LMFDB - Embedded newform 975.2.c.a.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:	$0$
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:	$\Z[a_1, a_2, a_3]$
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.a.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} +1.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} -1.00000i q^{17} -2.00000i q^{18} +2.00000 q^{19} -3.00000 q^{21} -2.00000i q^{22} +3.00000i q^{23} -2.00000 q^{26} -1.00000i q^{27} -6.00000i q^{28} +2.00000 q^{29} -6.00000 q^{31} -8.00000i q^{32} -1.00000i q^{33} +2.00000 q^{34} +2.00000 q^{36} +11.0000i q^{37} +4.00000i q^{38} -1.00000 q^{39} -5.00000 q^{41} -6.00000i q^{42} -4.00000i q^{43} +2.00000 q^{44} -6.00000 q^{46} -10.0000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} -11.0000i q^{53} +2.00000 q^{54} +2.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} +13.0000 q^{61} -12.0000i q^{62} -3.00000i q^{63} +8.00000 q^{64} +2.00000 q^{66} +12.0000i q^{67} +2.00000i q^{68} -3.00000 q^{69} -5.00000 q^{71} -10.0000i q^{73} -22.0000 q^{74} -4.00000 q^{76} -3.00000i q^{77} -2.00000i q^{78} +3.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +12.0000i q^{83} +6.00000 q^{84} +8.00000 q^{86} +2.00000i q^{87} +15.0000 q^{89} -3.00000 q^{91} -6.00000i q^{92} -6.00000i q^{93} +20.0000 q^{94} +8.00000 q^{96} +17.0000i q^{97} -4.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9	$ 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} - 12 q^{14} - 8 q^{16} + 4 q^{19} - 6 q^{21} - 4 q^{26} + 4 q^{29} - 12 q^{31} + 4 q^{34} + 4 q^{36} - 2 q^{39} - 10 q^{41} + 4 q^{44} - 12 q^{46} - 4 q^{49} + 2 q^{51} + 4 q^{54} - 16 q^{59} + 26 q^{61} + 16 q^{64} + 4 q^{66} - 6 q^{69} - 10 q^{71} - 44 q^{74} - 8 q^{76} + 6 q^{79} + 2 q^{81} + 12 q^{84} + 16 q^{86} + 30 q^{89} - 6 q^{91} + 40 q^{94} + 16 q^{96} + 2 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 - 2 * q^11 - 12 * q^14 - 8 * q^16 + 4 * q^19 - 6 * q^21 - 4 * q^26 + 4 * q^29 - 12 * q^31 + 4 * q^34 + 4 * q^36 - 2 * q^39 - 10 * q^41 + 4 * q^44 - 12 * q^46 - 4 * q^49 + 2 * q^51 + 4 * q^54 - 16 * q^59 + 26 * q^61 + 16 * q^64 + 4 * q^66 - 6 * q^69 - 10 * q^71 - 44 * q^74 - 8 * q^76 + 6 * q^79 + 2 * q^81 + 12 * q^84 + 16 * q^86 + 30 * q^89 - 6 * q^91 + 40 * q^94 + 16 * 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 $

$2$	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	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	0	0
$9$	−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	$0.548171\pi$
$12$	− 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
$17$	− 1.00000i	− 0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$	− 1.00000i	− 0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$	− 2.00000i	− 0.471405i
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	− 2.00000i	− 0.426401i
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	− 6.00000i	− 1.13389i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	− 8.00000i	− 1.41421i
$33$	− 1.00000i	− 0.174078i
$34$	2.00000	0.342997
$35$	0	0
$36$	2.00000	0.333333
$37$	11.0000i	1.80839i	0.427121	+	0.904194i	$0.359528\pi$
$37$	11.0000i	1.80839i	−0.427121	+	0.904194i	$0.640472\pi$
$38$	4.00000i	0.648886i
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	− 6.00000i	− 0.925820i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−6.00000	−0.884652
$47$	− 10.0000i	− 1.45865i	−0.684167	−	0.729325i	$-0.739834\pi$
$47$	− 10.0000i	− 1.45865i	0.684167	−	0.729325i	$-0.260166\pi$
$48$	− 4.00000i	− 0.577350i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	1.00000	0.140028
$52$	− 2.00000i	− 0.277350i
$53$	− 11.0000i	− 1.51097i	−0.655168	−	0.755483i	$-0.727402\pi$
$53$	− 11.0000i	− 1.51097i	0.655168	−	0.755483i	$-0.272598\pi$
$54$	2.00000	0.272166
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	4.00000i	0.525226i
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	− 12.0000i	− 1.52400i
$63$	− 3.00000i	− 0.377964i
$64$	8.00000	1.00000
$65$	0	0
$66$	2.00000	0.246183
