Properties

Label 810.2.i.e
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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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 1.86603 1.23205i 0 1.73205 1.00000i 1.00000i 0 −1.00000 + 2.00000i
109.2 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
379.1 −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
379.2 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
\(n\): 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.e 4
3.b odd 2 1 810.2.i.b 4
5.b even 2 1 inner 810.2.i.e 4
9.c even 3 1 30.2.c.a 2
9.c even 3 1 inner 810.2.i.e 4
9.d odd 6 1 90.2.c.a 2
9.d odd 6 1 810.2.i.b 4
15.d odd 2 1 810.2.i.b 4
36.f odd 6 1 240.2.f.a 2
36.h even 6 1 720.2.f.f 2
45.h odd 6 1 90.2.c.a 2
45.h odd 6 1 810.2.i.b 4
45.j even 6 1 30.2.c.a 2
45.j even 6 1 inner 810.2.i.e 4
45.k odd 12 1 150.2.a.a 1
45.k odd 12 1 150.2.a.c 1
45.l even 12 1 450.2.a.b 1
45.l even 12 1 450.2.a.f 1
63.g even 3 1 1470.2.n.h 4
63.h even 3 1 1470.2.n.h 4
63.k odd 6 1 1470.2.n.a 4
63.l odd 6 1 1470.2.g.g 2
63.t odd 6 1 1470.2.n.a 4
72.j odd 6 1 2880.2.f.e 2
72.l even 6 1 2880.2.f.c 2
72.n even 6 1 960.2.f.h 2
72.p odd 6 1 960.2.f.i 2
144.v odd 12 1 3840.2.d.j 2
144.v odd 12 1 3840.2.d.x 2
144.x even 12 1 3840.2.d.g 2
144.x even 12 1 3840.2.d.y 2
180.n even 6 1 720.2.f.f 2
180.p odd 6 1 240.2.f.a 2
180.v odd 12 1 3600.2.a.o 1
180.v odd 12 1 3600.2.a.bg 1
180.x even 12 1 1200.2.a.g 1
180.x even 12 1 1200.2.a.m 1
315.q odd 6 1 1470.2.n.a 4
315.r even 6 1 1470.2.n.h 4
315.bg odd 6 1 1470.2.g.g 2
315.bn odd 6 1 1470.2.n.a 4
315.bo even 6 1 1470.2.n.h 4
315.cb even 12 1 7350.2.a.bg 1
315.cb even 12 1 7350.2.a.cc 1
360.z odd 6 1 960.2.f.i 2
360.bd even 6 1 2880.2.f.c 2
360.bh odd 6 1 2880.2.f.e 2
360.bk even 6 1 960.2.f.h 2
360.bo even 12 1 4800.2.a.m 1
360.bo even 12 1 4800.2.a.cj 1
360.bu odd 12 1 4800.2.a.l 1
360.bu odd 12 1 4800.2.a.cg 1
720.ce even 12 1 3840.2.d.g 2
720.ce even 12 1 3840.2.d.y 2
720.cz odd 12 1 3840.2.d.j 2
720.cz odd 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.c even 3 1
30.2.c.a 2 45.j even 6 1
90.2.c.a 2 9.d odd 6 1
90.2.c.a 2 45.h odd 6 1
150.2.a.a 1 45.k odd 12 1
150.2.a.c 1 45.k odd 12 1
240.2.f.a 2 36.f odd 6 1
240.2.f.a 2 180.p odd 6 1
450.2.a.b 1 45.l even 12 1
450.2.a.f 1 45.l even 12 1
720.2.f.f 2 36.h even 6 1
720.2.f.f 2 180.n even 6 1
810.2.i.b 4 3.b odd 2 1
810.2.i.b 4 9.d odd 6 1
810.2.i.b 4 15.d odd 2 1
810.2.i.b 4 45.h odd 6 1
810.2.i.e 4 1.a even 1 1 trivial
810.2.i.e 4 5.b even 2 1 inner
810.2.i.e 4 9.c even 3 1 inner
810.2.i.e 4 45.j even 6 1 inner
960.2.f.h 2 72.n even 6 1
960.2.f.h 2 360.bk even 6 1
960.2.f.i 2 72.p odd 6 1
960.2.f.i 2 360.z odd 6 1
1200.2.a.g 1 180.x even 12 1
1200.2.a.m 1 180.x even 12 1
1470.2.g.g 2 63.l odd 6 1
1470.2.g.g 2 315.bg odd 6 1
1470.2.n.a 4 63.k odd 6 1
1470.2.n.a 4 63.t odd 6 1
1470.2.n.a 4 315.q odd 6 1
1470.2.n.a 4 315.bn odd 6 1
1470.2.n.h 4 63.g even 3 1
1470.2.n.h 4 63.h even 3 1
1470.2.n.h 4 315.r even 6 1
1470.2.n.h 4 315.bo even 6 1
2880.2.f.c 2 72.l even 6 1
2880.2.f.c 2 360.bd even 6 1
2880.2.f.e 2 72.j odd 6 1
2880.2.f.e 2 360.bh odd 6 1
3600.2.a.o 1 180.v odd 12 1
3600.2.a.bg 1 180.v odd 12 1
3840.2.d.g 2 144.x even 12 1
3840.2.d.g 2 720.ce even 12 1
3840.2.d.j 2 144.v odd 12 1
3840.2.d.j 2 720.cz odd 12 1
3840.2.d.x 2 144.v odd 12 1
3840.2.d.x 2 720.cz odd 12 1
3840.2.d.y 2 144.x even 12 1
3840.2.d.y 2 720.ce even 12 1
4800.2.a.l 1 360.bu odd 12 1
4800.2.a.m 1 360.bo even 12 1
4800.2.a.cg 1 360.bu odd 12 1
4800.2.a.cj 1 360.bo even 12 1
7350.2.a.bg 1 315.cb even 12 1
7350.2.a.cc 1 315.cb even 12 1

Hecke kernels

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

\( T_{7}^{4} - 4T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$71$ \( (T - 12)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T - 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
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