# Properties

 Label 1734.2.f.b Level $1734$ Weight $2$ Character orbit 1734.f Analytic conductor $13.846$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1734,2,Mod(829,1734)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1734, base_ring=CyclotomicField(4))

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("1734.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1734 = 2 \cdot 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1734.f (of order $$4$$, 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: $$13.8460597105$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 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: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{2} q^{2} + \zeta_{8}^{3} q^{3} - q^{4} + \zeta_{8} q^{6} + 2 \zeta_{8} q^{7} + \zeta_{8}^{2} q^{8} - \zeta_{8}^{2} q^{9} +O(q^{10})$$ q - z^2 * q^2 + z^3 * q^3 - q^4 + z * q^6 + 2*z * q^7 + z^2 * q^8 - z^2 * q^9 $$q - \zeta_{8}^{2} q^{2} + \zeta_{8}^{3} q^{3} - q^{4} + \zeta_{8} q^{6} + 2 \zeta_{8} q^{7} + \zeta_{8}^{2} q^{8} - \zeta_{8}^{2} q^{9} - \zeta_{8}^{3} q^{12} - 2 q^{13} - 2 \zeta_{8}^{3} q^{14} + q^{16} - q^{18} - 4 \zeta_{8}^{2} q^{19} - 2 q^{21} + 6 \zeta_{8} q^{23} - \zeta_{8} q^{24} + 5 \zeta_{8}^{2} q^{25} + 2 \zeta_{8}^{2} q^{26} + \zeta_{8} q^{27} - 2 \zeta_{8} q^{28} - 10 \zeta_{8}^{3} q^{31} - \zeta_{8}^{2} q^{32} + \zeta_{8}^{2} q^{36} + 8 \zeta_{8}^{3} q^{37} - 4 q^{38} - 2 \zeta_{8}^{3} q^{39} + 6 \zeta_{8} q^{41} + 2 \zeta_{8}^{2} q^{42} + 4 \zeta_{8}^{2} q^{43} - 6 \zeta_{8}^{3} q^{46} - 12 q^{47} + \zeta_{8}^{3} q^{48} - 3 \zeta_{8}^{2} q^{49} + 5 q^{50} + 2 q^{52} + 6 \zeta_{8}^{2} q^{53} - \zeta_{8}^{3} q^{54} + 2 \zeta_{8}^{3} q^{56} + 4 \zeta_{8} q^{57} + 12 \zeta_{8}^{2} q^{59} + 8 \zeta_{8} q^{61} - 10 \zeta_{8} q^{62} - 2 \zeta_{8}^{3} q^{63} - q^{64} - 4 q^{67} - 6 q^{69} + 6 \zeta_{8}^{3} q^{71} + q^{72} - 2 \zeta_{8}^{3} q^{73} + 8 \zeta_{8} q^{74} - 5 \zeta_{8} q^{75} + 4 \zeta_{8}^{2} q^{76} - 2 \zeta_{8} q^{78} + 10 \zeta_{8} q^{79} - q^{81} - 6 \zeta_{8}^{3} q^{82} + 12 \zeta_{8}^{2} q^{83} + 2 q^{84} + 4 q^{86} + 18 q^{89} - 4 \zeta_{8} q^{91} - 6 \zeta_{8} q^{92} + 10 \zeta_{8}^{2} q^{93} + 12 \zeta_{8}^{2} q^{94} + \zeta_{8} q^{96} - 14 \zeta_{8}^{3} q^{97} - 3 q^{98} +O(q^{100})$$ q - z^2 * q^2 + z^3 * q^3 - q^4 + z * q^6 + 2*z * q^7 + z^2 * q^8 - z^2 * q^9 - z^3 * q^12 - 2 * q^13 - 2*z^3 * q^14 + q^16 - q^18 - 4*z^2 * q^19 - 2 * q^21 + 6*z * q^23 - z * q^24 + 5*z^2 * q^25 + 2*z^2 * q^26 + z * q^27 - 2*z * q^28 - 10*z^3 * q^31 - z^2 * q^32 + z^2 * q^36 + 8*z^3 * q^37 - 4 * q^38 - 2*z^3 * q^39 + 6*z * q^41 + 2*z^2 * q^42 + 4*z^2 * q^43 - 6*z^3 * q^46 - 12 * q^47 + z^3 * q^48 - 3*z^2 * q^49 + 5 * q^50 + 2 * q^52 + 6*z^2 * q^53 - z^3 * q^54 + 2*z^3 * q^56 + 4*z * q^57 + 12*z^2 * q^59 + 8*z * q^61 - 10*z * q^62 - 2*z^3 * q^63 - q^64 - 4 * q^67 - 6 * q^69 + 6*z^3 * q^71 + q^72 - 2*z^3 * q^73 + 8*z * q^74 - 5*z * q^75 + 4*z^2 * q^76 - 2*z * q^78 + 10*z * q^79 - q^81 - 6*z^3 * q^82 + 12*z^2 * q^83 + 2 * q^84 + 4 * q^86 + 18 * q^89 - 4*z * q^91 - 6*z * q^92 + 10*z^2 * q^93 + 12*z^2 * q^94 + z * q^96 - 14*z^3 * q^97 - 3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} - 8 q^{13} + 4 q^{16} - 4 q^{18} - 8 q^{21} - 16 q^{38} - 48 q^{47} + 20 q^{50} + 8 q^{52} - 4 q^{64} - 16 q^{67} - 24 q^{69} + 4 q^{72} - 4 q^{81} + 8 q^{84} + 16 q^{86} + 72 q^{89} - 12 q^{98}+O(q^{100})$$ 4 * q - 4 * q^4 - 8 * q^13 + 4 * q^16 - 4 * q^18 - 8 * q^21 - 16 * q^38 - 48 * q^47 + 20 * q^50 + 8 * q^52 - 4 * q^64 - 16 * q^67 - 24 * q^69 + 4 * q^72 - 4 * q^81 + 8 * q^84 + 16 * q^86 + 72 * q^89 - 12 * q^98

