Properties

Label 3.65.b.a
Level $3$
Weight $65$
Character orbit 3.b
Analytic conductor $77.821$
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] = [3,65,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, 65, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 65);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 65 \)
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: \(77.8210007872\)
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: \( x^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{190}\cdot 3^{282}\cdot 5^{19}\cdot 7^{8}\cdot 11^{4}\cdot 13^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 13312 \beta_1 - 71037898477915) q^{3} + (\beta_{3} - 68 \beta_{2} - 34 \beta_1 - 78\!\cdots\!89) q^{4}+ \cdots + (\beta_{9} + 4 \beta_{8} - 69 \beta_{7} - 685 \beta_{6} + \cdots - 11\!\cdots\!52) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 13312 \beta_1 - 71037898477915) q^{3} + (\beta_{3} - 68 \beta_{2} - 34 \beta_1 - 78\!\cdots\!89) q^{4}+ \cdots + ( - 24\!\cdots\!45 \beta_{19} + \cdots + 15\!\cdots\!34) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 14\!\cdots\!92 q^{3}+ \cdots - 22\!\cdots\!00 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 14\!\cdots\!92 q^{3}+ \cdots + 31\!\cdots\!00 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + \cdots + 16\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!75 \nu^{19} + \cdots - 84\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!75 \nu^{19} + \cdots - 39\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!25 \nu^{19} + \cdots - 34\!\cdots\!00 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!61 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 66\!\cdots\!83 \nu^{19} + \cdots - 60\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!09 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!89 \nu^{19} + \cdots - 27\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!03 \nu^{19} + \cdots - 10\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!85 \nu^{19} + \cdots + 14\!\cdots\!00 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 37\!\cdots\!67 \nu^{19} + \cdots + 81\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31\!\cdots\!49 \nu^{19} + \cdots - 58\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 57\!\cdots\!81 \nu^{19} + \cdots + 53\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!29 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 60\!\cdots\!59 \nu^{19} + \cdots + 49\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 20\!\cdots\!81 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 59\!\cdots\!53 \nu^{19} + \cdots - 39\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 22\!\cdots\!13 \nu^{19} + \cdots + 26\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 17\!\cdots\!79 \nu^{19} + \cdots + 53\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 68\beta_{2} - 34\beta _1 - 26321833117958871105 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - 5 \beta_{7} - 22 \beta_{5} + 80800 \beta_{4} - 2512797 \beta_{3} - 4793992141335 \beta_{2} + \cdots + 1917597342933 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{19} + 4 \beta_{18} - 2 \beta_{16} - \beta_{15} - 384 \beta_{14} - 51 \beta_{13} - 3498 \beta_{12} + 3509 \beta_{11} - 173175 \beta_{10} + 298 \beta_{9} + 2121544 \beta_{8} + 1441158400 \beta_{7} + \cdots + 11\!\cdots\!15 ) / 1296 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 162140790 \beta_{19} - 324281580 \beta_{18} + 492987401 \beta_{17} - 7667037987 \beta_{16} - 4609393797 \beta_{15} + 1185831794550 \beta_{14} + \cdots - 20\!\cdots\!92 ) / 972 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 92\!\cdots\!55 \beta_{19} + \cdots - 69\!\cdots\!53 ) / 5832 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 11\!\cdots\!10 \beta_{19} + \cdots + 19\!\cdots\!56 ) / 4374 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 23\!\cdots\!15 \beta_{19} + \cdots + 14\!\cdots\!17 ) / 8748 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 16\!\cdots\!30 \beta_{19} + \cdots - 54\!\cdots\!84 ) / 6561 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 16\!\cdots\!05 \beta_{19} + \cdots - 95\!\cdots\!27 ) / 39366 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18\!\cdots\!00 \beta_{19} + \cdots + 88\!\cdots\!00 ) / 59049 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 38\!\cdots\!05 \beta_{19} + \cdots + 20\!\cdots\!47 ) / 59049 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 93\!