# Properties

 Label 5225.2.a.bb.1.8 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $22$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$0$$ Dimension: $$22$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.8 Character $$\chi$$ $$=$$ 5225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.28593 q^{2} -0.496540 q^{3} -0.346393 q^{4} +0.638514 q^{6} -3.51780 q^{7} +3.01729 q^{8} -2.75345 q^{9} +O(q^{10})$$ $$q-1.28593 q^{2} -0.496540 q^{3} -0.346393 q^{4} +0.638514 q^{6} -3.51780 q^{7} +3.01729 q^{8} -2.75345 q^{9} -1.00000 q^{11} +0.171998 q^{12} +2.55471 q^{13} +4.52364 q^{14} -3.18722 q^{16} -3.96000 q^{17} +3.54073 q^{18} -1.00000 q^{19} +1.74673 q^{21} +1.28593 q^{22} -7.91604 q^{23} -1.49821 q^{24} -3.28517 q^{26} +2.85682 q^{27} +1.21854 q^{28} -3.06230 q^{29} +1.12535 q^{31} -1.93604 q^{32} +0.496540 q^{33} +5.09227 q^{34} +0.953776 q^{36} -6.39797 q^{37} +1.28593 q^{38} -1.26852 q^{39} +1.65223 q^{41} -2.24617 q^{42} +8.34172 q^{43} +0.346393 q^{44} +10.1794 q^{46} -13.0685 q^{47} +1.58259 q^{48} +5.37494 q^{49} +1.96630 q^{51} -0.884934 q^{52} -7.33721 q^{53} -3.67366 q^{54} -10.6142 q^{56} +0.496540 q^{57} +3.93789 q^{58} -1.66706 q^{59} -3.64946 q^{61} -1.44712 q^{62} +9.68609 q^{63} +8.86406 q^{64} -0.638514 q^{66} -10.5579 q^{67} +1.37172 q^{68} +3.93064 q^{69} +13.9374 q^{71} -8.30795 q^{72} -9.76005 q^{73} +8.22731 q^{74} +0.346393 q^{76} +3.51780 q^{77} +1.63122 q^{78} -5.30165 q^{79} +6.84182 q^{81} -2.12465 q^{82} -3.85256 q^{83} -0.605056 q^{84} -10.7268 q^{86} +1.52056 q^{87} -3.01729 q^{88} -8.82721 q^{89} -8.98696 q^{91} +2.74207 q^{92} -0.558784 q^{93} +16.8051 q^{94} +0.961323 q^{96} -1.91792 q^{97} -6.91177 q^{98} +2.75345 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$22 q + 32 q^{4} - 12 q^{6} + 34 q^{9}+O(q^{10})$$ 22 * q + 32 * q^4 - 12 * q^6 + 34 * q^9 $$22 q + 32 q^{4} - 12 q^{6} + 34 q^{9} - 22 q^{11} - 8 q^{14} + 40 q^{16} - 22 q^{19} - 22 q^{21} - 22 q^{24} + 16 q^{26} - 10 q^{29} + 76 q^{31} + 56 q^{34} + 104 q^{36} - 8 q^{39} + 6 q^{41} - 32 q^{44} + 88 q^{46} + 28 q^{49} + 8 q^{51} + 38 q^{54} + 44 q^{56} + 40 q^{59} - 6 q^{61} + 140 q^{64} + 12 q^{66} + 74 q^{69} + 62 q^{71} - 26 q^{74} - 32 q^{76} + 102 q^{79} + 94 q^{81} - 38 q^{84} + 28 q^{86} + 54 q^{89} + 88 q^{91} + 36 q^{94} + 2 q^{96} - 34 q^{99}+O(q^{100})$$ 22 * q + 32 * q^4 - 12 * q^6 + 34 * q^9 - 22 * q^11 - 8 * q^14 + 40 * q^16 - 22 * q^19 - 22 * q^21 - 22 * q^24 + 16 * q^26 - 10 * q^29 + 76 * q^31 + 56 * q^34 + 104 * q^36 - 8 * q^39 + 6 * q^41 - 32 * q^44 + 88 * q^46 + 28 * q^49 + 8 * q^51 + 38 * q^54 + 44 * q^56 + 40 * q^59 - 6 * q^61 + 140 * q^64 + 12 * q^66 + 74 * q^69 + 62 * q^71 - 26 * q^74 - 32 * q^76 + 102 * q^79 + 94 * q^81 - 38 * q^84 + 28 * q^86 + 54 * q^89 + 88 * q^91 + 36 * q^94 + 2 * q^96 - 34 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.28593 −0.909287 −0.454644 0.890673i $$-0.650233\pi$$
−0.454644 + 0.890673i $$0.650233\pi$$
$$3$$ −0.496540 −0.286678 −0.143339 0.989674i $$-0.545784\pi$$
−0.143339 + 0.989674i $$0.545784\pi$$
$$4$$ −0.346393 −0.173197
$$5$$ 0 0
$$6$$ 0.638514 0.260672
$$7$$ −3.51780 −1.32960 −0.664802 0.747019i $$-0.731484\pi$$
−0.664802 + 0.747019i $$0.731484\pi$$
$$8$$ 3.01729 1.06677
$$9$$ −2.75345 −0.917816
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0.171998 0.0496516
$$13$$ 2.55471 0.708548 0.354274 0.935142i $$-0.384728\pi$$
0.354274 + 0.935142i $$0.384728\pi$$
$$14$$ 4.52364 1.20899
$$15$$ 0 0
$$16$$ −3.18722 −0.796806
$$17$$ −3.96000 −0.960440 −0.480220 0.877148i $$-0.659443\pi$$
−0.480220 + 0.877148i $$0.659443\pi$$
$$18$$ 3.54073 0.834558
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 1.74673 0.381168
$$22$$ 1.28593 0.274160
$$23$$ −7.91604 −1.65061 −0.825305 0.564688i $$-0.808997\pi$$
−0.825305 + 0.564688i $$0.808997\pi$$
$$24$$ −1.49821 −0.305820
$$25$$ 0 0
$$26$$ −3.28517 −0.644274
$$27$$ 2.85682 0.549795
$$28$$ 1.21854 0.230283
$$29$$ −3.06230 −0.568655 −0.284328 0.958727i $$-0.591770\pi$$
−0.284328 + 0.958727i $$0.591770\pi$$
$$30$$ 0 0
$$31$$ 1.12535 0.202120 0.101060 0.994880i $$-0.467777\pi$$
0.101060 + 0.994880i $$0.467777\pi$$
$$32$$ −1.93604 −0.342247
$$33$$ 0.496540 0.0864366
$$34$$ 5.09227 0.873316
$$35$$ 0 0
$$36$$ 0.953776 0.158963
$$37$$ −6.39797 −1.05182 −0.525910 0.850540i $$-0.676275\pi$$
−0.525910 + 0.850540i $$0.676275\pi$$
$$38$$ 1.28593 0.208605
$$39$$ −1.26852 −0.203125
$$40$$ 0 0
$$41$$ 1.65223 0.258036 0.129018 0.991642i $$-0.458818\pi$$
0.129018 + 0.991642i $$0.458818\pi$$
$$42$$ −2.24617 −0.346591
$$43$$ 8.34172 1.27210 0.636050 0.771647i $$-0.280567\pi$$
0.636050 + 0.771647i $$0.280567\pi$$
$$44$$ 0.346393 0.0522208
$$45$$ 0 0
$$46$$ 10.1794 1.50088
$$47$$ −13.0685 −1.90623 −0.953117 0.302601i $$-0.902145\pi$$
−0.953117 + 0.302601i $$0.902145\pi$$
$$48$$ 1.58259 0.228427
$$49$$ 5.37494 0.767848
$$50$$ 0 0
$$51$$ 1.96630 0.275337
