LMFDB - Embedded newform 650.2.b.h.599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,2,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, 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("650.599");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	650.b (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:	$5.19027613138$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.2.b.h.599.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +2.00000 q^{21} +3.00000i q^{22} -6.00000i q^{23} -2.00000 q^{24} -1.00000 q^{26} -4.00000i q^{27} -1.00000i q^{28} +3.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +4.00000i q^{38} +2.00000 q^{39} +2.00000i q^{42} -10.0000i q^{43} -3.00000 q^{44} +6.00000 q^{46} +3.00000i q^{47} -2.00000i q^{48} +6.00000 q^{49} -6.00000 q^{51} -1.00000i q^{52} +3.00000i q^{53} +4.00000 q^{54} +1.00000 q^{56} -8.00000i q^{57} +3.00000i q^{58} +15.0000 q^{59} -13.0000 q^{61} -1.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +13.0000i q^{67} +3.00000i q^{68} -12.0000 q^{69} +1.00000i q^{72} -10.0000i q^{73} +2.00000 q^{74} -4.00000 q^{76} +3.00000i q^{77} +2.00000i q^{78} +4.00000 q^{79} -11.0000 q^{81} +15.0000i q^{83} -2.00000 q^{84} +10.0000 q^{86} -6.00000i q^{87} -3.00000i q^{88} -6.00000 q^{89} -1.00000 q^{91} +6.00000i q^{92} +2.00000i q^{93} -3.00000 q^{94} +2.00000 q^{96} +4.00000i q^{97} +6.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{14} + 2 q^{16} + 8 q^{19} + 4 q^{21} - 4 q^{24} - 2 q^{26} + 6 q^{29} - 2 q^{31} + 6 q^{34} + 2 q^{36} + 4 q^{39} - 6 q^{44} + 12 q^{46} + 12 q^{49} - 12 q^{51} + 8 q^{54} + 2 q^{56} + 30 q^{59} - 26 q^{61} - 2 q^{64} + 12 q^{66} - 24 q^{69} + 4 q^{74} - 8 q^{76} + 8 q^{79} - 22 q^{81} - 4 q^{84} + 20 q^{86} - 12 q^{89} - 2 q^{91} - 6 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 6 * q^11 - 2 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^21 - 4 * q^24 - 2 * q^26 + 6 * q^29 - 2 * q^31 + 6 * q^34 + 2 * q^36 + 4 * q^39 - 6 * q^44 + 12 * q^46 + 12 * q^49 - 12 * q^51 + 8 * q^54 + 2 * q^56 + 30 * q^59 - 26 * q^61 - 2 * q^64 + 12 * q^66 - 24 * q^69 + 4 * q^74 - 8 * q^76 + 8 * q^79 - 22 * q^81 - 4 * q^84 + 20 * q^86 - 12 * q^89 - 2 * q^91 - 6 * q^94 + 4 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	− 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
$13$	1.00000i	0.277350i
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	− 1.00000i	− 0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	3.00000i	0.639602i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	−1.00000	−0.196116
$27$	− 4.00000i	− 0.769800i
$28$	− 1.00000i	− 0.188982i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	1.00000i	0.176777i
$33$	− 6.00000i	− 1.04447i
$34$	3.00000	0.514496
$35$	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	$-0.0525685\pi$
$38$	4.00000i	0.648886i
$39$	2.00000	0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	2.00000i	0.308607i
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	6.00000	0.884652
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	− 2.00000i	− 0.288675i
$49$	6.00000	0.857143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	− 1.00000i	− 0.138675i
$53$	3.00000i	0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$	3.00000i	0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	− 8.00000i	− 1.05963i
$58$	3.00000i	0.393919i
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	− 1.00000i	− 0.127000i
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	13.0000i	1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$	13.0000i	1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$	3.00000i	0.363803i
$69$	−12.0000	−1.44463
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000i	0.117851i
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	3.00000i	0.341882i
$78$	2.00000i	0.226455i
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	15.0000i	1.64646i	0.567705	+	0.823232i	$0.307831\pi$
