# Properties

 Label 2175.2.a.r.1.2 Level $2175$ Weight $2$ Character 2175.1 Self dual yes Analytic conductor $17.367$ 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] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.79129 q^{2} -1.00000 q^{3} +5.79129 q^{4} -2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.79129 q^{2} -1.00000 q^{3} +5.79129 q^{4} -2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} +5.00000 q^{11} -5.79129 q^{12} -4.58258 q^{13} -2.79129 q^{14} +17.9564 q^{16} +3.00000 q^{17} +2.79129 q^{18} -5.58258 q^{19} +1.00000 q^{21} +13.9564 q^{22} +4.00000 q^{23} -10.5826 q^{24} -12.7913 q^{26} -1.00000 q^{27} -5.79129 q^{28} +1.00000 q^{29} +4.00000 q^{31} +28.9564 q^{32} -5.00000 q^{33} +8.37386 q^{34} +5.79129 q^{36} +4.00000 q^{37} -15.5826 q^{38} +4.58258 q^{39} +9.16515 q^{41} +2.79129 q^{42} +0.417424 q^{43} +28.9564 q^{44} +11.1652 q^{46} -1.41742 q^{47} -17.9564 q^{48} -6.00000 q^{49} -3.00000 q^{51} -26.5390 q^{52} -9.58258 q^{53} -2.79129 q^{54} -10.5826 q^{56} +5.58258 q^{57} +2.79129 q^{58} +1.58258 q^{59} -14.7477 q^{61} +11.1652 q^{62} -1.00000 q^{63} +44.9129 q^{64} -13.9564 q^{66} -14.1652 q^{67} +17.3739 q^{68} -4.00000 q^{69} -0.417424 q^{71} +10.5826 q^{72} -4.00000 q^{73} +11.1652 q^{74} -32.3303 q^{76} -5.00000 q^{77} +12.7913 q^{78} -1.58258 q^{79} +1.00000 q^{81} +25.5826 q^{82} +2.41742 q^{83} +5.79129 q^{84} +1.16515 q^{86} -1.00000 q^{87} +52.9129 q^{88} +10.5826 q^{89} +4.58258 q^{91} +23.1652 q^{92} -4.00000 q^{93} -3.95644 q^{94} -28.9564 q^{96} -2.41742 q^{97} -16.7477 q^{98} +5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 - 2 * q^3 + 7 * q^4 - q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 $$2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 10 q^{11} - 7 q^{12} - q^{14} + 13 q^{16} + 6 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 5 q^{22} + 8 q^{23} - 12 q^{24} - 21 q^{26} - 2 q^{27} - 7 q^{28} + 2 q^{29} + 8 q^{31} + 35 q^{32} - 10 q^{33} + 3 q^{34} + 7 q^{36} + 8 q^{37} - 22 q^{38} + q^{42} + 10 q^{43} + 35 q^{44} + 4 q^{46} - 12 q^{47} - 13 q^{48} - 12 q^{49} - 6 q^{51} - 21 q^{52} - 10 q^{53} - q^{54} - 12 q^{56} + 2 q^{57} + q^{58} - 6 q^{59} - 2 q^{61} + 4 q^{62} - 2 q^{63} + 44 q^{64} - 5 q^{66} - 10 q^{67} + 21 q^{68} - 8 q^{69} - 10 q^{71} + 12 q^{72} - 8 q^{73} + 4 q^{74} - 28 q^{76} - 10 q^{77} + 21 q^{78} + 6 q^{79} + 2 q^{81} + 42 q^{82} + 14 q^{83} + 7 q^{84} - 16 q^{86} - 2 q^{87} + 60 q^{88} + 12 q^{89} + 28 q^{92} - 8 q^{93} + 15 q^{94} - 35 q^{96} - 14 q^{97} - 6 q^{98} + 10 q^{99}+O(q^{100})$$ 2 * q + q^2 - 2 * q^3 + 7 * q^4 - q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 + 10 * q^11 - 7 * q^12 - q^14 + 13 * q^16 + 6 * q^17 + q^18 - 2 * q^19 + 2 * q^21 + 5 * q^22 + 8 * q^23 - 12 * q^24 - 21 * q^26 - 2 * q^27 - 7 * q^28 + 2 * q^29 + 8 * q^31 + 35 * q^32 - 10 * q^33 + 3 * q^34 + 7 * q^36 + 8 * q^37 - 22 * q^38 + q^42 + 10 * q^43 + 35 * q^44 + 4 * q^46 - 12 * q^47 - 13 * q^48 - 12 * q^49 - 6 * q^51 - 21 * q^52 - 10 * q^53 - q^54 - 12 * q^56 + 2 * q^57 + q^58 - 6 * q^59 - 2 * q^61 + 4 * q^62 - 2 * q^63 + 44 * q^64 - 5 * q^66 - 10 * q^67 + 21 * q^68 - 8 * q^69 - 10 * q^71 + 12 * q^72 - 8 * q^73 + 4 * q^74 - 28 * q^76 - 10 * q^77 + 21 * q^78 + 6 * q^79 + 2 * q^81 + 42 * q^82 + 14 * q^83 + 7 * q^84 - 16 * q^86 - 2 * q^87 + 60 * q^88 + 12 * q^89 + 28 * q^92 - 8 * q^93 + 15 * q^94 - 35 * q^96 - 14 * q^97 - 6 * q^98 + 10 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.79129 1.97374 0.986869 0.161521i $$-0.0516399\pi$$
0.986869 + 0.161521i $$0.0516399\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 5.79129 2.89564
$$5$$ 0 0
$$6$$ −2.79129 −1.13954
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 10.5826 3.74151
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ −5.79129 −1.67180
$$13$$ −4.58258 −1.27098 −0.635489 0.772110i $$-0.719201\pi$$
−0.635489 + 0.772110i $$0.719201\pi$$
$$14$$ −2.79129 −0.746003
$$15$$ 0 0
$$16$$ 17.9564 4.48911
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 2.79129 0.657913
$$19$$ −5.58258 −1.28073 −0.640365 0.768070i $$-0.721217\pi$$
−0.640365 + 0.768070i $$0.721217\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 13.9564 2.97552
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −10.5826 −2.16016
$$25$$ 0 0
$$26$$ −12.7913 −2.50858
$$27$$ −1.00000 −0.192450
$$28$$ −5.79129 −1.09445
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 28.9564 5.11882
$$33$$ −5.00000 −0.870388
$$34$$ 8.37386 1.43611
$$35$$ 0 0
$$36$$ 5.79129 0.965215
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ −15.5826 −2.52783
$$39$$ 4.58258 0.733799
$$40$$ 0 0
$$41$$ 9.16515 1.43136 0.715678 0.698430i $$-0.246118\pi$$
0.715678 + 0.698430i $$0.246118\pi$$
$$42$$ 2.79129 0.430705
$$43$$ 0.417424 0.0636566 0.0318283 0.999493i $$-0.489867\pi$$
