# Properties

 Label 5225.2.a.j.1.4 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# 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: $$5$$ Coefficient field: $$\Q(\zeta_{22})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1$$ x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$1.30972$$ of defining polynomial 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)

 $$f(q)$$ $$=$$ $$q+1.54620 q^{2} -1.51334 q^{3} +0.390736 q^{4} -2.33992 q^{6} +4.16140 q^{7} -2.48825 q^{8} -0.709811 q^{9} +O(q^{10})$$ $$q+1.54620 q^{2} -1.51334 q^{3} +0.390736 q^{4} -2.33992 q^{6} +4.16140 q^{7} -2.48825 q^{8} -0.709811 q^{9} -1.00000 q^{11} -0.591315 q^{12} -1.82862 q^{13} +6.43436 q^{14} -4.62880 q^{16} -4.80511 q^{17} -1.09751 q^{18} +1.00000 q^{19} -6.29760 q^{21} -1.54620 q^{22} +5.53843 q^{23} +3.76555 q^{24} -2.82741 q^{26} +5.61419 q^{27} +1.62601 q^{28} +3.98933 q^{29} -8.51574 q^{31} -2.18056 q^{32} +1.51334 q^{33} -7.42966 q^{34} -0.277348 q^{36} +7.11059 q^{37} +1.54620 q^{38} +2.76732 q^{39} -3.85750 q^{41} -9.73735 q^{42} +12.0581 q^{43} -0.390736 q^{44} +8.56352 q^{46} -5.75936 q^{47} +7.00493 q^{48} +10.3172 q^{49} +7.27175 q^{51} -0.714506 q^{52} +2.82914 q^{53} +8.68067 q^{54} -10.3546 q^{56} -1.51334 q^{57} +6.16831 q^{58} -10.8553 q^{59} +7.35724 q^{61} -13.1670 q^{62} -2.95381 q^{63} +5.88602 q^{64} +2.33992 q^{66} +11.7680 q^{67} -1.87753 q^{68} -8.38151 q^{69} -4.55334 q^{71} +1.76618 q^{72} +6.03445 q^{73} +10.9944 q^{74} +0.390736 q^{76} -4.16140 q^{77} +4.27882 q^{78} +9.68391 q^{79} -6.36674 q^{81} -5.96447 q^{82} -0.508463 q^{83} -2.46070 q^{84} +18.6442 q^{86} -6.03720 q^{87} +2.48825 q^{88} +6.86161 q^{89} -7.60961 q^{91} +2.16406 q^{92} +12.8872 q^{93} -8.90513 q^{94} +3.29992 q^{96} +3.72641 q^{97} +15.9525 q^{98} +0.709811 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10})$$ 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 $$5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26} + 10 q^{27} + 22 q^{28} + 11 q^{29} - 5 q^{31} + 2 q^{32} - 7 q^{33} + 4 q^{34} - 3 q^{36} + 9 q^{37} + 3 q^{38} - 8 q^{39} + 15 q^{41} - 11 q^{42} + 13 q^{43} - 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{48} + 20 q^{49} + 24 q^{51} - q^{52} + 5 q^{53} + 17 q^{54} + 7 q^{57} + 33 q^{58} - 17 q^{59} + 3 q^{61} - 14 q^{62} + 22 q^{63} - 17 q^{64} - 2 q^{66} + 28 q^{67} + 25 q^{68} - 2 q^{69} - 6 q^{71} + 26 q^{72} + 16 q^{73} - 21 q^{74} + 5 q^{76} - 11 q^{77} - 29 q^{78} + 3 q^{79} + q^{81} - 2 q^{82} + 33 q^{83} + 33 q^{84} + 10 q^{86} + 3 q^{88} - 16 q^{89} - 22 q^{91} + 19 q^{92} + 26 q^{93} - 10 q^{94} + 5 q^{96} + 14 q^{97} - 10 q^{98} - 8 q^{99}+O(q^{100})$$ 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 - 5 * q^11 + 7 * q^12 - q^13 - 3 * q^16 + 3 * q^17 + 7 * q^18 + 5 * q^19 + 11 * q^21 - 3 * q^22 + 8 * q^23 + 9 * q^24 - 16 * q^26 + 10 * q^27 + 22 * q^28 + 11 * q^29 - 5 * q^31 + 2 * q^32 - 7 * q^33 + 4 * q^34 - 3 * q^36 + 9 * q^37 + 3 * q^38 - 8 * q^39 + 15 * q^41 - 11 * q^42 + 13 * q^43 - 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^48 + 20 * q^49 + 24 * q^51 - q^52 + 5 * q^53 + 17 * q^54 + 7 * q^57 + 33 * q^58 - 17 * q^59 + 3 * q^61 - 14 * q^62 + 22 * q^63 - 17 * q^64 - 2 * q^66 + 28 * q^67 + 25 * q^68 - 2 * q^69 - 6 * q^71 + 26 * q^72 + 16 * q^73 - 21 * q^74 + 5 * q^76 - 11 * q^77 - 29 * q^78 + 3 * q^79 + q^81 - 2 * q^82 + 33 * q^83 + 33 * q^84 + 10 * q^86 + 3 * q^88 - 16 * q^89 - 22 * q^91 + 19 * q^92 + 26 * q^93 - 10 * q^94 + 5 * q^96 + 14 * q^97 - 10 * q^98 - 8 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.54620 1.09333 0.546664 0.837352i $$-0.315897\pi$$
0.546664 + 0.837352i $$0.315897\pi$$
$$3$$ −1.51334 −0.873726 −0.436863 0.899528i $$-0.643910\pi$$
−0.436863 + 0.899528i $$0.643910\pi$$
$$4$$ 0.390736 0.195368
$$5$$ 0 0
$$6$$ −2.33992 −0.955269
$$7$$ 4.16140 1.57286 0.786430 0.617679i $$-0.211927\pi$$
0.786430 + 0.617679i $$0.211927\pi$$
$$8$$ −2.48825 −0.879728
$$9$$ −0.709811 −0.236604
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −0.591315 −0.170698
$$13$$ −1.82862 −0.507167 −0.253584 0.967313i $$-0.581609\pi$$
−0.253584 + 0.967313i $$0.581609\pi$$
$$14$$ 6.43436 1.71965
$$15$$ 0 0
$$16$$ −4.62880 −1.15720
$$17$$ −4.80511 −1.16541 −0.582705 0.812684i $$-0.698006\pi$$
−0.582705 + 0.812684i $$0.698006\pi$$
$$18$$ −1.09751 −0.258686
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −6.29760 −1.37425
$$22$$ −1.54620 −0.329651
$$23$$ 5.53843 1.15484 0.577421 0.816446i $$-0.304059\pi$$
0.577421 + 0.816446i $$0.304059\pi$$
$$24$$ 3.76555 0.768640
$$25$$ 0 0
$$26$$ −2.82741 −0.554501
$$27$$ 5.61419 1.08045
$$28$$ 1.62601 0.307286
$$29$$ 3.98933 0.740800 0.370400 0.928872i $$-0.379221\pi$$
0.370400 + 0.928872i $$0.379221\pi$$
$$30$$ 0 0
$$31$$ −8.51574 −1.52947 −0.764736 0.644343i $$-0.777131\pi$$
−0.764736 + 0.644343i $$0.777131\pi$$
$$32$$ −2.18056 −0.385472
$$33$$ 1.51334 0.263438
$$34$$ −7.42966 −1.27418
