Properties

Label 930.2.v.a
Level $930$
Weight $2$
Character orbit 930.v
Analytic conductor $7.426$
Analytic rank $0$
Dimension $80$
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.v (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80q + 20q^{4} + 12q^{7} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80q + 20q^{4} + 12q^{7} - 8q^{9} + 20q^{10} - 10q^{13} - 20q^{16} - 12q^{18} - 12q^{19} + 30q^{21} + 10q^{22} - 80q^{25} + 18q^{28} + 16q^{31} + 72q^{33} + 10q^{34} + 8q^{36} + 54q^{39} - 20q^{40} + 50q^{43} + 8q^{45} - 10q^{46} + 8q^{49} - 44q^{51} - 10q^{52} - 10q^{55} + 30q^{58} + 28q^{63} + 20q^{64} + 66q^{66} + 60q^{67} + 20q^{69} - 12q^{70} + 12q^{72} - 20q^{73} - 18q^{76} + 8q^{78} + 40q^{79} + 16q^{81} - 4q^{82} - 60q^{84} - 72q^{87} - 12q^{90} + 10q^{91} + 104q^{93} - 12q^{94} - 178q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1 −0.587785 + 0.809017i −1.73130 0.0510207i −0.309017 0.951057i 1.00000i 1.05891 1.37066i −0.425822 1.31054i 0.951057 + 0.309017i 2.99479 + 0.176664i 0.809017 + 0.587785i
401.2 −0.587785 + 0.809017i −1.71414 0.248453i −0.309017 0.951057i 1.00000i 1.20855 1.24073i 1.53704 + 4.73053i 0.951057 + 0.309017i 2.87654 + 0.851766i 0.809017 + 0.587785i
401.3 −0.587785 + 0.809017i −0.950729 + 1.44780i −0.309017 0.951057i 1.00000i −0.612467 1.62015i −0.531003 1.63426i 0.951057 + 0.309017i −1.19223 2.75292i 0.809017 + 0.587785i
401.4 −0.587785 + 0.809017i −0.922996 1.46563i −0.309017 0.951057i 1.00000i 1.72824 + 0.114757i 0.0841834 + 0.259090i 0.951057 + 0.309017i −1.29616 + 2.70555i 0.809017 + 0.587785i
401.5 −0.587785 + 0.809017i −0.438530 1.67562i −0.309017 0.951057i 1.00000i 1.61336 + 0.630124i 1.20094 + 3.69613i 0.951057 + 0.309017i −2.61538 + 1.46962i 0.809017 + 0.587785i
401.6 −0.587785 + 0.809017i 0.106510 + 1.72877i −0.309017 0.951057i 1.00000i −1.46121 0.929979i 0.417905 + 1.28618i 0.951057 + 0.309017i −2.97731 + 0.368264i 0.809017 + 0.587785i
401.7 −0.587785 + 0.809017i 1.13234 1.31065i −0.309017 0.951057i 1.00000i 0.394770 + 1.68646i 0.208734 + 0.642416i 0.951057 + 0.309017i −0.435626 2.96820i 0.809017 + 0.587785i
401.8 −0.587785 + 0.809017i 1.35636 + 1.07717i −0.309017 0.951057i 1.00000i −1.66869 + 0.464177i 0.584024 + 1.79744i 0.951057 + 0.309017i 0.679429 + 2.92205i 0.809017 + 0.587785i
401.9 −0.587785 + 0.809017i 1.47839 + 0.902413i −0.309017 0.951057i 1.00000i −1.59905 + 0.665622i −1.34568 4.14157i 0.951057 + 0.309017i 1.37130 + 2.66824i 0.809017 + 0.587785i
401.10 −0.587785 + 0.809017i 1.68409 0.404771i −0.309017 0.951057i 1.00000i −0.662417 + 1.60038i −0.230328 0.708876i 0.951057 + 0.309017i 2.67232 1.36334i 0.809017 + 0.587785i
