# Properties

 Label 429.2.y.b Level $429$ Weight $2$ Character orbit 429.y Analytic conductor $3.426$ Analytic rank $0$ Dimension $176$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$176$$ Relative dimension: $$44$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$176q - 6q^{3} + 20q^{4} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$176q - 6q^{3} + 20q^{4} + 6q^{9} + 12q^{12} - 10q^{13} + 36q^{16} - 96q^{22} - 20q^{25} + 6q^{27} - 100q^{30} - 116q^{36} + 50q^{39} + 60q^{40} + 52q^{42} + 32q^{48} - 88q^{49} - 90q^{51} + 210q^{52} - 20q^{55} - 92q^{64} - 62q^{66} + 96q^{69} + 166q^{75} - 112q^{78} - 60q^{79} + 70q^{81} - 144q^{82} - 148q^{88} + 100q^{90} + 62q^{91} - 180q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1 −2.50188 + 0.812912i −0.234485 + 1.71611i 3.98057 2.89205i 0.944274 2.90618i −0.808388 4.48411i −2.69270 + 1.95636i −4.51544 + 6.21497i −2.89003 0.804801i 8.03853i
116.2 −2.50188 + 0.812912i 1.55965 0.753314i 3.98057 2.89205i 0.944274 2.90618i −3.28969 + 3.15256i 2.69270 1.95636i −4.51544 + 6.21497i 1.86504 2.34982i 8.03853i
116.3 −2.12303 + 0.689813i −1.71440 + 0.246677i 2.41336 1.75341i −0.370527 + 1.14036i 3.46954 1.70631i −1.00158 + 0.727691i −1.28990 + 1.77539i 2.87830 0.845805i 2.67661i
116.4 −2.12303 + 0.689813i −0.295173 1.70671i 2.41336 1.75341i −0.370527 + 1.14036i 1.80397 + 3.41978i 1.00158 0.727691i −1.28990 + 1.77539i −2.82575 + 1.00755i 2.67661i
116.5 −2.06588 + 0.671246i −1.48049 + 0.898973i 2.19926 1.59786i 0.803773 2.47376i 2.45508 2.85094i 3.30516 2.40134i −0.917289 + 1.26254i 1.38370 2.66184i 5.65002i
116.6 −2.06588 + 0.671246i 0.397478 1.68583i 2.19926 1.59786i 0.803773 2.47376i 0.310462 + 3.74952i −3.30516 + 2.40134i −0.917289 + 1.26254i −2.68402 1.34016i 5.65002i
116.7 −2.03433 + 0.660995i −0.778983 + 1.54699i 2.08356 1.51379i −0.766534 + 2.35915i 0.562157 3.66200i −2.59152 + 1.88285i −0.723470 + 0.995771i −1.78637 2.41016i 5.30597i
116.8 −2.03433 + 0.660995i 1.23056 1.21890i 2.08356 1.51379i −0.766534 + 2.35915i −1.69767 + 3.29305i 2.59152 1.88285i −0.723470 + 0.995771i 0.0285456 2.99986i 5.30597i
116.9 −1.39922 + 0.454633i 0.606125 + 1.62253i 0.133079 0.0966876i 0.724806 2.23072i −1.58576 1.99471i −0.197187 + 0.143265i 1.58728 2.18470i −2.26523 + 1.96691i 3.45079i
116.10 −1.39922 + 0.454633i 1.73042 + 0.0750685i 0.133079 0.0966876i 0.724806 2.23072i −2.45536 + 0.681670i 0.197187 0.143265i 1.58728 2.18470i 2.98873 + 0.259801i 3.45079i
116.11 −1.39675 + 0.453832i −0.128417 + 1.72728i 0.126919 0.0922120i −0.121706 + 0.374573i −0.604530 2.47087i 2.86530 2.08177i 1.59106 2.18990i −2.96702 0.443626i 0.578420i
116.12 −1.39675 + 0.453832i 1.60306 0.655892i 0.126919 0.0922120i −0.121706 + 0.374573i −1.94141 + 1.64364i −2.86530 + 2.08177i 1.59106 2.18990i 2.13961 2.10287i 0.578420i
116.13 −1.15602 + 0.375612i −1.73161 0.0391064i −0.422745 + 0.307142i −1.20121 + 3.69693i 2.01646 0.605206i 3.46009 2.51391i 1.80225 2.48058i 2.99694 + 0.135434i 4.72490i
116.14 −1.15602 + 0.375612i −0.572289 1.63477i −0.422745 + 0.307142i −1.20121 + 3.69693i 1.27562 + 1.67487i −3.46009 + 2.51391i 1.80225 2.48058i −2.34497 + 1.87113i 4.72490i
