Properties

Label 1764.2.b.i
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.

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

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

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + T^{2} + 2 T^{3} - 6 T^{4} + 4 T^{5} + 4 T^{6} - 16 T^{7} + 16 T^{8} \)
$3$ 1
$5$ \( 1 - 18 T^{2} + 153 T^{4} - 906 T^{6} + 4676 T^{8} - 22650 T^{10} + 95625 T^{12} - 281250 T^{14} + 390625 T^{16} \)
$7$ 1
$11$ \( ( 1 - 10 T + 25 T^{2} + 94 T^{3} - 684 T^{4} + 1034 T^{5} + 3025 T^{6} - 13310 T^{7} + 14641 T^{8} )( 1 + 10 T + 25 T^{2} - 94 T^{3} - 684 T^{4} - 1034 T^{5} + 3025 T^{6} + 13310 T^{7} + 14641 T^{8} ) \)
$13$ \( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 8342178 T^{10} + 64005201 T^{12} - 318569394 T^{14} + 815730721 T^{16} \)
$17$ \( 1 - 80 T^{2} + 3228 T^{4} - 87472 T^{6} + 1728326 T^{8} - 25279408 T^{10} + 269605788 T^{12} - 1931005520 T^{14} + 6975757441 T^{16} \)
$19$ \( ( 1 - 6 T + 69 T^{2} - 282 T^{3} + 1904 T^{4} - 5358 T^{5} + 24909 T^{6} - 41154 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( 1 - 104 T^{2} + 5628 T^{4} - 203992 T^{6} + 5416646 T^{8} - 107911768 T^{10} + 1574945148 T^{12} - 15395732456 T^{14} + 78310985281 T^{16} \)
$29$ \( ( 1 - 8 T + 71 T^{2} - 344 T^{3} + 1924 T^{4} - 9976 T^{5} + 59711 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 + 6 T + 40 T^{2} + 288 T^{3} + 2601 T^{4} + 8928 T^{5} + 38440 T^{6} + 178746 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 6 T + 105 T^{2} + 306 T^{3} + 4436 T^{4} + 11322 T^{5} + 143745 T^{6} + 303918 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 1020662856 T^{10} + 28110670428 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} \)
$43$ \( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 3198300354 T^{10} + 81343532193 T^{12} - 1327486240290 T^{14} + 11688200277601 T^{16} \)
$47$ \( ( 1 - 4 T + 160 T^{2} - 548 T^{3} + 10686 T^{4} - 25756 T^{5} + 353440 T^{6} - 415292 T^{7} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 4 T + 151 T^{2} + 752 T^{3} + 10380 T^{4} + 39856 T^{5} + 424159 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 14 T + 209 T^{2} + 2018 T^{3} + 18892 T^{4} + 119062 T^{5} + 727529 T^{6} + 2875306 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( 1 - 216 T^{2} + 27420 T^{4} - 2531496 T^{6} + 176195558 T^{8} - 9419696616 T^{10} + 379652960220 T^{12} - 11128400861976 T^{14} + 191707312997281 T^{16} \)
$67$ \( 1 - 282 T^{2} + 45393 T^{4} - 4904034 T^{6} + 383059316 T^{8} - 22014208626 T^{10} + 914719835553 T^{12} - 25509263771658 T^{14} + 406067677556641 T^{16} \)
$71$ \( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 21860840928 T^{10} + 1060276978044 T^{12} - 36892881769248 T^{14} + 645753531245761 T^{16} \)
$73$ \( 1 - 370 T^{2} + 70161 T^{4} - 8626802 T^{6} + 743765828 T^{8} - 45972227858 T^{10} + 1992448986801 T^{12} - 55993663726930 T^{14} + 806460091894081 T^{16} \)
$79$ \( 1 - 228 T^{2} + 23202 T^{4} - 2215104 T^{6} + 207545771 T^{8} - 13824464064 T^{10} + 903719779362 T^{12} - 55423939858788 T^{14} + 1517108809906561 T^{16} \)
$83$ \( ( 1 + 2 T + 229 T^{2} + 802 T^{3} + 24432 T^{4} + 66566 T^{5} + 1577581 T^{6} + 1143574 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( 1 - 528 T^{2} + 131868 T^{4} - 20568048 T^{6} + 2191026566 T^{8} - 162919508208 T^{10} + 8273693836188 T^{12} - 262406121627408 T^{14} + 3936588805702081 T^{16} \)
$97$ \( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 268509845586 T^{10} + 14676118616337 T^{12} - 494785370927826 T^{14} + 7837433594376961 T^{16} \)
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