LMFDB - Embedded newform 975.2.c.d.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.d.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} -7.00000i q^{17} +1.00000i q^{18} -1.00000 q^{21} +1.00000i q^{22} -3.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} -5.00000 q^{29} -1.00000 q^{31} -5.00000i q^{32} +1.00000i q^{33} -7.00000 q^{34} -1.00000 q^{36} -8.00000i q^{37} +1.00000 q^{39} +6.00000 q^{41} +1.00000i q^{42} +8.00000i q^{43} -1.00000 q^{44} -5.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -7.00000 q^{51} +1.00000i q^{52} +1.00000i q^{53} +1.00000 q^{54} -3.00000 q^{56} +5.00000i q^{58} +3.00000 q^{59} -7.00000 q^{61} +1.00000i q^{62} +1.00000i q^{63} -7.00000 q^{64} +1.00000 q^{66} -7.00000i q^{67} -7.00000i q^{68} +3.00000i q^{72} +12.0000i q^{73} -8.00000 q^{74} +1.00000i q^{77} -1.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -11.0000i q^{83} -1.00000 q^{84} +8.00000 q^{86} +5.00000i q^{87} +3.00000i q^{88} -8.00000 q^{89} +1.00000 q^{91} +1.00000i q^{93} -5.00000 q^{94} -5.00000 q^{96} +10.0000i q^{97} -6.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{14} - 2 q^{16} - 2 q^{21} - 6 q^{24} + 2 q^{26} - 10 q^{29} - 2 q^{31} - 14 q^{34} - 2 q^{36} + 2 q^{39} + 12 q^{41} - 2 q^{44} + 12 q^{49} - 14 q^{51} + 2 q^{54} - 6 q^{56} + 6 q^{59} - 14 q^{61} - 14 q^{64} + 2 q^{66} - 16 q^{74} + 24 q^{79} + 2 q^{81} - 2 q^{84} + 16 q^{86} - 16 q^{89} + 2 q^{91} - 10 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 - 2 * q^14 - 2 * q^16 - 2 * q^21 - 6 * q^24 + 2 * q^26 - 10 * q^29 - 2 * q^31 - 14 * q^34 - 2 * q^36 + 2 * q^39 + 12 * q^41 - 2 * q^44 + 12 * q^49 - 14 * q^51 + 2 * q^54 - 6 * q^56 + 6 * q^59 - 14 * q^61 - 14 * q^64 + 2 * q^66 - 16 * q^74 + 24 * q^79 + 2 * q^81 - 2 * q^84 + 16 * q^86 - 16 * q^89 + 2 * q^91 - 10 * 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
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	− 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
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 7.00000i	− 1.69775i	−0.528594	−	0.848875i	$-0.677281\pi$
$17$	− 7.00000i	− 1.69775i	0.528594	−	0.848875i	$-0.322719\pi$
$18$	1.00000i	0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	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
$28$	− 1.00000i	− 0.188982i
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	− 5.00000i	− 0.883883i
$33$	1.00000i	0.174078i
$34$	−7.00000	−1.20049
$35$	0	0
$36$	−1.00000	−0.166667
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	1.00000i	0.154303i
$43$	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
$47$	− 5.00000i	− 0.729325i	−0.931140	−	0.364662i	$-0.881184\pi$
$47$	− 5.00000i	− 0.729325i	0.931140	−	0.364662i	$-0.118816\pi$
$48$	1.00000i	0.144338i
$49$	6.00000	0.857143
$50$	0	0
$51$	−7.00000	−0.980196
$52$	1.00000i	0.138675i
$53$	1.00000i	0.137361i	0.997639	+	0.0686803i	$0.0218788\pi$
$53$	1.00000i	0.137361i	−0.997639	+	0.0686803i	$0.978121\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	5.00000i	0.656532i
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	1.00000i	0.127000i
$63$	1.00000i	0.125988i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	1.00000	0.123091
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	− 7.00000i	− 0.848875i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000i	0.353553i
$73$	12.0000i	1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$	12.0000i	1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	0	0
$77$	1.00000i	0.113961i
$78$	− 1.00000i	− 0.113228i
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	− 11.0000i	− 1.20741i	−0.797209	−	0.603703i	$-0.793691\pi$
$83$	− 11.0000i	− 1.20741i	0.797209	−	0.603703i	$-0.206309\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	8.00000	0.862662
$87$	5.00000i	0.536056i
$88$	3.00000i	0.319801i
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.00000i	0.103695i
$94$	−5.00000	−0.515711
$95$	0	0
