# Properties

 Label 525.4.d.g Level $525$ Weight $4$ Character orbit 525.d Analytic conductor $30.976$ Analytic rank $0$ Dimension $4$ CM no 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: $$\Q(i, \sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 29x^{2} + 196$$ x^4 + 29*x^2 + 196 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{2} - 3 \beta_{2} q^{3} + (3 \beta_{3} - 10) q^{4} + (3 \beta_{3} - 6) q^{6} + 7 \beta_{2} q^{7} + (41 \beta_{2} - 5 \beta_1) q^{8} - 9 q^{9}+O(q^{10})$$ q + (-b2 + b1) * q^2 - 3*b2 * q^3 + (3*b3 - 10) * q^4 + (3*b3 - 6) * q^6 + 7*b2 * q^7 + (41*b2 - 5*b1) * q^8 - 9 * q^9 $$q + ( - \beta_{2} + \beta_1) q^{2} - 3 \beta_{2} q^{3} + (3 \beta_{3} - 10) q^{4} + (3 \beta_{3} - 6) q^{6} + 7 \beta_{2} q^{7} + (41 \beta_{2} - 5 \beta_1) q^{8} - 9 q^{9} + ( - 10 \beta_{3} + 2) q^{11} + (21 \beta_{2} - 9 \beta_1) q^{12} + ( - 14 \beta_{2} - 12 \beta_1) q^{13} + ( - 7 \beta_{3} + 14) q^{14} + ( - 27 \beta_{3} + 82) q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + (9 \beta_{2} - 9 \beta_1) q^{18} + ( - 24 \beta_{3} - 20) q^{19} + 21 q^{21} + ( - 132 \beta_{2} + 12 \beta_1) q^{22} + ( - 20 \beta_{2} - 34 \beta_1) q^{23} + ( - 15 \beta_{3} + 138) q^{24} + ( - 10 \beta_{3} + 164) q^{26} + 27 \beta_{2} q^{27} + ( - 49 \beta_{2} + 21 \beta_1) q^{28} + (24 \beta_{3} + 114) q^{29} + ( - 72 \beta_{3} + 56) q^{31} + ( - 105 \beta_{2} + 69 \beta_1) q^{32} + (24 \beta_{2} + 30 \beta_1) q^{33} + (6 \beta_{3} - 36) q^{34} + ( - 27 \beta_{3} + 90) q^{36} + ( - 106 \beta_{2} + 36 \beta_1) q^{37} + ( - 292 \beta_{2} + 4 \beta_1) q^{38} + ( - 36 \beta_{3} - 6) q^{39} + (30 \beta_{3} - 240) q^{41} + ( - 21 \beta_{2} + 21 \beta_1) q^{42} + ( - 212 \beta_{2} - 48 \beta_1) q^{43} + (76 \beta_{3} - 440) q^{44} + ( - 48 \beta_{3} + 504) q^{46} + ( - 40 \beta_{2} - 68 \beta_1) q^{47} + ( - 165 \beta_{2} + 81 \beta_1) q^{48} - 49 q^{49} + (6 \beta_{3} - 12) q^{51} + ( - 406 \beta_{2} + 78 \beta_1) q^{52} + (554 \beta_{2} + 4 \beta_1) q^{53} + ( - 27 \beta_{3} + 54) q^{54} + (35 \beta_{3} - 322) q^{56} + (132 \beta_{2} + 72 \beta_1) q^{57} + (198 \beta_{2} + 90 \beta_1) q^{58} + ( - 116 \beta_{3} - 344) q^{59} + ( - 72 \beta_{3} - 178) q^{61} + ( - 992 \beta_{2} + 128 \beta_1) q^{62} - 63 \beta_{2} q^{63} + (27 \beta_{3} - 658) q^{64} + (36 \beta_{3} - 432) q^{66} + (20 \beta_{2} - 108 \beta_1) q^{67} + (98 \beta_{2} - 26 \beta_1) q^{68} + ( - 102 \beta_{3} + 42) q^{69} + (30 \beta_{3} + 462) q^{71} + ( - 369 \beta_{2} + 45 \beta_1) q^{72} + ( - 530 \beta_{2} + 12 \beta_1) q^{73} + (178 \beta_{3} - 788) q^{74} + (108 \beta_{3} - 808) q^{76} + ( - 56 \beta_{2} - 70 \beta_1) q^{77} + ( - 462 \beta_{2} + 30 \beta_1) q^{78} + ( - 108 \beta_{3} + 340) q^{79} + 81 q^{81} + (630 \beta_{2} - 270 \beta_1) q^{82} + ( - 924 \beta_{2} + 96 \beta_1) q^{83} + (63 \beta_{3} - 210) q^{84} + (116 \beta_{3} + 344) q^{86} + ( - 414 \beta_{2} - 72 \beta_1) q^{87} + (372 \beta_{2} - 420 \beta_1) q^{88} + ( - 142 \beta_{3} - 112) q^{89} + (84 \beta_{3} + 14) q^{91} + ( - 1288 \beta_{2} + 280 \beta_1) q^{92} + (48 \beta_{2} + 216 \beta_1) q^{93} + ( - 96 \beta_{3} + 1008) q^{94} + (207 \beta_{3} - 522) q^{96} + (266 \beta_{2} - 276 \beta_1) q^{97} + (49 \beta_{2} - 49 \beta_1) q^{98} + (90 \beta_{3} - 18) q^{99}+O(q^{100})$$ q + (-b2 + b1) * q^2 - 3*b2 * q^3 + (3*b3 - 10) * q^4 + (3*b3 - 6) * q^6 + 7*b2 * q^7 + (41*b2 - 5*b1) * q^8 - 9 * q^9 + (-10*b3 + 2) * q^11 + (21*b2 - 9*b1) * q^12 + (-14*b2 - 12*b1) * q^13 + (-7*b3 + 14) * q^14 + (-27*b3 + 82) * q^16 + (-2*b2 + 2*b1) * q^17 + (9*b2 - 9*b1) * q^18 + (-24*b3 - 20) * q^19 + 21 * q^21 + (-132*b2 + 12*b1) * q^22 + (-20*b2 - 34*b1) * q^23 + (-15*b3 + 138) * q^24 + (-10*b3 + 164) * q^26 + 27*b2 * q^27 + (-49*b2 + 21*b1) * q^28 + (24*b3 + 114) * q^29 + (-72*b3 + 56) * q^31 + (-105*b2 + 69*b1) * q^32 + (24*b2 + 30*b1) * q^33 + (6*b3 - 36) * q^34 + (-27*b3 + 90) * q^36 + (-106*b2 + 36*b1) * q^37 + (-292*b2 + 4*b1) * q^38 + (-36*b3 - 6) * q^39 + (30*b3 - 240) * q^41 + (-21*b2 + 21*b1) * q^42 + (-212*b2 - 48*b1) * q^43 + (76*b3 - 440) * q^44 + (-48*b3 + 504) * q^46 + (-40*b2 - 68*b1) * q^47 + (-165*b2 + 81*b1) * q^48 - 49 * q^49 + (6*b3 - 12) * q^51 + (-406*b2 + 78*b1) * q^52 + (554*b2 + 4*b1) * q^53 + (-27*b3 + 54) * q^54 + (35*b3 - 322) * q^56 + (132*b2 + 72*b1) * q^57 + (198*b2 + 90*b1) * q^58 + (-116*b3 - 344) * q^59 + (-72*b3 - 178) * q^61 + (-992*b2 + 128*b1) * q^62 - 63*b2 * q^63 + (27*b3 - 658) * q^64 + (36*b3 - 432) * q^66 + (20*b2 - 108*b1) * q^67 + (98*b2 - 26*b1) * q^68 + (-102*b3 + 42) * q^69 + (30*b3 + 462) * q^71 + (-369*b2 + 45*b1) * q^72 + (-530*b2 + 12*b1) * q^73 + (178*b3 - 788) * q^74 + (108*b3 - 808) * q^76 + (-56*b2 - 70*b1) * q^77 + (-462*b2 + 30*b1) * q^78 + (-108*b3 + 340) * q^79 + 81 * q^81 + (630*b2 - 270*b1) * q^82 + (-924*b2 + 96*b1) * q^83 + (63*b3 - 210) * q^84 + (116*b3 + 344) * q^86 + (-414*b2 - 72*b1) * q^87 + (372*b2 - 420*b1) * q^88 + (-142*b3 - 112) * q^89 + (84*b3 + 14) * q^91 + (-1288*b2 + 280*b1) * q^92 + (48*b2 + 216*b1) * q^93 + (-96*b3 + 1008) * q^94 + (207*b3 - 522) * q^96 + (266*b2 - 276*b1) * q^97 + (49*b2 - 49*b1) * q^98 + (90*b3 - 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 34 q^{4} - 18 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 34 * q^4 - 18 * q^6 - 36 * q^9 $$4 q - 34 q^{4} - 18 q^{6} - 36 q^{9} - 12 q^{11} + 42 q^{14} + 274 q^{16} - 128 q^{19} + 84 q^{21} + 522 q^{24} + 636 q^{26} + 504 q^{29} + 80 q^{31} - 132 q^{34} + 306 q^{36} - 96 q^{39} - 900 q^{41} - 1608 q^{44} + 1920 q^{46} - 196 q^{49} - 36 q^{51} + 162 q^{54} - 1218 q^{56} - 1608 q^{59} - 856 q^{61} - 2578 q^{64} - 1656 q^{66} - 36 q^{69} + 1908 q^{71} - 2796 q^{74} - 3016 q^{76} + 1144 q^{79} + 324 q^{81} - 714 q^{84} + 1608 q^{86} - 732 q^{89} + 224 q^{91} + 3840 q^{94} - 1674 q^{96} + 108 q^{99}+O(q^{100})$$ 4 * q - 34 * q^4 - 18 * q^6 - 36 * q^9 - 12 * q^11 + 42 * q^14 + 274 * q^16 - 128 * q^19 + 84 * q^21 + 522 * q^24 + 636 * q^26 + 504 * q^29 + 80 * q^31 - 132 * q^34 + 306 * q^36 - 96 * q^39 - 900 * q^41 - 1608 * q^44 + 1920 * q^46 - 196 * q^49 - 36 * q^51 + 162 * q^54 - 1218 * q^56 - 1608 * q^59 - 856 * q^61 - 2578 * q^64 - 1656 * q^66 - 36 * q^69 + 1908 * q^71 - 2796 * q^74 - 3016 * q^76 + 1144 * q^79 + 324 * q^81 - 714 * q^84 + 1608 * q^86 - 732 * q^89 + 224 * q^91 + 3840 * q^94 - 1674 * q^96 + 108 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 29x^{2} + 196$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 15\nu ) / 14$$ (v^3 + 15*v) / 14 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 15$$ v^2 + 15
