# Properties

 Label 369.2.a.e Level 369 Weight 2 Character orbit 369.a Self dual yes Analytic conductor 2.946 Analytic rank 0 Dimension 3 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$369 = 3^{2} \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 369.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.94647983459$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 123) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} + 2 q^{10} + ( 1 + \beta_{1} ) q^{11} + ( 3 - \beta_{1} + \beta_{2} ) q^{13} + ( 4 + 2 \beta_{2} ) q^{14} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{16} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} ) q^{19} + ( 2 - 2 \beta_{2} ) q^{20} + ( -3 - \beta_{1} - \beta_{2} ) q^{22} + ( 3 + \beta_{1} - \beta_{2} ) q^{23} + ( 1 + 2 \beta_{1} - 4 \beta_{2} ) q^{25} + ( 2 - 4 \beta_{1} ) q^{26} + ( -4 - 4 \beta_{1} ) q^{28} + ( 1 + 3 \beta_{1} ) q^{29} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{31} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{32} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{34} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{35} + ( 6 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( 2 - 2 \beta_{1} ) q^{38} + ( -2 + 2 \beta_{2} ) q^{40} - q^{41} + ( 4 - 2 \beta_{1} + 5 \beta_{2} ) q^{43} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( -2 - 2 \beta_{1} ) q^{46} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{47} + ( 3 + 2 \beta_{1} ) q^{49} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{50} + ( 6 + 2 \beta_{2} ) q^{52} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -3 - \beta_{1} + \beta_{2} ) q^{55} + ( 4 + 4 \beta_{1} ) q^{56} + ( -9 - \beta_{1} - 3 \beta_{2} ) q^{58} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{61} + ( -13 + \beta_{1} - 5 \beta_{2} ) q^{62} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{64} + ( 2 - 2 \beta_{1} ) q^{65} + ( 2 + 6 \beta_{1} - 4 \beta_{2} ) q^{67} + ( -8 - 2 \beta_{1} ) q^{68} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{70} + ( 11 - \beta_{1} ) q^{71} + ( 2 - 2 \beta_{1} - 3 \beta_{2} ) q^{73} + ( -5 - 5 \beta_{1} - \beta_{2} ) q^{74} + 4 q^{76} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{77} + ( -8 + 4 \beta_{1} - 2 \beta_{2} ) q^{79} + ( -6 + 2 \beta_{2} ) q^{80} + \beta_{1} q^{82} + ( 5 - \beta_{1} + 3 \beta_{2} ) q^{83} + ( -7 - \beta_{1} + 5 \beta_{2} ) q^{85} + ( 1 - 9 \beta_{1} - 3 \beta_{2} ) q^{86} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{88} + ( -6 + 4 \beta_{1} ) q^{89} + ( 2 - 6 \beta_{1} ) q^{91} + 4 \beta_{2} q^{92} + ( 5 + 3 \beta_{1} + 3 \beta_{2} ) q^{94} + ( 4 - 2 \beta_{2} ) q^{95} + ( -3 - 3 \beta_{1} + \beta_{2} ) q^{97} + ( -6 - 3 \beta_{1} - 2 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} + 3q^{4} - 4q^{5} + 2q^{7} - 3q^{8} + O(q^{10})$$ $$3q - q^{2} + 3q^{4} - 4q^{5} + 2q^{7} - 3q^{8} + 6q^{10} + 4q^{11} + 8q^{13} + 12q^{14} - q^{16} - 2q^{17} + 2q^{19} + 6q^{20} - 10q^{22} + 10q^{23} + 5q^{25} + 2q^{26} - 16q^{28} + 6q^{29} - 2q^{31} - 7q^{32} - 8q^{35} + 20q^{37} + 4q^{38} - 6q^{40} - 3q^{41} + 10q^{43} + 8q^{44} - 8q^{46} - 4q^{47} + 11q^{49} - 3q^{50} + 18q^{52} - 14q^{53} - 10q^{55} + 16q^{56} - 28q^{58} + 8q^{59} - 8q^{61} - 38q^{62} - 17q^{64} + 4q^{65} + 12q^{67} - 26q^{68} + 4q^{70} + 32q^{71} + 4q^{73} - 20q^{74} + 12q^{76} - 10q^{77} - 20q^{79} - 18q^{80} + q^{82} + 14q^{83} - 22q^{85} - 6q^{86} - 8q^{88} - 14q^{89} + 18q^{94} + 12q^{95} - 12q^{97} - 21q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 2.34292 0.470683 −1.81361
