LMFDB - Embedded newform 75.12.b.d.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,12,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	75.b (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:	$57.6257385420$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1801})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 901x^{2} + 202500 $ x^4 + 901*x^2 + 202500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$20.7191i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.12.b.d.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+14.7191i q^{2} +243.000i q^{3} +1831.35 q^{4} -3576.74 q^{6} +79941.2i q^{7} +57100.5i q^{8} -59049.0 q^{9} +805067. q^{11} +445018. i q^{12} +1.19767e6i q^{13} -1.17666e6 q^{14} +2.91013e6 q^{16} +2.63319e6i q^{17} -869148. i q^{18} -1.16061e7 q^{19} -1.94257e7 q^{21} +1.18499e7i q^{22} -1.84216e7i q^{23} -1.38754e7 q^{24} -1.76286e7 q^{26} -1.43489e7i q^{27} +1.46400e8i q^{28} +1.90527e8 q^{29} +1.01127e8 q^{31} +1.59776e8i q^{32} +1.95631e8i q^{33} -3.87581e7 q^{34} -1.08139e8 q^{36} +8.06675e7i q^{37} -1.70831e8i q^{38} -2.91034e8 q^{39} +2.26316e8 q^{41} -2.85929e8i q^{42} -1.67149e9i q^{43} +1.47436e9 q^{44} +2.71150e8 q^{46} +8.58507e8i q^{47} +7.07162e8i q^{48} -4.41327e9 q^{49} -6.39864e8 q^{51} +2.19335e9i q^{52} +3.52750e9i q^{53} +2.11203e8 q^{54} -4.56468e9 q^{56} -2.82028e9i q^{57} +2.80438e9i q^{58} -4.35760e9 q^{59} -1.65393e9 q^{61} +1.48849e9i q^{62} -4.72045e9i q^{63} +3.60819e9 q^{64} -2.87951e9 q^{66} +7.58610e9i q^{67} +4.82228e9i q^{68} +4.47645e9 q^{69} -2.75809e10 q^{71} -3.37173e9i q^{72} -3.22368e10i q^{73} -1.18735e9 q^{74} -2.12548e10 q^{76} +6.43580e10i q^{77} -4.28376e9i q^{78} +2.43149e9 q^{79} +3.48678e9 q^{81} +3.33116e9i q^{82} -1.20729e10i q^{83} -3.55753e10 q^{84} +2.46028e10 q^{86} +4.62981e10i q^{87} +4.59697e10i q^{88} -4.44073e9 q^{89} -9.57433e10 q^{91} -3.37364e10i q^{92} +2.45737e10i q^{93} -1.26364e10 q^{94} -3.88257e10 q^{96} -2.04453e10i q^{97} -6.49594e10i q^{98} -4.75384e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6222 q^{4} + 6318 q^{6} - 236196 q^{9} + 591136 q^{11} - 6353592 q^{14} + 5948130 q^{16} - 35255952 q^{19} - 3783024 q^{21} + 17077554 q^{24} - 138104132 q^{26} + 403763896 q^{29} - 142114016 q^{31}+ \cdots - 34905989664 q^{99}+O(q^{100}) $ 4 * q + 6222 * q^4 + 6318 * q^6 - 236196 * q^9 + 591136 * q^11 - 6353592 * q^14 + 5948130 * q^16 - 35255952 * q^19 - 3783024 * q^21 + 17077554 * q^24 - 138104132 * q^26 + 403763896 * q^29 - 142114016 * q^31 + 252161404 * q^34 - 367402878 * q^36 + 319541112 * q^39 - 655310296 * q^41 + 1644753136 * q^44 + 3217895664 * q^46 - 15285229540 * q^49 - 4169854728 * q^51 - 373071582 * q^54 - 22440862920 * q^56 + 2957541152 * q^59 - 16529782920 * q^61 - 3092633362 * q^64 - 12622730256 * q^66 - 14502760992 * q^69 - 40437776512 * q^71 + 32284748068 * q^74 - 57921409528 * q^76 - 44649990880 * q^79 + 13947137604 * q^81 - 26275156488 * q^84 - 35119396184 * q^86 - 158419366152 * q^89 - 459213424096 * q^91 + 40951225552 * q^94 + 15018498846 * q^96 - 34905989664 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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 $

$2$	14.7191i	0.325249i	0.986688	+	0.162625i	$0.0519959\pi$
$2$	14.7191i	0.325249i	−0.986688	+	0.162625i	$0.948004\pi$
$3$	243.000i	0.577350i
$3$	243.000i	0.577350i
$4$	1831.35	0.894213
$5$	0	0
$5$	0	0
$6$	−3576.74	−0.187783
$7$	79941.2i	1.79776i	0.438195	+	0.898880i	$0.355618\pi$
$7$	79941.2i	1.79776i	−0.438195	+	0.898880i	$0.644382\pi$
$8$	57100.5i	0.616091i
$9$	−59049.0	−0.333333
$10$	0	0
$11$	805067.	1.50720	0.753602	−	0.657331i	$-0.228315\pi$
$11$	805067.	1.50720	0.753602	+	0.657331i	$0.228315\pi$
$12$	445018.i	0.516274i
$13$	1.19767e6i	0.894641i	0.894374	+	0.447321i	$0.147622\pi$
$13$	1.19767e6i	0.894641i	−0.894374	+	0.447321i	$0.852378\pi$
$14$	−1.17666e6	−0.584720
$15$	0	0
$16$	2.91013e6	0.693830
$17$	2.63319e6i	0.449793i	0.974383	+	0.224896i	$0.0722044\pi$
$17$	2.63319e6i	0.449793i	−0.974383	+	0.224896i	$0.927796\pi$
$18$	− 869148.i	− 0.108416i
$19$	−1.16061e7	−1.07533	−0.537664	−	0.843159i	$-0.680693\pi$
$19$	−1.16061e7	−1.07533	−0.537664	+	0.843159i	$0.680693\pi$
$20$	0	0
$21$	−1.94257e7	−1.03794
$22$	1.18499e7i	0.490217i
$23$	− 1.84216e7i	− 0.596794i	−0.954442	−	0.298397i	$-0.903548\pi$
$23$	− 1.84216e7i	− 0.596794i	0.954442	−	0.298397i	$-0.0964520\pi$
$24$	−1.38754e7	−0.355700
$25$	0	0
$26$	−1.76286e7	−0.290981
$27$	− 1.43489e7i	− 0.192450i
$28$	1.46400e8i	1.60758i
$29$	1.90527e8	1.72491	0.862457	−	0.506130i	$-0.168924\pi$
$29$	1.90527e8	1.72491	0.862457	+	0.506130i	$0.168924\pi$
$30$	0	0
$31$	1.01127e8	0.634419	0.317209	−	0.948356i	$-0.397254\pi$
$31$	1.01127e8	0.634419	0.317209	+	0.948356i	$0.397254\pi$
$32$	1.59776e8i	0.841759i
$33$	1.95631e8i	0.870185i
$34$	−3.87581e7	−0.146295
$35$	0	0
$36$	−1.08139e8	−0.298071
$37$	8.06675e7i	0.191245i	0.995418	+	0.0956223i	$0.0304841\pi$
$37$	8.06675e7i	0.191245i	−0.995418	+	0.0956223i	$0.969516\pi$
$38$	− 1.70831e8i	− 0.349749i
$39$	−2.91034e8	−0.516522
$40$	0	0
$41$	2.26316e8	0.305073	0.152537	−	0.988298i	$-0.451256\pi$
$41$	2.26316e8	0.305073	0.152537	+	0.988298i	$0.451256\pi$
$42$	− 2.85929e8i	− 0.337588i
$43$	− 1.67149e9i	− 1.73391i	−0.498385	−	0.866956i	$-0.666073\pi$
$43$	− 1.67149e9i	− 1.73391i	0.498385	−	0.866956i	$-0.333927\pi$
$44$	1.47436e9	1.34776
$45$	0	0
$46$	2.71150e8	0.194107
$47$	8.58507e8i	0.546016i	0.962012	+	0.273008i	$0.0880186\pi$
$47$	8.58507e8i	0.546016i	−0.962012	+	0.273008i	$0.911981\pi$
$48$	7.07162e8i	0.400583i
$49$	−4.41327e9	−2.23194
$50$	0	0
$51$	−6.39864e8	−0.259688
$52$	2.19335e9i	0.800000i
$53$	3.52750e9i	1.15864i	0.815099	+	0.579322i	$0.196683\pi$
$53$	3.52750e9i	1.15864i	−0.815099	+	0.579322i	$0.803317\pi$
$54$	2.11203e8	0.0625942
$55$	0	0
$56$	−4.56468e9	−1.10758
$57$	− 2.82028e9i	− 0.620841i
$58$	2.80438e9i	0.561027i
$59$	−4.35760e9	−0.793527	−0.396763	−	0.917921i	$-0.629867\pi$
$59$	−4.35760e9	−0.793527	−0.396763	+	0.917921i	$0.629867\pi$
$60$	0	0