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 15$$ b3 - 15 $$\nu^{3}$$ $$=$$ $$14\beta_{2} - 15\beta_1$$ 14*b2 - 15*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
 − 4.27492i − 3.27492i 3.27492i 4.27492i
5.27492i 3.00000i −19.8248 0 −15.8248 7.00000i 62.3746i −9.00000 0
274.2 2.27492i 3.00000i 2.82475 0 6.82475 7.00000i 24.6254i −9.00000 0
274.3 2.27492i 3.00000i 2.82475 0 6.82475 7.00000i 24.6254i −9.00000 0
274.4 5.27492i 3.00000i −19.8248 0 −15.8248 7.00000i 62.3746i −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.g 4
5.b even 2 1 inner 525.4.d.g 4
5.c odd 4 1 21.4.a.c 2
5.c odd 4 1 525.4.a.n 2
15.e even 4 1 63.4.a.e 2
15.e even 4 1 1575.4.a.p 2
20.e even 4 1 336.4.a.m 2
35.f even 4 1 147.4.a.i 2
35.k even 12 2 147.4.e.m 4
35.l odd 12 2 147.4.e.l 4
40.i odd 4 1 1344.4.a.bg 2
40.k even 4 1 1344.4.a.bo 2
60.l odd 4 1 1008.4.a.ba 2
105.k odd 4 1 441.4.a.r 2
105.w odd 12 2 441.4.e.p 4
105.x even 12 2 441.4.e.q 4
140.j odd 4 1 2352.4.a.bz 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 5.c odd 4 1
63.4.a.e 2 15.e even 4 1
147.4.a.i 2 35.f even 4 1
147.4.e.l 4 35.l odd 12 2
147.4.e.m 4 35.k even 12 2
336.4.a.m 2 20.e even 4 1
441.4.a.r 2 105.k odd 4 1
441.4.e.p 4 105.w odd 12 2
441.4.e.q 4 105.x even 12 2
525.4.a.n 2 5.c odd 4 1
525.4.d.g 4 1.a even 1 1 trivial
525.4.d.g 4 5.b even 2 1 inner
1008.4.a.ba 2 60.l odd 4 1
1344.4.a.bg 2 40.i odd 4 1
1344.4.a.bo 2 40.k even 4 1
1575.4.a.p 2 15.e even 4 1
2352.4.a.bz 2 140.j odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 33T_{2}^{2} + 144$$ T2^4 + 33*T2^2 + 144 $$T_{11}^{2} + 6T_{11} - 1416$$ T11^2 + 6*T11 - 1416

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 33T^{2} + 144$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 49)^{2}$$
$11$ $$(T^{2} + 6 T - 1416)^{2}$$
$13$ $$T^{4} + 4232 T^{2} + \cdots + 3952144$$
$17$ $$T^{4} + 132T^{2} + 2304$$
$19$ $$(T^{2} + 64 T - 7184)^{2}$$
$23$ $$T^{4} + 32964 T^{2} + \cdots + 271063296$$
$29$ $$(T^{2} - 252 T + 7668)^{2}$$
$31$ $$(T^{2} - 40 T - 73472)^{2}$$
$37$ $$T^{4} + 67688 T^{2} + \cdots + 9560464$$
$41$ $$(T^{2} + 450 T + 37800)^{2}$$
$43$ $$T^{4} + 136352 T^{2} + \cdots + 6310144$$
$47$ $$T^{4} + 131856 T^{2} + \cdots + 4337012736$$
$53$ $$T^{4} + 609864 T^{2} + \cdots + 92705634576$$
$59$ $$(T^{2} + 804 T - 30144)^{2}$$
$61$ $$(T^{2} + 428 T - 28076)^{2}$$
$67$ $$T^{4} + 343376 T^{2} + \cdots + 25836061696$$
$71$ $$(T^{2} - 954 T + 214704)^{2}$$
$73$ $$T^{4} + 578696 T^{2} + \cdots + 81364139536$$
$79$ $$(T^{2} - 572 T - 84416)^{2}$$
$83$ $$T^{4} + 2152224 T^{2} + \cdots + 661710663936$$
$89$ $$(T^{2} + 366 T - 253848)^{2}$$
$97$ $$T^{4} + 2497448 T^{2} + \cdots + 850622533264$$