# Properties

 Label 5225.2.a.z.1.13 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $16$ 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: $$1$$ 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 1.13 Root $$1.78099$$ 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)

 $$f(q)$$ $$=$$ $$q+1.78099 q^{2} +0.213501 q^{3} +1.17191 q^{4} +0.380243 q^{6} +3.50934 q^{7} -1.47481 q^{8} -2.95442 q^{9} +O(q^{10})$$ $$q+1.78099 q^{2} +0.213501 q^{3} +1.17191 q^{4} +0.380243 q^{6} +3.50934 q^{7} -1.47481 q^{8} -2.95442 q^{9} -1.00000 q^{11} +0.250205 q^{12} -1.11716 q^{13} +6.25008 q^{14} -4.97045 q^{16} -3.28198 q^{17} -5.26178 q^{18} +1.00000 q^{19} +0.749247 q^{21} -1.78099 q^{22} +0.303981 q^{23} -0.314874 q^{24} -1.98966 q^{26} -1.27127 q^{27} +4.11264 q^{28} -5.23841 q^{29} -0.126980 q^{31} -5.90267 q^{32} -0.213501 q^{33} -5.84516 q^{34} -3.46232 q^{36} -5.03915 q^{37} +1.78099 q^{38} -0.238516 q^{39} +0.293793 q^{41} +1.33440 q^{42} +0.180719 q^{43} -1.17191 q^{44} +0.541385 q^{46} -3.64879 q^{47} -1.06120 q^{48} +5.31545 q^{49} -0.700706 q^{51} -1.30922 q^{52} -1.17201 q^{53} -2.26412 q^{54} -5.17562 q^{56} +0.213501 q^{57} -9.32953 q^{58} +5.73363 q^{59} -3.61056 q^{61} -0.226150 q^{62} -10.3680 q^{63} -0.571685 q^{64} -0.380243 q^{66} -5.70818 q^{67} -3.84619 q^{68} +0.0649002 q^{69} -10.9423 q^{71} +4.35721 q^{72} -12.3731 q^{73} -8.97465 q^{74} +1.17191 q^{76} -3.50934 q^{77} -0.424794 q^{78} -2.32533 q^{79} +8.59183 q^{81} +0.523242 q^{82} -2.66052 q^{83} +0.878052 q^{84} +0.321858 q^{86} -1.11841 q^{87} +1.47481 q^{88} +0.0652748 q^{89} -3.92051 q^{91} +0.356239 q^{92} -0.0271104 q^{93} -6.49844 q^{94} -1.26023 q^{96} -17.5012 q^{97} +9.46675 q^{98} +2.95442 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 6 q^{4} - 8 q^{6} + 8 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.78099 1.25935 0.629674 0.776860i $$-0.283189\pi$$
0.629674 + 0.776860i $$0.283189\pi$$
$$3$$ 0.213501 0.123265 0.0616325 0.998099i $$-0.480369\pi$$
0.0616325 + 0.998099i $$0.480369\pi$$
$$4$$ 1.17191 0.585956
$$5$$ 0 0
$$6$$ 0.380243 0.155233
$$7$$ 3.50934 1.32640 0.663202 0.748440i $$-0.269197\pi$$
0.663202 + 0.748440i $$0.269197\pi$$
$$8$$ −1.47481 −0.521425
$$9$$ −2.95442 −0.984806
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0.250205 0.0722278
$$13$$ −1.11716 −0.309846 −0.154923 0.987927i $$-0.549513\pi$$
−0.154923 + 0.987927i $$0.549513\pi$$
$$14$$ 6.25008 1.67040
$$15$$ 0 0
$$16$$ −4.97045 −1.24261
$$17$$ −3.28198 −0.795997 −0.397999 0.917386i $$-0.630295\pi$$
−0.397999 + 0.917386i $$0.630295\pi$$
$$18$$ −5.26178 −1.24021
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0.749247 0.163499
$$22$$ −1.78099 −0.379708
$$23$$ 0.303981 0.0633844 0.0316922 0.999498i $$-0.489910\pi$$
