LMFDB - Embedded newform 200.14.a.l.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,14,Mod(1,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("200.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	200.a (trivial)

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:	yes
Analytic conductor:	$214.461857904$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 2792237 x^{8} + 94050736 x^{7} + 2615193085270 x^{6} - 30081688120700 x^{5} + \cdots + 14\!\cdots\!75 $ x^10 - 2x^9 - 2792237x^8 + 94050736x^7 + 2615193085270x^6 - 30081688120700x^5 - 919603429545956750x^4 - 94224520282915730000x^3 + 105082623154370946015625x^2 + 25030138583264107436718750*x + 1405679570677464988623046875
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{55}\cdot 3^{5}\cdot 5^{16} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-92.2624$ of defining polynomial
Character	$\chi$	$=$	200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+312.525 q^{3} -246082. q^{7} -1.49665e6 q^{9} +O(q^{10})$	$q+312.525 q^{3} -246082. q^{7} -1.49665e6 q^{9} +6.63065e6 q^{11} +4.61650e6 q^{13} -1.66414e7 q^{17} -1.93054e6 q^{19} -7.69067e7 q^{21} +6.44482e8 q^{23} -9.66006e8 q^{27} -1.09609e9 q^{29} +4.70409e7 q^{31} +2.07224e9 q^{33} -1.35328e10 q^{37} +1.44277e9 q^{39} -4.67201e10 q^{41} -1.34584e10 q^{43} -5.80336e10 q^{47} -3.63327e10 q^{49} -5.20084e9 q^{51} +1.59365e11 q^{53} -6.03342e8 q^{57} +6.28423e11 q^{59} -4.84129e11 q^{61} +3.68299e11 q^{63} -6.05549e10 q^{67} +2.01417e11 q^{69} -1.30944e11 q^{71} +1.57381e12 q^{73} -1.63168e12 q^{77} +2.27623e12 q^{79} +2.08424e12 q^{81} -4.65420e12 q^{83} -3.42557e11 q^{87} -2.58885e12 q^{89} -1.13604e12 q^{91} +1.47014e10 q^{93} +2.71425e12 q^{97} -9.92378e12 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 1276 q^{3} + 341068 q^{7} + 6557498 q^{9}+O(q^{10}) $ 10 * q + 1276 * q^3 + 341068 * q^7 + 6557498 * q^9	$ 10 q + 1276 q^{3} + 341068 q^{7} + 6557498 q^{9} - 4074536 q^{11} - 7720792 q^{13} + 29092704 q^{17} - 82552936 q^{19} - 140909032 q^{21} + 448872532 q^{23} + 6653011048 q^{27} + 4118753908 q^{29} - 161162304 q^{31} + 6635864336 q^{33} + 37799831624 q^{37} + 2651467216 q^{39} + 31205287876 q^{41} + 48886150956 q^{43} - 54068936100 q^{47} + 313063319346 q^{49} - 179707160640 q^{51} - 116117700520 q^{53} - 863281677424 q^{57} - 658612309176 q^{59} + 405225797804 q^{61} - 169179586580 q^{63} - 291232794812 q^{67} + 1989925462312 q^{69} - 956854398704 q^{71} + 262407355312 q^{73} + 3471232133584 q^{77} - 3176139465248 q^{79} + 3191121792770 q^{81} + 5123747754524 q^{83} + 9347096848952 q^{87} + 453451956924 q^{89} + 6935945759120 q^{91} - 806946503040 q^{93} - 11480198579200 q^{97} + 18677113053976 q^{99}+O(q^{100}) $ 10 * q + 1276 * q^3 + 341068 * q^7 + 6557498 * q^9 - 4074536 * q^11 - 7720792 * q^13 + 29092704 * q^17 - 82552936 * q^19 - 140909032 * q^21 + 448872532 * q^23 + 6653011048 * q^27 + 4118753908 * q^29 - 161162304 * q^31 + 6635864336 * q^33 + 37799831624 * q^37 + 2651467216 * q^39 + 31205287876 * q^41 + 48886150956 * q^43 - 54068936100 * q^47 + 313063319346 * q^49 - 179707160640 * q^51 - 116117700520 * q^53 - 863281677424 * q^57 - 658612309176 * q^59 + 405225797804 * q^61 - 169179586580 * q^63 - 291232794812 * q^67 + 1989925462312 * q^69 - 956854398704 * q^71 + 262407355312 * q^73 + 3471232133584 * q^77 - 3176139465248 * q^79 + 3191121792770 * q^81 + 5123747754524 * q^83 + 9347096848952 * q^87 + 453451956924 * q^89 + 6935945759120 * q^91 - 806946503040 * q^93 - 11480198579200 * q^97 + 18677113053976 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	312.525	0.247512	0.123756	−	0.992313i	$-0.460506\pi$
$3$	312.525	0.247512	0.123756	+	0.992313i	$0.460506\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−246082.	−0.790574	−0.395287	−	0.918558i	$-0.629355\pi$
$7$	−246082.	−0.790574	−0.395287	+	0.918558i	$0.629355\pi$
$8$	0	0
$9$	−1.49665e6	−0.938738
$10$	0	0
$11$	6.63065e6	1.12851	0.564253	−	0.825602i	$-0.309164\pi$
$11$	6.63065e6	1.12851	0.564253	+	0.825602i	$0.309164\pi$
$12$	0	0
$13$	4.61650e6	0.265266	0.132633	−	0.991165i	$-0.457657\pi$
$13$	4.61650e6	0.265266	0.132633	+	0.991165i	$0.457657\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.66414e7	−0.167213	−0.0836067	−	0.996499i	$-0.526644\pi$
$17$	−1.66414e7	−0.167213	−0.0836067	+	0.996499i	$0.526644\pi$
$18$	0	0
$19$	−1.93054e6	−0.00941414	−0.00470707	−	0.999989i	$-0.501498\pi$
$19$	−1.93054e6	−0.00941414	−0.00470707	+	0.999989i	$0.501498\pi$
$20$	0	0
$21$	−7.69067e7	−0.195677
$22$	0	0
$23$	6.44482e8	0.907778	0.453889	−	0.891058i	$-0.350036\pi$
$23$	6.44482e8	0.907778	0.453889	+	0.891058i	$0.350036\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−9.66006e8	−0.479861
$28$	0	0
$29$	−1.09609e9	−0.342185	−0.171093	−	0.985255i	$-0.554730\pi$
$29$	−1.09609e9	−0.342185	−0.171093	+	0.985255i	$0.554730\pi$
$30$	0	0
$31$	4.70409e7	0.00951973	0.00475987	−	0.999989i	$-0.498485\pi$
$31$	4.70409e7	0.00951973	0.00475987	+	0.999989i	$0.498485\pi$
$32$	0	0
$33$	2.07224e9	0.279319
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.35328e10	−0.867114	−0.433557	−	0.901126i	$-0.642742\pi$
$37$	−1.35328e10	−0.867114	−0.433557	+	0.901126i	$0.642742\pi$
