LMFDB - Embedded newform 80.20.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,20,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	80.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:	$183.053357245$
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} - 2x^{3} - 214954323x^{2} - 341671644076x + 8077617181385444 $ x^4 - 2x^3 - 214954323x^2 - 341671644076*x + 8077617181385444
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-9914.47$ of defining polynomial
Character	$\chi$	$=$	80.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+40429.9 q^{3} +1.95312e6 q^{5} -2.02472e8 q^{7} +4.72312e8 q^{9} +O(q^{10})$	$q+40429.9 q^{3} +1.95312e6 q^{5} -2.02472e8 q^{7} +4.72312e8 q^{9} -4.72798e9 q^{11} -4.99657e10 q^{13} +7.89646e10 q^{15} -6.49790e10 q^{17} +5.24975e11 q^{19} -8.18591e12 q^{21} -1.87788e12 q^{23} +3.81470e12 q^{25} -2.78945e13 q^{27} +7.67450e13 q^{29} -3.34473e13 q^{31} -1.91152e14 q^{33} -3.95453e14 q^{35} -2.32717e14 q^{37} -2.02010e15 q^{39} -8.39662e14 q^{41} +6.08205e15 q^{43} +9.22485e14 q^{45} -7.49620e15 q^{47} +2.95959e16 q^{49} -2.62709e15 q^{51} +4.53022e16 q^{53} -9.23434e15 q^{55} +2.12247e16 q^{57} +5.12413e16 q^{59} +1.23195e17 q^{61} -9.56299e16 q^{63} -9.75892e16 q^{65} -3.13637e17 q^{67} -7.59225e16 q^{69} +2.04136e17 q^{71} +6.98158e17 q^{73} +1.54228e17 q^{75} +9.57283e17 q^{77} -1.01295e18 q^{79} -1.67672e18 q^{81} +3.26963e18 q^{83} -1.26912e17 q^{85} +3.10279e18 q^{87} +1.84193e18 q^{89} +1.01166e19 q^{91} -1.35227e18 q^{93} +1.02534e18 q^{95} +8.70387e18 q^{97} -2.23308e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3080 q^{3} + 7812500 q^{5} - 148222040 q^{7} + 2231864116 q^{9}+O(q^{10}) $ 4 * q + 3080 * q^3 + 7812500 * q^5 - 148222040 * q^7 + 2231864116 * q^9	$ 4 q + 3080 q^{3} + 7812500 q^{5} - 148222040 q^{7} + 2231864116 q^{9} + 2334973920 q^{11} + 57423224120 q^{13} + 6015625000 q^{15} - 465961763160 q^{17} + 1624818160624 q^{19} + 5400395639744 q^{21} + 10101669670680 q^{23} + 15258789062500 q^{25} - 56910507730480 q^{27} - 21802908180264 q^{29} + 69517593805936 q^{31} - 204641262341280 q^{33} - 289496171875000 q^{35} + 21\!\cdots\!60 q^{37}+ \cdots + 27\!\cdots\!80 q^{99}+O(q^{100}) $ 4 * q + 3080 * q^3 + 7812500 * q^5 - 148222040 * q^7 + 2231864116 * q^9 + 2334973920 * q^11 + 57423224120 * q^13 + 6015625000 * q^15 - 465961763160 * q^17 + 1624818160624 * q^19 + 5400395639744 * q^21 + 10101669670680 * q^23 + 15258789062500 * q^25 - 56910507730480 * q^27 - 21802908180264 * q^29 + 69517593805936 * q^31 - 204641262341280 * q^33 - 289496171875000 * q^35 + 2156700568331960 * q^37 - 2385459176165456 * q^39 + 1367657291239944 * q^41 - 2895469359165560 * q^43 + 4359109601562500 * q^45 - 1740670005708600 * q^47 + 53534973320692932 * q^49 - 62580957868768368 * q^51 + 84253756978305240 * q^53 + 4560495937500000 * q^55 + 188957630874753440 * q^57 - 152838143056415952 * q^59 + 211425306591350168 * q^61 - 417796572994057400 * q^63 + 112154734609375000 * q^65 - 656919981274544360 * q^67 + 93524018475363648 * q^69 - 883122322400344752 * q^71 + 325226727237880520 * q^73 + 11749267578125000 * q^75 - 620955288586656480 * q^77 + 850256855296398112 * q^79 - 3380709519213288764 * q^81 + 3602925283549230600 * q^83 - 910081568671875000 * q^85 + 8179923434985534000 * q^87 + 418548299964421224 * q^89 + 10309921756152683504 * q^91 - 14102167255291233280 * q^93 + 3173472969968750000 * q^95 - 12135517460163032920 * q^97 + 27941442307774944480 * 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
$3$	40429.9	1.18591	0.592953	−	0.805237i	$-0.297962\pi$
$3$	40429.9	1.18591	0.592953	+	0.805237i	$0.297962\pi$
$4$	0	0
$5$	1.95312e6	0.447214
$5$	1.95312e6	0.447214
$6$	0	0
$7$	−2.02472e8	−1.89641	−0.948207	−	0.317653i	$-0.897105\pi$
$7$	−2.02472e8	−1.89641	−0.948207	+	0.317653i	$0.897105\pi$
$8$	0	0
$9$	4.72312e8	0.406374
$10$	0	0
$11$	−4.72798e9	−0.604568	−0.302284	−	0.953218i	$-0.597749\pi$
$11$	−4.72798e9	−0.604568	−0.302284	+	0.953218i	$0.597749\pi$
$12$	0	0
$13$	−4.99657e10	−1.30680	−0.653401	−	0.757012i	$-0.726658\pi$
$13$	−4.99657e10	−1.30680	−0.653401	+	0.757012i	$0.726658\pi$
$14$	0	0
$15$	7.89646e10	0.530353
$16$	0	0
$17$	−6.49790e10	−0.132895	−0.0664474	−	0.997790i	$-0.521166\pi$
$17$	−6.49790e10	−0.132895	−0.0664474	+	0.997790i	$0.521166\pi$
$18$	0	0
$19$	5.24975e11	0.373232	0.186616	−	0.982433i	$-0.440248\pi$
$19$	5.24975e11	0.373232	0.186616	+	0.982433i	$0.440248\pi$
$20$	0	0
$21$	−8.18591e12	−2.24897
$22$	0	0
$23$	−1.87788e12	−0.217397	−0.108699	−	0.994075i	$-0.534668\pi$
$23$	−1.87788e12	−0.217397	−0.108699	+	0.994075i	$0.534668\pi$
$24$	0	0
$25$	3.81470e12	0.200000
$26$	0	0
$27$	−2.78945e13	−0.703985
$28$	0	0
$29$	7.67450e13	0.982356	0.491178	−	0.871059i	$-0.336566\pi$
$29$	7.67450e13	0.982356	0.491178	+	0.871059i	$0.336566\pi$
$30$	0	0
$31$	−3.34473e13	−0.227209	−0.113604	−	0.993526i	$-0.536240\pi$
$31$	−3.34473e13	−0.227209	−0.113604	+	0.993526i	$0.536240\pi$
$32$	0	0
$33$	−1.91152e14	−0.716961
$34$	0	0
$35$	−3.95453e14	−0.848102
$36$	0	0
$37$	−2.32717e14	−0.294383	−0.147191	−	0.989108i	$-0.547023\pi$
$37$	−2.32717e14	−0.294383	−0.147191	+	0.989108i	$0.547023\pi$
$38$	0	0
$39$	−2.02010e15	−1.54974
$40$	0	0
$41$	−8.39662e14	−0.400551	−0.200276	−	0.979740i	$-0.564184\pi$
