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Properties

Label 2304.4.c.j
Level $2304$
Weight $4$
Character orbit 2304.c
Analytic conductor $135.940$
Analytic rank $0$
Dimension $6$
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] = [2304,4,Mod(2303,2304)]
 
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, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.2303");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.c (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: \(6\)
Coefficient field: 6.0.16269193728.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 39x^{4} + 50x^{3} + 533x^{2} - 264x + 1314 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{5} + 12) q^{11} + ( - \beta_{5} + \beta_{4} + 2) q^{13} + (4 \beta_{3} + \beta_1) q^{17} + ( - 2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{19} + ( - 3 \beta_{5} - 2 \beta_{4} - 36) q^{23} + ( - 4 \beta_{4} - 53) q^{25} + (4 \beta_{3} + 3 \beta_{2} - 7 \beta_1) q^{29} + ( - 5 \beta_{3} + 11 \beta_{2} - 27 \beta_1) q^{31} + ( - \beta_{5} - 8 \beta_{4} - 172) q^{35} + ( - 2 \beta_{4} - 56) q^{37} + (4 \beta_{3} - 16 \beta_{2} + 19 \beta_1) q^{41} + ( - 12 \beta_{3} + 4 \beta_{2} + 28 \beta_1) q^{43} + (\beta_{5} + 6 \beta_{4} - 52) q^{47} + ( - 8 \beta_{5} - 8 \beta_{4} - 177) q^{49} + ( - 4 \beta_{3} + 7 \beta_{2} - 35 \beta_1) q^{53} + ( - 2 \beta_{3} - 2 \beta_{2} + 66 \beta_1) q^{55} + (8 \beta_{5} - 8 \beta_{4} + 192) q^{59} + (8 \beta_{5} - 6 \beta_{4} - 120) q^{61} + ( - 20 \beta_{3} - 48 \beta_{2} + 42 \beta_1) q^{65} + ( - 18 \beta_{3} - 14 \beta_{2} - 94 \beta_1) q^{67} + (5 \beta_{5} + 6 \beta_{4} + 476) q^{71} + ( - 24 \beta_{5} - 200) q^{73} + ( - 32 \beta_{3} - 24 \beta_{2} - 56 \beta_1) q^{77} + (25 \beta_{3} - 23 \beta_{2} - 121 \beta_1) q^{79} + (13 \beta_{5} + 8 \beta_{4} - 324) q^{83} + (3 \beta_{5} + 15 \beta_{4} - 46) q^{85} + ( - 24 \beta_{3} - 32 \beta_{2} + 9 \beta_1) q^{89} + (18 \beta_{3} - 58 \beta_{2} + 270 \beta_1) q^{91} + (2 \beta_{5} - 28 \beta_{4} - 968) q^{95} + ( - 8 \beta_{5} - 12 \beta_{4} - 256) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 72 q^{11} + 12 q^{13} - 216 q^{23} - 318 q^{25} - 1032 q^{35} - 336 q^{37} - 312 q^{47} - 1062 q^{49} + 1152 q^{59} - 720 q^{61} + 2856 q^{71} - 1200 q^{73} - 1944 q^{83} - 276 q^{85} - 5808 q^{95} - 1536 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 39x^{4} + 50x^{3} + 533x^{2} - 264x + 1314 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{5} - 7\nu^{4} - 124\nu^{3} + 124\nu^{2} + 1803\nu - 414 ) / 705 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\nu^{5} - 76\nu^{4} - 1102\nu^{3} + 1552\nu^{2} + 17694\nu - 3132 ) / 2115 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -22\nu^{5} - 59\nu^{4} + 1282\nu^{3} - 112\nu^{2} - 18459\nu + 6822 ) / 2115 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{5} - 4\nu^{4} + 272\nu^{3} + 508\nu^{2} + 240\nu - 7704 ) / 423 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{5} - 208\nu^{4} - 256\nu^{3} + 8056\nu^{2} - 5088\nu - 70056 ) / 2115 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + 8\beta _1 + 8 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{5} + 4\beta_{4} - 12\beta_{3} - 12\beta_{2} + 4\beta _1 + 328 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{5} + 19\beta_{4} + 9\beta_{3} - 45\beta_{2} + 169\beta _1 + 184 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 67\beta_{5} + 82\beta_{4} - 480\beta_{3} - 624\beta_{2} + 448\beta _1 + 4600 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 41\beta_{5} + 296\beta_{4} + 90\beta_{3} - 3510\beta_{2} + 11762\beta _1 + 8168 ) / 24 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2303.1
