# Properties

 Label 405.4.a.n Level $405$ Weight $4$ Character orbit 405.a Self dual yes Analytic conductor $23.896$ Analytic rank $0$ Dimension $7$ 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: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288$$ x^7 - 2*x^6 - 44*x^5 + 74*x^4 + 479*x^3 - 460*x^2 - 1200*x + 288 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + 5 q^{5} + ( - \beta_{5} + 3) q^{7} + (\beta_{3} + 7 \beta_1 + 1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 5) * q^4 + 5 * q^5 + (-b5 + 3) * q^7 + (b3 + 7*b1 + 1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + 5 q^{5} + ( - \beta_{5} + 3) q^{7} + (\beta_{3} + 7 \beta_1 + 1) q^{8} + 5 \beta_1 q^{10} + (\beta_{4} + 3 \beta_1 + 2) q^{11} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - 4 \beta_1 + 15) q^{13} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 7 \beta_1 - 4) q^{14} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 7 \beta_{2} - 6 \beta_1 + 46) q^{16} + ( - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 22) q^{17} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 40) q^{19} + (5 \beta_{2} + 25) q^{20} + (\beta_{6} + 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 5 \beta_{2} - \beta_1 + 44) q^{22} + (\beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 11) q^{23} + 25 q^{25} + ( - 4 \beta_{5} - 2 \beta_{4} - \beta_{3} - 10 \beta_{2} + 32 \beta_1 - 59) q^{26} + ( - \beta_{6} + \beta_{5} - 4 \beta_{4} - 5 \beta_{3} - \beta_{2} - 14 \beta_1 + 52) q^{28} + (3 \beta_{6} - \beta_{5} - 5 \beta_{2} + 15 \beta_1 - 46) q^{29} + (\beta_{6} - \beta_{5} - \beta_{4} - 4 \beta_{3} - 9 \beta_{2} - 8 \beta_1 + 37) q^{31} + (2 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 53 \beta_1 - 58) q^{32} + (3 \beta_{6} + \beta_{5} - 2 \beta_{4} + 7 \beta_{3} - 7 \beta_{2} + 45 \beta_1 + 10) q^{34} + ( - 5 \beta_{5} + 15) q^{35} + ( - \beta_{6} - \beta_{5} + 5 \beta_{4} - 11 \beta_{2} - 14 \beta_1 + 61) q^{37} + ( - 5 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 17 \beta_{2} + 49 \beta_1 + 26) q^{38} + (5 \beta_{3} + 35 \beta_1 + 5) q^{40} + ( - \beta_{6} - 9 \beta_{5} + \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 8) q^{41} + (5 \beta_{4} - 2 \beta_{3} - 12 \beta_{2} - 29 \beta_1 + 82) q^{43} + ( - 3 \beta_{6} + 11 \beta_{5} + 6 \beta_{4} + 5 \beta_{3} + 15 \beta_{2} + \cdots - 6) q^{44}+ \cdots + (5 \beta_{6} - 9 \beta_{5} - 34 \beta_{4} - 11 \beta_{3} + 25 \beta_{2} + \cdots + 188) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 5) * q^4 + 5 * q^5 + (-b5 + 3) * q^7 + (b3 + 7*b1 + 1) * q^8 + 5*b1 * q^10 + (b4 + 3*b1 + 2) * q^11 + (b6 - b5 - b4 + b2 - 4*b1 + 15) * q^13 + (b6 - b5 - 2*b4 - 2*b2 + 7*b1 - 4) * q^14 + (-2*b6 + 2*b5 + 2*b4 - b3 + 7*b2 - 6*b1 + 46) * q^16 + (-2*b6 + 2*b5 + b4 - 2*b3 + 2*b2 - b1 + 22) * q^17 + (-b4 + 2*b3 + 