LMFDB - Embedded newform 2450.2.c.q.99.2

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +3.00000i q^{22} +9.00000i q^{23} -2.00000 q^{24} +1.00000 q^{26} -4.00000i q^{27} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +7.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} +3.00000 q^{41} +2.00000i q^{43} -3.00000 q^{44} -9.00000 q^{46} -9.00000i q^{47} -2.00000i q^{48} +12.0000 q^{51} +1.00000i q^{52} +9.00000i q^{53} +4.00000 q^{54} -2.00000i q^{57} -6.00000i q^{58} +8.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} +18.0000 q^{69} +1.00000i q^{72} -4.00000i q^{73} -7.00000 q^{74} -1.00000 q^{76} -2.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +3.00000i q^{82} -2.00000 q^{86} +12.0000i q^{87} -3.00000i q^{88} -6.00000 q^{89} -9.00000i q^{92} -16.0000i q^{93} +9.00000 q^{94} +2.00000 q^{96} +10.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{26} - 12 q^{29} + 16 q^{31} - 12 q^{34} + 2 q^{36} - 4 q^{39} + 6 q^{41} - 6 q^{44} - 18 q^{46} + 24 q^{51} + 8 q^{54} + 16 q^{61} - 2 q^{64} + 12 q^{66} + 36 q^{69} - 14 q^{74} - 2 q^{76} + 20 q^{79} - 22 q^{81} - 4 q^{86} - 12 q^{89} + 18 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 + 2 * q^26 - 12 * q^29 + 16 * q^31 - 12 * q^34 + 2 * q^36 - 4 * q^39 + 6 * q^41 - 6 * q^44 - 18 * q^46 + 24 * q^51 + 8 * q^54 + 16 * q^61 - 2 * q^64 + 12 * q^66 + 36 * q^69 - 14 * q^74 - 2 * q^76 + 20 * q^79 - 22 * q^81 - 4 * q^86 - 12 * q^89 + 18 * 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
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	− 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
$23$	9.00000i	1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$	9.00000i	1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	1.00000	0.196116
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000i	0.176777i
$33$	− 6.00000i	− 1.04447i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	1.00000i	0.162221i
$39$	−2.00000	−0.320256
$40$	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
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−9.00000	−1.32698
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	− 2.00000i	− 0.288675i
$49$	0	0
$50$	0	0
$51$	12.0000	1.68034
$52$	1.00000i	0.138675i
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	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
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	− 6.00000i	− 0.727607i
$69$	18.0000	2.16695
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	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	$-0.0752085\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	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
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	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
$92$	− 9.00000i	− 0.938315i
$93$	− 16.0000i	− 1.65912i
$94$	9.00000	0.928279
$95$	0	0
$96$	2.00000	0.204124
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	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	$-0.0631413\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−9.00000	−0.874157
$107$	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
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	14.0000	1.32882
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	6.00000	0.557086
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	8.00000i	0.724286i
$123$	− 6.00000i	− 0.541002i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	1.00000i	0.0887357i	0.999015	+	0.0443678i	$0.0141274\pi$
$127$	1.00000i	0.0887357i	−0.999015	+	0.0443678i	$0.985873\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	4.00000	0.352180
$130$	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
$134$	8.00000	0.691095
$135$	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	$-0.171323\pi$
