Properties

Label 144.7.e.c
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 36)
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.

\(f(q)\) \(=\) \( q + 5 \beta q^{5} + 244 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \beta q^{5} + 244 q^{7} - 188 \beta q^{11} - 2728 q^{13} + 463 \beta q^{17} + 5392 q^{19} - 196 \beta q^{23} + 11575 q^{25} - 3301 \beta q^{29} - 10172 q^{31} + 1220 \beta q^{35} + 65006 q^{37} - 5479 \beta q^{41} + 49480 q^{43} - 12948 \beta q^{47} - 58113 q^{49} + 1955 \beta q^{53} + 152280 q^{55} - 28216 \beta q^{59} + 100610 q^{61} - 13640 \beta q^{65} + 435736 q^{67} + 4724 \beta q^{71} + 619568 q^{73} - 45872 \beta q^{77} - 514340 q^{79} + 19164 \beta q^{83} - 375030 q^{85} - 33647 \beta q^{89} - 665632 q^{91} + 26960 \beta q^{95} + 42704 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 488 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 488 q^{7} - 5456 q^{13} + 10784 q^{19} + 23150 q^{25} - 20344 q^{31} + 130012 q^{37} + 98960 q^{43} - 116226 q^{49} + 304560 q^{55} + 201220 q^{61} + 871472 q^{67} + 1239136 q^{73} - 1028680 q^{79} - 750060 q^{85} - 1331264 q^{91} + 85408 q^{97}+O(q^{100}) \) 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 0 244.000 0 0 0
17.2 0 0 0 63.6396i 0 244.000 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.c 2
3.b odd 2 1 inner 144.7.e.c 2
4.b odd 2 1 36.7.c.a 2
8.b even 2 1 576.7.e.j 2
8.d odd 2 1 576.7.e.c 2
12.b even 2 1 36.7.c.a 2
24.f even 2 1 576.7.e.c 2
24.h odd 2 1 576.7.e.j 2
36.f odd 6 2 324.7.g.e 4
36.h even 6 2 324.7.g.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.7.c.a 2 4.b odd 2 1
36.7.c.a 2 12.b even 2 1
144.7.e.c 2 1.a even 1 1 trivial
144.7.e.c 2 3.b odd 2 1 inner
324.7.g.e 4 36.f odd 6 2
324.7.g.e 4 36.h even 6 2
576.7.e.c 2 8.d odd 2 1
576.7.e.c 2 24.f even 2 1
576.7.e.j 2 8.b even 2 1
576.7.e.j 2 24.h odd 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
\( T_{7} - 244 \) 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
$7$ \( (T - 244)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5725728 \) Copy content Toggle raw display
$13$ \( (T + 2728)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 34727778 \) Copy content Toggle raw display
$19$ \( (T - 5392)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6223392 \) Copy content Toggle raw display
$29$ \( T^{2} + 1765249362 \) Copy content Toggle raw display
$31$ \( (T + 10172)^{2} \) Copy content Toggle raw display
$37$ \( (T - 65006)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4863149442 \) Copy content Toggle raw display
$43$ \( (T - 49480)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 27159414048 \) Copy content Toggle raw display
$53$ \( T^{2} + 619168050 \) Copy content Toggle raw display
$59$ \( T^{2} + 128975110272 \) Copy content Toggle raw display
$61$ \( (T - 100610)^{2} \) Copy content Toggle raw display
$67$ \( (T - 435736)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 3615220512 \) Copy content Toggle raw display
$73$ \( (T - 619568)^{2} \) Copy content Toggle raw display
$79$ \( (T + 514340)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 59495941152 \) Copy content Toggle raw display
$89$ \( T^{2} + 183403538658 \) Copy content Toggle raw display
$97$ \( (T - 42704)^{2} \) Copy content Toggle raw display
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