LMFDB - Embedded newform 80.18.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,18,Mod(1,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	80.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6308] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$146.577669876$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{36061}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9015 $ x^2 - x - 9015
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5}\cdot 5 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$95.4487$ 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)

$f(q)$	$=$	$q-12037.8 q^{3} -390625. q^{5} +9.53475e6 q^{7} +1.57682e7 q^{9} -4.01191e8 q^{11} +8.56166e8 q^{13} +4.70226e9 q^{15} -3.89127e10 q^{17} +1.13839e11 q^{19} -1.14777e11 q^{21} -1.64834e10 q^{23} +1.52588e11 q^{25} +1.36475e12 q^{27} -2.27472e12 q^{29} +1.63788e12 q^{31} +4.82946e12 q^{33} -3.72451e12 q^{35} -1.75967e13 q^{37} -1.03064e13 q^{39} -2.95532e13 q^{41} -1.37690e14 q^{43} -6.15944e12 q^{45} +1.65452e14 q^{47} -1.41719e14 q^{49} +4.68422e14 q^{51} -7.25259e14 q^{53} +1.56715e14 q^{55} -1.37037e15 q^{57} -1.62177e15 q^{59} +2.46915e15 q^{61} +1.50346e14 q^{63} -3.34440e14 q^{65} +2.03244e14 q^{67} +1.98423e14 q^{69} -9.39117e15 q^{71} +1.54865e15 q^{73} -1.83682e15 q^{75} -3.82526e15 q^{77} -8.30977e15 q^{79} -1.84648e16 q^{81} +6.14697e15 q^{83} +1.52003e16 q^{85} +2.73826e16 q^{87} +4.67428e15 q^{89} +8.16334e15 q^{91} -1.97164e16 q^{93} -4.44684e16 q^{95} +1.01799e17 q^{97} -6.32605e15 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6308 q^{3} - 781250 q^{5} - 6543844 q^{7} + 223195906 q^{9} - 1189408704 q^{11} - 2017919228 q^{13} - 2464062500 q^{15} - 18755639436 q^{17} + 136704830600 q^{19} - 409751898376 q^{21} + 649234170708 q^{23}+ \cdots - 16\!\cdots\!12 q^{99}+O(q^{100}) $ 2 * q + 6308 * q^3 - 781250 * q^5 - 6543844 * q^7 + 223195906 * q^9 - 1189408704 * q^11 - 2017919228 * q^13 - 2464062500 * q^15 - 18755639436 * q^17 + 136704830600 * q^19 - 409751898376 * q^21 + 649234170708 * q^23 + 305175781250 * q^25 + 2800995425720 * q^27 - 4696543420020 * q^29 - 7120867378744 * q^31 - 9631011282816 * q^33 + 2556189062500 * q^35 + 9933114637444 * q^37 - 63033715222712 * q^39 + 9622711880844 * q^41 - 7708436011852 * q^43 - 87185900781250 * q^45 + 466072575837996 * q^47 - 115828146887046 * q^49 + 838218717084456 * q^51 - 623333120284428 * q^53 + 464612775000000 * q^55 - 950883425062000 * q^57 - 654616071995160 * q^59 + 1329040385162884 * q^61 - 3184801966786532 * q^63 + 788249698437500 * q^65 + 4249016411877596 * q^67 + 12411535659353832 * q^69 - 2035131949347624 * q^71 + 2248593461347972 * q^73 + 962524414062500 * q^75 + 8848168987836288 * q^77 - 19188427678495120 * q^79 - 18903014015714198 * q^81 + 18474649086611988 * q^83 + 7326421654687500 * q^85 - 17047661699498280 * q^87 - 17458912916395020 * q^89 + 54374608211433016 * q^91 - 180402465813627376 * q^93 - 53400324453125000 * q^95 + 195934325822490244 * q^97 - 169824195828146112 * q^99

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
$3$	−12037.8	−1.05929	−0.529646	−	0.848219i	$-0.677675\pi$
$3$	−12037.8	−1.05929	−0.529646	+	0.848219i	$0.677675\pi$
$4$	0	0
$5$	−390625.	−0.447214
$5$	−390625.	−0.447214
$6$	0	0
$7$	9.53475e6	0.625138	0.312569	−	0.949895i	$-0.398810\pi$
$7$	9.53475e6	0.625138	0.312569	+	0.949895i	$0.398810\pi$
$8$	0	0
$9$	1.57682e7	0.122101
$10$	0	0
$11$	−4.01191e8	−0.564305	−0.282152	−	0.959370i	$-0.591048\pi$
$11$	−4.01191e8	−0.564305	−0.282152	+	0.959370i	$0.591048\pi$
$12$	0	0
$13$	8.56166e8	0.291098	0.145549	−	0.989351i	$-0.453505\pi$
$13$	8.56166e8	0.291098	0.145549	+	0.989351i	$0.453505\pi$
$14$	0	0
$15$	4.70226e9	0.473730
$16$	0	0
$17$	−3.89127e10	−1.35293	−0.676465	−	0.736475i	$-0.736489\pi$
$17$	−3.89127e10	−1.35293	−0.676465	+	0.736475i	$0.736489\pi$
$18$	0	0
$19$	1.13839e11	1.53775	0.768876	−	0.639398i	$-0.220816\pi$
$19$	1.13839e11	1.53775	0.768876	+	0.639398i	$0.220816\pi$
$20$	0	0
$21$	−1.14777e11	−0.662205
$22$	0	0
$23$	−1.64834e10	−0.0438894	−0.0219447	−	0.999759i	$-0.506986\pi$
$23$	−1.64834e10	−0.0438894	−0.0219447	+	0.999759i	$0.506986\pi$
$24$	0	0
$25$	1.52588e11	0.200000
$26$	0	0
$27$	1.36475e12	0.929952
$28$	0	0
$29$	−2.27472e12	−0.844394	−0.422197	−	0.906504i	$-0.638741\pi$
$29$	−2.27472e12	−0.844394	−0.422197	+	0.906504i	$0.638741\pi$
$30$	0	0
$31$	1.63788e12	0.344911	0.172455	−	0.985017i	$-0.444830\pi$
$31$	1.63788e12	0.344911	0.172455	+	0.985017i	$0.444830\pi$
$32$	0	0
$33$	4.82946e12	0.597764
$34$	0	0
$35$	−3.72451e12	−0.279570
$36$	0	0
$37$	−1.75967e13	−0.823599	−0.411799	−	0.911275i	$-0.635100\pi$
$37$	−1.75967e13	−0.823599	−0.411799	+	0.911275i	$0.635100\pi$
$38$	0	0
$39$	−1.03064e13	−0.308358
$40$	0	0
$41$	−2.95532e13	−0.578018	−0.289009	−	0.957326i	$-0.593326\pi$
$41$	−2.95532e13	−0.578018	−0.289009	+	0.957326i	$0.593326\pi$
$42$	0	0
$43$	−1.37690e14	−1.79647	−0.898233	−	0.439519i	$-0.855149\pi$
$43$	−1.37690e14	−1.79647	−0.898233	+	0.439519i	$0.855149\pi$
$44$	0	0
$45$	−6.15944e12	−0.0546053
$46$	0	0
$47$	1.65452e14	1.01354	0.506770	−	0.862081i	$-0.330839\pi$
