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Properties

Label 959.2.e.b
Level $959$
Weight $2$
Character orbit 959.e
Analytic conductor $7.658$
Analytic rank $0$
Dimension $90$
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] = [959,2,Mod(275,959)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(959, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("959.275");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 959 = 7 \cdot 137 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 959.e (of order \(3\), degree \(2\), 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: \(7.65765355384\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(45\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 90 q + 11 q^{3} - 44 q^{4} + 4 q^{5} - 20 q^{6} - q^{7} - 6 q^{8} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 90 q + 11 q^{3} - 44 q^{4} + 4 q^{5} - 20 q^{6} - q^{7} - 6 q^{8} - 44 q^{9} + 25 q^{10} + 33 q^{12} - 72 q^{13} - 38 q^{16} + 18 q^{17} + 5 q^{18} + 43 q^{19} - 20 q^{20} + 8 q^{21} + q^{23} + 20 q^{24} - 43 q^{25} + 2 q^{26} - 106 q^{27} + 7 q^{28} - 8 q^{29} + 12 q^{30} + 59 q^{31} + 11 q^{32} + 37 q^{33} - 96 q^{34} + 2 q^{35} + 28 q^{36} + 39 q^{38} - 16 q^{39} + 56 q^{40} - 30 q^{41} + 26 q^{42} - 2 q^{43} + 2 q^{44} + 28 q^{45} - 31 q^{46} + 58 q^{47} - 24 q^{48} + 15 q^{49} - 148 q^{50} + 5 q^{51} + 115 q^{52} - 10 q^{53} + 39 q^{54} - 162 q^{55} + 63 q^{56} - 36 q^{57} + 11 q^{58} + 41 q^{59} - 90 q^{60} + 40 q^{61} + 58 q^{62} + 53 q^{63} + 30 q^{64} + 9 q^{65} + 42 q^{66} + 56 q^{68} - 10 q^{69} + 84 q^{70} - 84 q^{71} + 11 q^{72} + 67 q^{73} - 39 q^{74} + 40 q^{75} - 136 q^{76} + 21 q^{77} - 156 q^{78} + 9 q^{79} + 14 q^{80} - 73 q^{81} + 34 q^{82} - 96 q^{83} + 28 q^{84} + 12 q^{85} + 13 q^{86} + 135 q^{87} - 47 q^{88} + 17 q^{89} + 88 q^{90} + 26 q^{91} - 122 q^{92} + q^{93} + 28 q^{94} - 47 q^{95} + 50 q^{96} - 162 q^{97} + 128 q^{98} - 70 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
275.1 −1.33533 + 2.31287i 0.806168 + 1.39632i −2.56623 − 4.44485i 0.614637 − 1.06458i −4.30602 −2.33380 − 1.24635i 8.36579 0.200186 − 0.346732i 1.64149 + 2.84315i
275.2 −1.32902 + 2.30193i −0.499205 − 0.864648i −2.53258 − 4.38657i 1.48792 − 2.57716i 2.65381 2.47099 + 0.945615i 8.14734 1.00159 − 1.73480i 3.95496 + 6.85018i
275.3 −1.28329 + 2.22272i 0.941272 + 1.63033i −2.29367 − 3.97275i 0.0442449 − 0.0766345i −4.83170 2.54365 − 0.727894i 6.64060 −0.271985 + 0.471091i 0.113558 + 0.196689i
275.4 −1.20235 + 2.08253i 1.70170 + 2.94742i −1.89130 − 3.27582i −0.805237 + 1.39471i −8.18415 −0.550071 + 2.58794i 4.28661 −4.29154 + 7.43316i −1.93636 − 3.35387i
275.5 −1.18726 + 2.05640i −0.229506 − 0.397515i −1.81919 − 3.15093i −1.16760 + 2.02235i 1.08993 −0.814930 + 2.51712i 3.89037 1.39465 − 2.41561i −2.77251 − 4.80212i
275.6 −1.15213 + 1.99555i −0.629165 − 1.08975i −1.65481 − 2.86622i 1.70246 − 2.94875i 2.89952 −2.32503 + 1.26263i 3.01771 0.708304 − 1.22682i 3.92291 + 6.79468i
