Properties

Label 1530.2.d.b
Level $1530$
Weight $2$
Character orbit 1530.d
Analytic conductor $12.217$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2171115093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + ( -1 - 2 i ) q^{5} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + ( -1 - 2 i ) q^{5} + i q^{8} + ( -2 + i ) q^{10} + 6 q^{11} + 3 i q^{13} + q^{16} + i q^{17} + 7 q^{19} + ( 1 + 2 i ) q^{20} -6 i q^{22} + 8 i q^{23} + ( -3 + 4 i ) q^{25} + 3 q^{26} -5 q^{29} + 5 q^{31} -i q^{32} + q^{34} -8 i q^{37} -7 i q^{38} + ( 2 - i ) q^{40} -4 i q^{43} -6 q^{44} + 8 q^{46} + 3 i q^{47} + 7 q^{49} + ( 4 + 3 i ) q^{50} -3 i q^{52} -9 i q^{53} + ( -6 - 12 i ) q^{55} + 5 i q^{58} + 5 q^{59} -3 q^{61} -5 i q^{62} - q^{64} + ( 6 - 3 i ) q^{65} + 2 i q^{67} -i q^{68} + 15 q^{71} -11 i q^{73} -8 q^{74} -7 q^{76} -8 q^{79} + ( -1 - 2 i ) q^{80} -4 i q^{83} + ( 2 - i ) q^{85} -4 q^{86} + 6 i q^{88} - q^{89} -8 i q^{92} + 3 q^{94} + ( -7 - 14 i ) q^{95} + 9 i q^{97} -7 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{5} - 4q^{10} + 12q^{11} + 2q^{16} + 14q^{19} + 2q^{20} - 6q^{25} + 6q^{26} - 10q^{29} + 10q^{31} + 2q^{34} + 4q^{40} - 12q^{44} + 16q^{46} + 14q^{49} + 8q^{50} - 12q^{55} + 10q^{59} - 6q^{61} - 2q^{64} + 12q^{65} + 30q^{71} - 16q^{74} - 14q^{76} - 16q^{79} - 2q^{80} + 4q^{85} - 8q^{86} - 2q^{89} + 6q^{94} - 14q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times\).

\(n\) \(307\) \(1261\) \(1361\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
919.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −1.00000 2.00000i 0 0 1.00000i 0 −2.00000 + 1.00000i
919.2 1.00000i 0 −1.00000 −1.00000 + 2.00000i 0 0 1.00000i 0 −2.00000 1.00000i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.d.b 2
3.b odd 2 1 170.2.c.a 2
5.b even 2 1 inner 1530.2.d.b 2
5.c odd 4 1 7650.2.a.s 1
5.c odd 4 1 7650.2.a.cb 1
12.b even 2 1 1360.2.e.b 2
15.d odd 2 1 170.2.c.a 2
15.e even 4 1 850.2.a.d 1
15.e even 4 1 850.2.a.h 1
60.h even 2 1 1360.2.e.b 2
60.l odd 4 1 6800.2.a.g 1
60.l odd 4 1 6800.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.a 2 3.b odd 2 1
170.2.c.a 2 15.d odd 2 1
850.2.a.d 1 15.e even 4 1
850.2.a.h 1 15.e even 4 1
1360.2.e.b 2 12.b even 2 1
1360.2.e.b 2 60.h even 2 1
1530.2.d.b 2 1.a even 1 1 trivial
1530.2.d.b 2 5.b even 2 1 inner
6800.2.a.g 1 60.l odd 4 1
6800.2.a.r 1 60.l odd 4 1
7650.2.a.s 1 5.c odd 4 1
7650.2.a.cb 1 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\):

\( T_{7} \)
\( T_{11} - 6 \)
\( T_{29} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 9 + T^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( ( -7 + T )^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( 81 + T^{2} \)
$59$ \( ( -5 + T )^{2} \)
$61$ \( ( 3 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( -15 + T )^{2} \)
$73$ \( 121 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( 1 + T )^{2} \)
$97$ \( 81 + T^{2} \)
show more
show less