$138$	18.0000i	1.53226i
$139$	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
$141$	−18.0000	−1.51587
$142$	0	0
$143$	− 3.00000i	− 0.250873i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	− 7.00000i	− 0.575396i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	− 23.0000i	− 1.83560i	−0.397043	−	0.917800i	$-0.629964\pi$
$157$	− 23.0000i	− 1.83560i	0.397043	−	0.917800i	$-0.370036\pi$
$158$	10.0000i	0.795557i
$159$	18.0000	1.42749
$160$	0	0
$161$	0	0
$162$	− 11.0000i	− 0.864242i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	0	0
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 2.00000i	− 0.152499i
$173$	9.00000i	0.684257i	0.939653	+	0.342129i	$0.111148\pi$
$173$	9.00000i	0.684257i	−0.939653	+	0.342129i	$0.888852\pi$
$174$	−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
$183$	− 16.0000i	− 1.18275i
$184$	9.00000	0.663489
$185$	0	0
$186$	16.0000	1.17318
$187$	18.0000i	1.31629i
$188$	9.00000i	0.656392i
$189$	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
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	0	0
$197$	− 15.0000i	− 1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$197$	− 15.0000i	− 1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$198$	− 3.00000i	− 0.213201i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	12.0000i	0.844317i
$203$	0	0
$204$	−12.0000	−0.840168
$205$	0	0
$206$	4.00000	0.278693
$207$	− 9.00000i	− 0.625543i
$208$	− 1.00000i	− 0.0693375i
$209$	3.00000	0.207514
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	− 9.00000i	− 0.618123i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−4.00000	−0.272166
$217$	0	0
$218$	16.0000i	1.08366i
$219$	−8.00000	−0.540590
$220$	0	0
$221$	6.00000	0.403604
$222$	14.0000i	0.939618i
$223$	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
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	2.00000i	0.132453i
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	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	$-0.0629694\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	0	0
$237$	− 20.0000i	− 1.29914i
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	− 2.00000i	− 0.128565i
$243$	10.0000i	0.641500i
$244$	−8.00000	−0.512148
$245$	0	0
$246$	6.00000	0.382546
$247$	− 1.00000i	− 0.0636285i
$248$	− 8.00000i	− 0.508001i
$249$	0	0
$250$	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
$253$	27.0000i	1.69748i
$254$	−1.00000	−0.0627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	4.00000i	0.249029i
$259$	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
$268$	8.00000i	0.488678i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	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
$276$	−18.0000	−1.08347
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	4.00000i	0.239904i
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−27.0000	−1.61068	−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000	−1.61068	−0.805342	+	0.592810i	$0.798019\pi$
$282$	− 18.0000i	− 1.07188i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	0	0
$285$	0	0
$286$	3.00000	0.177394
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	20.0000	1.17242
$292$	4.00000i	0.234082i
$293$	− 9.00000i	− 0.525786i	−0.964825	−	0.262893i	$-0.915323\pi$
$293$	− 9.00000i	− 0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$	0	0
$295$	0	0
$296$	7.00000	0.406867
$297$	− 12.0000i	− 0.696311i
$298$	6.00000i	0.347571i
$299$	9.00000	0.520483
$300$	0	0
$301$	0	0
$302$	− 10.0000i	− 0.575435i
$303$	− 24.0000i	− 1.37876i
$304$	1.00000	0.0573539
$305$	0	0
$306$	6.00000	0.342997
$307$	− 14.0000i	− 0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$	− 14.0000i	− 0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	2.00000i	0.113228i
$313$	− 28.0000i	− 1.58265i	−0.611393	−	0.791327i	$-0.709391\pi$
