# Properties

 Label 845.2.c.f Level $845$ Weight $2$ Character orbit 845.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} - \beta_{4} ) q^{3} + \beta_{2} q^{4} -\beta_{5} q^{5} + ( -\beta_{1} - \beta_{5} ) q^{6} + ( -\beta_{3} - 2 \beta_{5} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( -1 + 4 \beta_{2} + 3 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} - \beta_{4} ) q^{3} + \beta_{2} q^{4} -\beta_{5} q^{5} + ( -\beta_{1} - \beta_{5} ) q^{6} + ( -\beta_{3} - 2 \beta_{5} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( -1 + 4 \beta_{2} + 3 \beta_{4} ) q^{9} + \beta_{4} q^{10} + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{11} + ( -3 \beta_{2} - \beta_{4} ) q^{12} + ( 1 + \beta_{4} ) q^{14} + ( \beta_{3} + 2 \beta_{5} ) q^{15} + ( -3 + 3 \beta_{2} + \beta_{4} ) q^{16} + ( 3 - 3 \beta_{2} - 3 \beta_{4} ) q^{17} + ( -2 \beta_{1} + \beta_{3} + 3 \beta_{5} ) q^{18} + ( -4 \beta_{1} + \beta_{3} - \beta_{5} ) q^{19} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{20} + ( -\beta_{1} + 4 \beta_{3} + 6 \beta_{5} ) q^{21} + ( -3 + \beta_{2} + 2 \beta_{4} ) q^{22} + ( -1 + 5 \beta_{4} ) q^{23} + ( -2 \beta_{3} - 3 \beta_{5} ) q^{24} - q^{25} + ( 1 - 8 \beta_{2} - 3 \beta_{4} ) q^{27} + ( 2 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} ) q^{28} + ( -2 - 3 \beta_{2} + 3 \beta_{4} ) q^{29} + ( -1 - \beta_{4} ) q^{30} + ( 4 \beta_{1} - 5 \beta_{3} - 4 \beta_{5} ) q^{31} + ( -3 \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{32} + ( -\beta_{3} - \beta_{5} ) q^{33} + ( 3 \beta_{1} - 3 \beta_{5} ) q^{34} + ( -1 - \beta_{2} - \beta_{4} ) q^{35} + ( 1 + 6 \beta_{2} + 4 \beta_{4} ) q^{36} + ( 2 \beta_{1} + \beta_{5} ) q^{37} + ( 7 - 4 \beta_{2} + 2 \beta_{4} ) q^{38} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{40} + ( -\beta_{1} + 4 \beta_{3} - 5 \beta_{5} ) q^{41} + ( -2 - \beta_{2} - 2 \beta_{4} ) q^{42} + ( 8 - \beta_{2} ) q^{43} + \beta_{3} q^{44} + ( \beta_{1} - 4 \beta_{3} - 3 \beta_{5} ) q^{45} + ( 4 \beta_{1} - 5 \beta_{3} + 5 \beta_{5} ) q^{46} + ( \beta_{1} - 4 \beta_{3} - 7 \beta_{5} ) q^{47} + ( 2 - 6 \beta_{2} - \beta_{4} ) q^{48} + ( 5 - 4 \beta_{2} - 3 \beta_{4} ) q^{49} -\beta_{1} q^{50} + ( 6 \beta_{2} + 3 \beta_{4} ) q^{51} + ( -1 + \beta_{2} ) q^{53} + ( 6 \beta_{1} - 5 \beta_{3} - 3 \beta_{5} ) q^{54} + ( -2 + \beta_{2} + 2 \beta_{4} ) q^{55} + ( 1 + 2 \beta_{2} + 2 \beta_{4} ) q^{56} + ( 5 \beta_{1} - \beta_{3} + 4 \beta_{5} ) q^{57} + ( 4 \beta_{1} - 6 \beta_{3} + 3 \beta_{5} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{3} - 5 \beta_{5} ) q^{59} + ( -2 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} ) q^{60} + ( -6 + \beta_{2} ) q^{61} + ( -3 + 4 \beta_{2} - \beta_{4} ) q^{62} + ( 5 \beta_{1} - 11 \beta_{3} - 13 \beta_{5} ) q^{63} + ( -4 + 3 \beta_{2} + 5 \beta_{4} ) q^{64} + q^{66} + ( 6 \beta_{1} - 6 \beta_{3} - 9 \beta_{5} ) q^{67} + ( -3 \beta_{2} - 3 \beta_{4} ) q^{68} + ( -4 + \beta_{2} - 4 \beta_{4} ) q^{69} + ( -\beta_{1} - \beta_{5} ) q^{70} + ( 7 \beta_{1} - 4 \beta_{3} - 8 \beta_{5} ) q^{71} + ( -5 \beta_{1} + 4 \beta_{3} + 10 \beta_{5} ) q^{72} + ( -8 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{73} + ( -4 + 2 \beta_{2} - \beta_{4} ) q^{74} + ( 1 + \beta_{2} + \beta_{4} ) q^{75} + ( 5 \beta_{1} - 4 \beta_{3} ) q^{76} + ( \beta_{2} + \beta_{4} ) q^{77} + ( -12 - 3 \beta_{2} + 2 \beta_{4} ) q^{79} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{80} + ( 5 + 11 \beta_{2} + \beta_{4} ) q^{81} + ( -2 - \beta_{2} + 9 \beta_{4} ) q^{82} + ( -3 \beta_{1} - 2 \beta_{3} + 10 \beta_{5} ) q^{83} + ( -5 \beta_{1} + 9 \beta_{3} + 10 \beta_{5} ) q^{84} + 3 \beta_{3} q^{85} + ( 9 \beta_{1} - \beta_{3} ) q^{86} + ( -1 + 11 \beta_{2} + 2 \beta_{4} ) q^{87} + ( -7 + 2 \beta_{2} + 5 \beta_{4} ) q^{88} + ( -10 \beta_{1} + 8 \beta_{3} + 5 \beta_{5} ) q^{89} + ( 2 + \beta_{2} - \beta_{4} ) q^{90} + ( -5 + 4 \beta_{2} ) q^{92} + ( -9 \beta_{1} + 14 \beta_{3} + 14 \beta_{5} ) q^{93} + ( 2 + \beta_{2} + 3 \beta_{4} ) q^{94} + ( -2 + \beta_{2} - 3 \beta_{4} ) q^{95} + ( 7 \beta_{1} - 9 \beta_{3} - 7 \beta_{5} ) q^{96} + ( 4 \beta_{1} - 5 \beta_{3} + \beta_{5} ) q^{97} + ( 6 \beta_{1} - \beta_{3} - 3 \beta_{5} ) q^{98} + ( -4 \beta_{1} + 7 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 10q^{3} + 2q^{4} + 8q^{9} + O(q^{10})$$ $$6q - 10q^{3} + 2q^{4} + 8q^{9} + 2q^{10} - 8q^{12} + 8q^{14} - 10q^{16} + 6q^{17} - 12q^{22} + 4q^{23} - 6q^{25} - 16q^{27} - 12q^{29} - 8q^{30} - 10q^{35} + 26q^{36} + 38q^{38} - 18q^{42} + 46q^{43} - 2q^{48} + 16q^{49} + 18q^{51} - 4q^{53} - 6q^{55} + 14q^{56} - 34q^{61} - 12q^{62} - 8q^{64} + 6q^{66} - 12q^{68} - 30q^{69} - 22q^{74} + 10q^{75} + 4q^{77} - 74q^{79} + 54q^{81} + 4q^{82} + 20q^{87} - 28q^{88} + 12q^{90} - 22q^{92} + 20q^{94} - 16q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4 \nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + 9 \beta_{1}$$

## 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
 − 1.80194i − 1.24698i − 0.445042i 0.445042i 1.24698i 1.80194i
