# Properties

 Label 845.2.l.b Level $845$ Weight $2$ Character orbit 845.l Analytic conductor $6.747$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{2} - \beta_1 q^{3} + \beta_{2} q^{4} + (\beta_{3} - 1) q^{5} + (\beta_{3} - \beta_1) q^{6} + 3 q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + (-b2 + 1) * q^2 - b1 * q^3 + b2 * q^4 + (b3 - 1) * q^5 + (b3 - b1) * q^6 + 3 * q^8 + b2 * q^9 $$q + ( - \beta_{2} + 1) q^{2} - \beta_1 q^{3} + \beta_{2} q^{4} + (\beta_{3} - 1) q^{5} + (\beta_{3} - \beta_1) q^{6} + 3 q^{8} + \beta_{2} q^{9} + (\beta_{2} + \beta_1 - 1) q^{10} - \beta_1 q^{11} - \beta_{3} q^{12} + ( - 4 \beta_{2} + \beta_1 + 4) q^{15} + ( - \beta_{2} + 1) q^{16} + q^{18} + (3 \beta_{3} - 3 \beta_1) q^{19} + (\beta_{3} - \beta_{2} - \beta_1) q^{20} + (\beta_{3} - \beta_1) q^{22} - 3 \beta_1 q^{23} - 3 \beta_1 q^{24} + ( - 2 \beta_{3} - 3) q^{25} + 2 \beta_{3} q^{27} + (6 \beta_{2} - 6) q^{29} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{30} + 3 \beta_{3} q^{31} + 5 \beta_{2} q^{32} + 4 \beta_{2} q^{33} + (\beta_{2} - 1) q^{36} + (6 \beta_{2} - 6) q^{37} + 3 \beta_{3} q^{38} + (3 \beta_{3} - 3) q^{40} + 4 \beta_1 q^{41} + (3 \beta_{3} - 3 \beta_1) q^{43} - \beta_{3} q^{44} + (\beta_{3} - \beta_{2} - \beta_1) q^{45} + (3 \beta_{3} - 3 \beta_1) q^{46} - 8 q^{47} + (\beta_{3} - \beta_1) q^{48} + ( - 7 \beta_{2} + 7) q^{49} + (3 \beta_{2} - 2 \beta_1 - 3) q^{50} - 6 \beta_{3} q^{53} + 2 \beta_1 q^{54} + ( - 4 \beta_{2} + \beta_1 + 4) q^{55} + 12 q^{57} + 6 \beta_{2} q^{58} + ( - \beta_{3} + \beta_1) q^{59} + (\beta_{3} + 4) q^{60} - 6 \beta_{2} q^{61} + 3 \beta_1 q^{62} + 7 q^{64} + 4 q^{66} + (12 \beta_{2} - 12) q^{67} + 12 \beta_{2} q^{69} + (\beta_{3} - \beta_1) q^{71} + 3 \beta_{2} q^{72} - 6 q^{73} + 6 \beta_{2} q^{74} + (8 \beta_{2} + 3 \beta_1 - 8) q^{75} - 3 \beta_1 q^{76} + (\beta_{2} + \beta_1 - 1) q^{80} + ( - 11 \beta_{2} + 11) q^{81} + ( - 4 \beta_{3} + 4 \beta_1) q^{82} - 4 q^{83} + 3 \beta_{3} q^{86} + ( - 6 \beta_{3} + 6 \beta_1) q^{87} - 3 \beta_1 q^{88} + 4 \beta_1 q^{89} + (\beta_{3} - 1) q^{90} - 3 \beta_{3} q^{92} + ( - 12 \beta_{2} + 12) q^{93} + (8 \beta_{2} - 8) q^{94} + ( - 3 \beta_{3} - 12 \beta_{2} + 3 \beta_1) q^{95} - 5 \beta_{3} q^{96} + 6 \beta_{2} q^{97} - 7 \beta_{2} q^{98} - \beta_{3} q^{99}+O(q^{100})$$ q + (-b2 + 1) * q^2 - b1 * q^3 + b2 * q^4 + (b3 - 1) * q^5 + (b3 - b1) * q^6 + 3 * q^8 + b2 * q^9 + (b2 + b1 - 1) * q^10 - b1 * q^11 - b3 * q^12 + (-4*b2 + b1 + 4) * q^15 + (-b2 + 1) * q^16 + q^18 + (3*b3 - 3*b1) * q^19 + (b3 - b2 - b1) * q^20 + (b3 - b1) * q^22 - 3*b1 * q^23 - 3*b1 * q^24 + (-2*b3 - 3) * q^25 + 2*b3 * q^27 + (6*b2 - 6) * q^29 + (-b3 - 4*b2 + b1) * q^30 + 3*b3 * q^31 + 5*b2 * q^32 + 4*b2 * q^33 + (b2 - 1) * q^36 + (6*b2 - 6) * q^37 + 3*b3 * q^38 + (3*b3 - 3) * q^40 + 4*b1 * q^41 + (3*b3 - 3*b1) * q^43 - b3 * q^44 + (b3 - b2 - b1) * q^45 + (3*b3 - 3*b1) * q^46 - 8 * q^47 + (b3 - b1) * q^48 + (-7*b2 + 7) * q^49 + (3*b2 - 2*b1 - 3) * q^50 - 6*b3 * q^53 + 2*b1 * q^54 + (-4*b2 + b1 + 4) * q^55 + 12 * q^57 + 6*b2 * q^58 + (-b3 + b1) * q^59 + (b3 + 4) * q^60 - 6*b2 * q^61 + 3*b1 * q^62 + 7 * q^64 + 4 * q^66 + (12*b2 - 12) * q^67 + 12*b2 * q^69 + (b3 - b1) * q^71 + 3*b2 * q^72 - 6 * q^73 + 6*b2 * q^74 + (8*b2 + 3*b1 - 8) * q^75 - 3*b1 * q^76 + (b2 + b1 - 1) * q^80 + (-11*b2 + 11) * q^81 + (-4*b3 + 4*b1) * q^82 - 4 * q^83 + 3*b3 * q^86 + (-6*b3 + 6*b1) * q^87 - 3*b1 * q^88 + 4*b1 * q^89 + (b3 - 1) * q^90 - 3*b3 * q^92 + (-12*b2 + 12) * q^93 + (8*b2 - 8) * q^94 + (-3*b3 - 12*b2 + 3*b1) * q^95 - 5*b3 * q^96 + 6*b2 * q^97 - 7*b2 * q^98 - b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 12 q^{8} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 12 * q^8 + 2 * q^9 $$4 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 12 q^{8} + 2 q^{9} - 2 q^{10} + 8 q^{15} + 2 q^{16} + 4 q^{18} - 2 q^{20} - 12 q^{25} - 12 q^{29} - 8 q^{30} + 10 q^{32} + 8 q^{33} - 2 q^{36} - 12 q^{37} - 12 q^{40} - 2 q^{45} - 32 q^{47} + 14 q^{49} - 6 q^{50} + 8 q^{55} + 48 q^{57} + 12 q^{58} + 16 q^{60} - 12 q^{61} + 28 q^{64} + 16 q^{66} - 24 q^{67} + 24 q^{69} + 6 q^{72} - 24 q^{73} + 12 q^{74} - 16 q^{75} - 2 q^{80} + 22 q^{81} - 16 q^{83} - 4 q^{90} + 24 q^{93} - 16 q^{94} - 24 q^{95} + 12 q^{97} - 14 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 12 * q^8 + 2 * q^9 - 2 * q^10 + 8 * q^15 + 2 * q^16 + 4 * q^18 - 2 * q^20 - 12 * q^25 - 12 * q^29 - 8 * q^30 + 10 * q^32 + 8 * q^33 - 2 * q^36 - 12 * q^37 - 12 * q^40 - 2 * q^45 - 32 * q^47 + 14 * q^49 - 6 * q^50 + 8 * q^55 + 48 * q^57 + 12 * q^58 + 16 * q^60 - 12 * q^61 + 28 * q^64 + 16 * q^66 - 24 * q^67 + 24 * q^69 + 6 * q^72 - 24 * q^73 + 12 * q^74 - 16 * q^75 - 2 * q^80 + 22 * q^81 - 16 * q^83 - 4 * q^90 + 24 * q^93 - 16 * q^94 - 24 * q^95 + 12 * q^97 - 14 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3
 $$\zeta_{12}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

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

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
