LMFDB - Embedded newform 4788.2.i.f.3457.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(3457,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4788.3457");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.i (of order $2$, degree $1$, 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:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 37x^{12} + 415x^{10} + 1423x^{8} + 2053x^{6} + 1321x^{4} + 355x^{2} + 27 $ x^14 + 37x^12 + 415x^10 + 1423x^8 + 2053x^6 + 1321x^4 + 355x^2 + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 1596)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3457.13
Root			$-0.348386i$ of defining polynomial
Character	$\chi$	$=$	4788.3457
Dual form			4788.2.i.f.3457.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.31434i q^{5} +(1.88089 - 1.86072i) q^{7} +4.03330 q^{11} -4.35648 q^{13} +3.96420i q^{17} +(-4.28615 - 0.793025i) q^{19} -5.24251 q^{23} -5.98483 q^{25} -6.11303i q^{29} -5.77562 q^{31} +(6.16704 + 6.23389i) q^{35} -4.33521i q^{37} -6.66165 q^{41} -6.98483 q^{43} -8.19086i q^{47} +(0.0754726 - 6.99959i) q^{49} -1.52777i q^{53} +13.3677i q^{55} -11.1249 q^{59} -1.72101i q^{61} -14.4388i q^{65} +3.12615i q^{67} +11.5125i q^{71} -7.60095i q^{73} +(7.58618 - 7.50482i) q^{77} +10.1692i q^{79} -2.51607i q^{83} -13.1387 q^{85} +3.82338 q^{89} +(-8.19405 + 8.10617i) q^{91} +(2.62835 - 14.2058i) q^{95} +9.44840 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - q^{7} - 6 q^{11} - 8 q^{13} - 8 q^{25} - 16 q^{31} - q^{35} - 8 q^{41} - 22 q^{43} + 15 q^{49} + 8 q^{59} + 7 q^{77} - 10 q^{85} - 8 q^{89} - 28 q^{91} + 14 q^{95} + 16 q^{97}+O(q^{100}) $ 14 * q - q^7 - 6 * q^11 - 8 * q^13 - 8 * q^25 - 16 * q^31 - q^35 - 8 * q^41 - 22 * q^43 + 15 * q^49 + 8 * q^59 + 7 * q^77 - 10 * q^85 - 8 * q^89 - 28 * q^91 + 14 * q^95 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4788\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$2395$	$4105$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$			3.31434i			1.48222i	0.671385	+	0.741108i	$0.265700\pi$
$5$			3.31434i			1.48222i	−0.671385	+	0.741108i	$0.734300\pi$
$6$		0			0
$7$	1.88089	−	1.86072i	0.710909	−	0.703285i
$7$	1.88089	−	1.86072i	0.710909	−	0.703285i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.03330			1.21608			0.608042	−	0.793905i	$-0.291955\pi$
$11$	4.03330			1.21608			0.608042	+	0.793905i	$0.291955\pi$
$12$		0			0
$13$	−4.35648			−1.20827			−0.604135	−	0.796882i	$-0.706481\pi$
$13$	−4.35648			−1.20827			−0.604135	+	0.796882i	$0.706481\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.96420i			0.961461i	0.876869	+	0.480730i	$0.159628\pi$
$17$			3.96420i			0.961461i	−0.876869	+	0.480730i	$0.840372\pi$
$18$		0			0
$19$	−4.28615	−	0.793025i	−0.983311	−	0.181932i
$19$	−4.28615	−	0.793025i	−0.983311	−	0.181932i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.24251			−1.09314			−0.546570	−	0.837414i	$-0.684067\pi$
$23$	−5.24251			−1.09314			−0.546570	+	0.837414i	$0.684067\pi$
$24$		0			0
$25$	−5.98483			−1.19697
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.11303i		−	1.13516i	−0.823318	−	0.567581i	$-0.807880\pi$
$29$		−	6.11303i		−	1.13516i	0.823318	−	0.567581i	$-0.192120\pi$
$30$		0			0
$31$	−5.77562			−1.03733			−0.518666	−	0.854977i	$-0.673571\pi$
$31$	−5.77562			−1.03733			−0.518666	+	0.854977i	$0.673571\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.16704	+	6.23389i	1.04242	+	1.05372i
$36$		0			0
$37$		−	4.33521i		−	0.712704i	−0.934352	−	0.356352i	$-0.884020\pi$
$37$		−	4.33521i		−	0.712704i	0.934352	−	0.356352i	$-0.115980\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.66165			−1.04037			−0.520187	−	0.854052i	$-0.674138\pi$
$41$	−6.66165			−1.04037			−0.520187	+	0.854052i	$0.674138\pi$
$42$		0			0
$43$	−6.98483			−1.06518			−0.532589	−	0.846374i	$-0.678781\pi$
$43$	−6.98483			−1.06518			−0.532589	+	0.846374i	$0.678781\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	8.19086i		−	1.19476i	−0.801958	−	0.597380i	$-0.796208\pi$
$47$		−	8.19086i		−	1.19476i	0.801958	−	0.597380i	$-0.203792\pi$
$48$		0			0
$49$	0.0754726	−	6.99959i	0.0107818	−	0.999942i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.52777i		−	0.209855i	−0.994480	−	0.104927i	$-0.966539\pi$
