# Properties

 Label 430.2.e.e 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: 6.0.954288.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - \nu^{3} + 9 \nu^{2} - 21 \nu - 9$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - \nu^{3} - 18 \nu^{2} + 33 \nu - 9$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 36$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} + 7 \nu^{3} + 9 \nu^{2} + 12 \nu + 9$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 18 \nu^{2} + 3 \nu - 72$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{5} + 7 \beta_{4} - 11 \beta_{3} + \beta_{2} - \beta_{1} + 7$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 20$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$7 \beta_{5} - 2 \beta_{4} - 20 \beta_{3} + \beta_{2} - 10 \beta_{1} + 43$$$$)/3$$

## Character values

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

 $$n$$ $$87$$ $$261$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{3}$$

## 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
 1.71903 + 0.211943i −1.62241 + 0.606458i 0.403374 − 1.68443i 1.71903 − 0.211943i −1.62241 − 0.606458i 0.403374 + 1.68443i
1.00000 −1.04307 + 1.80664i 1.00000 0.500000 0.866025i −1.04307 + 1.80664i 0.324030 + 0.561237i 1.00000 −0.675970 1.17081i 0.500000 0.866025i
221.2 1.00000 0.285997 0.495361i 1.00000 0.500000 0.866025i 0.285997 0.495361i 2.33641 + 4.04678i 1.00000 1.33641 + 2.31473i 0.500000 0.866025i
221.3 1.00000 1.25707 2.17731i 1.00000 0.500000 0.866025i 1.25707 2.17731i −0.660442 1.14392i 1.00000 −1.66044 2.87597i 0.500000 0.866025i
251.1 1.00000 −1.04307 1.80664i 1.00000 0.500000 + 0.866025i −1.04307 1.80664i 0.324030 0.561237i 1.00000 −0.675970 + 1.17081i 0.500000 + 0.866025i
251.2 1.00000 0.285997 + 0.495361i 1.00000 0.500000 + 0.866025i 0.285997 + 0.495361i 2.33641 4.04678i 1.00000 1.33641 2.31473i 0.500000 + 0.866025i
251.3 1.00000 1.25707 + 2.17731i 1.00000 0.500000 + 0.866025i 1.25707 + 2.17731i −0.660442 + 1.14392i 1.00000 −1.66044 + 2.87597i 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.e 6
43.c even 3 1 inner 430.2.e.e 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - T_{3}^{5} + 6 T_{3}^{4} - T_{3}^{3} + 28 T_{3}^{2} - 15 T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{6}$$
$3$ $$( 1 - 2 T + T^{2} + 3 T^{4} - 18 T^{5} + 27 T^{6} )( 1 + T - 2 T^{2} - 3 T^{3} - 6 T^{4} + 9 T^{5} + 27 T^{6} )$$
$5$ $$( 1 - T + T^{2} )^{3}$$
$7$ $$1 - 4 T - T^{2} + 36 T^{3} - 38 T^{4} - 128 T^{5} + 583 T^{6} - 896 T^{7} - 1862 T^{8} + 12348 T^{9} - 2401 T^{10} - 67228 T^{11} + 117649 T^{12}$$
$11$ $$( 1 + 4 T + 33 T^{2} + 82 T^{3} + 363 T^{4} + 484 T^{5} + 1331 T^{6} )^{2}$$
$13$ $$1 - 6 T + 21 T^{2} - 70 T^{3} - 30 T^{4} + 1176 T^{5} - 4515 T^{6} + 15288 T^{7} - 5070 T^{8} - 153790 T^{9} + 599781 T^{10} - 2227758 T^{11} + 4826809 T^{12}$$
$17$ $$( 1 - 17 T^{2} + 289 T^{4} )^{3}$$
