# Properties

 Label 525.4.d.j Level $525$ Weight $4$ Character orbit 525.d Analytic conductor $30.976$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(274,525)]

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, 1, 0]))

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

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 21x^{2} + 100$$ x^4 + 21*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + (3 \beta_{3} - 6) q^{4} + ( - 3 \beta_{3} + 6) q^{6} - 7 \beta_{2} q^{7} + (25 \beta_{2} - \beta_1) q^{8} - 9 q^{9}+O(q^{10})$$ q + (-b2 + b1) * q^2 + 3*b2 * q^3 + (3*b3 - 6) * q^4 + (-3*b3 + 6) * q^6 - 7*b2 * q^7 + (25*b2 - b1) * q^8 - 9 * q^9 $$q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + (3 \beta_{3} - 6) q^{4} + ( - 3 \beta_{3} + 6) q^{6} - 7 \beta_{2} q^{7} + (25 \beta_{2} - \beta_1) q^{8} - 9 q^{9} + (2 \beta_{3} + 30) q^{11} + ( - 9 \beta_{2} + 9 \beta_1) q^{12} + (2 \beta_{2} + 10 \beta_1) q^{13} + (7 \beta_{3} - 14) q^{14} + ( - 3 \beta_{3} + 14) q^{16} + ( - 26 \beta_{2} - 12 \beta_1) q^{17} + (9 \beta_{2} - 9 \beta_1) q^{18} + ( - 2 \beta_{3} + 62) q^{19} + 21 q^{21} + ( - 12 \beta_{2} + 28 \beta_1) q^{22} + (32 \beta_{2} + 48 \beta_1) q^{23} + (3 \beta_{3} - 78) q^{24} + (18 \beta_{3} - 116) q^{26} - 27 \beta_{2} q^{27} + (21 \beta_{2} - 21 \beta_1) q^{28} + (12 \beta_{3} - 182) q^{29} + (38 \beta_{3} + 14) q^{31} + (159 \beta_{2} + 9 \beta_1) q^{32} + (96 \beta_{2} + 6 \beta_1) q^{33} + (2 \beta_{3} + 92) q^{34} + ( - 27 \beta_{3} + 54) q^{36} + (134 \beta_{2} + 80 \beta_1) q^{37} + ( - 80 \beta_{2} + 64 \beta_1) q^{38} + ( - 30 \beta_{3} + 24) q^{39} + (108 \beta_{3} - 46) q^{41} + ( - 21 \beta_{2} + 21 \beta_1) q^{42} + ( - 248 \beta_{2} - 100 \beta_1) q^{43} + (84 \beta_{3} - 120) q^{44} + (64 \beta_{3} - 512) q^{46} + (164 \beta_{2} + 140 \beta_1) q^{47} + (33 \beta_{2} - 9 \beta_1) q^{48} - 49 q^{49} + (36 \beta_{3} + 42) q^{51} + (294 \beta_{2} - 54 \beta_1) q^{52} + (462 \beta_{2} - 58 \beta_1) q^{53} + (27 \beta_{3} - 54) q^{54} + ( - 7 \beta_{3} + 182) q^{56} + (180 \beta_{2} - 6 \beta_1) q^{57} + (290 \beta_{2} - 194 \beta_1) q^{58} + (76 \beta_{3} - 296) q^{59} + (84 \beta_{3} - 482) q^{61} + (328 \beta_{2} - 24 \beta_1) q^{62} + 63 \beta_{2} q^{63} + ( - 165 \beta_{3} + 322) q^{64} + ( - 84 \beta_{3} + 120) q^{66} + (64 \beta_{2} - 228 \beta_1) q^{67} + ( - 282 \beta_{2} - 6 \beta_1) q^{68} + ( - 144 \beta_{3} + 48) q^{69} + ( - 86 \beta_{3} + 198) q^{71} + ( - 225 \beta_{2} + 9 \beta_1) q^{72} + (182 \beta_{2} + 38 \beta_1) q^{73} + (26 \beta_{3} - 692) q^{74} + (192 \beta_{3} - 432) q^{76} + ( - 224 \beta_{2} - 14 \beta_1) q^{77} + ( - 294 \beta_{2} + 54 \beta_1) q^{78} + ( - 8 \beta_{3} - 912) q^{79} + 81 q^{81} + (1018 \beta_{2} - 154 \beta_1) q^{82} + ( - 228 \beta_{2} + 224 \beta_1) q^{83} + (63 \beta_{3} - 126) q^{84} + (48 \beta_{3} + 704) q^{86} + ( - 510 \beta_{2} + 36 \beta_1) q^{87} + (780 \beta_{2} + 20 \beta_1) q^{88} + (336 \beta_{3} - 566) q^{89} + (70 \beta_{3} - 56) q^{91} + (1344 \beta_{2} - 192 \beta_1) q^{92} + (156 \beta_{2} + 114 \beta_1) q^{93} + (116 \beta_{3} - 1352) q^{94} + ( - 27 \beta_{3} - 450) q^{96} + (474 \beta_{2} + 278 \beta_1) q^{97} + (49 \beta_{2} - 49 \beta_1) q^{98} + ( - 18 \beta_{3} - 270) q^{99}+O(q^{100})$$ q + (-b2 + b1) * q^2 + 3*b2 * q^3 + (3*b3 - 6) * q^4 + (-3*b3 + 6) * q^6 - 7*b2 * q^7 + (25*b2 - b1) * q^8 - 9 * q^9 + (2*b3 + 30) * q^11 + (-9*b2 + 9*b1) * q^12 + (2*b2 + 10*b1) * q^13 + (7*b3 - 14) * q^14 + (-3*b3 + 14) * q^16 + (-26*b2 - 12*b1) * q^17 + (9*b2 - 