# Properties

 Label 6664.2.a.o.1.4 Level $6664$ Weight $2$ Character 6664.1 Self dual yes Analytic conductor $53.212$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6664,2,Mod(1,6664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6664, 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("6664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6664 = 2^{3} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6664.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: $$53.2123079070$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.5225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + x + 11$$ x^4 - x^3 - 8*x^2 + x + 11 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 952) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$3.10522$$ of defining polynomial Character $$\chi$$ $$=$$ 6664.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.10522 q^{3} +3.40632 q^{5} +1.43195 q^{9} +O(q^{10})$$ $$q+2.10522 q^{3} +3.40632 q^{5} +1.43195 q^{9} -1.23607 q^{11} +6.47214 q^{13} +7.17104 q^{15} -1.00000 q^{17} +2.21044 q^{19} +1.39781 q^{23} +6.60299 q^{25} -3.30110 q^{27} -0.633874 q^{29} -0.965861 q^{31} -2.60219 q^{33} -8.31040 q^{37} +13.6253 q^{39} +5.60299 q^{41} +11.0796 q^{43} +4.87766 q^{45} -7.67652 q^{47} -2.10522 q^{51} +11.7167 q^{53} -4.21044 q^{55} +4.65345 q^{57} +0.602193 q^{59} +9.84352 q^{61} +22.0461 q^{65} +5.30636 q^{67} +2.94269 q^{69} -3.33603 q^{71} -14.0239 q^{73} +13.9007 q^{75} -0.323477 q^{79} -11.2454 q^{81} -13.8870 q^{83} -3.40632 q^{85} -1.33444 q^{87} +14.7509 q^{89} -2.03335 q^{93} +7.52945 q^{95} +5.90855 q^{97} -1.76998 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{3} + q^{5} + 7 q^{9}+O(q^{10})$$ 4 * q - 3 * q^3 + q^5 + 7 * q^9 $$4 q - 3 q^{3} + q^{5} + 7 q^{9} + 4 q^{11} + 8 q^{13} + 12 q^{15} - 4 q^{17} - 14 q^{19} + 8 q^{23} + 11 q^{25} - 12 q^{27} + 4 q^{29} - 5 q^{31} - 8 q^{33} - 4 q^{37} + 4 q^{39} + 7 q^{41} + 19 q^{43} + 2 q^{45} - 8 q^{47} + 3 q^{51} + 5 q^{53} + 6 q^{55} + 44 q^{57} + 23 q^{61} - 8 q^{65} + 15 q^{67} + 2 q^{69} + 2 q^{71} + 5 q^{73} - 10 q^{75} - 24 q^{79} - 8 q^{81} - 10 q^{83} - q^{85} - 16 q^{87} + 16 q^{89} - 20 q^{93} + 22 q^{95} + 15 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q - 3 * q^3 + q^5 + 7 * q^9 + 4 * q^11 + 8 * q^13 + 12 * q^15 - 4 * q^17 - 14 * q^19 + 8 * q^23 + 11 * q^25 - 12 * q^27 + 4 * q^29 - 5 * q^31 - 8 * q^33 - 4 * q^37 + 4 * q^39 + 7 * q^41 + 19 * q^43 + 2 * q^45 - 8 * q^47 + 3 * q^51 + 5 * q^53 + 6 * q^55 + 44 * q^57 + 23 * q^61 - 8 * q^65 + 15 * q^67 + 2 * q^69 + 2 * q^71 + 5 * q^73 - 10 * q^75 - 24 * q^79 - 8 * q^81 - 10 * q^83 - q^85 - 16 * q^87 + 16 * q^89 - 20 * q^93 + 22 * q^95 + 15 * q^97 + 2 * q^99

## 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
$$3$$ 2.10522 1.21545 0.607724 0.794148i $$-0.292083\pi$$
0.607724 + 0.794148i $$0.292083\pi$$
$$4$$ 0 0
$$5$$ 3.40632 1.52335 0.761675 0.647959i $$-0.224377\pi$$
0.761675 + 0.647959i $$0.224377\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.43195 0.477315
$$10$$ 0 0
$$11$$ −1.23607 −0.372689 −0.186344 0.982485i $$-0.559664\pi$$
−0.186344 + 0.982485i $$0.559664\pi$$
$$12$$ 0 0
$$13$$ 6.47214 1.79505 0.897524 0.440966i $$-0.145364\pi$$
0.897524 + 0.440966i $$0.145364\pi$$
$$14$$ 0 0
$$15$$ 7.17104 1.85155
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ 2.21044 0.507109 0.253555 0.967321i $$-0.418400\pi$$
0.253555 + 0.967321i $$0.418400\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.39781 0.291463 0.145731 0.989324i $$-0.453446\pi$$
0.145731 + 0.989324i $$0.453446\pi$$
$$24$$ 0 0
$$25$$ 6.60299 1.32060
$$26$$ 0 0
$$27$$ −3.30110 −0.635296
$$28$$ 0 0
$$29$$ −0.633874 −0.117708 −0.0588538 0.998267i $$-0.518745\pi$$
−0.0588538 + 0.998267i $$0.518745\pi$$
$$30$$ 0 0
$$31$$ −0.965861 −0.173474 −0.0867368 0.996231i $$-0.527644\pi$$
−0.0867368 + 0.996231i $$0.527644\pi$$
$$32$$ 0 0
$$33$$ −2.60219 −0.452984
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.31040 −1.36622 −0.683110 0.730315i $$-0.739373\pi$$
−0.683110 + 0.730315i $$0.739373\pi$$
$$38$$ 0 0
$$39$$ 13.6253 2.18179
$$40$$ 0 0
$$41$$ 5.60299 0.875039 0.437520 0.899209i $$-0.355857\pi$$
0.437520 + 0.899209i $$0.355857\pi$$
$$42$$ 0 0
$$43$$ 11.0796 1.68962 0.844811 0.535065i $$-0.179713\pi$$