$19$ $$1 + 2 T - 49 T^{2} - 42 T^{3} + 1654 T^{4} + 616 T^{5} - 35621 T^{6} + 11704 T^{7} + 597094 T^{8} - 288078 T^{9} - 6385729 T^{10} + 4952198 T^{11} + 47045881 T^{12}$$
$23$ $$1 + 8 T - 17 T^{2} - 64 T^{3} + 1850 T^{4} + 3260 T^{5} - 33689 T^{6} + 74980 T^{7} + 978650 T^{8} - 778688 T^{9} - 4757297 T^{10} + 51490744 T^{11} + 148035889 T^{12}$$
$29$ $$1 + 7 T + 13 T^{2} + 46 T^{3} - 469 T^{4} - 5021 T^{5} - 19406 T^{6} - 145609 T^{7} - 394429 T^{8} + 1121894 T^{9} + 9194653 T^{10} + 143578043 T^{11} + 594823321 T^{12}$$
$31$ $$1 - 8 T - 39 T^{2} + 156 T^{3} + 3496 T^{4} - 7070 T^{5} - 90701 T^{6} - 219170 T^{7} + 3359656 T^{8} + 4647396 T^{9} - 36017319 T^{10} - 229033208 T^{11} + 887503681 T^{12}$$
$37$ $$1 - 16 T + 65 T^{2} - 444 T^{3} + 8518 T^{4} - 45206 T^{5} + 106681 T^{6} - 1672622 T^{7} + 11661142 T^{8} - 22489932 T^{9} + 121820465 T^{10} - 1109503312 T^{11} + 2565726409 T^{12}$$
$41$ $$( 1 + T + 78 T^{2} + 109 T^{3} + 3198 T^{4} + 1681 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$1 - 5 T - 22 T^{2} + 163 T^{3} - 946 T^{4} - 9245 T^{5} + 79507 T^{6}$$
$47$ $$( 1 + 9 T + 72 T^{2} + 297 T^{3} + 3384 T^{4} + 19881 T^{5} + 103823 T^{6} )^{2}$$
$53$ $$1 - 8 T - 71 T^{2} + 328 T^{3} + 5438 T^{4} - 3032 T^{5} - 352331 T^{6} - 160696 T^{7} + 15275342 T^{8} + 48831656 T^{9} - 560224151 T^{10} - 3345563944 T^{11} + 22164361129 T^{12}$$
$59$ $$( 1 + 12 T + 153 T^{2} + 984 T^{3} + 9027 T^{4} + 41772 T^{5} + 205379 T^{6} )^{2}$$
$61$ $$1 - 12 T - 63 T^{2} + 308 T^{3} + 13470 T^{4} - 30576 T^{5} - 746835 T^{6} - 1865136 T^{7} + 50121870 T^{8} + 69910148 T^{9} - 872287983 T^{10} - 10135155612 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + 7 T - 27 T^{2} + 474 T^{3} + 481 T^{4} - 37661 T^{5} - 4586 T^{6} - 2523287 T^{7} + 2159209 T^{8} + 142561662 T^{9} - 544080267 T^{10} + 9450875749 T^{11} + 90458382169 T^{12}$$
$71$ $$1 + 12 T - 99 T^{2} - 528 T^{3} + 21132 T^{4} + 64866 T^{5} - 1237277 T^{6} + 4605486 T^{7} + 106526412 T^{8} - 188977008 T^{9} - 2515756419 T^{10} + 21650752212 T^{11} + 128100283921 T^{12}$$
$73$ $$1 + 14 T + 11 T^{2} - 702 T^{3} - 2324 T^{4} + 976 T^{5} - 55559 T^{6} + 71248 T^{7} - 12384596 T^{8} - 273089934 T^{9} + 312380651 T^{10} + 29023002302 T^{11} + 151334226289 T^{12}$$
$79$ $$1 + 4 T - 45 T^{2} + 132 T^{3} - 1322 T^{4} - 10172 T^{5} + 595399 T^{6} - 803588 T^{7} - 8250602 T^{8} + 65081148 T^{9} - 1752753645 T^{10} + 12308225596 T^{11} + 243087455521 T^{12}$$
$83$ $$1 - 15 T + 3 T^{2} + 1164 T^{3} - 2847 T^{4} - 49389 T^{5} + 551986 T^{6} - 4099287 T^{7} - 19612983 T^{8} + 665560068 T^{9} + 142374963 T^{10} - 59085609645 T^{11} + 326940373369 T^{12}$$
$89$ $$1 + 15 T + 75 T^{2} - 876 T^{3} - 14295 T^{4} - 47235 T^{5} + 68182 T^{6} - 4203915 T^{7} - 113230695 T^{8} - 617552844 T^{9} + 4705668075 T^{10} + 83760891735 T^{11} + 496981290961 T^{12}$$
$97$ $$( 1 + 10 T + 239 T^{2} + 1612 T^{3} + 23183 T^{4} + 94090 T^{5} + 912673 T^{6} )^{2}$$