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Properties

Label 115.2.b.b
Level $115$
Weight $2$
Character orbit 115.b
Analytic conductor $0.918$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,2,Mod(24,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.527896576.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + (\beta_{5} - \beta_{3} - \beta_1) q^{3} + ( - \beta_{6} + \beta_{4} - \beta_{2} - 1) q^{4} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{5} + ( - \beta_{6} + \beta_{4} - \beta_{2} - 2) q^{6}+ \cdots + (4 \beta_{6} - 4 \beta_{4} - 2 \beta_{3} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} - 6 q^{5} - 12 q^{6} - 8 q^{9} + 6 q^{10} + 4 q^{11} + 8 q^{14} + 6 q^{15} + 4 q^{16} + 8 q^{19} - 8 q^{20} - 4 q^{21} + 24 q^{24} - 16 q^{25} + 12 q^{26} - 8 q^{29} - 2 q^{30} - 28 q^{34}+ \cdots - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -27\nu^{7} + 31\nu^{6} + 7\nu^{5} - 221\nu^{4} - 187\nu^{3} + 128\nu^{2} + 66\nu - 882 ) / 467 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 52\nu^{7} - 77\nu^{6} + 73\nu^{5} + 97\nu^{4} + 585\nu^{3} - 333\nu^{2} + 288\nu + 142 ) / 467 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 138\nu^{7} - 366\nu^{6} + 535\nu^{5} - 12\nu^{4} + 852\nu^{3} - 1692\nu^{2} + 2309\nu - 162 ) / 467 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -142\nu^{7} + 336\nu^{6} - 361\nu^{5} - 211\nu^{4} - 897\nu^{3} + 2005\nu^{2} - 1469\nu - 280 ) / 467 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 273\nu^{7} - 521\nu^{6} + 500\nu^{5} + 626\nu^{4} + 1787\nu^{3} - 2332\nu^{2} + 1979\nu + 979 ) / 467 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -284\nu^{7} + 672\nu^{6} - 722\nu^{5} - 422\nu^{4} - 1794\nu^{3} + 3543\nu^{2} - 2938\nu - 560 ) / 467 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + 2\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{4} - 5\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{7} - 6\beta_{5} + 5\beta_{4} - 6\beta_{2} - 5\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{7} + 6\beta_{6} - 27\beta_{5} + 6\beta_{4} - \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 29\beta_{7} + 22\beta_{6} - 30\beta_{5} - 15\beta_{3} + 29\beta_{2} + 30 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
1.47984 + 1.47984i
−0.199724 + 0.199724i
0.790245 − 0.790245i
−1.07037 − 1.07037i
−1.07037 + 1.07037i
0.790245 + 0.790245i
−0.199724 − 0.199724i
1.47984 − 1.47984i
− 2.37988i − 1.95969i −3.66382 0.479844 + 2.18398i −4.66382 − 2.28394i 3.95969i −0.840379 5.19760 − 1.14197i
24.2 − 1.92022i − 1.39945i −1.68725 −1.19972 − 1.88697i −2.68725 4.60747i − 0.600553i 1.04155 −3.62340 + 2.30373i
24.3 − 0.751024i 0.580491i 1.43596 −0.209755 + 2.22621i 0.435963 0.315061i − 2.58049i 2.66303 1.67194 + 0.157531i
24.4 − 0.291367i 3.14073i 1.91511 −2.07037 − 0.844739i 0.915105 1.20647i − 1.14073i −6.86420 −0.246129 + 0.603236i
24.5 0.291367i − 3.14073i 1.91511 −2.07037 + 0.844739i 0.915105 − 1.20647i 1.14073i −6.86420 −0.246129 − 0.603236i
24.6 0.751024i − 0.580491i 1.43596 −0.209755 − 2.22621i 0.435963 − 0.315061i 2.58049i 2.66303 1.67194 − 0.157531i
24.7 1.92022i 1.39945i −1.68725 −1.19972 + 1.88697i −2.68725 − 4.60747i 0.600553i 1.04155 −3.62340 − 2.30373i
24.8 2.37988i 1.95969i −3.66382 0.479844 − 2.18398i −4.66382 2.28394i − 3.95969i −0.840379 5.19760 + 1.14197i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.b.b ✓ 8
3.b odd 2 1 1035.2.b.e 8
4.b odd 2 1 1840.2.e.d 8
5.b even 2 1 inner 115.2.b.b ✓ 8
5.c odd 4 1 575.2.a.i 4
5.c odd 4 1 575.2.a.j 4
15.d odd 2 1 1035.2.b.e 8
15.e even 4 1 5175.2.a.bv 4
15.e even 4 1 5175.2.a.bw 4
20.d odd 2 1 1840.2.e.d 8
20.e even 4 1 9200.2.a.ck 4
20.e even 4 1 9200.2.a.cq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.b ✓ 8 1.a even 1 1 trivial
115.2.b.b ✓ 8 5.b even 2 1 inner
575.2.a.i 4 5.c odd 4 1
575.2.a.j 4 5.c odd 4 1
1035.2.b.e 8 3.b odd 2 1
1035.2.b.e 8 15.d odd 2 1
1840.2.e.d 8 4.b odd 2 1
1840.2.e.d 8 20.d odd 2 1
5175.2.a.bv 4 15.e even 4 1
5175.2.a.bw 4 15.e even 4 1
9200.2.a.ck 4 20.e even 4 1
9200.2.a.cq 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 10T_{2}^{6} + 27T_{2}^{4} + 14T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(115, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 10 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 16 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 16 T^{2} + \cdots - 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 64 T^{6} + \cdots + 3721 \) Copy content Toggle raw display
$17$ \( T^{8} + 80 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} - 6 T^{2} + \cdots - 20)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 22 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 74 T^{2} + \cdots - 167)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 148 T^{6} + \cdots + 226576 \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} + \cdots + 2485)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 252 T^{6} + \cdots + 3857296 \) Copy content Toggle raw display
$47$ \( T^{8} + 280 T^{6} + \cdots + 20367169 \) Copy content Toggle raw display
$53$ \( T^{8} + 224 T^{6} + \cdots + 4804864 \) Copy content Toggle raw display
$59$ \( (T^{4} - 20 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8 T^{3} + \cdots - 2756)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 20 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( (T^{4} + 24 T^{3} + \cdots - 7435)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 368 T^{6} + \cdots + 69538921 \) Copy content Toggle raw display
$79$ \( (T^{4} + 24 T^{3} + \cdots + 28)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 576 T^{6} + \cdots + 223442704 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} + \cdots + 2380)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 460 T^{6} + \cdots + 21864976 \) Copy content Toggle raw display
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