# Properties

 Label 495.4.a.d Level $495$ Weight $4$ Character orbit 495.a Self dual yes Analytic conductor $29.206$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,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, 4, names="a")

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

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$29.2059454528$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} + (4 \beta - 4) q^{7} + ( - 11 \beta + 4) q^{8}+O(q^{10})$$ q + b * q^2 + (b - 4) * q^4 + 5 * q^5 + (4*b - 4) * q^7 + (-11*b + 4) * q^8 $$q + \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} + (4 \beta - 4) q^{7} + ( - 11 \beta + 4) q^{8} + 5 \beta q^{10} + 11 q^{11} + ( - 2 \beta - 44) q^{13} + 16 q^{14} + ( - 15 \beta - 12) q^{16} + ( - 44 \beta + 30) q^{17} + ( - 22 \beta - 74) q^{19} + (5 \beta - 20) q^{20} + 11 \beta q^{22} + (60 \beta + 32) q^{23} + 25 q^{25} + ( - 46 \beta - 8) q^{26} + ( - 16 \beta + 32) q^{28} + ( - 34 \beta + 96) q^{29} + ( - 12 \beta + 36) q^{31} + (61 \beta - 92) q^{32} + ( - 14 \beta - 176) q^{34} + (20 \beta - 20) q^{35} + ( - 112 \beta - 130) q^{37} + ( - 96 \beta - 88) q^{38} + ( - 55 \beta + 20) q^{40} + (154 \beta - 96) q^{41} + ( - 124 \beta - 196) q^{43} + (11 \beta - 44) q^{44} + (92 \beta + 240) q^{46} + ( - 216 \beta - 4) q^{47} + ( - 16 \beta - 263) q^{49} + 25 \beta q^{50} + ( - 38 \beta + 168) q^{52} + (196 \beta - 334) q^{53} + 55 q^{55} + (16 \beta - 192) q^{56} + (62 \beta - 136) q^{58} + ( - 240 \beta - 4) q^{59} + (364 \beta - 146) q^{61} + (24 \beta - 48) q^{62} + (89 \beta + 340) q^{64} + ( - 10 \beta - 220) q^{65} + (16 \beta - 380) q^{67} + (162 \beta - 296) q^{68} + 80 q^{70} + ( - 44 \beta - 1008) q^{71} + (58 \beta - 272) q^{73} + ( - 242 \beta - 448) q^{74} + ( - 8 \beta + 208) q^{76} + (44 \beta - 44) q^{77} + ( - 306 \beta + 474) q^{79} + ( - 75 \beta - 60) q^{80} + (58 \beta + 616) q^{82} + (426 \beta - 70) q^{83} + ( - 220 \beta + 150) q^{85} + ( - 320 \beta - 496) q^{86} + ( - 121 \beta + 44) q^{88} + (128 \beta - 186) q^{89} + ( - 176 \beta + 144) q^{91} + ( - 148 \beta + 112) q^{92} + ( - 220 \beta - 864) q^{94} + ( - 110 \beta - 370) q^{95} + (428 \beta - 298) q^{97} + ( - 279 \beta - 64) q^{98}+O(q^{100})$$ q + b * q^2 + (b - 4) * q^4 + 5 * q^5 + (4*b - 4) * q^7 + (-11*b + 4) * q^8 + 5*b * q^10 + 11 * q^11 + (-2*b - 44) * q^13 + 16 * q^14 + (-15*b - 12) * q^16 + (-44*b + 30) * q^17 + (-22*b - 74) * q^19 + (5*b - 20) * q^20 + 11*b * q^22 + (60*b + 32) * q^23 + 25 * q^25 + (-46*b - 8) * q^26 + (-16*b + 32) * q^28 + (-34*b + 96) * q^29 + (-12*b + 36) * q^31 + (61*b - 92) * q^32 + (-14*b - 176) * q^34 + (20*b - 20) * q^35 + (-112*b - 130) * q^37 + (-96*b - 88) * q^38 + (-55*b + 20) * q^40 + (154*b - 96) * q^41 + (-124*b - 196) * q^43 + (11*b - 44) * q^44 + (92*b + 240) * q^46 + (-216*b - 4) * q^47 + (-16*b - 263) * q^49 + 25*b * q^50 + (-38*b + 168) * q^52 + (196*b - 334) * q^53 + 55 * q^55 + (16*b - 192) * q^56 + (62*b - 136) * q^58 + (-240*b - 4) * q^59 + (364*b - 146) * q^61 + (24*b - 48) * q^62 + (89*b + 340) * q^64 + (-10*b - 220) * q^65 + (16*b - 380) * q^67 + (162*b - 296) * q^68 + 80 * q^70 + (-44*b - 1008) * q^71 + (58*b - 272) * q^73 + (-242*b - 448) * q^74 + (-8*b + 208) * q^76 + (44*b - 44) * q^77 + (-306*b + 474) * q^79 + (-75*b - 60) * q^80 + (58*b + 616) * q^82 + (426*b - 70) * q^83 + (-220*b + 150) * q^85 + (-320*b - 496) * q^86 + (-121*b + 44) * q^88 + (128*b - 186) * q^89 + (-176*b + 144) * q^91 + (-148*b + 112) * q^92 + (-220*b - 864) * q^94 + (-110*b - 370) * q^95 + (428*b - 298) * q^97 + (-279*b - 64) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10})$$ 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 4 * q^7 - 3 * q^8 $$2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8} + 5 q^{10} + 22 q^{11} - 90 q^{13} + 32 q^{14} - 39 q^{16} + 16 q^{17} - 170 q^{19} - 35 q^{20} + 11 q^{22} + 124 q^{23} + 50 q^{25} - 62 q^{26} + 48 q^{28} + 158 q^{29} + 60 q^{31} - 123 q^{32} - 366 q^{34} - 20 q^{35} - 372 q^{37} - 272 q^{38} - 15 q^{40} - 38 q^{41} - 516 q^{43} - 77 q^{44} + 572 q^{46} - 224 q^{47} - 542 q^{49} + 25 q^{50} + 298 q^{52} - 472 q^{53} + 110 q^{55} - 368 q^{56} - 210 q^{58} - 248 q^{59} + 72 q^{61} - 72 q^{62} + 769 q^{64} - 450 q^{65} - 744 q^{67} - 430 q^{68} + 160 q^{70} - 2060 q^{71} - 486 q^{73} - 1138 q^{74} + 408 q^{76} - 44 q^{77} + 642 q^{79} - 195 q^{80} + 1290 q^{82} + 286 q^{83} + 80 q^{85} - 1312 q^{86} - 33 q^{88} - 244 q^{89} + 112 q^{91} + 76 q^{92} - 1948 q^{94} - 850 q^{95} - 168 q^{97} - 407 q^{98}+O(q^{100})$$ 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 4 * q^7 - 3 * q^8 + 5 * q^10 + 22 * q^11 - 90 * q^13 + 32 * q^14 - 39 * q^16 + 16 * q^17 - 170 * q^19 - 35 * q^20 + 11 * q^22 + 124 * q^23 + 50 * q^25 - 62 * q^26 + 48 * q^28 + 158 * q^29 + 60 * q^31 - 123 * q^32 - 366 * q^34 - 20 * q^35 - 372 * q^37 - 272 * q^38 - 15 * q^40 - 38 * q^41 - 516 * q^43 - 77 * q^44 + 572 * q^46 - 224 * q^47 - 542 * q^49 + 25 * q^50 + 298 * q^52 - 472 * q^53 + 110 * q^55 - 368 * q^56 - 210 * q^58 - 248 * q^59 + 72 * q^61 - 72 * q^62 + 769 * q^64 - 450 * q^65 - 744 * q^67 - 430 * q^68 + 160 * q^70 - 2060 * q^71 - 486 * q^73 - 1138 * q^74 + 408 * q^76 - 44 * q^77 + 642 * q^79 - 195 * q^80 + 1290 * q^82 + 286 * q^83 + 80 * q^85 - 1312 * q^86 - 33 * q^88 - 244 * q^89 + 112 * q^91 + 76 * q^92 - 1948 * q^94 - 850 * q^95 - 168 * q^97 - 407 * q^98