$61$	−1.65393e9	−0.250727	−0.125364	−	0.992111i	$-0.540010\pi$
$61$	−1.65393e9	−0.250727	−0.125364	+	0.992111i	$0.540010\pi$
$62$	1.48849e9i	0.206344i
$63$	− 4.72045e9i	− 0.599253i
$64$	3.60819e9	0.420049
$65$	0	0
$66$	−2.87951e9	−0.283027
$67$	7.58610e9i	0.686447i	0.939254	+	0.343224i	$0.111519\pi$
$67$	7.58610e9i	0.686447i	−0.939254	+	0.343224i	$0.888481\pi$
$68$	4.82228e9i	0.402211i
$69$	4.47645e9	0.344559
$70$	0	0
$71$	−2.75809e10	−1.81421	−0.907104	−	0.420905i	$-0.861712\pi$
$71$	−2.75809e10	−1.81421	−0.907104	+	0.420905i	$0.861712\pi$
$72$	− 3.37173e9i	− 0.205364i
$73$	− 3.22368e10i	− 1.82002i	−0.414584	−	0.910011i	$-0.636073\pi$
$73$	− 3.22368e10i	− 1.82002i	0.414584	−	0.910011i	$-0.363927\pi$
$74$	−1.18735e9	−0.0622022
$75$	0	0
$76$	−2.12548e10	−0.961572
$77$	6.43580e10i	2.70959i
$78$	− 4.28376e9i	− 0.167998i
$79$	2.43149e9	0.0889046	0.0444523	−	0.999012i	$-0.485846\pi$
$79$	2.43149e9	0.0889046	0.0444523	+	0.999012i	$0.485846\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	3.33116e9i	0.0992247i
$83$	− 1.20729e10i	− 0.336421i	−0.985751	−	0.168210i	$-0.946201\pi$
$83$	− 1.20729e10i	− 0.336421i	0.985751	−	0.168210i	$-0.0537988\pi$
$84$	−3.55753e10	−0.928137
$85$	0	0
$86$	2.46028e10	0.563953
$87$	4.62981e10i	0.995880i
$88$	4.59697e10i	0.928575i
$89$	−4.44073e9	−0.0842964	−0.0421482	−	0.999111i	$-0.513420\pi$
$89$	−4.44073e9	−0.0842964	−0.0421482	+	0.999111i	$0.513420\pi$
$90$	0	0
$91$	−9.57433e10	−1.60835
$92$	− 3.37364e10i	− 0.533661i
$93$	2.45737e10i	0.366282i
$94$	−1.26364e10	−0.177591
$95$	0	0
$96$	−3.88257e10	−0.485990
$97$	− 2.04453e10i	− 0.241740i	−0.992668	−	0.120870i	$-0.961432\pi$
$97$	− 2.04453e10i	− 0.241740i	0.992668	−	0.120870i	$-0.0385684\pi$
$98$	− 6.49594e10i	− 0.725937i
$99$	−4.75384e10	−0.502401
$100$	0	0
$101$	−1.55947e11	−1.47642	−0.738209	−	0.674572i	$-0.764328\pi$
$101$	−1.55947e11	−1.47642	−0.738209	+	0.674572i	$0.764328\pi$
$102$	− 9.41823e9i	− 0.0844633i
$103$	5.10325e10i	0.433752i	0.976199	+	0.216876i	$0.0695868\pi$
$103$	5.10325e10i	0.433752i	−0.976199	+	0.216876i	$0.930413\pi$
$104$	−6.83876e10	−0.551181
$105$	0	0
$106$	−5.19217e10	−0.376848
$107$	1.06665e10i	0.0735207i	0.999324	+	0.0367603i	$0.0117038\pi$
$107$	1.06665e10i	0.0735207i	−0.999324	+	0.0367603i	$0.988296\pi$
$108$	− 2.62778e10i	− 0.172091i
$109$	3.12513e10	0.194546	0.0972729	−	0.995258i	$-0.468988\pi$
$109$	3.12513e10	0.194546	0.0972729	+	0.995258i	$0.468988\pi$
$110$	0	0
$111$	−1.96022e10	−0.110415
$112$	2.32640e11i	1.24734i
$113$	3.96346e10i	0.202369i	0.994868	+	0.101184i	$0.0322632\pi$
$113$	3.96346e10i	0.202369i	−0.994868	+	0.101184i	$0.967737\pi$
$114$	4.15119e10	0.201928
$115$	0	0
$116$	3.48921e11	1.54244
$117$	− 7.07213e10i	− 0.298214i
$118$	− 6.41400e10i	− 0.258094i
$119$	−2.10500e11	−0.808620
$120$	0	0
$121$	3.62821e11	1.27166
$122$	− 2.43443e10i	− 0.0815489i
$123$	5.49948e10i	0.176134i
$124$	1.85198e11	0.567305
$125$	0	0
$126$	6.94808e10	0.194907
$127$	2.63460e10i	0.0707611i	0.999374	+	0.0353806i	$0.0112643\pi$
$127$	2.63460e10i	0.0707611i	−0.999374	+	0.0353806i	$0.988736\pi$
$128$	3.80331e11i	0.978379i
$129$	4.06172e11	1.00107
$130$	0	0
$131$	−2.19917e11	−0.498044	−0.249022	−	0.968498i	$-0.580109\pi$
$131$	−2.19917e11	−0.498044	−0.249022	+	0.968498i	$0.580109\pi$
$132$	3.58269e11i	0.778131i
$133$	− 9.27805e11i	− 1.93318i
$134$	−1.11661e11	−0.223266
$135$	0	0
$136$	−1.50356e11	−0.277113
$137$	− 5.54041e11i	− 0.980795i	−0.871499	−	0.490398i	$-0.836852\pi$
$137$	− 5.54041e11i	− 0.980795i	0.871499	−	0.490398i	$-0.163148\pi$
$138$	6.58894e10i	0.112068i
$139$	6.17540e11	1.00945	0.504723	−	0.863281i	$-0.331594\pi$
$139$	6.17540e11	1.00945	0.504723	+	0.863281i	$0.331594\pi$
$140$	0	0
$141$	−2.08617e11	−0.315243
$142$	− 4.05966e11i	− 0.590070i
$143$	9.64205e11i	1.34841i
$144$	−1.71840e11	−0.231277
$145$	0	0
$146$	4.74497e11	0.591961
$147$	− 1.07243e12i	− 1.28861i
$148$	1.47730e11i	0.171013i
$149$	6.68742e11	0.745992	0.372996	−	0.927833i	$-0.378330\pi$
$149$	6.68742e11	0.745992	0.372996	+	0.927833i	$0.378330\pi$
$150$	0	0
$151$	1.38243e12	1.43308	0.716541	−	0.697545i	$-0.245724\pi$
$151$	1.38243e12	1.43308	0.716541	+	0.697545i	$0.245724\pi$
$152$	− 6.62713e11i	− 0.662500i
$153$	− 1.55487e11i	− 0.149931i
$154$	−9.47292e11	−0.881292
$155$	0	0
$156$	−5.32985e11	−0.461880
$157$	− 7.93661e11i	− 0.664029i	−0.943274	−	0.332014i	$-0.892272\pi$
$157$	− 7.93661e11i	− 0.664029i	0.943274	−	0.332014i	$-0.107728\pi$
$158$	3.57894e10i	0.0289161i
$159$	−8.57184e11	−0.668944
$160$	0	0
$161$	1.47265e12	1.07289
$162$	5.13223e10i	0.0361388i
$163$	4.67399e11i	0.318168i	0.987265	+	0.159084i	$0.0508540\pi$
$163$	4.67399e11i	0.318168i	−0.987265	+	0.159084i	$0.949146\pi$
$164$	4.14463e11	0.272800
$165$	0	0
$166$	1.77702e11	0.109421
$167$	2.87482e12i	1.71266i	0.516432	+	0.856328i	$0.327260\pi$
$167$	2.87482e12i	1.71266i	−0.516432	+	0.856328i	$0.672740\pi$
$168$	− 1.10922e12i	− 0.639464i
$169$	3.57745e11	0.199617
$170$	0	0
$171$	6.85328e11	0.358443
$172$	− 3.06108e12i	− 1.55049i
$173$	− 1.53085e12i	− 0.751068i	−0.926809	−	0.375534i	$-0.877459\pi$
$173$	− 1.53085e12i	− 0.751068i	0.926809	−	0.375534i	$-0.122541\pi$
$174$	−6.81465e11	−0.323909
$175$	0	0
$176$	2.34285e12	1.04574
$177$	− 1.05890e12i	− 0.458143i
$178$	− 6.53635e10i	− 0.0274173i
$179$	1.91337e12	0.778228	0.389114	−	0.921190i	$-0.372781\pi$
$179$	1.91337e12	0.778228	0.389114	+	0.921190i	$0.372781\pi$
$180$	0	0
$181$	−1.70819e12	−0.653587	−0.326794	−	0.945096i	$-0.605968\pi$
$181$	−1.70819e12	−0.653587	−0.326794	+	0.945096i	$0.605968\pi$
$182$	− 1.40925e12i	− 0.523115i
$183$	− 4.01904e11i	− 0.144758i
$184$	1.05188e12	0.367680
$185$	0	0
$186$	−3.61703e11	−0.119133
$187$	2.11989e12i	0.677930i
$188$	1.57223e12i	0.488255i
$189$	1.14707e12	0.345979
$190$	0	0
$191$	3.16020e12	0.899562	0.449781	−	0.893139i	$-0.351502\pi$
$191$	3.16020e12	0.899562	0.449781	+	0.893139i	$0.351502\pi$
$192$	8.76790e11i	0.242515i
$193$	− 2.77296e12i	− 0.745382i	−0.927956	−	0.372691i	$-0.878435\pi$
$193$	− 2.77296e12i	− 0.745382i	0.927956	−	0.372691i	$-0.121565\pi$
