# Properties

 Label 1008.2.t.h Level $1008$ Weight $2$ Character orbit 1008.t Analytic conductor $8.049$ Analytic rank $0$ Dimension $6$ 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.t (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Defining polynomial: $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{4} - \beta_{5} ) q^{3} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} ) q^{7} + ( -1 - 2 \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{4} - \beta_{5} ) q^{3} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} ) q^{7} + ( -1 - 2 \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{9} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{11} + ( 3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{13} + ( -2 + 2 \beta_{1} - \beta_{3} + 3 \beta_{4} ) q^{15} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{17} + ( -3 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} ) q^{19} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{21} + ( -2 + \beta_{2} + \beta_{3} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{25} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{27} + ( -2 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} ) q^{29} + ( -1 - \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{31} + ( -2 + 2 \beta_{1} - \beta_{3} + 3 \beta_{4} ) q^{33} + ( 3 + \beta_{1} + 4 \beta_{4} + 2 \beta_{5} ) q^{35} -\beta_{4} q^{37} + ( 2 - 5 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{39} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{41} + ( -3 + 6 \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{43} + ( -2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{45} + ( 3 - 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} ) q^{47} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{49} + ( 2 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{51} + ( 5 - \beta_{1} - \beta_{3} + 5 \beta_{4} ) q^{53} + ( 6 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{55} + ( 6 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 6 \beta_{4} - 5 \beta_{5} ) q^{57} + ( -2 + 3 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{59} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{61} + ( -8 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{63} + ( -2 + \beta_{1} + 6 \beta_{2} + \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{65} + ( 4 - 3 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} ) q^{67} + ( 1 - \beta_{1} - \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{69} + ( -6 + 2 \beta_{1} - 5 \beta_{2} - 9 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 8 + \beta_{1} + 5 \beta_{2} + \beta_{3} + 8 \beta_{4} - 5 \beta_{5} ) q^{73} + ( 5 - 5 \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{75} + ( 3 + \beta_{1} + 4 \beta_{4} + 2 \beta_{5} ) q^{77} + ( 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - 5 \beta_{5} ) q^{79} + ( -10 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{81} + ( 3 - 5 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{83} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{85} + ( 10 + 5 \beta_{1} - \beta_{3} + 12 \beta_{4} - 3 \beta_{5} ) q^{87} + ( 1 - 5 \beta_{1} + \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{89} + ( 11 + 3 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - 7 \beta_{5} ) q^{91} + ( 4 + 2 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{93} + ( -2 - \beta_{1} - 2 \beta_{3} - \beta_{4} + 5 \beta_{5} ) q^{95} + ( 2 + 2 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} ) q^{97} + ( -2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{3} - 2q^{5} + 4q^{7} - 4q^{9} + O(q^{10})$$ $$6q - 2q^{3} - 2q^{5} + 4q^{7} - 4q^{9} - 2q^{11} + 8q^{13} - 12q^{15} - 4q^{17} + 3q^{19} - 7q^{21} - 14q^{23} - 4q^{25} + 7q^{27} - 5q^{29} - 20q^{31} - 12q^{33} + 13q^{35} + 3q^{37} - q^{39} + 6q^{43} + 3q^{45} + 9q^{47} - 12q^{49} + 18q^{51} + 15q^{53} + 26q^{55} + 22q^{57} + 14q^{59} + 8q^{61} - 26q^{63} - 12q^{65} - q^{67} + 3q^{69} - 14q^{71} + 19q^{73} + 25q^{75} + 13q^{77} - 5q^{79} - 40q^{81} - 2q^{83} - 2q^{85} + 36q^{87} - 9q^{89} + 46q^{91} + 37q^{93} + 4q^{95} + 28q^{97} + 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + 5 \nu^{3} + \nu^{2} + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 5 \nu^{3} + 2 \nu^{2} - 3 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 33 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 6 \beta_{1} + 6$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{5} - 2 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} + 7$$

