# Properties

 Label 210.3.f.a Level $210$ Weight $3$ Character orbit 210.f Analytic conductor $5.722$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3317760000.3 Defining polynomial: $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{5} q^{3} + 2 q^{4} + \beta_{6} q^{5} + \beta_{3} q^{6} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{7} -2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{5} q^{3} + 2 q^{4} + \beta_{6} q^{5} + \beta_{3} q^{6} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{7} -2 \beta_{1} q^{8} -3 q^{9} + \beta_{7} q^{10} + ( -2 - 6 \beta_{1} + \beta_{2} ) q^{11} + 2 \beta_{5} q^{12} + ( 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{13} + ( 4 + \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{14} -\beta_{4} q^{15} + 4 q^{16} + ( 6 \beta_{6} - 2 \beta_{7} ) q^{17} + 3 \beta_{1} q^{18} + ( 7 \beta_{3} - 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{19} + 2 \beta_{6} q^{20} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} ) q^{21} + ( 12 + 2 \beta_{1} - 2 \beta_{4} ) q^{22} + ( -8 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} ) q^{23} + 2 \beta_{3} q^{24} -5 q^{25} + ( 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{26} -3 \beta_{5} q^{27} + ( -4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} ) q^{28} + ( -18 - 14 \beta_{1} + 4 \beta_{4} ) q^{29} + \beta_{2} q^{30} + ( \beta_{3} - 14 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{31} -4 \beta_{1} q^{32} + ( 6 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} ) q^{33} + ( -4 \beta_{6} + 6 \beta_{7} ) q^{34} + ( -10 + \beta_{2} - 5 \beta_{5} + 2 \beta_{7} ) q^{35} -6 q^{36} + ( -6 + 10 \beta_{1} - 4 \beta_{2} - 8 \beta_{4} ) q^{37} + ( -6 \beta_{3} + 14 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{38} + ( -6 + 6 \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{39} + 2 \beta_{7} q^{40} + ( -4 \beta_{3} - 4 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -6 + 2 \beta_{2} + 4 \beta_{5} - 3 \beta_{7} ) q^{42} + ( -8 + 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{4} ) q^{43} + ( -4 - 12 \beta_{1} + 2 \beta_{2} ) q^{44} -3 \beta_{6} q^{45} + ( 16 + 2 \beta_{2} + 8 \beta_{4} ) q^{46} + ( -14 \beta_{3} - 6 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{47} + 4 \beta_{5} q^{48} + ( -3 + 8 \beta_{2} - 12 \beta_{5} + 2 \beta_{7} ) q^{49} + 5 \beta_{1} q^{50} + ( -2 \beta_{2} - 6 \beta_{4} ) q^{51} + ( 4 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{52} + ( 16 - 11 \beta_{1} + 2 \beta_{2} + 14 \beta_{4} ) q^{53} -3 \beta_{3} q^{54} + ( -5 \beta_{3} - 2 \beta_{6} + 6 \beta_{7} ) q^{55} + ( 8 + 2 \beta_{2} + 4 \beta_{5} + 4 \beta_{7} ) q^{56} + ( 18 + 21 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{57} + ( 28 + 18 \beta_{1} - 4 \beta_{2} ) q^{58} + ( -14 \beta_{3} - 24 \beta_{5} + 8 \beta_{7} ) q^{59} -2 \beta_{4} q^{60} + ( -16 \beta_{3} + 28 \beta_{5} - 16 \beta_{6} - 10 \beta_{7} ) q^{61} + ( -14 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{62} + ( 6 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} - 