# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(930, [\chi])$$.