# Properties

 Label 5225.2.a.z.1.8 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $16$ Inner twists $2$

# Learn more

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: $$1$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 19x^{14} + 144x^{12} - 552x^{10} + 1119x^{8} - 1146x^{6} + 524x^{4} - 83x^{2} + 4$$ x^16 - 19*x^14 + 144*x^12 - 552*x^10 + 1119*x^8 - 1146*x^6 + 524*x^4 - 83*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.8 Root $$-0.301154$$ 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)

 $$f(q)$$ $$=$$ $$q-0.301154 q^{2} -1.85377 q^{3} -1.90931 q^{4} +0.558272 q^{6} +1.94668 q^{7} +1.17730 q^{8} +0.436480 q^{9} +O(q^{10})$$ $$q-0.301154 q^{2} -1.85377 q^{3} -1.90931 q^{4} +0.558272 q^{6} +1.94668 q^{7} +1.17730 q^{8} +0.436480 q^{9} -1.00000 q^{11} +3.53942 q^{12} -3.75832 q^{13} -0.586251 q^{14} +3.46406 q^{16} -2.16098 q^{17} -0.131448 q^{18} +1.00000 q^{19} -3.60871 q^{21} +0.301154 q^{22} +4.03747 q^{23} -2.18246 q^{24} +1.13183 q^{26} +4.75219 q^{27} -3.71681 q^{28} -2.18930 q^{29} +3.21529 q^{31} -3.39783 q^{32} +1.85377 q^{33} +0.650787 q^{34} -0.833374 q^{36} -5.64900 q^{37} -0.301154 q^{38} +6.96707 q^{39} -3.52926 q^{41} +1.08678 q^{42} +2.74094 q^{43} +1.90931 q^{44} -1.21590 q^{46} -1.87707 q^{47} -6.42159 q^{48} -3.21044 q^{49} +4.00596 q^{51} +7.17578 q^{52} +4.59639 q^{53} -1.43114 q^{54} +2.29184 q^{56} -1.85377 q^{57} +0.659317 q^{58} +3.10100 q^{59} -10.0971 q^{61} -0.968298 q^{62} +0.849687 q^{63} -5.90485 q^{64} -0.558272 q^{66} +12.0622 q^{67} +4.12596 q^{68} -7.48457 q^{69} +6.42317 q^{71} +0.513870 q^{72} +10.7452 q^{73} +1.70122 q^{74} -1.90931 q^{76} -1.94668 q^{77} -2.09816 q^{78} +10.8068 q^{79} -10.1189 q^{81} +1.06285 q^{82} +8.12591 q^{83} +6.89012 q^{84} -0.825447 q^{86} +4.05847 q^{87} -1.17730 q^{88} -11.1151 q^{89} -7.31624 q^{91} -7.70877 q^{92} -5.96042 q^{93} +0.565288 q^{94} +6.29880 q^{96} +11.4492 q^{97} +0.966837 q^{98} -0.436480 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 6 q^{4} - 8 q^{6} + 8 q^{9}+O(q^{10})$$ 16 * q + 6 * q^4 - 8 * q^6 + 8 * q^9 $$16 q + 6 q^{4} - 8 q^{6} + 8 q^{9} - 16 q^{11} - 4 q^{14} - 18 q^{16} + 16 q^{19} - 10 q^{21} - 10 q^{24} - 24 q^{26} - 2 q^{29} - 32 q^{31} + 16 q^{34} + 18 q^{36} - 40 q^{39} + 6 q^{41} - 6 q^{44} - 38 q^{49} - 16 q^{51} - 18 q^{54} + 12 q^{56} - 24 q^{59} - 42 q^{61} - 62 q^{64} + 8 q^{66} - 30 q^{69} - 46 q^{71} + 2 q^{74} + 6 q^{76} - 74 q^{79} - 56 q^{81} - 34 q^{84} + 8 q^{86} - 14 q^{89} - 24 q^{91} - 64 q^{94} + 54 q^{96} - 8 q^{99}+O(q^{100})$$ 16 * q + 6 * q^4 - 8 * q^6 + 8 * q^9 - 16 * q^11 - 4 * q^14 - 18 * q^16 + 16 * q^19 - 10 * q^21 - 10 * q^24 - 24 * q^26 - 2 * q^29 - 32 * q^31 + 16 * q^34 + 18 * q^36 - 40 * q^39 + 6 * q^41 - 6 * q^44 - 38 * q^49 - 16 * q^51 - 18 * q^54 + 12 * q^56 - 24 * q^59 - 42 * q^61 - 62 * q^64 + 8 * q^66 - 30 * q^69 - 46 * q^71 + 2 * q^74 + 6 * q^76 - 74 * q^79 - 56 * q^81 - 34 * q^84 + 8 * q^86 - 14 * q^89 - 24 * q^91 - 64 * q^94 + 54 * q^96 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.301154 −0.212948 −0.106474 0.994315i $$-0.533956\pi$$
−0.106474 + 0.994315i $$0.533956\pi$$
$$3$$ −1.85377 −1.07028 −0.535139 0.844764i $$-0.679741\pi$$
−0.535139 + 0.844764i $$0.679741\pi$$
$$4$$ −1.90931 −0.954653
$$5$$ 0 0
$$6$$ 0.558272 0.227914
$$7$$ 1.94668 0.735776 0.367888 0.929870i $$-0.380081\pi$$
0.367888 + 0.929870i $$0.380081\pi$$
$$8$$ 1.17730 0.416240
$$9$$ 0.436480 0.145493
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 3.53942 1.02174
$$13$$ −3.75832 −1.04237 −0.521185 0.853444i $$-0.674510\pi$$
−0.521185 + 0.853444i $$0.674510\pi$$
$$14$$ −0.586251 −0.156682
$$15$$ 0 0
$$16$$ 3.46406 0.866015
$$17$$ −2.16098 −0.524114 −0.262057 0.965052i $$-0.584401\pi$$
−0.262057 + 0.965052i $$0.584401\pi$$
$$18$$ −0.131448 −0.0309825
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −3.60871 −0.787484
$$22$$ 0.301154 0.0642063
$$23$$ 4.03747 0.841872 0.420936 0.907090i $$-0.361702\pi$$
0.420936 + 0.907090i $$0.361702\pi$$
$$24$$ −2.18246 −0.445492
$$25$$ 0 0
$$26$$ 1.13183 0.221971
$$27$$ 4.75219 0.914559
$$28$$ −3.71681 −0.702411
$$29$$ −2.18930 −0.406543 −0.203271 0.979122i $$-0.565157\pi$$
−0.203271 + 0.979122i $$0.565157\pi$$
$$30$$ 0 0
$$31$$ 3.21529 0.577483 0.288741 0.957407i $$-0.406763\pi$$
0.288741 + 0.957407i $$0.406763\pi$$
$$32$$ −3.39783 −0.600656
$$33$$ 1.85377 0.322701
$$34$$ 0.650787 0.111609
$$35$$ 0 0
$$36$$ −0.833374 −0.138896
$$37$$ −5.64900 −0.928690 −0.464345 0.885654i $$-0.653710\pi$$
−0.464345 + 0.885654i $$0.653710\pi$$
$$38$$ −0.301154 −0.0488537
$$39$$ 6.96707 1.11562
$$40$$ 0 0
$$41$$ −3.52926 −0.551177 −0.275589 0.961276i $$-0.588873\pi$$
−0.275589 + 0.961276i $$0.588873\pi$$
$$42$$ 1.08678 0.167693
$$43$$ 2.74094 0.417990 0.208995 0.977917i $$-0.432981\pi$$
0.208995 + 0.977917i $$0.432981\pi$$
$$44$$ 1.90931 0.287839
$$45$$ 0 0
$$46$$ −1.21590 −0.179275
