# Properties

 Label 2070.2.a.u.1.2 Level $2070$ Weight $2$ Character 2070.1 Self dual yes Analytic conductor $16.529$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2070.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: $$16.5290332184$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 2070.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.85410 q^{11} +4.09017 q^{13} -1.61803 q^{14} +1.00000 q^{16} +5.09017 q^{17} -4.85410 q^{19} -1.00000 q^{20} +3.85410 q^{22} -1.00000 q^{23} +1.00000 q^{25} -4.09017 q^{26} +1.61803 q^{28} +4.76393 q^{29} -2.09017 q^{31} -1.00000 q^{32} -5.09017 q^{34} -1.61803 q^{35} -2.47214 q^{37} +4.85410 q^{38} +1.00000 q^{40} +12.3262 q^{41} -3.85410 q^{44} +1.00000 q^{46} -9.70820 q^{47} -4.38197 q^{49} -1.00000 q^{50} +4.09017 q^{52} +8.47214 q^{53} +3.85410 q^{55} -1.61803 q^{56} -4.76393 q^{58} +11.7082 q^{59} +6.32624 q^{61} +2.09017 q^{62} +1.00000 q^{64} -4.09017 q^{65} +5.52786 q^{67} +5.09017 q^{68} +1.61803 q^{70} -7.09017 q^{71} -1.23607 q^{73} +2.47214 q^{74} -4.85410 q^{76} -6.23607 q^{77} +10.4721 q^{79} -1.00000 q^{80} -12.3262 q^{82} -10.9443 q^{83} -5.09017 q^{85} +3.85410 q^{88} +1.52786 q^{89} +6.61803 q^{91} -1.00000 q^{92} +9.70820 q^{94} +4.85410 q^{95} +14.6180 q^{97} +4.38197 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - 3 q^{13} - q^{14} + 2 q^{16} - q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} + q^{28} + 14 q^{29} + 7 q^{31} - 2 q^{32} + q^{34} - q^{35} + 4 q^{37} + 3 q^{38} + 2 q^{40} + 9 q^{41} - q^{44} + 2 q^{46} - 6 q^{47} - 11 q^{49} - 2 q^{50} - 3 q^{52} + 8 q^{53} + q^{55} - q^{56} - 14 q^{58} + 10 q^{59} - 3 q^{61} - 7 q^{62} + 2 q^{64} + 3 q^{65} + 20 q^{67} - q^{68} + q^{70} - 3 q^{71} + 2 q^{73} - 4 q^{74} - 3 q^{76} - 8 q^{77} + 12 q^{79} - 2 q^{80} - 9 q^{82} - 4 q^{83} + q^{85} + q^{88} + 12 q^{89} + 11 q^{91} - 2 q^{92} + 6 q^{94} + 3 q^{95} + 27 q^{97} + 11 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 - 3 * q^13 - q^14 + 2 * q^16 - q^17 - 3 * q^19 - 2 * q^20 + q^22 - 2 * q^23 + 2 * q^25 + 3 * q^26 + q^28 + 14 * q^29 + 7 * q^31 - 2 * q^32 + q^34 - q^35 + 4 * q^37 + 3 * q^38 + 2 * q^40 + 9 * q^41 - q^44 + 2 * q^46 - 6 * q^47 - 11 * q^49 - 2 * q^50 - 3 * q^52 + 8 * q^53 + q^55 - q^56 - 14 * q^58 + 10 * q^59 - 3 * q^61 - 7 * q^62 + 2 * q^64 + 3 * q^65 + 20 * q^67 - q^68 + q^70 - 3 * q^71 + 2 * q^73 - 4 * q^74 - 3 * q^76 - 8 * q^77 + 12 * q^79 - 2 * q^80 - 9 * q^82 - 4 * q^83 + q^85 + q^88 + 12 * q^89 + 11 * q^91 - 2 * q^92 + 6 * q^94 + 3 * q^95 + 27 * q^97 + 11 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.61803 0.611559 0.305780 0.952102i $$-0.401083\pi$$
0.305780 + 0.952102i $$0.401083\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ −3.85410 −1.16206 −0.581028 0.813884i $$-0.697349\pi$$
−0.581028 + 0.813884i $$0.697349\pi$$
$$12$$ 0 0
$$13$$ 4.09017 1.13441 0.567205 0.823577i $$-0.308025\pi$$
0.567205 + 0.823577i $$0.308025\pi$$
$$14$$ −1.61803 −0.432438
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.09017 1.23455 0.617274 0.786748i $$-0.288237\pi$$
0.617274 + 0.786748i $$0.288237\pi$$
$$18$$ 0 0
$$19$$ −4.85410 −1.11361 −0.556804 0.830644i $$-0.687972\pi$$
−0.556804 + 0.830644i $$0.687972\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 3.85410 0.821697
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −4.09017 −0.802148
$$27$$ 0 0
$$28$$ 1.61803 0.305780
$$29$$ 4.76393 0.884640 0.442320 0.896857i $$-0.354156\pi$$
0.442320 + 0.896857i $$0.354156\pi$$
$$30$$ 0 0
$$31$$ −2.09017 −0.375406 −0.187703 0.982226i $$-0.560104\pi$$
−0.187703 + 0.982226i $$0.560104\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −5.09017 −0.872957
$$35$$ −1.61803 −0.273498
$$36$$ 0 0
$$37$$ −2.47214 −0.406417 −0.203208 0.979136i $$-0.565137\pi$$
−0.203208 + 0.979136i $$0.565137\pi$$
$$38$$ 4.85410 0.787439
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 12.3262 1.92503 0.962517 0.271220i $$-0.0874270\pi$$
0.962517 + 0.271220i $$0.0874270\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ −3.85410 −0.581028
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ −9.70820 −1.41609 −0.708044 0.706169i $$-0.750422\pi$$
−0.708044 + 0.706169i $$0.750422\pi$$
$$48$$ 0 0
$$49$$ −4.38197 −0.625995
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 4.09017 0.567205
$$53$$ 8.47214 1.16374 0.581869 0.813283i $$-0.302322\pi$$
0.581869 + 0.813283i $$0.302322\pi$$
$$54$$ 0 0
$$55$$ 3.85410 0.519687
$$56$$ −1.61803 −0.216219
$$57$$ 0 0
$$58$$ −4.76393 −0.625535
$$59$$ 11.7082 1.52428 0.762139 0.647413i $$-0.224149\pi$$