0.844811 + 0.535065i $$0.179713\pi$$
$$44$$ 0 0
$$45$$ 4.87766 0.727119
$$46$$ 0 0
$$47$$ −7.67652 −1.11974 −0.559868 0.828582i $$-0.689148\pi$$
−0.559868 + 0.828582i $$0.689148\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −2.10522 −0.294790
$$52$$ 0 0
$$53$$ 11.7167 1.60941 0.804707 0.593672i $$-0.202322\pi$$
0.804707 + 0.593672i $$0.202322\pi$$
$$54$$ 0 0
$$55$$ −4.21044 −0.567735
$$56$$ 0 0
$$57$$ 4.65345 0.616365
$$58$$ 0 0
$$59$$ 0.602193 0.0783989 0.0391995 0.999231i $$-0.487519\pi$$
0.0391995 + 0.999231i $$0.487519\pi$$
$$60$$ 0 0
$$61$$ 9.84352 1.26033 0.630167 0.776460i $$-0.282986\pi$$
0.630167 + 0.776460i $$0.282986\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 22.0461 2.73449
$$66$$ 0 0
$$67$$ 5.30636 0.648275 0.324137 0.946010i $$-0.394926\pi$$
0.324137 + 0.946010i $$0.394926\pi$$
$$68$$ 0 0
$$69$$ 2.94269 0.354258
$$70$$ 0 0
$$71$$ −3.33603 −0.395914 −0.197957 0.980211i $$-0.563431\pi$$
−0.197957 + 0.980211i $$0.563431\pi$$
$$72$$ 0 0
$$73$$ −14.0239 −1.64137 −0.820684 0.571382i $$-0.806408\pi$$
−0.820684 + 0.571382i $$0.806408\pi$$
$$74$$ 0 0
$$75$$ 13.9007 1.60512
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −0.323477 −0.0363940 −0.0181970 0.999834i $$-0.505793\pi$$
−0.0181970 + 0.999834i $$0.505793\pi$$
$$80$$ 0 0
$$81$$ −11.2454 −1.24949
$$82$$ 0 0
$$83$$ −13.8870 −1.52429 −0.762146 0.647405i $$-0.775854\pi$$
−0.762146 + 0.647405i $$0.775854\pi$$
$$84$$ 0 0
$$85$$ −3.40632 −0.369467
$$86$$ 0 0
$$87$$ −1.33444 −0.143067
$$88$$ 0 0
$$89$$ 14.7509 1.56359 0.781794 0.623537i $$-0.214305\pi$$
0.781794 + 0.623537i $$0.214305\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −2.03335 −0.210848
$$94$$ 0 0
$$95$$ 7.52945 0.772505
$$96$$ 0 0
$$97$$ 5.90855 0.599922 0.299961 0.953951i $$-0.403026\pi$$
0.299961 + 0.953951i $$0.403026\pi$$
$$98$$ 0 0
$$99$$ −1.76998 −0.177890
$$100$$ 0 0
$$101$$ −5.08485 −0.505961 −0.252981 0.967471i $$-0.581411\pi$$
−0.252981 + 0.967471i $$0.581411\pi$$
$$102$$ 0 0
$$103$$ 5.28477 0.520724 0.260362 0.965511i $$-0.416158\pi$$
0.260362 + 0.965511i $$0.416158\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −9.25309 −0.894530 −0.447265 0.894402i $$-0.647602\pi$$
−0.447265 + 0.894402i $$0.647602\pi$$
$$108$$ 0 0
$$109$$ −16.1000 −1.54210 −0.771048 0.636777i $$-0.780267\pi$$
−0.771048 + 0.636777i $$0.780267\pi$$
$$110$$ 0 0
$$111$$ −17.4952 −1.66057
$$112$$ 0 0
$$113$$ 10.5509 0.992548 0.496274 0.868166i $$-0.334701\pi$$
0.496274 + 0.868166i $$0.334701\pi$$
$$114$$ 0 0
$$115$$ 4.76137 0.444000
$$116$$ 0 0
$$117$$ 9.26775 0.856804
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −9.47214 −0.861103
$$122$$ 0 0
$$123$$ 11.7955 1.06357
$$124$$ 0 0
$$125$$ 5.46027 0.488382
$$126$$ 0 0
$$127$$ −17.9272 −1.59078 −0.795389 0.606100i $$-0.792733\pi$$
−0.795389 + 0.606100i $$0.792733\pi$$
$$128$$ 0 0
$$129$$ 23.3250 2.05365
$$130$$ 0 0
$$131$$ −1.49777 −0.130860 −0.0654302 0.997857i $$-0.520842\pi$$
−0.0654302 + 0.997857i $$0.520842\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −11.2446 −0.967779
$$136$$ 0 0
$$137$$ 13.6030 1.16218 0.581091 0.813839i $$-0.302626\pi$$
0.581091 + 0.813839i $$0.302626\pi$$
$$138$$ 0 0
$$139$$ 6.28073 0.532724 0.266362 0.963873i $$-0.414178\pi$$
0.266362 + 0.963873i $$0.414178\pi$$
$$140$$ 0 0
$$141$$ −16.1608 −1.36098
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ −2.15918 −0.179310
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 21.7004 1.77776 0.888882 0.458136i $$-0.151483\pi$$
0.888882 + 0.458136i $$0.151483\pi$$
$$150$$ 0 0
$$151$$ 4.60745 0.374949 0.187475 0.982269i $$-0.439970\pi$$
0.187475 + 0.982269i $$0.439970\pi$$
$$152$$ 0 0
$$153$$ −1.43195 −0.115766
$$154$$ 0 0
$$155$$ −3.29003 −0.264261
$$156$$ 0 0
$$157$$ 4.53391 0.361846 0.180923 0.983497i $$-0.442092\pi$$
0.180923 + 0.983497i $$0.442092\pi$$
$$158$$ 0 0
$$159$$ 24.6662 1.95616
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −12.2421 −0.958877 −0.479438 0.877576i $$-0.659160\pi$$
−0.479438 + 0.877576i $$0.659160\pi$$
$$164$$ 0 0
$$165$$ −8.86389 −0.690053
$$166$$ 0 0
$$167$$ −1.34129 −0.103792 −0.0518959 0.998652i $$-0.516526\pi$$
−0.0518959 + 0.998652i $$0.516526\pi$$
$$168$$ 0 0
$$169$$ 28.8885 2.22220
$$170$$ 0 0
