Properties

Label 387.2.h.a
Level 387
Weight 2
Character orbit 387.h
Analytic conductor 3.090
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.09021055822\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + ( -1 + \zeta_{6} ) q^{5} -3 \zeta_{6} q^{7} + 3 q^{8} +O(q^{10})\) \( q - q^{2} - q^{4} + ( -1 + \zeta_{6} ) q^{5} -3 \zeta_{6} q^{7} + 3 q^{8} + ( 1 - \zeta_{6} ) q^{10} + 5 \zeta_{6} q^{13} + 3 \zeta_{6} q^{14} - q^{16} + 3 \zeta_{6} q^{17} + ( -1 + \zeta_{6} ) q^{19} + ( 1 - \zeta_{6} ) q^{20} + ( -7 + 7 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} -5 \zeta_{6} q^{26} + 3 \zeta_{6} q^{28} + 3 \zeta_{6} q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} -5 q^{32} -3 \zeta_{6} q^{34} + 3 q^{35} + ( 9 - 9 \zeta_{6} ) q^{37} + ( 1 - \zeta_{6} ) q^{38} + ( -3 + 3 \zeta_{6} ) q^{40} + 10 q^{41} + ( -7 + 6 \zeta_{6} ) q^{43} + ( 7 - 7 \zeta_{6} ) q^{46} + 8 q^{47} + ( -2 + 2 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{50} -5 \zeta_{6} q^{52} + ( -5 + 5 \zeta_{6} ) q^{53} -9 \zeta_{6} q^{56} -3 \zeta_{6} q^{58} -12 q^{59} + 13 \zeta_{6} q^{61} + ( 5 - 5 \zeta_{6} ) q^{62} + 7 q^{64} -5 q^{65} + ( 3 - 3 \zeta_{6} ) q^{67} -3 \zeta_{6} q^{68} -3 q^{70} -\zeta_{6} q^{71} -11 \zeta_{6} q^{73} + ( -9 + 9 \zeta_{6} ) q^{74} + ( 1 - \zeta_{6} ) q^{76} + 5 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} -10 q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} -3 q^{85} + ( 7 - 6 \zeta_{6} ) q^{86} + ( -1 + \zeta_{6} ) q^{89} + ( 15 - 15 \zeta_{6} ) q^{91} + ( 7 - 7 \zeta_{6} ) q^{92} -8 q^{94} -\zeta_{6} q^{95} -2 q^{97} + ( 2 - 2 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{4} - q^{5} - 3q^{7} + 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{4} - q^{5} - 3q^{7} + 6q^{8} + q^{10} + 5q^{13} + 3q^{14} - 2q^{16} + 3q^{17} - q^{19} + q^{20} - 7q^{23} + 4q^{25} - 5q^{26} + 3q^{28} + 3q^{29} - 5q^{31} - 10q^{32} - 3q^{34} + 6q^{35} + 9q^{37} + q^{38} - 3q^{40} + 20q^{41} - 8q^{43} + 7q^{46} + 16q^{47} - 2q^{49} - 4q^{50} - 5q^{52} - 5q^{53} - 9q^{56} - 3q^{58} - 24q^{59} + 13q^{61} + 5q^{62} + 14q^{64} - 10q^{65} + 3q^{67} - 3q^{68} - 6q^{70} - q^{71} - 11q^{73} - 9q^{74} + q^{76} + 5q^{79} + q^{80} - 20q^{82} + 9q^{83} - 6q^{85} + 8q^{86} - q^{89} + 15q^{91} + 7q^{92} - 16q^{94} - q^{95} - 4q^{97} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(173\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 0 −1.00000 −0.500000 0.866025i 0 −1.50000 + 2.59808i 3.00000 0 0.500000 + 0.866025i
307.1 −1.00000 0 −1.00000 −0.500000 + 0.866025i 0 −1.50000 2.59808i 3.00000 0 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 387.2.h.a 2
3.b odd 2 1 43.2.c.a 2
12.b even 2 1 688.2.i.d 2
43.c even 3 1 inner 387.2.h.a 2
129.f odd 6 1 43.2.c.a 2
129.f odd 6 1 1849.2.a.c 1
129.h even 6 1 1849.2.a.a 1
516.p even 6 1 688.2.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.a 2 3.b odd 2 1
43.2.c.a 2 129.f odd 6 1
387.2.h.a 2 1.a even 1 1 trivial
387.2.h.a 2 43.c even 3 1 inner
688.2.i.d 2 12.b even 2 1
688.2.i.d 2 516.p even 6 1
1849.2.a.a 1 129.h even 6 1
1849.2.a.c 1 129.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(387, [\chi])\).

Hecke characteristic polynomials

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