Properties

Label 576.7.e.k
Level $576$
Weight $7$
Character orbit 576.e
Analytic conductor $132.511$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(132.511152165\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 41 \beta q^{5} + 484 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 41 \beta q^{5} + 484 q^{7} - 316 \beta q^{11} - 3368 q^{13} - 3 \beta q^{17} + 5744 q^{19} - 796 \beta q^{23} - 14633 q^{25} + 6919 \beta q^{29} + 39796 q^{31} + 19844 \beta q^{35} - 52526 q^{37} + 8731 \beta q^{41} + 3800 q^{43} + 18100 \beta q^{47} + 116607 q^{49} + 56271 \beta q^{53} + 233208 q^{55} + 58888 \beta q^{59} - 13250 q^{61} - 138088 \beta q^{65} + 168968 q^{67} - 125268 \beta q^{71} + 236144 q^{73} - 152944 \beta q^{77} + 35116 q^{79} + 2588 \beta q^{83} + 2214 q^{85} + 30483 \beta q^{89} - 1630112 q^{91} + 235504 \beta q^{95} - 321424 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 968 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 968 q^{7} - 6736 q^{13} + 11488 q^{19} - 29266 q^{25} + 79592 q^{31} - 105052 q^{37} + 7600 q^{43} + 233214 q^{49} + 466416 q^{55} - 26500 q^{61} + 337936 q^{67} + 472288 q^{73} + 70232 q^{79} + 4428 q^{85} - 3260224 q^{91} - 642848 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.41421i
1.41421i
0 0 0 173.948i 0 484.000 0 0 0
449.2 0 0 0 173.948i 0 484.000 0 0 0
\(n\): e.g. 2-40 or 990-1000
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 576.7.e.k 2
3.b odd 2 1 inner 576.7.e.k 2
4.b odd 2 1 576.7.e.b 2
8.b even 2 1 144.7.e.d 2
8.d odd 2 1 18.7.b.a 2
12.b even 2 1 576.7.e.b 2
24.f even 2 1 18.7.b.a 2
24.h odd 2 1 144.7.e.d 2
40.e odd 2 1 450.7.d.a 2
40.k even 4 2 450.7.b.a 4
72.l even 6 2 162.7.d.d 4
72.p odd 6 2 162.7.d.d 4
120.m even 2 1 450.7.d.a 2
120.q odd 4 2 450.7.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.b.a 2 8.d odd 2 1
18.7.b.a 2 24.f even 2 1
144.7.e.d 2 8.b even 2 1
144.7.e.d 2 24.h odd 2 1
162.7.d.d 4 72.l even 6 2
162.7.d.d 4 72.p odd 6 2
450.7.b.a 4 40.k even 4 2
450.7.b.a 4 120.q odd 4 2
450.7.d.a 2 40.e odd 2 1
450.7.d.a 2 120.m even 2 1
576.7.e.b 2 4.b odd 2 1
576.7.e.b 2 12.b even 2 1
576.7.e.k 2 1.a even 1 1 trivial
576.7.e.k 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} + 30258 \) Copy content Toggle raw display
\( T_{7} - 484 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 30258 \) Copy content Toggle raw display
$7$ \( (T - 484)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1797408 \) Copy content Toggle raw display
$13$ \( (T + 3368)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 162 \) Copy content Toggle raw display
$19$ \( (T - 5744)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 11405088 \) Copy content Toggle raw display
$29$ \( T^{2} + 861706098 \) Copy content Toggle raw display
$31$ \( (T - 39796)^{2} \) Copy content Toggle raw display
$37$ \( (T + 52526)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1372146498 \) Copy content Toggle raw display
$43$ \( (T - 3800)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 5896980000 \) Copy content Toggle raw display
$53$ \( T^{2} + 56995657938 \) Copy content Toggle raw display
$59$ \( T^{2} + 62420337792 \) Copy content Toggle raw display
$61$ \( (T + 13250)^{2} \) Copy content Toggle raw display
$67$ \( (T - 168968)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 282457292832 \) Copy content Toggle raw display
$73$ \( (T - 236144)^{2} \) Copy content Toggle raw display
$79$ \( (T - 35116)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 120559392 \) Copy content Toggle raw display
$89$ \( T^{2} + 16725839202 \) Copy content Toggle raw display
$97$ \( (T + 321424)^{2} \) Copy content Toggle raw display
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