Properties

Label 847.2.f.r
Level $847$
Weight $2$
Character orbit 847.f
Analytic conductor $6.763$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.446265625.1
Defining polynomial: \( x^{8} - x^{7} + 4x^{6} - 7x^{5} + 19x^{4} + 21x^{3} + 36x^{2} + 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{3} + (\beta_{3} - \beta_1) q^{4} + (2 \beta_{6} + \beta_{4}) q^{5} + (\beta_{6} - 3 \beta_{4}) q^{6} + \beta_{3} q^{7} + (3 \beta_{7} - 3 \beta_{4} + 3 \beta_{3} + 3) q^{8} + \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{3} + (\beta_{3} - \beta_1) q^{4} + (2 \beta_{6} + \beta_{4}) q^{5} + (\beta_{6} - 3 \beta_{4}) q^{6} + \beta_{3} q^{7} + (3 \beta_{7} - 3 \beta_{4} + 3 \beta_{3} + 3) q^{8} + \beta_{5} q^{9} + ( - \beta_{2} - 6) q^{10} + ( - 2 \beta_{2} - 3) q^{12} + ( - 2 \beta_{7} - 2 \beta_{5}) q^{13} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{14} + ( - 6 \beta_{3} + \beta_1) q^{15} + (\beta_{6} + 2 \beta_{4}) q^{16} + (\beta_{6} + 5 \beta_{4}) q^{17} + ( - 3 \beta_{3} + \beta_1) q^{18} + ( - 3 \beta_{7} + 3 \beta_{4} - 3 \beta_{3} - 3) q^{19} + (5 \beta_{7} + 3 \beta_{5}) q^{20} - \beta_{2} q^{21} + (\beta_{2} - 5) q^{23} + 3 \beta_{5} q^{24} + ( - 8 \beta_{7} + 8 \beta_{4} - 8 \beta_{3} - 8) q^{25} + (6 \beta_{3} - 4 \beta_1) q^{26} + (2 \beta_{6} + 3 \beta_{4}) q^{27} + (\beta_{6} - \beta_{4}) q^{28} + ( - 7 \beta_{3} - \beta_1) q^{29} + ( - 3 \beta_{7} - 7 \beta_{6} - 7 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} + \cdots - 3) q^{30}+ \cdots - \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + q^{3} - 3 q^{4} - 7 q^{6} - 2 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + q^{3} - 3 q^{4} - 7 q^{6} - 2 q^{7} + 6 q^{8} - q^{9} - 52 q^{10} - 32 q^{12} + 6 q^{13} + q^{14} + 13 q^{15} + 3 q^{16} + 9 q^{17} + 7 q^{18} - 6 q^{19} - 13 q^{20} - 4 q^{21} - 36 q^{23} - 3 q^{24} - 16 q^{25} - 16 q^{26} + 4 q^{27} - 3 q^{28} + 13 q^{29} - 13 q^{30} - 2 q^{31} + 28 q^{32} - 8 q^{34} - 8 q^{36} - 4 q^{37} + 3 q^{38} - 16 q^{39} + 14 q^{41} - 7 q^{42} - 28 q^{43} + 52 q^{45} + 2 q^{46} - 7 q^{47} + 5 q^{48} - 2 q^{49} + 8 q^{50} + 2 q^{51} + 22 q^{52} + 15 q^{53} - 44 q^{54} - 24 q^{56} + 3 q^{57} - 17 q^{59} + 26 q^{60} - 5 q^{61} + q^{62} - q^{63} + 4 q^{64} - 104 q^{65} - 4 q^{67} + 7 q^{68} + 2 q^{69} + 13 q^{70} + 5 q^{71} + 3 q^{72} - 10 q^{73} - 24 q^{74} + 8 q^{75} + 36 q^{76} - 136 q^{78} - 13 q^{79} - 13 q^{80} + 14 q^{81} - 7 q^{82} + 6 q^{83} + 8 q^{84} - 13 q^{85} + 3 q^{86} + 12 q^{89} + 13 q^{90} + 6 q^{91} + 7 q^{92} + q^{93} - 16 q^{94} + 10 q^{96} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 4x^{6} - 7x^{5} + 19x^{4} + 21x^{3} + 36x^{2} + 27x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 21 ) / 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 40\nu ) / 57 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 97\nu^{2} ) / 171 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 7\nu^{5} + 19\nu^{4} - 76\nu^{3} + 36\nu^{2} + 27\nu + 81 ) / 171 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 40\nu^{2} ) / 57 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} - 16\nu^{6} + 28\nu^{5} - 76\nu^{4} + 133\nu^{3} - 144\nu^{2} - 108\nu - 324 ) / 513 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{7} - 7\beta_{6} - 7\beta_{5} + 12\beta_{4} - 12\beta_{3} - 7\beta_{2} + 7\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{2} - 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 57\beta_{3} - 40\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -97\beta_{6} + 120\beta_{4} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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
1.86298 1.35354i
−1.05397 + 0.765752i
0.402580 + 1.23901i
−0.711597 2.19007i
1.86298 + 1.35354i
−1.05397 0.765752i
0.402580 1.23901i
−0.711597 + 2.19007i
−0.711597 2.19007i 1.86298 + 1.35354i −2.67200 + 1.94132i 1.11418 3.42908i 1.63865 5.04324i −0.809017 + 0.587785i 2.42705 + 1.76336i 0.711597 + 2.19007i −8.30278
148.2 0.402580 + 1.23901i −1.05397 0.765752i 0.244951 0.177967i −1.11418 + 3.42908i 0.524471 1.61416i −0.809017 + 0.587785i 2.42705 + 1.76336i −0.402580 1.23901i −4.69722
323.1 −1.05397 0.765752i 0.402580 1.23901i −0.0935628 0.287957i 2.91695 2.11929i −1.37308 + 0.997603i 0.309017 + 0.951057i −0.927051 + 2.85317i 1.05397 + 0.765752i −4.69722
323.2 1.86298 + 1.35354i −0.711597 + 2.19007i 1.02061 + 3.14113i −2.91695 + 2.11929i −4.29004 + 3.11689i 0.309017 + 0.951057i −0.927051 + 2.85317i −1.86298 1.35354i −8.30278
372.1 −0.711597 + 2.19007i 1.86298 1.35354i −2.67200 1.94132i 1.11418 + 3.42908i 1.63865 + 5.04324i −0.809017 0.587785i 2.42705 1.76336i 0.711597 2.19007i −8.30278
372.2 0.402580 1.23901i −1.05397 + 0.765752i 0.244951 + 0.177967i −1.11418 3.42908i 0.524471 + 1.61416i −0.809017 0.587785i 2.42705 1.76336i −0.402580 + 1.23901i −4.69722
729.1 −1.05397 + 0.765752i 0.402580 + 1.23901i −0.0935628 + 0.287957i 2.91695 + 2.11929i −1.37308 0.997603i 0.309017 0.951057i −0.927051 2.85317i 1.05397 0.765752i −4.69722
729.2 1.86298 1.35354i −0.711597 2.19007i 1.02061 3.14113i −2.91695 2.11929i −4.29004 3.11689i 0.309017 0.951057i −0.927051 2.85317i −1.86298 + 1.35354i −8.30278
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 729.2
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.r 8
11.b odd 2 1 847.2.f.o 8
11.c even 5 1 847.2.a.e 2
11.c even 5 3 inner 847.2.f.r 8
11.d odd 10 1 847.2.a.g yes 2
11.d odd 10 3 847.2.f.o 8
33.f even 10 1 7623.2.a.bc 2
33.h odd 10 1 7623.2.a.bs 2
77.j odd 10 1 5929.2.a.k 2
77.l even 10 1 5929.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.e 2 11.c even 5 1
847.2.a.g yes 2 11.d odd 10 1
847.2.f.o 8 11.b odd 2 1
847.2.f.o 8 11.d odd 10 3
847.2.f.r 8 1.a even 1 1 trivial
847.2.f.r 8 11.c even 5 3 inner
5929.2.a.k 2 77.j odd 10 1
5929.2.a.p 2 77.l even 10 1
7623.2.a.bc 2 33.f even 10 1
7623.2.a.bs 2 33.h odd 10 1

