# Properties

 Label 3509.2.a.j Level $3509$ Weight $2$ Character orbit 3509.a Self dual yes Analytic conductor $28.020$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3509 = 11^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3509.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.0195060693$$ Analytic rank: $$0$$ 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 29) 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (\beta + 1) q^{3} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta + 3) q^{6} + 2 \beta q^{7} + (\beta + 3) q^{8} + 2 \beta q^{9}+O(q^{10})$$ q + (b + 1) * q^2 + (b + 1) * q^3 + (2*b + 1) * q^4 - q^5 + (2*b + 3) * q^6 + 2*b * q^7 + (b + 3) * q^8 + 2*b * q^9 $$q + (\beta + 1) q^{2} + (\beta + 1) q^{3} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta + 3) q^{6} + 2 \beta q^{7} + (\beta + 3) q^{8} + 2 \beta q^{9} + ( - \beta - 1) q^{10} + (3 \beta + 5) q^{12} + (2 \beta + 1) q^{13} + (2 \beta + 4) q^{14} + ( - \beta - 1) q^{15} + 3 q^{16} + ( - 2 \beta + 2) q^{17} + (2 \beta + 4) q^{18} - 6 q^{19} + ( - 2 \beta - 1) q^{20} + (2 \beta + 4) q^{21} + (4 \beta - 2) q^{23} + (4 \beta + 5) q^{24} - 4 q^{25} + (3 \beta + 5) q^{26} + ( - \beta + 1) q^{27} + (2 \beta + 8) q^{28} - q^{29} + ( - 2 \beta - 3) q^{30} + (5 \beta + 3) q^{31} + (\beta - 3) q^{32} - 2 q^{34} - 2 \beta q^{35} + (2 \beta + 8) q^{36} - 4 q^{37} + ( - 6 \beta - 6) q^{38} + (3 \beta + 5) q^{39} + ( - \beta - 3) q^{40} + (6 \beta - 4) q^{41} + (6 \beta + 8) q^{42} + (\beta - 5) q^{43} - 2 \beta q^{45} + (2 \beta + 6) q^{46} + ( - 3 \beta + 1) q^{47} + (3 \beta + 3) q^{48} + q^{49} + ( - 4 \beta - 4) q^{50} - 2 q^{51} + (4 \beta + 9) q^{52} + (6 \beta + 1) q^{53} - q^{54} + (6 \beta + 4) q^{56} + ( - 6 \beta - 6) q^{57} + ( - \beta - 1) q^{58} + ( - 4 \beta + 2) q^{59} + ( - 3 \beta - 5) q^{60} + (2 \beta + 2) q^{61} + (8 \beta + 13) q^{62} + 8 q^{63} + ( - 2 \beta - 7) q^{64} + ( - 2 \beta - 1) q^{65} + 4 \beta q^{67} + (2 \beta - 6) q^{68} + (2 \beta + 6) q^{69} + ( - 2 \beta - 4) q^{70} + ( - 2 \beta - 6) q^{71} + (6 \beta + 4) q^{72} - 4 q^{73} + ( - 4 \beta - 4) q^{74} + ( - 4 \beta - 4) q^{75} + ( - 12 \beta - 6) q^{76} + (8 \beta + 11) q^{78} + (\beta + 1) q^{79} - 3 q^{80} + ( - 6 \beta - 1) q^{81} + (2 \beta + 8) q^{82} + ( - 4 \beta - 2) q^{83} + (10 \beta + 12) q^{84} + (2 \beta - 2) q^{85} + ( - 4 \beta - 3) q^{86} + ( - \beta - 1) q^{87} + ( - 6 \beta - 4) q^{89} + ( - 2 \beta - 4) q^{90} + (2 \beta + 8) q^{91} + 14 q^{92} + (8 \beta + 13) q^{93} + ( - 2 \beta - 5) q^{94} + 6 q^{95} + ( - 2 \beta - 1) q^{96} + (6 \beta - 4) q^{97} + (\beta + 1) q^{98} +O(q^{100})$$ q + (b + 1) * q^2 + (b + 1) * q^3 + (2*b + 1) * q^4 - q^5 + (2*b + 3) * q^6 + 2*b * q^7 + (b + 3) * q^8 + 2*b * q^9 + (-b - 1) * q^10 + (3*b + 5) * q^12 + (2*b + 