# Properties

 Label 735.2.i.g 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} + \beta_{2} q^{3} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} + 1) q^{6} + (\beta_{3} + 3) q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^2 + b2 * q^3 + (-2*b3 + b2 - 2*b1) * q^4 + (-b2 - 1) * q^5 + (b3 + 1) * q^6 + (b3 + 3) * q^8 + (-b2 - 1) * q^9 $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} + \beta_{2} q^{3} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} + 1) q^{6} + (\beta_{3} + 3) q^{8} + ( - \beta_{2} - 1) q^{9} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{10} + ( - 2 \beta_{3} - 2 \beta_1) q^{11} + ( - \beta_{2} + 2 \beta_1 - 1) q^{12} + ( - 2 \beta_{3} - 2) q^{13} + q^{15} + ( - 3 \beta_{2} - 3) q^{16} + (4 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{17} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{18} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{19} + (2 \beta_{3} + 1) q^{20} + (2 \beta_{3} + 4) q^{22} + (2 \beta_{2} - 4 \beta_1 + 2) q^{23} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{24} + \beta_{2} q^{25} + (6 \beta_{2} - 4 \beta_1 + 6) q^{26} + q^{27} + 6 q^{29} + ( - \beta_{2} + \beta_1 - 1) q^{30} + ( - 6 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{31} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{32} + 2 \beta_1 q^{33} + ( - 6 \beta_{3} - 10) q^{34} + (2 \beta_{3} + 1) q^{36} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{37} - 2 \beta_{2} q^{38} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{39} + ( - 3 \beta_{2} + \beta_1 - 3) q^{40} + (4 \beta_{3} + 6) q^{41} + 8 q^{43} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{44} + \beta_{2} q^{45} + (6 \beta_{3} - 10 \beta_{2} + 6 \beta_1) q^{46} + 4 \beta_1 q^{47} + 3 q^{48} + (\beta_{3} + 1) q^{50} + (2 \beta_{2} - 4 \beta_1 + 2) q^{51} + (6 \beta_{3} - 10 \beta_{2} + 6 \beta_1) q^{52} + (6 \beta_{3} + 6 \beta_1) q^{53} + ( - \beta_{2} + \beta_1 - 1) q^{54} + 2 \beta_{3} q^{55} + ( - 2 \beta_{3} + 2) q^{57} + ( - 6 \beta_{2} + 6 \beta_1 - 6) q^{58} + (4 \beta_{3} + 8 \beta_{2} + 4 \beta_1) q^{59} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{60} + (4 \beta_{3} + 10) q^{62} + ( - 2 \beta_{3} - 7) q^{64} + (2 \beta_{2} - 2 \beta_1 + 2) q^{65} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{66} + (18 \beta_{2} - 8 \beta_1 + 18) q^{68} + ( - 4 \beta_{3} - 2) q^{69} - 2 \beta_{3} q^{71} + ( - 3 \beta_{2} + \beta_1 - 3) q^{72} + ( - 2 \beta_{3} - 14 \beta_{2} - 2 \beta_1) q^{73} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{74} + ( - \beta_{2} - 1) q^{75} + (2 \beta_{3} - 6) q^{76} + ( - 4 \beta_{3} - 6) q^{78} + (8 \beta_{2} + 8) q^{79} + 3 \beta_{2} q^{80} + \beta_{2} q^{81} + ( - 14 \beta_{2} + 10 \beta_1 - 14) q^{82} + ( - 4 \beta_{3} + 4) q^{83} + ( - 4 \beta_{3} - 2) q^{85} + ( - 8 \beta_{2} + 8 \beta_1 - 8) q^{86} + 6 \beta_{2} q^{87} + ( - 6 \beta_{3} + 4 \beta_{2} - 6 \beta_1) q^{88} + ( - 6 \beta_{2} - 8 \beta_1 - 6) q^{89} + (\beta_{3} + 1) q^{90} + ( - 8 \beta_{3} - 18) q^{92} + (2 \beta_{2} + 6 \beta_1 + 2) q^{93} + ( - 4 \beta_{3} + 8 \beta_{2} - 4 \beta_1) q^{94} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{95} + (3 \beta_{2} + \beta_1 + 3) q^{96} + (6 \beta_{3} - 2) q^{97} + 2 \beta_{3} q^{99}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^2 + b2 * q^3 + (-2*b3 + b2 - 2*b1) * q^4 + (-b2 - 1) * q^5 + (b3 + 1) * q^6 + (b3 + 3) * q^8 + (-b2 - 1) * q^9 + (-b3 + b2 - b1) * q^10 + (-2*b3 - 2*b1) * q^11 + (-b2 + 2*b1 - 1) * q^12 + (-2*b3 - 2) * q^13 + q^15 + (-3*b2 - 3) * q^16 + (4*b3 - 2*b2 + 4*b1) * q^17 + (-b3 + b2 - b1) * q^18 + (-2*b2 - 2*b1 - 2) * q^19 + (2*b3 + 1) * q^20 + (2*b3 + 4) * q^22 + (2*b2 - 4*b1 + 2) * q^23 + (-b3 + 3*b2 - b1) * q^24 + b2 * q^25 + (6*b2 - 4*b1 + 6) * q^26 + q^27 + 6 * q^29 + (-b2 + b1 - 1) * q^30 + (-6*b3 - 2*b2 - 6*b1) * q^31 + (-b3 - 3*b2 - b1) * q^32 + 2*b1 * q^33 + (-6*b3 - 10) * q^34 + (2*b3 + 1) * q^36 + (-2*b2 - 4*b1 - 2) * q^37 - 2*b2 * q^38 + (2*b3 - 2*b2 + 2*b1) * q^39 + (-3*b2 + b1 - 3) * q^40 + (4*b3 + 6) * q^41 + 8 * q^43 + (-8*b2 + 2*b1 - 8) * q^44 + b2 * q^45 + (6*b3 - 10*b2 + 6*b1) * q^46 + 4*b1 * q^47 + 3 * q^48 + (b3 + 1) * q^50 + (2*b2 - 4*b1 + 2) * q^51 + (6*b3 - 10*b2 + 6*b1) * q^52 + (6*b3 + 6*b1) * q^53 + (-b2 + b1 - 1) * q^54 + 2*b3 * q^55 + (-2*b3 + 2) * q^57 + (-6*b2 + 6*b1 - 6) * q^58 + (4*b3 + 8*b2 + 4*b1) * q^59 + (-2*b3 + b2 - 2*b1) * q^60 + (4*b3 + 10) * q^62 + (-2*b3 - 7) * q^64 + (2*b2 - 2*b1 + 2) * q^65 + (-2*b3 + 4*b2 - 2*b1) * q^66 + (18*b2 - 8*b1 + 18) * q^68 + (-4*b3 - 2) * q^69 - 2*b3 * q^71 + (-3*b2 + b1 - 3) * q^72 + (-2*b3 - 14*b2 - 2*b1) * q^73 + (2*b3 - 6*b2 + 2*b1) * q^74 + (-b2 - 1) * q^75 + (2*b3 - 6) * q^76 + (-4*b3 - 6) * q^78 + (8*b2 + 8) * q^79 + 3*b2 * q^80 + b2 * q^81 + (-14*b2 + 10*b1 - 14) * q^82 + (-4*b3 + 4) * q^83 + (-4*b3 - 2) * q^85 + (-8*b2 + 8*b1 - 8) * q^86 + 6*b2 * q^87 + (-6*b3 + 4*b2 - 6*b1) * q^88 + (-6*b2 - 8*b1 - 6) * q^89 + (b3 + 1) * q^90 + (-8*b3 - 18) * q^92 + (2*b2 + 6*b1 + 2) * q^93 + (-4*b3 + 8*b2 - 4*b1) * q^94 + (2*b3 + 2*b2 + 2*b1) * q^95 + (3*b2 + b1 + 3) * q^96 + (6*b3 - 2) * q^97 + 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 4 q^{6} + 12 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 4 * q^6 + 12 * q^8 - 2 * q^9 $$4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 4 q^{6} + 12 q^{8} - 2 q^{9} - 2 q^{10} - 2 q^{12} - 8 q^{13} + 4 q^{15} - 6 q^{16} + 4 q^{17} - 2 q^{18} - 4 q^{19} + 4 q^{20} + 16 q^{22} + 4 q^{23} - 6 q^{24} - 2 q^{25} + 12 q^{26} + 4 q^{27} + 24 q^{29} - 2 q^{30} + 4 q^{31} + 6 q^{32} - 40 q^{34} + 4 q^{36} - 4 q^{37} + 4 q^{38} + 4 q^{39} - 6 q^{40} + 24 q^{41} + 32 q^{43} - 16 q^{44} - 2 q^{45} + 20 q^{46} + 12 q^{48} + 4 q^{50} + 4 q^{51} + 20 q^{52} - 2 q^{54} + 8 q^{57} - 12 q^{58} - 16 q^{59} - 2 q^{60} + 40 q^{62} - 28 q^{64} + 4 q^{65} - 8 q^{66} + 36 q^{68} - 8 q^{69} - 6 q^{72} + 28 q^{73} + 12 q^{74} - 2 q^{75} - 24 q^{76} - 24 q^{78} + 16 q^{79} - 6 q^{80} - 2 q^{81} - 28 q^{82} + 16 q^{83} - 8 q^{85} - 16 q^{86} - 12 q^{87} - 8 q^{88} - 12 q^{89} + 4 q^{90} - 72 q^{92} + 4 q^{93} - 16 q^{94} - 4 q^{95} + 6 q^{96} - 8 q^{97}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 4 * q^6 + 12 * q^8 - 2 * q^9 - 2 * q^10 - 2 * q^12 - 8 * q^13 + 4 * q^15 - 6 * q^16 + 4 * q^17 - 2 * q^18 - 4 * q^19 + 4 * q^20 + 16 * q^22 + 4 * q^23 - 6 * q^24 - 2 * q^25 + 12 * q^26 + 4 * q^27 + 24 * q^29 - 2 * q^30 + 4 * q^31 + 6 * q^32 - 40 * q^34 + 4 * q^36 - 4 * q^37 + 4 * q^38 + 4 * q^39 - 6 * q^40 + 24 * q^41 + 32 * q^43 - 16 * q^44 - 2 * q^45 + 20 * q^46 + 12 * q^48 + 4 * q^50 + 4 * q^51 + 20 * q^52 - 2 * q^54 + 8 * q^57 - 12 * q^58 - 16 * q^59 - 2 * q^60 + 40 * q^62 - 28 * q^64 + 4 * q^65 - 8 * q^66 + 36 * q^68 - 8 * q^69 - 6 * q^72 + 28 * q^73 + 12 * q^74 - 2 * q^75 - 24 * q^76 - 24 * q^78 + 16 * q^79 - 6 * q^80 - 2 * q^81 - 28 * q^82 + 16 * q^83 - 8 * q^85 - 16 * q^86 - 12 * q^87 - 8 * q^88 - 12 * q^89 + 4 * q^90 - 72 * q^92 + 4 * q^93 - 16 * q^94 - 4 * q^95 + 6 * q^96 - 8 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$ $$1$$

## 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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−1.20711 + 2.09077i −0.500000 0.866025i −1.91421 3.31552i −0.500000 + 0.866025i 2.41421 0 4.41421 −0.500000 + 0.866025i −1.20711 2.09077i
226.2 0.207107 0.358719i −0.500000 0.866025i 0.914214 + 1.58346i −0.500000 + 0.866025i −0.414214 0 1.58579 −0.500000 + 0.866025i 0.207107 + 0.358719i
361.1 −1.20711 2.09077i −0.500000 + 0.866025i −1.91421 + 3.31552i −0.500000 0.866025i 2.41421 0 4.41421 −0.500000 0.866025i −1.20711 + 2.09077i
361.2 0.207107 + 0.358719i −0.500000 + 0.866025i 0.914214 1.58346i −0.500000 0.866025i −0.414214 0 1.58579 −0.500000 0.866025i 0.207107 0.358719i
 $$n$$: 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.g 4
7.b odd 2 1 735.2.i.h 4
7.c even 3 1 735.2.a.m yes 2
7.c even 3 1 inner 735.2.i.g 4
7.d odd 6 1 735.2.a.l 2
7.d odd 6 1 735.2.i.h 4
21.g even 6 1 2205.2.a.r 2
21.h odd 6 1 2205.2.a.o 2
35.i odd 6 1 3675.2.a.t 2
35.j even 6 1 3675.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.a.l 2 7.d odd 6 1
735.2.a.m yes 2 7.c even 3 1
735.2.i.g 4 1.a even 1 1 trivial
735.2.i.g 4 7.c even 3 1 inner
735.2.i.h 4 7.b odd 2 1
735.2.i.h 4 7.d odd 6 1
2205.2.a.o 2 21.h odd 6 1
2205.2.a.r 2 21.g even 6 1
3675.2.a.s 2 35.j even 6 1
3675.2.a.t 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 $$S_{2}^{\mathrm{new}}(735, [\chi])$$:

 $$T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1$$ T2^4 + 2*T2^3 + 5*T2^2 - 2*T2 + 1 $$T_{13}^{2} + 4T_{13} - 4$$ T13^2 + 4*T13 - 4 $$T_{17}^{4} - 4T_{17}^{3} + 44T_{17}^{2} + 112T_{17} + 784$$ T17^4 - 4*T17^3 + 44*T17^2 + 112*T17 + 784

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 8T^{2} + 64$$
$13$ $$(T^{2} + 4 T - 4)^{2}$$
$17$ $$T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784$$
$19$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$23$ $$T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784$$
$29$ $$(T - 6)^{4}$$
$31$ $$T^{4} - 4 T^{3} + 84 T^{2} + \cdots + 4624$$
$37$ $$T^{4} + 4 T^{3} + 44 T^{2} - 112 T + 784$$
$41$ $$(T^{2} - 12 T + 4)^{2}$$
$43$ $$(T - 8)^{4}$$
$47$ $$T^{4} + 32T^{2} + 1024$$
$53$ $$T^{4} + 72T^{2} + 5184$$
$59$ $$T^{4} + 16 T^{3} + 224 T^{2} + \cdots + 1024$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$(T^{2} - 8)^{2}$$
$73$ $$T^{4} - 28 T^{3} + 596 T^{2} + \cdots + 35344$$
$79$ $$(T^{2} - 8 T + 64)^{2}$$
$83$ $$(T^{2} - 8 T - 16)^{2}$$
$89$ $$T^{4} + 12 T^{3} + 236 T^{2} + \cdots + 8464$$
$97$ $$(T^{2} + 4 T - 68)^{2}$$