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Properties

Label 930.2.z.d
Level $930$
Weight $2$
Character orbit 930.z
Analytic conductor $7.426$
Analytic rank $0$
Dimension $72$
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: \(72\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72q + 18q^{4} + 4q^{5} + 72q^{6} + 18q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q + 18q^{4} + 4q^{5} + 72q^{6} + 18q^{9} - 6q^{11} - 4q^{14} - 18q^{16} + 12q^{19} - 4q^{20} - 16q^{21} + 18q^{24} - 4q^{25} + 12q^{26} + 24q^{29} + 4q^{30} + 22q^{31} + 14q^{34} - 22q^{35} + 72q^{36} + 8q^{39} - 28q^{41} - 4q^{44} - 4q^{45} - 22q^{46} + 42q^{49} - 32q^{50} - 4q^{51} + 18q^{54} + 14q^{55} + 24q^{56} - 28q^{59} + 56q^{61} + 18q^{64} + 46q^{65} - 6q^{66} - 8q^{69} + 46q^{70} - 68q^{71} + 8q^{74} + 32q^{75} - 12q^{76} + 26q^{79} - 6q^{80} - 18q^{81} - 4q^{84} - 32q^{85} + 80q^{86} + 32q^{89} - 44q^{91} + 12q^{94} + 74q^{95} - 18q^{96} - 4q^{99} + O(q^{100}) \)

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.

