# Properties

 Label 845.2.n.b Level $845$ Weight $2$ Character orbit 845.n 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.n (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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
484.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i −1.73205 + 1.00000i −0.500000 + 0.866025i 2.00000 1.00000i 1.00000 1.73205i 0 3.00000i 0.500000 0.866025i −1.23205 + 1.86603i
484.2 0.866025 0.500000i 1.73205 1.00000i −0.500000 + 0.866025i 2.00000 + 1.00000i 1.00000 1.73205i 0 3.00000i 0.500000 0.866025i 2.23205 0.133975i
529.1 −0.866025 0.500000i −1.73205 1.00000i −0.500000 0.866025i 2.00000 + 1.00000i 1.00000 + 1.73205i 0 3.00000i 0.500000 + 0.866025i −1.23205 1.86603i
529.2 0.866025 + 0.500000i 1.73205 + 1.00000i −0.500000 0.866025i 2.00000 1.00000i 1.00000 + 1.73205i 0 3.00000i 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
5.b even 2 1 inner
13.c even 3 1 inner
65.n 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.n.b 4
5.b even 2 1 inner 845.2.n.b 4
13.b even 2 1 845.2.n.a 4
13.c even 3 1 845.2.b.b 2
13.c even 3 1 inner 845.2.n.b 4
13.d odd 4 1 845.2.l.a 4
13.d odd 4 1 845.2.l.b 4
13.e even 6 1 845.2.b.a 2
13.e even 6 1 845.2.n.a 4
13.f odd 12 1 65.2.d.a 2
13.f odd 12 1 65.2.d.b yes 2
13.f odd 12 1 845.2.l.a 4
13.f odd 12 1 845.2.l.b 4
39.k even 12 1 585.2.h.b 2
39.k even 12 1 585.2.h.c 2
52.l even 12 1 1040.2.f.a 2
52.l even 12 1 1040.2.f.b 2
65.d even 2 1 845.2.n.a 4
65.g odd 4 1 845.2.l.a 4
65.g odd 4 1 845.2.l.b 4
65.l even 6 1 845.2.b.a 2
65.l even 6 1 845.2.n.a 4
65.n even 6 1 845.2.b.b 2
65.n even 6 1 inner 845.2.n.b 4
65.o even 12 1 325.2.c.b 2
65.o even 12 1 325.2.c.e 2
65.q odd 12 1 4225.2.a.h 1
65.q odd 12 1 4225.2.a.k 1
65.r odd 12 1 4225.2.a.e 1
65.r odd 12 1 4225.2.a.m 1
65.s odd 12 1 65.2.d.a 2
65.s odd 12 1 65.2.d.b yes 2
65.s odd 12 1 845.2.l.a 4
65.s odd 12 1 845.2.l.b 4
65.t even 12 1 325.2.c.b 2
65.t even 12 1 325.2.c.e 2
195.bh even 12 1 585.2.h.b 2
195.bh even 12 1 585.2.h.c 2
260.bc even 12 1 1040.2.f.a 2
260.bc even 12 1 1040.2.f.b 2

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

## Hecke kernels

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

 $$T_{2}^{4} - T_{2}^{2} + 1$$ T2^4 - T2^2 + 1 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - 4T^{2} + 16$$
$5$ $$(T^{2} - 4 T + 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} - 6 T + 36)^{2}$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} + 6 T + 36)^{2}$$
$31$ $$(T - 6)^{4}$$
$37$ $$T^{4} - 36T^{2} + 1296$$
$41$ $$(T^{2} - 8 T + 64)^{2}$$
$43$ $$T^{4} - 36T^{2} + 1296$$
$47$ $$(T^{2} + 64)^{2}$$
$53$ $$(T^{2} + 144)^{2}$$
$59$ $$(T^{2} - 2 T + 4)^{2}$$
$61$ $$(T^{2} + 6 T + 36)^{2}$$
$67$ $$T^{4} - 144 T^{2} + 20736$$
$71$ $$(T^{2} - 2 T + 4)^{2}$$
$73$ $$(T^{2} + 36)^{2}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 16)^{2}$$
$89$ $$(T^{2} + 8 T + 64)^{2}$$
$97$ $$T^{4} - 36T^{2} + 1296$$