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Properties

Label 616.4.a.j
Level $616$
Weight $4$
Character orbit 616.a
Self dual yes
Analytic conductor $36.345$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [616,4,Mod(1,616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(616, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("616.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 616.a (trivial)

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: yes
Analytic conductor: \(36.3451765635\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 145x^{5} - 10x^{4} + 4790x^{3} - 2452x^{2} - 1496x + 320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} + 1) q^{5} - 7 q^{7} + (\beta_{3} - \beta_{2} + 15) q^{9} - 11 q^{11} + ( - \beta_{6} - 2 \beta_{2} + \cdots + 13) q^{13} + (\beta_{5} - \beta_{3} - \beta_{2} + \cdots - 15) q^{15}+ \cdots + ( - 11 \beta_{3} + 11 \beta_{2} - 165) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 6 q^{5} - 49 q^{7} + 101 q^{9} - 77 q^{11} + 88 q^{13} - 106 q^{15} + 134 q^{17} + 14 q^{19} + 42 q^{23} + 343 q^{25} - 30 q^{27} + 482 q^{29} + 50 q^{31} - 42 q^{35} + 152 q^{37} + 72 q^{39} + 234 q^{41}+ \cdots - 1111 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 145x^{5} - 10x^{4} + 4790x^{3} - 2452x^{2} - 1496x + 320 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 41\nu^{6} + 14\nu^{5} - 5961\nu^{4} - 2300\nu^{3} + 196706\nu^{2} - 44544\nu - 43900 ) / 2556 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41\nu^{6} + 14\nu^{5} - 5961\nu^{4} - 2300\nu^{3} + 199262\nu^{2} - 44544\nu - 151252 ) / 2556 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -59\nu^{6} - 34\nu^{5} + 8391\nu^{4} + 6032\nu^{3} - 266094\nu^{2} - 49036\nu + 236 ) / 2556 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8\nu^{6} + \nu^{5} - 1151\nu^{4} - 357\nu^{3} + 37663\nu^{2} - 4906\nu - 14161 ) / 213 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{6} + 6\nu^{5} + 3318\nu^{4} - 580\nu^{3} - 109000\nu^{2} + 81040\nu + 23451 ) / 639 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 79\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{6} - 8\beta_{5} - 16\beta_{4} + 94\beta_{3} - 114\beta_{2} + 48\beta _1 + 3331 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -102\beta_{6} - 285\beta_{5} - 165\beta_{4} + 349\beta_{3} - 148\beta_{2} + 6709\beta _1 + 2921 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1039\beta_{6} - 1178\beta_{5} - 2326\beta_{4} + 8862\beta_{3} - 11720\beta_{2} + 10206\beta _1 + 283145 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
10.0610
6.46220
0.812616
0.179568
−0.466684
−7.90093
−9.14775
0 −10.0610 0 14.3419 0 −7.00000 0 74.2235 0
1.2 0 −6.46220 0 −3.00579 0 −7.00000 0 14.7600 0
1.3 0 −0.812616 0 −16.9890 0 −7.00000 0 −26.3397 0
1.4 0 −0.179568 0 18.8308 0 −7.00000 0 −26.9678 0
1.5 0 0.466684 0 −6.69970 0 −7.00000 0 −26.7822 0
1.6 0 7.90093 0 −12.7900 0 −7.00000 0 35.4247 0
1.7 0 9.14775 0 12.3118 0 −7.00000 0 56.6814 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( +1 \)
\(11\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 616.4.a.j ✓ 7
4.b odd 2 1 1232.4.a.bd 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.4.a.j ✓ 7 1.a even 1 1 trivial
1232.4.a.bd 7 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(616))\):

\( T_{3}^{7} - 145T_{3}^{5} + 10T_{3}^{4} + 4790T_{3}^{3} + 2452T_{3}^{2} - 1496T_{3} - 320 \) Copy content Toggle raw display
\( T_{5}^{7} - 6T_{5}^{6} - 591T_{5}^{5} + 2228T_{5}^{4} + 107076T_{5}^{3} - 111752T_{5}^{2} - 6037280T_{5} - 14549536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - 145 T^{5} + \cdots - 320 \) Copy content Toggle raw display
$5$ \( T^{7} - 6 T^{6} + \cdots - 14549536 \) Copy content Toggle raw display
$7$ \( (T + 7)^{7} \) Copy content Toggle raw display
$11$ \( (T + 11)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 79161659136 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 151755555840 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 15743217343488 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 120191659904000 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 2542483952512 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 5980457097664 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 16\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 45\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 32\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 62\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 55\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 90\!\cdots\!40 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 26\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 17\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 93\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 18\!\cdots\!96 \) Copy content Toggle raw display
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