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Properties

Label 76.9.c.b
Level $76$
Weight $9$
Character orbit 76.c
Analytic conductor $30.961$
Analytic rank $0$
Dimension $12$
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] = [76,9,Mod(37,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.37");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 76.c (of order \(2\), degree \(1\), 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: \(30.9607743646\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 62471 x^{10} + 1417247091 x^{8} + 14731726590693 x^{6} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{4}\cdot 7 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} - 47) q^{5} + ( - \beta_{5} + 144) q^{7} + ( - \beta_{3} + \beta_{2} - 3851) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} - 47) q^{5} + ( - \beta_{5} + 144) q^{7} + ( - \beta_{3} + \beta_{2} - 3851) q^{9} + ( - \beta_{9} + 4 \beta_{5} + \cdots + 162) q^{11}+ \cdots + ( - 5159 \beta_{9} - 41954 \beta_{5} + \cdots + 23048826) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 568 q^{5} + 1734 q^{7} - 46210 q^{9} + 1976 q^{11} - 47446 q^{17} - 113178 q^{19} + 378950 q^{23} - 502656 q^{25} - 1876832 q^{35} - 1463394 q^{39} + 6234396 q^{43} - 507200 q^{45} + 4902224 q^{47} - 16736298 q^{49} - 38366664 q^{55} - 19396170 q^{57} + 16602912 q^{61} - 11274020 q^{63} + 111268614 q^{73} - 134425928 q^{77} + 193590584 q^{81} + 34030388 q^{83} + 387605748 q^{85} - 217889946 q^{87} - 474880524 q^{93} - 256537036 q^{95} + 276136420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 62471 x^{10} + 1417247091 x^{8} + 14731726590693 x^{6} + \cdots + 18\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16443748187107 \nu^{10} + \cdots - 50\!\cdots\!60 ) / 53\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16443748187107 \nu^{10} + \cdots - 56\!\cdots\!20 ) / 53\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23375276921927 \nu^{10} + \cdots + 84\!\cdots\!80 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 532936767080041 \nu^{10} + \cdots - 17\!\cdots\!40 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 611053058969 \nu^{11} + \cdots - 16\!\cdots\!72 \nu ) / 43\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7895840912173 \nu^{11} + \cdots - 31\!\cdots\!80 \nu ) / 46\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 953247345957751 \nu^{11} + \cdots - 39\!\cdots\!40 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4798690520645 \nu^{10} + \cdots + 13\!\cdots\!60 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8759582122261 \nu^{11} + \cdots + 24\!\cdots\!40 \nu ) / 31\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 45\!\cdots\!57 \nu^{11} + \cdots + 15\!\cdots\!80 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 10412 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} - 4\beta_{8} - \beta_{7} - 7\beta_{6} - 17099\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -1480\beta_{9} - 10420\beta_{5} - 10130\beta_{4} + 49389\beta_{3} - 26771\beta_{2} + 178030176 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 35711\beta_{11} - 26771\beta_{10} + 121424\beta_{8} + 61289\beta_{7} + 276275\beta_{6} + 358396969\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 53863148 \beta_{9} + 451252808 \beta_{5} + 370187836 \beta_{4} - 1914899685 \beta_{3} + \cdots - 3731292206784 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1026134197 \beta_{11} + 628744537 \beta_{10} - 3039901228 \beta_{8} - 2420015641 \beta_{7} + \cdots - 7998063218279 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1571051491600 \beta_{9} - 14655274094140 \beta_{5} - 10615131164390 \beta_{4} + 62196831285933 \beta_{3} + \cdots + 83\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 27620728529531 \beta_{11} - 14536505926991 \beta_{10} + 73092080070344 \beta_{8} + \cdots + 18\!\cdots\!89 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 42\!\cdots\!32 \beta_{9} + \cdots - 19\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 72\!\cdots\!05 \beta_{11} + \cdots - 42\!\cdots\!99 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
− 155.915i
− 140.007i
− 76.7657i
− 74.3715i
− 67.8708i
− 50.2872i
50.2872i
67.8708i
74.3715i
76.7657i
140.007i
155.915i
0 − 155.915i 0 582.973 0 1753.47 0 −17748.6 0
37.2 0 − 140.007i 0 −899.569 0 −1723.79 0 −13041.1 0
37.3 0 − 76.7657i 0 510.215 0 −1605.51 0 668.032 0
37.4 0 − 74.3715i 0 −290.660 0 3406.67 0 1029.88 0
37.5 0 − 67.8708i 0 445.529 0 −2144.30 0 1954.55 0
37.6 0 − 50.2872i 0 −632.489 0 1180.45 0 4032.19 0
37.7 0 50.2872i 0 −632.489 0 1180.45 0 4032.19 0
37.8 0 67.8708i 0 445.529 0 −2144.30 0 1954.55 0
37.9 0 74.3715i 0 −290.660 0 3406.67 0 1029.88 0
37.10 0 76.7657i 0 510.215 0 −1605.51 0 668.032 0
37.11 0 140.007i 0 −899.569 0 −1723.79 0 −13041.1 0
37.12 0 155.915i 0 582.973 0 1753.47 0 −17748.6 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.9.c.b ✓ 12
4.b odd 2 1 304.9.e.c 12
19.b odd 2 1 inner 76.9.c.b ✓ 12
76.d even 2 1 304.9.e.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.9.c.b ✓ 12 1.a even 1 1 trivial
76.9.c.b ✓ 12 19.b odd 2 1 inner
304.9.e.c 12 4.b odd 2 1
304.9.e.c 12 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 62471 T_{3}^{10} + 1417247091 T_{3}^{8} + 14731726590693 T_{3}^{6} + \cdots + 18\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(76, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 41\!\cdots\!54)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 11\!\cdots\!66)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 23\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 34\!\cdots\!08)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 85\!\cdots\!94)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
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