# Properties

 Label 475.2.b.d.324.1 Level $475$ Weight $2$ Character 475.324 Analytic conductor $3.793$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 324.1 Root $$-0.854638 - 0.854638i$$ of defining polynomial Character $$\chi$$ $$=$$ 475.324 Dual form 475.2.b.d.324.6

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.17009i q^{2} -1.70928i q^{3} -2.70928 q^{4} -3.70928 q^{6} -1.07838i q^{7} +1.53919i q^{8} +0.0783777 q^{9} +O(q^{10})$$ $$q-2.17009i q^{2} -1.70928i q^{3} -2.70928 q^{4} -3.70928 q^{6} -1.07838i q^{7} +1.53919i q^{8} +0.0783777 q^{9} -6.34017 q^{11} +4.63090i q^{12} +1.36910i q^{13} -2.34017 q^{14} -2.07838 q^{16} -3.26180i q^{17} -0.170086i q^{18} +1.00000 q^{19} -1.84324 q^{21} +13.7587i q^{22} +2.34017i q^{23} +2.63090 q^{24} +2.97107 q^{26} -5.26180i q^{27} +2.92162i q^{28} -1.41855 q^{29} +8.68035 q^{31} +7.58864i q^{32} +10.8371i q^{33} -7.07838 q^{34} -0.212347 q^{36} -5.36910i q^{37} -2.17009i q^{38} +2.34017 q^{39} -3.26180 q^{41} +4.00000i q^{42} -11.9155i q^{43} +17.1773 q^{44} +5.07838 q^{46} -1.07838i q^{47} +3.55252i q^{48} +5.83710 q^{49} -5.57531 q^{51} -3.70928i q^{52} +6.63090i q^{53} -11.4186 q^{54} +1.65983 q^{56} -1.70928i q^{57} +3.07838i q^{58} +11.4186 q^{59} +5.60197 q^{61} -18.8371i q^{62} -0.0845208i q^{63} +12.3112 q^{64} +23.5174 q^{66} -10.3896i q^{67} +8.83710i q^{68} +4.00000 q^{69} -10.8371 q^{71} +0.120638i q^{72} +5.41855i q^{73} -11.6514 q^{74} -2.70928 q^{76} +6.83710i q^{77} -5.07838i q^{78} -14.2557 q^{79} -8.75872 q^{81} +7.07838i q^{82} -14.3402i q^{83} +4.99386 q^{84} -25.8576 q^{86} +2.42469i q^{87} -9.75872i q^{88} -7.57531 q^{89} +1.47641 q^{91} -6.34017i q^{92} -14.8371i q^{93} -2.34017 q^{94} +12.9711 q^{96} +8.88655i q^{97} -12.6670i q^{98} -0.496928 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 $$6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} - 16 q^{11} + 8 q^{14} - 6 q^{16} + 6 q^{19} - 24 q^{21} + 8 q^{24} - 12 q^{26} + 20 q^{29} + 8 q^{31} - 36 q^{34} - 22 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} + 24 q^{46} - 22 q^{49} + 8 q^{51} - 40 q^{54} + 32 q^{56} + 40 q^{59} - 4 q^{61} + 22 q^{64} + 40 q^{66} + 24 q^{69} - 8 q^{71} + 4 q^{74} - 2 q^{76} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 4 q^{89} + 40 q^{91} + 8 q^{94} + 48 q^{96} + 32 q^{99}+O(q^{100})$$ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 - 16 * q^11 + 8 * q^14 - 6 * q^16 + 6 * q^19 - 24 * q^21 + 8 * q^24 - 12 * q^26 + 20 * q^29 + 8 * q^31 - 36 * q^34 - 22 * q^36 - 8 * q^39 - 4 * q^41 + 24 * q^44 + 24 * q^46 - 22 * q^49 + 8 * q^51 - 40 * q^54 + 32 * q^56 + 40 * q^59 - 4 * q^61 + 22 * q^64 + 40 * q^66 + 24 * q^69 - 8 * q^71 + 4 * q^74 - 2 * q^76 - 2 * q^81 - 40 * q^84 - 32 * q^86 - 4 * q^89 + 40 * q^91 + 8 * q^94 + 48 * q^96 + 32 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.17009i − 1.53448i −0.641358 0.767241i $$-0.721629\pi$$
0.641358 0.767241i $$-0.278371\pi$$
$$3$$ − 1.70928i − 0.986851i −0.869788 0.493425i $$-0.835745\pi$$
0.869788 0.493425i $$-0.164255\pi$$
$$4$$ −2.70928 −1.35464
$$5$$ 0 0
$$6$$ −3.70928 −1.51431
$$7$$ − 1.07838i − 0.407588i −0.979014 0.203794i $$-0.934673\pi$$
0.979014 0.203794i $$-0.0653274\pi$$
$$8$$ 1.53919i 0.544185i
$$9$$ 0.0783777 0.0261259
$$10$$ 0 0
$$11$$ −6.34017 −1.91163 −0.955817 0.293962i $$-0.905026\pi$$
−0.955817 + 0.293962i $$0.905026\pi$$
$$12$$ 4.63090i 1.33682i
$$13$$ 1.36910i 0.379721i 0.981811 + 0.189860i $$0.0608035\pi$$
−0.981811 + 0.189860i $$0.939196\pi$$
$$14$$ −2.34017 −0.625438
$$15$$ 0 0
$$16$$ −2.07838 −0.519594
$$17$$ − 3.26180i − 0.791102i −0.918444 0.395551i $$-0.870554\pi$$
0.918444 0.395551i $$-0.129446\pi$$
$$18$$ − 0.170086i − 0.0400898i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −1.84324 −0.402229
$$22$$ 13.7587i 2.93337i
$$23$$ 2.34017i 0.487960i 0.969780 + 0.243980i $$0.0784531\pi$$
−0.969780 + 0.243980i $$0.921547\pi$$
$$24$$ 2.63090 0.537030
$$25$$ 0 0
$$26$$ 2.97107 0.582675
$$27$$ − 5.26180i − 1.01263i
$$28$$ 2.92162i 0.552135i
$$29$$ −1.41855 −0.263418 −0.131709 0.991288i $$-0.542046\pi$$
−0.131709 + 0.991288i $$0.542046\pi$$
$$30$$ 0 0
$$31$$ 8.68035 1.55904 0.779518 0.626380i $$-0.215464\pi$$
0.779518 + 0.626380i $$0.215464\pi$$
$$32$$ 7.58864i 1.34149i
$$33$$ 10.8371i 1.88650i
$$34$$ −7.07838 −1.21393
$$35$$ 0 0
$$36$$ −0.212347 −0.0353911
$$37$$ − 5.36910i − 0.882675i −0.897341 0.441337i $$-0.854504\pi$$
0.897341 0.441337i $$-0.145496\pi$$
$$38$$ − 2.17009i − 0.352035i
$$39$$ 2.34017 0.374728
$$40$$ 0 0
$$41$$ −3.26180 −0.509407 −0.254703 0.967019i $$-0.581978\pi$$
−0.254703 + 0.967019i $$0.581978\pi$$
$$42$$ 4.00000i 0.617213i
$$43$$ − 11.9155i − 1.81709i −0.417783 0.908547i $$-0.637193\pi$$
0.417783 0.908547i $$-0.362807\pi$$
$$44$$ 17.1773 2.58957
$$45$$ 0 0
$$46$$ 5.07838 0.748766
$$47$$ − 1.07838i − 0.157298i −0.996902 0.0786488i $$-0.974939\pi$$
0.996902 0.0786488i $$-0.0250606\pi$$
$$48$$ 3.55252i 0.512762i
$$49$$ 5.83710 0.833872
$$50$$ 0 0
$$51$$ −5.57531 −0.780699
$$52$$ − 3.70928i − 0.514384i
$$53$$ 6.63090i 0.910824i 0.890281 + 0.455412i $$0.150508\pi$$
