Properties

Label 810.2.i.b
Level $810$
Weight $2$
Character orbit 810.i
Analytic conductor $6.468$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( -1 - 2 \zeta_{12}^{3} ) q^{10} + ( 2 - 2 \zeta_{12}^{2} ) q^{11} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{13} + 2 \zeta_{12}^{2} q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 2 \zeta_{12}^{3} q^{17} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{20} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{22} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} + ( -3 + 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + 6 q^{26} + 2 \zeta_{12}^{3} q^{28} + 8 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( -2 + 2 \zeta_{12}^{2} ) q^{34} + ( -2 - 4 \zeta_{12}^{3} ) q^{35} -2 \zeta_{12}^{3} q^{37} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{40} + 2 \zeta_{12}^{2} q^{41} + 4 \zeta_{12} q^{43} + 2 q^{44} + 4 q^{46} + 8 \zeta_{12} q^{47} -3 \zeta_{12}^{2} q^{49} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{50} + 6 \zeta_{12} q^{52} -6 \zeta_{12}^{3} q^{53} + ( -4 + 2 \zeta_{12}^{3} ) q^{55} + ( -2 + 2 \zeta_{12}^{2} ) q^{56} -10 \zeta_{12}^{2} q^{59} + ( -2 + 2 \zeta_{12}^{2} ) q^{61} + 8 \zeta_{12}^{3} q^{62} - q^{64} + ( -6 - 12 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{65} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{67} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{68} + ( 4 - 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{70} -12 q^{71} -4 \zeta_{12}^{3} q^{73} + ( 2 - 2 \zeta_{12}^{2} ) q^{74} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{77} + ( 2 - \zeta_{12}^{3} ) q^{80} + 2 \zeta_{12}^{3} q^{82} -4 \zeta_{12} q^{83} + ( 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{85} + 4 \zeta_{12}^{2} q^{86} + 2 \zeta_{12} q^{88} -10 q^{89} + 12 q^{91} + 4 \zeta_{12} q^{92} + 8 \zeta_{12}^{2} q^{94} -8 \zeta_{12} q^{97} -3 \zeta_{12}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 4q^{5} + O(q^{10}) \) \( 4q + 2q^{4} - 4q^{5} - 4q^{10} + 4q^{11} + 4q^{14} - 2q^{16} + 4q^{20} - 6q^{25} + 24q^{26} + 16q^{31} - 4q^{34} - 8q^{35} - 2q^{40} + 4q^{41} + 8q^{44} + 16q^{46} - 6q^{49} + 8q^{50} - 16q^{55} - 4q^{56} - 20q^{59} - 4q^{61} - 4q^{64} - 12q^{65} + 8q^{70} - 48q^{71} + 4q^{74} + 8q^{80} - 4q^{85} + 8q^{86} - 40q^{89} + 48q^{91} + 16q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i −0.133975 + 2.23205i 0 −1.73205 + 1.00000i 1.00000i 0 −1.00000 2.00000i
109.2 0.866025 0.500000i 0 0.500000 0.866025i −1.86603 + 1.23205i 0 1.73205 1.00000i 1.00000i 0 −1.00000 + 2.00000i
379.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.133975 2.23205i 0 −1.73205 1.00000i 1.00000i 0 −1.00000 + 2.00000i
379.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.86603 1.23205i 0 1.73205 + 1.00000i 1.00000i 0 −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 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 810.2.i.b 4
3.b odd 2 1 810.2.i.e 4
5.b even 2 1 inner 810.2.i.b 4
9.c even 3 1 90.2.c.a 2
9.c even 3 1 inner 810.2.i.b 4
9.d odd 6 1 30.2.c.a 2
9.d odd 6 1 810.2.i.e 4
15.d odd 2 1 810.2.i.e 4
36.f odd 6 1 720.2.f.f 2
36.h even 6 1 240.2.f.a 2
45.h odd 6 1 30.2.c.a 2
45.h odd 6 1 810.2.i.e 4
45.j even 6 1 90.2.c.a 2
45.j even 6 1 inner 810.2.i.b 4
45.k odd 12 1 450.2.a.b 1
45.k odd 12 1 450.2.a.f 1
45.l even 12 1 150.2.a.a 1
45.l even 12 1 150.2.a.c 1
63.i even 6 1 1470.2.n.a 4
63.j odd 6 1 1470.2.n.h 4
