# Properties

 Label 3822.2.a.bs.1.1 Level $3822$ Weight $2$ Character 3822.1 Self dual yes Analytic conductor $30.519$ 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] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.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: $$30.5188236525$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3822.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.41421 q^{10} +0.414214 q^{11} +1.00000 q^{12} +1.00000 q^{13} -1.41421 q^{15} +1.00000 q^{16} +5.24264 q^{17} +1.00000 q^{18} +3.00000 q^{19} -1.41421 q^{20} +0.414214 q^{22} -9.07107 q^{23} +1.00000 q^{24} -3.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +5.00000 q^{29} -1.41421 q^{30} +6.82843 q^{31} +1.00000 q^{32} +0.414214 q^{33} +5.24264 q^{34} +1.00000 q^{36} -5.07107 q^{37} +3.00000 q^{38} +1.00000 q^{39} -1.41421 q^{40} +4.58579 q^{41} +4.82843 q^{43} +0.414214 q^{44} -1.41421 q^{45} -9.07107 q^{46} +9.00000 q^{47} +1.00000 q^{48} -3.00000 q^{50} +5.24264 q^{51} +1.00000 q^{52} +12.3137 q^{53} +1.00000 q^{54} -0.585786 q^{55} +3.00000 q^{57} +5.00000 q^{58} +7.58579 q^{59} -1.41421 q^{60} -5.24264 q^{61} +6.82843 q^{62} +1.00000 q^{64} -1.41421 q^{65} +0.414214 q^{66} -14.6569 q^{67} +5.24264 q^{68} -9.07107 q^{69} -1.00000 q^{71} +1.00000 q^{72} -1.07107 q^{73} -5.07107 q^{74} -3.00000 q^{75} +3.00000 q^{76} +1.00000 q^{78} +13.3137 q^{79} -1.41421 q^{80} +1.00000 q^{81} +4.58579 q^{82} -7.65685 q^{83} -7.41421 q^{85} +4.82843 q^{86} +5.00000 q^{87} +0.414214 q^{88} -6.24264 q^{89} -1.41421 q^{90} -9.07107 q^{92} +6.82843 q^{93} +9.00000 q^{94} -4.24264 q^{95} +1.00000 q^{96} +7.89949 q^{97} +0.414214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 6 q^{19} - 2 q^{22} - 4 q^{23} + 2 q^{24} - 6 q^{25} + 2 q^{26} + 2 q^{27} + 10 q^{29} + 8 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{34} + 2 q^{36} + 4 q^{37} + 6 q^{38} + 2 q^{39} + 12 q^{41} + 4 q^{43} - 2 q^{44} - 4 q^{46} + 18 q^{47} + 2 q^{48} - 6 q^{50} + 2 q^{51} + 2 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} + 6 q^{57} + 10 q^{58} + 18 q^{59} - 2 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{66} - 18 q^{67} + 2 q^{68} - 4 q^{69} - 2 q^{71} + 2 q^{72} + 12 q^{73} + 4 q^{74} - 6 q^{75} + 6 q^{76} + 2 q^{78} + 4 q^{79} + 2 q^{81} + 12 q^{82} - 4 q^{83} - 12 q^{85} + 4 q^{86} + 10 q^{87} - 2 q^{88} - 4 q^{89} - 4 q^{92} + 8 q^{93} + 18 q^{94} + 2 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^16 + 2 * q^17 + 2 * q^18 + 6 * q^19 - 2 * q^22 - 4 * q^23 + 2 * q^24 - 6 * q^25 + 2 * q^26 + 2 * q^27 + 10 * q^29 + 8 * q^31 + 2 * q^32 - 2 * q^33 + 2 * q^34 + 2 * q^36 + 4 * q^37 + 6 * q^38 + 2 * q^39 + 12 * q^41 + 4 * q^43 - 2 * q^44 - 4 * q^46 + 18 * q^47 + 2 * q^48 - 6 * q^50 + 2 * q^51 + 2 * q^52 + 2 * q^53 + 2 * q^54 - 4 * q^55 + 6 * q^57 + 10 * q^58 + 18 * q^59 - 2 * q^61 + 8 * q^62 + 2 * q^64 - 2 * q^66 - 18 * q^67 + 2 * q^68 - 4 * q^69 - 2 * q^71 + 2 * q^72 + 12 * q^73 + 4 * q^74 - 6 * q^75 + 6 * q^76 + 2 * q^78 + 4 * q^79 + 2 * q^81 + 12 * q^82 - 4 * q^83 - 12 * q^85 + 4 * q^86 + 10 * q^87 - 2 * q^88 - 4 * q^89 - 4 * q^92 + 8 * q^93 + 18 * q^94 + 2 * q^96 - 4 * q^97 - 2 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.41421 −0.632456 −0.316228 0.948683i $$-0.602416\pi$$
−0.316228 + 0.948683i $$0.602416\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.41421 −0.447214
$$11$$ 0.414214 0.124890 0.0624450 0.998048i $$-0.480110\pi$$
0.0624450 + 0.998048i $$0.480110\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −1.41421 −0.365148
$$16$$ 1.00000 0.250000
$$17$$ 5.24264 1.27153 0.635764 0.771884i $$-0.280685\pi$$
0.635764 + 0.771884i $$0.280685\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ −1.41421 −0.316228
$$21$$ 0 0
$$22$$ 0.414214 0.0883106
$$23$$ −9.07107 −1.89145 −0.945724 0.324970i $$-0.894646\pi$$
−0.945724 + 0.324970i $$0.894646\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −3.00000 −0.600000
$$26$$ 1.00000 0.196116
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ −1.41421 −0.258199
$$31$$ 6.82843 1.22642 0.613211 0.789919i $$-0.289878\pi$$
0.613211 + 0.789919i $$0.289878\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0.414214 0.0721053
$$34$$ 5.24264 0.899105
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −5.07107 −0.833678 −0.416839 0.908980i $$-0.636862\pi$$
−0.416839 + 0.908980i $$0.636862\pi$$
$$38$$ 3.00000 0.486664
$$39$$ 1.00000 0.160128
$$40$$ −1.41421 −0.223607
$$41$$ 4.58579 0.716180 0.358090 0.933687i $$-0.383428\pi$$
0.358090 + 0.933687i $$0.383428\pi$$
$$42$$ 0 0
$$43$$ 4.82843 0.736328 0.368164 0.929761i $$-0.379986\pi$$
0.368164 + 0.929761i $$0.379986\pi$$
$$44$$ 0.414214 0.0624450
$$45$$ −1.41421 −0.210819
$$46$$ −9.07107 −1.33746
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ −3.00000 −0.424264
$$51$$ 5.24264 0.734117
$$52$$ 1.00000 0.138675
$$53$$ 12.3137 1.69142 0.845709 0.533644i $$-0.179178\pi$$
0.845709 + 0.533644i $$0.179178\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −0.585786 −0.0789874