−0.890281 + 0.455412i $$0.849492\pi$$
$$54$$ −11.4186 −1.55387
$$55$$ 0 0
$$56$$ 1.65983 0.221804
$$57$$ − 1.70928i − 0.226399i
$$58$$ 3.07838i 0.404211i
$$59$$ 11.4186 1.48657 0.743284 0.668976i $$-0.233267\pi$$
0.743284 + 0.668976i $$0.233267\pi$$
$$60$$ 0 0
$$61$$ 5.60197 0.717259 0.358629 0.933480i $$-0.383244\pi$$
0.358629 + 0.933480i $$0.383244\pi$$
$$62$$ − 18.8371i − 2.39231i
$$63$$ − 0.0845208i − 0.0106486i
$$64$$ 12.3112 1.53891
$$65$$ 0 0
$$66$$ 23.5174 2.89480
$$67$$ − 10.3896i − 1.26929i −0.772802 0.634647i $$-0.781145\pi$$
0.772802 0.634647i $$-0.218855\pi$$
$$68$$ 8.83710i 1.07166i
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −10.8371 −1.28613 −0.643064 0.765813i $$-0.722337\pi$$
−0.643064 + 0.765813i $$0.722337\pi$$
$$72$$ 0.120638i 0.0142173i
$$73$$ 5.41855i 0.634193i 0.948393 + 0.317097i $$0.102708\pi$$
−0.948393 + 0.317097i $$0.897292\pi$$
$$74$$ −11.6514 −1.35445
$$75$$ 0 0
$$76$$ −2.70928 −0.310775
$$77$$ 6.83710i 0.779160i
$$78$$ − 5.07838i − 0.575013i
$$79$$ −14.2557 −1.60389 −0.801943 0.597400i $$-0.796200\pi$$
−0.801943 + 0.597400i $$0.796200\pi$$
$$80$$ 0 0
$$81$$ −8.75872 −0.973192
$$82$$ 7.07838i 0.781676i
$$83$$ − 14.3402i − 1.57404i −0.616928 0.787019i $$-0.711623\pi$$
0.616928 0.787019i $$-0.288377\pi$$
$$84$$ 4.99386 0.544874
$$85$$ 0 0
$$86$$ −25.8576 −2.78830
$$87$$ 2.42469i 0.259954i
$$88$$ − 9.75872i − 1.04028i
$$89$$ −7.57531 −0.802981 −0.401490 0.915863i $$-0.631508\pi$$
−0.401490 + 0.915863i $$0.631508\pi$$
$$90$$ 0 0
$$91$$ 1.47641 0.154770
$$92$$ − 6.34017i − 0.661009i
$$93$$ − 14.8371i − 1.53854i
$$94$$ −2.34017 −0.241370
$$95$$ 0 0
$$96$$ 12.9711 1.32385
$$97$$ 8.88655i 0.902292i 0.892450 + 0.451146i $$0.148985\pi$$
−0.892450 + 0.451146i $$0.851015\pi$$
$$98$$ − 12.6670i − 1.27956i
$$99$$ −0.496928 −0.0499432
$$100$$ 0 0
$$101$$ −4.92162 −0.489720 −0.244860 0.969558i $$-0.578742\pi$$
−0.244860 + 0.969558i $$0.578742\pi$$
$$102$$ 12.0989i 1.19797i
$$103$$ 6.38962i 0.629588i 0.949160 + 0.314794i $$0.101935\pi$$
−0.949160 + 0.314794i $$0.898065\pi$$
$$104$$ −2.10731 −0.206638
$$105$$ 0 0
$$106$$ 14.3896 1.39764
$$107$$ − 2.29072i − 0.221453i −0.993851 0.110726i $$-0.964682\pi$$
0.993851 0.110726i $$-0.0353177\pi$$
$$108$$ 14.2557i 1.37175i
$$109$$ 12.8371 1.22957 0.614786 0.788694i $$-0.289243\pi$$
0.614786 + 0.788694i $$0.289243\pi$$
$$110$$ 0 0
$$111$$ −9.17727 −0.871068
$$112$$ 2.24128i 0.211781i
$$113$$ − 12.8865i − 1.21226i −0.795364 0.606132i $$-0.792720\pi$$
0.795364 0.606132i $$-0.207280\pi$$
$$114$$ −3.70928 −0.347405
$$115$$ 0 0
$$116$$ 3.84324 0.356836
$$117$$ 0.107307i 0.00992055i
$$118$$ − 24.7792i − 2.28111i
$$119$$ −3.51745 −0.322444
$$120$$ 0 0
$$121$$ 29.1978 2.65434
$$122$$ − 12.1568i − 1.10062i
$$123$$ 5.57531i 0.502708i
$$124$$ −23.5174 −2.11193
$$125$$ 0 0
$$126$$ −0.183417 −0.0163401
$$127$$ 8.23287i 0.730549i 0.930900 + 0.365274i $$0.119025\pi$$
−0.930900 + 0.365274i $$0.880975\pi$$
$$128$$ − 11.5392i − 1.01993i
$$129$$ −20.3668 −1.79320
$$130$$ 0 0
$$131$$ 1.47641 0.128995 0.0644973 0.997918i $$-0.479456\pi$$
0.0644973 + 0.997918i $$0.479456\pi$$
$$132$$ − 29.3607i − 2.55552i
$$133$$ − 1.07838i − 0.0935072i
$$134$$ −22.5464 −1.94771
$$135$$ 0 0
$$136$$ 5.02052 0.430506
$$137$$ 3.94214i 0.336800i 0.985719 + 0.168400i $$0.0538600\pi$$
−0.985719 + 0.168400i $$0.946140\pi$$
$$138$$ − 8.68035i − 0.738920i
$$139$$ −8.86376 −0.751815 −0.375907 0.926657i $$-0.622669\pi$$
−0.375907 + 0.926657i $$0.622669\pi$$
$$140$$ 0 0
$$141$$ −1.84324 −0.155229
$$142$$ 23.5174i 1.97354i
$$143$$ − 8.68035i − 0.725887i
$$144$$ −0.162899 −0.0135749
$$145$$ 0 0
$$146$$ 11.7587 0.973159
$$147$$ − 9.97721i − 0.822907i
$$148$$ 14.5464i 1.19570i
$$149$$ 19.7587 1.61870 0.809349 0.587328i $$-0.199820\pi$$
0.809349 + 0.587328i $$0.199820\pi$$
$$150$$ 0 0
$$151$$ 3.41855 0.278198 0.139099 0.990279i $$-0.455579\pi$$
0.139099 + 0.990279i $$0.455579\pi$$
$$152$$ 1.53919i 0.124845i
$$153$$ − 0.255652i − 0.0206683i
$$154$$ 14.8371 1.19561
$$155$$ 0 0
$$156$$ −6.34017 −0.507620
$$157$$ − 9.41855i − 0.751682i −0.926684 0.375841i $$-0.877354\pi$$
0.926684 0.375841i $$-0.122646\pi$$
$$158$$ 30.9360i 2.46114i
$$159$$ 11.3340 0.898847
$$160$$ 0 0
$$161$$ 2.52359 0.198887
$$162$$ 19.0072i 1.49335i
$$163$$ − 2.92162i − 0.228839i −0.993433 0.114420i $$-0.963499\pi$$
0.993433 0.114420i $$-0.0365008\pi$$
$$164$$ 8.83710 0.690062
$$165$$ 0 0
$$166$$ −31.1194 −2.41534
$$167$$ − 20.9132i − 1.61831i −0.587593 0.809156i $$-0.699924\pi$$
0.587593 0.809156i $$-0.300076\pi$$
$$168$$ − 2.83710i − 0.218887i
$$169$$ 11.1256 0.855812
$$170$$ 0 0
$$171$$ 0.0783777 0.00599370
$$172$$ 32.2823i 2.46150i
$$173$$ − 1.05559i − 0.0802551i −0.999195 0.0401276i $$-0.987224\pi$$
0.999195 0.0401276i $$-0.0127764\pi$$
$$174$$ 5.26180 0.398896
$$175$$ 0 0
$$176$$ 13.1773 0.993274
$$177$$ − 19.5174i − 1.46702i
$$178$$ 16.4391i 1.23216i
$$179$$ −0.894960 −0.0668925 −0.0334462 0.999441i $$-0.510648\pi$$
−0.0334462 + 0.999441i $$0.510648\pi$$
$$180$$ 0 0
$$181$$ −0.837101 −0.0622213 −0.0311106 0.999516i $$-0.509904\pi$$