0.762139 + 0.647413i $$0.224149\pi$$
$$60$$ 0 0
$$61$$ 6.32624 0.809992 0.404996 0.914319i $$-0.367273\pi$$
0.404996 + 0.914319i $$0.367273\pi$$
$$62$$ 2.09017 0.265452
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −4.09017 −0.507323
$$66$$ 0 0
$$67$$ 5.52786 0.675336 0.337668 0.941265i $$-0.390362\pi$$
0.337668 + 0.941265i $$0.390362\pi$$
$$68$$ 5.09017 0.617274
$$69$$ 0 0
$$70$$ 1.61803 0.193392
$$71$$ −7.09017 −0.841448 −0.420724 0.907189i $$-0.638224\pi$$
−0.420724 + 0.907189i $$0.638224\pi$$
$$72$$ 0 0
$$73$$ −1.23607 −0.144671 −0.0723354 0.997380i $$-0.523045\pi$$
−0.0723354 + 0.997380i $$0.523045\pi$$
$$74$$ 2.47214 0.287380
$$75$$ 0 0
$$76$$ −4.85410 −0.556804
$$77$$ −6.23607 −0.710666
$$78$$ 0 0
$$79$$ 10.4721 1.17821 0.589104 0.808057i $$-0.299481\pi$$
0.589104 + 0.808057i $$0.299481\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ −12.3262 −1.36121
$$83$$ −10.9443 −1.20129 −0.600645 0.799516i $$-0.705089\pi$$
−0.600645 + 0.799516i $$0.705089\pi$$
$$84$$ 0 0
$$85$$ −5.09017 −0.552106
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 3.85410 0.410849
$$89$$ 1.52786 0.161953 0.0809766 0.996716i $$-0.474196\pi$$
0.0809766 + 0.996716i $$0.474196\pi$$
$$90$$ 0 0
$$91$$ 6.61803 0.693758
$$92$$ −1.00000 −0.104257
$$93$$ 0 0
$$94$$ 9.70820 1.00132
$$95$$ 4.85410 0.498020
$$96$$ 0 0
$$97$$ 14.6180 1.48424 0.742118 0.670269i $$-0.233821\pi$$
0.742118 + 0.670269i $$0.233821\pi$$
$$98$$ 4.38197 0.442645
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 13.7082 1.36402 0.682009 0.731344i $$-0.261107\pi$$
0.682009 + 0.731344i $$0.261107\pi$$
$$102$$ 0 0
$$103$$ −3.56231 −0.351004 −0.175502 0.984479i $$-0.556155\pi$$
−0.175502 + 0.984479i $$0.556155\pi$$
$$104$$ −4.09017 −0.401074
$$105$$ 0 0
$$106$$ −8.47214 −0.822887
$$107$$ 4.18034 0.404129 0.202064 0.979372i $$-0.435235\pi$$
0.202064 + 0.979372i $$0.435235\pi$$
$$108$$ 0 0
$$109$$ 8.56231 0.820120 0.410060 0.912059i $$-0.365508\pi$$
0.410060 + 0.912059i $$0.365508\pi$$
$$110$$ −3.85410 −0.367474
$$111$$ 0 0
$$112$$ 1.61803 0.152890
$$113$$ −18.9443 −1.78213 −0.891064 0.453878i $$-0.850040\pi$$
−0.891064 + 0.453878i $$0.850040\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 4.76393 0.442320
$$117$$ 0 0
$$118$$ −11.7082 −1.07783
$$119$$ 8.23607 0.754999
$$120$$ 0 0
$$121$$ 3.85410 0.350373
$$122$$ −6.32624 −0.572751
$$123$$ 0 0
$$124$$ −2.09017 −0.187703
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −6.18034 −0.548416 −0.274208 0.961670i $$-0.588416\pi$$
−0.274208 + 0.961670i $$0.588416\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 4.09017 0.358732
$$131$$ 14.9443 1.30569 0.652844 0.757493i $$-0.273576\pi$$
0.652844 + 0.757493i $$0.273576\pi$$
$$132$$ 0 0
$$133$$ −7.85410 −0.681037
$$134$$ −5.52786 −0.477535
$$135$$ 0 0
$$136$$ −5.09017 −0.436478
$$137$$ −5.32624 −0.455051 −0.227526 0.973772i $$-0.573064\pi$$
−0.227526 + 0.973772i $$0.573064\pi$$
$$138$$ 0 0
$$139$$ 17.2361 1.46194 0.730972 0.682407i $$-0.239067\pi$$
0.730972 + 0.682407i $$0.239067\pi$$
$$140$$ −1.61803 −0.136749
$$141$$ 0 0
$$142$$ 7.09017 0.594994
$$143$$ −15.7639 −1.31825
$$144$$ 0 0
$$145$$ −4.76393 −0.395623
$$146$$ 1.23607 0.102298
$$147$$ 0 0
$$148$$ −2.47214 −0.203208
$$149$$ 1.14590 0.0938756 0.0469378 0.998898i $$-0.485054\pi$$
0.0469378 + 0.998898i $$0.485054\pi$$
$$150$$ 0 0
$$151$$ 17.5623 1.42920 0.714600 0.699533i $$-0.246609\pi$$
0.714600 + 0.699533i $$0.246609\pi$$
$$152$$ 4.85410 0.393720
$$153$$ 0 0
$$154$$ 6.23607 0.502517
$$155$$ 2.09017 0.167886
$$156$$ 0 0
$$157$$ 9.70820 0.774799 0.387400 0.921912i $$-0.373373\pi$$
0.387400 + 0.921912i $$0.373373\pi$$
$$158$$ −10.4721 −0.833118
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ −1.61803 −0.127519
$$162$$ 0 0
$$163$$ 3.61803 0.283386 0.141693 0.989911i $$-0.454745\pi$$
0.141693 + 0.989911i $$0.454745\pi$$
$$164$$ 12.3262 0.962517
$$165$$ 0 0
$$166$$ 10.9443 0.849440
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 3.72949 0.286884
$$170$$ 5.09017 0.390398
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 21.5623 1.63935 0.819676 0.572828i $$-0.194154\pi$$
0.819676 + 0.572828i $$0.194154\pi$$
$$174$$ 0 0
$$175$$ 1.61803 0.122312
$$176$$ −3.85410 −0.290514
$$177$$ 0 0
$$178$$ −1.52786 −0.114518
$$179$$ 20.1803 1.50835 0.754175 0.656674i $$-0.228037\pi$$
0.754175 + 0.656674i $$0.228037\pi$$
$$180$$ 0 0
$$181$$ −18.8541 −1.40141 −0.700707 0.713449i $$-0.747132\pi$$
−0.700707 + 0.713449i $$0.747132\pi$$
$$182$$ −6.61803 −0.490561
$$183$$ 0 0
$$184$$ 1.00000 0.0737210
$$185$$ 2.47214 0.181755