$$47$$ −1.87707 −0.273799 −0.136899 0.990585i $$-0.543714\pi$$
−0.136899 + 0.990585i $$0.543714\pi$$
$$48$$ −6.42159 −0.926877
$$49$$ −3.21044 −0.458634
$$50$$ 0 0
$$51$$ 4.00596 0.560947
$$52$$ 7.17578 0.995101
$$53$$ 4.59639 0.631362 0.315681 0.948865i $$-0.397767\pi$$
0.315681 + 0.948865i $$0.397767\pi$$
$$54$$ −1.43114 −0.194754
$$55$$ 0 0
$$56$$ 2.29184 0.306259
$$57$$ −1.85377 −0.245538
$$58$$ 0.659317 0.0865726
$$59$$ 3.10100 0.403715 0.201858 0.979415i $$-0.435302\pi$$
0.201858 + 0.979415i $$0.435302\pi$$
$$60$$ 0 0
$$61$$ −10.0971 −1.29281 −0.646403 0.762996i $$-0.723728\pi$$
−0.646403 + 0.762996i $$0.723728\pi$$
$$62$$ −0.968298 −0.122974
$$63$$ 0.849687 0.107050
$$64$$ −5.90485 −0.738107
$$65$$ 0 0
$$66$$ −0.558272 −0.0687186
$$67$$ 12.0622 1.47363 0.736814 0.676096i $$-0.236329\pi$$
0.736814 + 0.676096i $$0.236329\pi$$
$$68$$ 4.12596 0.500347
$$69$$ −7.48457 −0.901036
$$70$$ 0 0
$$71$$ 6.42317 0.762291 0.381145 0.924515i $$-0.375530\pi$$
0.381145 + 0.924515i $$0.375530\pi$$
$$72$$ 0.513870 0.0605601
$$73$$ 10.7452 1.25763 0.628814 0.777556i $$-0.283541\pi$$
0.628814 + 0.777556i $$0.283541\pi$$
$$74$$ 1.70122 0.197763
$$75$$ 0 0
$$76$$ −1.90931 −0.219012
$$77$$ −1.94668 −0.221845
$$78$$ −2.09816 −0.237570
$$79$$ 10.8068 1.21586 0.607932 0.793989i $$-0.291999\pi$$
0.607932 + 0.793989i $$0.291999\pi$$
$$80$$ 0 0
$$81$$ −10.1189 −1.12432
$$82$$ 1.06285 0.117372
$$83$$ 8.12591 0.891934 0.445967 0.895049i $$-0.352860\pi$$
0.445967 + 0.895049i $$0.352860\pi$$
$$84$$ 6.89012 0.751774
$$85$$ 0 0
$$86$$ −0.825447 −0.0890103
$$87$$ 4.05847 0.435114
$$88$$ −1.17730 −0.125501
$$89$$ −11.1151 −1.17820 −0.589101 0.808060i $$-0.700518\pi$$
−0.589101 + 0.808060i $$0.700518\pi$$
$$90$$ 0 0
$$91$$ −7.31624 −0.766950
$$92$$ −7.70877 −0.803695
$$93$$ −5.96042 −0.618066
$$94$$ 0.565288 0.0583050
$$95$$ 0 0
$$96$$ 6.29880 0.642869
$$97$$ 11.4492 1.16249 0.581246 0.813728i $$-0.302565\pi$$
0.581246 + 0.813728i $$0.302565\pi$$
$$98$$ 0.966837 0.0976653
$$99$$ −0.436480 −0.0438679
$$100$$ 0 0
$$101$$ 6.86482 0.683075 0.341537 0.939868i $$-0.389052\pi$$
0.341537 + 0.939868i $$0.389052\pi$$
$$102$$ −1.20641 −0.119453
$$103$$ −13.7258 −1.35244 −0.676220 0.736699i $$-0.736383\pi$$
−0.676220 + 0.736699i $$0.736383\pi$$
$$104$$ −4.42468 −0.433876
$$105$$ 0 0
$$106$$ −1.38422 −0.134447
$$107$$ 14.8545 1.43603 0.718017 0.696025i $$-0.245050\pi$$
0.718017 + 0.696025i $$0.245050\pi$$
$$108$$ −9.07338 −0.873087
$$109$$ −7.13713 −0.683613 −0.341806 0.939770i $$-0.611039\pi$$
−0.341806 + 0.939770i $$0.611039\pi$$
$$110$$ 0 0
$$111$$ 10.4720 0.993956
$$112$$ 6.74342 0.637193
$$113$$ −1.22524 −0.115261 −0.0576303 0.998338i $$-0.518354\pi$$
−0.0576303 + 0.998338i $$0.518354\pi$$
$$114$$ 0.558272 0.0522870
$$115$$ 0 0
$$116$$ 4.18004 0.388107
$$117$$ −1.64043 −0.151658
$$118$$ −0.933879 −0.0859705
$$119$$ −4.20673 −0.385630
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 3.04080 0.275301
$$123$$ 6.54245 0.589913
$$124$$ −6.13897 −0.551296
$$125$$ 0 0
$$126$$ −0.255887 −0.0227962
$$127$$ −1.78182 −0.158111 −0.0790554 0.996870i $$-0.525190\pi$$
−0.0790554 + 0.996870i $$0.525190\pi$$
$$128$$ 8.57392 0.757835
$$129$$ −5.08109 −0.447365
$$130$$ 0 0
$$131$$ 18.4204 1.60939 0.804697 0.593685i $$-0.202328\pi$$
0.804697 + 0.593685i $$0.202328\pi$$
$$132$$ −3.53942 −0.308067
$$133$$ 1.94668 0.168799
$$134$$ −3.63257 −0.313807
$$135$$ 0 0
$$136$$ −2.54413 −0.218157
$$137$$ −13.8926 −1.18693 −0.593463 0.804861i $$-0.702240\pi$$
−0.593463 + 0.804861i $$0.702240\pi$$
$$138$$ 2.25401 0.191874
$$139$$ −4.35165 −0.369102 −0.184551 0.982823i $$-0.559083\pi$$
−0.184551 + 0.982823i $$0.559083\pi$$
$$140$$ 0 0
$$141$$ 3.47967 0.293041
$$142$$ −1.93437 −0.162328
$$143$$ 3.75832 0.314286
$$144$$ 1.51199 0.125999
$$145$$ 0 0
$$146$$ −3.23596 −0.267810
$$147$$ 5.95142 0.490865
$$148$$ 10.7857 0.886577
$$149$$ −11.6917 −0.957818 −0.478909 0.877864i $$-0.658968\pi$$
−0.478909 + 0.877864i $$0.658968\pi$$
$$150$$ 0 0
$$151$$ −23.0439 −1.87529 −0.937644 0.347596i $$-0.886998\pi$$
−0.937644 + 0.347596i $$0.886998\pi$$
$$152$$ 1.17730 0.0954920
$$153$$ −0.943222 −0.0762550
$$154$$ 0.586251 0.0472415
$$155$$ 0 0
$$156$$ −13.3023 −1.06503
$$157$$ 12.2578 0.978277 0.489138 0.872206i $$-0.337311\pi$$
0.489138 + 0.872206i $$0.337311\pi$$
$$158$$ −3.25452 −0.258916
$$159$$ −8.52066 −0.675732
$$160$$ 0 0
$$161$$ 7.85967 0.619429
$$162$$ 3.04736 0.239423
$$163$$ 8.00570 0.627055 0.313527 0.949579i $$-0.398489\pi$$
0.313527 + 0.949579i $$0.398489\pi$$
$$164$$ 6.73843 0.526183
$$165$$ 0 0
$$166$$ −2.44715 −0.189936
$$167$$ 5.00463 0.387270 0.193635 0.981074i $$-0.437972\pi$$
0.193635 + 0.981074i $$0.437972\pi$$
$$168$$ −4.24855 −0.327782
$$169$$ 1.12494 0.0865341
$$170$$ 0 0
$$171$$ 0.436480 0.0333784
$$172$$ −5.23330 −0.399035
$$173$$ 7.15833 0.544237 0.272119 0.962264i $$-0.412276\pi$$
0.272119 + 0.962264i $$0.412276\pi$$