$38$	0	0
$39$	1.44277e9	0.0656565
$40$	0	0
$41$	−4.67201e10	−1.53606	−0.768031	−	0.640413i	$-0.778763\pi$
$41$	−4.67201e10	−1.53606	−0.768031	+	0.640413i	$0.778763\pi$
$42$	0	0
$43$	−1.34584e10	−0.324676	−0.162338	−	0.986735i	$-0.551903\pi$
$43$	−1.34584e10	−0.324676	−0.162338	+	0.986735i	$0.551903\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.80336e10	−0.785314	−0.392657	−	0.919685i	$-0.628444\pi$
$47$	−5.80336e10	−0.785314	−0.392657	+	0.919685i	$0.628444\pi$
$48$	0	0
$49$	−3.63327e10	−0.374993
$50$	0	0
$51$	−5.20084e9	−0.0413873
$52$	0	0
$53$	1.59365e11	0.987644	0.493822	−	0.869563i	$-0.335599\pi$
$53$	1.59365e11	0.987644	0.493822	+	0.869563i	$0.335599\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.03342e8	−0.00233011
$58$	0	0
$59$	6.28423e11	1.93961	0.969804	−	0.243887i	$-0.0784225\pi$
$59$	6.28423e11	1.93961	0.969804	+	0.243887i	$0.0784225\pi$
$60$	0	0
$61$	−4.84129e11	−1.20314	−0.601571	−	0.798819i	$-0.705458\pi$
$61$	−4.84129e11	−1.20314	−0.601571	+	0.798819i	$0.705458\pi$
$62$	0	0
$63$	3.68299e11	0.742142
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.05549e10	−0.0817830	−0.0408915	−	0.999164i	$-0.513020\pi$
$67$	−6.05549e10	−0.0817830	−0.0408915	+	0.999164i	$0.513020\pi$
$68$	0	0
$69$	2.01417e11	0.224686
$70$	0	0
$71$	−1.30944e11	−0.121313	−0.0606563	−	0.998159i	$-0.519319\pi$
$71$	−1.30944e11	−0.121313	−0.0606563	+	0.998159i	$0.519319\pi$
$72$	0	0
$73$	1.57381e12	1.21718	0.608588	−	0.793486i	$-0.291736\pi$
$73$	1.57381e12	1.21718	0.608588	+	0.793486i	$0.291736\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.63168e12	−0.892168
$78$	0	0
$79$	2.27623e12	1.05351	0.526756	−	0.850016i	$-0.323408\pi$
$79$	2.27623e12	1.05351	0.526756	+	0.850016i	$0.323408\pi$
$80$	0	0
$81$	2.08424e12	0.819967
$82$	0	0
$83$	−4.65420e12	−1.56256	−0.781281	−	0.624179i	$-0.785434\pi$
$83$	−4.65420e12	−1.56256	−0.781281	+	0.624179i	$0.785434\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.42557e11	−0.0846949
$88$	0	0
$89$	−2.58885e12	−0.552169	−0.276085	−	0.961133i	$-0.589037\pi$
$89$	−2.58885e12	−0.552169	−0.276085	+	0.961133i	$0.589037\pi$
$90$	0	0
$91$	−1.13604e12	−0.209712
$92$	0	0
$93$	1.47014e10	0.00235625
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.71425e12	0.330852	0.165426	−	0.986222i	$-0.447100\pi$
$97$	2.71425e12	0.330852	0.165426	+	0.986222i	$0.447100\pi$
$98$	0	0
$99$	−9.92378e12	−1.05937
$100$	0	0
$101$	1.80624e13	1.69311	0.846555	−	0.532301i	$-0.178672\pi$
$101$	1.80624e13	1.69311	0.846555	+	0.532301i	$0.178672\pi$
$102$	0	0
$103$	−1.02300e13	−0.844180	−0.422090	−	0.906554i	$-0.638703\pi$
$103$	−1.02300e13	−0.844180	−0.422090	+	0.906554i	$0.638703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.80875e13	1.80933	0.904667	−	0.426119i	$-0.140120\pi$
$107$	2.80875e13	1.80933	0.904667	+	0.426119i	$0.140120\pi$
$108$	0	0
$109$	−1.26626e13	−0.723185	−0.361592	−	0.932336i	$-0.617767\pi$
$109$	−1.26626e13	−0.723185	−0.361592	+	0.932336i	$0.617767\pi$
$110$	0	0
$111$	−4.22933e12	−0.214621
$112$	0	0
$113$	2.50178e13	1.13042	0.565210	−	0.824947i	$-0.308795\pi$
$113$	2.50178e13	1.13042	0.565210	+	0.824947i	$0.308795\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.90930e12	−0.249015
$118$	0	0
$119$	4.09514e12	0.132195
$120$	0	0
$121$	9.44287e12	0.273526
$122$	0	0
$123$	−1.46012e13	−0.380194
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.76133e12	0.0583974	0.0291987	−	0.999574i	$-0.490704\pi$
$127$	2.76133e12	0.0583974	0.0291987	+	0.999574i	$0.490704\pi$
$128$	0	0
$129$	−4.20610e12	−0.0803611
$130$	0	0
$131$	−7.04396e12	−0.121774	−0.0608868	−	0.998145i	$-0.519393\pi$
$131$	−7.04396e12	−0.121774	−0.0608868	+	0.998145i	$0.519393\pi$
$132$	0	0
$133$	4.75071e11	0.00744257
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.10081e14	−1.42242	−0.711211	−	0.702978i	$-0.751853\pi$
$137$	−1.10081e14	−1.42242	−0.711211	+	0.702978i	$0.751853\pi$
$138$	0	0
$139$	1.31958e14	1.55181	0.775907	−	0.630847i	$-0.217293\pi$
$139$	1.31958e14	1.55181	0.775907	+	0.630847i	$0.217293\pi$
$140$	0	0
$141$	−1.81369e13	−0.194375
$142$	0	0
$143$	3.06104e13	0.299354
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.13549e13	−0.0928153
$148$	0	0
$149$	6.82529e13	0.510988	0.255494	−	0.966811i	$-0.417762\pi$
$149$	6.82529e13	0.510988	0.255494	+	0.966811i	$0.417762\pi$
$150$	0	0
$151$	−1.07050e14	−0.734911	−0.367455	−	0.930041i	$-0.619771\pi$
$151$	−1.07050e14	−0.734911	−0.367455	+	0.930041i	$0.619771\pi$
$152$	0	0
$153$	2.49063e13	0.156970
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.39300e13	−0.287397	−0.143699	−	0.989621i	$-0.545900\pi$
$157$	−5.39300e13	−0.287397	−0.143699	+	0.989621i	$0.545900\pi$
$158$	0	0
$159$	4.98056e13	0.244454
$160$	0	0
$161$	−1.58595e14	−0.717666
$162$	0	0
$163$	3.48137e14	1.45389	0.726944	−	0.686697i	$-0.240940\pi$
$163$	3.48137e14	1.45389	0.726944	+	0.686697i	$0.240940\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.98560e14	1.06506	0.532530	−	0.846411i	$-0.321241\pi$
$167$	2.98560e14	1.06506	0.532530	+	0.846411i	$0.321241\pi$
$168$	0	0
$169$	−2.81563e14	−0.929634
$170$	0	0
$171$	2.88935e12	0.00883741
$172$	0	0
$173$	2.57303e14	0.729700	0.364850	−	0.931066i	$-0.381120\pi$
