# Properties

 Label 9680.2.a.by.1.1 Level $9680$ Weight $2$ Character 9680.1 Self dual yes Analytic conductor $77.295$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$77.2951891566$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.788.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x - 3$$ x^3 - x^2 - 7*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4840) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.35386$$ of defining polynomial Character $$\chi$$ $$=$$ 9680.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.35386 q^{3} -1.00000 q^{5} -3.81322 q^{7} +8.24835 q^{9} +O(q^{10})$$ $$q-3.35386 q^{3} -1.00000 q^{5} -3.81322 q^{7} +8.24835 q^{9} -0.459363 q^{13} +3.35386 q^{15} +7.78899 q^{17} +1.54064 q^{19} +12.7890 q^{21} +8.24835 q^{23} +1.00000 q^{25} -17.6022 q^{27} +6.70771 q^{29} +7.54064 q^{31} +3.81322 q^{35} +4.70771 q^{37} +1.54064 q^{39} +1.45936 q^{41} +3.97577 q^{43} -8.24835 q^{45} -0.272582 q^{47} +7.54064 q^{49} -26.1231 q^{51} -2.62191 q^{53} -5.16707 q^{57} +8.24835 q^{59} -0.751651 q^{61} -31.4528 q^{63} +0.459363 q^{65} +4.06157 q^{67} -27.6638 q^{69} +10.4109 q^{71} -0.918726 q^{73} -3.35386 q^{75} +5.78899 q^{79} +34.2902 q^{81} -9.16707 q^{83} -7.78899 q^{85} -22.4967 q^{87} +12.7890 q^{89} +1.75165 q^{91} -25.2902 q^{93} -1.54064 q^{95} -0.459363 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 $$3 q - q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 2 q^{13} + q^{15} + 4 q^{17} + 4 q^{19} + 19 q^{21} + 6 q^{23} + 3 q^{25} - 25 q^{27} + 2 q^{29} + 22 q^{31} + 3 q^{35} - 4 q^{37} + 4 q^{39} + 5 q^{41} + q^{43} - 6 q^{45} + 7 q^{47} + 22 q^{49} - 24 q^{51} - 6 q^{53} + 2 q^{57} + 6 q^{59} - 21 q^{61} - 20 q^{63} + 2 q^{65} - 15 q^{67} - 28 q^{69} + 10 q^{71} - 4 q^{73} - q^{75} - 2 q^{79} + 31 q^{81} - 10 q^{83} - 4 q^{85} - 30 q^{87} + 19 q^{89} + 24 q^{91} - 4 q^{93} - 4 q^{95} - 2 q^{97}+O(q^{100})$$ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 2 * q^13 + q^15 + 4 * q^17 + 4 * q^19 + 19 * q^21 + 6 * q^23 + 3 * q^25 - 25 * q^27 + 2 * q^29 + 22 * q^31 + 3 * q^35 - 4 * q^37 + 4 * q^39 + 5 * q^41 + q^43 - 6 * q^45 + 7 * q^47 + 22 * q^49 - 24 * q^51 - 6 * q^53 + 2 * q^57 + 6 * q^59 - 21 * q^61 - 20 * q^63 + 2 * q^65 - 15 * q^67 - 28 * q^69 + 10 * q^71 - 4 * q^73 - q^75 - 2 * q^79 + 31 * q^81 - 10 * q^83 - 4 * q^85 - 30 * q^87 + 19 * q^89 + 24 * q^91 - 4 * q^93 - 4 * q^95 - 2 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.35386 −1.93635 −0.968175 0.250275i $$-0.919479\pi$$
−0.968175 + 0.250275i $$0.919479\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.81322 −1.44126 −0.720631 0.693319i $$-0.756148\pi$$
−0.720631 + 0.693319i $$0.756148\pi$$
$$8$$ 0 0
$$9$$ 8.24835 2.74945
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −0.459363 −0.127404 −0.0637022 0.997969i $$-0.520291\pi$$
−0.0637022 + 0.997969i $$0.520291\pi$$
$$14$$ 0 0
$$15$$ 3.35386 0.865962
$$16$$ 0 0
$$17$$ 7.78899 1.88911 0.944553 0.328358i $$-0.106495\pi$$
0.944553 + 0.328358i $$0.106495\pi$$
$$18$$ 0 0
$$19$$ 1.54064 0.353446 0.176723 0.984261i $$-0.443450\pi$$
0.176723 + 0.984261i $$0.443450\pi$$
$$20$$ 0 0
$$21$$ 12.7890 2.79079
$$22$$ 0 0
$$23$$ 8.24835 1.71990 0.859950 0.510379i $$-0.170495\pi$$
0.859950 + 0.510379i $$0.170495\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −17.6022 −3.38755
$$28$$ 0 0
$$29$$ 6.70771 1.24559 0.622795 0.782385i $$-0.285997\pi$$
0.622795 + 0.782385i $$0.285997\pi$$
$$30$$ 0 0
$$31$$ 7.54064 1.35434 0.677169 0.735827i $$-0.263207\pi$$
0.677169 + 0.735827i $$0.263207\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.81322 0.644552
$$36$$ 0 0
$$37$$ 4.70771 0.773943 0.386972 0.922092i $$-0.373521\pi$$
0.386972 + 0.922092i $$0.373521\pi$$
$$38$$ 0 0
$$39$$ 1.54064 0.246699
$$40$$ 0 0
$$41$$ 1.45936 0.227914 0.113957 0.993486i $$-0.463647\pi$$
0.113957 + 0.993486i $$0.463647\pi$$
$$42$$ 0 0
$$43$$ 3.97577 0.606299 0.303149 0.952943i $$-0.401962\pi$$
0.303149 + 0.952943i $$0.401962\pi$$
$$44$$ 0 0
$$45$$ −8.24835 −1.22959
$$46$$ 0 0
$$47$$ −0.272582 −0.0397601 −0.0198801 0.999802i $$-0.506328\pi$$
−0.0198801 + 0.999802i $$0.506328\pi$$
$$48$$ 0 0
$$49$$ 7.54064 1.07723
$$50$$ 0 0
$$51$$ −26.1231 −3.65797
$$52$$ 0 0