$$171$$ 3.16523 0.242051
$$172$$ 0 0
$$173$$ −1.83300 −0.139361 −0.0696803 0.997569i $$-0.522198\pi$$
−0.0696803 + 0.997569i $$0.522198\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1.26775 0.0952898
$$178$$ 0 0
$$179$$ 20.2343 1.51238 0.756191 0.654351i $$-0.227058\pi$$
0.756191 + 0.654351i $$0.227058\pi$$
$$180$$ 0 0
$$181$$ 6.03010 0.448214 0.224107 0.974565i $$-0.428054\pi$$
0.224107 + 0.974565i $$0.428054\pi$$
$$182$$ 0 0
$$183$$ 20.7228 1.53187
$$184$$ 0 0
$$185$$ −28.3078 −2.08123
$$186$$ 0 0
$$187$$ 1.23607 0.0903902
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.9146 −0.789753 −0.394876 0.918734i $$-0.629213\pi$$
−0.394876 + 0.918734i $$0.629213\pi$$
$$192$$ 0 0
$$193$$ −19.1717 −1.38001 −0.690006 0.723804i $$-0.742392\pi$$
−0.690006 + 0.723804i $$0.742392\pi$$
$$194$$ 0 0
$$195$$ 46.4119 3.32363
$$196$$ 0 0
$$197$$ 2.89557 0.206301 0.103151 0.994666i $$-0.467108\pi$$
0.103151 + 0.994666i $$0.467108\pi$$
$$198$$ 0 0
$$199$$ −26.1830 −1.85607 −0.928033 0.372498i $$-0.878501\pi$$
−0.928033 + 0.372498i $$0.878501\pi$$
$$200$$ 0 0
$$201$$ 11.1710 0.787944
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 19.0855 1.33299
$$206$$ 0 0
$$207$$ 2.00158 0.139120
$$208$$ 0 0
$$209$$ −2.73225 −0.188994
$$210$$ 0 0
$$211$$ −2.11048 −0.145291 −0.0726456 0.997358i $$-0.523144\pi$$
−0.0726456 + 0.997358i $$0.523144\pi$$
$$212$$ 0 0
$$213$$ −7.02307 −0.481213
$$214$$ 0 0
$$215$$ 37.7406 2.57389
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −29.5233 −1.99500
$$220$$ 0 0
$$221$$ −6.47214 −0.435363
$$222$$ 0 0
$$223$$ 18.8126 1.25979 0.629893 0.776682i $$-0.283099\pi$$
0.629893 + 0.776682i $$0.283099\pi$$
$$224$$ 0 0
$$225$$ 9.45512 0.630341
$$226$$ 0 0
$$227$$ 17.3737 1.15313 0.576565 0.817051i $$-0.304393\pi$$
0.576565 + 0.817051i $$0.304393\pi$$
$$228$$ 0 0
$$229$$ 2.71523 0.179428 0.0897138 0.995968i $$-0.471405\pi$$
0.0897138 + 0.995968i $$0.471405\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0.715233 0.0468565 0.0234282 0.999726i $$-0.492542\pi$$
0.0234282 + 0.999726i $$0.492542\pi$$
$$234$$ 0 0
$$235$$ −26.1487 −1.70575
$$236$$ 0 0
$$237$$ −0.680990 −0.0442351
$$238$$ 0 0
$$239$$ 8.60745 0.556770 0.278385 0.960470i $$-0.410201\pi$$
0.278385 + 0.960470i $$0.410201\pi$$
$$240$$ 0 0
$$241$$ −6.31241 −0.406618 −0.203309 0.979115i $$-0.565170\pi$$
−0.203309 + 0.979115i $$0.565170\pi$$
$$242$$ 0 0
$$243$$ −13.7707 −0.883389
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 14.3063 0.910285
$$248$$ 0 0
$$249$$ −29.2351 −1.85270
$$250$$ 0 0
$$251$$ 26.5711 1.67715 0.838577 0.544783i $$-0.183388\pi$$
0.838577 + 0.544783i $$0.183388\pi$$
$$252$$ 0 0
$$253$$ −1.72778 −0.108625
$$254$$ 0 0
$$255$$ −7.17104 −0.449068
$$256$$ 0 0
$$257$$ −6.75085 −0.421107 −0.210553 0.977582i $$-0.567527\pi$$
−0.210553 + 0.977582i $$0.567527\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −0.907674 −0.0561836
$$262$$ 0 0
$$263$$ 1.20439 0.0742657 0.0371328 0.999310i $$-0.488178\pi$$
0.0371328 + 0.999310i $$0.488178\pi$$
$$264$$ 0 0
$$265$$ 39.9108 2.45170
$$266$$ 0 0
$$267$$ 31.0538 1.90046
$$268$$ 0 0
$$269$$ −18.0673 −1.10158 −0.550791 0.834643i $$-0.685674\pi$$
−0.550791 + 0.834643i $$0.685674\pi$$
$$270$$ 0 0
$$271$$ −19.7464 −1.19951 −0.599754 0.800185i $$-0.704735\pi$$
−0.599754 + 0.800185i $$0.704735\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −8.16174 −0.492171
$$276$$ 0 0
$$277$$ −7.99744 −0.480519 −0.240260 0.970709i $$-0.577233\pi$$
−0.240260 + 0.970709i $$0.577233\pi$$
$$278$$ 0 0
$$279$$ −1.38306 −0.0828016
$$280$$ 0 0
$$281$$ 3.07890 0.183672 0.0918359 0.995774i $$-0.470726\pi$$
0.0918359 + 0.995774i $$0.470726\pi$$
$$282$$ 0 0
$$283$$ −28.6927 −1.70560 −0.852801 0.522236i $$-0.825098\pi$$
−0.852801 + 0.522236i $$0.825098\pi$$
$$284$$ 0 0
$$285$$ 15.8511 0.938940
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 12.4388 0.729175
$$292$$ 0 0
$$293$$ 2.14866 0.125526 0.0627630 0.998028i $$-0.480009\pi$$
0.0627630 + 0.998028i $$0.480009\pi$$
$$294$$ 0 0
$$295$$ 2.05126 0.119429
$$296$$ 0 0
$$297$$ 4.08038 0.236768
$$298$$ 0 0
$$299$$ 9.04679 0.523190
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −10.7047 −0.614970
$$304$$ 0 0
$$305$$ 33.5301 1.91993
$$306$$ 0 0
$$307$$ −16.0291 −0.914830 −0.457415 0.889253i $$-0.651225\pi$$