$96$	−5.00000	−0.510310
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 6.00000i	− 0.606092i
$99$	1.00000	0.100504
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	7.00000i	0.693103i
$103$	10.0000i	0.985329i	0.870219	+	0.492665i	$0.163977\pi$
$103$	10.0000i	0.985329i	−0.870219	+	0.492665i	$0.836023\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	1.00000	0.0971286
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	1.00000i	0.0962250i
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	1.00000i	0.0944911i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	−5.00000	−0.464238
$117$	− 1.00000i	− 0.0924500i
$118$	− 3.00000i	− 0.276172i
$119$	−7.00000	−0.641689
$120$	0	0
$121$	−10.0000	−0.909091
$122$	7.00000i	0.633750i
$123$	− 6.00000i	− 0.541002i
$124$	−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	$0.748747\pi$
$128$	− 3.00000i	− 0.265165i
$129$	8.00000	0.704361
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	1.00000i	0.0870388i
$133$	0	0
$134$	−7.00000	−0.604708
$135$	0	0
$136$	−21.0000	−1.80074
$137$	4.00000i	0.341743i	0.985293	+	0.170872i	$0.0546583\pi$
$137$	4.00000i	0.341743i	−0.985293	+	0.170872i	$0.945342\pi$
$138$	0	0
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	0	0
$141$	−5.00000	−0.421076
$142$	0	0
$143$	− 1.00000i	− 0.0836242i
$144$	1.00000	0.0833333
$145$	0	0
$146$	12.0000	0.993127
$147$	− 6.00000i	− 0.494872i
$148$	− 8.00000i	− 0.657596i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	0	0
$153$	7.00000i	0.565916i
$154$	1.00000	0.0805823
$155$	0	0
$156$	1.00000	0.0800641
$157$	3.00000i	0.239426i	0.992809	+	0.119713i	$0.0381975\pi$
$157$	3.00000i	0.239426i	−0.992809	+	0.119713i	$0.961803\pi$
$158$	− 12.0000i	− 0.954669i
$159$	1.00000	0.0793052
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−11.0000	−0.853766
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	3.00000i	0.231455i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	− 9.00000i	− 0.684257i	−0.939653	−	0.342129i	$-0.888852\pi$
$173$	− 9.00000i	− 0.684257i	0.939653	−	0.342129i	$-0.111148\pi$
$174$	5.00000	0.379049
$175$	0	0
$176$	1.00000	0.0753778
$177$	− 3.00000i	− 0.225494i
$178$	8.00000i	0.599625i
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	− 1.00000i	− 0.0741249i
$183$	7.00000i	0.517455i
$184$	0	0
$185$	0	0
$186$	1.00000	0.0733236
$187$	7.00000i	0.511891i
$188$	− 5.00000i	− 0.364662i
$189$	1.00000	0.0727393
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	7.00000i	0.505181i
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	6.00000	0.428571
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	− 1.00000i	− 0.0710669i
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	3.00000i	0.211079i
$203$	5.00000i	0.350931i
$204$	−7.00000	−0.490098
$205$	0	0
$206$	10.0000	0.696733
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	1.00000i	0.0686803i
$213$	0	0
$214$	−2.00000	−0.136717
$215$	0	0
$216$	3.00000	0.204124
$217$	1.00000i	0.0678844i
$218$	− 2.00000i	− 0.135457i
$219$	12.0000	0.810885
$220$	0	0
$221$	7.00000	0.470871
$222$	8.00000i	0.536925i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	14.0000	0.931266
$227$	25.0000i	1.65931i	0.558278	+	0.829654i	$0.311462\pi$
$227$	25.0000i	1.65931i	−0.558278	+	0.829654i	$0.688538\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	15.0000i	0.984798i
$233$	26.0000i	1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$	26.0000i	1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	3.00000	0.195283
$237$	− 12.0000i	− 0.779484i
$238$	7.00000i	0.453743i
$239$	−21.0000	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$240$	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
$244$	−7.00000	−0.448129
$245$	0	0
$246$	−6.00000	−0.382546
$247$	0	0
$248$	3.00000i	0.190500i
$249$	−11.0000	−0.697097
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	1.00000i	0.0629941i
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 1.00000i	− 0.0623783i	−0.999514	−	0.0311891i	$-0.990071\pi$