$47$	1.65452e14	1.01354	0.506770	+	0.862081i	$0.330839\pi$
$48$	0	0
$49$	−1.41719e14	−0.609202
$50$	0	0
$51$	4.68422e14	1.43315
$52$	0	0
$53$	−7.25259e14	−1.60010	−0.800052	−	0.599931i	$-0.795195\pi$
$53$	−7.25259e14	−1.60010	−0.800052	+	0.599931i	$0.795195\pi$
$54$	0	0
$55$	1.56715e14	0.252365
$56$	0	0
$57$	−1.37037e15	−1.62893
$58$	0	0
$59$	−1.62177e15	−1.43796	−0.718981	−	0.695030i	$-0.755391\pi$
$59$	−1.62177e15	−1.43796	−0.718981	+	0.695030i	$0.755391\pi$
$60$	0	0
$61$	2.46915e15	1.64909	0.824545	−	0.565796i	$-0.191431\pi$
$61$	2.46915e15	1.64909	0.824545	+	0.565796i	$0.191431\pi$
$62$	0	0
$63$	1.50346e14	0.0763301
$64$	0	0
$65$	−3.34440e14	−0.130183
$66$	0	0
$67$	2.03244e14	0.0611479	0.0305739	−	0.999533i	$-0.490266\pi$
$67$	2.03244e14	0.0611479	0.0305739	+	0.999533i	$0.490266\pi$
$68$	0	0
$69$	1.98423e14	0.0464917
$70$	0	0
$71$	−9.39117e15	−1.72593	−0.862966	−	0.505262i	$-0.831396\pi$
$71$	−9.39117e15	−1.72593	−0.862966	+	0.505262i	$0.831396\pi$
$72$	0	0
$73$	1.54865e15	0.224754	0.112377	−	0.993666i	$-0.464153\pi$
$73$	1.54865e15	0.224754	0.112377	+	0.993666i	$0.464153\pi$
$74$	0	0
$75$	−1.83682e15	−0.211859
$76$	0	0
$77$	−3.82526e15	−0.352769
$78$	0	0
$79$	−8.30977e15	−0.616252	−0.308126	−	0.951345i	$-0.599702\pi$
$79$	−8.30977e15	−0.616252	−0.308126	+	0.951345i	$0.599702\pi$
$80$	0	0
$81$	−1.84648e16	−1.10719
$82$	0	0
$83$	6.14697e15	0.299569	0.149785	−	0.988719i	$-0.452142\pi$
$83$	6.14697e15	0.299569	0.149785	+	0.988719i	$0.452142\pi$
$84$	0	0
$85$	1.52003e16	0.605048
$86$	0	0
$87$	2.73826e16	0.894460
$88$	0	0
$89$	4.67428e15	0.125863	0.0629317	−	0.998018i	$-0.479955\pi$
$89$	4.67428e15	0.125863	0.0629317	+	0.998018i	$0.479955\pi$
$90$	0	0
$91$	8.16334e15	0.181977
$92$	0	0
$93$	−1.97164e16	−0.365362
$94$	0	0
$95$	−4.44684e16	−0.687703
$96$	0	0
$97$	1.01799e17	1.31881	0.659407	−	0.751786i	$-0.270808\pi$
$97$	1.01799e17	1.31881	0.659407	+	0.751786i	$0.270808\pi$
$98$	0	0
$99$	−6.32605e15	−0.0689023
$100$	0	0
$101$	3.11898e16	0.286603	0.143302	−	0.989679i	$-0.454228\pi$
$101$	3.11898e16	0.286603	0.143302	+	0.989679i	$0.454228\pi$
$102$	0	0
$103$	1.71633e17	1.33501	0.667505	−	0.744606i	$-0.267362\pi$
$103$	1.71633e17	1.33501	0.667505	+	0.744606i	$0.267362\pi$
$104$	0	0
$105$	4.48349e16	0.296147
$106$	0	0
$107$	1.64074e17	0.923159	0.461579	−	0.887099i	$-0.347283\pi$
$107$	1.64074e17	0.923159	0.461579	+	0.887099i	$0.347283\pi$
$108$	0	0
$109$	2.80332e17	1.34756	0.673779	−	0.738933i	$-0.264670\pi$
$109$	2.80332e17	1.34756	0.673779	+	0.738933i	$0.264670\pi$
$110$	0	0
$111$	2.11825e17	0.872432
$112$	0	0
$113$	−5.30659e17	−1.87780	−0.938899	−	0.344193i	$-0.888153\pi$
$113$	−5.30659e17	−1.87780	−0.938899	+	0.344193i	$0.888153\pi$
$114$	0	0
$115$	6.43882e15	0.0196279
$116$	0	0
$117$	1.35002e16	0.0355434
$118$	0	0
$119$	−3.71023e17	−0.845768
$120$	0	0
$121$	−3.44493e17	−0.681560
$122$	0	0
$123$	3.55755e17	0.612291
$124$	0	0
$125$	−5.96046e16	−0.0894427
$126$	0	0
$127$	−3.45518e17	−0.453043	−0.226522	−	0.974006i	$-0.572735\pi$
$127$	−3.45518e17	−0.453043	−0.226522	+	0.974006i	$0.572735\pi$
$128$	0	0
$129$	1.65748e18	1.90298
$130$	0	0
$131$	5.92983e17	0.597360	0.298680	−	0.954353i	$-0.403454\pi$
$131$	5.92983e17	0.597360	0.298680	+	0.954353i	$0.403454\pi$
$132$	0	0
$133$	1.08543e18	0.961307
$134$	0	0
$135$	−5.33105e17	−0.415887
$136$	0	0
$137$	−5.64525e17	−0.388650	−0.194325	−	0.980937i	$-0.562252\pi$
$137$	−5.64525e17	−0.388650	−0.194325	+	0.980937i	$0.562252\pi$
$138$	0	0
$139$	−2.13511e18	−1.29956	−0.649778	−	0.760124i	$-0.725138\pi$
$139$	−2.13511e18	−1.29956	−0.649778	+	0.760124i	$0.725138\pi$
$140$	0	0
$141$	−1.99168e18	−1.07364
$142$	0	0
$143$	−3.43487e17	−0.164268
$144$	0	0
$145$	8.88563e17	0.377624
$146$	0	0
$147$	1.70598e18	0.645323
$148$	0	0
$149$	2.11167e18	0.712103	0.356051	−	0.934466i	$-0.384123\pi$
$149$	2.11167e18	0.712103	0.356051	+	0.934466i	$0.384123\pi$
$150$	0	0
$151$	−1.09446e18	−0.329532	−0.164766	−	0.986333i	$-0.552687\pi$
$151$	−1.09446e18	−0.329532	−0.164766	+	0.986333i	$0.552687\pi$
$152$	0	0
$153$	−6.13581e17	−0.165194
$154$	0	0
$155$	−6.39796e17	−0.154249
$156$	0	0
$157$	2.67975e18	0.579359	0.289680	−	0.957124i	$-0.406451\pi$
$157$	2.67975e18	0.579359	0.289680	+	0.957124i	$0.406451\pi$
$158$	0	0
$159$	8.73051e18	1.69498
$160$	0	0
$161$	−1.57165e17	−0.0274369
$162$	0	0
$163$	9.47539e18	1.48937	0.744685	−	0.667416i	$-0.232600\pi$
$163$	9.47539e18	1.48937	0.744685	+	0.667416i	$0.232600\pi$
$164$	0	0
$165$	−1.88651e18	−0.267328
$166$	0	0
$167$	6.70665e18	0.857859	0.428929	−	0.903338i	$-0.358891\pi$
$167$	6.70665e18	0.857859	0.428929	+	0.903338i	$0.358891\pi$
$168$	0	0
$169$	−7.91739e18	−0.915262
$170$	0	0
$171$	1.79503e18	0.187761
$172$	0	0
$173$	1.05695e19	1.00152	0.500762	−	0.865585i	$-0.333053\pi$
$173$	1.05695e19	1.00152	0.500762	+	0.865585i	$0.333053\pi$
$174$	0	0
$175$	1.45489e18	0.125028
$176$	0	0
$177$	1.95225e19	1.52322
$178$	0	0
$179$	1.98718e19	1.40924	0.704621	−	0.709583i	$-0.251117\pi$
$179$	1.98718e19	1.40924	0.704621	+	0.709583i	$0.251117\pi$
$180$	0	0
$181$	2.50176e19	1.61428	0.807138	−	0.590363i	$-0.201015\pi$
$181$	2.50176e19	1.61428	0.807138	+	0.590363i	$0.201015\pi$
