LMFDB - Embedded newform 200.14.a.f.1.1

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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7121x^{2} - 128406x + 3057138 $ x^4 - x^3 - 7121x^2 - 128406x + 3057138
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{2}\cdot 5^{2}\cdot 17 $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$90.8559$ 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)

$f(q)$	$=$	$q-2181.88 q^{3} +131313. q^{7} +3.16628e6 q^{9} +O(q^{10})$	$q-2181.88 q^{3} +131313. q^{7} +3.16628e6 q^{9} -8.87378e6 q^{11} -2.01584e7 q^{13} -1.78364e8 q^{17} +2.82799e8 q^{19} -2.86509e8 q^{21} +1.20367e7 q^{23} -3.42983e9 q^{27} +3.37678e9 q^{29} -4.55254e9 q^{31} +1.93615e10 q^{33} +1.43561e10 q^{37} +4.39831e10 q^{39} +2.52626e9 q^{41} -3.27564e10 q^{43} -4.42374e10 q^{47} -7.96460e10 q^{49} +3.89170e11 q^{51} -2.53214e11 q^{53} -6.17033e11 q^{57} -2.82005e11 q^{59} -2.77414e11 q^{61} +4.15773e11 q^{63} -5.64945e11 q^{67} -2.62626e10 q^{69} -1.10430e12 q^{71} -2.30939e12 q^{73} -1.16524e12 q^{77} +1.04667e12 q^{79} +2.43540e12 q^{81} +5.71638e12 q^{83} -7.36774e12 q^{87} -5.08589e12 q^{89} -2.64705e12 q^{91} +9.93309e12 q^{93} -2.63085e12 q^{97} -2.80969e13 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 936 q^{3} - 112424 q^{7} + 2737236 q^{9}+O(q^{10}) $ 4 * q - 936 * q^3 - 112424 * q^7 + 2737236 * q^9	$ 4 q - 936 q^{3} - 112424 q^{7} + 2737236 q^{9} - 85280 q^{11} - 22276696 q^{13} - 92966696 q^{17} - 52003280 q^{19} - 124736832 q^{21} + 498172904 q^{23} - 1984218768 q^{27} + 5868734200 q^{29} + 1204156144 q^{31} + 2413789344 q^{33} + 1227431400 q^{37} + 47183929776 q^{39} + 17822906952 q^{41} - 92192384488 q^{43} - 187731312904 q^{47} + 218812853892 q^{49} + 436920457680 q^{51} - 617748697656 q^{53} - 962043447456 q^{57} + 155305995248 q^{59} + 918726826936 q^{61} - 453684626952 q^{63} - 1172014952696 q^{67} + 2007735561792 q^{69} + 1499341271376 q^{71} - 3669705031176 q^{73} - 2543134824608 q^{77} + 1474589017632 q^{79} + 1358685231876 q^{81} + 10007128523992 q^{83} + 6056521948368 q^{87} - 2840927507160 q^{89} - 16479691138768 q^{91} + 4603403842560 q^{93} - 7763871939496 q^{97} - 50298890417952 q^{99}+O(q^{100}) $ 4 * q - 936 * q^3 - 112424 * q^7 + 2737236 * q^9 - 85280 * q^11 - 22276696 * q^13 - 92966696 * q^17 - 52003280 * q^19 - 124736832 * q^21 + 498172904 * q^23 - 1984218768 * q^27 + 5868734200 * q^29 + 1204156144 * q^31 + 2413789344 * q^33 + 1227431400 * q^37 + 47183929776 * q^39 + 17822906952 * q^41 - 92192384488 * q^43 - 187731312904 * q^47 + 218812853892 * q^49 + 436920457680 * q^51 - 617748697656 * q^53 - 962043447456 * q^57 + 155305995248 * q^59 + 918726826936 * q^61 - 453684626952 * q^63 - 1172014952696 * q^67 + 2007735561792 * q^69 + 1499341271376 * q^71 - 3669705031176 * q^73 - 2543134824608 * q^77 + 1474589017632 * q^79 + 1358685231876 * q^81 + 10007128523992 * q^83 + 6056521948368 * q^87 - 2840927507160 * q^89 - 16479691138768 * q^91 + 4603403842560 * q^93 - 7763871939496 * q^97 - 50298890417952 * 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
$3$	−2181.88	−1.72800	−0.863998	−	0.503494i	$-0.832047\pi$
$3$	−2181.88	−1.72800	−0.863998	+	0.503494i	$0.832047\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	131313.	0.421861	0.210931	−	0.977501i	$-0.432351\pi$
$7$	131313.	0.421861	0.210931	+	0.977501i	$0.432351\pi$
$8$	0	0
$9$	3.16628e6	1.98597
$10$	0	0
$11$	−8.87378e6	−1.51028	−0.755138	−	0.655566i	$-0.772430\pi$
$11$	−8.87378e6	−1.51028	−0.755138	+	0.655566i	$0.772430\pi$
$12$	0	0
$13$	−2.01584e7	−1.15831	−0.579153	−	0.815219i	$-0.696617\pi$
$13$	−2.01584e7	−1.15831	−0.579153	+	0.815219i	$0.696617\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.78364e8	−1.79222	−0.896108	−	0.443836i	$-0.853617\pi$
$17$	−1.78364e8	−1.79222	−0.896108	+	0.443836i	$0.853617\pi$
$18$	0	0
$19$	2.82799e8	1.37905	0.689523	−	0.724264i	$-0.257820\pi$
$19$	2.82799e8	1.37905	0.689523	+	0.724264i	$0.257820\pi$
$20$	0	0
$21$	−2.86509e8	−0.728975
$22$	0	0
$23$	1.20367e7	0.0169541	0.00847707	−	0.999964i	$-0.497302\pi$
$23$	1.20367e7	0.0169541	0.00847707	+	0.999964i	$0.497302\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.42983e9	−1.70376
$28$	0	0
$29$	3.37678e9	1.05418	0.527092	−	0.849808i	$-0.323282\pi$
$29$	3.37678e9	1.05418	0.527092	+	0.849808i	$0.323282\pi$
$30$	0	0
$31$	−4.55254e9	−0.921303	−0.460651	−	0.887581i	$-0.652384\pi$
$31$	−4.55254e9	−0.921303	−0.460651	+	0.887581i	$0.652384\pi$
$32$	0	0
$33$	1.93615e10	2.60975
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.43561e10	0.919869	0.459934	−	0.887953i	$-0.347873\pi$
$37$	1.43561e10	0.919869	0.459934	+	0.887953i	$0.347873\pi$
$38$	0	0
$39$	4.39831e10	2.00155
$40$	0	0
$41$	2.52626e9	0.0830584	0.0415292	−	0.999137i	$-0.486777\pi$
$41$	2.52626e9	0.0830584	0.0415292	+	0.999137i	$0.486777\pi$
$42$	0	0
$43$	−3.27564e10	−0.790227	−0.395113	−	0.918632i	$-0.629295\pi$
$43$	−3.27564e10	−0.790227	−0.395113	+	0.918632i	$0.629295\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.42374e10	−0.598623	−0.299311	−	0.954155i	$-0.596757\pi$
$47$	−4.42374e10	−0.598623	−0.299311	+	0.954155i	$0.596757\pi$
