LMFDB - Embedded newform 975.2.c.g.274.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(274,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, 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("975.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.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:	$7.78541419707$
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			274.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	975.274
Dual form			975.2.c.g.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} +3.00000 q^{14} -1.00000 q^{16} +5.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} -3.00000 q^{21} +1.00000i q^{22} +3.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +3.00000i q^{28} -1.00000 q^{29} +3.00000 q^{31} -5.00000i q^{32} -1.00000i q^{33} +5.00000 q^{34} -1.00000 q^{36} +8.00000i q^{37} -8.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} +3.00000i q^{42} +8.00000i q^{43} -1.00000 q^{44} +11.0000i q^{47} -1.00000i q^{48} -2.00000 q^{49} -5.00000 q^{51} +1.00000i q^{52} -11.0000i q^{53} -1.00000 q^{54} +9.00000 q^{56} +8.00000i q^{57} +1.00000i q^{58} -5.00000 q^{59} +1.00000 q^{61} -3.00000i q^{62} -3.00000i q^{63} -7.00000 q^{64} -1.00000 q^{66} -3.00000i q^{67} +5.00000i q^{68} +16.0000 q^{71} +3.00000i q^{72} -4.00000i q^{73} +8.00000 q^{74} +8.00000 q^{76} -3.00000i q^{77} +1.00000i q^{78} -12.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -3.00000i q^{83} -3.00000 q^{84} +8.00000 q^{86} -1.00000i q^{87} +3.00000i q^{88} -3.00000 q^{91} +3.00000i q^{93} +11.0000 q^{94} +5.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{14} - 2 q^{16} + 16 q^{19} - 6 q^{21} + 6 q^{24} + 2 q^{26} - 2 q^{29} + 6 q^{31} + 10 q^{34} - 2 q^{36} - 2 q^{39} - 4 q^{41} - 2 q^{44} - 4 q^{49} - 10 q^{51} - 2 q^{54} + 18 q^{56} - 10 q^{59} + 2 q^{61} - 14 q^{64} - 2 q^{66} + 32 q^{71} + 16 q^{74} + 16 q^{76} - 24 q^{79} + 2 q^{81} - 6 q^{84} + 16 q^{86} - 6 q^{91} + 22 q^{94} + 10 q^{96} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 6 * q^14 - 2 * q^16 + 16 * q^19 - 6 * q^21 + 6 * q^24 + 2 * q^26 - 2 * q^29 + 6 * q^31 + 10 * q^34 - 2 * q^36 - 2 * q^39 - 4 * q^41 - 2 * q^44 - 4 * q^49 - 10 * q^51 - 2 * q^54 + 18 * q^56 - 10 * q^59 + 2 * q^61 - 14 * q^64 - 2 * q^66 + 32 * q^71 + 16 * q^74 + 16 * q^76 - 24 * q^79 + 2 * q^81 - 6 * q^84 + 16 * q^86 - 6 * q^91 + 22 * q^94 + 10 * q^96 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$1$	$1$	$-1$

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	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	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
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	− 3.00000i	− 1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000i	0.288675i
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	3.00000	0.801784
$15$	0	0
$16$	−1.00000	−0.250000
$17$	5.00000i	1.21268i	0.795206	+	0.606339i	$0.207363\pi$
$17$	5.00000i	1.21268i	−0.795206	+	0.606339i	$0.792637\pi$
$18$	1.00000i	0.235702i
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	1.00000i	0.213201i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	1.00000	0.196116
$27$	− 1.00000i	− 0.192450i
$28$	3.00000i	0.566947i
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	− 5.00000i	− 0.883883i
$33$	− 1.00000i	− 0.174078i
$34$	5.00000	0.857493
$35$	0	0
$36$	−1.00000	−0.166667
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	− 8.00000i	− 1.29777i
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	3.00000i	0.462910i
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0	0
$47$	11.0000i	1.60451i	0.596978	+	0.802257i	$0.296368\pi$
$47$	11.0000i	1.60451i	−0.596978	+	0.802257i	$0.703632\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−5.00000	−0.700140
$52$	1.00000i	0.138675i
$53$	− 11.0000i	− 1.51097i	−0.655168	−	0.755483i	$-0.727402\pi$
$53$	− 11.0000i	− 1.51097i	0.655168	−	0.755483i	$-0.272598\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	9.00000	1.20268
$57$	8.00000i	1.05963i
$58$	1.00000i	0.131306i
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	− 3.00000i	− 0.381000i
$63$	− 3.00000i	− 0.377964i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	5.00000i	0.606339i
