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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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.f (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
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

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.

\(f(q)\) \(=\) \( q -3 \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q -3 \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{9} + 6 q^{12} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{13} -3 \zeta_{10}^{3} q^{15} -4 \zeta_{10} q^{16} -2 \zeta_{10} q^{17} -6 \zeta_{10}^{2} q^{19} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{20} + 3 q^{21} -5 q^{23} -4 \zeta_{10}^{2} q^{25} + 9 \zeta_{10} q^{27} -2 \zeta_{10} q^{28} -10 \zeta_{10}^{3} q^{29} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{31} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{35} -12 \zeta_{10}^{2} q^{36} + 5 \zeta_{10}^{3} q^{37} -12 \zeta_{10} q^{39} -2 \zeta_{10}^{2} q^{41} -8 q^{43} -6 q^{45} + 8 \zeta_{10}^{2} q^{47} + 12 \zeta_{10}^{3} q^{48} -\zeta_{10} q^{49} + 6 \zeta_{10}^{3} q^{51} + 8 \zeta_{10}^{2} q^{52} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{53} + ( -18 + 18 \zeta_{10} - 18 \zeta_{10}^{2} + 18 \zeta_{10}^{3} ) q^{57} -3 \zeta_{10}^{3} q^{59} + 6 \zeta_{10} q^{60} + 2 \zeta_{10} q^{61} -6 \zeta_{10}^{2} q^{63} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{64} + 4 q^{65} -3 q^{67} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{68} + 15 \zeta_{10}^{2} q^{69} -\zeta_{10} q^{71} -10 \zeta_{10}^{3} q^{73} + ( -12 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{75} + 12 q^{76} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{79} -4 \zeta_{10}^{2} q^{80} -9 \zeta_{10}^{3} q^{81} -12 \zeta_{10} q^{83} + 6 \zeta_{10}^{3} q^{84} -2 \zeta_{10}^{2} q^{85} -30 q^{87} -15 q^{89} + 4 \zeta_{10}^{2} q^{91} -10 \zeta_{10}^{3} q^{92} + 3 \zeta_{10} q^{93} -6 \zeta_{10}^{3} q^{95} + ( 5 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + 2 q^{4} + q^{5} + q^{7} - 6 q^{9} + O(q^{10}) \) \( 4 q + 3 q^{3} + 2 q^{4} + q^{5} + q^{7} - 6 q^{9} + 24 q^{12} + 4 q^{13} - 3 q^{15} - 4 q^{16} - 2 q^{17} + 6 q^{19} - 2 q^{20} + 12 q^{21} - 20 q^{23} + 4 q^{25} + 9 q^{27} - 2 q^{28} - 10 q^{29} - q^{31} - q^{35} + 12 q^{36} + 5 q^{37} - 12 q^{39} + 2 q^{41} - 32 q^{43} - 24 q^{45} - 8 q^{47} + 12 q^{48} - q^{49} + 6 q^{51} - 8 q^{52} + 6 q^{53} - 18 q^{57} - 3 q^{59} + 6 q^{60} + 2 q^{61} + 6 q^{63} + 8 q^{64} + 16 q^{65} - 12 q^{67} + 4 q^{68} - 15 q^{69} - q^{71} - 10 q^{73} - 12 q^{75} + 48 q^{76} - 6 q^{79} + 4 q^{80} - 9 q^{81} - 12 q^{83} + 6 q^{84} + 2 q^{85} - 120 q^{87} - 60 q^{89} - 4 q^{91} - 10 q^{92} + 3 q^{93} - 6 q^{95} + 5 q^{97} + O(q^{100}) \)

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.

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 \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2} \)
\( T_{3}^{4} - 3 T_{3}^{3} + 9 T_{3}^{2} - 27 T_{3} + 81 \)
\( T_{13}^{4} - 4 T_{13}^{3} + 16 T_{13}^{2} - 64 T_{13} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$5$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$7$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( ( 5 + T )^{4} \)
$29$ \( 10000 + 1000 T + 100 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$37$ \( 625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4} \)
$41$ \( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$43$ \( ( 8 + T )^{4} \)
$47$ \( 4096 + 512 T + 64 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( 81 + 27 T + 9 T^{2} + 3 T^{3} + T^{4} \)
$61$ \( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$67$ \( ( 3 + T )^{4} \)
$71$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$73$ \( 10000 + 1000 T + 100 T^{2} + 10 T^{3} + T^{4} \)
$79$ \( 1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4} \)
$83$ \( 20736 + 1728 T + 144 T^{2} + 12 T^{3} + T^{4} \)
$89$ \( ( 15 + T )^{4} \)
$97$ \( 625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4} \)
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