Properties

Label 308.2.c.b
Level $308$
Weight $2$
Character orbit 308.c
Analytic conductor $2.459$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,2,Mod(153,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.153");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.c (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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 9x^{2} - 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - 2 \beta_{2} q^{5} - \beta_{3} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - 2 \beta_{2} q^{5} - \beta_{3} q^{7} + q^{9} + ( - \beta_{3} + 2) q^{11} + \beta_1 q^{13} + 4 q^{15} + \beta_1 q^{17} - 2 \beta_1 q^{19} - \beta_1 q^{21} + 4 q^{23} - 3 q^{25} + 4 \beta_{2} q^{27} + \beta_{2} q^{31} + (2 \beta_{2} - \beta_1) q^{33} + 2 \beta_1 q^{35} - 4 q^{37} - 2 \beta_{3} q^{39} - \beta_1 q^{41} + 4 \beta_{3} q^{43} - 2 \beta_{2} q^{45} - 7 \beta_{2} q^{47} - 7 q^{49} - 2 \beta_{3} q^{51} - 4 q^{53} + ( - 4 \beta_{2} + 2 \beta_1) q^{55} + 4 \beta_{3} q^{57} - \beta_{2} q^{59} + 3 \beta_1 q^{61} - \beta_{3} q^{63} + 4 \beta_{3} q^{65} - 12 q^{67} + 4 \beta_{2} q^{69} + 8 q^{71} - 3 \beta_1 q^{73} - 3 \beta_{2} q^{75} + ( - 2 \beta_{3} - 7) q^{77} + 4 \beta_{3} q^{79} - 5 q^{81} - 2 \beta_1 q^{83} + 4 \beta_{3} q^{85} + 12 \beta_{2} q^{89} + 7 \beta_{2} q^{91} - 2 q^{93} - 8 \beta_{3} q^{95} - 4 \beta_{2} q^{97} + ( - \beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} + 8 q^{11} + 16 q^{15} + 16 q^{23} - 12 q^{25} - 16 q^{37} - 28 q^{49} - 16 q^{53} - 48 q^{67} + 32 q^{71} - 28 q^{77} - 20 q^{81} - 8 q^{93} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 9x^{2} - 8x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} - 3\nu^{2} + 17\nu - 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\nu^{3} + 6\nu^{2} - 32\nu + 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 2\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{3} - 13\beta_{2} + 3\beta _1 - 11 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
153.1
0.500000 0.0913379i
0.500000 2.73709i
0.500000 + 2.73709i
0.500000 + 0.0913379i
0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
153.2 0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
153.3 0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
153.4 0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.c.b 4
3.b odd 2 1 2772.2.i.a 4
4.b odd 2 1 1232.2.e.d 4
7.b odd 2 1 inner 308.2.c.b 4
7.c even 3 2 2156.2.q.a 8
7.d odd 6 2 2156.2.q.a 8
11.b odd 2 1 inner 308.2.c.b 4
21.c even 2 1 2772.2.i.a 4
28.d even 2 1 1232.2.e.d 4
33.d even 2 1 2772.2.i.a 4
44.c even 2 1 1232.2.e.d 4
77.b even 2 1 inner 308.2.c.b 4
77.h odd 6 2 2156.2.q.a 8
77.i even 6 2 2156.2.q.a 8
231.h odd 2 1 2772.2.i.a 4
308.g odd 2 1 1232.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.c.b 4 1.a even 1 1 trivial
308.2.c.b 4 7.b odd 2 1 inner
308.2.c.b 4 11.b odd 2 1 inner
308.2.c.b 4 77.b even 2 1 inner
1232.2.e.d 4 4.b odd 2 1
1232.2.e.d 4 28.d even 2 1
1232.2.e.d 4 44.c even 2 1
1232.2.e.d 4 308.g odd 2 1
2156.2.q.a 8 7.c even 3 2
2156.2.q.a 8 7.d odd 6 2
2156.2.q.a 8 77.h odd 6 2
2156.2.q.a 8 77.i even 6 2
2772.2.i.a 4 3.b odd 2 1
2772.2.i.a 4 21.c even 2 1
2772.2.i.a 4 33.d even 2 1
2772.2.i.a 4 231.h odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T + 11)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 14)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 14)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 56)^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 14)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$53$ \( (T + 4)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 126)^{2} \) Copy content Toggle raw display
$67$ \( (T + 12)^{4} \) Copy content Toggle raw display
$71$ \( (T - 8)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 126)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
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