$48$	0	0
$49$	−7.96460e10	−0.822033
$50$	0	0
$51$	3.89170e11	3.09694
$52$	0	0
$53$	−2.53214e11	−1.56926	−0.784630	−	0.619965i	$-0.787147\pi$
$53$	−2.53214e11	−1.56926	−0.784630	+	0.619965i	$0.787147\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.17033e11	−2.38299
$58$	0	0
$59$	−2.82005e11	−0.870398	−0.435199	−	0.900334i	$-0.643322\pi$
$59$	−2.82005e11	−0.870398	−0.435199	+	0.900334i	$0.643322\pi$
$60$	0	0
$61$	−2.77414e11	−0.689421	−0.344711	−	0.938709i	$-0.612023\pi$
$61$	−2.77414e11	−0.689421	−0.344711	+	0.938709i	$0.612023\pi$
$62$	0	0
$63$	4.15773e11	0.837805
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.64945e11	−0.762992	−0.381496	−	0.924370i	$-0.624591\pi$
$67$	−5.64945e11	−0.762992	−0.381496	+	0.924370i	$0.624591\pi$
$68$	0	0
$69$	−2.62626e10	−0.0292967
$70$	0	0
$71$	−1.10430e12	−1.02307	−0.511537	−	0.859261i	$-0.670924\pi$
$71$	−1.10430e12	−1.02307	−0.511537	+	0.859261i	$0.670924\pi$
$72$	0	0
$73$	−2.30939e12	−1.78607	−0.893037	−	0.449983i	$-0.851430\pi$
$73$	−2.30939e12	−1.78607	−0.893037	+	0.449983i	$0.851430\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.16524e12	−0.637127
$78$	0	0
$79$	1.04667e12	0.484435	0.242217	−	0.970222i	$-0.422125\pi$
$79$	1.04667e12	0.484435	0.242217	+	0.970222i	$0.422125\pi$
$80$	0	0
$81$	2.43540e12	0.958117
$82$	0	0
$83$	5.71638e12	1.91917	0.959585	−	0.281420i	$-0.0908053\pi$
$83$	5.71638e12	1.91917	0.959585	+	0.281420i	$0.0908053\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.36774e12	−1.82163
$88$	0	0
$89$	−5.08589e12	−1.08476	−0.542378	−	0.840135i	$-0.682476\pi$
$89$	−5.08589e12	−1.08476	−0.542378	+	0.840135i	$0.682476\pi$
$90$	0	0
$91$	−2.64705e12	−0.488645
$92$	0	0
$93$	9.93309e12	1.59201
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.63085e12	−0.320685	−0.160343	−	0.987061i	$-0.551260\pi$
$97$	−2.63085e12	−0.320685	−0.160343	+	0.987061i	$0.551260\pi$
$98$	0	0
$99$	−2.80969e13	−2.99937
$100$	0	0
$101$	−3.22417e12	−0.302224	−0.151112	−	0.988517i	$-0.548285\pi$
$101$	−3.22417e12	−0.302224	−0.151112	+	0.988517i	$0.548285\pi$
$102$	0	0
$103$	7.70821e12	0.636080	0.318040	−	0.948077i	$-0.396975\pi$
$103$	7.70821e12	0.636080	0.318040	+	0.948077i	$0.396975\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.27624e12	−0.533137	−0.266568	−	0.963816i	$-0.585890\pi$
$107$	−8.27624e12	−0.533137	−0.266568	+	0.963816i	$0.585890\pi$
$108$	0	0
$109$	−2.92699e13	−1.67167	−0.835833	−	0.548983i	$-0.815015\pi$
$109$	−2.92699e13	−1.67167	−0.835833	+	0.548983i	$0.815015\pi$
$110$	0	0
$111$	−3.13234e13	−1.58953
$112$	0	0
$113$	−8.53668e12	−0.385726	−0.192863	−	0.981226i	$-0.561777\pi$
$113$	−8.53668e12	−0.385726	−0.192863	+	0.981226i	$0.561777\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.38271e13	−2.30037
$118$	0	0
$119$	−2.34215e13	−0.756067
$120$	0	0
$121$	4.42213e13	1.28093
$122$	0	0
$123$	−5.51201e12	−0.143525
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.62503e13	1.18960	0.594800	−	0.803874i	$-0.297231\pi$
$127$	5.62503e13	1.18960	0.594800	+	0.803874i	$0.297231\pi$
$128$	0	0
$129$	7.14707e13	1.36551
$130$	0	0
$131$	−1.24755e13	−0.215672	−0.107836	−	0.994169i	$-0.534392\pi$
$131$	−1.24755e13	−0.215672	−0.107836	+	0.994169i	$0.534392\pi$
$132$	0	0
$133$	3.71351e13	0.581766
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.53658e13	−1.10306	−0.551531	−	0.834154i	$-0.685956\pi$
$137$	−8.53658e13	−1.10306	−0.551531	+	0.834154i	$0.685956\pi$
$138$	0	0
$139$	9.20051e13	1.08197	0.540986	−	0.841032i	$-0.318051\pi$
$139$	9.20051e13	1.08197	0.540986	+	0.841032i	$0.318051\pi$
$140$	0	0
$141$	9.65207e13	1.03442
$142$	0	0
$143$	1.78881e14	1.74936
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.73778e14	1.42047
$148$	0	0
$149$	−2.12310e14	−1.58950	−0.794748	−	0.606939i	$-0.792397\pi$
$149$	−2.12310e14	−1.58950	−0.794748	+	0.606939i	$0.792397\pi$
$150$	0	0
$151$	−8.09034e13	−0.555414	−0.277707	−	0.960666i	$-0.589574\pi$
$151$	−8.09034e13	−0.555414	−0.277707	+	0.960666i	$0.589574\pi$
$152$	0	0
$153$	−5.64752e14	−3.55929
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.11230e14	1.65857	0.829285	−	0.558825i	$-0.188748\pi$
$157$	3.11230e14	1.65857	0.829285	+	0.558825i	$0.188748\pi$
$158$	0	0
$159$	5.52483e14	2.71168
$160$	0	0
$161$	1.58057e12	0.00715230
$162$	0	0
$163$	−5.72157e13	−0.238944	−0.119472	−	0.992838i	$-0.538120\pi$
$163$	−5.72157e13	−0.238944	−0.119472	+	0.992838i	$0.538120\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.76892e14	−1.34450	−0.672249	−	0.740326i	$-0.734671\pi$
$167$	−3.76892e14	−1.34450	−0.672249	+	0.740326i	$0.734671\pi$
$168$	0	0
$169$	1.03484e14	0.341673
$170$	0	0
$171$	8.95420e14	2.73875
$172$	0	0
$173$	−1.56783e14	−0.444631	−0.222316	−	0.974975i	$-0.571362\pi$
$173$	−1.56783e14	−0.444631	−0.222316	+	0.974975i	$0.571362\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.15300e14	1.50405
$178$	0	0
$179$	7.07072e13	0.160664	0.0803321	−	0.996768i	$-0.474402\pi$
$179$	7.07072e13	0.160664	0.0803321	+	0.996768i	$0.474402\pi$
$180$	0	0
$181$	−2.08063e14	−0.439830	−0.219915	−	0.975519i	$-0.570578\pi$
$181$	−2.08063e14	−0.439830	−0.219915	+	0.975519i	$0.570578\pi$