2*b2 + 3*b1 + 40) * q^19 + (5*b2 + 25) * q^20 + (b6 + 3*b5 + 2*b4 + b3 + 5*b2 - b1 + 44) * q^22 + (b5 - 2*b3 + 2*b2 + 6*b1 + 11) * q^23 + 25 * q^25 + (-4*b5 - 2*b4 - b3 - 10*b2 + 32*b1 - 59) * q^26 + (-b6 + b5 - 4*b4 - 5*b3 - b2 - 14*b1 + 52) * q^28 + (3*b6 - b5 - 5*b2 + 15*b1 - 46) * q^29 + (b6 - b5 - b4 - 4*b3 - 9*b2 - 8*b1 + 37) * q^31 + (2*b6 + 6*b5 + 2*b4 + 4*b3 - 2*b2 + 53*b1 - 58) * q^32 + (3*b6 + b5 - 2*b4 + 7*b3 - 7*b2 + 45*b1 + 10) * q^34 + (-5*b5 + 15) * q^35 + (-b6 - b5 + 5*b4 - 11*b2 - 14*b1 + 61) * q^37 + (-5*b6 + b5 + 2*b4 - b3 + 17*b2 + 49*b1 + 26) * q^38 + (5*b3 + 35*b1 + 5) * q^40 + (-b6 - 9*b5 + b4 + 2*b3 - 5*b2 + 4*b1 - 8) * q^41 + (5*b4 - 2*b3 - 12*b2 - 29*b1 + 82) * q^43 + (-3*b6 + 11*b5 + 6*b4 + 5*b3 + 15*b2 + 45*b1 - 6) * q^44 + (3*b6 - 3*b5 - 2*b4 + 4*b3 - 8*b2 + 41*b1 + 94) * q^46 + (b6 + 8*b5 - 5*b4 + 2*b3 - 9*b2 - 4*b1 + 30) * q^47 + (-10*b6 - 6*b5 - b4 - 20*b2 + 19*b1 + 172) * q^49 + 25*b1 * q^50 + (-4*b6 - 4*b5 - 6*b4 - 11*b3 + 4*b2 - 98*b1 + 265) * q^52 + (10*b6 + 4*b5 - 4*b4 + 4*b3 + 24*b2 - 4*b1 + 74) * q^53 + (5*b4 + 15*b1 + 10) * q^55 + (-3*b6 - 13*b5 - 2*b4 + b3 - 42*b2 + 29*b1 - 143) * q^56 + (b6 - b5 + 4*b4 - 8*b3 + 7*b2 - 92*b1 + 189) * q^58 + (-13*b6 + 3*b5 - 4*b4 - 10*b3 - b2 - 53*b1 + 1) * q^59 + (6*b6 - 8*b5 - 7*b4 + 2*b3 - 20*b2 + 101*b1 + 95) * q^61 + (8*b6 - 12*b5 - 10*b4 - 7*b3 - 46*b2 - 18*b1 - 101) * q^62 + (4*b6 + 4*b5 + 12*b4 + 2*b3 + 41*b2 - 88*b1 + 335) * q^64 + (5*b6 - 5*b5 - 5*b4 + 5*b2 - 20*b1 + 75) * q^65 + (5*b5 - 7*b4 + 8*b3 - 12*b2 - 21*b1 + 171) * q^67 + (-b6 - 7*b5 + 10*b4 - 3*b3 + 77*b2 - 99*b1 + 364) * q^68 + (5*b6 - 5*b5 - 10*b4 - 10*b2 + 35*b1 - 20) * q^70 + (7*b6 + 7*b5 + b4 - 3*b2 - 144*b1 + 89) * q^71 + (-3*b6 + 3*b5 + 4*b4 + 10*b3 - 3*b2 + 29*b1 + 293) * q^73 + (6*b6 + 14*b5 + 6*b4 - 5*b3 - 4*b2 - 60*b1 - 173) * q^74 + (3*b6 + 5*b5 + 2*b4 + 9*b3 + 41*b2 + 169*b1 + 348) * q^76 + (-3*b6 + 5*b5 + 9*b4 + 4*b3 - 37*b2 - 104*b1 + 37) * q^77 + (16*b6 + 16*b5 - 8*b4 - 6*b3 - 6*b2 - 58*b1 + 322) * q^79 + (-10*b6 + 10*b5 + 10*b4 - 5*b3 + 35*b2 - 30*b1 + 230) * q^80 + (6*b6 - 2*b5 - 14*b4 - 5*b3 + 6*b2 - 39*b1 + 5) * q^82 + (-8*b6 - 3*b5 - 4*b4 + 14*b2 - 138*b1 + 223) * q^83 + (-10*b6 + 10*b5 + 5*b4 - 10*b3 + 10*b2 - 5*b1 + 110) * q^85 + (9*b6 + 11*b5 + 6*b4 - 5*b3 - 35*b2 - 39*b1 - 354) * q^86 + (-23*b6 + 15*b5 + 22*b4 + 11*b3 + 85*b2 + 55*b1 + 294) * q^88 + (-13*b6 - 19*b5 + 20*b4 + 2*b3 - 13*b2 - 133*b1 - 406) * q^89 + (-15*b6 + 11*b5 - 9*b4 - 22*b3 + 13*b2 + 98*b1 + 443) * q^91 + (-7*b6 - 9*b5 + 4*b4 - b3 + 41*b2 - 44*b1 + 398) * q^92 + (-17*b6 - 3*b5 + 12*b4 - 17*b3 + 16*b2 - 91*b1 - 63) * q^94 + (-5*b4 + 10*b3 + 10*b2 + 15*b1 + 200) * q^95 + (3*b6 + 7*b5 + 8*b4 + 22*b3 + 37*b2 - 175*b1 + 307) * q^97 + (5*b6 - 9*b5 - 34*b4 - 11*b3 + 25*b2 - b1 + 188) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8}+O(q^{10})$$ 