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Properties

Label 185.2.bc.a
Level $185$
Weight $2$
Character orbit 185.bc
Analytic conductor $1.477$
Analytic rank $0$
Dimension $204$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [185,2,Mod(2,185)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("185.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.bc (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 204 q - 12 q^{2} - 18 q^{3} - 12 q^{5} - 24 q^{6} - 12 q^{7} - 18 q^{8} - 6 q^{10} - 36 q^{11} - 12 q^{12} - 12 q^{13} + 24 q^{14} - 12 q^{15} - 24 q^{16} + 12 q^{17} + 66 q^{18} - 108 q^{20} - 24 q^{21}+ \cdots - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.74035 − 2.07407i −0.0347175 − 0.396822i −0.925649 + 5.24962i −1.27953 + 1.83379i −0.762617 + 0.762617i 0.475342 + 0.221655i 7.80950 − 4.50882i 2.79816 − 0.493391i 6.03026 − 0.537613i
2.2 −1.53786 − 1.83275i −0.271823 − 3.10695i −0.646659 + 3.66738i 1.92559 − 1.13670i −5.27623 + 5.27623i −2.82679 − 1.31816i 3.57195 − 2.06227i −6.62484 + 1.16814i −5.04457 − 1.78104i
2.3 −1.30898 − 1.55998i 0.184529 + 2.10918i −0.372815 + 2.11434i −1.14827 − 1.91872i 3.04873 − 3.04873i 1.79514 + 0.837086i 0.259166 − 0.149629i −1.46016 + 0.257466i −1.49010 + 4.30283i
2.4 −1.19441 − 1.42344i −0.0188574 − 0.215541i −0.252278 + 1.43074i 2.12483 + 0.696504i −0.284287 + 0.284287i 3.11286 + 1.45155i −0.880549 + 0.508385i 2.90832 − 0.512815i −1.54648 − 3.85648i
2.5 −1.11108 − 1.32414i 0.235130 + 2.68755i −0.171537 + 0.972832i −0.481419 + 2.18363i 3.29744 − 3.29744i −2.16065 − 1.00753i −1.51516 + 0.874778i −4.21323 + 0.742906i 3.42632 − 1.78873i
2.6 −0.797170 − 0.950030i −0.154208 − 1.76261i 0.0802189 − 0.454944i −2.09957 + 0.769290i −1.55160 + 1.55160i −1.73346 − 0.808325i −2.64421 + 1.52663i −0.128591 + 0.0226740i 2.40456 + 1.38140i
2.7 −0.505704 − 0.602675i 0.0729968 + 0.834357i 0.239816 − 1.36007i 1.65695 − 1.50151i 0.465931 − 0.465931i −4.33280 − 2.02042i −2.30362 + 1.33000i 2.26360 − 0.399134i −1.74285 − 0.239279i
2.8 −0.499796 − 0.595634i −0.196021 − 2.24053i 0.242313 − 1.37422i −0.0112386 − 2.23604i −1.23656 + 1.23656i 3.94366 + 1.83896i −2.28639 + 1.32005i −2.02711 + 0.357435i −1.32624 + 1.12426i
2.9 0.128419 + 0.153044i 0.258566 + 2.95543i 0.340365 − 1.93031i 1.57458 − 1.58767i −0.419106 + 0.419106i 3.05837 + 1.42614i 0.685170 − 0.395583i −5.71326 + 1.00740i 0.445191 + 0.0370924i
2.10 0.171183 + 0.204009i 0.0720915 + 0.824010i 0.334981 − 1.89977i 1.04398 + 1.97740i −0.155764 + 0.155764i −0.107802 − 0.0502691i 0.906182 − 0.523185i 2.28063 − 0.402136i −0.224694 + 0.551479i
2.11 0.212330 + 0.253046i −0.252851 − 2.89010i 0.328349 − 1.86216i 0.971024 + 2.01423i 0.677638 − 0.677638i 0.0918740 + 0.0428415i 1.11307 − 0.642633i −5.33430 + 0.940582i −0.303513 + 0.673395i
2.12 0.441667 + 0.526358i −0.0239257 − 0.273472i 0.265313 − 1.50467i −1.63021 − 1.53050i 0.133377 − 0.133377i −0.845431 − 0.394231i 2.09928 − 1.21202i 2.88021 − 0.507859i 0.0855789 − 1.53404i
2.13 0.908898 + 1.08318i 0.244719 + 2.79715i 0.000107931 0 0.000612105i −2.01164 + 0.976375i −2.80739 + 2.80739i −0.722962 − 0.337123i 2.44987 − 1.41443i −4.80971 + 0.848082i −2.88597 − 1.29155i
2.14 1.18598 + 1.41340i −0.149584 − 1.70975i −0.243847 + 1.38292i 2.04948 − 0.894229i 2.23916 − 2.23916i −2.02620 − 0.944833i 0.951920 − 0.549591i 0.0535513 − 0.00944254i 3.69455 + 1.83619i
2.15 1.21107 + 1.44330i −0.0783802 − 0.895890i −0.269121 + 1.52626i −1.54620 + 1.61532i 1.19811 − 1.19811i 3.73685 + 1.74252i 0.734568 − 0.424103i 2.15795 − 0.380504i −4.20395 − 0.275360i
2.16 1.55339 + 1.85126i 0.138107 + 1.57857i −0.666845 + 3.78187i −0.404683 − 2.19914i −2.70782 + 2.70782i −0.414297 − 0.193190i −3.85134 + 2.22357i 0.481602 − 0.0849194i 3.44256 − 4.16531i
2.17 1.71403 + 2.04270i −0.0989638 − 1.13116i −0.887435 + 5.03290i −0.169989 + 2.22960i 2.14100 − 2.14100i −3.45253 − 1.60994i −7.18319 + 4.14722i 1.68469 − 0.297056i −4.84577 + 3.47436i
13.1 −2.61056 − 0.460312i 1.96213 + 1.37390i 4.72375 + 1.71930i −1.56554 − 1.59658i −4.48984 − 4.48984i 0.424696 − 4.85430i −6.94883 − 4.01191i 0.936303 + 2.57247i 3.35201 + 4.88861i
13.2 −2.39802 − 0.422835i −1.70390 − 1.19308i 3.69232 + 1.34389i −2.23270 + 0.122714i 3.58150 + 3.58150i −0.271128 + 3.09901i −4.06844 − 2.34892i 0.453762 + 1.24670i 5.40594 + 0.649795i
13.3 −2.31445 − 0.408100i 1.05300 + 0.737320i 3.31076 + 1.20502i 2.18694 + 0.466162i −2.13622 − 2.13622i −0.255760 + 2.92335i −3.10023 − 1.78992i −0.460888 − 1.26628i −4.87132 − 1.97140i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.bc even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.bc.a yes 204
5.b even 2 1 925.2.bq.b 204
5.c odd 4 1 185.2.z.a ✓ 204
5.c odd 4 1 925.2.bn.b 204
37.i odd 36 1 185.2.z.a ✓ 204
185.z even 36 1 925.2.bq.b 204
185.ba odd 36 1 925.2.bn.b 204
185.bc even 36 1 inner 185.2.bc.a yes 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.z.a ✓ 204 5.c odd 4 1
185.2.z.a ✓ 204 37.i odd 36 1
185.2.bc.a yes 204 1.a even 1 1 trivial
185.2.bc.a yes 204 185.bc even 36 1 inner
925.2.bn.b 204 5.c odd 4 1
925.2.bn.b 204 185.ba odd 36 1
925.2.bq.b 204 5.b even 2 1
925.2.bq.b 204 185.z even 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(185, [\chi])\).