$182$	0	0
$183$	6.05285e14	1.19132
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.58277e15	2.70674
$188$	0	0
$189$	−4.50380e14	−0.718750
$190$	0	0
$191$	−7.91854e14	−1.18013	−0.590063	−	0.807358i	$-0.700897\pi$
$191$	−7.91854e14	−1.18013	−0.590063	+	0.807358i	$0.700897\pi$
$192$	0	0
$193$	−2.77110e14	−0.385949	−0.192974	−	0.981204i	$-0.561813\pi$
$193$	−2.77110e14	−0.385949	−0.192974	+	0.981204i	$0.561813\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.01802e15	1.24087	0.620433	−	0.784259i	$-0.286957\pi$
$197$	1.01802e15	1.24087	0.620433	+	0.784259i	$0.286957\pi$
$198$	0	0
$199$	1.91904e14	0.219048	0.109524	−	0.993984i	$-0.465067\pi$
$199$	1.91904e14	0.219048	0.109524	+	0.993984i	$0.465067\pi$
$200$	0	0
$201$	1.23264e15	1.31845
$202$	0	0
$203$	4.43415e14	0.444719
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.81115e13	0.0336705
$208$	0	0
$209$	−2.50949e15	−2.08274
$210$	0	0
$211$	−1.49871e15	−1.16918	−0.584591	−	0.811328i	$-0.698745\pi$
$211$	−1.49871e15	−1.16918	−0.584591	+	0.811328i	$0.698745\pi$
$212$	0	0
$213$	2.40945e15	1.76787
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.97806e14	−0.388662
$218$	0	0
$219$	5.03882e15	3.08633
$220$	0	0
$221$	3.59554e15	2.07594
$222$	0	0
$223$	−4.03033e14	−0.219462	−0.109731	−	0.993961i	$-0.534999\pi$
$223$	−4.03033e14	−0.219462	−0.109731	+	0.993961i	$0.534999\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.68690e13	0.0469912	0.0234956	−	0.999724i	$-0.492520\pi$
$227$	9.68690e13	0.0469912	0.0234956	+	0.999724i	$0.492520\pi$
$228$	0	0
$229$	−2.56706e15	−1.17627	−0.588133	−	0.808764i	$-0.700137\pi$
$229$	−2.56706e15	−1.17627	−0.588133	+	0.808764i	$0.700137\pi$
$230$	0	0
$231$	2.54242e15	1.10095
$232$	0	0
$233$	−4.20841e15	−1.72308	−0.861538	−	0.507694i	$-0.830498\pi$
$233$	−4.20841e15	−1.72308	−0.861538	+	0.507694i	$0.830498\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.28372e15	−0.837101
$238$	0	0
$239$	−5.54453e15	−1.92432	−0.962162	−	0.272477i	$-0.912157\pi$
$239$	−5.54453e15	−1.92432	−0.962162	+	0.272477i	$0.912157\pi$
$240$	0	0
$241$	−3.45459e15	−1.13576	−0.567879	−	0.823112i	$-0.692236\pi$
$241$	−3.45459e15	−1.13576	−0.567879	+	0.823112i	$0.692236\pi$
$242$	0	0
$243$	1.54494e14	0.0481362
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.70075e15	−1.59736
$248$	0	0
$249$	−1.24725e16	−3.31632
$250$	0	0
$251$	6.15729e15	1.55421	0.777107	−	0.629369i	$-0.216687\pi$
$251$	6.15729e15	1.55421	0.777107	+	0.629369i	$0.216687\pi$
$252$	0	0
$253$	−1.06811e14	−0.0256054
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.83079e13	0.0147879	0.00739393	−	0.999973i	$-0.497646\pi$
$257$	6.83079e13	0.0147879	0.00739393	+	0.999973i	$0.497646\pi$
$258$	0	0
$259$	1.88514e15	0.388057
$260$	0	0
$261$	1.06919e16	2.09358
$262$	0	0
$263$	6.50715e15	1.21249	0.606245	−	0.795278i	$-0.292675\pi$
$263$	6.50715e15	1.21249	0.606245	+	0.795278i	$0.292675\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.10968e16	1.87445
$268$	0	0
$269$	9.17224e15	1.47600	0.737999	−	0.674802i	$-0.235771\pi$
$269$	9.17224e15	1.47600	0.737999	+	0.674802i	$0.235771\pi$
$270$	0	0
$271$	6.57894e15	1.00892	0.504458	−	0.863436i	$-0.331692\pi$
$271$	6.57894e15	1.00892	0.504458	+	0.863436i	$0.331692\pi$
$272$	0	0
$273$	5.77555e15	0.844376
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.86197e15	−0.247659	−0.123829	−	0.992304i	$-0.539518\pi$
$277$	−1.86197e15	−0.247659	−0.123829	+	0.992304i	$0.539518\pi$
$278$	0	0
$279$	−1.44146e16	−1.82968
$280$	0	0
$281$	4.92604e15	0.596908	0.298454	−	0.954424i	$-0.403529\pi$
$281$	4.92604e15	0.596908	0.298454	+	0.954424i	$0.403529\pi$
$282$	0	0
$283$	8.33516e15	0.964500	0.482250	−	0.876034i	$-0.339820\pi$
$283$	8.33516e15	0.964500	0.482250	+	0.876034i	$0.339820\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.31731e14	0.0350391
$288$	0	0
$289$	2.19093e16	2.21204
$290$	0	0
$291$	5.74019e15	0.554143
$292$	0	0
$293$	−1.36965e16	−1.26465	−0.632326	−	0.774702i	$-0.717900\pi$
$293$	−1.36965e16	−1.26465	−0.632326	+	0.774702i	$0.717900\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.04356e16	2.57315
$298$	0	0
$299$	−2.42640e14	−0.0196381
$300$	0	0
$301$	−4.30134e15	−0.333366
$302$	0	0
$303$	7.03475e15	0.522242
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.84606e16	1.94019	0.970095	−	0.242726i	$-0.0780416\pi$
$307$	2.84606e16	1.94019	0.970095	+	0.242726i	$0.0780416\pi$
$308$	0	0
$309$	−1.68184e16	−1.09914
$310$	0	0
$311$	−1.30039e16	−0.814949	−0.407474	−	0.913217i	$-0.633590\pi$
$311$	−1.30039e16	−0.814949	−0.407474	+	0.913217i	$0.633590\pi$
$312$	0	0
$313$	3.70874e15	0.222940	0.111470	−	0.993768i	$-0.464444\pi$
$313$	3.70874e15	0.222940	0.111470	+	0.993768i	$0.464444\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.18963e15	0.397944	0.198972	−	0.980005i	$-0.436240\pi$
$317$	7.18963e15	0.397944	0.198972	+	0.980005i	$0.436240\pi$
$318$	0	0
$319$	−2.99648e16	−1.59211
$320$	0	0
$321$	1.80578e16	0.921259
$322$	0	0
$323$	−5.04412e16	−2.47155
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.38635e16	2.88863
$328$	0	0
$329$	−5.80893e15	−0.252536
$330$	0	0