0.0318283 + 0.999493i $$0.489867\pi$$
$$44$$ 28.9564 4.36535
$$45$$ 0 0
$$46$$ 11.1652 1.64621
$$47$$ −1.41742 −0.206753 −0.103376 0.994642i $$-0.532965\pi$$
−0.103376 + 0.994642i $$0.532965\pi$$
$$48$$ −17.9564 −2.59179
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ −26.5390 −3.68030
$$53$$ −9.58258 −1.31627 −0.658134 0.752901i $$-0.728654\pi$$
−0.658134 + 0.752901i $$0.728654\pi$$
$$54$$ −2.79129 −0.379846
$$55$$ 0 0
$$56$$ −10.5826 −1.41416
$$57$$ 5.58258 0.739430
$$58$$ 2.79129 0.366514
$$59$$ 1.58258 0.206034 0.103017 0.994680i $$-0.467150\pi$$
0.103017 + 0.994680i $$0.467150\pi$$
$$60$$ 0 0
$$61$$ −14.7477 −1.88825 −0.944126 0.329583i $$-0.893092\pi$$
−0.944126 + 0.329583i $$0.893092\pi$$
$$62$$ 11.1652 1.41798
$$63$$ −1.00000 −0.125988
$$64$$ 44.9129 5.61411
$$65$$ 0 0
$$66$$ −13.9564 −1.71792
$$67$$ −14.1652 −1.73055 −0.865274 0.501299i $$-0.832856\pi$$
−0.865274 + 0.501299i $$0.832856\pi$$
$$68$$ 17.3739 2.10689
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −0.417424 −0.0495392 −0.0247696 0.999693i $$-0.507885\pi$$
−0.0247696 + 0.999693i $$0.507885\pi$$
$$72$$ 10.5826 1.24717
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 11.1652 1.29792
$$75$$ 0 0
$$76$$ −32.3303 −3.70854
$$77$$ −5.00000 −0.569803
$$78$$ 12.7913 1.44833
$$79$$ −1.58258 −0.178054 −0.0890268 0.996029i $$-0.528376\pi$$
−0.0890268 + 0.996029i $$0.528376\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 25.5826 2.82512
$$83$$ 2.41742 0.265347 0.132673 0.991160i $$-0.457644\pi$$
0.132673 + 0.991160i $$0.457644\pi$$
$$84$$ 5.79129 0.631881
$$85$$ 0 0
$$86$$ 1.16515 0.125642
$$87$$ −1.00000 −0.107211
$$88$$ 52.9129 5.64053
$$89$$ 10.5826 1.12175 0.560875 0.827900i $$-0.310465\pi$$
0.560875 + 0.827900i $$0.310465\pi$$
$$90$$ 0 0
$$91$$ 4.58258 0.480384
$$92$$ 23.1652 2.41513
$$93$$ −4.00000 −0.414781
$$94$$ −3.95644 −0.408076
$$95$$ 0 0
$$96$$ −28.9564 −2.95535
$$97$$ −2.41742 −0.245452 −0.122726 0.992441i $$-0.539164\pi$$
−0.122726 + 0.992441i $$0.539164\pi$$
$$98$$ −16.7477 −1.69178
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ 8.58258 0.853998 0.426999 0.904252i $$-0.359571\pi$$
0.426999 + 0.904252i $$0.359571\pi$$
$$102$$ −8.37386 −0.829136
$$103$$ 3.16515 0.311872 0.155936 0.987767i $$-0.450161\pi$$
0.155936 + 0.987767i $$0.450161\pi$$
$$104$$ −48.4955 −4.75537
$$105$$ 0 0
$$106$$ −26.7477 −2.59797
$$107$$ 13.1652 1.27272 0.636362 0.771391i $$-0.280439\pi$$
0.636362 + 0.771391i $$0.280439\pi$$
$$108$$ −5.79129 −0.557267
$$109$$ −4.16515 −0.398949 −0.199475 0.979903i $$-0.563924\pi$$
−0.199475 + 0.979903i $$0.563924\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ −17.9564 −1.69672
$$113$$ −4.16515 −0.391824 −0.195912 0.980621i $$-0.562767\pi$$
−0.195912 + 0.980621i $$0.562767\pi$$
$$114$$ 15.5826 1.45944
$$115$$ 0 0
$$116$$ 5.79129 0.537708
$$117$$ −4.58258 −0.423659
$$118$$ 4.41742 0.406657
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ −41.1652 −3.72692
$$123$$ −9.16515 −0.826394
$$124$$ 23.1652 2.08029
$$125$$ 0 0
$$126$$ −2.79129 −0.248668
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 67.4519 5.96196
$$129$$ −0.417424 −0.0367522
$$130$$ 0 0
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ −28.9564 −2.52033
$$133$$ 5.58258 0.484071
$$134$$ −39.5390 −3.41565
$$135$$ 0 0
$$136$$ 31.7477 2.72235
$$137$$ 20.3303 1.73693 0.868467 0.495746i $$-0.165105\pi$$
0.868467 + 0.495746i $$0.165105\pi$$
$$138$$ −11.1652 −0.950441
$$139$$ −18.5826 −1.57615 −0.788077 0.615577i $$-0.788923\pi$$
−0.788077 + 0.615577i $$0.788923\pi$$
$$140$$ 0 0
$$141$$ 1.41742 0.119369
$$142$$ −1.16515 −0.0977773
$$143$$ −22.9129 −1.91607
$$144$$ 17.9564 1.49637
$$145$$ 0 0
$$146$$ −11.1652 −0.924035
$$147$$ 6.00000 0.494872
$$148$$ 23.1652 1.90416
$$149$$ −10.7477 −0.880488 −0.440244 0.897878i $$-0.645108\pi$$
−0.440244 + 0.897878i $$0.645108\pi$$
$$150$$ 0 0
$$151$$ −11.1652 −0.908607 −0.454304 0.890847i $$-0.650112\pi$$
−0.454304 + 0.890847i $$0.650112\pi$$
$$152$$ −59.0780 −4.79186
$$153$$ 3.00000 0.242536
$$154$$ −13.9564 −1.12464
$$155$$ 0 0
$$156$$ 26.5390 2.12482
$$157$$ 16.7477 1.33661 0.668307 0.743886i $$-0.267019\pi$$
0.668307 + 0.743886i $$0.267019\pi$$
$$158$$ −4.41742 −0.351431
$$159$$ 9.58258 0.759948
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 2.79129 0.219304
$$163$$ 1.58258 0.123957 0.0619784 0.998077i $$-0.480259\pi$$
0.0619784 + 0.998077i $$0.480259\pi$$
$$164$$ 53.0780 4.14470
$$165$$ 0 0
$$166$$ 6.74773 0.523725
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 10.5826 0.816463
$$169$$ 8.00000 0.615385
$$170$$ 0 0
$$171$$ −5.58258 −0.426910
$$172$$ 2.41742 0.184327
$$173$$ −15.1652 −1.15299 −0.576493 0.817102i $$-0.695579\pi$$
−0.576493 + 0.817102i $$0.695579\pi$$
$$174$$ −2.79129 −0.211607