−0.457415 + 0.889253i $$0.651225\pi$$
$$308$$ 0 0
$$309$$ 11.1256 0.632913
$$310$$ 0 0
$$311$$ −34.4505 −1.95351 −0.976756 0.214356i $$-0.931235\pi$$
−0.976756 + 0.214356i $$0.931235\pi$$
$$312$$ 0 0
$$313$$ −32.5585 −1.84031 −0.920157 0.391551i $$-0.871939\pi$$
−0.920157 + 0.391551i $$0.871939\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.82571 0.158708 0.0793539 0.996847i $$-0.474714\pi$$
0.0793539 + 0.996847i $$0.474714\pi$$
$$318$$ 0 0
$$319$$ 0.783512 0.0438682
$$320$$ 0 0
$$321$$ −19.4798 −1.08725
$$322$$ 0 0
$$323$$ −2.21044 −0.122992
$$324$$ 0 0
$$325$$ 42.7354 2.37053
$$326$$ 0 0
$$327$$ −33.8939 −1.87434
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 16.5822 0.911439 0.455720 0.890123i $$-0.349382\pi$$
0.455720 + 0.890123i $$0.349382\pi$$
$$332$$ 0 0
$$333$$ −11.9000 −0.652118
$$334$$ 0 0
$$335$$ 18.0751 0.987549
$$336$$ 0 0
$$337$$ 15.7553 0.858247 0.429123 0.903246i $$-0.358823\pi$$
0.429123 + 0.903246i $$0.358823\pi$$
$$338$$ 0 0
$$339$$ 22.2120 1.20639
$$340$$ 0 0
$$341$$ 1.19387 0.0646516
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 10.0237 0.539659
$$346$$ 0 0
$$347$$ −12.6630 −0.679785 −0.339893 0.940464i $$-0.610391\pi$$
−0.339893 + 0.940464i $$0.610391\pi$$
$$348$$ 0 0
$$349$$ 25.1547 1.34650 0.673250 0.739415i $$-0.264898\pi$$
0.673250 + 0.739415i $$0.264898\pi$$
$$350$$ 0 0
$$351$$ −21.3651 −1.14039
$$352$$ 0 0
$$353$$ −28.5801 −1.52116 −0.760581 0.649243i $$-0.775086\pi$$
−0.760581 + 0.649243i $$0.775086\pi$$
$$354$$ 0 0
$$355$$ −11.3636 −0.603115
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −25.4038 −1.34076 −0.670379 0.742018i $$-0.733869\pi$$
−0.670379 + 0.742018i $$0.733869\pi$$
$$360$$ 0 0
$$361$$ −14.1140 −0.742840
$$362$$ 0 0
$$363$$ −19.9409 −1.04663
$$364$$ 0 0
$$365$$ −47.7697 −2.50038
$$366$$ 0 0
$$367$$ 4.87073 0.254250 0.127125 0.991887i $$-0.459425\pi$$
0.127125 + 0.991887i $$0.459425\pi$$
$$368$$ 0 0
$$369$$ 8.02317 0.417670
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 23.2388 1.20326 0.601629 0.798776i $$-0.294519\pi$$
0.601629 + 0.798776i $$0.294519\pi$$
$$374$$ 0 0
$$375$$ 11.4951 0.593603
$$376$$ 0 0
$$377$$ −4.10252 −0.211291
$$378$$ 0 0
$$379$$ 15.8658 0.814971 0.407486 0.913212i $$-0.366406\pi$$
0.407486 + 0.913212i $$0.366406\pi$$
$$380$$ 0 0
$$381$$ −37.7406 −1.93351
$$382$$ 0 0
$$383$$ 2.06828 0.105684 0.0528421 0.998603i $$-0.483172\pi$$
0.0528421 + 0.998603i $$0.483172\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 15.8654 0.806482
$$388$$ 0 0
$$389$$ 3.51679 0.178308 0.0891542 0.996018i $$-0.471584\pi$$
0.0891542 + 0.996018i $$0.471584\pi$$
$$390$$ 0 0
$$391$$ −1.39781 −0.0706901
$$392$$ 0 0
$$393$$ −3.15313 −0.159054
$$394$$ 0 0
$$395$$ −1.10187 −0.0554408
$$396$$ 0 0
$$397$$ −5.47459 −0.274762 −0.137381 0.990518i $$-0.543868\pi$$
−0.137381 + 0.990518i $$0.543868\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.76137 −0.337647 −0.168823 0.985646i $$-0.553997\pi$$
−0.168823 + 0.985646i $$0.553997\pi$$
$$402$$ 0 0
$$403$$ −6.25118 −0.311393
$$404$$ 0 0
$$405$$ −38.3053 −1.90340
$$406$$ 0 0
$$407$$ 10.2722 0.509175
$$408$$ 0 0
$$409$$ 14.6378 0.723793 0.361897 0.932218i $$-0.382129\pi$$
0.361897 + 0.932218i $$0.382129\pi$$
$$410$$ 0 0
$$411$$ 28.6373 1.41257
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −47.3034 −2.32203
$$416$$ 0 0
$$417$$ 13.2223 0.647499
$$418$$ 0 0
$$419$$ −29.5736 −1.44476 −0.722382 0.691494i $$-0.756953\pi$$
−0.722382 + 0.691494i $$0.756953\pi$$
$$420$$ 0 0
$$421$$ 12.0223 0.585930 0.292965 0.956123i $$-0.405358\pi$$
0.292965 + 0.956123i $$0.405358\pi$$
$$422$$ 0 0
$$423$$ −10.9924 −0.534467
$$424$$ 0 0
$$425$$ −6.60299 −0.320292
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −16.8417 −0.813127
$$430$$ 0 0
$$431$$ −11.1034 −0.534834 −0.267417 0.963581i $$-0.586170\pi$$
−0.267417 + 0.963581i $$0.586170\pi$$
$$432$$ 0 0
$$433$$ −17.2165 −0.827372 −0.413686 0.910420i $$-0.635759\pi$$
−0.413686 + 0.910420i $$0.635759\pi$$
$$434$$ 0 0
$$435$$ −4.54554 −0.217942
$$436$$ 0 0
$$437$$ 3.08976 0.147803
$$438$$ 0 0
$$439$$ −25.1681 −1.20121 −0.600603 0.799548i $$-0.705073\pi$$
−0.600603 + 0.799548i $$0.705073\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 31.1531 1.48013 0.740065 0.672536i $$-0.234795\pi$$
