LMFDB - Embedded newform 975.2.c.f.274.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,2,Mod(274,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,2,0,-2,0,0,-2,0,8,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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:	no (minimal twist has level 39)
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.f.274.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{21} -4.00000i q^{22} -3.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} +10.0000 q^{29} +4.00000 q^{31} -5.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} +1.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} -12.0000i q^{43} +4.00000 q^{44} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +1.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +12.0000 q^{56} -10.0000i q^{58} -12.0000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -4.00000i q^{63} -7.00000 q^{64} -4.00000 q^{66} +8.00000i q^{67} -2.00000i q^{68} +3.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} +16.0000i q^{77} -1.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{84} -12.0000 q^{86} -10.0000i q^{87} -12.0000i q^{88} +2.00000 q^{89} -4.00000 q^{91} -4.00000i q^{93} -5.00000 q^{96} -10.0000i q^{97} +9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 8 q^{14} - 2 q^{16} + 8 q^{21} - 6 q^{24} + 2 q^{26} + 20 q^{29} + 8 q^{31} - 4 q^{34} - 2 q^{36} + 2 q^{39} + 12 q^{41} + 8 q^{44} - 18 q^{49} - 4 q^{51}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 8 * q^14 - 2 * q^16 + 8 * q^21 - 6 * q^24 + 2 * q^26 + 20 * q^29 + 8 * q^31 - 4 * q^34 - 2 * q^36 + 2 * q^39 + 12 * q^41 + 8 * q^44 - 18 * q^49 - 4 * q^51 + 2 * q^54 + 24 * q^56 - 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 + 4 * q^74 - 16 * q^79 + 2 * q^81 + 8 * q^84 - 24 * q^86 + 4 * q^89 - 8 * q^91 - 10 * q^96 - 8 * 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 $

$2$	− 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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 3.00000i	− 1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	− 1.00000i	− 0.288675i
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	− 4.00000i	− 0.852803i
$23$	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
$28$	4.00000i	0.755929i
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	− 5.00000i	− 0.883883i
$33$	− 4.00000i	− 0.696311i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	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	$0.344787\pi$
$42$	− 4.00000i	− 0.617213i
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	1.00000i	0.138675i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	12.0000	1.60357
$57$	0	0
$58$	− 10.0000i	− 1.31306i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	− 4.00000i	− 0.508001i
$63$	− 4.00000i	− 0.503953i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	− 2.00000i	− 0.242536i
$69$	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
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	0	0
$77$	16.0000i	1.82337i
$78$	− 1.00000i	− 0.113228i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−12.0000	−1.29399
$87$	− 10.0000i	− 1.07211i
$88$	− 12.0000i	− 1.27920i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	−5.00000	−0.510310
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	9.00000i	0.909137i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	2.00000i	0.198030i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	6.00000	0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	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
$111$	2.00000	0.189832
$112$	− 4.00000i	− 0.377964i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	− 1.00000i	− 0.0924500i
$118$	12.0000i	1.10469i
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000i	0.181071i
$123$	− 6.00000i	− 0.541002i
$124$	4.00000	0.359211
$125$	0	0
$126$	−4.00000	−0.356348
$127$	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
$129$	−12.0000	−1.05654
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	− 4.00000i	− 0.348155i
$133$	0	0
$134$	8.00000	0.691095
$135$	0	0
$136$	−6.00000	−0.514496
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000i	0.334497i
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	9.00000i	0.742307i
$148$	2.00000i	0.164399i
$149$	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
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	16.0000	1.28932
$155$	0	0
$156$	1.00000	0.0800641
