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Properties

Label 875.2.q.a
Level $875$
Weight $2$
Character orbit 875.q
Analytic conductor $6.987$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [875,2,Mod(51,875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(875, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([24, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("875.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.q (of order \(15\), degree \(8\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 3 q^{2} + 3 q^{3} + 13 q^{4} - 12 q^{6} + 22 q^{7} + 2 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 3 q^{2} + 3 q^{3} + 13 q^{4} - 12 q^{6} + 22 q^{7} + 2 q^{8} + 11 q^{9} - 6 q^{11} + 11 q^{12} + 12 q^{13} - 6 q^{14} + 13 q^{16} - 9 q^{17} + 18 q^{18} - 11 q^{19} - 3 q^{21} + 52 q^{22} + 17 q^{23} + 46 q^{24} + 44 q^{26} + 84 q^{27} - 62 q^{28} - 24 q^{29} - 21 q^{31} + 16 q^{32} + 18 q^{33} - 36 q^{34} - 104 q^{36} + 5 q^{37} + 12 q^{38} + 25 q^{39} + 38 q^{41} + 58 q^{42} - 20 q^{43} - 7 q^{44} + 21 q^{46} + q^{47} + 12 q^{48} - 38 q^{49} - 8 q^{51} - 50 q^{52} - 37 q^{53} + 15 q^{54} - 60 q^{56} - 136 q^{57} - 53 q^{58} - 39 q^{59} - 13 q^{61} - 124 q^{62} - 75 q^{63} + 42 q^{64} + 7 q^{66} + 13 q^{67} + 110 q^{68} + 50 q^{69} + 22 q^{71} + 18 q^{72} + 41 q^{73} - 10 q^{74} - 276 q^{76} - 37 q^{77} - 2 q^{78} + 9 q^{79} + 57 q^{81} + 108 q^{82} - 86 q^{83} - 29 q^{84} - 17 q^{86} + 7 q^{87} + 26 q^{88} - 42 q^{89} - 34 q^{91} + 62 q^{92} - 98 q^{93} - 11 q^{94} + 13 q^{96} - 96 q^{97} + 86 q^{98} - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −0.259568 − 2.46962i −1.51910 + 0.322895i −4.07537 + 0.866247i 0 1.19174 + 3.66780i −1.63141 − 2.08291i 1.66242 + 5.11641i −0.537226 + 0.239189i 0
51.2 −0.258406 − 2.45857i 1.04203 − 0.221490i −4.02149 + 0.854794i 0 −0.813815 − 2.50467i 2.33148 + 1.25068i 1.61290 + 4.96398i −1.70387 + 0.758611i 0
51.3 −0.247315 − 2.35304i 3.06662 − 0.651830i −3.51935 + 0.748061i 0 −2.29220 − 7.05468i 0.0714197 − 2.64479i 1.16834 + 3.59577i 6.23862 − 2.77761i 0
51.4 −0.222354 − 2.11556i −2.27352 + 0.483252i −2.46986 + 0.524984i 0 1.52788 + 4.70232i 2.64421 + 0.0903750i 0.345128 + 1.06219i 2.19474 − 0.977159i 0
51.5 −0.160768 − 1.52961i 1.30629 − 0.277660i −0.357556 + 0.0760009i 0 −0.634720 − 1.95347i −2.64452 + 0.0807455i −0.776821 − 2.39081i −1.11134 + 0.494802i 0
51.6 −0.144326 − 1.37317i −0.825232 + 0.175409i 0.0915245 − 0.0194541i 0 0.359969 + 1.10787i −2.36555 + 1.18498i −0.893265 − 2.74919i −2.09040 + 0.930704i 0
51.7 −0.0976327 − 0.928913i −1.90954 + 0.405885i 1.10295 − 0.234439i 0 0.563466 + 1.73417i 0.508729 + 2.59638i −0.902719 − 2.77828i 0.740961 − 0.329897i 0
51.8 −0.0774861 − 0.737231i 1.45155 − 0.308536i 1.41879 − 0.301573i 0 −0.339937 − 1.04622i 1.08929 − 2.41111i −0.790409 − 2.43263i −0.728840 + 0.324501i 0
