LMFDB - Embedded newform 1650.2.c.l.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1650,2,Mod(199,1650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1650.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1650.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:	$13.1753163335$
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 330)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1650.199
Dual form			1650.2.c.l.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} -1.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} +1.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +8.00000i q^{38} +2.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} -1.00000 q^{44} -4.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} +2.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} -8.00000i q^{57} +2.00000i q^{58} -4.00000 q^{59} -6.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +1.00000 q^{66} -12.0000i q^{67} +2.00000i q^{68} +4.00000 q^{69} -12.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -2.00000 q^{74} +8.00000 q^{76} -2.00000i q^{78} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} -8.00000 q^{86} -2.00000i q^{87} +1.00000i q^{88} +6.00000 q^{89} +4.00000i q^{92} +8.00000i q^{93} -4.00000 q^{94} +1.00000 q^{96} -14.0000i q^{97} -7.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 2 q^{11} + 2 q^{16} - 16 q^{19} - 2 q^{24} - 4 q^{26} - 4 q^{29} + 16 q^{31} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 12 q^{41} - 2 q^{44} - 8 q^{46} + 14 q^{49} + 4 q^{51} - 2 q^{54} - 8 q^{59} - 12 q^{61} - 2 q^{64} + 2 q^{66} + 8 q^{69} - 24 q^{71} - 4 q^{74} + 16 q^{76} + 2 q^{81} - 16 q^{86} + 12 q^{89} - 8 q^{94} + 2 q^{96} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 2 * q^11 + 2 * q^16 - 16 * q^19 - 2 * q^24 - 4 * q^26 - 4 * q^29 + 16 * q^31 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 12 * q^41 - 2 * q^44 - 8 * q^46 + 14 * q^49 + 4 * q^51 - 2 * q^54 - 8 * q^59 - 12 * q^61 - 2 * q^64 + 2 * q^66 + 8 * q^69 - 24 * q^71 - 4 * q^74 + 16 * q^76 + 2 * q^81 - 16 * q^86 + 12 * q^89 - 8 * q^94 + 2 * q^96 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times$.

$n$	$551$	$727$	$1201$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	− 1.00000i	− 0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	8.00000	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
$33$	1.00000i	0.174078i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	8.00000i	1.29777i
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	2.00000	0.280056
$52$	2.00000i	0.277350i
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	− 8.00000i	− 1.05963i
$58$	2.00000i	0.262613i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	2.00000i	0.242536i
$69$	4.00000	0.481543
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	8.00000	0.917663
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	− 2.00000i	− 0.214423i
$88$	1.00000i	0.106600i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	4.00000i	0.417029i
$93$	8.00000i	0.829561i
$94$	−4.00000	−0.412568
$95$	0	0
$96$	1.00000	0.102062
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	− 7.00000i	− 0.707107i
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−2.00000	−0.194257
$107$	20.0000i	1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$	20.0000i	1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$	1.00000i	0.0962250i
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	2.00000	0.185695
$117$	2.00000i	0.184900i
$118$	4.00000i	0.368230i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.00000i	0.543214i
$123$	6.00000i	0.541002i
$124$	−8.00000	−0.718421
$125$	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
$129$	8.00000	0.704361
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	− 1.00000i	− 0.0870388i
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	2.00000	0.171499
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	− 4.00000i	− 0.340503i
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	12.0000i	1.00702i
