Properties

Label 76.2.e.a
Level $76$
Weight $2$
Character orbit 76.e
Analytic conductor $0.607$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 76.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.606863055362\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} -4 q^{11} + \zeta_{6} q^{13} -\zeta_{6} q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + ( -5 + 2 \zeta_{6} ) q^{19} -5 \zeta_{6} q^{23} + 4 \zeta_{6} q^{25} + 5 q^{27} -7 \zeta_{6} q^{29} + 4 q^{31} + ( -4 + 4 \zeta_{6} ) q^{33} + 10 q^{37} + q^{39} + ( 5 - 5 \zeta_{6} ) q^{41} + ( 5 - 5 \zeta_{6} ) q^{43} + 2 q^{45} + 7 \zeta_{6} q^{47} -7 q^{49} + 3 \zeta_{6} q^{51} -11 \zeta_{6} q^{53} + ( -4 + 4 \zeta_{6} ) q^{55} + ( -3 + 5 \zeta_{6} ) q^{57} + ( -3 + 3 \zeta_{6} ) q^{59} -11 \zeta_{6} q^{61} + q^{65} + 3 \zeta_{6} q^{67} -5 q^{69} + ( -11 + 11 \zeta_{6} ) q^{71} + ( -15 + 15 \zeta_{6} ) q^{73} + 4 q^{75} + ( 13 - 13 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 3 \zeta_{6} q^{85} -7 q^{87} -3 \zeta_{6} q^{89} + ( 4 - 4 \zeta_{6} ) q^{93} + ( -3 + 5 \zeta_{6} ) q^{95} + ( 5 - 5 \zeta_{6} ) q^{97} -8 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + q^{3} + q^{5} + 2q^{9} - 8q^{11} + q^{13} - q^{15} - 3q^{17} - 8q^{19} - 5q^{23} + 4q^{25} + 10q^{27} - 7q^{29} + 8q^{31} - 4q^{33} + 20q^{37} + 2q^{39} + 5q^{41} + 5q^{43} + 4q^{45} + 7q^{47} - 14q^{49} + 3q^{51} - 11q^{53} - 4q^{55} - q^{57} - 3q^{59} - 11q^{61} + 2q^{65} + 3q^{67} - 10q^{69} - 11q^{71} - 15q^{73} + 8q^{75} + 13q^{79} - q^{81} + 3q^{85} - 14q^{87} - 3q^{89} + 4q^{93} - q^{95} + 5q^{97} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 1.00000 1.73205i 0
49.1 0 0.500000 0.866025i 0 0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.2.e.a 2
3.b odd 2 1 684.2.k.b 2
4.b odd 2 1 304.2.i.a 2
5.b even 2 1 1900.2.i.a 2
5.c odd 4 2 1900.2.s.a 4
8.b even 2 1 1216.2.i.c 2
8.d odd 2 1 1216.2.i.g 2
12.b even 2 1 2736.2.s.g 2
19.b odd 2 1 1444.2.e.b 2
19.c even 3 1 inner 76.2.e.a 2
19.c even 3 1 1444.2.a.b 1
19.d odd 6 1 1444.2.a.c 1
19.d odd 6 1 1444.2.e.b 2
57.h odd 6 1 684.2.k.b 2
76.f even 6 1 5776.2.a.f 1
76.g odd 6 1 304.2.i.a 2
76.g odd 6 1 5776.2.a.k 1
95.i even 6 1 1900.2.i.a 2
95.m odd 12 2 1900.2.s.a 4
152.k odd 6 1 1216.2.i.g 2
152.p even 6 1 1216.2.i.c 2
228.m even 6 1 2736.2.s.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 1.a even 1 1 trivial
76.2.e.a 2 19.c even 3 1 inner
304.2.i.a 2 4.b odd 2 1
304.2.i.a 2 76.g odd 6 1
684.2.k.b 2 3.b odd 2 1
684.2.k.b 2 57.h odd 6 1
1216.2.i.c 2 8.b even 2 1
1216.2.i.c 2 152.p even 6 1
1216.2.i.g 2 8.d odd 2 1
1216.2.i.g 2 152.k odd 6 1
1444.2.a.b 1 19.c even 3 1
1444.2.a.c 1 19.d odd 6 1
1444.2.e.b 2 19.b odd 2 1
1444.2.e.b 2 19.d odd 6 1
1900.2.i.a 2 5.b even 2 1
1900.2.i.a 2 95.i even 6 1
1900.2.s.a 4 5.c odd 4 2
1900.2.s.a 4 95.m odd 12 2
2736.2.s.g 2 12.b even 2 1
2736.2.s.g 2 228.m even 6 1
5776.2.a.f 1 76.f even 6 1
5776.2.a.k 1 76.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 1 - T + T^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 19 + 8 T + T^{2} \)
$23$ \( 25 + 5 T + T^{2} \)
$29$ \( 49 + 7 T + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( 25 - 5 T + T^{2} \)
$43$ \( 25 - 5 T + T^{2} \)
$47$ \( 49 - 7 T + T^{2} \)
$53$ \( 121 + 11 T + T^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( 121 + 11 T + T^{2} \)
$67$ \( 9 - 3 T + T^{2} \)
$71$ \( 121 + 11 T + T^{2} \)
$73$ \( 225 + 15 T + T^{2} \)
$79$ \( 169 - 13 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( 9 + 3 T + T^{2} \)
$97$ \( 25 - 5 T + T^{2} \)
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