## Character values

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

 $$n$$ $$1157$$ $$1159$$ $$\chi(n)$$ $$1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
829.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
1.00000i −0.707107 0.707107i −1.00000 0 0.707107 0.707107i 1.41421 1.41421i 1.00000i 1.00000i 0
829.2 1.00000i 0.707107 + 0.707107i −1.00000 0 −0.707107 + 0.707107i −1.41421 + 1.41421i 1.00000i 1.00000i 0
1483.1 1.00000i −0.707107 + 0.707107i −1.00000 0 0.707107 + 0.707107i 1.41421 + 1.41421i 1.00000i 1.00000i 0
1483.2 1.00000i 0.707107 0.707107i −1.00000 0 −0.707107 0.707107i −1.41421 1.41421i 1.00000i 1.00000i 0
 $$n$$: 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
17.c even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1734.2.f.b 4
17.b even 2 1 inner 1734.2.f.b 4
17.c even 4 2 inner 1734.2.f.b 4
17.d even 8 1 102.2.a.b 1
17.d even 8 1 1734.2.a.b 1
17.d even 8 2 1734.2.b.f 2
51.g odd 8 1 306.2.a.c 1
51.g odd 8 1 5202.2.a.j 1
68.g odd 8 1 816.2.a.d 1
85.k odd 8 1 2550.2.d.g 2
85.m even 8 1 2550.2.a.u 1
85.n odd 8 1 2550.2.d.g 2
119.l odd 8 1 4998.2.a.d 1
136.o even 8 1 3264.2.a.i 1
136.p odd 8 1 3264.2.a.w 1
204.p even 8 1 2448.2.a.i 1
255.y odd 8 1 7650.2.a.j 1
408.bd even 8 1 9792.2.a.ba 1
408.be odd 8 1 9792.2.a.bg 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.b 1 17.d even 8 1
306.2.a.c 1 51.g odd 8 1
816.2.a.d 1 68.g odd 8 1
1734.2.a.b 1 17.d even 8 1
1734.2.b.f 2 17.d even 8 2
1734.2.f.b 4 1.a even 1 1 trivial
1734.2.f.b 4 17.b even 2 1 inner
1734.2.f.b 4 17.c even 4 2 inner
2448.2.a.i 1 204.p even 8 1
2550.2.a.u 1 85.m even 8 1
2550.2.d.g 2 85.k odd 8 1
2550.2.d.g 2 85.n odd 8 1
3264.2.a.i 1 136.o even 8 1
3264.2.a.w 1 136.p odd 8 1
4998.2.a.d 1 119.l odd 8 1
5202.2.a.j 1 51.g odd 8 1
7650.2.a.j 1 255.y odd 8 1
9792.2.a.ba 1 408.bd even 8 1
9792.2.a.bg 1 408.be odd 8 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{7}^{4} + 16$$ T7^4 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 16$$
$11$ $$T^{4}$$
$13$ $$(T + 2)^{4}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 16)^{2}$$
$23$ $$T^{4} + 1296$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 10000$$
$37$ $$T^{4} + 4096$$
$41$ $$T^{4} + 1296$$
$43$ $$(T^{2} + 16)^{2}$$
$47$ $$(T + 12)^{4}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} + 144)^{2}$$
$61$ $$T^{4} + 4096$$
$67$ $$(T + 4)^{4}$$
$71$ $$T^{4} + 1296$$
$73$ $$T^{4} + 16$$
$79$ $$T^{4} + 10000$$
$83$ $$(T^{2} + 144)^{2}$$
$89$ $$(T - 18)^{4}$$
$97$ $$T^{4} + 38416$$