$173$	2.57303e14	0.729700	0.364850	+	0.931066i	$0.381120\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.96398e14	0.480076
$178$	0	0
$179$	−3.83096e14	−0.870488	−0.435244	−	0.900313i	$-0.643338\pi$
$179$	−3.83096e14	−0.870488	−0.435244	+	0.900313i	$0.643338\pi$
$180$	0	0
$181$	2.11643e14	0.447397	0.223698	−	0.974658i	$-0.428187\pi$
$181$	2.11643e14	0.447397	0.223698	+	0.974658i	$0.428187\pi$
$182$	0	0
$183$	−1.51302e14	−0.297792
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.10343e14	−0.188701
$188$	0	0
$189$	2.37717e14	0.379365
$190$	0	0
$191$	1.16537e14	0.173679	0.0868395	−	0.996222i	$-0.472323\pi$
$191$	1.16537e14	0.173679	0.0868395	+	0.996222i	$0.472323\pi$
$192$	0	0
$193$	−5.69250e14	−0.792831	−0.396415	−	0.918071i	$-0.629746\pi$
$193$	−5.69250e14	−0.792831	−0.396415	+	0.918071i	$0.629746\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.03893e15	1.26635	0.633177	−	0.774007i	$-0.281750\pi$
$197$	1.03893e15	1.26635	0.633177	+	0.774007i	$0.281750\pi$
$198$	0	0
$199$	−9.30518e14	−1.06213	−0.531067	−	0.847330i	$-0.678209\pi$
$199$	−9.30518e14	−1.06213	−0.531067	+	0.847330i	$0.678209\pi$
$200$	0	0
$201$	−1.89249e13	−0.0202423
$202$	0	0
$203$	2.69729e14	0.270523
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−9.64565e14	−0.852166
$208$	0	0
$209$	−1.28008e13	−0.0106239
$210$	0	0
$211$	1.39070e15	1.08492	0.542462	−	0.840081i	$-0.317492\pi$
$211$	1.39070e15	1.08492	0.542462	+	0.840081i	$0.317492\pi$
$212$	0	0
$213$	−4.09232e13	−0.0300263
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.15759e13	−0.00752605
$218$	0	0
$219$	4.91854e14	0.301266
$220$	0	0
$221$	−7.68250e13	−0.0443560
$222$	0	0
$223$	3.12371e15	1.70094	0.850469	−	0.526025i	$-0.176318\pi$
$223$	3.12371e15	1.70094	0.850469	+	0.526025i	$0.176318\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.28842e15	1.11011	0.555057	−	0.831812i	$-0.312697\pi$
$227$	2.28842e15	1.11011	0.555057	+	0.831812i	$0.312697\pi$
$228$	0	0
$229$	3.19182e15	1.46254	0.731270	−	0.682088i	$-0.238928\pi$
$229$	3.19182e15	1.46254	0.731270	+	0.682088i	$0.238928\pi$
$230$	0	0
$231$	−5.09942e14	−0.220822
$232$	0	0
$233$	4.39266e15	1.79852	0.899258	−	0.437419i	$-0.144107\pi$
$233$	4.39266e15	1.79852	0.899258	+	0.437419i	$0.144107\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.11378e14	0.260757
$238$	0	0
$239$	−3.38429e14	−0.117458	−0.0587288	−	0.998274i	$-0.518705\pi$
$239$	−3.38429e14	−0.117458	−0.0587288	+	0.998274i	$0.518705\pi$
$240$	0	0
$241$	−2.96923e15	−0.976188	−0.488094	−	0.872791i	$-0.662308\pi$
$241$	−2.96923e15	−0.976188	−0.488094	+	0.872791i	$0.662308\pi$
$242$	0	0
$243$	2.19150e15	0.682812
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.91235e12	−0.00249725
$248$	0	0
$249$	−1.45455e15	−0.386753
$250$	0	0
$251$	6.93506e14	0.175054	0.0875268	−	0.996162i	$-0.472104\pi$
$251$	6.93506e14	0.175054	0.0875268	+	0.996162i	$0.472104\pi$
$252$	0	0
$253$	4.27334e15	1.02443
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.06762e14	0.0664104	0.0332052	−	0.999449i	$-0.489429\pi$
$257$	3.06762e14	0.0664104	0.0332052	+	0.999449i	$0.489429\pi$
$258$	0	0
$259$	3.33018e15	0.685518
$260$	0	0
$261$	1.64047e15	0.321222
$262$	0	0
$263$	5.86465e15	1.09277	0.546386	−	0.837533i	$-0.316003\pi$
$263$	5.86465e15	1.09277	0.546386	+	0.837533i	$0.316003\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.09080e14	−0.136669
$268$	0	0
$269$	−1.15110e15	−0.185235	−0.0926173	−	0.995702i	$-0.529523\pi$
$269$	−1.15110e15	−0.185235	−0.0926173	+	0.995702i	$0.529523\pi$
$270$	0	0
$271$	−9.06285e15	−1.38984	−0.694919	−	0.719088i	$-0.744560\pi$
$271$	−9.06285e15	−1.38984	−0.694919	+	0.719088i	$0.744560\pi$
$272$	0	0
$273$	−3.55040e14	−0.0519063
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.23667e15	−0.164489	−0.0822444	−	0.996612i	$-0.526209\pi$
$277$	−1.23667e15	−0.164489	−0.0822444	+	0.996612i	$0.526209\pi$
$278$	0	0
$279$	−7.04038e13	−0.00893653
$280$	0	0
$281$	1.02370e16	1.24046	0.620228	−	0.784422i	$-0.287040\pi$
$281$	1.02370e16	1.24046	0.620228	+	0.784422i	$0.287040\pi$
$282$	0	0
$283$	5.19284e15	0.600888	0.300444	−	0.953800i	$-0.402865\pi$
$283$	5.19284e15	0.600888	0.300444	+	0.953800i	$0.402865\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.14970e16	1.21437
$288$	0	0
$289$	−9.62764e15	−0.972040
$290$	0	0
$291$	8.48272e14	0.0818899
$292$	0	0
$293$	1.65966e16	1.53243	0.766213	−	0.642586i	$-0.222139\pi$
$293$	1.65966e16	1.53243	0.766213	+	0.642586i	$0.222139\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−6.40525e15	−0.541526
$298$	0	0
$299$	2.97525e15	0.240803
$300$	0	0
$301$	3.31188e15	0.256680
$302$	0	0
$303$	5.64493e15	0.419065
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.09638e16	1.42912	0.714562	−	0.699572i	$-0.246626\pi$
$307$	2.09638e16	1.42912	0.714562	+	0.699572i	$0.246626\pi$
$308$	0	0
$309$	−3.19714e15	−0.208945
$310$	0	0
$311$	6.39642e15	0.400861	0.200431	−	0.979708i	$-0.435766\pi$
$311$	6.39642e15	0.400861	0.200431	+	0.979708i	$0.435766\pi$
$312$	0	0
$313$	1.78550e16	1.07330	0.536652	−	0.843804i	$-0.319689\pi$
$313$	1.78550e16	1.07330	0.536652	+	0.843804i	$0.319689\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.41863e16	1.89220	0.946100	−	0.323875i	$-0.104986\pi$
