Properties

 Label 1050.3.l.b Level $1050$ Weight $3$ Character orbit 1050.l Analytic conductor $28.610$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.l (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.12745506816.1 Defining polynomial: $$x^{8} + 23 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{3} ) q^{2} -\beta_{6} q^{3} + 2 \beta_{3} q^{4} + ( \beta_{1} + \beta_{6} ) q^{6} + \beta_{2} q^{7} + ( 2 - 2 \beta_{3} ) q^{8} -3 \beta_{3} q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{3} ) q^{2} -\beta_{6} q^{3} + 2 \beta_{3} q^{4} + ( \beta_{1} + \beta_{6} ) q^{6} + \beta_{2} q^{7} + ( 2 - 2 \beta_{3} ) q^{8} -3 \beta_{3} q^{9} + ( -1 + 5 \beta_{1} + \beta_{2} + \beta_{4} + 5 \beta_{6} - \beta_{7} ) q^{11} -2 \beta_{1} q^{12} + ( -1 + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{13} + ( -\beta_{2} - \beta_{7} ) q^{14} -4 q^{16} + ( 4 - 3 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{17} + ( -3 + 3 \beta_{3} ) q^{18} + ( -\beta_{1} - 4 \beta_{2} - 2 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{19} -\beta_{4} q^{21} + ( 1 - 10 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{22} + ( 5 - 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{24} + ( 2 + 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} ) q^{26} + 3 \beta_{1} q^{27} + 2 \beta_{7} q^{28} + ( -2 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 18 + 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{4} + 2 \beta_{6} + 3 \beta_{7} ) q^{31} + ( 4 + 4 \beta_{3} ) q^{32} + ( -15 + 15 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{33} + ( 3 \beta_{1} - \beta_{2} - 8 \beta_{3} - 3 \beta_{6} - \beta_{7} ) q^{34} + 6 q^{36} + ( -20 + 8 \beta_{1} - 12 \beta_{2} - 20 \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} ) q^{38} + ( -\beta_{1} - 3 \beta_{2} - 6 \beta_{3} + \beta_{6} - 3 \beta_{7} ) q^{39} + ( -40 - 15 \beta_{1} - 6 \beta_{2} + 4 \beta_{4} - 15 \beta_{6} + 6 \beta_{7} ) q^{41} + ( \beta_{4} - \beta_{5} ) q^{42} + ( 4 - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 14 \beta_{7} ) q^{43} + ( 10 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 10 \beta_{6} + 2 \beta_{7} ) q^{44} + ( -10 - 2 \beta_{2} - 4 \beta_{4} + 2 \beta_{7} ) q^{46} + ( 18 + 10 \beta_{1} - 2 \beta_{2} + 18 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{47} + 4 \beta_{6} q^{48} + 7 \beta_{3} q^{49} + ( 9 - 4 \beta_{1} - \beta_{4} - 4 \beta_{6} ) q^{51} + ( -2 - 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{52} + ( 25 - 25 \beta_{3} - \beta_{4} - \beta_{5} + 12 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -3 \beta_{1} + 3 \beta_{6} ) q^{54} + ( 2 \beta_{2} - 2 \beta_{7} ) q^{56} + ( 3 - 6 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{57} + ( 8 - 8 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{58} + ( -12 \beta_{1} + 6 \beta_{2} + 8 \beta_{3} - 16 \beta_{5} + 12 \beta_{6} + 6 \beta_{7} ) q^{59} + ( 36 - 4 \beta_{1} + 2 \beta_{2} + 14 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{61} + ( -18 - 4 \beta_{1} + 6 \beta_{2} - 18 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{62} -3 \beta_{7} q^{63} -8 \beta_{3} q^{64} + ( 30 - \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - \beta_{6} - 3 \beta_{7} ) q^{66} + ( -10 + 12 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} ) q^{67} + ( -8 + 8 \beta_{3} + 6 \beta_{6} + 2 \beta_{7} ) q^{68} + ( 5 \beta_{1} + 6 \beta_{2} - 2 \beta_{5} - 5 \beta_{6} + 6 \beta_{7} ) q^{69} + ( -35 + \beta_{1} - \beta_{2} + 5 \beta_{4} + \beta_{6} + \beta_{7} ) q^{71} + ( -6 - 6 \beta_{3} ) q^{72} + ( -39 + 39 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} + 22 \beta_{6} - 