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Properties

Label 308.2.y.b
Level $308$
Weight $2$
Character orbit 308.y
Analytic conductor $2.459$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,2,Mod(9,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 10, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.y (of order \(15\), degree \(8\), 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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - q^{3} + 7 q^{5} - 7 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - q^{3} + 7 q^{5} - 7 q^{7} + 13 q^{9} - 5 q^{11} - 8 q^{13} - 48 q^{15} + 19 q^{17} + 13 q^{19} - 22 q^{21} - 30 q^{23} - 7 q^{25} + 26 q^{27} - 12 q^{29} + 3 q^{31} - 28 q^{33} + 37 q^{35} + 19 q^{37} + 3 q^{39} - 10 q^{41} + 84 q^{43} - 24 q^{45} + 11 q^{47} + 51 q^{49} - 21 q^{51} + 33 q^{53} + 48 q^{55} - 108 q^{57} + 23 q^{59} + 46 q^{61} - 90 q^{63} - 54 q^{65} - 8 q^{67} - 82 q^{69} - 64 q^{71} + 4 q^{73} - 15 q^{75} + 6 q^{77} - 14 q^{79} + 7 q^{81} - 28 q^{83} + 50 q^{85} - 66 q^{87} - 58 q^{89} + 70 q^{91} - 42 q^{93} + 4 q^{95} - 100 q^{97} + 140 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −2.83413 − 1.26184i 0 1.32307 + 0.281228i 0 1.68941 + 2.03615i 0 4.43267 + 4.92298i 0
9.2 0 −1.59083 − 0.708282i 0 3.53897 + 0.752231i 0 −1.44841 − 2.21407i 0 0.0216755 + 0.0240731i 0
9.3 0 −0.416782 − 0.185563i 0 −0.887136 − 0.188567i 0 −2.58171 − 0.578578i 0 −1.86812 − 2.07476i 0
9.4 0 0.298336 + 0.132828i 0 −3.45675 − 0.734755i 0 2.29997 − 1.30772i 0 −1.93603 − 2.15018i 0
9.5 0 2.27473 + 1.01277i 0 −1.34506 − 0.285901i 0 1.38806 + 2.25240i 0 2.14129 + 2.37814i 0
9.6 0 2.61762 + 1.16544i 0 1.05782 + 0.224846i 0 −2.30739 − 1.29459i 0 3.48627 + 3.87190i 0
25.1 0 −3.17936 − 0.675793i 0 −0.0774019 + 0.736430i 0 −2.58614 + 0.558460i 0 6.91098 + 3.07697i 0
25.2 0 −1.94235 − 0.412859i 0 −0.0302385 + 0.287700i 0 2.64441 + 0.0843530i 0 0.861626 + 0.383621i 0
25.3 0 −1.23333 − 0.262153i 0 −0.367897 + 3.50030i 0 0.745250 − 2.53862i 0 −1.28825 − 0.573565i 0
25.4 0 −0.876178 − 0.186237i 0 0.461396 − 4.38989i 0 −2.16649 + 1.51865i 0 −2.00763 − 0.893856i 0
25.5 0 1.54096 + 0.327542i 0 −0.0703085 + 0.668941i 0 1.10028 + 2.40611i 0 −0.473354 − 0.210751i 0
25.6 0 3.12943 + 0.665181i 0 −0.358340 + 3.40938i 0 −2.48911 − 0.896852i 0 6.61023 + 2.94307i 0
37.1 0 −3.17936 + 0.675793i 0 −0.0774019 − 0.736430i 0 −2.58614 − 0.558460i 0 6.91098 − 3.07697i 0
37.2 0 −1.94235 + 0.412859i 0 −0.0302385 − 0.287700i 0 2.64441 − 0.0843530i 0 0.861626 − 0.383621i 0
37.3 0 −1.23333 + 0.262153i 0 −0.367897 − 3.50030i 0 0.745250 + 2.53862i 0 −1.28825 + 0.573565i 0
37.4 0 −0.876178 + 0.186237i 0 0.461396 + 4.38989i 0 −2.16649 − 1.51865i 0 −2.00763 + 0.893856i 0
37.5 0 1.54096 − 0.327542i 0 −0.0703085 − 0.668941i 0 1.10028 − 2.40611i 0 −0.473354 + 0.210751i 0
37.6 0 3.12943 − 0.665181i 0 −0.358340 − 3.40938i 0 −2.48911 + 0.896852i 0 6.61023 − 2.94307i 0
53.1 0 −0.299509 − 2.84964i 0 −0.723630 + 0.803673i 0 −1.94425 − 1.79440i 0 −5.09630 + 1.08325i 0
53.2 0 −0.260276 − 2.47636i 0 0.920128 − 1.02191i 0 2.57109 + 0.624089i 0 −3.13018 + 0.665340i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.y.b ✓ 48
7.c even 3 1 inner 308.2.y.b ✓ 48
11.c even 5 1 inner 308.2.y.b ✓ 48
77.m even 15 1 inner 308.2.y.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.y.b ✓ 48 1.a even 1 1 trivial
308.2.y.b ✓ 48 7.c even 3 1 inner
308.2.y.b ✓ 48 11.c even 5 1 inner
308.2.y.b ✓ 48 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + T_{3}^{47} - 15 T_{3}^{46} - 8 T_{3}^{45} + 73 T_{3}^{44} - 24 T_{3}^{43} + \cdots + 1982119441 \) acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\). Copy content Toggle raw display