LMFDB - Embedded newform 45.18.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,18,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	45.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:	$82.4499393051$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{849}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 212 $ x^2 - x - 212
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$15.0688$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-346.477 q^{2} -11025.8 q^{4} -390625. q^{5} -1.80797e7 q^{7} +4.92336e7 q^{8} +1.35343e8 q^{10} +1.45296e8 q^{11} -3.26155e9 q^{13} +6.26421e9 q^{14} -1.56131e10 q^{16} -7.81194e9 q^{17} -7.18160e10 q^{19} +4.30694e9 q^{20} -5.03416e10 q^{22} -1.67169e11 q^{23} +1.52588e11 q^{25} +1.13005e12 q^{26} +1.99343e11 q^{28} -1.63211e11 q^{29} -5.38766e12 q^{31} -1.04356e12 q^{32} +2.70666e12 q^{34} +7.06240e12 q^{35} -3.68759e13 q^{37} +2.48826e13 q^{38} -1.92319e13 q^{40} +7.52018e12 q^{41} -5.73935e13 q^{43} -1.60200e12 q^{44} +5.79201e13 q^{46} -1.42516e14 q^{47} +9.42466e13 q^{49} -5.28682e13 q^{50} +3.59611e13 q^{52} -8.01583e14 q^{53} -5.67561e13 q^{55} -8.90131e14 q^{56} +5.65490e13 q^{58} -9.64437e14 q^{59} -1.29792e15 q^{61} +1.86670e15 q^{62} +2.40801e15 q^{64} +1.27404e15 q^{65} +4.71635e15 q^{67} +8.61327e13 q^{68} -2.44696e15 q^{70} -9.29884e15 q^{71} -6.86407e15 q^{73} +1.27767e16 q^{74} +7.91827e14 q^{76} -2.62691e15 q^{77} +1.21446e16 q^{79} +6.09888e15 q^{80} -2.60557e15 q^{82} +3.23986e16 q^{83} +3.05154e15 q^{85} +1.98855e16 q^{86} +7.15343e15 q^{88} +4.91787e16 q^{89} +5.89681e16 q^{91} +1.84316e15 q^{92} +4.93784e16 q^{94} +2.80531e16 q^{95} +1.26124e17 q^{97} -3.26543e16 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 356 q^{2} + 351376 q^{4} - 781250 q^{5} - 20754552 q^{7} + 211737408 q^{8} - 139062500 q^{10} + 1131629912 q^{11} - 446672524 q^{13} + 4385222016 q^{14} + 51041317120 q^{16} - 20662021036 q^{17} - 194376127216 q^{19}+ \cdots - 19\!\cdots\!04 q^{98}+O(q^{100}) $ 2 * q + 356 * q^2 + 351376 * q^4 - 781250 * q^5 - 20754552 * q^7 + 211737408 * q^8 - 139062500 * q^10 + 1131629912 * q^11 - 446672524 * q^13 + 4385222016 * q^14 + 51041317120 * q^16 - 20662021036 * q^17 - 194376127216 * q^19 - 137256250000 * q^20 + 642535396112 * q^22 - 281815042584 * q^23 + 305175781250 * q^25 + 3107442504536 * q^26 - 770012242944 * q^28 + 4737062625892 * q^29 + 2256407074872 * q^31 + 24479953196032 * q^32 - 6320225301928 * q^34 + 8107246875000 * q^35 - 33686114979452 * q^37 - 61213036677904 * q^38 - 82709925000000 * q^40 + 13051545859228 * q^41 - 33149929851400 * q^43 + 355847272731712 * q^44 - 22616426690688 * q^46 - 84299585768440 * q^47 - 131229302667470 * q^49 + 54321289062500 * q^50 + 1056079273719136 * q^52 - 155836696419964 * q^53 - 442042934375000 * q^55 - 1324797309480960 * q^56 + 3498878281127368 * q^58 - 963392423116456 * q^59 - 2858793214972660 * q^61 + 7236477385534944 * q^62 + 11601155857485824 * q^64 + 174481454687500 * q^65 + 6027859197776136 * q^67 - 4570758410249888 * q^68 - 1712977350000000 * q^70 - 2206536655060304 * q^71 - 18626002136017276 * q^73 + 15017443813698616 * q^74 - 43624167666234944 * q^76 - 5265163905377376 * q^77 + 4997891097934440 * q^79 - 19938014500000000 * q^80 + 1280087848985176 * q^82 + 23351553482511288 * q^83 + 8071101967187500 * q^85 + 36916031436093872 * q^86 + 167436503858533632 * q^88 + 60857257652216796 * q^89 + 51438794903148912 * q^91 - 39704913484919808 * q^92 + 90273897275494496 * q^94 + 75928174693750000 * q^95 + 42869862659782756 * q^97 - 191045888549711804 * q^98

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 $

$2$	−346.477	−0.957016	−0.478508	−	0.878083i	$-0.658822\pi$
$2$	−346.477	−0.957016	−0.478508	+	0.878083i	$0.658822\pi$
$3$	0	0
$3$	0	0
$4$	−11025.8	−0.0841199
$5$	−390625.	−0.447214
$5$	−390625.	−0.447214
$6$	0	0
$7$	−1.80797e7	−1.18538	−0.592692	−	0.805429i	$-0.701935\pi$
$7$	−1.80797e7	−1.18538	−0.592692	+	0.805429i	$0.701935\pi$
$8$	4.92336e7	1.03752
$9$	0	0
$10$	1.35343e8	0.427991
$11$	1.45296e8	0.204369	0.102184	−	0.994765i	$-0.467417\pi$
$11$	1.45296e8	0.204369	0.102184	+	0.994765i	$0.467417\pi$
$12$	0	0
$13$	−3.26155e9	−1.10894	−0.554468	−	0.832205i	$-0.687078\pi$
$13$	−3.26155e9	−1.10894	−0.554468	+	0.832205i	$0.687078\pi$
$14$	6.26421e9	1.13443
$15$	0	0
$16$	−1.56131e10	−0.908804
$17$	−7.81194e9	−0.271608	−0.135804	−	0.990736i	$-0.543362\pi$
$17$	−7.81194e9	−0.271608	−0.135804	+	0.990736i	$0.543362\pi$
$18$	0	0
$19$	−7.18160e10	−0.970098	−0.485049	−	0.874487i	$-0.661198\pi$
$19$	−7.18160e10	−0.970098	−0.485049	+	0.874487i	$0.661198\pi$
$20$	4.30694e9	0.0376196
$21$	0	0
$22$	−5.03416e10	−0.195584
$23$	−1.67169e11	−0.445111	−0.222555	−	0.974920i	$-0.571440\pi$
$23$	−1.67169e11	−0.445111	−0.222555	+	0.974920i	$0.571440\pi$
$24$	0	0
$25$	1.52588e11	0.200000
$26$	1.13005e12	1.06127
$27$	0	0
$28$	1.99343e11	0.0997144
$29$	−1.63211e11	−0.0605854	−0.0302927	−	0.999541i	$-0.509644\pi$
$29$	−1.63211e11	−0.0605854	−0.0302927	+	0.999541i	$0.509644\pi$
$30$	0	0
$31$	−5.38766e12	−1.13456	−0.567278	−	0.823527i	$-0.692003\pi$
$31$	−5.38766e12	−1.13456	−0.567278	+	0.823527i	$0.692003\pi$
$32$	−1.04356e12	−0.167780
$33$	0	0
$34$	2.70666e12	0.259934
$35$	7.06240e12	0.530120
$36$	0	0
$37$	−3.68759e13	−1.72595	−0.862976	−	0.505246i	$-0.831402\pi$
$37$	−3.68759e13	−1.72595	−0.862976	+	0.505246i	$0.831402\pi$
$38$	2.48826e13	0.928400
$39$	0	0
$40$	−1.92319e13	−0.463993
$41$	7.52018e12	0.147084	0.0735420	−	0.997292i	$-0.476570\pi$
$41$	7.52018e12	0.147084	0.0735420	+	0.997292i	$0.476570\pi$
$42$	0	0
$43$	−5.73935e13	−0.748825	−0.374413	−	0.927262i	$-0.622156\pi$
