# Properties

 Label 361.2.a.g Level $361$ Weight $2$ Character orbit 361.a Self dual yes Analytic conductor $2.883$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 361.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.88259951297$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2) q^{6} - \beta_1 q^{7} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8} + ( - 3 \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b2 - 1) * q^3 + (b2 - 2*b1 + 1) * q^4 + (-b2 - 1) * q^5 + (-b2 + 2) * q^6 - b1 * q^7 + (-3*b2 + 2*b1 - 2) * q^8 + (-3*b2 + b1) * q^9 $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2) q^{6} - \beta_1 q^{7} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8} + ( - 3 \beta_{2} + \beta_1) q^{9} + (\beta_{2} - 2 \beta_1) q^{10} + (\beta_{2} + \beta_1) q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{12} + ( - \beta_{2} - 2 \beta_1) q^{13} + ( - \beta_{2} + \beta_1 - 2) q^{14} + (\beta_{2} - \beta_1 - 1) q^{15} + (3 \beta_{2} - 3 \beta_1 + 1) q^{16} + (\beta_{2} - \beta_1 + 2) q^{17} + (4 \beta_{2} - 4 \beta_1 - 1) q^{18} + ( - \beta_{2} + 3 \beta_1 - 1) q^{20} - q^{21} + 3 q^{22} + (2 \beta_1 - 2) q^{23} + (4 \beta_{2} - 3 \beta_1 - 2) q^{24} + (\beta_{2} + \beta_1 - 2) q^{25} + ( - \beta_{2} + \beta_1 - 5) q^{26} + (3 \beta_{2} - 3 \beta_1 - 2) q^{27} + (2 \beta_{2} - 2 \beta_1 + 3) q^{28} + (\beta_{2} - 5) q^{29} + ( - 2 \beta_{2} + \beta_1) q^{30} + (2 \beta_{2} + \beta_1 - 3) q^{31} + 3 \beta_1 q^{32} + ( - 2 \beta_{2} + \beta_1 + 3) q^{33} + ( - 2 \beta_{2} + 4 \beta_1 - 3) q^{34} + (2 \beta_1 + 1) q^{35} + ( - 2 \beta_{2} + 5 \beta_1 - 3) q^{36} + ( - 3 \beta_{2} + 2 \beta_1) q^{37} + (2 \beta_{2} - \beta_1 - 4) q^{39} + (2 \beta_{2} - \beta_1 + 6) q^{40} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{41} + ( - \beta_1 + 1) q^{42} + ( - 3 \beta_{2} + 5 \beta_1) q^{43} + ( - 2 \beta_{2} + \beta_1 - 3) q^{44} + (\beta_1 + 5) q^{45} + (2 \beta_{2} - 4 \beta_1 + 6) q^{46} + (3 \beta_{2} - \beta_1 - 2) q^{47} + ( - 5 \beta_{2} + 3 \beta_1 + 2) q^{48} + (\beta_{2} - 5) q^{49} + ( - 2 \beta_1 + 5) q^{50} + (\beta_1 - 1) q^{51} + (4 \beta_{2} - 3 \beta_1 + 6) q^{52} + (\beta_{2} - 3 \beta_1 - 2) q^{53} + ( - 6 \beta_{2} + 4 \beta_1 - 1) q^{54} + ( - 3 \beta_1 - 3) q^{55} + ( - 2 \beta_{2} + 5 \beta_1 - 1) q^{56} + ( - \beta_{2} - 4 \beta_1 + 6) q^{58} + ( - 2 \beta_1 - 7) q^{59} + (\beta_{2} - \beta_1 + 2) q^{60} + ( - 4 \beta_{2} + 4 \beta_1 + 3) q^{61} + ( - \beta_{2} - 2 \beta_1 + 7) q^{62} + ( - \beta_{2} + 3 \beta_1 + 1) q^{63} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{64} + (5 \beta_1 + 4) q^{65} + (3 \beta_{2} - 3) q^{66} + ( - 2 \beta_{2} + 6 \beta_1 + 6) q^{67} + (4 \beta_{2} - 7 \beta_1 + 5) q^{68} + ( - 2 \beta_{2} + 4) q^{69} + (2 \beta_{2} - \beta_1 + 3) q^{70} + ( - 2 \beta_1 - 10) q^{71} + ( - \beta_{2} - 2 \beta_1 + 13) q^{72} - 4 \beta_1 q^{73} + (5 \beta_{2} - 5 \beta_1 + 1) q^{74} + ( - 4 \beta_{2} + \beta_1 + 5) q^{75} + ( - \beta_{2} - \beta_1 - 3) q^{77} + ( - 3 \beta_{2} - \beta_1 + 4) q^{78} + (6 \beta_{2} - 7 \beta_1 - 3) q^{79} + ( - \beta_{2} + 3 \beta_1 - 4) q^{80} + (\beta_{2} + 5) q^{81} + (7 \beta_{2} - 11 \beta_1 + 6) q^{82} + (9 \beta_{2} - 6 \beta_1) q^{83} + ( - \beta_{2} + 2 \beta_1 - 1) q^{84} + ( - 2 \beta_{2} + \beta_1 - 3) q^{85} + (8 \beta_{2} - 8 \beta_1 + 7) q^{86} + ( - 7 \beta_{2} + \beta_1 + 7) q^{87} + (3 \beta_{2} - 6 \beta_1 - 3) q^{88} + (3 \beta_{2} - \beta_1 - 5) q^{89} + (\beta_{2} + 4 \beta_1 - 3) q^{90} + (2 \beta_{2} + \beta_1 + 5) q^{91} + ( - 6 \beta_{2} + 8 \beta_1 - 8) q^{92} + ( - 7 \beta_{2} + 2 \beta_1 + 8) q^{93} + ( - 4 \beta_{2} + 2 \beta_1 + 3) q^{94} + 3 q^{96} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{97} + ( - \beta_{2} - 4 \beta_1 + 6) q^{98} + (4 \beta_{2} - 5 \beta_1 - 6) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b2 - 1) * q^3 + (b2 - 2*b1 + 1) * q^4 + (-b2 - 1) * q^5 + (-b2 + 2) * q^6 - b1 * q^7 + (-3*b2 + 2*b1 - 2) * q^8 + (-3*b2 + b1) * q^9 + (b2 - 2*b1) * q^10 + (b2 + b1) * q^11 + (-b2 + b1 - 1) * q^12 + (-b2 - 2*b1) * q^13 + (-b2 + b1 - 2) * q^14 + (b2 - b1 - 1) * q^15 + (3*b2 - 3*b1 + 1) * q^16 + (b2 - b1 + 2) * q^17 + (4*b2 - 4*b1 - 1) * q^18 + (-b2 + 3*b1 - 1) * q^20 - q^21 + 3 * q^22 + (2*b1 - 2) * q^23 + (4*b2 - 3*b1 - 2) * q^24 + (b2 + b1 - 2) * q^25 + (-b2 + b1 - 5) * q^26 + (3*b2 - 3*b1 - 2) * q^27 + (2*b2 - 2*b1 + 3) * q^28 + (b2 - 5) * q^29 + (-2*b2 + b1) * q^30 + (2*b2 + b1 - 3) * q^31 + 3*b1 * q^32 + (-2*b2 + b1 + 3) * q^33 + (-2*b2 + 4*b1 - 3) * q^34 + (2*b1 + 1) * q^35 + (-2*b2 + 5*b1 - 3) * q^36 + (-3*b2 + 2*b1) * q^37 + (2*b2 - b1 - 4) * q^39 + (2*b2 - b1 + 6) * q^40 + (-4*b2 + 3*b1 - 4) * q^41 + (-b1 + 1) * q^42 + (-3*b2 + 5*b1) * q^43 + (-2*b2 + b1 - 3) * q^44 + (b1 + 5) * q^45 + (2*b2 - 4*b1 + 6) * q^46 + (3*b2 - b1 - 2) * q^47 + (-5*b2 + 3*b1 + 2) * q^48 + (b2 - 5) * q^49 + (-2*b1 + 5) * q^50 + (b1 - 1) * q^51 + (4*b2 - 3*b1 + 6) * q^52 + (b2 - 3*b1 - 2) * q^53 + (-6*b2 + 4*b1 - 1) * q^54 + (-3*b1 - 3) * q^55 + (-2*b2 + 5*b1 - 1) * q^56 + (-b2 - 4*b1 + 6) * q^58 + (-2*b1 - 7) * q^59 + (b2 - b1 + 2) * q^60 + (-4*b2 + 4*b1 + 3) * q^61 + (-b2 - 2*b1 + 7) * q^62 + (-b2 + 3*b1 + 1) * q^63 + (-3*b2 + 3*b1 + 4) * q^64 + (5*b1 + 4) * q^65 + (3*b2 - 3) * q^66 + (-2*b2 + 6*b1 + 6) * q^67 + (4*b2 - 7*b1 + 5) * q^68 + (-2*b2 + 4) * q^69 + (2*b2 - b1 + 3) * q^70 + (-2*b1 - 10) * q^71 + (-b2 - 2*b1 + 13) * q^72 - 4*b1 * q^73 + (5*b2 - 5*b1 + 1) * q^74 + (-4*b2 + b1 + 5) * q^75 + (-b2 - b1 - 3) * q^77 + (-3*b2 - b1 + 4) * q^78 + (6*b2 - 7*b1 - 3) * q^79 + (-b2 + 3*b1 - 4) * q^80 + (b2 + 5) * q^81 + (7*b2 - 11*b1 + 6) * q^82 + (9*b2 - 6*b1) * q^83 + (-b2 + 2*b1 - 1) * q^84 + (-2*b2 + b1 - 3) * q^85 + (8*b2 - 8*b1 + 7) * q^86 + (-7*b2 + b1 + 7) * q^87 + (3*b2 - 6*b1 - 3) * q^88 + (3*b2 - b1 - 5) * q^89 + (b2 + 4*b1 - 3) * q^90 + (2*b2 + b1 + 5) * q^91 + (-6*b2 + 8*b1 - 8) * q^92 + (-7*b2 + 2*b1 + 8) * q^93 + (-4*b2 + 2*b1 + 3) * q^94 + 3 * q^96 + (-2*b2 - 2*b1 + 5) * q^97 + (-b2 - 4*b1 + 6) * q^98 + (4*b2 - 5*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} + 6 q^{6} - 6 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 + 6 * q^6 - 6 * q^8 $$3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} + 6 q^{6} - 6 q^{8} - 3 q^{12} - 6 q^{14} - 3 q^{15} + 3 q^{16} + 6 q^{17} - 3 q^{18} - 3 q^{20} - 3 q^{21} + 9 q^{22} - 6 q^{23} - 6 q^{24} - 6 q^{25} - 15 q^{26} - 6 q^{27} + 9 q^{28} - 15 q^{29} - 9 q^{31} + 9 q^{33} - 9 q^{34} + 3 q^{35} - 9 q^{36} - 12 q^{39} + 18 q^{40} - 12 q^{41} + 3 q^{42} - 9 q^{44} + 15 q^{45} + 18 q^{46} - 6 q^{47} + 6 q^{48} - 15 q^{49} + 15 q^{50} - 3 q^{51} + 18 q^{52} - 6 q^{53} - 3 q^{54} - 9 q^{55} - 3 q^{56} + 18 q^{58} - 21 q^{59} + 6 q^{60} + 9 q^{61} + 21 q^{62} + 3 q^{63} + 12 q^{64} + 12 q^{65} - 9 q^{66} + 18 q^{67} + 15 q^{68} + 12 q^{69} + 9 q^{70} - 30 q^{71} + 39 q^{72} + 3 q^{74} + 15 q^{75} - 9 q^{77} + 12 q^{78} - 9 q^{79} - 12 q^{80} + 15 q^{81} + 18 q^{82} - 3 q^{84} - 9 q^{85} + 21 q^{86} + 21 q^{87} - 9 q^{88} - 15 q^{89} - 9 q^{90} + 15 q^{91} - 24 q^{92} + 24 q^{93} + 9 q^{94} + 9 q^{96} + 15 q^{97} + 18 q^{98} - 18 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 + 6 * q^6 - 6 * q^8 - 3 * q^12 - 6 * q^14 - 3 * q^15 + 3 * q^16 + 6 * q^17 - 3 * q^18 - 3 * q^20 - 3 * q^21 + 9 * q^22 - 6 * q^23 - 6 * q^24 - 6 * q^25 - 15 * q^26 - 6 * q^27 + 9 * q^28 - 15 * q^29 - 9 * q^31 + 9 * q^33 - 9 * q^34 + 3 * q^35 - 9 * q^36 - 12 * q^39 + 18 * q^40 - 12 * q^41 + 3 * q^42 - 9 * q^44 + 15 * q^45 + 18 * q^46 - 6 * q^47 + 6 * q^48 - 15 * q^49 + 15 * q^50 - 3 * q^51 + 18 * q^52 - 6 * q^53 - 3 * q^54 - 9 * q^55 - 3 * q^56 + 18 * q^58 - 21 * q^59 + 6 * q^60 + 9 * q^61 + 21 * q^62 + 3 * q^63 + 12 * q^64 + 12 * q^65 - 9 * q^66 + 18 * q^67 + 15 * q^68 + 12 * q^69 + 9 * q^70 - 30 * q^71 + 39 * q^72 + 3 * q^74 + 15 * q^75 - 9 * q^77 + 12 * q^78 - 9 * q^79 - 12 * q^80 + 15 * q^81 + 18 * q^82 - 3 * q^84 - 9 * q^85 + 21 * q^86 + 21 * q^87 - 9 * q^88 - 15 * q^89 - 9 * q^90 + 15 * q^91 - 24 * q^92 + 24 * q^93 + 9 * q^94 + 9 * q^96 + 15 * q^97 + 18 * q^98 - 18 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 −1.53209 −0.347296 1.87939
