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Properties

Label 364.2.g.a
Level $364$
Weight $2$
Character orbit 364.g
Analytic conductor $2.907$
Analytic rank $0$
Dimension $8$
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] = [364,2,Mod(337,364)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(364, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("364.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 364.g (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: \(2.90655463357\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 16x^{6} + 80x^{4} + 132x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{4} - \beta_{3}) q^{5} + \beta_{3} q^{7} + ( - \beta_{7} - \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{4} - \beta_{3}) q^{5} + \beta_{3} q^{7} + ( - \beta_{7} - \beta_{5} + 1) q^{9} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{11} + (\beta_{7} + \beta_{6} + \beta_{3}) q^{13} + ( - \beta_{6} - \beta_{4} + \beta_1) q^{15} + ( - 2 \beta_{5} - \beta_{2}) q^{17} + (\beta_{6} - \beta_{3} + 2 \beta_1) q^{19} - \beta_1 q^{21} + ( - \beta_{5} - \beta_{2} - 1) q^{23} + ( - \beta_{5} - \beta_{2}) q^{25} + ( - 2 \beta_{7} + 2) q^{27} + (3 \beta_{7} - \beta_{2} - 1) q^{29} + (\beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{31} + (\beta_{6} - \beta_{4} + \cdots + 2 \beta_1) q^{33}+ \cdots + ( - \beta_{6} + \beta_{4} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 2 q^{13} + 4 q^{17} - 6 q^{23} + 2 q^{25} + 12 q^{27} - 2 q^{29} + 6 q^{35} - 6 q^{43} - 8 q^{49} - 8 q^{51} + 22 q^{53} - 20 q^{55} + 8 q^{61} - 6 q^{65} - 20 q^{69} - 20 q^{75} - 26 q^{79} - 24 q^{81} - 32 q^{87} - 10 q^{91} - 18 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 16x^{6} + 80x^{4} + 132x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 8\nu^{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 16\nu^{5} - 72\nu^{3} - 68\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 16\nu^{5} + 80\nu^{3} + 116\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 14\nu^{4} - 52\nu^{2} - 36 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + 40\nu^{5} + 144\nu^{3} + 92\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 56\nu^{2} + 52 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} + 2\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{7} - 8\beta_{5} + 2\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{6} - 18\beta_{4} - 24\beta_{3} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60\beta_{7} + 56\beta_{5} - 28\beta_{2} - 164 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32\beta_{6} + 144\beta_{4} + 224\beta_{3} - 276\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/364\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(183\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
2.40538i
− 2.40538i
1.29363i
− 1.29363i
0.927719i
− 0.927719i
− 2.77129i
2.77129i
0 −2.40538 0 − 0.257562i 0 1.00000i 0 2.78585 0
337.2 0 −2.40538 0 0.257562i 0 − 1.00000i 0 2.78585 0
337.3 0 −1.29363 0 − 2.79846i 0 1.00000i 0 −1.32653 0
337.4 0 −1.29363 0 2.79846i 0 − 1.00000i 0 −1.32653 0
337.5 0 0.927719 0 − 2.38393i 0 − 1.00000i 0 −2.13934 0
337.6 0 0.927719 0 2.38393i 0 1.00000i 0 −2.13934 0
337.7 0 2.77129 0 − 2.32791i 0 1.00000i 0 4.68002 0
337.8 0 2.77129 0 2.32791i 0 − 1.00000i 0 4.68002 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 364.2.g.a ✓ 8
3.b odd 2 1 3276.2.e.f 8
4.b odd 2 1 1456.2.k.d 8
7.b odd 2 1 2548.2.g.g 8
7.c even 3 2 2548.2.y.f 16
7.d odd 6 2 2548.2.y.e 16
13.b even 2 1 inner 364.2.g.a ✓ 8
13.d odd 4 1 4732.2.a.o 4
13.d odd 4 1 4732.2.a.p 4
39.d odd 2 1 3276.2.e.f 8
52.b odd 2 1 1456.2.k.d 8
91.b odd 2 1 2548.2.g.g 8
91.r even 6 2 2548.2.y.f 16
91.s odd 6 2 2548.2.y.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.g.a ✓ 8 1.a even 1 1 trivial
364.2.g.a ✓ 8 13.b even 2 1 inner
1456.2.k.d 8 4.b odd 2 1
1456.2.k.d 8 52.b odd 2 1
2548.2.g.g 8 7.b odd 2 1
2548.2.g.g 8 91.b odd 2 1
2548.2.y.e 16 7.d odd 6 2
2548.2.y.e 16 91.s odd 6 2
2548.2.y.f 16 7.c even 3 2
2548.2.y.f 16 91.r even 6 2
3276.2.e.f 8 3.b odd 2 1
3276.2.e.f 8 39.d odd 2 1
4732.2.a.o 4 13.d odd 4 1
4732.2.a.p 4 13.d odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(364, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 8 T^{2} - 2 T + 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 19 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 36 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} - 52 T^{2} + \cdots + 44)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 71 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$23$ \( (T^{4} + 3 T^{3} - 13 T^{2} + \cdots - 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + T^{3} - 83 T^{2} + \cdots + 1182)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 123 T^{6} + \cdots + 322624 \) Copy content Toggle raw display
$37$ \( T^{8} + 212 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$41$ \( T^{8} + 156 T^{6} + \cdots + 123904 \) Copy content Toggle raw display
$43$ \( (T^{4} + 3 T^{3} + \cdots + 232)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 239 T^{6} + \cdots + 4032064 \) Copy content Toggle raw display
$53$ \( (T^{4} - 11 T^{3} + \cdots - 54)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 180 T^{6} + \cdots + 147456 \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} - 112 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 368 T^{6} + \cdots + 11943936 \) Copy content Toggle raw display
$71$ \( T^{8} + 408 T^{6} + \cdots + 2214144 \) Copy content Toggle raw display
$73$ \( T^{8} + 279 T^{6} + \cdots + 2421136 \) Copy content Toggle raw display
$79$ \( (T^{4} + 13 T^{3} + \cdots - 36)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 547 T^{6} + \cdots + 75481344 \) Copy content Toggle raw display
$89$ \( T^{8} + 439 T^{6} + \cdots + 57335184 \) Copy content Toggle raw display
$97$ \( T^{8} + 447 T^{6} + \cdots + 75134224 \) Copy content Toggle raw display
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