Properties

Label 810.3.j.c
Level $810$
Weight $3$
Character orbit 810.j
Analytic conductor $22.071$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.0709014132\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.443364212736.6
Defining polynomial: \( x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{2}) q^{2} + (2 \beta_1 - 2) q^{4} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{5} - \beta_{4} q^{7} + 2 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{2}) q^{2} + (2 \beta_1 - 2) q^{4} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{5} - \beta_{4} q^{7} + 2 \beta_{6} q^{8} + ( - \beta_{7} + \beta_{4} - 4) q^{10} + 4 \beta_{3} q^{11} + (2 \beta_{5} + 2 \beta_{3}) q^{14} - 4 \beta_1 q^{16} - 8 \beta_{6} q^{17} + 12 q^{19} + (4 \beta_{6} - 2 \beta_{3} + 4 \beta_{2}) q^{20} - 4 \beta_{7} q^{22} + 17 \beta_{2} q^{23} + (4 \beta_{4} + 9 \beta_1) q^{25} + ( - 2 \beta_{7} + 2 \beta_{4}) q^{28} + ( - 32 \beta_1 + 32) q^{31} + 4 \beta_{2} q^{32} + 16 \beta_1 q^{34} + ( - 17 \beta_{6} + 4 \beta_{5}) q^{35} + ( - 4 \beta_{7} + 4 \beta_{4}) q^{37} + ( - 12 \beta_{6} - 12 \beta_{2}) q^{38} + (2 \beta_{7} - 8 \beta_1 + 8) q^{40} + (14 \beta_{5} + 14 \beta_{3}) q^{41} + 7 \beta_{4} q^{43} + 8 \beta_{5} q^{44} + 34 q^{46} + (25 \beta_{6} + 25 \beta_{2}) q^{47} + (15 \beta_1 - 15) q^{49} + ( - 8 \beta_{5} - 8 \beta_{3} - 9 \beta_{2}) q^{50} + 48 \beta_{6} q^{53} + ( - 8 \beta_{7} + 8 \beta_{4} + 68) q^{55} - 4 \beta_{3} q^{56} + ( - 4 \beta_{5} - 4 \beta_{3}) q^{59} + 16 \beta_1 q^{61} - 32 \beta_{6} q^{62} + 8 q^{64} - \beta_{7} q^{67} - 16 \beta_{2} q^{68} + (4 \beta_{4} + 34 \beta_1) q^{70} + ( - 20 \beta_{7} + 20 \beta_{4}) q^{73} - 8 \beta_{3} q^{74} + (24 \beta_1 - 24) q^{76} + 68 \beta_{2} q^{77} + 72 \beta_1 q^{79} + ( - 8 \beta_{6} - 4 \beta_{5}) q^{80} + ( - 14 \beta_{7} + 14 \beta_{4}) q^{82} + (31 \beta_{6} + 31 \beta_{2}) q^{83} + ( - 8 \beta_{7} + 32 \beta_1 - 32) q^{85} + ( - 14 \beta_{5} - 14 \beta_{3}) q^{86} + 8 \beta_{4} q^{88} - 16 \beta_{5} q^{89} + ( - 34 \beta_{6} - 34 \beta_{2}) q^{92} + ( - 50 \beta_1 + 50) q^{94} + (12 \beta_{5} + 12 \beta_{3} - 24 \beta_{2}) q^{95} + 28 \beta_{4} q^{97} + 15 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 32 q^{10} - 16 q^{16} + 96 q^{19} + 36 q^{25} + 128 q^{31} + 64 q^{34} + 32 q^{40} + 272 q^{46} - 60 q^{49} + 544 q^{55} + 64 q^{61} + 64 q^{64} + 136 q^{70} - 96 q^{76} + 288 q^{79} - 128 q^{85} + 200 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -16\nu^{6} + 175\nu^{4} - 2800\nu^{2} + 20736 ) / 14175 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31\nu^{7} - 1225\nu^{5} + 5425\nu^{3} - 40176\nu ) / 127575 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -47\nu^{6} + 1400\nu^{4} - 8225\nu^{2} + 60912 ) / 14175 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 3079\nu ) / 1575 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 104 ) / 175 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -16\nu^{7} + 175\nu^{5} - 775\nu^{3} + 6561\nu ) / 18225 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -319\nu^{7} + 4375\nu^{5} - 55825\nu^{3} + 413424\nu ) / 127575 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} - 8\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 25\beta_{6} + 7\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 16\beta_{3} - 47\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -31\beta_{7} - 319\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -175\beta_{5} - 104 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3079\beta_{6} + 71\beta_{4} - 3079\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
−2.17132 2.07011i
2.87843 + 0.845366i
2.17132 + 2.07011i
−2.87843 0.845366i
−2.17132 + 2.07011i
2.87843 0.845366i
2.17132 2.07011i
−2.87843 + 0.845366i
−0.707107 + 1.22474i 0 −1.00000 1.73205i −2.15650 + 4.51104i 0 5.04975 + 2.91548i 2.82843 0 −4.00000 5.83095i
269.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 4.98493 + 0.387937i 0 −5.04975 2.91548i 2.82843 0 −4.00000 + 5.83095i
