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Properties

Label 2304.4.f.e
Level $2304$
Weight $4$
Character orbit 2304.f
Analytic conductor $135.940$
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] = [2304,4,Mod(1151,2304)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1151");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.f (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: \(135.940400653\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.77720518656.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 161x^{4} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 288)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{3}) q^{5} + (\beta_{4} + 4 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{3}) q^{5} + (\beta_{4} + 4 \beta_1) q^{7} + ( - \beta_{7} - 16 \beta_{2}) q^{11} + (\beta_{4} + 16 \beta_1) q^{13} + ( - \beta_{7} + 33 \beta_{2}) q^{17} + ( - 2 \beta_{5} - 48) q^{19} + (6 \beta_{6} - \beta_{3}) q^{23} + (\beta_{5} + 141) q^{25} + (7 \beta_{6} - 38 \beta_{3}) q^{29} + ( - \beta_{4} - 148 \beta_1) q^{31} + ( - 3 \beta_{7} + 256 \beta_{2}) q^{35} + (8 \beta_{4} - 111 \beta_1) q^{37} + ( - 7 \beta_{7} - 83 \beta_{2}) q^{41} + ( - 7 \beta_{5} + 80) q^{43} + (26 \beta_{6} - 31 \beta_{3}) q^{47} + (8 \beta_{5} - 249) q^{49} + (17 \beta_{6} + 124 \beta_{3}) q^{53} + ( - 18 \beta_{4} + 280 \beta_1) q^{55} + ( - 2 \beta_{7} - 400 \beta_{2}) q^{59} + (24 \beta_{4} + 157 \beta_1) q^{61} + ( - 15 \beta_{7} + 232 \beta_{2}) q^{65} + (\beta_{5} + 352) q^{67} + (18 \beta_{6} + 181 \beta_{3}) q^{71} + (22 \beta_{5} + 144) q^{73} + (16 \beta_{6} + 208 \beta_{3}) q^{77} + ( - 15 \beta_{4} - 236 \beta_1) q^{79} + ( - 13 \beta_{7} + 512 \beta_{2}) q^{83} + (31 \beta_{4} + 231 \beta_1) q^{85} + ( - 2 \beta_{7} + 285 \beta_{2}) q^{89} + (20 \beta_{5} - 784) q^{91} + ( - 56 \beta_{6} - 580 \beta_{3}) q^{95} + (15 \beta_{5} + 928) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 384 q^{19} + 1128 q^{25} + 640 q^{43} - 1992 q^{49} + 2816 q^{67} + 1152 q^{73} - 6272 q^{91} + 7424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 161x^{4} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 225\nu^{2} ) / 544 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{7} + 64\nu^{5} + 937\nu^{3} + 5696\nu ) / 8704 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{7} - 64\nu^{5} + 937\nu^{3} - 5696\nu ) / 4352 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 97\nu^{2} ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{4} + 1288 ) / 17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41\nu^{7} - 64\nu^{5} + 8137\nu^{3} - 40512\nu ) / 8704 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\nu^{7} + 64\nu^{5} + 4537\nu^{3} + 23104\nu ) / 2176 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{3} - 4\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 34\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{7} + 18\beta_{6} - 41\beta_{3} - 100\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17\beta_{5} - 1288 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -89\beta_{7} + 178\beta_{6} - 633\beta_{3} + 1444\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -225\beta_{4} - 3298\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -937\beta_{7} - 1874\beta_{6} + 8137\beta_{3} + 18148\beta_{2} ) / 16 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
1.67746 + 1.67746i
1.67746 − 1.67746i
2.38456 − 2.38456i
2.38456 + 2.38456i
−2.38456 + 2.38456i
−2.38456 − 2.38456i
−1.67746 − 1.67746i
−1.67746 + 1.67746i
0 0 0 −17.6623 0 − 14.9783i 0 0 0
1151.2 0 0 0 −17.6623 0 14.9783i 0 0 0
1151.3 0 0 0 −14.8339 0 − 30.9783i 0 0 0
1151.4 0 0 0 −14.8339 0 30.9783i 0 0 0
1151.5 0 0 0 14.8339 0 − 30.9783i 0 0 0
1151.6 0 0 0 14.8339 0 30.9783i 0 0 0
1151.7 0 0 0 17.6623 0 − 14.9783i 0 0 0
1151.8 0 0 0 17.6623 0 14.9783i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.4.f.e 8
3.b odd 2 1 inner 2304.4.f.e 8
4.b odd 2 1 2304.4.f.h 8
8.b even 2 1 2304.4.f.h 8
8.d odd 2 1 inner 2304.4.f.e 8
12.b even 2 1 2304.4.f.h 8
16.e even 4 1 288.4.c.b ✓ 8
16.e even 4 1 576.4.c.f 8
16.f odd 4 1 288.4.c.b ✓ 8
16.f odd 4 1 576.4.c.f 8
24.f even 2 1 inner 2304.4.f.e 8
24.h odd 2 1 2304.4.f.h 8
48.i odd 4 1 288.4.c.b ✓ 8
48.i odd 4 1 576.4.c.f 8
48.k even 4 1 288.4.c.b ✓ 8
48.k even 4 1 576.4.c.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.4.c.b ✓ 8 16.e even 4 1
288.4.c.b ✓ 8 16.f odd 4 1
288.4.c.b ✓ 8 48.i odd 4 1
288.4.c.b ✓ 8 48.k even 4 1
576.4.c.f 8 16.e even 4 1
576.4.c.f 8 16.f odd 4 1
576.4.c.f 8 48.i odd 4 1
576.4.c.f 8 48.k even 4 1
2304.4.f.e 8 1.a even 1 1 trivial
2304.4.f.e 8 3.b odd 2 1 inner
2304.4.f.e 8 8.d odd 2 1 inner
2304.4.f.e 8 24.f even 2 1 inner
2304.4.f.h 8 4.b odd 2 1
2304.4.f.h 8 8.b even 2 1
2304.4.f.h 8 12.b even 2 1
2304.4.f.h 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} - 532T_{5}^{2} + 68644 \) Copy content Toggle raw display
\( T_{19}^{2} + 96T_{19} - 6144 \) 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
$5$ \( (T^{4} - 532 T^{2} + 68644)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1184 T^{2} + 215296)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 3136 T^{2} + 295936)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 3104 T^{2} + 246016)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 6468 T^{2} + 1258884)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 96 T - 6144)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 19264 T^{2} + 87909376)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 53428 T^{2} + 708964)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 176288 T^{2} + 7584319744)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 166152 T^{2} + 240002064)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 131044 T^{2} + 1441417156)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 160 T - 97088)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 387904 T^{2} + 26561176576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 366036 T^{2} + 925741476)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 648448 T^{2} + 99714482176)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 805448 T^{2} + 42243403024)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 704 T + 121792)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 644416 T^{2} + 22842090496)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 288 T - 1001472)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 683168 T^{2} + 10812672256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1405504 T^{2} + 119594238976)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 333348 T^{2} + 25035467076)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 1856 T + 385984)^{4} \) Copy content Toggle raw display
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