$41$	−8.39662e14	−0.400551	−0.200276	+	0.979740i	$0.564184\pi$
$42$	0	0
$43$	6.08205e15	1.84544	0.922719	−	0.385473i	$-0.125962\pi$
$43$	6.08205e15	1.84544	0.922719	+	0.385473i	$0.125962\pi$
$44$	0	0
$45$	9.22485e14	0.181736
$46$	0	0
$47$	−7.49620e15	−0.977039	−0.488519	−	0.872553i	$-0.662463\pi$
$47$	−7.49620e15	−0.977039	−0.488519	+	0.872553i	$0.662463\pi$
$48$	0	0
$49$	2.95959e16	2.59639
$50$	0	0
$51$	−2.62709e15	−0.157601
$52$	0	0
$53$	4.53022e16	1.88581	0.942906	−	0.333060i	$-0.108081\pi$
$53$	4.53022e16	1.88581	0.942906	+	0.333060i	$0.108081\pi$
$54$	0	0
$55$	−9.23434e15	−0.270371
$56$	0	0
$57$	2.12247e16	0.442618
$58$	0	0
$59$	5.12413e16	0.770063	0.385031	−	0.922903i	$-0.374191\pi$
$59$	5.12413e16	0.770063	0.385031	+	0.922903i	$0.374191\pi$
$60$	0	0
$61$	1.23195e17	1.34883	0.674417	−	0.738351i	$-0.264395\pi$
$61$	1.23195e17	1.34883	0.674417	+	0.738351i	$0.264395\pi$
$62$	0	0
$63$	−9.56299e16	−0.770653
$64$	0	0
$65$	−9.75892e16	−0.584420
$66$	0	0
$67$	−3.13637e17	−1.40837	−0.704186	−	0.710016i	$-0.748688\pi$
$67$	−3.13637e17	−1.40837	−0.704186	+	0.710016i	$0.748688\pi$
$68$	0	0
$69$	−7.59225e16	−0.257813
$70$	0	0
$71$	2.04136e17	0.528402	0.264201	−	0.964468i	$-0.414892\pi$
$71$	2.04136e17	0.528402	0.264201	+	0.964468i	$0.414892\pi$
$72$	0	0
$73$	6.98158e17	1.38799	0.693995	−	0.719980i	$-0.255849\pi$
$73$	6.98158e17	1.38799	0.693995	+	0.719980i	$0.255849\pi$
$74$	0	0
$75$	1.54228e17	0.237181
$76$	0	0
$77$	9.57283e17	1.14651
$78$	0	0
$79$	−1.01295e18	−0.950890	−0.475445	−	0.879746i	$-0.657713\pi$
$79$	−1.01295e18	−0.950890	−0.475445	+	0.879746i	$0.657713\pi$
$80$	0	0
$81$	−1.67672e18	−1.24123
$82$	0	0
$83$	3.26963e18	1.91980	0.959900	−	0.280341i	$-0.0904477\pi$
$83$	3.26963e18	1.91980	0.959900	+	0.280341i	$0.0904477\pi$
$84$	0	0
$85$	−1.26912e17	−0.0594324
$86$	0	0
$87$	3.10279e18	1.16498
$88$	0	0
$89$	1.84193e18	0.557272	0.278636	−	0.960397i	$-0.410118\pi$
$89$	1.84193e18	0.557272	0.278636	+	0.960397i	$0.410118\pi$
$90$	0	0
$91$	1.01166e19	2.47824
$92$	0	0
$93$	−1.35227e18	−0.269448
$94$	0	0
$95$	1.02534e18	0.166915
$96$	0	0
$97$	8.70387e18	1.16247	0.581234	−	0.813737i	$-0.302570\pi$
$97$	8.70387e18	1.16247	0.581234	+	0.813737i	$0.302570\pi$
$98$	0	0
$99$	−2.23308e18	−0.245681
$100$	0	0
$101$	−1.27369e19	−1.15880	−0.579401	−	0.815042i	$-0.696714\pi$
$101$	−1.27369e19	−1.15880	−0.579401	+	0.815042i	$0.696714\pi$
$102$	0	0
$103$	8.23029e17	0.0621529	0.0310765	−	0.999517i	$-0.490106\pi$
$103$	8.23029e17	0.0621529	0.0310765	+	0.999517i	$0.490106\pi$
$104$	0	0
$105$	−1.59881e19	−1.00577
$106$	0	0
$107$	−3.57154e19	−1.87806	−0.939030	−	0.343835i	$-0.888274\pi$
$107$	−3.57154e19	−1.87806	−0.939030	+	0.343835i	$0.888274\pi$
$108$	0	0
$109$	−1.49228e19	−0.658111	−0.329055	−	0.944311i	$-0.606730\pi$
$109$	−1.49228e19	−0.658111	−0.329055	+	0.944311i	$0.606730\pi$
$110$	0	0
$111$	−9.40873e18	−0.349110
$112$	0	0
$113$	4.96684e19	1.55537	0.777687	−	0.628652i	$-0.216393\pi$
$113$	4.96684e19	1.55537	0.777687	+	0.628652i	$0.216393\pi$
$114$	0	0
$115$	−3.66774e18	−0.0972229
$116$	0	0
$117$	−2.35994e19	−0.531050
$118$	0	0
$119$	1.31564e19	0.252024
$120$	0	0
$121$	−3.88053e19	−0.634497
$122$	0	0
$123$	−3.39474e19	−0.475016
$124$	0	0
$125$	7.45058e18	0.0894427
$126$	0	0
$127$	−5.66085e19	−0.584449	−0.292224	−	0.956350i	$-0.594395\pi$
$127$	−5.66085e19	−0.584449	−0.292224	+	0.956350i	$0.594395\pi$
$128$	0	0
$129$	2.45896e20	2.18852
$130$	0	0
$131$	−1.94458e20	−1.49537	−0.747683	−	0.664056i	$-0.768834\pi$
$131$	−1.94458e20	−1.49537	−0.747683	+	0.664056i	$0.768834\pi$
$132$	0	0
$133$	−1.06293e20	−0.707803
$134$	0	0
$135$	−5.44815e19	−0.314832
$136$	0	0
$137$	1.24972e20	0.628009	0.314004	−	0.949422i	$-0.398329\pi$
$137$	1.24972e20	0.628009	0.314004	+	0.949422i	$0.398329\pi$
$138$	0	0
$139$	1.53410e19	0.0671756	0.0335878	−	0.999436i	$-0.489307\pi$
$139$	1.53410e19	0.0671756	0.0335878	+	0.999436i	$0.489307\pi$
$140$	0	0
$141$	−3.03070e20	−1.15868
$142$	0	0
$143$	2.36237e20	0.790051
$144$	0	0
$145$	1.49893e20	0.439323
$146$	0	0
$147$	1.19656e21	3.07907
$148$	0	0
$149$	−5.46064e20	−1.23587	−0.617937	−	0.786228i	$-0.712031\pi$
$149$	−5.46064e20	−1.23587	−0.617937	+	0.786228i	$0.712031\pi$
$150$	0	0
$151$	−7.82983e20	−1.56125	−0.780623	−	0.625002i	$-0.785098\pi$
$151$	−7.82983e20	−1.56125	−0.780623	+	0.625002i	$0.785098\pi$
$152$	0	0
$153$	−3.06904e19	−0.0540049
$154$	0	0
$155$	−6.53268e19	−0.101611
$156$	0	0
$157$	3.48872e20	0.480418	0.240209	−	0.970721i	$-0.422784\pi$
$157$	3.48872e20	0.480418	0.240209	+	0.970721i	$0.422784\pi$
$158$	0	0
$159$	1.83156e21	2.23640
$160$	0	0
$161$	3.80218e20	0.412275
$162$	0	0
$163$	−6.36403e20	−0.613692	−0.306846	−	0.951759i	$-0.599274\pi$
$163$	−6.36403e20	−0.613692	−0.306846	+	0.951759i	$0.599274\pi$
$164$	0	0
$165$	−3.73343e20	−0.320635
$166$	0	0
$167$	1.77809e21	1.36190	0.680952	−	0.732328i	$-0.261566\pi$
$167$	1.77809e21	1.36190	0.680952	+	0.732328i	$0.261566\pi$
$168$	0	0
$169$	1.03465e21	0.707731
$170$	0	0
$171$	2.47952e20	0.151672
$172$	0	0
$173$	2.52510e21	1.38306	0.691530	−	0.722347i	$-0.256937\pi$
$173$	2.52510e21	1.38306	0.691530	+	0.722347i	$0.256937\pi$
$174$	0	0
$175$	−7.72369e20	−0.379283
$176$	0	0
$177$	2.07168e21	0.913222
$178$	0	0
$179$	−6.75892e20	−0.267777	−0.133888	−	0.990996i	$-0.542746\pi$
