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Properties

Label 4.35.b.a
Level $4$
Weight $35$
Character orbit 4.b
Analytic conductor $29.290$
Analytic rank $0$
Dimension $16$
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] = [4,35,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, 35, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 35);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 35 \)
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: \(29.2902616171\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 72626617369828 x^{14} - 508386321588656 x^{13} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{240}\cdot 3^{26}\cdot 5^{6}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1711) q^{2} + ( - \beta_{2} - 76 \beta_1 + 19) q^{3} + (\beta_{4} + 7 \beta_{2} + \cdots - 1252054245) q^{4}+ \cdots + (\beta_{7} + 50 \beta_{6} + \cdots - 42\!\cdots\!21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1711) q^{2} + ( - \beta_{2} - 76 \beta_1 + 19) q^{3} + (\beta_{4} + 7 \beta_{2} + \cdots - 1252054245) q^{4}+ \cdots + (615042423583632 \beta_{15} + \cdots - 66\!\cdots\!69) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 27372 q^{2} - 20032875248 q^{4} - 21372255840 q^{5} + 20836736461728 q^{6} + 22\!\cdots\!12 q^{8}+ \cdots - 67\!\cdots\!80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 27372 q^{2} - 20032875248 q^{4} - 21372255840 q^{5} + 20836736461728 q^{6} + 22\!\cdots\!12 q^{8}+ \cdots + 16\!\cdots\!32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 72626617369828 x^{14} - 508386321588656 x^{13} + \cdots + 10\!\cdots\!04 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 26\!\cdots\!75 \nu^{15} + \cdots - 89\!\cdots\!28 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 49\!\cdots\!25 \nu^{15} + \cdots + 17\!\cdots\!32 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!45 \nu^{15} + \cdots - 28\!\cdots\!92 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\!\cdots\!45 \nu^{15} + \cdots + 25\!\cdots\!44 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!41 \nu^{15} + \cdots + 26\!\cdots\!48 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 49\!\cdots\!15 \nu^{15} + \cdots + 19\!\cdots\!00 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 88\!\cdots\!21 \nu^{15} + \cdots + 21\!\cdots\!16 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 59\!\cdots\!77 \nu^{15} + \cdots + 32\!\cdots\!76 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!75 \nu^{15} + \cdots - 45\!\cdots\!12 ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!93 \nu^{15} + \cdots - 45\!\cdots\!32 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43\!\cdots\!87 \nu^{15} + \cdots - 77\!\cdots\!76 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21\!\cdots\!39 \nu^{15} + \cdots + 16\!\cdots\!72 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 84\!\cdots\!53 \nu^{15} + \cdots + 17\!\cdots\!28 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16\!\cdots\!15 \nu^{15} + \cdots + 64\!\cdots\!84 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!65 \nu^{15} + \cdots + 70\!\cdots\!12 ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 76\beta _1 + 5 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 50 \beta_{6} + 23 \beta_{5} + 101670 \beta_{4} + 2103 \beta_{3} + 691967 \beta_{2} + \cdots - 20\!\cdots\!26 ) / 2304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 134640 \beta_{15} + 452577 \beta_{14} + 195496 \beta_{13} - 516800 \beta_{12} + \cdots - 87\!\cdots\!62 ) / 110592 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 548919743877 \beta_{15} + 3051470439708 \beta_{14} - 265916899456 \beta_{13} + \cdots + 62\!\cdots\!26 ) / 442368 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 46\!\cdots\!80 \beta_{15} + \cdots + 58\!\cdots\!35 ) / 2359296 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 13\!\cdots\!19 \beta_{15} + \cdots - 68\!\cdots\!76 ) / 28311552 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 39\!\cdots\!04 \beta_{15} + \cdots - 96\!\cdots\!41 ) / 150994944 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 24\!\cdots\!84 \beta_{15} + \cdots + 96\!\cdots\!73 ) / 226492416 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 63\!\cdots\!08 \beta_{15} + \cdots + 33\!\cdots\!25 ) / 226492416 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 23\!\cdots\!48 \beta_{15} + \cdots - 87\!\cdots\!21 ) / 113246208 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 20\!\cdots\!24 \beta_{15} + \cdots - 50\!\cdots\!59 ) / 150994944 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 27\!\cdots\!20 \beta_{15} + \cdots + 10\!\cdots\!77 ) / 75497472 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 50\!\cdots\!56 \beta_{15} + \cdots + 83\!\cdots\!25 ) / 113246208 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 33\!\cdots\!86 \beta_{15} + \cdots - 14\!\cdots\!59 ) / 56623104 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 94\!\cdots\!60 \beta_{15} + \cdots - 71\!\cdots\!91 ) / 452984832 \) 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