## 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
 −1.56155 2.56155
−1.56155 0 −5.56155 5.00000 0 −10.2462 21.1771 0 −7.80776
1.2 2.56155 0 −1.43845 5.00000 0 6.24621 −24.1771 0 12.8078
 $$n$$: 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 495.4.a.d 2
3.b odd 2 1 165.4.a.c 2
5.b even 2 1 2475.4.a.n 2
15.d odd 2 1 825.4.a.m 2
15.e even 4 2 825.4.c.j 4
33.d even 2 1 1815.4.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 3.b odd 2 1
495.4.a.d 2 1.a even 1 1 trivial
825.4.a.m 2 15.d odd 2 1
825.4.c.j 4 15.e even 4 2
1815.4.a.n 2 33.d even 2 1
2475.4.a.n 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(495))$$:

 $$T_{2}^{2} - T_{2} - 4$$ T2^2 - T2 - 4 $$T_{7}^{2} + 4T_{7} - 64$$ T7^2 + 4*T7 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$T^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2} + 4T - 64$$
$11$ $$(T - 11)^{2}$$
$13$ $$T^{2} + 90T + 2008$$
$17$ $$T^{2} - 16T - 8164$$
$19$ $$T^{2} + 170T + 5168$$
$23$ $$T^{2} - 124T - 11456$$
$29$ $$T^{2} - 158T + 1328$$
$31$ $$T^{2} - 60T + 288$$
$37$ $$T^{2} + 372T - 18716$$
$41$ $$T^{2} + 38T - 100432$$
$43$ $$T^{2} + 516T + 1216$$
$47$ $$T^{2} + 224T - 185744$$
$53$ $$T^{2} + 472T - 107572$$
$59$ $$T^{2} + 248T - 229424$$
$61$ $$T^{2} - 72T - 561812$$
$67$ $$T^{2} + 744T + 137296$$
$71$ $$T^{2} + 2060 T + 1052672$$
$73$ $$T^{2} + 486T + 44752$$
$79$ $$T^{2} - 642T - 294912$$
$83$ $$T^{2} - 286T - 750824$$
$89$ $$T^{2} + 244T - 54748$$
$97$ $$T^{2} + 168T - 771476$$