Properties

Label 847.2.f.e
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} + 2 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} - 2 \zeta_{10} q^{6} + \zeta_{10}^{3} q^{7} - 3 \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} + 2 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} - 2 \zeta_{10} q^{6} + \zeta_{10}^{3} q^{7} - 3 \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} - 2 q^{10} - 2 q^{12} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{13} - \zeta_{10}^{2} q^{14} + 4 \zeta_{10}^{3} q^{15} + \zeta_{10} q^{16} - 4 \zeta_{10} q^{17} - \zeta_{10}^{3} q^{18} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{20} - 2 q^{21} - 4 q^{23} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{24} - \zeta_{10}^{2} q^{25} - 4 \zeta_{10}^{3} q^{26} + 4 \zeta_{10} q^{27} - \zeta_{10} q^{28} + 6 \zeta_{10}^{3} q^{29} - 4 \zeta_{10}^{2} q^{30} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} - 10) q^{31} + 5 q^{32} + 4 q^{34} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{35} - \zeta_{10}^{2} q^{36} + 6 \zeta_{10}^{3} q^{37} - 8 \zeta_{10} q^{39} - 6 \zeta_{10}^{3} q^{40} + 4 \zeta_{10}^{2} q^{41} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{42} + 12 q^{43} - 2 q^{45} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{46} - 10 \zeta_{10}^{2} q^{47} + 2 \zeta_{10}^{3} q^{48} - \zeta_{10} q^{49} + \zeta_{10} q^{50} - 8 \zeta_{10}^{3} q^{51} - 4 \zeta_{10}^{2} q^{52} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{53} - 4 q^{54} + 3 q^{56} - 6 \zeta_{10}^{2} q^{58} - 2 \zeta_{10}^{3} q^{59} - 4 \zeta_{10} q^{60} - 10 \zeta_{10}^{3} q^{62} - \zeta_{10}^{2} q^{63} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 7) q^{64} - 8 q^{65} + 8 q^{67} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{68} - 8 \zeta_{10}^{2} q^{69} - 2 \zeta_{10}^{3} q^{70} + 12 \zeta_{10} q^{71} + 3 \zeta_{10} q^{72} + 8 \zeta_{10}^{3} q^{73} - 6 \zeta_{10}^{2} q^{74} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{75} + 8 q^{78} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{79} + 2 \zeta_{10}^{2} q^{80} + 11 \zeta_{10}^{3} q^{81} - 4 \zeta_{10} q^{82} - 2 \zeta_{10}^{3} q^{84} - 8 \zeta_{10}^{2} q^{85} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 12) q^{86} - 12 q^{87} - 6 q^{89} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{90} - 4 \zeta_{10}^{2} q^{91} - 4 \zeta_{10}^{3} q^{92} - 20 \zeta_{10} q^{93} + 10 \zeta_{10} q^{94} + 10 \zeta_{10}^{2} q^{96} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 10) q^{97} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 2 q^{3} + q^{4} + 2 q^{5} - 2 q^{6} + q^{7} + 3 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 2 q^{3} + q^{4} + 2 q^{5} - 2 q^{6} + q^{7} + 3 q^{8} - q^{9} - 8 q^{10} - 8 q^{12} - 4 q^{13} + q^{14} + 4 q^{15} + q^{16} - 4 q^{17} - q^{18} - 2 q^{20} - 8 q^{21} - 16 q^{23} + 6 q^{24} + q^{25} - 4 q^{26} + 4 q^{27} - q^{28} + 6 q^{29} + 4 q^{30} - 10 q^{31} + 20 q^{32} + 16 q^{34} - 2 q^{35} + q^{36} + 6 q^{37} - 8 q^{39} - 6 q^{40} - 4 q^{41} + 2 q^{42} + 48 q^{43} - 8 