LMFDB - Embedded newform 975.2.c.b.274.2

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:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	975.274
Dual form			975.2.c.b.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} -5.00000 q^{11} +2.00000i q^{12} -1.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} +5.00000i q^{17} -2.00000i q^{18} -2.00000 q^{19} -3.00000 q^{21} -10.0000i q^{22} +1.00000i q^{23} +2.00000 q^{26} +1.00000i q^{27} +6.00000i q^{28} -10.0000 q^{29} -2.00000 q^{31} -8.00000i q^{32} +5.00000i q^{33} -10.0000 q^{34} +2.00000 q^{36} -3.00000i q^{37} -4.00000i q^{38} -1.00000 q^{39} -9.00000 q^{41} -6.00000i q^{42} +4.00000i q^{43} +10.0000 q^{44} -2.00000 q^{46} +10.0000i q^{47} +4.00000i q^{48} -2.00000 q^{49} +5.00000 q^{51} +2.00000i q^{52} -9.00000i q^{53} -2.00000 q^{54} +2.00000i q^{57} -20.0000i q^{58} -11.0000 q^{61} -4.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} -10.0000 q^{66} -4.00000i q^{67} -10.0000i q^{68} +1.00000 q^{69} +15.0000 q^{71} -6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +15.0000i q^{77} -2.00000i q^{78} +11.0000 q^{79} +1.00000 q^{81} -18.0000i q^{82} -8.00000i q^{83} +6.00000 q^{84} -8.00000 q^{86} +10.0000i q^{87} +11.0000 q^{89} -3.00000 q^{91} -2.00000i q^{92} +2.00000i q^{93} -20.0000 q^{94} -8.00000 q^{96} -9.00000i q^{97} -4.00000i q^{98} +5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9} - 10 q^{11} + 12 q^{14} - 8 q^{16} - 4 q^{19} - 6 q^{21} + 4 q^{26} - 20 q^{29} - 4 q^{31} - 20 q^{34} + 4 q^{36} - 2 q^{39} - 18 q^{41} + 20 q^{44} - 4 q^{46} - 4 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 - 10 * q^11 + 12 * q^14 - 8 * q^16 - 4 * q^19 - 6 * q^21 + 4 * q^26 - 20 * q^29 - 4 * q^31 - 20 * q^34 + 4 * q^36 - 2 * q^39 - 18 * q^41 + 20 * q^44 - 4 * q^46 - 4 * q^49 + 10 * q^51 - 4 * q^54 - 22 * q^61 + 16 * q^64 - 20 * q^66 + 2 * q^69 + 30 * q^71 + 12 * q^74 + 8 * q^76 + 22 * q^79 + 2 * q^81 + 12 * q^84 - 16 * q^86 + 22 * q^89 - 6 * q^91 - 40 * q^94 - 16 * q^96 + 10 * q^99

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 $

$2$	2.00000i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$	2.00000i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	2.00000i	0.577350i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	6.00000	1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	5.00000i	1.21268i	0.795206	+	0.606339i	$0.207363\pi$
$17$	5.00000i	1.21268i	−0.795206	+	0.606339i	$0.792637\pi$
$18$	− 2.00000i	− 0.471405i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	− 10.0000i	− 2.13201i
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000i	0.192450i
$28$	6.00000i	1.13389i
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	− 8.00000i	− 1.41421i
$33$	5.00000i	0.870388i
$34$	−10.0000	−1.71499
$35$	0	0
$36$	2.00000	0.333333
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	− 6.00000i	− 0.925820i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	10.0000	1.50756
$45$	0	0
$46$	−2.00000	−0.294884
$47$	10.0000i	1.45865i	0.684167	+	0.729325i	$0.260166\pi$
$47$	10.0000i	1.45865i	−0.684167	+	0.729325i	$0.739834\pi$
$48$	4.00000i	0.577350i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	5.00000	0.700140
$52$	2.00000i	0.277350i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	−2.00000	−0.272166
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	− 20.0000i	− 2.62613i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	− 4.00000i	− 0.508001i
$63$	3.00000i	0.377964i
$64$	8.00000	1.00000
$65$	0	0
$66$	−10.0000	−1.23091
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 10.0000i	− 1.21268i
$69$	1.00000	0.120386
$70$	0	0
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	15.0000i	1.70941i
$78$	− 2.00000i	− 0.226455i
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 18.0000i	− 1.98777i
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	6.00000	0.654654
$85$	0	0
$86$	−8.00000	−0.862662
$87$	10.0000i	1.07211i
$88$	0	0
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	− 2.00000i	− 0.208514i
$93$	2.00000i	0.207390i
$94$	−20.0000	−2.06284
$95$	0	0
$96$	−8.00000	−0.816497