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 −2.20868 − 0.348916i 1.00000 0.804232 + 0.261311i 0.951057 − 0.309017i −0.309017 − 0.951057i 1.01595 + 1.99195i
109.2 −0.587785 − 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i −1.90693 + 1.16774i 1.00000 0.507889 + 0.165023i 0.951057 − 0.309017i −0.309017 − 0.951057i 2.06559 + 0.856357i
109.3 −0.587785 − 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i −0.635868 + 2.14375i 1.00000 2.45458 + 0.797540i 0.951057 − 0.309017i −0.309017 − 0.951057i 2.10809 − 0.745638i
109.4 −0.587785 − 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i −0.244369 − 2.22267i 1.00000 4.86212 + 1.57980i 0.951057 − 0.309017i −0.309017 − 0.951057i −1.65455 + 1.50415i
109.5 −0.587785 − 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 0.218072 + 2.22541i 1.00000 −0.288858 − 0.0938558i 0.951057 − 0.309017i −0.309017 − 0.951057i 1.67221 − 1.48449i
109.6 −0.587785 − 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 0.836694 − 2.07363i 1.00000 −0.595482 − 0.193484i 0.951057 − 0.309017i −0.309017 − 0.951057i −2.16940 + 0.541950i
109.7 −0.587785 − 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.25767 − 1.84886i 1.00000 −2.43606 − 0.791524i 0.951057 − 0.309017i −0.309017 − 0.951057i −2.23500 + 0.0692568i
109.8 −0.587785 − 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 2.06791 + 0.850729i 1.00000 −4.57919 − 1.48787i 0.951057 − 0.309017i −0.309017 − 0.951057i −0.527234 − 2.17302i
109.9 −0.587785 − 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 2.23353 + 0.106448i 1.00000 4.25057 + 1.38109i 0.951057 − 0.309017i −0.309017 − 0.951057i −1.22672 − 1.86953i
109.10 0.587785 + 0.809017i 0.587785 − 0.809017i −0.309017 + 0.951057i −2.20868 + 0.348916i 1.00000 −0.804232 − 0.261311i −0.951057 + 0.309017i −0.309017 − 0.951057i −1.58051 − 1.58177i
109.11 0.587785 + 0.809017i 0.587785 − 0.809017i −0.309017 + 0.951057i −1.90693 − 1.16774i 1.00000 −0.507889 − 0.165023i −0.951057 + 0.309017i −0.309017 − 0.951057i −0.176142 − 2.22912i
109.12 0.587785 + 0.809017i 0.587785 − 0.809017i −0.309017 + 0.951057i −0.635868 − 2.14375i 1.00000 −2.45458 − 0.797540i −0.951057 + 0.309017i −0.309017 − 0.951057i 1.36058 − 1.77449i
109.13 0.587785 + 0.809017i 0.587785 − 0.809017i −0.309017 + 0.951057i −0.244369 + 2.22267i 1.00000 −4.86212 − 1.57980i −0.951057 + 0.309017i −0.309017 − 0.951057i −1.94182 + 1.10876i
109.14 0.587785 + 0.809017i 0.587785 − 0.809017i −0.309017 + 0.951057i 0.218072 − 2.22541i 1.00000 0.288858 + 0.0938558i −0.951057 + 0.309017i −0.309017 − 0.951057i 1.92857 − 1.13164i
109.15 0.587785 + 0.809017i 0.587785 − 0.809017i −0.309017 + 0.951057i 0.836694 + 2.07363i 1.00000 0.595482 + 0.193484i −0.951057 + 0.309017i −0.309017 − 0.951057i −1.18581 + 1.89575i
109.16 0.587785 + 0.809017i 0.587785 − 0.809017i −0.309017 + 0.951057i 1.25767 + 1.84886i 1.00000 2.43606 + 0.791524i −0.951057 + 0.309017i −0.309017 − 0.951057i −0.756519 + 2.10421i
109.17 0.587785 + 0.809017i 0.587785 − 0.809017i −0.309017 + 0.951057i 2.06791 − 0.850729i 1.00000 4.57919 + 1.48787i −0.951057 + 0.309017i −0.309017 − 0.951057i 1.90374 + 1.17293i
109.18 0.587785 + 0.809017i 0.587785 − 0.809017i −0.309017 + 0.951057i 2.23353 − 0.106448i 1.00000 −4.25057 − 1.38109i −0.951057 + 0.309017i −0.309017 − 0.951057i 1.39896 + 1.74440i
349.1 −0.951057 − 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i −2.22153 − 0.254605i 1.00000 −2.23337 + 3.07398i −0.587785 − 0.809017i 0.809017 − 0.587785i 2.03412 + 0.928633i
349.2 −0.951057 − 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i −2.16020 + 0.577517i 1.00000 2.84298 − 3.91303i −0.587785 − 0.809017i 0.809017 − 0.587785i 2.23294 + 0.118288i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.18
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.d ✓ 72
5.b even 2 1 inner 930.2.z.d ✓ 72
31.d even 5 1 inner 930.2.z.d ✓ 72
155.n even 10 1 inner 930.2.z.d ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.z.d ✓ 72 1.a even 1 1 trivial
930.2.z.d ✓ 72 5.b even 2 1 inner
930.2.z.d ✓ 72 31.d even 5 1 inner
930.2.z.d ✓ 72 155.n even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(16\!\cdots\!55\)\( T_{7}^{56} - \)\(29\!\cdots\!80\)\( T_{7}^{54} + \)\(45\!\cdots\!30\)\( T_{7}^{52} - \)\(63\!\cdots\!00\)\( T_{7}^{50} + \)\(80\!\cdots\!85\)\( T_{7}^{48} - \)\(84\!\cdots\!20\)\( T_{7}^{46} + \)\(82\!\cdots\!90\)\( T_{7}^{44} - \)\(68\!\cdots\!70\)\( T_{7}^{42} + \)\(45\!\cdots\!65\)\( T_{7}^{40} - \)\(24\!\cdots\!20\)\( T_{7}^{38} + \)\(12\!\cdots\!65\)\( T_{7}^{36} - \)\(55\!\cdots\!20\)\( T_{7}^{34} + \)\(20\!\cdots\!10\)\( T_{7}^{32} - \)\(61\!\cdots\!00\)\( T_{7}^{30} + \)\(20\!\cdots\!55\)\( T_{7}^{28} - \)\(34\!\cdots\!80\)\( T_{7}^{26} + \)\(78\!\cdots\!50\)\( T_{7}^{24} - \)\(12\!\cdots\!76\)\( T_{7}^{22} + \)\(11\!\cdots\!39\)\( T_{7}^{20} - \)\(66\!\cdots\!92\)\( T_{7}^{18} + \)\(23\!\cdots\!83\)\( T_{7}^{16} - \)\(52\!\cdots\!28\)\( T_{7}^{14} + \)\(80\!\cdots\!47\)\( T_{7}^{12} - \)\(10\!\cdots\!34\)\( T_{7}^{10} + \)\(12\!\cdots\!41\)\( T_{7}^{8} - \)\(10\!\cdots\!80\)\( T_{7}^{6} + \)\(61\!\cdots\!00\)\( T_{7}^{4} - \)\(33\!\cdots\!00\)\( T_{7}^{2} + \)\(20\!\cdots\!00\)\( \)">\(T_{7}^{72} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).