# Properties

 Label 224.2.t.a Level $224$ Weight $2$ Character orbit 224.t Analytic conductor $1.789$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.951588245534976.1 Defining polynomial: $$x^{12} - x^{11} + 2 x^{10} - 9 x^{9} + 8 x^{8} - 13 x^{7} + 35 x^{6} - 26 x^{5} + 32 x^{4} - 72 x^{3} + 32 x^{2} - 32 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 56) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{10} q^{3} + ( -\beta_{4} + \beta_{7} ) q^{5} + ( \beta_{8} - \beta_{9} ) q^{7} + ( -\beta_{1} + \beta_{8} - \beta_{9} ) q^{9} +O(q^{10})$$ $$q -\beta_{10} q^{3} + ( -\beta_{4} + \beta_{7} ) q^{5} + ( \beta_{8} - \beta_{9} ) q^{7} + ( -\beta_{1} + \beta_{8} - \beta_{9} ) q^{9} + ( \beta_{10} - \beta_{11} ) q^{11} + ( -\beta_{6} - \beta_{11} ) q^{13} + ( 2 - \beta_{1} ) q^{15} + ( \beta_{3} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{17} + ( -2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{19} + ( -\beta_{2} + \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{21} -\beta_{8} q^{23} + ( -2 \beta_{3} + 2 \beta_{8} ) q^{25} + ( \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{27} + ( 2 \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{11} ) q^{29} + ( 2 \beta_{5} - \beta_{9} ) q^{31} + ( -3 + 4 \beta_{1} - 3 \beta_{5} - 2 \beta_{8} + 4 \beta_{9} ) q^{33} + ( -2 \beta_{2} - \beta_{7} + \beta_{11} ) q^{35} + ( \beta_{4} - \beta_{6} + \beta_{7} ) q^{37} + ( \beta_{3} - \beta_{8} + 3 \beta_{9} ) q^{39} + ( -3 \beta_{1} + \beta_{3} ) q^{41} + ( 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{7} - 2 \beta_{10} ) q^{43} + ( -2 \beta_{2} + \beta_{11} ) q^{45} + ( -6 + 3 \beta_{1} - 6 \beta_{5} + 3 \beta_{9} ) q^{47} + ( -1 + 3 \beta_{1} + \beta_{3} + 2 \beta_{5} - 2 \beta_{8} ) q^{49} + ( 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{51} + ( \beta_{2} + 3 \beta_{10} + \beta_{11} ) q^{53} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{55} + ( 1 - 4 \beta_{1} ) q^{57} + ( -4 \beta_{2} + 3 \beta_{10} ) q^{59} + ( -3 \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{61} + ( -2 + \beta_{1} + 4 \beta_{5} - \beta_{8} - \beta_{9} ) q^{63} + ( 2 + \beta_{1} + 2 \beta_{5} - 3 \beta_{8} + \beta_{9} ) q^{65} + ( 2 \beta_{2} - \beta_{10} ) q^{67} + ( \beta_{2} + \beta_{4} + \beta_{7} + \beta_{10} ) q^{69} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{71} + ( -4 \beta_{3} + \beta_{5} + 4 \beta_{8} - 2 \beta_{9} ) q^{73} + ( -2 \beta_{4} - 2 \beta_{7} ) q^{75} + ( -3 \beta_{2} - 2 \beta_{4} - \beta_{6} + 6 \beta_{7} + 3 \beta_{10} - \beta_{11} ) q^{77} + ( 4 - 2 \beta_{1} + 4 \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{79} + ( 3 \beta_{3} - 3 \beta_{5} - 3 \beta_{8} + \beta_{9} ) q^{81} + ( 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{10} ) q^{83} + ( 3 \beta_{2} + 3 \beta_{4} - \beta_{6} - 3 \beta_{7} - 3 \beta_{10} - \beta_{11} ) q^{85} + ( \beta_{3} - 2 \beta_{5} - \beta_{8} - 3 \beta_{9} ) q^{87} + ( -3 - 3 \beta_{5} + 4 \beta_{8} ) q^{89} + ( -2 \beta_{2} - 4 \beta_{4} - \beta_{6} + 2 \beta_{7} - 2 \beta_{10} - \beta_{11} ) q^{91} + ( \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{93} + ( -\beta_{3} - 