63.n odd 6 1 1470.2.n.h 4
63.o even 6 1 1470.2.g.g 2
63.s even 6 1 1470.2.n.a 4
72.j odd 6 1 960.2.f.h 2
72.l even 6 1 960.2.f.i 2
72.n even 6 1 2880.2.f.e 2
72.p odd 6 1 2880.2.f.c 2
144.u even 12 1 3840.2.d.j 2
144.u even 12 1 3840.2.d.x 2
144.w odd 12 1 3840.2.d.g 2
144.w odd 12 1 3840.2.d.y 2
180.n even 6 1 240.2.f.a 2
180.p odd 6 1 720.2.f.f 2
180.v odd 12 1 1200.2.a.g 1
180.v odd 12 1 1200.2.a.m 1
180.x even 12 1 3600.2.a.o 1
180.x even 12 1 3600.2.a.bg 1
315.u even 6 1 1470.2.n.a 4
315.v odd 6 1 1470.2.n.h 4
315.z even 6 1 1470.2.g.g 2
315.bq even 6 1 1470.2.n.a 4
315.br odd 6 1 1470.2.n.h 4
315.cf odd 12 1 7350.2.a.bg 1
315.cf odd 12 1 7350.2.a.cc 1
360.z odd 6 1 2880.2.f.c 2
360.bd even 6 1 960.2.f.i 2
360.bh odd 6 1 960.2.f.h 2
360.bk even 6 1 2880.2.f.e 2
360.br even 12 1 4800.2.a.l 1
360.br even 12 1 4800.2.a.cg 1
360.bt odd 12 1 4800.2.a.m 1
360.bt odd 12 1 4800.2.a.cj 1
720.ch odd 12 1 3840.2.d.g 2
720.ch odd 12 1 3840.2.d.y 2
720.da even 12 1 3840.2.d.j 2
720.da even 12 1 3840.2.d.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 9.d odd 6 1
30.2.c.a 2 45.h odd 6 1
90.2.c.a 2 9.c even 3 1
90.2.c.a 2 45.j even 6 1
150.2.a.a 1 45.l even 12 1
150.2.a.c 1 45.l even 12 1
240.2.f.a 2 36.h even 6 1
240.2.f.a 2 180.n even 6 1
450.2.a.b 1 45.k odd 12 1
450.2.a.f 1 45.k odd 12 1
720.2.f.f 2 36.f odd 6 1
720.2.f.f 2 180.p odd 6 1
810.2.i.b 4 1.a even 1 1 trivial
810.2.i.b 4 5.b even 2 1 inner
810.2.i.b 4 9.c even 3 1 inner
810.2.i.b 4 45.j even 6 1 inner
810.2.i.e 4 3.b odd 2 1
810.2.i.e 4 9.d odd 6 1
810.2.i.e 4 15.d odd 2 1
810.2.i.e 4 45.h odd 6 1
960.2.f.h 2 72.j odd 6 1
960.2.f.h 2 360.bh odd 6 1
960.2.f.i 2 72.l even 6 1
960.2.f.i 2 360.bd even 6 1
1200.2.a.g 1 180.v odd 12 1
1200.2.a.m 1 180.v odd 12 1
1470.2.g.g 2 63.o even 6 1
1470.2.g.g 2 315.z even 6 1
1470.2.n.a 4 63.i even 6 1
1470.2.n.a 4 63.s even 6 1
1470.2.n.a 4 315.u even 6 1
1470.2.n.a 4 315.bq even 6 1
1470.2.n.h 4 63.j odd 6 1
1470.2.n.h 4 63.n odd 6 1
1470.2.n.h 4 315.v odd 6 1
1470.2.n.h 4 315.br odd 6 1
2880.2.f.c 2 72.p odd 6 1
2880.2.f.c 2 360.z odd 6 1
2880.2.f.e 2 72.n even 6 1
2880.2.f.e 2 360.bk even 6 1
3600.2.a.o 1 180.x even 12 1
3600.2.a.bg 1 180.x even 12 1
3840.2.d.g 2 144.w odd 12 1
3840.2.d.g 2 720.ch odd 12 1
3840.2.d.j 2 144.u even 12 1
3840.2.d.j 2 720.da even 12 1
3840.2.d.x 2 144.u even 12 1
3840.2.d.x 2 720.da even 12 1
3840.2.d.y 2 144.w odd 12 1
3840.2.d.y 2 720.ch odd 12 1
4800.2.a.l 1 360.br even 12 1
4800.2.a.m 1 360.bt odd 12 1
4800.2.a.cg 1 360.br even 12 1
4800.2.a.cj 1 360.bt odd 12 1
7350.2.a.bg 1 315.cf odd 12 1
7350.2.a.cc 1 315.cf odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{4} - 4 T_{7}^{2} + 16 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ 1
$5$ \( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ \( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \)
$11$ \( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} ) \)
$17$ \( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( 1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 29 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8} \)
$47$ \( 1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 130 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 79 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 18 T + 227 T^{2} - 1746 T^{3} + 9409 T^{4} )( 1 + 18 T + 227 T^{2} + 1746 T^{3} + 9409 T^{4} ) \)
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