Properties

Label 1350.2.e.b
Level 1350
Weight 2
Character orbit 1350.e
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.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{7} + q^{8} + ( 6 - 6 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -4 q^{19} + 6 \zeta_{6} q^{22} -9 \zeta_{6} q^{23} -2 q^{26} + q^{28} + ( 3 - 3 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} -\zeta_{6} q^{32} -8 q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} -3 \zeta_{6} q^{41} + ( 8 - 8 \zeta_{6} ) q^{43} -6 q^{44} + 9 q^{46} + ( 3 - 3 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} + ( 2 - 2 \zeta_{6} ) q^{52} + 6 q^{53} + ( -1 + \zeta_{6} ) q^{56} + 3 \zeta_{6} q^{58} + 6 \zeta_{6} q^{59} + ( 13 - 13 \zeta_{6} ) q^{61} -4 q^{62} + q^{64} -13 \zeta_{6} q^{67} + 6 q^{71} + 4 q^{73} + ( 8 - 8 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{76} + 6 \zeta_{6} q^{77} + ( 10 - 10 \zeta_{6} ) q^{79} + 3 q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} + 8 \zeta_{6} q^{86} + ( 6 - 6 \zeta_{6} ) q^{88} -9 q^{89} -2 q^{91} + ( -9 + 9 \zeta_{6} ) q^{92} + 3 \zeta_{6} q^{94} + ( 2 - 2 \zeta_{6} ) q^{97} -6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - q^{7} + 2q^{8} + 6q^{11} + 2q^{13} - q^{14} - q^{16} - 8q^{19} + 6q^{22} - 9q^{23} - 4q^{26} + 2q^{28} + 3q^{29} + 4q^{31} - q^{32} - 16q^{37} + 4q^{38} - 3q^{41} + 8q^{43} - 12q^{44} + 18q^{46} + 3q^{47} + 6q^{49} + 2q^{52} + 12q^{53} - q^{56} + 3q^{58} + 6q^{59} + 13q^{61} - 8q^{62} + 2q^{64} - 13q^{67} + 12q^{71} + 8q^{73} + 8q^{74} + 4q^{76} + 6q^{77} + 10q^{79} + 6q^{82} + 9q^{83} + 8q^{86} + 6q^{88} - 18q^{89} - 4q^{91} - 9q^{92} + 3q^{94} + 2q^{97} - 12q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −0.500000 + 0.866025i 1.00000 0 0
901.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 0.866025i 1.00000 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
9.c even 3 1 inner

Twists

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

Hecke kernels

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

\( T_{7}^{2} + T_{7} + 1 \)
\( T_{11}^{2} - 6 T_{11} + 36 \)
\( T_{17} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( ( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \)
$11$ \( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \)
$29$ \( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \)
$37$ \( ( 1 + 8 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \)
$47$ \( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \)
$67$ \( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 4 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 9 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \)
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