$194$	3.00936e11	0.0786256
$195$	0	0
$196$	−8.08224e12	−1.99583
$197$	3.86504e12i	0.928089i	0.885812	+	0.464044i	$0.153602\pi$
$197$	3.86504e12i	0.928089i	−0.885812	+	0.464044i	$0.846398\pi$
$198$	− 6.99722e11i	− 0.163406i
$199$	1.01839e12	0.231325	0.115663	−	0.993289i	$-0.463101\pi$
$199$	1.01839e12	0.231325	0.115663	+	0.993289i	$0.463101\pi$
$200$	0	0
$201$	−1.84342e12	−0.396321
$202$	− 2.29540e12i	− 0.480204i
$203$	1.52310e13i	3.10098i
$204$	−1.17181e12	−0.232216
$205$	0	0
$206$	−7.51152e11	−0.141078
$207$	1.08778e12i	0.198931i
$208$	3.48538e12i	0.620729i
$209$	−9.34367e12	−1.62074
$210$	0	0
$211$	2.05668e12	0.338543	0.169271	−	0.985569i	$-0.445859\pi$
$211$	2.05668e12	0.338543	0.169271	+	0.985569i	$0.445859\pi$
$212$	6.46009e12i	1.03608i
$213$	− 6.70216e12i	− 1.04743i
$214$	−1.57001e11	−0.0239125
$215$	0	0
$216$	8.19330e11	0.118567
$217$	8.08418e12i	1.14053i
$218$	4.59990e11i	0.0632759i
$219$	7.83355e12	1.05079
$220$	0	0
$221$	−3.15369e12	−0.402403
$222$	− 2.88527e11i	− 0.0359124i
$223$	− 9.99986e12i	− 1.21428i	−0.794597	−	0.607138i	$-0.792318\pi$
$223$	− 9.99986e12i	− 1.21428i	0.794597	−	0.607138i	$-0.207682\pi$
$224$	−1.27727e13	−1.51328
$225$	0	0
$226$	−5.83386e11	−0.0658202
$227$	− 6.77123e12i	− 0.745634i	−0.927905	−	0.372817i	$-0.878392\pi$
$227$	− 6.77123e12i	− 0.745634i	0.927905	−	0.372817i	$-0.121608\pi$
$228$	− 5.16491e12i	− 0.555164i
$229$	−1.01933e13	−1.06959	−0.534796	−	0.844981i	$-0.679611\pi$
$229$	−1.01933e13	−1.06959	−0.534796	+	0.844981i	$0.679611\pi$
$230$	0	0
$231$	−1.56390e13	−1.56438
$232$	1.08792e13i	1.06270i
$233$	− 6.74446e12i	− 0.643412i	−0.946840	−	0.321706i	$-0.895744\pi$
$233$	− 6.74446e12i	− 0.643412i	0.946840	−	0.321706i	$-0.104256\pi$
$234$	1.04095e12	0.0969938
$235$	0	0
$236$	−7.98029e12	−0.709582
$237$	5.90853e11i	0.0513291i
$238$	− 3.09837e12i	− 0.263003i
$239$	1.76501e13	1.46406	0.732030	−	0.681272i	$-0.238573\pi$
$239$	1.76501e13	1.46406	0.732030	+	0.681272i	$0.238573\pi$
$240$	0	0
$241$	−1.37662e13	−1.09074	−0.545369	−	0.838196i	$-0.683611\pi$
$241$	−1.37662e13	−1.09074	−0.545369	+	0.838196i	$0.683611\pi$
$242$	5.34039e12i	0.413608i
$243$	8.47289e11i	0.0641500i
$244$	−3.02891e12	−0.224204
$245$	0	0
$246$	−8.09473e11	−0.0572874
$247$	− 1.39003e13i	− 0.962033i
$248$	5.77438e12i	0.390860i
$249$	2.93372e12	0.194233
$250$	0	0
$251$	3.02252e13	1.91498	0.957490	−	0.288467i	$-0.0931454\pi$
$251$	3.02252e13	1.91498	0.957490	+	0.288467i	$0.0931454\pi$
$252$	− 8.64479e12i	− 0.535860i
$253$	− 1.48306e13i	− 0.899491i
$254$	−3.87790e11	−0.0230150
$255$	0	0
$256$	1.79144e12	0.101832
$257$	6.88263e12i	0.382933i	0.981499	+	0.191466i	$0.0613243\pi$
$257$	6.88263e12i	0.382933i	−0.981499	+	0.191466i	$0.938676\pi$
$258$	5.97848e12i	0.325599i
$259$	−6.44866e12	−0.343812
$260$	0	0
$261$	−1.12504e13	−0.574971
$262$	− 3.23698e12i	− 0.161988i
$263$	3.29948e13i	1.61692i	0.588549	+	0.808462i	$0.299699\pi$
$263$	3.29948e13i	1.61692i	−0.588549	+	0.808462i	$0.700301\pi$
$264$	−1.11706e13	−0.536113
$265$	0	0
$266$	1.36564e13	0.628765
$267$	− 1.07910e12i	− 0.0486686i
$268$	1.38928e13i	0.613830i
$269$	2.97478e13	1.28771	0.643855	−	0.765148i	$-0.277334\pi$
$269$	2.97478e13	1.28771	0.643855	+	0.765148i	$0.277334\pi$
$270$	0	0
$271$	−4.30410e13	−1.78876	−0.894378	−	0.447312i	$-0.852381\pi$
$271$	−4.30410e13	−1.78876	−0.894378	+	0.447312i	$0.852381\pi$
$272$	7.66293e12i	0.312080i
$273$	− 2.32656e13i	− 0.928582i
$274$	8.15498e12	0.319003
$275$	0	0
$276$	8.19795e12	0.308110
$277$	1.46706e13i	0.540518i	0.962788	+	0.270259i	$0.0871094\pi$
$277$	1.46706e13i	0.540518i	−0.962788	+	0.270259i	$0.912891\pi$
$278$	9.08963e12i	0.328322i
$279$	−5.97142e12	−0.211473
$280$	0	0
$281$	4.18378e13	1.42457	0.712286	−	0.701890i	$-0.247660\pi$
$281$	4.18378e13	1.42457	0.712286	+	0.701890i	$0.247660\pi$
$282$	− 3.07066e12i	− 0.102532i
$283$	− 2.34242e13i	− 0.767078i	−0.923525	−	0.383539i	$-0.874705\pi$
$283$	− 2.34242e13i	− 0.767078i	0.923525	−	0.383539i	$-0.125295\pi$
$284$	−5.05102e13	−1.62229
$285$	0	0
$286$	−1.41922e13	−0.438568
$287$	1.80920e13i	0.548448i
$288$	− 9.43463e12i	− 0.280586i
$289$	2.73382e13	0.797686
$290$	0	0
$291$	4.96820e12	0.139568
$292$	− 5.90369e13i	− 1.62749i
$293$	7.31258e12i	0.197833i	0.995096	+	0.0989166i	$0.0315377\pi$
$293$	7.31258e12i	0.197833i	−0.995096	+	0.0989166i	$0.968462\pi$
$294$	1.57851e13	0.419120
$295$	0	0
$296$	−4.60616e12	−0.117824
$297$	− 1.15518e13i	− 0.290062i
$298$	9.84328e12i	0.242633i
$299$	2.20630e13	0.533917
$300$	0	0
$301$	1.33621e14	3.11716
$302$	2.03482e13i	0.466109i
$303$	− 3.78951e13i	− 0.852410i
$304$	−3.37752e13	−0.746094
$305$	0	0
$306$	2.28863e12	0.0487649
$307$	− 5.37239e13i	− 1.12436i	−0.827014	−	0.562181i	$-0.809962\pi$
$307$	− 5.37239e13i	− 1.12436i	0.827014	−	0.562181i	$-0.190038\pi$
$308$	1.17862e14i	2.42295i
$309$	−1.24009e13	−0.250427
$310$	0	0
$311$	5.41406e13	1.05521	0.527607	−	0.849489i	$-0.323089\pi$
$311$	5.41406e13	1.05521	0.527607	+	0.849489i	$0.323089\pi$
$312$	− 1.66182e13i	− 0.318224i
$313$	4.29721e12i	0.0808524i	0.999183	+	0.0404262i	$0.0128716\pi$
$313$	4.29721e12i	0.0808524i	−0.999183	+	0.0404262i	$0.987128\pi$
$314$	1.16820e13	0.215975
$315$	0	0
$316$	4.45291e12	0.0794996
$317$	− 2.81928e13i	− 0.494666i	−0.968931	−	0.247333i	$-0.920446\pi$
$317$	− 2.81928e13i	− 0.494666i	0.968931	−	0.247333i	$-0.0795541\pi$
$318$	− 1.26170e13i	− 0.217573i
$319$	1.53387e14	2.59980
$320$	0	0
$321$	−2.59195e12	−0.0424472
$322$	2.16760e13i	0.348958i
$323$	− 3.05610e13i	− 0.483675i
$324$	6.38552e12	0.0993570
$325$	0	0
$326$	−6.87969e12	−0.103484
$327$	7.59406e12i	0.112321i
$328$	1.29227e13i	0.187953i
$329$	−6.86301e13	−0.981606
$330$	0	0
$331$	−9.95469e13	−1.37713	−0.688563	−	0.725176i	$-0.741758\pi$
$331$	−9.95469e13	−1.37713	−0.688563	+	0.725176i	$0.741758\pi$
$332$	− 2.21097e13i	− 0.300832i
$333$	− 4.76334e12i	− 0.0637482i
$334$	−4.23148e13	−0.557040
$335$	0	0
$336$	−5.65314e13	−0.720152
$337$	− 9.05175e13i	− 1.13440i	−0.823579	−	0.567202i	$-0.808026\pi$