$$186$$ 0 0
$$187$$ −19.6180 −1.43461
$$188$$ −9.70820 −0.708044
$$189$$ 0 0
$$190$$ −4.85410 −0.352154
$$191$$ 0.291796 0.0211136 0.0105568 0.999944i $$-0.496640\pi$$
0.0105568 + 0.999944i $$0.496640\pi$$
$$192$$ 0 0
$$193$$ 5.23607 0.376900 0.188450 0.982083i $$-0.439654\pi$$
0.188450 + 0.982083i $$0.439654\pi$$
$$194$$ −14.6180 −1.04951
$$195$$ 0 0
$$196$$ −4.38197 −0.312998
$$197$$ 2.43769 0.173679 0.0868393 0.996222i $$-0.472323\pi$$
0.0868393 + 0.996222i $$0.472323\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −13.7082 −0.964506
$$203$$ 7.70820 0.541010
$$204$$ 0 0
$$205$$ −12.3262 −0.860902
$$206$$ 3.56231 0.248198
$$207$$ 0 0
$$208$$ 4.09017 0.283602
$$209$$ 18.7082 1.29407
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ 8.47214 0.581869
$$213$$ 0 0
$$214$$ −4.18034 −0.285762
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.38197 −0.229583
$$218$$ −8.56231 −0.579913
$$219$$ 0 0
$$220$$ 3.85410 0.259844
$$221$$ 20.8197 1.40048
$$222$$ 0 0
$$223$$ −3.05573 −0.204627 −0.102313 0.994752i $$-0.532624\pi$$
−0.102313 + 0.994752i $$0.532624\pi$$
$$224$$ −1.61803 −0.108109
$$225$$ 0 0
$$226$$ 18.9443 1.26015
$$227$$ 23.2361 1.54223 0.771116 0.636695i $$-0.219699\pi$$
0.771116 + 0.636695i $$0.219699\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ −1.00000 −0.0659380
$$231$$ 0 0
$$232$$ −4.76393 −0.312767
$$233$$ −19.7082 −1.29113 −0.645564 0.763706i $$-0.723378\pi$$
−0.645564 + 0.763706i $$0.723378\pi$$
$$234$$ 0 0
$$235$$ 9.70820 0.633293
$$236$$ 11.7082 0.762139
$$237$$ 0 0
$$238$$ −8.23607 −0.533865
$$239$$ −24.3607 −1.57576 −0.787881 0.615828i $$-0.788822\pi$$
−0.787881 + 0.615828i $$0.788822\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ −3.85410 −0.247751
$$243$$ 0 0
$$244$$ 6.32624 0.404996
$$245$$ 4.38197 0.279954
$$246$$ 0 0
$$247$$ −19.8541 −1.26329
$$248$$ 2.09017 0.132726
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ −12.8541 −0.811344 −0.405672 0.914019i $$-0.632962\pi$$
−0.405672 + 0.914019i $$0.632962\pi$$
$$252$$ 0 0
$$253$$ 3.85410 0.242305
$$254$$ 6.18034 0.387789
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 30.1803 1.88260 0.941299 0.337574i $$-0.109606\pi$$
0.941299 + 0.337574i $$0.109606\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ −4.09017 −0.253662
$$261$$ 0 0
$$262$$ −14.9443 −0.923260
$$263$$ 21.7426 1.34071 0.670354 0.742041i $$-0.266142\pi$$
0.670354 + 0.742041i $$0.266142\pi$$
$$264$$ 0 0
$$265$$ −8.47214 −0.520439
$$266$$ 7.85410 0.481566
$$267$$ 0 0
$$268$$ 5.52786 0.337668
$$269$$ −8.18034 −0.498764 −0.249382 0.968405i $$-0.580227\pi$$
−0.249382 + 0.968405i $$0.580227\pi$$
$$270$$ 0 0
$$271$$ −14.6738 −0.891368 −0.445684 0.895190i $$-0.647039\pi$$
−0.445684 + 0.895190i $$0.647039\pi$$
$$272$$ 5.09017 0.308637
$$273$$ 0 0
$$274$$ 5.32624 0.321770
$$275$$ −3.85410 −0.232411
$$276$$ 0 0
$$277$$ 2.58359 0.155233 0.0776165 0.996983i $$-0.475269\pi$$
0.0776165 + 0.996983i $$0.475269\pi$$
$$278$$ −17.2361 −1.03375
$$279$$ 0 0
$$280$$ 1.61803 0.0966960
$$281$$ −27.2361 −1.62477 −0.812384 0.583123i $$-0.801831\pi$$
−0.812384 + 0.583123i $$0.801831\pi$$
$$282$$ 0 0
$$283$$ −9.05573 −0.538307 −0.269154 0.963097i $$-0.586744\pi$$
−0.269154 + 0.963097i $$0.586744\pi$$
$$284$$ −7.09017 −0.420724
$$285$$ 0 0
$$286$$ 15.7639 0.932141
$$287$$ 19.9443 1.17727
$$288$$ 0 0
$$289$$ 8.90983 0.524108
$$290$$ 4.76393 0.279748
$$291$$ 0 0
$$292$$ −1.23607 −0.0723354
$$293$$ −15.8885 −0.928219 −0.464109 0.885778i $$-0.653626\pi$$
−0.464109 + 0.885778i $$0.653626\pi$$
$$294$$ 0 0
$$295$$ −11.7082 −0.681678
$$296$$ 2.47214 0.143690
$$297$$ 0 0
$$298$$ −1.14590 −0.0663801
$$299$$ −4.09017 −0.236541
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −17.5623 −1.01060
$$303$$ 0 0
$$304$$ −4.85410 −0.278402
$$305$$ −6.32624 −0.362239
$$306$$ 0 0
$$307$$ 27.4508 1.56670 0.783351 0.621579i $$-0.213509\pi$$
0.783351 + 0.621579i $$0.213509\pi$$
$$308$$ −6.23607 −0.355333
$$309$$ 0 0
$$310$$ −2.09017 −0.118714
$$311$$ −4.00000 −0.226819 −0.113410 0.993548i $$-0.536177\pi$$
−0.113410 + 0.993548i $$0.536177\pi$$
$$312$$ 0 0
$$313$$ −11.7984 −0.666884 −0.333442 0.942771i $$-0.608210\pi$$
−0.333442 + 0.942771i $$0.608210\pi$$
$$314$$ −9.70820 −0.547866
$$315$$ 0 0
$$316$$ 10.4721 0.589104
$$317$$ −0.0901699 −0.00506445 −0.00253222 0.999997i $$-0.500806\pi$$
−0.00253222 + 0.999997i $$0.500806\pi$$
$$318$$ 0 0
$$319$$ −18.3607 −1.02800
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 1.61803 0.0901695
$$323$$ −24.7082 −1.37480
$$324$$ 0 0
$$325$$ 4.09017 0.226882
$$326$$ −3.61803 −0.200384
