LMFDB - Embedded newform 2160.3.c.l.1889.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(1889,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2160.1889");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-11}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 15x^{2} + 4 $ x^4 + 15*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.1
Root			$-3.83776i$ of defining polynomial
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.l.1889.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.86421 - 4.09833i) q^{5} -7.15439i q^{7} +O(q^{10})$	$q+(-2.86421 - 4.09833i) q^{5} -7.15439i q^{7} +5.06984i q^{11} +3.12682i q^{13} +8.72842 q^{17} +20.1852 q^{19} +14.7284 q^{23} +(-8.59262 + 23.4769i) q^{25} +39.7995i q^{29} +39.3705 q^{31} +(-29.3210 + 20.4917i) q^{35} +34.8712i q^{37} +13.2665i q^{41} -66.6156i q^{43} +16.9137 q^{47} -2.18525 q^{49} -4.62950 q^{53} +(20.7779 - 14.5211i) q^{55} -25.7738i q^{59} -12.1852 q^{61} +(12.8148 - 8.95587i) q^{65} -106.415i q^{67} +101.487i q^{71} +23.2646i q^{73} +36.2716 q^{77} +66.5557 q^{79} +144.284 q^{83} +(-25.0000 - 35.7719i) q^{85} -154.553i q^{89} +22.3705 q^{91} +(-57.8148 - 82.7258i) q^{95} -175.733i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5}+O(q^{10}) $ 4 * q + 3 * q^5	$ 4 q + 3 q^{5} + 6 q^{17} - 6 q^{19} + 30 q^{23} + 9 q^{25} - 16 q^{31} - 45 q^{35} - 48 q^{47} + 78 q^{49} - 192 q^{53} - 47 q^{55} + 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} + 288 q^{83} - 100 q^{85} - 84 q^{91} - 318 q^{95}+O(q^{100}) $ 4 * q + 3 * q^5 + 6 * q^17 - 6 * q^19 + 30 * q^23 + 9 * q^25 - 16 * q^31 - 45 * q^35 - 48 * q^47 + 78 * q^49 - 192 * q^53 - 47 * q^55 + 38 * q^61 + 138 * q^65 + 174 * q^77 + 6 * q^79 + 288 * q^83 - 100 * q^85 - 84 * q^91 - 318 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.86421	−	4.09833i	−0.572842	−	0.819666i
$5$	−2.86421	−	4.09833i	−0.572842	−	0.819666i
$6$		0			0
$7$		−	7.15439i		−	1.02206i	−0.859564	−	0.511028i	$-0.829265\pi$
$7$		−	7.15439i		−	1.02206i	0.859564	−	0.511028i	$-0.170735\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.06984i			0.460894i	0.973085	+	0.230447i	$0.0740189\pi$
$11$			5.06984i			0.460894i	−0.973085	+	0.230447i	$0.925981\pi$
$12$		0			0
$13$			3.12682i			0.240525i	0.992742	+	0.120262i	$0.0383736\pi$
$13$			3.12682i			0.240525i	−0.992742	+	0.120262i	$0.961626\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	8.72842			0.513436			0.256718	−	0.966486i	$-0.417359\pi$
$17$	8.72842			0.513436			0.256718	+	0.966486i	$0.417359\pi$
$18$		0			0
$19$	20.1852			1.06238			0.531191	−	0.847252i	$-0.321745\pi$
$19$	20.1852			1.06238			0.531191	+	0.847252i	$0.321745\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	14.7284			0.640366			0.320183	−	0.947356i	$-0.396256\pi$
$23$	14.7284			0.640366			0.320183	+	0.947356i	$0.396256\pi$
$24$		0			0
$25$	−8.59262	+	23.4769i	−0.343705	+	0.939078i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			39.7995i			1.37240i	0.727415	+	0.686198i	$0.240722\pi$
$29$			39.7995i			1.37240i	−0.727415	+	0.686198i	$0.759278\pi$
$30$		0			0
$31$	39.3705			1.27002			0.635008	−	0.772506i	$-0.280997\pi$
$31$	39.3705			1.27002			0.635008	+	0.772506i	$0.280997\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−29.3210	+	20.4917i	−0.837744	+	0.585476i
$36$		0			0
$37$			34.8712i			0.942465i	0.882009	+	0.471232i	$0.156191\pi$
$37$			34.8712i			0.942465i	−0.882009	+	0.471232i	$0.843809\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			13.2665i			0.323573i	0.986826	+	0.161787i	$0.0517256\pi$
$41$			13.2665i			0.323573i	−0.986826	+	0.161787i	$0.948274\pi$
$42$		0			0
$43$		−	66.6156i		−	1.54920i	−0.632452	−	0.774600i	$-0.717951\pi$
$43$		−	66.6156i		−	1.54920i	0.632452	−	0.774600i	$-0.282049\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	16.9137			0.359865			0.179933	−	0.983679i	$-0.442412\pi$
$47$	16.9137			0.359865			0.179933	+	0.983679i	$0.442412\pi$
$48$		0			0
$49$	−2.18525			−0.0445969
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.62950			−0.0873491			−0.0436746	−	0.999046i	$-0.513906\pi$
$53$	−4.62950			−0.0873491			−0.0436746	+	0.999046i	$0.513906\pi$
$54$		0			0
