Properties

Label 810.3.j.e
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: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{7} - \zeta_{24}) q^{2} + (2 \zeta_{24}^{4} - 2) q^{4} + (5 \zeta_{24}^{7} - 5 \zeta_{24}^{3}) q^{5} + ( - 5 \zeta_{24}^{6} + 5 \zeta_{24}^{2}) q^{7} + ( - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{24}^{7} - \zeta_{24}) q^{2} + (2 \zeta_{24}^{4} - 2) q^{4} + (5 \zeta_{24}^{7} - 5 \zeta_{24}^{3}) q^{5} + ( - 5 \zeta_{24}^{6} + 5 \zeta_{24}^{2}) q^{7} + ( - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{8} + (5 \zeta_{24}^{6} + 5) q^{10} + (\zeta_{24}^{7} - \zeta_{24}) q^{11} - 9 \zeta_{24}^{2} q^{13} + (5 \zeta_{24}^{7} - 5 \zeta_{24}^{5} - 5 \zeta_{24}^{3}) q^{14} - 4 \zeta_{24}^{4} q^{16} + (8 \zeta_{24}^{5} - 8 \zeta_{24}^{3} - 8 \zeta_{24}) q^{17} + 21 q^{19} - 10 \zeta_{24}^{7} q^{20} + 2 \zeta_{24}^{2} q^{22} + ( - \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3}) q^{23} + ( - 25 \zeta_{24}^{6} + 25 \zeta_{24}^{2}) q^{25} + (9 \zeta_{24}^{5} + 9 \zeta_{24}^{3} - 9 \zeta_{24}) q^{26} + 10 \zeta_{24}^{6} q^{28} + (27 \zeta_{24}^{7} - 27 \zeta_{24}) q^{29} + (40 \zeta_{24}^{4} - 40) q^{31} + (4 \zeta_{24}^{7} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{3}) q^{32} + 16 \zeta_{24}^{4} q^{34} + (25 \zeta_{24}^{5} - 25 \zeta_{24}) q^{35} - 25 \zeta_{24}^{6} q^{37} + ( - 21 \zeta_{24}^{7} - 21 \zeta_{24}) q^{38} + (10 \zeta_{24}^{4} - 10 \zeta_{24}^{2} - 10) q^{40} + ( - 37 \zeta_{24}^{7} + 37 \zeta_{24}^{5} + 37 \zeta_{24}^{3}) q^{41} + ( - 64 \zeta_{24}^{6} + 64 \zeta_{24}^{2}) q^{43} + ( - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{44} - 2 q^{46} + (16 \zeta_{24}^{7} + 16 \zeta_{24}) q^{47} + (24 \zeta_{24}^{4} - 24) q^{49} + (25 \zeta_{24}^{7} - 25 \zeta_{24}^{5} - 25 \zeta_{24}^{3}) q^{50} + ( - 18 \zeta_{24}^{6} + 18 \zeta_{24}^{2}) q^{52} + (51 \zeta_{24}^{5} - 51 \zeta_{24}^{3} - 51 \zeta_{24}) q^{53} + ( - 5 \zeta_{24}^{6} + 5) q^{55} + ( - 10 \zeta_{24}^{7} + 10 \zeta_{24}) q^{56} + 54 \zeta_{24}^{2} q^{58} + ( - 64 \zeta_{24}^{7} + 64 \zeta_{24}^{5} + 64 \zeta_{24}^{3}) q^{59} + 97 \zeta_{24}^{4} q^{61} + ( - 40 \zeta_{24}^{5} + 40 \zeta_{24}^{3} + 40 \zeta_{24}) q^{62} + 8 q^{64} + 45 \zeta_{24} q^{65} + 131 \zeta_{24}^{2} q^{67} + ( - 16 \zeta_{24}^{7} - 16 \zeta_{24}^{5} + 16 \zeta_{24}^{3}) q^{68} + ( - 25 \zeta_{24}^{6} + 25 \zeta_{24}^{4} + 25 \zeta_{24}^{2}) q^{70} + (63 \zeta_{24}^{5} + 63 \zeta_{24}^{3} - 63 \zeta_{24}) q^{71} - 17 \zeta_{24}^{6} q^{73} + (25 \zeta_{24}^{7} - 25 \zeta_{24}) q^{74} + (42 \zeta_{24}^{4} - 42) q^{76} + (5 \zeta_{24}^{7} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{3}) q^{77} + 117 \zeta_{24}^{4} q^{79} + 20 \zeta_{24}^{3} q^{80} - 74 \zeta_{24}^{6} q^{82} + ( - 41 \zeta_{24}^{7} - 41 \zeta_{24}) q^{83} + ( - 40 \zeta_{24}^{4} + 40 \zeta_{24}^{2} + 40) q^{85} + (64 \zeta_{24}^{7} - 64 \zeta_{24}^{5} - 64 \zeta_{24}^{3}) q^{86} + (4 \zeta_{24}^{6} - 4 \zeta_{24}^{2}) q^{88} + ( - 104 \zeta_{24}^{5} - 104 \zeta_{24}^{3} + 104 \zeta_{24}) q^{89} - 45 q^{91} + (2 \zeta_{24}^{7} + 2 \zeta_{24}) q^{92} + ( - 32 \zeta_{24}^{4} + 32) q^{94} + (105 \zeta_{24}^{7} - 105 \zeta_{24}^{3}) q^{95} + (41 \zeta_{24}^{6} - 41 \zeta_{24}^{2}) q^{97} + ( - 24 \zeta_{24}^{5} + 24 \zeta_{24}^{3} + 24 \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 \( 8 q - 8 q^{4} + 40 q^{10} - 16 q^{16} + 168 q^{19} - 160 q^{31} + 64 q^{34} - 40 q^{40} - 16 q^{46} - 96 q^{49} + 40 q^{55} + 388 q^{61} + 64 q^{64} + 100 q^{70} - 168 q^{76} + 468 q^{79} + 160 q^{85} - 360 q^{91} + 128 q^{94}+O(q^{100}) \) 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
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
−0.707107 + 1.22474i 0 −1.00000 1.73205i −4.82963 1.29410i 0 4.33013 + 2.50000i 2.82843 0 5.00000 5.00000i
