# Properties

 Label 3822.2.a.bv.1.1 Level $3822$ Weight $2$ Character 3822.1 Self dual yes Analytic conductor $30.519$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.2700.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 15x - 20$$ x^3 - 15*x - 20 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 $$4.41883$$ 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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.41883 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.41883 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.41883 q^{10} -4.73042 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.41883 q^{15} +1.00000 q^{16} +7.79567 q^{17} -1.00000 q^{18} +2.68842 q^{19} -3.41883 q^{20} +4.73042 q^{22} -3.73042 q^{23} +1.00000 q^{24} +6.68842 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{29} -3.41883 q^{30} -5.68842 q^{31} -1.00000 q^{32} +4.73042 q^{33} -7.79567 q^{34} +1.00000 q^{36} -3.10725 q^{37} -2.68842 q^{38} -1.00000 q^{39} +3.41883 q^{40} +0.892750 q^{41} -12.8377 q^{43} -4.73042 q^{44} -3.41883 q^{45} +3.73042 q^{46} -10.1492 q^{47} -1.00000 q^{48} -6.68842 q^{50} -7.79567 q^{51} +1.00000 q^{52} -9.21450 q^{53} +1.00000 q^{54} +16.1725 q^{55} -2.68842 q^{57} +3.00000 q^{58} -6.10725 q^{59} +3.41883 q^{60} +3.04200 q^{61} +5.68842 q^{62} +1.00000 q^{64} -3.41883 q^{65} -4.73042 q^{66} -0.0652506 q^{67} +7.79567 q^{68} +3.73042 q^{69} -12.6884 q^{71} -1.00000 q^{72} +11.7304 q^{73} +3.10725 q^{74} -6.68842 q^{75} +2.68842 q^{76} +1.00000 q^{78} +14.5261 q^{79} -3.41883 q^{80} +1.00000 q^{81} -0.892750 q^{82} +0.934749 q^{83} -26.6521 q^{85} +12.8377 q^{86} +3.00000 q^{87} +4.73042 q^{88} +15.1072 q^{89} +3.41883 q^{90} -3.73042 q^{92} +5.68842 q^{93} +10.1492 q^{94} -9.19125 q^{95} +1.00000 q^{96} -4.79567 q^{97} -4.73042 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 - 3 * q^8 + 3 * q^9 $$3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 3 q^{12} + 3 q^{13} - 3 q^{15} + 3 q^{16} + 6 q^{17} - 3 q^{18} + 6 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{24} + 18 q^{25} - 3 q^{26} - 3 q^{27} - 9 q^{29} + 3 q^{30} - 15 q^{31} - 3 q^{32} + 3 q^{33} - 6 q^{34} + 3 q^{36} + 6 q^{37} - 6 q^{38} - 3 q^{39} - 3 q^{40} + 18 q^{41} - 12 q^{43} - 3 q^{44} + 3 q^{45} - 6 q^{47} - 3 q^{48} - 18 q^{50} - 6 q^{51} + 3 q^{52} + 3 q^{53} + 3 q^{54} + 27 q^{55} - 6 q^{57} + 9 q^{58} - 3 q^{59} - 3 q^{60} + 15 q^{62} + 3 q^{64} + 3 q^{65} - 3 q^{66} + 6 q^{67} + 6 q^{68} - 36 q^{71} - 3 q^{72} + 24 q^{73} - 6 q^{74} - 18 q^{75} + 6 q^{76} + 3 q^{78} + 15 q^{79} + 3 q^{80} + 3 q^{81} - 18 q^{82} + 9 q^{83} - 24 q^{85} + 12 q^{86} + 9 q^{87} + 3 q^{88} + 30 q^{89} - 3 q^{90} + 15 q^{93} + 6 q^{94} + 6 q^{95} + 3 q^{96} + 3 q^{97} - 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 - 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^11 - 3 * q^12 + 3 * q^13 - 3 * q^15 + 3 * q^16 + 6 * q^17 - 3 * q^18 + 6 * q^19 + 3 * q^20 + 3 * q^22 + 3 * q^24 + 18 * q^25 - 3 * q^26 - 3 * q^27 - 9 * q^29 + 3 * q^30 - 15 * q^31 - 3 * q^32 + 3 * q^33 - 6 * q^34 + 3 * q^36 + 6 * q^37 - 6 * q^38 - 3 * q^39 - 3 * q^40 + 18 * q^41 - 12 * q^43 - 3 * q^44 + 3 * q^45 - 6 * q^47 - 3 * q^48 - 18 * q^50 - 6 * q^51 + 3 * q^52 + 3 * q^53 + 3 * q^54 + 27 * q^55 - 6 * q^57 + 9 * q^58 - 3 * q^59 - 3 * q^60 + 15 * q^62 + 3 * q^64 + 3 * q^65 - 3 * q^66 + 6 * q^67 + 6 * q^68 - 36 * q^71 - 3 * q^72 + 24 * q^73 - 6 * q^74 - 18 * q^75 + 6 * q^76 + 3 * q^78 + 15 * q^79 + 3 * q^80 + 3 * q^81 - 18 * q^82 + 9 * q^83 - 24 * q^85 + 12 * q^86 + 9 * q^87 + 3 * q^88 + 30 * q^89 - 3 * q^90 + 15 * q^93 + 6 * q^94 + 6 * q^95 + 3 * q^96 + 3 * q^97 - 3 * 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
$$5$$ −3.41883 −1.52895 −0.764474 0.644654i $$-0.777001\pi$$
−0.764474 + 0.644654i $$0.777001\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 3.41883 1.08113
$$11$$ −4.73042 −1.42627 −0.713137 0.701025i $$-0.752726\pi$$
−0.713137 + 0.701025i $$0.752726\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 3.41883 0.882739
$$16$$ 1.00000 0.250000
$$17$$ 7.79567 1.89073 0.945363 0.326018i $$-0.105707\pi$$
0.945363 + 0.326018i $$0.105707\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 2.68842 0.616765 0.308383 0.951262i $$-0.400212\pi$$
0.308383 + 0.951262i $$0.400212\pi$$
$$20$$ −3.41883 −0.764474
$$21$$ 0 0
$$22$$ 4.73042 1.00853
$$23$$ −3.73042 −0.777845 −0.388923 0.921270i $$-0.627153\pi$$
−0.388923 + 0.921270i $$0.627153\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 6.68842 1.33768
$$26$$ −1.00000 −0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ −3.41883 −0.624191
$$31$$ −5.68842 −1.02167 −0.510835 0.859679i $$-0.670664\pi$$
−0.510835 + 0.859679i $$0.670664\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.73042 0.823460
$$34$$ −7.79567 −1.33695
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −3.10725 −0.510829 −0.255414 0.966832i $$-0.582212\pi$$
−0.255414 + 0.966832i $$0.582212\pi$$
$$38$$ −2.68842 −0.436119
$$39$$ −1.00000 −0.160128
$$40$$ 3.41883 0.540565
$$41$$ 0.892750 0.139424 0.0697121 0.997567i $$-0.477792\pi$$
