# Properties

 Label 405.4.a.d 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: $$x^{2} - x - 8$$ 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.

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

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

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