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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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

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