$$175$$ 0 0
$$176$$ 89.7822 6.76759
$$177$$ −1.58258 −0.118954
$$178$$ 29.5390 2.21404
$$179$$ −22.7477 −1.70024 −0.850122 0.526585i $$-0.823472\pi$$
−0.850122 + 0.526585i $$0.823472\pi$$
$$180$$ 0 0
$$181$$ −2.16515 −0.160934 −0.0804672 0.996757i $$-0.525641\pi$$
−0.0804672 + 0.996757i $$0.525641\pi$$
$$182$$ 12.7913 0.948153
$$183$$ 14.7477 1.09018
$$184$$ 42.3303 3.12063
$$185$$ 0 0
$$186$$ −11.1652 −0.818669
$$187$$ 15.0000 1.09691
$$188$$ −8.20871 −0.598682
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ −44.9129 −3.24131
$$193$$ −16.3303 −1.17548 −0.587740 0.809050i $$-0.699982\pi$$
−0.587740 + 0.809050i $$0.699982\pi$$
$$194$$ −6.74773 −0.484459
$$195$$ 0 0
$$196$$ −34.7477 −2.48198
$$197$$ −20.3303 −1.44847 −0.724237 0.689551i $$-0.757808\pi$$
−0.724237 + 0.689551i $$0.757808\pi$$
$$198$$ 13.9564 0.991841
$$199$$ 22.5826 1.60084 0.800418 0.599442i $$-0.204611\pi$$
0.800418 + 0.599442i $$0.204611\pi$$
$$200$$ 0 0
$$201$$ 14.1652 0.999133
$$202$$ 23.9564 1.68557
$$203$$ −1.00000 −0.0701862
$$204$$ −17.3739 −1.21641
$$205$$ 0 0
$$206$$ 8.83485 0.615553
$$207$$ 4.00000 0.278019
$$208$$ −82.2867 −5.70556
$$209$$ −27.9129 −1.93077
$$210$$ 0 0
$$211$$ −16.3303 −1.12422 −0.562112 0.827061i $$-0.690011\pi$$
−0.562112 + 0.827061i $$0.690011\pi$$
$$212$$ −55.4955 −3.81144
$$213$$ 0.417424 0.0286014
$$214$$ 36.7477 2.51202
$$215$$ 0 0
$$216$$ −10.5826 −0.720053
$$217$$ −4.00000 −0.271538
$$218$$ −11.6261 −0.787421
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ −13.7477 −0.924772
$$222$$ −11.1652 −0.749356
$$223$$ −7.00000 −0.468755 −0.234377 0.972146i $$-0.575305\pi$$
−0.234377 + 0.972146i $$0.575305\pi$$
$$224$$ −28.9564 −1.93473
$$225$$ 0 0
$$226$$ −11.6261 −0.773359
$$227$$ −1.58258 −0.105039 −0.0525196 0.998620i $$-0.516725\pi$$
−0.0525196 + 0.998620i $$0.516725\pi$$
$$228$$ 32.3303 2.14113
$$229$$ 1.16515 0.0769954 0.0384977 0.999259i $$-0.487743\pi$$
0.0384977 + 0.999259i $$0.487743\pi$$
$$230$$ 0 0
$$231$$ 5.00000 0.328976
$$232$$ 10.5826 0.694780
$$233$$ −5.16515 −0.338380 −0.169190 0.985583i $$-0.554115\pi$$
−0.169190 + 0.985583i $$0.554115\pi$$
$$234$$ −12.7913 −0.836193
$$235$$ 0 0
$$236$$ 9.16515 0.596601
$$237$$ 1.58258 0.102799
$$238$$ −8.37386 −0.542797
$$239$$ −26.0000 −1.68180 −0.840900 0.541190i $$-0.817974\pi$$
−0.840900 + 0.541190i $$0.817974\pi$$
$$240$$ 0 0
$$241$$ −7.33030 −0.472186 −0.236093 0.971730i $$-0.575867\pi$$
−0.236093 + 0.971730i $$0.575867\pi$$
$$242$$ 39.0780 2.51203
$$243$$ −1.00000 −0.0641500
$$244$$ −85.4083 −5.46771
$$245$$ 0 0
$$246$$ −25.5826 −1.63109
$$247$$ 25.5826 1.62778
$$248$$ 42.3303 2.68798
$$249$$ −2.41742 −0.153198
$$250$$ 0 0
$$251$$ −2.16515 −0.136663 −0.0683316 0.997663i $$-0.521768\pi$$
−0.0683316 + 0.997663i $$0.521768\pi$$
$$252$$ −5.79129 −0.364817
$$253$$ 20.0000 1.25739
$$254$$ −5.58258 −0.350282
$$255$$ 0 0
$$256$$ 98.4519 6.15324
$$257$$ 14.7477 0.919938 0.459969 0.887935i $$-0.347861\pi$$
0.459969 + 0.887935i $$0.347861\pi$$
$$258$$ −1.16515 −0.0725392
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ −41.8693 −2.58670
$$263$$ 6.33030 0.390343 0.195172 0.980769i $$-0.437474\pi$$
0.195172 + 0.980769i $$0.437474\pi$$
$$264$$ −52.9129 −3.25656
$$265$$ 0 0
$$266$$ 15.5826 0.955429
$$267$$ −10.5826 −0.647643
$$268$$ −82.0345 −5.01105
$$269$$ 13.4174 0.818075 0.409037 0.912518i $$-0.365865\pi$$
0.409037 + 0.912518i $$0.365865\pi$$
$$270$$ 0 0
$$271$$ −17.1652 −1.04271 −0.521354 0.853340i $$-0.674573\pi$$
−0.521354 + 0.853340i $$0.674573\pi$$
$$272$$ 53.8693 3.26631
$$273$$ −4.58258 −0.277350
$$274$$ 56.7477 3.42826
$$275$$ 0 0
$$276$$ −23.1652 −1.39438
$$277$$ −20.9129 −1.25653 −0.628267 0.777998i $$-0.716235\pi$$
−0.628267 + 0.777998i $$0.716235\pi$$
$$278$$ −51.8693 −3.11091
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −9.58258 −0.571649 −0.285824 0.958282i $$-0.592267\pi$$
−0.285824 + 0.958282i $$0.592267\pi$$
$$282$$ 3.95644 0.235603
$$283$$ 19.1652 1.13925 0.569625 0.821905i $$-0.307088\pi$$
0.569625 + 0.821905i $$0.307088\pi$$
$$284$$ −2.41742 −0.143448
$$285$$ 0 0
$$286$$ −63.9564 −3.78182
$$287$$ −9.16515 −0.541002
$$288$$ 28.9564 1.70627
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 2.41742 0.141712
$$292$$ −23.1652 −1.35564
$$293$$ −11.8348 −0.691399 −0.345700 0.938345i $$-0.612358\pi$$
−0.345700 + 0.938345i $$0.612358\pi$$
$$294$$ 16.7477 0.976747
$$295$$ 0 0
$$296$$ 42.3303 2.46040
$$297$$ −5.00000 −0.290129
$$298$$ −30.0000 −1.73785
$$299$$ −18.3303 −1.06007
$$300$$ 0 0
$$301$$ −0.417424 −0.0240599
$$302$$ −31.1652 −1.79335
$$303$$ −8.58258 −0.493056
$$304$$ −100.243 −5.74934
$$305$$ 0 0
$$306$$ 8.37386 0.478702
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ −28.9564 −1.64995
$$309$$ −3.16515 −0.180059
$$310$$ 0 0
$$311$$ −3.00000 −0.170114 −0.0850572 0.996376i $$-0.527107\pi$$
