# Properties

 Label 845.2.a.m Level $845$ Weight $2$ Character orbit 845.a Self dual yes Analytic conductor $6.747$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.4752.1 Defining polynomial: $$x^{4} - 2x^{3} - 3x^{2} + 4x + 1$$ x^4 - 2*x^3 - 3*x^2 + 4*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + \beta_1) q^{4} - q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{6} + ( - \beta_{3} + \beta_{2} + 3) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8} + 2 \beta_1 q^{9}+O(q^{10})$$ q + b1 * q^2 - b3 * q^3 + (b2 + b1) * q^4 - q^5 + (-b3 - b2 - b1) * q^6 + (-b3 + b2 + 3) * q^7 + (b3 + b2 + 1) * q^8 + 2*b1 * q^9 $$q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + \beta_1) q^{4} - q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{6} + ( - \beta_{3} + \beta_{2} + 3) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8} + 2 \beta_1 q^{9} - \beta_1 q^{10} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{11} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{12} + ( - \beta_{2} + 3 \beta_1 - 1) q^{14} + \beta_{3} q^{15} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{16} + ( - \beta_{3} + 2 \beta_{2}) q^{17} + (2 \beta_{2} + 2 \beta_1 + 4) q^{18} + (\beta_{2} + 4) q^{19} + ( - \beta_{2} - \beta_1) q^{20} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{21} + ( - \beta_{3} + \beta_1 + 3) q^{22} + (\beta_{3} + 2 \beta_1 - 4) q^{23} + ( - \beta_{2} - 4 \beta_1 - 4) q^{24} + q^{25} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{27} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{28} + (2 \beta_{2} + 2 \beta_1 + 1) q^{29} + (\beta_{3} + \beta_{2} + \beta_1) q^{30} + ( - 2 \beta_{2} + 2) q^{31} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{32} + ( - \beta_{3} - 3 \beta_{2} + 5) q^{33} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{34} + (\beta_{3} - \beta_{2} - 3) q^{35} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{36} + (3 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{37} + (\beta_{3} + 5 \beta_1 - 1) q^{38} + ( - \beta_{3} - \beta_{2} - 1) q^{40} + ( - \beta_{2} + 2) q^{41} + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{42} + ( - \beta_{3} + 2 \beta_{2}) q^{43} + (3 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{44} - 2 \beta_1 q^{45} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 4) q^{46} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{47} + ( - \beta_{3} - 3 \beta_1 - 5) q^{48} + ( - 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 6) q^{49} + \beta_1 q^{50} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{51} + ( - 2 \beta_1 - 2) q^{53} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{54} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{55} + (2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 3) q^{56} + ( - 3 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{57} + (2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 2) q^{58} + (\beta_{2} - 2 \beta_1 + 4) q^{59} + (2 \beta_{2} + 3 \beta_1 + 1) q^{60} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 7) q^{61} + ( - 