# Properties

 Label 1725.2.b.m Level $1725$ Weight $2$ Character orbit 1725.b Analytic conductor $13.774$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1725 = 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1725.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.7741943487$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 345) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} -4 q^{4} + \beta_{3} q^{6} -\beta_{1} q^{7} -2 \beta_{2} q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} -4 q^{4} + \beta_{3} q^{6} -\beta_{1} q^{7} -2 \beta_{2} q^{8} - q^{9} -\beta_{3} q^{11} + 4 \beta_{1} q^{12} + ( -2 \beta_{1} + \beta_{2} ) q^{13} + \beta_{3} q^{14} + 4 q^{16} + ( -3 \beta_{1} + \beta_{2} ) q^{17} -\beta_{2} q^{18} + ( -2 + \beta_{3} ) q^{19} - q^{21} -6 \beta_{1} q^{22} + \beta_{1} q^{23} -2 \beta_{3} q^{24} + ( -6 + 2 \beta_{3} ) q^{26} + \beta_{1} q^{27} + 4 \beta_{1} q^{28} + ( -3 + 3 \beta_{3} ) q^{29} + ( 5 + 2 \beta_{3} ) q^{31} + \beta_{2} q^{33} + ( -6 + 3 \beta_{3} ) q^{34} + 4 q^{36} + ( -\beta_{1} - 2 \beta_{2} ) q^{37} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{38} + ( -2 + \beta_{3} ) q^{39} + ( 3 - \beta_{3} ) q^{41} -\beta_{2} q^{42} -2 \beta_{1} q^{43} + 4 \beta_{3} q^{44} -\beta_{3} q^{46} + ( 6 \beta_{1} - \beta_{2} ) q^{47} -4 \beta_{1} q^{48} + 6 q^{49} + ( -3 + \beta_{3} ) q^{51} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{52} + ( -3 \beta_{1} - \beta_{2} ) q^{53} -\beta_{3} q^{54} -2 \beta_{3} q^{56} + ( 2 \beta_{1} - \beta_{2} ) q^{57} + ( 18 \beta_{1} - 3 \beta_{2} ) q^{58} + ( -3 + 3 \beta_{3} ) q^{59} + ( 8 + 3 \beta_{3} ) q^{61} + ( 12 \beta_{1} + 5 \beta_{2} ) q^{62} + \beta_{1} q^{63} + 8 q^{64} -6 q^{66} -7 \beta_{1} q^{67} + ( 12 \beta_{1} - 4 \beta_{2} ) q^{68} + q^{69} + ( 3 + 3 \beta_{3} ) q^{71} + 2 \beta_{2} q^{72} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{73} + ( 12 + \beta_{3} ) q^{74} + ( 8 - 4 \beta_{3} ) q^{76} + \beta_{2} q^{77} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{78} + 4 q^{79} + q^{81} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{82} + ( -3 \beta_{1} + 5 \beta_{2} ) q^{83} + 4 q^{84} + 2 \beta_{3} q^{86} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{87} + 12 \beta_{1} q^{88} + ( 12 - 2 \beta_{3} ) q^{89} + ( -2 + \beta_{3} ) q^{91} -4 \beta_{1} q^{92} + ( -5 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 6 - 6 \beta_{3} ) q^{94} + ( 8 \beta_{1} + 2 \beta_{2} ) q^{97} + 6 \beta_{2} q^{98} + \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{4} - 4q^{9} + O(q^{10})$$ $$4q - 16q^{4} - 4q^{9} + 16q^{16} - 8q^{19} - 4q^{21} - 24q^{26} - 12q^{29} + 20q^{31} - 24q^{34} + 16q^{36} - 8q^{39} + 12q^{41} + 24q^{49} - 12q^{51} - 12q^{59} + 32q^{61} + 32q^{64} - 24q^{66} + 4q^{69} + 12q^{71} + 48q^{74} + 32q^{76} + 16q^{79} + 4q^{81} + 16q^{84} + 48q^{89} - 8q^{91} + 24q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 3 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 3 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 3 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1174.1
 −1.22474 − 1.22474i 1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i
2.44949i 1.00000i −4.00000 0 −2.44949 1.00000i 4.89898i −1.00000 0
1174.2 2.44949i 1.00000i −4.00000 0 2.44949 1.00000i 4.89898i −1.00000 0
1174.3 2.44949i 1.00000i −4.00000 0 2.44949 1.00000i 4.89898i −1.00000 0
1174.4 2.44949i 1.00000i −4.00000 0 −2.44949 1.00000i 4.89898i −1.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1725.2.b.m 4
5.b even 2 1 inner 1725.2.b.m 4
5.c odd 4 1 345.2.a.i 2
5.c odd 4 1 1725.2.a.y 2
15.e even 4 1 1035.2.a.k 2
15.e even 4 1 5175.2.a.bl 2
20.e even 4 1 5520.2.a.bi 2
115.e even 4 1 7935.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.i 2 5.c odd 4 1
1035.2.a.k 2 15.e even 4 1
1725.2.a.y 2 5.c odd 4 1
1725.2.b.m 4 1.a even 1 1 trivial
1725.2.b.m 4 5.b even 2 1 inner
5175.2.a.bl 2 15.e even 4 1
5520.2.a.bi 2 20.e even 4 1
7935.2.a.t 2 115.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1725, [\chi])$$:

 $$T_{2}^{2} + 6$$ $$T_{7}^{2} + 1$$ $$T_{11}^{2} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 6 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -6 + T^{2} )^{2}$$
$13$ $$4 + 20 T^{2} + T^{4}$$
$17$ $$9 + 30 T^{2} + T^{4}$$
$19$ $$( -2 + 4 T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( -45 + 6 T + T^{2} )^{2}$$
$31$ $$( 1 - 10 T + T^{2} )^{2}$$
$37$ $$529 + 50 T^{2} + T^{4}$$
$41$ $$( 3 - 6 T + T^{2} )^{2}$$
$43$ $$( 4 + T^{2} )^{2}$$
$47$ $$900 + 84 T^{2} + T^{4}$$
$53$ $$9 + 30 T^{2} + T^{4}$$
$59$ $$( -45 + 6 T + T^{2} )^{2}$$
$61$ $$( 10 - 16 T + T^{2} )^{2}$$
$67$ $$( 49 + T^{2} )^{2}$$
$71$ $$( -45 - 6 T + T^{2} )^{2}$$
$73$ $$2500 + 116 T^{2} + T^{4}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$19881 + 318 T^{2} + T^{4}$$
$89$ $$( 120 - 24 T + T^{2} )^{2}$$
$97$ $$1600 + 176 T^{2} + T^{4}$$