Properties

Label 845.2.c.g
Level $845$
Weight $2$
Character orbit 845.c
Analytic conductor $6.747$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -1 + \beta_{2} + \beta_{5} ) q^{4} + \beta_{6} q^{5} + ( -\beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{6} + ( -\beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{7} + ( \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{8} + ( 2 - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -1 + \beta_{2} + \beta_{5} ) q^{4} + \beta_{6} q^{5} + ( -\beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{6} + ( -\beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{7} + ( \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{8} + ( 2 - 2 \beta_{2} ) q^{9} + ( 1 - \beta_{2} ) q^{10} + ( -2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{11} + ( 4 - 3 \beta_{2} - 2 \beta_{5} ) q^{12} + ( 2 - 3 \beta_{2} + \beta_{5} ) q^{14} + ( -\beta_{4} - \beta_{6} ) q^{15} + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{16} + ( 1 + \beta_{3} + 2 \beta_{5} ) q^{17} + ( 2 \beta_{1} - 4 \beta_{6} + 2 \beta_{7} ) q^{18} + ( -4 \beta_{6} + \beta_{7} ) q^{19} + ( \beta_{1} + \beta_{7} ) q^{20} + ( -2 \beta_{4} + \beta_{7} ) q^{21} + ( 3 - \beta_{2} - \beta_{3} ) q^{22} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{23} + ( 4 \beta_{1} - 4 \beta_{6} + \beta_{7} ) q^{24} - q^{25} + ( -1 + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{27} + ( \beta_{1} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{28} + ( 3 - 2 \beta_{2} - 2 \beta_{5} ) q^{29} + ( -2 + \beta_{2} - \beta_{3} + \beta_{5} ) q^{30} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{31} + ( \beta_{1} + \beta_{4} + \beta_{7} ) q^{32} + ( -\beta_{4} + 4 \beta_{6} + 3 \beta_{7} ) q^{33} + ( -\beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{34} + ( -2 + \beta_{3} + \beta_{5} ) q^{35} + ( -8 + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{36} + ( 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{37} + ( -5 + 5 \beta_{2} - \beta_{3} ) q^{38} + ( -2 - \beta_{3} + \beta_{5} ) q^{40} + ( -2 \beta_{6} - \beta_{7} ) q^{41} + ( -3 + \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{42} + ( 1 + \beta_{3} + 2 \beta_{5} ) q^{43} + ( -\beta_{1} - 3 \beta_{4} - 5 \beta_{6} - 2 \beta_{7} ) q^{44} -2 \beta_{1} q^{45} + ( \beta_{1} + \beta_{4} + 5 \beta_{6} - 3 \beta_{7} ) q^{46} + ( 4 \beta_{1} + 4 \beta_{6} + 2 \beta_{7} ) q^{47} + ( -9 + 3 \beta_{2} - \beta_{3} ) q^{48} + ( -2 \beta_{2} + 4 \beta_{3} + 4 \beta_{5} ) q^{49} -\beta_{1} q^{50} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{51} + ( -4 + 2 \beta_{2} ) q^{53} + ( -3 \beta_{1} + \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{54} + ( 2 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{55} + ( -3 - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} ) q^{56} + ( -2 \beta_{1} + 3 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{57} + ( 5 \beta_{1} - 2 \beta_{4} - 4 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 2 \beta_{1} + 4 \beta_{6} - \beta_{7} ) q^{59} + ( -3 \beta_{1} + \beta_{6} - 2 \beta_{7} ) q^{60} + ( 7 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{61} + 2 \beta_{3} q^{62} + ( -6 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{63} + ( 1 + 4 \beta_{3} + 2 \beta_{5} ) q^{64} + ( -\beta_{2} - 4 \beta_{3} + \beta_{5} ) q^{66} + ( -\beta_{4} - 8 \beta_{6} - \beta_{7} ) q^{67} + ( 4 + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{68} + ( 5 - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{69} + ( -3 \beta_{1} - \beta_{6} + \beta_{7} ) q^{70} + ( -6 \beta_{4} - 4 \beta_{6} - 3 \beta_{7} ) q^{71} + ( -6 \beta_{1} + 4 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{72} + ( 