$$56$$ 0 0
$$57$$ 3.00000 0.397360
$$58$$ 5.00000 0.656532
$$59$$ 7.58579 0.987585 0.493793 0.869580i $$-0.335610\pi$$
0.493793 + 0.869580i $$0.335610\pi$$
$$60$$ −1.41421 −0.182574
$$61$$ −5.24264 −0.671251 −0.335626 0.941995i $$-0.608948\pi$$
−0.335626 + 0.941995i $$0.608948\pi$$
$$62$$ 6.82843 0.867211
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.41421 −0.175412
$$66$$ 0.414214 0.0509862
$$67$$ −14.6569 −1.79062 −0.895310 0.445444i $$-0.853046\pi$$
−0.895310 + 0.445444i $$0.853046\pi$$
$$68$$ 5.24264 0.635764
$$69$$ −9.07107 −1.09203
$$70$$ 0 0
$$71$$ −1.00000 −0.118678 −0.0593391 0.998238i $$-0.518899\pi$$
−0.0593391 + 0.998238i $$0.518899\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −1.07107 −0.125359 −0.0626795 0.998034i $$-0.519965\pi$$
−0.0626795 + 0.998034i $$0.519965\pi$$
$$74$$ −5.07107 −0.589500
$$75$$ −3.00000 −0.346410
$$76$$ 3.00000 0.344124
$$77$$ 0 0
$$78$$ 1.00000 0.113228
$$79$$ 13.3137 1.49791 0.748955 0.662621i $$-0.230556\pi$$
0.748955 + 0.662621i $$0.230556\pi$$
$$80$$ −1.41421 −0.158114
$$81$$ 1.00000 0.111111
$$82$$ 4.58579 0.506415
$$83$$ −7.65685 −0.840449 −0.420224 0.907420i $$-0.638049\pi$$
−0.420224 + 0.907420i $$0.638049\pi$$
$$84$$ 0 0
$$85$$ −7.41421 −0.804184
$$86$$ 4.82843 0.520663
$$87$$ 5.00000 0.536056
$$88$$ 0.414214 0.0441553
$$89$$ −6.24264 −0.661719 −0.330859 0.943680i $$-0.607339\pi$$
−0.330859 + 0.943680i $$0.607339\pi$$
$$90$$ −1.41421 −0.149071
$$91$$ 0 0
$$92$$ −9.07107 −0.945724
$$93$$ 6.82843 0.708075
$$94$$ 9.00000 0.928279
$$95$$ −4.24264 −0.435286
$$96$$ 1.00000 0.102062
$$97$$ 7.89949 0.802072 0.401036 0.916062i $$-0.368650\pi$$
0.401036 + 0.916062i $$0.368650\pi$$
$$98$$ 0 0
$$99$$ 0.414214 0.0416300
$$100$$ −3.00000 −0.300000
$$101$$ −1.17157 −0.116576 −0.0582879 0.998300i $$-0.518564\pi$$
−0.0582879 + 0.998300i $$0.518564\pi$$
$$102$$ 5.24264 0.519099
$$103$$ 6.48528 0.639014 0.319507 0.947584i $$-0.396483\pi$$
0.319507 + 0.947584i $$0.396483\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 12.3137 1.19601
$$107$$ 13.6569 1.32026 0.660129 0.751152i $$-0.270502\pi$$
0.660129 + 0.751152i $$0.270502\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 1.65685 0.158698 0.0793489 0.996847i $$-0.474716\pi$$
0.0793489 + 0.996847i $$0.474716\pi$$
$$110$$ −0.585786 −0.0558525
$$111$$ −5.07107 −0.481324
$$112$$ 0 0
$$113$$ −6.07107 −0.571118 −0.285559 0.958361i $$-0.592179\pi$$
−0.285559 + 0.958361i $$0.592179\pi$$
$$114$$ 3.00000 0.280976
$$115$$ 12.8284 1.19626
$$116$$ 5.00000 0.464238
$$117$$ 1.00000 0.0924500
$$118$$ 7.58579 0.698328
$$119$$ 0 0
$$120$$ −1.41421 −0.129099
$$121$$ −10.8284 −0.984402
$$122$$ −5.24264 −0.474646
$$123$$ 4.58579 0.413486
$$124$$ 6.82843 0.613211
$$125$$ 11.3137 1.01193
$$126$$ 0 0
$$127$$ 9.89949 0.878438 0.439219 0.898380i $$-0.355255\pi$$
0.439219 + 0.898380i $$0.355255\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.82843 0.425119
$$130$$ −1.41421 −0.124035
$$131$$ 9.89949 0.864923 0.432461 0.901652i $$-0.357645\pi$$
0.432461 + 0.901652i $$0.357645\pi$$
$$132$$ 0.414214 0.0360527
$$133$$ 0 0
$$134$$ −14.6569 −1.26616
$$135$$ −1.41421 −0.121716
$$136$$ 5.24264 0.449553
$$137$$ −13.0711 −1.11674 −0.558368 0.829593i $$-0.688572\pi$$
−0.558368 + 0.829593i $$0.688572\pi$$
$$138$$ −9.07107 −0.772181
$$139$$ −21.2132 −1.79928 −0.899640 0.436632i $$-0.856171\pi$$
−0.899640 + 0.436632i $$0.856171\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ −1.00000 −0.0839181
$$143$$ 0.414214 0.0346383
$$144$$ 1.00000 0.0833333
$$145$$ −7.07107 −0.587220
$$146$$ −1.07107 −0.0886422
$$147$$ 0 0
$$148$$ −5.07107 −0.416839
$$149$$ 15.8995 1.30254 0.651269 0.758847i $$-0.274237\pi$$
0.651269 + 0.758847i $$0.274237\pi$$
$$150$$ −3.00000 −0.244949
$$151$$ −7.58579 −0.617323 −0.308661 0.951172i $$-0.599881\pi$$
−0.308661 + 0.951172i $$0.599881\pi$$
$$152$$ 3.00000 0.243332
$$153$$ 5.24264 0.423842
$$154$$ 0 0
$$155$$ −9.65685 −0.775657
$$156$$ 1.00000 0.0800641
$$157$$ 7.92893 0.632798 0.316399 0.948626i $$-0.397526\pi$$
0.316399 + 0.948626i $$0.397526\pi$$
$$158$$ 13.3137 1.05918
$$159$$ 12.3137 0.976541
$$160$$ −1.41421 −0.111803
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 17.1421 1.34268 0.671338 0.741151i $$-0.265720\pi$$
0.671338 + 0.741151i $$0.265720\pi$$
$$164$$ 4.58579 0.358090
$$165$$ −0.585786 −0.0456034
$$166$$ −7.65685 −0.594287
$$167$$ 12.3137 0.952863 0.476432 0.879211i $$-0.341930\pi$$
0.476432 + 0.879211i $$0.341930\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −7.41421 −0.568644
$$171$$ 3.00000 0.229416
$$172$$ 4.82843 0.368164
$$173$$ 20.3137 1.54442 0.772211 0.635366i $$-0.219151\pi$$
0.772211 + 0.635366i $$0.219151\pi$$
$$174$$ 5.00000 0.379049
$$175$$ 0 0
$$176$$ 0.414214 0.0312225
$$177$$ 7.58579 0.570183
$$178$$ −6.24264 −0.467906
$$179$$ −12.3431 −0.922570 −0.461285 0.887252i $$-0.652611\pi$$
−0.461285 + 0.887252i $$0.652611\pi$$
$$180$$ −1.41421 −0.105409
$$181$$ −11.5858 −0.861165 −0.430582 0.902551i $$-0.641692\pi$$