$97$	− 9.00000i	− 0.913812i	−0.889515	−	0.456906i	$-0.848958\pi$
$97$	− 9.00000i	− 0.913812i	0.889515	−	0.456906i	$-0.151042\pi$
$98$	− 4.00000i	− 0.404061i
$99$	5.00000	0.502519
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	10.0000i	0.990148i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	18.0000	1.74831
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	− 2.00000i	− 0.192450i
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	12.0000i	1.13389i
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	20.0000	1.85695
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	15.0000	1.37505
$120$	0	0
$121$	14.0000	1.27273
$122$	− 22.0000i	− 1.99179i
$123$	9.00000i	0.811503i
$124$	4.00000	0.359211
$125$	0	0
$126$	−6.00000	−0.534522
$127$	14.0000i	1.24230i	0.783692	+	0.621150i	$0.213334\pi$
$127$	14.0000i	1.24230i	−0.783692	+	0.621150i	$0.786666\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	− 10.0000i	− 0.870388i
$133$	6.00000i	0.520266i
$134$	8.00000	0.691095
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	2.00000i	0.170251i
$139$	17.0000	1.44192	0.720961	−	0.692976i	$-0.243701\pi$
$139$	17.0000	1.44192	0.720961	+	0.692976i	$0.243701\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	30.0000i	2.51754i
$143$	5.00000i	0.418121i
$144$	4.00000	0.333333
$145$	0	0
$146$	12.0000	0.993127
$147$	2.00000i	0.164957i
$148$	6.00000i	0.493197i
$149$	7.00000	0.573462	0.286731	−	0.958011i	$-0.407431\pi$
$149$	7.00000	0.573462	0.286731	+	0.958011i	$0.407431\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	− 5.00000i	− 0.404226i
$154$	−30.0000	−2.41747
$155$	0	0
$156$	2.00000	0.160128
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	22.0000i	1.75023i
$159$	−9.00000	−0.713746
$160$	0	0
$161$	3.00000	0.236433
$162$	2.00000i	0.157135i
$163$	11.0000i	0.861586i	0.902451	+	0.430793i	$0.141766\pi$
$163$	11.0000i	0.861586i	−0.902451	+	0.430793i	$0.858234\pi$
$164$	18.0000	1.40556
$165$	0	0
$166$	16.0000	1.24184
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	2.00000	0.152944
$172$	− 8.00000i	− 0.609994i
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	−20.0000	−1.51620
$175$	0	0
$176$	20.0000	1.50756
$177$	0	0
$178$	22.0000i	1.64897i
$179$	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
$181$	−23.0000	−1.70958	−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000	−1.70958	−0.854788	+	0.518977i	$0.826313\pi$
$182$	− 6.00000i	− 0.444750i
$183$	11.0000i	0.813143i
$184$	0	0
$185$	0	0
$186$	−4.00000	−0.293294
$187$	− 25.0000i	− 1.82818i
$188$	− 20.0000i	− 1.45865i
$189$	3.00000	0.218218
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	− 8.00000i	− 0.577350i
$193$	− 13.0000i	− 0.935760i	−0.883792	−	0.467880i	$-0.845018\pi$
$193$	− 13.0000i	− 0.935760i	0.883792	−	0.467880i	$-0.154982\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	4.00000	0.285714
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	10.0000i	0.710669i
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	− 24.0000i	− 1.68863i
$203$	30.0000i	2.10559i
$204$	−10.0000	−0.700140
$205$	0	0
$206$	8.00000	0.557386
$207$	− 1.00000i	− 0.0695048i
$208$	4.00000i	0.277350i
$209$	10.0000	0.691714
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	18.0000i	1.23625i
$213$	− 15.0000i	− 1.02778i
$214$	−6.00000	−0.410152
$215$	0	0
$216$	0	0
$217$	6.00000i	0.407307i
$218$	− 32.0000i	− 2.16731i
$219$	−6.00000	−0.405442
$220$	0	0
$221$	5.00000	0.336336
$222$	− 6.00000i	− 0.402694i
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−24.0000	−1.60357
$225$	0	0
$226$	4.00000	0.266076
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	− 4.00000i	− 0.264906i
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	15.0000	0.986928
$232$	0	0
$233$	25.0000i	1.63780i	0.573933	+	0.818902i	$0.305417\pi$
$233$	25.0000i	1.63780i	−0.573933	+	0.818902i	$0.694583\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	− 11.0000i	− 0.714527i
$238$	30.0000i	1.94461i
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	28.0000i	1.79991i
$243$	− 1.00000i	− 0.0641500i
$244$	22.0000	1.40841
$245$	0	0
$246$	−18.0000	−1.14764
$247$	2.00000i	0.127257i