$53$		−	1.52777i		−	0.209855i	0.994480	−	0.104927i	$-0.0334610\pi$
$54$		0			0
$55$			13.3677i			1.80250i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.1249			−1.44833			−0.724167	−	0.689625i	$-0.757776\pi$
$59$	−11.1249			−1.44833			−0.724167	+	0.689625i	$0.757776\pi$
$60$		0			0
$61$		−	1.72101i		−	0.220352i	−0.993912	−	0.110176i	$-0.964859\pi$
$61$		−	1.72101i		−	0.220352i	0.993912	−	0.110176i	$-0.0351415\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	14.4388i		−	1.79092i
$66$		0			0
$67$			3.12615i			0.381920i	0.981598	+	0.190960i	$0.0611601\pi$
$67$			3.12615i			0.381920i	−0.981598	+	0.190960i	$0.938840\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.5125i			1.36628i	0.730287	+	0.683140i	$0.239386\pi$
$71$			11.5125i			1.36628i	−0.730287	+	0.683140i	$0.760614\pi$
$72$		0			0
$73$		−	7.60095i		−	0.889623i	−0.895624	−	0.444812i	$-0.853271\pi$
$73$		−	7.60095i		−	0.889623i	0.895624	−	0.444812i	$-0.146729\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.58618	−	7.50482i	0.864525	−	0.855254i
$78$		0			0
$79$			10.1692i			1.14412i	0.820210	+	0.572062i	$0.193856\pi$
$79$			10.1692i			1.14412i	−0.820210	+	0.572062i	$0.806144\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	2.51607i		−	0.276175i	−0.990420	−	0.138087i	$-0.955905\pi$
$83$		−	2.51607i		−	0.276175i	0.990420	−	0.138087i	$-0.0440955\pi$
$84$		0			0
$85$	−13.1387			−1.42509
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.82338			0.405277			0.202639	−	0.979254i	$-0.435048\pi$
$89$	3.82338			0.405277			0.202639	+	0.979254i	$0.435048\pi$
$90$		0			0
$91$	−8.19405	+	8.10617i	−0.858970	+	0.849758i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.62835	−	14.2058i	0.269663	−	1.45748i
$96$		0			0
$97$	9.44840			0.959339			0.479670	−	0.877449i	$-0.340756\pi$
$97$	9.44840			0.959339			0.479670	+	0.877449i	$0.340756\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			16.2626i			1.61819i	0.587677	+	0.809095i	$0.300042\pi$
$101$			16.2626i			1.61819i	−0.587677	+	0.809095i	$0.699958\pi$
$102$		0			0
$103$	1.91272			0.188466			0.0942329	−	0.995550i	$-0.469960\pi$
$103$	1.91272			0.188466			0.0942329	+	0.995550i	$0.469960\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.8661i			1.53383i	0.641748	+	0.766916i	$0.278210\pi$
$107$			15.8661i			1.53383i	−0.641748	+	0.766916i	$0.721790\pi$
$108$		0			0
$109$		−	12.6780i		−	1.21433i	−0.794575	−	0.607166i	$-0.792306\pi$
$109$		−	12.6780i		−	1.21433i	0.794575	−	0.607166i	$-0.207694\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.64600i		−	0.154843i	−0.996998	−	0.0774213i	$-0.975331\pi$
$113$		−	1.64600i		−	0.154843i	0.996998	−	0.0774213i	$-0.0246687\pi$
$114$		0			0
$115$		−	17.3755i		−	1.62027i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.37626	+	7.45622i	0.676180	+	0.683510i
$120$		0			0
$121$	5.26749			0.478862
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	3.26407i		−	0.291947i
$126$		0			0
$127$			12.9730i			1.15117i	0.817742	+	0.575585i	$0.195225\pi$
$127$			12.9730i			1.15117i	−0.817742	+	0.575585i	$0.804775\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			17.8947i			1.56347i	0.623613	+	0.781733i	$0.285664\pi$
$131$			17.8947i			1.56347i	−0.623613	+	0.781733i	$0.714336\pi$
$132$		0			0
$133$	−9.53736	+	6.48372i	−0.826994	+	0.562210i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.9222			−1.01858			−0.509290	−	0.860595i	$-0.670092\pi$
$137$	−11.9222			−1.01858			−0.509290	+	0.860595i	$0.670092\pi$
$138$		0			0
$139$		−	5.84891i		−	0.496098i	−0.968748	−	0.248049i	$-0.920211\pi$
$139$		−	5.84891i		−	0.496098i	0.968748	−	0.248049i	$-0.0797894\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−17.5710			−1.46936
$144$		0			0
$145$	20.2606			1.68255
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.15913			0.586499			0.293250	−	0.956036i	$-0.405263\pi$
$149$	7.15913			0.586499			0.293250	+	0.956036i	$0.405263\pi$
$150$		0			0
$151$		−	23.1949i		−	1.88757i	−0.330555	−	0.943787i	$-0.607236\pi$
$151$		−	23.1949i		−	1.88757i	0.330555	−	0.943787i	$-0.392764\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	19.1423i		−	1.53755i
