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Properties

Label 312.2.h.a
Level $312$
Weight $2$
Character orbit 312.h
Analytic conductor $2.491$
Analytic rank $0$
Dimension $8$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [312,2,Mod(155,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("312.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.h (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: \(2.49133254306\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.151613669376.21
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} + ( - \beta_{7} - \beta_{4}) q^{5} - \beta_{7} q^{6} + (\beta_{7} - \beta_{6} + \beta_{4}) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} + ( - \beta_{7} - \beta_{4}) q^{5} - \beta_{7} q^{6} + (\beta_{7} - \beta_{6} + \beta_{4}) q^{8} + 3 q^{9} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} + 1) q^{10} + (\beta_{7} - \beta_{4} + 2 \beta_1) q^{11} + ( - \beta_{5} - 1) q^{12} + ( - \beta_{3} - 2 \beta_{2}) q^{13} + (\beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{15}+ \cdots + (3 \beta_{7} - 3 \beta_{4} + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} + 4 q^{10} - 12 q^{12} - 20 q^{16} + 28 q^{22} - 40 q^{25} - 36 q^{30} + 44 q^{40} - 32 q^{43} - 56 q^{49} + 52 q^{52} - 60 q^{66} + 96 q^{75} + 72 q^{81} + 68 q^{82} + 4 q^{88} + 12 q^{90} + 20 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + \nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 4\nu^{5} + 5\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 5\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - \nu^{3} ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + \beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{7} + \beta_{6} - \beta_{4} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
−1.19709 − 0.752986i
−1.19709 + 0.752986i
−0.752986 − 1.19709i
−0.752986 + 1.19709i
0.752986 − 1.19709i
0.752986 + 1.19709i
1.19709 − 0.752986i
1.19709 + 0.752986i
−1.19709 − 0.752986i −1.73205 0.866025 + 1.80278i 4.11439i 2.07341 + 1.30421i 0 0.320758 − 2.81018i 3.00000 3.09808 − 4.92527i
155.2 −1.19709 + 0.752986i −1.73205 0.866025 − 1.80278i − 4.11439i 2.07341 − 1.30421i 0 0.320758 + 2.81018i 3.00000 3.09808 + 4.92527i
155.3 −0.752986 − 1.19709i 1.73205 −0.866025 + 1.80278i − 1.75265i −1.30421 − 2.07341i 0 2.81018 − 0.320758i 3.00000 −2.09808 + 1.31972i
155.4 −0.752986 + 1.19709i 1.73205 −0.866025 − 1.80278i 1.75265i −1.30421 + 2.07341i 0 2.81018 + 0.320758i 3.00000 −2.09808 − 1.31972i
155.5 0.752986 − 1.19709i 1.73205 −0.866025 − 1.80278i − 1.75265i 1.30421 − 2.07341i 0 −2.81018 − 0.320758i 3.00000 −2.09808 − 1.31972i
155.6 0.752986 + 1.19709i 1.73205 −0.866025 + 1.80278i 1.75265i 1.30421 + 2.07341i 0 −2.81018 + 0.320758i 3.00000 −2.09808 + 1.31972i
155.7 1.19709 − 0.752986i −1.73205 0.866025 − 1.80278i 4.11439i −2.07341 + 1.30421i 0 −0.320758 − 2.81018i 3.00000 3.09808 + 4.92527i
155.8 1.19709 + 0.752986i −1.73205 0.866025 + 1.80278i − 4.11439i −2.07341 − 1.30421i 0 −0.320758 + 2.81018i 3.00000 3.09808 − 4.92527i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
8.d odd 2 1 inner
13.b even 2 1 inner
24.f even 2 1 inner
104.h odd 2 1 inner
312.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.h.a ✓ 8
3.b odd 2 1 inner 312.2.h.a ✓ 8
4.b odd 2 1 1248.2.h.a 8
8.b even 2 1 1248.2.h.a 8
8.d odd 2 1 inner 312.2.h.a ✓ 8
12.b even 2 1 1248.2.h.a 8
13.b even 2 1 inner 312.2.h.a ✓ 8
24.f even 2 1 inner 312.2.h.a ✓ 8
24.h odd 2 1 1248.2.h.a 8
39.d odd 2 1 CM 312.2.h.a ✓ 8
52.b odd 2 1 1248.2.h.a 8
104.e even 2 1 1248.2.h.a 8
104.h odd 2 1 inner 312.2.h.a ✓ 8
156.h even 2 1 1248.2.h.a 8
312.b odd 2 1 1248.2.h.a 8
312.h even 2 1 inner 312.2.h.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.h.a ✓ 8 1.a even 1 1 trivial
312.2.h.a ✓ 8 3.b odd 2 1 inner
312.2.h.a ✓ 8 8.d odd 2 1 inner
312.2.h.a ✓ 8 13.b even 2 1 inner
312.2.h.a ✓ 8 24.f even 2 1 inner
312.2.h.a ✓ 8 39.d odd 2 1 CM
312.2.h.a ✓ 8 104.h odd 2 1 inner
312.2.h.a ✓ 8 312.h even 2 1 inner
1248.2.h.a 8 4.b odd 2 1
1248.2.h.a 8 8.b even 2 1
1248.2.h.a 8 12.b even 2 1
1248.2.h.a 8 24.h odd 2 1
1248.2.h.a 8 52.b odd 2 1
1248.2.h.a 8 104.e even 2 1
1248.2.h.a 8 156.h even 2 1
1248.2.h.a 8 312.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 20T_{5}^{2} + 52 \) acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5T^{4} + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 20 T^{2} + 52)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 44 T^{2} + 52)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 164 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 188 T^{2} + 8788)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 236 T^{2} + 52)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 52)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 284 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 208)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 332 T^{2} + 27508)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 356 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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