$$52$$ −0.884934 −0.122718
$$53$$ −7.33721 −1.00784 −0.503922 0.863749i $$-0.668110\pi$$
−0.503922 + 0.863749i $$0.668110\pi$$
$$54$$ −3.67366 −0.499922
$$55$$ 0 0
$$56$$ −10.6142 −1.41839
$$57$$ 0.496540 0.0657684
$$58$$ 3.93789 0.517071
$$59$$ −1.66706 −0.217032 −0.108516 0.994095i $$-0.534610\pi$$
−0.108516 + 0.994095i $$0.534610\pi$$
$$60$$ 0 0
$$61$$ −3.64946 −0.467266 −0.233633 0.972325i $$-0.575061\pi$$
−0.233633 + 0.972325i $$0.575061\pi$$
$$62$$ −1.44712 −0.183785
$$63$$ 9.68609 1.22033
$$64$$ 8.86406 1.10801
$$65$$ 0 0
$$66$$ −0.638514 −0.0785957
$$67$$ −10.5579 −1.28985 −0.644927 0.764244i $$-0.723112\pi$$
−0.644927 + 0.764244i $$0.723112\pi$$
$$68$$ 1.37172 0.166345
$$69$$ 3.93064 0.473193
$$70$$ 0 0
$$71$$ 13.9374 1.65407 0.827034 0.562152i $$-0.190026\pi$$
0.827034 + 0.562152i $$0.190026\pi$$
$$72$$ −8.30795 −0.979101
$$73$$ −9.76005 −1.14233 −0.571164 0.820836i $$-0.693508\pi$$
−0.571164 + 0.820836i $$0.693508\pi$$
$$74$$ 8.22731 0.956406
$$75$$ 0 0
$$76$$ 0.346393 0.0397340
$$77$$ 3.51780 0.400891
$$78$$ 1.63122 0.184699
$$79$$ −5.30165 −0.596482 −0.298241 0.954491i $$-0.596400\pi$$
−0.298241 + 0.954491i $$0.596400\pi$$
$$80$$ 0 0
$$81$$ 6.84182 0.760202
$$82$$ −2.12465 −0.234629
$$83$$ −3.85256 −0.422874 −0.211437 0.977392i $$-0.567814\pi$$
−0.211437 + 0.977392i $$0.567814\pi$$
$$84$$ −0.605056 −0.0660170
$$85$$ 0 0
$$86$$ −10.7268 −1.15671
$$87$$ 1.52056 0.163021
$$88$$ −3.01729 −0.321644
$$89$$ −8.82721 −0.935682 −0.467841 0.883813i $$-0.654968\pi$$
−0.467841 + 0.883813i $$0.654968\pi$$
$$90$$ 0 0
$$91$$ −8.98696 −0.942089
$$92$$ 2.74207 0.285880
$$93$$ −0.558784 −0.0579432
$$94$$ 16.8051 1.73331
$$95$$ 0 0
$$96$$ 0.961323 0.0981146
$$97$$ −1.91792 −0.194736 −0.0973678 0.995248i $$-0.531042\pi$$
−0.0973678 + 0.995248i $$0.531042\pi$$
$$98$$ −6.91177 −0.698194
$$99$$ 2.75345 0.276732
$$100$$ 0 0
$$101$$ −2.98725 −0.297243 −0.148621 0.988894i $$-0.547484\pi$$
−0.148621 + 0.988894i $$0.547484\pi$$
$$102$$ −2.52852 −0.250360
$$103$$ −8.85180 −0.872194 −0.436097 0.899900i $$-0.643640\pi$$
−0.436097 + 0.899900i $$0.643640\pi$$
$$104$$ 7.70829 0.755860
$$105$$ 0 0
$$106$$ 9.43511 0.916419
$$107$$ 4.84194 0.468088 0.234044 0.972226i $$-0.424804\pi$$
0.234044 + 0.972226i $$0.424804\pi$$
$$108$$ −0.989583 −0.0952227
$$109$$ −4.68491 −0.448733 −0.224366 0.974505i $$-0.572031\pi$$
−0.224366 + 0.974505i $$0.572031\pi$$
$$110$$ 0 0
$$111$$ 3.17685 0.301533
$$112$$ 11.2120 1.05944
$$113$$ −18.8998 −1.77794 −0.888971 0.457963i $$-0.848579\pi$$
−0.888971 + 0.457963i $$0.848579\pi$$
$$114$$ −0.638514 −0.0598024
$$115$$ 0 0
$$116$$ 1.06076 0.0984892
$$117$$ −7.03425 −0.650317
$$118$$ 2.14371 0.197345
$$119$$ 13.9305 1.27701
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 4.69294 0.424879
$$123$$ −0.820401 −0.0739731
$$124$$ −0.389815 −0.0350065
$$125$$ 0 0
$$126$$ −12.4556 −1.10963
$$127$$ 13.2632 1.17692 0.588458 0.808528i $$-0.299735\pi$$
0.588458 + 0.808528i $$0.299735\pi$$
$$128$$ −7.52644 −0.665250
$$129$$ −4.14200 −0.364683
$$130$$ 0 0
$$131$$ −17.6530 −1.54235 −0.771176 0.636622i $$-0.780331\pi$$
−0.771176 + 0.636622i $$0.780331\pi$$
$$132$$ −0.171998 −0.0149705
$$133$$ 3.51780 0.305032
$$134$$ 13.5767 1.17285
$$135$$ 0 0
$$136$$ −11.9485 −1.02457
$$137$$ 4.42479 0.378035 0.189018 0.981974i $$-0.439470\pi$$
0.189018 + 0.981974i $$0.439470\pi$$
$$138$$ −5.05451 −0.430268
$$139$$ 12.4255 1.05392 0.526958 0.849892i $$-0.323333\pi$$
0.526958 + 0.849892i $$0.323333\pi$$
$$140$$ 0 0
$$141$$ 6.48903 0.546475
$$142$$ −17.9225 −1.50402
$$143$$ −2.55471 −0.213635
$$144$$ 8.77586 0.731321
$$145$$ 0 0
$$146$$ 12.5507 1.03870
$$147$$ −2.66887 −0.220125
$$148$$ 2.21621 0.182172
$$149$$ 10.0735 0.825251 0.412625 0.910901i $$-0.364612\pi$$
0.412625 + 0.910901i $$0.364612\pi$$
$$150$$ 0 0
$$151$$ −20.0465 −1.63136 −0.815680 0.578503i $$-0.803637\pi$$
−0.815680 + 0.578503i $$0.803637\pi$$
$$152$$ −3.01729 −0.244734
$$153$$ 10.9036 0.881507
$$154$$ −4.52364 −0.364525
$$155$$ 0 0
$$156$$ 0.439405 0.0351806
$$157$$ 13.2078 1.05410 0.527048 0.849836i $$-0.323299\pi$$
0.527048 + 0.849836i $$0.323299\pi$$
$$158$$ 6.81753 0.542374
$$159$$ 3.64322 0.288926
$$160$$ 0 0
$$161$$ 27.8471 2.19466
$$162$$ −8.79807 −0.691242
$$163$$ −14.4001 −1.12790 −0.563951 0.825808i $$-0.690719\pi$$
−0.563951 + 0.825808i $$0.690719\pi$$
$$164$$ −0.572323 −0.0446909
$$165$$ 0 0
$$166$$ 4.95411 0.384513
$$167$$ −20.5870 −1.59307 −0.796534 0.604593i $$-0.793336\pi$$
−0.796534 + 0.604593i $$0.793336\pi$$
$$168$$ 5.27039 0.406620
$$169$$ −6.47347 −0.497959
$$170$$ 0 0
$$171$$ 2.75345 0.210561
$$172$$ −2.88952 −0.220324
$$173$$ −3.10430 −0.236016 −0.118008 0.993013i $$-0.537651\pi$$
−0.118008 + 0.993013i $$0.537651\pi$$
$$174$$ −1.95532 −0.148233
$$175$$ 0 0
$$176$$ 3.18722 0.240246
$$177$$ 0.827761 0.0622183
$$178$$ 11.3511 0.850804
