Properties

Label 1170.2.e.h
Level $1170$
Weight $2$
Character orbit 1170.e
Analytic conductor $9.342$
Analytic rank $0$
Dimension $6$
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.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + (\beta_{4} - \beta_1) q^{7} - \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + (\beta_{4} - \beta_1) q^{7} - \beta_{4} q^{8} + ( - \beta_{3} + \beta_1) q^{10} + 2 q^{11} - \beta_{4} q^{13} + (\beta_{2} - 1) q^{14} + q^{16} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_1) q^{17} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{19} + (\beta_{5} - \beta_{2}) q^{20} + 2 \beta_{4} q^{22} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_1) q^{23} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 1) q^{25} + q^{26} + ( - \beta_{4} + \beta_1) q^{28} + ( - 2 \beta_{5} - 2 \beta_{3} - \beta_{2} - 5) q^{29} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} - 2) q^{31} + \beta_{4} q^{32} + ( - \beta_{5} - \beta_{3} + 3 \beta_{2} - 1) q^{34} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{35} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{37} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{38} + (\beta_{3} - \beta_1) q^{40} + ( - \beta_{5} - \beta_{3} + 4) q^{41} + (3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3}) q^{43} - 2 q^{44} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - 1) q^{46} + ( - 2 \beta_{4} - 2 \beta_1) q^{47} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 3) q^{49} + (\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 + 2) q^{50} + \beta_{4} q^{52} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 4 \beta_1) q^{53} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{55} + ( - \beta_{2} + 1) q^{56} + (2 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - \beta_1) q^{58} + (2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 4) q^{59} + ( - 2 \beta_{5} - 2 \beta_{3} + 6 \beta_{2}) q^{61} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_1) q^{62} - q^{64} + (\beta_{3} - \beta_1) q^{65} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{67} + (\beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_1) q^{68} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 2) q^{70} + ( - 3 \beta_{5} - 3 \beta_{3} + 2 \beta_{2}) q^{71} + (3 \beta_{4} - 3 \beta_1) q^{73} + (2 \beta_{5} + 2 \beta_{3} - 2) q^{74} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{76} + (2 \beta_{4} - 2 \beta_1) q^{77} + ( - 5 \beta_{5} - 5 \beta_{3} + 2 \beta_{2}) q^{79} + ( - \beta_{5} + \beta_{2}) q^{80} + (\beta_{5} + 4 \beta_{4} - \beta_{3}) q^{82} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 4 \beta_1) q^{83} + ( - 2 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{85} + (3 \beta_{5} + 3 \beta_{3} + 2) q^{86} - 2 \beta_{4} q^{88} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 8) q^{89} + ( - \beta_{2} + 1) q^{91} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_1) q^{92} + (2 \beta_{2} + 2) q^{94} + (2 \beta_{5} - 3 \beta_{4} + \beta_1 + 4) q^{95} + (9 \beta_{4} - \beta_1) q^{97} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 2 q^{10} + 12 q^{11} - 8 q^{14} + 6 q^{16} - 4 q^{19} + 2 q^{25} + 6 q^{26} - 20 q^{29} - 12 q^{31} - 8 q^{34} + 12 q^{35} - 2 q^{40} + 28 q^{41} - 12 q^{44} + 10 q^{49} + 8 q^{50} + 8 q^{56} - 28 q^{59} - 4 q^{61} - 6 q^{64} - 2 q^{65} + 12 q^{70} + 8 q^{71} - 20 q^{74} + 4 q^{76} + 16 q^{79} + 4 q^{85} - 44 q^{89} + 8 q^{91} + 8 q^{94} + 20 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1
0.403032 0.403032i
−0.854638 + 0.854638i
1.45161 1.45161i
0.403032 + 0.403032i
−0.854638 0.854638i
1.45161 + 1.45161i
1.00000i 0 −1.00000 −1.67513 1.48119i 0 1.19394i 1.00000i 0 −1.48119 + 1.67513i
469.2 1.00000i 0 −1.00000 −0.539189 + 2.17009i 0 3.70928i 1.00000i 0 2.17009 + 0.539189i
469.3 1.00000i 0 −1.00000 2.21432 + 0.311108i 0 0.903212i 1.00000i 0 0.311108 2.21432i
469.4 1.00000i 0 −1.00000 −1.67513 + 1.48119i 0 1.19394i 1.00000i 0 −1.48119 1.67513i
469.5 1.00000i 0 −1.00000 −0.539189 2.17009i 0 3.70928i 1.00000i 0 2.17009 0.539189i
469.6 1.00000i 0 −1.00000 2.21432 0.311108i 0 0.903212i 1.00000i 0 0.311108 + 2.21432i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.e.h yes 6
3.b odd 2 1 1170.2.e.g 6
5.b even 2 1 inner 1170.2.e.h yes 6
5.c odd 4 1 5850.2.a.co 3
5.c odd 4 1 5850.2.a.ct 3
15.d odd 2 1 1170.2.e.g 6
15.e even 4 1 5850.2.a.cq 3
15.e even 4 1 5850.2.a.cr 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.e.g 6 3.b odd 2 1
1170.2.e.g 6 15.d odd 2 1
1170.2.e.h yes 6 1.a even 1 1 trivial
1170.2.e.h yes 6 5.b even 2 1 inner
5850.2.a.co 3 5.c odd 4 1
5850.2.a.cq 3 15.e even 4 1
5850.2.a.cr 3 15.e even 4 1
5850.2.a.ct 3 5.c odd 4 1

Hecke kernels

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

\( T_{7}^{6} + 16T_{7}^{4} + 32T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} - 16 T^{3} - 5 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 16 T^{4} + 32 T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T - 2)^{6} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 72 T^{4} + 1712 T^{2} + \cdots + 13456 \) Copy content Toggle raw display
$19$ \( (T^{3} + 2 T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 80 T^{4} + 1600 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$29$ \( (T^{3} + 10 T^{2} - 44 T - 388)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 6 T^{2} - 4 T - 40)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 140 T^{4} + 3120 T^{2} + \cdots + 18496 \) Copy content Toggle raw display
$41$ \( (T^{3} - 14 T^{2} + 52 T - 40)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 240 T^{4} + 14400 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{6} + 272 T^{4} + 18496 T^{2} + \cdots + 246016 \) Copy content Toggle raw display
$59$ \( (T^{3} + 14 T^{2} + 28 T - 152)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 2 T^{2} - 132 T - 680)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 140 T^{4} + 2608 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( (T^{3} - 4 T^{2} - 80 T + 400)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 144 T^{4} + 2592 T^{2} + \cdots + 11664 \) Copy content Toggle raw display
$79$ \( (T^{3} - 8 T^{2} - 240 T + 1712)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 128 T^{4} + 2496 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$89$ \( (T^{3} + 22 T^{2} + 124 T + 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 272 T^{4} + 22752 T^{2} + \cdots + 583696 \) Copy content Toggle raw display
show more
show less