↑

Properties

Label 76.11.c.b
Level $76$
Weight $11$
Character orbit 76.c
Analytic conductor $48.287$
Analytic rank $0$
Dimension $14$
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] = [76,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.37");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(48.2871512032\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 604871 x^{12} + 143853611883 x^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{35}\cdot 3^{6}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + 362) q^{5} + ( - \beta_{4} - 2830) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2} - 27362) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + 362) q^{5} + ( - \beta_{4} - 2830) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2} - 27362) q^{9} + (\beta_{11} - 5 \beta_{2} + 4265) q^{11} + (\beta_{5} - 12 \beta_1) q^{13} + (\beta_{6} + 387 \beta_1) q^{15} + (\beta_{12} - 3 \beta_{11} + \cdots - 92046) q^{17}+ \cdots + ( - 2576 \beta_{12} + \cdots - 1450720269) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 5062 q^{5} - 39624 q^{7} - 383056 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 5062 q^{5} - 39624 q^{7} - 383056 q^{9} + 59746 q^{11} - 1287896 q^{17} + 4622604 q^{19} - 11539958 q^{23} - 417808 q^{25} - 77028922 q^{35} + 14606790 q^{39} + 71330230 q^{43} - 169810130 q^{45} - 394833110 q^{47} - 265204650 q^{49} - 612557042 q^{55} + 319590486 q^{57} - 1396847538 q^{61} - 3080375402 q^{63} - 7242583772 q^{73} - 922511302 q^{77} - 9163778710 q^{81} - 9328943264 q^{83} - 13063772714 q^{85} + 11593996398 q^{87} + 16259309364 q^{93} + 11710957630 q^{95} - 20299698578 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 604871 x^{12} + 143853611883 x^{10} + \cdots + 10\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 41\!\cdots\!07 \nu^{12} + \cdots - 25\!\cdots\!80 ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 36\!\cdots\!67 \nu^{12} + \cdots + 86\!\cdots\!20 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!97 \nu^{12} + \cdots + 37\!\cdots\!20 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 44\!\cdots\!77 \nu^{13} + \cdots + 10\!\cdots\!20 \nu ) / 56\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41\!\cdots\!07 \nu^{13} + \cdots - 25\!\cdots\!80 \nu ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!71 \nu^{12} + \cdots + 81\!\cdots\!40 ) / 67\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50\!\cdots\!13 \nu^{13} + \cdots - 59\!\cdots\!00 \nu ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!57 \nu^{13} + \cdots - 44\!\cdots\!20 \nu ) / 28\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 59\!\cdots\!67 \nu^{13} + \cdots + 18\!\cdots\!60 \nu ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 36\!\cdots\!33 \nu^{12} + \cdots + 76\!\cdots\!20 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 74\!\cdots\!57 \nu^{12} + \cdots - 36\!\cdots\!80 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!11 \nu^{13} + \cdots - 63\!\cdots\!60 \nu ) / 14\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - \beta_{2} - 86411 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{5} - 129220\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 178 \beta_{12} + 4430 \beta_{11} - 5640 \beta_{7} - 137651 \beta_{4} - 177313 \beta_{3} + \cdots + 11166050697 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 12222 \beta_{13} - 208199 \beta_{10} - 21168 \beta_{9} - 414227 \beta_{8} - 75306 \beta_{6} + \cdots + 18709372628 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 30401008 \beta_{12} - 936108916 \beta_{11} + 1464288168 \beta_{7} + 24103748905 \beta_{4} + \cdots - 16\!\cdots\!71 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2990732796 \beta_{13} + 36334752553 \beta_{10} + 11168560476 \beta_{9} + 92947826305 \beta_{8} + \cdots - 28\!\cdots\!12 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13707029835602 \beta_{12} + 158989831894682 \beta_{11} - 299927584021128 \beta_{7} + \cdots + 24\!\cdots\!97 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 559211674697658 \beta_{13} + \cdots + 44\!\cdots\!48 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 33\!\cdots\!28 \beta_{12} + \cdots - 38\!\cdots\!79 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 97\!\cdots\!48 \beta_{13} + \cdots - 70\!\cdots\!64 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 67\!\cdots\!54 \beta_{12} + \cdots + 60\!\cdots\!93 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 16\!\cdots\!38 \beta_{13} + \cdots + 11\!\cdots\!68 \beta_1 \) 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
− 413.548i
− 386.648i
− 367.380i
− 263.026i
− 236.595i
− 142.542i
− 62.5055i
62.5055i
142.542i
236.595i
263.026i
367.380i
386.648i
413.548i
0 − 413.548i 0 −832.398 0 21647.7 0 −111973. 0
37.2 0 − 386.648i 0 −2117.28 0 −23662.0 0 −90447.3 0
37.3 0 − 367.380i 0 5516.96 0 −5761.59 0 −75918.7 0
37.4 0 − 263.026i 0 1434.60 0 12851.9 0 −10133.6 0
37.5 0 − 236.595i 0 −3940.33 0 7909.30 0 3071.60 0
37.6 0 − 142.542i 0 3674.44 0 −13444.2 0 38730.8 0
37.7 0 − 62.5055i 0 −1205.00 0 −19353.1 0 55142.1 0
37.8 0 62.5055i 0 −1205.00 0 −19353.1 0 55142.1 0
37.9 0 142.542i 0 3674.44 0 −13444.2 0 38730.8 0
37.10 0 236.595i 0 −3940.33 0 7909.30 0 3071.60 0
37.11 0 263.026i 0 1434.60 0 12851.9 0 −10133.6 0
37.12 0 367.380i 0 5516.96 0 −5761.59 0 −75918.7 0
37.13 0 386.648i 0 −2117.28 0 −23662.0 0 −90447.3 0
37.14 0 413.548i 0 −832.398 0 21647.7 0 −111973. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.14
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 76.11.c.b ✓ 14
4.b odd 2 1 304.11.e.d 14
19.b odd 2 1 inner 76.11.c.b ✓ 14
76.d even 2 1 304.11.e.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.11.c.b ✓ 14 1.a even 1 1 trivial
76.11.c.b ✓ 14 19.b odd 2 1 inner
304.11.e.d 14 4.b odd 2 1
304.11.e.d 14 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 604871 T_{3}^{12} + 143853611883 T_{3}^{10} + \cdots + 10\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(76, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{7} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots - 78\!\cdots\!74)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots + 59\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 40\!\cdots\!82)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 32\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 49\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 34\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 11\!\cdots\!02)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 12\!\cdots\!48)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
show more
show less