Properties

Label 1350.2.c.f
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: yes
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} + i q^{7} -i q^{8} + 2 i q^{13} - q^{14} + q^{16} + 6 i q^{17} + q^{19} -6 i q^{23} -2 q^{26} -i q^{28} -6 q^{29} + 5 q^{31} + i q^{32} -6 q^{34} + 7 i q^{37} + i q^{38} -12 q^{41} + 11 i q^{43} + 6 q^{46} + 12 i q^{47} + 6 q^{49} -2 i q^{52} + q^{56} -6 i q^{58} -6 q^{59} -7 q^{61} + 5 i q^{62} - q^{64} + 4 i q^{67} -6 i q^{68} -6 q^{71} -7 i q^{73} -7 q^{74} - q^{76} + q^{79} -12 i q^{82} + 6 i q^{83} -11 q^{86} -6 q^{89} -2 q^{91} + 6 i q^{92} -12 q^{94} -5 i q^{97} + 6 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{14} + 2q^{16} + 2q^{19} - 4q^{26} - 12q^{29} + 10q^{31} - 12q^{34} - 24q^{41} + 12q^{46} + 12q^{49} + 2q^{56} - 12q^{59} - 14q^{61} - 2q^{64} - 12q^{71} - 14q^{74} - 2q^{76} + 2q^{79} - 22q^{86} - 12q^{89} - 4q^{91} - 24q^{94} + O(q^{100}) \)

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 1.00000i 1.00000i 0 0
649.2 1.00000i 0 −1.00000 0 0 1.00000i 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.f 2
3.b odd 2 1 1350.2.c.g 2
5.b even 2 1 inner 1350.2.c.f 2
5.c odd 4 1 1350.2.a.f 1
5.c odd 4 1 1350.2.a.s yes 1
15.d odd 2 1 1350.2.c.g 2
15.e even 4 1 1350.2.a.g yes 1
15.e even 4 1 1350.2.a.q yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.2.a.f 1 5.c odd 4 1
1350.2.a.g yes 1 15.e even 4 1
1350.2.a.q yes 1 15.e even 4 1
1350.2.a.s yes 1 5.c odd 4 1
1350.2.c.f 2 1.a even 1 1 trivial
1350.2.c.f 2 5.b even 2 1 inner
1350.2.c.g 2 3.b odd 2 1
1350.2.c.g 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 \(S_{2}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{2} + 1 \)
\( T_{11} \)
\( T_{13}^{2} + 4 \)
\( T_{17}^{2} + 36 \)
\( T_{29} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( 1 - 13 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )^{2} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 5 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 25 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 12 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 35 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 50 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 + 6 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 7 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 97 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 169 T^{2} + 9409 T^{4} \)
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