$156$		0			0
$157$		−	7.02849i		−	0.560934i	−0.959864	−	0.280467i	$-0.909511\pi$
$157$		−	7.02849i		−	0.560934i	0.959864	−	0.280467i	$-0.0904894\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.86058	+	9.75483i	−0.777122	+	0.768788i
$162$		0			0
$163$	11.7954			0.923884			0.461942	−	0.886910i	$-0.347153\pi$
$163$	11.7954			0.923884			0.461942	+	0.886910i	$0.347153\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.58195			0.664092			0.332046	−	0.943263i	$-0.392261\pi$
$167$	8.58195			0.664092			0.332046	+	0.943263i	$0.392261\pi$
$168$		0			0
$169$	5.97893			0.459918
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−13.2435			−1.00688			−0.503441	−	0.864030i	$-0.667933\pi$
$173$	−13.2435			−1.00688			−0.503441	+	0.864030i	$0.667933\pi$
$174$		0			0
$175$	−11.2568	+	11.1361i	−0.850934	+	0.841808i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	6.65878i		−	0.497700i	−0.968542	−	0.248850i	$-0.919947\pi$
$179$		−	6.65878i		−	0.497700i	0.968542	−	0.248850i	$-0.0800527\pi$
$180$		0			0
$181$	−16.0741			−1.19478			−0.597388	−	0.801952i	$-0.703795\pi$
$181$	−16.0741			−1.19478			−0.597388	+	0.801952i	$0.703795\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.3683			1.05638
$186$		0			0
$187$			15.9888i			1.16922i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.3476			−1.18287			−0.591435	−	0.806353i	$-0.701438\pi$
$191$	−16.3476			−1.18287			−0.591435	+	0.806353i	$0.701438\pi$
$192$		0			0
$193$		−	21.3685i		−	1.53814i	−0.639167	−	0.769068i	$-0.720721\pi$
$193$		−	21.3685i		−	1.53814i	0.639167	−	0.769068i	$-0.279279\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.65959			0.331982			0.165991	−	0.986127i	$-0.446918\pi$
$197$	4.65959			0.331982			0.165991	+	0.986127i	$0.446918\pi$
$198$		0			0
$199$			17.3410i			1.22927i	0.788810	+	0.614637i	$0.210697\pi$
$199$			17.3410i			1.22927i	−0.788810	+	0.614637i	$0.789303\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.3746	−	11.4979i	−0.798341	−	0.806996i
$204$		0			0
$205$		−	22.0790i		−	1.54206i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−17.2873	−	3.19851i	−1.19579	−	0.221245i
$210$		0			0
$211$		−	23.5438i		−	1.62082i	−0.585861	−	0.810411i	$-0.699244\pi$
$211$		−	23.5438i		−	1.62082i	0.585861	−	0.810411i	$-0.300756\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	23.1501i		−	1.57882i
$216$		0			0
$217$	−10.8633	+	10.7468i	−0.737448	+	0.729539i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	17.2700i		−	1.16170i
$222$		0			0
$223$	−12.6528			−0.847294			−0.423647	−	0.905827i	$-0.639250\pi$
$223$	−12.6528			−0.847294			−0.423647	+	0.905827i	$0.639250\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−23.5130			−1.56061			−0.780305	−	0.625399i	$-0.784936\pi$
$227$	−23.5130			−1.56061			−0.780305	+	0.625399i	$0.784936\pi$
$228$		0			0
$229$		−	4.17713i		−	0.276032i	−0.990430	−	0.138016i	$-0.955927\pi$
$229$		−	4.17713i		−	0.276032i	0.990430	−	0.138016i	$-0.0440726\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.25606			0.344336			0.172168	−	0.985068i	$-0.444923\pi$
$233$	5.25606			0.344336			0.172168	+	0.985068i	$0.444923\pi$
$234$		0			0
$235$	27.1473			1.77089
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.5845			−0.749341			−0.374671	−	0.927158i	$-0.622244\pi$
$239$	−11.5845			−0.749341			−0.374671	+	0.927158i	$0.622244\pi$
$240$		0			0
$241$	14.1321			0.910328			0.455164	−	0.890407i	$-0.349581\pi$
$241$	14.1321			0.910328			0.455164	+	0.890407i	$0.349581\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	23.1990	+	0.250142i	1.48213	+	0.0159810i
$246$		0			0
$247$	18.6725	+	3.45480i	1.18811	+	0.219824i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	15.8508i		−	1.00050i	−0.865882	−	0.500248i	$-0.833242\pi$
$251$		−	15.8508i		−	1.00050i	0.865882	−	0.500248i	$-0.166758\pi$
$252$		0			0
$253$	−21.1446			−1.32935
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.4714			1.21459			0.607296	−	0.794475i	$-0.292254\pi$
$257$	19.4714			1.21459			0.607296	+	0.794475i	$0.292254\pi$
$258$		0			0