$331$	4.26916e16	1.78427	0.892135	−	0.451769i	$-0.149207\pi$
$331$	4.26916e16	1.78427	0.892135	+	0.451769i	$0.149207\pi$
$332$	0	0
$333$	4.54555e16	1.82683
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.31508e15	−0.0489055	−0.0244528	−	0.999701i	$-0.507784\pi$
$337$	−1.31508e15	−0.0489055	−0.0244528	+	0.999701i	$0.507784\pi$
$338$	0	0
$339$	1.86260e16	0.666533
$340$	0	0
$341$	4.03982e16	1.39142
$342$	0	0
$343$	−2.31813e16	−0.768645
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.94193e15	−0.151969	−0.0759845	−	0.997109i	$-0.524210\pi$
$347$	−4.94193e15	−0.151969	−0.0759845	+	0.997109i	$0.524210\pi$
$348$	0	0
$349$	5.14124e16	1.52301	0.761505	−	0.648159i	$-0.224461\pi$
$349$	5.14124e16	1.52301	0.761505	+	0.648159i	$0.224461\pi$
$350$	0	0
$351$	6.91397e16	1.97347
$352$	0	0
$353$	3.41773e16	0.940159	0.470080	−	0.882624i	$-0.344225\pi$
$353$	3.41773e16	0.940159	0.470080	+	0.882624i	$0.344225\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.11030e16	1.30648
$358$	0	0
$359$	4.46360e16	1.10045	0.550226	−	0.835016i	$-0.314542\pi$
$359$	4.46360e16	1.10045	0.550226	+	0.835016i	$0.314542\pi$
$360$	0	0
$361$	3.79220e16	0.901767
$362$	0	0
$363$	−9.64856e16	−2.21345
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.34260e16	0.714092	0.357046	−	0.934087i	$-0.383784\pi$
$367$	3.34260e16	0.714092	0.357046	+	0.934087i	$0.383784\pi$
$368$	0	0
$369$	7.99887e15	0.164952
$370$	0	0
$371$	−3.32502e16	−0.662010
$372$	0	0
$373$	2.50412e16	0.481446	0.240723	−	0.970594i	$-0.422616\pi$
$373$	2.50412e16	0.481446	0.240723	+	0.970594i	$0.422616\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.80704e16	−1.22107
$378$	0	0
$379$	−5.85231e16	−1.01431	−0.507157	−	0.861854i	$-0.669304\pi$
$379$	−5.85231e16	−1.01431	−0.507157	+	0.861854i	$0.669304\pi$
$380$	0	0
$381$	−1.22732e17	−2.05563
$382$	0	0
$383$	8.06387e16	1.30542	0.652712	−	0.757606i	$-0.273631\pi$
$383$	8.06387e16	1.30542	0.652712	+	0.757606i	$0.273631\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.03716e17	−1.56937
$388$	0	0
$389$	1.84140e16	0.269449	0.134724	−	0.990883i	$-0.456985\pi$
$389$	1.84140e16	0.269449	0.134724	+	0.990883i	$0.456985\pi$
$390$	0	0
$391$	−2.14692e15	−0.0303855
$392$	0	0
$393$	2.72200e16	0.372680
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.03942e16	0.902398	0.451199	−	0.892423i	$-0.350996\pi$
$397$	7.03942e16	0.902398	0.451199	+	0.892423i	$0.350996\pi$
$398$	0	0
$399$	−8.10243e16	−1.00529
$400$	0	0
$401$	6.45416e16	0.775178	0.387589	−	0.921832i	$-0.373308\pi$
$401$	6.45416e16	0.775178	0.387589	+	0.921832i	$0.373308\pi$
$402$	0	0
$403$	9.17717e16	1.06715
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.27393e17	−1.38926
$408$	0	0
$409$	−9.70176e16	−1.02482	−0.512412	−	0.858740i	$-0.671248\pi$
$409$	−9.70176e16	−1.02482	−0.512412	+	0.858740i	$0.671248\pi$
$410$	0	0
$411$	1.86258e17	1.90609
$412$	0	0
$413$	−3.70308e16	−0.367187
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.00744e17	−1.86964
$418$	0	0
$419$	−6.08066e16	−0.548984	−0.274492	−	0.961589i	$-0.588510\pi$
$419$	−6.08066e16	−0.548984	−0.274492	+	0.961589i	$0.588510\pi$
$420$	0	0
$421$	−1.04637e17	−0.915905	−0.457953	−	0.888977i	$-0.651417\pi$
$421$	−1.04637e17	−0.915905	−0.457953	+	0.888977i	$0.651417\pi$
$422$	0	0
$423$	−1.40068e17	−1.18885
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.64280e16	−0.290840
$428$	0	0
$429$	−3.90297e17	−3.02289
$430$	0	0
$431$	−1.11280e17	−0.836206	−0.418103	−	0.908400i	$-0.637305\pi$
$431$	−1.11280e17	−0.836206	−0.418103	+	0.908400i	$0.637305\pi$
$432$	0	0
$433$	−8.39355e16	−0.612032	−0.306016	−	0.952026i	$-0.598996\pi$
$433$	−8.39355e16	−0.612032	−0.306016	+	0.952026i	$0.598996\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.40396e15	0.0233805
$438$	0	0
$439$	5.50274e16	0.366910	0.183455	−	0.983028i	$-0.441272\pi$
$439$	5.50274e16	0.366910	0.183455	+	0.983028i	$0.441272\pi$
$440$	0	0
$441$	−2.52182e17	−1.63254
$442$	0	0
$443$	3.97119e16	0.249630	0.124815	−	0.992180i	$-0.460166\pi$
$443$	3.97119e16	0.249630	0.124815	+	0.992180i	$0.460166\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.63235e17	2.74665
$448$	0	0
$449$	−5.61522e16	−0.323419	−0.161709	−	0.986838i	$-0.551701\pi$
$449$	−5.61522e16	−0.323419	−0.161709	+	0.986838i	$0.551701\pi$
$450$	0	0
$451$	−2.24175e16	−0.125441
$452$	0	0
$453$	1.76522e17	0.959754
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.83163e17	−0.940553	−0.470277	−	0.882519i	$-0.655846\pi$
$457$	−1.83163e17	−0.940553	−0.470277	+	0.882519i	$0.655846\pi$
$458$	0	0
$459$	6.11760e17	3.05350
$460$	0	0
$461$	1.64944e17	0.800352	0.400176	−	0.916438i	$-0.368949\pi$
$461$	1.64944e17	0.800352	0.400176	+	0.916438i	$0.368949\pi$
$462$	0	0
$463$	2.89656e17	1.36649	0.683245	−	0.730190i	$-0.260568\pi$
$463$	2.89656e17	1.36649	0.683245	+	0.730190i	$0.260568\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.92032e16	−0.264111	−0.132055	−	0.991242i	$-0.542158\pi$
$467$	−5.92032e16	−0.264111	−0.132055	+	0.991242i	$0.542158\pi$
$468$	0	0
$469$	−7.41845e16	−0.321877
$470$	0	0
$471$	−6.79067e17	−2.86601
$472$	0	0
$473$	2.90673e17	1.19346
