# Properties

 Label 525.4.d.a Level $525$ Weight $4$ Character orbit 525.d Analytic conductor $30.976$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 i q^{2} + 3 i q^{3} - 17 q^{4} - 15 q^{6} + 7 i q^{7} - 45 i q^{8} - 9 q^{9} +O(q^{10})$$ q + 5*i * q^2 + 3*i * q^3 - 17 * q^4 - 15 * q^6 + 7*i * q^7 - 45*i * q^8 - 9 * q^9 $$q + 5 i q^{2} + 3 i q^{3} - 17 q^{4} - 15 q^{6} + 7 i q^{7} - 45 i q^{8} - 9 q^{9} + 12 q^{11} - 51 i q^{12} - 30 i q^{13} - 35 q^{14} + 89 q^{16} - 134 i q^{17} - 45 i q^{18} + 92 q^{19} - 21 q^{21} + 60 i q^{22} - 112 i q^{23} + 135 q^{24} + 150 q^{26} - 27 i q^{27} - 119 i q^{28} + 58 q^{29} - 224 q^{31} + 85 i q^{32} + 36 i q^{33} + 670 q^{34} + 153 q^{36} - 146 i q^{37} + 460 i q^{38} + 90 q^{39} + 18 q^{41} - 105 i q^{42} - 340 i q^{43} - 204 q^{44} + 560 q^{46} + 208 i q^{47} + 267 i q^{48} - 49 q^{49} + 402 q^{51} + 510 i q^{52} + 754 i q^{53} + 135 q^{54} + 315 q^{56} + 276 i q^{57} + 290 i q^{58} - 380 q^{59} + 718 q^{61} - 1120 i q^{62} - 63 i q^{63} + 287 q^{64} - 180 q^{66} + 412 i q^{67} + 2278 i q^{68} + 336 q^{69} - 960 q^{71} + 405 i q^{72} - 1066 i q^{73} + 730 q^{74} - 1564 q^{76} + 84 i q^{77} + 450 i q^{78} - 896 q^{79} + 81 q^{81} + 90 i q^{82} - 436 i q^{83} + 357 q^{84} + 1700 q^{86} + 174 i q^{87} - 540 i q^{88} + 1038 q^{89} + 210 q^{91} + 1904 i q^{92} - 672 i q^{93} - 1040 q^{94} - 255 q^{96} - 702 i q^{97} - 245 i q^{98} - 108 q^{99} +O(q^{100})$$ q + 5*i * q^2 + 3*i * q^3 - 17 * q^4 - 15 * q^6 + 7*i * q^7 - 45*i * q^8 - 9 * q^9 + 12 * q^11 - 51*i * q^12 - 30*i * q^13 - 35 * q^14 + 89 * q^16 - 134*i * q^17 - 45*i * q^18 + 92 * q^19 - 21 * q^21 + 60*i * q^22 - 112*i * q^23 + 135 * q^24 + 150 * q^26 - 27*i * q^27 - 119*i * q^28 + 58 * q^29 - 224 * q^31 + 85*i * q^32 + 36*i * q^33 + 670 * q^34 + 153 * q^36 - 146*i * q^37 + 460*i * q^38 + 90 * q^39 + 18 * q^41 - 105*i * q^42 - 340*i * q^43 - 204 * q^44 + 560 * q^46 + 208*i * q^47 + 267*i * q^48 - 49 * q^49 + 402 * q^51 + 510*i * q^52 + 754*i * q^53 + 135 * q^54 + 315 * q^56 + 276*i * q^57 + 290*i * q^58 - 380 * q^59 + 718 * q^61 - 1120*i * q^62 - 63*i * q^63 + 287 * q^64 - 180 * q^66 + 412*i * q^67 + 2278*i * q^68 + 336 * q^69 - 960 * q^71 + 405*i * q^72 - 1066*i * q^73 + 730 * q^74 - 1564 * q^76 + 84*i * q^77 + 450*i * q^78 - 896 * q^79 + 81 * q^81 + 90*i * q^82 - 436*i * q^83 + 357 * q^84 + 1700 * q^86 + 174*i * q^87 - 540*i * q^88 + 1038 * q^89 + 210 * q^91 + 1904*i * q^92 - 672*i * q^93 - 1040 * q^94 - 255 * q^96 - 702*i * q^97 - 245*i * q^98 - 108 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 34 q^{4} - 30 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q - 34 * q^4 - 30 * q^6 - 18 * q^9 $$2 q - 34 q^{4} - 30 q^{6} - 18 q^{9} + 24 q^{11} - 70 q^{14} + 178 q^{16} + 184 q^{19} - 42 q^{21} + 270 q^{24} + 300 q^{26} + 116 q^{29} - 448 q^{31} + 1340 q^{34} + 306 q^{36} + 180 q^{39} + 36 q^{41} - 408 q^{44} + 1120 q^{46} - 98 q^{49} + 804 q^{51} + 270 q^{54} + 630 q^{56} - 760 q^{59} + 1436 q^{61} + 574 q^{64} - 360 q^{66} + 672 q^{69} - 1920 q^{71} + 1460 q^{74} - 3128 q^{76} - 1792 q^{79} + 162 q^{81} + 714 q^{84} + 3400 q^{86} + 2076 q^{89} + 420 q^{91} - 2080 q^{94} - 510 q^{96} - 216 q^{99}+O(q^{100})$$ 2 * q - 34 * q^4 - 30 * q^6 - 18 * q^9 + 24 * q^11 - 70 * q^14 + 178 * q^16 + 184 * q^19 - 42 * q^21 + 270 * q^24 + 300 * q^26 + 116 * q^29 - 448 * q^31 + 1340 * q^34 + 306 * q^36 + 180 * q^39 + 36 * q^41 - 408 * q^44 + 1120 * q^46 - 98 * q^49 + 804 * q^51 + 270 * q^54 + 630 * q^56 - 760 * q^59 + 1436 * q^61 + 574 * q^64 - 360 * q^66 + 672 * q^69 - 1920 * q^71 + 1460 * q^74 - 3128 * q^76 - 1792 * q^79 + 162 * q^81 + 714 * q^84 + 3400 * q^86 + 2076 * q^89 + 420 * q^91 - 2080 * q^94 - 510 * q^96 - 216 * q^99

## 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
 − 1.00000i 1.00000i
5.00000i 3.00000i −17.0000 0 −15.0000 7.00000i 45.0000i −9.00000 0
274.2 5.00000i 3.00000i −17.0000 0 −15.0000 7.00000i 45.0000i −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.a 2
5.b even 2 1 inner 525.4.d.a 2
5.c odd 4 1 105.4.a.b 1
5.c odd 4 1 525.4.a.a 1
15.e even 4 1 315.4.a.a 1
15.e even 4 1 1575.4.a.l 1
20.e even 4 1 1680.4.a.u 1
35.f even 4 1 735.4.a.j 1
105.k odd 4 1 2205.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.b 1 5.c odd 4 1
315.4.a.a 1 15.e even 4 1
525.4.a.a 1 5.c odd 4 1
525.4.d.a 2 1.a even 1 1 trivial
525.4.d.a 2 5.b even 2 1 inner
735.4.a.j 1 35.f even 4 1
1575.4.a.l 1 15.e even 4 1
1680.4.a.u 1 20.e even 4 1
2205.4.a.b 1 105.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 $$S_{4}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{2} + 25$$ T2^2 + 25 $$T_{11} - 12$$ T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 25$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 49$$
$11$ $$(T - 12)^{2}$$
$13$ $$T^{2} + 900$$
$17$ $$T^{2} + 17956$$
$19$ $$(T - 92)^{2}$$
$23$ $$T^{2} + 12544$$
$29$ $$(T - 58)^{2}$$
$31$ $$(T + 224)^{2}$$
$37$ $$T^{2} + 21316$$
$41$ $$(T - 18)^{2}$$
$43$ $$T^{2} + 115600$$
$47$ $$T^{2} + 43264$$
$53$ $$T^{2} + 568516$$
$59$ $$(T + 380)^{2}$$
$61$ $$(T - 718)^{2}$$
$67$ $$T^{2} + 169744$$
$71$ $$(T + 960)^{2}$$
$73$ $$T^{2} + 1136356$$
$79$ $$(T + 896)^{2}$$
$83$ $$T^{2} + 190096$$
$89$ $$(T - 1038)^{2}$$
$97$ $$T^{2} + 492804$$