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Properties

Label 637.2.bb.a
Level $637$
Weight $2$
Character orbit 637.bb
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(227,637)]
 
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([2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.227");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bb (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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} + 6 q^{3} + 6 q^{5} + 12 q^{6} - 4 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} + 6 q^{3} + 6 q^{5} + 12 q^{6} - 4 q^{8} + 6 q^{9} + 6 q^{10} + 2 q^{11} + 8 q^{12} + 10 q^{15} + 4 q^{16} + 12 q^{17} + 2 q^{18} - 14 q^{19} - 36 q^{20} - 8 q^{22} + 18 q^{24} - 24 q^{26} - 8 q^{29} - 30 q^{30} + 4 q^{31} + 10 q^{32} + 12 q^{33} + 12 q^{34} + 54 q^{36} - 10 q^{37} - 20 q^{39} - 48 q^{40} + 18 q^{41} + 48 q^{43} - 6 q^{44} + 6 q^{45} + 24 q^{46} + 6 q^{47} + 12 q^{48} + 10 q^{50} - 12 q^{51} + 26 q^{52} + 12 q^{53} + 30 q^{54} - 6 q^{55} + 12 q^{57} - 46 q^{58} - 42 q^{59} + 10 q^{60} - 30 q^{61} - 36 q^{62} + 28 q^{65} - 66 q^{66} - 10 q^{67} + 42 q^{69} - 42 q^{71} + 46 q^{72} - 40 q^{73} + 12 q^{74} + 40 q^{75} + 52 q^{76} - 62 q^{78} + 4 q^{79} - 30 q^{80} - 6 q^{81} + 54 q^{82} - 66 q^{83} - 54 q^{85} - 18 q^{86} - 6 q^{88} - 72 q^{90} - 156 q^{92} + 20 q^{93} + 18 q^{94} + 66 q^{96} + 62 q^{97} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
227.1 −1.56744 + 1.56744i −0.959879 + 0.554186i − 2.91373i 0.784926 − 2.92938i 0.635898 − 2.37321i 0 1.43221 + 1.43221i −0.885755 + 1.53417i 3.36131 + 5.82195i
227.2 −0.984398 + 0.984398i 1.25446 − 0.724265i 0.0619199i 0.172312 − 0.643078i −0.521927 + 1.94786i 0 −2.02975 − 2.02975i −0.450880 + 0.780947i 0.463421 + 0.802669i
227.3 −0.270646 + 0.270646i −0.792292 + 0.457430i 1.85350i −0.959617 + 3.58134i 0.0906291 − 0.338232i 0 −1.04293 − 1.04293i −1.08152 + 1.87324i −0.709559 − 1.22899i
227.4 −0.193244 + 0.193244i −2.17048 + 1.25313i 1.92531i 0.383199 − 1.43012i 0.177273 − 0.661591i 0 −0.758543 − 0.758543i 1.64065 − 2.84169i 0.202311 + 0.350412i
227.5 0.490988 − 0.490988i 2.71085 − 1.56511i 1.51786i 0.00962681 − 0.0359277i 0.562545 − 2.09944i 0 1.72723 + 1.72723i 3.39913 − 5.88747i −0.0129135 − 0.0223668i
227.6 1.14693 − 1.14693i 0.445073 − 0.256963i − 0.630890i −0.395109 + 1.47457i 0.215749 − 0.805186i 0 1.57027 + 1.57027i −1.36794 + 2.36934i 1.23806 + 2.14439i
227.7 1.74384 − 1.74384i 0.146239 − 0.0844309i − 4.08193i 0.638637 − 2.38343i 0.107782 − 0.402250i 0 −3.63054 − 3.63054i −1.48574 + 2.57338i −3.04263 − 5.26998i
362.1 −1.56744 − 1.56744i −0.959879 − 0.554186i 2.91373i 0.784926 + 2.92938i 0.635898 + 2.37321i 0 1.43221 − 1.43221i −0.885755 − 1.53417i 3.36131 − 5.82195i
362.2 −0.984398 − 0.984398i 1.25446 + 0.724265i − 0.0619199i 0.172312 + 0.643078i −0.521927 − 1.94786i 0 −2.02975 + 2.02975i −0.450880 − 0.780947i 0.463421 − 0.802669i
362.3 −0.270646 − 0.270646i −0.792292 − 0.457430i − 1.85350i −0.959617 − 3.58134i 0.0906291 + 0.338232i 0 −1.04293 + 1.04293i −1.08152 − 1.87324i −0.709559 + 1.22899i
362.4 −0.193244 − 0.193244i −2.17048 − 1.25313i − 1.92531i 0.383199 + 1.43012i 0.177273 + 0.661591i 0 −0.758543 + 0.758543i 1.64065 + 2.84169i 0.202311 − 0.350412i
362.5 0.490988 + 0.490988i 2.71085 + 1.56511i − 1.51786i 0.00962681 + 0.0359277i 0.562545 + 2.09944i 0 1.72723 − 1.72723i 3.39913 + 5.88747i −0.0129135 + 0.0223668i
362.6 1.14693 + 1.14693i 0.445073 + 0.256963i 0.630890i −0.395109 − 1.47457i 0.215749 + 0.805186i 0 1.57027 − 1.57027i −1.36794 − 2.36934i 1.23806 − 2.14439i
362.7 1.74384 + 1.74384i 0.146239 + 0.0844309i 4.08193i 0.638637 + 2.38343i 0.107782 + 0.402250i 0 −3.63054 + 3.63054i −1.48574 − 2.57338i −3.04263 + 5.26998i
423.1 −1.72056 − 1.72056i 1.70689 − 0.985473i 3.92067i −0.227080 − 0.0608458i −4.63238 − 1.24124i 0 3.30464 − 3.30464i 0.442313 − 0.766108i 0.286016 + 0.495394i
423.2 −1.42500 − 1.42500i −1.25027 + 0.721843i 2.06123i −3.16920 − 0.849184i 2.81025 + 0.753004i 0 0.0872533 − 0.0872533i −0.457887 + 0.793083i 3.30601 + 5.72618i
423.3 −0.876516 − 0.876516i −1.92717 + 1.11265i − 0.463441i 2.51660 + 0.674321i 2.66446 + 0.713939i 0 −2.15924 + 2.15924i 0.975997 − 1.69048i −1.61479 − 2.79689i
423.4 −0.347096 − 0.347096i 2.11812 − 1.22290i − 1.75905i 3.47544 + 0.931242i −1.15965 − 0.310728i 0 −1.30475 + 1.30475i 1.49096 − 2.58241i −0.883082 − 1.52954i
423.5 0.203761 + 0.203761i −0.923129 + 0.532969i − 1.91696i −0.499383 − 0.133809i −0.296695 − 0.0794993i 0 0.798123 − 0.798123i −0.931889 + 1.61408i −0.0744896 − 0.129020i
423.6 1.28453 + 1.28453i 2.65867 − 1.53499i 1.30006i −1.07541 − 0.288156i 5.38690 + 1.44342i 0 0.899098 − 0.899098i 3.21237 − 5.56398i −1.01126 − 1.75155i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 227.7
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 2 T_{2}^{27} + 2 T_{2}^{26} + 77 T_{2}^{24} + 152 T_{2}^{23} + 150 T_{2}^{22} + 6 T_{2}^{21} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\). Copy content Toggle raw display