Properties

Label 76.3.j.a
Level $76$
Weight $3$
Character orbit 76.j
Analytic conductor $2.071$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,3,Mod(13,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.j (of order \(18\), degree \(6\), 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: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 93 x^{16} + 3429 x^{14} + 64261 x^{12} + 647217 x^{10} + 3386277 x^{8} + 8232133 x^{6} + 8319228 x^{4} + 2467872 x^{2} + 69312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{12} - \beta_{10} - \beta_{8} - \beta_{6} - \beta_{4}) q^{3} - \beta_{17} q^{5} + (\beta_{16} + \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots + 2) q^{7}+ \cdots + (\beta_{16} + \beta_{14} - \beta_{12} + \beta_{11} + 3 \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} - \beta_{10} - \beta_{8} - \beta_{6} - \beta_{4}) q^{3} - \beta_{17} q^{5} + (\beta_{16} + \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots + 2) q^{7}+ \cdots + (2 \beta_{17} - 3 \beta_{16} - \beta_{15} + 2 \beta_{13} + 21 \beta_{11} - 7 \beta_{10} + \cdots - 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{3} + 9 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 6 q^{3} + 9 q^{7} + 6 q^{9} - 15 q^{11} + 51 q^{13} + 21 q^{15} - 45 q^{17} + 30 q^{19} - 63 q^{21} + 48 q^{23} - 54 q^{25} - 198 q^{27} - 39 q^{29} - 108 q^{31} - 105 q^{33} + 51 q^{35} + 48 q^{39} + 54 q^{41} + 75 q^{43} + 288 q^{45} + 339 q^{47} - 24 q^{49} + 360 q^{51} + 69 q^{53} - 51 q^{55} + 510 q^{57} - 483 q^{59} - 36 q^{61} - 267 q^{63} - 585 q^{65} - 87 q^{67} - 351 q^{69} - 234 q^{71} - 132 q^{73} + 108 q^{77} + 363 q^{79} + 258 q^{81} + 279 q^{83} + 666 q^{85} + 600 q^{89} + 270 q^{91} - 456 q^{93} - 39 q^{95} - 801 q^{97} - 267 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 93 x^{16} + 3429 x^{14} + 64261 x^{12} + 647217 x^{10} + 3386277 x^{8} + 8232133 x^{6} + 8319228 x^{4} + 2467872 x^{2} + 69312 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 209415 \nu^{16} + 21385203 \nu^{14} + 856101563 \nu^{12} + 16988391675 \nu^{10} + 173289673767 \nu^{8} + 838953142315 \nu^{6} + \cdots - 5641264464 ) / 174024457392 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 209415 \nu^{16} + 21385203 \nu^{14} + 856101563 \nu^{12} + 16988391675 \nu^{10} + 173289673767 \nu^{8} + 838953142315 \nu^{6} + \cdots - 5641264464 ) / 174024457392 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 103021303575977 \nu^{16} + \cdots + 56\!\cdots\!48 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6185597 \nu^{17} + 591176061 \nu^{15} + 22835687541 \nu^{13} + 462556367605 \nu^{11} + 5294541300849 \nu^{9} + \cdots + 6612929380896 ) / 13225858761792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\!\cdots\!51 \nu^{17} + \cdots + 18\!\cdots\!24 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21\!\cdots\!51 \nu^{17} + \cdots - 18\!\cdots\!24 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 356420916893601 \nu^{17} - 84585395212837 \nu^{16} + \cdots + 55\!\cdots\!44 ) / 22\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 356420916893601 \nu^{17} - 84585395212837 \nu^{16} + \cdots + 55\!\cdots\!44 ) / 22\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30\!\cdots\!17 \nu^{17} + \cdots + 17\!\cdots\!20 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30\!\cdots\!45 \nu^{17} + \cdots + 16\!\cdots\!20 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 30\!\cdots\!17 \nu^{17} + \cdots - 17\!\cdots\!20 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30\!\cdots\!45 \nu^{17} + \cdots - 16\!\cdots\!20 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29\!\cdots\!66 \nu^{17} + \cdots - 10\!\cdots\!12 ) / 70\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 40\!\cdots\!87 \nu^{17} + \cdots - 17\!\cdots\!84 ) / 42\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 40\!\cdots\!87 \nu^{17} + \cdots + 17\!\cdots\!84 ) / 42\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 89\!\cdots\!39 \nu^{17} + \cdots + 30\!\cdots\!40 ) / 84\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 89\!\cdots\!39 \nu^{17} + \cdots - 30\!\cdots\!40 ) / 84\!