$259$	−8.06659	−	8.15404i	−0.501234	−	0.506667i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−15.1924			−0.936802			−0.468401	−	0.883516i	$-0.655170\pi$
$263$	−15.1924			−0.936802			−0.468401	+	0.883516i	$0.655170\pi$
$264$		0			0
$265$	5.06353			0.311050
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.35810			−0.204747			−0.102373	−	0.994746i	$-0.532644\pi$
$269$	−3.35810			−0.204747			−0.102373	+	0.994746i	$0.532644\pi$
$270$		0			0
$271$		−	11.2025i		−	0.680502i	−0.940335	−	0.340251i	$-0.889488\pi$
$271$		−	11.2025i		−	0.680502i	0.940335	−	0.340251i	$-0.110512\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−24.1386			−1.45561
$276$		0			0
$277$	−24.0596			−1.44560			−0.722801	−	0.691056i	$-0.757146\pi$
$277$	−24.0596			−1.44560			−0.722801	+	0.691056i	$0.757146\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	13.3703i		−	0.797604i	−0.917037	−	0.398802i	$-0.869426\pi$
$281$		−	13.3703i		−	0.797604i	0.917037	−	0.398802i	$-0.130574\pi$
$282$		0			0
$283$			30.5046i			1.81331i	0.421874	+	0.906654i	$0.361372\pi$
$283$			30.5046i			1.81331i	−0.421874	+	0.906654i	$0.638628\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.5298	+	12.3954i	−0.739611	+	0.731680i
$288$		0			0
$289$	1.28509			0.0755936
$290$		0			0
$291$		0			0
$292$		0			0
$293$	26.8711			1.56982			0.784912	−	0.619607i	$-0.212708\pi$
$293$	26.8711			1.56982			0.784912	+	0.619607i	$0.212708\pi$
$294$		0			0
$295$		−	36.8716i		−	2.14675i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	22.8389			1.32081
$300$		0			0
$301$	−13.1377	+	12.9968i	−0.757243	+	0.749123i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.70400			0.326610
$306$		0			0
$307$	27.2887			1.55745			0.778725	−	0.627366i	$-0.215867\pi$
$307$	27.2887			1.55745			0.778725	+	0.627366i	$0.215867\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	20.7715i		−	1.17784i	−0.808191	−	0.588921i	$-0.799553\pi$
$311$		−	20.7715i		−	1.17784i	0.808191	−	0.588921i	$-0.200447\pi$
$312$		0			0
$313$		−	15.5300i		−	0.877806i	−0.898534	−	0.438903i	$-0.855367\pi$
$313$		−	15.5300i		−	0.877806i	0.898534	−	0.438903i	$-0.144633\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.94093i		−	0.165179i	−0.996584	−	0.0825895i	$-0.973681\pi$
$317$		−	2.94093i		−	0.165179i	0.996584	−	0.0825895i	$-0.0263190\pi$
$318$		0			0
$319$		−	24.6557i		−	1.38045i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.14371	−	16.9912i	0.174921	−	0.945415i
$324$		0			0
$325$	26.0728			1.44626
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−15.2409	−	15.4061i	−0.840256	−	0.849365i
$330$		0			0
$331$			26.0862i			1.43383i	0.697161	+	0.716914i	$0.254446\pi$
$331$			26.0862i			1.43383i	−0.697161	+	0.716914i	$0.745554\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.3611			−0.566089
$336$		0			0
$337$			0.747263i			0.0407060i	0.999793	+	0.0203530i	$0.00647901\pi$
$337$			0.747263i			0.0407060i	−0.999793	+	0.0203530i	$0.993521\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−23.2948			−1.26148
$342$		0			0
$343$	−12.8823	−	13.3059i	−0.695579	−	0.718450i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.00561103			0.000301216			0.000150608	−	1.00000i	$-0.499952\pi$
$347$	0.00561103			0.000301216			0.000150608		1.00000i	$0.499952\pi$
$348$		0			0
$349$			6.05622i			0.324182i	0.986776	+	0.162091i	$0.0518238\pi$
$349$			6.05622i			0.324182i	−0.986776	+	0.162091i	$0.948176\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	6.02908i		−	0.320895i	−0.987044	−	0.160448i	$-0.948706\pi$
$353$		−	6.02908i		−	0.320895i	0.987044	−	0.160448i	$-0.0512938\pi$
$354$		0			0
$355$	−38.1562			−2.02512
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−27.7208			−1.46305			−0.731525	−	0.681815i	$-0.761191\pi$
$359$	−27.7208			−1.46305			−0.731525	+	0.681815i	$0.761191\pi$
$360$		0			0
$361$	17.7422	+	6.79805i	0.933801	+	0.357792i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	25.1921			1.31861
$366$		0			0
$367$			16.2401i			0.847727i	0.905726	+	0.423863i	$0.139326\pi$
$367$			16.2401i			0.847727i	−0.905726	+	0.423863i	$0.860674\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.84274	−	2.87356i	−0.147588	−	0.149188i
$372$		0			0
$373$			32.2960i			1.67222i	0.548558	+	0.836112i	$0.315177\pi$