$182$	0	0
$183$	−2.97231e19	−1.74687
$184$	0	0
$185$	6.87370e18	0.368324
$186$	0	0
$187$	1.56114e19	0.763464
$188$	0	0
$189$	1.30125e19	0.581349
$190$	0	0
$191$	2.63385e19	1.07599	0.537995	−	0.842948i	$-0.319182\pi$
$191$	2.63385e19	1.07599	0.537995	+	0.842948i	$0.319182\pi$
$192$	0	0
$193$	3.68856e19	1.37918	0.689589	−	0.724201i	$-0.257791\pi$
$193$	3.68856e19	1.37918	0.689589	+	0.724201i	$0.257791\pi$
$194$	0	0
$195$	4.02592e18	0.137902
$196$	0	0
$197$	−4.04920e19	−1.27176	−0.635882	−	0.771786i	$-0.719364\pi$
$197$	−4.04920e19	−1.27176	−0.635882	+	0.771786i	$0.719364\pi$
$198$	0	0
$199$	3.32772e19	0.959171	0.479586	−	0.877495i	$-0.340787\pi$
$199$	3.32772e19	0.959171	0.479586	+	0.877495i	$0.340787\pi$
$200$	0	0
$201$	−2.44660e18	−0.0647735
$202$	0	0
$203$	−2.16889e19	−0.527863
$204$	0	0
$205$	1.15442e19	0.258498
$206$	0	0
$207$	−2.59912e17	−0.00535894
$208$	0	0
$209$	−4.56713e19	−0.867761
$210$	0	0
$211$	2.42162e19	0.424331	0.212166	−	0.977234i	$-0.431948\pi$
$211$	2.42162e19	0.424331	0.212166	+	0.977234i	$0.431948\pi$
$212$	0	0
$213$	1.13049e20	1.82827
$214$	0	0
$215$	5.37850e19	0.803404
$216$	0	0
$217$	1.56168e19	0.215617
$218$	0	0
$219$	−1.86423e19	−0.238081
$220$	0	0
$221$	−3.33157e19	−0.393835
$222$	0	0
$223$	−7.42273e19	−0.812779	−0.406390	−	0.913700i	$-0.633212\pi$
$223$	−7.42273e19	−0.812779	−0.406390	+	0.913700i	$0.633212\pi$
$224$	0	0
$225$	2.40603e18	0.0244202
$226$	0	0
$227$	−9.57531e18	−0.0901432	−0.0450716	−	0.998984i	$-0.514352\pi$
$227$	−9.57531e18	−0.0901432	−0.0450716	+	0.998984i	$0.514352\pi$
$228$	0	0
$229$	1.97753e20	1.72791	0.863955	−	0.503569i	$-0.167980\pi$
$229$	1.97753e20	1.72791	0.863955	+	0.503569i	$0.167980\pi$
$230$	0	0
$231$	4.60477e19	0.373685
$232$	0	0
$233$	−1.79778e20	−1.35585	−0.677925	−	0.735131i	$-0.737120\pi$
$233$	−1.79778e20	−1.35585	−0.677925	+	0.735131i	$0.737120\pi$
$234$	0	0
$235$	−6.46298e19	−0.453269
$236$	0	0
$237$	1.00031e20	0.652792
$238$	0	0
$239$	2.00207e20	1.21646	0.608229	−	0.793761i	$-0.291880\pi$
$239$	2.00207e20	1.21646	0.608229	+	0.793761i	$0.291880\pi$
$240$	0	0
$241$	−1.81616e20	−1.02804	−0.514018	−	0.857779i	$-0.671844\pi$
$241$	−1.81616e20	−1.02804	−0.514018	+	0.857779i	$0.671844\pi$
$242$	0	0
$243$	4.60321e19	0.242889
$244$	0	0
$245$	5.53590e19	0.272443
$246$	0	0
$247$	9.74653e19	0.447637
$248$	0	0
$249$	−7.39959e19	−0.317332
$250$	0	0
$251$	3.20192e20	1.28288	0.641438	−	0.767175i	$-0.278338\pi$
$251$	3.20192e20	1.28288	0.641438	+	0.767175i	$0.278338\pi$
$252$	0	0
$253$	6.61299e18	0.0247670
$254$	0	0
$255$	−1.82977e20	−0.640923
$256$	0	0
$257$	−3.34539e20	−1.09652	−0.548259	−	0.836309i	$-0.684709\pi$
$257$	−3.34539e20	−1.09652	−0.548259	+	0.836309i	$0.684709\pi$
$258$	0	0
$259$	−1.67780e20	−0.514863
$260$	0	0
$261$	−3.58681e19	−0.103101
$262$	0	0
$263$	−2.95034e20	−0.794782	−0.397391	−	0.917649i	$-0.630084\pi$
$263$	−2.95034e20	−0.794782	−0.397391	+	0.917649i	$0.630084\pi$
$264$	0	0
$265$	2.83304e20	0.715588
$266$	0	0
$267$	−5.62680e19	−0.133326
$268$	0	0
$269$	7.56967e20	1.68338	0.841690	−	0.539960i	$-0.181561\pi$
$269$	7.56967e20	1.68338	0.841690	+	0.539960i	$0.181561\pi$
$270$	0	0
$271$	1.18908e20	0.248298	0.124149	−	0.992264i	$-0.460380\pi$
$271$	1.18908e20	0.248298	0.124149	+	0.992264i	$0.460380\pi$
$272$	0	0
$273$	−9.82685e19	−0.192767
$274$	0	0
$275$	−6.12169e19	−0.112861
$276$	0	0
$277$	−1.96379e20	−0.340421	−0.170211	−	0.985408i	$-0.554445\pi$
$277$	−1.96379e20	−0.340421	−0.170211	+	0.985408i	$0.554445\pi$
$278$	0	0
$279$	2.58263e19	0.0421140
$280$	0	0
$281$	−3.39035e20	−0.520284	−0.260142	−	0.965570i	$-0.583769\pi$
$281$	−3.39035e20	−0.520284	−0.260142	+	0.965570i	$0.583769\pi$
$282$	0	0
$283$	−4.03445e20	−0.582908	−0.291454	−	0.956585i	$-0.594139\pi$
$283$	−4.03445e20	−0.582908	−0.291454	+	0.956585i	$0.594139\pi$
$284$	0	0
$285$	5.35301e20	0.728479
$286$	0	0
$287$	−2.81782e20	−0.361341
$288$	0	0
$289$	6.86955e20	0.830418
$290$	0	0
$291$	−1.22543e21	−1.39701
$292$	0	0
$293$	−1.05777e20	−0.113767	−0.0568836	−	0.998381i	$-0.518116\pi$
$293$	−1.05777e20	−0.113767	−0.0568836	+	0.998381i	$0.518116\pi$
$294$	0	0
$295$	6.33504e20	0.643076
$296$	0	0
$297$	−5.47525e20	−0.524776
$298$	0	0
$299$	−1.41125e19	−0.0127761
$300$	0	0
$301$	−1.31284e21	−1.12304
$302$	0	0
$303$	−3.75456e20	−0.303597
$304$	0	0
$305$	−9.64513e20	−0.737495
$306$	0	0
$307$	1.82911e21	1.32301	0.661504	−	0.749941i	$-0.269918\pi$
$307$	1.82911e21	1.32301	0.661504	+	0.749941i	$0.269918\pi$
$308$	0	0
$309$	−2.06608e21	−1.41417
$310$	0	0
$311$	−2.00748e21	−1.30073	−0.650366	−	0.759621i	$-0.725384\pi$
$311$	−2.00748e21	−1.30073	−0.650366	+	0.759621i	$0.725384\pi$
$312$	0	0
$313$	−2.84834e21	−1.74769	−0.873846	−	0.486203i	$-0.838381\pi$
$313$	−2.84834e21	−1.74769	−0.873846	+	0.486203i	$0.838381\pi$
$314$	0	0
$315$	−5.87287e19	−0.0341359
$316$	0	0
$317$	2.74347e20	0.151111	0.0755554	−	0.997142i	$-0.475927\pi$
$317$	2.74347e20	0.151111	0.0755554	+	0.997142i	$0.475927\pi$
$318$	0	0
$319$	9.12598e20	0.476495
$320$	0	0
$321$	−1.97508e21	−0.977895
$322$	0	0
$323$	−4.42979e21	−2.08047
$324$	0	0
$325$	1.30641e20	0.0582197
$326$	0	0
$327$	−3.37458e21	−1.42746
$328$	0	0
