# Properties

 Label 210.2.b.a Level $210$ Weight $2$ Character orbit 210.b Analytic conductor $1.677$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 210.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 - \beta_{1} ) q^{3} - q^{4} + q^{5} + ( -\beta_{2} - \beta_{3} ) q^{6} + ( 2 + \beta_{2} - \beta_{3} ) q^{7} -\beta_{2} q^{8} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 - \beta_{1} ) q^{3} - q^{4} + q^{5} + ( -\beta_{2} - \beta_{3} ) q^{6} + ( 2 + \beta_{2} - \beta_{3} ) q^{7} -\beta_{2} q^{8} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{9} + \beta_{2} q^{10} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( 1 + \beta_{1} ) q^{12} + ( -\beta_{1} - \beta_{3} ) q^{13} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{14} + ( -1 - \beta_{1} ) q^{15} + q^{16} + ( 4 + \beta_{1} - \beta_{3} ) q^{17} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{18} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{19} - q^{20} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{21} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{22} -4 \beta_{2} q^{23} + ( \beta_{2} + \beta_{3} ) q^{24} + q^{25} + ( \beta_{1} - \beta_{3} ) q^{26} + ( -3 - 4 \beta_{2} - \beta_{3} ) q^{27} + ( -2 - \beta_{2} + \beta_{3} ) q^{28} + ( -3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{29} + ( -\beta_{2} - \beta_{3} ) q^{30} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( 4 - 2 \beta_{1} - 6 \beta_{2} ) q^{33} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{34} + ( 2 + \beta_{2} - \beta_{3} ) q^{35} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{36} + ( 2 - \beta_{1} + \beta_{3} ) q^{37} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{38} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{39} -\beta_{2} q^{40} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{41} + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{42} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{43} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{44} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{45} + 4 q^{46} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{47} + ( -1 - \beta_{1} ) q^{48} + ( 3 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{49} + \beta_{2} q^{50} + ( -6 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{51} + ( \beta_{1} + \beta_{3} ) q^{52} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{54} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{55} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{56} + ( 4 - 2 \beta_{1} - 10 \beta_{2} - 4 \beta_{3} ) q^{57} + ( 4 + 3 \beta_{1} - 3 \beta_{3} ) q^{58} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( 1 + \beta_{1} ) q^{60} + ( \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{61} + ( 4 - \beta_{1} + \beta_{3} ) q^{62} + ( 2 + 5 \beta_{1} + 3 \beta_{2} ) q^{63} - q^{64} + ( -\beta_{1} - \beta_{3} ) q^{65} + ( 6 + 4 \beta_{2} - 2 \beta_{3} ) q^{66} -12 q^{67} + ( -4 - \beta_{1} + \beta_{3} ) q^{68} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{69} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{70} + ( -\beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{71} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{72} + ( -\beta_{1} - 10 \beta_{2} - \beta_{3} ) q^{73} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{74} + ( -1 - \beta_{1} ) q^{75} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 2 + 8 \beta_{2} + 6 \beta_{3} ) q^{77} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{78} + ( -4 - 4 \beta_{1} + 4 \beta_{3} ) q^{79} + q^{80} + ( 1 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{82} + ( 6 + 7 \beta_{1} - 7 \beta_{3} ) q^{83} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{84} + ( 4 + \beta_{1} - \beta_{3} ) q^{85} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{86} + ( -6 + 3 \beta_{1} + 10 \beta_{2} + \beta_{3} ) q^{87} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{88} + ( -8 + 2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{90} + ( -2 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{91} + 4 \beta_{2} q^{92} + ( 