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	2.00000i	0.242536i
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	−22.0000	−2.55745
$75$	0	0
$76$	−4.00000	−0.458831
$77$	− 3.00000i	− 0.341882i
$78$	− 2.00000i	− 0.226455i
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	6.00000	0.654654
$85$	0	0
$86$	8.00000	0.862662
$87$	2.00000i	0.214423i
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	− 6.00000i	− 0.625543i
$93$	− 6.00000i	− 0.622171i
$94$	20.0000	2.06284
$95$	0	0
$96$	8.00000	0.816497
$97$	17.0000i	1.72609i	0.505128	+	0.863044i	$0.331445\pi$
$97$	17.0000i	1.72609i	−0.505128	+	0.863044i	$0.668555\pi$
$98$	− 4.00000i	− 0.404061i
$99$	1.00000	0.100504
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	2.00000i	0.198030i
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	22.0000	2.13683
$107$	9.00000i	0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$	9.00000i	0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$	2.00000i	0.192450i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	− 12.0000i	− 1.13389i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−4.00000	−0.371391
$117$	− 1.00000i	− 0.0924500i
$118$	− 16.0000i	− 1.47292i
$119$	3.00000	0.275010
$120$	0	0
$121$	−10.0000	−0.909091
$122$	26.0000i	2.35393i
$123$	− 5.00000i	− 0.450835i
$124$	12.0000	1.07763
$125$	0	0
$126$	6.00000	0.534522
$127$	10.0000i	0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$	10.0000i	0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	2.00000i	0.174078i
$133$	6.00000i	0.520266i
$134$	−24.0000	−2.07328
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	− 6.00000i	− 0.510754i
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	− 10.0000i	− 0.839181i
$143$	− 1.00000i	− 0.0836242i
$144$	4.00000	0.333333
$145$	0	0
$146$	20.0000	1.65521
$147$	− 2.00000i	− 0.164957i
$148$	− 22.0000i	− 1.80839i
$149$	−13.0000	−1.06500	−0.532501	−	0.846430i	$-0.678748\pi$
$149$	−13.0000	−1.06500	−0.532501	+	0.846430i	$0.678748\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	1.00000i	0.0808452i
$154$	6.00000	0.483494
$155$	0	0
$156$	2.00000	0.160128
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	6.00000i	0.477334i
$159$	11.0000	0.872357
$160$	0	0
$161$	−9.00000	−0.709299
$162$	2.00000i	0.157135i
$163$	13.0000i	1.01824i	0.860696	+	0.509119i	$0.170029\pi$
$163$	13.0000i	1.01824i	−0.860696	+	0.509119i	$0.829971\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−24.0000	−1.86276
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	8.00000i	0.609994i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	4.00000	0.301511
$177$	− 8.00000i	− 0.601317i
$178$	30.0000i	2.24860i
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	− 6.00000i	− 0.444750i
$183$	13.0000i	0.960988i
$184$	0	0
$185$	0	0
$186$	12.0000	0.879883
$187$	1.00000i	0.0731272i
$188$	20.0000i	1.45865i
$189$	3.00000	0.218218
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	8.00000i	0.577350i
$193$	13.0000i	0.935760i	0.883792	+	0.467880i	$0.154982\pi$
$193$	13.0000i	0.935760i	−0.883792	+	0.467880i	$0.845018\pi$
$194$	−34.0000	−2.44106
$195$	0	0
$196$	4.00000	0.285714
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	2.00000i	0.142134i
$199$	−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
$201$	−12.0000	−0.846415
$202$	0	0
$203$	6.00000i	0.421117i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−32.0000	−2.22955
$207$	− 3.00000i	− 0.208514i
$208$	− 4.00000i	− 0.277350i
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	22.0000i	1.51097i
$213$	− 5.00000i	− 0.342594i
$214$	−18.0000	−1.23045
$215$	0	0
$216$	0	0
$217$	− 18.0000i	− 1.22192i
$218$	32.0000i	2.16731i
$219$	10.0000	0.675737
$220$	0	0
$221$	1.00000	0.0672673
$222$	− 22.0000i	− 1.47654i
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	24.0000	1.60357
$225$	0	0
$226$	28.0000	1.86253
$227$	22.0000i	1.46019i	0.683345	+	0.730096i	$0.260525\pi$
