# Properties

 Label 1512.2.s.p Level $1512$ Weight $2$ Character orbit 1512.s Analytic conductor $12.073$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1512.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.9391935744.3 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 12 x^{5} - 76 x^{4} + 84 x^{3} + 245 x^{2} - 1372 x + 2401$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} - \beta_{5} + \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{4} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{3} - \beta_{5} + \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{4} ) q^{7} + ( 1 - \beta_{1} + \beta_{3} + \beta_{7} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{13} + ( -1 - \beta_{3} - 3 \beta_{6} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{23} + ( 1 + \beta_{3} + 2 \beta_{6} ) q^{25} + ( -3 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{29} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{31} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{35} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{37} + ( 1 + 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{41} + ( -1 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{43} + ( -5 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{47} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{49} + ( 2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{53} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{55} + ( -5 - \beta_{1} - 5 \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{59} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{61} + ( 1 - \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{65} + ( \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{6} - \beta_{7} ) q^{67} + ( -6 + \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{71} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{73} + ( -7 + \beta_{1} + \beta_{3} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{77} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{7} ) q^{79} + ( 4 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{83} + ( 8 - 2 \beta_{5} ) q^{85} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - \beta_{5} + \beta_{6} ) q^{89} + ( -7 + \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + \beta_{4} - 2 \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{91} + ( 8 - \beta_{1} + 8 \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{95} + ( 1 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{5} - 2q^{7} + O(q^{10})$$ $$8q + 4q^{5} - 2q^{7} + 2q^{11} - 8q^{13} - 4q^{17} - 6q^{19} - 2q^{23} + 4q^{25} - 16q^{29} - 6q^{31} + 2q^{35} + 16q^{41} + 20q^{47} - 6q^{49} + 10q^{53} + 16q^{55} - 22q^{59} + 2q^{61} + 14q^{65} + 2q^{67} - 44q^{71} - 10q^{73} - 54q^{77} + 8q^{79} + 40q^{83} + 64q^{85} + 16q^{89} - 24q^{91} + 30q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 5 x^{6} + 12 x^{5} - 76 x^{4} + 84 x^{3} + 245 x^{2} - 1372 x + 2401$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 3 \nu^{6} - 26 \nu^{5} - 194 \nu^{4} - 1282 \nu^{3} + 3290 \nu^{2} - 1911 \nu + 343$$$$)/12348$$ $$\beta_{3}$$ $$=$$ $$($$$$13 \nu^{7} + 18 \nu^{6} + 128 \nu^{5} + 506 \nu^{4} - 1520 \nu^{3} + 574 \nu^{2} + 2205 \nu - 42532$$$$)/37044$$ $$\beta_{4}$$ $$=$$ $$($$$$19 \nu^{7} + 15 \nu^{6} + 74 \nu^{5} - 346 \nu^{4} + 334 \nu^{3} - 518 \nu^{2} - 8967 \nu + 3773$$$$)/12348$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} - 4 \nu^{6} - 10 \nu^{5} + 48 \nu^{4} - 146 \nu^{3} - 468 \nu^{2} + 1169 \nu - 1960$$$$)/1764$$ $$\beta_{6}$$ $$=$$ $$($$$$-43 \nu^{7} + 53 \nu^{6} + 114 \nu^{5} - 866 \nu^{4} + 762 \nu^{3} - 2506 \nu^{2} - 5929 \nu + 27783$$$$)/24696$$ $$\beta_{7}$$ $$=$$ $$($$$$-124 \nu^{7} + 405 \nu^{6} - 746 \nu^{5} - 2384 \nu^{4} + 5882 \nu^{3} + 224 \nu^{2} - 34398 \nu + 154693$$$$)/37044$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-2 \beta_{6} - 2 \beta_{5} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{6} - 4 \beta_{5} + \beta_{4} - 6 \beta_{3} - 5 \beta_{2} + 2 \beta_{1} - 7$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - 10 \beta_{6} - 4 \beta_{5} - 11 \beta_{4} + 25 \beta_{3} - 11 \beta_{2} - 10 \beta_{1} + 35$$ $$\nu^{5}$$ $$=$$ $$-17 \beta_{7} + 32 \beta_{6} - 28 \beta_{5} + 14 \beta_{4} + 67 \beta_{3} - 17 \beta_{2} + 14 \beta_{1} + 77$$ $$\nu^{6}$$ $$=$$ $$74 \beta_{7} - 24 \beta_{6} + 50 \beta_{4} + 362 \beta_{3} - 24 \beta_{2} + 74 \beta_{1} + 119$$ $$\nu^{7}$$ $$=$$ $$26 \beta_{7} - 272 \beta_{6} + 52 \beta_{5} + 338 \beta_{4} + 14 \beta_{3} + 169 \beta_{1} + 168$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 1.57052 + 2.12920i 0.295509 − 2.62920i 2.61033 − 0.431486i −2.47635 + 0.931486i 1.57052 − 2.12920i 0.295509 + 2.62920i 2.61033 + 0.431486i −2.47635 − 0.931486i
0 0 0 −0.366025 0.633975i 0 −2.62920 + 0.295509i 0 0 0
865.2 0 0 0 −0.366025 0.633975i 0 2.12920 + 1.57052i 0 0 0
865.3 0 0 0 1.36603 + 2.36603i 0 −0.931486 + 2.47635i 0 0 0
865.4 0 0 0 1.36603 + 2.36603i 0 0.431486 2.61033i 0 0 0
1297.1 0 0 0 −0.366025 + 0.633975i 0 −2.62920 0.295509i 0 0 0
1297.2 0 0 0 −0.366025 + 0.633975i 0 2.12920 1.57052i 0 0 0
1297.3 0 0 0 1.36603 2.36603i 0 −0.931486 2.47635i 0 0 0
1297.4 0 0 0 1.36603 2.36603i 0 0.431486 + 2.61033i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1297.4 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.s.p yes 8
3.b odd 2 1 1512.2.s.m 8
7.c even 3 1 inner 1512.2.s.p yes 8
21.h odd 6 1 1512.2.s.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.s.m 8 3.b odd 2 1
1512.2.s.m 8 21.h odd 6 1
1512.2.s.p yes 8 1.a even 1 1 trivial
1512.2.s.p yes 8 7.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$:

 $$T_{5}^{4} - 2 T_{5}^{3} + 6 T_{5}^{2} + 4 T_{5} + 4$$ $$T_{11}^{8} - \cdots$$ $$T_{13}^{4} + 4 T_{13}^{3} - 35 T_{13}^{2} - 204 T_{13} - 264$$