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Properties

Label 847.2.e.a
Level $847$
Weight $2$
Character orbit 847.e
Analytic conductor $6.763$
Analytic rank $0$
Dimension $4$
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] = [847,2,Mod(485,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.485");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.e (of order \(3\), degree \(2\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
0.809017 − 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 − 0.535233i
−0.809017 + 1.40126i −0.500000 − 0.866025i −0.309017 − 0.535233i −1.11803 + 1.93649i 1.61803 2.50000 − 0.866025i −2.23607 1.00000 − 1.73205i −1.80902 − 3.13331i
485.2 0.309017 − 0.535233i −0.500000 − 0.866025i 0.809017 + 1.40126i 1.11803 − 1.93649i −0.618034 2.50000 − 0.866025i 2.23607 1.00000 − 1.73205i −0.690983 − 1.19682i
606.1 −0.809017 − 1.40126i −0.500000 + 0.866025i −0.309017 + 0.535233i −1.11803 − 1.93649i 1.61803 2.50000 + 0.866025i −2.23607 1.00000 + 1.73205i −1.80902 + 3.13331i
606.2 0.309017 + 0.535233i −0.500000 + 0.866025i 0.809017 − 1.40126i 1.11803 + 1.93649i −0.618034 2.50000 + 0.866025i 2.23607 1.00000 + 1.73205i −0.690983 + 1.19682i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.e.a 4
7.c even 3 1 inner 847.2.e.a 4
7.c even 3 1 5929.2.a.q 2
7.d odd 6 1 5929.2.a.o 2
11.b odd 2 1 847.2.e.b 4
11.c even 5 2 77.2.m.a ✓ 8
11.c even 5 2 847.2.n.c 8
11.d odd 10 2 847.2.n.a 8
11.d odd 10 2 847.2.n.b 8
33.h odd 10 2 693.2.by.a 8
77.h odd 6 1 847.2.e.b 4
77.h odd 6 1 5929.2.a.l 2
77.i even 6 1 5929.2.a.j 2
77.j odd 10 2 539.2.q.a 8
77.m even 15 2 77.2.m.a ✓ 8
77.m even 15 2 539.2.f.a 4
77.m even 15 2 847.2.n.c 8
77.o odd 30 2 847.2.n.a 8
77.o odd 30 2 847.2.n.b 8
77.p odd 30 2 539.2.f.b 4
77.p odd 30 2 539.2.q.a 8
231.z odd 30 2 693.2.by.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.a ✓ 8 11.c even 5 2
77.2.m.a ✓ 8 77.m even 15 2
539.2.f.a 4 77.m even 15 2
539.2.f.b 4 77.p odd 30 2
539.2.q.a 8 77.j odd 10 2
539.2.q.a 8 77.p odd 30 2
693.2.by.a 8 33.h odd 10 2
693.2.by.a 8 231.z odd 30 2
847.2.e.a 4 1.a even 1 1 trivial
847.2.e.a 4 7.c even 3 1 inner
847.2.e.b 4 11.b odd 2 1
847.2.e.b 4 77.h odd 6 1
847.2.n.a 8 11.d odd 10 2
847.2.n.a 8 77.o odd 30 2
847.2.n.b 8 11.d odd 10 2
847.2.n.b 8 77.o odd 30 2
847.2.n.c 8 11.c even 5 2
847.2.n.c 8 77.m even 15 2
5929.2.a.j 2 77.i even 6 1
5929.2.a.l 2 77.h odd 6 1
5929.2.a.o 2 7.d odd 6 1
5929.2.a.q 2 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + 2 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$29$ \( (T^{2} + 9 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 11 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$37$ \( T^{4} - T^{3} + \cdots + 961 \) Copy content Toggle raw display
$41$ \( (T^{2} + 7 T + 11)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 7 T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + T^{3} + \cdots + 961 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + \cdots + 5776 \) Copy content Toggle raw display
$67$ \( T^{4} + 19 T^{3} + \cdots + 7921 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 15 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$83$ \( (T + 9)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$97$ \( (T^{2} + T - 11)^{2} \) Copy content Toggle raw display
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