$55$	20.7779	−	14.5211i	0.377780	−	0.264019i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	25.7738i		−	0.436844i	−0.975854	−	0.218422i	$-0.929909\pi$
$59$		−	25.7738i		−	0.436844i	0.975854	−	0.218422i	$-0.0700909\pi$
$60$		0			0
$61$	−12.1852			−0.199758			−0.0998791	−	0.995000i	$-0.531846\pi$
$61$	−12.1852			−0.199758			−0.0998791	+	0.995000i	$0.531846\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12.8148	−	8.95587i	0.197150	−	0.137783i
$66$		0			0
$67$		−	106.415i		−	1.58828i	−0.607732	−	0.794142i	$-0.707920\pi$
$67$		−	106.415i		−	1.58828i	0.607732	−	0.794142i	$-0.292080\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			101.487i			1.42939i	0.699436	+	0.714695i	$0.253435\pi$
$71$			101.487i			1.42939i	−0.699436	+	0.714695i	$0.746565\pi$
$72$		0			0
$73$			23.2646i			0.318694i	0.987223	+	0.159347i	$0.0509388\pi$
$73$			23.2646i			0.318694i	−0.987223	+	0.159347i	$0.949061\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	36.2716			0.471060
$78$		0			0
$79$	66.5557			0.842478			0.421239	−	0.906950i	$-0.361595\pi$
$79$	66.5557			0.842478			0.421239	+	0.906950i	$0.361595\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	144.284			1.73836			0.869182	−	0.494493i	$-0.164646\pi$
$83$	144.284			1.73836			0.869182	+	0.494493i	$0.164646\pi$
$84$		0			0
$85$	−25.0000	−	35.7719i	−0.294118	−	0.420846i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	154.553i		−	1.73655i	−0.496085	−	0.868274i	$-0.665230\pi$
$89$		−	154.553i		−	1.73655i	0.496085	−	0.868274i	$-0.334770\pi$
$90$		0			0
$91$	22.3705			0.245830
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−57.8148	−	82.7258i	−0.608576	−	0.870798i
$96$		0			0
$97$		−	175.733i		−	1.81168i	−0.423621	−	0.905839i	$-0.639241\pi$
$97$		−	175.733i		−	1.81168i	0.423621	−	0.905839i	$-0.360759\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			149.483i			1.48003i	0.672591	+	0.740014i	$0.265181\pi$
$101$			149.483i			1.48003i	−0.672591	+	0.740014i	$0.734819\pi$
$102$		0			0
$103$		−	28.6175i		−	0.277840i	−0.990304	−	0.138920i	$-0.955637\pi$
$103$		−	28.6175i		−	0.277840i	0.990304	−	0.138920i	$-0.0443631\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−156.297			−1.46072			−0.730359	−	0.683064i	$-0.760647\pi$
$107$	−156.297			−1.46072			−0.730359	+	0.683064i	$0.760647\pi$
$108$		0			0
$109$	8.55575			0.0784931			0.0392465	−	0.999230i	$-0.487504\pi$
$109$	8.55575			0.0784931			0.0392465	+	0.999230i	$0.487504\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−184.370			−1.63160			−0.815799	−	0.578336i	$-0.803702\pi$
$113$	−184.370			−1.63160			−0.815799	+	0.578336i	$0.803702\pi$
$114$		0			0
$115$	−42.1852	−	60.3619i	−0.366828	−	0.524886i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	62.4465i		−	0.524760i
$120$		0			0
$121$	95.2967			0.787576
$122$		0			0
$123$		0			0
$124$		0			0
$125$	120.827	−	32.0274i	0.966619	−	0.256219i
$126$		0			0
$127$		−	40.7002i		−	0.320474i	−0.987079	−	0.160237i	$-0.948774\pi$
$127$		−	40.7002i		−	0.320474i	0.987079	−	0.160237i	$-0.0512259\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	80.0236i		−	0.610867i	−0.952213	−	0.305434i	$-0.901199\pi$
$131$		−	80.0236i		−	0.610867i	0.952213	−	0.305434i	$-0.0988014\pi$
$132$		0			0
$133$		−	144.413i		−	1.08581i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.0738			0.0954289			0.0477144	−	0.998861i	$-0.484806\pi$
$137$	13.0738			0.0954289			0.0477144	+	0.998861i	$0.484806\pi$
$138$		0			0
$139$	−144.370			−1.03864			−0.519318	−	0.854581i	$-0.673814\pi$
$139$	−144.370			−1.03864			−0.519318	+	0.854581i	$0.673814\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−15.8525			−0.110857
$144$		0			0
$145$	163.111	−	113.994i	1.12491	−	0.786166i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	54.2498i		−	0.364093i	−0.983290	−	0.182046i	$-0.941728\pi$
$149$		−	54.2498i		−	0.364093i	0.983290	−	0.182046i	$-0.0582721\pi$
$150$		0			0
$151$	−21.1852			−0.140300			−0.0701498	−	0.997536i	$-0.522348\pi$
$151$	−21.1852			−0.140300			−0.0701498	+	0.997536i	$0.522348\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−112.765	−	161.353i	−0.727518	−	1.04099i
$156$		0			0
$157$		−	105.939i		−	0.674770i	−0.941367	−	0.337385i	$-0.890458\pi$
