# Properties

 Label 1225.1.j.a Level $1225$ Weight $1$ Character orbit 1225.j Analytic conductor $0.611$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1225.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.611354640475$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + z^5 * q^2 - z^3 * q^8 + z^2 * q^9 $$q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} - \zeta_{12}^{4} q^{11} + \zeta_{12}^{2} q^{16} - \zeta_{12} q^{18} + \zeta_{12}^{3} q^{22} - \zeta_{12}^{5} q^{23} + q^{29} + \zeta_{12} q^{32} + \zeta_{12}^{5} q^{37} + \zeta_{12}^{3} q^{43} + \zeta_{12}^{4} q^{46} + \zeta_{12} q^{53} + \zeta_{12}^{5} q^{58} - q^{64} + \zeta_{12} q^{67} - q^{71} - \zeta_{12}^{5} q^{72} - \zeta_{12}^{4} q^{74} - \zeta_{12}^{2} q^{79} + \zeta_{12}^{4} q^{81} - \zeta_{12}^{2} q^{86} - \zeta_{12} q^{88} + q^{99} +O(q^{100})$$ q + z^5 * q^2 - z^3 * q^8 + z^2 * q^9 - z^4 * q^11 + z^2 * q^16 - z * q^18 + z^3 * q^22 - z^5 * q^23 + q^29 + z * q^32 + z^5 * q^37 + z^3 * q^43 + z^4 * q^46 + z * q^53 + z^5 * q^58 - q^64 + z * q^67 - q^71 - z^5 * q^72 - z^4 * q^74 - z^2 * q^79 + z^4 * q^81 - z^2 * q^86 - z * q^88 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^9 $$4 q + 2 q^{9} + 2 q^{11} + 2 q^{16} + 4 q^{29} - 2 q^{46} - 4 q^{64} - 4 q^{71} + 2 q^{74} - 2 q^{79} - 2 q^{81} - 2 q^{86} + 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^9 + 2 * q^11 + 2 * q^16 + 4 * q^29 - 2 * q^46 - 4 * q^64 - 4 * q^71 + 2 * q^74 - 2 * q^79 - 2 * q^81 - 2 * q^86 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
374.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0 0 0 0 1.00000i 0.500000 + 0.866025i 0
374.2 0.866025 0.500000i 0 0 0 0 0 1.00000i 0.500000 + 0.866025i 0
999.1 −0.866025 0.500000i 0 0 0 0 0 1.00000i 0.500000 0.866025i 0
999.2 0.866025 + 0.500000i 0 0 0 0 0 1.00000i 0.500000 0.866025i 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
7.c even 3 1 inner
7.d odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 7.c even 3 1
175.1.c.a 2 7.d odd 6 1
175.1.c.a 2 35.i odd 6 1
175.1.c.a 2 35.j even 6 1
175.1.d.a 1 35.k even 12 1
175.1.d.a 1 35.l odd 12 1
175.1.d.b yes 1 35.k even 12 1
175.1.d.b yes 1 35.l odd 12 1
1225.1.i.a 2 5.c odd 4 1
1225.1.i.a 2 35.f even 4 1
1225.1.i.a 2 35.k even 12 1
1225.1.i.a 2 35.l odd 12 1
1225.1.i.b 2 5.c odd 4 1
1225.1.i.b 2 35.f even 4 1
1225.1.i.b 2 35.k even 12 1
1225.1.i.b 2 35.l odd 12 1
1225.1.j.a 4 1.a even 1 1 trivial
1225.1.j.a 4 5.b even 2 1 inner
1225.1.j.a 4 7.b odd 2 1 CM
1225.1.j.a 4 7.c even 3 1 inner
1225.1.j.a 4 7.d odd 6 1 inner
1225.1.j.a 4 35.c odd 2 1 inner
1225.1.j.a 4 35.i odd 6 1 inner
1225.1.j.a 4 35.j even 6 1 inner
1575.1.e.a 2 21.g even 6 1
1575.1.e.a 2 21.h odd 6 1
1575.1.e.a 2 105.o odd 6 1
1575.1.e.a 2 105.p even 6 1
1575.1.h.a 1 105.w odd 12 1
1575.1.h.a 1 105.x even 12 1
1575.1.h.c 1 105.w odd 12 1
1575.1.h.c 1 105.x even 12 1
2800.1.f.a 1 140.w even 12 1
2800.1.f.a 1 140.x odd 12 1
2800.1.f.b 1 140.w even 12 1
2800.1.f.b 1 140.x odd 12 1
2800.1.p.a 2 28.f even 6 1
2800.1.p.a 2 28.g odd 6 1
2800.1.p.a 2 140.p odd 6 1
2800.1.p.a 2 140.s even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T - 1)^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4} - T^{2} + 1$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 1)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4} - 4T^{2} + 16$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4} - T^{2} + 1$$
$71$ $$(T + 1)^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + T + 1)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$