# Properties

 Label 672.2.j.d Level 672 Weight 2 Character orbit 672.j Analytic conductor 5.366 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 672.j (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.2593100598870016.2 Defining polynomial: $$x^{12} - 2 x^{10} + x^{8} + 4 x^{6} + 4 x^{4} - 32 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( -\beta_{6} - \beta_{7} ) q^{5} + \beta_{1} q^{7} + ( -\beta_{2} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( -\beta_{6} - \beta_{7} ) q^{5} + \beta_{1} q^{7} + ( -\beta_{2} - \beta_{8} ) q^{9} + ( -\beta_{4} - \beta_{5} ) q^{13} + ( 3 \beta_{1} - \beta_{4} - \beta_{7} ) q^{15} + \beta_{9} q^{17} + ( 4 + \beta_{2} + \beta_{3} ) q^{19} + \beta_{6} q^{21} + ( \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{23} + ( 1 + \beta_{2} + \beta_{3} + \beta_{8} - \beta_{10} ) q^{25} + ( -2 + \beta_{2} + \beta_{9} ) q^{27} + ( 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -\beta_{2} + \beta_{3} ) q^{35} + ( 4 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{37} + ( -\beta_{1} - \beta_{4} + 3 \beta_{7} - \beta_{11} ) q^{39} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{41} + ( -4 - \beta_{2} - \beta_{3} - \beta_{8} + \beta_{10} ) q^{43} + ( 2 \beta_{1} + 3 \beta_{6} + \beta_{7} - \beta_{11} ) q^{45} + ( 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{47} - q^{49} + ( -\beta_{2} + \beta_{3} - \beta_{8} - 3 \beta_{10} ) q^{51} -2 \beta_{11} q^{53} + ( 3 - \beta_{2} + 4 \beta_{3} - \beta_{8} ) q^{57} + ( \beta_{2} - \beta_{3} + 2 \beta_{8} + 2 \beta_{10} ) q^{59} + ( 4 \beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{61} + ( \beta_{4} + \beta_{7} ) q^{63} + ( -3 \beta_{2} + 3 \beta_{3} + \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{65} + ( 4 - \beta_{2} - \beta_{3} + \beta_{8} - \beta_{10} ) q^{67} + ( -2 \beta_{1} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{69} + \beta_{11} q^{71} -6 q^{73} + ( 2 - 4 \beta_{2} + \beta_{3} - \beta_{9} ) q^{75} + ( 4 \beta_{6} - 4 \beta_{7} ) q^{79} + ( 3 - \beta_{2} - \beta_{3} - \beta_{8} - 3 \beta_{10} ) q^{81} + ( 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{83} + ( 2 \beta_{4} + 2 \beta_{5} ) q^{85} + ( 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{11} ) q^{87} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{89} + ( -\beta_{8} + \beta_{10} ) q^{91} + ( -\beta_{4} + \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{95} + ( -6 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{8} + 2 \beta_{10} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 4q^{9} + O(q^{10})$$ $$12q + 4q^{9} + 48q^{19} + 4q^{25} - 24q^{27} - 40q^{43} - 12q^{49} - 8q^{51} + 40q^{57} + 40q^{67} - 72q^{73} + 24q^{75} + 28q^{81} + 8q^{91} - 56q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{10} + x^{8} + 4 x^{6} + 4 x^{4} - 32 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{8} - \nu^{4} + 2 \nu^{2} - 8$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} - 2 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 9 \nu^{7} + 14 \nu^{6} + 8 \nu^{5} + 16 \nu^{4} + 4 \nu^{3} - 72 \nu^{2} + 48 \nu + 32$$$$)/128$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} - 2 \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 9 \nu^{7} + 14 \nu^{6} - 8 \nu^{5} + 16 \nu^{4} - 4 \nu^{3} - 72 \nu^{2} - 48 \nu + 32$$$$)/128$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} + 6 \nu^{10} + 6 \nu^{9} - 4 \nu^{8} - 9 \nu^{7} + 22 \nu^{6} - 