Properties

Label 760.2.bl.c
Level $760$
Weight $2$
Character orbit 760.bl
Analytic conductor $6.069$
Analytic rank $0$
Dimension $152$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [760,2,Mod(501,760)] 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("760.501"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.bl (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [152,-4,0,-2,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(76\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 152 q - 4 q^{2} - 2 q^{4} - 8 q^{6} + 20 q^{7} - 16 q^{8} + 74 q^{9} + 2 q^{10} - 24 q^{12} + 10 q^{14} - 14 q^{15} - 14 q^{16} - 4 q^{17} - 12 q^{18} - 4 q^{22} + 22 q^{23} - 24 q^{24} + 76 q^{25} + 20 q^{26}+ \cdots - 92 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
501.1 −1.40850 − 0.126977i −2.13983 + 1.23543i 1.96775 + 0.357695i −0.866025 + 0.500000i 3.17082 − 1.46840i 1.34571 −2.72617 − 0.753673i 1.55258 − 2.68914i 1.28329 − 0.594286i
501.2 −1.40803 + 0.132144i 2.22509 − 1.28466i 1.96508 − 0.372126i 0.866025 − 0.500000i −2.96323 + 2.10287i 2.09670 −2.71770 + 0.783636i 1.80069 − 3.11889i −1.15331 + 0.818454i
501.3 −1.40632 + 0.149183i −2.63386 + 1.52066i 1.95549 − 0.419599i 0.866025 − 0.500000i 3.47720 − 2.53147i 3.21815 −2.68745 + 0.881818i 3.12482 − 5.41235i −1.14332 + 0.832358i
501.4 −1.39842 + 0.210780i 1.27651 − 0.736995i 1.91114 − 0.589516i −0.866025 + 0.500000i −1.62975 + 1.29969i −3.14316 −2.54832 + 1.22722i −0.413677 + 0.716510i 1.10568 − 0.881749i
501.5 −1.39717 − 0.218887i −0.891397 + 0.514648i 1.90418 + 0.611646i 0.866025 − 0.500000i 1.35808 − 0.523937i −2.85681 −2.52658 − 1.27137i −0.970274 + 1.68056i −1.31943 + 0.509024i
501.6 −1.39400 − 0.238270i 0.574419 − 0.331641i 1.88645 + 0.664296i −0.866025 + 0.500000i −0.879758 + 0.325439i 4.00884 −2.47143 − 1.37551i −1.28003 + 2.21707i 1.32637 − 0.490650i
501.7 −1.38951 + 0.263200i −0.112105 + 0.0647237i 1.86145 − 0.731437i −0.866025 + 0.500000i 0.138735 − 0.119440i −2.48461 −2.39398 + 1.50627i −1.49162 + 2.58356i 1.07175 − 0.922691i
501.8 −1.34173 + 0.446942i −2.27924 + 1.31592i 1.60049 − 1.19935i 0.866025 − 0.500000i 2.46998 − 2.78429i −5.07442 −1.61138 + 2.32453i 1.96328 − 3.40050i −0.938503 + 1.05793i
501.9 −1.33536 − 0.465634i 2.82773 − 1.63259i 1.56637 + 1.24358i −0.866025 + 0.500000i −4.53622 + 0.863407i 0.186142 −1.51262 − 2.38998i 3.83069 − 6.63494i 1.38927 − 0.264429i
501.10 −1.28268 + 0.595590i −0.804944 + 0.464735i 1.29055 − 1.52790i 0.866025 − 0.500000i 0.755696 − 1.07552i 4.26572 −0.745356 + 2.72845i −1.06804 + 1.84991i −0.813040 + 1.15714i
501.11 −1.24873 − 0.663825i −0.325806 + 0.188104i 1.11867 + 1.65788i 0.866025 − 0.500000i 0.531713 − 0.0186141i −0.198767 −0.296384 − 2.81286i −1.42923 + 2.47551i −1.41335 + 0.0494781i
501.12 −1.24029 + 0.679467i −2.11459 + 1.22086i 1.07665 − 1.68548i −0.866025 + 0.500000i 1.79317 − 2.95101i −1.03246 −0.190135 + 2.82203i 1.48098 − 2.56513i 0.734391 − 1.20858i
501.13 −1.23789 − 0.683841i 1.20842 − 0.697684i 1.06472 + 1.69303i 0.866025 − 0.500000i −1.97300 + 0.0372826i 3.59014 −0.160236 − 2.82388i −0.526473 + 0.911878i −1.41396 + 0.0267188i
501.14 −1.22732 + 0.702634i 0.523251 − 0.302099i 1.01261 − 1.72471i −0.866025 + 0.500000i −0.429929 + 0.738425i 3.23945 −0.0309549 + 2.82826i −1.31747 + 2.28193i 0.711570 − 1.22216i
501.15 −1.21846 − 0.717878i 1.26904 − 0.732681i 0.969303 + 1.74941i −0.866025 + 0.500000i −2.07225 − 0.0182717i −0.262355 0.0748067 − 2.82744i −0.426358 + 0.738473i 1.41416 + 0.0124691i
501.16 −1.20670 − 0.737482i −1.13741 + 0.656685i 0.912241 + 1.77984i −0.866025 + 0.500000i 1.85681 + 0.0464000i −5.00422 0.211797 − 2.82049i −0.637529 + 1.10423i 1.41377 + 0.0353290i
501.17 −1.15820 + 0.811526i 0.979259 − 0.565376i 0.682851 − 1.87982i 0.866025 − 0.500000i −0.675360 + 1.44951i −1.85600 0.734643 + 2.73135i −0.860701 + 1.49078i −0.597267 + 1.28190i
501.18 −1.12190 + 0.861016i 1.08330 − 0.625444i 0.517304 − 1.93194i 0.866025 − 0.500000i −0.676834 + 1.63442i −2.42648 1.08307 + 2.61284i −0.717640 + 1.24299i −0.541083 + 1.30661i
501.19 −1.10388 − 0.883991i −1.89277 + 1.09279i 0.437121 + 1.95165i −0.866025 + 0.500000i 3.05542 + 0.466877i 3.22110 1.24271 − 2.54080i 0.888397 − 1.53875i 1.39799 + 0.213616i
501.20 −0.988373 + 1.01149i 2.74962 − 1.58750i −0.0462366 − 1.99947i −0.866025 + 0.500000i −1.11191 + 4.35026i −3.98801 2.06814 + 1.92945i 3.54029 − 6.13195i 0.350210 − 1.37017i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 501.76
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
19.c even 3 1 inner
152.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.bl.c ✓ 152
8.b even 2 1 inner 760.2.bl.c ✓ 152
19.c even 3 1 inner 760.2.bl.c ✓ 152
152.p even 6 1 inner 760.2.bl.c ✓ 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.bl.c ✓ 152 1.a even 1 1 trivial
760.2.bl.c ✓ 152 8.b even 2 1 inner
760.2.bl.c ✓ 152 19.c even 3 1 inner
760.2.bl.c ✓ 152 152.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{152} - 151 T_{3}^{150} + 11918 T_{3}^{148} - 646299 T_{3}^{146} + 26858556 T_{3}^{144} + \cdots + 13\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display