# Properties

 Label 405.4.a.i Level $405$ Weight $4$ Character orbit 405.a Self dual yes Analytic conductor $23.896$ Analytic rank $0$ Dimension $3$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.7032.1 Defining polynomial: $$x^{3} - x^{2} - 14x + 18$$ x^3 - x^2 - 14*x + 18 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} - 5 q^{5} + (3 \beta_{2} + 5 \beta_1 - 10) q^{7} + (\beta_{2} - 3 \beta_1 - 8) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 - b1 + 2) * q^4 - 5 * q^5 + (3*b2 + 5*b1 - 10) * q^7 + (b2 - 3*b1 - 8) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} - 5 q^{5} + (3 \beta_{2} + 5 \beta_1 - 10) q^{7} + (\beta_{2} - 3 \beta_1 - 8) q^{8} - 5 \beta_1 q^{10} + (5 \beta_{2} + 7 \beta_1 + 17) q^{11} + ( - 13 \beta_{2} + 13 \beta_1 - 20) q^{13} + (11 \beta_{2} - 9 \beta_1 + 56) q^{14} + ( - 9 \beta_{2} + 5 \beta_1 - 44) q^{16} + (8 \beta_{2} - 8 \beta_1 + 14) q^{17} + (14 \beta_{2} + 10 \beta_1 - 5) q^{19} + ( - 5 \beta_{2} + 5 \beta_1 - 10) q^{20} + (17 \beta_{2} + 20 \beta_1 + 80) q^{22} + ( - 19 \beta_{2} + 15 \beta_1 - 22) q^{23} + 25 q^{25} + ( - 13 \beta_{2} - 59 \beta_1 + 104) q^{26} + ( - 11 \beta_{2} + 47 \beta_1 + 12) q^{28} + ( - 13 \beta_{2} + 65 \beta_1 + 95) q^{29} + ( - \beta_{2} + 5 \beta_1 + 211) q^{31} + ( - 21 \beta_{2} - 43 \beta_1 + 96) q^{32} + (8 \beta_{2} + 38 \beta_1 - 64) q^{34} + ( - 15 \beta_{2} - 25 \beta_1 + 50) q^{35} + ( - 2 \beta_{2} + 54 \beta_1 - 156) q^{37} + (38 \beta_{2} + 13 \beta_1 + 128) q^{38} + ( - 5 \beta_{2} + 15 \beta_1 + 40) q^{40} + (62 \beta_{2} + 2 \beta_1 + 59) q^{41} + ( - 8 \beta_{2} + 28 \beta_1 - 288) q^{43} + (14 \beta_{2} + 38 \beta_1 + 98) q^{44} + ( - 23 \beta_{2} - 75 \beta_1 + 112) q^{46} + ( - 31 \beta_{2} + 67 \beta_1 + 56) q^{47} + (7 \beta_{2} - 11 \beta_1 + 301) q^{49} + 25 \beta_1 q^{50} + (19 \beta_{2} + 33 \beta_1 - 456) q^{52} + (21 \beta_{2} - 53 \beta_1 + 186) q^{53} + ( - 25 \beta_{2} - 35 \beta_1 - 85) q^{55} + ( - 63 \beta_{2} + 15 \beta_1) q^{56} + (39 \beta_{2} + 4 \beta_1 + 624) q^{58} + ( - 90 \beta_{2} + 58 \beta_1 + 159) q^{59} + (58 \beta_{2} - 122 \beta_1 + 6) q^{61} + (3 \beta_{2} + 204 \beta_1 + 48) q^{62} + ( - 13 \beta_{2} + 57 \beta_1 - 120) q^{64} + (65 \beta_{2} - 65 \beta_1 + 100) q^{65} + ( - 18 \beta_{2} - 154 \beta_1 + 38) q^{67} + ( - 10 \beta_{2} - 22 \beta_1 + 284) q^{68} + ( - 55 \beta_{2} + 45 \beta_1 - 280) q^{70} + ( - 69 \beta_{2} - 79 \beta_1 + 177) q^{71} + ( - 12 \beta_{2} - 32 \beta_1 - 226) q^{73} + (50 \beta_{2} - 214 \beta_1 + 536) q^{74} + ( - 23 \beta_{2} + 111 \beta_1 + 246) q^{76} + (98 \beta_{2} + 162 \beta_1 + 662) q^{77} + ( - 54 \beta_{2} - 82 \beta_1 - 184) q^{79} + (45 \beta_{2} - 25 \beta_1 + 220) q^{80} + (126 \beta_{2} + 181 \beta_1 + 144) q^{82} + ( - 30 \beta_{2} - 210 \beta_1 + 648) q^{83} + ( - 40 \beta_{2} + 40 \beta_1 - 70) q^{85} + (12 \beta_{2} - 332 \beta_1 + 264) q^{86} + ( - 70 \beta_{2} - 72 \beta_1 - 232) q^{88} + ( - 3 \beta_{2} + 