Properties

Label 1170.2.i.o
Level $1170$
Weight $2$
Character orbit 1170.i
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

Character values

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

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
1.28078 + 2.21837i
−0.780776 1.35234i
1.28078 2.21837i
−0.780776 + 1.35234i
0.500000 0.866025i 0 −0.500000 0.866025i −1.00000 0 −1.78078 3.08440i −1.00000 0 −0.500000 + 0.866025i
451.2 0.500000 0.866025i 0 −0.500000 0.866025i −1.00000 0 0.280776 + 0.486319i −1.00000 0 −0.500000 + 0.866025i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 −1.78078 + 3.08440i −1.00000 0 −0.500000 0.866025i
991.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 0.280776 0.486319i −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.o 4
3.b odd 2 1 390.2.i.g 4
13.c even 3 1 inner 1170.2.i.o 4
15.d odd 2 1 1950.2.i.bi 4
15.e even 4 2 1950.2.z.n 8
39.h odd 6 1 5070.2.a.bb 2
39.i odd 6 1 390.2.i.g 4
39.i odd 6 1 5070.2.a.bi 2
39.k even 12 2 5070.2.b.r 4
195.x odd 6 1 1950.2.i.bi 4
195.bl even 12 2 1950.2.z.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g 4 3.b odd 2 1
390.2.i.g 4 39.i odd 6 1
1170.2.i.o 4 1.a even 1 1 trivial
1170.2.i.o 4 13.c even 3 1 inner
1950.2.i.bi 4 15.d odd 2 1
1950.2.i.bi 4 195.x odd 6 1
1950.2.z.n 8 15.e even 4 2
1950.2.z.n 8 195.bl even 12 2
5070.2.a.bb 2 39.h odd 6 1
5070.2.a.bi 2 39.i odd 6 1
5070.2.b.r 4 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 \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{4} + 3 T_{7}^{3} + 11 T_{7}^{2} - 6 T_{7} + 4 \)
\( T_{11}^{4} + 17 T_{11}^{2} + 289 \)
\( T_{29}^{4} - 9 T_{29}^{3} + 65 T_{29}^{2} - 144 T_{29} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 4 - 6 T + 11 T^{2} + 3 T^{3} + T^{4} \)
$11$ \( 289 + 17 T^{2} + T^{4} \)
$13$ \( 169 + 13 T - 12 T^{2} + T^{3} + T^{4} \)
$17$ \( 256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 4 + 6 T + 11 T^{2} - 3 T^{3} + T^{4} \)
$23$ \( 1296 - 108 T + 45 T^{2} + 3 T^{3} + T^{4} \)
$29$ \( 256 - 144 T + 65 T^{2} - 9 T^{3} + T^{4} \)
$31$ \( ( -38 + T + T^{2} )^{2} \)
$37$ \( 289 + 17 T^{2} + T^{4} \)
$41$ \( 2704 + 416 T + 116 T^{2} - 8 T^{3} + T^{4} \)
$43$ \( 4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4} \)
$47$ \( ( 7 + T )^{4} \)
$53$ \( ( 38 + 13 T + T^{2} )^{2} \)
$59$ \( 4624 - 1156 T + 221 T^{2} - 17 T^{3} + T^{4} \)
$61$ \( ( 36 + 6 T + T^{2} )^{2} \)
$67$ \( 1024 - 384 T + 176 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 4096 + 1152 T + 260 T^{2} + 18 T^{3} + T^{4} \)
$73$ \( ( -144 + 6 T + T^{2} )^{2} \)
$79$ \( ( 86 - 19 T + T^{2} )^{2} \)
$83$ \( ( -8 - 6 T + T^{2} )^{2} \)
$89$ \( 1156 - 578 T + 323 T^{2} + 17 T^{3} + T^{4} \)
$97$ \( 64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4} \)
show more
show less