Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} + q^{8} + q^{10} -2 \zeta_{6} q^{13} + 4 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -6 q^{17} -4 q^{19} + ( -1 + \zeta_{6} ) q^{20} + ( -1 + \zeta_{6} ) q^{25} + 2 q^{26} -4 q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} -8 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( 6 - 6 \zeta_{6} ) q^{34} -4 q^{35} + 2 q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} -\zeta_{6} q^{40} -6 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} -9 \zeta_{6} q^{49} -\zeta_{6} q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} + 6 q^{53} + ( 4 - 4 \zeta_{6} ) q^{56} -6 \zeta_{6} q^{58} + ( 10 - 10 \zeta_{6} ) q^{61} + 8 q^{62} + q^{64} + ( -2 + 2 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} + ( 4 - 4 \zeta_{6} ) q^{70} + 2 q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{76} + ( -8 + 8 \zeta_{6} ) q^{79} + q^{80} + 6 q^{82} + ( 12 - 12 \zeta_{6} ) q^{83} + 6 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} -18 q^{89} -8 q^{91} + 4 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} + 9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - q^{5} + 4q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - q^{5} + 4q^{7} + 2q^{8} + 2q^{10} - 2q^{13} + 4q^{14} - q^{16} - 12q^{17} - 8q^{19} - q^{20} - q^{25} + 4q^{26} - 8q^{28} - 6q^{29} - 8q^{31} - q^{32} + 6q^{34} - 8q^{35} + 4q^{37} + 4q^{38} - q^{40} - 6q^{41} + 4q^{43} - 9q^{49} - q^{50} - 2q^{52} + 12q^{53} + 4q^{56} - 6q^{58} + 10q^{61} + 16q^{62} + 2q^{64} - 2q^{65} + 4q^{67} + 6q^{68} + 4q^{70} + 4q^{73} - 2q^{74} + 4q^{76} - 8q^{79} + 2q^{80} + 12q^{82} + 12q^{83} + 6q^{85} + 4q^{86} - 36q^{89} - 16q^{91} + 4q^{95} - 2q^{97} + 18q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(487\) \(731\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 2.00000 + 3.46410i 1.00000 0 1.00000
541.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 0.866025i 0 2.00000 3.46410i 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.e.b 2
3.b odd 2 1 810.2.e.l 2
9.c even 3 1 90.2.a.c 1
9.c even 3 1 inner 810.2.e.b 2
9.d odd 6 1 30.2.a.a 1
9.d odd 6 1 810.2.e.l 2
36.f odd 6 1 720.2.a.j 1
36.h even 6 1 240.2.a.b 1
45.h odd 6 1 150.2.a.b 1
45.j even 6 1 450.2.a.d 1
45.k odd 12 2 450.2.c.b 2
45.l even 12 2 150.2.c.a 2
63.i even 6 1 1470.2.i.q 2
63.j odd 6 1 1470.2.i.o 2
63.l odd 6 1 4410.2.a.z 1
63.n odd 6 1 1470.2.i.o 2
63.o even 6 1 1470.2.a.d 1
63.s even 6 1 1470.2.i.q 2
72.j odd 6 1 960.2.a.e 1
72.l even 6 1 960.2.a.p 1
72.n even 6 1 2880.2.a.a 1
72.p odd 6 1 2880.2.a.q 1
99.g even 6 1 3630.2.a.w 1
117.n odd 6 1 5070.2.a.w 1
117.z even 12 2 5070.2.b.k 2
144.u even 12 2 3840.2.k.f 2
144.w odd 12 2 3840.2.k.y 2
153.i odd 6 1 8670.2.a.g 1
180.n even 6 1 1200.2.a.k 1
180.p odd 6 1 3600.2.a.f 1
180.v odd 12 2 1200.2.f.e 2
180.x even 12 2 3600.2.f.i 2
315.z even 6 1 7350.2.a.ct 1
360.bd even 6 1 4800.2.a.d 1
360.bh odd 6 1 4800.2.a.cq 1
360.br even 12 2 4800.2.f.p 2
360.bt odd 12 2 4800.2.f.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 9.d odd 6 1
90.2.a.c 1 9.c even 3 1
150.2.a.b 1 45.h odd 6 1
150.2.c.a 2 45.l even 12 2
240.2.a.b 1 36.h even 6 1
450.2.a.d 1 45.j even 6 1
450.2.c.b 2 45.k odd 12 2
720.2.a.j 1 36.f odd 6 1
810.2.e.b 2 1.a even 1 1 trivial
810.2.e.b 2 9.c even 3 1 inner
810.2.e.l 2 3.b odd 2 1
810.2.e.l 2 9.d odd 6 1
960.2.a.e 1 72.j odd 6 1
960.2.a.p 1 72.l even 6 1
1200.2.a.k 1 180.n even 6 1
1200.2.f.e 2 180.v odd 12 2
1470.2.a.d 1 63.o even 6 1
1470.2.i.o 2 63.j odd 6 1
1470.2.i.o 2 63.n odd 6 1
1470.2.i.q 2 63.i even 6 1
1470.2.i.q 2 63.s even 6 1
2880.2.a.a 1 72.n even 6 1
2880.2.a.q 1 72.p odd 6 1
3600.2.a.f 1 180.p odd 6 1
3600.2.f.i 2 180.x even 12 2
3630.2.a.w 1 99.g even 6 1
3840.2.k.f 2 144.u even 12 2
3840.2.k.y 2 144.w odd 12 2
4410.2.a.z 1 63.l odd 6 1
4800.2.a.d 1 360.bd even 6 1
4800.2.a.cq 1 360.bh odd 6 1
4800.2.f.p 2 360.br even 12 2
4800.2.f.w 2 360.bt odd 12 2
5070.2.a.w 1 117.n odd 6 1
5070.2.b.k 2 117.z even 12 2
7350.2.a.ct 1 315.z even 6 1
8670.2.a.g 1 153.i odd 6 1

Hecke kernels

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

\( T_{7}^{2} - 4 T_{7} + 16 \)
\( T_{11} \)
\( T_{13}^{2} + 2 T_{13} + 4 \)
\( T_{17} + 6 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 16 - 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + 2 T + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 64 + 8 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 36 + 6 T + T^{2} \)
$43$ \( 16 - 4 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 100 - 10 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( 144 - 12 T + T^{2} \)
$89$ \( ( 18 + T )^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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