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	− 6.00000i	− 0.377964i
$253$	− 5.00000i	− 0.314347i
$254$	−28.0000	−1.75688
$255$	0	0
$256$	16.0000	1.00000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	8.00000i	0.498058i
$259$	−9.00000	−0.559233
$260$	0	0
$261$	10.0000	0.618984
$262$	12.0000i	0.741362i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−12.0000	−0.735767
$267$	− 11.0000i	− 0.673189i
$268$	8.00000i	0.488678i
$269$	−32.0000	−1.95107	−0.975537	−	0.219834i	$-0.929448\pi$
$269$	−32.0000	−1.95107	−0.975537	+	0.219834i	$0.929448\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	− 20.0000i	− 1.21268i
$273$	3.00000i	0.181568i
$274$	−12.0000	−0.724947
$275$	0	0
$276$	−2.00000	−0.120386
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	34.0000i	2.03918i
$279$	2.00000	0.119737
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	20.0000i	1.19098i
$283$	8.00000i	0.475551i	0.971320	+	0.237775i	$0.0764182\pi$
$283$	8.00000i	0.475551i	−0.971320	+	0.237775i	$0.923582\pi$
$284$	−30.0000	−1.78017
$285$	0	0
$286$	−10.0000	−0.591312
$287$	27.0000i	1.59376i
$288$	8.00000i	0.471405i
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−9.00000	−0.527589
$292$	12.0000i	0.702247i
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	−4.00000	−0.233285
$295$	0	0
$296$	0	0
$297$	− 5.00000i	− 0.290129i
$298$	14.0000i	0.810998i
$299$	1.00000	0.0578315
$300$	0	0
$301$	12.0000	0.691669
$302$	− 24.0000i	− 1.38104i
$303$	12.0000i	0.689382i
$304$	8.00000	0.458831
$305$	0	0
$306$	10.0000	0.571662
$307$	− 19.0000i	− 1.08439i	−0.840254	−	0.542194i	$-0.817594\pi$
$307$	− 19.0000i	− 1.08439i	0.840254	−	0.542194i	$-0.182406\pi$
$308$	− 30.0000i	− 1.70941i
$309$	−4.00000	−0.227552
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	−44.0000	−2.48306
$315$	0	0
$316$	−22.0000	−1.23760
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	− 18.0000i	− 1.00939i
$319$	50.0000	2.79946
$320$	0	0
$321$	3.00000	0.167444
$322$	6.00000i	0.334367i
$323$	− 10.0000i	− 0.556415i
$324$	−2.00000	−0.111111
$325$	0	0
$326$	−22.0000	−1.21847
$327$	16.0000i	0.884802i
$328$	0	0
$329$	30.0000	1.65395
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	16.0000i	0.878114i
$333$	3.00000i	0.164399i
$334$	16.0000	0.875481
$335$	0	0
$336$	12.0000	0.654654
$337$	4.00000i	0.217894i	0.994048	+	0.108947i	$0.0347479\pi$
$337$	4.00000i	0.217894i	−0.994048	+	0.108947i	$0.965252\pi$
$338$	− 2.00000i	− 0.108786i
$339$	−2.00000	−0.108625
$340$	0	0
$341$	10.0000	0.541530
$342$	4.00000i	0.216295i
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	−4.00000	−0.215041
$347$	− 1.00000i	− 0.0536828i	−0.999640	−	0.0268414i	$-0.991455\pi$
$347$	− 1.00000i	− 0.0536828i	0.999640	−	0.0268414i	$-0.00854491\pi$
$348$	− 20.0000i	− 1.07211i
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	40.0000i	2.13201i
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	0	0
$355$	0	0
$356$	−22.0000	−1.16600
$357$	− 15.0000i	− 0.793884i
$358$	12.0000i	0.634220i
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 46.0000i	− 2.41771i
$363$	− 14.0000i	− 0.734809i
$364$	6.00000	0.314485
$365$	0	0
$366$	−22.0000	−1.14996
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	− 4.00000i	− 0.208514i
$369$	9.00000	0.468521
$370$	0	0
$371$	−27.0000	−1.40177
$372$	− 4.00000i	− 0.207390i
$373$	− 16.0000i	− 0.828449i	−0.910175	−	0.414224i	$-0.864053\pi$
$373$	− 16.0000i	− 0.828449i	0.910175	−	0.414224i	$-0.135947\pi$
$374$	50.0000	2.58544
$375$	0	0
$376$	0	0
$377$	10.0000i	0.515026i
$378$	6.00000i	0.308607i
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	14.0000	0.717242
$382$	− 40.0000i	− 2.04658i
$383$	18.0000i	0.919757i	0.887982	+	0.459879i	$0.152107\pi$
$383$	18.0000i	0.919757i	−0.887982	+	0.459879i	$0.847893\pi$
$384$	0	0
$385$	0	0
$386$	26.0000	1.32337
$387$	− 4.00000i	− 0.203331i
$388$	18.0000i	0.913812i
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	0	0
$393$	− 6.00000i	− 0.302660i
$394$	24.0000	1.20910
$395$	0	0
$396$	−10.0000	−0.502519
$397$	− 19.0000i	− 0.953583i	−0.879017	−	0.476791i	$-0.841800\pi$
$397$	− 19.0000i	− 0.953583i	0.879017	−	0.476791i	$-0.158200\pi$
