Properties

Label 930.2.z.c
Level $930$
Weight $2$
Character orbit 930.z
Analytic conductor $7.426$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

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

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.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 8q^{4} - 16q^{5} - 32q^{6} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 8q^{4} - 16q^{5} - 32q^{6} + 8q^{9} - 4q^{11} - 4q^{14} - 8q^{16} - 16q^{19} + 16q^{20} - 4q^{21} - 8q^{24} - 64q^{25} - 16q^{26} + 4q^{29} + 16q^{30} - 24q^{31} - 12q^{34} + 32q^{36} + 4q^{39} - 48q^{41} + 4q^{44} + 16q^{45} + 8q^{46} - 28q^{49} + 8q^{51} - 8q^{54} - 8q^{55} - 16q^{56} - 8q^{59} + 32q^{61} + 8q^{64} - 8q^{65} + 4q^{66} - 12q^{69} - 8q^{70} + 32q^{71} + 4q^{74} - 24q^{76} + 64q^{79} + 24q^{80} - 8q^{81} + 4q^{84} + 28q^{85} + 84q^{86} - 128q^{89} - 28q^{91} + 40q^{94} + 8q^{95} + 8q^{96} - 16q^{99} + O(q^{100}) \)

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.

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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.8
Significant digits:
Format:

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 \(27\!\cdots\!45\)\( T_{7}^{8} - \)\(10\!\cdots\!10\)\( T_{7}^{6} + \)\(25\!\cdots\!26\)\( T_{7}^{4} - \)\(24\!\cdots\!44\)\( T_{7}^{2} + \)\(95\!\cdots\!61\)\( \)">\(T_{7}^{32} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).