0.740065 + 0.672536i $$0.234795\pi$$
$$444$$ 0 0
$$445$$ 50.2461 2.38189
$$446$$ 0 0
$$447$$ 45.6841 2.16078
$$448$$ 0 0
$$449$$ 10.7444 0.507057 0.253529 0.967328i $$-0.418409\pi$$
0.253529 + 0.967328i $$0.418409\pi$$
$$450$$ 0 0
$$451$$ −6.92567 −0.326117
$$452$$ 0 0
$$453$$ 9.69969 0.455731
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −30.0863 −1.40738 −0.703690 0.710508i $$-0.748465\pi$$
−0.703690 + 0.710508i $$0.748465\pi$$
$$458$$ 0 0
$$459$$ 3.30110 0.154082
$$460$$ 0 0
$$461$$ −8.29082 −0.386142 −0.193071 0.981185i $$-0.561845\pi$$
−0.193071 + 0.981185i $$0.561845\pi$$
$$462$$ 0 0
$$463$$ 11.2795 0.524203 0.262102 0.965040i $$-0.415584\pi$$
0.262102 + 0.965040i $$0.415584\pi$$
$$464$$ 0 0
$$465$$ −6.92622 −0.321196
$$466$$ 0 0
$$467$$ −38.2582 −1.77038 −0.885188 0.465233i $$-0.845971\pi$$
−0.885188 + 0.465233i $$0.845971\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 9.54488 0.439805
$$472$$ 0 0
$$473$$ −13.6951 −0.629702
$$474$$ 0 0
$$475$$ 14.5955 0.669687
$$476$$ 0 0
$$477$$ 16.7777 0.768198
$$478$$ 0 0
$$479$$ 33.0812 1.51152 0.755759 0.654850i $$-0.227268\pi$$
0.755759 + 0.654850i $$0.227268\pi$$
$$480$$ 0 0
$$481$$ −53.7860 −2.45243
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 20.1264 0.913892
$$486$$ 0 0
$$487$$ −25.3757 −1.14988 −0.574941 0.818195i $$-0.694975\pi$$
−0.574941 + 0.818195i $$0.694975\pi$$
$$488$$ 0 0
$$489$$ −25.7723 −1.16547
$$490$$ 0 0
$$491$$ 30.4505 1.37421 0.687107 0.726556i $$-0.258880\pi$$
0.687107 + 0.726556i $$0.258880\pi$$
$$492$$ 0 0
$$493$$ 0.633874 0.0285483
$$494$$ 0 0
$$495$$ −6.02912 −0.270989
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 11.9016 0.532790 0.266395 0.963864i $$-0.414167\pi$$
0.266395 + 0.963864i $$0.414167\pi$$
$$500$$ 0 0
$$501$$ −2.82370 −0.126154
$$502$$ 0 0
$$503$$ −8.90567 −0.397084 −0.198542 0.980092i $$-0.563621\pi$$
−0.198542 + 0.980092i $$0.563621\pi$$
$$504$$ 0 0
$$505$$ −17.3206 −0.770756
$$506$$ 0 0
$$507$$ 60.8167 2.70096
$$508$$ 0 0
$$509$$ 27.1838 1.20490 0.602451 0.798156i $$-0.294191\pi$$
0.602451 + 0.798156i $$0.294191\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −7.29687 −0.322165
$$514$$ 0 0
$$515$$ 18.0016 0.793245
$$516$$ 0 0
$$517$$ 9.48870 0.417313
$$518$$ 0 0
$$519$$ −3.85887 −0.169386
$$520$$ 0 0
$$521$$ −7.10981 −0.311487 −0.155743 0.987798i $$-0.549777\pi$$
−0.155743 + 0.987798i $$0.549777\pi$$
$$522$$ 0 0
$$523$$ 30.4581 1.33184 0.665919 0.746024i $$-0.268039\pi$$
0.665919 + 0.746024i $$0.268039\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0.965861 0.0420735
$$528$$ 0 0
$$529$$ −21.0461 −0.915049
$$530$$ 0 0
$$531$$ 0.862309 0.0374210
$$532$$ 0 0
$$533$$ 36.2633 1.57074
$$534$$ 0 0
$$535$$ −31.5189 −1.36268
$$536$$ 0 0
$$537$$ 42.5976 1.83822
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1.88794 −0.0811688 −0.0405844 0.999176i $$-0.512922\pi$$
−0.0405844 + 0.999176i $$0.512922\pi$$
$$542$$ 0 0
$$543$$ 12.6947 0.544781
$$544$$ 0 0
$$545$$ −54.8415 −2.34915
$$546$$ 0 0
$$547$$ 13.8844 0.593654 0.296827 0.954931i $$-0.404072\pi$$
0.296827 + 0.954931i $$0.404072\pi$$
$$548$$ 0 0
$$549$$ 14.0954 0.601577
$$550$$ 0 0
$$551$$ −1.40114 −0.0596906
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −59.5942 −2.52963
$$556$$ 0 0
$$557$$ −18.5183 −0.784644 −0.392322 0.919828i $$-0.628328\pi$$
−0.392322 + 0.919828i $$0.628328\pi$$
$$558$$ 0 0
$$559$$ 71.7086 3.03295
$$560$$ 0 0
$$561$$ 2.60219 0.109865
$$562$$ 0 0
$$563$$ 36.1932 1.52536 0.762681 0.646775i $$-0.223883\pi$$
0.762681 + 0.646775i $$0.223883\pi$$
$$564$$ 0 0
$$565$$ 35.9398 1.51200
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 28.9137 1.21213 0.606063 0.795417i $$-0.292748\pi$$
0.606063 + 0.795417i $$0.292748\pi$$
$$570$$ 0 0
$$571$$ 15.1214 0.632813 0.316406 0.948624i $$-0.397524\pi$$
0.316406 + 0.948624i $$0.397524\pi$$
$$572$$ 0 0
$$573$$ −22.9776 −0.959904
$$574$$ 0 0
$$575$$ 9.22970 0.384905
$$576$$ 0 0
$$577$$ 18.1608 0.756042 0.378021 0.925797i $$-0.376605\pi$$
0.378021 + 0.925797i $$0.376605\pi$$
$$578$$ 0 0
$$579$$ −40.3607 −1.67733
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −14.4827 −0.599810
$$584$$ 0 0
$$585$$ 31.5689 1.30521
$$586$$ 0 0
$$587$$ −45.0538 −1.85957 −0.929784 0.368105i $$-0.880007\pi$$