$69$	0	0
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	3.00000i	0.353553i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	8.00000	0.917663
$77$	− 3.00000i	− 0.341882i
$78$	1.00000i	0.113228i
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	− 3.00000i	− 0.329293i	−0.986353	−	0.164646i	$-0.947352\pi$
$83$	− 3.00000i	− 0.329293i	0.986353	−	0.164646i	$-0.0526483\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	8.00000	0.862662
$87$	− 1.00000i	− 0.107211i
$88$	3.00000i	0.319801i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	3.00000i	0.311086i
$94$	11.0000	1.13456
$95$	0	0
$96$	5.00000	0.510310
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	2.00000i	0.202031i
$99$	1.00000	0.100504
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	5.00000i	0.495074i
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−11.0000	−1.06841
$107$	− 18.0000i	− 1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$	− 18.0000i	− 1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	− 3.00000i	− 0.283473i
$113$	− 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
$116$	−1.00000	−0.0928477
$117$	− 1.00000i	− 0.0924500i
$118$	5.00000i	0.460287i
$119$	−15.0000	−1.37505
$120$	0	0
$121$	−10.0000	−0.909091
$122$	− 1.00000i	− 0.0905357i
$123$	− 2.00000i	− 0.180334i
$124$	3.00000	0.269408
$125$	0	0
$126$	−3.00000	−0.267261
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 3.00000i	− 0.265165i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	− 1.00000i	− 0.0870388i
$133$	24.0000i	2.08106i
$134$	−3.00000	−0.259161
$135$	0	0
$136$	15.0000	1.28624
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−11.0000	−0.926367
$142$	− 16.0000i	− 1.34269i
$143$	− 1.00000i	− 0.0836242i
$144$	1.00000	0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	− 2.00000i	− 0.164957i
$148$	8.00000i	0.657596i
$149$	−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
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	− 24.0000i	− 1.94666i
$153$	− 5.00000i	− 0.404226i
$154$	−3.00000	−0.241747
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	11.0000i	0.877896i	0.898513	+	0.438948i	$0.144649\pi$
$157$	11.0000i	0.877896i	−0.898513	+	0.438948i	$0.855351\pi$
$158$	12.0000i	0.954669i
$159$	11.0000	0.872357
$160$	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	$-0.286445\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−3.00000	−0.232845
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	9.00000i	0.694365i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−8.00000	−0.611775
$172$	8.00000i	0.609994i
$173$	− 21.0000i	− 1.59660i	−0.602260	−	0.798300i	$-0.705733\pi$
$173$	− 21.0000i	− 1.59660i	0.602260	−	0.798300i	$-0.294267\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	1.00000	0.0753778
$177$	− 5.00000i	− 0.375823i
$178$	0	0
$179$	−26.0000	−1.94333	−0.971666	−	0.236360i	$-0.924046\pi$
$179$	−26.0000	−1.94333	−0.971666	+	0.236360i	$0.924046\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	3.00000i	0.222375i
$183$	1.00000i	0.0739221i
$184$	0	0
$185$	0	0
$186$	3.00000	0.219971
$187$	− 5.00000i	− 0.365636i
$188$	11.0000i	0.802257i
$189$	3.00000	0.218218
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	− 7.00000i	− 0.505181i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−2.00000	−0.142857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	− 1.00000i	− 0.0710669i
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	− 9.00000i	− 0.633238i
$203$	− 3.00000i	− 0.210559i
$204$	−5.00000	−0.350070
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	− 11.0000i	− 0.755483i
$213$	16.0000i	1.09630i
$214$	−18.0000	−1.23045
$215$	0	0
$216$	−3.00000	−0.204124
$217$	9.00000i	0.610960i
$218$	− 10.0000i	− 0.677285i
$219$	4.00000	0.270295
$220$	0	0
$221$	−5.00000	−0.336336
$222$	8.00000i	0.536925i
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	15.0000	1.00223
$225$	0	0
$226$	−2.00000	−0.133038
$227$	− 23.0000i	− 1.52656i	−0.646066	−	0.763282i	$-0.723587\pi$
$227$	− 23.0000i	− 1.52656i	0.646066	−	0.763282i	$-0.276413\pi$
$228$	8.00000i	0.529813i
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	3.00000i	0.196960i
$233$	− 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
