# Properties

 Label 735.2.i.k 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2$$ (-v^3 + 2*v^2 - 2*v - 1) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2$$ (v^3 - 2*v^2 + 6*v - 1) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2$$ v^3 + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2$$ (b3 + b2 + 3*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2$$ b3 - 2

## Character values

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

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

## 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
 0.809017 − 1.40126i −0.309017 + 0.535233i 0.809017 + 1.40126i −0.309017 − 0.535233i
−1.11803 + 1.93649i 0.500000 + 0.866025i −1.50000 2.59808i 0.500000 0.866025i −2.23607 0 2.23607 −0.500000 + 0.866025i 1.11803 + 1.93649i
226.2 1.11803 1.93649i 0.500000 + 0.866025i −1.50000 2.59808i 0.500000 0.866025i 2.23607 0 −2.23607 −0.500000 + 0.866025i −1.11803 1.93649i
361.1 −1.11803 1.93649i 0.500000 0.866025i −1.50000 + 2.59808i 0.500000 + 0.866025i −2.23607 0 2.23607 −0.500000 0.866025i 1.11803 1.93649i
361.2 1.11803 + 1.93649i 0.500000 0.866025i −1.50000 + 2.59808i 0.500000 + 0.866025i 2.23607 0 −2.23607 −0.500000 0.866025i −1.11803 + 1.93649i
 $$n$$: 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.k 4
7.b odd 2 1 735.2.i.i 4
7.c even 3 1 105.2.a.b 2
7.c even 3 1 inner 735.2.i.k 4
7.d odd 6 1 735.2.a.k 2
7.d odd 6 1 735.2.i.i 4
21.g even 6 1 2205.2.a.w 2
21.h odd 6 1 315.2.a.d 2
28.g odd 6 1 1680.2.a.v 2
35.i odd 6 1 3675.2.a.y 2
35.j even 6 1 525.2.a.g 2
35.l odd 12 2 525.2.d.c 4
56.k odd 6 1 6720.2.a.cs 2
56.p even 6 1 6720.2.a.cx 2
84.n even 6 1 5040.2.a.bw 2
105.o odd 6 1 1575.2.a.r 2
105.x even 12 2 1575.2.d.d 4
140.p odd 6 1 8400.2.a.cx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b 2 7.c even 3 1
315.2.a.d 2 21.h odd 6 1
525.2.a.g 2 35.j even 6 1
525.2.d.c 4 35.l odd 12 2
735.2.a.k 2 7.d odd 6 1
735.2.i.i 4 7.b odd 2 1
735.2.i.i 4 7.d odd 6 1
735.2.i.k 4 1.a even 1 1 trivial
735.2.i.k 4 7.c even 3 1 inner
1575.2.a.r 2 105.o odd 6 1
1575.2.d.d 4 105.x even 12 2
1680.2.a.v 2 28.g odd 6 1
2205.2.a.w 2 21.g even 6 1
3675.2.a.y 2 35.i odd 6 1
5040.2.a.bw 2 84.n even 6 1
6720.2.a.cs 2 56.k odd 6 1
6720.2.a.cx 2 56.p even 6 1
8400.2.a.cx 2 140.p odd 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 5T_{2}^{2} + 25$$ T2^4 + 5*T2^2 + 25 $$T_{13}^{2} - 20$$ T13^2 - 20 $$T_{17}^{2} - 2T_{17} + 4$$ T17^2 - 2*T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 5T^{2} + 25$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 4 T^{3} + 32 T^{2} - 64 T + 256$$
$13$ $$(T^{2} - 20)^{2}$$
$17$ $$(T^{2} - 2 T + 4)^{2}$$
$19$ $$T^{4} + 4 T^{3} + 32 T^{2} - 64 T + 256$$
$23$ $$(T^{2} + 4 T + 16)^{2}$$
$29$ $$(T + 2)^{4}$$
$31$ $$T^{4} + 12 T^{3} + 128 T^{2} + \cdots + 256$$
$37$ $$T^{4} + 4 T^{3} + 92 T^{2} + \cdots + 5776$$
$41$ $$(T + 2)^{4}$$
$43$ $$(T^{2} - 80)^{2}$$
$47$ $$T^{4} + 8 T^{3} + 128 T^{2} + \cdots + 4096$$
$53$ $$T^{4} - 16 T^{3} + 212 T^{2} + \cdots + 1936$$
$59$ $$T^{4} + 80T^{2} + 6400$$
$61$ $$(T^{2} - 2 T + 4)^{2}$$
$67$ $$(T^{2} - 4 T + 16)^{2}$$
$71$ $$(T^{2} - 20 T + 80)^{2}$$
$73$ $$T^{4} - 16 T^{3} + 212 T^{2} + \cdots + 1936$$
$79$ $$T^{4} + 8 T^{3} + 128 T^{2} + \cdots + 4096$$
$83$ $$(T^{2} + 16 T - 16)^{2}$$
$89$ $$(T^{2} - 2 T + 4)^{2}$$
$97$ $$(T^{2} - 8 T - 4)^{2}$$