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Properties

Label 4.45.b.b
Level $4$
Weight $45$
Character orbit 4.b
Analytic conductor $49.048$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,45,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 45, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 45);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 45 \)
Character orbit: \([\chi]\) \(=\) 4.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: \(49.0478939917\)
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: \( x^{20} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{380}\cdot 3^{44}\cdot 5^{12}\cdot 7^{2}\cdot 11^{4} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 239996) q^{2} + (\beta_{2} + 431 \beta_1 + 86) q^{3} + ( - \beta_{3} + 37 \beta_{2} + \cdots - 1531938272161) q^{4} + ( - \beta_{4} + 16 \beta_{3} + \cdots - 64326974711593) q^{5} + ( - \beta_{5} + \beta_{4} + \cdots - 75\!\cdots\!08) q^{6}+ \cdots + ( - 23\!\cdots\!96 \beta_{19} + \cdots + 18\!\cdots\!78) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4799916 q^{2} - 30638766490096 q^{4} - 12\!\cdots\!00 q^{5} - 15\!\cdots\!24 q^{6} + 17\!\cdots\!96 q^{8} - 67\!\cdots\!92 q^{9} - 12\!\cdots\!00 q^{10} - 10\!\cdots\!80 q^{12} - 50\!\cdots\!88 q^{13}+ \cdots - 38\!\cdots\!44 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + \cdots + 47\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 21\!\cdots\!75 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 92\!\cdots\!25 \nu^{19} + \cdots + 56\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 86\!\cdots\!65 \nu^{19} + \cdots - 12\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 65\!\cdots\!15 \nu^{19} + \cdots - 81\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 97\!\cdots\!49 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!15 \nu^{19} + \cdots + 85\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77\!\cdots\!43 \nu^{19} + \cdots + 30\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!05 \nu^{19} + \cdots - 89\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22\!\cdots\!95 \nu^{19} + \cdots + 94\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47\!\cdots\!39 \nu^{19} + \cdots - 32\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!65 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!89 \nu^{19} + \cdots - 36\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!83 \nu^{19} + \cdots + 68\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15\!\cdots\!53 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10\!\cdots\!77 \nu^{19} + \cdots + 89\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 81\!\cdots\!55 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 47\!\cdots\!13 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 51\!\cdots\!03 \nu^{19} + \cdots + 81\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 19\!\cdots\!99 \nu^{19} + \cdots - 26\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 431\beta _1 + 86 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - 3 \beta_{8} - 2 \beta_{7} - 465 \beta_{6} - 485 \beta_{5} - 4808 \beta_{4} + \cdots - 13\!\cdots\!44 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 46576 \beta_{19} - 5961920 \beta_{18} + 10435980 \beta_{17} + 699168 \beta_{16} + \cdots - 21\!\cdots\!80 ) / 4096 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 72\!\cdots\!82 \beta_{17} + \cdots + 13\!\cdots\!34 ) / 32768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 50\!\cdots\!60 \beta_{19} + \cdots + 35\!\cdots\!36 ) / 524288 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 27\!\cdots\!17 \beta_{17} + \cdots - 41\!\cdots\!76 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 30\!\cdots\!36 \beta_{19} + \cdots - 13\!\cdots\!51 ) / 16777216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 98\!\cdots\!32 \beta_{17} + \cdots + 12\!\cdots\!54 ) / 33554432 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 14\!\cdots\!20 \beta_{19} + \cdots + 51\!\cdots\!34 ) / 536870912 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 34\!\cdots\!64 \beta_{17} + \cdots - 40\!\cdots\!94 ) / 1073741824 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 17\!\cdots\!12 \beta_{19} + \cdots - 58\!\cdots\!76 ) / 536870912 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 39\!\cdots\!08 \beta_{17} + \cdots + 41\!\cdots\!74 ) / 1073741824 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 21\!\cdots\!56 \beta_{19} + \cdots + 67\!\cdots\!94 ) / 536870912 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 22\!\cdots\!00 \beta_{17} + \cdots - 21\!\cdots\!94 ) / 536870912 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 13\!\cdots\!68 \beta_{19} + \cdots - 38\!\cdots\!68 ) / 268435456 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 25\!\cdots\!56 \beta_{17} + \cdots + 21\!\cdots\!90 ) / 536870912 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 30\!\cdots\!72 \beta_{19} + \cdots + 86\!\cdots\!74 ) / 536870912 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 58\!\cdots\!56 \beta_{17} + \cdots - 45\!\cdots\!22 ) / 1073741824 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 35\!\cdots\!20 \beta_{19} + \cdots - 98\!\cdots\!76 ) / 536870912 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