−0.0850572 + 0.996376i $$0.527107\pi$$
$$312$$ 48.4955 2.74551
$$313$$ −1.41742 −0.0801176 −0.0400588 0.999197i $$-0.512755\pi$$
−0.0400588 + 0.999197i $$0.512755\pi$$
$$314$$ 46.7477 2.63813
$$315$$ 0 0
$$316$$ −9.16515 −0.515580
$$317$$ 25.0000 1.40414 0.702070 0.712108i $$-0.252259\pi$$
0.702070 + 0.712108i $$0.252259\pi$$
$$318$$ 26.7477 1.49994
$$319$$ 5.00000 0.279946
$$320$$ 0 0
$$321$$ −13.1652 −0.734807
$$322$$ −11.1652 −0.622210
$$323$$ −16.7477 −0.931868
$$324$$ 5.79129 0.321738
$$325$$ 0 0
$$326$$ 4.41742 0.244659
$$327$$ 4.16515 0.230333
$$328$$ 96.9909 5.35543
$$329$$ 1.41742 0.0781451
$$330$$ 0 0
$$331$$ 28.3303 1.55717 0.778587 0.627537i $$-0.215937\pi$$
0.778587 + 0.627537i $$0.215937\pi$$
$$332$$ 14.0000 0.768350
$$333$$ 4.00000 0.219199
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 17.9564 0.979604
$$337$$ 2.83485 0.154424 0.0772120 0.997015i $$-0.475398\pi$$
0.0772120 + 0.997015i $$0.475398\pi$$
$$338$$ 22.3303 1.21461
$$339$$ 4.16515 0.226220
$$340$$ 0 0
$$341$$ 20.0000 1.08306
$$342$$ −15.5826 −0.842609
$$343$$ 13.0000 0.701934
$$344$$ 4.41742 0.238172
$$345$$ 0 0
$$346$$ −42.3303 −2.27569
$$347$$ 11.1652 0.599377 0.299688 0.954037i $$-0.403117\pi$$
0.299688 + 0.954037i $$0.403117\pi$$
$$348$$ −5.79129 −0.310446
$$349$$ 32.3303 1.73060 0.865301 0.501253i $$-0.167127\pi$$
0.865301 + 0.501253i $$0.167127\pi$$
$$350$$ 0 0
$$351$$ 4.58258 0.244600
$$352$$ 144.782 7.71692
$$353$$ 33.1652 1.76520 0.882601 0.470122i $$-0.155790\pi$$
0.882601 + 0.470122i $$0.155790\pi$$
$$354$$ −4.41742 −0.234783
$$355$$ 0 0
$$356$$ 61.2867 3.24819
$$357$$ 3.00000 0.158777
$$358$$ −63.4955 −3.35584
$$359$$ −8.83485 −0.466285 −0.233143 0.972443i $$-0.574901\pi$$
−0.233143 + 0.972443i $$0.574901\pi$$
$$360$$ 0 0
$$361$$ 12.1652 0.640271
$$362$$ −6.04356 −0.317643
$$363$$ −14.0000 −0.734809
$$364$$ 26.5390 1.39102
$$365$$ 0 0
$$366$$ 41.1652 2.15174
$$367$$ −17.4955 −0.913255 −0.456628 0.889658i $$-0.650943\pi$$
−0.456628 + 0.889658i $$0.650943\pi$$
$$368$$ 71.8258 3.74418
$$369$$ 9.16515 0.477119
$$370$$ 0 0
$$371$$ 9.58258 0.497503
$$372$$ −23.1652 −1.20106
$$373$$ 8.33030 0.431327 0.215663 0.976468i $$-0.430809\pi$$
0.215663 + 0.976468i $$0.430809\pi$$
$$374$$ 41.8693 2.16501
$$375$$ 0 0
$$376$$ −15.0000 −0.773566
$$377$$ −4.58258 −0.236015
$$378$$ 2.79129 0.143568
$$379$$ −26.0000 −1.33553 −0.667765 0.744372i $$-0.732749\pi$$
−0.667765 + 0.744372i $$0.732749\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ −11.1652 −0.571259
$$383$$ 5.58258 0.285256 0.142628 0.989776i $$-0.454445\pi$$
0.142628 + 0.989776i $$0.454445\pi$$
$$384$$ −67.4519 −3.44214
$$385$$ 0 0
$$386$$ −45.5826 −2.32009
$$387$$ 0.417424 0.0212189
$$388$$ −14.0000 −0.710742
$$389$$ 2.58258 0.130942 0.0654709 0.997854i $$-0.479145\pi$$
0.0654709 + 0.997854i $$0.479145\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ −63.4955 −3.20700
$$393$$ 15.0000 0.756650
$$394$$ −56.7477 −2.85891
$$395$$ 0 0
$$396$$ 28.9564 1.45512
$$397$$ −29.1652 −1.46376 −0.731878 0.681435i $$-0.761356\pi$$
−0.731878 + 0.681435i $$0.761356\pi$$
$$398$$ 63.0345 3.15963
$$399$$ −5.58258 −0.279478
$$400$$ 0 0
$$401$$ 21.5826 1.07778 0.538891 0.842375i $$-0.318843\pi$$
0.538891 + 0.842375i $$0.318843\pi$$
$$402$$ 39.5390 1.97203
$$403$$ −18.3303 −0.913097
$$404$$ 49.7042 2.47287
$$405$$ 0 0
$$406$$ −2.79129 −0.138529
$$407$$ 20.0000 0.991363
$$408$$ −31.7477 −1.57175
$$409$$ 24.7477 1.22370 0.611848 0.790975i $$-0.290426\pi$$
0.611848 + 0.790975i $$0.290426\pi$$
$$410$$ 0 0
$$411$$ −20.3303 −1.00282
$$412$$ 18.3303 0.903069
$$413$$ −1.58258 −0.0778735
$$414$$ 11.1652 0.548737
$$415$$ 0 0
$$416$$ −132.695 −6.50591
$$417$$ 18.5826 0.909993
$$418$$ −77.9129 −3.81084
$$419$$ −18.8348 −0.920143 −0.460071 0.887882i $$-0.652176\pi$$
−0.460071 + 0.887882i $$0.652176\pi$$
$$420$$ 0 0
$$421$$ 13.5826 0.661974 0.330987 0.943635i $$-0.392618\pi$$
0.330987 + 0.943635i $$0.392618\pi$$
$$422$$ −45.5826 −2.21893
$$423$$ −1.41742 −0.0689175
$$424$$ −101.408 −4.92482
$$425$$ 0 0
$$426$$ 1.16515 0.0564518
$$427$$ 14.7477 0.713693
$$428$$ 76.2432 3.68535
$$429$$ 22.9129 1.10624
$$430$$ 0 0
$$431$$ −20.8348 −1.00358 −0.501790 0.864990i $$-0.667325\pi$$
−0.501790 + 0.864990i $$0.667325\pi$$
$$432$$ −17.9564 −0.863930
$$433$$ 39.0780 1.87797 0.938985 0.343958i $$-0.111768\pi$$
0.938985 + 0.343958i $$0.111768\pi$$
$$434$$ −11.1652 −0.535944
$$435$$ 0 0
$$436$$ −24.1216 −1.15521
$$437$$ −22.3303 −1.06820
$$438$$ 11.1652 0.533492
$$439$$ 28.9129 1.37994 0.689968 0.723840i $$-0.257624\pi$$
0.689968 + 0.723840i $$0.257624\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ −38.3739 −1.82526
$$443$$ 8.58258 0.407770 0.203885 0.978995i $$-0.434643\pi$$
0.203885 + 0.978995i $$0.434643\pi$$
$$444$$ −23.1652 −1.09937
