# Properties

 Label 1352.1.g.a Level $1352$ Weight $1$ Character orbit 1352.g Analytic conductor $0.675$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -104 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1352 = 2^{3} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1352.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.674735897080$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 104) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.104.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of 12.0.43436029431808.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{3} - q^{4} + i q^{5} + i q^{6} - i q^{7} + i q^{8} +O(q^{10})$$ q - z * q^2 - q^3 - q^4 + z * q^5 + z * q^6 - z * q^7 + z * q^8 $$q - i q^{2} - q^{3} - q^{4} + i q^{5} + i q^{6} - i q^{7} + i q^{8} + q^{10} + q^{12} - q^{14} - i q^{15} + q^{16} + q^{17} - i q^{20} + i q^{21} - i q^{24} + q^{27} + i q^{28} - q^{30} - i q^{31} - i q^{32} - i q^{34} + q^{35} - i q^{37} - q^{40} + q^{42} + q^{43} - i q^{47} - q^{48} - q^{51} - i q^{54} + q^{56} + i q^{60} - 2 q^{62} - q^{64} - q^{68} - i q^{70} + i q^{71} - q^{74} + i q^{80} - q^{81} - i q^{84} + i q^{85} - i q^{86} + 2 i q^{93} - q^{94} + i q^{96} +O(q^{100})$$ q - z * q^2 - q^3 - q^4 + z * q^5 + z * q^6 - z * q^7 + z * q^8 + q^10 + q^12 - q^14 - z * q^15 + q^16 + q^17 - z * q^20 + z * q^21 - z * q^24 + q^27 + z * q^28 - q^30 - z * q^31 - z * q^32 - z * q^34 + q^35 - z * q^37 - q^40 + q^42 + q^43 - z * q^47 - q^48 - q^51 - z * q^54 + q^56 + z * q^60 - 2 * q^62 - q^64 - q^68 - z * q^70 + z * q^71 - q^74 + z * q^80 - q^81 - z * q^84 + z * q^85 - z * q^86 + 2*z * q^93 - q^94 + z * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{10} + 2 q^{12} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{27} - 2 q^{30} + 2 q^{35} - 2 q^{40} + 2 q^{42} + 2 q^{43} - 2 q^{48} - 2 q^{51} + 2 q^{56} - 4 q^{62} - 2 q^{64} - 2 q^{68} - 2 q^{74} - 2 q^{81} - 2 q^{94}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^10 + 2 * q^12 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 2 * q^27 - 2 * q^30 + 2 * q^35 - 2 * q^40 + 2 * q^42 + 2 * q^43 - 2 * q^48 - 2 * q^51 + 2 * q^56 - 4 * q^62 - 2 * q^64 - 2 * q^68 - 2 * q^74 - 2 * q^81 - 2 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1015$$ $$1185$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
339.1
 1.00000i − 1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.00000i 1.00000i 0 1.00000
339.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.00000i 1.00000i 0 1.00000
 $$n$$: 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
104.h odd 2 1 CM by $$\Q(\sqrt{-26})$$
8.d odd 2 1 inner
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.1.g.a 2
8.d odd 2 1 inner 1352.1.g.a 2
13.b even 2 1 inner 1352.1.g.a 2
13.c even 3 2 1352.1.n.a 4
