LMFDB - Embedded newform 64.22.e.a.17.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [64,22,Mod(17,64)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("64.17"); S:= CuspForms(chi, 22); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 22, names="a")

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	64.e (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$178.865500344$
Analytic rank:	$0$
Dimension:	$82$
Relative dimension:	$41$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.10
Character	$\chi$	$=$	64.17
Dual form			64.22.e.a.49.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-76248.0 - 76248.0i) q^{3} +(1.87162e7 - 1.87162e7i) q^{5} +8.00917e7i q^{7} +1.16716e9i q^{9} +(-5.81865e10 + 5.81865e10i) q^{11} +(-1.14156e11 - 1.14156e11i) q^{13} -2.85415e12 q^{15} -1.08847e13 q^{17} +(2.05483e13 + 2.05483e13i) q^{19} +(6.10683e12 - 6.10683e12i) q^{21} -2.73452e14i q^{23} -2.23758e14i q^{25} +(-7.08587e14 + 7.08587e14i) q^{27} +(1.57763e15 + 1.57763e15i) q^{29} -1.20723e15 q^{31} +8.87321e15 q^{33} +(1.49902e15 + 1.49902e15i) q^{35} +(2.16994e16 - 2.16994e16i) q^{37} +1.74084e16i q^{39} +1.20240e17i q^{41} +(4.92672e16 - 4.92672e16i) q^{43} +(2.18449e16 + 2.18449e16i) q^{45} -1.05625e16 q^{47} +5.52131e17 q^{49} +(8.29933e17 + 8.29933e17i) q^{51} +(1.46692e17 - 1.46692e17i) q^{53} +2.17806e18i q^{55} -3.13354e18i q^{57} +(-5.24717e18 + 5.24717e18i) q^{59} +(-1.31997e18 - 1.31997e18i) q^{61} -9.34800e16 q^{63} -4.27315e18 q^{65} +(-4.67014e18 - 4.67014e18i) q^{67} +(-2.08502e19 + 2.08502e19i) q^{69} -9.38208e18i q^{71} +5.24082e19i q^{73} +(-1.70611e19 + 1.70611e19i) q^{75} +(-4.66025e18 - 4.66025e18i) q^{77} -8.76083e19 q^{79} +1.20266e20 q^{81} +(9.44999e19 + 9.44999e19i) q^{83} +(-2.03720e20 + 2.03720e20i) q^{85} -2.40583e20i q^{87} +3.49801e20i q^{89} +(9.14296e18 - 9.14296e18i) q^{91} +(9.20489e19 + 9.20489e19i) q^{93} +7.69175e20 q^{95} +4.43573e20 q^{97} +(-6.79131e19 - 6.79131e19i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 2 q^{3} - 2 q^{5} - 67333320738 q^{11} - 2 q^{13} - 4613203124996 q^{15} - 4 q^{17} + 46007763621434 q^{19} + 20920706404 q^{21} - 11\!\cdots\!20 q^{27} - 24\!\cdots\!02 q^{29} + 98\!\cdots\!16 q^{31}+ \cdots - 27\!\cdots\!38 q^{99}+O(q^{100}) $ 82 * q + 2 * q^3 - 2 * q^5 - 67333320738 * q^11 - 2 * q^13 - 4613203124996 * q^15 - 4 * q^17 + 46007763621434 * q^19 + 20920706404 * q^21 - 1147760716556920 * q^27 - 2433173450216602 * q^29 + 9835539443769616 * q^31 - 4 * q^33 + 19253478556034884 * q^35 - 7245708527119386 * q^37 - 215332019803969386 * q^43 + 953653395699842 * q^45 - 1051982644716600976 * q^47 - 5585458640832840074 * q^49 - 3238772068175213124 * q^51 + 1889987679210147606 * q^53 - 3631979197965016922 * q^59 + 10385362452353654062 * q^61 + 17464422381302680956 * q^63 - 9364161130572538716 * q^65 + 17102217044572669130 * q^67 - 3951842785111207532 * q^69 - 145960939174166395366 * q^75 - 82041801652678580764 * q^77 - 204448210818924484480 * q^79 - 802405920297757300870 * q^81 - 317463236552669319678 * q^83 + 323656241286464843748 * q^85 - 144326756239046079748 * q^91 - 669717759565183008560 * q^93 + 1411330138695550130700 * q^95 - 4 * q^97 - 2710306841526615010238 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$.

