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Properties

Label 507.2.p.b
Level $507$
Weight $2$
Character orbit 507.p
Analytic conductor $4.048$
Analytic rank $0$
Dimension $192$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [507,2,Mod(25,507)] 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("507.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.p (of order \(26\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 16 q^{3} + 18 q^{4} - 16 q^{9} - 18 q^{12} - 63 q^{13} - 10 q^{14} - 6 q^{16} + 12 q^{17} + 16 q^{22} - 52 q^{23} + 58 q^{25} + 51 q^{26} + 16 q^{27} - 49 q^{29} - 26 q^{31} - 13 q^{33} - 65 q^{34}+ \cdots + 104 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −2.52295 − 0.956827i −0.568065 − 0.822984i 3.95272 + 3.50180i 4.16354 + 0.505545i 0.645743 + 2.61988i 0.140914 + 0.268489i −4.11396 − 7.83849i −0.354605 + 0.935016i −10.0207 − 5.25925i
25.2 −2.42142 − 0.918325i −0.568065 − 0.822984i 3.52295 + 3.12106i −2.78849 − 0.338584i 0.619759 + 2.51446i 0.357250 + 0.680683i −3.25740 − 6.20646i −0.354605 + 0.935016i 6.44118 + 3.38059i
25.3 −2.13008 − 0.807834i −0.568065 − 0.822984i 2.38764 + 2.11526i −1.00892 − 0.122506i 0.545191 + 2.21193i 1.86515 + 3.55375i −1.25969 − 2.40014i −0.354605 + 0.935016i 2.05013 + 1.07599i
25.4 −1.50260 − 0.569861i −0.568065 − 0.822984i 0.436045 + 0.386302i 2.17889 + 0.264565i 0.384588 + 1.56033i 0.728343 + 1.38774i 1.05859 + 2.01697i −0.354605 + 0.935016i −3.12324 − 1.63920i
25.5 −1.31279 − 0.497877i −0.568065 − 0.822984i −0.0214779 − 0.0190278i 1.11474 + 0.135354i 0.336007 + 1.36323i −0.369513 − 0.704048i 1.32369 + 2.52209i −0.354605 + 0.935016i −1.39603 − 0.732692i
25.6 −1.13089 − 0.428890i −0.568065 − 0.822984i −0.402056 − 0.356191i −4.00621 − 0.486442i 0.289449 + 1.17434i −2.12665 − 4.05199i 1.42607 + 2.71715i −0.354605 + 0.935016i 4.32195 + 2.26833i
25.7 −0.468954 − 0.177851i −0.568065 − 0.822984i −1.30873 − 1.15944i −1.41049 − 0.171264i 0.120028 + 0.486973i 2.16100 + 4.11744i 0.873690 + 1.66468i −0.354605 + 0.935016i 0.630994 + 0.331171i
25.8 −0.103579 − 0.0392824i −0.568065 − 0.822984i −1.48784 − 1.31811i −1.67597 − 0.203500i 0.0265109 + 0.107559i 0.366680 + 0.698651i 0.205293 + 0.391152i −0.354605 + 0.935016i 0.165602 + 0.0869145i
25.9 0.283929 + 0.107680i −0.568065 − 0.822984i −1.42800 − 1.26510i −0.879635 − 0.106807i −0.0726712 − 0.294839i −0.935242 − 1.78196i −0.551463 − 1.05073i −0.354605 + 0.935016i −0.238253 − 0.125045i
25.10 0.494430 + 0.187512i −0.568065 − 0.822984i −1.28772 − 1.14082i 3.75871 + 0.456390i −0.126548 − 0.513427i 1.26721 + 2.41446i −0.914254 − 1.74197i −0.354605 + 0.935016i 1.77284 + 0.930459i
25.11 0.917259 + 0.347870i −0.568065 − 0.822984i −0.776671 − 0.688071i 0.986534 + 0.119787i −0.234771 − 0.952502i −0.506792 − 0.965611i −1.38484 − 2.63860i −0.354605 + 0.935016i 0.863237 + 0.453062i
25.12 1.43480 + 0.544146i −0.568065 − 0.822984i 0.265521 + 0.235231i −3.88465 − 0.471682i −0.367233 − 1.48992i 2.17406 + 4.14233i −1.17328 − 2.23550i −0.354605 + 0.935016i −5.31702 − 2.79059i
25.13 1.78796 + 0.678085i −0.568065 − 0.822984i 1.23999 + 1.09854i −2.39960 − 0.291364i −0.457626 − 1.85666i −1.52079 − 2.89763i −0.305152 − 0.581418i −0.354605 + 0.935016i −4.09283 − 2.14808i
25.14 1.98795 + 0.753931i −0.568065 − 0.822984i 1.88652 + 1.67131i 3.81359 + 0.463053i −0.508813 − 2.06433i −2.20899 − 4.20888i 0.514149 + 0.979628i −0.354605 + 0.935016i 7.23212 + 3.79571i
25.15 2.12140 + 0.804542i −0.568065 − 0.822984i 2.35604 + 2.08727i 2.11775 + 0.257141i −0.542969 − 2.20291i 1.12326 + 2.14019i 1.21005 + 2.30557i −0.354605 + 0.935016i 4.28571 + 2.24932i
25.16 2.56554 + 0.972980i −0.568065 − 0.822984i 4.13827 + 3.66618i −1.46104 − 0.177402i −0.656644 − 2.66411i 0.543789 + 1.03610i 4.49950 + 8.57307i −0.354605 + 0.935016i −3.57574 − 1.87669i
64.1 −2.62119 − 0.318270i 0.748511 − 0.663123i 4.82745 + 1.18986i −0.880115 − 1.67692i −2.17304 + 1.49994i −2.02452 − 0.767800i −7.33724 − 2.78265i 0.120537 − 0.992709i 1.77323 + 4.67563i
64.2 −2.60940 − 0.316839i 0.748511 − 0.663123i 4.76672 + 1.17489i 1.60429 + 3.05673i −2.16327 + 1.49320i 2.76033 + 1.04686i −7.15054 − 2.71184i 0.120537 − 0.992709i −3.21776 − 8.48454i
64.3 −2.10986 − 0.256183i 0.748511 − 0.663123i 2.44400 + 0.602391i 1.00545 + 1.91572i −1.74913 + 1.20734i −2.58799 − 0.981496i −1.02769 − 0.389750i 0.120537 − 0.992709i −1.63057 − 4.29947i
64.4 −1.61341 − 0.195904i 0.748511 − 0.663123i 0.622835 + 0.153515i −1.54944 − 2.95221i −1.33756 + 0.923254i −1.00058 − 0.379471i 2.06448 + 0.782953i 0.120537 − 0.992709i 1.92153 + 5.06666i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.h even 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.p.b ✓ 192
169.h even 26 1 inner 507.2.p.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.p.b ✓ 192 1.a even 1 1 trivial
507.2.p.b ✓ 192 169.h even 26 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{192} - 25 T_{2}^{190} + 373 T_{2}^{188} - 4435 T_{2}^{186} + 130 T_{2}^{185} + \cdots + 2565164201769 \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\). Copy content Toggle raw display