6 \beta_{5} + \beta_{8} ) q^{95} + ( 2 + 3 \beta_{1} + \beta_{3} ) q^{97} + ( -2 \beta_{2} - 2 \beta_{4} - \beta_{6} + 6 \beta_{7} + 6 \beta_{10} - \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{7} + O(q^{10})$$ $$12 q + 4 q^{7} + 20 q^{15} - 2 q^{17} - 2 q^{23} - 4 q^{25} - 10 q^{31} - 14 q^{33} - 4 q^{39} - 8 q^{41} - 30 q^{47} - 12 q^{49} - 4 q^{55} - 4 q^{57} - 44 q^{63} + 8 q^{65} - 32 q^{71} - 10 q^{73} + 22 q^{79} + 22 q^{81} + 20 q^{87} - 10 q^{89} + 34 q^{95} + 40 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 2 x^{10} - 9 x^{9} + 8 x^{8} - 13 x^{7} + 35 x^{6} - 26 x^{5} + 32 x^{4} - 72 x^{3} + 32 x^{2} - 32 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{11} + 3 \nu^{10} + 9 \nu^{8} - 18 \nu^{7} - 7 \nu^{6} - 29 \nu^{5} + 44 \nu^{4} + 24 \nu^{3} + 32 \nu^{2} - 48 \nu - 64$$$$)/32$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{11} - 5 \nu^{10} - 6 \nu^{9} + 11 \nu^{8} + 24 \nu^{7} + 15 \nu^{6} - 25 \nu^{5} - 82 \nu^{4} - 32 \nu^{3} + 64 \nu^{2} + 160 \nu + 96$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{10} + 2 \nu^{9} - \nu^{8} + 5 \nu^{7} - 9 \nu^{6} + 3 \nu^{5} - 12 \nu^{4} + 19 \nu^{3} + 16 \nu - 16$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} - 2 \nu^{10} - 5 \nu^{9} + 5 \nu^{8} + 3 \nu^{7} + 27 \nu^{6} - 24 \nu^{5} + 7 \nu^{4} - 80 \nu^{3} + 48 \nu^{2} - 8 \nu + 128$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-4 \nu^{11} - \nu^{10} - 9 \nu^{9} + 20 \nu^{8} - 3 \nu^{7} + 42 \nu^{6} - 63 \nu^{5} + 7 \nu^{4} - 100 \nu^{3} + 104 \nu^{2} + 32 \nu + 144$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{11} + 7 \nu^{10} + 26 \nu^{9} - 25 \nu^{8} - 4 \nu^{7} - 141 \nu^{6} + 123 \nu^{5} - 10 \nu^{4} + 420 \nu^{3} - 248 \nu^{2} - 48 \nu - 672$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-9 \nu^{11} - 7 \nu^{10} - 22 \nu^{9} + 45 \nu^{8} + 8 \nu^{7} + 105 \nu^{6} - 147 \nu^{5} - 18 \nu^{4} - 260 \nu^{3} + 248 \nu^{2} + 96 \nu + 416$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-9 \nu^{11} - 7 \nu^{10} - 22 \nu^{9} + 45 \nu^{8} + 8 \nu^{7} + 105 \nu^{6} - 147 \nu^{5} - 18 \nu^{4} - 260 \nu^{3} + 248 \nu^{2} + 160 \nu + 416$$$$)/32$$ $$\beta_{9}$$ $$=$$ $$($$$$13 \nu^{11} + 5 \nu^{10} + 28 \nu^{9} - 69 \nu^{8} - 2 \nu^{7} - 125 \nu^{6} + 221 \nu^{5} + 16 \nu^{4} + 296 \nu^{3} - 352 \nu^{2} - 160 \nu - 448$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-11 \nu^{11} - \nu^{10} - 34 \nu^{9} + 67 \nu^{8} - 28 \nu^{7} + 167 \nu^{6} - 229 \nu^{5} + 82 \nu^{4} - 400 \nu^{3} + 392 \nu^{2} + 32 \nu + 544$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$7 \nu^{11} + 2 \nu^{10} + 15 \nu^{9} - 39 \nu^{8} + 3 \nu^{7} - 69 \nu^{6} + 112 \nu^{5} - 5 \nu^{4} + 164 \nu^{3} - 180 \nu^{2} - 64 \nu - 240$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{8} - \beta_{7}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{10} + \beta_{9} - \beta_{8} + \beta_{5} + \beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} - \beta_{1} + 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{8} - 3 \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{11} + \beta_{10} + 4 \beta_{9} - \beta_{8} + 2 \beta_{5} + \beta_{3} + 2 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{11} + 3 \beta_{10} + 3 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} + \beta_{3} - 3 \beta_{2} - 7 \beta_{1} + 3$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-5 \beta_{9} - 9 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 10 \beta_{4} - 5 \beta_{1} + 2$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-9 \beta_{11} + \beta_{10} + 4 \beta_{9} - 11 \beta_{5} + \beta_{2}$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-2 \beta_{11} + 3 \beta_{10} + 3 \beta_{7} - 2 \beta_{6} - 16 \beta_{4} + 11 \beta_{3} - 16 \beta_{2} - 18 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-17 \beta_{9} - 11 \beta_{8} - 27 \beta_{7} + 15 \beta_{5} + 9 \beta_{4} - 17 \beta_{1} + 15$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-13 \beta_{11} + 15 \beta_{10} - 9 \beta_{9} + 4 \beta_{8} - 72 \beta_{5} - 4 \beta_{3} + 4 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 1.41417 − 0.0105323i 1.26950 + 0.623187i −0.390636 − 1.35919i −0.981777 − 1.01790i −0.0950561 + 1.41102i −0.716208 + 1.21944i 1.41417 + 0.0105323i 1.26950 − 0.623187i −0.390636 + 1.35919i −0.981777 + 1.01790i −0.0950561 − 1.41102i −0.716208 − 1.21944i
0 −2.13038 1.22998i 0 −1.28690 + 0.742990i 0 −0.129755 + 2.64257i 0 1.52569 + 2.64257i 0
81.2 0 −1.36456 0.787829i 0 0.476087 0.274869i 0 2.60755 0.447998i 0 −0.258652 0.447998i 0
81.3 0 −0.591141 0.341295i 0 −2.80486 + 1.61939i 0 −1.47779 2.19457i 0 −1.26704 2.19457i 0
81.4 0 0.591141 + 0.341295i 0 2.80486 1.61939i 0 −1.47779 2.19457i 0 −1.26704 2.19457i 0
81.5 0 1.36456 + 0.787829i 0 −0.476087 + 0.274869i 0 2.60755 0.447998i 0 −0.258652 0.447998i 0
81.6 0 2.13038 + 1.22998i 0 1.28690 0.742990i 0 −0.129755 + 2.64257i 0 1.52569 + 2.64257i 0
177.1 0 −2.13038 + 1.22998i 0 −1.28690 0.742990i 0 −0.129755 2.64257i 0 1.52569 2.64257i 0
177.2 0 −1.36456 + 0.787829i 0 0.476087 + 0.274869i 0 2.60755 + 0.447998i 0 −0.258652 + 0.447998i 0
177.3 0 −0.591141 + 0.341295i 0 −2.80486 1.61939i 0 −1.47779 + 2.19457i 0 −1.26704 + 2.19457i 0
177.4 0 0.591141 0.341295i 0 2.80486 + 1.61939i 0 −1.47779 + 2.19457i 0 −1.26704 + 2.19457i 0
177.5 0 1.36456 0.787829i 0 −0.476087 0.274869i 0 2.60755 + 0.447998i 0 −0.258652 + 0.447998i 0
177.6 0 2.13038 1.22998i 0 1.28690 + 0.742990i 0 −0.129755 2.64257i 0 1.52569 2.64257i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 177.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.b even 2 1 inner
56.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.t.a 12
3.b odd 2 1 2016.2.cr.c 12
4.b odd 2 1 56.2.p.a 12
7.b odd 2 1 1568.2.t.g 12
7.c even 3 1 inner 224.2.t.a 12
7.c even 3 1 1568.2.b.f 6
7.d odd 6 1 1568.2.b.e 6
7.d odd 6 1 1568.2.t.g 12
8.b even 2 1 inner 224.2.t.a 12
8.d odd 2 1 56.2.p.a 12
12.b even 2 1 504.2.cj.c 12
21.h odd 6 1 2016.2.cr.c 12
24.f even 2 1 504.2.cj.c 12
24.h odd 2 1 2016.2.cr.c 12
28.d even 2 1 392.2.p.g 12
28.f even 6 1 392.2.b.f 6
28.f even 6 1 392.2.p.g 12
28.g odd 6 1 56.2.p.a 12
28.g odd 6 1 392.2.b.e 6
56.e even 2 1 392.2.p.g 12
56.h odd 2 1 1568.2.t.g 12
56.j odd 6 1 1568.2.b.e 6