$313$	− 28.0000i	− 1.58265i	0.611393	−	0.791327i	$-0.290609\pi$
$314$	23.0000	1.29797
$315$	0	0
$316$	−10.0000	−0.562544
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	18.0000i	1.00939i
$319$	−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
$326$	−20.0000	−1.10770
$327$	− 32.0000i	− 1.76960i
$328$	− 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	$0.561620\pi$
$332$	0	0
$333$	− 7.00000i	− 0.383598i
$334$	3.00000	0.164153
$335$	0	0
$336$	0	0
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	12.0000i	0.652714i
$339$	0	0
$340$	0	0
$341$	24.0000	1.29967
$342$	− 1.00000i	− 0.0540738i
$343$	0	0
$344$	2.00000	0.107833
$345$	0	0
$346$	−9.00000	−0.483843
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	− 12.0000i	− 0.643268i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	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
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	2.00000i	0.105118i
$363$	4.00000i	0.209946i
$364$	0	0
$365$	0	0
$366$	16.0000	0.836333
$367$	19.0000i	0.991792i	0.868382	+	0.495896i	$0.165160\pi$
$367$	19.0000i	0.991792i	−0.868382	+	0.495896i	$0.834840\pi$
$368$	9.00000i	0.469157i
$369$	−3.00000	−0.156174
$370$	0	0
$371$	0	0
$372$	16.0000i	0.829561i
$373$	2.00000i	0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$	2.00000i	0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	−9.00000	−0.464140
$377$	6.00000i	0.309016i
$378$	0	0
$379$	−23.0000	−1.18143	−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000	−1.18143	−0.590715	+	0.806880i	$0.701154\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	12.0000i	0.613973i
$383$	21.0000i	1.07305i	0.843884	+	0.536525i	$0.180263\pi$
$383$	21.0000i	1.07305i	−0.843884	+	0.536525i	$0.819737\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	16.0000	0.814379
$387$	− 2.00000i	− 0.101666i
$388$	− 10.0000i	− 0.507673i
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−54.0000	−2.73090
$392$	0	0
$393$	− 6.00000i	− 0.302660i
$394$	15.0000	0.755689
$395$	0	0
$396$	3.00000	0.150756
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	16.0000i	0.802008i
$399$	0	0
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	− 16.0000i	− 0.798007i
$403$	− 8.00000i	− 0.398508i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	0	0
$407$	21.0000i	1.04093i
$408$	− 12.0000i	− 0.594089i
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−24.0000	−1.18383
$412$	4.00000i	0.197066i
$413$	0	0
$414$	9.00000	0.442326
$415$	0	0
$416$	1.00000	0.0490290
$417$	− 8.00000i	− 0.391762i
$418$	3.00000i	0.146735i
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	23.0000i	1.11962i
$423$	9.00000i	0.437595i
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 12.0000i	− 0.580042i
$429$	−6.00000	−0.289683
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	− 4.00000i	− 0.192450i
$433$	− 40.0000i	− 1.92228i	−0.276066	−	0.961139i	$-0.589031\pi$
$433$	− 40.0000i	− 1.92228i	0.276066	−	0.961139i	$-0.410969\pi$
$434$	0	0
$435$	0	0
$436$	−16.0000	−0.766261
$437$	9.00000i	0.430528i
$438$	− 8.00000i	− 0.382255i
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	0	0
$442$	6.00000i	0.285391i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	−14.0000	−0.664411
$445$	0	0
$446$	−8.00000	−0.378811
$447$	− 12.0000i	− 0.567581i
$448$	0	0
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	0	0
$453$	20.0000i	0.939682i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	− 14.0000i	− 0.654892i	−0.944870	−	0.327446i	$-0.893812\pi$
$457$	− 14.0000i	− 0.654892i	0.944870	−	0.327446i	$-0.106188\pi$
$458$	4.00000i	0.186908i
$459$	24.0000	1.12022
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	− 1.00000i	− 0.0464739i	−0.999730	−	0.0232370i	$-0.992603\pi$
