# Properties

 Label 210.4.i.i Level $210$ Weight $4$ Character orbit 210.i 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.3904011012$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{295})$$ Defining polynomial: $$x^{4} + 295 x^{2} + 87025$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -2 - 2 \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + 4 \beta_{2} q^{4} + ( 5 + 5 \beta_{2} ) q^{5} -6 q^{6} + ( 4 + \beta_{1} - 4 \beta_{2} ) q^{7} + 8 q^{8} + ( -9 - 9 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -2 - 2 \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + 4 \beta_{2} q^{4} + ( 5 + 5 \beta_{2} ) q^{5} -6 q^{6} + ( 4 + \beta_{1} - 4 \beta_{2} ) q^{7} + 8 q^{8} + ( -9 - 9 \beta_{2} ) q^{9} -10 \beta_{2} q^{10} + ( \beta_{1} + 25 \beta_{2} + \beta_{3} ) q^{11} + ( 12 + 12 \beta_{2} ) q^{12} + ( -50 + \beta_{3} ) q^{13} + ( -16 - 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{14} + 15 q^{15} + ( -16 - 16 \beta_{2} ) q^{16} + ( 5 \beta_{1} + 17 \beta_{2} + 5 \beta_{3} ) q^{17} + 18 \beta_{2} q^{18} + ( 23 + 4 \beta_{1} + 23 \beta_{2} ) q^{19} -20 q^{20} + ( -12 - 24 \beta_{2} - 3 \beta_{3} ) q^{21} + ( 50 - 2 \beta_{3} ) q^{22} + ( -63 + 9 \beta_{1} - 63 \beta_{2} ) q^{23} -24 \beta_{2} q^{24} + 25 \beta_{2} q^{25} + ( 100 + 2 \beta_{1} + 100 \beta_{2} ) q^{26} -27 q^{27} + ( 16 + 32 \beta_{2} + 4 \beta_{3} ) q^{28} + ( -75 - 3 \beta_{3} ) q^{29} + ( -30 - 30 \beta_{2} ) q^{30} + ( -12 \beta_{1} - 37 \beta_{2} - 12 \beta_{3} ) q^{31} + 32 \beta_{2} q^{32} + ( 75 + 3 \beta_{1} + 75 \beta_{2} ) q^{33} + ( 34 - 10 \beta_{3} ) q^{34} + ( 40 + 5 \beta_{1} + 20 \beta_{2} + 5 \beta_{3} ) q^{35} + 36 q^{36} + ( 88 - 9 \beta_{1} + 88 \beta_{2} ) q^{37} + ( -8 \beta_{1} - 46 \beta_{2} - 8 \beta_{3} ) q^{38} + ( 3 \beta_{1} + 150 \beta_{2} + 3 \beta_{3} ) q^{39} + ( 40 + 40 \beta_{2} ) q^{40} + ( 5 - 17 \beta_{3} ) q^{41} + ( -24 - 6 \beta_{1} + 24 \beta_{2} ) q^{42} + ( 66 - 7 \beta_{3} ) q^{43} + ( -100 - 4 \beta_{1} - 100 \beta_{2} ) q^{44} -45 \beta_{2} q^{45} + ( -18 \beta_{1} + 126 \beta_{2} - 18 \beta_{3} ) q^{46} + ( -158 + 4 \beta_{1} - 158 \beta_{2} ) q^{47} -48 q^{48} + ( 8 \beta_{1} + 247 \beta_{2} - 8 \beta_{3} ) q^{49} + 50 q^{50} + ( 51 + 15 \beta_{1} + 51 \beta_{2} ) q^{51} + ( -4 \beta_{1} - 200 \beta_{2} - 4 \beta_{3} ) q^{52} + ( 10 \beta_{1} + 118 \beta_{2} + 10 \beta_{3} ) q^{53} + ( 54 + 54 \beta_{2} ) q^{54} + ( -125 + 5 \beta_{3} ) q^{55} + ( 32 + 8 \beta_{1} - 32 \beta_{2} ) q^{56} + ( 69 - 12 \beta_{3} ) q^{57} + ( 150 - 6 \beta_{1} + 150 \beta_{2} ) q^{58} + ( 49 \beta_{1} - 29 \beta_{2} + 49 \beta_{3} ) q^{59} + 60 \beta_{2} q^{60} + ( 396 - 4 \beta_{1} + 396 \beta_{2} ) q^{61} + ( -74 + 24 \beta_{3} ) q^{62} + ( -72 - 9 \beta_{1} - 36 \beta_{2} - 9 \beta_{3} ) q^{63} + 64 q^{64} + ( -250 - 5 \beta_{1} - 250 \beta_{2} ) q^{65} + ( -6 \beta_{1} - 150 \beta_{2} - 