$179$	−6.75892e20	−0.267777	−0.133888	+	0.990996i	$0.542746\pi$
$180$	0	0
$181$	−1.62859e21	−0.580584	−0.290292	−	0.956938i	$-0.593752\pi$
$181$	−1.62859e21	−0.580584	−0.290292	+	0.956938i	$0.593752\pi$
$182$	0	0
$183$	4.98074e21	1.59959
$184$	0	0
$185$	−4.54526e20	−0.131652
$186$	0	0
$187$	3.07219e20	0.0803440
$188$	0	0
$189$	5.64786e21	1.33505
$190$	0	0
$191$	−1.52150e21	−0.325428	−0.162714	−	0.986673i	$-0.552025\pi$
$191$	−1.52150e21	−0.325428	−0.162714	+	0.986673i	$0.552025\pi$
$192$	0	0
$193$	2.73472e21	0.529808	0.264904	−	0.964275i	$-0.414660\pi$
$193$	2.73472e21	0.529808	0.264904	+	0.964275i	$0.414660\pi$
$194$	0	0
$195$	−3.94552e21	−0.693067
$196$	0	0
$197$	8.06279e21	1.28545	0.642727	−	0.766095i	$-0.277803\pi$
$197$	8.06279e21	1.28545	0.642727	+	0.766095i	$0.277803\pi$
$198$	0	0
$199$	−5.71359e21	−0.827570	−0.413785	−	0.910375i	$-0.635793\pi$
$199$	−5.71359e21	−0.827570	−0.413785	+	0.910375i	$0.635793\pi$
$200$	0	0
$201$	−1.26803e22	−1.67020
$202$	0	0
$203$	−1.55387e22	−1.86295
$204$	0	0
$205$	−1.63996e21	−0.179132
$206$	0	0
$207$	−8.86947e20	−0.0883445
$208$	0	0
$209$	−2.48207e21	−0.225644
$210$	0	0
$211$	2.08299e22	1.72983	0.864917	−	0.501916i	$-0.167371\pi$
$211$	2.08299e22	1.72983	0.864917	+	0.501916i	$0.167371\pi$
$212$	0	0
$213$	8.25317e21	0.626635
$214$	0	0
$215$	1.18790e22	0.825305
$216$	0	0
$217$	6.77214e21	0.430882
$218$	0	0
$219$	2.82264e22	1.64603
$220$	0	0
$221$	3.24672e21	0.173667
$222$	0	0
$223$	6.53506e21	0.320888	0.160444	−	0.987045i	$-0.448707\pi$
$223$	6.53506e21	0.320888	0.160444	+	0.987045i	$0.448707\pi$
$224$	0	0
$225$	1.80173e21	0.0812747
$226$	0	0
$227$	3.40997e22	1.41418	0.707090	−	0.707124i	$-0.250008\pi$
$227$	3.40997e22	1.41418	0.707090	+	0.707124i	$0.250008\pi$
$228$	0	0
$229$	−5.03033e21	−0.191937	−0.0959686	−	0.995384i	$-0.530595\pi$
$229$	−5.03033e21	−0.191937	−0.0959686	+	0.995384i	$0.530595\pi$
$230$	0	0
$231$	3.87028e22	1.35966
$232$	0	0
$233$	2.99059e22	0.968000	0.484000	−	0.875068i	$-0.339183\pi$
$233$	2.99059e22	0.968000	0.484000	+	0.875068i	$0.339183\pi$
$234$	0	0
$235$	−1.46410e22	−0.436945
$236$	0	0
$237$	−4.09534e22	−1.12767
$238$	0	0
$239$	3.56580e22	0.906520	0.453260	−	0.891378i	$-0.350261\pi$
$239$	3.56580e22	0.906520	0.453260	+	0.891378i	$0.350261\pi$
$240$	0	0
$241$	2.21671e22	0.520652	0.260326	−	0.965521i	$-0.416170\pi$
$241$	2.21671e22	0.520652	0.260326	+	0.965521i	$0.416170\pi$
$242$	0	0
$243$	−3.53689e22	−0.768002
$244$	0	0
$245$	5.78045e22	1.16114
$246$	0	0
$247$	−2.62307e22	−0.487741
$248$	0	0
$249$	1.32191e23	2.27670
$250$	0	0
$251$	6.40819e22	1.02290	0.511452	−	0.859312i	$-0.329108\pi$
$251$	6.40819e22	1.02290	0.511452	+	0.859312i	$0.329108\pi$
$252$	0	0
$253$	8.87859e21	0.131431
$254$	0	0
$255$	−5.13104e21	−0.0704812
$256$	0	0
$257$	−8.41382e22	−1.07307	−0.536535	−	0.843878i	$-0.680267\pi$
$257$	−8.41382e22	−1.07307	−0.536535	+	0.843878i	$0.680267\pi$
$258$	0	0
$259$	4.71187e22	0.558272
$260$	0	0
$261$	3.62476e22	0.399204
$262$	0	0
$263$	−3.25249e22	−0.333147	−0.166574	−	0.986029i	$-0.553270\pi$
$263$	−3.25249e22	−0.333147	−0.166574	+	0.986029i	$0.553270\pi$
$264$	0	0
$265$	8.84808e22	0.843361
$266$	0	0
$267$	7.44688e22	0.660872
$268$	0	0
$269$	−9.90607e22	−0.818945	−0.409473	−	0.912322i	$-0.634287\pi$
$269$	−9.90607e22	−0.818945	−0.409473	+	0.912322i	$0.634287\pi$
$270$	0	0
$271$	8.52768e22	0.657087	0.328543	−	0.944489i	$-0.393442\pi$
$271$	8.52768e22	0.657087	0.328543	+	0.944489i	$0.393442\pi$
$272$	0	0
$273$	4.09014e23	2.93896
$274$	0	0
$275$	−1.80358e22	−0.120914
$276$	0	0
$277$	−4.24843e22	−0.265871	−0.132935	−	0.991125i	$-0.542440\pi$
$277$	−4.24843e22	−0.265871	−0.132935	+	0.991125i	$0.542440\pi$
$278$	0	0
$279$	−1.57976e22	−0.0923316
$280$	0	0
$281$	2.75771e23	1.50605	0.753025	−	0.657992i	$-0.228594\pi$
$281$	2.75771e23	1.50605	0.753025	+	0.657992i	$0.228594\pi$
$282$	0	0
$283$	−3.19999e23	−1.63372	−0.816860	−	0.576836i	$-0.804287\pi$
$283$	−3.19999e23	−1.63372	−0.816860	+	0.576836i	$0.804287\pi$
$284$	0	0
$285$	4.14544e22	0.197945
$286$	0	0
$287$	1.70008e23	0.759611
$288$	0	0
$289$	−2.34850e23	−0.982339
$290$	0	0
$291$	3.51896e23	1.37858
$292$	0	0
$293$	−6.04992e22	−0.222079	−0.111039	−	0.993816i	$-0.535418\pi$
$293$	−6.04992e22	−0.222079	−0.111039	+	0.993816i	$0.535418\pi$
$294$	0	0
$295$	1.00081e23	0.344383
$296$	0	0
$297$	1.31885e23	0.425607
$298$	0	0
$299$	9.38296e22	0.284095
$300$	0	0
$301$	−1.23144e24	−3.49971
$302$	0	0
$303$	−5.14950e23	−1.37423
$304$	0	0
$305$	2.40614e23	0.603217
$306$	0	0
$307$	−6.02744e22	−0.142010	−0.0710049	−	0.997476i	$-0.522621\pi$
$307$	−6.02744e22	−0.142010	−0.0710049	+	0.997476i	$0.522621\pi$
$308$	0	0
$309$	3.32750e22	0.0737076
$310$	0	0
$311$	7.10121e23	1.47948	0.739739	−	0.672894i	$-0.234949\pi$
$311$	7.10121e23	1.47948	0.739739	+	0.672894i	$0.234949\pi$
$312$	0	0
$313$	5.03304e22	0.0986641	0.0493321	−	0.998782i	$-0.484291\pi$
$313$	5.03304e22	0.0986641	0.0493321	+	0.998782i	$0.484291\pi$
$314$	0	0
$315$	−1.86777e23	−0.344646
$316$	0	0
$317$	−8.46304e23	−1.47049	−0.735247	−	0.677799i	$-0.762934\pi$
$317$	−8.46304e23	−1.47049	−0.735247	+	0.677799i	$0.762934\pi$
$318$	0	0
$319$	−3.62849e23	−0.593901
$320$	0	0
$321$	−1.44397e24	−2.22720
$322$	0	0
$323$	−3.41123e22	−0.0496006
$324$	0	0
