# Properties

 Label 1575.4.a.y 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$92.9280082590$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (3 \beta + 3) q^{4} - 7 q^{7} + (\beta + 25) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (3*b + 3) * q^4 - 7 * q^7 + (b + 25) * q^8 $$q + (\beta + 1) q^{2} + (3 \beta + 3) q^{4} - 7 q^{7} + (\beta + 25) q^{8} + (2 \beta - 32) q^{11} + (10 \beta - 2) q^{13} + ( - 7 \beta - 7) q^{14} + (3 \beta + 11) q^{16} + ( - 12 \beta + 26) q^{17} + ( - 2 \beta - 60) q^{19} + ( - 28 \beta - 12) q^{22} + ( - 48 \beta + 32) q^{23} + (18 \beta + 98) q^{26} + ( - 21 \beta - 21) q^{28} + ( - 12 \beta - 170) q^{29} + ( - 38 \beta + 52) q^{31} + (9 \beta - 159) q^{32} + (2 \beta - 94) q^{34} + ( - 80 \beta + 134) q^{37} + ( - 64 \beta - 80) q^{38} + (108 \beta - 62) q^{41} + ( - 100 \beta + 248) q^{43} + ( - 84 \beta - 36) q^{44} + ( - 64 \beta - 448) q^{46} + (140 \beta - 164) q^{47} + 49 q^{49} + (54 \beta + 294) q^{52} + (58 \beta + 462) q^{53} + ( - 7 \beta - 175) q^{56} + ( - 194 \beta - 290) q^{58} + ( - 76 \beta - 220) q^{59} + ( - 84 \beta - 398) q^{61} + ( - 24 \beta - 328) q^{62} + ( - 165 \beta - 157) q^{64} + (228 \beta + 64) q^{67} + (6 \beta - 282) q^{68} + ( - 86 \beta - 112) q^{71} + (38 \beta - 182) q^{73} + ( - 26 \beta - 666) q^{74} + ( - 192 \beta - 240) q^{76} + ( - 14 \beta + 224) q^{77} + ( - 8 \beta + 920) q^{79} + (154 \beta + 1018) q^{82} + ( - 224 \beta - 228) q^{83} + (48 \beta - 752) q^{86} + (20 \beta - 780) q^{88} + ( - 336 \beta - 230) q^{89} + ( - 70 \beta + 14) q^{91} + ( - 192 \beta - 1344) q^{92} + (116 \beta + 1236) q^{94} + ( - 278 \beta + 474) q^{97} + (49 \beta + 49) q^{98}+O(q^{100})$$ q + (b + 1) * q^2 + (3*b + 3) * q^4 - 7 * q^7 + (b + 25) * q^8 + (2*b - 32) * q^11 + (10*b - 2) * q^13 + (-7*b - 7) * q^14 + (3*b + 11) * q^16 + (-12*b + 26) * q^17 + (-2*b - 60) * q^19 + (-28*b - 12) * q^22 + (-48*b + 32) * q^23 + (18*b + 98) * q^26 + (-21*b - 21) * q^28 + (-12*b - 170) * q^29 + (-38*b + 52) * q^31 + (9*b - 159) * q^32 + (2*b - 94) * q^34 + (-80*b + 134) * q^37 + (-64*b - 80) * q^38 + (108*b - 62) * q^41 + (-100*b + 248) * q^43 + (-84*b - 36) * q^44 + (-64*b - 448) * q^46 + (140*b - 164) * q^47 + 49 * q^49 + (54*b + 294) * q^52 + (58*b + 462) * q^53 + (-7*b - 175) * q^56 + (-194*b - 290) * q^58 + (-76*b - 220) * q^59 + (-84*b - 398) * q^61 + (-24*b - 328) * q^62 + (-165*b - 157) * q^64 + (228*b + 64) * q^67 + (6*b - 282) * q^68 + (-86*b - 112) * q^71 + (38*b - 182) * q^73 + (-26*b - 666) * q^74 + (-192*b - 240) * q^76 + (-14*b + 224) * q^77 + (-8*b + 920) * q^79 + (154*b + 1018) * q^82 + (-224*b - 228) * q^83 + (48*b - 752) * q^86 + (20*b - 780) * q^88 + (-336*b - 230) * q^89 + (-70*b + 14) * q^91 + (-192*b - 1344) * q^92 + (116*b + 1236) * q^94 + (-278*b + 474) * q^97 + (49*b + 49) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 9 q^{4} - 14 q^{7} + 51 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 9 * q^4 - 14 * q^7 + 51 * q^8 $$2 q + 3 q^{2} + 9 q^{4} - 14 q^{7} + 51 q^{8} - 62 q^{11} + 6 q^{13} - 21 q^{14} + 25 q^{16} + 40 q^{17} - 122 q^{19} - 52 q^{22} + 16 q^{23} + 214 q^{26} - 63 q^{28} - 352 q^{29} + 66 q^{31} - 309 q^{32} - 186 q^{34} + 188 q^{37} - 224 q^{38} - 16 q^{41} + 396 q^{43} - 156 q^{44} - 960 q^{46} - 188 q^{47} + 98 q^{49} + 642 q^{52} + 982 q^{53} - 357 q^{56} - 774 q^{58} - 516 q^{59} - 880 q^{61} - 680 q^{62} - 479 q^{64} + 356 q^{67} - 558 q^{68} - 310 q^{71} - 326 q^{73} - 1358 q^{74} - 672 q^{76} + 434 q^{77} + 1832 q^{79} + 2190 q^{82} - 680 q^{83} - 1456 q^{86} - 1540 q^{88} - 796 q^{89} - 42 q^{91} - 2880 q^{92} + 2588 q^{94} + 670 q^{97} + 147 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 + 9 * q^4 - 14 * q^7 + 51 * q^8 - 62 * q^11 + 6 * q^13 - 21 * q^14 + 25 * q^16 + 40 * q^17 - 122 * q^19 - 52 * q^22 + 16 * q^23 + 214 * q^26 - 63 * q^28 - 352 * q^29 + 66 * q^31 - 309 * q^32 - 186 * q^34 + 188 * q^37 - 224 * q^38 - 16 * q^41 + 396 * q^43 - 156 * q^44 - 960 * q^46 - 188 * q^47 + 98 * q^49 + 642 * q^52 + 982 * q^53 - 357 * q^56 - 774 * q^58 - 516 * q^59 - 880 * q^61 - 680 * q^62 - 479 * q^64 + 356 * q^67 - 558 * q^68 - 310 * q^71 - 326 * q^73 - 1358 * q^74 - 672 * q^76 + 434 * q^77 + 1832 * q^79 + 2190 * q^82 - 680 * q^83 - 1456 * q^86 - 1540 * q^88 - 796 * q^89 - 42 * q^91 - 2880 * q^92 + 2588 * q^94 + 670 * q^97 + 147 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.70156 3.70156
−1.70156 0 −5.10469 0 0 −7.00000 22.2984 0 0
1.2 4.70156 0 14.1047 0 0 −7.00000 28.7016 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.y 2
3.b odd 2 1 525.4.a.i 2
5.b even 2 1 315.4.a.g 2
15.d odd 2 1 105.4.a.g 2
15.e even 4 2 525.4.d.j 4
35.c odd 2 1 2205.4.a.v 2
60.h even 2 1 1680.4.a.y 2
105.g even 2 1 735.4.a.q 2