$$179$$ 13.5070 1.00956 0.504782 0.863247i $$-0.331573\pi$$
0.504782 + 0.863247i $$0.331573\pi$$
$$180$$ 0 0
$$181$$ 11.0629 0.822300 0.411150 0.911568i $$-0.365127\pi$$
0.411150 + 0.911568i $$0.365127\pi$$
$$182$$ 11.5566 0.856630
$$183$$ 1.81210 0.133955
$$184$$ −23.8850 −1.76082
$$185$$ 0 0
$$186$$ 0.718555 0.0526870
$$187$$ 3.96000 0.289584
$$188$$ 4.52684 0.330154
$$189$$ −10.0497 −0.731010
$$190$$ 0 0
$$191$$ 19.8472 1.43610 0.718048 0.695994i $$-0.245036\pi$$
0.718048 + 0.695994i $$0.245036\pi$$
$$192$$ −4.40136 −0.317641
$$193$$ −17.1693 −1.23587 −0.617936 0.786228i $$-0.712031\pi$$
−0.617936 + 0.786228i $$0.712031\pi$$
$$194$$ 2.46631 0.177071
$$195$$ 0 0
$$196$$ −1.86184 −0.132989
$$197$$ 7.76076 0.552931 0.276466 0.961024i $$-0.410837\pi$$
0.276466 + 0.961024i $$0.410837\pi$$
$$198$$ −3.54073 −0.251629
$$199$$ 13.9454 0.988560 0.494280 0.869303i $$-0.335432\pi$$
0.494280 + 0.869303i $$0.335432\pi$$
$$200$$ 0 0
$$201$$ 5.24243 0.369773
$$202$$ 3.84139 0.270279
$$203$$ 10.7726 0.756086
$$204$$ −0.681113 −0.0476874
$$205$$ 0 0
$$206$$ 11.3828 0.793075
$$207$$ 21.7964 1.51496
$$208$$ −8.14243 −0.564576
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ 6.11231 0.420789 0.210394 0.977617i $$-0.432525\pi$$
0.210394 + 0.977617i $$0.432525\pi$$
$$212$$ 2.54156 0.174555
$$213$$ −6.92049 −0.474184
$$214$$ −6.22637 −0.425626
$$215$$ 0 0
$$216$$ 8.61985 0.586506
$$217$$ −3.95878 −0.268739
$$218$$ 6.02445 0.408027
$$219$$ 4.84626 0.327480
$$220$$ 0 0
$$221$$ −10.1166 −0.680519
$$222$$ −4.08519 −0.274180
$$223$$ 15.2282 1.01975 0.509877 0.860247i $$-0.329691\pi$$
0.509877 + 0.860247i $$0.329691\pi$$
$$224$$ 6.81061 0.455053
$$225$$ 0 0
$$226$$ 24.3037 1.61666
$$227$$ 23.3830 1.55199 0.775993 0.630741i $$-0.217249\pi$$
0.775993 + 0.630741i $$0.217249\pi$$
$$228$$ −0.171998 −0.0113909
$$229$$ −0.0884448 −0.00584460 −0.00292230 0.999996i $$-0.500930\pi$$
−0.00292230 + 0.999996i $$0.500930\pi$$
$$230$$ 0 0
$$231$$ −1.74673 −0.114926
$$232$$ −9.23985 −0.606626
$$233$$ −13.5633 −0.888562 −0.444281 0.895888i $$-0.646541\pi$$
−0.444281 + 0.895888i $$0.646541\pi$$
$$234$$ 9.04553 0.591325
$$235$$ 0 0
$$236$$ 0.577458 0.0375893
$$237$$ 2.63248 0.170998
$$238$$ −17.9136 −1.16117
$$239$$ −14.8357 −0.959642 −0.479821 0.877366i $$-0.659298\pi$$
−0.479821 + 0.877366i $$0.659298\pi$$
$$240$$ 0 0
$$241$$ −0.466928 −0.0300775 −0.0150388 0.999887i $$-0.504787\pi$$
−0.0150388 + 0.999887i $$0.504787\pi$$
$$242$$ −1.28593 −0.0826625
$$243$$ −11.9677 −0.767728
$$244$$ 1.26415 0.0809289
$$245$$ 0 0
$$246$$ 1.05498 0.0672628
$$247$$ −2.55471 −0.162552
$$248$$ 3.39552 0.215616
$$249$$ 1.91295 0.121228
$$250$$ 0 0
$$251$$ 1.71829 0.108458 0.0542288 0.998529i $$-0.482730\pi$$
0.0542288 + 0.998529i $$0.482730\pi$$
$$252$$ −3.35520 −0.211357
$$253$$ 7.91604 0.497677
$$254$$ −17.0555 −1.07015
$$255$$ 0 0
$$256$$ −8.04967 −0.503104
$$257$$ 0.514845 0.0321151 0.0160576 0.999871i $$-0.494889\pi$$
0.0160576 + 0.999871i $$0.494889\pi$$
$$258$$ 5.32631 0.331602
$$259$$ 22.5068 1.39850
$$260$$ 0 0
$$261$$ 8.43189 0.521921
$$262$$ 22.7005 1.40244
$$263$$ 25.7582 1.58832 0.794158 0.607711i $$-0.207912\pi$$
0.794158 + 0.607711i $$0.207912\pi$$
$$264$$ 1.49821 0.0922082
$$265$$ 0 0
$$266$$ −4.52364 −0.277362
$$267$$ 4.38307 0.268239
$$268$$ 3.65719 0.223399
$$269$$ −12.5554 −0.765513 −0.382757 0.923849i $$-0.625025\pi$$
−0.382757 + 0.923849i $$0.625025\pi$$
$$270$$ 0 0
$$271$$ −10.0377 −0.609745 −0.304873 0.952393i $$-0.598614\pi$$
−0.304873 + 0.952393i $$0.598614\pi$$
$$272$$ 12.6214 0.765285
$$273$$ 4.46239 0.270076
$$274$$ −5.68996 −0.343743
$$275$$ 0 0
$$276$$ −1.36155 −0.0819555
$$277$$ −1.86246 −0.111904 −0.0559522 0.998433i $$-0.517819\pi$$
−0.0559522 + 0.998433i $$0.517819\pi$$
$$278$$ −15.9782 −0.958312
$$279$$ −3.09861 −0.185509
$$280$$ 0 0
$$281$$ 13.0311 0.777372 0.388686 0.921370i $$-0.372929\pi$$
0.388686 + 0.921370i $$0.372929\pi$$
$$282$$ −8.34442 −0.496903
$$283$$ 22.4810 1.33636 0.668179 0.744001i $$-0.267074\pi$$
0.668179 + 0.744001i $$0.267074\pi$$
$$284$$ −4.82783 −0.286479
$$285$$ 0 0
$$286$$ 3.28517 0.194256
$$287$$ −5.81224 −0.343085
$$288$$ 5.33079 0.314120
$$289$$ −1.31842 −0.0775541
$$290$$ 0 0
$$291$$ 0.952327 0.0558264
$$292$$ 3.38082 0.197847
$$293$$ −24.4894 −1.43069 −0.715343 0.698774i $$-0.753729\pi$$
−0.715343 + 0.698774i $$0.753729\pi$$
$$294$$ 3.43197 0.200157
$$295$$ 0 0
$$296$$ −19.3045 −1.12205
$$297$$ −2.85682 −0.165769
$$298$$ −12.9537 −0.750390
$$299$$ −20.2232 −1.16954
$$300$$ 0 0
$$301$$ −29.3445 −1.69139
$$302$$ 25.7783 1.48338
$$303$$ 1.48329 0.0852129
$$304$$ 3.18722 0.182800
$$305$$ 0 0
$$306$$ −14.0213 −0.801544
$$307$$ −7.93720 −0.453000 −0.226500 0.974011i $$-0.572728\pi$$
−0.226500 + 0.974011i $$0.572728\pi$$
$$308$$ −1.21854 −0.0694330
$$309$$ 4.39528 0.250039
$$310$$ 0 0
$$311$$ −24.3499 −1.38075 −0.690377 0.723450i $$-0.742555\pi$$