$143$	− 2.00000i	− 0.167248i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	7.00000i	0.577350i
$148$	2.00000i	0.164399i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	− 8.00000i	− 0.648886i
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	8.00000	0.611775
$172$	8.00000i	0.609994i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	1.00000	0.0753778
$177$	− 4.00000i	− 0.300658i
$178$	− 6.00000i	− 0.449719i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	− 6.00000i	− 0.443533i
$184$	4.00000	0.294884
$185$	0	0
$186$	8.00000	0.586588
$187$	− 2.00000i	− 0.146254i
$188$	4.00000i	0.291730i
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	22.0000i	1.58359i	0.610784	+	0.791797i	$0.290854\pi$
$193$	22.0000i	1.58359i	−0.610784	+	0.791797i	$0.709146\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	1.00000i	0.0710669i
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	6.00000i	0.422159i
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−16.0000	−1.11477
$207$	4.00000i	0.278019i
$208$	− 2.00000i	− 0.138675i
$209$	−8.00000	−0.553372
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	2.00000i	0.137361i
$213$	− 12.0000i	− 0.822226i
$214$	20.0000	1.36717
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	− 6.00000i	− 0.406371i
$219$	2.00000	0.135147
$220$	0	0
$221$	−4.00000	−0.269069
$222$	− 2.00000i	− 0.134231i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	8.00000i	0.529813i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	− 2.00000i	− 0.131306i
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	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
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	1.00000i	0.0641500i
$244$	6.00000	0.384111
$245$	0	0
$246$	6.00000	0.382546
$247$	16.0000i	1.01806i
$248$	8.00000i	0.508001i
$249$	4.00000	0.253490
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.0000i	1.62184i	0.585160	+	0.810918i	$0.301032\pi$
$257$	26.0000i	1.62184i	−0.585160	+	0.810918i	$0.698968\pi$
$258$	− 8.00000i	− 0.498058i
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	− 12.0000i	− 0.741362i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	0	0
$267$	6.00000i	0.367194i
$268$	12.0000i	0.733017i
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	2.00000	0.120824
$275$	0	0
$276$	−4.00000	−0.240772
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	16.0000i	0.959616i
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	− 4.00000i	− 0.238197i
$283$	8.00000i	0.475551i	0.971320	+	0.237775i	$0.0764182\pi$
$283$	8.00000i	0.475551i	−0.971320	+	0.237775i	$0.923582\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	14.0000	0.820695
$292$	2.00000i	0.117041i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	2.00000	0.116248
$297$	− 1.00000i	− 0.0580259i
$298$	10.0000i	0.579284i
$299$	−8.00000	−0.462652
$300$	0	0
$301$	0	0
$302$	8.00000i	0.460348i
$303$	− 6.00000i	− 0.344691i
$304$	−8.00000	−0.458831
$305$	0	0
$306$	2.00000	0.114332
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	2.00000i	0.113228i
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	− 2.00000i	− 0.112154i
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−20.0000	−1.11629
$322$	0	0
$323$	16.0000i	0.890264i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	6.00000i	0.331801i
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	4.00000i	0.219529i
$333$	2.00000i	0.109599i
$334$	16.0000	0.875481
$335$	0	0
$336$	0	0
$337$	10.0000i	0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$	10.0000i	0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$	− 9.00000i	− 0.489535i
$339$	2.00000	0.108625
$340$	0	0
$341$	8.00000	0.433224
$342$	− 8.00000i	− 0.432590i
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	2.00000i	0.107211i
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	− 1.00000i	− 0.0533002i
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	2.00000i	0.105118i
$363$	1.00000i	0.0524864i
$364$	0	0
$365$	0	0
$366$	−6.00000	−0.313625
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	− 8.00000i	− 0.414781i
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	4.00000	0.206284
