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Properties

Label 288.4.d.d
Level $288$
Weight $4$
Character orbit 288.d
Analytic conductor $16.993$
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] = [288,4,Mod(145,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.145");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 288.d (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: \(16.9925500817\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8248384.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - \beta_1 - 5) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + ( - \beta_1 - 5) q^{7} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{11} + ( - 2 \beta_{4} + \beta_{3} + 3 \beta_{2}) q^{13} + (\beta_{5} + 3 \beta_1 - 8) q^{17} + ( - 3 \beta_{4} - 2 \beta_{3} + 6 \beta_{2}) q^{19} + ( - 2 \beta_1 + 54) q^{23} + (2 \beta_{5} - 2 \beta_1 - 19) q^{25} + (4 \beta_{4} + 2 \beta_{3} + 5 \beta_{2}) q^{29} + (2 \beta_{5} - \beta_1 + 105) q^{31} + ( - 7 \beta_{4} - 2 \beta_{3} + 14 \beta_{2}) q^{35} + (6 \beta_{4} - 11 \beta_{3} - 3 \beta_{2}) q^{37} + (\beta_{5} - 13 \beta_1 - 44) q^{41} + ( - 5 \beta_{4} - 10 \beta_{3} - 18 \beta_{2}) q^{43} + (2 \beta_{5} - 4 \beta_1 - 70) q^{47} + (4 \beta_{5} + 4 \beta_1 + 109) q^{49} + ( - 36 \beta_{4} - 10 \beta_{3} + 11 \beta_{2}) q^{53} + (6 \beta_{5} + 2 \beta_1 - 172) q^{55} + ( - \beta_{4} + 8 \beta_{3} - 8 \beta_{2}) q^{59} + ( - 46 \beta_{4} - 9 \beta_{3} + 3 \beta_{2}) q^{61} + ( - 3 \beta_{5} + 7 \beta_1 + 294) q^{65} + (5 \beta_{4} + 48 \beta_{2}) q^{67} + (6 \beta_{5} + 12 \beta_1 - 282) q^{71} + ( - 4 \beta_{5} - 20 \beta_1 + 154) q^{73} + (64 \beta_{4} - 20 \beta_{3} - 20 \beta_{2}) q^{77} + ( - 2 \beta_{5} - 9 \beta_1 + 5) q^{79} + (36 \beta_{4} + 2 \beta_{3} - 14 \beta_{2}) q^{83} + (72 \beta_{4} + 28 \beta_{3} + 42 \beta_{2}) q^{85} + (2 \beta_{5} + 6 \beta_1 + 38) q^{89} + ( - 2 \beta_{4} + 38 \beta_{3} - 18 \beta_{2}) q^{91} + ( - 4 \beta_{5} + 24 \beta_1 + 860) q^{95} + ( - 6 \beta_{5} - 2 \beta_1 - 406) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 28 q^{7} - 52 q^{17} + 328 q^{23} - 106 q^{25} + 636 q^{31} - 236 q^{41} - 408 q^{47} + 654 q^{49} - 1024 q^{55} + 1744 q^{65} - 1704 q^{71} + 956 q^{73} + 44 q^{79} + 220 q^{89} + 5104 q^{95} - 2444 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} - 3\nu^{2} - 12\nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 6\nu^{4} - 13\nu^{3} + 42\nu^{2} - 104\nu + 96 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 6\nu^{4} - 51\nu^{3} - 42\nu^{2} + 40\nu + 288 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} + 6\nu^{4} - 9\nu^{3} - 6\nu^{2} - 24\nu + 96 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{5} + 5\nu^{4} - 3\nu^{3} + 21\nu^{2} - 24\nu + 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + 6\beta_{4} - 12\beta_{2} - 5\beta _1 - 2 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{4} - 3\beta_{3} + 9\beta_{2} - 6\beta _1 - 18 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{5} - 6\beta_{4} - 24\beta_{3} - 12\beta_{2} + 5\beta _1 + 578 ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 14\beta_{4} - 3\beta_{3} - 15\beta_{2} - 38 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{5} + 126\beta_{4} - 48\beta_{3} + 84\beta_{2} - 49\beta _1 - 970 ) / 96 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−1.24181 − 1.56777i
