# Properties

 Label 1575.4.a.m Level $1575$ Weight $4$ Character orbit 1575.a Self dual yes Analytic conductor $92.928$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 - \beta ) q^{2} + ( 5 + 7 \beta ) q^{4} + 7 q^{7} + ( -19 - 25 \beta ) q^{8} +O(q^{10})$$ $$q + ( -3 - \beta ) q^{2} + ( 5 + 7 \beta ) q^{4} + 7 q^{7} + ( -19 - 25 \beta ) q^{8} + ( 8 + 10 \beta ) q^{11} + ( -18 + 22 \beta ) q^{13} + ( -21 - 7 \beta ) q^{14} + ( 117 + 63 \beta ) q^{16} + ( -6 + 28 \beta ) q^{17} + ( 100 - 26 \beta ) q^{19} + ( -64 - 48 \beta ) q^{22} + ( 64 + 56 \beta ) q^{23} + ( -34 - 70 \beta ) q^{26} + ( 35 + 49 \beta ) q^{28} + ( -26 + 84 \beta ) q^{29} + ( 156 + 18 \beta ) q^{31} + ( -451 - 169 \beta ) q^{32} + ( -94 - 106 \beta ) q^{34} + ( 78 - 24 \beta ) q^{37} + ( -196 + 4 \beta ) q^{38} + ( -30 - 140 \beta ) q^{41} + ( -216 + 68 \beta ) q^{43} + ( 320 + 176 \beta ) q^{44} + ( -416 - 288 \beta ) q^{46} + ( 92 + 108 \beta ) q^{47} + 49 q^{49} + ( 526 + 138 \beta ) q^{52} + ( -90 + 214 \beta ) q^{53} + ( -133 - 175 \beta ) q^{56} + ( -258 - 310 \beta ) q^{58} + ( -164 - 36 \beta ) q^{59} + ( 522 - 252 \beta ) q^{61} + ( -540 - 228 \beta ) q^{62} + ( 1093 + 623 \beta ) q^{64} + ( 408 - 28 \beta ) q^{67} + ( 754 + 294 \beta ) q^{68} + ( -392 + 330 \beta ) q^{71} + ( -478 + 178 \beta ) q^{73} + ( -138 + 18 \beta ) q^{74} + ( -228 + 388 \beta ) q^{76} + ( 56 + 70 \beta ) q^{77} + ( 160 + 88 \beta ) q^{79} + ( 650 + 590 \beta ) q^{82} + ( 700 - 264 \beta ) q^{83} + ( 376 - 56 \beta ) q^{86} + ( -1152 - 640 \beta ) q^{88} + ( -382 + 728 \beta ) q^{89} + ( -126 + 154 \beta ) q^{91} + ( 1888 + 1120 \beta ) q^{92} + ( -708 - 524 \beta ) q^{94} + ( 322 - 146 \beta ) q^{97} + ( -147 - 49 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + O(q^{10})$$ $$2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + 26 q^{11} - 14 q^{13} - 49 q^{14} + 297 q^{16} + 16 q^{17} + 174 q^{19} - 176 q^{22} + 184 q^{23} - 138 q^{26} + 119 q^{28} + 32 q^{29} + 330 q^{31} - 1071 q^{32} - 294 q^{34} + 132 q^{37} - 388 q^{38} - 200 q^{41} - 364 q^{43} + 816 q^{44} - 1120 q^{46} + 292 q^{47} + 98 q^{49} + 1190 q^{52} + 34 q^{53} - 441 q^{56} - 826 q^{58} - 364 q^{59} + 792 q^{61} - 1308 q^{62} + 2809 q^{64} + 788 q^{67} + 1802 q^{68} - 454 q^{71} - 778 q^{73} - 258 q^{74} - 68 q^{76} + 182 q^{77} + 408 q^{79} + 1890 q^{82} + 1136 q^{83} + 696 q^{86} - 2944 q^{88} - 36 q^{89} - 98 q^{91} + 4896 q^{92} - 1940 q^{94} + 498 q^{97} - 343 q^{98} + O(q^{100})$$

## 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
 2.56155 −1.56155
−5.56155 0 22.9309 0 0 7.00000 −83.0388 0 0
1.2 −1.43845 0 −5.93087 0 0 7.00000 20.0388 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

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 1575.4.a.m 2
3.b odd 2 1 525.4.a.p 2
5.b even 2 1 315.4.a.m 2
15.d odd 2 1 105.4.a.c 2
15.e even 4 2 525.4.d.i 4
35.c odd 2 1 2205.4.a.bh 2
60.h even 2 1 1680.4.a.bk 2
105.g even 2 1 735.4.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c 2 15.d odd 2 1
315.4.a.m 2 5.b even 2 1
525.4.a.p 2 3.b odd 2 1
525.4.d.i 4 15.e even 4 2
735.4.a.k 2 105.g even 2 1
1575.4.a.m 2 1.a even 1 1 trivial
1680.4.a.bk 2 60.h even 2 1
2205.4.a.bh 2 35.c odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 7 T_{2} + 8$$ $$T_{11}^{2} - 26 T_{11} - 256$$ $$T_{13}^{2} + 14 T_{13} - 2008$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$8 + 7 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-256 - 26 T + T^{2}$$
$13$ $$-2008 + 14 T + T^{2}$$
$17$ $$-3268 - 16 T + T^{2}$$
$19$ $$4696 - 174 T + T^{2}$$
$23$ $$-4864 - 184 T + T^{2}$$
$29$ $$-29732 - 32 T + T^{2}$$
$31$ $$25848 - 330 T + T^{2}$$
$37$ $$1908 - 132 T + T^{2}$$
$41$ $$-73300 + 200 T + T^{2}$$
$43$ $$13472 + 364 T + T^{2}$$
$47$ $$-28256 - 292 T + T^{2}$$
$53$ $$-194344 - 34 T + T^{2}$$
$59$ $$27616 + 364 T + T^{2}$$
$61$ $$-113076 - 792 T + T^{2}$$
$67$ $$151904 - 788 T + T^{2}$$
$71$ $$-411296 + 454 T + T^{2}$$
$73$ $$16664 + 778 T + T^{2}$$
$79$ $$8704 - 408 T + T^{2}$$
$83$ $$26416 - 1136 T + T^{2}$$
$89$ $$-2252108 + 36 T + T^{2}$$
$97$ $$-28592 - 498 T + T^{2}$$