$83$	15.0000i	1.64646i	−0.567705	+	0.823232i	$0.692169\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	10.0000	1.07833
$87$	− 6.00000i	− 0.643268i
$88$	− 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
$91$	−1.00000	−0.104828
$92$	6.00000i	0.625543i
$93$	2.00000i	0.207390i
$94$	−3.00000	−0.309426
$95$	0	0
$96$	2.00000	0.204124
$97$	4.00000i	0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$	4.00000i	0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$	6.00000i	0.606092i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	− 6.00000i	− 0.594089i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−3.00000	−0.291386
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	4.00000i	0.384900i
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	1.00000i	0.0944911i
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	−3.00000	−0.278543
$117$	− 1.00000i	− 0.0924500i
$118$	15.0000i	1.38086i
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 13.0000i	− 1.17696i
$123$	0	0
$124$	1.00000	0.0898027
$125$	0	0
$126$	1.00000	0.0890871
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−20.0000	−1.76090
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	6.00000i	0.522233i
$133$	4.00000i	0.346844i
$134$	−13.0000	−1.12303
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	− 12.0000i	− 1.02151i
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	3.00000i	0.250873i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	− 12.0000i	− 0.989743i
$148$	2.00000i	0.164399i
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	− 4.00000i	− 0.324443i
$153$	3.00000i	0.242536i
$154$	−3.00000	−0.241747
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 5.00000i	− 0.399043i	−0.979893	−	0.199522i	$-0.936061\pi$
$157$	− 5.00000i	− 0.399043i	0.979893	−	0.199522i	$-0.0639388\pi$
$158$	4.00000i	0.318223i
$159$	6.00000	0.475831
$160$	0	0
$161$	6.00000	0.472866
$162$	− 11.0000i	− 0.864242i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	−15.0000	−1.16423
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−4.00000	−0.305888
$172$	10.0000i	0.762493i
$173$	− 15.0000i	− 1.14043i	−0.821496	−	0.570214i	$-0.806860\pi$
$173$	− 15.0000i	− 1.14043i	0.821496	−	0.570214i	$-0.193140\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	3.00000	0.226134
$177$	− 30.0000i	− 2.25494i
$178$	− 6.00000i	− 0.449719i
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	− 1.00000i	− 0.0741249i
$183$	26.0000i	1.92198i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−2.00000	−0.146647
$187$	− 9.00000i	− 0.658145i
$188$	− 3.00000i	− 0.218797i
$189$	4.00000	0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	2.00000i	0.144338i
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	−6.00000	−0.428571
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	− 3.00000i	− 0.213201i
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	26.0000	1.83390
$202$	9.00000i	0.633238i
$203$	3.00000i	0.210559i
$204$	6.00000	0.420084
$205$	0	0
$206$	−8.00000	−0.557386
$207$	6.00000i	0.417029i
$208$	1.00000i	0.0693375i
$209$	12.0000	0.830057
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	− 3.00000i	− 0.206041i
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	−4.00000	−0.272166
$217$	− 1.00000i	− 0.0678844i
$218$	− 14.0000i	− 0.948200i
$219$	−20.0000	−1.35147
$220$	0	0
$221$	3.00000	0.201802
$222$	− 4.00000i	− 0.268462i
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−18.0000	−1.19734
$227$	− 9.00000i	− 0.597351i	−0.954355	−	0.298675i	$-0.903455\pi$
$227$	− 9.00000i	− 0.597351i	0.954355	−	0.298675i	$-0.0965448\pi$
$228$	8.00000i	0.529813i
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	− 3.00000i	− 0.196960i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−15.0000	−0.976417
$237$	− 8.00000i	− 0.519656i
$238$	3.00000i	0.194461i
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	− 2.00000i	− 0.128565i
$243$	10.0000i	0.641500i
$244$	13.0000	0.832240
$245$	0	0
$246$	0	0
$247$	4.00000i	0.254514i
