# Properties

 Label 1352.1.p.a Level $1352$ Weight $1$ Character orbit 1352.p 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.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.674735897080$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 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: $C_3\times S_3$ Artin field: Galois closure of 6.0.190102016.3

## $q$-expansion

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

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

## Character values

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

 $$n$$ $$677$$ $$1015$$ $$1185$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
147.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −1.00000 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 0 0.500000 0.866025i
699.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 0.500000 0.866025i 0.500000 0.866025i 1.00000 0 0.500000 + 0.866025i
 $$n$$: 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 $$\Q(\sqrt{-26})$$
13.c even 3 1 inner
104.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.1.p.a 2
8.d odd 2 1 1352.1.p.b 2
13.b even 2 1 1352.1.p.b 2
13.c even 3 1 104.1.h.b yes 1
13.c even 3 1 inner 1352.1.p.a 2
13.d odd 4 2 1352.1.n.a 4
13.e even 6 1 104.1.h.a 1
13.e even 6 1 1352.1.p.b 2
13.f odd 12 2 1352.1.g.a 2
13.f odd 12 2 1352.1.n.a 4
39.h odd 6 1 936.1.o.b 1
39.i odd 6 1 936.1.o.a 1
52.i odd 6 1 416.1.h.b 1
52.j odd 6 1 416.1.h.a 1
65.l even 6 1 2600.1.o.d 1
65.n even 6 1 2600.1.o.b 1
65.q odd 12 2 2600.1.b.a 2
65.r odd 12 2 2600.1.b.b 2
104.h odd 2 1 CM 1352.1.p.a 2
104.m even 4 2 1352.1.n.a 4
104.n odd 6 1 104.1.h.a 1
104.n odd 6 1 1352.1.p.b 2
104.p odd 6 1 104.1.h.b yes 1
104.p odd 6 1 inner 1352.1.p.a 2
104.r even 6 1 416.1.h.b 1
104.s even 6 1 416.1.h.a 1
104.u even 12 2 1352.1.g.a 2
104.u even 12 2 1352.1.n.a 4
156.p even 6 1 3744.1.o.b 1
156.r even 6 1 3744.1.o.a 1
208.bg odd 12 2 3328.1.c.a 2
208.bh even 12 2 3328.1.c.a 2
208.bi odd 12 2 3328.1.c.e 2
208.bj even 12 2 3328.1.c.e 2
312.ba even 6 1 936.1.o.a 1
312.bg odd 6 1 3744.1.o.b 1
312.bh odd 6 1 3744.1.o.a 1
312.bn even 6 1 936.1.o.b 1
520.bx odd 6 1 2600.1.o.d 1
520.cd odd 6 1 2600.1.o.b 1
520.cm even 12 2 2600.1.b.b 2
520.cs even 12 2 2600.1.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 13.e even 6 1
104.1.h.a 1 104.n odd 6 1
104.1.h.b yes 1 13.c even 3 1
104.1.h.b yes 1 104.p odd 6 1
416.1.h.a 1 52.j odd 6 1
416.1.h.a 1 104.s even 6 1
416.1.h.b 1 52.i odd 6 1
416.1.h.b 1 104.r even 6 1
936.1.o.a 1 39.i odd 6 1
936.1.o.a 1 312.ba even 6 1
936.1.o.b 1 39.h odd 6 1
936.1.o.b 1 312.bn even 6 1
1352.1.g.a 2 13.f odd 12 2
1352.1.g.a 2 104.u even 12 2
1352.1.n.a 4 13.d odd 4 2
1352.1.n.a 4 13.f odd 12 2
1352.1.n.a 4 104.m even 4 2
1352.1.n.a 4 104.u even 12 2
1352.1.p.a 2 1.a even 1 1 trivial
1352.1.p.a 2 13.c even 3 1 inner
1352.1.p.a 2 104.h odd 2 1 CM
1352.1.p.a 2 104.p odd 6 1 inner
1352.1.p.b 2 8.d odd 2 1
1352.1.p.b 2 13.b even 2 1
1352.1.p.b 2 13.e even 6 1
1352.1.p.b 2 104.n odd 6 1
2600.1.b.a 2 65.q odd 12 2
2600.1.b.a 2 520.cs even 12 2
2600.1.b.b 2 65.r odd 12 2
2600.1.b.b 2 520.cm even 12 2
2600.1.o.b 1 65.n even 6 1
2600.1.o.b 1 520.cd odd 6 1
2600.1.o.d 1 65.l even 6 1
2600.1.o.d 1 520.bx odd 6 1
3328.1.c.a 2 208.bg odd 12 2
3328.1.c.a 2 208.bh even 12 2
3328.1.c.e 2 208.bi odd 12 2
3328.1.c.e 2 208.bj even 12 2
3744.1.o.a 1 156.r even 6 1
3744.1.o.a 1 312.bh odd 6 1
3744.1.o.b 1 156.p even 6 1
3744.1.o.b 1 312.bg odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{5} + 1$$ T5 + 1

## Hecke characteristic polynomials

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