Properties

Label 847.2.f.n
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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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.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{2} + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{3} - 3 \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} + (2 \zeta_{10}^{2} + 4 \zeta_{10} + 2) q^{6} - \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{8} + (5 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 5) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{2} + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{3} - 3 \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} + (2 \zeta_{10}^{2} + 4 \zeta_{10} + 2) q^{6} - \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{8} + (5 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 5) q^{9} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 2) q^{10} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 6) q^{12} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{13} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{14} + (4 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{15} + \zeta_{10} q^{16} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{17} + (13 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{18} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{19} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{20} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{21} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 4) q^{23} + ( - 6 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 6) q^{24} - \zeta_{10}^{2} q^{25} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{26} + ( - 4 \zeta_{10}^{2} - 8 \zeta_{10} - 4) q^{27} - 3 \zeta_{10} q^{28} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{29} + (4 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 4 \zeta_{10}) q^{30} + ( - 6 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 6) q^{31} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 3) q^{32} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4) q^{34} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{35} + (12 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 12 \zeta_{10}) q^{36} + (2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{37} + (4 \zeta_{10}^{2} - 12 \zeta_{10} + 4) q^{38} + 4 \zeta_{10} q^{39} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{40} + (2 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 2 \zeta_{10}) q^{41} + ( - 6 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 6) q^{42} + 8 q^{43} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 10) q^{45} + (12 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 12) q^{46} + (2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 2 \zeta_{10}) q^{47} + (2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{48} - \zeta_{10} q^{49} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{50} - 4 \zeta_{10}^{3} q^{51} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10}) q^{52} + (6 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 6) q^{53} + (20 \zeta_{10}^{3} - 20 \zeta_{10}^{2} - 20) q^{54} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 1) q^{56} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{57} + ( - 8 \zeta_{10}^{3} + 14 \zeta_{10}^{2} - 8 \zeta_{10}) q^{58} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{59} + (12 \zeta_{10}^{2} + 12) q^{60} + ( - 2 \zeta_{10}^{2} + 6 \zeta_{10} - 2) q^{61} + ( - 10 \zeta_{10}^{3} - 10 \zeta_{10} + 10) q^{62} + (4 \zeta_{10}^{3} + \zeta_{10}^{2} + 4 \zeta_{10}) q^{63} + ( - 13 \zeta_{10}^{3} + 13 \zeta_{10}^{2} - 13 \zeta_{10} + 13) q^{64} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2}) q^{65} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 8) q^{67} + (6 \zeta_{10}^{2} - 6 \zeta_{10}) q^{68} + ( - 8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 8 \zeta_{10}) q^{69} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{70} + ( - 4 \zeta_{10}^{2} + 8 \zeta_{10} - 4) q^{71} + (6 \zeta_{10}^{2} + 7 \zeta_{10} + 6) q^{72} + (4 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{73} + (8 \zeta_{10}^{3} - 14 \zeta_{10}^{2} + 8 \zeta_{10}) q^{74} + ( - 2 \zeta_{10}^{3} + 2) q^{75} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2}) q^{76} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 4) q^{78} + (4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 4) q^{79} + 2 \zeta_{10}^{2} q^{80} + ( - 17 \zeta_{10}^{3} - 12 \zeta_{10} + 12) q^{81} + ( - 18 \zeta_{10}^{2} + 14 \zeta_{10} - 18) q^{82} + ( - 12 \zeta_{10}^{2} + 4 \zeta_{10} - 12) q^{83} + ( - 6 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{84} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{85} + ( - 8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} + 8) q^{86} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4) q^{87} + 2 q^{89} + (26 \zeta_{10}^{3} - 14 \zeta_{10}^{2} + 14 \zeta_{10} - 