# Properties

 Label 525.2.bc.e Level 525 Weight 2 Character orbit 525.bc Analytic conductor 4.192 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{7} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{7} + 24q^{8} - 8q^{11} - 8q^{21} + 8q^{22} + 8q^{23} + 24q^{26} + 24q^{28} + 24q^{31} - 24q^{32} + 36q^{33} - 32q^{36} - 4q^{37} - 12q^{38} - 16q^{42} - 40q^{43} - 40q^{46} + 60q^{47} - 8q^{51} + 108q^{52} + 24q^{53} - 48q^{56} - 16q^{57} - 4q^{58} - 24q^{61} - 4q^{63} + 72q^{66} - 8q^{67} - 132q^{68} - 16q^{71} - 12q^{72} - 36q^{73} - 60q^{77} - 80q^{78} + 16q^{81} - 12q^{82} - 16q^{86} + 24q^{87} + 32q^{88} - 24q^{91} + 56q^{92} + 24q^{93} + 72q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1 −2.17399 + 0.582519i 0.258819 0.965926i 2.65485 1.53278i 0 2.25068i 0.660211 + 2.56205i −1.69581 + 1.69581i −0.866025 0.500000i 0
82.2 −1.72112 + 0.461174i −0.258819 + 0.965926i 1.01754 0.587476i 0 1.78184i −1.68076 2.04329i 1.03952 1.03952i −0.866025 0.500000i 0
82.3 −1.49657 + 0.401003i −0.258819 + 0.965926i 0.346853 0.200256i 0 1.54936i 2.44171 + 1.01885i 1.75234 1.75234i −0.866025 0.500000i 0
82.4 −0.648264 + 0.173702i 0.258819 0.965926i −1.34198 + 0.774791i 0 0.671132i −2.57939 + 0.588837i 1.68450 1.68450i −0.866025 0.500000i 0
82.5 0.394487 0.105703i 0.258819 0.965926i −1.58760 + 0.916603i 0 0.408404i 0.605712 2.57548i −1.10697 + 1.10697i −0.866025 0.500000i 0
82.6 0.969545 0.259789i −0.258819 + 0.965926i −0.859523 + 0.496246i 0 1.00375i −2.42328 + 1.06195i −2.12394 + 2.12394i −0.866025 0.500000i 0
82.7 2.24814 0.602389i −0.258819 + 0.965926i 2.95923 1.70851i 0 2.32745i 2.59417 0.519864i 2.33208 2.33208i −0.866025 0.500000i 0
82.8 2.42777 0.650518i 0.258819 0.965926i 3.73883 2.15861i 0 2.51341i −1.61838 2.09305i 4.11829 4.11829i −0.866025 0.500000i 0
157.1 −0.650518 + 2.42777i 0.965926 0.258819i −3.73883 2.15861i 0 2.51341i −2.09305 1.61838i 4.11829 4.11829i 0.866025 0.500000i 0
157.2 −0.602389 + 2.24814i −0.965926 + 0.258819i −2.95923 1.70851i 0 2.32745i −0.519864 + 2.59417i 2.33208 2.33208i 0.866025 0.500000i 0
157.3 −0.259789 + 0.969545i −0.965926 + 0.258819i 0.859523 + 0.496246i 0 1.00375i 1.06195 2.42328i −2.12394 + 2.12394i 0.866025 0.500000i 0
157.4 −0.105703 + 0.394487i 0.965926 0.258819i 1.58760 + 0.916603i 0 0.408404i −2.57548 + 0.605712i −1.10697 + 1.10697i 0.866025 0.500000i 0
157.5 0.173702 0.648264i 0.965926 0.258819i 1.34198 + 0.774791i 0 0.671132i 0.588837 2.57939i 1.68450 1.68450i 0.866025 0.500000i 0
157.6 0.401003 1.49657i −0.965926 + 0.258819i −0.346853 0.200256i 0 1.54936i 1.01885 + 2.44171i 1.75234 1.75234i 0.866025 0.500000i 0
157.7 0.461174 1.72112i −0.965926 + 0.258819i −1.01754 0.587476i 0 1.78184i −2.04329 1.68076i 1.03952 1.03952i 0.866025 0.500000i 0
157.8 0.582519 2.17399i 0.965926 0.258819i −2.65485 1.53278i 0 2.25068i 2.56205 + 0.660211i −1.69581 + 1.69581i 0.866025 0.500000i 0
418.1 −0.650518 2.42777i 0.965926 + 0.258819i −3.73883 + 2.15861i 0 2.51341i −2.09305 + 1.61838i 4.11829 + 4.11829i 0.866025 + 0.500000i 0
418.2 −0.602389 2.24814i −0.965926 0.258819i −2.95923 + 1.70851i 0 2.32745i −0.519864 2.59417i 2.33208 + 2.33208i 0.866025 + 0.500000i 0
418.3 −0.259789 0.969545i −0.965926 0.258819i 0.859523 0.496246i 0 1.00375i 1.06195 + 2.42328i −2.12394 2.12394i 0.866025 + 0.500000i 0
418.4 −0.105703 0.394487i 0.965926 + 0.258819i 1.58760 0.916603i 0 0.408404i −2.57548 0.605712i −1.10697 1.10697i 0.866025 + 0.500000i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.8 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.c odd 4 1 inner
7.d odd 6 1 inner
35.k 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.bc.e 32
5.b even 2 1 105.2.u.a 32
5.c odd 4 1 105.2.u.a 32
5.c odd 4 1 inner 525.2.bc.e 32
7.d odd 6 1 inner 525.2.bc.e 32
15.d odd 2 1 315.2.bz.d 32
15.e even 4 1 315.2.bz.d 32
35.c odd 2 1 735.2.v.b 32
35.f even 4 1 735.2.v.b 32
35.i odd 6 1 105.2.u.a 32
35.i odd 6 1 735.2.m.c 32
35.j even 6 1 735.2.m.c 32
35.j even 6 1 735.2.v.b 32
35.k even 12 1 105.2.u.a 32
35.k even 12 1 inner 525.2.bc.e 32
35.k even 12 1 735.2.m.c 32
35.l odd 12 1 735.2.m.c 32
35.l odd 12 1 735.2.v.b 32
105.p even 6 1 315.2.bz.d 32
105.w odd 12 1 315.2.bz.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.u.a 32 5.b even 2 1
105.2.u.a 32 5.c odd 4 1
105.2.u.a 32 35.i odd 6 1
105.2.u.a 32 35.k even 12 1
315.2.bz.d 32 15.d odd 2 1
315.2.bz.d 32 15.e even 4 1
315.2.bz.d 32 105.p even 6 1
315.2.bz.d 32 105.w odd 12 1
525.2.bc.e 32 1.a even 1 1 trivial
525.2.bc.e 32 5.c odd 4 1 inner
525.2.bc.e 32 7.d odd 6 1 inner
525.2.bc.e 32 35.k even 12 1 inner
735.2.m.c 32 35.i odd 6 1
735.2.m.c 32 35.j even 6 1
735.2.m.c 32 35.k even 12 1
735.2.m.c 32 35.l odd 12 1
735.2.v.b 32 35.c odd 2 1
735.2.v.b 32 35.f even 4 1
735.2.v.b 32 35.j even 6 1
735.2.v.b 32 35.l odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database