−0.430582 + 0.902551i $$0.641692\pi$$
$$182$$ 0 0
$$183$$ −5.24264 −0.387547
$$184$$ −9.07107 −0.668728
$$185$$ 7.17157 0.527265
$$186$$ 6.82843 0.500685
$$187$$ 2.17157 0.158801
$$188$$ 9.00000 0.656392
$$189$$ 0 0
$$190$$ −4.24264 −0.307794
$$191$$ 2.34315 0.169544 0.0847720 0.996400i $$-0.472984\pi$$
0.0847720 + 0.996400i $$0.472984\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −15.7574 −1.13424 −0.567120 0.823635i $$-0.691942\pi$$
−0.567120 + 0.823635i $$0.691942\pi$$
$$194$$ 7.89949 0.567151
$$195$$ −1.41421 −0.101274
$$196$$ 0 0
$$197$$ 9.55635 0.680862 0.340431 0.940270i $$-0.389427\pi$$
0.340431 + 0.940270i $$0.389427\pi$$
$$198$$ 0.414214 0.0294369
$$199$$ 10.5858 0.750407 0.375203 0.926943i $$-0.377573\pi$$
0.375203 + 0.926943i $$0.377573\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ −14.6569 −1.03381
$$202$$ −1.17157 −0.0824316
$$203$$ 0 0
$$204$$ 5.24264 0.367058
$$205$$ −6.48528 −0.452952
$$206$$ 6.48528 0.451851
$$207$$ −9.07107 −0.630483
$$208$$ 1.00000 0.0693375
$$209$$ 1.24264 0.0859553
$$210$$ 0 0
$$211$$ −28.3848 −1.95409 −0.977044 0.213036i $$-0.931665\pi$$
−0.977044 + 0.213036i $$0.931665\pi$$
$$212$$ 12.3137 0.845709
$$213$$ −1.00000 −0.0685189
$$214$$ 13.6569 0.933563
$$215$$ −6.82843 −0.465695
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 1.65685 0.112216
$$219$$ −1.07107 −0.0723761
$$220$$ −0.585786 −0.0394937
$$221$$ 5.24264 0.352658
$$222$$ −5.07107 −0.340348
$$223$$ 21.3848 1.43203 0.716015 0.698085i $$-0.245964\pi$$
0.716015 + 0.698085i $$0.245964\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ −6.07107 −0.403841
$$227$$ −26.1421 −1.73511 −0.867557 0.497337i $$-0.834311\pi$$
−0.867557 + 0.497337i $$0.834311\pi$$
$$228$$ 3.00000 0.198680
$$229$$ −7.17157 −0.473911 −0.236955 0.971521i $$-0.576150\pi$$
−0.236955 + 0.971521i $$0.576150\pi$$
$$230$$ 12.8284 0.845881
$$231$$ 0 0
$$232$$ 5.00000 0.328266
$$233$$ −3.72792 −0.244224 −0.122112 0.992516i $$-0.538967\pi$$
−0.122112 + 0.992516i $$0.538967\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ −12.7279 −0.830278
$$236$$ 7.58579 0.493793
$$237$$ 13.3137 0.864818
$$238$$ 0 0
$$239$$ −2.51472 −0.162664 −0.0813318 0.996687i $$-0.525917\pi$$
−0.0813318 + 0.996687i $$0.525917\pi$$
$$240$$ −1.41421 −0.0912871
$$241$$ −16.4853 −1.06191 −0.530955 0.847400i $$-0.678167\pi$$
−0.530955 + 0.847400i $$0.678167\pi$$
$$242$$ −10.8284 −0.696078
$$243$$ 1.00000 0.0641500
$$244$$ −5.24264 −0.335626
$$245$$ 0 0
$$246$$ 4.58579 0.292379
$$247$$ 3.00000 0.190885
$$248$$ 6.82843 0.433606
$$249$$ −7.65685 −0.485233
$$250$$ 11.3137 0.715542
$$251$$ 12.8284 0.809723 0.404862 0.914378i $$-0.367320\pi$$
0.404862 + 0.914378i $$0.367320\pi$$
$$252$$ 0 0
$$253$$ −3.75736 −0.236223
$$254$$ 9.89949 0.621150
$$255$$ −7.41421 −0.464296
$$256$$ 1.00000 0.0625000
$$257$$ −21.7990 −1.35978 −0.679892 0.733312i $$-0.737973\pi$$
−0.679892 + 0.733312i $$0.737973\pi$$
$$258$$ 4.82843 0.300605
$$259$$ 0 0
$$260$$ −1.41421 −0.0877058
$$261$$ 5.00000 0.309492
$$262$$ 9.89949 0.611593
$$263$$ −7.55635 −0.465944 −0.232972 0.972483i $$-0.574845\pi$$
−0.232972 + 0.972483i $$0.574845\pi$$
$$264$$ 0.414214 0.0254931
$$265$$ −17.4142 −1.06975
$$266$$ 0 0
$$267$$ −6.24264 −0.382043
$$268$$ −14.6569 −0.895310
$$269$$ −3.68629 −0.224757 −0.112379 0.993665i $$-0.535847\pi$$
−0.112379 + 0.993665i $$0.535847\pi$$
$$270$$ −1.41421 −0.0860663
$$271$$ −15.3848 −0.934559 −0.467279 0.884110i $$-0.654766\pi$$
−0.467279 + 0.884110i $$0.654766\pi$$
$$272$$ 5.24264 0.317882
$$273$$ 0 0
$$274$$ −13.0711 −0.789652
$$275$$ −1.24264 −0.0749341
$$276$$ −9.07107 −0.546014
$$277$$ −16.4142 −0.986235 −0.493117 0.869963i $$-0.664143\pi$$
−0.493117 + 0.869963i $$0.664143\pi$$
$$278$$ −21.2132 −1.27228
$$279$$ 6.82843 0.408807
$$280$$ 0 0
$$281$$ 7.31371 0.436299 0.218150 0.975915i $$-0.429998\pi$$
0.218150 + 0.975915i $$0.429998\pi$$
$$282$$ 9.00000 0.535942
$$283$$ 17.0711 1.01477 0.507385 0.861720i $$-0.330612\pi$$
0.507385 + 0.861720i $$0.330612\pi$$
$$284$$ −1.00000 −0.0593391
$$285$$ −4.24264 −0.251312
$$286$$ 0.414214 0.0244930
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 10.4853 0.616781
$$290$$ −7.07107 −0.415227
$$291$$ 7.89949 0.463077
$$292$$ −1.07107 −0.0626795
$$293$$ −18.7279 −1.09410 −0.547048 0.837101i $$-0.684249\pi$$
−0.547048 + 0.837101i $$0.684249\pi$$
$$294$$ 0 0
$$295$$ −10.7279 −0.624604
$$296$$ −5.07107 −0.294750
$$297$$ 0.414214 0.0240351
$$298$$ 15.8995 0.921033
$$299$$ −9.07107 −0.524593
$$300$$ −3.00000 −0.173205
$$301$$ 0 0
$$302$$ −7.58579 −0.436513
$$303$$ −1.17157 −0.0673051
$$304$$ 3.00000 0.172062
$$305$$ 7.41421 0.424537
$$306$$ 5.24264 0.299702
$$307$$ −19.6274 −1.12020 −0.560098 0.828426i $$-0.689237\pi$$
−0.560098 + 0.828426i $$0.689237\pi$$
$$308$$ 0 0
$$309$$ 6.48528 0.368935
$$310$$ −9.65685 −0.548472
$$311$$ −6.58579 −0.373446 −0.186723 0.982413i $$-0.559787\pi$$
−0.186723 + 0.982413i $$0.559787\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ 18.1421 1.02545 0.512727 0.858552i $$-0.328635\pi$$
