Properties

Label 1782.2.e.s
Level $1782$
Weight $2$
Character orbit 1782.e
Analytic conductor $14.229$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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.s 2
3.b odd 2 1 1782.2.e.f 2
9.c even 3 1 66.2.a.a 1
9.c even 3 1 inner 1782.2.e.s 2
9.d odd 6 1 198.2.a.e 1
9.d odd 6 1 1782.2.e.f 2
36.f odd 6 1 528.2.a.d 1
36.h even 6 1 1584.2.a.h 1
45.h odd 6 1 4950.2.a.g 1
45.j even 6 1 1650.2.a.m 1
45.k odd 12 2 1650.2.c.d 2
45.l even 12 2 4950.2.c.r 2
63.l odd 6 1 3234.2.a.d 1
63.o even 6 1 9702.2.a.bu 1
72.j odd 6 1 6336.2.a.bj 1
72.l even 6 1 6336.2.a.bf 1
72.n even 6 1 2112.2.a.i 1
72.p odd 6 1 2112.2.a.v 1
99.g even 6 1 2178.2.a.b 1
99.h odd 6 1 726.2.a.i 1
99.m even 15 4 726.2.e.k 4
99.o odd 30 4 726.2.e.b 4
396.k even 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.c even 3 1
198.2.a.e 1 9.d odd 6 1
528.2.a.d 1 36.f odd 6 1
726.2.a.i 1 99.h odd 6 1
726.2.e.b 4 99.o odd 30 4
726.2.e.k 4 99.m even 15 4
1584.2.a.h 1 36.h even 6 1
1650.2.a.m 1 45.j even 6 1
1650.2.c.d 2 45.k odd 12 2
1782.2.e.f 2 3.b odd 2 1
1782.2.e.f 2 9.d odd 6 1
1782.2.e.s 2 1.a even 1 1 trivial
1782.2.e.s 2 9.c even 3 1 inner
2112.2.a.i 1 72.n even 6 1
2112.2.a.v 1 72.p odd 6 1
2178.2.a.b 1 99.g even 6 1
3234.2.a.d 1 63.l odd 6 1
4950.2.a.g 1 45.h odd 6 1
4950.2.c.r 2 45.l even 12 2
5808.2.a.l 1 396.k even 6 1
6336.2.a.bf 1 72.l even 6 1
6336.2.a.bj 1 72.j odd 6 1
9702.2.a.bu 1 63.o even 6 1

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} + 16 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display
\( T_{23}^{2} + 6T_{23} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
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