$257$	− 1.00000i	− 0.0623783i	0.999514	−	0.0311891i	$-0.00992942\pi$
$258$	− 8.00000i	− 0.498058i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	5.00000	0.309492
$262$	− 14.0000i	− 0.864923i
$263$	− 26.0000i	− 1.60323i	−0.597841	−	0.801614i	$-0.703975\pi$
$263$	− 26.0000i	− 1.60323i	0.597841	−	0.801614i	$-0.296025\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	0	0
$267$	8.00000i	0.489592i
$268$	− 7.00000i	− 0.427593i
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	7.00000i	0.424437i
$273$	− 1.00000i	− 0.0605228i
$274$	4.00000	0.241649
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	− 18.0000i	− 1.07957i
$279$	1.00000	0.0598684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	5.00000i	0.297746i
$283$	22.0000i	1.30776i	0.756596	+	0.653882i	$0.226861\pi$
$283$	22.0000i	1.30776i	−0.756596	+	0.653882i	$0.773139\pi$
$284$	0	0
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	− 6.00000i	− 0.354169i
$288$	5.00000i	0.294628i
$289$	−32.0000	−1.88235
$290$	0	0
$291$	10.0000	0.586210
$292$	12.0000i	0.702247i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	−24.0000	−1.39497
$297$	− 1.00000i	− 0.0580259i
$298$	− 6.00000i	− 0.347571i
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	− 19.0000i	− 1.09333i
$303$	3.00000i	0.172345i
$304$	0	0
$305$	0	0
$306$	7.00000	0.400163
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	1.00000i	0.0569803i
$309$	10.0000	0.568880
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	− 3.00000i	− 0.169842i
$313$	19.0000i	1.07394i	0.843600	+	0.536972i	$0.180432\pi$
$313$	19.0000i	1.07394i	−0.843600	+	0.536972i	$0.819568\pi$
$314$	3.00000	0.169300
$315$	0	0
$316$	12.0000	0.675053
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	− 1.00000i	− 0.0560772i
$319$	5.00000	0.279946
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	− 2.00000i	− 0.110600i
$328$	− 18.0000i	− 0.993884i
$329$	−5.00000	−0.275659
$330$	0	0
$331$	−36.0000	−1.97874	−0.989369	−	0.145424i	$-0.953545\pi$
$331$	−36.0000	−1.97874	−0.989369	+	0.145424i	$0.953545\pi$
$332$	− 11.0000i	− 0.603703i
$333$	8.00000i	0.438397i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	1.00000	0.0545545
$337$	− 13.0000i	− 0.708155i	−0.935216	−	0.354078i	$-0.884795\pi$
$337$	− 13.0000i	− 0.708155i	0.935216	−	0.354078i	$-0.115205\pi$
$338$	1.00000i	0.0543928i
$339$	14.0000	0.760376
$340$	0	0
$341$	1.00000	0.0541530
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	24.0000	1.29399
$345$	0	0
$346$	−9.00000	−0.483843
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	5.00000i	0.268028i
$349$	−24.0000	−1.28469	−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000	−1.28469	−0.642345	+	0.766415i	$0.722038\pi$
$350$	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	$0.840992\pi$
$354$	−3.00000	−0.159448
$355$	0	0
$356$	−8.00000	−0.423999
$357$	7.00000i	0.370479i
$358$	− 6.00000i	− 0.317110i
$359$	1.00000	0.0527780	0.0263890	−	0.999652i	$-0.491599\pi$
$359$	1.00000	0.0527780	0.0263890	+	0.999652i	$0.491599\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 5.00000i	− 0.262794i
$363$	10.0000i	0.524864i
$364$	1.00000	0.0524142
$365$	0	0
$366$	7.00000	0.365896
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	1.00000	0.0519174
$372$	1.00000i	0.0518476i
$373$	− 31.0000i	− 1.60512i	−0.596572	−	0.802560i	$-0.703471\pi$
$373$	− 31.0000i	− 1.60512i	0.596572	−	0.802560i	$-0.296529\pi$
$374$	7.00000	0.361961
$375$	0	0
$376$	−15.0000	−0.773566
$377$	− 5.00000i	− 0.257513i
$378$	− 1.00000i	− 0.0514344i
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	− 18.0000i	− 0.920960i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−22.0000	−1.11977
$387$	− 8.00000i	− 0.406663i
$388$	10.0000i	0.507673i
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	0	0
$392$	− 18.0000i	− 0.909137i
$393$	− 14.0000i	− 0.706207i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	1.00000	0.0502519
