# Properties

 Label 735.2.i.d Level $735$ Weight $2$ Character orbit 735.i Analytic conductor $5.869$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$1$$ Twist minimal: no (minimal twist has level 15) 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −1.00000 0 3.00000 −0.500000 + 0.866025i −0.500000 0.866025i
361.1 0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i −1.00000 0 3.00000 −0.500000 0.866025i −0.500000 + 0.866025i
 $$n$$: 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.d 2
7.b odd 2 1 735.2.i.e 2
7.c even 3 1 735.2.a.c 1
7.c even 3 1 inner 735.2.i.d 2
7.d odd 6 1 15.2.a.a 1
7.d odd 6 1 735.2.i.e 2
21.g even 6 1 45.2.a.a 1
21.h odd 6 1 2205.2.a.i 1
28.f even 6 1 240.2.a.d 1
35.i odd 6 1 75.2.a.b 1
35.j even 6 1 3675.2.a.j 1
35.k even 12 2 75.2.b.b 2
56.j odd 6 1 960.2.a.l 1
56.m even 6 1 960.2.a.a 1
63.i even 6 1 405.2.e.c 2
63.k odd 6 1 405.2.e.f 2
63.s even 6 1 405.2.e.c 2
63.t odd 6 1 405.2.e.f 2
77.i even 6 1 1815.2.a.d 1
84.j odd 6 1 720.2.a.c 1
91.s odd 6 1 2535.2.a.j 1
105.p even 6 1 225.2.a.b 1
105.w odd 12 2 225.2.b.b 2
112.v even 12 2 3840.2.k.r 2
112.x odd 12 2 3840.2.k.m 2
119.h odd 6 1 4335.2.a.c 1
133.o even 6 1 5415.2.a.j 1
140.s even 6 1 1200.2.a.e 1
140.x odd 12 2 1200.2.f.h 2
161.g even 6 1 7935.2.a.d 1
168.ba even 6 1 2880.2.a.y 1
168.be odd 6 1 2880.2.a.bc 1
231.k odd 6 1 5445.2.a.c 1
273.ba even 6 1 7605.2.a.g 1
280.ba even 6 1 4800.2.a.bz 1
280.bk odd 6 1 4800.2.a.t 1
280.bp odd 12 2 4800.2.f.c 2
280.bv even 12 2 4800.2.f.bf 2
385.o even 6 1 9075.2.a.g 1
420.be odd 6 1 3600.2.a.u 1
420.br even 12 2 3600.2.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 7.d odd 6 1
45.2.a.a 1 21.g even 6 1
75.2.a.b 1 35.i odd 6 1
75.2.b.b 2 35.k even 12 2
225.2.a.b 1 105.p even 6 1
225.2.b.b 2 105.w odd 12 2
240.2.a.d 1 28.f even 6 1
405.2.e.c 2 63.i even 6 1
405.2.e.c 2 63.s even 6 1
405.2.e.f 2 63.k odd 6 1
405.2.e.f 2 63.t odd 6 1
720.2.a.c 1 84.j odd 6 1
735.2.a.c 1 7.c even 3 1
735.2.i.d 2 1.a even 1 1 trivial
735.2.i.d 2 7.c even 3 1 inner
735.2.i.e 2 7.b odd 2 1
735.2.i.e 2 7.d odd 6 1
960.2.a.a 1 56.m even 6 1
960.2.a.l 1 56.j odd 6 1
1200.2.a.e 1 140.s even 6 1
1200.2.f.h 2 140.x odd 12 2
1815.2.a.d 1 77.i even 6 1
2205.2.a.i 1 21.h odd 6 1
2535.2.a.j 1 91.s odd 6 1
2880.2.a.y 1 168.ba even 6 1
2880.2.a.bc 1 168.be odd 6 1
3600.2.a.u 1 420.be odd 6 1
3600.2.f.e 2 420.br even 12 2
3675.2.a.j 1 35.j even 6 1
3840.2.k.m 2 112.x odd 12 2
3840.2.k.r 2 112.v even 12 2
4335.2.a.c 1 119.h odd 6 1
4800.2.a.t 1 280.bk odd 6 1
4800.2.a.bz 1 280.ba even 6 1
4800.2.f.c 2 280.bp odd 12 2
4800.2.f.bf 2 280.bv even 12 2
5415.2.a.j 1 133.o even 6 1
5445.2.a.c 1 231.k odd 6 1
7605.2.a.g 1 273.ba even 6 1
7935.2.a.d 1 161.g even 6 1
9075.2.a.g 1 385.o even 6 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{13} - 2$$ T13 - 2 $$T_{17}^{2} - 2T_{17} + 4$$ T17^2 - 2*T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T + 16$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2}$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 10T + 100$$
$41$ $$(T + 10)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} - 10T + 100$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} + 12T + 144$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 10T + 100$$
$79$ $$T^{2}$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T + 2)^{2}$$