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Properties

Label 77.2.f.a
Level $77$
Weight $2$
Character orbit 77.f
Analytic conductor $0.615$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [77,2,Mod(15,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.15");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.f (of order \(5\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.159390625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{4} + (\beta_{7} + \beta_{5} + \beta_{2} - \beta_1) q^{5} + (\beta_{7} + \beta_{5} - \beta_{4}) q^{6} + \beta_{3} q^{7} + (2 \beta_{6} + \beta_{5} + 3 \beta_{3} + \cdots - 2) q^{8}+ \cdots + (2 \beta_{6} + \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{4} + (\beta_{7} + \beta_{5} + \beta_{2} - \beta_1) q^{5} + (\beta_{7} + \beta_{5} - \beta_{4}) q^{6} + \beta_{3} q^{7} + (2 \beta_{6} + \beta_{5} + 3 \beta_{3} + \cdots - 2) q^{8}+ \cdots + (3 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - 4 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} - 4 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} - 2 q^{9} - 28 q^{10} + 5 q^{11} - 14 q^{12} + 5 q^{13} + q^{14} + 6 q^{15} - 3 q^{16} - 11 q^{17} + 4 q^{18} - 9 q^{19} + 21 q^{20} + 4 q^{21} - q^{22} - 16 q^{23} + 21 q^{24} + 5 q^{25} + 21 q^{26} - 22 q^{27} + 7 q^{28} - 9 q^{29} + 14 q^{30} - 11 q^{31} - 20 q^{32} + 10 q^{33} - 24 q^{34} - 3 q^{35} - 2 q^{36} + 6 q^{37} + 35 q^{38} - 5 q^{39} - 16 q^{40} - 22 q^{41} - 3 q^{42} + 16 q^{43} + 29 q^{44} + 18 q^{45} + 29 q^{46} + 7 q^{47} + 4 q^{48} - 2 q^{49} - 34 q^{50} + 3 q^{51} + 21 q^{52} + 2 q^{53} + 4 q^{54} + 26 q^{55} - 18 q^{56} - 3 q^{57} - 39 q^{58} + 25 q^{59} - 38 q^{60} + 7 q^{61} - 5 q^{62} - 8 q^{63} + q^{64} + 24 q^{65} + 18 q^{66} - 30 q^{67} + 8 q^{68} + 8 q^{69} - 2 q^{70} - 14 q^{71} + 3 q^{72} + 3 q^{73} - 9 q^{74} + 5 q^{75} - 52 q^{76} - 5 q^{77} - 18 q^{78} - 9 q^{79} - 33 q^{80} - 28 q^{81} + 31 q^{82} + 23 q^{83} + 4 q^{84} - 10 q^{85} - 17 q^{86} + 12 q^{87} - 7 q^{88} - 34 q^{89} + 2 q^{90} + 5 q^{91} - 34 q^{92} + 8 q^{93} - 30 q^{94} + 24 q^{95} + 10 q^{96} + 30 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 555\nu^{7} - 2159\nu^{6} + 7489\nu^{5} - 18164\nu^{4} + 40069\nu^{3} - 84434\nu^{2} + 43855\nu + 375 ) / 94655 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -970\nu^{7} - 1002\nu^{6} - 6608\nu^{5} + 9063\nu^{4} - 14943\nu^{3} + 27673\nu^{2} - 68120\nu + 35160 ) / 94655 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1604\nu^{7} + 4159\nu^{6} - 12059\nu^{5} + 28414\nu^{4} - 81659\nu^{3} + 38305\nu^{2} - 13500\nu - 13875 ) / 94655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2052\nu^{7} + 2252\nu^{6} - 19912\nu^{5} + 21007\nu^{4} - 82042\nu^{3} + 35785\nu^{2} - 19395\nu - 90925 ) / 94655 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2667\nu^{7} + 6691\nu^{6} - 17466\nu^{5} + 50856\nu^{4} - 82441\nu^{3} + 72554\nu^{2} - 4035\nu - 12035 ) / 94655 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4024\nu^{7} - 1464\nu^{6} + 21519\nu^{5} - 26434\nu^{4} + 59219\nu^{3} + 22635\nu^{2} + 54640\nu + 66675 ) / 94655 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + 7\beta_{6} + 2\beta_{5} + 13\beta_{3} + 13\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 11\beta_{6} - 20\beta_{5} + 20\beta_{4} - 11\beta_{2} + 8\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -19\beta_{7} - 19\beta_{4} - 68\beta_{3} - 36\beta_{2} - 24\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 111\beta_{7} + 81\beta_{6} + 111\beta_{5} - 55\beta_{4} + 81\beta_{3} + 148\beta_{2} - 56\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(45\) \(57\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−0.762262 − 2.34600i
0.453245 + 1.39494i
−0.762262 + 2.34600i
0.453245 − 1.39494i
−0.628998 − 0.456994i
1.43801 + 1.04478i
−0.628998 + 0.456994i
1.43801 − 1.04478i
−1.99563 − 1.44991i −0.500000 + 1.53884i 1.26226 + 3.88484i 2.80464 − 2.03769i 3.22899 − 2.34600i −0.309017 − 0.951057i 1.58914 − 4.89086i 0.309017 + 0.224514i −8.55150
