Properties

Label 77.2.b.b
Level $77$
Weight $2$
Character orbit 77.b
Analytic conductor $0.615$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{3} q^{3} - \beta_{3} q^{5} + ( - \beta_{2} + 2 \beta_1) q^{6} + (\beta_{2} + \beta_1) q^{7} - 2 \beta_{2} q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{3} q^{3} - \beta_{3} q^{5} + ( - \beta_{2} + 2 \beta_1) q^{6} + (\beta_{2} + \beta_1) q^{7} - 2 \beta_{2} q^{8} - 2 q^{9} + (\beta_{2} - 2 \beta_1) q^{10} + (\beta_{2} - 3) q^{11} + (2 \beta_{2} - 4 \beta_1) q^{13} + ( - \beta_{3} + 3) q^{14} + 5 q^{15} - 4 q^{16} + 2 \beta_{2} q^{18} + ( - \beta_{2} + 2 \beta_1) q^{19} + (4 \beta_{2} - 3 \beta_1) q^{21} + (3 \beta_{2} + 2) q^{22} - 3 q^{23} + ( - 2 \beta_{2} + 4 \beta_1) q^{24} + 4 \beta_{3} q^{26} + \beta_{3} q^{27} - \beta_{2} q^{29} - 5 \beta_{2} q^{30} - 3 \beta_{3} q^{31} + ( - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{33} + ( - 4 \beta_{2} + 3 \beta_1) q^{35} - q^{37} - 2 \beta_{3} q^{38} - 10 \beta_{2} q^{39} + (2 \beta_{2} - 4 \beta_1) q^{40} + ( - 3 \beta_{2} + 6 \beta_1) q^{41} + (3 \beta_{3} + 5) q^{42} + 3 \beta_{2} q^{43} + 2 \beta_{3} q^{45} + 3 \beta_{2} q^{46} + 2 \beta_{3} q^{47} - 4 \beta_{3} q^{48} + (3 \beta_{3} - 2) q^{49} + ( - \beta_{2} + 2 \beta_1) q^{54} + (3 \beta_{3} - \beta_{2} + 2 \beta_1) q^{55} + ( - 2 \beta_{3} + 6) q^{56} + 5 \beta_{2} q^{57} - 2 q^{58} - \beta_{3} q^{59} + ( - \beta_{2} + 2 \beta_1) q^{61} + (3 \beta_{2} - 6 \beta_1) q^{62} + ( - 2 \beta_{2} - 2 \beta_1) q^{63} - 8 q^{64} + 10 \beta_{2} q^{65} + (2 \beta_{3} + 3 \beta_{2} - 6 \beta_1) q^{66} + 11 q^{67} - 3 \beta_{3} q^{69} + ( - 3 \beta_{3} - 5) q^{70} + 9 q^{71} + 4 \beta_{2} q^{72} + ( - \beta_{2} + 2 \beta_1) q^{73} + \beta_{2} q^{74} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{77} - 20 q^{78} - 6 \beta_{2} q^{79} + 4 \beta_{3} q^{80} - 11 q^{81} - 6 \beta_{3} q^{82} + (3 \beta_{2} - 6 \beta_1) q^{83} + 6 q^{86} + ( - \beta_{2} + 2 \beta_1) q^{87} + (6 \beta_{2} + 4) q^{88} - \beta_{3} q^{89} + ( - 2 \beta_{2} + 4 \beta_1) q^{90} + ( - 6 \beta_{3} - 10) q^{91} + 15 q^{93} + ( - 2 \beta_{2} + 4 \beta_1) q^{94} - 5 \beta_{2} q^{95} + 3 \beta_{3} q^{97} + ( - \beta_{2} + 6 \beta_1) q^{98} + ( - 2 \beta_{2} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} - 12 q^{11} + 12 q^{14} + 20 q^{15} - 16 q^{16} + 8 q^{22} - 12 q^{23} - 4 q^{37} + 20 q^{42} - 8 q^{49} + 24 q^{56} - 8 q^{58} - 32 q^{64} + 44 q^{67} - 20 q^{70} + 36 q^{71} - 12 q^{77} - 80 q^{78} - 44 q^{81} + 24 q^{86} + 16 q^{88} - 40 q^{91} + 60 q^{93} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + \beta_1 \) 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
−1.58114 + 0.707107i
1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 0.707107i
1.41421i 2.23607i 0 2.23607i −3.16228 −1.58114 + 2.12132i 2.82843i −2.00000 3.16228
76.2 1.41421i 2.23607i 0 2.23607i 3.16228 1.58114 + 2.12132i 2.82843i −2.00000 −3.16228
76.3 1.41421i 2.23607i 0 2.23607i 3.16228 1.58114 2.12132i 2.82843i −2.00000 −3.16228
76.4 1.41421i 2.23607i 0 2.23607i −3.16228 −1.58114 2.12132i 2.82843i −2.00000 3.16228
\(n\): 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 inner
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.b 4
3.b odd 2 1 693.2.c.b 4
4.b odd 2 1 1232.2.e.c 4
7.b odd 2 1 inner 77.2.b.b 4
7.c even 3 2 539.2.i.b 8
7.d odd 6 2 539.2.i.b 8
11.b odd 2 1 inner 77.2.b.b 4
11.c even 5 4 847.2.l.g 16
11.d odd 10 4 847.2.l.g 16
21.c even 2 1 693.2.c.b 4
28.d even 2 1 1232.2.e.c 4
33.d even 2 1 693.2.c.b 4
44.c even 2 1 1232.2.e.c 4
77.b even 2 1 inner 77.2.b.b 4
77.h odd 6 2 539.2.i.b 8
77.i even 6 2 539.2.i.b 8
77.j odd 10 4 847.2.l.g 16
77.l even 10 4 847.2.l.g 16
231.h odd 2 1 693.2.c.b 4
308.g odd 2 1 1232.2.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.b.b 4 1.a even 1 1 trivial
77.2.b.b 4 7.b odd 2 1 inner
77.2.b.b 4 11.b odd 2 1 inner
77.2.b.b 4 77.b even 2 1 inner
539.2.i.b 8 7.c even 3 2
539.2.i.b 8 7.d odd 6 2
539.2.i.b 8 77.h odd 6 2
539.2.i.b 8 77.i even 6 2
693.2.c.b 4 3.b odd 2 1
693.2.c.b 4 21.c even 2 1
693.2.c.b 4 33.d even 2 1
693.2.c.b 4 231.h odd 2 1
847.2.l.g 16 11.c even 5 4
847.2.l.g 16 11.d odd 10 4
847.2.l.g 16 77.j odd 10 4
847.2.l.g 16 77.l even 10 4
1232.2.e.c 4 4.b odd 2 1
1232.2.e.c 4 28.d even 2 1
1232.2.e.c 4 44.c even 2 1
1232.2.e.c 4 308.g odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 4T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 11)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$23$ \( (T + 3)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$67$ \( (T - 11)^{4} \) Copy content Toggle raw display
$71$ \( (T - 9)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
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