$248$	1.00000i	0.0635001i
$249$	30.0000	1.90117
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	1.00000i	0.0629941i
$253$	− 18.0000i	− 1.13165i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.0000i	0.935674i	0.883815	+	0.467837i	$0.154967\pi$
$257$	15.0000i	0.935674i	−0.883815	+	0.467837i	$0.845033\pi$
$258$	− 20.0000i	− 1.24515i
$259$	2.00000	0.124274
$260$	0	0
$261$	−3.00000	−0.185695
$262$	− 6.00000i	− 0.370681i
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	−4.00000	−0.245256
$267$	12.0000i	0.734388i
$268$	− 13.0000i	− 0.794101i
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	− 3.00000i	− 0.181902i
$273$	2.00000i	0.121046i
$274$	18.0000	1.08742
$275$	0	0
$276$	12.0000	0.722315
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 2.00000i	− 0.119952i
$279$	1.00000	0.0598684
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	6.00000i	0.357295i
$283$	26.0000i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$	26.0000i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	0	0
$285$	0	0
$286$	−3.00000	−0.177394
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	8.00000	0.468968
$292$	10.0000i	0.585206i
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	12.0000	0.699854
$295$	0	0
$296$	−2.00000	−0.116248
$297$	− 12.0000i	− 0.696311i
$298$	− 12.0000i	− 0.695141i
$299$	6.00000	0.346989
$300$	0	0
$301$	10.0000	0.576390
$302$	11.0000i	0.632979i
$303$	− 18.0000i	− 1.03407i
$304$	4.00000	0.229416
$305$	0	0
$306$	−3.00000	−0.171499
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	− 3.00000i	− 0.170941i
$309$	16.0000	0.910208
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	− 2.00000i	− 0.113228i
$313$	− 19.0000i	− 1.07394i	−0.843600	−	0.536972i	$-0.819568\pi$
$313$	− 19.0000i	− 1.07394i	0.843600	−	0.536972i	$-0.180432\pi$
$314$	5.00000	0.282166
$315$	0	0
$316$	−4.00000	−0.225018
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	6.00000i	0.336463i
$319$	9.00000	0.503903
$320$	0	0
$321$	12.0000	0.669775
$322$	6.00000i	0.334367i
$323$	− 12.0000i	− 0.667698i
$324$	11.0000	0.611111
$325$	0	0
$326$	16.0000	0.886158
$327$	28.0000i	1.54840i
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	− 15.0000i	− 0.823232i
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	2.00000	0.109109
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	36.0000	1.95525
$340$	0	0
$341$	−3.00000	−0.162459
$342$	− 4.00000i	− 0.216295i
$343$	13.0000i	0.701934i
$344$	−10.0000	−0.539164
$345$	0	0
$346$	15.0000	0.806405
$347$	6.00000i	0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	6.00000i	0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$	6.00000i	0.321634i
$349$	−32.0000	−1.71292	−0.856460	−	0.516213i	$-0.827341\pi$
$349$	−32.0000	−1.71292	−0.856460	+	0.516213i	$0.827341\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	3.00000i	0.159901i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	30.0000	1.59448
$355$	0	0
$356$	6.00000	0.317999
$357$	− 6.00000i	− 0.317554i
$358$	24.0000i	1.26844i
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 25.0000i	− 1.31397i
$363$	4.00000i	0.209946i
$364$	1.00000	0.0524142
$365$	0	0
$366$	−26.0000	−1.35904
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	0	0
$371$	−3.00000	−0.155752
$372$	− 2.00000i	− 0.103695i
$373$	− 31.0000i	− 1.60512i	−0.596572	−	0.802560i	$-0.703471\pi$
$373$	− 31.0000i	− 1.60512i	0.596572	−	0.802560i	$-0.296529\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	3.00000	0.154713
$377$	3.00000i	0.154508i
$378$	4.00000i	0.205738i
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	0	0
$381$	32.0000	1.63941
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	4.00000	0.203595
$387$	10.0000i	0.508329i
$388$	− 4.00000i	− 0.203069i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	− 6.00000i	− 0.303046i
$393$	12.0000i	0.605320i
$394$	−24.0000	−1.20910
$395$	0	0
$396$	3.00000	0.150756
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	− 20.0000i	− 1.00251i