13.d odd 4 1 104.1.h.a 1
13.d odd 4 1 104.1.h.b yes 1
13.e even 6 2 1352.1.n.a 4
13.f odd 12 2 1352.1.p.a 2
13.f odd 12 2 1352.1.p.b 2
39.f even 4 1 936.1.o.a 1
39.f even 4 1 936.1.o.b 1
52.f even 4 1 416.1.h.a 1
52.f even 4 1 416.1.h.b 1
65.f even 4 1 2600.1.b.a 2
65.f even 4 1 2600.1.b.b 2
65.g odd 4 1 2600.1.o.b 1
65.g odd 4 1 2600.1.o.d 1
65.k even 4 1 2600.1.b.a 2
65.k even 4 1 2600.1.b.b 2
104.h odd 2 1 CM 1352.1.g.a 2
104.j odd 4 1 416.1.h.a 1
104.j odd 4 1 416.1.h.b 1
104.m even 4 1 104.1.h.a 1
104.m even 4 1 104.1.h.b yes 1
104.n odd 6 2 1352.1.n.a 4
104.p odd 6 2 1352.1.n.a 4
104.u even 12 2 1352.1.p.a 2
104.u even 12 2 1352.1.p.b 2
156.l odd 4 1 3744.1.o.a 1
156.l odd 4 1 3744.1.o.b 1
208.l even 4 1 3328.1.c.a 2
208.l even 4 1 3328.1.c.e 2
208.m odd 4 1 3328.1.c.a 2
208.m odd 4 1 3328.1.c.e 2
208.r odd 4 1 3328.1.c.a 2
208.r odd 4 1 3328.1.c.e 2
208.s even 4 1 3328.1.c.a 2
208.s even 4 1 3328.1.c.e 2
312.w odd 4 1 936.1.o.a 1
312.w odd 4 1 936.1.o.b 1
312.y even 4 1 3744.1.o.a 1
312.y even 4 1 3744.1.o.b 1
520.t even 4 1 2600.1.o.b 1
520.t even 4 1 2600.1.o.d 1
520.x odd 4 1 2600.1.b.a 2
520.x odd 4 1 2600.1.b.b 2
520.bk odd 4 1 2600.1.b.a 2
520.bk odd 4 1 2600.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 13.d odd 4 1
104.1.h.a 1 104.m even 4 1
104.1.h.b yes 1 13.d odd 4 1
104.1.h.b yes 1 104.m even 4 1
416.1.h.a 1 52.f even 4 1
416.1.h.a 1 104.j odd 4 1
416.1.h.b 1 52.f even 4 1
416.1.h.b 1 104.j odd 4 1
936.1.o.a 1 39.f even 4 1
936.1.o.a 1 312.w odd 4 1
936.1.o.b 1 39.f even 4 1
936.1.o.b 1 312.w odd 4 1
1352.1.g.a 2 1.a even 1 1 trivial
1352.1.g.a 2 8.d odd 2 1 inner
1352.1.g.a 2 13.b even 2 1 inner
1352.1.g.a 2 104.h odd 2 1 CM
1352.1.n.a 4 13.c even 3 2
1352.1.n.a 4 13.e even 6 2
1352.1.n.a 4 104.n odd 6 2
1352.1.n.a 4 104.p odd 6 2
1352.1.p.a 2 13.f odd 12 2
1352.1.p.a 2 104.u even 12 2
1352.1.p.b 2 13.f odd 12 2
1352.1.p.b 2 104.u even 12 2
2600.1.b.a 2 65.f even 4 1
2600.1.b.a 2 65.k even 4 1
2600.1.b.a 2 520.x odd 4 1
2600.1.b.a 2 520.bk odd 4 1
2600.1.b.b 2 65.f even 4 1
2600.1.b.b 2 65.k even 4 1
2600.1.b.b 2 520.x odd 4 1
2600.1.b.b 2 520.bk odd 4 1
2600.1.o.b 1 65.g odd 4 1
2600.1.o.b 1 520.t even 4 1
2600.1.o.d 1 65.g odd 4 1
2600.1.o.d 1 520.t even 4 1
3328.1.c.a 2 208.l even 4 1
3328.1.c.a 2 208.m odd 4 1
3328.1.c.a 2 208.r odd 4 1
3328.1.c.a 2 208.s even 4 1
3328.1.c.e 2 208.l even 4 1
3328.1.c.e 2 208.m odd 4 1
3328.1.c.e 2 208.r odd 4 1
3328.1.c.e 2 208.s even 4 1
3744.1.o.a 1 156.l odd 4 1
3744.1.o.a 1 312.y even 4 1
3744.1.o.b 1 156.l odd 4 1
3744.1.o.b 1 312.y even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1352, [\chi])$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{11}$$ T11

## Hecke characteristic polynomials

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