# Properties

 Label 735.2.i.l Level $735$ Weight $2$ Character orbit 735.i Analytic conductor $5.869$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(226,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("735.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.i (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$5.86900454856$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + \beta_1 q^{5} + (\beta_{3} - 1) q^{6} + (2 \beta_{3} - 6) q^{8} - \beta_1 q^{9}+O(q^{10})$$ q + (-b2 + b1) * q^2 + (b1 - 1) * q^3 + (2*b3 - 2*b2 + 2*b1 - 2) * q^4 + b1 * q^5 + (b3 - 1) * q^6 + (2*b3 - 6) * q^8 - b1 * q^9 $$q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + \beta_1 q^{5} + (\beta_{3} - 1) q^{6} + (2 \beta_{3} - 6) q^{8} - \beta_1 q^{9} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{10} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + (2 \beta_{2} - 2 \beta_1) q^{12} + ( - \beta_{3} - 4) q^{13} - q^{15} + (4 \beta_{2} - 8 \beta_1) q^{16} + (\beta_{3} - \beta_{2} - 5 \beta_1 + 5) q^{17} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{18} + ( - 2 \beta_{2} + \beta_1) q^{19} + (2 \beta_{3} - 2) q^{20} - 2 q^{22} + ( - \beta_{2} + 3 \beta_1) q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 6) q^{24} + (\beta_1 - 1) q^{25} + (3 \beta_{2} - \beta_1) q^{26} + q^{27} + (3 \beta_{3} + 1) q^{29} + (\beta_{2} - \beta_1) q^{30} + (2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{31} + ( - 8 \beta_{3} + 8 \beta_{2} - 8 \beta_1 + 8) q^{32} + (\beta_{2} + \beta_1) q^{33} + ( - 4 \beta_{3} + 2) q^{34} + ( - 2 \beta_{3} + 2) q^{36} + ( - 3 \beta_{2} - 2 \beta_1) q^{37} + (3 \beta_{3} - 3 \beta_{2} + 7 \beta_1 - 7) q^{38} + (\beta_{3} - \beta_{2} - 4 \beta_1 + 4) q^{39} + (2 \beta_{2} - 6 \beta_1) q^{40} + ( - \beta_{3} - 1) q^{41} + ( - 3 \beta_{3} - 2) q^{43} - 4 \beta_1 q^{44} + ( - \beta_1 + 1) q^{45} + (4 \beta_{3} - 4 \beta_{2} + 6 \beta_1 - 6) q^{46} + 2 \beta_1 q^{47} + ( - 4 \beta_{3} + 8) q^{48} + (\beta_{3} - 1) q^{50} + (\beta_{2} + 5 \beta_1) q^{51} + ( - 6 \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 2) q^{52} + (6 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 2) q^{53} + ( - \beta_{2} + \beta_1) q^{54} + (\beta_{3} + 1) q^{55} + (2 \beta_{3} - 1) q^{57} + (2 \beta_{2} - 8 \beta_1) q^{58} + (3 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 5) q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{60} + 4 \beta_1 q^{61} + ( - \beta_{3} - 3) q^{62} + ( - 8 \beta_{3} + 16) q^{64} + ( - \beta_{2} - 4 \beta_1) q^{65} + ( - 2 \beta_1 + 2) q^{66} + ( - 5 \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 6) q^{67} + ( - 8 \beta_{2} + 4 \beta_1) q^{68} + (\beta_{3} - 3) q^{69} + ( - 3 \beta_{3} + 1) q^{71} + ( - 2 \beta_{2} + 6 \beta_1) q^{72} + ( - 5 \beta_{3} + 5 \beta_{2} - 4 \beta_1 + 4) q^{73} + (\beta_{3} - \beta_{2} + 7 \beta_1 - 7) q^{74} - \beta_1 q^{75} + (6 \beta_{3} - 14) q^{76} + ( - 3 \beta_{3} + 1) q^{78} + ( - 6 \beta_{2} - 3 \beta_1) q^{79} + ( - 4 \beta_{3} + 4 \beta_{2} - 8 \beta_1 + 8) q^{80} + (\beta_1 - 1) q^{81} + 2 \beta_1 q^{82} + (7 \beta_{3} - 3) q^{83} + (\beta_{3} + 5) q^{85} + ( - \beta_{2} + 7 \beta_1) q^{86} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{87} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{88} + ( - 