56.j odd 6 1 1568.2.t.g 12
56.k odd 6 1 56.2.p.a 12
56.k odd 6 1 392.2.b.e 6
56.m even 6 1 392.2.b.f 6
56.m even 6 1 392.2.p.g 12
56.p even 6 1 inner 224.2.t.a 12
56.p even 6 1 1568.2.b.f 6
84.n even 6 1 504.2.cj.c 12
168.s odd 6 1 2016.2.cr.c 12
168.v even 6 1 504.2.cj.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.p.a 12 4.b odd 2 1
56.2.p.a 12 8.d odd 2 1
56.2.p.a 12 28.g odd 6 1
56.2.p.a 12 56.k odd 6 1
224.2.t.a 12 1.a even 1 1 trivial
224.2.t.a 12 7.c even 3 1 inner
224.2.t.a 12 8.b even 2 1 inner
224.2.t.a 12 56.p even 6 1 inner
392.2.b.e 6 28.g odd 6 1
392.2.b.e 6 56.k odd 6 1
392.2.b.f 6 28.f even 6 1
392.2.b.f 6 56.m even 6 1
392.2.p.g 12 28.d even 2 1
392.2.p.g 12 28.f even 6 1
392.2.p.g 12 56.e even 2 1
392.2.p.g 12 56.m even 6 1
504.2.cj.c 12 12.b even 2 1
504.2.cj.c 12 24.f even 2 1
504.2.cj.c 12 84.n even 6 1
504.2.cj.c 12 168.v even 6 1
1568.2.b.e 6 7.d odd 6 1
1568.2.b.e 6 56.j odd 6 1
1568.2.b.f 6 7.c even 3 1
1568.2.b.f 6 56.p even 6 1
1568.2.t.g 12 7.b odd 2 1
1568.2.t.g 12 7.d odd 6 1
1568.2.t.g 12 56.h odd 2 1
1568.2.t.g 12 56.j odd 6 1
2016.2.cr.c 12 3.b odd 2 1
2016.2.cr.c 12 21.h odd 6 1
2016.2.cr.c 12 24.h odd 2 1
2016.2.cr.c 12 168.s odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$49 - 133 T^{2} + 298 T^{4} - 157 T^{6} + 62 T^{8} - 9 T^{10} + T^{12}$$
$5$ $$49 - 189 T^{2} + 638 T^{4} - 337 T^{6} + 142 T^{8} - 13 T^{10} + T^{12}$$
$7$ $$( 343 - 98 T + 35 T^{2} - 32 T^{3} + 5 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$11$ $$717409 - 276969 T^{2} + 75590 T^{4} - 10405 T^{6} + 1042 T^{8} - 37 T^{10} + T^{12}$$
$13$ $$( 1008 + 320 T^{2} + 32 T^{4} + T^{6} )^{2}$$
$17$ $$( 441 + 357 T + 310 T^{2} + 25 T^{3} + 18 T^{4} + T^{5} + T^{6} )^{2}$$
$19$ $$9529569 - 2515905 T^{2} + 475918 T^{4} - 43541 T^{6} + 2906 T^{8} - 61 T^{10} + T^{12}$$
$23$ $$( 81 + 63 T + 58 T^{2} + 11 T^{3} + 8 T^{4} + T^{5} + T^{6} )^{2}$$
$29$ $$( 112 + 304 T^{2} + 72 T^{4} + T^{6} )^{2}$$
$31$ $$( 49 - 21 T + 44 T^{2} + 29 T^{3} + 22 T^{4} + 5 T^{5} + T^{6} )^{2}$$
$37$ $$3969 - 56637 T^{2} + 804358 T^{4} - 54713 T^{6} + 2822 T^{8} - 61 T^{10} + T^{12}$$
$41$ $$( -84 - 40 T + 2 T^{2} + T^{3} )^{4}$$
$43$ $$( 4032 + 4016 T^{2} + 164 T^{4} + T^{6} )^{2}$$
$47$ $$( 35721 - 5103 T + 3564 T^{2} + 783 T^{3} + 198 T^{4} + 15 T^{5} + T^{6} )^{2}$$
$53$ $$678446209 - 127604253 T^{2} + 19910822 T^{4} - 717049 T^{6} + 19750 T^{8} - 157 T^{10} + T^{12}$$
$59$ $$678446209 - 108016909 T^{2} + 12587290 T^{4} - 681925 T^{6} + 27182 T^{8} - 177 T^{10} + T^{12}$$
$61$ $$413015731569 - 16032513861 T^{2} + 434052550 T^{4} - 6024145 T^{6} + 60902 T^{8} - 293 T^{10} + T^{12}$$
$67$ $$3969 - 15813 T^{2} + 60418 T^{4} - 10165 T^{6} + 1430 T^{8} - 41 T^{10} + T^{12}$$
$71$ $$( -432 - 56 T + 8 T^{2} + T^{3} )^{4}$$
$73$ $$( 194481 + 41013 T + 10854 T^{2} + 417 T^{3} + 118 T^{4} + 5 T^{5} + T^{6} )^{2}$$
$79$ $$( 81 - 189 T + 342 T^{2} - 213 T^{3} + 100 T^{4} - 11 T^{5} + T^{6} )^{2}$$
$83$ $$( 448 + 432 T^{2} + 52 T^{4} + T^{6} )^{2}$$
$89$ $$( 53361 - 25179 T + 10726 T^{2} - 1007 T^{3} + 134 T^{4} + 5 T^{5} + T^{6} )^{2}$$
$97$ $$( -28 - 36 T - 10 T^{2} + T^{3} )^{4}$$