$157$		−	105.939i		−	0.674770i	0.941367	−	0.337385i	$-0.109542\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	105.373i		−	0.654489i
$162$		0			0
$163$			154.746i			0.949361i	0.880158	+	0.474680i	$0.157436\pi$
$163$			154.746i			0.949361i	−0.880158	+	0.474680i	$0.842564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	151.667			0.908187			0.454094	−	0.890954i	$-0.349963\pi$
$167$	151.667			0.908187			0.454094	+	0.890954i	$0.349963\pi$
$168$		0			0
$169$	159.223			0.942148
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−311.198			−1.79883			−0.899416	−	0.437094i	$-0.856008\pi$
$173$	−311.198			−1.79883			−0.899416	+	0.437094i	$0.856008\pi$
$174$		0			0
$175$	167.963	+	61.4750i	0.959789	+	0.351285i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			211.170i			1.17972i	0.807505	+	0.589861i	$0.200817\pi$
$179$			211.170i			1.17972i	−0.807505	+	0.589861i	$0.799183\pi$
$180$		0			0
$181$	149.297			0.824844			0.412422	−	0.910993i	$-0.364683\pi$
$181$	149.297			0.824844			0.412422	+	0.910993i	$0.364683\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	142.914	−	99.8783i	0.772506	−	0.539883i
$186$		0			0
$187$			44.2517i			0.236640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			24.1654i			0.126520i	0.997997	+	0.0632602i	$0.0201498\pi$
$191$			24.1654i			0.126520i	−0.997997	+	0.0632602i	$0.979850\pi$
$192$		0			0
$193$		−	332.653i		−	1.72359i	−0.507255	−	0.861796i	$-0.669340\pi$
$193$		−	332.653i		−	1.72359i	0.507255	−	0.861796i	$-0.330660\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	279.568			1.41913			0.709564	−	0.704641i	$-0.248892\pi$
$197$	279.568			1.41913			0.709564	+	0.704641i	$0.248892\pi$
$198$		0			0
$199$	161.185			0.809976			0.404988	−	0.914322i	$-0.367276\pi$
$199$	161.185			0.809976			0.404988	+	0.914322i	$0.367276\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	284.741			1.40266
$204$		0			0
$205$	54.3705	−	37.9980i	0.265222	−	0.185356i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			102.336i			0.489646i
$210$		0			0
$211$	92.1852			0.436897			0.218448	−	0.975848i	$-0.429900\pi$
$211$	92.1852			0.436897			0.218448	+	0.975848i	$0.429900\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−273.013	+	190.801i	−1.26983	+	0.887446i
$216$		0			0
$217$		−	281.672i		−	1.29803i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			27.2922i			0.123494i
$222$		0			0
$223$		−	197.672i		−	0.886422i	−0.896417	−	0.443211i	$-0.853839\pi$
$223$		−	197.672i		−	0.886422i	0.896417	−	0.443211i	$-0.146161\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−32.7284			−0.144178			−0.0720890	−	0.997398i	$-0.522967\pi$
$227$	−32.7284			−0.144178			−0.0720890	+	0.997398i	$0.522967\pi$
$228$		0			0
$229$	350.926			1.53243			0.766215	−	0.642585i	$-0.222138\pi$
$229$	350.926			1.53243			0.766215	+	0.642585i	$0.222138\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−399.827			−1.71600			−0.857999	−	0.513652i	$-0.828292\pi$
$233$	−399.827			−1.71600			−0.857999	+	0.513652i	$0.828292\pi$
$234$		0			0
$235$	−48.4443	−	69.3178i	−0.206146	−	0.294969i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	363.690i		−	1.52172i	−0.648919	−	0.760858i	$-0.724779\pi$
$239$		−	363.690i		−	1.52172i	0.648919	−	0.760858i	$-0.275221\pi$
$240$		0			0
$241$	227.667			0.944677			0.472339	−	0.881417i	$-0.343410\pi$
$241$	227.667			0.944677			0.472339	+	0.881417i	$0.343410\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.25901	+	8.95587i	0.0255470	+	0.0365546i
$246$		0			0
$247$			63.1157i			0.255529i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			227.988i			0.908319i	0.890920	+	0.454160i	$0.150060\pi$
$251$			227.988i			0.908319i	−0.890920	+	0.454160i	$0.849940\pi$
$252$		0			0
$253$			74.6707i			0.295141i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	456.038			1.77447			0.887233	−	0.461322i	$-0.152625\pi$
$257$	456.038			1.77447			0.887233	+	0.461322i	$0.152625\pi$
$258$		0			0
$259$	249.482			0.963251
$260$		0			0
$261$		0			0
$262$		0			0
$263$	186.543			0.709290			0.354645	−	0.935001i	$-0.384602\pi$
$263$	186.543			0.709290			0.354645	+	0.935001i	$0.384602\pi$
$264$		0			0
