# Properties

 Label 3525.1.l.d Level $3525$ Weight $1$ Character orbit 3525.l Analytic conductor $1.759$ Analytic rank $0$ Dimension $32$ Projective image $D_{30}$ CM discriminant -47 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3525.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.75920416953$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{120})$$ Defining polynomial: $$x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{30}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{30} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{120}^{41} - \zeta_{120}^{49} ) q^{2} + \zeta_{120}^{43} q^{3} + ( -\zeta_{120}^{22} - \zeta_{120}^{30} - \zeta_{120}^{38} ) q^{4} + ( \zeta_{120}^{24} + \zeta_{120}^{32} ) q^{6} + ( -\zeta_{120} - \zeta_{120}^{29} ) q^{7} + ( -\zeta_{120}^{3} - \zeta_{120}^{11} - \zeta_{120}^{19} - \zeta_{120}^{27} ) q^{8} -\zeta_{120}^{26} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{120}^{41} - \zeta_{120}^{49} ) q^{2} + \zeta_{120}^{43} q^{3} + ( -\zeta_{120}^{22} - \zeta_{120}^{30} - \zeta_{120}^{38} ) q^{4} + ( \zeta_{120}^{24} + \zeta_{120}^{32} ) q^{6} + ( -\zeta_{120} - \zeta_{120}^{29} ) q^{7} + ( -\zeta_{120}^{3} - \zeta_{120}^{11} - \zeta_{120}^{19} - \zeta_{120}^{27} ) q^{8} -\zeta_{120}^{26} q^{9} + ( \zeta_{120}^{5} + \zeta_{120}^{13} + \zeta_{120}^{21} ) q^{12} + ( -\zeta_{120}^{10} - \zeta_{120}^{18} + \zeta_{120}^{42} + \zeta_{120}^{50} ) q^{14} + ( -1 - \zeta_{120}^{8} - \zeta_{120}^{16} + \zeta_{120}^{44} + \zeta_{120}^{52} ) q^{16} + ( \zeta_{120}^{37} + \zeta_{120}^{53} ) q^{17} + ( -\zeta_{120}^{7} - \zeta_{120}^{15} ) q^{18} + ( \zeta_{120}^{12} - \zeta_{120}^{44} ) q^{21} + ( \zeta_{120}^{2} + \zeta_{120}^{10} - \zeta_{120}^{46} - \zeta_{120}^{54} ) q^{24} + \zeta_{120}^{9} q^{27} + ( -\zeta_{120}^{7} + \zeta_{120}^{23} + \zeta_{120}^{31} + \zeta_{120}^{39} + \zeta_{120}^{51} + \zeta_{120}^{59} ) q^{28} + ( -\zeta_{120}^{5} + \zeta_{120}^{25} + \zeta_{120}^{33} + \zeta_{120}^{41} + \zeta_{120}^{49} + \zeta_{120}^{57} ) q^{32} + ( \zeta_{120}^{18} + \zeta_{120}^{26} + \zeta_{120}^{34} + \zeta_{120}^{42} ) q^{34} + ( -\zeta_{120}^{4} + \zeta_{120}^{48} + \zeta_{120}^{56} ) q^{36} + ( -\zeta_{120}^{33} + \zeta_{120}^{57} ) q^{37} + ( \zeta_{120} - \zeta_{120}^{25} - \zeta_{120}^{33} - \zeta_{120}^{53} ) q^{42} -\zeta_{120}^{45} q^{47} + ( -\zeta_{120}^{27} - \zeta_{120}^{35} - \zeta_{120}^{43} - \zeta_{120}^{51} - \zeta_{120}^{59} ) q^{48} + ( \zeta_{120}^{2} + \zeta_{120}^{30} + \zeta_{120}^{58} ) q^{49} + ( -\zeta_{120}^{20} - \zeta_{120}^{36} ) q^{51} + ( -\zeta_{120}^{39} + \zeta_{120}^{51} ) q^{53} + ( -\zeta_{120}^{50} - \zeta_{120}^{58} ) q^{54} + ( \zeta_{120}^{4} + \zeta_{120}^{12} + \zeta_{120}^{20} + \zeta_{120}^{28} + \zeta_{120}^{32} + \zeta_{120}^{40} + \zeta_{120}^{48} + \zeta_{120}^{56} ) q^{56} + ( \zeta_{120}^{14} - \zeta_{120}^{46} ) q^{59} + ( -\zeta_{120}^{24} + \zeta_{120}^{36} ) q^{61} + ( \zeta_{120}^{27} + \zeta_{120}^{55} ) q^{63} + ( \zeta_{120}^{6} + \zeta_{120}^{14} + \zeta_{120}^{22} + \zeta_{120}^{30} + \zeta_{120}^{38} + \zeta_{120}^{46} + \zeta_{120}^{54} ) q^{64} + ( \zeta_{120}^{7} + 2 \zeta_{120}^{15} + \zeta_{120}^{23} + \zeta_{120}^{31} - \zeta_{120}^{59} ) q^{68} + ( -\zeta_{120}^{28} - \zeta_{120}^{32} ) q^{71} + ( \zeta_{120}^{29} + \zeta_{120}^{37} + \zeta_{120}^{45} + \zeta_{120}^{53} ) q^{72} + ( -\zeta_{120}^{14} - \zeta_{120}^{22} + \zeta_{120}^{38} + \zeta_{120}^{46} ) q^{74} + ( -\zeta_{120}^{18} - \zeta_{120}^{42} ) q^{79} + \zeta_{120}^{52} q^{81} + \zeta_{120}^{15} q^{83} + ( -\zeta_{120}^{6} - \zeta_{120}^{14} - \zeta_{120}^{22} - \zeta_{120}^{34} - \zeta_{120}^{42} - \zeta_{120}^{50} ) q^{84} + ( -\zeta_{120}^{6} + \zeta_{120}^{54} ) q^{89} + ( -\zeta_{120}^{26} - \zeta_{120}^{34} ) q^{94} + ( -\zeta_{120}^{8} - \zeta_{120}^{16} - \zeta_{120}^{24} - \zeta_{120}^{32} - \zeta_{120}^{40} - \zeta_{120}^{48} ) q^{96} + ( -\zeta_{120}^{9} - \zeta_{120}^{21} ) q^{97} + ( \zeta_{120}^{11} + \zeta_{120}^{19} + \zeta_{120}^{39} - \zeta_{120}^{43} + \zeta_{120}^{47} - \zeta_{120}^{51} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$32q - 4q^{6} + O(q^{10})$$ $$32q - 4q^{6} - 48q^{16} + 12q^{21} - 24q^{51} + 16q^{61} - 4q^{81} + 20q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times$$.

 $$n$$ $$1552$$ $$2026$$ $$2351$$ $$\chi(n)$$ $$\zeta_{120}^{30}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1268.1
 0.0523360 − 0.998630i −0.998630 + 0.0523360i −0.777146 + 0.629320i 0.629320 − 0.777146i −0.838671 − 0.544639i −0.544639 − 0.838671i −0.358368 − 0.933580i −0.933580 − 0.358368i 0.933580 + 0.358368i 0.358368 + 0.933580i 0.544639 + 0.838671i 0.838671 + 0.544639i −0.629320 + 0.777146i 0.777146 − 0.629320i 0.998630 − 0.0523360i −0.0523360 + 0.998630i 0.0523360 + 0.998630i −0.998630 − 0.0523360i −0.777146 − 0.629320i 0.629320 + 0.777146i
−1.38331 1.38331i −0.777146 0.629320i 2.82709i 0 0.204489 + 1.94558i −1.05097 + 1.05097i 2.52743 2.52743i 0.207912 + 0.978148i 0
1268.2 −1.38331 1.38331i 0.629320 + 0.777146i 2.82709i 0 0.204489 1.94558i 1.05097 1.05097i 2.52743 2.52743i −0.207912 + 0.978148i 0
1268.3 −1.29195 1.29195i 0.544639 0.838671i 2.33826i 0 −1.78716 + 0.379874i 1.40647 1.40647i 1.72896 1.72896i −0.406737 0.913545i 0
1268.4 −1.29195 1.29195i 0.838671 0.544639i 2.33826i 0 −1.78716 0.379874i −1.40647 + 1.40647i 1.72896 1.72896i 0.406737 0.913545i 0
1268.5 −0.946294 0.946294i −0.933580 + 0.358368i 0.790943i 0 1.22256 + 0.544320i 0.294032 0.294032i −0.197829 + 0.197829i 0.743145 0.669131i 0
1268.6 −0.946294 0.946294i −0.358368 + 0.933580i 0.790943i 0 1.22256 0.544320i −0.294032 + 0.294032i −0.197829 + 0.197829i −0.743145 0.669131i 0
1268.7 −0.147826 0.147826i −0.0523360 0.998630i 0.956295i 0 −0.139886 + 0.155360i −0.575212 + 0.575212i −0.289190 + 0.289190i −0.994522 + 0.104528i 0
1268.8 −0.147826 0.147826i 0.998630 + 0.0523360i 0.956295i 0 −0.139886 0.155360i 0.575212 0.575212i −0.289190 + 0.289190i 0.994522 + 0.104528i 0
1268.9 0.147826 + 0.147826i −0.998630 0.0523360i 0.956295i 0 −0.139886 0.155360i −0.575212 + 0.575212i 0.289190 0.289190i 0.994522 + 0.104528i 0
1268.10 0.147826 + 0.147826i 0.0523360 + 0.998630i 0.956295i 0 −0.139886 + 0.155360i 0.575212 0.575212i 0.289190 0.289190i −0.994522 + 0.104528i 0
1268.11 0.946294 + 0.946294i 0.358368 0.933580i 0.790943i 0 1.22256 0.544320i 0.294032 0.294032i 0.197829 0.197829i −0.743145 0.669131i 0
1268.12 0.946294 + 0.946294i 0.933580 0.358368i 0.790943i 0 1.22256 + 0.544320i −0.294032 + 0.294032i 0.197829 0.197829i 0.743145 0.669131i 0
