Properties

Label 930.2.br.b
Level $930$
Weight $2$
Character orbit 930.br
Analytic conductor $7.426$
Analytic rank $0$
Dimension $176$
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.br (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(22\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176q + 44q^{4} + 4q^{7} + 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 176q + 44q^{4} + 4q^{7} + 4q^{9} + 22q^{10} + 38q^{13} - 44q^{16} + 4q^{18} + 8q^{19} - 42q^{21} + 4q^{22} + 88q^{25} + 30q^{27} + 36q^{28} + 32q^{31} - 70q^{33} + 14q^{34} - 4q^{36} + 42q^{37} + 58q^{39} - 22q^{40} - 12q^{42} - 46q^{43} + 16q^{45} + 10q^{46} + 38q^{49} + 38q^{51} + 2q^{52} + 4q^{55} + 78q^{57} - 40q^{58} + 16q^{63} + 44q^{64} + 34q^{66} - 76q^{67} + 148q^{69} - 8q^{70} - 4q^{72} - 52q^{73} + 12q^{76} + 60q^{78} + 8q^{79} - 108q^{81} - 40q^{82} - 8q^{84} + 28q^{87} + 6q^{88} + 24q^{90} - 20q^{91} - 28q^{93} - 20q^{94} - 112q^{97} - 132q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.951057 + 0.309017i −1.72275 + 0.179283i 0.809017 0.587785i −0.866025 + 0.500000i 1.58303 0.702867i −0.113415 1.07907i −0.587785 + 0.809017i 2.93571 0.617720i 0.669131 0.743145i
11.2 −0.951057 + 0.309017i −1.26144 1.18692i 0.809017 0.587785i −0.866025 + 0.500000i 1.56648 + 0.739021i 0.426617 + 4.05899i −0.587785 + 0.809017i 0.182449 + 2.99445i 0.669131 0.743145i
11.3 −0.951057 + 0.309017i −1.15627 1.28959i 0.809017 0.587785i −0.866025 + 0.500000i 1.49818 + 0.869165i −0.480398 4.57068i −0.587785 + 0.809017i −0.326079 + 2.98223i 0.669131 0.743145i
11.4 −0.951057 + 0.309017i −1.05992 + 1.36988i 0.809017 0.587785i −0.866025 + 0.500000i 0.584727 1.63037i 0.140936 + 1.34092i −0.587785 + 0.809017i −0.753142 2.90392i 0.669131 0.743145i
11.5 −0.951057 + 0.309017i 0.0316079 1.73176i 0.809017 0.587785i −0.866025 + 0.500000i 0.505083 + 1.65677i −0.0802420 0.763452i −0.587785 + 0.809017i −2.99800 0.109475i 0.669131 0.743145i
11.6 −0.951057 + 0.309017i 0.0457608 + 1.73145i 0.809017 0.587785i −0.866025 + 0.500000i −0.578567 1.63256i 0.394896 + 3.75719i −0.587785 + 0.809017i −2.99581 + 0.158465i 0.669131 0.743145i
11.7 −0.951057 + 0.309017i 0.145925 + 1.72589i 0.809017 0.587785i −0.866025 + 0.500000i −0.672113 1.59633i −0.228500 2.17403i −0.587785 + 0.809017i −2.95741 + 0.503701i 0.669131 0.743145i
11.8 −0.951057 + 0.309017i 0.668302 1.59793i 0.809017 0.587785i −0.866025 + 0.500000i −0.141806 + 1.72624i 0.349465 + 3.32494i −0.587785 + 0.809017i −2.10675 2.13580i 0.669131 0.743145i
11.9 −0.951057 + 0.309017i 1.38456 1.04067i 0.809017 0.587785i −0.866025 + 0.500000i −0.995214 + 1.41759i −0.249897 2.37761i −0.587785 + 0.809017i 0.834029 2.88173i 0.669131 0.743145i
