Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.bs (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} -\zeta_{12}^{3} q^{5} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} -\zeta_{12}^{3} q^{5} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{10} + ( -4 + 3 \zeta_{12} + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{11} + ( -1 + 4 \zeta_{12}^{2} ) q^{13} -2 q^{14} -\zeta_{12}^{2} q^{16} + ( -4 + 4 \zeta_{12}^{2} ) q^{17} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{19} -\zeta_{12} q^{20} + ( -3 + 2 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{22} + ( -\zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{23} - q^{25} + ( -3 \zeta_{12} - \zeta_{12}^{3} ) q^{26} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} + ( \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{29} + ( -1 + 2 \zeta_{12}^{2} ) q^{31} + \zeta_{12} q^{32} -4 \zeta_{12}^{3} q^{34} + ( 2 - 2 \zeta_{12}^{2} ) q^{35} + ( 6 - 4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{37} + ( 4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{38} + q^{40} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} + ( -5 + 4 \zeta_{12} + 5 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{43} + ( -2 + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{44} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{46} + ( -2 + 4 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{47} -3 \zeta_{12}^{2} q^{49} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{50} + ( 3 + \zeta_{12}^{2} ) q^{52} + ( 6 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{53} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{55} + ( -2 + 2 \zeta_{12}^{2} ) q^{56} + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{58} + ( -2 + 5 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{59} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{61} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{62} - q^{64} + ( 4 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{65} + ( -4 + 8 \zeta_{12} + 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{67} + 4 \zeta_{12}^{2} q^{68} + 2 \zeta_{12}^{3} q^{70} + ( 6 - 2 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{71} + 2 \zeta_{12}^{3} q^{73} + ( 4 - 3 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{74} + ( 4 - 4 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{76} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{77} + ( -7 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{79} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{80} + ( -2 + 2 \zeta_{12}^{2} ) q^{82} + ( -4 + 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{83} + 4 \zeta_{12} q^{85} + ( -4 + 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{86} + ( -2 \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{88} + ( -4 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{89} + ( -2 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{91} + ( 2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{92} + ( -2 \zeta_{12} - 7 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{94} + ( -4 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{95} + ( -2 - 4 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{97} + 3 \zeta_{12} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} + 2q^{10} - 12q^{11} + 4q^{13} - 8q^{14} - 2q^{16} - 8q^{17} + 12q^{19} - 6q^{22} + 4q^{23} - 4q^{25} + 4q^{29} + 4q^{35} + 18q^{37} + 16q^{38} + 4q^{40} - 10q^{43} + 6q^{46} - 6q^{49} + 14q^{52} + 24q^{53} - 6q^{55} - 4q^{56} - 6q^{58} - 12q^{59} - 4q^{64} - 12q^{67} + 8q^{68} + 36q^{71} + 8q^{74} + 12q^{76} + 24q^{77} - 28q^{79} - 4q^{82} + 6q^{88} - 12q^{89} + 8q^{92} - 14q^{94} - 8q^{95} - 12q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1 - \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 1.00000i 0 1.73205 + 1.00000i 1.00000i 0 0.500000 + 0.866025i
361.2 0.866025 0.500000i 0 0.500000 0.866025i 1.00000i 0 −1.73205 1.00000i 1.00000i 0 0.500000 + 0.866025i
901.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 1.73205 1.00000i 1.00000i 0 0.500000 0.866025i
901.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 −1.73205 + 1.00000i 1.00000i 0 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.bs.e 4
3.b odd 2 1 390.2.bb.b 4
13.e even 6 1 inner 1170.2.bs.e 4
15.d odd 2 1 1950.2.bc.b 4
15.e even 4 1 1950.2.y.c 4
15.e even 4 1 1950.2.y.f 4
39.h odd 6 1 390.2.bb.b 4
39.h odd 6 1 5070.2.b.o 4
39.i odd 6 1 5070.2.b.o 4
39.k even 12 1 5070.2.a.y 2
39.k even 12 1 5070.2.a.bg 2
195.y odd 6 1 1950.2.bc.b 4
195.bf even 12 1 1950.2.y.c 4
195.bf even 12 1 1950.2.y.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.b 4 3.b odd 2 1
390.2.bb.b 4 39.h odd 6 1
1170.2.bs.e 4 1.a even 1 1 trivial
1170.2.bs.e 4 13.e even 6 1 inner
1950.2.y.c 4 15.e even 4 1
1950.2.y.c 4 195.bf even 12 1
1950.2.y.f 4 15.e even 4 1
1950.2.y.f 4 195.bf even 12 1
1950.2.bc.b 4 15.d odd 2 1
1950.2.bc.b 4 195.y odd 6 1
5070.2.a.y 2 39.k even 12 1
5070.2.a.bg 2 39.k even 12 1
5070.2.b.o 4 39.h odd 6 1
5070.2.b.o 4 39.i odd 6 1

Hecke kernels

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

\( T_{7}^{4} - 4 T_{7}^{2} + 16 \)
\( T_{11}^{4} + 12 T_{11}^{3} + 51 T_{11}^{2} + 36 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( 9 + 36 T + 51 T^{2} + 12 T^{3} + T^{4} \)
$13$ \( ( 13 - 2 T + T^{2} )^{2} \)
$17$ \( ( 16 + 4 T + T^{2} )^{2} \)
$19$ \( 16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4} \)
$23$ \( 1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( 1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( ( 3 + T^{2} )^{2} \)
$37$ \( 121 - 198 T + 119 T^{2} - 18 T^{3} + T^{4} \)
$41$ \( 16 - 4 T^{2} + T^{4} \)
$43$ \( 529 - 230 T + 123 T^{2} + 10 T^{3} + T^{4} \)
$47$ \( 1369 + 122 T^{2} + T^{4} \)
$53$ \( ( -12 - 12 T + T^{2} )^{2} \)
$59$ \( 169 - 156 T + 35 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 11664 + 108 T^{2} + T^{4} \)
$67$ \( 2704 - 624 T - 4 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 10816 - 3744 T + 536 T^{2} - 36 T^{3} + T^{4} \)
$73$ \( ( 4 + T^{2} )^{2} \)
$79$ \( ( 1 + 14 T + T^{2} )^{2} \)
$83$ \( 1936 + 104 T^{2} + T^{4} \)
$89$ \( 16 - 48 T + 44 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( 16 - 48 T + 44 T^{2} + 12 T^{3} + T^{4} \)
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