2 \beta_{3} + 2) q^{62} + ( - 2 \beta_{2} + 6 \beta_1 - 2) q^{63} + ( - 4 \beta_{3} + 2 \beta_{2} + 3) q^{64} + ( - 4 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{66} + (\beta_{3} - \beta_{2} + 7) q^{67} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{68} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 3) q^{69} + (\beta_{2} - 3 \beta_1 + 1) q^{70} + (6 \beta_{3} - 3 \beta_{2} - 2) q^{71} + (4 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 2) q^{72} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{73} + (2 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{74} - \beta_{3} q^{75} + (\beta_{3} + 4 \beta_{2} + 5 \beta_1 + 2) q^{76} + ( - 3 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{77} + (2 \beta_{3} - 6 \beta_1) q^{79} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{80} + (4 \beta_{2} - 2 \beta_1 - 1) q^{81} + ( - \beta_{3} + \beta_1 + 1) q^{82} + (2 \beta_{3} + 2 \beta_{2} - 4) q^{83} + ( - \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 4) q^{84} + (\beta_{3} - 2 \beta_{2}) q^{85} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{86} + ( - \beta_{3} - 4 \beta_{2} - 6 \beta_1 - 2) q^{87} + (3 \beta_{3} + 2 \beta_{2} - 6) q^{88} + (2 \beta_{3} + \beta_{2} - 8 \beta_1) q^{89} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{90} + (2 \beta_{3} + 3 \beta_1 + 3) q^{92} + ( - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{93} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 6) q^{94} + ( - \beta_{2} - 4) q^{95} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{96} + ( - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{97} + ( - 6 \beta_{2} + 4 \beta_1 - 8) q^{98} + ( - 2 \beta_{3} + 2 \beta_1 + 6) q^{99}+O(q^{100})$$ q + b1 * q^2 - b3 * q^3 + (b2 + b1) * q^4 - q^5 + (-b3 - b2 - b1) * q^6 + (-b3 + b2 + 3) * q^7 + (b3 + b2 + 1) * q^8 + 2*b1 * q^9 - b1 * q^10 + (-2*b3 + b2 + 2*b1) * q^11 + (-2*b2 - 3*b1 - 1) * q^12 + (-b2 + 3*b1 - 1) * q^14 + b3 * q^15 + (2*b3 - b2 + b1 - 1) * q^16 + (-b3 + 2*b2) * q^17 + (2*b2 + 2*b1 + 4) * q^18 + (b2 + 4) * q^19 + (-b2 - b1) * q^20 + (-2*b3 - b2 + 2) * q^21 + (-b3 + b1 + 3) * q^22 + (b3 + 2*b1 - 4) * q^23 + (-b2 - 4*b1 - 4) * q^24 + q^25 + (b3 - 2*b2 - 2*b1) * q^27 + (b3 + b2 + b1 + 1) * q^28 + (2*b2 + 2*b1 + 1) * q^29 + (b3 + b2 + b1) * q^30 + (-2*b2 + 2) * q^31 + (-b3 + b2 + b1 + 1) * q^32 + (-b3 - 3*b2 + 5) * q^33 + (b3 - b2 + b1 - 2) * q^34 + (b3 - b2 - 3) * q^35 + (2*b3 + 2*b2 + 4*b1 + 2) * q^36 + (3*b3 - b2 - 2*b1 - 1) * q^37 + (b3 + 5*b1 - 1) * q^38 + (-b3 - b2 - 1) * q^40 + (-b2 + 2) * q^41 + (-3*b3 - 2*b2 - b1 + 1) * q^42 + (-b3 + 2*b2) * q^43 + (3*b3 - 2*b2 - b1 + 2) * q^44 - 2*b1 * q^45 + (b3 + 3*b2 - b1 + 4) * q^46 + (-2*b2 - 4*b1 + 4) * q^47 + (-b3 - 3*b1 - 5) * q^48 + (-4*b3 + 4*b2 - 2*b1 + 6) * q^49 + b1 * q^50 + (2*b3 - 2*b2 - 2*b1 + 1) * q^51 + (-2*b1 - 2) * q^53 + (-b3 - b2 - 3*b1 - 2) * q^54 + (2*b3 - b2 - 2*b1) * q^55 + (2*b3 + 4*b2 - 2*b1 + 3) * q^56 + (-3*b3 - b2 - 2*b1 - 1) * q^57 + (2*b3 + 2*b2 + 5*b1 + 2) * q^58 + (b2 - 2*b1 + 4) * q^59 + (2*b2 + 3*b1 + 