Properties

Label 1350.2.c.k
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + i q^{7} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + i q^{7} + i q^{8} + 3 q^{11} -4 i q^{13} + q^{14} + q^{16} -2 q^{19} -3 i q^{22} + 6 i q^{23} -4 q^{26} -i q^{28} + 6 q^{29} + 5 q^{31} -i q^{32} -2 i q^{37} + 2 i q^{38} + 6 q^{41} -10 i q^{43} -3 q^{44} + 6 q^{46} + 6 i q^{47} + 6 q^{49} + 4 i q^{52} -9 i q^{53} - q^{56} -6 i q^{58} + 12 q^{59} + 8 q^{61} -5 i q^{62} - q^{64} -14 i q^{67} -7 i q^{73} -2 q^{74} + 2 q^{76} + 3 i q^{77} -8 q^{79} -6 i q^{82} + 3 i q^{83} -10 q^{86} + 3 i q^{88} -18 q^{89} + 4 q^{91} -6 i q^{92} + 6 q^{94} + i q^{97} -6 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 6q^{11} + 2q^{14} + 2q^{16} - 4q^{19} - 8q^{26} + 12q^{29} + 10q^{31} + 12q^{41} - 6q^{44} + 12q^{46} + 12q^{49} - 2q^{56} + 24q^{59} + 16q^{61} - 2q^{64} - 4q^{74} + 4q^{76} - 16q^{79} - 20q^{86} - 36q^{89} + 8q^{91} + 12q^{94} + O(q^{100}) \)

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.k 2
3.b odd 2 1 1350.2.c.b 2
5.b even 2 1 inner 1350.2.c.k 2
5.c odd 4 1 54.2.a.b yes 1
5.c odd 4 1 1350.2.a.h 1
15.d odd 2 1 1350.2.c.b 2
15.e even 4 1 54.2.a.a 1
15.e even 4 1 1350.2.a.r 1
20.e even 4 1 432.2.a.b 1
35.f even 4 1 2646.2.a.bd 1
40.i odd 4 1 1728.2.a.y 1
40.k even 4 1 1728.2.a.z 1
45.k odd 12 2 162.2.c.b 2
45.l even 12 2 162.2.c.c 2
55.e even 4 1 6534.2.a.b 1
60.l odd 4 1 432.2.a.g 1
65.h odd 4 1 9126.2.a.r 1
105.k odd 4 1 2646.2.a.a 1
120.q odd 4 1 1728.2.a.d 1
120.w even 4 1 1728.2.a.c 1
165.l odd 4 1 6534.2.a.bc 1
180.v odd 12 2 1296.2.i.c 2
180.x even 12 2 1296.2.i.o 2
195.s even 4 1 9126.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 15.e even 4 1
54.2.a.b yes 1 5.c odd 4 1
162.2.c.b 2 45.k odd 12 2
162.2.c.c 2 45.l even 12 2
432.2.a.b 1 20.e even 4 1
432.2.a.g 1 60.l odd 4 1
1296.2.i.c 2 180.v odd 12 2
1296.2.i.o 2 180.x even 12 2
1350.2.a.h 1 5.c odd 4 1
1350.2.a.r 1 15.e even 4 1
1350.2.c.b 2 3.b odd 2 1
1350.2.c.b 2 15.d odd 2 1
1350.2.c.k 2 1.a even 1 1 trivial
1350.2.c.k 2 5.b even 2 1 inner
1728.2.a.c 1 120.w even 4 1
1728.2.a.d 1 120.q odd 4 1
1728.2.a.y 1 40.i odd 4 1
1728.2.a.z 1 40.k even 4 1
2646.2.a.a 1 105.k odd 4 1
2646.2.a.bd 1 35.f even 4 1
6534.2.a.b 1 55.e even 4 1
6534.2.a.bc 1 165.l odd 4 1
9126.2.a.r 1 65.h odd 4 1
9126.2.a.u 1 195.s even 4 1

Hecke kernels

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

\( T_{7}^{2} + 1 \)
\( T_{11} - 3 \)
\( T_{13}^{2} + 16 \)
\( T_{17} \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 100 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 81 + T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( 196 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 49 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 9 + T^{2} \)
$89$ \( ( 18 + T )^{2} \)
$97$ \( 1 + T^{2} \)
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