275.7 −1.11057 + 1.92356i 1.46256 + 2.53322i −1.46672 − 2.54043i 1.54265 − 2.67195i −6.49707 0.863357 − 2.50092i 2.07329 −2.77814 + 4.81188i 3.42644 + 5.93476i
275.8 −0.964429 + 1.67044i −1.14869 − 1.98958i −0.860245 − 1.48999i 0.545938 − 0.945592i 4.43130 −1.37640 − 2.25954i −0.539134 −1.13896 + 1.97273i 1.05304 + 1.82391i
275.9 −0.950099 + 1.64562i 0.436708 + 0.756401i −0.805377 − 1.39495i −0.667817 + 1.15669i −1.65966 0.715515 − 2.54716i −0.739644 1.11857 − 1.93742i −1.26899 − 2.19795i
275.10 −0.890111 + 1.54172i −0.193526 − 0.335198i −0.584597 − 1.01255i −1.22332 + 2.11886i 0.689040 2.07172 − 1.64559i −1.47902 1.42510 − 2.46834i −2.17779 − 3.77204i
275.11 −0.853919 + 1.47903i −1.14527 − 1.98367i −0.458354 − 0.793893i −0.121834 + 0.211023i 3.91187 2.58468 − 0.565184i −1.85009 −1.12329 + 1.94559i −0.208073 − 0.360394i
275.12 −0.809668 + 1.40239i 1.09654 + 1.89926i −0.311125 − 0.538884i 0.209785 − 0.363358i −3.55133 −1.06146 + 2.42349i −2.23104 −0.904790 + 1.56714i 0.339712 + 0.588399i
275.13 −0.668058 + 1.15711i 0.581691 + 1.00752i 0.107397 + 0.186017i 2.11108 − 3.65650i −1.55441 2.62118 + 0.359744i −2.95922 0.823271 − 1.42595i 2.82065 + 4.88551i
275.14 −0.643220 + 1.11409i −0.682903 − 1.18282i 0.172535 + 0.298840i 0.595764 − 1.03189i 1.75703 0.253702 + 2.63356i −3.01679 0.567288 − 0.982571i 0.766415 + 1.32747i
275.15 −0.588727 + 1.01971i 1.53187 + 2.65327i 0.306801 + 0.531394i 1.41809 − 2.45621i −3.60741 −2.64142 − 0.151251i −3.07740 −3.19323 + 5.53084i 1.66974 + 2.89207i
275.16 −0.566169 + 0.980633i 0.776826 + 1.34550i 0.358906 + 0.621644i −1.72303 + 2.98438i −1.75926 −2.00079 + 1.73114i −3.07748 0.293084 − 0.507636i −1.95105 − 3.37933i
275.17 −0.563020 + 0.975178i −0.371199 − 0.642936i 0.366018 + 0.633962i −0.321276 + 0.556466i 0.835969 −2.36881 − 1.17844i −3.07638 1.22442 − 2.12076i −0.361769 − 0.626602i
275.18 −0.287230 + 0.497497i 0.763111 + 1.32175i 0.834998 + 1.44626i 0.400375 − 0.693470i −0.876754 0.934548 − 2.47520i −2.10827 0.335323 − 0.580797i 0.230000 + 0.398371i
275.19 −0.283548 + 0.491119i −1.37098 − 2.37460i 0.839201 + 1.45354i −1.04647 + 1.81254i 1.55495 −2.08396 + 1.63006i −2.08601 −2.25915 + 3.91297i −0.593448 − 1.02788i
275.20 −0.215926 + 0.373995i 1.53180 + 2.65316i 0.906752 + 1.57054i −0.825800 + 1.43033i −1.32302 −2.42683 − 1.05381i −1.64687 −3.19283 + 5.53014i −0.356623 − 0.617690i
See all 90 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 275.45
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 959.2.e.b ✓ 90
7.c even 3 1 inner 959.2.e.b ✓ 90
7.c even 3 1 6713.2.a.k 45
7.d odd 6 1 6713.2.a.l 45
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
959.2.e.b ✓ 90 1.a even 1 1 trivial
959.2.e.b ✓ 90 7.c even 3 1 inner
6713.2.a.k 45 7.c even 3 1
6713.2.a.l 45 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{90} + 67 T_{2}^{88} + 2 T_{2}^{87} + 2410 T_{2}^{86} + 127 T_{2}^{85} + 59958 T_{2}^{84} + \cdots + 5625 \) acting on \(S_{2}^{\mathrm{new}}(959, [\chi])\). Copy content Toggle raw display