# Properties

 Label 65.4.a.c Level $65$ Weight $4$ Character orbit 65.a Self dual yes Analytic conductor $3.835$ Analytic rank $0$ 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] = [65,4,Mod(1,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("65.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 65.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: $$3.83512415037$$ Analytic rank: $$0$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (4 \beta - 2) q^{3} + (\beta - 4) q^{4} - 5 q^{5} + (2 \beta + 16) q^{6} + ( - 2 \beta + 30) q^{7} + ( - 11 \beta + 4) q^{8} + 41 q^{9}+O(q^{10})$$ q + b * q^2 + (4*b - 2) * q^3 + (b - 4) * q^4 - 5 * q^5 + (2*b + 16) * q^6 + (-2*b + 30) * q^7 + (-11*b + 4) * q^8 + 41 * q^9 $$q + \beta q^{2} + (4 \beta - 2) q^{3} + (\beta - 4) q^{4} - 5 q^{5} + (2 \beta + 16) q^{6} + ( - 2 \beta + 30) q^{7} + ( - 11 \beta + 4) q^{8} + 41 q^{9} - 5 \beta q^{10} + (2 \beta - 44) q^{11} + ( - 14 \beta + 24) q^{12} - 13 q^{13} + (28 \beta - 8) q^{14} + ( - 20 \beta + 10) q^{15} + ( - 15 \beta - 12) q^{16} + ( - 32 \beta + 2) q^{17} + 41 \beta q^{18} + (22 \beta + 72) q^{19} + ( - 5 \beta + 20) q^{20} + (116 \beta - 92) q^{21} + ( - 42 \beta + 8) q^{22} + ( - 64 \beta + 62) q^{23} + ( - 6 \beta - 184) q^{24} + 25 q^{25} - 13 \beta q^{26} + (56 \beta - 28) q^{27} + (36 \beta - 128) q^{28} + (28 \beta + 46) q^{29} + ( - 10 \beta - 80) q^{30} + ( - 126 \beta + 24) q^{31} + (61 \beta - 92) q^{32} + ( - 172 \beta + 120) q^{33} + ( - 30 \beta - 128) q^{34} + (10 \beta - 150) q^{35} + (41 \beta - 164) q^{36} + (36 \beta + 162) q^{37} + (94 \beta + 88) q^{38} + ( - 52 \beta + 26) q^{39} + (55 \beta - 20) q^{40} + (124 \beta - 98) q^{41} + (24 \beta + 464) q^{42} + ( - 40 \beta - 2) q^{43} + ( - 50 \beta + 184) q^{44} - 205 q^{45} + ( - 2 \beta - 256) q^{46} + (62 \beta + 150) q^{47} + ( - 78 \beta - 216) q^{48} + ( - 116 \beta + 573) q^{49} + 25 \beta q^{50} + ( - 56 \beta - 516) q^{51} + ( - 13 \beta + 52) q^{52} + (96 \beta - 246) q^{53} + (28 \beta + 224) q^{54} + ( - 10 \beta + 220) q^{55} + ( - 316 \beta + 208) q^{56} + (332 \beta + 208) q^{57} + (74 \beta + 112) q^{58} + ( - 174 \beta + 96) q^{59} + (70 \beta - 120) q^{60} + 442 q^{61} + ( - 102 \beta - 504) q^{62} + ( - 82 \beta + 1230) q^{63} + (89 \beta + 340) q^{64} + 65 q^{65} + ( - 52 \beta - 688) q^{66} + ( - 210 \beta + 766) q^{67} + (98 \beta - 136) q^{68} + (120 \beta - 1148) q^{69} + ( - 140 \beta + 40) q^{70} + (314 \beta + 160) q^{71} + ( - 451 \beta + 164) q^{72} + (336 \beta - 198) q^{73} + (198 \beta + 144) q^{74} + (100 \beta - 50) q^{75} + (6 \beta - 200) q^{76} + (144 \beta - 1336) q^{77} + ( - 26 \beta - 208) q^{78} + (12 \beta - 96) q^{79} + (75 \beta + 60) q^{80} - 155 q^{81} + (26 \beta + 496) q^{82} + ( - 218 \beta + 466) q^{83} + ( - 440 \beta + 832) q^{84} + (160 \beta - 10) q^{85} + ( - 42 \beta - 160) q^{86} + (240 \beta + 356) q^{87} + (470 \beta - 264) q^{88} + (120 \beta - 486) q^{89} - 205 \beta q^{90} + (26 \beta - 390) q^{91} + (254 \beta - 504) q^{92} + ( - 156 \beta - 2064) q^{93} + (212 \beta + 248) q^{94} + ( - 110 \beta - 360) q^{95} + ( - 246 \beta + 1160) q^{96} + ( - 836 \beta + 434) q^{97} + (457 \beta - 464) q^{98} + (82 \beta - 1804) q^{99} +O(q^{100})$$ q + b * q^2 + (4*b - 2) * q^3 + (b - 4) * q^4 - 5 * q^5 + (2*b + 16) * q^6 + (-2*b + 30) * q^7 + (-11*b + 4) * q^8 + 41 * q^9 - 5*b * q^10 + (2*b - 44) * q^11 + (-14*b + 24) * q^12 - 13 * q^13 + (28*b - 8) * q^14 + (-20*b + 10) * q^15 + (-15*b - 12) * q^16 + (-32*b + 2) * q^17 + 41*b * q^18 + (22*b + 72) * q^19 + (-5*b + 20) * q^20 + (116*b - 92) * q^21 + (-42*b + 8) * q^22 + (-64*b + 62) * q^23 + (-6*b - 184) * q^24 + 25 * q^25 - 13*b * q^26 + (56*b - 28) * q^27 + (36*b - 128) * q^28 + (28*b + 46) * q^29 + (-10*b - 80) * q^30 + (-126*b + 24) * q^31 + (61*b - 92) * q^32 + (-172*b + 120) * q^33 + (-30*b - 128) * q^34 + (10*b - 150) * q^35 + (41*b - 164) * q^36 + (36*b + 162) * q^37 + (94*b + 88) * q^38 + (-52*b + 26) * q^39 + (55*b - 20) * q^40 + (124*b - 98) * q^41 + (24*b + 464) * q^42 + (-40*b - 2) * q^43 + (-50*b + 184) * q^44 - 205 * q^45 + (-2*b - 256) * q^46 + (62*b + 150) * q^47 + (-78*b - 216) * q^48 + (-116*b + 573) * q^49 + 25*b * q^50 + (-56*b - 516) * q^51 + (-13*b + 52) * q^52 + (96*b - 246) * q^53 + (28*b + 224) * q^54 + (-10*b + 220) * q^55 + (-316*b + 208) * q^56 + (332*b + 208) * q^57 + (74*b + 112) * q^58 + (-174*b + 96) * q^59 + (70*b - 120) * q^60 + 442 * q^61 + (-102*b - 504) * q^62 + (-82*b + 1230) * q^63 + (89*b + 340) * q^64 + 65 * q^65 + (-52*b - 688) * q^66 + (-210*b + 766) * q^67 + (98*b - 136) * q^68 + (120*b - 1148) * q^69 + (-140*b + 40) * q^70 + (314*b + 160) * q^71 + (-451*b + 164) * q^72 + (336*b - 198) * q^73 + (198*b + 144) * q^74 + (100*b - 50) * q^75 + (6*b - 200) * q^76 + (144*b - 1336) * q^77 + (-26*b - 208) * q^78 + (12*b - 96) * q^79 + (75*b + 60) * q^80 - 155 * q^81 + (26*b + 496) * q^82 + (-218*b + 466) * q^83 + (-440*b + 832) * q^84 + (160*b - 10) * q^85 + (-42*b - 160) * q^86 + (240*b + 356) * q^87 + (470*b - 264) * q^88 + (120*b - 486) * q^89 - 205*b * q^90 + (26*b - 390) * q^91 + (254*b - 504) * q^92 + (-156*b - 2064) * q^93 + (212*b + 248) * q^94 + (-110*b - 360) * q^95 + (-246*b + 1160) * q^96 + (-836*b + 434) * q^97 + (457*b - 464) * q^98 + (82*b - 1804) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 7 q^{4} - 10 q^{5} + 34 q^{6} + 58 q^{7} - 3 q^{8} + 82 q^{9}+O(q^{10})$$ 2 * q + q^2 - 7 * q^4 - 10 * q^5 + 34 * q^6 + 58 * q^7 - 3 * q^8 + 82 * q^9 $$2 q + q^{2} - 7 q^{4} - 10 q^{5} + 34 q^{6} + 58 q^{7} - 3 q^{8} + 82 q^{9} - 5 q^{10} - 86 q^{11} + 34 q^{12} - 26 q^{13} + 12 q^{14} - 39 q^{16} - 28 q^{17} + 41 q^{18} + 166 q^{19} + 35 q^{20} - 68 q^{21} - 26 q^{22} + 60 q^{23} - 374 q^{24} + 50 q^{25} - 13 q^{26} - 220 q^{28} + 120 q^{29} - 170 q^{30} - 78 q^{31} - 123 q^{32} + 68 q^{33} - 286 q^{34} - 290 q^{35} - 287 q^{36} + 360 q^{37} + 270 q^{38} + 15 q^{40} - 72 q^{41} + 952 q^{42} - 44 q^{43} + 318 q^{44} - 410 q^{45} - 514 q^{46} + 362 q^{47} - 510 q^{48} + 1030 q^{49} + 25 q^{50} - 1088 q^{51} + 91 q^{52} - 396 q^{53} + 476 q^{54} + 430 q^{55} + 100 q^{56} + 748 q^{57} + 298 q^{58} + 18 q^{59} - 170 q^{60} + 884 q^{61} - 1110 q^{62} + 2378 q^{63} + 769 q^{64} + 130 q^{65} - 1428 q^{66} + 