−0.929784 + 0.368105i $$0.880007\pi$$
$$588$$ 0 0
$$589$$ −2.13497 −0.0879701
$$590$$ 0 0
$$591$$ 6.09581 0.250748
$$592$$ 0 0
$$593$$ −6.43139 −0.264106 −0.132053 0.991243i $$-0.542157\pi$$
−0.132053 + 0.991243i $$0.542157\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −55.1210 −2.25595
$$598$$ 0 0
$$599$$ −41.0655 −1.67789 −0.838946 0.544215i $$-0.816828\pi$$
−0.838946 + 0.544215i $$0.816828\pi$$
$$600$$ 0 0
$$601$$ −39.2626 −1.60156 −0.800778 0.598961i $$-0.795580\pi$$
−0.800778 + 0.598961i $$0.795580\pi$$
$$602$$ 0 0
$$603$$ 7.59841 0.309431
$$604$$ 0 0
$$605$$ −32.2651 −1.31176
$$606$$ 0 0
$$607$$ 11.9954 0.486879 0.243440 0.969916i $$-0.421724\pi$$
0.243440 + 0.969916i $$0.421724\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −49.6835 −2.00998
$$612$$ 0 0
$$613$$ 28.0774 1.13404 0.567018 0.823706i $$-0.308097\pi$$
0.567018 + 0.823706i $$0.308097\pi$$
$$614$$ 0 0
$$615$$ 40.1792 1.62018
$$616$$ 0 0
$$617$$ −6.08688 −0.245049 −0.122524 0.992466i $$-0.539099\pi$$
−0.122524 + 0.992466i $$0.539099\pi$$
$$618$$ 0 0
$$619$$ −3.24659 −0.130491 −0.0652456 0.997869i $$-0.520783\pi$$
−0.0652456 + 0.997869i $$0.520783\pi$$
$$620$$ 0 0
$$621$$ −4.61429 −0.185165
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −14.4155 −0.576621
$$626$$ 0 0
$$627$$ −5.75199 −0.229712
$$628$$ 0 0
$$629$$ 8.31040 0.331357
$$630$$ 0 0
$$631$$ −6.53093 −0.259992 −0.129996 0.991515i $$-0.541496\pi$$
−0.129996 + 0.991515i $$0.541496\pi$$
$$632$$ 0 0
$$633$$ −4.44302 −0.176594
$$634$$ 0 0
$$635$$ −61.0655 −2.42331
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −4.77701 −0.188976
$$640$$ 0 0
$$641$$ −6.98745 −0.275988 −0.137994 0.990433i $$-0.544065\pi$$
−0.137994 + 0.990433i $$0.544065\pi$$
$$642$$ 0 0
$$643$$ −21.1348 −0.833475 −0.416737 0.909027i $$-0.636827\pi$$
−0.416737 + 0.909027i $$0.636827\pi$$
$$644$$ 0 0
$$645$$ 79.4522 3.12843
$$646$$ 0 0
$$647$$ −34.2892 −1.34805 −0.674024 0.738709i $$-0.735436\pi$$
−0.674024 + 0.738709i $$0.735436\pi$$
$$648$$ 0 0
$$649$$ −0.744352 −0.0292184
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 20.4234 0.799231 0.399615 0.916683i $$-0.369144\pi$$
0.399615 + 0.916683i $$0.369144\pi$$
$$654$$ 0 0
$$655$$ −5.10187 −0.199346
$$656$$ 0 0
$$657$$ −20.0814 −0.783450
$$658$$ 0 0
$$659$$ −9.16578 −0.357048 −0.178524 0.983936i $$-0.557132\pi$$
−0.178524 + 0.983936i $$0.557132\pi$$
$$660$$ 0 0
$$661$$ −33.1919 −1.29102 −0.645508 0.763754i $$-0.723354\pi$$
−0.645508 + 0.763754i $$0.723354\pi$$
$$662$$ 0 0
$$663$$ −13.6253 −0.529161
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −0.886034 −0.0343074
$$668$$ 0 0
$$669$$ 39.6047 1.53121
$$670$$ 0 0
$$671$$ −12.1673 −0.469712
$$672$$ 0 0
$$673$$ 44.8277 1.72798 0.863990 0.503509i $$-0.167958\pi$$
0.863990 + 0.503509i $$0.167958\pi$$
$$674$$ 0 0
$$675$$ −21.7971 −0.838970
$$676$$ 0 0
$$677$$ 17.9276 0.689013 0.344506 0.938784i $$-0.388046\pi$$
0.344506 + 0.938784i $$0.388046\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 36.5753 1.40157
$$682$$ 0 0
$$683$$ −35.3742 −1.35356 −0.676778 0.736187i $$-0.736624\pi$$
−0.676778 + 0.736187i $$0.736624\pi$$
$$684$$ 0 0
$$685$$ 46.3361 1.77041
$$686$$ 0 0
$$687$$ 5.71616 0.218085
$$688$$ 0 0
$$689$$ 75.8322 2.88898
$$690$$ 0 0
$$691$$ 32.6537 1.24221 0.621103 0.783729i $$-0.286685\pi$$
0.621103 + 0.783729i $$0.286685\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 21.3941 0.811526
$$696$$ 0 0
$$697$$ −5.60299 −0.212228
$$698$$ 0 0
$$699$$ 1.50572 0.0569516
$$700$$ 0 0
$$701$$ −12.3696 −0.467194 −0.233597 0.972334i $$-0.575050\pi$$
−0.233597 + 0.972334i $$0.575050\pi$$
$$702$$ 0 0
$$703$$ −18.3696 −0.692823
$$704$$ 0 0
$$705$$ −55.0486 −2.07325
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 20.9482 0.786727 0.393363 0.919383i $$-0.371311\pi$$
0.393363 + 0.919383i $$0.371311\pi$$
$$710$$ 0 0
$$711$$ −0.463202 −0.0173714
$$712$$ 0 0
$$713$$ −1.35009 −0.0505611
$$714$$ 0 0
$$715$$ −27.2505 −1.01911
$$716$$ 0 0
$$717$$ 18.1206 0.676725
$$718$$ 0 0
$$719$$ 28.0239 1.04511 0.522557 0.852604i $$-0.324978\pi$$
0.522557 + 0.852604i $$0.324978\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −13.2890 −0.494223
$$724$$ 0 0
$$725$$ −4.18546 −0.155444
$$726$$ 0 0