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	8.00000i	0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	8.00000i	0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$	8.00000i	0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	− 12.0000i	− 0.925820i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 12.0000i	− 0.914991i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−10.0000	−0.758098
$175$	0	0
$176$	−4.00000	−0.301511
$177$	12.0000i	0.901975i
$178$	− 2.00000i	− 0.149906i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	4.00000i	0.296500i
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	−4.00000	−0.293294
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	7.00000i	0.505181i
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−9.00000	−0.642857
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	4.00000i	0.284268i
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	18.0000i	1.26648i
$203$	40.0000i	2.80745i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	3.00000	0.204124
$217$	16.0000i	1.08615i
$218$	− 2.00000i	− 0.135457i
$219$	2.00000	0.135147
$220$	0	0
$221$	2.00000	0.134535
$222$	− 2.00000i	− 0.134231i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	20.0000	1.33631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	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
$231$	16.0000	1.05272
$232$	− 30.0000i	− 1.96960i
$233$	− 14.0000i	− 0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$	− 14.0000i	− 0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−12.0000	−0.781133
$237$	8.00000i	0.519656i
$238$	− 8.00000i	− 0.518563i
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	0	0
$248$	− 12.0000i	− 0.762001i
$249$	4.00000	0.253490
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	− 4.00000i	− 0.251976i
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 26.0000i	− 1.62184i	−0.585160	−	0.810918i	$-0.698968\pi$
$257$	− 26.0000i	− 1.62184i	0.585160	−	0.810918i	$-0.301032\pi$
$258$	12.0000i	0.747087i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−10.0000	−0.618984
$262$	− 4.00000i	− 0.247121i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	0	0
$267$	− 2.00000i	− 0.122398i
$268$	8.00000i	0.488678i
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	2.00000i	0.121268i
$273$	4.00000i	0.242091i
$274$	−6.00000	−0.362473
$275$	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	$0.902873\pi$
$278$	12.0000i	0.719712i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	0	0
$285$	0	0
$286$	4.00000	0.236525
$287$	24.0000i	1.41668i
$288$	5.00000i	0.294628i
$289$	13.0000	0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	2.00000i	0.117041i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	6.00000	0.348743
$297$	4.00000i	0.232104i
$298$	− 6.00000i	− 0.347571i
$299$	0	0
$300$	0	0
$301$	48.0000	2.76667
$302$	− 4.00000i	− 0.230174i
$303$	18.0000i	1.03407i
$304$	0	0
$305$	0	0
$306$	2.00000	0.114332
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	16.0000i	0.911685i
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	− 3.00000i	− 0.169842i
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 26.0000i	− 1.46031i	−0.683284	−	0.730153i	$-0.739449\pi$
$317$	− 26.0000i	− 1.46031i	0.683284	−	0.730153i	$-0.260551\pi$
$318$	− 6.00000i	− 0.336463i
$319$	40.0000	2.23957
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	8.00000	0.443079
$327$	− 2.00000i	− 0.110600i
$328$	− 18.0000i	− 0.993884i
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	4.00000i	0.219529i
$333$	− 2.00000i	− 0.109599i
$334$	8.00000	0.437741
$335$	0	0
$336$	−4.00000	−0.218218
$337$	− 18.0000i	− 0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	− 18.0000i	− 0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	1.00000i	0.0543928i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	−36.0000	−1.94099
$345$	0	0
$346$	6.00000	0.322562
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	− 10.0000i	− 0.536056i
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	− 20.0000i	− 1.06600i
$353$	− 2.00000i	− 0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$	− 2.00000i	− 0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	2.00000	0.106000
$357$	− 8.00000i	− 0.423405i
$358$	4.00000i	0.211407i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	10.0000i	0.525588i
$363$	− 5.00000i	− 0.262432i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	2.00000	0.104542
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	− 4.00000i	− 0.207390i
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	0	0
