Properties

 Label 65.4.a.d Level $65$ Weight $4$ Character orbit 65.a Self dual yes Analytic conductor $3.835$ 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] = [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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 2) q^{2} + ( - 3 \beta - 5) q^{3} + (4 \beta - 1) q^{4} - 5 q^{5} + ( - 11 \beta - 19) q^{6} + ( - 4 \beta - 18) q^{7} + ( - \beta - 6) q^{8} + (30 \beta + 25) q^{9}+O(q^{10})$$ q + (b + 2) * q^2 + (-3*b - 5) * q^3 + (4*b - 1) * q^4 - 5 * q^5 + (-11*b - 19) * q^6 + (-4*b - 18) * q^7 + (-b - 6) * q^8 + (30*b + 25) * q^9 $$q + (\beta + 2) q^{2} + ( - 3 \beta - 5) q^{3} + (4 \beta - 1) q^{4} - 5 q^{5} + ( - 11 \beta - 19) q^{6} + ( - 4 \beta - 18) q^{7} + ( - \beta - 6) q^{8} + (30 \beta + 25) q^{9} + ( - 5 \beta - 10) q^{10} + (3 \beta - 5) q^{11} + ( - 17 \beta - 31) q^{12} + 13 q^{13} + ( - 26 \beta - 48) q^{14} + (15 \beta + 25) q^{15} + ( - 40 \beta - 7) q^{16} + ( - 2 \beta - 52) q^{17} + (85 \beta + 140) q^{18} + (17 \beta - 7) q^{19} + ( - 20 \beta + 5) q^{20} + (74 \beta + 126) q^{21} + (\beta - 1) q^{22} + (3 \beta + 17) q^{23} + (23 \beta + 39) q^{24} + 25 q^{25} + (13 \beta + 26) q^{26} + ( - 144 \beta - 260) q^{27} + ( - 68 \beta - 30) q^{28} + ( - 26 \beta + 58) q^{29} + (55 \beta + 95) q^{30} + ( - 121 \beta - 53) q^{31} + ( - 79 \beta - 86) q^{32} - 2 q^{33} + ( - 56 \beta - 110) q^{34} + (20 \beta + 90) q^{35} + (70 \beta + 335) q^{36} + (180 \beta + 48) q^{37} + (27 \beta + 37) q^{38} + ( - 39 \beta - 65) q^{39} + (5 \beta + 30) q^{40} + ( - 142 \beta - 60) q^{41} + (274 \beta + 474) q^{42} + (33 \beta - 361) q^{43} + ( - 23 \beta + 41) q^{44} + ( - 150 \beta - 125) q^{45} + (23 \beta + 43) q^{46} + (84 \beta - 230) q^{47} + (221 \beta + 395) q^{48} + (144 \beta + 29) q^{49} + (25 \beta + 50) q^{50} + (166 \beta + 278) q^{51} + (52 \beta - 13) q^{52} + ( - 50 \beta - 276) q^{53} + ( - 548 \beta - 952) q^{54} + ( - 15 \beta + 25) q^{55} + (42 \beta + 120) q^{56} + ( - 64 \beta - 118) q^{57} + (6 \beta + 38) q^{58} + (211 \beta + 171) q^{59} + (85 \beta + 155) q^{60} + ( - 90 \beta - 134) q^{61} + ( - 295 \beta - 469) q^{62} + ( - 640 \beta - 810) q^{63} + (76 \beta - 353) q^{64} - 65 q^{65} + ( - 2 \beta - 4) q^{66} + (378 \beta - 4) q^{67} + ( - 206 \beta + 28) q^{68} + ( - 66 \beta - 112) q^{69} + (130 \beta + 240) q^{70} + ( - 139 \beta + 117) q^{71} + ( - 205 \beta - 240) q^{72} + (100 \beta - 168) q^{73} + (408 \beta + 636) q^{74} + ( - 75 \beta - 125) q^{75} + ( - 45 \beta + 211) q^{76} + ( - 34 \beta + 54) q^{77} + ( - 143 \beta - 247) q^{78} + (330 \beta - 434) q^{79} + (200 \beta + 35) q^{80} + (690 \beta + 1921) q^{81} + ( - 344 \beta - 546) q^{82} + ( - 92 \beta + 566) q^{83} + (430 \beta + 762) q^{84} + (10 \beta + 260) q^{85} + ( - 295 \beta - 623) q^{86} + ( - 44 \beta - 56) q^{87} + ( - 13 \beta + 21) q^{88} + ( - 292 \beta + 1090) q^{89} + ( - 425 \beta - 700) q^{90} + ( - 52 \beta - 234) q^{91} + (65 \beta + 19) q^{92} + (764 \beta + 1354) q^{93} + ( - 62 \beta - 208) q^{94} + ( - 85 \beta + 35) q^{95} + (653 \beta + 1141) q^{96} + ( - 672 \beta - 