$399$	8.00000	0.400501
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	26.0000i	1.29676i
$403$	− 1.00000i	− 0.0498135i
$404$	−9.00000	−0.447767
$405$	0	0
$406$	−3.00000	−0.148888
$407$	− 6.00000i	− 0.297409i
$408$	6.00000i	0.297044i
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	− 8.00000i	− 0.394132i
$413$	15.0000i	0.738102i
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	4.00000i	0.195881i
$418$	12.0000i	0.586939i
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	− 22.0000i	− 1.07094i
$423$	− 3.00000i	− 0.145865i
$424$	3.00000	0.145693
$425$	0	0
$426$	0	0
$427$	− 13.0000i	− 0.629114i
$428$	− 6.00000i	− 0.290021i
$429$	6.00000	0.289683
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	− 4.00000i	− 0.192450i
$433$	− 22.0000i	− 1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$	− 22.0000i	− 1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	1.00000	0.0480015
$435$	0	0
$436$	14.0000	0.670478
$437$	− 24.0000i	− 1.14808i
$438$	− 20.0000i	− 0.955637i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	3.00000i	0.142695i
$443$	− 6.00000i	− 0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$	− 6.00000i	− 0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	4.00000	0.189405
$447$	24.0000i	1.13516i
$448$	− 1.00000i	− 0.0472456i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	− 18.0000i	− 0.846649i
$453$	− 22.0000i	− 1.03365i
$454$	9.00000	0.422391
$455$	0	0
$456$	−8.00000	−0.374634
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	− 20.0000i	− 0.934539i
$459$	−12.0000	−0.560112
$460$	0	0
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	6.00000i	0.279145i
$463$	29.0000i	1.34774i	0.738848	+	0.673872i	$0.235370\pi$
$463$	29.0000i	1.34774i	−0.738848	+	0.673872i	$0.764630\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 30.0000i	− 1.38823i	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	− 30.0000i	− 1.38823i	0.719862	−	0.694117i	$-0.244205\pi$
$468$	1.00000i	0.0462250i
$469$	−13.0000	−0.600284
$470$	0	0
$471$	−10.0000	−0.460776
$472$	− 15.0000i	− 0.690431i
$473$	− 30.0000i	− 1.37940i
$474$	8.00000	0.367452
$475$	0	0
$476$	−3.00000	−0.137505
$477$	− 3.00000i	− 0.137361i
$478$	21.0000i	0.960518i
$479$	−27.0000	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$479$	−27.0000	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	8.00000i	0.364390i
$483$	− 12.0000i	− 0.546019i
$484$	2.00000	0.0909091
$485$	0	0
$486$	−10.0000	−0.453609
$487$	− 11.0000i	− 0.498458i	−0.968445	−	0.249229i	$-0.919823\pi$
$487$	− 11.0000i	− 0.498458i	0.968445	−	0.249229i	$-0.0801771\pi$
$488$	13.0000i	0.588482i
$489$	−32.0000	−1.44709
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	− 9.00000i	− 0.405340i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	30.0000i	1.34433i
$499$	7.00000	0.313363	0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000	0.313363	0.156682	+	0.987649i	$0.449920\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	18.0000	0.800198
$507$	2.00000i	0.0888231i
$508$	− 16.0000i	− 0.709885i
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	1.00000i	0.0441942i
$513$	− 16.0000i	− 0.706417i
$514$	−15.0000	−0.661622
$515$	0	0
$516$	20.0000	0.880451
$517$	9.00000i	0.395820i
$518$	2.00000i	0.0878750i
$519$	−30.0000	−1.31685
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	− 3.00000i	− 0.131306i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−18.0000	−0.784837
$527$	3.00000i	0.130682i
$528$	− 6.00000i	− 0.261116i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−15.0000	−0.650945
$532$	− 4.00000i	− 0.173422i
$533$	0	0
$534$	−12.0000	−0.519291
$535$	0	0
$536$	13.0000	0.561514
$537$	− 48.0000i	− 2.07135i
$538$	− 21.0000i	− 0.905374i
$539$	18.0000	0.775315
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	23.0000i	0.987935i
$543$	50.0000i	2.14571i
$544$	3.00000	0.128624
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	18.0000i	0.768922i
$549$	13.0000	0.554826
$550$	0	0
$551$	12.0000	0.511217