$317$	3.41863e16	1.89220	0.946100	+	0.323875i	$0.104986\pi$
$318$	0	0
$319$	−7.26782e15	−0.386158
$320$	0	0
$321$	8.77804e15	0.447832
$322$	0	0
$323$	3.21269e13	0.00157417
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.95736e15	−0.178997
$328$	0	0
$329$	1.42810e16	0.620849
$330$	0	0
$331$	−1.45014e16	−0.606075	−0.303038	−	0.952979i	$-0.598001\pi$
$331$	−1.45014e16	−0.606075	−0.303038	+	0.952979i	$0.598001\pi$
$332$	0	0
$333$	2.02539e16	0.813993
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.16925e15	−0.155047	−0.0775235	−	0.996991i	$-0.524701\pi$
$337$	−4.16925e15	−0.155047	−0.0775235	+	0.996991i	$0.524701\pi$
$338$	0	0
$339$	7.81869e15	0.279792
$340$	0	0
$341$	3.11912e14	0.0107431
$342$	0	0
$343$	3.27835e16	1.08703
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.85438e15	−0.0570239	−0.0285120	−	0.999593i	$-0.509077\pi$
$347$	−1.85438e15	−0.0570239	−0.0285120	+	0.999593i	$0.509077\pi$
$348$	0	0
$349$	−1.29245e16	−0.382867	−0.191434	−	0.981506i	$-0.561314\pi$
$349$	−1.29245e16	−0.382867	−0.191434	+	0.981506i	$0.561314\pi$
$350$	0	0
$351$	−4.45957e15	−0.127291
$352$	0	0
$353$	9.23710e15	0.254097	0.127049	−	0.991896i	$-0.459450\pi$
$353$	9.23710e15	0.254097	0.127049	+	0.991896i	$0.459450\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.27983e15	0.0327197
$358$	0	0
$359$	−4.89854e16	−1.20768	−0.603842	−	0.797104i	$-0.706364\pi$
$359$	−4.89854e16	−1.20768	−0.603842	+	0.797104i	$0.706364\pi$
$360$	0	0
$361$	−4.20493e16	−0.999911
$362$	0	0
$363$	2.95113e15	0.0677010
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.57601e16	−1.19123	−0.595613	−	0.803272i	$-0.703091\pi$
$367$	−5.57601e16	−1.19123	−0.595613	+	0.803272i	$0.703091\pi$
$368$	0	0
$369$	6.99237e16	1.44196
$370$	0	0
$371$	−3.92169e16	−0.780806
$372$	0	0
$373$	6.38239e16	1.22709	0.613545	−	0.789660i	$-0.289743\pi$
$373$	6.38239e16	1.22709	0.613545	+	0.789660i	$0.289743\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.06012e15	−0.0907700
$378$	0	0
$379$	2.96086e16	0.513172	0.256586	−	0.966521i	$-0.417402\pi$
$379$	2.96086e16	0.513172	0.256586	+	0.966521i	$0.417402\pi$
$380$	0	0
$381$	8.62983e14	0.0144541
$382$	0	0
$383$	7.34558e16	1.18914	0.594571	−	0.804043i	$-0.297322\pi$
$383$	7.34558e16	1.18914	0.594571	+	0.804043i	$0.297322\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.01426e16	0.304785
$388$	0	0
$389$	−6.58054e15	−0.0962918	−0.0481459	−	0.998840i	$-0.515331\pi$
$389$	−6.58054e15	−0.0962918	−0.0481459	+	0.998840i	$0.515331\pi$
$390$	0	0
$391$	−1.07251e16	−0.151793
$392$	0	0
$393$	−2.20141e15	−0.0301404
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.79850e16	1.12790	0.563950	−	0.825809i	$-0.309281\pi$
$397$	8.79850e16	1.12790	0.563950	+	0.825809i	$0.309281\pi$
$398$	0	0
$399$	1.48472e14	0.00184213
$400$	0	0
$401$	1.30089e17	1.56243	0.781216	−	0.624260i	$-0.214600\pi$
$401$	1.30089e17	1.56243	0.781216	+	0.624260i	$0.214600\pi$
$402$	0	0
$403$	2.17165e14	0.00252526
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.97313e16	−0.978544
$408$	0	0
$409$	3.23174e16	0.341377	0.170689	−	0.985325i	$-0.445401\pi$
$409$	3.23174e16	0.341377	0.170689	+	0.985325i	$0.445401\pi$
$410$	0	0
$411$	−3.44030e16	−0.352067
$412$	0	0
$413$	−1.54643e17	−1.53340
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.12402e16	0.384093
$418$	0	0
$419$	−5.09385e16	−0.459891	−0.229945	−	0.973204i	$-0.573855\pi$
$419$	−5.09385e16	−0.459891	−0.229945	+	0.973204i	$0.573855\pi$
$420$	0	0
$421$	1.55630e15	0.0136226	0.00681128	−	0.999977i	$-0.497832\pi$
$421$	1.55630e15	0.0136226	0.00681128	+	0.999977i	$0.497832\pi$
$422$	0	0
$423$	8.68560e16	0.737204
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.19135e17	0.951173
$428$	0	0
$429$	9.56652e15	0.0740938
$430$	0	0
$431$	2.24115e17	1.68410	0.842051	−	0.539399i	$-0.181348\pi$
$431$	2.24115e17	1.68410	0.842051	+	0.539399i	$0.181348\pi$
$432$	0	0
$433$	−1.17225e17	−0.854769	−0.427385	−	0.904070i	$-0.640565\pi$
$433$	−1.17225e17	−0.854769	−0.427385	+	0.904070i	$0.640565\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.24420e15	−0.00854595
$438$	0	0
$439$	−5.12537e16	−0.341748	−0.170874	−	0.985293i	$-0.554659\pi$
$439$	−5.12537e16	−0.341748	−0.170874	+	0.985293i	$0.554659\pi$
$440$	0	0
$441$	5.43774e16	0.352020
$442$	0	0
$443$	−3.14935e17	−1.97969	−0.989845	−	0.142149i	$-0.954599\pi$
$443$	−3.14935e17	−1.97969	−0.989845	+	0.142149i	$0.954599\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.13307e16	0.126476
$448$	0	0
$449$	−8.70899e16	−0.501611	−0.250805	−	0.968038i	$-0.580695\pi$
$449$	−8.70899e16	−0.501611	−0.250805	+	0.968038i	$0.580695\pi$
$450$	0	0
$451$	−3.09785e17	−1.73346
$452$	0	0
$453$	−3.34556e16	−0.181899
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.48379e17	−1.27544	−0.637719	−	0.770269i	$-0.720122\pi$
$457$	−2.48379e17	−1.27544	−0.637719	+	0.770269i	$0.720122\pi$
$458$	0	0
$459$	1.60757e16	0.0802392
$460$	0	0
$461$	2.81438e17	1.36561	0.682806	−	0.730600i	$-0.260760\pi$
$461$	2.81438e17	1.36561	0.682806	+	0.730600i	$0.260760\pi$
$462$	0	0
$463$	−2.56031e16	−0.120786	−0.0603929	−	0.998175i	$-0.519235\pi$
$463$	−2.56031e16	−0.120786	−0.0603929	+	0.998175i	$0.519235\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.67869e17	0.748878	0.374439	−	0.927252i	$-0.377835\pi$
