Properties

Label 1350.2.c.j
Level 1350
Weight 2
Character orbit 1350.c
Analytic conductor 10.780
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} + 2 i q^{7} -i q^{8} + 3 q^{11} -5 i q^{13} -2 q^{14} + q^{16} -3 i q^{17} + 4 q^{19} + 3 i q^{22} + 9 i q^{23} + 5 q^{26} -2 i q^{28} + 3 q^{29} + 5 q^{31} + i q^{32} + 3 q^{34} -10 i q^{37} + 4 i q^{38} + i q^{43} -3 q^{44} -9 q^{46} + 9 i q^{47} + 3 q^{49} + 5 i q^{52} + 12 i q^{53} + 2 q^{56} + 3 i q^{58} -12 q^{59} + 2 q^{61} + 5 i q^{62} - q^{64} -4 i q^{67} + 3 i q^{68} + 12 q^{71} + 10 i q^{73} + 10 q^{74} -4 q^{76} + 6 i q^{77} + 13 q^{79} -6 i q^{83} - q^{86} -3 i q^{88} + 12 q^{89} + 10 q^{91} -9 i q^{92} -9 q^{94} + 2 i q^{97} + 3 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 6q^{11} - 4q^{14} + 2q^{16} + 8q^{19} + 10q^{26} + 6q^{29} + 10q^{31} + 6q^{34} - 6q^{44} - 18q^{46} + 6q^{49} + 4q^{56} - 24q^{59} + 4q^{61} - 2q^{64} + 24q^{71} + 20q^{74} - 8q^{76} + 26q^{79} - 2q^{86} + 24q^{89} + 20q^{91} - 18q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(1\) \(-1\)

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
1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 0 0
649.2 1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 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 1350.2.c.j 2
3.b odd 2 1 1350.2.c.c 2
5.b even 2 1 inner 1350.2.c.j 2
5.c odd 4 1 270.2.a.c yes 1
5.c odd 4 1 1350.2.a.d 1
15.d odd 2 1 1350.2.c.c 2
15.e even 4 1 270.2.a.b 1
15.e even 4 1 1350.2.a.o 1
20.e even 4 1 2160.2.a.b 1
40.i odd 4 1 8640.2.a.bx 1
40.k even 4 1 8640.2.a.bn 1
45.k odd 12 2 810.2.e.d 2
45.l even 12 2 810.2.e.i 2
60.l odd 4 1 2160.2.a.q 1
120.q odd 4 1 8640.2.a.e 1
120.w even 4 1 8640.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.a.b 1 15.e even 4 1
270.2.a.c yes 1 5.c odd 4 1
810.2.e.d 2 45.k odd 12 2
810.2.e.i 2 45.l even 12 2
1350.2.a.d 1 5.c odd 4 1
1350.2.a.o 1 15.e even 4 1
1350.2.c.c 2 3.b odd 2 1
1350.2.c.c 2 15.d odd 2 1
1350.2.c.j 2 1.a even 1 1 trivial
1350.2.c.j 2 5.b even 2 1 inner
2160.2.a.b 1 20.e even 4 1
2160.2.a.q 1 60.l odd 4 1
8640.2.a.e 1 120.q odd 4 1
8640.2.a.y 1 120.w even 4 1
8640.2.a.bn 1 40.k even 4 1
8640.2.a.bx 1 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11} - 3 \)
\( T_{13}^{2} + 25 \)
\( T_{17}^{2} + 9 \)
\( T_{29} - 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 - T^{2} + 169 T^{4} \)
$17$ \( 1 - 25 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 35 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 3 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 5 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( 1 - 85 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 13 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 38 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 12 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 46 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 13 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 12 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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