20 \beta_{7} ) q^{73} + ( -8 \beta_{1} + 12 \beta_{2} + 40 \beta_{3} - 2 \beta_{5} + 8 \beta_{6} + 12 \beta_{7} ) q^{74} + ( 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} - 8 \beta_{7} ) q^{76} + ( 7 + 7 \beta_{1} - \beta_{2} + 7 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} ) q^{77} + ( -6 + 6 \beta_{3} - 2 \beta_{6} + 6 \beta_{7} ) q^{78} + ( -6 \beta_{1} - 20 \beta_{2} - 24 \beta_{3} - 16 \beta_{5} + 6 \beta_{6} - 20 \beta_{7} ) q^{79} -9 q^{81} + ( 40 + 30 \beta_{1} + 12 \beta_{2} + 40 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{82} + ( 40 - 40 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 38 \beta_{6} + 10 \beta_{7} ) q^{83} + 2 \beta_{5} q^{84} + ( -8 + 6 \beta_{1} - 14 \beta_{2} + 4 \beta_{4} + 6 \beta_{6} + 14 \beta_{7} ) q^{86} + ( 6 - 8 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{87} + ( -2 + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 20 \beta_{6} - 4 \beta_{7} ) q^{88} + ( 10 \beta_{1} + 15 \beta_{2} + 4 \beta_{3} - 16 \beta_{5} - 10 \beta_{6} + 15 \beta_{7} ) q^{89} + ( -7 \beta_{1} - \beta_{2} - 2 \beta_{4} - 7 \beta_{6} + \beta_{7} ) q^{91} + ( 10 + 4 \beta_{2} + 10 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{92} + ( -6 + 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 18 \beta_{6} + 12 \beta_{7} ) q^{93} + ( -10 \beta_{1} + 2 \beta_{2} - 36 \beta_{3} - 8 \beta_{5} + 10 \beta_{6} + 2 \beta_{7} ) q^{94} + ( -4 \beta_{1} - 4 \beta_{6} ) q^{96} + ( 3 + 6 \beta_{1} - 8 \beta_{2} + 3 \beta_{3} + 17 \beta_{4} - 17 \beta_{5} ) q^{97} + ( 7 - 7 \beta_{3} ) q^{98} + ( -15 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + 15 \beta_{6} - 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} + 16 q^{8} + O(q^{10})$$ $$8 q - 8 q^{2} + 16 q^{8} - 8 q^{11} - 8 q^{13} - 32 q^{16} + 32 q^{17} - 24 q^{18} + 8 q^{22} + 40 q^{23} + 16 q^{26} + 144 q^{31} + 32 q^{32} - 120 q^{33} + 48 q^{36} - 160 q^{37} - 320 q^{41} + 32 q^{43} - 80 q^{46} + 144 q^{47} + 72 q^{51} - 16 q^{52} + 200 q^{53} + 24 q^{57} + 64 q^{58} + 288 q^{61} - 144 q^{62} + 240 q^{66} - 80 q^{67} - 64 q^{68} - 280 q^{71} - 48 q^{72} - 312 q^{73} + 56 q^{77} - 48 q^{78} - 72 q^{81} + 320 q^{82} + 320 q^{83} - 64 q^{86} + 48 q^{87} - 16 q^{88} + 80 q^{92} - 48 q^{93} + 24 q^{97} + 56 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 23 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 19 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 29 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 24 \nu^{2}$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{4} + 23$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$-\nu^{6} - 22 \nu^{2}$$ $$\beta_{6}$$ $$=$$ $$($$$$-4 \nu^{7} - 91 \nu^{3}$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$6 \nu^{7} + 139 \nu^{3}$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 5 \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} + 3 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{4} - 23$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-19 \beta_{2} + 29 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-12 \beta_{5} - 55 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$($$$$-91 \beta_{7} - 139 \beta_{6}$$$$)/2$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −1.54779 + 1.54779i 0.323042 − 0.323042i −0.323042 + 0.323042i 1.54779 − 1.54779i −1.54779 − 1.54779i 0.323042 + 0.323042i −0.323042 − 0.323042i 1.54779 + 1.54779i
−1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 0 2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i 0
43.2 −1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 0 2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i 0
