# Properties

 Label 672.2.q.l Level 672 Weight 2 Character orbit 672.q Analytic conductor 5.366 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 672.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1156923.1 Defining polynomial: $$x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 27 x^{2} - 18 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{7} + ( -1 - \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{7} + ( -1 - \beta_{2} ) q^{9} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{11} + ( 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{13} -\beta_{1} q^{15} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{17} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{21} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{23} + ( -1 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{25} - q^{27} + ( 4 - \beta_{1} ) q^{29} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{31} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{33} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{35} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{37} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{39} + ( -4 + 2 \beta_{3} - 2 \beta_{5} ) q^{41} + ( -4 - \beta_{3} + \beta_{5} ) q^{43} -\beta_{4} q^{45} + ( 4 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{47} + ( -1 + 2 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{5} ) q^{49} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{51} + ( 2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{53} + ( 4 - 3 \beta_{1} ) q^{55} + ( -2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{57} + ( 1 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{59} + ( 6 + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( -1 + \beta_{3} ) q^{63} + ( -8 - 6 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{65} + ( -1 - \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{67} + ( -2 - 2 \beta_{1} ) q^{69} + 2 \beta_{1} q^{71} + ( -1 - \beta_{1} + 10 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{75} + ( -8 + \beta_{1} - 12 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{77} + ( -9 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{79} + \beta_{2} q^{81} + ( -5 + 4 \beta_{1} - \beta_{3} + \beta_{5} ) q^{83} + ( 14 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{85} + ( -4 \beta_{2} - \beta_{4} ) q^{87} + ( 4 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{89} + ( 7 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{91} + ( -1 + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{93} + ( -2 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{95} + ( 1 - \beta_{3} + \beta_{5} ) q^{97} + ( -1 + \beta_{3} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{3} + 3q^{7} - 3q^{9} + O(q^{10})$$ $$6q + 3q^{3} + 3q^{7} - 3q^{9} - 6q^{13} - 6q^{17} + 3q^{19} - 6q^{23} - 3q^{25} - 6q^{27} + 24q^{29} + 3q^{31} + 12q^{35} - 3q^{37} - 3q^{39} - 12q^{41} - 30q^{43} + 12q^{47} + 9q^{49} + 6q^{51} - 6q^{53} + 24q^{55} + 6q^{57} + 12q^{59} + 18q^{61} - 3q^{63} - 24q^{65} + 9q^{67} - 12q^{69} - 33q^{73} + 3q^{75} - 12q^{77} - 27q^{79} - 3q^{81} - 36q^{83} + 72q^{85} + 12q^{87} + 12q^{89} + 51q^{91} - 3q^{93} - 24q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 27 x^{2} - 