0.512727 + 0.858552i $$0.328635\pi$$
$$314$$ 7.92893 0.447456
$$315$$ 0 0
$$316$$ 13.3137 0.748955
$$317$$ 0.828427 0.0465291 0.0232646 0.999729i $$-0.492594\pi$$
0.0232646 + 0.999729i $$0.492594\pi$$
$$318$$ 12.3137 0.690518
$$319$$ 2.07107 0.115958
$$320$$ −1.41421 −0.0790569
$$321$$ 13.6569 0.762251
$$322$$ 0 0
$$323$$ 15.7279 0.875125
$$324$$ 1.00000 0.0555556
$$325$$ −3.00000 −0.166410
$$326$$ 17.1421 0.949415
$$327$$ 1.65685 0.0916242
$$328$$ 4.58579 0.253208
$$329$$ 0 0
$$330$$ −0.585786 −0.0322465
$$331$$ −22.4853 −1.23590 −0.617951 0.786216i $$-0.712037\pi$$
−0.617951 + 0.786216i $$0.712037\pi$$
$$332$$ −7.65685 −0.420224
$$333$$ −5.07107 −0.277893
$$334$$ 12.3137 0.673776
$$335$$ 20.7279 1.13249
$$336$$ 0 0
$$337$$ 28.6569 1.56104 0.780519 0.625132i $$-0.214955\pi$$
0.780519 + 0.625132i $$0.214955\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ −6.07107 −0.329735
$$340$$ −7.41421 −0.402092
$$341$$ 2.82843 0.153168
$$342$$ 3.00000 0.162221
$$343$$ 0 0
$$344$$ 4.82843 0.260331
$$345$$ 12.8284 0.690659
$$346$$ 20.3137 1.09207
$$347$$ 5.75736 0.309071 0.154536 0.987987i $$-0.450612\pi$$
0.154536 + 0.987987i $$0.450612\pi$$
$$348$$ 5.00000 0.268028
$$349$$ 5.41421 0.289816 0.144908 0.989445i $$-0.453711\pi$$
0.144908 + 0.989445i $$0.453711\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0.414214 0.0220777
$$353$$ 12.6274 0.672090 0.336045 0.941846i $$-0.390911\pi$$
0.336045 + 0.941846i $$0.390911\pi$$
$$354$$ 7.58579 0.403180
$$355$$ 1.41421 0.0750587
$$356$$ −6.24264 −0.330859
$$357$$ 0 0
$$358$$ −12.3431 −0.652356
$$359$$ −15.4558 −0.815728 −0.407864 0.913043i $$-0.633726\pi$$
−0.407864 + 0.913043i $$0.633726\pi$$
$$360$$ −1.41421 −0.0745356
$$361$$ −10.0000 −0.526316
$$362$$ −11.5858 −0.608935
$$363$$ −10.8284 −0.568345
$$364$$ 0 0
$$365$$ 1.51472 0.0792840
$$366$$ −5.24264 −0.274037
$$367$$ 14.0000 0.730794 0.365397 0.930852i $$-0.380933\pi$$
0.365397 + 0.930852i $$0.380933\pi$$
$$368$$ −9.07107 −0.472862
$$369$$ 4.58579 0.238727
$$370$$ 7.17157 0.372832
$$371$$ 0 0
$$372$$ 6.82843 0.354037
$$373$$ −11.1005 −0.574762 −0.287381 0.957816i $$-0.592785\pi$$
−0.287381 + 0.957816i $$0.592785\pi$$
$$374$$ 2.17157 0.112289
$$375$$ 11.3137 0.584237
$$376$$ 9.00000 0.464140
$$377$$ 5.00000 0.257513
$$378$$ 0 0
$$379$$ −33.6569 −1.72884 −0.864418 0.502773i $$-0.832313\pi$$
−0.864418 + 0.502773i $$0.832313\pi$$
$$380$$ −4.24264 −0.217643
$$381$$ 9.89949 0.507166
$$382$$ 2.34315 0.119886
$$383$$ 6.48528 0.331382 0.165691 0.986178i $$-0.447015\pi$$
0.165691 + 0.986178i $$0.447015\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −15.7574 −0.802028
$$387$$ 4.82843 0.245443
$$388$$ 7.89949 0.401036
$$389$$ 23.0000 1.16615 0.583073 0.812420i $$-0.301850\pi$$
0.583073 + 0.812420i $$0.301850\pi$$
$$390$$ −1.41421 −0.0716115
$$391$$ −47.5563 −2.40503
$$392$$ 0 0
$$393$$ 9.89949 0.499363
$$394$$ 9.55635 0.481442
$$395$$ −18.8284 −0.947361
$$396$$ 0.414214 0.0208150
$$397$$ −19.7574 −0.991593 −0.495797 0.868439i $$-0.665124\pi$$
−0.495797 + 0.868439i $$0.665124\pi$$
$$398$$ 10.5858 0.530618
$$399$$ 0 0
$$400$$ −3.00000 −0.150000
$$401$$ 19.0711 0.952364 0.476182 0.879347i $$-0.342020\pi$$
0.476182 + 0.879347i $$0.342020\pi$$
$$402$$ −14.6569 −0.731017
$$403$$ 6.82843 0.340148
$$404$$ −1.17157 −0.0582879
$$405$$ −1.41421 −0.0702728
$$406$$ 0 0
$$407$$ −2.10051 −0.104118
$$408$$ 5.24264 0.259549
$$409$$ −2.58579 −0.127859 −0.0639295 0.997954i $$-0.520363\pi$$
−0.0639295 + 0.997954i $$0.520363\pi$$
$$410$$ −6.48528 −0.320285
$$411$$ −13.0711 −0.644748
$$412$$ 6.48528 0.319507
$$413$$ 0 0
$$414$$ −9.07107 −0.445819
$$415$$ 10.8284 0.531547
$$416$$ 1.00000 0.0490290
$$417$$ −21.2132 −1.03882
$$418$$ 1.24264 0.0607795
$$419$$ −27.4558 −1.34131 −0.670653 0.741771i $$-0.733986\pi$$
−0.670653 + 0.741771i $$0.733986\pi$$
$$420$$ 0 0
$$421$$ 10.5858 0.515920 0.257960 0.966156i $$-0.416950\pi$$
0.257960 + 0.966156i $$0.416950\pi$$
$$422$$ −28.3848 −1.38175
$$423$$ 9.00000 0.437595
$$424$$ 12.3137 0.598007
$$425$$ −15.7279 −0.762916
$$426$$ −1.00000 −0.0484502
$$427$$ 0 0
$$428$$ 13.6569 0.660129
$$429$$ 0.414214 0.0199984
$$430$$ −6.82843 −0.329296
$$431$$ 14.4853 0.697731 0.348866 0.937173i $$-0.386567\pi$$
0.348866 + 0.937173i $$0.386567\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −19.0000 −0.913082 −0.456541 0.889702i $$-0.650912\pi$$
−0.456541 + 0.889702i $$0.650912\pi$$
$$434$$ 0 0
$$435$$ −7.07107 −0.339032
$$436$$ 1.65685 0.0793489
$$437$$ −27.2132 −1.30178
$$438$$ −1.07107 −0.0511776
$$439$$ −10.5858 −0.505232 −0.252616 0.967567i $$-0.581291\pi$$
−0.252616 + 0.967567i $$0.581291\pi$$
$$440$$ −0.585786 −0.0279263
$$441$$ 0 0
$$442$$ 5.24264 0.249367
$$443$$ 22.7279 1.07984 0.539918 0.841718i $$-0.318455\pi$$
0.539918 + 0.841718i $$0.318455\pi$$
$$444$$ −5.07107 −0.240662
$$445$$ 8.82843 0.418508
$$446$$ 21.3848 1.01260
$$447$$ 15.8995 0.752020
$$448$$ 0 0
$$449$$ −33.9411 −1.60178 −0.800890 0.598811i $$-0.795640\pi$$
−0.800890 + 0.598811i $$0.795640\pi$$