$236$	−5.00000	−0.325472
$237$	− 12.0000i	− 0.779484i
$238$	15.0000i	0.972306i
$239$	19.0000	1.22901	0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000	1.22901	0.614504	+	0.788914i	$0.289356\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	10.0000i	0.642824i
$243$	1.00000i	0.0641500i
$244$	1.00000	0.0640184
$245$	0	0
$246$	−2.00000	−0.127515
$247$	8.00000i	0.509028i
$248$	− 9.00000i	− 0.571501i
$249$	3.00000	0.190117
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	− 3.00000i	− 0.188982i
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	3.00000i	0.187135i	0.995613	+	0.0935674i	$0.0298271\pi$
$257$	3.00000i	0.187135i	−0.995613	+	0.0935674i	$0.970173\pi$
$258$	8.00000i	0.498058i
$259$	−24.0000	−1.49129
$260$	0	0
$261$	1.00000	0.0618984
$262$	− 6.00000i	− 0.370681i
$263$	14.0000i	0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$	14.0000i	0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	24.0000	1.47153
$267$	0	0
$268$	− 3.00000i	− 0.183254i
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	0	0
$271$	−5.00000	−0.303728	−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000	−0.303728	−0.151864	+	0.988401i	$0.548528\pi$
$272$	− 5.00000i	− 0.303170i
$273$	− 3.00000i	− 0.181568i
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	− 10.0000i	− 0.599760i
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	11.0000i	0.655040i
$283$	− 18.0000i	− 1.06999i	−0.844856	−	0.534994i	$-0.820314\pi$
$283$	− 18.0000i	− 1.06999i	0.844856	−	0.534994i	$-0.179686\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	− 6.00000i	− 0.354169i
$288$	5.00000i	0.294628i
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	− 4.00000i	− 0.234082i
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	24.0000	1.39497
$297$	1.00000i	0.0580259i
$298$	10.0000i	0.579284i
$299$	0	0
$300$	0	0
$301$	−24.0000	−1.38334
$302$	17.0000i	0.978240i
$303$	9.00000i	0.517036i
$304$	−8.00000	−0.458831
$305$	0	0
$306$	−5.00000	−0.285831
$307$	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
$309$	14.0000	0.796432
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	3.00000i	0.169842i
$313$	− 29.0000i	− 1.63918i	−0.572953	−	0.819588i	$-0.694202\pi$
$313$	− 29.0000i	− 1.63918i	0.572953	−	0.819588i	$-0.305798\pi$
$314$	11.0000	0.620766
$315$	0	0
$316$	−12.0000	−0.675053
$317$	16.0000i	0.898650i	0.893368	+	0.449325i	$0.148335\pi$
$317$	16.0000i	0.898650i	−0.893368	+	0.449325i	$0.851665\pi$
$318$	− 11.0000i	− 0.616849i
$319$	1.00000	0.0559893
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	40.0000i	2.22566i
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	10.0000i	0.553001i
$328$	6.00000i	0.331295i
$329$	−33.0000	−1.81935
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	− 3.00000i	− 0.164646i
$333$	− 8.00000i	− 0.438397i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	3.00000	0.163663
$337$	− 5.00000i	− 0.272367i	−0.990684	−	0.136184i	$-0.956516\pi$
$337$	− 5.00000i	− 0.272367i	0.990684	−	0.136184i	$-0.0434837\pi$
$338$	1.00000i	0.0543928i
$339$	2.00000	0.108625
$340$	0	0
$341$	−3.00000	−0.162459
$342$	8.00000i	0.432590i
$343$	15.0000i	0.809924i
$344$	24.0000	1.29399
$345$	0	0
$346$	−21.0000	−1.12897
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	− 1.00000i	− 0.0536056i
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	5.00000i	0.266501i
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	−5.00000	−0.265747
$355$	0	0
$356$	0	0
$357$	− 15.0000i	− 0.793884i
$358$	26.0000i	1.37414i
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 5.00000i	− 0.262794i
$363$	− 10.0000i	− 0.524864i
$364$	−3.00000	−0.157243
$365$	0	0
$366$	1.00000	0.0522708
$367$	12.0000i	0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$	12.0000i	0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	33.0000	1.71327
$372$	3.00000i	0.155543i
$373$	− 31.0000i	− 1.60512i	−0.596572	−	0.802560i	$-0.703471\pi$
$373$	− 31.0000i	− 1.60512i	0.596572	−	0.802560i	$-0.296529\pi$
$374$	−5.00000	−0.258544
$375$	0	0
$376$	33.0000	1.70185
$377$	− 1.00000i	− 0.0515026i
$378$	− 3.00000i	− 0.154303i
$379$	35.0000	1.79783	0.898915	−	0.438124i	$-0.144357\pi$
