Properties

Label 1584.3.i.b
Level $1584$
Weight $3$
Character orbit 1584.i
Analytic conductor $43.161$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1584.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(43.1608738747\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.65306824704.6
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{7} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{7} - 2) q^{7} - \beta_{5} q^{11} + (2 \beta_{7} + \beta_{4} - \beta_{3} - 1) q^{13} + ( - \beta_{6} - 3 \beta_{5} - 5 \beta_{2} - \beta_1) q^{17} + (\beta_{7} + 3 \beta_{4} - \beta_{3} - 5) q^{19} + (\beta_{6} + 6 \beta_{5} - \beta_1) q^{23} + ( - 2 \beta_{4} + \beta_{3} - 14) q^{25} + (6 \beta_{6} - 6 \beta_{2} - 2 \beta_1) q^{29} + ( - 4 \beta_{7} + 4 \beta_{4} - 3 \beta_{3} + 7) q^{31} + ( - 3 \beta_{6} + 9 \beta_{5} - 5 \beta_{2} - 3 \beta_1) q^{35} + (6 \beta_{7} - 2 \beta_{4} + \beta_{3} + 17) q^{37} + (5 \beta_{6} - 3 \beta_{5} + 13 \beta_{2} + \beta_1) q^{41} + ( - \beta_{7} - 5 \beta_{4} - 7 \beta_{3} + 13) q^{43} + (11 \beta_{6} - 6 \beta_{5} - 2 \beta_{2} - \beta_1) q^{47} + ( - 8 \beta_{7} - 2 \beta_{4} - 3 \beta_{3} - 12) q^{49} + ( - 8 \beta_{6} + \beta_{2} - 5 \beta_1) q^{53} + (3 \beta_{7} - 2 \beta_{3}) q^{55} + ( - 8 \beta_{6} + 6 \beta_{5} - 12 \beta_{2} + 2 \beta_1) q^{59} + ( - 2 \beta_{7} + 9 \beta_{4} + 3 \beta_{3} - 1) q^{61} + ( - 14 \beta_{6} + 24 \beta_{5} - 6 \beta_{2} + 2 \beta_1) q^{65} + ( - 6 \beta_{7} - 2 \beta_{4} - 4 \beta_{3} - 14) q^{67} + (\beta_{6} + 6 \beta_{5} + 4 \beta_{2} + 11 \beta_1) q^{71} + ( - 6 \beta_{7} + 10 \beta_{4} - 6 \beta_{3} + 56) q^{73} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{2} - 3 \beta_1) q^{77} + (\beta_{7} + 4 \beta_{4} - 6 \beta_{3} - 56) q^{79} + (7 \beta_{6} - 9 \beta_{5} + 25 \beta_{2} - 9 \beta_1) q^{83} + (2 \beta_{7} - 2 \beta_{4} - 18 \beta_{3} + 6) q^{85} + (9 \beta_{6} + 12 \beta_{5} + 27 \beta_{2} + 14 \beta_1) q^{89} + ( - 12 \beta_{7} - 6 \beta_{3} + 68) q^{91} + ( - 27 \beta_{6} + 15 \beta_{5} - 3 \beta_{2} + 3 \beta_1) q^{95} + ( - 16 \beta_{7} - 2 \beta_{4} + \beta_{3} - 19) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{7} - 8 q^{13} - 40 q^{19} - 112 q^{25} + 56 q^{31} + 136 q^{37} + 104 q^{43} - 96 q^{49} - 8 q^{61} - 112 q^{67} + 448 q^{73} - 448 q^{79} + 48 q^{85} + 544 q^{91} - 152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 6\nu^{4} - 17\nu^{3} - 20\nu^{2} + 29\nu - 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{7} + 11\nu^{6} + 60\nu^{5} - 83\nu^{4} - 170\nu^{3} - 21\nu^{2} + 21\nu + 54 ) / 324 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{7} + 11\nu^{6} + 60\nu^{5} - 83\nu^{4} - 170\nu^{3} + 303\nu^{2} + 21\nu - 756 ) / 162 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\nu^{7} - 53\nu^{6} - 186\nu^{5} + 503\nu^{4} + 716\nu^{3} - 1563\nu^{2} - 219\nu + 1080 ) / 324 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8\nu^{7} - 28\nu^{6} - 30\nu^{5} + 145\nu^{4} + 40\nu^{3} - 219\nu^{2} + 300\nu - 108 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -35\nu^{7} + 136\nu^{6} + 57\nu^{5} - 577\nu^{4} - 121\nu^{3} + 1191\nu^{2} - 1596\nu + 513 ) / 324 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\nu^{7} - 178\nu^{6} - 183\nu^{5} + 997\nu^{4} + 667\nu^{3} - 2127\nu^{2} - 546\nu - 27 ) / 324 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 7\beta_{6} - 9\beta_{5} + 4\beta_{4} + 7\beta_{3} - 16\beta_{2} + 21 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{6} - 6\beta_{5} + 2\beta_{4} + 5\beta_{3} - 22\beta_{2} + 2\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} - 