# Properties

 Label 525.6.d.b 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: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 6 i q^{2} - 9 i q^{3} - 4 q^{4} + 54 q^{6} - 49 i q^{7} + 168 i q^{8} - 81 q^{9} +O(q^{10})$$ q + 6*i * q^2 - 9*i * q^3 - 4 * q^4 + 54 * q^6 - 49*i * q^7 + 168*i * q^8 - 81 * q^9 $$q + 6 i q^{2} - 9 i q^{3} - 4 q^{4} + 54 q^{6} - 49 i q^{7} + 168 i q^{8} - 81 q^{9} + 444 q^{11} + 36 i q^{12} - 442 i q^{13} + 294 q^{14} - 1136 q^{16} + 126 i q^{17} - 486 i q^{18} - 2684 q^{19} - 441 q^{21} + 2664 i q^{22} + 4200 i q^{23} + 1512 q^{24} + 2652 q^{26} + 729 i q^{27} + 196 i q^{28} + 5442 q^{29} + 80 q^{31} - 1440 i q^{32} - 3996 i q^{33} - 756 q^{34} + 324 q^{36} + 5434 i q^{37} - 16104 i q^{38} - 3978 q^{39} + 7962 q^{41} - 2646 i q^{42} - 11524 i q^{43} - 1776 q^{44} - 25200 q^{46} + 13920 i q^{47} + 10224 i q^{48} - 2401 q^{49} + 1134 q^{51} + 1768 i q^{52} - 9594 i q^{53} - 4374 q^{54} + 8232 q^{56} + 24156 i q^{57} + 32652 i q^{58} - 27492 q^{59} + 49478 q^{61} + 480 i q^{62} + 3969 i q^{63} - 27712 q^{64} + 23976 q^{66} + 59356 i q^{67} - 504 i q^{68} + 37800 q^{69} + 32040 q^{71} - 13608 i q^{72} - 61846 i q^{73} - 32604 q^{74} + 10736 q^{76} - 21756 i q^{77} - 23868 i q^{78} + 65776 q^{79} + 6561 q^{81} + 47772 i q^{82} + 40188 i q^{83} + 1764 q^{84} + 69144 q^{86} - 48978 i q^{87} + 74592 i q^{88} + 7974 q^{89} - 21658 q^{91} - 16800 i q^{92} - 720 i q^{93} - 83520 q^{94} - 12960 q^{96} + 143662 i q^{97} - 14406 i q^{98} - 35964 q^{99} +O(q^{100})$$ q + 6*i * q^2 - 9*i * q^3 - 4 * q^4 + 54 * q^6 - 49*i * q^7 + 168*i * q^8 - 81 * q^9 + 444 * q^11 + 36*i * q^12 - 442*i * q^13 + 294 * q^14 - 1136 * q^16 + 126*i * q^17 - 486*i * q^18 - 2684 * q^19 - 441 * q^21 + 2664*i * q^22 + 4200*i * q^23 + 1512 * q^24 + 2652 * q^26 + 729*i * q^27 + 196*i * q^28 + 5442 * q^29 + 80 * q^31 - 1440*i * q^32 - 3996*i * q^33 - 756 * q^34 + 324 * q^36 + 5434*i * q^37 - 16104*i * q^38 - 3978 * q^39 + 7962 * q^41 - 2646*i * q^42 - 11524*i * q^43 - 1776 * q^44 - 25200 * q^46 + 13920*i * q^47 + 10224*i * q^48 - 2401 * q^49 + 1134 * q^51 + 1768*i * q^52 - 9594*i * q^53 - 4374 * q^54 + 8232 * q^56 + 24156*i * q^57 + 32652*i * q^58 - 27492 * q^59 + 49478 * q^61 + 480*i * q^62 + 3969*i * q^63 - 27712 * q^64 + 23976 * q^66 + 59356*i * q^67 - 504*i * q^68 + 37800 * q^69 + 32040 * q^71 - 13608*i * q^72 - 61846*i * q^73 - 32604 * q^74 + 10736 * q^76 - 21756*i * q^77 - 23868*i * q^78 + 65776 * q^79 + 6561 * q^81 + 47772*i * q^82 + 40188*i * q^83 + 1764 * q^84 + 69144 * q^86 - 48978*i * q^87 + 74592*i * q^88 + 7974 * q^89 - 21658 * q^91 - 16800*i * q^92 - 720*i * q^93 - 83520 * q^94 - 12960 * q^96 + 143662*i * q^97 - 14406*i * q^98 - 35964 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} + 108 q^{6} - 162 q^{9}+O(q^{10})$$ 2 * q - 8 * q^4 + 108 * q^6 - 162 * q^9 $$2 q - 8 q^{4} + 108 q^{6} - 162 q^{9} + 888 q^{11} + 588 q^{14} - 2272 q^{16} - 5368 q^{19} - 882 q^{21} + 3024 q^{24} + 5304 q^{26} + 10884 q^{29} + 160 q^{31} - 1512 q^{34} + 648 q^{36} - 7956 q^{39} + 15924 q^{41} - 3552 q^{44} - 50400 q^{46} - 4802 q^{49} + 2268 q^{51} - 8748 q^{54} + 16464 q^{56} - 54984 q^{59} + 98956 q^{61} - 55424 q^{64} + 47952 q^{66} + 75600 q^{69} + 64080 q^{71} - 65208 q^{74} + 21472 q^{76} + 131552 q^{79} + 13122 q^{81} + 3528 q^{84} + 138288 q^{86} + 15948 q^{89} - 43316 q^{91} - 167040 q^{94} - 25920 q^{96} - 71928 q^{99}+O(q^{100})$$ 2 * q - 8 * q^4 + 108 * q^6 - 162 * q^9 + 888 * q^11 + 588 * q^14 - 2272 * q^16 - 5368 * q^19 - 882 * q^21 + 3024 * q^24 + 5304 * q^26 + 10884 * q^29 + 160 * q^31 - 1512 * q^34 + 648 * q^36 - 7956 * q^39 + 15924 * q^41 - 3552 * q^44 - 50400 * q^46 - 4802 * q^49 + 2268 * q^51 - 8748 * q^54 + 16464 * q^56 - 54984 * q^59 + 98956 * q^61 - 55424 * q^64 + 47952 * q^66 + 75600 * q^69 + 64080 * q^71 - 65208 * q^74 + 21472 * q^76 + 131552 * q^79 + 13122 * q^81 + 3528 * q^84 + 138288 * q^86 + 15948 * q^89 - 43316 * q^91 - 167040 * q^94 - 25920 * q^96 - 71928 * 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
6.00000i 9.00000i −4.00000 0 54.0000 49.0000i 168.000i −81.0000 0
274.2 6.00000i 9.00000i −4.00000 0 54.0000 49.0000i 168.000i −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.b 2
5.b even 2 1 inner 525.6.d.b 2
5.c odd 4 1 21.6.a.a 1
5.c odd 4 1 525.6.a.d 1
15.e even 4 1 63.6.a.d 1
20.e even 4 1 336.6.a.r 1
35.f even 4 1 147.6.a.b 1
35.k even 12 2 147.6.e.i 2
35.l odd 12 2 147.6.e.j 2
60.l odd 4 1 1008.6.a.c 1
105.k odd 4 1 441.6.a.j 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 36$$ acting on $$S_{6}^{\mathrm{new}}(525, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 36$$
$3$ $$T^{2} + 81$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T - 444)^{2}$$
$13$ $$T^{2} + 195364$$
$17$ $$T^{2} + 15876$$
$19$ $$(T + 2684)^{2}$$
$23$ $$T^{2} + 17640000$$
$29$ $$(T - 5442)^{2}$$
$31$ $$(T - 80)^{2}$$
$37$ $$T^{2} + 29528356$$
$41$ $$(T - 7962)^{2}$$
$43$ $$T^{2} + 132802576$$
$47$ $$T^{2} + 193766400$$
$53$ $$T^{2} + 92044836$$
$59$ $$(T + 27492)^{2}$$
$61$ $$(T - 49478)^{2}$$
$67$ $$T^{2} + 3523134736$$
$71$ $$(T - 32040)^{2}$$
$73$ $$T^{2} + 3824927716$$
$79$ $$(T - 65776)^{2}$$
$83$ $$T^{2} + 1615075344$$
$89$ $$(T - 7974)^{2}$$
$97$ $$T^{2} + 20638770244$$