# Properties

 Label 3185.2.a.j Level $3185$ Weight $2$ Character orbit 3185.a Self dual yes Analytic conductor $25.432$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3185 = 5 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3185.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.4323530438$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} - \beta q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + (\beta - 2) q^{6} + (\beta - 3) q^{8} - q^{9} +O(q^{10})$$ q + (b - 1) * q^2 - b * q^3 + (-2*b + 1) * q^4 - q^5 + (b - 2) * q^6 + (b - 3) * q^8 - q^9 $$q + (\beta - 1) q^{2} - \beta q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + (\beta - 2) q^{6} + (\beta - 3) q^{8} - q^{9} + ( - \beta + 1) q^{10} + ( - \beta + 2) q^{11} + ( - \beta + 4) q^{12} + q^{13} + \beta q^{15} + 3 q^{16} + (2 \beta + 2) q^{17} + ( - \beta + 1) q^{18} + ( - \beta - 2) q^{19} + (2 \beta - 1) q^{20} + (3 \beta - 4) q^{22} - \beta q^{23} + (3 \beta - 2) q^{24} + q^{25} + (\beta - 1) q^{26} + 4 \beta q^{27} + 4 \beta q^{29} + ( - \beta + 2) q^{30} + ( - 3 \beta - 6) q^{31} + (\beta + 3) q^{32} + ( - 2 \beta + 2) q^{33} + 2 q^{34} + (2 \beta - 1) q^{36} + 6 \beta q^{37} - \beta q^{38} - \beta q^{39} + ( - \beta + 3) q^{40} + (2 \beta + 6) q^{41} + (5 \beta - 4) q^{43} + ( - 5 \beta + 6) q^{44} + q^{45} + (\beta - 2) q^{46} + ( - 2 \beta + 2) q^{47} - 3 \beta q^{48} + (\beta - 1) q^{50} + ( - 2 \beta - 4) q^{51} + ( - 2 \beta + 1) q^{52} + ( - 6 \beta - 6) q^{53} + ( - 4 \beta + 8) q^{54} + (\beta - 2) q^{55} + (2 \beta + 2) q^{57} + ( - 4 \beta + 8) q^{58} + ( - 3 \beta - 6) q^{59} + (\beta - 4) q^{60} + 8 q^{61} - 3 \beta q^{62} + (2 \beta - 7) q^{64} - q^{65} + (4 \beta - 6) q^{66} - 2 q^{67} + ( - 2 \beta - 6) q^{68} + 2 q^{69} + ( - 7 \beta + 2) q^{71} + ( - \beta + 3) q^{72} + 6 \beta q^{73} + ( - 6 \beta + 12) q^{74} - \beta q^{75} + (3 \beta + 2) q^{76} + (\beta - 2) q^{78} + 6 \beta q^{79} - 3 q^{80} - 5 q^{81} + (4 \beta - 2) q^{82} + (2 \beta + 6) q^{83} + ( - 2 \beta - 2) q^{85} + ( - 9 \beta + 14) q^{86} - 8 q^{87} + (5 \beta - 8) q^{88} - 6 q^{89} + (\beta - 1) q^{90} + ( - \beta + 4) q^{92} + (6 \beta + 6) q^{93} + (4 \beta - 6) q^{94} + (\beta + 2) q^{95} + ( - 3 \beta - 2) q^{96} + ( - 4 \beta + 2) q^{97} + (\beta - 2) q^{99} +O(q^{100})$$ q + (b - 1) * q^2 - b * q^3 + (-2*b + 1) * q^4 - q^5 + (b - 2) * q^6 + (b - 3) * q^8 - q^9 + (-b + 1) * q^10 + (-b + 2) * q^11 + (-b + 4) * q^12 + q^13 + b * q^15 + 3 * q^16 + (2*b + 2) * q^17 + (-b + 1) * q^18 + (-b - 2) * q^19 + (2*b - 1) * q^20 + (3*b - 4) * q^22 - b * q^23 + (3*b - 2) * q^24 + q^25 + (b - 1) * q^26 + 4*b * q^27 + 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 - 1) * q^36 + 6*b * q^37 - b * q^38 - b * q^39 + (-b + 3) * q^40 + (2*b + 6) * q^41 + (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 + (b - 1) * q^50 + (-2*b - 4) * q^51 + (-2*b + 1) * q^52 + (-6*b - 6) * q^53 + (-4*b + 8) * q^54 + (b - 2) * q^55 + (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 - 7) * q^64 - q^65 + (4*b - 6) * q^66 - 2 * q^67 + (-2*b - 6) * q^68 + 2 * q^69 + (-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 + (b - 2) * q^78 + 6*b * q^79 - 3 * q^80 - 5 * q^81 + (4*b - 2) * q^82 + (2*b + 6) * q^83 + (-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 + (b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 6 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^6 - 6 * q^8 - 2 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 6 q^{8} - 2 q^{9} + 2 q^{10} + 4 q^{11} + 8 q^{12} + 2 q^{13} + 6 q^{16} + 4 q^{17} + 2 q^{18} - 4 q^{19} - 2 q^{20} - 8 q^{22} - 4 q^{24} + 2 q^{25} - 2 q^{26} + 4 q^{30} - 12 q^{31} + 6 q^{32} + 4 q^{33} + 4 q^{34} - 2 q^{36} + 6 q^{40} + 12 q^{41} - 8 q^{43} + 12 q^{44} + 2 q^{45} - 4 q^{46} + 4 q^{47} - 2 q^{50} - 8 q^{51} + 2 q^{52} - 12 q^{53} + 16 q^{54} - 4 q^{55} + 4 q^{57} + 16 q^{58} - 12 q^{59} - 8 q^{60} + 16 q^{61} - 14 q^{64} - 2 q^{65} - 12 q^{66} - 4 q^{67} - 12 q^{68} + 4 q^{69} + 4 q^{71} + 6 q^{72} + 24 q^{74} + 4 q^{76} - 4 q^{78} - 6 q^{80} - 10 q^{81} - 4 q^{82} + 12 q^{83} - 4 q^{85} + 28 q^{86} - 16 q^{87} - 16 q^{88} - 12 q^{89} - 2 q^{90} + 8 q^{92} + 12 q^{93} - 12 q^{94} + 4 q^{95} - 4 q^{96} + 4 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^6 - 6 * q^8 - 2 * q^9 + 2 * q^10 + 4 * q^11 + 8 * q^12 + 2 * q^13 + 6 * q^16 + 4 * q^17 + 2 * q^18 - 4 * q^19 - 2 * q^20 - 8 * q^22 - 4 * q^24 + 2 * q^25 - 2 * q^26 + 4 * q^30 - 12 * q^31 + 6 * q^32 + 4 * q^33 + 4 * q^34 - 2 * q^36 + 6 * q^40 + 12 * q^41 - 8 * q^43 + 12 * q^44 + 2 * q^45 - 4 * q^46 + 4 * q^47 - 2 * q^50 - 8 * q^51 + 2 * q^52 - 12 * q^53 + 16 * q^54 - 4 * q^55 + 4 * q^57 + 16 * q^58 - 12 * q^59 - 8 * q^60 + 16 * q^61 - 14 * q^64 - 2 * q^65 - 12 * q^66 - 4 * q^67 - 12 * q^68 + 4 * q^69 + 4 * q^71 + 6 * q^72 + 24 * q^74 + 4 * q^76 - 4 * q^78 - 6 * q^80 - 10 * q^81 - 4 * q^82 + 12 * q^83 - 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 - 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
−2.41421 1.41421 3.82843 −1.00000 −3.41421 0 −4.41421 −1.00000 2.41421
1.2 0.414214 −1.41421 −1.82843 −1.00000 −0.585786 0 −1.58579 −1.00000 −0.414214
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 3185.2.a.j 2
7.b odd 2 1 65.2.a.b 2
21.c even 2 1 585.2.a.m 2
28.d even 2 1 1040.2.a.j 2
35.c odd 2 1 325.2.a.i 2
35.f even 4 2 325.2.b.f 4
56.e even 2 1 4160.2.a.z 2
56.h odd 2 1 4160.2.a.bf 2
77.b even 2 1 7865.2.a.j 2
84.h odd 2 1 9360.2.a.cd 2
91.b odd 2 1 845.2.a.g 2
91.i even 4 2 845.2.c.b 4
91.n odd 6 2 845.2.e.h 4
91.t odd 6 2 845.2.e.c 4
91.bc even 12 4 845.2.m.f 8
105.g even 2 1 2925.2.a.u 2
105.k odd 4 2 2925.2.c.r 4
140.c even 2 1 5200.2.a.bu 2
273.g even 2 1 7605.2.a.x 2
455.h odd 2 1 4225.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 7.b odd 2 1
325.2.a.i 2 35.c odd 2 1
325.2.b.f 4 35.f even 4 2
585.2.a.m 2 21.c even 2 1
845.2.a.g 2 91.b odd 2 1
845.2.c.b 4 91.i even 4 2
845.2.e.c 4 91.t odd 6 2
845.2.e.h 4 91.n odd 6 2
845.2.m.f 8 91.bc even 12 4
1040.2.a.j 2 28.d even 2 1
2925.2.a.u 2 105.g even 2 1
2925.2.c.r 4 105.k odd 4 2
3185.2.a.j 2 1.a even 1 1 trivial
4160.2.a.z 2 56.e even 2 1
4160.2.a.bf 2 56.h odd 2 1
4225.2.a.r 2 455.h odd 2 1
5200.2.a.bu 2 140.c even 2 1
7605.2.a.x 2 273.g even 2 1
7865.2.a.j 2 77.b even 2 1
9360.2.a.cd 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3185))$$:

 $$T_{2}^{2} + 2T_{2} - 1$$ T2^2 + 2*T2 - 1 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{11}^{2} - 4T_{11} + 2$$ T11^2 - 4*T11 + 2

## Hecke characteristic polynomials

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