\cdots\!88 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{12} - 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{15} + \beta_{14} - 2 \beta_{13} + 13 \beta_{12} + 13 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} + 8 \beta_{4} - \beta_{3} - 19 \beta_{2} + 19 \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + 18 \beta_{12} + 22 \beta_{11} - 18 \beta_{10} - 22 \beta_{9} - 58 \beta_{8} - 58 \beta_{7} - 26 \beta_{6} + 26 \beta_{5} - 18 \beta_{3} + 31 \beta_{2} + 31 \beta _1 + 132 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 29 \beta_{17} - 29 \beta_{16} - 38 \beta_{15} - 38 \beta_{14} + 60 \beta_{13} - 459 \beta_{12} + 40 \beta_{11} - 459 \beta_{10} + 40 \beta_{9} + 85 \beta_{8} - 85 \beta_{7} + 225 \beta_{6} + 225 \beta_{5} - 252 \beta_{4} + 30 \beta_{3} + \cdots + 126 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 89 \beta_{17} + 89 \beta_{16} + 77 \beta_{15} - 77 \beta_{14} - 296 \beta_{12} - 20 \beta_{11} + 296 \beta_{10} + 20 \beta_{9} + 1680 \beta_{8} + 1680 \beta_{7} + 618 \beta_{6} - 618 \beta_{5} + 379 \beta_{3} - 910 \beta_{2} + \cdots - 2544 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 896 \beta_{17} + 896 \beta_{16} + 1224 \beta_{15} + 1224 \beta_{14} - 1642 \beta_{13} + 14421 \beta_{12} - 2380 \beta_{11} + 14421 \beta_{10} - 2380 \beta_{9} - 2827 \beta_{8} + 2827 \beta_{7} - 7835 \beta_{6} + \cdots - 3588 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3694 \beta_{17} - 3694 \beta_{16} - 2424 \beta_{15} + 2424 \beta_{14} + 4566 \beta_{12} - 9828 \beta_{11} - 4566 \beta_{10} + 9828 \beta_{9} - 49104 \beta_{8} - 49104 \beta_{7} - 14868 \beta_{6} + 14868 \beta_{5} + \cdots + 54666 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 28118 \beta_{17} - 28118 \beta_{16} - 37658 \beta_{15} - 37658 \beta_{14} + 45666 \beta_{13} - 438743 \beta_{12} + 97194 \beta_{11} - 438743 \beta_{10} + 97194 \beta_{9} + 88198 \beta_{8} + \cdots + 101772 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 126447 \beta_{17} + 126447 \beta_{16} + 73373 \beta_{15} - 73373 \beta_{14} - 56837 \beta_{12} + 459788 \beta_{11} + 56837 \beta_{10} - 459788 \beta_{9} + 1443824 \beta_{8} + 1443824 \beta_{7} + \cdots - 1291752 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 871475 \beta_{17} + 871475 \beta_{16} + 1136400 \beta_{15} + 1136400 \beta_{14} - 1299616 \beta_{13} + 13174994 \beta_{12} - 3409206 \beta_{11} + 13174994 \beta_{10} - 3409206 \beta_{9} + \cdots - 2922752 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4014631 \beta_{17} - 4014631 \beta_{16} - 2203840 \beta_{15} + 2203840 \beta_{14} + 174619 \beta_{12} - 16329884 \beta_{11} - 174619 \beta_{10} + 16329884 \beta_{9} - 42611265 \beta_{8} + \cdots + 32971146 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 26622829 \beta_{17} - 26622829 \beta_{16} - 33988119 \beta_{15} - 33988119 \beta_{14} + 37619138 \beta_{13} - 393303354 \beta_{12} + 110991596 \beta_{11} - 393303354 \beta_{10} + \cdots + 84934128 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 123188846 \beta_{17} + 123188846 \beta_{16} + 66046944 \beta_{15} - 66046944 \beta_{14} + 23991495 \beta_{12} + 527304288 \beta_{11} - 23991495 \beta_{10} + \cdots - 890242260 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 804684652 \beta_{17} + 804684652 \beta_{16} + 1012343518 \beta_{15} + 1012343518 \beta_{14} - 1100650188 \beta_{13} + 11708483164 \beta_{12} - 3472747716 \beta_{11} + \cdots - 2488644354 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 3718260648 \beta_{17} - 3718260648 \beta_{16} - 1975813312 \beta_{15} + 1975813312 \beta_{14} - 1270453589 \beta_{12} - 16326763294 \beta_{11} + 1270453589 \beta_{10} + \cdots + 24955550316 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 24153101263 \beta_{17} - 24153101263 \beta_{16} - 30096307983 \beta_{15} - 30096307983 \beta_{14} + 32406252302 \beta_{13} - 348091131067 \beta_{12} + \cdots + 73301139148 \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(39\)
\(\chi(n)\) \(\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
3.09175i
0.656794i
4.19502i
1.57014i
0.176873i
5.44868i
1.57014i
0.176873i
5.44868i
