# Properties

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

# Related objects

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: $$\Q(\sqrt{41})$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} - 7 q^{7} + ( - \beta - 25) q^{8} + 9 q^{9}+O(q^{10})$$ q + (-b - 1) * q^2 - 3 * q^3 + (3*b + 3) * q^4 + (3*b + 3) * q^6 - 7 * q^7 + (-b - 25) * q^8 + 9 * q^9 $$q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} - 7 q^{7} + ( - \beta - 25) q^{8} + 9 q^{9} + ( - 2 \beta + 32) q^{11} + ( - 9 \beta - 9) q^{12} + (10 \beta - 2) q^{13} + (7 \beta + 7) q^{14} + (3 \beta + 11) q^{16} + (12 \beta - 26) q^{17} + ( - 9 \beta - 9) q^{18} + ( - 2 \beta - 60) q^{19} + 21 q^{21} + ( - 28 \beta - 12) q^{22} + (48 \beta - 32) q^{23} + (3 \beta + 75) q^{24} + ( - 18 \beta - 98) q^{26} - 27 q^{27} + ( - 21 \beta - 21) q^{28} + (12 \beta + 170) q^{29} + ( - 38 \beta + 52) q^{31} + ( - 9 \beta + 159) q^{32} + (6 \beta - 96) q^{33} + (2 \beta - 94) q^{34} + (27 \beta + 27) q^{36} + ( - 80 \beta + 134) q^{37} + (64 \beta + 80) q^{38} + ( - 30 \beta + 6) q^{39} + ( - 108 \beta + 62) q^{41} + ( - 21 \beta - 21) q^{42} + ( - 100 \beta + 248) q^{43} + (84 \beta + 36) q^{44} + ( - 64 \beta - 448) q^{46} + ( - 140 \beta + 164) q^{47} + ( - 9 \beta - 33) q^{48} + 49 q^{49} + ( - 36 \beta + 78) q^{51} + (54 \beta + 294) q^{52} + ( - 58 \beta - 462) q^{53} + (27 \beta + 27) q^{54} + (7 \beta + 175) q^{56} + (6 \beta + 180) q^{57} + ( - 194 \beta - 290) q^{58} + (76 \beta + 220) q^{59} + ( - 84 \beta - 398) q^{61} + (24 \beta + 328) q^{62} - 63 q^{63} + ( - 165 \beta - 157) q^{64} + (84 \beta + 36) q^{66} + (228 \beta + 64) q^{67} + ( - 6 \beta + 282) q^{68} + ( - 144 \beta + 96) q^{69} + (86 \beta + 112) q^{71} + ( - 9 \beta - 225) q^{72} + (38 \beta - 182) q^{73} + (26 \beta + 666) q^{74} + ( - 192 \beta - 240) q^{76} + (14 \beta - 224) q^{77} + (54 \beta + 294) q^{78} + ( - 8 \beta + 920) q^{79} + 81 q^{81} + (154 \beta + 1018) q^{82} + (224 \beta + 228) q^{83} + (63 \beta + 63) q^{84} + ( - 48 \beta + 752) q^{86} + ( - 36 \beta - 510) q^{87} + (20 \beta - 780) q^{88} + (336 \beta + 230) q^{89} + ( - 70 \beta + 14) q^{91} + (192 \beta + 1344) q^{92} + (114 \beta - 156) q^{93} + (116 \beta + 1236) q^{94} + (27 \beta - 477) q^{96} + ( - 278 \beta + 474) q^{97} + ( - 49 \beta - 49) q^{98} + ( - 18 \beta + 288) q^{99}+O(q^{100})$$ q + (-b - 1) * q^2 - 3 * q^3 + (3*b + 3) * q^4 + (3*b + 3) * q^6 - 7 * q^7 + (-b - 25) * q^8 + 9 * q^9 + (-2*b + 32) * q^11 + (-9*b - 9) * q^12 + (10*b - 2) * q^13 + (7*b + 7) * q^14 + (3*b + 11) * q^16 + (12*b - 26) * q^17 + (-9*b - 9) * q^18 + (-2*b - 60) * q^19 + 21 * q^21 + (-28*b - 12) * q^22 + (48*b - 32) * q^23 + (3*b + 75) * q^24 + (-18*b - 98) * q^26 - 27 * q^27 + (-21*b - 21) * q^28 + (12*b + 170) * q^29 + (-38*b + 52) * q^31 + (-9*b + 159) * q^32 + (6*b - 96) * q^33 + (2*b - 94) * q^34 + (27*b + 27) * q^36 + (-80*b + 134) * q^37 + (64*b + 80) * q^38 + (-30*b + 6) * q^39 + (-108*b + 62) * q^41 + (-21*b - 21) * q^42 + (-100*b + 248) * q^43 + (84*b + 36) * q^44 + (-64*b - 448) * q^46 + (-140*b + 164) * q^47 + (-9*b - 33) * q^48 + 49 * q^49 + (-36*b + 78) * q^51 + (54*b + 294) * q^52 + (-58*b - 462) * q^53 + (27*b + 27) * q^54 + (7*b + 175) * q^56 + (6*b + 180) * q^57 + (-194*b - 290) * q^58 + (76*b + 220) * q^59 + (-84*b - 398) * q^61 + (24*b + 328) * q^62 - 63 * q^63 + (-165*b - 157) * q^64 + (84*b + 36) * q^66 + (228*b + 64) * q^67 + (-6*b + 282) * q^68 + (-144*b + 96) * q^69 + (86*b + 112) * q^71 + (-9*b - 225) * q^72 + (38*b - 182) * q^73 + (26*b + 666) * q^74 + (-192*b - 240) * q^76 + (14*b - 224) * q^77 + (54*b + 294) * q^78 + (-8*b + 920) * q^79 + 81 * q^81 + (154*b + 1018) * q^82 + (224*b + 228) * q^83 + (63*b + 63) * q^84 + (-48*b + 752) * q^86 + (-36*b - 510) * q^87 + (20*b - 780) * q^88 + (336*b + 230) * q^89 + (-70*b + 14) * q^91 + (192*b + 1344) * q^92 + (114*b - 156) * q^93 + (116*b + 1236) * q^94 + (27*b - 477) * q^96 + (-278*b + 474) * q^97 + (-49*b - 49) * q^98 + (-18*b + 288) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 14 q^{7} - 51 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 6 * q^3 + 9 * q^4 + 9 * q^6 - 14 * q^7 - 51 * q^8 + 18 * q^9 $$2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 14 q^{7} - 51 q^{8} + 18 q^{9} + 62 q^{11} - 27 q^{12} + 6 q^{13} + 21 q^{14} + 25 q^{16} - 40 q^{17} - 27 q^{18} - 122 q^{19} + 42 q^{21} - 52 q^{22} - 16 q^{23} + 153 q^{24} - 214 q^{26} - 54 q^{27} - 63 q^{28} + 352 q^{29} + 66 q^{31} + 309 q^{32} - 186 q^{33} - 186 q^{34} + 81 q^{36} + 188 q^{37} + 224 q^{38} - 18 q^{39} + 16 q^{41} - 63 q^{42} + 396 q^{43} + 156 q^{44} - 960 q^{46} + 188 q^{47} - 75 q^{48} + 98 q^{49} + 120 q^{51} + 642 q^{52} - 982 q^{53} + 81 q^{54} + 357 q^{56} + 366 q^{57} - 774 q^{58} + 516 q^{59} - 880 q^{61} + 680 q^{62} - 126 q^{63} - 479 q^{64} + 156 q^{66} + 356 q^{67} + 558 q^{68} + 48 q^{69} + 310 q^{71} - 459 q^{72} - 326 q^{73} + 1358 q^{74} - 672 q^{76} - 434 q^{77} + 642 q^{78} + 1832 q^{79} + 162 q^{81} + 2190 q^{82} + 680 q^{83} + 189 q^{84} + 1456 q^{86} - 1056 q^{87} - 1540 q^{88} + 796 q^{89} - 42 q^{91} + 2880 q^{92} - 198 q^{93} + 2588 q^{94} - 927 q^{96} + 670 q^{97} - 147 q^{98} + 558 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 6 * q^3 + 9 * q^4 + 9 * q^6 - 14 * q^7 - 51 * q^8 + 18 * q^9 + 62 * q^11 - 27 * q^12 + 6 * q^13 + 21 * q^14 + 25 * q^16 - 40 * q^17 - 27 * q^18 - 122 * q^19 + 42 * q^21 - 52 * q^22 - 16 * q^23 + 153 * q^24 - 214 * q^26 - 54 * q^27 - 63 * q^28 + 352 * q^29 + 66 * q^31 + 309 * q^32 - 186 * q^33 - 186 * q^34 + 81 * q^36 + 188 * q^37 + 224 * q^38 - 18 * q^39 + 16 * q^41 - 63 * q^42 + 396 * q^43 + 156 * q^44 - 960 * q^46 + 188 * q^47 - 75 * q^48 + 98 * q^49 + 120 * q^51 + 642 * q^52 - 982 * q^53 + 81 * q^54 + 357 * q^56 + 366 * q^57 - 774 * q^58 + 516 * q^59 - 880 * q^61 + 680 * q^62 - 126 * q^63 - 479 * q^64 + 156 * q^66 + 356 * q^67 + 558 * q^68 + 48 * q^69 + 310 * q^71 - 459 * q^72 - 326 * q^73 + 1358 * q^74 - 672 * q^76 - 434 * q^77 + 642 * q^78 + 1832 * q^79 + 162 * q^81 + 2190 * q^82 + 680 * q^83 + 189 * q^84 + 1456 * q^86 - 1056 * q^87 - 1540 * q^88 + 796 * q^89 - 42 * q^91 + 2880 * q^92 - 198 * q^93 + 2588 * q^94 - 927 * q^96 + 670 * q^97 - 147 * q^98 + 558 * 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
 3.70156 −2.70156
−4.70156 −3.00000 14.1047 0 14.1047 −7.00000 −28.7016 9.00000 0
1.2 1.70156 −3.00000 −5.10469 0 −5.10469 −7.00000 −22.2984 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.i 2
3.b odd 2 1 1575.4.a.y 2
5.b even 2 1 105.4.a.g 2
5.c odd 4 2 525.4.d.j 4
15.d odd 2 1 315.4.a.g 2
20.d odd 2 1 1680.4.a.y 2
35.c odd 2 1 735.4.a.q 2
105.g even 2 1 2205.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 5.b even 2 1
315.4.a.g 2 15.d odd 2 1
525.4.a.i 2 1.a even 1 1 trivial
525.4.d.j 4 5.c odd 4 2
735.4.a.q 2 35.c odd 2 1
1575.4.a.y 2 3.b odd 2 1
1680.4.a.y 2 20.d odd 2 1
2205.4.a.v 2 105.g even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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