Hecke kernels

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

\( T_{2}^{8} - T_{2}^{7} + 4T_{2}^{6} - 7T_{2}^{5} + 19T_{2}^{4} + 21T_{2}^{3} + 36T_{2}^{2} + 27T_{2} + 81 \) Copy content Toggle raw display
\( T_{3}^{8} - T_{3}^{7} + 4T_{3}^{6} - 7T_{3}^{5} + 19T_{3}^{4} + 21T_{3}^{3} + 36T_{3}^{2} + 27T_{3} + 81 \) Copy content Toggle raw display
\( T_{13}^{8} - 6T_{13}^{7} + 40T_{13}^{6} - 264T_{13}^{5} + 1744T_{13}^{4} + 1056T_{13}^{3} + 640T_{13}^{2} + 384T_{13} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 4 T^{6} - 7 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + 4 T^{6} - 7 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 13 T^{6} + 169 T^{4} + \cdots + 28561 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 6 T^{7} + 40 T^{6} - 264 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} - 9 T^{7} + 64 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( (T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 9 T + 17)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} - 13 T^{7} + 130 T^{6} + \cdots + 2313441 \) Copy content Toggle raw display
$31$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + 64 T^{6} + \cdots + 5308416 \) Copy content Toggle raw display
$41$ \( (T^{4} - 7 T^{3} + 49 T^{2} - 343 T + 2401)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 7 T + 9)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 7 T^{7} + 66 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$53$ \( T^{8} - 15 T^{7} + 198 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$59$ \( T^{8} + 17 T^{7} + 220 T^{6} + \cdots + 22667121 \) Copy content Toggle raw display
$61$ \( T^{8} + 5 T^{7} + 22 T^{6} + 95 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 81)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 5 T^{7} + 22 T^{6} - 95 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$73$ \( (T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 13 T^{7} + 130 T^{6} + \cdots + 2313441 \) Copy content Toggle raw display
$83$ \( (T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T - 261)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 13 T^{6} + 169 T^{4} + \cdots + 28561 \) Copy content Toggle raw display
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