Properties

Label 1350.2.j.e
Level 1350
Weight 2
Character orbit 1350.j
Analytic conductor 10.780
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1350.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( 6 - 6 \zeta_{12}^{2} ) q^{11} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + \zeta_{12}^{2} q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 4 q^{19} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{22} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{23} -2 q^{26} + \zeta_{12}^{3} q^{28} + ( -3 + 3 \zeta_{12}^{2} ) q^{29} + 4 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 8 \zeta_{12}^{3} q^{37} + 4 \zeta_{12} q^{38} -3 \zeta_{12}^{2} q^{41} + 8 \zeta_{12} q^{43} + 6 q^{44} + 9 q^{46} -3 \zeta_{12} q^{47} -6 \zeta_{12}^{2} q^{49} -2 \zeta_{12} q^{52} + 6 \zeta_{12}^{3} q^{53} + ( -1 + \zeta_{12}^{2} ) q^{56} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{58} -6 \zeta_{12}^{2} q^{59} + ( 13 - 13 \zeta_{12}^{2} ) q^{61} + 4 \zeta_{12}^{3} q^{62} - q^{64} + ( -13 \zeta_{12} + 13 \zeta_{12}^{3} ) q^{67} + 6 q^{71} + 4 \zeta_{12}^{3} q^{73} + ( -8 + 8 \zeta_{12}^{2} ) q^{74} + 4 \zeta_{12}^{2} q^{76} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} -3 \zeta_{12}^{3} q^{82} + 9 \zeta_{12} q^{83} + 8 \zeta_{12}^{2} q^{86} + 6 \zeta_{12} q^{88} + 9 q^{89} -2 q^{91} + 9 \zeta_{12} q^{92} -3 \zeta_{12}^{2} q^{94} -2 \zeta_{12} q^{97} -6 \zeta_{12}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} + 12q^{11} + 2q^{14} - 2q^{16} + 16q^{19} - 8q^{26} - 6q^{29} + 8q^{31} - 6q^{41} + 24q^{44} + 36q^{46} - 12q^{49} - 2q^{56} - 12q^{59} + 26q^{61} - 4q^{64} + 24q^{71} - 16q^{74} + 8q^{76} - 20q^{79} + 16q^{86} + 36q^{89} - 8q^{91} - 6q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −0.866025 + 0.500000i 1.00000i 0 0
199.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0.866025 0.500000i 1.00000i 0 0
1099.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −0.866025 0.500000i 1.00000i 0 0
1099.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0.866025 + 0.500000i 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
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.j.e 4
3.b odd 2 1 450.2.j.c 4
5.b even 2 1 inner 1350.2.j.e 4
5.c odd 4 1 270.2.e.b 2
5.c odd 4 1 1350.2.e.b 2
9.c even 3 1 inner 1350.2.j.e 4
9.c even 3 1 4050.2.c.a 2
9.d odd 6 1 450.2.j.c 4
9.d odd 6 1 4050.2.c.t 2
15.d odd 2 1 450.2.j.c 4
15.e even 4 1 90.2.e.a 2
15.e even 4 1 450.2.e.e 2
20.e even 4 1 2160.2.q.b 2
45.h odd 6 1 450.2.j.c 4
45.h odd 6 1 4050.2.c.t 2
45.j even 6 1 inner 1350.2.j.e 4
45.j even 6 1 4050.2.c.a 2
45.k odd 12 1 270.2.e.b 2
45.k odd 12 1 810.2.a.b 1
45.k odd 12 1 1350.2.e.b 2
45.k odd 12 1 4050.2.a.ba 1
45.l even 12 1 90.2.e.a 2
45.l even 12 1 450.2.e.e 2
45.l even 12 1 810.2.a.g 1
45.l even 12 1 4050.2.a.n 1
60.l odd 4 1 720.2.q.b 2
180.v odd 12 1 720.2.q.b 2
180.v odd 12 1 6480.2.a.g 1
180.x even 12 1 2160.2.q.b 2
180.x even 12 1 6480.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.a 2 15.e even 4 1
90.2.e.a 2 45.l even 12 1
270.2.e.b 2 5.c odd 4 1
270.2.e.b 2 45.k odd 12 1
450.2.e.e 2 15.e even 4 1
450.2.e.e 2 45.l even 12 1
450.2.j.c 4 3.b odd 2 1
450.2.j.c 4 9.d odd 6 1
450.2.j.c 4 15.d odd 2 1
450.2.j.c 4 45.h odd 6 1
720.2.q.b 2 60.l odd 4 1
720.2.q.b 2 180.v odd 12 1
810.2.a.b 1 45.k odd 12 1
810.2.a.g 1 45.l even 12 1
1350.2.e.b 2 5.c odd 4 1
1350.2.e.b 2 45.k odd 12 1
1350.2.j.e 4 1.a even 1 1 trivial
1350.2.j.e 4 5.b even 2 1 inner
1350.2.j.e 4 9.c even 3 1 inner
1350.2.j.e 4 45.j even 6 1 inner
2160.2.q.b 2 20.e even 4 1
2160.2.q.b 2 180.x even 12 1
4050.2.a.n 1 45.l even 12 1
4050.2.a.ba 1 45.k odd 12 1
4050.2.c.a 2 9.c even 3 1
4050.2.c.a 2 45.j even 6 1
4050.2.c.t 2 9.d odd 6 1
4050.2.c.t 2 45.h odd 6 1
6480.2.a.g 1 180.v odd 12 1
6480.2.a.v 1 180.x even 12 1

Hecke kernels

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

\( T_{7}^{4} - T_{7}^{2} + 1 \)
\( T_{11}^{2} - 6 T_{11} + 36 \)
\( T_{19} - 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( \)
$5$ \( \)
$7$ \( ( 1 + 2 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} ) \)
$11$ \( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} ) \)
$17$ \( ( 1 - 17 T^{2} )^{4} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{4} \)
$23$ \( 1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 61 T^{2} + 1849 T^{4} )( 1 + 83 T^{2} + 1849 T^{4} ) \)
$47$ \( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \)
$67$ \( 1 - 35 T^{2} - 3264 T^{4} - 157115 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 130 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 + 85 T^{2} + 336 T^{4} + 585565 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 - 9 T + 89 T^{2} )^{4} \)
$97$ \( 1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8} \)
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