# Properties

 Label 1456.2.r.p Level $1456$ Weight $2$ Character orbit 1456.r Analytic conductor $11.626$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1456 = 2^{4} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1456.r (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.6262185343$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - x^{9} + 8 x^{8} + 7 x^{7} + 41 x^{6} + 18 x^{5} + 58 x^{4} + 28 x^{3} + 64 x^{2} + 16 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{9} q^{3} + ( -\beta_{3} - \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{5} + \beta_{9} ) q^{7} + ( \beta_{3} - \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{9} q^{3} + ( -\beta_{3} - \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{5} + \beta_{9} ) q^{7} + ( \beta_{3} - \beta_{6} + \beta_{7} ) q^{9} + ( 2 + \beta_{5} - 2 \beta_{6} ) q^{11} + q^{13} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{15} + ( 1 + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{17} + ( \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{19} + ( -2 - \beta_{3} - \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{21} + ( \beta_{2} + 2 \beta_{6} + \beta_{9} ) q^{23} + ( -1 - 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{25} + ( \beta_{2} - \beta_{3} - \beta_{4} ) q^{27} + ( -1 - \beta_{4} ) q^{29} + ( -2 + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{31} + ( -2 \beta_{2} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{33} + ( 2 - \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - \beta_{8} - 2 \beta_{9} ) q^{37} -\beta_{9} q^{39} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{41} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{43} + ( 6 + 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{9} ) q^{45} + ( -\beta_{1} - 4 \beta_{2} - \beta_{5} - 4 \beta_{9} ) q^{47} + ( -1 + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{49} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{51} + ( -3 + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} ) q^{53} + ( 2 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{55} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{57} + ( 2 + \beta_{5} - 2 \beta_{6} ) q^{59} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{9} ) q^{61} + ( -2 + 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{63} + ( -\beta_{3} - \beta_{7} ) q^{65} + ( 4 - \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} - \beta_{8} ) q^{67} + ( 4 + 2 \beta_{2} - \beta_{3} ) q^{69} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{71} + ( -2 \beta_{5} - \beta_{7} ) q^{73} + ( \beta_{3} - 4 \beta_{6} + \beta_{7} - \beta_{8} ) q^{75} + ( -5 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{77} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{79} + ( 5 - 2 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{81} + ( -2 + 3 \beta_{3} + \beta_{4} ) q^{83} + ( -4 + 4 \beta_{2} - \beta_{3} ) q^{85} + ( -2 - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{87} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - \beta_{8} - 4 \beta_{9} ) q^{89} + ( \beta_{1} + \beta_{5} + \beta_{9} ) q^{91} + ( 4 \beta_{2} + 3 \beta_{3} - 4 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} ) q^{93} + ( -2 + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - 5 \beta_{9} ) q^{95} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{97} + ( -4 - \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 2q^{5} - q^{7} - 3q^{9} + O(q^{10})$$ $$10q - 2q^{5} - q^{7} - 3q^{9} + 11q^{11} + 10q^{13} + 5q^{17} + 9q^{19} + 2q^{21} + 10q^{23} - 9q^{25} - 6q^{29} - 6q^{31} - 8q^{33} + 4q^{35} - 4q^{37} + 28q^{41} - 4q^{43} + 32q^{45} + q^{47} - 11q^{49} - 8q^{51} - 17q^{53} - 32q^{57} + 11q^{59} + 11q^{61} - 5q^{63} - 2q^{65} + 13q^{67} + 36q^{69} - 30q^{71} - 20q^{75} - 46q^{77} + 2q^{79} + 19q^{81} - 12q^{83} - 44q^{85} - 8q^{87} + 4q^{89} - q^{91} - 18q^{93} - 12q^{95} - 24q^{97} - 22q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 8 x^{8} + 7 x^{7} + 41 x^{6} + 18 x^{5} + 58 x^{4} + 28 x^{3} + 64 x^{2} + 16 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$983 \nu^{9} + 7328 \nu^{8} - 9160 \nu^{7} + 87336 \nu^{6} + 36640 \nu^{5} + 287624 \nu^{4} + 39747 \nu^{3} + 392048 \nu^{2} + 98928 \nu + 22604$$$$)/118350$$ $$\beta_{2}$$ $$=$$ $$($$$$-1159 \nu^{9} - 9844 \nu^{8} + 12305 \nu^{7} - 109053 \nu^{6} - 49220 \nu^{5} - 386377 \nu^{4} + 25194 \nu^{3} - 526654 \nu^{2} - 132894 \nu - 348592$$$$)/118350$$ $$\beta_{3}$$ $$=$$ $$($$$$916 \nu^{9} - 3044 \nu^{8} + 3805 \nu^{7} - 3978 \nu^{6} - 15220 \nu^{5} - 119477 \nu^{4} - 129531 \nu^{3} - 162854 \nu^{2} - 41094 \nu - 180842$$$$)/59175$$ $$\beta_{4}$$ $$=$$ $$($$$$-629 \nu^{9} - 3164 \nu^{8} + 3955 \nu^{7} - 40293 \nu^{6} - 15820 \nu^{5} - 124187 \nu^{4} + 64764 \nu^{3} - 169274 \nu^{2} - 42714 \nu - 95852$$$$)/39450$$ $$\beta_{5}$$ $$=$$ $$($$$$-8713 \nu^{9} + 13967 \nu^{8} - 61840 \nu^{7} - 70821 \nu^{6} - 166865 \nu^{5} - 117514 \nu^{4} - 196692 \nu^{3} - 643378 \nu^{2} - 136908 \nu - 33244$$$$)/236700$$ $$\beta_{6}$$ $$=$$ $$($$$$-15259 \nu^{9} + 14531 \nu^{8} - 121720 \nu^{7} - 107253 \nu^{6} - 637445 \nu^{5} - 272902 \nu^{4} - 871206 \nu^{3} - 258154 \nu^{2} - 957744 \nu - 2692$$$$)/236700$$ $$\beta_{7}$$ $$=$$ $$($$$$16307 \nu^{9} - 15688 \nu^{8} + 137960 \nu^{7} + 104769 \nu^{6} + 750010 \nu^{5} + 331046 \nu^{4} + 1240338 \nu^{3} + 580892 \nu^{2} + 1445112 \nu + 362216$$$$)/118350$$ $$\beta_{8}$$ $$=$$ $$($$$$-1058 \nu^{9} + 1552 \nu^{8} - 9041 \nu^{7} - 3726 \nu^{6} - 39580 \nu^{5} - 2993 \nu^{4} - 53832 \nu^{3} - 11648 \nu^{2} - 40590 \nu + 136$$$$)/4734$$ $$\beta_{9}$$ $$=$$ $$($$$$9203 \nu^{9} - 9652 \nu^{8} + 71240 \nu^{7} + 67401 \nu^{6} + 346240 \nu^{5} + 153734 \nu^{4} + 440202 \nu^{3} + 272618 \nu^{2} + 461448 \nu + 115064$$$$)/39450$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{8} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{9} + \beta_{8} + 6 \beta_{6} + 2 \beta_{5} + \beta_{4} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{4} + \beta_{3} - 5 \beta_{2} - 2 \beta_{1} - 4$$ $$\nu^{4}$$ $$=$$ $$($$$$-29 \beta_{9} - 13 \beta_{8} + 4 \beta_{7} - 38 \beta_{6} - 18 \beta_{5} + 4 \beta_{3} - 29 \beta_{2} - 18 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-91 \beta_{9} - 47 \beta_{8} + 18 \beta_{7} - 90 \beta_{6} - 46 \beta_{5} - 47 \beta_{4} + 90$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-64 \beta_{4} - 23 \beta_{3} + 134 \beta_{2} + 78 \beta_{1} + 154$$ $$\nu^{7}$$ $$=$$ $$($$$$819 \beta_{9} + 407 \beta_{8} - 156 \beta_{7} + 862 \beta_{6} + 442 \beta_{5} - 156 \beta_{3} + 819 \beta_{2} + 442 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$2443 \beta_{9} + 1187 \beta_{8} - 442 \beta_{7} + 2698 \beta_{6} + 1382 \beta_{5} + 1187 \beta_{4} - 2698$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$1817 \beta_{4} + 691 \beta_{3} - 3695 \beta_{2} - 2036 \beta_{1} - 3968$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$$.