$379$	35.0000	1.79783	0.898915	+	0.438124i	$0.144357\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	− 10.0000i	− 0.511645i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	10.0000	0.508987
$387$	− 8.00000i	− 0.406663i
$388$	2.00000i	0.101535i
$389$	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
$391$	0	0
$392$	6.00000i	0.303046i
$393$	6.00000i	0.302660i
$394$	6.00000	0.302276
$395$	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	$-0.0159822\pi$
$398$	10.0000i	0.501255i
$399$	−24.0000	−1.20150
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	− 3.00000i	− 0.149626i
$403$	3.00000i	0.149441i
$404$	9.00000	0.447767
$405$	0	0
$406$	−3.00000	−0.148888
$407$	− 8.00000i	− 0.396545i
$408$	15.0000i	0.742611i
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	− 14.0000i	− 0.689730i
$413$	− 15.0000i	− 0.738102i
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	10.0000i	0.489702i
$418$	8.00000i	0.391293i
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	16.0000i	0.778868i
$423$	− 11.0000i	− 0.534838i
$424$	−33.0000	−1.60262
$425$	0	0
$426$	16.0000	0.775203
$427$	3.00000i	0.145180i
$428$	− 18.0000i	− 0.870063i
$429$	1.00000	0.0482805
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	1.00000i	0.0481125i
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	9.00000	0.432014
$435$	0	0
$436$	10.0000	0.478913
$437$	0	0
$438$	− 4.00000i	− 0.191127i
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	5.00000i	0.237826i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	−8.00000	−0.378811
$447$	− 10.0000i	− 0.472984i
$448$	− 21.0000i	− 0.992157i
$449$	28.0000	1.32140	0.660701	−	0.750649i	$-0.270259\pi$
$449$	28.0000	1.32140	0.660701	+	0.750649i	$0.270259\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	− 2.00000i	− 0.0940721i
$453$	− 17.0000i	− 0.798730i
$454$	−23.0000	−1.07944
$455$	0	0
$456$	24.0000	1.12390
$457$	34.0000i	1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$	34.0000i	1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$	− 18.0000i	− 0.841085i
$459$	5.00000	0.233380
$460$	0	0
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	− 3.00000i	− 0.139573i
$463$	− 9.00000i	− 0.418265i	−0.977887	−	0.209133i	$-0.932936\pi$
$463$	− 9.00000i	− 0.418265i	0.977887	−	0.209133i	$-0.0670641\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 32.0000i	− 1.48078i	−0.672176	−	0.740392i	$-0.734640\pi$
$467$	− 32.0000i	− 1.48078i	0.672176	−	0.740392i	$-0.265360\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	9.00000	0.415581
$470$	0	0
$471$	−11.0000	−0.506853
$472$	15.0000i	0.690431i
$473$	− 8.00000i	− 0.367840i
$474$	−12.0000	−0.551178
$475$	0	0
$476$	−15.0000	−0.687524
$477$	11.0000i	0.503655i
$478$	− 19.0000i	− 0.869040i
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	− 10.0000i	− 0.455488i
$483$	0	0
$484$	−10.0000	−0.454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 37.0000i	− 1.67663i	−0.545186	−	0.838315i	$-0.683541\pi$
$487$	− 37.0000i	− 1.67663i	0.545186	−	0.838315i	$-0.316459\pi$
$488$	− 3.00000i	− 0.135804i
$489$	20.0000	0.904431
$490$	0	0
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	− 5.00000i	− 0.225189i
$494$	8.00000	0.359937
$495$	0	0
$496$	−3.00000	−0.134704
$497$	48.0000i	2.15309i
$498$	− 3.00000i	− 0.134433i
$499$	41.0000	1.83541	0.917706	−	0.397260i	$-0.130039\pi$
$499$	41.0000	1.83541	0.917706	+	0.397260i	$0.130039\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	30.0000i	1.33897i
$503$	26.0000i	1.15928i	0.814872	+	0.579641i	$0.196807\pi$
$503$	26.0000i	1.15928i	−0.814872	+	0.579641i	$0.803193\pi$
$504$	−9.00000	−0.400892
$505$	0	0
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	− 8.00000i	− 0.354943i
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	11.0000i	0.486136i
$513$	− 8.00000i	− 0.353209i
$514$	3.00000	0.132324
$515$	0	0
$516$	−8.00000	−0.352180
$517$	− 11.0000i	− 0.483779i
$518$	24.0000i	1.05450i
$519$	21.0000	0.921798
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	− 1.00000i	− 0.0437688i
$523$	26.0000i	1.13690i	0.822718	+	0.568450i	$0.192457\pi$
$523$	26.0000i	1.13690i	−0.822718	+	0.568450i	$0.807543\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	14.0000	0.610429
$527$	15.0000i	0.653410i
$528$	1.00000i	0.0435194i
