# Properties

 Label 1734.2.b.b Level $1734$ Weight $2$ Character orbit 1734.b Analytic conductor $13.846$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1734,2,Mod(577,1734)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1734, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1734.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1734 = 2 \cdot 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1734.b (of order $$2$$, degree $$1$$, 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: $$13.8460597105$$ Analytic rank: $$1$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} - i q^{3} + q^{4} + 2 i q^{5} + i q^{6} - q^{8} - q^{9} +O(q^{10})$$ q - q^2 - i * q^3 + q^4 + 2*i * q^5 + i * q^6 - q^8 - q^9 $$q - q^{2} - i q^{3} + q^{4} + 2 i q^{5} + i q^{6} - q^{8} - q^{9} - 2 i q^{10} - 4 i q^{11} - i q^{12} - 2 q^{13} + 2 q^{15} + q^{16} + q^{18} - 4 q^{19} + 2 i q^{20} + 4 i q^{22} + i q^{24} + q^{25} + 2 q^{26} + i q^{27} + 10 i q^{29} - 2 q^{30} - 8 i q^{31} - q^{32} - 4 q^{33} - q^{36} + 2 i q^{37} + 4 q^{38} + 2 i q^{39} - 2 i q^{40} + 10 i q^{41} - 12 q^{43} - 4 i q^{44} - 2 i q^{45} - i q^{48} + 7 q^{49} - q^{50} - 2 q^{52} - 6 q^{53} - i q^{54} + 8 q^{55} + 4 i q^{57} - 10 i q^{58} - 12 q^{59} + 2 q^{60} - 10 i q^{61} + 8 i q^{62} + q^{64} - 4 i q^{65} + 4 q^{66} - 12 q^{67} + q^{72} - 10 i q^{73} - 2 i q^{74} - i q^{75} - 4 q^{76} - 2 i q^{78} - 8 i q^{79} + 2 i q^{80} + q^{81} - 10 i q^{82} - 4 q^{83} + 12 q^{86} + 10 q^{87} + 4 i q^{88} - 6 q^{89} + 2 i q^{90} - 8 q^{93} - 8 i q^{95} + i q^{96} + 14 i q^{97} - 7 q^{98} + 4 i q^{99} +O(q^{100})$$ q - q^2 - i * q^3 + q^4 + 2*i * q^5 + i * q^6 - q^8 - q^9 - 2*i * q^10 - 4*i * q^11 - i * q^12 - 2 * q^13 + 2 * q^15 + q^16 + q^18 - 4 * q^19 + 2*i * q^20 + 4*i * q^22 + i * q^24 + q^25 + 2 * q^26 + i * q^27 + 10*i * q^29 - 2 * q^30 - 8*i * q^31 - q^32 - 4 * q^33 - q^36 + 2*i * q^37 + 4 * q^38 + 2*i * q^39 - 2*i * q^40 + 10*i * q^41 - 12 * q^43 - 4*i * q^44 - 2*i * q^45 - i * q^48 + 7 * q^49 - q^50 - 2 * q^52 - 6 * q^53 - i * q^54 + 8 * q^55 + 4*i * q^57 - 10*i * q^58 - 12 * q^59 + 2 * q^60 - 10*i * q^61 + 8*i * q^62 + q^64 - 4*i * q^65 + 4 * q^66 - 12 * q^67 + q^72 - 10*i * q^73 - 2*i * q^74 - i * q^75 - 4 * q^76 - 2*i * q^78 - 8*i * q^79 + 2*i * q^80 + q^81 - 10*i * q^82 - 4 * q^83 + 12 * q^86 + 10 * q^87 + 4*i * q^88 - 6 * q^89 + 2*i * q^90 - 8 * q^93 - 8*i * q^95 + i * q^96 + 14*i * q^97 - 7 * q^98 + 4*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 4 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{18} - 8 q^{19} + 2 q^{25} + 4 q^{26} - 4 q^{30} - 2 q^{32} - 8 q^{33} - 2 q^{36} + 8 q^{38} - 24 q^{43} + 14 q^{49} - 2 q^{50} - 4 q^{52} - 12 q^{53} + 16 q^{55} - 24 q^{59} + 4 q^{60} + 2 q^{64} + 8 q^{66} - 24 q^{67} + 2 q^{72} - 8 q^{76} + 2 q^{81} - 8 q^{83} + 24 q^{86} + 20 q^{87} - 12 q^{89} - 16 q^{93} - 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 - 4 * q^13 + 4 * q^15 + 2 * q^16 + 2 * q^18 - 8 * q^19 + 2 * q^25 + 4 * q^26 - 4 * q^30 - 2 * q^32 - 8 * q^33 - 2 * q^36 + 8 * q^38 - 24 * q^43 + 14 * q^49 - 2 * q^50 - 4 * q^52 - 12 * q^53 + 16 * q^55 - 24 * q^59 + 4 * q^60 + 2 * q^64 + 8 * q^66 - 24 * q^67 + 2 * q^72 - 8 * q^76 + 2 * q^81 - 8 * q^83 + 24 * q^86 + 20 * q^87 - 12 * q^89 - 16 * q^93 - 14 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times$$.

 $$n$$ $$1157$$ $$1159$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 1.00000i − 1.00000i
−1.00000 1.00000i 1.00000 2.00000i 1.00000i 0 −1.00000 −1.00000 2.00000i
577.2 −1.00000 1.00000i 1.00000 2.00000i 1.00000i 0 −1.00000 −1.00000 2.00000i
 $$n$$: 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
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1734.2.b.b 2
17.b even 2 1 inner 1734.2.b.b 2
17.c even 4 1 102.2.a.c 1
17.c even 4 1 1734.2.a.j 1
17.d even 8 4 1734.2.f.e 4
51.f odd 4 1 306.2.a.b 1
51.f odd 4 1 5202.2.a.c 1
68.f odd 4 1 816.2.a.b 1
85.f odd 4 1 2550.2.d.m 2
85.i odd 4 1 2550.2.d.m 2
85.j even 4 1 2550.2.a.c 1
119.f odd 4 1 4998.2.a.be 1
136.i even 4 1 3264.2.a.m 1
136.j odd 4 1 3264.2.a.bc 1
204.l even 4 1 2448.2.a.p 1
255.i odd 4 1 7650.2.a.ca 1
408.q even 4 1 9792.2.a.l 1
408.t odd 4 1 9792.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 17.c even 4 1
306.2.a.b 1 51.f odd 4 1
816.2.a.b 1 68.f odd 4 1
1734.2.a.j 1 17.c even 4 1
1734.2.b.b 2 1.a even 1 1 trivial
1734.2.b.b 2 17.b even 2 1 inner
1734.2.f.e 4 17.d even 8 4
2448.2.a.p 1 204.l even 4 1
2550.2.a.c 1 85.j even 4 1
2550.2.d.m 2 85.f odd 4 1
2550.2.d.m 2 85.i odd 4 1
3264.2.a.m 1 136.i even 4 1
3264.2.a.bc 1 136.j odd 4 1
4998.2.a.be 1 119.f odd 4 1
5202.2.a.c 1 51.f odd 4 1
7650.2.a.ca 1 255.i odd 4 1
9792.2.a.k 1 408.t odd 4 1
9792.2.a.l 1 408.q even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1734, [\chi])$$:

 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 16$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 100$$
$31$ $$T^{2} + 64$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2} + 100$$
$43$ $$(T + 12)^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$(T + 12)^{2}$$
$61$ $$T^{2} + 100$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$T^{2} + 64$$
$83$ $$(T + 4)^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 196$$