$467$	1.67869e17	0.748878	0.374439	+	0.927252i	$0.377835\pi$
$468$	0	0
$469$	1.49015e16	0.0646555
$470$	0	0
$471$	−1.68545e16	−0.0711343
$472$	0	0
$473$	−8.92383e16	−0.366399
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.38514e17	−0.927139
$478$	0	0
$479$	−3.36107e17	−1.27144	−0.635722	−	0.771918i	$-0.719298\pi$
$479$	−3.36107e17	−1.27144	−0.635722	+	0.771918i	$0.719298\pi$
$480$	0	0
$481$	−6.24742e16	−0.230016
$482$	0	0
$483$	−4.95650e16	−0.177631
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.14667e17	−1.06884	−0.534421	−	0.845219i	$-0.679470\pi$
$487$	−3.14667e17	−1.06884	−0.534421	+	0.845219i	$0.679470\pi$
$488$	0	0
$489$	1.08801e17	0.359855
$490$	0	0
$491$	−5.51204e17	−1.77535	−0.887673	−	0.460475i	$-0.847679\pi$
$491$	−5.51204e17	−1.77535	−0.887673	+	0.460475i	$0.847679\pi$
$492$	0	0
$493$	1.82405e16	0.0572179
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.22229e16	0.0959065
$498$	0	0
$499$	4.01378e17	1.16386	0.581930	−	0.813239i	$-0.302298\pi$
$499$	4.01378e17	1.16386	0.581930	+	0.813239i	$0.302298\pi$
$500$	0	0
$501$	9.33073e16	0.263615
$502$	0	0
$503$	−4.69895e17	−1.29363	−0.646813	−	0.762649i	$-0.723898\pi$
$503$	−4.69895e17	−1.29363	−0.646813	+	0.762649i	$0.723898\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8.79954e16	−0.230096
$508$	0	0
$509$	2.34703e17	0.598210	0.299105	−	0.954220i	$-0.403312\pi$
$509$	2.34703e17	0.598210	0.299105	+	0.954220i	$0.403312\pi$
$510$	0	0
$511$	−3.87286e17	−0.962267
$512$	0	0
$513$	1.86491e15	0.00451748
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.84801e17	−0.886232
$518$	0	0
$519$	8.04134e16	0.180610
$520$	0	0
$521$	7.31281e17	1.60191	0.800957	−	0.598721i	$-0.204324\pi$
$521$	7.31281e17	1.60191	0.800957	+	0.598721i	$0.204324\pi$
$522$	0	0
$523$	−5.20172e17	−1.11144	−0.555720	−	0.831369i	$-0.687557\pi$
$523$	−5.20172e17	−1.11144	−0.555720	+	0.831369i	$0.687557\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.82826e14	−0.00159183
$528$	0	0
$529$	−8.86793e16	−0.175938
$530$	0	0
$531$	−9.40530e17	−1.82078
$532$	0	0
$533$	−2.15684e17	−0.407465
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.19727e17	−0.215456
$538$	0	0
$539$	−2.40910e17	−0.423182
$540$	0	0
$541$	3.98618e17	0.683557	0.341779	−	0.939781i	$-0.388971\pi$
$541$	3.98618e17	0.683557	0.341779	+	0.939781i	$0.388971\pi$
$542$	0	0
$543$	6.61436e16	0.110736
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.26274e17	0.361175	0.180587	−	0.983559i	$-0.442200\pi$
$547$	2.26274e17	0.361175	0.180587	+	0.983559i	$0.442200\pi$
$548$	0	0
$549$	7.24572e17	1.12943
$550$	0	0
$551$	2.11606e15	0.00322138
$552$	0	0
$553$	−5.60139e17	−0.832880
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.43488e15	0.0119680	0.00598398	−	0.999982i	$-0.498095\pi$
$557$	8.43488e15	0.0119680	0.00598398	+	0.999982i	$0.498095\pi$
$558$	0	0
$559$	−6.21309e16	−0.0861254
$560$	0	0
$561$	−3.44850e16	−0.0467059
$562$	0	0
$563$	−1.45126e18	−1.92062	−0.960311	−	0.278932i	$-0.910019\pi$
$563$	−1.45126e18	−1.92062	−0.960311	+	0.278932i	$0.910019\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5.12895e17	−0.648244
$568$	0	0
$569$	−6.31803e17	−0.780462	−0.390231	−	0.920717i	$-0.627605\pi$
$569$	−6.31803e17	−0.780462	−0.390231	+	0.920717i	$0.627605\pi$
$570$	0	0
$571$	3.25857e17	0.393453	0.196726	−	0.980458i	$-0.436969\pi$
$571$	3.25857e17	0.393453	0.196726	+	0.980458i	$0.436969\pi$
$572$	0	0
$573$	3.64207e16	0.0429876
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.94542e17	0.219467	0.109734	−	0.993961i	$-0.465000\pi$
$577$	1.94542e17	0.219467	0.109734	+	0.993961i	$0.465000\pi$
$578$	0	0
$579$	−1.77905e17	−0.196235
$580$	0	0
$581$	1.14531e18	1.23532
$582$	0	0
$583$	1.05670e18	1.11456
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.12947e17	0.618408	0.309204	−	0.950996i	$-0.399937\pi$
$587$	6.12947e17	0.618408	0.309204	+	0.950996i	$0.399937\pi$
$588$	0	0
$589$	−9.08144e13	−8.96201e−5 0
$590$	0	0
$591$	3.24691e17	0.313438
$592$	0	0
$593$	−1.05886e18	−0.999958	−0.499979	−	0.866038i	$-0.666659\pi$
$593$	−1.05886e18	−0.999958	−0.499979	+	0.866038i	$0.666659\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.90810e17	−0.262891
$598$	0	0
$599$	−2.14973e18	−1.90156	−0.950781	−	0.309864i	$-0.899717\pi$
$599$	−2.14973e18	−1.90156	−0.950781	+	0.309864i	$0.899717\pi$
$600$	0	0
$601$	5.94673e17	0.514747	0.257374	−	0.966312i	$-0.417143\pi$
$601$	5.94673e17	0.514747	0.257374	+	0.966312i	$0.417143\pi$
$602$	0	0
$603$	9.06296e16	0.0767728
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.86146e18	−1.51052	−0.755260	−	0.655425i	$-0.772489\pi$
$607$	−1.86146e18	−1.51052	−0.755260	+	0.655425i	$0.772489\pi$
$608$	0	0
$609$	8.42970e16	0.0669576
$610$	0	0
$611$	−2.67912e17	−0.208317
$612$	0	0
$613$	−2.10098e18	−1.59930	−0.799648	−	0.600469i	$-0.794981\pi$
$613$	−2.10098e18	−1.59930	−0.799648	+	0.600469i	$0.794981\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.55090e18	1.86140	0.930699	−	0.365786i	$-0.119200\pi$
$617$	2.55090e18	1.86140	0.930699	+	0.365786i	$0.119200\pi$
$618$	0	0
$619$	1.44223e17	0.103049	0.0515245	−	0.998672i	$-0.483592\pi$
