Properties

Label 1350.2.a.x
Level 1350
Weight 2
Character orbit 1350.a
Self dual yes
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta q^{7} + q^{8} -\beta q^{11} + \beta q^{14} + q^{16} + 4 q^{17} + 6 q^{19} -\beta q^{22} -2 q^{23} + \beta q^{28} + 7 q^{31} + q^{32} + 4 q^{34} -2 \beta q^{37} + 6 q^{38} + 2 \beta q^{41} -2 \beta q^{43} -\beta q^{44} -2 q^{46} + 2 q^{47} + 12 q^{49} -3 q^{53} + \beta q^{56} -2 \beta q^{59} -4 q^{61} + 7 q^{62} + q^{64} + 2 \beta q^{67} + 4 q^{68} + \beta q^{73} -2 \beta q^{74} + 6 q^{76} -19 q^{77} + 2 \beta q^{82} -5 q^{83} -2 \beta q^{86} -\beta q^{88} + 2 \beta q^{89} -2 q^{92} + 2 q^{94} -\beta q^{97} + 12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + 2q^{16} + 8q^{17} + 12q^{19} - 4q^{23} + 14q^{31} + 2q^{32} + 8q^{34} + 12q^{38} - 4q^{46} + 4q^{47} + 24q^{49} - 6q^{53} - 8q^{61} + 14q^{62} + 2q^{64} + 8q^{68} + 12q^{76} - 38q^{77} - 10q^{83} - 4q^{92} + 4q^{94} + 24q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.35890
4.35890
1.00000 0 1.00000 0 0 −4.35890 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 4.35890 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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.a.x 2
3.b odd 2 1 1350.2.a.w 2
5.b even 2 1 1350.2.a.w 2
5.c odd 4 2 270.2.c.c 4
15.d odd 2 1 inner 1350.2.a.x 2
15.e even 4 2 270.2.c.c 4
20.e even 4 2 2160.2.f.m 4
45.k odd 12 4 810.2.i.h 8
45.l even 12 4 810.2.i.h 8
60.l odd 4 2 2160.2.f.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.c 4 5.c odd 4 2
270.2.c.c 4 15.e even 4 2
810.2.i.h 8 45.k odd 12 4
810.2.i.h 8 45.l even 12 4
1350.2.a.w 2 3.b odd 2 1
1350.2.a.w 2 5.b even 2 1
1350.2.a.x 2 1.a even 1 1 trivial
1350.2.a.x 2 15.d odd 2 1 inner
2160.2.f.m 4 20.e even 4 2
2160.2.f.m 4 60.l odd 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\):

\( T_{7}^{2} - 19 \)
\( T_{11}^{2} - 19 \)
\( T_{13} \)
\( T_{17} - 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( 1 - 5 T^{2} + 49 T^{4} \)
$11$ \( 1 + 3 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 4 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 6 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 2 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 2 T^{2} + 1369 T^{4} \)
$41$ \( 1 + 6 T^{2} + 1681 T^{4} \)
$43$ \( 1 + 10 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 2 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 3 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 42 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 + 4 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 58 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 + 127 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 5 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 102 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 175 T^{2} + 9409 T^{4} \)
show more
show less