$325$	−1.90604e23	−0.261360
$326$	0	0
$327$	−6.03327e23	−0.780458
$328$	0	0
$329$	1.51777e24	1.85287
$330$	0	0
$331$	−1.12699e24	−1.29883	−0.649415	−	0.760434i	$-0.724986\pi$
$331$	−1.12699e24	−1.29883	−0.649415	+	0.760434i	$0.724986\pi$
$332$	0	0
$333$	−1.09915e23	−0.119629
$334$	0	0
$335$	−6.12573e23	−0.629843
$336$	0	0
$337$	8.82894e23	0.857876	0.428938	−	0.903334i	$-0.358888\pi$
$337$	8.82894e23	0.857876	0.428938	+	0.903334i	$0.358888\pi$
$338$	0	0
$339$	2.00809e24	1.84453
$340$	0	0
$341$	1.58138e23	0.137363
$342$	0	0
$343$	−3.68439e24	−3.02741
$344$	0	0
$345$	−1.48286e23	−0.115297
$346$	0	0
$347$	1.57915e24	1.16223	0.581117	−	0.813820i	$-0.302616\pi$
$347$	1.57915e24	1.16223	0.581117	+	0.813820i	$0.302616\pi$
$348$	0	0
$349$	1.61428e24	1.12496	0.562481	−	0.826810i	$-0.309847\pi$
$349$	1.61428e24	1.12496	0.562481	+	0.826810i	$0.309847\pi$
$350$	0	0
$351$	1.39377e24	0.919969
$352$	0	0
$353$	1.27385e24	0.796634	0.398317	−	0.917248i	$-0.369594\pi$
$353$	1.27385e24	0.796634	0.398317	+	0.917248i	$0.369594\pi$
$354$	0	0
$355$	3.98702e23	0.236309
$356$	0	0
$357$	5.31912e23	0.298876
$358$	0	0
$359$	3.63003e24	1.93425	0.967126	−	0.254296i	$-0.0818438\pi$
$359$	3.63003e24	1.93425	0.967126	+	0.254296i	$0.0818438\pi$
$360$	0	0
$361$	−1.70282e24	−0.860698
$362$	0	0
$363$	−1.56889e24	−0.752454
$364$	0	0
$365$	1.36359e24	0.620728
$366$	0	0
$367$	−6.44799e23	−0.278674	−0.139337	−	0.990245i	$-0.544497\pi$
$367$	−6.44799e23	−0.278674	−0.139337	+	0.990245i	$0.544497\pi$
$368$	0	0
$369$	−3.96583e23	−0.162773
$370$	0	0
$371$	−9.17241e24	−3.57628
$372$	0	0
$373$	−3.12964e24	−1.15947	−0.579736	−	0.814805i	$-0.696844\pi$
$373$	−3.12964e24	−1.15947	−0.579736	+	0.814805i	$0.696844\pi$
$374$	0	0
$375$	3.01226e23	0.106071
$376$	0	0
$377$	−3.83461e24	−1.28374
$378$	0	0
$379$	−2.60010e24	−0.827785	−0.413892	−	0.910326i	$-0.635831\pi$
$379$	−2.60010e24	−0.827785	−0.413892	+	0.910326i	$0.635831\pi$
$380$	0	0
$381$	−2.28867e24	−0.693101
$382$	0	0
$383$	5.74210e23	0.165456	0.0827279	−	0.996572i	$-0.473637\pi$
$383$	5.74210e23	0.165456	0.0827279	+	0.996572i	$0.473637\pi$
$384$	0	0
$385$	1.86969e24	0.512736
$386$	0	0
$387$	2.87263e24	0.749937
$388$	0	0
$389$	−2.58485e24	−0.642560	−0.321280	−	0.946984i	$-0.604113\pi$
$389$	−2.58485e24	−0.642560	−0.321280	+	0.946984i	$0.604113\pi$
$390$	0	0
$391$	1.22023e23	0.0288909
$392$	0	0
$393$	−7.86189e24	−1.77336
$394$	0	0
$395$	−1.97842e24	−0.425251
$396$	0	0
$397$	−1.42273e24	−0.291482	−0.145741	−	0.989323i	$-0.546557\pi$
$397$	−1.42273e24	−0.291482	−0.145741	+	0.989323i	$0.546557\pi$
$398$	0	0
$399$	−4.29739e24	−0.839388
$400$	0	0
$401$	−5.17650e23	−0.0964195	−0.0482098	−	0.998837i	$-0.515352\pi$
$401$	−5.17650e23	−0.0964195	−0.0482098	+	0.998837i	$0.515352\pi$
$402$	0	0
$403$	1.67122e24	0.296917
$404$	0	0
$405$	−3.27485e24	−0.555097
$406$	0	0
$407$	1.10028e24	0.177974
$408$	0	0
$409$	2.03074e24	0.313533	0.156766	−	0.987636i	$-0.449893\pi$
$409$	2.03074e24	0.313533	0.156766	+	0.987636i	$0.449893\pi$
$410$	0	0
$411$	5.05258e24	0.744760
$412$	0	0
$413$	−1.03749e25	−1.46036
$414$	0	0
$415$	6.38599e24	0.858561
$416$	0	0
$417$	6.20233e23	0.0796640
$418$	0	0
$419$	1.27162e25	1.56072	0.780359	−	0.625332i	$-0.215036\pi$
$419$	1.27162e25	1.56072	0.780359	+	0.625332i	$0.215036\pi$
$420$	0	0
$421$	4.78299e24	0.561073	0.280536	−	0.959843i	$-0.409488\pi$
$421$	4.78299e24	0.561073	0.280536	+	0.959843i	$0.409488\pi$
$422$	0	0
$423$	−3.54055e24	−0.397043
$424$	0	0
$425$	−2.47875e23	−0.0265790
$426$	0	0
$427$	−2.49434e25	−2.55795
$428$	0	0
$429$	9.55102e24	0.936926
$430$	0	0
$431$	−1.11637e25	−1.04779	−0.523894	−	0.851784i	$-0.675521\pi$
$431$	−1.11637e25	−1.04779	−0.523894	+	0.851784i	$0.675521\pi$
$432$	0	0
$433$	1.35368e25	1.21585	0.607926	−	0.793994i	$-0.292002\pi$
$433$	1.35368e25	1.21585	0.607926	+	0.793994i	$0.292002\pi$
$434$	0	0
$435$	6.06013e24	0.520996
$436$	0	0
$437$	−9.85840e23	−0.0811396
$438$	0	0
$439$	−3.88685e24	−0.306327	−0.153163	−	0.988201i	$-0.548946\pi$
$439$	−3.88685e24	−0.306327	−0.153163	+	0.988201i	$0.548946\pi$
$440$	0	0
$441$	1.39785e25	1.05510
$442$	0	0
$443$	−7.62006e23	−0.0550964	−0.0275482	−	0.999620i	$-0.508770\pi$
$443$	−7.62006e23	−0.0550964	−0.0275482	+	0.999620i	$0.508770\pi$
$444$	0	0
$445$	3.59751e24	0.249220
$446$	0	0
$447$	−2.20773e25	−1.46563
$448$	0	0
$449$	−2.98183e24	−0.189733	−0.0948663	−	0.995490i	$-0.530242\pi$
$449$	−2.98183e24	−0.189733	−0.0948663	+	0.995490i	$0.530242\pi$
$450$	0	0
$451$	3.96991e24	0.242160
$452$	0	0
$453$	−3.16559e25	−1.85149
$454$	0	0
$455$	1.97591e25	1.10830
$456$	0	0
$457$	1.94645e25	1.04723	0.523613	−	0.851956i	$-0.324584\pi$
$457$	1.94645e25	1.04723	0.523613	+	0.851956i	$0.324584\pi$
$458$	0	0
$459$	1.81256e24	0.0935560
$460$	0	0
$461$	1.67921e25	0.831659	0.415829	−	0.909443i	$-0.363491\pi$
$461$	1.67921e25	0.831659	0.415829	+	0.909443i	$0.363491\pi$
$462$	0	0
$463$	2.86320e25	1.36092	0.680460	−	0.732786i	$-0.261780\pi$
$463$	2.86320e25	1.36092	0.680460	+	0.732786i	$0.261780\pi$
$464$	0	0
$465$	−2.64115e24	−0.120501
$466$	0	0
$467$	1.88200e25	0.824347	0.412173	−	0.911105i	$-0.364770\pi$
$467$	1.88200e25	0.824347	0.412173	+	0.911105i	$0.364770\pi$
$468$	0	0
$469$	6.35027e25	2.67085
$470$	0	0
$471$	1.41049e25	0.569731
$472$	0	0
$473$	−2.87558e25	−1.11569
