Properties

Label 1782.2.e.n
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: \(\Q(\sqrt{-3}) \)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} - q^{8} -4 q^{10} + ( 1 - \zeta_{6} ) q^{11} -4 \zeta_{6} q^{13} -2 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 2 q^{17} + ( -4 + 4 \zeta_{6} ) q^{20} -\zeta_{6} q^{22} -6 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} -4 q^{26} -2 q^{28} + ( 10 - 10 \zeta_{6} ) q^{29} + 8 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{34} -8 q^{35} -2 q^{37} + 4 \zeta_{6} q^{40} + 2 \zeta_{6} q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} - q^{44} -6 q^{46} + ( -2 + 2 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + 11 \zeta_{6} q^{50} + ( -4 + 4 \zeta_{6} ) q^{52} -4 q^{53} -4 q^{55} + ( -2 + 2 \zeta_{6} ) q^{56} -10 \zeta_{6} q^{58} + ( 8 - 8 \zeta_{6} ) q^{61} + 8 q^{62} + q^{64} + ( -16 + 16 \zeta_{6} ) q^{65} + 12 \zeta_{6} q^{67} -2 \zeta_{6} q^{68} + ( -8 + 8 \zeta_{6} ) q^{70} -2 q^{71} -6 q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} -2 \zeta_{6} q^{77} + ( -10 + 10 \zeta_{6} ) q^{79} + 4 q^{80} + 2 q^{82} + ( 4 - 4 \zeta_{6} ) q^{83} -8 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + ( -1 + \zeta_{6} ) q^{88} -10 q^{89} -8 q^{91} + ( -6 + 6 \zeta_{6} ) q^{92} + 2 \zeta_{6} q^{94} + ( 2 - 2 \zeta_{6} ) q^{97} + 3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 4q^{5} + 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 4q^{5} + 2q^{7} - 2q^{8} - 8q^{10} + q^{11} - 4q^{13} - 2q^{14} - q^{16} + 4q^{17} - 4q^{20} - q^{22} - 6q^{23} - 11q^{25} - 8q^{26} - 4q^{28} + 10q^{29} + 8q^{31} + q^{32} + 2q^{34} - 16q^{35} - 4q^{37} + 4q^{40} + 2q^{41} - 4q^{43} - 2q^{44} - 12q^{46} - 2q^{47} + 3q^{49} + 11q^{50} - 4q^{52} - 8q^{53} - 8q^{55} - 2q^{56} - 10q^{58} + 8q^{61} + 16q^{62} + 2q^{64} - 16q^{65} + 12q^{67} - 2q^{68} - 8q^{70} - 4q^{71} - 12q^{73} - 2q^{74} - 2q^{77} - 10q^{79} + 8q^{80} + 4q^{82} + 4q^{83} - 8q^{85} + 4q^{86} - q^{88} - 20q^{89} - 16q^{91} - 6q^{92} + 2q^{94} + 2q^{97} + 6q^{98} + O(q^{100}) \)

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 −2.00000 + 3.46410i 0 1.00000 + 1.73205i −1.00000 0 −4.00000
1189.1 0.500000 0.866025i 0 −0.500000 0.866025i −2.00000 3.46410i 0 1.00000 1.73205i −1.00000 0 −4.00000
\(n\): 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.n 2
3.b odd 2 1 1782.2.e.l 2
9.c even 3 1 198.2.a.c 1
9.c even 3 1 inner 1782.2.e.n 2
9.d odd 6 1 66.2.a.c 1
9.d odd 6 1 1782.2.e.l 2
36.f odd 6 1 1584.2.a.s 1
36.h even 6 1 528.2.a.a 1
45.h odd 6 1 1650.2.a.c 1
45.j even 6 1 4950.2.a.bo 1
45.k odd 12 2 4950.2.c.d 2
45.l even 12 2 1650.2.c.m 2
63.l odd 6 1 9702.2.a.a 1
63.o even 6 1 3234.2.a.s 1
72.j odd 6 1 2112.2.a.n 1
72.l even 6 1 2112.2.a.bd 1
72.n even 6 1 6336.2.a.c 1
72.p odd 6 1 6336.2.a.d 1
99.g even 6 1 726.2.a.d 1
99.h odd 6 1 2178.2.a.m 1
99.n odd 30 4 726.2.e.e 4
99.p even 30 4 726.2.e.m 4
396.o odd 6 1 5808.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.c 1 9.d odd 6 1
198.2.a.c 1 9.c even 3 1
528.2.a.a 1 36.h even 6 1
726.2.a.d 1 99.g even 6 1
726.2.e.e 4 99.n odd 30 4
726.2.e.m 4 99.p even 30 4
1584.2.a.s 1 36.f odd 6 1
1650.2.a.c 1 45.h odd 6 1
1650.2.c.m 2 45.l even 12 2
1782.2.e.l 2 3.b odd 2 1
1782.2.e.l 2 9.d odd 6 1
1782.2.e.n 2 1.a even 1 1 trivial
1782.2.e.n 2 9.c even 3 1 inner
2112.2.a.n 1 72.j odd 6 1
2112.2.a.bd 1 72.l even 6 1
2178.2.a.m 1 99.h odd 6 1
3234.2.a.s 1 63.o even 6 1
4950.2.a.bo 1 45.j even 6 1
4950.2.c.d 2 45.k odd 12 2
5808.2.a.b 1 396.o odd 6 1
6336.2.a.c 1 72.n even 6 1
6336.2.a.d 1 72.p odd 6 1
9702.2.a.a 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 \(S_{2}^{\mathrm{new}}(1782, [\chi])\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ 1
$5$ \( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ \( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \)
$29$ \( 1 - 10 T + 71 T^{2} - 290 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 4 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 2 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 6 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 4 T - 67 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \)
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