# Properties

 Label 1617.2.a.n Level $1617$ Weight $2$ Character orbit 1617.a Self dual yes Analytic conductor $12.912$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1617.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.9118100068$$ 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 231) 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - 2 q^{5} + (\beta - 1) q^{6} + (\beta - 3) q^{8} + q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + q^3 + (-2*b + 1) * q^4 - 2 * q^5 + (b - 1) * q^6 + (b - 3) * q^8 + q^9 $$q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - 2 q^{5} + (\beta - 1) q^{6} + (\beta - 3) q^{8} + q^{9} + ( - 2 \beta + 2) q^{10} + q^{11} + ( - 2 \beta + 1) q^{12} + (2 \beta + 2) q^{13} - 2 q^{15} + 3 q^{16} + (\beta - 3) q^{17} + (\beta - 1) q^{18} + ( - 3 \beta + 3) q^{19} + (4 \beta - 2) q^{20} + (\beta - 1) q^{22} - 7 q^{23} + (\beta - 3) q^{24} - q^{25} + 2 q^{26} + q^{27} + ( - 3 \beta - 1) q^{29} + ( - 2 \beta + 2) q^{30} - 4 \beta q^{31} + (\beta + 3) q^{32} + q^{33} + ( - 4 \beta + 5) q^{34} + ( - 2 \beta + 1) q^{36} + (6 \beta - 1) q^{37} + (6 \beta - 9) q^{38} + (2 \beta + 2) q^{39} + ( - 2 \beta + 6) q^{40} + ( - 2 \beta - 4) q^{41} + ( - 3 \beta - 7) q^{43} + ( - 2 \beta + 1) q^{44} - 2 q^{45} + ( - 7 \beta + 7) q^{46} + ( - 2 \beta + 7) q^{47} + 3 q^{48} + ( - \beta + 1) q^{50} + (\beta - 3) q^{51} + ( - 2 \beta - 6) q^{52} + ( - 2 \beta - 10) q^{53} + (\beta - 1) q^{54} - 2 q^{55} + ( - 3 \beta + 3) q^{57} + (2 \beta - 5) q^{58} + (4 \beta - 3) q^{59} + (4 \beta - 2) q^{60} - 4 q^{61} + (4 \beta - 8) q^{62} + (2 \beta - 7) q^{64} + ( - 4 \beta - 4) q^{65} + (\beta - 1) q^{66} + ( - 2 \beta - 6) q^{67} + (7 \beta - 7) q^{68} - 7 q^{69} + ( - 2 \beta - 7) q^{71} + (\beta - 3) q^{72} + (4 \beta + 6) q^{73} + ( - 7 \beta + 13) q^{74} - q^{75} + ( - 9 \beta + 15) q^{76} + 2 q^{78} + ( - 8 \beta + 2) q^{79} - 6 q^{80} + q^{81} - 2 \beta q^{82} + 2 \beta q^{83} + ( - 2 \beta + 6) q^{85} + ( - 4 \beta + 1) q^{86} + ( - 3 \beta - 1) q^{87} + (\beta - 3) q^{88} + 10 \beta q^{89} + ( - 2 \beta + 2) q^{90} + (14 \beta - 7) q^{92} - 4 \beta q^{93} + (9 \beta - 11) q^{94} + (6 \beta - 6) q^{95} + (\beta + 3) q^{96} + (6 \beta - 3) q^{97} + q^{99} +O(q^{100})$$ q + (b - 1) * q^2 + q^3 + (-2*b + 1) * q^4 - 2 * q^5 + (b - 1) * q^6 + (b - 3) * q^8 + q^9 + (-2*b + 2) * q^10 + q^11 + (-2*b + 1) * q^12 + (2*b + 2) * q^13 - 2 * q^15 + 3 * q^16 + (b - 3) * q^17 + (b - 1) * q^18 + (-3*b + 3) * q^19 + (4*b - 2) * q^20 + (b - 1) * q^22 - 7 * q^23 + (b - 3) * q^24 - q^25 + 2 * q^26 + q^27 + (-3*b - 1) * q^29 + (-2*b + 2) * q^30 - 4*b * q^31 + (b + 3) * q^32 + q^33 + (-4*b + 5) * q^34 + (-2*b + 1) * q^36 + (6*b - 1) * q^37 + (6*b - 9) * q^38 + (2*b + 2) * q^39 + (-2*b + 6) * q^40 + (-2*b - 4) * q^41 + (-3*b - 7) * q^43 + (-2*b + 1) * q^44 - 2 * q^45 + (-7*b + 7) * q^46 + (-2*b + 7) * q^47 + 3 * q^48 + (-b + 1) * q^50 + (b - 3) * q^51 + (-2*b - 6) * q^52 + (-2*b - 10) * q^53 + (b - 1) * q^54 - 2 * q^55 + (-3*b + 3) * q^57 + (2*b - 5) * q^58 + (4*b - 3) * q^59 + (4*b - 2) * q^60 - 4 * q^61 + (4*b - 8) * q^62 + (2*b - 7) * q^64 + (-4*b - 4) * q^65 + (b - 1) * q^66 + (-2*b - 6) * q^67 + (7*b - 7) * q^68 - 7 * q^69 + (-2*b - 7) * q^71 + (b - 3) * q^72 + (4*b + 6) * q^73 + (-7*b + 13) * q^74 - q^75 + (-9*b + 15) * q^76 + 2 * q^78 + (-8*b + 2) * q^79 - 6 * q^80 + q^81 - 2*b * q^82 + 2*b * q^83 + (-2*b + 6) * q^85 + (-4*b + 1) * q^86 + (-3*b - 1) * q^87 + (b - 3) * q^88 + 10*b * q^89 + (-2*b + 2) * q^90 + (14*b - 7) * q^92 - 4*b * q^93 + (9*b - 11) * q^94 + (6*b - 6) * q^95 + (b + 3) * q^96 + (6*b - 3) * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 6 q^{8} + 2 q^{9} + 4 q^{10} + 2 q^{11} + 2 q^{12} + 4 q^{13} - 4 q^{15} + 6 q^{16} - 6 q^{17} - 2 q^{18} + 6 q^{19} - 4 q^{20} - 2 q^{22} - 14 q^{23} - 6 q^{24} - 2 q^{25} + 4 q^{26} + 2 q^{27} - 2 q^{29} + 4 q^{30} + 6 q^{32} + 2 q^{33} + 10 q^{34} + 2 q^{36} - 2 q^{37} - 18 q^{38} + 4 q^{39} + 12 q^{40} - 8 q^{41} - 14 q^{43} + 2 q^{44} - 4 q^{45} + 14 q^{46} + 14 q^{47} + 6 q^{48} + 2 q^{50} - 6 q^{51} - 12 q^{52} - 20 q^{53} - 2 q^{54} - 4 q^{55} + 6 q^{57} - 10 q^{58} - 6 q^{59} - 4 q^{60} - 8 q^{61} - 16 q^{62} - 14 q^{64} - 8 q^{65} - 2 q^{66} - 12 q^{67} - 14 q^{68} - 14 q^{69} - 14 q^{71} - 6 q^{72} + 12 q^{73} + 26 q^{74} - 2 q^{75} + 30 q^{76} + 4 q^{78} + 4 q^{79} - 12 q^{80} + 2 q^{81} + 12 q^{85} + 2 q^{86} - 2 q^{87} - 6 q^{88} + 4 q^{90} - 14 q^{92} - 22 q^{94} - 12 q^{95} + 6 q^{96} - 6 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 6 * q^8 + 2 * q^9 + 4 * q^10 + 2 * q^11 + 2 * q^12 + 4 * q^13 - 4 * q^15 + 6 * q^16 - 6 * q^17 - 2 * q^18 + 6 * q^19 - 4 * q^20 - 2 * q^22 - 14 * q^23 - 6 * q^24 - 2 * q^25 + 4 * q^26 + 2 * q^27 - 2 * q^29 + 4 * q^30 + 6 * q^32 + 2 * q^33 + 10 * q^34 + 2 * q^36 - 2 * q^37 - 18 * q^38 + 4 * q^39 + 12 * q^40 - 8 * q^41 - 14 * q^43 + 2 * q^44 - 4 * q^45 + 14 * q^46 + 14 * q^47 + 6 * q^48 + 2 * q^50 - 6 * q^51 - 12 * q^52 - 20 * q^53 - 2 * q^54 - 4 * q^55 + 6 * q^57 - 10 * q^58 - 6 * q^59 - 4 * q^60 - 8 * q^61 - 16 * q^62 - 14 * q^64 - 8 * q^65 - 2 * q^66 - 12 * q^67 - 14 * q^68 - 14 * q^69 - 14 * q^71 - 6 * q^72 + 12 * q^73 + 26 * q^74 - 2 * q^75 + 30 * q^76 + 4 * q^78 + 4 * q^79 - 12 * q^80 + 2 * q^81 + 12 * q^85 + 2 * q^86 - 2 * q^87 - 6 * q^88 + 4 * q^90 - 14 * q^92 - 22 * q^94 - 12 * q^95 + 6 * q^96 - 6 * q^97 + 2 * 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 1.00000 3.82843 −2.00000 −2.41421 0 −4.41421 1.00000 4.82843
1.2 0.414214 1.00000 −1.82843 −2.00000 0.414214 0 −1.58579 1.00000 −0.828427
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## 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 1617.2.a.n 2
3.b odd 2 1 4851.2.a.be 2
7.b odd 2 1 1617.2.a.m 2
7.c even 3 2 231.2.i.d 4
21.c even 2 1 4851.2.a.bd 2
21.h odd 6 2 693.2.i.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.d 4 7.c even 3 2
693.2.i.f 4 21.h odd 6 2
1617.2.a.m 2 7.b odd 2 1
1617.2.a.n 2 1.a even 1 1 trivial
4851.2.a.bd 2 21.c even 2 1
4851.2.a.be 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2}^{2} + 2T_{2} - 1$$ T2^2 + 2*T2 - 1 $$T_{5} + 2$$ T5 + 2 $$T_{13}^{2} - 4T_{13} - 4$$ T13^2 - 4*T13 - 4 $$T_{17}^{2} + 6T_{17} + 7$$ T17^2 + 6*T17 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 4T - 4$$
$17$ $$T^{2} + 6T + 7$$
$19$ $$T^{2} - 6T - 9$$
$23$ $$(T + 7)^{2}$$
$29$ $$T^{2} + 2T - 17$$
$31$ $$T^{2} - 32$$
$37$ $$T^{2} + 2T - 71$$
$41$ $$T^{2} + 8T + 8$$
$43$ $$T^{2} + 14T + 31$$
$47$ $$T^{2} - 14T + 41$$
$53$ $$T^{2} + 20T + 92$$
$59$ $$T^{2} + 6T - 23$$
$61$ $$(T + 4)^{2}$$
$67$ $$T^{2} + 12T + 28$$
$71$ $$T^{2} + 14T + 41$$
$73$ $$T^{2} - 12T + 4$$
$79$ $$T^{2} - 4T - 124$$
$83$ $$T^{2} - 8$$
$89$ $$T^{2} - 200$$
$97$ $$T^{2} + 6T - 63$$
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