$n$	$5$	$63$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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
$3$	−76248.0	−	76248.0i	−0.745513	−	0.745513i	0.228120	−	0.973633i	$-0.426742\pi$
$3$	−76248.0	−	76248.0i	−0.745513	−	0.745513i	−0.973633	+	0.228120i	$0.926742\pi$
$4$		0			0
$5$	1.87162e7	−	1.87162e7i	0.857104	−	0.857104i	−0.133892	−	0.990996i	$-0.542747\pi$
$5$	1.87162e7	−	1.87162e7i	0.857104	−	0.857104i	0.990996	+	0.133892i	$0.0427475\pi$
$6$		0			0
$7$			8.00917e7i			0.107166i	0.998563	+	0.0535831i	$0.0170642\pi$
$7$			8.00917e7i			0.107166i	−0.998563	+	0.0535831i	$0.982936\pi$
$8$		0			0
$9$			1.16716e9i			0.111580i
$10$		0			0
$11$	−5.81865e10	+	5.81865e10i	−0.676393	+	0.676393i	−0.959182	−	0.282789i	$-0.908740\pi$
$11$	−5.81865e10	+	5.81865e10i	−0.676393	+	0.676393i	0.282789	+	0.959182i	$0.408740\pi$
$12$		0			0
$13$	−1.14156e11	−	1.14156e11i	−0.229665	−	0.229665i	0.582888	−	0.812553i	$-0.301923\pi$
$13$	−1.14156e11	−	1.14156e11i	−0.229665	−	0.229665i	−0.812553	+	0.582888i	$0.801923\pi$
$14$		0			0
$15$	−2.85415e12			−1.27796
$16$		0			0
$17$	−1.08847e13			−1.30949			−0.654743	−	0.755851i	$-0.727223\pi$
$17$	−1.08847e13			−1.30949			−0.654743	+	0.755851i	$0.727223\pi$
$18$		0			0
$19$	2.05483e13	+	2.05483e13i	0.768889	+	0.768889i	0.977911	−	0.209022i	$-0.0670280\pi$
$19$	2.05483e13	+	2.05483e13i	0.768889	+	0.768889i	−0.209022	+	0.977911i	$0.567028\pi$
$20$		0			0
$21$	6.10683e12	−	6.10683e12i	0.0798938	−	0.0798938i
$22$		0			0
$23$		−	2.73452e14i		−	1.37638i	−0.725529	−	0.688192i	$-0.758405\pi$
$23$		−	2.73452e14i		−	1.37638i	0.725529	−	0.688192i	$-0.241595\pi$
$24$		0			0
$25$		−	2.23758e14i		−	0.469255i
$26$		0			0
$27$	−7.08587e14	+	7.08587e14i	−0.662329	+	0.662329i
$28$		0			0
$29$	1.57763e15	+	1.57763e15i	0.696349	+	0.696349i	0.963621	−	0.267272i	$-0.0861223\pi$
$29$	1.57763e15	+	1.57763e15i	0.696349	+	0.696349i	−0.267272	+	0.963621i	$0.586122\pi$
$30$		0			0
$31$	−1.20723e15			−0.264541			−0.132270	−	0.991214i	$-0.542227\pi$
$31$	−1.20723e15			−0.264541			−0.132270	+	0.991214i	$0.542227\pi$
$32$		0			0
$33$	8.87321e15			1.00852
$34$		0			0
$35$	1.49902e15	+	1.49902e15i	0.0918526	+	0.0918526i
$36$		0			0
$37$	2.16994e16	−	2.16994e16i	0.741872	−	0.741872i	−0.231066	−	0.972938i	$-0.574221\pi$
$37$	2.16994e16	−	2.16994e16i	0.741872	−	0.741872i	0.972938	+	0.231066i	$0.0742213\pi$
$38$		0			0
$39$			1.74084e16i			0.342436i
$40$		0			0
$41$			1.20240e17i			1.39900i	0.714634	+	0.699499i	$0.246593\pi$
$41$			1.20240e17i			1.39900i	−0.714634	+	0.699499i	$0.753407\pi$
$42$		0			0
$43$	4.92672e16	−	4.92672e16i	0.347647	−	0.347647i	−0.511585	−	0.859232i	$-0.670942\pi$
$43$	4.92672e16	−	4.92672e16i	0.347647	−	0.347647i	0.859232	+	0.511585i	$0.170942\pi$
$44$		0			0
$45$	2.18449e16	+	2.18449e16i	0.0956353	+	0.0956353i
$46$		0			0
$47$	−1.05625e16			−0.0292913			−0.0146456	−	0.999893i	$-0.504662\pi$
