# Properties

 Label 845.2.b.d Level $845$ Weight $2$ Character orbit 845.b Analytic conductor $6.747$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.49843600.1 Defining polynomial: $$x^{6} + 8 x^{4} + 10 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{5} ) q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{4} q^{5} + ( 2 - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + ( -1 - \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{5} ) q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{4} q^{5} + ( 2 - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + ( -1 - \beta_{3} - \beta_{4} ) q^{9} + ( -2 + \beta_{2} - \beta_{3} ) q^{10} + ( -\beta_{3} - \beta_{4} ) q^{11} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{12} + ( -4 + \beta_{2} ) q^{14} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{5} ) q^{15} + ( 3 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{17} + ( -1 - \beta_{3} + \beta_{4} ) q^{18} + ( -2 + \beta_{3} + \beta_{4} ) q^{19} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{20} + ( -2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{21} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{22} + ( \beta_{1} - \beta_{5} ) q^{23} + ( -6 + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{24} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{25} + ( -\beta_{1} - \beta_{5} ) q^{27} + ( -1 - 4 \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{28} -3 q^{29} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{30} + ( -2 + 2 \beta_{2} ) q^{31} + ( 2 \beta_{1} + \beta_{5} ) q^{32} + ( \beta_{1} - 3 \beta_{5} ) q^{33} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{34} + ( -3 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{35} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{36} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{37} + ( 1 - 3 \beta_{1} + \beta_{3} - \beta_{4} ) q^{38} + ( 6 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{40} + ( -3 + 2 \beta_{2} ) q^{41} + ( 1 + 5 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{42} + ( -\beta_{1} + 3 \beta_{5} ) q^{43} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{44} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{45} + ( -2 + \beta_{2} ) q^{46} + ( -2 + 6 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -1 - 5 \beta_{1} - \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{48} + ( -1 - \beta_{3} - \beta_{4} ) q^{49} + ( 5 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{50} + ( 2 + \beta_{3} + \beta_{4} ) q^{51} + ( 1 - 2 \beta_{1} + \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{53} + ( 4 - \beta_{2} ) q^{54} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{55} + ( 4 - 4 \beta_{2} + \beta_{3} + \beta_{4} ) q^{56} + ( \beta_{1} + \beta_{5} ) q^{57} -3 \beta_{1} q^{58} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{59} + ( -10 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} ) q^{60} + ( -1 - 2 \beta_{3} - 2 \beta_{4} ) q^{61} + ( -2 - 6 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{62} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{63} + ( -1 + 2 \beta_{3} + 2 \beta_{4} ) q^{64} + \beta_{2} q^{66} + ( -3 \beta_{1} - \beta_{5} ) q^{67} + ( -2 \beta_{1} - \beta_{5} ) q^{68} + ( 4 + \beta_{3} + \beta_{4} ) q^{69} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{70} + ( 2 + \beta_{3} + \beta_{4} ) q^{71} + ( -1 + 6 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{72} + ( 1 - 6 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{73} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{74} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} ) q^{75} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{76} + ( -2 + 3 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{77} + ( -8 - 2 \beta_{2} ) q^{79} + ( -5 + 5 \beta_{1} - \beta_{4} ) q^{80} + ( -3 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{81} + ( -2 - 7 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{82} + ( 2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{83} + ( -16 + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{84} + ( -2 - 3 \beta_{1} + 3 \beta_{3} + \beta_{5} ) q^{85} -\beta_{2} q^{86} + ( 3 \beta_{1} - 3 \beta_{5} ) q^{87} + ( -2 + 5 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{88} + ( -2 - 4 \beta_{2} + \beta_{3} + \beta_{4} ) q^{89} + ( 5 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{90} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{92} + ( 2 + 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{93} + ( -10 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{94} + ( -5 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{95} + ( 