# Properties

 Label 29.10.b.a Level $29$ Weight $10$ Character orbit 29.b Analytic conductor $14.936$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 29.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.9360392488$$ Analytic rank: $$0$$ Dimension: $$22$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22q - 5804q^{4} - 1374q^{5} - 8304q^{6} - 4956q^{7} - 112244q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q - 5804q^{4} - 1374q^{5} - 8304q^{6} - 4956q^{7} - 112244q^{9} - 244222q^{13} + 1246804q^{16} - 1658748q^{20} + 822328q^{22} - 874956q^{23} + 8668172q^{24} + 5307748q^{25} - 620352q^{28} - 2425374q^{29} - 8942448q^{30} + 10134274q^{33} - 37785784q^{34} - 20790348q^{35} + 34550680q^{36} - 30663552q^{38} + 56872008q^{42} - 43877176q^{45} - 131743922q^{49} - 6194732q^{51} + 342496580q^{52} + 34886610q^{53} + 116488784q^{54} - 308361676q^{57} + 342193888q^{58} + 175799052q^{59} - 484313328q^{62} - 190643424q^{63} - 419498924q^{64} - 149739966q^{65} - 508277640q^{67} + 263144256q^{71} + 435201408q^{74} + 1065897336q^{78} + 2990464236q^{80} - 129895134q^{81} - 527065064q^{82} + 1555989756q^{83} - 3422424120q^{86} + 2176720604q^{87} - 387386068q^{88} - 1493579244q^{91} - 1262849472q^{92} + 2042413382q^{93} + 166226488q^{94} - 6686432820q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1 43.1405i 202.470i −1349.11 1768.78 −8734.66 −2939.41 36113.2i −21311.1 76306.1i
28.2 39.9552i 171.812i −1084.42 917.921 6864.79 835.627 22871.2i −9836.35 36675.8i
28.3 37.8978i 104.304i −924.245 −2682.94 −3952.89 4793.67 15623.2i 8803.72 101678.i
28.4 34.5490i 59.9553i −681.634 −636.317 2071.40 −7630.94 5860.68i 16088.4 21984.1i
28.5 27.8229i 61.9183i −262.115 1301.46 −1722.75 8294.02 6952.53i 15849.1 36210.3i
28.6 25.3186i 226.177i −129.033 −952.179 5726.51 6455.42 9696.20i −31473.3 24107.9i
28.7 20.3455i 80.4219i 98.0611 535.165 −1636.22 −10636.4 12412.0i 13215.3 10888.2i
28.8 19.5556i 267.976i 129.577 −595.738 −5240.44 −893.851 12546.4i −52128.1 11650.0i
28.9 10.3669i 137.845i 404.527 −2029.73 1429.03 −4871.43 9501.57i 681.666 21042.0i
28.10 9.11441i 171.314i 428.927 2218.10 1561.43 −1871.46 8576.00i −9665.49 20216.7i
28.11 6.67374i 77.6458i 467.461 −531.517 −518.188 5986.73 6536.67i 13654.1 3547.20i
28.12 6.67374i 77.6458i 467.461 −531.517 −518.188 5986.73 6536.67i 13654.1 3547.20i
28.13 9.11441i 171.314i 428.927 2218.10 1561.43 −1871.46 8576.00i −9665.49 20216.7i
28.14 10.3669i 137.845i 404.527 −2029.73 1429.03 −4871.43 9501.57i 681.666 21042.0i
28.15 19.5556i 267.976i 129.577 −595.738 −5240.44 −893.851 12546.4i −52128.1 11650.0i
28.16 20.3455i 80.4219i 98.0611 535.165 −1636.22 −10636.4 12412.0i 13215.3 10888.2i
28.17 25.3186i 226.177i −129.033 −952.179 5726.51 6455.42 9696.20i −31473.3 24107.9i
28.18 27.8229i 61.9183i −262.115 1301.46 −1722.75 8294.02 6952.53i 15849.1 36210.3i
28.19 34.5490i 59.9553i −681.634 −636.317 2071.40 −7630.94 5860.68i 16088.4 21984.1i
28.20 37.8978i 104.304i −924.245 −2682.94 −3952.89 4793.67 15623.2i 8803.72 101678.i
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.10.b.a 22
29.b even 2 1 inner 29.10.b.a 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.10.b.a 22 1.a even 1 1 trivial
29.10.b.a 22 29.b even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(29, [\chi])$$.