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	− 22.0000i	− 1.10276i
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	7.00000i	0.349128i
$403$	− 1.00000i	− 0.0498135i
$404$	−3.00000	−0.149256
$405$	0	0
$406$	5.00000	0.248146
$407$	8.00000i	0.396545i
$408$	21.0000i	1.03965i
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	10.0000i	0.492665i
$413$	− 3.00000i	− 0.147620i
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	− 18.0000i	− 0.881464i
$418$	0	0
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	0	0
$423$	5.00000i	0.243108i
$424$	3.00000	0.145693
$425$	0	0
$426$	0	0
$427$	7.00000i	0.338754i
$428$	− 2.00000i	− 0.0966736i
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	1.00000	0.0480015
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	− 12.0000i	− 0.573382i
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	− 7.00000i	− 0.332956i
$443$	16.0000i	0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$	16.0000i	0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	−16.0000	−0.757622
$447$	− 6.00000i	− 0.283790i
$448$	7.00000i	0.330719i
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	14.0000i	0.658505i
$453$	− 19.0000i	− 0.892698i
$454$	25.0000	1.17331
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	− 10.0000i	− 0.467269i
$459$	7.00000	0.326732
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	− 1.00000i	− 0.0465242i
$463$	− 21.0000i	− 0.975953i	−0.872857	−	0.487976i	$-0.837735\pi$
$463$	− 21.0000i	− 0.975953i	0.872857	−	0.487976i	$-0.162265\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	26.0000	1.20443
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	−7.00000	−0.323230
$470$	0	0
$471$	3.00000	0.138233
$472$	− 9.00000i	− 0.414259i
$473$	− 8.00000i	− 0.367840i
$474$	−12.0000	−0.551178
$475$	0	0
$476$	−7.00000	−0.320844
$477$	− 1.00000i	− 0.0457869i
$478$	21.0000i	0.960518i
$479$	39.0000	1.78196	0.890978	−	0.454047i	$-0.150020\pi$
$479$	39.0000	1.78196	0.890978	+	0.454047i	$0.150020\pi$
$480$	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
$487$	23.0000i	1.04223i	0.853487	+	0.521115i	$0.174484\pi$
$487$	23.0000i	1.04223i	−0.853487	+	0.521115i	$0.825516\pi$
$488$	21.0000i	0.950625i
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	− 6.00000i	− 0.270501i
$493$	35.0000i	1.57632i
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	0	0
$498$	11.0000i	0.492922i
$499$	29.0000	1.29822	0.649109	−	0.760695i	$-0.275142\pi$
$499$	29.0000	1.29822	0.649109	+	0.760695i	$0.275142\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	− 18.0000i	− 0.803379i
$503$	42.0000i	1.87269i	0.351085	+	0.936344i	$0.385813\pi$
$503$	42.0000i	1.87269i	−0.351085	+	0.936344i	$0.614187\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	16.0000i	0.709885i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	11.0000i	0.486136i
$513$	0	0
$514$	−1.00000	−0.0441081
$515$	0	0
$516$	8.00000	0.352180
$517$	5.00000i	0.219900i
$518$	8.00000i	0.351500i
$519$	−9.00000	−0.395056
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	− 5.00000i	− 0.218844i
$523$	− 6.00000i	− 0.262362i	−0.991358	−	0.131181i	$-0.958123\pi$
$523$	− 6.00000i	− 0.262362i	0.991358	−	0.131181i	$-0.0418769\pi$
$524$	14.0000	0.611593
$525$	0	0
$526$	−26.0000	−1.13365
$527$	7.00000i	0.304925i
$528$	− 1.00000i	− 0.0435194i
$529$	23.0000	1.00000
$530$	0	0
$531$	−3.00000	−0.130189
$532$	0	0
$533$	6.00000i	0.259889i
$534$	8.00000	0.346194
$535$	0	0
$536$	−21.0000	−0.907062
$537$	− 6.00000i	− 0.258919i
$538$	− 3.00000i	− 0.129339i
$539$	−6.00000	−0.258438
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	− 23.0000i	− 0.987935i
$543$	− 5.00000i	− 0.214571i
$544$	−35.0000	−1.50061
$545$	0	0
$546$	−1.00000	−0.0427960
$547$	6.00000i	0.256541i	0.991739	+	0.128271i	$0.0409426\pi$
$547$	6.00000i	0.256541i	−0.991739	+	0.128271i	$0.959057\pi$
$548$	4.00000i	0.170872i
$549$	7.00000	0.298753
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 12.0000i	− 0.510292i
