# Properties

 Label 1008.2.q.i Level $1008$ Weight $2$ Character orbit 1008.q Analytic conductor $8.049$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.991381711347.1 Defining polynomial: $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + ( 1 - \beta_{1} + \beta_{5} - \beta_{9} ) q^{7} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{8} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + ( 1 - \beta_{1} + \beta_{5} - \beta_{9} ) q^{7} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{9} + ( -1 + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{11} + ( -2 + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{13} + ( \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{15} + ( -\beta_{1} - \beta_{2} + 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{17} + ( -\beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{19} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{9} ) q^{21} + ( \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{23} + ( 2 + \beta_{2} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} ) q^{25} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{27} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{29} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{31} + ( 3 - \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{9} ) q^{33} + ( 1 - \beta_{2} - 3 \beta_{4} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} ) q^{35} + ( -2 \beta_{5} - 2 \beta_{9} ) q^{37} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} ) q^{39} + ( 2 + \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{41} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{7} - \beta_{8} ) q^{43} + ( -1 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + 3 \beta_{8} ) q^{45} + ( 5 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{47} + ( 2 - 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{49} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{51} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 3 \beta_{6} ) q^{53} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{8} - \beta_{9} ) q^{55} + ( -5 + \beta_{1} - \beta_{3} + \beta_{4} - 4 \beta_{5} + 6 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{57} + ( 7 - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{59} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{61} + ( 6 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{63} + ( 2 - \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} ) q^{65} + ( 2 - 5 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{67} + ( 2 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{9} ) q^{69} + ( -2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{71} + ( \beta_{2} + 3 \beta_{3} + 4 \beta_{6} - \beta_{7} - \beta_{8} ) q^{73} + ( -1 + 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{75} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + 5 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{77} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{5} - \beta_{7} - \beta_{8} ) q^{79} + ( 1 + 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} ) q^{81} + ( -2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{83} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{85} + ( 1 + 4 \beta_{1} - \beta_{3} + 3 \beta_{4} - \beta_{5} - 6 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{87} + ( 8 + 3 \beta_{5} - 8 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{89} + ( 