# Properties

 Label 525.6.d.c Level $525$ Weight $6$ Character orbit 525.d Analytic conductor $84.202$ 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] = [525,6,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, 6, names="a")

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$84.2015054018$$ 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 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 i q^{2} - 9 i q^{3} + 7 q^{4} + 45 q^{6} - 49 i q^{7} + 195 i q^{8} - 81 q^{9} +O(q^{10})$$ q + 5*i * q^2 - 9*i * q^3 + 7 * q^4 + 45 * q^6 - 49*i * q^7 + 195*i * q^8 - 81 * q^9 $$q + 5 i q^{2} - 9 i q^{3} + 7 q^{4} + 45 q^{6} - 49 i q^{7} + 195 i q^{8} - 81 q^{9} + 52 q^{11} - 63 i q^{12} + 770 i q^{13} + 245 q^{14} - 751 q^{16} - 2022 i q^{17} - 405 i q^{18} - 1732 q^{19} - 441 q^{21} + 260 i q^{22} + 576 i q^{23} + 1755 q^{24} - 3850 q^{26} + 729 i q^{27} - 343 i q^{28} - 5518 q^{29} + 6336 q^{31} + 2485 i q^{32} - 468 i q^{33} + 10110 q^{34} - 567 q^{36} - 7338 i q^{37} - 8660 i q^{38} + 6930 q^{39} - 3262 q^{41} - 2205 i q^{42} - 5420 i q^{43} + 364 q^{44} - 2880 q^{46} + 864 i q^{47} + 6759 i q^{48} - 2401 q^{49} - 18198 q^{51} + 5390 i q^{52} - 4182 i q^{53} - 3645 q^{54} + 9555 q^{56} + 15588 i q^{57} - 27590 i q^{58} + 11220 q^{59} - 45602 q^{61} + 31680 i q^{62} + 3969 i q^{63} - 36457 q^{64} + 2340 q^{66} + 1396 i q^{67} - 14154 i q^{68} + 5184 q^{69} + 18720 q^{71} - 15795 i q^{72} - 46362 i q^{73} + 36690 q^{74} - 12124 q^{76} - 2548 i q^{77} + 34650 i q^{78} - 97424 q^{79} + 6561 q^{81} - 16310 i q^{82} + 81228 i q^{83} - 3087 q^{84} + 27100 q^{86} + 49662 i q^{87} + 10140 i q^{88} + 3182 q^{89} + 37730 q^{91} + 4032 i q^{92} - 57024 i q^{93} - 4320 q^{94} + 22365 q^{96} + 4914 i q^{97} - 12005 i q^{98} - 4212 q^{99} +O(q^{100})$$ q + 5*i * q^2 - 9*i * q^3 + 7 * q^4 + 45 * q^6 - 49*i * q^7 + 195*i * q^8 - 81 * q^9 + 52 * q^11 - 63*i * q^12 + 770*i * q^13 + 245 * q^14 - 751 * q^16 - 2022*i * q^17 - 405*i * q^18 - 1732 * q^19 - 441 * q^21 + 260*i * q^22 + 576*i * q^23 + 1755 * q^24 - 3850 * q^26 + 729*i * q^27 - 343*i * q^28 - 5518 * q^29 + 6336 * q^31 + 2485*i * q^32 - 468*i * q^33 + 10110 * q^34 - 567 * q^36 - 7338*i * q^37 - 8660*i * q^38 + 6930 * q^39 - 3262 * q^41 - 2205*i * q^42 - 5420*i * q^43 + 364 * q^44 - 2880 * q^46 + 864*i * q^47 + 6759*i * q^48 - 2401 * q^49 - 18198 * q^51 + 5390*i * q^52 - 4182*i * q^53 - 3645 * q^54 + 9555 * q^56 + 15588*i * q^57 - 27590*i * q^58 + 11220 * q^59 - 45602 * q^61 + 31680*i * q^62 + 3969*i * q^63 - 36457 * q^64 + 2340 * q^66 + 1396*i * q^67 - 14154*i * q^68 + 5184 * q^69 + 18720 * q^71 - 15795*i * q^72 - 46362*i * q^73 + 36690 * q^74 - 12124 * q^76 - 2548*i * q^77 + 34650*i * q^78 - 97424 * q^79 + 6561 * q^81 - 16310*i * q^82 + 81228*i * q^83 - 3087 * q^84 + 27100 * q^86 + 49662*i * q^87 + 10140*i * q^88 + 3182 * q^89 + 37730 * q^91 + 4032*i * q^92 - 57024*i * q^93 - 