# Properties

 Label 448.2.q.b Level $448$ Weight $2$ Character orbit 448.q Analytic conductor $3.577$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 448.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.57729801055$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 4 x^{9} - 16 x^{8} + 8 x^{7} + 8 x^{6} + 32 x^{5} + 240 x^{4} + 120 x^{3} + 32 x^{2} + 16 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} + ( \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{4} - \beta_{10} ) q^{7} + ( \beta_{7} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} + ( \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{4} - \beta_{10} ) q^{7} + ( \beta_{7} - \beta_{11} ) q^{9} + ( \beta_{4} + 2 \beta_{9} ) q^{11} + ( 1 + \beta_{2} ) q^{13} + ( \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{15} + ( -1 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{17} + ( \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{19} + ( -4 + \beta_{2} + \beta_{6} + 2 \beta_{7} - \beta_{11} ) q^{21} + 3 \beta_{10} q^{23} + ( -1 - 3 \beta_{1} - \beta_{2} - 3 \beta_{6} + \beta_{7} + \beta_{11} ) q^{25} + ( 2 \beta_{3} + \beta_{4} - 2 \beta_{8} + \beta_{9} ) q^{27} + ( -2 + \beta_{1} + 2 \beta_{6} + 4 \beta_{7} ) q^{29} + ( -\beta_{3} - \beta_{4} + 2 \beta_{8} - 2 \beta_{9} ) q^{31} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + \beta_{6} + 2 \beta_{7} - \beta_{11} ) q^{33} + ( -3 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{8} - \beta_{10} ) q^{35} + ( 2 - \beta_{7} ) q^{37} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{39} + ( 2 - \beta_{1} - 2 \beta_{6} - 4 \beta_{7} ) q^{41} + ( -2 \beta_{3} - 2 \beta_{8} ) q^{43} + ( 2 - \beta_{1} - \beta_{6} - 2 \beta_{7} ) q^{45} + ( -2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{47} + ( 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} + \beta_{11} ) q^{49} + ( 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{51} + ( -2 + \beta_{1} + \beta_{2} - \beta_{6} - 2 \beta_{7} + \beta_{11} ) q^{53} + ( 2 \beta_{3} + \beta_{5} + 2 \beta_{8} - \beta_{10} ) q^{55} + ( 7 - 2 \beta_{2} ) q^{57} + ( 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{59} + ( -5 \beta_{7} - 2 \beta_{11} ) q^{61} + ( \beta_{3} - 2 \beta_{4} - 3 \beta_{8} - \beta_{9} - 3 \beta_{10} ) q^{63} + 3 \beta_{6} q^{65} + ( -\beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} ) q^{67} + ( 12 - 3 \beta_{2} ) q^{69} + ( 3 \beta_{3} - 3 \beta_{8} ) q^{71} + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{11} ) q^{73} + ( 5 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} ) q^{75} + ( -8 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + 2 \beta_{11} ) q^{77} + ( 2 \beta_{9} + \beta_{10} ) q^{79} + ( -1 + 3 \beta_{1} + 3 \beta_{6} + \beta_{7} ) q^{81} + ( 5 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{83} + ( -6 - 3 \beta_{1} - \beta_{2} - 6 \beta_{6} + 12 \beta_{7} + 2 \beta_{11} ) q^{85} + ( \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{87} + ( -6 + 3 \beta_{7} ) q^{89} + ( -3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{91} + ( 