q^{45} + 4 q^{46} + 10 q^{47} + 2 q^{48} - q^{49} + q^{50} - 8 q^{51} + 4 q^{52} + 6 q^{53} - 16 q^{54} + 12 q^{56} + 6 q^{58} - 2 q^{59} - 4 q^{60} - 10 q^{62} + q^{63} - 7 q^{64} - 32 q^{65} + 32 q^{67} + 4 q^{68} + 8 q^{69} - 2 q^{70} + 12 q^{71} + 3 q^{72} + 8 q^{73} + 6 q^{74} + 2 q^{75} + 32 q^{78} - 8 q^{79} - 2 q^{80} + 11 q^{81} - 4 q^{82} - 2 q^{84} + 8 q^{85} - 12 q^{86} - 48 q^{87} - 24 q^{89} + 2 q^{90} + 4 q^{91} - 4 q^{92} - 20 q^{93} + 10 q^{94} - 10 q^{96} + 10 q^{97} + 4 q^{98}+O(q^{100}) \) 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
0.309017 + 0.951057i −1.61803 1.17557i 0.809017 0.587785i −0.618034 + 1.90211i 0.618034 1.90211i 0.809017 0.587785i 2.42705 + 1.76336i 0.309017 + 0.951057i −2.00000
323.1 −0.809017 0.587785i 0.618034 1.90211i −0.309017 0.951057i 1.61803 1.17557i −1.61803 + 1.17557i −0.309017 0.951057i −0.927051 + 2.85317i −0.809017 0.587785i −2.00000
372.1 0.309017 0.951057i −1.61803 + 1.17557i 0.809017 + 0.587785i −0.618034 1.90211i 0.618034 + 1.90211i 0.809017 + 0.587785i 2.42705 1.76336i 0.309017 0.951057i −2.00000
729.1 −0.809017 + 0.587785i 0.618034 + 1.90211i −0.309017 + 0.951057i 1.61803 + 1.17557i −1.61803 1.17557i −0.309017 + 0.951057i −0.927051 2.85317i −0.809017 + 0.587785i −2.00000
\(n\): 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.e 4
11.b odd 2 1 847.2.f.k 4
11.c even 5 1 77.2.a.c 1
11.c even 5 3 inner 847.2.f.e 4
11.d odd 10 1 847.2.a.a 1
11.d odd 10 3 847.2.f.k 4
33.f even 10 1 7623.2.a.n 1
33.h odd 10 1 693.2.a.a 1
44.h odd 10 1 1232.2.a.a 1
55.j even 10 1 1925.2.a.c 1
55.k odd 20 2 1925.2.b.d 2
77.j odd 10 1 539.2.a.d 1
77.l even 10 1 5929.2.a.b 1
77.m even 15 2 539.2.e.a 2
77.p odd 30 2 539.2.e.b 2
88.l odd 10 1 4928.2.a.bi 1
88.o even 10 1 4928.2.a.g 1
231.u even 10 1 4851.2.a.a 1
308.t even 10 1 8624.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.c 1 11.c even 5 1
539.2.a.d 1 77.j odd 10 1
539.2.e.a 2 77.m even 15 2
539.2.e.b 2 77.p odd 30 2
693.2.a.a 1 33.h odd 10 1
847.2.a.a 1 11.d odd 10 1
847.2.f.e 4 1.a even 1 1 trivial
847.2.f.e 4 11.c even 5 3 inner
847.2.f.k 4 11.b odd 2 1
847.2.f.k 4 11.d odd 10 3
1232.2.a.a 1 44.h odd 10 1
1925.2.a.c 1 55.j even 10 1
1925.2.b.d 2 55.k odd 20 2
4851.2.a.a 1 231.u even 10 1
4928.2.a.g 1 88.o even 10 1
4928.2.a.bi 1 88.l odd 10 1
5929.2.a.b 1 77.l even 10 1
7623.2.a.n 1 33.f even 10 1
8624.2.a.bc 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 \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} + 4T_{3}^{2} + 8T_{3} + 16 \) 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

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) 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
$17$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T + 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$43$ \( (T - 12)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T - 8)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736 \) Copy content Toggle raw display
$73$ \( T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
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