Properties

Label 1764.2.b.k
Level 1764
Weight 2
Character orbit 1764.b
Analytic conductor 14.086
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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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: 8.0.1212153856.10
Defining polynomial: \(x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 196)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{2} + ( 1 + \beta_{3} + \beta_{5} - \beta_{6} ) q^{4} + ( \beta_{2} + \beta_{7} ) q^{5} + ( 1 - \beta_{3} - 2 \beta_{6} ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{2} + ( 1 + \beta_{3} + \beta_{5} - \beta_{6} ) q^{4} + ( \beta_{2} + \beta_{7} ) q^{5} + ( 1 - \beta_{3} - 2 \beta_{6} ) q^{8} + ( \beta_{1} + \beta_{2} - \beta_{4} ) q^{10} + ( 1 - \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{11} + ( \beta_{2} - \beta_{7} ) q^{13} + ( -1 - \beta_{3} - 3 \beta_{5} - \beta_{6} ) q^{16} + ( -2 \beta_{2} + \beta_{7} ) q^{17} + ( -\beta_{2} + 2 \beta_{4} ) q^{19} + ( \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{7} ) q^{20} + ( -2 \beta_{3} - 3 \beta_{5} - \beta_{6} ) q^{22} + ( 2 - 4 \beta_{6} ) q^{23} + ( 3 + 2 \beta_{3} + 2 \beta_{5} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{26} + ( 2 - 2 \beta_{3} - 2 \beta_{5} ) q^{29} + ( 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{7} ) q^{31} + ( -5 + \beta_{3} - 2 \beta_{5} ) q^{32} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{34} -4 q^{37} + ( \beta_{1} + \beta_{2} + 4 \beta_{7} ) q^{38} + ( \beta_{1} - \beta_{2} + 3 \beta_{4} + 2 \beta_{7} ) q^{40} -3 \beta_{2} q^{41} + ( 1 + 3 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{43} + ( -4 + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{44} + ( -2 - 2 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} ) q^{46} + ( -4 \beta_{1} - 2 \beta_{7} ) q^{47} + ( 5 + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{50} + ( 3 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{7} ) q^{52} + ( 4 + 6 \beta_{3} + 6 \beta_{5} ) q^{53} + ( -2 \beta_{2} + 4 \beta_{4} ) q^{55} + ( 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{58} + ( 4 \beta_{1} - 3 \beta_{2} + 6 \beta_{4} + 2 \beta_{7} ) q^{59} + ( 5 \beta_{2} - 5 \beta_{7} ) q^{61} + ( 6 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 8 \beta_{7} ) q^{62} + ( -7 - 3 \beta_{3} + \beta_{5} - \beta_{6} ) q^{64} + ( -2 - 2 \beta_{3} - 2 \beta_{5} ) q^{65} + ( -2 - 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{67} + ( -5 \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{7} ) q^{68} + ( -\beta_{2} + 8 \beta_{7} ) q^{73} + ( -4 - 4 \beta_{3} ) q^{74} + ( 2 \beta_{1} - 3 \beta_{4} ) q^{76} + ( 2 - 2 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} ) q^{79} + ( 3 \beta_{1} + \beta_{2} - \beta_{4} + 6 \beta_{7} ) q^{80} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{82} + ( -4 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{7} ) q^{83} + ( 4 + 2 \beta_{3} + 2 \beta_{5} ) q^{85} + ( -4 + 2 \beta_{3} + \beta_{5} - 5 \beta_{6} ) q^{86} + ( -6 + 3 \beta_{5} - \beta_{6} ) q^{88} + ( 3 \beta_{2} + 2 \beta_{7} ) q^{89} + ( -10 - 2 \beta_{3} - 6 \beta_{5} - 2 \beta_{6} ) q^{92} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{94} + ( -2 + 4 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} ) q^{95} -5 \beta_{7} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} - 4q^{4} + 4q^{8} + O(q^{10}) \) \( 8q + 4q^{2} - 4q^{4} + 4q^{8} + 4q^{16} + 16q^{22} + 8q^{25} + 32q^{29} - 36q^{32} - 32q^{37} - 24q^{44} - 8q^{46} + 20q^{50} - 16q^{53} - 52q^{64} - 16q^{74} + 16q^{85} - 64q^{86} - 64q^{88} - 56q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{3} \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} - 2 \nu^{2} \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 2 \nu^{3} \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{2} + 2 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{4} + 6 \nu^{2} - 4 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 8 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{4} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}\)
\(\nu^{5}\)\(=\)\(2 \beta_{4} + 2 \beta_{2}\)
\(\nu^{6}\)\(=\)\(2 \beta_{6} + 4 \beta_{5} - 2 \beta_{3} - 2\)
\(\nu^{7}\)\(=\)\(2 \beta_{7} - 2 \beta_{4} + 8 \beta_{2} - 2 \beta_{1}\)