3.84460i
1.29756i
4.09415i
3.09175i
0.656794i
4.19502i
3.84460i
1.29756i
4.09415i
0 −0.617749 + 1.69725i 0 0.757040 + 4.29338i 0 −2.41011 + 4.17443i 0 4.39535 + 3.68814i 0
13.2 0 0.215056 0.590861i 0 −1.30596 7.40644i 0 3.03221 5.25195i 0 6.59153 + 5.53095i 0
13.3 0 1.87447 5.15008i 0 0.722564 + 4.09786i 0 2.33853 4.05046i 0 −16.1152 13.5223i 0
21.1 0 −2.27531 + 2.71161i 0 −5.68966 + 2.07087i 0 −6.17078 10.6881i 0 −0.612961 3.47627i 0
21.2 0 −1.37974 + 1.64431i 0 3.12714 1.13819i 0 5.96525 + 10.3321i 0 0.762765 + 4.32585i 0
21.3 0 2.23630 2.66512i 0 1.62283 0.590661i 0 −1.87959 3.25555i 0 −0.538989 3.05676i 0
29.1 0 −2.27531 2.71161i 0 −5.68966 2.07087i 0 −6.17078 + 10.6881i 0 −0.612961 + 3.47627i 0
29.2 0 −1.37974 1.64431i 0 3.12714 + 1.13819i 0 5.96525 10.3321i 0 0.762765 4.32585i 0
29.3 0 2.23630 + 2.66512i 0 1.62283 + 0.590661i 0 −1.87959 + 3.25555i 0 −0.538989 + 3.05676i 0
33.1 0 −4.45984 0.786390i 0 5.56146 + 4.66662i 0 4.24641 + 7.35499i 0 10.8145 + 3.93617i 0
33.2 0 −1.95150 0.344102i 0 −6.26982 5.26101i 0 −0.733695 1.27080i 0 −4.76730 1.73516i 0
33.3 0 3.35830 + 0.592159i 0 1.47441 + 1.23717i 0 0.111774 + 0.193599i 0 2.47031 + 0.899120i 0
41.1 0 −0.617749 1.69725i 0 0.757040 4.29338i 0 −2.41011 4.17443i 0 4.39535 3.68814i 0
41.2 0 0.215056 + 0.590861i 0 −1.30596 + 7.40644i 0 3.03221 + 5.25195i 0 6.59153 5.53095i 0
41.3 0 1.87447 + 5.15008i 0 0.722564 4.09786i 0 2.33853 + 4.05046i 0 −16.1152 + 13.5223i 0
53.1 0 −4.45984 + 0.786390i 0 5.56146 4.66662i 0 4.24641 7.35499i 0 10.8145 3.93617i 0
53.2 0 −1.95150 + 0.344102i 0 −6.26982 + 5.26101i 0 −0.733695 + 1.27080i 0 −4.76730 + 1.73516i 0
53.3 0 3.35830 0.592159i 0 1.47441 1.23717i 0 0.111774 0.193599i 0 2.47031 0.899120i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.j.a 18
4.b odd 2 1 304.3.z.b 18
19.e even 9 1 1444.3.c.c 18
19.f odd 18 1 inner 76.3.j.a 18
19.f odd 18 1 1444.3.c.c 18
76.k even 18 1 304.3.z.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.j.a 18 1.a even 1 1 trivial
76.3.j.a 18 19.f odd 18 1 inner
304.3.z.b 18 4.b odd 2 1
304.3.z.b 18 76.k even 18 1
1444.3.c.c 18 19.e even 9 1
1444.3.c.c 18 19.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + 6 T^{17} + 15 T^{16} + \cdots + 25351947 \) Copy content Toggle raw display
$5$ \( T^{18} + 27 T^{16} + \cdots + 294796874304 \) Copy content Toggle raw display
$7$ \( T^{18} - 9 T^{17} + \cdots + 44446758976 \) Copy content Toggle raw display
$11$ \( T^{18} + 15 T^{17} + \cdots + 423492482169 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 645464049285312 \) Copy content Toggle raw display
$17$ \( T^{18} + 45 T^{17} + \cdots + 84\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{18} - 30 T^{17} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{18} - 48 T^{17} + \cdots + 73\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{18} + 39 T^{17} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{18} + 108 T^{17} + \cdots + 96\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{18} + 13410 T^{16} + \cdots + 21\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{18} - 54 T^{17} + \cdots + 48\!\cdots\!67 \) Copy content Toggle raw display
$43$ \( T^{18} - 75 T^{17} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( T^{18} - 339 T^{17} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{18} - 69 T^{17} + \cdots + 19\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{18} + 483 T^{17} + \cdots + 34\!\cdots\!43 \) Copy content Toggle raw display
$61$ \( T^{18} + 36 T^{17} + \cdots + 70\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{18} + 87 T^{17} + \cdots + 58\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{18} + 234 T^{17} + \cdots + 29\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{18} + 132 T^{17} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{18} - 363 T^{17} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{18} - 279 T^{17} + \cdots + 61\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{18} - 600 T^{17} + \cdots + 26\!\cdots\!03 \) Copy content Toggle raw display
$97$ \( T^{18} + 801 T^{17} + \cdots + 14\!\cdots\!07 \) Copy content Toggle raw display
show more
show less