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Properties

Label 637.2.x.a
Level $637$
Weight $2$
Character orbit 637.x
Analytic conductor $5.086$
Analytic rank $0$
Dimension $28$
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] = [637,2,Mod(19,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.x (of order \(12\), degree \(4\), 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} - 6 q^{4} + 6 q^{5} - 12 q^{6} - 4 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} - 6 q^{4} + 6 q^{5} - 12 q^{6} - 4 q^{8} - 12 q^{9} + 12 q^{10} + 2 q^{11} - 8 q^{12} + 10 q^{15} - 2 q^{16} + 6 q^{17} - 4 q^{18} + 8 q^{19} + 36 q^{20} - 8 q^{22} - 6 q^{23} - 12 q^{24} - 24 q^{26} - 8 q^{29} + 38 q^{31} - 20 q^{32} - 18 q^{33} - 12 q^{34} + 54 q^{36} - 16 q^{37} + 28 q^{39} - 48 q^{40} - 18 q^{41} + 48 q^{43} - 6 q^{44} - 12 q^{45} + 18 q^{46} + 42 q^{47} - 12 q^{48} + 10 q^{50} + 12 q^{51} + 28 q^{52} + 12 q^{53} + 30 q^{54} + 6 q^{55} + 12 q^{57} + 62 q^{58} + 6 q^{59} + 16 q^{60} + 36 q^{62} - 2 q^{65} - 66 q^{66} - 4 q^{67} - 30 q^{68} - 42 q^{69} - 42 q^{71} - 38 q^{72} - 14 q^{73} - 6 q^{74} + 20 q^{75} - 52 q^{76} - 62 q^{78} + 4 q^{79} - 12 q^{80} + 12 q^{81} + 108 q^{82} + 66 q^{83} - 54 q^{85} - 30 q^{86} - 42 q^{87} + 30 q^{89} + 72 q^{90} - 156 q^{92} + 14 q^{93} - 6 q^{95} - 18 q^{96} - 62 q^{97} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.38212 − 0.638288i − 0.168862i 3.53505 + 2.04096i 2.38343 − 0.638637i −0.107782 + 0.402250i 0 −3.63054 − 3.63054i 2.97149 −6.08525
19.2 −1.56673 − 0.419805i − 0.513926i 0.546366 + 0.315445i −1.47457 + 0.395109i −0.215749 + 0.805186i 0 1.57027 + 1.57027i 2.73588 2.47612
19.3 −0.670702 − 0.179714i − 3.13022i −1.31451 − 0.758931i 0.0359277 − 0.00962681i −0.562545 + 2.09944i 0 1.72723 + 1.72723i −6.79826 −0.0258269
19.4 0.263976 + 0.0707322i 2.50625i −1.66737 − 0.962657i 1.43012 − 0.383199i −0.177273 + 0.661591i 0 −0.758543 − 0.758543i −3.28130 0.404621
19.5 0.369709 + 0.0990633i 0.914861i −1.60518 − 0.926751i −3.58134 + 0.959617i −0.0906291 + 0.338232i 0 −1.04293 − 1.04293i 2.16303 −1.41912
19.6 1.34471 + 0.360315i − 1.44853i −0.0536242 − 0.0309600i 0.643078 − 0.172312i 0.521927 − 1.94786i 0 −2.02975 − 2.02975i 0.901760 0.926843
19.7 2.14116 + 0.573722i 1.10837i 2.52336 + 1.45686i 2.92938 − 0.784926i −0.635898 + 2.37321i 0 1.43221 + 1.43221i 1.77151 6.72261
80.1 −0.629770 − 2.35033i 1.97095i −3.39540 + 1.96034i −0.0608458 + 0.227080i 4.63238 − 1.24124i 0 3.30464 + 3.30464i −0.884626 0.572032
80.2 −0.521585 − 1.94658i − 1.44369i −1.78508 + 1.03062i −0.849184 + 3.16920i −2.81025 + 0.753004i 0 0.0872533 + 0.0872533i 0.915773 6.61202
