Properties

Label 1782.2.e.f
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -2 + 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -2 + 2 \zeta_{6} ) q^{7} + q^{8} + ( -1 + \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} -2 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 6 q^{17} -4 q^{19} -\zeta_{6} q^{22} + 6 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} -4 q^{26} + 2 q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} -8 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( -6 + 6 \zeta_{6} ) q^{34} -10 q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + 6 \zeta_{6} q^{41} + ( -8 + 8 \zeta_{6} ) q^{43} + q^{44} -6 q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + 5 \zeta_{6} q^{50} + ( 4 - 4 \zeta_{6} ) q^{52} + ( -2 + 2 \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{58} + ( -8 + 8 \zeta_{6} ) q^{61} + 8 q^{62} + q^{64} + 4 \zeta_{6} q^{67} -6 \zeta_{6} q^{68} -6 q^{71} + 2 q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{76} -2 \zeta_{6} q^{77} + ( -14 + 14 \zeta_{6} ) q^{79} -6 q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} -8 \zeta_{6} q^{86} + ( -1 + \zeta_{6} ) q^{88} + 6 q^{89} -8 q^{91} + ( 6 - 6 \zeta_{6} ) q^{92} -6 \zeta_{6} q^{94} + ( -14 + 14 \zeta_{6} ) q^{97} -3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 2q^{7} + 2q^{8} - q^{11} + 4q^{13} - 2q^{14} - q^{16} + 12q^{17} - 8q^{19} - q^{22} + 6q^{23} + 5q^{25} - 8q^{26} + 4q^{28} + 6q^{29} - 8q^{31} - q^{32} - 6q^{34} - 20q^{37} + 4q^{38} + 6q^{41} - 8q^{43} + 2q^{44} - 12q^{46} - 6q^{47} + 3q^{49} + 5q^{50} + 4q^{52} - 2q^{56} + 6q^{58} - 8q^{61} + 16q^{62} + 2q^{64} + 4q^{67} - 6q^{68} - 12q^{71} + 4q^{73} + 10q^{74} + 4q^{76} - 2q^{77} - 14q^{79} - 12q^{82} - 12q^{83} - 8q^{86} - q^{88} + 12q^{89} - 16q^{91} + 6q^{92} - 6q^{94} - 14q^{97} - 6q^{98} + O(q^{100}) \)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ 1
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ \( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 10 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{2} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 2 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 5 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \)
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