# Properties

 Label 430.2.e.f Level 430 Weight 2 Character orbit 430.e Analytic conductor 3.434 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.27870912.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 7 x^{4} + 2 x^{3} + 38 x^{2} - 12 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 7 \nu^{4} + 49 \nu^{3} - 38 \nu^{2} + 12 \nu - 84$$$$)/254$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{5} + 49 \nu^{4} - 89 \nu^{3} + 266 \nu^{2} - 84 \nu + 1096$$$$)/254$$ $$\beta_{4}$$ $$=$$ $$($$$$21 \nu^{5} - 20 \nu^{4} + 140 \nu^{3} + 91 \nu^{2} + 760 \nu + 14$$$$)/254$$ $$\beta_{5}$$ $$=$$ $$($$$$-85 \nu^{5} + 87 \nu^{4} - 609 \nu^{3} - 72 \nu^{2} - 3306 \nu + 1044$$$$)/254$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 4 \beta_{4} + \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 7 \beta_{2} - 2$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{5} - 26 \beta_{4} + 7 \beta_{3} - 11 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-11 \beta_{5} - 30 \beta_{4} - 51 \beta_{2} - 51 \beta_{1} + 30$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/430\mathbb{Z}\right)^\times$$.

 $$n$$ $$87$$ $$261$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1
 1.42789 − 2.47317i 0.160819 − 0.278546i −1.08870 + 1.88569i 1.42789 + 2.47317i 0.160819 + 0.278546i −1.08870 − 1.88569i
1.00000 −0.927886 + 1.60715i 1.00000 −0.500000 + 0.866025i −0.927886 + 1.60715i 1.00000 + 1.73205i 1.00000 −0.221946 0.384421i −0.500000 + 0.866025i
221.2 1.00000 0.339181 0.587479i 1.00000 −0.500000 + 0.866025i 0.339181 0.587479i 1.00000 + 1.73205i 1.00000 1.26991 + 2.19955i −0.500000 + 0.866025i
221.3 1.00000 1.58870 2.75172i 1.00000 −0.500000 + 0.866025i 1.58870 2.75172i 1.00000 + 1.73205i 1.00000 −3.54797 6.14526i −0.500000 + 0.866025i
251.1 1.00000 −0.927886 1.60715i 1.00000 −0.500000 0.866025i −0.927886 1.60715i 1.00000 1.73205i 1.00000 −0.221946 + 0.384421i −0.500000 0.866025i
251.2 1.00000 0.339181 + 0.587479i 1.00000 −0.500000 0.866025i 0.339181 + 0.587479i 1.00000 1.73205i 1.00000 1.26991 2.19955i −0.500000 0.866025i
251.3 1.00000 1.58870 + 2.75172i 1.00000 −0.500000 0.866025i 1.58870 + 2.75172i 1.00000 1.73205i 1.00000 −3.54797 + 6.14526i −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.e.f 6
43.c even 3 1 inner 430.2.e.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.e.f 6 1.a even 1 1 trivial
430.2.e.f 6 43.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 2 T_{3}^{5} + 9 T_{3}^{4} + 2 T_{3}^{3} + 33 T_{3}^{2} - 20 T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{6}$$
$3$ $$1 - 2 T + 8 T^{3} - 12 T^{4} - 2 T^{5} + 34 T^{6} - 6 T^{7} - 108 T^{8} + 216 T^{9} - 486 T^{11} + 729 T^{12}$$
$5$ $$( 1 + T + T^{2} )^{3}$$
$7$ $$( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{3}$$
$11$ $$( 1 + T + 27 T^{2} + 20 T^{3} + 297 T^{4} + 121 T^{5} + 1331 T^{6} )^{2}$$
$13$ $$1 - 25 T^{2} + 32 T^{3} + 300 T^{4} - 400 T^{5} - 3475 T^{6} - 5200 T^{7} + 50700 T^{8} + 70304 T^{9} - 714025 T^{10} + 4826809 T^{12}$$
