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Properties

Label 2304.3.h.j
Level $2304$
Weight $3$
Character orbit 2304.h
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(2177,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.2177");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (of order \(2\), degree \(1\), not 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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.959512576.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1152)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3}) q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3}) q^{5} - 2 q^{7} + (2 \beta_{5} - 4 \beta_{3}) q^{11} + ( - \beta_{6} + 4 \beta_1) q^{13} + (\beta_{4} - \beta_{2}) q^{17} + (2 \beta_{6} + 4 \beta_1) q^{19} + ( - 2 \beta_{4} - 6 \beta_{2}) q^{23} + ( - \beta_{7} + 21) q^{25} + (\beta_{5} + 6 \beta_{3}) q^{29} + (2 \beta_{7} + 2) q^{31} + (2 \beta_{5} - 2 \beta_{3}) q^{35} - 13 \beta_1 q^{37} + ( - \beta_{4} + 11 \beta_{2}) q^{41} + (4 \beta_{6} - 14 \beta_1) q^{43} + ( - 2 \beta_{4} + 38 \beta_{2}) q^{47} - 45 q^{49} + (3 \beta_{5} - 12 \beta_{3}) q^{53} + (4 \beta_{7} - 100) q^{55} - 26 \beta_{3} q^{59} + (4 \beta_{6} + 15 \beta_1) q^{61} + ( - 5 \beta_{4} - 52 \beta_{2}) q^{65} + (2 \beta_{6} + 42 \beta_1) q^{67} + (4 \beta_{4} + 22 \beta_{2}) q^{71} + (6 \beta_{7} + 28) q^{73} + ( - 4 \beta_{5} + 8 \beta_{3}) q^{77} + ( - 2 \beta_{7} + 38) q^{79} + (6 \beta_{5} - 6 \beta_{3}) q^{83} + (\beta_{6} - 43 \beta_1) q^{85} + (6 \beta_{4} + 47 \beta_{2}) q^{89} + (2 \beta_{6} - 8 \beta_1) q^{91} + ( - 2 \beta_{4} + 80 \beta_{2}) q^{95} + (\beta_{7} - 88) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{7} + 168 q^{25} + 16 q^{31} - 360 q^{49} - 800 q^{55} + 224 q^{73} + 304 q^{79} - 704 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} + 32\nu^{2} ) / 45 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{7} - 9\nu^{5} - 13\nu^{3} - 9\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{7} - 18\nu^{5} + 26\nu^{3} - 18\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{4} + 28 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} - 30\nu^{6} - 9\nu^{5} + 13\nu^{3} + 60\nu^{2} - 9\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\nu^{7} + 18\nu^{5} + 166\nu^{3} + 558\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -32\nu^{7} + 36\nu^{5} - 332\nu^{3} + 1116\nu ) / 135 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} + 2\beta_{3} + 4\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - \beta_{3} + 10\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + 8\beta_{3} - 16\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{4} - 28 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} - 62\beta_{3} - 124\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -8\beta_{5} + 4\beta_{3} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} + 26\beta_{6} - 166\beta_{3} + 332\beta_{2} ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2177.1
−1.52616 + 0.819051i
−1.52616 − 0.819051i
1.52616 + 0.819051i
1.52616 − 0.819051i
0.819051 + 1.52616i
0.819051 − 1.52616i
−0.819051 + 1.52616i
−0.819051 − 1.52616i
0 0 0 −8.04746 0 −2.00000 0 0 0
2177.2 0 0 0 −8.04746 0 −2.00000 0 0 0
2177.3 0 0 0 −5.21904 0 −2.00000 0 0 0
2177.4 0 0 0 −5.21904 0 −2.00000 0 0 0
2177.5 0 0 0 5.21904 0 −2.00000 0 0 0
2177.6 0 0 0 5.21904 0 −2.00000 0 0 0
2177.7 0 0 0 8.04746 0 −2.00000 0 0 0
2177.8 0 0 0 8.04746 0 −2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2177.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.h.j 8
3.b odd 2 1 inner 2304.3.h.j 8
4.b odd 2 1 2304.3.h.l 8
8.b even 2 1 inner 2304.3.h.j 8
8.d odd 2 1 2304.3.h.l 8
12.b even 2 1 2304.3.h.l 8
16.e even 4 1 1152.3.e.e yes 4
16.e even 4 1 1152.3.e.g yes 4
16.f odd 4 1 1152.3.e.a ✓ 4
16.f odd 4 1 1152.3.e.c yes 4
24.f even 2 1 2304.3.h.l 8
24.h odd 2 1 inner 2304.3.h.j 8
48.i odd 4 1 1152.3.e.e yes 4
48.i odd 4 1 1152.3.e.g yes 4
48.k even 4 1 1152.3.e.a ✓ 4
48.k even 4 1 1152.3.e.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.e.a ✓ 4 16.f odd 4 1
1152.3.e.a ✓ 4 48.k even 4 1
1152.3.e.c yes 4 16.f odd 4 1
1152.3.e.c yes 4 48.k even 4 1
1152.3.e.e yes 4 16.e even 4 1
1152.3.e.e yes 4 48.i odd 4 1
1152.3.e.g yes 4 16.e even 4 1
1152.3.e.g yes 4 48.i odd 4 1
2304.3.h.j 8 1.a even 1 1 trivial
2304.3.h.j 8 3.b odd 2 1 inner
2304.3.h.j 8 8.b even 2 1 inner
2304.3.h.j 8 24.h odd 2 1 inner
2304.3.h.l 8 4.b odd 2 1
2304.3.h.l 8 8.d odd 2 1
2304.3.h.l 8 12.b even 2 1
2304.3.h.l 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} - 92T_{5}^{2} + 1764 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 92 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 496 T^{2} + 10816)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 304 T^{2} + 576)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 356 T^{2} + 30276)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 832 T^{2} + 82944)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1552 T^{2} + 399424)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 764 T^{2} + 86436)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 1404)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 676)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 836 T^{2} + 4356)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4384 T^{2} + 389376)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 7184 T^{2} + 4769856)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2556 T^{2} + 236196)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 5408)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4616 T^{2} + 258064)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 14816 T^{2} + 44943616)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 7568 T^{2} + 3415104)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 56 T - 11888)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 76 T + 36)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 3312 T^{2} + 2286144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 21508 T^{2} + 3678724)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 176 T + 7392)^{4} \) Copy content Toggle raw display
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