# Properties

 Label 525.2.q.g Level 525 Weight 2 Character orbit 525.q Analytic conductor 4.192 Analytic rank 0 Dimension 40 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.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 28q^{4} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 28q^{4} + 14q^{9} - 36q^{16} - 18q^{21} - 36q^{24} + 84q^{31} - 72q^{36} - 16q^{46} + 8q^{49} + 42q^{51} + 150q^{54} - 180q^{61} + 240q^{64} + 12q^{66} - 92q^{79} + 58q^{81} - 150q^{84} - 60q^{91} - 12q^{94} + 270q^{96} - 188q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1 −1.36696 + 2.36764i −1.70405 0.310170i −2.73715 4.74088i 0 3.06374 3.61059i 1.39384 + 2.24882i 9.49844 2.80759 + 1.05709i 0
299.2 −1.36696 + 2.36764i −0.583411 + 1.63084i −2.73715 4.74088i 0 −3.06374 3.61059i −1.39384 2.24882i 9.49844 −2.31926 1.90290i 0
299.3 −1.12521 + 1.94891i 1.42544 0.983931i −1.53217 2.65380i 0 0.313682 + 3.88518i −2.22667 1.42897i 2.39522 1.06376 2.80507i 0
299.4 −1.12521 + 1.94891i 1.56483 0.742502i −1.53217 2.65380i 0 −0.313682 + 3.88518i 2.22667 + 1.42897i 2.39522 1.89738 2.32378i 0
299.5 −0.846473 + 1.46613i −1.53466 0.803015i −0.433034 0.750036i 0 2.47637 1.57028i 2.01688 1.71236i −1.91969 1.71033 + 2.46470i 0
299.6 −0.846473 + 1.46613i −0.0718963 + 1.73056i −0.433034 0.750036i 0 −2.47637 1.57028i −2.01688 + 1.71236i −1.91969 −2.98966 0.248842i 0
299.7 −0.450666 + 0.780577i 0.248842 1.71408i 0.593800 + 1.02849i 0 1.22583 + 0.966719i −2.64365 0.105498i −2.87309 −2.87616 0.853070i 0
299.8 −0.450666 + 0.780577i 1.60886 + 0.641538i 0.593800 + 1.02849i 0 −1.22583 + 0.966719i 2.64365 + 0.105498i −2.87309 2.17686 + 2.06429i 0
299.9 −0.442404 + 0.766266i −1.72165 + 0.189492i 0.608557 + 1.05405i 0 0.616465 1.40308i 0.206062 2.63771i −2.84653 2.92819 0.652481i 0
299.10 −0.442404 + 0.766266i −1.02493 + 1.39625i 0.608557 + 1.05405i 0 −0.616465 1.40308i −0.206062 + 2.63771i −2.84653 −0.899028 2.86212i 0
299.11 0.442404 0.766266i 1.02493 1.39625i 0.608557 + 1.05405i 0 −0.616465 1.40308i 0.206062 2.63771i 2.84653 −0.899028 2.86212i 0
299.12 0.442404 0.766266i 1.72165 0.189492i 0.608557 + 1.05405i 0 0.616465 1.40308i −0.206062 + 2.63771i 2.84653 2.92819 0.652481i 0
299.13 0.450666 0.780577i −1.60886 0.641538i 0.593800 + 1.02849i 0 −1.22583 + 0.966719i −2.64365 0.105498i 2.87309 2.17686 + 2.06429i 0
299.14 0.450666 0.780577i −0.248842 + 1.71408i 0.593800 + 1.02849i 0 1.22583 + 0.966719i 2.64365 + 0.105498i 2.87309 −2.87616 0.853070i 0
299.15 0.846473 1.46613i 0.0718963 1.73056i −0.433034 0.750036i 0 −2.47637 1.57028i 2.01688 1.71236i 1.91969 −2.98966 0.248842i 0
299.16 0.846473 1.46613i 1.53466 + 0.803015i −0.433034 0.750036i 0 2.47637 1.57028i −2.01688 + 1.71236i 1.91969 1.71033 + 2.46470i 0
299.17 1.12521 1.94891i −1.56483 + 0.742502i −1.53217 2.65380i 0 −0.313682 + 3.88518i −2.22667 1.42897i −2.39522 1.89738 2.32378i 0
299.18 1.12521 1.94891i −1.42544 + 0.983931i −1.53217 2.65380i 0 0.313682 + 3.88518i 2.22667 + 1.42897i −2.39522 1.06376 2.80507i 0
299.19 1.36696 2.36764i 0.583411 1.63084i −2.73715 4.74088i 0 −3.06374 3.61059i 1.39384 + 2.24882i −9.49844 −2.31926 1.90290i 0
299.20 1.36696 2.36764i 1.70405 + 0.310170i −2.73715 4.74088i 0 3.06374 3.61059i −1.39384 2.24882i −9.49844 2.80759 + 1.05709i 0
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 374.20 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.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.q.g 40
3.b odd 2 1 inner 525.2.q.g 40
5.b even 2 1 inner 525.2.q.g 40
5.c odd 4 1 525.2.t.h 20
5.c odd 4 1 525.2.t.i yes 20
7.d odd 6 1 inner 525.2.q.g 40
15.d odd 2 1 inner 525.2.q.g 40
15.e even 4 1 525.2.t.h 20
15.e even 4 1 525.2.t.i yes 20
21.g even 6 1 inner 525.2.q.g 40
35.i odd 6 1 inner 525.2.q.g 40
35.k even 12 1 525.2.t.h 20
35.k even 12 1 525.2.t.i yes 20
105.p even 6 1 inner 525.2.q.g 40
105.w odd 12 1 525.2.t.h 20
105.w odd 12 1 525.2.t.i yes 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.q.g 40 1.a even 1 1 trivial
525.2.q.g 40 3.b odd 2 1 inner
525.2.q.g 40 5.b even 2 1 inner
525.2.q.g 40 7.d odd 6 1 inner
525.2.q.g 40 15.d odd 2 1 inner
525.2.q.g 40 21.g even 6 1 inner
525.2.q.g 40 35.i odd 6 1 inner
525.2.q.g 40 105.p even 6 1 inner
525.2.t.h 20 5.c odd 4 1
525.2.t.h 20 15.e even 4 1
525.2.t.h 20 35.k even 12 1
525.2.t.h 20 105.w odd 12 1
525.2.t.i yes 20 5.c odd 4 1
525.2.t.i yes 20 15.e even 4 1
525.2.t.i yes 20 35.k even 12 1
525.2.t.i yes 20 105.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 $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{20} + \cdots$$ $$T_{11}^{20} - \cdots$$ $$T_{13}^{10} - 72 T_{13}^{8} + 1950 T_{13}^{6} - 24651 T_{13}^{4} + 142515 T_{13}^{2} - 286443$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database