7 \beta_{2} + 3 \beta_1) q^{89} + ( - \beta_{3} + 1) q^{90} + (8 \beta_{3} - 12) q^{92} + (2 \beta_{2} + 3 \beta_1) q^{93} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{94} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{95} + ( - 8 \beta_{2} + 8 \beta_1) q^{96} + (4 \beta_{3} - 8) q^{97} + ( - \beta_{3} - 1) q^{99}+O(q^{100})$$ q + (-b2 + b1) * q^2 + (b1 - 1) * q^3 + (2*b3 - 2*b2 + 2*b1 - 2) * q^4 + b1 * q^5 + (b3 - 1) * q^6 + (2*b3 - 6) * q^8 - b1 * q^9 + (b3 - b2 + b1 - 1) * q^10 + (b3 - b2 - b1 + 1) * q^11 + (2*b2 - 2*b1) * q^12 + (-b3 - 4) * q^13 - q^15 + (4*b2 - 8*b1) * q^16 + (b3 - b2 - 5*b1 + 5) * q^17 + (-b3 + b2 - b1 + 1) * q^18 + (-2*b2 + b1) * q^19 + (2*b3 - 2) * q^20 - 2 * q^22 + (-b2 + 3*b1) * q^23 + (-2*b3 + 2*b2 - 6*b1 + 6) * q^24 + (b1 - 1) * q^25 + (3*b2 - b1) * q^26 + q^27 + (3*b3 + 1) * q^29 + (b2 - b1) * q^30 + (2*b3 - 2*b2 - 3*b1 + 3) * q^31 + (-8*b3 + 8*b2 - 8*b1 + 8) * q^32 + (b2 + b1) * q^33 + (-4*b3 + 2) * q^34 + (-2*b3 + 2) * q^36 + (-3*b2 - 2*b1) * q^37 + (3*b3 - 3*b2 + 7*b1 - 7) * q^38 + (b3 - b2 - 4*b1 + 4) * q^39 + (2*b2 - 6*b1) * q^40 + (-b3 - 1) * q^41 + (-3*b3 - 2) * q^43 - 4*b1 * q^44 + (-b1 + 1) * q^45 + (4*b3 - 4*b2 + 6*b1 - 6) * q^46 + 2*b1 * q^47 + (-4*b3 + 8) * q^48 + (b3 - 1) * q^50 + (b2 + 5*b1) * q^51 + (-6*b3 + 6*b2 - 2*b1 + 2) * q^52 + (6*b3 - 6*b2 + 2*b1 - 2) * q^53 + (-b2 + b1) * q^54 + (b3 + 1) * q^55 + (2*b3 - 1) * q^57 + (2*b2 - 8*b1) * q^58 + (3*b3 - 3*b2 - 5*b1 + 5) * q^59 + (-2*b3 + 2*b2 - 2*b1 + 2) * q^60 + 4*b1 * q^61 + (-b3 - 3) * q^62 + (-8*b3 + 16) * q^64 + (-b2 - 4*b1) * q^65 + (-2*b1 + 2) * q^66 + (-5*b3 + 5*b2 - 6*b1 + 6) * q^67 + (-8*b2 + 4*b1) * q^68 + (b3 - 3) * q^69 + (-3*b3 + 1) * q^71 + (-2*b2 + 6*b1) * q^72 + (-5*b3 + 5*b2 - 4*b1 + 4) * q^73 + (b3 - b2 + 7*b1 - 7) * q^74 - b1 * q^75 + (6*b3 - 14) * q^76 + (-3*b3 + 1) * q^78 + (-6*b2 - 3*b1) * q^79 + (-4*b3 + 4*b2 - 8*b1 + 8) * q^80 + (b1 - 1) * q^81 + 2*b1 * q^82 + (7*b3 - 3) * q^83 + (b3 + 5) * q^85 + (-b2 + 7*b1) * q^86 + (-3*b3 + 3*b2 + b1 - 1) * q^87 + (-4*b3 + 4*b2) * q^88 + (-7*b2 + 3*b1) * q^89 + (-b3 + 1) * q^90 + (8*b3 - 12) * q^92 + (2*b2 + 3*b1) * q^93 + (2*b3 - 2*b2 + 2*b1 - 2) * q^94 + (2*b3 - 2*b2 + b1 - 1) * q^95 + (-8*b2 + 8*b1) * q^96 + (4*b3 - 8) * q^97 + (-b3 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} - 4 q^{4} + 2 q^{5} - 4 q^{6} - 24 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 - 4 * q^4 + 2 * q^5 - 4 * q^6 - 24 * q^8 - 2 * q^9 $$4 q + 2 q^{2} - 2 q^{3} - 4 q^{4} + 2 q^{5} - 4 q^{6} - 24 q^{8} - 2 q^{9} - 2 q^{10} + 2 q^{11} - 4 q^{12} - 16 q^{13} - 4 q^{15} - 16 q^{16} + 10 q^{17} + 2 q^{18} + 2 q^{19} - 8 q^{20} - 8 q^{22} + 6 q^{23} + 12 q^{24} - 2 q^{25} - 2 q^{26} + 4 q^{27} + 4 q^{29} - 2 q^{30} + 6 q^{31} + 16 q^{32} + 2 q^{33} + 8 q^{34} + 8 q^{36} - 4 q^{37} - 14 q^{38} + 8 q^{39} - 12 q^{40} - 4 q^{41} - 8 q^{43} - 8 q^{44} + 2 q^{45} - 12 q^{46} + 4 q^{47} + 32 q^{48} - 4 q^{50} + 10 q^{51} + 4 q^{52} - 4 q^{53} + 2 q^{54} + 4 q^{55} - 4 q^{57} - 16 q^{58} + 10 q^{59} + 4 q^{60} + 8 q^{61} - 12 q^{62} + 64 q^{64} - 8 q^{65} + 4 q^{66} + 12 q^{67} + 8 q^{68} - 12 q^{69} + 4 q^{71} + 12 q^{72} + 8 q^{73} - 14 q^{74} - 2 q^{75} - 