## Character values

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

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$ $$1$$ $$-1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 + 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 − 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i
0 −1.29418 1.15113i 0 1.58836 0 2.64400 + 0.0963576i 0 0.349814 + 2.97954i 0
193.2 0 −0.796790 + 1.53790i 0 0.593579 0 0.0665372 2.64491i 0 −1.73025 2.45076i 0
193.3 0 1.09097 + 1.34528i 0 −3.18194 0 −0.710533 + 2.54856i 0 −0.619562 + 2.93533i 0
961.1 0 −1.29418 + 1.15113i 0 1.58836 0 2.64400 0.0963576i 0 0.349814 2.97954i 0
961.2 0 −0.796790 1.53790i 0 0.593579 0 0.0665372 + 2.64491i 0 −1.73025 + 2.45076i 0
961.3 0 1.09097 1.34528i 0 −3.18194 0 −0.710533 2.54856i 0 −0.619562 2.93533i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.3 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.g 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.t.h 6
3.b odd 2 1 3024.2.t.h 6
4.b odd 2 1 126.2.h.d yes 6
7.c even 3 1 1008.2.q.g 6
9.c even 3 1 1008.2.q.g 6
9.d odd 6 1 3024.2.q.g 6
12.b even 2 1 378.2.h.c 6
21.h odd 6 1 3024.2.q.g 6
28.d even 2 1 882.2.h.p 6
28.f even 6 1 882.2.e.o 6
28.f even 6 1 882.2.f.o 6
28.g odd 6 1 126.2.e.c 6
28.g odd 6 1 882.2.f.n 6
36.f odd 6 1 126.2.e.c 6
36.f odd 6 1 1134.2.g.m 6
36.h even 6 1 378.2.e.d 6
36.h even 6 1 1134.2.g.l 6
63.g even 3 1 inner 1008.2.t.h 6
63.n odd 6 1 3024.2.t.h 6
84.h odd 2 1 2646.2.h.o 6
84.j odd 6 1 2646.2.e.p 6
84.j odd 6 1 2646.2.f.m 6
84.n even 6 1 378.2.e.d 6
84.n even 6 1 2646.2.f.l 6
252.n even 6 1 882.2.h.p 6
252.n even 6 1 7938.2.a.bw 3
252.o even 6 1 378.2.h.c 6
252.o even 6 1 7938.2.a.ca 3
252.r odd 6 1 2646.2.f.m 6
252.s odd 6 1 2646.2.e.p 6
252.u odd 6 1 882.2.f.n 6
252.u odd 6 1 1134.2.g.m 6
252.bb even 6 1 1134.2.g.l 6
252.bb even 6 1 2646.2.f.l 6
252.bi even 6 1 882.2.e.o 6
252.bj even 6 1 882.2.f.o 6
252.bl odd 6 1 126.2.h.d yes 6
252.bl odd 6 1 7938.2.a.bv 3
252.bn odd 6 1 2646.2.h.o 6
252.bn odd 6 1 7938.2.a.bz 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 28.g odd 6 1
126.2.e.c 6 36.f odd 6 1
126.2.h.d yes 6 4.b odd 2 1
126.2.h.d yes 6 252.bl odd 6 1
378.2.e.d 6 36.h even 6 1
378.2.e.d 6 84.n even 6 1
378.2.h.c 6 12.b even 2 1
378.2.h.c 6 252.o even 6 1
882.2.e.o 6 28.f even 6 1
882.2.e.o 6 252.bi even 6 1
882.2.f.n 6 28.g odd 6 1
882.2.f.n 6 252.u odd 6 1
882.2.f.o 6 28.f even 6 1
882.2.f.o 6 252.bj even 6 1
882.2.h.p 6 28.d even 2 1
882.2.h.p 6 252.n even 6 1
1008.2.q.g 6 7.c even 3 1
1008.2.q.g 6 9.c even 3 1
1008.2.t.h 6 1.a even 1 1 trivial
1008.2.t.h 6 63.g even 3 1 inner
1134.2.g.l 6 36.h even 6 1
1134.2.g.l 6 252.bb even 6 1
1134.2.g.m 6 36.f odd 6 1
1134.2.g.m 6 252.u odd 6 1
2646.2.e.p 6 84.j odd 6 1
2646.2.e.p 6 252.s odd 6 1
2646.2.f.l 6 84.n even 6 1
2646.2.f.l 6 252.bb even 6 1
2646.2.f.m 6 84.j odd 6 1
2646.2.f.m 6 252.r odd 6 1
2646.2.h.o 6 84.h odd 2 1
2646.2.h.o 6 252.bn odd 6 1
3024.2.q.g 6 9.d odd 6 1
3024.2.q.g 6 21.h odd 6 1
3024.2.t.h 6 3.b odd 2 1
3024.2.t.h 6 63.n odd 6 1
7938.2.a.bv 3 252.bl odd 6 1
7938.2.a.bw 3 252.n even 6 1
7938.2.a.bz 3 252.bn odd 6 1
7938.2.a.ca 3 252.o even 6 1