$$327$$ 0 0
$$328$$ −12.3262 −0.680603
$$329$$ −15.7082 −0.866021
$$330$$ 0 0
$$331$$ 14.7639 0.811499 0.405750 0.913984i $$-0.367011\pi$$
0.405750 + 0.913984i $$0.367011\pi$$
$$332$$ −10.9443 −0.600645
$$333$$ 0 0
$$334$$ −8.00000 −0.437741
$$335$$ −5.52786 −0.302019
$$336$$ 0 0
$$337$$ 29.3262 1.59750 0.798751 0.601662i $$-0.205494\pi$$
0.798751 + 0.601662i $$0.205494\pi$$
$$338$$ −3.72949 −0.202858
$$339$$ 0 0
$$340$$ −5.09017 −0.276053
$$341$$ 8.05573 0.436242
$$342$$ 0 0
$$343$$ −18.4164 −0.994393
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −21.5623 −1.15920
$$347$$ 8.61803 0.462640 0.231320 0.972878i $$-0.425696\pi$$
0.231320 + 0.972878i $$0.425696\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ −1.61803 −0.0864876
$$351$$ 0 0
$$352$$ 3.85410 0.205424
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ 7.09017 0.376307
$$356$$ 1.52786 0.0809766
$$357$$ 0 0
$$358$$ −20.1803 −1.06656
$$359$$ 18.3607 0.969040 0.484520 0.874780i $$-0.338994\pi$$
0.484520 + 0.874780i $$0.338994\pi$$
$$360$$ 0 0
$$361$$ 4.56231 0.240121
$$362$$ 18.8541 0.990950
$$363$$ 0 0
$$364$$ 6.61803 0.346879
$$365$$ 1.23607 0.0646988
$$366$$ 0 0
$$367$$ −2.47214 −0.129044 −0.0645222 0.997916i $$-0.520552\pi$$
−0.0645222 + 0.997916i $$0.520552\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 0 0
$$370$$ −2.47214 −0.128520
$$371$$ 13.7082 0.711694
$$372$$ 0 0
$$373$$ −2.18034 −0.112894 −0.0564469 0.998406i $$-0.517977\pi$$
−0.0564469 + 0.998406i $$0.517977\pi$$
$$374$$ 19.6180 1.01442
$$375$$ 0 0
$$376$$ 9.70820 0.500662
$$377$$ 19.4853 1.00354
$$378$$ 0 0
$$379$$ 33.4508 1.71825 0.859127 0.511762i $$-0.171007\pi$$
0.859127 + 0.511762i $$0.171007\pi$$
$$380$$ 4.85410 0.249010
$$381$$ 0 0
$$382$$ −0.291796 −0.0149296
$$383$$ −17.8885 −0.914062 −0.457031 0.889451i $$-0.651087\pi$$
−0.457031 + 0.889451i $$0.651087\pi$$
$$384$$ 0 0
$$385$$ 6.23607 0.317819
$$386$$ −5.23607 −0.266509
$$387$$ 0 0
$$388$$ 14.6180 0.742118
$$389$$ −5.67376 −0.287671 −0.143836 0.989602i $$-0.545944\pi$$
−0.143836 + 0.989602i $$0.545944\pi$$
$$390$$ 0 0
$$391$$ −5.09017 −0.257421
$$392$$ 4.38197 0.221323
$$393$$ 0 0
$$394$$ −2.43769 −0.122809
$$395$$ −10.4721 −0.526910
$$396$$ 0 0
$$397$$ −8.32624 −0.417882 −0.208941 0.977928i $$-0.567002\pi$$
−0.208941 + 0.977928i $$0.567002\pi$$
$$398$$ −2.00000 −0.100251
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −11.7082 −0.584680 −0.292340 0.956314i $$-0.594434\pi$$
−0.292340 + 0.956314i $$0.594434\pi$$
$$402$$ 0 0
$$403$$ −8.54915 −0.425864
$$404$$ 13.7082 0.682009
$$405$$ 0 0
$$406$$ −7.70820 −0.382552
$$407$$ 9.52786 0.472279
$$408$$ 0 0
$$409$$ 21.2148 1.04900 0.524502 0.851409i $$-0.324252\pi$$
0.524502 + 0.851409i $$0.324252\pi$$
$$410$$ 12.3262 0.608750
$$411$$ 0 0
$$412$$ −3.56231 −0.175502
$$413$$ 18.9443 0.932187
$$414$$ 0 0
$$415$$ 10.9443 0.537233
$$416$$ −4.09017 −0.200537
$$417$$ 0 0
$$418$$ −18.7082 −0.915048
$$419$$ −5.52786 −0.270054 −0.135027 0.990842i $$-0.543112\pi$$
−0.135027 + 0.990842i $$0.543112\pi$$
$$420$$ 0 0
$$421$$ −28.7426 −1.40083 −0.700415 0.713735i $$-0.747002\pi$$
−0.700415 + 0.713735i $$0.747002\pi$$
$$422$$ −14.0000 −0.681509
$$423$$ 0 0
$$424$$ −8.47214 −0.411443
$$425$$ 5.09017 0.246910
$$426$$ 0 0
$$427$$ 10.2361 0.495358
$$428$$ 4.18034 0.202064
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.6525 −1.66915 −0.834576 0.550894i $$-0.814287\pi$$
−0.834576 + 0.550894i $$0.814287\pi$$
$$432$$ 0 0
$$433$$ −29.5066 −1.41800 −0.708998 0.705211i $$-0.750852\pi$$
−0.708998 + 0.705211i $$0.750852\pi$$
$$434$$ 3.38197 0.162340
$$435$$ 0 0
$$436$$ 8.56231 0.410060
$$437$$ 4.85410 0.232203
$$438$$ 0 0
$$439$$ −15.6180 −0.745408 −0.372704 0.927950i $$-0.621569\pi$$
−0.372704 + 0.927950i $$0.621569\pi$$
$$440$$ −3.85410 −0.183737
$$441$$ 0 0
$$442$$ −20.8197 −0.990290
$$443$$ −13.9098 −0.660876 −0.330438 0.943828i $$-0.607196\pi$$
−0.330438 + 0.943828i $$0.607196\pi$$
$$444$$ 0 0
$$445$$ −1.52786 −0.0724277
$$446$$ 3.05573 0.144693
$$447$$ 0 0
$$448$$ 1.61803 0.0764449
$$449$$ −18.5623 −0.876009 −0.438005 0.898973i $$-0.644315\pi$$
−0.438005 + 0.898973i $$0.644315\pi$$
$$450$$ 0 0
$$451$$ −47.5066 −2.23700
$$452$$ −18.9443 −0.891064
$$453$$ 0 0
$$454$$ −23.2361 −1.09052
$$455$$ −6.61803 −0.310258
$$456$$ 0 0
$$457$$ 33.7771 1.58003 0.790013 0.613090i $$-0.210074\pi$$
0.790013 + 0.613090i $$0.210074\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 0 0
$$460$$ 1.00000 0.0466252
$$461$$ 34.7639 1.61912 0.809559 0.587039i $$-0.199706\pi$$
0.809559 + 0.587039i $$0.199706\pi$$