$377$	10.0000i	0.515026i
$378$	4.00000i	0.205738i
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	− 8.00000i	− 0.409316i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	18.0000	0.916176
$387$	12.0000i	0.609994i
$388$	− 10.0000i	− 0.507673i
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	0	0
$392$	27.0000i	1.36371i
$393$	− 4.00000i	− 0.201773i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	−4.00000	−0.201008
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	− 8.00000i	− 0.399004i
$403$	4.00000i	0.199254i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	40.0000	1.98517
$407$	8.00000i	0.396545i
$408$	6.00000i	0.297044i
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	− 48.0000i	− 2.36193i
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	12.0000i	0.587643i
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	20.0000i	0.973585i
$423$	0	0
$424$	18.0000	0.874157
$425$	0	0
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	− 12.0000i	− 0.580042i
$429$	4.00000	0.193122
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	− 2.00000i	− 0.0955637i
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	− 2.00000i	− 0.0951303i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	4.00000	0.189405
$447$	− 6.00000i	− 0.283790i
$448$	− 28.0000i	− 1.32288i
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	− 6.00000i	− 0.282216i
$453$	− 4.00000i	− 0.187936i
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	− 10.0000i	− 0.467269i
$459$	2.00000	0.0933520
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	− 16.0000i	− 0.744387i
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	−14.0000	−0.648537
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	−32.0000	−1.47762
$470$	0	0
$471$	18.0000	0.829396
$472$	36.0000i	1.65703i
$473$	− 48.0000i	− 2.20704i
$474$	8.00000	0.367452
$475$	0	0
$476$	8.00000	0.366679
$477$	− 6.00000i	− 0.274721i
$478$	− 24.0000i	− 1.09773i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	− 10.0000i	− 0.455488i
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 12.0000i	− 0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$	− 12.0000i	− 0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	6.00000i	0.271607i
$489$	8.00000	0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	− 6.00000i	− 0.270501i
$493$	− 20.0000i	− 0.900755i
$494$	0	0
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	− 4.00000i	− 0.179244i
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	12.0000i	0.535586i
$503$	− 8.00000i	− 0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$	− 8.00000i	− 0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$	−12.0000	−0.534522
$505$	0	0
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	16.0000i	0.709885i
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	11.0000i	0.486136i
$513$	0	0
$514$	−26.0000	−1.14681
$515$	0	0
$516$	−12.0000	−0.528271
$517$	0	0
$518$	8.00000i	0.351500i
$519$	6.00000	0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	10.0000i	0.437688i
$523$	44.0000i	1.92399i	0.273075	+	0.961993i	$0.411959\pi$
$523$	44.0000i	1.92399i	−0.273075	+	0.961993i	$0.588041\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	24.0000	1.04645
$527$	− 8.00000i	− 0.348485i
$528$	4.00000i	0.174078i
$529$	23.0000	1.00000
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	6.00000i	0.259889i
$534$	−2.00000	−0.0865485
$535$	0	0
$536$	24.0000	1.03664
$537$	4.00000i	0.172613i
$538$	22.0000i	0.948487i
$539$	−36.0000	−1.55063
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	12.0000i	0.515444i
$543$	10.0000i	0.429141i
$544$	−10.0000	−0.428746
$545$	0	0
$546$	4.00000	0.171184
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	− 6.00000i	− 0.256307i
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	10.0000	0.424859
$555$	0	0
$556$	−12.0000	−0.508913
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	4.00000i	0.169334i
$559$	12.0000	0.507546
$560$	0	0
$561$	−8.00000	−0.337760
$562$	10.0000i	0.421825i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	12.0000	0.504398
$567$	4.00000i	0.167984i
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	4.00000i	0.167248i
$573$	− 8.00000i	− 0.334205i
$574$	24.0000	1.00174
$575$	0	0
$576$	7.00000	0.291667
$577$	46.0000i	1.91501i	0.288425	+	0.957503i	$0.406868\pi$
$577$	46.0000i	1.91501i	−0.288425	+	0.957503i	$0.593132\pi$
$578$	− 13.0000i	− 0.540729i
$579$	18.0000	0.748054
$580$	0	0
$581$	−16.0000	−0.663792
$582$	10.0000i	0.414513i
