Properties

Label 5400.2.f.e
Level $5400$
Weight $2$
Character orbit 5400.f
Analytic conductor $43.119$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5400.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(43.1192170915\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{7} +O(q^{10})\) \( q + 3 i q^{7} -4 q^{11} + i q^{13} -4 i q^{17} + q^{19} -4 i q^{23} -4 q^{31} + 9 i q^{37} -8 i q^{43} -12 i q^{47} -2 q^{49} + 8 i q^{53} + 4 q^{59} -5 q^{61} -11 i q^{67} -8 q^{71} + i q^{73} -12 i q^{77} + 5 q^{79} -8 i q^{83} + 12 q^{89} -3 q^{91} -5 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 8q^{11} + 2q^{19} - 8q^{31} - 4q^{49} + 8q^{59} - 10q^{61} - 16q^{71} + 10q^{79} + 24q^{89} - 6q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\) \(2701\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 0 0 3.00000i 0 0 0
649.2 0 0 0 0 0 3.00000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5400.2.f.e 2
3.b odd 2 1 5400.2.f.v 2
5.b even 2 1 inner 5400.2.f.e 2
5.c odd 4 1 216.2.a.a 1
5.c odd 4 1 5400.2.a.bn 1
15.d odd 2 1 5400.2.f.v 2
15.e even 4 1 216.2.a.d yes 1
15.e even 4 1 5400.2.a.bp 1
20.e even 4 1 432.2.a.a 1
40.i odd 4 1 1728.2.a.ba 1
40.k even 4 1 1728.2.a.bb 1
45.k odd 12 2 648.2.i.h 2
45.l even 12 2 648.2.i.a 2
60.l odd 4 1 432.2.a.h 1
120.q odd 4 1 1728.2.a.b 1
120.w even 4 1 1728.2.a.a 1
180.v odd 12 2 1296.2.i.a 2
180.x even 12 2 1296.2.i.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.a 1 5.c odd 4 1
216.2.a.d yes 1 15.e even 4 1
432.2.a.a 1 20.e even 4 1
432.2.a.h 1 60.l odd 4 1
648.2.i.a 2 45.l even 12 2
648.2.i.h 2 45.k odd 12 2
1296.2.i.a 2 180.v odd 12 2
1296.2.i.q 2 180.x even 12 2
1728.2.a.a 1 120.w even 4 1
1728.2.a.b 1 120.q odd 4 1
1728.2.a.ba 1 40.i odd 4 1
1728.2.a.bb 1 40.k even 4 1
5400.2.a.bn 1 5.c odd 4 1
5400.2.a.bp 1 15.e even 4 1
5400.2.f.e 2 1.a even 1 1 trivial
5400.2.f.e 2 5.b even 2 1 inner
5400.2.f.v 2 3.b odd 2 1
5400.2.f.v 2 15.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 \(S_{2}^{\mathrm{new}}(5400, [\chi])\):

\( T_{7}^{2} + 9 \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 1 \)
\( T_{29} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 16 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 81 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( 64 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( 5 + T )^{2} \)
$67$ \( 121 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 1 + T^{2} \)
$79$ \( ( -5 + T )^{2} \)
$83$ \( 64 + T^{2} \)
$89$ \( ( -12 + T )^{2} \)
$97$ \( 25 + T^{2} \)
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