80.3 −0.320827 − 1.19734i − 2.22531i 0.401352 − 0.231720i 0.674321 − 2.51660i −2.66446 + 0.713939i 0 −2.15924 − 2.15924i −1.95199 −3.22957
80.4 −0.127046 − 0.474142i 2.44579i 1.52338 − 0.879524i 0.931242 − 3.47544i 1.15965 − 0.310728i 0 −1.30475 − 1.30475i −2.98191 −1.76616
80.5 0.0745816 + 0.278342i − 1.06594i 1.66014 − 0.958482i −0.133809 + 0.499383i 0.296695 − 0.0794993i 0 0.798123 + 0.798123i 1.86378 −0.148979
80.6 0.470172 + 1.75471i 3.06997i −1.12588 + 0.650030i −0.288156 + 1.07541i −5.38690 + 1.44342i 0 0.899098 + 0.899098i −6.42473 −2.02252
80.7 0.554474 + 2.06932i − 0.0197323i −2.24261 + 1.29477i 0.360406 − 1.34505i 0.0408324 − 0.0109410i 0 −0.893066 − 0.893066i 2.99961 2.98319
215.1 −0.629770 + 2.35033i − 1.97095i −3.39540 − 1.96034i −0.0608458 − 0.227080i 4.63238 + 1.24124i 0 3.30464 − 3.30464i −0.884626 0.572032
215.2 −0.521585 + 1.94658i 1.44369i −1.78508 − 1.03062i −0.849184 − 3.16920i −2.81025 − 0.753004i 0 0.0872533 − 0.0872533i 0.915773 6.61202
215.3 −0.320827 + 1.19734i 2.22531i 0.401352 + 0.231720i 0.674321 + 2.51660i −2.66446 − 0.713939i 0 −2.15924 + 2.15924i −1.95199 −3.22957
215.4 −0.127046 + 0.474142i − 2.44579i 1.52338 + 0.879524i 0.931242 + 3.47544i 1.15965 + 0.310728i 0 −1.30475 + 1.30475i −2.98191 −1.76616
215.5 0.0745816 − 0.278342i 1.06594i 1.66014 + 0.958482i −0.133809 − 0.499383i 0.296695 + 0.0794993i 0 0.798123 − 0.798123i 1.86378 −0.148979
215.6 0.470172 − 1.75471i − 3.06997i −1.12588 − 0.650030i −0.288156 − 1.07541i −5.38690 − 1.44342i 0 0.899098 − 0.899098i −6.42473 −2.02252
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.x.a 28
7.b odd 2 1 91.2.w.a ✓ 28
7.c even 3 1 91.2.ba.a yes 28
7.c even 3 1 637.2.bd.b 28
7.d odd 6 1 637.2.bb.a 28
7.d odd 6 1 637.2.bd.a 28
13.f odd 12 1 637.2.bb.a 28
21.c even 2 1 819.2.gh.b 28
21.h odd 6 1 819.2.et.b 28
91.w even 12 1 inner 637.2.x.a 28
91.x odd 12 1 637.2.bd.a 28
91.ba even 12 1 637.2.bd.b 28
91.bc even 12 1 91.2.ba.a yes 28
91.bd odd 12 1 91.2.w.a ✓ 28
273.bw even 12 1 819.2.gh.b 28
273.ca odd 12 1 819.2.et.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.w.a ✓ 28 7.b odd 2 1
91.2.w.a ✓ 28 91.bd odd 12 1
91.2.ba.a yes 28 7.c even 3 1
91.2.ba.a yes 28 91.bc even 12 1
637.2.x.a 28 1.a even 1 1 trivial
637.2.x.a 28 91.w even 12 1 inner
637.2.bb.a 28 7.d odd 6 1
637.2.bb.a 28 13.f odd 12 1
637.2.bd.a 28 7.d odd 6 1
637.2.bd.a 28 91.x odd 12 1
637.2.bd.b 28 7.c even 3 1
637.2.bd.b 28 91.ba even 12 1
819.2.et.b 28 21.h odd 6 1
819.2.et.b 28 273.ca odd 12 1
819.2.gh.b 28 21.c even 2 1
819.2.gh.b 28 273.bw even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 2 T_{2}^{27} + 5 T_{2}^{26} + 6 T_{2}^{25} - 31 T_{2}^{24} - 46 T_{2}^{23} - 138 T_{2}^{22} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\). Copy content Toggle raw display