0.0697121 + 0.997567i $$0.477792\pi$$
$$42$$ 0 0
$$43$$ −12.8377 −1.95773 −0.978863 0.204518i $$-0.934437\pi$$
−0.978863 + 0.204518i $$0.934437\pi$$
$$44$$ −4.73042 −0.713137
$$45$$ −3.41883 −0.509649
$$46$$ 3.73042 0.550020
$$47$$ −10.1492 −1.48042 −0.740210 0.672376i $$-0.765274\pi$$
−0.740210 + 0.672376i $$0.765274\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ −6.68842 −0.945885
$$51$$ −7.79567 −1.09161
$$52$$ 1.00000 0.138675
$$53$$ −9.21450 −1.26571 −0.632854 0.774271i $$-0.718117\pi$$
−0.632854 + 0.774271i $$0.718117\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 16.1725 2.18070
$$56$$ 0 0
$$57$$ −2.68842 −0.356090
$$58$$ 3.00000 0.393919
$$59$$ −6.10725 −0.795096 −0.397548 0.917581i $$-0.630139\pi$$
−0.397548 + 0.917581i $$0.630139\pi$$
$$60$$ 3.41883 0.441369
$$61$$ 3.04200 0.389488 0.194744 0.980854i $$-0.437612\pi$$
0.194744 + 0.980854i $$0.437612\pi$$
$$62$$ 5.68842 0.722430
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −3.41883 −0.424054
$$66$$ −4.73042 −0.582274
$$67$$ −0.0652506 −0.00797163 −0.00398581 0.999992i $$-0.501269\pi$$
−0.00398581 + 0.999992i $$0.501269\pi$$
$$68$$ 7.79567 0.945363
$$69$$ 3.73042 0.449089
$$70$$ 0 0
$$71$$ −12.6884 −1.50584 −0.752919 0.658113i $$-0.771355\pi$$
−0.752919 + 0.658113i $$0.771355\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 11.7304 1.37294 0.686471 0.727158i $$-0.259159\pi$$
0.686471 + 0.727158i $$0.259159\pi$$
$$74$$ 3.10725 0.361210
$$75$$ −6.68842 −0.772312
$$76$$ 2.68842 0.308383
$$77$$ 0 0
$$78$$ 1.00000 0.113228
$$79$$ 14.5261 1.63431 0.817156 0.576417i $$-0.195549\pi$$
0.817156 + 0.576417i $$0.195549\pi$$
$$80$$ −3.41883 −0.382237
$$81$$ 1.00000 0.111111
$$82$$ −0.892750 −0.0985878
$$83$$ 0.934749 0.102602 0.0513010 0.998683i $$-0.483663\pi$$
0.0513010 + 0.998683i $$0.483663\pi$$
$$84$$ 0 0
$$85$$ −26.6521 −2.89082
$$86$$ 12.8377 1.38432
$$87$$ 3.00000 0.321634
$$88$$ 4.73042 0.504264
$$89$$ 15.1072 1.60137 0.800683 0.599089i $$-0.204470\pi$$
0.800683 + 0.599089i $$0.204470\pi$$
$$90$$ 3.41883 0.360377
$$91$$ 0 0
$$92$$ −3.73042 −0.388923
$$93$$ 5.68842 0.589861
$$94$$ 10.1492 1.04682
$$95$$ −9.19125 −0.943002
$$96$$ 1.00000 0.102062
$$97$$ −4.79567 −0.486926 −0.243463 0.969910i $$-0.578283\pi$$
−0.243463 + 0.969910i $$0.578283\pi$$
$$98$$ 0 0
$$99$$ −4.73042 −0.475425
$$100$$ 6.68842 0.668842
$$101$$ 0.623166 0.0620074 0.0310037 0.999519i $$-0.490130\pi$$
0.0310037 + 0.999519i $$0.490130\pi$$
$$102$$ 7.79567 0.771886
$$103$$ 8.83767 0.870801 0.435401 0.900237i $$-0.356607\pi$$
0.435401 + 0.900237i $$0.356607\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 9.21450 0.894991
$$107$$ 5.77241 0.558040 0.279020 0.960285i $$-0.409990\pi$$
0.279020 + 0.960285i $$0.409990\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ −16.1725 −1.54199
$$111$$ 3.10725 0.294927
$$112$$ 0 0
$$113$$ 10.6333 1.00030 0.500150 0.865939i $$-0.333278\pi$$
0.500150 + 0.865939i $$0.333278\pi$$
$$114$$ 2.68842 0.251793
$$115$$ 12.7537 1.18929
$$116$$ −3.00000 −0.278543
$$117$$ 1.00000 0.0924500
$$118$$ 6.10725 0.562218
$$119$$ 0 0
$$120$$ −3.41883 −0.312095
$$121$$ 11.3768 1.03426
$$122$$ −3.04200 −0.275410
$$123$$ −0.892750 −0.0804966
$$124$$ −5.68842 −0.510835
$$125$$ −5.77241 −0.516300
$$126$$ 0 0
$$127$$ −6.17250 −0.547721 −0.273860 0.961769i $$-0.588301\pi$$
−0.273860 + 0.961769i $$0.588301\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 12.8377 1.13029
$$130$$ 3.41883 0.299851
$$131$$ −20.4710 −1.78856 −0.894280 0.447509i $$-0.852311\pi$$
−0.894280 + 0.447509i $$0.852311\pi$$
$$132$$ 4.73042 0.411730
$$133$$ 0 0
$$134$$ 0.0652506 0.00563679
$$135$$ 3.41883 0.294246
$$136$$ −7.79567 −0.668473
$$137$$ −11.1072 −0.948956 −0.474478 0.880267i $$-0.657363\pi$$
−0.474478 + 0.880267i $$0.657363\pi$$
$$138$$ −3.73042 −0.317554
$$139$$ −2.26958 −0.192504 −0.0962518 0.995357i $$-0.530685\pi$$
−0.0962518 + 0.995357i $$0.530685\pi$$
$$140$$ 0 0
$$141$$ 10.1492 0.854721
$$142$$ 12.6884 1.06479
$$143$$ −4.73042 −0.395577
$$144$$ 1.00000 0.0833333
$$145$$ 10.2565 0.851756
$$146$$ −11.7304 −0.970816
$$147$$ 0 0
$$148$$ −3.10725 −0.255414
$$149$$ −8.26958 −0.677471 −0.338735 0.940882i $$-0.609999\pi$$
−0.338735 + 0.940882i $$0.609999\pi$$
$$150$$ 6.68842 0.546107
$$151$$ −5.26958 −0.428833 −0.214416 0.976742i $$-0.568785\pi$$
−0.214416 + 0.976742i $$0.568785\pi$$
$$152$$ −2.68842 −0.218059
$$153$$ 7.79567 0.630242
$$154$$ 0 0
$$155$$ 19.4477 1.56208
$$156$$ −1.00000 −0.0800641
$$157$$ −11.2565 −0.898366 −0.449183 0.893440i $$-0.648285\pi$$
−0.449183 + 0.893440i $$0.648285\pi$$
$$158$$ −14.5261 −1.15563
$$159$$ 9.21450 0.730757
$$160$$ 3.41883 0.270282
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 16.3637 1.28171 0.640854 0.767663i $$-0.278580\pi$$
0.640854 + 0.767663i $$0.278580\pi$$
$$164$$ 0.892750 0.0697121
$$165$$ −16.1725 −1.25903
$$166$$ −0.934749 −0.0725506
$$167$$ 22.9869 1.77878 0.889390 0.457148i $$-0.151129\pi$$
0.889390 + 0.457148i $$0.151129\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 26.6521 2.04412
$$171$$ 2.68842 0.205588