$$462$$ 0 0
$$463$$ 2.00000 0.0929479 0.0464739 0.998920i $$-0.485202\pi$$
0.0464739 + 0.998920i $$0.485202\pi$$
$$464$$ 4.76393 0.221160
$$465$$ 0 0
$$466$$ 19.7082 0.912965
$$467$$ −23.1246 −1.07008 −0.535040 0.844827i $$-0.679703\pi$$
−0.535040 + 0.844827i $$0.679703\pi$$
$$468$$ 0 0
$$469$$ 8.94427 0.413008
$$470$$ −9.70820 −0.447806
$$471$$ 0 0
$$472$$ −11.7082 −0.538914
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −4.85410 −0.222721
$$476$$ 8.23607 0.377500
$$477$$ 0 0
$$478$$ 24.3607 1.11423
$$479$$ −3.88854 −0.177672 −0.0888361 0.996046i $$-0.528315\pi$$
−0.0888361 + 0.996046i $$0.528315\pi$$
$$480$$ 0 0
$$481$$ −10.1115 −0.461043
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 3.85410 0.175186
$$485$$ −14.6180 −0.663771
$$486$$ 0 0
$$487$$ −42.1803 −1.91137 −0.955687 0.294385i $$-0.904885\pi$$
−0.955687 + 0.294385i $$0.904885\pi$$
$$488$$ −6.32624 −0.286375
$$489$$ 0 0
$$490$$ −4.38197 −0.197957
$$491$$ 16.1803 0.730209 0.365104 0.930967i $$-0.381033\pi$$
0.365104 + 0.930967i $$0.381033\pi$$
$$492$$ 0 0
$$493$$ 24.2492 1.09213
$$494$$ 19.8541 0.893278
$$495$$ 0 0
$$496$$ −2.09017 −0.0938514
$$497$$ −11.4721 −0.514596
$$498$$ 0 0
$$499$$ −32.3607 −1.44866 −0.724331 0.689452i $$-0.757851\pi$$
−0.724331 + 0.689452i $$0.757851\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 12.8541 0.573707
$$503$$ −20.6738 −0.921797 −0.460899 0.887453i $$-0.652473\pi$$
−0.460899 + 0.887453i $$0.652473\pi$$
$$504$$ 0 0
$$505$$ −13.7082 −0.610007
$$506$$ −3.85410 −0.171336
$$507$$ 0 0
$$508$$ −6.18034 −0.274208
$$509$$ −5.34752 −0.237025 −0.118512 0.992953i $$-0.537813\pi$$
−0.118512 + 0.992953i $$0.537813\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −30.1803 −1.33120
$$515$$ 3.56231 0.156974
$$516$$ 0 0
$$517$$ 37.4164 1.64557
$$518$$ 4.00000 0.175750
$$519$$ 0 0
$$520$$ 4.09017 0.179366
$$521$$ 24.4721 1.07214 0.536072 0.844172i $$-0.319908\pi$$
0.536072 + 0.844172i $$0.319908\pi$$
$$522$$ 0 0
$$523$$ −26.0000 −1.13690 −0.568450 0.822718i $$-0.692457\pi$$
−0.568450 + 0.822718i $$0.692457\pi$$
$$524$$ 14.9443 0.652844
$$525$$ 0 0
$$526$$ −21.7426 −0.948024
$$527$$ −10.6393 −0.463456
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 8.47214 0.368006
$$531$$ 0 0
$$532$$ −7.85410 −0.340519
$$533$$ 50.4164 2.18378
$$534$$ 0 0
$$535$$ −4.18034 −0.180732
$$536$$ −5.52786 −0.238767
$$537$$ 0 0
$$538$$ 8.18034 0.352679
$$539$$ 16.8885 0.727441
$$540$$ 0 0
$$541$$ −30.8328 −1.32561 −0.662803 0.748794i $$-0.730633\pi$$
−0.662803 + 0.748794i $$0.730633\pi$$
$$542$$ 14.6738 0.630292
$$543$$ 0 0
$$544$$ −5.09017 −0.218239
$$545$$ −8.56231 −0.366769
$$546$$ 0 0
$$547$$ −36.9230 −1.57871 −0.789356 0.613935i $$-0.789586\pi$$
−0.789356 + 0.613935i $$0.789586\pi$$
$$548$$ −5.32624 −0.227526
$$549$$ 0 0
$$550$$ 3.85410 0.164339
$$551$$ −23.1246 −0.985142
$$552$$ 0 0
$$553$$ 16.9443 0.720544
$$554$$ −2.58359 −0.109766
$$555$$ 0 0
$$556$$ 17.2361 0.730972
$$557$$ −30.8328 −1.30643 −0.653214 0.757173i $$-0.726580\pi$$
−0.653214 + 0.757173i $$0.726580\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −1.61803 −0.0683744
$$561$$ 0 0
$$562$$ 27.2361 1.14888
$$563$$ −21.8885 −0.922492 −0.461246 0.887272i $$-0.652597\pi$$
−0.461246 + 0.887272i $$0.652597\pi$$
$$564$$ 0 0
$$565$$ 18.9443 0.796992
$$566$$ 9.05573 0.380641
$$567$$ 0 0
$$568$$ 7.09017 0.297497
$$569$$ 2.00000 0.0838444 0.0419222 0.999121i $$-0.486652\pi$$
0.0419222 + 0.999121i $$0.486652\pi$$
$$570$$ 0 0
$$571$$ −30.9787 −1.29642 −0.648209 0.761462i $$-0.724482\pi$$
−0.648209 + 0.761462i $$0.724482\pi$$
$$572$$ −15.7639 −0.659123
$$573$$ 0 0
$$574$$ −19.9443 −0.832458
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −12.4721 −0.519222 −0.259611 0.965713i $$-0.583594\pi$$
−0.259611 + 0.965713i $$0.583594\pi$$
$$578$$ −8.90983 −0.370600
$$579$$ 0 0
$$580$$ −4.76393 −0.197812
$$581$$ −17.7082 −0.734660
$$582$$ 0 0
$$583$$ −32.6525 −1.35233
$$584$$ 1.23607 0.0511489
$$585$$ 0 0
$$586$$ 15.8885 0.656350
$$587$$ −11.3820 −0.469784 −0.234892 0.972021i $$-0.575474\pi$$
−0.234892 + 0.972021i $$0.575474\pi$$
$$588$$ 0 0
$$589$$ 10.1459 0.418054
$$590$$ 11.7082 0.482019
$$591$$ 0 0
$$592$$ −2.47214 −0.101604
$$593$$ −34.7639 −1.42758 −0.713792 0.700358i $$-0.753024\pi$$
−0.713792 + 0.700358i $$0.753024\pi$$
$$594$$ 0 0
$$595$$ −8.23607 −0.337646
$$596$$ 1.14590 0.0469378
$$597$$ 0 0
$$598$$ 4.09017 0.167259
$$599$$ 20.6180 0.842430 0.421215 0.906961i $$-0.361604\pi$$
0.421215 + 0.906961i $$0.361604\pi$$
$$600$$ 0 0
$$601$$ 0.270510 0.0110343 0.00551716 0.999985i $$-0.498244\pi$$