$398$	− 8.00000i	− 0.401004i
$399$	6.00000	0.300376
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	− 8.00000i	− 0.399004i
$403$	2.00000i	0.0996271i
$404$	24.0000	1.19404
$405$	0	0
$406$	−60.0000	−2.97775
$407$	15.0000i	0.743522i
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	8.00000i	0.394132i
$413$	0	0
$414$	2.00000	0.0982946
$415$	0	0
$416$	−8.00000	−0.392232
$417$	− 17.0000i	− 0.832494i
$418$	20.0000i	0.978232i
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	− 8.00000i	− 0.389434i
$423$	− 10.0000i	− 0.486217i
$424$	0	0
$425$	0	0
$426$	30.0000	1.45350
$427$	33.0000i	1.59698i
$428$	− 6.00000i	− 0.290021i
$429$	5.00000	0.241402
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	− 4.00000i	− 0.192450i
$433$	− 24.0000i	− 1.15337i	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	− 24.0000i	− 1.15337i	0.816968	−	0.576683i	$-0.195653\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	32.0000	1.53252
$437$	− 2.00000i	− 0.0956730i
$438$	− 12.0000i	− 0.573382i
$439$	33.0000	1.57500	0.787502	−	0.616312i	$-0.211374\pi$
$439$	33.0000	1.57500	0.787502	+	0.616312i	$0.211374\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	10.0000i	0.475651i
$443$	− 35.0000i	− 1.66290i	−0.555599	−	0.831450i	$-0.687511\pi$
$443$	− 35.0000i	− 1.66290i	0.555599	−	0.831450i	$-0.312489\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	0	0
$447$	− 7.00000i	− 0.331089i
$448$	− 24.0000i	− 1.13389i
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	45.0000	2.11897
$452$	4.00000i	0.188144i
$453$	12.0000i	0.563809i
$454$	36.0000	1.68956
$455$	0	0
$456$	0	0
$457$	− 13.0000i	− 0.608114i	−0.952654	−	0.304057i	$-0.901659\pi$
$457$	− 13.0000i	− 0.608114i	0.952654	−	0.304057i	$-0.0983414\pi$
$458$	28.0000i	1.30835i
$459$	−5.00000	−0.233380
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	30.0000i	1.39573i
$463$	− 5.00000i	− 0.232370i	−0.993228	−	0.116185i	$-0.962933\pi$
$463$	− 5.00000i	− 0.232370i	0.993228	−	0.116185i	$-0.0370665\pi$
$464$	40.0000	1.85695
$465$	0	0
$466$	−50.0000	−2.31621
$467$	− 29.0000i	− 1.34196i	−0.741475	−	0.670980i	$-0.765874\pi$
$467$	− 29.0000i	− 1.34196i	0.741475	−	0.670980i	$-0.234126\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−12.0000	−0.554109
$470$	0	0
$471$	22.0000	1.01371
$472$	0	0
$473$	− 20.0000i	− 0.919601i
$474$	22.0000	1.01049
$475$	0	0
$476$	−30.0000	−1.37505
$477$	9.00000i	0.412082i
$478$	− 30.0000i	− 1.37217i
$479$	−5.00000	−0.228456	−0.114228	−	0.993455i	$-0.536439\pi$
$479$	−5.00000	−0.228456	−0.114228	+	0.993455i	$0.536439\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	− 28.0000i	− 1.27537i
$483$	− 3.00000i	− 0.136505i
$484$	−28.0000	−1.27273
$485$	0	0
$486$	2.00000	0.0907218
$487$	7.00000i	0.317200i	0.987343	+	0.158600i	$0.0506981\pi$
$487$	7.00000i	0.317200i	−0.987343	+	0.158600i	$0.949302\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	− 18.0000i	− 0.811503i
$493$	− 50.0000i	− 2.25189i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	− 45.0000i	− 2.01853i
$498$	− 16.0000i	− 0.716977i
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	40.0000i	1.78529i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	0	0
$505$	0	0
$506$	10.0000	0.444554
$507$	1.00000i	0.0444116i
$508$	− 28.0000i	− 1.24230i
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	32.0000i	1.41421i
$513$	− 2.00000i	− 0.0883022i
$514$	−36.0000	−1.58789
$515$	0	0
$516$	−8.00000	−0.352180
$517$	− 50.0000i	− 2.19900i
$518$	− 18.0000i	− 0.790875i
$519$	2.00000	0.0877903
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	20.0000i	0.875376i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	0	0
$527$	− 10.0000i	− 0.435607i
$528$	− 20.0000i	− 0.870388i
$529$	22.0000	0.956522
$530$	0	0
$531$	0	0
$532$	− 12.0000i	− 0.520266i
$533$	9.00000i	0.389833i
$534$	22.0000	0.952033
$535$	0	0
$536$	0	0
$537$	− 6.00000i	− 0.258919i
$538$	− 64.0000i	− 2.75924i
$539$	10.0000	0.430730
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	− 4.00000i	− 0.171815i
$543$	23.0000i	0.987024i
$544$	40.0000	1.71499
$545$	0	0
$546$	−6.00000	−0.256776