18 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 22 \nu^{3} - 28 \nu^{2} + 43 \nu - 18$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{5} + 7 \nu^{4} - 31 \nu^{3} + 37 \nu^{2} - 56 \nu + 22$$ $$\beta_{4}$$ $$=$$ $$($$$$-6 \nu^{5} + 15 \nu^{4} - 64 \nu^{3} + 82 \nu^{2} - 121 \nu + 50$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$-3 \nu^{5} + 8 \nu^{4} - 33 \nu^{3} + 44 \nu^{2} - 62 \nu + 24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{1} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{5} + 8 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - \beta_{1} - 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{5} + 18 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} - 17 \beta_{1} + 27$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$13 \beta_{5} - 28 \beta_{4} + 3 \beta_{3} - 34 \beta_{2} - 11 \beta_{1} + 49$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 + 0.0585812i 0.5 + 1.51496i 0.5 − 2.43956i 0.5 − 0.0585812i 0.5 − 1.51496i 0.5 + 2.43956i
0 0.500000 0.866025i 0 −1.37328 2.37860i 0 2.64510 + 0.0585812i 0 −0.500000 0.866025i 0
193.2 0 0.500000 0.866025i 0 −0.227452 0.393958i 0 −2.16908 + 1.51496i 0 −0.500000 0.866025i 0
193.3 0 0.500000 0.866025i 0 1.60074 + 2.77256i 0 1.02398 2.43956i 0 −0.500000 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 −1.37328 + 2.37860i 0 2.64510 0.0585812i 0 −0.500000 + 0.866025i 0
289.2 0 0.500000 + 0.866025i 0 −0.227452 + 0.393958i 0 −2.16908 1.51496i 0 −0.500000 + 0.866025i 0
289.3 0 0.500000 + 0.866025i 0 1.60074 2.77256i 0 1.02398 + 2.43956i 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 672.2.q.l yes 6
3.b odd 2 1 2016.2.s.v 6
4.b odd 2 1 672.2.q.k 6
7.c even 3 1 inner 672.2.q.l yes 6
7.c even 3 1 4704.2.a.bs 3
7.d odd 6 1 4704.2.a.bv 3
8.b even 2 1 1344.2.q.y 6
8.d odd 2 1 1344.2.q.z 6
12.b even 2 1 2016.2.s.u 6
21.h odd 6 1 2016.2.s.v 6
28.f even 6 1 4704.2.a.bt 3
28.g odd 6 1 672.2.q.k 6
28.g odd 6 1 4704.2.a.bu 3
56.j odd 6 1 9408.2.a.eg 3
56.k odd 6 1 1344.2.q.z 6
56.k odd 6 1 9408.2.a.eh 3
56.m even 6 1 9408.2.a.ei 3
56.p even 6 1 1344.2.q.y 6
56.p even 6 1 9408.2.a.ej 3
84.n even 6 1 2016.2.s.u 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.k 6 4.b odd 2 1
672.2.q.k 6 28.g odd 6 1
672.2.q.l yes 6 1.a even 1 1 trivial
672.2.q.l yes 6 7.c even 3 1 inner
1344.2.q.y 6 8.b even 2 1
1344.2.q.y 6 56.p even 6 1
1344.2.q.z 6 8.d odd 2 1
1344.2.q.z 6 56.k odd 6 1
2016.2.s.u 6 12.b even 2 1
2016.2.s.u 6 84.n even 6 1
2016.2.s.v 6 3.b odd 2 1
2016.2.s.v 6 21.h odd 6 1
4704.2.a.bs 3 7.c even 3 1
4704.2.a.bt 3 28.f even 6 1
4704.2.a.bu 3 28.g odd 6 1
4704.2.a.bv 3 7.d odd 6 1
9408.2.a.eg 3 56.j odd 6 1
9408.2.a.eh 3 56.k odd 6 1
9408.2.a.ei 3 56.m even 6 1
9408.2.a.ej 3 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(672, [\chi])$$:

 $$T_{5}^{6} + 9 T_{5}^{4} + 8 T_{5}^{3} + 81 T_{5}^{2} + 36 T_{5} + 16$$ $$T_{11}^{6} + 27 T_{11}^{4} - 76 T_{11}^{3} + 729 T_{11}^{2} - 1026 T_{11} + 1444$$ $$T_{13}^{3} + 3 T_{13}^{2} - 36 T_{13} - 112$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T + T^{2} )^{3}$$
$5$ $$1 - 6 T^{2} + 8 T^{3} + 6 T^{4} - 24 T^{5} + 86 T^{6} - 120 T^{7} + 150 T^{8} + 1000 T^{9} - 3750 T^{10} + 15625 T^{12}$$
$7$ $$1 - 3 T + 5 T^{3} - 147 T^{5} + 343 T^{6}$$
