# Properties

 Label 105.4.a.f Level $105$ Weight $4$ Character orbit 105.a Self dual yes Analytic conductor $6.195$ 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] = [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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 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 $$\beta = \frac{1}{2}(1 + \sqrt{65})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 3 q^{3} + (\beta + 8) q^{4} + 5 q^{5} + 3 \beta q^{6} - 7 q^{7} + (\beta + 16) q^{8} + 9 q^{9}+O(q^{10})$$ q + b * q^2 + 3 * q^3 + (b + 8) * q^4 + 5 * q^5 + 3*b * q^6 - 7 * q^7 + (b + 16) * q^8 + 9 * q^9 $$q + \beta q^{2} + 3 q^{3} + (\beta + 8) q^{4} + 5 q^{5} + 3 \beta q^{6} - 7 q^{7} + (\beta + 16) q^{8} + 9 q^{9} + 5 \beta q^{10} + ( - 2 \beta - 10) q^{11} + (3 \beta + 24) q^{12} + (2 \beta - 12) q^{13} - 7 \beta q^{14} + 15 q^{15} + (9 \beta - 48) q^{16} + ( - 16 \beta + 66) q^{17} + 9 \beta q^{18} + ( - 14 \beta + 58) q^{19} + (5 \beta + 40) q^{20} - 21 q^{21} + ( - 12 \beta - 32) q^{22} + ( - 20 \beta + 140) q^{23} + (3 \beta + 48) q^{24} + 25 q^{25} + ( - 10 \beta + 32) q^{26} + 27 q^{27} + ( - 7 \beta - 56) q^{28} + ( - 48 \beta - 74) q^{29} + 15 \beta q^{30} + (42 \beta + 54) q^{31} + ( - 47 \beta + 16) q^{32} + ( - 6 \beta - 30) q^{33} + (50 \beta - 256) q^{34} - 35 q^{35} + (9 \beta + 72) q^{36} + ( - 36 \beta - 30) q^{37} + (44 \beta - 224) q^{38} + (6 \beta - 36) q^{39} + (5 \beta + 80) q^{40} + (100 \beta - 138) q^{41} - 21 \beta q^{42} + ( - 32 \beta - 156) q^{43} + ( - 28 \beta - 112) q^{44} + 45 q^{45} + (120 \beta - 320) q^{46} + ( - 48 \beta + 304) q^{47} + (27 \beta - 144) q^{48} + 49 q^{49} + 25 \beta q^{50} + ( - 48 \beta + 198) q^{51} + (6 \beta - 64) q^{52} + (86 \beta + 120) q^{53} + 27 \beta q^{54} + ( - 10 \beta - 50) q^{55} + ( - 7 \beta - 112) q^{56} + ( - 42 \beta + 174) q^{57} + ( - 122 \beta - 768) q^{58} + (84 \beta - 464) q^{59} + (15 \beta + 120) q^{60} + (24 \beta - 114) q^{61} + (96 \beta + 672) q^{62} - 63 q^{63} + ( - 103 \beta - 368) q^{64} + (10 \beta - 60) q^{65} + ( - 36 \beta - 96) q^{66} + (64 \beta - 84) q^{67} + ( - 78 \beta + 272) q^{68} + ( - 60 \beta + 420) q^{69} - 35 \beta q^{70} + (42 \beta + 814) q^{71} + (9 \beta + 144) q^{72} + ( - 202 \beta - 92) q^{73} + ( - 66 \beta - 576) q^{74} + 75 q^{75} + ( - 68 \beta + 240) q^{76} + (14 \beta + 70) q^{77} + ( - 30 \beta + 96) q^{78} + ( - 104 \beta - 392) q^{79} + (45 \beta - 240) q^{80} + 81 q^{81} + ( - 38 \beta + 1600) q^{82} + (216 \beta + 356) q^{83} + ( - 21 \beta - 168) q^{84} + ( - 80 \beta + 330) q^{85} + ( - 188 \beta - 512) q^{86} + ( - 144 \beta - 222) q^{87} + ( - 44 \beta - 192) q^{88} + (8 \beta + 290) q^{89} + 45 \beta q^{90} + ( - 14 \beta + 84) q^{91} + ( - 40 \beta + 800) q^{92} + (126 \beta + 162) q^{93} + (256 \beta - 768) q^{94} + ( - 70 \beta + 290) q^{95} + ( - 141 \beta + 48) q^{96} + (314 \beta + 104) q^{97} + 49 \beta q^{98} + ( - 18 \beta - 90) q^{99} +O(q^{100})$$ q + b * q^2 + 3 * q^3 + (b + 8) * q^4 + 5 * q^5 + 3*b * q^6 - 7 * q^7 + (b + 16) * q^8 + 9 * q^9 + 5*b * q^10 + (-2*b - 10) * q^11 + (3*b + 24) * q^12 + (2*b - 12) * q^13 - 7*b * q^14 + 15 * q^15 + (9*b - 48) * q^16 + (-16*b + 66) * q^17 + 9*b * q^18 + (-14*b + 58) * q^19 + (5*b + 40) * q^20 - 21 * q^21 + (-12*b - 32) * q^22 + (-20*b + 140) * q^23 + (3*b + 48) * q^24 + 25 * q^25 + (-10*b + 32) * q^26 + 27 * q^27 + (-7*b - 56) * q^28 + (-48*b - 74) * q^29 + 15*b * q^30 + (42*b + 54) * q^31 + (-47*b + 16) * q^32 + (-6*b - 30) * q^33 + (50*b - 256) * q^34 - 35 * q^35 + (9*b + 72) * q^36 + (-36*b - 30) * q^37 + (44*b - 224) * q^38 + (6*b - 36) * q^39 + (5*b + 80) * q^40 + (100*b - 138) * q^41 - 21*b * q^42 + (-32*b - 156) * q^43 + (-28*b - 112) * q^44 + 45 * q^45 + (120*b - 320) * q^46 + (-48*b + 304) * q^47 + (27*b - 144) * q^48 + 49 * q^49 + 25*b * q^50 + (-48*b + 198) * q^51 + (6*b - 64) * q^52 + (86*b + 120) * q^53 + 27*b * q^54 + (-10*b - 50) * q^55 + (-7*b - 112) * q^56 + (-42*b + 174) * q^57 + (-122*b - 768) * q^58 + (84*b - 464) * q^59 + (15*b + 120) * q^60 + (24*b - 114) * q^61 + (96*b + 672) * q^62 - 63 * q^63 + (-103*b - 368) * q^64 + (10*b - 60) * q^65 + (-36*b - 96) * q^66 + (64*b - 84) * q^67 + (-78*b + 272) * q^68 + (-60*b + 420) * q^69 - 35*b * q^70 + (42*b + 814) * q^71 + (9*b + 144) * q^72 + (-202*b - 92) * q^73 + (-66*b - 576) * q^74 + 75 * q^75 + (-68*b + 240) * q^76 + (14*b + 70) * q^77 + (-30*b + 96) * q^78 + (-104*b - 392) * q^79 + (45*b - 240) * q^80 + 81 * q^81 + (-38*b + 1600) * q^82 + (216*b + 356) * q^83 + (-21*b - 168) * q^84 + (-80*b + 330) * q^85 + (-188*b - 512) * q^86 + (-144*b - 222) * q^87 + (-44*b - 192) * q^88 + (8*b + 290) * q^89 + 45*b * q^90 + (-14*b + 84) * q^91 + (-40*b + 800) * q^92 + (126*b + 162) * q^93 + (256*b - 768) * q^94 + (-70*b + 290) * q^95 + (-141*b + 48) * q^96 + (314*b + 104) * q^97 + 49*b * q^98 + (-18*b - 90) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 3 q^{6} - 14 q^{7} + 33 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 + 3 * q^6 - 14 * q^7 + 33 * q^8 + 18 * q^9 $$2 q + q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 3 q^{6} - 14 q^{7} + 33 q^{8} + 18 q^{9} + 5 q^{10} - 22 q^{11} + 51 q^{12} - 22 q^{13} - 7 q^{14} + 30 q^{15} - 87 q^{16} + 116 q^{17} + 9 q^{18} + 102 q^{19} + 85 q^{20} - 42 q^{21} - 76 q^{22} + 260 q^{23} + 99 q^{24} + 50 q^{25} + 54 q^{26} + 54 q^{27} - 119 q^{28} - 196 q^{29} + 15 q^{30} + 150 q^{31} - 15 q^{32} - 66 q^{33} - 462 q^{34} - 70 q^{35} + 153 q^{36} - 96 q^{37} - 404 q^{38} - 66 q^{39} + 165 q^{40} - 176 q^{41} - 21 q^{42} - 344 q^{43} - 252 q^{44} + 90 q^{45} - 520 q^{46} + 560 q^{47} - 261 q^{48} + 98 q^{49} + 25 q^{50} + 348 q^{51} - 122 q^{52} + 326 q^{53} + 27 q^{54} - 110 q^{55} - 231 q^{56} + 306 q^{57} - 1658 q^{58} - 844 q^{59} + 255 q^{60} - 204 q^{61} + 1440 q^{62} - 126 q^{63} - 839 q^{64} - 110 q^{65} - 228 q^{66} - 104 q^{67} + 466 q^{68} + 780 q^{69} - 35 q^{70} + 1670 q^{71} + 297 q^{72} - 386 q^{73} - 1218 q^{74} + 150 q^{75} + 412 q^{76} + 154 q^{77} + 162 q^{78} - 888 q^{79} - 435 q^{80} + 162 q^{81} + 3162 q^{82} + 