$$172$$ −12.8377 −0.978863
$$173$$ 10.7724 0.819012 0.409506 0.912308i $$-0.365701\pi$$
0.409506 + 0.912308i $$0.365701\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ 0 0
$$176$$ −4.73042 −0.356569
$$177$$ 6.10725 0.459049
$$178$$ −15.1072 −1.13234
$$179$$ 19.0522 1.42403 0.712013 0.702166i $$-0.247784\pi$$
0.712013 + 0.702166i $$0.247784\pi$$
$$180$$ −3.41883 −0.254825
$$181$$ 13.7957 1.02542 0.512712 0.858561i $$-0.328641\pi$$
0.512712 + 0.858561i $$0.328641\pi$$
$$182$$ 0 0
$$183$$ −3.04200 −0.224871
$$184$$ 3.73042 0.275010
$$185$$ 10.6232 0.781031
$$186$$ −5.68842 −0.417095
$$187$$ −36.8767 −2.69669
$$188$$ −10.1492 −0.740210
$$189$$ 0 0
$$190$$ 9.19125 0.666803
$$191$$ 10.2985 0.745173 0.372587 0.927997i $$-0.378471\pi$$
0.372587 + 0.927997i $$0.378471\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 17.6333 1.26927 0.634637 0.772810i $$-0.281149\pi$$
0.634637 + 0.772810i $$0.281149\pi$$
$$194$$ 4.79567 0.344309
$$195$$ 3.41883 0.244828
$$196$$ 0 0
$$197$$ −8.26958 −0.589183 −0.294592 0.955623i $$-0.595184\pi$$
−0.294592 + 0.955623i $$0.595184\pi$$
$$198$$ 4.73042 0.336176
$$199$$ −5.02325 −0.356089 −0.178044 0.984022i $$-0.556977\pi$$
−0.178044 + 0.984022i $$0.556977\pi$$
$$200$$ −6.68842 −0.472942
$$201$$ 0.0652506 0.00460242
$$202$$ −0.623166 −0.0438458
$$203$$ 0 0
$$204$$ −7.79567 −0.545806
$$205$$ −3.05216 −0.213172
$$206$$ −8.83767 −0.615749
$$207$$ −3.73042 −0.259282
$$208$$ 1.00000 0.0693375
$$209$$ −12.7173 −0.879676
$$210$$ 0 0
$$211$$ 14.5681 1.00291 0.501454 0.865184i $$-0.332799\pi$$
0.501454 + 0.865184i $$0.332799\pi$$
$$212$$ −9.21450 −0.632854
$$213$$ 12.6884 0.869396
$$214$$ −5.77241 −0.394594
$$215$$ 43.8898 2.99326
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −4.00000 −0.270914
$$219$$ −11.7304 −0.792668
$$220$$ 16.1725 1.09035
$$221$$ 7.79567 0.524393
$$222$$ −3.10725 −0.208545
$$223$$ −10.3217 −0.691195 −0.345598 0.938383i $$-0.612324\pi$$
−0.345598 + 0.938383i $$0.612324\pi$$
$$224$$ 0 0
$$225$$ 6.68842 0.445894
$$226$$ −10.6333 −0.707319
$$227$$ 4.44208 0.294831 0.147416 0.989075i $$-0.452904\pi$$
0.147416 + 0.989075i $$0.452904\pi$$
$$228$$ −2.68842 −0.178045
$$229$$ −21.4608 −1.41817 −0.709086 0.705122i $$-0.750892\pi$$
−0.709086 + 0.705122i $$0.750892\pi$$
$$230$$ −12.7537 −0.840952
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ −7.66517 −0.502162 −0.251081 0.967966i $$-0.580786\pi$$
−0.251081 + 0.967966i $$0.580786\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 34.6986 2.26349
$$236$$ −6.10725 −0.397548
$$237$$ −14.5261 −0.943570
$$238$$ 0 0
$$239$$ −9.52608 −0.616191 −0.308096 0.951355i $$-0.599692\pi$$
−0.308096 + 0.951355i $$0.599692\pi$$
$$240$$ 3.41883 0.220685
$$241$$ −7.36375 −0.474341 −0.237170 0.971468i $$-0.576220\pi$$
−0.237170 + 0.971468i $$0.576220\pi$$
$$242$$ −11.3768 −0.731331
$$243$$ −1.00000 −0.0641500
$$244$$ 3.04200 0.194744
$$245$$ 0 0
$$246$$ 0.892750 0.0569197
$$247$$ 2.68842 0.171060
$$248$$ 5.68842 0.361215
$$249$$ −0.934749 −0.0592373
$$250$$ 5.77241 0.365080
$$251$$ 12.7406 0.804178 0.402089 0.915601i $$-0.368284\pi$$
0.402089 + 0.915601i $$0.368284\pi$$
$$252$$ 0 0
$$253$$ 17.6464 1.10942
$$254$$ 6.17250 0.387297
$$255$$ 26.6521 1.66902
$$256$$ 1.00000 0.0625000
$$257$$ −13.4608 −0.839664 −0.419832 0.907602i $$-0.637911\pi$$
−0.419832 + 0.907602i $$0.637911\pi$$
$$258$$ −12.8377 −0.799238
$$259$$ 0 0
$$260$$ −3.41883 −0.212027
$$261$$ −3.00000 −0.185695
$$262$$ 20.4710 1.26470
$$263$$ 24.0289 1.48169 0.740843 0.671678i $$-0.234426\pi$$
0.740843 + 0.671678i $$0.234426\pi$$
$$264$$ −4.73042 −0.291137
$$265$$ 31.5028 1.93520
$$266$$ 0 0
$$267$$ −15.1072 −0.924549
$$268$$ −0.0652506 −0.00398581
$$269$$ 15.9217 0.970761 0.485380 0.874303i $$-0.338681\pi$$
0.485380 + 0.874303i $$0.338681\pi$$
$$270$$ −3.41883 −0.208064
$$271$$ −19.6521 −1.19378 −0.596889 0.802324i $$-0.703597\pi$$
−0.596889 + 0.802324i $$0.703597\pi$$
$$272$$ 7.79567 0.472682
$$273$$ 0 0
$$274$$ 11.1072 0.671013
$$275$$ −31.6390 −1.90790
$$276$$ 3.73042 0.224545
$$277$$ −5.17250 −0.310785 −0.155393 0.987853i $$-0.549664\pi$$
−0.155393 + 0.987853i $$0.549664\pi$$
$$278$$ 2.26958 0.136121
$$279$$ −5.68842 −0.340557
$$280$$ 0 0
$$281$$ −24.4290 −1.45731 −0.728656 0.684880i $$-0.759855\pi$$
−0.728656 + 0.684880i $$0.759855\pi$$
$$282$$ −10.1492 −0.604379
$$283$$ −3.19125 −0.189700 −0.0948500 0.995492i $$-0.530237\pi$$
−0.0948500 + 0.995492i $$0.530237\pi$$
$$284$$ −12.6884 −0.752919
$$285$$ 9.19125 0.544443
$$286$$ 4.73042 0.279715
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 43.7724 2.57485
$$290$$ −10.2565 −0.602282
$$291$$ 4.79567 0.281127
$$292$$ 11.7304 0.686471
$$293$$ 2.58117 0.150793 0.0753967 0.997154i $$-0.475978\pi$$
0.0753967 + 0.997154i $$0.475978\pi$$
$$294$$ 0 0
$$295$$ 20.8797 1.21566
$$296$$ 3.10725 0.180605
$$297$$ 4.73042 0.274487
$$298$$ 8.26958 0.479044
$$299$$ −3.73042 −0.215736
$$300$$ −6.68842 −0.386156
$$301$$ 0 0
$$302$$ 5.26958 0.303230
$$303$$ −0.623166 −0.0358000
$$304$$ 2.68842 0.154191
$$305$$ −10.4001 −0.595507
$$306$$ −7.79567 −0.445649
