# Properties

 Label 735.4.a.q Level $735$ Weight $4$ Character orbit 735.a Self dual yes Analytic conductor $43.366$ 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] = [735,4,Mod(1,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, 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("735.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 735.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: $$43.3664038542$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) 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 $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} - 3 q^{3} + (3 \beta + 3) q^{4} + 5 q^{5} + ( - 3 \beta - 3) q^{6} + (\beta + 25) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b + 1) * q^2 - 3 * q^3 + (3*b + 3) * q^4 + 5 * q^5 + (-3*b - 3) * q^6 + (b + 25) * q^8 + 9 * q^9 $$q + (\beta + 1) q^{2} - 3 q^{3} + (3 \beta + 3) q^{4} + 5 q^{5} + ( - 3 \beta - 3) q^{6} + (\beta + 25) q^{8} + 9 q^{9} + (5 \beta + 5) q^{10} + ( - 2 \beta + 32) q^{11} + ( - 9 \beta - 9) q^{12} + (10 \beta - 2) q^{13} - 15 q^{15} + (3 \beta + 11) q^{16} + (12 \beta - 26) q^{17} + (9 \beta + 9) q^{18} + (2 \beta + 60) q^{19} + (15 \beta + 15) q^{20} + (28 \beta + 12) q^{22} + ( - 48 \beta + 32) q^{23} + ( - 3 \beta - 75) q^{24} + 25 q^{25} + (18 \beta + 98) q^{26} - 27 q^{27} + (12 \beta + 170) q^{29} + ( - 15 \beta - 15) q^{30} + (38 \beta - 52) q^{31} + (9 \beta - 159) q^{32} + (6 \beta - 96) q^{33} + ( - 2 \beta + 94) q^{34} + (27 \beta + 27) q^{36} + (80 \beta - 134) q^{37} + (64 \beta + 80) q^{38} + ( - 30 \beta + 6) q^{39} + (5 \beta + 125) q^{40} + (108 \beta - 62) q^{41} + (100 \beta - 248) q^{43} + (84 \beta + 36) q^{44} + 45 q^{45} + ( - 64 \beta - 448) q^{46} + ( - 140 \beta + 164) q^{47} + ( - 9 \beta - 33) q^{48} + (25 \beta + 25) q^{50} + ( - 36 \beta + 78) q^{51} + (54 \beta + 294) q^{52} + (58 \beta + 462) q^{53} + ( - 27 \beta - 27) q^{54} + ( - 10 \beta + 160) q^{55} + ( - 6 \beta - 180) q^{57} + (194 \beta + 290) q^{58} + ( - 76 \beta - 220) q^{59} + ( - 45 \beta - 45) q^{60} + (84 \beta + 398) q^{61} + (24 \beta + 328) q^{62} + ( - 165 \beta - 157) q^{64} + (50 \beta - 10) q^{65} + ( - 84 \beta - 36) q^{66} + ( - 228 \beta - 64) q^{67} + ( - 6 \beta + 282) q^{68} + (144 \beta - 96) q^{69} + (86 \beta + 112) q^{71} + (9 \beta + 225) q^{72} + (38 \beta - 182) q^{73} + (26 \beta + 666) q^{74} - 75 q^{75} + (192 \beta + 240) q^{76} + ( - 54 \beta - 294) q^{78} + ( - 8 \beta + 920) q^{79} + (15 \beta + 55) q^{80} + 81 q^{81} + (154 \beta + 1018) q^{82} + (224 \beta + 228) q^{83} + (60 \beta - 130) q^{85} + ( - 48 \beta + 752) q^{86} + ( - 36 \beta - 510) q^{87} + ( - 20 \beta + 780) q^{88} + ( - 336 \beta - 230) q^{89} + (45 \beta + 45) q^{90} + ( - 192 \beta - 1344) q^{92} + ( - 114 \beta + 156) q^{93} + ( - 116 \beta - 1236) q^{94} + (10 \beta + 300) q^{95} + ( - 27 \beta + 477) q^{96} + ( - 278 \beta + 474) q^{97} + ( - 