# Properties

 Label 210.2.d.b Level 210 Weight 2 Character orbit 210.d Analytic conductor 1.677 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 210.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.10070523904.11 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 10 x^{4} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - \nu^{2}$$$$)/18$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 7 \nu$$$$)/6$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 19 \nu^{2}$$$$)/18$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + 3 \nu^{5} + 9 \nu^{4} + \nu^{3} - \nu^{2} - 3 \nu - 45$$$$)/36$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 3 \nu^{5} + 9 \nu^{4} - \nu^{3} + \nu^{2} + 3 \nu - 45$$$$)/36$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} - 10 \nu^{3}$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{5} + 5$$ $$\nu^{5}$$ $$=$$ $$6 \beta_{3} + 7 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{4} + 19 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{7} + 20 \beta_{6} - 20 \beta_{5} + 20 \beta_{3} + 20 \beta_{2} + 20 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 −1.68014 − 0.420861i −1.68014 + 0.420861i −0.420861 − 1.68014i −0.420861 + 1.68014i 0.420861 − 1.68014i 0.420861 + 1.68014i 1.68014 − 0.420861i 1.68014 + 0.420861i
1.00000 −1.68014 0.420861i 1.00000 1.08495 + 1.95522i −1.68014 0.420861i 0.595188 + 2.57794i 1.00000 2.64575 + 1.41421i 1.08495 + 1.95522i
209.2 1.00000 −1.68014 + 0.420861i 1.00000 1.08495 1.95522i −1.68014 + 0.420861i 0.595188 2.57794i 1.00000 2.64575 1.41421i 1.08495 1.95522i
209.3 1.00000 −0.420861 1.68014i 1.00000 −1.95522 1.08495i −0.420861 1.68014i 2.37608 1.16372i 1.00000 −2.64575 + 1.41421i −1.95522 1.08495i
209.4 1.00000 −0.420861 + 1.68014i 1.00000 −1.95522 + 1.08495i −0.420861 + 1.68014i 2.37608 + 1.16372i 1.00000 −2.64575 1.41421i −1.95522 + 1.08495i
209.5 1.00000 0.420861 1.68014i 1.00000 1.95522 1.08495i 0.420861 1.68014i −2.37608 + 1.16372i 1.00000 −2.64575 1.41421i 1.95522 1.08495i
209.6 1.00000 0.420861 + 1.68014i 1.00000 1.95522 + 1.08495i 0.420861 + 1.68014i −2.37608 1.16372i 1.00000 −2.64575 + 1.41421i 1.95522 + 1.08495i
209.7 1.00000 1.68014 0.420861i 1.00000 −1.08495 + 1.95522i 1.68014 0.420861i −0.595188 2.57794i 1.00000 2.64575 1.41421i −1.08495 + 1.95522i
209.8 1.00000 1.68014 + 0.420861i 1.00000 −1.08495 1.95522i 1.68014 + 0.420861i −0.595188 + 2.57794i 1.00000 2.64575 + 1.41421i −1.08495 1.95522i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.8 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.b odd 2 1 inner
15.d odd 2 1 inner
105.g 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.d.b yes 8
3.b odd 2 1 210.2.d.a 8
4.b odd 2 1 1680.2.k.f 8
5.b even 2 1 210.2.d.a 8
5.c odd 4 2 1050.2.b.f 16
7.b odd 2 1 inner 210.2.d.b yes 8
12.b even 2 1 1680.2.k.e 8
15.d odd 2 1 inner 210.2.d.b yes 8
15.e even 4 2 1050.2.b.f 16
20.d odd 2 1 1680.2.k.e 8
21.c even 2 1 210.2.d.a 8
28.d even 2 1 1680.2.k.f 8
35.c odd 2 1 210.2.d.a 8
35.f even 4 2 1050.2.b.f 16
60.h even 2 1 1680.2.k.f 8
84.h odd 2 1 1680.2.k.e 8
105.g even 2 1 inner 210.2.d.b yes 8
105.k odd 4 2 1050.2.b.f 16
140.c even 2 1 1680.2.k.e 8
420.o odd 2 1 1680.2.k.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.d.a 8 3.b odd 2 1
210.2.d.a 8 5.b even 2 1
210.2.d.a 8 21.c even 2 1
210.2.d.a 8 35.c odd 2 1
210.2.d.b yes 8 1.a even 1 1 trivial
210.2.d.b yes 8 7.b odd 2 1 inner
210.2.d.b yes 8 15.d odd 2 1 inner
210.2.d.b yes 8 105.g even 2 1 inner
1050.2.b.f 16 5.c odd 4 2
1050.2.b.f 16 15.e even 4 2
1050.2.b.f 16 35.f even 4 2
1050.2.b.f 16 105.k odd 4 2
1680.2.k.e 8 12.b even 2 1
1680.2.k.e 8 20.d odd 2 1
1680.2.k.e 8 84.h odd 2 1
1680.2.k.e 8 140.c even 2 1
1680.2.k.f 8 4.b odd 2 1
1680.2.k.f 8 28.d even 2 1
1680.2.k.f 8 60.h even 2 1
1680.2.k.f 8 420.o odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{2} + 4 T_{23} - 24$$ acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{8}$$
$3$ $$1 - 10 T^{4} + 81 T^{8}$$
$5$ $$1 + 22 T^{4} + 625 T^{8}$$
$7$ $$1 + 4 T^{2} - 10 T^{4} + 196 T^{6} + 2401 T^{8}$$
$11$ $$( 1 - 6 T + 11 T^{2} )^{4}( 1 + 6 T + 11 T^{2} )^{4}$$
$13$ $$( 1 + 40 T^{2} + 710 T^{4} + 6760 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 44 T^{2} + 950 T^{4} - 12716 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 24 T^{2} + 838 T^{4} - 8664 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 4 T + 22 T^{2} + 92 T^{3} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 52 T^{2} + 1350 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 84 T^{2} + 3574 T^{4} - 80724 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 20 T^{2} + 38 T^{4} - 27380 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 84 T^{2} + 4678 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 44 T^{2} + 2390 T^{4} - 81356 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 108 T^{2} + 6886 T^{4} - 238572 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 4 T - 2 T^{2} + 212 T^{3} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 216 T^{2} + 18598 T^{4} + 751896 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 136 T^{2} + 9798 T^{4} - 506056 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 140 T^{2} + 12086 T^{4} - 628460 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 28 T^{2} + 9270 T^{4} - 141148 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 252 T^{2} + 26422 T^{4} + 1342908 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 4 T + 134 T^{2} + 316 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 224 T^{2} + 26294 T^{4} - 1543136 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 60 T^{2} + 14950 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 108 T^{2} + 16246 T^{4} + 1016172 T^{6} + 88529281 T^{8} )^{2}$$