# Properties

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

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$13.8460597105$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ 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

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.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{2} + \zeta_{8} q^{3} - q^{4} + 2 \zeta_{8} q^{5} -\zeta_{8}^{3} q^{6} + \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{2} + \zeta_{8} q^{3} - q^{4} + 2 \zeta_{8} q^{5} -\zeta_{8}^{3} q^{6} + \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} -2 \zeta_{8}^{3} q^{10} + 4 \zeta_{8}^{3} q^{11} -\zeta_{8} q^{12} + 2 q^{13} + 2 \zeta_{8}^{2} q^{15} + q^{16} + q^{18} -4 \zeta_{8}^{2} q^{19} -2 \zeta_{8} q^{20} + 4 \zeta_{8} q^{22} + \zeta_{8}^{3} q^{24} -\zeta_{8}^{2} q^{25} -2 \zeta_{8}^{2} q^{26} + \zeta_{8}^{3} q^{27} + 10 \zeta_{8} q^{29} + 2 q^{30} + 8 \zeta_{8} q^{31} -\zeta_{8}^{2} q^{32} -4 q^{33} -\zeta_{8}^{2} q^{36} -2 \zeta_{8} q^{37} -4 q^{38} + 2 \zeta_{8} q^{39} + 2 \zeta_{8}^{3} q^{40} + 10 \zeta_{8}^{3} q^{41} + 12 \zeta_{8}^{2} q^{43} -4 \zeta_{8}^{3} q^{44} + 2 \zeta_{8}^{3} q^{45} + \zeta_{8} q^{48} + 7 \zeta_{8}^{2} q^{49} - q^{50} -2 q^{52} -6 \zeta_{8}^{2} q^{53} + \zeta_{8} q^{54} -8 q^{55} -4 \zeta_{8}^{3} q^{57} -10 \zeta_{8}^{3} q^{58} + 12 \zeta_{8}^{2} q^{59} -2 \zeta_{8}^{2} q^{60} -10 \zeta_{8}^{3} q^{61} -8 \zeta_{8}^{3} q^{62} - q^{64} + 4 \zeta_{8} q^{65} + 4 \zeta_{8}^{2} q^{66} -12 q^{67} - q^{72} -10 \zeta_{8} q^{73} + 2 \zeta_{8}^{3} q^{74} -\zeta_{8}^{3} q^{75} + 4 \zeta_{8}^{2} q^{76} -2 \zeta_{8}^{3} q^{78} + 8 \zeta_{8}^{3} q^{79} + 2 \zeta_{8} q^{80} - q^{81} + 10 \zeta_{8} q^{82} -4 \zeta_{8}^{2} q^{83} + 12 q^{86} + 10 \zeta_{8}^{2} q^{87} -4 \zeta_{8} q^{88} + 6 q^{89} + 2 \zeta_{8} q^{90} + 8 \zeta_{8}^{2} q^{93} -8 \zeta_{8}^{3} q^{95} -\zeta_{8}^{3} q^{96} + 14 \zeta_{8} q^{97} + 7 q^{98} -4 \zeta_{8} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} + 8q^{13} + 4q^{16} + 4q^{18} + 8q^{30} - 16q^{33} - 16q^{38} - 4q^{50} - 8q^{52} - 32q^{55} - 4q^{64} - 48q^{67} - 4q^{72} - 4q^{81} + 48q^{86} + 24q^{89} + 28q^{98} + O(q^{100})$$

## Character values

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

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

## 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.

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 −1.41421 1.41421i −0.707107 + 0.707107i 0 1.00000i 1.00000i −1.41421 + 1.41421i
829.2 1.00000i 0.707107 + 0.707107i −1.00000 1.41421 + 1.41421i 0.707107 0.707107i 0 1.00000i 1.00000i 1.41421 1.41421i
1483.1 1.00000i −0.707107 + 0.707107i −1.00000 −1.41421 + 1.41421i −0.707107 0.707107i 0 1.00000i 1.00000i −1.41421 1.41421i
1483.2 1.00000i 0.707107 0.707107i −1.00000 1.41421 1.41421i 0.707107 + 0.707107i 0 1.00000i 1.00000i 1.41421 + 1.41421i
 $$n$$: 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.e 4
17.b even 2 1 inner 1734.2.f.e 4
17.c even 4 2 inner 1734.2.f.e 4
17.d even 8 1 102.2.a.c 1
17.d even 8 1 1734.2.a.j 1
17.d even 8 2 1734.2.b.b 2
51.g odd 8 1 306.2.a.b 1
51.g odd 8 1 5202.2.a.c 1
68.g odd 8 1 816.2.a.b 1
85.k odd 8 1 2550.2.d.m 2
85.m even 8 1 2550.2.a.c 1
85.n odd 8 1 2550.2.d.m 2
119.l odd 8 1 4998.2.a.be 1
136.o even 8 1 3264.2.a.m 1
136.p odd 8 1 3264.2.a.bc 1
204.p even 8 1 2448.2.a.p 1
255.y odd 8 1 7650.2.a.ca 1
408.bd even 8 1 9792.2.a.l 1
408.be odd 8 1 9792.2.a.k 1

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$1 + T^{4}$$
$5$ $$16 + T^{4}$$
$7$ $$T^{4}$$
$11$ $$256 + T^{4}$$
$13$ $$( -2 + T )^{4}$$
$17$ $$T^{4}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$10000 + T^{4}$$
$31$ $$4096 + T^{4}$$
$37$ $$16 + T^{4}$$
$41$ $$10000 + T^{4}$$
$43$ $$( 144 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 144 + T^{2} )^{2}$$
$61$ $$10000 + T^{4}$$
$67$ $$( 12 + T )^{4}$$
$71$ $$T^{4}$$
$73$ $$10000 + T^{4}$$
$79$ $$4096 + T^{4}$$
$83$ $$( 16 + T^{2} )^{2}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$38416 + T^{4}$$