$529$	23.0000	1.00000
$530$	0	0
$531$	5.00000	0.216982
$532$	24.0000i	1.04053i
$533$	− 2.00000i	− 0.0866296i
$534$	0	0
$535$	0	0
$536$	−9.00000	−0.388741
$537$	− 26.0000i	− 1.12198i
$538$	17.0000i	0.732922i
$539$	2.00000	0.0861461
$540$	0	0
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	5.00000i	0.214768i
$543$	5.00000i	0.214571i
$544$	25.0000	1.07187
$545$	0	0
$546$	−3.00000	−0.128388
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	12.0000i	0.512615i
$549$	−1.00000	−0.0426790
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	− 36.0000i	− 1.53088i
$554$	−6.00000	−0.254916
$555$	0	0
$556$	10.0000	0.424094
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	3.00000i	0.127000i
$559$	−8.00000	−0.338364
$560$	0	0
$561$	5.00000	0.211100
$562$	18.0000i	0.759284i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	−11.0000	−0.463184
$565$	0	0
$566$	−18.0000	−0.756596
$567$	3.00000i	0.125988i
$568$	− 48.0000i	− 2.01404i
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	− 1.00000i	− 0.0418121i
$573$	10.0000i	0.417756i
$574$	−6.00000	−0.250435
$575$	0	0
$576$	7.00000	0.291667
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	8.00000i	0.332756i
$579$	−10.0000	−0.415586
$580$	0	0
$581$	9.00000	0.373383
$582$	2.00000i	0.0829027i
$583$	11.0000i	0.455573i
$584$	−12.0000	−0.496564
$585$	0	0
$586$	30.0000	1.23929
$587$	45.0000i	1.85735i	0.370896	+	0.928674i	$0.379051\pi$
$587$	45.0000i	1.85735i	−0.370896	+	0.928674i	$0.620949\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	24.0000	0.988903
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 8.00000i	− 0.328798i
$593$	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
$597$	− 10.0000i	− 0.409273i
$598$	0	0
$599$	−46.0000	−1.87951	−0.939755	−	0.341850i	$-0.888947\pi$
$599$	−46.0000	−1.87951	−0.939755	+	0.341850i	$0.888947\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	24.0000i	0.978167i
$603$	3.00000i	0.122169i
$604$	−17.0000	−0.691720
$605$	0	0
$606$	9.00000	0.365600
$607$	− 38.0000i	− 1.54237i	−0.636610	−	0.771186i	$-0.719664\pi$
$607$	− 38.0000i	− 1.54237i	0.636610	−	0.771186i	$-0.280336\pi$
$608$	− 40.0000i	− 1.62221i
$609$	3.00000	0.121566
$610$	0	0
$611$	−11.0000	−0.445012
$612$	− 5.00000i	− 0.202113i
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	−9.00000	−0.362620
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	− 14.0000i	− 0.563163i
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	− 6.00000i	− 0.240578i
$623$	0	0
$624$	1.00000	0.0400320
$625$	0	0
$626$	−29.0000	−1.15907
$627$	− 8.00000i	− 0.319489i
$628$	11.0000i	0.438948i
$629$	−40.0000	−1.59490
$630$	0	0
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	36.0000i	1.43200i
$633$	− 16.0000i	− 0.635943i
$634$	16.0000	0.635441
$635$	0	0
$636$	11.0000	0.436178
$637$	− 2.00000i	− 0.0792429i
$638$	− 1.00000i	− 0.0395904i
$639$	−16.0000	−0.632950
$640$	0	0
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	− 18.0000i	− 0.710403i
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	0	0
$646$	40.0000	1.57378
$647$	− 16.0000i	− 0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$	− 16.0000i	− 0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$	− 3.00000i	− 0.117851i
$649$	5.00000	0.196267
$650$	0	0
$651$	−9.00000	−0.352738
$652$	− 20.0000i	− 0.783260i
$653$	35.0000i	1.36966i	0.728705	+	0.684828i	$0.240123\pi$
$653$	35.0000i	1.36966i	−0.728705	+	0.684828i	$0.759877\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	2.00000	0.0780869
$657$	4.00000i	0.156055i
$658$	33.0000i	1.28647i
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	12.0000i	0.466393i
$663$	− 5.00000i	− 0.194184i
$664$	−9.00000	−0.349268
$665$	0	0
$666$	−8.00000	−0.309994
$667$	0	0
$668$	− 12.0000i	− 0.464294i
$669$	8.00000	0.309298
$670$	0	0
$671$	−1.00000	−0.0386046
$672$	15.0000i	0.578638i
$673$	45.0000i	1.73462i	0.497766	+	0.867311i	$0.334154\pi$
$673$	45.0000i	1.73462i	−0.497766	+	0.867311i	$0.665846\pi$
$674$	−5.00000	−0.192593
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	− 2.00000i	− 0.0768095i
$679$	−6.00000	−0.230259
$680$	0	0
$681$	23.0000	0.881362
$682$	3.00000i	0.114876i
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	15.0000	0.572703