$373$			32.2960i			1.67222i	−0.548558	+	0.836112i	$0.684823\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			26.6313i			1.37158i
$378$		0			0
$379$			21.1706i			1.08746i	0.839259	+	0.543732i	$0.182989\pi$
$379$			21.1706i			1.08746i	−0.839259	+	0.543732i	$0.817011\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.68238			−0.290356			−0.145178	−	0.989406i	$-0.546375\pi$
$383$	−5.68238			−0.290356			−0.145178	+	0.989406i	$0.546375\pi$
$384$		0			0
$385$	24.8735	+	25.1432i	1.26767	+	1.28141i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.13814			−0.463322			−0.231661	−	0.972797i	$-0.574416\pi$
$389$	−9.13814			−0.463322			−0.231661	+	0.972797i	$0.574416\pi$
$390$		0			0
$391$		−	20.7824i		−	1.05101i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−33.7042			−1.69584
$396$		0			0
$397$			22.9841i			1.15354i	0.816906	+	0.576770i	$0.195687\pi$
$397$			22.9841i			1.15354i	−0.816906	+	0.576770i	$0.804313\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4.41910i		−	0.220679i	−0.993894	−	0.110340i	$-0.964806\pi$
$401$		−	4.41910i		−	0.220679i	0.993894	−	0.110340i	$-0.0351939\pi$
$402$		0			0
$403$	25.1614			1.25338
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	17.4852i		−	0.866709i
$408$		0			0
$409$	−22.9984			−1.13720			−0.568598	−	0.822616i	$-0.692514\pi$
$409$	−22.9984			−1.13720			−0.568598	+	0.822616i	$0.692514\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−20.9246	+	20.7002i	−1.02963	+	1.01859i
$414$		0			0
$415$	8.33911			0.409351
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	13.4329i		−	0.656240i	−0.944636	−	0.328120i	$-0.893585\pi$
$419$		−	13.4329i		−	0.656240i	0.944636	−	0.328120i	$-0.106415\pi$
$420$		0			0
$421$		−	11.0245i		−	0.537303i	−0.963237	−	0.268652i	$-0.913422\pi$
$421$		−	11.0245i		−	0.537303i	0.963237	−	0.268652i	$-0.0865780\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	23.7251i		−	1.15084i
$426$		0			0
$427$	−3.20230	−	3.23702i	−0.154970	−	0.156650i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	19.6845i		−	0.948167i	−0.880480	−	0.474084i	$-0.842779\pi$
$431$		−	19.6845i		−	0.948167i	0.880480	−	0.474084i	$-0.157221\pi$
$432$		0			0
$433$	−25.5912			−1.22984			−0.614918	−	0.788591i	$-0.710811\pi$
$433$	−25.5912			−1.22984			−0.614918	+	0.788591i	$0.710811\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	22.4702	+	4.15744i	1.07490	+	0.198877i
$438$		0			0
$439$	−5.60570			−0.267546			−0.133773	−	0.991012i	$-0.542709\pi$
$439$	−5.60570			−0.267546			−0.133773	+	0.991012i	$0.542709\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−39.2207			−1.86343			−0.931716	−	0.363188i	$-0.881688\pi$
$443$	−39.2207			−1.86343			−0.931716	+	0.363188i	$0.881688\pi$
$444$		0			0
$445$			12.6720i			0.600709i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			27.4698i			1.29638i	0.761478	+	0.648191i	$0.224474\pi$
$449$			27.4698i			1.29638i	−0.761478	+	0.648191i	$0.775526\pi$
$450$		0			0
$451$	−26.8684			−1.26518
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−26.8666	−	27.1578i	−1.25953	−	1.27318i
$456$		0			0
$457$	32.0041			1.49709			0.748545	−	0.663084i	$-0.230753\pi$
$457$	32.0041			1.49709			0.748545	+	0.663084i	$0.230753\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			12.9913i			0.605064i	0.953139	+	0.302532i	$0.0978319\pi$
$461$			12.9913i			0.605064i	−0.953139	+	0.302532i	$0.902168\pi$
$462$		0			0
$463$	10.9542			0.509084			0.254542	−	0.967062i	$-0.418075\pi$
$463$	10.9542			0.509084			0.254542	+	0.967062i	$0.418075\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			31.1520i			1.44154i	0.693172	+	0.720772i	$0.256213\pi$
$467$			31.1520i			1.44154i	−0.693172	+	0.720772i	$0.743787\pi$
$468$		0			0
$469$	5.81688	+	5.87994i	0.268599	+	0.271510i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−28.1719			−1.29535
$474$		0			0
$475$	25.6519	+	4.74612i	1.17699	+	0.217767i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	20.6941i		−	0.945535i	−0.881187	−	0.472768i	$-0.843255\pi$
$479$		−	20.6941i		−	0.945535i	0.881187	−	0.472768i	$-0.156745\pi$
$480$		0			0
$481$			18.8863i			0.861139i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			31.3152i			1.42195i
