Properties

Label 930.2.be.b
Level $930$
Weight $2$
Character orbit 930.be
Analytic conductor $7.426$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64q + 4q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 64q + 4q^{7} - 4q^{10} - 24q^{14} - 8q^{15} - 64q^{16} - 4q^{17} + 12q^{20} - 24q^{21} - 4q^{22} - 32q^{24} + 28q^{25} + 8q^{28} - 16q^{29} + 8q^{31} + 4q^{33} + 24q^{35} + 32q^{36} + 16q^{37} - 28q^{38} - 8q^{41} + 4q^{42} - 40q^{43} + 4q^{44} - 12q^{45} + 8q^{47} + 60q^{49} - 8q^{50} - 24q^{53} + 64q^{54} + 44q^{55} + 4q^{57} - 52q^{58} - 24q^{59} + 20q^{62} - 4q^{63} + 44q^{65} + 8q^{66} - 44q^{67} - 4q^{68} + 12q^{69} - 44q^{70} + 8q^{71} + 4q^{73} - 12q^{74} + 8q^{75} - 8q^{76} + 104q^{77} - 56q^{79} + 32q^{81} - 16q^{82} - 48q^{83} - 32q^{85} - 24q^{86} - 32q^{87} + 8q^{88} + 176q^{89} + 16q^{93} + 64q^{95} - 68q^{97} + 32q^{98} + 4q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −0.707107 0.707107i 0.258819 0.965926i 1.00000i −2.15172 + 0.608374i −0.866025 + 0.500000i 0.345988 1.29125i 0.707107 0.707107i −0.866025 0.500000i 1.95168 + 1.09131i
37.2 −0.707107 0.707107i 0.258819 0.965926i 1.00000i −1.90333 1.17360i −0.866025 + 0.500000i −0.365018 + 1.36227i 0.707107 0.707107i −0.866025 0.500000i 0.515994 + 2.17572i
37.3 −0.707107 0.707107i 0.258819 0.965926i 1.00000i −1.31828 1.80614i −0.866025 + 0.500000i 1.13468 4.23469i 0.707107 0.707107i −0.866025 0.500000i −0.344964 + 2.20930i
37.4 −0.707107 0.707107i 0.258819 0.965926i 1.00000i −0.0950827 2.23405i −0.866025 + 0.500000i −0.582615 + 2.17435i 0.707107 0.707107i −0.866025 0.500000i −1.51248 + 1.64694i
37.5 −0.707107 0.707107i 0.258819 0.965926i 1.00000i 0.0920628 + 2.23417i −0.866025 + 0.500000i 0.111124 0.414720i 0.707107 0.707107i −0.866025 0.500000i 1.51470 1.64490i
37.6 −0.707107 0.707107i 0.258819 0.965926i 1.00000i 1.76107 + 1.37790i −0.866025 + 0.500000i 1.28122 4.78156i 0.707107 0.707107i −0.866025 0.500000i −0.270942 2.21959i
37.7 −0.707107 0.707107i 0.258819 0.965926i 1.00000i 2.10363 0.758108i −0.866025 + 0.500000i −1.03988 + 3.88090i 0.707107 0.707107i −0.866025 0.500000i −2.02356 0.951430i
37.8 −0.707107 0.707107i 0.258819 0.965926i 1.00000i 2.11885 0.714484i −0.866025 + 0.500000i 0.511082 1.90738i 0.707107 0.707107i −0.866025 0.500000i −2.00347 0.993035i
37.9 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i −2.08483 + 0.808396i −0.866025 + 0.500000i −0.0488226 + 0.182208i −0.707107 + 0.707107i −0.866025 0.500000i −2.04582 0.902572i
37.10 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i −2.06875 0.848677i −0.866025 + 0.500000i 0.339513 1.26708i −0.707107 + 0.707107i −0.866025 0.500000i −0.862725 2.06294i
37.11 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i −0.604368 + 2.15284i −0.866025 + 0.500000i −0.350136 + 1.30673i −0.707107 + 0.707107i −0.866025 0.500000i −1.94964 + 1.09494i
37.12 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i −0.0918774 2.23418i −0.866025 + 0.500000i −0.264810 + 0.988286i −0.707107 + 0.707107i −0.866025 0.500000i 1.51484 1.64477i
37.13 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 0.260478 2.22084i −0.866025 + 0.500000i 0.587501 2.19258i −0.707107 + 0.707107i −0.866025 0.500000i 1.75456 1.38619i
37.14 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 1.24237 + 1.85917i −0.866025 + 0.500000i −0.725743 + 2.70851i −0.707107 + 0.707107i −0.866025 0.500000i −0.436149 + 2.19312i