126) q^{97} + (317 \beta + 490) q^{98} + ( - 75 \beta + 145) q^{99}+O(q^{100})$$ q + (b + 2) * q^2 + (-3*b - 5) * q^3 + (4*b - 1) * q^4 - 5 * q^5 + (-11*b - 19) * q^6 + (-4*b - 18) * q^7 + (-b - 6) * q^8 + (30*b + 25) * q^9 + (-5*b - 10) * q^10 + (3*b - 5) * q^11 + (-17*b - 31) * q^12 + 13 * q^13 + (-26*b - 48) * q^14 + (15*b + 25) * q^15 + (-40*b - 7) * q^16 + (-2*b - 52) * q^17 + (85*b + 140) * q^18 + (17*b - 7) * q^19 + (-20*b + 5) * q^20 + (74*b + 126) * q^21 + (b - 1) * q^22 + (3*b + 17) * q^23 + (23*b + 39) * q^24 + 25 * q^25 + (13*b + 26) * q^26 + (-144*b - 260) * q^27 + (-68*b - 30) * q^28 + (-26*b + 58) * q^29 + (55*b + 95) * q^30 + (-121*b - 53) * q^31 + (-79*b - 86) * q^32 - 2 * q^33 + (-56*b - 110) * q^34 + (20*b + 90) * q^35 + (70*b + 335) * q^36 + (180*b + 48) * q^37 + (27*b + 37) * q^38 + (-39*b - 65) * q^39 + (5*b + 30) * q^40 + (-142*b - 60) * q^41 + (274*b + 474) * q^42 + (33*b - 361) * q^43 + (-23*b + 41) * q^44 + (-150*b - 125) * q^45 + (23*b + 43) * q^46 + (84*b - 230) * q^47 + (221*b + 395) * q^48 + (144*b + 29) * q^49 + (25*b + 50) * q^50 + (166*b + 278) * q^51 + (52*b - 13) * q^52 + (-50*b - 276) * q^53 + (-548*b - 952) * q^54 + (-15*b + 25) * q^55 + (42*b + 120) * q^56 + (-64*b - 118) * q^57 + (6*b + 38) * q^58 + (211*b + 171) * q^59 + (85*b + 155) * q^60 + (-90*b - 134) * q^61 + (-295*b - 469) * q^62 + (-640*b - 810) * q^63 + (76*b - 353) * q^64 - 65 * q^65 + (-2*b - 4) * q^66 + (378*b - 4) * q^67 + (-206*b + 28) * q^68 + (-66*b - 112) * q^69 + (130*b + 240) * q^70 + (-139*b + 117) * q^71 + (-205*b - 240) * q^72 + (100*b - 168) * q^73 + (408*b + 636) * q^74 + (-75*b - 125) * q^75 + (-45*b + 211) * q^76 + (-34*b + 54) * q^77 + (-143*b - 247) * q^78 + (330*b - 434) * q^79 + (200*b + 35) * q^80 + (690*b + 1921) * q^81 + (-344*b - 546) * q^82 + (-92*b + 566) * q^83 + (430*b + 762) * q^84 + (10*b + 260) * q^85 + (-295*b - 623) * q^86 + (-44*b - 56) * q^87 + (-13*b + 21) * q^88 + (-292*b + 1090) * q^89 + (-425*b - 700) * q^90 + (-52*b - 234) * q^91 + (65*b + 19) * q^92 + (764*b + 1354) * q^93 + (-62*b - 208) * q^94 + (-85*b + 35) * q^95 + (653*b + 1141) * q^96 + (-672*b - 126) * q^97 + (317*b + 490) * q^98 + (-75*b + 145) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 10 q^{3} - 2 q^{4} - 10 q^{5} - 38 q^{6} - 36 q^{7} - 12 q^{8} + 50 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 10 * q^3 - 2 * q^4 - 10 * q^5 - 38 * q^6 - 36 * q^7 - 12 * q^8 + 50 * q^9 $$2 q + 4 q^{2} - 10 q^{3} - 2 q^{4} - 10 q^{5} - 38 q^{6} - 36 q^{7} - 12 q^{8} + 50 q^{9} - 20 q^{10} - 10 q^{11} - 62 q^{12} + 26 q^{13} - 96 q^{14} + 50 q^{15} - 14 q^{16} - 104 q^{17} + 280 q^{18} - 14 q^{19} + 10 q^{20} + 252 q^{21} - 2 q^{22} + 34 q^{23} + 78 q^{24} + 50 q^{25} + 52 q^{26} - 520 q^{27} - 60 q^{28} + 116 q^{29} + 190 q^{30} - 106 q^{31} - 172 q^{32} - 4 q^{33} - 220 q^{34} + 180 q^{35} + 670 q^{36} + 96 q^{37} + 74 q^{38} - 130 q^{39} + 60 q^{40} - 120 q^{41} + 948 q^{42} - 722 q^{43} + 82 q^{44} - 250 q^{45} + 86 q^{46} - 460 q^{47} + 790 q^{48} + 58 q^{49} + 100 q^{50} + 556 q^{51} - 26 q^{52} - 552 q^{53} - 1904 q^{54} + 50 q^{55} + 240 q^{56} - 236 q^{57} + 