$$35$$ 0 0
$$36$$ −0.277348 −0.0462247
$$37$$ 7.11059 1.16897 0.584487 0.811403i $$-0.301296\pi$$
0.584487 + 0.811403i $$0.301296\pi$$
$$38$$ 1.54620 0.250827
$$39$$ 2.76732 0.443125
$$40$$ 0 0
$$41$$ −3.85750 −0.602441 −0.301220 0.953555i $$-0.597394\pi$$
−0.301220 + 0.953555i $$0.597394\pi$$
$$42$$ −9.73735 −1.50251
$$43$$ 12.0581 1.83884 0.919420 0.393277i $$-0.128659\pi$$
0.919420 + 0.393277i $$0.128659\pi$$
$$44$$ −0.390736 −0.0589056
$$45$$ 0 0
$$46$$ 8.56352 1.26262
$$47$$ −5.75936 −0.840090 −0.420045 0.907503i $$-0.637986\pi$$
−0.420045 + 0.907503i $$0.637986\pi$$
$$48$$ 7.00493 1.01107
$$49$$ 10.3172 1.47389
$$50$$ 0 0
$$51$$ 7.27175 1.01825
$$52$$ −0.714506 −0.0990842
$$53$$ 2.82914 0.388612 0.194306 0.980941i $$-0.437754\pi$$
0.194306 + 0.980941i $$0.437754\pi$$
$$54$$ 8.68067 1.18129
$$55$$ 0 0
$$56$$ −10.3546 −1.38369
$$57$$ −1.51334 −0.200446
$$58$$ 6.16831 0.809938
$$59$$ −10.8553 −1.41324 −0.706619 0.707594i $$-0.749781\pi$$
−0.706619 + 0.707594i $$0.749781\pi$$
$$60$$ 0 0
$$61$$ 7.35724 0.941998 0.470999 0.882134i $$-0.343893\pi$$
0.470999 + 0.882134i $$0.343893\pi$$
$$62$$ −13.1670 −1.67222
$$63$$ −2.95381 −0.372145
$$64$$ 5.88602 0.735752
$$65$$ 0 0
$$66$$ 2.33992 0.288025
$$67$$ 11.7680 1.43769 0.718845 0.695170i $$-0.244671\pi$$
0.718845 + 0.695170i $$0.244671\pi$$
$$68$$ −1.87753 −0.227684
$$69$$ −8.38151 −1.00902
$$70$$ 0 0
$$71$$ −4.55334 −0.540382 −0.270191 0.962807i $$-0.587087\pi$$
−0.270191 + 0.962807i $$0.587087\pi$$
$$72$$ 1.76618 0.208147
$$73$$ 6.03445 0.706278 0.353139 0.935571i $$-0.385114\pi$$
0.353139 + 0.935571i $$0.385114\pi$$
$$74$$ 10.9944 1.27807
$$75$$ 0 0
$$76$$ 0.390736 0.0448204
$$77$$ −4.16140 −0.474235
$$78$$ 4.27882 0.484481
$$79$$ 9.68391 1.08953 0.544763 0.838590i $$-0.316620\pi$$
0.544763 + 0.838590i $$0.316620\pi$$
$$80$$ 0 0
$$81$$ −6.36674 −0.707415
$$82$$ −5.96447 −0.658666
$$83$$ −0.508463 −0.0558110 −0.0279055 0.999611i $$-0.508884\pi$$
−0.0279055 + 0.999611i $$0.508884\pi$$
$$84$$ −2.46070 −0.268484
$$85$$ 0 0
$$86$$ 18.6442 2.01046
$$87$$ −6.03720 −0.647256
$$88$$ 2.48825 0.265248
$$89$$ 6.86161 0.727329 0.363665 0.931530i $$-0.381525\pi$$
0.363665 + 0.931530i $$0.381525\pi$$
$$90$$ 0 0
$$91$$ −7.60961 −0.797704
$$92$$ 2.16406 0.225619
$$93$$ 12.8872 1.33634
$$94$$ −8.90513 −0.918494
$$95$$ 0 0
$$96$$ 3.29992 0.336796
$$97$$ 3.72641 0.378360 0.189180 0.981942i $$-0.439417\pi$$
0.189180 + 0.981942i $$0.439417\pi$$
$$98$$ 15.9525 1.61145
$$99$$ 0.709811 0.0713387
$$100$$ 0 0
$$101$$ −14.0934 −1.40234 −0.701171 0.712994i $$-0.747339\pi$$
−0.701171 + 0.712994i $$0.747339\pi$$
$$102$$ 11.2436 1.11328
$$103$$ −10.8474 −1.06883 −0.534414 0.845223i $$-0.679468\pi$$
−0.534414 + 0.845223i $$0.679468\pi$$
$$104$$ 4.55005 0.446169
$$105$$ 0 0
$$106$$ 4.37442 0.424881
$$107$$ 9.49711 0.918120 0.459060 0.888405i $$-0.348186\pi$$
0.459060 + 0.888405i $$0.348186\pi$$
$$108$$ 2.19367 0.211086
$$109$$ 10.2467 0.981456 0.490728 0.871313i $$-0.336731\pi$$
0.490728 + 0.871313i $$0.336731\pi$$
$$110$$ 0 0
$$111$$ −10.7607 −1.02136
$$112$$ −19.2623 −1.82011
$$113$$ 7.19036 0.676412 0.338206 0.941072i $$-0.390180\pi$$
0.338206 + 0.941072i $$0.390180\pi$$
$$114$$ −2.33992 −0.219154
$$115$$ 0 0
$$116$$ 1.55877 0.144728
$$117$$ 1.29797 0.119998
$$118$$ −16.7845 −1.54513
$$119$$ −19.9960 −1.83303
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 11.3758 1.02991
$$123$$ 5.83770 0.526368
$$124$$ −3.32740 −0.298810
$$125$$ 0 0
$$126$$ −4.56718 −0.406876
$$127$$ 3.35345 0.297570 0.148785 0.988870i $$-0.452464\pi$$
0.148785 + 0.988870i $$0.452464\pi$$
$$128$$ 13.4621 1.18989
$$129$$ −18.2479 −1.60664
$$130$$ 0 0
$$131$$ −5.15388 −0.450297 −0.225148 0.974324i $$-0.572287\pi$$
−0.225148 + 0.974324i $$0.572287\pi$$
$$132$$ 0.591315 0.0514673
$$133$$ 4.16140 0.360839
$$134$$ 18.1957 1.57187
$$135$$ 0 0
$$136$$ 11.9563 1.02524
$$137$$ 11.3104 0.966316 0.483158 0.875533i $$-0.339490\pi$$
0.483158 + 0.875533i $$0.339490\pi$$
$$138$$ −12.9595 −1.10319
$$139$$ 5.14614 0.436490 0.218245 0.975894i $$-0.429967\pi$$
0.218245 + 0.975894i $$0.429967\pi$$
$$140$$ 0 0
$$141$$ 8.71586 0.734008
$$142$$ −7.04038 −0.590816
$$143$$ 1.82862 0.152917
$$144$$ 3.28557 0.273798
$$145$$ 0 0
$$146$$ 9.33046 0.772195
$$147$$ −15.6135 −1.28778
$$148$$ 2.77836 0.228380
$$149$$ 10.3400 0.847089 0.423545 0.905875i $$-0.360786\pi$$
0.423545 + 0.905875i $$0.360786\pi$$
$$150$$ 0 0
$$151$$ −4.86952 −0.396276 −0.198138 0.980174i $$-0.563489\pi$$
−0.198138 + 0.980174i $$0.563489\pi$$
$$152$$ −2.48825 −0.201823
$$153$$ 3.41072 0.275740
$$154$$ −6.43436 −0.518495
$$155$$ 0 0
$$156$$ 1.08129 0.0865724
$$157$$ 17.3831 1.38732 0.693661 0.720302i $$-0.255997\pi$$
0.693661 + 0.720302i $$0.255997\pi$$
$$158$$ 14.9733 1.19121
$$159$$ −4.28144 −0.339541
$$160$$ 0 0
$$161$$ 23.0476 1.81641
$$162$$ −9.84425 −0.773437
$$163$$ −17.0128 −1.33255 −0.666274 0.745707i $$-0.732112\pi$$
−0.666274 + 0.745707i $$0.732112\pi$$
