# Properties

 Label 1575.4.a.t Level $1575$ Weight $4$ Character orbit 1575.a Self dual yes Analytic conductor $92.928$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.9280082590$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ Defining polynomial: $$x^{2} - 19$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 11 q^{4} + 7 q^{7} + 3 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + 11 q^{4} + 7 q^{7} + 3 \beta q^{8} + 10 \beta q^{11} -82 q^{13} + 7 \beta q^{14} -31 q^{16} -18 \beta q^{17} -20 q^{19} + 190 q^{22} -30 \beta q^{23} -82 \beta q^{26} + 77 q^{28} -56 \beta q^{29} + 156 q^{31} -55 \beta q^{32} -342 q^{34} -186 q^{37} -20 \beta q^{38} + 38 \beta q^{41} -164 q^{43} + 110 \beta q^{44} -570 q^{46} + 108 \beta q^{47} + 49 q^{49} -902 q^{52} -36 \beta q^{53} + 21 \beta q^{56} -1064 q^{58} -36 \beta q^{59} + 790 q^{61} + 156 \beta q^{62} -797 q^{64} + 44 q^{67} -198 \beta q^{68} -102 \beta q^{71} -126 q^{73} -186 \beta q^{74} -220 q^{76} + 70 \beta q^{77} -712 q^{79} + 722 q^{82} -336 \beta q^{83} -164 \beta q^{86} + 570 q^{88} + 334 \beta q^{89} -574 q^{91} -330 \beta q^{92} + 2052 q^{94} -798 q^{97} + 49 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 22 q^{4} + 14 q^{7} + O(q^{10})$$ $$2 q + 22 q^{4} + 14 q^{7} - 164 q^{13} - 62 q^{16} - 40 q^{19} + 380 q^{22} + 154 q^{28} + 312 q^{31} - 684 q^{34} - 372 q^{37} - 328 q^{43} - 1140 q^{46} + 98 q^{49} - 1804 q^{52} - 2128 q^{58} + 1580 q^{61} - 1594 q^{64} + 88 q^{67} - 252 q^{73} - 440 q^{76} - 1424 q^{79} + 1444 q^{82} + 1140 q^{88} - 1148 q^{91} + 4104 q^{94} - 1596 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.35890 4.35890
−4.35890 0 11.0000 0 0 7.00000 −13.0767 0 0
1.2 4.35890 0 11.0000 0 0 7.00000 13.0767 0 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
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.4.a.t 2
3.b odd 2 1 inner 1575.4.a.t 2
5.b even 2 1 63.4.a.d 2
15.d odd 2 1 63.4.a.d 2
20.d odd 2 1 1008.4.a.be 2
35.c odd 2 1 441.4.a.q 2
35.i odd 6 2 441.4.e.s 4
35.j even 6 2 441.4.e.r 4
60.h even 2 1 1008.4.a.be 2
105.g even 2 1 441.4.a.q 2
105.o odd 6 2 441.4.e.r 4
105.p even 6 2 441.4.e.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d 2 5.b even 2 1
63.4.a.d 2 15.d odd 2 1
441.4.a.q 2 35.c odd 2 1
441.4.a.q 2 105.g even 2 1
441.4.e.r 4 35.j even 6 2
441.4.e.r 4 105.o odd 6 2
441.4.e.s 4 35.i odd 6 2
441.4.e.s 4 105.p even 6 2
1008.4.a.be 2 20.d odd 2 1
1008.4.a.be 2 60.h even 2 1
1575.4.a.t 2 1.a even 1 1 trivial
1575.4.a.t 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{2} - 19$$ $$T_{11}^{2} - 1900$$ $$T_{13} + 82$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-19 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-1900 + T^{2}$$
$13$ $$( 82 + T )^{2}$$
$17$ $$-6156 + T^{2}$$
$19$ $$( 20 + T )^{2}$$
$23$ $$-17100 + T^{2}$$
$29$ $$-59584 + T^{2}$$
$31$ $$( -156 + T )^{2}$$
$37$ $$( 186 + T )^{2}$$
$41$ $$-27436 + T^{2}$$
$43$ $$( 164 + T )^{2}$$
$47$ $$-221616 + T^{2}$$
$53$ $$-24624 + T^{2}$$
$59$ $$-24624 + T^{2}$$
$61$ $$( -790 + T )^{2}$$
$67$ $$( -44 + T )^{2}$$
$71$ $$-197676 + T^{2}$$
$73$ $$( 126 + T )^{2}$$
$79$ $$( 712 + T )^{2}$$
$83$ $$-2145024 + T^{2}$$
$89$ $$-2119564 + T^{2}$$
$97$ $$( 798 + T )^{2}$$