Properties

Label 810.3.j.b
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: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{7} - \zeta_{24}) q^{2} + (2 \zeta_{24}^{4} - 2) q^{4} + ( - 4 \zeta_{24}^{7} - 3 \zeta_{24}^{5} + 4 \zeta_{24}^{3}) q^{5} + ( - 11 \zeta_{24}^{6} + 11 \zeta_{24}^{2}) q^{7} + ( - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{24}^{7} - \zeta_{24}) q^{2} + (2 \zeta_{24}^{4} - 2) q^{4} + ( - 4 \zeta_{24}^{7} - 3 \zeta_{24}^{5} + 4 \zeta_{24}^{3}) q^{5} + ( - 11 \zeta_{24}^{6} + 11 \zeta_{24}^{2}) q^{7} + ( - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{8} + ( - \zeta_{24}^{6} - 7) q^{10} + ( - 5 \zeta_{24}^{7} + 5 \zeta_{24}) q^{11} - 15 \zeta_{24}^{2} q^{13} + (11 \zeta_{24}^{7} - 11 \zeta_{24}^{5} - 11 \zeta_{24}^{3}) q^{14} - 4 \zeta_{24}^{4} q^{16} + ( - 16 \zeta_{24}^{5} + 16 \zeta_{24}^{3} + 16 \zeta_{24}) q^{17} - 3 q^{19} + (8 \zeta_{24}^{7} + 6 \zeta_{24}) q^{20} - 10 \zeta_{24}^{2} q^{22} + ( - 13 \zeta_{24}^{7} - 13 \zeta_{24}^{5} + 13 \zeta_{24}^{3}) q^{23} + ( - 7 \zeta_{24}^{6} - 24 \zeta_{24}^{4} + 7 \zeta_{24}^{2}) q^{25} + (15 \zeta_{24}^{5} + 15 \zeta_{24}^{3} - 15 \zeta_{24}) q^{26} + 22 \zeta_{24}^{6} q^{28} + (9 \zeta_{24}^{7} - 9 \zeta_{24}) q^{29} + ( - 8 \zeta_{24}^{4} + 8) q^{31} + (4 \zeta_{24}^{7} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{3}) q^{32} - 32 \zeta_{24}^{4} q^{34} + ( - 44 \zeta_{24}^{5} - 33 \zeta_{24}^{3} + 44 \zeta_{24}) q^{35} + 65 \zeta_{24}^{6} q^{37} + (3 \zeta_{24}^{7} + 3 \zeta_{24}) q^{38} + ( - 14 \zeta_{24}^{4} + 2 \zeta_{24}^{2} + 14) q^{40} + ( - 55 \zeta_{24}^{7} + 55 \zeta_{24}^{5} + 55 \zeta_{24}^{3}) q^{41} + (32 \zeta_{24}^{6} - 32 \zeta_{24}^{2}) q^{43} + (10 \zeta_{24}^{5} + 10 \zeta_{24}^{3} - 10 \zeta_{24}) q^{44} - 26 q^{46} + (40 \zeta_{24}^{7} + 40 \zeta_{24}) q^{47} + ( - 72 \zeta_{24}^{4} + 72) q^{49} + (31 \zeta_{24}^{7} + 17 \zeta_{24}^{5} - 31 \zeta_{24}^{3}) q^{50} + ( - 30 \zeta_{24}^{6} + 30 \zeta_{24}^{2}) q^{52} + ( - 9 \zeta_{24}^{5} + 9 \zeta_{24}^{3} + 9 \zeta_{24}) q^{53} + ( - 35 \zeta_{24}^{6} + 5) q^{55} + ( - 22 \zeta_{24}^{7} + 22 \zeta_{24}) q^{56} + 18 \zeta_{24}^{2} q^{58} + (56 \zeta_{24}^{7} - 56 \zeta_{24}^{5} - 56 \zeta_{24}^{3}) q^{59} - 95 \zeta_{24}^{4} q^{61} + (8 \zeta_{24}^{5} - 8 \zeta_{24}^{3} - 8 \zeta_{24}) q^{62} + 8 q^{64} + (45 \zeta_{24}^{7} - 60 \zeta_{24}) q^{65} - 19 \zeta_{24}^{2} q^{67} + (32 \zeta_{24}^{7} + 32 \zeta_{24}^{5} - 32 \zeta_{24}^{3}) q^{68} + (77 \zeta_{24}^{6} - 11 \zeta_{24}^{4} - 77 \zeta_{24}^{2}) q^{70} + ( - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{3} + 3 \zeta_{24}) q^{71} - 119 \zeta_{24}^{6} q^{73} + ( - 65 \zeta_{24}^{7} + 65 \zeta_{24}) q^{74} + ( - 6 \zeta_{24}^{4} + 6) q^{76} + ( - 55 \zeta_{24}^{7} - 55 \zeta_{24}^{5} + 55 \zeta_{24}^{3}) q^{77} - 99 \zeta_{24}^{4} q^{79} + (12 \zeta_{24}^{5} - 16 \zeta_{24}^{3} - 12 \zeta_{24}) q^{80} - 110 \zeta_{24}^{6} q^{82} + ( - 77 \zeta_{24}^{7} - 77 \zeta_{24}) q^{83} + ( - 112 \zeta_{24}^{4} + 16 \zeta_{24}^{2} + 112) q^{85} + ( - 32 \zeta_{24}^{7} + 32 \zeta_{24}^{5} + 32 \zeta_{24}^{3}) q^{86} + ( - 20 \zeta_{24}^{6} + 20 \zeta_{24}^{2}) q^{88} + (64 \zeta_{24}^{5} + 64 \zeta_{24}^{3} - 64 \zeta_{24}) q^{89} - 165 q^{91} + (26 \zeta_{24}^{7} + 26 \zeta_{24}) q^{92} + ( - 80 \zeta_{24}^{4} + 80) q^{94} + (12 \zeta_{24}^{7} + 9 \zeta_{24}^{5} - 12 \zeta_{24}^{3}) q^{95} + (95 \zeta_{24}^{6} - 95 \zeta_{24}^{2}) q^{97} + (72 \zeta_{24}^{5} - 72 \zeta_{24}^{3} - 72 \zeta_{24}) 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 \( 8 q - 8 q^{4} - 56 q^{10} - 16 q^{16} - 24 q^{19} - 96 q^{25} + 32 q^{31} - 128 q^{34} + 56 q^{40} - 208 q^{46} + 288 q^{49} + 40 q^{55} - 380 q^{61} + 64 q^{64} - 44 q^{70} + 24 q^{76} - 396 q^{79} + 448 q^{85} - 1320 q^{91} + 320 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{24}^{4}\)