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{73} + ( -2 + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{74} + ( 1 + \beta_{3} ) q^{75} + ( -5 \beta_{1} + \beta_{4} + 3 \beta_{6} - 4 \beta_{7} ) q^{76} + ( -5 + 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{77} + ( -4 + 6 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -\beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{80} + ( -3 + 2 \beta_{2} - 4 \beta_{5} ) q^{81} + ( -1 + \beta_{2} + \beta_{3} ) q^{82} + ( -2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( -5 \beta_{1} + \beta_{4} + 5 \beta_{6} - 2 \beta_{7} ) q^{84} + ( \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{85} + ( -\beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{86} + ( -9 + 6 \beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{87} + ( 3 - 3 \beta_{3} + 2 \beta_{5} ) q^{88} + ( 8 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{89} + ( 6 - 2 \beta_{2} - 2 \beta_{5} ) q^{90} + ( 8 - 3 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 4 \beta_{1} + 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{93} + ( -10 + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} ) q^{94} + ( 4 - \beta_{5} ) q^{95} + ( -\beta_{1} + \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{96} + ( 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{97} + ( -4 \beta_{1} - 8 \beta_{6} + 6 \beta_{7} ) q^{98} + ( -2 \beta_{1} - 2 \beta_{4} + 4 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} - 4q^{4} + 8q^{9} + O(q^{10}) \) \( 8q - 4q^{3} - 4q^{4} + 8q^{9} + 4q^{10} + 20q^{12} + 4q^{14} + 4q^{16} + 4q^{17} + 24q^{22} + 20q^{23} - 8q^{25} - 4q^{27} + 16q^{29} - 8q^{30} - 20q^{35} - 40q^{36} - 16q^{38} - 12q^{40} - 8q^{42} + 4q^{43} - 56q^{48} - 24q^{49} - 8q^{51} - 24q^{53} - 24q^{56} + 56q^{61} - 8q^{62} - 8q^{64} + 12q^{66} + 28q^{68} + 32q^{69} - 20q^{74} + 4q^{75} - 36q^{77} - 16q^{79} - 16q^{81} - 8q^{82} - 44q^{87} + 36q^{88} + 40q^{90} + 44q^{92} - 64q^{94} + 32q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 5 \nu^{5} - 2 \nu^{4} + 11 \nu^{3} - 4 \nu^{2} - 28 \nu + 32 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} - \nu^{5} + 4 \nu^{4} - \nu^{3} - 6 \nu^{2} + 10 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{6} - 3 \nu^{5} - 10 \nu^{4} + 13 \nu^{3} + 8 \nu^{2} - 32 \nu + 24 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} + 12 \nu^{6} - 5 \nu^{5} - 18 \nu^{4} + 19 \nu^{3} + 28 \nu^{2} - 64 \nu + 40 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 9 \nu^{6} - 5 \nu^{5} - 13 \nu^{4} + 21 \nu^{3} + 13 \nu^{2} - 54 \nu + 40 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{7} + 11 \nu^{6} - 6 \nu^{5} - 17 \nu^{4} + 24 \nu^{3} + 15 \nu^{2} - 62 \nu + 48 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} + 7 \nu^{6} - 3 \nu^{5} - 11 \nu^{4} + 15 \nu^{3} + 11 \nu^{2} - 40 \nu + 30 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_{1} - 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} - 5 \beta_{6} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{7} - 4 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - \beta_{1} + 6\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-3 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + 8 \beta_{3} - 2 \beta_{2} + 5\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-8 \beta_{7} + 4 \beta_{6} + 8 \beta_{5} - 2 \beta_{4} + \beta_{3} - 5 \beta_{2} - 3 \beta_{1} + 11\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
506.1
0.665665 + 1.24775i
1.20036 + 0.747754i
−1.27597 + 0.609843i
1.40994 + 0.109843i
1.40994 0.109843i
−1.27597 0.609843i
1.20036 0.747754i
0.665665 1.24775i
2.49551i −2.82684 −4.22756 1.00000i 7.05440i 1.90521i 5.55889i 4.99102 2.49551
