# Properties

 Label 495.6.a.e Level $495$ Weight $6$ Character orbit 495.a Self dual yes Analytic conductor $79.390$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 495.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$79.3899908074$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.34253.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 52x + 48$$ x^3 - x^2 - 52*x + 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 2) q^{2} + (\beta_{2} + 4 \beta_1 + 7) q^{4} - 25 q^{5} + (\beta_{2} + 11 \beta_1 - 61) q^{7} + (7 \beta_{2} + 77) q^{8}+O(q^{10})$$ q + (b1 + 2) * q^2 + (b2 + 4*b1 + 7) * q^4 - 25 * q^5 + (b2 + 11*b1 - 61) * q^7 + (7*b2 + 77) * q^8 $$q + (\beta_1 + 2) q^{2} + (\beta_{2} + 4 \beta_1 + 7) q^{4} - 25 q^{5} + (\beta_{2} + 11 \beta_1 - 61) q^{7} + (7 \beta_{2} + 77) q^{8} + ( - 25 \beta_1 - 50) q^{10} + 121 q^{11} + ( - 28 \beta_{2} - 24 \beta_1 - 210) q^{13} + (14 \beta_{2} - 22 \beta_1 + 250) q^{14} + ( - 11 \beta_{2} + 68 \beta_1 - 161) q^{16} + ( - 39 \beta_{2} - 37 \beta_1 + 801) q^{17} + ( - 29 \beta_{2} - 67 \beta_1 - 935) q^{19} + ( - 25 \beta_{2} - 100 \beta_1 - 175) q^{20} + (121 \beta_1 + 242) q^{22} + (20 \beta_{2} - 220 \beta_1 - 684) q^{23} + 625 q^{25} + ( - 108 \beta_{2} - 734 \beta_1 - 896) q^{26} + ( - 12 \beta_{2} + 92 \beta_1 + 1500) q^{28} + ( - 77 \beta_{2} - 107 \beta_1 + 2615) q^{29} + (38 \beta_{2} + 130 \beta_1 + 146) q^{31} + ( - 189 \beta_{2} - 212 \beta_1 - 263) q^{32} + ( - 154 \beta_{2} + 64 \beta_1 + 814) q^{34} + ( - 25 \beta_{2} - 275 \beta_1 + 1525) q^{35} + (272 \beta_{2} - 528 \beta_1 - 2866) q^{37} + ( - 154 \beta_{2} - 1562 \beta_1 - 3838) q^{38} + ( - 175 \beta_{2} - 1925) q^{40} + (473 \beta_{2} - 977 \beta_1 + 3245) q^{41} + (341 \beta_{2} - 809 \beta_1 - 4621) q^{43} + (121 \beta_{2} + 484 \beta_1 + 847) q^{44} + ( - 160 \beta_{2} - 784 \beta_1 - 9328) q^{46} + (422 \beta_{2} - 222 \beta_1 + 6538) q^{47} + (4 \beta_{2} - 964 \beta_1 - 8555) q^{49} + (625 \beta_1 + 1250) q^{50} + ( - 162 \beta_{2} - 3432 \beta_1 - 19358) q^{52} + ( - 586 \beta_{2} + 962 \beta_1 + 1212) q^{53} - 3025 q^{55} + ( - 392 \beta_{2} + 2184 \beta_1 - 1624) q^{56} + ( - 338 \beta_{2} + 1092 \beta_1 + 2486) q^{58} + ( - 356 \beta_{2} + 2748 \beta_1 + 2200) q^{59} + (364 \beta_{2} - 3300 \beta_1 - 18926) q^{61} + (244 \beta_{2} + 1052 \beta_1 + 4348) q^{62} + ( - 427 \beta_{2} - 6076 \beta_1 - 337) q^{64} + (700 \beta_{2} + 600 \beta_1 + 5250) q^{65} + (680 \beta_{2} - 2144 \beta_1 - 12108) q^{67} + (850 \beta_{2} - 492 \beta_1 - 19762) q^{68} + ( - 350 \beta_{2} + 550 \beta_1 - 6250) q^{70} + ( - 980 \beta_{2} - 2612 \beta_1 + 25548) q^{71} + (428 \beta_{2} + 2808 \beta_1 - 15750) q^{73} + (288 \beta_{2} + 702 \beta_1 - 27748) q^{74} + ( - 1096 \beta_{2} - 7436 \beta_1 - 30424) q^{76} + (121 \beta_{2} + 1331 \beta_1 - 7381) q^{77} + ( - 775 \beta_{2} - 5417 \beta_1 - 34233) q^{79} + (275 \beta_{2} - 1700 \beta_1 + 4025) q^{80} + (442 \beta_{2} + 9332 \beta_1 - 33854) q^{82} + ( - 1474 \beta_{2} - 14234 \beta_1 + 32042) q^{83} + (975 \beta_{2} + 925 \beta_1 - 20025) q^{85} + (214 \beta_{2} - 442 \beta_1 - 41990) q^{86} + (847 \beta_{2} + 9317) q^{88} + (132 \beta_{2} + 2140 \beta_1 - 56494) q^{89} + (1378 \beta_{2} - 6602 \beta_1 - 8410) q^{91} + ( - 1904 \beta_{2} - 6576 \beta_1 - 22128) q^{92} + (1044 \beta_{2} + 13268 \beta_1 - 180) q^{94} + (725 \beta_{2} + 1675 \beta_1 + 23375) q^{95} + ( - 1664 \beta_{2} - 8760 \beta_1 + 56154) q^{97} + ( - 952 \beta_{2} - 10415 \beta_1 - 50902) q^{98}+O(q^{100})$$ q + (b1 + 2) * q^2 + (b2 + 4*b1 + 7) * q^4 - 25 * q^5 + (b2 + 11*b1 - 61) * q^7 + (7*b2 + 77) * q^8 + (-25*b1 - 50) * q^10 + 121 * q^11 + (-28*b2 - 24*b1 - 210) * q^13 + (14*b2 - 22*b1 + 250) * q^14 + (-11*b2 + 68*b1 - 161) * q^16 + (-39*b2 - 37*b1 + 801) * q^17 + (-29*b2 - 67*b1 - 935) * q^19 + (-25*b2 - 100*b1 - 175) * q^20 + (121*b1 + 242) * q^22 + (20*b2 - 220*b1 - 684) * q^23 + 625 * q^25 + (-108*b2 - 734*b1 - 896) * q^26 + (-12*b2 + 92*b1 + 1500) * q^28 + (-77*b2 - 107*b1 + 2615) * q^29 + (38*b2 + 130*b1 + 146) * q^31 + (-189*b2 - 212*b1 - 263) * q^32 + (-154*b2 + 64*b1 + 814) * q^34 + (-25*b2 - 275*b1 + 1525) * q^35 + (272*b2 - 528*b1 - 2866) * q^37 + (-154*b2 - 1562*b1 - 3838) * q^38 + (-175*b2 - 1925) * q^40 + (473*b2 - 977*b1 + 3245) * q^41 + (341*b2 - 809*b1 - 4621) * q^43 + (121*b2 + 484*b1 + 847) * q^44 + (-160*b2 - 784*b1 - 9328) * q^46 + (422*b2 - 222*b1 + 6538) * q^47 + (4*b2 - 964*b1 - 8555) * q^49 + (625*b1 + 1250) * q^50 + (-162*b2 - 3432*b1 - 19358) * q^52 + (-586*b2 + 962*b1 + 1212) * q^53 - 3025 * q^55 + (-392*b2 + 2184*b1 - 1624) * q^56 + (-338*b2 + 1092*b1 + 2486) * q^58 + (-356*b2 + 2748*b1 + 2200) * q^59 + (364*b2 - 3300*b1 - 18926) * q^61 + (244*b2 + 1052*b1 + 4348) * q^62 + (-427*b2 - 6076*b1 - 337) * q^64 + (700*b2 + 600*b1 + 5250) * q^65 + (680*b2 - 2144*b1 - 12108) * q^67 + (850*b2 - 492*b1 - 19762) * q^68 + (-350*b2 + 550*b1 - 6250) * q^70 + (-980*b2 - 2612*b1 + 25548) * q^71 + (428*b2 + 2808*b1 - 15750) * q^73 + (288*b2 + 702*b1 - 27748) * q^74 + (-1096*b2 - 7436*b1 - 30424) * q^76 + (121*b2 + 1331*b1 - 7381) * q^77 + (-775*b2 - 5417*b1 - 34233) * q^79 + (275*b2 - 1700*b1 + 4025) * q^80 + (442*b2 + 9332*b1 - 33854) * q^82 + (-1474*b2 - 14234*b1 + 32042) * q^83 + (975*b2 + 925*b1 - 20025) * q^85 + (214*b2 - 442*b1 - 41990) * q^86 + (847*b2 + 9317) * q^88 + (132*b2 + 2140*b1 - 56494) * q^89 + (1378*b2 - 6602*b1 - 8410) * q^91 + (-1904*b2 - 6576*b1 - 22128) * q^92 + (1044*b2 + 13268*b1 - 180) * q^94 + (725*b2 + 1675*b1 + 23375) * q^95 + (-1664*b2 - 8760*b1 + 56154) * q^97 + (-952*b2 - 10415*b1 - 50902) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8}+O(q^{10})$$ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 $$3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8} - 175 q^{10} + 363 q^{11} - 654 q^{13} + 728 q^{14} - 415 q^{16} + 2366 q^{17} - 2872 q^{19} - 625 q^{20} + 847 q^{22} - 2272 q^{23} + 1875 q^{25} - 3422 q^{26} + 4592 q^{28} + 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 2506 q^{34} + 4300 q^{35} - 9126 q^{37} - 13076 q^{38} - 5775 q^{40} + 8758 q^{41} - 14672 q^{43} + 3025 q^{44} - 28768 q^{46} + 19392 q^{47} - 26629 q^{49} + 4375 q^{50} - 61506 q^{52} + 4598 q^{53} - 9075 q^{55} - 2688 q^{56} + 8550 q^{58} + 9348 q^{59} - 60078 q^{61} + 14096 q^{62} - 7087 q^{64} + 16350 q^{65} - 38468 q^{67} - 59778 q^{68} - 18200 q^{70} + 74032 q^{71} - 44442 q^{73} - 