$$450$$ −3.00000 −0.141421
$$451$$ 1.89949 0.0894437
$$452$$ −6.07107 −0.285559
$$453$$ −7.58579 −0.356411
$$454$$ −26.1421 −1.22691
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ −15.2132 −0.711644 −0.355822 0.934554i $$-0.615799\pi$$
−0.355822 + 0.934554i $$0.615799\pi$$
$$458$$ −7.17157 −0.335106
$$459$$ 5.24264 0.244706
$$460$$ 12.8284 0.598128
$$461$$ −33.4558 −1.55819 −0.779097 0.626903i $$-0.784322\pi$$
−0.779097 + 0.626903i $$0.784322\pi$$
$$462$$ 0 0
$$463$$ −37.9411 −1.76327 −0.881637 0.471929i $$-0.843558\pi$$
−0.881637 + 0.471929i $$0.843558\pi$$
$$464$$ 5.00000 0.232119
$$465$$ −9.65685 −0.447826
$$466$$ −3.72792 −0.172693
$$467$$ −0.242641 −0.0112281 −0.00561404 0.999984i $$-0.501787\pi$$
−0.00561404 + 0.999984i $$0.501787\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 0 0
$$470$$ −12.7279 −0.587095
$$471$$ 7.92893 0.365346
$$472$$ 7.58579 0.349164
$$473$$ 2.00000 0.0919601
$$474$$ 13.3137 0.611519
$$475$$ −9.00000 −0.412948
$$476$$ 0 0
$$477$$ 12.3137 0.563806
$$478$$ −2.51472 −0.115021
$$479$$ −31.1421 −1.42292 −0.711460 0.702726i $$-0.751966\pi$$
−0.711460 + 0.702726i $$0.751966\pi$$
$$480$$ −1.41421 −0.0645497
$$481$$ −5.07107 −0.231221
$$482$$ −16.4853 −0.750884
$$483$$ 0 0
$$484$$ −10.8284 −0.492201
$$485$$ −11.1716 −0.507275
$$486$$ 1.00000 0.0453609
$$487$$ −23.7279 −1.07521 −0.537607 0.843195i $$-0.680672\pi$$
−0.537607 + 0.843195i $$0.680672\pi$$
$$488$$ −5.24264 −0.237323
$$489$$ 17.1421 0.775194
$$490$$ 0 0
$$491$$ −26.2843 −1.18619 −0.593096 0.805132i $$-0.702095\pi$$
−0.593096 + 0.805132i $$0.702095\pi$$
$$492$$ 4.58579 0.206743
$$493$$ 26.2132 1.18058
$$494$$ 3.00000 0.134976
$$495$$ −0.585786 −0.0263291
$$496$$ 6.82843 0.306605
$$497$$ 0 0
$$498$$ −7.65685 −0.343112
$$499$$ −30.0000 −1.34298 −0.671492 0.741012i $$-0.734346\pi$$
−0.671492 + 0.741012i $$0.734346\pi$$
$$500$$ 11.3137 0.505964
$$501$$ 12.3137 0.550136
$$502$$ 12.8284 0.572561
$$503$$ 35.7990 1.59620 0.798099 0.602526i $$-0.205839\pi$$
0.798099 + 0.602526i $$0.205839\pi$$
$$504$$ 0 0
$$505$$ 1.65685 0.0737290
$$506$$ −3.75736 −0.167035
$$507$$ 1.00000 0.0444116
$$508$$ 9.89949 0.439219
$$509$$ 12.9706 0.574910 0.287455 0.957794i $$-0.407191\pi$$
0.287455 + 0.957794i $$0.407191\pi$$
$$510$$ −7.41421 −0.328307
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 3.00000 0.132453
$$514$$ −21.7990 −0.961512
$$515$$ −9.17157 −0.404148
$$516$$ 4.82843 0.212560
$$517$$ 3.72792 0.163954
$$518$$ 0 0
$$519$$ 20.3137 0.891673
$$520$$ −1.41421 −0.0620174
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 5.00000 0.218844
$$523$$ −3.85786 −0.168693 −0.0843463 0.996437i $$-0.526880\pi$$
−0.0843463 + 0.996437i $$0.526880\pi$$
$$524$$ 9.89949 0.432461
$$525$$ 0 0
$$526$$ −7.55635 −0.329472
$$527$$ 35.7990 1.55943
$$528$$ 0.414214 0.0180263
$$529$$ 59.2843 2.57758
$$530$$ −17.4142 −0.756425
$$531$$ 7.58579 0.329195
$$532$$ 0 0
$$533$$ 4.58579 0.198632
$$534$$ −6.24264 −0.270145
$$535$$ −19.3137 −0.835004
$$536$$ −14.6569 −0.633080
$$537$$ −12.3431 −0.532646
$$538$$ −3.68629 −0.158927
$$539$$ 0 0
$$540$$ −1.41421 −0.0608581
$$541$$ 12.8701 0.553327 0.276663 0.960967i $$-0.410771\pi$$
0.276663 + 0.960967i $$0.410771\pi$$
$$542$$ −15.3848 −0.660833
$$543$$ −11.5858 −0.497194
$$544$$ 5.24264 0.224776
$$545$$ −2.34315 −0.100369
$$546$$ 0 0
$$547$$ −5.75736 −0.246167 −0.123083 0.992396i $$-0.539278\pi$$
−0.123083 + 0.992396i $$0.539278\pi$$
$$548$$ −13.0711 −0.558368
$$549$$ −5.24264 −0.223750
$$550$$ −1.24264 −0.0529864
$$551$$ 15.0000 0.639021
$$552$$ −9.07107 −0.386090
$$553$$ 0 0
$$554$$ −16.4142 −0.697373
$$555$$ 7.17157 0.304416
$$556$$ −21.2132 −0.899640
$$557$$ −10.9706 −0.464838 −0.232419 0.972616i $$-0.574664\pi$$
−0.232419 + 0.972616i $$0.574664\pi$$
$$558$$ 6.82843 0.289070
$$559$$ 4.82843 0.204221
$$560$$ 0 0
$$561$$ 2.17157 0.0916839
$$562$$ 7.31371 0.308510
$$563$$ −0.485281 −0.0204522 −0.0102261 0.999948i $$-0.503255\pi$$
−0.0102261 + 0.999948i $$0.503255\pi$$
$$564$$ 9.00000 0.378968
$$565$$ 8.58579 0.361207
$$566$$ 17.0711 0.717551
$$567$$ 0 0
$$568$$ −1.00000 −0.0419591
$$569$$ −25.5858 −1.07261 −0.536306 0.844024i $$-0.680181\pi$$
−0.536306 + 0.844024i $$0.680181\pi$$
$$570$$ −4.24264 −0.177705
$$571$$ −1.65685 −0.0693372 −0.0346686 0.999399i $$-0.511038\pi$$
−0.0346686 + 0.999399i $$0.511038\pi$$
$$572$$ 0.414214 0.0173191
$$573$$ 2.34315 0.0978863
$$574$$ 0 0
$$575$$ 27.2132 1.13487
$$576$$ 1.00000 0.0416667
$$577$$ −18.0000 −0.749350 −0.374675 0.927156i $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ 10.4853 0.436130
$$579$$ −15.7574 −0.654854
$$580$$ −7.07107 −0.293610
$$581$$ 0 0
$$582$$ 7.89949 0.327445
$$583$$ 5.10051 0.211241
$$584$$ −1.07107 −0.0443211
$$585$$ −1.41421 −0.0584705
$$586$$ −18.7279 −0.773643
$$587$$ 11.2426 0.464033 0.232017 0.972712i $$-0.425468\pi$$
0.232017 + 0.972712i $$0.425468\pi$$
$$588$$ 0 0
$$589$$ 20.4853 0.844081
$$590$$ −10.7279 −0.441662
$$591$$ 9.55635 0.393096
$$592$$ −5.07107 −0.208420