1) * q^60 + (2*b3 - 2*b2 - 2*b1 + 7) * q^61 + (-2*b3 + 2) * q^62 + (-2*b2 + 6*b1 - 2) * q^63 + (-4*b3 + 2*b2 + 3) * q^64 + (-4*b3 - b2 + b1 + 3) * q^66 + (b3 - b2 + 7) * q^67 + (2*b3 - 2*b2 - b1 + 3) * q^68 + (2*b3 - 2*b2 - 4*b1 - 3) * q^69 + (b2 - 3*b1 + 1) * q^70 + (6*b3 - 3*b2 - 2) * q^71 + (4*b3 + 2*b2 + 6*b1 - 2) * q^72 + (-2*b3 + 4*b2 - 2*b1) * q^73 + (2*b3 + b2 - b1 - 3) * q^74 - b3 * q^75 + (b3 + 4*b2 + 5*b1 + 2) * q^76 + (-3*b3 - 2*b2 + 4*b1 + 4) * q^77 + (2*b3 - 6*b1) * q^79 + (-2*b3 + b2 - b1 + 1) * q^80 + (4*b2 - 2*b1 - 1) * q^81 + (-b3 + b1 + 1) * q^82 + (2*b3 + 2*b2 - 4) * q^83 + (-b3 - 2*b2 - 5*b1 - 4) * q^84 + (b3 - 2*b2) * q^85 + (b3 - b2 + b1 - 2) * q^86 + (-b3 - 4*b2 - 6*b1 - 2) * q^87 + (3*b3 + 2*b2 - 6) * q^88 + (2*b3 + b2 - 8*b1) * q^89 + (-2*b2 - 2*b1 - 4) * q^90 + (2*b3 + 3*b1 + 3) * q^92 + (-4*b3 + 2*b2 + 4*b1 + 2) * q^93 + (-2*b3 - 4*b2 - 2*b1 - 6) * q^94 + (-b2 - 4) * q^95 + (-b3 - 2*b2 - b1 + 2) * q^96 + (-3*b3 - b2 + 2*b1 + 1) * q^97 + (-6*b2 + 4*b1 - 8) * q^98 + (-2*b3 + 2*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 4 q^{6} + 10 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 4 * q^6 + 10 * q^7 + 6 * q^8 + 4 * q^9 $$4 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 4 q^{6} + 10 q^{7} + 6 q^{8} + 4 q^{9} - 2 q^{10} - 10 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 20 q^{18} + 16 q^{19} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 10 q^{23} - 24 q^{24} + 4 q^{25} - 2 q^{27} + 8 q^{28} + 8 q^{29} + 4 q^{30} + 8 q^{31} + 4 q^{32} + 18 q^{33} - 4 q^{34} - 10 q^{35} + 20 q^{36} - 2 q^{37} + 8 q^{38} - 6 q^{40} + 8 q^{41} - 4 q^{42} - 2 q^{43} + 12 q^{44} - 4 q^{45} + 16 q^{46} + 8 q^{47} - 28 q^{48} + 12 q^{49} + 2 q^{50} + 4 q^{51} - 12 q^{53} - 16 q^{54} + 12 q^{56} - 14 q^{57} + 22 q^{58} + 12 q^{59} + 10 q^{60} + 28 q^{61} + 4 q^{62} + 4 q^{63} + 4 q^{64} + 6 q^{66} + 30 q^{67} + 14 q^{68} - 16 q^{69} - 2 q^{70} + 4 q^{71} + 12 q^{72} - 8 q^{73} - 10 q^{74} - 2 q^{75} + 20 q^{76} + 18 q^{77} - 8 q^{79} - 2 q^{80} - 8 q^{81} + 4 q^{82} - 12 q^{83} - 28 q^{84} + 2 q^{85} - 4 q^{86} - 22 q^{87} - 18 q^{88} - 12 q^{89} - 20 q^{90} + 22 q^{92} + 8 q^{93} - 32 q^{94} - 16 q^{95} + 4 q^{96} + 2 q^{97} - 24 q^{98} + 24 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 4 * q^6 + 10 * q^7 + 6 * q^8 + 4 * q^9 - 2 * q^10 - 10 * q^12 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 20 * q^18 + 16 * q^19 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 10 * q^23 - 24 * q^24 + 4 * q^25 - 2 * q^27 + 8 * q^28 + 8 * q^29 + 4 * q^30 + 8 * q^31 + 4 * q^32 + 18 * q^33 - 4 * q^34 - 10 * q^35 + 20 * q^36 - 2 * q^37 + 8 * q^38 - 6 * q^40 + 8 * q^41 - 4 * q^42 - 2 * q^43 + 12 * q^44 - 4 * q^45 + 16 * q^46 + 8 * q^47 - 28 * q^48 + 12 * q^49 + 2 * q^50 + 4 * q^51 - 12 * q^53 - 16 * q^54 + 12 * q^56 - 14 * q^57 + 22 * q^58 + 12 * q^59 + 10 * q^60 + 28 * q^61 + 4 * q^62 + 4 * q^63 + 4 * q^64 + 6 * q^66 + 30 * q^67 + 14 * q^68 - 16 * q^69 - 2 * q^70 + 4 * q^71 + 12 * q^72 - 8 * q^73 - 10 * q^74 - 2 * q^75 + 20 * q^76 + 18 * q^77 - 8 * q^79 - 2 * q^80 - 8 * q^81 + 4 * q^82 - 12 * q^83 - 28 * q^84 + 2 * q^85 - 4 * q^86 - 22 * q^87 - 18 * q^88 - 12 * q^89 - 20 * q^90 + 22 * q^92 + 8 * q^93 - 32 * q^94 - 16 * q^95 + 4 * q^96 + 2 * q^97 - 24 * q^98 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 3x^{2} + 4x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 1$$ v^3 - v^2 - 3*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 1$$ b3 + b2 + 4*b1 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.49551 −0.219687 1.21969 2.49551
−1.49551 0.0947876 0.236543 −1.00000 −0.141756 4.82684 2.63726 −2.99102 1.49551
1.2 −0.219687 −1.60020 −1.95174 −1.00000 0.351542 −0.332247 0.868145 −0.439374 0.219687
1.3 1.21969 2.33225 −0.512364 −1.00000 2.84461 3.60020 −3.06430 2.43937 −1.21969
1.4 2.49551 −2.82684 4.22756 −1.00000 −7.05440 1.90521 5.55889 4.99102 −2.49551
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.a.m 4
3.b odd 2 1 7605.2.a.cf 4
5.b even 2 1 4225.2.a.bi 4
13.b even 2 1 845.2.a.l 4
13.c even 3 2 845.2.e.m 8
13.d odd 4 2 845.2.c.g 8
13.e even 6 2 845.2.e.n 8
13.f odd 12 2 65.2.m.a 8
13.f odd 12 2 845.2.m.g 8
39.d odd 2 1 7605.2.a.cj 4
39.k even 12 2 585.2.bu.c 8
52.l even 12 2 1040.2.da.b 8
65.d even 2 1 4225.2.a.bl 4
65.o even 12 2 325.2.m.b 8
65.s odd 12 2 325.2.n.d 8
65.t even 12 2 325.2.m.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 13.f odd 12 2
325.2.m.b 8 65.o even 12 2
325.2.m.c 8 65.t even 12 2
325.2.n.d 8 65.s odd 12 2
585.2.bu.c 8 39.k even 12 2
845.2.a.l 4 13.b even 2 1
845.2.a.m 4 1.a even 1 1 trivial
845.2.c.g 8 13.d odd 4 2
845.2.e.m 8 13.c even 3 2
845.2.e.n 8 13.e even 6 2
845.2.m.g 8 13.f odd 12 2
1040.2.da.b 8 52.l even 12 2
4225.2.a.bi 4 5.b even 2 1
4225.2.a.bl 4 65.d even 2 1
7605.2.a.cf 4 3.b odd 2 1
7605.2.a.cj 4 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} - 3T_{2}^{2} + 4T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(845))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} - 3 T^{2} + 4 T + 1$$
$3$ $$T^{4} + 2 T^{3} - 6 T^{2} - 10 T + 1$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4} - 10 T^{3} + 30 T^{2} - 22 T - 11$$
$11$ $$T^{4} - 30T^{2} + 33$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 2 T^{3} - 18 T^{2} - 10 T + 13$$
$19$ $$(T^{2} - 8 T + 13)^{2}$$
$23$ $$T^{4} + 10 T^{3} + 6 T^{2} - 146 T - 299$$
$29$ $$T^{4} - 8 T^{3} - 18 T^{2} + 40 T + 1$$
$31$ $$(T^{2} - 4 T - 8)^{2}$$
$37$ $$T^{4} + 2 T^{3} - 54 T^{2} + 38 T + 1$$
$41$ $$(T^{2} - 4 T + 1)^{2}$$
$43$ $$T^{4} + 2 T^{3} - 18 T^{2} - 10 T + 13$$
$47$ $$T^{4} - 8 T^{3} - 72 T^{2} + \cdots - 1328$$
$53$ $$T^{4} + 12 T^{3} + 36 T^{2} - 48$$
$59$ $$T^{4} - 12 T^{3} + 30 T^{2} - 12 T - 3$$
$61$ $$T^{4} - 28 T^{3} + 258 T^{2} + \cdots + 1261$$
$67$ $$T^{4} - 30 T^{3} + 330 T^{2} + \cdots + 2769$$
$71$ $$T^{4} - 4 T^{3} - 210 T^{2} + \cdots + 10477$$
$73$ $$T^{4} + 8 T^{3} - 84 T^{2} + \cdots - 1712$$
$79$ $$T^{4} + 8 T^{3} - 132 T^{2} + \cdots + 4432$$
$83$ $$T^{4} + 12 T^{3} - 24 T^{2} + \cdots - 192$$
$89$ $$T^{4} + 12 T^{3} - 234 T^{2} + \cdots + 8853$$
$97$ $$T^{4} - 2 T^{3} - 90 T^{2} - 374 T - 443$$