$486$		0			0
$487$		−	9.43866i		−	0.427707i	−0.976866	−	0.213853i	$-0.931399\pi$
$487$		−	9.43866i		−	0.427707i	0.976866	−	0.213853i	$-0.0686015\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.7595			0.575830			0.287915	−	0.957656i	$-0.407038\pi$
$491$	12.7595			0.575830			0.287915	+	0.957656i	$0.407038\pi$
$492$		0			0
$493$	24.2333			1.09141
$494$		0			0
$495$		0			0
$496$		0			0
$497$	21.4215	+	21.6537i	0.960884	+	0.971300i
$498$		0			0
$499$	29.4787			1.31965			0.659824	−	0.751420i	$-0.270631\pi$
$499$	29.4787			1.31965			0.659824	+	0.751420i	$0.270631\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			17.8223i			0.794659i	0.917676	+	0.397329i	$0.130063\pi$
$503$			17.8223i			0.794659i	−0.917676	+	0.397329i	$0.869937\pi$
$504$		0			0
$505$	−53.8998			−2.39851
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.40660			0.372616			0.186308	−	0.982491i	$-0.440348\pi$
$509$	8.40660			0.372616			0.186308	+	0.982491i	$0.440348\pi$
$510$		0			0
$511$	−14.1432	−	14.2965i	−0.625658	−	0.632441i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			6.33940i			0.279347i
$516$		0			0
$517$		−	33.0362i		−	1.45293i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.3828			−1.15585			−0.577926	−	0.816089i	$-0.696138\pi$
$521$	−26.3828			−1.15585			−0.577926	+	0.816089i	$0.696138\pi$
$522$		0			0
$523$	2.13998			0.0935747			0.0467874	−	0.998905i	$-0.485102\pi$
$523$	2.13998			0.0935747			0.0467874	+	0.998905i	$0.485102\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	22.8957i		−	0.997353i
$528$		0			0
$529$	4.48394			0.194954
$530$		0			0
$531$		0			0
$532$		0			0
$533$	29.0213			1.25705
$534$		0			0
$535$	−52.5855			−2.27347
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.304403	−	28.2314i	0.0131116	−	1.21601i
$540$		0			0
$541$	34.7311			1.49321			0.746604	−	0.665269i	$-0.231683\pi$
$541$	34.7311			1.49321			0.746604	+	0.665269i	$0.231683\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	42.0191			1.79990
$546$		0			0
$547$			37.5317i			1.60474i	0.596827	+	0.802370i	$0.296428\pi$
$547$			37.5317i			1.60474i	−0.596827	+	0.802370i	$0.703572\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.84778	+	26.2014i	−0.206523	+	1.11622i
$552$		0			0
$553$	18.9220	+	19.1271i	0.804645	+	0.813368i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25.2915			1.07163			0.535817	−	0.844334i	$-0.320004\pi$
$557$	25.2915			1.07163			0.535817	+	0.844334i	$0.320004\pi$
$558$		0			0
$559$	30.4293			1.28702
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.05337			0.255119			0.127560	−	0.991831i	$-0.459286\pi$
$563$	6.05337			0.255119			0.127560	+	0.991831i	$0.459286\pi$
$564$		0			0
$565$	5.45540			0.229510
$566$		0			0
$567$		0			0
$568$		0			0
$569$			35.3530i			1.48207i	0.671465	+	0.741036i	$0.265665\pi$
$569$			35.3530i			1.48207i	−0.671465	+	0.741036i	$0.734335\pi$
$570$		0			0
$571$	−40.7930			−1.70713			−0.853566	−	0.520985i	$-0.825565\pi$
$571$	−40.7930			−1.70713			−0.853566	+	0.520985i	$0.825565\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	31.3756			1.30845
$576$		0			0
$577$		−	39.0205i		−	1.62444i	−0.583348	−	0.812222i	$-0.698258\pi$
$577$		−	39.0205i		−	1.62444i	0.583348	−	0.812222i	$-0.301742\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.68169	−	4.73245i	−0.194229	−	0.196335i
$582$		0			0
$583$		−	6.16194i		−	0.255201i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			18.4455i			0.761327i	0.924714	+	0.380663i	$0.124304\pi$
$587$			18.4455i			0.761327i	−0.924714	+	0.380663i	$0.875696\pi$
$588$		0			0
$589$	24.7552	+	4.58021i	1.02002	+	0.188724i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	13.9098i		−	0.571205i	−0.958348	−	0.285603i	$-0.907806\pi$
$593$		−	13.9098i		−	0.571205i	0.958348	−	0.285603i	$-0.0921938\pi$
$594$		0			0
$595$	−24.7124	+	24.4474i	−1.01311	+	1.00225i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			3.26958i			0.133591i	0.997767	+	0.0667956i	$0.0212775\pi$
$599$			3.26958i			0.133591i	−0.997767	+	0.0667956i	$0.978722\pi$
$600$		0			0
$601$	18.1246			0.739319			0.369660	−	0.929167i	$-0.379474\pi$