$$164$$ −1.50726 −0.117698
$$165$$ 0 0
$$166$$ −0.786185 −0.0610198
$$167$$ 10.3330 0.799590 0.399795 0.916604i $$-0.369081\pi$$
0.399795 + 0.916604i $$0.369081\pi$$
$$168$$ 15.6700 1.20896
$$169$$ −9.65616 −0.742781
$$170$$ 0 0
$$171$$ −0.709811 −0.0542806
$$172$$ 4.71152 0.359250
$$173$$ −15.1982 −1.15550 −0.577748 0.816215i $$-0.696068\pi$$
−0.577748 + 0.816215i $$0.696068\pi$$
$$174$$ −9.33473 −0.707664
$$175$$ 0 0
$$176$$ 4.62880 0.348909
$$177$$ 16.4277 1.23478
$$178$$ 10.6094 0.795210
$$179$$ −10.2858 −0.768794 −0.384397 0.923168i $$-0.625591\pi$$
−0.384397 + 0.923168i $$0.625591\pi$$
$$180$$ 0 0
$$181$$ 23.0997 1.71699 0.858495 0.512823i $$-0.171400\pi$$
0.858495 + 0.512823i $$0.171400\pi$$
$$182$$ −11.7660 −0.872152
$$183$$ −11.1340 −0.823048
$$184$$ −13.7810 −1.01595
$$185$$ 0 0
$$186$$ 19.9262 1.46106
$$187$$ 4.80511 0.351384
$$188$$ −2.25039 −0.164126
$$189$$ 23.3629 1.69940
$$190$$ 0 0
$$191$$ 13.6788 0.989766 0.494883 0.868960i $$-0.335211\pi$$
0.494883 + 0.868960i $$0.335211\pi$$
$$192$$ −8.90753 −0.642845
$$193$$ −9.48835 −0.682987 −0.341493 0.939884i $$-0.610933\pi$$
−0.341493 + 0.939884i $$0.610933\pi$$
$$194$$ 5.76178 0.413672
$$195$$ 0 0
$$196$$ 4.03131 0.287951
$$197$$ 17.2237 1.22713 0.613567 0.789643i $$-0.289734\pi$$
0.613567 + 0.789643i $$0.289734\pi$$
$$198$$ 1.09751 0.0779966
$$199$$ 14.4608 1.02510 0.512549 0.858658i $$-0.328701\pi$$
0.512549 + 0.858658i $$0.328701\pi$$
$$200$$ 0 0
$$201$$ −17.8090 −1.25615
$$202$$ −21.7911 −1.53322
$$203$$ 16.6012 1.16518
$$204$$ 2.84133 0.198933
$$205$$ 0 0
$$206$$ −16.7723 −1.16858
$$207$$ −3.93124 −0.273240
$$208$$ 8.46430 0.586894
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 1.14231 0.0786401 0.0393200 0.999227i $$-0.487481\pi$$
0.0393200 + 0.999227i $$0.487481\pi$$
$$212$$ 1.10545 0.0759223
$$213$$ 6.89074 0.472146
$$214$$ 14.6844 1.00381
$$215$$ 0 0
$$216$$ −13.9695 −0.950504
$$217$$ −35.4374 −2.40565
$$218$$ 15.8434 1.07305
$$219$$ −9.13215 −0.617094
$$220$$ 0 0
$$221$$ 8.78671 0.591058
$$222$$ −16.6382 −1.11668
$$223$$ −6.86812 −0.459923 −0.229962 0.973200i $$-0.573860\pi$$
−0.229962 + 0.973200i $$0.573860\pi$$
$$224$$ −9.07417 −0.606293
$$225$$ 0 0
$$226$$ 11.1177 0.739541
$$227$$ 0.104970 0.00696713 0.00348356 0.999994i $$-0.498891\pi$$
0.00348356 + 0.999994i $$0.498891\pi$$
$$228$$ −0.591315 −0.0391608
$$229$$ 14.9679 0.989106 0.494553 0.869147i $$-0.335332\pi$$
0.494553 + 0.869147i $$0.335332\pi$$
$$230$$ 0 0
$$231$$ 6.29760 0.414352
$$232$$ −9.92644 −0.651702
$$233$$ 16.8004 1.10063 0.550316 0.834956i $$-0.314507\pi$$
0.550316 + 0.834956i $$0.314507\pi$$
$$234$$ 2.00693 0.131197
$$235$$ 0 0
$$236$$ −4.24155 −0.276101
$$237$$ −14.6550 −0.951946
$$238$$ −30.9178 −2.00410
$$239$$ −3.44343 −0.222737 −0.111369 0.993779i $$-0.535523\pi$$
−0.111369 + 0.993779i $$0.535523\pi$$
$$240$$ 0 0
$$241$$ −14.8135 −0.954222 −0.477111 0.878843i $$-0.658316\pi$$
−0.477111 + 0.878843i $$0.658316\pi$$
$$242$$ 1.54620 0.0993935
$$243$$ −7.20757 −0.462366
$$244$$ 2.87474 0.184036
$$245$$ 0 0
$$246$$ 9.02626 0.575493
$$247$$ −1.82862 −0.116352
$$248$$ 21.1893 1.34552
$$249$$ 0.769475 0.0487635
$$250$$ 0 0
$$251$$ 8.76253 0.553086 0.276543 0.961002i $$-0.410811\pi$$
0.276543 + 0.961002i $$0.410811\pi$$
$$252$$ −1.15416 −0.0727050
$$253$$ −5.53843 −0.348198
$$254$$ 5.18510 0.325342
$$255$$ 0 0
$$256$$ 9.04303 0.565189
$$257$$ −2.52960 −0.157792 −0.0788960 0.996883i $$-0.525140\pi$$
−0.0788960 + 0.996883i $$0.525140\pi$$
$$258$$ −28.2150 −1.75659
$$259$$ 29.5900 1.83863
$$260$$ 0 0
$$261$$ −2.83167 −0.175276
$$262$$ −7.96893 −0.492322
$$263$$ 15.3262 0.945053 0.472527 0.881316i $$-0.343342\pi$$
0.472527 + 0.881316i $$0.343342\pi$$
$$264$$ −3.76555 −0.231754
$$265$$ 0 0
$$266$$ 6.43436 0.394516
$$267$$ −10.3839 −0.635486
$$268$$ 4.59818 0.280878
$$269$$ 19.1624 1.16835 0.584176 0.811627i $$-0.301418\pi$$
0.584176 + 0.811627i $$0.301418\pi$$
$$270$$ 0 0
$$271$$ 30.2567 1.83796 0.918982 0.394299i $$-0.129012\pi$$
0.918982 + 0.394299i $$0.129012\pi$$
$$272$$ 22.2419 1.34861
$$273$$ 11.5159 0.696974
$$274$$ 17.4882 1.05650
$$275$$ 0 0
$$276$$ −3.27495 −0.197129
$$277$$ −4.07877 −0.245069 −0.122535 0.992464i $$-0.539102\pi$$
−0.122535 + 0.992464i $$0.539102\pi$$
$$278$$ 7.95697 0.477227
$$279$$ 6.04457 0.361879
$$280$$ 0 0
$$281$$ 3.04066 0.181390 0.0906952 0.995879i $$-0.471091\pi$$
0.0906952 + 0.995879i $$0.471091\pi$$
$$282$$ 13.4765 0.802512
$$283$$ −2.24468 −0.133432 −0.0667162 0.997772i $$-0.521252\pi$$
−0.0667162 + 0.997772i $$0.521252\pi$$
$$284$$ −1.77915 −0.105573
$$285$$ 0 0
$$286$$ 2.82741 0.167188
$$287$$ −16.0526 −0.947556
$$288$$ 1.54778 0.0912040
$$289$$ 6.08907 0.358181
$$290$$ 0 0
$$291$$ −5.63932 −0.330583
$$292$$ 2.35787 0.137984
$$293$$ −18.3882 −1.07425 −0.537124 0.843503i $$-0.680489\pi$$
−0.537124 + 0.843503i $$0.680489\pi$$
$$294$$ −24.1415 −1.40796
$$295$$ 0 0
$$296$$ −17.6929 −1.02838