$$445$$ 0 0
$$446$$ −19.5390 −0.925199
$$447$$ 10.7477 0.508350
$$448$$ −44.9129 −2.12193
$$449$$ 34.0780 1.60824 0.804121 0.594466i $$-0.202636\pi$$
0.804121 + 0.594466i $$0.202636\pi$$
$$450$$ 0 0
$$451$$ 45.8258 2.15785
$$452$$ −24.1216 −1.13458
$$453$$ 11.1652 0.524585
$$454$$ −4.41742 −0.207320
$$455$$ 0 0
$$456$$ 59.0780 2.76658
$$457$$ −23.7477 −1.11087 −0.555436 0.831559i $$-0.687449\pi$$
−0.555436 + 0.831559i $$0.687449\pi$$
$$458$$ 3.25227 0.151969
$$459$$ −3.00000 −0.140028
$$460$$ 0 0
$$461$$ 9.16515 0.426864 0.213432 0.976958i $$-0.431536\pi$$
0.213432 + 0.976958i $$0.431536\pi$$
$$462$$ 13.9564 0.649312
$$463$$ −18.1652 −0.844206 −0.422103 0.906548i $$-0.638708\pi$$
−0.422103 + 0.906548i $$0.638708\pi$$
$$464$$ 17.9564 0.833607
$$465$$ 0 0
$$466$$ −14.4174 −0.667874
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ −26.5390 −1.22677
$$469$$ 14.1652 0.654086
$$470$$ 0 0
$$471$$ −16.7477 −0.771695
$$472$$ 16.7477 0.770877
$$473$$ 2.08712 0.0959659
$$474$$ 4.41742 0.202899
$$475$$ 0 0
$$476$$ −17.3739 −0.796330
$$477$$ −9.58258 −0.438756
$$478$$ −72.5735 −3.31943
$$479$$ −0.834849 −0.0381452 −0.0190726 0.999818i $$-0.506071\pi$$
−0.0190726 + 0.999818i $$0.506071\pi$$
$$480$$ 0 0
$$481$$ −18.3303 −0.835790
$$482$$ −20.4610 −0.931972
$$483$$ 4.00000 0.182006
$$484$$ 81.0780 3.68536
$$485$$ 0 0
$$486$$ −2.79129 −0.126615
$$487$$ 34.3303 1.55565 0.777827 0.628478i $$-0.216322\pi$$
0.777827 + 0.628478i $$0.216322\pi$$
$$488$$ −156.069 −7.06491
$$489$$ −1.58258 −0.0715665
$$490$$ 0 0
$$491$$ 16.0000 0.722070 0.361035 0.932552i $$-0.382424\pi$$
0.361035 + 0.932552i $$0.382424\pi$$
$$492$$ −53.0780 −2.39294
$$493$$ 3.00000 0.135113
$$494$$ 71.4083 3.21281
$$495$$ 0 0
$$496$$ 71.8258 3.22507
$$497$$ 0.417424 0.0187240
$$498$$ −6.74773 −0.302373
$$499$$ −22.5826 −1.01093 −0.505467 0.862846i $$-0.668680\pi$$
−0.505467 + 0.862846i $$0.668680\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −6.04356 −0.269737
$$503$$ −22.9129 −1.02163 −0.510817 0.859689i $$-0.670657\pi$$
−0.510817 + 0.859689i $$0.670657\pi$$
$$504$$ −10.5826 −0.471385
$$505$$ 0 0
$$506$$ 55.8258 2.48176
$$507$$ −8.00000 −0.355292
$$508$$ −11.5826 −0.513894
$$509$$ −0.747727 −0.0331424 −0.0165712 0.999863i $$-0.505275\pi$$
−0.0165712 + 0.999863i $$0.505275\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 139.904 6.18293
$$513$$ 5.58258 0.246477
$$514$$ 41.1652 1.81572
$$515$$ 0 0
$$516$$ −2.41742 −0.106421
$$517$$ −7.08712 −0.311691
$$518$$ −11.1652 −0.490569
$$519$$ 15.1652 0.665676
$$520$$ 0 0
$$521$$ −21.0780 −0.923445 −0.461723 0.887024i $$-0.652768\pi$$
−0.461723 + 0.887024i $$0.652768\pi$$
$$522$$ 2.79129 0.122171
$$523$$ −3.33030 −0.145624 −0.0728120 0.997346i $$-0.523197\pi$$
−0.0728120 + 0.997346i $$0.523197\pi$$
$$524$$ −86.8693 −3.79490
$$525$$ 0 0
$$526$$ 17.6697 0.770435
$$527$$ 12.0000 0.522728
$$528$$ −89.7822 −3.90727
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 1.58258 0.0686779
$$532$$ 32.3303 1.40170
$$533$$ −42.0000 −1.81922
$$534$$ −29.5390 −1.27828
$$535$$ 0 0
$$536$$ −149.904 −6.47486
$$537$$ 22.7477 0.981637
$$538$$ 37.4519 1.61467
$$539$$ −30.0000 −1.29219
$$540$$ 0 0
$$541$$ 23.4955 1.01015 0.505074 0.863076i $$-0.331465\pi$$
0.505074 + 0.863076i $$0.331465\pi$$
$$542$$ −47.9129 −2.05803
$$543$$ 2.16515 0.0929155
$$544$$ 86.8693 3.72449
$$545$$ 0 0
$$546$$ −12.7913 −0.547417
$$547$$ 14.1652 0.605658 0.302829 0.953045i $$-0.402069\pi$$
0.302829 + 0.953045i $$0.402069\pi$$
$$548$$ 117.739 5.02955
$$549$$ −14.7477 −0.629418
$$550$$ 0 0
$$551$$ −5.58258 −0.237826
$$552$$ −42.3303 −1.80170
$$553$$ 1.58258 0.0672980
$$554$$ −58.3739 −2.48007
$$555$$ 0 0
$$556$$ −107.617 −4.56398
$$557$$ −4.74773 −0.201168 −0.100584 0.994929i $$-0.532071\pi$$
−0.100584 + 0.994929i $$0.532071\pi$$
$$558$$ 11.1652 0.472659
$$559$$ −1.91288 −0.0809061
$$560$$ 0 0
$$561$$ −15.0000 −0.633300
$$562$$ −26.7477 −1.12828
$$563$$ 5.41742 0.228317 0.114159 0.993463i $$-0.463583\pi$$
0.114159 + 0.993463i $$0.463583\pi$$
$$564$$ 8.20871 0.345649
$$565$$ 0 0
$$566$$ 53.4955 2.24858
$$567$$ −1.00000 −0.0419961
$$568$$ −4.41742 −0.185351
$$569$$ 19.4174 0.814021 0.407010 0.913424i $$-0.366571\pi$$
0.407010 + 0.913424i $$0.366571\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −132.695 −5.54826
$$573$$ 4.00000 0.167102
$$574$$ −25.5826 −1.06780
$$575$$ 0 0
$$576$$ 44.9129 1.87137
$$577$$ 0.834849 0.0347552 0.0173776 0.999849i $$-0.494468\pi$$
0.0173776 + 0.999849i $$0.494468\pi$$
$$578$$ −22.3303 −0.928818
$$579$$ 16.3303 0.678664
$$580$$ 0 0
$$581$$ −2.41742 −0.100292
$$582$$ 6.74773 0.279702
$$583$$ −47.9129 −1.98435
$$584$$ −42.3303 −1.75164
$$585$$ 0 0
$$586$$ −33.0345 −1.36464
$$587$$ 2.41742 0.0997778 0.0498889 0.998755i $$-0.484113\pi$$
0.0498889 + 0.998755i $$0.484113\pi$$