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

 $$T_{2}^{2} - 3T_{2} - 8$$ T2^2 - 3*T2 - 8 $$T_{11}^{2} + 62T_{11} + 920$$ T11^2 + 62*T11 + 920 $$T_{13}^{2} - 6T_{13} - 1016$$ T13^2 - 6*T13 - 1016

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T - 8$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} + 62T + 920$$
$13$ $$T^{2} - 6T - 1016$$
$17$ $$T^{2} - 40T - 1076$$
$19$ $$T^{2} + 122T + 3680$$
$23$ $$T^{2} - 16T - 23552$$
$29$ $$T^{2} + 352T + 29500$$
$31$ $$T^{2} - 66T - 13712$$
$37$ $$T^{2} - 188T - 56764$$
$41$ $$T^{2} + 16T - 119492$$
$43$ $$T^{2} - 396T - 63296$$
$47$ $$T^{2} + 188T - 192064$$
$53$ $$T^{2} - 982T + 206600$$
$59$ $$T^{2} + 516T + 7360$$
$61$ $$T^{2} + 880T + 121276$$
$67$ $$T^{2} - 356T - 501152$$
$71$ $$T^{2} + 310T - 51784$$
$73$ $$T^{2} + 326T + 11768$$
$79$ $$T^{2} - 1832 T + 838400$$
$83$ $$T^{2} + 680T - 398704$$
$89$ $$T^{2} + 796T - 998780$$
$97$ $$T^{2} - 670T - 679936$$