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
−0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
−0.707107 + 1.22474i 0 −1.00000 1.73205i 1.86250 + 4.64016i 0 −9.52628 5.50000i 2.82843 0 −7.00000 1.00000i
269.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 3.08725 + 3.93305i 0 9.52628 + 5.50000i 2.82843 0 −7.00000 + 1.00000i
269.3 0.707107 1.22474i 0 −1.00000 1.73205i −3.08725 3.93305i 0 9.52628 + 5.50000i −2.82843 0 −7.00000 + 1.00000i
269.4 0.707107 1.22474i 0 −1.00000 1.73205i −1.86250 4.64016i 0 −9.52628 5.50000i −2.82843 0 −7.00000 1.00000i
539.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i 1.86250 4.64016i 0 −9.52628 + 5.50000i 2.82843 0 −7.00000 + 1.00000i
539.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 3.08725 3.93305i 0 9.52628 5.50000i 2.82843 0 −7.00000 1.00000i
539.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −3.08725 + 3.93305i 0 9.52628 5.50000i −2.82843 0 −7.00000 1.00000i
539.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −1.86250 + 4.64016i 0 −9.52628 + 5.50000i −2.82843 0 −7.00000 + 1.00000i
\(n\): 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.b 8
3.b odd 2 1 inner 810.3.j.b 8
5.b even 2 1 inner 810.3.j.b 8
9.c even 3 1 270.3.b.b 4
9.c even 3 1 inner 810.3.j.b 8
9.d odd 6 1 270.3.b.b 4
9.d odd 6 1 inner 810.3.j.b 8
15.d odd 2 1 inner 810.3.j.b 8
36.f odd 6 1 2160.3.c.j 4
36.h even 6 1 2160.3.c.j 4
45.h odd 6 1 270.3.b.b 4
45.h odd 6 1 inner 810.3.j.b 8
45.j even 6 1 270.3.b.b 4
45.j even 6 1 inner 810.3.j.b 8
45.k odd 12 1 1350.3.d.b 2
45.k odd 12 1 1350.3.d.j 2
45.l even 12 1 1350.3.d.b 2
45.l even 12 1 1350.3.d.j 2
180.n even 6 1 2160.3.c.j 4
180.p odd 6 1 2160.3.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.b 4 9.c even 3 1
270.3.b.b 4 9.d odd 6 1
270.3.b.b 4 45.h odd 6 1
270.3.b.b 4 45.j even 6 1
810.3.j.b 8 1.a even 1 1 trivial
810.3.j.b 8 3.b odd 2 1 inner
810.3.j.b 8 5.b even 2 1 inner
810.3.j.b 8 9.c even 3 1 inner
810.3.j.b 8 9.d odd 6 1 inner
810.3.j.b 8 15.d odd 2 1 inner
810.3.j.b 8 45.h odd 6 1 inner
810.3.j.b 8 45.j even 6 1 inner
1350.3.d.b 2 45.k odd 12 1
1350.3.d.b 2 45.l even 12 1
1350.3.d.j 2 45.k odd 12 1
1350.3.d.j 2 45.l even 12 1
2160.3.c.j 4 36.f odd 6 1
2160.3.c.j 4 36.h even 6 1
2160.3.c.j 4 180.n even 6 1
2160.3.c.j 4 180.p odd 6 1

Hecke kernels

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

\( T_{7}^{4} - 121T_{7}^{2} + 14641 \) Copy content Toggle raw display
\( T_{17}^{2} - 512 \) 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
$5$ \( T^{8} + 48 T^{6} + 1679 T^{4} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 121 T^{2} + 14641)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 50 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 225 T^{2} + 50625)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 512)^{4} \) Copy content Toggle raw display
$19$ \( (T + 3)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 338 T^{2} + 114244)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 162 T^{2} + 26244)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4225)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 6050 T^{2} + 36602500)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 1024 T^{2} + 1048576)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3200 T^{2} + 10240000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 162)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 6272 T^{2} + 39337984)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 95 T + 9025)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 361 T^{2} + 130321)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 14161)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 99 T + 9801)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 11858 T^{2} + \cdots + 140612164)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 8192)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 9025 T^{2} + 81450625)^{2} \) Copy content Toggle raw display
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