7 * q + 2 * q^2 + 36 * q^4 + 35 * q^5 + 22 * q^7 + 18 * q^8 $$7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + 10 q^{10} + 23 q^{11} + 96 q^{13} - 21 q^{14} + 324 q^{16} + 161 q^{17} + 279 q^{19} + 180 q^{20} + 311 q^{22} + 96 q^{23} + 175 q^{25} - 358 q^{26} + 337 q^{28} - 296 q^{29} + 244 q^{31} - 314 q^{32} + 125 q^{34} + 110 q^{35} + 404 q^{37} + 305 q^{38} + 90 q^{40} - 47 q^{41} + 525 q^{43} + 55 q^{44} + 717 q^{46} + 164 q^{47} + 1225 q^{49} + 50 q^{50} + 1682 q^{52} + 506 q^{53} + 115 q^{55} - 981 q^{56} + 1183 q^{58} - 85 q^{59} + 828 q^{61} - 786 q^{62} + 2236 q^{64} + 480 q^{65} + 1093 q^{67} + 2473 q^{68} - 105 q^{70} + 328 q^{71} + 2085 q^{73} - 1316 q^{74} + 2789 q^{76} + 24 q^{77} + 2110 q^{79} + 1620 q^{80} - 62 q^{82} + 1290 q^{83} + 805 q^{85} - 2569 q^{86} + 2271 q^{88} - 3048 q^{89} + 3338 q^{91} + 2763 q^{92} - 517 q^{94} + 1395 q^{95} + 1787 q^{97} + 1279 q^{98}+O(q^{100})$$ 7 * q + 2 * q^2 + 36 * q^4 + 35 * q^5 + 22 * q^7 + 18 * q^8 + 10 * q^10 + 23 * q^11 + 96 * q^13 - 21 * q^14 + 324 * q^16 + 161 * q^17 + 279 * q^19 + 180 * q^20 + 311 * q^22 + 96 * q^23 + 175 * q^25 - 358 * q^26 + 337 * q^28 - 296 * q^29 + 244 * q^31 - 314 * q^32 + 125 * q^34 + 110 * q^35 + 404 * q^37 + 305 * q^38 + 90 * q^40 - 47 * q^41 + 525 * q^43 + 55 * q^44 + 717 * q^46 + 164 * q^47 + 1225 * q^49 + 50 * q^50 + 1682 * q^52 + 506 * q^53 + 115 * q^55 - 981 * q^56 + 1183 * q^58 - 85 * q^59 + 828 * q^61 - 786 * q^62 + 2236 * q^64 + 480 * q^65 + 1093 * q^67 + 2473 * q^68 - 105 * q^70 + 328 * q^71 + 2085 * q^73 - 1316 * q^74 + 2789 * q^76 + 24 * q^77 + 2110 * q^79 + 1620 * q^80 - 62 * q^82 + 1290 * q^83 + 805 * q^85 - 2569 * q^86 + 2271 * q^88 - 3048 * q^89 + 3338 * q^91 + 2763 * q^92 - 517 * q^94 + 1395 * q^95 + 1787 * q^97 + 1279 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 13$$ v^2 - 13 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 23\nu - 1$$ v^3 - 23*v - 1 $$\beta_{4}$$ $$=$$ $$( \nu^{6} - \nu^{5} - 39\nu^{4} + 35\nu^{3} + 310\nu^{2} - 114\nu - 240 ) / 12$$ (v^6 - v^5 - 39*v^4 + 35*v^3 + 310*v^2 - 114*v - 240) / 12 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 4\nu^{5} + 42\nu^{4} - 140\nu^{3} - 397\nu^{2} + 756\nu + 672 ) / 24$$ (-v^6 + 4*v^5 + 42*v^4 - 140*v^3 - 397*v^2 + 756*v + 672) / 24 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 2\nu^{5} - 48\nu^{4} - 82\nu^{3} + 595\nu^{2} + 732\nu - 1104 ) / 24$$ (v^6 + 2*v^5 - 48*v^4 - 82*v^3 + 595*v^2 + 732*v - 1104) / 24
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 13$$ b2 + 13 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 23\beta _1 + 1$$ b3 + 23*b1 + 1 $$\nu^{4}$$ $$=$$ $$-2\beta_{6} + 2\beta_{5} + 2\beta_{4} - \beta_{3} + 31\beta_{2} - 6\beta _1 + 294$$ -2*b6 + 2*b5 + 2*b4 - b3 + 31*b2 - 6*b1 + 294 $$\nu^{5}$$ $$=$$ $$2\beta_{6} + 6\beta_{5} + 2\beta_{4} + 36\beta_{3} - 2\beta_{2} + 597\beta _1 - 26$$ 2*b6 + 6*b5 + 2*b4 + 36*b3 - 2*b2 + 597*b1 - 26 $$\nu^{6}$$ $$=$$ $$-76\beta_{6} + 84\beta_{5} + 92\beta_{4} - 38\beta_{3} + 897\beta_{2} - 328\beta _1 + 7615$$ -76*b6 + 84*b5 + 92*b4 - 38*b3 + 897*b2 - 328*b1 + 7615