49\beta_{6} - 90\beta_{5} + 7\beta_{4} + 25\beta_{3} - 184\beta_{2} + 15\beta _1 + 93 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{7} - 31\beta_{6} - 75\beta_{5} + 6\beta_{4} + 5\beta_{3} - 195\beta_{2} + 19\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{7} - 328\beta_{6} - 693\beta_{5} - 77\beta_{4} - 116\beta_{3} - 1588\beta_{2} + 147\beta _1 - 312 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(353\) \(991\) \(1189\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−0.136233 0.707107i
1.13623 + 0.707107i
2.75726 0.707107i
−1.75726 + 0.707107i
−1.75726 0.707107i
2.75726 + 0.707107i
1.13623 0.707107i
−0.136233 + 0.707107i
0 0 0 7.62670i 0 2.45638 0 0 0
881.2 0 0 0 6.21249i 0 −11.1468 0 0 0
881.3 0 0 0 6.10332i 0 −2.61086 0 0 0
881.4 0 0 0 4.68911i 0 3.30128 0 0 0
881.5 0 0 0 4.68911i 0 3.30128 0 0 0
881.6 0 0 0 6.10332i 0 −2.61086 0 0 0
881.7 0 0 0 6.21249i 0 −11.1468 0 0 0
881.8 0 0 0 7.62670i 0 2.45638 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.3.i.b 8
3.b odd 2 1 inner 1584.3.i.b 8
4.b odd 2 1 99.3.b.a 8
12.b even 2 1 99.3.b.a 8
44.c even 2 1 1089.3.b.g 8
132.d odd 2 1 1089.3.b.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.b.a 8 4.b odd 2 1
99.3.b.a 8 12.b even 2 1
1089.3.b.g 8 44.c even 2 1
1089.3.b.g 8 132.d odd 2 1
1584.3.i.b 8 1.a even 1 1 trivial
1584.3.i.b 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 156T_{5}^{6} + 8796T_{5}^{4} + 212240T_{5}^{2} + 1838736 \) acting on \(S_{3}^{\mathrm{new}}(1584, [\chi])\). 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
$5$ \( T^{8} + 156 T^{6} + 8796 T^{4} + \cdots + 1838736 \) Copy content Toggle raw display
$7$ \( (T^{4} + 8 T^{3} - 42 T^{2} - 56 T + 236)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} - 390 T^{2} - 2416 T + 2828)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 1080 T^{6} + \cdots + 472801536 \) Copy content Toggle raw display
$19$ \( (T^{4} + 20 T^{3} - 576 T^{2} + \cdots - 37908)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 1788 T^{6} + \cdots + 11052737424 \) Copy content Toggle raw display
$29$ \( T^{8} + 4848 T^{6} + \cdots + 125180100864 \) Copy content Toggle raw display
$31$ \( (T^{4} - 28 T^{3} - 2412 T^{2} + \cdots + 244016)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 68 T^{3} - 972 T^{2} + \cdots - 914256)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 4824 T^{6} + \cdots + 85142571264 \) Copy content Toggle raw display
$43$ \( (T^{4} - 52 T^{3} - 3936 T^{2} + \cdots - 3241332)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 11052 T^{6} + \cdots + 12929518743696 \) Copy content Toggle raw display
$53$ \( T^{8} + 8124 T^{6} + \cdots + 26983975824 \) Copy content Toggle raw display
$59$ \( T^{8} + 7872 T^{6} + \cdots + 1028100096 \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} - 6198 T^{2} + \cdots + 1712844)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 56 T^{3} - 2520 T^{2} + \cdots - 2846576)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 17916 T^{6} + \cdots + 13312019262096 \) Copy content Toggle raw display
$73$ \( (T^{4} - 224 T^{3} + 7464 T^{2} + \cdots - 37468144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 224 T^{3} + 15318 T^{2} + \cdots - 4606964)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 177188471398656 \) Copy content Toggle raw display
$89$ \( T^{8} + 49704 T^{6} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{4} + 76 T^{3} - 15060 T^{2} + \cdots - 9076976)^{2} \) Copy content Toggle raw display
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