$$53$$ −2.62191 −0.360147 −0.180074 0.983653i $$-0.557634\pi$$
−0.180074 + 0.983653i $$0.557634\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −5.16707 −0.684396
$$58$$ 0 0
$$59$$ 8.24835 1.07384 0.536922 0.843632i $$-0.319587\pi$$
0.536922 + 0.843632i $$0.319587\pi$$
$$60$$ 0 0
$$61$$ −0.751651 −0.0962391 −0.0481195 0.998842i $$-0.515323\pi$$
−0.0481195 + 0.998842i $$0.515323\pi$$
$$62$$ 0 0
$$63$$ −31.4528 −3.96267
$$64$$ 0 0
$$65$$ 0.459363 0.0569770
$$66$$ 0 0
$$67$$ 4.06157 0.496199 0.248100 0.968735i $$-0.420194\pi$$
0.248100 + 0.968735i $$0.420194\pi$$
$$68$$ 0 0
$$69$$ −27.6638 −3.33033
$$70$$ 0 0
$$71$$ 10.4109 1.23555 0.617773 0.786356i $$-0.288035\pi$$
0.617773 + 0.786356i $$0.288035\pi$$
$$72$$ 0 0
$$73$$ −0.918726 −0.107529 −0.0537644 0.998554i $$-0.517122\pi$$
−0.0537644 + 0.998554i $$0.517122\pi$$
$$74$$ 0 0
$$75$$ −3.35386 −0.387270
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.78899 0.651312 0.325656 0.945488i $$-0.394415\pi$$
0.325656 + 0.945488i $$0.394415\pi$$
$$80$$ 0 0
$$81$$ 34.2902 3.81002
$$82$$ 0 0
$$83$$ −9.16707 −1.00622 −0.503109 0.864223i $$-0.667810\pi$$
−0.503109 + 0.864223i $$0.667810\pi$$
$$84$$ 0 0
$$85$$ −7.78899 −0.844834
$$86$$ 0 0
$$87$$ −22.4967 −2.41190
$$88$$ 0 0
$$89$$ 12.7890 1.35563 0.677815 0.735233i $$-0.262927\pi$$
0.677815 + 0.735233i $$0.262927\pi$$
$$90$$ 0 0
$$91$$ 1.75165 0.183623
$$92$$ 0 0
$$93$$ −25.2902 −2.62247
$$94$$ 0 0
$$95$$ −1.54064 −0.158066
$$96$$ 0 0
$$97$$ −0.459363 −0.0466412 −0.0233206 0.999728i $$-0.507424\pi$$
−0.0233206 + 0.999728i $$0.507424\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 9.54516 0.949779 0.474890 0.880045i $$-0.342488\pi$$
0.474890 + 0.880045i $$0.342488\pi$$
$$102$$ 0 0
$$103$$ −6.41090 −0.631684 −0.315842 0.948812i $$-0.602287\pi$$
−0.315842 + 0.948812i $$0.602287\pi$$
$$104$$ 0 0
$$105$$ −12.7890 −1.24808
$$106$$ 0 0
$$107$$ 0.435130 0.0420656 0.0210328 0.999779i $$-0.493305\pi$$
0.0210328 + 0.999779i $$0.493305\pi$$
$$108$$ 0 0
$$109$$ −8.66377 −0.829839 −0.414919 0.909858i $$-0.636190\pi$$
−0.414919 + 0.909858i $$0.636190\pi$$
$$110$$ 0 0
$$111$$ −15.7890 −1.49862
$$112$$ 0 0
$$113$$ −17.2529 −1.62301 −0.811507 0.584343i $$-0.801352\pi$$
−0.811507 + 0.584343i $$0.801352\pi$$
$$114$$ 0 0
$$115$$ −8.24835 −0.769162
$$116$$ 0 0
$$117$$ −3.78899 −0.350292
$$118$$ 0 0
$$119$$ −29.7011 −2.72270
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −4.89449 −0.441322
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 3.14284 0.278882 0.139441 0.990230i $$-0.455469\pi$$
0.139441 + 0.990230i $$0.455469\pi$$
$$128$$ 0 0
$$129$$ −13.3341 −1.17401
$$130$$ 0 0
$$131$$ 16.7935 1.46726 0.733628 0.679551i $$-0.237826\pi$$
0.733628 + 0.679551i $$0.237826\pi$$
$$132$$ 0 0
$$133$$ −5.87479 −0.509409
$$134$$ 0 0
$$135$$ 17.6022 1.51496
$$136$$ 0 0
$$137$$ 11.0045 0.940180 0.470090 0.882618i $$-0.344221\pi$$
0.470090 + 0.882618i $$0.344221\pi$$
$$138$$ 0 0
$$139$$ −20.9561 −1.77747 −0.888735 0.458421i $$-0.848415\pi$$
−0.888735 + 0.458421i $$0.848415\pi$$
$$140$$ 0 0
$$141$$ 0.914200 0.0769895
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −6.70771 −0.557045
$$146$$ 0 0
$$147$$ −25.2902 −2.08590
$$148$$ 0 0
$$149$$ −1.62191 −0.132872 −0.0664361 0.997791i $$-0.521163\pi$$
−0.0664361 + 0.997791i $$0.521163\pi$$
$$150$$ 0 0
$$151$$ 7.00453 0.570020 0.285010 0.958525i $$-0.408003\pi$$
0.285010 + 0.958525i $$0.408003\pi$$
$$152$$ 0 0
$$153$$ 64.2463 5.19400
$$154$$ 0 0
$$155$$ −7.54064 −0.605679
$$156$$ 0 0
$$157$$ −10.3341 −0.824755 −0.412377 0.911013i $$-0.635301\pi$$
−0.412377 + 0.911013i $$0.635301\pi$$
$$158$$ 0 0
$$159$$ 8.79351 0.697371
$$160$$ 0 0
$$161$$ −31.4528 −2.47882
$$162$$ 0 0
$$163$$ 10.1868 0.797890 0.398945 0.916975i $$-0.369376\pi$$
0.398945 + 0.916975i $$0.369376\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 22.8066 1.76483 0.882414 0.470473i $$-0.155917\pi$$
0.882414 + 0.470473i $$0.155917\pi$$
$$168$$ 0 0
$$169$$ −12.7890 −0.983768
$$170$$ 0 0
$$171$$ 12.7077 0.971783
$$172$$ 0 0
$$173$$ 17.8748 1.35899 0.679497 0.733678i $$-0.262198\pi$$
0.679497 + 0.733678i $$0.262198\pi$$
$$174$$ 0 0
$$175$$ −3.81322 −0.288252
$$176$$ 0 0
$$177$$ −27.6638 −2.07934
$$178$$ 0 0
