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Properties

Label 847.2.f.l
Level $847$
Weight $2$
Character orbit 847.f
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(148,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.148");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.f (of order \(5\), degree \(4\), not 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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) 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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(122\) \(365\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−0.309017 + 0.951057i
0.809017 − 0.587785i
−0.309017 − 0.951057i
0.809017 + 0.587785i
−0.118034 − 0.363271i 1.30902 + 0.951057i 1.50000 − 1.08981i 0.309017 − 0.951057i 0.190983 − 0.587785i −0.809017 + 0.587785i −1.19098 − 0.865300i −0.118034 − 0.363271i −0.381966
323.1 2.11803 + 1.53884i 0.190983 − 0.587785i 1.50000 + 4.61653i −0.809017 + 0.587785i 1.30902 − 0.951057i 0.309017 + 0.951057i −2.30902 + 7.10642i 2.11803 + 1.53884i −2.61803
372.1 −0.118034 + 0.363271i 1.30902 − 0.951057i 1.50000 + 1.08981i 0.309017 + 0.951057i 0.190983 + 0.587785i −0.809017 − 0.587785i −1.19098 + 0.865300i −0.118034 + 0.363271i −0.381966
729.1 2.11803 − 1.53884i 0.190983 + 0.587785i 1.50000 − 4.61653i −0.809017 − 0.587785i 1.30902 + 0.951057i 0.309017 − 0.951057i −2.30902 − 7.10642i 2.11803 − 1.53884i −2.61803
\(n\): e.g. 2-40 or 990-1000
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 847.2.f.l 4
11.b odd 2 1 847.2.f.c 4
11.c even 5 1 847.2.a.d ✓ 2
11.c even 5 2 847.2.f.d 4
11.c even 5 1 inner 847.2.f.l 4
11.d odd 10 1 847.2.a.h yes 2
11.d odd 10 1 847.2.f.c 4
11.d odd 10 2 847.2.f.j 4
33.f even 10 1 7623.2.a.t 2
33.h odd 10 1 7623.2.a.bx 2
77.j odd 10 1 5929.2.a.i 2
77.l even 10 1 5929.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.d ✓ 2 11.c even 5 1
847.2.a.h yes 2 11.d odd 10 1
847.2.f.c 4 11.b odd 2 1
847.2.f.c 4 11.d odd 10 1
847.2.f.d 4 11.c even 5 2
847.2.f.j 4 11.d odd 10 2
847.2.f.l 4 1.a even 1 1 trivial
847.2.f.l 4 11.c even 5 1 inner
5929.2.a.i 2 77.j odd 10 1
5929.2.a.s 2 77.l even 10 1
7623.2.a.t 2 33.f even 10 1
7623.2.a.bx 2 33.h odd 10 1

Hecke kernels

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

\( T_{2}^{4} - 4T_{2}^{3} + 6T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - 3T_{3}^{3} + 4T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 6T_{13}^{3} + 16T_{13}^{2} + 16T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 15 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( (T^{2} + T - 31)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$31$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{4} + 12 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{4} - 25 T^{3} + \cdots + 15625 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T - 95)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 61)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} + \cdots + 6241 \) Copy content Toggle raw display
$73$ \( T^{4} - 17 T^{3} + \cdots + 19321 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$83$ \( T^{4} - 19 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$89$ \( (T^{2} - 7 T + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 7 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
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