# Properties

 Label 735.2.a.g Level $735$ Weight $2$ Character orbit 735.a Self dual yes Analytic conductor $5.869$ 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] = [735,2,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, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("735.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$5.86900454856$$ Analytic rank: $$0$$ 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: 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} - q^{3} + ( - 2 \beta + 2) q^{4} + q^{5} + ( - \beta + 1) q^{6} + (2 \beta - 6) q^{8} + q^{9}+O(q^{10})$$ q + (b - 1) * q^2 - q^3 + (-2*b + 2) * q^4 + q^5 + (-b + 1) * q^6 + (2*b - 6) * q^8 + q^9 $$q + (\beta - 1) q^{2} - q^{3} + ( - 2 \beta + 2) q^{4} + q^{5} + ( - \beta + 1) q^{6} + (2 \beta - 6) q^{8} + q^{9} + (\beta - 1) q^{10} + ( - \beta - 1) q^{11} + (2 \beta - 2) q^{12} + (\beta + 4) q^{13} - q^{15} + ( - 4 \beta + 8) q^{16} + (\beta + 5) q^{17} + (\beta - 1) q^{18} + ( - 2 \beta + 1) q^{19} + ( - 2 \beta + 2) q^{20} - 2 q^{22} + (\beta - 3) q^{23} + ( - 2 \beta + 6) q^{24} + q^{25} + (3 \beta - 1) q^{26} - q^{27} + (3 \beta + 1) q^{29} + ( - \beta + 1) q^{30} + (2 \beta + 3) q^{31} + (8 \beta - 8) q^{32} + (\beta + 1) q^{33} + (4 \beta - 2) q^{34} + ( - 2 \beta + 2) q^{36} + (3 \beta + 2) q^{37} + (3 \beta - 7) q^{38} + ( - \beta - 4) q^{39} + (2 \beta - 6) q^{40} + (\beta + 1) q^{41} + ( - 3 \beta - 2) q^{43} + 4 q^{44} + q^{45} + ( - 4 \beta + 6) q^{46} + 2 q^{47} + (4 \beta - 8) q^{48} + (\beta - 1) q^{50} + ( - \beta - 5) q^{51} + ( - 6 \beta + 2) q^{52} + ( - 6 \beta + 2) q^{53} + ( - \beta + 1) q^{54} + ( - \beta - 1) q^{55} + (2 \beta - 1) q^{57} + ( - 2 \beta + 8) q^{58} + (3 \beta + 5) q^{59} + (2 \beta - 2) q^{60} + 4 q^{61} + (\beta + 3) q^{62} + ( - 8 \beta + 16) q^{64} + (\beta + 4) q^{65} + 2 q^{66} + (5 \beta - 6) q^{67} + ( - 8 \beta + 4) q^{68} + ( - \beta + 3) q^{69} + ( - 3 \beta + 1) q^{71} + (2 \beta - 6) q^{72} + ( - 5 \beta + 4) q^{73} + ( - \beta + 7) q^{74} - q^{75} + ( - 6 \beta + 14) q^{76} + ( - 3 \beta + 1) q^{78} + (6 \beta + 3) q^{79} + ( - 4 \beta + 8) q^{80} + q^{81} + 2 q^{82} + ( - 7 \beta + 3) q^{83} + (\beta + 5) q^{85} + (\beta - 7) q^{86} + ( - 3 \beta - 1) q^{87} + 4 \beta q^{88} + ( - 7 \beta + 3) q^{89} + (\beta - 1) q^{90} + (8 \beta - 12) q^{92} + ( - 2 \beta - 3) q^{93} + (2 \beta - 2) q^{94} + ( - 2 \beta + 1) q^{95} + ( - 8 \beta + 8) q^{96} + ( - 4 \beta + 8) q^{97} + ( - \beta - 1) q^{99}+O(q^{100})$$ q + (b - 1) * q^2 - q^3 + (-2*b + 2) * q^4 + q^5 + (-b + 1) * q^6 + (2*b - 6) * q^8 + q^9 + (b - 1) * q^10 + (-b - 1) * q^11 + (2*b - 2) * q^12 + (b + 4) * q^13 - q^15 + (-4*b + 8) * q^16 + (b + 5) * q^17 + (b - 1) * q^18 + (-2*b + 1) * q^19 + (-2*b + 2) * q^20 - 2 * q^22 + (b - 3) * q^23 + (-2*b + 6) * q^24 + q^25 + (3*b - 1) * q^26 - q^27 + (3*b + 1) * q^29 + (-b + 1) * q^30 + (2*b + 3) * q^31 + (8*b - 8) * q^32 + (b + 1) * q^33 + (4*b - 2) * q^34 + (-2*b + 2) * q^36 + (3*b + 2) * q^37 + (3*b - 7) * q^38 + (-b - 4) * q^39 + (2*b - 6) * q^40 + (b + 1) * q^41 + (-3*b - 2) * q^43 + 4 * q^44 + q^45 + (-4*b + 6) * q^46 + 2 * q^47 + (4*b - 8) * q^48 + (b - 1) * q^50 + (-b - 5) * q^51 + (-6*b + 2) * q^52 + (-6*b + 2) * q^53 + (-b + 1) * q^54 + (-b - 1) * q^55 + (2*b - 1) * q^57 + (-2*b + 8) * q^58 + (3*b + 5) * q^59 + (2*b - 2) * q^60 + 4 * q^61 + (b + 3) * q^62 + (-8*b + 16) * q^64 + (b + 4) * q^65 + 2 * q^66 + (5*b - 6) * q^67 + (-8*b + 4) * q^68 + (-b + 3) * q^69 + (-3*b + 1) * q^71 + (2*b - 6) * q^72 + (-5*b + 4) * q^73 + (-b + 7) * q^74 - q^75 + (-6*b + 14) * q^76 + (-3*b + 1) * q^78 + (6*b + 3) * q^79 + (-4*b + 8) * q^80 + q^81 + 2 * q^82 + (-7*b + 3) * q^83 + (b + 5) * q^85 + (b - 7) * q^86 + (-3*b - 1) * q^87 + 4*b * q^88 + (-7*b + 3) * q^89 + (b - 1) * q^90 + (8*b - 12) * q^92 + (-2*b - 3) * q^93 + (2*b - 2) * q^94 + (-2*b + 1) * q^95 + (-8*b + 8) * q^96 + (-4*b + 8) * q^97 + (-b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 12 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 12 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 8 q^{13} - 2 q^{15} + 16 q^{16} + 10 q^{17} - 2 q^{18} + 2 q^{19} + 4 q^{20} - 4 q^{22} - 6 q^{23} + 12 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} + 2 q^{29} + 2 q^{30} + 6 q^{31} - 16 q^{32} + 2 q^{33} - 4 q^{34} + 4 q^{36} + 4 q^{37} - 14 q^{38} - 8 q^{39} - 12 q^{40} + 2 q^{41} - 4 q^{43} + 8 q^{44} + 2 q^{45} + 12 q^{46} + 4 q^{47} - 16 q^{48} - 2 q^{50} - 10 q^{51} + 4 q^{52} + 4 q^{53} + 2 q^{54} - 2 q^{55} - 2 q^{57} + 16 q^{58} + 10 q^{59} - 4 q^{60} + 8 q^{61} + 6 q^{62} + 32 q^{64} + 8 q^{65} + 4 q^{66} - 12 q^{67} + 8 q^{68} + 6 q^{69} + 2 q^{71} - 12 q^{72} + 8 q^{73} + 14 q^{74} - 2 q^{75} + 28 q^{76} + 2 q^{78} + 6 q^{79} + 16 q^{80} + 2 q^{81} + 4 q^{82} + 6 q^{83} + 10 q^{85} - 14 q^{86} - 2 q^{87} + 6 q^{89} - 2 q^{90} - 24 q^{92} - 6 q^{93} - 4 q^{94} + 2 q^{95} + 16 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 12 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 8 * q^13 - 2 * q^15 + 16 * q^16 + 10 * q^17 - 2 * q^18 + 2 * q^19 + 4 * q^20 - 4 * q^22 - 6 * q^23 + 12 * q^24 + 2 * q^25 - 2 * q^26 - 2 * q^27 + 2 * q^29 + 2 * q^30 + 6 * q^31 - 16 * q^32 + 2 * q^33 - 4 * q^34 + 4 * q^36 + 4 * q^37 - 14 * q^38 - 8 * q^39 - 12 * q^40 + 2 * q^41 - 4 * q^43 + 8 * q^44 + 2 * q^45 + 12 * q^46 + 4 * q^47 - 16 * q^48 - 2 * q^50 - 10 * q^51 + 4 * q^52 + 4 * q^53 + 2 * q^54 - 2 * q^55 - 2 * q^57 + 16 * q^58 + 10 * q^59 - 4 * q^60 + 8 * q^61 + 6 * q^62 + 32 * q^64 + 8 * q^65 + 4 * q^66 - 12 * q^67 + 8 * q^68 + 6 * q^69 + 2 * q^71 - 12 * q^72 + 8 * q^73 + 14 * q^74 - 2 * q^75 + 28 * q^76 + 2 * q^78 + 6 * q^79 + 16 * q^80 + 2 * q^81 + 4 * q^82 + 6 * q^83 + 10 * q^85 - 14 * q^86 - 2 * q^87 + 6 * q^89 - 2 * q^90 - 24 * q^92 - 6 * q^93 - 4 * q^94 + 2 * q^95 + 16 * q^96 + 16 * q^97 - 2 * 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
−2.73205 −1.00000 5.46410 1.00000 2.73205 0 −9.46410 1.00000 −2.73205
1.2 0.732051 −1.00000 −1.46410 1.00000 −0.732051 0 −2.53590 1.00000 0.732051
 $$n$$: 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 735.2.a.g 2
3.b odd 2 1 2205.2.a.z 2
5.b even 2 1 3675.2.a.bg 2
7.b odd 2 1 735.2.a.h 2
7.c even 3 2 105.2.i.d 4
7.d odd 6 2 735.2.i.l 4
21.c even 2 1 2205.2.a.ba 2
21.h odd 6 2 315.2.j.c 4
28.g odd 6 2 1680.2.bg.o 4
35.c odd 2 1 3675.2.a.be 2
35.j even 6 2 525.2.i.f 4
35.l odd 12 2 525.2.r.a 4
35.l odd 12 2 525.2.r.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 7.c even 3 2
315.2.j.c 4 21.h odd 6 2
525.2.i.f 4 35.j even 6 2
525.2.r.a 4 35.l odd 12 2
525.2.r.f 4 35.l odd 12 2
735.2.a.g 2 1.a even 1 1 trivial
735.2.a.h 2 7.b odd 2 1
735.2.i.l 4 7.d odd 6 2
1680.2.bg.o 4 28.g odd 6 2
2205.2.a.z 2 3.b odd 2 1
2205.2.a.ba 2 21.c even 2 1
3675.2.a.be 2 35.c odd 2 1
3675.2.a.bg 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(735))$$:

 $$T_{2}^{2} + 2T_{2} - 2$$ T2^2 + 2*T2 - 2 $$T_{13}^{2} - 8T_{13} + 13$$ T13^2 - 8*T13 + 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 2$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 2T - 2$$
$13$ $$T^{2} - 8T + 13$$
$17$ $$T^{2} - 10T + 22$$
$19$ $$T^{2} - 2T - 11$$
$23$ $$T^{2} + 6T + 6$$
$29$ $$T^{2} - 2T - 26$$
$31$ $$T^{2} - 6T - 3$$
$37$ $$T^{2} - 4T - 23$$
$41$ $$T^{2} - 2T - 2$$
$43$ $$T^{2} + 4T - 23$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} - 4T - 104$$
$59$ $$T^{2} - 10T - 2$$
$61$ $$(T - 4)^{2}$$
$67$ $$T^{2} + 12T - 39$$
$71$ $$T^{2} - 2T - 26$$
$73$ $$T^{2} - 8T - 59$$
$79$ $$T^{2} - 6T - 99$$
$83$ $$T^{2} - 6T - 138$$
$89$ $$T^{2} - 6T - 138$$
$97$ $$T^{2} - 16T + 16$$