$337$	− 9.05175e13i	− 1.13440i	0.823579	−	0.567202i	$-0.191974\pi$
$338$	5.26568e12i	0.0649251i
$339$	−9.63121e12	−0.116838
$340$	0	0
$341$	8.14136e13	0.956198
$342$	1.00874e13i	0.116583i
$343$	− 1.94733e14i	− 2.21473i
$344$	9.54428e13	1.06825
$345$	0	0
$346$	2.25327e13	0.244284
$347$	− 6.36892e13i	− 0.679601i	−0.940498	−	0.339801i	$-0.889640\pi$
$347$	− 6.36892e13i	− 0.679601i	0.940498	−	0.339801i	$-0.110360\pi$
$348$	8.47879e13i	0.890529i
$349$	−7.08598e13	−0.732588	−0.366294	−	0.930499i	$-0.619374\pi$
$349$	−7.08598e13	−0.732588	−0.366294	+	0.930499i	$0.619374\pi$
$350$	0	0
$351$	1.71853e13	0.172174
$352$	1.28631e14i	1.26870i
$353$	− 6.62893e13i	− 0.643699i	−0.946791	−	0.321849i	$-0.895696\pi$
$353$	− 6.62893e13i	− 0.643699i	0.946791	−	0.321849i	$-0.104304\pi$
$354$	1.55860e13	0.149011
$355$	0	0
$356$	−8.13252e12	−0.0753790
$357$	− 5.11516e13i	− 0.466857i
$358$	2.81630e13i	0.253118i
$359$	−6.80217e13	−0.602044	−0.301022	−	0.953617i	$-0.597328\pi$
$359$	−6.80217e13	−0.602044	−0.301022	+	0.953617i	$0.597328\pi$
$360$	0	0
$361$	1.82109e13	0.156330
$362$	− 2.51430e13i	− 0.212579i
$363$	8.81654e13i	0.734196i
$364$	−1.75339e14	−1.43821
$365$	0	0
$366$	5.91566e12	0.0470823
$367$	2.19771e14i	1.72309i	0.507685	+	0.861543i	$0.330501\pi$
$367$	2.19771e14i	1.72309i	−0.507685	+	0.861543i	$0.669499\pi$
$368$	− 5.36094e13i	− 0.414074i
$369$	−1.33637e13	−0.101691
$370$	0	0
$371$	−2.81993e14	−2.08296
$372$	4.50031e13i	0.327534i
$373$	1.67232e14i	1.19928i	0.800271	+	0.599639i	$0.204689\pi$
$373$	1.67232e14i	1.19928i	−0.800271	+	0.599639i	$0.795311\pi$
$374$	−3.12029e13	−0.220496
$375$	0	0
$376$	−4.90212e13	−0.336396
$377$	2.28189e14i	1.54318i
$378$	1.68838e13i	0.112529i
$379$	1.57778e14	1.03641	0.518205	−	0.855256i	$-0.326600\pi$
$379$	1.57778e14	1.03641	0.518205	+	0.855256i	$0.326600\pi$
$380$	0	0
$381$	−6.40209e12	−0.0408540
$382$	4.65153e13i	0.292582i
$383$	− 5.49255e13i	− 0.340550i	−0.985397	−	0.170275i	$-0.945534\pi$
$383$	− 5.49255e13i	− 0.340550i	0.985397	−	0.170275i	$-0.0544655\pi$
$384$	−9.24205e13	−0.564867
$385$	0	0
$386$	4.08155e13	0.242435
$387$	9.86997e13i	0.577971i
$388$	− 3.74424e13i	− 0.216167i
$389$	2.32940e14	1.32593	0.662965	−	0.748650i	$-0.269298\pi$
$389$	2.32940e14	1.32593	0.662965	+	0.748650i	$0.269298\pi$
$390$	0	0
$391$	4.85076e13	0.268434
$392$	− 2.52000e14i	− 1.37508i
$393$	− 5.34399e13i	− 0.287546i
$394$	−5.68899e13	−0.301860
$395$	0	0
$396$	−8.70593e13	−0.449254
$397$	− 4.76134e13i	− 0.242316i	−0.992633	−	0.121158i	$-0.961339\pi$
$397$	− 4.76134e13i	− 0.242316i	0.992633	−	0.121158i	$-0.0386607\pi$
$398$	1.49898e13i	0.0752383i
$399$	2.25456e14	1.11612
$400$	0	0
$401$	−2.99074e14	−1.44040	−0.720202	−	0.693764i	$-0.755951\pi$
$401$	−2.99074e14	−1.44040	−0.720202	+	0.693764i	$0.755951\pi$
$402$	− 2.71335e13i	− 0.128903i
$403$	1.21116e14i	0.567577i
$404$	−2.85593e14	−1.32023
$405$	0	0
$406$	−2.24186e14	−1.00859
$407$	6.49428e13i	0.288245i
$408$	− 3.65366e13i	− 0.159992i
$409$	2.55879e14	1.10550	0.552748	−	0.833349i	$-0.313579\pi$
$409$	2.55879e14	1.10550	0.552748	+	0.833349i	$0.313579\pi$
$410$	0	0
$411$	1.34632e14	0.566262
$412$	9.34582e13i	0.387867i
$413$	− 3.48352e14i	− 1.42657i
$414$	−1.60111e13	−0.0647023
$415$	0	0
$416$	−1.91359e14	−0.753072
$417$	1.50062e14i	0.582804i
$418$	− 1.37530e14i	− 0.527144i
$419$	−2.21825e14	−0.839139	−0.419570	−	0.907723i	$-0.637819\pi$
$419$	−2.21825e14	−0.839139	−0.419570	+	0.907723i	$0.637819\pi$
$420$	0	0
$421$	9.78949e13	0.360752	0.180376	−	0.983598i	$-0.442269\pi$
$421$	9.78949e13	0.360752	0.180376	+	0.983598i	$0.442269\pi$
$422$	3.02725e13i	0.110111i
$423$	− 5.06940e13i	− 0.182005i
$424$	−2.01422e14	−0.713831
$425$	0	0
$426$	9.86497e13	0.340677
$427$	− 1.32217e14i	− 0.450748i
$428$	1.95340e13i	0.0657432i
$429$	−2.34302e14	−0.778503
$430$	0	0
$431$	1.11415e14	0.360842	0.180421	−	0.983589i	$-0.442254\pi$
$431$	1.11415e14	0.360842	0.180421	+	0.983589i	$0.442254\pi$
$432$	− 4.17572e13i	− 0.133528i
$433$	3.42045e14i	1.07994i	0.841684	+	0.539970i	$0.181564\pi$
$433$	3.42045e14i	1.07994i	−0.841684	+	0.539970i	$0.818436\pi$
$434$	−1.18992e14	−0.370957
$435$	0	0
$436$	5.72320e13	0.173965
$437$	2.13803e14i	0.641750i
$438$	1.15303e14i	0.341769i
$439$	−1.20947e14	−0.354029	−0.177015	−	0.984208i	$-0.556644\pi$
$439$	−1.20947e14	−0.354029	−0.177015	+	0.984208i	$0.556644\pi$
$440$	0	0
$441$	2.60599e14	0.743980
$442$	− 4.64195e13i	− 0.130881i
$443$	1.27316e14i	0.354539i	0.984162	+	0.177270i	$0.0567264\pi$
$443$	1.27316e14i	0.354539i	−0.984162	+	0.177270i	$0.943274\pi$
$444$	−3.58985e13	−0.0987347
$445$	0	0
$446$	1.47189e14	0.394942
$447$	1.62504e14i	0.430699i
$448$	2.88443e14i	0.755147i
$449$	1.61837e14	0.418526	0.209263	−	0.977859i	$-0.432894\pi$
$449$	1.61837e14	0.418526	0.209263	+	0.977859i	$0.432894\pi$
$450$	0	0
$451$	1.82199e14	0.459807
$452$	7.25848e13i	0.180961i
$453$	3.35931e14i	0.827390i
$454$	9.96664e13	0.242517
$455$	0	0
$456$	1.61039e14	0.382494
$457$	− 3.49165e14i	− 0.819391i	−0.912222	−	0.409696i	$-0.865635\pi$
$457$	− 3.49165e14i	− 0.819391i	0.912222	−	0.409696i	$-0.134365\pi$
$458$	− 1.50036e14i	− 0.347884i
$459$	3.77834e13	0.0865627
$460$	0	0
$461$	−4.49619e14	−1.00575	−0.502875	−	0.864359i	$-0.667724\pi$
$461$	−4.49619e14	−1.00575	−0.502875	+	0.864359i	$0.667724\pi$
$462$	− 2.30192e14i	− 0.508814i
$463$	− 4.21098e13i	− 0.0919787i	−0.998942	−	0.0459894i	$-0.985356\pi$
$463$	− 4.21098e13i	− 0.0919787i	0.998942	−	0.0459894i	$-0.0146440\pi$
$464$	5.54459e14	1.19680
$465$	0	0
$466$	9.92723e13	0.209269
$467$	− 7.91358e13i	− 0.164866i	−0.996597	−	0.0824328i	$-0.973731\pi$
$467$	− 7.91358e13i	− 0.164866i	0.996597	−	0.0824328i	$-0.0262690\pi$
$468$	− 1.29515e14i	− 0.266667i
$469$	−6.06442e14	−1.23407
$470$	0	0
$471$	1.92860e14	0.383377
$472$	− 2.48821e14i	− 0.488885i
$473$	− 1.34566e15i	− 2.61336i
$474$	−8.69682e12	−0.0166947
$475$	0	0
$476$	−3.85499e14	−0.723078
$477$	− 2.08296e14i	− 0.386215i
$478$	2.59794e14i	0.476184i
$479$	9.62318e14	1.74371	0.871853	−	0.489768i	$-0.162919\pi$