$463$	− 1.00000i	− 0.0464739i	0.999730	−	0.0232370i	$-0.00739722\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	6.00000i	0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$	6.00000i	0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	0	0
$470$	0	0
$471$	−46.0000	−2.11957
$472$	0	0
$473$	6.00000i	0.275880i
$474$	20.0000	0.918630
$475$	0	0
$476$	0	0
$477$	− 9.00000i	− 0.412082i
$478$	6.00000i	0.274434i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	− 1.00000i	− 0.0455488i
$483$	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	$0.881918\pi$
$488$	− 8.00000i	− 0.362143i
$489$	40.0000	1.80886
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	6.00000i	0.270501i
$493$	− 36.0000i	− 1.62136i
$494$	1.00000	0.0449921
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	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
$501$	−6.00000	−0.268060
$502$	− 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
$506$	−27.0000	−1.20030
$507$	− 24.0000i	− 1.06588i
$508$	− 1.00000i	− 0.0443678i
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	−4.00000	−0.176090
$517$	− 27.0000i	− 1.18746i
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−15.0000	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$522$	6.00000i	0.262613i
$523$	− 28.0000i	− 1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$	− 28.0000i	− 1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	0	0
$527$	48.0000i	2.09091i
$528$	− 6.00000i	− 0.261116i
$529$	−58.0000	−2.52174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.00000i	− 0.129944i
$534$	−12.0000	−0.519291
$535$	0	0
$536$	−8.00000	−0.345547
$537$	− 6.00000i	− 0.258919i
$538$	0	0
$539$	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
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	12.0000i	0.512615i
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−6.00000	−0.255609
$552$	− 18.0000i	− 0.766131i
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	9.00000i	0.381342i	0.981654	+	0.190671i	$0.0610664\pi$
$557$	9.00000i	0.381342i	−0.981654	+	0.190671i	$0.938934\pi$
$558$	− 8.00000i	− 0.338667i
$559$	2.00000	0.0845910
$560$	0	0
$561$	36.0000	1.51992
$562$	− 27.0000i	− 1.13893i
$563$	42.0000i	1.77009i	0.465506	+	0.885044i	$0.345872\pi$
$563$	42.0000i	1.77009i	−0.465506	+	0.885044i	$0.654128\pi$
$564$	18.0000	0.757937
$565$	0	0
$566$	−14.0000	−0.588464
$567$	0	0
$568$	0	0
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	3.00000i	0.125436i
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 44.0000i	− 1.83174i	−0.401470	−	0.915872i	$-0.631501\pi$
$577$	− 44.0000i	− 1.83174i	0.401470	−	0.915872i	$-0.368499\pi$
$578$	− 19.0000i	− 0.790296i
$579$	−32.0000	−1.32987
$580$	0	0
$581$	0	0
$582$	20.0000i	0.829027i
$583$	27.0000i	1.11823i
$584$	−4.00000	−0.165521
$585$	0	0
$586$	9.00000	0.371787
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−30.0000	−1.23404
$592$	7.00000i	0.287698i
$593$	24.0000i	0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$	24.0000i	0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$	12.0000	0.492366
$595$	0	0
$596$	−6.00000	−0.245770
$597$	− 32.0000i	− 1.30967i
$598$	9.00000i	0.368037i
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	8.00000i	0.325785i
$604$	10.0000	0.406894
$605$	0	0
$606$	24.0000	0.974933
$607$	1.00000i	0.0405887i	0.999794	+	0.0202944i	$0.00646034\pi$
$607$	1.00000i	0.0405887i	−0.999794	+	0.0202944i	$0.993540\pi$
$608$	1.00000i	0.0405554i
$609$	0	0
$610$	0	0
$611$	−9.00000	−0.364101
$612$	6.00000i	0.242536i
$613$	29.0000i	1.17130i	0.810564	+	0.585649i	$0.199160\pi$
$613$	29.0000i	1.17130i	−0.810564	+	0.585649i	$0.800840\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	− 8.00000i	− 0.321807i
$619$	−23.0000	−0.924448	−0.462224	−	0.886763i	$-0.652948\pi$
$619$	−23.0000	−0.924448	−0.462224	+	0.886763i	$0.652948\pi$
$620$	0	0
$621$	36.0000	1.44463
$622$	− 24.0000i	− 0.962312i