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	− 12.0000i	− 0.512615i
$549$	11.0000	0.469469
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	− 33.0000i	− 1.40330i
$554$	−52.0000	−2.20927
$555$	0	0
$556$	−34.0000	−1.44192
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	4.00000i	0.169334i
$559$	4.00000	0.169182
$560$	0	0
$561$	−25.0000	−1.05550
$562$	− 20.0000i	− 0.843649i
$563$	41.0000i	1.72794i	0.503540	+	0.863972i	$0.332031\pi$
$563$	41.0000i	1.72794i	−0.503540	+	0.863972i	$0.667969\pi$
$564$	−20.0000	−0.842152
$565$	0	0
$566$	−16.0000	−0.672530
$567$	− 3.00000i	− 0.125988i
$568$	0	0
$569$	−16.0000	−0.670755	−0.335377	−	0.942084i	$-0.608864\pi$
$569$	−16.0000	−0.670755	−0.335377	+	0.942084i	$0.608864\pi$
$570$	0	0
$571$	17.0000	0.711428	0.355714	−	0.934595i	$-0.384238\pi$
$571$	17.0000	0.711428	0.355714	+	0.934595i	$0.384238\pi$
$572$	− 10.0000i	− 0.418121i
$573$	20.0000i	0.835512i
$574$	−54.0000	−2.25392
$575$	0	0
$576$	−8.00000	−0.333333
$577$	− 21.0000i	− 0.874241i	−0.899403	−	0.437121i	$-0.855998\pi$
$577$	− 21.0000i	− 0.874241i	0.899403	−	0.437121i	$-0.144002\pi$
$578$	− 16.0000i	− 0.665512i
$579$	−13.0000	−0.540262
$580$	0	0
$581$	−24.0000	−0.995688
$582$	− 18.0000i	− 0.746124i
$583$	45.0000i	1.86371i
$584$	0	0
$585$	0	0
$586$	48.0000	1.98286
$587$	42.0000i	1.73353i	0.498721	+	0.866763i	$0.333803\pi$
$587$	42.0000i	1.73353i	−0.498721	+	0.866763i	$0.666197\pi$
$588$	− 4.00000i	− 0.164957i
$589$	4.00000	0.164817
$590$	0	0
$591$	−12.0000	−0.493614
$592$	12.0000i	0.493197i
$593$	36.0000i	1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$	36.0000i	1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$	10.0000	0.410305
$595$	0	0
$596$	−14.0000	−0.573462
$597$	4.00000i	0.163709i
$598$	2.00000i	0.0817861i
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	24.0000i	0.978167i
$603$	4.00000i	0.162893i
$604$	24.0000	0.976546
$605$	0	0
$606$	−24.0000	−0.974933
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	16.0000i	0.648886i
$609$	30.0000	1.21566
$610$	0	0
$611$	10.0000	0.404557
$612$	10.0000i	0.404226i
$613$	− 3.00000i	− 0.121169i	−0.998163	−	0.0605844i	$-0.980704\pi$
$613$	− 3.00000i	− 0.121169i	0.998163	−	0.0605844i	$-0.0192964\pi$
$614$	38.0000	1.53356
$615$	0	0
$616$	0	0
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	− 8.00000i	− 0.321807i
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	48.0000i	1.92462i
$623$	− 33.0000i	− 1.32212i
$624$	4.00000	0.160128
$625$	0	0
$626$	−20.0000	−0.799361
$627$	− 10.0000i	− 0.399362i
$628$	− 44.0000i	− 1.75579i
$629$	15.0000	0.598089
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	−24.0000	−0.953162
$635$	0	0
$636$	18.0000	0.713746
$637$	2.00000i	0.0792429i
$638$	100.000i	3.95904i
$639$	−15.0000	−0.593391
$640$	0	0
$641$	−36.0000	−1.42191	−0.710957	−	0.703235i	$-0.751738\pi$
$641$	−36.0000	−1.42191	−0.710957	+	0.703235i	$0.751738\pi$
$642$	6.00000i	0.236801i
$643$	− 1.00000i	− 0.0394362i	−0.999806	−	0.0197181i	$-0.993723\pi$
$643$	− 1.00000i	− 0.0394362i	0.999806	−	0.0197181i	$-0.00627687\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	20.0000	0.786889
$647$	21.0000i	0.825595i	0.910823	+	0.412798i	$0.135448\pi$
$647$	21.0000i	0.825595i	−0.910823	+	0.412798i	$0.864552\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.00000	0.235159
$652$	− 22.0000i	− 0.861586i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	−32.0000	−1.25130
$655$	0	0
$656$	36.0000	1.40556
$657$	6.00000i	0.234082i
$658$	60.0000i	2.33904i
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	64.0000i	2.48743i
$663$	− 5.00000i	− 0.194184i
$664$	0	0
$665$	0	0
$666$	−6.00000	−0.232495
$667$	− 10.0000i	− 0.387202i
$668$	16.0000i	0.619059i
$669$	0	0
$670$	0	0
$671$	55.0000	2.12325
$672$	24.0000i	0.925820i
$673$	− 22.0000i	− 0.848038i	−0.905653	−	0.424019i	$-0.860619\pi$
$673$	− 22.0000i	− 0.848038i	0.905653	−	0.424019i	$-0.139381\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	2.00000	0.0769231
$677$	− 7.00000i	− 0.269032i	−0.990911	−	0.134516i	$-0.957052\pi$
$677$	− 7.00000i	− 0.269032i	0.990911	−	0.134516i	$-0.0429479\pi$
$678$	− 4.00000i	− 0.153619i
