# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{2} + \beta_1 q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{6} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{7} + (\beta_{3} - 3) q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^2 + b1 * q^3 + (2*b3 + b2 + 2*b1) * q^4 + q^5 + (b3 + 2*b2 + b1) * q^6 + (2*b3 + 2*b2 + 2*b1) * q^7 + (b3 - 3) * q^8 - b2 * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{2} + \beta_1 q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{6} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{7} + (\beta_{3} - 3) q^{8} - \beta_{2} q^{9} + (\beta_{2} + \beta_1 + 1) q^{10} + ( - 2 \beta_{2} - \beta_1 - 2) q^{11} + (\beta_{3} - 4) q^{12} + (4 \beta_{3} - 6) q^{14} + \beta_1 q^{15} + ( - 3 \beta_{2} - 3) q^{16} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{17} + ( - \beta_{3} + 1) q^{18} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{19} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{20} + (2 \beta_{3} - 4) q^{21} + ( - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{22} - \beta_1 q^{23} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{24} + q^{25} - 4 \beta_{3} q^{27} + ( - 10 \beta_{2} - 6 \beta_1 - 10) q^{28} + 4 \beta_1 q^{29} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{30} + (3 \beta_{3} + 6) q^{31} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{32} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{33} - 2 q^{34} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{35} + (\beta_{2} + 2 \beta_1 + 1) q^{36} + 6 \beta_1 q^{37} + \beta_{3} q^{38} + (\beta_{3} - 3) q^{40} + (6 \beta_{2} - 2 \beta_1 + 6) q^{41} + ( - 8 \beta_{2} - 6 \beta_1 - 8) q^{42} + ( - 5 \beta_{3} - 4 \beta_{2} - 5 \beta_1) q^{43} + ( - 5 \beta_{3} + 6) q^{44} - \beta_{2} q^{45} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{46} + (2 \beta_{3} - 2) q^{47} + ( - 3 \beta_{3} - 3 \beta_1) q^{48} + ( - 5 \beta_{2} - 8 \beta_1 - 5) q^{49} + (\beta_{2} + \beta_1 + 1) q^{50} + ( - 2 \beta_{3} - 4) q^{51} + ( - 6 \beta_{3} - 6) q^{53} + (8 \beta_{2} + 4 \beta_1 + 8) q^{54} + ( - 2 \beta_{2} - \beta_1 - 2) q^{55} + ( - 8 \beta_{3} - 10 \beta_{2} - 8 \beta_1) q^{56} + (2 \beta_{3} + 2) q^{57} + (4 \beta_{3} + 8 \beta_{2} + 4 \beta_1) q^{58} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1) q^{59} + (\beta_{3} - 4) q^{60} - 8 \beta_{2} q^{61} + 3 \beta_1 q^{62} + (2 \beta_{2} + 2 \beta_1 + 2) q^{63} + (2 \beta_{3} - 7) q^{64} + ( - 4 \beta_{3} + 6) q^{66} + (2 \beta_{2} + 2) q^{67} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{68} - 2 \beta_{2} q^{69} + (4 \beta_{3} - 6) q^{70} + (7 \beta_{3} + 2 \beta_{2} + 7 \beta_1) q^{71} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{72} - 