$$174$$ −1.22223 −0.0926567
$$175$$ 0 0
$$176$$ −3.46406 −0.261113
$$177$$ −5.74855 −0.432087
$$178$$ 3.34737 0.250896
$$179$$ −4.12217 −0.308106 −0.154053 0.988063i $$-0.549233\pi$$
−0.154053 + 0.988063i $$0.549233\pi$$
$$180$$ 0 0
$$181$$ 17.5288 1.30291 0.651454 0.758688i $$-0.274159\pi$$
0.651454 + 0.758688i $$0.274159\pi$$
$$182$$ 2.20332 0.163321
$$183$$ 18.7178 1.38366
$$184$$ 4.75334 0.350421
$$185$$ 0 0
$$186$$ 1.79501 0.131616
$$187$$ 2.16098 0.158026
$$188$$ 3.58390 0.261383
$$189$$ 9.25099 0.672911
$$190$$ 0 0
$$191$$ −27.5080 −1.99041 −0.995205 0.0978155i $$-0.968814\pi$$
−0.995205 + 0.0978155i $$0.968814\pi$$
$$192$$ 10.9463 0.789979
$$193$$ 13.1923 0.949603 0.474802 0.880093i $$-0.342520\pi$$
0.474802 + 0.880093i $$0.342520\pi$$
$$194$$ −3.44798 −0.247551
$$195$$ 0 0
$$196$$ 6.12970 0.437836
$$197$$ −11.4532 −0.816006 −0.408003 0.912981i $$-0.633775\pi$$
−0.408003 + 0.912981i $$0.633775\pi$$
$$198$$ 0.131448 0.00934159
$$199$$ −19.7562 −1.40048 −0.700240 0.713907i $$-0.746924\pi$$
−0.700240 + 0.713907i $$0.746924\pi$$
$$200$$ 0 0
$$201$$ −22.3605 −1.57719
$$202$$ −2.06737 −0.145460
$$203$$ −4.26187 −0.299124
$$204$$ −7.64861 −0.535510
$$205$$ 0 0
$$206$$ 4.13358 0.288000
$$207$$ 1.76228 0.122487
$$208$$ −13.0190 −0.902708
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −22.4594 −1.54617 −0.773085 0.634303i $$-0.781287\pi$$
−0.773085 + 0.634303i $$0.781287\pi$$
$$212$$ −8.77591 −0.602732
$$213$$ −11.9071 −0.815862
$$214$$ −4.47349 −0.305801
$$215$$ 0 0
$$216$$ 5.59477 0.380676
$$217$$ 6.25914 0.424898
$$218$$ 2.14938 0.145574
$$219$$ −19.9191 −1.34601
$$220$$ 0 0
$$221$$ 8.12163 0.546320
$$222$$ −3.15368 −0.211661
$$223$$ 10.3113 0.690499 0.345249 0.938511i $$-0.387794\pi$$
0.345249 + 0.938511i $$0.387794\pi$$
$$224$$ −6.61448 −0.441949
$$225$$ 0 0
$$226$$ 0.368986 0.0245446
$$227$$ −19.9756 −1.32583 −0.662914 0.748696i $$-0.730680\pi$$
−0.662914 + 0.748696i $$0.730680\pi$$
$$228$$ 3.53942 0.234404
$$229$$ −15.7027 −1.03767 −0.518833 0.854876i $$-0.673633\pi$$
−0.518833 + 0.854876i $$0.673633\pi$$
$$230$$ 0 0
$$231$$ 3.60871 0.237435
$$232$$ −2.57747 −0.169219
$$233$$ 21.1823 1.38770 0.693850 0.720120i $$-0.255913\pi$$
0.693850 + 0.720120i $$0.255913\pi$$
$$234$$ 0.494022 0.0322953
$$235$$ 0 0
$$236$$ −5.92075 −0.385408
$$237$$ −20.0334 −1.30131
$$238$$ 1.26687 0.0821193
$$239$$ −0.429031 −0.0277517 −0.0138759 0.999904i $$-0.504417\pi$$
−0.0138759 + 0.999904i $$0.504417\pi$$
$$240$$ 0 0
$$241$$ 11.9657 0.770781 0.385391 0.922753i $$-0.374067\pi$$
0.385391 + 0.922753i $$0.374067\pi$$
$$242$$ −0.301154 −0.0193589
$$243$$ 4.50164 0.288780
$$244$$ 19.2785 1.23418
$$245$$ 0 0
$$246$$ −1.97029 −0.125621
$$247$$ −3.75832 −0.239136
$$248$$ 3.78537 0.240371
$$249$$ −15.0636 −0.954617
$$250$$ 0 0
$$251$$ 7.13854 0.450581 0.225290 0.974292i $$-0.427667\pi$$
0.225290 + 0.974292i $$0.427667\pi$$
$$252$$ −1.62231 −0.102196
$$253$$ −4.03747 −0.253834
$$254$$ 0.536602 0.0336694
$$255$$ 0 0
$$256$$ 9.22763 0.576727
$$257$$ −6.86772 −0.428397 −0.214198 0.976790i $$-0.568714\pi$$
−0.214198 + 0.976790i $$0.568714\pi$$
$$258$$ 1.53019 0.0952657
$$259$$ −10.9968 −0.683308
$$260$$ 0 0
$$261$$ −0.955585 −0.0591492
$$262$$ −5.54737 −0.342718
$$263$$ −0.473180 −0.0291775 −0.0145888 0.999894i $$-0.504644\pi$$
−0.0145888 + 0.999894i $$0.504644\pi$$
$$264$$ 2.18246 0.134321
$$265$$ 0 0
$$266$$ −0.586251 −0.0359454
$$267$$ 20.6049 1.26100
$$268$$ −23.0304 −1.40680
$$269$$ 2.26859 0.138318 0.0691591 0.997606i $$-0.477968\pi$$
0.0691591 + 0.997606i $$0.477968\pi$$
$$270$$ 0 0
$$271$$ −13.6384 −0.828474 −0.414237 0.910169i $$-0.635952\pi$$
−0.414237 + 0.910169i $$0.635952\pi$$
$$272$$ −7.48575 −0.453891
$$273$$ 13.5627 0.820850
$$274$$ 4.18382 0.252754
$$275$$ 0 0
$$276$$ 14.2903 0.860177
$$277$$ −1.64693 −0.0989544 −0.0494772 0.998775i $$-0.515756\pi$$
−0.0494772 + 0.998775i $$0.515756\pi$$
$$278$$ 1.31052 0.0785997
$$279$$ 1.40341 0.0840198
$$280$$ 0 0
$$281$$ 16.7390 0.998565 0.499282 0.866439i $$-0.333597\pi$$
0.499282 + 0.866439i $$0.333597\pi$$
$$282$$ −1.04792 −0.0624025
$$283$$ −9.91741 −0.589529 −0.294764 0.955570i $$-0.595241\pi$$
−0.294764 + 0.955570i $$0.595241\pi$$
$$284$$ −12.2638 −0.727723
$$285$$ 0 0
$$286$$ −1.13183 −0.0669267
$$287$$ −6.87034 −0.405543
$$288$$ −1.48308 −0.0873915
$$289$$ −12.3302 −0.725305
$$290$$ 0 0
$$291$$ −21.2243 −1.24419
$$292$$ −20.5158 −1.20060
$$293$$ −14.1780 −0.828289 −0.414144 0.910211i $$-0.635919\pi$$
−0.414144 + 0.910211i $$0.635919\pi$$
$$294$$ −1.79230 −0.104529
$$295$$ 0 0
$$296$$ −6.65059 −0.386558
$$297$$ −4.75219 −0.275750
$$298$$ 3.52100 0.203966
$$299$$ −15.1741 −0.877541
$$300$$ 0 0
$$301$$ 5.33574 0.307547
$$302$$ 6.93978 0.399340
$$303$$ −12.7258 −0.731079
$$304$$ 3.46406 0.198678
$$305$$ 0 0
$$306$$ 0.284056 0.0162384
$$307$$ −7.49870 −0.427974 −0.213987 0.976837i $$-0.568645\pi$$
−0.213987 + 0.976837i $$0.568645\pi$$