5.20067 − 1.41421i
0.258712 + 1.41421i
−4.45938 − 1.41421i
−4.45938 + 1.41421i
0.258712 − 1.41421i
5.20067 + 1.41421i
0 0 0 − 20.5039i 0 − 32.2479i 0 0 0
2303.2 0 0 0 − 9.59247i 0 4.29193i 0 0 0
2303.3 0 0 0 − 4.64495i 0 22.3977i 0 0 0
2303.4 0 0 0 4.64495i 0 − 22.3977i 0 0 0
2303.5 0 0 0 9.59247i 0 − 4.29193i 0 0 0
2303.6 0 0 0 20.5039i 0 32.2479i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2303.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b 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.c.j 6
3.b odd 2 1 2304.4.c.h 6
4.b odd 2 1 2304.4.c.h 6
8.b even 2 1 2304.4.c.g 6
8.d odd 2 1 2304.4.c.i 6
12.b even 2 1 inner 2304.4.c.j 6
16.e even 4 2 1152.4.f.g yes 12
16.f odd 4 2 1152.4.f.f ✓ 12
24.f even 2 1 2304.4.c.g 6
24.h odd 2 1 2304.4.c.i 6
48.i odd 4 2 1152.4.f.f ✓ 12
48.k even 4 2 1152.4.f.g yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.4.f.f ✓ 12 16.f odd 4 2
1152.4.f.f ✓ 12 48.i odd 4 2
1152.4.f.g yes 12 16.e even 4 2
1152.4.f.g yes 12 48.k even 4 2
2304.4.c.g 6 8.b even 2 1
2304.4.c.g 6 24.f even 2 1
2304.4.c.h 6 3.b odd 2 1
2304.4.c.h 6 4.b odd 2 1
2304.4.c.i 6 8.d odd 2 1
2304.4.c.i 6 24.h odd 2 1
2304.4.c.j 6 1.a even 1 1 trivial
2304.4.c.j 6 12.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{6} + 534T_{5}^{4} + 49740T_{5}^{2} + 834632 \) Copy content Toggle raw display
\( T_{11}^{3} - 36T_{11}^{2} - 1104T_{11} + 23360 \) Copy content Toggle raw display
\( T_{13}^{3} - 6T_{13}^{2} - 4644T_{13} + 93400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 534 T^{4} + \cdots + 834632 \) Copy content Toggle raw display
$7$ \( T^{6} + 1560 T^{4} + \cdots + 9609728 \) Copy content Toggle raw display
$11$ \( (T^{3} - 36 T^{2} + \cdots + 23360)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 6 T^{2} + \cdots + 93400)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 60717761288 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 68420403200 \) Copy content Toggle raw display
$23$ \( (T^{3} + 108 T^{2} + \cdots - 1518016)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 179695329032 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 40793425555968 \) Copy content Toggle raw display
$37$ \( (T^{3} + 168 T^{2} + \cdots - 51200)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 126758035353672 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 189148449210368 \) Copy content Toggle raw display
$47$ \( (T^{3} + 156 T^{2} + \cdots - 17539776)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 47270745288 \) Copy content Toggle raw display
$59$ \( (T^{3} - 576 T^{2} + \cdots + 7077888)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 360 T^{2} + \cdots - 58836992)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 147152639393792 \) Copy content Toggle raw display
$71$ \( (T^{3} - 1428 T^{2} + \cdots - 51686848)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 600 T^{2} + \cdots - 260959744)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 77\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( (T^{3} + 972 T^{2} + \cdots - 35573184)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{3} + 768 T^{2} + \cdots - 27426816)^{2} \) Copy content Toggle raw display
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