$329$	1.57755e21	0.633603
$330$	0	0
$331$	−1.83521e20	−0.0700082	−0.0350041	−	0.999387i	$-0.511144\pi$
$331$	−1.83521e20	−0.0700082	−0.0350041	+	0.999387i	$0.511144\pi$
$332$	0	0
$333$	−2.77467e20	−0.100562
$334$	0	0
$335$	−7.93921e19	−0.0273462
$336$	0	0
$337$	−6.21202e20	−0.203413	−0.101706	−	0.994814i	$-0.532430\pi$
$337$	−6.21202e20	−0.203413	−0.101706	+	0.994814i	$0.532430\pi$
$338$	0	0
$339$	6.38796e21	1.98914
$340$	0	0
$341$	−6.57102e20	−0.194635
$342$	0	0
$343$	−3.56933e21	−1.00597
$344$	0	0
$345$	−7.75091e19	−0.0207917
$346$	0	0
$347$	−5.08978e21	−1.29987	−0.649933	−	0.759992i	$-0.725203\pi$
$347$	−5.08978e21	−1.29987	−0.649933	+	0.759992i	$0.725203\pi$
$348$	0	0
$349$	−3.81066e20	−0.0926796	−0.0463398	−	0.998926i	$-0.514756\pi$
$349$	−3.81066e20	−0.0926796	−0.0463398	+	0.998926i	$0.514756\pi$
$350$	0	0
$351$	1.16845e21	0.270707
$352$	0	0
$353$	2.34699e21	0.518115	0.259058	−	0.965862i	$-0.416588\pi$
$353$	2.34699e21	0.518115	0.259058	+	0.965862i	$0.416588\pi$
$354$	0	0
$355$	3.66842e21	0.771860
$356$	0	0
$357$	4.46629e21	0.895916
$358$	0	0
$359$	6.19792e19	0.0118561	0.00592807	−	0.999982i	$-0.498113\pi$
$359$	6.19792e19	0.0118561	0.00592807	+	0.999982i	$0.498113\pi$
$360$	0	0
$361$	7.47897e21	1.36468
$362$	0	0
$363$	4.14693e21	0.721972
$364$	0	0
$365$	−6.04941e20	−0.100513
$366$	0	0
$367$	3.40565e21	0.540180	0.270090	−	0.962835i	$-0.412946\pi$
$367$	3.40565e21	0.540180	0.270090	+	0.962835i	$0.412946\pi$
$368$	0	0
$369$	−4.65999e20	−0.0705767
$370$	0	0
$371$	−6.91517e21	−1.00029
$372$	0	0
$373$	−6.44986e21	−0.891302	−0.445651	−	0.895207i	$-0.647028\pi$
$373$	−6.44986e21	−0.891302	−0.445651	+	0.895207i	$0.647028\pi$
$374$	0	0
$375$	7.17508e20	0.0947460
$376$	0	0
$377$	−1.94754e21	−0.245802
$378$	0	0
$379$	5.68235e21	0.685638	0.342819	−	0.939401i	$-0.388618\pi$
$379$	5.68235e21	0.685638	0.342819	+	0.939401i	$0.388618\pi$
$380$	0	0
$381$	4.15928e21	0.479905
$382$	0	0
$383$	1.28548e22	1.41865	0.709327	−	0.704879i	$-0.248999\pi$
$383$	1.28548e22	1.41865	0.709327	+	0.704879i	$0.248999\pi$
$384$	0	0
$385$	1.49424e21	0.157763
$386$	0	0
$387$	−2.17111e21	−0.219351
$388$	0	0
$389$	1.70189e22	1.64573	0.822867	−	0.568234i	$-0.192373\pi$
$389$	1.70189e22	1.64573	0.822867	+	0.568234i	$0.192373\pi$
$390$	0	0
$391$	6.41412e20	0.0593792
$392$	0	0
$393$	−7.13820e21	−0.632779
$394$	0	0
$395$	3.24600e21	0.275596
$396$	0	0
$397$	−1.81182e22	−1.47365	−0.736827	−	0.676081i	$-0.763677\pi$
$397$	−1.81182e22	−1.47365	−0.736827	+	0.676081i	$0.763677\pi$
$398$	0	0
$399$	−1.30662e22	−1.01831
$400$	0	0
$401$	1.75008e22	1.30717	0.653585	−	0.756853i	$-0.273264\pi$
$401$	1.75008e22	1.30717	0.653585	+	0.756853i	$0.273264\pi$
$402$	0	0
$403$	1.40230e21	0.100403
$404$	0	0
$405$	7.21283e21	0.495152
$406$	0	0
$407$	7.05963e21	0.464761
$408$	0	0
$409$	9.06225e21	0.572253	0.286127	−	0.958192i	$-0.407632\pi$
$409$	9.06225e21	0.572253	0.286127	+	0.958192i	$0.407632\pi$
$410$	0	0
$411$	6.79563e21	0.411694
$412$	0	0
$413$	−1.54632e22	−0.898925
$414$	0	0
$415$	−2.40116e21	−0.133971
$416$	0	0
$417$	2.57020e22	1.37661
$418$	0	0
$419$	−7.49706e21	−0.385542	−0.192771	−	0.981244i	$-0.561747\pi$
$419$	−7.49706e21	−0.385542	−0.192771	+	0.981244i	$0.561747\pi$
$420$	0	0
$421$	−2.95580e22	−1.45975	−0.729873	−	0.683583i	$-0.760421\pi$
$421$	−2.95580e22	−1.45975	−0.729873	+	0.683583i	$0.760421\pi$
$422$	0	0
$423$	2.60888e21	0.123754
$424$	0	0
$425$	−5.93760e21	−0.270586
$426$	0	0
$427$	2.35428e22	1.03091
$428$	0	0
$429$	4.13482e21	0.174008
$430$	0	0
$431$	7.37016e21	0.298140	0.149070	−	0.988827i	$-0.452372\pi$
$431$	7.37016e21	0.298140	0.149070	+	0.988827i	$0.452372\pi$
$432$	0	0
$433$	1.77584e21	0.0690650	0.0345325	−	0.999404i	$-0.489006\pi$
$433$	1.77584e21	0.0690650	0.0345325	+	0.999404i	$0.489006\pi$
$434$	0	0
$435$	−1.06963e22	−0.400015
$436$	0	0
$437$	−1.87645e21	−0.0674909
$438$	0	0
$439$	2.92011e22	1.01030	0.505151	−	0.863031i	$-0.331437\pi$
$439$	2.92011e22	1.01030	0.505151	+	0.863031i	$0.331437\pi$
$440$	0	0
$441$	−2.23465e21	−0.0743842
$442$	0	0
$443$	4.06554e22	1.30223	0.651113	−	0.758980i	$-0.274302\pi$
$443$	4.06554e22	1.30223	0.651113	+	0.758980i	$0.274302\pi$
$444$	0	0
$445$	−1.82589e21	−0.0562879
$446$	0	0
$447$	−2.54198e22	−0.754325
$448$	0	0
$449$	4.62783e21	0.132216	0.0661079	−	0.997812i	$-0.478942\pi$
$449$	4.62783e21	0.132216	0.0661079	+	0.997812i	$0.478942\pi$
$450$	0	0
$451$	1.18565e22	0.326179
$452$	0	0
$453$	1.31749e22	0.349071
$454$	0	0
$455$	−3.18880e21	−0.0813825
$456$	0	0
$457$	6.30841e22	1.55107	0.775536	−	0.631303i	$-0.217480\pi$
$457$	6.30841e22	1.55107	0.775536	+	0.631303i	$0.217480\pi$
$458$	0	0
$459$	−5.31060e22	−1.25816
$460$	0	0
$461$	5.62982e22	1.28539	0.642697	−	0.766120i	$-0.277815\pi$
$461$	5.62982e22	1.28539	0.642697	+	0.766120i	$0.277815\pi$
$462$	0	0
$463$	−5.90180e21	−0.129881	−0.0649405	−	0.997889i	$-0.520686\pi$
$463$	−5.90180e21	−0.129881	−0.0649405	+	0.997889i	$0.520686\pi$
$464$	0	0
$465$	7.70172e21	0.163395
$466$	0	0
$467$	−2.51115e22	−0.513663	−0.256832	−	0.966456i	$-0.582679\pi$
$467$	−2.51115e22	−0.513663	−0.256832	+	0.966456i	$0.582679\pi$
$468$	0	0
$469$	1.93788e21	0.0382259
$470$	0	0