$$308$$ 3.71681 0.211785
$$309$$ 25.4445 1.44749
$$310$$ 0 0
$$311$$ −22.6904 −1.28666 −0.643328 0.765591i $$-0.722447\pi$$
−0.643328 + 0.765591i $$0.722447\pi$$
$$312$$ 8.20236 0.464367
$$313$$ −16.3070 −0.921723 −0.460862 0.887472i $$-0.652460\pi$$
−0.460862 + 0.887472i $$0.652460\pi$$
$$314$$ −3.69148 −0.208322
$$315$$ 0 0
$$316$$ −20.6336 −1.16073
$$317$$ −26.5837 −1.49309 −0.746544 0.665336i $$-0.768288\pi$$
−0.746544 + 0.665336i $$0.768288\pi$$
$$318$$ 2.56603 0.143896
$$319$$ 2.18930 0.122577
$$320$$ 0 0
$$321$$ −27.5368 −1.53696
$$322$$ −2.36697 −0.131906
$$323$$ −2.16098 −0.120240
$$324$$ 19.3201 1.07334
$$325$$ 0 0
$$326$$ −2.41095 −0.133530
$$327$$ 13.2306 0.731655
$$328$$ −4.15501 −0.229422
$$329$$ −3.65406 −0.201455
$$330$$ 0 0
$$331$$ 12.6060 0.692887 0.346444 0.938071i $$-0.387389\pi$$
0.346444 + 0.938071i $$0.387389\pi$$
$$332$$ −15.5148 −0.851487
$$333$$ −2.46567 −0.135118
$$334$$ −1.50717 −0.0824685
$$335$$ 0 0
$$336$$ −12.5008 −0.681974
$$337$$ −6.18984 −0.337182 −0.168591 0.985686i $$-0.553922\pi$$
−0.168591 + 0.985686i $$0.553922\pi$$
$$338$$ −0.338782 −0.0184273
$$339$$ 2.27131 0.123361
$$340$$ 0 0
$$341$$ −3.21529 −0.174118
$$342$$ −0.131448 −0.00710788
$$343$$ −19.8765 −1.07323
$$344$$ 3.22693 0.173984
$$345$$ 0 0
$$346$$ −2.15576 −0.115894
$$347$$ −12.5724 −0.674921 −0.337460 0.941340i $$-0.609568\pi$$
−0.337460 + 0.941340i $$0.609568\pi$$
$$348$$ −7.74886 −0.415382
$$349$$ 17.0337 0.911793 0.455897 0.890033i $$-0.349319\pi$$
0.455897 + 0.890033i $$0.349319\pi$$
$$350$$ 0 0
$$351$$ −17.8602 −0.953308
$$352$$ 3.39783 0.181105
$$353$$ −8.51838 −0.453387 −0.226694 0.973966i $$-0.572792\pi$$
−0.226694 + 0.973966i $$0.572792\pi$$
$$354$$ 1.73120 0.0920123
$$355$$ 0 0
$$356$$ 21.2222 1.12477
$$357$$ 7.79833 0.412731
$$358$$ 1.24141 0.0656106
$$359$$ −6.43010 −0.339367 −0.169684 0.985499i $$-0.554275\pi$$
−0.169684 + 0.985499i $$0.554275\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −5.27889 −0.277452
$$363$$ −1.85377 −0.0972979
$$364$$ 13.9689 0.732172
$$365$$ 0 0
$$366$$ −5.63696 −0.294648
$$367$$ −29.9633 −1.56407 −0.782037 0.623233i $$-0.785819\pi$$
−0.782037 + 0.623233i $$0.785819\pi$$
$$368$$ 13.9861 0.729074
$$369$$ −1.54045 −0.0801926
$$370$$ 0 0
$$371$$ 8.94769 0.464541
$$372$$ 11.3803 0.590039
$$373$$ −34.6611 −1.79469 −0.897343 0.441334i $$-0.854505\pi$$
−0.897343 + 0.441334i $$0.854505\pi$$
$$374$$ −0.650787 −0.0336514
$$375$$ 0 0
$$376$$ −2.20988 −0.113966
$$377$$ 8.22808 0.423768
$$378$$ −2.78598 −0.143295
$$379$$ 1.82041 0.0935083 0.0467542 0.998906i $$-0.485112\pi$$
0.0467542 + 0.998906i $$0.485112\pi$$
$$380$$ 0 0
$$381$$ 3.30309 0.169222
$$382$$ 8.28415 0.423854
$$383$$ 17.0353 0.870463 0.435232 0.900319i $$-0.356666\pi$$
0.435232 + 0.900319i $$0.356666\pi$$
$$384$$ −15.8941 −0.811094
$$385$$ 0 0
$$386$$ −3.97292 −0.202216
$$387$$ 1.19637 0.0608147
$$388$$ −21.8601 −1.10978
$$389$$ −21.7778 −1.10418 −0.552090 0.833784i $$-0.686170\pi$$
−0.552090 + 0.833784i $$0.686170\pi$$
$$390$$ 0 0
$$391$$ −8.72488 −0.441236
$$392$$ −3.77966 −0.190902
$$393$$ −34.1472 −1.72250
$$394$$ 3.44918 0.173767
$$395$$ 0 0
$$396$$ 0.833374 0.0418786
$$397$$ 8.81816 0.442571 0.221285 0.975209i $$-0.428975\pi$$
0.221285 + 0.975209i $$0.428975\pi$$
$$398$$ 5.94967 0.298230
$$399$$ −3.60871 −0.180661
$$400$$ 0 0
$$401$$ −11.0380 −0.551210 −0.275605 0.961271i $$-0.588878\pi$$
−0.275605 + 0.961271i $$0.588878\pi$$
$$402$$ 6.73397 0.335860
$$403$$ −12.0841 −0.601950
$$404$$ −13.1070 −0.652099
$$405$$ 0 0
$$406$$ 1.28348 0.0636980
$$407$$ 5.64900 0.280011
$$408$$ 4.71624 0.233489
$$409$$ −25.9583 −1.28355 −0.641777 0.766891i $$-0.721803\pi$$
−0.641777 + 0.766891i $$0.721803\pi$$
$$410$$ 0 0
$$411$$ 25.7538 1.27034
$$412$$ 26.2067 1.29111
$$413$$ 6.03665 0.297044
$$414$$ −0.530717 −0.0260833
$$415$$ 0 0
$$416$$ 12.7701 0.626106
$$417$$ 8.06698 0.395042
$$418$$ 0.301154 0.0147299
$$419$$ 8.14823 0.398067 0.199034 0.979993i $$-0.436220\pi$$
0.199034 + 0.979993i $$0.436220\pi$$
$$420$$ 0 0
$$421$$ −14.7467 −0.718711 −0.359356 0.933201i $$-0.617003\pi$$
−0.359356 + 0.933201i $$0.617003\pi$$
$$422$$ 6.76375 0.329254
$$423$$ −0.819304 −0.0398359
$$424$$ 5.41134 0.262798
$$425$$ 0 0
$$426$$ 3.58588 0.173736
$$427$$ −19.6559 −0.951216
$$428$$ −28.3617 −1.37092
$$429$$ −6.96707 −0.336373
$$430$$ 0 0
$$431$$ −28.0178 −1.34957 −0.674785 0.738014i $$-0.735764\pi$$
−0.674785 + 0.738014i $$0.735764\pi$$
$$432$$ 16.4619 0.792022
$$433$$ −2.14430 −0.103049 −0.0515243 0.998672i $$-0.516408\pi$$
−0.0515243 + 0.998672i $$0.516408\pi$$
$$434$$ −1.88497 −0.0904813
$$435$$ 0 0
$$436$$ 13.6270 0.652613
$$437$$ 4.03747 0.193139
$$438$$ 5.99873 0.286631
$$439$$ 4.98343 0.237846 0.118923 0.992903i $$-0.462056\pi$$
0.118923 + 0.992903i $$0.462056\pi$$
$$440$$ 0 0
$$441$$ −1.40129 −0.0667281
$$442$$ −2.44586 −0.116338
$$443$$ −10.3754 −0.492952 −0.246476 0.969149i $$-0.579273\pi$$
