Properties

Label 2368.2.a.bj
Level $2368$
Weight $2$
Character orbit 2368.a
Self dual yes
Analytic conductor $18.909$
Analytic rank $0$
Dimension $8$
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] = [2368,2,Mod(1,2368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.a (trivial)

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: yes
Analytic conductor: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 20x^{6} + 116x^{4} - 221x^{2} + 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{6} q^{5} - \beta_{4} q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{6} q^{5} - \beta_{4} q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{7} + \beta_1) q^{11} + ( - \beta_{6} - \beta_{5} + \beta_{2}) q^{13} - \beta_{7} q^{15} + 2 q^{17} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{19} + (\beta_{6} + 2 \beta_{5} - \beta_{2} - 1) q^{21} + \beta_{3} q^{23} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 3) q^{25} + (\beta_{7} + \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{27} + ( - \beta_{6} - \beta_{5} + \beta_{2}) q^{29} + (\beta_{7} - \beta_{3}) q^{31} + ( - \beta_{6} + \beta_{5} + 3) q^{33} + (2 \beta_{7} - 2 \beta_{4}) q^{35} - q^{37} + (\beta_{7} + 3 \beta_{4} - 2 \beta_{3} + \beta_1) q^{39} + ( - \beta_{5} + 5) q^{41} + (\beta_{4} + \beta_{3} - \beta_1) q^{43} + (2 \beta_{6} + \beta_{5} - \beta_{2} - 2) q^{45} + (2 \beta_{7} - \beta_{4}) q^{47} + ( - \beta_{6} + 2 \beta_{5} - \beta_{2} + 4) q^{49} + 2 \beta_1 q^{51} + ( - \beta_{6} - 2 \beta_{5} - \beta_{2} + 1) q^{53} + ( - 2 \beta_{7} + \beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{55} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + 4) q^{57} + ( - 2 \beta_{7} - \beta_{4} + \beta_{3} - \beta_1) q^{59} + (\beta_{6} - 4) q^{61} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3}) q^{63} + ( - \beta_{5} - \beta_{2} + 2) q^{65} + ( - \beta_{7} + 3 \beta_{4} - 2 \beta_{3} + \beta_1) q^{67} + (\beta_{6} - 2 \beta_{2}) q^{69} + (2 \beta_{7} - \beta_{4} + 2 \beta_1) q^{71} + (\beta_{2} + 7) q^{73} + ( - \beta_{7} + \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{75} + (5 \beta_{6} + 4 \beta_{5} - \beta_{2} + 1) q^{77} + \beta_{3} q^{79} + ( - 2 \beta_{6} - 3 \beta_{5} + 5 \beta_{2} + 7) q^{81} + (\beta_{4} - 2 \beta_1) q^{83} - 2 \beta_{6} q^{85} + (\beta_{7} + 3 \beta_{4} - 2 \beta_{3} + \beta_1) q^{87} + (2 \beta_{6} - 2 \beta_{5} + 4) q^{89} + (2 \beta_{7} + 2 \beta_{4} - 6 \beta_1) q^{91} + ( - \beta_{5} + 3 \beta_{2} + 2) q^{93} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{95} + 2 q^{97} + (\beta_{7} - 2 \beta_{4} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} + 16 q^{9} - 2 q^{13} + 16 q^{17} - 2 q^{21} + 22 q^{25} - 2 q^{29} + 30 q^{33} - 8 q^{37} + 36 q^{41} - 16 q^{45} + 42 q^{49} + 2 q^{53} + 36 q^{57} - 34 q^{61} + 12 q^{65} - 2 q^{69} + 56 q^{73} + 14 q^{77} + 48 q^{81} + 4 q^{85} + 20 q^{89} + 12 q^{93} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 20x^{6} + 116x^{4} - 221x^{2} + 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} - 88\nu^{5} + 360\nu^{3} - 153\nu ) / 44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 22\nu^{5} - 138\nu^{3} + 211\nu ) / 22 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} + 33\nu^{4} - 122\nu^{2} + 81 ) / 11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{6} - 55\nu^{4} + 260\nu^{2} - 270 ) / 11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 55\nu^{5} + 260\nu^{3} - 270\nu ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{4} - 2\beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} - 3\beta_{5} + 14\beta_{2} + 43 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{7} + 20\beta_{4} - 28\beta_{3} + 79\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -33\beta_{6} - 55\beta_{5} + 170\beta_{2} + 445 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 192\beta_{7} + 280\beta_{4} - 340\beta_{3} + 845\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.39115
−2.36494
−1.34313
−1.05032
1.05032
1.34313
2.36494
3.39115
0 −3.39115 0 −0.805051 0 1.94652 0 8.49990 0
1.2 0 −2.36494 0 1.03927 0 −5.11652 0 2.59295 0
1.3 0 −1.34313 0 −3.42359 0 1.69567 0 −1.19601 0
1.4 0 −1.05032 0 4.18937 0 4.01957 0 −1.89684 0
1.5 0 1.05032 0 4.18937 0 −4.01957 0 −1.89684 0
1.6 0 1.34313 0 −3.42359 0 −1.69567 0 −1.19601 0
1.7 0 2.36494 0 1.03927 0 5.11652 0 2.59295 0
1.8 0 3.39115 0 −0.805051 0 −1.94652 0 8.49990 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(37\) \(1\)