2 - \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{93} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{94} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -\beta_{2} - \beta_{3} ) q^{96} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{97} + ( -2 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{98} + ( -4 - 4 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} - 4q^{4} + 4q^{5} - 2q^{6} + 6q^{7} + O(q^{10})$$ $$4q - 2q^{3} - 4q^{4} + 4q^{5} - 2q^{6} + 6q^{7} + 2q^{12} - 6q^{14} - 2q^{15} + 4q^{16} + 12q^{17} - 8q^{18} - 4q^{20} - 16q^{21} + 2q^{24} + 4q^{25} - 4q^{26} - 14q^{27} - 6q^{28} - 2q^{30} + 20q^{33} + 6q^{35} + 12q^{37} - 16q^{38} - 12q^{39} + 8q^{41} + 16q^{43} + 16q^{46} - 8q^{47} - 2q^{48} - 16q^{51} + 14q^{54} + 6q^{56} + 12q^{57} + 4q^{58} + 2q^{60} + 20q^{62} - 2q^{63} - 4q^{64} + 20q^{66} - 48q^{67} - 12q^{68} + 8q^{69} - 6q^{70} + 8q^{72} - 2q^{75} + 20q^{77} - 8q^{78} + 4q^{80} + 4q^{81} - 4q^{83} + 16q^{84} + 12q^{85} - 28q^{87} - 40q^{89} - 8q^{90} - 16q^{91} + 20q^{93} - 2q^{96} - 8q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 − 0.618034i 1.61803i 0.618034i − 1.61803i
1.00000i −1.61803 + 0.618034i −1.00000 1.00000 0.618034 + 1.61803i 2.61803 0.381966i 1.00000i 2.23607 2.00000i 1.00000i
41.2 1.00000i 0.618034 1.61803i −1.00000 1.00000 −1.61803 0.618034i 0.381966 2.61803i 1.00000i −2.23607 2.00000i 1.00000i
41.3 1.00000i −1.61803 0.618034i −1.00000 1.00000 0.618034 1.61803i 2.61803 + 0.381966i 1.00000i 2.23607 + 2.00000i 1.00000i
41.4 1.00000i 0.618034 + 1.61803i −1.00000 1.00000 −1.61803 + 0.618034i 0.381966 + 2.61803i 1.00000i −2.23607 + 2.00000i 1.00000i
 $$n$$: 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
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.b.a 4
3.b odd 2 1 210.2.b.b yes 4
4.b odd 2 1 1680.2.f.i 4
5.b even 2 1 1050.2.b.c 4
5.c odd 4 1 1050.2.d.c 4
5.c odd 4 1 1050.2.d.d 4
7.b odd 2 1 210.2.b.b yes 4
12.b even 2 1 1680.2.f.e 4
15.d odd 2 1 1050.2.b.a 4
15.e even 4 1 1050.2.d.a 4
15.e even 4 1 1050.2.d.f 4
21.c even 2 1 inner 210.2.b.a 4
28.d even 2 1 1680.2.f.e 4
35.c odd 2 1 1050.2.b.a 4
35.f even 4 1 1050.2.d.a 4
35.f even 4 1 1050.2.d.f 4
84.h odd 2 1 1680.2.f.i 4
105.g even 2 1 1050.2.b.c 4
105.k odd 4 1 1050.2.d.c 4
105.k odd 4 1 1050.2.d.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.b.a 4 1.a even 1 1 trivial
210.2.b.a 4 21.c even 2 1 inner
210.2.b.b yes 4 3.b odd 2 1
210.2.b.b yes 4 7.b odd 2 1
1050.2.b.a 4 15.d odd 2 1
1050.2.b.a 4 35.c odd 2 1
1050.2.b.c 4 5.b even 2 1
1050.2.b.c 4 105.g even 2 1
1050.2.d.a 4 15.e even 4 1
1050.2.d.a 4 35.f even 4 1
1050.2.d.c 4 5.c odd 4 1
1050.2.d.c 4 105.k odd 4 1
1050.2.d.d 4 5.c odd 4 1
1050.2.d.d 4 105.k odd 4 1
1050.2.d.f 4 15.e even 4 1
1050.2.d.f 4 35.f even 4 1
1680.2.f.e 4 12.b even 2 1
1680.2.f.e 4 28.d even 2 1
1680.2.f.i 4 4.b odd 2 1
1680.2.f.i 4 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{2} - 6 T_{17} + 4$$ acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$5$ $$( 1 - T )^{4}$$
$7$ $$1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4}$$
$11$ $$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$
$13$ $$1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$
$29$ $$1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8}$$
$31$ $$1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8}$$
$37$ $$( 1 - 6 T + 78 T^{2} - 222 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 8 T + 22 T^{2} - 344 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 4 T + 78 T^{2} + 188 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 98 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 + 12 T + 67 T^{2} )^{4}$$
$71$ $$1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8}$$
$73$ $$1 - 120 T^{2} + 12638 T^{4} - 639480 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 78 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 2 T - 78 T^{2} + 166 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 20 T + 258 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 360 T^{2} + 51038 T^{4} - 3387240 T^{6} + 88529281 T^{8}$$