$$179$$ 6.91873 0.517130 0.258565 0.965994i $$-0.416750\pi$$
0.258565 + 0.965994i $$0.416750\pi$$
$$180$$ 0 0
$$181$$ 14.9515 1.11134 0.555669 0.831403i $$-0.312462\pi$$
0.555669 + 0.831403i $$0.312462\pi$$
$$182$$ 0 0
$$183$$ 2.52093 0.186353
$$184$$ 0 0
$$185$$ −4.70771 −0.346118
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 67.1211 4.88234
$$190$$ 0 0
$$191$$ 5.32962 0.385638 0.192819 0.981234i $$-0.438237\pi$$
0.192819 + 0.981234i $$0.438237\pi$$
$$192$$ 0 0
$$193$$ −5.29229 −0.380947 −0.190474 0.981692i $$-0.561002\pi$$
−0.190474 + 0.981692i $$0.561002\pi$$
$$194$$ 0 0
$$195$$ −1.54064 −0.110327
$$196$$ 0 0
$$197$$ −12.1231 −0.863738 −0.431869 0.901936i $$-0.642146\pi$$
−0.431869 + 0.901936i $$0.642146\pi$$
$$198$$ 0 0
$$199$$ −12.3341 −0.874345 −0.437172 0.899378i $$-0.644020\pi$$
−0.437172 + 0.899378i $$0.644020\pi$$
$$200$$ 0 0
$$201$$ −13.6219 −0.960816
$$202$$ 0 0
$$203$$ −25.5780 −1.79522
$$204$$ 0 0
$$205$$ −1.45936 −0.101926
$$206$$ 0 0
$$207$$ 68.0353 4.72878
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.95606 −0.341189 −0.170595 0.985341i $$-0.554569\pi$$
−0.170595 + 0.985341i $$0.554569\pi$$
$$212$$ 0 0
$$213$$ −34.9166 −2.39245
$$214$$ 0 0
$$215$$ −3.97577 −0.271145
$$216$$ 0 0
$$217$$ −28.7541 −1.95196
$$218$$ 0 0
$$219$$ 3.08127 0.208213
$$220$$ 0 0
$$221$$ −3.57797 −0.240680
$$222$$ 0 0
$$223$$ 8.39780 0.562358 0.281179 0.959655i $$-0.409275\pi$$
0.281179 + 0.959655i $$0.409275\pi$$
$$224$$ 0 0
$$225$$ 8.24835 0.549890
$$226$$ 0 0
$$227$$ −15.2681 −1.01338 −0.506688 0.862129i $$-0.669130\pi$$
−0.506688 + 0.862129i $$0.669130\pi$$
$$228$$ 0 0
$$229$$ 8.62644 0.570051 0.285026 0.958520i $$-0.407998\pi$$
0.285026 + 0.958520i $$0.407998\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 26.4109 1.73024 0.865118 0.501569i $$-0.167244\pi$$
0.865118 + 0.501569i $$0.167244\pi$$
$$234$$ 0 0
$$235$$ 0.272582 0.0177813
$$236$$ 0 0
$$237$$ −19.4154 −1.26117
$$238$$ 0 0
$$239$$ −1.21554 −0.0786268 −0.0393134 0.999227i $$-0.512517\pi$$
−0.0393134 + 0.999227i $$0.512517\pi$$
$$240$$ 0 0
$$241$$ −14.1186 −0.909460 −0.454730 0.890629i $$-0.650264\pi$$
−0.454730 + 0.890629i $$0.650264\pi$$
$$242$$ 0 0
$$243$$ −62.1978 −3.98999
$$244$$ 0 0
$$245$$ −7.54064 −0.481754
$$246$$ 0 0
$$247$$ −0.707712 −0.0450306
$$248$$ 0 0
$$249$$ 30.7450 1.94839
$$250$$ 0 0
$$251$$ −9.16707 −0.578621 −0.289310 0.957235i $$-0.593426\pi$$
−0.289310 + 0.957235i $$0.593426\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 26.1231 1.63589
$$256$$ 0 0
$$257$$ −10.4594 −0.652437 −0.326219 0.945294i $$-0.605775\pi$$
−0.326219 + 0.945294i $$0.605775\pi$$
$$258$$ 0 0
$$259$$ −17.9515 −1.11545
$$260$$ 0 0
$$261$$ 55.3275 3.42469
$$262$$ 0 0
$$263$$ −10.6219 −0.654975 −0.327488 0.944855i $$-0.606202\pi$$
−0.327488 + 0.944855i $$0.606202\pi$$
$$264$$ 0 0
$$265$$ 2.62191 0.161063
$$266$$ 0 0
$$267$$ −42.8924 −2.62497
$$268$$ 0 0
$$269$$ 10.0045 0.609987 0.304993 0.952354i $$-0.401346\pi$$
0.304993 + 0.952354i $$0.401346\pi$$
$$270$$ 0 0
$$271$$ −2.54516 −0.154608 −0.0773038 0.997008i $$-0.524631\pi$$
−0.0773038 + 0.997008i $$0.524631\pi$$
$$272$$ 0 0
$$273$$ −5.87479 −0.355558
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 17.2923 1.03899 0.519496 0.854473i $$-0.326120\pi$$
0.519496 + 0.854473i $$0.326120\pi$$
$$278$$ 0 0
$$279$$ 62.1978 3.72369
$$280$$ 0 0
$$281$$ 16.8308 1.00404 0.502022 0.864855i $$-0.332590\pi$$
0.502022 + 0.864855i $$0.332590\pi$$
$$282$$ 0 0
$$283$$ 1.30539 0.0775974 0.0387987 0.999247i $$-0.487647\pi$$
0.0387987 + 0.999247i $$0.487647\pi$$
$$284$$ 0 0
$$285$$ 5.16707 0.306071
$$286$$ 0 0
$$287$$ −5.56487 −0.328484
$$288$$ 0 0
$$289$$ 43.6683 2.56872
$$290$$ 0 0
$$291$$ 1.54064 0.0903137
$$292$$ 0 0
$$293$$ 3.91420 0.228670 0.114335 0.993442i $$-0.463526\pi$$
0.114335 + 0.993442i $$0.463526\pi$$
$$294$$ 0 0
$$295$$ −8.24835 −0.480237
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −3.78899 −0.219123
$$300$$ 0 0
$$301$$ −15.1605 −0.873835
$$302$$ 0 0
$$303$$ −32.0131 −1.83910
$$304$$ 0 0
$$305$$ 0.751651 0.0430394
$$306$$ 0 0
$$307$$ −21.1671 −1.20807 −0.604034 0.796958i $$-0.706441\pi$$
−0.604034 + 0.796958i $$0.706441\pi$$
$$308$$ 0 0
$$309$$ 21.5012 1.22316
$$310$$ 0 0