$619$	1.44223e17	0.103049	0.0515245	+	0.998672i	$0.483592\pi$
$620$	0	0
$621$	−6.22573e17	−0.435607
$622$	0	0
$623$	6.37070e17	0.436531
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.00055e15	−0.00262955
$628$	0	0
$629$	2.25204e17	0.144993
$630$	0	0
$631$	−2.06644e18	−1.30326	−0.651632	−	0.758535i	$-0.725915\pi$
$631$	−2.06644e18	−1.30326	−0.651632	+	0.758535i	$0.725915\pi$
$632$	0	0
$633$	4.34630e17	0.268531
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.67730e17	−0.0994728
$638$	0	0
$639$	1.95977e17	0.113881
$640$	0	0
$641$	1.12686e18	0.641643	0.320822	−	0.947140i	$-0.396041\pi$
$641$	1.12686e18	0.641643	0.320822	+	0.947140i	$0.396041\pi$
$642$	0	0
$643$	1.43099e18	0.798480	0.399240	−	0.916846i	$-0.369274\pi$
$643$	1.43099e18	0.798480	0.399240	+	0.916846i	$0.369274\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.22593e18	0.657032	0.328516	−	0.944498i	$-0.393451\pi$
$647$	1.22593e18	0.657032	0.328516	+	0.944498i	$0.393451\pi$
$648$	0	0
$649$	4.16685e18	2.18886
$650$	0	0
$651$	−3.61776e15	−0.00186279
$652$	0	0
$653$	1.58892e18	0.801983	0.400991	−	0.916082i	$-0.368666\pi$
$653$	1.58892e18	0.801983	0.400991	+	0.916082i	$0.368666\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.35544e18	−1.14261
$658$	0	0
$659$	−6.84487e17	−0.325544	−0.162772	−	0.986664i	$-0.552044\pi$
$659$	−6.84487e17	−0.325544	−0.162772	+	0.986664i	$0.552044\pi$
$660$	0	0
$661$	2.02505e18	0.944335	0.472168	−	0.881509i	$-0.343472\pi$
$661$	2.02505e18	0.944335	0.472168	+	0.881509i	$0.343472\pi$
$662$	0	0
$663$	−2.40097e16	−0.0109786
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.06413e17	−0.310628
$668$	0	0
$669$	9.76235e17	0.421003
$670$	0	0
$671$	−3.21009e18	−1.35775
$672$	0	0
$673$	3.93757e18	1.63354	0.816771	−	0.576961i	$-0.195762\pi$
$673$	3.93757e18	1.63354	0.816771	+	0.576961i	$0.195762\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.57479e18	0.628632	0.314316	−	0.949318i	$-0.398225\pi$
$677$	1.57479e18	0.628632	0.314316	+	0.949318i	$0.398225\pi$
$678$	0	0
$679$	−6.67929e17	−0.261563
$680$	0	0
$681$	7.15187e17	0.274766
$682$	0	0
$683$	3.38311e18	1.27521	0.637604	−	0.770364i	$-0.279926\pi$
$683$	3.38311e18	1.27521	0.637604	+	0.770364i	$0.279926\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.97523e17	0.361996
$688$	0	0
$689$	7.35710e17	0.261988
$690$	0	0
$691$	−2.49197e18	−0.870836	−0.435418	−	0.900228i	$-0.643399\pi$
$691$	−2.49197e18	−0.870836	−0.435418	+	0.900228i	$0.643399\pi$
$692$	0	0
$693$	2.44206e18	0.837511
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.77487e17	0.256850
$698$	0	0
$699$	1.37282e18	0.445154
$700$	0	0
$701$	1.91092e18	0.608239	0.304120	−	0.952634i	$-0.401638\pi$
$701$	1.91092e18	0.608239	0.304120	+	0.952634i	$0.401638\pi$
$702$	0	0
$703$	2.61256e16	0.00816314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.44482e18	−1.33853
$708$	0	0
$709$	−1.60149e18	−0.473504	−0.236752	−	0.971570i	$-0.576083\pi$
$709$	−1.60149e18	−0.473504	−0.236752	+	0.971570i	$0.576083\pi$
$710$	0	0
$711$	−3.40672e18	−0.988972
$712$	0	0
$713$	3.03170e16	0.00864181
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.05767e17	−0.0290722
$718$	0	0
$719$	2.60406e18	0.702932	0.351466	−	0.936201i	$-0.385683\pi$
$719$	2.60406e18	0.702932	0.351466	+	0.936201i	$0.385683\pi$
$720$	0	0
$721$	2.51743e18	0.667387
$722$	0	0
$723$	−9.27959e17	−0.241618
$724$	0	0
$725$	0	0
$726$	0	0
$727$	5.98317e18	1.50300	0.751498	−	0.659736i	$-0.229332\pi$
$727$	5.98317e18	1.50300	0.751498	+	0.659736i	$0.229332\pi$
$728$	0	0
$729$	−2.63806e18	−0.650962
$730$	0	0
$731$	2.23967e17	0.0542902
$732$	0	0
$733$	4.80459e18	1.14414	0.572072	−	0.820204i	$-0.306140\pi$
$733$	4.80459e18	1.14414	0.572072	+	0.820204i	$0.306140\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.01519e17	−0.0922927
$738$	0	0
$739$	−5.59789e18	−1.26426	−0.632129	−	0.774863i	$-0.717819\pi$
$739$	−5.59789e18	−1.26426	−0.632129	+	0.774863i	$0.717819\pi$
$740$	0	0
$741$	−2.78533e15	−0.000618099 0
$742$	0	0
$743$	−3.85273e18	−0.840119	−0.420060	−	0.907496i	$-0.637991\pi$
$743$	−3.85273e18	−0.840119	−0.420060	+	0.907496i	$0.637991\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.96571e18	1.46684
$748$	0	0
$749$	−6.91183e18	−1.43041
$750$	0	0
$751$	2.60685e18	0.530221	0.265110	−	0.964218i	$-0.414592\pi$
$751$	2.60685e18	0.530221	0.265110	+	0.964218i	$0.414592\pi$
$752$	0	0
$753$	2.16738e17	0.0433279
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.24876e18	0.241189	0.120594	−	0.992702i	$-0.461520\pi$
$757$	1.24876e18	0.241189	0.120594	+	0.992702i	$0.461520\pi$
$758$	0	0
$759$	1.33552e18	0.253560
$760$	0	0
$761$	2.60181e18	0.485595	0.242798	−	0.970077i	$-0.421935\pi$
$761$	2.60181e18	0.485595	0.242798	+	0.970077i	$0.421935\pi$
$762$	0	0
$763$	3.11603e18	0.571731
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.90112e18	0.514512
$768$	0	0
$769$	−1.46996e18	−0.256322	−0.128161	−	0.991753i	$-0.540907\pi$
$769$	−1.46996e18	−0.256322	−0.128161	+	0.991753i	$0.540907\pi$
$770$	0	0
$771$	9.58707e16	0.0164374
$772$	0	0
$773$	9.45928e18	1.59475	0.797373	−	0.603487i	$-0.206223\pi$
$773$	9.45928e18	1.59475	0.797373	+	0.603487i	$0.206223\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.04076e18	0.169674
