LMFDB - Embedded newform 1024.2.e.j.769.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(257,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("1024.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.e (of order $4$, degree $2$, 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:	$8.17668116698$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			769.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1024.769
Dual form			1024.2.e.j.257.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 1.41421i) q^{5} +3.00000i q^{9} +O(q^{10})$	$q+(-1.41421 - 1.41421i) q^{5} +3.00000i q^{9} +(4.24264 - 4.24264i) q^{13} -2.00000 q^{17} -1.00000i q^{25} +(7.07107 - 7.07107i) q^{29} +(-1.41421 - 1.41421i) q^{37} -10.0000i q^{41} +(4.24264 - 4.24264i) q^{45} +7.00000 q^{49} +(-9.89949 - 9.89949i) q^{53} +(7.07107 - 7.07107i) q^{61} -12.0000 q^{65} +6.00000i q^{73} -9.00000 q^{81} +(2.82843 + 2.82843i) q^{85} +10.0000i q^{89} +18.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{17} + 28 q^{49} - 48 q^{65} - 36 q^{81} + 72 q^{97}+O(q^{100}) $ 4 * q - 8 * q^17 + 28 * q^49 - 48 * q^65 - 36 * q^81 + 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$	−1.41421	−	1.41421i	−0.632456	−	0.632456i	0.316228	−	0.948683i	$-0.397584\pi$
$5$	−1.41421	−	1.41421i	−0.632456	−	0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$	4.24264	−	4.24264i	1.17670	−	1.17670i	0.196116	−	0.980581i	$-0.437167\pi$
$13$	4.24264	−	4.24264i	1.17670	−	1.17670i	0.980581	−	0.196116i	$-0.0628330\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.07107	−	7.07107i	1.31306	−	1.31306i	0.393919	−	0.919145i	$-0.371119\pi$
$29$	7.07107	−	7.07107i	1.31306	−	1.31306i	0.919145	−	0.393919i	$-0.128881\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.41421	−	1.41421i	−0.232495	−	0.232495i	0.581238	−	0.813733i	$-0.302568\pi$
$37$	−1.41421	−	1.41421i	−0.232495	−	0.232495i	−0.813733	+	0.581238i	$0.802568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	4.24264	−	4.24264i	0.632456	−	0.632456i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.89949	−	9.89949i	−1.35980	−	1.35980i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	−9.89949	−	9.89949i	−1.35980	−	1.35980i	−0.485643	−	0.874157i	$-0.661414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	7.07107	−	7.07107i	0.905357	−	0.905357i	−0.0905357	−	0.995893i	$-0.528858\pi$
$61$	7.07107	−	7.07107i	0.905357	−	0.905357i	0.995893	+	0.0905357i	$0.0288579\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.0000			−1.48842
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	2.82843	+	2.82843i	0.306786	+	0.306786i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			10.0000i			1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$			10.0000i			1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	18.0000			1.82762			0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000			1.82762			0.913812	+	0.406138i	$0.133125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.41421	−	1.41421i	−0.140720	−	0.140720i	0.633238	−	0.773957i	$-0.281726\pi$
$101$	−1.41421	−	1.41421i	−0.140720	−	0.140720i	−0.773957	+	0.633238i	$0.781726\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$	4.24264	−	4.24264i	0.406371	−	0.406371i	−0.474100	−	0.880471i	$-0.657226\pi$
$109$	4.24264	−	4.24264i	0.406371	−	0.406371i	0.880471	+	0.474100i	$0.157226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	12.7279	+	12.7279i	1.17670	+	1.17670i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	11.0000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.48528	+	8.48528i	−0.758947	+	0.758947i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			22.0000i			1.87959i	0.341743	+	0.939793i	$0.388983\pi$
$137$			22.0000i			1.87959i	−0.341743	+	0.939793i	$0.611017\pi$
$138$		0			0
$139$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$139$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−20.0000			−1.66091
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.89949	−	9.89949i	−0.810998	−	0.810998i	0.173785	−	0.984784i	$-0.444400\pi$
$149$	−9.89949	−	9.89949i	−0.810998	−	0.810998i	−0.984784	+	0.173785i	$0.944400\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		−	6.00000i		−	0.485071i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−15.5563	+	15.5563i	−1.24153	+	1.24153i	−0.282166	+	0.959366i	$0.591053\pi$