$$297$$ −5.61419 −0.325769
$$298$$ 15.9878 0.926147
$$299$$ −10.1277 −0.585698
$$300$$ 0 0
$$301$$ 50.1785 2.89224
$$302$$ −7.52925 −0.433260
$$303$$ 21.3280 1.22526
$$304$$ −4.62880 −0.265480
$$305$$ 0 0
$$306$$ 5.27365 0.301475
$$307$$ −13.5196 −0.771607 −0.385804 0.922581i $$-0.626076\pi$$
−0.385804 + 0.922581i $$0.626076\pi$$
$$308$$ −1.62601 −0.0926503
$$309$$ 16.4158 0.933862
$$310$$ 0 0
$$311$$ 27.3678 1.55188 0.775941 0.630805i $$-0.217275\pi$$
0.775941 + 0.630805i $$0.217275\pi$$
$$312$$ −6.88576 −0.389829
$$313$$ 21.1505 1.19550 0.597748 0.801684i $$-0.296063\pi$$
0.597748 + 0.801684i $$0.296063\pi$$
$$314$$ 26.8777 1.51680
$$315$$ 0 0
$$316$$ 3.78385 0.212858
$$317$$ 4.68075 0.262897 0.131448 0.991323i $$-0.458037\pi$$
0.131448 + 0.991323i $$0.458037\pi$$
$$318$$ −6.61997 −0.371230
$$319$$ −3.98933 −0.223360
$$320$$ 0 0
$$321$$ −14.3723 −0.802185
$$322$$ 35.6362 1.98593
$$323$$ −4.80511 −0.267363
$$324$$ −2.48771 −0.138206
$$325$$ 0 0
$$326$$ −26.3053 −1.45691
$$327$$ −15.5067 −0.857523
$$328$$ 9.59842 0.529984
$$329$$ −23.9670 −1.32134
$$330$$ 0 0
$$331$$ 24.3025 1.33578 0.667892 0.744258i $$-0.267197\pi$$
0.667892 + 0.744258i $$0.267197\pi$$
$$332$$ −0.198674 −0.0109037
$$333$$ −5.04717 −0.276583
$$334$$ 15.9769 0.874215
$$335$$ 0 0
$$336$$ 29.1503 1.59028
$$337$$ −11.6901 −0.636803 −0.318401 0.947956i $$-0.603146\pi$$
−0.318401 + 0.947956i $$0.603146\pi$$
$$338$$ −14.9304 −0.812104
$$339$$ −10.8814 −0.590999
$$340$$ 0 0
$$341$$ 8.51574 0.461153
$$342$$ −1.09751 −0.0593465
$$343$$ 13.8043 0.745365
$$344$$ −30.0035 −1.61768
$$345$$ 0 0
$$346$$ −23.4994 −1.26334
$$347$$ −9.85130 −0.528845 −0.264423 0.964407i $$-0.585181\pi$$
−0.264423 + 0.964407i $$0.585181\pi$$
$$348$$ −2.35895 −0.126453
$$349$$ −3.99813 −0.214015 −0.107007 0.994258i $$-0.534127\pi$$
−0.107007 + 0.994258i $$0.534127\pi$$
$$350$$ 0 0
$$351$$ −10.2662 −0.547970
$$352$$ 2.18056 0.116224
$$353$$ −7.73116 −0.411488 −0.205744 0.978606i $$-0.565961\pi$$
−0.205744 + 0.978606i $$0.565961\pi$$
$$354$$ 25.4005 1.35002
$$355$$ 0 0
$$356$$ 2.68107 0.142097
$$357$$ 30.2606 1.60156
$$358$$ −15.9038 −0.840545
$$359$$ 4.83542 0.255204 0.127602 0.991825i $$-0.459272\pi$$
0.127602 + 0.991825i $$0.459272\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 35.7168 1.87723
$$363$$ −1.51334 −0.0794296
$$364$$ −2.97334 −0.155846
$$365$$ 0 0
$$366$$ −17.2154 −0.899862
$$367$$ 29.8171 1.55644 0.778220 0.627992i $$-0.216123\pi$$
0.778220 + 0.627992i $$0.216123\pi$$
$$368$$ −25.6363 −1.33638
$$369$$ 2.73810 0.142540
$$370$$ 0 0
$$371$$ 11.7732 0.611233
$$372$$ 5.03548 0.261078
$$373$$ 28.2081 1.46056 0.730280 0.683148i $$-0.239390\pi$$
0.730280 + 0.683148i $$0.239390\pi$$
$$374$$ 7.42966 0.384179
$$375$$ 0 0
$$376$$ 14.3307 0.739050
$$377$$ −7.29496 −0.375710
$$378$$ 36.1237 1.85800
$$379$$ −33.7427 −1.73325 −0.866624 0.498962i $$-0.833715\pi$$
−0.866624 + 0.498962i $$0.833715\pi$$
$$380$$ 0 0
$$381$$ −5.07490 −0.259995
$$382$$ 21.1502 1.08214
$$383$$ 29.4579 1.50523 0.752615 0.658461i $$-0.228792\pi$$
0.752615 + 0.658461i $$0.228792\pi$$
$$384$$ −20.3727 −1.03964
$$385$$ 0 0
$$386$$ −14.6709 −0.746729
$$387$$ −8.55896 −0.435076
$$388$$ 1.45604 0.0739193
$$389$$ −24.5828 −1.24640 −0.623199 0.782063i $$-0.714167\pi$$
−0.623199 + 0.782063i $$0.714167\pi$$
$$390$$ 0 0
$$391$$ −26.6128 −1.34586
$$392$$ −25.6718 −1.29662
$$393$$ 7.79956 0.393436
$$394$$ 26.6312 1.34166
$$395$$ 0 0
$$396$$ 0.277348 0.0139373
$$397$$ −22.5952 −1.13402 −0.567009 0.823711i $$-0.691900\pi$$
−0.567009 + 0.823711i $$0.691900\pi$$
$$398$$ 22.3593 1.12077
$$399$$ −6.29760 −0.315274
$$400$$ 0 0
$$401$$ −10.9449 −0.546562 −0.273281 0.961934i $$-0.588109\pi$$
−0.273281 + 0.961934i $$0.588109\pi$$
$$402$$ −27.5362 −1.37338
$$403$$ 15.5720 0.775699
$$404$$ −5.50677 −0.273972
$$405$$ 0 0
$$406$$ 25.6688 1.27392
$$407$$ −7.11059 −0.352459
$$408$$ −18.0939 −0.895781
$$409$$ 30.7338 1.51969 0.759844 0.650106i $$-0.225275\pi$$
0.759844 + 0.650106i $$0.225275\pi$$
$$410$$ 0 0
$$411$$ −17.1165 −0.844295
$$412$$ −4.23847 −0.208814
$$413$$ −45.1732 −2.22283
$$414$$ −6.07848 −0.298741
$$415$$ 0 0
$$416$$ 3.98741 0.195499
$$417$$ −7.78785 −0.381373
$$418$$ −1.54620 −0.0756271
$$419$$ −35.3598 −1.72744 −0.863720 0.503972i $$-0.831871\pi$$
−0.863720 + 0.503972i $$0.831871\pi$$
$$420$$ 0 0
$$421$$ −38.7689 −1.88948 −0.944740 0.327820i $$-0.893686\pi$$
−0.944740 + 0.327820i $$0.893686\pi$$
$$422$$ 1.76624 0.0859795
$$423$$ 4.08806 0.198768
$$424$$ −7.03960 −0.341873
$$425$$ 0 0
$$426$$ 10.6545 0.516211
$$427$$ 30.6164 1.48163
$$428$$ 3.71086 0.179371
$$429$$ −2.76732 −0.133607
$$430$$ 0 0
$$431$$ 29.7370 1.43238 0.716191 0.697904i $$-0.245884\pi$$
0.716191 + 0.697904i $$0.245884\pi$$
$$432$$ −25.9870 −1.25030
$$433$$ 8.54319 0.410559 0.205280 0.978703i $$-0.434190\pi$$
0.205280 + 0.978703i $$0.434190\pi$$
$$434$$ −54.7933 −2.63016
$$435$$ 0 0
$$436$$ 4.00375 0.191745
$$437$$ 5.53843 0.264939
