Properties

Label 72.7.e.a
Level $72$
Weight $7$
Character orbit 72.e
Analytic conductor $16.564$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 72.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.5638940206\)
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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 5 \beta q^{5} + 60 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \beta q^{5} + 60 q^{7} + 236 \beta q^{11} + 1192 q^{13} + 2935 \beta q^{17} + 8432 q^{19} + 4532 \beta q^{23} + 15575 q^{25} + 23787 \beta q^{29} + 46892 q^{31} + 300 \beta q^{35} + 10926 q^{37} + 51921 \beta q^{41} + 59416 q^{43} - 83004 \beta q^{47} - 114049 q^{49} + 58259 \beta q^{53} - 2360 q^{55} - 199336 \beta q^{59} - 339902 q^{61} + 5960 \beta q^{65} - 148024 q^{67} - 291044 \beta q^{71} - 401552 q^{73} + 14160 \beta q^{77} + 79156 q^{79} - 70988 \beta q^{83} - 29350 q^{85} + 265785 \beta q^{89} + 71520 q^{91} + 42160 \beta q^{95} + 663920 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 120 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 120 q^{7} + 2384 q^{13} + 16864 q^{19} + 31150 q^{25} + 93784 q^{31} + 21852 q^{37} + 118832 q^{43} - 228098 q^{49} - 4720 q^{55} - 679804 q^{61} - 296048 q^{67} - 803104 q^{73} + 158312 q^{79} - 58700 q^{85} + 143040 q^{91} + 1327840 q^{97}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.41421i
1.41421i
0 0 0 7.07107i 0 60.0000 0 0 0
17.2 0 0 0 7.07107i 0 60.0000 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 72.7.e.a 2
3.b odd 2 1 inner 72.7.e.a 2
4.b odd 2 1 144.7.e.b 2
8.b even 2 1 576.7.e.h 2
8.d odd 2 1 576.7.e.e 2
12.b even 2 1 144.7.e.b 2
24.f even 2 1 576.7.e.e 2
24.h odd 2 1 576.7.e.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.e.a 2 1.a even 1 1 trivial
72.7.e.a 2 3.b odd 2 1 inner
144.7.e.b 2 4.b odd 2 1
144.7.e.b 2 12.b even 2 1
576.7.e.e 2 8.d odd 2 1
576.7.e.e 2 24.f even 2 1
576.7.e.h 2 8.b even 2 1
576.7.e.h 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 50 \) acting on \(S_{7}^{\mathrm{new}}(72, [\chi])\). 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
$5$ \( T^{2} + 50 \) Copy content Toggle raw display
$7$ \( (T - 60)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 111392 \) Copy content Toggle raw display
$13$ \( (T - 1192)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 17228450 \) Copy content Toggle raw display
$19$ \( (T - 8432)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 41078048 \) Copy content Toggle raw display
$29$ \( T^{2} + 1131642738 \) Copy content Toggle raw display
$31$ \( (T - 46892)^{2} \) Copy content Toggle raw display
$37$ \( (T - 10926)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 5391580482 \) Copy content Toggle raw display
$43$ \( (T - 59416)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 13779328032 \) Copy content Toggle raw display
$53$ \( T^{2} + 6788222162 \) Copy content Toggle raw display
$59$ \( T^{2} + 79469681792 \) Copy content Toggle raw display
$61$ \( (T + 339902)^{2} \) Copy content Toggle raw display
$67$ \( (T + 148024)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 169413219872 \) Copy content Toggle raw display
$73$ \( (T + 401552)^{2} \) Copy content Toggle raw display
$79$ \( (T - 79156)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 10078592288 \) Copy content Toggle raw display
$89$ \( T^{2} + 141283332450 \) Copy content Toggle raw display
$97$ \( (T - 663920)^{2} \) Copy content Toggle raw display
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