−0.690377 + 0.723450i $$0.742555\pi$$
$$312$$ −3.82748 −0.216688
$$313$$ −24.7333 −1.39801 −0.699005 0.715116i $$-0.746374\pi$$
−0.699005 + 0.715116i $$0.746374\pi$$
$$314$$ −16.9842 −0.958475
$$315$$ 0 0
$$316$$ 1.83646 0.103309
$$317$$ 31.8563 1.78923 0.894615 0.446838i $$-0.147450\pi$$
0.894615 + 0.446838i $$0.147450\pi$$
$$318$$ −4.68492 −0.262717
$$319$$ 3.06230 0.171456
$$320$$ 0 0
$$321$$ −2.40422 −0.134190
$$322$$ −35.8093 −1.99557
$$323$$ 3.96000 0.220340
$$324$$ −2.36996 −0.131664
$$325$$ 0 0
$$326$$ 18.5175 1.02559
$$327$$ 2.32625 0.128642
$$328$$ 4.98527 0.275265
$$329$$ 45.9724 2.53454
$$330$$ 0 0
$$331$$ −0.532253 −0.0292552 −0.0146276 0.999893i $$-0.504656\pi$$
−0.0146276 + 0.999893i $$0.504656\pi$$
$$332$$ 1.33450 0.0732403
$$333$$ 17.6165 0.965376
$$334$$ 26.4733 1.44856
$$335$$ 0 0
$$336$$ −5.56723 −0.303717
$$337$$ −21.0568 −1.14704 −0.573519 0.819192i $$-0.694422\pi$$
−0.573519 + 0.819192i $$0.694422\pi$$
$$338$$ 8.32441 0.452788
$$339$$ 9.38451 0.509696
$$340$$ 0 0
$$341$$ −1.12535 −0.0609414
$$342$$ −3.54073 −0.191461
$$343$$ 5.71665 0.308670
$$344$$ 25.1694 1.35704
$$345$$ 0 0
$$346$$ 3.99190 0.214606
$$347$$ 29.1458 1.56463 0.782316 0.622882i $$-0.214038\pi$$
0.782316 + 0.622882i $$0.214038\pi$$
$$348$$ −0.526711 −0.0282347
$$349$$ 22.4161 1.19991 0.599953 0.800035i $$-0.295186\pi$$
0.599953 + 0.800035i $$0.295186\pi$$
$$350$$ 0 0
$$351$$ 7.29834 0.389556
$$352$$ 1.93604 0.103191
$$353$$ −16.4937 −0.877869 −0.438935 0.898519i $$-0.644644\pi$$
−0.438935 + 0.898519i $$0.644644\pi$$
$$354$$ −1.06444 −0.0565743
$$355$$ 0 0
$$356$$ 3.05769 0.162057
$$357$$ −6.91705 −0.366089
$$358$$ −17.3691 −0.917983
$$359$$ 11.3098 0.596910 0.298455 0.954424i $$-0.403529\pi$$
0.298455 + 0.954424i $$0.403529\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −14.2261 −0.747707
$$363$$ −0.496540 −0.0260616
$$364$$ 3.11302 0.163167
$$365$$ 0 0
$$366$$ −2.33023 −0.121803
$$367$$ −4.45032 −0.232305 −0.116152 0.993231i $$-0.537056\pi$$
−0.116152 + 0.993231i $$0.537056\pi$$
$$368$$ 25.2302 1.31522
$$369$$ −4.54934 −0.236829
$$370$$ 0 0
$$371$$ 25.8109 1.34003
$$372$$ 0.193559 0.0100356
$$373$$ −4.45578 −0.230712 −0.115356 0.993324i $$-0.536801\pi$$
−0.115356 + 0.993324i $$0.536801\pi$$
$$374$$ −5.09227 −0.263315
$$375$$ 0 0
$$376$$ −39.4314 −2.03352
$$377$$ −7.82328 −0.402920
$$378$$ 12.9232 0.664698
$$379$$ 15.8233 0.812787 0.406393 0.913698i $$-0.366786\pi$$
0.406393 + 0.913698i $$0.366786\pi$$
$$380$$ 0 0
$$381$$ −6.58570 −0.337396
$$382$$ −25.5221 −1.30582
$$383$$ −1.83670 −0.0938509 −0.0469255 0.998898i $$-0.514942\pi$$
−0.0469255 + 0.998898i $$0.514942\pi$$
$$384$$ 3.73718 0.190712
$$385$$ 0 0
$$386$$ 22.0784 1.12376
$$387$$ −22.9685 −1.16755
$$388$$ 0.664356 0.0337276
$$389$$ −17.5818 −0.891433 −0.445717 0.895174i $$-0.647051\pi$$
−0.445717 + 0.895174i $$0.647051\pi$$
$$390$$ 0 0
$$391$$ 31.3475 1.58531
$$392$$ 16.2177 0.819119
$$393$$ 8.76544 0.442158
$$394$$ −9.97977 −0.502774
$$395$$ 0 0
$$396$$ −0.953776 −0.0479291
$$397$$ 8.70488 0.436885 0.218443 0.975850i $$-0.429902\pi$$
0.218443 + 0.975850i $$0.429902\pi$$
$$398$$ −17.9327 −0.898885
$$399$$ −1.74673 −0.0874459
$$400$$ 0 0
$$401$$ 28.1153 1.40401 0.702006 0.712171i $$-0.252288\pi$$
0.702006 + 0.712171i $$0.252288\pi$$
$$402$$ −6.74138 −0.336230
$$403$$ 2.87495 0.143212
$$404$$ 1.03476 0.0514815
$$405$$ 0 0
$$406$$ −13.8527 −0.687500
$$407$$ 6.39797 0.317135
$$408$$ 5.93289 0.293722
$$409$$ −11.2277 −0.555172 −0.277586 0.960701i $$-0.589534\pi$$
−0.277586 + 0.960701i $$0.589534\pi$$
$$410$$ 0 0
$$411$$ −2.19709 −0.108374
$$412$$ 3.06621 0.151061
$$413$$ 5.86438 0.288567
$$414$$ −28.0286 −1.37753
$$415$$ 0 0
$$416$$ −4.94602 −0.242499
$$417$$ −6.16975 −0.302134
$$418$$ −1.28593 −0.0628967
$$419$$ −10.3008 −0.503228 −0.251614 0.967828i $$-0.580961\pi$$
−0.251614 + 0.967828i $$0.580961\pi$$
$$420$$ 0 0
$$421$$ 16.7955 0.818564 0.409282 0.912408i $$-0.365779\pi$$
0.409282 + 0.912408i $$0.365779\pi$$
$$422$$ −7.85998 −0.382618
$$423$$ 35.9834 1.74957
$$424$$ −22.1385 −1.07514
$$425$$ 0 0
$$426$$ 8.89925 0.431170
$$427$$ 12.8381 0.621278
$$428$$ −1.67721 −0.0810712
$$429$$ 1.26852 0.0612445
$$430$$ 0 0
$$431$$ 14.2315 0.685506 0.342753 0.939425i $$-0.388641\pi$$
0.342753 + 0.939425i $$0.388641\pi$$
$$432$$ −9.10533 −0.438080
$$433$$ 28.8922 1.38847 0.694236 0.719747i $$-0.255742\pi$$
0.694236 + 0.719747i $$0.255742\pi$$
$$434$$ 5.09069 0.244361
$$435$$ 0 0
$$436$$ 1.62282 0.0777190
$$437$$ 7.91604 0.378676
$$438$$ −6.23193 −0.297773
$$439$$ 28.1483 1.34344 0.671722 0.740803i $$-0.265555\pi$$
0.671722 + 0.740803i $$0.265555\pi$$
$$440$$ 0 0
$$441$$ −14.7996 −0.704743
$$442$$ 13.0092 0.618787
$$443$$ 19.5365 0.928205 0.464102 0.885782i $$-0.346377\pi$$
0.464102 + 0.885782i $$0.346377\pi$$
$$444$$ −1.10044 −0.0522245
$$445$$ 0 0
$$446$$ −19.5823 −0.927249
$$447$$ −5.00189 −0.236581
$$448$$ −31.1820 −1.47321
