# Properties

 Label 950.2.bd.a Level $950$ Weight $2$ Character orbit 950.bd Analytic conductor $7.586$ Analytic rank $0$ Dimension $800$ 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.bd (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$800$$ Relative dimension: $$50$$ over $$\Q(\zeta_{60})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$800q - 4q^{5} - 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$800q - 4q^{5} - 8q^{7} - 36q^{15} - 100q^{16} - 4q^{17} + 40q^{19} - 96q^{22} - 44q^{23} - 4q^{25} + 4q^{28} + 120q^{29} - 64q^{30} + 132q^{33} + 28q^{35} + 100q^{36} - 4q^{38} - 80q^{39} - 32q^{42} - 36q^{43} + 216q^{45} + 76q^{47} + 36q^{53} + 88q^{55} - 124q^{57} + 120q^{59} + 48q^{60} - 8q^{62} - 72q^{63} - 24q^{67} + 88q^{68} + 24q^{70} - 24q^{73} - 256q^{77} + 24q^{78} + 4q^{80} - 100q^{81} - 8q^{82} - 24q^{83} + 28q^{85} - 112q^{87} - 12q^{90} + 4q^{92} - 108q^{93} - 28q^{95} + 12q^{97} - 120q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1 −0.777146 0.629320i −0.174328 + 3.32637i 0.207912 + 0.978148i −2.18227 + 0.487544i 2.22883 2.47537i 2.37586 + 2.37586i 0.453990 0.891007i −8.05081 0.846175i 2.00276 + 0.994454i
27.2 −0.777146 0.629320i −0.165111 + 3.15050i 0.207912 + 0.978148i 1.29881 1.82019i 2.11099 2.34449i −1.30050 1.30050i 0.453990 0.891007i −6.91481 0.726775i −2.15485 + 0.597183i
27.3 −0.777146 0.629320i −0.146876 + 2.80257i 0.207912 + 0.978148i 0.359551 + 2.20697i 1.87786 2.08557i −2.28843 2.28843i 0.453990 0.891007i −4.84924 0.509675i 1.10947 1.94141i
27.4 −0.777146 0.629320i −0.125909 + 2.40249i 0.207912 + 0.978148i 2.21427 0.311457i 1.60978 1.78784i 1.52376 + 1.52376i 0.453990 0.891007i −2.77252 0.291403i −1.91682 1.15144i
27.5 −0.777146 0.629320i −0.102836 + 1.96222i 0.207912 + 0.978148i −1.42891 1.71995i 1.31478 1.46022i −2.76852 2.76852i 0.453990 0.891007i −0.856176 0.0899877i 0.0280713 + 2.23589i
27.6 −0.777146 0.629320i −0.0968046 + 1.84714i 0.207912 + 0.978148i 1.09978 + 1.94692i 1.23768 1.37458i −0.632973 0.632973i 0.453990 0.891007i −0.418995 0.0440382i 0.370549 2.20515i
27.7 −0.777146 0.629320i −0.0959375 + 1.83060i 0.207912 + 0.978148i −1.73058 + 1.41601i 1.22659 1.36227i 1.20076 + 1.20076i 0.453990 0.891007i −0.358316 0.0376605i 2.23604 0.0113576i
27.8 −0.777146 0.629320i −0.0775294 + 1.47935i 0.207912 + 0.978148i −1.34338 1.78755i 0.991237 1.10088i 1.75517 + 1.75517i 0.453990 0.891007i 0.801101 + 0.0841991i −0.0809353 + 2.23460i
27.9 −0.777146 0.629320i −0.0710372 + 1.35547i 0.207912 + 0.978148i −1.70399 + 1.44791i 0.908231 1.00869i −0.514487 0.514487i 0.453990 0.891007i 1.15131 + 0.121008i 2.23544 0.0528825i
27.10 −0.777146 0.629320i −0.0494370 + 0.943315i 0.207912 + 0.978148i 1.73843 1.40637i 0.632067 0.701982i 3.25050 + 3.25050i 0.453990 0.891007i 2.09617 + 0.220316i −2.23607 0.00107286i
27.11 −0.777146 0.629320i −0.0107868 + 0.205824i 0.207912 + 0.978148i 0.365512 2.20599i 0.137912 0.153167i −1.79103 1.79103i 0.453990 0.891007i 2.94132 + 0.309145i −1.67233 + 1.48435i
27.12 −0.777146 0.629320i 0.00690609 0.131776i 0.207912 + 0.978148i −2.11597 0.722972i −0.0882964 + 0.0980631i 0.298792 + 0.298792i 0.453990 0.891007i 2.96625 + 0.311765i 1.18943 + 1.89348i
27.13 −0.777146 0.629320i 0.00693568 0.132341i 0.207912 + 0.978148i 1.23454 1.86438i −0.0886748 + 0.0984833i −2.47768 2.47768i 0.453990 0.891007i 2.96610 + 0.311750i −2.13271 + 0.671975i
27.14 −0.777146 0.629320i 0.0138662 0.264584i 0.207912 + 0.978148i 0.889244 + 2.05164i −0.177284 + 0.196894i 2.41122 + 2.41122i 0.453990 0.891007i 2.91375 + 0.306248i 0.600069 2.15405i
27.15 −0.777146 0.629320i 0.0178542 0.340678i 0.207912 + 0.978148i 1.74069 + 1.40356i −0.228271 + 0.253521i −0.826843 0.826843i 0.453990 0.891007i 2.86782 + 0.301420i −0.469482 2.18623i
27.16 −0.777146 0.629320i 0.0239777 0.457521i 0.207912 + 0.978148i 0.165650 2.22992i −0.306561 + 0.340471i 1.81231 + 1.81231i 0.453990 0.891007i 2.77482 + 0.291645i −1.53207 + 1.62873i
27.17 −0.777146 0.629320i 0.0402384 0.767794i 0.207912 + 0.978148i −1.26479 + 1.84399i −0.514459 + 0.571365i −3.24113 3.24113i 0.453990 0.891007i 2.39568 + 0.251796i 2.14339 0.637094i
27.18 −0.777146 0.629320i 0.0828442 1.58076i 0.207912 + 0.978148i −2.09573 0.779702i −1.05919 + 1.17635i 0.596283 + 0.596283i 0.453990 0.891007i 0.491621 + 0.0516714i 1.13800 + 1.92483i
27.19 −0.777146 0.629320i 0.0945439 1.80400i 0.207912 + 0.978148i 2.05987 0.870013i −1.20877 + 1.34248i 0.220705 + 0.220705i 0.453990 0.891007i −0.261928 0.0275297i −2.14834 0.620193i
27.20 −0.777146 0.629320i 0.108889 2.07773i 0.207912 + 0.978148i −1.28740 + 1.82828i −1.39218 + 1.54617i 0.660854 + 0.660854i 0.453990 0.891007i −1.32154 0.138899i 2.15107 0.610647i
See next 80 embeddings (of 800 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 867.50 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.d odd 6 1 inner
25.f odd 20 1 inner
475.be even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bd.a 800
19.d odd 6 1 inner 950.2.bd.a 800
25.f odd 20 1 inner 950.2.bd.a 800
475.be even 60 1 inner 950.2.bd.a 800

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bd.a 800 1.a even 1 1 trivial
950.2.bd.a 800 19.d odd 6 1 inner
950.2.bd.a 800 25.f odd 20 1 inner
950.2.bd.a 800 475.be even 60 1 inner

## Hecke kernels

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