q^{12} +1.00000 q^{16} +4.00000i q^{17} +6.00000i q^{18} +6.00000 q^{19} +2.00000i q^{22} +3.00000i q^{23} -3.00000 q^{24} -9.00000i q^{27} -9.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +4.00000 q^{34} +6.00000 q^{36} +4.00000i q^{37} -6.00000i q^{38} -7.00000 q^{41} -5.00000i q^{43} +2.00000 q^{44} +3.00000 q^{46} -8.00000i q^{47} +3.00000i q^{48} -12.0000 q^{51} -2.00000i q^{53} -9.00000 q^{54} +18.0000i q^{57} +9.00000i q^{58} -10.0000 q^{59} +1.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -6.00000 q^{66} +9.00000i q^{67} -4.00000i q^{68} -9.00000 q^{69} +2.00000 q^{71} -6.00000i q^{72} -4.00000i q^{73} +4.00000 q^{74} -6.00000 q^{76} -10.0000 q^{79} +9.00000 q^{81} +7.00000i q^{82} -7.00000i q^{83} -5.00000 q^{86} -27.0000i q^{87} -2.00000i q^{88} -1.00000 q^{89} -3.00000i q^{92} -12.0000i q^{93} -8.00000 q^{94} +3.00000 q^{96} -14.0000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9	\( 2 q - 2 q^{4} + 6 q^{6} - 12 q^{9} - 4 q^{11} + 2 q^{16} + 12 q^{19} - 6 q^{24} - 18 q^{29} - 8 q^{31} + 8 q^{34} + 12 q^{36} - 14 q^{41} + 4 q^{44} + 6 q^{46} - 24 q^{51} - 18 q^{54} - 20 q^{59} + 2 q^{61} - 2 q^{64} - 12 q^{66} - 18 q^{69} + 4 q^{71} + 8 q^{74} - 12 q^{76} - 20 q^{79} + 18 q^{81} - 10 q^{86} - 2 q^{89} - 16 q^{94} + 6 q^{96} + 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9 - 4 * q^11 + 2 * q^16 + 12 * q^19 - 6 * q^24 - 18 * q^29 - 8 * q^31 + 8 * q^34 + 12 * q^36 - 14 * q^41 + 4 * q^44 + 6 * q^46 - 24 * q^51 - 18 * q^54 - 20 * q^59 + 2 * q^61 - 2 * q^64 - 12 * q^66 - 18 * q^69 + 4 * q^71 + 8 * q^74 - 12 * q^76 - 20 * q^79 + 18 * q^81 - 10 * q^86 - 2 * q^89 - 16 * q^94 + 6 * q^96 + 24 * q^99

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