$471$	−3.22583e22	−0.613711
$472$	0	0
$473$	5.52399e22	1.01375
$474$	0	0
$475$	1.73705e22	0.307550
$476$	0	0
$477$	−1.14360e22	−0.195375
$478$	0	0
$479$	3.54309e22	0.584158	0.292079	−	0.956394i	$-0.405653\pi$
$479$	3.54309e22	0.584158	0.292079	+	0.956394i	$0.405653\pi$
$480$	0	0
$481$	−1.50657e22	−0.239748
$482$	0	0
$483$	1.89192e21	0.0290637
$484$	0	0
$485$	−3.97652e22	−0.589791
$486$	0	0
$487$	1.51696e22	0.217259	0.108629	−	0.994082i	$-0.465354\pi$
$487$	1.51696e22	0.217259	0.108629	+	0.994082i	$0.465354\pi$
$488$	0	0
$489$	−1.14063e23	−1.57768
$490$	0	0
$491$	6.96906e22	0.931068	0.465534	−	0.885030i	$-0.345862\pi$
$491$	6.96906e22	0.931068	0.465534	+	0.885030i	$0.345862\pi$
$492$	0	0
$493$	8.85154e22	1.14240
$494$	0	0
$495$	2.47111e21	0.0308140
$496$	0	0
$497$	−8.95425e22	−1.07895
$498$	0	0
$499$	5.10974e22	0.595037	0.297518	−	0.954716i	$-0.403841\pi$
$499$	5.10974e22	0.595037	0.297518	+	0.954716i	$0.403841\pi$
$500$	0	0
$501$	−8.07332e22	−0.908723
$502$	0	0
$503$	−6.26474e21	−0.0681671	−0.0340836	−	0.999419i	$-0.510851\pi$
$503$	−6.26474e21	−0.0681671	−0.0340836	+	0.999419i	$0.510851\pi$
$504$	0	0
$505$	−1.21835e22	−0.128173
$506$	0	0
$507$	9.53079e22	0.969530
$508$	0	0
$509$	−7.41555e22	−0.729528	−0.364764	−	0.931100i	$-0.618850\pi$
$509$	−7.41555e22	−0.729528	−0.364764	+	0.931100i	$0.618850\pi$
$510$	0	0
$511$	1.47660e22	0.140503
$512$	0	0
$513$	1.55362e23	1.43003
$514$	0	0
$515$	−6.70441e22	−0.597034
$516$	0	0
$517$	−6.63780e22	−0.571945
$518$	0	0
$519$	−1.27233e23	−1.06091
$520$	0	0
$521$	9.53386e21	0.0769393	0.0384696	−	0.999260i	$-0.487752\pi$
$521$	9.53386e21	0.0769393	0.0384696	+	0.999260i	$0.487752\pi$
$522$	0	0
$523$	−1.49123e23	−1.16487	−0.582437	−	0.812876i	$-0.697901\pi$
$523$	−1.49123e23	−1.16487	−0.582437	+	0.812876i	$0.697901\pi$
$524$	0	0
$525$	−1.75136e22	−0.132441
$526$	0	0
$527$	−6.37341e22	−0.466640
$528$	0	0
$529$	−1.40778e23	−0.998074
$530$	0	0
$531$	−2.55723e22	−0.175577
$532$	0	0
$533$	−2.53025e22	−0.168260
$534$	0	0
$535$	−6.40912e22	−0.412849
$536$	0	0
$537$	−2.39212e23	−1.49280
$538$	0	0
$539$	5.68564e22	0.343776
$540$	0	0
$541$	8.46775e22	0.496125	0.248063	−	0.968744i	$-0.420206\pi$
$541$	8.46775e22	0.496125	0.248063	+	0.968744i	$0.420206\pi$
$542$	0	0
$543$	−3.01157e23	−1.70999
$544$	0	0
$545$	−1.09505e23	−0.602646
$546$	0	0
$547$	−4.72359e22	−0.251988	−0.125994	−	0.992031i	$-0.540212\pi$
$547$	−4.72359e22	−0.251988	−0.125994	+	0.992031i	$0.540212\pi$
$548$	0	0
$549$	3.89340e22	0.201356
$550$	0	0
$551$	−2.58952e23	−1.29847
$552$	0	0
$553$	−7.92316e22	−0.385243
$554$	0	0
$555$	−8.27441e22	−0.390163
$556$	0	0
$557$	3.77820e23	1.72789	0.863945	−	0.503587i	$-0.167987\pi$
$557$	3.77820e23	1.72789	0.863945	+	0.503587i	$0.167987\pi$
$558$	0	0
$559$	−1.17885e23	−0.522949
$560$	0	0
$561$	−1.87927e23	−0.808732
$562$	0	0
$563$	4.35959e23	1.82022	0.910112	−	0.414363i	$-0.135995\pi$
$563$	4.35959e23	1.82022	0.910112	+	0.414363i	$0.135995\pi$
$564$	0	0
$565$	2.07289e23	0.839777
$566$	0	0
$567$	−1.76058e23	−0.692149
$568$	0	0
$569$	−1.25095e23	−0.477293	−0.238647	−	0.971106i	$-0.576704\pi$
$569$	−1.25095e23	−0.477293	−0.238647	+	0.971106i	$0.576704\pi$
$570$	0	0
$571$	3.10316e23	1.14920	0.574602	−	0.818433i	$-0.305157\pi$
$571$	3.10316e23	1.14920	0.574602	+	0.818433i	$0.305157\pi$
$572$	0	0
$573$	−3.17058e23	−1.13979
$574$	0	0
$575$	−2.51516e21	−0.00877787
$576$	0	0
$577$	−2.11578e23	−0.716928	−0.358464	−	0.933544i	$-0.616699\pi$
$577$	−2.11578e23	−0.716928	−0.358464	+	0.933544i	$0.616699\pi$
$578$	0	0
$579$	−4.44021e23	−1.46095
$580$	0	0
$581$	5.86099e22	0.187272
$582$	0	0
$583$	2.90968e23	0.902947
$584$	0	0
$585$	−5.27350e21	−0.0158955
$586$	0	0
$587$	−2.15996e23	−0.632443	−0.316222	−	0.948685i	$-0.602414\pi$
$587$	−2.15996e23	−0.632443	−0.316222	+	0.948685i	$0.602414\pi$
$588$	0	0
$589$	1.86455e23	0.530387
$590$	0	0
$591$	4.87434e23	1.34717
$592$	0	0
$593$	3.56471e23	0.957326	0.478663	−	0.877999i	$-0.341122\pi$
$593$	3.56471e23	0.957326	0.478663	+	0.877999i	$0.341122\pi$
$594$	0	0
$595$	1.44931e23	0.378239
$596$	0	0
$597$	−4.00584e23	−1.01604
$598$	0	0
$599$	−3.17118e23	−0.781796	−0.390898	−	0.920434i	$-0.627835\pi$
$599$	−3.17118e23	−0.781796	−0.390898	+	0.920434i	$0.627835\pi$
$600$	0	0
$601$	−2.47473e23	−0.593055	−0.296528	−	0.955024i	$-0.595829\pi$
$601$	−2.47473e23	−0.593055	−0.296528	+	0.955024i	$0.595829\pi$
$602$	0	0
$603$	3.20478e21	0.00746622
$604$	0	0
$605$	1.34567e23	0.304803
$606$	0	0
$607$	−1.27464e23	−0.280726	−0.140363	−	0.990100i	$-0.544827\pi$
$607$	−1.27464e23	−0.280726	−0.140363	+	0.990100i	$0.544827\pi$
$608$	0	0
$609$	2.61086e23	0.559161
$610$	0	0
$611$	1.41655e23	0.295040
$612$	0	0
$613$	2.08992e23	0.423366	0.211683	−	0.977338i	$-0.432105\pi$
$613$	2.08992e23	0.423366	0.211683	+	0.977338i	$0.432105\pi$
$614$	0	0
$615$	−1.38967e23	−0.273825
$616$	0	0
$617$	2.52173e23	0.483364	0.241682	−	0.970356i	$-0.422301\pi$
$617$	2.52173e23	0.483364	0.241682	+	0.970356i	$0.422301\pi$
$618$	0	0
$619$	8.32171e23	1.55182	0.775911	−	0.630842i	$-0.217291\pi$
$619$	8.32171e23	1.55182	0.775911	+	0.630842i	$0.217291\pi$
$620$	0	0
$621$	−2.24956e22	−0.0408150
$622$	0	0
