Properties

 Label 1575.4.a.p 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: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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 $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + (3 \beta + 7) q^{4} - 7 q^{7} + ( - 5 \beta - 41) q^{8}+O(q^{10})$$ q + (-b - 1) * q^2 + (3*b + 7) * q^4 - 7 * q^7 + (-5*b - 41) * q^8 $$q + ( - \beta - 1) q^{2} + (3 \beta + 7) q^{4} - 7 q^{7} + ( - 5 \beta - 41) q^{8} + ( - 10 \beta + 8) q^{11} + (12 \beta - 14) q^{13} + (7 \beta + 7) q^{14} + (27 \beta + 55) q^{16} + ( - 2 \beta - 2) q^{17} + ( - 24 \beta + 44) q^{19} + (12 \beta + 132) q^{22} + ( - 34 \beta + 20) q^{23} + ( - 10 \beta - 154) q^{26} + ( - 21 \beta - 49) q^{28} + ( - 24 \beta + 138) q^{29} + (72 \beta - 16) q^{31} + ( - 69 \beta - 105) q^{32} + (6 \beta + 30) q^{34} + (36 \beta + 106) q^{37} + (4 \beta + 292) q^{38} + (30 \beta + 210) q^{41} + (48 \beta - 212) q^{43} + ( - 76 \beta - 364) q^{44} + (48 \beta + 456) q^{46} + (68 \beta - 40) q^{47} + 49 q^{49} + (78 \beta + 406) q^{52} + (4 \beta - 554) q^{53} + (35 \beta + 287) q^{56} + ( - 90 \beta + 198) q^{58} + (116 \beta - 460) q^{59} + (72 \beta - 250) q^{61} + ( - 128 \beta - 992) q^{62} + (27 \beta + 631) q^{64} + ( - 108 \beta - 20) q^{67} + ( - 26 \beta - 98) q^{68} + (30 \beta - 492) q^{71} + ( - 12 \beta - 530) q^{73} + ( - 178 \beta - 610) q^{74} + ( - 108 \beta - 700) q^{76} + (70 \beta - 56) q^{77} + ( - 108 \beta - 232) q^{79} + ( - 270 \beta - 630) q^{82} + (96 \beta + 924) q^{83} + (116 \beta - 460) q^{86} + (420 \beta + 372) q^{88} + (142 \beta - 254) q^{89} + ( - 84 \beta + 98) q^{91} + ( - 280 \beta - 1288) q^{92} + ( - 96 \beta - 912) q^{94} + ( - 276 \beta - 266) q^{97} + ( - 49 \beta - 49) q^{98}+O(q^{100})$$ q + (-b - 1) * q^2 + (3*b + 7) * q^4 - 7 * q^7 + (-5*b - 41) * q^8 + (-10*b + 8) * q^11 + (12*b - 14) * q^13 + (7*b + 7) * q^14 + (27*b + 55) * q^16 + (-2*b - 2) * q^17 + (-24*b + 44) * q^19 + (12*b + 132) * q^22 + (-34*b + 20) * q^23 + (-10*b - 154) * q^26 + (-21*b - 49) * q^28 + (-24*b + 138) * q^29 + (72*b - 16) * q^31 + (-69*b - 105) * q^32 + (6*b + 30) * q^34 + (36*b + 106) * q^37 + (4*b + 292) * q^38 + (30*b + 210) * q^41 + (48*b - 212) * q^43 + (-76*b - 364) * q^44 + (48*b + 456) * q^46 + (68*b - 40) * q^47 + 49 * q^49 + (78*b + 406) * q^52 + (4*b - 554) * q^53 + (35*b + 287) * q^56 + (-90*b + 198) * q^58 + (116*b - 460) * q^59 + (72*b - 250) * q^61 + (-128*b - 992) * q^62 + (27*b + 631) * q^64 + (-108*b - 20) * q^67 + (-26*b - 98) * q^68 + (30*b - 492) * q^71 + (-12*b - 530) * q^73 + (-178*b - 610) * q^74 + (-108*b - 700) * q^76 + (70*b - 56) * q^77 + (-108*b - 232) * q^79 + (-270*b - 630) * q^82 + (96*b + 924) * q^83 + (116*b - 460) * q^86 + (420*b + 372) * q^88 + (142*b - 254) * q^89 + (-84*b + 98) * q^91 + (-280*b - 1288) * q^92 + (-96*b - 912) * q^94 + (-276*b - 266) * q^97 + (-49*b - 49) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8 $$2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + 6 q^{11} - 16 q^{13} + 21 q^{14} + 137 q^{16} - 6 q^{17} + 64 q^{19} + 276 q^{22} + 6 q^{23} - 318 q^{26} - 119 q^{28} + 252 q^{29} + 40 q^{31} - 279 q^{32} + 66 q^{34} + 248 q^{37} + 588 q^{38} + 450 q^{41} - 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 98 q^{49} + 890 q^{52} - 1104 q^{53} + 609 q^{56} + 306 q^{58} - 804 q^{59} - 428 q^{61} - 2112 q^{62} + 1289 q^{64} - 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} - 1508 q^{76} - 42 q^{77} - 572 q^{79} - 1530 q^{82} + 1944 q^{83} - 804 q^{86} + 1164 q^{88} - 366 q^{89} + 112 q^{91} - 2856 q^{92} - 1920 q^{94} - 808 q^{97} - 147 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8 + 6 * q^11 - 16 * q^13 + 21 * q^14 + 137 * q^16 - 6 * q^17 + 64 * q^19 + 276 * q^22 + 6 * q^23 - 318 * q^26 - 119 * q^28 + 252 * q^29 + 40 * q^31 - 279 * q^32 + 66 * q^34 + 248 * q^37 + 588 * q^38 + 450 * q^41 - 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 98 * q^49 + 890 * q^52 - 1104 * q^53 + 609 * q^56 + 306 * q^58 - 804 * q^59 - 428 * q^61 - 2112 * q^62 + 1289 * q^64 - 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 - 1508 * q^76 - 42 * q^77 - 572 * q^79 - 1530 * q^82 + 1944 * q^83 - 804 * q^86 + 1164 * q^88 - 366 * q^89 + 112 * q^91 - 2856 * q^92 - 1920 * q^94 - 808 * 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
 4.27492 −3.27492
−5.27492 0 19.8248 0 0 −7.00000 −62.3746 0 0
1.2 2.27492 0 −2.82475 0 0 −7.00000 −24.6254 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.p 2
3.b odd 2 1 525.4.a.n 2
5.b even 2 1 63.4.a.e 2
15.d odd 2 1 21.4.a.c 2
15.e even 4 2 525.4.d.g 4
20.d odd 2 1 1008.4.a.ba 2
35.c odd 2 1 441.4.a.r 2
35.i odd 6 2 441.4.e.p 4
35.j even 6 2 441.4.e.q 4
60.h even 2 1 336.4.a.m 2
105.g even 2 1 147.4.a.i 2
105.o odd 6 2 147.4.e.l 4
105.p even 6 2 147.4.e.m 4
120.i odd 2 1 1344.4.a.bg 2
120.m even 2 1 1344.4.a.bo 2
420.o odd 2 1 2352.4.a.bz 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 15.d odd 2 1
63.4.a.e 2 5.b even 2 1
147.4.a.i 2 105.g even 2 1
147.4.e.l 4 105.o odd 6 2
147.4.e.m 4 105.p even 6 2
336.4.a.m 2 60.h even 2 1
441.4.a.r 2 35.c odd 2 1
441.4.e.p 4 35.i odd 6 2
441.4.e.q 4 35.j even 6 2
525.4.a.n 2 3.b odd 2 1
525.4.d.g 4 15.e even 4 2
1008.4.a.ba 2 20.d odd 2 1
1344.4.a.bg 2 120.i odd 2 1
1344.4.a.bo 2 120.m even 2 1
1575.4.a.p 2 1.a even 1 1 trivial
2352.4.a.bz 2 420.o odd 2 1

