# Properties

 Label 1575.4.a.o Level $1575$ Weight $4$ Character orbit 1575.a Self dual yes Analytic conductor $92.928$ Analytic rank $1$ 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: $$1$$ 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 525) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{2} + ( -3 + 3 \beta ) q^{4} -7 q^{7} + ( -1 + 5 \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{2} + ( -3 + 3 \beta ) q^{4} -7 q^{7} + ( -1 + 5 \beta ) q^{8} + ( 18 - 5 \beta ) q^{11} + ( 11 + 17 \beta ) q^{13} + ( 7 + 7 \beta ) q^{14} + ( 5 - 33 \beta ) q^{16} + ( 53 - 27 \beta ) q^{17} + ( -56 + 56 \beta ) q^{19} + ( 2 - 8 \beta ) q^{22} + ( -115 - 24 \beta ) q^{23} + ( -79 - 45 \beta ) q^{26} + ( 21 - 21 \beta ) q^{28} + ( 73 - 84 \beta ) q^{29} + ( -61 - 13 \beta ) q^{31} + ( 135 + 21 \beta ) q^{32} + ( 55 + \beta ) q^{34} + ( 66 - 19 \beta ) q^{37} + ( -168 - 56 \beta ) q^{38} + ( -95 - 45 \beta ) q^{41} + ( 373 + 58 \beta ) q^{43} + ( -114 + 54 \beta ) q^{44} + ( 211 + 163 \beta ) q^{46} + ( -120 + 88 \beta ) q^{47} + 49 q^{49} + ( 171 + 33 \beta ) q^{52} + ( -119 + 89 \beta ) q^{53} + ( 7 - 35 \beta ) q^{56} + ( 263 + 95 \beta ) q^{58} + ( 325 - 209 \beta ) q^{59} + ( 25 - 273 \beta ) q^{61} + ( 113 + 87 \beta ) q^{62} + ( -259 + 87 \beta ) q^{64} + ( 640 - 123 \beta ) q^{67} + ( -483 + 159 \beta ) q^{68} + ( -192 - 235 \beta ) q^{71} + ( 90 + 88 \beta ) q^{73} + ( 10 - 28 \beta ) q^{74} + ( 840 - 168 \beta ) q^{76} + ( -126 + 35 \beta ) q^{77} + ( -162 - 103 \beta ) q^{79} + ( 275 + 185 \beta ) q^{82} + ( -821 + 431 \beta ) q^{83} + ( -605 - 489 \beta ) q^{86} + ( -118 + 70 \beta ) q^{88} + ( -494 + 522 \beta ) q^{89} + ( -77 - 119 \beta ) q^{91} + ( 57 - 345 \beta ) q^{92} + ( -232 - 56 \beta ) q^{94} + ( -266 + 704 \beta ) q^{97} + ( -49 - 49 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 3 q^{4} - 14 q^{7} + 3 q^{8} + O(q^{10})$$ $$2 q - 3 q^{2} - 3 q^{4} - 14 q^{7} + 3 q^{8} + 31 q^{11} + 39 q^{13} + 21 q^{14} - 23 q^{16} + 79 q^{17} - 56 q^{19} - 4 q^{22} - 254 q^{23} - 203 q^{26} + 21 q^{28} + 62 q^{29} - 135 q^{31} + 291 q^{32} + 111 q^{34} + 113 q^{37} - 392 q^{38} - 235 q^{41} + 804 q^{43} - 174 q^{44} + 585 q^{46} - 152 q^{47} + 98 q^{49} + 375 q^{52} - 149 q^{53} - 21 q^{56} + 621 q^{58} + 441 q^{59} - 223 q^{61} + 313 q^{62} - 431 q^{64} + 1157 q^{67} - 807 q^{68} - 619 q^{71} + 268 q^{73} - 8 q^{74} + 1512 q^{76} - 217 q^{77} - 427 q^{79} + 735 q^{82} - 1211 q^{83} - 1699 q^{86} - 166 q^{88} - 466 q^{89} - 273 q^{91} - 231 q^{92} - 520 q^{94} + 172 q^{97} - 147 q^{98} + O(q^{100})$$

## 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
−3.56155 0 4.68466 0 0 −7.00000 11.8078 0 0
1.2 0.561553 0 −7.68466 0 0 −7.00000 −8.80776 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.o 2
3.b odd 2 1 525.4.a.m yes 2
5.b even 2 1 1575.4.a.x 2
15.d odd 2 1 525.4.a.j 2
15.e even 4 2 525.4.d.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.j 2 15.d odd 2 1
525.4.a.m yes 2 3.b odd 2 1
525.4.d.m 4 15.e even 4 2
1575.4.a.o 2 1.a even 1 1 trivial
1575.4.a.x 2 5.b even 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 3 T_{2} - 2$$ $$T_{11}^{2} - 31 T_{11} + 134$$ $$T_{13}^{2} - 39 T_{13} - 848$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 7 + T )^{2}$$
$11$ $$134 - 31 T + T^{2}$$
$13$ $$-848 - 39 T + T^{2}$$
$17$ $$-1538 - 79 T + T^{2}$$
$19$ $$-12544 + 56 T + T^{2}$$
$23$ $$13681 + 254 T + T^{2}$$
$29$ $$-29027 - 62 T + T^{2}$$
$31$ $$3838 + 135 T + T^{2}$$
$37$ $$1658 - 113 T + T^{2}$$
$41$ $$5200 + 235 T + T^{2}$$
$43$ $$147307 - 804 T + T^{2}$$
$47$ $$-27136 + 152 T + T^{2}$$
$53$ $$-28114 + 149 T + T^{2}$$
$59$ $$-137024 - 441 T + T^{2}$$
$61$ $$-304316 + 223 T + T^{2}$$
$67$ $$270364 - 1157 T + T^{2}$$
$71$ $$-138916 + 619 T + T^{2}$$
$73$ $$-14956 - 268 T + T^{2}$$
$79$ $$494 + 427 T + T^{2}$$
$83$ $$-422854 + 1211 T + T^{2}$$
$89$ $$-1103768 + 466 T + T^{2}$$
$97$ $$-2098972 - 172 T + T^{2}$$