−0.0311106 + 0.999516i $$0.509904\pi$$
$$182$$ − 3.20394i − 0.237492i
$$183$$ − 9.57531i − 0.707827i
$$184$$ −3.60197 −0.265541
$$185$$ 0 0
$$186$$ −32.1978 −2.36086
$$187$$ 20.6803i 1.51230i
$$188$$ 2.92162i 0.213081i
$$189$$ −5.67420 −0.412738
$$190$$ 0 0
$$191$$ 22.0410 1.59483 0.797417 0.603429i $$-0.206199\pi$$
0.797417 + 0.603429i $$0.206199\pi$$
$$192$$ − 21.0433i − 1.51867i
$$193$$ 12.7877i 0.920475i 0.887796 + 0.460238i $$0.152236\pi$$
−0.887796 + 0.460238i $$0.847764\pi$$
$$194$$ 19.2846 1.38455
$$195$$ 0 0
$$196$$ −15.8143 −1.12959
$$197$$ 9.20394i 0.655753i 0.944721 + 0.327877i $$0.106333\pi$$
−0.944721 + 0.327877i $$0.893667\pi$$
$$198$$ 1.07838i 0.0766370i
$$199$$ 16.1978 1.14823 0.574116 0.818774i $$-0.305346\pi$$
0.574116 + 0.818774i $$0.305346\pi$$
$$200$$ 0 0
$$201$$ −17.7587 −1.25260
$$202$$ 10.6803i 0.751467i
$$203$$ 1.52973i 0.107366i
$$204$$ 15.1050 1.05756
$$205$$ 0 0
$$206$$ 13.8660 0.966092
$$207$$ 0.183417i 0.0127484i
$$208$$ − 2.84551i − 0.197301i
$$209$$ −6.34017 −0.438559
$$210$$ 0 0
$$211$$ 7.78539 0.535968 0.267984 0.963423i $$-0.413643\pi$$
0.267984 + 0.963423i $$0.413643\pi$$
$$212$$ − 17.9649i − 1.23384i
$$213$$ 18.5236i 1.26922i
$$214$$ −4.97107 −0.339815
$$215$$ 0 0
$$216$$ 8.09890 0.551060
$$217$$ − 9.36069i − 0.635445i
$$218$$ − 27.8576i − 1.88676i
$$219$$ 9.26180 0.625854
$$220$$ 0 0
$$221$$ 4.46573 0.300398
$$222$$ 19.9155i 1.33664i
$$223$$ 12.5464i 0.840168i 0.907485 + 0.420084i $$0.137999\pi$$
−0.907485 + 0.420084i $$0.862001\pi$$
$$224$$ 8.18342 0.546778
$$225$$ 0 0
$$226$$ −27.9649 −1.86020
$$227$$ − 2.29072i − 0.152041i −0.997106 0.0760204i $$-0.975779\pi$$
0.997106 0.0760204i $$-0.0242214\pi$$
$$228$$ 4.63090i 0.306689i
$$229$$ 5.91548 0.390906 0.195453 0.980713i $$-0.437382\pi$$
0.195453 + 0.980713i $$0.437382\pi$$
$$230$$ 0 0
$$231$$ 11.6865 0.768915
$$232$$ − 2.18342i − 0.143348i
$$233$$ 13.5174i 0.885557i 0.896631 + 0.442779i $$0.146007\pi$$
−0.896631 + 0.442779i $$0.853993\pi$$
$$234$$ 0.232866 0.0152229
$$235$$ 0 0
$$236$$ −30.9360 −2.01376
$$237$$ 24.3668i 1.58280i
$$238$$ 7.63317i 0.494785i
$$239$$ −13.8432 −0.895445 −0.447723 0.894173i $$-0.647765\pi$$
−0.447723 + 0.894173i $$0.647765\pi$$
$$240$$ 0 0
$$241$$ 7.26180 0.467773 0.233887 0.972264i $$-0.424856\pi$$
0.233887 + 0.972264i $$0.424856\pi$$
$$242$$ − 63.3617i − 4.07305i
$$243$$ − 0.814315i − 0.0522383i
$$244$$ −15.1773 −0.971625
$$245$$ 0 0
$$246$$ 12.0989 0.771397
$$247$$ 1.36910i 0.0871139i
$$248$$ 13.3607i 0.848405i
$$249$$ −24.5113 −1.55334
$$250$$ 0 0
$$251$$ 10.4703 0.660877 0.330439 0.943827i $$-0.392803\pi$$
0.330439 + 0.943827i $$0.392803\pi$$
$$252$$ 0.228990i 0.0144250i
$$253$$ − 14.8371i − 0.932801i
$$254$$ 17.8660 1.12101
$$255$$ 0 0
$$256$$ −0.418551 −0.0261594
$$257$$ 23.6248i 1.47367i 0.676072 + 0.736836i $$0.263681\pi$$
−0.676072 + 0.736836i $$0.736319\pi$$
$$258$$ 44.1978i 2.75163i
$$259$$ −5.78992 −0.359768
$$260$$ 0 0
$$261$$ −0.111183 −0.00688204
$$262$$ − 3.20394i − 0.197940i
$$263$$ 5.65983i 0.349000i 0.984657 + 0.174500i $$0.0558309\pi$$
−0.984657 + 0.174500i $$0.944169\pi$$
$$264$$ −16.6803 −1.02660
$$265$$ 0 0
$$266$$ −2.34017 −0.143485
$$267$$ 12.9483i 0.792422i
$$268$$ 28.1483i 1.71943i
$$269$$ 2.31351 0.141057 0.0705286 0.997510i $$-0.477531\pi$$
0.0705286 + 0.997510i $$0.477531\pi$$
$$270$$ 0 0
$$271$$ −19.7009 −1.19674 −0.598371 0.801219i $$-0.704185\pi$$
−0.598371 + 0.801219i $$0.704185\pi$$
$$272$$ 6.77924i 0.411052i
$$273$$ − 2.52359i − 0.152735i
$$274$$ 8.55479 0.516814
$$275$$ 0 0
$$276$$ −10.8371 −0.652317
$$277$$ 25.7321i 1.54609i 0.634351 + 0.773045i $$0.281267\pi$$
−0.634351 + 0.773045i $$0.718733\pi$$
$$278$$ 19.2351i 1.15365i
$$279$$ 0.680346 0.0407312
$$280$$ 0 0
$$281$$ −6.58145 −0.392616 −0.196308 0.980542i $$-0.562895\pi$$
−0.196308 + 0.980542i $$0.562895\pi$$
$$282$$ 4.00000i 0.238197i
$$283$$ − 0.496928i − 0.0295393i −0.999891 0.0147697i $$-0.995298\pi$$
0.999891 0.0147697i $$-0.00470150\pi$$
$$284$$ 29.3607 1.74224
$$285$$ 0 0
$$286$$ −18.8371 −1.11386
$$287$$ 3.51745i 0.207628i
$$288$$ 0.594780i 0.0350478i
$$289$$ 6.36069 0.374158
$$290$$ 0 0
$$291$$ 15.1896 0.890428
$$292$$ − 14.6803i − 0.859102i
$$293$$ 6.63090i 0.387381i 0.981063 + 0.193691i $$0.0620458\pi$$
−0.981063 + 0.193691i $$0.937954\pi$$
$$294$$ −21.6514 −1.26274
$$295$$ 0 0
$$296$$ 8.26406 0.480339
$$297$$ 33.3607i 1.93578i
$$298$$ − 42.8781i − 2.48386i
$$299$$ −3.20394 −0.185288
$$300$$ 0 0
$$301$$ −12.8494 −0.740626
$$302$$ − 7.41855i − 0.426890i
$$303$$ 8.41241i 0.483280i
$$304$$ −2.07838 −0.119203
$$305$$ 0 0
$$306$$ −0.554787 −0.0317151
$$307$$ − 14.6042i − 0.833508i −0.909019 0.416754i $$-0.863168\pi$$
0.909019 0.416754i $$-0.136832\pi$$
$$308$$ − 18.5236i − 1.05548i
$$309$$ 10.9216 0.621309
$$310$$ 0 0
$$311$$ 19.3340 1.09633 0.548166 0.836369i $$-0.315326\pi$$
0.548166 + 0.836369i $$0.315326\pi$$
$$312$$ 3.60197i 0.203921i
$$313$$ 30.6803i 1.73416i 0.498173 + 0.867078i $$0.334005\pi$$
−0.498173 + 0.867078i $$0.665995\pi$$
$$314$$ −20.4391 −1.15344
$$315$$ 0 0