76 q^{58} + 342 q^{59} + 310 q^{60} - 268 q^{61} - 938 q^{62} - 1620 q^{63} - 706 q^{64} - 130 q^{65} - 8 q^{66} - 8 q^{67} + 56 q^{68} - 224 q^{69} + 480 q^{70} + 234 q^{71} - 480 q^{72} - 336 q^{73} + 1272 q^{74} - 250 q^{75} + 422 q^{76} + 108 q^{77} - 494 q^{78} - 868 q^{79} + 70 q^{80} + 3842 q^{81} - 1092 q^{82} + 1132 q^{83} + 1524 q^{84} + 520 q^{85} - 1246 q^{86} - 112 q^{87} + 42 q^{88} + 2180 q^{89} - 1400 q^{90} - 468 q^{91} + 38 q^{92} + 2708 q^{93} - 416 q^{94} + 70 q^{95} + 2282 q^{96} - 252 q^{97} + 980 q^{98} + 290 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - 10 * q^3 - 2 * q^4 - 10 * q^5 - 38 * q^6 - 36 * q^7 - 12 * q^8 + 50 * q^9 - 20 * q^10 - 10 * q^11 - 62 * q^12 + 26 * q^13 - 96 * q^14 + 50 * q^15 - 14 * q^16 - 104 * q^17 + 280 * q^18 - 14 * q^19 + 10 * q^20 + 252 * q^21 - 2 * q^22 + 34 * q^23 + 78 * q^24 + 50 * q^25 + 52 * q^26 - 520 * q^27 - 60 * q^28 + 116 * q^29 + 190 * q^30 - 106 * q^31 - 172 * q^32 - 4 * q^33 - 220 * q^34 + 180 * q^35 + 670 * q^36 + 96 * q^37 + 74 * q^38 - 130 * q^39 + 60 * q^40 - 120 * q^41 + 948 * q^42 - 722 * q^43 + 82 * q^44 - 250 * q^45 + 86 * q^46 - 460 * q^47 + 790 * q^48 + 58 * q^49 + 100 * q^50 + 556 * q^51 - 26 * q^52 - 552 * q^53 - 1904 * q^54 + 50 * q^55 + 240 * q^56 - 236 * q^57 + 76 * q^58 + 342 * q^59 + 310 * q^60 - 268 * q^61 - 938 * q^62 - 1620 * q^63 - 706 * q^64 - 130 * q^65 - 8 * q^66 - 8 * q^67 + 56 * q^68 - 224 * q^69 + 480 * q^70 + 234 * q^71 - 480 * q^72 - 336 * q^73 + 1272 * q^74 - 250 * q^75 + 422 * q^76 + 108 * q^77 - 494 * q^78 - 868 * q^79 + 70 * q^80 + 3842 * q^81 - 1092 * q^82 + 1132 * q^83 + 1524 * q^84 + 520 * q^85 - 1246 * q^86 - 112 * q^87 + 42 * q^88 + 2180 * q^89 - 1400 * q^90 - 468 * q^91 + 38 * q^92 + 2708 * q^93 - 416 * q^94 + 70 * q^95 + 2282 * q^96 - 252 * q^97 + 980 * q^98 + 290 * 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
 −1.73205 1.73205
0.267949 0.196152 −7.92820 −5.00000 0.0525589 −11.0718 −4.26795 −26.9615 −1.33975
1.2 3.73205 −10.1962 5.92820 −5.00000 −38.0526 −24.9282 −7.73205 76.9615 −18.6603
 $$n$$: 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.d 2
3.b odd 2 1 585.4.a.f 2
4.b odd 2 1 1040.4.a.o 2
5.b even 2 1 325.4.a.e 2
5.c odd 4 2 325.4.b.c 4
13.b even 2 1 845.4.a.c 2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 4T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(65))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 1$$
$3$ $$T^{2} + 10T - 2$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2} + 36T + 276$$
$11$ $$T^{2} + 10T - 2$$
$13$ $$(T - 13)^{2}$$
$17$ $$T^{2} + 104T + 2692$$
$19$ $$T^{2} + 14T - 818$$
$23$ $$T^{2} - 34T + 262$$
$29$ $$T^{2} - 116T + 1336$$
$31$ $$T^{2} + 106T - 41114$$
$37$ $$T^{2} - 96T - 94896$$
$41$ $$T^{2} + 120T - 56892$$
$43$ $$T^{2} + 722T + 127054$$
$47$ $$T^{2} + 460T + 31732$$
$53$ $$T^{2} + 552T + 68676$$
$59$ $$T^{2} - 342T - 104322$$
$61$ $$T^{2} + 268T - 6344$$
$67$ $$T^{2} + 8T - 428636$$
$71$ $$T^{2} - 234T - 44274$$
$73$ $$T^{2} + 336T - 1776$$
$79$ $$T^{2} + 868T - 138344$$
$83$ $$T^{2} - 1132 T + 294964$$
$89$ $$T^{2} - 2180 T + 932308$$
$97$ $$T^{2} + 252 T - 1338876$$