$$311$$ 5.96267 0.338112 0.169056 0.985606i $$-0.445928\pi$$
0.169056 + 0.985606i $$0.445928\pi$$
$$312$$ 0 0
$$313$$ −28.3230 −1.60091 −0.800456 0.599392i $$-0.795409\pi$$
−0.800456 + 0.599392i $$0.795409\pi$$
$$314$$ 0 0
$$315$$ 31.4528 1.77216
$$316$$ 0 0
$$317$$ −1.57797 −0.0886277 −0.0443139 0.999018i $$-0.514110\pi$$
−0.0443139 + 0.999018i $$0.514110\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −1.45936 −0.0814537
$$322$$ 0 0
$$323$$ 12.0000 0.667698
$$324$$ 0 0
$$325$$ −0.459363 −0.0254809
$$326$$ 0 0
$$327$$ 29.0570 1.60686
$$328$$ 0 0
$$329$$ 1.03941 0.0573047
$$330$$ 0 0
$$331$$ −17.9233 −0.985151 −0.492576 0.870270i $$-0.663944\pi$$
−0.492576 + 0.870270i $$0.663944\pi$$
$$332$$ 0 0
$$333$$ 38.8308 2.12792
$$334$$ 0 0
$$335$$ −4.06157 −0.221907
$$336$$ 0 0
$$337$$ −26.3715 −1.43655 −0.718273 0.695761i $$-0.755067\pi$$
−0.718273 + 0.695761i $$0.755067\pi$$
$$338$$ 0 0
$$339$$ 57.8637 3.14272
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −2.06157 −0.111314
$$344$$ 0 0
$$345$$ 27.6638 1.48937
$$346$$ 0 0
$$347$$ 8.80661 0.472764 0.236382 0.971660i $$-0.424038\pi$$
0.236382 + 0.971660i $$0.424038\pi$$
$$348$$ 0 0
$$349$$ −12.3736 −0.662342 −0.331171 0.943571i $$-0.607444\pi$$
−0.331171 + 0.943571i $$0.607444\pi$$
$$350$$ 0 0
$$351$$ 8.08580 0.431588
$$352$$ 0 0
$$353$$ 13.5012 0.718598 0.359299 0.933223i $$-0.383016\pi$$
0.359299 + 0.933223i $$0.383016\pi$$
$$354$$ 0 0
$$355$$ −10.4109 −0.552553
$$356$$ 0 0
$$357$$ 99.6132 5.27209
$$358$$ 0 0
$$359$$ 25.0792 1.32363 0.661815 0.749668i $$-0.269787\pi$$
0.661815 + 0.749668i $$0.269787\pi$$
$$360$$ 0 0
$$361$$ −16.6264 −0.875076
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0.918726 0.0480883
$$366$$ 0 0
$$367$$ 22.5977 1.17959 0.589795 0.807553i $$-0.299209\pi$$
0.589795 + 0.807553i $$0.299209\pi$$
$$368$$ 0 0
$$369$$ 12.0373 0.626639
$$370$$ 0 0
$$371$$ 9.99792 0.519066
$$372$$ 0 0
$$373$$ −17.5012 −0.906179 −0.453090 0.891465i $$-0.649678\pi$$
−0.453090 + 0.891465i $$0.649678\pi$$
$$374$$ 0 0
$$375$$ 3.35386 0.173192
$$376$$ 0 0
$$377$$ −3.08127 −0.158694
$$378$$ 0 0
$$379$$ 33.1165 1.70108 0.850541 0.525909i $$-0.176275\pi$$
0.850541 + 0.525909i $$0.176275\pi$$
$$380$$ 0 0
$$381$$ −10.5406 −0.540013
$$382$$ 0 0
$$383$$ −28.5340 −1.45802 −0.729010 0.684503i $$-0.760019\pi$$
−0.729010 + 0.684503i $$0.760019\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 32.7935 1.66699
$$388$$ 0 0
$$389$$ 24.2812 1.23110 0.615552 0.788096i $$-0.288933\pi$$
0.615552 + 0.788096i $$0.288933\pi$$
$$390$$ 0 0
$$391$$ 64.2463 3.24907
$$392$$ 0 0
$$393$$ −56.3230 −2.84112
$$394$$ 0 0
$$395$$ −5.78899 −0.291275
$$396$$ 0 0
$$397$$ −26.3341 −1.32167 −0.660837 0.750530i $$-0.729798\pi$$
−0.660837 + 0.750530i $$0.729798\pi$$
$$398$$ 0 0
$$399$$ 19.7032 0.986393
$$400$$ 0 0
$$401$$ 17.8308 0.890430 0.445215 0.895424i $$-0.353127\pi$$
0.445215 + 0.895424i $$0.353127\pi$$
$$402$$ 0 0
$$403$$ −3.46389 −0.172549
$$404$$ 0 0
$$405$$ −34.2902 −1.70389
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −6.37809 −0.315376 −0.157688 0.987489i $$-0.550404\pi$$
−0.157688 + 0.987489i $$0.550404\pi$$
$$410$$ 0 0
$$411$$ −36.9076 −1.82052
$$412$$ 0 0
$$413$$ −31.4528 −1.54769
$$414$$ 0 0
$$415$$ 9.16707 0.449994
$$416$$ 0 0
$$417$$ 70.2836 3.44180
$$418$$ 0 0
$$419$$ −27.6638 −1.35146 −0.675732 0.737148i $$-0.736172\pi$$
−0.675732 + 0.737148i $$0.736172\pi$$
$$420$$ 0 0
$$421$$ −7.78446 −0.379391 −0.189696 0.981843i $$-0.560750\pi$$
−0.189696 + 0.981843i $$0.560750\pi$$
$$422$$ 0 0
$$423$$ −2.24835 −0.109318
$$424$$ 0 0
$$425$$ 7.78899 0.377821
$$426$$ 0 0
$$427$$ 2.86621 0.138706
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −21.3670 −1.02921 −0.514605 0.857427i $$-0.672061\pi$$
−0.514605 + 0.857427i $$0.672061\pi$$
$$432$$ 0 0
$$433$$ −16.3715 −0.786763 −0.393382 0.919375i $$-0.628695\pi$$
−0.393382 + 0.919375i $$0.628695\pi$$
$$434$$ 0 0
$$435$$ 22.4967 1.07863
$$436$$ 0 0
$$437$$ 12.7077 0.607892
$$438$$ 0 0
$$439$$ −34.6198 −1.65231 −0.826157 0.563440i $$-0.809478\pi$$
−0.826157 + 0.563440i $$0.809478\pi$$
$$440$$ 0 0
$$441$$ 62.1978 2.96180
$$442$$ 0 0
$$443$$ 10.4725 0.497562 0.248781 0.968560i $$-0.419970\pi$$
0.248781 + 0.968560i $$0.419970\pi$$
$$444$$ 0 0
