Properties

 Label 1575.1.e.a Level $1575$ Weight $1$ Character orbit 1575.e Analytic conductor $0.786$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -7 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1575.e (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.786027394897$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 175) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.175.1

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q -i q^{2} -i q^{7} -i q^{8} +O(q^{10})$$ $$q -i q^{2} -i q^{7} -i q^{8} + q^{11} - q^{14} - q^{16} -i q^{22} + i q^{23} - q^{29} + i q^{37} -i q^{43} + q^{46} - q^{49} -2 i q^{53} - q^{56} + i q^{58} - q^{64} + i q^{67} + q^{71} + q^{74} -i q^{77} + q^{79} - q^{86} -i q^{88} + i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 2 q^{11} - 2 q^{14} - 2 q^{16} - 2 q^{29} + 2 q^{46} - 2 q^{49} - 2 q^{56} - 2 q^{64} + 2 q^{71} + 2 q^{74} + 2 q^{79} - 2 q^{86} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
874.1
 1.00000i − 1.00000i
1.00000i 0 0 0 0 1.00000i 1.00000i 0 0
874.2 1.00000i 0 0 0 0 1.00000i 1.00000i 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-7})$$
5.b even 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 1575.1.e.a 2
3.b odd 2 1 175.1.c.a 2
5.b even 2 1 inner 1575.1.e.a 2
5.c odd 4 1 1575.1.h.a 1
5.c odd 4 1 1575.1.h.c 1
7.b odd 2 1 CM 1575.1.e.a 2
12.b even 2 1 2800.1.p.a 2
15.d odd 2 1 175.1.c.a 2
15.e even 4 1 175.1.d.a 1
15.e even 4 1 175.1.d.b yes 1
21.c even 2 1 175.1.c.a 2
21.g even 6 2 1225.1.j.a 4
21.h odd 6 2 1225.1.j.a 4
35.c odd 2 1 inner 1575.1.e.a 2
35.f even 4 1 1575.1.h.a 1
35.f even 4 1 1575.1.h.c 1
60.h even 2 1 2800.1.p.a 2
60.l odd 4 1 2800.1.f.a 1
60.l odd 4 1 2800.1.f.b 1
84.h odd 2 1 2800.1.p.a 2
105.g even 2 1 175.1.c.a 2
105.k odd 4 1 175.1.d.a 1
105.k odd 4 1 175.1.d.b yes 1
105.o odd 6 2 1225.1.j.a 4
105.p even 6 2 1225.1.j.a 4
105.w odd 12 2 1225.1.i.a 2
105.w odd 12 2 1225.1.i.b 2
105.x even 12 2 1225.1.i.a 2
105.x even 12 2 1225.1.i.b 2
420.o odd 2 1 2800.1.p.a 2
420.w even 4 1 2800.1.f.a 1
420.w even 4 1 2800.1.f.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 3.b odd 2 1
175.1.c.a 2 15.d odd 2 1
175.1.c.a 2 21.c even 2 1
175.1.c.a 2 105.g even 2 1
175.1.d.a 1 15.e even 4 1
175.1.d.a 1 105.k odd 4 1
175.1.d.b yes 1 15.e even 4 1
175.1.d.b yes 1 105.k odd 4 1
1225.1.i.a 2 105.w odd 12 2
1225.1.i.a 2 105.x even 12 2
1225.1.i.b 2 105.w odd 12 2
1225.1.i.b 2 105.x even 12 2
1225.1.j.a 4 21.g even 6 2
1225.1.j.a 4 21.h odd 6 2
1225.1.j.a 4 105.o odd 6 2
1225.1.j.a 4 105.p even 6 2
1575.1.e.a 2 1.a even 1 1 trivial
1575.1.e.a 2 5.b even 2 1 inner
1575.1.e.a 2 7.b odd 2 1 CM
1575.1.e.a 2 35.c odd 2 1 inner
1575.1.h.a 1 5.c odd 4 1
1575.1.h.a 1 35.f even 4 1
1575.1.h.c 1 5.c odd 4 1
1575.1.h.c 1 35.f even 4 1
2800.1.f.a 1 60.l odd 4 1
2800.1.f.a 1 420.w even 4 1
2800.1.f.b 1 60.l odd 4 1
2800.1.f.b 1 420.w even 4 1
2800.1.p.a 2 12.b even 2 1
2800.1.p.a 2 60.h even 2 1
2800.1.p.a 2 84.h odd 2 1
2800.1.p.a 2 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1575, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$1 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$1 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$1 + T^{2}$$
$71$ $$( -1 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$( -1 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$