Properties

Label 931.2.u.a
Level $931$
Weight $2$
Character orbit 931.u
Analytic conductor $7.434$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(134,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.134");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.u (of order \(7\), degree \(6\), 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 3 q^{2} - 37 q^{4} - 2 q^{5} + 8 q^{6} + 9 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 3 q^{2} - 37 q^{4} - 2 q^{5} + 8 q^{6} + 9 q^{8} - 22 q^{9} - 4 q^{10} + 51 q^{12} + 31 q^{14} + 18 q^{15} - 39 q^{16} - 19 q^{17} - 42 q^{18} - 240 q^{19} - 14 q^{20} - 6 q^{21} + 2 q^{22} + 31 q^{23} + 2 q^{24} - 28 q^{25} + 2 q^{26} + 21 q^{27} + 46 q^{28} + 52 q^{29} - 60 q^{30} - 8 q^{31} - 64 q^{32} - 22 q^{33} + 64 q^{34} - 2 q^{35} + 13 q^{36} - 37 q^{37} - 3 q^{38} + 16 q^{39} + 102 q^{40} - 26 q^{41} + 29 q^{42} + 2 q^{43} + 66 q^{44} + 7 q^{45} + 60 q^{46} + 44 q^{47} + 16 q^{48} - 32 q^{49} + 8 q^{50} + 36 q^{51} - 36 q^{52} + 67 q^{53} + 30 q^{54} + 76 q^{55} - 6 q^{56} + 58 q^{58} + 2 q^{59} - 116 q^{60} - 2 q^{61} - 38 q^{62} - 210 q^{63} - 51 q^{64} - 58 q^{65} + 70 q^{66} - 60 q^{67} - 176 q^{68} - 83 q^{69} - 27 q^{70} + 7 q^{71} + 43 q^{72} + 5 q^{73} - 14 q^{74} + 24 q^{75} + 37 q^{76} + 13 q^{77} + 4 q^{78} - 48 q^{79} + 58 q^{80} - 10 q^{81} + 70 q^{82} + 31 q^{83} - 8 q^{84} + 28 q^{85} - 15 q^{86} - 24 q^{87} + 70 q^{88} + 23 q^{89} - 31 q^{90} + 116 q^{91} + 31 q^{92} + 81 q^{93} + 19 q^{94} + 2 q^{95} + 38 q^{96} + 56 q^{97} + 23 q^{98} - 164 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1 −0.621156 2.72146i 0.938242 + 1.17652i −5.21859 + 2.51314i −0.812455 1.01879i 2.61906 3.28419i −2.54602 0.719575i 6.60010 + 8.27626i 0.163665 0.717065i −2.26793 + 2.84389i
134.2 −0.581996 2.54989i −1.65819 2.07931i −4.36129 + 2.10028i −0.0437147 0.0548165i −4.33694 + 5.43835i 1.78836 + 1.94981i 4.63232 + 5.80874i −0.906351 + 3.97098i −0.114334 + 0.143371i
134.3 −0.529571 2.32020i −1.19064 1.49302i −3.30095 + 1.58965i 0.356947 + 0.447597i −2.83357 + 3.55319i 1.46934 2.20024i 2.46875 + 3.09571i −0.143913 + 0.630525i 0.849486 1.06522i
134.4 −0.508395 2.22742i 0.725631 + 0.909913i −2.90101 + 1.39705i 1.38146 + 1.73229i 1.65785 2.07888i −1.57351 2.12698i 1.73771 + 2.17902i 0.366162 1.60426i 3.15622 3.95778i
134.5 −0.503302 2.20511i −0.283663 0.355702i −2.80727 + 1.35191i −1.98407 2.48794i −0.641595 + 0.804534i −2.09834 + 1.61151i 1.57357 + 1.97319i 0.621503 2.72298i −4.48761 + 5.62728i
134.6 −0.494193 2.16520i 1.80659 + 2.26540i −2.64192 + 1.27228i 0.467889 + 0.586714i 4.01223 5.03118i 2.41463 1.08146i 1.29097 + 1.61883i −1.20068 + 5.26053i 1.03912 1.30302i
134.7 −0.388681 1.70292i 0.250559 + 0.314191i −0.946929 + 0.456017i −1.61480 2.02489i 0.437655 0.548802i 0.187958 2.63907i −1.03350 1.29597i 0.631627 2.76734i −2.82059 + 3.53690i