6 \beta_{6} ) q^{63} + 8 q^{64} + ( 10 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{65} + ( -2 \beta_{3} + 12 \beta_{5} - 6 \beta_{6} ) q^{66} + ( -24 + 36 \beta_{1} + 4 \beta_{2} + 6 \beta_{4} ) q^{67} + ( 12 \beta_{6} - 4 \beta_{7} ) q^{68} + ( 8 \beta_{3} - 6 \beta_{6} + 12 \beta_{7} ) q^{69} + ( 10 \beta_{1} - 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} ) q^{70} + ( 22 + 14 \beta_{1} - 15 \beta_{2} + 4 \beta_{4} ) q^{71} + 6 \beta_{1} q^{72} + ( -2 \beta_{3} + 38 \beta_{5} - 14 \beta_{6} - 9 \beta_{7} ) q^{73} + ( -20 + 6 \beta_{1} + 8 \beta_{2} + 8 \beta_{4} ) q^{74} -5 \beta_{5} q^{75} + ( 14 \beta_{3} - 12 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{76} + ( 24 - 11 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} - 2 \beta_{4} + 12 \beta_{5} - 10 \beta_{6} + 12 \beta_{7} ) q^{77} + ( -12 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{78} + ( -36 + 16 \beta_{1} + 8 \beta_{2} ) q^{79} + 4 \beta_{6} q^{80} + 9 q^{81} + ( -4 \beta_{3} - 8 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} ) q^{82} + ( -6 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} - 20 \beta_{7} ) q^{83} + ( 6 \beta_{1} + 4 \beta_{3} - 4 \beta_{4} - 6 \beta_{6} ) q^{84} + ( -30 - 10 \beta_{1} ) q^{85} + ( -8 + 8 \beta_{1} - 4 \beta_{2} + 16 \beta_{4} ) q^{86} + ( 14 \beta_{3} - 18 \beta_{5} + 12 \beta_{6} ) q^{87} + ( 24 + 4 \beta_{1} - 4 \beta_{4} ) q^{88} + ( -4 \beta_{3} - 10 \beta_{6} + 32 \beta_{7} ) q^{89} -3 \beta_{7} q^{90} + ( 8 + 16 \beta_{1} + 2 \beta_{2} - \beta_{3} - 6 \beta_{4} + 18 \beta_{5} - 2 \beta_{6} - 10 \beta_{7} ) q^{91} + ( -16 \beta_{1} - 8 \beta_{2} - 4 \beta_{4} ) q^{92} + ( 42 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{93} + ( -6 \beta_{3} - 28 \beta_{5} - 16 \beta_{6} + 4 \beta_{7} ) q^{94} + ( 10 - 10 \beta_{1} + 7 \beta_{2} + 6 \beta_{4} ) q^{95} + 4 \beta_{3} q^{96} + ( 6 \beta_{3} + 30 \beta_{5} + 34 \beta_{6} + 3 \beta_{7} ) q^{97} + ( 3 \beta_{1} - 12 \beta_{3} - 16 \beta_{4} + 4 \beta_{6} ) q^{98} + ( 6 + 18 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{4} - 24q^{9} + O(q^{10})$$ $$8q + 16q^{4} - 24q^{9} - 16q^{11} + 32q^{14} + 32q^{16} + 96q^{22} - 40q^{25} - 144q^{29} - 80q^{35} - 48q^{36} - 48q^{37} - 48q^{39} - 48q^{42} - 64q^{43} - 32q^{44} + 128q^{46} - 24q^{49} + 128q^{53} + 64q^{56} + 144q^{57} + 224q^{58} + 64q^{64} + 80q^{65} - 192q^{67} + 176q^{71} - 160q^{74} + 192q^{77} - 96q^{78} - 288q^{79} + 72q^{81} - 240q^{85} - 64q^{86} + 192q^{88} + 64q^{91} + 336q^{93} + 80q^{95} + 48q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu$$$$)/189$$ $$\beta_{2}$$ $$=$$ $$($$$$-10 \nu^{7} + 49 \nu^{5} - 133 \nu^{3} + 801 \nu$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{7} + \nu^{5} + 5 \nu^{3} - 63 \nu$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} + 2 \nu^{2} + 18$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{6} + 14 \nu^{4} - 56 \nu^{2} + 225$$$$)/63$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 22$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} - 7 \nu^{5} + 