$43$	−5.73935e13	−0.748825	−0.374413	+	0.927262i	$0.622156\pi$
$44$	−1.60200e12	−0.0171915
$45$	0	0
$46$	5.79201e13	0.425978
$47$	−1.42516e14	−0.873034	−0.436517	−	0.899696i	$-0.643788\pi$
$47$	−1.42516e14	−0.873034	−0.436517	+	0.899696i	$0.643788\pi$
$48$	0	0
$49$	9.42466e13	0.405134
$50$	−5.28682e13	−0.191403
$51$	0	0
$52$	3.59611e13	0.0932836
$53$	−8.01583e14	−1.76849	−0.884247	−	0.467020i	$-0.845328\pi$
$53$	−8.01583e14	−1.76849	−0.884247	+	0.467020i	$0.845328\pi$
$54$	0	0
$55$	−5.67561e13	−0.0913966
$56$	−8.90131e14	−1.22986
$57$	0	0
$58$	5.65490e13	0.0579812
$59$	−9.64437e14	−0.855129	−0.427565	−	0.903985i	$-0.640628\pi$
$59$	−9.64437e14	−0.855129	−0.427565	+	0.903985i	$0.640628\pi$
$60$	0	0
$61$	−1.29792e15	−0.866849	−0.433424	−	0.901190i	$-0.642695\pi$
$61$	−1.29792e15	−0.866849	−0.433424	+	0.901190i	$0.642695\pi$
$62$	1.86670e15	1.08579
$63$	0	0
$64$	2.40801e15	1.06937
$65$	1.27404e15	0.495931
$66$	0	0
$67$	4.71635e15	1.41896	0.709480	−	0.704726i	$-0.248930\pi$
$67$	4.71635e15	1.41896	0.709480	+	0.704726i	$0.248930\pi$
$68$	8.61327e13	0.0228477
$69$	0	0
$70$	−2.44696e15	−0.507333
$71$	−9.29884e15	−1.70896	−0.854482	−	0.519482i	$-0.826125\pi$
$71$	−9.29884e15	−1.70896	−0.854482	+	0.519482i	$0.826125\pi$
$72$	0	0
$73$	−6.86407e15	−0.996180	−0.498090	−	0.867125i	$-0.665965\pi$
$73$	−6.86407e15	−0.996180	−0.498090	+	0.867125i	$0.665965\pi$
$74$	1.27767e16	1.65176
$75$	0	0
$76$	7.91827e14	0.0816046
$77$	−2.62691e15	−0.242256
$78$	0	0
$79$	1.21446e16	0.900646	0.450323	−	0.892866i	$-0.351309\pi$
$79$	1.21446e16	0.900646	0.450323	+	0.892866i	$0.351309\pi$
$80$	6.09888e15	0.406429
$81$	0	0
$82$	−2.60557e15	−0.140762
$83$	3.23986e16	1.57893	0.789465	−	0.613796i	$-0.210358\pi$
$83$	3.23986e16	1.57893	0.789465	+	0.613796i	$0.210358\pi$
$84$	0	0
$85$	3.05154e15	0.121467
$86$	1.98855e16	0.716638
$87$	0	0
$88$	7.15343e15	0.212037
$89$	4.91787e16	1.32423	0.662113	−	0.749404i	$-0.269660\pi$
$89$	4.91787e16	1.32423	0.662113	+	0.749404i	$0.269660\pi$
$90$	0	0
$91$	5.89681e16	1.31451
$92$	1.84316e15	0.0374427
$93$	0	0
$94$	4.93784e16	0.835507
$95$	2.80531e16	0.433841
$96$	0	0
$97$	1.26124e17	1.63394	0.816972	−	0.576678i	$-0.195651\pi$
$97$	1.26124e17	1.63394	0.816972	+	0.576678i	$0.195651\pi$
$98$	−3.26543e16	−0.387720
$99$	0	0
$100$	−1.68240e15	−0.0168240
$101$	1.89692e15	0.0174308	0.00871538	−	0.999962i	$-0.497226\pi$
$101$	1.89692e15	0.0174308	0.00871538	+	0.999962i	$0.497226\pi$
$102$	0	0
$103$	−8.39711e16	−0.653150	−0.326575	−	0.945171i	$-0.605895\pi$
$103$	−8.39711e16	−0.653150	−0.326575	+	0.945171i	$0.605895\pi$
$104$	−1.60578e17	−1.15054
$105$	0	0
$106$	2.77730e17	1.69248
$107$	−1.85086e17	−1.04139	−0.520694	−	0.853744i	$-0.674327\pi$
$107$	−1.85086e17	−1.04139	−0.520694	+	0.853744i	$0.674327\pi$
$108$	0	0
$109$	−6.78355e15	−0.0326086	−0.0163043	−	0.999867i	$-0.505190\pi$
$109$	−6.78355e15	−0.0326086	−0.0163043	+	0.999867i	$0.505190\pi$
$110$	1.96647e16	0.0874680
$111$	0	0
$112$	2.82281e17	1.07728
$113$	4.78376e17	1.69279	0.846395	−	0.532556i	$-0.178769\pi$
$113$	4.78376e17	1.69279	0.846395	+	0.532556i	$0.178769\pi$
$114$	0	0
$115$	6.53002e16	0.199060
$116$	1.79953e15	0.00509644
$117$	0	0
$118$	3.34155e17	0.818372
$119$	1.41238e17	0.321960
$120$	0	0
$121$	−4.84336e17	−0.958233
$122$	4.49698e17	0.829588
$123$	0	0
$124$	5.94031e16	0.0954387
$125$	−5.96046e16	−0.0894427
$126$	0	0
$127$	−9.82303e17	−1.28799	−0.643997	−	0.765028i	$-0.722725\pi$
$127$	−9.82303e17	−1.28799	−0.643997	+	0.765028i	$0.722725\pi$
$128$	−6.97540e17	−0.855626
$129$	0	0
$130$	−4.41427e17	−0.474614
$131$	1.52030e18	1.53153	0.765763	−	0.643123i	$-0.222362\pi$
$131$	1.52030e18	1.53153	0.765763	+	0.643123i	$0.222362\pi$
$132$	0	0
$133$	1.29842e18	1.14994
$134$	−1.63411e18	−1.35797
$135$	0	0
$136$	−3.84610e17	−0.281799
$137$	1.70259e17	0.117215	0.0586077	−	0.998281i	$-0.481334\pi$
$137$	1.70259e17	0.117215	0.0586077	+	0.998281i	$0.481334\pi$
$138$	0	0
$139$	1.42834e18	0.869372	0.434686	−	0.900582i	$-0.356859\pi$
$139$	1.42834e18	0.869372	0.434686	+	0.900582i	$0.356859\pi$
$140$	−7.78684e16	−0.0445936
$141$	0	0
$142$	3.22183e18	1.63551
$143$	−4.73890e17	−0.226632
$144$	0	0
$145$	6.37545e16	0.0270946
$146$	2.37824e18	0.953360
$147$	0	0
$148$	4.06586e17	0.145187
$149$	1.68597e18	0.568549	0.284274	−	0.958743i	$-0.408247\pi$
$149$	1.68597e18	0.568549	0.284274	+	0.958743i	$0.408247\pi$
$150$	0	0
$151$	−6.03579e18	−1.81732	−0.908658	−	0.417542i	$-0.862892\pi$
$151$	−6.03579e18	−1.81732	−0.908658	+	0.417542i	$0.862892\pi$
$152$	−3.53576e18	−1.00650
$153$	0	0
$154$	9.10163e17	0.231843
$155$	2.10455e18	0.507388
$156$	0	0
$157$	2.55069e18	0.551456	0.275728	−	0.961236i	$-0.411081\pi$
$157$	2.55069e18	0.551456	0.275728	+	0.961236i	$0.411081\pi$
$158$	−4.20784e18	−0.861933
$159$	0	0
$160$	4.07639e17	0.0750336
$161$	3.02237e18	0.527627
$162$	0	0
$163$	−1.79794e18	−0.282606	−0.141303	−	0.989966i	$-0.545129\pi$
$163$	−1.79794e18	−0.282606	−0.141303	+	0.989966i	$0.545129\pi$
$164$	−8.29158e16	−0.0123727
$165$	0	0
$166$	−1.12254e19	−1.51106
$167$	−1.86888e18	−0.239052	−0.119526	−	0.992831i	$-0.538137\pi$
$167$	−1.86888e18	−0.239052	−0.119526	+	0.992831i	$0.538137\pi$
$168$	0	0
$169$	1.98732e18	0.229737
$170$	−1.05729e18	−0.116246
$171$	0	0
$172$	6.32807e17	0.0629912
$173$	−6.66325e18	−0.631385	−0.315692	−	0.948862i	$-0.602237\pi$
$173$	−6.66325e18	−0.631385	−0.315692	+	0.948862i	$0.602237\pi$
$174$	0	0
$175$	−2.75875e18	−0.237077
$176$	−2.26852e18	−0.185731
$177$	0	0
$178$	−1.70393e19	−1.26731
$179$	9.91594e18	0.703207	0.351603	−	0.936149i	$-0.385637\pi$
$179$	9.91594e18	0.703207	0.351603	+	0.936149i	$0.385637\pi$
$180$	0	0
$181$	−1.63607e19	−1.05568	−0.527841	−	0.849343i	$-0.676998\pi$