− 3.35197e9i
3.35197e9i
2.38483e9i
− 2.38483e9i
1.23618e9i
− 1.23618e9i
− 1.22466e9i
1.22466e9i
− 1.35492e9i
1.35492e9i
3.08594e9i
− 3.08594e9i
− 3.00773e9i
3.00773e9i
3.19621e8i
− 3.19621e8i
1.49929e9i
− 1.49929e9i
− 2.98896e9i
2.98896e9i
−3.68465e6 − 2.00389e6i − 5.36316e10i 9.56104e12 + 1.47672e13i 7.16734e14 −1.07472e17 + 1.97613e17i 1.83098e18i −5.63715e18 − 7.35713e19i −1.89157e21 −2.64091e21 − 1.43626e21i
3.2 −3.68465e6 + 2.00389e6i 5.36316e10i 9.56104e12 − 1.47672e13i 7.16734e14 −1.07472e17 − 1.97613e17i − 1.83098e18i −5.63715e18 + 7.35713e19i −1.89157e21 −2.64091e21 + 1.43626e21i
3.3 −3.36765e6 − 2.50023e6i 3.81573e10i 5.08992e12 + 1.68398e13i −2.81252e15 9.54019e16 − 1.28500e17i 3.86238e18i 2.49622e19 − 6.94364e19i −4.71210e20 9.47159e21 + 7.03195e21i
3.4 −3.36765e6 + 2.50023e6i − 3.81573e10i 5.08992e12 − 1.68398e13i −2.81252e15 9.54019e16 + 1.28500e17i − 3.86238e18i 2.49622e19 + 6.94364e19i −4.71210e20 9.47159e21 − 7.03195e21i
3.5 −2.67504e6 − 3.23054e6i 1.97789e10i −3.28055e12 + 1.72836e13i 4.32114e15 6.38964e16 − 5.29092e16i − 4.05908e18i 6.46109e19 − 3.56363e19i 5.93567e20 −1.15592e22 − 1.39596e22i
3.6 −2.67504e6 + 3.23054e6i − 1.97789e10i −3.28055e12 − 1.72836e13i 4.32114e15 6.38964e16 + 5.29092e16i 4.05908e18i 6.46109e19 + 3.56363e19i 5.93567e20 −1.15592e22 + 1.39596e22i
3.7 −1.53066e6 − 3.90503e6i − 1.95946e10i −1.29064e13 + 1.19545e13i −3.24586e15 −7.65175e16 + 2.99926e16i − 5.50179e18i 6.64381e19 + 3.21014e19i 6.00823e20 4.96831e21 + 1.26752e22i
3.8 −1.53066e6 + 3.90503e6i 1.95946e10i −1.29064e13 − 1.19545e13i −3.24586e15 −7.65175e16 − 2.99926e16i 5.50179e18i 6.64381e19 − 3.21014e19i 6.00823e20 4.96831e21 − 1.26752e22i
3.9 −174802. − 4.19066e6i − 2.16788e10i −1.75311e13 + 1.46508e12i 1.03983e15 −9.08484e16 + 3.78950e15i 7.00373e18i 9.20411e18 + 7.32107e19i 5.14802e20 −1.81765e20 − 4.35758e21i
3.10 −174802. + 4.19066e6i 2.16788e10i −1.75311e13 − 1.46508e12i 1.03983e15 −9.08484e16 − 3.78950e15i − 7.00373e18i 9.20411e18 − 7.32107e19i 5.14802e20 −1.81765e20 + 4.35758e21i
3.11 692907. − 4.13667e6i 4.93751e10i −1.66319e13 − 5.73266e12i 3.31017e14 2.04249e17 + 3.42123e16i − 3.30487e17i −3.52385e19 + 6.48287e19i −1.45313e21 2.29364e20 − 1.36931e21i
3.12 692907. + 4.13667e6i − 4.93751e10i −1.66319e13 + 5.73266e12i 3.31017e14 2.04249e17 − 3.42123e16i 3.30487e17i −3.52385e19 − 6.48287e19i −1.45313e21 2.29364e20 + 1.36931e21i
3.13 2.45341e6 − 3.40190e6i − 4.81236e10i −5.55370e12 − 1.66926e13i 2.94946e15 −1.63712e17 − 1.18067e17i − 5.91201e18i −7.04120e19 − 2.20606e19i −1.33111e21 7.23624e21 − 1.00338e22i
3.14 2.45341e6 + 3.40190e6i 4.81236e10i −5.55370e12 + 1.66926e13i 2.94946e15 −1.63712e17 + 1.18067e17i 5.91201e18i −7.04120e19 + 2.20606e19i −1.33111e21 7.23624e21 + 1.00338e22i
3.15 2.63672e6 − 3.26188e6i 5.11393e9i −3.68760e12 − 1.72014e13i −1.83599e15 1.66811e16 + 1.34840e16i − 1.24564e18i −6.58320e19 − 3.33266e19i 9.58619e20 −4.84100e21 + 5.98879e21i
3.16 2.63672e6 + 3.26188e6i − 5.11393e9i −3.68760e12 + 1.72014e13i −1.83599e15 1.66811e16 − 1.34840e16i 1.24564e18i −6.58320e19 + 3.33266e19i 9.58619e20 −4.84100e21 − 5.98879e21i
3.17 3.98169e6 − 1.31845e6i 2.39886e10i 1.41156e13 − 1.04993e13i 1.91424e15 3.16279e16 + 9.55154e16i 2.07168e18i 4.23609e19 − 6.04158e19i 4.09316e20 7.62190e21 − 2.52383e21i
3.18 3.98169e6 + 1.31845e6i − 2.39886e10i 1.41156e13 + 1.04993e13i 1.91424e15 3.16279e16 − 9.55154e16i − 2.07168e18i 4.23609e19 + 6.04158e19i 4.09316e20 7.62190e21 + 2.52383e21i
3.19 4.06802e6 − 1.02149e6i − 4.78233e10i 1.55053e13 − 8.31084e12i −4.02131e15 −4.88508e16 − 1.94546e17i 5.21410e18i 5.45865e19 − 4.96471e19i −1.30230e21 −1.63587e22 + 4.10771e21i
3.20 4.06802e6 + 1.02149e6i 4.78233e10i 1.55053e13 + 8.31084e12i −4.02131e15 −4.88508e16 + 1.94546e17i − 5.21410e18i 5.45865e19 + 4.96471e19i −1.30230e21 −1.63587e22 − 4.10771e21i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.45.b.b ✓ 20
4.b odd 2 1 inner 4.45.b.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.45.b.b ✓ 20 1.a even 1 1 trivial
4.45.b.b ✓ 20 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + \cdots + 57\!\cdots\!00 \) acting on \(S_{45}^{\mathrm{new}}(4, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 60\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 24\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 14\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
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