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Properties

Label 3.67.b.b
Level $3$
Weight $67$
Character orbit 3.b
Analytic conductor $82.760$
Analytic rank $0$
Dimension $20$
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] = [3,67,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 67, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 67);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 67 \)
Character orbit: \([\chi]\) \(=\) 3.b (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: \(82.7604085389\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{216}\cdot 3^{291}\cdot 5^{20}\cdot 7^{10}\cdot 11^{6}\cdot 13^{4} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 15513 \beta_1 + 243548779847726) q^{3} + (\beta_{3} - 1519 \beta_{2} + \cdots - 43\!\cdots\!40) q^{4}+ \cdots + (\beta_{9} - 5 \beta_{8} + \cdots + 75\!\cdots\!72) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 15513 \beta_1 + 243548779847726) q^{3} + (\beta_{3} - 1519 \beta_{2} + \cdots - 43\!\cdots\!40) q^{4}+ \cdots + (30\!\cdots\!50 \beta_{19} + \cdots + 65\!\cdots\!98) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 48\!\cdots\!16 q^{3}+ \cdots + 15\!\cdots\!00 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 48\!\cdots\!16 q^{3}+ \cdots + 13\!\cdots\!00 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + \cdots + 56\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 36\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 70\!\cdots\!75 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 70\!\cdots\!75 \nu^{19} + \cdots + 59\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 44\!\cdots\!05 \nu^{19} + \cdots - 42\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\!\cdots\!59 \nu^{19} + \cdots - 54\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 33\!\cdots\!31 \nu^{19} + \cdots + 80\!\cdots\!00 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 93\!\cdots\!67 \nu^{19} + \cdots + 19\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!11 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 55\!\cdots\!33 \nu^{19} + \cdots - 96\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!99 \nu^{19} + \cdots - 70\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 84\!\cdots\!43 \nu^{19} + \cdots - 83\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 34\!\cdots\!15 \nu^{19} + \cdots - 57\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 55\!\cdots\!03 \nu^{19} + \cdots - 85\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 28\!\cdots\!69 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20\!\cdots\!87 \nu^{19} + \cdots + 78\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 42\!\cdots\!01 \nu^{19} + \cdots + 13\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 99\!\cdots\!49 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 84\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 16\!\cdots\!53 \nu^{19} + \cdots + 53\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 25\!\cdots\!29 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 1519\beta_{2} + 141\beta _1 - 117510611961387580904 ) / 1296 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} + 7 \beta_{7} + 227 \beta_{5} + 671787 \beta_{4} - 90503341 \beta_{3} + \cdots + 3272193665213 ) / 46656 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 13575 \beta_{16} + 29517 \beta_{15} - 17257 \beta_{13} + 26976 \beta_{12} + \cdots + 22\!\cdots\!07 ) / 1679616 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2575287648 \beta_{19} - 13548022232 \beta_{18} - 5827627322860 \beta_{17} + \cdots - 97\!\cdots\!57 ) / 3779136 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 39\!\cdots\!73 \beta_{16} + \cdots - 31\!\cdots\!01 ) / 136048896 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 73\!\cdots\!60 \beta_{19} + \cdots + 23\!\cdots\!03 ) / 306110016 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 28\!\cdots\!65 \beta_{16} + \cdots + 15\!\cdots\!37 ) / 3673320192 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 51\!\cdots\!52 \beta_{19} + \cdots - 16\!\cdots\!15 ) / 8264970432 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 54\!\cdots\!27 \beta_{16} + \cdots - 25\!\cdots\!99 ) / 297538935552 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 97\!\cdots\!72 \beta_{19} + \cdots + 32\!\cdots\!41 ) / 669462604992 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 33\!\cdots\!39 \beta_{16} + \cdots + 14\!\cdots\!11 ) / 8033551259904 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 58\!\cdots\!80 \beta_{19} + \cdots - 20\!\cdots\!73 ) / 18075490334784 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 22\!\cdots\!75 \beta_{16} + \cdots - 88\!