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 108116 q^{79} + 10375 q^{80} - 92230 q^{82} + 81892 q^{83} - 59150 q^{85} - 126412 q^{86} + 27951 q^{88} - 167342 q^{89} - 31832 q^{91} - 72960 q^{92} + 12728 q^{94} + 71800 q^{95} + 159702 q^{97} - 163121 q^{98}+O(q^{100})$$ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 - 175 * q^10 + 363 * q^11 - 654 * q^13 + 728 * q^14 - 415 * q^16 + 2366 * q^17 - 2872 * q^19 - 625 * q^20 + 847 * q^22 - 2272 * q^23 + 1875 * q^25 - 3422 * q^26 + 4592 * q^28 + 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 2506 * q^34 + 4300 * q^35 - 9126 * q^37 - 13076 * q^38 - 5775 * q^40 + 8758 * q^41 - 14672 * q^43 + 3025 * q^44 - 28768 * q^46 + 19392 * q^47 - 26629 * q^49 + 4375 * q^50 - 61506 * q^52 + 4598 * q^53 - 9075 * q^55 - 2688 * q^56 + 8550 * q^58 + 9348 * q^59 - 60078 * q^61 + 14096 * q^62 - 7087 * q^64 + 16350 * q^65 - 38468 * q^67 - 59778 * q^68 - 18200 * q^70 + 74032 * q^71 - 44442 * q^73 - 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 108116 * q^79 + 10375 * q^80 - 92230 * q^82 + 81892 * q^83 - 59150 * q^85 - 126412 * q^86 + 27951 * q^88 - 167342 * q^89 - 31832 * q^91 - 72960 * q^92 + 12728 * q^94 + 71800 * q^95 + 159702 * q^97 - 163121 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52x + 48$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 35$$ v^2 - 35
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 35$$ b2 + 35

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −7.17710 0.921799 7.25531
−5.17710 0 −5.19759 −25.0000 0 −123.437 192.576 0 129.428
1.2 2.92180 0 −23.4631 −25.0000 0 −85.0105 −162.052 0 −73.0450
1.3 9.25531 0 53.6607 −25.0000 0 36.4478 200.476 0 −231.383
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 495.6.a.e 3
3.b odd 2 1 165.6.a.a 3
15.d odd 2 1 825.6.a.j 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.a 3 3.b odd 2 1
495.6.a.e 3 1.a even 1 1 trivial
825.6.a.j 3 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 7T_{2}^{2} - 36T_{2} + 140$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(495))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 7 T^{2} - 36 T + 140$$
$3$ $$T^{3}$$
$5$ $$(T + 25)^{3}$$
$7$ $$T^{3} + 172 T^{2} + 2896 T - 382464$$
$11$ $$(T - 121)^{3}$$
$13$ $$T^{3} + 654 T^{2} + \cdots - 317918392$$
$17$ $$T^{3} - 2366 T^{2} + \cdots + 137826264$$
$19$ $$T^{3} + 2872 T^{2} + \cdots + 11543616$$
$23$ $$T^{3} + 2272 T^{2} + \cdots - 3706904576$$
$29$ $$T^{3} - 7738 T^{2} + \cdots - 5220625848$$
$31$ $$T^{3} - 568 T^{2} + \cdots - 289787904$$
$37$ $$T^{3} + 9126 T^{2} + \cdots - 129972509048$$
$41$ $$T^{3} - 8758 T^{2} + \cdots + 1122652557432$$
$43$ $$T^{3} + 14672 T^{2} + \cdots - 518908872384$$
$47$ $$T^{3} - 19392 T^{2} + \cdots + 1508908531200$$
$53$ $$T^{3} - 4598 T^{2} + \cdots - 728896505288$$
$59$ $$T^{3} - 9348 T^{2} + \cdots + 6267836310080$$
$61$ $$T^{3} + 60078 T^{2} + \cdots - 13500896397400$$
$67$ $$T^{3} + 38468 T^{2} + \cdots - 8479952260160$$
$71$ $$T^{3} - 74032 T^{2} + \cdots + 17011990639616$$
$73$ $$T^{3} + 44442 T^{2} + \cdots - 9750515676328$$
$79$ $$T^{3} + 108116 T^{2} + \cdots + 9069370346752$$
$83$ $$T^{3} + \cdots + 739830059345664$$
$89$ $$T^{3} + \cdots + 158914472576552$$
$97$ $$T^{3} + \cdots + 352998320493112$$