$$316$$ 38.6225 2.17268
$$317$$ − 18.7298i − 1.05197i −0.850494 0.525985i $$-0.823697\pi$$
0.850494 0.525985i $$-0.176303\pi$$
$$318$$ − 24.5958i − 1.37927i
$$319$$ 8.99386 0.503559
$$320$$ 0 0
$$321$$ −3.91548 −0.218541
$$322$$ − 5.47641i − 0.305188i
$$323$$ − 3.26180i − 0.181491i
$$324$$ 23.7298 1.31832
$$325$$ 0 0
$$326$$ −6.34017 −0.351150
$$327$$ − 21.9421i − 1.21340i
$$328$$ − 5.02052i − 0.277212i
$$329$$ −1.16290 −0.0641127
$$330$$ 0 0
$$331$$ −2.73820 −0.150505 −0.0752527 0.997164i $$-0.523976\pi$$
−0.0752527 + 0.997164i $$0.523976\pi$$
$$332$$ 38.8515i 2.13225i
$$333$$ − 0.420818i − 0.0230607i
$$334$$ −45.3835 −2.48327
$$335$$ 0 0
$$336$$ 3.83096 0.208996
$$337$$ 6.04945i 0.329534i 0.986332 + 0.164767i $$0.0526873\pi$$
−0.986332 + 0.164767i $$0.947313\pi$$
$$338$$ − 24.1434i − 1.31323i
$$339$$ −22.0267 −1.19632
$$340$$ 0 0
$$341$$ −55.0349 −2.98031
$$342$$ − 0.170086i − 0.00919722i
$$343$$ − 13.8432i − 0.747465i
$$344$$ 18.3402 0.988836
$$345$$ 0 0
$$346$$ −2.29072 −0.123150
$$347$$ 5.97334i 0.320666i 0.987063 + 0.160333i $$0.0512567\pi$$
−0.987063 + 0.160333i $$0.948743\pi$$
$$348$$ − 6.56916i − 0.352144i
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 7.20394 0.384518
$$352$$ − 48.1133i − 2.56445i
$$353$$ − 14.0989i − 0.750409i −0.926942 0.375204i $$-0.877573\pi$$
0.926942 0.375204i $$-0.122427\pi$$
$$354$$ −42.3545 −2.25112
$$355$$ 0 0
$$356$$ 20.5236 1.08775
$$357$$ 6.01229i 0.318204i
$$358$$ 1.94214i 0.102645i
$$359$$ −6.02666 −0.318075 −0.159038 0.987273i $$-0.550839\pi$$
−0.159038 + 0.987273i $$0.550839\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 1.81658i 0.0954775i
$$363$$ − 49.9071i − 2.61944i
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ −20.7792 −1.08615
$$367$$ − 1.07838i − 0.0562909i −0.999604 0.0281454i $$-0.991040\pi$$
0.999604 0.0281454i $$-0.00896015\pi$$
$$368$$ − 4.86376i − 0.253541i
$$369$$ −0.255652 −0.0133087
$$370$$ 0 0
$$371$$ 7.15061 0.371241
$$372$$ 40.1978i 2.08416i
$$373$$ 12.3051i 0.637134i 0.947900 + 0.318567i $$0.103202\pi$$
−0.947900 + 0.318567i $$0.896798\pi$$
$$374$$ 44.8781 2.32059
$$375$$ 0 0
$$376$$ 1.65983 0.0855990
$$377$$ − 1.94214i − 0.100025i
$$378$$ 12.3135i 0.633339i
$$379$$ −1.04718 −0.0537901 −0.0268950 0.999638i $$-0.508562\pi$$
−0.0268950 + 0.999638i $$0.508562\pi$$
$$380$$ 0 0
$$381$$ 14.0722 0.720942
$$382$$ − 47.8310i − 2.44724i
$$383$$ 0.0806452i 0.00412078i 0.999998 + 0.00206039i $$0.000655842\pi$$
−0.999998 + 0.00206039i $$0.999344\pi$$
$$384$$ −19.7237 −1.00652
$$385$$ 0 0
$$386$$ 27.7503 1.41245
$$387$$ − 0.933908i − 0.0474732i
$$388$$ − 24.0761i − 1.22228i
$$389$$ 20.5236 1.04059 0.520294 0.853987i $$-0.325822\pi$$
0.520294 + 0.853987i $$0.325822\pi$$
$$390$$ 0 0
$$391$$ 7.63317 0.386026
$$392$$ 8.98440i 0.453781i
$$393$$ − 2.52359i − 0.127298i
$$394$$ 19.9733 1.00624
$$395$$ 0 0
$$396$$ 1.34632 0.0676549
$$397$$ − 39.4596i − 1.98042i −0.139586 0.990210i $$-0.544577\pi$$
0.139586 0.990210i $$-0.455423\pi$$
$$398$$ − 35.1506i − 1.76194i
$$399$$ −1.84324 −0.0922776
$$400$$ 0 0
$$401$$ 0.470266 0.0234840 0.0117420 0.999931i $$-0.496262\pi$$
0.0117420 + 0.999931i $$0.496262\pi$$
$$402$$ 38.5380i 1.92210i
$$403$$ 11.8843i 0.591998i
$$404$$ 13.3340 0.663393
$$405$$ 0 0
$$406$$ 3.31965 0.164752
$$407$$ 34.0410i 1.68735i
$$408$$ − 8.58145i − 0.424845i
$$409$$ −10.4826 −0.518329 −0.259164 0.965833i $$-0.583447\pi$$
−0.259164 + 0.965833i $$0.583447\pi$$
$$410$$ 0 0
$$411$$ 6.73820 0.332371
$$412$$ − 17.3112i − 0.852864i
$$413$$ − 12.3135i − 0.605908i
$$414$$ 0.398032 0.0195622
$$415$$ 0 0
$$416$$ −10.3896 −0.509393
$$417$$ 15.1506i 0.741929i
$$418$$ 13.7587i 0.672961i
$$419$$ −34.6681 −1.69365 −0.846823 0.531875i $$-0.821488\pi$$
−0.846823 + 0.531875i $$0.821488\pi$$
$$420$$ 0 0
$$421$$ −34.6102 −1.68680 −0.843399 0.537288i $$-0.819449\pi$$
−0.843399 + 0.537288i $$0.819449\pi$$
$$422$$ − 16.8950i − 0.822434i
$$423$$ − 0.0845208i − 0.00410954i
$$424$$ −10.2062 −0.495657
$$425$$ 0 0
$$426$$ 40.1978 1.94759
$$427$$ − 6.04104i − 0.292346i
$$428$$ 6.20620i 0.299988i
$$429$$ −14.8371 −0.716342
$$430$$ 0 0
$$431$$ 6.73820 0.324568 0.162284 0.986744i $$-0.448114\pi$$
0.162284 + 0.986744i $$0.448114\pi$$
$$432$$ 10.9360i 0.526158i
$$433$$ 20.4741i 0.983924i 0.870617 + 0.491962i $$0.163720\pi$$
−0.870617 + 0.491962i $$0.836280\pi$$
$$434$$ −20.3135 −0.975080
$$435$$ 0 0
$$436$$ −34.7792 −1.66562
$$437$$ 2.34017i 0.111946i
$$438$$ − 20.0989i − 0.960362i
$$439$$ 21.4596 1.02421 0.512105 0.858923i $$-0.328866\pi$$
0.512105 + 0.858923i $$0.328866\pi$$
$$440$$ 0 0
$$441$$ 0.457499 0.0217857
$$442$$ − 9.69102i − 0.460955i
$$443$$ − 21.5441i − 1.02359i −0.859107 0.511796i $$-0.828980\pi$$
0.859107 0.511796i $$-0.171020\pi$$
$$444$$ 24.8638 1.17998
$$445$$ 0 0
$$446$$ 27.2267 1.28922
$$447$$ − 33.7731i − 1.59741i
$$448$$ − 13.2762i − 0.627240i
$$449$$ 8.47027 0.399737 0.199868 0.979823i $$-0.435949\pi$$
0.199868 + 0.979823i $$0.435949\pi$$
$$450$$ 0 0
$$451$$ 20.6803 0.973799
$$452$$ 34.9132i 1.64218i
$$453$$ − 5.84324i − 0.274540i
$$454$$ −4.97107 −0.233304