 $$n$$ $$561$$ $$911$$ $$1093$$ $$1249$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
417.1
 −0.132804 − 0.230024i −0.862625 − 1.49411i 0.597828 + 1.03547i 1.50426 + 2.60546i −0.606661 − 1.05077i −0.132804 + 0.230024i −0.862625 + 1.49411i 0.597828 − 1.03547i 1.50426 − 2.60546i −0.606661 + 1.05077i
0 −1.31364 + 2.27529i 0 1.45130 + 2.51373i 0 1.29536 2.30696i 0 −1.95130 3.37975i 0
417.2 0 −0.673208 + 1.16603i 0 −1.09358 1.89414i 0 2.19729 + 1.47375i 0 0.593582 + 1.02811i 0
417.3 0 −0.129894 + 0.224983i 0 −1.96625 3.40565i 0 −1.12324 2.39548i 0 1.46625 + 2.53963i 0
417.4 0 0.879528 1.52339i 0 −0.452861 0.784378i 0 −0.237709 + 2.63505i 0 −0.0471392 0.0816475i 0
417.5 0 1.23721 2.14292i 0 1.06140 + 1.83839i 0 −2.63169 0.272389i 0 −1.56140 2.70442i 0
625.1 0 −1.31364 2.27529i 0 1.45130 2.51373i 0 1.29536 + 2.30696i 0 −1.95130 + 3.37975i 0
625.2 0 −0.673208 1.16603i 0 −1.09358 + 1.89414i 0 2.19729 1.47375i 0 0.593582 1.02811i 0
625.3 0 −0.129894 0.224983i 0 −1.96625 + 3.40565i 0 −1.12324 + 2.39548i 0 1.46625 2.53963i 0
625.4 0 0.879528 + 1.52339i 0 −0.452861 + 0.784378i 0 −0.237709 2.63505i 0 −0.0471392 + 0.0816475i 0
625.5 0 1.23721 + 2.14292i 0 1.06140 1.83839i 0 −2.63169 + 0.272389i 0 −1.56140 + 2.70442i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 625.5 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.r.p 10
4.b odd 2 1 91.2.e.c 10
7.c even 3 1 inner 1456.2.r.p 10
12.b even 2 1 819.2.j.h 10
28.d even 2 1 637.2.e.m 10
28.f even 6 1 637.2.a.k 5
28.f even 6 1 637.2.e.m 10
28.g odd 6 1 91.2.e.c 10
28.g odd 6 1 637.2.a.l 5
52.b odd 2 1 1183.2.e.f 10
84.j odd 6 1 5733.2.a.bm 5
84.n even 6 1 819.2.j.h 10
84.n even 6 1 5733.2.a.bl 5
364.x even 6 1 8281.2.a.bx 5
364.bl odd 6 1 1183.2.e.f 10
364.bl odd 6 1 8281.2.a.bw 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c 10 4.b odd 2 1
91.2.e.c 10 28.g odd 6 1
637.2.a.k 5 28.f even 6 1
637.2.a.l 5 28.g odd 6 1
637.2.e.m 10 28.d even 2 1
637.2.e.m 10 28.f even 6 1
819.2.j.h 10 12.b even 2 1
819.2.j.h 10 84.n even 6 1
1183.2.e.f 10 52.b odd 2 1
1183.2.e.f 10 364.bl odd 6 1
1456.2.r.p 10 1.a even 1 1 trivial
1456.2.r.p 10 7.c even 3 1 inner
5733.2.a.bl 5 84.n even 6 1
5733.2.a.bm 5 84.j odd 6 1
8281.2.a.bw 5 364.bl odd 6 1
8281.2.a.bx 5 364.x even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1456, [\chi])$$:

 $$T_{3}^{10} + \cdots$$ $$T_{5}^{10} + \cdots$$ $$T_{11}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$16 + 64 T + 256 T^{2} + 72 T^{3} + 144 T^{4} + 4 T^{5} + 65 T^{6} + 9 T^{8} + T^{10}$$
$5$ $$2304 + 2304 T + 3264 T^{2} + 480 T^{3} + 1024 T^{4} + 156 T^{5} + 217 T^{6} + 10 T^{7} + 19 T^{8} + 2 T^{9} + T^{10}$$