Hecke kernels

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

 $$T_{2}^{2} + 3T_{2} - 12$$ T2^2 + 3*T2 - 12 $$T_{11}^{2} - 6T_{11} - 1416$$ T11^2 - 6*T11 - 1416 $$T_{13}^{2} + 16T_{13} - 1988$$ T13^2 + 16*T13 - 1988

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T - 12$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} - 6T - 1416$$
$13$ $$T^{2} + 16T - 1988$$
$17$ $$T^{2} + 6T - 48$$
$19$ $$T^{2} - 64T - 7184$$
$23$ $$T^{2} - 6T - 16464$$
$29$ $$T^{2} - 252T + 7668$$
$31$ $$T^{2} - 40T - 73472$$
$37$ $$T^{2} - 248T - 3092$$
$41$ $$T^{2} - 450T + 37800$$
$43$ $$T^{2} + 376T + 2512$$
$47$ $$T^{2} + 12T - 65856$$
$53$ $$T^{2} + 1104 T + 304476$$
$59$ $$T^{2} + 804T - 30144$$
$61$ $$T^{2} + 428T - 28076$$
$67$ $$T^{2} + 148T - 160736$$
$71$ $$T^{2} + 954T + 214704$$
$73$ $$T^{2} + 1072 T + 285244$$
$79$ $$T^{2} + 572T - 84416$$
$83$ $$T^{2} - 1944 T + 813456$$
$89$ $$T^{2} + 366T - 253848$$
$97$ $$T^{2} + 808T - 922292$$