Properties

 Label 1575.4.a.s 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: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 175) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( 2 + \beta ) q^{4} + 7 q^{7} + ( -10 + 5 \beta ) q^{8} +O(q^{10})$$ $$q -\beta q^{2} + ( 2 + \beta ) q^{4} + 7 q^{7} + ( -10 + 5 \beta ) q^{8} + ( 23 + 10 \beta ) q^{11} + ( 30 - 8 \beta ) q^{13} -7 \beta q^{14} + ( -66 - 3 \beta ) q^{16} + ( 54 - 5 \beta ) q^{17} + ( -32 + 7 \beta ) q^{19} + ( -100 - 33 \beta ) q^{22} + ( 13 + 5 \beta ) q^{23} + ( 80 - 22 \beta ) q^{26} + ( 14 + 7 \beta ) q^{28} + ( 193 + 27 \beta ) q^{29} + ( -114 + 66 \beta ) q^{31} + ( 110 + 29 \beta ) q^{32} + ( 50 - 49 \beta ) q^{34} + ( 9 + 57 \beta ) q^{37} + ( -70 + 25 \beta ) q^{38} + ( 222 + 61 \beta ) q^{41} + ( -75 + 77 \beta ) q^{43} + ( 146 + 53 \beta ) q^{44} + ( -50 - 18 \beta ) q^{46} + ( -58 - 108 \beta ) q^{47} + 49 q^{49} + ( -20 + 6 \beta ) q^{52} + ( 174 - 86 \beta ) q^{53} + ( -70 + 35 \beta ) q^{56} + ( -270 - 220 \beta ) q^{58} + ( 10 - 210 \beta ) q^{59} + ( 468 + 54 \beta ) q^{61} + ( -660 + 48 \beta ) q^{62} + ( 238 - 115 \beta ) q^{64} + ( 537 - 166 \beta ) q^{67} + ( 58 + 39 \beta ) q^{68} + ( -215 + 303 \beta ) q^{71} + ( -34 - 269 \beta ) q^{73} + ( -570 - 66 \beta ) q^{74} + ( 6 - 11 \beta ) q^{76} + ( 161 + 70 \beta ) q^{77} + ( -539 - 41 \beta ) q^{79} + ( -610 - 283 \beta ) q^{82} + ( 724 + 69 \beta ) q^{83} + ( -770 - 2 \beta ) q^{86} + ( 270 + 65 \beta ) q^{88} + ( 848 + 17 \beta ) q^{89} + ( 210 - 56 \beta ) q^{91} + ( 76 + 28 \beta ) q^{92} + ( 1080 + 166 \beta ) q^{94} + ( 1018 - 272 \beta ) q^{97} -49 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 5 q^{4} + 14 q^{7} - 15 q^{8} + O(q^{10})$$ $$2 q - q^{2} + 5 q^{4} + 14 q^{7} - 15 q^{8} + 56 q^{11} + 52 q^{13} - 7 q^{14} - 135 q^{16} + 103 q^{17} - 57 q^{19} - 233 q^{22} + 31 q^{23} + 138 q^{26} + 35 q^{28} + 413 q^{29} - 162 q^{31} + 249 q^{32} + 51 q^{34} + 75 q^{37} - 115 q^{38} + 505 q^{41} - 73 q^{43} + 345 q^{44} - 118 q^{46} - 224 q^{47} + 98 q^{49} - 34 q^{52} + 262 q^{53} - 105 q^{56} - 760 q^{58} - 190 q^{59} + 990 q^{61} - 1272 q^{62} + 361 q^{64} + 908 q^{67} + 155 q^{68} - 127 q^{71} - 337 q^{73} - 1206 q^{74} + q^{76} + 392 q^{77} - 1119 q^{79} - 1503 q^{82} + 1517 q^{83} - 1542 q^{86} + 605 q^{88} + 1713 q^{89} + 364 q^{91} + 180 q^{92} + 2326 q^{94} + 1764 q^{97} - 49 q^{98} + O(q^{100})$$

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
 3.70156 −2.70156
−3.70156 0 5.70156 0 0 7.00000 8.50781 0 0
1.2 2.70156 0 −0.701562 0 0 7.00000 −23.5078 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.s 2
3.b odd 2 1 175.4.a.e yes 2
5.b even 2 1 1575.4.a.v 2
15.d odd 2 1 175.4.a.d 2
15.e even 4 2 175.4.b.d 4
21.c even 2 1 1225.4.a.t 2
105.g even 2 1 1225.4.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.a.d 2 15.d odd 2 1
175.4.a.e yes 2 3.b odd 2 1
175.4.b.d 4 15.e even 4 2
1225.4.a.r 2 105.g even 2 1
1225.4.a.t 2 21.c even 2 1
1575.4.a.s 2 1.a even 1 1 trivial
1575.4.a.v 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{2} + T_{2} - 10$$ $$T_{11}^{2} - 56 T_{11} - 241$$ $$T_{13}^{2} - 52 T_{13} + 20$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-10 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-241 - 56 T + T^{2}$$
$13$ $$20 - 52 T + T^{2}$$
$17$ $$2396 - 103 T + T^{2}$$
$19$ $$310 + 57 T + T^{2}$$
$23$ $$-16 - 31 T + T^{2}$$
$29$ $$35170 - 413 T + T^{2}$$
$31$ $$-38088 + 162 T + T^{2}$$
$37$ $$-31896 - 75 T + T^{2}$$
$41$ $$25616 - 505 T + T^{2}$$
$43$ $$-59440 + 73 T + T^{2}$$
$47$ $$-107012 + 224 T + T^{2}$$
$53$ $$-58648 - 262 T + T^{2}$$
$59$ $$-443000 + 190 T + T^{2}$$
$61$ $$215136 - 990 T + T^{2}$$
$67$ $$-76333 - 908 T + T^{2}$$
$71$ $$-937010 + 127 T + T^{2}$$
$73$ $$-713308 + 337 T + T^{2}$$
$79$ $$295810 + 1119 T + T^{2}$$
$83$ $$526522 - 1517 T + T^{2}$$
$89$ $$730630 - 1713 T + T^{2}$$
$97$ $$19588 - 1764 T + T^{2}$$