# Properties

 Label 405.4.a.h 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: 3.3.2292.1 Defining polynomial: $$x^{3} - x^{2} - 13x + 1$$ x^3 - x^2 - 13*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 + 3) q^{4} + 5 q^{5} + ( - \beta_{2} + 2 \beta_1 - 14) q^{7} + (2 \beta_{2} + \beta_1 - 8) q^{8}+O(q^{10})$$ q + b2 * q^2 + (-b2 - b1 + 3) * q^4 + 5 * q^5 + (-b2 + 2*b1 - 14) * q^7 + (2*b2 + b1 - 8) * q^8 $$q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 + 3) q^{4} + 5 q^{5} + ( - \beta_{2} + 2 \beta_1 - 14) q^{7} + (2 \beta_{2} + \beta_1 - 8) q^{8} + 5 \beta_{2} q^{10} + ( - 11 \beta_{2} - 3 \beta_1) q^{11} + (13 \beta_{2} + \beta_1 + 18) q^{13} + ( - 25 \beta_{2} + \beta_1 - 17) q^{14} + ( - 8 \beta_{2} + 6 \beta_1 - 5) q^{16} + ( - 3 \beta_{2} + \beta_1 - 56) q^{17} + ( - 7 \beta_{2} - 3 \beta_1 - 58) q^{19} + ( - 5 \beta_{2} - 5 \beta_1 + 15) q^{20} + (29 \beta_{2} + 11 \beta_1 - 112) q^{22} + ( - 3 \beta_{2} + 12 \beta_1 + 60) q^{23} + 25 q^{25} + ( - \beta_{2} - 13 \beta_1 + 140) q^{26} + (10 \beta_{2} + 9 \beta_1 - 166) q^{28} + (4 \beta_{2} + 10 \beta_1 - 107) q^{29} + ( - 51 \beta_{2} - 11 \beta_1 - 138) q^{31} + ( - 49 \beta_{2} - 42) q^{32} + ( - 59 \beta_{2} + 3 \beta_1 - 36) q^{34} + ( - 5 \beta_{2} + 10 \beta_1 - 70) q^{35} + ( - 7 \beta_{2} - 17 \beta_1 + 126) q^{37} + ( - 33 \beta_{2} + 7 \beta_1 - 68) q^{38} + (10 \beta_{2} + 5 \beta_1 - 40) q^{40} + ( - 17 \beta_{2} - 23 \beta_1 + 49) q^{41} + (37 \beta_{2} - 29 \beta_1 - 198) q^{43} + ( - 119 \beta_{2} - 5 \beta_1 + 286) q^{44} + ( - 9 \beta_{2} + 3 \beta_1 - 69) q^{46} + ( - 54 \beta_{2} - 5 \beta_1 + 202) q^{47} + (63 \beta_{2} - 37 \beta_1 + 152) q^{49} + 25 \beta_{2} q^{50} + (115 \beta_{2} - 7 \beta_1 - 116) q^{52} + (62 \beta_{2} + 8 \beta_1 - 220) q^{53} + ( - 55 \beta_{2} - 15 \beta_1) q^{55} + ( - 30 \beta_{2} - 18 \beta_1 + 219) q^{56} + ( - 171 \beta_{2} - 4 \beta_1 + 14) q^{58} + (118 \beta_{2} - 36 \beta_1 - 72) q^{59} + (45 \beta_{2} - 25 \beta_1 - 473) q^{61} + ( - 21 \beta_{2} + 51 \beta_1 - 528) q^{62} + (71 \beta_{2} + \beta_1 - 499) q^{64} + (65 \beta_{2} + 5 \beta_1 + 90) q^{65} + ( - 48 \beta_{2} + 7 \beta_1 - 630) q^{67} + (29 \beta_{2} + 51 \beta_1 - 210) q^{68} + ( - 125 \beta_{2} + 5 \beta_1 - 85) q^{70} + (7 \beta_{2} - 83 \beta_1 - 2) q^{71} + ( - 20 \beta_{2} + 64 \beta_1 - 108) q^{73} + (235 \beta_{2} + 7 \beta_1 - 26) q^{74} + ( - 21 \beta_{2} + 57 \beta_1 + 80) q^{76} + (239 \beta_{2} + \beta_1 - 236) q^{77} + (48 \beta_{2} + 64 \beta_1 - 90) q^{79} + ( - 40 \beta_{2} + 30 \beta_1 - 25) q^{80} + (204 \beta_{2} + 17 \beta_1 - 118) q^{82} + ( - 13 \beta_{2} + 118 \beta_1 - 242) q^{83} + ( - 15 \beta_{2} + 5 \beta_1 - 280) q^{85} + ( - 61 \beta_{2} - 37 \beta_1 + 494) q^{86} + (203 \beta_{2} + 31 \beta_1 - 398) q^{88} + (88 \beta_{2} + 80 \beta_1 + 629) q^{89} + ( - 331 \beta_{2} + 45 \beta_1 - 332) q^{91} + ( - 54 \beta_{2} - 87 \beta_1 - 588) q^{92} + (286 \beta_{2} + 54 \beta_1 - 579) q^{94} + ( - 35 \beta_{2} - 15 \beta_1 - 