# Properties

 Label 1050.3.l.a 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: yes 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} + ( -4 + 3 \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{11} + 2 \beta_{1} q^{12} + ( -5 + 5 \beta_{3} - 5 \beta_{6} - \beta_{7} ) q^{13} + ( -\beta_{2} - \beta_{7} ) q^{14} -4 q^{16} + ( 5 + \beta_{1} - 9 \beta_{2} + 5 \beta_{3} ) q^{17} + ( -3 + 3 \beta_{3} ) q^{18} + ( -2 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} - \beta_{5} + 2 \beta_{6} + 7 \beta_{7} ) q^{19} + \beta_{4} q^{21} + ( 4 - 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{22} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + 8 \beta_{6} - \beta_{7} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{6} ) q^{24} + ( 10 + 5 \beta_{1} - \beta_{2} + 5 \beta_{6} + \beta_{7} ) q^{26} -3 \beta_{1} q^{27} + 2 \beta_{7} q^{28} + ( 4 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} + \beta_{5} - 4 \beta_{6} + 9 \beta_{7} ) q^{29} + ( 12 - 6 \beta_{1} - 2 \beta_{2} - 6 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} ) q^{31} + ( 4 + 4 \beta_{3} ) q^{32} + ( 9 - 9 \beta_{3} + \beta_{4} + \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{33} + ( -\beta_{1} + 9 \beta_{2} - 10 \beta_{3} + \beta_{6} + 9 \beta_{7} ) q^{34} + 6 q^{36} + ( -14 + 15 \beta_{1} - 4 \beta_{2} - 14 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( 3 - 3 \beta_{3} + \beta_{4} + \beta_{5} - 4 \beta_{6} - 14 \beta_{7} ) q^{38} + ( 5 \beta_{1} + 15 \beta_{3} + \beta_{5} - 5 \beta_{6} ) q^{39} + ( -20 - 10 \beta_{1} + 2 \beta_{2} - 10 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -\beta_{4} + \beta_{5} ) q^{42} + ( 8 - 8 \beta_{3} - \beta_{4} - \beta_{5} + 13 \beta_{6} - 2 \beta_{7} ) q^{43} + ( 6 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} ) q^{44} + ( -2 - 8 \beta_{1} - \beta_{2} + 2 \beta_{4} - 8 \beta_{6} + \beta_{7} ) q^{46} + ( -8 - 13 \beta_{1} - 7 \beta_{2} - 8 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} ) q^{47} -4 \beta_{6} q^{48} + 7 \beta_{3} q^{49} + ( 3 + 5 \beta_{1} - 9 \beta_{4} + 5 \beta_{6} ) q^{51} + ( -10 - 10 \beta_{1} + 2 \beta_{2} - 10 \beta_{3} ) q^{52} + ( -10 + 10 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} ) q^{53} + ( 3 \beta_{1} - 3 \beta_{6} ) q^{54} + ( 2 \beta_{2} - 2 \beta_{7} ) q^{56} + ( -6 + 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} ) q^{57} + ( -4 + 4 \beta_{3} - \beta_{4} - \beta_{5} + 8 \beta_{6} - 18 \beta_{7} ) q^{58} + ( -20 \beta_{1} + 5 \beta_{2} + 15 \beta_{3} - 11 \beta_{5} + 20 \beta_{6} + 5 \beta_{7} ) q^{59} + ( -16 + 16 \beta_{1} - 16 \beta_{2} - 8 \beta_{4} + 16 \beta_{6} + 16 \beta_{7} ) q^{61} + ( -12 + 12 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{62} -3 \beta_{7} q^{63} -8 \beta_{3} q^{64} + ( -18 