# Properties

 Label 210.4.i.h 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: $$\Q(\sqrt{-3}, \sqrt{46})$$ Defining polynomial: $$x^{4} + 46 x^{2} + 2116$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + ( 3 + 3 \beta_{2} ) q^{3} + ( -4 - 4 \beta_{2} ) q^{4} + 5 \beta_{2} q^{5} -6 q^{6} + ( -4 - \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{7} + 8 q^{8} + 9 \beta_{2} q^{9} +O(q^{10})$$ $$q + 2 \beta_{2} q^{2} + ( 3 + 3 \beta_{2} ) q^{3} + ( -4 - 4 \beta_{2} ) q^{4} + 5 \beta_{2} q^{5} -6 q^{6} + ( -4 - \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{7} + 8 q^{8} + 9 \beta_{2} q^{9} + ( -10 - 10 \beta_{2} ) q^{10} + ( 10 + 5 \beta_{1} + 10 \beta_{2} ) q^{11} -12 \beta_{2} q^{12} + ( 21 + \beta_{3} ) q^{13} + ( 10 + 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{14} -15 q^{15} + 16 \beta_{2} q^{16} + ( -38 + 11 \beta_{1} - 38 \beta_{2} ) q^{17} + ( -18 - 18 \beta_{2} ) q^{18} + ( 2 \beta_{1} + 45 \beta_{2} + 2 \beta_{3} ) q^{19} + 20 q^{20} + ( 3 + 6 \beta_{1} - 12 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -20 + 10 \beta_{3} ) q^{22} + ( 11 \beta_{1} + 22 \beta_{2} + 11 \beta_{3} ) q^{23} + ( 24 + 24 \beta_{2} ) q^{24} + ( -25 - 25 \beta_{2} ) q^{25} + ( -2 \beta_{1} + 42 \beta_{2} - 2 \beta_{3} ) q^{26} -27 q^{27} + ( -4 - 8 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} ) q^{28} + ( -80 - 5 \beta_{3} ) q^{29} -30 \beta_{2} q^{30} + ( -31 + 8 \beta_{1} - 31 \beta_{2} ) q^{31} + ( -32 - 32 \beta_{2} ) q^{32} + ( 15 \beta_{1} + 30 \beta_{2} + 15 \beta_{3} ) q^{33} + ( 76 + 22 \beta_{3} ) q^{34} + ( 25 + 15 \beta_{1} + 5 \beta_{2} + 10 \beta_{3} ) q^{35} + 36 q^{36} + ( 31 \beta_{1} - 179 \beta_{2} + 31 \beta_{3} ) q^{37} + ( -90 - 4 \beta_{1} - 90 \beta_{2} ) q^{38} + ( 63 - 3 \beta_{1} + 63 \beta_{2} ) q^{39} + 40 \beta_{2} q^{40} + ( 18 - 27 \beta_{3} ) q^{41} + ( 24 + 6 \beta_{1} + 30 \beta_{2} + 18 \beta_{3} ) q^{42} + ( -67 + 27 \beta_{3} ) q^{43} + ( -20 \beta_{1} - 40 \beta_{2} - 20 \beta_{3} ) q^{44} + ( -45 - 45 \beta_{2} ) q^{45} + ( -44 - 22 \beta_{1} - 44 \beta_{2} ) q^{46} + ( 36 \beta_{1} + 342 \beta_{2} + 36 \beta_{3} ) q^{47} -48 q^{48} + ( 129 - 22 \beta_{1} - 215 \beta_{2} + 4 \beta_{3} ) q^{49} + 50 q^{50} + ( 33 \beta_{1} - 114 \beta_{2} + 33 \beta_{3} ) q^{51} + ( -84 + 4 \beta_{1} - 84 \beta_{2} ) q^{52} + ( -8 + 50 \beta_{1} - 8 \beta_{2} ) q^{53} -54 \beta_{2} q^{54} + ( -50 + 25 \beta_{3} ) q^{55} + ( -32 - 8 \beta_{1} - 40 \beta_{2} - 24 \beta_{3} ) q^{56} + ( -135 + 6 \beta_{3} ) q^{57} + ( 10 \beta_{1} - 160 \beta_{2} + 10 \beta_{3} ) q^{58} + ( -276 - 15 \beta_{1} - 276 \beta_{2} ) q^{59} + ( 60 + 60 \beta_{2} ) q^{60} + ( 18 \beta_{1} + 656 \beta_{2} + 18 \beta_{3} ) q^{61} + ( 62 + 16 \beta_{3} ) q^{62} + ( 45 + 27 \beta_{1} + 9 \beta_{2} + 18 \beta_{3} ) q^{63} + 64 q^{64} + ( -5 \beta_{1} + 105 \beta_{2} - 5 \beta_{3} ) q^{65} + ( -60 - 30 \beta_{1} - 60 \beta_{2} ) q^{66} + ( -97 + 41 \beta_{1} - 97 \beta_{2} ) q^{67} + ( -44 \beta_{1} + 152 \beta_{2} - 44 \beta_{3} ) q^{68} + ( -66 + 33 \beta_{3} ) q^{69} + ( -10 - 20 \beta_{1} + 40 \beta_{2} + 10 \beta_{3} ) q^{70} + ( 190 - 137 \beta_{3} ) q^{71} + 72 \beta_{2} q^{72} + ( -99 - 11 \beta_{1} - 99 \beta_{2} ) q^{73} + ( 358 - 62 \beta_{1} + 358 \beta_{2} ) q^{74} -75 \beta_{2} q^{75} + ( 180 - 8 \beta_{3} ) q^{76} + ( 700 + 420 \beta_{2} - 35 \beta_{3} ) q^{77} + ( -126 - 6 \beta_{3} ) q^{78} + ( -166 \beta_{1} + 63 \beta_{2} - 166 \beta_{3} ) q^{79} + ( -80 - 80 \beta_{2} ) q^{80} + ( -81 - 81 \beta_{2} ) q^{81} + ( 54 \beta_{1} + 36 \beta_{2} + 54 \beta_{3} ) q^{82} + ( 432 - 147 \beta_{3} ) q^{83} + ( -60 - 36 \beta_{1} - 12 \beta_{2} - 24 \beta_{3} ) q^{84} + ( 190 + 55 \beta_{3} ) q^{85} + ( -54 \beta_{1} - 134 \beta_{2} - 54 \beta_{3} ) q^{86} + ( -240 + 15 \beta_{1} - 240 \beta_{2} ) q^{87} + ( 80 + 40 \beta_{1} + 80 \beta_{2} ) q^{88} + ( -157 \beta_{1} - 92 \beta_{2} - 157 \beta_{3} ) q^{89} + 90 q^{90} + ( -176 - 16 \beta_{1} - 59 \beta_{2} - 62 \beta_{3} ) q^{91} + ( 88 - 44 \beta_{3} ) q^{92} + ( 24 \beta_{1} - 93 \beta_{2} + 24 \beta_{3} ) q^{93} + ( -684 - 72 \beta_{1} - 684 \beta_{2} ) q^{94} + ( -225 - 10 \beta_{1} - 225 \beta_{2} ) q^{95} -96 \beta_{2} q^{96} + ( 284 - 30 \beta_{3} ) q^{97} + ( 430 - 8 \beta_{1} + 688 \beta_{2} - 52 \beta_{3} ) q^{98} + ( -90 + 45 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 6q^{3} - 8q^{4} - 10q^{5} - 24q^{6} - 6q^{7} + 32q^{8} - 18q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 6q^{3} - 8q^{4} - 10q^{5} - 24q^{6} - 6q^{7} + 32q^{8} - 18q^{9} - 20q^{10} + 20q^{11} + 24q^{12} + 84q^{13} + 36q^{14} - 60q^{15} - 32q^{16} - 76q^{17} - 36q^{18} - 90q^{19} + 80q^{20} + 36q^{21} - 80q^{22} - 44q^{23} + 48q^{24} - 50q^{25} - 84q^{26} - 108q^{27} - 48q^{28} - 320q^{29} + 60q^{30} - 62q^{31} - 64q^{32} - 60q^{33} + 304q^{34} + 90q^{35} + 144q^{36} + 358q^{37} - 180q^{38} + 126q^{39} - 80q^{40} + 72q^{41} + 36q^{42} - 268q^{43} + 80q^{44} - 90q^{45} - 88q^{46} - 684q^{47} - 192q^{48} + 946q^{49} + 200q^{50} + 228q^{51} - 168q^{52} - 16q^{53} + 108q^{54} - 200q^{55} - 48q^{56} - 540q^{57} + 320q^{58} - 552q^{59} + 120q^{60} - 1312q^{61} + 248q^{62} + 162q^{63} + 256q^{64} - 210q^{65} - 120q^{66} - 194q^{67} - 304q^{68} - 264q^{69} - 120q^{70} + 760q^{71} - 144q^{72} - 198q^{73} + 716q^{74} + 150q^{75} + 720q^{76} + 1960q^{77} - 504q^{78} - 126q^{79} - 160q^{80} - 162q^{81} - 72q^{82} + 1728q^{83} - 216q^{84} + 760q^{85} + 268q^{86} - 480q^{87} + 160q^{88} + 184q^{89} + 360q^{90} - 586q^{91} + 352q^{92} + 186q^{93} - 1368q^{94} - 450q^{95} + 192q^{96} + 1136q^{97} + 344q^{98} - 360q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 46 x^{2} + 2116$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/46$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/46$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$46 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$46 \beta_{3}$$