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	28.0000	1.11911
$627$	− 6.00000i	− 0.239617i
$628$	23.0000i	0.917800i
$629$	−42.0000	−1.67465
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	− 10.0000i	− 0.397779i
$633$	− 46.0000i	− 1.82834i
$634$	6.00000	0.238290
$635$	0	0
$636$	−18.0000	−0.713746
$637$	0	0
$638$	− 18.0000i	− 0.712627i
$639$	0	0
$640$	0	0
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	24.0000i	0.947204i
$643$	2.00000i	0.0788723i	0.999222	+	0.0394362i	$0.0125562\pi$
$643$	2.00000i	0.0788723i	−0.999222	+	0.0394362i	$0.987444\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	− 33.0000i	− 1.29736i	−0.761060	−	0.648682i	$-0.775321\pi$
$647$	− 33.0000i	− 1.29736i	0.761060	−	0.648682i	$-0.224679\pi$
$648$	11.0000i	0.432121i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	− 9.00000i	− 0.352197i	−0.984373	−	0.176099i	$-0.943652\pi$
$653$	− 9.00000i	− 0.352197i	0.984373	−	0.176099i	$-0.0563478\pi$
$654$	32.0000	1.25130
$655$	0	0
$656$	3.00000	0.117130
$657$	4.00000i	0.156055i
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	− 7.00000i	− 0.272063i
$663$	− 12.0000i	− 0.466041i
$664$	0	0
$665$	0	0
$666$	7.00000	0.271244
$667$	− 54.0000i	− 2.09089i
$668$	3.00000i	0.116073i
$669$	16.0000	0.618596
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	−12.0000	−0.461538
$677$	9.00000i	0.345898i	0.984931	+	0.172949i	$0.0553296\pi$
$677$	9.00000i	0.345898i	−0.984931	+	0.172949i	$0.944670\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	24.0000	0.919682
$682$	24.0000i	0.919007i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	0	0
$687$	− 8.00000i	− 0.305219i
$688$	2.00000i	0.0762493i
$689$	9.00000	0.342873
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	− 9.00000i	− 0.342129i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	12.0000	0.454859
$697$	18.0000i	0.681799i
$698$	− 26.0000i	− 0.984115i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	− 4.00000i	− 0.150970i
$703$	7.00000i	0.264010i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−12.0000	−0.451626
$707$	0	0
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	6.00000i	0.224860i
$713$	72.0000i	2.69642i
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	− 12.0000i	− 0.448148i
$718$	− 18.0000i	− 0.671754i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 18.0000i	− 0.669891i
$723$	2.00000i	0.0743808i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−4.00000	−0.148454
$727$	1.00000i	0.0370879i	0.999828	+	0.0185440i	$0.00590307\pi$
$727$	1.00000i	0.0370879i	−0.999828	+	0.0185440i	$0.994097\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−12.0000	−0.443836
$732$	16.0000i	0.591377i
$733$	− 43.0000i	− 1.58824i	−0.607760	−	0.794121i	$-0.707932\pi$
$733$	− 43.0000i	− 1.58824i	0.607760	−	0.794121i	$-0.292068\pi$
$734$	−19.0000	−0.701303
$735$	0	0
$736$	−9.00000	−0.331744
$737$	− 24.0000i	− 0.884051i
$738$	− 3.00000i	− 0.110432i
$739$	−35.0000	−1.28750	−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000	−1.28750	−0.643748	+	0.765238i	$0.722621\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	− 45.0000i	− 1.65089i	−0.564483	−	0.825445i	$-0.690924\pi$
$743$	− 45.0000i	− 1.65089i	0.564483	−	0.825445i	$-0.309076\pi$
$744$	−16.0000	−0.586588
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	0	0
$748$	− 18.0000i	− 0.658145i
$749$	0	0
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	− 9.00000i	− 0.328196i
$753$	30.0000i	1.09326i
$754$	−6.00000	−0.218507
$755$	0	0
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	− 23.0000i	− 0.835398i
$759$	54.0000	1.96008
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	2.00000i	0.0724524i
$763$	0	0
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−21.0000	−0.758761
$767$	0	0
$768$	− 2.00000i	− 0.0721688i