$377$	4.00000i	0.206010i
$378$	0	0
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	12.0000i	0.613973i
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	22.0000	1.11977
$387$	8.00000i	0.406663i
$388$	14.0000i	0.710742i
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	7.00000i	0.353553i
$393$	12.0000i	0.605320i
$394$	10.0000	0.503793
$395$	0	0
$396$	1.00000	0.0502519
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	− 24.0000i	− 1.20301i
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	− 12.0000i	− 0.598506i
$403$	− 16.0000i	− 0.797017i
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	− 2.00000i	− 0.0991363i
$408$	2.00000i	0.0990148i
$409$	−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
$411$	−2.00000	−0.0986527
$412$	16.0000i	0.788263i
$413$	0	0
$414$	4.00000	0.196589
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 16.0000i	− 0.783523i
$418$	8.00000i	0.391293i
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	0	0
$423$	4.00000i	0.194487i
$424$	2.00000	0.0971286
$425$	0	0
$426$	−12.0000	−0.581402
$427$	0	0
$428$	− 20.0000i	− 0.966736i
$429$	2.00000	0.0965609
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	32.0000i	1.53077i
$438$	− 2.00000i	− 0.0955637i
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	4.00000i	0.190261i
$443$	28.0000i	1.33032i	0.746701	+	0.665160i	$0.231637\pi$
$443$	28.0000i	1.33032i	−0.746701	+	0.665160i	$0.768363\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	16.0000	0.757622
$447$	− 10.0000i	− 0.472984i
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	2.00000i	0.0940721i
$453$	− 8.00000i	− 0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	8.00000	0.374634
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	14.0000i	0.654177i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	18.0000	0.833834
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	− 4.00000i	− 0.184115i
$473$	− 8.00000i	− 0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	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
$481$	−4.00000	−0.182384
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	− 6.00000i	− 0.271607i
$489$	20.0000	0.904431
$490$	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
$493$	4.00000i	0.180151i
$494$	16.0000	0.719874
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	− 4.00000i	− 0.179244i
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	20.0000i	0.892644i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	9.00000i	0.399704i
$508$	16.0000i	0.709885i
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	8.00000i	0.353209i
$514$	26.0000	1.14681
$515$	0	0
$516$	−8.00000	−0.352180
$517$	− 4.00000i	− 0.175920i
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	− 24.0000i	− 1.04945i	−0.851273	−	0.524723i	$-0.824169\pi$
$523$	− 24.0000i	− 1.04945i	0.851273	−	0.524723i	$-0.175831\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	− 16.0000i	− 0.696971i
$528$	1.00000i	0.0435194i
$529$	7.00000	0.304348
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	6.00000	0.259645
$535$	0	0
$536$	12.0000	0.518321
$537$	12.0000i	0.517838i
$538$	− 22.0000i	− 0.948487i
$539$	7.00000	0.301511
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	0	0
$543$	− 2.00000i	− 0.0858282i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	6.00000	0.256074
$550$	0	0
$551$	16.0000	0.681623
$552$	4.00000i	0.170251i
$553$	0	0
$554$	−14.0000	−0.594803
$555$	0	0
$556$	16.0000	0.678551
$557$	− 46.0000i	− 1.94908i	−0.224208	−	0.974541i	$-0.571980\pi$
$557$	− 46.0000i	− 1.94908i	0.224208	−	0.974541i	$-0.428020\pi$
$558$	8.00000i	0.338667i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	2.00000	0.0844401
$562$	10.0000i	0.421825i
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	8.00000	0.336265
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	2.00000i	0.0836242i
$573$	− 12.0000i	− 0.501307i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−22.0000	−0.914289
$580$	0	0
$581$	0	0
$582$	− 14.0000i	− 0.580319i
$583$	− 2.00000i	− 0.0828315i