$11$ $$1 - 6 T^{2} - 76 T^{3} - 30 T^{4} + 228 T^{5} + 3710 T^{6} + 2508 T^{7} - 3630 T^{8} - 101156 T^{9} - 87846 T^{10} + 1771561 T^{12}$$
$13$ $$( 1 + 3 T + 3 T^{2} - 34 T^{3} + 39 T^{4} + 507 T^{5} + 2197 T^{6} )^{2}$$
$17$ $$1 + 6 T + 9 T^{2} - 54 T^{3} - 378 T^{4} - 858 T^{5} - 1307 T^{6} - 14586 T^{7} - 109242 T^{8} - 265302 T^{9} + 751689 T^{10} + 8519142 T^{11} + 24137569 T^{12}$$
$19$ $$1 - 3 T - 12 T^{2} - 59 T^{3} + 36 T^{4} + 1269 T^{5} + 2094 T^{6} + 24111 T^{7} + 12996 T^{8} - 404681 T^{9} - 1563852 T^{10} - 7428297 T^{11} + 47045881 T^{12}$$
$23$ $$1 + 6 T - 9 T^{2} - 90 T^{3} - 90 T^{4} - 1146 T^{5} - 8885 T^{6} - 26358 T^{7} - 47610 T^{8} - 1095030 T^{9} - 2518569 T^{10} + 38618058 T^{11} + 148035889 T^{12}$$
$29$ $$( 1 - 12 T + 126 T^{2} - 728 T^{3} + 3654 T^{4} - 10092 T^{5} + 24389 T^{6} )^{2}$$
$31$ $$1 - 3 T - 63 T^{2} + 62 T^{3} + 2535 T^{4} + 501 T^{5} - 90450 T^{6} + 15531 T^{7} + 2436135 T^{8} + 1847042 T^{9} - 58181823 T^{10} - 85887453 T^{11} + 887503681 T^{12}$$
$37$ $$1 + 3 T - 18 T^{2} + 373 T^{3} + 168 T^{4} - 5829 T^{5} + 83328 T^{6} - 215673 T^{7} + 229992 T^{8} + 18893569 T^{9} - 33734898 T^{10} + 208031871 T^{11} + 2565726409 T^{12}$$
$41$ $$( 1 + 6 T + 27 T^{2} - 20 T^{3} + 1107 T^{4} + 10086 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$( 1 + 15 T + 177 T^{2} + 1318 T^{3} + 7611 T^{4} + 27735 T^{5} + 79507 T^{6} )^{2}$$
$47$ $$1 - 12 T - 9 T^{2} + 196 T^{3} + 4026 T^{4} - 9372 T^{5} - 178417 T^{6} - 440484 T^{7} + 8893434 T^{8} + 20349308 T^{9} - 43917129 T^{10} - 2752140084 T^{11} + 10779215329 T^{12}$$
$53$ $$1 + 6 T - 42 T^{2} - 1136 T^{3} - 2862 T^{4} + 29802 T^{5} + 517382 T^{6} + 1579506 T^{7} - 8039358 T^{8} - 169124272 T^{9} - 331400202 T^{10} + 2509172958 T^{11} + 22164361129 T^{12}$$
$59$ $$1 - 12 T - 54 T^{2} + 292 T^{3} + 11514 T^{4} - 25536 T^{5} - 623986 T^{6} - 1506624 T^{7} + 40080234 T^{8} + 59970668 T^{9} - 654337494 T^{10} - 8579091588 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 18 T + 129 T^{2} - 222 T^{3} - 3462 T^{4} + 40662 T^{5} - 368431 T^{6} + 2480382 T^{7} - 12882102 T^{8} - 50389782 T^{9} + 1786113489 T^{10} - 15202733418 T^{11} + 51520374361 T^{12}$$
$67$ $$1 - 9 T - 108 T^{2} + 383 T^{3} + 13680 T^{4} - 9405 T^{5} - 1069626 T^{6} - 630135 T^{7} + 61409520 T^{8} + 115192229 T^{9} - 2176321068 T^{10} - 12151125963 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 + 177 T^{2} + 32 T^{3} + 12567 T^{4} + 357911 T^{6} )^{2}$$
$73$ $$1 + 33 T + 534 T^{2} + 6687 T^{3} + 75648 T^{4} + 728637 T^{5} + 6291524 T^{6} + 53190501 T^{7} + 403128192 T^{8} + 2601356679 T^{9} + 15164660694 T^{10} + 68411362569 T^{11} + 151334226289 T^{12}$$
$79$ $$1 + 27 T + 297 T^{2} + 2426 T^{3} + 27783 T^{4} + 321003 T^{5} + 3028254 T^{6} + 25359237 T^{7} + 173393703 T^{8} + 1196112614 T^{9} + 11568174057 T^{10} + 83080522773 T^{11} + 243087455521 T^{12}$$
$83$ $$( 1 + 18 T + 234 T^{2} + 2040 T^{3} + 19422 T^{4} + 124002 T^{5} + 571787 T^{6} )^{2}$$
$89$ $$1 - 12 T - 135 T^{2} + 700 T^{3} + 28722 T^{4} - 63804 T^{5} - 2561959 T^{6} - 5678556 T^{7} + 227506962 T^{8} + 493478300 T^{9} - 8470202535 T^{10} - 67008713388 T^{11} + 496981290961 T^{12}$$
$97$ $$( 1 + 264 T^{2} + 38 T^{3} + 25608 T^{4} + 912673 T^{6} )^{2}$$