0.0316922 + 0.999498i $$0.489910\pi$$
$$24$$ −0.314874 −0.0642734
$$25$$ 0 0
$$26$$ −1.98966 −0.390203
$$27$$ −1.27127 −0.244657
$$28$$ 4.11264 0.777215
$$29$$ −5.23841 −0.972748 −0.486374 0.873751i $$-0.661681\pi$$
−0.486374 + 0.873751i $$0.661681\pi$$
$$30$$ 0 0
$$31$$ −0.126980 −0.0228063 −0.0114031 0.999935i $$-0.503630\pi$$
−0.0114031 + 0.999935i $$0.503630\pi$$
$$32$$ −5.90267 −1.04345
$$33$$ −0.213501 −0.0371658
$$34$$ −5.84516 −1.00244
$$35$$ 0 0
$$36$$ −3.46232 −0.577053
$$37$$ −5.03915 −0.828431 −0.414215 0.910179i $$-0.635944\pi$$
−0.414215 + 0.910179i $$0.635944\pi$$
$$38$$ 1.78099 0.288914
$$39$$ −0.238516 −0.0381931
$$40$$ 0 0
$$41$$ 0.293793 0.0458828 0.0229414 0.999737i $$-0.492697\pi$$
0.0229414 + 0.999737i $$0.492697\pi$$
$$42$$ 1.33440 0.205902
$$43$$ 0.180719 0.0275594 0.0137797 0.999905i $$-0.495614\pi$$
0.0137797 + 0.999905i $$0.495614\pi$$
$$44$$ −1.17191 −0.176672
$$45$$ 0 0
$$46$$ 0.541385 0.0798229
$$47$$ −3.64879 −0.532230 −0.266115 0.963941i $$-0.585740\pi$$
−0.266115 + 0.963941i $$0.585740\pi$$
$$48$$ −1.06120 −0.153170
$$49$$ 5.31545 0.759350
$$50$$ 0 0
$$51$$ −0.700706 −0.0981185
$$52$$ −1.30922 −0.181556
$$53$$ −1.17201 −0.160988 −0.0804939 0.996755i $$-0.525650\pi$$
−0.0804939 + 0.996755i $$0.525650\pi$$
$$54$$ −2.26412 −0.308108
$$55$$ 0 0
$$56$$ −5.17562 −0.691621
$$57$$ 0.213501 0.0282789
$$58$$ −9.32953 −1.22503
$$59$$ 5.73363 0.746454 0.373227 0.927740i $$-0.378251\pi$$
0.373227 + 0.927740i $$0.378251\pi$$
$$60$$ 0 0
$$61$$ −3.61056 −0.462285 −0.231142 0.972920i $$-0.574246\pi$$
−0.231142 + 0.972920i $$0.574246\pi$$
$$62$$ −0.226150 −0.0287210
$$63$$ −10.3680 −1.30625
$$64$$ −0.571685 −0.0714606
$$65$$ 0 0
$$66$$ −0.380243 −0.0468046
$$67$$ −5.70818 −0.697365 −0.348682 0.937241i $$-0.613371\pi$$
−0.348682 + 0.937241i $$0.613371\pi$$
$$68$$ −3.84619 −0.466419
$$69$$ 0.0649002 0.00781307
$$70$$ 0 0
$$71$$ −10.9423 −1.29861 −0.649306 0.760527i $$-0.724941\pi$$
−0.649306 + 0.760527i $$0.724941\pi$$
$$72$$ 4.35721 0.513502
$$73$$ −12.3731 −1.44816 −0.724080 0.689716i $$-0.757735\pi$$
−0.724080 + 0.689716i $$0.757735\pi$$
$$74$$ −8.97465 −1.04328
$$75$$ 0 0
$$76$$ 1.17191 0.134428
$$77$$ −3.50934 −0.399926
$$78$$ −0.424794 −0.0480984
$$79$$ −2.32533 −0.261620 −0.130810 0.991407i $$-0.541758\pi$$
−0.130810 + 0.991407i $$0.541758\pi$$
$$80$$ 0 0
$$81$$ 8.59183 0.954648
$$82$$ 0.523242 0.0577824
$$83$$ −2.66052 −0.292031 −0.146015 0.989282i $$-0.546645\pi$$
−0.146015 + 0.989282i $$0.546645\pi$$
$$84$$ 0.878052 0.0958034
$$85$$ 0 0
$$86$$ 0.321858 0.0347069
$$87$$ −1.11841 −0.119906
$$88$$ 1.47481 0.157216
$$89$$ 0.0652748 0.00691912 0.00345956 0.999994i $$-0.498899\pi$$
0.00345956 + 0.999994i $$0.498899\pi$$
$$90$$ 0 0
$$91$$ −3.92051 −0.410981
