# Properties

 Label 495.6.a.d 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: 3.3.3368.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 15x + 11$$ x^3 - x^2 - 15*x + 11 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{2} + (4 \beta_{2} + 8) q^{4} + 25 q^{5} + ( - 11 \beta_{2} + 14 \beta_1 - 69) q^{7} + (8 \beta_{2} - 4 \beta_1 - 12) q^{8}+O(q^{10})$$ q + (b1 + 1) * q^2 + (4*b2 + 8) * q^4 + 25 * q^5 + (-11*b2 + 14*b1 - 69) * q^7 + (8*b2 - 4*b1 - 12) * q^8 $$q + (\beta_1 + 1) q^{2} + (4 \beta_{2} + 8) q^{4} + 25 q^{5} + ( - 11 \beta_{2} + 14 \beta_1 - 69) q^{7} + (8 \beta_{2} - 4 \beta_1 - 12) q^{8} + (25 \beta_1 + 25) q^{10} - 121 q^{11} + ( - 31 \beta_{2} - 25 \beta_1 + 152) q^{13} + (34 \beta_{2} - 138 \beta_1 + 444) q^{14} + ( - 128 \beta_{2} + 32 \beta_1 - 400) q^{16} + (57 \beta_{2} - 34 \beta_1 + 81) q^{17} + (136 \beta_{2} + 119 \beta_1 - 1351) q^{19} + (100 \beta_{2} + 200) q^{20} + ( - 121 \beta_1 - 121) q^{22} + ( - 164 \beta_{2} - 372 \beta_1 + 2284) q^{23} + 625 q^{25} + ( - 162 \beta_{2} + 22 \beta_1 - 916) q^{26} + ( - 132 \beta_{2} + 304 \beta_1 - 2628) q^{28} + ( - 580 \beta_{2} - 93 \beta_1 - 1185) q^{29} + (764 \beta_{2} - 920 \beta_1 - 764) q^{31} + ( - 384 \beta_{2} - 944 \beta_1 + 848) q^{32} + ( - 22 \beta_{2} + 400 \beta_1 - 1074) q^{34} + ( - 275 \beta_{2} + 350 \beta_1 - 1725) q^{35} + ( - 1312 \beta_{2} + 410 \beta_1 + 1324) q^{37} + (748 \beta_{2} - 790 \beta_1 + 3698) q^{38} + (200 \beta_{2} - 100 \beta_1 - 300) q^{40} + ( - 140 \beta_{2} + 521 \beta_1 - 3331) q^{41} + ( - 37 \beta_{2} - 2 \beta_1 - 11831) q^{43} + ( - 484 \beta_{2} - 968) q^{44} + ( - 1816 \beta_{2} + 1836 \beta_1 - 12716) q^{46} + ( - 1004 \beta_{2} + 640 \beta_1 + 1248) q^{47} + (1510 \beta_{2} - 3622 \beta_1 + 845) q^{49} + (625 \beta_1 + 625) q^{50} + (756 \beta_{2} - 948 \beta_1 - 5408) q^{52} + (2746 \beta_{2} + 2226 \beta_1 + 4102) q^{53} - 3025 q^{55} + ( - 136 \beta_{2} + 824 \beta_1 - 5376) q^{56} + ( - 1532 \beta_{2} - 3992 \beta_1 - 6552) q^{58} + ( - 844 \beta_{2} - 3292 \beta_1 + 26476) q^{59} + (916 \beta_{2} - 3326 \beta_1 - 22180) q^{61} + ( - 2152 \beta_{2} + 3976 \beta_1 - 34352) q^{62} + ( - 448 \beta_{2} - 1152 \beta_1 - 24320) q^{64} + ( - 775 \beta_{2} - 625 \beta_1 + 3800) q^{65} + ( - 4474 \beta_{2} + 496 \beta_1 - 13726) q^{67} + ( - 268 \beta_{2} - 496 \beta_1 + 11868) q^{68} + (850 \beta_{2} - 3450 \beta_1 + 11100) q^{70} + (26 \beta_{2} - 1080 \beta_1 + 21206) q^{71} + (3773 \beta_{2} - 8443 \beta_1 - 3802) q^{73} + ( - 984 \beta_{2} - 5646 \beta_1 + 13378) q^{74} + ( - 6016 \beta_{2} + 4420 \beta_1 + 18364) q^{76} + (1331 \beta_{2} - 1694 \beta_1 + 8349) q^{77} + (10742 \beta_{2} - 5831 \beta_1 + 9101) q^{79} + ( - 3200 \beta_{2} + 800 \beta_1 - 10000) q^{80} + (1804 \beta_{2} - 4552 \beta_1 + 16568) q^{82} + (2095 \beta_{2} + 4559 \beta_1 + 34496) q^{83} + (1425 \beta_{2} - 850 \beta_1 + 2025) q^{85} + ( - 82 \beta_{2} - 12014 \beta_1 - 12020) q^{86} + ( - 968 \beta_{2} + 484 \beta_1 + 1452) q^{88} + (12390 \beta_{2} - 2204 \beta_1 - 24872) q^{89} + ( - 2456 \beta_{2} + 4440 \beta_1 - 