11.10 −0.951057 + 0.309017i 1.39578 + 1.02558i 0.809017 0.587785i −0.866025 + 0.500000i −1.64438 0.544062i 0.359918 + 3.42439i −0.587785 + 0.809017i 0.896389 + 2.86295i 0.669131 0.743145i
11.11 −0.951057 + 0.309017i 1.52844 + 0.814785i 0.809017 0.587785i −0.866025 + 0.500000i −1.70541 0.302593i −0.239608 2.27972i −0.587785 + 0.809017i 1.67225 + 2.49070i 0.669131 0.743145i
11.12 0.951057 0.309017i −1.71718 0.226496i 0.809017 0.587785i 0.866025 0.500000i −1.70312 + 0.315227i 0.394896 + 3.75719i 0.587785 0.809017i 2.89740 + 0.777866i 0.669131 0.743145i
11.13 0.951057 0.309017i −1.70118 0.325530i 0.809017 0.587785i 0.866025 0.500000i −1.71852 + 0.216097i −0.228500 2.17403i 0.587785 0.809017i 2.78806 + 1.10757i 0.669131 0.743145i
11.14 0.951057 0.309017i −1.47317 + 0.910922i 0.809017 0.587785i 0.866025 0.500000i −1.11958 + 1.32157i 0.140936 + 1.34092i 0.587785 0.809017i 1.34044 2.68388i 0.669131 0.743145i
11.15 0.951057 0.309017i −0.874059 1.49533i 0.809017 0.587785i 0.866025 0.500000i −1.29336 1.15205i 0.359918 + 3.42439i 0.587785 0.809017i −1.47204 + 2.61402i 0.669131 0.743145i
11.16 0.951057 0.309017i −0.650557 1.60523i 0.809017 0.587785i 0.866025 0.500000i −1.11476 1.32564i −0.239608 2.27972i 0.587785 0.809017i −2.15355 + 2.08859i 0.669131 0.743145i
11.17 0.951057 0.309017i −0.358377 + 1.69457i 0.809017 0.587785i 0.866025 0.500000i 0.182814 + 1.72238i −0.113415 1.07907i 0.587785 0.809017i −2.74313 1.21459i 0.669131 0.743145i
11.18 0.951057 0.309017i 1.04856 + 1.37859i 0.809017 0.587785i 0.866025 0.500000i 1.42325 + 0.987098i 0.426617 + 4.05899i 0.587785 0.809017i −0.801042 + 2.89108i 0.669131 0.743145i
11.19 0.951057 0.309017i 1.16166 + 1.28473i 0.809017 0.587785i 0.866025 0.500000i 1.50181 + 0.862882i −0.480398 4.57068i 0.587785 0.809017i −0.301087 + 2.98485i 0.669131 0.743145i
11.20 0.951057 0.309017i 1.17969 1.26820i 0.809017 0.587785i 0.866025 0.500000i 0.730058 1.57067i −0.249897 2.37761i 0.587785 0.809017i −0.216657 2.99217i 0.669131 0.743145i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 911.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.h odd 30 1 inner
93.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.br.b 176
3.b odd 2 1 inner 930.2.br.b 176
31.h odd 30 1 inner 930.2.br.b 176
93.p even 30 1 inner 930.2.br.b 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.br.b 176 1.a even 1 1 trivial
930.2.br.b 176 3.b odd 2 1 inner
930.2.br.b 176 31.h odd 30 1 inner