6 \beta_{3} ) q^{66} + ( -13 \beta_{1} - 434 \beta_{2} - 13 \beta_{3} ) q^{67} + ( -68 - 20 \beta_{1} - 68 \beta_{2} ) q^{68} + ( -189 - 27 \beta_{3} ) q^{69} + ( -40 - 80 \beta_{2} - 10 \beta_{3} ) q^{70} + ( -193 + 25 \beta_{3} ) q^{71} + ( -72 - 72 \beta_{2} ) q^{72} + ( -15 \beta_{1} - 610 \beta_{2} - 15 \beta_{3} ) q^{73} + ( 18 \beta_{1} - 176 \beta_{2} + 18 \beta_{3} ) q^{74} + ( 75 + 75 \beta_{2} ) q^{75} + ( -92 + 16 \beta_{3} ) q^{76} + ( -195 + 8 \beta_{1} + 200 \beta_{2} + 29 \beta_{3} ) q^{77} + ( 300 - 6 \beta_{3} ) q^{78} + ( -27 + 44 \beta_{1} - 27 \beta_{2} ) q^{79} -80 \beta_{2} q^{80} + 81 \beta_{2} q^{81} + ( -10 - 34 \beta_{1} - 10 \beta_{2} ) q^{82} + ( -91 + 61 \beta_{3} ) q^{83} + ( 96 + 12 \beta_{1} + 48 \beta_{2} + 12 \beta_{3} ) q^{84} + ( -85 + 25 \beta_{3} ) q^{85} + ( -132 - 14 \beta_{1} - 132 \beta_{2} ) q^{86} + ( -9 \beta_{1} + 225 \beta_{2} - 9 \beta_{3} ) q^{87} + ( 8 \beta_{1} + 200 \beta_{2} + 8 \beta_{3} ) q^{88} + ( 209 + 23 \beta_{1} + 209 \beta_{2} ) q^{89} -90 q^{90} + ( -495 - 46 \beta_{1} - 95 \beta_{2} + 8 \beta_{3} ) q^{91} + ( 252 + 36 \beta_{3} ) q^{92} + ( -111 - 36 \beta_{1} - 111 \beta_{2} ) q^{93} + ( -8 \beta_{1} + 316 \beta_{2} - 8 \beta_{3} ) q^{94} + ( 20 \beta_{1} + 115 \beta_{2} + 20 \beta_{3} ) q^{95} + ( 96 + 96 \beta_{2} ) q^{96} + ( -196 + 12 \beta_{3} ) q^{97} + ( 494 - 32 \beta_{1} - 16 \beta_{3} ) q^{98} + ( 225 - 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 6q^{3} - 8q^{4} + 10q^{5} - 24q^{6} + 24q^{7} + 32q^{8} - 18q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 6q^{3} - 8q^{4} + 10q^{5} - 24q^{6} + 24q^{7} + 32q^{8} - 18q^{9} + 20q^{10} - 50q^{11} + 24q^{12} - 200q^{13} - 48q^{14} + 60q^{15} - 32q^{16} - 34q^{17} - 36q^{18} + 46q^{19} - 80q^{20} + 200q^{22} - 126q^{23} + 48q^{24} - 50q^{25} + 200q^{26} - 108q^{27} - 300q^{29} - 60q^{30} + 74q^{31} - 64q^{32} + 150q^{33} + 136q^{34} + 120q^{35} + 144q^{36} + 176q^{37} + 92q^{38} - 300q^{39} + 80q^{40} + 20q^{41} - 144q^{42} + 264q^{43} - 200q^{44} + 90q^{45} - 252q^{46} - 316q^{47} - 192q^{48} - 494q^{49} + 200q^{50} + 102q^{51} + 400q^{52} - 236q^{53} + 108q^{54} - 500q^{55} + 192q^{56} + 276q^{57} + 300q^{58} + 58q^{59} - 120q^{60} + 792q^{61} - 296q^{62} - 216q^{63} + 256q^{64} - 500q^{65} + 300q^{66} + 868q^{67} - 136q^{68} - 756q^{69} - 772q^{71} - 144q^{72} + 1220q^{73} + 352q^{74} + 150q^{75} - 368q^{76} - 1180q^{77} + 1200q^{78} - 54q^{79} + 160q^{80} - 162q^{81} - 20q^{82} - 364q^{83} + 288q^{84} - 340q^{85} - 264q^{86} - 450q^{87} - 400q^{88} + 418q^{89} - 360q^{90} - 1790q^{91} + 1008q^{92} - 222q^{93} - 632q^{94} - 230q^{95} + 192q^{96} - 784q^{97} + 1976q^{98} + 900q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 295 x^{2} + 87025$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/295$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/295$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$295 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$295 \beta_{3}$$

