# Properties

 Label 105.4.a.d Level $105$ Weight $4$ Character orbit 105.a Self dual yes Analytic conductor $6.195$ 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] = [105,4,Mod(1,105)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(105, 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("105.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.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: $$6.19520055060$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 2) q^{2} + 3 q^{3} + (4 \beta + 1) q^{4} - 5 q^{5} + ( - 3 \beta - 6) q^{6} - 7 q^{7} + ( - \beta - 6) q^{8} + 9 q^{9}+O(q^{10})$$ q + (-b - 2) * q^2 + 3 * q^3 + (4*b + 1) * q^4 - 5 * q^5 + (-3*b - 6) * q^6 - 7 * q^7 + (-b - 6) * q^8 + 9 * q^9 $$q + ( - \beta - 2) q^{2} + 3 q^{3} + (4 \beta + 1) q^{4} - 5 q^{5} + ( - 3 \beta - 6) q^{6} - 7 q^{7} + ( - \beta - 6) q^{8} + 9 q^{9} + (5 \beta + 10) q^{10} + (2 \beta - 46) q^{11} + (12 \beta + 3) q^{12} + (38 \beta + 4) q^{13} + (7 \beta + 14) q^{14} - 15 q^{15} + ( - 24 \beta + 9) q^{16} + ( - 44 \beta - 22) q^{17} + ( - 9 \beta - 18) q^{18} + ( - 26 \beta - 54) q^{19} + ( - 20 \beta - 5) q^{20} - 21 q^{21} + (42 \beta + 82) q^{22} + (20 \beta - 160) q^{23} + ( - 3 \beta - 18) q^{24} + 25 q^{25} + ( - 80 \beta - 198) q^{26} + 27 q^{27} + ( - 28 \beta - 7) q^{28} + ( - 12 \beta - 118) q^{29} + (15 \beta + 30) q^{30} + ( - 102 \beta - 30) q^{31} + (47 \beta + 150) q^{32} + (6 \beta - 138) q^{33} + (110 \beta + 264) q^{34} + 35 q^{35} + (36 \beta + 9) q^{36} + ( - 24 \beta + 102) q^{37} + (106 \beta + 238) q^{38} + (114 \beta + 12) q^{39} + (5 \beta + 30) q^{40} + (80 \beta + 22) q^{41} + (21 \beta + 42) q^{42} + ( - 128 \beta + 68) q^{43} + ( - 182 \beta - 6) q^{44} - 45 q^{45} + (120 \beta + 220) q^{46} + (168 \beta + 200) q^{47} + ( - 72 \beta + 27) q^{48} + 49 q^{49} + ( - 25 \beta - 50) q^{50} + ( - 132 \beta - 66) q^{51} + (54 \beta + 764) q^{52} + ( - 86 \beta + 8) q^{53} + ( - 27 \beta - 54) q^{54} + ( - 10 \beta + 230) q^{55} + (7 \beta + 42) q^{56} + ( - 78 \beta - 162) q^{57} + (142 \beta + 296) q^{58} + (36 \beta - 232) q^{59} + ( - 60 \beta - 15) q^{60} + ( - 84 \beta - 342) q^{61} + (234 \beta + 570) q^{62} - 63 q^{63} + ( - 52 \beta - 607) q^{64} + ( - 190 \beta - 20) q^{65} + (126 \beta + 246) q^{66} + ( - 164 \beta + 368) q^{67} + ( - 132 \beta - 902) q^{68} + (60 \beta - 480) q^{69} + ( - 35 \beta - 70) q^{70} + (138 \beta - 370) q^{71} + ( - 9 \beta - 54) q^{72} + (122 \beta + 212) q^{73} + ( - 54 \beta - 84) q^{74} + 75 q^{75} + ( - 242 \beta - 574) q^{76} + ( - 14 \beta + 322) q^{77} + ( - 240 \beta - 594) q^{78} + (484 \beta - 204) q^{79} + (120 \beta - 45) q^{80} + 81 q^{81} + ( - 182 \beta - 444) q^{82} + (84 \beta + 304) q^{83} + ( - 84 \beta - 21) q^{84} + (220 \beta + 110) q^{85} + (188 \beta + 504) q^{86} + ( - 36 \beta - 354) q^{87} + (34 \beta + 266) q^{88} + (112 \beta - 666) q^{89} + (45 \beta + 90) q^{90} + ( - 266 \beta - 28) q^{91} + ( - 620 \beta + 240) q^{92} + ( - 306 \beta - 90) q^{93} + ( - 536 \beta - 1240) q^{94} + (130 \beta + 270) q^{95} + (141 \beta + 450) q^{96} + (86 \beta - 1224) q^{97} + ( - 49 \beta - 98) q^{98} + (18 \beta - 414) q^{99}+O(q^{100})$$ q + (-b - 2) * q^2 + 3 * q^3 + (4*b + 1) * q^4 - 5 * q^5 + (-3*b - 6) * q^6 - 7 * q^7 + (-b - 6) * q^8 + 9 * q^9 + (5*b + 10) * q^10 + (2*b - 46) * q^11 + (12*b + 3) * q^12 + (38*b + 4) * q^13 + (7*b + 14) * q^14 - 15 * q^15 + (-24*b + 9) * q^16 + (-44*b - 22) * q^17 + (-9*b - 18) * q^18 + (-26*b - 54) * q^19 + (-20*b - 5) * q^20 - 21 * q^21 + (42*b + 82) * q^22 + (20*b - 160) * q^23 + (-3*b - 18) * q^24 + 25 * q^25 + (-80*b - 198) * q^26 + 27 * q^27 + (-28*b - 7) * q^28 + (-12*b - 118) * q^29 + (15*b + 30) * q^30 + (-102*b - 30) * q^31 + (47*b + 150) * q^32 + (6*b - 138) * q^33 + (110*b + 264) * q^34 + 35 * q^35 + (36*b + 9) * q^36 + (-24*b + 102) * q^37 + (106*b + 238) * q^38 + (114*b + 12) * q^39 + (5*b + 30) * q^40 + (80*b + 22) * q^41 + (21*b + 42) * q^42 + (-128*b + 68) * q^43 + (-182*b - 6) * q^44 - 45 * q^45 + (120*b + 220) * q^46 + (168*b + 200) * q^47 + (-72*b + 27) * q^48 + 49 * q^49 + (-25*b - 50) * q^50 + (-132*b - 66) * q^51 + (54*b + 764) * q^52 + (-86*b + 8) * q^53 + (-27*b - 54) * q^54 + (-10*b + 230) * q^55 + (7*b + 42) * q^56 + (-78*b - 162) * q^57 + (142*b + 296) * q^58 + (36*b - 232) * q^59 + (-60*b - 15) * q^60 + (-84*b - 342) * q^61 + (234*b + 570) * q^62 - 63 * q^63 + (-52*b - 607) * q^64 + (-190*b - 20) * q^65 + (126*b + 246) * q^66 + (-164*b + 368) * q^67 + (-132*b - 902) * q^68 + (60*b - 480) * q^69 + (-35*b - 70) * q^70 + (138*b - 370) * q^71 + (-9*b - 54) * q^72 + (122*b + 212) * q^73 + (-54*b - 84) * q^74 + 75 * q^75 + (-242*b - 574) * q^76 + (-14*b + 322) * q^77 + (-240*b - 594) * q^78 + (484*b - 204) * q^79 + (120*b - 45) * q^80 + 81 * q^81 + (-182*b - 444) * q^82 + (84*b + 304) * q^83 + (-84*b - 21) * q^84 + (220*b + 110) * q^85 + (188*b + 504) * q^86 + (-36*b - 354) * q^87 + (34*b + 266) * q^88 + (112*b - 666) * q^89 + (45*b + 90) * q^90 + (-266*b - 28) * q^91 + (-620*b + 240) * q^92 + (-306*b - 90) * q^93 + (-536*b - 1240) * q^94 + (130*b + 270) * q^95 + (141*b + 450) * q^96 + (86*b - 1224) * q^97 + (-49*b - 98) * q^98 + (18*b - 414) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 12 q^{6} - 14 q^{7} - 12 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 12 * q^6 - 14 * q^7 - 12 * q^8 + 18 * q^9 $$2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 12 q^{6} - 14 q^{7} - 12 q^{8} + 18 q^{9} + 20 q^{10} - 92 q^{11} + 6 q^{12} + 8 q^{13} + 28 q^{14} - 30 q^{15} + 18 q^{16} - 44 q^{17} - 36 q^{18} - 108 q^{19} - 10 q^{20} - 42 q^{21} + 164 q^{22} - 320 q^{23} - 36 q^{24} + 50 q^{25} - 396 q^{26} + 54 q^{27} - 14 q^{28} - 236 q^{29} + 60 q^{30} - 60 q^{31} + 300 q^{32} - 276 q^{33} + 528 q^{34} + 70 q^{35} + 18 q^{36} + 204 q^{37} + 476 q^{38} + 24 q^{39} + 60 q^{40} + 44 q^{41} + 84 q^{42} + 136 q^{43} - 12 q^{44} - 90 q^{45} + 440 q^{46} + 400 q^{47} + 54 q^{48} + 98 q^{49} - 100 q^{50} - 132 q^{51} + 1528 q^{52} + 16 q^{53} - 108 q^{54} + 460 q^{55} + 84 q^{56} - 324 q^{57} + 592 q^{58} - 464 q^{59} - 30 q^{60} - 684 q^{61} + 1140 q^{62} - 126 q^{63} - 1214 q^{64} - 40 q^{65} + 492 q^{66} + 736 q^{67} - 1804 q^{68} - 960 q^{69} - 140 q^{70} - 740 q^{71} - 108 q^{72} + 424 q^{73} - 168 q^{74} + 150 