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	0	0
$772$	16.0000i	0.575853i
$773$	− 51.0000i	− 1.83434i	−0.398493	−	0.917171i	$-0.630467\pi$
$773$	− 51.0000i	− 1.83434i	0.398493	−	0.917171i	$-0.369533\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	12.0000i	0.430221i
$779$	3.00000	0.107486
$780$	0	0
$781$	0	0
$782$	− 54.0000i	− 1.93104i
$783$	24.0000i	0.857690i
$784$	0	0
$785$	0	0
$786$	6.00000	0.214013
$787$	22.0000i	0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$	22.0000i	0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$	15.0000i	0.534353i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	3.00000i	0.106600i
$793$	− 8.00000i	− 0.284088i
$794$	14.0000	0.496841
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	54.0000	1.91038
$800$	0	0
$801$	6.00000	0.212000
$802$	− 27.0000i	− 0.953403i
$803$	− 12.0000i	− 0.423471i
$804$	16.0000	0.564276
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	− 12.0000i	− 0.422159i
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	0	0
$811$	−25.0000	−0.877869	−0.438934	−	0.898519i	$-0.644644\pi$
$811$	−25.0000	−0.877869	−0.438934	+	0.898519i	$0.644644\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	−21.0000	−0.736050
$815$	0	0
$816$	12.0000	0.420084
$817$	2.00000i	0.0699711i
$818$	− 26.0000i	− 0.909069i
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	− 24.0000i	− 0.837096i
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	− 6.00000i	− 0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$	− 6.00000i	− 0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$	9.00000i	0.312772i
$829$	−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
$831$	20.0000	0.693792
$832$	1.00000i	0.0346688i
$833$	0	0
$834$	8.00000	0.277017
$835$	0	0
$836$	−3.00000	−0.103757
$837$	− 32.0000i	− 1.10608i
$838$	9.00000i	0.310900i
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	2.00000i	0.0689246i
$843$	54.0000i	1.85986i
$844$	−23.0000	−0.791693
$845$	0	0
$846$	−9.00000	−0.309426
$847$	0	0
$848$	9.00000i	0.309061i
$849$	28.0000	0.960958
$850$	0	0
$851$	−63.0000	−2.15961
$852$	0	0
$853$	− 19.0000i	− 0.650548i	−0.945620	−	0.325274i	$-0.894544\pi$
$853$	− 19.0000i	− 0.650548i	0.945620	−	0.325274i	$-0.105456\pi$
$854$	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	$0.900530\pi$
$858$	− 6.00000i	− 0.204837i
$859$	−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
$862$	12.0000i	0.408722i
$863$	− 3.00000i	− 0.102121i	−0.998696	−	0.0510606i	$-0.983740\pi$
$863$	− 3.00000i	− 0.102121i	0.998696	−	0.0510606i	$-0.0162602\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	40.0000	1.35926
$867$	38.0000i	1.29055i
$868$	0	0
$869$	30.0000	1.01768
$870$	0	0
$871$	−8.00000	−0.271070
$872$	− 16.0000i	− 0.541828i
$873$	− 10.0000i	− 0.338449i
$874$	−9.00000	−0.304430
$875$	0	0
$876$	8.00000	0.270295
$877$	13.0000i	0.438979i	0.975615	+	0.219489i	$0.0704391\pi$
$877$	13.0000i	0.438979i	−0.975615	+	0.219489i	$0.929561\pi$
$878$	− 26.0000i	− 0.877457i
$879$	−18.0000	−0.607125
$880$	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	$0.957022\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	−12.0000	−0.403148
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	− 14.0000i	− 0.469809i
$889$	0	0
$890$	0	0
$891$	−33.0000	−1.10554
$892$	− 8.00000i	− 0.267860i
$893$	− 9.00000i	− 0.301174i
$894$	12.0000	0.401340
$895$	0	0
$896$	0	0
$897$	− 18.0000i	− 0.601003i
$898$	− 21.0000i	− 0.700779i
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−54.0000	−1.79900
$902$	9.00000i	0.299667i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−20.0000	−0.664455
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−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
$913$	0	0
$914$	14.0000	0.463079
$915$	0	0
$916$	−4.00000	−0.132164
$917$	0	0
$918$	24.0000i	0.792118i