−0.246476 + 0.969149i $$0.579273\pi$$
$$444$$ −19.9942 −0.948883
$$445$$ 0 0
$$446$$ −3.10531 −0.147041
$$447$$ 21.6737 1.02513
$$448$$ −11.4949 −0.543081
$$449$$ 35.9683 1.69745 0.848725 0.528835i $$-0.177371\pi$$
0.848725 + 0.528835i $$0.177371\pi$$
$$450$$ 0 0
$$451$$ 3.52926 0.166186
$$452$$ 2.33935 0.110034
$$453$$ 42.7183 2.00708
$$454$$ 6.01574 0.282333
$$455$$ 0 0
$$456$$ −2.18246 −0.102203
$$457$$ −27.3842 −1.28098 −0.640490 0.767966i $$-0.721269\pi$$
−0.640490 + 0.767966i $$0.721269\pi$$
$$458$$ 4.72895 0.220969
$$459$$ −10.2694 −0.479333
$$460$$ 0 0
$$461$$ −9.58866 −0.446589 −0.223294 0.974751i $$-0.571681\pi$$
−0.223294 + 0.974751i $$0.571681\pi$$
$$462$$ −1.08678 −0.0505615
$$463$$ −27.4154 −1.27410 −0.637050 0.770822i $$-0.719846\pi$$
−0.637050 + 0.770822i $$0.719846\pi$$
$$464$$ −7.58387 −0.352072
$$465$$ 0 0
$$466$$ −6.37915 −0.295508
$$467$$ 30.6386 1.41778 0.708892 0.705317i $$-0.249195\pi$$
0.708892 + 0.705317i $$0.249195\pi$$
$$468$$ 3.13208 0.144781
$$469$$ 23.4812 1.08426
$$470$$ 0 0
$$471$$ −22.7231 −1.04703
$$472$$ 3.65082 0.168043
$$473$$ −2.74094 −0.126029
$$474$$ 6.03315 0.277112
$$475$$ 0 0
$$476$$ 8.03193 0.368143
$$477$$ 2.00623 0.0918589
$$478$$ 0.129205 0.00590968
$$479$$ 5.55489 0.253809 0.126905 0.991915i $$-0.459496\pi$$
0.126905 + 0.991915i $$0.459496\pi$$
$$480$$ 0 0
$$481$$ 21.2307 0.968038
$$482$$ −3.60354 −0.164137
$$483$$ −14.5701 −0.662961
$$484$$ −1.90931 −0.0867866
$$485$$ 0 0
$$486$$ −1.35569 −0.0614953
$$487$$ 15.4839 0.701642 0.350821 0.936442i $$-0.385902\pi$$
0.350821 + 0.936442i $$0.385902\pi$$
$$488$$ −11.8874 −0.538118
$$489$$ −14.8408 −0.671122
$$490$$ 0 0
$$491$$ −25.8915 −1.16847 −0.584233 0.811586i $$-0.698605\pi$$
−0.584233 + 0.811586i $$0.698605\pi$$
$$492$$ −12.4915 −0.563162
$$493$$ 4.73103 0.213075
$$494$$ 1.13183 0.0509236
$$495$$ 0 0
$$496$$ 11.1380 0.500109
$$497$$ 12.5039 0.560875
$$498$$ 4.53647 0.203284
$$499$$ −33.9769 −1.52102 −0.760508 0.649329i $$-0.775050\pi$$
−0.760508 + 0.649329i $$0.775050\pi$$
$$500$$ 0 0
$$501$$ −9.27746 −0.414486
$$502$$ −2.14980 −0.0959504
$$503$$ −11.0065 −0.490756 −0.245378 0.969428i $$-0.578912\pi$$
−0.245378 + 0.969428i $$0.578912\pi$$
$$504$$ 1.00034 0.0445587
$$505$$ 0 0
$$506$$ 1.21590 0.0540535
$$507$$ −2.08539 −0.0926155
$$508$$ 3.40204 0.150941
$$509$$ −20.0893 −0.890443 −0.445221 0.895420i $$-0.646875\pi$$
−0.445221 + 0.895420i $$0.646875\pi$$
$$510$$ 0 0
$$511$$ 20.9174 0.925332
$$512$$ −19.9268 −0.880648
$$513$$ 4.75219 0.209814
$$514$$ 2.06824 0.0912264
$$515$$ 0 0
$$516$$ 9.70136 0.427079
$$517$$ 1.87707 0.0825535
$$518$$ 3.31173 0.145509
$$519$$ −13.2699 −0.582485
$$520$$ 0 0
$$521$$ −5.13542 −0.224987 −0.112493 0.993652i $$-0.535884\pi$$
−0.112493 + 0.993652i $$0.535884\pi$$
$$522$$ 0.287779 0.0125957
$$523$$ 11.4021 0.498578 0.249289 0.968429i $$-0.419803\pi$$
0.249289 + 0.968429i $$0.419803\pi$$
$$524$$ −35.1701 −1.53641
$$525$$ 0 0
$$526$$ 0.142500 0.00621330
$$527$$ −6.94816 −0.302667
$$528$$ 6.42159 0.279464
$$529$$ −6.69880 −0.291252
$$530$$ 0 0
$$531$$ 1.35352 0.0587379
$$532$$ −3.71681 −0.161144
$$533$$ 13.2641 0.574530
$$534$$ −6.20527 −0.268528
$$535$$ 0 0
$$536$$ 14.2008 0.613383
$$537$$ 7.64158 0.329758
$$538$$ −0.683195 −0.0294546
$$539$$ 3.21044 0.138283
$$540$$ 0 0
$$541$$ 31.1731 1.34024 0.670118 0.742254i $$-0.266243\pi$$
0.670118 + 0.742254i $$0.266243\pi$$
$$542$$ 4.10726 0.176422
$$543$$ −32.4945 −1.39447
$$544$$ 7.34262 0.314812
$$545$$ 0 0
$$546$$ −4.08445 −0.174798
$$547$$ 36.3618 1.55472 0.777360 0.629056i $$-0.216558\pi$$
0.777360 + 0.629056i $$0.216558\pi$$
$$548$$ 26.5252 1.13310
$$549$$ −4.40720 −0.188095
$$550$$ 0 0
$$551$$ −2.18930 −0.0932673
$$552$$ −8.81161 −0.375047
$$553$$ 21.0374 0.894603
$$554$$ 0.495980 0.0210722
$$555$$ 0 0
$$556$$ 8.30863 0.352364
$$557$$ 0.847753 0.0359204 0.0179602 0.999839i $$-0.494283\pi$$
0.0179602 + 0.999839i $$0.494283\pi$$
$$558$$ −0.422642 −0.0178919
$$559$$ −10.3013 −0.435700
$$560$$ 0 0
$$561$$ −4.00596 −0.169132
$$562$$ −5.04102 −0.212643
$$563$$ −39.5533 −1.66697 −0.833486 0.552541i $$-0.813658\pi$$
−0.833486 + 0.552541i $$0.813658\pi$$
$$564$$ −6.64375 −0.279752
$$565$$ 0 0
$$566$$ 2.98667 0.125539
$$567$$ −19.6983 −0.827251
$$568$$ 7.56203 0.317296
$$569$$ 18.3932 0.771082 0.385541 0.922691i $$-0.374015\pi$$
0.385541 + 0.922691i $$0.374015\pi$$
$$570$$ 0 0
$$571$$ 33.4442 1.39959 0.699797 0.714341i $$-0.253274\pi$$
0.699797 + 0.714341i $$0.253274\pi$$
$$572$$ −7.17578 −0.300034
$$573$$ 50.9936 2.13029
$$574$$ 2.06903 0.0863597
$$575$$ 0 0
$$576$$ −2.57735 −0.107390
$$577$$ −32.8041 −1.36565 −0.682826 0.730581i $$-0.739249\pi$$
−0.682826 + 0.730581i $$0.739249\pi$$
$$578$$ 3.71329 0.154452
$$579$$ −24.4556 −1.01634
$$580$$ 0 0
$$581$$ 15.8185 0.656264
$$582$$ 6.39178 0.264948
$$583$$ −4.59639 −0.190363
$$584$$ 12.6503 0.523475
$$585$$ 0 0
$$586$$ 4.26977 0.176383