37.15 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 2.23583 0.0323803i −0.866025 + 0.500000i 0.933337 3.48326i −0.707107 + 0.707107i −0.866025 0.500000i 1.60387 + 1.55808i
37.16 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 2.23599 0.0184061i −0.866025 + 0.500000i −0.867414 + 3.23723i −0.707107 + 0.707107i −0.866025 0.500000i 1.59410 + 1.56807i
223.1 −0.707107 + 0.707107i −0.965926 0.258819i 1.00000i −2.21911 + 0.274898i 0.866025 0.500000i −4.03706 1.08173i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.37476 1.76353i
223.2 −0.707107 + 0.707107i −0.965926 0.258819i 1.00000i −2.13285 0.671543i 0.866025 0.500000i 0.713991 + 0.191313i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.98300 1.03330i
223.3 −0.707107 + 0.707107i −0.965926 0.258819i 1.00000i −1.41990 + 1.72739i 0.866025 0.500000i 4.37058 + 1.17109i 0.707107 + 0.707107i 0.866025 + 0.500000i −0.217429 2.22547i
223.4 −0.707107 + 0.707107i −0.965926 0.258819i 1.00000i −0.0340238 2.23581i 0.866025 0.500000i −1.26918 0.340075i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.60501 + 1.55690i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 553.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
155.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.be.b yes 64
5.c odd 4 1 930.2.be.a 64
31.e odd 6 1 930.2.be.a 64
155.p even 12 1 inner 930.2.be.b yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.be.a 64 5.c odd 4 1
930.2.be.a 64 31.e odd 6 1
930.2.be.b yes 64 1.a even 1 1 trivial
930.2.be.b yes 64 155.p even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(19\!\cdots\!60\)\( T_{7}^{43} + \)\(12\!\cdots\!32\)\( T_{7}^{42} + \)\(13\!\cdots\!96\)\( T_{7}^{41} - \)\(77\!\cdots\!71\)\( T_{7}^{40} - \)\(20\!\cdots\!08\)\( T_{7}^{39} - \)\(92\!\cdots\!50\)\( T_{7}^{38} - \)\(10\!\cdots\!08\)\( T_{7}^{37} + \)\(32\!\cdots\!15\)\( T_{7}^{36} + \)\(10\!\cdots\!04\)\( T_{7}^{35} + \)\(45\!\cdots\!58\)\( T_{7}^{34} + \)\(71\!\cdots\!68\)\( T_{7}^{33} + \)\(53\!\cdots\!57\)\( T_{7}^{32} - \)\(22\!\cdots\!92\)\( T_{7}^{31} - \)\(11\!\cdots\!24\)\( T_{7}^{30} - \)\(22\!\cdots\!44\)\( T_{7}^{29} - \)\(26\!\cdots\!82\)\( T_{7}^{28} + \)\(15\!\cdots\!92\)\( T_{7}^{27} + \)\(19\!\cdots\!56\)\( T_{7}^{26} + \)\(53\!\cdots\!44\)\( T_{7}^{25} + \)\(11\!\cdots\!73\)\( T_{7}^{24} + \)\(17\!\cdots\!80\)\( T_{7}^{23} + \)\(19\!\cdots\!26\)\( T_{7}^{22} + \)\(17\!\cdots\!36\)\( T_{7}^{21} + \)\(56\!\cdots\!75\)\( T_{7}^{20} - \)\(11\!\cdots\!60\)\( T_{7}^{19} - \)\(22\!\cdots\!26\)\( T_{7}^{18} - \)\(26\!\cdots\!64\)\( T_{7}^{17} - \)\(12\!\cdots\!52\)\( T_{7}^{16} + \)\(10\!\cdots\!52\)\( T_{7}^{15} + \)\(22\!\cdots\!94\)\( T_{7}^{14} + \)\(29\!\cdots\!72\)\( T_{7}^{13} + \)\(15\!\cdots\!85\)\( T_{7}^{12} - \)\(69\!\cdots\!80\)\( T_{7}^{11} - \)\(11\!\cdots\!06\)\( T_{7}^{10} - \)\(11\!\cdots\!60\)\( T_{7}^{9} - \)\(11\!\cdots\!43\)\( T_{7}^{8} + \)\(60\!\cdots\!04\)\( T_{7}^{7} + \)\(94\!\cdots\!92\)\( T_{7}^{6} + \)\(66\!\cdots\!72\)\( T_{7}^{5} + \)\(80\!\cdots\!64\)\( T_{7}^{4} - \)\(19\!\cdots\!12\)\( T_{7}^{3} + \)\(26\!\cdots\!60\)\( T_{7}^{2} - \)\(72\!\cdots\!12\)\( T_{7} + \)\(86\!\cdots\!76\)\( \)">\(T_{7}^{64} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).