$47$	−1.05625e16			−0.0292913			−0.0146456	+	0.999893i	$0.504662\pi$
$48$		0			0
$49$	5.52131e17			0.988515
$50$		0			0
$51$	8.29933e17	+	8.29933e17i	0.976239	+	0.976239i
$52$		0			0
$53$	1.46692e17	−	1.46692e17i	0.115215	−	0.115215i	−0.647149	−	0.762364i	$-0.724039\pi$
$53$	1.46692e17	−	1.46692e17i	0.115215	−	0.115215i	0.762364	+	0.647149i	$0.224039\pi$
$54$		0			0
$55$			2.17806e18i			1.15948i
$56$		0			0
$57$		−	3.13354e18i		−	1.14643i
$58$		0			0
$59$	−5.24717e18	+	5.24717e18i	−1.33653	+	1.33653i	−0.437135	+	0.899396i	$0.644007\pi$
$59$	−5.24717e18	+	5.24717e18i	−1.33653	+	1.33653i	−0.899396	+	0.437135i	$0.855993\pi$
$60$		0			0
$61$	−1.31997e18	−	1.31997e18i	−0.236919	−	0.236919i	0.578654	−	0.815573i	$-0.303578\pi$
$61$	−1.31997e18	−	1.31997e18i	−0.236919	−	0.236919i	−0.815573	+	0.578654i	$0.803578\pi$
$62$		0			0
$63$	−9.34800e16			−0.0119576
$64$		0			0
$65$	−4.27315e18			−0.393693
$66$		0			0
$67$	−4.67014e18	−	4.67014e18i	−0.313000	−	0.313000i	0.533071	−	0.846071i	$-0.321038\pi$
$67$	−4.67014e18	−	4.67014e18i	−0.313000	−	0.313000i	−0.846071	+	0.533071i	$0.821038\pi$
$68$		0			0
$69$	−2.08502e19	+	2.08502e19i	−1.02611	+	1.02611i
$70$		0			0
$71$		−	9.38208e18i		−	0.342047i	−0.985267	−	0.171024i	$-0.945293\pi$
$71$		−	9.38208e18i		−	0.342047i	0.985267	−	0.171024i	$-0.0547075\pi$
$72$		0			0
$73$			5.24082e19i			1.42728i	0.700513	+	0.713640i	$0.252955\pi$
$73$			5.24082e19i			1.42728i	−0.700513	+	0.713640i	$0.747045\pi$
$74$		0			0
$75$	−1.70611e19	+	1.70611e19i	−0.349836	+	0.349836i
$76$		0			0
$77$	−4.66025e18	−	4.66025e18i	−0.0724865	−	0.0724865i
$78$		0			0
$79$	−8.76083e19			−1.04102			−0.520512	−	0.853854i	$-0.674259\pi$
$79$	−8.76083e19			−1.04102			−0.520512	+	0.853854i	$0.674259\pi$
$80$		0			0
$81$	1.20266e20			1.09913
$82$		0			0
$83$	9.44999e19	+	9.44999e19i	0.668515	+	0.668515i	0.957372	−	0.288857i	$-0.0932752\pi$
$83$	9.44999e19	+	9.44999e19i	0.668515	+	0.668515i	−0.288857	+	0.957372i	$0.593275\pi$
$84$		0			0
$85$	−2.03720e20	+	2.03720e20i	−1.12237	+	1.12237i
$86$		0			0
$87$		−	2.40583e20i		−	1.03827i
$88$		0			0
$89$			3.49801e20i			1.18912i	0.804051	+	0.594561i	$0.202674\pi$
$89$			3.49801e20i			1.18912i	−0.804051	+	0.594561i	$0.797326\pi$
$90$		0			0
$91$	9.14296e18	−	9.14296e18i	0.0246123	−	0.0246123i
$92$		0			0
$93$	9.20489e19	+	9.20489e19i	0.197219	+	0.197219i
$94$		0			0
$95$	7.69175e20			1.31804
$96$		0			0
$97$	4.43573e20			0.610748			0.305374	−	0.952232i	$-0.401218\pi$
$97$	4.43573e20			0.610748			0.305374	+	0.952232i	$0.401218\pi$
$98$		0			0
$99$	−6.79131e19	−	6.79131e19i	−0.0754716	−	0.0754716i

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.22.e.a.17.10		82
4.3	odd	2		16.22.e.a.13.36	yes	82
16.5	even	4	inner	64.22.e.a.49.10		82
16.11	odd	4		16.22.e.a.5.36	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.22.e.a.5.36	✓	82	16.11	odd	4