928 q^{83} - 357 q^{84} + 580 q^{85} - 1212 q^{86} - 588 q^{87} - 428 q^{88} + 588 q^{89} + 45 q^{90} + 154 q^{91} + 1560 q^{92} + 450 q^{93} - 1280 q^{94} + 510 q^{95} - 45 q^{96} + 522 q^{97} + 49 q^{98} - 198 q^{99}+O(q^{100})$$ 2 * q + q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 + 3 * q^6 - 14 * q^7 + 33 * q^8 + 18 * q^9 + 5 * q^10 - 22 * q^11 + 51 * q^12 - 22 * q^13 - 7 * q^14 + 30 * q^15 - 87 * q^16 + 116 * q^17 + 9 * q^18 + 102 * q^19 + 85 * q^20 - 42 * q^21 - 76 * q^22 + 260 * q^23 + 99 * q^24 + 50 * q^25 + 54 * q^26 + 54 * q^27 - 119 * q^28 - 196 * q^29 + 15 * q^30 + 150 * q^31 - 15 * q^32 - 66 * q^33 - 462 * q^34 - 70 * q^35 + 153 * q^36 - 96 * q^37 - 404 * q^38 - 66 * q^39 + 165 * q^40 - 176 * q^41 - 21 * q^42 - 344 * q^43 - 252 * q^44 + 90 * q^45 - 520 * q^46 + 560 * q^47 - 261 * q^48 + 98 * q^49 + 25 * q^50 + 348 * q^51 - 122 * q^52 + 326 * q^53 + 27 * q^54 - 110 * q^55 - 231 * q^56 + 306 * q^57 - 1658 * q^58 - 844 * q^59 + 255 * q^60 - 204 * q^61 + 1440 * q^62 - 126 * q^63 - 839 * q^64 - 110 * q^65 - 228 * q^66 - 104 * q^67 + 466 * q^68 + 780 * q^69 - 35 * q^70 + 1670 * q^71 + 297 * q^72 - 386 * q^73 - 1218 * q^74 + 150 * q^75 + 412 * q^76 + 154 * q^77 + 162 * q^78 - 888 * q^79 - 435 * q^80 + 162 * q^81 + 3162 * q^82 + 928 * q^83 - 357 * q^84 + 580 * q^85 - 1212 * q^86 - 588 * q^87 - 428 * q^88 + 588 * q^89 + 45 * q^90 + 154 * q^91 + 1560 * q^92 + 450 * q^93 - 1280 * q^94 + 510 * q^95 - 45 * q^96 + 522 * q^97 + 49 * q^98 - 198 * 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
 −3.53113 4.53113
−3.53113 3.00000 4.46887 5.00000 −10.5934 −7.00000 12.4689 9.00000 −17.6556
1.2 4.53113 3.00000 12.5311 5.00000 13.5934 −7.00000 20.5311 9.00000 22.6556
 $$n$$: 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.f 2
3.b odd 2 1 315.4.a.i 2
4.b odd 2 1 1680.4.a.bg 2
5.b even 2 1 525.4.a.k 2
5.c odd 4 2 525.4.d.h 4
7.b odd 2 1 735.4.a.p 2
15.d odd 2 1 1575.4.a.w 2
21.c even 2 1 2205.4.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.f 2 1.a even 1 1 trivial
315.4.a.i 2 3.b odd 2 1
525.4.a.k 2 5.b even 2 1
525.4.d.h 4 5.c odd 4 2
735.4.a.p 2 7.b odd 2 1
1575.4.a.w 2 15.d odd 2 1
1680.4.a.bg 2 4.b odd 2 1
2205.4.a.z 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 16$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(105))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 16$$
$3$ $$(T - 3)^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} + 22T + 56$$
$13$ $$T^{2} + 22T + 56$$
$17$ $$T^{2} - 116T - 796$$
$19$ $$T^{2} - 102T - 584$$
$23$ $$T^{2} - 260T + 10400$$
$29$ $$T^{2} + 196T - 27836$$
$31$ $$T^{2} - 150T - 23040$$
$37$ $$T^{2} + 96T - 18756$$
$41$ $$T^{2} + 176T - 154756$$
$43$ $$T^{2} + 344T + 12944$$
$47$ $$T^{2} - 560T + 40960$$
$53$ $$T^{2} - 326T - 93616$$
$59$ $$T^{2} + 844T + 63424$$
$61$ $$T^{2} + 204T + 1044$$
$67$ $$T^{2} + 104T - 63856$$
$71$ $$T^{2} - 1670 T + 668560$$
$73$ $$T^{2} + 386T - 625816$$
$79$ $$T^{2} + 888T + 21376$$
$83$ $$T^{2} - 928T - 542864$$
$89$ $$T^{2} - 588T + 85396$$
$97$ $$T^{2} - 522 T - 1534064$$