$554$	10.0000	0.424859
$555$	0	0
$556$	18.0000	0.763370
$557$	− 38.0000i	− 1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	− 38.0000i	− 1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	−8.00000	−0.338364
$560$	0	0
$561$	7.00000	0.295540
$562$	− 30.0000i	− 1.26547i
$563$	− 32.0000i	− 1.34864i	−0.738440	−	0.674320i	$-0.764437\pi$
$563$	− 32.0000i	− 1.34864i	0.738440	−	0.674320i	$-0.235563\pi$
$564$	−5.00000	−0.210538
$565$	0	0
$566$	22.0000	0.924729
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	−19.0000	−0.796521	−0.398261	−	0.917272i	$-0.630386\pi$
$569$	−19.0000	−0.796521	−0.398261	+	0.917272i	$0.630386\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	− 1.00000i	− 0.0418121i
$573$	− 18.0000i	− 0.751961i
$574$	−6.00000	−0.250435
$575$	0	0
$576$	7.00000	0.291667
$577$	− 34.0000i	− 1.41544i	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	− 34.0000i	− 1.41544i	0.706494	−	0.707719i	$-0.250276\pi$
$578$	32.0000i	1.33102i
$579$	−22.0000	−0.914289
$580$	0	0
$581$	−11.0000	−0.456357
$582$	− 10.0000i	− 0.414513i
$583$	− 1.00000i	− 0.0414158i
$584$	36.0000	1.48969
$585$	0	0
$586$	14.0000	0.578335
$587$	37.0000i	1.52715i	0.645717	+	0.763577i	$0.276559\pi$
$587$	37.0000i	1.52715i	−0.645717	+	0.763577i	$0.723441\pi$
$588$	− 6.00000i	− 0.247436i
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	8.00000i	0.328798i
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	6.00000	0.245770
$597$	− 22.0000i	− 0.900400i
$598$	0	0
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	0	0
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	− 8.00000i	− 0.326056i
$603$	7.00000i	0.285062i
$604$	19.0000	0.773099
$605$	0	0
$606$	3.00000	0.121867
$607$	− 14.0000i	− 0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$	− 14.0000i	− 0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$	0	0
$609$	5.00000	0.202610
$610$	0	0
$611$	5.00000	0.202278
$612$	7.00000i	0.282958i
$613$	− 42.0000i	− 1.69636i	−0.529705	−	0.848182i	$-0.677697\pi$
$613$	− 42.0000i	− 1.69636i	0.529705	−	0.848182i	$-0.322303\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	3.00000	0.120873
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	− 10.0000i	− 0.402259i
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	10.0000i	0.400963i
$623$	8.00000i	0.320513i
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	19.0000	0.759393
$627$	0	0
$628$	3.00000i	0.119713i
$629$	−56.0000	−2.23287
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	− 36.0000i	− 1.43200i
$633$	0	0
$634$	24.0000	0.953162
$635$	0	0
$636$	1.00000	0.0396526
$637$	6.00000i	0.237729i
$638$	− 5.00000i	− 0.197952i
$639$	0	0
$640$	0	0
$641$	−43.0000	−1.69840	−0.849199	−	0.528073i	$-0.822915\pi$
$641$	−43.0000	−1.69840	−0.849199	+	0.528073i	$0.822915\pi$
$642$	2.00000i	0.0789337i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	− 3.00000i	− 0.117851i
$649$	−3.00000	−0.117760
$650$	0	0
$651$	1.00000	0.0391931
$652$	− 12.0000i	− 0.469956i
$653$	− 9.00000i	− 0.352197i	−0.984373	−	0.176099i	$-0.943652\pi$
$653$	− 9.00000i	− 0.352197i	0.984373	−	0.176099i	$-0.0563478\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 12.0000i	− 0.468165i
$658$	5.00000i	0.194920i
$659$	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	$0.562301\pi$
$662$	36.0000i	1.39918i
$663$	− 7.00000i	− 0.271857i
$664$	−33.0000	−1.28065
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	− 12.0000i	− 0.464294i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	7.00000	0.270232
$672$	5.00000i	0.192879i
$673$	21.0000i	0.809491i	0.914429	+	0.404745i	$0.132640\pi$
$673$	21.0000i	0.809491i	−0.914429	+	0.404745i	$0.867360\pi$
$674$	−13.0000	−0.500741
$675$	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	$0.887574\pi$
$678$	− 14.0000i	− 0.537667i
$679$	10.0000	0.383765
$680$	0	0
$681$	25.0000	0.958002
$682$	− 1.00000i	− 0.0382920i
$683$	− 7.00000i	− 0.267848i	−0.990992	−	0.133924i	$-0.957242\pi$