6 \beta_{3} q^{73} + (6 \beta_{3} + 12 \beta_{2} + 6 \beta_1) q^{74} + \beta_1 q^{75} + (2 \beta_{2} - 3 \beta_1 + 2) q^{76} + ( - 6 \beta_{3} + 8) q^{77} + 6 \beta_{3} q^{79} + ( - 3 \beta_{2} - 3) q^{80} + (5 \beta_{2} + 5) q^{81} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{82} + ( - 2 \beta_{3} - 6) q^{83} + ( - 10 \beta_{3} - 12 \beta_{2} - 10 \beta_1) q^{84} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{85} + ( - 9 \beta_{3} + 14) q^{86} + 8 \beta_{2} q^{87} + (8 \beta_{2} + 5 \beta_1 + 8) q^{88} + ( - 6 \beta_{2} - 6) q^{89} + ( - \beta_{3} + 1) q^{90} + ( - \beta_{3} + 4) q^{92} + ( - 6 \beta_{2} + 6 \beta_1 - 6) q^{93} + ( - 6 \beta_{2} - 4 \beta_1 - 6) q^{94} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{95} + (3 \beta_{3} + 2) q^{96} + ( - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{97} + ( - 13 \beta_{3} - 21 \beta_{2} - 13 \beta_1) q^{98} + (\beta_{3} - 2) q^{99}+O(q^{100})$$ q + (b2 + b1 + 1) * q^2 + b1 * q^3 + (2*b3 + b2 + 2*b1) * q^4 + q^5 + (b3 + 2*b2 + b1) * q^6 + (2*b3 + 2*b2 + 2*b1) * q^7 + (b3 - 3) * q^8 - b2 * q^9 + (b2 + b1 + 1) * q^10 + (-2*b2 - b1 - 2) * q^11 + (b3 - 4) * q^12 + (4*b3 - 6) * q^14 + b1 * q^15 + (-3*b2 - 3) * q^16 + (2*b3 - 2*b2 + 2*b1) * q^17 + (-b3 + 1) * q^18 + (-b3 + 2*b2 - b1) * q^19 + (2*b3 + b2 + 2*b1) * q^20 + (2*b3 - 4) * q^21 + (-3*b3 - 4*b2 - 3*b1) * q^22 - b1 * q^23 + (-2*b2 - 3*b1 - 2) * q^24 + q^25 - 4*b3 * q^27 + (-10*b2 - 6*b1 - 10) * q^28 + 4*b1 * q^29 + (b3 + 2*b2 + b1) * q^30 + (3*b3 + 6) * q^31 + (-b3 + 3*b2 - b1) * q^32 + (-2*b3 - 2*b2 - 2*b1) * q^33 - 2 * q^34 + (2*b3 + 2*b2 + 2*b1) * q^35 + (b2 + 2*b1 + 1) * q^36 + 6*b1 * q^37 + b3 * q^38 + (b3 - 3) * q^40 + (6*b2 - 2*b1 + 6) * q^41 + (-8*b2 - 6*b1 - 8) * q^42 + (-5*b3 - 4*b2 - 5*b1) * q^43 + (-5*b3 + 6) * q^44 - b2 * q^45 + (-b3 - 2*b2 - b1) * q^46 + (2*b3 - 2) * q^47 + (-3*b3 - 3*b1) * q^48 + (-5*b2 - 8*b1 - 5) * q^49 + (b2 + b1 + 1) * q^50 + (-2*b3 - 4) * q^51 + (-6*b3 - 6) * q^53 + (8*b2 + 4*b1 + 8) * q^54 + (-2*b2 - b1 - 2) * q^55 + (-8*b3 - 10*b2 - 8*b1) * q^56 + (2*b3 + 2) * q^57 + (4*b3 + 8*b2 + 4*b1) * q^58 + (-3*b3 + 6*b2 - 3*b1) * q^59 + (b3 - 4) * q^60 - 8*b2 * q^61 + 3*b1 * q^62 + (2*b2 + 2*b1 + 2) * q^63 + (2*b3 - 7) * q^64 + (-4*b3 + 6) * q^66 + (2*b2 + 2) * q^67 + (-6*b2 + 2*b1 - 6) * q^68 - 2*b2 * q^69 + (4*b3 - 6) * q^70 + (7*b3 + 2*b2 + 7*b1) * q^71 + (b3 + 3*b2 + b1) * q^72 - 6*b3 * q^73 + (6*b3 + 12*b2 + 6*b1) * q^74 + b1 * q^75 + (2*b2 - 3*b1 + 2) * q^76 + (-6*b3 + 8) * q^77 + 6*b3 * q^79 + (-3*b2 - 3) * q^80 + (5*b2 + 5) * q^81 + (4*b3 + 2*b2 + 4*b1) * q^82 + (-2*b3 - 6) * q^83 + (-10*b3 - 12*b2 - 10*b1) * q^84 + (2*b3 - 2*b2 + 2*b1) * q^85 + (-9*b3 + 14) * q^86 + 8*b2 * q^87 + (8*b2 + 5*b1 + 8) * q^88 + (-6*b2 - 6) * q^89 + (-b3 + 1) * q^90 + (-b3 + 4) * q^92 + (-6*b2 + 6*b1 - 6) * q^93 + (-6*b2 - 4*b1 - 6) * q^94 + (-b3 + 2*b2 - b1) * q^95 + (3*b3 + 2) * q^96 + (-4*b3 - 2*b2 - 4*b1) * q^97 + (-13*b3 - 21*b2 - 13*b1) * q^98 + (b3 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 - 12 * q^8 + 2 * q^9 $$4 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} - 12 q^{8} + 2 q^{9} + 2 q^{10} - 4 q^{11} - 16 q^{12} - 24 q^{14} - 6 q^{16} + 4 q^{17} + 4 q^{18} - 4 q^{19} - 2 q^{20} - 16 q^{21} + 8 q^{22} - 4 q^{24} + 4 q^{25} - 20 q^{28} - 4 q^{30} + 24 q^{31} - 6 q^{32} + 4 q^{33} - 8 q^{34} - 4 q^{35} + 2 q^{36} - 12 q^{40} + 12 q^{41} - 16 q^{42} + 8 q^{43} + 24 q^{44} + 2 q^{45} + 4 q^{46} - 8 q^{47} - 10 q^{49} + 2 q^{50} - 16 q^{51} - 24 q^{53} + 16 q^{54} - 4 q^{55} + 20 q^{56} + 8 q^{57} - 16 q^{58} - 12 q^{59} - 16 q^{60} + 16 q^{61} + 4 q^{63} - 28 q^{64} + 24 q^{66} + 4 q^{67} - 12 q^{68} + 4 q^{69} - 24 q^{70} - 4 q^{71} - 6 q^{72} - 24 q^{74} + 4 q^{76} + 32 q^{77} - 6 q^{80} + 10 q^{81} - 4 q^{82} - 24 q^{83} + 24 q^{84} + 4 q^{85} + 56 q^{86} - 16 q^{87} + 16 q^{88} - 12 q^{89} + 4 q^{90} + 16 q^{92} - 12 q^{93} - 12 q^{94} - 4 q^{95} + 8 q^{96} + 4 q^{97} + 42 q^{98} - 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 - 12 * q^8 + 2 * q^9 + 2 * q^10 - 4 * q^11 - 16 * q^12 - 24 * q^14 - 6 * q^16 + 4 * q^17 + 4 * q^18 - 4 * q^19 - 2 * q^20 - 16 * q^21 + 8 * q^22 - 4 * q^24 + 4 * q^25 - 20 * q^28 - 4 * q^30 + 24 * q^31 - 6 * q^32 + 4 * q^33 - 8 * q^34 - 4 * q^35 + 2 * q^36 - 12 * q^40 + 12 * q^41 - 16 * q^42 + 8 * q^43 + 24 * q^44 + 2 * q^45 + 4 * q^46 - 8 * q^47 - 10 * q^49 + 2 * q^50 - 16 * q^51 - 24 * q^53 + 16 * q^54 - 4 * q^55 + 20 * q^56 + 8 * q^57 - 16 * q^58 - 12 * q^59 - 16 * q^60 + 16 * q^61 + 4 * q^63 - 28 * q^64 + 24 * q^66 + 4 * q^67 - 12 * q^68 + 4 * q^69 - 24 * q^70 - 4 * q^71 - 6 * q^72 - 24 * q^74 + 4 * q^76 + 32 * q^77 - 6 * q^80 + 10 * q^81 - 4 * q^82 - 24 * q^83 + 24 * q^84 + 4 * q^85 + 56 * q^86 - 16 * q^87 + 16 * q^88 - 12 * q^89 + 4 * q^90 + 16 * q^92 - 12 * q^93 - 12 * q^94 - 4 * q^95 + 8 * q^96 + 4 * q^97 + 42 * q^98 - 8 * q^99

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.207107 0.358719i −0.707107 1.22474i 0.914214 1.58346i 1.00000 −0.292893 + 0.507306i 0.414214 0.717439i −1.58579 0.500000 0.866025i −0.207107 0.358719i
146.2 1.20711 + 2.09077i 0.707107 + 1.22474i −1.91421 + 3.31552i 1.00000 −1.70711 + 2.95680i −2.41421 + 4.18154i −4.41421 0.500000 0.866025i 1.20711 + 2.09077i
191.1 −0.207107 + 0.358719i −0.707107 + 1.22474i 0.914214 + 1.58346i 1.00000 −0.292893 0.507306i 0.414214 + 0.717439i −1.58579 0.500000 + 0.866025i −0.207107 + 0.358719i