$$307$$ 23.3116 1.33046 0.665231 0.746637i $$-0.268333\pi$$
0.665231 + 0.746637i $$0.268333\pi$$
$$308$$ 0 0
$$309$$ −8.83767 −0.502757
$$310$$ −19.4477 −1.10456
$$311$$ 10.7826 0.611424 0.305712 0.952124i $$-0.401106\pi$$
0.305712 + 0.952124i $$0.401106\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ 14.8508 0.839414 0.419707 0.907660i $$-0.362133\pi$$
0.419707 + 0.907660i $$0.362133\pi$$
$$314$$ 11.2565 0.635241
$$315$$ 0 0
$$316$$ 14.5261 0.817156
$$317$$ 23.0653 1.29547 0.647737 0.761864i $$-0.275716\pi$$
0.647737 + 0.761864i $$0.275716\pi$$
$$318$$ −9.21450 −0.516723
$$319$$ 14.1912 0.794557
$$320$$ −3.41883 −0.191119
$$321$$ −5.77241 −0.322185
$$322$$ 0 0
$$323$$ 20.9580 1.16613
$$324$$ 1.00000 0.0555556
$$325$$ 6.68842 0.371007
$$326$$ −16.3637 −0.906304
$$327$$ −4.00000 −0.221201
$$328$$ −0.892750 −0.0492939
$$329$$ 0 0
$$330$$ 16.1725 0.890267
$$331$$ 0.130501 0.00717299 0.00358650 0.999994i $$-0.498858\pi$$
0.00358650 + 0.999994i $$0.498858\pi$$
$$332$$ 0.934749 0.0513010
$$333$$ −3.10725 −0.170276
$$334$$ −22.9869 −1.25779
$$335$$ 0.223081 0.0121882
$$336$$ 0 0
$$337$$ 16.0522 0.874417 0.437209 0.899360i $$-0.355967\pi$$
0.437209 + 0.899360i $$0.355967\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ −10.6333 −0.577523
$$340$$ −26.6521 −1.44541
$$341$$ 26.9086 1.45718
$$342$$ −2.68842 −0.145373
$$343$$ 0 0
$$344$$ 12.8377 0.692161
$$345$$ −12.7537 −0.686634
$$346$$ −10.7724 −0.579129
$$347$$ 21.3217 1.14461 0.572306 0.820040i $$-0.306049\pi$$
0.572306 + 0.820040i $$0.306049\pi$$
$$348$$ 3.00000 0.160817
$$349$$ 6.35358 0.340099 0.170050 0.985435i $$-0.445607\pi$$
0.170050 + 0.985435i $$0.445607\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 4.73042 0.252132
$$353$$ 30.4290 1.61957 0.809786 0.586725i $$-0.199583\pi$$
0.809786 + 0.586725i $$0.199583\pi$$
$$354$$ −6.10725 −0.324597
$$355$$ 43.3796 2.30235
$$356$$ 15.1072 0.800683
$$357$$ 0 0
$$358$$ −19.0522 −1.00694
$$359$$ −18.0000 −0.950004 −0.475002 0.879985i $$-0.657553\pi$$
−0.475002 + 0.879985i $$0.657553\pi$$
$$360$$ 3.41883 0.180188
$$361$$ −11.7724 −0.619601
$$362$$ −13.7957 −0.725084
$$363$$ −11.3768 −0.597129
$$364$$ 0 0
$$365$$ −40.1043 −2.09916
$$366$$ 3.04200 0.159008
$$367$$ 4.22759 0.220678 0.110339 0.993894i $$-0.464806\pi$$
0.110339 + 0.993894i $$0.464806\pi$$
$$368$$ −3.73042 −0.194461
$$369$$ 0.892750 0.0464747
$$370$$ −10.6232 −0.552272
$$371$$ 0 0
$$372$$ 5.68842 0.294931
$$373$$ −35.6855 −1.84772 −0.923862 0.382725i $$-0.874986\pi$$
−0.923862 + 0.382725i $$0.874986\pi$$
$$374$$ 36.8767 1.90685
$$375$$ 5.77241 0.298086
$$376$$ 10.1492 0.523408
$$377$$ −3.00000 −0.154508
$$378$$ 0 0
$$379$$ 9.67533 0.496988 0.248494 0.968633i $$-0.420064\pi$$
0.248494 + 0.968633i $$0.420064\pi$$
$$380$$ −9.19125 −0.471501
$$381$$ 6.17250 0.316227
$$382$$ −10.2985 −0.526917
$$383$$ −3.67533 −0.187801 −0.0939003 0.995582i $$-0.529933\pi$$
−0.0939003 + 0.995582i $$0.529933\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −17.6333 −0.897513
$$387$$ −12.8377 −0.652575
$$388$$ −4.79567 −0.243463
$$389$$ 6.47392 0.328241 0.164120 0.986440i $$-0.447521\pi$$
0.164120 + 0.986440i $$0.447521\pi$$
$$390$$ −3.41883 −0.173119
$$391$$ −29.0811 −1.47069
$$392$$ 0 0
$$393$$ 20.4710 1.03263
$$394$$ 8.26958 0.416616
$$395$$ −49.6622 −2.49878
$$396$$ −4.73042 −0.237712
$$397$$ −7.86092 −0.394528 −0.197264 0.980350i $$-0.563206\pi$$
−0.197264 + 0.980350i $$0.563206\pi$$
$$398$$ 5.02325 0.251793
$$399$$ 0 0
$$400$$ 6.68842 0.334421
$$401$$ −5.64642 −0.281969 −0.140984 0.990012i $$-0.545027\pi$$
−0.140984 + 0.990012i $$0.545027\pi$$
$$402$$ −0.0652506 −0.00325440
$$403$$ −5.68842 −0.283360
$$404$$ 0.623166 0.0310037
$$405$$ −3.41883 −0.169883
$$406$$ 0 0
$$407$$ 14.6986 0.728582
$$408$$ 7.79567 0.385943
$$409$$ 25.7173 1.27164 0.635820 0.771837i $$-0.280662\pi$$
0.635820 + 0.771837i $$0.280662\pi$$
$$410$$ 3.05216 0.150736
$$411$$ 11.1072 0.547880
$$412$$ 8.83767 0.435401
$$413$$ 0 0
$$414$$ 3.73042 0.183340
$$415$$ −3.19575 −0.156873
$$416$$ −1.00000 −0.0490290
$$417$$ 2.26958 0.111142
$$418$$ 12.7173 0.622025
$$419$$ 8.83767 0.431748 0.215874 0.976421i $$-0.430740\pi$$
0.215874 + 0.976421i $$0.430740\pi$$
$$420$$ 0 0
$$421$$ −10.1594 −0.495140 −0.247570 0.968870i $$-0.579632\pi$$
−0.247570 + 0.968870i $$0.579632\pi$$
$$422$$ −14.5681 −0.709163
$$423$$ −10.1492 −0.493473
$$424$$ 9.21450 0.447496
$$425$$ 52.1407 2.52919
$$426$$ −12.6884 −0.614756
$$427$$ 0 0
$$428$$ 5.77241 0.279020
$$429$$ 4.73042 0.228387
$$430$$ −43.8898 −2.11656
$$431$$ −11.6753 −0.562381 −0.281190 0.959652i $$-0.590729\pi$$
−0.281190 + 0.959652i $$0.590729\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 6.60442 0.317388 0.158694 0.987328i $$-0.449272\pi$$
0.158694 + 0.987328i $$0.449272\pi$$
$$434$$ 0 0
$$435$$ −10.2565 −0.491761
$$436$$ 4.00000 0.191565
$$437$$ −10.0289 −0.479748
$$438$$ 11.7304 0.560501
$$439$$ −25.0102 −1.19367 −0.596835 0.802364i $$-0.703575\pi$$
−0.596835 + 0.802364i $$0.703575\pi$$
$$440$$ −16.1725 −0.770994
$$441$$ 0 0
$$442$$ −7.79567 −0.370802
