Properties

Label 5400.2.f.b
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.

\(f(q)\) \(=\) \( q + 3 i q^{7} +O(q^{10})\) \( q + 3 i q^{7} -5 q^{11} -4 i q^{13} + 8 i q^{17} -2 q^{19} + 2 i q^{23} + 6 q^{29} -7 q^{31} -6 i q^{37} + 6 q^{41} + 2 i q^{43} -6 i q^{47} -2 q^{49} + 5 i q^{53} -4 q^{59} -8 q^{61} -10 i q^{67} + 8 q^{71} -i q^{73} -15 i q^{77} -16 q^{79} -11 i q^{83} + 6 q^{89} + 12 q^{91} -i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 10q^{11} - 4q^{19} + 12q^{29} - 14q^{31} + 12q^{41} - 4q^{49} - 8q^{59} - 16q^{61} + 16q^{71} - 32q^{79} + 12q^{89} + 24q^{91} + O(q^{100}) \)

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

\( T_{7}^{2} + 9 \)
\( T_{11} + 5 \)
\( T_{13}^{2} + 16 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 64 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 4 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 7 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 25 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( 100 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 1 + T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( 121 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 1 + T^{2} \)
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