## Hecke kernels

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

 $$T_{5}^{3} + T_{5}^{2} - 6 T_{5} + 3$$ $$T_{11}^{3} + T_{11}^{2} - 6 T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 2 T + 4 T^{2} + 3 T^{3} + 12 T^{4} + 18 T^{5} + 27 T^{6}$$
$5$ $$( 1 + T + 9 T^{2} + 13 T^{3} + 45 T^{4} + 25 T^{5} + 125 T^{6} )^{2}$$
$7$ $$1 - 4 T + 14 T^{2} - 55 T^{3} + 98 T^{4} - 196 T^{5} + 343 T^{6}$$
$11$ $$( 1 + T + 27 T^{2} + 25 T^{3} + 297 T^{4} + 121 T^{5} + 1331 T^{6} )^{2}$$
$13$ $$1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 18304 T^{7} - 5408 T^{8} - 92274 T^{9} + 685464 T^{10} - 2970344 T^{11} + 4826809 T^{12}$$
$17$ $$1 + 4 T - 23 T^{2} - 68 T^{3} + 410 T^{4} + 220 T^{5} - 8111 T^{6} + 3740 T^{7} + 118490 T^{8} - 334084 T^{9} - 1920983 T^{10} + 5679428 T^{11} + 24137569 T^{12}$$
$19$ $$1 - 3 T - 12 T^{2} + 67 T^{3} - 153 T^{4} - 54 T^{5} + 6315 T^{6} - 1026 T^{7} - 55233 T^{8} + 459553 T^{9} - 1563852 T^{10} - 7428297 T^{11} + 47045881 T^{12}$$
$23$ $$( 1 + 7 T + 81 T^{2} + 325 T^{3} + 1863 T^{4} + 3703 T^{5} + 12167 T^{6} )^{2}$$
$29$ $$1 + 5 T + 4 T^{2} + 251 T^{3} + 197 T^{4} - 3418 T^{5} + 20293 T^{6} - 99122 T^{7} + 165677 T^{8} + 6121639 T^{9} + 2829124 T^{10} + 102555745 T^{11} + 594823321 T^{12}$$
$31$ $$1 + 20 T + 186 T^{2} + 1398 T^{3} + 10342 T^{4} + 62234 T^{5} + 331987 T^{6} + 1929254 T^{7} + 9938662 T^{8} + 41647818 T^{9} + 171774906 T^{10} + 572583020 T^{11} + 887503681 T^{12}$$
$37$ $$( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3}$$
$41$ $$1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 33210 T^{7} + 7413210 T^{8} + 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12}$$
$43$ $$( 1 - 18 T + 198 T^{2} - 1519 T^{3} + 8514 T^{4} - 33282 T^{5} + 79507 T^{6} )( 1 + 12 T - 6 T^{2} - 547 T^{3} - 258 T^{4} + 22188 T^{5} + 79507 T^{6} )$$
$47$ $$1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 161586 T^{7} - 5374497 T^{8} + 55130013 T^{9} - 29278086 T^{10} - 2064105063 T^{11} + 10779215329 T^{12}$$
$53$ $$1 - 15 T - 33 T^{3} + 13635 T^{4} - 60360 T^{5} - 225155 T^{6} - 3199080 T^{7} + 38300715 T^{8} - 4912941 T^{9} - 6272932395 T^{11} + 22164361129 T^{12}$$
$59$ $$1 - 14 T - 20 T^{2} + 154 T^{3} + 11666 T^{4} - 35126 T^{5} - 499301 T^{6} - 2072434 T^{7} + 40609346 T^{8} + 31628366 T^{9} - 242347220 T^{10} - 10008940186 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 806054 T^{7} + 51208402 T^{8} + 77627502 T^{9} - 1578425874 T^{10} - 6756770408 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + T - 88 T^{2} + 243 T^{3} + 2035 T^{4} - 14290 T^{5} + 72259 T^{6} - 957430 T^{7} + 9135115 T^{8} + 73085409 T^{9} - 1773298648 T^{10} + 1350125107 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 + 7 T + 15 T^{2} - 599 T^{3} + 1065 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12}$$
$79$ $$1 + 5 T - 138 T^{2} - 123 T^{3} + 11347 T^{4} - 21118 T^{5} - 1048937 T^{6} - 1668322 T^{7} + 70816627 T^{8} - 60643797 T^{9} - 5375111178 T^{10} + 15385281995 T^{11} + 243087455521 T^{12}$$
$83$ $$1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 1125812 T^{7} + 129430532 T^{8} + 1143574 T^{9} - 8637414422 T^{10} + 7878081286 T^{11} + 326940373369 T^{12}$$
$89$ $$1 + 9 T - 144 T^{2} - 1197 T^{3} + 16101 T^{4} + 73314 T^{5} - 1141967 T^{6} + 6524946 T^{7} + 127536021 T^{8} - 843847893 T^{9} - 9034882704 T^{10} + 50256535041 T^{11} + 496981290961 T^{12}$$
$97$ $$1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 37655012 T^{7} + 425079802 T^{8} - 2486121252 T^{9} + 24876727961 T^{10} - 240445527196 T^{11} + 832972004929 T^{12}$$