## Character values

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

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −8.58778 + 14.8745i 8.58778 − 14.8745i −8.58778 − 14.8745i 8.58778 + 14.8745i
−1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 2.50000 4.33013i −6.00000 −2.58778 + 18.3386i 8.00000 −4.50000 + 7.79423i 5.00000 + 8.66025i
121.2 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 2.50000 4.33013i −6.00000 14.5878 11.4104i 8.00000 −4.50000 + 7.79423i 5.00000 + 8.66025i
151.1 −1.00000 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i 2.50000 + 4.33013i −6.00000 −2.58778 18.3386i 8.00000 −4.50000 7.79423i 5.00000 8.66025i
151.2 −1.00000 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i 2.50000 + 4.33013i −6.00000 14.5878 + 11.4104i 8.00000 −4.50000 7.79423i 5.00000 8.66025i
 $$n$$: 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
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 210.4.i.i 4
3.b odd 2 1 630.4.k.m 4
7.c even 3 1 inner 210.4.i.i 4
7.c even 3 1 1470.4.a.bn 2
7.d odd 6 1 1470.4.a.bs 2
21.h odd 6 1 630.4.k.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.i.i 4 1.a even 1 1 trivial
210.4.i.i 4 7.c even 3 1 inner
630.4.k.m 4 3.b odd 2 1
630.4.k.m 4 21.h odd 6 1
1470.4.a.bn 2 7.c even 3 1
1470.4.a.bs 2 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 50 T_{11}^{3} + 2170 T_{11}^{2} + 16500 T_{11} + 108900$$ acting on $$S_{4}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$( 25 - 5 T + T^{2} )^{2}$$
$7$ $$117649 - 8232 T + 535 T^{2} - 24 T^{3} + T^{4}$$
$11$ $$108900 + 16500 T + 2170 T^{2} + 50 T^{3} + T^{4}$$
$13$ $$( 2205 + 100 T + T^{2} )^{2}$$
$17$ $$50211396 - 240924 T + 8242 T^{2} + 34 T^{3} + T^{4}$$
$19$ $$17564481 + 192786 T + 6307 T^{2} - 46 T^{3} + T^{4}$$
$23$ $$397045476 - 2510676 T + 35802 T^{2} + 126 T^{3} + T^{4}$$
$29$ $$( 2970 + 150 T + T^{2} )^{2}$$
$31$ $$1690114321 + 3042214 T + 46587 T^{2} - 74 T^{3} + T^{4}$$
$37$ $$260854801 + 2842576 T + 47127 T^{2} - 176 T^{3} + T^{4}$$
$41$ $$( -85230 - 10 T + T^{2} )^{2}$$
$43$ $$( -10099 - 132 T + T^{2} )^{2}$$
$47$ $$409819536 + 6397104 T + 79612 T^{2} + 316 T^{3} + T^{4}$$
$53$ $$242611776 - 3675936 T + 71272 T^{2} + 236 T^{3} + T^{4}$$
$59$ $$500491162116 + 41032332 T + 710818 T^{2} - 58 T^{3} + T^{4}$$
$61$ $$23133193216 - 120460032 T + 475168 T^{2} - 792 T^{3} + T^{4}$$
$67$ $$19182527001 - 120218868 T + 614923 T^{2} - 868 T^{3} + T^{4}$$
$71$ $$( -147126 + 386 T + T^{2} )^{2}$$
$73$ $$93467775625 - 372984500 T + 1182675 T^{2} - 1220 T^{3} + T^{4}$$
$79$ $$325345892881 - 30801114 T + 573307 T^{2} + 54 T^{3} + T^{4}$$
$83$ $$( -1089414 + 182 T + T^{2} )^{2}$$
$89$ $$12627915876 + 46972332 T + 287098 T^{2} - 418 T^{3} + T^{4}$$
$97$ $$( -4064 + 392 T + T^{2} )^{2}$$