$181$	−1.63607e19	−1.05568	−0.527841	+	0.849343i	$0.676998\pi$
$182$	−2.04311e19	−1.25801
$183$	0	0
$184$	−8.23031e18	−0.461811
$185$	1.44047e19	0.771869
$186$	0	0
$187$	−1.13504e18	−0.0555083
$188$	1.57135e18	0.0734395
$189$	0	0
$190$	−9.71976e18	−0.415193
$191$	1.25239e19	0.511630	0.255815	−	0.966726i	$-0.417656\pi$
$191$	1.25239e19	0.511630	0.255815	+	0.966726i	$0.417656\pi$
$192$	0	0
$193$	−4.14210e19	−1.54876	−0.774379	−	0.632722i	$-0.781937\pi$
$193$	−4.14210e19	−1.54876	−0.774379	+	0.632722i	$0.781937\pi$
$194$	−4.36989e19	−1.56371
$195$	0	0
$196$	−1.03914e18	−0.0340799
$197$	−5.58901e19	−1.75538	−0.877691	−	0.479226i	$-0.840917\pi$
$197$	−5.58901e19	−1.75538	−0.877691	+	0.479226i	$0.840917\pi$
$198$	0	0
$199$	4.36216e19	1.25733	0.628667	−	0.777675i	$-0.283601\pi$
$199$	4.36216e19	1.25733	0.628667	+	0.777675i	$0.283601\pi$
$200$	7.51245e18	0.207504
$201$	0	0
$202$	−6.57237e17	−0.0166815
$203$	2.95082e18	0.0718169
$204$	0	0
$205$	−2.93757e18	−0.0657780
$206$	2.90940e19	0.625076
$207$	0	0
$208$	5.09231e19	1.00780
$209$	−1.04346e19	−0.198258
$210$	0	0
$211$	4.00768e19	0.702251	0.351125	−	0.936328i	$-0.385799\pi$
$211$	4.00768e19	0.702251	0.351125	+	0.936328i	$0.385799\pi$
$212$	8.83807e18	0.148766
$213$	0	0
$214$	6.41281e19	0.996624
$215$	2.24193e19	0.334885
$216$	0	0
$217$	9.74075e19	1.34488
$218$	2.35034e18	0.0312069
$219$	0	0
$220$	6.25780e17	0.00768827
$221$	2.54791e19	0.301196
$222$	0	0
$223$	−1.47056e19	−0.161024	−0.0805120	−	0.996754i	$-0.525656\pi$
$223$	−1.47056e19	−0.161024	−0.0805120	+	0.996754i	$0.525656\pi$
$224$	1.88672e19	0.198884
$225$	0	0
$226$	−1.65746e20	−1.62003
$227$	9.28542e19	0.874142	0.437071	−	0.899427i	$-0.356016\pi$
$227$	9.28542e19	0.874142	0.437071	+	0.899427i	$0.356016\pi$
$228$	0	0
$229$	8.55435e19	0.747456	0.373728	−	0.927538i	$-0.378079\pi$
$229$	8.55435e19	0.747456	0.373728	+	0.927538i	$0.378079\pi$
$230$	−2.26250e19	−0.190503
$231$	0	0
$232$	−8.03549e18	−0.0628585
$233$	1.52132e20	1.14735	0.573673	−	0.819085i	$-0.305518\pi$
$233$	1.52132e20	1.14735	0.573673	+	0.819085i	$0.305518\pi$
$234$	0	0
$235$	5.56702e19	0.390433
$236$	1.06337e19	0.0719334
$237$	0	0
$238$	−4.89357e19	−0.308121
$239$	1.99392e20	1.21151	0.605753	−	0.795653i	$-0.292872\pi$
$239$	1.99392e20	1.21151	0.605753	+	0.795653i	$0.292872\pi$
$240$	0	0
$241$	4.27174e19	0.241802	0.120901	−	0.992665i	$-0.461422\pi$
$241$	4.27174e19	0.241802	0.120901	+	0.992665i	$0.461422\pi$
$242$	1.67811e20	0.917045
$243$	0	0
$244$	1.43105e19	0.0729193
$245$	−3.68151e19	−0.181182
$246$	0	0
$247$	2.34232e20	1.07578
$248$	−2.65254e20	−1.17712
$249$	0	0
$250$	2.06516e19	0.0855981
$251$	7.43273e19	0.297798	0.148899	−	0.988852i	$-0.452427\pi$
$251$	7.43273e19	0.297798	0.148899	+	0.988852i	$0.452427\pi$
$252$	0	0
$253$	−2.42889e19	−0.0909668
$254$	3.40345e20	1.23263
$255$	0	0
$256$	−7.39416e19	−0.250524
$257$	−2.45390e20	−0.804314	−0.402157	−	0.915571i	$-0.631739\pi$
$257$	−2.45390e20	−0.804314	−0.402157	+	0.915571i	$0.631739\pi$
$258$	0	0
$259$	6.66708e20	2.04591
$260$	−1.40473e19	−0.0417177
$261$	0	0
$262$	−5.26750e20	−1.46570
$263$	−6.82308e20	−1.83805	−0.919023	−	0.394203i	$-0.871021\pi$
$263$	−6.82308e20	−1.83805	−0.919023	+	0.394203i	$0.871021\pi$
$264$	0	0
$265$	3.13118e20	0.790895
$266$	−4.49871e20	−1.10051
$267$	0	0
$268$	−5.20014e19	−0.119363
$269$	6.63222e19	0.147491	0.0737453	−	0.997277i	$-0.476505\pi$
$269$	6.63222e19	0.147491	0.0737453	+	0.997277i	$0.476505\pi$
$270$	0	0
$271$	2.71020e20	0.565930	0.282965	−	0.959130i	$-0.408682\pi$
$271$	2.71020e20	0.565930	0.282965	+	0.959130i	$0.408682\pi$
$272$	1.21969e20	0.246839
$273$	0	0
$274$	−5.89908e19	−0.112177
$275$	2.21704e19	0.0408738
$276$	0	0
$277$	−8.88228e20	−1.53974	−0.769868	−	0.638203i	$-0.779678\pi$
$277$	−8.88228e20	−1.53974	−0.769868	+	0.638203i	$0.779678\pi$
$278$	−4.94887e20	−0.832003
$279$	0	0
$280$	3.47707e20	0.550010
$281$	−6.79681e19	−0.104304	−0.0521521	−	0.998639i	$-0.516608\pi$
$281$	−6.79681e19	−0.104304	−0.0521521	+	0.998639i	$0.516608\pi$
$282$	0	0
$283$	−1.19124e21	−1.72113	−0.860566	−	0.509339i	$-0.829890\pi$
$283$	−1.19124e21	−1.72113	−0.860566	+	0.509339i	$0.829890\pi$
$284$	1.02527e20	0.143758
$285$	0	0
$286$	1.64192e20	0.216890
$287$	−1.35963e20	−0.174351
$288$	0	0
$289$	−7.66214e20	−0.926229
$290$	−2.20895e19	−0.0259300
$291$	0	0
$292$	7.56817e19	0.0837986
$293$	9.27187e20	0.997223	0.498612	−	0.866825i	$-0.333843\pi$
$293$	9.27187e20	0.997223	0.498612	+	0.866825i	$0.333843\pi$
$294$	0	0
$295$	3.76733e20	0.382425
$296$	−1.81554e21	−1.79071
$297$	0	0
$298$	−5.84151e20	−0.544110
$299$	5.45229e20	0.493599
$300$	0	0
$301$	1.03766e21	0.887645
$302$	2.09126e21	1.73920
$303$	0	0
$304$	1.12127e21	0.881629
$305$	5.06999e20	0.387666
$306$	0	0
$307$	−1.13596e21	−0.821650	−0.410825	−	0.911714i	$-0.634759\pi$
$307$	−1.13596e21	−0.821650	−0.410825	+	0.911714i	$0.634759\pi$
$308$	2.89637e19	0.0203785
$309$	0	0
$310$	−7.29179e20	−0.485579
$311$	−6.77029e19	−0.0438676	−0.0219338	−	0.999759i	$-0.506982\pi$
$311$	−6.77029e19	−0.0438676	−0.0219338	+	0.999759i	$0.506982\pi$
$312$	0	0
$313$	−2.90716e21	−1.78379	−0.891893	−	0.452247i	$-0.850623\pi$
$313$	−2.90716e21	−1.78379	−0.891893	+	0.452247i	$0.850623\pi$
$314$	−8.83755e20	−0.527752
$315$	0	0
$316$	−1.33904e20	−0.0757623
$317$	5.72332e20	0.315242	0.157621	−	0.987500i	$-0.449618\pi$
$317$	5.72332e20	0.315242	0.157621	+	0.987500i	$0.449618\pi$
$318$	0	0
$319$	−2.37139e19	−0.0123818
$320$	−9.40630e20	−0.478238
$321$	0	0
$322$	−1.04718e21	−0.504948
$323$	5.61023e20	0.263487
$324$	0	0
$325$	−4.97674e20	−0.221787
$326$	6.22945e20	0.270458
$327$	0	0
$328$	3.70245e20	0.152603