$$445$$ −12.7890 −0.606256
$$446$$ 0 0
$$447$$ 5.43966 0.257287
$$448$$ 0 0
$$449$$ 39.5825 1.86801 0.934007 0.357255i $$-0.116287\pi$$
0.934007 + 0.357255i $$0.116287\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −23.4922 −1.10376
$$454$$ 0 0
$$455$$ −1.75165 −0.0821187
$$456$$ 0 0
$$457$$ −34.5340 −1.61543 −0.807717 0.589570i $$-0.799297\pi$$
−0.807717 + 0.589570i $$0.799297\pi$$
$$458$$ 0 0
$$459$$ −137.103 −6.39943
$$460$$ 0 0
$$461$$ −0.788986 −0.0367467 −0.0183734 0.999831i $$-0.505849\pi$$
−0.0183734 + 0.999831i $$0.505849\pi$$
$$462$$ 0 0
$$463$$ 6.56034 0.304885 0.152443 0.988312i $$-0.451286\pi$$
0.152443 + 0.988312i $$0.451286\pi$$
$$464$$ 0 0
$$465$$ 25.2902 1.17281
$$466$$ 0 0
$$467$$ −3.19131 −0.147676 −0.0738381 0.997270i $$-0.523525\pi$$
−0.0738381 + 0.997270i $$0.523525\pi$$
$$468$$ 0 0
$$469$$ −15.4876 −0.715153
$$470$$ 0 0
$$471$$ 34.6592 1.59701
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.54064 0.0706893
$$476$$ 0 0
$$477$$ −21.6264 −0.990207
$$478$$ 0 0
$$479$$ 16.4482 0.751539 0.375769 0.926713i $$-0.377378\pi$$
0.375769 + 0.926713i $$0.377378\pi$$
$$480$$ 0 0
$$481$$ −2.16255 −0.0986037
$$482$$ 0 0
$$483$$ 105.488 4.79987
$$484$$ 0 0
$$485$$ 0.459363 0.0208586
$$486$$ 0 0
$$487$$ 36.9166 1.67285 0.836426 0.548079i $$-0.184641\pi$$
0.836426 + 0.548079i $$0.184641\pi$$
$$488$$ 0 0
$$489$$ −34.1650 −1.54499
$$490$$ 0 0
$$491$$ −23.0328 −1.03946 −0.519728 0.854332i $$-0.673967\pi$$
−0.519728 + 0.854332i $$0.673967\pi$$
$$492$$ 0 0
$$493$$ 52.2463 2.35305
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −39.6990 −1.78074
$$498$$ 0 0
$$499$$ 27.0903 1.21273 0.606365 0.795187i $$-0.292627\pi$$
0.606365 + 0.795187i $$0.292627\pi$$
$$500$$ 0 0
$$501$$ −76.4901 −3.41733
$$502$$ 0 0
$$503$$ 30.0222 1.33862 0.669311 0.742983i $$-0.266589\pi$$
0.669311 + 0.742983i $$0.266589\pi$$
$$504$$ 0 0
$$505$$ −9.54516 −0.424754
$$506$$ 0 0
$$507$$ 42.8924 1.90492
$$508$$ 0 0
$$509$$ 19.1141 0.847217 0.423608 0.905845i $$-0.360763\pi$$
0.423608 + 0.905845i $$0.360763\pi$$
$$510$$ 0 0
$$511$$ 3.50330 0.154977
$$512$$ 0 0
$$513$$ −27.1186 −1.19732
$$514$$ 0 0
$$515$$ 6.41090 0.282498
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −59.9495 −2.63149
$$520$$ 0 0
$$521$$ −5.28568 −0.231570 −0.115785 0.993274i $$-0.536938\pi$$
−0.115785 + 0.993274i $$0.536938\pi$$
$$522$$ 0 0
$$523$$ 24.0373 1.05108 0.525540 0.850769i $$-0.323863\pi$$
0.525540 + 0.850769i $$0.323863\pi$$
$$524$$ 0 0
$$525$$ 12.7890 0.558157
$$526$$ 0 0
$$527$$ 58.7339 2.55849
$$528$$ 0 0
$$529$$ 45.0353 1.95805
$$530$$ 0 0
$$531$$ 68.0353 2.95248
$$532$$ 0 0
$$533$$ −0.670377 −0.0290373
$$534$$ 0 0
$$535$$ −0.435130 −0.0188123
$$536$$ 0 0
$$537$$ −23.2044 −1.00134
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −24.2529 −1.04271 −0.521356 0.853339i $$-0.674574\pi$$
−0.521356 + 0.853339i $$0.674574\pi$$
$$542$$ 0 0
$$543$$ −50.1453 −2.15194
$$544$$ 0 0
$$545$$ 8.66377 0.371115
$$546$$ 0 0
$$547$$ 37.8354 1.61772 0.808862 0.587999i $$-0.200084\pi$$
0.808862 + 0.587999i $$0.200084\pi$$
$$548$$ 0 0
$$549$$ −6.19988 −0.264605
$$550$$ 0 0
$$551$$ 10.3341 0.440250
$$552$$ 0 0
$$553$$ −22.0747 −0.938710
$$554$$ 0 0
$$555$$ 15.7890 0.670205
$$556$$ 0 0
$$557$$ −6.54516 −0.277327 −0.138664 0.990340i $$-0.544281\pi$$
−0.138664 + 0.990340i $$0.544281\pi$$
$$558$$ 0 0
$$559$$ −1.82632 −0.0772451
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −33.1892 −1.39876 −0.699380 0.714750i $$-0.746540\pi$$
−0.699380 + 0.714750i $$0.746540\pi$$
$$564$$ 0 0
$$565$$ 17.2529 0.725834
$$566$$ 0 0
$$567$$ −130.756 −5.49124
$$568$$ 0 0
$$569$$ −12.0045 −0.503256 −0.251628 0.967824i $$-0.580966\pi$$
−0.251628 + 0.967824i $$0.580966\pi$$
$$570$$ 0 0
$$571$$ −21.0419 −0.880574 −0.440287 0.897857i $$-0.645123\pi$$
−0.440287 + 0.897857i $$0.645123\pi$$
$$572$$ 0 0
$$573$$ −17.8748 −0.746730
$$574$$ 0 0
$$575$$ 8.24835 0.343980
$$576$$ 0 0
$$577$$ 35.4922 1.47756 0.738779 0.673948i $$-0.235403\pi$$
0.738779 + 0.673948i $$0.235403\pi$$
$$578$$ 0 0
$$579$$ 17.7496 0.737647
$$580$$ 0 0
$$581$$ 34.9561 1.45022
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 3.78899 0.156655
$$586$$ 0 0
$$587$$ 9.10551 0.375825 0.187912 0.982186i $$-0.439828\pi$$