$$593$$ −28.5858 −1.17388 −0.586939 0.809631i $$-0.699667\pi$$
−0.586939 + 0.809631i $$0.699667\pi$$
$$594$$ 0.414214 0.0169954
$$595$$ 0 0
$$596$$ 15.8995 0.651269
$$597$$ 10.5858 0.433247
$$598$$ −9.07107 −0.370944
$$599$$ −14.8701 −0.607574 −0.303787 0.952740i $$-0.598251\pi$$
−0.303787 + 0.952740i $$0.598251\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ −17.8284 −0.727237 −0.363618 0.931548i $$-0.618459\pi$$
−0.363618 + 0.931548i $$0.618459\pi$$
$$602$$ 0 0
$$603$$ −14.6569 −0.596873
$$604$$ −7.58579 −0.308661
$$605$$ 15.3137 0.622591
$$606$$ −1.17157 −0.0475919
$$607$$ 27.0711 1.09878 0.549390 0.835566i $$-0.314860\pi$$
0.549390 + 0.835566i $$0.314860\pi$$
$$608$$ 3.00000 0.121666
$$609$$ 0 0
$$610$$ 7.41421 0.300193
$$611$$ 9.00000 0.364101
$$612$$ 5.24264 0.211921
$$613$$ −35.7990 −1.44591 −0.722954 0.690896i $$-0.757216\pi$$
−0.722954 + 0.690896i $$0.757216\pi$$
$$614$$ −19.6274 −0.792098
$$615$$ −6.48528 −0.261512
$$616$$ 0 0
$$617$$ −6.48528 −0.261088 −0.130544 0.991443i $$-0.541672\pi$$
−0.130544 + 0.991443i $$0.541672\pi$$
$$618$$ 6.48528 0.260876
$$619$$ 43.4558 1.74664 0.873319 0.487149i $$-0.161963\pi$$
0.873319 + 0.487149i $$0.161963\pi$$
$$620$$ −9.65685 −0.387829
$$621$$ −9.07107 −0.364009
$$622$$ −6.58579 −0.264066
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ −1.00000 −0.0400000
$$626$$ 18.1421 0.725106
$$627$$ 1.24264 0.0496263
$$628$$ 7.92893 0.316399
$$629$$ −26.5858 −1.06004
$$630$$ 0 0
$$631$$ 36.6274 1.45811 0.729057 0.684453i $$-0.239959\pi$$
0.729057 + 0.684453i $$0.239959\pi$$
$$632$$ 13.3137 0.529591
$$633$$ −28.3848 −1.12819
$$634$$ 0.828427 0.0329010
$$635$$ −14.0000 −0.555573
$$636$$ 12.3137 0.488270
$$637$$ 0 0
$$638$$ 2.07107 0.0819944
$$639$$ −1.00000 −0.0395594
$$640$$ −1.41421 −0.0559017
$$641$$ 33.1716 1.31020 0.655099 0.755543i $$-0.272627\pi$$
0.655099 + 0.755543i $$0.272627\pi$$
$$642$$ 13.6569 0.538993
$$643$$ 36.3137 1.43207 0.716036 0.698063i $$-0.245954\pi$$
0.716036 + 0.698063i $$0.245954\pi$$
$$644$$ 0 0
$$645$$ −6.82843 −0.268869
$$646$$ 15.7279 0.618807
$$647$$ 28.6274 1.12546 0.562730 0.826641i $$-0.309751\pi$$
0.562730 + 0.826641i $$0.309751\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 3.14214 0.123340
$$650$$ −3.00000 −0.117670
$$651$$ 0 0
$$652$$ 17.1421 0.671338
$$653$$ 15.1127 0.591406 0.295703 0.955280i $$-0.404446\pi$$
0.295703 + 0.955280i $$0.404446\pi$$
$$654$$ 1.65685 0.0647881
$$655$$ −14.0000 −0.547025
$$656$$ 4.58579 0.179045
$$657$$ −1.07107 −0.0417863
$$658$$ 0 0
$$659$$ 30.7279 1.19699 0.598495 0.801127i $$-0.295766\pi$$
0.598495 + 0.801127i $$0.295766\pi$$
$$660$$ −0.585786 −0.0228017
$$661$$ −13.5563 −0.527281 −0.263640 0.964621i $$-0.584923\pi$$
−0.263640 + 0.964621i $$0.584923\pi$$
$$662$$ −22.4853 −0.873915
$$663$$ 5.24264 0.203607
$$664$$ −7.65685 −0.297144
$$665$$ 0 0
$$666$$ −5.07107 −0.196500
$$667$$ −45.3553 −1.75617
$$668$$ 12.3137 0.476432
$$669$$ 21.3848 0.826783
$$670$$ 20.7279 0.800789
$$671$$ −2.17157 −0.0838326
$$672$$ 0 0
$$673$$ −1.17157 −0.0451608 −0.0225804 0.999745i $$-0.507188\pi$$
−0.0225804 + 0.999745i $$0.507188\pi$$
$$674$$ 28.6569 1.10382
$$675$$ −3.00000 −0.115470
$$676$$ 1.00000 0.0384615
$$677$$ 28.1716 1.08272 0.541361 0.840790i $$-0.317909\pi$$
0.541361 + 0.840790i $$0.317909\pi$$
$$678$$ −6.07107 −0.233158
$$679$$ 0 0
$$680$$ −7.41421 −0.284322
$$681$$ −26.1421 −1.00177
$$682$$ 2.82843 0.108306
$$683$$ 5.45584 0.208762 0.104381 0.994537i $$-0.466714\pi$$
0.104381 + 0.994537i $$0.466714\pi$$
$$684$$ 3.00000 0.114708
$$685$$ 18.4853 0.706286
$$686$$ 0 0
$$687$$ −7.17157 −0.273613
$$688$$ 4.82843 0.184082
$$689$$ 12.3137 0.469115
$$690$$ 12.8284 0.488370
$$691$$ 11.1421 0.423867 0.211933 0.977284i $$-0.432024\pi$$
0.211933 + 0.977284i $$0.432024\pi$$
$$692$$ 20.3137 0.772211
$$693$$ 0 0
$$694$$ 5.75736 0.218546
$$695$$ 30.0000 1.13796
$$696$$ 5.00000 0.189525
$$697$$ 24.0416 0.910642
$$698$$ 5.41421 0.204931
$$699$$ −3.72792 −0.141003
$$700$$ 0 0
$$701$$ 48.4853 1.83126 0.915632 0.402018i $$-0.131691\pi$$
0.915632 + 0.402018i $$0.131691\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ −15.2132 −0.573777
$$704$$ 0.414214 0.0156113
$$705$$ −12.7279 −0.479361
$$706$$ 12.6274 0.475239
$$707$$ 0 0
$$708$$ 7.58579 0.285091
$$709$$ −32.0416 −1.20335 −0.601674 0.798741i $$-0.705499\pi$$
−0.601674 + 0.798741i $$0.705499\pi$$
$$710$$ 1.41421 0.0530745
$$711$$ 13.3137 0.499303
$$712$$ −6.24264 −0.233953
$$713$$ −61.9411 −2.31971
$$714$$ 0 0
$$715$$ −0.585786 −0.0219072
$$716$$ −12.3431 −0.461285
$$717$$ −2.51472 −0.0939139
$$718$$ −15.4558 −0.576807
$$719$$ −22.2843 −0.831063 −0.415532 0.909579i $$-0.636404\pi$$
−0.415532 + 0.909579i $$0.636404\pi$$
$$720$$ −1.41421 −0.0527046
$$721$$ 0 0
$$722$$ −10.0000 −0.372161
$$723$$ −16.4853 −0.613094
$$724$$ −11.5858 −0.430582
$$725$$ −15.0000 −0.557086
$$726$$ −10.8284 −0.401881
$$727$$ −38.8284 −1.44007 −0.720033 0.693939i $$-0.755874\pi$$
−0.720033 + 0.693939i $$0.755874\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 1.51472 0.0560623