$552$	12.0000i	0.510754i
$553$	4.00000i	0.170097i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	2.00000	0.0848189
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	1.00000i	0.0423334i
$559$	10.0000	0.422955
$560$	0	0
$561$	−18.0000	−0.759961
$562$	24.0000i	1.01238i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−26.0000	−1.09286
$567$	− 11.0000i	− 0.461957i
$568$	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
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	− 3.00000i	− 0.125436i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 32.0000i	− 1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$	− 32.0000i	− 1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	8.00000i	0.332756i
$579$	−8.00000	−0.332469
$580$	0	0
$581$	−15.0000	−0.622305
$582$	8.00000i	0.331611i
$583$	9.00000i	0.372742i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	12.0000	0.495715
$587$	− 33.0000i	− 1.36206i	−0.732257	−	0.681028i	$-0.761533\pi$
$587$	− 33.0000i	− 1.36206i	0.732257	−	0.681028i	$-0.238467\pi$
$588$	12.0000i	0.494872i
$589$	−4.00000	−0.164817
$590$	0	0
$591$	48.0000	1.97446
$592$	− 2.00000i	− 0.0821995i
$593$	42.0000i	1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$	42.0000i	1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$	12.0000	0.492366
$595$	0	0
$596$	12.0000	0.491539
$597$	40.0000i	1.63709i
$598$	6.00000i	0.245358i
$599$	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
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	10.0000i	0.407570i
$603$	− 13.0000i	− 0.529401i
$604$	−11.0000	−0.447584
$605$	0	0
$606$	18.0000	0.731200
$607$	− 14.0000i	− 0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$	− 14.0000i	− 0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$	4.00000i	0.162221i
$609$	6.00000	0.243132
$610$	0	0
$611$	−3.00000	−0.121367
$612$	− 3.00000i	− 0.121268i
$613$	44.0000i	1.77714i	0.458738	+	0.888572i	$0.348302\pi$
$613$	44.0000i	1.77714i	−0.458738	+	0.888572i	$0.651698\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	3.00000	0.120873
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	16.0000i	0.643614i
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	18.0000i	0.721734i
$623$	− 6.00000i	− 0.240385i
$624$	2.00000	0.0800641
$625$	0	0
$626$	19.0000	0.759393
$627$	− 24.0000i	− 0.958468i
$628$	5.00000i	0.199522i
$629$	−6.00000	−0.239236
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	− 4.00000i	− 0.159111i
$633$	44.0000i	1.74884i
$634$	−12.0000	−0.476581
$635$	0	0
$636$	−6.00000	−0.237915
$637$	6.00000i	0.237729i
$638$	9.00000i	0.356313i
$639$	0	0
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	12.0000i	0.473602i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	12.0000	0.472134
$647$	− 18.0000i	− 0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$	− 18.0000i	− 0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$	11.0000i	0.432121i
$649$	45.0000	1.76640
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	16.0000i	0.626608i
$653$	21.0000i	0.821794i	0.911682	+	0.410897i	$0.134784\pi$
$653$	21.0000i	0.821794i	−0.911682	+	0.410897i	$0.865216\pi$
$654$	−28.0000	−1.09489
$655$	0	0
$656$	0	0
$657$	10.0000i	0.390137i
$658$	− 3.00000i	− 0.116952i
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	− 4.00000i	− 0.155464i
$663$	− 6.00000i	− 0.233021i
$664$	15.0000	0.582113
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	− 18.0000i	− 0.696963i
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−39.0000	−1.50558
$672$	2.00000i	0.0771517i
$673$	23.0000i	0.886585i	0.896377	+	0.443292i	$0.146190\pi$
$673$	23.0000i	0.886585i	−0.896377	+	0.443292i	$0.853810\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	36.0000i	1.38257i
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−18.0000	−0.689761
$682$	− 3.00000i	− 0.114876i
$683$	39.0000i	1.49229i	0.665782	+	0.746147i	$0.268098\pi$
$683$	39.0000i	1.49229i	−0.665782	+	0.746147i	$0.731902\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−13.0000	−0.496342