$$455$$ 0 0
$$456$$ 2.63090 0.123203
$$457$$ − 11.3607i − 0.531431i −0.964052 0.265715i $$-0.914392\pi$$
0.964052 0.265715i $$-0.0856081\pi$$
$$458$$ − 12.8371i − 0.599838i
$$459$$ −17.1629 −0.801096
$$460$$ 0 0
$$461$$ 3.04718 0.141921 0.0709607 0.997479i $$-0.477394\pi$$
0.0709607 + 0.997479i $$0.477394\pi$$
$$462$$ − 25.3607i − 1.17989i
$$463$$ − 9.97334i − 0.463500i −0.972775 0.231750i $$-0.925555\pi$$
0.972775 0.231750i $$-0.0744452\pi$$
$$464$$ 2.94828 0.136871
$$465$$ 0 0
$$466$$ 29.3340 1.35887
$$467$$ − 1.49079i − 0.0689853i −0.999405 0.0344927i $$-0.989018\pi$$
0.999405 0.0344927i $$-0.0109815\pi$$
$$468$$ − 0.290725i − 0.0134388i
$$469$$ −11.2039 −0.517350
$$470$$ 0 0
$$471$$ −16.0989 −0.741798
$$472$$ 17.5753i 0.808969i
$$473$$ 75.5462i 3.47362i
$$474$$ 52.8781 2.42877
$$475$$ 0 0
$$476$$ 9.52973 0.436795
$$477$$ 0.519715i 0.0237961i
$$478$$ 30.0410i 1.37405i
$$479$$ 10.1711 0.464731 0.232365 0.972629i $$-0.425353\pi$$
0.232365 + 0.972629i $$0.425353\pi$$
$$480$$ 0 0
$$481$$ 7.35085 0.335170
$$482$$ − 15.7587i − 0.717790i
$$483$$ − 4.31351i − 0.196272i
$$484$$ −79.1049 −3.59568
$$485$$ 0 0
$$486$$ −1.76713 −0.0801588
$$487$$ 24.2784i 1.10016i 0.835112 + 0.550081i $$0.185403\pi$$
−0.835112 + 0.550081i $$0.814597\pi$$
$$488$$ 8.62249i 0.390322i
$$489$$ −4.99386 −0.225830
$$490$$ 0 0
$$491$$ 19.2039 0.866662 0.433331 0.901235i $$-0.357338\pi$$
0.433331 + 0.901235i $$0.357338\pi$$
$$492$$ − 15.1050i − 0.680988i
$$493$$ 4.62702i 0.208391i
$$494$$ 2.97107 0.133675
$$495$$ 0 0
$$496$$ −18.0410 −0.810067
$$497$$ 11.6865i 0.524211i
$$498$$ 53.1917i 2.38357i
$$499$$ −7.33403 −0.328316 −0.164158 0.986434i $$-0.552491\pi$$
−0.164158 + 0.986434i $$0.552491\pi$$
$$500$$ 0 0
$$501$$ −35.7464 −1.59703
$$502$$ − 22.7214i − 1.01410i
$$503$$ 29.0616i 1.29579i 0.761729 + 0.647895i $$0.224351\pi$$
−0.761729 + 0.647895i $$0.775649\pi$$
$$504$$ 0.130094 0.00579483
$$505$$ 0 0
$$506$$ −32.1978 −1.43137
$$507$$ − 19.0166i − 0.844559i
$$508$$ − 22.3051i − 0.989629i
$$509$$ −26.0456 −1.15445 −0.577225 0.816585i $$-0.695864\pi$$
−0.577225 + 0.816585i $$0.695864\pi$$
$$510$$ 0 0
$$511$$ 5.84324 0.258490
$$512$$ − 22.1701i − 0.979789i
$$513$$ − 5.26180i − 0.232314i
$$514$$ 51.2678 2.26132
$$515$$ 0 0
$$516$$ 55.1794 2.42914
$$517$$ 6.83710i 0.300695i
$$518$$ 12.5646i 0.552058i
$$519$$ −1.80430 −0.0791998
$$520$$ 0 0
$$521$$ −38.8248 −1.70095 −0.850473 0.526019i $$-0.823684\pi$$
−0.850473 + 0.526019i $$0.823684\pi$$
$$522$$ 0.241276i 0.0105604i
$$523$$ 4.59970i 0.201131i 0.994930 + 0.100565i $$0.0320652\pi$$
−0.994930 + 0.100565i $$0.967935\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 12.2823 0.535534
$$527$$ − 28.3135i − 1.23336i
$$528$$ − 22.5236i − 0.980213i
$$529$$ 17.5236 0.761895
$$530$$ 0 0
$$531$$ 0.894960 0.0388380
$$532$$ 2.92162i 0.126668i
$$533$$ − 4.46573i − 0.193432i
$$534$$ 28.0989 1.21596
$$535$$ 0 0
$$536$$ 15.9916 0.690731
$$537$$ 1.52973i 0.0660129i
$$538$$ − 5.02052i − 0.216450i
$$539$$ −37.0082 −1.59406
$$540$$ 0 0
$$541$$ −12.1256 −0.521318 −0.260659 0.965431i $$-0.583940\pi$$
−0.260659 + 0.965431i $$0.583940\pi$$
$$542$$ 42.7526i 1.83638i
$$543$$ 1.43084i 0.0614031i
$$544$$ 24.7526 1.06126
$$545$$ 0 0
$$546$$ −5.47641 −0.234369
$$547$$ 9.54023i 0.407911i 0.978980 + 0.203955i $$0.0653798\pi$$
−0.978980 + 0.203955i $$0.934620\pi$$
$$548$$ − 10.6803i − 0.456242i
$$549$$ 0.439070 0.0187390
$$550$$ 0 0
$$551$$ −1.41855 −0.0604323
$$552$$ 6.15676i 0.262049i
$$553$$ 15.3730i 0.653726i
$$554$$ 55.8408 2.37245
$$555$$ 0 0
$$556$$ 24.0144 1.01844
$$557$$ − 19.9421i − 0.844976i −0.906369 0.422488i $$-0.861157\pi$$
0.906369 0.422488i $$-0.138843\pi$$
$$558$$ − 1.47641i − 0.0625014i
$$559$$ 16.3135 0.689988
$$560$$ 0 0
$$561$$ 35.3484 1.49241
$$562$$ 14.2823i 0.602463i
$$563$$ 23.5525i 0.992620i 0.868145 + 0.496310i $$0.165312\pi$$
−0.868145 + 0.496310i $$0.834688\pi$$
$$564$$ 4.99386 0.210279
$$565$$ 0 0
$$566$$ −1.07838 −0.0453276
$$567$$ 9.44521i 0.396662i
$$568$$ − 16.6803i − 0.699892i
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −23.5031 −0.983573 −0.491786 0.870716i $$-0.663656\pi$$
−0.491786 + 0.870716i $$0.663656\pi$$
$$572$$ 23.5174i 0.983314i
$$573$$ − 37.6742i − 1.57386i
$$574$$ 7.63317 0.318602
$$575$$ 0 0
$$576$$ 0.964928 0.0402053
$$577$$ − 16.4703i − 0.685666i −0.939396 0.342833i $$-0.888613\pi$$
0.939396 0.342833i $$-0.111387\pi$$
$$578$$ − 13.8033i − 0.574140i
$$579$$ 21.8576 0.908372
$$580$$ 0 0
$$581$$ −15.4641 −0.641560
$$582$$ − 32.9627i − 1.36635i
$$583$$ − 42.0410i − 1.74116i
$$584$$ −8.34017 −0.345119
$$585$$ 0 0
$$586$$ 14.3896 0.594430
$$587$$ 28.8104i 1.18913i 0.804046 + 0.594567i $$0.202677\pi$$
−0.804046 + 0.594567i $$0.797323\pi$$
$$588$$ 27.0310i 1.11474i
$$589$$ 8.68035 0.357667
$$590$$ 0 0
$$591$$ 15.7321 0.647131
$$592$$ 11.1590i 0.458633i
$$593$$ − 43.2450i − 1.77586i −0.459980 0.887929i $$-0.652144\pi$$
0.459980 0.887929i $$-0.347856\pi$$
$$594$$ 72.3956 2.97043
$$595$$ 0 0
$$596$$ −53.5318 −2.19275
$$597$$ − 27.6865i − 1.13313i
$$598$$ 6.95282i 0.284322i
