Properties

 Label 405.4.a.g Level $405$ Weight $4$ Character orbit 405.a Self dual yes Analytic conductor $23.896$ Analytic rank $1$ 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: $$1$$ 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
 3.52348 1.32681 −3.85028
−3.52348 0 4.41489 5.00000 0 25.4325 12.6321 0 −17.6174
1.2 −1.32681 0 −6.23958 5.00000 0 −24.1043 18.8932 0 −6.63404
1.3 3.85028 0 6.82469 5.00000 0 −26.3282 −4.52526 0 19.2514
 $$n$$: 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.g 3
3.b odd 2 1 405.4.a.i yes 3
5.b even 2 1 2025.4.a.r 3
9.c even 3 2 405.4.e.u 6
9.d odd 6 2 405.4.e.s 6
15.d odd 2 1 2025.4.a.p 3

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

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