$679$	−27.0000	−1.03616
$680$	0	0
$681$	−18.0000	−0.689761
$682$	20.0000i	0.765840i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	30.0000	1.14541
$687$	− 14.0000i	− 0.534133i
$688$	− 16.0000i	− 0.609994i
$689$	−9.00000	−0.342873
$690$	0	0
$691$	6.00000	0.228251	0.114125	−	0.993466i	$-0.463593\pi$
$691$	6.00000	0.228251	0.114125	+	0.993466i	$0.463593\pi$
$692$	− 4.00000i	− 0.152057i
$693$	− 15.0000i	− 0.569803i
$694$	2.00000	0.0759190
$695$	0	0
$696$	0	0
$697$	− 45.0000i	− 1.70450i
$698$	− 40.0000i	− 1.51402i
$699$	25.0000	0.945587
$700$	0	0
$701$	−4.00000	−0.151078	−0.0755390	−	0.997143i	$-0.524068\pi$
$701$	−4.00000	−0.151078	−0.0755390	+	0.997143i	$0.524068\pi$
$702$	2.00000i	0.0754851i
$703$	6.00000i	0.226294i
$704$	−40.0000	−1.50756
$705$	0	0
$706$	−28.0000	−1.05379
$707$	36.0000i	1.35392i
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	−11.0000	−0.412532
$712$	0	0
$713$	− 2.00000i	− 0.0749006i
$714$	30.0000	1.12272
$715$	0	0
$716$	−12.0000	−0.448461
$717$	15.0000i	0.560185i
$718$	− 32.0000i	− 1.19423i
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	− 30.0000i	− 1.11648i
$723$	14.0000i	0.520666i
$724$	46.0000	1.70958
$725$	0	0
$726$	28.0000	1.03918
$727$	6.00000i	0.222528i	0.993791	+	0.111264i	$0.0354899\pi$
$727$	6.00000i	0.222528i	−0.993791	+	0.111264i	$0.964510\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−20.0000	−0.739727
$732$	− 22.0000i	− 0.813143i
$733$	15.0000i	0.554038i	0.960864	+	0.277019i	$0.0893464\pi$
$733$	15.0000i	0.554038i	−0.960864	+	0.277019i	$0.910654\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	8.00000	0.294884
$737$	20.0000i	0.736709i
$738$	18.0000i	0.662589i
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	− 54.0000i	− 1.98240i
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	0	0
$745$	0	0
$746$	32.0000	1.17160
$747$	8.00000i	0.292705i
$748$	50.0000i	1.82818i
$749$	9.00000	0.328853
$750$	0	0
$751$	−45.0000	−1.64207	−0.821037	−	0.570875i	$-0.806604\pi$
$751$	−45.0000	−1.64207	−0.821037	+	0.570875i	$0.806604\pi$
$752$	− 40.0000i	− 1.45865i
$753$	− 20.0000i	− 0.728841i
$754$	−20.0000	−0.728357
$755$	0	0
$756$	−6.00000	−0.218218
$757$	36.0000i	1.30844i	0.756303	+	0.654221i	$0.227003\pi$
$757$	36.0000i	1.30844i	−0.756303	+	0.654221i	$0.772997\pi$
$758$	− 12.0000i	− 0.435860i
$759$	−5.00000	−0.181489
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	28.0000i	1.01433i
$763$	48.0000i	1.73772i
$764$	40.0000	1.44715
$765$	0	0
$766$	−36.0000	−1.30073
$767$	0	0
$768$	− 16.0000i	− 0.577350i
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	26.0000i	0.935760i
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	0	0
$777$	9.00000i	0.322873i
$778$	48.0000i	1.72088i
$779$	18.0000	0.644917
$780$	0	0
$781$	−75.0000	−2.68371
$782$	− 10.0000i	− 0.357599i
$783$	− 10.0000i	− 0.357371i
$784$	8.00000	0.285714
$785$	0	0
$786$	12.0000	0.428026
$787$	44.0000i	1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$	44.0000i	1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$	24.0000i	0.854965i
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	11.0000i	0.390621i
$794$	38.0000	1.34857
$795$	0	0
$796$	8.00000	0.283552
$797$	− 5.00000i	− 0.177109i	−0.996071	−	0.0885545i	$-0.971775\pi$
$797$	− 5.00000i	− 0.177109i	0.996071	−	0.0885545i	$-0.0282248\pi$
$798$	12.0000i	0.424795i
$799$	−50.0000	−1.76887
$800$	0	0
$801$	−11.0000	−0.388666
$802$	− 36.0000i	− 1.27120i
$803$	30.0000i	1.05868i
$804$	8.00000	0.282138
$805$	0	0
$806$	−4.00000	−0.140894
$807$	32.0000i	1.12645i
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	− 60.0000i	− 2.10559i
$813$	2.00000i	0.0701431i
$814$	−30.0000	−1.05150
$815$	0	0
$816$	−20.0000	−0.700140
$817$	− 8.00000i	− 0.279885i
$818$	52.0000i	1.81814i
$819$	3.00000	0.104828
$820$	0	0
$821$	−41.0000	−1.43091	−0.715455	−	0.698659i	$-0.753781\pi$
$821$	−41.0000	−1.43091	−0.715455	+	0.698659i	$0.753781\pi$
$822$	12.0000i	0.418548i
$823$	48.0000i	1.67317i	0.547833	+	0.836587i	$0.315453\pi$
$823$	48.0000i	1.67317i	−0.547833	+	0.836587i	$0.684547\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 42.0000i	− 1.46048i	−0.683189	−	0.730242i	$-0.739408\pi$