269.3 0.707107 1.22474i 0 −1.00000 1.73205i −4.98493 0.387937i 0 −5.04975 2.91548i −2.82843 0 −4.00000 + 5.83095i
269.4 0.707107 1.22474i 0 −1.00000 1.73205i 2.15650 4.51104i 0 5.04975 + 2.91548i −2.82843 0 −4.00000 5.83095i
539.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −2.15650 4.51104i 0 5.04975 2.91548i 2.82843 0 −4.00000 + 5.83095i
539.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 4.98493 0.387937i 0 −5.04975 + 2.91548i 2.82843 0 −4.00000 5.83095i
539.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −4.98493 + 0.387937i 0 −5.04975 + 2.91548i −2.82843 0 −4.00000 5.83095i
539.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 2.15650 + 4.51104i 0 5.04975 2.91548i −2.82843 0 −4.00000 + 5.83095i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 539.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.3.j.c 8
3.b odd 2 1 inner 810.3.j.c 8
5.b even 2 1 inner 810.3.j.c 8
9.c even 3 1 30.3.b.a 4
9.c even 3 1 inner 810.3.j.c 8
9.d odd 6 1 30.3.b.a 4
9.d odd 6 1 inner 810.3.j.c 8
15.d odd 2 1 inner 810.3.j.c 8
36.f odd 6 1 240.3.c.c 4
36.h even 6 1 240.3.c.c 4
45.h odd 6 1 30.3.b.a 4
45.h odd 6 1 inner 810.3.j.c 8
45.j even 6 1 30.3.b.a 4
45.j even 6 1 inner 810.3.j.c 8
45.k odd 12 2 150.3.d.d 4
45.l even 12 2 150.3.d.d 4
72.j odd 6 1 960.3.c.f 4
72.l even 6 1 960.3.c.e 4
72.n even 6 1 960.3.c.f 4
72.p odd 6 1 960.3.c.e 4
180.n even 6 1 240.3.c.c 4
180.p odd 6 1 240.3.c.c 4
180.v odd 12 2 1200.3.l.t 4
180.x even 12 2 1200.3.l.t 4
360.z odd 6 1 960.3.c.e 4
360.bd even 6 1 960.3.c.e 4
360.bh odd 6 1 960.3.c.f 4
360.bk even 6 1 960.3.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.b.a 4 9.c even 3 1
30.3.b.a 4 9.d odd 6 1
30.3.b.a 4 45.h odd 6 1
30.3.b.a 4 45.j even 6 1
150.3.d.d 4 45.k odd 12 2
150.3.d.d 4 45.l even 12 2
240.3.c.c 4 36.f odd 6 1
240.3.c.c 4 36.h even 6 1
240.3.c.c 4 180.n even 6 1
240.3.c.c 4 180.p odd 6 1
810.3.j.c 8 1.a even 1 1 trivial
810.3.j.c 8 3.b odd 2 1 inner
810.3.j.c 8 5.b even 2 1 inner
810.3.j.c 8 9.c even 3 1 inner
810.3.j.c 8 9.d odd 6 1 inner
810.3.j.c 8 15.d odd 2 1 inner
810.3.j.c 8 45.h odd 6 1 inner
810.3.j.c 8 45.j even 6 1 inner
960.3.c.e 4 72.l even 6 1
960.3.c.e 4 72.p odd 6 1
960.3.c.e 4 360.z odd 6 1
960.3.c.e 4 360.bd even 6 1
960.3.c.f 4 72.j odd 6 1
960.3.c.f 4 72.n even 6 1
960.3.c.f 4 360.bh odd 6 1
960.3.c.f 4 360.bk even 6 1
1200.3.l.t 4 180.v odd 12 2
1200.3.l.t 4 180.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{4} - 34T_{7}^{2} + 1156 \) Copy content Toggle raw display
\( T_{17}^{2} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 18 T^{6} - 301 T^{4} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 34 T^{2} + 1156)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 272 T^{2} + 73984)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} - 128)^{4} \) Copy content Toggle raw display
$19$ \( (T - 12)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 578 T^{2} + 334084)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} - 32 T + 1024)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 544)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 3332 T^{2} + 11102224)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 1666 T^{2} + 2775556)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1250 T^{2} + 1562500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4608)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 272 T^{2} + 73984)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 16 T + 256)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 34 T^{2} + 1156)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 13600)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 72 T + 5184)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1922 T^{2} + 3694084)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4352)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 26656 T^{2} + \cdots + 710542336)^{2} \) Copy content Toggle raw display
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