Properties

Label 810.4.e.h
Level $810$
Weight $4$
Character orbit 810.e
Analytic conductor $47.792$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.7915471046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + 8 q^{8} - 10 q^{10} + (42 \zeta_{6} - 42) q^{11} - 20 \zeta_{6} q^{13} + 8 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + 93 q^{17} + 59 q^{19} + ( - 20 \zeta_{6} + 20) q^{20} - 84 \zeta_{6} q^{22} - 9 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 40 q^{26} - 16 q^{28} + (120 \zeta_{6} - 120) q^{29} - 47 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + (186 \zeta_{6} - 186) q^{34} + 20 q^{35} - 262 q^{37} + (118 \zeta_{6} - 118) q^{38} + 40 \zeta_{6} q^{40} - 126 \zeta_{6} q^{41} + ( - 178 \zeta_{6} + 178) q^{43} + 168 q^{44} + 18 q^{46} + (144 \zeta_{6} - 144) q^{47} + 327 \zeta_{6} q^{49} - 50 \zeta_{6} q^{50} + (80 \zeta_{6} - 80) q^{52} + 741 q^{53} - 210 q^{55} + ( - 32 \zeta_{6} + 32) q^{56} - 240 \zeta_{6} q^{58} + 444 \zeta_{6} q^{59} + (221 \zeta_{6} - 221) q^{61} + 94 q^{62} + 64 q^{64} + ( - 100 \zeta_{6} + 100) q^{65} + 538 \zeta_{6} q^{67} - 372 \zeta_{6} q^{68} + (40 \zeta_{6} - 40) q^{70} + 690 q^{71} - 1126 q^{73} + ( - 524 \zeta_{6} + 524) q^{74} - 236 \zeta_{6} q^{76} + 168 \zeta_{6} q^{77} + (665 \zeta_{6} - 665) q^{79} - 80 q^{80} + 252 q^{82} + (75 \zeta_{6} - 75) q^{83} + 465 \zeta_{6} q^{85} + 356 \zeta_{6} q^{86} + (336 \zeta_{6} - 336) q^{88} - 1086 q^{89} - 80 q^{91} + (36 \zeta_{6} - 36) q^{92} - 288 \zeta_{6} q^{94} + 295 \zeta_{6} q^{95} + (1544 \zeta_{6} - 1544) q^{97} - 654 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 5 q^{5} + 4 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} + 5 q^{5} + 4 q^{7} + 16 q^{8} - 20 q^{10} - 42 q^{11} - 20 q^{13} + 8 q^{14} - 16 q^{16} + 186 q^{17} + 118 q^{19} + 20 q^{20} - 84 q^{22} - 9 q^{23} - 25 q^{25} + 80 q^{26} - 32 q^{28} - 120 q^{29} - 47 q^{31} - 32 q^{32} - 186 q^{34} + 40 q^{35} - 524 q^{37} - 118 q^{38} + 40 q^{40} - 126 q^{41} + 178 q^{43} + 336 q^{44} + 36 q^{46} - 144 q^{47} + 327 q^{49} - 50 q^{50} - 80 q^{52} + 1482 q^{53} - 420 q^{55} + 32 q^{56} - 240 q^{58} + 444 q^{59} - 221 q^{61} + 188 q^{62} + 128 q^{64} + 100 q^{65} + 538 q^{67} - 372 q^{68} - 40 q^{70} + 1380 q^{71} - 2252 q^{73} + 524 q^{74} - 236 q^{76} + 168 q^{77} - 665 q^{79} - 160 q^{80} + 504 q^{82} - 75 q^{83} + 465 q^{85} + 356 q^{86} - 336 q^{88} - 2172 q^{89} - 160 q^{91} - 36 q^{92} - 288 q^{94} + 295 q^{95} - 1544 q^{97} - 1308 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
−1.00000 1.73205i 0 −2.00000 + 3.46410i 2.50000 4.33013i 0 2.00000 + 3.46410i 8.00000 0 −10.0000
541.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 2.50000 + 4.33013i 0 2.00000 3.46410i 8.00000 0 −10.0000
\(n\): 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 810.4.e.h 2
3.b odd 2 1 810.4.e.q 2
9.c even 3 1 270.4.a.i yes 1
9.c even 3 1 inner 810.4.e.h 2
9.d odd 6 1 270.4.a.e 1
9.d odd 6 1 810.4.e.q 2
36.f odd 6 1 2160.4.a.e 1
36.h even 6 1 2160.4.a.o 1
45.h odd 6 1 1350.4.a.u 1
45.j even 6 1 1350.4.a.g 1
45.k odd 12 2 1350.4.c.q 2
45.l even 12 2 1350.4.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.e 1 9.d odd 6 1
270.4.a.i yes 1 9.c even 3 1
810.4.e.h 2 1.a even 1 1 trivial
810.4.e.h 2 9.c even 3 1 inner
810.4.e.q 2 3.b odd 2 1
810.4.e.q 2 9.d odd 6 1
1350.4.a.g 1 45.j even 6 1
1350.4.a.u 1 45.h odd 6 1
1350.4.c.d 2 45.l even 12 2
1350.4.c.q 2 45.k odd 12 2
2160.4.a.e 1 36.f odd 6 1
2160.4.a.o 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 42T_{11} + 1764 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 42T + 1764 \) Copy content Toggle raw display
$13$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$17$ \( (T - 93)^{2} \) Copy content Toggle raw display
$19$ \( (T - 59)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$29$ \( T^{2} + 120T + 14400 \) Copy content Toggle raw display
$31$ \( T^{2} + 47T + 2209 \) Copy content Toggle raw display
$37$ \( (T + 262)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 126T + 15876 \) Copy content Toggle raw display
$43$ \( T^{2} - 178T + 31684 \) Copy content Toggle raw display
$47$ \( T^{2} + 144T + 20736 \) Copy content Toggle raw display
$53$ \( (T - 741)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 444T + 197136 \) Copy content Toggle raw display
$61$ \( T^{2} + 221T + 48841 \) Copy content Toggle raw display
$67$ \( T^{2} - 538T + 289444 \) Copy content Toggle raw display
$71$ \( (T - 690)^{2} \) Copy content Toggle raw display
$73$ \( (T + 1126)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 665T + 442225 \) Copy content Toggle raw display
$83$ \( T^{2} + 75T + 5625 \) Copy content Toggle raw display
$89$ \( (T + 1086)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1544 T + 2383936 \) Copy content Toggle raw display
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