$329$	2.57665e21	1.03488
$330$	0	0
$331$	−1.51278e21	−0.577082	−0.288541	−	0.957467i	$-0.593170\pi$
$331$	−1.51278e21	−0.577082	−0.288541	+	0.957467i	$0.593170\pi$
$332$	−3.57220e20	−0.132819
$333$	0	0
$334$	6.47524e20	0.228776
$335$	−1.84232e21	−0.634578
$336$	0	0
$337$	4.17496e21	1.36709	0.683547	−	0.729906i	$-0.260436\pi$
$337$	4.17496e21	1.36709	0.683547	+	0.729906i	$0.260436\pi$
$338$	−6.88561e20	−0.219862
$339$	0	0
$340$	−3.36456e19	−0.0102178
$341$	−7.82803e20	−0.231868
$342$	0	0
$343$	2.50195e21	0.705144
$344$	−2.82569e21	−0.776922
$345$	0	0
$346$	2.30866e21	0.604245
$347$	−6.74224e21	−1.72188	−0.860942	−	0.508703i	$-0.830125\pi$
$347$	−6.74224e21	−1.72188	−0.860942	+	0.508703i	$0.830125\pi$
$348$	0	0
$349$	1.37021e21	0.333250	0.166625	−	0.986020i	$-0.446713\pi$
$349$	1.37021e21	0.333250	0.166625	+	0.986020i	$0.446713\pi$
$350$	9.55843e20	0.226886
$351$	0	0
$352$	−1.51624e20	−0.0342891
$353$	−5.52770e21	−1.22028	−0.610140	−	0.792294i	$-0.708887\pi$
$353$	−5.52770e21	−1.22028	−0.610140	+	0.792294i	$0.708887\pi$
$354$	0	0
$355$	3.63236e21	0.764272
$356$	−5.42233e20	−0.111394
$357$	0	0
$358$	−3.43564e21	−0.672980
$359$	3.91350e20	0.0748621	0.0374311	−	0.999299i	$-0.488083\pi$
$359$	3.91350e20	0.0748621	0.0374311	+	0.999299i	$0.488083\pi$
$360$	0	0
$361$	−3.22844e20	−0.0589090
$362$	5.66860e21	1.01031
$363$	0	0
$364$	−6.50168e20	−0.110577
$365$	2.68128e21	0.445505
$366$	0	0
$367$	5.83834e21	0.926036	0.463018	−	0.886349i	$-0.346766\pi$
$367$	5.83834e21	0.926036	0.463018	+	0.886349i	$0.346766\pi$
$368$	2.61003e21	0.404518
$369$	0	0
$370$	−4.99088e21	−0.738691
$371$	1.44924e22	2.09634
$372$	0	0
$373$	7.61840e21	1.05278	0.526391	−	0.850243i	$-0.323545\pi$
$373$	7.61840e21	1.05278	0.526391	+	0.850243i	$0.323545\pi$
$374$	3.93266e20	0.0531224
$375$	0	0
$376$	−7.01656e21	−0.905790
$377$	5.32323e20	0.0671852
$378$	0	0
$379$	−5.37453e21	−0.648496	−0.324248	−	0.945972i	$-0.605111\pi$
$379$	−5.37453e21	−0.648496	−0.324248	+	0.945972i	$0.605111\pi$
$380$	−3.09307e20	−0.0364947
$381$	0	0
$382$	−4.33924e21	−0.489638
$383$	−1.25019e22	−1.37971	−0.689854	−	0.723949i	$-0.742325\pi$
$383$	−1.25019e22	−1.37971	−0.689854	+	0.723949i	$0.742325\pi$
$384$	0	0
$385$	1.02614e21	0.108340
$386$	1.43514e22	1.48219
$387$	0	0
$388$	−1.39061e21	−0.137447
$389$	6.28931e21	0.608179	0.304089	−	0.952643i	$-0.401648\pi$
$389$	6.28931e21	0.608179	0.304089	+	0.952643i	$0.401648\pi$
$390$	0	0
$391$	1.30591e21	0.120896
$392$	4.64010e21	0.420335
$393$	0	0
$394$	1.93646e22	1.67993
$395$	−4.74400e21	−0.402781
$396$	0	0
$397$	9.76024e21	0.793855	0.396927	−	0.917850i	$-0.370076\pi$
$397$	9.76024e21	0.793855	0.396927	+	0.917850i	$0.370076\pi$
$398$	−1.51139e22	−1.20329
$399$	0	0
$400$	−2.38237e21	−0.181761
$401$	1.49385e22	1.11578	0.557891	−	0.829914i	$-0.311611\pi$
$401$	1.49385e22	1.11578	0.557891	+	0.829914i	$0.311611\pi$
$402$	0	0
$403$	1.75721e22	1.25815
$404$	−2.09150e19	−0.00146627
$405$	0	0
$406$	−1.02239e21	−0.0687299
$407$	−5.35792e21	−0.352731
$408$	0	0
$409$	3.26206e21	0.205989	0.102994	−	0.994682i	$-0.467158\pi$
$409$	3.26206e21	0.205989	0.102994	+	0.994682i	$0.467158\pi$
$410$	1.01780e21	0.0629506
$411$	0	0
$412$	9.25846e20	0.0549430
$413$	1.74368e22	1.01366
$414$	0	0
$415$	−1.26557e22	−0.706119
$416$	3.40362e21	0.186057
$417$	0	0
$418$	3.61533e21	0.189736
$419$	−1.20674e22	−0.620575	−0.310288	−	0.950643i	$-0.600425\pi$
$419$	−1.20674e22	−0.620575	−0.310288	+	0.950643i	$0.600425\pi$
$420$	0	0
$421$	2.42720e22	1.19869	0.599347	−	0.800489i	$-0.295427\pi$
$421$	2.42720e22	1.19869	0.599347	+	0.800489i	$0.295427\pi$
$422$	−1.38857e22	−0.672066
$423$	0	0
$424$	−3.94648e22	−1.83485
$425$	−1.19201e21	−0.0543217
$426$	0	0
$427$	2.34660e22	1.02755
$428$	2.04072e21	0.0876014
$429$	0	0
$430$	−7.76778e21	−0.320490
$431$	4.48255e21	0.181330	0.0906648	−	0.995881i	$-0.471101\pi$
$431$	4.48255e21	0.181330	0.0906648	+	0.995881i	$0.471101\pi$
$432$	0	0
$433$	−2.52232e22	−0.980964	−0.490482	−	0.871451i	$-0.663179\pi$
$433$	−2.52232e22	−0.980964	−0.490482	+	0.871451i	$0.663179\pi$
$434$	−3.37494e22	−1.28707
$435$	0	0
$436$	7.47939e19	0.00274303
$437$	1.20054e22	0.431801
$438$	0	0
$439$	−1.27343e22	−0.440583	−0.220292	−	0.975434i	$-0.570701\pi$
$439$	−1.27343e22	−0.440583	−0.220292	+	0.975434i	$0.570701\pi$
$440$	−2.79431e21	−0.0948258
$441$	0	0
$442$	−8.82791e21	−0.288250
$443$	−3.44286e21	−0.110278	−0.0551388	−	0.998479i	$-0.517560\pi$
$443$	−3.44286e21	−0.110278	−0.0551388	+	0.998479i	$0.517560\pi$
$444$	0	0
$445$	−1.92104e22	−0.592212
$446$	5.09514e21	0.154103
$447$	0	0
$448$	−4.35362e22	−1.26762
$449$	−3.26192e22	−0.931923	−0.465961	−	0.884805i	$-0.654291\pi$
$449$	−3.26192e22	−0.931923	−0.465961	+	0.884805i	$0.654291\pi$
$450$	0	0
$451$	1.09265e21	0.0300594
$452$	−5.27447e21	−0.142397
$453$	0	0
$454$	−3.21718e22	−0.836568
$455$	−2.30344e22	−0.587868
$456$	0	0
$457$	4.34642e22	1.06867	0.534335	−	0.845273i	$-0.320562\pi$
$457$	4.34642e22	1.06867	0.534335	+	0.845273i	$0.320562\pi$
$458$	−2.96389e22	−0.715328
$459$	0	0
$460$	−7.19985e20	−0.0167449
$461$	1.80871e22	0.412963	0.206482	−	0.978450i	$-0.433799\pi$
$461$	1.80871e22	0.412963	0.206482	+	0.978450i	$0.433799\pi$
$462$	0	0
$463$	4.20915e22	0.926308	0.463154	−	0.886278i	$-0.346718\pi$
$463$	4.20915e22	0.926308	0.463154	+	0.886278i	$0.346718\pi$
$464$	2.54824e21	0.0550602
$465$	0	0
$466$	−5.27101e22	−1.09803
$467$	2.75126e22	0.562779	0.281390	−	0.959594i	$-0.409205\pi$
$467$	2.75126e22	0.562779	0.281390	+	0.959594i	$0.409205\pi$
$468$	0	0
$469$	−8.52704e22	−1.68201
$470$	−1.92884e22	−0.373650
$471$	0	0
$472$	−4.74827e22	−0.887214
$473$	−8.33902e21	−0.153037
$474$	0	0
$475$	−1.09583e22	−0.194020