4 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{91} + ( -2 \beta_{1} + \beta_{2} + 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{93} + ( -2 + \beta_{3} - 2 \beta_{4} + 2 \beta_{7} - \beta_{9} ) q^{95} + ( \beta_{1} + 6 \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} ) q^{97} + ( -5 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + q^{3} + 4q^{5} + 4q^{7} + 11q^{9} + O(q^{10})$$ $$10q + q^{3} + 4q^{5} + 4q^{7} + 11q^{9} - 4q^{11} - 8q^{13} + 19q^{15} + 12q^{17} - q^{19} + 13q^{21} - 3q^{23} - q^{25} + 7q^{27} + 7q^{29} - 6q^{31} + 14q^{33} - 5q^{35} - 2q^{39} + 5q^{41} + 7q^{43} - 16q^{45} + 54q^{47} - 8q^{49} + 9q^{51} - 21q^{53} - 4q^{55} - 4q^{57} + 60q^{59} + 28q^{61} + 59q^{63} + 22q^{65} - 4q^{67} + 15q^{69} + 6q^{71} + 15q^{73} + 14q^{75} + 11q^{77} - 8q^{79} + 23q^{81} - 9q^{83} - 6q^{85} - 2q^{87} + 28q^{89} + 4q^{91} - 6q^{93} - 28q^{95} - 12q^{97} - 35q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{9} + 9 \nu^{8} - 3 \nu^{7} + 95 \nu^{6} + 18 \nu^{5} + 402 \nu^{4} - 87 \nu^{3} + 936 \nu^{2} + 342 \nu + 72$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{9} + \nu^{8} - 12 \nu^{7} - 8 \nu^{6} - 68 \nu^{5} - 30 \nu^{4} - 123 \nu^{3} - 204 \nu^{2} - 270 \nu - 63$$$$)/63$$ $$\beta_{4}$$ $$=$$ $$($$$$17 \nu^{9} - 24 \nu^{8} + 159 \nu^{7} - 106 \nu^{6} + 786 \nu^{5} - 417 \nu^{4} + 1893 \nu^{3} - 27 \nu^{2} + 1395 \nu + 639$$$$)/567$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180$$$$)/567$$ $$\beta_{6}$$ $$=$$ $$($$$$20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu + 504$$$$)/567$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{9} - 12 \nu^{8} + 69 \nu^{7} - 43 \nu^{6} + 330 \nu^{5} - 219 \nu^{4} + 732 \nu^{3} - 45 \nu^{2} + 477 \nu - 306$$$$)/189$$ $$\beta_{8}$$ $$=$$ $$($$$$-71 \nu^{9} + 123 \nu^{8} - 591 \nu^{7} + 403 \nu^{6} - 2604 \nu^{5} + 1794 \nu^{4} - 5214 \nu^{3} - 1458 \nu^{2} - 1476 \nu - 234$$$$)/567$$ $$\beta_{9}$$ $$=$$ $$($$$$-82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720$$$$)/567$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} + 3 \beta_{6} + \beta_{4} - \beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} + 4 \beta_{5} - \beta_{3} - 4 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{8} - 5 \beta_{7} - 13 \beta_{6} - \beta_{4} - \beta_{3} + 4 \beta_{2} - \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{9} + \beta_{8} - 2 \beta_{7} - 7 \beta_{6} - 19 \beta_{5} - \beta_{4} + \beta_{2} + 7$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{9} + 9 \beta_{8} + 15 \beta_{7} - 10 \beta_{5} - 15 \beta_{4} + 10 \beta_{3} + 9 \beta_{2} + 10 \beta_{1} + 61$$ $$\nu^{7}$$ $$=$$ $$11 \beta_{8} + 11 \beta_{7} + 46 \beta_{6} + 19 \beta_{4} + 43 \beta_{3} + 8 \beta_{2} + 94 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$73 \beta_{9} + 56 \beta_{8} + 62 \beta_{7} + 298 \beta_{6} + 76 \beta_{5} + 118 \beta_{4} - 118 \beta_{2} - 298$$ $$\nu^{9}$$ $$=$$ $$253 \beta_{9} - 135 \beta_{8} + 48 \beta_{7} + 478 \beta_{5} - 48 \beta_{4} - 253 \beta_{3} - 135 \beta_{2} - 478 \beta_{1} - 295$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{6}$$ $$1$$ $$-1 + \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1
 −0.335166 + 0.580525i 0.920620 − 1.59456i 0.247934 − 0.429435i 1.19343 − 2.06709i −1.02682 + 1.77851i −0.335166 − 0.580525i 0.920620 + 1.59456i 0.247934 + 0.429435i 1.19343 + 2.06709i −1.02682 − 1.77851i