7224) q^{91} + (8960 \beta_{2} - 11728 \beta_1 - 19648) q^{92} + (552 \beta_{2} - 4412 \beta_1 + 23196) q^{94} + (3400 \beta_{2} + 2975 \beta_1 - 33775) q^{95} + (3142 \beta_{2} + 10816 \beta_1 - 104024) q^{97} + ( - 11468 \beta_{2} + 12017 \beta_1 - 135883) q^{98}+O(q^{100})$$ q + (b1 + 1) * q^2 + (4*b2 + 8) * q^4 + 25 * q^5 + (-11*b2 + 14*b1 - 69) * q^7 + (8*b2 - 4*b1 - 12) * q^8 + (25*b1 + 25) * q^10 - 121 * q^11 + (-31*b2 - 25*b1 + 152) * q^13 + (34*b2 - 138*b1 + 444) * q^14 + (-128*b2 + 32*b1 - 400) * q^16 + (57*b2 - 34*b1 + 81) * q^17 + (136*b2 + 119*b1 - 1351) * q^19 + (100*b2 + 200) * q^20 + (-121*b1 - 121) * q^22 + (-164*b2 - 372*b1 + 2284) * q^23 + 625 * q^25 + (-162*b2 + 22*b1 - 916) * q^26 + (-132*b2 + 304*b1 - 2628) * q^28 + (-580*b2 - 93*b1 - 1185) * q^29 + (764*b2 - 920*b1 - 764) * q^31 + (-384*b2 - 944*b1 + 848) * q^32 + (-22*b2 + 400*b1 - 1074) * q^34 + (-275*b2 + 350*b1 - 1725) * q^35 + (-1312*b2 + 410*b1 + 1324) * q^37 + (748*b2 - 790*b1 + 3698) * q^38 + (200*b2 - 100*b1 - 300) * q^40 + (-140*b2 + 521*b1 - 3331) * q^41 + (-37*b2 - 2*b1 - 11831) * q^43 + (-484*b2 - 968) * q^44 + (-1816*b2 + 1836*b1 - 12716) * q^46 + (-1004*b2 + 640*b1 + 1248) * q^47 + (1510*b2 - 3622*b1 + 845) * q^49 + (625*b1 + 625) * q^50 + (756*b2 - 948*b1 - 5408) * q^52 + (2746*b2 + 2226*b1 + 4102) * q^53 - 3025 * q^55 + (-136*b2 + 824*b1 - 5376) * q^56 + (-1532*b2 - 3992*b1 - 6552) * q^58 + (-844*b2 - 3292*b1 + 26476) * q^59 + (916*b2 - 3326*b1 - 22180) * q^61 + (-2152*b2 + 3976*b1 - 34352) * q^62 + (-448*b2 - 1152*b1 - 24320) * q^64 + (-775*b2 - 625*b1 + 3800) * q^65 + (-4474*b2 + 496*b1 - 13726) * q^67 + (-268*b2 - 496*b1 + 11868) * q^68 + (850*b2 - 3450*b1 + 11100) * q^70 + (26*b2 - 1080*b1 + 21206) * q^71 + (3773*b2 - 8443*b1 - 3802) * q^73 + (-984*b2 - 5646*b1 + 13378) * q^74 + (-6016*b2 + 4420*b1 + 18364) * q^76 + (1331*b2 - 1694*b1 + 8349) * q^77 + (10742*b2 - 5831*b1 + 9101) * q^79 + (-3200*b2 + 800*b1 - 10000) * q^80 + (1804*b2 - 4552*b1 + 16568) * q^82 + (2095*b2 + 4559*b1 + 34496) * q^83 + (1425*b2 - 850*b1 + 2025) * q^85 + (-82*b2 - 12014*b1 - 12020) * q^86 + (-968*b2 + 484*b1 + 1452) * q^88 + (12390*b2 - 2204*b1 - 24872) * q^89 + (-2456*b2 + 4440*b1 - 7224) * q^91 + (8960*b2 - 11728*b1 - 19648) * q^92 + (552*b2 - 4412*b1 + 23196) * q^94 + (3400*b2 + 2975*b1 - 33775) * q^95 + (3142*b2 + 10816*b1 - 104024) * q^97 + (-11468*b2 + 12017*b1 - 135883) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8}+O(q^{10})$$ 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8 $$3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8} + 50 q^{10} - 363 q^{11} + 450 q^{13} + 1504 q^{14} - 1360 q^{16} + 334 q^{17} - 4036 q^{19} + 700 q^{20} - 242 q^{22} + 7060 q^{23} + 1875 q^{25} - 2932 q^{26} - 8320 q^{28} - 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3644 q^{34} - 5800 q^{35} + 2250 q^{37} + 12632 q^{38} - 600 q^{40} - 10654 q^{41} - 35528 q^{43} - 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 7667 q^{49} + 1250 q^{50} - 14520 q^{52} + 12826 q^{53} - 9075 q^{55} - 17088 q^{56} - 17196 q^{58} + 81876 q^{59} - 62298 q^{61} - 109184 