$476$	−1.55726e21	−0.0270833
$477$	0	0
$478$	−6.90847e22	−1.15943
$479$	9.42350e22	1.55368	0.776838	−	0.629701i	$-0.216823\pi$
$479$	9.42350e22	1.55368	0.776838	+	0.629701i	$0.216823\pi$
$480$	0	0
$481$	1.20273e23	1.91397
$482$	−1.48006e22	−0.231408
$483$	0	0
$484$	5.34018e21	0.0806065
$485$	−4.92671e22	−0.730722
$486$	0	0
$487$	−6.21562e22	−0.890202	−0.445101	−	0.895480i	$-0.646832\pi$
$487$	−6.21562e22	−0.890202	−0.445101	+	0.895480i	$0.646832\pi$
$488$	−6.39011e22	−0.899373
$489$	0	0
$490$	1.27556e22	0.173394
$491$	5.23525e21	0.0699431	0.0349715	−	0.999388i	$-0.488866\pi$
$491$	5.23525e21	0.0699431	0.0349715	+	0.999388i	$0.488866\pi$
$492$	0	0
$493$	1.27500e21	0.0164555
$494$	−8.11559e22	−1.02954
$495$	0	0
$496$	8.41182e22	1.03109
$497$	1.68121e23	2.02578
$498$	0	0
$499$	1.25380e23	1.46007	0.730035	−	0.683410i	$-0.239504\pi$
$499$	1.25380e23	1.46007	0.730035	+	0.683410i	$0.239504\pi$
$500$	6.57187e20	0.00752392
$501$	0	0
$502$	−2.57527e22	−0.284998
$503$	−9.43437e22	−1.02656	−0.513281	−	0.858221i	$-0.671570\pi$
$503$	−9.43437e22	−1.02656	−0.513281	+	0.858221i	$0.671570\pi$
$504$	0	0
$505$	−7.40983e20	−0.00779527
$506$	8.41553e21	0.0870567
$507$	0	0
$508$	1.08306e22	0.108346
$509$	1.78875e23	1.75974	0.879868	−	0.475218i	$-0.157631\pi$
$509$	1.78875e23	1.75974	0.879868	+	0.475218i	$0.157631\pi$
$510$	0	0
$511$	1.24101e23	1.18085
$512$	1.17047e23	1.09538
$513$	0	0
$514$	8.50220e22	0.769741
$515$	3.28012e22	0.292098
$516$	0	0
$517$	−2.07069e22	−0.178421
$518$	−2.30999e23	−1.95797
$519$	0	0
$520$	6.27258e22	0.514538
$521$	8.97435e22	0.724240	0.362120	−	0.932132i	$-0.382053\pi$
$521$	8.97435e22	0.724240	0.362120	+	0.932132i	$0.382053\pi$
$522$	0	0
$523$	8.40919e22	0.656886	0.328443	−	0.944524i	$-0.393476\pi$
$523$	8.40919e22	0.656886	0.328443	+	0.944524i	$0.393476\pi$
$524$	−1.67625e22	−0.128832
$525$	0	0
$526$	2.36404e23	1.75904
$527$	4.20881e22	0.308155
$528$	0	0
$529$	−1.13105e23	−0.801876
$530$	−1.08488e23	−0.756899
$531$	0	0
$532$	−1.43160e22	−0.0967328
$533$	−2.45275e22	−0.163107
$534$	0	0
$535$	7.22993e22	0.465722
$536$	2.32203e23	1.47220
$537$	0	0
$538$	−2.29791e22	−0.141151
$539$	1.36936e22	0.0827969
$540$	0	0
$541$	−3.23321e23	−1.89434	−0.947168	−	0.320737i	$-0.896070\pi$
$541$	−3.23321e23	−1.89434	−0.947168	+	0.320737i	$0.896070\pi$
$542$	−9.39023e22	−0.541604
$543$	0	0
$544$	8.15220e21	0.0455705
$545$	2.64982e21	0.0145830
$546$	0	0
$547$	1.95314e23	1.04193	0.520967	−	0.853577i	$-0.325571\pi$
$547$	1.95314e23	1.04193	0.520967	+	0.853577i	$0.325571\pi$
$548$	−1.87724e21	−0.00986016
$549$	0	0
$550$	−7.68152e21	−0.0391169
$551$	1.17212e22	0.0587738
$552$	0	0
$553$	−2.19572e23	−1.06761
$554$	3.07751e23	1.47355
$555$	0	0
$556$	−1.57485e22	−0.0731315
$557$	2.79561e23	1.27852	0.639260	−	0.768991i	$-0.279241\pi$
$557$	2.79561e23	1.27852	0.639260	+	0.768991i	$0.279241\pi$
$558$	0	0
$559$	1.87192e23	0.830399
$560$	−1.10266e23	−0.481775
$561$	0	0
$562$	2.35494e22	0.0998208
$563$	−4.27440e23	−1.78465	−0.892326	−	0.451391i	$-0.850928\pi$
$563$	−4.27440e23	−1.78465	−0.892326	+	0.451391i	$0.850928\pi$
$564$	0	0
$565$	−1.86866e23	−0.757038
$566$	4.12737e23	1.64715
$567$	0	0
$568$	−4.57815e23	−1.77308
$569$	−1.78853e22	−0.0682404	−0.0341202	−	0.999418i	$-0.510863\pi$
$569$	−1.78853e22	−0.0682404	−0.0341202	+	0.999418i	$0.510863\pi$
$570$	0	0
$571$	4.80795e23	1.78055	0.890273	−	0.455428i	$-0.150514\pi$
$571$	4.80795e23	1.78055	0.890273	+	0.455428i	$0.150514\pi$
$572$	5.22500e21	0.0190643
$573$	0	0
$574$	4.71080e22	0.166857
$575$	−2.55079e22	−0.0890221
$576$	0	0
$577$	−6.04913e22	−0.204974	−0.102487	−	0.994734i	$-0.532680\pi$
$577$	−6.04913e22	−0.204974	−0.102487	+	0.994734i	$0.532680\pi$
$578$	2.65475e23	0.886416
$579$	0	0
$580$	−7.02942e20	−0.00227920
$581$	−5.85759e23	−1.87164
$582$	0	0
$583$	−1.16467e23	−0.361425
$584$	−3.37943e23	−1.03356
$585$	0	0
$586$	−3.21249e23	−0.954359
$587$	−2.44376e23	−0.715542	−0.357771	−	0.933809i	$-0.616463\pi$
$587$	−2.44376e23	−0.715542	−0.357771	+	0.933809i	$0.616463\pi$
$588$	0	0
$589$	3.86920e23	1.10063
$590$	−1.30529e23	−0.365987
$591$	0	0
$592$	5.75749e23	1.56855
$593$	−4.28495e23	−1.15075	−0.575375	−	0.817890i	$-0.695144\pi$
$593$	−4.28495e23	−1.15075	−0.575375	+	0.817890i	$0.695144\pi$
$594$	0	0
$595$	−5.51711e22	−0.143985
$596$	−1.85892e22	−0.0478263
$597$	0	0
$598$	−1.88909e23	−0.472382
$599$	−3.13787e23	−0.773583	−0.386791	−	0.922167i	$-0.626417\pi$
$599$	−3.13787e23	−0.773583	−0.386791	+	0.922167i	$0.626417\pi$
$600$	0	0
$601$	−4.34182e23	−1.04049	−0.520246	−	0.854017i	$-0.674160\pi$
$601$	−4.34182e23	−1.04049	−0.520246	+	0.854017i	$0.674160\pi$
$602$	−3.59525e23	−0.849491
$603$	0	0
$604$	6.65493e22	0.152872
$605$	1.89194e23	0.428535
$606$	0	0
$607$	−1.44167e23	−0.317514	−0.158757	−	0.987318i	$-0.550749\pi$
$607$	−1.44167e23	−0.317514	−0.158757	+	0.987318i	$0.550749\pi$
$608$	7.49441e22	0.162763
$609$	0	0
$610$	−1.75663e23	−0.371003
$611$	4.64823e23	0.968138
$612$	0	0
$613$	−3.77031e23	−0.763770	−0.381885	−	0.924210i	$-0.624725\pi$
$613$	−3.77031e23	−0.763770	−0.381885	+	0.924210i	$0.624725\pi$
$614$	3.93584e23	0.786332
$615$	0	0
$616$	−1.29332e23	−0.251345
$617$	−1.22488e23	−0.234784	−0.117392	−	0.993086i	$-0.537453\pi$
$617$	−1.22488e23	−0.234784	−0.117392	+	0.993086i	$0.537453\pi$
$618$	0	0
$619$	3.03696e23	0.566329	0.283164	−	0.959071i	$-0.408616\pi$
$619$	3.03696e23	0.566329	0.283164	+	0.959071i	$0.408616\pi$
$620$	−2.32043e22	−0.0426815
$621$	0	0
$622$	2.34575e22	0.0419820
$623$	−8.89138e23	−1.56971
$624$	0	0
$625$	2.32831e22	0.0400000
$626$	1.00727e24	1.70711
$627$	0	0
$628$	−2.81233e22	−0.0463884
$629$	2.88073e23	0.468783
$630$	0	0
