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Properties

Label 371.2.t.a
Level $371$
Weight $2$
Character orbit 371.t
Analytic conductor $2.962$
Analytic rank $0$
Dimension $816$
Inner twists $4$

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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] = [371,2,Mod(4,371)] 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("371.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(78)) chi = DirichletCharacter(H, H._module([52, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 371.t (of order \(78\), degree \(24\), 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: \(2.96244991499\)
Analytic rank: \(0\)
Dimension: \(816\)
Relative dimension: \(34\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 816 q - 13 q^{2} - 13 q^{3} - 45 q^{4} - 13 q^{5} - 28 q^{6} - 26 q^{7} - 41 q^{9} - 3 q^{10} - 11 q^{11} - 13 q^{12} - 52 q^{13} - 26 q^{14} - 28 q^{15} + 7 q^{16} + 18 q^{17} - 13 q^{18} - 13 q^{19} - 52 q^{20}+ \cdots + 196 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.08865 − 2.55516i −1.72920 − 2.73452i −3.95821 + 4.12093i −1.11785 − 0.530425i −5.10462 + 7.39532i 0.729863 − 2.54309i 9.64489 + 3.65782i −3.20136 + 6.74673i −0.138374 + 3.43372i
4.2 −1.03256 − 2.42351i 1.01761 + 1.60923i −3.42177 + 3.56244i −0.439425 − 0.208510i 2.84923 − 4.12783i −2.52032 − 0.804985i 7.24057 + 2.74599i −0.267999 + 0.564797i −0.0515924 + 1.28025i
4.3 −0.976818 − 2.29268i −0.557412 − 0.881477i −2.91674 + 3.03665i 2.26215 + 1.07340i −1.47645 + 2.13901i 1.27785 + 2.31670i 5.15089 + 1.95348i 0.819785 − 1.72766i 0.251258 − 6.23489i
4.4 −0.975507 − 2.28960i 1.05773 + 1.67267i −2.90520 + 3.02464i −2.44004 − 1.15782i 2.79792 − 4.05349i 2.31200 + 1.28634i 5.10521 + 1.93615i −0.392956 + 0.828138i −0.270653 + 6.71618i
4.5 −0.838788 − 1.96871i −0.583820 − 0.923237i −1.78680 + 1.86025i 1.79422 + 0.851367i −1.32788 + 1.92377i −2.51532 − 0.820470i 1.15927 + 0.439654i 0.774556 − 1.63234i 0.171124 − 4.24641i
4.6 −0.825964 − 1.93861i −0.670689 − 1.06061i −1.69054 + 1.76004i −3.61008 − 1.71301i −1.50214 + 2.17623i −1.76787 + 1.96841i 0.867756 + 0.329097i 0.611009 − 1.28767i −0.339049 + 8.41342i
4.7 −0.813029 − 1.90825i 1.70105 + 2.69000i −1.59495 + 1.66052i 2.75517 + 1.30734i 3.75018 − 5.43307i 1.87143 − 1.87023i 0.586548 + 0.222448i −3.05643 + 6.44129i 0.254706 − 6.32046i
4.8 −0.721591 − 1.69364i −0.531237 − 0.840084i −0.962259 + 1.00182i −2.10544 − 0.999043i −1.03946 + 1.50592i 0.284351 − 2.63043i −1.05156 − 0.398805i 0.862549 − 1.81778i −0.172750 + 4.28675i
4.9 −0.651166 − 1.52834i 0.966499 + 1.52840i −0.526364 + 0.548002i 2.48980 + 1.18142i 1.70656 − 2.47238i −0.0294282 + 2.64559i −1.92636 − 0.730572i −0.115797 + 0.244037i 0.184349 − 4.57456i