1.80194i −1.55496 −1.24698 1.00000i 2.80194i 1.55496i 1.35690i −0.582105 1.80194
506.2 1.24698i −0.198062 0.445042 1.00000i 0.246980i 0.198062i 3.04892i −2.96077 −1.24698
506.3 0.445042i −3.24698 1.80194 1.00000i 1.44504i 3.24698i 1.69202i 7.54288 0.445042
506.4 0.445042i −3.24698 1.80194 1.00000i 1.44504i 3.24698i 1.69202i 7.54288 0.445042
506.5 1.24698i −0.198062 0.445042 1.00000i 0.246980i 0.198062i 3.04892i −2.96077 −1.24698
506.6 1.80194i −1.55496 −1.24698 1.00000i 2.80194i 1.55496i 1.35690i −0.582105 1.80194
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 506.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.f 6
13.b even 2 1 inner 845.2.c.f 6
13.c even 3 2 845.2.m.i 12
13.d odd 4 1 845.2.a.h 3
13.d odd 4 1 845.2.a.j yes 3
13.e even 6 2 845.2.m.i 12
13.f odd 12 2 845.2.e.j 6
13.f odd 12 2 845.2.e.l 6
39.f even 4 1 7605.2.a.br 3
39.f even 4 1 7605.2.a.by 3
65.g odd 4 1 4225.2.a.bd 3
65.g odd 4 1 4225.2.a.bf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.a.h 3 13.d odd 4 1
845.2.a.j yes 3 13.d odd 4 1
845.2.c.f 6 1.a even 1 1 trivial
845.2.c.f 6 13.b even 2 1 inner
845.2.e.j 6 13.f odd 12 2
845.2.e.l 6 13.f odd 12 2
845.2.m.i 12 13.c even 3 2
845.2.m.i 12 13.e even 6 2
4225.2.a.bd 3 65.g odd 4 1
4225.2.a.bf 3 65.g odd 4 1
7605.2.a.br 3 39.f even 4 1
7605.2.a.by 3 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 5 T_{2}^{4} + 6 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$3$ $$( 1 + 6 T + 5 T^{2} + T^{3} )^{2}$$
$5$ $$( 1 + T^{2} )^{3}$$
$7$ $$1 + 26 T^{2} + 13 T^{4} + T^{6}$$
$11$ $$1 + 10 T^{2} + 17 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$( 27 - 18 T - 3 T^{2} + T^{3} )^{2}$$
$19$ $$841 + 545 T^{2} + 82 T^{4} + T^{6}$$
$23$ $$( 71 - 57 T - 2 T^{2} + T^{3} )^{2}$$
$29$ $$( -307 - 51 T + 6 T^{2} + T^{3} )^{2}$$
$31$ $$1 + 2110 T^{2} + 101 T^{4} + T^{6}$$
$37$ $$169 + 131 T^{2} + 27 T^{4} + T^{6}$$
$41$ $$1849 + 8889 T^{2} + 194 T^{4} + T^{6}$$
$43$ $$( -433 + 174 T - 23 T^{2} + T^{3} )^{2}$$
$47$ $$1681 + 1713 T^{2} + 146 T^{4} + T^{6}$$
$53$ $$( -1 - T + 2 T^{2} + T^{3} )^{2}$$
$59$ $$169 + 675 T^{2} + 59 T^{4} + T^{6}$$
$61$ $$( 169 + 94 T + 17 T^{2} + T^{3} )^{2}$$
$67$ $$123201 + 10611 T^{2} + 243 T^{4} + T^{6}$$
$71$ $$361201 + 16526 T^{2} + 229 T^{4} + T^{6}$$
$73$ $$2096704 + 56656 T^{2} + 440 T^{4} + T^{6}$$
$79$ $$( 1217 + 412 T + 37 T^{2} + T^{3} )^{2}$$
$83$ $$344569 + 21650 T^{2} + 369 T^{4} + T^{6}$$
$89$ $$284089 + 34051 T^{2} + 395 T^{4} + T^{6}$$
$97$ $$32761 + 4345 T^{2} + 146 T^{4} + T^{6}$$