$227$	22.0000i	1.46019i	−0.683345	+	0.730096i	$0.739475\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	27.0000i	1.76883i	0.466702	+	0.884414i	$0.345442\pi$
$233$	27.0000i	1.76883i	−0.466702	+	0.884414i	$0.654558\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	16.0000	1.04151
$237$	3.00000i	0.194871i
$238$	6.00000i	0.388922i
$239$	13.0000	0.840900	0.420450	−	0.907316i	$-0.361872\pi$
$239$	13.0000	0.840900	0.420450	+	0.907316i	$0.361872\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	− 20.0000i	− 1.28565i
$243$	1.00000i	0.0641500i
$244$	−26.0000	−1.66448
$245$	0	0
$246$	10.0000	0.637577
$247$	2.00000i	0.127257i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	6.00000i	0.377964i
$253$	− 3.00000i	− 0.188608i
$254$	−20.0000	−1.25491
$255$	0	0
$256$	16.0000	1.00000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	8.00000i	0.498058i
$259$	−33.0000	−2.05052
$260$	0	0
$261$	−2.00000	−0.123797
$262$	− 12.0000i	− 0.741362i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	0	0
$265$	0	0
$266$	−12.0000	−0.735767
$267$	15.0000i	0.917985i
$268$	− 24.0000i	− 1.46603i
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	4.00000i	0.242536i
$273$	− 3.00000i	− 0.181568i
$274$	−36.0000	−2.17484
$275$	0	0
$276$	6.00000	0.361158
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	2.00000i	0.119952i
$279$	6.00000	0.359211
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	20.0000i	1.19098i
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	2.00000	0.118262
$287$	− 15.0000i	− 0.885422i
$288$	8.00000i	0.471405i
$289$	16.0000	0.941176
$290$	0	0
$291$	−17.0000	−0.996558
$292$	20.0000i	1.17041i
$293$	− 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
$297$	1.00000i	0.0580259i
$298$	− 26.0000i	− 1.50614i
$299$	−3.00000	−0.173494
$300$	0	0
$301$	12.0000	0.691669
$302$	32.0000i	1.84139i
$303$	0	0
$304$	−8.00000	−0.458831
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 5.00000i	− 0.285365i	−0.989769	−	0.142683i	$-0.954427\pi$
$307$	− 5.00000i	− 0.285365i	0.989769	−	0.142683i	$-0.0455728\pi$
$308$	6.00000i	0.341882i
$309$	−16.0000	−0.910208
$310$	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	$0.908798\pi$
$314$	20.0000	1.12867
$315$	0	0
$316$	−6.00000	−0.337526
$317$	28.0000i	1.57264i	0.617822	+	0.786318i	$0.288015\pi$
$317$	28.0000i	1.57264i	−0.617822	+	0.786318i	$0.711985\pi$
$318$	22.0000i	1.23370i
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−9.00000	−0.502331
$322$	− 18.0000i	− 1.00310i
$323$	− 2.00000i	− 0.111283i
$324$	−2.00000	−0.111111
$325$	0	0
$326$	−26.0000	−1.44001
$327$	16.0000i	0.884802i
$328$	0	0
$329$	30.0000	1.65395
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	− 24.0000i	− 1.31717i
$333$	− 11.0000i	− 0.602796i
$334$	24.0000	1.31322
$335$	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
$339$	14.0000	0.760376
$340$	0	0
$341$	6.00000	0.324918
$342$	− 4.00000i	− 0.216295i
$343$	15.0000i	0.809924i
$344$	0	0
$345$	0	0
$346$	−12.0000	−0.645124
$347$	− 19.0000i	− 1.01997i	−0.860182	−	0.509987i	$-0.829650\pi$
$347$	− 19.0000i	− 1.01997i	0.860182	−	0.509987i	$-0.170350\pi$
$348$	− 4.00000i	− 0.214423i
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	8.00000i	0.426401i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	16.0000	0.850390
$355$	0	0
$356$	−30.0000	−1.59000
$357$	3.00000i	0.158777i
$358$	− 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
$361$	−15.0000	−0.789474
$362$	− 14.0000i	− 0.735824i
$363$	− 10.0000i	− 0.524864i
$364$	6.00000	0.314485
$365$	0	0
$366$	−26.0000	−1.35904
$367$	36.0000i	1.87918i	0.342296	+	0.939592i	$0.388796\pi$
$367$	36.0000i	1.87918i	−0.342296	+	0.939592i	$0.611204\pi$
$368$	− 12.0000i	− 0.625543i
$369$	5.00000	0.260290
$370$	0	0
$371$	33.0000	1.71327
$372$	12.0000i	0.622171i
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	0	0
$377$	2.00000i	0.103005i
$378$	6.00000i	0.308607i
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	−10.0000	−0.512316
$382$	− 16.0000i	− 0.818631i
$383$	30.0000i	1.53293i	0.642287	+	0.766464i	$0.277986\pi$
