Properties

Label 230.6.b.b
Level $230$
Weight $6$
Character orbit 230.b
Analytic conductor $36.888$
Analytic rank $0$
Dimension $30$
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] = [230,6,Mod(139,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.139");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.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: \(36.8882785570\)
Analytic rank: \(0\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 480 q^{4} - 30 q^{5} + 216 q^{6} - 3020 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 480 q^{4} - 30 q^{5} + 216 q^{6} - 3020 q^{9} + 80 q^{10} + 2074 q^{11} - 808 q^{14} - 3056 q^{15} + 7680 q^{16} - 1750 q^{19} + 480 q^{20} + 524 q^{21} - 3456 q^{24} + 12218 q^{25} + 10440 q^{26} - 27548 q^{29} + 16280 q^{30} + 31170 q^{31} - 25080 q^{34} + 20664 q^{35} + 48320 q^{36} - 47176 q^{39} - 1280 q^{40} + 71758 q^{41} - 33184 q^{44} - 9476 q^{45} + 63480 q^{46} - 212504 q^{49} + 50768 q^{50} - 129106 q^{51} - 52656 q^{54} + 137582 q^{55} + 12928 q^{56} - 212568 q^{59} + 48896 q^{60} + 7814 q^{61} - 122880 q^{64} + 50144 q^{65} - 54264 q^{66} - 28566 q^{69} + 11320 q^{70} + 268238 q^{71} - 41264 q^{74} + 61368 q^{75} + 28000 q^{76} + 93452 q^{79} - 7680 q^{80} + 412886 q^{81} - 8384 q^{84} + 67716 q^{85} + 248368 q^{86} - 349096 q^{89} - 40304 q^{90} + 348102 q^{91} - 98864 q^{94} + 60058 q^{95} + 55296 q^{96} - 578634 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1 4.00000i 28.7267i −16.0000 −53.1448 17.3388i −114.907 115.551i 64.0000i −582.223 −69.3553 + 212.579i
139.2 4.00000i 24.7103i −16.0000 44.5324 + 33.7915i −98.8414 141.158i 64.0000i −367.601 135.166 178.130i
139.3 4.00000i 18.9029i −16.0000 −54.8757 + 10.6611i −75.6115 198.078i 64.0000i −114.319 42.6446 + 219.503i
139.4 4.00000i 16.5762i −16.0000 35.3277 43.3238i −66.3049 169.564i 64.0000i −31.7715 −173.295 141.311i
139.5 4.00000i 10.9586i −16.0000 −39.6919 39.3644i −43.8345 12.6560i 64.0000i 122.909 −157.458 + 158.767i
139.6 4.00000i 4.21065i −16.0000 53.1232 17.4048i −16.8426 167.782i 64.0000i 225.270 −69.6193 212.493i
139.7 4.00000i 2.12202i −16.0000 19.1932 + 52.5036i −8.48807 64.0472i 64.0000i 238.497 210.014 76.7726i
139.8 4.00000i 0.0569538i −16.0000 −30.6335 46.7609i 0.227815 149.136i 64.0000i 242.997 −187.044 + 122.534i
139.9 4.00000i 8.39668i −16.0000 −49.8703 + 25.2578i 33.5867 95.8251i 64.0000i 172.496 101.031 + 199.481i
139.10 4.00000i 10.0068i −16.0000 7.97591 + 55.3298i 40.0274 231.737i 64.0000i 142.863 221.319 31.9036i
139.11 4.00000i 14.1177i −16.0000 13.1286 54.3382i 56.4709 69.8012i 64.0000i 43.6900 −217.353 52.5144i
139.12 4.00000i 19.5360i −16.0000 −18.7601 + 52.6598i 78.1441 153.006i 64.0000i −138.657 210.639 + 75.0405i
139.13 4.00000i 26.2766i −16.0000 54.8977 + 10.5471i 105.107 233.425i 64.0000i −447.461 42.1883 219.591i
139.14 4.00000i 27.0471i −16.0000 54.9312 + 10.3712i 108.188 238.280i 64.0000i −488.545 41.4850 219.725i
139.15 4.00000i 27.7695i −16.0000 −51.1336 22.5909i 111.078 2.41627i 64.0000i −528.145 −90.3638 + 204.535i
139.16 4.00000i 27.7695i −16.0000 −51.1336 + 22.5909i 111.078 2.41627i 64.0000i −528.145 −90.3638 204.535i
139.17 4.00000i 27.0471i −16.0000 54.9312 10.3712i 108.188 238.280i 64.0000i −488.545 41.4850 + 219.725i
139.18 4.00000i 26.2766i −16.0000 54.8977 10.5471i 105.107 233.425i 64.0000i −447.461 42.1883 + 219.591i
139.19 4.00000i 19.5360i −16.0000 −18.7601 52.6598i 78.1441 153.006i 64.0000i −138.657 210.639 75.0405i
139.20 4.00000i 14.1177i −16.0000 13.1286 + 54.3382i 56.4709 69.8012i 64.0000i 43.6900 −217.353 + 52.5144i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.30
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.6.b.b 30
5.b even 2 1 inner 230.6.b.b 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.b.b 30 1.a even 1 1 trivial
230.6.b.b 30 5.b even 2 1 inner