$687$	18.0000i	0.686743i
$688$	− 8.00000i	− 0.304997i
$689$	11.0000	0.419067
$690$	0	0
$691$	−29.0000	−1.10321	−0.551606	−	0.834105i	$-0.685985\pi$
$691$	−29.0000	−1.10321	−0.551606	+	0.834105i	$0.685985\pi$
$692$	− 21.0000i	− 0.798300i
$693$	3.00000i	0.113961i
$694$	2.00000	0.0759190
$695$	0	0
$696$	−3.00000	−0.113715
$697$	− 10.0000i	− 0.378777i
$698$	16.0000i	0.605609i
$699$	6.00000	0.226941
$700$	0	0
$701$	31.0000	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	64.0000i	2.41381i
$704$	7.00000	0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	27.0000i	1.01544i
$708$	− 5.00000i	− 0.187912i
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	0	0
$714$	−15.0000	−0.561361
$715$	0	0
$716$	−26.0000	−0.971666
$717$	19.0000i	0.709568i
$718$	15.0000i	0.559795i
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	− 45.0000i	− 1.67473i
$723$	10.0000i	0.371904i
$724$	5.00000	0.185824
$725$	0	0
$726$	−10.0000	−0.371135
$727$	2.00000i	0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$	2.00000i	0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$	9.00000i	0.333562i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−40.0000	−1.47945
$732$	1.00000i	0.0369611i
$733$	− 20.0000i	− 0.738717i	−0.929287	−	0.369358i	$-0.879577\pi$
$733$	− 20.0000i	− 0.738717i	0.929287	−	0.369358i	$-0.120423\pi$
$734$	12.0000	0.442928
$735$	0	0
$736$	0	0
$737$	3.00000i	0.110506i
$738$	− 2.00000i	− 0.0736210i
$739$	−7.00000	−0.257499	−0.128750	−	0.991677i	$-0.541096\pi$
$739$	−7.00000	−0.257499	−0.128750	+	0.991677i	$0.541096\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	− 33.0000i	− 1.21147i
$743$	13.0000i	0.476924i	0.971152	+	0.238462i	$0.0766432\pi$
$743$	13.0000i	0.476924i	−0.971152	+	0.238462i	$0.923357\pi$
$744$	9.00000	0.329956
$745$	0	0
$746$	−31.0000	−1.13499
$747$	3.00000i	0.109764i
$748$	− 5.00000i	− 0.182818i
$749$	54.0000	1.97312
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	− 11.0000i	− 0.401129i
$753$	− 30.0000i	− 1.09326i
$754$	−1.00000	−0.0364179
$755$	0	0
$756$	3.00000	0.109109
$757$	− 23.0000i	− 0.835949i	−0.908459	−	0.417975i	$-0.862740\pi$
$757$	− 23.0000i	− 0.835949i	0.908459	−	0.417975i	$-0.137260\pi$
$758$	− 35.0000i	− 1.27126i
$759$	0	0
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	− 8.00000i	− 0.289809i
$763$	30.0000i	1.08607i
$764$	10.0000	0.361787
$765$	0	0
$766$	−24.0000	−0.867155
$767$	− 5.00000i	− 0.180540i
$768$	− 17.0000i	− 0.613435i
$769$	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
$771$	−3.00000	−0.108042
$772$	10.0000i	0.359908i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	6.00000	0.215387
$777$	− 24.0000i	− 0.860995i
$778$	− 30.0000i	− 1.07555i
$779$	−16.0000	−0.573259
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	1.00000i	0.0357371i
$784$	2.00000	0.0714286
$785$	0	0
$786$	6.00000	0.214013
$787$	− 17.0000i	− 0.605985i	−0.952993	−	0.302992i	$-0.902014\pi$
$787$	− 17.0000i	− 0.605985i	0.952993	−	0.302992i	$-0.0979856\pi$
$788$	6.00000i	0.213741i
$789$	−14.0000	−0.498413
$790$	0	0
$791$	6.00000	0.213335
$792$	− 3.00000i	− 0.106600i
$793$	1.00000i	0.0355110i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−10.0000	−0.354441
$797$	43.0000i	1.52314i	0.648084	+	0.761569i	$0.275571\pi$
$797$	43.0000i	1.52314i	−0.648084	+	0.761569i	$0.724429\pi$
$798$	24.0000i	0.849591i
$799$	−55.0000	−1.94576
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.00000i	0.141157i
$804$	3.00000	0.105802
$805$	0	0
$806$	3.00000	0.105670
$807$	− 17.0000i	− 0.598428i
$808$	− 27.0000i	− 0.949857i
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	−33.0000	−1.15879	−0.579393	−	0.815048i	$-0.696710\pi$
$811$	−33.0000	−1.15879	−0.579393	+	0.815048i	$0.696710\pi$
$812$	− 3.00000i	− 0.105279i
$813$	− 5.00000i	− 0.175358i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	5.00000	0.175035
$817$	64.0000i	2.23908i
$818$	2.00000i	0.0699284i
$819$	3.00000	0.104828
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	12.0000i	0.418548i
$823$	28.0000i	0.976019i	0.872838	+	0.488009i	$0.162277\pi$
$823$	28.0000i	0.976019i	−0.872838	+	0.488009i	$0.837723\pi$
$824$	−42.0000	−1.46314
$825$	0	0
$826$	−15.0000	−0.521917
