Properties

Label 1170.2.bj.b
Level $1170$
Weight $2$
Character orbit 1170.bj
Analytic conductor $9.342$
Analytic rank $0$
Dimension $8$
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.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.50027374224.1
Defining polynomial: \( x^{8} + 20x^{6} + 132x^{4} + 332x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{3} - 1) q^{4} - \beta_{6} q^{5} + (\beta_{7} - \beta_{4} - \beta_{3} + 2) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{3} - 1) q^{4} - \beta_{6} q^{5} + (\beta_{7} - \beta_{4} - \beta_{3} + 2) q^{7} - q^{8} + (\beta_{4} - \beta_{3}) q^{10} + (\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{11} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{13} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{14} - \beta_{3} q^{16} + ( - \beta_{7} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{17} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{19} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{20} + ( - \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2}) q^{22} + (\beta_{7} - \beta_{6} - \beta_{4} + 2) q^{23} + (\beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{2} + 1) q^{25} + (\beta_{6} - \beta_{5} + \beta_1 - 1) q^{26} + ( - \beta_{6} - \beta_{5} + \beta_{3}) q^{28} + ( - \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{29} + (\beta_{5} - \beta_{4} - \beta_1 + 1) q^{31} + ( - \beta_{3} + 1) q^{32} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} - \beta_1 - 1) q^{34} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 + 5) q^{35} + (5 \beta_{3} + \beta_{2} + \beta_1) q^{37} + ( - \beta_{7} - \beta_{6} + 2 \beta_{3} - \beta_1 - 1) q^{38} + \beta_{6} q^{40} + (2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2) q^{43} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_1) q^{44} + (\beta_{7} - \beta_{6} - \beta_{5} + 1) q^{46} + (\beta_{7} - \beta_{6} - 3 \beta_{5} - 3 \beta_{4}) q^{47} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{49} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1) q^{50} + ( - \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{52} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 6 \beta_{3} - \beta_1 - 4) q^{53} + (\beta_{7} + \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 - 4) q^{55} + ( - \beta_{7} + \beta_{4} + \beta_{3} - 2) q^{56} + ( - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{58} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2) q^{59} + (\beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{2} + 6 \beta_1 - 2) q^{61} + (\beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{62} + q^{64} + ( - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + 7 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{65} - 4 \beta_{3} q^{67} + ( - \beta_{6} + \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{68} + (\beta_{7} - \beta_{6} + 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{70} - 4 \beta_{2} q^{71} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{73} + (5 \beta_{3} - \beta_{2} + 2 \beta_1 - 5) q^{74} + ( - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{76} + ( - \beta_{7} - \beta_{6} + 2 \beta_{3} - 3 \beta_1 - 1) q^{77} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 - 1) q^{79} + ( - \beta_{4} + \beta_{3}) q^{80} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{82} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + \beta_1 + 6) q^{83} + (2 \beta_{7} - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{85} + ( - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{3} - 2 \beta_1 - 2) q^{86} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{88} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + 4) q^{89} + (3 \beta_{7} - 4 \beta_{5} - 3 \beta_{3} - 3 \beta_{2} - \beta_1 + 3) q^{91} + ( - \beta_{5} + \beta_{4} - 1) q^{92} + ( - 2 \beta_{7} - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{94} + (3 \beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{95} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} - 2 \beta_1 - 1) q^{97} + (2 \beta_{7} - \beta_{6} + \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 3 q^{5} + 5 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} - 3 q^{5} + 5 q^{7} - 8 q^{8} + 3 q^{11} + 4 q^{13} + 10 q^{14} - 4 q^{16} + 15 q^{17} + 9 q^{19} + 3 q^{20} + 3 q^{22} + 6 q^{23} + 5 q^{25} - q^{26} + 5 q^{28} + 3 q^{29} + 4 q^{32} + 33 q^{35} + 20 q^{37} + 3 q^{40} - 21 q^{41} + 18 q^{43} + 6 q^{46} - 6 q^{47} - 15 q^{49} + q^{50} - 5 q^{52} - 23 q^{55} - 5 q^{56} - 3 q^{58} - 30 q^{59} - 5 q^{61} + 6 q^{62} + 8 q^{64} + 6 q^{65} - 16 q^{67} - 15 q^{68} + 18 q^{70} + 26 q^{73} - 20 q^{74} - 9 q^{76} + 4 q^{79} - 21 q^{82} + 48 q^{83} - 34 q^{85} - 3 q^{88} + 39 q^{89} + 19 q^{91} - 3 q^{94} - 9 q^{95} - 4 q^{97} + 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 20x^{6} + 132x^{4} + 332x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 10\nu^{2} + 2\nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 20\nu^{5} + 116\nu^{3} + 172\nu + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 16\nu^{5} + 68\nu^{3} + 8\nu^{2} + 60\nu + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 16\nu^{5} - 68\nu^{3} + 8\nu^{2} - 60\nu + 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 8\nu^{6} + 20\nu^{5} + 128\nu^{4} + 132\nu^{3} + 544\nu^{2} + 268\nu + 544 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 8\nu^{6} + 20\nu^{5} - 128\nu^{4} + 132\nu^{3} - 544\nu^{2} + 268\nu - 544 ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 2\beta_{6} - 4\beta_{3} - 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{5} - 10\beta_{4} + 4\beta_{2} - 2\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -24\beta_{7} - 24\beta_{6} + 2\beta_{5} - 2\beta_{4} + 64\beta_{3} + 44\beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{7} + 4\beta_{6} + 92\beta_{5} + 92\beta_{4} - 64\beta_{2} + 32\beta _1 - 272 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 248\beta_{7} + 248\beta_{6} - 40\beta_{5} + 40\beta_{4} - 752\beta_{3} - 356\beta _1 + 336 \) Copy content Toggle raw display