6 - 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{11} ) q^{93} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{95} + ( 5 + 2 \beta_{1} - \beta_{2} + 4 \beta_{6} - 10 \beta_{7} + 2 \beta_{11} ) q^{97} + ( -3 \beta_{3} + \beta_{4} + 3 \beta_{5} - 3 \beta_{8} - \beta_{9} - 3 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 6q^{5} + 4q^{9} + O(q^{10})$$ $$12q - 6q^{5} + 4q^{9} + 16q^{13} - 18q^{17} - 34q^{21} - 8q^{25} - 30q^{33} + 18q^{37} + 12q^{45} + 12q^{49} - 30q^{53} + 76q^{57} - 34q^{61} + 132q^{69} - 6q^{73} - 90q^{77} - 6q^{81} - 54q^{89} + 42q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{9} - 16 x^{8} + 8 x^{7} + 8 x^{6} + 32 x^{5} + 240 x^{4} + 120 x^{3} + 32 x^{2} + 16 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-76 \nu^{11} + 38 \nu^{10} - 241 \nu^{9} + 480 \nu^{8} + 976 \nu^{7} - 430 \nu^{6} - 504 \nu^{5} - 4844 \nu^{4} - 2498 \nu^{3} - 656 \nu^{2} - 328 \nu - 29024$$$$)/55944$$ $$\beta_{2}$$ $$=$$ $$($$$$80 \nu^{11} - 40 \nu^{10} + 131 \nu^{9} - 996 \nu^{8} - 782 \nu^{7} + 698 \nu^{6} + 1512 \nu^{5} + 12460 \nu^{4} + 6310 \nu^{3} + 1672 \nu^{2} + 836 \nu - 78392$$$$)/18648$$ $$\beta_{3}$$ $$=$$ $$($$$$545 \nu^{11} - 3880 \nu^{10} + 1940 \nu^{9} - 2706 \nu^{8} + 6952 \nu^{7} + 58604 \nu^{6} - 55134 \nu^{5} + 9376 \nu^{4} + 16000 \nu^{3} - 835040 \nu^{2} + 1616 \nu + 808$$$$)/55944$$ $$\beta_{4}$$ $$=$$ $$($$$$1483 \nu^{11} - 9788 \nu^{10} + 4894 \nu^{9} - 7158 \nu^{8} + 17480 \nu^{7} + 147868 \nu^{6} - 145098 \nu^{5} + 23768 \nu^{4} + 41228 \nu^{3} - 1961200 \nu^{2} + 4192 \nu + 2096$$$$)/111888$$ $$\beta_{5}$$ $$=$$ $$($$$$-545 \nu^{11} + 3880 \nu^{10} - 1940 \nu^{9} + 2706 \nu^{8} - 6952 \nu^{7} - 58604 \nu^{6} + 55134 \nu^{5} - 9376 \nu^{4} - 16000 \nu^{3} + 797744 \nu^{2} - 1616 \nu - 808$$$$)/37296$$ $$\beta_{6}$$ $$=$$ $$($$$$-970 \nu^{11} + 263 \nu^{10} - 76 \nu^{9} + 3918 \nu^{8} + 14227 \nu^{7} - 11488 \nu^{6} - 4680 \nu^{5} - 29366 \nu^{4} - 224888 \nu^{3} - 58124 \nu^{2} - 29950 \nu - 7760$$$$)/13986$$ $$\beta_{7}$$ $$=$$ $$($$$$-239 \nu^{11} + 64 \nu^{10} - 32 \nu^{9} + 972 \nu^{8} + 3560 \nu^{7} - 2804 \nu^{6} - 954 \nu^{5} - 7504 \nu^{4} - 55384 \nu^{3} - 14308 \nu^{2} - 7376 \nu - 1912$$$$)/1776$$ $$\beta_{8}$$ $$=$$ $$($$$$-14975 \nu^{11} + 3880 \nu^{10} - 164 \nu^{9} + 59982 \nu^{8} + 223928 \nu^{7} - 179372 \nu^{6} - 87390 \nu^{5} - 449824 \nu^{4} - 3473872 \nu^{3} - 898336 \nu^{2} - 30032 \nu - 119800$$$$)/55944$$ $$\beta_{9}$$ $$=$$ $$($$$$35149 \nu^{11} - 8528 \nu^{10} - 620 \nu^{9} - 140286 \nu^{8} - 529870 \nu^{7} + 423028 \nu^{6} + 220194 \nu^{5} + 1052300 \nu^{4} + 8155808 \nu^{3} + 2109560 \nu^{2} + 69988 \nu + 281192$$$$)/111888$$ $$\beta_{10}$$ $$=$$ $$($$$$-14303 \nu^{11} + 3544 \nu^{10} + 4 \nu^{9} + 57210 \nu^{8} + 214562 \nu^{7} - 171644 \nu^{6} - 85878 \nu^{5} - 429076 \nu^{4} - 3318304 \nu^{3} - 858184 \nu^{2} - 28604 \nu - 114424$$$$)/37296$$ $$\beta_{11}$$ $$=$$ $$($$$$21215 \nu^{11} - 5668 \nu^{10} + 3056 \nu^{9} - 86388 \nu^{8} - 316892 \nu^{7} + 248468 \nu^{6} + 81450 \nu^{5} + 689104 \nu^{4} + 4915768 \nu^{3} + 1269844 \nu^{2} + 654680 \nu + 169720$$$$)/37296$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} + \beta_{9} - \beta_{6}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{5} - 3 \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{10} - 4 \beta_{9} + \beta_{8} - 4 \beta_{5} - 4 \beta_{4} - \beta_{3} + 4 \beta_{1} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{11} + 10 \beta_{7} - 3 \beta_{6}$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{11} + 13 \beta_{7} - 17 \beta_{6} - 18 \beta_{5} - 16 \beta_{4} - 7 \beta_{3} - \beta_{2} - 17 \beta_{1} - 13$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-24 \beta_{10} - 8 \beta_{9} + 25 \beta_{8} - 24 \beta_{5} - 8 \beta_{4} - 25 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$4 \beta_{11} - 41 \beta_{10} - 33 \beta_{9} + 20 \beta_{8} + 36 \beta_{7} - 37 \beta_{6}$$ $$\nu^{8}$$ $$=$$ $$33 \beta_{11} + 181 \beta_{7} - 81 \beta_{6} - 33 \beta_{2} - 81 \beta_{1} - 181$$ $$\nu^{9}$$ $$=$$ $$-188 \beta_{10} - 140 \beta_{9} + 105 \beta_{8} - 188 \beta_{5} - 140 \beta_{4} - 105 \beta_{3} - 24 \beta_{2} - 164 \beta_{1} - 186$$ $$\nu^{10}$$ $$=$$ $$-538 \beta_{10} - 258 \beta_{9} + 468 \beta_{8}$$ $$\nu^{11}$$ $$=$$ $$129 \beta_{11} + 925 \beta_{7} - 737 \beta_{6} + 866 \beta_{5} + 608 \beta_{4} + 527 \beta_{3} - 129 \beta_{2} - 737 \beta_{1} - 925$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$\beta_{7}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.353325 − 0.0946732i −0.558440 + 2.08413i 0.463767 − 1.73080i −1.73080 − 0.463767i 2.08413 + 0.558440i 0.0946732 − 0.353325i −0.353325 + 0.0946732i −0.558440 − 2.08413i 0.463767 + 1.73080i −1.73080 + 0.463767i 2.08413 − 0.558440i 0.0946732 + 0.353325i
0 −2.48220 1.43310i 0 −0.241348 0.418027i 0 0.447998 2.60755i 0 2.60755 + 4.51640i 0
31.2 0 −1.43366 0.827721i 0 −2.02569 3.50859i 0 2.64257 0.129755i 0 −0.129755 0.224743i 0
31.3 0 −0.182520 0.105378i 0 0.767035 + 1.32854i 0 −2.19457 1.47779i 0 −1.47779 2.55961i 0
31.4 0 0.182520 + 0.105378i 0 0.767035 + 1.32854i 0 2.19457 + 1.47779i 0 −1.47779 2.55961i 0
31.5 0 1.43366 + 0.827721i 0 −2.02569 3.50859i 0 −2.64257 + 0.129755i 0 −0.129755 0.224743i 0
31.6 0 2.48220 + 1.43310i 0 −0.241348 0.418027i 0 −0.447998 + 2.60755i 0 2.60755 + 4.51640i 0
159.1 0 −2.48220 + 1.43310i 0 −0.241348 + 0.418027i 0 0.447998 + 2.60755i 0 2.60755 4.51640i 0
159.2 0 −1.43366 + 0.827721i 0 −2.02569 + 3.50859i 0 2.64257 + 0.129755i 0 −0.129755 + 0.224743i 0
159.3 0 −0.182520 + 0.105378i 0 0.767035 1.32854i 0 −2.19457 + 1.47779i 0 −1.47779 + 2.55961i 0
159.4 0 0.182520 0.105378i 0 0.767035 1.32854i 0 2.19457 1.47779i 0 −1.47779 + 2.55961i 0
159.5 0 1.43366 0.827721i 0 −2.02569 + 3.50859i 0 −2.64257 0.129755i 0 −0.129755 + 0.224743i 0
159.6 0 2.48220 1.43310i 0 −0.241348 + 0.418027i 0 −0.447998 2.60755i 0 2.60755 4.51640i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 159.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.j odd 6 1 inner
56.m even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.q.b 12
