# Properties

 Label 210.4.g.a Level $210$ Weight $4$ Character orbit 210.g Analytic conductor $12.390$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 210.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.3904011012$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{1} q^{2} + 3 \beta_{1} q^{3} -4 q^{4} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{5} -6 q^{6} + 7 \beta_{1} q^{7} -8 \beta_{1} q^{8} -9 q^{9} +O(q^{10})$$ $$q + 2 \beta_{1} q^{2} + 3 \beta_{1} q^{3} -4 q^{4} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{5} -6 q^{6} + 7 \beta_{1} q^{7} -8 \beta_{1} q^{8} -9 q^{9} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{10} + ( 6 - 4 \beta_{3} ) q^{11} -12 \beta_{1} q^{12} + ( 50 \beta_{1} - 8 \beta_{2} ) q^{13} -14 q^{14} + ( 6 - 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{15} + 16 q^{16} + ( -40 \beta_{1} - 12 \beta_{2} ) q^{17} -18 \beta_{1} q^{18} + ( 2 - 2 \beta_{3} ) q^{19} + ( 4 + 8 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{20} -21 q^{21} + ( 12 \beta_{1} - 8 \beta_{2} ) q^{22} + ( 32 \beta_{1} - 26 \beta_{2} ) q^{23} + 24 q^{24} + ( 69 - 92 \beta_{1} + 6 \beta_{2} + 8 \beta_{3} ) q^{25} + ( -100 + 16 \beta_{3} ) q^{26} -27 \beta_{1} q^{27} -28 \beta_{1} q^{28} + ( -10 + 2 \beta_{3} ) q^{29} + ( 6 + 12 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} ) q^{30} + ( -146 - 4 \beta_{3} ) q^{31} + 32 \beta_{1} q^{32} + ( 18 \beta_{1} - 12 \beta_{2} ) q^{33} + ( 80 + 24 \beta_{3} ) q^{34} + ( 14 - 7 \beta_{1} - 14 \beta_{2} - 7 \beta_{3} ) q^{35} + 36 q^{36} + ( 136 \beta_{1} - 30 \beta_{2} ) q^{37} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{38} + ( -150 + 24 \beta_{3} ) q^{39} + ( -16 + 8 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} ) q^{40} + ( -66 + 16 \beta_{3} ) q^{41} -42 \beta_{1} q^{42} + ( 16 \beta_{1} + 70 \beta_{2} ) q^{43} + ( -24 + 16 \beta_{3} ) q^{44} + ( 9 + 18 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} ) q^{45} + ( -64 + 52 \beta_{3} ) q^{46} + ( -284 \beta_{1} - 34 \beta_{2} ) q^{47} + 48 \beta_{1} q^{48} -49 q^{49} + ( 184 + 138 \beta_{1} + 16 \beta_{2} - 12 \beta_{3} ) q^{50} + ( 120 + 36 \beta_{3} ) q^{51} + ( -200 \beta_{1} + 32 \beta_{2} ) q^{52} + ( 174 \beta_{1} - 94 \beta_{2} ) q^{53} + 54 q^{54} + ( 186 - 108 \beta_{1} + 14 \beta_{2} - 8 \beta_{3} ) q^{55} + 56 q^{56} + ( 6 \beta_{1} - 6 \beta_{2} ) q^{57} + ( -20 \beta_{1} + 4 \beta_{2} ) q^{58} + ( -112 - 128 \beta_{3} ) q^{59} + ( -24 + 12 \beta_{1} + 24 \beta_{2} + 12 \beta_{3} ) q^{60} + ( -6 - 82 \beta_{3} ) q^{61} + ( -292 \beta_{1} - 8 \beta_{2} ) q^{62} -63 \beta_{1} q^{63} -64 q^{64} + ( 292 + 334 \beta_{1} - 92 \beta_{2} - 66 \beta_{3} ) q^{65} + ( -36 + 24 \beta_{3} ) q^{66} + ( -460 \beta_{1} + 78 \beta_{2} ) q^{67} + ( 160 \beta_{1} + 48 \beta_{2} ) q^{68} + ( -96 + 78 \beta_{3} ) q^{69} + ( 14 + 28 \beta_{1} - 14 \beta_{2} + 28 \beta_{3} ) q^{70} + ( -78 - 98 \beta_{3} ) q^{71} + 72 \beta_{1} q^{72} + ( 166 \beta_{1} - 28 \beta_{2} ) q^{73} + ( -272 + 60 \beta_{3} ) q^{74} + ( 276 + 207 \beta_{1} + 24 \beta_{2} - 18 \beta_{3} ) q^{75} + ( -8 + 8 \beta_{3} ) q^{76} + ( 42 \beta_{1} - 28 \beta_{2} ) q^{77} + ( -300 \beta_{1} + 48 \beta_{2} ) q^{78} + ( -188 + 16 \beta_{3} ) q^{79} + ( -16 - 32 \beta_{1} + 16 \beta_{2} - 32 \beta_{3} ) q^{80} + 81 q^{81} + ( -132 \beta_{1} + 32 \beta_{2} ) q^{82} + ( 1008 \beta_{1} + 52 \beta_{2} ) q^{83} + 84 q^{84} + ( 208 + 616 \beta_{1} + 92 \beta_{2} + 16 \beta_{3} ) q^{85} + ( -32 - 140 \beta_{3} ) q^{86} + ( -30 \beta_{1} + 6 \beta_{2} ) q^{87} + ( -48 \beta_{1} + 32 \beta_{2} ) q^{88} + ( -274 - 228 \beta_{3} ) q^{89} + ( -36 + 18 \beta_{1} + 36 \beta_{2} + 18 \beta_{3} ) q^{90} + ( -350 + 56 \beta_{3} ) q^{91} + ( -128 \beta_{1} + 104 \beta_{2} ) q^{92} + ( -438 \beta_{1} - 12 \beta_{2} ) q^{93} + ( 568 + 68 \beta_{3} ) q^{94} + ( 94 - 52 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{95} -96 q^{96} + ( -810 \beta_{1} + 80 \beta_{2} ) q^{97} -98 \beta_{1} q^{98} + ( -54 + 36 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{4} - 4q^{5} - 24q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 16q^{4} - 4q^{5} - 24q^{6} - 36q^{9} + 16q^{10} + 24q^{11} - 56q^{14} + 24q^{15} + 64q^{16} + 8q^{19} + 16q^{20} - 84q^{21} + 96q^{24} + 276q^{25} - 400q^{26} - 40q^{29} + 24q^{30} - 584q^{31} + 320q^{34} + 56q^{35} + 144q^{36} - 600q^{39} - 64q^{40} - 264q^{41} - 96q^{44} + 36q^{45} - 256q^{46} - 196q^{49} + 736q^{50} + 480q^{51} + 216q^{54} + 744q^{55} + 224q^{56} - 448q^{59} - 96q^{60} - 24q^{61} - 256q^{64} + 1168q^{65} - 144q^{66} - 384q^{69} + 56q^{70} - 312q^{71} - 1088q^{74} + 1104q^{75} - 32q^{76} - 752q^{79} - 64q^{80} + 324q^{81} + 336q^{84} + 832q^{85} - 128q^{86} - 1096q^{89} - 144q^{90} - 1400q^{91} + 2272q^{94} + 376q^{95} - 384q^{96} - 216q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i −1.22474 − 1.22474i
2.00000i 3.00000i −4.00000 −10.7980 2.89898i −6.00000 7.00000i 8.00000i −9.00000 −5.79796 + 21.5959i
169.2 2.00000i 3.00000i −4.00000 8.79796 + 6.89898i −6.00000 7.00000i 8.00000i −9.00000 13.7980 17.5959i
169.3 2.00000i 3.00000i −4.00000 −10.7980 + 2.89898i −6.00000 7.00000i 8.00000i −9.00000 −5.79796 21.5959i
169.4 2.00000i 3.00000i −4.00000 8.79796 6.89898i −6.00000 7.00000i 8.00000i −9.00000 13.7980 + 17.5959i
 $$n$$: e.g. 2-40 or 990-1000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.4.g.a 4
3.b odd 2 1 630.4.g.e 4
5.b even 2 1 inner 210.4.g.a 4
5.c odd 4 1 1050.4.a.bc 2
5.c odd 4 1 1050.4.a.bg 2
15.d odd 2 1 630.4.g.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.g.a 4 1.a even 1 1 trivial
210.4.g.a 4 5.b even 2 1 inner
630.4.g.e 4 3.b odd 2 1
630.4.g.e 4 15.d odd 2 1
1050.4.a.bc 2 5.c odd 4 1
1050.4.a.bg 2 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} - 12 T_{11} - 348$$ acting on $$S_{4}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + T^{2} )^{2}$$
$3$ $$( 9 + T^{2} )^{2}$$
$5$ $$15625 + 500 T - 130 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$( 49 + T^{2} )^{2}$$
$11$ $$( -348 - 12 T + T^{2} )^{2}$$
$13$ $$929296 + 8072 T^{2} + T^{4}$$
$17$ $$3444736 + 10112 T^{2} + T^{4}$$
$19$ $$( -92 - 4 T + T^{2} )^{2}$$
$23$ $$231040000 + 34496 T^{2} + T^{4}$$
$29$ $$( 4 + 20 T + T^{2} )^{2}$$
$31$ $$( 20932 + 292 T + T^{2} )^{2}$$
$37$ $$9634816 + 80192 T^{2} + T^{4}$$
$41$ $$( -1788 + 132 T + T^{2} )^{2}$$
$43$ $$13769614336 + 235712 T^{2} + T^{4}$$
$47$ $$2799679744 + 216800 T^{2} + T^{4}$$
$53$ $$33046876944 + 484680 T^{2} + T^{4}$$
$59$ $$( -380672 + 224 T + T^{2} )^{2}$$
$61$ $$( -161340 + 12 T + T^{2} )^{2}$$
$67$ $$4301261056 + 715232 T^{2} + T^{4}$$
$71$ $$( -224412 + 156 T + T^{2} )^{2}$$
$73$ $$76387600 + 92744 T^{2} + T^{4}$$
$79$ $$( 29200 + 376 T + T^{2} )^{2}$$
$83$ $$904720564224 + 2161920 T^{2} + T^{4}$$
$89$ $$( -1172540 + 548 T + T^{2} )^{2}$$
$97$ $$252506250000 + 1619400 T^{2} + T^{4}$$