# Properties

 Label 224.2.x.a Level 224 Weight 2 Character orbit 224.x Analytic conductor 1.789 Analytic rank 0 Dimension 8 CM discriminant -7 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.x (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{8}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{4} ) q^{2} + ( \beta_{3} + \beta_{5} ) q^{4} + ( 2 \beta_{2} + \beta_{4} ) q^{7} + ( 2 \beta_{6} + \beta_{7} ) q^{8} -3 \beta_{6} q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{4} ) q^{2} + ( \beta_{3} + \beta_{5} ) q^{4} + ( 2 \beta_{2} + \beta_{4} ) q^{7} + ( 2 \beta_{6} + \beta_{7} ) q^{8} -3 \beta_{6} q^{9} + ( -3 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{11} + ( 3 \beta_{3} + \beta_{5} ) q^{14} + ( -2 + 3 \beta_{1} ) q^{16} -3 \beta_{1} q^{18} + ( -4 - \beta_{1} + 4 \beta_{6} - 3 \beta_{7} ) q^{22} + ( 5 - 2 \beta_{1} - 5 \beta_{3} + 2 \beta_{5} ) q^{23} -5 \beta_{4} q^{25} + ( 2 \beta_{6} + 3 \beta_{7} ) q^{28} + ( -4 \beta_{2} + \beta_{3} - 3 \beta_{4} - 4 \beta_{5} ) q^{29} + ( \beta_{2} - 5 \beta_{4} ) q^{32} + ( -3 \beta_{2} + 3 \beta_{4} ) q^{36} + ( -\beta_{3} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{37} + ( -7 + 2 \beta_{1} + 2 \beta_{2} + 7 \beta_{4} ) q^{43} + ( 6 + \beta_{1} - 5 \beta_{2} - 3 \beta_{4} ) q^{44} + ( 3 \beta_{2} + 7 \beta_{4} + 4 \beta_{6} - 5 \beta_{7} ) q^{46} + 7 \beta_{3} q^{49} + ( 5 \beta_{3} - 5 \beta_{5} ) q^{50} + ( 3 + 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{4} ) q^{53} + ( -6 + 5 \beta_{1} ) q^{56} + ( -5 \beta_{3} - 3 \beta_{5} - 8 \beta_{6} + \beta_{7} ) q^{58} + ( 3 - 6 \beta_{1} ) q^{63} + ( 7 \beta_{3} - 5 \beta_{5} ) q^{64} + ( -1 + 6 \beta_{1} - 5 \beta_{6} + 6 \beta_{7} ) q^{67} + 16 \beta_{6} q^{71} + ( -9 \beta_{3} + 3 \beta_{5} ) q^{72} + ( 8 - 5 \beta_{1} - 8 \beta_{6} - \beta_{7} ) q^{74} + ( -5 - 4 \beta_{1} + 9 \beta_{6} - 4 \beta_{7} ) q^{77} + ( 6 \beta_{2} - \beta_{4} - \beta_{6} - 6 \beta_{7} ) q^{79} + 9 \beta_{3} q^{81} + ( -5 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} + 7 \beta_{5} ) q^{86} + ( 7 \beta_{2} - 7 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} ) q^{88} + ( 10 - \beta_{1} - \beta_{3} + 7 \beta_{5} ) q^{92} + 7 \beta_{7} q^{98} + ( -6 \beta_{2} + 3 \beta_{3} - 9 \beta_{4} + 6 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 4q^{16} - 12q^{18} - 36q^{22} + 32q^{23} - 48q^{43} + 52q^{44} + 40q^{53} - 28q^{56} + 16q^{67} + 44q^{74} - 56q^{77} + 76q^{92} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{4} + 2$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + \nu$$$$)/6$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 7 \nu^{2}$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 7 \nu^{3}$$$$)/24$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{3}$$$$)/12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{4} + \beta_{2}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$-5 \beta_{4} + \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$-5 \beta_{5} + 7 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$7 \beta_{7} - 10 \beta_{6}$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1