$$588$$ 34.7477 1.43297
$$589$$ −22.3303 −0.920104
$$590$$ 0 0
$$591$$ 20.3303 0.836277
$$592$$ 71.8258 2.95202
$$593$$ −17.5826 −0.722030 −0.361015 0.932560i $$-0.617570\pi$$
−0.361015 + 0.932560i $$0.617570\pi$$
$$594$$ −13.9564 −0.572640
$$595$$ 0 0
$$596$$ −62.2432 −2.54958
$$597$$ −22.5826 −0.924243
$$598$$ −51.1652 −2.09230
$$599$$ −18.1652 −0.742208 −0.371104 0.928591i $$-0.621021\pi$$
−0.371104 + 0.928591i $$0.621021\pi$$
$$600$$ 0 0
$$601$$ 4.33030 0.176637 0.0883184 0.996092i $$-0.471851\pi$$
0.0883184 + 0.996092i $$0.471851\pi$$
$$602$$ −1.16515 −0.0474880
$$603$$ −14.1652 −0.576850
$$604$$ −64.6606 −2.63100
$$605$$ 0 0
$$606$$ −23.9564 −0.973164
$$607$$ 5.58258 0.226590 0.113295 0.993561i $$-0.463860\pi$$
0.113295 + 0.993561i $$0.463860\pi$$
$$608$$ −161.652 −6.55583
$$609$$ 1.00000 0.0405220
$$610$$ 0 0
$$611$$ 6.49545 0.262778
$$612$$ 17.3739 0.702297
$$613$$ 16.2523 0.656423 0.328212 0.944604i $$-0.393554\pi$$
0.328212 + 0.944604i $$0.393554\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −52.9129 −2.13192
$$617$$ 39.4955 1.59003 0.795014 0.606592i $$-0.207464\pi$$
0.795014 + 0.606592i $$0.207464\pi$$
$$618$$ −8.83485 −0.355390
$$619$$ −5.16515 −0.207605 −0.103802 0.994598i $$-0.533101\pi$$
−0.103802 + 0.994598i $$0.533101\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ −8.37386 −0.335761
$$623$$ −10.5826 −0.423982
$$624$$ 82.2867 3.29411
$$625$$ 0 0
$$626$$ −3.95644 −0.158131
$$627$$ 27.9129 1.11473
$$628$$ 96.9909 3.87036
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −34.9129 −1.38986 −0.694930 0.719078i $$-0.744565\pi$$
−0.694930 + 0.719078i $$0.744565\pi$$
$$632$$ −16.7477 −0.666189
$$633$$ 16.3303 0.649071
$$634$$ 69.7822 2.77141
$$635$$ 0 0
$$636$$ 55.4955 2.20054
$$637$$ 27.4955 1.08941
$$638$$ 13.9564 0.552541
$$639$$ −0.417424 −0.0165131
$$640$$ 0 0
$$641$$ 2.58258 0.102006 0.0510028 0.998699i $$-0.483758\pi$$
0.0510028 + 0.998699i $$0.483758\pi$$
$$642$$ −36.7477 −1.45032
$$643$$ 29.6606 1.16970 0.584850 0.811141i $$-0.301153\pi$$
0.584850 + 0.811141i $$0.301153\pi$$
$$644$$ −23.1652 −0.912835
$$645$$ 0 0
$$646$$ −46.7477 −1.83926
$$647$$ 2.74773 0.108024 0.0540121 0.998540i $$-0.482799\pi$$
0.0540121 + 0.998540i $$0.482799\pi$$
$$648$$ 10.5826 0.415723
$$649$$ 7.91288 0.310608
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ 9.16515 0.358935
$$653$$ −7.83485 −0.306601 −0.153301 0.988180i $$-0.548990\pi$$
−0.153301 + 0.988180i $$0.548990\pi$$
$$654$$ 11.6261 0.454618
$$655$$ 0 0
$$656$$ 164.573 6.42552
$$657$$ −4.00000 −0.156055
$$658$$ 3.95644 0.154238
$$659$$ −0.165151 −0.00643338 −0.00321669 0.999995i $$-0.501024\pi$$
−0.00321669 + 0.999995i $$0.501024\pi$$
$$660$$ 0 0
$$661$$ 25.0000 0.972387 0.486194 0.873851i $$-0.338385\pi$$
0.486194 + 0.873851i $$0.338385\pi$$
$$662$$ 79.0780 3.07345
$$663$$ 13.7477 0.533917
$$664$$ 25.5826 0.992796
$$665$$ 0 0
$$666$$ 11.1652 0.432641
$$667$$ 4.00000 0.154881
$$668$$ 0 0
$$669$$ 7.00000 0.270636
$$670$$ 0 0
$$671$$ −73.7386 −2.84665
$$672$$ 28.9564 1.11702
$$673$$ 25.7477 0.992502 0.496251 0.868179i $$-0.334710\pi$$
0.496251 + 0.868179i $$0.334710\pi$$
$$674$$ 7.91288 0.304793
$$675$$ 0 0
$$676$$ 46.3303 1.78193
$$677$$ 46.8258 1.79966 0.899830 0.436241i $$-0.143690\pi$$
0.899830 + 0.436241i $$0.143690\pi$$
$$678$$ 11.6261 0.446499
$$679$$ 2.41742 0.0927722
$$680$$ 0 0
$$681$$ 1.58258 0.0606444
$$682$$ 55.8258 2.13768
$$683$$ 11.1652 0.427223 0.213611 0.976919i $$-0.431477\pi$$
0.213611 + 0.976919i $$0.431477\pi$$
$$684$$ −32.3303 −1.23618
$$685$$ 0 0
$$686$$ 36.2867 1.38543
$$687$$ −1.16515 −0.0444533
$$688$$ 7.49545 0.285762
$$689$$ 43.9129 1.67295
$$690$$ 0 0
$$691$$ −2.91288 −0.110811 −0.0554056 0.998464i $$-0.517645\pi$$
−0.0554056 + 0.998464i $$0.517645\pi$$
$$692$$ −87.8258 −3.33863
$$693$$ −5.00000 −0.189934
$$694$$ 31.1652 1.18301
$$695$$ 0 0
$$696$$ −10.5826 −0.401131
$$697$$ 27.4955 1.04146
$$698$$ 90.2432 3.41575
$$699$$ 5.16515 0.195364
$$700$$ 0 0
$$701$$ 41.0780 1.55150 0.775748 0.631043i $$-0.217373\pi$$
0.775748 + 0.631043i $$0.217373\pi$$
$$702$$ 12.7913 0.482776
$$703$$ −22.3303 −0.842203
$$704$$ 224.564 8.46359
$$705$$ 0 0
$$706$$ 92.5735 3.48405
$$707$$ −8.58258 −0.322781
$$708$$ −9.16515 −0.344447
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −1.58258 −0.0593512
$$712$$ 111.991 4.19704
$$713$$ 16.0000 0.599205
$$714$$ 8.37386 0.313384
$$715$$ 0 0
$$716$$ −131.739 −4.92330
$$717$$ 26.0000 0.970988
$$718$$ −24.6606 −0.920326
$$719$$ −37.9129 −1.41391 −0.706956 0.707258i $$-0.749932\pi$$
−0.706956 + 0.707258i $$0.749932\pi$$
$$720$$ 0 0
$$721$$ −3.16515 −0.117876
$$722$$ 33.9564 1.26373
$$723$$ 7.33030 0.272617
$$724$$ −12.5390 −0.466009
$$725$$ 0 0
$$726$$ −39.0780 −1.45032
$$727$$ −46.3303 −1.71830 −0.859148 0.511727i $$-0.829006\pi$$