−2.34292 0 3.48929 −0.853635 0 −3.83221 −3.48929 0 2.00000
1.2 −0.470683 0 −1.77846 −4.24914 0 3.30777 1.77846 0 2.00000
1.3 1.81361 0 1.28917 1.10278 0 2.52444 −1.28917 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 369.2.a.e 3
3.b odd 2 1 123.2.a.d 3
4.b odd 2 1 5904.2.a.bd 3
5.b even 2 1 9225.2.a.bx 3
12.b even 2 1 1968.2.a.w 3
15.d odd 2 1 3075.2.a.t 3
21.c even 2 1 6027.2.a.s 3
24.f even 2 1 7872.2.a.bs 3
24.h odd 2 1 7872.2.a.bx 3
123.b odd 2 1 5043.2.a.n 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.2.a.d 3 3.b odd 2 1
369.2.a.e 3 1.a even 1 1 trivial
1968.2.a.w 3 12.b even 2 1
3075.2.a.t 3 15.d odd 2 1
5043.2.a.n 3 123.b odd 2 1
5904.2.a.bd 3 4.b odd 2 1
6027.2.a.s 3 21.c even 2 1
7872.2.a.bs 3 24.f even 2 1
7872.2.a.bx 3 24.h odd 2 1
9225.2.a.bx 3 5.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$41$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(369))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 2 T^{2} + 2 T^{3} + 4 T^{4} + 4 T^{5} + 8 T^{6}$$
$3$ 
$5$ $$1 + 4 T + 13 T^{2} + 36 T^{3} + 65 T^{4} + 100 T^{5} + 125 T^{6}$$
$7$ $$1 - 2 T + 7 T^{2} + 4 T^{3} + 49 T^{4} - 98 T^{5} + 343 T^{6}$$
$11$ $$1 - 4 T + 34 T^{2} - 84 T^{3} + 374 T^{4} - 484 T^{5} + 1331 T^{6}$$
$13$ $$1 - 8 T + 53 T^{2} - 204 T^{3} + 689 T^{4} - 1352 T^{5} + 2197 T^{6}$$
$17$ $$1 + 2 T + 28 T^{2} + 6 T^{3} + 476 T^{4} + 578 T^{5} + 4913 T^{6}$$
$19$ $$1 - 2 T + 51 T^{2} - 68 T^{3} + 969 T^{4} - 722 T^{5} + 6859 T^{6}$$
$23$ $$1 - 10 T + 95 T^{2} - 476 T^{3} + 2185 T^{4} - 5290 T^{5} + 12167 T^{6}$$
$29$ $$1 - 6 T + 60 T^{2} - 262 T^{3} + 1740 T^{4} - 5046 T^{5} + 24389 T^{6}$$
$31$ $$1 + 2 T + 2 T^{2} - 132 T^{3} + 62 T^{4} + 1922 T^{5} + 29791 T^{6}$$
$37$ $$1 - 20 T + 228 T^{2} - 1646 T^{3} + 8436 T^{4} - 27380 T^{5} + 50653 T^{6}$$
$41$ $$( 1 + T )^{3}$$
$43$ $$1 - 10 T + 10 T^{2} + 296 T^{3} + 430 T^{4} - 18490 T^{5} + 79507 T^{6}$$
$47$ $$1 + 4 T + 106 T^{2} + 384 T^{3} + 4982 T^{4} + 8836 T^{5} + 103823 T^{6}$$
$53$ $$1 + 14 T + 159 T^{2} + 1452 T^{3} + 8427 T^{4} + 39326 T^{5} + 148877 T^{6}$$
$59$ $$1 - 8 T + 137 T^{2} - 976 T^{3} + 8083 T^{4} - 27848 T^{5} + 205379 T^{6}$$
$61$ $$1 + 8 T + 188 T^{2} + 930 T^{3} + 11468 T^{4} + 29768 T^{5} + 226981 T^{6}$$
$67$ $$1 - 12 T + 77 T^{2} - 632 T^{3} + 5159 T^{4} - 53868 T^{5} + 300763 T^{6}$$
$71$ $$1 - 32 T + 550 T^{2} - 5712 T^{3} + 39050 T^{4} - 161312 T^{5} + 357911 T^{6}$$
$73$ $$1 - 4 T + 120 T^{2} - 130 T^{3} + 8760 T^{4} - 21316 T^{5} + 389017 T^{6}$$
$79$ $$1 + 20 T + 305 T^{2} + 3192 T^{3} + 24095 T^{4} + 124820 T^{5} + 493039 T^{6}$$
$83$ $$1 - 14 T + 259 T^{2} - 2028 T^{3} + 21497 T^{4} - 96446 T^{5} + 571787 T^{6}$$
$89$ $$1 + 14 T + 263 T^{2} + 2308 T^{3} + 23407 T^{4} + 110894 T^{5} + 704969 T^{6}$$
$97$ $$1 + 12 T + 305 T^{2} + 2180 T^{3} + 29585 T^{4} + 112908 T^{5} + 912673 T^{6}$$