# Properties

 Label 672.2.b.b Level 672 Weight 2 Character orbit 672.b Analytic conductor 5.366 Analytic rank 0 Dimension 8 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.836829184.2 Defining polynomial: $$x^{8} + 14 x^{6} + 61 x^{4} + 84 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} -\beta_{1} q^{5} + \beta_{2} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} -\beta_{1} q^{5} + \beta_{2} q^{7} + q^{9} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{11} + \beta_{5} q^{13} -\beta_{1} q^{15} -\beta_{1} q^{17} + ( 2 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{19} + \beta_{2} q^{21} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{23} + ( -2 + \beta_{3} + \beta_{6} ) q^{25} + q^{27} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{29} + ( -2 - \beta_{6} ) q^{31} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{33} + ( -1 - \beta_{1} + \beta_{3} - \beta_{5} ) q^{35} + ( 1 - \beta_{3} + \beta_{6} ) q^{37} + \beta_{5} q^{39} + ( -\beta_{1} + 2 \beta_{5} ) q^{41} + ( \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{43} -\beta_{1} q^{45} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{47} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{49} -\beta_{1} q^{51} + ( 4 - 2 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{53} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{55} + ( 2 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{57} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{59} + ( \beta_{5} + 2 \beta_{7} ) q^{61} + \beta_{2} q^{63} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{65} + ( 3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{67} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{69} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{73} + ( -2 + \beta_{3} + \beta_{6} ) q^{75} + ( 4 + \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{77} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{79} + q^{81} + ( 5 - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{83} + ( -7 + \beta_{3} + \beta_{6} ) q^{85} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{87} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{89} + ( 2 + \beta_{1} - \beta_{6} + \beta_{7} ) q^{91} + ( -2 - \beta_{6} ) q^{93} + ( -4 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} ) q^{95} + ( 2 \beta_{5} + 2 \beta_{7} ) q^{97} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{3} + 4q^{7} + 8q^{9} + O(q^{10})$$ $$8q + 8q^{3} + 4q^{7} + 8q^{9} + 8q^{19} + 4q^{21} - 16q^{25} + 8q^{27} - 16q^{31} - 8q^{35} + 8q^{37} + 16q^{47} + 16q^{53} - 8q^{55} + 8q^{57} + 32q^{59} + 4q^{63} + 16q^{65} - 16q^{75} + 32q^{77} + 8q^{81} + 16q^{83} - 56q^{85} + 16q^{91} - 16q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 14 x^{6} + 61 x^{4} + 84 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 9 \nu^{3} + 16 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 11 \nu^{4} + 30 \nu^{2} + 4 \nu + 12$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 9 \nu^{4} - 16 \nu^{2} + 2$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 11 \nu^{4} - 30 \nu^{2} + 4 \nu - 12$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} + 11 \nu^{3} + 26 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 11 \nu^{4} - 26 \nu^{2} + 4$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu^{5} + 50 \nu^{3} + 54 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - \beta_{4} + \beta_{2} - 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{5} - 5 \beta_{4} - 5 \beta_{2} - 2 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 42$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-18 \beta_{5} + 29 \beta_{4} + 29 \beta_{2} + 22 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$47 \beta_{6} - 29 \beta_{4} - 22 \beta_{3} + 29 \beta_{2} - 246$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$4 \beta_{7} + 134 \beta_{5} - 181 \beta_{4} - 181 \beta_{2} - 186 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
223.1
 − 2.06644i 2.63640i 0.222191i − 1.65222i 1.65222i − 0.222191i − 2.63640i 2.06644i
0 1.00000 0 4.33660i 0 1.65222 2.06644i 0 1.00000 0
223.2 0 1.00000 0 2.31423i 0 −0.222191 + 2.63640i 0 1.00000 0
223.3 0 1.00000 0 1.72844i 0 2.63640 + 0.222191i 0 1.00000 0
223.4 0 1.00000 0 0.922382i 0 −2.06644 1.65222i 0 1.00000 0
223.5 0 1.00000 0 0.922382i 0 −2.06644 + 1.65222i 0 1.00000 0
223.6 0 1.00000 0 1.72844i 0 2.63640 0.222191i 0 1.00000 0