$479$	9.62318e14	1.74371	0.871853	+	0.489768i	$0.162919\pi$
$480$	0	0
$481$	−9.66132e13	−0.171095
$482$	− 2.02626e14i	− 0.354762i
$483$	3.57853e14i	0.619435i
$484$	6.64451e14	1.13714
$485$	0	0
$486$	−1.24713e13	−0.0208647
$487$	4.57117e14i	0.756168i	0.925771	+	0.378084i	$0.123417\pi$
$487$	4.57117e14i	0.756168i	−0.925771	+	0.378084i	$0.876583\pi$
$488$	− 9.44399e13i	− 0.154471i
$489$	−1.13578e14	−0.183694
$490$	0	0
$491$	4.40122e14	0.696025	0.348013	−	0.937490i	$-0.386857\pi$
$491$	4.40122e14	0.696025	0.348013	+	0.937490i	$0.386857\pi$
$492$	1.00715e14i	0.157501i
$493$	5.01693e14i	0.775854i
$494$	2.04599e14	0.312900
$495$	0	0
$496$	2.94292e14	0.440179
$497$	− 2.20485e15i	− 3.26151i
$498$	4.31817e13i	0.0631740i
$499$	−6.36085e14	−0.920369	−0.460185	−	0.887823i	$-0.652217\pi$
$499$	−6.36085e14	−0.920369	−0.460185	+	0.887823i	$0.652217\pi$
$500$	0	0
$501$	−6.98581e14	−0.988802
$502$	4.44888e14i	0.622846i
$503$	4.81639e12i	0.00666957i	0.999994	+	0.00333478i	$0.00106150\pi$
$503$	4.81639e12i	0.00666957i	−0.999994	+	0.00333478i	$0.998939\pi$
$504$	2.69540e14	0.369195
$505$	0	0
$506$	2.18294e14	0.292559
$507$	8.69320e13i	0.115249i
$508$	4.82488e13i	0.0632755i
$509$	5.73829e14	0.744449	0.372224	−	0.928143i	$-0.378595\pi$
$509$	5.73829e14	0.744449	0.372224	+	0.928143i	$0.378595\pi$
$510$	0	0
$511$	2.57705e15	3.27196
$512$	8.05287e14i	1.01150i
$513$	1.66535e14i	0.206947i
$514$	−1.01306e14	−0.124548
$515$	0	0
$516$	7.43842e14	0.895174
$517$	6.91155e14i	0.822958i
$518$	− 9.49185e13i	− 0.111825i
$519$	3.71997e14	0.433630
$520$	0	0
$521$	5.14705e14	0.587422	0.293711	−	0.955894i	$-0.405110\pi$
$521$	5.14705e14	0.587422	0.293711	+	0.955894i	$0.405110\pi$
$522$	− 1.65596e14i	− 0.187009i
$523$	− 1.20146e15i	− 1.34261i	−0.741182	−	0.671304i	$-0.765735\pi$
$523$	− 1.20146e15i	− 1.34261i	0.741182	−	0.671304i	$-0.234265\pi$
$524$	−4.02745e14	−0.445357
$525$	0	0
$526$	−4.85654e14	−0.525903
$527$	2.66285e14i	0.285357i
$528$	5.69313e14i	0.603760i
$529$	6.13454e14	0.643836
$530$	0	0
$531$	2.57312e14	0.264509
$532$	− 1.69913e15i	− 1.72868i
$533$	2.71052e14i	0.272931i
$534$	1.58833e13	0.0158294
$535$	0	0
$536$	−4.33170e14	−0.422914
$537$	4.64948e14i	0.449310i
$538$	4.37861e14i	0.418826i
$539$	−3.55298e15	−3.36399
$540$	0	0
$541$	−1.89107e15	−1.75438	−0.877190	−	0.480143i	$-0.840585\pi$
$541$	−1.89107e15	−1.75438	−0.877190	+	0.480143i	$0.840585\pi$
$542$	− 6.33524e14i	− 0.581791i
$543$	− 4.15090e14i	− 0.377349i
$544$	−4.20721e14	−0.378617
$545$	0	0
$546$	3.42449e14	0.302020
$547$	1.37400e14i	0.119966i	0.998199	+	0.0599830i	$0.0191046\pi$
$547$	1.37400e14i	0.119966i	−0.998199	+	0.0599830i	$0.980895\pi$
$548$	− 1.01464e15i	− 0.877040i
$549$	9.76626e13	0.0835758
$550$	0	0
$551$	−2.21127e15	−1.85485
$552$	2.55608e14i	0.212280i
$553$	1.94377e14i	0.159829i
$554$	−2.15939e14	−0.175803
$555$	0	0
$556$	1.13093e15	0.902661
$557$	− 1.18622e15i	− 0.937483i	−0.883335	−	0.468742i	$-0.844708\pi$
$557$	− 1.18622e15i	− 0.937483i	0.883335	−	0.468742i	$-0.155292\pi$
$558$	− 8.78939e13i	− 0.0687814i
$559$	2.00189e15	1.55123
$560$	0	0
$561$	−5.15134e14	−0.391403
$562$	6.15815e14i	0.463340i
$563$	2.58208e15i	1.92386i	0.273298	+	0.961929i	$0.411885\pi$
$563$	2.58208e15i	1.92386i	−0.273298	+	0.961929i	$0.588115\pi$
$564$	−3.82051e14	−0.281894
$565$	0	0
$566$	3.44783e14	0.249491
$567$	2.78738e14i	0.199751i
$568$	− 1.57488e15i	− 1.11772i
$569$	−6.86455e13	−0.0482497	−0.0241249	−	0.999709i	$-0.507680\pi$
$569$	−6.86455e13	−0.0482497	−0.0241249	+	0.999709i	$0.507680\pi$
$570$	0	0
$571$	−1.48521e15	−1.02397	−0.511986	−	0.858994i	$-0.671090\pi$
$571$	−1.48521e15	−1.02397	−0.511986	+	0.858994i	$0.671090\pi$
$572$	1.76579e15i	1.20576i
$573$	7.67929e14i	0.519363i
$574$	−2.66297e14	−0.178382
$575$	0	0
$576$	−2.13060e14	−0.140016
$577$	− 1.11682e14i	− 0.0726966i	−0.999339	−	0.0363483i	$-0.988427\pi$
$577$	− 1.11682e14i	− 0.0726966i	0.999339	−	0.0363483i	$-0.0115726\pi$
$578$	4.02394e14i	0.259447i
$579$	6.73830e14	0.430346
$580$	0	0
$581$	9.65124e14	0.604803
$582$	7.31274e13i	0.0453945i
$583$	2.83988e15i	1.74631i
$584$	1.84074e15	1.12130
$585$	0	0
$586$	−1.07635e14	−0.0643450
$587$	1.03228e15i	0.611347i	0.952136	+	0.305673i	$0.0988815\pi$
$587$	1.03228e15i	0.611347i	−0.952136	+	0.305673i	$0.901118\pi$
$588$	− 1.96398e15i	− 1.15229i
$589$	−1.17368e15	−0.682208
$590$	0	0
$591$	−9.39204e14	−0.535832
$592$	2.34753e14i	0.132691i
$593$	− 5.05946e14i	− 0.283337i	−0.989914	−	0.141668i	$-0.954753\pi$
$593$	− 5.05946e14i	− 0.283337i	0.989914	−	0.141668i	$-0.0452467\pi$
$594$	1.70032e14	0.0943423
$595$	0	0
$596$	1.22470e15	0.667076
$597$	2.47469e14i	0.133556i
$598$	3.24748e14i	0.173656i
$599$	1.86439e14	0.0987846	0.0493923	−	0.998779i	$-0.484272\pi$
$599$	1.86439e14	0.0987846	0.0493923	+	0.998779i	$0.484272\pi$
$600$	0	0
$601$	−6.39018e14	−0.332433	−0.166216	−	0.986089i	$-0.553155\pi$
$601$	−6.39018e14	−0.332433	−0.166216	+	0.986089i	$0.553155\pi$
$602$	1.96678e15i	1.01385i
$603$	− 4.47952e14i	− 0.228816i
$604$	2.53172e15	1.28148
$605$	0	0
$606$	5.57782e14	0.277246
$607$	3.07440e15i	1.51434i	0.653219	+	0.757169i	$0.273418\pi$
$607$	3.07440e15i	1.51434i	−0.653219	+	0.757169i	$0.726582\pi$
$608$	− 1.85438e15i	− 0.905166i
$609$	−3.70112e15	−1.79035
$610$	0	0
$611$	−1.02821e15	−0.488489
$612$	− 2.84751e14i	− 0.134070i
$613$	− 1.45823e15i	− 0.680447i	−0.940345	−	0.340223i	$-0.889497\pi$
$613$	− 1.45823e15i	− 0.680447i	0.940345	−	0.340223i	$-0.110503\pi$
$614$	7.90767e14	0.365698
$615$	0	0
$616$	−3.67488e15	−1.66935
$617$	2.52038e15i	1.13474i	0.823462	+	0.567372i	$0.192040\pi$
$617$	2.52038e15i	1.13474i	−0.823462	+	0.567372i	$0.807960\pi$
$618$	− 1.82530e14i	− 0.0814512i
$619$	−2.76117e15	−1.22122	−0.610612	−	0.791930i	$-0.709076\pi$
$619$	−2.76117e15	−1.22122	−0.610612	+	0.791930i	$0.709076\pi$
$620$	0	0
$621$	−2.64330e14	−0.114853
$622$	7.96900e14i	0.343207i
$623$	− 3.54997e14i	− 0.151545i
$624$	−8.46948e14	−0.358378
$625$	0	0
$626$	−6.32511e13	−0.0262972
$627$	− 2.27051e15i	− 0.935734i
$628$	− 1.45347e15i	− 0.593783i
$629$	−2.12413e14	−0.0860205