$$92$$ 0.356239 0.0371405
$$93$$ −0.0271104 −0.00281121
$$94$$ −6.49844 −0.670263
$$95$$ 0 0
$$96$$ −1.26023 −0.128621
$$97$$ −17.5012 −1.77698 −0.888489 0.458899i $$-0.848244\pi$$
−0.888489 + 0.458899i $$0.848244\pi$$
$$98$$ 9.46675 0.956286
$$99$$ 2.95442 0.296930
$$100$$ 0 0
$$101$$ 7.81681 0.777802 0.388901 0.921280i $$-0.372855\pi$$
0.388901 + 0.921280i $$0.372855\pi$$
$$102$$ −1.24795 −0.123565
$$103$$ 5.42056 0.534104 0.267052 0.963682i $$-0.413950\pi$$
0.267052 + 0.963682i $$0.413950\pi$$
$$104$$ 1.64761 0.161561
$$105$$ 0 0
$$106$$ −2.08733 −0.202740
$$107$$ 9.22338 0.891658 0.445829 0.895118i $$-0.352909\pi$$
0.445829 + 0.895118i $$0.352909\pi$$
$$108$$ −1.48982 −0.143358
$$109$$ 3.85920 0.369644 0.184822 0.982772i $$-0.440829\pi$$
0.184822 + 0.982772i $$0.440829\pi$$
$$110$$ 0 0
$$111$$ −1.07586 −0.102116
$$112$$ −17.4430 −1.64821
$$113$$ 0.909757 0.0855827 0.0427914 0.999084i $$-0.486375\pi$$
0.0427914 + 0.999084i $$0.486375\pi$$
$$114$$ 0.380243 0.0356130
$$115$$ 0 0
$$116$$ −6.13895 −0.569988
$$117$$ 3.30057 0.305138
$$118$$ 10.2115 0.940046
$$119$$ −11.5176 −1.05581
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −6.43036 −0.582177
$$123$$ 0.0627252 0.00565574
$$124$$ −0.148809 −0.0133635
$$125$$ 0 0
$$126$$ −18.4654 −1.64502
$$127$$ 1.00655 0.0893165 0.0446582 0.999002i $$-0.485780\pi$$
0.0446582 + 0.999002i $$0.485780\pi$$
$$128$$ 10.7872 0.953461
$$129$$ 0.0385837 0.00339711
$$130$$ 0 0
$$131$$ 8.34802 0.729370 0.364685 0.931131i $$-0.381177\pi$$
0.364685 + 0.931131i $$0.381177\pi$$
$$132$$ −0.250205 −0.0217775
$$133$$ 3.50934 0.304298
$$134$$ −10.1662 −0.878224
$$135$$ 0 0
$$136$$ 4.84031 0.415053
$$137$$ 14.8998 1.27297 0.636486 0.771288i $$-0.280387\pi$$
0.636486 + 0.771288i $$0.280387\pi$$
$$138$$ 0.115586 0.00983937
$$139$$ −3.03344 −0.257293 −0.128646 0.991691i $$-0.541063\pi$$
−0.128646 + 0.991691i $$0.541063\pi$$
$$140$$ 0 0
$$141$$ −0.779020 −0.0656053
$$142$$ −19.4881 −1.63540
$$143$$ 1.11716 0.0934220
$$144$$ 14.6848 1.22373
$$145$$ 0 0
$$146$$ −22.0363 −1.82374
$$147$$ 1.13485 0.0936012
$$148$$ −5.90544 −0.485424
$$149$$ −23.5095 −1.92597 −0.962986 0.269553i $$-0.913124\pi$$
−0.962986 + 0.269553i $$0.913124\pi$$
$$150$$ 0 0
$$151$$ −5.05226 −0.411147 −0.205574 0.978642i $$-0.565906\pi$$
−0.205574 + 0.978642i $$0.565906\pi$$
$$152$$ −1.47481 −0.119623
$$153$$ 9.69634 0.783902
$$154$$ −6.25008 −0.503646
$$155$$ 0 0
$$156$$ −0.279520 −0.0223795
$$157$$ −14.0292 −1.11965 −0.559827 0.828609i $$-0.689133\pi$$
−0.559827 + 0.828609i $$0.689133\pi$$
$$158$$ −4.14138 −0.329470
$$159$$ −0.250225 −0.0198441
$$160$$ 0 0
$$161$$ 1.06677 0.0840733
$$162$$ 15.3019 1.20223
$$163$$ 8.22502 0.644233 0.322117 0.946700i $$-0.395606\pi$$
0.322117 + 0.946700i $$0.395606\pi$$
$$164$$ 0.344300 0.0268853