43.3 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i 0 −2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i 0
43.4 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i 0 −2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i 0
757.1 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 0 2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i 0
757.2 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 0 2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i 0
757.3 −1.00000 1.00000i 1.22474 1.22474i 2.00000i 0 −2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i 0
757.4 −1.00000 1.00000i 1.22474 1.22474i 2.00000i 0 −2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 757.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.l.b 8
5.b even 2 1 210.3.l.a 8
5.c odd 4 1 210.3.l.a 8
5.c odd 4 1 inner 1050.3.l.b 8
15.d odd 2 1 630.3.o.b 8
15.e even 4 1 630.3.o.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.l.a 8 5.b even 2 1
210.3.l.a 8 5.c odd 4 1
630.3.o.b 8 15.d odd 2 1
630.3.o.b 8 15.e even 4 1
1050.3.l.b 8 1.a even 1 1 trivial
1050.3.l.b 8 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{4} + 4 T_{11}^{3} - 364 T_{11}^{2} - 2416 T_{11} + 10000$$ $$T_{13}^{8} + \cdots$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + 2 T + T^{2} )^{4}$$
$3$ $$( 9 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 49 + T^{4} )^{2}$$
$11$ $$( 10000 - 2416 T - 364 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$13$ $$473344 - 286208 T + 86528 T^{2} + 34432 T^{3} + 7840 T^{4} - 352 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$17$ $$565504 - 48128 T + 2048 T^{2} - 16128 T^{3} + 16880 T^{4} - 4032 T^{5} + 512 T^{6} - 32 T^{7} + T^{8}$$
$19$ $$254083600 + 27245984 T^{2} + 426264 T^{4} + 1256 T^{6} + T^{8}$$
$23$ $$6250000 + 14600000 T + 17052800 T^{2} - 437280 T^{3} + 3464 T^{4} - 2160 T^{5} + 800 T^{6} - 40 T^{7} + T^{8}$$
$29$ $$1914937600 + 39569664 T^{2} + 292448 T^{4} + 912 T^{6} + T^{8}$$
$31$ $$( -332604 + 19728 T + 972 T^{2} - 72 T^{3} + T^{4} )^{2}$$
$37$ $$2032050250000 + 146792288000 T + 5302028288 T^{2} - 10412416 T^{3} + 8214856 T^{4} + 473536 T^{5} + 12800 T^{6} + 160 T^{7} + T^{8}$$
$41$ $$( -9703676 - 215360 T + 5220 T^{2} + 160 T^{3} + T^{4} )^{2}$$
$43$ $$1562460000256 - 5919924224 T + 11214848 T^{2} + 54700032 T^{3} + 7134848 T^{4} + 104064 T^{5} + 512 T^{6} - 32 T^{7} + T^{8}$$
$47$ $$94568550400 - 7321436160 T + 283410432 T^{2} + 666624 T^{3} + 4179584 T^{4} - 295680 T^{5} + 10368 T^{6} - 144 T^{7} + T^{8}$$
$53$ $$53935417600 - 32870320640 T + 10016219648 T^{2} - 872969856 T^{3} + 41733536 T^{4} - 1157664 T^{5} + 20000 T^{6} - 200 T^{7} + T^{8}$$
$59$ $$107464902774784 + 632685058048 T^{2} + 230901120 T^{4} + 27232 T^{6} + T^{8}$$
$61$ $$( 4954000 + 465600 T - 760 T^{2} - 144 T^{3} + T^{4} )^{2}$$
$67$ $$653738987622400 - 18023006494720 T + 248439185408 T^{2} + 6402008064 T^{3} + 92479616 T^{4} - 253824 T^{5} + 3200 T^{6} + 80 T^{7} + T^{8}$$
$71$ $$( 478864 + 96880 T + 6260 T^{2} + 140 T^{3} + T^{4} )^{2}$$
$73$ $$3345696964000000 + 83542357440000 T + 1043030131200 T^{2} + 7000544640 T^{3} + 174175904 T^{4} + 3830496 T^{5} + 48672 T^{6} + 312 T^{7} + T^{8}$$
$79$ $$1068576474271744 + 1532269918208 T^{2} + 612428160 T^{4} + 47072 T^{6} + T^{8}$$
$83$ $$471997524054016 - 19452147261440 T + 400834764800 T^{2} + 927006720 T^{3} + 120891008 T^{4} - 3711360 T^{5} + 51200 T^{6} - 320 T^{7} + T^{8}$$
$89$ $$3929513290000 + 364463932000 T^{2} + 304535256 T^{4} + 36568 T^{6} + T^{8}$$
$97$ $$17601328900000000 + 76736328000000 T + 167273280000 T^{2} - 11437872000 T^{3} + 373738400 T^{4} + 28320 T^{5} + 288 T^{6} - 24 T^{7} + T^{8}$$