# Properties

 Label 525.2.bh.a Level 525 Weight 2 Character orbit 525.bh Analytic conductor 4.192 Analytic rank 0 Dimension 320 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.bh (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$320$$ Relative dimension: $$40$$ over $$\Q(\zeta_{20})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$320q + 8q^{7} - 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$320q + 8q^{7} - 24q^{8} - 8q^{15} + 80q^{16} - 24q^{22} + 36q^{28} - 40q^{29} - 32q^{30} - 48q^{32} + 28q^{35} + 80q^{36} - 32q^{37} + 16q^{42} - 144q^{43} - 288q^{50} + 136q^{53} - 8q^{57} - 32q^{58} - 40q^{60} - 8q^{63} - 120q^{64} - 40q^{65} + 32q^{67} - 40q^{70} - 24q^{72} + 24q^{77} + 8q^{78} + 80q^{81} - 80q^{84} - 48q^{85} - 56q^{88} + 40q^{92} + 96q^{93} - 88q^{95} - 184q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −2.65979 + 0.421270i −0.891007 0.453990i 4.99492 1.62295i 0.301881 + 2.21560i 2.56115 + 0.832167i 2.34376 1.22751i −7.80289 + 3.97577i 0.587785 + 0.809017i −1.73630 5.76586i
13.2 −2.65979 + 0.421270i 0.891007 + 0.453990i 4.99492 1.62295i −0.301881 2.21560i −2.56115 0.832167i 1.22751 2.34376i −7.80289 + 3.97577i 0.587785 + 0.809017i 1.73630 + 5.76586i
13.3 −2.65246 + 0.420109i −0.891007 0.453990i 4.95695 1.61061i 1.94004 1.11186i 2.55409 + 0.829873i −0.972375 + 2.46059i −7.68586 + 3.91614i 0.587785 + 0.809017i −4.67879 + 3.76420i
13.4 −2.65246 + 0.420109i 0.891007 + 0.453990i 4.95695 1.61061i −1.94004 + 1.11186i −2.55409 0.829873i −2.46059 + 0.972375i −7.68586 + 3.91614i 0.587785 + 0.809017i 4.67879 3.76420i
13.5 −2.18124 + 0.345475i −0.891007 0.453990i 2.73636 0.889096i −1.83877 + 1.27236i 2.10034 + 0.682443i −2.14582 1.54772i −1.72604 + 0.879463i 0.587785 + 0.809017i 3.57124 3.41059i
13.6 −2.18124 + 0.345475i 0.891007 + 0.453990i 2.73636 0.889096i 1.83877 1.27236i −2.10034 0.682443i 1.54772 + 2.14582i −1.72604 + 0.879463i 0.587785 + 0.809017i −3.57124 + 3.41059i
13.7 −1.83690 + 0.290936i −0.891007 0.453990i 1.38744 0.450806i −0.677854 2.13085i 1.76877 + 0.574708i −2.16215 + 1.52483i 0.896755 0.456919i 0.587785 + 0.809017i 1.86509 + 3.71694i
13.8 −1.83690 + 0.290936i 0.891007 + 0.453990i 1.38744 0.450806i 0.677854 + 2.13085i −1.76877 0.574708i −1.52483 + 2.16215i 0.896755 0.456919i 0.587785 + 0.809017i −1.86509 3.71694i
13.9 −1.60091 + 0.253560i −0.891007 0.453990i 0.596516 0.193820i 2.13278 + 0.671755i 1.54154 + 0.500876i −0.369115 2.61988i 1.98258 1.01018i 0.587785 + 0.809017i −3.58472 0.534634i
13.10 −1.60091 + 0.253560i 0.891007 + 0.453990i 0.596516 0.193820i −2.13278 0.671755i −1.54154 0.500876i 2.61988 + 0.369115i 1.98258 1.01018i 0.587785 + 0.809017i 3.58472 + 0.534634i
13.11 −1.56905 + 0.248513i −0.891007 0.453990i 0.498045 0.161825i −1.85206 + 1.25295i 1.51086 + 0.490907i 0.803473 + 2.52080i 2.08968 1.06474i 0.587785 + 0.809017i 2.59460 2.42620i
13.12 −1.56905 + 0.248513i 0.891007 + 0.453990i 0.498045 0.161825i 1.85206 1.25295i −1.51086 0.490907i −2.52080 0.803473i 2.08968 1.06474i 0.587785 + 0.809017i −2.59460 + 2.42620i
13.13 −1.44172 + 0.228346i −0.891007 0.453990i 0.124296 0.0403862i 1.71338 + 1.43678i 1.38825 + 0.451069i 1.34100 + 2.28073i 2.43121 1.23876i 0.587785 + 0.809017i −2.79829 1.68019i
13.14 −1.44172 + 0.228346i 0.891007 + 0.453990i 0.124296 0.0403862i −1.71338 1.43678i −1.38825 0.451069i −2.28073 1.34100i 2.43121 1.23876i 0.587785 + 0.809017i 2.79829 + 1.68019i
13.15 −0.809024 + 0.128137i −0.891007 0.453990i −1.26401 + 0.410703i −1.82130 1.29725i 0.779018 + 0.253118i 2.49866 0.869897i 2.42965 1.23797i 0.587785 + 0.809017i 1.63970 + 0.816128i
13.16 −0.809024 + 0.128137i 0.891007 + 0.453990i −1.26401 + 0.410703i 1.82130 + 1.29725i −0.779018 0.253118i 0.869897 2.49866i 2.42965 1.23797i 0.587785 + 0.809017i −1.63970 0.816128i
13.17 −0.112696 + 0.0178493i −0.891007 0.453990i −1.88973 + 0.614011i −2.18359 + 0.481591i 0.108517 + 0.0352592i −1.28646 2.31193i 0.405335 0.206529i 0.587785 + 0.809017i 0.237486 0.0932491i
13.18 −0.112696 + 0.0178493i 0.891007 + 0.453990i −1.88973 + 0.614011i 2.18359 0.481591i −0.108517 0.0352592i 2.31193 + 1.28646i 0.405335 0.206529i 0.587785 + 0.809017i −0.237486 + 0.0932491i
13.19 −0.0439745 + 0.00696487i −0.891007 0.453990i −1.90023 + 0.617421i 1.50515 1.65364i 0.0423435 + 0.0137582i 2.54086 + 0.737568i 0.158601 0.0808113i 0.587785 + 0.809017i −0.0546709 + 0.0832010i
13.20 −0.0439745 + 0.00696487i 0.891007 + 0.453990i −1.90023 + 0.617421i −1.50515 + 1.65364i −0.0423435 0.0137582i −0.737568 2.54086i 0.158601 0.0808113i 0.587785 + 0.809017i 0.0546709 0.0832010i
See next 80 embeddings (of 320 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 517.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
25.f odd 20 1 inner
175.s even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bh.a 320
7.b odd 2 1 inner 525.2.bh.a 320
25.f odd 20 1 inner 525.2.bh.a 320
175.s even 20 1 inner 525.2.bh.a 320

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bh.a 320 1.a even 1 1 trivial
525.2.bh.a 320 7.b odd 2 1 inner
525.2.bh.a 320 25.f odd 20 1 inner
525.2.bh.a 320 175.s even 20 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database