LMFDB - Embedded newform 64.22.e.a.17.6

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			17.6
Character	$\chi$	$=$	64.17
Dual form			64.22.e.a.49.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-109714. - 109714. i) q^{3} +(-1.97304e7 + 1.97304e7i) q^{5} -1.02336e9i q^{7} +1.36141e10i q^{9} +(7.32170e10 - 7.32170e10i) q^{11} +(-4.28208e11 - 4.28208e11i) q^{13} +4.32941e12 q^{15} +6.62050e12 q^{17} +(1.48987e13 + 1.48987e13i) q^{19} +(-1.12277e14 + 1.12277e14i) q^{21} -2.73784e14i q^{23} -3.01738e14i q^{25} +(3.46012e14 - 3.46012e14i) q^{27} +(1.70828e15 + 1.70828e15i) q^{29} -1.42001e15 q^{31} -1.60659e16 q^{33} +(2.01913e16 + 2.01913e16i) q^{35} +(-9.05710e15 + 9.05710e15i) q^{37} +9.39612e16i q^{39} -5.53453e16i q^{41} +(8.48059e16 - 8.48059e16i) q^{43} +(-2.68611e17 - 2.68611e17i) q^{45} -5.38091e17 q^{47} -4.88724e17 q^{49} +(-7.26363e17 - 7.26363e17i) q^{51} +(-6.11834e17 + 6.11834e17i) q^{53} +2.88920e18i q^{55} -3.26920e18i q^{57} +(-5.01134e18 + 5.01134e18i) q^{59} +(7.57981e18 + 7.57981e18i) q^{61} +1.39322e19 q^{63} +1.68974e19 q^{65} +(1.45969e19 + 1.45969e19i) q^{67} +(-3.00381e19 + 3.00381e19i) q^{69} +3.61689e19i q^{71} +1.74764e19i q^{73} +(-3.31049e19 + 3.31049e19i) q^{75} +(-7.49276e19 - 7.49276e19i) q^{77} -9.41739e18 q^{79} +6.64834e19 q^{81} +(1.22582e20 + 1.22582e20i) q^{83} +(-1.30625e20 + 1.30625e20i) q^{85} -3.74845e20i q^{87} +4.34193e20i q^{89} +(-4.38212e20 + 4.38212e20i) q^{91} +(1.55795e20 + 1.55795e20i) q^{93} -5.87913e20 q^{95} -4.29145e20 q^{97} +(9.96785e20 + 9.96785e20i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−109714.	−	109714.i	−1.07273	−	1.07273i	−0.997139	−	0.0755903i	$-0.975916\pi$
$3$	−109714.	−	109714.i	−1.07273	−	1.07273i	−0.0755903	−	0.997139i	$-0.524084\pi$
$4$		0			0
$5$	−1.97304e7	+	1.97304e7i	−0.903546	+	0.903546i	−0.995741	−	0.0921952i	$-0.970612\pi$
$5$	−1.97304e7	+	1.97304e7i	−0.903546	+	0.903546i	0.0921952	+	0.995741i	$0.470612\pi$
$6$		0			0
$7$		−	1.02336e9i		−	1.36930i	−0.728870	−	0.684652i	$-0.759954\pi$
$7$		−	1.02336e9i		−	1.36930i	0.728870	−	0.684652i	$-0.240046\pi$
$8$		0			0
$9$			1.36141e10i			1.30150i
$10$		0			0
$11$	7.32170e10	−	7.32170e10i	0.851116	−	0.851116i	−0.139154	−	0.990271i	$-0.544438\pi$
$11$	7.32170e10	−	7.32170e10i	0.851116	−	0.851116i	0.990271	+	0.139154i	$0.0444384\pi$
$12$		0			0
$13$	−4.28208e11	−	4.28208e11i	−0.861489	−	0.861489i	0.130022	−	0.991511i	$-0.458495\pi$
$13$	−4.28208e11	−	4.28208e11i	−0.861489	−	0.861489i	−0.991511	+	0.130022i	$0.958495\pi$
$14$		0			0
$15$	4.32941e12			1.93852
$16$		0			0
$17$	6.62050e12			0.796484			0.398242	−	0.917280i	$-0.369620\pi$
$17$	6.62050e12			0.796484			0.398242	+	0.917280i	$0.369620\pi$
$18$		0			0
$19$	1.48987e13	+	1.48987e13i	0.557488	+	0.557488i	0.928591	−	0.371104i	$-0.121021\pi$
$19$	1.48987e13	+	1.48987e13i	0.557488	+	0.557488i	−0.371104	+	0.928591i	$0.621021\pi$
$20$		0			0
$21$	−1.12277e14	+	1.12277e14i	−1.46889	+	1.46889i
$22$		0			0
$23$		−	2.73784e14i		−	1.37805i	−0.724736	−	0.689027i	$-0.758038\pi$