$687$	40.0000i	1.52610i
$688$	− 10.0000i	− 0.381246i
$689$	−3.00000	−0.114291
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	15.0000i	0.570214i
$693$	− 3.00000i	− 0.113961i
$694$	−6.00000	−0.227757
$695$	0	0
$696$	−6.00000	−0.227429
$697$	0	0
$698$	− 32.0000i	− 1.21122i
$699$	12.0000	0.453882
$700$	0	0
$701$	3.00000	0.113308	0.0566542	−	0.998394i	$-0.481957\pi$
$701$	3.00000	0.113308	0.0566542	+	0.998394i	$0.481957\pi$
$702$	4.00000i	0.150970i
$703$	− 8.00000i	− 0.301726i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	9.00000i	0.338480i
$708$	30.0000i	1.12747i
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	6.00000i	0.224860i
$713$	6.00000i	0.224702i
$714$	6.00000	0.224544
$715$	0	0
$716$	−24.0000	−0.896922
$717$	− 42.0000i	− 1.56852i
$718$	− 9.00000i	− 0.335877i
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 3.00000i	− 0.111648i
$723$	− 16.0000i	− 0.595046i
$724$	25.0000	0.929118
$725$	0	0
$726$	−4.00000	−0.148454
$727$	− 44.0000i	− 1.63187i	−0.578144	−	0.815935i	$-0.696223\pi$
$727$	− 44.0000i	− 1.63187i	0.578144	−	0.815935i	$-0.303777\pi$
$728$	1.00000i	0.0370625i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−30.0000	−1.10959
$732$	− 26.0000i	− 0.960988i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	6.00000	0.221163
$737$	39.0000i	1.43658i
$738$	0	0
$739$	43.0000	1.58178	0.790890	−	0.611958i	$-0.209618\pi$
$739$	43.0000	1.58178	0.790890	+	0.611958i	$0.209618\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	− 3.00000i	− 0.110133i
$743$	− 3.00000i	− 0.110059i	−0.998485	−	0.0550297i	$-0.982475\pi$
$743$	− 3.00000i	− 0.110059i	0.998485	−	0.0550297i	$-0.0175253\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	31.0000	1.13499
$747$	− 15.0000i	− 0.548821i
$748$	9.00000i	0.329073i
$749$	−6.00000	−0.219235
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	3.00000i	0.109399i
$753$	0	0
$754$	−3.00000	−0.109254
$755$	0	0
$756$	−4.00000	−0.145479
$757$	13.0000i	0.472493i	0.971693	+	0.236247i	$0.0759173\pi$
$757$	13.0000i	0.472493i	−0.971693	+	0.236247i	$0.924083\pi$
$758$	− 11.0000i	− 0.399538i
$759$	−36.0000	−1.30672
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	32.0000i	1.15924i
$763$	− 14.0000i	− 0.506834i
$764$	0	0
$765$	0	0
$766$	12.0000	0.433578
$767$	15.0000i	0.541619i
$768$	− 2.00000i	− 0.0721688i
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	4.00000i	0.143963i
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	4.00000	0.143592
$777$	− 4.00000i	− 0.143499i
$778$	30.0000i	1.07555i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 18.0000i	− 0.643679i
$783$	− 12.0000i	− 0.428845i
$784$	6.00000	0.214286
$785$	0	0
$786$	−12.0000	−0.428026
$787$	7.00000i	0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$	7.00000i	0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$	− 24.0000i	− 0.854965i
$789$	36.0000	1.28163
$790$	0	0
$791$	−18.0000	−0.640006
$792$	3.00000i	0.106600i
$793$	− 13.0000i	− 0.461644i
$794$	2.00000	0.0709773
$795$	0	0
$796$	20.0000	0.708881
$797$	33.0000i	1.16892i	0.811423	+	0.584460i	$0.198694\pi$
$797$	33.0000i	1.16892i	−0.811423	+	0.584460i	$0.801306\pi$
$798$	8.00000i	0.283197i
$799$	9.00000	0.318397
$800$	0	0
$801$	6.00000	0.212000
$802$	24.0000i	0.847469i
$803$	− 30.0000i	− 1.05868i
$804$	−26.0000	−0.916949
$805$	0	0
$806$	1.00000	0.0352235
$807$	42.0000i	1.47847i
$808$	− 9.00000i	− 0.316619i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	− 3.00000i	− 0.105279i
$813$	− 46.0000i	− 1.61329i
$814$	6.00000	0.210300
$815$	0	0
$816$	−6.00000	−0.210042
$817$	− 40.0000i	− 1.39942i
$818$	− 14.0000i	− 0.489499i
$819$	1.00000	0.0349428
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	− 36.0000i	− 1.25564i
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−15.0000	−0.521917
$827$	− 15.0000i	− 0.521601i	−0.965393	−	0.260801i	$-0.916014\pi$
$827$	− 15.0000i	− 0.521601i	0.965393	−	0.260801i	$-0.0839865\pi$
$828$	− 6.00000i	− 0.208514i