−0.641412 − 1.89436i
1.88322 + 0.673417i
1.88322 − 0.673417i
−0.641412 + 1.89436i
−1.24181 + 1.56777i
0 0 0 − 18.5422i 0 −9.32669 0 0 0
145.2 0 0 0 − 9.15486i 0 −27.4175 0 0 0
145.3 0 0 0 − 0.612661i 0 22.7441 0 0 0
145.4 0 0 0 0.612661i 0 22.7441 0 0 0
145.5 0 0 0 9.15486i 0 −27.4175 0 0 0
145.6 0 0 0 18.5422i 0 −9.32669 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.4.d.d 6
3.b odd 2 1 96.4.d.a 6
4.b odd 2 1 72.4.d.d 6
8.b even 2 1 inner 288.4.d.d 6
8.d odd 2 1 72.4.d.d 6
12.b even 2 1 24.4.d.a ✓ 6
16.e even 4 1 2304.4.a.bu 3
16.e even 4 1 2304.4.a.bw 3
16.f odd 4 1 2304.4.a.bt 3
16.f odd 4 1 2304.4.a.bv 3
24.f even 2 1 24.4.d.a ✓ 6
24.h odd 2 1 96.4.d.a 6
48.i odd 4 1 768.4.a.q 3
48.i odd 4 1 768.4.a.t 3
48.k even 4 1 768.4.a.r 3
48.k even 4 1 768.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a ✓ 6 12.b even 2 1
24.4.d.a ✓ 6 24.f even 2 1
72.4.d.d 6 4.b odd 2 1
72.4.d.d 6 8.d odd 2 1
96.4.d.a 6 3.b odd 2 1
96.4.d.a 6 24.h odd 2 1
288.4.d.d 6 1.a even 1 1 trivial
288.4.d.d 6 8.b even 2 1 inner
768.4.a.q 3 48.i odd 4 1
768.4.a.r 3 48.k even 4 1
768.4.a.s 3 48.k even 4 1
768.4.a.t 3 48.i odd 4 1
2304.4.a.bt 3 16.f odd 4 1
2304.4.a.bu 3 16.e even 4 1
2304.4.a.bv 3 16.f odd 4 1
2304.4.a.bw 3 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 428T_{5}^{4} + 28976T_{5}^{2} + 10816 \) acting on \(S_{4}^{\mathrm{new}}(288, [\chi])\). 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
$5$ \( T^{6} + 428 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$7$ \( (T^{3} + 14 T^{2} + \cdots - 5816)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 2415919104 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 3121680384 \) Copy content Toggle raw display
$17$ \( (T^{3} + 26 T^{2} + \cdots - 477576)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 75488661504 \) Copy content Toggle raw display
$23$ \( (T^{3} - 164 T^{2} + \cdots - 45504)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 3766031424 \) Copy content Toggle raw display
$31$ \( (T^{3} - 318 T^{2} + \cdots + 3749624)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 6879707136 \) Copy content Toggle raw display
$41$ \( (T^{3} + 118 T^{2} + \cdots - 19985976)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 73984219582464 \) Copy content Toggle raw display
$47$ \( (T^{3} + 204 T^{2} + \cdots - 1964736)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 427051482970176 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 72651484205056 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{3} + 852 T^{2} + \cdots - 85084992)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 478 T^{2} + \cdots + 120833304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 22 T^{2} + \cdots + 7902616)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{3} - 110 T^{2} + \cdots - 1423656)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 1222 T^{2} + \cdots - 74802424)^{2} \) Copy content Toggle raw display
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