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−8.01747e17	−3.11651
$478$	0	0
$479$	4.47107e17	1.69134	0.845669	−	0.533707i	$-0.179202\pi$
$479$	4.47107e17	1.69134	0.845669	+	0.533707i	$0.179202\pi$
$480$	0	0
$481$	−2.89396e17	−1.06549
$482$	0	0
$483$	−3.44862e15	−0.0123591
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.09392e17	1.05092	0.525462	−	0.850817i	$-0.323892\pi$
$487$	3.09392e17	1.05092	0.525462	+	0.850817i	$0.323892\pi$
$488$	0	0
$489$	1.24838e17	0.412894
$490$	0	0
$491$	−1.28158e16	−0.0412777	−0.0206389	−	0.999787i	$-0.506570\pi$
$491$	−1.28158e16	−0.0412777	−0.0206389	+	0.999787i	$0.506570\pi$
$492$	0	0
$493$	−6.02298e17	−1.88933
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.45008e17	−0.431595
$498$	0	0
$499$	1.11910e17	0.324502	0.162251	−	0.986750i	$-0.448125\pi$
$499$	1.11910e17	0.324502	0.162251	+	0.986750i	$0.448125\pi$
$500$	0	0
$501$	8.22334e17	2.32329
$502$	0	0
$503$	−5.73707e17	−1.57942	−0.789711	−	0.613478i	$-0.789770\pi$
$503$	−5.73707e17	−1.57942	−0.789711	+	0.613478i	$0.789770\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.25791e17	−0.590411
$508$	0	0
$509$	6.29892e17	1.60546	0.802732	−	0.596340i	$-0.203379\pi$
$509$	6.29892e17	1.60546	0.802732	+	0.596340i	$0.203379\pi$
$510$	0	0
$511$	−3.03253e17	−0.753476
$512$	0	0
$513$	−9.69951e17	−2.34956
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.92553e17	0.904086
$518$	0	0
$519$	3.42082e17	0.768321
$520$	0	0
$521$	2.89090e17	0.633268	0.316634	−	0.948548i	$-0.397447\pi$
$521$	2.89090e17	0.633268	0.316634	+	0.948548i	$0.397447\pi$
$522$	0	0
$523$	1.03076e17	0.220240	0.110120	−	0.993918i	$-0.464877\pi$
$523$	1.03076e17	0.220240	0.110120	+	0.993918i	$0.464877\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.12011e17	1.65117
$528$	0	0
$529$	−5.03891e17	−0.999713
$530$	0	0
$531$	−8.92906e17	−1.72859
$532$	0	0
$533$	−5.09253e16	−0.0962071
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.54275e17	−0.277627
$538$	0	0
$539$	7.06761e17	1.24150
$540$	0	0
$541$	−3.84676e17	−0.659649	−0.329824	−	0.944042i	$-0.606990\pi$
$541$	−3.84676e17	−0.659649	−0.329824	+	0.944042i	$0.606990\pi$
$542$	0	0
$543$	4.53969e17	0.760024
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.89793e16	−0.0941418	−0.0470709	−	0.998892i	$-0.514989\pi$
$547$	−5.89793e16	−0.0941418	−0.0470709	+	0.998892i	$0.514989\pi$
$548$	0	0
$549$	−8.78372e17	−1.36917
$550$	0	0
$551$	9.54950e17	1.45377
$552$	0	0
$553$	1.37442e17	0.204364
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.15394e18	−1.63728	−0.818642	−	0.574304i	$-0.805273\pi$
$557$	−1.15394e18	−1.63728	−0.818642	+	0.574304i	$0.805273\pi$
$558$	0	0
$559$	6.60316e17	0.915325
$560$	0	0
$561$	−3.45341e18	−4.67724
$562$	0	0
$563$	−1.11160e18	−1.47111	−0.735555	−	0.677465i	$-0.763078\pi$
$563$	−1.11160e18	−1.47111	−0.735555	+	0.677465i	$0.763078\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.19800e17	0.404192
$568$	0	0
$569$	1.38110e17	0.170607	0.0853033	−	0.996355i	$-0.472814\pi$
$569$	1.38110e17	0.170607	0.0853033	+	0.996355i	$0.472814\pi$
$570$	0	0
$571$	7.10232e17	0.857562	0.428781	−	0.903409i	$-0.358943\pi$
$571$	7.10232e17	0.857562	0.428781	+	0.903409i	$0.358943\pi$
$572$	0	0
$573$	1.72773e18	2.03925
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−9.63075e17	−1.08647	−0.543234	−	0.839581i	$-0.682801\pi$
$577$	−9.63075e17	−1.08647	−0.543234	+	0.839581i	$0.682801\pi$
$578$	0	0
$579$	6.04621e17	0.666918
$580$	0	0
$581$	7.50633e17	0.809623
$582$	0	0
$583$	2.24697e18	2.37001
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.49221e18	−1.50551	−0.752755	−	0.658301i	$-0.771276\pi$
$587$	−1.49221e18	−1.50551	−0.752755	+	0.658301i	$0.771276\pi$
$588$	0	0
$589$	−1.28745e18	−1.27052
$590$	0	0
$591$	−2.22120e18	−2.14421
$592$	0	0
$593$	−7.37046e16	−0.0696048	−0.0348024	−	0.999394i	$-0.511080\pi$
$593$	−7.37046e16	−0.0696048	−0.0348024	+	0.999394i	$0.511080\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.18712e17	−0.378514
$598$	0	0
$599$	−3.24788e17	−0.287293	−0.143647	−	0.989629i	$-0.545883\pi$
$599$	−3.24788e17	−0.287293	−0.143647	+	0.989629i	$0.545883\pi$
$600$	0	0
$601$	−3.36784e17	−0.291520	−0.145760	−	0.989320i	$-0.546563\pi$
$601$	−3.36784e17	−0.291520	−0.145760	+	0.989320i	$0.546563\pi$
$602$	0	0
$603$	−1.78878e18	−1.51528
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.08491e18	−0.880371	−0.440186	−	0.897907i	$-0.645087\pi$
$607$	−1.08491e18	−0.880371	−0.440186	+	0.897907i	$0.645087\pi$
$608$	0	0
$609$	−9.67479e17	−0.768474
$610$	0	0
$611$	8.91753e17	0.693389
$612$	0	0
$613$	7.03949e17	0.535856	0.267928	−	0.963439i	$-0.413661\pi$
$613$	7.03949e17	0.535856	0.267928	+	0.963439i	$0.413661\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.04062e18	−1.48905	−0.744525	−	0.667594i	$-0.767324\pi$
$617$	−2.04062e18	−1.48905	−0.744525	+	0.667594i	$0.767324\pi$
$618$	0	0
$619$	5.07476e17	0.362599	0.181299	−	0.983428i	$-0.441970\pi$
$619$	5.07476e17	0.362599	0.181299	+	0.983428i	$0.441970\pi$
$620$	0	0
$621$	−4.12838e16	−0.0288858
$622$	0	0
$623$	−6.67842e17	−0.457617
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.47541e18	3.59897
$628$	0	0