$829$	−17.0000	−0.590434	−0.295217	−	0.955430i	$-0.595392\pi$
$829$	−17.0000	−0.590434	−0.295217	+	0.955430i	$0.595392\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	− 1.00000i	− 0.0346688i
$833$	− 18.0000i	− 0.623663i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	−12.0000	−0.415029
$837$	4.00000i	0.138260i
$838$	− 6.00000i	− 0.207267i
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	20.0000i	0.689246i
$843$	− 48.0000i	− 1.65321i
$844$	22.0000	0.757271
$845$	0	0
$846$	3.00000	0.103142
$847$	− 2.00000i	− 0.0687208i
$848$	3.00000i	0.103020i
$849$	52.0000	1.78464
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	13.0000	0.444851
$855$	0	0
$856$	6.00000	0.205076
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	6.00000i	0.204837i
$859$	10.0000	0.341196	0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000	0.341196	0.170598	+	0.985341i	$0.445430\pi$
$860$	0	0
$861$	0	0
$862$	− 12.0000i	− 0.408722i
$863$	− 39.0000i	− 1.32758i	−0.747921	−	0.663788i	$-0.768948\pi$
$863$	− 39.0000i	− 1.32758i	0.747921	−	0.663788i	$-0.231052\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	22.0000	0.747590
$867$	− 16.0000i	− 0.543388i
$868$	1.00000i	0.0339422i
$869$	12.0000	0.407072
$870$	0	0
$871$	−13.0000	−0.440488
$872$	14.0000i	0.474100i
$873$	− 4.00000i	− 0.135379i
$874$	24.0000	0.811812
$875$	0	0
$876$	20.0000	0.675737
$877$	46.0000i	1.55331i	0.629926	+	0.776655i	$0.283085\pi$
$877$	46.0000i	1.55331i	−0.629926	+	0.776655i	$0.716915\pi$
$878$	− 8.00000i	− 0.269987i
$879$	−24.0000	−0.809500
$880$	0	0
$881$	−39.0000	−1.31394	−0.656972	−	0.753915i	$-0.728163\pi$
$881$	−39.0000	−1.31394	−0.656972	+	0.753915i	$0.728163\pi$
$882$	− 6.00000i	− 0.202031i
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	6.00000	0.201574
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	4.00000i	0.134231i
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−33.0000	−1.10554
$892$	4.00000i	0.133930i
$893$	12.0000i	0.401565i
$894$	−24.0000	−0.802680
$895$	0	0
$896$	1.00000	0.0334077
$897$	− 12.0000i	− 0.400668i
$898$	6.00000i	0.200223i
$899$	−3.00000	−0.100056
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	− 20.0000i	− 0.665558i
$904$	18.0000	0.598671
$905$	0	0
$906$	22.0000	0.730901
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	9.00000i	0.298675i
$909$	−9.00000	−0.298511
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	− 8.00000i	− 0.264906i
$913$	45.0000i	1.48928i
$914$	−10.0000	−0.330771
$915$	0	0
$916$	20.0000	0.660819
$917$	− 6.00000i	− 0.198137i
$918$	− 12.0000i	− 0.396059i
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	− 36.0000i	− 1.18560i
$923$	0	0
$924$	−6.00000	−0.197386
$925$	0	0
$926$	−29.0000	−0.952999
$927$	− 8.00000i	− 0.262754i
$928$	3.00000i	0.0984798i
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	− 6.00000i	− 0.196537i
$933$	− 36.0000i	− 1.17859i
$934$	30.0000	0.981630
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	13.0000i	0.424691i	0.977195	+	0.212346i	$0.0681103\pi$
$937$	13.0000i	0.424691i	−0.977195	+	0.212346i	$0.931890\pi$
$938$	− 13.0000i	− 0.424465i
$939$	−38.0000	−1.24008
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	− 10.0000i	− 0.325818i
$943$	0	0
$944$	15.0000	0.488208
$945$	0	0
$946$	30.0000	0.975384
$947$	− 3.00000i	− 0.0974869i	−0.998811	−	0.0487435i	$-0.984478\pi$
$947$	− 3.00000i	− 0.0974869i	0.998811	−	0.0487435i	$-0.0155217\pi$
$948$	8.00000i	0.259828i
$949$	10.0000	0.324614
$950$	0	0
$951$	24.0000	0.778253
$952$	− 3.00000i	− 0.0972306i
$953$	− 3.00000i	− 0.0971795i	−0.998819	−	0.0485898i	$-0.984527\pi$
$953$	− 3.00000i	− 0.0971795i	0.998819	−	0.0485898i	$-0.0154727\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	−21.0000	−0.679189
$957$	− 18.0000i	− 0.581857i
$958$	− 27.0000i	− 0.872330i
$959$	18.0000	0.581250
$960$	0	0
$961$	−30.0000	−0.967742
$962$	2.00000i	0.0644826i
$963$	− 6.00000i	− 0.193347i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	12.0000	0.386094
