# Properties

 Label 525.4.a.n Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, 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("525.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.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: $$30.9760027530$$ Analytic rank: $$0$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} - 3 q^{3} + (3 \beta + 7) q^{4} + ( - 3 \beta - 3) q^{6} - 7 q^{7} + (5 \beta + 41) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b + 1) * q^2 - 3 * q^3 + (3*b + 7) * q^4 + (-3*b - 3) * q^6 - 7 * q^7 + (5*b + 41) * q^8 + 9 * q^9 $$q + (\beta + 1) q^{2} - 3 q^{3} + (3 \beta + 7) q^{4} + ( - 3 \beta - 3) q^{6} - 7 q^{7} + (5 \beta + 41) q^{8} + 9 q^{9} + (10 \beta - 8) q^{11} + ( - 9 \beta - 21) q^{12} + (12 \beta - 14) q^{13} + ( - 7 \beta - 7) q^{14} + (27 \beta + 55) q^{16} + (2 \beta + 2) q^{17} + (9 \beta + 9) q^{18} + ( - 24 \beta + 44) q^{19} + 21 q^{21} + (12 \beta + 132) q^{22} + (34 \beta - 20) q^{23} + ( - 15 \beta - 123) q^{24} + (10 \beta + 154) q^{26} - 27 q^{27} + ( - 21 \beta - 49) q^{28} + (24 \beta - 138) q^{29} + (72 \beta - 16) q^{31} + (69 \beta + 105) q^{32} + ( - 30 \beta + 24) q^{33} + (6 \beta + 30) q^{34} + (27 \beta + 63) q^{36} + (36 \beta + 106) q^{37} + ( - 4 \beta - 292) q^{38} + ( - 36 \beta + 42) q^{39} + ( - 30 \beta - 210) q^{41} + (21 \beta + 21) q^{42} + (48 \beta - 212) q^{43} + (76 \beta + 364) q^{44} + (48 \beta + 456) q^{46} + ( - 68 \beta + 40) q^{47} + ( - 81 \beta - 165) q^{48} + 49 q^{49} + ( - 6 \beta - 6) q^{51} + (78 \beta + 406) q^{52} + ( - 4 \beta + 554) q^{53} + ( - 27 \beta - 27) q^{54} + ( - 35 \beta - 287) q^{56} + (72 \beta - 132) q^{57} + ( - 90 \beta + 198) q^{58} + ( - 116 \beta + 460) q^{59} + (72 \beta - 250) q^{61} + (128 \beta + 992) q^{62} - 63 q^{63} + (27 \beta + 631) q^{64} + ( - 36 \beta - 396) q^{66} + ( - 108 \beta - 20) q^{67} + (26 \beta + 98) q^{68} + ( - 102 \beta + 60) q^{69} + ( - 30 \beta + 492) q^{71} + (45 \beta + 369) q^{72} + ( - 12 \beta - 530) q^{73} + (178 \beta + 610) q^{74} + ( - 108 \beta - 700) q^{76} + ( - 70 \beta + 56) q^{77} + ( - 30 \beta - 462) q^{78} + ( - 108 \beta - 232) q^{79} + 81 q^{81} + ( - 270 \beta - 630) q^{82} + ( - 96 \beta - 924) q^{83} + (63 \beta + 147) q^{84} + ( - 116 \beta + 460) q^{86} + ( - 72 \beta + 414) q^{87} + (420 \beta + 372) q^{88} + ( - 142 \beta + 254) q^{89} + ( - 84 \beta + 98) q^{91} + (280 \beta + 1288) q^{92} + ( - 216 \beta + 48) q^{93} + ( - 96 \beta - 912) q^{94} + ( - 207 \beta - 315) q^{96} + ( - 276 \beta - 266) q^{97} + (49 \beta + 49) q^{98} + (90 \beta - 72) q^{99}+O(q^{100})$$ q + (b + 1) * q^2 - 3 * q^3 + (3*b + 7) * q^4 + (-3*b - 3) * q^6 - 7 * q^7 + (5*b + 41) * q^8 + 9 * q^9 + (10*b - 8) * q^11 + (-9*b - 21) * q^12 + (12*b - 14) * q^13 + (-7*b - 7) * q^14 + (27*b + 55) * q^16 + (2*b + 2) * q^17 + (9*b + 9) * q^18 + (-24*b + 44) * q^19 + 21 * q^21 + (12*b + 132) * q^22 + (34*b - 20) * q^23 + (-15*b - 123) * q^24 + (10*b + 154) * q^26 - 27 * q^27 + (-21*b - 49) * q^28 + (24*b - 138) * q^29 + (72*b - 16) * q^31 + (69*b + 105) * q^32 + (-30*b + 24) * q^33 + (6*b + 30) * q^34 + (27*b + 63) * q^36 + (36*b + 106) * q^37 + (-4*b - 292) * q^38 + (-36*b + 42) * q^39 + (-30*b - 210) * q^41 + (21*b + 21) * q^42 + (48*b - 212) * q^43 + (76*b + 364) * q^44 + (48*b + 456) * q^46 + (-68*b + 40) * q^47 + (-81*b - 165) * q^48 + 49 * q^49 + (-6*b - 6) * q^51 + (78*b + 406) * q^52 + (-4*b + 554) * q^53 + (-27*b - 27) * q^54 + (-35*b - 287) * q^56 + (72*b - 132) * q^57 + (-90*b + 198) * q^58 + (-116*b + 460) * q^59 + (72*b - 250) * q^61 + (128*b + 992) * q^62 - 63 * q^63 + (27*b + 631) * q^64 + (-36*b - 396) * q^66 + (-108*b - 20) * q^67 + (26*b + 98) * q^68 + (-102*b + 60) * q^69 + (-30*b + 492) * q^71 + (45*b + 369) * q^72 + (-12*b - 530) * q^73 + (178*b + 610) * q^74 + (-108*b - 700) * q^76 + (-70*b + 56) * q^77 + (-30*b - 462) * q^78 + (-108*b - 232) * q^79 + 81 * q^81 + (-270*b - 630) * q^82 + (-96*b - 924) * q^83 + (63*b + 147) * q^84 + (-116*b + 460) * q^86 + (-72*b + 414) * q^87 + (420*b + 372) * q^88 + (-142*b + 254) * q^89 + (-84*b + 98) * q^91 + (280*b + 1288) * q^92 + (-216*b + 48) * q^93 + (-96*b - 912) * q^94 + (-207*b - 315) * q^96 + (-276*b - 266) * q^97 + (49*b + 49) * q^98 + (90*b - 72) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - 6 q^{3} + 17 q^{4} - 9 q^{6} - 14 q^{7} + 87 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 - 6 * q^3 + 17 * q^4 - 9 * q^6 - 14 * q^7 + 87 * q^8 + 18 * q^9 $$2 q + 3 q^{2} - 6 q^{3} + 17 q^{4} - 9 q^{6} - 14 q^{7} + 87 q^{8} + 18 q^{9} - 6 q^{11} - 51 q^{12} - 16 q^{13} - 21 q^{14} + 137 q^{16} + 6 q^{17} + 27 q^{18} + 64 q^{19} + 42 q^{21} + 276 q^{22} - 6 q^{23} - 261 q^{24} + 318 q^{26} - 54 q^{27} - 119 q^{28} - 252 q^{29} + 40 q^{31} + 279 q^{32} + 18 q^{33} + 66 q^{34} + 153 q^{36} + 248 q^{37} - 588 q^{38} + 48 q^{39} - 450 q^{41} + 63 q^{42} - 376 q^{43} + 804 q^{44} + 960 q^{46} + 12 q^{47} - 411 q^{48} + 98 q^{49} - 18 q^{51} + 890 q^{52} + 1104 q^{53} - 81 q^{54} - 609 q^{56} - 192 q^{57} + 306 q^{58} + 804 q^{59} - 428 q^{61} + 2112 q^{62} - 126 q^{63} + 1289 q^{64} - 828 q^{66} - 148 q^{67} + 222 q^{68} + 18 q^{69} + 954 q^{71} + 783 q^{72} - 1072 q^{73} + 1398 q^{74} - 1508 q^{76} + 42 q^{77} - 954 q^{78} - 572 q^{79} + 162 q^{81} - 1530 q^{82} - 1944 q^{83} + 357 q^{84} + 804 q^{86} + 756 q^{87} + 1164 q^{88} + 366 q^{89} + 112 q^{91} + 2856 q^{92} - 120 q^{93} - 1920 q^{94} - 837 q^{96} - 808 q^{97} + 147 q^{98} - 54 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 - 6 * q^3 + 17 * q^4 - 9 * q^6 - 14 * q^7 + 87 * q^8 + 18 * q^9 - 6 * q^11 - 51 * q^12 - 16 * q^13 - 21 * q^14 + 137 * q^16 + 6 * q^17 + 27 * q^18 + 64 * q^19 + 42 * q^21 + 276 * q^22 - 6 * q^23 - 261 * q^24 + 318 * q^26 - 54 * q^27 - 119 * q^28 - 252 * q^29 + 40 * q^31 + 279 * q^32 + 18 * q^33 + 66 * q^34 + 153 * q^36 + 248 * q^37 - 588 * q^38 + 48 * q^39 - 450 * q^41 + 63 * q^42 - 376 * q^43 + 804 * q^44 + 960 * q^46 + 12 * q^47 - 411 * q^48 + 98 * q^49 - 18 * q^51 + 890 * q^52 + 1104 * q^53 - 81 * q^54 - 609 * q^56 - 192 * q^57 + 306 * q^58 + 804 * q^59 - 428 * q^61 + 2112 * q^62 - 126 * q^63 + 1289 * q^64 - 828 * q^66 - 148 * q^67 + 222 * q^68 + 18 * q^69 + 954 * q^71 + 783 * q^72 - 1072 * q^73 + 1398 * q^74 - 1508 * q^76 + 42 * q^77 - 954 * q^78 - 572 * q^79 + 162 * q^81 - 1530 * q^82 - 1944 * q^83 + 357 * q^84 + 804 * q^86 + 756 * q^87 + 1164 * q^88 + 366 * q^89 + 112 * q^91 + 2856 * q^92 - 120 * q^93 - 1920 * q^94 - 837 * q^96 - 808 * q^97 + 147 * q^98 - 54 * q^99