1322 q^{67} - 174 q^{68} - 2176 q^{69} - 60 q^{70} + 634 q^{71} - 123 q^{72} - 60 q^{73} + 486 q^{74} - 394 q^{76} - 2528 q^{77} - 442 q^{78} - 180 q^{79} + 195 q^{80} - 310 q^{81} + 1018 q^{82} + 714 q^{83} + 1224 q^{84} + 140 q^{85} - 362 q^{86} + 952 q^{87} - 58 q^{88} - 852 q^{89} - 205 q^{90} - 754 q^{91} - 754 q^{92} - 4284 q^{93} + 708 q^{94} - 830 q^{95} + 2074 q^{96} + 32 q^{97} - 471 q^{98} - 3526 q^{99}+O(q^{100})$$ 2 * q + q^2 - 7 * q^4 - 10 * q^5 + 34 * q^6 + 58 * q^7 - 3 * q^8 + 82 * q^9 - 5 * q^10 - 86 * q^11 + 34 * q^12 - 26 * q^13 + 12 * q^14 - 39 * q^16 - 28 * q^17 + 41 * q^18 + 166 * q^19 + 35 * q^20 - 68 * q^21 - 26 * q^22 + 60 * q^23 - 374 * q^24 + 50 * q^25 - 13 * q^26 - 220 * q^28 + 120 * q^29 - 170 * q^30 - 78 * q^31 - 123 * q^32 + 68 * q^33 - 286 * q^34 - 290 * q^35 - 287 * q^36 + 360 * q^37 + 270 * q^38 + 15 * q^40 - 72 * q^41 + 952 * q^42 - 44 * q^43 + 318 * q^44 - 410 * q^45 - 514 * q^46 + 362 * q^47 - 510 * q^48 + 1030 * q^49 + 25 * q^50 - 1088 * q^51 + 91 * q^52 - 396 * q^53 + 476 * q^54 + 430 * q^55 + 100 * q^56 + 748 * q^57 + 298 * q^58 + 18 * q^59 - 170 * q^60 + 884 * q^61 - 1110 * q^62 + 2378 * q^63 + 769 * q^64 + 130 * q^65 - 1428 * q^66 + 1322 * q^67 - 174 * q^68 - 2176 * q^69 - 60 * q^70 + 634 * q^71 - 123 * q^72 - 60 * q^73 + 486 * q^74 - 394 * q^76 - 2528 * q^77 - 442 * q^78 - 180 * q^79 + 195 * q^80 - 310 * q^81 + 1018 * q^82 + 714 * q^83 + 1224 * q^84 + 140 * q^85 - 362 * q^86 + 952 * q^87 - 58 * q^88 - 852 * q^89 - 205 * q^90 - 754 * q^91 - 754 * q^92 - 4284 * q^93 + 708 * q^94 - 830 * q^95 + 2074 * q^96 + 32 * q^97 - 471 * q^98 - 3526 * q^99

## 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 −8.24621 −5.56155 −5.00000 12.8769 33.1231 21.1771 41.0000 7.80776
1.2 2.56155 8.24621 −1.43845 −5.00000 21.1231 24.8769 −24.1771 41.0000 −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
$$5$$ $$1$$
$$13$$ $$1$$

## 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 65.4.a.c 2
3.b odd 2 1 585.4.a.h 2
4.b odd 2 1 1040.4.a.k 2
5.b even 2 1 325.4.a.g 2
5.c odd 4 2 325.4.b.f 4
13.b even 2 1 845.4.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.a.c 2 1.a even 1 1 trivial
325.4.a.g 2 5.b even 2 1
325.4.b.f 4 5.c odd 4 2
585.4.a.h 2 3.b odd 2 1
845.4.a.d 2 13.b even 2 1
1040.4.a.k 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 4$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(65))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$T^{2} - 68$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2} - 58T + 824$$
$11$ $$T^{2} + 86T + 1832$$
$13$ $$(T + 13)^{2}$$
$17$ $$T^{2} + 28T - 4156$$
$19$ $$T^{2} - 166T + 4832$$
$23$ $$T^{2} - 60T - 16508$$
$29$ $$T^{2} - 120T + 268$$
$31$ $$T^{2} + 78T - 65952$$
$37$ $$T^{2} - 360T + 26892$$
$41$ $$T^{2} + 72T - 64052$$
$43$ $$T^{2} + 44T - 6316$$
$47$ $$T^{2} - 362T + 16424$$
$53$ $$T^{2} + 396T + 36$$
$59$ $$T^{2} - 18T - 128592$$
$61$ $$(T - 442)^{2}$$
$67$ $$T^{2} - 1322 T + 249496$$
$71$ $$T^{2} - 634T - 318544$$
$73$ $$T^{2} + 60T - 478908$$
$79$ $$T^{2} + 180T + 7488$$
$83$ $$T^{2} - 714T - 74528$$
$89$ $$T^{2} + 852T + 120276$$
$97$ $$T^{2} - 32T - 2970052$$