Properties

Label 1170.2.b.d
Level $1170$
Weight $2$
Character orbit 1170.b
Analytic conductor $9.342$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} - q^{4} -\beta_{1} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} - q^{4} -\beta_{1} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + \beta_{1} q^{8} - q^{10} -\beta_{3} q^{13} + ( -1 + \beta_{3} ) q^{14} + q^{16} + ( -1 + \beta_{3} ) q^{17} + ( -\beta_{1} + \beta_{2} ) q^{19} + \beta_{1} q^{20} + ( 5 + \beta_{3} ) q^{23} - q^{25} + \beta_{2} q^{26} + ( \beta_{1} - \beta_{2} ) q^{28} + ( -1 + \beta_{3} ) q^{29} + 6 \beta_{1} q^{31} -\beta_{1} q^{32} + ( \beta_{1} - \beta_{2} ) q^{34} + ( -1 + \beta_{3} ) q^{35} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -1 + \beta_{3} ) q^{38} + q^{40} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{41} + 8 q^{43} + ( -5 \beta_{1} - \beta_{2} ) q^{46} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( -7 + 2 \beta_{3} ) q^{49} + \beta_{1} q^{50} + \beta_{3} q^{52} -6 q^{53} + ( 1 - \beta_{3} ) q^{56} + ( \beta_{1} - \beta_{2} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -4 + 2 \beta_{3} ) q^{61} + 6 q^{62} - q^{64} + \beta_{2} q^{65} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 1 - \beta_{3} ) q^{68} + ( \beta_{1} - \beta_{2} ) q^{70} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -5 \beta_{1} - \beta_{2} ) q^{73} + ( -2 + 2 \beta_{3} ) q^{74} + ( \beta_{1} - \beta_{2} ) q^{76} + 4 \beta_{3} q^{79} -\beta_{1} q^{80} + ( 4 + 2 \beta_{3} ) q^{82} + ( -10 \beta_{1} - 2 \beta_{2} ) q^{83} + ( \beta_{1} - \beta_{2} ) q^{85} -8 \beta_{1} q^{86} + ( -8 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -13 \beta_{1} + \beta_{2} ) q^{91} + ( -5 - \beta_{3} ) q^{92} + ( 2 - 2 \beta_{3} ) q^{94} + ( -1 + \beta_{3} ) q^{95} + ( 5 \beta_{1} + \beta_{2} ) q^{97} + ( 7 \beta_{1} - 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{10} - 4q^{14} + 4q^{16} - 4q^{17} + 20q^{23} - 4q^{25} - 4q^{29} - 4q^{35} - 4q^{38} + 4q^{40} + 32q^{43} - 28q^{49} - 24q^{53} + 4q^{56} - 16q^{61} + 24q^{62} - 4q^{64} + 4q^{68} - 8q^{74} + 16q^{82} - 20q^{92} + 8q^{94} - 4q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 10 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{2} + 5 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
2.30278i
1.30278i
1.30278i
2.30278i
1.00000i 0 −1.00000 1.00000i 0 4.60555i 1.00000i 0 −1.00000
181.2 1.00000i 0 −1.00000 1.00000i 0 2.60555i 1.00000i 0 −1.00000
181.3 1.00000i 0 −1.00000 1.00000i 0 2.60555i 1.00000i 0 −1.00000
181.4 1.00000i 0 −1.00000 1.00000i 0 4.60555i 1.00000i 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.b.d 4
3.b odd 2 1 390.2.b.c 4
12.b even 2 1 3120.2.g.q 4
13.b even 2 1 inner 1170.2.b.d 4
15.d odd 2 1 1950.2.b.k 4
15.e even 4 1 1950.2.f.m 4
15.e even 4 1 1950.2.f.n 4
39.d odd 2 1 390.2.b.c 4
39.f even 4 1 5070.2.a.z 2
39.f even 4 1 5070.2.a.bf 2
156.h even 2 1 3120.2.g.q 4
195.e odd 2 1 1950.2.b.k 4
195.s even 4 1 1950.2.f.m 4
195.s even 4 1 1950.2.f.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.c 4 3.b odd 2 1
390.2.b.c 4 39.d odd 2 1
1170.2.b.d 4 1.a even 1 1 trivial
1170.2.b.d 4 13.b even 2 1 inner
1950.2.b.k 4 15.d odd 2 1
1950.2.b.k 4 195.e odd 2 1
1950.2.f.m 4 15.e even 4 1
1950.2.f.m 4 195.s even 4 1
1950.2.f.n 4 15.e even 4 1
1950.2.f.n 4 195.s even 4 1
3120.2.g.q 4 12.b even 2 1
3120.2.g.q 4 156.h even 2 1
5070.2.a.z 2 39.f even 4 1
5070.2.a.bf 2 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{4} + 28 T_{7}^{2} + 144 \)
\( T_{11} \)
\( T_{17}^{2} + 2 T_{17} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 144 + 28 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( -13 + T^{2} )^{2} \)
$17$ \( ( -12 + 2 T + T^{2} )^{2} \)
$19$ \( 144 + 28 T^{2} + T^{4} \)
$23$ \( ( 12 - 10 T + T^{2} )^{2} \)
$29$ \( ( -12 + 2 T + T^{2} )^{2} \)
$31$ \( ( 36 + T^{2} )^{2} \)
$37$ \( 2304 + 112 T^{2} + T^{4} \)
$41$ \( 1296 + 136 T^{2} + T^{4} \)
$43$ \( ( -8 + T )^{4} \)
$47$ \( 2304 + 112 T^{2} + T^{4} \)
$53$ \( ( 6 + T )^{4} \)
$59$ \( 2304 + 112 T^{2} + T^{4} \)
$61$ \( ( -36 + 8 T + T^{2} )^{2} \)
$67$ \( 1296 + 136 T^{2} + T^{4} \)
$71$ \( 2304 + 112 T^{2} + T^{4} \)
$73$ \( 144 + 76 T^{2} + T^{4} \)
$79$ \( ( -208 + T^{2} )^{2} \)
$83$ \( 2304 + 304 T^{2} + T^{4} \)
$89$ \( 144 + 232 T^{2} + T^{4} \)
$97$ \( 144 + 76 T^{2} + T^{4} \)
show more
show less