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Properties

Label 847.2.a.k
Level $847$
Weight $2$
Character orbit 847.a
Self dual yes
Analytic conductor $6.763$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(1,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

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: yes
Analytic conductor: \(6.76332905120\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2525.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 3 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.46673
1.77748
−0.777484
−1.46673
−2.46673 −1.61803 4.08477 −3.46673 3.99126 1.00000 −5.14256 −0.381966 8.55150
1.2 −1.77748 0.618034 1.15945 −2.77748 −1.09855 1.00000 1.49406 −2.61803 4.93693
1.3 0.777484 0.618034 −1.39552 −0.222516 0.480512 1.00000 −2.63996 −2.61803 −0.173002
1.4 1.46673 −1.61803 0.151302 0.466732 −2.37322 1.00000 −2.71154 −0.381966 0.684570
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

\( T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 5T_{2} + 5 \) Copy content Toggle raw display
\( T_{3}^{2} + T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 32 T^{2} + \cdots - 29 \) Copy content Toggle raw display
$17$ \( T^{4} - 3 T^{3} + \cdots - 71 \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} + \cdots - 145 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} + \cdots - 205 \) Copy content Toggle raw display
$29$ \( T^{4} + 3 T^{3} + \cdots + 405 \) Copy content Toggle raw display
$31$ \( T^{4} + 3 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} + \cdots - 5 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + \cdots + 79 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 41)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 14 T^{3} + \cdots - 305 \) Copy content Toggle raw display
$53$ \( T^{4} + 9 T^{3} + \cdots - 869 \) Copy content Toggle raw display
$59$ \( T^{4} + 25 T^{3} + \cdots - 1189 \) Copy content Toggle raw display
$61$ \( T^{4} - 19 T^{3} + \cdots - 995 \) Copy content Toggle raw display
$67$ \( T^{4} + 15 T^{3} + \cdots - 199 \) Copy content Toggle raw display
$71$ \( T^{4} + 7 T^{3} + \cdots - 991 \) Copy content Toggle raw display
$73$ \( T^{4} - 11 T^{3} + \cdots + 4975 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + \cdots - 271 \) Copy content Toggle raw display
$83$ \( T^{4} - T^{3} + \cdots + 2245 \) Copy content Toggle raw display
$89$ \( T^{4} + 17 T^{3} + \cdots - 755 \) Copy content Toggle raw display
$97$ \( T^{4} + 15 T^{3} + \cdots - 4225 \) Copy content Toggle raw display
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