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: \(\Q(\sqrt{-7}) \)
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 7 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 11 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 112 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 28 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 252 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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