Properties

Label 7200.2.o.j
Level $7200$
Weight $2$
Character orbit 7200.o
Analytic conductor $57.492$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1440)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 2) q^{7} + (\beta_{3} - 4) q^{11} + ( - \beta_{2} - \beta_1) q^{13} + (2 \beta_{3} - 4) q^{17} + ( - 2 \beta_{2} - \beta_1) q^{23} + 3 \beta_1 q^{29} + (6 \beta_{2} - \beta_1) q^{31} + (5 \beta_{2} + \beta_1) q^{37} + ( - 5 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_{3} + 4) q^{43} - 4 \beta_{2} q^{47} + (4 \beta_{3} - 1) q^{49} + ( - 2 \beta_{3} + 4) q^{53} + ( - \beta_{3} + 8) q^{59} + (2 \beta_{3} + 10) q^{61} + 8 q^{67} - 4 \beta_{3} q^{71} + ( - 6 \beta_{2} - \beta_1) q^{73} + ( - 2 \beta_{3} - 6) q^{77} + ( - 6 \beta_{2} - 3 \beta_1) q^{79} + (2 \beta_{2} - 6 \beta_1) q^{83} - 3 \beta_{2} q^{89} + ( - 4 \beta_{2} - 3 \beta_1) q^{91} + ( - 6 \beta_{2} + 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} - 16 q^{11} - 16 q^{17} + 16 q^{43} - 4 q^{49} + 16 q^{53} + 32 q^{59} + 40 q^{61} + 32 q^{67} - 24 q^{77}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7199.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 0 0 0.585786 0 0 0
7199.2 0 0 0 0 0 0.585786 0 0 0
7199.3 0 0 0 0 0 3.41421 0 0 0
7199.4 0 0 0 0 0 3.41421 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
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.o.j 4
3.b odd 2 1 7200.2.o.m 4
4.b odd 2 1 7200.2.o.e 4
5.b even 2 1 7200.2.o.b 4
5.c odd 4 1 1440.2.h.a 4
5.c odd 4 1 7200.2.h.c 4
12.b even 2 1 7200.2.o.b 4
15.d odd 2 1 7200.2.o.e 4
15.e even 4 1 1440.2.h.d yes 4
15.e even 4 1 7200.2.h.i 4
20.d odd 2 1 7200.2.o.m 4
20.e even 4 1 1440.2.h.d yes 4
20.e even 4 1 7200.2.h.i 4
40.i odd 4 1 2880.2.h.d 4
40.k even 4 1 2880.2.h.a 4
60.h even 2 1 inner 7200.2.o.j 4
60.l odd 4 1 1440.2.h.a 4
60.l odd 4 1 7200.2.h.c 4
120.q odd 4 1 2880.2.h.d 4
120.w even 4 1 2880.2.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.h.a 4 5.c odd 4 1
1440.2.h.a 4 60.l odd 4 1
1440.2.h.d yes 4 15.e even 4 1
1440.2.h.d yes 4 20.e even 4 1
2880.2.h.a 4 40.k even 4 1
2880.2.h.a 4 120.w even 4 1
2880.2.h.d 4 40.i odd 4 1
2880.2.h.d 4 120.q odd 4 1
7200.2.h.c 4 5.c odd 4 1
7200.2.h.c 4 60.l odd 4 1
7200.2.h.i 4 15.e even 4 1
7200.2.h.i 4 20.e even 4 1
7200.2.o.b 4 5.b even 2 1
7200.2.o.b 4 12.b even 2 1
7200.2.o.e 4 4.b odd 2 1
7200.2.o.e 4 15.d odd 2 1
7200.2.o.j 4 1.a even 1 1 trivial
7200.2.o.j 4 60.h even 2 1 inner
7200.2.o.m 4 3.b odd 2 1
7200.2.o.m 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{2} - 4T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 8T_{11} + 14 \) Copy content Toggle raw display
\( T_{17}^{2} + 8T_{17} + 8 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T + 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 152T^{2} + 4624 \) Copy content Toggle raw display
$37$ \( T^{4} + 108T^{2} + 2116 \) Copy content Toggle raw display
$41$ \( T^{4} + 132T^{2} + 1156 \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 16 T + 62)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 20 T + 92)^{2} \) Copy content Toggle raw display
$67$ \( (T - 8)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 152T^{2} + 4624 \) Copy content Toggle raw display
$79$ \( T^{4} + 216T^{2} + 1296 \) Copy content Toggle raw display
$83$ \( T^{4} + 304 T^{2} + 18496 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 216T^{2} + 1296 \) Copy content Toggle raw display
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