26) q^{90} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{91} + (12 \zeta_{10}^{3} + 12 \zeta_{10} - 12) q^{92} + (12 \zeta_{10}^{2} + 4 \zeta_{10} + 12) q^{93} + (10 \zeta_{10}^{2} + 10) q^{94} + ( - 8 \zeta_{10} + 8) q^{95} + (6 \zeta_{10}^{3} + 12 \zeta_{10}^{2} + 6 \zeta_{10}) q^{96} + ( - 2 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} + 2) q^{97} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 4 q^{3} - 3 q^{4} + 2 q^{5} + 10 q^{6} - q^{7} - 5 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{2} + 4 q^{3} - 3 q^{4} + 2 q^{5} + 10 q^{6} - q^{7} - 5 q^{8} - 13 q^{9} + 12 q^{12} + 4 q^{13} - 5 q^{14} - 8 q^{15} + q^{16} - 4 q^{17} - 5 q^{18} - 12 q^{19} + 6 q^{20} + 4 q^{21} - 8 q^{23} + 10 q^{24} + q^{25} - 10 q^{26} - 20 q^{27} - 3 q^{28} - 14 q^{29} + 10 q^{31} - 20 q^{34} + 2 q^{35} + 21 q^{36} + 14 q^{37} + 4 q^{39} + 10 q^{40} + 14 q^{41} + 10 q^{42} + 32 q^{43} - 24 q^{45} - 20 q^{46} - 4 q^{48} - q^{49} - 5 q^{50} - 4 q^{51} - 18 q^{52} - 14 q^{53} - 40 q^{54} + 8 q^{57} - 30 q^{58} + 4 q^{59} + 36 q^{60} + 20 q^{62} + 7 q^{63} + 13 q^{64} - 8 q^{65} + 40 q^{67} - 12 q^{68} - 8 q^{69} + 10 q^{70} - 4 q^{71} + 25 q^{72} - 2 q^{73} + 30 q^{74} + 6 q^{75} + 24 q^{76} - 20 q^{79} - 2 q^{80} + 19 q^{81} - 40 q^{82} - 32 q^{83} + 12 q^{84} - 12 q^{85} + 40 q^{86} - 24 q^{87} + 8 q^{89} - 50 q^{90} - 6 q^{91} - 24 q^{92} + 40 q^{93} + 30 q^{94} + 24 q^{95} + 26 q^{97}+O(q^{100}) \) 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
0.690983 + 2.12663i 1.00000 + 0.726543i −2.42705 + 1.76336i −0.618034 + 1.90211i −0.854102 + 2.62866i −0.809017 + 0.587785i −1.80902 1.31433i −0.454915 1.40008i −4.47214
323.1 1.80902 + 1.31433i 1.00000 3.07768i 0.927051 + 2.85317i 1.61803 1.17557i 5.85410 4.25325i 0.309017 + 0.951057i −0.690983 + 2.12663i −6.04508 4.39201i 4.47214
372.1 0.690983 2.12663i 1.00000 0.726543i −2.42705 1.76336i −0.618034 1.90211i −0.854102 2.62866i −0.809017 0.587785i −1.80902 + 1.31433i −0.454915 + 1.40008i −4.47214
729.1 1.80902 1.31433i 1.00000 + 3.07768i 0.927051 2.85317i 1.61803 + 1.17557i 5.85410 + 4.25325i 0.309017 0.951057i −0.690983 2.12663i −6.04508 + 4.39201i 4.47214
\(n\): 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.n 4
11.b odd 2 1 847.2.f.b 4
11.c even 5 1 77.2.a.d 2
11.c even 5 2 847.2.f.a 4
11.c even 5 1 inner 847.2.f.n 4
11.d odd 10 1 847.2.a.f 2
11.d odd 10 1 847.2.f.b 4
11.d odd 10 2 847.2.f.m 4
33.f even 10 1 7623.2.a.bl 2
33.h odd 10 1 693.2.a.h 2
44.h odd 10 1 1232.2.a.m 2
55.j even 10 1 1925.2.a.r 2
55.k odd 20 2 1925.2.b.h 4
77.j odd 10 1 539.2.a.f 2
77.l even 10 1 5929.2.a.m 2
77.m even 15 2 539.2.e.i 4
77.p odd 30 2 539.2.e.j 4
88.l odd 10 1 4928.2.a.bv 2
88.o even 10 1 4928.2.a.bm 2
231.u even 10 1 4851.2.a.y 2
308.t even 10 1 8624.2.a.ce 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 11.c even 5 1
539.2.a.f 2 77.j odd 10 1
539.2.e.i 4 77.m even 15 2
539.2.e.j 4 77.p odd 30 2
693.2.a.h 2 33.h odd 10 1
847.2.a.f 2 11.d odd 10 1
847.2.f.a 4 11.c even 5 2
847.2.f.b 4 11.b odd 2 1
847.2.f.b 4 11.d odd 10 1
847.2.f.m 4 11.d odd 10 2
847.2.f.n 4 1.a even 1 1 trivial
847.2.f.n 4 11.c even 5 1 inner
1232.2.a.m 2 44.h odd 10 1
1925.2.a.r 2 55.j even 10 1
1925.2.b.h 4 55.k odd 20 2
4851.2.a.y 2 231.u even 10 1
4928.2.a.bm 2 88.o even 10 1
4928.2.a.bv 2 88.l odd 10 1
5929.2.a.m 2 77.l even 10 1
7623.2.a.bl 2 33.f even 10 1
8624.2.a.ce 2 308.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 \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{4} - 5T_{2}^{3} + 15T_{2}^{2} - 25T_{2} + 25 \) Copy content Toggle raw display
\( T_{3}^{4} - 4T_{3}^{3} + 16T_{3}^{2} - 24T_{3} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} - 4T_{13}^{3} + 16T_{13}^{2} - 24T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16 \) 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
$13$ \( T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 64 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 14 T^{3} + 76 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} + 40 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$37$ \( T^{4} - 14 T^{3} + 76 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$41$ \( T^{4} - 14 T^{3} + 96 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$43$ \( (T - 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 40 T^{2} + 200 T + 400 \) Copy content Toggle raw display
$53$ \( T^{4} + 14 T^{3} + 76 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$61$ \( T^{4} + 40 T^{2} - 200 T + 400 \) Copy content Toggle raw display
$67$ \( (T^{2} - 20 T + 80)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 4 T^{3} + 96 T^{2} - 256 T + 256 \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} + 20 T^{3} + 240 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$83$ \( T^{4} + 32 T^{3} + 544 T^{2} + \cdots + 30976 \) Copy content Toggle raw display
$89$ \( (T - 2)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 26 T^{3} + 556 T^{2} + \cdots + 26896 \) Copy content Toggle raw display
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