$474$	0	0
$475$	2.00262e24	0.0746464
$476$	0	0
$477$	2.13968e25	0.766344
$478$	0	0
$479$	1.65870e24	0.0570925	0.0285463	−	0.999592i	$-0.490912\pi$
$479$	1.65870e24	0.0570925	0.0285463	+	0.999592i	$0.490912\pi$
$480$	0	0
$481$	1.16279e25	0.384700
$482$	0	0
$483$	1.53722e25	0.488919
$484$	0	0
$485$	1.69997e25	0.519871
$486$	0	0
$487$	2.17283e25	0.639000	0.319500	−	0.947586i	$-0.396485\pi$
$487$	2.17283e25	0.639000	0.319500	+	0.947586i	$0.396485\pi$
$488$	0	0
$489$	−2.57297e25	−0.727781
$490$	0	0
$491$	1.11783e25	0.304161	0.152080	−	0.988368i	$-0.451403\pi$
$491$	1.11783e25	0.304161	0.152080	+	0.988368i	$0.451403\pi$
$492$	0	0
$493$	−4.98681e24	−0.130550
$494$	0	0
$495$	−4.36149e24	−0.109872
$496$	0	0
$497$	−4.13317e25	−1.00207
$498$	0	0
$499$	5.48869e25	1.28089	0.640447	−	0.768002i	$-0.278749\pi$
$499$	5.48869e25	1.28089	0.640447	+	0.768002i	$0.278749\pi$
$500$	0	0
$501$	7.18878e25	1.61509
$502$	0	0
$503$	5.35349e25	1.15809	0.579043	−	0.815297i	$-0.303426\pi$
$503$	5.35349e25	1.15809	0.579043	+	0.815297i	$0.303426\pi$
$504$	0	0
$505$	−2.48767e25	−0.518232
$506$	0	0
$507$	4.18306e25	0.839303
$508$	0	0
$509$	4.94476e24	0.0955710	0.0477855	−	0.998858i	$-0.484784\pi$
$509$	4.94476e24	0.0955710	0.0477855	+	0.998858i	$0.484784\pi$
$510$	0	0
$511$	−1.41357e26	−2.63220
$512$	0	0
$513$	−1.46439e25	−0.262750
$514$	0	0
$515$	1.60748e24	0.0277956
$516$	0	0
$517$	3.54419e25	0.590687
$518$	0	0
$519$	1.02090e26	1.64018
$520$	0	0
$521$	−5.22958e25	−0.810044	−0.405022	−	0.914307i	$-0.632736\pi$
$521$	−5.22958e25	−0.810044	−0.405022	+	0.914307i	$0.632736\pi$
$522$	0	0
$523$	−1.96180e25	−0.293015	−0.146507	−	0.989210i	$-0.546803\pi$
$523$	−1.96180e25	−0.293015	−0.146507	+	0.989210i	$0.546803\pi$
$524$	0	0
$525$	−3.12268e25	−0.449794
$526$	0	0
$527$	2.17337e24	0.0301949
$528$	0	0
$529$	−7.10890e25	−0.952738
$530$	0	0
$531$	2.42019e25	0.312933
$532$	0	0
$533$	4.19543e25	0.523441
$534$	0	0
$535$	−6.97566e25	−0.839894
$536$	0	0
$537$	−2.73262e25	−0.317558
$538$	0	0
$539$	−1.39929e26	−1.56969
$540$	0	0
$541$	1.70215e25	0.184342	0.0921712	−	0.995743i	$-0.470619\pi$
$541$	1.70215e25	0.184342	0.0921712	+	0.995743i	$0.470619\pi$
$542$	0	0
$543$	−6.58436e25	−0.688518
$544$	0	0
$545$	−2.91461e25	−0.294316
$546$	0	0
$547$	−6.45974e25	−0.629993	−0.314996	−	0.949093i	$-0.602003\pi$
$547$	−6.45974e25	−0.629993	−0.314996	+	0.949093i	$0.602003\pi$
$548$	0	0
$549$	5.81863e25	0.548130
$550$	0	0
$551$	4.02892e25	0.366647
$552$	0	0
$553$	2.05094e26	1.80328
$554$	0	0
$555$	−1.83764e25	−0.156127
$556$	0	0
$557$	1.78032e26	1.46175	0.730874	−	0.682512i	$-0.239113\pi$
$557$	1.78032e26	1.46175	0.730874	+	0.682512i	$0.239113\pi$
$558$	0	0
$559$	−3.03894e26	−2.41162
$560$	0	0
$561$	1.24208e25	0.0952804
$562$	0	0
$563$	1.96395e26	1.45646	0.728232	−	0.685331i	$-0.240342\pi$
$563$	1.96395e26	1.45646	0.728232	+	0.685331i	$0.240342\pi$
$564$	0	0
$565$	9.70085e25	0.695584
$566$	0	0
$567$	3.39489e26	2.35389
$568$	0	0
$569$	−7.29393e25	−0.489097	−0.244548	−	0.969637i	$-0.578640\pi$
$569$	−7.29393e25	−0.489097	−0.244548	+	0.969637i	$0.578640\pi$
$570$	0	0
$571$	−2.67931e25	−0.173772	−0.0868860	−	0.996218i	$-0.527692\pi$
$571$	−2.67931e25	−0.173772	−0.0868860	+	0.996218i	$0.527692\pi$
$572$	0	0
$573$	−6.15140e25	−0.385927
$574$	0	0
$575$	−7.16355e24	−0.0434794
$576$	0	0
$577$	1.55173e26	0.911267	0.455633	−	0.890168i	$-0.349413\pi$
$577$	1.55173e26	0.911267	0.455633	+	0.890168i	$0.349413\pi$
$578$	0	0
$579$	1.10564e26	0.628303
$580$	0	0
$581$	−6.62007e26	−3.64074
$582$	0	0
$583$	−2.14188e26	−1.14010
$584$	0	0
$585$	−4.60926e25	−0.237493
$586$	0	0
$587$	−1.66648e26	−0.831263	−0.415632	−	0.909533i	$-0.636439\pi$
$587$	−1.66648e26	−0.831263	−0.415632	+	0.909533i	$0.636439\pi$
$588$	0	0
$589$	−1.75590e25	−0.0848016
$590$	0	0
$591$	3.25978e26	1.52443
$592$	0	0
$593$	2.97683e26	1.34814	0.674068	−	0.738669i	$-0.264545\pi$
$593$	2.97683e26	1.34814	0.674068	+	0.738669i	$0.264545\pi$
$594$	0	0
$595$	2.56961e25	0.112708
$596$	0	0
$597$	−2.31000e26	−0.981420
$598$	0	0
$599$	−2.21228e24	−0.00910513	−0.00455256	−	0.999990i	$-0.501449\pi$
$599$	−2.21228e24	−0.00910513	−0.00455256	+	0.999990i	$0.501449\pi$
$600$	0	0
$601$	1.55205e26	0.618869	0.309434	−	0.950921i	$-0.399860\pi$
$601$	1.55205e26	0.618869	0.309434	+	0.950921i	$0.399860\pi$
$602$	0	0
$603$	−1.48135e26	−0.572325
$604$	0	0
$605$	−7.57915e25	−0.283756
$606$	0	0
$607$	−1.55704e26	−0.564945	−0.282473	−	0.959275i	$-0.591155\pi$
$607$	−1.55704e26	−0.564945	−0.282473	+	0.959275i	$0.591155\pi$
$608$	0	0
$609$	−6.28227e26	−2.20929
$610$	0	0
$611$	3.74553e26	1.27680
$612$	0	0
$613$	−4.55136e25	−0.150407	−0.0752033	−	0.997168i	$-0.523961\pi$
$613$	−4.55136e25	−0.150407	−0.0752033	+	0.997168i	$0.523961\pi$
$614$	0	0
$615$	−6.63036e25	−0.212434
$616$	0	0
$617$	−7.00378e25	−0.217582	−0.108791	−	0.994065i	$-0.534698\pi$
$617$	−7.00378e25	−0.217582	−0.108791	+	0.994065i	$0.534698\pi$
$618$	0	0
$619$	−1.28674e26	−0.387642	−0.193821	−	0.981037i	$-0.562088\pi$
$619$	−1.28674e26	−0.387642	−0.193821	+	0.981037i	$0.562088\pi$
$620$	0	0
$621$	5.23827e25	0.153044
$622$	0	0
$623$	−3.72938e26	−1.05682
$624$	0	0
$625$	1.45519e25	0.0400000
$626$	0	0
$627$	−1.00350e26	−0.267593
$628$	0	0
$629$	1.51217e25	0.0391219
$630$	0	0