$383$	30.0000i	1.53293i	−0.642287	+	0.766464i	$0.722014\pi$
$384$	0	0
$385$	0	0
$386$	−26.0000	−1.32337
$387$	4.00000i	0.203331i
$388$	− 34.0000i	− 1.72609i
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	− 6.00000i	− 0.302660i
$394$	0	0
$395$	0	0
$396$	−2.00000	−0.100504
$397$	− 29.0000i	− 1.45547i	−0.685859	−	0.727734i	$-0.740573\pi$
$397$	− 29.0000i	− 1.45547i	0.685859	−	0.727734i	$-0.259427\pi$
$398$	− 8.00000i	− 0.401004i
$399$	−6.00000	−0.300376
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	− 24.0000i	− 1.19701i
$403$	− 6.00000i	− 0.298881i
$404$	0	0
$405$	0	0
$406$	−12.0000	−0.595550
$407$	− 11.0000i	− 0.545250i
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	− 32.0000i	− 1.57653i
$413$	− 24.0000i	− 1.18096i
$414$	6.00000	0.294884
$415$	0	0
$416$	8.00000	0.392232
$417$	1.00000i	0.0489702i
$418$	− 4.00000i	− 0.195646i
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	− 8.00000i	− 0.389434i
$423$	10.0000i	0.486217i
$424$	0	0
$425$	0	0
$426$	10.0000	0.484502
$427$	39.0000i	1.88734i
$428$	− 18.0000i	− 0.870063i
$429$	1.00000	0.0482805
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	4.00000i	0.192450i
$433$	− 4.00000i	− 0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$	− 4.00000i	− 0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	36.0000	1.72806
$435$	0	0
$436$	−32.0000	−1.53252
$437$	6.00000i	0.287019i
$438$	20.0000i	0.955637i
$439$	17.0000	0.811366	0.405683	−	0.914014i	$-0.367034\pi$
$439$	17.0000	0.811366	0.405683	+	0.914014i	$0.367034\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	2.00000i	0.0951303i
$443$	− 9.00000i	− 0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$	− 9.00000i	− 0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$	22.0000	1.04407
$445$	0	0
$446$	16.0000	0.757622
$447$	− 13.0000i	− 0.614879i
$448$	24.0000i	1.13389i
$449$	13.0000	0.613508	0.306754	−	0.951789i	$-0.400757\pi$
$449$	13.0000	0.613508	0.306754	+	0.951789i	$0.400757\pi$
$450$	0	0
$451$	5.00000	0.235441
$452$	28.0000i	1.31701i
$453$	16.0000i	0.751746i
$454$	−44.0000	−2.06502
$455$	0	0
$456$	0	0
$457$	− 11.0000i	− 0.514558i	−0.966337	−	0.257279i	$-0.917174\pi$
$457$	− 11.0000i	− 0.514558i	0.966337	−	0.257279i	$-0.0828260\pi$
$458$	− 36.0000i	− 1.68217i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	6.00000i	0.279145i
$463$	− 27.0000i	− 1.25480i	−0.778699	−	0.627398i	$-0.784120\pi$
$463$	− 27.0000i	− 1.25480i	0.778699	−	0.627398i	$-0.215880\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−54.0000	−2.50150
$467$	− 23.0000i	− 1.06431i	−0.846646	−	0.532157i	$-0.821382\pi$
$467$	− 23.0000i	− 1.06431i	0.846646	−	0.532157i	$-0.178618\pi$
$468$	2.00000i	0.0924500i
$469$	−36.0000	−1.66233
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	4.00000i	0.183920i
$474$	−6.00000	−0.275589
$475$	0	0
$476$	−6.00000	−0.275010
$477$	11.0000i	0.503655i
$478$	26.0000i	1.18921i
$479$	−9.00000	−0.411220	−0.205610	−	0.978634i	$-0.565918\pi$
$479$	−9.00000	−0.411220	−0.205610	+	0.978634i	$0.565918\pi$
$480$	0	0
$481$	−11.0000	−0.501557
$482$	− 4.00000i	− 0.182195i
$483$	− 9.00000i	− 0.409514i
$484$	20.0000	0.909091
$485$	0	0
$486$	−2.00000	−0.0907218
$487$	− 7.00000i	− 0.317200i	−0.987343	−	0.158600i	$-0.949302\pi$
$487$	− 7.00000i	− 0.317200i	0.987343	−	0.158600i	$-0.0506981\pi$
$488$	0	0
$489$	−13.0000	−0.587880
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	10.0000i	0.450835i
$493$	− 2.00000i	− 0.0900755i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	24.0000	1.07763
$497$	− 15.0000i	− 0.672842i
$498$	− 24.0000i	− 1.07547i
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	− 28.0000i	− 1.24846i	−0.781241	−	0.624229i	$-0.785413\pi$
$503$	− 28.0000i	− 1.24846i	0.781241	−	0.624229i	$-0.214587\pi$
$504$	0	0
$505$	0	0
$506$	6.00000	0.266733
$507$	− 1.00000i	− 0.0444116i
$508$	− 20.0000i	− 0.887357i
$509$	7.00000	0.310270	0.155135	−	0.987893i	$-0.450419\pi$
$509$	7.00000	0.310270	0.155135	+	0.987893i	$0.450419\pi$
$510$	0	0
$511$	30.0000	1.32712
