# Properties

 Label 1734.2.a.b Level $1734$ Weight $2$ Character orbit 1734.a Self dual yes Analytic conductor $13.846$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1734,2,Mod(1,1734)]

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, 0]))

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

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

chi := DirichletCharacter("1734.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1734 = 2 \cdot 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1734.a (trivial)

## 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: yes Analytic conductor: $$13.8460597105$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 −1.00000 1.00000 0 1.00000 −2.00000 −1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$17$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.b 1 17.b even 2 1
306.2.a.c 1 51.c odd 2 1
816.2.a.d 1 68.d odd 2 1
1734.2.a.b 1 1.a even 1 1 trivial
1734.2.b.f 2 17.c even 4 2
1734.2.f.b 4 17.d even 8 4
2448.2.a.i 1 204.h even 2 1
2550.2.a.u 1 85.c even 2 1
2550.2.d.g 2 85.g odd 4 2
3264.2.a.i 1 136.h even 2 1
3264.2.a.w 1 136.e odd 2 1
4998.2.a.d 1 119.d odd 2 1
5202.2.a.j 1 3.b odd 2 1
7650.2.a.j 1 255.h odd 2 1
9792.2.a.ba 1 408.h even 2 1
9792.2.a.bg 1 408.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1734))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T$$
$19$ $$T + 4$$
$23$ $$T - 6$$
$29$ $$T$$
$31$ $$T - 10$$
$37$ $$T + 8$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T - 12$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T + 8$$
$67$ $$T + 4$$
$71$ $$T + 6$$
$73$ $$T + 2$$
$79$ $$T - 10$$
$83$ $$T - 12$$
$89$ $$T + 18$$
$97$ $$T + 14$$
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