 −1.28897 + 0.581861i 0.581861 − 1.28897i −1.28897 − 0.581861i 0.581861 + 1.28897i −0.581861 + 1.28897i 1.28897 − 0.581861i −0.581861 − 1.28897i 1.28897 + 0.581861i
−1.28897 + 0.581861i 0 1.32288 1.50000i 0 0 −1.87083 + 1.87083i −0.832353 + 2.70318i 2.12132 2.12132i 0
27.2 0.581861 1.28897i 0 −1.32288 1.50000i 0 0 1.87083 1.87083i −2.70318 + 0.832353i 2.12132 2.12132i 0
83.1 −1.28897 0.581861i 0 1.32288 + 1.50000i 0 0 −1.87083 1.87083i −0.832353 2.70318i 2.12132 + 2.12132i 0
83.2 0.581861 + 1.28897i 0 −1.32288 + 1.50000i 0 0 1.87083 + 1.87083i −2.70318 0.832353i 2.12132 + 2.12132i 0
139.1 −0.581861 + 1.28897i 0 −1.32288 1.50000i 0 0 −1.87083 + 1.87083i 2.70318 0.832353i −2.12132 + 2.12132i 0
139.2 1.28897 0.581861i 0 1.32288 1.50000i 0 0 1.87083 1.87083i 0.832353 2.70318i −2.12132 + 2.12132i 0
195.1 −0.581861 1.28897i 0 −1.32288 + 1.50000i 0 0 −1.87083 1.87083i 2.70318 + 0.832353i −2.12132 2.12132i 0
195.2 1.28897 + 0.581861i 0 1.32288 + 1.50000i 0 0 1.87083 + 1.87083i 0.832353 + 2.70318i −2.12132 2.12132i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 195.2 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.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
32.h odd 8 1 inner
224.x even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.x.a 8
4.b odd 2 1 896.2.x.a 8
7.b odd 2 1 CM 224.2.x.a 8
28.d even 2 1 896.2.x.a 8
32.g even 8 1 896.2.x.a 8
32.h odd 8 1 inner 224.2.x.a 8
224.v odd 8 1 896.2.x.a 8
224.x even 8 1 inner 224.2.x.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.x.a 8 1.a even 1 1 trivial
224.2.x.a 8 7.b odd 2 1 CM
224.2.x.a 8 32.h odd 8 1 inner
224.2.x.a 8 224.x even 8 1 inner
896.2.x.a 8 4.b odd 2 1
896.2.x.a 8 28.d even 2 1
896.2.x.a 8 32.g even 8 1
896.2.x.a 8 224.v odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4} + 16 T^{8}$$
$3$ $$( 1 + 81 T^{8} )^{2}$$
$5$ $$( 1 + 625 T^{8} )^{2}$$
$7$ $$( 1 + 49 T^{4} )^{2}$$
$11$ $$( 1 - 6 T^{2} + 121 T^{4} )^{2}( 1 - 206 T^{4} + 14641 T^{8} )$$
$13$ $$( 1 + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{8}$$
$19$ $$( 1 + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 8 T + 23 T^{2} )^{4}( 1 + 18 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 54 T^{2} + 841 T^{4} )^{2}( 1 + 1234 T^{4} + 707281 T^{8} )$$
$31$ $$( 1 + 31 T^{2} )^{8}$$
$37$ $$( 1 - 38 T^{2} + 1369 T^{4} )^{2}( 1 - 1294 T^{4} + 1874161 T^{8} )$$
$41$ $$( 1 + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 12 T + 43 T^{2} )^{4}( 1 - 334 T^{4} + 3418801 T^{8} )$$
$47$ $$( 1 - 47 T^{2} )^{8}$$
$53$ $$( 1 - 10 T + 53 T^{2} )^{4}( 1 - 5582 T^{4} + 7890481 T^{8} )$$
$59$ $$( 1 + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 4 T + 67 T^{2} )^{4}( 1 + 4946 T^{4} + 20151121 T^{8} )$$
$71$ $$( 1 + 2914 T^{4} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 3646 T^{4} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 97 T^{2} )^{8}$$