$583$	24.0000i	0.993978i
$584$	6.00000	0.248282
$585$	0	0
$586$	−6.00000	−0.247858
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	9.00000i	0.371154i
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	− 2.00000i	− 0.0821995i
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	8.00000i	0.327418i
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	− 48.0000i	− 1.95633i
$603$	− 8.00000i	− 0.325785i
$604$	4.00000	0.162758
$605$	0	0
$606$	18.0000	0.731200
$607$	16.0000i	0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$	16.0000i	0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$	0	0
$609$	40.0000	1.62088
$610$	0	0
$611$	0	0
$612$	2.00000i	0.0808452i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	48.0000	1.93398
$617$	− 22.0000i	− 0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$	− 22.0000i	− 0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.00000i	0.320513i
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	18.0000i	0.718278i
$629$	4.00000	0.159490
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	24.0000i	0.954669i
$633$	20.0000i	0.794929i
$634$	−26.0000	−1.03259
$635$	0	0
$636$	6.00000	0.237915
$637$	− 9.00000i	− 0.356593i
$638$	− 40.0000i	− 1.58362i
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	12.0000i	0.473602i
$643$	− 40.0000i	− 1.57745i	−0.614749	−	0.788723i	$-0.710743\pi$
$643$	− 40.0000i	− 1.57745i	0.614749	−	0.788723i	$-0.289257\pi$
$644$	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
$649$	−48.0000	−1.88416
$650$	0	0
$651$	16.0000	0.627089
$652$	8.00000i	0.313304i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	16.0000i	0.621858i
$663$	− 2.00000i	− 0.0776736i
$664$	12.0000	0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	8.00000i	0.309529i
$669$	4.00000	0.154649
$670$	0	0
$671$	−8.00000	−0.308837
$672$	− 20.0000i	− 0.771517i
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	− 22.0000i	− 0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$	− 22.0000i	− 0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	6.00000i	0.230429i
$679$	40.0000	1.53506
$680$	0	0
$681$	20.0000	0.766402
$682$	− 16.0000i	− 0.612672i
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	− 10.0000i	− 0.381524i
$688$	12.0000i	0.457496i
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−24.0000	−0.913003	−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000	−0.913003	−0.456502	+	0.889723i	$0.650898\pi$
$692$	6.00000i	0.228086i
$693$	− 16.0000i	− 0.607790i
$694$	12.0000	0.455514
$695$	0	0
$696$	−30.0000	−1.13715
$697$	− 12.0000i	− 0.454532i
$698$	− 26.0000i	− 0.984115i
$699$	−14.0000	−0.529529
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	1.00000i	0.0377426i
$703$	0	0
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	− 72.0000i	− 2.70784i
$708$	12.0000i	0.450988i
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 6.00000i	− 0.224860i
$713$	0	0
$714$	−8.00000	−0.299392
$715$	0	0
$716$	−4.00000	−0.149487
$717$	− 24.0000i	− 0.896296i
$718$	24.0000i	0.895672i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	19.0000i	0.707107i
$723$	− 10.0000i	− 0.371904i
$724$	−10.0000	−0.371647
$725$	0	0
$726$	−5.00000	−0.185567
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	12.0000i	0.444750i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	2.00000i	0.0739221i
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	0	0
$737$	32.0000i	1.17874i
$738$	6.00000i	0.220863i
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	0	0
$741$	0	0
$742$	24.0000i	0.881068i
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	−12.0000	−0.439941
$745$	0	0
$746$	−26.0000	−0.951928
$747$	− 4.00000i	− 0.146352i
$748$	− 8.00000i	− 0.292509i
$749$	48.0000	1.75388
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	10.0000	0.364179
$755$	0	0
$756$	−4.00000	−0.145479
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	− 24.0000i	− 0.871719i
$759$	0	0
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	− 16.0000i	− 0.579619i
$763$	8.00000i	0.289619i
$764$	8.00000	0.289430
$765$	0	0
$766$	−16.0000	−0.578103
$767$	− 12.0000i	− 0.433295i
$768$	17.0000i	0.613435i
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	−26.0000	−0.936367
$772$	18.0000i	0.647834i
$773$	10.0000i	0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$	10.0000i	0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	−30.0000	−1.07694
$777$	8.00000i	0.286998i
$778$	22.0000i	0.788738i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	10.0000i	0.357371i