9*b1) * q^18 + (-2*b3 + 62) * q^19 + 21 * q^21 + (-12*b2 + 28*b1) * q^22 + (32*b2 + 48*b1) * q^23 + (3*b3 - 78) * q^24 + (18*b3 - 116) * q^26 - 27*b2 * q^27 + (21*b2 - 21*b1) * q^28 + (12*b3 - 182) * q^29 + (38*b3 + 14) * q^31 + (159*b2 + 9*b1) * q^32 + (96*b2 + 6*b1) * q^33 + (2*b3 + 92) * q^34 + (-27*b3 + 54) * q^36 + (134*b2 + 80*b1) * q^37 + (-80*b2 + 64*b1) * q^38 + (-30*b3 + 24) * q^39 + (108*b3 - 46) * q^41 + (-21*b2 + 21*b1) * q^42 + (-248*b2 - 100*b1) * q^43 + (84*b3 - 120) * q^44 + (64*b3 - 512) * q^46 + (164*b2 + 140*b1) * q^47 + (33*b2 - 9*b1) * q^48 - 49 * q^49 + (36*b3 + 42) * q^51 + (294*b2 - 54*b1) * q^52 + (462*b2 - 58*b1) * q^53 + (27*b3 - 54) * q^54 + (-7*b3 + 182) * q^56 + (180*b2 - 6*b1) * q^57 + (290*b2 - 194*b1) * q^58 + (76*b3 - 296) * q^59 + (84*b3 - 482) * q^61 + (328*b2 - 24*b1) * q^62 + 63*b2 * q^63 + (-165*b3 + 322) * q^64 + (-84*b3 + 120) * q^66 + (64*b2 - 228*b1) * q^67 + (-282*b2 - 6*b1) * q^68 + (-144*b3 + 48) * q^69 + (-86*b3 + 198) * q^71 + (-225*b2 + 9*b1) * q^72 + (182*b2 + 38*b1) * q^73 + (26*b3 - 692) * q^74 + (192*b3 - 432) * q^76 + (-224*b2 - 14*b1) * q^77 + (-294*b2 + 54*b1) * q^78 + (-8*b3 - 912) * q^79 + 81 * q^81 + (1018*b2 - 154*b1) * q^82 + (-228*b2 + 224*b1) * q^83 + (63*b3 - 126) * q^84 + (48*b3 + 704) * q^86 + (-510*b2 + 36*b1) * q^87 + (780*b2 + 20*b1) * q^88 + (336*b3 - 566) * q^89 + (70*b3 - 56) * q^91 + (1344*b2 - 192*b1) * q^92 + (156*b2 + 114*b1) * q^93 + (116*b3 - 1352) * q^94 + (-27*b3 - 450) * q^96 + (474*b2 + 278*b1) * q^97 + (49*b2 - 49*b1) * q^98 + (-18*b3 - 270) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 18 q^{4} + 18 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 18 * q^4 + 18 * q^6 - 36 * q^9 $$4 q - 18 q^{4} + 18 q^{6} - 36 q^{9} + 124 q^{11} - 42 q^{14} + 50 q^{16} + 244 q^{19} + 84 q^{21} - 306 q^{24} - 428 q^{26} - 704 q^{29} + 132 q^{31} + 372 q^{34} + 162 q^{36} + 36 q^{39} + 32 q^{41} - 312 q^{44} - 1920 q^{46} - 196 q^{49} + 240 q^{51} - 162 q^{54} + 714 q^{56} - 1032 q^{59} - 1760 q^{61} + 958 q^{64} + 312 q^{66} - 96 q^{69} + 620 q^{71} - 2716 q^{74} - 1344 q^{76} - 3664 q^{79} + 324 q^{81} - 378 q^{84} + 2912 q^{86} - 1592 q^{89} - 84 q^{91} - 5176 q^{94} - 1854 q^{96} - 1116 q^{99}+O(q^{100})$$ 4 * q - 18 * q^4 + 18 * q^6 - 36 * q^9 + 124 * q^11 - 42 * q^14 + 50 * q^16 + 244 * q^19 + 84 * q^21 - 306 * q^24 - 428 * q^26 - 704 * q^29 + 132 * q^31 + 372 * q^34 + 162 * q^36 + 36 * q^39 + 32 * q^41 - 312 * q^44 - 1920 * q^46 - 196 * q^49 + 240 * q^51 - 162 * q^54 + 714 * q^56 - 1032 * q^59 - 1760 * q^61 + 958 * q^64 + 312 * q^66 - 96 * q^69 + 620 * q^71 - 2716 * q^74 - 1344 * q^76 - 3664 * q^79 + 324 * q^81 - 378 * q^84 + 2912 * q^86 - 1592 * q^89 - 84 * q^91 - 5176 * q^94 - 1854 * q^96 - 1116 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 11\nu ) / 10$$ (v^3 + 11*v) / 10 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 11$$ v^2 + 11
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 11$$ b3 - 11 $$\nu^{3}$$ $$=$$ $$10\beta_{2} - 11\beta_1$$ 10*b2 - 11*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 3.70156i − 2.70156i 2.70156i 3.70156i
4.70156i 3.00000i −14.1047 0 14.1047 7.00000i 28.7016i −9.00000 0
274.2 1.70156i 3.00000i 5.10469 0 −5.10469 7.00000i 22.2984i −9.00000 0
274.3 1.70156i 3.00000i 5.10469 0 −5.10469 7.00000i 22.2984i −9.00000 0
274.4 4.70156i 3.00000i −14.1047 0 14.1047 7.00000i 28.7016i −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} + 25T_{2}^{2} + 64$$ T2^4 + 25*T2^2 + 64 $$T_{11}^{2} - 62T_{11} + 920$$ T11^2 - 62*T11 + 920