$265$	13.2599	+	18.9732i	0.0500372	+	0.0715971i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			337.157i			1.25337i	0.779272	+	0.626686i	$0.215589\pi$
$269$			337.157i			1.25337i	−0.779272	+	0.626686i	$0.784411\pi$
$270$		0			0
$271$	−58.2590			−0.214978			−0.107489	−	0.994206i	$-0.534281\pi$
$271$	−58.2590			−0.214978			−0.107489	+	0.994206i	$0.534281\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−119.024	−	43.5632i	−0.432816	−	0.158412i
$276$		0			0
$277$			332.229i			1.19938i	0.800232	+	0.599691i	$0.204710\pi$
$277$			332.229i			1.19938i	−0.800232	+	0.599691i	$0.795290\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			112.296i			0.399629i	0.979834	+	0.199814i	$0.0640339\pi$
$281$			112.296i			0.399629i	−0.979834	+	0.199814i	$0.935966\pi$
$282$		0			0
$283$		−	325.975i		−	1.15185i	−0.817501	−	0.575927i	$-0.804641\pi$
$283$		−	325.975i		−	1.15185i	0.817501	−	0.575927i	$-0.195359\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	94.9137			0.330710
$288$		0			0
$289$	−212.815			−0.736383
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−71.9623			−0.245605			−0.122803	−	0.992431i	$-0.539188\pi$
$293$	−71.9623			−0.245605			−0.122803	+	0.992431i	$0.539188\pi$
$294$		0			0
$295$	−105.630	+	73.8215i	−0.358066	+	0.250242i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			46.0531i			0.154024i
$300$		0			0
$301$	−476.593			−1.58337
$302$		0			0
$303$		0			0
$304$		0			0
$305$	34.9011	+	49.9392i	0.114430	+	0.163735i
$306$		0			0
$307$		−	245.900i		−	0.800977i	−0.916302	−	0.400488i	$-0.868841\pi$
$307$		−	245.900i		−	0.800977i	0.916302	−	0.400488i	$-0.131159\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			69.3693i			0.223053i	0.993761	+	0.111526i	$0.0355739\pi$
$311$			69.3693i			0.223053i	−0.993761	+	0.111526i	$0.964426\pi$
$312$		0			0
$313$		−	218.659i		−	0.698592i	−0.937013	−	0.349296i	$-0.886421\pi$
$313$		−	218.659i		−	0.698592i	0.937013	−	0.349296i	$-0.113579\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	74.4317			0.234800			0.117400	−	0.993085i	$-0.462544\pi$
$317$	74.4317			0.234800			0.117400	+	0.993085i	$0.462544\pi$
$318$		0			0
$319$	−201.777			−0.632530
$320$		0			0
$321$		0			0
$322$		0			0
$323$	176.185			0.545465
$324$		0			0
$325$	−73.4082	−	26.8676i	−0.225871	−	0.0826696i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	121.007i		−	0.367802i
$330$		0			0
$331$	220.223			0.665326			0.332663	−	0.943046i	$-0.392053\pi$
$331$	220.223			0.665326			0.332663	+	0.943046i	$0.392053\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−436.124	+	304.795i	−1.30186	+	0.909835i
$336$		0			0
$337$			24.6415i			0.0731203i	0.999331	+	0.0365601i	$0.0116400\pi$
$337$			24.6415i			0.0731203i	−0.999331	+	0.0365601i	$0.988360\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			199.602i			0.585343i
$342$		0			0
$343$		−	334.931i		−	0.976475i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	147.556			0.425233			0.212616	−	0.977136i	$-0.431802\pi$
$347$	147.556			0.425233			0.212616	+	0.977136i	$0.431802\pi$
$348$		0			0
$349$	−503.149			−1.44169			−0.720844	−	0.693097i	$-0.756246\pi$
$349$	−503.149			−1.44169			−0.720844	+	0.693097i	$0.756246\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−236.716			−0.670583			−0.335292	−	0.942114i	$-0.608835\pi$
$353$	−236.716			−0.670583			−0.335292	+	0.942114i	$0.608835\pi$
$354$		0			0
$355$	415.926	−	290.679i	1.17162	−	0.818815i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			195.871i			0.545601i	0.962071	+	0.272800i	$0.0879499\pi$
$359$			195.871i			0.545601i	−0.962071	+	0.272800i	$0.912050\pi$
$360$		0			0
$361$	46.4443			0.128654
$362$		0			0
$363$		0			0
$364$		0			0
$365$	95.3462	−	66.6348i	0.261222	−	0.182561i
$366$		0			0
$367$		−	339.756i		−	0.925766i	−0.886419	−	0.462883i	$-0.846815\pi$
$367$		−	339.756i		−	0.925766i	0.886419	−	0.462883i	$-0.153185\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			33.1213i			0.0892756i
$372$		0			0
$373$			540.980i			1.45035i	0.688566	+	0.725174i	$0.258241\pi$
$373$			540.980i			1.45035i	−0.688566	+	0.725174i	$0.741759\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−124.446			−0.330095
$378$		0			0