116.15 −0.743149 + 0.241464i −1.46911 + 0.917452i −1.12407 + 0.816683i 0.877069 2.69934i 0.870236 1.03654i −1.52431 + 1.10747i 1.55673 2.14266i 1.31656 2.69567i 2.21779i
116.16 −0.743149 + 0.241464i 0.418569 1.68071i −1.12407 + 0.816683i 0.877069 2.69934i 0.0947722 + 1.35009i 1.52431 1.10747i 1.55673 2.14266i −2.64960 1.40699i 2.21779i
116.17 −0.687043 + 0.223234i −1.63884 0.560531i −1.19584 + 0.868828i −0.0885211 + 0.272440i 1.25108 + 0.0192637i −1.47481 + 1.07151i 1.47687 2.03274i 2.37161 + 1.83724i 0.206939i
116.18 −0.687043 + 0.223234i −1.03953 1.38542i −1.19584 + 0.868828i −0.0885211 + 0.272440i 1.02347 + 0.719784i 1.47481 1.07151i 1.47687 2.03274i −0.838768 + 2.88036i 0.206939i
116.19 −0.473784 + 0.153942i 0.663585 + 1.59989i −1.41726 + 1.02970i 0.269770 0.830268i −0.560687 0.655850i −3.14053 + 2.28173i 1.09859 1.51208i −2.11931 + 2.12333i 0.434897i
116.20 −0.473784 + 0.153942i 1.72665 + 0.136713i −1.41726 + 1.02970i 0.269770 0.830268i −0.839104 + 0.201031i 3.14053 2.28173i 1.09859 1.51208i 2.96262 + 0.472112i 0.434897i
See next 80 embeddings (of 176 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 272.44 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
13.b even 2 1 inner
33.f even 10 1 inner
39.d odd 2 1 inner
143.l odd 10 1 inner
429.y 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.y.b 176
3.b odd 2 1 inner 429.2.y.b 176
11.d odd 10 1 inner 429.2.y.b 176
13.b even 2 1 inner 429.2.y.b 176
33.f even 10 1 inner 429.2.y.b 176
39.d odd 2 1 inner 429.2.y.b 176
143.l odd 10 1 inner 429.2.y.b 176
429.y even 10 1 inner 429.2.y.b 176

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.y.b 176 1.a even 1 1 trivial
429.2.y.b 176 3.b odd 2 1 inner
429.2.y.b 176 11.d odd 10 1 inner
429.2.y.b 176 13.b even 2 1 inner
429.2.y.b 176 33.f even 10 1 inner
429.2.y.b 176 39.d odd 2 1 inner
429.2.y.b 176 143.l odd 10 1 inner
429.2.y.b 176 429.y even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$25\!\cdots\!10$$$$T_{2}^{60} -$$$$95\!\cdots\!36$$$$T_{2}^{58} +$$$$32\!\cdots\!58$$$$T_{2}^{56} -$$$$10\!\cdots\!18$$$$T_{2}^{54} +$$$$30\!\cdots\!93$$$$T_{2}^{52} -$$$$82\!\cdots\!04$$$$T_{2}^{50} +$$$$20\!\cdots\!85$$$$T_{2}^{48} -$$$$48\!\cdots\!75$$$$T_{2}^{46} +$$$$10\!\cdots\!30$$$$T_{2}^{44} -$$$$21\!\cdots\!50$$$$T_{2}^{42} +$$$$38\!\cdots\!45$$$$T_{2}^{40} -$$$$63\!\cdots\!34$$$$T_{2}^{38} +$$$$95\!\cdots\!28$$$$T_{2}^{36} -$$$$12\!\cdots\!16$$$$T_{2}^{34} +$$$$15\!\cdots\!19$$$$T_{2}^{32} -$$$$15\!\cdots\!40$$$$T_{2}^{30} +$$$$13\!\cdots\!47$$$$T_{2}^{28} -$$$$92\!\cdots\!26$$$$T_{2}^{26} +$$$$46\!\cdots\!51$$$$T_{2}^{24} -$$$$17\!\cdots\!66$$$$T_{2}^{22} +$$$$50\!\cdots\!58$$$$T_{2}^{20} -$$$$12\!\cdots\!88$$$$T_{2}^{18} +$$$$27\!\cdots\!86$$$$T_{2}^{16} -$$$$45\!\cdots\!37$$$$T_{2}^{14} +$$$$57\!\cdots\!28$$$$T_{2}^{12} - 668531051850 T_{2}^{10} + 90970504619 T_{2}^{8} - 856732217 T_{2}^{6} + 517302467 T_{2}^{4} - 4441547 T_{2}^{2} + 14641$$">$$T_{2}^{88} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.