0.187912 + 0.982186i $$0.439828\pi$$
$$588$$ 0 0
$$589$$ 11.6174 0.478686
$$590$$ 0 0
$$591$$ 40.6592 1.67250
$$592$$ 0 0
$$593$$ −13.7032 −0.562722 −0.281361 0.959602i $$-0.590786\pi$$
−0.281361 + 0.959602i $$0.590786\pi$$
$$594$$ 0 0
$$595$$ 29.7011 1.21763
$$596$$ 0 0
$$597$$ 41.3670 1.69304
$$598$$ 0 0
$$599$$ 27.1277 1.10841 0.554203 0.832381i $$-0.313023\pi$$
0.554203 + 0.832381i $$0.313023\pi$$
$$600$$ 0 0
$$601$$ 10.1141 0.412562 0.206281 0.978493i $$-0.433864\pi$$
0.206281 + 0.978493i $$0.433864\pi$$
$$602$$ 0 0
$$603$$ 33.5012 1.36428
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −8.63096 −0.350320 −0.175160 0.984540i $$-0.556044\pi$$
−0.175160 + 0.984540i $$0.556044\pi$$
$$608$$ 0 0
$$609$$ 85.7848 3.47618
$$610$$ 0 0
$$611$$ 0.125214 0.00506561
$$612$$ 0 0
$$613$$ 26.7824 1.08173 0.540865 0.841109i $$-0.318097\pi$$
0.540865 + 0.841109i $$0.318097\pi$$
$$614$$ 0 0
$$615$$ 4.89449 0.197365
$$616$$ 0 0
$$617$$ 19.2417 0.774643 0.387322 0.921945i $$-0.373400\pi$$
0.387322 + 0.921945i $$0.373400\pi$$
$$618$$ 0 0
$$619$$ 17.4901 0.702986 0.351493 0.936190i $$-0.385674\pi$$
0.351493 + 0.936190i $$0.385674\pi$$
$$620$$ 0 0
$$621$$ −145.189 −5.82624
$$622$$ 0 0
$$623$$ −48.7672 −1.95382
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 36.6683 1.46206
$$630$$ 0 0
$$631$$ 13.5870 0.540891 0.270445 0.962735i $$-0.412829\pi$$
0.270445 + 0.962735i $$0.412829\pi$$
$$632$$ 0 0
$$633$$ 16.6219 0.660662
$$634$$ 0 0
$$635$$ −3.14284 −0.124720
$$636$$ 0 0
$$637$$ −3.46389 −0.137244
$$638$$ 0 0
$$639$$ 85.8727 3.39707
$$640$$ 0 0
$$641$$ −20.0091 −0.790310 −0.395155 0.918614i $$-0.629309\pi$$
−0.395155 + 0.918614i $$0.629309\pi$$
$$642$$ 0 0
$$643$$ −40.8551 −1.61117 −0.805584 0.592482i $$-0.798148\pi$$
−0.805584 + 0.592482i $$0.798148\pi$$
$$644$$ 0 0
$$645$$ 13.3341 0.525032
$$646$$ 0 0
$$647$$ 21.3427 0.839069 0.419535 0.907739i $$-0.362193\pi$$
0.419535 + 0.907739i $$0.362193\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 96.4371 3.77967
$$652$$ 0 0
$$653$$ −23.5295 −0.920781 −0.460390 0.887717i $$-0.652291\pi$$
−0.460390 + 0.887717i $$0.652291\pi$$
$$654$$ 0 0
$$655$$ −16.7935 −0.656177
$$656$$ 0 0
$$657$$ −7.57797 −0.295645
$$658$$ 0 0
$$659$$ −22.7824 −0.887476 −0.443738 0.896157i $$-0.646348\pi$$
−0.443738 + 0.896157i $$0.646348\pi$$
$$660$$ 0 0
$$661$$ −26.8838 −1.04566 −0.522830 0.852437i $$-0.675124\pi$$
−0.522830 + 0.852437i $$0.675124\pi$$
$$662$$ 0 0
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ 5.87479 0.227814
$$666$$ 0 0
$$667$$ 55.3275 2.14229
$$668$$ 0 0
$$669$$ −28.1650 −1.08892
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −35.0307 −1.35034 −0.675168 0.737664i $$-0.735929\pi$$
−0.675168 + 0.737664i $$0.735929\pi$$
$$674$$ 0 0
$$675$$ −17.6022 −0.677509
$$676$$ 0 0
$$677$$ 29.7405 1.14302 0.571511 0.820595i $$-0.306358\pi$$
0.571511 + 0.820595i $$0.306358\pi$$
$$678$$ 0 0
$$679$$ 1.75165 0.0672222
$$680$$ 0 0
$$681$$ 51.2069 1.96225
$$682$$ 0 0
$$683$$ 14.6441 0.560340 0.280170 0.959950i $$-0.409609\pi$$
0.280170 + 0.959950i $$0.409609\pi$$
$$684$$ 0 0
$$685$$ −11.0045 −0.420461
$$686$$ 0 0
$$687$$ −28.9318 −1.10382
$$688$$ 0 0
$$689$$ 1.20441 0.0458843
$$690$$ 0 0
$$691$$ −32.5825 −1.23950 −0.619748 0.784801i $$-0.712765\pi$$
−0.619748 + 0.784801i $$0.712765\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 20.9561 0.794909
$$696$$ 0 0
$$697$$ 11.3670 0.430554
$$698$$ 0 0
$$699$$ −88.5783 −3.35034
$$700$$ 0 0
$$701$$ −31.5295 −1.19085 −0.595426 0.803410i $$-0.703017\pi$$
−0.595426 + 0.803410i $$0.703017\pi$$
$$702$$ 0 0
$$703$$ 7.25287 0.273547
$$704$$ 0 0
$$705$$ −0.914200 −0.0344307
$$706$$ 0 0
$$707$$ −36.3978 −1.36888
$$708$$ 0 0
$$709$$ 32.1650 1.20798 0.603991 0.796991i $$-0.293576\pi$$
0.603991 + 0.796991i $$0.293576\pi$$
$$710$$ 0 0
$$711$$ 47.7496 1.79075
$$712$$ 0 0
$$713$$ 62.1978 2.32933
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 4.07675 0.152249
$$718$$ 0 0
$$719$$ −17.8263 −0.664810 −0.332405 0.943137i $$-0.607860\pi$$
−0.332405 + 0.943137i $$0.607860\pi$$
$$720$$ 0 0
$$721$$ 24.4462 0.910422
$$722$$ 0 0
$$723$$ 47.3518 1.76103
$$724$$ 0 0
$$725$$ 6.70771 0.249118
$$726$$ 0 0
$$727$$ −0.511878 −0.0189845 −0.00949225 0.999955i $$-0.503022\pi$$