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 25T^{2} + 64$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 49)^{2}$$
$11$ $$(T^{2} - 62 T + 920)^{2}$$
$13$ $$T^{4} + 2068 T^{2} + 1032256$$
$17$ $$T^{4} + 3752 T^{2} + 1157776$$
$19$ $$(T^{2} - 122 T + 3680)^{2}$$
$23$ $$T^{4} + 47360 T^{2} + 554696704$$
$29$ $$(T^{2} + 352 T + 29500)^{2}$$
$31$ $$(T^{2} - 66 T - 13712)^{2}$$
$37$ $$T^{4} + \cdots + 3222151696$$
$41$ $$(T^{2} - 16 T - 119492)^{2}$$
$43$ $$T^{4} + \cdots + 4006383616$$
$47$ $$T^{4} + \cdots + 36888580096$$
$53$ $$T^{4} + \cdots + 42683560000$$
$59$ $$(T^{2} + 516 T + 7360)^{2}$$
$61$ $$(T^{2} + 880 T + 121276)^{2}$$
$67$ $$T^{4} + \cdots + 251153327104$$
$71$ $$(T^{2} - 310 T - 51784)^{2}$$
$73$ $$T^{4} + 82740 T^{2} + 138485824$$
$79$ $$(T^{2} + 1832 T + 838400)^{2}$$
$83$ $$T^{4} + \cdots + 158964879616$$
$89$ $$(T^{2} + 796 T - 998780)^{2}$$
$97$ $$T^{4} + \cdots + 462312964096$$