0 −1.65263 + 0.518475i 0 −0.712469 + 1.23403i 0 2.36039 + 1.19522i 0 2.46237 1.71369i 0
529.2 0 −1.39291 1.02946i 0 −0.667377 + 1.15593i 0 −1.90267 1.83844i 0 0.880416 + 2.86790i 0
529.3 0 0.221298 + 1.71786i 0 1.84629 3.19787i 0 −0.926641 2.47817i 0 −2.90205 + 0.760316i 0
529.4 0 1.61557 0.624446i 0 1.46043 2.52954i 0 0.138560 + 2.64212i 0 2.22013 2.01767i 0
529.5 0 1.70867 + 0.283604i 0 0.0731228 0.126652i 0 2.33035 1.25278i 0 2.83914 + 0.969173i 0
625.1 0 −1.65263 0.518475i 0 −0.712469 1.23403i 0 2.36039 1.19522i 0 2.46237 + 1.71369i 0
625.2 0 −1.39291 + 1.02946i 0 −0.667377 1.15593i 0 −1.90267 + 1.83844i 0 0.880416 2.86790i 0
625.3 0 0.221298 1.71786i 0 1.84629 + 3.19787i 0 −0.926641 + 2.47817i 0 −2.90205 0.760316i 0
625.4 0 1.61557 + 0.624446i 0 1.46043 + 2.52954i 0 0.138560 2.64212i 0 2.22013 + 2.01767i 0
625.5 0 1.70867 0.283604i 0 0.0731228 + 0.126652i 0 2.33035 + 1.25278i 0 2.83914 0.969173i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 625.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.q.i 10
3.b odd 2 1 3024.2.q.i 10
4.b odd 2 1 63.2.h.b yes 10
7.c even 3 1 1008.2.t.i 10
9.c even 3 1 1008.2.t.i 10
9.d odd 6 1 3024.2.t.i 10
12.b even 2 1 189.2.h.b 10
21.h odd 6 1 3024.2.t.i 10
28.d even 2 1 441.2.h.f 10
28.f even 6 1 441.2.f.f 10
28.f even 6 1 441.2.g.f 10
28.g odd 6 1 63.2.g.b 10
28.g odd 6 1 441.2.f.e 10
36.f odd 6 1 63.2.g.b 10
36.f odd 6 1 567.2.e.f 10
36.h even 6 1 189.2.g.b 10
36.h even 6 1 567.2.e.e 10
63.h even 3 1 inner 1008.2.q.i 10
63.j odd 6 1 3024.2.q.i 10
84.h odd 2 1 1323.2.h.f 10
84.j odd 6 1 1323.2.f.f 10
84.j odd 6 1 1323.2.g.f 10
84.n even 6 1 189.2.g.b 10
84.n even 6 1 1323.2.f.e 10
252.n even 6 1 441.2.f.f 10
252.o even 6 1 567.2.e.e 10
252.o even 6 1 1323.2.f.e 10
252.r odd 6 1 1323.2.h.f 10
252.r odd 6 1 3969.2.a.bb 5
252.s odd 6 1 1323.2.g.f 10
252.u odd 6 1 63.2.h.b yes 10
252.u odd 6 1 3969.2.a.z 5
252.bb even 6 1 189.2.h.b 10
252.bb even 6 1 3969.2.a.bc 5
252.bi even 6 1 441.2.g.f 10
252.bj even 6 1 441.2.h.f 10
252.bj even 6 1 3969.2.a.ba 5
252.bl odd 6 1 441.2.f.e 10
252.bl odd 6 1 567.2.e.f 10
252.bn odd 6 1 1323.2.f.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 28.g odd 6 1
63.2.g.b 10 36.f odd 6 1
63.2.h.b yes 10 4.b odd 2 1
63.2.h.b yes 10 252.u odd 6 1
189.2.g.b 10 36.h even 6 1
189.2.g.b 10 84.n even 6 1
189.2.h.b 10 12.b even 2 1
189.2.h.b 10 252.bb even 6 1
441.2.f.e 10 28.g odd 6 1
441.2.f.e 10 252.bl odd 6 1
441.2.f.f 10 28.f even 6 1
441.2.f.f 10 252.n even 6 1
441.2.g.f 10 28.f even 6 1
441.2.g.f 10 252.bi even 6 1
441.2.h.f 10 28.d even 2 1
441.2.h.f 10 252.bj even 6 1
567.2.e.e 10 36.h even 6 1
567.2.e.e 10 252.o even 6 1
567.2.e.f 10 36.f odd 6 1
567.2.e.f 10 252.bl odd 6 1
1008.2.q.i 10 1.a even 1 1 trivial
1008.2.q.i 10 63.h even 3 1 inner
1008.2.t.i 10 7.c even 3 1
1008.2.t.i 10 9.c even 3 1
1323.2.f.e 10 84.n even 6 1
1323.2.f.e 10 252.o even 6 1
1323.2.f.f 10 84.j odd 6 1
1323.2.f.f 10 252.bn odd 6 1
1323.2.g.f 10 84.j odd 6 1
1323.2.g.f 10 252.s odd 6 1
1323.2.h.f 10 84.h odd 2 1
1323.2.h.f 10 252.r odd 6 1
3024.2.q.i 10 3.b odd 2 1
3024.2.q.i 10 63.j odd 6 1
3024.2.t.i 10 9.d odd 6 1
3024.2.t.i 10 21.h odd 6 1
3969.2.a.z 5 252.u odd 6 1
3969.2.a.ba 5 252.bj even 6 1
3969.2.a.bb 5 252.r odd 6 1
3969.2.a.bc 5 252.bb even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{10} - \cdots$$ $$T_{11}^{10} + \cdots$$