$629$	−2.56062e18	−1.64860
$630$	0	0
$631$	1.75443e18	1.10649	0.553243	−	0.833020i	$-0.313390\pi$
$631$	1.75443e18	1.10649	0.553243	+	0.833020i	$0.313390\pi$
$632$	0	0
$633$	3.27001e18	2.02034
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.60553e18	0.952166
$638$	0	0
$639$	−3.49652e18	−2.03180
$640$	0	0
$641$	−9.57215e16	−0.0545045	−0.0272523	−	0.999629i	$-0.508676\pi$
$641$	−9.57215e16	−0.0545045	−0.0272523	+	0.999629i	$0.508676\pi$
$642$	0	0
$643$	−1.19310e18	−0.665742	−0.332871	−	0.942972i	$-0.608017\pi$
$643$	−1.19310e18	−0.665742	−0.332871	+	0.942972i	$0.608017\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.61861e18	−1.40344	−0.701719	−	0.712454i	$-0.747584\pi$
$647$	−2.61861e18	−1.40344	−0.701719	+	0.712454i	$0.747584\pi$
$648$	0	0
$649$	2.50245e18	1.31454
$650$	0	0
$651$	1.30434e18	0.671607
$652$	0	0
$653$	1.49031e18	0.752213	0.376107	−	0.926576i	$-0.377263\pi$
$653$	1.49031e18	0.752213	0.376107	+	0.926576i	$0.377263\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.31220e18	−3.54710
$658$	0	0
$659$	−2.17664e18	−1.03522	−0.517610	−	0.855617i	$-0.673178\pi$
$659$	−2.17664e18	−1.03522	−0.517610	+	0.855617i	$0.673178\pi$
$660$	0	0
$661$	2.93295e18	1.36771	0.683857	−	0.729616i	$-0.260301\pi$
$661$	2.93295e18	1.36771	0.683857	+	0.729616i	$0.260301\pi$
$662$	0	0
$663$	−7.84503e18	−3.58721
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.06453e16	0.0178728
$668$	0	0
$669$	8.79369e17	0.379229
$670$	0	0
$671$	2.46171e18	1.04122
$672$	0	0
$673$	2.80337e18	1.16301	0.581503	−	0.813544i	$-0.302465\pi$
$673$	2.80337e18	1.16301	0.581503	+	0.813544i	$0.302465\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.50989e18	1.40109	0.700546	−	0.713607i	$-0.252940\pi$
$677$	3.50989e18	1.40109	0.700546	+	0.713607i	$0.252940\pi$
$678$	0	0
$679$	−3.45464e17	−0.135285
$680$	0	0
$681$	−2.11357e17	−0.0812007
$682$	0	0
$683$	−1.80755e18	−0.681326	−0.340663	−	0.940186i	$-0.610651\pi$
$683$	−1.80755e18	−0.681326	−0.340663	+	0.940186i	$0.610651\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.60102e18	2.03258
$688$	0	0
$689$	5.10438e18	1.81768
$690$	0	0
$691$	−2.45978e18	−0.859584	−0.429792	−	0.902928i	$-0.641413\pi$
$691$	−2.45978e18	−0.859584	−0.429792	+	0.902928i	$0.641413\pi$
$692$	0	0
$693$	−3.68948e18	−1.26532
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.50596e17	−0.148859
$698$	0	0
$699$	9.18224e18	2.97747
$700$	0	0
$701$	−2.58559e18	−0.822987	−0.411493	−	0.911413i	$-0.634993\pi$
$701$	−2.58559e18	−0.822987	−0.411493	+	0.911413i	$0.634993\pi$
$702$	0	0
$703$	4.05989e18	1.26854
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.23374e17	−0.127496
$708$	0	0
$709$	−1.31832e18	−0.389781	−0.194890	−	0.980825i	$-0.562435\pi$
$709$	−1.31832e18	−0.389781	−0.194890	+	0.980825i	$0.562435\pi$
$710$	0	0
$711$	3.31406e18	0.962074
$712$	0	0
$713$	−5.47974e16	−0.0156199
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.20975e19	3.32523
$718$	0	0
$719$	1.19239e18	0.321869	0.160934	−	0.986965i	$-0.448549\pi$
$719$	1.19239e18	0.321869	0.160934	+	0.986965i	$0.448549\pi$
$720$	0	0
$721$	1.01219e18	0.268337
$722$	0	0
$723$	7.53751e18	1.96259
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.66215e18	0.668744	0.334372	−	0.942441i	$-0.391476\pi$
$727$	2.66215e18	0.668744	0.334372	+	0.942441i	$0.391476\pi$
$728$	0	0
$729$	−4.21991e18	−1.04130
$730$	0	0
$731$	5.84259e18	1.41626
$732$	0	0
$733$	2.22829e18	0.530635	0.265317	−	0.964161i	$-0.414523\pi$
$733$	2.22829e18	0.530635	0.265317	+	0.964161i	$0.414523\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.01320e18	1.15233
$738$	0	0
$739$	−5.93406e18	−1.34018	−0.670090	−	0.742280i	$-0.733744\pi$
$739$	−5.93406e18	−1.34018	−0.670090	+	0.742280i	$0.733744\pi$
$740$	0	0
$741$	1.24384e19	2.76023
$742$	0	0
$743$	4.47308e18	0.975393	0.487697	−	0.873013i	$-0.337837\pi$
$743$	4.47308e18	0.975393	0.487697	+	0.873013i	$0.337837\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.80997e19	3.81142
$748$	0	0
$749$	−1.08678e18	−0.224910
$750$	0	0
$751$	−1.45220e18	−0.295371	−0.147685	−	0.989034i	$-0.547182\pi$
$751$	−1.45220e18	−0.295371	−0.147685	+	0.989034i	$0.547182\pi$
$752$	0	0
$753$	−1.34345e19	−2.68568
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7.39859e18	−1.42898	−0.714489	−	0.699647i	$-0.753341\pi$
$757$	−7.39859e18	−1.42898	−0.714489	+	0.699647i	$0.753341\pi$
$758$	0	0
$759$	2.33049e17	0.0442461
$760$	0	0
$761$	−1.37956e18	−0.257479	−0.128739	−	0.991678i	$-0.541093\pi$
$761$	−1.37956e18	−0.257479	−0.128739	+	0.991678i	$0.541093\pi$
$762$	0	0
$763$	−3.84352e18	−0.705211
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.68475e18	1.00819
$768$	0	0
$769$	−3.26667e18	−0.569619	−0.284810	−	0.958584i	$-0.591930\pi$
$769$	−3.26667e18	−0.569619	−0.284810	+	0.958584i	$0.591930\pi$
$770$	0	0
$771$	−1.49040e17	−0.0255534
$772$	0	0
$773$	4.80868e18	0.810698	0.405349	−	0.914162i	$-0.367150\pi$
$773$	4.80868e18	0.810698	0.405349	+	0.914162i	$0.367150\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.11316e18	−0.670561
$778$	0	0
$779$	7.14424e17	0.114541
$780$	0	0
$781$	9.79929e18	1.54512
$782$	0	0
$783$	−1.15818e19	−1.79607