$$438$$ −14.1201 −0.674686
$$439$$ −2.07249 −0.0989146 −0.0494573 0.998776i $$-0.515749\pi$$
−0.0494573 + 0.998776i $$0.515749\pi$$
$$440$$ 0 0
$$441$$ −7.32329 −0.348728
$$442$$ 13.5860 0.646221
$$443$$ −23.8457 −1.13295 −0.566473 0.824081i $$-0.691692\pi$$
−0.566473 + 0.824081i $$0.691692\pi$$
$$444$$ −4.20459 −0.199541
$$445$$ 0 0
$$446$$ −10.6195 −0.502848
$$447$$ −15.6480 −0.740123
$$448$$ 24.4941 1.15724
$$449$$ 5.44413 0.256925 0.128462 0.991714i $$-0.458996\pi$$
0.128462 + 0.991714i $$0.458996\pi$$
$$450$$ 0 0
$$451$$ 3.85750 0.181643
$$452$$ 2.80953 0.132149
$$453$$ 7.36923 0.346236
$$454$$ 0.162305 0.00761736
$$455$$ 0 0
$$456$$ 3.76555 0.176338
$$457$$ 23.4484 1.09687 0.548435 0.836193i $$-0.315224\pi$$
0.548435 + 0.836193i $$0.315224\pi$$
$$458$$ 23.1434 1.08142
$$459$$ −26.9768 −1.25917
$$460$$ 0 0
$$461$$ −37.5982 −1.75112 −0.875560 0.483109i $$-0.839508\pi$$
−0.875560 + 0.483109i $$0.839508\pi$$
$$462$$ 9.73735 0.453022
$$463$$ −27.2719 −1.26743 −0.633716 0.773566i $$-0.718471\pi$$
−0.633716 + 0.773566i $$0.718471\pi$$
$$464$$ −18.4658 −0.857253
$$465$$ 0 0
$$466$$ 25.9768 1.20335
$$467$$ −14.2213 −0.658082 −0.329041 0.944316i $$-0.606725\pi$$
−0.329041 + 0.944316i $$0.606725\pi$$
$$468$$ 0.507164 0.0234437
$$469$$ 48.9713 2.26129
$$470$$ 0 0
$$471$$ −26.3065 −1.21214
$$472$$ 27.0106 1.24327
$$473$$ −12.0581 −0.554431
$$474$$ −22.6596 −1.04079
$$475$$ 0 0
$$476$$ −7.81314 −0.358115
$$477$$ −2.00816 −0.0919471
$$478$$ −5.32424 −0.243525
$$479$$ 1.56056 0.0713037 0.0356518 0.999364i $$-0.488649\pi$$
0.0356518 + 0.999364i $$0.488649\pi$$
$$480$$ 0 0
$$481$$ −13.0025 −0.592865
$$482$$ −22.9047 −1.04328
$$483$$ −34.8788 −1.58704
$$484$$ 0.390736 0.0177607
$$485$$ 0 0
$$486$$ −11.1443 −0.505518
$$487$$ 17.8046 0.806803 0.403402 0.915023i $$-0.367828\pi$$
0.403402 + 0.915023i $$0.367828\pi$$
$$488$$ −18.3066 −0.828702
$$489$$ 25.7462 1.16428
$$490$$ 0 0
$$491$$ 4.97956 0.224724 0.112362 0.993667i $$-0.464158\pi$$
0.112362 + 0.993667i $$0.464158\pi$$
$$492$$ 2.28100 0.102835
$$493$$ −19.1692 −0.863336
$$494$$ −2.82741 −0.127211
$$495$$ 0 0
$$496$$ 39.4176 1.76990
$$497$$ −18.9483 −0.849946
$$498$$ 1.18976 0.0533146
$$499$$ 7.14052 0.319654 0.159827 0.987145i $$-0.448906\pi$$
0.159827 + 0.987145i $$0.448906\pi$$
$$500$$ 0 0
$$501$$ −15.6373 −0.698623
$$502$$ 13.5486 0.604705
$$503$$ −39.9024 −1.77916 −0.889579 0.456781i $$-0.849002\pi$$
−0.889579 + 0.456781i $$0.849002\pi$$
$$504$$ 7.34979 0.327386
$$505$$ 0 0
$$506$$ −8.56352 −0.380695
$$507$$ 14.6130 0.648987
$$508$$ 1.31031 0.0581357
$$509$$ 2.18899 0.0970252 0.0485126 0.998823i $$-0.484552\pi$$
0.0485126 + 0.998823i $$0.484552\pi$$
$$510$$ 0 0
$$511$$ 25.1117 1.11088
$$512$$ −12.9418 −0.571953
$$513$$ 5.61419 0.247873
$$514$$ −3.91127 −0.172519
$$515$$ 0 0
$$516$$ −7.13012 −0.313886
$$517$$ 5.75936 0.253297
$$518$$ 45.7520 2.01023
$$519$$ 23.0000 1.00959
$$520$$ 0 0
$$521$$ 21.6375 0.947957 0.473979 0.880536i $$-0.342817\pi$$
0.473979 + 0.880536i $$0.342817\pi$$
$$522$$ −4.37833 −0.191634
$$523$$ −26.8411 −1.17368 −0.586840 0.809703i $$-0.699628\pi$$
−0.586840 + 0.809703i $$0.699628\pi$$
$$524$$ −2.01380 −0.0879735
$$525$$ 0 0
$$526$$ 23.6974 1.03325
$$527$$ 40.9191 1.78246
$$528$$ −7.00493 −0.304850
$$529$$ 7.67419 0.333661
$$530$$ 0 0
$$531$$ 7.70520 0.334377
$$532$$ 1.62601 0.0704963
$$533$$ 7.05390 0.305538
$$534$$ −16.0556 −0.694795
$$535$$ 0 0
$$536$$ −29.2817 −1.26478
$$537$$ 15.5658 0.671715
$$538$$ 29.6289 1.27739
$$539$$ −10.3172 −0.444395
$$540$$ 0 0
$$541$$ −44.0968 −1.89587 −0.947935 0.318465i $$-0.896833\pi$$
−0.947935 + 0.318465i $$0.896833\pi$$
$$542$$ 46.7829 2.00950
$$543$$ −34.9577 −1.50018
$$544$$ 10.4778 0.449232
$$545$$ 0 0
$$546$$ 17.8059 0.762022
$$547$$ 21.5021 0.919362 0.459681 0.888084i $$-0.347964\pi$$
0.459681 + 0.888084i $$0.347964\pi$$
$$548$$ 4.41939 0.188787
$$549$$ −5.22225 −0.222880
$$550$$ 0 0
$$551$$ 3.98933 0.169951
$$552$$ 20.8553 0.887658
$$553$$ 40.2986 1.71367
$$554$$ −6.30659 −0.267941
$$555$$ 0 0
$$556$$ 2.01078 0.0852761
$$557$$ 33.2260 1.40783 0.703915 0.710284i $$-0.251434\pi$$
0.703915 + 0.710284i $$0.251434\pi$$
$$558$$ 9.34611 0.395652
$$559$$ −22.0496 −0.932600
$$560$$ 0 0
$$561$$ −7.27175 −0.307013
$$562$$ 4.70146 0.198319
$$563$$ 7.31538 0.308306 0.154153 0.988047i $$-0.450735\pi$$
0.154153 + 0.988047i $$0.450735\pi$$
$$564$$ 3.40560 0.143401
$$565$$ 0 0
$$566$$ −3.47073 −0.145885
$$567$$ −26.4945 −1.11267
$$568$$ 11.3298 0.475389
$$569$$ 7.20515 0.302056 0.151028 0.988530i $$-0.451742\pi$$
0.151028 + 0.988530i $$0.451742\pi$$
$$570$$ 0 0
$$571$$ 15.0655 0.630472 0.315236 0.949013i $$-0.397916\pi$$
0.315236 + 0.949013i $$0.397916\pi$$
$$572$$ 0.714506 0.0298750
$$573$$ −20.7007 −0.864783
$$574$$ −24.8206 −1.03599
$$575$$ 0 0
$$576$$ −4.17796 −0.174082
$$577$$ −5.82824 −0.242633 −0.121316 0.992614i $$-0.538712\pi$$
−0.121316 + 0.992614i $$0.538712\pi$$
$$578$$ 9.41492 0.391609
