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Properties

Label 656.6.a.l
Level $656$
Weight $6$
Character orbit 656.a
Self dual yes
Analytic conductor $105.212$
Analytic rank $0$
Dimension $14$
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] = [656,6,Mod(1,656)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(656, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("656.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 656 = 2^{4} \cdot 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 656.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: \(105.211785797\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2328 x^{12} - 620 x^{11} + 1971272 x^{10} + 374128 x^{9} - 746872952 x^{8} + \cdots + 100072399345920 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 328)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{5} + 3) q^{5} + (\beta_{7} - 11) q^{7} + (\beta_{8} - \beta_{7} + 90) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{5} + 3) q^{5} + (\beta_{7} - 11) q^{7} + (\beta_{8} - \beta_{7} + 90) q^{9} + (\beta_{7} - \beta_{5} + \cdots + 2 \beta_1) q^{11}+ \cdots + (64 \beta_{13} - 84 \beta_{12} + \cdots - 9686) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 44 q^{5} - 148 q^{7} + 1254 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 44 q^{5} - 148 q^{7} + 1254 q^{9} + 4 q^{11} + 1324 q^{13} - 628 q^{15} + 2220 q^{17} + 2464 q^{19} - 2224 q^{21} + 360 q^{23} + 5026 q^{25} + 1860 q^{27} + 9196 q^{29} - 2776 q^{31} + 5552 q^{33} - 1164 q^{35} + 15660 q^{37} - 30904 q^{39} + 23534 q^{41} - 25240 q^{43} - 2844 q^{45} - 30256 q^{47} + 59582 q^{49} - 58120 q^{51} + 70860 q^{53} - 58980 q^{55} + 124896 q^{57} - 13496 q^{59} + 123532 q^{61} - 165536 q^{63} + 127544 q^{65} - 14068 q^{67} + 137520 q^{69} - 122896 q^{71} + 94412 q^{73} - 44928 q^{75} + 231912 q^{77} - 40800 q^{79} + 386542 q^{81} - 138208 q^{83} + 353800 q^{85} - 170232 q^{87} + 406620 q^{89} - 177720 q^{91} + 581568 q^{93} - 292068 q^{95} + 471740 q^{97} - 137696 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2328 x^{12} - 620 x^{11} + 1971272 x^{10} + 374128 x^{9} - 746872952 x^{8} + \cdots + 100072399345920 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 35\!\cdots\!05 \nu^{13} + \cdots - 32\!\cdots\!32 ) / 90\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!97 \nu^{13} + \cdots - 48\!\cdots\!56 ) / 90\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!61 \nu^{13} + \cdots + 90\!\cdots\!18 ) / 15\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 49\!\cdots\!51 \nu^{13} + \cdots - 15\!\cdots\!52 ) / 30\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\!\cdots\!31 \nu^{13} + \cdots - 47\!\cdots\!92 ) / 90\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 62\!\cdots\!94 \nu^{13} + \cdots - 25\!\cdots\!52 ) / 90\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 62\!\cdots\!94 \nu^{13} + \cdots - 55\!\cdots\!56 ) / 90\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!66 \nu^{13} + \cdots + 22\!\cdots\!00 ) / 15\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47\!\cdots\!49 \nu^{13} + \cdots + 12\!\cdots\!84 ) / 30\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 60\!\cdots\!81 \nu^{13} + \cdots - 68\!\cdots\!36 ) / 30\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\!\cdots\!20 \nu^{13} + \cdots - 19\!\cdots\!36 ) / 90\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 27\!\cdots\!06 \nu^{13} + \cdots - 13\!\cdots\!32 ) / 90\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + 333 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 5 \beta_{13} + 5 \beta_{12} - 2 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + \beta_{7} + \cdots + 129 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 22 \beta_{13} - 84 \beta_{12} + 16 \beta_{11} + 43 \beta_{10} + 9 \beta_{9} + 883 \beta_{8} + \cdots + 211359 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5959 \beta_{13} + 5276 \beta_{12} - 1471 \beta_{11} + 3154 \beta_{10} + 1957 \beta_{9} + \cdots + 376147 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36491 \beta_{13} - 107838 \beta_{12} + 16587 \beta_{11} + 48104 \beta_{10} + 9221 \beta_{9} + \cdots + 156114363 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5809764 \beta_{13} + 4975357 \beta_{12} - 1037427 \beta_{11} + 2816176 \beta_{10} + 1787834 \beta_{9} + \cdots + 388543280 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 45406794 \beta_{13} - 111056298 \beta_{12} + 14033084 \beta_{11} + 43428837 \beta_{10} + \cdots + 124096886117 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5321185820 \beta_{13} + 4569798201 \beta_{12} - 775112121 \beta_{11} + 2405815825 \beta_{10} + \cdots + 301686162886 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 50023794340 \beta_{13} - 106755043100 \beta_{12} + 11804842256 \beta_{11} + 36645027248 \beta_{10} + \cdots + 102509167914906 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4758772883284 \beta_{13} + 4154024322412 \beta_{12} - 612878399884 \beta_{11} + \cdots + 186675961559966 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 51766976458628 \beta_{13} - 99772325235318 \beta_{12} + 10233796138942 \beta_{11} + \cdots + 86\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 42\!\cdots\!92 \beta_{13} + \cdots + 75\!\cdots\!92 \) 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
−30.0793
−22.0913
−21.4937
−18.6500
−11.8942
−2.08304
0.722084
2.79910
5.04932
6.34120
15.0017
18.9314
28.6284
28.8183
0 −30.0793 0 −13.0332 0 −55.2397 0 661.766 0
1.2 0 −22.0913 0 92.4547 0 19.9949 0 245.024 0
1.3 0 −21.4937 0 27.1594 0 −215.851 0 218.980 0
1.4 0 −18.6500 0 −62.8950 0 139.790 0 104.823 0
1.5 0 −11.8942 0 −72.7125 0 120.153 0 −101.528 0
1.6 0 −2.08304 0 96.2877 0 −92.3678 0 −238.661 0
1.7 0 0.722084 0 −11.6972 0 −86.5800 0 −242.479 0
1.8 0 2.79910 0 70.7011 0 210.038 0 −235.165 0
1.9 0 5.04932 0 −46.1142 0 −153.002 0 −217.504 0
1.10 0 6.34120 0 −60.4615 0 139.365 0 −202.789 0
1.11 0 15.0017 0 14.7218 0 143.743 0 −17.9479 0
1.12 0 18.9314 0 29.4263 0 −217.311 0 115.398 0
1.13 0 28.6284 0 55.6916 0 81.7768 0 576.587 0
1.14 0 28.8183 0 −75.5290 0 −182.509 0 587.495 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(41\) \( -1 \)

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 656.6.a.l 14
4.b odd 2 1 328.6.a.d ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
328.6.a.d ✓ 14 4.b odd 2 1
656.6.a.l 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 2328 T_{3}^{12} - 620 T_{3}^{11} + 1971272 T_{3}^{10} + 374128 T_{3}^{9} + \cdots + 100072399345920 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(656))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 100072399345920 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 60\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 66\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 71\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 74\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 55\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 49\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 66\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 20\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T - 1681)^{14} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 88\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 17\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 76\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 89\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 61\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 49\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 40\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 18\!\cdots\!80 \) Copy content Toggle raw display
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