q^{75} - 1148 q^{76} + 644 q^{77} - 1188 q^{78} - 408 q^{79} - 90 q^{80} + 162 q^{81} - 888 q^{82} + 608 q^{83} - 42 q^{84} + 220 q^{85} + 1008 q^{86} - 708 q^{87} + 532 q^{88} - 1332 q^{89} + 180 q^{90} - 56 q^{91} + 480 q^{92} - 180 q^{93} - 2480 q^{94} + 540 q^{95} + 900 q^{96} - 2448 q^{97} - 196 q^{98} - 828 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 12 * q^6 - 14 * q^7 - 12 * q^8 + 18 * q^9 + 20 * q^10 - 92 * q^11 + 6 * q^12 + 8 * q^13 + 28 * q^14 - 30 * q^15 + 18 * q^16 - 44 * q^17 - 36 * q^18 - 108 * q^19 - 10 * q^20 - 42 * q^21 + 164 * q^22 - 320 * q^23 - 36 * q^24 + 50 * q^25 - 396 * q^26 + 54 * q^27 - 14 * q^28 - 236 * q^29 + 60 * q^30 - 60 * q^31 + 300 * q^32 - 276 * q^33 + 528 * q^34 + 70 * q^35 + 18 * q^36 + 204 * q^37 + 476 * q^38 + 24 * q^39 + 60 * q^40 + 44 * q^41 + 84 * q^42 + 136 * q^43 - 12 * q^44 - 90 * q^45 + 440 * q^46 + 400 * q^47 + 54 * q^48 + 98 * q^49 - 100 * q^50 - 132 * q^51 + 1528 * q^52 + 16 * q^53 - 108 * q^54 + 460 * q^55 + 84 * q^56 - 324 * q^57 + 592 * q^58 - 464 * q^59 - 30 * q^60 - 684 * q^61 + 1140 * q^62 - 126 * q^63 - 1214 * q^64 - 40 * q^65 + 492 * q^66 + 736 * q^67 - 1804 * q^68 - 960 * q^69 - 140 * q^70 - 740 * q^71 - 108 * q^72 + 424 * q^73 - 168 * q^74 + 150 * q^75 - 1148 * q^76 + 644 * q^77 - 1188 * q^78 - 408 * q^79 - 90 * q^80 + 162 * q^81 - 888 * q^82 + 608 * q^83 - 42 * q^84 + 220 * q^85 + 1008 * q^86 - 708 * q^87 + 532 * q^88 - 1332 * q^89 + 180 * q^90 - 56 * q^91 + 480 * q^92 - 180 * q^93 - 2480 * q^94 + 540 * q^95 + 900 * q^96 - 2448 * q^97 - 196 * q^98 - 828 * 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
 1.61803 −0.618034
−4.23607 3.00000 9.94427 −5.00000 −12.7082 −7.00000 −8.23607 9.00000 21.1803
1.2 0.236068 3.00000 −7.94427 −5.00000 0.708204 −7.00000 −3.76393 9.00000 −1.18034
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$+1$$
$$7$$ $$+1$$

## 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 105.4.a.d 2
3.b odd 2 1 315.4.a.l 2
4.b odd 2 1 1680.4.a.bd 2
5.b even 2 1 525.4.a.o 2
5.c odd 4 2 525.4.d.k 4
7.b odd 2 1 735.4.a.m 2
15.d odd 2 1 1575.4.a.n 2
21.c even 2 1 2205.4.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 1.a even 1 1 trivial
315.4.a.l 2 3.b odd 2 1
525.4.a.o 2 5.b even 2 1
525.4.d.k 4 5.c odd 4 2
735.4.a.m 2 7.b odd 2 1
1575.4.a.n 2 15.d odd 2 1
1680.4.a.bd 2 4.b odd 2 1
2205.4.a.be 2 21.c even 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(105))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4T - 1$$
$3$ $$(T - 3)^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} + 92T + 2096$$
$13$ $$T^{2} - 8T - 7204$$
$17$ $$T^{2} + 44T - 9196$$
$19$ $$T^{2} + 108T - 464$$
$23$ $$T^{2} + 320T + 23600$$
$29$ $$T^{2} + 236T + 13204$$
$31$ $$T^{2} + 60T - 51120$$
$37$ $$T^{2} - 204T + 7524$$
$41$ $$T^{2} - 44T - 31516$$
$43$ $$T^{2} - 136T - 77296$$
$47$ $$T^{2} - 400T - 101120$$
$53$ $$T^{2} - 16T - 36916$$
$59$ $$T^{2} + 464T + 47344$$
$61$ $$T^{2} + 684T + 81684$$
$67$ $$T^{2} - 736T + 944$$
$71$ $$T^{2} + 740T + 41680$$
$73$ $$T^{2} - 424T - 29476$$
$79$ $$T^{2} + 408 T - 1129664$$
$83$ $$T^{2} - 608T + 57136$$
$89$ $$T^{2} + 1332 T + 380836$$
$97$ $$T^{2} + 2448 T + 1461196$$