$$449$$ −20.5402 −0.969351 −0.484675 0.874694i $$-0.661062\pi$$
−0.484675 + 0.874694i $$0.661062\pi$$
$$450$$ 0 0
$$451$$ −1.65223 −0.0778007
$$452$$ 6.54676 0.307934
$$453$$ 9.95389 0.467675
$$454$$ −30.0689 −1.41120
$$455$$ 0 0
$$456$$ 1.49821 0.0701599
$$457$$ 12.1384 0.567812 0.283906 0.958852i $$-0.408370\pi$$
0.283906 + 0.958852i $$0.408370\pi$$
$$458$$ 0.113733 0.00531442
$$459$$ −11.3130 −0.528045
$$460$$ 0 0
$$461$$ −29.3582 −1.36735 −0.683673 0.729788i $$-0.739619\pi$$
−0.683673 + 0.729788i $$0.739619\pi$$
$$462$$ 2.24617 0.104501
$$463$$ 0.547114 0.0254265 0.0127133 0.999919i $$-0.495953\pi$$
0.0127133 + 0.999919i $$0.495953\pi$$
$$464$$ 9.76024 0.453108
$$465$$ 0 0
$$466$$ 17.4414 0.807958
$$467$$ −3.82267 −0.176892 −0.0884460 0.996081i $$-0.528190\pi$$
−0.0884460 + 0.996081i $$0.528190\pi$$
$$468$$ 2.43662 0.112633
$$469$$ 37.1407 1.71500
$$470$$ 0 0
$$471$$ −6.55819 −0.302186
$$472$$ −5.02999 −0.231524
$$473$$ −8.34172 −0.383553
$$474$$ −3.38518 −0.155486
$$475$$ 0 0
$$476$$ −4.82543 −0.221173
$$477$$ 20.2026 0.925015
$$478$$ 19.0776 0.872590
$$479$$ −0.914772 −0.0417970 −0.0208985 0.999782i $$-0.506653\pi$$
−0.0208985 + 0.999782i $$0.506653\pi$$
$$480$$ 0 0
$$481$$ −16.3449 −0.745265
$$482$$ 0.600436 0.0273491
$$483$$ −13.8272 −0.629159
$$484$$ −0.346393 −0.0157452
$$485$$ 0 0
$$486$$ 15.3896 0.698085
$$487$$ −36.8291 −1.66888 −0.834442 0.551095i $$-0.814210\pi$$
−0.834442 + 0.551095i $$0.814210\pi$$
$$488$$ −11.0115 −0.498466
$$489$$ 7.15023 0.323345
$$490$$ 0 0
$$491$$ −20.3717 −0.919360 −0.459680 0.888085i $$-0.652036\pi$$
−0.459680 + 0.888085i $$0.652036\pi$$
$$492$$ 0.284182 0.0128119
$$493$$ 12.1267 0.546159
$$494$$ 3.28517 0.147807
$$495$$ 0 0
$$496$$ −3.58676 −0.161050
$$497$$ −49.0291 −2.19926
$$498$$ −2.45992 −0.110231
$$499$$ 9.18801 0.411312 0.205656 0.978624i $$-0.434067\pi$$
0.205656 + 0.978624i $$0.434067\pi$$
$$500$$ 0 0
$$501$$ 10.2223 0.456697
$$502$$ −2.20960 −0.0986192
$$503$$ 40.2078 1.79278 0.896390 0.443267i $$-0.146181\pi$$
0.896390 + 0.443267i $$0.146181\pi$$
$$504$$ 29.2257 1.30182
$$505$$ 0 0
$$506$$ −10.1794 −0.452532
$$507$$ 3.21434 0.142754
$$508$$ −4.59427 −0.203838
$$509$$ 43.0322 1.90737 0.953684 0.300811i $$-0.0972575\pi$$
0.953684 + 0.300811i $$0.0972575\pi$$
$$510$$ 0 0
$$511$$ 34.3339 1.51884
$$512$$ 25.4042 1.12272
$$513$$ −2.85682 −0.126132
$$514$$ −0.662052 −0.0292019
$$515$$ 0 0
$$516$$ 1.43476 0.0631619
$$517$$ 13.0685 0.574751
$$518$$ −28.9421 −1.27164
$$519$$ 1.54141 0.0676604
$$520$$ 0 0
$$521$$ 34.9660 1.53189 0.765944 0.642908i $$-0.222272\pi$$
0.765944 + 0.642908i $$0.222272\pi$$
$$522$$ −10.8428 −0.474576
$$523$$ 17.5898 0.769147 0.384574 0.923094i $$-0.374348\pi$$
0.384574 + 0.923094i $$0.374348\pi$$
$$524$$ 6.11489 0.267130
$$525$$ 0 0
$$526$$ −33.1231 −1.44424
$$527$$ −4.45640 −0.194124
$$528$$ −1.58259 −0.0688732
$$529$$ 39.6637 1.72451
$$530$$ 0 0
$$531$$ 4.59015 0.199196
$$532$$ −1.21854 −0.0528306
$$533$$ 4.22098 0.182831
$$534$$ −5.63630 −0.243907
$$535$$ 0 0
$$536$$ −31.8563 −1.37598
$$537$$ −6.70679 −0.289419
$$538$$ 16.1453 0.696071
$$539$$ −5.37494 −0.231515
$$540$$ 0 0
$$541$$ 24.1969 1.04030 0.520152 0.854074i $$-0.325875\pi$$
0.520152 + 0.854074i $$0.325875\pi$$
$$542$$ 12.9077 0.554434
$$543$$ −5.49319 −0.235735
$$544$$ 7.66672 0.328708
$$545$$ 0 0
$$546$$ −5.73830 −0.245577
$$547$$ 11.6589 0.498498 0.249249 0.968439i $$-0.419816\pi$$
0.249249 + 0.968439i $$0.419816\pi$$
$$548$$ −1.53272 −0.0654745
$$549$$ 10.0486 0.428864
$$550$$ 0 0
$$551$$ 3.06230 0.130458
$$552$$ 11.8599 0.504789
$$553$$ 18.6502 0.793085
$$554$$ 2.39499 0.101753
$$555$$ 0 0
$$556$$ −4.30410 −0.182535
$$557$$ 0.691233 0.0292885 0.0146442 0.999893i $$-0.495338\pi$$
0.0146442 + 0.999893i $$0.495338\pi$$
$$558$$ 3.98458 0.168681
$$559$$ 21.3107 0.901345
$$560$$ 0 0
$$561$$ −1.96630 −0.0830172
$$562$$ −16.7571 −0.706854
$$563$$ −46.5572 −1.96215 −0.981075 0.193627i $$-0.937975\pi$$
−0.981075 + 0.193627i $$0.937975\pi$$
$$564$$ −2.24776 −0.0946477
$$565$$ 0 0
$$566$$ −28.9089 −1.21513
$$567$$ −24.0682 −1.01077
$$568$$ 42.0532 1.76451
$$569$$ −17.3675 −0.728083 −0.364041 0.931383i $$-0.618603\pi$$
−0.364041 + 0.931383i $$0.618603\pi$$
$$570$$ 0 0
$$571$$ 30.8085 1.28930 0.644649 0.764479i $$-0.277004\pi$$
0.644649 + 0.764479i $$0.277004\pi$$
$$572$$ 0.884934 0.0370009
$$573$$ −9.85496 −0.411697
$$574$$ 7.47411 0.311963
$$575$$ 0 0
$$576$$ −24.4067 −1.01695
$$577$$ 9.11130 0.379308 0.189654 0.981851i $$-0.439263\pi$$
0.189654 + 0.981851i $$0.439263\pi$$
$$578$$ 1.69539 0.0705190
$$579$$ 8.52525 0.354297
$$580$$ 0 0
$$581$$ 13.5525 0.562254
$$582$$ −1.22462 −0.0507622
$$583$$ 7.33721 0.303876
$$584$$ −29.4489 −1.21860
$$585$$ 0 0
$$586$$ 31.4916 1.30090
$$587$$ 26.4096 1.09004 0.545021 0.838422i $$-0.316522\pi$$
0.545021 + 0.838422i $$0.316522\pi$$
$$588$$ 0.924480 0.0381249