$967$	13.0000i	0.418052i	0.977910	+	0.209026i	$0.0670293\pi$
$967$	13.0000i	0.418052i	−0.977910	+	0.209026i	$0.932971\pi$
$968$	2.00000i	0.0642824i
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	− 10.0000i	− 0.320750i
$973$	− 2.00000i	− 0.0641171i
$974$	11.0000	0.352463
$975$	0	0
$976$	−13.0000	−0.416120
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	− 32.0000i	− 1.02325i
$979$	−18.0000	−0.575282
$980$	0	0
$981$	14.0000	0.446986
$982$	− 30.0000i	− 0.957338i
$983$	− 9.00000i	− 0.287055i	−0.989646	−	0.143528i	$-0.954155\pi$
$983$	− 9.00000i	− 0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$984$	0	0
$985$	0	0
$986$	9.00000	0.286618
$987$	6.00000i	0.190982i
$988$	− 4.00000i	− 0.127257i
$989$	−60.0000	−1.90789
$990$	0	0
$991$	50.0000	1.58830	0.794151	−	0.607720i	$-0.207916\pi$
$991$	50.0000	1.58830	0.794151	+	0.607720i	$0.207916\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	8.00000i	0.253872i
$994$	0	0
$995$	0	0
$996$	−30.0000	−0.950586
$997$	− 35.0000i	− 1.10846i	−0.832363	−	0.554231i	$-0.813013\pi$
$997$	− 35.0000i	− 1.10846i	0.832363	−	0.554231i	$-0.186987\pi$
$998$	7.00000i	0.221581i
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.2.b.h.599.2		2
3.2	odd	2		5850.2.e.j.5149.1		2
5.2	odd	4		650.2.a.b.1.1	✓	1
5.3	odd	4		650.2.a.k.1.1	yes	1
5.4	even	2	inner	650.2.b.h.599.1		2
15.2	even	4		5850.2.a.bm.1.1		1
15.8	even	4		5850.2.a.q.1.1		1
15.14	odd	2		5850.2.e.j.5149.2		2
20.3	even	4		5200.2.a.g.1.1		1
20.7	even	4		5200.2.a.bg.1.1		1
65.12	odd	4		8450.2.a.p.1.1		1
65.38	odd	4		8450.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.a.b.1.1	✓	1	5.2	odd	4
650.2.a.k.1.1	yes	1	5.3	odd	4
650.2.b.h.599.1		2	5.4	even	2	inner
650.2.b.h.599.2		2	1.1	even	1	trivial
5200.2.a.g.1.1		1	20.3	even	4
5200.2.a.bg.1.1		1	20.7	even	4
5850.2.a.q.1.1		1	15.8	even	4
5850.2.a.bm.1.1		1	15.2	even	4
5850.2.e.j.5149.1		2	3.2	odd	2
5850.2.e.j.5149.2		2	15.14	odd	2
8450.2.a.j.1.1		1	65.38	odd	4
8450.2.a.p.1.1		1	65.12	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +2.00000 q^{21} +3.00000i q^{22} -6.00000i q^{23} -2.00000 q^{24} -1.00000 q^{26} -4.00000i q^{27} -1.00000i q^{28} +3.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +4.00000i q^{38} +2.00000 q^{39} +2.00000i q^{42} -10.0000i q^{43} -3.00000 q^{44} +6.00000 q^{46} +3.00000i q^{47} -2.00000i q^{48} +6.00000 q^{49} -6.00000 q^{51} -1.00000i q^{52} +3.00000i q^{53} +4.00000 q^{54} +1.00000 q^{56} -8.00000i q^{57} +3.00000i q^{58} +15.0000 q^{59} -13.0000 q^{61} -1.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +13.0000i q^{67} +3.00000i q^{68} -12.0000 q^{69} +1.00000i q^{72} -10.0000i q^{73} +2.00000 q^{74} -4.00000 q^{76} +3.00000i q^{77} +2.00000i q^{78} +4.00000 q^{79} -11.0000 q^{81} +15.0000i q^{83} -2.00000 q^{84} +10.0000 q^{86} -6.00000i q^{87} -3.00000i q^{88} -6.00000 q^{89} -1.00000 q^{91} +6.00000i q^{92} +2.00000i q^{93} -3.00000 q^{94} +2.00000 q^{96} +4.00000i q^{97} +6.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 6 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{21} - 4 q^{24} - 2 q^{26} + 6 q^{29} - 2 q^{31} + 6 q^{34} + 2 q^{36} + 4 q^{39} - 6 q^{44} + 12 q^{46} + 12 q^{49} - 12 q^{51} + 8 q^{54} + 2 q^{56} + 30 q^{59} - 26 q^{61} - 2 q^{64} + 12 q^{66} - 24 q^{69} + 4 q^{74} - 8 q^{76} + 8 q^{79} - 22 q^{81} - 4 q^{84} + 20 q^{86} - 12 q^{89} - 2 q^{91} - 6 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 6 * q^11 - 2 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^21 - 4 * q^24 - 2 * q^26 + 6 * q^29 - 2 * q^31 + 6 * q^34 + 2 * q^36 + 4 * q^39 - 6 * q^44 + 12 * q^46 + 12 * q^49 - 12 * q^51 + 8 * q^54 + 2 * q^56 + 30 * q^59 - 26 * q^61 - 2 * q^64 + 12 * q^66 - 24 * q^69 + 4 * q^74 - 8 * q^76 + 8 * q^79 - 22 * q^81 - 4 * q^84 + 20 * q^86 - 12 * q^89 - 2 * q^91 - 6 * q^94 + 4 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

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Database

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