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 4 \beta_{6} - 3 \beta_{7} ) q^{66} + ( 38 + 7 \beta_{1} - 4 \beta_{2} + 38 \beta_{3} + \beta_{4} - \beta_{5} ) q^{67} + ( -10 + 10 \beta_{3} - 2 \beta_{6} - 18 \beta_{7} ) q^{68} + ( -\beta_{1} + 3 \beta_{2} - 24 \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{69} + ( 30 - \beta_{1} - 19 \beta_{2} + \beta_{4} - \beta_{6} + 19 \beta_{7} ) q^{71} + ( -6 - 6 \beta_{3} ) q^{72} + ( -3 + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 19 \beta_{6} + 27 \beta_{7} ) q^{73} + ( -15 \beta_{1} + 4 \beta_{2} + 28 \beta_{3} - 4 \beta_{5} + 15 \beta_{6} + 4 \beta_{7} ) q^{74} + ( -6 + 4 \beta_{1} - 14 \beta_{2} - 2 \beta_{4} + 4 \beta_{6} + 14 \beta_{7} ) q^{76} + ( 7 - 7 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{77} + ( 15 - 15 \beta_{3} - \beta_{4} - \beta_{5} + 10 \beta_{6} ) q^{78} + ( 6 \beta_{1} + 19 \beta_{2} - 29 \beta_{3} - 6 \beta_{5} - 6 \beta_{6} + 19 \beta_{7} ) q^{79} -9 q^{81} + ( 20 + 20 \beta_{1} - 4 \beta_{2} + 20 \beta_{3} ) q^{82} + ( -8 + 8 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} ) q^{83} -2 \beta_{5} q^{84} + ( -16 - 13 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 13 \beta_{6} + 2 \beta_{7} ) q^{86} + ( 12 - 4 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} ) q^{87} + ( -8 + 8 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 12 \beta_{6} - 4 \beta_{7} ) q^{88} + ( 29 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 5 \beta_{5} - 29 \beta_{6} + 2 \beta_{7} ) q^{89} + ( 7 - 5 \beta_{2} - 5 \beta_{4} + 5 \beta_{7} ) q^{91} + ( 2 + 16 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{92} + ( -18 + 18 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 12 \beta_{6} + 18 \beta_{7} ) q^{93} + ( 13 \beta_{1} + 7 \beta_{2} + 16 \beta_{3} + 18 \beta_{5} - 13 \beta_{6} + 7 \beta_{7} ) q^{94} + ( 4 \beta_{1} + 4 \beta_{6} ) q^{96} + ( -34 - 42 \beta_{1} - 12 \beta_{2} - 34 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{97} + ( 7 - 7 \beta_{3} ) q^{98} + ( -9 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} - 3 \beta_{5} + 9 \beta_{6} - 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} + 16 q^{8} + O(q^{10})$$ $$8 q - 8 q^{2} + 16 q^{8} - 32 q^{11} - 40 q^{13} - 32 q^{16} + 40 q^{17} - 24 q^{18} + 32 q^{22} + 8 q^{23} + 80 q^{26} + 96 q^{31} + 32 q^{32} + 72 q^{33} + 48 q^{36} - 112 q^{37} + 24 q^{38} - 160 q^{41} + 64 q^{43} - 16 q^{46} - 64 q^{47} + 24 q^{51} - 80 q^{52} - 80 q^{53} - 48 q^{57} - 32 q^{58} - 128 q^{61} - 96 q^{62} - 144 q^{66} + 304 q^{67} - 80 q^{68} + 240 q^{71} - 48 q^{72} - 24 q^{73} - 48 q^{76} + 56 q^{77} + 120 q^{78} - 72 q^{81} + 160 q^{82} - 64 q^{83} - 128 q^{86} + 96 q^{87} - 64 q^{88} + 56 q^{91} + 16 q^{92} - 144 q^{93} - 272 q^{97} + 56 q^{98} + O(q^{100})$$