1268.13 1.29195 + 1.29195i −0.838671 + 0.544639i 2.33826i 0 −1.78716 0.379874i 1.40647 1.40647i −1.72896 + 1.72896i 0.406737 0.913545i 0
1268.14 1.29195 + 1.29195i −0.544639 + 0.838671i 2.33826i 0 −1.78716 + 0.379874i −1.40647 + 1.40647i −1.72896 + 1.72896i −0.406737 0.913545i 0
1268.15 1.38331 + 1.38331i −0.629320 0.777146i 2.82709i 0 0.204489 1.94558i −1.05097 + 1.05097i −2.52743 + 2.52743i −0.207912 + 0.978148i 0
1268.16 1.38331 + 1.38331i 0.777146 + 0.629320i 2.82709i 0 0.204489 + 1.94558i 1.05097 1.05097i −2.52743 + 2.52743i 0.207912 + 0.978148i 0
1832.1 −1.38331 + 1.38331i −0.777146 + 0.629320i 2.82709i 0 0.204489 1.94558i −1.05097 1.05097i 2.52743 + 2.52743i 0.207912 0.978148i 0
1832.2 −1.38331 + 1.38331i 0.629320 0.777146i 2.82709i 0 0.204489 + 1.94558i 1.05097 + 1.05097i 2.52743 + 2.52743i −0.207912 0.978148i 0
1832.3 −1.29195 + 1.29195i 0.544639 + 0.838671i 2.33826i 0 −1.78716 0.379874i 1.40647 + 1.40647i 1.72896 + 1.72896i −0.406737 + 0.913545i 0
1832.4 −1.29195 + 1.29195i 0.838671 + 0.544639i 2.33826i 0 −1.78716 + 0.379874i −1.40647 1.40647i 1.72896 + 1.72896i 0.406737 + 0.913545i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1832.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by $$\Q(\sqrt{-47})$$
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner
141.c even 2 1 inner
235.b odd 2 1 inner
235.e even 4 2 inner
705.g even 2 1 inner
705.l odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3525.1.l.d 32
3.b odd 2 1 inner 3525.1.l.d 32
5.b even 2 1 inner 3525.1.l.d 32
5.c odd 4 2 inner 3525.1.l.d 32
15.d odd 2 1 inner 3525.1.l.d 32
15.e even 4 2 inner 3525.1.l.d 32
47.b odd 2 1 CM 3525.1.l.d 32
141.c even 2 1 inner 3525.1.l.d 32
235.b odd 2 1 inner 3525.1.l.d 32
235.e even 4 2 inner 3525.1.l.d 32
705.g even 2 1 inner 3525.1.l.d 32
705.l odd 4 2 inner 3525.1.l.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.1.l.d 32 1.a even 1 1 trivial
3525.1.l.d 32 3.b odd 2 1 inner
3525.1.l.d 32 5.b even 2 1 inner
3525.1.l.d 32 5.c odd 4 2 inner
3525.1.l.d 32 15.d odd 2 1 inner
3525.1.l.d 32 15.e even 4 2 inner
3525.1.l.d 32 47.b odd 2 1 CM
3525.1.l.d 32 141.c even 2 1 inner
3525.1.l.d 32 235.b odd 2 1 inner
3525.1.l.d 32 235.e even 4 2 inner
3525.1.l.d 32 705.g even 2 1 inner
3525.1.l.d 32 705.l odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 29 T_{2}^{12} + 246 T_{2}^{8} + 524 T_{2}^{4} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3525, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 524 T^{4} + 246 T^{8} + 29 T^{12} + T^{16} )^{2}$$
$3$ $$1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32}$$
$5$ $$T^{32}$$
$7$ $$( 1 + 36 T^{4} + 86 T^{8} + 21 T^{12} + T^{16} )^{2}$$
$11$ $$T^{32}$$
$13$ $$T^{32}$$
$17$ $$( 1 + 524 T^{4} + 246 T^{8} + 29 T^{12} + T^{16} )^{2}$$
$19$ $$T^{32}$$
$23$ $$T^{32}$$
$29$ $$T^{32}$$
$31$ $$T^{32}$$
$37$ $$( 25 + 15 T^{4} + T^{8} )^{4}$$
$41$ $$T^{32}$$
$43$ $$T^{32}$$
$47$ $$( 1 + T^{4} )^{8}$$
$53$ $$( 1 + 7 T^{4} + T^{8} )^{4}$$
$59$ $$( 1 - 8 T^{2} + 14 T^{4} - 7 T^{6} + T^{8} )^{4}$$
$61$ $$( -1 - T + T^{2} )^{16}$$
$67$ $$T^{32}$$
$71$ $$( 1 + 8 T^{2} + 14 T^{4} + 7 T^{6} + T^{8} )^{4}$$
$73$ $$T^{32}$$
$79$ $$( 1 + 3 T^{2} + T^{4} )^{8}$$
$83$ $$( 1 + T^{4} )^{8}$$
$89$ $$( 5 - 5 T^{2} + T^{4} )^{8}$$
$97$ $$( 25 + 15 T^{4} + T^{8} )^{4}$$