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
−1.07072 + 0.923880i
1.07072 0.923880i
1.07072 + 0.923880i
−1.07072 0.923880i
−1.36145 0.382683i
1.36145 + 0.382683i
1.36145 0.382683i
−1.36145 + 0.382683i
−0.207107 1.39897i 0 −1.91421 + 0.579471i 2.61313i 0 0 1.20711 + 2.55791i 0 −3.65568 + 0.541196i
1567.2 −0.207107 1.39897i 0 −1.91421 + 0.579471i 2.61313i 0 0 1.20711 + 2.55791i 0 3.65568 0.541196i
1567.3 −0.207107 + 1.39897i 0 −1.91421 0.579471i 2.61313i 0 0 1.20711 2.55791i 0 3.65568 + 0.541196i
1567.4 −0.207107 + 1.39897i 0 −1.91421 0.579471i 2.61313i 0 0 1.20711 2.55791i 0 −3.65568 0.541196i
1567.5 1.20711 0.736813i 0 0.914214 1.77882i 1.08239i 0 0 −0.207107 2.82083i 0 −0.797521 1.30656i
1567.6 1.20711 0.736813i 0 0.914214 1.77882i 1.08239i 0 0 −0.207107 2.82083i 0 0.797521 + 1.30656i
1567.7 1.20711 + 0.736813i 0 0.914214 + 1.77882i 1.08239i 0 0 −0.207107 + 2.82083i 0 0.797521 1.30656i
1567.8 1.20711 + 0.736813i 0 0.914214 + 1.77882i 1.08239i 0 0 −0.207107 + 2.82083i 0 −0.797521 + 1.30656i
\(n\): 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
4.b odd 2 1 inner
7.b odd 2 1 inner
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.k 8
3.b odd 2 1 196.2.d.c 8
4.b odd 2 1 inner 1764.2.b.k 8
7.b odd 2 1 inner 1764.2.b.k 8
12.b even 2 1 196.2.d.c 8
21.c even 2 1 196.2.d.c 8
21.g even 6 2 196.2.f.d 16
21.h odd 6 2 196.2.f.d 16
24.f even 2 1 3136.2.f.i 8
24.h odd 2 1 3136.2.f.i 8
28.d even 2 1 inner 1764.2.b.k 8
84.h odd 2 1 196.2.d.c 8
84.j odd 6 2 196.2.f.d 16
84.n even 6 2 196.2.f.d 16
168.e odd 2 1 3136.2.f.i 8
168.i even 2 1 3136.2.f.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.c 8 3.b odd 2 1
196.2.d.c 8 12.b even 2 1
196.2.d.c 8 21.c even 2 1
196.2.d.c 8 84.h odd 2 1
196.2.f.d 16 21.g even 6 2
196.2.f.d 16 21.h odd 6 2
196.2.f.d 16 84.j odd 6 2
196.2.f.d 16 84.n even 6 2
1764.2.b.k 8 1.a even 1 1 trivial
1764.2.b.k 8 4.b odd 2 1 inner
1764.2.b.k 8 7.b odd 2 1 inner
1764.2.b.k 8 28.d even 2 1 inner
3136.2.f.i 8 24.f even 2 1
3136.2.f.i 8 24.h odd 2 1
3136.2.f.i 8 168.e odd 2 1
3136.2.f.i 8 168.i even 2 1

Hecke kernels

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

\( T_{5}^{4} + 8 T_{5}^{2} + 8 \)
\( T_{11}^{4} + 20 T_{11}^{2} + 68 \)
\( T_{19}^{4} - 28 T_{19}^{2} + 34 \)
\( T_{29}^{2} - 8 T_{29} + 8 \)
\( T_{53}^{2} + 4 T_{53} - 68 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \)
$3$ 1
$5$ \( ( 1 - 12 T^{2} + 78 T^{4} - 300 T^{6} + 625 T^{8} )^{2} \)
$7$ 1
$11$ \( ( 1 - 24 T^{2} + 354 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 44 T^{2} + 814 T^{4} - 7436 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 48 T^{2} + 1056 T^{4} - 13872 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 48 T^{2} + 1136 T^{4} + 17328 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 36 T^{2} + 870 T^{4} - 19044 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4} )^{4} \)
$31$ \( ( 1 + 28 T^{2} + 1990 T^{4} + 26908 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 4 T + 37 T^{2} )^{8} \)
$41$ \( ( 1 - 128 T^{2} + 7296 T^{4} - 215168 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 56 T^{2} + 4450 T^{4} - 103544 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 140 T^{2} + 9286 T^{4} + 309260 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 4 T + 38 T^{2} + 212 T^{3} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 + 32 T^{2} + 6640 T^{4} + 111392 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 44 T^{2} + 2926 T^{4} - 163724 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 156 T^{2} + 13014 T^{4} - 700284 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{8} \)
$73$ \( ( 1 - 32 T^{2} + 6496 T^{4} - 170528 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 236 T^{2} + 25894 T^{4} - 1472876 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 224 T^{2} + 25072 T^{4} + 1543136 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 304 T^{2} + 38848 T^{4} - 2407984 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 288 T^{2} + 38304 T^{4} - 2709792 T^{6} + 88529281 T^{8} )^{2} \)
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