−2.53209 −0.652704 4.41147 −1.34730 1.65270 1.53209 −6.10607 −2.57398 3.41147
1.2 −1.34730 −2.87939 −0.184793 0.879385 3.87939 0.347296 2.94356 5.29086 −1.18479
1.3 0.879385 0.532089 −1.22668 −2.53209 0.467911 −1.87939 −2.83750 −2.71688 −2.22668
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$19$$ $$1$$

## 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 361.2.a.g 3
3.b odd 2 1 3249.2.a.z 3
4.b odd 2 1 5776.2.a.br 3
5.b even 2 1 9025.2.a.bd 3
19.b odd 2 1 361.2.a.h 3
19.c even 3 2 361.2.c.i 6
19.d odd 6 2 361.2.c.h 6
19.e even 9 2 19.2.e.a 6
19.e even 9 2 361.2.e.f 6
19.e even 9 2 361.2.e.g 6
19.f odd 18 2 361.2.e.a 6
19.f odd 18 2 361.2.e.b 6
19.f odd 18 2 361.2.e.h 6
57.d even 2 1 3249.2.a.s 3
57.l odd 18 2 171.2.u.c 6
76.d even 2 1 5776.2.a.bi 3
76.l odd 18 2 304.2.u.b 6
95.d odd 2 1 9025.2.a.x 3
95.p even 18 2 475.2.l.a 6
95.q odd 36 4 475.2.u.a 12
133.u even 9 2 931.2.v.b 6
133.w even 9 2 931.2.x.a 6
133.x odd 18 2 931.2.v.a 6
133.y odd 18 2 931.2.w.a 6
133.z odd 18 2 931.2.x.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 19.e even 9 2
171.2.u.c 6 57.l odd 18 2
304.2.u.b 6 76.l odd 18 2
361.2.a.g 3 1.a even 1 1 trivial
361.2.a.h 3 19.b odd 2 1
361.2.c.h 6 19.d odd 6 2
361.2.c.i 6 19.c even 3 2
361.2.e.a 6 19.f odd 18 2
361.2.e.b 6 19.f odd 18 2
361.2.e.f 6 19.e even 9 2
361.2.e.g 6 19.e even 9 2
361.2.e.h 6 19.f odd 18 2
475.2.l.a 6 95.p even 18 2
475.2.u.a 12 95.q odd 36 4
931.2.v.a 6 133.x odd 18 2
931.2.v.b 6 133.u even 9 2
931.2.w.a 6 133.y odd 18 2
931.2.x.a 6 133.w even 9 2
931.2.x.b 6 133.z odd 18 2
3249.2.a.s 3 57.d even 2 1
3249.2.a.z 3 3.b odd 2 1
5776.2.a.bi 3 76.d even 2 1
5776.2.a.br 3 4.b odd 2 1
9025.2.a.x 3 95.d odd 2 1
9025.2.a.bd 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(361))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 3T^{2} - 3$$
$3$ $$T^{3} + 3T^{2} - 1$$
$5$ $$T^{3} + 3T^{2} - 3$$
$7$ $$T^{3} - 3T + 1$$
$11$ $$T^{3} - 9T - 9$$
$13$ $$T^{3} - 21T + 37$$
$17$ $$T^{3} - 6 T^{2} + 9 T - 3$$
$19$ $$T^{3}$$
$23$ $$T^{3} + 6T^{2} - 24$$
$29$ $$T^{3} + 15 T^{2} + 72 T + 111$$
$31$ $$T^{3} + 9 T^{2} + 6 T - 53$$
$37$ $$T^{3} - 21T - 17$$
$41$ $$T^{3} + 12 T^{2} + 9 T - 111$$
$43$ $$T^{3} - 57T + 163$$
$47$ $$T^{3} + 6 T^{2} - 9 T + 3$$
$53$ $$T^{3} + 6 T^{2} - 9 T - 51$$
$59$ $$T^{3} + 21 T^{2} + 135 T + 267$$
$61$ $$T^{3} - 9 T^{2} - 21 T + 181$$
$67$ $$T^{3} - 18 T^{2} + 24 T + 424$$
$71$ $$T^{3} + 30 T^{2} + 288 T + 888$$
$73$ $$T^{3} - 48T + 64$$
$79$ $$T^{3} + 9 T^{2} - 102 T - 809$$
$83$ $$T^{3} - 189T + 459$$
$89$ $$T^{3} + 15 T^{2} + 54 T + 57$$
$97$ $$T^{3} - 15 T^{2} + 39 T + 127$$