$$587$$ 26.2752 1.08449 0.542246 0.840220i $$-0.317574\pi$$
0.542246 + 0.840220i $$0.317574\pi$$
$$588$$ −11.3631 −0.468606
$$589$$ 3.21529 0.132484
$$590$$ 0 0
$$591$$ 21.2316 0.873353
$$592$$ −19.5685 −0.804260
$$593$$ −19.5147 −0.801371 −0.400686 0.916216i $$-0.631228\pi$$
−0.400686 + 0.916216i $$0.631228\pi$$
$$594$$ 1.43114 0.0587205
$$595$$ 0 0
$$596$$ 22.3230 0.914384
$$597$$ 36.6236 1.49890
$$598$$ 4.56975 0.186871
$$599$$ 15.9690 0.652476 0.326238 0.945288i $$-0.394219\pi$$
0.326238 + 0.945288i $$0.394219\pi$$
$$600$$ 0 0
$$601$$ −18.0313 −0.735510 −0.367755 0.929923i $$-0.619874\pi$$
−0.367755 + 0.929923i $$0.619874\pi$$
$$602$$ −1.60688 −0.0654916
$$603$$ 5.26489 0.214403
$$604$$ 43.9979 1.79025
$$605$$ 0 0
$$606$$ 3.83244 0.155682
$$607$$ 9.02312 0.366237 0.183119 0.983091i $$-0.441381\pi$$
0.183119 + 0.983091i $$0.441381\pi$$
$$608$$ −3.39783 −0.137800
$$609$$ 7.90054 0.320146
$$610$$ 0 0
$$611$$ 7.05463 0.285400
$$612$$ 1.80090 0.0727971
$$613$$ −35.9969 −1.45390 −0.726950 0.686690i $$-0.759063\pi$$
−0.726950 + 0.686690i $$0.759063\pi$$
$$614$$ 2.25827 0.0911362
$$615$$ 0 0
$$616$$ −2.29184 −0.0923407
$$617$$ −44.8507 −1.80562 −0.902811 0.430037i $$-0.858501\pi$$
−0.902811 + 0.430037i $$0.858501\pi$$
$$618$$ −7.66272 −0.308240
$$619$$ 36.6485 1.47303 0.736514 0.676422i $$-0.236470\pi$$
0.736514 + 0.676422i $$0.236470\pi$$
$$620$$ 0 0
$$621$$ 19.1868 0.769941
$$622$$ 6.83331 0.273991
$$623$$ −21.6376 −0.866892
$$624$$ 24.1344 0.966148
$$625$$ 0 0
$$626$$ 4.91091 0.196279
$$627$$ 1.85377 0.0740326
$$628$$ −23.4038 −0.933915
$$629$$ 12.2074 0.486739
$$630$$ 0 0
$$631$$ −39.4249 −1.56948 −0.784739 0.619826i $$-0.787203\pi$$
−0.784739 + 0.619826i $$0.787203\pi$$
$$632$$ 12.7229 0.506091
$$633$$ 41.6347 1.65483
$$634$$ 8.00579 0.317950
$$635$$ 0 0
$$636$$ 16.2686 0.645090
$$637$$ 12.0658 0.478066
$$638$$ −0.659317 −0.0261026
$$639$$ 2.80359 0.110908
$$640$$ 0 0
$$641$$ −6.91478 −0.273117 −0.136559 0.990632i $$-0.543604\pi$$
−0.136559 + 0.990632i $$0.543604\pi$$
$$642$$ 8.29283 0.327292
$$643$$ 38.8033 1.53025 0.765127 0.643879i $$-0.222676\pi$$
0.765127 + 0.643879i $$0.222676\pi$$
$$644$$ −15.0065 −0.591340
$$645$$ 0 0
$$646$$ 0.650787 0.0256049
$$647$$ −18.4537 −0.725490 −0.362745 0.931888i $$-0.618160\pi$$
−0.362745 + 0.931888i $$0.618160\pi$$
$$648$$ −11.9131 −0.467989
$$649$$ −3.10100 −0.121725
$$650$$ 0 0
$$651$$ −11.6030 −0.454758
$$652$$ −15.2853 −0.598620
$$653$$ 12.6249 0.494052 0.247026 0.969009i $$-0.420547\pi$$
0.247026 + 0.969009i $$0.420547\pi$$
$$654$$ −3.98446 −0.155805
$$655$$ 0 0
$$656$$ −12.2256 −0.477328
$$657$$ 4.69005 0.182976
$$658$$ 1.10044 0.0428994
$$659$$ 37.4031 1.45702 0.728509 0.685036i $$-0.240214\pi$$
0.728509 + 0.685036i $$0.240214\pi$$
$$660$$ 0 0
$$661$$ −12.6089 −0.490431 −0.245215 0.969469i $$-0.578859\pi$$
−0.245215 + 0.969469i $$0.578859\pi$$
$$662$$ −3.79634 −0.147549
$$663$$ −15.0557 −0.584714
$$664$$ 9.56666 0.371259
$$665$$ 0 0
$$666$$ 0.742549 0.0287732
$$667$$ −8.83924 −0.342257
$$668$$ −9.55538 −0.369709
$$669$$ −19.1149 −0.739025
$$670$$ 0 0
$$671$$ 10.0971 0.389796
$$672$$ 12.2618 0.473008
$$673$$ −6.19541 −0.238815 −0.119408 0.992845i $$-0.538100\pi$$
−0.119408 + 0.992845i $$0.538100\pi$$
$$674$$ 1.86410 0.0718023
$$675$$ 0 0
$$676$$ −2.14786 −0.0826101
$$677$$ 26.5020 1.01855 0.509276 0.860603i $$-0.329913\pi$$
0.509276 + 0.860603i $$0.329913\pi$$
$$678$$ −0.684016 −0.0262695
$$679$$ 22.2880 0.855334
$$680$$ 0 0
$$681$$ 37.0303 1.41900
$$682$$ 0.968298 0.0370780
$$683$$ 44.3957 1.69876 0.849378 0.527786i $$-0.176978\pi$$
0.849378 + 0.527786i $$0.176978\pi$$
$$684$$ −0.833374 −0.0318648
$$685$$ 0 0
$$686$$ 5.98588 0.228542
$$687$$ 29.1093 1.11059
$$688$$ 9.49480 0.361986
$$689$$ −17.2747 −0.658113
$$690$$ 0 0
$$691$$ −15.0873 −0.573948 −0.286974 0.957938i $$-0.592649\pi$$
−0.286974 + 0.957938i $$0.592649\pi$$
$$692$$ −13.6674 −0.519558
$$693$$ −0.849687 −0.0322769
$$694$$ 3.78623 0.143723
$$695$$ 0 0
$$696$$ 4.77805 0.181112
$$697$$ 7.62664 0.288880
$$698$$ −5.12978 −0.194165
$$699$$ −39.2672 −1.48522
$$700$$ 0 0
$$701$$ 8.66946 0.327441 0.163720 0.986507i $$-0.447650\pi$$
0.163720 + 0.986507i $$0.447650\pi$$
$$702$$ 5.37869 0.203005
$$703$$ −5.64900 −0.213056
$$704$$ 5.90485 0.222548
$$705$$ 0 0
$$706$$ 2.56535 0.0965481
$$707$$ 13.3636 0.502590
$$708$$ 10.9757 0.412494
$$709$$ 42.7639 1.60603 0.803015 0.595958i $$-0.203228\pi$$
0.803015 + 0.595958i $$0.203228\pi$$
$$710$$ 0 0
$$711$$ 4.71696 0.176900
$$712$$ −13.0859 −0.490415
$$713$$ 12.9816 0.486166
$$714$$ −2.34850 −0.0878904
$$715$$ 0 0
$$716$$ 7.87049 0.294134
$$717$$ 0.795327 0.0297020
$$718$$ 1.93645 0.0722677
$$719$$ 27.6655 1.03175 0.515875 0.856664i $$-0.327467\pi$$
0.515875 + 0.856664i $$0.327467\pi$$
$$720$$ 0 0
$$721$$ −26.7197 −0.995093
$$722$$ −0.301154 −0.0112078
$$723$$ −22.1818 −0.824950
$$724$$ −33.4679 −1.24383
$$725$$ 0 0