2 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{96} + ( 2 + 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{97} + ( -1 - \beta_{3} + \beta_{4} ) q^{98} + ( 8 + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{4} - 3 q^{5} + 10 q^{6} - 6 q^{9} + O(q^{10})$$ $$6 q - 4 q^{4} - 3 q^{5} + 10 q^{6} - 6 q^{9} - 7 q^{10} - 22 q^{14} + 4 q^{15} + 16 q^{16} - 12 q^{19} + q^{20} - 4 q^{21} - 32 q^{24} - q^{25} - 18 q^{29} - 4 q^{30} - 8 q^{31} + 8 q^{34} - 10 q^{35} + 2 q^{36} + 35 q^{40} - 14 q^{41} - 2 q^{44} + 29 q^{45} - 10 q^{46} - 6 q^{49} + 31 q^{50} + 12 q^{51} + 22 q^{54} + 26 q^{55} + 16 q^{56} + 4 q^{59} - 48 q^{60} - 6 q^{61} - 6 q^{64} + 2 q^{66} + 24 q^{69} + 10 q^{70} + 12 q^{71} - 8 q^{74} - 2 q^{75} + 10 q^{76} - 52 q^{79} - 33 q^{80} - 14 q^{81} - 90 q^{84} - 21 q^{85} - 2 q^{86} - 20 q^{89} + 31 q^{90} - 56 q^{94} - 20 q^{95} + 6 q^{96} + 52 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 8 x^{4} + 10 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + \nu^{4} + 7 \nu^{3} + 7 \nu^{2} + 5 \nu + 3$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 7 \nu^{3} + 7 \nu^{2} - 5 \nu + 5$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 8 \nu^{3} + 10 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - \beta_{3} - 5 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} - 7 \beta_{2} + 17$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + 30 \beta_{1} + 8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
339.1
 − 2.54574i − 1.18733i − 0.330837i 0.330837i 1.18733i 2.54574i
2.54574i 2.15293i −4.48079 −0.817544 2.08125i 5.48079 2.93855i 6.31544i −1.63509 −5.29833 + 2.08125i
339.2 1.18733i 0.345110i 0.590239 1.44045 + 1.71029i 0.409761 2.02956i 3.07548i 2.88090 2.03069 1.71029i
339.3 0.330837i 2.69180i 1.89055 −2.12291 0.702335i −0.890547 3.35348i 1.28714i −4.24581 −0.232358 + 0.702335i
339.4 0.330837i 2.69180i 1.89055 −2.12291 + 0.702335i −0.890547 3.35348i 1.28714i −4.24581 −0.232358 0.702335i
339.5 1.18733i 0.345110i 0.590239 1.44045 1.71029i 0.409761 2.02956i 3.07548i 2.88090 2.03069 + 1.71029i
339.6 2.54574i 2.15293i −4.48079 −0.817544 + 2.08125i 5.48079 2.93855i 6.31544i −1.63509 −5.29833 2.08125i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 339.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.b.d 6
5.b even 2 1 inner 845.2.b.d 6
5.c odd 4 2 4225.2.a.br 6
13.b even 2 1 845.2.b.e 6
13.c even 3 2 65.2.n.a 12
13.d odd 4 2 845.2.d.d 12
13.e even 6 2 845.2.n.e 12
13.f odd 12 4 845.2.l.f 24
39.i odd 6 2 585.2.bs.a 12
52.j odd 6 2 1040.2.dh.a 12
65.d even 2 1 845.2.b.e 6
65.g odd 4 2 845.2.d.d 12
65.h odd 4 2 4225.2.a.bq 6
65.l even 6 2 845.2.n.e 12
65.n even 6 2 65.2.n.a 12
65.q odd 12 4 325.2.e.e 12
65.s odd 12 4 845.2.l.f 24
195.x odd 6 2 585.2.bs.a 12
260.v odd 6 2 1040.2.dh.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 13.c even 3 2
65.2.n.a 12 65.n even 6 2
325.2.e.e 12 65.q odd 12 4
585.2.bs.a 12 39.i odd 6 2
585.2.bs.a 12 195.x odd 6 2
845.2.b.d 6 1.a even 1 1 trivial
845.2.b.d 6 5.b even 2 1 inner
845.2.b.e 6 13.b even 2 1
845.2.b.e 6 65.d even 2 1
845.2.d.d 12 13.d odd 4 2
845.2.d.d 12 65.g odd 4 2
845.2.l.f 24 13.f odd 12 4
845.2.l.f 24 65.s odd 12 4
845.2.n.e 12 13.e even 6 2
845.2.n.e 12 65.l even 6 2
1040.2.dh.a 12 52.j odd 6 2
1040.2.dh.a 12 260.v odd 6 2
4225.2.a.bq 6 65.h odd 4 2
4225.2.a.br 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$:

 $$T_{2}^{6} + 8 T_{2}^{4} + 10 T_{2}^{2} + 1$$ $$T_{11}^{3} - 13 T_{11} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 10 T^{2} + 8 T^{4} + T^{6}$$
$3$ $$4 + 35 T^{2} + 12 T^{4} + T^{6}$$
$5$ $$125 + 75 T + 25 T^{2} + 10 T^{3} + 5 T^{4} + 3 T^{5} + T^{6}$$
$7$ $$400 + 179 T^{2} + 24 T^{4} + T^{6}$$
$11$ $$( -8 - 13 T + T^{3} )^{2}$$
$13$ $$T^{6}$$
$17$ $$169 + 163 T^{2} + 35 T^{4} + T^{6}$$
$19$ $$( -10 - T + 6 T^{2} + T^{3} )^{2}$$
$23$ $$4 + 35 T^{2} + 12 T^{4} + T^{6}$$
$29$ $$( 3 + T )^{6}$$
$31$ $$( 40 - 40 T + 4 T^{2} + T^{3} )^{2}$$
$37$ $$169 + 163 T^{2} + 35 T^{4} + T^{6}$$
$41$ $$( 5 - 29 T + 7 T^{2} + T^{3} )^{2}$$
$43$ $$256 + 283 T^{2} + 80 T^{4} + T^{6}$$
$47$ $$270400 + 14640 T^{2} + 236 T^{4} + T^{6}$$
$53$ $$400 + 1040 T^{2} + 171 T^{4} + T^{6}$$
$59$ $$( 136 - 55 T - 2 T^{2} + T^{3} )^{2}$$
$61$ $$( -115 - 49 T + 3 T^{2} + T^{3} )^{2}$$
$67$ $$20164 + 2603 T^{2} + 100 T^{4} + T^{6}$$
$71$ $$( 26 - T - 6 T^{2} + T^{3} )^{2}$$
$73$ $$250000 + 13900 T^{2} + 215 T^{4} + T^{6}$$
$79$ $$( 160 + 180 T + 26 T^{2} + T^{3} )^{2}$$
$83$ $$640000 + 23600 T^{2} + 276 T^{4} + T^{6}$$
$89$ $$( -1586 - 157 T + 10 T^{2} + T^{3} )^{2}$$
$97$ $$204304 + 14363 T^{2} + 280 T^{4} + T^{6}$$