Properties

Label 304.2.bg.a
Level $304$
Weight $2$
Character orbit 304.bg
Analytic conductor $2.427$
Analytic rank $0$
Dimension $456$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.42745222145\)
Analytic rank: \(0\)
Dimension: \(456\)
Relative dimension: \(38\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 456q - 12q^{2} - 12q^{3} - 12q^{4} - 12q^{5} - 12q^{6} - 12q^{7} - 18q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 456q - 12q^{2} - 12q^{3} - 12q^{4} - 12q^{5} - 12q^{6} - 12q^{7} - 18q^{8} - 42q^{10} - 6q^{11} - 18q^{12} - 12q^{13} - 24q^{16} - 24q^{17} - 12q^{19} - 24q^{20} + 6q^{21} - 12q^{22} - 24q^{23} - 12q^{24} - 54q^{26} - 18q^{27} + 12q^{28} - 12q^{29} - 48q^{30} + 18q^{32} - 24q^{33} + 48q^{34} + 18q^{35} - 60q^{36} - 66q^{38} - 48q^{39} - 42q^{40} + 144q^{42} - 12q^{43} + 54q^{44} - 6q^{45} - 108q^{46} - 12q^{48} - 168q^{49} + 36q^{50} + 12q^{51} - 60q^{52} - 12q^{53} - 126q^{54} - 24q^{55} - 24q^{58} - 12q^{59} + 30q^{60} - 12q^{61} - 6q^{64} - 36q^{65} - 72q^{66} - 12q^{67} - 42q^{68} + 126q^{69} + 102q^{70} - 24q^{71} - 48q^{72} + 72q^{74} + 36q^{76} + 60q^{77} - 108q^{78} + 48q^{80} - 24q^{81} - 72q^{82} - 6q^{83} - 18q^{84} - 108q^{85} - 12q^{86} - 12q^{87} - 18q^{88} + 96q^{90} + 30q^{91} - 12q^{92} + 6q^{93} - 132q^{96} - 24q^{97} - 66q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.41273 + 0.0647022i 0.459790 + 0.214404i 1.99163 0.182814i −1.87793 2.68196i −0.663433 0.273146i 0.233325 + 0.404130i −2.80181 + 0.387130i −1.76292 2.10097i 2.82654 + 3.66738i
3.2 −1.38797 + 0.271180i 2.35855 + 1.09981i 1.85292 0.752781i 1.12887 + 1.61220i −3.57184 0.886910i 0.814974 + 1.41158i −2.36766 + 1.54731i 2.42481 + 2.88978i −2.00404 1.93156i
3.3 −1.36996 0.351025i −1.52863 0.712812i 1.75356 + 0.961777i 2.13072 + 3.04299i 1.84394 + 1.51311i 0.591379 + 1.02430i −2.06470 1.93314i −0.0997514 0.118879i −1.85084 4.91670i
3.4 −1.27934 0.602732i −2.12965 0.993074i 1.27343 + 1.54220i −0.856388 1.22305i 2.12600 + 2.55409i 2.11839 + 3.66917i −0.699613 2.74054i 1.62087 + 1.93167i 0.358441 + 2.08087i
3.5 −1.25803 + 0.646026i −0.566911 0.264355i 1.16530 1.62545i 0.256415 + 0.366198i 0.883974 0.0336715i 1.25795 + 2.17884i −0.415910 + 2.79768i −1.67686 1.99840i −0.559152 0.295039i
3.6 −1.23391 + 0.690987i −2.50672 1.16890i 1.04507 1.70523i 1.02987 + 1.47081i 3.90077 0.289789i −0.843507 1.46100i −0.111235 + 2.82624i 2.98897 + 3.56211i −2.28708 1.10322i
3.7 −1.20851 0.734512i 1.75061 + 0.816321i 0.920986 + 1.77533i 0.635225 + 0.907195i −1.51603 2.27237i 0.220626 + 0.382136i 0.190979 2.82197i 0.469880 + 0.559981i −0.101329 1.56293i
3.8 −1.15700 + 0.813233i 2.20762 + 1.02943i 0.677304 1.88182i −1.33336 1.90424i −3.39139 + 0.604258i −2.13323 3.69486i 0.746720 + 2.72808i 1.88550 + 2.24705i 3.09129 + 1.11887i