18 \beta + 288) q^{99}+O(q^{100})$$ q + (b + 1) * q^2 - 3 * q^3 + (3*b + 3) * q^4 + 5 * q^5 + (-3*b - 3) * q^6 + (b + 25) * q^8 + 9 * q^9 + (5*b + 5) * q^10 + (-2*b + 32) * q^11 + (-9*b - 9) * q^12 + (10*b - 2) * q^13 - 15 * q^15 + (3*b + 11) * q^16 + (12*b - 26) * q^17 + (9*b + 9) * q^18 + (2*b + 60) * q^19 + (15*b + 15) * q^20 + (28*b + 12) * q^22 + (-48*b + 32) * q^23 + (-3*b - 75) * q^24 + 25 * q^25 + (18*b + 98) * q^26 - 27 * q^27 + (12*b + 170) * q^29 + (-15*b - 15) * q^30 + (38*b - 52) * q^31 + (9*b - 159) * q^32 + (6*b - 96) * q^33 + (-2*b + 94) * q^34 + (27*b + 27) * q^36 + (80*b - 134) * q^37 + (64*b + 80) * q^38 + (-30*b + 6) * q^39 + (5*b + 125) * q^40 + (108*b - 62) * q^41 + (100*b - 248) * q^43 + (84*b + 36) * q^44 + 45 * q^45 + (-64*b - 448) * q^46 + (-140*b + 164) * q^47 + (-9*b - 33) * q^48 + (25*b + 25) * q^50 + (-36*b + 78) * q^51 + (54*b + 294) * q^52 + (58*b + 462) * q^53 + (-27*b - 27) * q^54 + (-10*b + 160) * q^55 + (-6*b - 180) * q^57 + (194*b + 290) * q^58 + (-76*b - 220) * q^59 + (-45*b - 45) * q^60 + (84*b + 398) * q^61 + (24*b + 328) * q^62 + (-165*b - 157) * q^64 + (50*b - 10) * q^65 + (-84*b - 36) * q^66 + (-228*b - 64) * q^67 + (-6*b + 282) * q^68 + (144*b - 96) * q^69 + (86*b + 112) * q^71 + (9*b + 225) * q^72 + (38*b - 182) * q^73 + (26*b + 666) * q^74 - 75 * q^75 + (192*b + 240) * q^76 + (-54*b - 294) * q^78 + (-8*b + 920) * q^79 + (15*b + 55) * q^80 + 81 * q^81 + (154*b + 1018) * q^82 + (224*b + 228) * q^83 + (60*b - 130) * q^85 + (-48*b + 752) * q^86 + (-36*b - 510) * q^87 + (-20*b + 780) * q^88 + (-336*b - 230) * q^89 + (45*b + 45) * q^90 + (-192*b - 1344) * q^92 + (-114*b + 156) * q^93 + (-116*b - 1236) * q^94 + (10*b + 300) * q^95 + (-27*b + 477) * q^96 + (-278*b + 474) * q^97 + (-18*b + 288) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - 6 q^{3} + 9 q^{4} + 10 q^{5} - 9 q^{6} + 51 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 - 6 * q^3 + 9 * q^4 + 10 * q^5 - 9 * q^6 + 51 * q^8 + 18 * q^9 $$2 q + 3 q^{2} - 6 q^{3} + 9 q^{4} + 10 q^{5} - 9 q^{6} + 51 q^{8} + 18 q^{9} + 15 q^{10} + 62 q^{11} - 27 q^{12} + 6 q^{13} - 30 q^{15} + 25 q^{16} - 40 q^{17} + 27 q^{18} + 122 q^{19} + 45 q^{20} + 52 q^{22} + 16 q^{23} - 153 q^{24} + 50 q^{25} + 214 q^{26} - 54 q^{27} + 352 q^{29} - 45 q^{30} - 66 q^{31} - 309 q^{32} - 186 q^{33} + 186 q^{34} + 81 q^{36} - 188 q^{37} + 224 q^{38} - 18 q^{39} + 255 q^{40} - 16 q^{41} - 396 q^{43} + 156 q^{44} + 90 q^{45} - 960 q^{46} + 188 q^{47} - 75 q^{48} + 75 q^{50} + 120 q^{51} + 642 q^{52} + 982 q^{53} - 81 q^{54} + 310 q^{55} - 366 q^{57} + 774 q^{58} - 516 q^{59} - 135 q^{60} + 880 q^{61} + 680 q^{62} - 479 q^{64} + 30 q^{65} - 156 q^{66} - 356 q^{67} + 558 q^{68} - 48 q^{69} + 310 q^{71} + 459 q^{72} - 326 q^{73} + 1358 q^{74} - 150 q^{75} + 672 q^{76} - 642 q^{78} + 1832 q^{79} + 125 q^{80} + 162 q^{81} + 2190 q^{82} + 680 q^{83} - 200 q^{85} + 1456 q^{86} - 