$630$	0	0
$631$	−5.89126e14	−0.234448	−0.117224	−	0.993105i	$-0.537400\pi$
$631$	−5.89126e14	−0.234448	−0.117224	+	0.993105i	$0.537400\pi$
$632$	1.38839e14i	0.0547733i
$633$	4.99774e14i	0.195458i
$634$	4.14972e14	0.160890
$635$	0	0
$636$	−1.56980e15	−0.598178
$637$	− 5.28565e15i	− 1.99679i
$638$	2.25772e15i	0.845582i
$639$	1.62862e15	0.604736
$640$	0	0
$641$	1.59787e15	0.583208	0.291604	−	0.956539i	$-0.405811\pi$
$641$	1.59787e15	0.583208	0.291604	+	0.956539i	$0.405811\pi$
$642$	− 3.81512e13i	− 0.0138059i
$643$	− 4.44445e15i	− 1.59462i	−0.603569	−	0.797310i	$-0.706255\pi$
$643$	− 4.44445e15i	− 1.59462i	0.603569	−	0.797310i	$-0.293745\pi$
$644$	2.69693e15	0.959395
$645$	0	0
$646$	4.49830e14	0.157315
$647$	− 8.54247e14i	− 0.296217i	−0.988971	−	0.148108i	$-0.952682\pi$
$647$	− 8.54247e14i	− 0.296217i	0.988971	−	0.148108i	$-0.0473185\pi$
$648$	1.99097e14i	0.0684546i
$649$	−3.50816e15	−1.19601
$650$	0	0
$651$	−1.96446e15	−0.658486
$652$	8.55971e14i	0.284510i
$653$	2.86428e15i	0.944045i	0.881587	+	0.472022i	$0.156476\pi$
$653$	2.86428e15i	0.944045i	−0.881587	+	0.472022i	$0.843524\pi$
$654$	−1.11778e14	−0.0365323
$655$	0	0
$656$	6.58609e14	0.211669
$657$	1.90355e15i	0.606674i
$658$	− 1.01017e15i	− 0.319267i
$659$	−4.78135e15	−1.49858	−0.749291	−	0.662241i	$-0.769606\pi$
$659$	−4.78135e15	−1.49858	−0.749291	+	0.662241i	$0.769606\pi$
$660$	0	0
$661$	−5.52275e14	−0.170234	−0.0851172	−	0.996371i	$-0.527126\pi$
$661$	−5.52275e14	−0.170234	−0.0851172	+	0.996371i	$0.527126\pi$
$662$	− 1.46524e15i	− 0.447909i
$663$	− 7.66347e14i	− 0.232328i
$664$	6.89369e14	0.207266
$665$	0	0
$666$	7.01120e13	0.0207341
$667$	− 3.50982e15i	− 1.02942i
$668$	5.26480e15i	1.53148i
$669$	2.42997e15	0.701062
$670$	0	0
$671$	−1.33152e15	−0.377897
$672$	− 3.10377e15i	− 0.873693i
$673$	3.31175e15i	0.924645i	0.886712	+	0.462323i	$0.152984\pi$
$673$	3.31175e15i	0.924645i	−0.886712	+	0.462323i	$0.847016\pi$
$674$	1.33234e15	0.368964
$675$	0	0
$676$	6.55156e14	0.178500
$677$	6.69181e15i	1.80845i	0.427060	+	0.904223i	$0.359549\pi$
$677$	6.69181e15i	1.80845i	−0.427060	+	0.904223i	$0.640451\pi$
$678$	− 1.41763e14i	− 0.0380013i
$679$	1.63442e15	0.434590
$680$	0	0
$681$	1.64541e15	0.430492
$682$	1.19833e15i	0.311003i
$683$	7.20810e15i	1.85570i	0.372959	+	0.927848i	$0.378343\pi$
$683$	7.20810e15i	1.85570i	−0.372959	+	0.927848i	$0.621657\pi$
$684$	1.25507e15	0.320524
$685$	0	0
$686$	2.86629e15	0.720340
$687$	− 2.47696e15i	− 0.617529i
$688$	− 4.86426e15i	− 1.20304i
$689$	−4.22479e15	−1.03657
$690$	0	0
$691$	7.79467e15	1.88221	0.941106	−	0.338112i	$-0.109788\pi$
$691$	7.79467e15	1.88221	0.941106	+	0.338112i	$0.109788\pi$
$692$	− 2.80352e15i	− 0.671615i
$693$	− 3.80028e15i	− 0.903197i
$694$	9.37448e14	0.221040
$695$	0	0
$696$	−2.64364e15	−0.613553
$697$	5.95932e14i	0.137220i
$698$	− 1.04299e15i	− 0.238274i
$699$	1.63890e15	0.371474
$700$	0	0
$701$	6.71290e15	1.49782	0.748912	−	0.662669i	$-0.230577\pi$
$701$	6.71290e15	1.49782	0.748912	+	0.662669i	$0.230577\pi$
$702$	2.52952e14i	0.0559994i
$703$	− 9.36234e14i	− 0.205651i
$704$	2.90483e15	0.633099
$705$	0	0
$706$	9.75719e14	0.209362
$707$	− 1.24666e16i	− 2.65424i
$708$	− 1.93921e15i	− 0.409677i
$709$	4.10214e15	0.859917	0.429959	−	0.902849i	$-0.358528\pi$
$709$	4.10214e15	0.859917	0.429959	+	0.902849i	$0.358528\pi$
$710$	0	0
$711$	−1.43577e14	−0.0296349
$712$	− 2.53568e14i	− 0.0519343i
$713$	− 1.86291e15i	− 0.378617i
$714$	7.52905e14	0.151845
$715$	0	0
$716$	3.50404e15	0.695902
$717$	4.28898e15i	0.845276i
$718$	− 1.00122e15i	− 0.195814i
$719$	3.31818e15	0.644008	0.322004	−	0.946738i	$-0.395644\pi$
$719$	3.31818e15	0.644008	0.322004	+	0.946738i	$0.395644\pi$
$720$	0	0
$721$	−4.07960e15	−0.779783
$722$	2.68047e14i	0.0508460i
$723$	− 3.34519e15i	− 0.629738i
$724$	−3.12829e15	−0.584446
$725$	0	0
$726$	−1.29772e15	−0.238797
$727$	3.34273e15i	0.610467i	0.952278	+	0.305233i	$0.0987344\pi$
$727$	3.34273e15i	0.610467i	−0.952278	+	0.305233i	$0.901266\pi$
$728$	− 5.46699e15i	− 0.990890i
$729$	−2.05891e14	−0.0370370
$730$	0	0
$731$	4.40134e15	0.779901
$732$	− 7.36026e14i	− 0.129444i
$733$	6.14966e15i	1.07344i	0.843759	+	0.536722i	$0.180338\pi$
$733$	6.14966e15i	1.07344i	−0.843759	+	0.536722i	$0.819662\pi$
$734$	−3.23483e15	−0.560432
$735$	0	0
$736$	2.94334e15	0.502357
$737$	6.10732e15i	1.03462i
$738$	− 1.96702e14i	− 0.0330749i
$739$	4.15701e15	0.693804	0.346902	−	0.937901i	$-0.387234\pi$
$739$	4.15701e15	0.693804	0.346902	+	0.937901i	$0.387234\pi$
$740$	0	0
$741$	3.37776e15	0.555430
$742$	− 4.15068e15i	− 0.677482i
$743$	4.90364e15i	0.794474i	0.917716	+	0.397237i	$0.130031\pi$
$743$	4.90364e15i	0.794474i	−0.917716	+	0.397237i	$0.869969\pi$
$744$	−1.40317e15	−0.225663
$745$	0	0
$746$	−2.46150e15	−0.390064
$747$	7.12893e14i	0.112140i
$748$	3.88226e15i	0.606214i
$749$	−8.52690e14	−0.132173
$750$	0	0
$751$	−2.00580e15	−0.306385	−0.153192	−	0.988196i	$-0.548955\pi$
$751$	−2.00580e15	−0.306385	−0.153192	+	0.988196i	$0.548955\pi$
$752$	2.49837e15i	0.378842i
$753$	7.34473e15i	1.10561i
$754$	−3.35873e15	−0.501918
$755$	0	0
$756$	2.10068e15	0.309379
$757$	2.85016e15i	0.416718i	0.978052	+	0.208359i	$0.0668122\pi$
$757$	2.85016e15i	0.416718i	−0.978052	+	0.208359i	$0.933188\pi$
$758$	2.32235e15i	0.337091i
$759$	3.60384e15	0.519321
$760$	0	0
$761$	8.65962e15	1.22994	0.614969	−	0.788551i	$-0.289168\pi$
$761$	8.65962e15	1.22994	0.614969	+	0.788551i	$0.289168\pi$
$762$	− 9.42329e13i	− 0.0132877i
$763$	2.49827e15i	0.349747i
$764$	5.78743e15	0.804400
$765$	0	0
$766$	8.08453e14	0.110763
$767$	− 5.21897e15i	− 0.709922i
$768$	4.35320e14i	0.0587925i
$769$	7.49727e15	1.00533	0.502665	−	0.864481i	$-0.332353\pi$
$769$	7.49727e15	1.00533	0.502665	+	0.864481i	$0.332353\pi$
$770$	0	0
$771$	−1.67248e15	−0.221086
$772$	− 5.07826e15i	− 0.666530i
$773$	− 2.07867e15i	− 0.270894i	−0.990785	−	0.135447i	$-0.956753\pi$
$773$	− 2.07867e15i	− 0.270894i	0.990785	−	0.135447i	$-0.0432470\pi$
$774$	−1.45277e15	−0.187984
$775$	0	0
$776$	1.16743e15	0.148934
$777$	− 1.56703e15i	− 0.198500i
