Properties

Label 43.2.c.a
Level 43
Weight 2
Character orbit 43.c
Analytic conductor 0.343
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 43.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.343356728692\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 + \zeta_{6} ) q^{3} - q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( -1 + \zeta_{6} ) q^{6} -3 \zeta_{6} q^{7} -3 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 + \zeta_{6} ) q^{3} - q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( -1 + \zeta_{6} ) q^{6} -3 \zeta_{6} q^{7} -3 q^{8} + 2 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( 1 - \zeta_{6} ) q^{12} + 5 \zeta_{6} q^{13} -3 \zeta_{6} q^{14} + \zeta_{6} q^{15} - q^{16} -3 \zeta_{6} q^{17} + 2 \zeta_{6} q^{18} + ( -1 + \zeta_{6} ) q^{19} + ( -1 + \zeta_{6} ) q^{20} + 3 q^{21} + ( 7 - 7 \zeta_{6} ) q^{23} + ( 3 - 3 \zeta_{6} ) q^{24} + 4 \zeta_{6} q^{25} + 5 \zeta_{6} q^{26} -5 q^{27} + 3 \zeta_{6} q^{28} -3 \zeta_{6} q^{29} + \zeta_{6} q^{30} + ( -5 + 5 \zeta_{6} ) q^{31} + 5 q^{32} -3 \zeta_{6} q^{34} -3 q^{35} -2 \zeta_{6} q^{36} + ( 9 - 9 \zeta_{6} ) q^{37} + ( -1 + \zeta_{6} ) q^{38} -5 q^{39} + ( -3 + 3 \zeta_{6} ) q^{40} -10 q^{41} + 3 q^{42} + ( -7 + 6 \zeta_{6} ) q^{43} + 2 q^{45} + ( 7 - 7 \zeta_{6} ) q^{46} -8 q^{47} + ( 1 - \zeta_{6} ) q^{48} + ( -2 + 2 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{50} + 3 q^{51} -5 \zeta_{6} q^{52} + ( 5 - 5 \zeta_{6} ) q^{53} -5 q^{54} + 9 \zeta_{6} q^{56} -\zeta_{6} q^{57} -3 \zeta_{6} q^{58} + 12 q^{59} -\zeta_{6} q^{60} + 13 \zeta_{6} q^{61} + ( -5 + 5 \zeta_{6} ) q^{62} + ( 6 - 6 \zeta_{6} ) q^{63} + 7 q^{64} + 5 q^{65} + ( 3 - 3 \zeta_{6} ) q^{67} + 3 \zeta_{6} q^{68} + 7 \zeta_{6} q^{69} -3 q^{70} + \zeta_{6} q^{71} -6 \zeta_{6} q^{72} -11 \zeta_{6} q^{73} + ( 9 - 9 \zeta_{6} ) q^{74} -4 q^{75} + ( 1 - \zeta_{6} ) q^{76} -5 q^{78} + 5 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -10 q^{82} + ( -9 + 9 \zeta_{6} ) q^{83} -3 q^{84} -3 q^{85} + ( -7 + 6 \zeta_{6} ) q^{86} + 3 q^{87} + ( 1 - \zeta_{6} ) q^{89} + 2 q^{90} + ( 15 - 15 \zeta_{6} ) q^{91} + ( -7 + 7 \zeta_{6} ) q^{92} -5 \zeta_{6} q^{93} -8 q^{94} + \zeta_{6} q^{95} + ( -5 + 5 \zeta_{6} ) q^{96} -2 q^{97} + ( -2 + 2 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - q^{3} - 2q^{4} + q^{5} - q^{6} - 3q^{7} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - q^{3} - 2q^{4} + q^{5} - q^{6} - 3q^{7} - 6q^{8} + 2q^{9} + q^{10} + q^{12} + 5q^{13} - 3q^{14} + q^{15} - 2q^{16} - 3q^{17} + 2q^{18} - q^{19} - q^{20} + 6q^{21} + 7q^{23} + 3q^{24} + 4q^{25} + 5q^{26} - 10q^{27} + 3q^{28} - 3q^{29} + q^{30} - 5q^{31} + 10q^{32} - 3q^{34} - 6q^{35} - 2q^{36} + 9q^{37} - q^{38} - 10q^{39} - 3q^{40} - 20q^{41} + 6q^{42} - 8q^{43} + 4q^{45} + 7q^{46} - 16q^{47} + q^{48} - 2q^{49} + 4q^{50} + 6q^{51} - 5q^{52} + 5q^{53} - 10q^{54} + 9q^{56} - q^{57} - 3q^{58} + 24q^{59} - q^{60} + 13q^{61} - 5q^{62} + 6q^{63} + 14q^{64} + 10q^{65} + 3q^{67} + 3q^{68} + 7q^{69} - 6q^{70} + q^{71} - 6q^{72} - 11q^{73} + 9q^{74} - 8q^{75} + q^{76} - 10q^{78} + 5q^{79} - q^{80} - q^{81} - 20q^{82} - 9q^{83} - 6q^{84} - 6q^{85} - 8q^{86} + 6q^{87} + q^{89} + 4q^{90} + 15q^{91} - 7q^{92} - 5q^{93} - 16q^{94} + q^{95} - 5q^{96} - 4q^{97} - 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 −0.500000 + 0.866025i −1.00000 0.500000 0.866025i −0.500000 + 0.866025i −1.50000 2.59808i −3.00000 1.00000 + 1.73205i 0.500000 0.866025i
36.1 1.00000 −0.500000 0.866025i −1.00000 0.500000 + 0.866025i −0.500000 0.866025i −1.50000 + 2.59808i −3.00000 1.00000 1.73205i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.2.c.a 2
3.b odd 2 1 387.2.h.a 2
4.b odd 2 1 688.2.i.d 2
43.c even 3 1 inner 43.2.c.a 2
43.c even 3 1 1849.2.a.c 1
43.d odd 6 1 1849.2.a.a 1
129.f odd 6 1 387.2.h.a 2
172.g odd 6 1 688.2.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.a 2 1.a even 1 1 trivial
43.2.c.a 2 43.c even 3 1 inner
387.2.h.a 2 3.b odd 2 1
387.2.h.a 2 129.f odd 6 1
688.2.i.d 2 4.b odd 2 1
688.2.i.d 2 172.g odd 6 1
1849.2.a.a 1 43.d odd 6 1
1849.2.a.c 1 43.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + 2 T^{2} )^{2} \)
$3$ \( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \)
$29$ \( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \)
$37$ \( 1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 8 T + 43 T^{2} \)
$47$ \( ( 1 + 8 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 5 T - 28 T^{2} - 265 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 - 12 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \)
$67$ \( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \)
$71$ \( 1 - T - 70 T^{2} - 71 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 5 T - 54 T^{2} - 395 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \)
$89$ \( 1 - T - 88 T^{2} - 89 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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