# Properties

 Label 950.2.w.a Level $950$ Weight $2$ Character orbit 950.w 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.w (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$400$$ Relative dimension: $$50$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$400q + 4q^{5} - 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$400q + 4q^{5} - 4q^{7} + 100q^{16} + 40q^{17} - 40q^{19} - 76q^{23} - 20q^{25} - 4q^{28} + 64q^{30} + 8q^{35} - 100q^{36} + 4q^{38} + 80q^{39} + 32q^{42} - 84q^{43} - 144q^{45} + 20q^{47} - 16q^{55} - 32q^{57} - 16q^{62} + 108q^{63} - 100q^{68} - 12q^{73} + 256q^{77} - 4q^{80} + 100q^{81} - 16q^{82} + 116q^{85} - 272q^{87} - 4q^{92} - 24q^{93} - 56q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −0.987688 + 0.156434i −1.41733 + 2.78167i 0.951057 0.309017i 2.23282 0.120389i 0.964734 2.96915i 3.07932 + 3.07932i −0.891007 + 0.453990i −3.96551 5.45806i −2.18650 + 0.468198i
37.2 −0.987688 + 0.156434i −1.40212 + 2.75181i 0.951057 0.309017i −1.82971 1.28537i 0.954377 2.93727i −1.62377 1.62377i −0.891007 + 0.453990i −3.84317 5.28967i 2.00826 + 0.983314i
37.3 −0.987688 + 0.156434i −1.32806 + 2.60647i 0.951057 0.309017i 1.06510 + 1.96610i 0.903971 2.78214i −0.494215 0.494215i −0.891007 + 0.453990i −3.26659 4.49607i −1.35955 1.77528i
37.4 −0.987688 + 0.156434i −1.26480 + 2.48231i 0.951057 0.309017i −1.52838 + 1.63219i 0.860911 2.64961i −1.59067 1.59067i −0.891007 + 0.453990i −2.79880 3.85222i 1.25423 1.85119i
37.5 −0.987688 + 0.156434i −1.00551 + 1.97343i 0.951057 0.309017i 1.74220 1.40169i 0.684420 2.10643i −1.24411 1.24411i −0.891007 + 0.453990i −1.12001 1.54156i −1.50148 + 1.65697i
37.6 −0.987688 + 0.156434i −0.966228 + 1.89633i 0.951057 0.309017i −0.861084 2.06362i 0.657681 2.02413i 2.78834 + 2.78834i −0.891007 + 0.453990i −0.899114 1.23752i 1.17330 + 1.90351i
37.7 −0.987688 + 0.156434i −0.669280 + 1.31354i 0.951057 0.309017i −2.09747 + 0.775001i 0.455558 1.40206i 1.60624 + 1.60624i −0.891007 + 0.453990i 0.485913 + 0.668802i 1.95041 1.09358i
37.8 −0.987688 + 0.156434i −0.581458 + 1.14117i 0.951057 0.309017i 1.37992 + 1.75950i 0.395780 1.21809i 1.14962 + 1.14962i −0.891007 + 0.453990i 0.799169 + 1.09996i −1.63817 1.52197i
37.9 −0.987688 + 0.156434i −0.380740 + 0.747244i 0.951057 0.309017i 2.16957 + 0.541276i 0.259158 0.797606i −1.83916 1.83916i −0.891007 + 0.453990i 1.34994 + 1.85804i −2.22753 0.195217i
37.10 −0.987688 + 0.156434i −0.341841 + 0.670901i 0.951057 0.309017i 0.0404826 2.23570i 0.232680 0.716117i 2.21716 + 2.21716i −0.891007 + 0.453990i 1.43010 + 1.96837i 0.309757 + 2.21451i
37.11 −0.987688 + 0.156434i −0.172149 + 0.337861i 0.951057 0.309017i −1.08112 + 1.95734i 0.117176 0.360631i −2.52947 2.52947i −0.891007 + 0.453990i 1.67884 + 2.31073i 0.761613 2.10237i
37.12 −0.987688 + 0.156434i −0.140807 + 0.276349i 0.951057 0.309017i 0.842195 + 2.07140i 0.0958429 0.294974i 2.00698 + 2.00698i −0.891007 + 0.453990i 1.70681 + 2.34923i −1.15586 1.91415i
37.13 −0.987688 + 0.156434i −0.0204485 + 0.0401324i 0.951057 0.309017i −2.01849 0.962141i 0.0139186 0.0428372i −1.29403 1.29403i −0.891007 + 0.453990i 1.76216 + 2.42541i 2.14415 + 0.634534i
37.14 −0.987688 + 0.156434i 0.0539197 0.105823i 0.951057 0.309017i 0.164450 2.23001i −0.0367014 + 0.112955i −3.59050 3.59050i −0.891007 + 0.453990i 1.75506 + 2.41564i 0.186426 + 2.22828i
37.15 −0.987688 + 0.156434i 0.146430 0.287385i 0.951057 0.309017i 2.22675 0.203946i −0.0996703 + 0.306754i −0.119348 0.119348i −0.891007 + 0.453990i 1.70221 + 2.34289i −2.16743 + 0.549775i
37.16 −0.987688 + 0.156434i 0.299107 0.587030i 0.951057 0.309017i −2.03682 0.922698i −0.203592 + 0.626593i −0.375210 0.375210i −0.891007 + 0.453990i 1.50822 + 2.07588i 2.15608 + 0.592709i
37.17 −0.987688 + 0.156434i 0.531107 1.04236i 0.951057 0.309017i −1.52005 + 1.63996i −0.361507 + 1.11261i 3.11636 + 3.11636i −0.891007 + 0.453990i 0.958925 + 1.31985i 1.24479 1.85755i
37.18 −0.987688 + 0.156434i 0.672164 1.31920i 0.951057 0.309017i 0.922100 2.03709i −0.457521 + 1.40810i 1.03073 + 1.03073i −0.891007 + 0.453990i 0.474883 + 0.653621i −0.592076 + 2.15626i
37.19 −0.987688 + 0.156434i 0.888230 1.74325i 0.951057 0.309017i −1.88136 + 1.20850i −0.604590 + 1.86074i −1.93054 1.93054i −0.891007 + 0.453990i −0.486612 0.669764i 1.66915 1.48793i
37.20 −0.987688 + 0.156434i 0.966230 1.89633i 0.951057 0.309017i 0.684737 + 2.12865i −0.657682 + 2.02414i 0.498269 + 0.498269i −0.891007 + 0.453990i −0.899122 1.23753i −1.00930 1.99532i
See next 80 embeddings (of 400 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 873.50 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.b odd 2 1 inner
25.f odd 20 1 inner
475.v even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.w.a 400
19.b odd 2 1 inner 950.2.w.a 400
25.f odd 20 1 inner 950.2.w.a 400
475.v even 20 1 inner 950.2.w.a 400

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.w.a 400 1.a even 1 1 trivial
950.2.w.a 400 19.b odd 2 1 inner
950.2.w.a 400 25.f odd 20 1 inner
950.2.w.a 400 475.v even 20 1 inner

## Hecke kernels

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