# Properties

 Label 210.4.i.j 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: $$\Q(\sqrt{-3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \beta_{2} ) q^{2} + 3 \beta_{2} q^{3} -4 \beta_{2} q^{4} + ( 5 - 5 \beta_{2} ) q^{5} + 6 q^{6} + ( -6 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} -8 q^{8} + ( -9 + 9 \beta_{2} ) q^{9} +O(q^{10})$$ $$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} + ( -6 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} -8 q^{8} + ( -9 + 9 \beta_{2} ) q^{9} -10 \beta_{2} q^{10} + ( -3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{11} + ( 12 - 12 \beta_{2} ) q^{12} + ( 32 - 6 \beta_{1} + 3 \beta_{3} ) q^{13} + ( -8 - 6 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{14} + 15 q^{15} + ( -16 + 16 \beta_{2} ) q^{16} + ( -3 \beta_{1} - 51 \beta_{2} - 3 \beta_{3} ) q^{17} + 18 \beta_{2} q^{18} + ( 31 - 4 \beta_{1} - 31 \beta_{2} + 8 \beta_{3} ) q^{19} -20 q^{20} + ( -6 + 6 \beta_{1} - 12 \beta_{2} - 9 \beta_{3} ) q^{21} + ( 18 - 12 \beta_{1} + 6 \beta_{3} ) q^{22} + ( -21 + 7 \beta_{1} + 21 \beta_{2} - 14 \beta_{3} ) q^{23} -24 \beta_{2} q^{24} -25 \beta_{2} q^{25} + ( 64 - 6 \beta_{1} - 64 \beta_{2} + 12 \beta_{3} ) q^{26} -27 q^{27} + ( 8 - 8 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} ) q^{28} + ( 111 - 14 \beta_{1} + 7 \beta_{3} ) q^{29} + ( 30 - 30 \beta_{2} ) q^{30} + ( -24 \beta_{1} + 25 \beta_{2} - 24 \beta_{3} ) q^{31} + 32 \beta_{2} q^{32} + ( -27 + 9 \beta_{1} + 27 \beta_{2} - 18 \beta_{3} ) q^{33} + ( -102 - 12 \beta_{1} + 6 \beta_{3} ) q^{34} + ( -20 - 15 \beta_{1} + 30 \beta_{2} + 5 \beta_{3} ) q^{35} + 36 q^{36} + ( 286 - 7 \beta_{1} - 286 \beta_{2} + 14 \beta_{3} ) q^{37} + ( 8 \beta_{1} - 62 \beta_{2} + 8 \beta_{3} ) q^{38} + ( -9 \beta_{1} + 96 \beta_{2} - 9 \beta_{3} ) q^{39} + ( -40 + 40 \beta_{2} ) q^{40} + ( 27 + 6 \beta_{1} - 3 \beta_{3} ) q^{41} + ( -36 - 6 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{42} + ( -124 + 10 \beta_{1} - 5 \beta_{3} ) q^{43} + ( 36 - 12 \beta_{1} - 36 \beta_{2} + 24 \beta_{3} ) q^{44} + 45 \beta_{2} q^{45} + ( -14 \beta_{1} + 42 \beta_{2} - 14 \beta_{3} ) q^{46} + ( -222 + 4 \beta_{1} + 222 \beta_{2} - 8 \beta_{3} ) q^{47} -48 q^{48} + ( -328 + 20 \beta_{1} + 205 \beta_{2} + 12 \beta_{3} ) q^{49} -50 q^{50} + ( 153 + 9 \beta_{1} - 153 \beta_{2} - 18 \beta_{3} ) q^{51} + ( 12 \beta_{1} - 128 \beta_{2} + 12 \beta_{3} ) q^{52} + ( 34 \beta_{1} + 6 \beta_{2} + 34 \beta_{3} ) q^{53} + ( -54 + 54 \beta_{2} ) q^{54} + ( 45 - 30 \beta_{1} + 15 \beta_{3} ) q^{55} + ( 48 + 8 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{56} + ( 93 - 24 \beta_{1} + 12 \beta_{3} ) q^{57} + ( 222 - 14 \beta_{1} - 222 \beta_{2} + 28 \beta_{3} ) q^{58} + ( 45 \beta_{1} - 129 \beta_{2} + 45 \beta_{3} ) q^{59} -60 \beta_{2} q^{60} + ( -32 + 8 \beta_{1} + 32 \beta_{2} - 16 \beta_{3} ) q^{61} + ( 50 - 96 \beta_{1} + 48 \beta_{3} ) q^{62} + ( 36 + 27 \beta_{1} - 54 \beta_{2} - 9 \beta_{3} ) q^{63} + 64 q^{64} + ( 160 - 15 \beta_{1} - 160 \beta_{2} + 30 \beta_{3} ) q^{65} + ( -18 \beta_{1} + 54 \beta_{2} - 18 \beta_{3} ) q^{66} + ( -37 \beta_{1} + 316 \beta_{2} - 37 \beta_{3} ) q^{67} + ( -204 - 12 \beta_{1} + 204 \beta_{2} + 24 \beta_{3} ) q^{68} + ( -63 + 42 \beta_{1} - 21 \beta_{3} ) q^{69} + ( 20 - 20 \beta_{1} + 40 \beta_{2} + 30 \beta_{3} ) q^{70} + ( -39 + 90 \beta_{1} - 45 \beta_{3} ) q^{71} + ( 72 - 72 \beta_{2} ) q^{72} + ( -15 \beta_{1} + 232 \beta_{2} - 15 \beta_{3} ) q^{73} + ( 14 \beta_{1} - 572 \beta_{2} + 14 \beta_{3} ) q^{74} + ( 75 - 75 \beta_{2} ) q^{75} + ( -124 + 32 \beta_{1} - 16 \beta_{3} ) q^{76} + ( -693 + 42 \beta_{1} + 504 \beta_{2} - 21 \beta_{3} ) q^{77} + ( 192 - 36 \beta_{1} + 18 \beta_{3} ) q^{78} + ( 973 + 4 \beta_{1} - 973 \beta_{2} - 8 \beta_{3} ) q^{79} + 80 \beta_{2} q^{80} -81 \beta_{2} q^{81} + ( 54 + 6 \beta_{1} - 54 \beta_{2} - 12 \beta_{3} ) q^{82} + ( -117 + 170 \beta_{1} - 85 \beta_{3} ) q^{83} + ( -48 - 36 \beta_{1} + 72 \beta_{2} + 12 \beta_{3} ) q^{84} + ( -255 - 30 \beta_{1} + 15 \beta_{3} ) q^{85} + ( -248 + 10 \beta_{1} + 248 \beta_{2} - 20 \beta_{3} ) q^{86} + ( -21 \beta_{1} + 333 \beta_{2} - 21 \beta_{3} ) q^{87} + ( 24 \beta_{1} - 72 \beta_{2} + 24 \beta_{3} ) q^{88} + ( -837 - 11 \beta_{1} + 837 \beta_{2} + 22 \beta_{3} ) q^{89} + 90 q^{90} + ( -327 - 2 \beta_{1} + 739 \beta_{2} - 88 \beta_{3} ) q^{91} + ( 84 - 56 \beta_{1} + 28 \beta_{3} ) q^{92} + ( -75 + 72 \beta_{1} + 75 \beta_{2} - 144 \beta_{3} ) q^{93} + ( -8 \beta_{1} + 444 \beta_{2} - 8 \beta_{3} ) q^{94} + ( 20 \beta_{1} - 155 \beta_{2} + 20 \beta_{3} ) q^{95} + ( -96 + 96 \beta_{2} ) q^{96} + ( 416 + 80 \beta_{1} - 40 \beta_{3} ) q^{97} + ( -246 + 64 \beta_{1} + 656 \beta_{2} - 40 \beta_{3} ) q^{98} + ( -81 + 54 \beta_{1} - 27 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 6q^{3} - 8q^{4} + 10q^{5} + 24q^{6} - 20q^{7} - 32q^{8} - 18q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 6q^{3} - 8q^{4} + 10q^{5} + 24q^{6} - 20q^{7} - 32q^{8} - 18q^{9} - 20q^{10} + 18q^{11} + 24q^{12} + 128q^{13} - 8q^{14} + 60q^{15} - 32q^{16} - 102q^{17} + 36q^{18} + 62q^{19} - 80q^{20} - 48q^{21} + 72q^{22} - 42q^{23} - 48q^{24} - 50q^{25} + 128q^{26} - 108q^{27} + 64q^{28} + 444q^{29} + 60q^{30} + 50q^{31} + 64q^{32} - 54q^{33} - 408q^{34} - 20q^{35} + 144q^{36} + 572q^{37} - 124q^{38} + 192q^{39} - 80q^{40} + 108q^{41} - 120q^{42} - 496q^{43} + 72q^{44} + 90q^{45} + 84q^{46} - 444q^{47} - 192q^{48} - 902q^{49} - 200q^{50} + 306q^{51} - 256q^{52} + 12q^{53} - 108q^{54} + 180q^{55} + 160q^{56} + 372q^{57} + 444q^{58} - 258q^{59} - 120q^{60} - 64q^{61} + 200q^{62} + 