20 \nu^{3} + 120 \nu^{2} + 144 \nu - 96$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} + 6 \nu^{10} - 6 \nu^{9} - 4 \nu^{8} + 9 \nu^{7} + 22 \nu^{6} + 20 \nu^{3} + 120 \nu^{2} - 144 \nu - 96$$$$)/128$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 4 \nu^{8} - 5 \nu^{7} + 14 \nu^{6} + 16 \nu^{5} - 40 \nu^{4} - 4 \nu^{3} - 24 \nu^{2} - 112 \nu + 96$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{11} + 2 \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 5 \nu^{7} - 14 \nu^{6} + 16 \nu^{5} + 40 \nu^{4} - 4 \nu^{3} + 24 \nu^{2} - 112 \nu - 96$$$$)/128$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{11} - 10 \nu^{10} - 10 \nu^{9} + 12 \nu^{8} - 5 \nu^{7} + 6 \nu^{6} + 8 \nu^{5} - 16 \nu^{4} + 12 \nu^{3} + 24 \nu^{2} - 48 \nu + 160$$$$)/128$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{11} + 2 \nu^{9} - 7 \nu^{7} + 8 \nu^{5} + 52 \nu^{3} - 48 \nu$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{11} + 10 \nu^{10} - 10 \nu^{9} - 12 \nu^{8} - 5 \nu^{7} - 6 \nu^{6} + 8 \nu^{5} + 16 \nu^{4} + 12 \nu^{3} - 24 \nu^{2} - 48 \nu - 160$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{11} - \nu^{7} + 10 \nu^{5} - 16 \nu^{3} + 16 \nu$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_{1} + 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{10} + 2 \beta_{9} + \beta_{8} - 3 \beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{10} + \beta_{8} + 5 \beta_{7} - 5 \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 6 \beta_{1} + 2$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 5 \beta_{3} + 5 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-\beta_{10} + \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 5 \beta_{5} + 5 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} - 6 \beta_{1} - 6$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} + 9 \beta_{5} - 9 \beta_{4} - 13 \beta_{3} + 13 \beta_{2}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-\beta_{10} + \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 54 \beta_{1} - 30$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-4 \beta_{11} - 23 \beta_{10} + 2 \beta_{9} - 23 \beta_{8} + 17 \beta_{7} + 17 \beta_{6} - 7 \beta_{5} + 7 \beta_{4} - 5 \beta_{3} + 5 \beta_{2}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$23 \beta_{10} - 23 \beta_{8} - 11 \beta_{7} + 11 \beta_{6} + 5 \beta_{5} + 5 \beta_{4} - 9 \beta_{3} - 9 \beta_{2} - 54 \beta_{1} + 26$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-24 \beta_{11} + 13 \beta_{10} - 10 \beta_{9} + 13 \beta_{8} + 45 \beta_{7} + 45 \beta_{6} - 31 \beta_{5} + 31 \beta_{4} - 37 \beta_{3} + 37 \beta_{2}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 −1.19877 − 0.750295i 1.19877 − 0.750295i −1.19877 + 0.750295i 1.19877 + 0.750295i −1.37027 + 0.349801i 1.37027 + 0.349801i −1.37027 − 0.349801i 1.37027 − 0.349801i 0.430469 + 1.34711i −0.430469 + 1.34711i 0.430469 − 1.34711i −0.430469 − 1.34711i
0 −1.67298 0.448478i 0 −0.896956 0 1.00000i 0 2.59774 + 1.50059i 0
239.2 0 −1.67298 0.448478i 0 0.896956 0 1.00000i 0 2.59774 + 1.50059i 0
239.3 0 −1.67298 + 0.448478i 0 −0.896956 0 1.00000i 0 2.59774 1.50059i 0
239.4 0 −1.67298 + 0.448478i 0 0.896956 0 1.00000i 0 2.59774 1.50059i 0
239.5 0 0.203364 1.72007i 0 −3.44014 0 1.00000i 0 −2.91729 0.699602i 0
239.6 0 0.203364 1.72007i 0 3.44014 0 1.00000i 0 −2.91729 0.699602i 0
239.7 0 0.203364 + 1.72007i 0 −3.44014 0 1.00000i 0 −2.91729 + 0.699602i 0
