Properties

Label 76.6.a.b
Level $76$
Weight $6$
Character orbit 76.a
Self dual yes
Analytic conductor $12.189$
Analytic rank $0$
Dimension $4$
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] = [76,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(12.1891703058\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 140x^{2} - 84x + 3103 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 3) q^{3} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 - 26) q^{5} + (5 \beta_{3} + 3 \beta_{2} - \beta_1 + 10) q^{7} + ( - 3 \beta_{3} - 12 \beta_{2} + \beta_1 + 218) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 3) q^{3} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 - 26) q^{5} + (5 \beta_{3} + 3 \beta_{2} - \beta_1 + 10) q^{7} + ( - 3 \beta_{3} - 12 \beta_{2} + \beta_1 + 218) q^{9} + ( - \beta_{3} - 10 \beta_{2} - 3 \beta_1 + 176) q^{11} + (10 \beta_{3} + 15 \beta_{2} - 14 \beta_1 + 202) q^{13} + ( - 48 \beta_{3} - 6 \beta_{2} + 37 \beta_1 + 674) q^{15} + (30 \beta_{3} + 48 \beta_{2} - 36 \beta_1 + 75) q^{17} - 361 q^{19} + ( - 56 \beta_{3} - 75 \beta_{2} + 31 \beta_1 + 1754) q^{21} + ( - 42 \beta_{3} + 95 \beta_{2} + 4 \beta_1 + 1450) q^{23} + ( - 97 \beta_{3} - 132 \beta_{2} + 107 \beta_1 + 2895) q^{25} + (101 \beta_{3} + 24 \beta_{2} - 137 \beta_1 + 1157) q^{27} + (144 \beta_{3} + 19 \beta_{2} - 67 \beta_1 + 1382) q^{29} + (82 \beta_{3} + 150 \beta_{2} - 5 \beta_1 - 174) q^{31} + (226 \beta_{3} - 90 \beta_{2} - 151 \beta_1 + 2800) q^{33} + ( - 317 \beta_{3} - 186 \beta_{2} + 199 \beta_1 + 4922) q^{35} + (358 \beta_{3} + 330 \beta_{2} + 133 \beta_1 - 5004) q^{37} + ( - 146 \beta_{3} - 411 \beta_{2} + 50 \beta_1 + 3778) q^{39} + ( - 58 \beta_{3} - 320 \beta_{2} + 13 \beta_1 - 2580) q^{41} + ( - 15 \beta_{3} + 378 \beta_{2} - 81 \beta_1 - 3146) q^{43} + (725 \beta_{3} + 960 \beta_{2} + 7 \beta_1 - 17464) q^{45} + ( - 435 \beta_{3} + 32 \beta_{2} - 299 \beta_1 - 1424) q^{47} + ( - 516 \beta_{3} - 324 \beta_{2} + 144 \beta_1 - 5602) q^{49} + ( - 921 \beta_{3} - 1080 \beta_{2} + 246 \beta_1 + 8169) q^{51} + ( - 266 \beta_{3} + 251 \beta_{2} + 53 \beta_1 - 19254) q^{53} + (57 \beta_{3} + 648 \beta_{2} + 9 \beta_1 - 18200) q^{55} + ( - 361 \beta_{3} - 1083) q^{57} + ( - 391 \beta_{3} - 420 \beta_{2} - 661 \beta_1 + 5779) q^{59} + (97 \beta_{3} - 570 \beta_{2} - 611 \beta_1 - 8072) q^{61} + (1847 \beta_{3} + 462 \beta_{2} - 403 \beta_1 - 9968) q^{63} + ( - 1318 \beta_{3} - 734 \beta_{2} + 538 \beta_1 - 2178) q^{65} + (823 \beta_{3} - 660 \beta_{2} + 1426 \beta_1 + 1667) q^{67} + (990 \beta_{3} + 885 \beta_{2} + 1138 \beta_1 - 36886) q^{69} + ( - 426 \beta_{3} - 130 \beta_{2} + 376 \beta_1 + 30112) q^{71} + (160 \beta_{3} + 96 \beta_{2} - 1382 \beta_1 - 26095) q^{73} + (5817 \beta_{3} + 3336 \beta_{2} - 611 \beta_1 - 20899) q^{75} + (1021 \beta_{3} + 12 \beta_{2} - 467 \beta_1 - 1990) q^{77} + ( - 532 \beta_{3} + 1326 \beta_{2} + 17 \beta_1 + 29016) q^{79} + ( - 556 \beta_{3} - 1512 \beta_{2} - 1224 \beta_1 + 10953) q^{81} + (408 \beta_{3} + 834 \beta_{2} + 1344 \beta_1 + 23412) q^{83} + ( - 4797 \beta_{3} - 3186 \beta_{2} + 1179 \beta_1 + 17682) q^{85} + ( - 438 \beta_{3} - 3279 \beta_{2} - 298 \beta_1 + 74818) q^{87} + ( - 4478 \beta_{3} + 1494 \beta_{2} - 953 \beta_1 - 742) q^{89} + ( - 1655 \beta_{3} - 444 \beta_{2} - 383 \beta_1 + 40577) q^{91} + ( - 1926 \beta_{3} - 654 \beta_{2} + 1832 \beta_1 + 3082) q^{93} + ( - 1083 \beta_{3} - 722 \beta_{2} - 361 \beta_1 + 9386) q^{95} + ( - 4466 \beta_{3} + 1104 \beta_{2} - 908 \beta_1 + 3510) q^{97} + (595 \beta_{3} - 4176 \beta_{2} - 1635 \beta_1 + 110652) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} - 110 q^{5} + 30 q^{7} + 878 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{3} - 110 q^{5} + 30 q^{7} + 878 q^{9} + 706 q^{11} + 788 q^{13} + 