19 \nu^{3} - 81 \nu$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{3} + \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 2 \beta_{5} + \beta_{4} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 7 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{6} + \beta_{5} + 4 \beta_{4} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} + 19 \beta_{3} + 5 \beta_{2} + 19 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{6} + 22$$ $$\nu^{7}$$ $$=$$ $$($$$$29 \beta_{7} + 13 \beta_{3} + 29 \beta_{2} - 13 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 1.72286 + 0.178197i −1.01575 − 1.40294i −1.01575 + 1.40294i 1.72286 − 0.178197i −1.72286 − 0.178197i 1.01575 + 1.40294i 1.01575 − 1.40294i −1.72286 + 0.178197i
−1.41421 1.73205i 2.00000 2.23607i 2.44949i −6.70141 2.02265i −2.82843 −3.00000 3.16228i
181.2 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 1.04456 + 6.92163i −2.82843 −3.00000 3.16228i
181.3 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 1.04456 6.92163i −2.82843 −3.00000 3.16228i
181.4 −1.41421 1.73205i 2.00000 2.23607i 2.44949i −6.70141 + 2.02265i −2.82843 −3.00000 3.16228i
181.5 1.41421 1.73205i 2.00000 2.23607i 2.44949i −1.04456 6.92163i 2.82843 −3.00000 3.16228i
181.6 1.41421 1.73205i 2.00000 2.23607i 2.44949i 6.70141 + 2.02265i 2.82843 −3.00000 3.16228i
181.7 1.41421 1.73205i 2.00000 2.23607i 2.44949i 6.70141 2.02265i 2.82843 −3.00000 3.16228i
181.8 1.41421 1.73205i 2.00000 2.23607i 2.44949i −1.04456 + 6.92163i 2.82843 −3.00000 3.16228i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.8 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.f.a 8
3.b odd 2 1 630.3.f.c 8
4.b odd 2 1 1680.3.s.a 8
5.b even 2 1 1050.3.f.b 8
5.c odd 4 2 1050.3.h.b 16
7.b odd 2 1 inner 210.3.f.a 8
21.c even 2 1 630.3.f.c 8
28.d even 2 1 1680.3.s.a 8
35.c odd 2 1 1050.3.f.b 8
35.f even 4 2 1050.3.h.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.f.a 8 1.a even 1 1 trivial
210.3.f.a 8 7.b odd 2 1 inner
630.3.f.c 8 3.b odd 2 1
630.3.f.c 8 21.c even 2 1
1050.3.f.b 8 5.b even 2 1
1050.3.f.b 8 35.c odd 2 1
1050.3.h.b 16 5.c odd 4 2
1050.3.h.b 16 35.f even 4 2
1680.3.s.a 8 4.b odd 2 1
1680.3.s.a 8 28.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{4}$$
$3$ $$( 1 + 3 T^{2} )^{4}$$
$5$ $$( 1 + 5 T^{2} )^{4}$$
$7$ $$1 + 12 T^{2} - 2842 T^{4} + 28812 T^{6} + 5764801 T^{8}$$
$11$ $$( 1 + 8 T + 304 T^{2} + 2120 T^{3} + 45250 T^{4} + 256520 T^{5} + 4450864 T^{6} + 14172488 T^{7} + 214358881 T^{8} )^{2}$$
$13$ $$1 - 1088 T^{2} + 557124 T^{4} - 173160640 T^{6} + 35590649030 T^{8} - 4945641039040 T^{10} + 454463162206404 T^{12} - 25348316613259328 T^{14} + 665416609183179841 T^{16}$$
$17$ $$( 1 - 716 T^{2} + 266406 T^{4} - 59801036 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$1 - 1040 T^{2} + 715428 T^{4} - 373461040 T^{6} + 149925694598 T^{8} - 48669816193840 T^{10} + 12150516539296548 T^{12} - 2301847515828807440 T^{14} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$( 1 + 780 T^{2} + 15360 T^{3} + 320102 T^{4} + 8125440 T^{5} + 218275980 T^{6} + 78310985281 T^{8} )^{2}$$
