Properties

Label 847.2.f.c
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: yes
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{2} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{3} + ( - 3 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{4} - \zeta_{10} q^{5} + ( - \zeta_{10}^{2} - 1) q^{6} + \zeta_{10}^{3} q^{7} + (4 \zeta_{10}^{3} + \zeta_{10}^{2} + 4 \zeta_{10}) q^{8} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{2} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{3} + ( - 3 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{4} - \zeta_{10} q^{5} + ( - \zeta_{10}^{2} - 1) q^{6} + \zeta_{10}^{3} q^{7} + (4 \zeta_{10}^{3} + \zeta_{10}^{2} + 4 \zeta_{10}) q^{8} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{9} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 2) q^{10} + 3 q^{12} + ( - 2 \zeta_{10}^{3} + 2) q^{13} + ( - \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10}) q^{14} + ( - \zeta_{10} + 1) q^{15} + ( - 3 \zeta_{10}^{2} - 5 \zeta_{10} - 3) q^{16} + ( - 5 \zeta_{10}^{2} - 5) q^{17} + (5 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{18} + (2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 2 \zeta_{10}) q^{19} + (3 \zeta_{10}^{3} - 3) q^{20} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{21} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 3) q^{23} + (4 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{24} - 4 \zeta_{10}^{2} q^{25} + (6 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{26} + (4 \zeta_{10}^{2} - 3 \zeta_{10} + 4) q^{27} + (3 \zeta_{10}^{2} + 3) q^{28} + ( - 3 \zeta_{10}^{3} + \zeta_{10} - 1) q^{29} + (\zeta_{10}^{3} + \zeta_{10}) q^{30} + ( - \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{31} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 9) q^{32} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 15) q^{34} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{35} + ( - 6 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 6 \zeta_{10}) q^{36} + (4 \zeta_{10} - 4) q^{37} + ( - 7 \zeta_{10}^{2} - 5 \zeta_{10} - 7) q^{38} + 2 \zeta_{10} q^{39} + ( - 5 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{40} + ( - 10 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 10 \zeta_{10}) q^{41} + ( - \zeta_{10}^{3} + 1) q^{42} + (9 \zeta_{10}^{3} - 9 \zeta_{10}^{2} - 2) q^{43} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 2) q^{45} + ( - 11 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 11) q^{46} + (\zeta_{10}^{3} - 6 \zeta_{10}^{2} + \zeta_{10}) q^{47} + ( - 3 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{48} - \zeta_{10} q^{49} + (4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{50} - 5 \zeta_{10}^{3} q^{51} + ( - 6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 6 \zeta_{10}) q^{52} + ( - 4 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 4) q^{53} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 6) q^{54} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 5) q^{56} + (2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 2) q^{57} + (2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 2 \zeta_{10}) q^{58} + ( - 3 \zeta_{10}^{3} + 5 \zeta_{10} - 5) q^{59} - 3 \zeta_{10} q^{60} + (\zeta_{10}^{2} - 7 \zeta_{10} + 1) q^{61} + ( - \zeta_{10} + 1) q^{62} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{63} + (5 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 5) q^{64} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{65} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + 3) q^{67} + (30 \zeta_{10}^{3} - 15 \zeta_{10}^{2} + 15 \zeta_{10} - 30) q^{68} + (2 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 2 \zeta_{10}) q^{69} + (2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{70} + ( - 5 \zeta_{10}^{2} + 13 \zeta_{10} - 5) q^{71} + (9 \zeta_{10}^{2} + 5 \zeta_{10} + 9) q^{72} + ( - 11 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{73} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}) q^{74} + ( - 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{75} + ( - 15 \zeta_{10}^{3} + 15 \zeta_{10}^{2} + 21) q^{76} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 4) q^{78} + ( - 8 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 8) q^{79} + (3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 3 \zeta_{10}) q^{80} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{81} + (15 \zeta_{10}^{2} + 5 \zeta_{10} + 15) q^{82} + ( - 6 \zeta_{10}^{2} - \zeta_{10} - 6) q^{83} + 3 \zeta_{10}^{3} q^{84} + (5 \zeta_{10}^{3} + 5 \zeta_{10}) q^{85} + ( - 13 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 13) q^{86} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 1) q^{87} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 2) q^{89} + ( - 5 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 5) q^{90} + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{91} + (24 \zeta_{10}^{3} + 9 \zeta_{10} - 9) q^{92} + (3 \zeta_{10}^{2} - 5 \zeta_{10} + 3) q^{93} + (4 \zeta_{10}^{2} + 5 \zeta_{10} + 4) q^{94} + ( - 5 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{95} + (6 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 6 \zeta_{10}) q^{96} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 7) q^{97} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 2) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 3 q^{3} + 6 q^{4} - q^{5} - 3 q^{6} + q^{7} + 7 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 3 q^{3} + 6 q^{4} - q^{5} - 3 q^{6} + q^{7} + 7 q^{8} + 4 q^{9} + 6 q^{10} + 12 q^{12} + 6 q^{13} - q^{14} + 3 q^{15} - 14 q^{16} - 15 q^{17} - 4 q^{18} + q^{19} - 9 q^{20} + 2 q^{21} - 2 q^{23} - 6 q^{24} + 4 q^{25} - 6 q^{26} + 9 q^{27} + 9 q^{28} - 6 q^{29} + 2 q^{30} - 3 q^{31} + 30 q^{32} + 40 q^{34} + q^{35} - 9 q^{36} - 12 q^{37} - 26 q^{38} + 2 q^{39} + 7 q^{40} - 25 q^{41} + 3 q^{42} + 10 q^{43} - 6 q^{45} + 27 q^{46} + 8 q^{47} + 12 q^{48} - q^{49} + 16 q^{50} - 5 q^{51} - 6 q^{52} + 6 q^{53} - 14 q^{54} - 12 q^{56} - 8 q^{57} + q^{58} - 18 q^{59} - 3 q^{60} - 4 q^{61} + 3 q^{62} + q^{63} - 17 q^{64} - 4 q^{65} - 2 q^{67} - 60 q^{68} + 11 q^{69} - q^{70} - 2 q^{71} + 32 q^{72} - 17 q^{73} - 8 q^{74} + 8 q^{75} + 54 q^{76} - 12 q^{78} + 10 q^{79} + q^{80} + 16 q^{81} + 50 q^{82} - 19 q^{83} + 3 q^{84} + 10 q^{85} + 35 q^{86} - 12 q^{87} + 14 q^{89} + 11 q^{90} + 4 q^{91} - 3 q^{92} + 4 q^{93} + 17 q^{94} + q^{95} + 15 q^{96} - 7 q^{97} + 6 q^{98}+O(q^{100}) \) 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
0.118034 + 0.363271i 1.30902 + 0.951057i 1.50000 1.08981i 0.309017 0.951057i −0.190983 + 0.587785i 0.809017 0.587785i 1.19098 + 0.865300i −0.118034 0.363271i 0.381966
323.1 −2.11803 1.53884i 0.190983 0.587785i 1.50000 + 4.61653i −0.809017 + 0.587785i −1.30902 + 0.951057i −0.309017 0.951057i 2.30902 7.10642i 2.11803 + 1.53884i 2.61803
372.1 0.118034 0.363271i 1.30902 0.951057i 1.50000 + 1.08981i 0.309017 + 0.951057i −0.190983 0.587785i 0.809017 + 0.587785i 1.19098 0.865300i −0.118034 + 0.363271i 0.381966
729.1 −2.11803 + 1.53884i 0.190983 + 0.587785i 1.50000 4.61653i −0.809017 0.587785i −1.30902 0.951057i −0.309017 + 0.951057i 2.30902 + 7.10642i 2.11803 1.53884i 2.61803
\(n\): 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.c 4
11.b odd 2 1 847.2.f.l 4
11.c even 5 1 847.2.a.h yes 2
11.c even 5 1 inner 847.2.f.c 4
11.c even 5 2 847.2.f.j 4
11.d odd 10 1 847.2.a.d 2
11.d odd 10 2 847.2.f.d 4
11.d odd 10 1 847.2.f.l 4
33.f even 10 1 7623.2.a.bx 2
33.h odd 10 1 7623.2.a.t 2
77.j odd 10 1 5929.2.a.s 2
77.l even 10 1 5929.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.d 2 11.d odd 10 1
847.2.a.h yes 2 11.c even 5 1
847.2.f.c 4 1.a even 1 1 trivial
847.2.f.c 4 11.c even 5 1 inner
847.2.f.d 4 11.d odd 10 2
847.2.f.j 4 11.c even 5 2
847.2.f.l 4 11.b odd 2 1
847.2.f.l 4 11.d odd 10 1
5929.2.a.i 2 77.l even 10 1
5929.2.a.s 2 77.j odd 10 1
7623.2.a.t 2 33.h odd 10 1
7623.2.a.bx 2 33.f even 10 1