$$443$$ −16.8797 −0.801977 −0.400989 0.916083i $$-0.631333\pi$$
−0.400989 + 0.916083i $$0.631333\pi$$
$$444$$ 3.10725 0.147464
$$445$$ −51.6492 −2.44840
$$446$$ 10.3217 0.488749
$$447$$ 8.26958 0.391138
$$448$$ 0 0
$$449$$ −20.1680 −0.951787 −0.475893 0.879503i $$-0.657875\pi$$
−0.475893 + 0.879503i $$0.657875\pi$$
$$450$$ −6.68842 −0.315295
$$451$$ −4.22308 −0.198857
$$452$$ 10.6333 0.500150
$$453$$ 5.26958 0.247587
$$454$$ −4.44208 −0.208477
$$455$$ 0 0
$$456$$ 2.68842 0.125897
$$457$$ −16.2565 −0.760447 −0.380223 0.924895i $$-0.624153\pi$$
−0.380223 + 0.924895i $$0.624153\pi$$
$$458$$ 21.4608 1.00280
$$459$$ −7.79567 −0.363871
$$460$$ 12.7537 0.594643
$$461$$ −16.0840 −0.749106 −0.374553 0.927205i $$-0.622204\pi$$
−0.374553 + 0.927205i $$0.622204\pi$$
$$462$$ 0 0
$$463$$ 1.67533 0.0778592 0.0389296 0.999242i $$-0.487605\pi$$
0.0389296 + 0.999242i $$0.487605\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ −19.4477 −0.901868
$$466$$ 7.66517 0.355082
$$467$$ 31.9449 1.47823 0.739117 0.673577i $$-0.235243\pi$$
0.739117 + 0.673577i $$0.235243\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 0 0
$$470$$ −34.6986 −1.60053
$$471$$ 11.2565 0.518672
$$472$$ 6.10725 0.281109
$$473$$ 60.7275 2.79225
$$474$$ 14.5261 0.667205
$$475$$ 17.9813 0.825036
$$476$$ 0 0
$$477$$ −9.21450 −0.421903
$$478$$ 9.52608 0.435713
$$479$$ 37.6941 1.72229 0.861143 0.508362i $$-0.169749\pi$$
0.861143 + 0.508362i $$0.169749\pi$$
$$480$$ −3.41883 −0.156048
$$481$$ −3.10725 −0.141678
$$482$$ 7.36375 0.335410
$$483$$ 0 0
$$484$$ 11.3768 0.517129
$$485$$ 16.3956 0.744485
$$486$$ 1.00000 0.0453609
$$487$$ −4.94491 −0.224075 −0.112038 0.993704i $$-0.535738\pi$$
−0.112038 + 0.993704i $$0.535738\pi$$
$$488$$ −3.04200 −0.137705
$$489$$ −16.3637 −0.739994
$$490$$ 0 0
$$491$$ 28.2014 1.27271 0.636356 0.771396i $$-0.280441\pi$$
0.636356 + 0.771396i $$0.280441\pi$$
$$492$$ −0.892750 −0.0402483
$$493$$ −23.3870 −1.05330
$$494$$ −2.68842 −0.120958
$$495$$ 16.1725 0.726900
$$496$$ −5.68842 −0.255417
$$497$$ 0 0
$$498$$ 0.934749 0.0418871
$$499$$ 19.7593 0.884549 0.442275 0.896880i $$-0.354172\pi$$
0.442275 + 0.896880i $$0.354172\pi$$
$$500$$ −5.77241 −0.258150
$$501$$ −22.9869 −1.02698
$$502$$ −12.7406 −0.568640
$$503$$ −2.40867 −0.107397 −0.0536986 0.998557i $$-0.517101\pi$$
−0.0536986 + 0.998557i $$0.517101\pi$$
$$504$$ 0 0
$$505$$ −2.13050 −0.0948061
$$506$$ −17.6464 −0.784479
$$507$$ −1.00000 −0.0444116
$$508$$ −6.17250 −0.273860
$$509$$ 25.5782 1.13374 0.566868 0.823809i $$-0.308155\pi$$
0.566868 + 0.823809i $$0.308155\pi$$
$$510$$ −26.6521 −1.18017
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ −2.68842 −0.118697
$$514$$ 13.4608 0.593732
$$515$$ −30.2145 −1.33141
$$516$$ 12.8377 0.565147
$$517$$ 48.0102 2.11148
$$518$$ 0 0
$$519$$ −10.7724 −0.472857
$$520$$ 3.41883 0.149926
$$521$$ 14.7912 0.648013 0.324006 0.946055i $$-0.394970\pi$$
0.324006 + 0.946055i $$0.394970\pi$$
$$522$$ 3.00000 0.131306
$$523$$ −22.5130 −0.984425 −0.492212 0.870475i $$-0.663812\pi$$
−0.492212 + 0.870475i $$0.663812\pi$$
$$524$$ −20.4710 −0.894280
$$525$$ 0 0
$$526$$ −24.0289 −1.04771
$$527$$ −44.3450 −1.93170
$$528$$ 4.73042 0.205865
$$529$$ −9.08400 −0.394956
$$530$$ −31.5028 −1.36840
$$531$$ −6.10725 −0.265032
$$532$$ 0 0
$$533$$ 0.892750 0.0386693
$$534$$ 15.1072 0.653755
$$535$$ −19.7349 −0.853215
$$536$$ 0.0652506 0.00281840
$$537$$ −19.0522 −0.822162
$$538$$ −15.9217 −0.686432
$$539$$ 0 0
$$540$$ 3.41883 0.147123
$$541$$ 29.3217 1.26064 0.630320 0.776335i $$-0.282924\pi$$
0.630320 + 0.776335i $$0.282924\pi$$
$$542$$ 19.6521 0.844129
$$543$$ −13.7957 −0.592029
$$544$$ −7.79567 −0.334236
$$545$$ −13.6753 −0.585787
$$546$$ 0 0
$$547$$ 7.64642 0.326937 0.163469 0.986549i $$-0.447732\pi$$
0.163469 + 0.986549i $$0.447732\pi$$
$$548$$ −11.1072 −0.474478
$$549$$ 3.04200 0.129829
$$550$$ 31.6390 1.34909
$$551$$ −8.06525 −0.343591
$$552$$ −3.73042 −0.158777
$$553$$ 0 0
$$554$$ 5.17250 0.219758
$$555$$ −10.6232 −0.450928
$$556$$ −2.26958 −0.0962518
$$557$$ 12.2276 0.518099 0.259050 0.965864i $$-0.416591\pi$$
0.259050 + 0.965864i $$0.416591\pi$$
$$558$$ 5.68842 0.240810
$$559$$ −12.8377 −0.542975
$$560$$ 0 0
$$561$$ 36.8767 1.55694
$$562$$ 24.4290 1.03048
$$563$$ −30.3116 −1.27748 −0.638740 0.769422i $$-0.720544\pi$$
−0.638740 + 0.769422i $$0.720544\pi$$
$$564$$ 10.1492 0.427360
$$565$$ −36.3536 −1.52941
$$566$$ 3.19125 0.134138
$$567$$ 0 0
$$568$$ 12.6884 0.532394
$$569$$ 18.0942 0.758547 0.379273 0.925285i $$-0.376174\pi$$
0.379273 + 0.925285i $$0.376174\pi$$
$$570$$ −9.19125 −0.384979
$$571$$ −12.4290 −0.520137 −0.260069 0.965590i $$-0.583745\pi$$
−0.260069 + 0.965590i $$0.583745\pi$$
$$572$$ −4.73042 −0.197789
$$573$$ −10.2985 −0.430226
$$574$$ 0 0
$$575$$ −24.9506 −1.04051
$$576$$ 1.00000 0.0416667
$$577$$ −2.09708 −0.0873028 −0.0436514 0.999047i $$-0.513899\pi$$
−0.0436514 + 0.999047i $$0.513899\pi$$
$$578$$ −43.7724 −1.82069
$$579$$ −17.6333 −0.732816
$$580$$ 10.2565 0.425878
$$581$$ 0 0
$$582$$ −4.79567 −0.198787
$$583$$ 43.5884 1.80525
$$584$$ −11.7304 −0.485408
$$585$$ −3.41883 −0.141351