$584$	2.00000	0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	− 7.00000i	− 0.288675i
$589$	−64.0000	−2.63707
$590$	0	0
$591$	−10.0000	−0.411345
$592$	− 2.00000i	− 0.0821995i
$593$	26.0000i	1.06769i	0.845582	+	0.533846i	$0.179254\pi$
$593$	26.0000i	1.06769i	−0.845582	+	0.533846i	$0.820746\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	10.0000	0.409616
$597$	24.0000i	0.982255i
$598$	8.00000i	0.327144i
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	8.00000	0.325515
$605$	0	0
$606$	−6.00000	−0.243733
$607$	16.0000i	0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$	16.0000i	0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$	8.00000i	0.324443i
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	− 2.00000i	− 0.0808452i
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	− 16.0000i	− 0.643614i
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	20.0000i	0.801927i
$623$	0	0
$624$	2.00000	0.0800641
$625$	0	0
$626$	6.00000	0.239808
$627$	− 8.00000i	− 0.319489i
$628$	2.00000i	0.0798087i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	−6.00000	−0.238290
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	− 14.0000i	− 0.554700i
$638$	2.00000i	0.0791808i
$639$	12.0000	0.474713
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	20.0000i	0.789337i
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	0	0
$645$	0	0
$646$	16.0000	0.629512
$647$	− 44.0000i	− 1.72982i	−0.501928	−	0.864909i	$-0.667376\pi$
$647$	− 44.0000i	− 1.72982i	0.501928	−	0.864909i	$-0.332624\pi$
$648$	1.00000i	0.0392837i
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	20.0000i	0.783260i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	6.00000	0.234261
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	12.0000i	0.466393i
$663$	− 4.00000i	− 0.155347i
$664$	4.00000	0.155230
$665$	0	0
$666$	2.00000	0.0774984
$667$	8.00000i	0.309761i
$668$	− 16.0000i	− 0.619059i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	6.00000i	0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$	6.00000i	0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	− 2.00000i	− 0.0768095i
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	− 8.00000i	− 0.306336i
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	0	0
$687$	− 14.0000i	− 0.534133i
$688$	− 8.00000i	− 0.304997i
$689$	−4.00000	−0.152388
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	18.0000i	0.684257i
$693$	0	0
$694$	−20.0000	−0.759190
$695$	0	0
$696$	2.00000	0.0758098
$697$	− 12.0000i	− 0.454532i
$698$	− 30.0000i	− 1.13552i
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	2.00000i	0.0754851i
$703$	16.0000i	0.603451i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−26.0000	−0.978523
$707$	0	0
$708$	4.00000i	0.150329i
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	− 32.0000i	− 1.19841i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	− 16.0000i	− 0.597531i
$718$	− 32.0000i	− 1.19423i
$719$	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
$722$	− 45.0000i	− 1.67473i
$723$	18.0000i	0.669427i
$724$	2.00000	0.0743294
$725$	0	0
$726$	1.00000	0.0371135
$727$	− 40.0000i	− 1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$	− 40.0000i	− 1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	6.00000i	0.221766i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 12.0000i	− 0.442026i
$738$	6.00000i	0.220863i
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	0	0
$741$	−16.0000	−0.587775
$742$	0	0
$743$	8.00000i	0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$	8.00000i	0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	4.00000i	0.146352i
$748$	2.00000i	0.0731272i
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	− 4.00000i	− 0.145865i
$753$	− 20.0000i	− 0.728841i
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	36.0000i	1.30758i
$759$	4.00000	0.145191
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	− 16.0000i	− 0.579619i
$763$	0	0
$764$	12.0000	0.434145
$765$	0	0
$766$	−12.0000	−0.433578
$767$	8.00000i	0.288863i
$768$	1.00000i	0.0360844i