\cdots\!60 \beta_{19} + \cdots - 46\!\cdots\!48 ) / 177147 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 59\!\cdots\!30 \beta_{19} + \cdots - 30\!\cdots\!74 ) / 59049 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 29\!\cdots\!80 \beta_{19} + \cdots + 79\!\cdots\!68 ) / 177147 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 93\!\cdots\!00 \beta_{19} + \cdots + 45\!\cdots\!64 ) / 59049 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 68\!\cdots\!80 \beta_{19} + \cdots - 13\!\cdots\!72 ) / 177147 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 48\!\cdots\!60 \beta_{19} + \cdots - 22\!\cdots\!28 ) / 19683 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 45\!\cdots\!20 \beta_{19} + \cdots + 74\!\cdots\!04 ) / 59049 \) Copy content Toggle raw display

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
1.25990e9i
1.17774e9i
1.16631e9i
1.15913e9i
7.41715e8i
6.71940e8i
5.00631e8i
4.99832e8i
3.25526e8i
1.58953e8i
1.58953e8i
3.25526e8i
4.99832e8i
5.00631e8i
6.71940e8i
7.41715e8i
1.15913e9i
1.16631e9i
1.17774e9i
1.25990e9i
7.55940e9i −2.98618e14 1.82880e15i −3.86978e19 3.78912e22i −1.38246e25 + 2.25737e24i 1.61394e27 1.53086e29i −3.25534e30 + 1.09222e30i 2.86435e32
2.2 7.06644e9i −1.85210e15 5.82746e13i −3.14879e19 1.76003e22i −4.11794e23 + 1.30878e25i −6.88028e26 9.21545e28i 3.42689e30 + 2.15861e29i −1.24372e32
2.3 6.99784e9i 3.59786e14 + 1.81776e15i −3.05230e19 1.06729e22i 1.27204e25 2.51773e24i 9.75273e26 8.45080e28i −3.17479e30 + 1.30801e30i −7.46870e31
2.4 6.95479e9i 1.80867e15 4.02985e14i −2.99223e19 4.26608e21i −2.80268e24 1.25789e25i −1.59304e27 7.98101e28i 3.10889e30 1.45773e30i −2.96697e31
2.5 4.45029e9i −9.93580e14 + 1.56412e15i −1.35835e18 4.47337e22i 6.96081e24 + 4.42172e24i −9.42280e26 7.60483e28i −1.45928e30 3.10816e30i 1.99078e32
2.6 4.03164e9i 5.57989e14 1.76701e15i 2.19261e18 2.91915e22i −7.12396e24 2.24961e24i 6.92352e26 8.32105e28i −2.81098e30 1.97195e30i −1.17690e32
2.7 3.00379e9i 1.81065e15 + 3.93978e14i 9.42400e18 2.20086e22i 1.18343e24 5.43882e24i 9.43770e26 8.37178e28i 3.12325e30 + 1.42671e30i 6.61093e31
2.8 2.99899e9i −1.38295e15 1.23335e15i 9.45278e18 5.43489e21i −3.69880e24 + 4.14745e24i −9.58380e26 8.36705e28i 3.91402e29 + 3.41130e30i 1.62992e31
2.9 1.95316e9i −1.56985e15 + 9.84500e14i 1.46319e19 1.66808e22i 1.92288e24 + 3.06617e24i 1.45784e27 6.46078e28i 1.49520e30 3.09104e30i −3.25803e31
2.10 9.53717e8i 8.49626e14 + 1.64676e15i 1.75372e19 2.45445e22i 1.57054e24 8.10302e23i −1.16039e27 3.43185e28i −1.98996e30 + 2.79826e30i −2.34085e31
2.11 9.53717e8i 8.49626e14 1.64676e15i 1.75372e19 2.45445e22i 1.57054e24 + 8.10302e23i −1.16039e27 3.43185e28i −1.98996e30 2.79826e30i −2.34085e31
2.12 1.95316e9i −1.56985e15 9.84500e14i 1.46319e19 1.66808e22i 1.92288e24 3.06617e24i 1.45784e27 6.46078e28i 1.49520e30 + 3.09104e30i −3.25803e31
2.13 2.99899e9i −1.38295e15 + 1.23335e15i 9.45278e18 5.43489e21i −3.69880e24 4.14745e24i −9.58380e26 8.36705e28i 3.91402e29 3.41130e30i 1.62992e31
2.14 3.00379e9i 1.81065e15 3.93978e14i 9.42400e18 2.20086e22i 1.18343e24 + 5.43882e24i 9.43770e26 8.37178e28i 3.12325e30 1.42671e30i 6.61093e31
2.15 4.03164e9i 5.57989e14 + 1.76701e15i 2.19261e18 2.91915e22i −7.12396e24 + 2.24961e24i 6.92352e26 8.32105e28i −2.81098e30 + 1.97195e30i −1.17690e32
2.16 4.45029e9i −9.93580e14 1.56412e15i −1.35835e18 4.47337e22i 6.96081e24 4.42172e24i −9.42280e26 7.60483e28i −1.45928e30 + 3.10816e30i 1.99078e32
2.17 6.95479e9i 1.80867e15 + 4.02985e14i −2.99223e19 4.26608e21i −2.80268e24 + 1.25789e25i −1.59304e27 7.98101e28i 3.10889e30 + 1.45773e30i −2.96697e31
2.18 6.99784e9i 3.59786e14 1.81776e15i −3.05230e19 1.06729e22i 1.27204e25 + 2.51773e24i 9.75273e26 8.45080e28i −3.17479e30 1.30801e30i −7.46870e31
2.19 7.06644e9i −1.85210e15 + 5.82746e13i −3.14879e19 1.76003e22i −4.11794e23 1.30878e25i −6.88028e26 9.21545e28i 3.42689e30 2.15861e29i −1.24372e32
2.20 7.55940e9i −2.98618e14 + 1.82880e15i −3.86978e19 3.78912e22i −1.38246e25 2.25737e24i 1.61394e27 1.53086e29i −3.25534e30 1.09222e30i 2.86435e32
\(n\): 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 3.65.b.a 20
3.b odd 2 1 inner 3.65.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.65.b.a 20 1.a even 1 1 trivial
3.65.b.a 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{65}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 53\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 24\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
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