39 \beta_1 - 297) q^{89} + (161 \beta_{2} - 581 \beta_1 - 216) q^{91} + (31 \beta_{2} + 21 \beta_1 - 620) q^{92} + (5 \beta_{2} - 73 \beta_1 + 608) q^{94} + ( - 70 \beta_{2} - 50 \beta_1 + 25) q^{95} + (32 \beta_{2} - 336 \beta_1 + 112) q^{97} + (3 \beta_{2} + 326 \beta_1 - 96) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 - b1 + 2) * q^4 - 5 * q^5 + (3*b2 + 5*b1 - 10) * q^7 + (b2 - 3*b1 - 8) * q^8 - 5*b1 * q^10 + (5*b2 + 7*b1 + 17) * q^11 + (-13*b2 + 13*b1 - 20) * q^13 + (11*b2 - 9*b1 + 56) * q^14 + (-9*b2 + 5*b1 - 44) * q^16 + (8*b2 - 8*b1 + 14) * q^17 + (14*b2 + 10*b1 - 5) * q^19 + (-5*b2 + 5*b1 - 10) * q^20 + (17*b2 + 20*b1 + 80) * q^22 + (-19*b2 + 15*b1 - 22) * q^23 + 25 * q^25 + (-13*b2 - 59*b1 + 104) * q^26 + (-11*b2 + 47*b1 + 12) * q^28 + (-13*b2 + 65*b1 + 95) * q^29 + (-b2 + 5*b1 + 211) * q^31 + (-21*b2 - 43*b1 + 96) * q^32 + (8*b2 + 38*b1 - 64) * q^34 + (-15*b2 - 25*b1 + 50) * q^35 + (-2*b2 + 54*b1 - 156) * q^37 + (38*b2 + 13*b1 + 128) * q^38 + (-5*b2 + 15*b1 + 40) * q^40 + (62*b2 + 2*b1 + 59) * q^41 + (-8*b2 + 28*b1 - 288) * q^43 + (14*b2 + 38*b1 + 98) * q^44 + (-23*b2 - 75*b1 + 112) * q^46 + (-31*b2 + 67*b1 + 56) * q^47 + (7*b2 - 11*b1 + 301) * q^49 + 25*b1 * q^50 + (19*b2 + 33*b1 - 456) * q^52 + (21*b2 - 53*b1 + 186) * q^53 + (-25*b2 - 35*b1 - 85) * q^55 + (-63*b2 + 15*b1) * q^56 + (39*b2 + 4*b1 + 624) * q^58 + (-90*b2 + 58*b1 + 159) * q^59 + (58*b2 - 122*b1 + 6) * q^61 + (3*b2 + 204*b1 + 48) * q^62 + (-13*b2 + 57*b1 - 120) * q^64 + (65*b2 - 65*b1 + 100) * q^65 + (-18*b2 - 154*b1 + 38) * q^67 + (-10*b2 - 22*b1 + 284) * q^68 + (-55*b2 + 45*b1 - 280) * q^70 + (-69*b2 - 79*b1 + 177) * q^71 + (-12*b2 - 32*b1 - 226) * q^73 + (50*b2 - 214*b1 + 536) * q^74 + (-23*b2 + 111*b1 + 246) * q^76 + (98*b2 + 162*b1 + 662) * q^77 + (-54*b2 - 82*b1 - 184) * q^79 + (45*b2 - 25*b1 + 220) * q^80 + (126*b2 + 181*b1 + 144) * q^82 + (-30*b2 - 210*b1 + 648) * q^83 + (-40*b2 + 40*b1 - 70) * q^85 + (12*b2 - 332*b1 + 264) * q^86 + (-70*b2 - 72*b1 - 232) * q^88 + (-3*b2 + 39*b1 - 297) * q^89 + (161*b2 - 581*b1 - 216) * q^91 + (31*b2 + 21*b1 - 620) * q^92 + (5*b2 - 73*b1 + 608) * q^94 + (-70*b2 - 50*b1 + 25) * q^95 + (32*b2 - 336*b1 + 112) * q^97 + (3*b2 + 326*b1 - 96) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8}+O(q^{10})$$ 3 * q + q^2 + 5 * q^4 - 15 * q^5 - 25 * q^7 - 27 * q^8 $$3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8} - 5 q^{10} + 58 q^{11} - 47 q^{13} + 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 25 q^{20} + 260 q^{22} - 51 q^{23} + 75 q^{25} + 253 q^{26} + 83 q^{28} + 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 125 q^{35} - 414 q^{37} + 397 q^{38} + 135 q^{40} + 179 q^{41} - 836 q^{43} + 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 25 q^{50} - 1335 q^{52} + 505 q^{53} - 290 q^{55} + 15 q^{56} + 1876 q^{58} + 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 235 q^{65} - 40 q^{67} + 830 q^{68} - 795 q^{70} + 452 q^{71} - 710 q^{73} + 