Properties

Label 847.2.f.a
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: \(\Q(\zeta_{10})\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{3} -3 \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} + ( -2 + 6 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{6} -\zeta_{10}^{3} q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} + ( -1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{3} -3 \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} + ( -2 + 6 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{6} -\zeta_{10}^{3} q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} + ( -1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + ( -2 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{10} + ( -6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{12} + ( -2 + 2 \zeta_{10}^{3} ) q^{13} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{14} + ( 4 - 4 \zeta_{10} ) q^{15} + \zeta_{10} q^{16} + ( 2 + 2 \zeta_{10}^{2} ) q^{17} + ( 6 - 6 \zeta_{10} + 7 \zeta_{10}^{3} ) q^{18} + ( 4 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{19} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{20} + ( -2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{21} + ( 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{23} + ( 4 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{24} -\zeta_{10}^{2} q^{25} + ( 2 - 2 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{26} + ( 4 - 12 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{27} -3 \zeta_{10} q^{28} + ( 4 - 4 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{29} + ( -4 \zeta_{10} + 12 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{30} + ( 4 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{31} + ( -3 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{32} + ( -6 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{34} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{35} + ( -12 \zeta_{10} + 15 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{36} + ( -4 + 4 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{37} + ( -4 - 8 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{38} + 4 \zeta_{10} q^{39} + ( -4 + 4 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{40} + ( -2 \zeta_{10} - 8 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{41} + ( 4 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{42} + 8 q^{43} + ( -2 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{45} + ( -8 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{46} + ( -2 \zeta_{10} + 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{47} + ( 2 - 2 \zeta_{10} ) q^{48} -\zeta_{10} q^{49} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{50} -4 \zeta_{10}^{3} q^{51} + ( 6 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{52} + ( -2 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{53} + ( 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{54} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{56} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{57} + ( 8 \zeta_{10} + 6 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{58} + ( -2 + 2 \zeta_{10} ) q^{59} + ( -12 + 12 \zeta_{10} - 12 \zeta_{10}^{2} ) q^{60} + ( 2 + 4 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{61} + ( -10 + 10 \zeta_{10} ) q^{62} + ( -4 \zeta_{10} + 5 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{63} + ( 13 - 13 \zeta_{10} + 13 \zeta_{10}^{2} - 13 \zeta_{10}^{3} ) q^{64} + ( -4 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{65} + ( 12 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{67} + ( 6 - 6 \zeta_{10}^{3} ) q^{68} + ( 8 \zeta_{10} - 16 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{69} + ( -4 + 4 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{70} + ( 4 + 4 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{71} + ( -6 + 13 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{72} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{73} + ( -8 \zeta_{10} - 6 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{74} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{75} + ( 12 + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{76} + ( -4 - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{78} + ( 4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{79} + 2 \zeta_{10}^{2} q^{80} + ( -12 + 12 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{81} + ( 18 - 4 \zeta_{10} + 18 \zeta_{10}^{2} ) q^{82} + ( 12 - 8 \zeta_{10} + 12 \zeta_{10}^{2} ) q^{83} + ( -6 + 6 \zeta_{10} ) q^{84} + ( 4 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{85} + ( -8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{86} + ( -8 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{87} + 2 q^{89} + ( -14 + 26 \zeta_{10} - 26 \zeta_{10}^{2} + 14 \zeta_{10}^{3} ) q^{90} + ( 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{91} + ( 12 - 12 \zeta_{10} ) q^{92} + ( -12 + 16 \zeta_{10} - 12 \zeta_{10}^{2} ) q^{93} + ( -10 + 10 \zeta_{10} - 10 \zeta_{10}^{2} ) q^{94} + ( -8 + 8 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{95} + ( -6 \zeta_{10} + 18 