$631$	2.61906e23	0.414855	0.207427	−	0.978250i	$-0.433491\pi$
$631$	2.61906e23	0.414855	0.207427	+	0.978250i	$0.433491\pi$
$632$	5.97924e23	0.934439
$633$	0	0
$634$	−1.98300e23	−0.301692
$635$	3.83712e23	0.576009
$636$	0	0
$637$	−3.07390e23	−0.449268
$638$	8.21633e21	0.0118496
$639$	0	0
$640$	2.72476e23	0.382648
$641$	1.36869e24	1.89675	0.948376	−	0.317148i	$-0.102725\pi$
$641$	1.36869e24	1.89675	0.948376	+	0.317148i	$0.102725\pi$
$642$	0	0
$643$	−9.20999e23	−1.24299	−0.621493	−	0.783420i	$-0.713473\pi$
$643$	−9.20999e23	−1.24299	−0.621493	+	0.783420i	$0.713473\pi$
$644$	−3.33239e22	−0.0443840
$645$	0	0
$646$	−1.94381e23	−0.252161
$647$	−8.42397e23	−1.07853	−0.539263	−	0.842137i	$-0.681297\pi$
$647$	−8.42397e23	−1.07853	−0.539263	+	0.842137i	$0.681297\pi$
$648$	0	0
$649$	−1.40128e23	−0.174762
$650$	1.72432e23	0.212254
$651$	0	0
$652$	1.98237e22	0.0237728
$653$	−1.66957e24	−1.97625	−0.988127	−	0.153636i	$-0.950902\pi$
$653$	−1.66957e24	−1.97625	−0.988127	+	0.153636i	$0.950902\pi$
$654$	0	0
$655$	−5.93869e23	−0.684919
$656$	−1.17414e23	−0.133671
$657$	0	0
$658$	−8.92749e23	−0.990397
$659$	2.95184e23	0.323271	0.161635	−	0.986851i	$-0.448323\pi$
$659$	2.95184e23	0.323271	0.161635	+	0.986851i	$0.448323\pi$
$660$	0	0
$661$	−4.40451e23	−0.470094	−0.235047	−	0.971984i	$-0.575524\pi$
$661$	−4.40451e23	−0.470094	−0.235047	+	0.971984i	$0.575524\pi$
$662$	5.24143e23	0.552277
$663$	0	0
$664$	1.59510e24	1.63817
$665$	−5.07194e23	−0.514268
$666$	0	0
$667$	2.72838e22	0.0269672
$668$	2.06058e22	0.0201090
$669$	0	0
$670$	6.38323e23	0.607301
$671$	−1.88582e23	−0.177157
$672$	0	0
$673$	1.24648e24	1.14171	0.570857	−	0.821050i	$-0.306611\pi$
$673$	1.24648e24	1.14171	0.570857	+	0.821050i	$0.306611\pi$
$674$	−1.44653e24	−1.30833
$675$	0	0
$676$	−2.19117e22	−0.0193255
$677$	4.62880e23	0.403148	0.201574	−	0.979473i	$-0.435394\pi$
$677$	4.62880e23	0.403148	0.201574	+	0.979473i	$0.435394\pi$
$678$	0	0
$679$	−2.28028e24	−1.93685
$680$	1.50238e23	0.126024
$681$	0	0
$682$	2.71223e23	0.221901
$683$	−2.48612e23	−0.200884	−0.100442	−	0.994943i	$-0.532026\pi$
$683$	−2.48612e23	−0.200884	−0.100442	+	0.994943i	$0.532026\pi$
$684$	0	0
$685$	−6.65074e22	−0.0524204
$686$	−8.66866e23	−0.674834
$687$	0	0
$688$	8.96092e23	0.680535
$689$	2.61441e24	1.96114
$690$	0	0
$691$	−1.35462e24	−0.991410	−0.495705	−	0.868491i	$-0.665090\pi$
$691$	−1.35462e24	−0.991410	−0.495705	+	0.868491i	$0.665090\pi$
$692$	7.34674e22	0.0531120
$693$	0	0
$694$	2.33603e24	1.64787
$695$	−5.57945e23	−0.388795
$696$	0	0
$697$	−5.87472e22	−0.0399493
$698$	−4.74746e23	−0.318926
$699$	0	0
$700$	3.04173e22	0.0199429
$701$	1.81829e23	0.117777	0.0588885	−	0.998265i	$-0.481244\pi$
$701$	1.81829e23	0.117777	0.0588885	+	0.998265i	$0.481244\pi$
$702$	0	0
$703$	2.64828e24	1.67434
$704$	3.49874e23	0.218546
$705$	0	0
$706$	1.91522e24	1.16783
$707$	−3.42957e22	−0.0206621
$708$	0	0
$709$	−2.23286e24	−1.31331	−0.656657	−	0.754190i	$-0.728030\pi$
$709$	−2.23286e24	−1.31331	−0.656657	+	0.754190i	$0.728030\pi$
$710$	−1.25853e24	−0.731420
$711$	0	0
$712$	2.42124e24	1.37391
$713$	9.00647e23	0.505003
$714$	0	0
$715$	1.85113e23	0.101353
$716$	−1.09331e23	−0.0591537
$717$	0	0
$718$	−1.35594e23	−0.0716443
$719$	−3.16369e24	−1.65196	−0.825978	−	0.563703i	$-0.809376\pi$
$719$	−3.16369e24	−1.65196	−0.825978	+	0.563703i	$0.809376\pi$
$720$	0	0
$721$	1.51818e24	0.774234
$722$	1.11858e23	0.0563769
$723$	0	0
$724$	1.80389e23	0.0888040
$725$	−2.49041e22	−0.0121171
$726$	0	0
$727$	6.61806e23	0.314549	0.157275	−	0.987555i	$-0.449729\pi$
$727$	6.61806e23	0.314549	0.157275	+	0.987555i	$0.449729\pi$
$728$	2.90321e24	1.36383
$729$	0	0
$730$	−9.29001e23	−0.426356
$731$	4.48354e23	0.203387
$732$	0	0
$733$	4.28033e23	0.189711	0.0948556	−	0.995491i	$-0.469761\pi$
$733$	4.28033e23	0.189711	0.0948556	+	0.995491i	$0.469761\pi$
$734$	−2.02285e24	−0.886232
$735$	0	0
$736$	1.74450e23	0.0746808
$737$	6.85265e23	0.289991
$738$	0	0
$739$	−1.82465e24	−0.754575	−0.377288	−	0.926096i	$-0.623143\pi$
$739$	−1.82465e24	−0.754575	−0.377288	+	0.926096i	$0.623143\pi$
$740$	−1.58823e23	−0.0649296
$741$	0	0
$742$	−5.02129e24	−2.00623
$743$	3.46988e24	1.37059	0.685297	−	0.728263i	$-0.259672\pi$
$743$	3.46988e24	1.37059	0.685297	+	0.728263i	$0.259672\pi$
$744$	0	0
$745$	−6.58584e23	−0.254263
$746$	−2.63960e24	−1.00753
$747$	0	0
$748$	1.25147e22	0.00466936
$749$	3.34631e24	1.23444
$750$	0	0
$751$	4.39459e24	1.58481	0.792407	−	0.609992i	$-0.208828\pi$
$751$	4.39459e24	1.58481	0.792407	+	0.609992i	$0.208828\pi$
$752$	2.22512e24	0.793416
$753$	0	0
$754$	−1.84438e23	−0.0642974
$755$	2.35773e24	0.812728
$756$	0	0
$757$	1.47842e24	0.498291	0.249146	−	0.968466i	$-0.419850\pi$
$757$	1.47842e24	0.498291	0.249146	+	0.968466i	$0.419850\pi$
$758$	1.86215e24	0.620622
$759$	0	0
$760$	1.38116e24	0.450119
$761$	−3.72845e24	−1.20160	−0.600799	−	0.799400i	$-0.705151\pi$
$761$	−3.72845e24	−1.20160	−0.600799	+	0.799400i	$0.705151\pi$
$762$	0	0
$763$	1.22645e23	0.0386537
$764$	−1.38086e23	−0.0430383
$765$	0	0
$766$	4.33163e24	1.32040
$767$	3.14556e24	0.948283
$768$	0	0
$769$	−5.86779e24	−1.73022	−0.865108	−	0.501585i	$-0.832750\pi$
$769$	−5.86779e24	−1.73022	−0.865108	+	0.501585i	$0.832750\pi$
$770$	−3.55532e23	−0.103683
$771$	0	0
$772$	4.56698e23	0.130281
$773$	2.76247e24	0.779422	0.389711	−	0.920937i	$-0.372575\pi$
$773$	2.76247e24	0.779422	0.389711	+	0.920937i	$0.372575\pi$
$774$	0	0
$775$	−8.22091e23	−0.226911
$776$	6.20952e24	1.69525
$777$	0	0
$778$	−2.17910e24	−0.582037
$779$	−5.40070e23	−0.142686
$780$	0	0
$781$	−1.35108e24	−0.349259
$782$	−4.52468e23	−0.115699
$783$	0	0
$784$	−1.47148e24	−0.368188
$785$	−9.96363e23	−0.246618
$786$	0	0