$$589$$ −1.12535 −0.0463694
$$590$$ 0 0
$$591$$ −3.85353 −0.158513
$$592$$ 20.3918 0.838096
$$593$$ −37.1161 −1.52417 −0.762087 0.647475i $$-0.775825\pi$$
−0.762087 + 0.647475i $$0.775825\pi$$
$$594$$ 3.67366 0.150732
$$595$$ 0 0
$$596$$ −3.48938 −0.142931
$$597$$ −6.92443 −0.283398
$$598$$ 26.0055 1.06344
$$599$$ −44.1959 −1.80580 −0.902899 0.429853i $$-0.858565\pi$$
−0.902899 + 0.429853i $$0.858565\pi$$
$$600$$ 0 0
$$601$$ 26.0883 1.06416 0.532081 0.846693i $$-0.321410\pi$$
0.532081 + 0.846693i $$0.321410\pi$$
$$602$$ 37.7349 1.53796
$$603$$ 29.0707 1.18385
$$604$$ 6.94397 0.282546
$$605$$ 0 0
$$606$$ −1.90740 −0.0774830
$$607$$ −17.9808 −0.729819 −0.364910 0.931043i $$-0.618900\pi$$
−0.364910 + 0.931043i $$0.618900\pi$$
$$608$$ 1.93604 0.0785169
$$609$$ −5.34902 −0.216753
$$610$$ 0 0
$$611$$ −33.3862 −1.35066
$$612$$ −3.77695 −0.152674
$$613$$ 7.45345 0.301042 0.150521 0.988607i $$-0.451905\pi$$
0.150521 + 0.988607i $$0.451905\pi$$
$$614$$ 10.2067 0.411907
$$615$$ 0 0
$$616$$ 10.6142 0.427659
$$617$$ −22.5986 −0.909785 −0.454892 0.890546i $$-0.650322\pi$$
−0.454892 + 0.890546i $$0.650322\pi$$
$$618$$ −5.65200 −0.227357
$$619$$ 34.2496 1.37661 0.688303 0.725423i $$-0.258356\pi$$
0.688303 + 0.725423i $$0.258356\pi$$
$$620$$ 0 0
$$621$$ −22.6147 −0.907497
$$622$$ 31.3121 1.25550
$$623$$ 31.0524 1.24409
$$624$$ 4.04304 0.161851
$$625$$ 0 0
$$626$$ 31.8052 1.27119
$$627$$ −0.496540 −0.0198299
$$628$$ −4.57509 −0.182566
$$629$$ 25.3359 1.01021
$$630$$ 0 0
$$631$$ −20.4459 −0.813940 −0.406970 0.913442i $$-0.633415\pi$$
−0.406970 + 0.913442i $$0.633415\pi$$
$$632$$ −15.9966 −0.636311
$$633$$ −3.03501 −0.120631
$$634$$ −40.9649 −1.62692
$$635$$ 0 0
$$636$$ −1.26199 −0.0500411
$$637$$ 13.7314 0.544057
$$638$$ −3.93789 −0.155903
$$639$$ −38.3760 −1.51813
$$640$$ 0 0
$$641$$ −32.1759 −1.27087 −0.635436 0.772154i $$-0.719180\pi$$
−0.635436 + 0.772154i $$0.719180\pi$$
$$642$$ 3.09165 0.122018
$$643$$ 6.17037 0.243336 0.121668 0.992571i $$-0.461176\pi$$
0.121668 + 0.992571i $$0.461176\pi$$
$$644$$ −9.64604 −0.380107
$$645$$ 0 0
$$646$$ −5.09227 −0.200352
$$647$$ 20.2691 0.796861 0.398430 0.917199i $$-0.369555\pi$$
0.398430 + 0.917199i $$0.369555\pi$$
$$648$$ 20.6437 0.810963
$$649$$ 1.66706 0.0654377
$$650$$ 0 0
$$651$$ 1.96569 0.0770415
$$652$$ 4.98810 0.195349
$$653$$ 49.9790 1.95583 0.977916 0.209000i $$-0.0670210\pi$$
0.977916 + 0.209000i $$0.0670210\pi$$
$$654$$ −2.99138 −0.116972
$$655$$ 0 0
$$656$$ −5.26604 −0.205604
$$657$$ 26.8738 1.04845
$$658$$ −59.1171 −2.30462
$$659$$ −39.7429 −1.54816 −0.774082 0.633085i $$-0.781788\pi$$
−0.774082 + 0.633085i $$0.781788\pi$$
$$660$$ 0 0
$$661$$ −1.09749 −0.0426876 −0.0213438 0.999772i $$-0.506794\pi$$
−0.0213438 + 0.999772i $$0.506794\pi$$
$$662$$ 0.684438 0.0266014
$$663$$ 5.02332 0.195090
$$664$$ −11.6243 −0.451110
$$665$$ 0 0
$$666$$ −22.6535 −0.877804
$$667$$ 24.2413 0.938627
$$668$$ 7.13120 0.275914
$$669$$ −7.56140 −0.292341
$$670$$ 0 0
$$671$$ 3.64946 0.140886
$$672$$ −3.38174 −0.130454
$$673$$ −14.4864 −0.558410 −0.279205 0.960231i $$-0.590071\pi$$
−0.279205 + 0.960231i $$0.590071\pi$$
$$674$$ 27.0775 1.04299
$$675$$ 0 0
$$676$$ 2.24237 0.0862449
$$677$$ 21.4670 0.825045 0.412522 0.910947i $$-0.364648\pi$$
0.412522 + 0.910947i $$0.364648\pi$$
$$678$$ −12.0678 −0.463460
$$679$$ 6.74688 0.258921
$$680$$ 0 0
$$681$$ −11.6106 −0.444920
$$682$$ 1.44712 0.0554132
$$683$$ −13.1753 −0.504140 −0.252070 0.967709i $$-0.581111\pi$$
−0.252070 + 0.967709i $$0.581111\pi$$
$$684$$ −0.953776 −0.0364685
$$685$$ 0 0
$$686$$ −7.35120 −0.280670
$$687$$ 0.0439164 0.00167552
$$688$$ −26.5869 −1.01362
$$689$$ −18.7444 −0.714106
$$690$$ 0 0
$$691$$ 44.3448 1.68695 0.843477 0.537165i $$-0.180505\pi$$
0.843477 + 0.537165i $$0.180505\pi$$
$$692$$ 1.07531 0.0408771
$$693$$ −9.68609 −0.367944
$$694$$ −37.4794 −1.42270
$$695$$ 0 0
$$696$$ 4.58796 0.173906
$$697$$ −6.54284 −0.247828
$$698$$ −28.8255 −1.09106
$$699$$ 6.73473 0.254731
$$700$$ 0 0
$$701$$ −25.5155 −0.963707 −0.481853 0.876252i $$-0.660036\pi$$
−0.481853 + 0.876252i $$0.660036\pi$$
$$702$$ −9.38512 −0.354219
$$703$$ 6.39797 0.241304
$$704$$ −8.86406 −0.334077
$$705$$ 0 0
$$706$$ 21.2096 0.798235
$$707$$ 10.5086 0.395215
$$708$$ −0.286731 −0.0107760
$$709$$ −7.94123 −0.298239 −0.149120 0.988819i $$-0.547644\pi$$
−0.149120 + 0.988819i $$0.547644\pi$$
$$710$$ 0 0
$$711$$ 14.5978 0.547461
$$712$$ −26.6342 −0.998161
$$713$$ −8.90836 −0.333621
$$714$$ 8.89482 0.332880
$$715$$ 0 0
$$716$$ −4.67875 −0.174853
$$717$$ 7.36653 0.275108
$$718$$ −14.5436 −0.542762
$$719$$ 18.7440 0.699035 0.349517 0.936930i $$-0.386346\pi$$
0.349517 + 0.936930i $$0.386346\pi$$
$$720$$ 0 0
$$721$$ 31.1389 1.15967
$$722$$ −1.28593 −0.0478572
$$723$$ 0.231849 0.00862255
$$724$$ −3.83212 −0.142420
$$725$$ 0 0
$$726$$ 0.638514 0.0236975
$$727$$ 40.5490 1.50388 0.751940 0.659232i $$-0.229119\pi$$
