# Properties

 Label 1575.4.a.q 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + 7 q^{7} + ( - 5 \beta - 9) q^{8}+O(q^{10})$$ q + (b - 1) * q^2 + (-2*b + 1) * q^4 + 7 * q^7 + (-5*b - 9) * q^8 $$q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + 7 q^{7} + ( - 5 \beta - 9) q^{8} + (20 \beta + 8) q^{11} + ( - 2 \beta + 38) q^{13} + (7 \beta - 7) q^{14} + (12 \beta - 39) q^{16} + (2 \beta - 62) q^{17} + (16 \beta - 48) q^{19} + ( - 12 \beta + 152) q^{22} + (34 \beta - 8) q^{23} + (40 \beta - 54) q^{26} + ( - 14 \beta + 7) q^{28} + ( - 54 \beta - 94) q^{29} + ( - 18 \beta - 60) q^{31} + ( - 11 \beta + 207) q^{32} + ( - 64 \beta + 78) q^{34} + ( - 66 \beta + 66) q^{37} + ( - 64 \beta + 176) q^{38} + (80 \beta - 50) q^{41} + (62 \beta + 268) q^{43} + (4 \beta - 312) q^{44} + ( - 42 \beta + 280) q^{46} + (42 \beta - 464) q^{47} + 49 q^{49} + ( - 78 \beta + 70) q^{52} + ( - 64 \beta + 442) q^{53} + ( - 35 \beta - 63) q^{56} + ( - 40 \beta - 338) q^{58} + ( - 204 \beta - 52) q^{59} + (42 \beta - 234) q^{61} + ( - 42 \beta - 84) q^{62} + (122 \beta + 17) q^{64} + (38 \beta + 844) q^{67} + (126 \beta - 94) q^{68} + ( - 150 \beta + 68) q^{71} + (322 \beta - 254) q^{73} + (132 \beta - 594) q^{74} + (112 \beta - 304) q^{76} + (140 \beta + 56) q^{77} + (232 \beta - 216) q^{79} + ( - 130 \beta + 690) q^{82} + (84 \beta - 292) q^{83} + (206 \beta + 228) q^{86} + ( - 220 \beta - 872) q^{88} + (112 \beta + 702) q^{89} + ( - 14 \beta + 266) q^{91} + (50 \beta - 552) q^{92} + ( - 506 \beta + 800) q^{94} + (46 \beta + 594) q^{97} + (49 \beta - 49) q^{98}+O(q^{100})$$ q + (b - 1) * q^2 + (-2*b + 1) * q^4 + 7 * q^7 + (-5*b - 9) * q^8 + (20*b + 8) * q^11 + (-2*b + 38) * q^13 + (7*b - 7) * q^14 + (12*b - 39) * q^16 + (2*b - 62) * q^17 + (16*b - 48) * q^19 + (-12*b + 152) * q^22 + (34*b - 8) * q^23 + (40*b - 54) * q^26 + (-14*b + 7) * q^28 + (-54*b - 94) * q^29 + (-18*b - 60) * q^31 + (-11*b + 207) * q^32 + (-64*b + 78) * q^34 + (-66*b + 66) * q^37 + (-64*b + 176) * q^38 + (80*b - 50) * q^41 + (62*b + 268) * q^43 + (4*b - 312) * q^44 + (-42*b + 280) * q^46 + (42*b - 464) * q^47 + 49 * q^49 + (-78*b + 70) * q^52 + (-64*b + 442) * q^53 + (-35*b - 63) * q^56 + (-40*b - 338) * q^58 + (-204*b - 52) * q^59 + (42*b - 234) * q^61 + (-42*b - 84) * q^62 + (122*b + 17) * q^64 + (38*b + 844) * q^67 + (126*b - 94) * q^68 + (-150*b + 68) * q^71 + (322*b - 254) * q^73 + (132*b - 594) * q^74 + (112*b - 304) * q^76 + (140*b + 56) * q^77 + (232*b - 216) * q^79 + (-130*b + 690) * q^82 + (84*b - 292) * q^83 + (206*b + 228) * q^86 + (-220*b - 872) * q^88 + (112*b + 702) * q^89 + (-14*b + 266) * q^91 + (50*b - 552) * q^92 + (-506*b + 800) * q^94 + (46*b + 594) * q^97 + (49*b - 49) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 14 q^{7} - 18 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 14 * q^7 - 18 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 14 q^{7} - 18 q^{8} + 16 q^{11} + 76 q^{13} - 14 q^{14} - 78 q^{16} - 124 q^{17} - 96 q^{19} + 304 q^{22} - 16 q^{23} - 108 q^{26} + 14 q^{28} - 188 q^{29} - 120 q^{31} + 414 q^{32} + 156 q^{34} + 132 q^{37} + 352 q^{38} - 100 q^{41} + 536 q^{43} - 624 q^{44} + 560 q^{46} - 928 q^{47} + 98 q^{49} + 140 q^{52} + 884 q^{53} - 126 q^{56} - 676 q^{58} - 104 q^{59} - 468 q^{61} - 168 q^{62} + 34 q^{64} + 1688 q^{67} - 188 q^{68} + 136 q^{71} - 508 q^{73} - 1188 q^{74} - 608 q^{76} + 112 q^{77} - 432 q^{79} + 1380 q^{82} - 584 q^{83} + 456 q^{86} - 1744 q^{88} + 1404 q^{89} + 532 q^{91} - 1104 q^{92} + 1600 q^{94} + 1188 q^{97} - 98 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 14 * q^7 - 18 * q^8 + 16 * q^11 + 76 * q^13 - 14 * q^14 - 78 * q^16 - 124 * q^17 - 96 * q^19 + 304 * q^22 - 16 * q^23 - 108 * q^26 + 14 * q^28 - 188 * q^29 - 120 * q^31 + 414 * q^32 + 156 * q^34 + 132 * q^37 + 352 * q^38 - 100 * q^41 + 536 * q^43 - 624 * q^44 + 560 * q^46 - 928 * q^47 + 98 * q^49 + 140 * q^52 + 884 * q^53 - 126 * q^56 - 676 * q^58 - 104 * q^59 - 468 * q^61 - 168 * q^62 + 34 * q^64 + 1688 * q^67 - 188 * q^68 + 136 * q^71 - 508 * q^73 - 1188 * q^74 - 608 * q^76 + 112 * q^77 - 432 * q^79 + 1380 * q^82 - 584 * q^83 + 456 * q^86 - 1744 * q^88 + 1404 * q^89 + 532 * q^91 - 1104 * q^92 + 1600 * q^94 + 1188 * q^97 - 98 * q^98

