Properties

Label 1170.2.bs.d
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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + \zeta_{12}^{3} q^{5} + 3 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + \zeta_{12}^{3} q^{5} + 3 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{10} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{11} + ( - 4 \zeta_{12}^{3} + \zeta_{12}) q^{13} - 3 q^{14} - \zeta_{12}^{2} q^{16} + ( - 4 \zeta_{12}^{2} + 4) q^{17} + (\zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{19} + \zeta_{12} q^{20} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{22} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{23} - q^{25} + (4 \zeta_{12}^{2} - 1) q^{26} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{28} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{29} + (2 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{31} + \zeta_{12} q^{32} + 4 \zeta_{12}^{3} q^{34} + (3 \zeta_{12}^{2} - 3) q^{35} + ( - \zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12} - 8) q^{37} + (\zeta_{12}^{3} - 2 \zeta_{12} - 4) q^{38} - q^{40} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{41} + (6 \zeta_{12}^{2} - 6) q^{43} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{44} + ( - 2 \zeta_{12}^{2} - 2) q^{46} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{47} + 2 \zeta_{12}^{2} q^{49} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{50} + ( - \zeta_{12}^{3} - 3 \zeta_{12}) q^{52} + (\zeta_{12}^{3} - 2 \zeta_{12} + 2) q^{53} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12}) q^{55} + (3 \zeta_{12}^{2} - 3) q^{56} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{58} + (2 \zeta_{12}^{2} + 8 \zeta_{12} + 2) q^{59} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{61} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12}) q^{62} - q^{64} + (\zeta_{12}^{2} + 3) q^{65} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{67} - 4 \zeta_{12}^{2} q^{68} - 3 \zeta_{12}^{3} q^{70} + (4 \zeta_{12}^{2} + 6 \zeta_{12} + 4) q^{71} + (8 \zeta_{12}^{2} - 4) q^{73} + ( - 8 \zeta_{12}^{3} + \zeta_{12}^{2} + 4 \zeta_{12} - 1) q^{74} + ( - 4 \zeta_{12}^{3} - \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{76} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 6) q^{77} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 10) q^{79} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{80} + (4 \zeta_{12}^{2} - 4) q^{82} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{83} + 4 \zeta_{12} q^{85} - 6 \zeta_{12}^{3} q^{86} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12}) q^{88} + ( - 2 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 2 \zeta_{12} + 14) q^{89} + ( - 9 \zeta_{12}^{2} + 12) q^{91} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{92} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 2 \zeta_{12}) q^{94} + (2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - \zeta_{12} - 4) q^{95} + (6 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{97} - 2 \zeta_{12} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{10} + 6 q^{11} - 12 q^{14} - 2 q^{16} + 8 q^{17} + 6 q^{19} + 4 q^{22} - 4 q^{25} + 4 q^{26} - 4 q^{29} - 6 q^{35} - 24 q^{37} - 16 q^{38} - 4 q^{40} - 12 q^{43} - 12 q^{46} + 4 q^{49} + 8 q^{53} - 4 q^{55} - 6 q^{56} - 12 q^{58} + 12 q^{59} + 8 q^{61} - 4 q^{62} - 4 q^{64} + 14 q^{65} + 12 q^{67} - 8 q^{68} + 24 q^{71} - 2 q^{74} + 6 q^{76} - 24 q^{77} + 40 q^{79} - 8 q^{82} - 4 q^{88} + 42 q^{89} + 30 q^{91} + 6 q^{94} - 8 q^{95} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 2.59808 + 1.50000i 1.00000i 0 −0.500000 0.866025i
361.2 0.866025 0.500000i 0 0.500000 0.866025i 1.00000i 0 −2.59808 1.50000i 1.00000i 0 −0.500000 0.866025i
901.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 2.59808 1.50000i 1.00000i 0 −0.500000 + 0.866025i
901.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 −2.59808 + 1.50000i 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.d 4
3.b odd 2 1 390.2.bb.a 4
13.e even 6 1 inner 1170.2.bs.d 4
15.d odd 2 1 1950.2.bc.a 4
15.e even 4 1 1950.2.y.d 4
15.e even 4 1 1950.2.y.e 4
39.h odd 6 1 390.2.bb.a 4
39.h odd 6 1 5070.2.b.p 4
39.i odd 6 1 5070.2.b.p 4
39.k even 12 1 5070.2.a.ba 2
39.k even 12 1 5070.2.a.be 2
195.y odd 6 1 1950.2.bc.a 4
195.bf even 12 1 1950.2.y.d 4
195.bf even 12 1 1950.2.y.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 3.b odd 2 1
390.2.bb.a 4 39.h odd 6 1
1170.2.bs.d 4 1.a even 1 1 trivial
1170.2.bs.d 4 13.e even 6 1 inner
1950.2.y.d 4 15.e even 4 1
1950.2.y.d 4 195.bf even 12 1
1950.2.y.e 4 15.e even 4 1
1950.2.y.e 4 195.bf even 12 1
1950.2.bc.a 4 15.d odd 2 1
1950.2.bc.a 4 195.y odd 6 1
5070.2.a.ba 2 39.k even 12 1
5070.2.a.be 2 39.k even 12 1
5070.2.b.p 4 39.h odd 6 1
5070.2.b.p 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 \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{4} - 9T_{7}^{2} + 81 \) Copy content Toggle raw display
\( T_{11}^{4} - 6T_{11}^{3} + 11T_{11}^{2} + 6T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} - T^{2} + 78 T + 169 \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + 24 T^{2} - 32 T + 64 \) Copy content Toggle raw display
$31$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$37$ \( T^{4} + 24 T^{3} + 239 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$41$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$53$ \( (T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} - 4 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + 56 T^{2} - 96 T + 64 \) Copy content Toggle raw display
$71$ \( T^{4} - 24 T^{3} + 204 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 20 T + 52)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$89$ \( T^{4} - 42 T^{3} + 731 T^{2} + \cdots + 20449 \) Copy content Toggle raw display
$97$ \( T^{4} - 36 T^{3} + 536 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
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