$379$	432.926			1.14229			0.571143	−	0.820851i	$-0.306500\pi$
$379$	432.926			1.14229			0.571143	+	0.820851i	$0.306500\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	478.977			1.25059			0.625296	−	0.780388i	$-0.284978\pi$
$383$	478.977			1.25059			0.625296	+	0.780388i	$0.284978\pi$
$384$		0			0
$385$	−103.889	−	148.653i	−0.269843	−	0.386112i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	339.190i		−	0.871954i	−0.899958	−	0.435977i	$-0.856403\pi$
$389$		−	339.190i		−	0.871954i	0.899958	−	0.435977i	$-0.143597\pi$
$390$		0			0
$391$	128.556			0.328787
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−190.630	−	272.767i	−0.482606	−	0.690550i
$396$		0			0
$397$			113.042i			0.284740i	0.989814	+	0.142370i	$0.0454723\pi$
$397$			113.042i			0.284740i	−0.989814	+	0.142370i	$0.954528\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	477.774i		−	1.19146i	−0.803186	−	0.595728i	$-0.796864\pi$
$401$		−	477.774i		−	1.19146i	0.803186	−	0.595728i	$-0.203136\pi$
$402$		0			0
$403$			123.105i			0.305470i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−176.791			−0.434377
$408$		0			0
$409$	677.370			1.65616			0.828081	−	0.560608i	$-0.189433\pi$
$409$	677.370			1.65616			0.828081	+	0.560608i	$0.189433\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−184.396			−0.446479
$414$		0			0
$415$	−413.260	−	591.324i	−0.995807	−	1.42488i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			344.170i			0.821408i	0.911769	+	0.410704i	$0.134717\pi$
$419$			344.170i			0.821408i	−0.911769	+	0.410704i	$0.865283\pi$
$420$		0			0
$421$	29.5918			0.0702892			0.0351446	−	0.999382i	$-0.488811\pi$
$421$	29.5918			0.0702892			0.0351446	+	0.999382i	$0.488811\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−75.0000	+	204.917i	−0.176471	+	0.482157i
$426$		0			0
$427$			87.1780i			0.204164i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	724.253i		−	1.68040i	−0.542276	−	0.840201i	$-0.682437\pi$
$431$		−	724.253i		−	1.68040i	0.542276	−	0.840201i	$-0.317563\pi$
$432$		0			0
$433$			96.6100i			0.223118i	0.993758	+	0.111559i	$0.0355844\pi$
$433$			96.6100i			0.223118i	−0.993758	+	0.111559i	$0.964416\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	297.297			0.680313
$438$		0			0
$439$	−101.223			−0.230576			−0.115288	−	0.993332i	$-0.536779\pi$
$439$	−101.223			−0.230576			−0.115288	+	0.993332i	$0.536779\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−377.520			−0.852189			−0.426094	−	0.904679i	$-0.640111\pi$
$443$	−377.520			−0.852189			−0.426094	+	0.904679i	$0.640111\pi$
$444$		0			0
$445$	−633.408	+	442.671i	−1.42339	+	0.994767i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			654.974i			1.45874i	0.684120	+	0.729369i	$0.260186\pi$
$449$			654.974i			1.45874i	−0.684120	+	0.729369i	$0.739814\pi$
$450$		0			0
$451$	−67.2590			−0.149133
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−64.0738	−	91.6817i	−0.140821	−	0.201498i
$456$		0			0
$457$			321.574i			0.703664i	0.936063	+	0.351832i	$0.114441\pi$
$457$			321.574i			0.703664i	−0.936063	+	0.351832i	$0.885559\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	465.511i		−	1.00979i	−0.863182	−	0.504893i	$-0.831532\pi$
$461$		−	465.511i		−	1.00979i	0.863182	−	0.504893i	$-0.168468\pi$
$462$		0			0
$463$			250.880i			0.541857i	0.962600	+	0.270928i	$0.0873307\pi$
$463$			250.880i			0.541857i	−0.962600	+	0.270928i	$0.912669\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	246.259			0.527321			0.263661	−	0.964616i	$-0.415070\pi$
$467$	246.259			0.527321			0.263661	+	0.964616i	$0.415070\pi$
$468$		0			0
$469$	−761.334			−1.62331
$470$		0			0
$471$		0			0
$472$		0			0
$473$	337.730			0.714017
$474$		0			0
$475$	−173.444	+	473.888i	−0.365146	+	0.997659i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			175.012i			0.365370i	0.983172	+	0.182685i	$0.0584788\pi$
$479$			175.012i			0.365370i	−0.983172	+	0.182685i	$0.941521\pi$
$480$		0			0
$481$	−109.036			−0.226686
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−720.211	+	503.335i	−1.48497	+	1.03781i
$486$		0			0
$487$			28.9906i			0.0595289i	0.999557	+	0.0297645i	$0.00947573\pi$
