# Properties

 Label 1575.4.a.n 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 - \beta ) q^{2} + ( 1 + 4 \beta ) q^{4} + 7 q^{7} + ( -6 - \beta ) q^{8} +O(q^{10})$$ $$q + ( -2 - \beta ) q^{2} + ( 1 + 4 \beta ) q^{4} + 7 q^{7} + ( -6 - \beta ) q^{8} + ( 46 - 2 \beta ) q^{11} + ( -4 - 38 \beta ) q^{13} + ( -14 - 7 \beta ) q^{14} + ( 9 - 24 \beta ) q^{16} + ( -22 - 44 \beta ) q^{17} + ( -54 - 26 \beta ) q^{19} + ( -82 - 42 \beta ) q^{22} + ( -160 + 20 \beta ) q^{23} + ( 198 + 80 \beta ) q^{26} + ( 7 + 28 \beta ) q^{28} + ( 118 + 12 \beta ) q^{29} + ( -30 - 102 \beta ) q^{31} + ( 150 + 47 \beta ) q^{32} + ( 264 + 110 \beta ) q^{34} + ( -102 + 24 \beta ) q^{37} + ( 238 + 106 \beta ) q^{38} + ( -22 - 80 \beta ) q^{41} + ( -68 + 128 \beta ) q^{43} + ( 6 + 182 \beta ) q^{44} + ( 220 + 120 \beta ) q^{46} + ( 200 + 168 \beta ) q^{47} + 49 q^{49} + ( -764 - 54 \beta ) q^{52} + ( 8 - 86 \beta ) q^{53} + ( -42 - 7 \beta ) q^{56} + ( -296 - 142 \beta ) q^{58} + ( 232 - 36 \beta ) q^{59} + ( -342 - 84 \beta ) q^{61} + ( 570 + 234 \beta ) q^{62} + ( -607 - 52 \beta ) q^{64} + ( -368 + 164 \beta ) q^{67} + ( -902 - 132 \beta ) q^{68} + ( 370 - 138 \beta ) q^{71} + ( -212 - 122 \beta ) q^{73} + ( 84 + 54 \beta ) q^{74} + ( -574 - 242 \beta ) q^{76} + ( 322 - 14 \beta ) q^{77} + ( -204 + 484 \beta ) q^{79} + ( 444 + 182 \beta ) q^{82} + ( 304 + 84 \beta ) q^{83} + ( -504 - 188 \beta ) q^{86} + ( -266 - 34 \beta ) q^{88} + ( 666 - 112 \beta ) q^{89} + ( -28 - 266 \beta ) q^{91} + ( 240 - 620 \beta ) q^{92} + ( -1240 - 536 \beta ) q^{94} + ( 1224 - 86 \beta ) q^{97} + ( -98 - 49 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 2 q^{4} + 14 q^{7} - 12 q^{8} + O(q^{10})$$ $$2 q - 4 q^{2} + 2 q^{4} + 14 q^{7} - 12 q^{8} + 92 q^{11} - 8 q^{13} - 28 q^{14} + 18 q^{16} - 44 q^{17} - 108 q^{19} - 164 q^{22} - 320 q^{23} + 396 q^{26} + 14 q^{28} + 236 q^{29} - 60 q^{31} + 300 q^{32} + 528 q^{34} - 204 q^{37} + 476 q^{38} - 44 q^{41} - 136 q^{43} + 12 q^{44} + 440 q^{46} + 400 q^{47} + 98 q^{49} - 1528 q^{52} + 16 q^{53} - 84 q^{56} - 592 q^{58} + 464 q^{59} - 684 q^{61} + 1140 q^{62} - 1214 q^{64} - 736 q^{67} - 1804 q^{68} + 740 q^{71} - 424 q^{73} + 168 q^{74} - 1148 q^{76} + 644 q^{77} - 408 q^{79} + 888 q^{82} + 608 q^{83} - 1008 q^{86} - 532 q^{88} + 1332 q^{89} - 56 q^{91} + 480 q^{92} - 2480 q^{94} + 2448 q^{97} - 196 q^{98} + O(q^{100})$$

## 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
 1.61803 −0.618034
−4.23607 0 9.94427 0 0 7.00000 −8.23607 0 0
1.2 0.236068 0 −7.94427 0 0 7.00000 −3.76393 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.n 2
3.b odd 2 1 525.4.a.o 2
5.b even 2 1 315.4.a.l 2
15.d odd 2 1 105.4.a.d 2
15.e even 4 2 525.4.d.k 4
35.c odd 2 1 2205.4.a.be 2
60.h even 2 1 1680.4.a.bd 2
105.g even 2 1 735.4.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 15.d odd 2 1
315.4.a.l 2 5.b even 2 1
525.4.a.o 2 3.b odd 2 1
525.4.d.k 4 15.e even 4 2
735.4.a.m 2 105.g even 2 1
1575.4.a.n 2 1.a even 1 1 trivial
1680.4.a.bd 2 60.h even 2 1
2205.4.a.be 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{2} + 4 T_{2} - 1$$ $$T_{11}^{2} - 92 T_{11} + 2096$$ $$T_{13}^{2} + 8 T_{13} - 7204$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 4 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$2096 - 92 T + T^{2}$$
$13$ $$-7204 + 8 T + T^{2}$$
$17$ $$-9196 + 44 T + T^{2}$$
$19$ $$-464 + 108 T + T^{2}$$
$23$ $$23600 + 320 T + T^{2}$$
$29$ $$13204 - 236 T + T^{2}$$
$31$ $$-51120 + 60 T + T^{2}$$
$37$ $$7524 + 204 T + T^{2}$$
$41$ $$-31516 + 44 T + T^{2}$$
$43$ $$-77296 + 136 T + T^{2}$$
$47$ $$-101120 - 400 T + T^{2}$$
$53$ $$-36916 - 16 T + T^{2}$$
$59$ $$47344 - 464 T + T^{2}$$
$61$ $$81684 + 684 T + T^{2}$$
$67$ $$944 + 736 T + T^{2}$$
$71$ $$41680 - 740 T + T^{2}$$
$73$ $$-29476 + 424 T + T^{2}$$
$79$ $$-1129664 + 408 T + T^{2}$$
$83$ $$57136 - 608 T + T^{2}$$
$89$ $$380836 - 1332 T + T^{2}$$
$97$ $$1461196 - 2448 T + T^{2}$$