Hecke kernels

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

\( T_{2}^{4} + 4T_{2}^{3} + 6T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - 3T_{3}^{3} + 4T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} - 6T_{13}^{3} + 16T_{13}^{2} - 16T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + 6 T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + T^{2} + T + 1 \) 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
$13$ \( T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 15 T^{3} + 100 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( T^{4} - T^{3} + 31 T^{2} + 99 T + 121 \) Copy content Toggle raw display
$23$ \( (T^{2} + T - 31)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$31$ \( T^{4} + 3 T^{3} + 19 T^{2} + 7 T + 1 \) Copy content Toggle raw display
$37$ \( T^{4} + 12 T^{3} + 64 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{4} + 25 T^{3} + 375 T^{2} + \cdots + 15625 \) Copy content Toggle raw display
$43$ \( (T^{2} - 5 T - 95)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 16 T^{2} - 11 T + 121 \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + 124 T^{2} + 7 T + 1 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + 46 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 61)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} + 204 T^{2} + \cdots + 6241 \) Copy content Toggle raw display
$73$ \( T^{4} + 17 T^{3} + 159 T^{2} + \cdots + 19321 \) Copy content Toggle raw display
$79$ \( T^{4} - 10 T^{3} + 60 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$83$ \( T^{4} + 19 T^{3} + 151 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$89$ \( (T^{2} - 7 T + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 7 T^{3} + 49 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
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