## 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
 −5.38503 −3.04174 −1.57021 0.225250 2.19444 4.26178 5.31551
−5.38503 0 20.9986 5.00000 0 12.5702 −69.9976 0 −26.9252
1.2 −3.04174 0 1.25219 5.00000 0 −13.7122 20.5251 0 −15.2087
1.3 −1.57021 0 −5.53444 5.00000 0 34.2398 21.2519 0 −7.85104
1.4 0.225250 0 −7.94926 5.00000 0 −31.1940 −3.59257 0 1.12625
1.5 2.19444 0 −3.18442 5.00000 0 2.76605 −24.5436 0 10.9722
1.6 4.26178 0 10.1628 5.00000 0 30.7639 9.21718 0 21.3089
1.7 5.31551 0 20.2546 5.00000 0 −13.4337 65.1396 0 26.5775
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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.n 7
3.b odd 2 1 405.4.a.m 7
5.b even 2 1 2025.4.a.ba 7
9.c even 3 2 135.4.e.c 14
9.d odd 6 2 45.4.e.c 14
15.d odd 2 1 2025.4.a.bb 7
45.h odd 6 2 225.4.e.d 14
45.l even 12 4 225.4.k.d 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.c 14 9.d odd 6 2
135.4.e.c 14 9.c even 3 2
225.4.e.d 14 45.h odd 6 2
225.4.k.d 28 45.l even 12 4
405.4.a.m 7 3.b odd 2 1
405.4.a.n 7 1.a even 1 1 trivial
2025.4.a.ba 7 5.b even 2 1
2025.4.a.bb 7 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} - 2T_{2}^{6} - 44T_{2}^{5} + 74T_{2}^{4} + 479T_{2}^{3} - 460T_{2}^{2} - 1200T_{2} + 288$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(405))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} - 2 T^{6} - 44 T^{5} + 74 T^{4} + \cdots + 288$$
$3$ $$T^{7}$$
$5$ $$(T - 5)^{7}$$
$7$ $$T^{7} - 22 T^{6} + \cdots + 210450744$$
$11$ $$T^{7} - 23 T^{6} + \cdots - 20019444768$$
$13$ $$T^{7} - 96 T^{6} + \cdots + 159234392576$$
$17$ $$T^{7} - 161 T^{6} + \cdots + 60588009792$$
$19$ $$T^{7} - 279 T^{6} + \cdots - 1377989598400$$
$23$ $$T^{7} - 96 T^{6} + \cdots - 4061042235738$$
$29$ $$T^{7} + 296 T^{6} + \cdots - 12128150971026$$
$31$ $$T^{7} - 244 T^{6} + \cdots + 29481260805504$$
$37$ $$T^{7} - 404 T^{6} + \cdots - 83646911884544$$
$41$ $$T^{7} + 47 T^{6} + \cdots + 15\!\cdots\!61$$
$43$ $$T^{7} - 525 T^{6} + \cdots - 13\!\cdots\!84$$
$47$ $$T^{7} - 164 T^{6} + \cdots + 65\!\cdots\!72$$
$53$ $$T^{7} - 506 T^{6} + \cdots + 11\!\cdots\!92$$
$59$ $$T^{7} + 85 T^{6} + \cdots - 59\!\cdots\!08$$
$61$ $$T^{7} - 828 T^{6} + \cdots + 12\!\cdots\!46$$
$67$ $$T^{7} - 1093 T^{6} + \cdots + 29\!\cdots\!03$$
$71$ $$T^{7} - 328 T^{6} + \cdots + 32\!\cdots\!28$$
$73$ $$T^{7} - 2085 T^{6} + \cdots + 82\!\cdots\!12$$
$79$ $$T^{7} - 2110 T^{6} + \cdots + 16\!\cdots\!08$$
$83$ $$T^{7} - 1290 T^{6} + \cdots - 66\!\cdots\!72$$
$89$ $$T^{7} + 3048 T^{6} + \cdots + 32\!\cdots\!50$$
$97$ $$T^{7} - 1787 T^{6} + \cdots + 47\!\cdots\!76$$