1056 q^{87} + 1540 q^{88} - 796 q^{89} + 135 q^{90} - 2880 q^{92} + 198 q^{93} - 2588 q^{94} + 610 q^{95} + 927 q^{96} + 670 q^{97} + 558 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 - 6 * q^3 + 9 * q^4 + 10 * q^5 - 9 * q^6 + 51 * q^8 + 18 * q^9 + 15 * q^10 + 62 * q^11 - 27 * q^12 + 6 * q^13 - 30 * q^15 + 25 * q^16 - 40 * q^17 + 27 * q^18 + 122 * q^19 + 45 * q^20 + 52 * q^22 + 16 * q^23 - 153 * q^24 + 50 * q^25 + 214 * q^26 - 54 * q^27 + 352 * q^29 - 45 * q^30 - 66 * q^31 - 309 * q^32 - 186 * q^33 + 186 * q^34 + 81 * q^36 - 188 * q^37 + 224 * q^38 - 18 * q^39 + 255 * q^40 - 16 * q^41 - 396 * q^43 + 156 * q^44 + 90 * q^45 - 960 * q^46 + 188 * q^47 - 75 * q^48 + 75 * q^50 + 120 * q^51 + 642 * q^52 + 982 * q^53 - 81 * q^54 + 310 * q^55 - 366 * q^57 + 774 * q^58 - 516 * q^59 - 135 * q^60 + 880 * q^61 + 680 * q^62 - 479 * q^64 + 30 * q^65 - 156 * q^66 - 356 * q^67 + 558 * q^68 - 48 * q^69 + 310 * q^71 + 459 * q^72 - 326 * q^73 + 1358 * q^74 - 150 * q^75 + 672 * q^76 - 642 * q^78 + 1832 * q^79 + 125 * q^80 + 162 * q^81 + 2190 * q^82 + 680 * q^83 - 200 * q^85 + 1456 * q^86 - 1056 * q^87 + 1540 * q^88 - 796 * q^89 + 135 * q^90 - 2880 * q^92 + 198 * q^93 - 2588 * q^94 + 610 * q^95 + 927 * q^96 + 670 * q^97 + 558 * 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
 −2.70156 3.70156
−1.70156 −3.00000 −5.10469 5.00000 5.10469 0 22.2984 9.00000 −8.50781
1.2 4.70156 −3.00000 14.1047 5.00000 −14.1047 0 28.7016 9.00000 23.5078
 $$n$$: 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 735.4.a.q 2
3.b odd 2 1 2205.4.a.v 2
7.b odd 2 1 105.4.a.g 2
21.c even 2 1 315.4.a.g 2
28.d even 2 1 1680.4.a.y 2
35.c odd 2 1 525.4.a.i 2
35.f even 4 2 525.4.d.j 4
105.g even 2 1 1575.4.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 7.b odd 2 1
315.4.a.g 2 21.c even 2 1
525.4.a.i 2 35.c odd 2 1
525.4.d.j 4 35.f even 4 2
735.4.a.q 2 1.a even 1 1 trivial
1575.4.a.y 2 105.g even 2 1
1680.4.a.y 2 28.d even 2 1
2205.4.a.v 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(735))$$:

 $$T_{2}^{2} - 3T_{2} - 8$$ T2^2 - 3*T2 - 8 $$T_{11}^{2} - 62T_{11} + 920$$ T11^2 - 62*T11 + 920 $$T_{13}^{2} - 6T_{13} - 1016$$ T13^2 - 6*T13 - 1016

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T - 8$$
$3$ $$(T + 3)^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 62T + 920$$
$13$ $$T^{2} - 6T - 1016$$
$17$ $$T^{2} + 40T - 1076$$
$19$ $$T^{2} - 122T + 3680$$
$23$ $$T^{2} - 16T - 23552$$
$29$ $$T^{2} - 352T + 29500$$
$31$ $$T^{2} + 66T - 13712$$
$37$ $$T^{2} + 188T - 56764$$
$41$ $$T^{2} + 16T - 119492$$
$43$ $$T^{2} + 396T - 63296$$
$47$ $$T^{2} - 188T - 192064$$
$53$ $$T^{2} - 982T + 206600$$
$59$ $$T^{2} + 516T + 7360$$
$61$ $$T^{2} - 880T + 121276$$
$67$ $$T^{2} + 356T - 501152$$
$71$ $$T^{2} - 310T - 51784$$
$73$ $$T^{2} + 326T + 11768$$
$79$ $$T^{2} - 1832 T + 838400$$
$83$ $$T^{2} - 680T - 398704$$
$89$ $$T^{2} + 796T - 998780$$
$97$ $$T^{2} - 670T - 679936$$