$778$	0	0
$779$	9.01951e16	0.0144607
$780$	0	0
$781$	−8.68243e17	−0.136902
$782$	0	0
$783$	1.05883e18	0.164201
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.52285e18	0.228466	0.114233	−	0.993454i	$-0.463559\pi$
$787$	1.52285e18	0.228466	0.114233	+	0.993454i	$0.463559\pi$
$788$	0	0
$789$	1.83285e18	0.270474
$790$	0	0
$791$	−6.15643e18	−0.893680
$792$	0	0
$793$	−2.23498e18	−0.319153
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.94016e18	0.406342	0.203171	−	0.979143i	$-0.434875\pi$
$797$	2.94016e18	0.406342	0.203171	+	0.979143i	$0.434875\pi$
$798$	0	0
$799$	9.65759e17	0.131315
$800$	0	0
$801$	3.87461e18	0.518342
$802$	0	0
$803$	1.04354e19	1.37359
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.59746e17	−0.0458478
$808$	0	0
$809$	4.21346e18	0.528413	0.264207	−	0.964466i	$-0.414890\pi$
$809$	4.21346e18	0.528413	0.264207	+	0.964466i	$0.414890\pi$
$810$	0	0
$811$	9.15267e18	1.12957	0.564784	−	0.825239i	$-0.308960\pi$
$811$	9.15267e18	1.12957	0.564784	+	0.825239i	$0.308960\pi$
$812$	0	0
$813$	−2.83236e18	−0.344002
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.59821e16	0.00305654
$818$	0	0
$819$	1.70025e18	0.196865
$820$	0	0
$821$	−2.02305e18	−0.230556	−0.115278	−	0.993333i	$-0.536776\pi$
$821$	−2.02305e18	−0.230556	−0.115278	+	0.993333i	$0.536776\pi$
$822$	0	0
$823$	8.86487e18	0.994428	0.497214	−	0.867628i	$-0.334356\pi$
$823$	8.86487e18	0.994428	0.497214	+	0.867628i	$0.334356\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.10500e19	−1.20110	−0.600548	−	0.799589i	$-0.705051\pi$
$827$	−1.10500e19	−1.20110	−0.600548	+	0.799589i	$0.705051\pi$
$828$	0	0
$829$	1.20756e19	1.29212	0.646062	−	0.763285i	$-0.276415\pi$
$829$	1.20756e19	1.29212	0.646062	+	0.763285i	$0.276415\pi$
$830$	0	0
$831$	−3.86491e17	−0.0407129
$832$	0	0
$833$	6.04626e17	0.0627039
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.54418e16	−0.00456815
$838$	0	0
$839$	7.15175e18	0.707880	0.353940	−	0.935268i	$-0.384842\pi$
$839$	7.15175e18	0.707880	0.353940	+	0.935268i	$0.384842\pi$
$840$	0	0
$841$	−9.05921e18	−0.882909
$842$	0	0
$843$	3.19931e18	0.307028
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.32372e18	−0.216243
$848$	0	0
$849$	1.62289e18	0.148727
$850$	0	0
$851$	−8.72164e18	−0.787148
$852$	0	0
$853$	−7.60357e18	−0.675848	−0.337924	−	0.941173i	$-0.609725\pi$
$853$	−7.60357e18	−0.675848	−0.337924	+	0.941173i	$0.609725\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.96681e19	−1.69585	−0.847925	−	0.530115i	$-0.822149\pi$
$857$	−1.96681e19	−1.69585	−0.847925	+	0.530115i	$0.822149\pi$
$858$	0	0
$859$	−1.88342e19	−1.59952	−0.799762	−	0.600317i	$-0.795041\pi$
$859$	−1.88342e19	−1.59952	−0.799762	+	0.600317i	$0.795041\pi$
$860$	0	0
$861$	3.59309e18	0.300571
$862$	0	0
$863$	−1.31278e19	−1.08174	−0.540868	−	0.841107i	$-0.681904\pi$
$863$	−1.31278e19	−1.08174	−0.540868	+	0.841107i	$0.681904\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.00888e18	−0.240591
$868$	0	0
$869$	1.50929e19	1.18890
$870$	0	0
$871$	−2.79552e17	−0.0216943
$872$	0	0
$873$	−4.06229e18	−0.310584
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.26889e19	0.941729	0.470864	−	0.882206i	$-0.343942\pi$
$877$	1.26889e19	0.941729	0.470864	+	0.882206i	$0.343942\pi$
$878$	0	0
$879$	5.18685e18	0.379294
$880$	0	0
$881$	−1.42763e19	−1.02866	−0.514329	−	0.857593i	$-0.671959\pi$
$881$	−1.42763e19	−1.02866	−0.514329	+	0.857593i	$0.671959\pi$
$882$	0	0
$883$	8.22615e18	0.584053	0.292026	−	0.956410i	$-0.405670\pi$
$883$	8.22615e18	0.584053	0.292026	+	0.956410i	$0.405670\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.44660e18	0.513398	0.256699	−	0.966491i	$-0.417365\pi$
$887$	7.44660e18	0.513398	0.256699	+	0.966491i	$0.417365\pi$
$888$	0	0
$889$	−6.79513e17	−0.0461675
$890$	0	0
$891$	1.38199e19	0.925337
$892$	0	0
$893$	1.12036e17	0.00739306
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.29840e17	0.0596015
$898$	0	0
$899$	−5.15613e16	−0.00325751
$900$	0	0
$901$	−2.65206e18	−0.165147
$902$	0	0
$903$	1.03504e18	0.0635314
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.15552e19	−0.689176	−0.344588	−	0.938754i	$-0.611981\pi$
$907$	−1.15552e19	−0.689176	−0.344588	+	0.938754i	$0.611981\pi$
$908$	0	0
$909$	−2.70330e19	−1.58939
$910$	0	0
$911$	2.59128e19	1.50191	0.750955	−	0.660353i	$-0.229593\pi$
$911$	2.59128e19	1.50191	0.750955	+	0.660353i	$0.229593\pi$
$912$	0	0
$913$	−3.08604e19	−1.76336
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.73339e18	0.0962710
$918$	0	0
$919$	−3.41319e18	−0.186900	−0.0934500	−	0.995624i	$-0.529790\pi$
$919$	−3.41319e18	−0.186900	−0.0934500	+	0.995624i	$0.529790\pi$
$920$	0	0
$921$	6.55170e18	0.353725
$922$	0	0
$923$	−6.04502e17	−0.0321801
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.53108e19	0.792464
$928$	0	0
$929$	−3.09722e19	−1.58077	−0.790387	−	0.612607i	$-0.790121\pi$
$929$	−3.09722e19	−1.58077	−0.790387	+	0.612607i	$0.790121\pi$
$930$	0	0
$931$	7.01418e16	0.00353024
$932$	0	0
$933$	1.99904e18	0.0992180
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.41201e19	1.16432	0.582158	−	0.813075i	$-0.302208\pi$
$937$	2.41201e19	1.16432	0.582158	+	0.813075i	$0.302208\pi$
$938$	0	0
$939$	5.58014e18	0.265656
$940$	0	0