−0.859148 + 0.511727i $$0.829006\pi$$
$$728$$ 48.4955 1.79736
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.25227 0.0463170
$$732$$ 85.4083 3.15678
$$733$$ 47.5826 1.75750 0.878751 0.477280i $$-0.158377\pi$$
0.878751 + 0.477280i $$0.158377\pi$$
$$734$$ −48.8348 −1.80253
$$735$$ 0 0
$$736$$ 115.826 4.26939
$$737$$ −70.8258 −2.60890
$$738$$ 25.5826 0.941708
$$739$$ 46.7477 1.71964 0.859821 0.510595i $$-0.170575\pi$$
0.859821 + 0.510595i $$0.170575\pi$$
$$740$$ 0 0
$$741$$ −25.5826 −0.939799
$$742$$ 26.7477 0.981940
$$743$$ −2.25227 −0.0826279 −0.0413139 0.999146i $$-0.513154\pi$$
−0.0413139 + 0.999146i $$0.513154\pi$$
$$744$$ −42.3303 −1.55190
$$745$$ 0 0
$$746$$ 23.2523 0.851326
$$747$$ 2.41742 0.0884489
$$748$$ 86.8693 3.17626
$$749$$ −13.1652 −0.481044
$$750$$ 0 0
$$751$$ 37.4955 1.36823 0.684114 0.729375i $$-0.260189\pi$$
0.684114 + 0.729375i $$0.260189\pi$$
$$752$$ −25.4519 −0.928135
$$753$$ 2.16515 0.0789025
$$754$$ −12.7913 −0.465831
$$755$$ 0 0
$$756$$ 5.79129 0.210627
$$757$$ −34.3303 −1.24776 −0.623878 0.781522i $$-0.714444\pi$$
−0.623878 + 0.781522i $$0.714444\pi$$
$$758$$ −72.5735 −2.63599
$$759$$ −20.0000 −0.725954
$$760$$ 0 0
$$761$$ 45.5826 1.65237 0.826184 0.563401i $$-0.190507\pi$$
0.826184 + 0.563401i $$0.190507\pi$$
$$762$$ 5.58258 0.202235
$$763$$ 4.16515 0.150789
$$764$$ −23.1652 −0.838086
$$765$$ 0 0
$$766$$ 15.5826 0.563021
$$767$$ −7.25227 −0.261864
$$768$$ −98.4519 −3.55258
$$769$$ 16.4174 0.592027 0.296014 0.955184i $$-0.404343\pi$$
0.296014 + 0.955184i $$0.404343\pi$$
$$770$$ 0 0
$$771$$ −14.7477 −0.531126
$$772$$ −94.5735 −3.40377
$$773$$ −13.1652 −0.473518 −0.236759 0.971568i $$-0.576085\pi$$
−0.236759 + 0.971568i $$0.576085\pi$$
$$774$$ 1.16515 0.0418805
$$775$$ 0 0
$$776$$ −25.5826 −0.918361
$$777$$ 4.00000 0.143499
$$778$$ 7.20871 0.258445
$$779$$ −51.1652 −1.83318
$$780$$ 0 0
$$781$$ −2.08712 −0.0746831
$$782$$ 33.4955 1.19779
$$783$$ −1.00000 −0.0357371
$$784$$ −107.739 −3.84781
$$785$$ 0 0
$$786$$ 41.8693 1.49343
$$787$$ 24.0000 0.855508 0.427754 0.903895i $$-0.359305\pi$$
0.427754 + 0.903895i $$0.359305\pi$$
$$788$$ −117.739 −4.19427
$$789$$ −6.33030 −0.225365
$$790$$ 0 0
$$791$$ 4.16515 0.148096
$$792$$ 52.9129 1.88018
$$793$$ 67.5826 2.39993
$$794$$ −81.4083 −2.88907
$$795$$ 0 0
$$796$$ 130.782 4.63545
$$797$$ 4.33030 0.153387 0.0766936 0.997055i $$-0.475564\pi$$
0.0766936 + 0.997055i $$0.475564\pi$$
$$798$$ −15.5826 −0.551617
$$799$$ −4.25227 −0.150435
$$800$$ 0 0
$$801$$ 10.5826 0.373917
$$802$$ 60.2432 2.12726
$$803$$ −20.0000 −0.705785
$$804$$ 82.0345 2.89313
$$805$$ 0 0
$$806$$ −51.1652 −1.80222
$$807$$ −13.4174 −0.472316
$$808$$ 90.8258 3.19524
$$809$$ −20.0780 −0.705906 −0.352953 0.935641i $$-0.614822\pi$$
−0.352953 + 0.935641i $$0.614822\pi$$
$$810$$ 0 0
$$811$$ 23.4174 0.822297 0.411148 0.911568i $$-0.365128\pi$$
0.411148 + 0.911568i $$0.365128\pi$$
$$812$$ −5.79129 −0.203234
$$813$$ 17.1652 0.602008
$$814$$ 55.8258 1.95669
$$815$$ 0 0
$$816$$ −53.8693 −1.88580
$$817$$ −2.33030 −0.0815270
$$818$$ 69.0780 2.41526
$$819$$ 4.58258 0.160128
$$820$$ 0 0
$$821$$ 23.4955 0.819997 0.409999 0.912086i $$-0.365529\pi$$
0.409999 + 0.912086i $$0.365529\pi$$
$$822$$ −56.7477 −1.97930
$$823$$ 13.0780 0.455871 0.227936 0.973676i $$-0.426802\pi$$
0.227936 + 0.973676i $$0.426802\pi$$
$$824$$ 33.4955 1.16687
$$825$$ 0 0
$$826$$ −4.41742 −0.153702
$$827$$ −47.8258 −1.66306 −0.831532 0.555476i $$-0.812536\pi$$
−0.831532 + 0.555476i $$0.812536\pi$$
$$828$$ 23.1652 0.805045
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ 20.9129 0.725460
$$832$$ −205.817 −7.13541
$$833$$ −18.0000 −0.623663
$$834$$ 51.8693 1.79609
$$835$$ 0 0
$$836$$ −161.652 −5.59083
$$837$$ −4.00000 −0.138260
$$838$$ −52.5735 −1.81612
$$839$$ 6.49545 0.224248 0.112124 0.993694i $$-0.464235\pi$$
0.112124 + 0.993694i $$0.464235\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 37.9129 1.30656
$$843$$ 9.58258 0.330041
$$844$$ −94.5735 −3.25535
$$845$$ 0 0
$$846$$ −3.95644 −0.136025
$$847$$ −14.0000 −0.481046
$$848$$ −172.069 −5.90887
$$849$$ −19.1652 −0.657746
$$850$$ 0 0
$$851$$ 16.0000 0.548473
$$852$$ 2.41742 0.0828196
$$853$$ 6.74773 0.231038 0.115519 0.993305i $$-0.463147\pi$$
0.115519 + 0.993305i $$0.463147\pi$$
$$854$$ 41.1652 1.40864
$$855$$ 0 0
$$856$$ 139.321 4.76190
$$857$$ −26.6606 −0.910709 −0.455354 0.890310i $$-0.650487\pi$$
−0.455354 + 0.890310i $$0.650487\pi$$
$$858$$ 63.9564 2.18344
$$859$$ −38.4174 −1.31079 −0.655393 0.755288i $$-0.727497\pi$$
−0.655393 + 0.755288i $$0.727497\pi$$
$$860$$ 0 0
$$861$$ 9.16515 0.312348
$$862$$ −58.1561 −1.98080
$$863$$ 15.5826 0.530437 0.265219 0.964188i $$-0.414556\pi$$
0.265219 + 0.964188i $$0.414556\pi$$
$$864$$ −28.9564 −0.985118
$$865$$ 0 0
$$866$$ 109.078 3.70662
$$867$$ 8.00000 0.271694