\cdots\!99 ) / 24100653779712 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 12\!\cdots\!12 \beta_{19} + \cdots + 46\!\cdots\!23 ) / 18075490334784 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 51\!\cdots\!81 \beta_{16} + \cdots + 18\!\cdots\!17 ) / 24100653779712 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 92\!\cdots\!40 \beta_{19} + \cdots - 35\!\cdots\!59 ) / 6025163444928 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 37\!\cdots\!85 \beta_{16} + \cdots - 12\!\cdots\!29 ) / 8033551259904 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 59\!\cdots\!84 \beta_{19} + \cdots + 23\!\cdots\!55 ) / 18075490334784 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 4.67878e8i
− 4.43955e8i
− 3.58413e8i
− 3.39530e8i
− 3.00969e8i
− 2.56585e8i
− 2.39059e8i
− 1.32287e8i
− 1.26111e8i
− 96378.2i
96378.2i
1.26111e8i
1.32287e8i
2.39059e8i
2.56585e8i
3.00969e8i
3.39530e8i
3.58413e8i
4.43955e8i
4.67878e8i
− 1.68436e10i −3.79959e15 + 4.05786e15i −2.09920e20 9.39164e22i 6.83490e25 + 6.39988e25i −5.61847e27 2.29297e30i −2.02937e30 − 3.08364e31i 1.58189e33
2.2 − 1.59824e10i 5.28212e15 − 1.73274e15i −1.81649e20 − 8.56960e22i −2.76933e25 − 8.44208e25i 1.24551e28 1.72390e30i 2.48984e31 − 1.83050e31i −1.36963e33
2.3 − 1.29029e10i −3.61968e15 − 4.21913e15i −9.26974e19 − 1.27679e23i −5.44390e25 + 4.67043e25i 1.78063e27 2.43999e29i −4.69903e30 + 3.05438e31i −1.64743e33
2.4 − 1.22231e10i 1.27232e15 − 5.41150e15i −7.56171e19 1.11898e23i −6.61453e25 − 1.55517e25i −1.34063e28 2.23699e28i −2.76656e31 − 1.37703e31i 1.36774e33
2.5 − 1.08349e10i 3.60729e15 + 4.22973e15i −4.36076e19 5.11119e22i 4.58286e25 − 3.90845e25i −1.55423e27 − 3.26990e29i −4.87811e30 + 3.05157e31i 5.53791e32
2.6 − 9.23704e9i −3.04579e15 + 4.65041e15i −1.15360e19 − 1.92130e23i 4.29561e25 + 2.81341e25i 2.12741e26 − 5.75015e29i −1.23495e31 − 2.83283e31i −1.77471e33
2.7 − 8.60612e9i −5.53379e15 − 5.29477e14i −2.78318e17 1.34902e23i −4.55674e24 + 4.76244e25i 8.41299e27 − 6.32624e29i 3.03425e31 + 5.86002e30i 1.16098e33
2.8 − 4.76234e9i 4.09693e15 − 3.75743e15i 5.11071e19 1.11626e23i −1.78942e25 − 1.95110e25i 1.00296e28 − 5.94788e29i 2.66653e30 − 3.07879e31i 5.31601e32
2.9 − 4.54000e9i 5.46530e15 − 1.01669e15i 5.31753e19 − 1.75307e23i −4.61577e24 − 2.48125e25i −8.19169e27 − 5.76409e29i 2.88358e31 − 1.11130e31i −7.95893e32
2.10 − 3.46962e6i −1.28962e15 − 5.40740e15i 7.37870e19 − 7.05043e22i −1.87616e22 + 4.47450e21i 2.25837e27 − 5.12025e26i −2.75769e31 + 1.39470e31i −2.44623e29
2.11 3.46962e6i −1.28962e15 + 5.40740e15i 7.37870e19 7.05043e22i −1.87616e22 − 4.47450e21i 2.25837e27 5.12025e26i −2.75769e31 − 1.39470e31i −2.44623e29
2.12 4.54000e9i 5.46530e15 + 1.01669e15i 5.31753e19 1.75307e23i −4.61577e24 + 2.48125e25i −8.19169e27 5.76409e29i 2.88358e31 + 1.11130e31i −7.95893e32
2.13 4.76234e9i 4.09693e15 + 3.75743e15i 5.11071e19 − 1.11626e23i −1.78942e25 + 1.95110e25i 1.00296e28 5.94788e29i 2.66653e30 + 3.07879e31i 5.31601e32
2.14 8.60612e9i −5.53379e15 + 5.29477e14i −2.78318e17 − 1.34902e23i −4.55674e24 − 4.76244e25i 8.41299e27 6.32624e29i 3.03425e31 − 5.86002e30i 1.16098e33
2.15 9.23704e9i −3.04579e15 − 4.65041e15i −1.15360e19 1.92130e23i 4.29561e25 − 2.81341e25i 2.12741e26 5.75015e29i −1.23495e31 + 2.83283e31i −1.77471e33
2.16 1.08349e10i 3.60729e15 − 4.22973e15i −4.36076e19 − 5.11119e22i 4.58286e25 + 3.90845e25i −1.55423e27 3.26990e29i −4.87811e30 − 3.05157e31i 5.53791e32
2.17 1.22231e10i 1.27232e15 + 5.41150e15i −7.56171e19 − 1.11898e23i −6.61453e25 + 1.55517e25i −1.34063e28 − 2.23699e28i −2.76656e31 + 1.37703e31i 1.36774e33
2.18 1.29029e10i −3.61968e15 + 4.21913e15i −9.26974e19 1.27679e23i −5.44390e25 − 4.67043e25i 1.78063e27 − 2.43999e29i −4.69903e30 − 3.05438e31i −1.64743e33
2.19 1.59824e10i 5.28212e15 + 1.73274e15i −1.81649e20 8.56960e22i −2.76933e25 + 8.44208e25i 1.24551e28 − 1.72390e30i 2.48984e31 + 1.83050e31i −1.36963e33
2.20 1.68436e10i −3.79959e15 − 4.05786e15i −2.09920e20 − 9.39164e22i 6.83490e25 − 6.39988e25i −5.61847e27 − 2.29297e30i −2.02937e30 + 3.08364e31i 1.58189e33
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.67.b.b ✓ 20
3.b odd 2 1 inner 3.67.b.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.67.b.b ✓ 20 1.a even 1 1 trivial
3.67.b.b ✓ 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + \cdots + 75\!\cdots\!00 \) acting on \(S_{67}^{\mathrm{new}}(3, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 86\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 37\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 33\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 28\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
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