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Properties

Label 409.3.p.a
Level $409$
Weight $3$
Character orbit 409.p
Analytic conductor $11.144$
Analytic rank $0$
Dimension $8576$
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(21,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(408)) chi = DirichletCharacter(H, H._module([1])) 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("409.21"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 409 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 409.p (of order \(408\), degree \(128\), 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: \(8576\)
Relative dimension: \(67\) over \(\Q(\zeta_{408})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{408}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 8576 q - 124 q^{2} - 112 q^{3} - 124 q^{4} - 120 q^{5} - 120 q^{6} - 144 q^{7} + 8 q^{8} - 292 q^{9} - 96 q^{10} - 160 q^{11} + 36 q^{12} - 164 q^{13} - 16 q^{14} - 80 q^{15} - 204 q^{16} - 148 q^{17} - 148 q^{18}+ \cdots + 1180 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −3.14834 − 2.30225i 3.28582 + 2.56174i 3.39906 + 10.6847i −1.47962 − 0.276588i −4.44712 − 15.6300i 2.60078 − 1.99565i 8.93960 − 26.6722i 2.03891 + 8.10667i 4.02156 + 4.27724i
21.2 −3.09961 − 2.26662i −0.127755 − 0.0996022i 3.25741 + 10.2395i −7.59618 − 1.41997i 0.170231 + 0.598300i 1.43711 − 1.10273i 8.23105 − 24.5581i −2.18882 − 8.70272i 20.3266 + 21.6190i
21.3 −2.99391 − 2.18932i −0.00398819 − 0.00310933i 2.95775 + 9.29750i 9.08269 + 1.69785i 0.00513296 + 0.0180405i 7.60459 − 5.83521i 6.78526 − 20.2445i −2.19522 − 8.72815i −23.4756 − 24.9681i
21.4 −2.97290 − 2.17396i −1.71585 − 1.33773i 2.89942 + 9.11414i 3.00382 + 0.561512i 2.19287 + 7.70713i −8.04167 + 6.17059i 6.51243 − 19.4305i −1.04062 − 4.13748i −7.70936 − 8.19950i
21.5 −2.95014 − 2.15731i −3.38474 − 2.63886i 2.83669 + 8.91697i 0.831514 + 0.155437i 4.29261 + 15.0869i −4.35489 + 3.34163i 6.22226 − 18.5647i 2.29768 + 9.13554i −2.11775 − 2.25240i
21.6 −2.76952 − 2.02524i 1.19744 + 0.933567i 2.35606 + 7.40613i −0.653831 − 0.122222i −1.42565 − 5.01064i −4.20606 + 3.22742i 4.11263 − 12.2704i −1.63290 − 6.49239i 1.56327 + 1.66266i
21.7 −2.68967 − 1.96684i −2.96732 − 2.31342i 2.15322 + 6.76852i 3.99314 + 0.746448i 3.43096 + 12.0586i 2.64255 − 2.02770i 3.28554 − 9.80272i 1.25784 + 5.00114i −9.27208 − 9.86158i
21.8 −2.68542 − 1.96373i 4.21566 + 3.28667i 2.14260 + 6.73512i 1.36686 + 0.255510i −4.86665 − 17.1045i 0.936382 − 0.718511i 3.24329 − 9.67665i 4.77435 + 18.9828i −3.16882 − 3.37029i
21.9 −2.58268 − 1.88861i 1.62520 + 1.26706i 1.89079 + 5.94357i 3.85111 + 0.719898i −1.80439 − 6.34179i 5.56108 − 4.26716i 2.27463 − 6.78657i −1.15939 − 4.60970i −8.58659 − 9.13250i
21.10 −2.55140 − 1.86573i −4.16183 − 3.24470i 1.81607 + 5.70871i −7.26226 − 1.35755i 4.56475 + 16.0434i 0.255082 − 0.195731i 1.99952 − 5.96576i 4.59751 + 18.2796i 15.9961 + 17.0131i
21.11 −2.40558 − 1.75910i −0.841559 − 0.656107i 1.47977 + 4.65157i −4.57291 − 0.854825i 0.870278 + 3.05871i 5.95986 − 4.57316i 0.834620 − 2.49017i −1.91748 − 7.62387i 9.49679 + 10.1006i
21.12 −2.34533 − 1.71505i 2.89471 + 2.25681i 1.34660 + 4.23295i −6.42035 − 1.20017i −2.91852 − 10.2575i −9.12845 + 7.00451i 0.408099 − 1.21760i 1.09092 + 4.33747i 12.9995 + 13.8260i
21.13 −2.11486 − 1.54651i −4.28019 − 3.33698i 0.868321 + 2.72951i 7.37549 + 1.37872i 3.89132 + 13.6766i 1.03165 − 0.791617i −0.945591 + 2.82126i 4.98936 + 19.8376i −13.4659 − 14.3220i
21.14 −2.01731 − 1.47517i 4.06090 + 3.16601i 0.680781 + 2.13999i 9.09221 + 1.69963i −3.52166 − 12.3774i −2.63166 + 2.01934i −1.39330 + 4.15703i 4.27204 + 16.9856i −15.8345 − 16.8413i
21.15 −2.00582 − 1.46677i 1.28816 + 1.00430i 0.659278 + 2.07240i 4.28592 + 0.801176i −1.11075 − 3.90388i −4.78569 + 3.67219i −1.44137 + 4.30048i −1.54447 − 6.14077i −7.42163 − 7.89348i
21.16 −1.97622 − 1.44513i 3.65349 + 2.84838i 0.604430 + 1.89999i −9.18551 − 1.71707i −3.10381 − 10.9088i 7.41791 − 5.69196i −1.56086 + 4.65699i 3.03947 + 12.0849i 15.6712 + 16.6675i
21.17 −1.93068 − 1.41183i −0.650907 − 0.507469i 0.521667 + 1.63983i 7.25030 + 1.35532i 0.540236 + 1.89873i −5.16938 + 3.96660i −1.73242 + 5.16882i −2.02907 − 8.06754i −12.0846 − 12.8529i
21.18 −1.87616 − 1.37196i −2.27472 − 1.77344i 0.425102 + 1.33628i −5.80963 − 1.08601i 1.83464 + 6.44810i −7.01076 + 5.37954i −1.91878 + 5.72485i −0.165999 − 0.660008i 9.40985 + 10.0081i
21.19 −1.80222 − 1.31789i −2.33941 − 1.82388i 0.298547 + 0.938465i 0.349559 + 0.0653439i 1.81245 + 6.37011i 9.64109 − 7.39787i −2.13935 + 6.38294i −0.0489291 − 0.194541i −0.543865 − 0.578443i
21.20 −1.59279 − 1.16475i −2.07047 − 1.61421i −0.0322475 − 0.101368i −2.55276 − 0.477193i 1.41770 + 4.98268i −6.76777 + 5.19309i −2.57500 + 7.68276i −0.514037 − 2.04380i 3.51021 + 3.73338i
See next 80 embeddings (of 8576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.67
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
409.p odd 408 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 409.3.p.a ✓ 8576
409.p odd 408 1 inner 409.3.p.a ✓ 8576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
409.3.p.a ✓ 8576 1.a even 1 1 trivial
409.3.p.a ✓ 8576 409.p odd 408 1 inner

Hecke kernels

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