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: \(x^{4} - x^{3} + x^{2} - x + 1\)
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + 2 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} -2 \zeta_{10} q^{6} + \zeta_{10}^{3} q^{7} -3 \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) 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} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} -2 q^{10} -2 q^{12} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) 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 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{20} -2 q^{21} -4 q^{23} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) 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 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{31} + 5 q^{32} + 4 q^{34} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) 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 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{42} + 12 q^{43} -2 q^{45} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) 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 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) 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 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{64} -8 q^{65} + 8 q^{67} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) 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 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{75} + 8 q^{78} + ( -8 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) 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 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{86} -12 q^{87} -6 q^{89} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) 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 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} - 2q^{3} + q^{4} + 2q^{5} - 2q^{6} + q^{7} + 3q^{8} - q^{9} + O(q^{10}) \) \( 4q - q^{2} - 2q^{3} + q^{4} + 2q^{5} - 2q^{6} + q^{7} + 3q^{8} - q^{9} - 8q^{10} - 8q^{12} - 4q^{13} + q^{14} + 4q^{15} + q^{16} - 4q^{17} - q^{18} - 2q^{20} - 8q^{21} - 16q^{23} + 6q^{24} + q^{25} - 4q^{26} + 4q^{27} - q^{28} + 6q^{29} + 4q^{30} - 10q^{31} + 20q^{32} + 16q^{34} - 2q^{35} + q^{36} + 6q^{37} - 8q^{39} - 6q^{40} - 4q^{41} + 2q^{42} + 48q^{43} - 8q^{45} + 4q^{46} + 10q^{47} + 2q^{48} - q^{49} + q^{50} - 8q^{51} + 4q^{52} + 6q^{53} - 16q^{54} + 12q^{56} + 6q^{58} - 2q^{59} - 4q^{60} - 10q^{62} + q^{63} - 7q^{64} - 32q^{65} + 32q^{67} + 4q^{68} + 8q^{69} - 2q^{70} + 12q^{71} + 3q^{72} + 8q^{73} + 6q^{74} + 2q^{75} + 32q^{78} - 8q^{79} - 2q^{80} + 11q^{81} - 4q^{82} - 2q^{84} + 8q^{85} - 12q^{86} - 48q^{87} - 24q^{89} + 2q^{90} + 4q^{91} - 4q^{92} - 20q^{93} + 10q^{94} - 10q^{96} + 10q^{97} + 4q^{98} + O(q^{100}) \)

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
0.309017 + 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 \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1 \)
\( T_{3}^{4} + 2 T_{3}^{3} + 4 T_{3}^{2} + 8 T_{3} + 16 \)
\( T_{13}^{4} + 4 T_{13}^{3} + 16 T_{13}^{2} + 64 T_{13} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} \)
$3$ \( 1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 12 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8} \)
$5$ \( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 60 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \)
$7$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$11$ 1
$13$ \( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 520 T^{5} + 507 T^{6} + 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 4 T - T^{2} - 72 T^{3} - 271 T^{4} - 1224 T^{5} - 289 T^{6} + 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{4} \)
$29$ \( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 10 T + 69 T^{2} + 380 T^{3} + 1661 T^{4} + 11780 T^{5} + 66309 T^{6} + 297910 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 8436 T^{5} - 1369 T^{6} - 303918 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 4 T - 25 T^{2} - 264 T^{3} - 31 T^{4} - 10824 T^{5} - 42025 T^{6} + 275684 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )^{4} \)
$47$ \( 1 - 10 T + 53 T^{2} - 60 T^{3} - 1891 T^{4} - 2820 T^{5} + 117077 T^{6} - 1038230 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 22260 T^{5} - 47753 T^{6} - 893262 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 2 T - 55 T^{2} - 228 T^{3} + 2789 T^{4} - 13452 T^{5} - 191455 T^{6} + 410758 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{4} \)
$71$ \( 1 - 12 T + 73 T^{2} - 24 T^{3} - 4895 T^{4} - 1704 T^{5} + 367993 T^{6} - 4294932 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 8 T - 9 T^{2} + 656 T^{3} - 4591 T^{4} + 47888 T^{5} - 47961 T^{6} - 3112136 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} - 59408 T^{5} - 93615 T^{6} + 3944312 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{4} \)
$97$ \( 1 - 10 T + 3 T^{2} + 940 T^{3} - 9691 T^{4} + 91180 T^{5} + 28227 T^{6} - 9126730 T^{7} + 88529281 T^{8} \)
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