$827$	− 1.00000i	− 0.0347734i	−0.999849	−	0.0173867i	$-0.994465\pi$
$827$	− 1.00000i	− 0.0347734i	0.999849	−	0.0173867i	$-0.00553464\pi$
$828$	0	0
$829$	−51.0000	−1.77130	−0.885652	−	0.464350i	$-0.846288\pi$
$829$	−51.0000	−1.77130	−0.885652	+	0.464350i	$0.846288\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	− 7.00000i	− 0.242681i
$833$	− 10.0000i	− 0.346479i
$834$	10.0000	0.346272
$835$	0	0
$836$	−8.00000	−0.276686
$837$	− 3.00000i	− 0.103695i
$838$	− 26.0000i	− 0.898155i
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	− 28.0000i	− 0.964944i
$843$	− 18.0000i	− 0.619953i
$844$	−16.0000	−0.550743
$845$	0	0
$846$	−11.0000	−0.378188
$847$	− 30.0000i	− 1.03081i
$848$	11.0000i	0.377742i
$849$	18.0000	0.617758
$850$	0	0
$851$	0	0
$852$	16.0000i	0.548151i
$853$	24.0000i	0.821744i	0.911693	+	0.410872i	$0.134776\pi$
$853$	24.0000i	0.821744i	−0.911693	+	0.410872i	$0.865224\pi$
$854$	3.00000	0.102658
$855$	0	0
$856$	−54.0000	−1.84568
$857$	38.0000i	1.29806i	0.760765	+	0.649028i	$0.224824\pi$
$857$	38.0000i	1.29806i	−0.760765	+	0.649028i	$0.775176\pi$
$858$	− 1.00000i	− 0.0341394i
$859$	−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
$861$	6.00000	0.204479
$862$	12.0000i	0.408722i
$863$	− 7.00000i	− 0.238283i	−0.992877	−	0.119141i	$-0.961986\pi$
$863$	− 7.00000i	− 0.238283i	0.992877	−	0.119141i	$-0.0380142\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−34.0000	−1.15537
$867$	− 8.00000i	− 0.271694i
$868$	9.00000i	0.305480i
$869$	12.0000	0.407072
$870$	0	0
$871$	3.00000	0.101651
$872$	− 30.0000i	− 1.01593i
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	− 24.0000i	− 0.810422i	−0.914223	−	0.405211i	$-0.867198\pi$
$877$	− 24.0000i	− 0.810422i	0.914223	−	0.405211i	$-0.132802\pi$
$878$	4.00000i	0.134993i
$879$	−30.0000	−1.01187
$880$	0	0
$881$	−25.0000	−0.842271	−0.421136	−	0.906998i	$-0.638368\pi$
$881$	−25.0000	−0.842271	−0.421136	+	0.906998i	$0.638368\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	−5.00000	−0.168168
$885$	0	0
$886$	0	0
$887$	− 12.0000i	− 0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$	− 12.0000i	− 0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$	24.0000i	0.805387i
$889$	24.0000	0.804934
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	− 8.00000i	− 0.267860i
$893$	88.0000i	2.94481i
$894$	−10.0000	−0.334450
$895$	0	0
$896$	9.00000	0.300669
$897$	0	0
$898$	− 28.0000i	− 0.934372i
$899$	−3.00000	−0.100056
$900$	0	0
$901$	55.0000	1.83232
$902$	− 2.00000i	− 0.0665927i
$903$	− 24.0000i	− 0.798670i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−17.0000	−0.564787
$907$	− 24.0000i	− 0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$	− 24.0000i	− 0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$	− 23.0000i	− 0.763282i
$909$	−9.00000	−0.298511
$910$	0	0
$911$	56.0000	1.85536	0.927681	−	0.373373i	$-0.121799\pi$
$911$	56.0000	1.85536	0.927681	+	0.373373i	$0.121799\pi$
$912$	− 8.00000i	− 0.264906i
$913$	3.00000i	0.0992855i
$914$	34.0000	1.12462
$915$	0	0
$916$	18.0000	0.594737
$917$	18.0000i	0.594412i
$918$	− 5.00000i	− 0.165025i
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	− 42.0000i	− 1.38320i
$923$	16.0000i	0.526646i
$924$	3.00000	0.0986928
$925$	0	0
$926$	−9.00000	−0.295758
$927$	14.0000i	0.459820i
$928$	5.00000i	0.164133i
$929$	26.0000	0.853032	0.426516	−	0.904480i	$-0.359741\pi$
$929$	26.0000	0.853032	0.426516	+	0.904480i	$0.359741\pi$
$930$	0	0
$931$	−16.0000	−0.524379
$932$	− 6.00000i	− 0.196537i
$933$	6.00000i	0.196431i
$934$	−32.0000	−1.04707
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	− 45.0000i	− 1.47009i	−0.678021	−	0.735043i	$-0.737162\pi$
$937$	− 45.0000i	− 1.47009i	0.678021	−	0.735043i	$-0.262838\pi$
$938$	− 9.00000i	− 0.293860i
$939$	29.0000	0.946379
$940$	0	0
$941$	−56.0000	−1.82555	−0.912774	−	0.408465i	$-0.866064\pi$
$941$	−56.0000	−1.82555	−0.912774	+	0.408465i	$0.866064\pi$
$942$	11.0000i	0.358399i
$943$	0	0
$944$	5.00000	0.162736
$945$	0	0
$946$	−8.00000	−0.260102
$947$	− 21.0000i	− 0.682408i	−0.939989	−	0.341204i	$-0.889165\pi$
$947$	− 21.0000i	− 0.682408i	0.939989	−	0.341204i	$-0.110835\pi$
$948$	− 12.0000i	− 0.389742i
$949$	4.00000	0.129845
$950$	0	0
$951$	−16.0000	−0.518836
$952$	45.0000i	1.45846i