$17$ $$1 + 5 T - 18 T^{2} - 93 T^{3} + 348 T^{4} + 757 T^{5} - 5116 T^{6} + 12869 T^{7} + 100572 T^{8} - 456909 T^{9} - 1503378 T^{10} + 7099285 T^{11} + 24137569 T^{12}$$
$19$ $$1 + 7 T + 30 T^{2} + 117 T^{3} - 56 T^{4} - 2375 T^{5} - 10766 T^{6} - 45125 T^{7} - 20216 T^{8} + 802503 T^{9} + 3909630 T^{10} + 17332693 T^{11} + 47045881 T^{12}$$
$23$ $$1 - 8 T - T^{2} + 120 T^{3} + 170 T^{4} - 1112 T^{5} - 4297 T^{6} - 25576 T^{7} + 89930 T^{8} + 1460040 T^{9} - 279841 T^{10} - 51490744 T^{11} + 148035889 T^{12}$$
$29$ $$1 - 10 T + 34 T^{2} - 76 T^{3} - 478 T^{4} + 10598 T^{5} - 76382 T^{6} + 307342 T^{7} - 401998 T^{8} - 1853564 T^{9} + 24047554 T^{10} - 205111490 T^{11} + 594823321 T^{12}$$
$31$ $$1 + 6 T - 55 T^{2} - 190 T^{3} + 3252 T^{4} + 5402 T^{5} - 96301 T^{6} + 167462 T^{7} + 3125172 T^{8} - 5660290 T^{9} - 50793655 T^{10} + 171774906 T^{11} + 887503681 T^{12}$$
$37$ $$1 + 10 T - 13 T^{2} - 102 T^{3} + 2428 T^{4} - 322 T^{5} - 128591 T^{6} - 11914 T^{7} + 3323932 T^{8} - 5166606 T^{9} - 24364093 T^{10} + 693439570 T^{11} + 2565726409 T^{12}$$
$41$ $$( 1 + 5 T + 30 T^{2} + 241 T^{3} + 1230 T^{4} + 8405 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$1 + 7 T + 14 T^{2} - 167 T^{3} + 602 T^{4} + 12943 T^{5} + 79507 T^{6}$$
$47$ $$( 1 - 4 T + 130 T^{2} - 350 T^{3} + 6110 T^{4} - 8836 T^{5} + 103823 T^{6} )^{2}$$
$53$ $$1 - 8 T - 51 T^{2} + 360 T^{3} + 2010 T^{4} + 1448 T^{5} - 173347 T^{6} + 76744 T^{7} + 5646090 T^{8} + 53595720 T^{9} - 402414531 T^{10} - 3345563944 T^{11} + 22164361129 T^{12}$$
$59$ $$( 1 - 3 T + 97 T^{2} + 2 T^{3} + 5723 T^{4} - 10443 T^{5} + 205379 T^{6} )^{2}$$
$61$ $$1 + 4 T + 17 T^{2} + 1516 T^{3} + 3102 T^{4} + 16132 T^{5} + 971965 T^{6} + 984052 T^{7} + 11542542 T^{8} + 344103196 T^{9} + 235379297 T^{10} + 3378385204 T^{11} + 51520374361 T^{12}$$
$67$ $$1 - 9 T - 21 T^{2} - 72 T^{3} - 237 T^{4} + 39465 T^{5} - 280222 T^{6} + 2644155 T^{7} - 1063893 T^{8} - 21654936 T^{9} - 423173541 T^{10} - 12151125963 T^{11} + 90458382169 T^{12}$$
$71$ $$1 - 4 T - 155 T^{2} + 580 T^{3} + 14572 T^{4} - 31108 T^{5} - 1041521 T^{6} - 2208668 T^{7} + 73457452 T^{8} + 207588380 T^{9} - 3938810555 T^{10} - 7216917404 T^{11} + 128100283921 T^{12}$$
$73$ $$1 - 21 T + 132 T^{2} - 473 T^{3} + 9756 T^{4} - 93987 T^{5} + 505896 T^{6} - 6861051 T^{7} + 51989724 T^{8} - 184005041 T^{9} + 3748567812 T^{10} - 43534503453 T^{11} + 151334226289 T^{12}$$
$79$ $$1 - 4 T - 125 T^{2} + 444 T^{3} + 6726 T^{4} - 6916 T^{5} - 430361 T^{6} - 546364 T^{7} + 41976966 T^{8} + 218909316 T^{9} - 4868760125 T^{10} - 12308225596 T^{11} + 243087455521 T^{12}$$
$83$ $$1 - 10 T - 112 T^{2} + 816 T^{3} + 12668 T^{4} - 28186 T^{5} - 1120654 T^{6} - 2339438 T^{7} + 87269852 T^{8} + 466578192 T^{9} - 5315331952 T^{10} - 39390406430 T^{11} + 326940373369 T^{12}$$
$89$ $$1 - 7 T - 169 T^{2} + 792 T^{3} + 20009 T^{4} - 36481 T^{5} - 1864546 T^{6} - 3246809 T^{7} + 158491289 T^{8} + 558335448 T^{9} - 10603438729 T^{10} - 39088416143 T^{11} + 496981290961 T^{12}$$
$97$ $$( 1 - 5 T + 155 T^{2} - 538 T^{3} + 15035 T^{4} - 47045 T^{5} + 912673 T^{6} )^{2}$$