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{96} + ( -10 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{97} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 5q^{2} - 6q^{3} - 3q^{4} + 2q^{5} - q^{7} + 5q^{8} + 7q^{9} + O(q^{10}) \) \( 4q - 5q^{2} - 6q^{3} - 3q^{4} + 2q^{5} - q^{7} + 5q^{8} + 7q^{9} + 12q^{12} - 6q^{13} + 5q^{14} + 12q^{15} + q^{16} + 6q^{17} + 25q^{18} + 8q^{19} + 6q^{20} + 4q^{21} - 8q^{23} + q^{25} - 3q^{28} + 6q^{29} - 20q^{30} - 20q^{34} + 2q^{35} - 39q^{36} - 6q^{37} - 20q^{38} + 4q^{39} - 10q^{40} + 4q^{41} + 32q^{43} - 24q^{45} - 10q^{47} + 6q^{48} - q^{49} + 5q^{50} - 4q^{51} + 12q^{52} + 6q^{53} - 40q^{54} + 8q^{57} + 10q^{58} - 6q^{59} - 24q^{60} + 10q^{61} - 30q^{62} - 13q^{63} + 13q^{64} - 8q^{65} + 40q^{67} + 18q^{68} + 32q^{69} - 10q^{70} + 16q^{71} - 5q^{72} + 8q^{73} - 10q^{74} - 4q^{75} + 24q^{76} + 20q^{79} - 2q^{80} - 41q^{81} + 50q^{82} + 28q^{83} - 18q^{84} + 8q^{85} - 40q^{86} - 24q^{87} + 8q^{89} + 10q^{90} + 4q^{91} + 36q^{92} - 20q^{93} - 20q^{94} - 16q^{95} - 30q^{96} - 34q^{97} + O(q^{100}) \)

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
−0.690983 2.12663i −2.61803 1.90211i −2.42705 + 1.76336i −0.618034 + 1.90211i −2.23607 + 6.88191i −0.809017 + 0.587785i 1.80902 + 1.31433i 2.30902 + 7.10642i 4.47214
323.1 −1.80902 1.31433i −0.381966 + 1.17557i 0.927051 + 2.85317i 1.61803 1.17557i 2.23607 1.62460i 0.309017 + 0.951057i 0.690983 2.12663i 1.19098 + 0.865300i −4.47214
372.1 −0.690983 + 2.12663i −2.61803 + 1.90211i −2.42705 1.76336i −0.618034 1.90211i −2.23607 6.88191i −0.809017 0.587785i 1.80902 1.31433i 2.30902 7.10642i 4.47214
729.1 −1.80902 + 1.31433i −0.381966 1.17557i 0.927051 2.85317i 1.61803 + 1.17557i 2.23607 + 1.62460i 0.309017 0.951057i 0.690983 + 2.12663i 1.19098 0.865300i −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.a 4
11.b odd 2 1 847.2.f.m 4
11.c even 5 1 77.2.a.d 2
11.c even 5 1 inner 847.2.f.a 4
11.c even 5 2 847.2.f.n 4
11.d odd 10 1 847.2.a.f 2
11.d odd 10 2 847.2.f.b 4
11.d odd 10 1 847.2.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 1.a even 1 1 trivial
847.2.f.a 4 11.c even 5 1 inner
847.2.f.b 4 11.d odd 10 2
847.2.f.m 4 11.b odd 2 1
847.2.f.m 4 11.d odd 10 1
847.2.f.n 4 11.c even 5 2
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 \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{4} + 5 T_{2}^{3} + 15 T_{2}^{2} + 25 T_{2} + 25 \)
\( T_{3}^{4} + 6 T_{3}^{3} + 16 T_{3}^{2} + 16 T_{3} + 16 \)
\( T_{13}^{4} + 6 T_{13}^{3} + 16 T_{13}^{2} + 16 T_{13} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 50 T^{5} + 52 T^{6} + 40 T^{7} + 16 T^{8} \)
$3$ \( 1 + 6 T + 13 T^{2} + 10 T^{3} + T^{4} + 30 T^{5} + 117 T^{6} + 162 T^{7} + 81 T^{8} \)
$5$ \( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 60 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$11$ \( \)
$13$ \( 1 + 6 T + 3 T^{2} - 10 T^{3} + 81 T^{4} - 130 T^{5} + 507 T^{6} + 13182 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 6 T - T^{2} + 18 T^{3} + 169 T^{4} + 306 T^{5} - 289 T^{6} - 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 8 T + 45 T^{2} - 268 T^{3} + 1529 T^{4} - 5092 T^{5} + 16245 T^{6} - 54872 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 + 4 T + 30 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 - 6 T + 47 T^{2} - 288 T^{3} + 2365 T^{4} - 8352 T^{5} + 39527 T^{6} - 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 9 T^{2} + 110 T^{3} + 741 T^{4} + 3410 T^{5} + 8649 T^{6} + 923521 T^{8} \)
$37$ \( 1 + 6 T + 39 T^{2} + 352 T^{3} + 3309 T^{4} + 13024 T^{5} + 53391 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 4 T + 55 T^{2} - 326 T^{3} + 1389 T^{4} - 13366 T^{5} + 92455 T^{6} - 275684 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{4} \)
$47$ \( 1 + 10 T - 7 T^{2} - 470 T^{3} - 3031 T^{4} - 22090 T^{5} - 15463 T^{6} + 1038230 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 6 T + 23 T^{2} - 480 T^{3} + 5581 T^{4} - 25440 T^{5} + 64607 T^{6} - 893262 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 6 T - 43 T^{2} - 102 T^{3} + 3025 T^{4} - 6018 T^{5} - 149683 T^{6} + 1232274 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 10 T - 21 T^{2} + 610 T^{3} - 3199 T^{4} + 37210 T^{5} - 78141 T^{6} - 2269810 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 - 20 T + 214 T^{2} - 1340 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 - 16 T + 25 T^{2} + 916 T^{3} - 9471 T^{4} + 65036 T^{5} + 126025 T^{6} - 5726576 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 8 T - 49 T^{2} + 446 T^{3} + 1549 T^{4} + 32558 T^{5} - 261121 T^{6} - 3112136 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 20 T + 161 T^{2} - 1600 T^{3} + 18961 T^{4} - 126400 T^{5} + 1004801 T^{6} - 9860780 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 28 T + 461 T^{2} - 5964 T^{3} + 61769 T^{4} - 495012 T^{5} + 3175829 T^{6} - 16010036 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{4} \)
$97$ \( 1 + 34 T + 459 T^{2} + 3488 T^{3} + 25829 T^{4} + 338336 T^{5} + 4318731 T^{6} + 31030882 T^{7} + 88529281 T^{8} \)
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