56 q^{76} + 4 q^{78} - 6 q^{79} + 16 q^{80} - 2 q^{81} + 4 q^{82} - 12 q^{83} + 20 q^{85} + 14 q^{86} - 2 q^{87} + 6 q^{89} + 4 q^{90} - 48 q^{92} + 6 q^{93} - 4 q^{94} - 2 q^{95} + 16 q^{96} - 32 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 - 4 * q^4 + 2 * q^5 - 4 * q^6 - 24 * q^8 - 2 * q^9 - 2 * q^10 + 2 * q^11 - 4 * q^12 - 16 * q^13 - 4 * q^15 - 16 * q^16 + 10 * q^17 + 2 * q^18 + 2 * q^19 - 8 * q^20 - 8 * q^22 + 6 * q^23 + 12 * q^24 - 2 * q^25 - 2 * q^26 + 4 * q^27 + 4 * q^29 - 2 * q^30 + 6 * q^31 + 16 * q^32 + 2 * q^33 + 8 * q^34 + 8 * q^36 - 4 * q^37 - 14 * q^38 + 8 * q^39 - 12 * q^40 - 4 * q^41 - 8 * q^43 - 8 * q^44 + 2 * q^45 - 12 * q^46 + 4 * q^47 + 32 * q^48 - 4 * q^50 + 10 * q^51 + 4 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 4 * q^57 - 16 * q^58 + 10 * q^59 + 4 * q^60 + 8 * q^61 - 12 * q^62 + 64 * q^64 - 8 * q^65 + 4 * q^66 + 12 * q^67 + 8 * q^68 - 12 * q^69 + 4 * q^71 + 12 * q^72 + 8 * q^73 - 14 * q^74 - 2 * q^75 - 56 * q^76 + 4 * q^78 - 6 * q^79 + 16 * q^80 - 2 * q^81 + 4 * q^82 - 12 * q^83 + 20 * q^85 + 14 * q^86 - 2 * q^87 + 6 * q^89 + 4 * q^90 - 48 * q^92 + 6 * q^93 - 4 * q^94 - 2 * q^95 + 16 * q^96 - 32 * q^97 - 4 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$$.

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$-\beta_{1}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.366025 + 0.633975i −0.500000 0.866025i 0.732051 + 1.26795i 0.500000 0.866025i 0.732051 0 −2.53590 −0.500000 + 0.866025i 0.366025 + 0.633975i
226.2 1.36603 2.36603i −0.500000 0.866025i −2.73205 4.73205i 0.500000 0.866025i −2.73205 0 −9.46410 −0.500000 + 0.866025i −1.36603 2.36603i
361.1 −0.366025 0.633975i −0.500000 + 0.866025i 0.732051 1.26795i 0.500000 + 0.866025i 0.732051 0 −2.53590 −0.500000 0.866025i 0.366025 0.633975i
361.2 1.36603 + 2.36603i −0.500000 + 0.866025i −2.73205 + 4.73205i 0.500000 + 0.866025i −2.73205 0 −9.46410 −0.500000 0.866025i −1.36603 + 2.36603i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} - 2T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 4$$ T2^4 - 2*T2^3 + 6*T2^2 + 4*T2 + 4 $$T_{13}^{2} + 8T_{13} + 13$$ T13^2 + 8*T13 + 13 $$T_{17}^{4} - 10T_{17}^{3} + 78T_{17}^{2} - 220T_{17} + 484$$ T17^4 - 10*T17^3 + 78*T17^2 - 220*T17 + 484

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4$$
$13$ $$(T^{2} + 8 T + 13)^{2}$$
$17$ $$T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484$$
$19$ $$T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121$$
$23$ $$T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36$$
$29$ $$(T^{2} - 2 T - 26)^{2}$$
$31$ $$T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9$$
$37$ $$T^{4} + 4 T^{3} + 39 T^{2} - 92 T + 529$$
$41$ $$(T^{2} + 2 T - 2)^{2}$$
$43$ $$(T^{2} + 4 T - 23)^{2}$$
$47$ $$(T^{2} - 2 T + 4)^{2}$$
$53$ $$T^{4} + 4 T^{3} + 120 T^{2} + \cdots + 10816$$
$59$ $$T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4$$
$61$ $$(T^{2} - 4 T + 16)^{2}$$
$67$ $$T^{4} - 12 T^{3} + 183 T^{2} + \cdots + 1521$$
$71$ $$(T^{2} - 2 T - 26)^{2}$$
$73$ $$T^{4} - 8 T^{3} + 123 T^{2} + \cdots + 3481$$
$79$ $$T^{4} + 6 T^{3} + 135 T^{2} + \cdots + 9801$$
$83$ $$(T^{2} + 6 T - 138)^{2}$$
$89$ $$T^{4} - 6 T^{3} + 174 T^{2} + \cdots + 19044$$
$97$ $$(T^{2} + 16 T + 16)^{2}$$