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−26.0000	−0.936367
$772$	− 22.0000i	− 0.791797i
$773$	− 34.0000i	− 1.22290i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	− 34.0000i	− 1.22290i	0.791285	−	0.611448i	$-0.209412\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	− 22.0000i	− 0.788738i
$779$	−48.0000	−1.71978
$780$	0	0
$781$	−12.0000	−0.429394
$782$	8.00000i	0.286079i
$783$	2.00000i	0.0714742i
$784$	7.00000	0.250000
$785$	0	0
$786$	12.0000	0.428026
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 10.0000i	− 0.356235i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	0	0
$792$	− 1.00000i	− 0.0355335i
$793$	12.0000i	0.426132i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−24.0000	−0.850657
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 18.0000i	− 0.635602i
$803$	− 2.00000i	− 0.0705785i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−16.0000	−0.563576
$807$	22.0000i	0.774437i
$808$	− 6.00000i	− 0.211079i
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	0	0
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	2.00000	0.0700140
$817$	64.0000i	2.23908i
$818$	26.0000i	0.909069i
$819$	0	0
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	2.00000i	0.0697580i
$823$	24.0000i	0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$	24.0000i	0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	− 4.00000i	− 0.139010i
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	2.00000i	0.0693375i
$833$	− 14.0000i	− 0.485071i
$834$	−16.0000	−0.554035
$835$	0	0
$836$	8.00000	0.276686
$837$	− 8.00000i	− 0.276520i
$838$	4.00000i	0.138178i
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 38.0000i	− 1.30957i
$843$	− 10.0000i	− 0.344418i
$844$	0	0
$845$	0	0
$846$	4.00000	0.137523
$847$	0	0
$848$	− 2.00000i	− 0.0686803i
$849$	−8.00000	−0.274559
$850$	0	0
$851$	−8.00000	−0.274236
$852$	12.0000i	0.411113i
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	0	0
$856$	−20.0000	−0.683586
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	− 2.00000i	− 0.0682789i
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	24.0000i	0.817443i
$863$	− 4.00000i	− 0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$	− 4.00000i	− 0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	13.0000i	0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	6.00000i	0.203186i
$873$	14.0000i	0.473828i
$874$	32.0000	1.08242
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	− 16.0000i	− 0.539974i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	58.0000	1.95407	0.977035	−	0.213080i	$-0.0683494\pi$
$881$	58.0000	1.95407	0.977035	+	0.213080i	$0.0683494\pi$
$882$	7.00000i	0.235702i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	28.0000	0.940678
$887$	40.0000i	1.34307i	0.740973	+	0.671534i	$0.234364\pi$
$887$	40.0000i	1.34307i	−0.740973	+	0.671534i	$0.765636\pi$
$888$	2.00000i	0.0671156i
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	− 16.0000i	− 0.535720i
$893$	32.0000i	1.07084i
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	− 8.00000i	− 0.267112i
$898$	10.0000i	0.333704i
$899$	−16.0000	−0.533630
$900$	0	0
$901$	−4.00000	−0.133259
$902$	− 6.00000i	− 0.199778i
$903$	0	0
$904$	2.00000	0.0665190
$905$	0	0
$906$	−8.00000	−0.265782
$907$	52.0000i	1.72663i	0.504664	+	0.863316i	$0.331616\pi$
$907$	52.0000i	1.72663i	−0.504664	+	0.863316i	$0.668384\pi$
$908$	− 12.0000i	− 0.398234i
$909$	6.00000	0.199007
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	− 8.00000i	− 0.264906i
$913$	− 4.00000i	− 0.132381i
$914$	−38.0000	−1.25693
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	0	0
$922$	− 18.0000i	− 0.592798i
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	16.0000i	0.525509i
$928$	2.00000i	0.0656532i
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	−56.0000	−1.83533
$932$	− 18.0000i	− 0.589610i
$933$	− 20.0000i	− 0.654771i
$934$	36.0000	1.17796
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 54.0000i	− 1.76410i	−0.471153	−	0.882052i	$-0.656162\pi$