$623$	4.45681e22	0.0786821
$624$	0	0
$625$	2.32831e22	0.0400000
$626$	0	0
$627$	5.49781e23	0.919212
$628$	0	0
$629$	6.84733e23	1.11427
$630$	0	0
$631$	−2.44878e23	−0.387883	−0.193942	−	0.981013i	$-0.562127\pi$
$631$	−2.44878e23	−0.387883	−0.193942	+	0.981013i	$0.562127\pi$
$632$	0	0
$633$	−2.91510e23	−0.449491
$634$	0	0
$635$	1.34968e23	0.202607
$636$	0	0
$637$	−1.21335e23	−0.177338
$638$	0	0
$639$	−1.48081e23	−0.210738
$640$	0	0
$641$	8.76132e23	1.21416	0.607081	−	0.794640i	$-0.292340\pi$
$641$	8.76132e23	1.21416	0.607081	+	0.794640i	$0.292340\pi$
$642$	0	0
$643$	3.14015e23	0.423796	0.211898	−	0.977292i	$-0.432036\pi$
$643$	3.14015e23	0.423796	0.211898	+	0.977292i	$0.432036\pi$
$644$	0	0
$645$	−6.47452e23	−0.851040
$646$	0	0
$647$	7.19139e22	0.0920718	0.0460359	−	0.998940i	$-0.485341\pi$
$647$	7.19139e22	0.0920718	0.0460359	+	0.998940i	$0.485341\pi$
$648$	0	0
$649$	6.50640e23	0.811448
$650$	0	0
$651$	−1.87991e23	−0.228402
$652$	0	0
$653$	−7.53337e23	−0.891717	−0.445858	−	0.895103i	$-0.647102\pi$
$653$	−7.53337e23	−0.891717	−0.445858	+	0.895103i	$0.647102\pi$
$654$	0	0
$655$	−2.31634e23	−0.267148
$656$	0	0
$657$	2.44193e22	0.0274428
$658$	0	0
$659$	1.78954e24	1.95981	0.979906	−	0.199459i	$-0.0639186\pi$
$659$	1.78954e24	1.95981	0.979906	+	0.199459i	$0.0639186\pi$
$660$	0	0
$661$	3.55263e23	0.379173	0.189586	−	0.981864i	$-0.439285\pi$
$661$	3.55263e23	0.379173	0.189586	+	0.981864i	$0.439285\pi$
$662$	0	0
$663$	4.01047e23	0.417187
$664$	0	0
$665$	−4.23996e23	−0.429910
$666$	0	0
$667$	3.74951e22	0.0370599
$668$	0	0
$669$	8.93533e23	0.860971
$670$	0	0
$671$	−9.90603e23	−0.930589
$672$	0	0
$673$	5.15130e23	0.471833	0.235917	−	0.971773i	$-0.424191\pi$
$673$	5.15130e23	0.471833	0.235917	+	0.971773i	$0.424191\pi$
$674$	0	0
$675$	2.08244e23	0.185990
$676$	0	0
$677$	1.69205e23	0.147370	0.0736851	−	0.997282i	$-0.476524\pi$
$677$	1.69205e23	0.147370	0.0736851	+	0.997282i	$0.476524\pi$
$678$	0	0
$679$	9.70628e23	0.824441
$680$	0	0
$681$	1.15265e23	0.0954880
$682$	0	0
$683$	−1.97763e23	−0.159797	−0.0798987	−	0.996803i	$-0.525460\pi$
$683$	−1.97763e23	−0.159797	−0.0798987	+	0.996803i	$0.525460\pi$
$684$	0	0
$685$	2.20518e23	0.173809
$686$	0	0
$687$	−2.38051e24	−1.83036
$688$	0	0
$689$	−6.20943e23	−0.465788
$690$	0	0
$691$	−2.62293e24	−1.91966	−0.959828	−	0.280589i	$-0.909470\pi$
$691$	−2.62293e24	−1.91966	−0.959828	+	0.280589i	$0.909470\pi$
$692$	0	0
$693$	−6.03173e22	−0.0430734
$694$	0	0
$695$	8.34028e23	0.581179
$696$	0	0
$697$	1.14999e24	0.782018
$698$	0	0
$699$	2.16413e24	1.43624
$700$	0	0
$701$	−4.05441e22	−0.0262618	−0.0131309	−	0.999914i	$-0.504180\pi$
$701$	−4.05441e22	−0.0262618	−0.0131309	+	0.999914i	$0.504180\pi$
$702$	0	0
$703$	−2.00319e24	−1.26649
$704$	0	0
$705$	7.78000e23	0.480144
$706$	0	0
$707$	2.97387e23	0.179167
$708$	0	0
$709$	−7.92278e23	−0.465999	−0.232999	−	0.972477i	$-0.574854\pi$
$709$	−7.92278e23	−0.465999	−0.232999	+	0.972477i	$0.574854\pi$
$710$	0	0
$711$	−1.31030e23	−0.0752451
$712$	0	0
$713$	−2.69977e22	−0.0151379
$714$	0	0
$715$	1.34174e23	0.0734630
$716$	0	0
$717$	−2.41005e24	−1.28859
$718$	0	0
$719$	7.47670e23	0.390404	0.195202	−	0.980763i	$-0.437464\pi$
$719$	7.47670e23	0.390404	0.195202	+	0.980763i	$0.437464\pi$
$720$	0	0
$721$	1.63648e24	0.834566
$722$	0	0
$723$	2.18625e24	1.08899
$724$	0	0
$725$	−3.47095e23	−0.168879
$726$	0	0
$727$	−5.40980e23	−0.257122	−0.128561	−	0.991702i	$-0.541036\pi$
$727$	−5.40980e23	−0.257122	−0.128561	+	0.991702i	$0.541036\pi$
$728$	0	0
$729$	1.83043e24	0.849902
$730$	0	0
$731$	5.35787e24	2.43049
$732$	0	0
$733$	2.02014e24	0.895361	0.447680	−	0.894194i	$-0.352250\pi$
$733$	2.02014e24	0.895361	0.447680	+	0.894194i	$0.352250\pi$
$734$	0	0
$735$	−6.66400e23	−0.288597
$736$	0	0
$737$	−8.15396e22	−0.0345060
$738$	0	0
$739$	2.11021e24	0.872666	0.436333	−	0.899785i	$-0.356277\pi$
$739$	2.11021e24	0.872666	0.436333	+	0.899785i	$0.356277\pi$
$740$	0	0
$741$	−1.17327e24	−0.474179
$742$	0	0
$743$	−1.34376e24	−0.530784	−0.265392	−	0.964141i	$-0.585501\pi$
$743$	−1.34376e24	−0.530784	−0.265392	+	0.964141i	$0.585501\pi$
$744$	0	0
$745$	−8.24871e23	−0.318462
$746$	0	0
$747$	9.69264e22	0.0365777
$748$	0	0
$749$	1.56440e24	0.577102
$750$	0	0
$751$	−2.72368e24	−0.982236	−0.491118	−	0.871093i	$-0.663412\pi$
$751$	−2.72368e24	−0.982236	−0.491118	+	0.871093i	$0.663412\pi$
$752$	0	0
$753$	−3.85441e24	−1.35894
$754$	0	0
$755$	4.27525e23	0.147371
$756$	0	0
$757$	−3.29400e24	−1.11022	−0.555109	−	0.831777i	$-0.687324\pi$
$757$	−3.29400e24	−1.11022	−0.555109	+	0.831777i	$0.687324\pi$
$758$	0	0
$759$	−7.96057e22	−0.0262355
$760$	0	0
$761$	−4.29576e24	−1.38443	−0.692215	−	0.721692i	$-0.743365\pi$
$761$	−4.29576e24	−1.38443	−0.692215	+	0.721692i	$0.743365\pi$
$762$	0	0
$763$	2.67290e24	0.842410
$764$	0	0
$765$	2.39680e23	0.0738771
$766$	0	0
$767$	−1.38851e24	−0.418588
$768$	0	0
$769$	5.93781e24	1.75086	0.875432	−	0.483341i	$-0.160577\pi$
$769$	5.93781e24	1.75086	0.875432	+	0.483341i	$0.160577\pi$
$770$	0	0
$771$	4.02711e24	1.16153
$772$	0	0
$773$	−1.98352e24	−0.559642	−0.279821	−	0.960052i	$-0.590275\pi$
$773$	−1.98352e24	−0.559642	−0.279821	+	0.960052i	$0.590275\pi$
$774$	0	0