51.9 −0.0490101 − 0.466300i 2.53014 − 0.537798i 1.74126 − 0.370117i 0 −0.374778 − 1.15345i 1.07014 + 2.41967i −0.547701 − 1.68565i 3.37176 − 1.50120i 0
51.10 0.0216612 + 0.206093i −0.890484 + 0.189278i 1.91429 − 0.406895i 0 −0.0582978 − 0.179422i 2.20881 − 1.45642i 0.253398 + 0.779878i −1.98350 + 0.883111i 0
51.11 0.0605794 + 0.576375i −0.335766 + 0.0713692i 1.62776 − 0.345990i 0 −0.0614760 − 0.189204i −1.74489 − 1.98881i 0.656210 + 2.01961i −2.63299 + 1.17228i 0
51.12 0.0765496 + 0.728321i 2.20730 − 0.469177i 1.43170 − 0.304318i 0 0.510679 + 1.57171i −1.21370 + 2.35094i 0.783844 + 2.41242i 1.91142 − 0.851019i 0
51.13 0.109431 + 1.04116i −1.78822 + 0.380098i 0.884249 − 0.187953i 0 −0.591430 − 1.82023i 2.07580 + 1.64044i 0.939473 + 2.89140i 0.312618 − 0.139187i 0
51.14 0.157297 + 1.49658i 2.99129 − 0.635819i −0.258728 + 0.0549943i 0 1.42208 + 4.37671i −0.781664 − 2.52765i 0.807034 + 2.48380i 5.80292 − 2.58363i 0
51.15 0.175660 + 1.67130i −2.41719 + 0.513790i −0.806078 + 0.171337i 0 −1.28330 − 3.94959i −2.14197 − 1.55304i 0.610656 + 1.87941i 2.83820 − 1.26365i 0
51.16 0.213815 + 2.03432i −0.146584 + 0.0311574i −2.13643 + 0.454112i 0 −0.0947260 − 0.291537i −1.42653 + 2.22823i −0.116408 − 0.358267i −2.72012 + 1.21108i 0
51.17 0.253171 + 2.40877i 0.706586 − 0.150189i −3.78176 + 0.803838i 0 0.540658 + 1.66398i 2.59815 + 0.499637i −1.39679 − 4.29888i −2.26393 + 1.00797i 0
51.18 0.279570 + 2.65993i −2.59163 + 0.550869i −5.04079 + 1.07145i 0 −2.18982 − 6.73956i 1.54320 − 2.14908i −2.60627 − 8.02126i 3.67247 − 1.63509i 0
151.1 −1.77578 + 1.97221i −0.0898168 − 0.854549i −0.527138 − 5.01538i 0 1.84484 + 1.34036i −0.281635 − 2.63072i 6.53340 + 4.74679i 2.21226 − 0.470229i 0
151.2 −1.57980 + 1.75455i −0.239157 − 2.27542i −0.373607 − 3.55463i 0 4.37016 + 3.17511i −0.862134 + 2.50134i 3.00686 + 2.18461i −2.18591 + 0.464630i 0
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.d even 5 1 inner
175.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 875.2.q.a 144
5.b even 2 1 175.2.q.a ✓ 144
5.c odd 4 2 875.2.u.b 288
7.c even 3 1 inner 875.2.q.a 144
25.d even 5 1 inner 875.2.q.a 144
25.e even 10 1 175.2.q.a ✓ 144
25.f odd 20 2 875.2.u.b 288
35.j even 6 1 175.2.q.a ✓ 144
35.l odd 12 2 875.2.u.b 288
175.q even 15 1 inner 875.2.q.a 144
175.t even 30 1 175.2.q.a ✓ 144
175.w odd 60 2 875.2.u.b 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.q.a ✓ 144 5.b even 2 1
175.2.q.a ✓ 144 25.e even 10 1
175.2.q.a ✓ 144 35.j even 6 1
175.2.q.a ✓ 144 175.t even 30 1
875.2.q.a 144 1.a even 1 1 trivial
875.2.q.a 144 7.c even 3 1 inner
875.2.q.a 144 25.d even 5 1 inner
875.2.q.a 144 175.q even 15 1 inner
875.2.u.b 288 5.c odd 4 2
875.2.u.b 288 25.f odd 20 2
875.2.u.b 288 35.l odd 12 2
875.2.u.b 288 175.w odd 60 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} - 3 T_{2}^{143} - 20 T_{2}^{142} + 63 T_{2}^{141} + 145 T_{2}^{140} - 459 T_{2}^{139} + \cdots + 5907237891361 \) acting on \(S_{2}^{\mathrm{new}}(875, [\chi])\). Copy content Toggle raw display