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Properties

Label 624.6.c.f
Level $624$
Weight $6$
Character orbit 624.c
Analytic conductor $100.080$
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] = [624,6,Mod(337,624)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.337");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 624.c (of order \(2\), degree \(1\), 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: \(100.079503563\)
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: \( x^{8} + 4385x^{6} + 4890448x^{4} + 396656640x^{2} + 3049690176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 78)
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.

\(f(q)\) \(=\) \( q - 9 q^{3} + \beta_{3} q^{5} + (\beta_{7} + \beta_{3}) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + \beta_{3} q^{5} + (\beta_{7} + \beta_{3}) q^{7} + 81 q^{9} + ( - \beta_{7} - \beta_{6} + \cdots + 2 \beta_{4}) q^{11}+ \cdots + ( - 81 \beta_{7} + \cdots + 162 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{3} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 72 q^{3} + 648 q^{9} - 52 q^{13} + 600 q^{17} + 1656 q^{23} - 10128 q^{25} - 5832 q^{27} - 3360 q^{29} - 27144 q^{35} + 468 q^{39} - 16944 q^{43} - 59480 q^{49} - 5400 q^{51} + 9576 q^{53} - 5424 q^{55} + 58344 q^{61} - 192888 q^{65} - 14904 q^{69} + 91152 q^{75} + 243960 q^{77} - 155968 q^{79} + 52488 q^{81} + 30240 q^{87} + 159280 q^{91} + 368232 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4385x^{6} + 4890448x^{4} + 396656640x^{2} + 3049690176 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 16817\nu^{4} - 58831796\nu^{2} - 98920316037 ) / 113662995 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7567\nu^{6} - 36323843\nu^{4} - 44025330148\nu^{2} - 2523873114792 ) / 2045933910 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1828\nu^{7} + 8008877\nu^{5} + 8823651193\nu^{3} + 1758896749296\nu ) / 627692523588 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7312\nu^{7} + 32035508\nu^{5} + 35294604772\nu^{3} + 2014046808480\nu ) / 52307710299 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2785207 \nu^{7} - 49832757 \nu^{6} + 11830623659 \nu^{5} - 210344950881 \nu^{4} + \cdots - 73\!\cdots\!24 ) / 18830775707640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2785207 \nu^{7} + 49832757 \nu^{6} + 11830623659 \nu^{5} + 210344950881 \nu^{4} + \cdots + 73\!\cdots\!24 ) / 18830775707640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8558159\nu^{7} + 37352135791\nu^{5} + 41414444082176\nu^{3} + 3255885627467184\nu ) / 37661551415280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + 48\beta_{3} ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} + 2\beta_{5} - 3\beta_{2} - 58\beta _1 - 52618 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 780\beta_{7} - 75\beta_{6} - 75\beta_{5} - 844\beta_{4} - 12732\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2690\beta_{6} + 2690\beta_{5} - 3765\beta_{2} + 35495\beta _1 + 28344677 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2587980\beta_{7} - 54885\beta_{6} - 54885\beta_{5} + 3732178\beta_{4} + 28366476\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5273944\beta_{6} - 5273944\beta_{5} + 6397386\beta_{2} - 28674839\beta _1 - 21177167049 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7573504140 \beta_{7} + 602484165 \beta_{6} + 602484165 \beta_{5} - 12157292584 \beta_{4} - 64475908332 \beta_{3} ) / 12 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 49.6941i
− 42.7585i
− 8.86675i
− 2.93114i
2.93114i
8.86675i
42.7585i
49.6941i
0 −9.00000 0 − 101.388i 0 49.3291i 0 81.0000 0
337.2 0 −9.00000 0 − 83.5169i 0 − 187.367i 0 81.0000 0
337.3 0 −9.00000 0 − 15.7335i 0 − 69.0882i 0 81.0000 0
337.4 0 −9.00000 0 − 7.86228i 0 − 233.784i 0 81.0000 0
337.5 0 −9.00000 0 7.86228i 0 233.784i 0 81.0000 0
337.6 0 −9.00000 0 15.7335i 0 69.0882i 0 81.0000 0
337.7 0 −9.00000 0 83.5169i 0 187.367i 0 81.0000 0
337.8 0 −9.00000 0 101.388i 0 − 49.3291i 0 81.0000 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 624.6.c.f 8
4.b odd 2 1 78.6.b.b ✓ 8
12.b even 2 1 234.6.b.d 8
13.b even 2 1 inner 624.6.c.f 8
52.b odd 2 1 78.6.b.b ✓ 8
52.f even 4 1 1014.6.a.t 4
52.f even 4 1 1014.6.a.u 4
156.h even 2 1 234.6.b.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.6.b.b ✓ 8 4.b odd 2 1
78.6.b.b ✓ 8 52.b odd 2 1
234.6.b.d 8 12.b even 2 1
234.6.b.d 8 156.h even 2 1
624.6.c.f 8 1.a even 1 1 trivial
624.6.c.f 8 13.b even 2 1 inner
1014.6.a.t 4 52.f even 4 1
1014.6.a.u 4 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 17564T_{5}^{6} + 77053936T_{5}^{4} + 22445285184T_{5}^{2} + 1097164071936 \) acting on \(S_{6}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T + 9)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 1097164071936 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( (T^{4} - 300 T^{3} + \cdots + 223697296368)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 17976140163072)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 12705499286928)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 47\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 53\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 53\!\cdots\!60)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 46\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
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