$$727$$ 23.9844 0.889531 0.444765 0.895647i $$-0.353287\pi$$
0.444765 + 0.895647i $$0.353287\pi$$
$$728$$ 0 0
$$729$$ 4.74583 0.175772
$$730$$ 0 0
$$731$$ −11.0796 −0.409793
$$732$$ 0 0
$$733$$ 31.0864 1.14820 0.574102 0.818784i $$-0.305351\pi$$
0.574102 + 0.818784i $$0.305351\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −6.55902 −0.241604
$$738$$ 0 0
$$739$$ −12.0841 −0.444519 −0.222260 0.974988i $$-0.571343\pi$$
−0.222260 + 0.974988i $$0.571343\pi$$
$$740$$ 0 0
$$741$$ 30.1178 1.10640
$$742$$ 0 0
$$743$$ 52.8588 1.93920 0.969600 0.244695i $$-0.0786879\pi$$
0.969600 + 0.244695i $$0.0786879\pi$$
$$744$$ 0 0
$$745$$ 73.9184 2.70816
$$746$$ 0 0
$$747$$ −19.8854 −0.727568
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 30.6499 1.11843 0.559216 0.829022i $$-0.311102\pi$$
0.559216 + 0.829022i $$0.311102\pi$$
$$752$$ 0 0
$$753$$ 55.9380 2.03849
$$754$$ 0 0
$$755$$ 15.6944 0.571179
$$756$$ 0 0
$$757$$ −3.57276 −0.129854 −0.0649271 0.997890i $$-0.520681\pi$$
−0.0649271 + 0.997890i $$0.520681\pi$$
$$758$$ 0 0
$$759$$ −3.63736 −0.132028
$$760$$ 0 0
$$761$$ 4.64537 0.168395 0.0841973 0.996449i $$-0.473167\pi$$
0.0841973 + 0.996449i $$0.473167\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −4.87766 −0.176352
$$766$$ 0 0
$$767$$ 3.89748 0.140730
$$768$$ 0 0
$$769$$ 13.1361 0.473700 0.236850 0.971546i $$-0.423885\pi$$
0.236850 + 0.971546i $$0.423885\pi$$
$$770$$ 0 0
$$771$$ −14.2120 −0.511833
$$772$$ 0 0
$$773$$ 44.0372 1.58391 0.791954 0.610581i $$-0.209064\pi$$
0.791954 + 0.610581i $$0.209064\pi$$
$$774$$ 0 0
$$775$$ −6.37756 −0.229089
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.3850 0.443740
$$780$$ 0 0
$$781$$ 4.12356 0.147552
$$782$$ 0 0
$$783$$ 2.09248 0.0747792
$$784$$ 0 0
$$785$$ 15.4439 0.551218
$$786$$ 0 0
$$787$$ −39.3178 −1.40153 −0.700765 0.713393i $$-0.747158\pi$$
−0.700765 + 0.713393i $$0.747158\pi$$
$$788$$ 0 0
$$789$$ 2.53550 0.0902661
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 63.7086 2.26236
$$794$$ 0 0
$$795$$ 84.0210 2.97992
$$796$$ 0 0
$$797$$ 32.5987 1.15470 0.577352 0.816496i $$-0.304086\pi$$
0.577352 + 0.816496i $$0.304086\pi$$
$$798$$ 0 0
$$799$$ 7.67652 0.271576
$$800$$ 0 0
$$801$$ 21.1224 0.746324
$$802$$ 0 0
$$803$$ 17.3344 0.611719
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −38.0356 −1.33892
$$808$$ 0 0
$$809$$ −56.0461 −1.97048 −0.985239 0.171187i $$-0.945240\pi$$
−0.985239 + 0.171187i $$0.945240\pi$$
$$810$$ 0 0
$$811$$ −9.39490 −0.329900 −0.164950 0.986302i $$-0.552746\pi$$
−0.164950 + 0.986302i $$0.552746\pi$$
$$812$$ 0 0
$$813$$ −41.5705 −1.45794
$$814$$ 0 0
$$815$$ −41.7005 −1.46071
$$816$$ 0 0
$$817$$ 24.4907 0.856822
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −7.40016 −0.258267 −0.129134 0.991627i $$-0.541220\pi$$
−0.129134 + 0.991627i $$0.541220\pi$$
$$822$$ 0 0
$$823$$ −39.2675 −1.36878 −0.684390 0.729116i $$-0.739932\pi$$
−0.684390 + 0.729116i $$0.739932\pi$$
$$824$$ 0 0
$$825$$ −17.1822 −0.598209
$$826$$ 0 0
$$827$$ 30.5052 1.06077 0.530385 0.847757i $$-0.322047\pi$$
0.530385 + 0.847757i $$0.322047\pi$$
$$828$$ 0 0
$$829$$ 46.0461 1.59925 0.799624 0.600501i $$-0.205032\pi$$
0.799624 + 0.600501i $$0.205032\pi$$
$$830$$ 0 0
$$831$$ −16.8364 −0.584047
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −4.56885 −0.158111
$$836$$ 0 0
$$837$$ 3.18840 0.110207
$$838$$ 0 0
$$839$$ −43.1903 −1.49110 −0.745548 0.666452i $$-0.767812\pi$$
−0.745548 + 0.666452i $$0.767812\pi$$
$$840$$ 0 0
$$841$$ −28.5982 −0.986145
$$842$$ 0 0
$$843$$ 6.48176 0.223244
$$844$$ 0 0
$$845$$ 98.4035 3.38518
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −60.4043 −2.07307
$$850$$ 0 0
$$851$$ −11.6163 −0.398203
$$852$$ 0 0
$$853$$ −31.9186 −1.09287 −0.546437 0.837500i $$-0.684016\pi$$
−0.546437 + 0.837500i $$0.684016\pi$$
$$854$$ 0 0
$$855$$ 10.7818 0.368728
$$856$$ 0 0
$$857$$ −11.6363 −0.397490 −0.198745 0.980051i $$-0.563686\pi$$
−0.198745 + 0.980051i $$0.563686\pi$$
$$858$$ 0 0
$$859$$ −44.9205 −1.53267 −0.766335 0.642442i $$-0.777922\pi$$
−0.766335 + 0.642442i $$0.777922\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 22.2676 0.757999 0.379000 0.925397i $$-0.376268\pi$$
0.379000 + 0.925397i $$0.376268\pi$$
$$864$$ 0 0
$$865$$ −6.24379 −0.212295
$$866$$ 0 0
$$867$$ 2.10522 0.0714970