$683$	− 7.00000i	− 0.267848i	0.990992	−	0.133924i	$-0.0427577\pi$
$684$	0	0
$685$	0	0
$686$	−13.0000	−0.496342
$687$	− 10.0000i	− 0.381524i
$688$	− 8.00000i	− 0.304997i
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	31.0000	1.17930	0.589648	−	0.807661i	$-0.299267\pi$
$691$	31.0000	1.17930	0.589648	+	0.807661i	$0.299267\pi$
$692$	− 9.00000i	− 0.342129i
$693$	− 1.00000i	− 0.0379869i
$694$	2.00000	0.0759190
$695$	0	0
$696$	15.0000	0.568574
$697$	− 42.0000i	− 1.59086i
$698$	24.0000i	0.908413i
$699$	26.0000	0.983410
$700$	0	0
$701$	−29.0000	−1.09531	−0.547657	−	0.836703i	$-0.684480\pi$
$701$	−29.0000	−1.09531	−0.547657	+	0.836703i	$0.684480\pi$
$702$	1.00000i	0.0377426i
$703$	0	0
$704$	7.00000	0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	3.00000i	0.112827i
$708$	− 3.00000i	− 0.112747i
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	24.0000i	0.899438i
$713$	0	0
$714$	7.00000	0.261968
$715$	0	0
$716$	6.00000	0.224231
$717$	21.0000i	0.784259i
$718$	− 1.00000i	− 0.0373197i
$719$	−46.0000	−1.71551	−0.857755	−	0.514058i	$-0.828142\pi$
$719$	−46.0000	−1.71551	−0.857755	+	0.514058i	$0.828142\pi$
$720$	0	0
$721$	10.0000	0.372419
$722$	19.0000i	0.707107i
$723$	− 10.0000i	− 0.371904i
$724$	5.00000	0.185824
$725$	0	0
$726$	10.0000	0.371135
$727$	42.0000i	1.55769i	0.627214	+	0.778847i	$0.284195\pi$
$727$	42.0000i	1.55769i	−0.627214	+	0.778847i	$0.715805\pi$
$728$	− 3.00000i	− 0.111187i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	56.0000	2.07123
$732$	7.00000i	0.258727i
$733$	20.0000i	0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$	20.0000i	0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	0	0
$737$	7.00000i	0.257848i
$738$	6.00000i	0.220863i
$739$	−27.0000	−0.993211	−0.496606	−	0.867976i	$-0.665420\pi$
$739$	−27.0000	−0.993211	−0.496606	+	0.867976i	$0.665420\pi$
$740$	0	0
$741$	0	0
$742$	− 1.00000i	− 0.0367112i
$743$	− 27.0000i	− 0.990534i	−0.868741	−	0.495267i	$-0.835070\pi$
$743$	− 27.0000i	− 0.990534i	0.868741	−	0.495267i	$-0.164930\pi$
$744$	3.00000	0.109985
$745$	0	0
$746$	−31.0000	−1.13499
$747$	11.0000i	0.402469i
$748$	7.00000i	0.255945i
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	5.00000i	0.182331i
$753$	− 18.0000i	− 0.655956i
$754$	−5.00000	−0.182089
$755$	0	0
$756$	1.00000	0.0363696
$757$	− 7.00000i	− 0.254419i	−0.991876	−	0.127210i	$-0.959398\pi$
$757$	− 7.00000i	− 0.254419i	0.991876	−	0.127210i	$-0.0406021\pi$
$758$	1.00000i	0.0363216i
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	− 16.0000i	− 0.579619i
$763$	− 2.00000i	− 0.0724049i
$764$	18.0000	0.651217
$765$	0	0
$766$	24.0000	0.867155
$767$	3.00000i	0.108324i
$768$	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
$771$	−1.00000	−0.0360141
$772$	− 22.0000i	− 0.791797i
$773$	50.0000i	1.79838i	0.437564	+	0.899188i	$0.355842\pi$
$773$	50.0000i	1.79838i	−0.437564	+	0.899188i	$0.644158\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	30.0000	1.07694
$777$	8.00000i	0.286998i
$778$	2.00000i	0.0717035i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 5.00000i	− 0.178685i
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−14.0000	−0.499363
$787$	− 13.0000i	− 0.463400i	−0.972787	−	0.231700i	$-0.925571\pi$
$787$	− 13.0000i	− 0.463400i	0.972787	−	0.231700i	$-0.0744288\pi$
$788$	− 18.0000i	− 0.641223i
$789$	−26.0000	−0.925625
$790$	0	0
$791$	14.0000	0.497783
$792$	− 3.00000i	− 0.106600i
$793$	− 7.00000i	− 0.248577i
$794$	−18.0000	−0.638796
$795$	0	0
$796$	22.0000	0.779769
$797$	39.0000i	1.38145i	0.723117	+	0.690725i	$0.242709\pi$
$797$	39.0000i	1.38145i	−0.723117	+	0.690725i	$0.757291\pi$
$798$	0	0
$799$	−35.0000	−1.23821
$800$	0	0
$801$	8.00000	0.282666
$802$	8.00000i	0.282490i
$803$	− 12.0000i	− 0.423471i
$804$	−7.00000	−0.246871
$805$	0	0
$806$	−1.00000	−0.0352235
$807$	− 3.00000i	− 0.105605i
$808$	9.00000i	0.316619i
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	0	0