$601$	18.1246			0.739319			0.369660	+	0.929167i	$0.379474\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			17.4582i			0.709778i
$606$		0			0
$607$	10.1642			0.412553			0.206276	−	0.978494i	$-0.433865\pi$
$607$	10.1642			0.412553			0.206276	+	0.978494i	$0.433865\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			35.6833i			1.44359i
$612$		0			0
$613$	26.1018			1.05424			0.527120	−	0.849791i	$-0.323272\pi$
$613$	26.1018			1.05424			0.527120	+	0.849791i	$0.323272\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.8728			0.719533			0.359766	−	0.933042i	$-0.382856\pi$
$617$	17.8728			0.719533			0.359766	+	0.933042i	$0.382856\pi$
$618$		0			0
$619$			19.5703i			0.786598i	0.919411	+	0.393299i	$0.128666\pi$
$619$			19.5703i			0.786598i	−0.919411	+	0.393299i	$0.871334\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.19134	−	7.11422i	0.288115	−	0.285025i
$624$		0			0
$625$	−19.1059			−0.764237
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17.1857			0.685237
$630$		0			0
$631$	13.4699			0.536227			0.268113	−	0.963387i	$-0.413600\pi$
$631$	13.4699			0.536227			0.268113	+	0.963387i	$0.413600\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−42.9970			−1.70628
$636$		0			0
$637$	−0.328795	+	30.4936i	−0.0130273	+	1.20820i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			16.3001i			0.643814i	0.946771	+	0.321907i	$0.104324\pi$
$641$			16.3001i			0.643814i	−0.946771	+	0.321907i	$0.895676\pi$
$642$		0			0
$643$			44.9878i			1.77415i	0.461628	+	0.887074i	$0.347266\pi$
$643$			44.9878i			1.77415i	−0.461628	+	0.887074i	$0.652734\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.57675i		−	0.0619884i	−0.999520	−	0.0309942i	$-0.990133\pi$
$647$		−	1.57675i		−	0.0619884i	0.999520	−	0.0309942i	$-0.00986735\pi$
$648$		0			0
$649$	−44.8699			−1.76130
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−36.8857			−1.44345			−0.721725	−	0.692180i	$-0.756650\pi$
$653$	−36.8857			−1.44345			−0.721725	+	0.692180i	$0.756650\pi$
$654$		0			0
$655$	−59.3090			−2.31740
$656$		0			0
$657$		0			0
$658$		0			0
$659$			16.6676i			0.649277i	0.945838	+	0.324639i	$0.105243\pi$
$659$			16.6676i			0.649277i	−0.945838	+	0.324639i	$0.894757\pi$
$660$		0			0
$661$	−37.8693			−1.47295			−0.736473	−	0.676467i	$-0.763510\pi$
$661$	−37.8693			−1.47295			−0.736473	+	0.676467i	$0.763510\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−21.4892	−	31.6100i	−0.833317	−	1.22579i
$666$		0			0
$667$			32.0476i			1.24089i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	6.94133i		−	0.267967i
$672$		0			0
$673$			27.2338i			1.04979i	0.851168	+	0.524894i	$0.175895\pi$
$673$			27.2338i			1.04979i	−0.851168	+	0.524894i	$0.824105\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.2186			0.469598			0.234799	−	0.972044i	$-0.424557\pi$
$677$	12.2186			0.469598			0.234799	+	0.972044i	$0.424557\pi$
$678$		0			0
$679$	17.7714	−	17.5808i	0.682003	−	0.674689i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	34.4046i		−	1.31646i	−0.752818	−	0.658229i	$-0.771306\pi$
$683$		−	34.4046i		−	1.31646i	0.752818	−	0.658229i	$-0.228694\pi$
$684$		0			0
$685$		−	39.5141i		−	1.50976i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			6.65569i			0.253561i
$690$		0			0
$691$		−	17.0960i		−	0.650364i	−0.945651	−	0.325182i	$-0.894574\pi$
$691$		−	17.0960i		−	0.650364i	0.945651	−	0.325182i	$-0.105426\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	19.3853			0.735325
$696$		0			0
$697$		−	26.4081i		−	1.00028i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.24710			−0.235950			−0.117975	−	0.993017i	$-0.537640\pi$
$701$	−6.24710			−0.235950			−0.117975	+	0.993017i	$0.537640\pi$
$702$		0			0
$703$	−3.43793	+	18.5814i	−0.129664	+	0.700810i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	30.2601	+	30.5881i	1.13805	+	1.15039i
$708$		0			0
$709$	−5.84304			−0.219440			−0.109720	−	0.993963i	$-0.534995\pi$
$709$	−5.84304			−0.219440			−0.109720	+	0.993963i	$0.534995\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	30.2787			1.13395
$714$		0			0
$715$		−	58.2362i		−	2.17791i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			4.68298i			0.174646i	0.996180	+	0.0873228i	$0.0278312\pi$