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.55991e18	0.234026	0.117013	−	0.993130i	$-0.462668\pi$
$787$	1.55991e18	0.234026	0.117013	+	0.993130i	$0.462668\pi$
$788$	0	0
$789$	−1.41978e19	−2.09518
$790$	0	0
$791$	−1.12097e18	−0.162723
$792$	0	0
$793$	5.59222e18	0.798561
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.08687e18	−0.426618	−0.213309	−	0.976985i	$-0.568424\pi$
$797$	−3.08687e18	−0.426618	−0.213309	+	0.976985i	$0.568424\pi$
$798$	0	0
$799$	7.89038e18	1.07286
$800$	0	0
$801$	−1.61034e19	−2.15430
$802$	0	0
$803$	2.04931e19	2.69746
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.00127e19	−2.55052
$808$	0	0
$809$	4.14005e18	0.519207	0.259603	−	0.965715i	$-0.416408\pi$
$809$	4.14005e18	0.519207	0.259603	+	0.965715i	$0.416408\pi$
$810$	0	0
$811$	−2.84031e18	−0.350534	−0.175267	−	0.984521i	$-0.556079\pi$
$811$	−2.84031e18	−0.350534	−0.175267	+	0.984521i	$0.556079\pi$
$812$	0	0
$813$	−1.43545e19	−1.74340
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−9.26347e18	−1.08976
$818$	0	0
$819$	−8.38131e18	−0.970435
$820$	0	0
$821$	1.24975e19	1.42427	0.712133	−	0.702045i	$-0.247729\pi$
$821$	1.24975e19	1.42427	0.712133	+	0.702045i	$0.247729\pi$
$822$	0	0
$823$	2.53931e18	0.284851	0.142425	−	0.989806i	$-0.454510\pi$
$823$	2.53931e18	0.284851	0.142425	+	0.989806i	$0.454510\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.16070e18	0.778340	0.389170	−	0.921166i	$-0.372762\pi$
$827$	7.16070e18	0.778340	0.389170	+	0.921166i	$0.372762\pi$
$828$	0	0
$829$	5.34539e18	0.571972	0.285986	−	0.958234i	$-0.407679\pi$
$829$	5.34539e18	0.571972	0.285986	+	0.958234i	$0.407679\pi$
$830$	0	0
$831$	4.06259e18	0.427953
$832$	0	0
$833$	1.42060e19	1.47326
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.56144e19	1.56968
$838$	0	0
$839$	1.56118e19	1.54526	0.772628	−	0.634860i	$-0.218942\pi$
$839$	1.56118e19	1.54526	0.772628	+	0.634860i	$0.218942\pi$
$840$	0	0
$841$	1.14204e18	0.111304
$842$	0	0
$843$	−1.07480e19	−1.03145
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.80682e18	0.540376
$848$	0	0
$849$	−1.81863e19	−1.66665
$850$	0	0
$851$	1.72800e17	0.0155956
$852$	0	0
$853$	1.26340e19	1.12298	0.561491	−	0.827483i	$-0.310228\pi$
$853$	1.26340e19	1.12298	0.561491	+	0.827483i	$0.310228\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.12954e19	−0.973931	−0.486965	−	0.873421i	$-0.661896\pi$
$857$	−1.12954e19	−0.973931	−0.486965	+	0.873421i	$0.661896\pi$
$858$	0	0
$859$	−1.54999e19	−1.31636	−0.658178	−	0.752862i	$-0.728673\pi$
$859$	−1.54999e19	−1.31636	−0.658178	+	0.752862i	$0.728673\pi$
$860$	0	0
$861$	−7.23797e17	−0.0605475
$862$	0	0
$863$	−1.51357e18	−0.124719	−0.0623595	−	0.998054i	$-0.519863\pi$
$863$	−1.51357e18	−0.124719	−0.0623595	+	0.998054i	$0.519863\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.78035e19	−3.82240
$868$	0	0
$869$	−9.28795e18	−0.731630
$870$	0	0
$871$	1.13884e19	0.883779
$872$	0	0
$873$	−8.33000e18	−0.636873
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.17715e18	0.458449	0.229225	−	0.973374i	$-0.426381\pi$
$877$	6.17715e18	0.458449	0.229225	+	0.973374i	$0.426381\pi$
$878$	0	0
$879$	2.98842e19	2.18532
$880$	0	0
$881$	−1.45977e19	−1.05182	−0.525908	−	0.850541i	$-0.676274\pi$
$881$	−1.45977e19	−1.05182	−0.525908	+	0.850541i	$0.676274\pi$
$882$	0	0
$883$	3.89337e18	0.276427	0.138214	−	0.990402i	$-0.455864\pi$
$883$	3.89337e18	0.276427	0.138214	+	0.990402i	$0.455864\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.23861e18	−0.499059	−0.249529	−	0.968367i	$-0.580276\pi$
$887$	−7.23861e18	−0.499059	−0.249529	+	0.968367i	$0.580276\pi$
$888$	0	0
$889$	7.38639e18	0.501846
$890$	0	0
$891$	−2.16112e19	−1.44702
$892$	0	0
$893$	−1.25103e19	−0.825528
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.29411e17	0.0339346
$898$	0	0
$899$	−1.53729e19	−0.971223
$900$	0	0
$901$	4.51644e19	2.81245
$902$	0	0
$903$	9.38501e18	0.576056
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.59817e19	−0.953179	−0.476590	−	0.879126i	$-0.658127\pi$
$907$	−1.59817e19	−0.953179	−0.476590	+	0.879126i	$0.658127\pi$
$908$	0	0
$909$	−1.02086e19	−0.600208
$910$	0	0
$911$	−9.90005e17	−0.0573810	−0.0286905	−	0.999588i	$-0.509134\pi$
$911$	−9.90005e17	−0.0573810	−0.0286905	+	0.999588i	$0.509134\pi$
$912$	0	0
$913$	−5.07259e19	−2.89848
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.63819e18	−0.0909836
$918$	0	0
$919$	−1.19291e19	−0.653217	−0.326609	−	0.945160i	$-0.605906\pi$
$919$	−1.19291e19	−0.653217	−0.326609	+	0.945160i	$0.605906\pi$
$920$	0	0
$921$	−6.20976e19	−3.35264
$922$	0	0
$923$	2.22608e19	1.18503
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.44064e19	1.26324
$928$	0	0
$929$	−1.59991e19	−0.816571	−0.408285	−	0.912854i	$-0.633873\pi$
$929$	−1.59991e19	−0.816571	−0.408285	+	0.912854i	$0.633873\pi$
$930$	0	0
$931$	−2.25238e19	−1.13362
$932$	0	0
$933$	2.83729e19	1.40823
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.20830e19	−1.06598	−0.532992	−	0.846120i	$-0.678932\pi$
$937$	−2.20830e19	−1.06598	−0.532992	+	0.846120i	$0.678932\pi$
$938$	0	0
$939$	−8.09204e18	−0.385240
$940$	0	0
$941$	−1.29867e18	−0.0609768	−0.0304884	−	0.999535i	$-0.509706\pi$