$$726$$ 0.558272 0.0207194
$$727$$ 20.9256 0.776088 0.388044 0.921641i $$-0.373151\pi$$
0.388044 + 0.921641i $$0.373151\pi$$
$$728$$ −8.61344 −0.319235
$$729$$ 22.0118 0.815250
$$730$$ 0 0
$$731$$ −5.92311 −0.219074
$$732$$ −35.7381 −1.32092
$$733$$ 27.8171 1.02745 0.513723 0.857956i $$-0.328266\pi$$
0.513723 + 0.857956i $$0.328266\pi$$
$$734$$ 9.02359 0.333067
$$735$$ 0 0
$$736$$ −13.7186 −0.505676
$$737$$ −12.0622 −0.444316
$$738$$ 0.463913 0.0170769
$$739$$ 9.45503 0.347809 0.173904 0.984763i $$-0.444362\pi$$
0.173904 + 0.984763i $$0.444362\pi$$
$$740$$ 0 0
$$741$$ 6.96707 0.255942
$$742$$ −2.69464 −0.0989232
$$743$$ −5.01586 −0.184014 −0.0920070 0.995758i $$-0.529328\pi$$
−0.0920070 + 0.995758i $$0.529328\pi$$
$$744$$ −7.01723 −0.257264
$$745$$ 0 0
$$746$$ 10.4384 0.382175
$$747$$ 3.54679 0.129770
$$748$$ −4.12596 −0.150860
$$749$$ 28.9169 1.05660
$$750$$ 0 0
$$751$$ −42.5515 −1.55273 −0.776363 0.630286i $$-0.782938\pi$$
−0.776363 + 0.630286i $$0.782938\pi$$
$$752$$ −6.50229 −0.237114
$$753$$ −13.2332 −0.482246
$$754$$ −2.47792 −0.0902406
$$755$$ 0 0
$$756$$ −17.6630 −0.642396
$$757$$ 6.77125 0.246105 0.123053 0.992400i $$-0.460732\pi$$
0.123053 + 0.992400i $$0.460732\pi$$
$$758$$ −0.548225 −0.0199124
$$759$$ 7.48457 0.271673
$$760$$ 0 0
$$761$$ 31.5134 1.14236 0.571180 0.820825i $$-0.306486\pi$$
0.571180 + 0.820825i $$0.306486\pi$$
$$762$$ −0.994739 −0.0360356
$$763$$ −13.8937 −0.502986
$$764$$ 52.5212 1.90015
$$765$$ 0 0
$$766$$ −5.13025 −0.185364
$$767$$ −11.6545 −0.420821
$$768$$ −17.1059 −0.617258
$$769$$ 10.0557 0.362617 0.181308 0.983426i $$-0.441967\pi$$
0.181308 + 0.983426i $$0.441967\pi$$
$$770$$ 0 0
$$771$$ 12.7312 0.458503
$$772$$ −25.1882 −0.906541
$$773$$ 4.14250 0.148995 0.0744977 0.997221i $$-0.476265\pi$$
0.0744977 + 0.997221i $$0.476265\pi$$
$$774$$ −0.360291 −0.0129504
$$775$$ 0 0
$$776$$ 13.4792 0.483876
$$777$$ 20.3856 0.731329
$$778$$ 6.55849 0.235133
$$779$$ −3.52926 −0.126449
$$780$$ 0 0
$$781$$ −6.42317 −0.229839
$$782$$ 2.62754 0.0939605
$$783$$ −10.4040 −0.371807
$$784$$ −11.1211 −0.397184
$$785$$ 0 0
$$786$$ 10.2836 0.366803
$$787$$ −6.39855 −0.228084 −0.114042 0.993476i $$-0.536380\pi$$
−0.114042 + 0.993476i $$0.536380\pi$$
$$788$$ 21.8677 0.779003
$$789$$ 0.877168 0.0312280
$$790$$ 0 0
$$791$$ −2.38515 −0.0848060
$$792$$ −0.513870 −0.0182596
$$793$$ 37.9483 1.34758
$$794$$ −2.65563 −0.0942447
$$795$$ 0 0
$$796$$ 37.7207 1.33697
$$797$$ 31.6848 1.12233 0.561166 0.827703i $$-0.310353\pi$$
0.561166 + 0.827703i $$0.310353\pi$$
$$798$$ 1.08678 0.0384715
$$799$$ 4.05631 0.143502
$$800$$ 0 0
$$801$$ −4.85153 −0.171420
$$802$$ 3.32413 0.117379
$$803$$ −10.7452 −0.379189
$$804$$ 42.6931 1.50567
$$805$$ 0 0
$$806$$ 3.63917 0.128184
$$807$$ −4.20545 −0.148039
$$808$$ 8.08198 0.284323
$$809$$ 34.9827 1.22993 0.614963 0.788556i $$-0.289171\pi$$
0.614963 + 0.788556i $$0.289171\pi$$
$$810$$ 0 0
$$811$$ −27.4687 −0.964556 −0.482278 0.876018i $$-0.660190\pi$$
−0.482278 + 0.876018i $$0.660190\pi$$
$$812$$ 8.13721 0.285560
$$813$$ 25.2825 0.886697
$$814$$ −1.70122 −0.0596278
$$815$$ 0 0
$$816$$ 13.8769 0.485789
$$817$$ 2.74094 0.0958935
$$818$$ 7.81745 0.273331
$$819$$ −3.19339 −0.111586
$$820$$ 0 0
$$821$$ 17.9999 0.628200 0.314100 0.949390i $$-0.398297\pi$$
0.314100 + 0.949390i $$0.398297\pi$$
$$822$$ −7.75586 −0.270517
$$823$$ 8.49995 0.296290 0.148145 0.988966i $$-0.452670\pi$$
0.148145 + 0.988966i $$0.452670\pi$$
$$824$$ −16.1594 −0.562940
$$825$$ 0 0
$$826$$ −1.81796 −0.0632550
$$827$$ 10.1919 0.354406 0.177203 0.984174i $$-0.443295\pi$$
0.177203 + 0.984174i $$0.443295\pi$$
$$828$$ −3.36472 −0.116932
$$829$$ −4.33629 −0.150605 −0.0753027 0.997161i $$-0.523992\pi$$
−0.0753027 + 0.997161i $$0.523992\pi$$
$$830$$ 0 0
$$831$$ 3.05304 0.105909
$$832$$ 22.1923 0.769380
$$833$$ 6.93768 0.240376
$$834$$ −2.42941 −0.0841234
$$835$$ 0 0
$$836$$ 1.90931 0.0660347
$$837$$ 15.2796 0.528142
$$838$$ −2.45388 −0.0847677
$$839$$ −42.6097 −1.47105 −0.735525 0.677498i $$-0.763064\pi$$
−0.735525 + 0.677498i $$0.763064\pi$$
$$840$$ 0 0
$$841$$ −24.2070 −0.834723
$$842$$ 4.44104 0.153048
$$843$$ −31.0303 −1.06874
$$844$$ 42.8819 1.47606
$$845$$ 0 0
$$846$$ 0.246737 0.00848299
$$847$$ 1.94668 0.0668887
$$848$$ 15.9222 0.546769
$$849$$ 18.3846 0.630959
$$850$$ 0 0
$$851$$ −22.8077 −0.781838
$$852$$ 22.7343 0.778865
$$853$$ 26.8673 0.919917 0.459959 0.887940i $$-0.347864\pi$$
0.459959 + 0.887940i $$0.347864\pi$$
$$854$$ 5.91946 0.202560
$$855$$ 0 0
$$856$$ 17.4882 0.597735
$$857$$ −56.6619 −1.93553 −0.967767 0.251846i $$-0.918962\pi$$
−0.967767 + 0.251846i $$0.918962\pi$$
$$858$$ 2.09816 0.0716301
$$859$$ 51.0188 1.74074 0.870370 0.492398i $$-0.163880\pi$$
0.870370 + 0.492398i $$0.163880\pi$$
$$860$$ 0 0
$$861$$ 12.7361 0.434043
$$862$$ 8.43768 0.287389
$$863$$ 33.1838 1.12959 0.564795 0.825231i $$-0.308955\pi$$
0.564795 + 0.825231i $$0.308955\pi$$
