Properties

Label 1350.2.e.i
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 1 - \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 1 - \zeta_{6} ) q^{7} - q^{8} + ( -2 + 2 \zeta_{6} ) q^{11} + 6 \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 2 q^{17} + 6 q^{19} + 2 \zeta_{6} q^{22} + \zeta_{6} q^{23} + 6 q^{26} - q^{28} + ( 9 - 9 \zeta_{6} ) q^{29} + 2 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{34} + 2 q^{37} + ( 6 - 6 \zeta_{6} ) q^{38} -11 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} + 2 q^{44} + q^{46} + ( -7 + 7 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} + ( 6 - 6 \zeta_{6} ) q^{52} + ( -1 + \zeta_{6} ) q^{56} -9 \zeta_{6} q^{58} -4 \zeta_{6} q^{59} + ( 7 - 7 \zeta_{6} ) q^{61} + 2 q^{62} + q^{64} + 11 \zeta_{6} q^{67} -2 \zeta_{6} q^{68} + 6 q^{71} -4 q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} -6 \zeta_{6} q^{76} + 2 \zeta_{6} q^{77} + ( 12 - 12 \zeta_{6} ) q^{79} -11 q^{82} + ( -11 + 11 \zeta_{6} ) q^{83} -4 \zeta_{6} q^{86} + ( 2 - 2 \zeta_{6} ) q^{88} - q^{89} + 6 q^{91} + ( 1 - \zeta_{6} ) q^{92} + 7 \zeta_{6} q^{94} + ( 8 - 8 \zeta_{6} ) q^{97} + 6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + q^{7} - 2q^{8} - 2q^{11} + 6q^{13} - q^{14} - q^{16} + 4q^{17} + 12q^{19} + 2q^{22} + q^{23} + 12q^{26} - 2q^{28} + 9q^{29} + 2q^{31} + q^{32} + 2q^{34} + 4q^{37} + 6q^{38} - 11q^{41} + 4q^{43} + 4q^{44} + 2q^{46} - 7q^{47} + 6q^{49} + 6q^{52} - q^{56} - 9q^{58} - 4q^{59} + 7q^{61} + 4q^{62} + 2q^{64} + 11q^{67} - 2q^{68} + 12q^{71} - 8q^{73} + 2q^{74} - 6q^{76} + 2q^{77} + 12q^{79} - 22q^{82} - 11q^{83} - 4q^{86} + 2q^{88} - 2q^{89} + 12q^{91} + q^{92} + 7q^{94} + 8q^{97} + 12q^{98} + O(q^{100}) \)

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

Hecke kernels

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

\( T_{7}^{2} - T_{7} + 1 \)
\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{17} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \)
$11$ \( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 6 T + 19 T^{2} )^{2} \)
$23$ \( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \)
$29$ \( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \)
$31$ \( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 11 T + 80 T^{2} + 451 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{2} \)
$59$ \( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 4 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 11 T + 38 T^{2} + 913 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + T + 89 T^{2} )^{2} \)
$97$ \( 1 - 8 T - 33 T^{2} - 776 T^{3} + 9409 T^{4} \)
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