$775$	2.49920e23	0.0689822
$776$	0	0
$777$	2.01970e24	0.545391
$778$	0	0
$779$	−3.36431e24	−0.888849
$780$	0	0
$781$	3.76766e24	0.973952
$782$	0	0
$783$	−3.10442e24	−0.785246
$784$	0	0
$785$	−1.04678e24	−0.259097
$786$	0	0
$787$	−3.61371e24	−0.875322	−0.437661	−	0.899140i	$-0.644193\pi$
$787$	−3.61371e24	−0.875322	−0.437661	+	0.899140i	$0.644193\pi$
$788$	0	0
$789$	3.55156e24	0.841907
$790$	0	0
$791$	−5.05970e24	−1.17388
$792$	0	0
$793$	2.11401e24	0.480047
$794$	0	0
$795$	−3.41036e24	−0.758018
$796$	0	0
$797$	−4.63064e24	−1.00750	−0.503751	−	0.863849i	$-0.668047\pi$
$797$	−4.63064e24	−1.00750	−0.503751	+	0.863849i	$0.668047\pi$
$798$	0	0
$799$	−6.43819e24	−1.37125
$800$	0	0
$801$	7.37048e22	0.0153681
$802$	0	0
$803$	−6.21304e23	−0.126830
$804$	0	0
$805$	6.13925e22	0.0122702
$806$	0	0
$807$	−9.11220e24	−1.78319
$808$	0	0
$809$	−1.22642e24	−0.235005	−0.117502	−	0.993073i	$-0.537489\pi$
$809$	−1.22642e24	−0.235005	−0.117502	+	0.993073i	$0.537489\pi$
$810$	0	0
$811$	2.42686e24	0.455373	0.227686	−	0.973735i	$-0.426884\pi$
$811$	2.42686e24	0.455373	0.227686	+	0.973735i	$0.426884\pi$
$812$	0	0
$813$	−1.43139e24	−0.263020
$814$	0	0
$815$	−3.70132e24	−0.666066
$816$	0	0
$817$	−1.56745e25	−2.76252
$818$	0	0
$819$	1.28721e23	0.0222196
$820$	0	0
$821$	−9.87401e24	−1.66946	−0.834731	−	0.550658i	$-0.814377\pi$
$821$	−9.87401e24	−1.66946	−0.834731	+	0.550658i	$0.814377\pi$
$822$	0	0
$823$	3.09810e24	0.513094	0.256547	−	0.966532i	$-0.417415\pi$
$823$	3.09810e24	0.513094	0.256547	+	0.966532i	$0.417415\pi$
$824$	0	0
$825$	7.36917e23	0.119553
$826$	0	0
$827$	3.89248e24	0.618628	0.309314	−	0.950960i	$-0.399901\pi$
$827$	3.89248e24	0.618628	0.309314	+	0.950960i	$0.399901\pi$
$828$	0	0
$829$	−1.04208e25	−1.62251	−0.811256	−	0.584691i	$-0.801216\pi$
$829$	−1.04208e25	−1.62251	−0.811256	+	0.584691i	$0.801216\pi$
$830$	0	0
$831$	2.36397e24	0.360606
$832$	0	0
$833$	5.51466e24	0.824207
$834$	0	0
$835$	−2.61979e24	−0.383646
$836$	0	0
$837$	2.23529e24	0.320750
$838$	0	0
$839$	6.67595e24	0.938721	0.469360	−	0.883007i	$-0.344485\pi$
$839$	6.67595e24	0.938721	0.469360	+	0.883007i	$0.344485\pi$
$840$	0	0
$841$	−2.08280e24	−0.286999
$842$	0	0
$843$	4.08123e24	0.551133
$844$	0	0
$845$	3.09273e24	0.409318
$846$	0	0
$847$	−3.28465e24	−0.426069
$848$	0	0
$849$	4.85659e24	0.617470
$850$	0	0
$851$	2.90052e23	0.0361472
$852$	0	0
$853$	9.79573e24	1.19666	0.598329	−	0.801250i	$-0.295831\pi$
$853$	9.79573e24	1.19666	0.598329	+	0.801250i	$0.295831\pi$
$854$	0	0
$855$	−7.01185e23	−0.0839694
$856$	0	0
$857$	−4.38126e24	−0.514354	−0.257177	−	0.966364i	$-0.582792\pi$
$857$	−4.38126e24	−0.514354	−0.257177	+	0.966364i	$0.582792\pi$
$858$	0	0
$859$	1.50626e25	1.73364	0.866821	−	0.498620i	$-0.166159\pi$
$859$	1.50626e25	1.73364	0.866821	+	0.498620i	$0.166159\pi$
$860$	0	0
$861$	3.39204e24	0.382766
$862$	0	0
$863$	−1.37694e25	−1.52344	−0.761718	−	0.647909i	$-0.775644\pi$
$863$	−1.37694e25	−1.52344	−0.761718	+	0.647909i	$0.775644\pi$
$864$	0	0
$865$	−4.12870e24	−0.447896
$866$	0	0
$867$	−8.26942e24	−0.879655
$868$	0	0
$869$	3.33381e24	0.347754
$870$	0	0
$871$	1.74010e23	0.0178000
$872$	0	0
$873$	1.60518e24	0.161029
$874$	0	0
$875$	−5.68316e23	−0.0559141
$876$	0	0
$877$	−7.56082e24	−0.729579	−0.364790	−	0.931090i	$-0.618859\pi$
$877$	−7.56082e24	−0.729579	−0.364790	+	0.931090i	$0.618859\pi$
$878$	0	0
$879$	1.27332e24	0.120513
$880$	0	0
$881$	5.11196e24	0.474561	0.237281	−	0.971441i	$-0.423744\pi$
$881$	5.11196e24	0.474561	0.237281	+	0.971441i	$0.423744\pi$
$882$	0	0
$883$	1.37334e25	1.25058	0.625289	−	0.780393i	$-0.284981\pi$
$883$	1.37334e25	1.25058	0.625289	+	0.780393i	$0.284981\pi$
$884$	0	0
$885$	−7.62599e24	−0.681206
$886$	0	0
$887$	−1.15450e25	−1.01168	−0.505838	−	0.862628i	$-0.668817\pi$
$887$	−1.15450e25	−1.01168	−0.505838	+	0.862628i	$0.668817\pi$
$888$	0	0
$889$	−3.29443e24	−0.283215
$890$	0	0
$891$	7.40794e24	0.624794
$892$	0	0
$893$	1.88349e25	1.55857
$894$	0	0
$895$	−7.76241e24	−0.630233
$896$	0	0
$897$	1.69883e23	0.0135337
$898$	0	0
$899$	−3.72571e24	−0.291240
$900$	0	0
$901$	2.82218e25	2.16483
$902$	0	0
$903$	1.58036e25	1.18963
$904$	0	0
$905$	−9.77250e24	−0.721926
$906$	0	0
$907$	1.74990e25	1.26868	0.634338	−	0.773056i	$-0.281273\pi$
$907$	1.74990e25	1.26868	0.634338	+	0.773056i	$0.281273\pi$
$908$	0	0
$909$	4.91806e23	0.0349946
$910$	0	0
$911$	1.12560e25	0.786102	0.393051	−	0.919517i	$-0.371420\pi$
$911$	1.12560e25	0.786102	0.393051	+	0.919517i	$0.371420\pi$
$912$	0	0
$913$	−2.46611e24	−0.169048
$914$	0	0
$915$	1.16106e25	0.781224
$916$	0	0
$917$	5.65395e24	0.373433
$918$	0	0
$919$	2.33947e24	0.151682	0.0758412	−	0.997120i	$-0.475836\pi$
$919$	2.33947e24	0.151682	0.0758412	+	0.997120i	$0.475836\pi$
$920$	0	0
$921$	−2.20184e25	−1.40145
$922$	0	0
$923$	−8.04040e24	−0.502416
$924$	0	0
$925$	−2.68504e24	−0.164720
$926$	0	0
$927$	2.70634e24	0.163006
$928$	0	0
$929$	3.62091e24	0.214133	0.107067	−	0.994252i	$-0.465854\pi$
$929$	3.62091e24	0.214133	0.107067	+	0.994252i	$0.465854\pi$
$930$	0	0
$931$	−1.61332e25	−0.936801
$932$	0	0
$933$	2.41656e25	1.37786
$934$	0	0
$935$	−6.09821e24	−0.341432
$936$	0	0