$787$	−5.28513e24	−1.28018	−0.640089	−	0.768300i	$-0.721103\pi$
$787$	−5.28513e24	−1.28018	−0.640089	+	0.768300i	$0.721103\pi$
$788$	6.16231e23	0.147663
$789$	0	0
$790$	1.64369e24	0.385468
$791$	−8.64892e24	−2.00660
$792$	0	0
$793$	4.23323e24	0.961279
$794$	−3.38170e24	−0.759732
$795$	0	0
$796$	−4.80962e23	−0.105767
$797$	−6.74168e24	−1.46680	−0.733402	−	0.679795i	$-0.762069\pi$
$797$	−6.74168e24	−1.46680	−0.733402	+	0.679795i	$0.762069\pi$
$798$	0	0
$799$	1.11332e24	0.237123
$800$	−1.59234e23	−0.0335561
$801$	0	0
$802$	−5.17584e24	−1.06782
$803$	−9.97320e23	−0.203588
$804$	0	0
$805$	−1.18061e24	−0.235962
$806$	−6.08834e24	−1.20407
$807$	0	0
$808$	9.33920e22	0.0180848
$809$	9.91715e23	0.190031	0.0950155	−	0.995476i	$-0.469710\pi$
$809$	9.91715e23	0.190031	0.0950155	+	0.995476i	$0.469710\pi$
$810$	0	0
$811$	−2.93030e24	−0.549839	−0.274919	−	0.961467i	$-0.588651\pi$
$811$	−2.93030e24	−0.549839	−0.274919	+	0.961467i	$0.588651\pi$
$812$	−3.25351e22	−0.00604123
$813$	0	0
$814$	1.85639e24	0.337569
$815$	7.02321e23	0.126385
$816$	0	0
$817$	4.12177e24	0.726434
$818$	−1.13023e24	−0.197135
$819$	0	0
$820$	3.23890e22	0.00553324
$821$	−6.65129e24	−1.12458	−0.562288	−	0.826941i	$-0.690079\pi$
$821$	−6.65129e24	−1.12458	−0.562288	+	0.826941i	$0.690079\pi$
$822$	0	0
$823$	5.08673e24	0.842442	0.421221	−	0.906958i	$-0.361602\pi$
$823$	5.08673e24	0.842442	0.421221	+	0.906958i	$0.361602\pi$
$824$	−4.13420e24	−0.677657
$825$	0	0
$826$	−6.04144e24	−0.970085
$827$	−6.33488e24	−1.00680	−0.503398	−	0.864055i	$-0.667917\pi$
$827$	−6.33488e24	−1.00680	−0.503398	+	0.864055i	$0.667917\pi$
$828$	0	0
$829$	−6.00803e23	−0.0935445	−0.0467723	−	0.998906i	$-0.514894\pi$
$829$	−6.00803e23	−0.0935445	−0.0467723	+	0.998906i	$0.514894\pi$
$830$	4.38491e24	0.675767
$831$	0	0
$832$	−7.85386e24	−1.18586
$833$	−7.36249e23	−0.110038
$834$	0	0
$835$	7.30031e23	0.106907
$836$	1.15049e23	0.0166775
$837$	0	0
$838$	4.18107e24	0.593900
$839$	9.85513e24	1.38575	0.692876	−	0.721056i	$-0.256343\pi$
$839$	9.85513e24	1.38575	0.692876	+	0.721056i	$0.256343\pi$
$840$	0	0
$841$	−7.23051e24	−0.996329
$842$	−8.40970e24	−1.14717
$843$	0	0
$844$	−4.41878e23	−0.0590733
$845$	−7.76297e23	−0.102741
$846$	0	0
$847$	8.75667e24	1.13587
$848$	1.25152e25	1.60721
$849$	0	0
$850$	4.13003e23	0.0519867
$851$	6.16450e24	0.768239
$852$	0	0
$853$	−9.21992e24	−1.12632	−0.563158	−	0.826349i	$-0.690414\pi$
$853$	−9.21992e24	−1.12632	−0.563158	+	0.826349i	$0.690414\pi$
$854$	−8.13043e24	−0.983380
$855$	0	0
$856$	−9.11246e24	−1.08046
$857$	3.70909e24	0.435442	0.217721	−	0.976011i	$-0.430138\pi$
$857$	3.70909e24	0.435442	0.217721	+	0.976011i	$0.430138\pi$
$858$	0	0
$859$	−8.47452e24	−0.975379	−0.487689	−	0.873017i	$-0.662160\pi$
$859$	−8.47452e24	−0.975379	−0.487689	+	0.873017i	$0.662160\pi$
$860$	−2.47190e23	−0.0281705
$861$	0	0
$862$	−1.55310e24	−0.173535
$863$	−7.39487e24	−0.818162	−0.409081	−	0.912498i	$-0.634151\pi$
$863$	−7.39487e24	−0.818162	−0.409081	+	0.912498i	$0.634151\pi$
$864$	0	0
$865$	2.60283e24	0.282364
$866$	8.73925e24	0.938798
$867$	0	0
$868$	−1.07399e24	−0.113131
$869$	1.76456e24	0.184064
$870$	0	0
$871$	−1.53826e25	−1.57353
$872$	−3.33978e23	−0.0338321
$873$	0	0
$874$	−4.15959e24	−0.413241
$875$	1.07764e24	0.106024
$876$	0	0
$877$	1.70613e25	1.64633	0.823164	−	0.567804i	$-0.192207\pi$
$877$	1.70613e25	1.64633	0.823164	+	0.567804i	$0.192207\pi$
$878$	4.41215e24	0.421645
$879$	0	0
$880$	8.86141e23	0.0830616
$881$	−6.09883e24	−0.566176	−0.283088	−	0.959094i	$-0.591359\pi$
$881$	−6.09883e24	−0.566176	−0.283088	+	0.959094i	$0.591359\pi$
$882$	0	0
$883$	1.48222e25	1.34973	0.674866	−	0.737940i	$-0.264201\pi$
$883$	1.48222e25	1.34973	0.674866	+	0.737940i	$0.264201\pi$
$884$	−2.80926e23	−0.0253366
$885$	0	0
$886$	1.19287e24	0.105537
$887$	8.84878e24	0.775412	0.387706	−	0.921783i	$-0.373268\pi$
$887$	8.84878e24	0.775412	0.387706	+	0.921783i	$0.373268\pi$
$888$	0	0
$889$	1.77598e25	1.52677
$890$	6.65597e24	0.566756
$891$	0	0
$892$	1.62140e23	0.0135453
$893$	1.02349e25	0.846929
$894$	0	0
$895$	−3.87341e24	−0.314484
$896$	1.26113e25	1.01425
$897$	0	0
$898$	1.13018e25	0.891865
$899$	8.79328e23	0.0687374
$900$	0	0
$901$	6.26192e24	0.480338
$902$	−3.78578e23	−0.0287673
$903$	0	0
$904$	2.35522e25	1.75630
$905$	6.39089e24	0.472116
$906$	0	0
$907$	1.72279e24	0.124902	0.0624510	−	0.998048i	$-0.480108\pi$
$907$	1.72279e24	0.124902	0.0624510	+	0.998048i	$0.480108\pi$
$908$	−1.02379e24	−0.0735327
$909$	0	0
$910$	7.98089e24	0.562599
$911$	2.38368e24	0.166473	0.0832363	−	0.996530i	$-0.473474\pi$
$911$	2.38368e24	0.166473	0.0832363	+	0.996530i	$0.473474\pi$
$912$	0	0
$913$	4.70738e24	0.322684
$914$	−1.50593e25	−1.02273
$915$	0	0
$916$	−9.43183e23	−0.0628760
$917$	−2.74867e25	−1.81545
$918$	0	0
$919$	1.50053e25	0.972886	0.486443	−	0.873712i	$-0.338294\pi$
$919$	1.50053e25	0.972886	0.486443	+	0.873712i	$0.338294\pi$
$920$	3.21497e24	0.206528
$921$	0	0
$922$	−6.26677e24	−0.395213
$923$	3.03287e25	1.89513
$924$	0	0
$925$	−5.62682e24	−0.345190
$926$	−1.45837e25	−0.886492
$927$	0	0
$928$	1.70320e23	0.0101650
$929$	−8.99800e23	−0.0532124	−0.0266062	−	0.999646i	$-0.508470\pi$
$929$	−8.99800e23	−0.0532124	−0.0266062	+	0.999646i	$0.508470\pi$
$930$	0	0
$931$	−6.76842e24	−0.393020
$932$	−1.67737e24	−0.0965146
$933$	0	0
$934$	−9.53248e24	−0.538589
$935$	4.43376e23	0.0248241
$936$	0	0
$937$	3.51867e23	0.0193460	0.00967302	−	0.999953i	$-0.496921\pi$
$937$	3.51867e23	0.0193460	0.00967302	+	0.999953i	$0.496921\pi$
$938$	2.95442e25	1.60971
$939$	0	0
$940$	−6.13807e23	−0.0328432
$941$	−3.32160e25	−1.76131	−0.880654	−	0.473761i	$-0.842896\pi$
$941$	−3.32160e25	−1.76131	−0.880654	+	0.473761i	$0.842896\pi$
$942$	0	0