15.2 1.18661 + 0.862123i −0.500000 + 1.53884i 0.0467549 + 0.143897i −0.377594 + 0.274338i −1.91998 + 1.39494i −0.309017 − 0.951057i 0.837913 − 2.57883i 0.309017 + 0.224514i −0.684570
36.1 −1.99563 + 1.44991i −0.500000 − 1.53884i 1.26226 − 3.88484i 2.80464 + 2.03769i 3.22899 + 2.34600i −0.309017 + 0.951057i 1.58914 + 4.89086i 0.309017 − 0.224514i −8.55150
36.2 1.18661 − 0.862123i −0.500000 − 1.53884i 0.0467549 − 0.143897i −0.377594 − 0.274338i −1.91998 − 1.39494i −0.309017 + 0.951057i 0.837913 + 2.57883i 0.309017 − 0.224514i −0.684570
64.1 −0.240256 + 0.739431i −0.500000 + 0.363271i 1.12900 + 0.820265i −0.0687611 − 0.211625i −0.148486 − 0.456994i 0.809017 + 0.587785i −2.13577 + 1.55173i −0.809017 + 2.48990i 0.173002
64.2 0.549273 − 1.69049i −0.500000 + 0.363271i −0.938015 − 0.681508i −0.858290 − 2.64154i 0.339469 + 1.04478i 0.809017 + 0.587785i 1.20872 − 0.878189i −0.809017 + 2.48990i −4.93693
71.1 −0.240256 − 0.739431i −0.500000 − 0.363271i 1.12900 − 0.820265i −0.0687611 + 0.211625i −0.148486 + 0.456994i 0.809017 − 0.587785i −2.13577 − 1.55173i −0.809017 − 2.48990i 0.173002
71.2 0.549273 + 1.69049i −0.500000 − 0.363271i −0.938015 + 0.681508i −0.858290 + 2.64154i 0.339469 − 1.04478i 0.809017 − 0.587785i 1.20872 + 0.878189i −0.809017 − 2.48990i −4.93693
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.f.a ✓ 8
3.b odd 2 1 693.2.m.g 8
7.b odd 2 1 539.2.f.d 8
7.c even 3 2 539.2.q.c 16
7.d odd 6 2 539.2.q.b 16
11.b odd 2 1 847.2.f.q 8
11.c even 5 1 inner 77.2.f.a ✓ 8
11.c even 5 1 847.2.a.l 4
11.c even 5 2 847.2.f.p 8
11.d odd 10 1 847.2.a.k 4
11.d odd 10 1 847.2.f.q 8
11.d odd 10 2 847.2.f.s 8
33.f even 10 1 7623.2.a.co 4
33.h odd 10 1 693.2.m.g 8
33.h odd 10 1 7623.2.a.ch 4
77.j odd 10 1 539.2.f.d 8
77.j odd 10 1 5929.2.a.bi 4
77.l even 10 1 5929.2.a.bb 4
77.m even 15 2 539.2.q.c 16
77.p odd 30 2 539.2.q.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a ✓ 8 1.a even 1 1 trivial
77.2.f.a ✓ 8 11.c even 5 1 inner
539.2.f.d 8 7.b odd 2 1
539.2.f.d 8 77.j odd 10 1
539.2.q.b 16 7.d odd 6 2
539.2.q.b 16 77.p odd 30 2
539.2.q.c 16 7.c even 3 2
539.2.q.c 16 77.m even 15 2
693.2.m.g 8 3.b odd 2 1
693.2.m.g 8 33.h odd 10 1
847.2.a.k 4 11.d odd 10 1
847.2.a.l 4 11.c even 5 1
847.2.f.p 8 11.c even 5 2
847.2.f.q 8 11.b odd 2 1
847.2.f.q 8 11.d odd 10 1
847.2.f.s 8 11.d odd 10 2
5929.2.a.bb 4 77.l even 10 1
5929.2.a.bi 4 77.j odd 10 1
7623.2.a.ch 4 33.h odd 10 1
7623.2.a.co 4 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + T_{2}^{7} + T_{2}^{6} + T_{2}^{5} + 16T_{2}^{4} - 25T_{2}^{3} + 35T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 5 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$17$ \( T^{8} + 11 T^{7} + \cdots + 5041 \) Copy content Toggle raw display
$19$ \( T^{8} + 9 T^{7} + \cdots + 21025 \) Copy content Toggle raw display
$23$ \( (T^{4} + 8 T^{3} + \cdots - 205)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 9 T^{7} + \cdots + 164025 \) Copy content Toggle raw display
$31$ \( T^{8} + 11 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$41$ \( T^{8} + 22 T^{7} + \cdots + 6241 \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 41)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 7 T^{7} + \cdots + 93025 \) Copy content Toggle raw display
$53$ \( T^{8} - 2 T^{7} + \cdots + 755161 \) Copy content Toggle raw display
$59$ \( T^{8} - 25 T^{7} + \cdots + 1413721 \) Copy content Toggle raw display
$61$ \( T^{8} - 7 T^{7} + \cdots + 990025 \) Copy content Toggle raw display
$67$ \( (T^{4} + 15 T^{3} + \cdots - 199)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 14 T^{7} + \cdots + 982081 \) Copy content Toggle raw display
$73$ \( T^{8} - 3 T^{7} + \cdots + 24750625 \) Copy content Toggle raw display
$79$ \( T^{8} + 9 T^{7} + \cdots + 73441 \) Copy content Toggle raw display
$83$ \( T^{8} - 23 T^{7} + \cdots + 5040025 \) Copy content Toggle raw display
$89$ \( (T^{4} + 17 T^{3} + \cdots - 755)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 30 T^{7} + \cdots + 17850625 \) Copy content Toggle raw display
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