$784$	9.00000	0.321429
$785$	0	0
$786$	−4.00000	−0.142675
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	− 18.0000i	− 0.641223i
$789$	24.0000	0.854423
$790$	0	0
$791$	24.0000	0.853342
$792$	12.0000i	0.426401i
$793$	− 2.00000i	− 0.0710221i
$794$	−38.0000	−1.34857
$795$	0	0
$796$	−8.00000	−0.283552
$797$	− 46.0000i	− 1.62940i	−0.579880	−	0.814702i	$-0.696901\pi$
$797$	− 46.0000i	− 1.62940i	0.579880	−	0.814702i	$-0.303099\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	− 22.0000i	− 0.776847i
$803$	8.00000i	0.282314i
$804$	8.00000	0.282138
$805$	0	0
$806$	4.00000	0.140894
$807$	22.0000i	0.774437i
$808$	54.0000i	1.89971i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	40.0000i	1.40372i
$813$	12.0000i	0.420858i
$814$	8.00000	0.280400
$815$	0	0
$816$	2.00000	0.0700140
$817$	0	0
$818$	34.0000i	1.18878i
$819$	4.00000	0.139771
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	6.00000i	0.209274i
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	−48.0000	−1.67013
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	− 7.00000i	− 0.242681i
$833$	18.0000i	0.623663i
$834$	12.0000	0.415526
$835$	0	0
$836$	0	0
$837$	4.00000i	0.138260i
$838$	4.00000i	0.138178i
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	10.0000i	0.344623i
$843$	10.0000i	0.344418i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	20.0000i	0.687208i
$848$	− 6.00000i	− 0.206041i
$849$	12.0000	0.411839
$850$	0	0
$851$	0	0
$852$	0	0
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−36.0000	−1.23045
$857$	46.0000i	1.57133i	0.618652	+	0.785665i	$0.287679\pi$
$857$	46.0000i	1.57133i	−0.618652	+	0.785665i	$0.712321\pi$
$858$	− 4.00000i	− 0.136558i
$859$	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
$861$	24.0000	0.817918
$862$	0	0
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	34.0000	1.15537
$867$	− 13.0000i	− 0.441503i
$868$	16.0000i	0.543075i
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−8.00000	−0.271070
$872$	− 6.00000i	− 0.203186i
$873$	10.0000i	0.338449i
$874$	0	0
$875$	0	0
$876$	2.00000	0.0675737
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	32.0000i	1.07995i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	58.0000	1.95407	0.977035	−	0.213080i	$-0.0683494\pi$
$881$	58.0000	1.95407	0.977035	+	0.213080i	$0.0683494\pi$
$882$	− 9.00000i	− 0.303046i
$883$	− 44.0000i	− 1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$	− 44.0000i	− 1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−4.00000	−0.134383
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	− 6.00000i	− 0.201347i
$889$	−64.0000	−2.14649
$890$	0	0
$891$	4.00000	0.134005
$892$	4.00000i	0.133930i
$893$	0	0
$894$	−6.00000	−0.200670
$895$	0	0
$896$	12.0000	0.400892
$897$	0	0
$898$	22.0000i	0.734150i
$899$	40.0000	1.33407
$900$	0	0
$901$	12.0000	0.399778
$902$	− 24.0000i	− 0.799113i
$903$	− 48.0000i	− 1.59734i
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−4.00000	−0.132891
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	20.0000i	0.663723i
$909$	18.0000	0.597022
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	16.0000i	0.529523i
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	10.0000	0.330409
$917$	16.0000i	0.528367i
$918$	− 2.00000i	− 0.0660098i
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	38.0000i	1.25146i
$923$	0	0
$924$	16.0000	0.526361
$925$	0	0
$926$	4.00000	0.131448
$927$	0	0
$928$	− 50.0000i	− 1.64133i
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	− 14.0000i	− 0.458585i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	32.0000i	1.04484i
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	− 18.0000i	− 0.586472i
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	−48.0000	−1.56061
$947$	60.0000i	1.94974i	0.222779	+	0.974869i	$0.428487\pi$
$947$	60.0000i	1.94974i	−0.222779	+	0.974869i	$0.571513\pi$
$948$	8.00000i	0.259828i
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	−26.0000	−0.843108
$952$	− 24.0000i	− 0.777844i
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	− 40.0000i	− 1.29302i
$958$	− 24.0000i	− 0.775405i
$959$	24.0000	0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	2.00000i	0.0644826i
$963$	12.0000i	0.386695i
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	52.0000i	1.67221i	0.548572	+	0.836104i	$0.315172\pi$
$967$	52.0000i	1.67221i	−0.548572	+	0.836104i	$0.684828\pi$
$968$	− 15.0000i	− 0.482118i
$969$	0	0
$970$	0	0
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 48.0000i	− 1.53881i
$974$	−12.0000	−0.384505
$975$	0	0
