# Properties

 Label 210.8.i.b Level $210$ Weight $8$ Character orbit 210.i Analytic conductor $65.601$ 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: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 210.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$65.6008553517$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} + 5593 x^{2} + 5592 x + 31270464$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -8 + 8 \beta_{1} ) q^{2} -27 \beta_{1} q^{3} -64 \beta_{1} q^{4} + ( 125 - 125 \beta_{1} ) q^{5} + 216 q^{6} + ( 7 - 42 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} ) q^{7} + 512 q^{8} + ( -729 + 729 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( -8 + 8 \beta_{1} ) q^{2} -27 \beta_{1} q^{3} -64 \beta_{1} q^{4} + ( 125 - 125 \beta_{1} ) q^{5} + 216 q^{6} + ( 7 - 42 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} ) q^{7} + 512 q^{8} + ( -729 + 729 \beta_{1} ) q^{9} + 1000 \beta_{1} q^{10} + ( 2651 \beta_{1} + 23 \beta_{2} - 23 \beta_{3} ) q^{11} + ( -1728 + 1728 \beta_{1} ) q^{12} + ( -3499 + 46 \beta_{1} + 46 \beta_{2} + 92 \beta_{3} ) q^{13} + ( 280 + 56 \beta_{2} ) q^{14} -3375 q^{15} + ( -4096 + 4096 \beta_{1} ) q^{16} + ( -10860 \beta_{1} - 66 \beta_{2} + 66 \beta_{3} ) q^{17} -5832 \beta_{1} q^{18} + ( -22205 + 22299 \beta_{1} - 188 \beta_{2} - 94 \beta_{3} ) q^{19} -8000 q^{20} + ( -1134 + 1134 \beta_{1} + 189 \beta_{3} ) q^{21} + ( -21576 + 184 \beta_{1} + 184 \beta_{2} + 368 \beta_{3} ) q^{22} + ( 22989 - 23044 \beta_{1} + 110 \beta_{2} + 55 \beta_{3} ) q^{23} -13824 \beta_{1} q^{24} -15625 \beta_{1} q^{25} + ( 27992 - 27624 \beta_{1} - 736 \beta_{2} - 368 \beta_{3} ) q^{26} + 19683 q^{27} + ( -2688 + 2688 \beta_{1} + 448 \beta_{3} ) q^{28} + ( 120024 + 202 \beta_{1} + 202 \beta_{2} + 404 \beta_{3} ) q^{29} + ( 27000 - 27000 \beta_{1} ) q^{30} + ( 171012 \beta_{1} - 97 \beta_{2} + 97 \beta_{3} ) q^{31} -32768 \beta_{1} q^{32} + ( 72819 - 72198 \beta_{1} - 1242 \beta_{2} - 621 \beta_{3} ) q^{33} + ( 87936 - 528 \beta_{1} - 528 \beta_{2} - 1056 \beta_{3} ) q^{34} + ( -4375 - 875 \beta_{2} ) q^{35} + 46656 q^{36} + ( -129671 + 129509 \beta_{1} + 324 \beta_{2} + 162 \beta_{3} ) q^{37} + ( -179144 \beta_{1} + 752 \beta_{2} - 752 \beta_{3} ) q^{38} + ( 91989 \beta_{1} + 1242 \beta_{2} - 1242 \beta_{3} ) q^{39} + ( 64000 - 64000 \beta_{1} ) q^{40} + ( -159435 - 703 \beta_{1} - 703 \beta_{2} - 1406 \beta_{3} ) q^{41} + ( 1512 - 9072 \beta_{1} - 1512 \beta_{2} - 1512 \beta_{3} ) q^{42} + ( -496912 + 1243 \beta_{1} + 1243 \beta_{2} + 2486 \beta_{3} ) q^{43} + ( 172608 - 171136 \beta_{1} - 2944 \beta_{2} - 1472 \beta_{3} ) q^{44} + 91125 \beta_{1} q^{45} + ( 184792 \beta_{1} - 440 \beta_{2} + 440 \beta_{3} ) q^{46} + ( -860559 + 861152 \beta_{1} - 1186 \beta_{2} - 593 \beta_{3} ) q^{47} + 110592 q^{48} + ( 820309 - 820309 \beta_{1} + 539 \beta_{3} ) q^{49} + 125000 q^{50} + ( -296784 + 295002 \beta_{1} + 3564 \beta_{2} + 1782 \beta_{3} ) q^{51} + ( 218048 \beta_{1} + 2944 \beta_{2} - 2944 \beta_{3} ) q^{52} + ( -1497067 \beta_{1} - 547 \beta_{2} + 547 \beta_{3} ) q^{53} + ( -157464 + 157464 \beta_{1} ) q^{54} + ( 337125 - 2875 \beta_{1} - 2875 \beta_{2} - 5750 \beta_{3} ) q^{55} + ( 3584 - 21504 \beta_{1} - 3584 \beta_{2} - 3584 \beta_{3} ) q^{56} + ( 599535 + 2538 \beta_{1} + 2538 \beta_{2} + 5076 \beta_{3} ) q^{57} + ( -960192 + 961808 \beta_{1} - 3232 \beta_{2} - 1616 \beta_{3} ) q^{58} + ( 424224 \beta_{1} + 10002 \beta_{2} - 10002 \beta_{3} ) q^{59} + 216000 \beta_{1} q^{60} + ( 1957936 - 1959498 \beta_{1} + 3124 \beta_{2} + 1562 \beta_{3} ) q^{61} + ( -1366544 - 776 \beta_{1} - 776 \beta_{2} - 1552 \beta_{3} ) q^{62} + ( 25515 + 5103 \beta_{2} ) q^{63} + 262144 q^{64} + ( -437375 + 431625 \beta_{1} + 11500 \beta_{2} + 5750 \beta_{3} ) q^{65} + ( 572616 \beta_{1} + 4968 \beta_{2} - 4968 \beta_{3} ) q^{66} + ( 2054190 \beta_{1} + 1541 \beta_{2} - 1541 \beta_{3} ) q^{67} + ( -703488 + 699264 \beta_{1} + 8448 \beta_{2} + 4224 \beta_{3} ) q^{68} + ( -620703 - 1485 \beta_{1} - 1485 \beta_{2} - 2970 \beta_{3} ) q^{69} + ( 42000 - 42000 \beta_{1} - 7000 \beta_{3} ) q^{70} + ( -3155742 + 8458 \beta_{1} + 8458 \beta_{2} + 16916 \beta_{3} ) q^{71} + ( -373248 + 373248 \beta_{1} ) q^{72} + ( -2062112 \beta_{1} + 9159 \beta_{2} - 9159 \beta_{3} ) q^{73} + ( -1034776 \beta_{1} - 1296 \beta_{2} + 1296 \beta_{3} ) q^{74} + ( -421875 + 421875 \beta_{1} ) q^{75} + ( 1421120 + 6016 \beta_{1} + 6016 \beta_{2} + 12032 \beta_{3} ) q^{76} + ( 2814210 + 2588789 \beta_{1} - 1771 \beta_{2} - 19523 \beta_{3} ) q^{77} + ( -755784 + 9936 \beta_{1} + 9936 \beta_{2} + 19872 \beta_{3} ) q^{78} + ( 851584 - 868233 \beta_{1} + 33298 \beta_{2} + 16649 \beta_{3} ) q^{79} + 512000 \beta_{1} q^{80} -531441 \beta_{1} q^{81} + ( 1275480 - 1281104 \beta_{1} + 11248 \beta_{2} + 5624 \beta_{3} ) q^{82} + ( 7766712 + 3136 \beta_{1} + 3136 \beta_{2} + 6272 \beta_{3} ) q^{83} + ( 60480 + 12096 \beta_{2} ) q^{84} + ( -1374000 + 8250 \beta_{1} + 8250 \beta_{2} + 16500 \beta_{3} ) q^{85} + ( 3975296 - 3965352 \beta_{1} - 19888 \beta_{2} - 9944 \beta_{3} ) q^{86} + ( -3251556 \beta_{1} + 5454 \beta_{2} - 5454 \beta_{3} ) q^{87} + ( 1357312 \beta_{1} + 11776 \beta_{2} - 11776 \beta_{3} ) q^{88} + ( 2924346 - 2935918 \beta_{1} + 23144 \beta_{2} + 11572 \beta_{3} ) q^{89} -729000 q^{90} + ( -10828237 + 5545288 \beta_{1} + 25781 \beta_{2} + 22239 \beta_{3} ) q^{91} + ( -1471296 - 3520 \beta_{1} - 3520 \beta_{2} - 7040 \beta_{3} ) q^{92} + ( 4612086 - 4614705 \beta_{1} + 5238 \beta_{2} + 2619 \beta_{3} ) q^{93} + ( -6893960 \beta_{1} + 4744 \beta_{2} - 4744 \beta_{3} ) q^{94} + ( 2799125 \beta_{1} - 11750 \beta_{2} + 11750 \beta_{3} ) q^{95} + ( -884736 + 884736 \beta_{1} ) q^{96} + ( -5421856 + 27922 \beta_{1} + 27922 \beta_{2} + 55844 \beta_{3} ) q^{97} + ( 4312 + 6562472 \beta_{1} - 4312 \beta_{2} - 4312 \beta_{3} ) q^{98} + ( -1966113 + 16767 \beta_{1} + 16767 \beta_{2} + 33534 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{2} - 54q^{3} - 128q^{4} + 250q^{5} + 864q^{6} - 77q^{7} + 2048q^{8} - 1458q^{9} + O(q^{10})$$ $$4q - 16q^{2} - 54q^{3} - 128q^{4} + 250q^{5} + 864q^{6} - 77q^{7} + 2048q^{8} - 1458q^{9} + 2000q^{10} + 5325q^{11} - 3456q^{12} - 13720q^{13} + 1232q^{14} - 13500q^{15} - 8192q^{16} - 21786q^{17} - 11664q^{18} - 44692q^{19} - 32000q^{20} - 2079q^{21} - 85200q^{22} + 46143q^{23} - 27648q^{24} - 31250q^{25} + 54880q^{26} + 78732q^{27} - 4928q^{28} + 481308q^{29} + 54000q^{30} + 341927q^{31} - 65536q^{32} + 143775q^{33} + 348576q^{34} - 19250q^{35} + 186624q^{36} - 258856q^{37} - 357536q^{38} + 185220q^{39} + 128000q^{40} - 641958q^{41} - 16632q^{42} - 1980190q^{43} + 340800q^{44} + 182250q^{45} + 369144q^{46} - 1722897q^{47} + 442368q^{48} + 1641157q^{49} + 500000q^{50} - 588222q^{51} + 439040q^{52} - 2994681q^{53} - 314928q^{54} + 1331250q^{55} - 39424q^{56} + 2413368q^{57} - 1925232q^{58} + 858450q^{59} + 432000q^{60} + 3920558q^{61} - 5470832q^{62} + 112266q^{63} + 1048576q^{64} - 857500q^{65} + 1150200q^{66} + 4109921q^{67} - 1394304q^{68} - 2491722q^{69} + 77000q^{70} - 12572220q^{71} - 746496q^{72} - 4115065q^{73} - 2070848q^{74} - 843750q^{75} + 5720576q^{76} + 16411353q^{77} - 2963520q^{78} + 1753115q^{79} + 1024000q^{80} - 1062882q^{81} + 2567832q^{82} + 31085664q^{83} + 266112q^{84} - 5446500q^{85} + 7920760q^{86} - 6497658q^{87} + 2726400q^{88} + 5883408q^{89} - 2916000q^{90} - 32148571q^{91} - 5906304q^{92} + 9232029q^{93} - 13783176q^{94} + 5586500q^{95} - 1769472q^{96} - 21519892q^{97} + 13129256q^{98} - 7763850q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5593 x^{2} + 5592 x + 31270464$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 5593 \nu^{2} - 5593 \nu + 31270464$$$$)/31276056$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} + 11185 \nu + 5592$$$$)/5592$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 5593 \nu + 11185$$$$)/5593$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + 16778 \beta_{1} - 16779$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$11186 \beta_{3} + 5593 \beta_{2} + 5593 \beta_{1} - 33555$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −37.1407 − 64.3295i 37.6407 + 65.1956i −37.1407 + 64.3295i 37.6407 − 65.1956i
−4.00000 + 6.92820i −13.5000 23.3827i −32.0000 55.4256i 62.5000 108.253i 216.000 −804.454 + 419.996i 512.000 −364.500 + 631.333i 500.000 + 866.025i