$811$	3.00000	0.105344	0.0526721	−	0.998612i	$-0.483226\pi$
$811$	3.00000	0.105344	0.0526721	+	0.998612i	$0.483226\pi$
$812$	5.00000i	0.175466i
$813$	− 23.0000i	− 0.806645i
$814$	8.00000	0.280400
$815$	0	0
$816$	7.00000	0.245049
$817$	0	0
$818$	− 6.00000i	− 0.209785i
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	− 4.00000i	− 0.139516i
$823$	− 20.0000i	− 0.697156i	−0.937280	−	0.348578i	$-0.886665\pi$
$823$	− 20.0000i	− 0.697156i	0.937280	−	0.348578i	$-0.113335\pi$
$824$	30.0000	1.04510
$825$	0	0
$826$	−3.00000	−0.104383
$827$	31.0000i	1.07798i	0.842314	+	0.538988i	$0.181193\pi$
$827$	31.0000i	1.07798i	−0.842314	+	0.538988i	$0.818807\pi$
$828$	0	0
$829$	29.0000	1.00721	0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000	1.00721	0.503606	+	0.863934i	$0.332006\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	− 7.00000i	− 0.242681i
$833$	− 42.0000i	− 1.45521i
$834$	−18.0000	−0.623289
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	14.0000i	0.483622i
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	20.0000i	0.689246i
$843$	− 30.0000i	− 1.03325i
$844$	0	0
$845$	0	0
$846$	5.00000	0.171904
$847$	10.0000i	0.343604i
$848$	− 1.00000i	− 0.0343401i
$849$	22.0000	0.755038
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	7.00000	0.239535
$855$	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	$0.967323\pi$
$858$	1.00000i	0.0341394i
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	− 20.0000i	− 0.681203i
$863$	− 39.0000i	− 1.32758i	−0.747921	−	0.663788i	$-0.768948\pi$
$863$	− 39.0000i	− 1.32758i	0.747921	−	0.663788i	$-0.231052\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	14.0000	0.475739
$867$	32.0000i	1.08678i
$868$	1.00000i	0.0339422i
$869$	−12.0000	−0.407072
$870$	0	0
$871$	7.00000	0.237186
$872$	− 6.00000i	− 0.203186i
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	12.0000	0.405442
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	12.0000i	0.404980i
$879$	14.0000	0.472208
$880$	0	0
$881$	−37.0000	−1.24656	−0.623281	−	0.781998i	$-0.714201\pi$
$881$	−37.0000	−1.24656	−0.623281	+	0.781998i	$0.714201\pi$
$882$	6.00000i	0.202031i
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	7.00000	0.235435
$885$	0	0
$886$	16.0000	0.537531
$887$	28.0000i	0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$	28.0000i	0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$	24.0000i	0.805387i
$889$	16.0000	0.536623
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	−6.00000	−0.200670
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	12.0000i	0.400445i
$899$	5.00000	0.166759
$900$	0	0
$901$	7.00000	0.233204
$902$	6.00000i	0.199778i
$903$	− 8.00000i	− 0.266223i
$904$	42.0000	1.39690
$905$	0	0
$906$	−19.0000	−0.631233
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	25.0000i	0.829654i
$909$	3.00000	0.0995037
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	11.0000i	0.364047i
$914$	−22.0000	−0.727695
$915$	0	0
$916$	10.0000	0.330409
$917$	− 14.0000i	− 0.462321i
$918$	− 7.00000i	− 0.231034i
$919$	58.0000	1.91324	0.956622	−	0.291333i	$-0.0940987\pi$
$919$	58.0000	1.91324	0.956622	+	0.291333i	$0.0940987\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	38.0000i	1.25146i
$923$	0	0
$924$	1.00000	0.0328976
$925$	0	0
$926$	−21.0000	−0.690103
$927$	− 10.0000i	− 0.328443i
$928$	25.0000i	0.820665i
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	0	0
$932$	26.0000i	0.851658i
$933$	10.0000i	0.327385i
$934$	8.00000	0.261768
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	− 21.0000i	− 0.686040i	−0.939328	−	0.343020i	$-0.888550\pi$
$937$	− 21.0000i	− 0.686040i	0.939328	−	0.343020i	$-0.111450\pi$
$938$	7.00000i	0.228558i
$939$	19.0000	0.620042
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	− 3.00000i	− 0.0977453i
$943$	0	0
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	−8.00000	−0.260102
$947$	− 45.0000i	− 1.46230i	−0.682215	−	0.731152i	$-0.738983\pi$