$$579$$ 14.3591 0.596743
$$580$$ 0 0
$$581$$ −2.11592 −0.0877830
$$582$$ −8.71952 −0.361436
$$583$$ −2.82914 −0.117171
$$584$$ −15.0152 −0.621333
$$585$$ 0 0
$$586$$ −28.4318 −1.17451
$$587$$ 40.6364 1.67724 0.838622 0.544714i $$-0.183362\pi$$
0.838622 + 0.544714i $$0.183362\pi$$
$$588$$ −6.10073 −0.251590
$$589$$ −8.51574 −0.350885
$$590$$ 0 0
$$591$$ −26.0652 −1.07218
$$592$$ −32.9135 −1.35273
$$593$$ 32.0839 1.31753 0.658764 0.752350i $$-0.271080\pi$$
0.658764 + 0.752350i $$0.271080\pi$$
$$594$$ −8.68067 −0.356172
$$595$$ 0 0
$$596$$ 4.04022 0.165494
$$597$$ −21.8840 −0.895654
$$598$$ −15.6594 −0.640361
$$599$$ −47.1879 −1.92805 −0.964023 0.265820i $$-0.914357\pi$$
−0.964023 + 0.265820i $$0.914357\pi$$
$$600$$ 0 0
$$601$$ −6.75179 −0.275411 −0.137706 0.990473i $$-0.543973\pi$$
−0.137706 + 0.990473i $$0.543973\pi$$
$$602$$ 77.5860 3.16217
$$603$$ −8.35305 −0.340163
$$604$$ −1.90269 −0.0774195
$$605$$ 0 0
$$606$$ 32.9774 1.33961
$$607$$ −37.4849 −1.52147 −0.760733 0.649065i $$-0.775160\pi$$
−0.760733 + 0.649065i $$0.775160\pi$$
$$608$$ −2.18056 −0.0884332
$$609$$ −25.1232 −1.01804
$$610$$ 0 0
$$611$$ 10.5317 0.426066
$$612$$ 1.33269 0.0538708
$$613$$ −35.4094 −1.43017 −0.715085 0.699037i $$-0.753612\pi$$
−0.715085 + 0.699037i $$0.753612\pi$$
$$614$$ −20.9041 −0.843620
$$615$$ 0 0
$$616$$ 10.3546 0.417198
$$617$$ 27.4510 1.10513 0.552567 0.833469i $$-0.313648\pi$$
0.552567 + 0.833469i $$0.313648\pi$$
$$618$$ 25.3821 1.02102
$$619$$ −7.15830 −0.287716 −0.143858 0.989598i $$-0.545951\pi$$
−0.143858 + 0.989598i $$0.545951\pi$$
$$620$$ 0 0
$$621$$ 31.0938 1.24775
$$622$$ 42.3160 1.69672
$$623$$ 28.5539 1.14399
$$624$$ −12.8093 −0.512784
$$625$$ 0 0
$$626$$ 32.7029 1.30707
$$627$$ 1.51334 0.0604369
$$628$$ 6.79219 0.271038
$$629$$ −34.1671 −1.36233
$$630$$ 0 0
$$631$$ −28.7158 −1.14316 −0.571578 0.820547i $$-0.693669\pi$$
−0.571578 + 0.820547i $$0.693669\pi$$
$$632$$ −24.0959 −0.958485
$$633$$ −1.72870 −0.0687098
$$634$$ 7.23737 0.287433
$$635$$ 0 0
$$636$$ −1.67291 −0.0663353
$$637$$ −18.8663 −0.747509
$$638$$ −6.16831 −0.244206
$$639$$ 3.23201 0.127856
$$640$$ 0 0
$$641$$ −42.9018 −1.69452 −0.847259 0.531180i $$-0.821749\pi$$
−0.847259 + 0.531180i $$0.821749\pi$$
$$642$$ −22.2225 −0.877052
$$643$$ 26.1604 1.03167 0.515834 0.856689i $$-0.327482\pi$$
0.515834 + 0.856689i $$0.327482\pi$$
$$644$$ 9.00552 0.354867
$$645$$ 0 0
$$646$$ −7.42966 −0.292316
$$647$$ −32.5031 −1.27783 −0.638915 0.769277i $$-0.720616\pi$$
−0.638915 + 0.769277i $$0.720616\pi$$
$$648$$ 15.8420 0.622333
$$649$$ 10.8553 0.426107
$$650$$ 0 0
$$651$$ 53.6287 2.10188
$$652$$ −6.64752 −0.260337
$$653$$ −18.9389 −0.741137 −0.370569 0.928805i $$-0.620837\pi$$
−0.370569 + 0.928805i $$0.620837\pi$$
$$654$$ −23.9765 −0.937554
$$655$$ 0 0
$$656$$ 17.8556 0.697144
$$657$$ −4.28331 −0.167108
$$658$$ −37.0578 −1.44466
$$659$$ 43.5740 1.69740 0.848701 0.528873i $$-0.177385\pi$$
0.848701 + 0.528873i $$0.177385\pi$$
$$660$$ 0 0
$$661$$ −34.9291 −1.35858 −0.679292 0.733868i $$-0.737713\pi$$
−0.679292 + 0.733868i $$0.737713\pi$$
$$662$$ 37.5765 1.46045
$$663$$ −13.2973 −0.516422
$$664$$ 1.26518 0.0490985
$$665$$ 0 0
$$666$$ −7.80394 −0.302396
$$667$$ 22.0946 0.855507
$$668$$ 4.03746 0.156214
$$669$$ 10.3938 0.401847
$$670$$ 0 0
$$671$$ −7.35724 −0.284023
$$672$$ 13.7323 0.529734
$$673$$ −39.0030 −1.50346 −0.751728 0.659473i $$-0.770780\pi$$
−0.751728 + 0.659473i $$0.770780\pi$$
$$674$$ −18.0753 −0.696235
$$675$$ 0 0
$$676$$ −3.77300 −0.145116
$$677$$ −27.3864 −1.05255 −0.526273 0.850316i $$-0.676411\pi$$
−0.526273 + 0.850316i $$0.676411\pi$$
$$678$$ −16.8249 −0.646156
$$679$$ 15.5071 0.595107
$$680$$ 0 0
$$681$$ −0.158856 −0.00608736
$$682$$ 13.1670 0.504192
$$683$$ −44.9382 −1.71951 −0.859755 0.510706i $$-0.829384\pi$$
−0.859755 + 0.510706i $$0.829384\pi$$
$$684$$ −0.277348 −0.0106047
$$685$$ 0 0
$$686$$ 21.3443 0.814929
$$687$$ −22.6515 −0.864207
$$688$$ −55.8144 −2.12790
$$689$$ −5.17342 −0.197092
$$690$$ 0 0
$$691$$ 25.4509 0.968199 0.484100 0.875013i $$-0.339147\pi$$
0.484100 + 0.875013i $$0.339147\pi$$
$$692$$ −5.93847 −0.225747
$$693$$ 2.95381 0.112206
$$694$$ −15.2321 −0.578202
$$695$$ 0 0
$$696$$ 15.0220 0.569409
$$697$$ 18.5357 0.702091
$$698$$ −6.18191 −0.233989
$$699$$ −25.4247 −0.961651
$$700$$ 0 0
$$701$$ 50.6146 1.91169 0.955844 0.293873i $$-0.0949444\pi$$
0.955844 + 0.293873i $$0.0949444\pi$$
$$702$$ −15.8736 −0.599111
$$703$$ 7.11059 0.268181
$$704$$ −5.88602 −0.221838
$$705$$ 0 0
$$706$$ −11.9539 −0.449892
$$707$$ −58.6481 −2.20569
$$708$$ 6.41889 0.241237
$$709$$ 25.8266 0.969938 0.484969 0.874531i $$-0.338831\pi$$
0.484969 + 0.874531i $$0.338831\pi$$
$$710$$ 0 0
$$711$$ −6.87374 −0.257786
$$712$$ −17.0734 −0.639852
$$713$$ −47.1638 −1.76630
$$714$$ 46.7890 1.75104
$$715$$ 0 0
$$716$$ −4.01901 −0.150198
$$717$$ 5.21107 0.194611
$$718$$ 7.47653 0.279021
$$719$$ −26.1220 −0.974187 −0.487093 0.873350i $$-0.661943\pi$$