191.2 1.20711 2.09077i 0.707107 1.22474i −1.91421 3.31552i 1.00000 −1.70711 2.95680i −2.41421 4.18154i −4.41421 0.500000 + 0.866025i 1.20711 2.09077i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.e.h 4
13.b even 2 1 845.2.e.c 4
13.c even 3 1 65.2.a.b 2
13.c even 3 1 inner 845.2.e.h 4
13.d odd 4 2 845.2.m.f 8
13.e even 6 1 845.2.a.g 2
13.e even 6 1 845.2.e.c 4
13.f odd 12 2 845.2.c.b 4
13.f odd 12 2 845.2.m.f 8
39.h odd 6 1 7605.2.a.x 2
39.i odd 6 1 585.2.a.m 2
52.j odd 6 1 1040.2.a.j 2
65.l even 6 1 4225.2.a.r 2
65.n even 6 1 325.2.a.i 2
65.q odd 12 2 325.2.b.f 4
91.n odd 6 1 3185.2.a.j 2
104.n odd 6 1 4160.2.a.z 2
104.r even 6 1 4160.2.a.bf 2
143.k odd 6 1 7865.2.a.j 2
156.p even 6 1 9360.2.a.cd 2
195.x odd 6 1 2925.2.a.u 2
195.bl even 12 2 2925.2.c.r 4
260.v odd 6 1 5200.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 13.c even 3 1
325.2.a.i 2 65.n even 6 1
325.2.b.f 4 65.q odd 12 2
585.2.a.m 2 39.i odd 6 1
845.2.a.g 2 13.e even 6 1
845.2.c.b 4 13.f odd 12 2
845.2.e.c 4 13.b even 2 1
845.2.e.c 4 13.e even 6 1
845.2.e.h 4 1.a even 1 1 trivial
845.2.e.h 4 13.c even 3 1 inner
845.2.m.f 8 13.d odd 4 2
845.2.m.f 8 13.f odd 12 2
1040.2.a.j 2 52.j odd 6 1
2925.2.a.u 2 195.x odd 6 1
2925.2.c.r 4 195.bl even 12 2
3185.2.a.j 2 91.n odd 6 1
4160.2.a.z 2 104.n odd 6 1
4160.2.a.bf 2 104.r even 6 1
4225.2.a.r 2 65.l even 6 1
5200.2.a.bu 2 260.v odd 6 1
7605.2.a.x 2 39.h odd 6 1
7865.2.a.j 2 143.k odd 6 1
9360.2.a.cd 2 156.p even 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1$$ T2^4 - 2*T2^3 + 5*T2^2 + 2*T2 + 1 $$T_{7}^{4} + 4T_{7}^{3} + 20T_{7}^{2} - 16T_{7} + 16$$ T7^4 + 4*T7^3 + 20*T7^2 - 16*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$3$ $$T^{4} + 2T^{2} + 4$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$11$ $$T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4$$
$13$ $$T^{4}$$
$17$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$19$ $$T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4$$
$23$ $$T^{4} + 2T^{2} + 4$$
$29$ $$T^{4} + 32T^{2} + 1024$$
$31$ $$(T^{2} - 12 T + 18)^{2}$$
$37$ $$T^{4} + 72T^{2} + 5184$$
$41$ $$T^{4} - 12 T^{3} + 116 T^{2} + \cdots + 784$$
$43$ $$T^{4} - 8 T^{3} + 98 T^{2} + \cdots + 1156$$
$47$ $$(T^{2} + 4 T - 4)^{2}$$
$53$ $$(T^{2} + 12 T - 36)^{2}$$
$59$ $$T^{4} + 12 T^{3} + 126 T^{2} + \cdots + 324$$
$61$ $$(T^{2} - 8 T + 64)^{2}$$
$67$ $$(T^{2} - 2 T + 4)^{2}$$
$71$ $$T^{4} + 4 T^{3} + 110 T^{2} + \cdots + 8836$$
$73$ $$(T^{2} - 72)^{2}$$
$79$ $$(T^{2} - 72)^{2}$$
$83$ $$(T^{2} + 12 T + 28)^{2}$$
$89$ $$(T^{2} + 6 T + 36)^{2}$$
$97$ $$T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784$$
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