1) * q^13 + (2*b + 4) * q^14 + (-b - 1) * q^15 + 3 * q^16 + (-2*b + 2) * q^17 + (2*b + 4) * q^18 - 6 * q^19 + (-2*b - 1) * q^20 + (2*b + 4) * q^21 + (4*b - 2) * q^23 + (4*b + 5) * q^24 - 4 * q^25 + (3*b + 5) * q^26 + (-b + 1) * q^27 + (2*b + 8) * q^28 - q^29 + (-2*b - 3) * q^30 + (5*b + 3) * q^31 + (b - 3) * q^32 - 2 * q^34 - 2*b * q^35 + (2*b + 8) * q^36 - 4 * q^37 + (-6*b - 6) * q^38 + (3*b + 5) * q^39 + (-b - 3) * q^40 + (6*b - 4) * q^41 + (6*b + 8) * q^42 + (b - 5) * q^43 - 2*b * q^45 + (2*b + 6) * q^46 + (-3*b + 1) * q^47 + (3*b + 3) * q^48 + q^49 + (-4*b - 4) * q^50 - 2 * q^51 + (4*b + 9) * q^52 + (6*b + 1) * q^53 - q^54 + (6*b + 4) * q^56 + (-6*b - 6) * q^57 + (-b - 1) * q^58 + (-4*b + 2) * q^59 + (-3*b - 5) * q^60 + (2*b + 2) * q^61 + (8*b + 13) * q^62 + 8 * q^63 + (-2*b - 7) * q^64 + (-2*b - 1) * q^65 + 4*b * q^67 + (2*b - 6) * q^68 + (2*b + 6) * q^69 + (-2*b - 4) * q^70 + (-2*b - 6) * q^71 + (6*b + 4) * q^72 - 4 * q^73 + (-4*b - 4) * q^74 + (-4*b - 4) * q^75 + (-12*b - 6) * q^76 + (8*b + 11) * q^78 + (b + 1) * q^79 - 3 * q^80 + (-6*b - 1) * q^81 + (2*b + 8) * q^82 + (-4*b - 2) * q^83 + (10*b + 12) * q^84 + (2*b - 2) * q^85 + (-4*b - 3) * q^86 + (-b - 1) * q^87 + (-6*b - 4) * q^89 + (-2*b - 4) * q^90 + (2*b + 8) * q^91 + 14 * q^92 + (8*b + 13) * q^93 + (-2*b - 5) * q^94 + 6 * q^95 + (-2*b - 1) * q^96 + (6*b - 4) * q^97 + (b + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 6 q^{6} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 6 * q^6 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 6 q^{6} + 6 q^{8} - 2 q^{10} + 10 q^{12} + 2 q^{13} + 8 q^{14} - 2 q^{15} + 6 q^{16} + 4 q^{17} + 8 q^{18} - 12 q^{19} - 2 q^{20} + 8 q^{21} - 4 q^{23} + 10 q^{24} - 8 q^{25} + 10 q^{26} + 2 q^{27} + 16 q^{28} - 2 q^{29} - 6 q^{30} + 6 q^{31} - 6 q^{32} - 4 q^{34} + 16 q^{36} - 8 q^{37} - 12 q^{38} + 10 q^{39} - 6 q^{40} - 8 q^{41} + 16 q^{42} - 10 q^{43} + 12 q^{46} + 2 q^{47} + 6 q^{48} + 2 q^{49} - 8 q^{50} - 4 q^{51} + 18 q^{52} + 2 q^{53} - 2 q^{54} + 8 q^{56} - 12 q^{57} - 2 q^{58} + 4 q^{59} - 10 q^{60} + 4 q^{61} + 26 q^{62} + 16 q^{63} - 14 q^{64} - 2 q^{65} - 12 q^{68} + 12 q^{69} - 8 q^{70} - 12 q^{71} + 8 q^{72} - 8 q^{73} - 8 q^{74} - 8 q^{75} - 12 q^{76} + 22 q^{78} + 2 q^{79} - 6 q^{80} - 2 q^{81} + 16 q^{82} - 4 q^{83} + 24 q^{84} - 4 q^{85} - 6 q^{86} - 2 q^{87} - 8 q^{89} - 8 q^{90} + 16 q^{91} + 28 q^{92} + 26 q^{93} - 10 q^{94} + 12 q^{95} - 2 q^{96} - 8 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 6 * q^6 + 6 * q^8 - 2 * q^10 + 10 * q^12 + 2 * q^13 + 8 * q^14 - 2 * q^15 + 6 * q^16 + 4 * q^17 + 8 * q^18 - 12 * q^19 - 2 * q^20 + 8 * q^21 - 4 * q^23 + 10 * q^24 - 8 * q^25 + 10 * q^26 + 2 * q^27 + 16 * q^28 - 2 * q^29 - 6 * q^30 + 6 * q^31 - 6 * q^32 - 4 * q^34 + 16 * q^36 - 8 * q^37 - 12 * q^38 + 10 * q^39 - 6 * q^40 - 8 * q^41 + 16 * q^42 - 10 * q^43 + 12 * q^46 + 2 * q^47 + 6 * q^48 + 2 * q^49 - 8 * q^50 - 4 * q^51 + 18 * q^52 + 2 * q^53 - 2 * q^54 + 8 * q^56 - 12 * q^57 - 2 * q^58 + 4 * q^59 - 10 * q^60 + 4 * q^61 + 26 * q^62 + 16 * q^63 - 14 * q^64 - 2 * q^65 - 12 * q^68 + 12 * q^69 - 8 * q^70 - 12 * q^71 + 8 * q^72 - 8 * q^73 - 8 * q^74 - 8 * q^75 - 12 * q^76 + 22 * q^78 + 2 * q^79 - 6 * q^80 - 2 * q^81 + 16 * q^82 - 4 * q^83 + 24 * q^84 - 4 * q^85 - 6 * q^86 - 2 * q^87 - 8 * q^89 - 8 * q^90 + 16 * q^91 + 28 * q^92 + 26 * q^93 - 10 * q^94 + 12 * q^95 - 2 * q^96 - 8 * q^97 + 2 * q^98

## 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 −0.414214 −1.82843 −1.00000 0.171573 −2.82843 1.58579 −2.82843 0.414214
1.2 2.41421 2.41421 3.82843 −1.00000 5.82843 2.82843 4.41421 2.82843 −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
$$11$$ $$-1$$
$$29$$ $$1$$

## 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 3509.2.a.j 2
11.b odd 2 1 29.2.a.a 2
33.d even 2 1 261.2.a.d 2
44.c even 2 1 464.2.a.h 2
55.d odd 2 1 725.2.a.b 2
55.e even 4 2 725.2.b.b 4
77.b even 2 1 1421.2.a.j 2
88.b odd 2 1 1856.2.a.r 2
88.g even 2 1 1856.2.a.w 2
132.d odd 2 1 4176.2.a.bq 2
143.d odd 2 1 4901.2.a.g 2
165.d even 2 1 6525.2.a.o 2
187.b odd 2 1 8381.2.a.e 2
319.d odd 2 1 841.2.a.d 2
319.f even 4 2 841.2.b.a 4
319.l odd 14 6 841.2.d.f 12
319.m odd 14 6 841.2.d.j 12
319.q even 28 12 841.2.e.k 24
957.b even 2 1 7569.2.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 11.b odd 2 1
261.2.a.d 2 33.d even 2 1
464.2.a.h 2 44.c even 2 1
725.2.a.b 2 55.d odd 2 1
725.2.b.b 4 55.e even 4 2
841.2.a.d 2 319.d odd 2 1
841.2.b.a 4 319.f even 4 2
841.2.d.f 12 319.l odd 14 6
841.2.d.j 12 319.m odd 14 6
841.2.e.k 24 319.q even 28 12
1421.2.a.j 2 77.b even 2 1
1856.2.a.r 2 88.b odd 2 1
1856.2.a.w 2 88.g even 2 1
3509.2.a.j 2 1.a even 1 1 trivial
4176.2.a.bq 2 132.d odd 2 1
4901.2.a.g 2 143.d odd 2 1
6525.2.a.o 2 165.d even 2 1
7569.2.a.c 2 957.b even 2 1
8381.2.a.e 2 187.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3509))$$:

 $$T_{2}^{2} - 2T_{2} - 1$$ T2^2 - 2*T2 - 1 $$T_{3}^{2} - 2T_{3} - 1$$ T3^2 - 2*T3 - 1 $$T_{7}^{2} - 8$$ T7^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2} - 2T - 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 2T - 7$$
$17$ $$T^{2} - 4T - 4$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} + 4T - 28$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2} - 6T - 41$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} + 8T - 56$$
$43$ $$T^{2} + 10T + 23$$
$47$ $$T^{2} - 2T - 17$$
$53$ $$T^{2} - 2T - 71$$
$59$ $$T^{2} - 4T - 28$$
$61$ $$T^{2} - 4T - 4$$
$67$ $$T^{2} - 32$$
$71$ $$T^{2} + 12T + 28$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} - 2T - 1$$
$83$ $$T^{2} + 4T - 28$$
$89$ $$T^{2} + 8T - 56$$
$97$ $$T^{2} + 8T - 56$$