$157$	−15.5563	+	15.5563i	−1.24153	+	1.24153i	−0.959366	+	0.282166i	$0.908947\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$		−	23.0000i		−	1.76923i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−18.3848	+	18.3848i	−1.39777	+	1.39777i	−0.591364	+	0.806405i	$0.701410\pi$
$173$	−18.3848	+	18.3848i	−1.39777	+	1.39777i	−0.806405	+	0.591364i	$0.798590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$179$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$	12.7279	+	12.7279i	0.946059	+	0.946059i	0.998618	−	0.0525588i	$-0.0167377\pi$
$181$	12.7279	+	12.7279i	0.946059	+	0.946059i	−0.0525588	+	0.998618i	$0.516738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			4.00000i			0.294086i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.41421	−	1.41421i	−0.100759	−	0.100759i	0.654931	−	0.755689i	$-0.272698\pi$
$197$	−1.41421	−	1.41421i	−0.100759	−	0.100759i	−0.755689	+	0.654931i	$0.772698\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−14.1421	+	14.1421i	−0.987730	+	0.987730i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$211$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.48528	+	8.48528i	−0.570782	+	0.570782i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	3.00000			0.200000
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$	21.2132	+	21.2132i	1.40181	+	1.40181i	0.794358	+	0.607450i	$0.207808\pi$
$229$	21.2132	+	21.2132i	1.40181	+	1.40181i	0.607450	+	0.794358i	$0.292192\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	26.0000i		−	1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$		−	26.0000i		−	1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	30.0000			1.93247			0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000			1.93247			0.966235	+	0.257663i	$0.0829523\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.89949	−	9.89949i	−0.632456	−	0.632456i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	21.2132	+	21.2132i	1.31306	+	1.31306i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$			28.0000i			1.72003i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.3848	+	18.3848i	−1.12094	+	1.12094i	−0.129339	+	0.991600i	$0.541286\pi$
$269$	−18.3848	+	18.3848i	−1.12094	+	1.12094i	−0.991600	+	0.129339i	$0.958714\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	12.7279	+	12.7279i	0.764747	+	0.764747i	0.977176	−	0.212430i	$-0.0681376\pi$
$277$	12.7279	+	12.7279i	0.764747	+	0.764747i	−0.212430	+	0.977176i	$0.568138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.0416	−	24.0416i	−1.40453	−	1.40453i	−0.784883	−	0.619644i	$-0.787277\pi$
$293$	−24.0416	−	24.0416i	−1.40453	−	1.40453i	−0.619644	−	0.784883i	$-0.712723\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−20.0000			−1.14520
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			26.0000i			1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$			26.0000i			1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.5563	+	15.5563i	−0.873732	+	0.873732i	−0.992877	−	0.119145i	$-0.961985\pi$
$317$	−15.5563	+	15.5563i	−0.873732	+	0.873732i	0.119145	+	0.992877i	$0.461985\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−4.24264	−	4.24264i	−0.235339	−	0.235339i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$		0			0
$333$	4.24264	−	4.24264i	0.232495	−	0.232495i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−18.0000			−0.980522			−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000			−0.980522			−0.490261	+	0.871576i	$0.663099\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$	7.07107	−	7.07107i	0.378506	−	0.378506i	−0.492057	−	0.870563i	$-0.663755\pi$
$349$	7.07107	−	7.07107i	0.378506	−	0.378506i	0.870563	+	0.492057i	$0.163755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	34.0000			1.80964			0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000			1.80964			0.904819	+	0.425797i	$0.140006\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.48528	−	8.48528i	0.444140	−	0.444140i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	30.0000			1.56174
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−9.89949	−	9.89949i	−0.512576	−	0.512576i	0.402739	−	0.915315i	$-0.368058\pi$
$373$	−9.89949	−	9.89949i	−0.512576	−	0.512576i	−0.915315	+	0.402739i	$0.868058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	60.0000i		−	3.09016i
$378$		0			0
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−24.0416	−	24.0416i	−1.21896	−	1.21896i	−0.967997	−	0.250962i	$-0.919253\pi$