239.8 0 0.203364 + 1.72007i 0 3.44014 0 1.00000i 0 −2.91729 + 0.699602i 0
239.9 0 1.46962 0.916638i 0 −1.83328 0 1.00000i 0 1.31955 2.69421i 0
239.10 0 1.46962 0.916638i 0 1.83328 0 1.00000i 0 1.31955 2.69421i 0
239.11 0 1.46962 + 0.916638i 0 −1.83328 0 1.00000i 0 1.31955 + 2.69421i 0
239.12 0 1.46962 + 0.916638i 0 1.83328 0 1.00000i 0 1.31955 + 2.69421i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f 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.j.d 12
3.b odd 2 1 inner 672.2.j.d 12
4.b odd 2 1 168.2.j.d 12
8.b even 2 1 168.2.j.d 12
8.d odd 2 1 inner 672.2.j.d 12
12.b even 2 1 168.2.j.d 12
24.f even 2 1 inner 672.2.j.d 12
24.h odd 2 1 168.2.j.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.j.d 12 4.b odd 2 1
168.2.j.d 12 8.b even 2 1
168.2.j.d 12 12.b even 2 1
168.2.j.d 12 24.h odd 2 1
672.2.j.d 12 1.a even 1 1 trivial
672.2.j.d 12 3.b odd 2 1 inner
672.2.j.d 12 8.d odd 2 1 inner
672.2.j.d 12 24.f even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 16 T_{5}^{4} + 52 T_{5}^{2} - 32$$ acting on $$S_{2}^{\mathrm{new}}(672, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T^{2} + 4 T^{3} - 3 T^{4} + 27 T^{6} )^{2}$$
$5$ $$( 1 + 14 T^{2} + 107 T^{4} + 588 T^{6} + 2675 T^{8} + 8750 T^{10} + 15625 T^{12} )^{2}$$
$7$ $$( 1 + T^{2} )^{6}$$
$11$ $$( 1 - 11 T^{2} )^{12}$$
$13$ $$( 1 - 14 T^{2} + 331 T^{4} - 5132 T^{6} + 55939 T^{8} - 399854 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 - 38 T^{2} + 1151 T^{4} - 20724 T^{6} + 332639 T^{8} - 3173798 T^{10} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 - 12 T + 95 T^{2} - 476 T^{3} + 1805 T^{4} - 4332 T^{5} + 6859 T^{6} )^{4}$$
$23$ $$( 1 + 66 T^{2} + 2399 T^{4} + 64348 T^{6} + 1269071 T^{8} + 18469506 T^{10} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 + 46 T^{2} + 2695 T^{4} + 76516 T^{6} + 2266495 T^{8} + 32534926 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{12}$$
$37$ $$( 1 - 94 T^{2} + 3191 T^{4} - 75972 T^{6} + 4368479 T^{8} - 176171134 T^{10} + 2565726409 T^{12} )^{2}$$
$41$ $$( 1 - 86 T^{2} + 5647 T^{4} - 231764 T^{6} + 9492607 T^{8} - 243015446 T^{10} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 + 10 T + 129 T^{2} + 844 T^{3} + 5547 T^{4} + 18490 T^{5} + 79507 T^{6} )^{4}$$
$47$ $$( 1 + 58 T^{2} + 4591 T^{4} + 251884 T^{6} + 10141519 T^{8} + 283021498 T^{10} + 10779215329 T^{12} )^{2}$$
$53$ $$( 1 + 62 T^{2} + 6551 T^{4} + 242436 T^{6} + 18401759 T^{8} + 489209822 T^{10} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 - 226 T^{2} + 25867 T^{4} - 1888084 T^{6} + 90043027 T^{8} - 2738523586 T^{10} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 - 222 T^{2} + 25163 T^{4} - 1831724 T^{6} + 93631523 T^{8} - 3073776702 T^{10} + 51520374361 T^{12} )^{2}$$
$67$ $$( 1 - 10 T + 185 T^{2} - 1116 T^{3} + 12395 T^{4} - 44890 T^{5} + 300763 T^{6} )^{4}$$
$71$ $$( 1 + 362 T^{2} + 58607 T^{4} + 5382060 T^{6} + 295437887 T^{8} + 9199028522 T^{10} + 128100283921 T^{12} )^{2}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{12}$$
$79$ $$( 1 - 154 T^{2} + 18095 T^{4} - 1857324 T^{6} + 112930895 T^{8} - 5998312474 T^{10} + 243087455521 T^{12} )^{2}$$
$83$ $$( 1 - 162 T^{2} + 19931 T^{4} - 2214676 T^{6} + 137304659 T^{8} - 7688248002 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 - 278 T^{2} + 40879 T^{4} - 4202324 T^{6} + 323802559 T^{8} - 17442342998 T^{10} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 + 14 T + 159 T^{2} + 1124 T^{3} + 15423 T^{4} + 131726 T^{5} + 912673 T^{6} )^{4}$$