$$599$$ 44.2967 1.80991 0.904957 0.425503i $$-0.139903\pi$$
0.904957 + 0.425503i $$0.139903\pi$$
$$600$$ 0 0
$$601$$ −24.3090 −0.991584 −0.495792 0.868441i $$-0.665122\pi$$
−0.495792 + 0.868441i $$0.665122\pi$$
$$602$$ 27.8843i 1.13648i
$$603$$ − 0.814315i − 0.0331615i
$$604$$ −9.26180 −0.376857
$$605$$ 0 0
$$606$$ 18.2557 0.741585
$$607$$ 6.29072i 0.255333i 0.991817 + 0.127666i $$0.0407487\pi$$
−0.991817 + 0.127666i $$0.959251\pi$$
$$608$$ 7.58864i 0.307760i
$$609$$ 2.61474 0.105954
$$610$$ 0 0
$$611$$ 1.47641 0.0597291
$$612$$ 0.692632i 0.0279980i
$$613$$ − 12.7915i − 0.516645i −0.966059 0.258322i $$-0.916830\pi$$
0.966059 0.258322i $$-0.0831697\pi$$
$$614$$ −31.6925 −1.27900
$$615$$ 0 0
$$616$$ −10.5236 −0.424008
$$617$$ − 17.9299i − 0.721829i −0.932599 0.360914i $$-0.882465\pi$$
0.932599 0.360914i $$-0.117535\pi$$
$$618$$ − 23.7009i − 0.953389i
$$619$$ −26.8515 −1.07925 −0.539626 0.841905i $$-0.681434\pi$$
−0.539626 + 0.841905i $$0.681434\pi$$
$$620$$ 0 0
$$621$$ 12.3135 0.494124
$$622$$ − 41.9565i − 1.68230i
$$623$$ 8.16904i 0.327286i
$$624$$ −4.86376 −0.194706
$$625$$ 0 0
$$626$$ 66.5790 2.66103
$$627$$ 10.8371i 0.432792i
$$628$$ 25.5174i 1.01826i
$$629$$ −17.5129 −0.698286
$$630$$ 0 0
$$631$$ 35.5318 1.41450 0.707250 0.706964i $$-0.249936\pi$$
0.707250 + 0.706964i $$0.249936\pi$$
$$632$$ − 21.9421i − 0.872812i
$$633$$ − 13.3074i − 0.528920i
$$634$$ −40.6453 −1.61423
$$635$$ 0 0
$$636$$ −30.7070 −1.21761
$$637$$ 7.99159i 0.316638i
$$638$$ − 19.5174i − 0.772703i
$$639$$ −0.849388 −0.0336013
$$640$$ 0 0
$$641$$ 12.9360 0.510941 0.255471 0.966817i $$-0.417770\pi$$
0.255471 + 0.966817i $$0.417770\pi$$
$$642$$ 8.49693i 0.335347i
$$643$$ 8.49693i 0.335086i 0.985865 + 0.167543i $$0.0535833\pi$$
−0.985865 + 0.167543i $$0.946417\pi$$
$$644$$ −6.83710 −0.269420
$$645$$ 0 0
$$646$$ −7.07838 −0.278495
$$647$$ 45.4908i 1.78843i 0.447640 + 0.894214i $$0.352264\pi$$
−0.447640 + 0.894214i $$0.647736\pi$$
$$648$$ − 13.4813i − 0.529597i
$$649$$ −72.3956 −2.84178
$$650$$ 0 0
$$651$$ −16.0000 −0.627089
$$652$$ 7.91548i 0.309994i
$$653$$ 35.9421i 1.40652i 0.710930 + 0.703262i $$0.248274\pi$$
−0.710930 + 0.703262i $$0.751726\pi$$
$$654$$ −47.6163 −1.86195
$$655$$ 0 0
$$656$$ 6.77924 0.264685
$$657$$ 0.424694i 0.0165689i
$$658$$ 2.52359i 0.0983798i
$$659$$ −30.9360 −1.20510 −0.602548 0.798083i $$-0.705848\pi$$
−0.602548 + 0.798083i $$0.705848\pi$$
$$660$$ 0 0
$$661$$ 10.0989 0.392802 0.196401 0.980524i $$-0.437075\pi$$
0.196401 + 0.980524i $$0.437075\pi$$
$$662$$ 5.94214i 0.230948i
$$663$$ − 7.63317i − 0.296448i
$$664$$ 22.0722 0.856569
$$665$$ 0 0
$$666$$ −0.913212 −0.0353862
$$667$$ − 3.31965i − 0.128538i
$$668$$ 56.6596i 2.19223i
$$669$$ 21.4452 0.829120
$$670$$ 0 0
$$671$$ −35.5174 −1.37114
$$672$$ − 13.9877i − 0.539588i
$$673$$ − 8.67194i − 0.334279i −0.985933 0.167139i $$-0.946547\pi$$
0.985933 0.167139i $$-0.0534530\pi$$
$$674$$ 13.1278 0.505665
$$675$$ 0 0
$$676$$ −30.1422 −1.15932
$$677$$ − 41.9793i − 1.61340i −0.590964 0.806698i $$-0.701253\pi$$
0.590964 0.806698i $$-0.298747\pi$$
$$678$$ 47.7998i 1.83574i
$$679$$ 9.58306 0.367764
$$680$$ 0 0
$$681$$ −3.91548 −0.150041
$$682$$ 119.430i 4.57323i
$$683$$ 46.3896i 1.77505i 0.460760 + 0.887525i $$0.347577\pi$$
−0.460760 + 0.887525i $$0.652423\pi$$
$$684$$ −0.212347 −0.00811929
$$685$$ 0 0
$$686$$ −30.0410 −1.14697
$$687$$ − 10.1112i − 0.385766i
$$688$$ 24.7649i 0.944152i
$$689$$ −9.07838 −0.345859
$$690$$ 0 0
$$691$$ −34.8515 −1.32581 −0.662906 0.748702i $$-0.730677\pi$$
−0.662906 + 0.748702i $$0.730677\pi$$
$$692$$ 2.85989i 0.108717i
$$693$$ 0.535877i 0.0203563i
$$694$$ 12.9627 0.492056
$$695$$ 0 0
$$696$$ −3.73206 −0.141463
$$697$$ 10.6393i 0.402993i
$$698$$ − 21.7009i − 0.821390i
$$699$$ 23.1050 0.873913
$$700$$ 0 0
$$701$$ 35.6430 1.34622 0.673109 0.739543i $$-0.264959\pi$$
0.673109 + 0.739543i $$0.264959\pi$$
$$702$$ − 15.6332i − 0.590036i
$$703$$ − 5.36910i − 0.202500i
$$704$$ −78.0554 −2.94182
$$705$$ 0 0
$$706$$ −30.5958 −1.15149
$$707$$ 5.30737i 0.199604i
$$708$$ 52.8781i 1.98728i
$$709$$ −16.7214 −0.627985 −0.313992 0.949426i $$-0.601667\pi$$
−0.313992 + 0.949426i $$0.601667\pi$$
$$710$$ 0 0
$$711$$ −1.11733 −0.0419030
$$712$$ − 11.6598i − 0.436970i
$$713$$ 20.3135i 0.760747i
$$714$$ 13.0472 0.488278
$$715$$ 0 0
$$716$$ 2.42469 0.0906151
$$717$$ 23.6619i 0.883670i
$$718$$ 13.0784i 0.488081i
$$719$$ 6.85148 0.255517 0.127758 0.991805i $$-0.459222\pi$$
0.127758 + 0.991805i $$0.459222\pi$$
$$720$$ 0 0
$$721$$ 6.89043 0.256613
$$722$$ − 2.17009i − 0.0807623i
$$723$$ − 12.4124i − 0.461622i
$$724$$ 2.26794 0.0842873
$$725$$ 0 0
$$726$$ −108.303 −4.01949
$$727$$ − 34.4391i − 1.27727i −0.769508 0.638637i $$-0.779498\pi$$
0.769508 0.638637i $$-0.220502\pi$$
$$728$$ 2.27247i 0.0842235i
$$729$$ −27.6681 −1.02474
$$730$$ 0 0
$$731$$ −38.8659 −1.43751
$$732$$ 25.9421i 0.958849i
$$733$$ − 19.3607i − 0.715103i −0.933893 0.357552i $$-0.883612\pi$$
0.933893 0.357552i $$-0.116388\pi$$
$$734$$ −2.34017 −0.0863774
$$735$$ 0 0
$$736$$ −17.7587 −0.654595
$$737$$ 65.8720i 2.42643i
$$738$$ 0.554787i 0.0204220i
