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Properties

Label 56.7.c.a
Level $56$
Weight $7$
Character orbit 56.c
Analytic conductor $12.883$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [56,7,Mod(41,56)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("56.41"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 56.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.8830286827\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 448x^{10} + 72800x^{8} + 5676816x^{6} + 227194912x^{4} + 4440310336x^{2} + 32999832608 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{51}\cdot 7^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_{8} q^{5} + (\beta_{5} + \beta_{4} - 47) q^{7} + (\beta_{7} + \beta_{5} - \beta_1 - 346) q^{9} + ( - \beta_{7} - \beta_{5} - \beta_{2} + 136) q^{11} + (\beta_{8} + 23 \beta_{4} + \beta_{3}) q^{13}+ \cdots + (110 \beta_{11} - 220 \beta_{8} + \cdots - 537696) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 564 q^{7} - 4148 q^{9} + 1624 q^{11} + 928 q^{15} + 12448 q^{21} - 28104 q^{23} - 81780 q^{25} - 51464 q^{29} - 91104 q^{35} + 76728 q^{37} + 301024 q^{39} + 104472 q^{43} + 46860 q^{49} + 242688 q^{51}+ \cdots - 6448424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 448x^{10} + 72800x^{8} + 5676816x^{6} + 227194912x^{4} + 4440310336x^{2} + 32999832608 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5211328 \nu^{10} + 2183756256 \nu^{8} + 312853278944 \nu^{6} + 19424829697536 \nu^{4} + \cdots + 44\!\cdots\!44 ) / 1116361705529 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 168244949 \nu^{10} - 67781937262 \nu^{8} - 9213443697985 \nu^{6} - 547771254084112 \nu^{4} + \cdots - 13\!\cdots\!45 ) / 1116361705529 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33499945660 \nu^{11} + 499295544755 \nu^{9} + \cdots + 22\!\cdots\!76 \nu ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 41461377948 \nu^{11} - 16282001342723 \nu^{9} + \cdots - 22\!\cdots\!36 \nu ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 130612008413 \nu^{11} + 21129486949 \nu^{10} + 53444125411297 \nu^{9} + \cdots - 50\!\cdots\!20 ) / 71\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 130612008413 \nu^{11} - 2449995920819 \nu^{10} + 53444125411297 \nu^{9} + \cdots - 31\!\cdots\!76 ) / 71\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 130612008413 \nu^{11} + 6537697776837 \nu^{10} - 53444125411297 \nu^{9} + \cdots + 36\!\cdots\!86 ) / 71\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 280529600418 \nu^{11} - 113160907131017 \nu^{9} + \cdots - 23\!\cdots\!88 \nu ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 323051250814 \nu^{11} - 122425054856931 \nu^{9} + \cdots + 45\!\cdots\!64 \nu ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2258141142092 \nu^{11} + 542628582562 \nu^{10} - 837737853408739 \nu^{9} + \cdots + 87\!\cdots\!08 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1194813659309 \nu^{11} - 7080326359399 \nu^{10} - 487269785010096 \nu^{9} + \cdots - 45\!\cdots\!94 ) / 71\!\cdots\!54 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 6 \beta_{11} - 7 \beta_{10} - 4 \beta_{9} - 59 \beta_{8} - 6 \beta_{7} + 15 \beta_{5} + \cdots - 5 \beta_1 ) / 6272 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -12\beta_{11} + 24\beta_{8} - 12\beta_{7} + 4\beta_{6} - 100\beta_{5} - 4\beta_{2} - 113\beta _1 - 66900 ) / 896 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 58 \beta_{11} + 133 \beta_{10} + 524 \beta_{9} + 7617 \beta_{8} + 58 \beta_{7} + 51 \beta_{5} + \cdots - 17 \beta_1 ) / 3136 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 297 \beta_{11} - 594 \beta_{8} + 325 \beta_{7} - 171 \beta_{6} + 2575 \beta_{5} + 143 \beta_{2} + \cdots + 1028551 ) / 112 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 50696 \beta_{11} - 2121 \beta_{10} - 222195 \beta_{9} - 3498648 \beta_{8} + 50696 \beta_{7} + \cdots + 103513 \beta_1 ) / 6272 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 110036 \beta_{11} + 220072 \beta_{8} - 132660 \beta_{7} + 80044 \beta_{6} - 982956 \beta_{5} + \cdots - 339801292 ) / 224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7618432 \beta_{11} - 1096011 \beta_{10} + 22847599 \beta_{9} + 367402800 \beta_{8} + \cdots - 14140853 \beta_1 ) / 3136 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5248059 \beta_{11} - 10496118 \beta_{8} + 6684067 \beta_{7} - 4234453 \beta_{6} + 47654933 \beta_{5} + \cdots + 15872818701 ) / 56 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 213681274 \beta_{11} + 37220043 \beta_{10} - 580274654 \beta_{9} - 9364009563 \beta_{8} + \cdots + 390142505 \beta_1 ) / 392 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 256772474 \beta_{11} + 513544948 \beta_{8} - 335367648 \beta_{7} + 216766595 \beta_{6} + \cdots - 775834987948 ) / 14 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 89164670264 \beta_{11} - 16277564863 \beta_{10} + 234369333617 \beta_{9} + \cdots - 162051775665 \beta_1 ) / 784 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
7.63204i
4.43280i
14.1721i
5.79470i
9.26925i
− 7.05389i
7.05389i
− 9.26925i
− 5.79470i
− 14.1721i
− 4.43280i
− 7.63204i
0 − 52.3774i 0 180.145i 0 −136.924 + 314.485i 0 −2014.40 0
41.2 0 − 39.5424i 0 − 231.698i 0 −334.862 − 74.2723i 0 −834.602 0
41.3 0 − 31.4304i 0 − 148.536i 0 315.340 − 134.943i 0 −258.867 0
41.4 0 − 24.1687i 0 135.157i 0 −211.893 − 269.723i 0 144.872 0
41.5 0 − 22.9686i 0 47.8215i 0 250.191 + 234.635i 0 201.444 0
41.6 0 − 6.43832i 0 76.7232i 0 −163.853 − 301.332i 0 687.548 0
41.7 0 6.43832i 0 − 76.7232i 0 −163.853 + 301.332i 0 687.548 0
41.8 0 22.9686i 0 − 47.8215i 0 250.191 − 234.635i 0 201.444 0
41.9 0 24.1687i 0 − 135.157i 0 −211.893 + 269.723i 0 144.872 0
41.10 0 31.4304i 0 148.536i 0 315.340 + 134.943i 0 −258.867 0
41.11 0 39.5424i 0 231.698i 0 −334.862 + 74.2723i 0 −834.602 0
41.12 0 52.3774i 0 − 180.145i 0 −136.924 − 314.485i 0 −2014.40 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.7.c.a ✓ 12
3.b odd 2 1 504.7.f.a 12
4.b odd 2 1 112.7.c.e 12
7.b odd 2 1 inner 56.7.c.a ✓ 12
8.b even 2 1 448.7.c.i 12
8.d odd 2 1 448.7.c.j 12
21.c even 2 1 504.7.f.a 12
28.d even 2 1 112.7.c.e 12
56.e even 2 1 448.7.c.j 12
56.h odd 2 1 448.7.c.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.7.c.a ✓ 12 1.a even 1 1 trivial
56.7.c.a ✓ 12 7.b odd 2 1 inner
112.7.c.e 12 4.b odd 2 1
112.7.c.e 12 28.d even 2 1
448.7.c.i 12 8.b even 2 1
448.7.c.i 12 56.h odd 2 1
448.7.c.j 12 8.d odd 2 1
448.7.c.j 12 56.e even 2 1
504.7.f.a 12 3.b odd 2 1
504.7.f.a 12 21.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(56, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 54\!\cdots\!92 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 19\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 26\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 27\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 68\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 60\!\cdots\!12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 53\!\cdots\!48)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 16\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 31\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 62\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 93\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 14\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 42\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 18\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 16\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 23\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 44\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 41\!\cdots\!72 \) Copy content Toggle raw display
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