Properties

Label 77.2.e.b
Level $77$
Weight $2$
Character orbit 77.e
Analytic conductor $0.615$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1783323.2
Defining polynomial: \(x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{3} + \beta_{5} ) q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( \beta_{2} - \beta_{3} ) q^{6} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( -3 + \beta_{3} ) q^{8} + ( -\beta_{3} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{3} + \beta_{5} ) q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( \beta_{2} - \beta_{3} ) q^{6} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( -3 + \beta_{3} ) q^{8} + ( -\beta_{3} + \beta_{5} ) q^{9} + ( 3 - 3 \beta_{4} - \beta_{5} ) q^{10} + ( 1 - \beta_{4} ) q^{11} + 3 \beta_{4} q^{12} + ( -4 - \beta_{2} ) q^{13} + ( 3 + \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{14} + ( -3 - 2 \beta_{2} ) q^{15} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{16} + ( 1 - \beta_{4} + \beta_{5} ) q^{17} + ( -3 - \beta_{1} + 3 \beta_{4} + \beta_{5} ) q^{18} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{19} + ( 5 + \beta_{2} - 2 \beta_{3} ) q^{20} + ( \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{21} -\beta_{3} q^{22} + ( \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{23} + ( -2 \beta_{1} + \beta_{5} ) q^{24} + ( -2 + 3 \beta_{1} + 2 \beta_{4} ) q^{25} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{26} + ( -2 \beta_{2} - \beta_{3} ) q^{27} + ( 3 - \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{28} + ( -2 + 3 \beta_{2} - \beta_{3} ) q^{29} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{30} + ( 2 - 3 \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{31} + 3 \beta_{1} q^{32} + ( \beta_{1} + \beta_{2} ) q^{33} + ( -3 + \beta_{2} ) q^{34} + ( 1 + 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{35} + ( -3 + 3 \beta_{3} ) q^{36} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( -3 + \beta_{1} + 3 \beta_{4} + 6 \beta_{5} ) q^{38} + ( 3 - 4 \beta_{1} - 3 \beta_{4} + \beta_{5} ) q^{39} + ( -3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{40} + ( -2 - \beta_{2} + 4 \beta_{3} ) q^{41} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{4} + \beta_{5} ) q^{42} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{43} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{44} + ( 3 - 3 \beta_{4} - \beta_{5} ) q^{45} + ( 3 + \beta_{1} - 3 \beta_{4} - 5 \beta_{5} ) q^{46} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{47} + ( 3 - \beta_{2} + 3 \beta_{3} ) q^{48} + ( -3 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{49} + ( 3 \beta_{2} - \beta_{3} ) q^{50} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{51} + ( 1 + 4 \beta_{1} - \beta_{4} - 4 \beta_{5} ) q^{52} + ( -6 + \beta_{1} + 6 \beta_{4} - 2 \beta_{5} ) q^{53} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{54} + ( 