930.2.br.b 176 93.p even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(27\!\cdots\!98\)\( T_{7}^{70} - \)\(63\!\cdots\!48\)\( T_{7}^{69} + \)\(15\!\cdots\!87\)\( T_{7}^{68} + \)\(70\!\cdots\!70\)\( T_{7}^{67} + \)\(37\!\cdots\!69\)\( T_{7}^{66} + \)\(55\!\cdots\!40\)\( T_{7}^{65} - \)\(82\!\cdots\!55\)\( T_{7}^{64} - \)\(18\!\cdots\!26\)\( T_{7}^{63} + \)\(10\!\cdots\!65\)\( T_{7}^{62} + \)\(44\!\cdots\!26\)\( T_{7}^{61} + \)\(30\!\cdots\!25\)\( T_{7}^{60} - \)\(53\!\cdots\!06\)\( T_{7}^{59} - \)\(38\!\cdots\!31\)\( T_{7}^{58} - \)\(68\!\cdots\!72\)\( T_{7}^{57} + \)\(38\!\cdots\!71\)\( T_{7}^{56} + \)\(26\!\cdots\!14\)\( T_{7}^{55} + \)\(72\!\cdots\!34\)\( T_{7}^{54} - \)\(13\!\cdots\!26\)\( T_{7}^{53} - \)\(89\!\cdots\!16\)\( T_{7}^{52} - \)\(38\!\cdots\!96\)\( T_{7}^{51} - \)\(64\!\cdots\!14\)\( T_{7}^{50} + \)\(11\!\cdots\!94\)\( T_{7}^{49} + \)\(10\!\cdots\!05\)\( T_{7}^{48} + \)\(29\!\cdots\!18\)\( T_{7}^{47} + \)\(30\!\cdots\!74\)\( T_{7}^{46} - \)\(96\!\cdots\!82\)\( T_{7}^{45} - \)\(47\!\cdots\!86\)\( T_{7}^{44} - \)\(78\!\cdots\!44\)\( T_{7}^{43} + \)\(81\!\cdots\!73\)\( T_{7}^{42} + \)\(79\!\cdots\!94\)\( T_{7}^{41} + \)\(21\!\cdots\!31\)\( T_{7}^{40} + \)\(29\!\cdots\!38\)\( T_{7}^{39} - \)\(13\!\cdots\!00\)\( T_{7}^{38} - \)\(94\!\cdots\!42\)\( T_{7}^{37} - \)\(16\!\cdots\!44\)\( T_{7}^{36} + \)\(10\!\cdots\!22\)\( T_{7}^{35} + \)\(11\!\cdots\!09\)\( T_{7}^{34} + \)\(33\!\cdots\!46\)\( T_{7}^{33} + \)\(60\!\cdots\!27\)\( T_{7}^{32} + \)\(79\!\cdots\!04\)\( T_{7}^{31} + \)\(74\!\cdots\!18\)\( T_{7}^{30} + \)\(45\!\cdots\!60\)\( T_{7}^{29} + \)\(13\!\cdots\!10\)\( T_{7}^{28} + \)\(99\!\cdots\!10\)\( T_{7}^{27} + \)\(45\!\cdots\!37\)\( T_{7}^{26} + \)\(97\!\cdots\!70\)\( T_{7}^{25} + \)\(12\!\cdots\!46\)\( T_{7}^{24} + \)\(11\!\cdots\!96\)\( T_{7}^{23} + \)\(74\!\cdots\!03\)\( T_{7}^{22} + \)\(43\!\cdots\!72\)\( T_{7}^{21} + \)\(32\!\cdots\!46\)\( T_{7}^{20} + \)\(28\!\cdots\!92\)\( T_{7}^{19} + \)\(19\!\cdots\!53\)\( T_{7}^{18} + \)\(56\!\cdots\!26\)\( T_{7}^{17} - \)\(15\!\cdots\!18\)\( T_{7}^{16} - \)\(11\!\cdots\!36\)\( T_{7}^{15} + \)\(13\!\cdots\!51\)\( T_{7}^{14} + \)\(22\!\cdots\!08\)\( T_{7}^{13} + \)\(13\!\cdots\!72\)\( T_{7}^{12} + \)\(91\!\cdots\!34\)\( T_{7}^{11} - \)\(11\!\cdots\!73\)\( T_{7}^{10} + \)\(32\!\cdots\!36\)\( T_{7}^{9} + \)\(19\!\cdots\!29\)\( T_{7}^{8} - \)\(18\!\cdots\!44\)\( T_{7}^{7} - \)\(15\!\cdots\!44\)\( T_{7}^{6} + \)\(25\!\cdots\!12\)\( T_{7}^{5} + \)\(96\!\cdots\!96\)\( T_{7}^{4} - \)\(39\!\cdots\!68\)\( T_{7}^{3} + \)\(54\!\cdots\!52\)\( T_{7}^{2} - \)\(34\!\cdots\!80\)\( T_{7} + \)\(85\!\cdots\!96\)\( \)">\(T_{7}^{88} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).