Properties

Label 847.2.f.g
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: \( x^{4} - x^{3} + x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} - 3 \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} - 3 \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} - 2 q^{12} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{13} - 3 \zeta_{10}^{3} q^{15} - 4 \zeta_{10} q^{16} - 6 \zeta_{10} q^{17} - 2 \zeta_{10}^{2} q^{19} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{20} - q^{21} + 3 q^{23} + 4 \zeta_{10}^{2} q^{25} + 5 \zeta_{10} q^{27} - 2 \zeta_{10} q^{28} - 6 \zeta_{10}^{3} q^{29} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{31} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{35} + 4 \zeta_{10}^{2} q^{36} - 11 \zeta_{10}^{3} q^{37} - 4 \zeta_{10} q^{39} - 6 \zeta_{10}^{2} q^{41} - 8 q^{43} - 6 q^{45} - 4 \zeta_{10}^{3} q^{48} - \zeta_{10} q^{49} - 6 \zeta_{10}^{3} q^{51} - 8 \zeta_{10}^{2} q^{52} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{53} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{57} + 9 \zeta_{10}^{3} q^{59} + 6 \zeta_{10} q^{60} - 10 \zeta_{10} q^{61} + 2 \zeta_{10}^{2} q^{63} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{64} + 12 q^{65} + 5 q^{67} + ( - 12 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} + 12) q^{68} + 3 \zeta_{10}^{2} q^{69} - 9 \zeta_{10} q^{71} + 2 \zeta_{10}^{3} q^{73} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{75} + 4 q^{76} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} - 10) q^{79} + 12 \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} + 12 \zeta_{10} q^{83} - 2 \zeta_{10}^{3} q^{84} + 18 \zeta_{10}^{2} q^{85} + 6 q^{87} - 3 q^{89} - 4 \zeta_{10}^{2} q^{91} + 6 \zeta_{10}^{3} q^{92} - 5 \zeta_{10} q^{93} + 6 \zeta_{10}^{3} q^{95} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 2 q^{4} - 3 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 2 q^{4} - 3 q^{5} + q^{7} + 2 q^{9} - 8 q^{12} - 4 q^{13} - 3 q^{15} - 4 q^{16} - 6 q^{17} + 2 q^{19} + 6 q^{20} - 4 q^{21} + 12 q^{23} - 4 q^{25} + 5 q^{27} - 2 q^{28} - 6 q^{29} - 5 q^{31} + 3 q^{35} - 4 q^{36} - 11 q^{37} - 4 q^{39} + 6 q^{41} - 32 q^{43} - 24 q^{45} - 4 q^{48} - q^{49} - 6 q^{51} + 8 q^{52} + 6 q^{53} + 2 q^{57} + 9 q^{59} + 6 q^{60} - 10 q^{61} - 2 q^{63} + 8 q^{64} + 48 q^{65} + 20 q^{67} + 12 q^{68} - 3 q^{69} - 9 q^{71} + 2 q^{73} - 4 q^{75} + 16 q^{76} - 10 q^{79} - 12 q^{80} - q^{81} + 12 q^{83} - 2 q^{84} - 18 q^{85} + 24 q^{87} - 12 q^{89} + 4 q^{91} + 6 q^{92} - 5 q^{93} + 6 q^{95} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −0.809017 0.587785i 1.61803 1.17557i 0.927051 2.85317i 0 0.809017 0.587785i 0 −0.618034 1.90211i 0
323.1 0 0.309017 0.951057i −0.618034 1.90211i −2.42705 + 1.76336i 0 −0.309017 0.951057i 0 1.61803 + 1.17557i 0
372.1 0 −0.809017 + 0.587785i 1.61803 + 1.17557i 0.927051 + 2.85317i 0 0.809017 + 0.587785i 0 −0.618034 + 1.90211i 0
729.1 0 0.309017 + 0.951057i −0.618034 + 1.90211i −2.42705 1.76336i 0 −0.309017 + 0.951057i 0 1.61803 1.17557i 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.g 4
11.b odd 2 1 847.2.f.f 4
11.c even 5 1 847.2.a.c 1
11.c even 5 3 inner 847.2.f.g 4
11.d odd 10 1 77.2.a.b 1
11.d odd 10 3 847.2.f.f 4
33.f even 10 1 693.2.a.b 1
33.h odd 10 1 7623.2.a.i 1
44.g even 10 1 1232.2.a.d 1
55.h odd 10 1 1925.2.a.f 1
55.l even 20 2 1925.2.b.g 2
77.j odd 10 1 5929.2.a.d 1
77.l even 10 1 539.2.a.b 1
77.n even 30 2 539.2.e.e 2
77.o odd 30 2 539.2.e.d 2
88.k even 10 1 4928.2.a.x 1
88.p odd 10 1 4928.2.a.i 1
231.r odd 10 1 4851.2.a.k 1
308.s odd 10 1 8624.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.b 1 11.d odd 10 1
539.2.a.b 1 77.l even 10 1
539.2.e.d 2 77.o odd 30 2
539.2.e.e 2 77.n even 30 2
693.2.a.b 1 33.f even 10 1
847.2.a.c 1 11.c even 5 1
847.2.f.f 4 11.b odd 2 1
847.2.f.f 4 11.d odd 10 3
847.2.f.g 4 1.a even 1 1 trivial
847.2.f.g 4 11.c even 5 3 inner
1232.2.a.d 1 44.g even 10 1
1925.2.a.f 1 55.h odd 10 1
1925.2.b.g 2 55.l even 20 2
4851.2.a.k 1 231.r odd 10 1
4928.2.a.i 1 88.p odd 10 1
4928.2.a.x 1 88.k even 10 1
5929.2.a.d 1 77.j odd 10 1
7623.2.a.i 1 33.h odd 10 1
8624.2.a.s 1 308.s odd 10 1

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{4} + T_{3}^{3} + T_{3}^{2} + T_{3} + 1 \) 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
$3$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$23$ \( (T - 3)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625 \) Copy content Toggle raw display
$37$ \( T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$43$ \( (T + 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} - 9 T^{3} + 81 T^{2} + \cdots + 6561 \) Copy content Toggle raw display
$61$ \( T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$67$ \( (T - 5)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 9 T^{3} + 81 T^{2} + \cdots + 6561 \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736 \) Copy content Toggle raw display
$89$ \( (T + 3)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
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