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Properties

Label 624.2.bv.g
Level $624$
Weight $2$
Character orbit 624.bv
Analytic conductor $4.983$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(49,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.bv (of order \(6\), degree \(2\), not 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.649638144.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + ( - \beta_{5} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{3} + \cdots + 1) q^{7} + \beta_{2} q^{9} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 2) q^{13}+ \cdots + (\beta_{7} + \beta_{6} + \cdots - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{9} + 6 q^{11} - 6 q^{13} - 6 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{23} - 4 q^{25} - 8 q^{27} + 6 q^{29} + 6 q^{33} - 10 q^{35} - 24 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{49} + 24 q^{51}+ \cdots - 66 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 25\nu^{5} + 315\nu^{3} + 182\nu^{2} - 740\nu - 546 ) / 364 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 62\nu^{3} + 79\nu - 26 ) / 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{7} + 13\nu^{6} + 98\nu^{5} - 130\nu^{4} - 616\nu^{3} + 546\nu^{2} + 1463\nu - 845 ) / 364 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} + 39\nu^{6} - 82\nu^{5} - 481\nu^{4} + 560\nu^{3} + 1911\nu^{2} - 1226\nu - 2080 ) / 364 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - 13\nu^{6} + 57\nu^{5} + 130\nu^{4} - 245\nu^{3} - 546\nu^{2} + 304\nu + 663 ) / 182 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -11\nu^{7} + 89\nu^{5} + 91\nu^{4} - 175\nu^{3} - 819\nu^{2} - 314\nu + 1729 ) / 364 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\nu^{7} - 13\nu^{6} - 146\nu^{5} + 130\nu^{4} + 420\nu^{3} - 546\nu^{2} + 192\nu + 663 ) / 364 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_{2} + 3\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{7} + 5\beta_{6} + 2\beta_{5} + 5\beta_{4} + 3\beta_{3} + 19\beta_{2} + 15\beta _1 + 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{7} - \beta_{6} - 6\beta_{5} - 5\beta_{4} + 3\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 63\beta_{7} + 18\beta_{6} - 11\beta_{5} + 18\beta_{4} - 13\beta_{3} + 147\beta_{2} + 54\beta _1 + 67 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -25\beta_{7} + 22\beta_{6} - 99\beta_{5} - 58\beta_{4} + 7\beta_{3} + 7\beta_{2} + 18\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 251\beta_{7} + 21\beta_{6} - 199\beta_{5} + 21\beta_{4} - 210\beta_{3} + 830\beta_{2} + 63\beta _1 + 310 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.42055 − 0.500000i
−2.34138 + 0.500000i
−1.42055 − 0.500000i
2.34138 + 0.500000i
2.34138 − 0.500000i
−1.42055 + 0.500000i
−2.34138 − 0.500000i
1.42055 + 0.500000i
0 0.500000 + 0.866025i 0 − 1.55452i 0 2.56383 + 1.48023i 0 −0.500000 + 0.866025i 0
49.2 0 0.500000 + 0.866025i 0 − 0.475353i 0 −3.94508 − 2.27769i 0 −0.500000 + 0.866025i 0
49.3 0 0.500000 + 0.866025i 0 1.28657i 0 −1.69781 − 0.980228i 0 −0.500000 + 0.866025i 0
49.4 0 0.500000 + 0.866025i 0 4.20740i 0 3.07905 + 1.77769i 0 −0.500000 + 0.866025i 0
433.1 0 0.500000 − 0.866025i 0 − 4.20740i 0 3.07905 − 1.77769i 0 −0.500000 − 0.866025i 0
433.2 0 0.500000 − 0.866025i 0 − 1.28657i 0 −1.69781 + 0.980228i 0 −0.500000 − 0.866025i 0
433.3 0 0.500000 − 0.866025i 0 0.475353i 0 −3.94508 + 2.27769i 0 −0.500000 − 0.866025i 0
433.4 0 0.500000 − 0.866025i 0 1.55452i 0 2.56383 − 1.48023i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.bv.g 8
3.b odd 2 1 1872.2.by.m 8
4.b odd 2 1 312.2.bf.b ✓ 8
12.b even 2 1 936.2.bi.c 8
13.e even 6 1 inner 624.2.bv.g 8
13.f odd 12 1 8112.2.a.cq 4
13.f odd 12 1 8112.2.a.cs 4
39.h odd 6 1 1872.2.by.m 8
52.i odd 6 1 312.2.bf.b ✓ 8
52.i odd 6 1 4056.2.c.p 8
52.j odd 6 1 4056.2.c.p 8
52.l even 12 1 4056.2.a.bd 4
52.l even 12 1 4056.2.a.be 4
156.r even 6 1 936.2.bi.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bf.b ✓ 8 4.b odd 2 1
312.2.bf.b ✓ 8 52.i odd 6 1
624.2.bv.g 8 1.a even 1 1 trivial
624.2.bv.g 8 13.e even 6 1 inner
936.2.bi.c 8 12.b even 2 1
936.2.bi.c 8 156.r even 6 1
1872.2.by.m 8 3.b odd 2 1
1872.2.by.m 8 39.h odd 6 1
4056.2.a.bd 4 52.l even 12 1
4056.2.a.be 4 52.l even 12 1
4056.2.c.p 8 52.i odd 6 1
4056.2.c.p 8 52.j odd 6 1
8112.2.a.cq 4 13.f odd 12 1
8112.2.a.cs 4 13.f odd 12 1

Hecke kernels

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

\( T_{5}^{8} + 22T_{5}^{6} + 81T_{5}^{4} + 88T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 23T_{7}^{6} + 435T_{7}^{4} - 414T_{7}^{3} - 2054T_{7}^{2} + 1692T_{7} + 8836 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 22 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} - 23 T^{6} + \cdots + 8836 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 141376 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{8} - 6 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{8} + 118 T^{6} + \cdots + 141376 \) Copy content Toggle raw display
$37$ \( T^{8} - 41 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( T^{8} + 24 T^{7} + \cdots + 3154176 \) Copy content Toggle raw display
$43$ \( T^{8} - 8 T^{7} + \cdots + 481636 \) Copy content Toggle raw display
$47$ \( T^{8} + 160 T^{6} + \cdots + 2096704 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} + \cdots + 208)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 36 T^{7} + \cdots + 262144 \) Copy content Toggle raw display
$61$ \( T^{8} - 2 T^{7} + \cdots + 42003361 \) Copy content Toggle raw display
$67$ \( T^{8} + 36 T^{7} + \cdots + 45796 \) Copy content Toggle raw display
$71$ \( T^{8} + 6 T^{7} + \cdots + 1272384 \) Copy content Toggle raw display
$73$ \( T^{8} + 124 T^{6} + \cdots + 185761 \) Copy content Toggle raw display
$79$ \( (T^{4} - 6 T^{3} + \cdots + 216)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 312 T^{6} + \cdots + 1557504 \) Copy content Toggle raw display
$89$ \( T^{8} + 12 T^{7} + \cdots + 589824 \) Copy content Toggle raw display
$97$ \( T^{8} + 66 T^{7} + \cdots + 33039504 \) Copy content Toggle raw display
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