$512$	32.0000i	1.41421i
$513$	− 2.00000i	− 0.0883022i
$514$	36.0000	1.58789
$515$	0	0
$516$	−8.00000	−0.352180
$517$	10.0000i	0.439799i
$518$	− 66.0000i	− 2.89987i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	− 4.00000i	− 0.175075i
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	6.00000i	0.261364i
$528$	4.00000i	0.174078i
$529$	14.0000	0.608696
$530$	0	0
$531$	8.00000	0.347170
$532$	− 12.0000i	− 0.520266i
$533$	− 5.00000i	− 0.216574i
$534$	−30.0000	−1.29823
$535$	0	0
$536$	0	0
$537$	− 2.00000i	− 0.0863064i
$538$	8.00000i	0.344904i
$539$	2.00000	0.0861461
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	44.0000i	1.88996i
$543$	− 7.00000i	− 0.300399i
$544$	−8.00000	−0.342997
$545$	0	0
$546$	6.00000	0.256776
$547$	− 32.0000i	− 1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$	− 32.0000i	− 1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$	− 36.0000i	− 1.53784i
$549$	−13.0000	−0.554826
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	9.00000i	0.382719i
$554$	36.0000	1.52949
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	12.0000i	0.508001i
$559$	4.00000	0.169182
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	60.0000i	2.53095i
$563$	− 21.0000i	− 0.885044i	−0.896758	−	0.442522i	$-0.854084\pi$
$563$	− 21.0000i	− 0.885044i	0.896758	−	0.442522i	$-0.145916\pi$
$564$	−20.0000	−0.842152
$565$	0	0
$566$	−24.0000	−1.00880
$567$	3.00000i	0.125988i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	2.00000i	0.0836242i
$573$	− 8.00000i	− 0.334205i
$574$	30.0000	1.25218
$575$	0	0
$576$	−8.00000	−0.333333
$577$	− 19.0000i	− 0.790980i	−0.918470	−	0.395490i	$-0.870575\pi$
$577$	− 19.0000i	− 0.790980i	0.918470	−	0.395490i	$-0.129425\pi$
$578$	32.0000i	1.33102i
$579$	−13.0000	−0.540262
$580$	0	0
$581$	−36.0000	−1.49353
$582$	− 34.0000i	− 1.40935i
$583$	11.0000i	0.455573i
$584$	0	0
$585$	0	0
$586$	48.0000	1.98286
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	4.00000i	0.164957i
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	− 44.0000i	− 1.80839i
$593$	− 4.00000i	− 0.164260i	−0.996622	−	0.0821302i	$-0.973828\pi$
$593$	− 4.00000i	− 0.164260i	0.996622	−	0.0821302i	$-0.0261723\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	26.0000	1.06500
$597$	− 4.00000i	− 0.163709i
$598$	− 6.00000i	− 0.245358i
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	24.0000i	0.978167i
$603$	− 12.0000i	− 0.488678i
$604$	−32.0000	−1.30206
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	− 16.0000i	− 0.648886i
$609$	−6.00000	−0.243132
$610$	0	0
$611$	10.0000	0.404557
$612$	− 2.00000i	− 0.0808452i
$613$	− 13.0000i	− 0.525065i	−0.964923	−	0.262533i	$-0.915442\pi$
$613$	− 13.0000i	− 0.525065i	0.964923	−	0.262533i	$-0.0845577\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	0	0
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	− 32.0000i	− 1.28723i
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	48.0000i	1.92462i
$623$	45.0000i	1.80289i
$624$	4.00000	0.160128
$625$	0	0
$626$	−20.0000	−0.799361
$627$	− 2.00000i	− 0.0798723i
$628$	20.0000i	0.798087i
$629$	11.0000	0.438599
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	− 4.00000i	− 0.158986i
$634$	−56.0000	−2.22404
$635$	0	0
$636$	−22.0000	−0.872357
$637$	− 2.00000i	− 0.0792429i
$638$	− 4.00000i	− 0.158362i
$639$	5.00000	0.197797
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	− 18.0000i	− 0.710403i
$643$	− 15.0000i	− 0.591542i	−0.955259	−	0.295771i	$-0.904423\pi$
$643$	− 15.0000i	− 0.591542i	0.955259	−	0.295771i	$-0.0955766\pi$
$644$	18.0000	0.709299
$645$	0	0
$646$	4.00000	0.157378
$647$	47.0000i	1.84776i	0.382682	+	0.923880i	$0.375001\pi$
$647$	47.0000i	1.84776i	−0.382682	+	0.923880i	$0.624999\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	18.0000	0.705476
$652$	− 26.0000i	− 1.01824i
$653$	− 22.0000i	− 0.860927i	−0.902608	−	0.430463i	$-0.858350\pi$
$653$	− 22.0000i	− 0.860927i	0.902608	−	0.430463i	$-0.141650\pi$
$654$	−32.0000	−1.25130
$655$	0	0
$656$	20.0000	0.780869