$$586$$ −2.58117 −0.106627
$$587$$ 47.7826 1.97220 0.986099 0.166158i $$-0.0531363\pi$$
0.986099 + 0.166158i $$0.0531363\pi$$
$$588$$ 0 0
$$589$$ −15.2928 −0.630130
$$590$$ −20.8797 −0.859602
$$591$$ 8.26958 0.340165
$$592$$ −3.10725 −0.127707
$$593$$ −9.32175 −0.382798 −0.191399 0.981512i $$-0.561302\pi$$
−0.191399 + 0.981512i $$0.561302\pi$$
$$594$$ −4.73042 −0.194091
$$595$$ 0 0
$$596$$ −8.26958 −0.338735
$$597$$ 5.02325 0.205588
$$598$$ 3.73042 0.152548
$$599$$ −5.64642 −0.230706 −0.115353 0.993325i $$-0.536800\pi$$
−0.115353 + 0.993325i $$0.536800\pi$$
$$600$$ 6.68842 0.273053
$$601$$ −19.3450 −0.789099 −0.394550 0.918875i $$-0.629099\pi$$
−0.394550 + 0.918875i $$0.629099\pi$$
$$602$$ 0 0
$$603$$ −0.0652506 −0.00265721
$$604$$ −5.26958 −0.214416
$$605$$ −38.8955 −1.58133
$$606$$ 0.623166 0.0253144
$$607$$ 42.9637 1.74384 0.871921 0.489647i $$-0.162874\pi$$
0.871921 + 0.489647i $$0.162874\pi$$
$$608$$ −2.68842 −0.109030
$$609$$ 0 0
$$610$$ 10.4001 0.421087
$$611$$ −10.1492 −0.410595
$$612$$ 7.79567 0.315121
$$613$$ 27.8898 1.12646 0.563230 0.826300i $$-0.309559\pi$$
0.563230 + 0.826300i $$0.309559\pi$$
$$614$$ −23.3116 −0.940779
$$615$$ 3.05216 0.123075
$$616$$ 0 0
$$617$$ 41.4347 1.66810 0.834048 0.551691i $$-0.186017\pi$$
0.834048 + 0.551691i $$0.186017\pi$$
$$618$$ 8.83767 0.355503
$$619$$ −30.6232 −1.23085 −0.615424 0.788196i $$-0.711015\pi$$
−0.615424 + 0.788196i $$0.711015\pi$$
$$620$$ 19.4477 0.781040
$$621$$ 3.73042 0.149696
$$622$$ −10.7826 −0.432342
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ −13.7072 −0.548287
$$626$$ −14.8508 −0.593555
$$627$$ 12.7173 0.507881
$$628$$ −11.2565 −0.449183
$$629$$ −24.2231 −0.965837
$$630$$ 0 0
$$631$$ −12.9347 −0.514924 −0.257462 0.966288i $$-0.582886\pi$$
−0.257462 + 0.966288i $$0.582886\pi$$
$$632$$ −14.5261 −0.577817
$$633$$ −14.5681 −0.579029
$$634$$ −23.0653 −0.916038
$$635$$ 21.1027 0.837437
$$636$$ 9.21450 0.365379
$$637$$ 0 0
$$638$$ −14.1912 −0.561837
$$639$$ −12.6884 −0.501946
$$640$$ 3.41883 0.135141
$$641$$ −6.92166 −0.273389 −0.136695 0.990613i $$-0.543648\pi$$
−0.136695 + 0.990613i $$0.543648\pi$$
$$642$$ 5.77241 0.227819
$$643$$ −19.2276 −0.758262 −0.379131 0.925343i $$-0.623777\pi$$
−0.379131 + 0.925343i $$0.623777\pi$$
$$644$$ 0 0
$$645$$ −43.8898 −1.72816
$$646$$ −20.9580 −0.824582
$$647$$ 7.05216 0.277249 0.138625 0.990345i $$-0.455732\pi$$
0.138625 + 0.990345i $$0.455732\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 28.8898 1.13403
$$650$$ −6.68842 −0.262341
$$651$$ 0 0
$$652$$ 16.3637 0.640854
$$653$$ −3.01875 −0.118133 −0.0590664 0.998254i $$-0.518812\pi$$
−0.0590664 + 0.998254i $$0.518812\pi$$
$$654$$ 4.00000 0.156412
$$655$$ 69.9869 2.73462
$$656$$ 0.892750 0.0348560
$$657$$ 11.7304 0.457647
$$658$$ 0 0
$$659$$ −12.3536 −0.481227 −0.240614 0.970621i $$-0.577349\pi$$
−0.240614 + 0.970621i $$0.577349\pi$$
$$660$$ −16.1725 −0.629514
$$661$$ 40.5884 1.57871 0.789353 0.613939i $$-0.210416\pi$$
0.789353 + 0.613939i $$0.210416\pi$$
$$662$$ −0.130501 −0.00507207
$$663$$ −7.79567 −0.302759
$$664$$ −0.934749 −0.0362753
$$665$$ 0 0
$$666$$ 3.10725 0.120403
$$667$$ 11.1912 0.433327
$$668$$ 22.9869 0.889390
$$669$$ 10.3217 0.399062
$$670$$ −0.223081 −0.00861837
$$671$$ −14.3899 −0.555517
$$672$$ 0 0
$$673$$ 28.5261 1.09960 0.549800 0.835296i $$-0.314704\pi$$
0.549800 + 0.835296i $$0.314704\pi$$
$$674$$ −16.0522 −0.618306
$$675$$ −6.68842 −0.257437
$$676$$ 1.00000 0.0384615
$$677$$ −18.1362 −0.697029 −0.348515 0.937303i $$-0.613314\pi$$
−0.348515 + 0.937303i $$0.613314\pi$$
$$678$$ 10.6333 0.408371
$$679$$ 0 0
$$680$$ 26.6521 1.02206
$$681$$ −4.44208 −0.170221
$$682$$ −26.9086 −1.03038
$$683$$ 11.1958 0.428394 0.214197 0.976791i $$-0.431287\pi$$
0.214197 + 0.976791i $$0.431287\pi$$
$$684$$ 2.68842 0.102794
$$685$$ 37.9738 1.45091
$$686$$ 0 0
$$687$$ 21.4608 0.818782
$$688$$ −12.8377 −0.489431
$$689$$ −9.21450 −0.351044
$$690$$ 12.7537 0.485524
$$691$$ −0.641914 −0.0244195 −0.0122098 0.999925i $$-0.503887\pi$$
−0.0122098 + 0.999925i $$0.503887\pi$$
$$692$$ 10.7724 0.409506
$$693$$ 0 0
$$694$$ −21.3217 −0.809363
$$695$$ 7.75933 0.294328
$$696$$ −3.00000 −0.113715
$$697$$ 6.95958 0.263613
$$698$$ −6.35358 −0.240487
$$699$$ 7.66517 0.289923
$$700$$ 0 0
$$701$$ 30.0334 1.13435 0.567173 0.823599i $$-0.308037\pi$$
0.567173 + 0.823599i $$0.308037\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ −8.35358 −0.315061
$$704$$ −4.73042 −0.178284
$$705$$ −34.6986 −1.30682
$$706$$ −30.4290 −1.14521
$$707$$ 0 0
$$708$$ 6.10725 0.229524
$$709$$ 30.2434 1.13582 0.567908 0.823092i $$-0.307753\pi$$
0.567908 + 0.823092i $$0.307753\pi$$
$$710$$ −43.3796 −1.62801
$$711$$ 14.5261 0.544771
$$712$$ −15.1072 −0.566168
$$713$$ 21.2202 0.794701
$$714$$ 0 0
$$715$$ 16.1725 0.604817
$$716$$ 19.0522 0.712013
$$717$$ 9.52608 0.355758
$$718$$ 18.0000 0.671754
$$719$$ 42.5970 1.58860 0.794300 0.607526i $$-0.207838\pi$$
0.794300 + 0.607526i $$0.207838\pi$$
$$720$$ −3.41883 −0.127412
$$721$$ 0 0
$$722$$ 11.7724 0.438124
$$723$$ 7.36375 0.273861
$$724$$ 13.7957 0.512712
$$725$$ −20.0653 −0.745205
