Properties

Label 1782.2.e.v
Level $1782$
Weight $2$
Character orbit 1782.e
Analytic conductor $14.229$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1782 = 2 \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1782.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.2293416402\)
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 66)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} - q^{8} + 2 q^{10} + ( -1 + \zeta_{6} ) q^{11} + 6 \zeta_{6} q^{13} -4 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -2 q^{17} + 4 q^{19} + ( 2 - 2 \zeta_{6} ) q^{20} + \zeta_{6} q^{22} + 4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + 6 q^{26} -4 q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} + \zeta_{6} q^{32} + ( -2 + 2 \zeta_{6} ) q^{34} + 8 q^{35} + 6 q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} -2 \zeta_{6} q^{40} -6 \zeta_{6} q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} + q^{44} + 4 q^{46} + ( -12 + 12 \zeta_{6} ) q^{47} -9 \zeta_{6} q^{49} -\zeta_{6} q^{50} + ( 6 - 6 \zeta_{6} ) q^{52} -2 q^{53} -2 q^{55} + ( -4 + 4 \zeta_{6} ) q^{56} -6 \zeta_{6} q^{58} + 12 \zeta_{6} q^{59} + ( 14 - 14 \zeta_{6} ) q^{61} + q^{64} + ( -12 + 12 \zeta_{6} ) q^{65} -4 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + ( 8 - 8 \zeta_{6} ) q^{70} + 12 q^{71} -6 q^{73} + ( 6 - 6 \zeta_{6} ) q^{74} -4 \zeta_{6} q^{76} + 4 \zeta_{6} q^{77} + ( 4 - 4 \zeta_{6} ) q^{79} -2 q^{80} -6 q^{82} + ( 4 - 4 \zeta_{6} ) q^{83} -4 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + ( 1 - \zeta_{6} ) q^{88} -10 q^{89} + 24 q^{91} + ( 4 - 4 \zeta_{6} ) q^{92} + 12 \zeta_{6} q^{94} + 8 \zeta_{6} q^{95} + ( 14 - 14 \zeta_{6} ) q^{97} -9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + 2q^{5} + 4q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + 2q^{5} + 4q^{7} - 2q^{8} + 4q^{10} - q^{11} + 6q^{13} - 4q^{14} - q^{16} - 4q^{17} + 8q^{19} + 2q^{20} + q^{22} + 4q^{23} + q^{25} + 12q^{26} - 8q^{28} + 6q^{29} + q^{32} - 2q^{34} + 16q^{35} + 12q^{37} + 4q^{38} - 2q^{40} - 6q^{41} - 4q^{43} + 2q^{44} + 8q^{46} - 12q^{47} - 9q^{49} - q^{50} + 6q^{52} - 4q^{53} - 4q^{55} - 4q^{56} - 6q^{58} + 12q^{59} + 14q^{61} + 2q^{64} - 12q^{65} - 4q^{67} + 2q^{68} + 8q^{70} + 24q^{71} - 12q^{73} + 6q^{74} - 4q^{76} + 4q^{77} + 4q^{79} - 4q^{80} - 12q^{82} + 4q^{83} - 4q^{85} + 4q^{86} + q^{88} - 20q^{89} + 48q^{91} + 4q^{92} + 12q^{94} + 8q^{95} + 14q^{97} - 18q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1782\mathbb{Z}\right)^\times\).

\(n\) \(1135\) \(1541\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
595.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 1.73205i 0 2.00000 + 3.46410i −1.00000 0 2.00000
1189.1 0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 + 1.73205i 0 2.00000 3.46410i −1.00000 0 2.00000
\(n\): 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 1782.2.e.v 2
3.b odd 2 1 1782.2.e.e 2
9.c even 3 1 198.2.a.a 1
9.c even 3 1 inner 1782.2.e.v 2
9.d odd 6 1 66.2.a.b 1
9.d odd 6 1 1782.2.e.e 2
36.f odd 6 1 1584.2.a.f 1
36.h even 6 1 528.2.a.j 1
45.h odd 6 1 1650.2.a.k 1
45.j even 6 1 4950.2.a.bu 1
45.k odd 12 2 4950.2.c.p 2
45.l even 12 2 1650.2.c.e 2
63.l odd 6 1 9702.2.a.x 1
63.o even 6 1 3234.2.a.t 1
72.j odd 6 1 2112.2.a.r 1
72.l even 6 1 2112.2.a.e 1
72.n even 6 1 6336.2.a.bw 1
72.p odd 6 1 6336.2.a.cj 1
99.g even 6 1 726.2.a.c 1
99.h odd 6 1 2178.2.a.g 1
99.n odd 30 4 726.2.e.g 4
99.p even 30 4 726.2.e.o 4
396.o odd 6 1 5808.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 9.d odd 6 1
198.2.a.a 1 9.c even 3 1
528.2.a.j 1 36.h even 6 1
726.2.a.c 1 99.g even 6 1
726.2.e.g 4 99.n odd 30 4
726.2.e.o 4 99.p even 30 4
1584.2.a.f 1 36.f odd 6 1
1650.2.a.k 1 45.h odd 6 1
1650.2.c.e 2 45.l even 12 2
1782.2.e.e 2 3.b odd 2 1
1782.2.e.e 2 9.d odd 6 1
1782.2.e.v 2 1.a even 1 1 trivial
1782.2.e.v 2 9.c even 3 1 inner
2112.2.a.e 1 72.l even 6 1
2112.2.a.r 1 72.j odd 6 1
2178.2.a.g 1 99.h odd 6 1
3234.2.a.t 1 63.o even 6 1
4950.2.a.bu 1 45.j even 6 1
4950.2.c.p 2 45.k odd 12 2
5808.2.a.bc 1 396.o odd 6 1
6336.2.a.bw 1 72.n even 6 1
6336.2.a.cj 1 72.p odd 6 1
9702.2.a.x 1 63.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1782, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{7}^{2} - 4 T_{7} + 16 \)
\( T_{13}^{2} - 6 T_{13} + 36 \)
\( T_{17} + 2 \)
\( T_{23}^{2} - 4 T_{23} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( 16 - 4 T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( 36 - 6 T + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( 36 - 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( 36 + 6 T + T^{2} \)
$43$ \( 16 + 4 T + T^{2} \)
$47$ \( 144 + 12 T + T^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( 144 - 12 T + T^{2} \)
$61$ \( 196 - 14 T + T^{2} \)
$67$ \( 16 + 4 T + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( 16 - 4 T + T^{2} \)
$83$ \( 16 - 4 T + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 196 - 14 T + T^{2} \)
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