$23$		−	2.73784e14i		−	1.37805i	0.724736	−	0.689027i	$-0.241962\pi$
$24$		0			0
$25$		−	3.01738e14i		−	0.632790i
$26$		0			0
$27$	3.46012e14	−	3.46012e14i	0.323424	−	0.323424i
$28$		0			0
$29$	1.70828e15	+	1.70828e15i	0.754013	+	0.754013i	0.975226	−	0.221212i	$-0.0710014\pi$
$29$	1.70828e15	+	1.70828e15i	0.754013	+	0.754013i	−0.221212	+	0.975226i	$0.571001\pi$
$30$		0			0
$31$	−1.42001e15			−0.311166			−0.155583	−	0.987823i	$-0.549726\pi$
$31$	−1.42001e15			−0.311166			−0.155583	+	0.987823i	$0.549726\pi$
$32$		0			0
$33$	−1.60659e16			−1.82604
$34$		0			0
$35$	2.01913e16	+	2.01913e16i	1.23723	+	1.23723i
$36$		0			0
$37$	−9.05710e15	+	9.05710e15i	−0.309650	+	0.309650i	−0.844774	−	0.535124i	$-0.820265\pi$
$37$	−9.05710e15	+	9.05710e15i	−0.309650	+	0.309650i	0.535124	+	0.844774i	$0.320265\pi$
$38$		0			0
$39$			9.39612e16i			1.84829i
$40$		0			0
$41$		−	5.53453e16i		−	0.643947i	−0.946749	−	0.321973i	$-0.895654\pi$
$41$		−	5.53453e16i		−	0.643947i	0.946749	−	0.321973i	$-0.104346\pi$
$42$		0			0
$43$	8.48059e16	−	8.48059e16i	0.598422	−	0.598422i	−0.341471	−	0.939892i	$-0.610925\pi$
$43$	8.48059e16	−	8.48059e16i	0.598422	−	0.598422i	0.939892	+	0.341471i	$0.110925\pi$
$44$		0			0
$45$	−2.68611e17	−	2.68611e17i	−1.17596	−	1.17596i
$46$		0			0
$47$	−5.38091e17			−1.49220			−0.746102	−	0.665832i	$-0.768077\pi$
$47$	−5.38091e17			−1.49220			−0.746102	+	0.665832i	$0.768077\pi$
$48$		0			0
$49$	−4.88724e17			−0.874994
$50$		0			0
$51$	−7.26363e17	−	7.26363e17i	−0.854411	−	0.854411i
$52$		0			0
$53$	−6.11834e17	+	6.11834e17i	−0.480548	+	0.480548i	−0.905307	−	0.424758i	$-0.860359\pi$
$53$	−6.11834e17	+	6.11834e17i	−0.480548	+	0.480548i	0.424758	+	0.905307i	$0.360359\pi$
$54$		0			0
$55$			2.88920e18i			1.53805i
$56$		0			0
$57$		−	3.26920e18i		−	1.19607i
$58$		0			0
$59$	−5.01134e18	+	5.01134e18i	−1.27646	+	1.27646i	−0.333827	+	0.942634i	$0.608340\pi$
$59$	−5.01134e18	+	5.01134e18i	−1.27646	+	1.27646i	−0.942634	+	0.333827i	$0.891660\pi$
$60$		0			0
$61$	7.57981e18	+	7.57981e18i	1.36049	+	1.36049i	0.873292	+	0.487198i	$0.161981\pi$
$61$	7.57981e18	+	7.57981e18i	1.36049	+	1.36049i	0.487198	+	0.873292i	$0.338019\pi$
$62$		0			0
$63$	1.39322e19			1.78214
$64$		0			0
$65$	1.68974e19			1.55679
$66$		0			0
$67$	1.45969e19	+	1.45969e19i	0.978307	+	0.978307i	0.999770	−	0.0214631i	$-0.00683245\pi$
$67$	1.45969e19	+	1.45969e19i	0.978307	+	0.978307i	−0.0214631	+	0.999770i	$0.506832\pi$
$68$		0			0
$69$	−3.00381e19	+	3.00381e19i	−1.47828	+	1.47828i
$70$		0			0
$71$			3.61689e19i			1.31863i	0.751867	+	0.659315i	$0.229154\pi$
$71$			3.61689e19i			1.31863i	−0.751867	+	0.659315i	$0.770846\pi$
$72$		0			0
$73$			1.74764e19i			0.475950i	0.971271	+	0.237975i	$0.0764836\pi$
$73$			1.74764e19i			0.475950i	−0.971271	+	0.237975i	$0.923516\pi$
$74$		0			0
$75$	−3.31049e19	+	3.31049e19i	−0.678812	+	0.678812i
$76$		0			0
$77$	−7.49276e19	−	7.49276e19i	−1.16544	−	1.16544i
$78$		0			0
$79$	−9.41739e18			−0.111904			−0.0559521	−	0.998433i	$-0.517819\pi$
$79$	−9.41739e18			−0.111904			−0.0559521	+	0.998433i	$0.517819\pi$
