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Properties

Label 51.5.b.a
Level $51$
Weight $5$
Character orbit 51.b
Analytic conductor $5.272$
Analytic rank $0$
Dimension $22$
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] = [51,5,Mod(35,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.35");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 51.b (of order \(2\), degree \(1\), 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: \(5.27186811728\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 10 q^{3} - 192 q^{4} - 2 q^{6} - 4 q^{7} - 102 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 10 q^{3} - 192 q^{4} - 2 q^{6} - 4 q^{7} - 102 q^{9} + 216 q^{10} + 290 q^{12} - 374 q^{13} + 234 q^{15} + 2008 q^{16} - 328 q^{18} + 238 q^{19} - 208 q^{21} - 1160 q^{22} + 378 q^{24} - 3752 q^{25} - 2782 q^{27} + 16 q^{28} + 3820 q^{30} + 552 q^{31} - 2966 q^{33} - 1872 q^{36} + 3120 q^{37} + 4536 q^{39} + 1220 q^{40} + 4824 q^{42} - 178 q^{43} + 1584 q^{45} - 5820 q^{46} - 2846 q^{48} + 10666 q^{49} + 12392 q^{52} - 20258 q^{54} - 4082 q^{55} + 10088 q^{57} - 15236 q^{58} + 12572 q^{60} - 1672 q^{61} + 1664 q^{63} - 12816 q^{64} - 18184 q^{66} - 9260 q^{67} - 19566 q^{69} - 3968 q^{70} + 17136 q^{72} + 20416 q^{73} + 6010 q^{75} - 19768 q^{76} - 9420 q^{78} + 13092 q^{79} + 14934 q^{81} + 33268 q^{82} + 45776 q^{84} + 4046 q^{85} + 21624 q^{87} + 57236 q^{88} - 30712 q^{90} - 54508 q^{91} + 16960 q^{93} - 28824 q^{94} - 53478 q^{96} - 16192 q^{97} - 44068 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 − 7.62776i −8.93748 − 1.05902i −42.1827 − 31.8328i −8.07792 + 68.1729i 49.9993 199.715i 78.7570 + 18.9299i −242.813
35.2 − 7.45921i 6.25778 − 6.46840i −39.6399 17.7671i −48.2492 − 46.6781i −57.5407 176.335i −2.68045 − 80.9556i 132.529
35.3 − 6.76844i 1.24326 + 8.91372i −29.8117 39.6318i 60.3319 − 8.41489i 61.1772 93.4839i −77.9086 + 22.1640i 268.245
35.4 − 5.43986i −0.930028 + 8.95182i −13.5920 − 28.7349i 48.6966 + 5.05922i −83.4208 − 13.0990i −79.2701 − 16.6509i −156.314
35.5 − 5.10522i −8.36812 − 3.31280i −10.0633 37.2642i −16.9126 + 42.7211i −61.5862 − 30.3083i 59.0508 + 55.4437i 190.242
35.6 − 4.85809i 0.169940 − 8.99840i −7.60106 − 18.5162i −43.7150 − 0.825585i 22.1650 − 40.8028i −80.9422 − 3.05838i −89.9535
35.7 − 4.67750i 8.94434 + 0.999348i −5.87898 − 8.92904i 4.67445 − 41.8371i 26.2181 − 47.3410i 79.0026 + 17.8770i −41.7656
35.8 − 2.90018i −7.69923 + 4.66068i 7.58896 − 1.06905i 13.5168 + 22.3291i 50.8090 − 68.4122i 37.5562 − 71.7672i −3.10045
35.9 − 1.09697i −5.76322 − 6.91269i 14.7967 − 15.3908i −7.58302 + 6.32208i −37.9635 − 33.7830i −14.5706 + 79.6787i −16.8832
35.10 − 1.00762i 3.43330 − 8.31940i 14.9847 49.2545i −8.38278 − 3.45945i 68.1136 − 31.2208i −57.4249 − 57.1260i 49.6297
35.11 − 0.775055i 6.64945 + 6.06505i 15.3993 23.4607i 4.70074 − 5.15369i −39.9710 − 24.3362i 7.43044 + 80.6585i 18.1834
35.12 0.775055i 6.64945 − 6.06505i 15.3993 − 23.4607i 4.70074 + 5.15369i −39.9710 24.3362i 7.43044 − 80.6585i 18.1834
35.13 1.00762i 3.43330 + 8.31940i 14.9847 − 49.2545i −8.38278 + 3.45945i 68.1136 31.2208i −57.4249 + 57.1260i 49.6297
35.14 1.09697i −5.76322 + 6.91269i 14.7967 15.3908i −7.58302 − 6.32208i −37.9635 33.7830i −14.5706 − 79.6787i −16.8832
35.15 2.90018i −7.69923 − 4.66068i 7.58896 1.06905i 13.5168 − 22.3291i 50.8090 68.4122i 37.5562 + 71.7672i −3.10045
35.16 4.67750i 8.94434 − 0.999348i −5.87898 8.92904i 4.67445 + 41.8371i 26.2181 47.3410i 79.0026 − 17.8770i −41.7656
35.17 4.85809i 0.169940 + 8.99840i −7.60106 18.5162i −43.7150 + 0.825585i 22.1650 40.8028i −80.9422 + 3.05838i −89.9535
35.18 5.10522i −8.36812 + 3.31280i −10.0633 − 37.2642i −16.9126 − 42.7211i −61.5862 30.3083i 59.0508 − 55.4437i 190.242
35.19 5.43986i −0.930028 − 8.95182i −13.5920 28.7349i 48.6966 − 5.05922i −83.4208 13.0990i −79.2701 + 16.6509i −156.314
35.20 6.76844i 1.24326 − 8.91372i −29.8117 − 39.6318i 60.3319 + 8.41489i 61.1772 − 93.4839i −77.9086 − 22.1640i 268.245
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.5.b.a ✓ 22
3.b odd 2 1 inner 51.5.b.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.5.b.a ✓ 22 1.a even 1 1 trivial
51.5.b.a ✓ 22 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(51, [\chi])\).