1 + \beta_{2} + \beta_{3} ) q^{55} + ( -3 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} ) q^{56} + ( 6 - 2 \beta_{2} + \beta_{3} ) q^{57} + ( -4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{58} + ( -3 + \beta_{1} + 3 \beta_{4} - 6 \beta_{5} ) q^{59} + ( -3 + 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{60} + ( -2 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 3 - 4 \beta_{2} ) q^{62} + ( 3 + \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{63} + ( -2 + \beta_{2} + \beta_{3} ) q^{64} + ( -6 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} ) q^{65} + ( -\beta_{1} - \beta_{5} ) q^{66} + ( 6 - 2 \beta_{1} - 6 \beta_{4} ) q^{67} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{68} + ( 3 \beta_{2} + \beta_{3} ) q^{69} + ( -3 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{70} + ( 1 - 4 \beta_{2} - 3 \beta_{3} ) q^{71} + ( \beta_{1} + \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} ) q^{72} + ( 5 + 5 \beta_{1} - 5 \beta_{4} ) q^{73} + ( -6 - 4 \beta_{1} + 6 \beta_{4} + 2 \beta_{5} ) q^{74} + ( -2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} ) q^{75} + ( -12 + 3 \beta_{2} + 6 \beta_{3} ) q^{76} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{77} + ( -3 - 3 \beta_{2} + 2 \beta_{3} ) q^{78} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} ) q^{79} + ( -2 - 3 \beta_{1} + 2 \beta_{4} + 3 \beta_{5} ) q^{80} + ( 6 - \beta_{1} - 6 \beta_{4} + 4 \beta_{5} ) q^{81} + ( 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - 12 \beta_{4} - 5 \beta_{5} ) q^{82} + ( -3 + 2 \beta_{2} - \beta_{3} ) q^{83} + ( 3 + 3 \beta_{2} - 3 \beta_{5} ) q^{84} + ( 4 + \beta_{2} ) q^{85} + ( -3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{86} + ( -9 - 3 \beta_{1} + 9 \beta_{4} - 4 \beta_{5} ) q^{87} + ( -3 + 3 \beta_{4} + \beta_{5} ) q^{88} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{89} + ( 3 - \beta_{2} - 4 \beta_{3} ) q^{90} + ( -7 + 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 7 - 4 \beta_{2} - 7 \beta_{3} ) q^{92} + ( \beta_{1} + \beta_{2} + 4 \beta_{3} + 9 \beta_{4} - 4 \beta_{5} ) q^{93} + ( 3 + \beta_{1} - 3 \beta_{4} ) q^{94} + ( 6 - \beta_{1} - 6 \beta_{4} - 4 \beta_{5} ) q^{95} + ( -3 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} ) q^{96} + ( 4 + 3 \beta_{2} ) q^{97} + ( -6 - \beta_{1} - 2 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{98} -\beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + q^{3} - 4q^{4} + 2q^{5} - 2q^{6} + 2q^{7} - 18q^{8} + O(q^{10}) \) \( 6q + q^{3} - 4q^{4} + 2q^{5} - 2q^{6} + 2q^{7} - 18q^{8} + 9q^{10} + 3q^{11} + 9q^{12} - 22q^{13} + 12q^{14} - 14q^{15} - 2q^{16} + 3q^{17} - 10q^{18} + 11q^{19} + 28q^{20} + 10q^{21} - 12q^{23} - 2q^{24} - 3q^{25} - q^{26} + 4q^{27} + 13q^{28} - 18q^{29} - 2q^{30} + 3q^{31} + 3q^{32} - q^{33} - 20q^{34} + 