$389$	−24.0416	−	24.0416i	−1.21896	−	1.21896i	−0.250962	−	0.967997i	$-0.580747\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	26.8701	−	26.8701i	1.34857	−	1.34857i	0.461353	−	0.887217i	$-0.347364\pi$
$397$	26.8701	−	26.8701i	1.34857	−	1.34857i	0.887217	−	0.461353i	$-0.152636\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.00000			−0.0998752			−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000			−0.0998752			−0.0499376	+	0.998752i	$0.515902\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	12.7279	+	12.7279i	0.632456	+	0.632456i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	6.00000i		−	0.296681i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$		−	6.00000i		−	0.296681i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	21.2132	+	21.2132i	1.03387	+	1.03387i	0.999406	+	0.0344623i	$0.0109719\pi$
$421$	21.2132	+	21.2132i	1.03387	+	1.03387i	0.0344623	+	0.999406i	$0.489028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			2.00000i			0.0970143i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	14.1421	−	14.1421i	0.670402	−	0.670402i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.0000			−0.660701			−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000			−0.660701			−0.330350	+	0.943858i	$0.607167\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	42.0000i		−	1.96468i	−0.187112	−	0.982339i	$-0.559913\pi$
$457$		−	42.0000i		−	1.96468i	0.187112	−	0.982339i	$-0.440087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	26.8701	−	26.8701i	1.25146	−	1.25146i	0.296399	−	0.955064i	$-0.404214\pi$
$461$	26.8701	−	26.8701i	1.25146	−	1.25146i	0.955064	−	0.296399i	$-0.0957859\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	29.6985	−	29.6985i	1.35980	−	1.35980i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−25.4558	−	25.4558i	−1.15589	−	1.15589i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$491$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$	−14.1421	+	14.1421i	−0.636930	+	0.636930i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$			4.00000i			0.177998i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.07107	−	7.07107i	0.313420	−	0.313420i	−0.532813	−	0.846233i	$-0.678865\pi$
$509$	7.07107	−	7.07107i	0.313420	−	0.313420i	0.846233	+	0.532813i	$0.178865\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			22.0000i			0.963837i	0.876216	+	0.481919i	$0.160060\pi$
$521$			22.0000i			0.963837i	−0.876216	+	0.481919i	$0.839940\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−42.4264	−	42.4264i	−1.83769	−	1.83769i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	29.6985	−	29.6985i	1.27684	−	1.27684i	0.334410	−	0.942428i	$-0.391463\pi$
$541$	29.6985	−	29.6985i	1.27684	−	1.27684i	0.942428	−	0.334410i	$-0.108537\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000			−0.514024
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$	21.2132	+	21.2132i	0.905357	+	0.905357i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	26.8701	−	26.8701i	1.13852	−	1.13852i	0.149805	−	0.988716i	$-0.452135\pi$
$557$	26.8701	−	26.8701i	1.13852	−	1.13852i	0.988716	−	0.149805i	$-0.0478647\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	−19.7990	−	19.7990i	−0.832950	−	0.832950i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			26.0000i			1.08998i	0.838444	+	0.544988i	$0.183466\pi$
$569$			26.0000i			1.08998i	−0.838444	+	0.544988i	$0.816534\pi$
$570$		0			0
$571$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$571$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		−	36.0000i		−	1.48842i
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	46.0000			1.88899			0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000			1.88899			0.944497	+	0.328521i	$0.106550\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			10.0000i			0.407909i	0.978980	+	0.203954i	$0.0653794\pi$
$601$			10.0000i			0.407909i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.5563	+	15.5563i	−0.632456	+	0.632456i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−24.0416	−	24.0416i	−0.971032	−	0.971032i	0.0285598	−	0.999592i	$-0.490908\pi$