## 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
 −1.41421 1.41421
−3.82843 0 6.65685 0 0 7.00000 5.14214 0 0
1.2 1.82843 0 −4.65685 0 0 7.00000 −23.1421 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.q 2
3.b odd 2 1 525.4.a.l 2
5.b even 2 1 315.4.a.k 2
15.d odd 2 1 105.4.a.e 2
15.e even 4 2 525.4.d.l 4
35.c odd 2 1 2205.4.a.bb 2
60.h even 2 1 1680.4.a.bo 2
105.g even 2 1 735.4.a.o 2

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

 $$T_{2}^{2} + 2T_{2} - 7$$ T2^2 + 2*T2 - 7 $$T_{11}^{2} - 16T_{11} - 3136$$ T11^2 - 16*T11 - 3136 $$T_{13}^{2} - 76T_{13} + 1412$$ T13^2 - 76*T13 + 1412

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 7$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 7)^{2}$$
$11$ $$T^{2} - 16T - 3136$$
$13$ $$T^{2} - 76T + 1412$$
$17$ $$T^{2} + 124T + 3812$$
$19$ $$T^{2} + 96T + 256$$
$23$ $$T^{2} + 16T - 9184$$
$29$ $$T^{2} + 188T - 14492$$
$31$ $$T^{2} + 120T + 1008$$
$37$ $$T^{2} - 132T - 30492$$
$41$ $$T^{2} + 100T - 48700$$
$43$ $$T^{2} - 536T + 41072$$
$47$ $$T^{2} + 928T + 201184$$
$53$ $$T^{2} - 884T + 162596$$
$59$ $$T^{2} + 104T - 330224$$
$61$ $$T^{2} + 468T + 40644$$
$67$ $$T^{2} - 1688 T + 700784$$
$71$ $$T^{2} - 136T - 175376$$
$73$ $$T^{2} + 508T - 764956$$
$79$ $$T^{2} + 432T - 383936$$
$83$ $$T^{2} + 584T + 28816$$
$89$ $$T^{2} - 1404 T + 392452$$
$97$ $$T^{2} - 1188 T + 335908$$