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 2.34017 0.0859684
$$742$$ − 15.5174i − 0.569663i
$$743$$ − 12.7154i − 0.466483i −0.972419 0.233242i $$-0.925067\pi$$
0.972419 0.233242i $$-0.0749333\pi$$
$$744$$ 22.8371 0.837249
$$745$$ 0 0
$$746$$ 26.7031 0.977671
$$747$$ − 1.12395i − 0.0411232i
$$748$$ − 56.0288i − 2.04861i
$$749$$ −2.47027 −0.0902616
$$750$$ 0 0
$$751$$ 3.15836 0.115250 0.0576252 0.998338i $$-0.481647\pi$$
0.0576252 + 0.998338i $$0.481647\pi$$
$$752$$ 2.24128i 0.0817309i
$$753$$ − 17.8966i − 0.652187i
$$754$$ −4.21461 −0.153487
$$755$$ 0 0
$$756$$ 15.3730 0.559110
$$757$$ − 28.0410i − 1.01917i −0.860421 0.509584i $$-0.829799\pi$$
0.860421 0.509584i $$-0.170201\pi$$
$$758$$ 2.27247i 0.0825399i
$$759$$ −25.3607 −0.920535
$$760$$ 0 0
$$761$$ −16.4924 −0.597849 −0.298924 0.954277i $$-0.596628\pi$$
−0.298924 + 0.954277i $$0.596628\pi$$
$$762$$ − 30.5380i − 1.10627i
$$763$$ − 13.8432i − 0.501159i
$$764$$ −59.7152 −2.16042
$$765$$ 0 0
$$766$$ 0.175007 0.00632326
$$767$$ 15.6332i 0.564481i
$$768$$ 0.715418i 0.0258154i
$$769$$ −26.9627 −0.972298 −0.486149 0.873876i $$-0.661599\pi$$
−0.486149 + 0.873876i $$0.661599\pi$$
$$770$$ 0 0
$$771$$ 40.3812 1.45429
$$772$$ − 34.6453i − 1.24691i
$$773$$ 42.1939i 1.51761i 0.651318 + 0.758805i $$0.274216\pi$$
−0.651318 + 0.758805i $$0.725784\pi$$
$$774$$ −2.02666 −0.0728469
$$775$$ 0 0
$$776$$ −13.6781 −0.491014
$$777$$ 9.89657i 0.355037i
$$778$$ − 44.5380i − 1.59676i
$$779$$ −3.26180 −0.116866
$$780$$ 0 0
$$781$$ 68.7091 2.45860
$$782$$ − 16.5646i − 0.592350i
$$783$$ 7.46412i 0.266746i
$$784$$ −12.1317 −0.433275
$$785$$ 0 0
$$786$$ −5.47641 −0.195337
$$787$$ 35.4368i 1.26319i 0.775300 + 0.631593i $$0.217599\pi$$
−0.775300 + 0.631593i $$0.782401\pi$$
$$788$$ − 24.9360i − 0.888308i
$$789$$ 9.67420 0.344411
$$790$$ 0 0
$$791$$ −13.8966 −0.494105
$$792$$ − 0.764867i − 0.0271784i
$$793$$ 7.66967i 0.272358i
$$794$$ −85.6307 −3.03892
$$795$$ 0 0
$$796$$ −43.8843 −1.55544
$$797$$ − 15.2579i − 0.540463i −0.962795 0.270232i $$-0.912900\pi$$
0.962795 0.270232i $$-0.0871003\pi$$
$$798$$ 4.00000i 0.141598i
$$799$$ −3.51745 −0.124438
$$800$$ 0 0
$$801$$ −0.593735 −0.0209786
$$802$$ − 1.02052i − 0.0360358i
$$803$$ − 34.3545i − 1.21235i
$$804$$ 48.1133 1.69682
$$805$$ 0 0
$$806$$ 25.7899 0.908411
$$807$$ − 3.95443i − 0.139202i
$$808$$ − 7.57531i − 0.266498i
$$809$$ 7.16290 0.251834 0.125917 0.992041i $$-0.459813\pi$$
0.125917 + 0.992041i $$0.459813\pi$$
$$810$$ 0 0
$$811$$ −50.3545 −1.76819 −0.884094 0.467310i $$-0.845223\pi$$
−0.884094 + 0.467310i $$0.845223\pi$$
$$812$$ − 4.14447i − 0.145442i
$$813$$ 33.6742i 1.18101i
$$814$$ 73.8720 2.58921
$$815$$ 0 0
$$816$$ 11.5876 0.405647
$$817$$ − 11.9155i − 0.416870i
$$818$$ 22.7480i 0.795367i
$$819$$ 0.115718 0.00404350
$$820$$ 0 0
$$821$$ 0.952819 0.0332536 0.0166268 0.999862i $$-0.494707\pi$$
0.0166268 + 0.999862i $$0.494707\pi$$
$$822$$ − 14.6225i − 0.510018i
$$823$$ − 44.2290i − 1.54173i −0.637001 0.770863i $$-0.719825\pi$$
0.637001 0.770863i $$-0.280175\pi$$
$$824$$ −9.83483 −0.342613
$$825$$ 0 0
$$826$$ −26.7214 −0.929756
$$827$$ 37.8615i 1.31657i 0.752767 + 0.658287i $$0.228719\pi$$
−0.752767 + 0.658287i $$0.771281\pi$$
$$828$$ − 0.496928i − 0.0172695i
$$829$$ 56.7214 1.97002 0.985008 0.172511i $$-0.0551881\pi$$
0.985008 + 0.172511i $$0.0551881\pi$$
$$830$$ 0 0
$$831$$ 43.9832 1.52576
$$832$$ 16.8554i 0.584354i
$$833$$ − 19.0394i − 0.659677i
$$834$$ 32.8781 1.13848
$$835$$ 0 0
$$836$$ 17.1773 0.594088
$$837$$ − 45.6742i − 1.57873i
$$838$$ 75.2327i 2.59887i
$$839$$ 28.3591 0.979064 0.489532 0.871985i $$-0.337168\pi$$
0.489532 + 0.871985i $$0.337168\pi$$
$$840$$ 0 0
$$841$$ −26.9877 −0.930611
$$842$$ 75.1071i 2.58836i
$$843$$ 11.2495i 0.387454i
$$844$$ −21.0928 −0.726043
$$845$$ 0 0
$$846$$ −0.183417 −0.00630602
$$847$$ − 31.4863i − 1.08188i
$$848$$ − 13.7815i − 0.473259i
$$849$$ −0.849388 −0.0291509
$$850$$ 0 0
$$851$$ 12.5646 0.430710
$$852$$ − 50.1855i − 1.71933i
$$853$$ − 17.0061i − 0.582279i −0.956681 0.291140i $$-0.905966\pi$$
0.956681 0.291140i $$-0.0940344\pi$$
$$854$$ −13.1096 −0.448600
$$855$$ 0 0
$$856$$ 3.52586 0.120511
$$857$$ − 15.4101i − 0.526400i −0.964741 0.263200i $$-0.915222\pi$$
0.964741 0.263200i $$-0.0847780\pi$$
$$858$$ 32.1978i 1.09921i
$$859$$ 37.7275 1.28725 0.643623 0.765342i $$-0.277430\pi$$
0.643623 + 0.765342i $$0.277430\pi$$
$$860$$ 0 0
$$861$$ 6.01229 0.204898
$$862$$ − 14.6225i − 0.498044i
$$863$$ 32.1340i 1.09385i 0.837181 + 0.546927i $$0.184202\pi$$
−0.837181 + 0.546927i $$0.815798\pi$$
$$864$$ 39.9299 1.35844
$$865$$ 0 0
$$866$$ 44.4307 1.50982
$$867$$ − 10.8722i − 0.369238i
$$868$$ 25.3607i 0.860798i
$$869$$ 90.3833 3.06604
$$870$$ 0 0
$$871$$ 14.2245 0.481977
$$872$$ 19.7587i 0.669115i
$$873$$ 0.696508i 0.0235732i
$$874$$ 5.07838 0.171779
$$875$$ 0 0
$$876$$ −25.0928 −0.847806
$$877$$ 19.8927i 0.671729i 0.941910 + 0.335864i $$0.109028\pi$$
−0.941910 + 0.335864i $$0.890972\pi$$
$$878$$ − 46.5692i − 1.57163i
$$879$$ 11.3340 0.382287
$$880$$ 0 0
$$881$$ 24.0722 0.811014 0.405507 0.914092i $$-0.367095\pi$$