0.500000 + 3.36022e6i
0.500000 − 3.36022e6i
0.500000 + 182748.i
0.500000 − 182748.i
0.500000 − 2.05624e6i
0.500000 + 2.05624e6i
0.500000 + 4.30163e6i
0.500000 − 4.30163e6i
0.500000 − 4.39438e6i
0.500000 + 4.39438e6i
0.500000 − 190511.i
0.500000 + 190511.i
0.500000 + 1.72842e6i
0.500000 − 1.72842e6i
0.500000 − 4.02935e6i
0.500000 + 4.02935e6i
−130415. − 13105.5i − 1.61291e8i 1.68364e10 + 3.41831e9i −2.13902e10 −2.11380e12 + 2.10348e13i − 7.96031e13i −2.15092e15 − 6.66449e14i −9.33752e15 2.78961e15 + 2.80330e14i
3.2 −130415. + 13105.5i 1.61291e8i 1.68364e10 − 3.41831e9i −2.13902e10 −2.11380e12 − 2.10348e13i 7.96031e13i −2.15092e15 + 6.66449e14i −9.33752e15 2.78961e15 − 2.80330e14i
3.3 −88896.6 − 96318.6i − 8.77192e6i −1.37468e9 + 1.71248e10i 8.59192e11 −8.44899e11 + 7.79793e11i 3.22173e14i 1.77164e15 − 1.38993e15i 1.66002e16 −7.63792e16 − 8.27562e16i
3.4 −88896.6 + 96318.6i 8.77192e6i −1.37468e9 − 1.71248e10i 8.59192e11 −8.44899e11 − 7.79793e11i − 3.22173e14i 1.77164e15 + 1.38993e15i 1.66002e16 −7.63792e16 + 8.27562e16i
3.5 −83785.2 − 100796.i 9.86995e7i −3.13996e9 + 1.68905e10i −1.17209e12 9.94856e12 − 8.26956e12i − 2.80726e14i 1.96558e15 − 1.09867e15i 6.93558e15 9.82041e16 + 1.18143e17i
3.6 −83785.2 + 100796.i − 9.86995e7i −3.13996e9 − 1.68905e10i −1.17209e12 9.94856e12 + 8.26956e12i 2.80726e14i 1.96558e15 + 1.09867e15i 6.93558e15 9.82041e16 − 1.18143e17i
3.7 −13913.7 − 130331.i − 2.06478e8i −1.67927e10 + 3.62680e9i −9.83896e10 −2.69106e13 + 2.87289e12i − 2.44416e14i 7.06335e14 + 2.13815e15i −2.59561e16 1.36897e15 + 1.28233e16i
3.8 −13913.7 + 130331.i 2.06478e8i −1.67927e10 − 3.62680e9i −9.83896e10 −2.69106e13 − 2.87289e12i 2.44416e14i 7.06335e14 − 2.13815e15i −2.59561e16 1.36897e15 − 1.28233e16i
3.9 13540.0 − 130371.i 2.10930e8i −1.68132e10 − 3.53043e9i 1.26172e12 2.74991e13 + 2.85599e12i − 2.66990e14i −6.87916e14 + 2.14415e15i −2.78144e16 1.70837e16 − 1.64492e17i
3.10 13540.0 + 130371.i − 2.10930e8i −1.68132e10 + 3.53043e9i 1.26172e12 2.74991e13 − 2.85599e12i 2.66990e14i −6.87916e14 − 2.14415e15i −2.78144e16 1.70837e16 + 1.64492e17i
3.11 49362.6 − 121422.i 9.14451e6i −1.23065e10 − 1.19874e10i −6.20904e11 1.11034e12 + 4.51397e11i 3.03953e14i −2.06301e15 + 9.02552e14i 1.65936e16 −3.06494e16 + 7.53911e16i
3.12 49362.6 + 121422.i − 9.14451e6i −1.23065e10 + 1.19874e10i −6.20904e11 1.11034e12 − 4.51397e11i − 3.03953e14i −2.06301e15 − 9.02552e14i 1.65936e16 −3.06494e16 − 7.53911e16i
3.13 114483. − 63824.3i − 8.29643e7i 9.03280e9 − 1.46136e10i 6.47366e11 −5.29513e12 − 9.49799e12i − 1.29684e14i 1.01400e14 − 2.24952e15i 9.79411e15 7.41124e16 − 4.13177e16i
3.14 114483. + 63824.3i 8.29643e7i 9.03280e9 + 1.46136e10i 6.47366e11 −5.29513e12 + 9.49799e12i 1.29684e14i 1.01400e14 + 2.24952e15i 9.79411e15 7.41124e16 + 4.13177e16i
3.15 125939. − 36320.7i 1.93409e8i 1.45415e10 − 9.14841e9i −8.66188e11 7.02475e12 + 2.43577e13i − 8.12413e13i 1.49906e15 − 1.68030e15i −2.07298e16 −1.09087e17 + 3.14606e16i
3.16 125939. + 36320.7i − 1.93409e8i 1.45415e10 + 9.14841e9i −8.66188e11 7.02475e12 − 2.43577e13i 8.12413e13i 1.49906e15 + 1.68030e15i −2.07298e16 −1.09087e17 − 3.14606e16i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.16
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 4.35.b.a ✓ 16
4.b odd 2 1 inner 4.35.b.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.35.b.a ✓ 16 1.a even 1 1 trivial
4.35.b.a ✓ 16 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{35}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 75\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots - 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 38\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 35\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 23\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 44\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
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