# Properties

 Label 525.2.bf.f Level 525 Weight 2 Character orbit 525.bf Analytic conductor 4.192 Analytic rank 0 Dimension 48 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.bf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 2q^{3} - 24q^{6} + 12q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 2q^{3} - 24q^{6} + 12q^{7} + 10q^{12} + 16q^{13} - 8q^{16} - 14q^{18} - 28q^{21} + 8q^{22} - 40q^{27} + 60q^{28} - 24q^{31} + 4q^{33} + 8q^{36} - 4q^{37} - 14q^{42} - 16q^{43} - 32q^{46} - 44q^{48} + 8q^{51} - 36q^{52} + 88q^{57} - 56q^{58} - 8q^{61} - 44q^{63} + 76q^{66} - 12q^{67} + 34q^{72} - 52q^{73} + 64q^{76} + 120q^{78} + 20q^{81} - 104q^{82} + 46q^{87} + 72q^{91} + 44q^{93} + 12q^{96} + 120q^{97} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1 −2.35640 0.631395i 1.54878 0.775426i 3.42191 + 1.97564i 0 −4.13914 + 0.849321i 1.91891 1.82148i −3.36596 3.36596i 1.79743 2.40193i 0
32.2 −2.17249 0.582118i 0.245221 + 1.71460i 2.64881 + 1.52929i 0 0.465359 3.86771i 2.38227 + 1.15099i −1.68355 1.68355i −2.87973 + 0.840915i 0
32.3 −1.46015 0.391246i 0.852980 1.50746i 0.246919 + 0.142558i 0 −1.83527 + 1.86739i −2.36949 + 1.17707i 1.83305 + 1.83305i −1.54485 2.57166i 0
32.4 −1.26950 0.340162i −1.71749 + 0.224146i −0.236127 0.136328i 0 2.25660 + 0.299670i −1.25943 2.32676i 2.11207 + 2.11207i 2.89952 0.769934i 0
32.5 −0.907300 0.243110i −1.12459 1.31730i −0.967960 0.558852i 0 0.700094 + 1.46859i 2.64571 0.0144144i 2.07075 + 2.07075i −0.470578 + 2.96286i 0
32.6 −0.298314 0.0799329i 1.64547 + 0.540759i −1.64945 0.952310i 0 −0.447643 0.292843i −0.951942 + 2.46856i 0.852694 + 0.852694i 2.41516 + 1.77961i 0
32.7 0.298314 + 0.0799329i −1.15464 + 1.29105i −1.64945 0.952310i 0 −0.447643 + 0.292843i −0.951942 + 2.46856i −0.852694 0.852694i −0.333606 2.98139i 0
32.8 0.907300 + 0.243110i 0.315275 1.70312i −0.967960 0.558852i 0 0.700094 1.46859i 2.64571 0.0144144i −2.07075 2.07075i −2.80120 1.07390i 0
32.9 1.26950 + 0.340162i 1.59946 0.664627i −0.236127 0.136328i 0 2.25660 0.299670i −1.25943 2.32676i −2.11207 2.11207i 2.11654 2.12609i 0
32.10 1.46015 + 0.391246i −1.49243 0.879005i 0.246919 + 0.142558i 0 −1.83527 1.86739i −2.36949 + 1.17707i −1.83305 1.83305i 1.45470 + 2.62371i 0
32.11 2.17249 + 0.582118i 0.644934 + 1.60750i 2.64881 + 1.52929i 0 0.465359 + 3.86771i 2.38227 + 1.15099i 1.68355 + 1.68355i −2.16812 + 2.07346i 0
32.12 2.35640 + 0.631395i −1.72899 + 0.102851i 3.42191 + 1.97564i 0 −4.13914 0.849321i 1.91891 1.82148i 3.36596 + 3.36596i 2.97884 0.355658i 0
107.1 −0.631395 2.35640i 0.775426 1.54878i −3.42191 + 1.97564i 0 −4.13914 0.849321i −1.82148 + 1.91891i 3.36596 + 3.36596i −1.79743 2.40193i 0
107.2 −0.582118 2.17249i −1.71460 0.245221i −2.64881 + 1.52929i 0 0.465359 + 3.86771i 1.15099 + 2.38227i 1.68355 + 1.68355i 2.87973 + 0.840915i 0
107.3 −0.391246 1.46015i 1.50746 0.852980i −0.246919 + 0.142558i 0 −1.83527 1.86739i 1.17707 2.36949i −1.83305 1.83305i 1.54485 2.57166i 0