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
 −0.323042 + 0.323042i 1.54779 − 1.54779i −1.54779 + 1.54779i 0.323042 − 0.323042i −0.323042 − 0.323042i 1.54779 + 1.54779i −1.54779 − 1.54779i 0.323042 + 0.323042i
−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.a 8
5.b even 2 1 1050.3.l.e yes 8
5.c odd 4 1 inner 1050.3.l.a 8
5.c odd 4 1 1050.3.l.e yes 8

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

## Hecke kernels

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

 $$T_{11}^{4} + 16 T_{11}^{3} - 82 T_{11}^{2} - 160 T_{11} + 625$$ $$T_{13}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + 2 T + T^{2} )^{4}$$
$3$ $$( 9 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 49 + T^{4} )^{2}$$
$11$ $$( 625 - 160 T - 82 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$13$ $$1157776 + 1377280 T + 819200 T^{2} - 131040 T^{3} + 20648 T^{4} + 6720 T^{5} + 800 T^{6} + 40 T^{7} + T^{8}$$
$17$ $$69482851216 + 5482796800 T + 216320000 T^{2} - 6928160 T^{3} + 178408 T^{4} + 12800 T^{5} + 800 T^{6} - 40 T^{7} + T^{8}$$
$19$ $$145676568976 + 1253503040 T^{2} + 3066648 T^{4} + 2960 T^{6} + T^{8}$$
$23$ $$244140625 + 71875000 T + 10580000 T^{2} - 2037000 T^{3} + 189650 T^{4} - 840 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$29$ $$1070879337889 + 5038353292 T^{2} + 8103750 T^{4} + 5068 T^{6} + T^{8}$$
$31$ $$( -379760 + 66624 T - 1192 T^{2} - 48 T^{3} + T^{4} )^{2}$$
$37$ $$1163755500625 + 111519444400 T + 5343298688 T^{2} + 75130608 T^{3} + 2352914 T^{4} + 152880 T^{5} + 6272 T^{6} + 112 T^{7} + T^{8}$$
$41$ $$( -68864 - 20480 T + 1088 T^{2} + 80 T^{3} + T^{4} )^{2}$$
$43$ $$223532401 - 560243872 T + 702075392 T^{2} - 15421056 T^{3} + 178898 T^{4} - 12768 T^{5} + 2048 T^{6} - 64 T^{7} + T^{8}$$
$47$ $$6469392250000 - 1371251720000 T + 145325187200 T^{2} + 4007848320 T^{3} + 54758696 T^{4} + 44016 T^{5} + 2048 T^{6} + 64 T^{7} + T^{8}$$
$53$ $$383805030400 + 45358776320 T + 2680291328 T^{2} + 65961984 T^{3} + 1289216 T^{4} + 55296 T^{5} + 3200 T^{6} + 80 T^{7} + T^{8}$$
$59$ $$125230513134736 + 247059022784 T^{2} + 143675736 T^{4} + 22064 T^{6} + T^{8}$$
$61$ $$( -22016000 - 1085440 T - 11392 T^{2} + 64 T^{3} + T^{4} )^{2}$$
$67$ $$38759303135809 - 4837615596880 T + 301895580800 T^{2} - 11104159152 T^{3} + 267307682 T^{4} - 4307664 T^{5} + 46208 T^{6} - 304 T^{7} + T^{8}$$
$71$ $$( 17125945 + 495336 T - 4762 T^{2} - 120 T^{3} + T^{4} )^{2}$$
$73$ $$165058256250000 - 6382638000000 T + 123405120000 T^{2} + 5951340000 T^{3} + 133065000 T^{4} + 194400 T^{5} + 288 T^{6} + 24 T^{7} + T^{8}$$
$79$ $$7023613249 + 689858784972 T^{2} + 241774310 T^{4} + 27468 T^{6} + T^{8}$$
$83$ $$66382491904 + 8442609664 T + 536870912 T^{2} + 16751616 T^{3} + 515360 T^{4} + 32256 T^{5} + 2048 T^{6} + 64 T^{7} + T^{8}$$
$89$ $$362929170490000 + 466959816000 T^{2} + 169779416 T^{4} + 22608 T^{6} + T^{8}$$
$97$ $$392594596000000 + 29109143680000 T + 1079156787200 T^{2} + 4014311680 T^{3} + 40504096 T^{4} + 1723712 T^{5} + 36992 T^{6} + 272 T^{7} + T^{8}$$