# Properties

 Label 429.2.b.a Level $429$ Weight $2$ Character orbit 429.b Analytic conductor $3.426$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 13 x^{8} + 54 x^{6} + 74 x^{4} + 21 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 13 x^{8} + 54 x^{6} + 74 x^{4} + 21 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 7 \nu^{4} + 9 \nu^{2} + 1$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{8} - 12 \nu^{6} - 44 \nu^{4} - 46 \nu^{2} - 3$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{9} + 13 \nu^{7} + 53 \nu^{5} + 67 \nu^{3} + 12 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{8} + 13 \nu^{6} + 53 \nu^{4} + 67 \nu^{2} + 10$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{9} - 13 \nu^{7} - 53 \nu^{5} - 65 \nu^{3} - 2 \nu$$$$)/2$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{9} + 13 \nu^{7} + 55 \nu^{5} + 81 \nu^{3} + 30 \nu$$$$)/2$$ $$\beta_{9}$$ $$=$$ $$($$$$-3 \nu^{9} - 38 \nu^{7} - 152 \nu^{5} - 192 \nu^{3} - 35 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{5} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{4} - \beta_{3} - 6 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$\beta_{8} - 7 \beta_{7} - 8 \beta_{5} + 26 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{6} - 7 \beta_{4} + 9 \beta_{3} + 33 \beta_{2} - 79$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{9} - 7 \beta_{8} + 40 \beta_{7} + 53 \beta_{5} - 138 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$40 \beta_{6} + 38 \beta_{4} - 64 \beta_{3} - 178 \beta_{2} + 423$$ $$\nu^{9}$$ $$=$$ $$-26 \beta_{9} + 38 \beta_{8} - 216 \beta_{7} - 330 \beta_{5} + 739 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
298.1
 − 2.35969i − 2.25835i − 1.40449i − 0.547285i − 0.244130i 0.244130i 0.547285i 1.40449i 2.25835i 2.35969i
2.35969i −1.00000 −3.56813 1.80373i 2.35969i 4.78347i 3.70030i 1.00000 −4.25625
298.2 2.25835i −1.00000 −3.10015 3.76912i 2.25835i 0.701152i 2.48452i 1.00000 8.51198
298.3 1.40449i −1.00000 0.0273977 3.56736i 1.40449i 0.116494i 2.84747i 1.00000 −5.01033
298.4 0.547285i −1.00000 1.70048 0.955178i 0.547285i 4.37449i 2.02522i 1.00000 0.522755
298.5 0.244130i −1.00000 1.94040 0.949685i 0.244130i 2.34031i 0.961969i 1.00000 0.231846
298.6 0.244130i −1.00000 1.94040 0.949685i 0.244130i 2.34031i 0.961969i 1.00000 0.231846
298.7 0.547285i −1.00000 1.70048 0.955178i 0.547285i 4.37449i 2.02522i 1.00000 0.522755
298.8 1.40449i −1.00000 0.0273977 3.56736i 1.40449i 0.116494i 2.84747i 1.00000 −5.01033
298.9 2.25835i −1.00000 −3.10015 3.76912i 2.25835i 0.701152i 2.48452i 1.00000 8.51198
298.10 2.35969i −1.00000 −3.56813 1.80373i 2.35969i 4.78347i 3.70030i 1.00000 −4.25625
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 298.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.b.a 10
3.b odd 2 1 1287.2.b.a 10
13.b even 2 1 inner 429.2.b.a 10
13.d odd 4 1 5577.2.a.q 5
13.d odd 4 1 5577.2.a.t 5
39.d odd 2 1 1287.2.b.a 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.b.a 10 1.a even 1 1 trivial
429.2.b.a 10 13.b even 2 1 inner
1287.2.b.a 10 3.b odd 2 1
1287.2.b.a 10 39.d odd 2 1
5577.2.a.q 5 13.d odd 4 1
5577.2.a.t 5 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 13 T_{2}^{8} + 54 T_{2}^{6} + 74 T_{2}^{4} + 21 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 21 T^{2} + 74 T^{4} + 54 T^{6} + 13 T^{8} + T^{10}$$
$3$ $$( 1 + T )^{10}$$
$5$ $$484 + 1288 T^{2} + 1100 T^{4} + 324 T^{6} + 32 T^{8} + T^{10}$$
$7$ $$16 + 1216 T^{2} + 2736 T^{4} + 692 T^{6} + 48 T^{8} + T^{10}$$
$11$ $$( 1 + T^{2} )^{5}$$
$13$ $$371293 - 342732 T + 125229 T^{2} - 10816 T^{3} - 9126 T^{4} + 4248 T^{5} - 702 T^{6} - 64 T^{7} + 57 T^{8} - 12 T^{9} + T^{10}$$
$17$ $$( -186 + 102 T + 148 T^{2} - 2 T^{3} - 10 T^{4} + T^{5} )^{2}$$
$19$ $$4443664 + 1581008 T^{2} + 170544 T^{4} + 7676 T^{6} + 148 T^{8} + T^{10}$$
$23$ $$( 1276 + 1104 T + 112 T^{2} - 68 T^{3} - 4 T^{4} + T^{5} )^{2}$$
$29$ $$( 2214 + 1854 T + 90 T^{2} - 86 T^{3} - 4 T^{4} + T^{5} )^{2}$$
$31$ $$10797796 + 3484968 T^{2} + 349916 T^{4} + 13452 T^{6} + 208 T^{8} + T^{10}$$
$37$ $$1827904 + 3115008 T^{2} + 333152 T^{4} + 12672 T^{6} + 196 T^{8} + T^{10}$$
$41$ $$4963984 + 2291696 T^{2} + 289104 T^{4} + 12860 T^{6} + 208 T^{8} + T^{10}$$
$43$ $$( 2194 - 746 T - 648 T^{2} - 46 T^{3} + 12 T^{4} + T^{5} )^{2}$$
$47$ $$107584 + 175296 T^{2} + 41504 T^{4} + 3648 T^{6} + 124 T^{8} + T^{10}$$
$53$ $$( 848 - 912 T - 680 T^{2} - 108 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$59$ $$1024 + 13568 T^{2} + 15744 T^{4} + 4448 T^{6} + 196 T^{8} + T^{10}$$
$61$ $$( 6024 - 8856 T + 2428 T^{2} - 72 T^{3} - 18 T^{4} + T^{5} )^{2}$$
$67$ $$1150159396 + 163319224 T^{2} + 6258468 T^{4} + 86132 T^{6} + 492 T^{8} + T^{10}$$
$71$ $$2585664 + 1686144 T^{2} + 333952 T^{4} + 19200 T^{6} + 272 T^{8} + T^{10}$$
$73$ $$5438224 + 1629248 T^{2} + 166752 T^{4} + 7508 T^{6} + 148 T^{8} + T^{10}$$
$79$ $$( -6014 - 1442 T + 1320 T^{2} - 54 T^{3} - 18 T^{4} + T^{5} )^{2}$$
$83$ $$1673791744 + 232027136 T^{2} + 7880896 T^{4} + 102208 T^{6} + 548 T^{8} + T^{10}$$
$89$ $$3978834084 + 520640544 T^{2} + 14550856 T^{4} + 154556 T^{6} + 672 T^{8} + T^{10}$$
$97$ $$7929546304 + 674364800 T^{2} + 16553088 T^{4} + 159632 T^{6} + 664 T^{8} + T^{10}$$