3.9 −1.09186 0.898798i 0.945637 + 0.440958i 0.384323 + 1.96273i −1.51828 2.16833i −0.636173 1.33140i −0.308642 0.534584i 1.34447 2.48845i −1.22858 1.46416i −0.291138 + 3.73214i
3.10 −1.01722 0.982474i −2.34554 1.09375i 0.0694880 + 1.99879i −0.667915 0.953881i 1.31137 + 3.41702i −2.50561 4.33984i 1.89308 2.10149i 2.37694 + 2.83272i −0.257745 + 1.62652i
3.11 −0.843947 + 1.13479i 0.0677865 + 0.0316094i −0.575508 1.91541i 0.591033 + 0.844083i −0.0930783 + 0.0502470i −2.10365 3.64363i 2.65929 + 0.963421i −1.92477 2.29385i −1.45666 0.0416610i
3.12 −0.799968 + 1.16621i −2.34714 1.09449i −0.720104 1.86586i −2.31438 3.30527i 3.15404 1.86171i 0.352854 + 0.611162i 2.75205 + 0.652837i 2.38278 + 2.83969i 5.70607 0.0549447i
3.13 −0.610064 + 1.27586i 0.759059 + 0.353955i −1.25564 1.55671i 2.32234 + 3.31664i −0.914672 + 0.752519i 0.950093 + 1.64561i 2.75217 0.652334i −1.47748 1.76079i −5.64835 + 0.939618i
3.14 −0.573578 1.29267i −1.00251 0.467479i −1.34202 + 1.48290i 0.566420 + 0.808932i −0.0292792 + 1.56406i 0.913291 + 1.58187i 2.68666 + 0.884231i −1.14187 1.36083i 0.720799 1.19618i
3.15 −0.463438 1.33612i 2.94375 + 1.37269i −1.57045 + 1.23842i −1.38143 1.97288i 0.469841 4.56937i 1.46995 + 2.54603i 2.38249 + 1.52438i 4.85302 + 5.78360i −1.99581 + 2.76006i
3.16 −0.443631 + 1.34283i 2.49171 + 1.16190i −1.60638 1.19144i −0.636818 0.909471i −2.66564 + 2.83049i 1.63512 + 2.83211i 2.31255 1.62854i 2.93025 + 3.49213i 1.50378 0.451669i
3.17 −0.372821 1.36419i 2.26327 + 1.05538i −1.72201 + 1.01720i 2.33412 + 3.33347i 0.595940 3.48099i −1.68527 2.91898i 2.02965 + 1.96991i 2.08020 + 2.47909i 3.67727 4.42697i
3.18 −0.302259 + 1.38154i 0.00958574 + 0.00446990i −1.81728 0.835162i −1.14263 1.63184i −0.00907270 + 0.0118920i 0.290298 + 0.502811i 1.70309 2.25820i −1.92829 2.29805i 2.59982 1.08534i
3.19 −0.256606 1.39074i −0.0196774 0.00917574i −1.86831 + 0.713745i −0.252165 0.360129i −0.00771170 + 0.0297207i −1.27538 2.20902i 1.47205 + 2.41517i −1.92806 2.29777i −0.436138 + 0.443107i
3.20 0.0596919 1.41295i −2.91030 1.35709i −1.99287 0.168684i 1.72144 + 2.45847i −2.09123 + 4.03111i 0.331324 + 0.573871i −0.357301 + 2.80577i 4.69976 + 5.60096i 3.57646 2.28556i
See next 80 embeddings (of 456 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
19.f odd 18 1 inner
304.bg even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.bg.a 456
16.f odd 4 1 inner 304.2.bg.a 456
19.f odd 18 1 inner 304.2.bg.a 456
304.bg even 36 1 inner 304.2.bg.a 456
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.bg.a 456 1.a even 1 1 trivial
304.2.bg.a 456 16.f odd 4 1 inner
304.2.bg.a 456 19.f odd 18 1 inner
304.2.bg.a 456 304.bg even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(304, [\chi])\).