# Properties

 Label 525.3.e.c Level 525 Weight 3 Character orbit 525.e Analytic conductor 14.305 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 525.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3052138789$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 88q^{4} + 72q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 88q^{4} + 72q^{9} - 32q^{11} + 80q^{14} + 184q^{16} + 72q^{21} - 208q^{29} - 264q^{36} + 48q^{39} - 384q^{44} + 400q^{46} - 120q^{49} + 48q^{51} - 736q^{56} + 40q^{64} + 64q^{71} - 368q^{74} - 240q^{79} + 216q^{81} - 216q^{84} + 800q^{86} + 48q^{91} - 96q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1 1.71214i −1.73205 1.06857 0 2.96552i −6.15534 3.33344i 8.67811i 3.00000 0
349.2 1.71214i −1.73205 1.06857 0 2.96552i −6.15534 + 3.33344i 8.67811i 3.00000 0
349.3 3.80460i 1.73205 −10.4750 0 6.58976i 6.55866 + 2.44621i 24.6348i 3.00000 0
349.4 3.80460i 1.73205 −10.4750 0 6.58976i 6.55866 2.44621i 24.6348i 3.00000 0
349.5 0.112974i −1.73205 3.98724 0 0.195676i −1.98374 6.71303i 0.902349i 3.00000 0
349.6 0.112974i −1.73205 3.98724 0 0.195676i −1.98374 + 6.71303i 0.902349i 3.00000 0
349.7 2.79155i −1.73205 −3.79273 0 4.83510i 5.63139 + 4.15782i 0.578591i 3.00000 0
349.8 2.79155i −1.73205 −3.79273 0 4.83510i 5.63139 4.15782i 0.578591i 3.00000 0
349.9 3.50369i −1.73205 −8.27584 0 6.06857i 2.03600 + 6.69736i 14.9812i 3.00000 0
349.10 3.50369i −1.73205 −8.27584 0 6.06857i 2.03600 6.69736i 14.9812i 3.00000 0
349.11 3.80460i −1.73205 −10.4750 0 6.58976i −6.55866 + 2.44621i 24.6348i 3.00000 0
349.12 3.80460i −1.73205 −10.4750 0 6.58976i −6.55866 2.44621i 24.6348i 3.00000 0
349.13 2.91758i 1.73205 −4.51225 0 5.05339i 3.36195 6.13981i 1.49451i 3.00000 0
349.14 2.91758i 1.73205 −4.51225 0 5.05339i 3.36195 + 6.13981i 1.49451i 3.00000 0
349.15 1.71214i 1.73205 1.06857 0 2.96552i 6.15534 3.33344i 8.67811i 3.00000 0
349.16 1.71214i 1.73205 1.06857 0 2.96552i 6.15534 + 3.33344i 8.67811i 3.00000 0
349.17 0.112974i 1.73205 3.98724 0 0.195676i 1.98374 6.71303i 0.902349i 3.00000 0
349.18 0.112974i 1.73205 3.98724 0 0.195676i 1.98374 + 6.71303i 0.902349i 3.00000 0
349.19 2.91758i −1.73205 −4.51225 0 5.05339i −3.36195 6.13981i 1.49451i 3.00000 0
349.20 2.91758i −1.73205 −4.51225 0 5.05339i −3.36195 + 6.13981i 1.49451i 3.00000 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.24 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
7.b odd 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.e.c 24
5.b even 2 1 inner 525.3.e.c 24
5.c odd 4 1 105.3.h.a 12
5.c odd 4 1 525.3.h.d 12
7.b odd 2 1 inner 525.3.e.c 24
15.e even 4 1 315.3.h.d 12
20.e even 4 1 1680.3.s.c 12
35.c odd 2 1 inner 525.3.e.c 24
35.f even 4 1 105.3.h.a 12
35.f even 4 1 525.3.h.d 12
105.k odd 4 1 315.3.h.d 12
140.j odd 4 1 1680.3.s.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.h.a 12 5.c odd 4 1
105.3.h.a 12 35.f even 4 1
315.3.h.d 12 15.e even 4 1
315.3.h.d 12 105.k odd 4 1
525.3.e.c 24 1.a even 1 1 trivial
525.3.e.c 24 5.b even 2 1 inner
525.3.e.c 24 7.b odd 2 1 inner
525.3.e.c 24 35.c odd 2 1 inner
525.3.h.d 12 5.c odd 4 1
525.3.h.d 12 35.f even 4 1
1680.3.s.c 12 20.e even 4 1
1680.3.s.c 12 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 46 T_{2}^{10} + 807 T_{2}^{8} + 6676 T_{2}^{6} + 25567 T_{2}^{4} + 34878 T_{2}^{2} + 441$$ acting on $$S_{3}^{\mathrm{new}}(525, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database