Properties

 Label 735.2.d.a Level $735$ Weight $2$ Character orbit 735.d 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(589,735)]

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, 1, 0]))

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

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

chi := DirichletCharacter("735.589");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.d (of order $$2$$, degree $$1$$, 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: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

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

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 −1.00000 2.00000i 1.00000 0 3.00000i −1.00000 −2.00000 + 1.00000i
589.2 1.00000i 1.00000i 1.00000 −1.00000 + 2.00000i 1.00000 0 3.00000i −1.00000 −2.00000 1.00000i
 $$n$$: 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
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.d.a 2
3.b odd 2 1 2205.2.d.f 2
5.b even 2 1 inner 735.2.d.a 2
5.c odd 4 1 3675.2.a.d 1
5.c odd 4 1 3675.2.a.l 1
7.b odd 2 1 105.2.d.a 2
7.c even 3 2 735.2.q.b 4
7.d odd 6 2 735.2.q.a 4
15.d odd 2 1 2205.2.d.f 2
21.c even 2 1 315.2.d.c 2
28.d even 2 1 1680.2.t.f 2
35.c odd 2 1 105.2.d.a 2
35.f even 4 1 525.2.a.b 1
35.f even 4 1 525.2.a.c 1
35.i odd 6 2 735.2.q.a 4
35.j even 6 2 735.2.q.b 4
84.h odd 2 1 5040.2.t.e 2
105.g even 2 1 315.2.d.c 2
105.k odd 4 1 1575.2.a.e 1
105.k odd 4 1 1575.2.a.i 1
140.c even 2 1 1680.2.t.f 2
140.j odd 4 1 8400.2.a.bj 1
140.j odd 4 1 8400.2.a.ch 1
420.o odd 2 1 5040.2.t.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 7.b odd 2 1
105.2.d.a 2 35.c odd 2 1
315.2.d.c 2 21.c even 2 1
315.2.d.c 2 105.g even 2 1
525.2.a.b 1 35.f even 4 1
525.2.a.c 1 35.f even 4 1
735.2.d.a 2 1.a even 1 1 trivial
735.2.d.a 2 5.b even 2 1 inner
735.2.q.a 4 7.d odd 6 2
735.2.q.a 4 35.i odd 6 2
735.2.q.b 4 7.c even 3 2
735.2.q.b 4 35.j even 6 2
1575.2.a.e 1 105.k odd 4 1
1575.2.a.i 1 105.k odd 4 1
1680.2.t.f 2 28.d even 2 1
1680.2.t.f 2 140.c even 2 1
2205.2.d.f 2 3.b odd 2 1
2205.2.d.f 2 15.d odd 2 1
3675.2.a.d 1 5.c odd 4 1
3675.2.a.l 1 5.c odd 4 1
5040.2.t.e 2 84.h odd 2 1
5040.2.t.e 2 420.o odd 2 1
8400.2.a.bj 1 140.j odd 4 1
8400.2.a.ch 1 140.j odd 4 1

Hecke kernels

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{19} + 6$$ T19 + 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2}$$
$11$ $$(T + 6)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 16$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$(T - 10)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 8)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 256$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 4$$