$937$	− 54.0000i	− 1.76410i	0.471153	−	0.882052i	$-0.343838\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	− 24.0000i	− 0.781548i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−8.00000	−0.260102
$947$	28.0000i	0.909878i	0.890523	+	0.454939i	$0.150339\pi$
$947$	28.0000i	0.909878i	−0.890523	+	0.454939i	$0.849661\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	− 38.0000i	− 1.23094i	−0.788160	−	0.615470i	$-0.788966\pi$
$953$	− 38.0000i	− 1.23094i	0.788160	−	0.615470i	$-0.211034\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	16.0000	0.517477
$957$	− 2.00000i	− 0.0646508i
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	4.00000i	0.128965i
$963$	− 20.0000i	− 0.644491i
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	1.00000i	0.0321412i
$969$	−16.0000	−0.513994
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	8.00000	0.256337
$975$	0	0
$976$	−6.00000	−0.192055
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	− 20.0000i	− 0.639529i
$979$	6.00000	0.191761
$980$	0	0
$981$	−6.00000	−0.191565
$982$	12.0000i	0.382935i
$983$	− 4.00000i	− 0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$	− 4.00000i	− 0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	4.00000	0.127386
$987$	0	0
$988$	− 16.0000i	− 0.509028i
$989$	−32.0000	−1.01754
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	− 8.00000i	− 0.254000i
$993$	− 12.0000i	− 0.380808i
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	4.00000i	0.126618i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1650.2.c.l.199.1		2
3.2	odd	2		4950.2.c.g.199.2		2
5.2	odd	4		1650.2.a.r.1.1		1
5.3	odd	4		330.2.a.a.1.1	✓	1
5.4	even	2	inner	1650.2.c.l.199.2		2
15.2	even	4		4950.2.a.k.1.1		1
15.8	even	4		990.2.a.j.1.1		1
15.14	odd	2		4950.2.c.g.199.1		2
20.3	even	4		2640.2.a.n.1.1		1
55.43	even	4		3630.2.a.n.1.1		1
60.23	odd	4		7920.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
330.2.a.a.1.1	✓	1	5.3	odd	4
990.2.a.j.1.1		1	15.8	even	4
1650.2.a.r.1.1		1	5.2	odd	4
1650.2.c.l.199.1		2	1.1	even	1	trivial
1650.2.c.l.199.2		2	5.4	even	2	inner
2640.2.a.n.1.1		1	20.3	even	4
3630.2.a.n.1.1		1	55.43	even	4
4950.2.a.k.1.1		1	15.2	even	4
4950.2.c.g.199.1		2	15.14	odd	2
4950.2.c.g.199.2		2	3.2	odd	2
7920.2.a.bb.1.1		1	60.23	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:	\( N \)	\(=\)	\( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1650.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} -1.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} +1.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +8.00000i q^{38} +2.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} -1.00000 q^{44} -4.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} +2.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} -8.00000i q^{57} +2.00000i q^{58} -4.00000 q^{59} -6.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +1.00000 q^{66} -12.0000i q^{67} +2.00000i q^{68} +4.00000 q^{69} -12.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -2.00000 q^{74} +8.00000 q^{76} -2.00000i q^{78} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} -8.00000 q^{86} -2.00000i q^{87} +1.00000i q^{88} +6.00000 q^{89} +4.00000i q^{92} +8.00000i q^{93} -4.00000 q^{94} +1.00000 q^{96} -14.0000i q^{97} -7.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 2 q^{11} + 2 q^{16} - 16 q^{19} - 2 q^{24} - 4 q^{26} - 4 q^{29} + 16 q^{31} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 12 q^{41} - 2 q^{44} - 8 q^{46} + 14 q^{49} + 4 q^{51} - 2 q^{54} - 8 q^{59} - 12 q^{61} - 2 q^{64} + 2 q^{66} + 8 q^{69} - 24 q^{71} - 4 q^{74} + 16 q^{76} + 2 q^{81} - 16 q^{86} + 12 q^{89} - 8 q^{94} + 2 q^{96} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 2 * q^11 + 2 * q^16 - 16 * q^19 - 2 * q^24 - 4 * q^26 - 4 * q^29 + 16 * q^31 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 12 * q^41 - 2 * q^44 - 8 * q^46 + 14 * q^49 + 4 * q^51 - 2 * q^54 - 8 * q^59 - 12 * q^61 - 2 * q^64 + 2 * q^66 + 8 * q^69 - 24 * q^71 - 4 * q^74 + 16 * q^76 + 2 * q^81 - 16 * q^86 + 12 * q^89 - 8 * q^94 + 2 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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