16.22.e.a.13.36	yes	82	4.3	odd	2
64.22.e.a.17.10		82	1.1	even	1	trivial
64.22.e.a.49.10		82	16.5	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	64.e (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-76248.0 - 76248.0i) q^{3} +(1.87162e7 - 1.87162e7i) q^{5} +8.00917e7i q^{7} +1.16716e9i q^{9} +(-5.81865e10 + 5.81865e10i) q^{11} +(-1.14156e11 - 1.14156e11i) q^{13} -2.85415e12 q^{15} -1.08847e13 q^{17} +(2.05483e13 + 2.05483e13i) q^{19} +(6.10683e12 - 6.10683e12i) q^{21} -2.73452e14i q^{23} -2.23758e14i q^{25} +(-7.08587e14 + 7.08587e14i) q^{27} +(1.57763e15 + 1.57763e15i) q^{29} -1.20723e15 q^{31} +8.87321e15 q^{33} +(1.49902e15 + 1.49902e15i) q^{35} +(2.16994e16 - 2.16994e16i) q^{37} +1.74084e16i q^{39} +1.20240e17i q^{41} +(4.92672e16 - 4.92672e16i) q^{43} +(2.18449e16 + 2.18449e16i) q^{45} -1.05625e16 q^{47} +5.52131e17 q^{49} +(8.29933e17 + 8.29933e17i) q^{51} +(1.46692e17 - 1.46692e17i) q^{53} +2.17806e18i q^{55} -3.13354e18i q^{57} +(-5.24717e18 + 5.24717e18i) q^{59} +(-1.31997e18 - 1.31997e18i) q^{61} -9.34800e16 q^{63} -4.27315e18 q^{65} +(-4.67014e18 - 4.67014e18i) q^{67} +(-2.08502e19 + 2.08502e19i) q^{69} -9.38208e18i q^{71} +5.24082e19i q^{73} +(-1.70611e19 + 1.70611e19i) q^{75} +(-4.66025e18 - 4.66025e18i) q^{77} -8.76083e19 q^{79} +1.20266e20 q^{81} +(9.44999e19 + 9.44999e19i) q^{83} +(-2.03720e20 + 2.03720e20i) q^{85} -2.40583e20i q^{87} +3.49801e20i q^{89} +(9.14296e18 - 9.14296e18i) q^{91} +(9.20489e19 + 9.20489e19i) q^{93} +7.69175e20 q^{95} +4.43573e20 q^{97} +(-6.79131e19 - 6.79131e19i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 2 q^{3} - 2 q^{5} - 67333320738 q^{11} - 2 q^{13} - 4613203124996 q^{15} - 4 q^{17} + 46007763621434 q^{19} + 20920706404 q^{21} - 11\!\cdots\!20 q^{27} - 24\!\cdots\!02 q^{29} + 98\!\cdots\!16 q^{31}+ \cdots - 27\!\cdots\!38 q^{99}+O(q^{100}) \) 82 * q + 2 * q^3 - 2 * q^5 - 67333320738 * q^11 - 2 * q^13 - 4613203124996 * q^15 - 4 * q^17 + 46007763621434 * q^19 + 20920706404 * q^21 - 1147760716556920 * q^27 - 2433173450216602 * q^29 + 9835539443769616 * q^31 - 4 * q^33 + 19253478556034884 * q^35 - 7245708527119386 * q^37 - 215332019803969386 * q^43 + 953653395699842 * q^45 - 1051982644716600976 * q^47 - 5585458640832840074 * q^49 - 3238772068175213124 * q^51 + 1889987679210147606 * q^53 - 3631979197965016922 * q^59 + 10385362452353654062 * q^61 + 17464422381302680956 * q^63 - 9364161130572538716 * q^65 + 17102217044572669130 * q^67 - 3951842785111207532 * q^69 - 145960939174166395366 * q^75 - 82041801652678580764 * q^77 - 204448210818924484480 * q^79 - 802405920297757300870 * q^81 - 317463236552669319678 * q^83 + 323656241286464843748 * q^85 - 144326756239046079748 * q^91 - 669717759565183008560 * q^93 + 1411330138695550130700 * q^95 - 4 * q^97 - 2710306841526615010238 * q^99

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