$613$	−24.0416	−	24.0416i	−0.971032	−	0.971032i	−0.999592	+	0.0285598i	$0.990908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			38.0000i			1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$			38.0000i			1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	19.0000			0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.82843	+	2.82843i	0.112777	+	0.112777i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	29.6985	−	29.6985i	1.17670	−	1.17670i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	50.0000			1.97488			0.987441	−	0.157991i	$-0.0505015\pi$
$641$	50.0000			1.97488			0.987441	+	0.157991i	$0.0505015\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.3848	+	18.3848i	−0.719452	+	0.719452i	−0.968493	−	0.249041i	$-0.919885\pi$
$653$	−18.3848	+	18.3848i	−0.719452	+	0.719452i	0.249041	+	0.968493i	$0.419885\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−18.0000			−0.702247
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$	35.3553	+	35.3553i	1.37516	+	1.37516i	0.852601	+	0.522562i	$0.175024\pi$
$661$	35.3553	+	35.3553i	1.37516	+	1.37516i	0.522562	+	0.852601i	$0.324976\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.41421	−	1.41421i	−0.0543526	−	0.0543526i	0.679408	−	0.733761i	$-0.262237\pi$
$677$	−1.41421	−	1.41421i	−0.0543526	−	0.0543526i	−0.733761	+	0.679408i	$0.762237\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	31.1127	−	31.1127i	1.18876	−	1.18876i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−84.0000			−3.20015
$690$		0			0
$691$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$691$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			20.0000i			0.757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7.07107	−	7.07107i	0.267071	−	0.267071i	−0.560848	−	0.827919i	$-0.689525\pi$
$701$	7.07107	−	7.07107i	0.267071	−	0.267071i	0.827919	+	0.560848i	$0.189525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	21.2132	+	21.2132i	0.796679	+	0.796679i	0.982570	−	0.185892i	$-0.0595174\pi$
$709$	21.2132	+	21.2132i	0.796679	+	0.796679i	−0.185892	+	0.982570i	$0.559517\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.07107	−	7.07107i	−0.262613	−	0.262613i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−38.1838	+	38.1838i	−1.41035	+	1.41035i	−0.652940	+	0.757410i	$0.726464\pi$
$733$	−38.1838	+	38.1838i	−1.41035	+	1.41035i	−0.757410	+	0.652940i	$0.773536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$			28.0000i			1.02584i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	12.7279	+	12.7279i	0.462604	+	0.462604i	0.899508	−	0.436904i	$-0.143925\pi$
$757$	12.7279	+	12.7279i	0.462604	+	0.462604i	−0.436904	+	0.899508i	$0.643925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	38.0000i		−	1.37750i	−0.724999	−	0.688749i	$-0.758160\pi$
$761$		−	38.0000i		−	1.37750i	0.724999	−	0.688749i	$-0.241840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−8.48528	+	8.48528i	−0.306786	+	0.306786i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	50.0000			1.80305			0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000			1.80305			0.901523	+	0.432731i	$0.142450\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−24.0416	−	24.0416i	−0.864717	−	0.864717i	0.127164	−	0.991882i	$-0.459412\pi$
$773$	−24.0416	−	24.0416i	−0.864717	−	0.864717i	−0.991882	+	0.127164i	$0.959412\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	44.0000			1.57043
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	60.0000i		−	2.13066i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.5563	+	15.5563i	−0.551034	+	0.551034i	−0.926739	−	0.375705i	$-0.877401\pi$
$797$	−15.5563	+	15.5563i	−0.551034	+	0.551034i	0.375705	+	0.926739i	$0.377401\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−30.0000			−1.06000
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.e (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 1.41421i) q^{5} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.41421 - 1.41421i) q^{5} +3.00000i q^{9} +(4.24264 - 4.24264i) q^{13} -2.00000 q^{17} -1.00000i q^{25} +(7.07107 - 7.07107i) q^{29} +(-1.41421 - 1.41421i) q^{37} -10.0000i q^{41} +(4.24264 - 4.24264i) q^{45} +7.00000 q^{49} +(-9.89949 - 9.89949i) q^{53} +(7.07107 - 7.07107i) q^{61} -12.0000 q^{65} +6.00000i q^{73} -9.00000 q^{81} +(2.82843 + 2.82843i) q^{85} +10.0000i q^{89} +18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{17} + 28 q^{49} - 48 q^{65} - 36 q^{81} + 72 q^{97}+O(q^{100}) \) 4 * q - 8 * q^17 + 28 * q^49 - 48 * q^65 - 36 * q^81 + 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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