$657$	10.0000i	0.390137i
$658$	60.0000i	2.33904i
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	0	0
$663$	1.00000i	0.0388368i
$664$	0	0
$665$	0	0
$666$	22.0000	0.852483
$667$	6.00000i	0.232321i
$668$	24.0000i	0.928588i
$669$	8.00000	0.309298
$670$	0	0
$671$	−13.0000	−0.501859
$672$	24.0000i	0.925820i
$673$	6.00000i	0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$	6.00000i	0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	2.00000	0.0769231
$677$	3.00000i	0.115299i	0.998337	+	0.0576497i	$0.0183606\pi$
$677$	3.00000i	0.115299i	−0.998337	+	0.0576497i	$0.981639\pi$
$678$	28.0000i	1.07533i
$679$	−51.0000	−1.95720
$680$	0	0
$681$	−22.0000	−0.843042
$682$	12.0000i	0.459504i
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−30.0000	−1.14541
$687$	− 18.0000i	− 0.686743i
$688$	16.0000i	0.609994i
$689$	11.0000	0.419067
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	− 12.0000i	− 0.456172i
$693$	3.00000i	0.113961i
$694$	38.0000	1.44246
$695$	0	0
$696$	0	0
$697$	5.00000i	0.189389i
$698$	16.0000i	0.605609i
$699$	−27.0000	−1.02123
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	2.00000i	0.0754851i
$703$	22.0000i	0.829746i
$704$	−8.00000	−0.301511
$705$	0	0
$706$	12.0000	0.451626
$707$	0	0
$708$	16.0000i	0.601317i
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	0	0
$713$	− 18.0000i	− 0.674105i
$714$	−6.00000	−0.224544
$715$	0	0
$716$	4.00000	0.149487
$717$	13.0000i	0.485494i
$718$	− 48.0000i	− 1.79134i
$719$	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
$721$	−48.0000	−1.78761
$722$	− 30.0000i	− 1.11648i
$723$	− 2.00000i	− 0.0743808i
$724$	14.0000	0.520306
$725$	0	0
$726$	20.0000	0.742270
$727$	38.0000i	1.40934i	0.709534	+	0.704671i	$0.248905\pi$
$727$	38.0000i	1.40934i	−0.709534	+	0.704671i	$0.751095\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	− 26.0000i	− 0.960988i
$733$	49.0000i	1.80986i	0.425564	+	0.904928i	$0.360076\pi$
$733$	49.0000i	1.80986i	−0.425564	+	0.904928i	$0.639924\pi$
$734$	−72.0000	−2.65757
$735$	0	0
$736$	24.0000	0.884652
$737$	− 12.0000i	− 0.442026i
$738$	10.0000i	0.368105i
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	66.0000i	2.42294i
$743$	34.0000i	1.24734i	0.781688	+	0.623670i	$0.214359\pi$
$743$	34.0000i	1.24734i	−0.781688	+	0.623670i	$0.785641\pi$
$744$	0	0
$745$	0	0
$746$	8.00000	0.292901
$747$	− 12.0000i	− 0.439057i
$748$	− 2.00000i	− 0.0731272i
$749$	−27.0000	−0.986559
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	40.0000i	1.45865i
$753$	0	0
$754$	−4.00000	−0.145671
$755$	0	0
$756$	−6.00000	−0.218218
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	28.0000i	1.01701i
$759$	3.00000	0.108893
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	− 20.0000i	− 0.724524i
$763$	48.0000i	1.73772i
$764$	16.0000	0.578860
$765$	0	0
$766$	−60.0000	−2.16789
$767$	− 8.00000i	− 0.288863i
$768$	16.0000i	0.577350i
$769$	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
$771$	18.0000	0.648254
$772$	− 26.0000i	− 0.935760i
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	0	0
$777$	− 33.0000i	− 1.18387i
$778$	0	0
$779$	−10.0000	−0.358287
$780$	0	0
$781$	5.00000	0.178914
$782$	6.00000i	0.214560i
$783$	− 2.00000i	− 0.0714742i
$784$	8.00000	0.285714
$785$	0	0
$786$	12.0000	0.428026
$787$	52.0000i	1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$	52.0000i	1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	42.0000	1.49335
$792$	0	0
$793$	13.0000i	0.461644i
$794$	58.0000	2.05834
$795$	0	0
$796$	8.00000	0.283552
$797$	− 47.0000i	− 1.66483i	−0.554156	−	0.832413i	$-0.686959\pi$
$797$	− 47.0000i	− 1.66483i	0.554156	−	0.832413i	$-0.313041\pi$
$798$	− 12.0000i	− 0.424795i
$799$	−10.0000	−0.353775
$800$	0	0
$801$	−15.0000	−0.529999
$802$	60.0000i	2.11867i
$803$	10.0000i	0.352892i
$804$	24.0000	0.846415
$805$	0	0
$806$	12.0000	0.422682
$807$	4.00000i	0.140807i
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	− 12.0000i	− 0.421117i
$813$	22.0000i	0.771574i
$814$	22.0000	0.771100
$815$	0	0
$816$	−4.00000	−0.140028
