Properties

Label 72.8.d.a
Level $72$
Weight $8$
Character orbit 72.d
Analytic conductor $22.492$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 4 \beta q^{2} - 128 q^{4} + 197 \beta q^{5} - 754 q^{7} - 512 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta q^{2} - 128 q^{4} + 197 \beta q^{5} - 754 q^{7} - 512 \beta q^{8} - 6304 q^{10} + 2378 \beta q^{11} - 3016 \beta q^{14} + 16384 q^{16} - 25216 \beta q^{20} - 76096 q^{22} - 232347 q^{25} + 96512 q^{28} - 89011 \beta q^{29} + 331370 q^{31} + 65536 \beta q^{32} - 148538 \beta q^{35} + 806912 q^{40} - 304384 \beta q^{44} - 255027 q^{49} - 929388 \beta q^{50} + 84695 \beta q^{53} - 3747728 q^{55} + 386048 \beta q^{56} + 2848352 q^{58} + 523436 \beta q^{59} + 1325480 \beta q^{62} - 2097152 q^{64} + 4753216 q^{70} - 2819362 q^{73} - 1793012 \beta q^{77} - 7561270 q^{79} + 3227648 \beta q^{80} + 2956826 \beta q^{83} + 9740288 q^{88} - 11745214 q^{97} - 1020108 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 256 q^{4} - 1508 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 256 q^{4} - 1508 q^{7} - 12608 q^{10} + 32768 q^{16} - 152192 q^{22} - 464694 q^{25} + 193024 q^{28} + 662740 q^{31} + 1613824 q^{40} - 510054 q^{49} - 7495456 q^{55} + 5696704 q^{58} - 4194304 q^{64} + 9506432 q^{70} - 5638724 q^{73} - 15122540 q^{79} + 19480576 q^{88} - 23490428 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.41421i
1.41421i
11.3137i 0 −128.000 557.200i 0 −754.000 1448.15i 0 −6304.00
37.2 11.3137i 0 −128.000 557.200i 0 −754.000 1448.15i 0 −6304.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
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 72.8.d.a 2
3.b odd 2 1 inner 72.8.d.a 2
4.b odd 2 1 288.8.d.a 2
8.b even 2 1 inner 72.8.d.a 2
8.d odd 2 1 288.8.d.a 2
12.b even 2 1 288.8.d.a 2
24.f even 2 1 288.8.d.a 2
24.h odd 2 1 CM 72.8.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.d.a 2 1.a even 1 1 trivial
72.8.d.a 2 3.b odd 2 1 inner
72.8.d.a 2 8.b even 2 1 inner
72.8.d.a 2 24.h odd 2 1 CM
288.8.d.a 2 4.b odd 2 1
288.8.d.a 2 8.d odd 2 1
288.8.d.a 2 12.b even 2 1
288.8.d.a 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 310472 \) acting on \(S_{8}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 128 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 310472 \) Copy content Toggle raw display
$7$ \( (T + 754)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 45239072 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 63383664968 \) Copy content Toggle raw display
$31$ \( (T - 331370)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 57385944200 \) Copy content Toggle raw display
$59$ \( T^{2} + 2191881968768 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 2819362)^{2} \) Copy content Toggle raw display
$79$ \( (T + 7561270)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 69942559954208 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 11745214)^{2} \) Copy content Toggle raw display
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