$$726$$ 11.3768 0.422234
$$727$$ 19.3637 0.718162 0.359081 0.933306i $$-0.383090\pi$$
0.359081 + 0.933306i $$0.383090\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 40.1043 1.48433
$$731$$ −100.078 −3.70152
$$732$$ −3.04200 −0.112436
$$733$$ 26.0754 0.963117 0.481559 0.876414i $$-0.340071\pi$$
0.481559 + 0.876414i $$0.340071\pi$$
$$734$$ −4.22759 −0.156043
$$735$$ 0 0
$$736$$ 3.73042 0.137505
$$737$$ 0.308662 0.0113697
$$738$$ −0.892750 −0.0328626
$$739$$ −20.6435 −0.759383 −0.379692 0.925113i $$-0.623970\pi$$
−0.379692 + 0.925113i $$0.623970\pi$$
$$740$$ 10.6232 0.390515
$$741$$ −2.68842 −0.0987615
$$742$$ 0 0
$$743$$ −2.81892 −0.103416 −0.0517080 0.998662i $$-0.516467\pi$$
−0.0517080 + 0.998662i $$0.516467\pi$$
$$744$$ −5.68842 −0.208547
$$745$$ 28.2723 1.03582
$$746$$ 35.6855 1.30654
$$747$$ 0.934749 0.0342007
$$748$$ −36.8767 −1.34835
$$749$$ 0 0
$$750$$ −5.77241 −0.210779
$$751$$ −16.0709 −0.586436 −0.293218 0.956046i $$-0.594726\pi$$
−0.293218 + 0.956046i $$0.594726\pi$$
$$752$$ −10.1492 −0.370105
$$753$$ −12.7406 −0.464293
$$754$$ 3.00000 0.109254
$$755$$ 18.0158 0.655663
$$756$$ 0 0
$$757$$ 5.34050 0.194104 0.0970518 0.995279i $$-0.469059\pi$$
0.0970518 + 0.995279i $$0.469059\pi$$
$$758$$ −9.67533 −0.351424
$$759$$ −17.6464 −0.640524
$$760$$ 9.19125 0.333402
$$761$$ −36.3188 −1.31656 −0.658278 0.752775i $$-0.728715\pi$$
−0.658278 + 0.752775i $$0.728715\pi$$
$$762$$ −6.17250 −0.223606
$$763$$ 0 0
$$764$$ 10.2985 0.372587
$$765$$ −26.6521 −0.963608
$$766$$ 3.67533 0.132795
$$767$$ −6.10725 −0.220520
$$768$$ −1.00000 −0.0360844
$$769$$ 39.9783 1.44166 0.720828 0.693114i $$-0.243762\pi$$
0.720828 + 0.693114i $$0.243762\pi$$
$$770$$ 0 0
$$771$$ 13.4608 0.484780
$$772$$ 17.6333 0.634637
$$773$$ 18.5970 0.668887 0.334444 0.942416i $$-0.391452\pi$$
0.334444 + 0.942416i $$0.391452\pi$$
$$774$$ 12.8377 0.461440
$$775$$ −38.0465 −1.36667
$$776$$ 4.79567 0.172154
$$777$$ 0 0
$$778$$ −6.47392 −0.232101
$$779$$ 2.40009 0.0859920
$$780$$ 3.41883 0.122414
$$781$$ 60.0215 2.14774
$$782$$ 29.0811 1.03994
$$783$$ 3.00000 0.107211
$$784$$ 0 0
$$785$$ 38.4841 1.37356
$$786$$ −20.4710 −0.730176
$$787$$ 0.772415 0.0275336 0.0137668 0.999905i $$-0.495618\pi$$
0.0137668 + 0.999905i $$0.495618\pi$$
$$788$$ −8.26958 −0.294592
$$789$$ −24.0289 −0.855452
$$790$$ 49.6622 1.76690
$$791$$ 0 0
$$792$$ 4.73042 0.168088
$$793$$ 3.04200 0.108025
$$794$$ 7.86092 0.278974
$$795$$ −31.5028 −1.11729
$$796$$ −5.02325 −0.178044
$$797$$ 7.10275 0.251592 0.125796 0.992056i $$-0.459851\pi$$
0.125796 + 0.992056i $$0.459851\pi$$
$$798$$ 0 0
$$799$$ −79.1202 −2.79907
$$800$$ −6.68842 −0.236471
$$801$$ 15.1072 0.533788
$$802$$ 5.64642 0.199382
$$803$$ −55.4897 −1.95819
$$804$$ 0.0652506 0.00230121
$$805$$ 0 0
$$806$$ 5.68842 0.200366
$$807$$ −15.9217 −0.560469
$$808$$ −0.623166 −0.0219229
$$809$$ −20.6798 −0.727064 −0.363532 0.931582i $$-0.618429\pi$$
−0.363532 + 0.931582i $$0.618429\pi$$
$$810$$ 3.41883 0.120126
$$811$$ 36.2985 1.27461 0.637306 0.770611i $$-0.280049\pi$$
0.637306 + 0.770611i $$0.280049\pi$$
$$812$$ 0 0
$$813$$ 19.6521 0.689229
$$814$$ −14.6986 −0.515185
$$815$$ −55.9449 −1.95966
$$816$$ −7.79567 −0.272903
$$817$$ −34.5130 −1.20746
$$818$$ −25.7173 −0.899185
$$819$$ 0 0
$$820$$ −3.05216 −0.106586
$$821$$ −52.1928 −1.82154 −0.910771 0.412911i $$-0.864512\pi$$
−0.910771 + 0.412911i $$0.864512\pi$$
$$822$$ −11.1072 −0.387410
$$823$$ 39.1072 1.36319 0.681597 0.731728i $$-0.261286\pi$$
0.681597 + 0.731728i $$0.261286\pi$$
$$824$$ −8.83767 −0.307875
$$825$$ 31.6390 1.10153
$$826$$ 0 0
$$827$$ −51.6724 −1.79683 −0.898413 0.439152i $$-0.855279\pi$$
−0.898413 + 0.439152i $$0.855279\pi$$
$$828$$ −3.73042 −0.129641
$$829$$ −31.9840 −1.11085 −0.555425 0.831567i $$-0.687444\pi$$
−0.555425 + 0.831567i $$0.687444\pi$$
$$830$$ 3.19575 0.110926
$$831$$ 5.17250 0.179432
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ −2.26958 −0.0785893
$$835$$ −78.5884 −2.71966
$$836$$ −12.7173 −0.439838
$$837$$ 5.68842 0.196620
$$838$$ −8.83767 −0.305292
$$839$$ −18.0187 −0.622076 −0.311038 0.950397i $$-0.600677\pi$$
−0.311038 + 0.950397i $$0.600677\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 10.1594 0.350117
$$843$$ 24.4290 0.841379
$$844$$ 14.5681 0.501454
$$845$$ −3.41883 −0.117611
$$846$$ 10.1492 0.348938
$$847$$ 0 0
$$848$$ −9.21450 −0.316427
$$849$$ 3.19125 0.109523
$$850$$ −52.1407 −1.78841
$$851$$ 11.5913 0.397346
$$852$$ 12.6884 0.434698
$$853$$ 49.1651 1.68338 0.841690 0.539961i $$-0.181561\pi$$
0.841690 + 0.539961i $$0.181561\pi$$
$$854$$ 0 0
$$855$$ −9.19125 −0.314334
$$856$$ −5.77241 −0.197297
$$857$$ 37.6015 1.28444 0.642221 0.766519i $$-0.278013\pi$$
0.642221 + 0.766519i $$0.278013\pi$$
$$858$$ −4.73042 −0.161494
$$859$$ −5.78550 −0.197399 −0.0986994 0.995117i $$-0.531468\pi$$
−0.0986994 + 0.995117i $$0.531468\pi$$
$$860$$ 43.8898 1.49663
$$861$$ 0 0
$$862$$ 11.6753 0.397663
$$863$$ 18.9217 0.644101 0.322050 0.946723i $$-0.395628\pi$$
0.322050 + 0.946723i $$0.395628\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −36.8291 −1.25223