$778$	3.42866e15i	0.431258i
$779$	−2.62664e15	−0.328053
$780$	0	0
$781$	−2.22045e16	−2.73438
$782$	7.13988e14i	0.0873079i
$783$	− 2.73385e15i	− 0.331960i
$784$	−1.28432e16	−1.54859
$785$	0	0
$786$	7.86587e14	0.0935239
$787$	− 1.58421e16i	− 1.87047i	−0.354026	−	0.935235i	$-0.615188\pi$
$787$	− 1.58421e16i	− 1.87047i	0.354026	−	0.935235i	$-0.384812\pi$
$788$	7.07823e15i	0.829909i
$789$	−8.01774e15	−0.933531
$790$	0	0
$791$	−3.16844e15	−0.363810
$792$	− 2.71447e15i	− 0.309525i
$793$	− 1.98086e15i	− 0.224311i
$794$	7.00826e14	0.0788129
$795$	0	0
$796$	1.86503e15	0.206854
$797$	− 4.58469e15i	− 0.504997i	−0.967597	−	0.252499i	$-0.918748\pi$
$797$	− 4.58469e15i	− 0.504997i	0.967597	−	0.252499i	$-0.0812523\pi$
$798$	3.31852e15i	0.363018i
$799$	−2.26061e15	−0.245594
$800$	0	0
$801$	2.62221e14	0.0280988
$802$	− 4.40210e15i	− 0.468490i
$803$	− 2.59528e16i	− 2.74315i
$804$	−3.37595e15	−0.354395
$805$	0	0
$806$	−1.78272e15	−0.184604
$807$	7.22872e15i	0.743459i
$808$	− 8.90465e15i	− 0.909608i
$809$	1.72813e15	0.175331	0.0876656	−	0.996150i	$-0.472059\pi$
$809$	1.72813e15	0.175331	0.0876656	+	0.996150i	$0.472059\pi$
$810$	0	0
$811$	4.24638e14	0.0425015	0.0212507	−	0.999774i	$-0.493235\pi$
$811$	4.24638e14	0.0425015	0.0212507	+	0.999774i	$0.493235\pi$
$812$	2.78932e16i	2.77294i
$813$	− 1.04590e16i	− 1.03274i
$814$	−9.55899e14	−0.0937514
$815$	0	0
$816$	−1.86209e15	−0.180179
$817$	1.93994e16i	1.86452i
$818$	3.76631e15i	0.359561i
$819$	5.65355e15	0.536117
$820$	0	0
$821$	−1.96517e16	−1.83871	−0.919354	−	0.393431i	$-0.871288\pi$
$821$	−1.96517e16	−1.83871	−0.919354	+	0.393431i	$0.871288\pi$
$822$	1.98166e15i	0.184176i
$823$	− 5.95701e14i	− 0.0549957i	−0.999622	−	0.0274979i	$-0.991246\pi$
$823$	− 5.95701e14i	− 0.0549957i	0.999622	−	0.0274979i	$-0.00875394\pi$
$824$	−2.91398e15	−0.267231
$825$	0	0
$826$	5.12743e15	0.463991
$827$	− 1.40975e16i	− 1.26725i	−0.773639	−	0.633626i	$-0.781566\pi$
$827$	− 1.40975e16i	− 1.26725i	0.773639	−	0.633626i	$-0.218434\pi$
$828$	1.99210e15i	0.177887i
$829$	7.87187e15	0.698277	0.349139	−	0.937071i	$-0.386474\pi$
$829$	7.87187e15	0.698277	0.349139	+	0.937071i	$0.386474\pi$
$830$	0	0
$831$	−3.56497e15	−0.312068
$832$	4.32142e15i	0.375793i
$833$	− 1.16210e16i	− 1.00391i
$834$	−2.20878e15	−0.189557
$835$	0	0
$836$	−1.71115e16	−1.44929
$837$	− 1.45106e15i	− 0.122094i
$838$	− 3.26507e15i	− 0.272929i
$839$	−6.12117e15	−0.508327	−0.254164	−	0.967161i	$-0.581800\pi$
$839$	−6.12117e15	−0.508327	−0.254164	+	0.967161i	$0.581800\pi$
$840$	0	0
$841$	2.41000e16	1.97533
$842$	1.44092e15i	0.117334i
$843$	1.01666e16i	0.822476i
$844$	3.76650e15	0.302729
$845$	0	0
$846$	7.46170e14	0.0591971
$847$	2.90043e16i	2.28615i
$848$	1.02655e16i	0.803902i
$849$	5.69208e15	0.442873
$850$	0	0
$851$	1.48603e15	0.114134
$852$	− 1.22740e16i	− 0.936629i
$853$	− 2.83032e15i	− 0.214594i	−0.994227	−	0.107297i	$-0.965781\pi$
$853$	− 2.83032e15i	− 0.214594i	0.994227	−	0.107297i	$-0.0342195\pi$
$854$	1.94611e15	0.146605
$855$	0	0
$856$	−6.09060e14	−0.0452954
$857$	− 4.83193e14i	− 0.0357048i	−0.999841	−	0.0178524i	$-0.994317\pi$
$857$	− 4.83193e14i	− 0.0357048i	0.999841	−	0.0178524i	$-0.00568289\pi$
$858$	− 3.44871e15i	− 0.253208i
$859$	−1.23886e16	−0.903775	−0.451888	−	0.892075i	$-0.649249\pi$
$859$	−1.23886e16	−0.903775	−0.451888	+	0.892075i	$0.649249\pi$
$860$	0	0
$861$	−4.39635e15	−0.316647
$862$	1.63992e15i	0.117364i
$863$	2.26458e16i	1.61038i	0.593016	+	0.805191i	$0.297937\pi$
$863$	2.26458e16i	1.61038i	−0.593016	+	0.805191i	$0.702063\pi$
$864$	2.29262e15	0.161997
$865$	0	0
$866$	−5.03459e15	−0.351249
$867$	6.64319e15i	0.460544i
$868$	1.48050e16i	1.01988i
$869$	1.95751e15	0.133997
$870$	0	0
$871$	−9.08565e15	−0.614124
$872$	1.78446e15i	0.119858i
$873$	1.20727e15i	0.0805799i
$874$	−3.14698e15	−0.208728
$875$	0	0
$876$	1.43460e16	0.939630
$877$	− 8.89622e15i	− 0.579039i	−0.957172	−	0.289519i	$-0.906505\pi$
$877$	− 8.89622e15i	− 0.579039i	0.957172	−	0.289519i	$-0.0934955\pi$
$878$	− 1.78023e15i	− 0.115148i
$879$	−1.77696e15	−0.114219
$880$	0	0
$881$	−1.82720e16	−1.15989	−0.579946	−	0.814655i	$-0.696927\pi$
$881$	−1.82720e16	−1.15989	−0.579946	+	0.814655i	$0.696927\pi$
$882$	3.83579e15i	0.241979i
$883$	− 8.75758e15i	− 0.549035i	−0.961582	−	0.274518i	$-0.911482\pi$
$883$	− 8.75758e15i	− 0.549035i	0.961582	−	0.274518i	$-0.0885182\pi$
$884$	−5.77551e15	−0.359834
$885$	0	0
$886$	−1.87398e15	−0.115314
$887$	8.52159e15i	0.521124i	0.965457	+	0.260562i	$0.0839078\pi$
$887$	8.52159e15i	0.521124i	−0.965457	+	0.260562i	$0.916092\pi$
$888$	− 1.11930e15i	− 0.0680258i
$889$	−2.10613e15	−0.127212
$890$	0	0
$891$	2.80709e15	0.167467
$892$	− 1.83132e16i	− 1.08582i
$893$	− 9.96390e15i	− 0.587146i
$894$	−2.39192e15	−0.140084
$895$	0	0
$896$	−3.04042e16	−1.75889
$897$	5.36132e15i	0.308257i
$898$	2.38209e15i	0.136125i
$899$	1.92673e16	1.09432
$900$	0	0
$901$	−9.28858e15	−0.521150
$902$	2.68181e15i	0.149552i
$903$	3.24699e16i	1.79969i
$904$	−2.26316e15	−0.124677
$905$	0	0
$906$	−4.94461e15	−0.269108
$907$	2.10550e16i	1.13898i	0.821998	+	0.569490i	$0.192859\pi$
$907$	2.10550e16i	1.13898i	−0.821998	+	0.569490i	$0.807141\pi$
$908$	− 1.24005e16i	− 0.666756i
$909$	9.20851e15	0.492139
$910$	0	0
$911$	3.04479e16	1.60771	0.803854	−	0.594827i	$-0.202780\pi$
$911$	3.04479e16	1.60771	0.803854	+	0.594827i	$0.202780\pi$
$912$	− 8.20739e15i	− 0.430758i
$913$	− 9.71950e15i	− 0.507055i
$914$	5.13939e15	0.266506
$915$	0	0
$916$	−1.86674e16	−0.956443
$917$	− 1.75805e16i	− 0.895363i
$918$	5.56137e14i	0.0281544i
$919$	4.47695e15	0.225293	0.112646	−	0.993635i	$-0.464067\pi$
$919$	4.47695e15	0.225293	0.112646	+	0.993635i	$0.464067\pi$
$920$	0	0
$921$	1.30549e16	0.649151
$922$	− 6.61799e15i	− 0.327119i
$923$	− 3.30328e16i	− 1.62307i
$924$	−2.86405e16	−1.39889
$925$	0	0
$926$	6.19818e14	0.0299160
$927$	− 3.01342e15i	− 0.144584i
$928$	3.04417e16i	1.45196i
$929$	−2.63310e16	−1.24848	−0.624238	−	0.781234i	$-0.714591\pi$
$929$	−2.63310e16	−1.24848	−0.624238	+	0.781234i	$0.714591\pi$
$930$	0	0
$931$	5.12208e16	2.40007