9q^{35} - 18q^{36} + 4q^{37} - 8q^{38} + 5q^{39} + 3q^{40} - 10q^{41} - 2q^{42} + 4q^{43} + 4q^{44} + 9q^{45} + 10q^{46} + 3q^{47} + 20q^{48} - 24q^{49} - 6q^{50} - 2q^{51} + 7q^{52} - 17q^{53} + 8q^{54} + 4q^{55} + 3q^{56} + 40q^{57} + 13q^{58} - 8q^{59} - 6q^{60} + 24q^{61} + 26q^{62} + 12q^{63} - 14q^{64} - 15q^{65} - q^{66} + 16q^{67} - 5q^{68} - 6q^{69} - 27q^{70} + 14q^{71} - 10q^{72} + 20q^{73} - 22q^{74} - 25q^{75} - 78q^{76} - 2q^{77} - 12q^{78} - 3q^{79} - 9q^{80} + 17q^{81} - 41q^{82} - 22q^{83} + 12q^{84} + 22q^{85} + 21q^{86} - 30q^{87} - 9q^{88} - q^{89} + 20q^{90} - 15q^{91} + 50q^{92} + 26q^{93} + 10q^{94} + 17q^{95} - 27q^{96} + 18q^{97} - 24q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 5 \nu^{4} + 25 \nu^{3} - 19 \nu^{2} + 12 \nu - 60 \)\()/83\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{5} - 20 \nu^{4} + 17 \nu^{3} - 76 \nu^{2} + 48 \nu - 240 \)\()/83\)
\(\beta_{4}\)\(=\)\((\)\( -20 \nu^{5} + 17 \nu^{4} - 85 \nu^{3} - 35 \nu^{2} - 323 \nu + 204 \)\()/249\)
\(\beta_{5}\)\(=\)\((\)\( -16 \nu^{5} - 3 \nu^{4} - 68 \nu^{3} - 28 \nu^{2} - 275 \nu - 36 \)\()/83\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} - \beta_{3}\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + 4 \beta_{2}\)
\(\nu^{4}\)\(=\)\(-5 \beta_{5} + 12 \beta_{4} - \beta_{1} - 12\)
\(\nu^{5}\)\(=\)\(-6 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} - 17 \beta_{2} - 17 \beta_{1}\)

Character values

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

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
1.09935 + 1.90412i
−0.956115 1.65604i
0.356769 + 0.617942i
1.09935 1.90412i
−0.956115 + 1.65604i
0.356769 0.617942i
−0.917122 + 1.58850i 1.09935 + 1.90412i −0.682224 1.18165i 0.317776 0.550404i −4.03293 0.317776 2.62660i −1.16576 −0.917122 + 1.58850i 0.582878 + 1.00958i
23.2 −0.328310 + 0.568650i −0.956115 1.65604i 0.784425 + 1.35866i 1.78442 3.09071i 1.25561 1.78442 + 1.95341i −2.34338 −0.328310 + 0.568650i 1.17169 + 2.02943i
23.3 1.24543 2.15715i 0.356769 + 0.617942i −2.10220 3.64112i −1.10220 + 1.90907i 1.77733 −1.10220 + 2.40523i −5.49086 1.24543 2.15715i 2.74543 + 4.75523i
67.1 −0.917122 1.58850i 1.09935 1.90412i −0.682224 + 1.18165i 0.317776 + 0.550404i −4.03293 0.317776 + 2.62660i −1.16576 −0.917122 1.58850i 0.582878 1.00958i
67.2 −0.328310 0.568650i −0.956115 + 1.65604i 0.784425 1.35866i 1.78442 + 3.09071i 1.25561 1.78442 1.95341i −2.34338 −0.328310 0.568650i 1.17169 2.02943i
67.3 1.24543 + 2.15715i 0.356769 0.617942i −2.10220 + 3.64112i −1.10220 1.90907i 1.77733 −1.10220 2.40523i −5.49086 1.24543 + 2.15715i 2.74543 4.75523i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.e.b 6
3.b odd 2 1 693.2.i.g 6
4.b odd 2 1 1232.2.q.k 6
7.b odd 2 1 539.2.e.l 6
7.c even 3 1 inner 77.2.e.b 6
7.c even 3 1 539.2.a.h 3
7.d odd 6 1 539.2.a.i 3
7.d odd 6 1 539.2.e.l 6
11.b odd 2 1 847.2.e.d 6
