Properties

Label 810.4.e.f
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (14 \zeta_{6} - 14) q^{7} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (14 \zeta_{6} - 14) q^{7} + 8 q^{8} - 10 q^{10} + ( - 3 \zeta_{6} + 3) q^{11} - 47 \zeta_{6} q^{13} - 28 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + 39 q^{17} + 32 q^{19} + ( - 20 \zeta_{6} + 20) q^{20} + 6 \zeta_{6} q^{22} - 99 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 94 q^{26} + 56 q^{28} + ( - 51 \zeta_{6} + 51) q^{29} - 83 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + (78 \zeta_{6} - 78) q^{34} - 70 q^{35} + 314 q^{37} + (64 \zeta_{6} - 64) q^{38} + 40 \zeta_{6} q^{40} - 108 \zeta_{6} q^{41} + (299 \zeta_{6} - 299) q^{43} - 12 q^{44} + 198 q^{46} + ( - 531 \zeta_{6} + 531) q^{47} + 147 \zeta_{6} q^{49} - 50 \zeta_{6} q^{50} + (188 \zeta_{6} - 188) q^{52} - 564 q^{53} + 15 q^{55} + (112 \zeta_{6} - 112) q^{56} + 102 \zeta_{6} q^{58} + 12 \zeta_{6} q^{59} + (230 \zeta_{6} - 230) q^{61} + 166 q^{62} + 64 q^{64} + ( - 235 \zeta_{6} + 235) q^{65} + 268 \zeta_{6} q^{67} - 156 \zeta_{6} q^{68} + ( - 140 \zeta_{6} + 140) q^{70} - 120 q^{71} + 1106 q^{73} + (628 \zeta_{6} - 628) q^{74} - 128 \zeta_{6} q^{76} + 42 \zeta_{6} q^{77} + ( - 739 \zeta_{6} + 739) q^{79} - 80 q^{80} + 216 q^{82} + ( - 1086 \zeta_{6} + 1086) q^{83} + 195 \zeta_{6} q^{85} - 598 \zeta_{6} q^{86} + ( - 24 \zeta_{6} + 24) q^{88} + 120 q^{89} + 658 q^{91} + (396 \zeta_{6} - 396) q^{92} + 1062 \zeta_{6} q^{94} + 160 \zeta_{6} q^{95} + ( - 1642 \zeta_{6} + 1642) q^{97} - 294 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 5 q^{5} - 14 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} + 5 q^{5} - 14 q^{7} + 16 q^{8} - 20 q^{10} + 3 q^{11} - 47 q^{13} - 28 q^{14} - 16 q^{16} + 78 q^{17} + 64 q^{19} + 20 q^{20} + 6 q^{22} - 99 q^{23} - 25 q^{25} + 188 q^{26} + 112 q^{28} + 51 q^{29} - 83 q^{31} - 32 q^{32} - 78 q^{34} - 140 q^{35} + 628 q^{37} - 64 q^{38} + 40 q^{40} - 108 q^{41} - 299 q^{43} - 24 q^{44} + 396 q^{46} + 531 q^{47} + 147 q^{49} - 50 q^{50} - 188 q^{52} - 1128 q^{53} + 30 q^{55} - 112 q^{56} + 102 q^{58} + 12 q^{59} - 230 q^{61} + 332 q^{62} + 128 q^{64} + 235 q^{65} + 268 q^{67} - 156 q^{68} + 140 q^{70} - 240 q^{71} + 2212 q^{73} - 628 q^{74} - 128 q^{76} + 42 q^{77} + 739 q^{79} - 160 q^{80} + 432 q^{82} + 1086 q^{83} + 195 q^{85} - 598 q^{86} + 24 q^{88} + 240 q^{89} + 1316 q^{91} - 396 q^{92} + 1062 q^{94} + 160 q^{95} + 1642 q^{97} - 588 q^{98}+O(q^{100}) \) 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 −7.00000 12.1244i 8.00000 0 −10.0000
541.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 2.50000 + 4.33013i 0 −7.00000 + 12.1244i 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.f 2
3.b odd 2 1 810.4.e.n 2
9.c even 3 1 270.4.a.j yes 1
9.c even 3 1 inner 810.4.e.f 2
9.d odd 6 1 270.4.a.f 1
9.d odd 6 1 810.4.e.n 2
36.f odd 6 1 2160.4.a.b 1
36.h even 6 1 2160.4.a.l 1
45.h odd 6 1 1350.4.a.r 1
45.j even 6 1 1350.4.a.e 1
45.k odd 12 2 1350.4.c.j 2
45.l even 12 2 1350.4.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.f 1 9.d odd 6 1
270.4.a.j yes 1 9.c even 3 1
810.4.e.f 2 1.a even 1 1 trivial
810.4.e.f 2 9.c even 3 1 inner
810.4.e.n 2 3.b odd 2 1
810.4.e.n 2 9.d odd 6 1
1350.4.a.e 1 45.j even 6 1
1350.4.a.r 1 45.h odd 6 1
1350.4.c.j 2 45.k odd 12 2
1350.4.c.k 2 45.l even 12 2
2160.4.a.b 1 36.f odd 6 1
2160.4.a.l 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 \(S_{4}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{2} + 14T_{7} + 196 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) 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
$7$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 47T + 2209 \) Copy content Toggle raw display
$17$ \( (T - 39)^{2} \) Copy content Toggle raw display
$19$ \( (T - 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 99T + 9801 \) Copy content Toggle raw display
$29$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$31$ \( T^{2} + 83T + 6889 \) Copy content Toggle raw display
$37$ \( (T - 314)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 108T + 11664 \) Copy content Toggle raw display
$43$ \( T^{2} + 299T + 89401 \) Copy content Toggle raw display
$47$ \( T^{2} - 531T + 281961 \) Copy content Toggle raw display
$53$ \( (T + 564)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 230T + 52900 \) Copy content Toggle raw display
$67$ \( T^{2} - 268T + 71824 \) Copy content Toggle raw display
$71$ \( (T + 120)^{2} \) Copy content Toggle raw display
$73$ \( (T - 1106)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 739T + 546121 \) Copy content Toggle raw display
$83$ \( T^{2} - 1086 T + 1179396 \) Copy content Toggle raw display
$89$ \( (T - 120)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1642 T + 2696164 \) Copy content Toggle raw display
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