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 2368.2.a.bj 8
4.b odd 2 1 inner 2368.2.a.bj 8
8.b even 2 1 1184.2.a.p 8
8.d odd 2 1 1184.2.a.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1184.2.a.p 8 8.b even 2 1
1184.2.a.p 8 8.d odd 2 1
2368.2.a.bj 8 1.a even 1 1 trivial
2368.2.a.bj 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2368))\):

\( T_{3}^{8} - 20T_{3}^{6} + 116T_{3}^{4} - 221T_{3}^{2} + 128 \) Copy content Toggle raw display
\( T_{5}^{4} - T_{5}^{3} - 15T_{5}^{2} + 4T_{5} + 12 \) Copy content Toggle raw display
\( T_{7}^{8} - 49T_{7}^{6} + 716T_{7}^{4} - 3280T_{7}^{2} + 4608 \) Copy content Toggle raw display
\( T_{11}^{8} - 64T_{11}^{6} + 1280T_{11}^{4} - 7869T_{11}^{2} + 3200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 20 T^{6} + 116 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} - 15 T^{2} + 4 T + 12)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 49 T^{6} + 716 T^{4} + \cdots + 4608 \) Copy content Toggle raw display
$11$ \( T^{8} - 64 T^{6} + 1280 T^{4} + \cdots + 3200 \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} - 37 T^{2} - 30 T + 152)^{2} \) Copy content Toggle raw display
$17$ \( (T - 2)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} - 96 T^{6} + 2512 T^{4} + \cdots + 2048 \) Copy content Toggle raw display
$23$ \( T^{8} - 79 T^{6} + 1541 T^{4} + \cdots + 7200 \) Copy content Toggle raw display
$29$ \( (T^{4} + T^{3} - 37 T^{2} - 30 T + 152)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 111 T^{6} + 3449 T^{4} + \cdots + 41472 \) Copy content Toggle raw display
$37$ \( (T + 1)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 18 T^{3} + 86 T^{2} + 31 T - 678)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 196 T^{6} + 13808 T^{4} + \cdots + 4333568 \) Copy content Toggle raw display
$47$ \( T^{8} - 233 T^{6} + 11980 T^{4} + \cdots + 4608 \) Copy content Toggle raw display
$53$ \( (T^{4} - T^{3} - 218 T^{2} + 292 T + 8856)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 328 T^{6} + \cdots + 11674112 \) Copy content Toggle raw display
$61$ \( (T^{4} + 17 T^{3} + 93 T^{2} + 180 T + 76)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 539 T^{6} + \cdots + 259555328 \) Copy content Toggle raw display
$71$ \( T^{8} - 349 T^{6} + \cdots + 30732800 \) Copy content Toggle raw display
$73$ \( (T^{4} - 28 T^{3} + 260 T^{2} - 957 T + 1210)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 79 T^{6} + 1541 T^{4} + \cdots + 7200 \) Copy content Toggle raw display
$83$ \( T^{8} - 125 T^{6} + 2832 T^{4} + \cdots + 8192 \) Copy content Toggle raw display
$89$ \( (T^{4} - 10 T^{3} - 224 T^{2} + 1024 T + 1920)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{8} \) Copy content Toggle raw display
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