2792 q^{15} + 240 q^{17} - 1444 q^{19} + 7128 q^{21} + 5884 q^{23} + 11774 q^{25} + 4426 q^{27} + 5240 q^{29} - 860 q^{31} + 10748 q^{33} + 20322 q^{35} - 20732 q^{37} + 15404 q^{39} - 10204 q^{41} - 12554 q^{43} - 71306 q^{45} - 4826 q^{47} - 21376 q^{49} + 34518 q^{51} - 76484 q^{53} - 72914 q^{55} - 3610 q^{57} + 23898 q^{59} - 32482 q^{61} - 43566 q^{63} - 6076 q^{65} + 5022 q^{67} - 149524 q^{69} + 121300 q^{71} - 104700 q^{73} - 95230 q^{75} - 10002 q^{77} + 117128 q^{79} + 44924 q^{81} + 92832 q^{83} + 80322 q^{85} + 300148 q^{87} + 5988 q^{89} + 165618 q^{91} + 16180 q^{93} + 39710 q^{95} + 22972 q^{97} + 441418 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 140x^{2} - 84x + 3103 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 5\nu^{2} - 79\nu + 287 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} - 2\nu - 71 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} + \beta _1 + 142 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 40\beta_{3} + 48\beta_{2} + 89\beta _1 + 272 ) / 4 \) 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
4.80512
−5.81615
11.0547
−10.0437
0 −25.7605 0 −109.245 0 −177.299 0 420.606 0
1.2 0 −9.77006 0 −24.1428 0 64.5622 0 −147.546 0
1.3 0 17.5489 0 87.4671 0 76.9277 0 64.9636 0
1.4 0 27.9817 0 −64.0791 0 65.8093 0 539.976 0
\(n\): e.g. 2-40 or 990-1000
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 76.6.a.b 4
3.b odd 2 1 684.6.a.e 4
4.b odd 2 1 304.6.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.a.b 4 1.a even 1 1 trivial
304.6.a.k 4 4.b odd 2 1
684.6.a.e 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 10T_{3}^{3} - 875T_{3}^{2} + 5988T_{3} + 123588 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(76))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 10 T^{3} - 875 T^{2} + \cdots + 123588 \) Copy content Toggle raw display
$5$ \( T^{4} + 110 T^{3} + \cdots - 14782624 \) Copy content Toggle raw display
$7$ \( T^{4} - 30 T^{3} - 22476 T^{2} + \cdots - 57950181 \) Copy content Toggle raw display
$11$ \( T^{4} - 706 T^{3} + \cdots + 842790380 \) Copy content Toggle raw display
$13$ \( T^{4} - 788 T^{3} + \cdots + 15521087536 \) Copy content Toggle raw display
$17$ \( T^{4} - 240 T^{3} + \cdots + 933010219809 \) Copy content Toggle raw display
$19$ \( (T + 361)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 5884 T^{3} + \cdots - 69330313326016 \) Copy content Toggle raw display
$29$ \( T^{4} - 5240 T^{3} + \cdots - 37858523058292 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 129248370446848 \) Copy content Toggle raw display
$37$ \( T^{4} + 20732 T^{3} + \cdots - 47\!\cdots\!40 \) Copy content Toggle raw display
$41$ \( T^{4} + 10204 T^{3} + \cdots + 11\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{4} + 12554 T^{3} + \cdots - 23\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{4} + 4826 T^{3} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} + 76484 T^{3} + \cdots + 75\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{4} - 23898 T^{3} + \cdots - 12\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{4} + 32482 T^{3} + \cdots - 38\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{4} - 5022 T^{3} + \cdots + 50\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{4} - 121300 T^{3} + \cdots + 40\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{4} + 104700 T^{3} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{4} - 117128 T^{3} + \cdots - 34\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{4} - 92832 T^{3} + \cdots - 20\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{4} - 5988 T^{3} + \cdots + 14\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{4} - 22972 T^{3} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display
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