401.11 0.587785 0.809017i −1.59642 + 0.671902i −0.309017 0.951057i 1.00000i −0.394770 + 1.68646i 0.208734 + 0.642416i −0.951057 0.309017i 2.09709 2.14527i 0.809017 + 0.587785i
401.12 0.587785 0.809017i −1.45809 0.934861i −0.309017 0.951057i 1.00000i −1.61336 + 0.630124i 1.20094 + 3.69613i −0.951057 0.309017i 1.25207 + 2.72623i 0.809017 + 0.587785i
401.13 0.587785 0.809017i −1.10868 1.33073i −0.309017 0.951057i 1.00000i −1.72824 + 0.114757i 0.0841834 + 0.259090i −0.951057 0.309017i −0.541668 + 2.95069i 0.809017 + 0.587785i
401.14 0.587785 0.809017i −0.905373 + 1.47658i −0.309017 0.951057i 1.00000i 0.662417 + 1.60038i −0.230328 0.708876i −0.951057 0.309017i −1.36060 2.67372i 0.809017 + 0.587785i
401.15 0.587785 0.809017i 0.293405 1.70702i −0.309017 0.951057i 1.00000i −1.20855 1.24073i 1.53704 + 4.73053i −0.951057 0.309017i −2.82783 1.00170i 0.809017 + 0.587785i
401.16 0.587785 0.809017i 0.401396 + 1.68490i −0.309017 0.951057i 1.00000i 1.59905 + 0.665622i −1.34568 4.14157i −0.951057 0.309017i −2.67776 + 1.35262i 0.809017 + 0.587785i
401.17 0.587785 0.809017i 0.486477 1.66233i −0.309017 0.951057i 1.00000i −1.05891 1.37066i −0.425822 1.31054i −0.951057 0.309017i −2.52668 1.61737i 0.809017 + 0.587785i
401.18 0.587785 0.809017i 0.605307 + 1.62284i −0.309017 0.951057i 1.00000i 1.66869 + 0.464177i 0.584024 + 1.79744i −0.951057 0.309017i −2.26721 + 1.96463i 0.809017 + 0.587785i
401.19 0.587785 0.809017i 1.61125 + 0.635517i −0.309017 0.951057i 1.00000i 1.46121 0.929979i 0.417905 + 1.28618i −0.951057 0.309017i 2.19224 + 2.04795i 0.809017 + 0.587785i
401.20 0.587785 0.809017i 1.67073 0.456803i −0.309017 0.951057i 1.00000i 0.612467 1.62015i −0.531003 1.63426i −0.951057 0.309017i 2.58266 1.52639i 0.809017 + 0.587785i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 821.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.f odd 10 1 inner
93.k 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.v.a 80
3.b odd 2 1 inner 930.2.v.a 80
31.f odd 10 1 inner 930.2.v.a 80
93.k even 10 1 inner 930.2.v.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.v.a 80 1.a even 1 1 trivial
930.2.v.a 80 3.b odd 2 1 inner
930.2.v.a 80 31.f odd 10 1 inner
930.2.v.a 80 93.k even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(25\!\cdots\!65\)\( T_{7}^{14} + 217726535038 T_{7}^{13} + \)\(53\!\cdots\!30\)\( T_{7}^{12} + 65440814015 T_{7}^{11} + \)\(84\!\cdots\!93\)\( T_{7}^{10} - 646762248628 T_{7}^{9} + \)\(11\!\cdots\!81\)\( T_{7}^{8} - \)\(14\!\cdots\!44\)\( T_{7}^{7} + \)\(15\!\cdots\!42\)\( T_{7}^{6} - \)\(21\!\cdots\!95\)\( T_{7}^{5} + \)\(94\!\cdots\!39\)\( T_{7}^{4} - \)\(15\!\cdots\!70\)\( T_{7}^{3} + \)\(27\!\cdots\!08\)\( T_{7}^{2} - 368852411048 T_{7} + 156850849936 \)">\(T_{7}^{40} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).