107.4 −0.340162 1.26950i −0.224146 + 1.71749i 0.236127 0.136328i 0 2.25660 0.299670i −2.32676 1.25943i −2.11207 2.11207i −2.89952 0.769934i 0
107.5 −0.243110 0.907300i 1.31730 + 1.12459i 0.967960 0.558852i 0 0.700094 1.46859i −0.0144144 + 2.64571i −2.07075 2.07075i 0.470578 + 2.96286i 0
107.6 −0.0799329 0.298314i −0.540759 1.64547i 1.64945 0.952310i 0 −0.447643 + 0.292843i 2.46856 0.951942i −0.852694 0.852694i −2.41516 + 1.77961i 0
107.7 0.0799329 + 0.298314i −1.29105 + 1.15464i 1.64945 0.952310i 0 −0.447643 0.292843i 2.46856 0.951942i 0.852694 + 0.852694i 0.333606 2.98139i 0
107.8 0.243110 + 0.907300i 1.70312 0.315275i 0.967960 0.558852i 0 0.700094 + 1.46859i −0.0144144 + 2.64571i 2.07075 + 2.07075i 2.80120 1.07390i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 443.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bf.f 48
3.b odd 2 1 inner 525.2.bf.f 48
5.b even 2 1 105.2.x.a 48
5.c odd 4 1 105.2.x.a 48
5.c odd 4 1 inner 525.2.bf.f 48
7.c even 3 1 inner 525.2.bf.f 48
15.d odd 2 1 105.2.x.a 48
15.e even 4 1 105.2.x.a 48
15.e even 4 1 inner 525.2.bf.f 48
21.h odd 6 1 inner 525.2.bf.f 48
35.c odd 2 1 735.2.y.i 48
35.f even 4 1 735.2.y.i 48
35.i odd 6 1 735.2.j.e 24
35.i odd 6 1 735.2.y.i 48
35.j even 6 1 105.2.x.a 48
35.j even 6 1 735.2.j.g 24
35.k even 12 1 735.2.j.e 24
35.k even 12 1 735.2.y.i 48
35.l odd 12 1 105.2.x.a 48
35.l odd 12 1 inner 525.2.bf.f 48
35.l odd 12 1 735.2.j.g 24
105.g even 2 1 735.2.y.i 48
105.k odd 4 1 735.2.y.i 48
105.o odd 6 1 105.2.x.a 48
105.o odd 6 1 735.2.j.g 24
105.p even 6 1 735.2.j.e 24
105.p even 6 1 735.2.y.i 48
105.w odd 12 1 735.2.j.e 24
105.w odd 12 1 735.2.y.i 48
105.x even 12 1 105.2.x.a 48
105.x even 12 1 inner 525.2.bf.f 48
105.x even 12 1 735.2.j.g 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.x.a 48 5.b even 2 1
105.2.x.a 48 5.c odd 4 1
105.2.x.a 48 15.d odd 2 1
105.2.x.a 48 15.e even 4 1
105.2.x.a 48 35.j even 6 1
105.2.x.a 48 35.l odd 12 1
105.2.x.a 48 105.o odd 6 1
105.2.x.a 48 105.x even 12 1
525.2.bf.f 48 1.a even 1 1 trivial
525.2.bf.f 48 3.b odd 2 1 inner
525.2.bf.f 48 5.c odd 4 1 inner
525.2.bf.f 48 7.c even 3 1 inner
525.2.bf.f 48 15.e even 4 1 inner
525.2.bf.f 48 21.h odd 6 1 inner
525.2.bf.f 48 35.l odd 12 1 inner
525.2.bf.f 48 105.x even 12 1 inner
735.2.j.e 24 35.i odd 6 1
735.2.j.e 24 35.k even 12 1
735.2.j.e 24 105.p even 6 1
735.2.j.e 24 105.w odd 12 1
735.2.j.g 24 35.j even 6 1
735.2.j.g 24 35.l odd 12 1
735.2.j.g 24 105.o odd 6 1
735.2.j.g 24 105.x even 12 1
735.2.y.i 48 35.c odd 2 1
735.2.y.i 48 35.f even 4 1
735.2.y.i 48 35.i odd 6 1
735.2.y.i 48 35.k even 12 1
735.2.y.i 48 105.g even 2 1
735.2.y.i 48 105.k odd 4 1
735.2.y.i 48 105.p even 6 1
735.2.y.i 48 105.w odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{48} - \cdots$$ $$T_{13}^{12} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database