$$731$$ 25.3137 0.936261
$$732$$ −5.24264 −0.193774
$$733$$ −29.0122 −1.07159 −0.535795 0.844348i $$-0.679988\pi$$
−0.535795 + 0.844348i $$0.679988\pi$$
$$734$$ 14.0000 0.516749
$$735$$ 0 0
$$736$$ −9.07107 −0.334364
$$737$$ −6.07107 −0.223631
$$738$$ 4.58579 0.168805
$$739$$ 6.97056 0.256416 0.128208 0.991747i $$-0.459077\pi$$
0.128208 + 0.991747i $$0.459077\pi$$
$$740$$ 7.17157 0.263632
$$741$$ 3.00000 0.110208
$$742$$ 0 0
$$743$$ 11.4853 0.421354 0.210677 0.977556i $$-0.432433\pi$$
0.210677 + 0.977556i $$0.432433\pi$$
$$744$$ 6.82843 0.250342
$$745$$ −22.4853 −0.823797
$$746$$ −11.1005 −0.406418
$$747$$ −7.65685 −0.280150
$$748$$ 2.17157 0.0794006
$$749$$ 0 0
$$750$$ 11.3137 0.413118
$$751$$ −34.3431 −1.25320 −0.626600 0.779341i $$-0.715554\pi$$
−0.626600 + 0.779341i $$0.715554\pi$$
$$752$$ 9.00000 0.328196
$$753$$ 12.8284 0.467494
$$754$$ 5.00000 0.182089
$$755$$ 10.7279 0.390429
$$756$$ 0 0
$$757$$ 23.4437 0.852074 0.426037 0.904706i $$-0.359909\pi$$
0.426037 + 0.904706i $$0.359909\pi$$
$$758$$ −33.6569 −1.22247
$$759$$ −3.75736 −0.136384
$$760$$ −4.24264 −0.153897
$$761$$ −17.8579 −0.647347 −0.323674 0.946169i $$-0.604918\pi$$
−0.323674 + 0.946169i $$0.604918\pi$$
$$762$$ 9.89949 0.358621
$$763$$ 0 0
$$764$$ 2.34315 0.0847720
$$765$$ −7.41421 −0.268061
$$766$$ 6.48528 0.234323
$$767$$ 7.58579 0.273907
$$768$$ 1.00000 0.0360844
$$769$$ −6.92893 −0.249864 −0.124932 0.992165i $$-0.539871\pi$$
−0.124932 + 0.992165i $$0.539871\pi$$
$$770$$ 0 0
$$771$$ −21.7990 −0.785071
$$772$$ −15.7574 −0.567120
$$773$$ −29.5980 −1.06457 −0.532283 0.846567i $$-0.678666\pi$$
−0.532283 + 0.846567i $$0.678666\pi$$
$$774$$ 4.82843 0.173554
$$775$$ −20.4853 −0.735853
$$776$$ 7.89949 0.283575
$$777$$ 0 0
$$778$$ 23.0000 0.824590
$$779$$ 13.7574 0.492909
$$780$$ −1.41421 −0.0506370
$$781$$ −0.414214 −0.0148217
$$782$$ −47.5563 −1.70061
$$783$$ 5.00000 0.178685
$$784$$ 0 0
$$785$$ −11.2132 −0.400216
$$786$$ 9.89949 0.353103
$$787$$ −8.17157 −0.291285 −0.145643 0.989337i $$-0.546525\pi$$
−0.145643 + 0.989337i $$0.546525\pi$$
$$788$$ 9.55635 0.340431
$$789$$ −7.55635 −0.269013
$$790$$ −18.8284 −0.669885
$$791$$ 0 0
$$792$$ 0.414214 0.0147184
$$793$$ −5.24264 −0.186172
$$794$$ −19.7574 −0.701162
$$795$$ −17.4142 −0.617619
$$796$$ 10.5858 0.375203
$$797$$ −36.9706 −1.30956 −0.654782 0.755818i $$-0.727240\pi$$
−0.654782 + 0.755818i $$0.727240\pi$$
$$798$$ 0 0
$$799$$ 47.1838 1.66924
$$800$$ −3.00000 −0.106066
$$801$$ −6.24264 −0.220573
$$802$$ 19.0711 0.673423
$$803$$ −0.443651 −0.0156561
$$804$$ −14.6569 −0.516907
$$805$$ 0 0
$$806$$ 6.82843 0.240521
$$807$$ −3.68629 −0.129764
$$808$$ −1.17157 −0.0412158
$$809$$ 35.0416 1.23200 0.615999 0.787747i $$-0.288753\pi$$
0.615999 + 0.787747i $$0.288753\pi$$
$$810$$ −1.41421 −0.0496904
$$811$$ 26.7696 0.940006 0.470003 0.882665i $$-0.344253\pi$$
0.470003 + 0.882665i $$0.344253\pi$$
$$812$$ 0 0
$$813$$ −15.3848 −0.539568
$$814$$ −2.10051 −0.0736227
$$815$$ −24.2426 −0.849183
$$816$$ 5.24264 0.183529
$$817$$ 14.4853 0.506776
$$818$$ −2.58579 −0.0904099
$$819$$ 0 0
$$820$$ −6.48528 −0.226476
$$821$$ −9.61522 −0.335574 −0.167787 0.985823i $$-0.553662\pi$$
−0.167787 + 0.985823i $$0.553662\pi$$
$$822$$ −13.0711 −0.455906
$$823$$ −8.10051 −0.282366 −0.141183 0.989984i $$-0.545091\pi$$
−0.141183 + 0.989984i $$0.545091\pi$$
$$824$$ 6.48528 0.225925
$$825$$ −1.24264 −0.0432632
$$826$$ 0 0
$$827$$ 13.5269 0.470377 0.235188 0.971950i $$-0.424429\pi$$
0.235188 + 0.971950i $$0.424429\pi$$
$$828$$ −9.07107 −0.315241
$$829$$ 18.6985 0.649425 0.324713 0.945813i $$-0.394732\pi$$
0.324713 + 0.945813i $$0.394732\pi$$
$$830$$ 10.8284 0.375860
$$831$$ −16.4142 −0.569403
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ −21.2132 −0.734553
$$835$$ −17.4142 −0.602644
$$836$$ 1.24264 0.0429776
$$837$$ 6.82843 0.236025
$$838$$ −27.4558 −0.948446
$$839$$ 2.51472 0.0868177 0.0434089 0.999057i $$-0.486178\pi$$
0.0434089 + 0.999057i $$0.486178\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 10.5858 0.364810
$$843$$ 7.31371 0.251898
$$844$$ −28.3848 −0.977044
$$845$$ −1.41421 −0.0486504
$$846$$ 9.00000 0.309426
$$847$$ 0 0
$$848$$ 12.3137 0.422854
$$849$$ 17.0711 0.585878
$$850$$ −15.7279 −0.539463
$$851$$ 46.0000 1.57686
$$852$$ −1.00000 −0.0342594
$$853$$ −17.6985 −0.605985 −0.302992 0.952993i $$-0.597986\pi$$
−0.302992 + 0.952993i $$0.597986\pi$$
$$854$$ 0 0
$$855$$ −4.24264 −0.145095
$$856$$ 13.6569 0.466782
$$857$$ −51.3848 −1.75527 −0.877635 0.479329i $$-0.840880\pi$$
−0.877635 + 0.479329i $$0.840880\pi$$
$$858$$ 0.414214 0.0141410
$$859$$ 0.201010 0.00685838 0.00342919 0.999994i $$-0.498908\pi$$
0.00342919 + 0.999994i $$0.498908\pi$$
$$860$$ −6.82843 −0.232847
$$861$$ 0 0
$$862$$ 14.4853 0.493371
$$863$$ 4.00000 0.136162 0.0680808 0.997680i $$-0.478312\pi$$
0.0680808 + 0.997680i $$0.478312\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −28.7279 −0.976779
$$866$$ −19.0000 −0.645646
$$867$$ 10.4853 0.356099
$$868$$ 0 0