$947$	− 45.0000i	− 1.46230i	0.682215	−	0.731152i	$-0.261017\pi$
$948$	− 12.0000i	− 0.389742i
$949$	−12.0000	−0.389536
$950$	0	0
$951$	24.0000	0.778253
$952$	21.0000i	0.680614i
$953$	− 15.0000i	− 0.485898i	−0.970039	−	0.242949i	$-0.921885\pi$
$953$	− 15.0000i	− 0.485898i	0.970039	−	0.242949i	$-0.0781147\pi$
$954$	−1.00000	−0.0323762
$955$	0	0
$956$	−21.0000	−0.679189
$957$	− 5.00000i	− 0.161627i
$958$	− 39.0000i	− 1.26003i
$959$	4.00000	0.129167
$960$	0	0
$961$	−30.0000	−0.967742
$962$	− 8.00000i	− 0.257930i
$963$	2.00000i	0.0644491i
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	7.00000i	0.225105i	0.993646	+	0.112552i	$0.0359026\pi$
$967$	7.00000i	0.225105i	−0.993646	+	0.112552i	$0.964097\pi$
$968$	30.0000i	0.964237i
$969$	0	0
$970$	0	0
$971$	−10.0000	−0.320915	−0.160458	−	0.987043i	$-0.551297\pi$
$971$	−10.0000	−0.320915	−0.160458	+	0.987043i	$0.551297\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 18.0000i	− 0.577054i
$974$	23.0000	0.736968
$975$	0	0
$976$	7.00000	0.224065
$977$	12.0000i	0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$	12.0000i	0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$	12.0000i	0.383718i
$979$	8.00000	0.255681
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	42.0000i	1.34027i
$983$	− 29.0000i	− 0.924956i	−0.886631	−	0.462478i	$-0.846960\pi$
$983$	− 29.0000i	− 0.924956i	0.886631	−	0.462478i	$-0.153040\pi$
$984$	−18.0000	−0.573819
$985$	0	0
$986$	35.0000	1.11463
$987$	5.00000i	0.159152i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	5.00000i	0.158750i
$993$	36.0000i	1.14243i
$994$	0	0
$995$	0	0
$996$	−11.0000	−0.348548
$997$	− 19.0000i	− 0.601736i	−0.953666	−	0.300868i	$-0.902724\pi$
$997$	− 19.0000i	− 0.601736i	0.953666	−	0.300868i	$-0.0972764\pi$
$998$	− 29.0000i	− 0.917979i
$999$	8.00000	0.253109

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.e.1.1	✓	1	5.3	odd	4
975.2.a.j.1.1	yes	1	5.2	odd	4
975.2.c.d.274.1		2	1.1	even	1	trivial
975.2.c.d.274.2		2	5.4	even	2	inner
2925.2.a.e.1.1		1	15.2	even	4
2925.2.a.n.1.1		1	15.8	even	4
2925.2.c.j.2224.1		2	15.14	odd	2
2925.2.c.j.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} -7.00000i q^{17} +1.00000i q^{18} -1.00000 q^{21} +1.00000i q^{22} -3.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} -5.00000 q^{29} -1.00000 q^{31} -5.00000i q^{32} +1.00000i q^{33} -7.00000 q^{34} -1.00000 q^{36} -8.00000i q^{37} +1.00000 q^{39} +6.00000 q^{41} +1.00000i q^{42} +8.00000i q^{43} -1.00000 q^{44} -5.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -7.00000 q^{51} +1.00000i q^{52} +1.00000i q^{53} +1.00000 q^{54} -3.00000 q^{56} +5.00000i q^{58} +3.00000 q^{59} -7.00000 q^{61} +1.00000i q^{62} +1.00000i q^{63} -7.00000 q^{64} +1.00000 q^{66} -7.00000i q^{67} -7.00000i q^{68} +3.00000i q^{72} +12.0000i q^{73} -8.00000 q^{74} +1.00000i q^{77} -1.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -11.0000i q^{83} -1.00000 q^{84} +8.00000 q^{86} +5.00000i q^{87} +3.00000i q^{88} -8.00000 q^{89} +1.00000 q^{91} +1.00000i q^{93} -5.00000 q^{94} -5.00000 q^{96} +10.0000i q^{97} -6.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 2 q^{11} - 2 q^{14} - 2 q^{16} - 2 q^{21} - 6 q^{24} + 2 q^{26} - 10 q^{29} - 2 q^{31} - 14 q^{34} - 2 q^{36} + 2 q^{39} + 12 q^{41} - 2 q^{44} + 12 q^{49} - 14 q^{51} + 2 q^{54} - 6 q^{56} + 6 q^{59} - 14 q^{61} - 14 q^{64} + 2 q^{66} - 16 q^{74} + 24 q^{79} + 2 q^{81} - 2 q^{84} + 16 q^{86} - 16 q^{89} + 2 q^{91} - 10 q^{94} - 10 q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 2 * q^11 - 2 * q^14 - 2 * q^16 - 2 * q^21 - 6 * q^24 + 2 * q^26 - 10 * q^29 - 2 * q^31 - 14 * q^34 - 2 * q^36 + 2 * q^39 + 12 * q^41 - 2 * q^44 + 12 * q^49 - 14 * q^51 + 2 * q^54 - 6 * q^56 + 6 * q^59 - 14 * q^61 - 14 * q^64 + 2 * q^66 - 16 * q^74 + 24 * q^79 + 2 * q^81 - 2 * q^84 + 16 * q^86 - 16 * q^89 + 2 * q^91 - 10 * q^94 - 10 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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