$$868$$ −23.1652 −0.786276
$$869$$ −7.91288 −0.268426
$$870$$ 0 0
$$871$$ 64.9129 2.19949
$$872$$ −44.0780 −1.49267
$$873$$ −2.41742 −0.0818174
$$874$$ −62.3303 −2.10835
$$875$$ 0 0
$$876$$ 23.1652 0.782678
$$877$$ −56.3303 −1.90214 −0.951070 0.308977i $$-0.900013\pi$$
−0.951070 + 0.308977i $$0.900013\pi$$
$$878$$ 80.7042 2.72363
$$879$$ 11.8348 0.399180
$$880$$ 0 0
$$881$$ 32.0780 1.08074 0.540368 0.841429i $$-0.318285\pi$$
0.540368 + 0.841429i $$0.318285\pi$$
$$882$$ −16.7477 −0.563925
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ −79.6170 −2.67781
$$885$$ 0 0
$$886$$ 23.9564 0.804832
$$887$$ −0.912878 −0.0306515 −0.0153257 0.999883i $$-0.504879\pi$$
−0.0153257 + 0.999883i $$0.504879\pi$$
$$888$$ −42.3303 −1.42051
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ 5.00000 0.167506
$$892$$ −40.5390 −1.35735
$$893$$ 7.91288 0.264794
$$894$$ 30.0000 1.00335
$$895$$ 0 0
$$896$$ −67.4519 −2.25341
$$897$$ 18.3303 0.612031
$$898$$ 95.1216 3.17425
$$899$$ 4.00000 0.133407
$$900$$ 0 0
$$901$$ −28.7477 −0.957726
$$902$$ 127.913 4.25903
$$903$$ 0.417424 0.0138910
$$904$$ −44.0780 −1.46601
$$905$$ 0 0
$$906$$ 31.1652 1.03539
$$907$$ −14.4174 −0.478723 −0.239361 0.970931i $$-0.576938\pi$$
−0.239361 + 0.970931i $$0.576938\pi$$
$$908$$ −9.16515 −0.304156
$$909$$ 8.58258 0.284666
$$910$$ 0 0
$$911$$ 40.8258 1.35262 0.676309 0.736618i $$-0.263578\pi$$
0.676309 + 0.736618i $$0.263578\pi$$
$$912$$ 100.243 3.31938
$$913$$ 12.0871 0.400025
$$914$$ −66.2867 −2.19257
$$915$$ 0 0
$$916$$ 6.74773 0.222951
$$917$$ 15.0000 0.495344
$$918$$ −8.37386 −0.276379
$$919$$ −26.9129 −0.887774 −0.443887 0.896083i $$-0.646401\pi$$
−0.443887 + 0.896083i $$0.646401\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 25.5826 0.842517
$$923$$ 1.91288 0.0629632
$$924$$ 28.9564 0.952597
$$925$$ 0 0
$$926$$ −50.7042 −1.66624
$$927$$ 3.16515 0.103957
$$928$$ 28.9564 0.950542
$$929$$ 52.3303 1.71690 0.858451 0.512896i $$-0.171427\pi$$
0.858451 + 0.512896i $$0.171427\pi$$
$$930$$ 0 0
$$931$$ 33.4955 1.09777
$$932$$ −29.9129 −0.979829
$$933$$ 3.00000 0.0982156
$$934$$ 22.3303 0.730670
$$935$$ 0 0
$$936$$ −48.4955 −1.58512
$$937$$ 9.41742 0.307654 0.153827 0.988098i $$-0.450840\pi$$
0.153827 + 0.988098i $$0.450840\pi$$
$$938$$ 39.5390 1.29099
$$939$$ 1.41742 0.0462559
$$940$$ 0 0
$$941$$ −25.9129 −0.844736 −0.422368 0.906425i $$-0.638801\pi$$
−0.422368 + 0.906425i $$0.638801\pi$$
$$942$$ −46.7477 −1.52312
$$943$$ 36.6606 1.19383
$$944$$ 28.4174 0.924908
$$945$$ 0 0
$$946$$ 5.82576 0.189412
$$947$$ 9.08712 0.295292 0.147646 0.989040i $$-0.452830\pi$$
0.147646 + 0.989040i $$0.452830\pi$$
$$948$$ 9.16515 0.297670
$$949$$ 18.3303 0.595027
$$950$$ 0 0
$$951$$ −25.0000 −0.810681
$$952$$ −31.7477 −1.02895
$$953$$ 28.4174 0.920531 0.460265 0.887781i $$-0.347754\pi$$
0.460265 + 0.887781i $$0.347754\pi$$
$$954$$ −26.7477 −0.865990
$$955$$ 0 0
$$956$$ −150.573 −4.86989
$$957$$ −5.00000 −0.161627
$$958$$ −2.33030 −0.0752887
$$959$$ −20.3303 −0.656500
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −51.1652 −1.64963
$$963$$ 13.1652 0.424241
$$964$$ −42.4519 −1.36728
$$965$$ 0 0
$$966$$ 11.1652 0.359233
$$967$$ −36.7477 −1.18173 −0.590864 0.806771i $$-0.701213\pi$$
−0.590864 + 0.806771i $$0.701213\pi$$
$$968$$ 148.156 4.76192
$$969$$ 16.7477 0.538015
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ −5.79129 −0.185756
$$973$$ 18.5826 0.595730
$$974$$ 95.8258 3.07046
$$975$$ 0 0
$$976$$ −264.817 −8.47657
$$977$$ 57.4955 1.83944 0.919721 0.392572i $$-0.128415\pi$$
0.919721 + 0.392572i $$0.128415\pi$$
$$978$$ −4.41742 −0.141254
$$979$$ 52.9129 1.69110
$$980$$ 0 0
$$981$$ −4.16515 −0.132983
$$982$$ 44.6606 1.42518
$$983$$ −36.8348 −1.17485 −0.587425 0.809279i $$-0.699858\pi$$
−0.587425 + 0.809279i $$0.699858\pi$$
$$984$$ −96.9909 −3.09196
$$985$$ 0 0
$$986$$ 8.37386 0.266678
$$987$$ −1.41742 −0.0451171
$$988$$ 148.156 4.71347
$$989$$ 1.66970 0.0530933
$$990$$ 0 0
$$991$$ −48.0780 −1.52725 −0.763624 0.645661i $$-0.776582\pi$$
−0.763624 + 0.645661i $$0.776582\pi$$
$$992$$ 115.826 3.67747
$$993$$ −28.3303 −0.899035
$$994$$ 1.16515 0.0369564
$$995$$ 0 0
$$996$$ −14.0000 −0.443607
$$997$$ 19.1652 0.606966 0.303483 0.952837i $$-0.401850\pi$$
0.303483 + 0.952837i $$0.401850\pi$$
$$998$$ −63.0345 −1.99532
$$999$$ −4.00000 −0.126554
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 2175.2.a.r.1.2 2
3.2 odd 2 6525.2.a.t.1.1 2
5.2 odd 4 2175.2.c.f.349.4 4
5.3 odd 4 2175.2.c.f.349.1 4
5.4 even 2 435.2.a.f.1.1 2
15.14 odd 2 1305.2.a.m.1.2 2
20.19 odd 2 6960.2.a.bw.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.1 2 5.4 even 2
1305.2.a.m.1.2 2 15.14 odd 2
2175.2.a.r.1.2 2 1.1 even 1 trivial
2175.2.c.f.349.1 4 5.3 odd 4
2175.2.c.f.349.4 4 5.2 odd 4
6525.2.a.t.1.1 2 3.2 odd 2
6960.2.a.bw.1.2 2 20.19 odd 2