## Character values

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

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

## 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
 −3.39116 − 5.87367i 3.39116 + 5.87367i −3.39116 + 5.87367i 3.39116 − 5.87367i
−1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i −2.50000 + 4.33013i −6.00000 −18.4558 + 1.54354i 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 15.4558 10.2038i 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 −18.4558 1.54354i 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 15.4558 + 10.2038i 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.h 4
3.b odd 2 1 630.4.k.n 4
7.c even 3 1 inner 210.4.i.h 4
7.c even 3 1 1470.4.a.bo 2
7.d odd 6 1 1470.4.a.bp 2
21.h odd 6 1 630.4.k.n 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} - 20 T_{11}^{3} + 1450 T_{11}^{2} + 21000 T_{11} + 1102500$$ acting on $$S_{4}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$( 25 + 5 T + T^{2} )^{2}$$
$7$ $$117649 + 2058 T - 455 T^{2} + 6 T^{3} + T^{4}$$
$11$ $$1102500 + 21000 T + 1450 T^{2} - 20 T^{3} + T^{4}$$
$13$ $$( 395 - 42 T + T^{2} )^{2}$$
$17$ $$16990884 - 313272 T + 9898 T^{2} + 76 T^{3} + T^{4}$$
$19$ $$3389281 + 165690 T + 6259 T^{2} + 90 T^{3} + T^{4}$$
$23$ $$25826724 - 223608 T + 7018 T^{2} + 44 T^{3} + T^{4}$$
$29$ $$( 5250 + 160 T + T^{2} )^{2}$$
$31$ $$3932289 - 122946 T + 5827 T^{2} + 62 T^{3} + T^{4}$$
$37$ $$147987225 + 4355070 T + 140329 T^{2} - 358 T^{3} + T^{4}$$
$41$ $$( -33210 - 36 T + T^{2} )^{2}$$
$43$ $$( -29045 + 134 T + T^{2} )^{2}$$
$47$ $$3288793104 + 39226032 T + 410508 T^{2} + 684 T^{3} + T^{4}$$
$53$ $$13210284096 - 1838976 T + 115192 T^{2} + 16 T^{3} + T^{4}$$
$59$ $$4333062276 + 36335952 T + 238878 T^{2} + 552 T^{3} + T^{4}$$
$61$ $$172583746624 + 545046784 T + 1305912 T^{2} + 1312 T^{3} + T^{4}$$
$67$ $$4612718889 - 13175898 T + 105553 T^{2} + 194 T^{3} + T^{4}$$
$71$ $$( -827274 - 380 T + T^{2} )^{2}$$
$73$ $$17935225 + 838530 T + 34969 T^{2} + 198 T^{3} + T^{4}$$
$79$ $$1596702650449 - 159214482 T + 1279483 T^{2} + 126 T^{3} + T^{4}$$
$83$ $$( -807390 - 864 T + T^{2} )^{2}$$
$89$ $$1266502652100 + 207071760 T + 1159246 T^{2} - 184 T^{3} + T^{4}$$
$97$ $$( 39256 - 568 T + T^{2} )^{2}$$