269.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 1.29410 4.82963i 0 −4.33013 2.50000i 2.82843 0 5.00000 + 5.00000i
269.3 0.707107 1.22474i 0 −1.00000 1.73205i −1.29410 + 4.82963i 0 −4.33013 2.50000i −2.82843 0 5.00000 + 5.00000i
269.4 0.707107 1.22474i 0 −1.00000 1.73205i 4.82963 + 1.29410i 0 4.33013 + 2.50000i −2.82843 0 5.00000 5.00000i
539.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −4.82963 + 1.29410i 0 4.33013 2.50000i 2.82843 0 5.00000 + 5.00000i
539.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 1.29410 + 4.82963i 0 −4.33013 + 2.50000i 2.82843 0 5.00000 5.00000i
539.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −1.29410 4.82963i 0 −4.33013 + 2.50000i −2.82843 0 5.00000 5.00000i
539.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 4.82963 1.29410i 0 4.33013 2.50000i −2.82843 0 5.00000 + 5.00000i
\(n\): 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.e 8
3.b odd 2 1 inner 810.3.j.e 8
5.b even 2 1 inner 810.3.j.e 8
9.c even 3 1 270.3.b.c 4
9.c even 3 1 inner 810.3.j.e 8
9.d odd 6 1 270.3.b.c 4
9.d odd 6 1 inner 810.3.j.e 8
15.d odd 2 1 inner 810.3.j.e 8
36.f odd 6 1 2160.3.c.i 4
36.h even 6 1 2160.3.c.i 4
45.h odd 6 1 270.3.b.c 4
45.h odd 6 1 inner 810.3.j.e 8
45.j even 6 1 270.3.b.c 4
45.j even 6 1 inner 810.3.j.e 8
45.k odd 12 1 1350.3.d.f 2
45.k odd 12 1 1350.3.d.g 2
45.l even 12 1 1350.3.d.f 2
45.l even 12 1 1350.3.d.g 2
180.n even 6 1 2160.3.c.i 4
180.p odd 6 1 2160.3.c.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.c 4 9.c even 3 1
270.3.b.c 4 9.d odd 6 1
270.3.b.c 4 45.h odd 6 1
270.3.b.c 4 45.j even 6 1
810.3.j.e 8 1.a even 1 1 trivial
810.3.j.e 8 3.b odd 2 1 inner
810.3.j.e 8 5.b even 2 1 inner
810.3.j.e 8 9.c even 3 1 inner
810.3.j.e 8 9.d odd 6 1 inner
810.3.j.e 8 15.d odd 2 1 inner
810.3.j.e 8 45.h odd 6 1 inner
810.3.j.e 8 45.j even 6 1 inner
1350.3.d.f 2 45.k odd 12 1
1350.3.d.f 2 45.l even 12 1
1350.3.d.g 2 45.k odd 12 1
1350.3.d.g 2 45.l even 12 1
2160.3.c.i 4 36.f odd 6 1
2160.3.c.i 4 36.h even 6 1
2160.3.c.i 4 180.n even 6 1
2160.3.c.i 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 \(S_{3}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{4} - 25T_{7}^{2} + 625 \) 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
$5$ \( T^{8} - 625 T^{4} + 390625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 25 T^{2} + 625)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 81 T^{2} + 6561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 128)^{4} \) Copy content Toggle raw display
$19$ \( (T - 21)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 1458 T^{2} + 2125764)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 40 T + 1600)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 625)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 2738 T^{2} + 7496644)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4096 T^{2} + 16777216)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 512 T^{2} + 262144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 5202)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 8192 T^{2} + 67108864)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 97 T + 9409)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 17161 T^{2} + \cdots + 294499921)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 7938)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 289)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 117 T + 13689)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 3362 T^{2} + 11303044)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 21632)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1681 T^{2} + 2825761)^{2} \) Copy content Toggle raw display
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