Properties

Label 76.8.a.b
Level $76$
Weight $8$
Character orbit 76.a
Self dual yes
Analytic conductor $23.741$
Analytic rank $0$
Dimension $6$
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] = [76,8,Mod(1,76)]
 
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, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.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: \(23.7412619368\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8376x^{4} + 135458x^{3} + 16275767x^{2} - 280013424x - 6276171312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 7) q^{3} + ( - \beta_{2} - \beta_1 + 47) q^{5} + ( - \beta_{4} - \beta_{2} + 5 \beta_1 - 262) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} - 34 \beta_1 + 662) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 7) q^{3} + ( - \beta_{2} - \beta_1 + 47) q^{5} + ( - \beta_{4} - \beta_{2} + 5 \beta_1 - 262) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} - 34 \beta_1 + 662) q^{9} + (2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} - 52 \beta_1 + 1346) q^{11} + (6 \beta_{5} - \beta_{4} + 2 \beta_{3} + 12 \beta_{2} - 52 \beta_1 + 221) q^{13} + ( - 9 \beta_{5} + 7 \beta_{4} - 13 \beta_{3} - 3 \beta_{2} - 217 \beta_1 + 4540) q^{15} + ( - 19 \beta_{5} - 7 \beta_{4} + 12 \beta_{3} + 16 \beta_{2} - 46 \beta_1 + 3647) q^{17} + 6859 q^{19} + (67 \beta_{5} - 44 \beta_{4} - 31 \beta_{3} + 81 \beta_{2} + 533 \beta_1 - 14601) q^{21} + (26 \beta_{5} - \beta_{4} + 12 \beta_{3} - 38 \beta_{2} + 494 \beta_1 + 16649) q^{23} + ( - 108 \beta_{5} + 42 \beta_{4} + 123 \beta_{3} - 93 \beta_{2} + \cdots + 62269) q^{25}+ \cdots + ( - 7190 \beta_{5} + 6832 \beta_{4} - 5047 \beta_{3} + \cdots + 4064986) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9} + 7983 q^{11} + 1250 q^{13} + 26790 q^{15} + 21735 q^{17} + 41154 q^{19} - 86346 q^{21} + 100920 q^{23} + 373305 q^{25} + 534790 q^{27} - 58656 q^{29} + 403808 q^{31} + 916430 q^{33} + 463497 q^{35} + 808780 q^{37} + 758704 q^{39} + 556944 q^{41} + 1220735 q^{43} + 3234843 q^{45} + 1915305 q^{47} + 2045883 q^{49} + 908816 q^{51} + 511650 q^{53} + 1813341 q^{55} + 274360 q^{57} + 1300572 q^{59} + 565335 q^{61} - 7170325 q^{63} - 6195012 q^{65} - 45010 q^{67} - 7381528 q^{69} - 1424106 q^{71} - 11153825 q^{73} + 3941974 q^{75} - 17515425 q^{77} - 6392144 q^{79} + 6187530 q^{81} - 3164160 q^{83} - 19479255 q^{85} - 25999500 q^{87} - 14502678 q^{89} - 9736226 q^{91} - 18344300 q^{93} + 1913661 q^{95} - 21377010 q^{97} + 24032935 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 8376x^{4} + 135458x^{3} + 16275767x^{2} - 280013424x - 6276171312 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -23\nu^{5} + 1337\nu^{4} + 331031\nu^{3} - 3798869\nu^{2} - 655626372\nu - 394340616 ) / 12622824 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 51\nu^{5} + 2117\nu^{4} - 357983\nu^{3} - 9662021\nu^{2} + 548036696\nu + 11111121348 ) / 2103804 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -269\nu^{5} - 12312\nu^{4} + 1628074\nu^{3} + 41365824\nu^{2} - 2202338405\nu - 33289998684 ) / 2103804 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -617\nu^{5} - 30195\nu^{4} + 3641083\nu^{3} + 101646951\nu^{2} - 4929275990\nu - 76626597048 ) / 4207608 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{3} - 3\beta_{2} - 20\beta _1 + 2800 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 78\beta_{5} - 90\beta_{4} + 12\beta_{3} + 198\beta_{2} + 4321\beta _1 - 60834 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9331\beta_{5} + 10219\beta_{4} - 4033\beta_{3} - 19821\beta_{2} - 212696\beta _1 + 12452470 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 745380\beta_{5} - 866472\beta_{4} + 103440\beta_{3} + 1644228\beta_{2} + 24624409\beta _1 - 631311360 \) Copy content Toggle raw display

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
52.7878
49.5941
44.4724
−13.7370
−46.2258
−84.8914
0 −45.7878 0 −327.289 0 −7.23734 0 −90.4805 0
1.2 0 −42.5941 0 51.7645 0 −1212.43 0 −372.745 0
1.3 0 −37.4724 0 534.866 0 1051.36 0 −782.818 0
1.4 0 20.7370 0 −501.411 0 −1008.02 0 −1756.98 0
1.5 0 53.2258 0 88.7692 0 1094.09 0 645.987 0
1.6 0 91.8914 0 432.300 0 −1482.77 0 6257.03 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 76.8.a.b 6
4.b odd 2 1 304.8.a.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.8.a.b 6 1.a even 1 1 trivial
304.8.a.i 6 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 40T_{3}^{5} - 7711T_{3}^{4} + 93190T_{3}^{3} + 16686996T_{3}^{2} + 43655400T_{3} - 7412317344 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(76))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 40 T^{5} + \cdots - 7412317344 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 174360864384000 \) Copy content Toggle raw display
$7$ \( T^{6} + 1565 T^{5} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{6} - 7983 T^{5} + \cdots - 47\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{6} - 1250 T^{5} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{6} - 21735 T^{5} + \cdots + 81\!\cdots\!86 \) Copy content Toggle raw display
$19$ \( (T - 6859)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 100920 T^{5} + \cdots - 10\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{6} + 58656 T^{5} + \cdots - 19\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{6} - 403808 T^{5} + \cdots - 59\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{6} - 808780 T^{5} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{6} - 556944 T^{5} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{6} - 1220735 T^{5} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{6} - 1915305 T^{5} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{6} - 511650 T^{5} + \cdots - 69\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{6} - 1300572 T^{5} + \cdots - 86\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{6} - 565335 T^{5} + \cdots - 23\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{6} + 45010 T^{5} + \cdots - 13\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{6} + 1424106 T^{5} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{6} + 11153825 T^{5} + \cdots + 96\!\cdots\!26 \) Copy content Toggle raw display
$79$ \( T^{6} + 6392144 T^{5} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{6} + 3164160 T^{5} + \cdots - 65\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{6} + 14502678 T^{5} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{6} + 21377010 T^{5} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
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