$$866$$ −6.60442 −0.224427
$$867$$ −43.7724 −1.48659
$$868$$ 0 0
$$869$$ −68.7144 −2.33098
$$870$$ 10.2565 0.347728
$$871$$ −0.0652506 −0.00221093
$$872$$ −4.00000 −0.135457
$$873$$ −4.79567 −0.162309
$$874$$ 10.0289 0.339233
$$875$$ 0 0
$$876$$ −11.7304 −0.396334
$$877$$ −30.3739 −1.02565 −0.512827 0.858492i $$-0.671402\pi$$
−0.512827 + 0.858492i $$0.671402\pi$$
$$878$$ 25.0102 0.844052
$$879$$ −2.58117 −0.0870606
$$880$$ 16.1725 0.545175
$$881$$ −42.0578 −1.41696 −0.708482 0.705729i $$-0.750620\pi$$
−0.708482 + 0.705729i $$0.750620\pi$$
$$882$$ 0 0
$$883$$ 8.84625 0.297700 0.148850 0.988860i $$-0.452443\pi$$
0.148850 + 0.988860i $$0.452443\pi$$
$$884$$ 7.79567 0.262197
$$885$$ −20.8797 −0.701862
$$886$$ 16.8797 0.567083
$$887$$ 44.5595 1.49616 0.748081 0.663608i $$-0.230976\pi$$
0.748081 + 0.663608i $$0.230976\pi$$
$$888$$ −3.10725 −0.104272
$$889$$ 0 0
$$890$$ 51.6492 1.73128
$$891$$ −4.73042 −0.158475
$$892$$ −10.3217 −0.345598
$$893$$ −27.2854 −0.913071
$$894$$ −8.26958 −0.276576
$$895$$ −65.1362 −2.17726
$$896$$ 0 0
$$897$$ 3.73042 0.124555
$$898$$ 20.1680 0.673015
$$899$$ 17.0653 0.569158
$$900$$ 6.68842 0.222947
$$901$$ −71.8332 −2.39311
$$902$$ 4.22308 0.140613
$$903$$ 0 0
$$904$$ −10.6333 −0.353659
$$905$$ −47.1651 −1.56782
$$906$$ −5.26958 −0.175070
$$907$$ −42.0289 −1.39555 −0.697774 0.716318i $$-0.745826\pi$$
−0.697774 + 0.716318i $$0.745826\pi$$
$$908$$ 4.44208 0.147416
$$909$$ 0.623166 0.0206691
$$910$$ 0 0
$$911$$ 46.0000 1.52405 0.762024 0.647549i $$-0.224206\pi$$
0.762024 + 0.647549i $$0.224206\pi$$
$$912$$ −2.68842 −0.0890224
$$913$$ −4.42175 −0.146339
$$914$$ 16.2565 0.537717
$$915$$ 10.4001 0.343816
$$916$$ −21.4608 −0.709086
$$917$$ 0 0
$$918$$ 7.79567 0.257295
$$919$$ 9.91600 0.327099 0.163549 0.986535i $$-0.447706\pi$$
0.163549 + 0.986535i $$0.447706\pi$$
$$920$$ −12.7537 −0.420476
$$921$$ −23.3116 −0.768143
$$922$$ 16.0840 0.529698
$$923$$ −12.6884 −0.417644
$$924$$ 0 0
$$925$$ −20.7826 −0.683327
$$926$$ −1.67533 −0.0550548
$$927$$ 8.83767 0.290267
$$928$$ 3.00000 0.0984798
$$929$$ −20.2985 −0.665972 −0.332986 0.942932i $$-0.608056\pi$$
−0.332986 + 0.942932i $$0.608056\pi$$
$$930$$ 19.4477 0.637717
$$931$$ 0 0
$$932$$ −7.66517 −0.251081
$$933$$ −10.7826 −0.353006
$$934$$ −31.9449 −1.04527
$$935$$ 126.075 4.12311
$$936$$ −1.00000 −0.0326860
$$937$$ −34.2667 −1.11944 −0.559722 0.828681i $$-0.689092\pi$$
−0.559722 + 0.828681i $$0.689092\pi$$
$$938$$ 0 0
$$939$$ −14.8508 −0.484636
$$940$$ 34.6986 1.13174
$$941$$ −25.1492 −0.819842 −0.409921 0.912121i $$-0.634444\pi$$
−0.409921 + 0.912121i $$0.634444\pi$$
$$942$$ −11.2565 −0.366757
$$943$$ −3.33033 −0.108450
$$944$$ −6.10725 −0.198774
$$945$$ 0 0
$$946$$ −60.7275 −1.97442
$$947$$ 27.2100 0.884206 0.442103 0.896964i $$-0.354233\pi$$
0.442103 + 0.896964i $$0.354233\pi$$
$$948$$ −14.5261 −0.471785
$$949$$ 11.7304 0.380785
$$950$$ −17.9813 −0.583389
$$951$$ −23.0653 −0.747942
$$952$$ 0 0
$$953$$ 54.5028 1.76552 0.882760 0.469824i $$-0.155683\pi$$
0.882760 + 0.469824i $$0.155683\pi$$
$$954$$ 9.21450 0.298330
$$955$$ −35.2088 −1.13933
$$956$$ −9.52608 −0.308096
$$957$$ −14.1912 −0.458738
$$958$$ −37.6941 −1.21784
$$959$$ 0 0
$$960$$ 3.41883 0.110342
$$961$$ 1.35809 0.0438092
$$962$$ 3.10725 0.100182
$$963$$ 5.77241 0.186013
$$964$$ −7.36375 −0.237170
$$965$$ −60.2854 −1.94066
$$966$$ 0 0
$$967$$ −13.9767 −0.449462 −0.224731 0.974421i $$-0.572150\pi$$
−0.224731 + 0.974421i $$0.572150\pi$$
$$968$$ −11.3768 −0.365665
$$969$$ −20.9580 −0.673268
$$970$$ −16.3956 −0.526430
$$971$$ −35.9783 −1.15460 −0.577300 0.816532i $$-0.695894\pi$$
−0.577300 + 0.816532i $$0.695894\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 4.94491 0.158445
$$975$$ −6.68842 −0.214201
$$976$$ 3.04200 0.0973720
$$977$$ 13.4608 0.430650 0.215325 0.976542i $$-0.430919\pi$$
0.215325 + 0.976542i $$0.430919\pi$$
$$978$$ 16.3637 0.523255
$$979$$ −71.4636 −2.28399
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ −28.2014 −0.899943
$$983$$ 2.06525 0.0658713 0.0329356 0.999457i $$-0.489514\pi$$
0.0329356 + 0.999457i $$0.489514\pi$$
$$984$$ 0.892750 0.0284598
$$985$$ 28.2723 0.900831
$$986$$ 23.3870 0.744794
$$987$$ 0 0
$$988$$ 2.68842 0.0855299
$$989$$ 47.8898 1.52281
$$990$$ −16.1725 −0.513996
$$991$$ −23.0187 −0.731215 −0.365607 0.930769i $$-0.619139\pi$$
−0.365607 + 0.930769i $$0.619139\pi$$
$$992$$ 5.68842 0.180607
$$993$$ −0.130501 −0.00414133
$$994$$ 0 0
$$995$$ 17.1737 0.544442
$$996$$ −0.934749 −0.0296187
$$997$$ −26.3927 −0.835864 −0.417932 0.908478i $$-0.637245\pi$$
−0.417932 + 0.908478i $$0.637245\pi$$
$$998$$ −19.7593 −0.625471
$$999$$ 3.10725 0.0983090
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.bv.1.1 3
7.2 even 3 546.2.i.k.235.3 yes 6
7.4 even 3 546.2.i.k.79.3 6
7.6 odd 2 3822.2.a.bw.1.3 3
21.2 odd 6 1638.2.j.q.235.1 6
21.11 odd 6 1638.2.j.q.1171.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.k.79.3 6 7.4 even 3
546.2.i.k.235.3 yes 6 7.2 even 3
1638.2.j.q.235.1 6 21.2 odd 6
1638.2.j.q.1171.1 6 21.11 odd 6
3822.2.a.bv.1.1 3 1.1 even 1 trivial
3822.2.a.bw.1.3 3 7.6 odd 2