Properties

Label 1170.2.i.d
Level $1170$
Weight $2$
Character orbit 1170.i
Analytic conductor $9.342$
Analytic rank $0$
Dimension $2$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + q^{5} -2 \zeta_{6} q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + q^{5} -2 \zeta_{6} q^{7} + q^{8} + ( -1 + \zeta_{6} ) q^{10} + ( -3 + 3 \zeta_{6} ) q^{11} + ( 3 - 4 \zeta_{6} ) q^{13} + 2 q^{14} + ( -1 + \zeta_{6} ) q^{16} + 6 \zeta_{6} q^{17} -2 \zeta_{6} q^{19} -\zeta_{6} q^{20} -3 \zeta_{6} q^{22} + ( 3 - 3 \zeta_{6} ) q^{23} + q^{25} + ( 1 + 3 \zeta_{6} ) q^{26} + ( -2 + 2 \zeta_{6} ) q^{28} + ( 3 - 3 \zeta_{6} ) q^{29} + 5 q^{31} -\zeta_{6} q^{32} -6 q^{34} -2 \zeta_{6} q^{35} + ( 7 - 7 \zeta_{6} ) q^{37} + 2 q^{38} + q^{40} + ( 6 - 6 \zeta_{6} ) q^{41} + \zeta_{6} q^{43} + 3 q^{44} + 3 \zeta_{6} q^{46} + 3 q^{47} + ( 3 - 3 \zeta_{6} ) q^{49} + ( -1 + \zeta_{6} ) q^{50} + ( -4 + \zeta_{6} ) q^{52} + 6 q^{53} + ( -3 + 3 \zeta_{6} ) q^{55} -2 \zeta_{6} q^{56} + 3 \zeta_{6} q^{58} -9 \zeta_{6} q^{59} -2 \zeta_{6} q^{61} + ( -5 + 5 \zeta_{6} ) q^{62} + q^{64} + ( 3 - 4 \zeta_{6} ) q^{65} + ( -8 + 8 \zeta_{6} ) q^{67} + ( 6 - 6 \zeta_{6} ) q^{68} + 2 q^{70} -12 \zeta_{6} q^{71} + 14 q^{73} + 7 \zeta_{6} q^{74} + ( -2 + 2 \zeta_{6} ) q^{76} + 6 q^{77} + 5 q^{79} + ( -1 + \zeta_{6} ) q^{80} + 6 \zeta_{6} q^{82} + 6 q^{83} + 6 \zeta_{6} q^{85} - q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} + ( -18 + 18 \zeta_{6} ) q^{89} + ( -8 + 2 \zeta_{6} ) q^{91} -3 q^{92} + ( -3 + 3 \zeta_{6} ) q^{94} -2 \zeta_{6} q^{95} -14 \zeta_{6} q^{97} + 3 \zeta_{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8} - q^{10} - 3 q^{11} + 2 q^{13} + 4 q^{14} - q^{16} + 6 q^{17} - 2 q^{19} - q^{20} - 3 q^{22} + 3 q^{23} + 2 q^{25} + 5 q^{26} - 2 q^{28} + 3 q^{29} + 10 q^{31} - q^{32} - 12 q^{34} - 2 q^{35} + 7 q^{37} + 4 q^{38} + 2 q^{40} + 6 q^{41} + q^{43} + 6 q^{44} + 3 q^{46} + 6 q^{47} + 3 q^{49} - q^{50} - 7 q^{52} + 12 q^{53} - 3 q^{55} - 2 q^{56} + 3 q^{58} - 9 q^{59} - 2 q^{61} - 5 q^{62} + 2 q^{64} + 2 q^{65} - 8 q^{67} + 6 q^{68} + 4 q^{70} - 12 q^{71} + 28 q^{73} + 7 q^{74} - 2 q^{76} + 12 q^{77} + 10 q^{79} - q^{80} + 6 q^{82} + 12 q^{83} + 6 q^{85} - 2 q^{86} - 3 q^{88} - 18 q^{89} - 14 q^{91} - 6 q^{92} - 3 q^{94} - 2 q^{95} - 14 q^{97} + 3 q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 0 −1.00000 1.73205i 1.00000 0 −0.500000 + 0.866025i
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 −1.00000 + 1.73205i 1.00000 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.i.d 2
3.b odd 2 1 390.2.i.c 2
13.c even 3 1 inner 1170.2.i.d 2
15.d odd 2 1 1950.2.i.n 2
15.e even 4 2 1950.2.z.k 4
39.h odd 6 1 5070.2.a.v 1
39.i odd 6 1 390.2.i.c 2
39.i odd 6 1 5070.2.a.j 1
39.k even 12 2 5070.2.b.j 2
195.x odd 6 1 1950.2.i.n 2
195.bl even 12 2 1950.2.z.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.c 2 3.b odd 2 1
390.2.i.c 2 39.i odd 6 1
1170.2.i.d 2 1.a even 1 1 trivial
1170.2.i.d 2 13.c even 3 1 inner
1950.2.i.n 2 15.d odd 2 1
1950.2.i.n 2 195.x odd 6 1
1950.2.z.k 4 15.e even 4 2
1950.2.z.k 4 195.bl even 12 2
5070.2.a.j 1 39.i odd 6 1
5070.2.a.v 1 39.h odd 6 1
5070.2.b.j 2 39.k even 12 2

Hecke kernels

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

\( T_{7}^{2} + 2 T_{7} + 4 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{29}^{2} - 3 T_{29} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 4 + 2 T + T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( 13 - 2 T + T^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( 9 - 3 T + T^{2} \)
$29$ \( 9 - 3 T + T^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 49 - 7 T + T^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( ( -3 + T )^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 81 + 9 T + T^{2} \)
$61$ \( 4 + 2 T + T^{2} \)
$67$ \( 64 + 8 T + T^{2} \)
$71$ \( 144 + 12 T + T^{2} \)
$73$ \( ( -14 + T )^{2} \)
$79$ \( ( -5 + T )^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 324 + 18 T + T^{2} \)
$97$ \( 196 + 14 T + T^{2} \)
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