4.10 −0.542824 − 1.27406i −1.73237 − 2.73952i 0.0568888 − 0.0592275i −0.378295 − 0.179503i −2.54993 + 3.69421i 0.556357 + 2.58659i −2.69610 − 1.02250i −3.21780 + 6.78137i −0.0233493 + 0.579407i
4.11 −0.535824 − 1.25763i −1.30830 − 2.06891i 0.0909347 − 0.0946730i 3.52229 + 1.67135i −1.90090 + 2.75393i 1.74152 − 1.99176i −2.72415 − 1.03313i −1.28267 + 2.70317i 0.214601 − 5.32527i
4.12 −0.515655 − 1.21029i 0.619229 + 0.979232i 0.186557 − 0.194226i −0.231070 − 0.109644i 0.865842 − 1.25439i 0.364558 − 2.62051i −2.79140 − 1.05864i 0.710627 − 1.49761i −0.0135484 + 0.336199i
4.13 −0.377123 − 0.885139i 1.37816 + 2.17939i 0.744198 − 0.774793i −3.43263 − 1.62880i 1.40933 − 2.04177i 2.50937 + 0.838493i −2.76567 − 1.04888i −1.56434 + 3.29677i −0.147195 + 3.65262i
4.14 −0.372312 − 0.873848i −0.147487 − 0.233233i 0.760454 − 0.791716i −0.208612 − 0.0989878i −0.148899 + 0.215717i 2.37041 + 1.17522i −2.75123 − 1.04340i 1.25343 − 2.64155i −0.00883144 + 0.219150i
4.15 −0.220268 − 0.516987i −1.27399 − 2.01466i 1.16669 − 1.21465i −0.0470833 − 0.0223413i −0.760934 + 1.10240i −2.60234 − 0.477312i −1.93582 − 0.734159i −1.14972 + 2.42297i −0.00117924 + 0.0292625i
4.16 −0.126644 − 0.297245i −0.465794 − 0.736595i 1.31313 − 1.36712i 0.779362 + 0.369812i −0.159959 + 0.231741i −2.14881 + 1.54357i −1.17688 − 0.446331i 0.960470 − 2.02415i 0.0112230 − 0.278497i
4.17 −0.0504332 − 0.118371i 0.605516 + 0.957547i 1.37398 − 1.43047i −3.56359 − 1.69095i 0.0828077 − 0.119968i −2.55269 − 0.695524i −0.479231 − 0.181748i 0.735831 − 1.55073i −0.0204357 + 0.507106i
4.18 −0.0195009 − 0.0457703i 1.64334 + 2.59873i 1.38373 − 1.44062i 0.502287 + 0.238338i 0.0868982 − 0.125894i −1.26893 + 2.32160i −0.185959 − 0.0705248i −2.76677 + 5.83084i 0.00111376 − 0.0276377i
4.19 −0.0190846 − 0.0447933i 0.285943 + 0.452183i 1.38381 − 1.44070i 3.21224 + 1.52423i 0.0147976 − 0.0214381i 2.64435 − 0.0861861i −0.181994 − 0.0690211i 1.16337 − 2.45175i 0.00697070 − 0.172976i
4.20 0.0385501 + 0.0904803i 1.23659 + 1.95552i 1.37875 − 1.43543i 1.28372 + 0.609131i −0.129265 + 0.187273i −1.16875 − 2.37361i 0.366947 + 0.139165i −1.00881 + 2.12602i −0.00562702 + 0.139633i
See next 80 embeddings (of 816 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 4.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
53.e even 26 1 inner
371.t even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 371.2.t.a ✓ 816
7.c even 3 1 inner 371.2.t.a ✓ 816
53.e even 26 1 inner 371.2.t.a ✓ 816
371.t even 78 1 inner 371.2.t.a ✓ 816
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
371.2.t.a ✓ 816 1.a even 1 1 trivial
371.2.t.a ✓ 816 7.c even 3 1 inner
371.2.t.a ✓ 816 53.e even 26 1 inner
371.2.t.a ✓ 816 371.t even 78 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(371, [\chi])\).