$976$	2.00000	0.0640184
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	− 8.00000i	− 0.255812i
$979$	8.00000	0.255681
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	12.0000i	0.382935i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	−18.0000	−0.573819
$985$	0	0
$986$	−20.0000	−0.636930
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	− 20.0000i	− 0.635001i
$993$	16.0000i	0.507745i
$994$	0	0
$995$	0	0
$996$	4.00000	0.126745
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	− 24.0000i	− 0.759707i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.c.f.274.1		2
3.2	odd	2		2925.2.c.e.2224.2		2
5.2	odd	4		39.2.a.a.1.1	✓	1
5.3	odd	4		975.2.a.f.1.1		1
5.4	even	2	inner	975.2.c.f.274.2		2
15.2	even	4		117.2.a.a.1.1		1
15.8	even	4		2925.2.a.p.1.1		1
15.14	odd	2		2925.2.c.e.2224.1		2
20.7	even	4		624.2.a.i.1.1		1
35.27	even	4		1911.2.a.f.1.1		1
40.27	even	4		2496.2.a.e.1.1		1
40.37	odd	4		2496.2.a.q.1.1		1
45.2	even	12		1053.2.e.d.352.1		2
45.7	odd	12		1053.2.e.b.352.1		2
45.22	odd	12		1053.2.e.b.703.1		2
45.32	even	12		1053.2.e.d.703.1		2
55.32	even	4		4719.2.a.c.1.1		1
60.47	odd	4		1872.2.a.h.1.1		1
65.2	even	12		507.2.j.e.316.2		4
65.7	even	12		507.2.j.e.361.2		4
65.12	odd	4		507.2.a.a.1.1		1
65.17	odd	12		507.2.e.b.484.1		2
65.22	odd	12		507.2.e.a.484.1		2
65.32	even	12		507.2.j.e.361.1		4
65.37	even	12		507.2.j.e.316.1		4
65.42	odd	12		507.2.e.a.22.1		2
65.47	even	4		507.2.b.a.337.2		2
65.57	even	4		507.2.b.a.337.1		2
65.62	odd	12		507.2.e.b.22.1		2
105.62	odd	4		5733.2.a.e.1.1		1
120.77	even	4		7488.2.a.bl.1.1		1
120.107	odd	4		7488.2.a.by.1.1		1
195.47	odd	4		1521.2.b.b.1351.1		2
195.77	even	4		1521.2.a.e.1.1		1
195.122	odd	4		1521.2.b.b.1351.2		2
260.207	even	4		8112.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.a.1.1	✓	1	5.2	odd	4
117.2.a.a.1.1		1	15.2	even	4
507.2.a.a.1.1		1	65.12	odd	4
507.2.b.a.337.1		2	65.57	even	4
507.2.b.a.337.2		2	65.47	even	4
507.2.e.a.22.1		2	65.42	odd	12
507.2.e.a.484.1		2	65.22	odd	12
507.2.e.b.22.1		2	65.62	odd	12
507.2.e.b.484.1		2	65.17	odd	12
507.2.j.e.316.1		4	65.37	even	12
507.2.j.e.316.2		4	65.2	even	12
507.2.j.e.361.1		4	65.32	even	12
507.2.j.e.361.2		4	65.7	even	12
624.2.a.i.1.1		1	20.7	even	4
975.2.a.f.1.1		1	5.3	odd	4
975.2.c.f.274.1		2	1.1	even	1	trivial
975.2.c.f.274.2		2	5.4	even	2	inner
1053.2.e.b.352.1		2	45.7	odd	12
1053.2.e.b.703.1		2	45.22	odd	12
1053.2.e.d.352.1		2	45.2	even	12
1053.2.e.d.703.1		2	45.32	even	12
1521.2.a.e.1.1		1	195.77	even	4
1521.2.b.b.1351.1		2	195.47	odd	4
1521.2.b.b.1351.2		2	195.122	odd	4
1872.2.a.h.1.1		1	60.47	odd	4
1911.2.a.f.1.1		1	35.27	even	4
2496.2.a.e.1.1		1	40.27	even	4
2496.2.a.q.1.1		1	40.37	odd	4
2925.2.a.p.1.1		1	15.8	even	4
2925.2.c.e.2224.1		2	15.14	odd	2
2925.2.c.e.2224.2		2	3.2	odd	2
4719.2.a.c.1.1		1	55.32	even	4
5733.2.a.e.1.1		1	105.62	odd	4
7488.2.a.bl.1.1		1	120.77	even	4
7488.2.a.by.1.1		1	120.107	odd	4
8112.2.a.s.1.1		1	260.207	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{21} -4.00000i q^{22} -3.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} +10.0000 q^{29} +4.00000 q^{31} -5.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} +1.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} -12.0000i q^{43} +4.00000 q^{44} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +1.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +12.0000 q^{56} -10.0000i q^{58} -12.0000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -4.00000i q^{63} -7.00000 q^{64} -4.00000 q^{66} +8.00000i q^{67} -2.00000i q^{68} +3.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} +16.0000i q^{77} -1.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{84} -12.0000 q^{86} -10.0000i q^{87} -12.0000i q^{88} +2.00000 q^{89} -4.00000 q^{91} -4.00000i q^{93} -5.00000 q^{96} -10.0000i q^{97} +9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 8 q^{14} - 2 q^{16} + 8 q^{21} - 6 q^{24} + 2 q^{26} + 20 q^{29} + 8 q^{31} - 4 q^{34} - 2 q^{36} + 2 q^{39} + 12 q^{41} + 8 q^{44} - 18 q^{49} - 4 q^{51}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 8 * q^14 - 2 * q^16 + 8 * q^21 - 6 * q^24 + 2 * q^26 + 20 * q^29 + 8 * q^31 - 4 * q^34 - 2 * q^36 + 2 * q^39 + 12 * q^41 + 8 * q^44 - 18 * q^49 - 4 * q^51 + 2 * q^54 + 24 * q^56 - 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 + 4 * q^74 - 16 * q^79 + 2 * q^81 + 8 * q^84 - 24 * q^86 + 4 * q^89 - 8 * q^91 - 10 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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