# Properties

 Label 845.2.a.g Level $845$ Weight $2$ Character orbit 845.a Self dual yes Analytic conductor $6.747$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−0.414214 1.41421 −1.82843 −1.00000 −0.585786 0.828427 1.58579 −1.00000 0.414214
1.2 2.41421 −1.41421 3.82843 −1.00000 −3.41421 −4.82843 4.41421 −1.00000 −2.41421
 $$n$$: 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.g 2
3.b odd 2 1 7605.2.a.x 2
5.b even 2 1 4225.2.a.r 2
13.b even 2 1 65.2.a.b 2
13.c even 3 2 845.2.e.c 4
13.d odd 4 2 845.2.c.b 4
13.e even 6 2 845.2.e.h 4
13.f odd 12 4 845.2.m.f 8
39.d odd 2 1 585.2.a.m 2
52.b odd 2 1 1040.2.a.j 2
65.d even 2 1 325.2.a.i 2
65.h odd 4 2 325.2.b.f 4
91.b odd 2 1 3185.2.a.j 2
104.e even 2 1 4160.2.a.bf 2
104.h odd 2 1 4160.2.a.z 2
143.d odd 2 1 7865.2.a.j 2
156.h even 2 1 9360.2.a.cd 2
195.e odd 2 1 2925.2.a.u 2
195.s even 4 2 2925.2.c.r 4
260.g odd 2 1 5200.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 13.b even 2 1
325.2.a.i 2 65.d even 2 1
325.2.b.f 4 65.h odd 4 2
585.2.a.m 2 39.d odd 2 1
845.2.a.g 2 1.a even 1 1 trivial
845.2.c.b 4 13.d odd 4 2
845.2.e.c 4 13.c even 3 2
845.2.e.h 4 13.e even 6 2
845.2.m.f 8 13.f odd 12 4
1040.2.a.j 2 52.b odd 2 1
2925.2.a.u 2 195.e odd 2 1
2925.2.c.r 4 195.s even 4 2
3185.2.a.j 2 91.b odd 2 1
4160.2.a.z 2 104.h odd 2 1
4160.2.a.bf 2 104.e even 2 1
4225.2.a.r 2 5.b even 2 1
5200.2.a.bu 2 260.g odd 2 1
7605.2.a.x 2 3.b odd 2 1
7865.2.a.j 2 143.d odd 2 1
9360.2.a.cd 2 156.h even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2} - 2$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 4T - 4$$
$11$ $$T^{2} + 4T + 2$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 4T - 4$$
$19$ $$T^{2} + 4T + 2$$
$23$ $$T^{2} - 2$$
$29$ $$T^{2} - 32$$
$31$ $$T^{2} + 12T + 18$$
$37$ $$T^{2} - 72$$
$41$ $$T^{2} - 12T + 28$$
$43$ $$T^{2} + 8T - 34$$
$47$ $$T^{2} - 4T - 4$$
$53$ $$T^{2} + 12T - 36$$
$59$ $$T^{2} + 12T + 18$$
$61$ $$(T + 8)^{2}$$
$67$ $$(T - 2)^{2}$$
$71$ $$T^{2} + 4T - 94$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} - 72$$
$83$ $$T^{2} - 12T + 28$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} - 4T - 28$$