11.c even 5 4 847.2.n.e 24
11.d odd 10 4 847.2.n.d 24
21.g even 6 1 4851.2.a.bn 3
21.h odd 6 1 693.2.i.g 6
21.h odd 6 1 4851.2.a.bo 3
28.f even 6 1 8624.2.a.ck 3
28.g odd 6 1 1232.2.q.k 6
28.g odd 6 1 8624.2.a.cl 3
77.h odd 6 1 847.2.e.d 6
77.h odd 6 1 5929.2.a.v 3
77.i even 6 1 5929.2.a.w 3
77.m even 15 4 847.2.n.e 24
77.o odd 30 4 847.2.n.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b 6 1.a even 1 1 trivial
77.2.e.b 6 7.c even 3 1 inner
539.2.a.h 3 7.c even 3 1
539.2.a.i 3 7.d odd 6 1
539.2.e.l 6 7.b odd 2 1
539.2.e.l 6 7.d odd 6 1
693.2.i.g 6 3.b odd 2 1
693.2.i.g 6 21.h odd 6 1
847.2.e.d 6 11.b odd 2 1
847.2.e.d 6 77.h odd 6 1
847.2.n.d 24 11.d odd 10 4
847.2.n.d 24 77.o odd 30 4
847.2.n.e 24 11.c even 5 4
847.2.n.e 24 77.m even 15 4
1232.2.q.k 6 4.b odd 2 1
1232.2.q.k 6 28.g odd 6 1
4851.2.a.bn 3 21.g even 6 1
4851.2.a.bo 3 21.h odd 6 1
5929.2.a.v 3 77.h odd 6 1
5929.2.a.w 3 77.i even 6 1
8624.2.a.ck 3 28.f even 6 1
8624.2.a.cl 3 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 5 T_{2}^{4} + 6 T_{2}^{3} + 25 T_{2}^{2} + 15 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 15 T + 25 T^{2} + 6 T^{3} + 5 T^{4} + T^{6} \)
$3$ \( 9 - 12 T + 19 T^{2} - 2 T^{3} + 5 T^{4} - T^{5} + T^{6} \)
$5$ \( 25 - 35 T + 59 T^{2} + 4 T^{3} + 11 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( 343 - 98 T + 98 T^{2} - 23 T^{3} + 14 T^{4} - 2 T^{5} + T^{6} \)
$11$ \( ( 1 - T + T^{2} )^{3} \)
$13$ \( ( 35 + 36 T + 11 T^{2} + T^{3} )^{2} \)
$17$ \( 49 - 14 T + 25 T^{2} - 8 T^{3} + 11 T^{4} - 3 T^{5} + T^{6} \)
$19$ \( 3249 + 1140 T + 1027 T^{2} - 334 T^{3} + 101 T^{4} - 11 T^{5} + T^{6} \)
$23$ \( 2209 + 2021 T + 1285 T^{2} + 422 T^{3} + 101 T^{4} + 12 T^{5} + T^{6} \)
$29$ \( ( -53 - 20 T + 9 T^{2} + T^{3} )^{2} \)
$31$ \( 11449 - 4708 T + 2257 T^{2} - 82 T^{3} + 53 T^{4} - 3 T^{5} + T^{6} \)
$37$ \( 23104 - 5472 T + 1904 T^{2} - 160 T^{3} + 52 T^{4} - 4 T^{5} + T^{6} \)
$41$ \( ( -109 - 80 T + 5 T^{2} + T^{3} )^{2} \)
$43$ \( ( 41 - 25 T - 2 T^{2} + T^{3} )^{2} \)
$47$ \( 49 - 14 T + 25 T^{2} - 8 T^{3} + 11 T^{4} - 3 T^{5} + T^{6} \)
$53$ \( 441 + 1554 T + 5119 T^{2} + 1216 T^{3} + 215 T^{4} + 17 T^{5} + T^{6} \)
$59$ \( 1750329 + 207711 T + 35233 T^{2} + 1390 T^{3} + 221 T^{4} + 8 T^{5} + T^{6} \)
$61$ \( 141376 - 64672 T + 20560 T^{2} - 3376 T^{3} + 404 T^{4} - 24 T^{5} + T^{6} \)
$67$ \( 5184 - 4896 T + 3472 T^{2} - 944 T^{3} + 188 T^{4} - 16 T^{5} + T^{6} \)
$71$ \( ( 419 - 86 T - 7 T^{2} + T^{3} )^{2} \)
$73$ \( 390625 + 15625 T + 13125 T^{2} - 1750 T^{3} + 375 T^{4} - 20 T^{5} + T^{6} \)
$79$ \( 19881 + 5358 T + 1867 T^{2} + 168 T^{3} + 47 T^{4} + 3 T^{5} + T^{6} \)
$83$ \( ( -3 + 16 T + 11 T^{2} + T^{3} )^{2} \)
$89$ \( 9 + 24 T + 67 T^{2} - 2 T^{3} + 9 T^{4} + T^{5} + T^{6} \)
$97$ \( ( 47 - 12 T - 9 T^{2} + T^{3} )^{2} \)
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