# Properties

 Label 950.2.x.a Level $950$ Weight $2$ Character orbit 950.x Analytic conductor $7.586$ Analytic rank $0$ Dimension $400$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.x (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$400$$ Relative dimension: $$50$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$400q - 50q^{4} + 2q^{5} - 50q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$400q - 50q^{4} + 2q^{5} - 50q^{9} - 12q^{11} + 8q^{14} - 18q^{15} + 50q^{16} - 10q^{17} - 12q^{19} + 4q^{20} - 32q^{21} + 20q^{22} + 30q^{23} - 14q^{25} + 60q^{27} + 24q^{29} - 52q^{30} + 20q^{33} + 8q^{34} - 8q^{35} + 50q^{36} - 24q^{39} - 16q^{41} + 4q^{44} - 144q^{45} - 32q^{46} + 120q^{47} - 440q^{49} - 40q^{50} + 52q^{51} + 40q^{53} - 12q^{54} - 88q^{55} + 16q^{56} + 48q^{59} - 2q^{60} - 28q^{61} + 10q^{63} + 100q^{64} + 152q^{65} + 16q^{66} - 80q^{67} - 16q^{69} + 8q^{70} - 14q^{71} + 80q^{73} - 104q^{75} - 8q^{76} - 80q^{77} + 60q^{78} - 8q^{79} + 2q^{80} + 106q^{81} + 80q^{83} + 56q^{84} + 78q^{85} - 20q^{86} + 80q^{87} + 12q^{89} + 78q^{90} - 8q^{91} - 20q^{92} + 4q^{95} - 30q^{97} - 40q^{98} - 12q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
159.1 −0.994522 0.104528i −2.37897 2.14203i 0.978148 + 0.207912i 2.19514 + 0.425845i 2.14203 + 2.37897i 3.35325i −0.951057 0.309017i 0.757599 + 7.20807i −2.13861 0.652967i
159.2 −0.994522 0.104528i −2.37883 2.14191i 0.978148 + 0.207912i −2.23592 + 0.0257798i 2.14191 + 2.37883i 0.582392i −0.951057 0.309017i 0.757479 + 7.20693i 2.22637 + 0.208079i
159.3 −0.994522 0.104528i −1.73756 1.56450i 0.978148 + 0.207912i −0.908640 2.04313i 1.56450 + 1.73756i 2.62111i −0.951057 0.309017i 0.257848 + 2.45326i 0.690097 + 2.12691i
159.4 −0.994522 0.104528i −1.62203 1.46048i 0.978148 + 0.207912i 0.144205 + 2.23141i 1.46048 + 1.62203i 0.133313i −0.951057 0.309017i 0.184386 + 1.75432i 0.0898316 2.23426i
159.5 −0.994522 0.104528i −1.46570 1.31972i 0.978148 + 0.207912i 2.21506 0.305758i 1.31972 + 1.46570i 2.84835i −0.951057 0.309017i 0.0930228 + 0.885053i −2.23489 + 0.0725453i
159.6 −0.994522 0.104528i −1.44658 1.30250i 0.978148 + 0.207912i −1.66215 + 1.49575i 1.30250 + 1.44658i 4.00690i −0.951057 0.309017i 0.0824840 + 0.784783i 1.80939 1.31382i
159.7 −0.994522 0.104528i −1.26268 1.13692i 0.978148 + 0.207912i 0.601083 2.15376i 1.13692 + 1.26268i 3.10293i −0.951057 0.309017i −0.0118172 0.112433i −0.822920 + 2.07913i
159.8 −0.994522 0.104528i −1.10634 0.996156i 0.978148 + 0.207912i 2.09315 + 0.786590i 0.996156 + 1.10634i 0.326434i −0.951057 0.309017i −0.0819168 0.779387i −1.99946 1.00107i
159.9 −0.994522 0.104528i −1.07277 0.965928i 0.978148 + 0.207912i −1.93765 1.11603i 0.965928 + 1.07277i 2.14291i −0.951057 0.309017i −0.0957633 0.911127i 1.81038 + 1.31246i
159.10 −0.994522 0.104528i −0.852655 0.767734i 0.978148 + 0.207912i −0.257892 + 2.22115i 0.767734 + 0.852655i 4.58392i −0.951057 0.309017i −0.175980 1.67434i 0.488652 2.18202i
159.11 −0.994522 0.104528i −0.455776 0.410383i 0.978148 + 0.207912i −0.905416 2.04456i 0.410383 + 0.455776i 3.69355i −0.951057 0.309017i −0.274267 2.60948i 0.686741 + 2.12800i
159.12 −0.994522 0.104528i −0.305771 0.275318i 0.978148 + 0.207912i 1.48246 1.67401i 0.275318 + 0.305771i 4.36047i −0.951057 0.309017i −0.295889 2.81520i −1.64932 + 1.50988i
159.13 −0.994522 0.104528i 0.0825043 + 0.0742872i 0.978148 + 0.207912i 0.751196 + 2.10611i −0.0742872 0.0825043i 0.0772685i −0.951057 0.309017i −0.312297 2.97131i −0.526932 2.17310i
159.14 −0.994522 0.104528i 0.353923 + 0.318674i 0.978148 + 0.207912i −2.22902 0.177361i −0.318674 0.353923i 2.02267i −0.951057 0.309017i −0.289877 2.75799i 2.19827 + 0.409386i
159.15 −0.994522 0.104528i 0.415578 + 0.374188i 0.978148 + 0.207912i −1.68108 + 1.47444i −0.374188 0.415578i 0.399754i −0.951057 0.309017i −0.280897 2.67256i 1.82599 1.29065i
159.16 −0.994522 0.104528i 0.650894 + 0.586067i 0.978148 + 0.207912i −2.14939 0.616527i −0.586067 0.650894i 3.13390i −0.951057 0.309017i −0.233398 2.22063i 2.07317 + 0.837823i
159.17 −0.994522 0.104528i 0.792127 + 0.713235i 0.978148 + 0.207912i 1.59520 1.56695i −0.713235 0.792127i 0.0485779i −0.951057 0.309017i −0.194823 1.85362i −1.75025 + 1.39162i
159.18 −0.994522 0.104528i 1.14808 + 1.03373i 0.978148 + 0.207912i 1.84118 + 1.26888i −1.03373 1.14808i 2.15604i −0.951057 0.309017i −0.0641093 0.609959i −1.69846 1.45439i
159.19 −0.994522 0.104528i 1.25394 + 1.12906i 0.978148 + 0.207912i −0.311300 2.21429i −1.12906 1.25394i 1.86474i −0.951057 0.309017i −0.0159781 0.152021i 0.0781379 + 2.23470i
159.20 −0.994522 0.104528i 1.37103 + 1.23448i 0.978148 + 0.207912i 2.22915 + 0.175792i −1.23448 1.37103i 1.39428i −0.951057 0.309017i 0.0421934 + 0.401443i −2.19856 0.407839i
See next 80 embeddings (of 400 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 919.50 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner
25.e even 10 1 inner
475.x even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.x.a 400
19.c even 3 1 inner 950.2.x.a 400
25.e even 10 1 inner 950.2.x.a 400
475.x even 30 1 inner 950.2.x.a 400

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.x.a 400 1.a even 1 1 trivial
950.2.x.a 400 19.c even 3 1 inner
950.2.x.a 400 25.e even 10 1 inner
950.2.x.a 400 475.x even 30 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(950, [\chi])$$.