$7$ $$16807 + 2401 T + 2058 T^{2} + 833 T^{3} + 119 T^{4} + 204 T^{5} + 17 T^{6} + 17 T^{7} + 6 T^{8} + T^{9} + T^{10}$$
$11$ $$1089 - 1485 T + 2751 T^{2} - 1386 T^{3} + 2467 T^{4} - 1749 T^{5} + 1099 T^{6} - 352 T^{7} + 85 T^{8} - 11 T^{9} + T^{10}$$
$13$ $$( -1 + T )^{10}$$
$17$ $$184041 - 39897 T + 54123 T^{2} - 9018 T^{3} + 11137 T^{4} - 1831 T^{5} + 921 T^{6} - 102 T^{7} + 47 T^{8} - 5 T^{9} + T^{10}$$
$19$ $$49729 - 38579 T + 69177 T^{2} + 24204 T^{3} + 31391 T^{4} + 427 T^{5} + 1607 T^{6} - 226 T^{7} + 95 T^{8} - 9 T^{9} + T^{10}$$
$23$ $$144 - 144 T + 456 T^{2} - 432 T^{3} + 1168 T^{4} - 1034 T^{5} + 713 T^{6} - 258 T^{7} + 69 T^{8} - 10 T^{9} + T^{10}$$
$29$ $$( -108 + 144 T - 19 T^{2} - 25 T^{3} + 3 T^{4} + T^{5} )^{2}$$
$31$ $$126736 - 180848 T + 221752 T^{2} - 95248 T^{3} + 43528 T^{4} - 230 T^{5} + 3825 T^{6} - 162 T^{7} + 97 T^{8} + 6 T^{9} + T^{10}$$
$37$ $$49505296 + 4643760 T + 5206008 T^{2} + 1114512 T^{3} + 504800 T^{4} + 77014 T^{5} + 14373 T^{6} + 912 T^{7} + 127 T^{8} + 4 T^{9} + T^{10}$$
$41$ $$( -1584 - 2544 T + 940 T^{2} - 28 T^{3} - 14 T^{4} + T^{5} )^{2}$$
$43$ $$( -64 - 288 T - 308 T^{2} - 72 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$47$ $$26718561 - 14530059 T + 8036115 T^{2} - 1208826 T^{3} + 344071 T^{4} - 2771 T^{5} + 12591 T^{6} + 72 T^{7} + 125 T^{8} - T^{9} + T^{10}$$
$53$ $$398361681 + 254656881 T + 114371547 T^{2} + 27999402 T^{3} + 5280613 T^{4} + 593371 T^{5} + 59477 T^{6} + 3594 T^{7} + 363 T^{8} + 17 T^{9} + T^{10}$$
$59$ $$1089 - 1485 T + 2751 T^{2} - 1386 T^{3} + 2467 T^{4} - 1749 T^{5} + 1099 T^{6} - 352 T^{7} + 85 T^{8} - 11 T^{9} + T^{10}$$
$61$ $$71588521 + 49759141 T + 28105035 T^{2} + 6569330 T^{3} + 1397309 T^{4} + 44391 T^{5} + 17429 T^{6} - 190 T^{7} + 243 T^{8} - 11 T^{9} + T^{10}$$
$67$ $$515244601 + 13415109 T + 49379121 T^{2} - 8631036 T^{3} + 4274771 T^{4} - 387985 T^{5} + 54915 T^{6} - 2214 T^{7} + 331 T^{8} - 13 T^{9} + T^{10}$$
$71$ $$( 6336 - 456 T - 853 T^{2} - 25 T^{3} + 15 T^{4} + T^{5} )^{2}$$
$73$ $$506944 + 498400 T + 519904 T^{2} + 77400 T^{3} + 54264 T^{4} + 3862 T^{5} + 4925 T^{6} + 84 T^{7} + 75 T^{8} + T^{10}$$
$79$ $$1000000 - 1500000 T + 2060000 T^{2} - 559000 T^{3} + 239600 T^{4} + 31030 T^{5} + 16889 T^{6} + 654 T^{7} + 141 T^{8} - 2 T^{9} + T^{10}$$
$83$ $$( 7488 + 2688 T - 308 T^{2} - 124 T^{3} + 6 T^{4} + T^{5} )^{2}$$
$89$ $$59166864 - 16522416 T + 9952152 T^{2} - 893808 T^{3} + 783808 T^{4} - 98078 T^{5} + 24653 T^{6} - 768 T^{7} + 171 T^{8} - 4 T^{9} + T^{10}$$
$97$ $$( -2384 - 2240 T - 612 T^{2} - 16 T^{3} + 12 T^{4} + T^{5} )^{2}$$