4.b odd 2 1 inner 448.2.q.b 12
7.c even 3 1 3136.2.e.e 12
7.d odd 6 1 448.2.q.c yes 12
7.d odd 6 1 3136.2.e.d 12
8.b even 2 1 448.2.q.c yes 12
8.d odd 2 1 448.2.q.c yes 12
28.f even 6 1 448.2.q.c yes 12
28.f even 6 1 3136.2.e.d 12
28.g odd 6 1 3136.2.e.e 12
56.j odd 6 1 inner 448.2.q.b 12
56.j odd 6 1 3136.2.e.e 12
56.k odd 6 1 3136.2.e.d 12
56.m even 6 1 inner 448.2.q.b 12
56.m even 6 1 3136.2.e.e 12
56.p even 6 1 3136.2.e.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.q.b 12 1.a even 1 1 trivial
448.2.q.b 12 4.b odd 2 1 inner
448.2.q.b 12 56.j odd 6 1 inner
448.2.q.b 12 56.m even 6 1 inner
448.2.q.c yes 12 7.d odd 6 1
448.2.q.c yes 12 8.b even 2 1
448.2.q.c yes 12 8.d odd 2 1
448.2.q.c yes 12 28.f even 6 1
3136.2.e.d 12 7.d odd 6 1
3136.2.e.d 12 28.f even 6 1
3136.2.e.d 12 56.k odd 6 1
3136.2.e.d 12 56.p even 6 1
3136.2.e.e 12 7.c even 3 1
3136.2.e.e 12 28.g odd 6 1
3136.2.e.e 12 56.j odd 6 1
3136.2.e.e 12 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(448, [\chi])$$:

 $$T_{3}^{12} - 11 T_{3}^{10} + 98 T_{3}^{8} - 251 T_{3}^{6} + 518 T_{3}^{4} - 23 T_{3}^{2} + 1$$ $$T_{5}^{6} + 3 T_{5}^{5} + 14 T_{5}^{4} - 9 T_{5}^{3} + 34 T_{5}^{2} + 15 T_{5} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$1 - 23 T^{2} + 518 T^{4} - 251 T^{6} + 98 T^{8} - 11 T^{10} + T^{12}$$
$5$ $$( 9 + 15 T + 34 T^{2} - 9 T^{3} + 14 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$7$ $$117649 - 14406 T^{2} - 1617 T^{4} + 380 T^{6} - 33 T^{8} - 6 T^{10} + T^{12}$$
$11$ $$4782969 + 1318761 T^{2} + 265194 T^{4} + 22761 T^{6} + 1422 T^{8} + 45 T^{10} + T^{12}$$
$13$ $$( 36 - 12 T - 4 T^{2} + T^{3} )^{4}$$
$17$ $$( 243 + 405 T + 144 T^{2} - 135 T^{3} + 12 T^{4} + 9 T^{5} + T^{6} )^{2}$$
$19$ $$15752961 - 3536379 T^{2} + 575586 T^{4} - 41067 T^{6} + 2134 T^{8} - 55 T^{10} + T^{12}$$
$23$ $$531441 - 1358127 T^{2} + 3398598 T^{4} - 182979 T^{6} + 7938 T^{8} - 99 T^{10} + T^{12}$$
$29$ $$( 3888 + 1440 T^{2} + 84 T^{4} + T^{6} )^{2}$$
$31$ $$10673289 + 6909705 T^{2} + 4169394 T^{4} + 190161 T^{6} + 6534 T^{8} + 93 T^{10} + T^{12}$$
$37$ $$( 3 - 3 T + T^{2} )^{6}$$
$41$ $$( 3888 + 1440 T^{2} + 84 T^{4} + T^{6} )^{2}$$
$43$ $$( -48 + T^{2} )^{6}$$
$47$ $$166858361289 + 7216669161 T^{2} + 215312418 T^{4} + 3370113 T^{6} + 38502 T^{8} + 237 T^{10} + T^{12}$$
$53$ $$( 2187 + 3159 T + 1116 T^{2} - 585 T^{3} + 36 T^{4} + 15 T^{5} + T^{6} )^{2}$$
$59$ $$96059601 - 20297871 T^{2} + 3161926 T^{4} - 218563 T^{6} + 11154 T^{8} - 115 T^{10} + T^{12}$$
$61$ $$( 110889 - 8991 T + 6390 T^{2} + 1125 T^{3} + 262 T^{4} + 17 T^{5} + T^{6} )^{2}$$
$67$ $$31381059609 + 1937102445 T^{2} + 83967678 T^{4} + 1843641 T^{6} + 29466 T^{8} + 201 T^{10} + T^{12}$$
$71$ $$( 36 + T^{2} )^{6}$$
$73$ $$( 49923 - 33669 T + 7956 T^{2} - 261 T^{3} - 84 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$79$ $$3418801 - 1314639 T^{2} + 411222 T^{4} - 32563 T^{6} + 1890 T^{8} - 51 T^{10} + T^{12}$$
$83$ $$( 419904 + 32944 T^{2} + 364 T^{4} + T^{6} )^{2}$$
$89$ $$( 27 + 9 T + T^{2} )^{6}$$
$97$ $$( 2834352 + 82224 T^{2} + 540 T^{4} + T^{6} )^{2}$$