Properties

Label 144.7.e.a
Level $144$
Weight $7$
Character orbit 144.e
Analytic conductor $33.128$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 144.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.1277880413\)
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: \( 3^{2} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 9\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \beta q^{5} - 524 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \beta q^{5} - 524 q^{7} + 68 \beta q^{11} + 344 q^{13} - 561 \beta q^{17} + 2320 q^{19} - 452 \beta q^{23} + 11575 q^{25} + 1819 \beta q^{29} + 10564 q^{31} - 2620 \beta q^{35} - 24082 q^{37} - 8551 \beta q^{41} + 90952 q^{43} - 10132 \beta q^{47} + 156927 q^{49} - 15453 \beta q^{53} - 55080 q^{55} - 3128 \beta q^{59} + 251138 q^{61} + 1720 \beta q^{65} + 216088 q^{67} - 4236 \beta q^{71} - 308176 q^{73} - 35632 \beta q^{77} + 540124 q^{79} - 73252 \beta q^{83} + 454410 q^{85} + 17553 \beta q^{89} - 180256 q^{91} + 11600 \beta q^{95} - 37168 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1048 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1048 q^{7} + 688 q^{13} + 4640 q^{19} + 23150 q^{25} + 21128 q^{31} - 48164 q^{37} + 181904 q^{43} + 313854 q^{49} - 110160 q^{55} + 502276 q^{61} + 432176 q^{67} - 616352 q^{73} + 1080248 q^{79} + 908820 q^{85} - 360512 q^{91} - 74336 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

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

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.

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 63.6396i 0 −524.000 0 0 0
17.2 0 0 0 63.6396i 0 −524.000 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 144.7.e.a 2
3.b odd 2 1 inner 144.7.e.a 2
4.b odd 2 1 9.7.b.a 2
8.b even 2 1 576.7.e.a 2
8.d odd 2 1 576.7.e.l 2
12.b even 2 1 9.7.b.a 2
20.d odd 2 1 225.7.c.a 2
20.e even 4 2 225.7.d.a 4
24.f even 2 1 576.7.e.l 2
24.h odd 2 1 576.7.e.a 2
28.d even 2 1 441.7.b.a 2
36.f odd 6 2 81.7.d.d 4
36.h even 6 2 81.7.d.d 4
60.h even 2 1 225.7.c.a 2
60.l odd 4 2 225.7.d.a 4
84.h odd 2 1 441.7.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.7.b.a 2 4.b odd 2 1
9.7.b.a 2 12.b even 2 1
81.7.d.d 4 36.f odd 6 2
81.7.d.d 4 36.h even 6 2
144.7.e.a 2 1.a even 1 1 trivial
144.7.e.a 2 3.b odd 2 1 inner
225.7.c.a 2 20.d odd 2 1
225.7.c.a 2 60.h even 2 1
225.7.d.a 4 20.e even 4 2
225.7.d.a 4 60.l odd 4 2
441.7.b.a 2 28.d even 2 1
441.7.b.a 2 84.h odd 2 1
576.7.e.a 2 8.b even 2 1
576.7.e.a 2 24.h odd 2 1
576.7.e.l 2 8.d odd 2 1
576.7.e.l 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(144, [\chi])\):

\( T_{5}^{2} + 4050 \) Copy content Toggle raw display
\( T_{7} + 524 \) 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
$5$ \( T^{2} + 4050 \) Copy content Toggle raw display
$7$ \( (T + 524)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 749088 \) Copy content Toggle raw display
$13$ \( (T - 344)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 50984802 \) Copy content Toggle raw display
$19$ \( (T - 2320)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 33097248 \) Copy content Toggle raw display
$29$ \( T^{2} + 536019282 \) Copy content Toggle raw display
$31$ \( (T - 10564)^{2} \) Copy content Toggle raw display
$37$ \( (T + 24082)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 11845375362 \) Copy content Toggle raw display
$43$ \( (T - 90952)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 16630502688 \) Copy content Toggle raw display
$53$ \( T^{2} + 38684823858 \) Copy content Toggle raw display
$59$ \( T^{2} + 1585070208 \) Copy content Toggle raw display
$61$ \( (T - 251138)^{2} \) Copy content Toggle raw display
$67$ \( (T - 216088)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 2906878752 \) Copy content Toggle raw display
$73$ \( (T + 308176)^{2} \) Copy content Toggle raw display
$79$ \( (T - 540124)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 869268591648 \) Copy content Toggle raw display
$89$ \( T^{2} + 49913465058 \) Copy content Toggle raw display
$97$ \( (T + 37168)^{2} \) Copy content Toggle raw display
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