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Properties

Label 425.3.t.f
Level $425$
Weight $3$
Character orbit 425.t
Analytic conductor $11.580$
Analytic rank $0$
Dimension $96$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,3,Mod(24,425)] 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("425.24"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 11])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 425.t (of order \(16\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,0,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5804112353\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 24 q^{13} - 32 q^{14} + 64 q^{17} + 24 q^{19} + 48 q^{22} + 72 q^{23} + 336 q^{24} - 224 q^{26} - 64 q^{31} + 400 q^{32} - 256 q^{33} - 64 q^{34} + 192 q^{36} + 72 q^{37} + 496 q^{38} - 16 q^{39}+ \cdots + 160 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −3.39707 − 1.40711i 4.59572 + 3.07076i 6.73167 + 6.73167i 0 −11.2911 − 16.8983i 8.54917 + 1.70054i −7.76727 − 18.7519i 8.24693 + 19.9099i 0
24.2 −3.25764 − 1.34936i −2.41173 − 1.61146i 5.96305 + 5.96305i 0 5.68210 + 8.50386i −0.642506 − 0.127802i −5.98175 − 14.4412i −0.224546 − 0.542101i 0
24.3 −2.25784 − 0.935226i 0.883115 + 0.590078i 1.39475 + 1.39475i 0 −1.44207 − 2.15821i −2.53703 − 0.504646i 1.89620 + 4.57783i −3.01245 − 7.27270i 0
24.4 −1.15640 − 0.478998i −4.57983 − 3.06014i −1.72060 − 1.72060i 0 3.83032 + 5.73248i −7.36497 − 1.46498i 3.08153 + 7.43948i 8.16619 + 19.7149i 0
24.5 −0.808326 − 0.334820i 2.20904 + 1.47604i −2.28714 − 2.28714i 0 −1.29142 − 1.93275i 11.7391 + 2.33505i 2.42226 + 5.84784i −0.742961 − 1.79367i 0
24.6 −0.387390 − 0.160462i −1.40425 − 0.938292i −2.70410 − 2.70410i 0 0.393433 + 0.588815i −1.84965 − 0.367918i 1.25549 + 3.03101i −2.35262 − 5.67972i 0
24.7 −0.380849 − 0.157753i 4.65541 + 3.11065i −2.70827 − 2.70827i 0 −1.28229 − 1.91909i −12.0922 − 2.40528i 1.23521 + 2.98207i 8.55259 + 20.6478i 0
24.8 1.36492 + 0.565370i −2.90475 − 1.94089i −1.28505 − 1.28505i 0 −2.86744 − 4.29143i 7.10611 + 1.41349i −3.28895 − 7.94023i 1.22636 + 2.96069i 0
24.9 1.38844 + 0.575111i 1.85190 + 1.23740i −1.23141 − 1.23141i 0 1.85961 + 2.78310i −1.16686 − 0.232103i −3.30199 − 7.97170i −1.54578 − 3.73184i 0
24.10 2.47229 + 1.02405i −0.332126 − 0.221920i 2.23508 + 2.23508i 0 −0.593853 − 0.888764i −10.0153 − 1.99217i −0.859298 − 2.07453i −3.38309 − 8.16751i 0
24.11 2.98291 + 1.23556i 3.49096 + 2.33259i 4.54273 + 4.54273i 0 7.53119 + 11.2712i 4.81975 + 0.958708i 2.99548 + 7.23174i 3.30172 + 7.97105i 0
24.12 3.43695 + 1.42363i −2.79150 − 1.86522i 6.95747 + 6.95747i 0 −6.93887 − 10.3848i 3.45437 + 0.687118i 8.31308 + 20.0696i 0.869284 + 2.09864i 0
74.1 −1.34277 + 3.24172i 0.149309 + 0.750626i −5.87732 − 5.87732i 0 −2.63381 − 0.523897i −10.0229 + 6.69710i 13.9776 − 5.78972i 7.77377 − 3.22000i 0
74.2 −1.30638 + 3.15387i −0.996642 − 5.01046i −5.41187 − 5.41187i 0 17.1043 + 3.40226i 5.94663 − 3.97341i 11.5228 − 4.77291i −15.7965 + 6.54310i 0
74.3 −1.07895 + 2.60482i 1.06398 + 5.34901i −2.79250 − 2.79250i 0 −15.0812 − 2.99983i 4.13356 − 2.76196i −0.132340 + 0.0548169i −19.1649 + 7.93836i 0
74.4 −0.794614 + 1.91837i 0.105417 + 0.529966i −0.220301 − 0.220301i 0 −1.10044 − 0.218890i 3.05451 − 2.04096i −7.07580 + 2.93089i 8.04516 − 3.33242i 0
74.5 −0.569348 + 1.37453i −0.626196 − 3.14810i 1.26326 + 1.26326i 0 4.68368 + 0.931641i −2.12616 + 1.42066i −7.95373 + 3.29454i −1.20351 + 0.498509i 0
74.6 0.0736350 − 0.177771i 0.806529 + 4.05469i 2.80225 + 2.80225i 0 0.780194 + 0.155190i −7.08373 + 4.73320i 1.41558 − 0.586354i −7.47514 + 3.09630i 0
74.7 0.175653 − 0.424064i 0.189421 + 0.952285i 2.67945 + 2.67945i 0 0.437102 + 0.0869449i 7.74055 − 5.17207i 3.30317 − 1.36822i 7.44395 − 3.08338i 0
74.8 0.343116 − 0.828354i −0.685429 − 3.44588i 2.25998 + 2.25998i 0 −3.08959 − 0.614558i −9.31471 + 6.22389i 5.96092 − 2.46909i −3.08938 + 1.27966i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.12
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.p odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.3.t.f 96
5.b even 2 1 425.3.t.g 96
5.c odd 4 1 425.3.u.c ✓ 96
5.c odd 4 1 425.3.u.d yes 96
17.e odd 16 1 425.3.t.g 96
85.o even 16 1 425.3.u.d yes 96
85.p odd 16 1 inner 425.3.t.f 96
85.r even 16 1 425.3.u.c ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.3.t.f 96 1.a even 1 1 trivial
425.3.t.f 96 85.p odd 16 1 inner
425.3.t.g 96 5.b even 2 1
425.3.t.g 96 17.e odd 16 1
425.3.u.c ✓ 96 5.c odd 4 1
425.3.u.c ✓ 96 85.r even 16 1
425.3.u.d yes 96 5.c odd 4 1
425.3.u.d yes 96 85.o even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 80 T_{2}^{91} + 472 T_{2}^{90} + 640 T_{2}^{89} + 104176 T_{2}^{88} + 16504 T_{2}^{87} + \cdots + 10\!\cdots\!61 \) acting on \(S_{3}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display