Character values

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

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(1 - \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
2.40987i
3.07108i
1.17644i
1.83766i
2.40987i
3.07108i
1.17644i
1.83766i
0.500000 0.866025i 0 −0.500000 0.866025i −2.10653 + 0.750022i 0 −0.702803 1.21729i −1.00000 0 −0.403726 + 2.19932i
199.2 0.500000 0.866025i 0 −0.500000 0.866025i −1.36822 1.76861i 0 1.84755 + 3.20005i −1.00000 0 −2.21577 + 0.300609i
199.3 0.500000 0.866025i 0 −0.500000 0.866025i −0.235468 + 2.22364i 0 −1.04346 1.80732i −1.00000 0 1.80799 + 1.31574i
199.4 0.500000 0.866025i 0 −0.500000 0.866025i 2.21022 0.339024i 0 2.39871 + 4.15469i −1.00000 0 0.811505 2.08362i
829.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.10653 0.750022i 0 −0.702803 + 1.21729i −1.00000 0 −0.403726 2.19932i
829.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.36822 + 1.76861i 0 1.84755 3.20005i −1.00000 0 −2.21577 0.300609i
829.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.235468 2.22364i 0 −1.04346 + 1.80732i −1.00000 0 1.80799 1.31574i
829.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.21022 + 0.339024i 0 2.39871 4.15469i −1.00000 0 0.811505 + 2.08362i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l 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.bj.b 8
3.b odd 2 1 130.2.m.a 8
5.b even 2 1 1170.2.bj.a 8
12.b even 2 1 1040.2.df.c 8
13.e even 6 1 1170.2.bj.a 8
15.d odd 2 1 130.2.m.b yes 8
15.e even 4 2 650.2.m.e 16
39.h odd 6 1 130.2.m.b yes 8
39.h odd 6 1 1690.2.c.e 8
39.i odd 6 1 1690.2.c.f 8
39.k even 12 2 1690.2.b.e 16
60.h even 2 1 1040.2.df.a 8
65.l even 6 1 inner 1170.2.bj.b 8
156.r even 6 1 1040.2.df.a 8
195.x odd 6 1 1690.2.c.e 8
195.y odd 6 1 130.2.m.a 8
195.y odd 6 1 1690.2.c.f 8
195.bc odd 12 2 8450.2.a.cr 8
195.bf even 12 2 650.2.m.e 16
195.bh even 12 2 1690.2.b.e 16
195.bn odd 12 2 8450.2.a.cs 8
780.cb even 6 1 1040.2.df.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.m.a 8 3.b odd 2 1
130.2.m.a 8 195.y odd 6 1
130.2.m.b yes 8 15.d odd 2 1
130.2.m.b yes 8 39.h odd 6 1
650.2.m.e 16 15.e even 4 2
650.2.m.e 16 195.bf even 12 2
1040.2.df.a 8 60.h even 2 1
1040.2.df.a 8 156.r even 6 1
1040.2.df.c 8 12.b even 2 1
1040.2.df.c 8 780.cb even 6 1
1170.2.bj.a 8 5.b even 2 1
1170.2.bj.a 8 13.e even 6 1
1170.2.bj.b 8 1.a even 1 1 trivial
1170.2.bj.b 8 65.l even 6 1 inner
1690.2.b.e 16 39.k even 12 2
1690.2.b.e 16 195.bh even 12 2
1690.2.c.e 8 39.h odd 6 1
1690.2.c.e 8 195.x odd 6 1
1690.2.c.f 8 39.i odd 6 1
1690.2.c.f 8 195.y odd 6 1
8450.2.a.cr 8 195.bc odd 12 2
8450.2.a.cs 8 195.bn odd 12 2