4320 * q^94 + 22365 * q^96 + 4914*i * q^97 - 12005*i * q^98 - 4212 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4} + 90 q^{6} - 162 q^{9}+O(q^{10})$$ 2 * q + 14 * q^4 + 90 * q^6 - 162 * q^9 $$2 q + 14 q^{4} + 90 q^{6} - 162 q^{9} + 104 q^{11} + 490 q^{14} - 1502 q^{16} - 3464 q^{19} - 882 q^{21} + 3510 q^{24} - 7700 q^{26} - 11036 q^{29} + 12672 q^{31} + 20220 q^{34} - 1134 q^{36} + 13860 q^{39} - 6524 q^{41} + 728 q^{44} - 5760 q^{46} - 4802 q^{49} - 36396 q^{51} - 7290 q^{54} + 19110 q^{56} + 22440 q^{59} - 91204 q^{61} - 72914 q^{64} + 4680 q^{66} + 10368 q^{69} + 37440 q^{71} + 73380 q^{74} - 24248 q^{76} - 194848 q^{79} + 13122 q^{81} - 6174 q^{84} + 54200 q^{86} + 6364 q^{89} + 75460 q^{91} - 8640 q^{94} + 44730 q^{96} - 8424 q^{99}+O(q^{100})$$ 2 * q + 14 * q^4 + 90 * q^6 - 162 * q^9 + 104 * q^11 + 490 * q^14 - 1502 * q^16 - 3464 * q^19 - 882 * q^21 + 3510 * q^24 - 7700 * q^26 - 11036 * q^29 + 12672 * q^31 + 20220 * q^34 - 1134 * q^36 + 13860 * q^39 - 6524 * q^41 + 728 * q^44 - 5760 * q^46 - 4802 * q^49 - 36396 * q^51 - 7290 * q^54 + 19110 * q^56 + 22440 * q^59 - 91204 * q^61 - 72914 * q^64 + 4680 * q^66 + 10368 * q^69 + 37440 * q^71 + 73380 * q^74 - 24248 * q^76 - 194848 * q^79 + 13122 * q^81 - 6174 * q^84 + 54200 * q^86 + 6364 * q^89 + 75460 * q^91 - 8640 * q^94 + 44730 * q^96 - 8424 * 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 9.00000i 7.00000 0 45.0000 49.0000i 195.000i −81.0000 0
274.2 5.00000i 9.00000i 7.00000 0 45.0000 49.0000i 195.000i −81.0000 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 525.6.d.c 2
5.b even 2 1 inner 525.6.d.c 2
5.c odd 4 1 21.6.a.c 1
5.c odd 4 1 525.6.a.b 1
15.e even 4 1 63.6.a.b 1
20.e even 4 1 336.6.a.i 1
35.f even 4 1 147.6.a.f 1
35.k even 12 2 147.6.e.d 2
35.l odd 12 2 147.6.e.c 2
60.l odd 4 1 1008.6.a.a 1
105.k odd 4 1 441.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 5.c odd 4 1
63.6.a.b 1 15.e even 4 1
147.6.a.f 1 35.f even 4 1
147.6.e.c 2 35.l odd 12 2
147.6.e.d 2 35.k even 12 2
336.6.a.i 1 20.e even 4 1
441.6.a.c 1 105.k odd 4 1
525.6.a.b 1 5.c odd 4 1
525.6.d.c 2 1.a even 1 1 trivial
525.6.d.c 2 5.b even 2 1 inner
1008.6.a.a 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 25$$ acting on $$S_{6}^{\mathrm{new}}(525, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 25$$
$3$ $$T^{2} + 81$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T - 52)^{2}$$
$13$ $$T^{2} + 592900$$
$17$ $$T^{2} + 4088484$$
$19$ $$(T + 1732)^{2}$$
$23$ $$T^{2} + 331776$$
$29$ $$(T + 5518)^{2}$$
$31$ $$(T - 6336)^{2}$$
$37$ $$T^{2} + 53846244$$
$41$ $$(T + 3262)^{2}$$
$43$ $$T^{2} + 29376400$$
$47$ $$T^{2} + 746496$$
$53$ $$T^{2} + 17489124$$
$59$ $$(T - 11220)^{2}$$
$61$ $$(T + 45602)^{2}$$
$67$ $$T^{2} + 1948816$$
$71$ $$(T - 18720)^{2}$$
$73$ $$T^{2} + 2149435044$$
$79$ $$(T + 97424)^{2}$$
$83$ $$T^{2} + 6597987984$$
$89$ $$(T - 3182)^{2}$$
$97$ $$T^{2} + 24147396$$