# Properties

 Label 1352.1.g.b Level $1352$ Weight $1$ Character orbit 1352.g Self dual yes Analytic conductor $0.675$ Analytic rank $0$ Dimension $3$ Projective image $D_{7}$ CM discriminant -8 Inner twists $2$

# 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: yes Analytic conductor: $$0.674735897080$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.2471326208.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10})$$ q - q^2 - b1 * q^3 + q^4 + b1 * q^6 - q^8 + (b2 + 1) * q^9 $$q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} - q^{8} + (\beta_{2} + 1) q^{9} + (\beta_{2} - \beta_1 + 1) q^{11} - \beta_1 q^{12} + q^{16} + ( - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{2} - 1) q^{18} - \beta_{2} q^{19} + ( - \beta_{2} + \beta_1 - 1) q^{22} + \beta_1 q^{24} + q^{25} + ( - \beta_{2} - 1) q^{27} - q^{32} + ( - \beta_1 + 1) q^{33} + (\beta_{2} - \beta_1 + 1) q^{34} + (\beta_{2} + 1) q^{36} + \beta_{2} q^{38} - \beta_{2} q^{41} + \beta_{2} q^{43} + (\beta_{2} - \beta_1 + 1) q^{44} - \beta_1 q^{48} + q^{49} - q^{50} + (\beta_1 - 1) q^{51} + (\beta_{2} + 1) q^{54} + (\beta_{2} + 1) q^{57} + \beta_1 q^{59} + q^{64} + (\beta_1 - 1) q^{66} + \beta_1 q^{67} + ( - \beta_{2} + \beta_1 - 1) q^{68} + ( - \beta_{2} - 1) q^{72} + \beta_1 q^{73} - \beta_1 q^{75} - \beta_{2} q^{76} + \beta_1 q^{81} + \beta_{2} q^{82} - \beta_{2} q^{83} - \beta_{2} q^{86} + ( - \beta_{2} + \beta_1 - 1) q^{88} + \beta_1 q^{89} + \beta_1 q^{96} + (\beta_{2} - \beta_1 + 1) q^{97} - q^{98} + q^{99}+O(q^{100})$$ q - q^2 - b1 * q^3 + q^4 + b1 * q^6 - q^8 + (b2 + 1) * q^9 + (b2 - b1 + 1) * q^11 - b1 * q^12 + q^16 + (-b2 + b1 - 1) * q^17 + (-b2 - 1) * q^18 - b2 * q^19 + (-b2 + b1 - 1) * q^22 + b1 * q^24 + q^25 + (-b2 - 1) * q^27 - q^32 + (-b1 + 1) * q^33 + (b2 - b1 + 1) * q^34 + (b2 + 1) * q^36 + b2 * q^38 - b2 * q^41 + b2 * q^43 + (b2 - b1 + 1) * q^44 - b1 * q^48 + q^49 - q^50 + (b1 - 1) * q^51 + (b2 + 1) * q^54 + (b2 + 1) * q^57 + b1 * q^59 + q^64 + (b1 - 1) * q^66 + b1 * q^67 + (-b2 + b1 - 1) * q^68 + (-b2 - 1) * q^72 + b1 * q^73 - b1 * q^75 - b2 * q^76 + b1 * q^81 + b2 * q^82 - b2 * q^83 - b2 * q^86 + (-b2 + b1 - 1) * q^88 + b1 * q^89 + b1 * q^96 + (b2 - b1 + 1) * q^97 - q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 3 q^{8} + 2 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^6 - 3 * q^8 + 2 * q^9 $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 3 q^{8} + 2 q^{9} + q^{11} - q^{12} + 3 q^{16} - q^{17} - 2 q^{18} + q^{19} - q^{22} + q^{24} + 3 q^{25} - 2 q^{27} - 3 q^{32} + 2 q^{33} + q^{34} + 2 q^{36} - q^{38} + q^{41} - q^{43} + q^{44} - q^{48} + 3 q^{49} - 3 q^{50} - 2 q^{51} + 2 q^{54} + 2 q^{57} + q^{59} + 3 q^{64} - 2 q^{66} + q^{67} - q^{68} - 2 q^{72} + q^{73} - q^{75} + q^{76} + q^{81} - q^{82} + q^{83} + q^{86} - q^{88} + q^{89} + q^{96} + q^{97} - 3 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^6 - 3 * q^8 + 2 * q^9 + q^11 - q^12 + 3 * q^16 - q^17 - 2 * q^18 + q^19 - q^22 + q^24 + 3 * q^25 - 2 * q^27 - 3 * q^32 + 2 * q^33 + q^34 + 2 * q^36 - q^38 + q^41 - q^43 + q^44 - q^48 + 3 * q^49 - 3 * q^50 - 2 * q^51 + 2 * q^54 + 2 * q^57 + q^59 + 3 * q^64 - 2 * q^66 + q^67 - q^68 - 2 * q^72 + q^73 - q^75 + q^76 + q^81 - q^82 + q^83 + q^86 - q^88 + q^89 + q^96 + q^97 - 3 * q^98 + 3 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 1.80194 0.445042 −1.24698
−1.00000 −1.80194 1.00000 0 1.80194 0 −1.00000 2.24698 0
339.2 −1.00000 −0.445042 1.00000 0 0.445042 0 −1.00000 −0.801938 0
339.3 −1.00000 1.24698 1.00000 0 −1.24698 0 −1.00000 0.554958 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.1.g.b 3
8.d odd 2 1 CM 1352.1.g.b 3
13.b even 2 1 1352.1.g.c yes 3
13.c even 3 2 1352.1.n.c 6
13.d odd 4 2 1352.1.h.a 6
13.e even 6 2 1352.1.n.b 6
13.f odd 12 4 1352.1.p.c 12
104.h odd 2 1 1352.1.g.c yes 3
104.m even 4 2 1352.1.h.a 6
104.n odd 6 2 1352.1.n.c 6
104.p odd 6 2 1352.1.n.b 6
104.u even 12 4 1352.1.p.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1352.1.g.b 3 1.a even 1 1 trivial
1352.1.g.b 3 8.d odd 2 1 CM
1352.1.g.c yes 3 13.b even 2 1
1352.1.g.c yes 3 104.h odd 2 1
1352.1.h.a 6 13.d odd 4 2
1352.1.h.a 6 104.m even 4 2
1352.1.n.b 6 13.e even 6 2
1352.1.n.b 6 104.p odd 6 2
1352.1.n.c 6 13.c even 3 2
1352.1.n.c 6 104.n odd 6 2
1352.1.p.c 12 13.f odd 12 4
1352.1.p.c 12 104.u even 12 4

## Hecke kernels

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

 $$T_{3}^{3} + T_{3}^{2} - 2T_{3} - 1$$ T3^3 + T3^2 - 2*T3 - 1 $$T_{11}^{3} - T_{11}^{2} - 2T_{11} + 1$$ T11^3 - T11^2 - 2*T11 + 1

## Hecke characteristic polynomials

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