$$868$$ 0 0
$$869$$ 0.399840 0.0135636
$$870$$ 0 0
$$871$$ 34.3435 1.16368
$$872$$ 0 0
$$873$$ 8.46072 0.286352
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 18.2095 0.614890 0.307445 0.951566i $$-0.400526\pi$$
0.307445 + 0.951566i $$0.400526\pi$$
$$878$$ 0 0
$$879$$ 4.52340 0.152570
$$880$$ 0 0
$$881$$ −31.7548 −1.06985 −0.534923 0.844901i $$-0.679659\pi$$
−0.534923 + 0.844901i $$0.679659\pi$$
$$882$$ 0 0
$$883$$ −57.7644 −1.94393 −0.971964 0.235130i $$-0.924448\pi$$
−0.971964 + 0.235130i $$0.924448\pi$$
$$884$$ 0 0
$$885$$ 4.31835 0.145160
$$886$$ 0 0
$$887$$ −21.9442 −0.736813 −0.368407 0.929665i $$-0.620097\pi$$
−0.368407 + 0.929665i $$0.620097\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 13.9000 0.465669
$$892$$ 0 0
$$893$$ −16.9685 −0.567828
$$894$$ 0 0
$$895$$ 68.9244 2.30389
$$896$$ 0 0
$$897$$ 19.0455 0.635910
$$898$$ 0 0
$$899$$ 0.612234 0.0204192
$$900$$ 0 0
$$901$$ −11.7167 −0.390340
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 20.5404 0.682786
$$906$$ 0 0
$$907$$ −10.0883 −0.334978 −0.167489 0.985874i $$-0.553566\pi$$
−0.167489 + 0.985874i $$0.553566\pi$$
$$908$$ 0 0
$$909$$ −7.28123 −0.241503
$$910$$ 0 0
$$911$$ −53.3155 −1.76642 −0.883210 0.468977i $$-0.844623\pi$$
−0.883210 + 0.468977i $$0.844623\pi$$
$$912$$ 0 0
$$913$$ 17.1652 0.568086
$$914$$ 0 0
$$915$$ 70.5883 2.33358
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 14.3247 0.472529 0.236264 0.971689i $$-0.424077\pi$$
0.236264 + 0.971689i $$0.424077\pi$$
$$920$$ 0 0
$$921$$ −33.7448 −1.11193
$$922$$ 0 0
$$923$$ −21.5912 −0.710684
$$924$$ 0 0
$$925$$ −54.8734 −1.80423
$$926$$ 0 0
$$927$$ 7.56750 0.248549
$$928$$ 0 0
$$929$$ −30.0091 −0.984567 −0.492284 0.870435i $$-0.663838\pi$$
−0.492284 + 0.870435i $$0.663838\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −72.5259 −2.37439
$$934$$ 0 0
$$935$$ 4.21044 0.137696
$$936$$ 0 0
$$937$$ 45.1807 1.47599 0.737994 0.674807i $$-0.235773\pi$$
0.737994 + 0.674807i $$0.235773\pi$$
$$938$$ 0 0
$$939$$ −68.5427 −2.23681
$$940$$ 0 0
$$941$$ 37.4598 1.22116 0.610578 0.791956i $$-0.290937\pi$$
0.610578 + 0.791956i $$0.290937\pi$$
$$942$$ 0 0
$$943$$ 7.83189 0.255041
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1.33347 −0.0433318 −0.0216659 0.999765i $$-0.506897\pi$$
−0.0216659 + 0.999765i $$0.506897\pi$$
$$948$$ 0 0
$$949$$ −90.7643 −2.94633
$$950$$ 0 0
$$951$$ 5.94874 0.192901
$$952$$ 0 0
$$953$$ 4.14271 0.134196 0.0670978 0.997746i $$-0.478626\pi$$
0.0670978 + 0.997746i $$0.478626\pi$$
$$954$$ 0 0
$$955$$ −37.1786 −1.20307
$$956$$ 0 0
$$957$$ 1.64946 0.0533196
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.0671 −0.969907
$$962$$ 0 0
$$963$$ −13.2499 −0.426973
$$964$$ 0 0
$$965$$ −65.3050 −2.10224
$$966$$ 0 0
$$967$$ 23.9265 0.769423 0.384712 0.923037i $$-0.374301\pi$$
0.384712 + 0.923037i $$0.374301\pi$$
$$968$$ 0 0
$$969$$ −4.65345 −0.149490
$$970$$ 0 0
$$971$$ −41.7634 −1.34025 −0.670126 0.742248i $$-0.733760\pi$$
−0.670126 + 0.742248i $$0.733760\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 89.9674 2.88126
$$976$$ 0 0
$$977$$ −35.4143 −1.13300 −0.566501 0.824061i $$-0.691703\pi$$
−0.566501 + 0.824061i $$0.691703\pi$$
$$978$$ 0 0
$$979$$ −18.2331 −0.582731
$$980$$ 0 0
$$981$$ −23.0543 −0.736066
$$982$$ 0 0
$$983$$ 45.7062 1.45780 0.728901 0.684620i $$-0.240032\pi$$
0.728901 + 0.684620i $$0.240032\pi$$
$$984$$ 0 0
$$985$$ 9.86324 0.314269
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 15.4871 0.492462
$$990$$ 0 0
$$991$$ 13.3586 0.424351 0.212176 0.977232i $$-0.431945\pi$$
0.212176 + 0.977232i $$0.431945\pi$$
$$992$$ 0 0
$$993$$ 34.9091 1.10781
$$994$$ 0 0
$$995$$ −89.1877 −2.82744
$$996$$ 0 0
$$997$$ 31.8346 1.00821 0.504106 0.863642i $$-0.331822\pi$$
0.504106 + 0.863642i $$0.331822\pi$$
$$998$$ 0 0
$$999$$ 27.4334 0.867955
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 6664.2.a.o.1.4 4
7.6 odd 2 952.2.a.g.1.1 4
21.20 even 2 8568.2.a.bj.1.4 4
28.27 even 2 1904.2.a.q.1.4 4
56.13 odd 2 7616.2.a.bj.1.4 4
56.27 even 2 7616.2.a.bp.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.1 4 7.6 odd 2
1904.2.a.q.1.4 4 28.27 even 2
6664.2.a.o.1.4 4 1.1 even 1 trivial
7616.2.a.bj.1.4 4 56.13 odd 2
7616.2.a.bp.1.1 4 56.27 even 2
8568.2.a.bj.1.4 4 21.20 even 2