$631$	2.70030e25	0.0677850	0.0338925	−	0.999425i	$-0.489210\pi$
$631$	2.70030e25	0.0677850	0.0338925	+	0.999425i	$0.489210\pi$
$632$	0	0
$633$	8.42151e26	2.05142
$634$	0	0
$635$	−1.10563e26	−0.261373
$636$	0	0
$637$	−1.47878e27	−3.39296
$638$	0	0
$639$	9.64158e25	0.214729
$640$	0	0
$641$	8.06107e25	0.174277	0.0871387	−	0.996196i	$-0.472228\pi$
$641$	8.06107e25	0.174277	0.0871387	+	0.996196i	$0.472228\pi$
$642$	0	0
$643$	2.71625e25	0.0570119	0.0285059	−	0.999594i	$-0.490925\pi$
$643$	2.71625e25	0.0570119	0.0285059	+	0.999594i	$0.490925\pi$
$644$	0	0
$645$	4.80266e26	0.978734
$646$	0	0
$647$	−1.60882e25	−0.0318360	−0.0159180	−	0.999873i	$-0.505067\pi$
$647$	−1.60882e25	−0.0318360	−0.0159180	+	0.999873i	$0.505067\pi$
$648$	0	0
$649$	−2.42268e26	−0.465556
$650$	0	0
$651$	2.73797e26	0.510985
$652$	0	0
$653$	−5.43708e26	−0.985576	−0.492788	−	0.870149i	$-0.664022\pi$
$653$	−5.43708e26	−0.985576	−0.492788	+	0.870149i	$0.664022\pi$
$654$	0	0
$655$	−3.79800e26	−0.668748
$656$	0	0
$657$	3.29749e26	0.564043
$658$	0	0
$659$	−8.97093e26	−1.49082	−0.745412	−	0.666604i	$-0.767747\pi$
$659$	−8.97093e26	−1.49082	−0.745412	+	0.666604i	$0.767747\pi$
$660$	0	0
$661$	1.51613e26	0.244807	0.122403	−	0.992480i	$-0.460940\pi$
$661$	1.51613e26	0.244807	0.122403	+	0.992480i	$0.460940\pi$
$662$	0	0
$663$	1.31264e26	0.205953
$664$	0	0
$665$	−2.07603e26	−0.316539
$666$	0	0
$667$	−1.44118e26	−0.213561
$668$	0	0
$669$	2.64211e26	0.380543
$670$	0	0
$671$	−5.82462e26	−0.815462
$672$	0	0
$673$	8.13858e25	0.110766	0.0553829	−	0.998465i	$-0.482362\pi$
$673$	8.13858e25	0.110766	0.0553829	+	0.998465i	$0.482362\pi$
$674$	0	0
$675$	−1.06409e26	−0.140797
$676$	0	0
$677$	3.42607e26	0.440762	0.220381	−	0.975414i	$-0.429270\pi$
$677$	3.42607e26	0.440762	0.220381	+	0.975414i	$0.429270\pi$
$678$	0	0
$679$	−1.76229e27	−2.20452
$680$	0	0
$681$	1.37865e27	1.67708
$682$	0	0
$683$	−4.56304e26	−0.539831	−0.269915	−	0.962884i	$-0.586996\pi$
$683$	−4.56304e26	−0.539831	−0.269915	+	0.962884i	$0.586996\pi$
$684$	0	0
$685$	2.44085e26	0.280854
$686$	0	0
$687$	−2.03375e26	−0.227620
$688$	0	0
$689$	−2.26355e27	−2.46438
$690$	0	0
$691$	7.01529e26	0.743026	0.371513	−	0.928428i	$-0.378839\pi$
$691$	7.01529e26	0.743026	0.371513	+	0.928428i	$0.378839\pi$
$692$	0	0
$693$	4.52137e26	0.465912
$694$	0	0
$695$	2.99628e25	0.0300419
$696$	0	0
$697$	5.45604e25	0.0532311
$698$	0	0
$699$	1.20909e27	1.14796
$700$	0	0
$701$	−8.36495e26	−0.772933	−0.386467	−	0.922303i	$-0.626305\pi$
$701$	−8.36495e26	−0.772933	−0.386467	+	0.922303i	$0.626305\pi$
$702$	0	0
$703$	−1.22171e26	−0.109873
$704$	0	0
$705$	−5.91935e26	−0.518176
$706$	0	0
$707$	2.57886e27	2.19757
$708$	0	0
$709$	−2.06889e26	−0.171632	−0.0858158	−	0.996311i	$-0.527350\pi$
$709$	−2.06889e26	−0.171632	−0.0858158	+	0.996311i	$0.527350\pi$
$710$	0	0
$711$	−4.78428e26	−0.386416
$712$	0	0
$713$	6.28101e25	0.0493945
$714$	0	0
$715$	4.61400e26	0.353322
$716$	0	0
$717$	1.44165e27	1.07505
$718$	0	0
$719$	7.70076e26	0.559254	0.279627	−	0.960109i	$-0.409789\pi$
$719$	7.70076e26	0.559254	0.279627	+	0.960109i	$0.409789\pi$
$720$	0	0
$721$	−1.66640e26	−0.117868
$722$	0	0
$723$	8.96214e26	0.617444
$724$	0	0
$725$	2.92759e26	0.196471
$726$	0	0
$727$	−1.20906e27	−0.790442	−0.395221	−	0.918586i	$-0.629332\pi$
$727$	−1.20906e27	−0.790442	−0.395221	+	0.918586i	$0.629332\pi$
$728$	0	0
$729$	5.18829e26	0.330456
$730$	0	0
$731$	−3.95205e26	−0.245249
$732$	0	0
$733$	−2.57122e27	−1.55472	−0.777358	−	0.629058i	$-0.783441\pi$
$733$	−2.57122e27	−1.55472	−0.777358	+	0.629058i	$0.783441\pi$
$734$	0	0
$735$	2.33703e27	1.37700
$736$	0	0
$737$	1.48287e27	0.851456
$738$	0	0
$739$	2.06120e27	1.15345	0.576725	−	0.816939i	$-0.304331\pi$
$739$	2.06120e27	1.15345	0.576725	+	0.816939i	$0.304331\pi$
$740$	0	0
$741$	−1.06050e27	−0.578415
$742$	0	0
$743$	2.84713e27	1.51361	0.756804	−	0.653642i	$-0.226760\pi$
$743$	2.84713e27	1.51361	0.756804	+	0.653642i	$0.226760\pi$
$744$	0	0
$745$	−1.06653e27	−0.552700
$746$	0	0
$747$	1.54428e27	0.780156
$748$	0	0
$749$	7.23136e27	3.56158
$750$	0	0
$751$	2.22975e27	1.07072	0.535361	−	0.844623i	$-0.320175\pi$
$751$	2.22975e27	1.07072	0.535361	+	0.844623i	$0.320175\pi$
$752$	0	0
$753$	2.59082e27	1.21307
$754$	0	0
$755$	−1.52926e27	−0.698210
$756$	0	0
$757$	4.07708e27	1.81526	0.907629	−	0.419772i	$-0.137890\pi$
$757$	4.07708e27	1.81526	0.907629	+	0.419772i	$0.137890\pi$
$758$	0	0
$759$	3.58960e26	0.155865
$760$	0	0
$761$	8.54605e26	0.361919	0.180959	−	0.983491i	$-0.442080\pi$
$761$	8.54605e26	0.361919	0.180959	+	0.983491i	$0.442080\pi$
$762$	0	0
$763$	3.02145e27	1.24805
$764$	0	0
$765$	−5.99421e25	−0.0241517
$766$	0	0
$767$	−2.56031e27	−1.00632
$768$	0	0
$769$	1.47566e27	0.565829	0.282914	−	0.959145i	$-0.408699\pi$
$769$	1.47566e27	0.565829	0.282914	+	0.959145i	$0.408699\pi$
$770$	0	0
$771$	−3.40169e27	−1.27256
$772$	0	0
$773$	−5.09798e27	−1.86077	−0.930385	−	0.366583i	$-0.880528\pi$
$773$	−5.09798e27	−1.86077	−0.930385	+	0.366583i	$0.880528\pi$
$774$	0	0
$775$	−1.27591e26	−0.0454417
$776$	0	0
$777$	1.90500e27	0.662058
$778$	0	0
$779$	−4.40801e26	−0.149499
$780$	0	0
$781$	−9.65149e26	−0.319455
$782$	0	0
$783$	−2.14077e27	−0.691564
$784$	0	0
$785$	6.81391e26	0.214850
$786$	0	0
$787$	4.05406e27	1.24776	0.623878	−	0.781522i	$-0.285556\pi$
$787$	4.05406e27	1.24776	0.623878	+	0.781522i	$0.285556\pi$
