# Properties

 Label 405.4.a.f Level $405$ Weight $4$ Character orbit 405.a Self dual yes Analytic conductor $23.896$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.8957735523$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + ( -4 + 2 \beta ) q^{4} + 5 q^{5} + 4 \beta q^{7} + ( -6 - 10 \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + ( -4 + 2 \beta ) q^{4} + 5 q^{5} + 4 \beta q^{7} + ( -6 - 10 \beta ) q^{8} + ( 5 + 5 \beta ) q^{10} + ( -11 - 28 \beta ) q^{11} + ( -32 + 4 \beta ) q^{13} + ( 12 + 4 \beta ) q^{14} + ( -4 - 32 \beta ) q^{16} + ( 16 + 56 \beta ) q^{17} + ( -5 - 68 \beta ) q^{19} + ( -20 + 10 \beta ) q^{20} + ( -95 - 39 \beta ) q^{22} + ( -42 - 32 \beta ) q^{23} + 25 q^{25} + ( -20 - 28 \beta ) q^{26} + ( 24 - 16 \beta ) q^{28} + ( -85 - 24 \beta ) q^{29} + ( -129 + 12 \beta ) q^{31} + ( -52 + 44 \beta ) q^{32} + ( 184 + 72 \beta ) q^{34} + 20 \beta q^{35} + ( 38 + 148 \beta ) q^{37} + ( -209 - 73 \beta ) q^{38} + ( -30 - 50 \beta ) q^{40} + ( -289 + 48 \beta ) q^{41} + ( 190 - 156 \beta ) q^{43} + ( -124 + 90 \beta ) q^{44} + ( -138 - 74 \beta ) q^{46} + ( -242 - 16 \beta ) q^{47} -295 q^{49} + ( 25 + 25 \beta ) q^{50} + ( 152 - 80 \beta ) q^{52} + ( -272 + 100 \beta ) q^{53} + ( -55 - 140 \beta ) q^{55} + ( -120 - 24 \beta ) q^{56} + ( -157 - 109 \beta ) q^{58} + ( -353 + 28 \beta ) q^{59} + ( 334 + 192 \beta ) q^{61} + ( -93 - 117 \beta ) q^{62} + ( 112 + 248 \beta ) q^{64} + ( -160 + 20 \beta ) q^{65} + ( -726 + 52 \beta ) q^{67} + ( 272 - 192 \beta ) q^{68} + ( 60 + 20 \beta ) q^{70} + ( -487 - 196 \beta ) q^{71} + ( 592 + 92 \beta ) q^{73} + ( 482 + 186 \beta ) q^{74} + ( -388 + 262 \beta ) q^{76} + ( -336 - 44 \beta ) q^{77} + ( 204 - 104 \beta ) q^{79} + ( -20 - 160 \beta ) q^{80} + ( -145 - 241 \beta ) q^{82} + ( -222 + 600 \beta ) q^{83} + ( 80 + 280 \beta ) q^{85} + ( -278 + 34 \beta ) q^{86} + ( 906 + 278 \beta ) q^{88} -513 q^{89} + ( 48 - 128 \beta ) q^{91} + ( -24 + 44 \beta ) q^{92} + ( -290 - 258 \beta ) q^{94} + ( -25 - 340 \beta ) q^{95} + ( -334 + 440 \beta ) q^{97} + ( -295 - 295 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 8 q^{4} + 10 q^{5} - 12 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} - 8 q^{4} + 10 q^{5} - 12 q^{8} + 10 q^{10} - 22 q^{11} - 64 q^{13} + 24 q^{14} - 8 q^{16} + 32 q^{17} - 10 q^{19} - 40 q^{20} - 190 q^{22} - 84 q^{23} + 50 q^{25} - 40 q^{26} + 48 q^{28} - 170 q^{29} - 258 q^{31} - 104 q^{32} + 368 q^{34} + 76 q^{37} - 418 q^{38} - 60 q^{40} - 578 q^{41} + 380 q^{43} - 248 q^{44} - 276 q^{46} - 484 q^{47} - 590 q^{49} + 50 q^{50} + 304 q^{52} - 544 q^{53} - 110 q^{55} - 240 q^{56} - 314 q^{58} - 706 q^{59} + 668 q^{61} - 186 q^{62} + 224 q^{64} - 320 q^{65} - 1452 q^{67} + 544 q^{68} + 120 q^{70} - 974 q^{71} + 1184 q^{73} + 964 q^{74} - 776 q^{76} - 672 q^{77} + 408 q^{79} - 40 q^{80} - 290 q^{82} - 444 q^{83} + 160 q^{85} - 556 q^{86} + 1812 q^{88} - 1026 q^{89} + 96 q^{91} - 48 q^{92} - 580 q^{94} - 50 q^{95} - 668 q^{97} - 590 q^{98} + O(q^{100})$$

## 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
 −1.73205 1.73205
−0.732051 0 −7.46410 5.00000 0 −6.92820 11.3205 0 −3.66025
1.2 2.73205 0 −0.535898 5.00000 0 6.92820 −23.3205 0 13.6603
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$

## 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 405.4.a.f yes 2
3.b odd 2 1 405.4.a.c 2
5.b even 2 1 2025.4.a.i 2
9.c even 3 2 405.4.e.o 4
9.d odd 6 2 405.4.e.p 4
15.d odd 2 1 2025.4.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.c 2 3.b odd 2 1
405.4.a.f yes 2 1.a even 1 1 trivial
405.4.e.o 4 9.c even 3 2
405.4.e.p 4 9.d odd 6 2
2025.4.a.i 2 5.b even 2 1
2025.4.a.m 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2 T_{2} - 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(405))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 - 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -5 + T )^{2}$$
$7$ $$-48 + T^{2}$$
$11$ $$-2231 + 22 T + T^{2}$$
$13$ $$976 + 64 T + T^{2}$$
$17$ $$-9152 - 32 T + T^{2}$$
$19$ $$-13847 + 10 T + T^{2}$$
$23$ $$-1308 + 84 T + T^{2}$$
$29$ $$5497 + 170 T + T^{2}$$
$31$ $$16209 + 258 T + T^{2}$$
$37$ $$-64268 - 76 T + T^{2}$$
$41$ $$76609 + 578 T + T^{2}$$
$43$ $$-36908 - 380 T + T^{2}$$
$47$ $$57796 + 484 T + T^{2}$$
$53$ $$43984 + 544 T + T^{2}$$
$59$ $$122257 + 706 T + T^{2}$$
$61$ $$964 - 668 T + T^{2}$$
$67$ $$518964 + 1452 T + T^{2}$$
$71$ $$121921 + 974 T + T^{2}$$
$73$ $$325072 - 1184 T + T^{2}$$
$79$ $$9168 - 408 T + T^{2}$$
$83$ $$-1030716 + 444 T + T^{2}$$
$89$ $$( 513 + T )^{2}$$
$97$ $$-469244 + 668 T + T^{2}$$