$487$			28.9906i			0.0595289i	−0.999557	+	0.0297645i	$0.990524\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	974.373i		−	1.98447i	−0.124388	−	0.992234i	$-0.539697\pi$
$491$		−	974.373i		−	1.98447i	0.124388	−	0.992234i	$-0.460303\pi$
$492$		0			0
$493$			347.387i			0.704638i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	726.075			1.46092
$498$		0			0
$499$	−626.185			−1.25488			−0.627440	−	0.778665i	$-0.715897\pi$
$499$	−626.185			−1.25488			−0.627440	+	0.778665i	$0.715897\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	71.5197			0.142186			0.0710932	−	0.997470i	$-0.477351\pi$
$503$	71.5197			0.142186			0.0710932	+	0.997470i	$0.477351\pi$
$504$		0			0
$505$	612.630	−	428.150i	1.21313	−	0.847822i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	932.296i		−	1.83162i	−0.401608	−	0.915812i	$-0.631549\pi$
$509$		−	932.296i		−	1.83162i	0.401608	−	0.915812i	$-0.368451\pi$
$510$		0			0
$511$	166.444			0.325723
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−117.284	+	81.9666i	−0.227736	+	0.159158i
$516$		0			0
$517$			85.7495i			0.165860i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			373.070i			0.716066i	0.933709	+	0.358033i	$0.116552\pi$
$521$			373.070i			0.716066i	−0.933709	+	0.358033i	$0.883448\pi$
$522$		0			0
$523$			828.480i			1.58409i	0.610461	+	0.792046i	$0.290984\pi$
$523$			828.480i			1.58409i	−0.610461	+	0.792046i	$0.709016\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	343.642			0.652072
$528$		0			0
$529$	−312.074			−0.589931
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−41.4820			−0.0778274
$534$		0			0
$535$	447.666	+	640.556i	0.836760	+	1.19730i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	11.0789i		−	0.0205545i
$540$		0			0
$541$	−50.5935			−0.0935184			−0.0467592	−	0.998906i	$-0.514889\pi$
$541$	−50.5935			−0.0935184			−0.0467592	+	0.998906i	$0.514889\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−24.5054	−	35.0643i	−0.0449641	−	0.0643381i
$546$		0			0
$547$			485.070i			0.886782i	0.896328	+	0.443391i	$0.146225\pi$
$547$			485.070i			0.886782i	−0.896328	+	0.443391i	$0.853775\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			803.363i			1.45801i
$552$		0			0
$553$		−	476.166i		−	0.861059i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−198.372			−0.356144			−0.178072	−	0.984017i	$-0.556986\pi$
$557$	−198.372			−0.356144			−0.178072	+	0.984017i	$0.556986\pi$
$558$		0			0
$559$	208.295			0.372621
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−660.840			−1.17378			−0.586892	−	0.809666i	$-0.699649\pi$
$563$	−660.840			−1.17378			−0.586892	+	0.809666i	$0.699649\pi$
$564$		0			0
$565$	528.075	+	755.611i	0.934647	+	1.33736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			162.994i			0.286457i	0.989690	+	0.143229i	$0.0457484\pi$
$569$			162.994i			0.286457i	−0.989690	+	0.143229i	$0.954252\pi$
$570$		0			0
$571$	−262.631			−0.459950			−0.229975	−	0.973197i	$-0.573864\pi$
$571$	−262.631			−0.459950			−0.229975	+	0.973197i	$0.573864\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−126.556	+	345.778i	−0.220097	+	0.601353i
$576$		0			0
$577$		−	532.551i		−	0.922966i	−0.887149	−	0.461483i	$-0.847318\pi$
$577$		−	532.551i		−	0.922966i	0.887149	−	0.461483i	$-0.152682\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	1032.26i		−	1.77670i
$582$		0			0
$583$		−	23.4708i		−	0.0402587i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	927.593			1.58023			0.790114	−	0.612960i	$-0.210021\pi$
$587$	927.593			1.58023			0.790114	+	0.612960i	$0.210021\pi$
$588$		0			0
$589$	794.703			1.34924
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−890.336			−1.50141			−0.750705	−	0.660638i	$-0.770286\pi$
$593$	−890.336			−1.50141			−0.750705	+	0.660638i	$0.770286\pi$
$594$		0			0
$595$	−255.926	+	178.860i	−0.430128	+	0.300604i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1036.76i		−	1.73081i	−0.501073	−	0.865405i	$-0.667061\pi$
$599$		−	1036.76i		−	1.73081i	0.501073	−	0.865405i	$-0.332939\pi$
$600$		0			0
$601$	865.816			1.44063			0.720313	−	0.693649i	$-0.243998\pi$
$601$	865.816			1.44063			0.720313	+	0.693649i	$0.243998\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−272.950	−	390.558i	−0.451157	−	0.645550i