$788$	0	0
$789$	−1.31498e27	−0.395081
$790$	0	0
$791$	−1.00564e28	−2.94963
$792$	0	0
$793$	−6.15550e27	−1.76266
$794$	0	0
$795$	3.57727e27	1.00015
$796$	0	0
$797$	−2.74905e27	−0.750461	−0.375230	−	0.926932i	$-0.622436\pi$
$797$	−2.74905e27	−0.750461	−0.375230	+	0.926932i	$0.622436\pi$
$798$	0	0
$799$	4.87096e26	0.129843
$800$	0	0
$801$	8.69965e26	0.226461
$802$	0	0
$803$	−3.30088e27	−0.839135
$804$	0	0
$805$	7.42613e26	0.184375
$806$	0	0
$807$	−4.00501e27	−0.971192
$808$	0	0
$809$	−2.02058e26	−0.0478592	−0.0239296	−	0.999714i	$-0.507618\pi$
$809$	−2.02058e26	−0.0478592	−0.0239296	+	0.999714i	$0.507618\pi$
$810$	0	0
$811$	−7.05690e26	−0.163274	−0.0816368	−	0.996662i	$-0.526015\pi$
$811$	−7.05690e26	−0.163274	−0.0816368	+	0.996662i	$0.526015\pi$
$812$	0	0
$813$	3.44773e27	0.779243
$814$	0	0
$815$	−1.24298e27	−0.274451
$816$	0	0
$817$	3.19292e27	0.688777
$818$	0	0
$819$	4.77821e27	1.00709
$820$	0	0
$821$	7.06583e27	1.45514	0.727568	−	0.686036i	$-0.240651\pi$
$821$	7.06583e27	1.45514	0.727568	+	0.686036i	$0.240651\pi$
$822$	0	0
$823$	−5.42742e27	−1.09218	−0.546091	−	0.837726i	$-0.683885\pi$
$823$	−5.42742e27	−1.09218	−0.546091	+	0.837726i	$0.683885\pi$
$824$	0	0
$825$	−7.29186e26	−0.143392
$826$	0	0
$827$	4.24973e27	0.816693	0.408346	−	0.912827i	$-0.366106\pi$
$827$	4.24973e27	0.816693	0.408346	+	0.912827i	$0.366106\pi$
$828$	0	0
$829$	−9.63265e26	−0.180916	−0.0904582	−	0.995900i	$-0.528833\pi$
$829$	−9.63265e26	−0.180916	−0.0904582	+	0.995900i	$0.528833\pi$
$830$	0	0
$831$	−1.71764e27	−0.315298
$832$	0	0
$833$	−1.92311e27	−0.345046
$834$	0	0
$835$	3.47283e27	0.609062
$836$	0	0
$837$	9.32997e26	0.159952
$838$	0	0
$839$	−3.94547e27	−0.661242	−0.330621	−	0.943764i	$-0.607258\pi$
$839$	−3.94547e27	−0.661242	−0.330621	+	0.943764i	$0.607258\pi$
$840$	0	0
$841$	−2.13471e26	−0.0349766
$842$	0	0
$843$	1.11494e28	1.78603
$844$	0	0
$845$	2.02079e27	0.316507
$846$	0	0
$847$	7.85697e27	1.20327
$848$	0	0
$849$	−1.29375e28	−1.93744
$850$	0	0
$851$	4.37016e26	0.0639980
$852$	0	0
$853$	−1.02200e28	−1.46364	−0.731821	−	0.681497i	$-0.761329\pi$
$853$	−1.02200e28	−1.46364	−0.731821	+	0.681497i	$0.761329\pi$
$854$	0	0
$855$	4.84281e26	0.0678297
$856$	0	0
$857$	−1.02850e28	−1.40893	−0.704463	−	0.709741i	$-0.748812\pi$
$857$	−1.02850e28	−1.40893	−0.704463	+	0.709741i	$0.748812\pi$
$858$	0	0
$859$	−1.83312e27	−0.245616	−0.122808	−	0.992430i	$-0.539190\pi$
$859$	−1.83312e27	−0.245616	−0.122808	+	0.992430i	$0.539190\pi$
$860$	0	0
$861$	6.87339e27	0.900827
$862$	0	0
$863$	3.00343e27	0.385048	0.192524	−	0.981292i	$-0.438333\pi$
$863$	3.00343e27	0.385048	0.192524	+	0.981292i	$0.438333\pi$
$864$	0	0
$865$	4.93184e27	0.618524
$866$	0	0
$867$	−9.49496e27	−1.16496
$868$	0	0
$869$	4.78920e27	0.574878
$870$	0	0
$871$	1.56711e28	1.84046
$872$	0	0
$873$	4.11095e27	0.472396
$874$	0	0
$875$	−1.50853e27	−0.169620
$876$	0	0
$877$	−9.50373e27	−1.04568	−0.522839	−	0.852432i	$-0.675127\pi$
$877$	−9.50373e27	−1.04568	−0.522839	+	0.852432i	$0.675127\pi$
$878$	0	0
$879$	−2.44597e27	−0.263365
$880$	0	0
$881$	−1.42753e28	−1.50423	−0.752115	−	0.659032i	$-0.770966\pi$
$881$	−1.42753e28	−1.50423	−0.752115	+	0.659032i	$0.770966\pi$
$882$	0	0
$883$	−3.38364e27	−0.348945	−0.174473	−	0.984662i	$-0.555822\pi$
$883$	−3.38364e27	−0.348945	−0.174473	+	0.984662i	$0.555822\pi$
$884$	0	0
$885$	4.04625e27	0.408405
$886$	0	0
$887$	−2.89735e27	−0.286238	−0.143119	−	0.989706i	$-0.545713\pi$
$887$	−2.89735e27	−0.286238	−0.143119	+	0.989706i	$0.545713\pi$
$888$	0	0
$889$	1.14616e28	1.10836
$890$	0	0
$891$	7.92752e27	0.750411
$892$	0	0
$893$	−3.93532e27	−0.364662
$894$	0	0
$895$	−1.32010e27	−0.119753
$896$	0	0
$897$	3.79352e27	0.336910
$898$	0	0
$899$	−2.56691e27	−0.223200
$900$	0	0
$901$	−2.94369e27	−0.250615
$902$	0	0
$903$	−4.97871e28	−4.15033
$904$	0	0
$905$	−3.18084e27	−0.259645
$906$	0	0
$907$	−2.05811e27	−0.164513	−0.0822563	−	0.996611i	$-0.526213\pi$
$907$	−2.05811e27	−0.164513	−0.0822563	+	0.996611i	$0.526213\pi$
$908$	0	0
$909$	−6.01578e27	−0.470907
$910$	0	0
$911$	−2.36302e27	−0.181152	−0.0905761	−	0.995890i	$-0.528871\pi$
$911$	−2.36302e27	−0.181152	−0.0905761	+	0.995890i	$0.528871\pi$
$912$	0	0
$913$	−1.54587e28	−1.16065
$914$	0	0
$915$	9.72801e27	0.715358
$916$	0	0
$917$	3.93722e28	2.83583
$918$	0	0
$919$	−1.61283e28	−1.13786	−0.568932	−	0.822384i	$-0.692643\pi$
$919$	−1.61283e28	−1.13786	−0.568932	+	0.822384i	$0.692643\pi$
$920$	0	0
$921$	−2.43689e27	−0.168410
$922$	0	0
$923$	−1.01998e28	−0.690517
$924$	0	0
$925$	−8.87746e26	−0.0588766
$926$	0	0
$927$	3.88727e26	0.0252573
$928$	0	0
$929$	−1.45492e28	−0.926166	−0.463083	−	0.886315i	$-0.653257\pi$
$929$	−1.45492e28	−0.926166	−0.463083	+	0.886315i	$0.653257\pi$
$930$	0	0
$931$	1.55371e28	0.969055
$932$	0	0
$933$	2.87101e28	1.75452
$934$	0	0
$935$	6.00038e26	0.0359309
$936$	0	0
$937$	−1.23803e28	−0.726448	−0.363224	−	0.931702i	$-0.618324\pi$
$937$	−1.23803e28	−0.726448	−0.363224	+	0.931702i	$0.618324\pi$
$938$	0	0
$939$	2.03485e27	0.117006
$940$	0	0
$941$	−2.51308e28	−1.41614	−0.708069	−	0.706143i	$-0.750434\pi$
$941$	−2.51308e28	−1.41614	−0.708069	+	0.706143i	$0.750434\pi$
$942$	0	0
$943$	1.57679e27	0.0870786
$944$	0	0
$945$	1.10310e28	0.597051
$946$	0	0
$947$	3.17547e28	1.68455	0.842274	−	0.539050i	$-0.181217\pi$