$937$	4.50917e24	0.247919	0.123960	−	0.992287i	$-0.460441\pi$
$937$	4.50917e24	0.247919	0.123960	+	0.992287i	$0.460441\pi$
$938$	0	0
$939$	3.42877e25	1.85132
$940$	0	0
$941$	2.46547e25	1.30734	0.653669	−	0.756780i	$-0.273229\pi$
$941$	2.46547e25	1.30734	0.653669	+	0.756780i	$0.273229\pi$
$942$	0	0
$943$	4.87136e23	0.0253689
$944$	0	0
$945$	−5.08302e24	−0.259987
$946$	0	0
$947$	−2.52153e25	−1.26675	−0.633373	−	0.773847i	$-0.718330\pi$
$947$	−2.52153e25	−1.26675	−0.633373	+	0.773847i	$0.718330\pi$
$948$	0	0
$949$	1.32590e24	0.0654257
$950$	0	0
$951$	−3.30253e24	−0.160071
$952$	0	0
$953$	2.22635e25	1.05999	0.529996	−	0.848000i	$-0.322193\pi$
$953$	2.22635e25	1.05999	0.529996	+	0.848000i	$0.322193\pi$
$954$	0	0
$955$	−1.02885e25	−0.481197
$956$	0	0
$957$	−1.09857e25	−0.504748
$958$	0	0
$959$	−5.38261e24	−0.242960
$960$	0	0
$961$	−1.98675e25	−0.881037
$962$	0	0
$963$	2.58714e24	0.112719
$964$	0	0
$965$	−1.44085e25	−0.616787
$966$	0	0
$967$	4.73776e24	0.199273	0.0996363	−	0.995024i	$-0.468232\pi$
$967$	4.73776e24	0.199273	0.0996363	+	0.995024i	$0.468232\pi$
$968$	0	0
$969$	5.33248e25	2.20383
$970$	0	0
$971$	3.41703e25	1.38767	0.693833	−	0.720136i	$-0.255921\pi$
$971$	3.41703e25	1.38767	0.693833	+	0.720136i	$0.255921\pi$
$972$	0	0
$973$	−2.03578e25	−0.812402
$974$	0	0
$975$	−1.57262e24	−0.0616717
$976$	0	0
$977$	6.32382e24	0.243711	0.121856	−	0.992548i	$-0.461116\pi$
$977$	6.32382e24	0.243711	0.121856	+	0.992548i	$0.461116\pi$
$978$	0	0
$979$	−1.87528e24	−0.0710254
$980$	0	0
$981$	4.42032e24	0.164538
$982$	0	0
$983$	5.13743e25	1.87949	0.939747	−	0.341870i	$-0.111060\pi$
$983$	5.13743e25	1.87949	0.939747	+	0.341870i	$0.111060\pi$
$984$	0	0
$985$	1.58172e25	0.568750
$986$	0	0
$987$	−1.89902e25	−0.671171
$988$	0	0
$989$	2.26959e24	0.0788458
$990$	0	0
$991$	−3.11936e25	−1.06522	−0.532610	−	0.846361i	$-0.678789\pi$
$991$	−3.11936e25	−1.06522	−0.532610	+	0.846361i	$0.678789\pi$
$992$	0	0
$993$	2.20919e24	0.0741592
$994$	0	0
$995$	−1.29989e25	−0.428954
$996$	0	0
$997$	−4.66217e25	−1.51244	−0.756222	−	0.654315i	$-0.772957\pi$
$997$	−4.66217e25	−1.51244	−0.756222	+	0.654315i	$0.772957\pi$
$998$	0	0
$999$	−2.40150e25	−0.765907

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.18.a.e.1.1		2
4.3	odd	2		10.18.a.b.1.2	✓	2
12.11	even	2		90.18.a.n.1.1		2
20.3	even	4		50.18.b.e.49.4		4
20.7	even	4		50.18.b.e.49.1		4
20.19	odd	2		50.18.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.a.b.1.2	✓	2	4.3	odd	2
50.18.a.g.1.1		2	20.19	odd	2
50.18.b.e.49.1		4	20.7	even	4
50.18.b.e.49.4		4	20.3	even	4
80.18.a.e.1.1		2	1.1	even	1	trivial
90.18.a.n.1.1		2	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q-12037.8 q^{3} -390625. q^{5} +9.53475e6 q^{7} +1.57682e7 q^{9} -4.01191e8 q^{11} +8.56166e8 q^{13} +4.70226e9 q^{15} -3.89127e10 q^{17} +1.13839e11 q^{19} -1.14777e11 q^{21} -1.64834e10 q^{23} +1.52588e11 q^{25} +1.36475e12 q^{27} -2.27472e12 q^{29} +1.63788e12 q^{31} +4.82946e12 q^{33} -3.72451e12 q^{35} -1.75967e13 q^{37} -1.03064e13 q^{39} -2.95532e13 q^{41} -1.37690e14 q^{43} -6.15944e12 q^{45} +1.65452e14 q^{47} -1.41719e14 q^{49} +4.68422e14 q^{51} -7.25259e14 q^{53} +1.56715e14 q^{55} -1.37037e15 q^{57} -1.62177e15 q^{59} +2.46915e15 q^{61} +1.50346e14 q^{63} -3.34440e14 q^{65} +2.03244e14 q^{67} +1.98423e14 q^{69} -9.39117e15 q^{71} +1.54865e15 q^{73} -1.83682e15 q^{75} -3.82526e15 q^{77} -8.30977e15 q^{79} -1.84648e16 q^{81} +6.14697e15 q^{83} +1.52003e16 q^{85} +2.73826e16 q^{87} +4.67428e15 q^{89} +8.16334e15 q^{91} -1.97164e16 q^{93} -4.44684e16 q^{95} +1.01799e17 q^{97} -6.32605e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6308 q^{3} - 781250 q^{5} - 6543844 q^{7} + 223195906 q^{9} - 1189408704 q^{11} - 2017919228 q^{13} - 2464062500 q^{15} - 18755639436 q^{17} + 136704830600 q^{19} - 409751898376 q^{21} + 649234170708 q^{23}+ \cdots - 16\!\cdots\!12 q^{99}+O(q^{100}) \) 2 * q + 6308 * q^3 - 781250 * q^5 - 6543844 * q^7 + 223195906 * q^9 - 1189408704 * q^11 - 2017919228 * q^13 - 2464062500 * q^15 - 18755639436 * q^17 + 136704830600 * q^19 - 409751898376 * q^21 + 649234170708 * q^23 + 305175781250 * q^25 + 2800995425720 * q^27 - 4696543420020 * q^29 - 7120867378744 * q^31 - 9631011282816 * q^33 + 2556189062500 * q^35 + 9933114637444 * q^37 - 63033715222712 * q^39 + 9622711880844 * q^41 - 7708436011852 * q^43 - 87185900781250 * q^45 + 466072575837996 * q^47 - 115828146887046 * q^49 + 838218717084456 * q^51 - 623333120284428 * q^53 + 464612775000000 * q^55 - 950883425062000 * q^57 - 654616071995160 * q^59 + 1329040385162884 * q^61 - 3184801966786532 * q^63 + 788249698437500 * q^65 + 4249016411877596 * q^67 + 12411535659353832 * q^69 - 2035131949347624 * q^71 + 2248593461347972 * q^73 + 962524414062500 * q^75 + 8848168987836288 * q^77 - 19188427678495120 * q^79 - 18903014015714198 * q^81 + 18474649086611988 * q^83 + 7326421654687500 * q^85 - 17047661699498280 * q^87 - 17458912916395020 * q^89 + 54374608211433016 * q^91 - 180402465813627376 * q^93 - 53400324453125000 * q^95 + 195934325822490244 * q^97 - 169824195828146112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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