q^{62} - 72256 q^{64} + 11250 q^{65} - 46148 q^{67} + 35832 q^{68} + 37600 q^{70} + 64724 q^{71} + 810 q^{73} + 44796 q^{74} + 44656 q^{76} + 28072 q^{77} + 43876 q^{79} - 34000 q^{80} + 56060 q^{82} + 101024 q^{83} + 8350 q^{85} - 24128 q^{86} + 2904 q^{88} - 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 74552 q^{94} - 100900 q^{95} - 319746 q^{97} - 431134 q^{98}+O(q^{100})$$ 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8 + 50 * q^10 - 363 * q^11 + 450 * q^13 + 1504 * q^14 - 1360 * q^16 + 334 * q^17 - 4036 * q^19 + 700 * q^20 - 242 * q^22 + 7060 * q^23 + 1875 * q^25 - 2932 * q^26 - 8320 * q^28 - 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3644 * q^34 - 5800 * q^35 + 2250 * q^37 + 12632 * q^38 - 600 * q^40 - 10654 * q^41 - 35528 * q^43 - 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 7667 * q^49 + 1250 * q^50 - 14520 * q^52 + 12826 * q^53 - 9075 * q^55 - 17088 * q^56 - 17196 * q^58 + 81876 * q^59 - 62298 * q^61 - 109184 * q^62 - 72256 * q^64 + 11250 * q^65 - 46148 * q^67 + 35832 * q^68 + 37600 * q^70 + 64724 * q^71 + 810 * q^73 + 44796 * q^74 + 44656 * q^76 + 28072 * q^77 + 43876 * q^79 - 34000 * q^80 + 56060 * q^82 + 101024 * q^83 + 8350 * q^85 - 24128 * q^86 + 2904 * q^88 - 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 74552 * q^94 - 100900 * q^95 - 319746 * q^97 - 431134 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 15x + 11$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 10$$ v^2 - 10
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 10$$ b2 + 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.

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
 −3.76300 0.723686 4.03932
−7.52601 0 24.6408 25.0000 0 −234.126 55.3856 0 −188.150
1.2 1.44737 0 −29.9051 25.0000 0 41.5023 −89.5997 0 36.1843
1.3 8.07863 0 33.2643 25.0000 0 −39.3760 10.2141 0 201.966
 $$n$$: 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.d 3
3.b odd 2 1 165.6.a.b 3
15.d odd 2 1 825.6.a.i 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 2T_{2}^{2} - 60T_{2} + 88$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(495))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2 T^{2} + \cdots + 88$$
$3$ $$T^{3}$$
$5$ $$(T - 25)^{3}$$
$7$ $$T^{3} + 232 T^{2} + \cdots - 382608$$
$11$ $$(T + 121)^{3}$$
$13$ $$T^{3} - 450 T^{2} + \cdots + 22659488$$
$17$ $$T^{3} - 334 T^{2} + \cdots + 57782448$$
$19$ $$T^{3} + \cdots - 1630951200$$
$23$ $$T^{3} + \cdots + 24275701568$$
$29$ $$T^{3} + \cdots - 65949214584$$
$31$ $$T^{3} + \cdots - 211578448896$$
$37$ $$T^{3} + \cdots - 431879868536$$
$41$ $$T^{3} + \cdots + 7803557208$$
$43$ $$T^{3} + \cdots + 1659712050000$$
$47$ $$T^{3} + \cdots - 52162385088$$
$53$ $$T^{3} + \cdots - 2687939232856$$
$59$ $$T^{3} + \cdots + 3633753791296$$
$61$ $$T^{3} + \cdots - 12904038746056$$
$67$ $$T^{3} + \cdots - 40648408406912$$
$71$ $$T^{3} + \cdots - 8578136735360$$
$73$ $$T^{3} + \cdots - 144432126809632$$
$79$ $$T^{3} + \cdots + 351884592248992$$
$83$ $$T^{3} + \cdots - 5794291383408$$
$89$ $$T^{3} + \cdots + 246103360939432$$
$97$ $$T^{3} + \cdots + 179909862970168$$