Properties

Label 1764.2.b.j
Level $1764$
Weight $2$
Character orbit 1764.b
Analytic conductor $14.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.562828176.1
Defining polynomial: \(x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + \beta_{6} q^{4} + ( \beta_{2} - \beta_{6} ) q^{5} + ( -1 + \beta_{3} + \beta_{7} ) q^{8} +O(q^{10})\) \( q -\beta_{4} q^{2} + \beta_{6} q^{4} + ( \beta_{2} - \beta_{6} ) q^{5} + ( -1 + \beta_{3} + \beta_{7} ) q^{8} + ( 1 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{10} + ( -\beta_{4} + \beta_{5} - \beta_{7} ) q^{11} + ( -\beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{13} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{16} + ( \beta_{1} + 2 \beta_{4} ) q^{17} + ( -1 - \beta_{2} - \beta_{6} ) q^{19} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{20} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{22} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{23} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{25} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{26} + ( 2 + 2 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{29} + ( 3 - 2 \beta_{1} - \beta_{2} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{31} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{32} + ( 4 - 2 \beta_{6} ) q^{34} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{37} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{38} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{40} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{43} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{44} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{46} + ( -2 + \beta_{1} - 2 \beta_{4} ) q^{47} + ( -1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{50} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{52} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{53} + ( 2 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{55} + ( -3 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{58} + ( 2 - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{59} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{61} + ( -6 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{62} + ( -1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{64} + ( 2 - \beta_{1} + 2 \beta_{4} ) q^{65} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{67} + ( 2 - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{7} ) q^{68} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{73} + ( -5 + \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{74} + ( -5 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{76} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + ( 1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{80} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - 4 \beta_{7} ) q^{82} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{83} + ( -4 - 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{85} + ( -1 + 2 \beta_{1} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{86} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{6} ) q^{89} + ( 4 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{92} + ( 4 + 2 \beta_{4} + 2 \beta_{6} ) q^{94} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{95} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} + 2q^{4} - 4q^{8} + O(q^{10}) \) \( 8q + 2q^{2} + 2q^{4} - 4q^{8} + 8q^{10} + 10q^{16} - 12q^{19} + 22q^{20} - 6q^{22} - 4q^{25} - 6q^{26} + 16q^{29} + 12q^{31} + 12q^{32} + 28q^{34} - 12q^{37} - 2q^{38} - 4q^{40} - 4q^{44} - 12q^{46} - 8q^{47} - 2q^{50} - 4q^{52} - 8q^{53} + 8q^{55} - 14q^{58} + 28q^{59} - 48q^{62} + 2q^{64} + 8q^{65} + 16q^{68} - 38q^{74} - 44q^{76} + 6q^{80} + 4q^{82} + 4q^{83} - 32q^{85} - 6q^{86} + 26q^{88} + 28q^{92} + 32q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + \nu^{5} + 4 \nu^{4} - 6 \nu^{3} + 4 \nu - 16 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + \nu^{5} + 2 \nu^{4} - 6 \nu^{3} + 4 \nu^{2} + 4 \nu - 16 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + \nu^{5} + 3 \nu^{4} - 2 \nu^{3} + 2 \nu^{2} + 4 \nu - 12 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + \nu^{5} - 3 \nu^{4} + 4 \nu^{3} + 2 \nu^{2} - 8 \nu + 12 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{7} + \nu^{6} - 3 \nu^{4} + 6 \nu^{3} + 2 \nu^{2} - 4 \nu + 20 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{5} - \beta_{4} - \beta_{3} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{1} + 1\)
\(\nu^{5}\)\(=\)\(2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(-\beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_{1} - 1\)