$953$	37.0000i	1.19855i	0.800544	+	0.599274i	$0.204544\pi$
$953$	37.0000i	1.19855i	−0.800544	+	0.599274i	$0.795456\pi$
$954$	11.0000	0.356138
$955$	0	0
$956$	19.0000	0.614504
$957$	1.00000i	0.0323254i
$958$	− 15.0000i	− 0.484628i
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	8.00000i	0.257930i
$963$	18.0000i	0.580042i
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	− 37.0000i	− 1.18984i	−0.803785	−	0.594920i	$-0.797184\pi$
$967$	− 37.0000i	− 1.18984i	0.803785	−	0.594920i	$-0.202816\pi$
$968$	30.0000i	0.964237i
$969$	−40.0000	−1.28499
$970$	0	0
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	1.00000i	0.0320750i
$973$	30.0000i	0.961756i
$974$	−37.0000	−1.18556
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	− 28.0000i	− 0.895799i	−0.894084	−	0.447900i	$-0.852172\pi$
$977$	− 28.0000i	− 0.895799i	0.894084	−	0.447900i	$-0.147828\pi$
$978$	− 20.0000i	− 0.639529i
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	26.0000i	0.829693i
$983$	− 21.0000i	− 0.669796i	−0.942254	−	0.334898i	$-0.891298\pi$
$983$	− 21.0000i	− 0.669796i	0.942254	−	0.334898i	$-0.108702\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	−5.00000	−0.159232
$987$	− 33.0000i	− 1.05040i
$988$	8.00000i	0.254514i
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	− 15.0000i	− 0.476250i
$993$	− 12.0000i	− 0.380808i
$994$	48.0000	1.52247
$995$	0	0
$996$	3.00000	0.0950586
$997$	− 3.00000i	− 0.0950110i	−0.998871	−	0.0475055i	$-0.984873\pi$
$997$	− 3.00000i	− 0.0950110i	0.998871	−	0.0475055i	$-0.0151272\pi$
$998$	− 41.0000i	− 1.29783i
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.c.g.274.1		2
3.2	odd	2		2925.2.c.i.2224.2		2
5.2	odd	4		975.2.a.k.1.1	yes	1
5.3	odd	4		975.2.a.d.1.1	✓	1
5.4	even	2	inner	975.2.c.g.274.2		2
15.2	even	4		2925.2.a.b.1.1		1
15.8	even	4		2925.2.a.o.1.1		1
15.14	odd	2		2925.2.c.i.2224.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.d.1.1	✓	1	5.3	odd	4
975.2.a.k.1.1	yes	1	5.2	odd	4
975.2.c.g.274.1		2	1.1	even	1	trivial
975.2.c.g.274.2		2	5.4	even	2	inner
2925.2.a.b.1.1		1	15.2	even	4
2925.2.a.o.1.1		1	15.8	even	4
2925.2.c.i.2224.1		2	15.14	odd	2
2925.2.c.i.2224.2		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} +3.00000 q^{14} -1.00000 q^{16} +5.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} -3.00000 q^{21} +1.00000i q^{22} +3.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +3.00000i q^{28} -1.00000 q^{29} +3.00000 q^{31} -5.00000i q^{32} -1.00000i q^{33} +5.00000 q^{34} -1.00000 q^{36} +8.00000i q^{37} -8.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} +3.00000i q^{42} +8.00000i q^{43} -1.00000 q^{44} +11.0000i q^{47} -1.00000i q^{48} -2.00000 q^{49} -5.00000 q^{51} +1.00000i q^{52} -11.0000i q^{53} -1.00000 q^{54} +9.00000 q^{56} +8.00000i q^{57} +1.00000i q^{58} -5.00000 q^{59} +1.00000 q^{61} -3.00000i q^{62} -3.00000i q^{63} -7.00000 q^{64} -1.00000 q^{66} -3.00000i q^{67} +5.00000i q^{68} +16.0000 q^{71} +3.00000i q^{72} -4.00000i q^{73} +8.00000 q^{74} +8.00000 q^{76} -3.00000i q^{77} +1.00000i q^{78} -12.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -3.00000i q^{83} -3.00000 q^{84} +8.00000 q^{86} -1.00000i q^{87} +3.00000i q^{88} -3.00000 q^{91} +3.00000i q^{93} +11.0000 q^{94} +5.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 6 q^{14} - 2 q^{16} + 16 q^{19} - 6 q^{21} + 6 q^{24} + 2 q^{26} - 2 q^{29} + 6 q^{31} + 10 q^{34} - 2 q^{36} - 2 q^{39} - 4 q^{41} - 2 q^{44} - 4 q^{49} - 10 q^{51} - 2 q^{54} + 18 q^{56} - 10 q^{59} + 2 q^{61} - 14 q^{64} - 2 q^{66} + 32 q^{71} + 16 q^{74} + 16 q^{76} - 24 q^{79} + 2 q^{81} - 6 q^{84} + 16 q^{86} - 6 q^{91} + 22 q^{94} + 10 q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 6 * q^14 - 2 * q^16 + 16 * q^19 - 6 * q^21 + 6 * q^24 + 2 * q^26 - 2 * q^29 + 6 * q^31 + 10 * q^34 - 2 * q^36 - 2 * q^39 - 4 * q^41 - 2 * q^44 - 4 * q^49 - 10 * q^51 - 2 * q^54 + 18 * q^56 - 10 * q^59 + 2 * q^61 - 14 * q^64 - 2 * q^66 + 32 * q^71 + 16 * q^74 + 16 * q^76 - 24 * q^79 + 2 * q^81 - 6 * q^84 + 16 * q^86 - 6 * q^91 + 22 * q^94 + 10 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

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Database

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