121.2 −4.00000 + 6.92820i −13.5000 23.3827i −32.0000 55.4256i 62.5000 108.253i 216.000 765.954 486.680i 512.000 −364.500 + 631.333i 500.000 + 866.025i
151.1 −4.00000 6.92820i −13.5000 + 23.3827i −32.0000 + 55.4256i 62.5000 + 108.253i 216.000 −804.454 419.996i 512.000 −364.500 631.333i 500.000 866.025i
151.2 −4.00000 6.92820i −13.5000 + 23.3827i −32.0000 + 55.4256i 62.5000 + 108.253i 216.000 765.954 + 486.680i 512.000 −364.500 631.333i 500.000 866.025i
 $$n$$: 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 210.8.i.b 4
7.c even 3 1 inner 210.8.i.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.8.i.b 4 1.a even 1 1 trivial
210.8.i.b 4 7.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} - 5325 T_{11}^{3} + 47891421 T_{11}^{2} + 104028113700 T_{11} +$$$$38\!\cdots\!16$$ acting on $$S_{8}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 64 + 8 T + T^{2} )^{2}$$
$3$ $$( 729 + 27 T + T^{2} )^{2}$$
$5$ $$( 15625 - 125 T + T^{2} )^{2}$$
$7$ $$678223072849 + 63412811 T - 817614 T^{2} + 77 T^{3} + T^{4}$$
$11$ $$381647325353616 + 104028113700 T + 47891421 T^{2} - 5325 T^{3} + T^{4}$$
$13$ $$( -94733909 + 6860 T + T^{2} )^{2}$$
$17$ $$10116561700454400 - 2191260280320 T + 575210916 T^{2} + 21786 T^{3} + T^{4}$$
$19$ $$2983959125143129 + 2441328521884 T + 1942749237 T^{2} + 44692 T^{3} + T^{4}$$
$23$ $$144434282594551236 - 17536421326158 T + 1749131343 T^{2} - 46143 T^{3} + T^{4}$$
$29$ $$( 12424911408 - 240654 T + T^{2} )^{2}$$
$31$ $$82\!\cdots\!00$$$$- 9832097553266270 T + 88159112319 T^{2} - 341927 T^{3} + T^{4}$$
$37$ $$23\!\cdots\!09$$$$+ 3994339660102168 T + 51575688633 T^{2} + 258856 T^{3} + T^{4}$$
$41$ $$( 883217088 + 320979 T + T^{2} )^{2}$$
$43$ $$( 167309324824 + 990095 T + T^{2} )^{2}$$
$47$ $$52\!\cdots\!00$$$$+ 1248057862170194790 T + 2243979136539 T^{2} + 1722897 T^{3} + T^{4}$$
$53$ $$49\!\cdots\!24$$$$+ 6669062677748535408 T + 6741144982593 T^{2} + 2994681 T^{3} + T^{4}$$
$59$ $$23\!\cdots\!16$$$$+ 4164172960516981200 T + 5587740713196 T^{2} - 858450 T^{3} + T^{4}$$
$61$ $$13\!\cdots\!00$$$$- 14584067235347242880 T + 11650879232004 T^{2} - 3920558 T^{3} + T^{4}$$
$67$ $$16\!\cdots\!00$$$$- 16864421188932515360 T + 12788106258081 T^{2} - 4109921 T^{3} + T^{4}$$
$71$ $$( 6278281186464 + 6286110 T + T^{2} )^{2}$$
$73$ $$12\!\cdots\!36$$$$+ 46797194075350390 T + 16922387790219 T^{2} + 4115065 T^{3} + T^{4}$$
$79$ $$17\!\cdots\!76$$$$+ 23110699233244746760 T + 16256060936049 T^{2} - 1753115 T^{3} + T^{4}$$
$83$ $$( 59899934002752 - 15542832 T + T^{2} )^{2}$$
$89$ $$36\!\cdots\!00$$$$- 11259896007487780800 T + 32700650639364 T^{2} - 5883408 T^{3} + T^{4}$$
$97$ $$( -10295270196512 + 10759946 T + T^{2} )^{2}$$