0.00551716 + 0.999985i $$0.498244\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 17.5623 0.714600
$$605$$ −3.85410 −0.156692
$$606$$ 0 0
$$607$$ 17.5279 0.711434 0.355717 0.934594i $$-0.384237\pi$$
0.355717 + 0.934594i $$0.384237\pi$$
$$608$$ 4.85410 0.196860
$$609$$ 0 0
$$610$$ 6.32624 0.256142
$$611$$ −39.7082 −1.60642
$$612$$ 0 0
$$613$$ −43.3050 −1.74907 −0.874535 0.484962i $$-0.838833\pi$$
−0.874535 + 0.484962i $$0.838833\pi$$
$$614$$ −27.4508 −1.10783
$$615$$ 0 0
$$616$$ 6.23607 0.251258
$$617$$ −22.9098 −0.922315 −0.461158 0.887318i $$-0.652566\pi$$
−0.461158 + 0.887318i $$0.652566\pi$$
$$618$$ 0 0
$$619$$ 21.7984 0.876151 0.438075 0.898938i $$-0.355660\pi$$
0.438075 + 0.898938i $$0.355660\pi$$
$$620$$ 2.09017 0.0839432
$$621$$ 0 0
$$622$$ 4.00000 0.160385
$$623$$ 2.47214 0.0990440
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 11.7984 0.471558
$$627$$ 0 0
$$628$$ 9.70820 0.387400
$$629$$ −12.5836 −0.501741
$$630$$ 0 0
$$631$$ −16.0689 −0.639692 −0.319846 0.947470i $$-0.603631\pi$$
−0.319846 + 0.947470i $$0.603631\pi$$
$$632$$ −10.4721 −0.416559
$$633$$ 0 0
$$634$$ 0.0901699 0.00358111
$$635$$ 6.18034 0.245259
$$636$$ 0 0
$$637$$ −17.9230 −0.710135
$$638$$ 18.3607 0.726906
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 44.3607 1.75214 0.876071 0.482183i $$-0.160156\pi$$
0.876071 + 0.482183i $$0.160156\pi$$
$$642$$ 0 0
$$643$$ −21.7082 −0.856088 −0.428044 0.903758i $$-0.640797\pi$$
−0.428044 + 0.903758i $$0.640797\pi$$
$$644$$ −1.61803 −0.0637595
$$645$$ 0 0
$$646$$ 24.7082 0.972131
$$647$$ −44.2492 −1.73962 −0.869808 0.493390i $$-0.835758\pi$$
−0.869808 + 0.493390i $$0.835758\pi$$
$$648$$ 0 0
$$649$$ −45.1246 −1.77130
$$650$$ −4.09017 −0.160430
$$651$$ 0 0
$$652$$ 3.61803 0.141693
$$653$$ −21.0344 −0.823141 −0.411571 0.911378i $$-0.635020\pi$$
−0.411571 + 0.911378i $$0.635020\pi$$
$$654$$ 0 0
$$655$$ −14.9443 −0.583921
$$656$$ 12.3262 0.481259
$$657$$ 0 0
$$658$$ 15.7082 0.612370
$$659$$ −34.2492 −1.33416 −0.667080 0.744986i $$-0.732456\pi$$
−0.667080 + 0.744986i $$0.732456\pi$$
$$660$$ 0 0
$$661$$ 34.3262 1.33514 0.667568 0.744549i $$-0.267335\pi$$
0.667568 + 0.744549i $$0.267335\pi$$
$$662$$ −14.7639 −0.573817
$$663$$ 0 0
$$664$$ 10.9443 0.424720
$$665$$ 7.85410 0.304569
$$666$$ 0 0
$$667$$ −4.76393 −0.184460
$$668$$ 8.00000 0.309529
$$669$$ 0 0
$$670$$ 5.52786 0.213560
$$671$$ −24.3820 −0.941255
$$672$$ 0 0
$$673$$ 6.94427 0.267682 0.133841 0.991003i $$-0.457269\pi$$
0.133841 + 0.991003i $$0.457269\pi$$
$$674$$ −29.3262 −1.12960
$$675$$ 0 0
$$676$$ 3.72949 0.143442
$$677$$ 33.0557 1.27043 0.635217 0.772333i $$-0.280910\pi$$
0.635217 + 0.772333i $$0.280910\pi$$
$$678$$ 0 0
$$679$$ 23.6525 0.907699
$$680$$ 5.09017 0.195199
$$681$$ 0 0
$$682$$ −8.05573 −0.308470
$$683$$ 11.4377 0.437651 0.218826 0.975764i $$-0.429777\pi$$
0.218826 + 0.975764i $$0.429777\pi$$
$$684$$ 0 0
$$685$$ 5.32624 0.203505
$$686$$ 18.4164 0.703142
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 34.6525 1.32015
$$690$$ 0 0
$$691$$ −24.7639 −0.942064 −0.471032 0.882116i $$-0.656118\pi$$
−0.471032 + 0.882116i $$0.656118\pi$$
$$692$$ 21.5623 0.819676
$$693$$ 0 0
$$694$$ −8.61803 −0.327136
$$695$$ −17.2361 −0.653801
$$696$$ 0 0
$$697$$ 62.7426 2.37655
$$698$$ 2.00000 0.0757011
$$699$$ 0 0
$$700$$ 1.61803 0.0611559
$$701$$ 48.3394 1.82575 0.912877 0.408235i $$-0.133856\pi$$
0.912877 + 0.408235i $$0.133856\pi$$
$$702$$ 0 0
$$703$$ 12.0000 0.452589
$$704$$ −3.85410 −0.145257
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ 22.1803 0.834178
$$708$$ 0 0
$$709$$ −14.9098 −0.559950 −0.279975 0.960007i $$-0.590326\pi$$
−0.279975 + 0.960007i $$0.590326\pi$$
$$710$$ −7.09017 −0.266089
$$711$$ 0 0
$$712$$ −1.52786 −0.0572591
$$713$$ 2.09017 0.0782775
$$714$$ 0 0
$$715$$ 15.7639 0.589538
$$716$$ 20.1803 0.754175
$$717$$ 0 0
$$718$$ −18.3607 −0.685214
$$719$$ −1.72949 −0.0644991 −0.0322495 0.999480i $$-0.510267\pi$$
−0.0322495 + 0.999480i $$0.510267\pi$$
$$720$$ 0 0
$$721$$ −5.76393 −0.214660
$$722$$ −4.56231 −0.169791
$$723$$ 0 0
$$724$$ −18.8541 −0.700707
$$725$$ 4.76393 0.176928
$$726$$ 0 0
$$727$$ 52.7984 1.95818 0.979092 0.203420i $$-0.0652056\pi$$
0.979092 + 0.203420i $$0.0652056\pi$$
$$728$$ −6.61803 −0.245281
$$729$$ 0 0
$$730$$ −1.23607 −0.0457489
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −2.58359 −0.0954272 −0.0477136 0.998861i $$-0.515193\pi$$
−0.0477136 + 0.998861i $$0.515193\pi$$
$$734$$ 2.47214 0.0912482
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ −21.3050 −0.784778
$$738$$ 0 0