−0.00949225 + 0.999955i $$0.503022\pi$$
$$728$$ 0 0
$$729$$ 105.732 3.91599
$$730$$ 0 0
$$731$$ 30.9672 1.14536
$$732$$ 0 0
$$733$$ 50.1978 1.85410 0.927049 0.374940i $$-0.122337\pi$$
0.927049 + 0.374940i $$0.122337\pi$$
$$734$$ 0 0
$$735$$ 25.2902 0.932843
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 39.2044 1.44216 0.721079 0.692853i $$-0.243647\pi$$
0.721079 + 0.692853i $$0.243647\pi$$
$$740$$ 0 0
$$741$$ 2.37356 0.0871950
$$742$$ 0 0
$$743$$ 2.38666 0.0875582 0.0437791 0.999041i $$-0.486060\pi$$
0.0437791 + 0.999041i $$0.486060\pi$$
$$744$$ 0 0
$$745$$ 1.62191 0.0594222
$$746$$ 0 0
$$747$$ −75.6132 −2.76654
$$748$$ 0 0
$$749$$ −1.65925 −0.0606275
$$750$$ 0 0
$$751$$ 8.25948 0.301393 0.150696 0.988580i $$-0.451848\pi$$
0.150696 + 0.988580i $$0.451848\pi$$
$$752$$ 0 0
$$753$$ 30.7450 1.12041
$$754$$ 0 0
$$755$$ −7.00453 −0.254921
$$756$$ 0 0
$$757$$ 15.0792 0.548063 0.274031 0.961721i $$-0.411643\pi$$
0.274031 + 0.961721i $$0.411643\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −17.2529 −0.625416 −0.312708 0.949849i $$-0.601236\pi$$
−0.312708 + 0.949849i $$0.601236\pi$$
$$762$$ 0 0
$$763$$ 33.0369 1.19601
$$764$$ 0 0
$$765$$ −64.2463 −2.32283
$$766$$ 0 0
$$767$$ −3.78899 −0.136812
$$768$$ 0 0
$$769$$ −42.1322 −1.51933 −0.759663 0.650317i $$-0.774636\pi$$
−0.759663 + 0.650317i $$0.774636\pi$$
$$770$$ 0 0
$$771$$ 35.0792 1.26335
$$772$$ 0 0
$$773$$ −46.8197 −1.68399 −0.841994 0.539487i $$-0.818618\pi$$
−0.841994 + 0.539487i $$0.818618\pi$$
$$774$$ 0 0
$$775$$ 7.54064 0.270868
$$776$$ 0 0
$$777$$ 60.2069 2.15991
$$778$$ 0 0
$$779$$ 2.24835 0.0805554
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −118.071 −4.21950
$$784$$ 0 0
$$785$$ 10.3341 0.368842
$$786$$ 0 0
$$787$$ −11.0682 −0.394538 −0.197269 0.980349i $$-0.563207\pi$$
−0.197269 + 0.980349i $$0.563207\pi$$
$$788$$ 0 0
$$789$$ 35.6244 1.26826
$$790$$ 0 0
$$791$$ 65.7890 2.33919
$$792$$ 0 0
$$793$$ 0.345281 0.0122613
$$794$$ 0 0
$$795$$ −8.79351 −0.311874
$$796$$ 0 0
$$797$$ −3.41750 −0.121054 −0.0605271 0.998167i $$-0.519278\pi$$
−0.0605271 + 0.998167i $$0.519278\pi$$
$$798$$ 0 0
$$799$$ −2.12313 −0.0751111
$$800$$ 0 0
$$801$$ 105.488 3.72724
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 31.4528 1.10856
$$806$$ 0 0
$$807$$ −33.5537 −1.18115
$$808$$ 0 0
$$809$$ 44.5452 1.56612 0.783062 0.621943i $$-0.213657\pi$$
0.783062 + 0.621943i $$0.213657\pi$$
$$810$$ 0 0
$$811$$ −3.63757 −0.127732 −0.0638662 0.997958i $$-0.520343\pi$$
−0.0638662 + 0.997958i $$0.520343\pi$$
$$812$$ 0 0
$$813$$ 8.53611 0.299374
$$814$$ 0 0
$$815$$ −10.1868 −0.356827
$$816$$ 0 0
$$817$$ 6.12521 0.214294
$$818$$ 0 0
$$819$$ 14.4482 0.504862
$$820$$ 0 0
$$821$$ −8.96267 −0.312799 −0.156400 0.987694i $$-0.549989\pi$$
−0.156400 + 0.987694i $$0.549989\pi$$
$$822$$ 0 0
$$823$$ 36.1756 1.26100 0.630502 0.776188i $$-0.282849\pi$$
0.630502 + 0.776188i $$0.282849\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −43.7536 −1.52146 −0.760731 0.649067i $$-0.775159\pi$$
−0.760731 + 0.649067i $$0.775159\pi$$
$$828$$ 0 0
$$829$$ 19.6310 0.681812 0.340906 0.940097i $$-0.389266\pi$$
0.340906 + 0.940097i $$0.389266\pi$$
$$830$$ 0 0
$$831$$ −57.9958 −2.01185
$$832$$ 0 0
$$833$$ 58.7339 2.03501
$$834$$ 0 0
$$835$$ −22.8066 −0.789255
$$836$$ 0 0
$$837$$ −132.732 −4.58788
$$838$$ 0 0
$$839$$ 4.00905 0.138408 0.0692039 0.997603i $$-0.477954\pi$$
0.0692039 + 0.997603i $$0.477954\pi$$
$$840$$ 0 0
$$841$$ 15.9934 0.551496
$$842$$ 0 0
$$843$$ −56.4482 −1.94418
$$844$$ 0 0
$$845$$ 12.7890 0.439954
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −4.37809 −0.150256
$$850$$ 0 0
$$851$$ 38.8308 1.33110
$$852$$ 0 0
$$853$$ 19.2529 0.659206 0.329603 0.944120i $$-0.393085\pi$$
0.329603 + 0.944120i $$0.393085\pi$$
$$854$$ 0 0
$$855$$ −12.7077 −0.434595
$$856$$ 0 0
$$857$$ −21.6174 −0.738436 −0.369218 0.929343i $$-0.620374\pi$$
−0.369218 + 0.929343i $$0.620374\pi$$
$$858$$ 0 0
$$859$$ 37.0045 1.26258 0.631289 0.775548i $$-0.282526\pi$$
0.631289 + 0.775548i $$0.282526\pi$$
$$860$$ 0 0
$$861$$ 18.6638 0.636060
$$862$$ 0 0
$$863$$ 28.8087 0.980659 0.490330 0.871537i $$-0.336876\pi$$
0.490330 + 0.871537i $$0.336876\pi$$
$$864$$ 0 0
$$865$$ −17.8748 −0.607761