$943$	−1.25714e24	−0.0654687
$944$	1.50579e25	0.777145
$945$	0	0
$946$	2.88928e24	0.146459
$947$	1.58693e25	0.797230	0.398615	−	0.917118i	$-0.369491\pi$
$947$	1.58693e25	0.797230	0.398615	+	0.917118i	$0.369491\pi$
$948$	0	0
$949$	2.23875e25	1.10470
$950$	3.79678e24	0.185680
$951$	0	0
$952$	6.95365e24	0.334040
$953$	−2.71609e25	−1.29317	−0.646583	−	0.762844i	$-0.723802\pi$
$953$	−2.71609e25	−1.29317	−0.646583	+	0.762844i	$0.723802\pi$
$954$	0	0
$955$	−4.89214e24	−0.228808
$956$	−2.19845e24	−0.101912
$957$	0	0
$958$	−3.26502e25	−1.48689
$959$	−3.07824e24	−0.138945
$960$	0	0
$961$	6.47674e24	0.287215
$962$	−4.16718e25	−1.83170
$963$	0	0
$964$	−4.70992e23	−0.0203404
$965$	1.61801e25	0.692625
$966$	0	0
$967$	−9.97959e24	−0.419747	−0.209873	−	0.977729i	$-0.567305\pi$
$967$	−9.97959e24	−0.419747	−0.209873	+	0.977729i	$0.567305\pi$
$968$	−2.38456e25	−0.994187
$969$	0	0
$970$	1.70699e25	0.699313
$971$	−9.30142e24	−0.377734	−0.188867	−	0.982003i	$-0.560481\pi$
$971$	−9.30142e24	−0.377734	−0.188867	+	0.982003i	$0.560481\pi$
$972$	0	0
$973$	−2.58240e25	−1.03054
$974$	2.15357e25	0.851938
$975$	0	0
$976$	2.02645e25	0.787795
$977$	8.09204e24	0.311856	0.155928	−	0.987768i	$-0.450163\pi$
$977$	8.09204e24	0.311856	0.155928	+	0.987768i	$0.450163\pi$
$978$	0	0
$979$	7.14545e24	0.270631
$980$	4.05915e23	0.0152410
$981$	0	0
$982$	−1.81389e24	−0.0669367
$983$	3.54828e25	1.29812	0.649058	−	0.760739i	$-0.275163\pi$
$983$	3.54828e25	1.29812	0.649058	+	0.760739i	$0.275163\pi$
$984$	0	0
$985$	2.18321e25	0.785031
$986$	−4.41758e23	−0.0157482
$987$	0	0
$988$	−2.58259e24	−0.0904942
$989$	9.59438e24	0.333310
$990$	0	0
$991$	−3.95976e25	−1.35220	−0.676102	−	0.736808i	$-0.736332\pi$
$991$	−3.95976e25	−1.35220	−0.676102	+	0.736808i	$0.736332\pi$
$992$	5.62232e24	0.190356
$993$	0	0
$994$	−5.82499e25	−1.93870
$995$	−1.70397e25	−0.562297
$996$	0	0
$997$	−3.02793e25	−0.982284	−0.491142	−	0.871079i	$-0.663420\pi$
$997$	−3.02793e25	−0.982284	−0.491142	+	0.871079i	$0.663420\pi$
$998$	−4.34413e25	−1.39731
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.18.a.b.1.1		2
3.2	odd	2		15.18.a.a.1.2	✓	2
15.2	even	4		75.18.b.b.49.3		4
15.8	even	4		75.18.b.b.49.2		4
15.14	odd	2		75.18.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.18.a.a.1.2	✓	2	3.2	odd	2
45.18.a.b.1.1		2	1.1	even	1	trivial
75.18.a.c.1.1		2	15.14	odd	2
75.18.b.b.49.2		4	15.8	even	4
75.18.b.b.49.3		4	15.2	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q-346.477 q^{2} -11025.8 q^{4} -390625. q^{5} -1.80797e7 q^{7} +4.92336e7 q^{8} +1.35343e8 q^{10} +1.45296e8 q^{11} -3.26155e9 q^{13} +6.26421e9 q^{14} -1.56131e10 q^{16} -7.81194e9 q^{17} -7.18160e10 q^{19} +4.30694e9 q^{20} -5.03416e10 q^{22} -1.67169e11 q^{23} +1.52588e11 q^{25} +1.13005e12 q^{26} +1.99343e11 q^{28} -1.63211e11 q^{29} -5.38766e12 q^{31} -1.04356e12 q^{32} +2.70666e12 q^{34} +7.06240e12 q^{35} -3.68759e13 q^{37} +2.48826e13 q^{38} -1.92319e13 q^{40} +7.52018e12 q^{41} -5.73935e13 q^{43} -1.60200e12 q^{44} +5.79201e13 q^{46} -1.42516e14 q^{47} +9.42466e13 q^{49} -5.28682e13 q^{50} +3.59611e13 q^{52} -8.01583e14 q^{53} -5.67561e13 q^{55} -8.90131e14 q^{56} +5.65490e13 q^{58} -9.64437e14 q^{59} -1.29792e15 q^{61} +1.86670e15 q^{62} +2.40801e15 q^{64} +1.27404e15 q^{65} +4.71635e15 q^{67} +8.61327e13 q^{68} -2.44696e15 q^{70} -9.29884e15 q^{71} -6.86407e15 q^{73} +1.27767e16 q^{74} +7.91827e14 q^{76} -2.62691e15 q^{77} +1.21446e16 q^{79} +6.09888e15 q^{80} -2.60557e15 q^{82} +3.23986e16 q^{83} +3.05154e15 q^{85} +1.98855e16 q^{86} +7.15343e15 q^{88} +4.91787e16 q^{89} +5.89681e16 q^{91} +1.84316e15 q^{92} +4.93784e16 q^{94} +2.80531e16 q^{95} +1.26124e17 q^{97} -3.26543e16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 356 q^{2} + 351376 q^{4} - 781250 q^{5} - 20754552 q^{7} + 211737408 q^{8} - 139062500 q^{10} + 1131629912 q^{11} - 446672524 q^{13} + 4385222016 q^{14} + 51041317120 q^{16} - 20662021036 q^{17} - 194376127216 q^{19}+ \cdots - 19\!\cdots\!04 q^{98}+O(q^{100}) \) 2 * q + 356 * q^2 + 351376 * q^4 - 781250 * q^5 - 20754552 * q^7 + 211737408 * q^8 - 139062500 * q^10 + 1131629912 * q^11 - 446672524 * q^13 + 4385222016 * q^14 + 51041317120 * q^16 - 20662021036 * q^17 - 194376127216 * q^19 - 137256250000 * q^20 + 642535396112 * q^22 - 281815042584 * q^23 + 305175781250 * q^25 + 3107442504536 * q^26 - 770012242944 * q^28 + 4737062625892 * q^29 + 2256407074872 * q^31 + 24479953196032 * q^32 - 6320225301928 * q^34 + 8107246875000 * q^35 - 33686114979452 * q^37 - 61213036677904 * q^38 - 82709925000000 * q^40 + 13051545859228 * q^41 - 33149929851400 * q^43 + 355847272731712 * q^44 - 22616426690688 * q^46 - 84299585768440 * q^47 - 131229302667470 * q^49 + 54321289062500 * q^50 + 1056079273719136 * q^52 - 155836696419964 * q^53 - 442042934375000 * q^55 - 1324797309480960 * q^56 + 3498878281127368 * q^58 - 963392423116456 * q^59 - 2858793214972660 * q^61 + 7236477385534944 * q^62 + 11601155857485824 * q^64 + 174481454687500 * q^65 + 6027859197776136 * q^67 - 4570758410249888 * q^68 - 1712977350000000 * q^70 - 2206536655060304 * q^71 - 18626002136017276 * q^73 + 15017443813698616 * q^74 - 43624167666234944 * q^76 - 5265163905377376 * q^77 + 4997891097934440 * q^79 - 19938014500000000 * q^80 + 1280087848985176 * q^82 + 23351553482511288 * q^83 + 8071101967187500 * q^85 + 36916031436093872 * q^86 + 167436503858533632 * q^88 + 60857257652216796 * q^89 + 51438794903148912 * q^91 - 39704913484919808 * q^92 + 90273897275494496 * q^94 + 75928174693750000 * q^95 + 42869862659782756 * q^97 - 191045888549711804 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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