$$869$$ 5.51472 0.187074
$$870$$ −7.07107 −0.239732
$$871$$ −14.6569 −0.496629
$$872$$ 1.65685 0.0561082
$$873$$ 7.89949 0.267357
$$874$$ −27.2132 −0.920500
$$875$$ 0 0
$$876$$ −1.07107 −0.0361880
$$877$$ 23.4142 0.790642 0.395321 0.918543i $$-0.370633\pi$$
0.395321 + 0.918543i $$0.370633\pi$$
$$878$$ −10.5858 −0.357253
$$879$$ −18.7279 −0.631677
$$880$$ −0.585786 −0.0197469
$$881$$ −41.1127 −1.38512 −0.692561 0.721359i $$-0.743518\pi$$
−0.692561 + 0.721359i $$0.743518\pi$$
$$882$$ 0 0
$$883$$ −0.928932 −0.0312611 −0.0156305 0.999878i $$-0.504976\pi$$
−0.0156305 + 0.999878i $$0.504976\pi$$
$$884$$ 5.24264 0.176329
$$885$$ −10.7279 −0.360615
$$886$$ 22.7279 0.763559
$$887$$ 41.3137 1.38718 0.693589 0.720371i $$-0.256028\pi$$
0.693589 + 0.720371i $$0.256028\pi$$
$$888$$ −5.07107 −0.170174
$$889$$ 0 0
$$890$$ 8.82843 0.295930
$$891$$ 0.414214 0.0138767
$$892$$ 21.3848 0.716015
$$893$$ 27.0000 0.903521
$$894$$ 15.8995 0.531759
$$895$$ 17.4558 0.583485
$$896$$ 0 0
$$897$$ −9.07107 −0.302874
$$898$$ −33.9411 −1.13263
$$899$$ 34.1421 1.13870
$$900$$ −3.00000 −0.100000
$$901$$ 64.5563 2.15068
$$902$$ 1.89949 0.0632463
$$903$$ 0 0
$$904$$ −6.07107 −0.201921
$$905$$ 16.3848 0.544648
$$906$$ −7.58579 −0.252021
$$907$$ −32.1005 −1.06588 −0.532940 0.846153i $$-0.678913\pi$$
−0.532940 + 0.846153i $$0.678913\pi$$
$$908$$ −26.1421 −0.867557
$$909$$ −1.17157 −0.0388586
$$910$$ 0 0
$$911$$ 54.0000 1.78910 0.894550 0.446968i $$-0.147496\pi$$
0.894550 + 0.446968i $$0.147496\pi$$
$$912$$ 3.00000 0.0993399
$$913$$ −3.17157 −0.104964
$$914$$ −15.2132 −0.503208
$$915$$ 7.41421 0.245106
$$916$$ −7.17157 −0.236955
$$917$$ 0 0
$$918$$ 5.24264 0.173033
$$919$$ −25.7990 −0.851030 −0.425515 0.904951i $$-0.639907\pi$$
−0.425515 + 0.904951i $$0.639907\pi$$
$$920$$ 12.8284 0.422941
$$921$$ −19.6274 −0.646745
$$922$$ −33.4558 −1.10181
$$923$$ −1.00000 −0.0329154
$$924$$ 0 0
$$925$$ 15.2132 0.500207
$$926$$ −37.9411 −1.24682
$$927$$ 6.48528 0.213005
$$928$$ 5.00000 0.164133
$$929$$ 18.2843 0.599887 0.299944 0.953957i $$-0.403032\pi$$
0.299944 + 0.953957i $$0.403032\pi$$
$$930$$ −9.65685 −0.316661
$$931$$ 0 0
$$932$$ −3.72792 −0.122112
$$933$$ −6.58579 −0.215609
$$934$$ −0.242641 −0.00793945
$$935$$ −3.07107 −0.100435
$$936$$ 1.00000 0.0326860
$$937$$ 40.5980 1.32628 0.663139 0.748496i $$-0.269224\pi$$
0.663139 + 0.748496i $$0.269224\pi$$
$$938$$ 0 0
$$939$$ 18.1421 0.592046
$$940$$ −12.7279 −0.415139
$$941$$ −20.9706 −0.683621 −0.341810 0.939769i $$-0.611040\pi$$
−0.341810 + 0.939769i $$0.611040\pi$$
$$942$$ 7.92893 0.258339
$$943$$ −41.5980 −1.35462
$$944$$ 7.58579 0.246896
$$945$$ 0 0
$$946$$ 2.00000 0.0650256
$$947$$ 17.2426 0.560311 0.280155 0.959955i $$-0.409614\pi$$
0.280155 + 0.959955i $$0.409614\pi$$
$$948$$ 13.3137 0.432409
$$949$$ −1.07107 −0.0347683
$$950$$ −9.00000 −0.291999
$$951$$ 0.828427 0.0268636
$$952$$ 0 0
$$953$$ −10.2132 −0.330838 −0.165419 0.986223i $$-0.552898\pi$$
−0.165419 + 0.986223i $$0.552898\pi$$
$$954$$ 12.3137 0.398671
$$955$$ −3.31371 −0.107229
$$956$$ −2.51472 −0.0813318
$$957$$ 2.07107 0.0669481
$$958$$ −31.1421 −1.00616
$$959$$ 0 0
$$960$$ −1.41421 −0.0456435
$$961$$ 15.6274 0.504110
$$962$$ −5.07107 −0.163498
$$963$$ 13.6569 0.440086
$$964$$ −16.4853 −0.530955
$$965$$ 22.2843 0.717356
$$966$$ 0 0
$$967$$ 9.38478 0.301794 0.150897 0.988549i $$-0.451784\pi$$
0.150897 + 0.988549i $$0.451784\pi$$
$$968$$ −10.8284 −0.348039
$$969$$ 15.7279 0.505254
$$970$$ −11.1716 −0.358698
$$971$$ 57.5563 1.84707 0.923536 0.383513i $$-0.125286\pi$$
0.923536 + 0.383513i $$0.125286\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −23.7279 −0.760292
$$975$$ −3.00000 −0.0960769
$$976$$ −5.24264 −0.167813
$$977$$ 2.48528 0.0795112 0.0397556 0.999209i $$-0.487342\pi$$
0.0397556 + 0.999209i $$0.487342\pi$$
$$978$$ 17.1421 0.548145
$$979$$ −2.58579 −0.0826421
$$980$$ 0 0
$$981$$ 1.65685 0.0528993
$$982$$ −26.2843 −0.838765
$$983$$ −22.1716 −0.707163 −0.353582 0.935404i $$-0.615036\pi$$
−0.353582 + 0.935404i $$0.615036\pi$$
$$984$$ 4.58579 0.146190
$$985$$ −13.5147 −0.430615
$$986$$ 26.2132 0.834798
$$987$$ 0 0
$$988$$ 3.00000 0.0954427
$$989$$ −43.7990 −1.39273
$$990$$ −0.585786 −0.0186175
$$991$$ −7.31371 −0.232328 −0.116164 0.993230i $$-0.537060\pi$$
−0.116164 + 0.993230i $$0.537060\pi$$
$$992$$ 6.82843 0.216803
$$993$$ −22.4853 −0.713549
$$994$$ 0 0
$$995$$ −14.9706 −0.474599
$$996$$ −7.65685 −0.242617
$$997$$ 6.69848 0.212143 0.106072 0.994358i $$-0.466173\pi$$
0.106072 + 0.994358i $$0.466173\pi$$
$$998$$ −30.0000 −0.949633
$$999$$ −5.07107 −0.160441
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 3822.2.a.bs.1.1 2
7.2 even 3 546.2.i.h.235.2 yes 4
7.4 even 3 546.2.i.h.79.2 4
7.6 odd 2 3822.2.a.bp.1.2 2
21.2 odd 6 1638.2.j.n.235.1 4
21.11 odd 6 1638.2.j.n.1171.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.h.79.2 4 7.4 even 3
546.2.i.h.235.2 yes 4 7.2 even 3
1638.2.j.n.235.1 4 21.2 odd 6
1638.2.j.n.1171.1 4 21.11 odd 6
3822.2.a.bp.1.2 2 7.6 odd 2
3822.2.a.bs.1.1 2 1.1 even 1 trivial