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} + 635 q^{80} + 613 q^{82} + 1734 q^{83} - 170 q^{85} + 460 q^{86} - 768 q^{88} - 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 25 q^{95} + 38 q^{98}+O(q^{100})$$ 3 * q + q^2 + 5 * q^4 - 15 * q^5 - 25 * q^7 - 27 * q^8 - 5 * q^10 + 58 * q^11 - 47 * q^13 + 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 25 * q^20 + 260 * q^22 - 51 * q^23 + 75 * q^25 + 253 * q^26 + 83 * q^28 + 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 125 * q^35 - 414 * q^37 + 397 * q^38 + 135 * q^40 + 179 * q^41 - 836 * q^43 + 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 25 * q^50 - 1335 * q^52 + 505 * q^53 - 290 * q^55 + 15 * q^56 + 1876 * q^58 + 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 235 * q^65 - 40 * q^67 + 830 * q^68 - 795 * q^70 + 452 * q^71 - 710 * q^73 + 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 + 635 * q^80 + 613 * q^82 + 1734 * q^83 - 170 * q^85 + 460 * q^86 - 768 * q^88 - 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 25 * q^95 + 38 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 14x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 10$$ v^2 + v - 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 10$$ b2 - b1 + 10

## 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
 −3.85028 1.32681 3.52348
−3.85028 0 6.82469 −5.00000 0 −26.3282 4.52526 0 19.2514
1.2 1.32681 0 −6.23958 −5.00000 0 −24.1043 −18.8932 0 −6.63404
1.3 3.52348 0 4.41489 −5.00000 0 25.4325 −12.6321 0 −17.6174
 $$n$$: 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.i yes 3
3.b odd 2 1 405.4.a.g 3
5.b even 2 1 2025.4.a.p 3
9.c even 3 2 405.4.e.s 6
9.d odd 6 2 405.4.e.u 6
15.d odd 2 1 2025.4.a.r 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - T_{2}^{2} - 14T_{2} + 18$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(405))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 14 T + 18$$
$3$ $$T^{3}$$
$5$ $$(T + 5)^{3}$$
$7$ $$T^{3} + 25 T^{2} - 648 T - 16140$$
$11$ $$T^{3} - 58 T^{2} - 911 T - 3000$$
$13$ $$T^{3} + 47 T^{2} - 7432 T - 370352$$
$17$ $$T^{3} - 34 T^{2} - 2708 T + 90984$$
$19$ $$T^{3} + 5 T^{2} - 10777 T - 299645$$
$23$ $$T^{3} + 51 T^{2} - 15240 T - 1041156$$
$29$ $$T^{3} - 350 T^{2} + \cdots + 11237760$$
$31$ $$T^{3} - 638 T^{2} + 135321 T - 9539064$$
$37$ $$T^{3} + 414 T^{2} + 16032 T - 577760$$
$41$ $$T^{3} - 179 T^{2} + \cdots + 17799627$$
$43$ $$T^{3} + 836 T^{2} + \cdots + 18692992$$
$47$ $$T^{3} - 235 T^{2} - 69680 T + 9005376$$
$53$ $$T^{3} - 505 T^{2} + 35128 T + 1500684$$
$59$ $$T^{3} - 535 T^{2} + \cdots - 22317657$$
$61$ $$T^{3} + 104 T^{2} + \cdots - 23542832$$
$67$ $$T^{3} + 40 T^{2} - 375180 T - 15716208$$
$71$ $$T^{3} - 452 T^{2} + \cdots + 116183454$$
$73$ $$T^{3} + 710 T^{2} + 144236 T + 8707528$$
$79$ $$T^{3} + 634 T^{2} - 120288 T + 5053056$$
$83$ $$T^{3} - 1734 T^{2} + \cdots + 222334848$$
$89$ $$T^{3} + 852 T^{2} + \cdots + 17926434$$
$97$ $$T^{3} - 1575168 T - 703275008$$