36q^{63} + 256q^{64} + 320q^{65} + 108q^{66} + 632q^{67} - 408q^{68} - 252q^{69} + 160q^{70} - 156q^{71} + 144q^{72} + 464q^{73} - 1144q^{74} + 150q^{75} - 496q^{76} - 1764q^{77} + 768q^{78} + 1946q^{79} + 160q^{80} - 162q^{81} + 108q^{82} - 468q^{83} - 48q^{84} - 1020q^{85} - 496q^{86} + 666q^{87} - 144q^{88} - 1674q^{89} + 360q^{90} + 170q^{91} + 336q^{92} - 150q^{93} + 888q^{94} - 310q^{95} - 192q^{96} + 1664q^{97} + 328q^{98} - 324q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{3}$$$$/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/3$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$$$/3$$

## 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
 1.93649 + 1.11803i −1.93649 − 1.11803i 1.93649 − 1.11803i −1.93649 + 1.11803i
1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 2.50000 4.33013i 6.00000 −10.8095 15.0385i −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 0.809475 + 18.5026i −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 −10.8095 + 15.0385i −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 0.809475 18.5026i −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.j 4
3.b odd 2 1 630.4.k.k 4
7.c even 3 1 inner 210.4.i.j 4
7.c even 3 1 1470.4.a.be 2
7.d odd 6 1 1470.4.a.bj 2
21.h odd 6 1 630.4.k.k 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} - 18 T_{11}^{3} + 1458 T_{11}^{2} + 20412 T_{11} + 1285956$$ acting on $$S_{4}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T + T^{2} )^{2}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$( 25 - 5 T + T^{2} )^{2}$$
$7$ $$117649 + 6860 T + 651 T^{2} + 20 T^{3} + T^{4}$$
$11$ $$1285956 + 20412 T + 1458 T^{2} - 18 T^{3} + T^{4}$$
$13$ $$( -191 - 64 T + T^{2} )^{2}$$
$17$ $$1920996 + 141372 T + 9018 T^{2} + 102 T^{3} + T^{4}$$
$19$ $$1437601 + 74338 T + 5043 T^{2} - 62 T^{3} + T^{4}$$
$23$ $$38118276 - 259308 T + 7938 T^{2} + 42 T^{3} + T^{4}$$
$29$ $$( 5706 - 222 T + T^{2} )^{2}$$
$31$ $$5949808225 + 3856750 T + 79635 T^{2} - 50 T^{3} + T^{4}$$
$37$ $$5652182761 - 43003532 T + 252003 T^{2} - 572 T^{3} + T^{4}$$
$41$ $$( -486 - 54 T + T^{2} )^{2}$$
$43$ $$( 12001 + 248 T + T^{2} )^{2}$$
$47$ $$2220671376 + 20923056 T + 150012 T^{2} + 444 T^{3} + T^{4}$$
$53$ $$24343488576 + 1872288 T + 156168 T^{2} - 12 T^{3} + T^{4}$$
$59$ $$65912346756 - 66237372 T + 323298 T^{2} + 258 T^{3} + T^{4}$$
$61$ $$58003456 - 487424 T + 11712 T^{2} + 64 T^{3} + T^{4}$$
$67$ $$7218031681 + 53694088 T + 484383 T^{2} - 632 T^{3} + T^{4}$$
$71$ $$( -271854 + 78 T + T^{2} )^{2}$$
$73$ $$549855601 - 10880336 T + 191847 T^{2} - 464 T^{3} + T^{4}$$
$79$ $$892210595761 - 1838131274 T + 2842347 T^{2} - 1946 T^{3} + T^{4}$$
$83$ $$( -961686 + 234 T + T^{2} )^{2}$$
$89$ $$468176166756 + 1145407716 T + 2118042 T^{2} + 1674 T^{3} + T^{4}$$
$97$ $$( -42944 - 832 T + T^{2} )^{2}$$