$$739$$ 21.8885 0.805183 0.402592 0.915380i $$-0.368110\pi$$
0.402592 + 0.915380i $$0.368110\pi$$
$$740$$ 2.47214 0.0908775
$$741$$ 0 0
$$742$$ −13.7082 −0.503244
$$743$$ 44.6312 1.63736 0.818680 0.574250i $$-0.194706\pi$$
0.818680 + 0.574250i $$0.194706\pi$$
$$744$$ 0 0
$$745$$ −1.14590 −0.0419825
$$746$$ 2.18034 0.0798279
$$747$$ 0 0
$$748$$ −19.6180 −0.717306
$$749$$ 6.76393 0.247149
$$750$$ 0 0
$$751$$ 29.0132 1.05871 0.529353 0.848402i $$-0.322435\pi$$
0.529353 + 0.848402i $$0.322435\pi$$
$$752$$ −9.70820 −0.354022
$$753$$ 0 0
$$754$$ −19.4853 −0.709612
$$755$$ −17.5623 −0.639158
$$756$$ 0 0
$$757$$ 17.8885 0.650170 0.325085 0.945685i $$-0.394607\pi$$
0.325085 + 0.945685i $$0.394607\pi$$
$$758$$ −33.4508 −1.21499
$$759$$ 0 0
$$760$$ −4.85410 −0.176077
$$761$$ 35.8673 1.30019 0.650094 0.759854i $$-0.274730\pi$$
0.650094 + 0.759854i $$0.274730\pi$$
$$762$$ 0 0
$$763$$ 13.8541 0.501552
$$764$$ 0.291796 0.0105568
$$765$$ 0 0
$$766$$ 17.8885 0.646339
$$767$$ 47.8885 1.72916
$$768$$ 0 0
$$769$$ −33.4164 −1.20503 −0.602513 0.798109i $$-0.705834\pi$$
−0.602513 + 0.798109i $$0.705834\pi$$
$$770$$ −6.23607 −0.224732
$$771$$ 0 0
$$772$$ 5.23607 0.188450
$$773$$ −11.0557 −0.397647 −0.198823 0.980035i $$-0.563712\pi$$
−0.198823 + 0.980035i $$0.563712\pi$$
$$774$$ 0 0
$$775$$ −2.09017 −0.0750811
$$776$$ −14.6180 −0.524757
$$777$$ 0 0
$$778$$ 5.67376 0.203414
$$779$$ −59.8328 −2.14373
$$780$$ 0 0
$$781$$ 27.3262 0.977810
$$782$$ 5.09017 0.182024
$$783$$ 0 0
$$784$$ −4.38197 −0.156499
$$785$$ −9.70820 −0.346501
$$786$$ 0 0
$$787$$ −43.1246 −1.53723 −0.768613 0.639714i $$-0.779053\pi$$
−0.768613 + 0.639714i $$0.779053\pi$$
$$788$$ 2.43769 0.0868393
$$789$$ 0 0
$$790$$ 10.4721 0.372582
$$791$$ −30.6525 −1.08988
$$792$$ 0 0
$$793$$ 25.8754 0.918862
$$794$$ 8.32624 0.295487
$$795$$ 0 0
$$796$$ 2.00000 0.0708881
$$797$$ −0.291796 −0.0103359 −0.00516797 0.999987i $$-0.501645\pi$$
−0.00516797 + 0.999987i $$0.501645\pi$$
$$798$$ 0 0
$$799$$ −49.4164 −1.74823
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 11.7082 0.413431
$$803$$ 4.76393 0.168116
$$804$$ 0 0
$$805$$ 1.61803 0.0570282
$$806$$ 8.54915 0.301131
$$807$$ 0 0
$$808$$ −13.7082 −0.482253
$$809$$ 4.25735 0.149681 0.0748403 0.997196i $$-0.476155\pi$$
0.0748403 + 0.997196i $$0.476155\pi$$
$$810$$ 0 0
$$811$$ 44.1803 1.55138 0.775691 0.631113i $$-0.217402\pi$$
0.775691 + 0.631113i $$0.217402\pi$$
$$812$$ 7.70820 0.270505
$$813$$ 0 0
$$814$$ −9.52786 −0.333951
$$815$$ −3.61803 −0.126734
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −21.2148 −0.741757
$$819$$ 0 0
$$820$$ −12.3262 −0.430451
$$821$$ 50.9443 1.77797 0.888984 0.457939i $$-0.151412\pi$$
0.888984 + 0.457939i $$0.151412\pi$$
$$822$$ 0 0
$$823$$ −1.41641 −0.0493729 −0.0246864 0.999695i $$-0.507859\pi$$
−0.0246864 + 0.999695i $$0.507859\pi$$
$$824$$ 3.56231 0.124099
$$825$$ 0 0
$$826$$ −18.9443 −0.659156
$$827$$ −8.29180 −0.288334 −0.144167 0.989553i $$-0.546050\pi$$
−0.144167 + 0.989553i $$0.546050\pi$$
$$828$$ 0 0
$$829$$ 1.05573 0.0366670 0.0183335 0.999832i $$-0.494164\pi$$
0.0183335 + 0.999832i $$0.494164\pi$$
$$830$$ −10.9443 −0.379881
$$831$$ 0 0
$$832$$ 4.09017 0.141801
$$833$$ −22.3050 −0.772821
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ 18.7082 0.647037
$$837$$ 0 0
$$838$$ 5.52786 0.190957
$$839$$ 43.0132 1.48498 0.742490 0.669858i $$-0.233645\pi$$
0.742490 + 0.669858i $$0.233645\pi$$
$$840$$ 0 0
$$841$$ −6.30495 −0.217412
$$842$$ 28.7426 0.990537
$$843$$ 0 0
$$844$$ 14.0000 0.481900
$$845$$ −3.72949 −0.128298
$$846$$ 0 0
$$847$$ 6.23607 0.214274
$$848$$ 8.47214 0.290934
$$849$$ 0 0
$$850$$ −5.09017 −0.174591
$$851$$ 2.47214 0.0847437
$$852$$ 0 0
$$853$$ 13.7984 0.472447 0.236224 0.971699i $$-0.424090\pi$$
0.236224 + 0.971699i $$0.424090\pi$$
$$854$$ −10.2361 −0.350271
$$855$$ 0 0
$$856$$ −4.18034 −0.142881
$$857$$ 33.4164 1.14148 0.570741 0.821130i $$-0.306656\pi$$
0.570741 + 0.821130i $$0.306656\pi$$
$$858$$ 0 0
$$859$$ 34.0689 1.16242 0.581208 0.813755i $$-0.302580\pi$$
0.581208 + 0.813755i $$0.302580\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 34.6525 1.18027
$$863$$ −37.2361 −1.26753 −0.633765 0.773525i $$-0.718491\pi$$
−0.633765 + 0.773525i $$0.718491\pi$$
$$864$$ 0 0
$$865$$ −21.5623 −0.733140
$$866$$ 29.5066 1.00267
$$867$$ 0 0
$$868$$ −3.38197 −0.114791
$$869$$ −40.3607 −1.36914
$$870$$ 0 0
$$871$$ 22.6099 0.766107
$$872$$ −8.56231 −0.289956
$$873$$ 0 0
$$874$$ −4.85410 −0.164192
$$875$$ −1.61803 −0.0546995
$$876$$ 0 0
$$877$$ −23.7426 −0.801732 −0.400866 0.916137i $$-0.631291\pi$$
−0.400866 + 0.916137i $$0.631291\pi$$
