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Properties

Label 409.3.n.a
Level $409$
Weight $3$
Character orbit 409.n
Analytic conductor $11.144$
Analytic rank $0$
Dimension $4352$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [409,3,Mod(11,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(136)) chi = DirichletCharacter(H, H._module([63])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("409.11"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 409 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 409.n (of order \(136\), degree \(64\), 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: \(11.1444428123\)
Analytic rank: \(0\)
Dimension: \(4352\)
Relative dimension: \(68\) over \(\Q(\zeta_{136})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{136}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 4352 q - 68 q^{2} - 56 q^{3} - 68 q^{4} - 60 q^{5} - 60 q^{6} - 48 q^{7} - 20 q^{8} - 68 q^{9} - 48 q^{10} - 80 q^{11} - 36 q^{12} - 28 q^{13} - 80 q^{14} - 4 q^{15} + 172 q^{16} - 92 q^{17} - 188 q^{18}+ \cdots - 220 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −3.94845 + 0.182548i −0.588736 + 0.327924i 11.5740 − 1.07249i −7.47952 − 2.89758i 2.26473 − 1.40226i −4.85758 + 11.7272i −29.8444 + 4.16312i −4.49881 + 7.26583i 30.0614 + 10.0756i
11.2 −3.85024 + 0.178008i −4.24324 + 2.36347i 10.8098 − 1.00167i 6.19198 + 2.39879i 15.9168 − 9.85527i −0.445733 + 1.07609i −26.1723 + 3.65088i 7.68123 − 12.4056i −24.2676 − 8.13369i
11.3 −3.75130 + 0.173433i 2.33784 − 1.30217i 10.0592 − 0.932125i 2.10260 + 0.814551i −8.54408 + 5.29027i −0.699538 + 1.68883i −22.6963 + 3.16600i −0.968047 + 1.56345i −8.02875 − 2.69096i
11.4 −3.64488 + 0.168513i 4.77679 − 2.66065i 9.27385 − 0.859349i −6.67381 − 2.58545i −16.9625 + 10.5027i 2.86289 − 6.91162i −19.2022 + 2.67859i 11.0007 − 17.7668i 24.7609 + 8.29903i
11.5 −3.57541 + 0.165301i −0.716795 + 0.399252i 8.77330 − 0.812966i 4.52848 + 1.75434i 2.49684 − 1.54598i 3.75575 − 9.06718i −17.0541 + 2.37895i −4.38350 + 7.07959i −16.4812 − 5.52394i
11.6 −3.39470 + 0.156946i −1.03242 + 0.575055i 7.51639 − 0.696496i −5.85749 − 2.26920i 3.41450 − 2.11417i 4.76659 − 11.5076i −11.9436 + 1.66606i −4.00268 + 6.46456i 20.2405 + 6.78394i
11.7 −3.37181 + 0.155888i 2.85075 − 1.58786i 7.36183 − 0.682174i 0.435768 + 0.168817i −9.36463 + 5.79834i 0.148229 − 0.357856i −11.3442 + 1.58244i 0.867579 − 1.40119i −1.49564 − 0.501288i
11.8 −3.25420 + 0.150451i −4.88051 + 2.71842i 6.58422 − 0.610118i −5.60785 − 2.17249i 15.4731 − 9.58056i 1.08129 − 2.61047i −8.42884 + 1.17577i 11.6916 − 18.8826i 18.5759 + 6.22602i
11.9 −3.15813 + 0.146009i −1.71713 + 0.956433i 5.96951 − 0.553156i −2.82066 − 1.09273i 5.28325 − 3.27125i −0.201059 + 0.485398i −6.24697 + 0.871415i −2.70413 + 4.36732i 9.06755 + 3.03914i
11.10 −3.14914 + 0.145594i 5.12161 − 2.85272i 5.91297 − 0.547917i 6.64265 + 2.57338i −15.7134 + 9.72930i −3.99096 + 9.63504i −6.05191 + 0.844205i 13.3550 − 21.5691i −21.2933 − 7.13680i
11.11 −3.13998 + 0.145170i −3.57707 + 1.99241i 5.85549 − 0.542591i 0.113194 + 0.0438515i 10.9427 − 6.77543i −2.02453 + 4.88764i −5.85459 + 0.816681i 4.08781 − 6.60203i −0.361792 − 0.121261i
11.12 −3.12914 + 0.144669i −1.35978 + 0.757392i 5.78765 − 0.536305i 7.32532 + 2.83785i 4.14537 − 2.56670i −3.92800 + 9.48302i −5.62300 + 0.784375i −3.46253 + 5.59218i −23.3325 − 7.82027i
11.13 −2.70393 + 0.125010i 3.05013 − 1.69891i 3.31269 − 0.306966i 7.96146 + 3.08429i −8.03497 + 4.97504i 3.44387 − 8.31423i 1.80454 − 0.251723i 1.67911 − 2.71185i −21.9128 − 7.34444i
11.14 −2.58312 + 0.119425i 0.980302 − 0.546025i 2.67532 − 0.247905i −1.99403 − 0.772493i −2.46703 + 1.52752i −2.02538 + 4.88969i 3.36326 − 0.469155i −4.07504 + 6.58142i 5.24309 + 1.75731i
11.15 −2.53276 + 0.117096i 3.83884 − 2.13822i 2.41820 − 0.224080i −6.59902 − 2.55647i −9.47246 + 5.86510i −4.19675 + 10.1319i 3.94611 − 0.550458i 5.42681 − 8.76461i 17.0131 + 5.70220i
11.16 −2.48980 + 0.115110i 0.859768 − 0.478888i 2.20292 − 0.204131i −8.70688 − 3.37306i −2.08553 + 1.29130i 1.04203 − 2.51568i 4.41290 − 0.615573i −4.22802 + 6.82849i 22.0667 + 7.39600i
11.17 −2.22478 + 0.102858i −3.77370 + 2.10194i 0.956140 − 0.0885994i 5.25614 + 2.03624i 8.17946 − 5.06451i 3.25482 − 7.85783i 6.70511 − 0.935324i 5.08479 − 8.21221i −11.9032 − 3.98956i
11.18 −2.15703 + 0.0997256i 4.01874 − 2.23842i 0.659908 − 0.0611495i −1.58675 − 0.614710i −8.44532 + 5.22912i 5.07836 − 12.2603i 7.13717 − 0.995593i 6.40182 − 10.3393i 3.48397 + 1.16771i
11.19 −2.09918 + 0.0970509i 2.68089 − 1.49324i 0.414200 − 0.0383813i 0.420051 + 0.162728i −5.48274 + 3.39477i 1.55324 − 3.74985i 7.45933 − 1.04053i 0.219488 − 0.354485i −0.897555 − 0.300830i
11.20 −2.04001 + 0.0943154i −0.934413 + 0.520464i 0.169815 − 0.0157357i 4.53708 + 1.75767i 1.85712 − 1.14988i 0.0302228 − 0.0729644i 7.74549 − 1.08045i −4.13565 + 6.67930i −9.42147 − 3.15776i
See next 80 embeddings (of 4352 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.68
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
409.n odd 136 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 409.3.n.a ✓ 4352
409.n odd 136 1 inner 409.3.n.a ✓ 4352
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
409.3.n.a ✓ 4352 1.a even 1 1 trivial
409.3.n.a ✓ 4352 409.n odd 136 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(409, [\chi])\).