$817$	− 8.00000i	− 0.279885i
$818$	− 4.00000i	− 0.139857i
$819$	3.00000	0.104828
$820$	0	0
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	− 36.0000i	− 1.25564i
$823$	− 20.0000i	− 0.697156i	−0.937280	−	0.348578i	$-0.886665\pi$
$823$	− 20.0000i	− 0.697156i	0.937280	−	0.348578i	$-0.113335\pi$
$824$	0	0
$825$	0	0
$826$	48.0000	1.67013
$827$	26.0000i	0.904109i	0.891990	+	0.452054i	$0.149309\pi$
$827$	26.0000i	0.904109i	−0.891990	+	0.452054i	$0.850691\pi$
$828$	6.00000i	0.208514i
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	8.00000i	0.277350i
$833$	2.00000i	0.0692959i
$834$	−2.00000	−0.0692543
$835$	0	0
$836$	4.00000	0.138343
$837$	6.00000i	0.207390i
$838$	52.0000i	1.79631i
$839$	5.00000	0.172619	0.0863096	−	0.996268i	$-0.472493\pi$
$839$	5.00000	0.172619	0.0863096	+	0.996268i	$0.472493\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	8.00000i	0.275698i
$843$	30.0000i	1.03325i
$844$	8.00000	0.275371
$845$	0	0
$846$	−20.0000	−0.687614
$847$	− 30.0000i	− 1.03081i
$848$	44.0000i	1.51097i
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−33.0000	−1.13123
$852$	10.0000i	0.342594i
$853$	− 45.0000i	− 1.54077i	−0.637579	−	0.770385i	$-0.720064\pi$
$853$	− 45.0000i	− 1.54077i	0.637579	−	0.770385i	$-0.279936\pi$
$854$	−78.0000	−2.66911
$855$	0	0
$856$	0	0
$857$	29.0000i	0.990621i	0.868716	+	0.495311i	$0.164946\pi$
$857$	29.0000i	0.990621i	−0.868716	+	0.495311i	$0.835054\pi$
$858$	2.00000i	0.0682789i
$859$	29.0000	0.989467	0.494734	−	0.869045i	$-0.335266\pi$
$859$	29.0000	0.989467	0.494734	+	0.869045i	$0.335266\pi$
$860$	0	0
$861$	15.0000	0.511199
$862$	− 48.0000i	− 1.63489i
$863$	− 34.0000i	− 1.15737i	−0.815550	−	0.578687i	$-0.803565\pi$
$863$	− 34.0000i	− 1.15737i	0.815550	−	0.578687i	$-0.196435\pi$
$864$	−8.00000	−0.272166
$865$	0	0
$866$	8.00000	0.271851
$867$	16.0000i	0.543388i
$868$	36.0000i	1.22192i
$869$	−3.00000	−0.101768
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	− 17.0000i	− 0.575363i
$874$	−12.0000	−0.405906
$875$	0	0
$876$	−20.0000	−0.675737
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	34.0000i	1.14744i
$879$	24.0000	0.809500
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	4.00000i	0.134687i
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	18.0000	0.604722
$887$	− 21.0000i	− 0.705111i	−0.935791	−	0.352555i	$-0.885313\pi$
$887$	− 21.0000i	− 0.705111i	0.935791	−	0.352555i	$-0.114687\pi$
$888$	0	0
$889$	−30.0000	−1.00617
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	16.0000i	0.535720i
$893$	− 20.0000i	− 0.669274i
$894$	26.0000	0.869570
$895$	0	0
$896$	0	0
$897$	− 3.00000i	− 0.100167i
$898$	26.0000i	0.867631i
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−11.0000	−0.366463
$902$	10.0000i	0.332964i
$903$	12.0000i	0.399335i
$904$	0	0
$905$	0	0
$906$	−32.0000	−1.06313
$907$	6.00000i	0.199227i	0.995026	+	0.0996134i	$0.0317606\pi$
$907$	6.00000i	0.199227i	−0.995026	+	0.0996134i	$0.968239\pi$
$908$	− 44.0000i	− 1.46019i
$909$	0	0
$910$	0	0
$911$	44.0000	1.45779	0.728893	−	0.684628i	$-0.240035\pi$
$911$	44.0000	1.45779	0.728893	+	0.684628i	$0.240035\pi$
$912$	− 8.00000i	− 0.264906i
$913$	− 12.0000i	− 0.397142i
$914$	22.0000	0.727695
$915$	0	0
$916$	36.0000	1.18947
$917$	− 18.0000i	− 0.594412i
$918$	− 2.00000i	− 0.0660098i
$919$	−37.0000	−1.22052	−0.610259	−	0.792202i	$-0.708935\pi$
$919$	−37.0000	−1.22052	−0.610259	+	0.792202i	$0.708935\pi$
$920$	0	0
$921$	5.00000	0.164756
$922$	30.0000i	0.987997i
$923$	− 5.00000i	− 0.164577i
$924$	−6.00000	−0.197386
$925$	0	0
$926$	54.0000	1.77455
$927$	− 16.0000i	− 0.525509i
$928$	− 16.0000i	− 0.525226i
$929$	−1.00000	−0.0328089	−0.0164045	−	0.999865i	$-0.505222\pi$
$929$	−1.00000	−0.0328089	−0.0164045	+	0.999865i	$0.505222\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	− 54.0000i	− 1.76883i
$933$	24.0000i	0.785725i
$934$	46.0000	1.50517
$935$	0	0
$936$	0	0
$937$	30.0000i	0.980057i	0.871706	+	0.490029i	$0.163014\pi$
$937$	30.0000i	0.980057i	−0.871706	+	0.490029i	$0.836986\pi$