$29$ $$( 1 + 72 T + 4044 T^{2} + 159480 T^{3} + 5105990 T^{4} + 134122680 T^{5} + 2860244364 T^{6} + 42827279112 T^{7} + 500246412961 T^{8} )^{2}$$
$31$ $$1 - 5072 T^{2} + 12940644 T^{4} - 21077526640 T^{6} + 23985269180870 T^{8} - 19465538480099440 T^{10} + 11036959286314652004 T^{12} -$$$$39\!\cdots\!92$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$( 1 + 24 T + 2412 T^{2} - 16728 T^{3} + 1972550 T^{4} - 22900632 T^{5} + 4520476332 T^{6} + 61577433816 T^{7} + 3512479453921 T^{8} )^{2}$$
$41$ $$1 - 11432 T^{2} + 60126684 T^{4} - 189254916760 T^{6} + 389111411703110 T^{8} - 534789162838654360 T^{10} +$$$$48\!\cdots\!64$$$$T^{12} -$$$$25\!\cdots\!92$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$( 1 + 32 T + 3396 T^{2} + 140128 T^{3} + 8375270 T^{4} + 259096672 T^{5} + 11610248196 T^{6} + 202283617568 T^{7} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 - 9656 T^{2} + 49532316 T^{4} - 174004173832 T^{6} + 446973571058246 T^{8} - 849084860968707592 T^{10} +$$$$11\!\cdots\!76$$$$T^{12} -$$$$11\!\cdots\!96$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 - 64 T + 6168 T^{2} - 270464 T^{3} + 22592978 T^{4} - 759733376 T^{5} + 48668486808 T^{6} - 1418519112256 T^{7} + 62259690411361 T^{8} )^{2}$$
$59$ $$1 - 13672 T^{2} + 87440284 T^{4} - 360721973080 T^{6} + 1260995942842630 T^{8} - 4370998368442641880 T^{10} +$$$$12\!\cdots\!64$$$$T^{12} -$$$$24\!\cdots\!32$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$1 - 5096 T^{2} + 58282524 T^{4} - 205728528856 T^{6} + 1227606280747910 T^{8} - 2848484499704087896 T^{10} +$$$$11\!\cdots\!44$$$$T^{12} -$$$$13\!\cdots\!16$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$( 1 + 96 T + 14188 T^{2} + 794016 T^{3} + 79706118 T^{4} + 3564337824 T^{5} + 285904104748 T^{6} + 8684004688224 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 88 T + 8304 T^{2} - 522200 T^{3} + 58604450 T^{4} - 2632410200 T^{5} + 211018599024 T^{6} - 11272824985048 T^{7} + 645753531245761 T^{8} )^{2}$$
$73$ $$1 - 18048 T^{2} + 179971204 T^{4} - 1281534583680 T^{6} + 7286987478858630 T^{8} - 36393327957179306880 T^{10} +$$$$14\!\cdots\!24$$$$T^{12} -$$$$41\!\cdots\!08$$$$T^{14} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$( 1 + 144 T + 27876 T^{2} + 2532528 T^{3} + 267406406 T^{4} + 15805507248 T^{5} + 1085772457956 T^{6} + 35004593595024 T^{7} + 1517108809906561 T^{8} )^{2}$$
$83$ $$1 - 37880 T^{2} + 719278428 T^{4} - 8629223884360 T^{6} + 71063717963313158 T^{8} -$$$$40\!\cdots\!60$$$$T^{10} +$$$$16\!\cdots\!48$$$$T^{12} -$$$$40\!\cdots\!80$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$1 - 20024 T^{2} + 352120860 T^{4} - 3746723082376 T^{6} + 35814387700257734 T^{8} -$$$$23\!\cdots\!16$$$$T^{10} +$$$$13\!\cdots\!60$$$$T^{12} -$$$$49\!\cdots\!04$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 - 40128 T^{2} + 785762308 T^{4} - 10615096489536 T^{6} + 111881094946850310 T^{8} -$$$$93\!\cdots\!16$$$$T^{10} +$$$$61\!\cdots\!88$$$$T^{12} -$$$$27\!\cdots\!48$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16}$$