Properties

 Label 224.3.v.a Level 224 Weight 3 Character orbit 224.v Analytic conductor 6.104 Analytic rank 0 Dimension 8 CM discriminant -7 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 224.v (of order $$8$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: 8.0.157351936.1 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{6} - \beta_{7} ) q^{2} + ( -\beta_{3} + 3 \beta_{5} ) q^{4} -7 \beta_{6} q^{7} + ( 5 \beta_{1} - 2 \beta_{4} ) q^{8} -9 \beta_{4} q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{6} - \beta_{7} ) q^{2} + ( -\beta_{3} + 3 \beta_{5} ) q^{4} -7 \beta_{6} q^{7} + ( 5 \beta_{1} - 2 \beta_{4} ) q^{8} -9 \beta_{4} q^{9} + ( 8 \beta_{1} - \beta_{3} + \beta_{4} + 8 \beta_{5} ) q^{11} + ( -7 \beta_{3} - 7 \beta_{5} ) q^{14} + ( 14 + 3 \beta_{2} ) q^{16} + ( 18 - 9 \beta_{2} ) q^{18} + ( 14 + 15 \beta_{1} + 9 \beta_{2} - 2 \beta_{4} ) q^{22} + ( -17 + 16 \beta_{2} + 17 \beta_{3} - 16 \beta_{5} ) q^{23} -25 \beta_{6} q^{25} + ( -21 \beta_{1} - 14 \beta_{4} ) q^{28} + ( 31 \beta_{3} - 8 \beta_{5} - 31 \beta_{6} + 8 \beta_{7} ) q^{29} + ( 34 \beta_{6} - 11 \beta_{7} ) q^{32} + ( 18 \beta_{6} - 27 \beta_{7} ) q^{36} + ( -24 \beta_{1} - 31 \beta_{3} - 31 \beta_{4} + 24 \beta_{5} ) q^{37} + ( -41 + 24 \beta_{2} - 17 \beta_{6} - 24 \beta_{7} ) q^{43} + ( 34 + 13 \beta_{2} + 46 \beta_{6} - 5 \beta_{7} ) q^{44} + ( -15 \beta_{1} + 34 \beta_{4} - 2 \beta_{6} + 33 \beta_{7} ) q^{46} + 49 \beta_{3} q^{49} + ( -25 \beta_{3} - 25 \beta_{5} ) q^{50} + ( 23 - 40 \beta_{2} - 17 \beta_{6} + 40 \beta_{7} ) q^{53} + ( -14 - 35 \beta_{2} ) q^{56} + ( 15 \beta_{1} - 7 \beta_{3} + 62 \beta_{4} - 39 \beta_{5} ) q^{58} -63 q^{63} + ( \beta_{3} + 45 \beta_{5} ) q^{64} + ( -71 - 24 \beta_{1} + 24 \beta_{2} + 47 \beta_{4} ) q^{67} + ( 64 \beta_{1} + 32 \beta_{4} ) q^{71} + ( -63 \beta_{3} + 45 \beta_{5} ) q^{72} + ( 14 + 17 \beta_{1} - 55 \beta_{2} - 62 \beta_{4} ) q^{74} + ( 7 - 56 \beta_{1} - 56 \beta_{2} - 49 \beta_{4} ) q^{77} + ( 48 \beta_{1} + 71 \beta_{4} + 23 \beta_{6} + 48 \beta_{7} ) q^{79} -81 \beta_{3} q^{81} + ( -89 \beta_{3} + 7 \beta_{5} - 34 \beta_{6} + 65 \beta_{7} ) q^{86} + ( 31 \beta_{3} + 51 \beta_{5} + 94 \beta_{6} - 21 \beta_{7} ) q^{88} + ( -98 + 19 \beta_{2} + 97 \beta_{3} - 35 \beta_{5} ) q^{92} + ( 49 \beta_{1} + 98 \beta_{4} ) q^{98} + ( -63 \beta_{3} + 72 \beta_{5} + 63 \beta_{6} - 72 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 124q^{16} + 108q^{18} + 148q^{22} - 72q^{23} - 232q^{43} + 324q^{44} + 24q^{53} - 252q^{56} - 504q^{63} - 472q^{67} - 108q^{74} - 168q^{77} - 708q^{92} + O(q^{100})$$

