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Properties

Label 77.2.b.a
Level 77
Weight 2
Character orbit 77.b
Analytic conductor 0.615
Analytic rank 0
Dimension 2
CM discriminant -7
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 77.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} -5 q^{4} -\beta q^{7} + 3 \beta q^{8} + 3 q^{9} +O(q^{10})\) \( q -\beta q^{2} -5 q^{4} -\beta q^{7} + 3 \beta q^{8} + 3 q^{9} + ( 2 + \beta ) q^{11} -7 q^{14} + 11 q^{16} -3 \beta q^{18} + ( 7 - 2 \beta ) q^{22} -8 q^{23} + 5 q^{25} + 5 \beta q^{28} + 4 \beta q^{29} -5 \beta q^{32} -15 q^{36} -6 q^{37} -2 \beta q^{43} + ( -10 - 5 \beta ) q^{44} + 8 \beta q^{46} -7 q^{49} -5 \beta q^{50} + 10 q^{53} + 21 q^{56} + 28 q^{58} -3 \beta q^{63} -13 q^{64} -4 q^{67} -16 q^{71} + 9 \beta q^{72} + 6 \beta q^{74} + ( 7 - 2 \beta ) q^{77} -6 \beta q^{79} + 9 q^{81} -14 q^{86} + ( -21 + 6 \beta ) q^{88} + 40 q^{92} + 7 \beta q^{98} + ( 6 + 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 10q^{4} + 6q^{9} + O(q^{10}) \) \( 2q - 10q^{4} + 6q^{9} + 4q^{11} - 14q^{14} + 22q^{16} + 14q^{22} - 16q^{23} + 10q^{25} - 30q^{36} - 12q^{37} - 20q^{44} - 14q^{49} + 20q^{53} + 42q^{56} + 56q^{58} - 26q^{64} - 8q^{67} - 32q^{71} + 14q^{77} + 18q^{81} - 28q^{86} - 42q^{88} + 80q^{92} + 12q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0.500000 + 1.32288i
0.500000 − 1.32288i
− 2.64575i 0 −5.00000 0 0 − 2.64575i 7.93725i 3.00000 0
76.2 2.64575i 0 −5.00000 0 0 2.64575i − 7.93725i 3.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.b.a ✓ 2
3.b odd 2 1 693.2.c.a 2
4.b odd 2 1 1232.2.e.a 2
7.b odd 2 1 CM 77.2.b.a ✓ 2
7.c even 3 2 539.2.i.a 4
7.d odd 6 2 539.2.i.a 4
11.b odd 2 1 inner 77.2.b.a ✓ 2
11.c even 5 4 847.2.l.b 8
11.d odd 10 4 847.2.l.b 8
21.c even 2 1 693.2.c.a 2
28.d even 2 1 1232.2.e.a 2
33.d even 2 1 693.2.c.a 2
44.c even 2 1 1232.2.e.a 2
77.b even 2 1 inner 77.2.b.a ✓ 2
77.h odd 6 2 539.2.i.a 4
77.i even 6 2 539.2.i.a 4
77.j odd 10 4 847.2.l.b 8
77.l even 10 4 847.2.l.b 8
231.h odd 2 1 693.2.c.a 2
308.g odd 2 1 1232.2.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.b.a ✓ 2 1.a even 1 1 trivial
77.2.b.a ✓ 2 7.b odd 2 1 CM
77.2.b.a ✓ 2 11.b odd 2 1 inner
77.2.b.a ✓ 2 77.b even 2 1 inner
539.2.i.a 4 7.c even 3 2
539.2.i.a 4 7.d odd 6 2
539.2.i.a 4 77.h odd 6 2
539.2.i.a 4 77.i even 6 2
693.2.c.a 2 3.b odd 2 1
693.2.c.a 2 21.c even 2 1
693.2.c.a 2 33.d even 2 1
693.2.c.a 2 231.h odd 2 1
847.2.l.b 8 11.c even 5 4
847.2.l.b 8 11.d odd 10 4
847.2.l.b 8 77.j odd 10 4
847.2.l.b 8 77.l even 10 4
1232.2.e.a 2 4.b odd 2 1
1232.2.e.a 2 28.d even 2 1
1232.2.e.a 2 44.c even 2 1
1232.2.e.a 2 308.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 7 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke Characteristic Polynomials

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