$606$		0			0
$607$			708.606i			1.16739i	0.811973	+	0.583695i	$0.198394\pi$
$607$			708.606i			1.16739i	−0.811973	+	0.583695i	$0.801606\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			52.8860i			0.0865565i
$612$		0			0
$613$			677.814i			1.10573i	0.833270	+	0.552866i	$0.186466\pi$
$613$			677.814i			1.10573i	−0.833270	+	0.552866i	$0.813534\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	474.063			0.768335			0.384168	−	0.923263i	$-0.374488\pi$
$617$	474.063			0.768335			0.384168	+	0.923263i	$0.374488\pi$
$618$		0			0
$619$	−323.187			−0.522111			−0.261056	−	0.965324i	$-0.584071\pi$
$619$	−323.187			−0.522111			−0.261056	+	0.965324i	$0.584071\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1105.73			−1.77485
$624$		0			0
$625$	−477.334	−	403.457i	−0.763734	−	0.645531i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			304.370i			0.483896i
$630$		0			0
$631$	−977.669			−1.54940			−0.774698	−	0.632331i	$-0.782098\pi$
$631$	−977.669			−1.54940			−0.774698	+	0.632331i	$0.782098\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−166.803	+	116.574i	−0.262682	+	0.183581i
$636$		0			0
$637$		−	6.83288i		−	0.0107267i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			245.231i			0.382575i	0.981534	+	0.191288i	$0.0612663\pi$
$641$			245.231i			0.382575i	−0.981534	+	0.191288i	$0.938734\pi$
$642$		0			0
$643$			244.471i			0.380204i	0.981764	+	0.190102i	$0.0608819\pi$
$643$			244.471i			0.380204i	−0.981764	+	0.190102i	$0.939118\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	866.779			1.33969			0.669844	−	0.742501i	$-0.266361\pi$
$647$	866.779			1.33969			0.669844	+	0.742501i	$0.266361\pi$
$648$		0			0
$649$	130.669			0.201339
$650$		0			0
$651$		0			0
$652$		0			0
$653$	641.198			0.981926			0.490963	−	0.871180i	$-0.336645\pi$
$653$	641.198			0.981926			0.490963	+	0.871180i	$0.336645\pi$
$654$		0			0
$655$	−327.963	+	229.204i	−0.500707	+	0.349930i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			771.490i			1.17070i	0.810781	+	0.585349i	$0.199042\pi$
$659$			771.490i			1.17070i	−0.810781	+	0.585349i	$0.800958\pi$
$660$		0			0
$661$	−796.075			−1.20435			−0.602175	−	0.798364i	$-0.705699\pi$
$661$	−796.075			−1.20435			−0.602175	+	0.798364i	$0.705699\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−591.852	+	413.629i	−0.890004	+	0.621999i
$666$		0			0
$667$			586.184i			0.878836i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	61.7772i		−	0.0920674i
$672$		0			0
$673$		−	698.325i		−	1.03763i	−0.854887	−	0.518815i	$-0.826373\pi$
$673$		−	698.325i		−	1.03763i	0.854887	−	0.518815i	$-0.173627\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	433.655			0.640553			0.320277	−	0.947324i	$-0.396224\pi$
$677$	433.655			0.640553			0.320277	+	0.947324i	$0.396224\pi$
$678$		0			0
$679$	−1257.26			−1.85164
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1112.83			−1.62933			−0.814663	−	0.579935i	$-0.803078\pi$
$683$	−1112.83			−1.62933			−0.814663	+	0.579935i	$0.803078\pi$
$684$		0			0
$685$	−37.4460	−	53.5806i	−0.0546656	−	0.0782198i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	14.4756i		−	0.0210096i
$690$		0			0
$691$	−497.743			−0.720322			−0.360161	−	0.932890i	$-0.617278\pi$
$691$	−497.743			−0.720322			−0.360161	+	0.932890i	$0.617278\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	413.507	+	591.678i	0.594974	+	0.851335i
$696$		0			0
$697$			115.796i			0.166134i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			660.713i			0.942529i	0.881992	+	0.471264i	$0.156202\pi$
$701$			660.713i			0.942529i	−0.881992	+	0.471264i	$0.843798\pi$
$702$		0			0
$703$			703.884i			1.00126i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1069.46			1.51267
$708$		0			0
$709$	−1013.56			−1.42956			−0.714780	−	0.699350i	$-0.753473\pi$
$709$	−1013.56			−1.42956			−0.714780	+	0.699350i	$0.753473\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	579.865			0.813275
$714$		0			0
$715$	45.4048	+	64.9687i	0.0635032	+	0.0908653i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1067.57i		−	1.48480i	−0.669955	−	0.742402i	$-0.733687\pi$
$719$		−	1067.57i		−	1.48480i	0.669955	−	0.742402i	$-0.266313\pi$