$947$	3.17547e28	1.68455	0.842274	+	0.539050i	$0.181217\pi$
$948$	0	0
$949$	−3.48839e28	−1.81383
$950$	0	0
$951$	−3.42160e28	−1.74387
$952$	0	0
$953$	3.36113e28	1.67920	0.839602	−	0.543203i	$-0.182788\pi$
$953$	3.36113e28	1.67920	0.839602	+	0.543203i	$0.182788\pi$
$954$	0	0
$955$	−2.97168e27	−0.145536
$956$	0	0
$957$	−1.46699e28	−0.704311
$958$	0	0
$959$	−2.53032e28	−1.19096
$960$	0	0
$961$	−2.05519e28	−0.948376
$962$	0	0
$963$	−1.68688e28	−0.763194
$964$	0	0
$965$	5.34125e27	0.236937
$966$	0	0
$967$	−1.31463e28	−0.571812	−0.285906	−	0.958258i	$-0.592295\pi$
$967$	−1.31463e28	−0.571812	−0.285906	+	0.958258i	$0.592295\pi$
$968$	0	0
$969$	−1.37916e27	−0.0588217
$970$	0	0
$971$	3.80889e28	1.59300	0.796500	−	0.604638i	$-0.206682\pi$
$971$	3.80889e28	1.59300	0.796500	+	0.604638i	$0.206682\pi$
$972$	0	0
$973$	−3.10611e27	−0.127393
$974$	0	0
$975$	−7.70609e27	−0.309949
$976$	0	0
$977$	4.67963e28	1.84592	0.922960	−	0.384895i	$-0.125762\pi$
$977$	4.67963e28	1.84592	0.922960	+	0.384895i	$0.125762\pi$
$978$	0	0
$979$	−8.70859e27	−0.336909
$980$	0	0
$981$	−7.04823e27	−0.267439
$982$	0	0
$983$	2.42051e28	0.900843	0.450421	−	0.892816i	$-0.351274\pi$
$983$	2.42051e28	0.900843	0.450421	+	0.892816i	$0.351274\pi$
$984$	0	0
$985$	1.57476e28	0.574872
$986$	0	0
$987$	6.13632e28	2.19733
$988$	0	0
$989$	−1.14214e28	−0.401193
$990$	0	0
$991$	−4.23429e28	−1.45909	−0.729543	−	0.683935i	$-0.760267\pi$
$991$	−4.23429e28	−1.45909	−0.729543	+	0.683935i	$0.760267\pi$
$992$	0	0
$993$	−4.55639e28	−1.54029
$994$	0	0
$995$	−1.11593e28	−0.370100
$996$	0	0
$997$	−4.42155e28	−1.43870	−0.719351	−	0.694647i	$-0.755561\pi$
$997$	−4.42155e28	−1.43870	−0.719351	+	0.694647i	$0.755561\pi$
$998$	0	0
$999$	6.49155e27	0.207241

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.20.a.h.1.3		4
4.3	odd	2		20.20.a.b.1.2	✓	4
20.3	even	4		100.20.c.c.49.3		8
20.7	even	4		100.20.c.c.49.6		8
20.19	odd	2		100.20.a.c.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.20.a.b.1.2	✓	4	4.3	odd	2
80.20.a.h.1.3		4	1.1	even	1	trivial
100.20.a.c.1.3		4	20.19	odd	2
100.20.c.c.49.3		8	20.3	even	4
100.20.c.c.49.6		8	20.7	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:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q+40429.9 q^{3} +1.95312e6 q^{5} -2.02472e8 q^{7} +4.72312e8 q^{9} +O(q^{10})\)	\(q+40429.9 q^{3} +1.95312e6 q^{5} -2.02472e8 q^{7} +4.72312e8 q^{9} -4.72798e9 q^{11} -4.99657e10 q^{13} +7.89646e10 q^{15} -6.49790e10 q^{17} +5.24975e11 q^{19} -8.18591e12 q^{21} -1.87788e12 q^{23} +3.81470e12 q^{25} -2.78945e13 q^{27} +7.67450e13 q^{29} -3.34473e13 q^{31} -1.91152e14 q^{33} -3.95453e14 q^{35} -2.32717e14 q^{37} -2.02010e15 q^{39} -8.39662e14 q^{41} +6.08205e15 q^{43} +9.22485e14 q^{45} -7.49620e15 q^{47} +2.95959e16 q^{49} -2.62709e15 q^{51} +4.53022e16 q^{53} -9.23434e15 q^{55} +2.12247e16 q^{57} +5.12413e16 q^{59} +1.23195e17 q^{61} -9.56299e16 q^{63} -9.75892e16 q^{65} -3.13637e17 q^{67} -7.59225e16 q^{69} +2.04136e17 q^{71} +6.98158e17 q^{73} +1.54228e17 q^{75} +9.57283e17 q^{77} -1.01295e18 q^{79} -1.67672e18 q^{81} +3.26963e18 q^{83} -1.26912e17 q^{85} +3.10279e18 q^{87} +1.84193e18 q^{89} +1.01166e19 q^{91} -1.35227e18 q^{93} +1.02534e18 q^{95} +8.70387e18 q^{97} -2.23308e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3080 q^{3} + 7812500 q^{5} - 148222040 q^{7} + 2231864116 q^{9}+O(q^{10}) \) 4 * q + 3080 * q^3 + 7812500 * q^5 - 148222040 * q^7 + 2231864116 * q^9	\( 4 q + 3080 q^{3} + 7812500 q^{5} - 148222040 q^{7} + 2231864116 q^{9} + 2334973920 q^{11} + 57423224120 q^{13} + 6015625000 q^{15} - 465961763160 q^{17} + 1624818160624 q^{19} + 5400395639744 q^{21} + 10101669670680 q^{23} + 15258789062500 q^{25} - 56910507730480 q^{27} - 21802908180264 q^{29} + 69517593805936 q^{31} - 204641262341280 q^{33} - 289496171875000 q^{35} + 21\!\cdots\!60 q^{37}+ \cdots + 27\!\cdots\!80 q^{99}+O(q^{100}) \) 4 * q + 3080 * q^3 + 7812500 * q^5 - 148222040 * q^7 + 2231864116 * q^9 + 2334973920 * q^11 + 57423224120 * q^13 + 6015625000 * q^15 - 465961763160 * q^17 + 1624818160624 * q^19 + 5400395639744 * q^21 + 10101669670680 * q^23 + 15258789062500 * q^25 - 56910507730480 * q^27 - 21802908180264 * q^29 + 69517593805936 * q^31 - 204641262341280 * q^33 - 289496171875000 * q^35 + 2156700568331960 * q^37 - 2385459176165456 * q^39 + 1367657291239944 * q^41 - 2895469359165560 * q^43 + 4359109601562500 * q^45 - 1740670005708600 * q^47 + 53534973320692932 * q^49 - 62580957868768368 * q^51 + 84253756978305240 * q^53 + 4560495937500000 * q^55 + 188957630874753440 * q^57 - 152838143056415952 * q^59 + 211425306591350168 * q^61 - 417796572994057400 * q^63 + 112154734609375000 * q^65 - 656919981274544360 * q^67 + 93524018475363648 * q^69 - 883122322400344752 * q^71 + 325226727237880520 * q^73 + 11749267578125000 * q^75 - 620955288586656480 * q^77 + 850256855296398112 * q^79 - 3380709519213288764 * q^81 + 3602925283549230600 * q^83 - 910081568671875000 * q^85 + 8179923434985534000 * q^87 + 418548299964421224 * q^89 + 10309921756152683504 * q^91 - 14102167255291233280 * q^93 + 3173472969968750000 * q^95 - 12135517460163032920 * q^97 + 27941442307774944480 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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