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

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

Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −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
−1.99607 + 0.125246i 0 3.96863 0.500000i 0 0 4.94975 4.94975i −7.85905 + 1.49509i −6.36396 6.36396i 0
13.2 −0.125246 + 1.99607i 0 −3.96863 0.500000i 0 0 4.94975 4.94975i 1.49509 7.85905i −6.36396 6.36396i 0
69.1 −1.99607 0.125246i 0 3.96863 + 0.500000i 0 0 4.94975 + 4.94975i −7.85905 1.49509i −6.36396 + 6.36396i 0
69.2 −0.125246 1.99607i 0 −3.96863 + 0.500000i 0 0 4.94975 + 4.94975i 1.49509 + 7.85905i −6.36396 + 6.36396i 0
125.1 0.125246 1.99607i 0 −3.96863 0.500000i 0 0 −4.94975 + 4.94975i −1.49509 + 7.85905i 6.36396 + 6.36396i 0
125.2 1.99607 0.125246i 0 3.96863 0.500000i 0 0 −4.94975 + 4.94975i 7.85905 1.49509i 6.36396 + 6.36396i 0
181.1 0.125246 + 1.99607i 0 −3.96863 + 0.500000i 0 0 −4.94975 4.94975i −1.49509 7.85905i 6.36396 6.36396i 0
181.2 1.99607 + 0.125246i 0 3.96863 + 0.500000i 0 0 −4.94975 4.94975i 7.85905 + 1.49509i 6.36396 6.36396i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.2 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.g even 8 1 inner
224.v odd 8 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.v.a 8
7.b odd 2 1 CM 224.3.v.a 8
32.g even 8 1 inner 224.3.v.a 8
224.v odd 8 1 inner 224.3.v.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.v.a 8 1.a even 1 1 trivial
224.3.v.a 8 7.b odd 2 1 CM
224.3.v.a 8 32.g even 8 1 inner
224.3.v.a 8 224.v odd 8 1 inner

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 31 T^{4} + 256 T^{8}$$
$3$ $$( 1 + 6561 T^{8} )^{2}$$
$5$ $$( 1 + 390625 T^{8} )^{2}$$
$7$ $$( 1 + 2401 T^{4} )^{2}$$
$11$ $$( 1 - 206 T^{2} + 14641 T^{4} )^{2}( 1 + 13154 T^{4} + 214358881 T^{8} )$$
$13$ $$( 1 + 815730721 T^{8} )^{2}$$
$17$ $$( 1 + 289 T^{2} )^{8}$$
$19$ $$( 1 + 16983563041 T^{8} )^{2}$$
$23$ $$( 1 + 18 T + 529 T^{2} )^{4}( 1 - 734 T^{2} + 279841 T^{4} )^{2}$$
$29$ $$( 1 + 1234 T^{2} + 707281 T^{4} )^{2}( 1 + 108194 T^{4} + 500246412961 T^{8} )$$
$31$ $$( 1 - 31 T )^{8}( 1 + 31 T )^{8}$$
$37$ $$( 1 - 1294 T^{2} + 1874161 T^{4} )^{2}( 1 - 2073886 T^{4} + 3512479453921 T^{8} )$$
$41$ $$( 1 + 2825761 T^{4} )^{4}$$
$43$ $$( 1 + 58 T + 1849 T^{2} )^{4}( 1 - 6726046 T^{4} + 11688200277601 T^{8} )$$
$47$ $$( 1 + 2209 T^{2} )^{8}$$
$53$ $$( 1 - 6 T + 2809 T^{2} )^{4}( 1 + 15377762 T^{4} + 62259690411361 T^{8} )$$
$59$ $$( 1 + 146830437604321 T^{8} )^{2}$$
$61$ $$( 1 + 191707312997281 T^{8} )^{2}$$
$67$ $$( 1 + 118 T + 4489 T^{2} )^{4}( 1 - 15839326 T^{4} + 406067677556641 T^{8} )$$
$71$ $$( 1 - 42331966 T^{4} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 + 28398241 T^{4} )^{4}$$
$79$ $$( 1 - 64606846 T^{4} + 1517108809906561 T^{8} )^{2}$$
$83$ $$( 1 + 2252292232139041 T^{8} )^{2}$$
$89$ $$( 1 + 62742241 T^{4} )^{4}$$
$97$ $$( 1 - 97 T )^{8}( 1 + 97 T )^{8}$$