$$878$$ 15.6180 0.527083
$$879$$ 0 0
$$880$$ 3.85410 0.129922
$$881$$ −35.4164 −1.19321 −0.596605 0.802535i $$-0.703484\pi$$
−0.596605 + 0.802535i $$0.703484\pi$$
$$882$$ 0 0
$$883$$ −4.56231 −0.153534 −0.0767669 0.997049i $$-0.524460\pi$$
−0.0767669 + 0.997049i $$0.524460\pi$$
$$884$$ 20.8197 0.700241
$$885$$ 0 0
$$886$$ 13.9098 0.467310
$$887$$ 58.8328 1.97541 0.987706 0.156321i $$-0.0499634\pi$$
0.987706 + 0.156321i $$0.0499634\pi$$
$$888$$ 0 0
$$889$$ −10.0000 −0.335389
$$890$$ 1.52786 0.0512141
$$891$$ 0 0
$$892$$ −3.05573 −0.102313
$$893$$ 47.1246 1.57697
$$894$$ 0 0
$$895$$ −20.1803 −0.674554
$$896$$ −1.61803 −0.0540547
$$897$$ 0 0
$$898$$ 18.5623 0.619432
$$899$$ −9.95743 −0.332099
$$900$$ 0 0
$$901$$ 43.1246 1.43669
$$902$$ 47.5066 1.58180
$$903$$ 0 0
$$904$$ 18.9443 0.630077
$$905$$ 18.8541 0.626732
$$906$$ 0 0
$$907$$ −33.1246 −1.09988 −0.549942 0.835203i $$-0.685350\pi$$
−0.549942 + 0.835203i $$0.685350\pi$$
$$908$$ 23.2361 0.771116
$$909$$ 0 0
$$910$$ 6.61803 0.219386
$$911$$ 22.0689 0.731175 0.365587 0.930777i $$-0.380868\pi$$
0.365587 + 0.930777i $$0.380868\pi$$
$$912$$ 0 0
$$913$$ 42.1803 1.39597
$$914$$ −33.7771 −1.11725
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 24.1803 0.798505
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ −1.00000 −0.0329690
$$921$$ 0 0
$$922$$ −34.7639 −1.14489
$$923$$ −29.0000 −0.954547
$$924$$ 0 0
$$925$$ −2.47214 −0.0812833
$$926$$ −2.00000 −0.0657241
$$927$$ 0 0
$$928$$ −4.76393 −0.156384
$$929$$ 12.4721 0.409198 0.204599 0.978846i $$-0.434411\pi$$
0.204599 + 0.978846i $$0.434411\pi$$
$$930$$ 0 0
$$931$$ 21.2705 0.697113
$$932$$ −19.7082 −0.645564
$$933$$ 0 0
$$934$$ 23.1246 0.756660
$$935$$ 19.6180 0.641578
$$936$$ 0 0
$$937$$ −12.2016 −0.398610 −0.199305 0.979938i $$-0.563868\pi$$
−0.199305 + 0.979938i $$0.563868\pi$$
$$938$$ −8.94427 −0.292041
$$939$$ 0 0
$$940$$ 9.70820 0.316647
$$941$$ 60.5066 1.97246 0.986229 0.165385i $$-0.0528868\pi$$
0.986229 + 0.165385i $$0.0528868\pi$$
$$942$$ 0 0
$$943$$ −12.3262 −0.401398
$$944$$ 11.7082 0.381070
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −5.68692 −0.184800 −0.0924000 0.995722i $$-0.529454\pi$$
−0.0924000 + 0.995722i $$0.529454\pi$$
$$948$$ 0 0
$$949$$ −5.05573 −0.164116
$$950$$ 4.85410 0.157488
$$951$$ 0 0
$$952$$ −8.23607 −0.266932
$$953$$ 20.7984 0.673725 0.336863 0.941554i $$-0.390634\pi$$
0.336863 + 0.941554i $$0.390634\pi$$
$$954$$ 0 0
$$955$$ −0.291796 −0.00944230
$$956$$ −24.3607 −0.787881
$$957$$ 0 0
$$958$$ 3.88854 0.125633
$$959$$ −8.61803 −0.278291
$$960$$ 0 0
$$961$$ −26.6312 −0.859071
$$962$$ 10.1115 0.326006
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −5.23607 −0.168555
$$966$$ 0 0
$$967$$ 50.5410 1.62529 0.812645 0.582759i $$-0.198027\pi$$
0.812645 + 0.582759i $$0.198027\pi$$
$$968$$ −3.85410 −0.123876
$$969$$ 0 0
$$970$$ 14.6180 0.469357
$$971$$ −0.729490 −0.0234105 −0.0117052 0.999931i $$-0.503726\pi$$
−0.0117052 + 0.999931i $$0.503726\pi$$
$$972$$ 0 0
$$973$$ 27.8885 0.894066
$$974$$ 42.1803 1.35155
$$975$$ 0 0
$$976$$ 6.32624 0.202498
$$977$$ 3.43769 0.109982 0.0549908 0.998487i $$-0.482487\pi$$
0.0549908 + 0.998487i $$0.482487\pi$$
$$978$$ 0 0
$$979$$ −5.88854 −0.188199
$$980$$ 4.38197 0.139977
$$981$$ 0 0
$$982$$ −16.1803 −0.516335
$$983$$ −19.2705 −0.614634 −0.307317 0.951607i $$-0.599431\pi$$
−0.307317 + 0.951607i $$0.599431\pi$$
$$984$$ 0 0
$$985$$ −2.43769 −0.0776714
$$986$$ −24.2492 −0.772253
$$987$$ 0 0
$$988$$ −19.8541 −0.631643
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −10.5066 −0.333752 −0.166876 0.985978i $$-0.553368\pi$$
−0.166876 + 0.985978i $$0.553368\pi$$
$$992$$ 2.09017 0.0663630
$$993$$ 0 0
$$994$$ 11.4721 0.363874
$$995$$ −2.00000 −0.0634043
$$996$$ 0 0
$$997$$ −41.1935 −1.30461 −0.652306 0.757956i $$-0.726198\pi$$
−0.652306 + 0.757956i $$0.726198\pi$$
$$998$$ 32.3607 1.02436
$$999$$ 0 0
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 2070.2.a.u.1.2 2
3.2 odd 2 230.2.a.c.1.1 2
12.11 even 2 1840.2.a.l.1.2 2
15.2 even 4 1150.2.b.i.599.4 4
15.8 even 4 1150.2.b.i.599.1 4
15.14 odd 2 1150.2.a.j.1.2 2
24.5 odd 2 7360.2.a.bh.1.2 2
24.11 even 2 7360.2.a.bn.1.1 2
60.59 even 2 9200.2.a.bu.1.1 2
69.68 even 2 5290.2.a.o.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.1 2 3.2 odd 2
1150.2.a.j.1.2 2 15.14 odd 2
1150.2.b.i.599.1 4 15.8 even 4
1150.2.b.i.599.4 4 15.2 even 4
1840.2.a.l.1.2 2 12.11 even 2
2070.2.a.u.1.2 2 1.1 even 1 trivial
5290.2.a.o.1.1 2 69.68 even 2
7360.2.a.bh.1.2 2 24.5 odd 2
7360.2.a.bn.1.1 2 24.11 even 2
9200.2.a.bu.1.1 2 60.59 even 2