506.2 1.49551i 0.0947876 −0.236543 1.00000i 0.141756i 4.82684i 2.63726i −2.99102 −1.49551
506.3 1.21969i 2.33225 0.512364 1.00000i 2.84461i 3.60020i 3.06430i 2.43937 1.21969
506.4 0.219687i −1.60020 1.95174 1.00000i 0.351542i 0.332247i 0.868145i −0.439374 −0.219687
506.5 0.219687i −1.60020 1.95174 1.00000i 0.351542i 0.332247i 0.868145i −0.439374 −0.219687
506.6 1.21969i 2.33225 0.512364 1.00000i 2.84461i 3.60020i 3.06430i 2.43937 1.21969
506.7 1.49551i 0.0947876 −0.236543 1.00000i 0.141756i 4.82684i 2.63726i −2.99102 −1.49551
506.8 2.49551i −2.82684 −4.22756 1.00000i 7.05440i 1.90521i 5.55889i 4.99102 2.49551
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 506.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.c.g 8
13.b even 2 1 inner 845.2.c.g 8
13.c even 3 1 65.2.m.a 8
13.c even 3 1 845.2.m.g 8
13.d odd 4 1 845.2.a.l 4
13.d odd 4 1 845.2.a.m 4
13.e even 6 1 65.2.m.a 8
13.e even 6 1 845.2.m.g 8
13.f odd 12 2 845.2.e.m 8
13.f odd 12 2 845.2.e.n 8
39.f even 4 1 7605.2.a.cf 4
39.f even 4 1 7605.2.a.cj 4
39.h odd 6 1 585.2.bu.c 8
39.i odd 6 1 585.2.bu.c 8
52.i odd 6 1 1040.2.da.b 8
52.j odd 6 1 1040.2.da.b 8
65.g odd 4 1 4225.2.a.bi 4
65.g odd 4 1 4225.2.a.bl 4
65.l even 6 1 325.2.n.d 8
65.n even 6 1 325.2.n.d 8
65.q odd 12 1 325.2.m.b 8
65.q odd 12 1 325.2.m.c 8
65.r odd 12 1 325.2.m.b 8
65.r odd 12 1 325.2.m.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 13.c even 3 1
65.2.m.a 8 13.e even 6 1
325.2.m.b 8 65.q odd 12 1
325.2.m.b 8 65.r odd 12 1
325.2.m.c 8 65.q odd 12 1
325.2.m.c 8 65.r odd 12 1
325.2.n.d 8 65.l even 6 1
325.2.n.d 8 65.n even 6 1
585.2.bu.c 8 39.h odd 6 1
585.2.bu.c 8 39.i odd 6 1
845.2.a.l 4 13.d odd 4 1
845.2.a.m 4 13.d odd 4 1
845.2.c.g 8 1.a even 1 1 trivial
845.2.c.g 8 13.b even 2 1 inner
845.2.e.m 8 13.f odd 12 2
845.2.e.n 8 13.f odd 12 2
845.2.m.g 8 13.c even 3 1
845.2.m.g 8 13.e even 6 1
1040.2.da.b 8 52.i odd 6 1
1040.2.da.b 8 52.j odd 6 1
4225.2.a.bi 4 65.g odd 4 1
4225.2.a.bl 4 65.g odd 4 1
7605.2.a.cf 4 39.f even 4 1
7605.2.a.cj 4 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 10 T_{2}^{6} + 27 T_{2}^{4} + 22 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 22 T^{2} + 27 T^{4} + 10 T^{6} + T^{8} \)
$3$ \( ( 1 - 10 T - 6 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( 121 + 1144 T^{2} + 438 T^{4} + 40 T^{6} + T^{8} \)
$11$ \( ( 33 + 30 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( ( 13 + 10 T - 18 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$19$ \( ( 169 + 38 T^{2} + T^{4} )^{2} \)
$23$ \( ( -299 + 146 T + 6 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$29$ \( ( 1 + 40 T - 18 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$31$ \( ( 64 + 32 T^{2} + T^{4} )^{2} \)
$37$ \( 1 + 1552 T^{2} + 2766 T^{4} + 112 T^{6} + T^{8} \)
$41$ \( ( 1 + 14 T^{2} + T^{4} )^{2} \)
$43$ \( ( 13 + 10 T - 18 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$47$ \( 1763584 + 350464 T^{2} + 14304 T^{4} + 208 T^{6} + T^{8} \)
$53$ \( ( -48 + 36 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$59$ \( 9 + 324 T^{2} + 606 T^{4} + 84 T^{6} + T^{8} \)
$61$ \( ( 1261 - 964 T + 258 T^{2} - 28 T^{3} + T^{4} )^{2} \)
$67$ \( 7667361 + 662544 T^{2} + 19758 T^{4} + 240 T^{6} + T^{8} \)
$71$ \( 109767529 + 4583524 T^{2} + 68478 T^{4} + 436 T^{6} + T^{8} \)
$73$ \( 2930944 + 404608 T^{2} + 16944 T^{4} + 232 T^{6} + T^{8} \)
$79$ \( ( 4432 - 640 T - 132 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$83$ \( 36864 + 73728 T^{2} + 7104 T^{4} + 192 T^{6} + T^{8} \)
$89$ \( 78375609 + 8757108 T^{2} + 124014 T^{4} + 612 T^{6} + T^{8} \)
$97$ \( 196249 + 60136 T^{2} + 5718 T^{4} + 184 T^{6} + T^{8} \)
show more
show less