# Properties

 Label 405.4.a.e 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: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + \beta q^{4} -5 q^{5} + ( 5 - \beta ) q^{7} + ( 8 - 7 \beta ) q^{8} +O(q^{10})$$ $$q + \beta q^{2} + \beta q^{4} -5 q^{5} + ( 5 - \beta ) q^{7} + ( 8 - 7 \beta ) q^{8} -5 \beta q^{10} + ( 16 + 5 \beta ) q^{11} + ( -60 + 8 \beta ) q^{13} + ( -8 + 4 \beta ) q^{14} + ( -56 - 7 \beta ) q^{16} + ( -26 - 25 \beta ) q^{17} + ( 30 - 25 \beta ) q^{19} -5 \beta q^{20} + ( 40 + 21 \beta ) q^{22} + ( 145 - 23 \beta ) q^{23} + 25 q^{25} + ( 64 - 52 \beta ) q^{26} + ( -8 + 4 \beta ) q^{28} + ( -143 - 39 \beta ) q^{29} -6 q^{31} + ( -120 - 7 \beta ) q^{32} + ( -200 - 51 \beta ) q^{34} + ( -25 + 5 \beta ) q^{35} + ( -314 - 10 \beta ) q^{37} + ( -200 + 5 \beta ) q^{38} + ( -40 + 35 \beta ) q^{40} + ( -89 - 60 \beta ) q^{41} + ( -92 + 87 \beta ) q^{43} + ( 40 + 21 \beta ) q^{44} + ( -184 + 122 \beta ) q^{46} + ( 445 + 11 \beta ) q^{47} + ( -310 - 9 \beta ) q^{49} + 25 \beta q^{50} + ( 64 - 52 \beta ) q^{52} + ( -162 + 100 \beta ) q^{53} + ( -80 - 25 \beta ) q^{55} + ( 96 - 36 \beta ) q^{56} + ( -312 - 182 \beta ) q^{58} + ( 18 + 49 \beta ) q^{59} + ( -221 + 195 \beta ) q^{61} -6 \beta q^{62} + ( 392 - 71 \beta ) q^{64} + ( 300 - 40 \beta ) q^{65} + ( -211 - 184 \beta ) q^{67} + ( -200 - 51 \beta ) q^{68} + ( 40 - 20 \beta ) q^{70} + ( -252 + 110 \beta ) q^{71} + ( -508 + 205 \beta ) q^{73} + ( -80 - 324 \beta ) q^{74} + ( -200 + 5 \beta ) q^{76} + ( 40 + 4 \beta ) q^{77} + ( -382 - 76 \beta ) q^{79} + ( 280 + 35 \beta ) q^{80} + ( -480 - 149 \beta ) q^{82} + ( -33 + 453 \beta ) q^{83} + ( 130 + 125 \beta ) q^{85} + ( 696 - 5 \beta ) q^{86} + ( -152 - 107 \beta ) q^{88} + ( 375 + 315 \beta ) q^{89} + ( -364 + 92 \beta ) q^{91} + ( -184 + 122 \beta ) q^{92} + ( 88 + 456 \beta ) q^{94} + ( -150 + 125 \beta ) q^{95} + ( -450 - 131 \beta ) q^{97} + ( -72 - 319 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} - 10 q^{5} + 9 q^{7} + 9 q^{8} + O(q^{10})$$ $$2 q + q^{2} + q^{4} - 10 q^{5} + 9 q^{7} + 9 q^{8} - 5 q^{10} + 37 q^{11} - 112 q^{13} - 12 q^{14} - 119 q^{16} - 77 q^{17} + 35 q^{19} - 5 q^{20} + 101 q^{22} + 267 q^{23} + 50 q^{25} + 76 q^{26} - 12 q^{28} - 325 q^{29} - 12 q^{31} - 247 q^{32} - 451 q^{34} - 45 q^{35} - 638 q^{37} - 395 q^{38} - 45 q^{40} - 238 q^{41} - 97 q^{43} + 101 q^{44} - 246 q^{46} + 901 q^{47} - 629 q^{49} + 25 q^{50} + 76 q^{52} - 224 q^{53} - 185 q^{55} + 156 q^{56} - 806 q^{58} + 85 q^{59} - 247 q^{61} - 6 q^{62} + 713 q^{64} + 560 q^{65} - 606 q^{67} - 451 q^{68} + 60 q^{70} - 394 q^{71} - 811 q^{73} - 484 q^{74} - 395 q^{76} + 84 q^{77} - 840 q^{79} + 595 q^{80} - 1109 q^{82} + 387 q^{83} + 385 q^{85} + 1387 q^{86} - 411 q^{88} + 1065 q^{89} - 636 q^{91} - 246 q^{92} + 632 q^{94} - 175 q^{95} - 1031 q^{97} - 463 q^{98} + O(q^{100})$$

## 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
 −2.37228 3.37228
−2.37228 0 −2.37228 −5.00000 0 7.37228 24.6060 0 11.8614
1.2 3.37228 0 3.37228 −5.00000 0 1.62772 −15.6060 0 −16.8614
 $$n$$: 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.e 2
3.b odd 2 1 405.4.a.d 2
5.b even 2 1 2025.4.a.j 2
9.c even 3 2 135.4.e.a 4
9.d odd 6 2 45.4.e.a 4
15.d odd 2 1 2025.4.a.l 2
45.h odd 6 2 225.4.e.a 4
45.l even 12 4 225.4.k.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.a 4 9.d odd 6 2
135.4.e.a 4 9.c even 3 2
225.4.e.a 4 45.h odd 6 2
225.4.k.a 8 45.l even 12 4
405.4.a.d 2 3.b odd 2 1
405.4.a.e 2 1.a even 1 1 trivial
2025.4.a.j 2 5.b even 2 1
2025.4.a.l 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 8$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(405))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-8 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 5 + T )^{2}$$
$7$ $$12 - 9 T + T^{2}$$
$11$ $$136 - 37 T + T^{2}$$
$13$ $$2608 + 112 T + T^{2}$$
$17$ $$-3674 + 77 T + T^{2}$$
$19$ $$-4850 - 35 T + T^{2}$$
$23$ $$13458 - 267 T + T^{2}$$
$29$ $$13858 + 325 T + T^{2}$$
$31$ $$( 6 + T )^{2}$$
$37$ $$100936 + 638 T + T^{2}$$
$41$ $$-15539 + 238 T + T^{2}$$
$43$ $$-60092 + 97 T + T^{2}$$
$47$ $$201952 - 901 T + T^{2}$$
$53$ $$-69956 + 224 T + T^{2}$$
$59$ $$-18002 - 85 T + T^{2}$$
$61$ $$-298454 + 247 T + T^{2}$$
$67$ $$-187503 + 606 T + T^{2}$$
$71$ $$-61016 + 394 T + T^{2}$$
$73$ $$-182276 + 811 T + T^{2}$$
$79$ $$128748 + 840 T + T^{2}$$
$83$ $$-1655532 - 387 T + T^{2}$$
$89$ $$-535050 - 1065 T + T^{2}$$
$97$ $$124162 + 1031 T + T^{2}$$