0.751940 + 0.659232i $$0.229119\pi$$
$$728$$ −27.1162 −1.00499
$$729$$ −14.5830 −0.540111
$$730$$ 0 0
$$731$$ −33.0332 −1.22178
$$732$$ −0.627701 −0.0232005
$$733$$ −25.5111 −0.942274 −0.471137 0.882060i $$-0.656156\pi$$
−0.471137 + 0.882060i $$0.656156\pi$$
$$734$$ 5.72278 0.211232
$$735$$ 0 0
$$736$$ 15.3258 0.564916
$$737$$ 10.5579 0.388906
$$738$$ 5.85012 0.215346
$$739$$ −39.4859 −1.45251 −0.726256 0.687424i $$-0.758741\pi$$
−0.726256 + 0.687424i $$0.758741\pi$$
$$740$$ 0 0
$$741$$ 1.26852 0.0466001
$$742$$ −33.1909 −1.21848
$$743$$ 3.90702 0.143335 0.0716674 0.997429i $$-0.477168\pi$$
0.0716674 + 0.997429i $$0.477168\pi$$
$$744$$ −1.68601 −0.0618122
$$745$$ 0 0
$$746$$ 5.72980 0.209783
$$747$$ 10.6078 0.388120
$$748$$ −1.37172 −0.0501549
$$749$$ −17.0330 −0.622371
$$750$$ 0 0
$$751$$ 17.1331 0.625194 0.312597 0.949886i $$-0.398801\pi$$
0.312597 + 0.949886i $$0.398801\pi$$
$$752$$ 41.6522 1.51890
$$753$$ −0.853202 −0.0310924
$$754$$ 10.0602 0.366370
$$755$$ 0 0
$$756$$ 3.48116 0.126609
$$757$$ −23.8080 −0.865315 −0.432657 0.901558i $$-0.642424\pi$$
−0.432657 + 0.901558i $$0.642424\pi$$
$$758$$ −20.3476 −0.739057
$$759$$ −3.93064 −0.142673
$$760$$ 0 0
$$761$$ −21.3690 −0.774627 −0.387313 0.921948i $$-0.626597\pi$$
−0.387313 + 0.921948i $$0.626597\pi$$
$$762$$ 8.46872 0.306789
$$763$$ 16.4806 0.596637
$$764$$ −6.87495 −0.248727
$$765$$ 0 0
$$766$$ 2.36186 0.0853374
$$767$$ −4.25884 −0.153778
$$768$$ 3.99698 0.144229
$$769$$ −34.2173 −1.23391 −0.616954 0.786999i $$-0.711634\pi$$
−0.616954 + 0.786999i $$0.711634\pi$$
$$770$$ 0 0
$$771$$ −0.255641 −0.00920669
$$772$$ 5.94733 0.214049
$$773$$ 48.0641 1.72874 0.864372 0.502852i $$-0.167716\pi$$
0.864372 + 0.502852i $$0.167716\pi$$
$$774$$ 29.5358 1.06164
$$775$$ 0 0
$$776$$ −5.78693 −0.207739
$$777$$ −11.1755 −0.400920
$$778$$ 22.6089 0.810569
$$779$$ −1.65223 −0.0591974
$$780$$ 0 0
$$781$$ −13.9374 −0.498720
$$782$$ −40.3106 −1.44150
$$783$$ −8.74844 −0.312644
$$784$$ −17.1311 −0.611826
$$785$$ 0 0
$$786$$ −11.2717 −0.402049
$$787$$ 42.8263 1.52659 0.763296 0.646049i $$-0.223580\pi$$
0.763296 + 0.646049i $$0.223580\pi$$
$$788$$ −2.68828 −0.0957659
$$789$$ −12.7900 −0.455335
$$790$$ 0 0
$$791$$ 66.4857 2.36396
$$792$$ 8.30795 0.295210
$$793$$ −9.32330 −0.331080
$$794$$ −11.1938 −0.397254
$$795$$ 0 0
$$796$$ −4.83058 −0.171215
$$797$$ −39.4762 −1.39832 −0.699160 0.714965i $$-0.746443\pi$$
−0.699160 + 0.714965i $$0.746443\pi$$
$$798$$ 2.24617 0.0795135
$$799$$ 51.7512 1.83082
$$800$$ 0 0
$$801$$ 24.3053 0.858784
$$802$$ −36.1542 −1.27665
$$803$$ 9.76005 0.344425
$$804$$ −1.81594 −0.0640434
$$805$$ 0 0
$$806$$ −3.69698 −0.130220
$$807$$ 6.23424 0.219456
$$808$$ −9.01341 −0.317091
$$809$$ −25.3312 −0.890596 −0.445298 0.895382i $$-0.646902\pi$$
−0.445298 + 0.895382i $$0.646902\pi$$
$$810$$ 0 0
$$811$$ 0.958500 0.0336575 0.0168287 0.999858i $$-0.494643\pi$$
0.0168287 + 0.999858i $$0.494643\pi$$
$$812$$ −3.73155 −0.130952
$$813$$ 4.98411 0.174800
$$814$$ −8.22731 −0.288367
$$815$$ 0 0
$$816$$ −6.26704 −0.219390
$$817$$ −8.34172 −0.291840
$$818$$ 14.4379 0.504810
$$819$$ 24.7451 0.864664
$$820$$ 0 0
$$821$$ −1.74252 −0.0608144 −0.0304072 0.999538i $$-0.509680\pi$$
−0.0304072 + 0.999538i $$0.509680\pi$$
$$822$$ 2.82529 0.0985434
$$823$$ 24.2792 0.846320 0.423160 0.906055i $$-0.360921\pi$$
0.423160 + 0.906055i $$0.360921\pi$$
$$824$$ −26.7084 −0.930433
$$825$$ 0 0
$$826$$ −7.54116 −0.262390
$$827$$ −24.7673 −0.861242 −0.430621 0.902533i $$-0.641705\pi$$
−0.430621 + 0.902533i $$0.641705\pi$$
$$828$$ −7.55013 −0.262385
$$829$$ 28.8962 1.00361 0.501803 0.864982i $$-0.332670\pi$$
0.501803 + 0.864982i $$0.332670\pi$$
$$830$$ 0 0
$$831$$ 0.924786 0.0320805
$$832$$ 22.6451 0.785077
$$833$$ −21.2847 −0.737472
$$834$$ 7.93385 0.274727
$$835$$ 0 0
$$836$$ −0.346393 −0.0119803
$$837$$ 3.21493 0.111124
$$838$$ 13.2461 0.457579
$$839$$ −27.4268 −0.946878 −0.473439 0.880827i $$-0.656988\pi$$
−0.473439 + 0.880827i $$0.656988\pi$$
$$840$$ 0 0
$$841$$ −19.6223 −0.676631
$$842$$ −21.5978 −0.744310
$$843$$ −6.47048 −0.222855
$$844$$ −2.11726 −0.0728792
$$845$$ 0 0
$$846$$ −46.2720 −1.59086
$$847$$ −3.51780 −0.120873
$$848$$ 23.3853 0.803056
$$849$$ −11.1627 −0.383104
$$850$$ 0 0
$$851$$ 50.6466 1.73614
$$852$$ 2.39721 0.0821272
$$853$$ 0.802929 0.0274918 0.0137459 0.999906i $$-0.495624\pi$$
0.0137459 + 0.999906i $$0.495624\pi$$
$$854$$ −16.5088 −0.564920
$$855$$ 0 0
$$856$$ 14.6095 0.499343
$$857$$ 14.5854 0.498228 0.249114 0.968474i $$-0.419861\pi$$
0.249114 + 0.968474i $$0.419861\pi$$
$$858$$ −1.63122 −0.0556888
$$859$$ −26.3712 −0.899775 −0.449888 0.893085i $$-0.648536\pi$$
−0.449888 + 0.893085i $$0.648536\pi$$
$$860$$ 0 0
$$861$$ 2.88601 0.0983549
$$862$$ −18.3006 −0.623322
$$863$$ −9.36128 −0.318662 −0.159331 0.987225i $$-0.550934\pi$$
−0.159331 + 0.987225i $$0.550934\pi$$
$$864$$ −5.53092 −0.188166
$$865$$ 0 0