223.7 0 1.00000 0 2.31423i 0 −0.222191 2.63640i 0 1.00000 0
223.8 0 1.00000 0 4.33660i 0 1.65222 + 2.06644i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.b.b yes 8
3.b odd 2 1 2016.2.b.c 8
4.b odd 2 1 672.2.b.a 8
7.b odd 2 1 672.2.b.a 8
8.b even 2 1 1344.2.b.g 8
8.d odd 2 1 1344.2.b.h 8
12.b even 2 1 2016.2.b.a 8
21.c even 2 1 2016.2.b.a 8
24.f even 2 1 4032.2.b.o 8
24.h odd 2 1 4032.2.b.q 8
28.d even 2 1 inner 672.2.b.b yes 8
56.e even 2 1 1344.2.b.g 8
56.h odd 2 1 1344.2.b.h 8
84.h odd 2 1 2016.2.b.c 8
168.e odd 2 1 4032.2.b.q 8
168.i even 2 1 4032.2.b.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.b.a 8 4.b odd 2 1
672.2.b.a 8 7.b odd 2 1
672.2.b.b yes 8 1.a even 1 1 trivial
672.2.b.b yes 8 28.d even 2 1 inner
1344.2.b.g 8 8.b even 2 1
1344.2.b.g 8 56.e even 2 1
1344.2.b.h 8 8.d odd 2 1
1344.2.b.h 8 56.h odd 2 1
2016.2.b.a 8 12.b even 2 1
2016.2.b.a 8 21.c even 2 1
2016.2.b.c 8 3.b odd 2 1
2016.2.b.c 8 84.h odd 2 1
4032.2.b.o 8 24.f even 2 1
4032.2.b.o 8 168.i even 2 1
4032.2.b.q 8 24.h odd 2 1
4032.2.b.q 8 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{4} - 4 T_{19}^{3} - 44 T_{19}^{2} + 96 T_{19} + 64$$ acting on $$S_{2}^{\mathrm{new}}(672, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T )^{8}$$
$5$ $$1 - 12 T^{2} + 56 T^{4} + 28 T^{6} - 1074 T^{8} + 700 T^{10} + 35000 T^{12} - 187500 T^{14} + 390625 T^{16}$$
$7$ $$1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 140 T^{5} + 392 T^{6} - 1372 T^{7} + 2401 T^{8}$$
$11$ $$1 - 20 T^{2} + 296 T^{4} - 2268 T^{6} + 20718 T^{8} - 274428 T^{10} + 4333736 T^{12} - 35431220 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 - 18 T^{2} + 169 T^{4} )^{4}$$
$17$ $$1 - 108 T^{2} + 5432 T^{4} - 166628 T^{6} + 3420078 T^{8} - 48155492 T^{10} + 453686072 T^{12} - 2606857452 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 - 4 T + 32 T^{2} - 132 T^{3} + 558 T^{4} - 2508 T^{5} + 11552 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 - 100 T^{2} + 5000 T^{4} - 174764 T^{6} + 4622926 T^{8} - 92450156 T^{10} + 1399205000 T^{12} - 14803588900 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 + 36 T^{2} + 96 T^{3} + 390 T^{4} + 2784 T^{5} + 30276 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 8 T + 100 T^{2} + 616 T^{3} + 4534 T^{4} + 19096 T^{5} + 96100 T^{6} + 238328 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 4 T + 40 T^{2} + 292 T^{3} - 866 T^{4} + 10804 T^{5} + 54760 T^{6} - 202612 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$1 - 204 T^{2} + 21176 T^{4} - 1439172 T^{6} + 69360622 T^{8} - 2419248132 T^{10} + 59838314936 T^{12} - 969021265164 T^{14} + 7984925229121 T^{16}$$
$43$ $$1 - 192 T^{2} + 18396 T^{4} - 1162560 T^{6} + 55940774 T^{8} - 2149573440 T^{10} + 62892263196 T^{12} - 1213701705408 T^{14} + 11688200277601 T^{16}$$
$47$ $$( 1 - 8 T + 108 T^{2} - 872 T^{3} + 6758 T^{4} - 40984 T^{5} + 238572 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 8 T + 132 T^{2} - 632 T^{3} + 7718 T^{4} - 33496 T^{5} + 370788 T^{6} - 1191016 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 8 T + 102 T^{2} - 472 T^{3} + 3481 T^{4} )^{4}$$
$61$ $$1 - 184 T^{2} + 16284 T^{4} - 814280 T^{6} + 38728934 T^{8} - 3029935880 T^{10} + 225465674844 T^{12} - 9479748882424 T^{14} + 191707312997281 T^{16}$$
$67$ $$1 - 240 T^{2} + 37436 T^{4} - 3815056 T^{6} + 299829030 T^{8} - 17125786384 T^{10} + 754377365756 T^{12} - 21710011720560 T^{14} + 406067677556641 T^{16}$$
$71$ $$1 - 484 T^{2} + 107912 T^{4} - 14382508 T^{6} + 1250123214 T^{8} - 72502222828 T^{10} + 2742225320072 T^{12} - 62000537417764 T^{14} + 645753531245761 T^{16}$$
$73$ $$1 - 408 T^{2} + 80380 T^{4} - 10037032 T^{6} + 869620166 T^{8} - 53487343528 T^{10} + 2282650611580 T^{12} - 61744364325912 T^{14} + 806460091894081 T^{16}$$
$79$ $$1 - 368 T^{2} + 73436 T^{4} - 9593744 T^{6} + 892827078 T^{8} - 59874556304 T^{10} + 2860338148316 T^{12} - 89456183631728 T^{14} + 1517108809906561 T^{16}$$
$83$ $$( 1 - 8 T + 124 T^{2} - 1480 T^{3} + 7318 T^{4} - 122840 T^{5} + 854236 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 + 52 T^{2} - 328 T^{4} + 280764 T^{6} + 120770670 T^{8} + 2223931644 T^{10} - 20579455048 T^{12} + 25843027129972 T^{14} + 3936588805702081 T^{16}$$
$97$ $$1 - 344 T^{2} + 58172 T^{4} - 6404840 T^{6} + 615495942 T^{8} - 60263139560 T^{10} + 5149925334332 T^{12} - 286542369695576 T^{14} + 7837433594376961 T^{16}$$