# Properties

 Label 210.4.g.b 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: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{1} q^{2} -3 \beta_{1} q^{3} -4 q^{4} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{5} + 6 q^{6} -7 \beta_{1} q^{7} -8 \beta_{1} q^{8} -9 q^{9} +O(q^{10})$$ $$q + 2 \beta_{1} q^{2} -3 \beta_{1} q^{3} -4 q^{4} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{5} + 6 q^{6} -7 \beta_{1} q^{7} -8 \beta_{1} q^{8} -9 q^{9} + ( 4 + 8 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{10} + ( -6 + 8 \beta_{3} ) q^{11} + 12 \beta_{1} q^{12} + ( 64 \beta_{1} - 2 \beta_{2} ) q^{13} + 14 q^{14} + ( -6 - 12 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{15} + 16 q^{16} + ( -34 \beta_{1} - 18 \beta_{2} ) q^{17} -18 \beta_{1} q^{18} + ( 14 + 20 \beta_{3} ) q^{19} + ( -16 + 8 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{20} -21 q^{21} + ( -12 \beta_{1} + 16 \beta_{2} ) q^{22} + ( 104 \beta_{1} + 20 \beta_{2} ) q^{23} -24 q^{24} + ( -51 + 68 \beta_{1} + 12 \beta_{2} + 16 \beta_{3} ) q^{25} + ( -128 + 4 \beta_{3} ) q^{26} + 27 \beta_{1} q^{27} + 28 \beta_{1} q^{28} + ( 112 + 14 \beta_{3} ) q^{29} + ( 24 - 12 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} ) q^{30} + ( 4 + 34 \beta_{3} ) q^{31} + 32 \beta_{1} q^{32} + ( 18 \beta_{1} - 24 \beta_{2} ) q^{33} + ( 68 + 36 \beta_{3} ) q^{34} + ( -14 - 28 \beta_{1} - 7 \beta_{2} + 14 \beta_{3} ) q^{35} + 36 q^{36} + ( -130 \beta_{1} + 18 \beta_{2} ) q^{37} + ( 28 \beta_{1} + 40 \beta_{2} ) q^{38} + ( 192 - 6 \beta_{3} ) q^{39} + ( -16 - 32 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} ) q^{40} + ( -72 + 34 \beta_{3} ) q^{41} -42 \beta_{1} q^{42} + ( 290 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 24 - 32 \beta_{3} ) q^{44} + ( -36 + 18 \beta_{1} - 18 \beta_{2} - 9 \beta_{3} ) q^{45} + ( -208 - 40 \beta_{3} ) q^{46} + ( -2 \beta_{1} + 118 \beta_{2} ) q^{47} -48 \beta_{1} q^{48} -49 q^{49} + ( -136 - 102 \beta_{1} + 32 \beta_{2} - 24 \beta_{3} ) q^{50} + ( -102 - 54 \beta_{3} ) q^{51} + ( -256 \beta_{1} + 8 \beta_{2} ) q^{52} + ( -354 \beta_{1} - 44 \beta_{2} ) q^{53} -54 q^{54} + ( 144 + 348 \beta_{1} - 28 \beta_{2} + 26 \beta_{3} ) q^{55} -56 q^{56} + ( -42 \beta_{1} - 60 \beta_{2} ) q^{57} + ( 224 \beta_{1} + 28 \beta_{2} ) q^{58} + ( -128 - 80 \beta_{3} ) q^{59} + ( 24 + 48 \beta_{1} + 12 \beta_{2} - 24 \beta_{3} ) q^{60} + ( -228 + 118 \beta_{3} ) q^{61} + ( 8 \beta_{1} + 68 \beta_{2} ) q^{62} + 63 \beta_{1} q^{63} -64 q^{64} + ( 212 + 214 \beta_{1} + 56 \beta_{2} - 132 \beta_{3} ) q^{65} + ( -36 + 48 \beta_{3} ) q^{66} + ( -374 \beta_{1} - 150 \beta_{2} ) q^{67} + ( 136 \beta_{1} + 72 \beta_{2} ) q^{68} + ( 312 + 60 \beta_{3} ) q^{69} + ( 56 - 28 \beta_{1} + 28 \beta_{2} + 14 \beta_{3} ) q^{70} + ( -492 - 134 \beta_{3} ) q^{71} + 72 \beta_{1} q^{72} + ( -124 \beta_{1} - 34 \beta_{2} ) q^{73} + ( 260 - 36 \beta_{3} ) q^{74} + ( 204 + 153 \beta_{1} - 48 \beta_{2} + 36 \beta_{3} ) q^{75} + ( -56 - 80 \beta_{3} ) q^{76} + ( 42 \beta_{1} - 56 \beta_{2} ) q^{77} + ( 384 \beta_{1} - 12 \beta_{2} ) q^{78} + ( 112 + 116 \beta_{3} ) q^{79} + ( 64 - 32 \beta_{1} + 32 \beta_{2} + 16 \beta_{3} ) q^{80} + 81 q^{81} + ( -144 \beta_{1} + 68 \beta_{2} ) q^{82} + ( -780 \beta_{1} + 44 \beta_{2} ) q^{83} + 84 q^{84} + ( 688 - 514 \beta_{1} - 106 \beta_{2} + 32 \beta_{3} ) q^{85} + ( -580 + 4 \beta_{3} ) q^{86} + ( -336 \beta_{1} - 42 \beta_{2} ) q^{87} + ( 48 \beta_{1} - 64 \beta_{2} ) q^{88} + ( -20 + 30 \beta_{3} ) q^{89} + ( -36 - 72 \beta_{1} - 18 \beta_{2} + 36 \beta_{3} ) q^{90} + ( 448 - 14 \beta_{3} ) q^{91} + ( -416 \beta_{1} - 80 \beta_{2} ) q^{92} + ( -12 \beta_{1} - 102 \beta_{2} ) q^{93} + ( 4 - 236 \beta_{3} ) q^{94} + ( 476 + 812 \beta_{1} - 12 \beta_{2} + 94 \beta_{3} ) q^{95} + 96 q^{96} + ( 552 \beta_{1} - 262 \beta_{2} ) q^{97} -98 \beta_{1} q^{98} + ( 54 - 72 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{4} + 16q^{5} + 24q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 16q^{4} + 16q^{5} + 24q^{6} - 36q^{9} + 16q^{10} - 24q^{11} + 56q^{14} - 24q^{15} + 64q^{16} + 56q^{19} - 64q^{20} - 84q^{21} - 96q^{24} - 204q^{25} - 512q^{26} + 448q^{29} + 96q^{30} + 16q^{31} + 272q^{34} - 56q^{35} + 144q^{36} + 768q^{39} - 64q^{40} - 288q^{41} + 96q^{44} - 144q^{45} - 832q^{46} - 196q^{49} - 544q^{50} - 408q^{51} - 216q^{54} + 576q^{55} - 224q^{56} - 512q^{59} + 96q^{60} - 912q^{61} - 256q^{64} + 848q^{65} - 144q^{66} + 1248q^{69} + 224q^{70} - 1968q^{71} + 1040q^{74} + 816q^{75} - 224q^{76} + 448q^{79} + 256q^{80} + 324q^{81} + 336q^{84} + 2752q^{85} - 2320q^{86} - 80q^{89} - 144q^{90} + 1792q^{91} + 16q^{94} + 1904q^{95} + 384q^{96} + 216q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 6 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 16 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 11$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{2} + 8 \beta_{1}$$

## 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
 2.79129i − 1.79129i − 2.79129i 1.79129i
2.00000i 3.00000i −4.00000 −0.582576 + 11.1652i 6.00000 7.00000i 8.00000i −9.00000 22.3303 + 1.16515i
169.2 2.00000i 3.00000i −4.00000 8.58258 7.16515i 6.00000 7.00000i 8.00000i −9.00000 −14.3303 17.1652i
169.3 2.00000i 3.00000i −4.00000 −0.582576 11.1652i 6.00000 7.00000i 8.00000i −9.00000 22.3303 1.16515i
169.4 2.00000i 3.00000i −4.00000 8.58258 + 7.16515i 6.00000 7.00000i 8.00000i −9.00000 −14.3303 + 17.1652i
 $$n$$: 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.b 4
3.b odd 2 1 630.4.g.d 4
5.b even 2 1 inner 210.4.g.b 4
5.c odd 4 1 1050.4.a.ba 2
5.c odd 4 1 1050.4.a.bh 2
15.d odd 2 1 630.4.g.d 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} + 12 T_{11} - 1308$$ acting on $$S_{4}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + T^{2} )^{2}$$
$3$ $$( 9 + T^{2} )^{2}$$
$5$ $$15625 - 2000 T + 230 T^{2} - 16 T^{3} + T^{4}$$
$7$ $$( 49 + T^{2} )^{2}$$
$11$ $$( -1308 + 12 T + T^{2} )^{2}$$
$13$ $$16096144 + 8360 T^{2} + T^{4}$$
$17$ $$31899904 + 15920 T^{2} + T^{4}$$
$19$ $$( -8204 - 28 T + T^{2} )^{2}$$
$23$ $$5837056 + 38432 T^{2} + T^{4}$$
$29$ $$( 8428 - 224 T + T^{2} )^{2}$$
$31$ $$( -24260 - 8 T + T^{2} )^{2}$$
$37$ $$101929216 + 47408 T^{2} + T^{4}$$
$41$ $$( -19092 + 144 T + T^{2} )^{2}$$
$43$ $$7058688256 + 168368 T^{2} + T^{4}$$
$47$ $$85497760000 + 584816 T^{2} + T^{4}$$
$53$ $$7167315600 + 331944 T^{2} + T^{4}$$
$59$ $$( -118016 + 256 T + T^{2} )^{2}$$
$61$ $$( -240420 + 456 T + T^{2} )^{2}$$
$67$ $$110638725376 + 1224752 T^{2} + T^{4}$$
$71$ $$( -135012 + 984 T + T^{2} )^{2}$$
$73$ $$79210000 + 79304 T^{2} + T^{4}$$
$79$ $$( -270032 - 224 T + T^{2} )^{2}$$
$83$ $$322333249536 + 1298112 T^{2} + T^{4}$$
$89$ $$( -18500 + 40 T + T^{2} )^{2}$$
$97$ $$1292359712400 + 3492456 T^{2} + T^{4}$$