LMFDB - Newform orbit 930.2.z.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(109,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 5, 2]))

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

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

chi := DirichletCharacter("930.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.z (of order $10$, degree $4$, 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:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$8$ over $\Q(\zeta_{10})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $q$-expansion, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) = $ $ 32 q + 8 q^{4} - 16 q^{5} - 32 q^{6} + 8 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 32 q + 8 q^{4} - 16 q^{5} - 32 q^{6} + 8 q^{9} - 4 q^{11} - 4 q^{14} - 8 q^{16} - 16 q^{19} + 16 q^{20} - 4 q^{21} - 8 q^{24} - 64 q^{25} - 16 q^{26} + 4 q^{29} + 16 q^{30} - 24 q^{31} - 12 q^{34} + 32 q^{36} + 4 q^{39} - 48 q^{41} + 4 q^{44} + 16 q^{45} + 8 q^{46} - 28 q^{49} + 8 q^{51} - 8 q^{54} - 8 q^{55} - 16 q^{56} - 8 q^{59} + 32 q^{61} + 8 q^{64} - 8 q^{65} + 4 q^{66} - 12 q^{69} - 8 q^{70} + 32 q^{71} + 4 q^{74} - 24 q^{76} + 64 q^{79} + 24 q^{80} - 8 q^{81} + 4 q^{84} + 28 q^{85} + 84 q^{86} - 128 q^{89} - 28 q^{91} + 40 q^{94} + 8 q^{95} + 8 q^{96} - 16 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label	$ a_{2} $			$ a_{3} $			$ a_{4} $			$ a_{5} $			$ a_{6} $	$ a_{7} $			$ a_{8} $			$ a_{9} $			$ a_{10} $
109.1	−0.587785	−	0.809017i	0.587785	−	0.809017i	−0.309017	+	0.951057i	−1.61803	−	1.54336i	−1.00000	−3.49122	−	1.13437i	0.951057	−	0.309017i	−0.309017	−	0.951057i	−0.297550	+	2.21618i
109.2	−0.587785	−	0.809017i	0.587785	−	0.809017i	−0.309017	+	0.951057i	−1.61803	−	1.54336i	−1.00000	2.54016	+	0.825349i	0.951057	−	0.309017i	−0.309017	−	0.951057i	−0.297550	+	2.21618i
109.3	−0.587785	−	0.809017i	0.587785	−	0.809017i	−0.309017	+	0.951057i	−1.61803	+	1.54336i	−1.00000	−1.81154	−	0.588604i	0.951057	−	0.309017i	−0.309017	−	0.951057i	2.19966	+	0.401852i
109.4	−0.587785	−	0.809017i	0.587785	−	0.809017i	−0.309017	+	0.951057i	−1.61803	+	1.54336i	−1.00000	0.860480	+	0.279587i	0.951057	−	0.309017i	−0.309017	−	0.951057i	2.19966	+	0.401852i
109.5	0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.309017	+	0.951057i	−1.61803	−	1.54336i	−1.00000	−0.860480	−	0.279587i	−0.951057	+	0.309017i	−0.309017	−	0.951057i	0.297550	−	2.21618i
109.6	0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.309017	+	0.951057i	−1.61803	−	1.54336i	−1.00000	1.81154	+	0.588604i	−0.951057	+	0.309017i	−0.309017	−	0.951057i	0.297550	−	2.21618i
109.7	0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.309017	+	0.951057i	−1.61803	+	1.54336i	−1.00000	−2.54016	−	0.825349i	−0.951057	+	0.309017i	−0.309017	−	0.951057i	−2.19966	−	0.401852i
109.8	0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.309017	+	0.951057i	−1.61803	+	1.54336i	−1.00000	3.49122	+	1.13437i	−0.951057	+	0.309017i	−0.309017	−	0.951057i	−2.19966	−	0.401852i
349.1	−0.951057	−	0.309017i	0.951057	−	0.309017i	0.809017	+	0.587785i	0.618034	−	2.14896i	−1.00000	−1.53347	+	2.11064i	−0.587785	−	0.809017i	0.809017	−	0.587785i	−1.25185	+	1.85280i
349.2	−0.951057	−	0.309017i	0.951057	−	0.309017i	0.809017	+	0.587785i	0.618034	−	2.14896i	−1.00000	2.12126	−	2.91966i	−0.587785	−	0.809017i	0.809017	−	0.587785i	−1.25185	+	1.85280i
349.3	−0.951057	−	0.309017i	0.951057	−	0.309017i	0.809017	+	0.587785i	0.618034	+	2.14896i	−1.00000	−1.68258	+	2.31587i	−0.587785	−	0.809017i	0.809017	−	0.587785i	0.0762803	−	2.23477i
349.4	−0.951057	−	0.309017i	0.951057	−	0.309017i	0.809017	+	0.587785i	0.618034	+	2.14896i	−1.00000	2.27036	−	3.12489i	−0.587785	−	0.809017i	0.809017	−	0.587785i	0.0762803	−	2.23477i
349.5	0.951057	+	0.309017i	−0.951057	+	0.309017i	0.809017	+	0.587785i	0.618034	−	2.14896i	−1.00000	−2.27036	+	3.12489i	0.587785	+	0.809017i	0.809017	−	0.587785i	1.25185	−	1.85280i
349.6	0.951057	+	0.309017i	−0.951057	+	0.309017i	0.809017	+	0.587785i	0.618034	−	2.14896i	−1.00000	1.68258	−	2.31587i	0.587785	+	0.809017i	0.809017	−	0.587785i	1.25185	−	1.85280i
349.7	0.951057	+	0.309017i	−0.951057	+	0.309017i	0.809017	+	0.587785i	0.618034	+	2.14896i	−1.00000	−2.12126	+	2.91966i	0.587785	+	0.809017i	0.809017	−	0.587785i	−0.0762803	+	2.23477i
349.8	0.951057	+	0.309017i	−0.951057	+	0.309017i	0.809017	+	0.587785i	0.618034	+	2.14896i	−1.00000	1.53347	−	2.11064i	0.587785	+	0.809017i	0.809017	−	0.587785i	−0.0762803	+	2.23477i
469.1	−0.951057	+	0.309017i	0.951057	+	0.309017i	0.809017	−	0.587785i	0.618034	−	2.14896i	−1.00000	−1.68258	−	2.31587i	−0.587785	+	0.809017i	0.809017	+	0.587785i	0.0762803	+	2.23477i
469.2	−0.951057	+	0.309017i	0.951057	+	0.309017i	0.809017	−	0.587785i	0.618034	−	2.14896i	−1.00000	2.27036	+	3.12489i	−0.587785	+	0.809017i	0.809017	+	0.587785i	0.0762803	+	2.23477i
469.3	−0.951057	+	0.309017i	0.951057	+	0.309017i	0.809017	−	0.587785i	0.618034	+	2.14896i	−1.00000	−1.53347	−	2.11064i	−0.587785	+	0.809017i	0.809017	+	0.587785i	−1.25185	−	1.85280i
469.4	−0.951057	+	0.309017i	0.951057	+	0.309017i	0.809017	−	0.587785i	0.618034	+	2.14896i	−1.00000	2.12126	+	2.91966i	−0.587785	+	0.809017i	0.809017	+	0.587785i	−1.25185	−	1.85280i

