# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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