−0.487093 + 0.873350i $$0.661943\pi$$
$$720$$ 0 0
$$721$$ −45.1404 −1.68112
$$722$$ 1.54620 0.0575436
$$723$$ 22.4178 0.833728
$$724$$ 9.02588 0.335444
$$725$$ 0 0
$$726$$ −2.33992 −0.0868427
$$727$$ 32.1853 1.19369 0.596844 0.802357i $$-0.296421\pi$$
0.596844 + 0.802357i $$0.296421\pi$$
$$728$$ 18.9346 0.701762
$$729$$ 30.0077 1.11140
$$730$$ 0 0
$$731$$ −57.9404 −2.14300
$$732$$ −4.35044 −0.160797
$$733$$ 24.9746 0.922457 0.461229 0.887281i $$-0.347409\pi$$
0.461229 + 0.887281i $$0.347409\pi$$
$$734$$ 46.1032 1.70170
$$735$$ 0 0
$$736$$ −12.0769 −0.445159
$$737$$ −11.7680 −0.433480
$$738$$ 4.23365 0.155843
$$739$$ −21.0642 −0.774859 −0.387429 0.921899i $$-0.626637\pi$$
−0.387429 + 0.921899i $$0.626637\pi$$
$$740$$ 0 0
$$741$$ 2.76732 0.101660
$$742$$ 18.2037 0.668279
$$743$$ 19.3902 0.711357 0.355679 0.934608i $$-0.384250\pi$$
0.355679 + 0.934608i $$0.384250\pi$$
$$744$$ −32.0665 −1.17561
$$745$$ 0 0
$$746$$ 43.6154 1.59687
$$747$$ 0.360912 0.0132051
$$748$$ 1.87753 0.0686492
$$749$$ 39.5213 1.44408
$$750$$ 0 0
$$751$$ −32.5618 −1.18820 −0.594098 0.804393i $$-0.702491\pi$$
−0.594098 + 0.804393i $$0.702491\pi$$
$$752$$ 26.6589 0.972151
$$753$$ −13.2607 −0.483245
$$754$$ −11.2795 −0.410774
$$755$$ 0 0
$$756$$ 9.12871 0.332008
$$757$$ −40.4497 −1.47017 −0.735084 0.677976i $$-0.762857\pi$$
−0.735084 + 0.677976i $$0.762857\pi$$
$$758$$ −52.1730 −1.89501
$$759$$ 8.38151 0.304230
$$760$$ 0 0
$$761$$ 20.6529 0.748666 0.374333 0.927294i $$-0.377872\pi$$
0.374333 + 0.927294i $$0.377872\pi$$
$$762$$ −7.84681 −0.284260
$$763$$ 42.6406 1.54369
$$764$$ 5.34481 0.193368
$$765$$ 0 0
$$766$$ 45.5479 1.64571
$$767$$ 19.8502 0.716749
$$768$$ −13.6852 −0.493820
$$769$$ −2.53650 −0.0914686 −0.0457343 0.998954i $$-0.514563\pi$$
−0.0457343 + 0.998954i $$0.514563\pi$$
$$770$$ 0 0
$$771$$ 3.82813 0.137867
$$772$$ −3.70744 −0.133434
$$773$$ −27.4490 −0.987273 −0.493637 0.869668i $$-0.664333\pi$$
−0.493637 + 0.869668i $$0.664333\pi$$
$$774$$ −13.2339 −0.475681
$$775$$ 0 0
$$776$$ −9.27223 −0.332854
$$777$$ −44.7796 −1.60646
$$778$$ −38.0100 −1.36272
$$779$$ −3.85750 −0.138209
$$780$$ 0 0
$$781$$ 4.55334 0.162931
$$782$$ −41.1486 −1.47147
$$783$$ 22.3969 0.800399
$$784$$ −47.7564 −1.70559
$$785$$ 0 0
$$786$$ 12.0597 0.430155
$$787$$ −39.4380 −1.40581 −0.702907 0.711281i $$-0.748115\pi$$
−0.702907 + 0.711281i $$0.748115\pi$$
$$788$$ 6.72989 0.239742
$$789$$ −23.1937 −0.825717
$$790$$ 0 0
$$791$$ 29.9220 1.06390
$$792$$ −1.76618 −0.0627586
$$793$$ −13.4536 −0.477751
$$794$$ −34.9367 −1.23986
$$795$$ 0 0
$$796$$ 5.65034 0.200271
$$797$$ −38.1840 −1.35255 −0.676273 0.736651i $$-0.736406\pi$$
−0.676273 + 0.736651i $$0.736406\pi$$
$$798$$ −9.73735 −0.344698
$$799$$ 27.6744 0.979049
$$800$$ 0 0
$$801$$ −4.87044 −0.172089
$$802$$ −16.9230 −0.597572
$$803$$ −6.03445 −0.212951
$$804$$ −6.95859 −0.245411
$$805$$ 0 0
$$806$$ 24.0775 0.848094
$$807$$ −28.9992 −1.02082
$$808$$ 35.0677 1.23368
$$809$$ −15.2742 −0.537011 −0.268506 0.963278i $$-0.586530\pi$$
−0.268506 + 0.963278i $$0.586530\pi$$
$$810$$ 0 0
$$811$$ 9.16787 0.321928 0.160964 0.986960i $$-0.448540\pi$$
0.160964 + 0.986960i $$0.448540\pi$$
$$812$$ 6.48668 0.227638
$$813$$ −45.7886 −1.60588
$$814$$ −10.9944 −0.385353
$$815$$ 0 0
$$816$$ −33.6594 −1.17832
$$817$$ 12.0581 0.421859
$$818$$ 47.5206 1.66152
$$819$$ 5.40138 0.188740
$$820$$ 0 0
$$821$$ −8.97266 −0.313148 −0.156574 0.987666i $$-0.550045\pi$$
−0.156574 + 0.987666i $$0.550045\pi$$
$$822$$ −26.4656 −0.923092
$$823$$ 11.8425 0.412805 0.206402 0.978467i $$-0.433824\pi$$
0.206402 + 0.978467i $$0.433824\pi$$
$$824$$ 26.9910 0.940277
$$825$$ 0 0
$$826$$ −69.8468 −2.43028
$$827$$ 35.1524 1.22237 0.611184 0.791488i $$-0.290693\pi$$
0.611184 + 0.791488i $$0.290693\pi$$
$$828$$ −1.53607 −0.0533823
$$829$$ −37.8166 −1.31343 −0.656713 0.754141i $$-0.728054\pi$$
−0.656713 + 0.754141i $$0.728054\pi$$
$$830$$ 0 0
$$831$$ 6.17255 0.214123
$$832$$ −10.7633 −0.373149
$$833$$ −49.5754 −1.71769
$$834$$ −12.0416 −0.416966
$$835$$ 0 0
$$836$$ −0.390736 −0.0135139
$$837$$ −47.8090 −1.65252
$$838$$ −54.6733 −1.88866
$$839$$ 25.0571 0.865067 0.432533 0.901618i $$-0.357620\pi$$
0.432533 + 0.901618i $$0.357620\pi$$
$$840$$ 0 0
$$841$$ −13.0852 −0.451215
$$842$$ −59.9445 −2.06582
$$843$$ −4.60154 −0.158485
$$844$$ 0.446342 0.0153637
$$845$$ 0 0
$$846$$ 6.32096 0.217319
$$847$$ 4.16140 0.142987
$$848$$ −13.0955 −0.449702
$$849$$ 3.39696 0.116583
$$850$$ 0 0
$$851$$ 39.3815 1.34998
$$852$$ 2.69246 0.0922421
$$853$$ −23.8418 −0.816329 −0.408164 0.912908i $$-0.633831\pi$$
−0.408164 + 0.912908i $$0.633831\pi$$
$$854$$ 47.3391 1.61991
$$855$$ 0 0
$$856$$ −23.6311 −0.807696
$$857$$ 34.6121 1.18233 0.591164 0.806551i $$-0.298669\pi$$
0.591164 + 0.806551i $$0.298669\pi$$
$$858$$ −4.27882 −0.146077
$$859$$ 32.6520 1.11407 0.557035 0.830489i $$-0.311939\pi$$
0.557035 + 0.830489i $$0.311939\pi$$
$$860$$ 0 0
$$861$$ 24.2930 0.827904