See all 32 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
31.d	even	5	1	inner
155.n	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.z.c	✓	32
5.b	even	2	1	inner	930.2.z.c	✓	32
31.d	even	5	1	inner	930.2.z.c	✓	32
155.n	even	10	1	inner	930.2.z.c	✓	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.z.c	✓	32	1.a	even	1	1	trivial
930.2.z.c	✓	32	5.b	even	2	1	inner
930.2.z.c	✓	32	31.d	even	5	1	inner
930.2.z.c	✓	32	155.n	even	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{32} - 14 T_{7}^{30} + 431 T_{7}^{28} - 7980 T_{7}^{26} + 131055 T_{7}^{24} - 1532936 T_{7}^{22} + \cdots + 9573337234561 $ T7^32 - 14*T7^30 + 431*T7^28 - 7980*T7^26 + 131055*T7^24 - 1532936*T7^22 + 22288045*T7^20 - 202450060*T7^18 + 1832945766*T7^16 - 14543525160*T7^14 + 101022596775*T7^12 - 513844519796*T7^10 + 2739921678345*T7^8 - 10809442099410*T7^6 + 25910329830226*T7^4 - 24070786805544*T7^2 + 9573337234561 acting on $S_{2}^{\mathrm{new}}(930, [\chi])$.

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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.z (of order \(10\), degree \(4\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 109.8
Significant digits:
Format:

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