$941$	−2.75937e19	−1.29562	−0.647809	−	0.761803i	$-0.724314\pi$
$941$	−2.75937e19	−1.29562	−0.647809	+	0.761803i	$0.724314\pi$
$942$	0	0
$943$	−3.01103e19	−1.39440
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.71058e19	−0.770670	−0.385335	−	0.922777i	$-0.625914\pi$
$947$	−1.71058e19	−0.770670	−0.385335	+	0.922777i	$0.625914\pi$
$948$	0	0
$949$	7.26549e18	0.322875
$950$	0	0
$951$	1.06841e19	0.468342
$952$	0	0
$953$	1.17877e19	0.509712	0.254856	−	0.966979i	$-0.417972\pi$
$953$	1.17877e19	0.509712	0.254856	+	0.966979i	$0.417972\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.27137e18	−0.0955787
$958$	0	0
$959$	2.70889e19	1.12453
$960$	0	0
$961$	−2.44153e19	−0.999909
$962$	0	0
$963$	−4.20372e19	−1.69849
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.70890e19	−0.672116	−0.336058	−	0.941841i	$-0.609094\pi$
$967$	−1.70890e19	−0.672116	−0.336058	+	0.941841i	$0.609094\pi$
$968$	0	0
$969$	1.00404e16	0.000389626 0
$970$	0	0
$971$	1.99227e19	0.762822	0.381411	−	0.924406i	$-0.375438\pi$
$971$	1.99227e19	0.762822	0.381411	+	0.924406i	$0.375438\pi$
$972$	0	0
$973$	−3.24725e19	−1.22682
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.06645e19	−0.760170	−0.380085	−	0.924951i	$-0.624105\pi$
$977$	−2.06645e19	−0.760170	−0.380085	+	0.924951i	$0.624105\pi$
$978$	0	0
$979$	−1.71658e19	−0.623127
$980$	0	0
$981$	1.89514e19	0.678881
$982$	0	0
$983$	−4.10331e19	−1.45056	−0.725281	−	0.688453i	$-0.758290\pi$
$983$	−4.10331e19	−1.45056	−0.725281	+	0.688453i	$0.758290\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.46317e18	0.153667
$988$	0	0
$989$	−8.67372e18	−0.294734
$990$	0	0
$991$	2.91021e19	0.975991	0.487995	−	0.872846i	$-0.337728\pi$
$991$	2.91021e19	0.975991	0.487995	+	0.872846i	$0.337728\pi$
$992$	0	0
$993$	−4.53203e18	−0.150011
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.70427e19	−1.19449	−0.597247	−	0.802057i	$-0.703739\pi$
$997$	−3.70427e19	−1.19449	−0.597247	+	0.802057i	$0.703739\pi$
$998$	0	0
$999$	1.30728e19	0.416094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.14.a.l.1.5		10
5.2	odd	4		40.14.c.a.9.10	✓	20
5.3	odd	4		40.14.c.a.9.11	yes	20
5.4	even	2		200.14.a.k.1.6		10
20.3	even	4		80.14.c.d.49.10		20
20.7	even	4		80.14.c.d.49.11		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.14.c.a.9.10	✓	20	5.2	odd	4
40.14.c.a.9.11	yes	20	5.3	odd	4
80.14.c.d.49.10		20	20.3	even	4
80.14.c.d.49.11		20	20.7	even	4
200.14.a.k.1.6		10	5.4	even	2
200.14.a.l.1.5		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)

\(f(q)\)	\(=\)	\(q+312.525 q^{3} -246082. q^{7} -1.49665e6 q^{9} +O(q^{10})\)	\(q+312.525 q^{3} -246082. q^{7} -1.49665e6 q^{9} +6.63065e6 q^{11} +4.61650e6 q^{13} -1.66414e7 q^{17} -1.93054e6 q^{19} -7.69067e7 q^{21} +6.44482e8 q^{23} -9.66006e8 q^{27} -1.09609e9 q^{29} +4.70409e7 q^{31} +2.07224e9 q^{33} -1.35328e10 q^{37} +1.44277e9 q^{39} -4.67201e10 q^{41} -1.34584e10 q^{43} -5.80336e10 q^{47} -3.63327e10 q^{49} -5.20084e9 q^{51} +1.59365e11 q^{53} -6.03342e8 q^{57} +6.28423e11 q^{59} -4.84129e11 q^{61} +3.68299e11 q^{63} -6.05549e10 q^{67} +2.01417e11 q^{69} -1.30944e11 q^{71} +1.57381e12 q^{73} -1.63168e12 q^{77} +2.27623e12 q^{79} +2.08424e12 q^{81} -4.65420e12 q^{83} -3.42557e11 q^{87} -2.58885e12 q^{89} -1.13604e12 q^{91} +1.47014e10 q^{93} +2.71425e12 q^{97} -9.92378e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 1276 q^{3} + 341068 q^{7} + 6557498 q^{9}+O(q^{10}) \) 10 * q + 1276 * q^3 + 341068 * q^7 + 6557498 * q^9	\( 10 q + 1276 q^{3} + 341068 q^{7} + 6557498 q^{9} - 4074536 q^{11} - 7720792 q^{13} + 29092704 q^{17} - 82552936 q^{19} - 140909032 q^{21} + 448872532 q^{23} + 6653011048 q^{27} + 4118753908 q^{29} - 161162304 q^{31} + 6635864336 q^{33} + 37799831624 q^{37} + 2651467216 q^{39} + 31205287876 q^{41} + 48886150956 q^{43} - 54068936100 q^{47} + 313063319346 q^{49} - 179707160640 q^{51} - 116117700520 q^{53} - 863281677424 q^{57} - 658612309176 q^{59} + 405225797804 q^{61} - 169179586580 q^{63} - 291232794812 q^{67} + 1989925462312 q^{69} - 956854398704 q^{71} + 262407355312 q^{73} + 3471232133584 q^{77} - 3176139465248 q^{79} + 3191121792770 q^{81} + 5123747754524 q^{83} + 9347096848952 q^{87} + 453451956924 q^{89} + 6935945759120 q^{91} - 806946503040 q^{93} - 11480198579200 q^{97} + 18677113053976 q^{99}+O(q^{100}) \) 10 * q + 1276 * q^3 + 341068 * q^7 + 6557498 * q^9 - 4074536 * q^11 - 7720792 * q^13 + 29092704 * q^17 - 82552936 * q^19 - 140909032 * q^21 + 448872532 * q^23 + 6653011048 * q^27 + 4118753908 * q^29 - 161162304 * q^31 + 6635864336 * q^33 + 37799831624 * q^37 + 2651467216 * q^39 + 31205287876 * q^41 + 48886150956 * q^43 - 54068936100 * q^47 + 313063319346 * q^49 - 179707160640 * q^51 - 116117700520 * q^53 - 863281677424 * q^57 - 658612309176 * q^59 + 405225797804 * q^61 - 169179586580 * q^63 - 291232794812 * q^67 + 1989925462312 * q^69 - 956854398704 * q^71 + 262407355312 * q^73 + 3471232133584 * q^77 - 3176139465248 * q^79 + 3191121792770 * q^81 + 5123747754524 * q^83 + 9347096848952 * q^87 + 453451956924 * q^89 + 6935945759120 * q^91 - 806946503040 * q^93 - 11480198579200 * q^97 + 18677113053976 * q^99

Introduction

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