Properties

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

Related objects

Downloads

Learn more about

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 450)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -2 + 2 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -2 + 2 \zeta_{6} ) q^{7} - q^{8} + 4 \zeta_{6} q^{13} + 2 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 6 q^{17} -7 q^{19} + 4 q^{26} + 2 q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + 10 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( 6 - 6 \zeta_{6} ) q^{34} + 2 q^{37} + ( -7 + 7 \zeta_{6} ) q^{38} + 9 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + ( 4 - 4 \zeta_{6} ) q^{52} + 12 q^{53} + ( 2 - 2 \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{58} -9 \zeta_{6} q^{59} + ( 4 - 4 \zeta_{6} ) q^{61} + 10 q^{62} + q^{64} + 13 \zeta_{6} q^{67} -6 \zeta_{6} q^{68} -6 q^{71} - q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + 7 \zeta_{6} q^{76} + ( -2 + 2 \zeta_{6} ) q^{79} + 9 q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} -\zeta_{6} q^{86} -15 q^{89} -8 q^{91} -6 \zeta_{6} q^{94} + ( -17 + 17 \zeta_{6} ) q^{97} + 3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 2q^{7} - 2q^{8} + 4q^{13} + 2q^{14} - q^{16} + 12q^{17} - 14q^{19} + 8q^{26} + 4q^{28} - 6q^{29} + 10q^{31} + q^{32} + 6q^{34} + 4q^{37} - 7q^{38} + 9q^{41} + q^{43} + 6q^{47} + 3q^{49} + 4q^{52} + 24q^{53} + 2q^{56} + 6q^{58} - 9q^{59} + 4q^{61} + 20q^{62} + 2q^{64} + 13q^{67} - 6q^{68} - 12q^{71} - 2q^{73} + 2q^{74} + 7q^{76} - 2q^{79} + 18q^{82} + 9q^{83} - q^{86} - 30q^{89} - 16q^{91} - 6q^{94} - 17q^{97} + 6q^{98} + O(q^{100}) \)

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 −1.00000 + 1.73205i −1.00000 0 0
901.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −1.00000 1.73205i −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.h 2
3.b odd 2 1 450.2.e.c 2
5.b even 2 1 1350.2.e.d 2
5.c odd 4 2 1350.2.j.b 4
9.c even 3 1 inner 1350.2.e.h 2
9.c even 3 1 4050.2.a.o 1
9.d odd 6 1 450.2.e.c 2
9.d odd 6 1 4050.2.a.bg 1
15.d odd 2 1 450.2.e.f yes 2
15.e even 4 2 450.2.j.b 4
45.h odd 6 1 450.2.e.f yes 2
45.h odd 6 1 4050.2.a.d 1
45.j even 6 1 1350.2.e.d 2
45.j even 6 1 4050.2.a.u 1
45.k odd 12 2 1350.2.j.b 4
45.k odd 12 2 4050.2.c.m 2
45.l even 12 2 450.2.j.b 4
45.l even 12 2 4050.2.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.c 2 3.b odd 2 1
450.2.e.c 2 9.d odd 6 1
450.2.e.f yes 2 15.d odd 2 1
450.2.e.f yes 2 45.h odd 6 1
450.2.j.b 4 15.e even 4 2
450.2.j.b 4 45.l even 12 2
1350.2.e.d 2 5.b even 2 1
1350.2.e.d 2 45.j even 6 1
1350.2.e.h 2 1.a even 1 1 trivial
1350.2.e.h 2 9.c even 3 1 inner
1350.2.j.b 4 5.c odd 4 2
1350.2.j.b 4 45.k odd 12 2
4050.2.a.d 1 45.h odd 6 1
4050.2.a.o 1 9.c even 3 1
4050.2.a.u 1 45.j even 6 1
4050.2.a.bg 1 9.d odd 6 1
4050.2.c.h 2 45.l even 12 2
4050.2.c.m 2 45.k odd 12 2

Hecke kernels

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

\( T_{7}^{2} + 2 T_{7} + 4 \)
\( T_{11} \)
\( T_{17} - 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \)
$43$ \( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 12 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + T + 73 T^{2} )^{2} \)
$79$ \( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 15 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4} \)
show more
show less