$$866$$ 0 0
$$867$$ −146.457 −4.97395
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −1.86573 −0.0632180
$$872$$ 0 0
$$873$$ −3.78899 −0.128238
$$874$$ 0 0
$$875$$ 3.81322 0.128910
$$876$$ 0 0
$$877$$ 22.9166 0.773840 0.386920 0.922113i $$-0.373539\pi$$
0.386920 + 0.922113i $$0.373539\pi$$
$$878$$ 0 0
$$879$$ −13.1277 −0.442785
$$880$$ 0 0
$$881$$ 8.76070 0.295156 0.147578 0.989050i $$-0.452852\pi$$
0.147578 + 0.989050i $$0.452852\pi$$
$$882$$ 0 0
$$883$$ 1.66377 0.0559904 0.0279952 0.999608i $$-0.491088\pi$$
0.0279952 + 0.999608i $$0.491088\pi$$
$$884$$ 0 0
$$885$$ 27.6638 0.929908
$$886$$ 0 0
$$887$$ −28.4815 −0.956316 −0.478158 0.878274i $$-0.658695\pi$$
−0.478158 + 0.878274i $$0.658695\pi$$
$$888$$ 0 0
$$889$$ −11.9843 −0.401942
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −0.419949 −0.0140531
$$894$$ 0 0
$$895$$ −6.91873 −0.231268
$$896$$ 0 0
$$897$$ 12.7077 0.424298
$$898$$ 0 0
$$899$$ 50.5804 1.68695
$$900$$ 0 0
$$901$$ −20.4220 −0.680356
$$902$$ 0 0
$$903$$ 50.8460 1.69205
$$904$$ 0 0
$$905$$ −14.9515 −0.497006
$$906$$ 0 0
$$907$$ 7.22864 0.240023 0.120012 0.992772i $$-0.461707\pi$$
0.120012 + 0.992772i $$0.461707\pi$$
$$908$$ 0 0
$$909$$ 78.7318 2.61137
$$910$$ 0 0
$$911$$ −11.9515 −0.395972 −0.197986 0.980205i $$-0.563440\pi$$
−0.197986 + 0.980205i $$0.563440\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −2.52093 −0.0833394
$$916$$ 0 0
$$917$$ −64.0373 −2.11470
$$918$$ 0 0
$$919$$ 8.58250 0.283110 0.141555 0.989930i $$-0.454790\pi$$
0.141555 + 0.989930i $$0.454790\pi$$
$$920$$ 0 0
$$921$$ 70.9913 2.33924
$$922$$ 0 0
$$923$$ −4.78238 −0.157414
$$924$$ 0 0
$$925$$ 4.70771 0.154789
$$926$$ 0 0
$$927$$ −52.8793 −1.73678
$$928$$ 0 0
$$929$$ −22.5936 −0.741273 −0.370636 0.928778i $$-0.620860\pi$$
−0.370636 + 0.928778i $$0.620860\pi$$
$$930$$ 0 0
$$931$$ 11.6174 0.380744
$$932$$ 0 0
$$933$$ −19.9979 −0.654703
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −31.5295 −1.03002 −0.515012 0.857183i $$-0.672213\pi$$
−0.515012 + 0.857183i $$0.672213\pi$$
$$938$$ 0 0
$$939$$ 94.9913 3.09992
$$940$$ 0 0
$$941$$ −59.5804 −1.94227 −0.971133 0.238538i $$-0.923332\pi$$
−0.971133 + 0.238538i $$0.923332\pi$$
$$942$$ 0 0
$$943$$ 12.0373 0.391990
$$944$$ 0 0
$$945$$ −67.1211 −2.18345
$$946$$ 0 0
$$947$$ −25.3781 −0.824677 −0.412339 0.911031i $$-0.635288\pi$$
−0.412339 + 0.911031i $$0.635288\pi$$
$$948$$ 0 0
$$949$$ 0.422029 0.0136996
$$950$$ 0 0
$$951$$ 5.29229 0.171614
$$952$$ 0 0
$$953$$ 3.13879 0.101675 0.0508377 0.998707i $$-0.483811\pi$$
0.0508377 + 0.998707i $$0.483811\pi$$
$$954$$ 0 0
$$955$$ −5.32962 −0.172463
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −41.9627 −1.35505
$$960$$ 0 0
$$961$$ 25.8612 0.834232
$$962$$ 0 0
$$963$$ 3.58910 0.115657
$$964$$ 0 0
$$965$$ 5.29229 0.170365
$$966$$ 0 0
$$967$$ 1.66377 0.0535033 0.0267516 0.999642i $$-0.491484\pi$$
0.0267516 + 0.999642i $$0.491484\pi$$
$$968$$ 0 0
$$969$$ −40.2463 −1.29290
$$970$$ 0 0
$$971$$ −21.3013 −0.683593 −0.341796 0.939774i $$-0.611035\pi$$
−0.341796 + 0.939774i $$0.611035\pi$$
$$972$$ 0 0
$$973$$ 79.9100 2.56180
$$974$$ 0 0
$$975$$ 1.54064 0.0493399
$$976$$ 0 0
$$977$$ −1.95153 −0.0624351 −0.0312176 0.999513i $$-0.509938\pi$$
−0.0312176 + 0.999513i $$0.509938\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −71.4618 −2.28160
$$982$$ 0 0
$$983$$ −41.0065 −1.30790 −0.653952 0.756536i $$-0.726890\pi$$
−0.653952 + 0.756536i $$0.726890\pi$$
$$984$$ 0 0
$$985$$ 12.1231 0.386275
$$986$$ 0 0
$$987$$ −3.48604 −0.110962
$$988$$ 0 0
$$989$$ 32.7935 1.04277
$$990$$ 0 0
$$991$$ −46.1120 −1.46480 −0.732398 0.680877i $$-0.761599\pi$$
−0.732398 + 0.680877i $$0.761599\pi$$
$$992$$ 0 0
$$993$$ 60.1120 1.90760
$$994$$ 0 0
$$995$$ 12.3341 0.391019
$$996$$ 0 0
$$997$$ −17.8354 −0.564852 −0.282426 0.959289i $$-0.591139\pi$$
−0.282426 + 0.959289i $$0.591139\pi$$
$$998$$ 0 0
$$999$$ −82.8661 −2.62177
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9680.2.a.by.1.1 3
4.3 odd 2 4840.2.a.v.1.3 yes 3
11.10 odd 2 9680.2.a.cd.1.1 3
44.43 even 2 4840.2.a.s.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.s.1.3 3 44.43 even 2
4840.2.a.v.1.3 yes 3 4.3 odd 2
9680.2.a.by.1.1 3 1.1 even 1 trivial
9680.2.a.cd.1.1 3 11.10 odd 2