# Properties

 Label 429.2.bb.b Level $429$ Weight $2$ Character orbit 429.bb Analytic conductor $3.426$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.bb (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 14q^{3} + 20q^{4} - 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 14q^{3} + 20q^{4} - 14q^{9} + 60q^{12} - 3q^{13} - 42q^{14} - 16q^{16} - 6q^{17} - 40q^{22} + 12q^{23} + 4q^{25} - 24q^{26} + 14q^{27} - 22q^{29} - 10q^{30} - 10q^{35} + 20q^{36} - 26q^{38} + 3q^{39} - 18q^{42} + 6q^{48} - 14q^{49} - 24q^{51} + 39q^{52} + 10q^{53} - 64q^{55} - 20q^{56} + 2q^{61} - 8q^{62} + 24q^{64} - 58q^{65} + 30q^{66} + 18q^{68} - 2q^{69} - 86q^{74} - 14q^{75} + 52q^{77} + 14q^{78} - 24q^{79} - 14q^{81} + 62q^{82} - 28q^{87} - 46q^{88} - 10q^{90} - 19q^{91} + 58q^{92} + 86q^{94} + 156q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 −1.59022 2.18875i −0.309017 0.951057i −1.64379 + 5.05905i 0.714239 0.983066i −1.59022 + 2.18875i 1.83379 + 0.595836i 8.54091 2.77511i −0.809017 + 0.587785i −3.28748
25.2 −1.28832 1.77322i −0.309017 0.951057i −0.866514 + 2.66686i −1.37901 + 1.89804i −1.28832 + 1.77322i −1.10854 0.360186i 1.67618 0.544625i −0.809017 + 0.587785i 5.14227
25.3 −1.20071 1.65263i −0.309017 0.951057i −0.671457 + 2.06653i −0.326611 + 0.449541i −1.20071 + 1.65263i −1.66332 0.540445i 0.335871 0.109131i −0.809017 + 0.587785i 1.13509
25.4 −0.812362 1.11812i −0.309017 0.951057i 0.0277724 0.0854748i 2.36802 3.25930i −0.812362 + 1.11812i −1.14719 0.372746i −2.74699 + 0.892552i −0.809017 + 0.587785i −5.56798
25.5 −0.774047 1.06538i −0.309017 0.951057i 0.0821387 0.252797i −1.70947 + 2.35289i −0.774047 + 1.06538i 4.34559 + 1.41197i −2.83777 + 0.922049i −0.809017 + 0.587785i 3.82995
25.6 −0.386931 0.532564i −0.309017 0.951057i 0.484125 1.48998i −0.663105 + 0.912685i −0.386931 + 0.532564i −3.82419 1.24256i −2.23297 + 0.725535i −0.809017 + 0.587785i 0.742639
25.7 −0.0523187 0.0720105i −0.309017 0.951057i 0.615586 1.89458i −1.59536 + 2.19582i −0.0523187 + 0.0720105i −0.897995 0.291776i −0.337943 + 0.109804i −0.809017 + 0.587785i 0.241589
25.8 0.0523187 + 0.0720105i −0.309017 0.951057i 0.615586 1.89458i 1.59536 2.19582i 0.0523187 0.0720105i 0.897995 + 0.291776i 0.337943 0.109804i −0.809017 + 0.587785i 0.241589
25.9 0.386931 + 0.532564i −0.309017 0.951057i 0.484125 1.48998i 0.663105 0.912685i 0.386931 0.532564i 3.82419 + 1.24256i 2.23297 0.725535i −0.809017 + 0.587785i 0.742639
25.10 0.774047 + 1.06538i −0.309017 0.951057i 0.0821387 0.252797i 1.70947 2.35289i 0.774047 1.06538i −4.34559 1.41197i 2.83777 0.922049i −0.809017 + 0.587785i 3.82995
25.11 0.812362 + 1.11812i −0.309017 0.951057i 0.0277724 0.0854748i −2.36802 + 3.25930i 0.812362 1.11812i 1.14719 + 0.372746i 2.74699 0.892552i −0.809017 + 0.587785i −5.56798
25.12 1.20071 + 1.65263i −0.309017 0.951057i −0.671457 + 2.06653i 0.326611 0.449541i 1.20071 1.65263i 1.66332 + 0.540445i −0.335871 + 0.109131i −0.809017 + 0.587785i 1.13509
25.13 1.28832 + 1.77322i −0.309017 0.951057i −0.866514 + 2.66686i 1.37901 1.89804i 1.28832 1.77322i 1.10854 + 0.360186i −1.67618 + 0.544625i −0.809017 + 0.587785i 5.14227
25.14 1.59022 + 2.18875i −0.309017 0.951057i −1.64379 + 5.05905i −0.714239 + 0.983066i 1.59022 2.18875i −1.83379 0.595836i −8.54091 + 2.77511i −0.809017 + 0.587785i −3.28748
64.1 −2.52137 0.819244i 0.809017 0.587785i 4.06813 + 2.95567i 1.72603 0.560820i −2.52137 + 0.819244i 2.49852 3.43892i −4.71928 6.49553i 0.309017 0.951057i −4.81141
64.2 −2.29028 0.744157i 0.809017 0.587785i 3.07358 + 2.23309i 0.810545 0.263362i −2.29028 + 0.744157i −1.77414 + 2.44189i −2.54665 3.50517i 0.309017 0.951057i −2.05236
64.3 −1.97843 0.642831i 0.809017 0.587785i 1.88292 + 1.36802i −3.50059 + 1.13741i −1.97843 + 0.642831i 2.06240 2.83865i −0.400343 0.551025i 0.309017 0.951057i 7.65683
64.4 −1.54582 0.502269i 0.809017 0.587785i 0.519265 + 0.377268i −1.17463 + 0.381659i −1.54582 + 0.502269i 0.262950 0.361920i 1.29754 + 1.78591i 0.309017 0.951057i 2.00746
64.5 −1.15287 0.374592i 0.809017 0.587785i −0.429233 0.311856i −0.0540551 + 0.0175636i −1.15287 + 0.374592i −1.35564 + 1.86588i 1.80306 + 2.48171i 0.309017 0.951057i 0.0688979
64.6 −0.975059 0.316816i 0.809017 0.587785i −0.767667 0.557743i 3.70008 1.20223i −0.975059 + 0.316816i 1.98998 2.73897i 1.77706 + 2.44591i 0.309017 0.951057i −3.98868
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.b even 2 1 inner
143.n even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bb.b 56
11.c even 5 1 inner 429.2.bb.b 56
13.b even 2 1 inner 429.2.bb.b 56
143.n even 10 1 inner 429.2.bb.b 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bb.b 56 1.a even 1 1 trivial
429.2.bb.b 56 11.c even 5 1 inner
429.2.bb.b 56 13.b even 2 1 inner
429.2.bb.b 56 143.n even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{56} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.