290) q^{95} + ( - 58 \beta_{2} - 20 \beta_1 - 120) q^{97} + (311 \beta_{2} - 63 \beta_1 + 804) q^{98}+O(q^{100})$$ q + b2 * q^2 + (-b2 - b1 + 3) * q^4 + 5 * q^5 + (-b2 + 2*b1 - 14) * q^7 + (2*b2 + b1 - 8) * q^8 + 5*b2 * q^10 + (-11*b2 - 3*b1) * q^11 + (13*b2 + b1 + 18) * q^13 + (-25*b2 + b1 - 17) * q^14 + (-8*b2 + 6*b1 - 5) * q^16 + (-3*b2 + b1 - 56) * q^17 + (-7*b2 - 3*b1 - 58) * q^19 + (-5*b2 - 5*b1 + 15) * q^20 + (29*b2 + 11*b1 - 112) * q^22 + (-3*b2 + 12*b1 + 60) * q^23 + 25 * q^25 + (-b2 - 13*b1 + 140) * q^26 + (10*b2 + 9*b1 - 166) * q^28 + (4*b2 + 10*b1 - 107) * q^29 + (-51*b2 - 11*b1 - 138) * q^31 + (-49*b2 - 42) * q^32 + (-59*b2 + 3*b1 - 36) * q^34 + (-5*b2 + 10*b1 - 70) * q^35 + (-7*b2 - 17*b1 + 126) * q^37 + (-33*b2 + 7*b1 - 68) * q^38 + (10*b2 + 5*b1 - 40) * q^40 + (-17*b2 - 23*b1 + 49) * q^41 + (37*b2 - 29*b1 - 198) * q^43 + (-119*b2 - 5*b1 + 286) * q^44 + (-9*b2 + 3*b1 - 69) * q^46 + (-54*b2 - 5*b1 + 202) * q^47 + (63*b2 - 37*b1 + 152) * q^49 + 25*b2 * q^50 + (115*b2 - 7*b1 - 116) * q^52 + (62*b2 + 8*b1 - 220) * q^53 + (-55*b2 - 15*b1) * q^55 + (-30*b2 - 18*b1 + 219) * q^56 + (-171*b2 - 4*b1 + 14) * q^58 + (118*b2 - 36*b1 - 72) * q^59 + (45*b2 - 25*b1 - 473) * q^61 + (-21*b2 + 51*b1 - 528) * q^62 + (71*b2 + b1 - 499) * q^64 + (65*b2 + 5*b1 + 90) * q^65 + (-48*b2 + 7*b1 - 630) * q^67 + (29*b2 + 51*b1 - 210) * q^68 + (-125*b2 + 5*b1 - 85) * q^70 + (7*b2 - 83*b1 - 2) * q^71 + (-20*b2 + 64*b1 - 108) * q^73 + (235*b2 + 7*b1 - 26) * q^74 + (-21*b2 + 57*b1 + 80) * q^76 + (239*b2 + b1 - 236) * q^77 + (48*b2 + 64*b1 - 90) * q^79 + (-40*b2 + 30*b1 - 25) * q^80 + (204*b2 + 17*b1 - 118) * q^82 + (-13*b2 + 118*b1 - 242) * q^83 + (-15*b2 + 5*b1 - 280) * q^85 + (-61*b2 - 37*b1 + 494) * q^86 + (203*b2 + 31*b1 - 398) * q^88 + (88*b2 + 80*b1 + 629) * q^89 + (-331*b2 + 45*b1 - 332) * q^91 + (-54*b2 - 87*b1 - 588) * q^92 + (286*b2 + 54*b1 - 579) * q^94 + (-35*b2 - 15*b1 - 290) * q^95 + (-58*b2 - 20*b1 - 120) * q^97 + (311*b2 - 63*b1 + 804) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 11 q^{4} + 15 q^{5} - 43 q^{7} - 27 q^{8}+O(q^{10})$$ 3 * q - q^2 + 11 * q^4 + 15 * q^5 - 43 * q^7 - 27 * q^8 $$3 q - q^{2} + 11 q^{4} + 15 q^{5} - 43 q^{7} - 27 q^{8} - 5 q^{10} + 14 q^{11} + 40 q^{13} - 27 q^{14} - 13 q^{16} - 166 q^{17} - 164 q^{19} + 55 q^{20} - 376 q^{22} + 171 q^{23} + 75 q^{25} + 434 q^{26} - 517 q^{28} - 335 q^{29} - 352 q^{31} - 77 q^{32} - 52 q^{34} - 215 q^{35} + 402 q^{37} - 178 q^{38} - 135 q^{40} + 187 q^{41} - 602 q^{43} + 982 q^{44} - 201 q^{46} + 665 q^{47} + 430 q^{49} - 25 q^{50} - 456 q^{52} - 730 q^{53} + 70 q^{55} + 705 q^{56} + 217 q^{58} - 298 q^{59} - 1439 q^{61} - 1614 q^{62} - 1569 q^{64} + 200 q^{65} - 1849 q^{67} - 710 q^{68} - 135 q^{70} + 70 q^{71} - 368 q^{73} - 320 q^{74} + 204 q^{76} - 948 q^{77} - 382 q^{79} - 65 q^{80} - 575 q^{82} - 831 q^{83} - 830 q^{85} + 1580 q^{86} - 1428 q^{88} + 1719 q^{89} - 710 q^{91} - 1623 q^{92} - 2077 q^{94} - 820 q^{95} - 282 q^{97} + 2164 q^{98}+O(q^{100})$$ 3 * q - q^2 + 11 * q^4 + 15 * q^5 - 43 * q^7 - 27 * q^8 - 5 * q^10 + 14 * q^11 + 40 * q^13 - 27 * q^14 - 13 * q^16 - 166 * q^17 - 164 * q^19 + 55 * q^20 - 376 * q^22 + 171 * q^23 + 75 * q^25 + 434 * q^26 - 517 * q^28 - 335 * q^29 - 352 * q^31 - 77 * q^32 - 52 * q^34 - 215 * q^35 + 402 * q^37 - 178 * q^38 - 135 * q^40 + 187 * q^41 - 602 * q^43 + 982 * q^44 - 201 * q^46 + 665 * q^47 + 430 * q^49 - 25 * q^50 - 456 * q^52 - 730 * q^53 + 70 * q^55 + 705 * q^56 + 217 * q^58 - 298 * q^59 - 1439 * q^61 - 1614 * q^62 - 1569 * q^64 + 200 * q^65 - 1849 * q^67 - 710 * q^68 - 135 * q^70 + 70 * q^71 - 368 * q^73 - 320 * q^74 + 204 * q^76 - 948 * q^77 - 382 * q^79 - 65 * q^80 - 575 * q^82 - 831 * q^83 - 830 * q^85 + 1580 * q^86 - 1428 * q^88 + 1719 * q^89 - 710 * q^91 - 1623 * q^92 - 2077 * q^94 - 820 * q^95 - 282 * q^97 + 2164 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 13x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} + 4\nu - 11 ) / 2$$ (v^2 + 4*v - 11) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 2\nu - 9 ) / 2$$ (v^2 - 2*v - 9) / 2
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta _1 + 1 ) / 3$$ (-b2 + b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( 4\beta_{2} + 2\beta _1 + 29 ) / 3$$ (4*b2 + 2*b1 + 29) / 3

## 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
 0.0765073 4.10645 −3.18296
−4.57358 0 12.9176 5.00000 0 −20.1145 −22.4912 0 −22.8679
1.2 −0.174985 0 −7.96938 5.00000 0 8.46371 2.79440 0 −0.874923
1.3 3.74857 0 6.05174 5.00000 0 −31.3492 −7.30318 0 18.7428
 $$n$$: 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.h 3
3.b odd 2 1 405.4.a.j 3
5.b even 2 1 2025.4.a.s 3
9.c even 3 2 45.4.e.b 6
9.d odd 6 2 135.4.e.b 6
15.d odd 2 1 2025.4.a.q 3
45.j even 6 2 225.4.e.c 6
45.k odd 12 4 225.4.k.c 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + T_{2}^{2} - 17T_{2} - 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(405))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 17T - 3$$
$3$ $$T^{3}$$
$5$ $$(T - 5)^{3}$$
$7$ $$T^{3} + 43 T^{2} + 195 T - 5337$$
$11$ $$T^{3} - 14 T^{2} - 2816 T - 43548$$
$13$ $$T^{3} - 40 T^{2} - 2452 T + 75364$$
$17$ $$T^{3} + 166 T^{2} + 8920 T + 156324$$
$19$ $$T^{3} + 164 T^{2} + 7292 T + 57316$$
$23$ $$T^{3} - 171 T^{2} - 4833 T + 61209$$
$29$ $$T^{3} + 335 T^{2} + 27331 T - 107067$$
$31$ $$T^{3} + 352 T^{2} - 13932 T - 9860940$$
$37$ $$T^{3} - 402 T^{2} + 24708 T + 3335284$$
$41$ $$T^{3} - 187 T^{2} - 44597 T + 7228275$$
$43$ $$T^{3} + 602 T^{2} + \cdots - 15444524$$
$47$ $$T^{3} - 665 T^{2} + 95281 T - 2487483$$
$53$ $$T^{3} + 730 T^{2} + 106300 T + 3250536$$
$59$ $$T^{3} + 298 T^{2} + \cdots - 127375896$$
$61$ $$T^{3} + 1439 T^{2} + \cdots + 55613497$$
$67$ $$T^{3} + 1849 T^{2} + \cdots + 208776159$$
$71$ $$T^{3} - 70 T^{2} + \cdots + 223775052$$
$73$ $$T^{3} + 368 T^{2} + \cdots - 134927744$$
$79$ $$T^{3} + 382 T^{2} + \cdots - 138322584$$
$83$ $$T^{3} + 831 T^{2} + \cdots - 955843821$$
$89$ $$T^{3} - 1719 T^{2} + \cdots + 125506395$$
$97$ $$T^{3} + 282 T^{2} + \cdots - 16898264$$