$941$	−1.29867e18	−0.0609768	−0.0304884	+	0.999535i	$0.509706\pi$
$942$	0	0
$943$	3.04078e16	0.00140818
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.02789e19	−0.463098	−0.231549	−	0.972823i	$-0.574379\pi$
$947$	−1.02789e19	−0.463098	−0.231549	+	0.972823i	$0.574379\pi$
$948$	0	0
$949$	4.65536e19	2.06882
$950$	0	0
$951$	−1.56869e19	−0.687645
$952$	0	0
$953$	3.89146e19	1.68271	0.841355	−	0.540483i	$-0.181759\pi$
$953$	3.89146e19	1.68271	0.841355	+	0.540483i	$0.181759\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.53797e19	2.75116
$958$	0	0
$959$	−1.12096e19	−0.465339
$960$	0	0
$961$	−3.69196e18	−0.151201
$962$	0	0
$963$	−2.62049e19	−1.05880
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.25024e19	0.885026	0.442513	−	0.896762i	$-0.354087\pi$
$967$	2.25024e19	0.885026	0.442513	+	0.896762i	$0.354087\pi$
$968$	0	0
$969$	1.10057e20	4.27083
$970$	0	0
$971$	9.28282e18	0.355431	0.177715	−	0.984082i	$-0.443129\pi$
$971$	9.28282e18	0.355431	0.177715	+	0.984082i	$0.443129\pi$
$972$	0	0
$973$	1.20814e19	0.456442
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.97250e19	0.725607	0.362804	−	0.931866i	$-0.381820\pi$
$977$	1.97250e19	0.725607	0.362804	+	0.931866i	$0.381820\pi$
$978$	0	0
$979$	4.51311e19	1.63828
$980$	0	0
$981$	−9.26769e19	−3.31988
$982$	0	0
$983$	−2.27864e19	−0.805522	−0.402761	−	0.915305i	$-0.631949\pi$
$983$	−2.27864e19	−0.805522	−0.402761	+	0.915305i	$0.631949\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.26744e19	0.436381
$988$	0	0
$989$	−3.94279e17	−0.0133976
$990$	0	0
$991$	9.73004e18	0.326314	0.163157	−	0.986600i	$-0.447832\pi$
$991$	9.73004e18	0.326314	0.163157	+	0.986600i	$0.447832\pi$
$992$	0	0
$993$	−9.31480e19	−3.08321
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.39785e18	0.0773220	0.0386610	−	0.999252i	$-0.487691\pi$
$997$	2.39785e18	0.0773220	0.0386610	+	0.999252i	$0.487691\pi$
$998$	0	0
$999$	−4.92391e19	−1.56723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.14.a.f.1.1		4
5.2	odd	4		200.14.c.f.49.8		8
5.3	odd	4		200.14.c.f.49.1		8
5.4	even	2		40.14.a.d.1.4	✓	4
20.19	odd	2		80.14.a.k.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.14.a.d.1.4	✓	4	5.4	even	2
80.14.a.k.1.1		4	20.19	odd	2
200.14.a.f.1.1		4	1.1	even	1	trivial
200.14.c.f.49.1		8	5.3	odd	4
200.14.c.f.49.8		8	5.2	odd	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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)

\(f(q)\)	\(=\)	\(q-2181.88 q^{3} +131313. q^{7} +3.16628e6 q^{9} +O(q^{10})\)	\(q-2181.88 q^{3} +131313. q^{7} +3.16628e6 q^{9} -8.87378e6 q^{11} -2.01584e7 q^{13} -1.78364e8 q^{17} +2.82799e8 q^{19} -2.86509e8 q^{21} +1.20367e7 q^{23} -3.42983e9 q^{27} +3.37678e9 q^{29} -4.55254e9 q^{31} +1.93615e10 q^{33} +1.43561e10 q^{37} +4.39831e10 q^{39} +2.52626e9 q^{41} -3.27564e10 q^{43} -4.42374e10 q^{47} -7.96460e10 q^{49} +3.89170e11 q^{51} -2.53214e11 q^{53} -6.17033e11 q^{57} -2.82005e11 q^{59} -2.77414e11 q^{61} +4.15773e11 q^{63} -5.64945e11 q^{67} -2.62626e10 q^{69} -1.10430e12 q^{71} -2.30939e12 q^{73} -1.16524e12 q^{77} +1.04667e12 q^{79} +2.43540e12 q^{81} +5.71638e12 q^{83} -7.36774e12 q^{87} -5.08589e12 q^{89} -2.64705e12 q^{91} +9.93309e12 q^{93} -2.63085e12 q^{97} -2.80969e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 936 q^{3} - 112424 q^{7} + 2737236 q^{9}+O(q^{10}) \) 4 * q - 936 * q^3 - 112424 * q^7 + 2737236 * q^9	\( 4 q - 936 q^{3} - 112424 q^{7} + 2737236 q^{9} - 85280 q^{11} - 22276696 q^{13} - 92966696 q^{17} - 52003280 q^{19} - 124736832 q^{21} + 498172904 q^{23} - 1984218768 q^{27} + 5868734200 q^{29} + 1204156144 q^{31} + 2413789344 q^{33} + 1227431400 q^{37} + 47183929776 q^{39} + 17822906952 q^{41} - 92192384488 q^{43} - 187731312904 q^{47} + 218812853892 q^{49} + 436920457680 q^{51} - 617748697656 q^{53} - 962043447456 q^{57} + 155305995248 q^{59} + 918726826936 q^{61} - 453684626952 q^{63} - 1172014952696 q^{67} + 2007735561792 q^{69} + 1499341271376 q^{71} - 3669705031176 q^{73} - 2543134824608 q^{77} + 1474589017632 q^{79} + 1358685231876 q^{81} + 10007128523992 q^{83} + 6056521948368 q^{87} - 2840927507160 q^{89} - 16479691138768 q^{91} + 4603403842560 q^{93} - 7763871939496 q^{97} - 50298890417952 q^{99}+O(q^{100}) \) 4 * q - 936 * q^3 - 112424 * q^7 + 2737236 * q^9 - 85280 * q^11 - 22276696 * q^13 - 92966696 * q^17 - 52003280 * q^19 - 124736832 * q^21 + 498172904 * q^23 - 1984218768 * q^27 + 5868734200 * q^29 + 1204156144 * q^31 + 2413789344 * q^33 + 1227431400 * q^37 + 47183929776 * q^39 + 17822906952 * q^41 - 92192384488 * q^43 - 187731312904 * q^47 + 218812853892 * q^49 + 436920457680 * q^51 - 617748697656 * q^53 - 962043447456 * q^57 + 155305995248 * q^59 + 918726826936 * q^61 - 453684626952 * q^63 - 1172014952696 * q^67 + 2007735561792 * q^69 + 1499341271376 * q^71 - 3669705031176 * q^73 - 2543134824608 * q^77 + 1474589017632 * q^79 + 1358685231876 * q^81 + 10007128523992 * q^83 + 6056521948368 * q^87 - 2840927507160 * q^89 - 16479691138768 * q^91 + 4603403842560 * q^93 - 7763871939496 * q^97 - 50298890417952 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more