$932$	− 1.23515e16i	− 0.575348i
$933$	1.31562e16i	0.609228i
$934$	1.16481e15	0.0536224
$935$	0	0
$936$	4.03822e15	0.183727
$937$	1.18259e16i	0.534894i	0.963573	+	0.267447i	$0.0861800\pi$
$937$	1.18259e16i	0.534894i	−0.963573	+	0.267447i	$0.913820\pi$
$938$	− 8.92628e15i	− 0.401379i
$939$	−1.04422e15	−0.0466802
$940$	0	0
$941$	6.96003e15	0.307517	0.153758	−	0.988108i	$-0.450862\pi$
$941$	6.96003e15	0.307517	0.153758	+	0.988108i	$0.450862\pi$
$942$	2.83872e15i	0.124693i
$943$	− 4.16911e15i	− 0.182066i
$944$	−1.26812e16	−0.550573
$945$	0	0
$946$	1.98069e16	0.849993
$947$	− 3.38501e16i	− 1.44423i	−0.691775	−	0.722113i	$-0.743171\pi$
$947$	− 3.38501e16i	− 1.44423i	0.691775	−	0.722113i	$-0.256829\pi$
$948$	1.08206e15i	0.0458991i
$949$	3.86091e16	1.62827
$950$	0	0
$951$	6.85084e15	0.285595
$952$	− 1.20197e16i	− 0.498183i
$953$	− 4.50062e16i	− 1.85465i	−0.374262	−	0.927323i	$-0.622104\pi$
$953$	− 4.50062e16i	− 1.85465i	0.374262	−	0.927323i	$-0.377896\pi$
$954$	3.06592e15	0.125616
$955$	0	0
$956$	3.23235e16	1.30918
$957$	3.72730e16i	1.50099i
$958$	1.41644e16i	0.567139i
$959$	4.42907e16	1.76323
$960$	0	0
$961$	−1.51819e16	−0.597513
$962$	− 1.42206e15i	− 0.0556486i
$963$	− 6.29844e14i	− 0.0245069i
$964$	−2.52107e16	−0.975353
$965$	0	0
$966$	−5.26728e15	−0.201471
$967$	− 1.48359e16i	− 0.564246i	−0.959378	−	0.282123i	$-0.908961\pi$
$967$	− 1.48359e16i	− 0.564246i	0.959378	−	0.282123i	$-0.0910386\pi$
$968$	2.07172e16i	0.783461i
$969$	7.42632e15	0.279250
$970$	0	0
$971$	1.14414e16	0.425378	0.212689	−	0.977120i	$-0.431778\pi$
$971$	1.14414e16	0.425378	0.212689	+	0.977120i	$0.431778\pi$
$972$	1.55168e15i	0.0573638i
$973$	4.93669e16i	1.81474i
$974$	−6.72835e15	−0.245943
$975$	0	0
$976$	−4.81314e15	−0.173962
$977$	− 2.64437e16i	− 0.950390i	−0.879881	−	0.475195i	$-0.842378\pi$
$977$	− 2.64437e16i	− 0.950390i	0.879881	−	0.475195i	$-0.157622\pi$
$978$	− 1.67177e15i	− 0.0597464i
$979$	−3.57508e15	−0.127052
$980$	0	0
$981$	−1.84536e15	−0.0648486
$982$	6.47820e15i	0.226382i
$983$	− 3.63656e16i	− 1.26371i	−0.775088	−	0.631854i	$-0.782294\pi$
$983$	− 3.63656e16i	− 1.26371i	0.775088	−	0.631854i	$-0.217706\pi$
$984$	−3.14023e15	−0.108515
$985$	0	0
$986$	−7.38447e15	−0.252346
$987$	− 1.66771e16i	− 0.566731i
$988$	− 2.54562e16i	− 0.860262i
$989$	−3.07915e16	−1.03479
$990$	0	0
$991$	5.09067e16	1.69188	0.845941	−	0.533276i	$-0.179039\pi$
$991$	5.09067e16	1.69188	0.845941	+	0.533276i	$0.179039\pi$
$992$	1.61576e16i	0.534027i
$993$	− 2.41899e16i	− 0.795084i
$994$	3.24534e16	1.06080
$995$	0	0
$996$	5.37266e15	0.173685
$997$	− 4.28291e16i	− 1.37694i	−0.725264	−	0.688471i	$-0.758282\pi$
$997$	− 4.28291e16i	− 1.37694i	0.725264	−	0.688471i	$-0.241718\pi$
$998$	− 9.36259e15i	− 0.299349i
$999$	1.15749e15	0.0368051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.12.b.d.49.3		4
3.2	odd	2		225.12.b.i.199.2		4
5.2	odd	4		75.12.a.c.1.1		2
5.3	odd	4		15.12.a.c.1.2	✓	2
5.4	even	2	inner	75.12.b.d.49.2		4
15.2	even	4		225.12.a.i.1.2		2
15.8	even	4		45.12.a.c.1.1		2
15.14	odd	2		225.12.b.i.199.3		4
20.3	even	4		240.12.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.c.1.2	✓	2	5.3	odd	4
45.12.a.c.1.1		2	15.8	even	4
75.12.a.c.1.1		2	5.2	odd	4
75.12.b.d.49.2		4	5.4	even	2	inner
75.12.b.d.49.3		4	1.1	even	1	trivial
225.12.a.i.1.2		2	15.2	even	4
225.12.b.i.199.2		4	3.2	odd	2
225.12.b.i.199.3		4	15.14	odd	2
240.12.a.m.1.1		2	20.3	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+14.7191i q^{2} +243.000i q^{3} +1831.35 q^{4} -3576.74 q^{6} +79941.2i q^{7} +57100.5i q^{8} -59049.0 q^{9} +805067. q^{11} +445018. i q^{12} +1.19767e6i q^{13} -1.17666e6 q^{14} +2.91013e6 q^{16} +2.63319e6i q^{17} -869148. i q^{18} -1.16061e7 q^{19} -1.94257e7 q^{21} +1.18499e7i q^{22} -1.84216e7i q^{23} -1.38754e7 q^{24} -1.76286e7 q^{26} -1.43489e7i q^{27} +1.46400e8i q^{28} +1.90527e8 q^{29} +1.01127e8 q^{31} +1.59776e8i q^{32} +1.95631e8i q^{33} -3.87581e7 q^{34} -1.08139e8 q^{36} +8.06675e7i q^{37} -1.70831e8i q^{38} -2.91034e8 q^{39} +2.26316e8 q^{41} -2.85929e8i q^{42} -1.67149e9i q^{43} +1.47436e9 q^{44} +2.71150e8 q^{46} +8.58507e8i q^{47} +7.07162e8i q^{48} -4.41327e9 q^{49} -6.39864e8 q^{51} +2.19335e9i q^{52} +3.52750e9i q^{53} +2.11203e8 q^{54} -4.56468e9 q^{56} -2.82028e9i q^{57} +2.80438e9i q^{58} -4.35760e9 q^{59} -1.65393e9 q^{61} +1.48849e9i q^{62} -4.72045e9i q^{63} +3.60819e9 q^{64} -2.87951e9 q^{66} +7.58610e9i q^{67} +4.82228e9i q^{68} +4.47645e9 q^{69} -2.75809e10 q^{71} -3.37173e9i q^{72} -3.22368e10i q^{73} -1.18735e9 q^{74} -2.12548e10 q^{76} +6.43580e10i q^{77} -4.28376e9i q^{78} +2.43149e9 q^{79} +3.48678e9 q^{81} +3.33116e9i q^{82} -1.20729e10i q^{83} -3.55753e10 q^{84} +2.46028e10 q^{86} +4.62981e10i q^{87} +4.59697e10i q^{88} -4.44073e9 q^{89} -9.57433e10 q^{91} -3.37364e10i q^{92} +2.45737e10i q^{93} -1.26364e10 q^{94} -3.88257e10 q^{96} -2.04453e10i q^{97} -6.49594e10i q^{98} -4.75384e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6222 q^{4} + 6318 q^{6} - 236196 q^{9} + 591136 q^{11} - 6353592 q^{14} + 5948130 q^{16} - 35255952 q^{19} - 3783024 q^{21} + 17077554 q^{24} - 138104132 q^{26} + 403763896 q^{29} - 142114016 q^{31}+ \cdots - 34905989664 q^{99}+O(q^{100}) \) 4 * q + 6222 * q^4 + 6318 * q^6 - 236196 * q^9 + 591136 * q^11 - 6353592 * q^14 + 5948130 * q^16 - 35255952 * q^19 - 3783024 * q^21 + 17077554 * q^24 - 138104132 * q^26 + 403763896 * q^29 - 142114016 * q^31 + 252161404 * q^34 - 367402878 * q^36 + 319541112 * q^39 - 655310296 * q^41 + 1644753136 * q^44 + 3217895664 * q^46 - 15285229540 * q^49 - 4169854728 * q^51 - 373071582 * q^54 - 22440862920 * q^56 + 2957541152 * q^59 - 16529782920 * q^61 - 3092633362 * q^64 - 12622730256 * q^66 - 14502760992 * q^69 - 40437776512 * q^71 + 32284748068 * q^74 - 57921409528 * q^76 - 44649990880 * q^79 + 13947137604 * q^81 - 26275156488 * q^84 - 35119396184 * q^86 - 158419366152 * q^89 - 459213424096 * q^91 + 40951225552 * q^94 + 15018498846 * q^96 - 34905989664 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more