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: \(\Q(\sqrt{-2}, \sqrt{-5})\)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + \beta_{3} q^{3} -\beta_{3} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} -2 \beta_{2} q^{8} -2 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + \beta_{3} q^{3} -\beta_{3} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} -2 \beta_{2} q^{8} -2 q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{10} + ( -3 + \beta_{2} ) q^{11} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 3 - \beta_{3} ) q^{14} + 5 q^{15} -4 q^{16} + 2 \beta_{2} q^{18} + ( 2 \beta_{1} - \beta_{2} ) q^{19} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{21} + ( 2 + 3 \beta_{2} ) q^{22} -3 q^{23} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{24} + 4 \beta_{3} q^{26} + \beta_{3} q^{27} -\beta_{2} q^{29} -5 \beta_{2} q^{30} -3 \beta_{3} q^{31} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{33} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{35} - q^{37} -2 \beta_{3} q^{38} -10 \beta_{2} q^{39} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{40} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{41} + ( 5 + 3 \beta_{3} ) q^{42} + 3 \beta_{2} q^{43} + 2 \beta_{3} q^{45} + 3 \beta_{2} q^{46} + 2 \beta_{3} q^{47} -4 \beta_{3} q^{48} + ( -2 + 3 \beta_{3} ) q^{49} + ( 2 \beta_{1} - \beta_{2} ) q^{54} + ( 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{55} + ( 6 - 2 \beta_{3} ) q^{56} + 5 \beta_{2} q^{57} -2 q^{58} -\beta_{3} q^{59} + ( 2 \beta_{1} - \beta_{2} ) q^{61} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{62} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{63} -8 q^{64} + 10 \beta_{2} q^{65} + ( -6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{66} + 11 q^{67} -3 \beta_{3} q^{69} + ( -5 - 3 \beta_{3} ) q^{70} + 9 q^{71} + 4 \beta_{2} q^{72} + ( 2 \beta_{1} - \beta_{2} ) q^{73} + \beta_{2} q^{74} + ( -3 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{77} -20 q^{78} -6 \beta_{2} q^{79} + 4 \beta_{3} q^{80} -11 q^{81} -6 \beta_{3} q^{82} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{83} + 6 q^{86} + ( 2 \beta_{1} - \beta_{2} ) q^{87} + ( 4 + 6 \beta_{2} ) q^{88} -\beta_{3} q^{89} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{90} + ( -10 - 6 \beta_{3} ) q^{91} + 15 q^{93} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{94} -5 \beta_{2} q^{95} + 3 \beta_{3} q^{97} + ( 6 \beta_{1} - \beta_{2} ) q^{98} + ( 6 - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{9} + O(q^{10}) \) \( 4q - 8q^{9} - 12q^{11} + 12q^{14} + 20q^{15} - 16q^{16} + 8q^{22} - 12q^{23} - 4q^{37} + 20q^{42} - 8q^{49} + 24q^{56} - 8q^{58} - 32q^{64} + 44q^{67} - 20q^{70} + 36q^{71} - 12q^{77} - 80q^{78} - 44q^{81} + 24q^{86} + 16q^{88} - 40q^{91} + 60q^{93} + 24q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + \beta_{1}\)

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 \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \)
$3$ \( ( 1 - T^{2} + 9 T^{4} )^{2} \)
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
$7$ \( 1 + 4 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 + 28 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 3 T + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 17 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 8 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 68 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 74 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 - 113 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 112 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 11 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 9 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 136 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 86 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 76 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 173 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 149 T^{2} + 9409 T^{4} )^{2} \)
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