$719$			4.68298i			0.174646i	−0.996180	+	0.0873228i	$0.972169\pi$
$720$		0			0
$721$	3.59761	−	3.55903i	0.133982	−	0.132545i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			36.5855i			1.35875i
$726$		0			0
$727$		−	50.2695i		−	1.86439i	−0.361955	−	0.932196i	$-0.617890\pi$
$727$		−	50.2695i		−	1.86439i	0.361955	−	0.932196i	$-0.382110\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	27.6893i		−	1.02413i
$732$		0			0
$733$		−	34.2636i		−	1.26555i	−0.774334	−	0.632777i	$-0.781915\pi$
$733$		−	34.2636i		−	1.26555i	0.774334	−	0.632777i	$-0.218085\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12.6087i			0.464448i
$738$		0			0
$739$	−25.4568			−0.936444			−0.468222	−	0.883611i	$-0.655105\pi$
$739$	−25.4568			−0.936444			−0.468222	+	0.883611i	$0.655105\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	35.4359i		−	1.30002i	−0.759926	−	0.650009i	$-0.774765\pi$
$743$		−	35.4359i		−	1.30002i	0.759926	−	0.650009i	$-0.225235\pi$
$744$		0			0
$745$			23.7278i			0.869319i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	29.5223	+	29.8423i	1.07872	+	1.09041i
$750$		0			0
$751$		−	53.8794i		−	1.96609i	−0.183375	−	0.983043i	$-0.558702\pi$
$751$		−	53.8794i		−	1.96609i	0.183375	−	0.983043i	$-0.441298\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	76.8757			2.79779
$756$		0			0
$757$	−39.0592			−1.41963			−0.709815	−	0.704388i	$-0.751222\pi$
$757$	−39.0592			−1.41963			−0.709815	+	0.704388i	$0.751222\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	0.555606i		−	0.0201407i	−0.999949	−	0.0100704i	$-0.996794\pi$
$761$		−	0.555606i		−	0.0201407i	0.999949	−	0.0100704i	$-0.00320555\pi$
$762$		0			0
$763$	−23.5901	−	23.8459i	−0.854021	−	0.863279i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	48.4653			1.74998
$768$		0			0
$769$			37.8868i			1.36623i	0.730310	+	0.683116i	$0.239376\pi$
$769$			37.8868i			1.36623i	−0.730310	+	0.683116i	$0.760624\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−25.1691			−0.905269			−0.452634	−	0.891696i	$-0.649516\pi$
$773$	−25.1691			−0.905269			−0.452634	+	0.891696i	$0.649516\pi$
$774$		0			0
$775$	34.5661			1.24165
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.5529	+	5.28285i	1.02301	+	0.189278i
$780$		0			0
$781$			46.4333i			1.66151i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.2948			0.831426
$786$		0			0
$787$	−0.527575			−0.0188060			−0.00940301	−	0.999956i	$-0.502993\pi$
$787$	−0.527575			−0.0188060			−0.00940301	+	0.999956i	$0.502993\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.06274	−	3.09594i	−0.108898	−	0.110079i
$792$		0			0
$793$			7.49753i			0.266245i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	34.4328			1.21967			0.609836	−	0.792528i	$-0.291235\pi$
$797$	34.4328			1.21967			0.609836	+	0.792528i	$0.291235\pi$
$798$		0			0
$799$	32.4702			1.14871
$800$		0			0
$801$		0			0
$802$		0			0
$803$

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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.31434i q^{5} +(1.88089 - 1.86072i) q^{7} +4.03330 q^{11} -4.35648 q^{13} +3.96420i q^{17} +(-4.28615 - 0.793025i) q^{19} -5.24251 q^{23} -5.98483 q^{25} -6.11303i q^{29} -5.77562 q^{31} +(6.16704 + 6.23389i) q^{35} -4.33521i q^{37} -6.66165 q^{41} -6.98483 q^{43} -8.19086i q^{47} +(0.0754726 - 6.99959i) q^{49} -1.52777i q^{53} +13.3677i q^{55} -11.1249 q^{59} -1.72101i q^{61} -14.4388i q^{65} +3.12615i q^{67} +11.5125i q^{71} -7.60095i q^{73} +(7.58618 - 7.50482i) q^{77} +10.1692i q^{79} -2.51607i q^{83} -13.1387 q^{85} +3.82338 q^{89} +(-8.19405 + 8.10617i) q^{91} +(2.62835 - 14.2058i) q^{95} +9.44840 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - q^{7} - 6 q^{11} - 8 q^{13} - 8 q^{25} - 16 q^{31} - q^{35} - 8 q^{41} - 22 q^{43} + 15 q^{49} + 8 q^{59} + 7 q^{77} - 10 q^{85} - 8 q^{89} - 28 q^{91} + 14 q^{95} + 16 q^{97}+O(q^{100}) \) 14 * q - q^7 - 6 * q^11 - 8 * q^13 - 8 * q^25 - 16 * q^31 - q^35 - 8 * q^41 - 22 * q^43 + 15 * q^49 + 8 * q^59 + 7 * q^77 - 10 * q^85 - 8 * q^89 - 28 * q^91 + 14 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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