134.8 −0.383603 1.68067i 1.60746 + 2.01570i −0.875573 + 0.421654i 1.78237 + 2.23502i 2.77110 3.47485i −2.03592 + 1.68969i −1.10513 1.38579i −0.811528 + 3.55554i 3.07261 3.85293i
134.9 −0.359451 1.57486i −0.0302877 0.0379796i −0.549034 + 0.264401i 2.58425 + 3.24054i −0.0489255 + 0.0613506i 2.61644 + 0.392732i −1.40057 1.75626i 0.667038 2.92248i 4.17448 5.23464i
134.10 −0.351113 1.53832i −1.54116 1.93255i −0.441225 + 0.212483i 2.19402 + 2.75122i −2.43177 + 3.04934i −0.964130 + 2.46383i −1.48580 1.86314i −0.692023 + 3.03195i 3.46192 4.34111i
134.11 −0.343403 1.50455i 1.31135 + 1.64438i −0.343796 + 0.165563i −1.06709 1.33809i 2.02372 2.53767i 2.13280 + 1.56562i −1.55723 1.95270i −0.316786 + 1.38793i −1.64678 + 2.06500i
134.12 −0.330941 1.44995i −1.08625 1.36212i −0.190886 + 0.0919259i −0.922678 1.15700i −1.61551 + 2.02579i −2.45589 + 0.984181i −1.65809 2.07918i −0.00785776 + 0.0344271i −1.37224 + 1.72073i
134.13 −0.210709 0.923178i −0.943131 1.18265i 0.994078 0.478723i 1.07640 + 1.34977i −0.893069 + 1.11987i −0.961706 2.46478i −1.83220 2.29750i 0.158400 0.693997i 1.01927 1.27812i
134.14 −0.188703 0.826764i 1.48823 + 1.86618i 1.15401 0.555741i −1.92466 2.41344i 1.26206 1.58257i −1.81896 1.92130i −1.73470 2.17525i −0.600238 + 2.62981i −1.63216 + 2.04666i
134.15 −0.132098 0.578759i −0.511059 0.640848i 1.48443 0.714862i 1.37186 + 1.72026i −0.303387 + 0.380435i 2.64006 + 0.173462i −1.35008 1.69295i 0.518058 2.26976i 0.814397 1.02122i
134.16 −0.131589 0.576531i −2.01272 2.52388i 1.48687 0.716037i −1.52326 1.91011i −1.19024 + 1.49251i 1.15626 2.37972i −1.34588 1.68768i −1.65133 + 7.23494i −0.900792 + 1.12956i
134.17 −0.123648 0.541737i −0.728257 0.913205i 1.52375 0.733798i −1.54879 1.94212i −0.404670 + 0.507440i 0.864780 + 2.50043i −1.27884 1.60362i 0.363977 1.59469i −0.860614 + 1.07918i
134.18 −0.103597 0.453889i 0.569314 + 0.713897i 1.60665 0.773724i 0.915572 + 1.14809i 0.265051 0.332363i −2.48325 0.912949i −1.09818 1.37707i 0.482032 2.11192i 0.426255 0.534507i
134.19 −0.0539220 0.236248i 1.42145 + 1.78245i 1.74903 0.842290i 1.01077 + 1.26746i 0.344452 0.431929i 2.25382 1.38575i −0.595473 0.746699i −0.489023 + 2.14255i 0.244933 0.307136i
134.20 −0.0169383 0.0742118i 2.00667 + 2.51628i 1.79672 0.865253i −1.00398 1.25895i 0.152748 0.191540i −1.08518 + 2.41296i −0.189566 0.237708i −1.63739 + 7.17388i −0.0764234 + 0.0958319i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 134.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.u.a 240
49.e even 7 1 inner 931.2.u.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.u.a 240 1.a even 1 1 trivial
931.2.u.a 240 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{240} - 3 T_{2}^{239} + 63 T_{2}^{238} - 187 T_{2}^{237} + 2144 T_{2}^{236} - 6289 T_{2}^{235} + 52257 T_{2}^{234} - 150606 T_{2}^{233} + 1019210 T_{2}^{232} - 2872776 T_{2}^{231} + 16873139 T_{2}^{230} + \cdots + 12\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\). Copy content Toggle raw display