$$862$$ 45.9794 1.56606
$$863$$ 10.1698 0.346182 0.173091 0.984906i $$-0.444624\pi$$
0.173091 + 0.984906i $$0.444624\pi$$
$$864$$ −12.2421 −0.416484
$$865$$ 0 0
$$866$$ 13.2095 0.448876
$$867$$ −9.21481 −0.312951
$$868$$ −13.8467 −0.469986
$$869$$ −9.68391 −0.328504
$$870$$ 0 0
$$871$$ −21.5192 −0.729150
$$872$$ −25.4963 −0.863414
$$873$$ −2.64505 −0.0895213
$$874$$ 8.56352 0.289665
$$875$$ 0 0
$$876$$ −3.56826 −0.120560
$$877$$ −8.91797 −0.301138 −0.150569 0.988599i $$-0.548111\pi$$
−0.150569 + 0.988599i $$0.548111\pi$$
$$878$$ −3.20449 −0.108146
$$879$$ 27.8275 0.938598
$$880$$ 0 0
$$881$$ −9.08968 −0.306239 −0.153120 0.988208i $$-0.548932\pi$$
−0.153120 + 0.988208i $$0.548932\pi$$
$$882$$ −11.3233 −0.381274
$$883$$ 45.0015 1.51442 0.757210 0.653171i $$-0.226562\pi$$
0.757210 + 0.653171i $$0.226562\pi$$
$$884$$ 3.43328 0.115474
$$885$$ 0 0
$$886$$ −36.8703 −1.23868
$$887$$ 21.4687 0.720850 0.360425 0.932788i $$-0.382632\pi$$
0.360425 + 0.932788i $$0.382632\pi$$
$$888$$ 26.7753 0.898520
$$889$$ 13.9550 0.468037
$$890$$ 0 0
$$891$$ 6.36674 0.213294
$$892$$ −2.68362 −0.0898542
$$893$$ −5.75936 −0.192730
$$894$$ −24.1949 −0.809198
$$895$$ 0 0
$$896$$ 56.0211 1.87153
$$897$$ 15.3266 0.511740
$$898$$ 8.41772 0.280903
$$899$$ −33.9721 −1.13303
$$900$$ 0 0
$$901$$ −13.5943 −0.452893
$$902$$ 5.96447 0.198595
$$903$$ −75.9370 −2.52702
$$904$$ −17.8914 −0.595059
$$905$$ 0 0
$$906$$ 11.3943 0.378550
$$907$$ −12.5080 −0.415322 −0.207661 0.978201i $$-0.566585\pi$$
−0.207661 + 0.978201i $$0.566585\pi$$
$$908$$ 0.0410156 0.00136115
$$909$$ 10.0036 0.331799
$$910$$ 0 0
$$911$$ −4.62780 −0.153326 −0.0766629 0.997057i $$-0.524427\pi$$
−0.0766629 + 0.997057i $$0.524427\pi$$
$$912$$ 7.00493 0.231956
$$913$$ 0.508463 0.0168277
$$914$$ 36.2559 1.19924
$$915$$ 0 0
$$916$$ 5.84849 0.193239
$$917$$ −21.4474 −0.708254
$$918$$ −41.7116 −1.37669
$$919$$ −11.1376 −0.367395 −0.183698 0.982983i $$-0.558807\pi$$
−0.183698 + 0.982983i $$0.558807\pi$$
$$920$$ 0 0
$$921$$ 20.4598 0.674173
$$922$$ −58.1343 −1.91455
$$923$$ 8.32632 0.274064
$$924$$ 2.46070 0.0809509
$$925$$ 0 0
$$926$$ −42.1678 −1.38572
$$927$$ 7.69961 0.252888
$$928$$ −8.69896 −0.285557
$$929$$ −27.3911 −0.898672 −0.449336 0.893363i $$-0.648339\pi$$
−0.449336 + 0.893363i $$0.648339\pi$$
$$930$$ 0 0
$$931$$ 10.3172 0.338134
$$932$$ 6.56452 0.215028
$$933$$ −41.4166 −1.35592
$$934$$ −21.9889 −0.719500
$$935$$ 0 0
$$936$$ −3.22968 −0.105565
$$937$$ −41.4439 −1.35391 −0.676957 0.736022i $$-0.736702\pi$$
−0.676957 + 0.736022i $$0.736702\pi$$
$$938$$ 75.7195 2.47233
$$939$$ −32.0078 −1.04453
$$940$$ 0 0
$$941$$ −43.5568 −1.41991 −0.709956 0.704246i $$-0.751285\pi$$
−0.709956 + 0.704246i $$0.751285\pi$$
$$942$$ −40.6751 −1.32527
$$943$$ −21.3645 −0.695724
$$944$$ 50.2469 1.63540
$$945$$ 0 0
$$946$$ −18.6442 −0.606176
$$947$$ 33.0491 1.07395 0.536976 0.843598i $$-0.319567\pi$$
0.536976 + 0.843598i $$0.319567\pi$$
$$948$$ −5.72624 −0.185980
$$949$$ −11.0347 −0.358201
$$950$$ 0 0
$$951$$ −7.08355 −0.229700
$$952$$ 49.7549 1.61257
$$953$$ −0.505736 −0.0163824 −0.00819120 0.999966i $$-0.502607\pi$$
−0.00819120 + 0.999966i $$0.502607\pi$$
$$954$$ −3.10501 −0.100528
$$955$$ 0 0
$$956$$ −1.34547 −0.0435157
$$957$$ 6.03720 0.195155
$$958$$ 2.41293 0.0779584
$$959$$ 47.0672 1.51988
$$960$$ 0 0
$$961$$ 41.5179 1.33929
$$962$$ −20.1045 −0.648196
$$963$$ −6.74115 −0.217231
$$964$$ −5.78817 −0.186424
$$965$$ 0 0
$$966$$ −53.9296 −1.73516
$$967$$ 46.1493 1.48406 0.742031 0.670366i $$-0.233863\pi$$
0.742031 + 0.670366i $$0.233863\pi$$
$$968$$ −2.48825 −0.0799752
$$969$$ 7.27175 0.233602
$$970$$ 0 0
$$971$$ −27.0101 −0.866795 −0.433398 0.901203i $$-0.642685\pi$$
−0.433398 + 0.901203i $$0.642685\pi$$
$$972$$ −2.81625 −0.0903313
$$973$$ 21.4152 0.686538
$$974$$ 27.5295 0.882101
$$975$$ 0 0
$$976$$ −34.0552 −1.09008
$$977$$ 30.5323 0.976814 0.488407 0.872616i $$-0.337578\pi$$
0.488407 + 0.872616i $$0.337578\pi$$
$$978$$ 39.8087 1.27294
$$979$$ −6.86161 −0.219298
$$980$$ 0 0
$$981$$ −7.27322 −0.232216
$$982$$ 7.69940 0.245698
$$983$$ −27.5882 −0.879926 −0.439963 0.898016i $$-0.645008\pi$$
−0.439963 + 0.898016i $$0.645008\pi$$
$$984$$ −14.5256 −0.463060
$$985$$ 0 0
$$986$$ −29.6394 −0.943910
$$987$$ 36.2702 1.15449
$$988$$ −0.714506 −0.0227315
$$989$$ 66.7828 2.12357
$$990$$ 0 0
$$991$$ −57.2710 −1.81927 −0.909636 0.415405i $$-0.863640\pi$$
−0.909636 + 0.415405i $$0.863640\pi$$
$$992$$ 18.5691 0.589568
$$993$$ −36.7778 −1.16711
$$994$$ −29.2978 −0.929271
$$995$$ 0 0
$$996$$ 0.300661 0.00952682
$$997$$ 50.4928 1.59912 0.799561 0.600584i $$-0.205065\pi$$
0.799561 + 0.600584i $$0.205065\pi$$
$$998$$ 11.0407 0.349486
$$999$$ 39.9202 1.26302
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.j.1.4 5
5.4 even 2 1045.2.a.d.1.2 5
15.14 odd 2 9405.2.a.v.1.4 5

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.d.1.2 5 5.4 even 2
5225.2.a.j.1.4 5 1.1 even 1 trivial
9405.2.a.v.1.4 5 15.14 odd 2