## 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
 −3.27492 4.27492
−2.27492 −3.00000 −2.82475 0 6.82475 −7.00000 24.6254 9.00000 0
1.2 5.27492 −3.00000 19.8248 0 −15.8248 −7.00000 62.3746 9.00000 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 525.4.a.n 2
3.b odd 2 1 1575.4.a.p 2
5.b even 2 1 21.4.a.c 2
5.c odd 4 2 525.4.d.g 4
15.d odd 2 1 63.4.a.e 2
20.d odd 2 1 336.4.a.m 2
35.c odd 2 1 147.4.a.i 2
35.i odd 6 2 147.4.e.m 4
35.j even 6 2 147.4.e.l 4
40.e odd 2 1 1344.4.a.bo 2
40.f even 2 1 1344.4.a.bg 2
60.h even 2 1 1008.4.a.ba 2
105.g even 2 1 441.4.a.r 2
105.o odd 6 2 441.4.e.q 4
105.p even 6 2 441.4.e.p 4
140.c even 2 1 2352.4.a.bz 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 5.b even 2 1
63.4.a.e 2 15.d odd 2 1
147.4.a.i 2 35.c odd 2 1
147.4.e.l 4 35.j even 6 2
147.4.e.m 4 35.i odd 6 2
336.4.a.m 2 20.d odd 2 1
441.4.a.r 2 105.g even 2 1
441.4.e.p 4 105.p even 6 2
441.4.e.q 4 105.o odd 6 2
525.4.a.n 2 1.a even 1 1 trivial
525.4.d.g 4 5.c odd 4 2
1008.4.a.ba 2 60.h even 2 1
1344.4.a.bg 2 40.f even 2 1
1344.4.a.bo 2 40.e odd 2 1
1575.4.a.p 2 3.b odd 2 1
2352.4.a.bz 2 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

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

## Hecke characteristic polynomials

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