Hecke kernels

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

\( T_{7}^{8} - 5T_{7}^{7} + 34T_{7}^{6} - 29T_{7}^{5} + 214T_{7}^{4} + 187T_{7}^{3} + 1837T_{7}^{2} + 1924T_{7} + 2704 \) Copy content Toggle raw display
\( T_{17}^{8} - 15T_{17}^{7} + 83T_{17}^{6} - 120T_{17}^{5} - 144T_{17}^{4} + 336T_{17}^{3} + 572T_{17}^{2} - 84T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{7} + 2 T^{6} - 15 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 5 T^{7} + 34 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} - 16 T^{6} + 57 T^{5} + \cdots + 784 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + 28 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 15 T^{7} + 83 T^{6} - 120 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{8} - 9 T^{7} + 18 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} - 4 T^{6} + 96 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{8} - 3 T^{7} + 27 T^{6} - 66 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( T^{8} + 60 T^{6} + 1176 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$37$ \( T^{8} - 20 T^{7} + 280 T^{6} + \cdots + 11881 \) Copy content Toggle raw display
$41$ \( T^{8} + 21 T^{7} + 119 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$43$ \( T^{8} - 18 T^{7} + 72 T^{6} + \cdots + 2359296 \) Copy content Toggle raw display
$47$ \( (T^{4} + 3 T^{3} - 117 T^{2} - 267 T - 96)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 356 T^{6} + \cdots + 17783089 \) Copy content Toggle raw display
$59$ \( T^{8} + 30 T^{7} + 344 T^{6} + \cdots + 369664 \) Copy content Toggle raw display
$61$ \( T^{8} + 5 T^{7} + 271 T^{6} + \cdots + 28751044 \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 160 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
$73$ \( (T^{4} - 13 T^{3} - 114 T^{2} + 1298 T - 1406)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} - 216 T^{2} + 502 T + 1384)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 24 T^{3} + 78 T^{2} + 1602 T - 9744)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 39 T^{7} + 548 T^{6} + \cdots + 59474944 \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{7} + 76 T^{6} + \cdots + 327184 \) Copy content Toggle raw display
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