$938$	− 72.0000i	− 2.35088i
$939$	−10.0000	−0.326338
$940$	0	0
$941$	37.0000	1.20617	0.603083	−	0.797679i	$-0.293939\pi$
$941$	37.0000	1.20617	0.603083	+	0.797679i	$0.293939\pi$
$942$	20.0000i	0.651635i
$943$	− 15.0000i	− 0.488467i
$944$	32.0000	1.04151
$945$	0	0
$946$	−8.00000	−0.260102
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	− 6.00000i	− 0.194871i
$949$	10.0000	0.324614
$950$	0	0
$951$	−28.0000	−0.907962
$952$	0	0
$953$	1.00000i	0.0323932i	0.999869	+	0.0161966i	$0.00515576\pi$
$953$	1.00000i	0.0323932i	−0.999869	+	0.0161966i	$0.994844\pi$
$954$	−22.0000	−0.712276
$955$	0	0
$956$	−26.0000	−0.840900
$957$	− 2.00000i	− 0.0646508i
$958$	− 18.0000i	− 0.581554i
$959$	−54.0000	−1.74375
$960$	0	0
$961$	5.00000	0.161290
$962$	− 22.0000i	− 0.709308i
$963$	− 9.00000i	− 0.290021i
$964$	4.00000	0.128831
$965$	0	0
$966$	18.0000	0.579141
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	0	0
$969$	2.00000	0.0642493
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 2.00000i	− 0.0641500i
$973$	3.00000i	0.0961756i
$974$	14.0000	0.448589
$975$	0	0
$976$	−52.0000	−1.66448
$977$	32.0000i	1.02377i	0.859054	+	0.511885i	$0.171053\pi$
$977$	32.0000i	1.02377i	−0.859054	+	0.511885i	$0.828947\pi$
$978$	− 26.0000i	− 0.831388i
$979$	−15.0000	−0.479402
$980$	0	0
$981$	−16.0000	−0.510841
$982$	56.0000i	1.78703i
$983$	12.0000i	0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$	12.0000i	0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$	0	0
$985$	0	0
$986$	4.00000	0.127386
$987$	30.0000i	0.954911i
$988$	− 4.00000i	− 0.127257i
$989$	12.0000	0.381578
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	48.0000i	1.52400i
$993$	0	0
$994$	30.0000	0.951542
$995$	0	0
$996$	24.0000	0.760469
$997$	− 36.0000i	− 1.14013i	−0.821599	−	0.570066i	$-0.806918\pi$
$997$	− 36.0000i	− 1.14013i	0.821599	−	0.570066i	$-0.193082\pi$
$998$	28.0000i	0.886325i
$999$	11.0000	0.348025

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} +1.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} -1.00000i q^{17} -2.00000i q^{18} +2.00000 q^{19} -3.00000 q^{21} -2.00000i q^{22} +3.00000i q^{23} -2.00000 q^{26} -1.00000i q^{27} -6.00000i q^{28} +2.00000 q^{29} -6.00000 q^{31} -8.00000i q^{32} -1.00000i q^{33} +2.00000 q^{34} +2.00000 q^{36} +11.0000i q^{37} +4.00000i q^{38} -1.00000 q^{39} -5.00000 q^{41} -6.00000i q^{42} -4.00000i q^{43} +2.00000 q^{44} -6.00000 q^{46} -10.0000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} -11.0000i q^{53} +2.00000 q^{54} +2.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} +13.0000 q^{61} -12.0000i q^{62} -3.00000i q^{63} +8.00000 q^{64} +2.00000 q^{66} +12.0000i q^{67} +2.00000i q^{68} -3.00000 q^{69} -5.00000 q^{71} -10.0000i q^{73} -22.0000 q^{74} -4.00000 q^{76} -3.00000i q^{77} -2.00000i q^{78} +3.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +12.0000i q^{83} +6.00000 q^{84} +8.00000 q^{86} +2.00000i q^{87} +15.0000 q^{89} -3.00000 q^{91} -6.00000i q^{92} -6.00000i q^{93} +20.0000 q^{94} +8.00000 q^{96} +17.0000i q^{97} -4.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} - 12 q^{14} - 8 q^{16} + 4 q^{19} - 6 q^{21} - 4 q^{26} + 4 q^{29} - 12 q^{31} + 4 q^{34} + 4 q^{36} - 2 q^{39} - 10 q^{41} + 4 q^{44} - 12 q^{46} - 4 q^{49} + 2 q^{51} + 4 q^{54} - 16 q^{59} + 26 q^{61} + 16 q^{64} + 4 q^{66} - 6 q^{69} - 10 q^{71} - 44 q^{74} - 8 q^{76} + 6 q^{79} + 2 q^{81} + 12 q^{84} + 16 q^{86} + 30 q^{89} - 6 q^{91} + 40 q^{94} + 16 q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 - 2 * q^11 - 12 * q^14 - 8 * q^16 + 4 * q^19 - 6 * q^21 - 4 * q^26 + 4 * q^29 - 12 * q^31 + 4 * q^34 + 4 * q^36 - 2 * q^39 - 10 * q^41 + 4 * q^44 - 12 * q^46 - 4 * q^49 + 2 * q^51 + 4 * q^54 - 16 * q^59 + 26 * q^61 + 16 * q^64 + 4 * q^66 - 6 * q^69 - 10 * q^71 - 44 * q^74 - 8 * q^76 + 6 * q^79 + 2 * q^81 + 12 * q^84 + 16 * q^86 + 30 * q^89 - 6 * q^91 + 40 * q^94 + 16 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

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