$827$	− 42.0000i	− 1.46048i	0.683189	−	0.730242i	$-0.260592\pi$
$828$	2.00000i	0.0695048i
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	− 8.00000i	− 0.277350i
$833$	− 10.0000i	− 0.346479i
$834$	34.0000	1.17732
$835$	0	0
$836$	−20.0000	−0.691714
$837$	− 2.00000i	− 0.0691301i
$838$	− 52.0000i	− 1.79631i
$839$	−7.00000	−0.241667	−0.120833	−	0.992673i	$-0.538557\pi$
$839$	−7.00000	−0.241667	−0.120833	+	0.992673i	$0.538557\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	64.0000i	2.20559i
$843$	10.0000i	0.344418i
$844$	8.00000	0.275371
$845$	0	0
$846$	20.0000	0.687614
$847$	− 42.0000i	− 1.44314i
$848$	36.0000i	1.23625i
$849$	8.00000	0.274559
$850$	0	0
$851$	3.00000	0.102839
$852$	30.0000i	1.02778i
$853$	− 51.0000i	− 1.74621i	−0.487535	−	0.873103i	$-0.662104\pi$
$853$	− 51.0000i	− 1.74621i	0.487535	−	0.873103i	$-0.337896\pi$
$854$	−66.0000	−2.25847
$855$	0	0
$856$	0	0
$857$	− 17.0000i	− 0.580709i	−0.956919	−	0.290354i	$-0.906227\pi$
$857$	− 17.0000i	− 0.580709i	0.956919	−	0.290354i	$-0.0937732\pi$
$858$	10.0000i	0.341394i
$859$	−35.0000	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$860$	0	0
$861$	27.0000	0.920158
$862$	− 32.0000i	− 1.08992i
$863$	22.0000i	0.748889i	0.927249	+	0.374444i	$0.122167\pi$
$863$	22.0000i	0.748889i	−0.927249	+	0.374444i	$0.877833\pi$
$864$	8.00000	0.272166
$865$	0	0
$866$	48.0000	1.63111
$867$	8.00000i	0.271694i
$868$	− 12.0000i	− 0.407307i
$869$	−55.0000	−1.86575
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	9.00000i	0.304604i
$874$	4.00000	0.135302
$875$	0	0
$876$	12.0000	0.405442
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	66.0000i	2.22739i
$879$	−24.0000	−0.809500
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	4.00000i	0.134687i
$883$	− 12.0000i	− 0.403832i	−0.979403	−	0.201916i	$-0.935283\pi$
$883$	− 12.0000i	− 0.403832i	0.979403	−	0.201916i	$-0.0647168\pi$
$884$	−10.0000	−0.336336
$885$	0	0
$886$	70.0000	2.35170
$887$	− 15.0000i	− 0.503651i	−0.967773	−	0.251825i	$-0.918969\pi$
$887$	− 15.0000i	− 0.503651i	0.967773	−	0.251825i	$-0.0810309\pi$
$888$	0	0
$889$	42.0000	1.40863
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	− 20.0000i	− 0.669274i
$894$	14.0000	0.468230
$895$	0	0
$896$	0	0
$897$	− 1.00000i	− 0.0333890i
$898$	− 30.0000i	− 1.00111i
$899$	20.0000	0.667037
$900$	0	0
$901$	45.0000	1.49917
$902$	90.0000i	2.99667i
$903$	− 12.0000i	− 0.399335i
$904$	0	0
$905$	0	0
$906$	−24.0000	−0.797347
$907$	2.00000i	0.0664089i	0.999449	+	0.0332045i	$0.0105712\pi$
$907$	2.00000i	0.0664089i	−0.999449	+	0.0332045i	$0.989429\pi$
$908$	36.0000i	1.19470i
$909$	12.0000	0.398015
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	− 8.00000i	− 0.264906i
$913$	40.0000i	1.32381i
$914$	26.0000	0.860004
$915$	0	0
$916$	−28.0000	−0.925146
$917$	− 18.0000i	− 0.594412i
$918$	− 10.0000i	− 0.330049i
$919$	−29.0000	−0.956622	−0.478311	−	0.878191i	$-0.658751\pi$
$919$	−29.0000	−0.956622	−0.478311	+	0.878191i	$0.658751\pi$
$920$	0	0
$921$	−19.0000	−0.626071
$922$	6.00000i	0.197599i
$923$	− 15.0000i	− 0.493731i
$924$	−30.0000	−0.986928
$925$	0	0
$926$	10.0000	0.328620
$927$	4.00000i	0.131377i
$928$	80.0000i	2.62613i
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	− 50.0000i	− 1.63780i
$933$	− 24.0000i	− 0.785725i
$934$	58.0000	1.89782
$935$	0	0
$936$	0	0
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	− 24.0000i	− 0.783628i
$939$	10.0000	0.326338
$940$	0	0
$941$	−23.0000	−0.749779	−0.374889	−	0.927070i	$-0.622319\pi$
$941$	−23.0000	−0.749779	−0.374889	+	0.927070i	$0.622319\pi$
$942$	44.0000i	1.43360i
$943$	− 9.00000i	− 0.293080i
$944$	0	0
$945$	0	0
$946$	40.0000	1.30051
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	22.0000i	0.714527i
$949$	−6.00000	−0.194768
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	11.0000i	0.356325i	0.984001	+	0.178162i	$0.0570153\pi$
$953$	11.0000i	0.356325i	−0.984001	+	0.178162i	$0.942985\pi$
$954$	−18.0000	−0.582772
$955$	0	0
$956$	30.0000	0.970269
$957$	− 50.0000i	− 1.61627i
$958$	− 10.0000i	− 0.323085i
$959$	18.0000	0.581250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 6.00000i	− 0.193448i
