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Properties

Label 624.2.n.f
Level $624$
Weight $2$
Character orbit 624.n
Analytic conductor $4.983$
Analytic rank $0$
Dimension $8$
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] = [624,2,Mod(623,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("624.623");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.n (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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.170772624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{5} + \beta_{3}) q^{5} + (\beta_{5} + \beta_{3}) q^{7} + (\beta_{7} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{5} + \beta_{3}) q^{5} + (\beta_{5} + \beta_{3}) q^{7} + (\beta_{7} + \beta_1 + 1) q^{9} + 2 \beta_{6} q^{11} + (\beta_{6} - \beta_{4} - \beta_1 + 1) q^{13} + ( - \beta_{6} - \beta_{5} - \beta_{2}) q^{15} + (2 \beta_{7} - \beta_{4} + \beta_1) q^{17} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{19} + ( - \beta_{6} - \beta_{5} - \beta_{2}) q^{21} + ( - \beta_{4} - \beta_1 + 3) q^{25} + (\beta_{7} + 2 \beta_{4} - 2 \beta_1 + 4) q^{27} + (4 \beta_{7} + 2 \beta_{4} - 2 \beta_1) q^{29} + 2 \beta_{2} q^{31} + (2 \beta_{6} + 2 \beta_{3}) q^{33} + ( - \beta_{4} - \beta_1 + 8) q^{35} + ( - 4 \beta_{6} - \beta_{5} + \beta_{3}) q^{37} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \cdots - 4) q^{39}+ \cdots + (4 \beta_{6} - 2 \beta_{5} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 10 q^{9} + 4 q^{13} + 20 q^{25} + 32 q^{27} + 60 q^{35} - 32 q^{39} + 4 q^{49} + 26 q^{51} + 16 q^{61} - 28 q^{75} - 14 q^{81} - 4 q^{87} - 120 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - \nu^{5} + \nu^{4} - 2\nu^{3} + 4\nu^{2} - 4\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 3\nu^{4} + 6\nu^{2} + 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 5\nu^{5} + 4\nu^{4} - 6\nu^{3} + 10\nu^{2} - 12\nu + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 5\nu^{5} + 4\nu^{4} - 4\nu^{3} + 10\nu^{2} - 16\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} + 7\nu^{5} - 5\nu^{4} + 6\nu^{3} - 20\nu^{2} + 24\nu - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 4\nu^{6} - 8\nu^{5} + 5\nu^{4} - 8\nu^{3} + 22\nu^{2} - 20\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 5\nu^{6} - 7\nu^{5} + 6\nu^{4} - 10\nu^{3} + 24\nu^{2} - 28\nu + 28 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{2} + 4\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + \beta_{6} - 4\beta_{5} - 4\beta_{4} - 2\beta_{3} + \beta_{2} + 8\beta _1 - 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 3\beta_{5} + 2\beta_{4} - 3\beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - 7\beta_{6} + 2\beta_{5} + 8\beta_{4} + 4\beta_{3} + 7\beta_{2} - 4\beta _1 - 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10\beta_{7} - 9\beta_{6} - 4\beta_{4} - 2\beta_{3} + 3\beta_{2} + 34 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{6} + 5\beta_{5} + 6\beta_{4} - 5\beta_{3} + 6\beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -14\beta_{7} - 3\beta_{6} - 10\beta_{5} - \beta_{2} + 52\beta _1 - 10 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
623.1
−1.02187 − 0.977642i
0.335728 + 1.37379i
−1.02187 + 0.977642i
0.335728 − 1.37379i
1.41203 + 0.0786378i
0.774115 + 1.18353i
1.41203 − 0.0786378i
0.774115 − 1.18353i
0 −1.18614 − 1.26217i 0 −3.22060 0 −3.22060 0 −0.186141 + 2.99422i 0
623.2 0 −1.18614 − 1.26217i 0 3.22060 0 3.22060 0 −0.186141 + 2.99422i 0
623.3 0 −1.18614 + 1.26217i 0 −3.22060 0 −3.22060 0 −0.186141 − 2.99422i 0
623.4 0 −1.18614 + 1.26217i 0 3.22060 0 3.22060 0 −0.186141 − 2.99422i 0
623.5 0 1.68614 − 0.396143i 0 −2.15121 0 −2.15121 0 2.68614 − 1.33591i 0
623.6 0 1.68614 − 0.396143i 0 2.15121 0 2.15121 0 2.68614 − 1.33591i 0
623.7 0 1.68614 + 0.396143i 0 −2.15121 0 −2.15121 0 2.68614 + 1.33591i 0
623.8 0 1.68614 + 0.396143i 0 2.15121 0 2.15121 0 2.68614 + 1.33591i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 623.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.n.f yes 8
3.b odd 2 1 624.2.n.e ✓ 8
4.b odd 2 1 624.2.n.e ✓ 8
12.b even 2 1 inner 624.2.n.f yes 8
13.b even 2 1 inner 624.2.n.f yes 8
39.d odd 2 1 624.2.n.e ✓ 8
52.b odd 2 1 624.2.n.e ✓ 8
156.h even 2 1 inner 624.2.n.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.n.e ✓ 8 3.b odd 2 1
624.2.n.e ✓ 8 4.b odd 2 1
624.2.n.e ✓ 8 39.d odd 2 1
624.2.n.e ✓ 8 52.b odd 2 1
624.2.n.f yes 8 1.a even 1 1 trivial
624.2.n.f yes 8 12.b even 2 1 inner
624.2.n.f yes 8 13.b even 2 1 inner
624.2.n.f yes 8 156.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{4} - 15T_{5}^{2} + 48 \) Copy content Toggle raw display
\( T_{7}^{4} - 15T_{7}^{2} + 48 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{107}^{2} + 12T_{107} - 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} - 2 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 15 T^{2} + 48)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 15 T^{2} + 48)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 36 T^{2} + 192)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 43 T^{2} + 256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 60 T^{2} + 768)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 108 T^{2} + 1728)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 165 T^{2} + 5808)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 108 T^{2} + 1728)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 87 T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 45 T^{2} + 432)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 112 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 192 T^{2} + 768)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 165 T^{2} + 5808)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 144 T^{2} + 3072)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 36 T^{2} + 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 300 T^{2} + 192)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 180 T^{2} + 6912)^{2} \) Copy content Toggle raw display
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