# Properties

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

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$13.8460597105$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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

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

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

## 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.

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 $$S_{2}^{\mathrm{new}}(1734, [\chi])$$:

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

## Hecke characteristic polynomials

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