Properties

Label 5328.2.h.m
Level $5328$
Weight $2$
Character orbit 5328.h
Analytic conductor $42.544$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5328.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.5442941969\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + 2 q^{7} + ( - \beta_{3} + 2) q^{11} + \beta_{2} q^{13} + 2 \beta_{2} q^{17} - 2 \beta_1 q^{19} + \beta_{2} q^{23} + (3 \beta_{3} - 4) q^{25} + \beta_{2} q^{29} + ( - 2 \beta_{2} + \beta_1) q^{31} - 2 \beta_1 q^{35} + (\beta_{2} - \beta_1 + 4) q^{37} + ( - \beta_{3} - 7) q^{41} + ( - 2 \beta_{2} - 2 \beta_1) q^{43} + ( - 2 \beta_{3} - 2) q^{47} - 3 q^{49} + (2 \beta_{3} - 4) q^{53} + ( - \beta_{2} - 4 \beta_1) q^{55} + 2 \beta_1 q^{59} + ( - 4 \beta_{2} - 3 \beta_1) q^{61} - 3 q^{65} + ( - 3 \beta_{3} + 2) q^{67} + ( - 4 \beta_{3} + 2) q^{71} + ( - 3 \beta_{3} + 4) q^{73} + ( - 2 \beta_{3} + 4) q^{77} + (3 \beta_{2} + 4 \beta_1) q^{79} + (4 \beta_{3} + 4) q^{83} - 6 q^{85} + (2 \beta_{2} + 2 \beta_1) q^{89} + 2 \beta_{2} q^{91} + (6 \beta_{3} - 18) q^{95} + (4 \beta_{2} + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} + 6 q^{11} - 10 q^{25} + 16 q^{37} - 30 q^{41} - 12 q^{47} - 12 q^{49} - 12 q^{53} - 12 q^{65} + 2 q^{67} + 10 q^{73} + 12 q^{77} + 24 q^{83} - 24 q^{85} - 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 17\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 17\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1333\) \(1999\) \(2369\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2737.1
2.79129i
1.79129i
1.79129i
2.79129i
0 0 0 3.79129i 0 2.00000 0 0 0
2737.2 0 0 0 0.791288i 0 2.00000 0 0 0
2737.3 0 0 0 0.791288i 0 2.00000 0 0 0
2737.4 0 0 0 3.79129i 0 2.00000 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
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5328.2.h.m 4
3.b odd 2 1 592.2.g.c 4
4.b odd 2 1 666.2.c.b 4
12.b even 2 1 74.2.b.a 4
24.f even 2 1 2368.2.g.j 4
24.h odd 2 1 2368.2.g.h 4
37.b even 2 1 inner 5328.2.h.m 4
60.h even 2 1 1850.2.d.e 4
60.l odd 4 1 1850.2.c.g 4
60.l odd 4 1 1850.2.c.h 4
111.d odd 2 1 592.2.g.c 4
148.b odd 2 1 666.2.c.b 4
444.g even 2 1 74.2.b.a 4
444.j odd 4 1 2738.2.a.h 2
444.j odd 4 1 2738.2.a.k 2
888.c even 2 1 2368.2.g.j 4
888.i odd 2 1 2368.2.g.h 4
2220.p even 2 1 1850.2.d.e 4
2220.bf odd 4 1 1850.2.c.g 4
2220.bf odd 4 1 1850.2.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.b.a 4 12.b even 2 1
74.2.b.a 4 444.g even 2 1
592.2.g.c 4 3.b odd 2 1
592.2.g.c 4 111.d odd 2 1
666.2.c.b 4 4.b odd 2 1
666.2.c.b 4 148.b odd 2 1
1850.2.c.g 4 60.l odd 4 1
1850.2.c.g 4 2220.bf odd 4 1
1850.2.c.h 4 60.l odd 4 1
1850.2.c.h 4 2220.bf odd 4 1
1850.2.d.e 4 60.h even 2 1
1850.2.d.e 4 2220.p even 2 1
2368.2.g.h 4 24.h odd 2 1
2368.2.g.h 4 888.i odd 2 1
2368.2.g.j 4 24.f even 2 1
2368.2.g.j 4 888.c even 2 1
2738.2.a.h 2 444.j odd 4 1
2738.2.a.k 2 444.j odd 4 1
5328.2.h.m 4 1.a even 1 1 trivial
5328.2.h.m 4 37.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5328, [\chi])\):

\( T_{5}^{4} + 15T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{13}^{4} + 15T_{13}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T - 2)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$19$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$23$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$29$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$31$ \( T^{4} + 99T^{2} + 2025 \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 37)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 15 T + 51)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{4} + 231 T^{2} + 11025 \) Copy content Toggle raw display
$67$ \( (T^{2} - T - 47)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 5 T - 41)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 231 T^{2} + 11025 \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T - 48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 204T^{2} + 3600 \) Copy content Toggle raw display
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