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Properties

Label 309.3.b.a
Level $309$
Weight $3$
Character orbit 309.b
Analytic conductor $8.420$
Analytic rank $0$
Dimension $68$
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] = [309,3,Mod(104,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.104");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 309.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: \(8.41964016873\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} - 140 q^{4} + 14 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} - 140 q^{4} + 14 q^{6} - 12 q^{9} + 36 q^{10} - 6 q^{12} - 32 q^{13} - 10 q^{15} + 244 q^{16} + 42 q^{18} - 8 q^{19} + 8 q^{21} + 32 q^{22} - 72 q^{24} - 276 q^{25} + 62 q^{27} - 128 q^{28} - 20 q^{30} + 80 q^{31} + 38 q^{33} + 96 q^{34} - 110 q^{36} + 16 q^{37} - 40 q^{39} - 168 q^{40} + 162 q^{42} - 136 q^{43} - 58 q^{45} + 92 q^{46} - 24 q^{48} + 612 q^{49} + 12 q^{51} + 16 q^{52} + 262 q^{54} - 160 q^{55} - 78 q^{57} + 156 q^{58} - 282 q^{60} - 224 q^{61} + 128 q^{63} - 280 q^{64} + 14 q^{66} + 12 q^{67} - 58 q^{69} - 100 q^{70} + 88 q^{72} + 140 q^{73} - 184 q^{75} - 500 q^{76} - 244 q^{78} + 300 q^{79} + 116 q^{81} + 324 q^{82} + 20 q^{84} - 84 q^{85} + 230 q^{87} + 484 q^{88} - 238 q^{90} - 204 q^{91} + 392 q^{93} - 584 q^{94} + 860 q^{96} - 408 q^{97} - 456 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
104.1 − 3.96847i −2.13057 + 2.11203i −11.7487 2.12273i 8.38152 + 8.45510i 11.5244 30.7506i 0.0786650 − 8.99966i 8.42399
104.2 − 3.89754i 2.74457 + 1.21133i −11.1908 5.08922i 4.72120 − 10.6971i −3.41140 28.0267i 6.06537 + 6.64916i 19.8355
104.3 − 3.66976i 1.16030 − 2.76653i −9.46715 − 2.06414i −10.1525 − 4.25801i −4.11053 20.0631i −6.30742 − 6.42000i −7.57491
104.4 − 3.59865i −2.47136 − 1.70070i −8.95030 4.57817i −6.12023 + 8.89357i −9.56470 17.8144i 3.21524 + 8.40608i 16.4753
104.5 − 3.50446i 2.97974 + 0.348055i −8.28124 − 9.62276i 1.21974 − 10.4424i 9.21342 15.0034i 8.75772 + 2.07423i −33.7226
104.6 − 3.41551i −1.83825 + 2.37083i −7.66569 − 6.75813i 8.09758 + 6.27856i −8.26128 12.5202i −2.24166 − 8.71636i −23.0824
104.7 − 3.30191i −1.60474 − 2.53472i −6.90262 6.35294i −8.36943 + 5.29869i 11.3267 9.58418i −3.84965 + 8.13512i 20.9768
104.8 − 3.29198i 1.16904 + 2.76285i −6.83714 − 2.71195i 9.09525 − 3.84846i −3.93087 9.33983i −6.26668 + 6.45977i −8.92770
104.9 − 3.16032i −2.99602 − 0.154505i −5.98764 − 2.26601i −0.488286 + 9.46839i −3.69585 6.28158i 8.95226 + 0.925801i −7.16133
104.10 − 3.09436i 2.50369 − 1.65273i −5.57504 6.46778i −5.11414 − 7.74732i 7.19911 4.87374i 3.53697 − 8.27586i 20.0136
104.11 − 3.05648i −0.543091 − 2.95043i −5.34209 − 4.27159i −9.01795 + 1.65995i 5.96777 4.10209i −8.41010 + 3.20471i −13.0560
104.12 − 3.00452i 0.682080 + 2.92143i −5.02712 − 1.28832i 8.77749 − 2.04932i 6.10945 3.08601i −8.06953 + 3.98530i −3.87078
104.13 − 2.87604i 0.280657 + 2.98684i −4.27160 8.31281i 8.59028 − 0.807179i −0.741437 0.781134i −8.84246 + 1.67655i 23.9080
104.14 − 2.75982i −2.79655 + 1.08596i −3.61660 6.08512i 2.99705 + 7.71797i −1.42947 − 1.05812i 6.64138 − 6.07388i 16.7938
104.15 − 2.62518i 2.99963 + 0.0472303i −2.89156 − 2.43090i 0.123988 − 7.87456i −12.3682 − 2.90984i 8.99554 + 0.283347i −6.38156
104.16 − 2.44062i −2.99508 − 0.171807i −1.95662 − 5.01388i −0.419315 + 7.30984i 8.64385 − 4.98710i 8.94096 + 1.02915i −12.2370
104.17 − 2.26836i 2.73590 + 1.23079i −1.14544 4.95528i 2.79187 − 6.20599i −1.27525 − 6.47516i 5.97030 + 6.73465i 11.2403
104.18 − 2.26791i 0.823172 − 2.88486i −1.14343 9.26315i −6.54260 − 1.86688i −12.8663 − 6.47845i −7.64478 − 4.74946i 21.0080
104.19 − 2.10878i −1.20338 − 2.74807i −0.446957 − 8.66756i −5.79507 + 2.53767i −9.02781 − 7.49259i −6.10374 + 6.61396i −18.2780
104.20 − 2.00952i 2.38507 − 1.81974i −0.0381697 − 3.49557i −3.65680 − 4.79284i 2.72916 − 7.96138i 2.37710 − 8.68040i −7.02441
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 104.68
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 309.3.b.a ✓ 68
3.b odd 2 1 inner 309.3.b.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.3.b.a ✓ 68 1.a even 1 1 trivial
309.3.b.a ✓ 68 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(309, [\chi])\).