$$866$$ −37.1533 −1.26252
$$867$$ 0.654649 0.0222330
$$868$$ 1.37129 0.0465447
$$869$$ 5.30165 0.179846
$$870$$ 0 0
$$871$$ −26.9724 −0.913925
$$872$$ −14.1357 −0.478696
$$873$$ 5.28090 0.178731
$$874$$ −10.1794 −0.344325
$$875$$ 0 0
$$876$$ −1.67871 −0.0567184
$$877$$ 41.8351 1.41267 0.706335 0.707878i $$-0.250347\pi$$
0.706335 + 0.707878i $$0.250347\pi$$
$$878$$ −36.1966 −1.22158
$$879$$ 12.1600 0.410146
$$880$$ 0 0
$$881$$ −42.1365 −1.41962 −0.709808 0.704396i $$-0.751218\pi$$
−0.709808 + 0.704396i $$0.751218\pi$$
$$882$$ 19.0312 0.640814
$$883$$ 35.4866 1.19422 0.597109 0.802160i $$-0.296316\pi$$
0.597109 + 0.802160i $$0.296316\pi$$
$$884$$ 3.50434 0.117864
$$885$$ 0 0
$$886$$ −25.1224 −0.844005
$$887$$ 9.49861 0.318932 0.159466 0.987203i $$-0.449023\pi$$
0.159466 + 0.987203i $$0.449023\pi$$
$$888$$ 9.58547 0.321667
$$889$$ −46.6572 −1.56483
$$890$$ 0 0
$$891$$ −6.84182 −0.229209
$$892$$ −5.27494 −0.176618
$$893$$ 13.0685 0.437320
$$894$$ 6.43206 0.215120
$$895$$ 0 0
$$896$$ 26.4765 0.884519
$$897$$ 10.0416 0.335280
$$898$$ 26.4132 0.881418
$$899$$ −3.44617 −0.114936
$$900$$ 0 0
$$901$$ 29.0553 0.967974
$$902$$ 2.12465 0.0707432
$$903$$ 14.5707 0.484884
$$904$$ −57.0261 −1.89666
$$905$$ 0 0
$$906$$ −12.8000 −0.425251
$$907$$ −23.5091 −0.780608 −0.390304 0.920686i $$-0.627630\pi$$
−0.390304 + 0.920686i $$0.627630\pi$$
$$908$$ −8.09973 −0.268799
$$909$$ 8.22524 0.272814
$$910$$ 0 0
$$911$$ −56.8014 −1.88192 −0.940958 0.338524i $$-0.890072\pi$$
−0.940958 + 0.338524i $$0.890072\pi$$
$$912$$ −1.58259 −0.0524047
$$913$$ 3.85256 0.127501
$$914$$ −15.6091 −0.516304
$$915$$ 0 0
$$916$$ 0.0306367 0.00101226
$$917$$ 62.0999 2.05072
$$918$$ 14.5477 0.480145
$$919$$ −28.7186 −0.947338 −0.473669 0.880703i $$-0.657071\pi$$
−0.473669 + 0.880703i $$0.657071\pi$$
$$920$$ 0 0
$$921$$ 3.94114 0.129865
$$922$$ 37.7524 1.24331
$$923$$ 35.6060 1.17199
$$924$$ 0.605056 0.0199049
$$925$$ 0 0
$$926$$ −0.703548 −0.0231200
$$927$$ 24.3730 0.800513
$$928$$ 5.92874 0.194621
$$929$$ 1.43519 0.0470871 0.0235436 0.999723i $$-0.492505\pi$$
0.0235436 + 0.999723i $$0.492505\pi$$
$$930$$ 0 0
$$931$$ −5.37494 −0.176156
$$932$$ 4.69824 0.153896
$$933$$ 12.0907 0.395831
$$934$$ 4.91567 0.160846
$$935$$ 0 0
$$936$$ −21.2244 −0.693740
$$937$$ −30.7588 −1.00485 −0.502423 0.864622i $$-0.667558\pi$$
−0.502423 + 0.864622i $$0.667558\pi$$
$$938$$ −47.7602 −1.55942
$$939$$ 12.2811 0.400779
$$940$$ 0 0
$$941$$ −29.2576 −0.953770 −0.476885 0.878966i $$-0.658234\pi$$
−0.476885 + 0.878966i $$0.658234\pi$$
$$942$$ 8.43335 0.274774
$$943$$ −13.0792 −0.425916
$$944$$ 5.31329 0.172933
$$945$$ 0 0
$$946$$ 10.7268 0.348760
$$947$$ 20.3395 0.660945 0.330472 0.943816i $$-0.392792\pi$$
0.330472 + 0.943816i $$0.392792\pi$$
$$948$$ −0.911875 −0.0296163
$$949$$ −24.9341 −0.809394
$$950$$ 0 0
$$951$$ −15.8180 −0.512932
$$952$$ 42.0323 1.36228
$$953$$ −44.0412 −1.42664 −0.713318 0.700841i $$-0.752808\pi$$
−0.713318 + 0.700841i $$0.752808\pi$$
$$954$$ −25.9791 −0.841104
$$955$$ 0 0
$$956$$ 5.13899 0.166207
$$957$$ −1.52056 −0.0491526
$$958$$ 1.17633 0.0380055
$$959$$ −15.5655 −0.502638
$$960$$ 0 0
$$961$$ −29.7336 −0.959148
$$962$$ 21.0184 0.677660
$$963$$ −13.3320 −0.429618
$$964$$ 0.161741 0.00520932
$$965$$ 0 0
$$966$$ 17.7808 0.572087
$$967$$ −37.4983 −1.20586 −0.602932 0.797793i $$-0.706001\pi$$
−0.602932 + 0.797793i $$0.706001\pi$$
$$968$$ 3.01729 0.0969793
$$969$$ −1.96630 −0.0631666
$$970$$ 0 0
$$971$$ −12.4049 −0.398094 −0.199047 0.979990i $$-0.563785\pi$$
−0.199047 + 0.979990i $$0.563785\pi$$
$$972$$ 4.14553 0.132968
$$973$$ −43.7104 −1.40129
$$974$$ 47.3595 1.51750
$$975$$ 0 0
$$976$$ 11.6317 0.372320
$$977$$ −51.5047 −1.64778 −0.823891 0.566748i $$-0.808201\pi$$
−0.823891 + 0.566748i $$0.808201\pi$$
$$978$$ −9.19467 −0.294013
$$979$$ 8.82721 0.282119
$$980$$ 0 0
$$981$$ 12.8996 0.411854
$$982$$ 26.1965 0.835963
$$983$$ −18.7339 −0.597518 −0.298759 0.954329i $$-0.596573\pi$$
−0.298759 + 0.954329i $$0.596573\pi$$
$$984$$ −2.47539 −0.0789125
$$985$$ 0 0
$$986$$ −15.5940 −0.496616
$$987$$ −22.8271 −0.726596
$$988$$ 0.884934 0.0281535
$$989$$ −66.0334 −2.09974
$$990$$ 0 0
$$991$$ −12.6275 −0.401125 −0.200563 0.979681i $$-0.564277\pi$$
−0.200563 + 0.979681i $$0.564277\pi$$
$$992$$ −2.17873 −0.0691749
$$993$$ 0.264285 0.00838683
$$994$$ 63.0478 1.99976
$$995$$ 0 0
$$996$$ −0.662634 −0.0209964
$$997$$ −37.1826 −1.17758 −0.588792 0.808284i $$-0.700396\pi$$
−0.588792 + 0.808284i $$0.700396\pi$$
$$998$$ −11.8151 −0.374001
$$999$$ −18.2778 −0.578285
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 5225.2.a.bb.1.8 22
5.2 odd 4 1045.2.b.d.419.8 22
5.3 odd 4 1045.2.b.d.419.15 yes 22
5.4 even 2 inner 5225.2.a.bb.1.15 22

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.d.419.8 22 5.2 odd 4
1045.2.b.d.419.15 yes 22 5.3 odd 4
5225.2.a.bb.1.8 22 1.1 even 1 trivial
5225.2.a.bb.1.15 22 5.4 even 2 inner