Properties

Label 308.2.i.a
Level $308$
Weight $2$
Character orbit 308.i
Analytic conductor $2.459$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,2,Mod(177,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.i (of order \(3\), degree \(2\), 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots - \beta_1) q^{9}+ \cdots + (2 \beta_{3} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + 2 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{3} + 2 q^{5} + 4 q^{7} - 4 q^{9} + 3 q^{11} + 6 q^{13} - 30 q^{15} + 5 q^{17} - 11 q^{19} - 10 q^{21} + 4 q^{23} - 11 q^{25} + 44 q^{27} - 2 q^{29} - 19 q^{31} + q^{33} - 17 q^{35} + 8 q^{37} - 17 q^{39} + 34 q^{41} + 20 q^{43} + 21 q^{45} - 13 q^{47} - 12 q^{49} - 9 q^{53} + 4 q^{55} + 4 q^{57} - 6 q^{59} - 4 q^{61} + 50 q^{63} + 37 q^{65} - 8 q^{67} - 6 q^{69} - 26 q^{71} - 22 q^{73} - 13 q^{75} + 2 q^{77} - 5 q^{79} - 19 q^{81} + 6 q^{83} - 14 q^{85} + 18 q^{87} + 3 q^{89} - 31 q^{91} - 14 q^{93} - 3 q^{95} + 50 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.500000 1.41036i
0.500000 + 2.05195i
0.500000 + 0.224437i
0.500000 + 1.41036i
0.500000 2.05195i
0.500000 0.224437i
0 −1.59097 2.75564i 0 1.71053 2.96273i 0 −0.710533 2.54856i 0 −3.56238 + 6.17023i 0
177.2 0 0.296790 + 0.514055i 0 0.933463 1.61680i 0 0.0665372 + 2.64491i 0 1.32383 2.29294i 0
177.3 0 0.794182 + 1.37556i 0 −1.64400 + 2.84748i 0 2.64400 0.0963576i 0 0.238550 0.413181i 0
221.1 0 −1.59097 + 2.75564i 0 1.71053 + 2.96273i 0 −0.710533 + 2.54856i 0 −3.56238 6.17023i 0
221.2 0 0.296790 0.514055i 0 0.933463 + 1.61680i 0 0.0665372 2.64491i 0 1.32383 + 2.29294i 0
221.3 0 0.794182 1.37556i 0 −1.64400 2.84748i 0 2.64400 + 0.0963576i 0 0.238550 + 0.413181i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.i.a 6
3.b odd 2 1 2772.2.s.f 6
4.b odd 2 1 1232.2.q.l 6
7.b odd 2 1 2156.2.i.l 6
7.c even 3 1 inner 308.2.i.a 6
7.c even 3 1 2156.2.a.i 3
7.d odd 6 1 2156.2.a.h 3
7.d odd 6 1 2156.2.i.l 6
21.h odd 6 1 2772.2.s.f 6
28.f even 6 1 8624.2.a.cn 3
28.g odd 6 1 1232.2.q.l 6
28.g odd 6 1 8624.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.i.a 6 1.a even 1 1 trivial
308.2.i.a 6 7.c even 3 1 inner
1232.2.q.l 6 4.b odd 2 1
1232.2.q.l 6 28.g odd 6 1
2156.2.a.h 3 7.d odd 6 1
2156.2.a.i 3 7.c even 3 1
2156.2.i.l 6 7.b odd 2 1
2156.2.i.l 6 7.d odd 6 1
2772.2.s.f 6 3.b odd 2 1
2772.2.s.f 6 21.h odd 6 1
8624.2.a.ci 3 28.g odd 6 1
8624.2.a.cn 3 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + T_{3}^{5} + 7T_{3}^{4} - 12T_{3}^{3} + 33T_{3}^{2} - 18T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + 7 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{6} - 4 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T^{3} - 3 T^{2} - 24 T + 79)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 5 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} + 11 T^{5} + \cdots + 1089 \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( (T^{3} + T^{2} - 50 T - 129)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 19 T^{5} + \cdots + 9409 \) Copy content Toggle raw display
$37$ \( T^{6} - 8 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( (T^{3} - 17 T^{2} + \cdots - 33)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 10 T^{2} + \cdots + 73)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 13 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$53$ \( T^{6} + 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + \cdots + 59049 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + \cdots + 222784 \) Copy content Toggle raw display
$67$ \( T^{6} + 8 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$71$ \( (T^{3} + 13 T^{2} + \cdots + 63)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 22 T^{5} + \cdots + 124609 \) Copy content Toggle raw display
$79$ \( T^{6} + 5 T^{5} + \cdots + 91809 \) Copy content Toggle raw display
$83$ \( (T^{3} - 3 T^{2} + \cdots + 477)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 3 T^{5} + \cdots + 59049 \) Copy content Toggle raw display
$97$ \( (T^{3} - 25 T^{2} + \cdots + 1171)^{2} \) Copy content Toggle raw display
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