0.405507 + 0.914092i $$0.367095\pi$$
$$882$$ − 0.992812i − 0.0334297i
$$883$$ − 31.1727i − 1.04905i −0.851396 0.524523i $$-0.824244\pi$$
0.851396 0.524523i $$-0.175756\pi$$
$$884$$ −12.0989 −0.406930
$$885$$ 0 0
$$886$$ −46.7526 −1.57068
$$887$$ − 4.86764i − 0.163439i −0.996655 0.0817197i $$-0.973959\pi$$
0.996655 0.0817197i $$-0.0260412\pi$$
$$888$$ − 14.1256i − 0.474023i
$$889$$ 8.87814 0.297763
$$890$$ 0 0
$$891$$ 55.5318 1.86039
$$892$$ − 33.9916i − 1.13812i
$$893$$ − 1.07838i − 0.0360865i
$$894$$ −73.2905 −2.45120
$$895$$ 0 0
$$896$$ −12.4436 −0.415712
$$897$$ 5.47641i 0.182852i
$$898$$ − 18.3812i − 0.613389i
$$899$$ −12.3135 −0.410679
$$900$$ 0 0
$$901$$ 21.6286 0.720554
$$902$$ − 44.8781i − 1.49428i
$$903$$ 21.9631i 0.730888i
$$904$$ 19.8348 0.659697
$$905$$ 0 0
$$906$$ −12.6803 −0.421276
$$907$$ 45.4778i 1.51007i 0.655686 + 0.755033i $$0.272379\pi$$
−0.655686 + 0.755033i $$0.727621\pi$$
$$908$$ 6.20620i 0.205960i
$$909$$ −0.385746 −0.0127944
$$910$$ 0 0
$$911$$ −20.9483 −0.694048 −0.347024 0.937856i $$-0.612808\pi$$
−0.347024 + 0.937856i $$0.612808\pi$$
$$912$$ 3.55252i 0.117636i
$$913$$ 90.9192i 3.00899i
$$914$$ −24.6537 −0.815471
$$915$$ 0 0
$$916$$ −16.0267 −0.529536
$$917$$ − 1.59213i − 0.0525767i
$$918$$ 37.2450i 1.22927i
$$919$$ 59.5174 1.96330 0.981650 0.190693i $$-0.0610735\pi$$
0.981650 + 0.190693i $$0.0610735\pi$$
$$920$$ 0 0
$$921$$ −24.9627 −0.822548
$$922$$ − 6.61265i − 0.217776i
$$923$$ − 14.8371i − 0.488369i
$$924$$ −31.6619 −1.04160
$$925$$ 0 0
$$926$$ −21.6430 −0.711233
$$927$$ 0.500804i 0.0164486i
$$928$$ − 10.7649i − 0.353374i
$$929$$ 16.7214 0.548611 0.274305 0.961643i $$-0.411552\pi$$
0.274305 + 0.961643i $$0.411552\pi$$
$$930$$ 0 0
$$931$$ 5.83710 0.191303
$$932$$ − 36.6225i − 1.19961i
$$933$$ − 33.0472i − 1.08192i
$$934$$ −3.23513 −0.105857
$$935$$ 0 0
$$936$$ −0.165166 −0.00539862
$$937$$ 32.4534i 1.06021i 0.847933 + 0.530104i $$0.177847\pi$$
−0.847933 + 0.530104i $$0.822153\pi$$
$$938$$ 24.3135i 0.793864i
$$939$$ 52.4412 1.71135
$$940$$ 0 0
$$941$$ −23.6742 −0.771757 −0.385878 0.922550i $$-0.626102\pi$$
−0.385878 + 0.922550i $$0.626102\pi$$
$$942$$ 34.9360i 1.13828i
$$943$$ − 7.63317i − 0.248570i
$$944$$ −23.7321 −0.772413
$$945$$ 0 0
$$946$$ 163.942 5.33021
$$947$$ − 21.9733i − 0.714038i −0.934097 0.357019i $$-0.883793\pi$$
0.934097 0.357019i $$-0.116207\pi$$
$$948$$ − 66.0165i − 2.14412i
$$949$$ −7.41855 −0.240816
$$950$$ 0 0
$$951$$ −32.0144 −1.03814
$$952$$ − 5.41402i − 0.175469i
$$953$$ − 53.2990i − 1.72652i −0.504757 0.863261i $$-0.668418\pi$$
0.504757 0.863261i $$-0.331582\pi$$
$$954$$ 1.12783 0.0365147
$$955$$ 0 0
$$956$$ 37.5052 1.21300
$$957$$ − 15.3730i − 0.496938i
$$958$$ − 22.0722i − 0.713122i
$$959$$ 4.25112 0.137276
$$960$$ 0 0
$$961$$ 44.3484 1.43059
$$962$$ − 15.9520i − 0.514313i
$$963$$ − 0.179542i − 0.00578565i
$$964$$ −19.6742 −0.633663
$$965$$ 0 0
$$966$$ −9.36069 −0.301175
$$967$$ − 15.8166i − 0.508627i −0.967122 0.254314i $$-0.918150\pi$$
0.967122 0.254314i $$-0.0818495\pi$$
$$968$$ 44.9409i 1.44446i
$$969$$ −5.57531 −0.179105
$$970$$ 0 0
$$971$$ −43.1506 −1.38477 −0.692385 0.721529i $$-0.743440\pi$$
−0.692385 + 0.721529i $$0.743440\pi$$
$$972$$ 2.20620i 0.0707640i
$$973$$ 9.55849i 0.306431i
$$974$$ 52.6863 1.68818
$$975$$ 0 0
$$976$$ −11.6430 −0.372684
$$977$$ 8.47414i 0.271112i 0.990770 + 0.135556i $$0.0432820\pi$$
−0.990770 + 0.135556i $$0.956718\pi$$
$$978$$ 10.8371i 0.346532i
$$979$$ 48.0288 1.53501
$$980$$ 0 0
$$981$$ 1.00614 0.0321237
$$982$$ − 41.6742i − 1.32988i
$$983$$ − 5.59356i − 0.178407i −0.996013 0.0892034i $$-0.971568\pi$$
0.996013 0.0892034i $$-0.0284321\pi$$
$$984$$ −8.58145 −0.273567
$$985$$ 0 0
$$986$$ 10.0410 0.319772
$$987$$ 1.98771i 0.0632696i
$$988$$ − 3.70928i − 0.118008i
$$989$$ 27.8843 0.886669
$$990$$ 0 0
$$991$$ 32.8950 1.04494 0.522471 0.852657i $$-0.325010\pi$$
0.522471 + 0.852657i $$0.325010\pi$$
$$992$$ 65.8720i 2.09144i
$$993$$ 4.68035i 0.148526i
$$994$$ 25.3607 0.804392
$$995$$ 0 0
$$996$$ 66.4079 2.10421
$$997$$ − 23.2618i − 0.736708i −0.929686 0.368354i $$-0.879921\pi$$
0.929686 0.368354i $$-0.120079\pi$$
$$998$$ 15.9155i 0.503796i
$$999$$ −28.2511 −0.893826
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 475.2.b.d.324.1 6
5.2 odd 4 95.2.a.a.1.3 3
5.3 odd 4 475.2.a.f.1.1 3
5.4 even 2 inner 475.2.b.d.324.6 6
15.2 even 4 855.2.a.i.1.1 3
15.8 even 4 4275.2.a.bk.1.3 3
20.3 even 4 7600.2.a.bx.1.1 3
20.7 even 4 1520.2.a.p.1.3 3
35.27 even 4 4655.2.a.u.1.3 3
40.27 even 4 6080.2.a.by.1.1 3
40.37 odd 4 6080.2.a.bo.1.3 3
95.18 even 4 9025.2.a.bb.1.3 3
95.37 even 4 1805.2.a.f.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.a.1.3 3 5.2 odd 4
475.2.a.f.1.1 3 5.3 odd 4
475.2.b.d.324.1 6 1.1 even 1 trivial
475.2.b.d.324.6 6 5.4 even 2 inner
855.2.a.i.1.1 3 15.2 even 4
1520.2.a.p.1.3 3 20.7 even 4
1805.2.a.f.1.1 3 95.37 even 4
4275.2.a.bk.1.3 3 15.8 even 4
4655.2.a.u.1.3 3 35.27 even 4
6080.2.a.bo.1.3 3 40.37 odd 4
6080.2.a.by.1.1 3 40.27 even 4
7600.2.a.bx.1.1 3 20.3 even 4
9025.2.a.bb.1.3 3 95.18 even 4