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	1.00000	0.0328620
$927$	4.00000i	0.131377i
$928$	− 6.00000i	− 0.196960i
$929$	−57.0000	−1.87011	−0.935055	−	0.354504i	$-0.884650\pi$
$929$	−57.0000	−1.87011	−0.935055	+	0.354504i	$0.884650\pi$
$930$	0	0
$931$	0	0
$932$	6.00000i	0.196537i
$933$	48.0000i	1.57145i
$934$	−6.00000	−0.196326
$935$	0	0
$936$	1.00000	0.0326860
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	0	0
$939$	−56.0000	−1.82749
$940$	0	0
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	− 46.0000i	− 1.49876i
$943$	27.0000i	0.879241i
$944$	0	0
$945$	0	0
$946$	−6.00000	−0.195077
$947$	− 6.00000i	− 0.194974i	−0.995237	−	0.0974869i	$-0.968920\pi$
$947$	− 6.00000i	− 0.194974i	0.995237	−	0.0974869i	$-0.0310804\pi$
$948$	20.0000i	0.649570i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	36.0000i	1.16615i	0.812417	+	0.583077i	$0.198151\pi$
$953$	36.0000i	1.16615i	−0.812417	+	0.583077i	$0.801849\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	−6.00000	−0.194054
$957$	36.0000i	1.16371i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	7.00000i	0.225689i
$963$	− 12.0000i	− 0.386695i
$964$	1.00000	0.0322078
$965$	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
$971$	−45.0000	−1.44412	−0.722059	−	0.691831i	$-0.756804\pi$
$971$	−45.0000	−1.44412	−0.722059	+	0.691831i	$0.756804\pi$
$972$	− 10.0000i	− 0.320750i
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	8.00000	0.256074
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	40.0000i	1.27906i
$979$	−18.0000	−0.575282
$980$	0	0
$981$	−16.0000	−0.510841
$982$	36.0000i	1.14881i
$983$	3.00000i	0.0956851i	0.998855	+	0.0478426i	$0.0152346\pi$
$983$	3.00000i	0.0956851i	−0.998855	+	0.0478426i	$0.984765\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	36.0000	1.14647
$987$	0	0
$988$	1.00000i	0.0318142i
$989$	−18.0000	−0.572367
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	8.00000i	0.254000i
$993$	14.0000i	0.444277i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	22.0000i	0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$	22.0000i	0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$	4.00000i	0.126618i
$999$	28.0000	0.885881

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +3.00000i q^{22} +9.00000i q^{23} -2.00000 q^{24} +1.00000 q^{26} -4.00000i q^{27} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +7.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} +3.00000 q^{41} +2.00000i q^{43} -3.00000 q^{44} -9.00000 q^{46} -9.00000i q^{47} -2.00000i q^{48} +12.0000 q^{51} +1.00000i q^{52} +9.00000i q^{53} +4.00000 q^{54} -2.00000i q^{57} -6.00000i q^{58} +8.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} +18.0000 q^{69} +1.00000i q^{72} -4.00000i q^{73} -7.00000 q^{74} -1.00000 q^{76} -2.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +3.00000i q^{82} -2.00000 q^{86} +12.0000i q^{87} -3.00000i q^{88} -6.00000 q^{89} -9.00000i q^{92} -16.0000i q^{93} +9.00000 q^{94} +2.00000 q^{96} +10.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 6 q^{11} + 2 q^{16} + 2 q^{19} - 4 q^{24} + 2 q^{26} - 12 q^{29} + 16 q^{31} - 12 q^{34} + 2 q^{36} - 4 q^{39} + 6 q^{41} - 6 q^{44} - 18 q^{46} + 24 q^{51} + 8 q^{54} + 16 q^{61} - 2 q^{64} + 12 q^{66} + 36 q^{69} - 14 q^{74} - 2 q^{76} + 20 q^{79} - 22 q^{81} - 4 q^{86} - 12 q^{89} + 18 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 6 * q^11 + 2 * q^16 + 2 * q^19 - 4 * q^24 + 2 * q^26 - 12 * q^29 + 16 * q^31 - 12 * q^34 + 2 * q^36 - 4 * q^39 + 6 * q^41 - 6 * q^44 - 18 * q^46 + 24 * q^51 + 8 * q^54 + 16 * q^61 - 2 * q^64 + 12 * q^66 + 36 * q^69 - 14 * q^74 - 2 * q^76 + 20 * q^79 - 22 * q^81 - 4 * q^86 - 12 * q^89 + 18 * q^94 + 4 * q^96 - 6 * q^99

Introduction

L-functions

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