$$864$$ −16.1471 −0.549336
$$865$$ 0 0
$$866$$ 0.645766 0.0219440
$$867$$ 22.8574 0.776277
$$868$$ −11.9506 −0.405630
$$869$$ −10.8068 −0.366597
$$870$$ 0 0
$$871$$ −45.3334 −1.53606
$$872$$ −8.40257 −0.284547
$$873$$ 4.99736 0.169135
$$874$$ −1.21590 −0.0411285
$$875$$ 0 0
$$876$$ 38.0317 1.28497
$$877$$ −43.4040 −1.46565 −0.732825 0.680418i $$-0.761798\pi$$
−0.732825 + 0.680418i $$0.761798\pi$$
$$878$$ −1.50078 −0.0506490
$$879$$ 26.2828 0.886499
$$880$$ 0 0
$$881$$ −34.1080 −1.14913 −0.574565 0.818459i $$-0.694829\pi$$
−0.574565 + 0.818459i $$0.694829\pi$$
$$882$$ 0.422005 0.0142096
$$883$$ 31.9448 1.07503 0.537514 0.843255i $$-0.319364\pi$$
0.537514 + 0.843255i $$0.319364\pi$$
$$884$$ −15.5067 −0.521546
$$885$$ 0 0
$$886$$ 3.12461 0.104973
$$887$$ 15.6331 0.524907 0.262453 0.964945i $$-0.415468\pi$$
0.262453 + 0.964945i $$0.415468\pi$$
$$888$$ 12.3287 0.413724
$$889$$ −3.46863 −0.116334
$$890$$ 0 0
$$891$$ 10.1189 0.338997
$$892$$ −19.6875 −0.659187
$$893$$ −1.87707 −0.0628138
$$894$$ −6.52713 −0.218300
$$895$$ 0 0
$$896$$ 16.6907 0.557597
$$897$$ 28.1294 0.939212
$$898$$ −10.8320 −0.361469
$$899$$ −7.03923 −0.234771
$$900$$ 0 0
$$901$$ −9.93268 −0.330905
$$902$$ −1.06285 −0.0353891
$$903$$ −9.89126 −0.329161
$$904$$ −1.44248 −0.0479761
$$905$$ 0 0
$$906$$ −12.8648 −0.427404
$$907$$ −15.2429 −0.506132 −0.253066 0.967449i $$-0.581439\pi$$
−0.253066 + 0.967449i $$0.581439\pi$$
$$908$$ 38.1395 1.26571
$$909$$ 2.99635 0.0993828
$$910$$ 0 0
$$911$$ 10.7020 0.354574 0.177287 0.984159i $$-0.443268\pi$$
0.177287 + 0.984159i $$0.443268\pi$$
$$912$$ −6.42159 −0.212640
$$913$$ −8.12591 −0.268928
$$914$$ 8.24688 0.272783
$$915$$ 0 0
$$916$$ 29.9813 0.990611
$$917$$ 35.8586 1.18415
$$918$$ 3.09266 0.102073
$$919$$ −59.1251 −1.95036 −0.975178 0.221421i $$-0.928931\pi$$
−0.975178 + 0.221421i $$0.928931\pi$$
$$920$$ 0 0
$$921$$ 13.9009 0.458050
$$922$$ 2.88767 0.0951003
$$923$$ −24.1403 −0.794588
$$924$$ −6.89012 −0.226668
$$925$$ 0 0
$$926$$ 8.25626 0.271318
$$927$$ −5.99102 −0.196771
$$928$$ 7.43886 0.244193
$$929$$ 41.4480 1.35986 0.679932 0.733276i $$-0.262009\pi$$
0.679932 + 0.733276i $$0.262009\pi$$
$$930$$ 0 0
$$931$$ −3.21044 −0.105218
$$932$$ −40.4435 −1.32477
$$933$$ 42.0629 1.37708
$$934$$ −9.22694 −0.301915
$$935$$ 0 0
$$936$$ −1.93128 −0.0631260
$$937$$ −2.64414 −0.0863803 −0.0431901 0.999067i $$-0.513752\pi$$
−0.0431901 + 0.999067i $$0.513752\pi$$
$$938$$ −7.07146 −0.230891
$$939$$ 30.2294 0.986500
$$940$$ 0 0
$$941$$ −55.5911 −1.81222 −0.906109 0.423045i $$-0.860961\pi$$
−0.906109 + 0.423045i $$0.860961\pi$$
$$942$$ 6.84317 0.222963
$$943$$ −14.2493 −0.464021
$$944$$ 10.7420 0.349624
$$945$$ 0 0
$$946$$ 0.825447 0.0268376
$$947$$ −30.2967 −0.984509 −0.492255 0.870451i $$-0.663827\pi$$
−0.492255 + 0.870451i $$0.663827\pi$$
$$948$$ 38.2500 1.24230
$$949$$ −40.3838 −1.31091
$$950$$ 0 0
$$951$$ 49.2801 1.59802
$$952$$ −4.95260 −0.160515
$$953$$ 0.456421 0.0147849 0.00739247 0.999973i $$-0.497647\pi$$
0.00739247 + 0.999973i $$0.497647\pi$$
$$954$$ −0.604185 −0.0195612
$$955$$ 0 0
$$956$$ 0.819152 0.0264933
$$957$$ −4.05847 −0.131192
$$958$$ −1.67288 −0.0540482
$$959$$ −27.0445 −0.873312
$$960$$ 0 0
$$961$$ −20.6619 −0.666514
$$962$$ −6.39373 −0.206142
$$963$$ 6.48367 0.208933
$$964$$ −22.8463 −0.735829
$$965$$ 0 0
$$966$$ 4.38784 0.141176
$$967$$ 34.6037 1.11278 0.556389 0.830922i $$-0.312186\pi$$
0.556389 + 0.830922i $$0.312186\pi$$
$$968$$ 1.17730 0.0378400
$$969$$ 4.00596 0.128690
$$970$$ 0 0
$$971$$ −20.0747 −0.644228 −0.322114 0.946701i $$-0.604393\pi$$
−0.322114 + 0.946701i $$0.604393\pi$$
$$972$$ −8.59501 −0.275685
$$973$$ −8.47127 −0.271576
$$974$$ −4.66304 −0.149414
$$975$$ 0 0
$$976$$ −34.9771 −1.11959
$$977$$ 8.02308 0.256681 0.128341 0.991730i $$-0.459035\pi$$
0.128341 + 0.991730i $$0.459035\pi$$
$$978$$ 4.46936 0.142914
$$979$$ 11.1151 0.355241
$$980$$ 0 0
$$981$$ −3.11521 −0.0994611
$$982$$ 7.79733 0.248823
$$983$$ 10.1782 0.324633 0.162317 0.986739i $$-0.448103\pi$$
0.162317 + 0.986739i $$0.448103\pi$$
$$984$$ 7.70245 0.245545
$$985$$ 0 0
$$986$$ −1.42477 −0.0453739
$$987$$ 6.77380 0.215612
$$988$$ 7.17578 0.228292
$$989$$ 11.0665 0.351894
$$990$$ 0 0
$$991$$ 25.7709 0.818641 0.409321 0.912391i $$-0.365766\pi$$
0.409321 + 0.912391i $$0.365766\pi$$
$$992$$ −10.9250 −0.346869
$$993$$ −23.3686 −0.741581
$$994$$ −3.76559 −0.119437
$$995$$ 0 0
$$996$$ 28.7610 0.911328
$$997$$ −33.3107 −1.05496 −0.527481 0.849567i $$-0.676863\pi$$
−0.527481 + 0.849567i $$0.676863\pi$$
$$998$$ 10.2323 0.323898
$$999$$ −26.8451 −0.849342
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.z.1.8 16
5.2 odd 4 1045.2.b.b.419.8 16
5.3 odd 4 1045.2.b.b.419.9 yes 16
5.4 even 2 inner 5225.2.a.z.1.9 16

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.b.419.8 16 5.2 odd 4
1045.2.b.b.419.9 yes 16 5.3 odd 4
5225.2.a.z.1.8 16 1.1 even 1 trivial
5225.2.a.z.1.9 16 5.4 even 2 inner