$80$		0			0
$81$	6.64834e19			0.607604
$82$		0			0
$83$	1.22582e20	+	1.22582e20i	0.867178	+	0.867178i	0.992159	−	0.124981i	$-0.0398869\pi$
$83$	1.22582e20	+	1.22582e20i	0.867178	+	0.867178i	−0.124981	+	0.992159i	$0.539887\pi$
$84$		0			0
$85$	−1.30625e20	+	1.30625e20i	−0.719659	+	0.719659i
$86$		0			0
$87$		−	3.74845e20i		−	1.61770i
$88$		0			0
$89$			4.34193e20i			1.47600i	0.674799	+	0.738002i	$0.264230\pi$
$89$			4.34193e20i			1.47600i	−0.674799	+	0.738002i	$0.735770\pi$
$90$		0			0
$91$	−4.38212e20	+	4.38212e20i	−1.17964	+	1.17964i
$92$		0			0
$93$	1.55795e20	+	1.55795e20i	0.333797	+	0.333797i
$94$		0			0
$95$	−5.87913e20			−1.00743
$96$		0			0
$97$	−4.29145e20			−0.590882			−0.295441	−	0.955361i	$-0.595467\pi$
$97$	−4.29145e20			−0.590882			−0.295441	+	0.955361i	$0.595467\pi$
$98$		0			0
$99$	9.96785e20	+	9.96785e20i	1.10772	+	1.10772i

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.17.6		82
4.3	odd	2		16.22.e.a.13.3	yes	82
16.5	even	4	inner	64.22.e.a.49.6		82
16.11	odd	4		16.22.e.a.5.3	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.22.e.a.5.3	✓	82	16.11	odd	4
16.22.e.a.13.3	yes	82	4.3	odd	2
64.22.e.a.17.6		82	1.1	even	1	trivial
64.22.e.a.49.6		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)

\(f(q)\)	\(=\)	\(q+(-109714. - 109714. i) q^{3} +(-1.97304e7 + 1.97304e7i) q^{5} -1.02336e9i q^{7} +1.36141e10i q^{9} +(7.32170e10 - 7.32170e10i) q^{11} +(-4.28208e11 - 4.28208e11i) q^{13} +4.32941e12 q^{15} +6.62050e12 q^{17} +(1.48987e13 + 1.48987e13i) q^{19} +(-1.12277e14 + 1.12277e14i) q^{21} -2.73784e14i q^{23} -3.01738e14i q^{25} +(3.46012e14 - 3.46012e14i) q^{27} +(1.70828e15 + 1.70828e15i) q^{29} -1.42001e15 q^{31} -1.60659e16 q^{33} +(2.01913e16 + 2.01913e16i) q^{35} +(-9.05710e15 + 9.05710e15i) q^{37} +9.39612e16i q^{39} -5.53453e16i q^{41} +(8.48059e16 - 8.48059e16i) q^{43} +(-2.68611e17 - 2.68611e17i) q^{45} -5.38091e17 q^{47} -4.88724e17 q^{49} +(-7.26363e17 - 7.26363e17i) q^{51} +(-6.11834e17 + 6.11834e17i) q^{53} +2.88920e18i q^{55} -3.26920e18i q^{57} +(-5.01134e18 + 5.01134e18i) q^{59} +(7.57981e18 + 7.57981e18i) q^{61} +1.39322e19 q^{63} +1.68974e19 q^{65} +(1.45969e19 + 1.45969e19i) q^{67} +(-3.00381e19 + 3.00381e19i) q^{69} +3.61689e19i q^{71} +1.74764e19i q^{73} +(-3.31049e19 + 3.31049e19i) q^{75} +(-7.49276e19 - 7.49276e19i) q^{77} -9.41739e18 q^{79} +6.64834e19 q^{81} +(1.22582e20 + 1.22582e20i) q^{83} +(-1.30625e20 + 1.30625e20i) q^{85} -3.74845e20i q^{87} +4.34193e20i q^{89} +(-4.38212e20 + 4.38212e20i) q^{91} +(1.55795e20 + 1.55795e20i) q^{93} -5.87913e20 q^{95} -4.29145e20 q^{97} +(9.96785e20 + 9.96785e20i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 2 q^{3} - 2 q^{5} - 67333320738 q^{11} - 2 q^{13} - 4613203124996 q^{15} - 4 q^{17} + 46007763621434 q^{19} + 20920706404 q^{21} - 11\!\cdots\!20 q^{27} - 24\!\cdots\!02 q^{29} + 98\!\cdots\!16 q^{31}+ \cdots - 27\!\cdots\!38 q^{99}+O(q^{100}) \) 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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