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Properties

Label 847.2.f.i
Level $847$
Weight $2$
Character orbit 847.f
Analytic conductor $6.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
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.

\(f(q)\) \(=\) \( q - 3 \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + \cdots - 6) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + \cdots - 6) q^{9} + \cdots + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + \cdots + 5) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + 2 q^{4} + q^{5} + q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} + 2 q^{4} + q^{5} + q^{7} - 6 q^{9} + 24 q^{12} + 4 q^{13} - 3 q^{15} - 4 q^{16} - 2 q^{17} + 6 q^{19} - 2 q^{20} + 12 q^{21} - 20 q^{23} + 4 q^{25} + 9 q^{27} - 2 q^{28} - 10 q^{29} - q^{31} - q^{35} + 12 q^{36} + 5 q^{37} - 12 q^{39} + 2 q^{41} - 32 q^{43} - 24 q^{45} - 8 q^{47} + 12 q^{48} - q^{49} + 6 q^{51} - 8 q^{52} + 6 q^{53} - 18 q^{57} - 3 q^{59} + 6 q^{60} + 2 q^{61} + 6 q^{63} + 8 q^{64} + 16 q^{65} - 12 q^{67} + 4 q^{68} - 15 q^{69} - q^{71} - 10 q^{73} - 12 q^{75} + 48 q^{76} - 6 q^{79} + 4 q^{80} - 9 q^{81} - 12 q^{83} + 6 q^{84} + 2 q^{85} - 120 q^{87} - 60 q^{89} - 4 q^{91} - 10 q^{92} + 3 q^{93} - 6 q^{95} + 5 q^{97}+O(q^{100}) \) 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 2.42705 + 1.76336i 1.61803 − 1.17557i −0.309017 + 0.951057i 0 0.809017 − 0.587785i 0 1.85410 + 5.70634i 0
323.1 0 −0.927051 + 2.85317i −0.618034 − 1.90211i 0.809017 − 0.587785i 0 −0.309017 − 0.951057i 0 −4.85410 − 3.52671i 0
372.1 0 2.42705 − 1.76336i 1.61803 + 1.17557i −0.309017 − 0.951057i 0 0.809017 + 0.587785i 0 1.85410 − 5.70634i 0
729.1 0 −0.927051 − 2.85317i −0.618034 + 1.90211i 0.809017 + 0.587785i 0 −0.309017 + 0.951057i 0 −4.85410 + 3.52671i 0
\(n\): 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 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.i 4
11.b odd 2 1 847.2.f.h 4
11.c even 5 1 77.2.a.a ✓ 1
11.c even 5 3 inner 847.2.f.i 4
11.d odd 10 1 847.2.a.b 1
11.d odd 10 3 847.2.f.h 4
33.f even 10 1 7623.2.a.j 1
33.h odd 10 1 693.2.a.c 1
44.h odd 10 1 1232.2.a.l 1
55.j even 10 1 1925.2.a.h 1
55.k odd 20 2 1925.2.b.e 2
77.j odd 10 1 539.2.a.c 1
77.l even 10 1 5929.2.a.f 1
77.m even 15 2 539.2.e.f 2
77.p odd 30 2 539.2.e.c 2
88.l odd 10 1 4928.2.a.a 1
88.o even 10 1 4928.2.a.bj 1
231.u even 10 1 4851.2.a.j 1
308.t even 10 1 8624.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a ✓ 1 11.c even 5 1
539.2.a.c 1 77.j odd 10 1
539.2.e.c 2 77.p odd 30 2
539.2.e.f 2 77.m even 15 2
693.2.a.c 1 33.h odd 10 1
847.2.a.b 1 11.d odd 10 1
847.2.f.h 4 11.b odd 2 1
847.2.f.h 4 11.d odd 10 3
847.2.f.i 4 1.a even 1 1 trivial
847.2.f.i 4 11.c even 5 3 inner
1232.2.a.l 1 44.h odd 10 1
1925.2.a.h 1 55.j even 10 1
1925.2.b.e 2 55.k odd 20 2
4851.2.a.j 1 231.u even 10 1
4928.2.a.a 1 88.l odd 10 1
4928.2.a.bj 1 88.o even 10 1
5929.2.a.f 1 77.l even 10 1
7623.2.a.j 1 33.f even 10 1
8624.2.a.a 1 308.t even 10 1

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{4} - 3T_{3}^{3} + 9T_{3}^{2} - 27T_{3} + 81 \) Copy content Toggle raw display
\( T_{13}^{4} - 4T_{13}^{3} + 16T_{13}^{2} - 64T_{13} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + \cdots + 81 \) 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
$13$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$23$ \( (T + 5)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + \cdots + 10000 \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{4} - 5 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( (T + 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} + 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$67$ \( (T + 3)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} + \cdots + 10000 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$89$ \( (T + 15)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 5 T^{3} + \cdots + 625 \) Copy content Toggle raw display
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