$720$		0			0
$721$	−204.741			−0.283968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−934.370	−	341.982i	−1.28879	−	0.471699i
$726$		0			0
$727$			878.561i			1.20847i	0.796804	+	0.604237i	$0.206522\pi$
$727$			878.561i			1.20847i	−0.796804	+	0.604237i	$0.793478\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	581.448i		−	0.795415i
$732$		0			0
$733$			76.9483i			0.104977i	0.998622	+	0.0524886i	$0.0167153\pi$
$733$			76.9483i			0.104977i	−0.998622	+	0.0524886i	$0.983285\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	539.507			0.732031
$738$		0			0
$739$	613.815			0.830602			0.415301	−	0.909684i	$-0.363676\pi$
$739$	613.815			0.830602			0.415301	+	0.909684i	$0.363676\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−331.036			−0.445540			−0.222770	−	0.974871i	$-0.571510\pi$
$743$	−331.036			−0.445540			−0.222770	+	0.974871i	$0.571510\pi$
$744$		0			0
$745$	−222.334	+	155.383i	−0.298434	+	0.208567i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1118.21i			1.49293i
$750$		0			0
$751$	496.852			0.661588			0.330794	−	0.943703i	$-0.392683\pi$
$751$	496.852			0.661588			0.330794	+	0.943703i	$0.392683\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	60.6790	+	86.8241i	0.0803695	+	0.114999i
$756$		0			0
$757$			1234.26i			1.63046i	0.579135	+	0.815232i	$0.303390\pi$
$757$			1234.26i			1.63046i	−0.579135	+	0.815232i	$0.696610\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	69.3693i		−	0.0911555i	−0.998961	−	0.0455777i	$-0.985487\pi$
$761$		−	69.3693i		−	0.0911555i	0.998961	−	0.0455777i	$-0.0145129\pi$
$762$		0			0
$763$		−	61.2111i		−	0.0802243i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	80.5901			0.105072
$768$		0			0
$769$	1110.85			1.44454			0.722271	−	0.691610i	$-0.243098\pi$
$769$	1110.85			1.44454			0.722271	+	0.691610i	$0.243098\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	733.074			0.948349			0.474174	−	0.880431i	$-0.342747\pi$
$773$	733.074			0.948349			0.474174	+	0.880431i	$0.342747\pi$
$774$		0			0
$775$	−338.296	+	924.299i	−0.436511	+	1.19264i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			267.788i			0.343758i
$780$		0			0
$781$	−514.521			−0.658798
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−434.173	+	303.431i	−0.553086	+	0.386536i
$786$		0			0
$787$		−	1004.84i		−	1.27680i	−0.769703	−	0.638402i	$-0.779596\pi$
$787$		−	1004.84i		−	1.27680i	0.769703	−	0.638402i	$-0.220404\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1319.06i			1.66758i
$792$		0			0
$793$		−	38.1011i		−	0.0480468i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	621.101			0.779298			0.389649	−	0.920963i	$-0.372596\pi$
$797$	621.101			0.779298			0.389649	+	0.920963i	$0.372596\pi$
$798$		0			0
$799$	147.630			0.184768
$800$		0			0

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.86421 - 4.09833i) q^{5} -7.15439i q^{7} +O(q^{10})\)	\(q+(-2.86421 - 4.09833i) q^{5} -7.15439i q^{7} +5.06984i q^{11} +3.12682i q^{13} +8.72842 q^{17} +20.1852 q^{19} +14.7284 q^{23} +(-8.59262 + 23.4769i) q^{25} +39.7995i q^{29} +39.3705 q^{31} +(-29.3210 + 20.4917i) q^{35} +34.8712i q^{37} +13.2665i q^{41} -66.6156i q^{43} +16.9137 q^{47} -2.18525 q^{49} -4.62950 q^{53} +(20.7779 - 14.5211i) q^{55} -25.7738i q^{59} -12.1852 q^{61} +(12.8148 - 8.95587i) q^{65} -106.415i q^{67} +101.487i q^{71} +23.2646i q^{73} +36.2716 q^{77} +66.5557 q^{79} +144.284 q^{83} +(-25.0000 - 35.7719i) q^{85} -154.553i q^{89} +22.3705 q^{91} +(-57.8148 - 82.7258i) q^{95} -175.733i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5}+O(q^{10}) \) 4 * q + 3 * q^5	\( 4 q + 3 q^{5} + 6 q^{17} - 6 q^{19} + 30 q^{23} + 9 q^{25} - 16 q^{31} - 45 q^{35} - 48 q^{47} + 78 q^{49} - 192 q^{53} - 47 q^{55} + 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} + 288 q^{83} - 100 q^{85} - 84 q^{91} - 318 q^{95}+O(q^{100}) \) 4 * q + 3 * q^5 + 6 * q^17 - 6 * q^19 + 30 * q^23 + 9 * q^25 - 16 * q^31 - 45 * q^35 - 48 * q^47 + 78 * q^49 - 192 * q^53 - 47 * q^55 + 38 * q^61 + 138 * q^65 + 174 * q^77 + 6 * q^79 + 288 * q^83 - 100 * q^85 - 84 * q^91 - 318 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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