654.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0.500000 0.866025i −1.73205 1.00000i 0.500000 + 0.866025i −1.00000 + 2.00000i −1.73205 + 1.00000i 0 3.00000 0.500000 + 0.866025i 1.23205 + 1.86603i
654.2 0.500000 0.866025i 1.73205 + 1.00000i 0.500000 + 0.866025i −1.00000 2.00000i 1.73205 1.00000i 0 3.00000 0.500000 + 0.866025i −2.23205 0.133975i
699.1 0.500000 + 0.866025i −1.73205 + 1.00000i 0.500000 0.866025i −1.00000 2.00000i −1.73205 1.00000i 0 3.00000 0.500000 0.866025i 1.23205 1.86603i
699.2 0.500000 + 0.866025i 1.73205 1.00000i 0.500000 0.866025i −1.00000 + 2.00000i 1.73205 + 1.00000i 0 3.00000 0.500000 0.866025i −2.23205 + 0.133975i
 $$n$$: e.g. 2-40 or 990-1000 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.c even 3 1 inner
65.d even 2 1 inner
65.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.l.b 4
5.b even 2 1 845.2.l.a 4
13.b even 2 1 845.2.l.a 4
13.c even 3 1 65.2.d.a 2
13.c even 3 1 inner 845.2.l.b 4
13.d odd 4 1 845.2.n.a 4
13.d odd 4 1 845.2.n.b 4
13.e even 6 1 65.2.d.b yes 2
13.e even 6 1 845.2.l.a 4
13.f odd 12 1 845.2.b.a 2
13.f odd 12 1 845.2.b.b 2
13.f odd 12 1 845.2.n.a 4
13.f odd 12 1 845.2.n.b 4
39.h odd 6 1 585.2.h.b 2
39.i odd 6 1 585.2.h.c 2
52.i odd 6 1 1040.2.f.b 2
52.j odd 6 1 1040.2.f.a 2
65.d even 2 1 inner 845.2.l.b 4
65.g odd 4 1 845.2.n.a 4
65.g odd 4 1 845.2.n.b 4
65.l even 6 1 65.2.d.a 2
65.l even 6 1 inner 845.2.l.b 4
65.n even 6 1 65.2.d.b yes 2
65.n even 6 1 845.2.l.a 4
65.o even 12 1 4225.2.a.e 1
65.o even 12 1 4225.2.a.h 1
65.q odd 12 1 325.2.c.b 2
65.q odd 12 1 325.2.c.e 2
65.r odd 12 1 325.2.c.b 2
65.r odd 12 1 325.2.c.e 2
65.s odd 12 1 845.2.b.a 2
65.s odd 12 1 845.2.b.b 2
65.s odd 12 1 845.2.n.a 4
65.s odd 12 1 845.2.n.b 4
65.t even 12 1 4225.2.a.k 1
65.t even 12 1 4225.2.a.m 1
195.x odd 6 1 585.2.h.b 2
195.y odd 6 1 585.2.h.c 2
260.v odd 6 1 1040.2.f.b 2
260.w odd 6 1 1040.2.f.a 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} - 4T^{2} + 16$$
$5$ $$(T^{2} + 2 T + 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 4T^{2} + 16$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4} - 36T^{2} + 1296$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} + 6 T + 36)^{2}$$
$31$ $$(T^{2} + 36)^{2}$$
$37$ $$(T^{2} + 6 T + 36)^{2}$$
$41$ $$T^{4} - 64T^{2} + 4096$$
$43$ $$T^{4} - 36T^{2} + 1296$$
$47$ $$(T + 8)^{4}$$
$53$ $$(T^{2} + 144)^{2}$$
$59$ $$T^{4} - 4T^{2} + 16$$
$61$ $$(T^{2} + 6 T + 36)^{2}$$
$67$ $$(T^{2} + 12 T + 144)^{2}$$
$71$ $$T^{4} - 4T^{2} + 16$$
$73$ $$(T + 6)^{4}$$
$79$ $$T^{4}$$
$83$ $$(T + 4)^{4}$$
$89$ $$T^{4} - 64T^{2} + 4096$$
$97$ $$(T^{2} - 6 T + 36)^{2}$$