\(\nu^{7}\)\(=\)\(-4 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \beta_{1} + 5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−1.33790 + 0.458297i
−1.33790 0.458297i
0.0777157 + 1.41208i
0.0777157 1.41208i
0.856419 + 1.12541i
0.856419 1.12541i
1.40376 + 0.171630i
1.40376 0.171630i
−1.33790 0.458297i 0 1.57993 + 1.22631i 2.45262i 0 0 −1.55176 2.36475i 0 −1.12403 + 3.28134i
1567.2 −1.33790 + 0.458297i 0 1.57993 1.22631i 2.45262i 0 0 −1.55176 + 2.36475i 0 −1.12403 3.28134i
1567.3 0.0777157 1.41208i 0 −1.98792 0.219481i 0.438962i 0 0 −0.464416 + 2.79004i 0 0.619848 + 0.0341142i
1567.4 0.0777157 + 1.41208i 0 −1.98792 + 0.219481i 0.438962i 0 0 −0.464416 2.79004i 0 0.619848 0.0341142i
1567.5 0.856419 1.12541i 0 −0.533092 1.92764i 3.85529i 0 0 −2.62594 1.05092i 0 4.33878 + 3.30174i
1567.6 0.856419 + 1.12541i 0 −0.533092 + 1.92764i 3.85529i 0 0 −2.62594 + 1.05092i 0 4.33878 3.30174i
1567.7 1.40376 0.171630i 0 1.94109 0.481855i 0.963711i 0 0 2.64212 1.00956i 0 0.165402 + 1.35282i
1567.8 1.40376 + 0.171630i 0 1.94109 + 0.481855i 0.963711i 0 0 2.64212 + 1.00956i 0 0.165402 1.35282i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.b.j 8
3.b odd 2 1 588.2.b.a 8
4.b odd 2 1 1764.2.b.i 8
7.b odd 2 1 1764.2.b.i 8
7.c even 3 1 252.2.bf.f 8
7.d odd 6 1 252.2.bf.g 8
12.b even 2 1 588.2.b.b 8
21.c even 2 1 588.2.b.b 8
21.g even 6 1 84.2.o.a 8
21.g even 6 1 588.2.o.b 8
21.h odd 6 1 84.2.o.b yes 8
21.h odd 6 1 588.2.o.d 8
28.d even 2 1 inner 1764.2.b.j 8
28.f even 6 1 252.2.bf.f 8
28.g odd 6 1 252.2.bf.g 8
84.h odd 2 1 588.2.b.a 8
84.j odd 6 1 84.2.o.b yes 8
84.j odd 6 1 588.2.o.d 8
84.n even 6 1 84.2.o.a 8
84.n even 6 1 588.2.o.b 8
168.s odd 6 1 1344.2.bl.i 8
168.v even 6 1 1344.2.bl.j 8
168.ba even 6 1 1344.2.bl.j 8
168.be odd 6 1 1344.2.bl.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.o.a 8 21.g even 6 1
84.2.o.a 8 84.n even 6 1
84.2.o.b yes 8 21.h odd 6 1
84.2.o.b yes 8 84.j odd 6 1
252.2.bf.f 8 7.c even 3 1
252.2.bf.f 8 28.f even 6 1
252.2.bf.g 8 7.d odd 6 1
252.2.bf.g 8 28.g odd 6 1
588.2.b.a 8 3.b odd 2 1
588.2.b.a 8 84.h odd 2 1
588.2.b.b 8 12.b even 2 1
588.2.b.b 8 21.c even 2 1
588.2.o.b 8 21.g even 6 1
588.2.o.b 8 84.n even 6 1
588.2.o.d 8 21.h odd 6 1
588.2.o.d 8 84.j odd 6 1
1344.2.bl.i 8 168.s odd 6 1
1344.2.bl.i 8 168.be odd 6 1
1344.2.bl.j 8 168.v even 6 1
1344.2.bl.j 8 168.ba even 6 1
1764.2.b.i 8 4.b odd 2 1
1764.2.b.i 8 7.b odd 2 1
1764.2.b.j 8 1.a even 1 1 trivial
1764.2.b.j 8 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{8} + 22 T_{5}^{6} + 113 T_{5}^{4} + 104 T_{5}^{2} + 16 \)
\( T_{11}^{8} + 38 T_{11}^{6} + 257 T_{11}^{4} + 568 T_{11}^{2} + 400 \)
\( T_{19}^{4} + 6 T_{19}^{3} - 7 T_{19}^{2} - 60 T_{19} + 4 \)
\( T_{29}^{4} - 8 T_{29}^{3} - 45 T_{29}^{2} + 352 T_{29} - 512 \)
\( T_{53}^{4} + 4 T_{53}^{3} - 61 T_{53}^{2} + 116 T_{53} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 16 T + 4 T^{2} + 4 T^{3} - 6 T^{4} + 2 T^{5} + T^{6} - 2 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 16 + 104 T^{2} + 113 T^{4} + 22 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 400 + 568 T^{2} + 257 T^{4} + 38 T^{6} + T^{8} \)
$13$ \( 256 + 1936 T^{2} + 473 T^{4} + 38 T^{6} + T^{8} \)
$17$ \( 1024 + 2560 T^{2} + 848 T^{4} + 56 T^{6} + T^{8} \)
$19$ \( ( 4 - 60 T - 7 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$23$ \( 16384 + 13312 T^{2} + 1856 T^{4} + 80 T^{6} + T^{8} \)
$29$ \( ( -512 + 352 T - 45 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$31$ \( ( 2043 + 270 T - 84 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$37$ \( ( -596 - 360 T - 43 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$41$ \( 350464 + 311552 T^{2} + 14048 T^{4} + 208 T^{6} + T^{8} \)
$43$ \( 1073296 + 140152 T^{2} + 6593 T^{4} + 134 T^{6} + T^{8} \)
$47$ \( ( 64 - 16 T - 28 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$53$ \( ( -8 + 116 T - 61 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$59$ \( ( 1192 + 460 T - 27 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$61$ \( 1048576 + 557056 T^{2} + 22784 T^{4} + 272 T^{6} + T^{8} \)
$67$ \( 4129024 + 680416 T^{2} + 21809 T^{4} + 254 T^{6} + T^{8} \)
$71$ \( 200704 + 173312 T^{2} + 19856 T^{4} + 280 T^{6} + T^{8} \)
$73$ \( 952576 + 338912 T^{2} + 14681 T^{4} + 214 T^{6} + T^{8} \)
$79$ \( 241081 + 198820 T^{2} + 39950 T^{4} + 404 T^{6} + T^{8} \)
$83$ \( ( 196 - 304 T - 103 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$89$ \( 4096 + 15872 T^{2} + 8336 T^{4} + 184 T^{6} + T^{8} \)
$97$ \( 246016 + 86176 T^{2} + 8249 T^{4} + 182 T^{6} + T^{8} \)
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