$963$	− 3.00000i	− 0.0966736i
$964$	28.0000	0.901819
$965$	0	0
$966$	6.00000	0.193047
$967$	− 40.0000i	− 1.28631i	−0.765735	−	0.643157i	$-0.777624\pi$
$967$	− 40.0000i	− 1.28631i	0.765735	−	0.643157i	$-0.222376\pi$
$968$	0	0
$969$	−10.0000	−0.321246
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	2.00000i	0.0641500i
$973$	− 51.0000i	− 1.63498i
$974$	−14.0000	−0.448589
$975$	0	0
$976$	44.0000	1.40841
$977$	12.0000i	0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$	12.0000i	0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$	22.0000i	0.703482i
$979$	−55.0000	−1.75781
$980$	0	0
$981$	16.0000	0.510841
$982$	32.0000i	1.02116i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	0	0
$986$	100.000	3.18465
$987$	− 30.0000i	− 0.954911i
$988$	− 4.00000i	− 0.127257i
$989$	−4.00000	−0.127193
$990$	0	0
$991$	39.0000	1.23888	0.619438	−	0.785046i	$-0.287361\pi$
$991$	39.0000	1.23888	0.619438	+	0.785046i	$0.287361\pi$
$992$	16.0000i	0.508001i
$993$	− 32.0000i	− 1.01549i
$994$	90.0000	2.85463
$995$	0	0
$996$	16.0000	0.506979
$997$	− 16.0000i	− 0.506725i	−0.967371	−	0.253363i	$-0.918463\pi$
$997$	− 16.0000i	− 0.506725i	0.967371	−	0.253363i	$-0.0815366\pi$
$998$	− 68.0000i	− 2.15250i
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.c.b.274.2		2
3.2	odd	2		2925.2.c.d.2224.1		2
5.2	odd	4		975.2.a.b.1.1		1
5.3	odd	4		195.2.a.d.1.1	✓	1
5.4	even	2	inner	975.2.c.b.274.1		2
15.2	even	4		2925.2.a.t.1.1		1
15.8	even	4		585.2.a.a.1.1		1
15.14	odd	2		2925.2.c.d.2224.2		2
20.3	even	4		3120.2.a.n.1.1		1
35.13	even	4		9555.2.a.t.1.1		1
60.23	odd	4		9360.2.a.w.1.1		1
65.38	odd	4		2535.2.a.b.1.1		1
195.38	even	4		7605.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.d.1.1	✓	1	5.3	odd	4
585.2.a.a.1.1		1	15.8	even	4
975.2.a.b.1.1		1	5.2	odd	4
975.2.c.b.274.1		2	5.4	even	2	inner
975.2.c.b.274.2		2	1.1	even	1	trivial
2535.2.a.b.1.1		1	65.38	odd	4
2925.2.a.t.1.1		1	15.2	even	4
2925.2.c.d.2224.1		2	3.2	odd	2
2925.2.c.d.2224.2		2	15.14	odd	2
3120.2.a.n.1.1		1	20.3	even	4
7605.2.a.v.1.1		1	195.38	even	4
9360.2.a.w.1.1		1	60.23	odd	4
9555.2.a.t.1.1		1	35.13	even	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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} -5.00000 q^{11} +2.00000i q^{12} -1.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} +5.00000i q^{17} -2.00000i q^{18} -2.00000 q^{19} -3.00000 q^{21} -10.0000i q^{22} +1.00000i q^{23} +2.00000 q^{26} +1.00000i q^{27} +6.00000i q^{28} -10.0000 q^{29} -2.00000 q^{31} -8.00000i q^{32} +5.00000i q^{33} -10.0000 q^{34} +2.00000 q^{36} -3.00000i q^{37} -4.00000i q^{38} -1.00000 q^{39} -9.00000 q^{41} -6.00000i q^{42} +4.00000i q^{43} +10.0000 q^{44} -2.00000 q^{46} +10.0000i q^{47} +4.00000i q^{48} -2.00000 q^{49} +5.00000 q^{51} +2.00000i q^{52} -9.00000i q^{53} -2.00000 q^{54} +2.00000i q^{57} -20.0000i q^{58} -11.0000 q^{61} -4.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} -10.0000 q^{66} -4.00000i q^{67} -10.0000i q^{68} +1.00000 q^{69} +15.0000 q^{71} -6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +15.0000i q^{77} -2.00000i q^{78} +11.0000 q^{79} +1.00000 q^{81} -18.0000i q^{82} -8.00000i q^{83} +6.00000 q^{84} -8.00000 q^{86} +10.0000i q^{87} +11.0000 q^{89} -3.00000 q^{91} -2.00000i q^{92} +2.00000i q^{93} -20.0000 q^{94} -8.00000 q^{96} -9.00000i q^{97} -4.00000i q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9} - 10 q^{11} + 12 q^{14} - 8 q^{16} - 4 q^{19} - 6 q^{21} + 4 q^{26} - 20 q^{29} - 4 q^{31} - 20 q^{34} + 4 q^{36} - 2 q^{39} - 18 q^{41} + 20 q^{44} - 4 q^{46} - 4 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 - 10 * q^11 + 12 * q^14 - 8 * q^16 - 4 * q^19 - 6 * q^21 + 4 * q^26 - 20 * q^29 - 4 * q^31 - 20 * q^34 + 4 * q^36 - 2 * q^39 - 18 * q^41 + 20 * q^44 - 4 * q^46 - 4 * q^49 + 10 * q^51 - 4 * q^54 - 22 * q^61 + 16 * q^64 - 20 * q^66 + 2 * q^69 + 30 * q^71 + 12 * q^74 + 8 * q^76 + 22 * q^79 + 2 * q^81 + 12 * q^84 - 16 * q^86 + 22 * q^89 - 6 * q^91 - 40 * q^94 - 16 * q^96 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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