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Properties

Label 930.2.k.a
Level $930$
Weight $2$
Character orbit 930.k
Analytic conductor $7.426$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [930,2,Mod(247,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.247");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.k (of order \(4\), degree \(2\), 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: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{7} + 4 q^{10} - 4 q^{15} - 32 q^{16} - 8 q^{17} + 4 q^{22} + 32 q^{24} + 8 q^{25} + 4 q^{28} - 8 q^{29} - 20 q^{31} - 4 q^{33} - 24 q^{35} - 32 q^{36} - 4 q^{37} + 16 q^{38} - 16 q^{41} - 4 q^{42} + 16 q^{43} + 8 q^{44} - 8 q^{47} - 16 q^{50} + 24 q^{53} + 32 q^{54} + 28 q^{55} - 16 q^{57} - 20 q^{58} + 16 q^{62} + 4 q^{63} - 56 q^{65} - 8 q^{66} + 32 q^{67} - 8 q^{68} - 28 q^{70} + 16 q^{71} + 20 q^{73} - 24 q^{74} + 16 q^{75} - 16 q^{76} + 40 q^{77} + 56 q^{79} - 32 q^{81} + 16 q^{82} - 72 q^{83} + 32 q^{85} + 20 q^{87} + 4 q^{88} + 64 q^{89} - 16 q^{93} + 32 q^{95} - 4 q^{97} + 16 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
247.1 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i −2.23600 + 0.0168647i − 1.00000i −2.83917 − 2.83917i 0.707107 − 0.707107i 1.00000i 1.59302 + 1.56917i
247.2 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i −2.14949 + 0.616200i − 1.00000i 0.0344550 + 0.0344550i 0.707107 − 0.707107i 1.00000i 1.95564 + 1.08420i
247.3 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i −1.64980 − 1.50935i − 1.00000i 2.19620 + 2.19620i 0.707107 − 0.707107i 1.00000i 0.0993121 + 2.23386i
247.4 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i −0.899839 + 2.04702i − 1.00000i 2.95191 + 2.95191i 0.707107 − 0.707107i 1.00000i 2.08374 − 0.811179i
247.5 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i 1.06571 + 1.96577i − 1.00000i −2.96361 − 2.96361i 0.707107 − 0.707107i 1.00000i 0.636441 − 2.14358i
247.6 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i 1.06824 − 1.96440i − 1.00000i −1.05833 − 1.05833i 0.707107 − 0.707107i 1.00000i −2.14440 + 0.633681i
247.7 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i 1.94894 − 1.09618i − 1.00000i −1.54434 − 1.54434i 0.707107 − 0.707107i 1.00000i −2.15323 − 0.602992i
247.8 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i 2.14513 + 0.631187i − 1.00000i 2.22289 + 2.22289i 0.707107 − 0.707107i 1.00000i −1.07052 − 1.96316i
247.9 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i −1.96413 − 1.06873i − 1.00000i −1.95397 − 1.95397i −0.707107 + 0.707107i 1.00000i −0.633142 − 2.14456i
247.10 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i −1.87799 + 1.21373i − 1.00000i −0.853491 − 0.853491i −0.707107 + 0.707107i 1.00000i −2.18618 − 0.469704i
247.11 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i −1.03553 − 1.98184i − 1.00000i 0.181215 + 0.181215i −0.707107 + 0.707107i 1.00000i 0.669145 − 2.13360i
247.12 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i 0.0490162 + 2.23553i − 1.00000i 3.32380 + 3.32380i −0.707107 + 0.707107i 1.00000i −1.54610 + 1.61542i
247.13 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i 0.433540 + 2.19364i − 1.00000i 0.130349 + 0.130349i −0.707107 + 0.707107i 1.00000i −1.24458 + 1.85769i
247.14 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i 0.975733 − 2.01195i − 1.00000i −3.17444 − 3.17444i −0.707107 + 0.707107i 1.00000i 2.11261 − 0.732717i
247.15 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i 1.89248 − 1.19101i − 1.00000i 2.48213 + 2.48213i −0.707107 + 0.707107i 1.00000i 2.18036 + 0.496012i
247.16 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i 2.23399 − 0.0964676i − 1.00000i −1.13560 − 1.13560i −0.707107 + 0.707107i 1.00000i 1.64788 + 1.51145i
433.1 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i −2.23600 − 0.0168647i 1.00000i −2.83917 + 2.83917i 0.707107 + 0.707107i − 1.00000i 1.59302 − 1.56917i
433.2 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i −2.14949 − 0.616200i 1.00000i 0.0344550 − 0.0344550i 0.707107 + 0.707107i − 1.00000i 1.95564 − 1.08420i
433.3 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i −1.64980 + 1.50935i 1.00000i 2.19620 − 2.19620i 0.707107 + 0.707107i − 1.00000i 0.0993121 − 2.23386i
433.4 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i −0.899839 − 2.04702i 1.00000i 2.95191 − 2.95191i 0.707107 + 0.707107i − 1.00000i 2.08374 + 0.811179i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 247.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
155.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.k.a ✓ 32
5.c odd 4 1 930.2.k.b yes 32
31.b odd 2 1 930.2.k.b yes 32
155.f even 4 1 inner 930.2.k.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.k.a ✓ 32 1.a even 1 1 trivial
930.2.k.a ✓ 32 155.f even 4 1 inner
930.2.k.b yes 32 5.c odd 4 1
930.2.k.b yes 32 31.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{32} - 52 T_{13}^{29} + 2474 T_{13}^{28} - 2428 T_{13}^{27} + 1352 T_{13}^{26} + \cdots + 6879707136 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display