$$165$$ 0 0
$$166$$ −4.73836 −0.367768
$$167$$ 4.03652 0.312356 0.156178 0.987729i $$-0.450083\pi$$
0.156178 + 0.987729i $$0.450083\pi$$
$$168$$ −1.10500 −0.0852526
$$169$$ −11.7519 −0.903996
$$170$$ 0 0
$$171$$ −2.95442 −0.225930
$$172$$ 0.211787 0.0161486
$$173$$ −3.01611 −0.229311 −0.114655 0.993405i $$-0.536576\pi$$
−0.114655 + 0.993405i $$0.536576\pi$$
$$174$$ −1.99187 −0.151003
$$175$$ 0 0
$$176$$ 4.97045 0.374661
$$177$$ 1.22414 0.0920116
$$178$$ 0.116254 0.00871357
$$179$$ 8.24312 0.616120 0.308060 0.951367i $$-0.400320\pi$$
0.308060 + 0.951367i $$0.400320\pi$$
$$180$$ 0 0
$$181$$ −18.1982 −1.35266 −0.676331 0.736598i $$-0.736431\pi$$
−0.676331 + 0.736598i $$0.736431\pi$$
$$182$$ −6.98237 −0.517568
$$183$$ −0.770858 −0.0569835
$$184$$ −0.448315 −0.0330502
$$185$$ 0 0
$$186$$ −0.0482832 −0.00354029
$$187$$ 3.28198 0.240002
$$188$$ −4.27606 −0.311864
$$189$$ −4.46133 −0.324514
$$190$$ 0 0
$$191$$ 22.9398 1.65986 0.829931 0.557866i $$-0.188380\pi$$
0.829931 + 0.557866i $$0.188380\pi$$
$$192$$ −0.122055 −0.00880858
$$193$$ 4.53030 0.326098 0.163049 0.986618i $$-0.447867\pi$$
0.163049 + 0.986618i $$0.447867\pi$$
$$194$$ −31.1694 −2.23783
$$195$$ 0 0
$$196$$ 6.22924 0.444946
$$197$$ −13.4791 −0.960344 −0.480172 0.877174i $$-0.659426\pi$$
−0.480172 + 0.877174i $$0.659426\pi$$
$$198$$ 5.26178 0.373938
$$199$$ −5.37014 −0.380679 −0.190340 0.981718i $$-0.560959\pi$$
−0.190340 + 0.981718i $$0.560959\pi$$
$$200$$ 0 0
$$201$$ −1.21870 −0.0859606
$$202$$ 13.9216 0.979523
$$203$$ −18.3833 −1.29026
$$204$$ −0.821166 −0.0574931
$$205$$ 0 0
$$206$$ 9.65395 0.672623
$$207$$ −0.898086 −0.0624213
$$208$$ 5.55281 0.385018
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 21.6683 1.49171 0.745855 0.666108i $$-0.232041\pi$$
0.745855 + 0.666108i $$0.232041\pi$$
$$212$$ −1.37349 −0.0943318
$$213$$ −2.33619 −0.160073
$$214$$ 16.4267 1.12291
$$215$$ 0 0
$$216$$ 1.87489 0.127570
$$217$$ −0.445616 −0.0302504
$$218$$ 6.87318 0.465510
$$219$$ −2.64167 −0.178507
$$220$$ 0 0
$$221$$ 3.66651 0.246636
$$222$$ −1.91610 −0.128600
$$223$$ 27.8124 1.86246 0.931228 0.364438i $$-0.118739\pi$$
0.931228 + 0.364438i $$0.118739\pi$$
$$224$$ −20.7145 −1.38404
$$225$$ 0 0
$$226$$ 1.62026 0.107778
$$227$$ 7.61682 0.505546 0.252773 0.967526i $$-0.418657\pi$$
0.252773 + 0.967526i $$0.418657\pi$$
$$228$$ 0.250205 0.0165702
$$229$$ −16.4932 −1.08990 −0.544952 0.838467i $$-0.683452\pi$$
−0.544952 + 0.838467i $$0.683452\pi$$
$$230$$ 0 0
$$231$$ −0.749247 −0.0492969
$$232$$ 7.72567 0.507215
$$233$$ −20.6984 −1.35599 −0.677997 0.735065i $$-0.737152\pi$$
−0.677997 + 0.735065i $$0.737152\pi$$
$$234$$ 5.87827 0.384275
$$235$$ 0 0
$$236$$ 6.71931 0.437390
$$237$$ −0.496460 −0.0322486
$$238$$ −20.5126 −1.32964
$$239$$ 30.2075 1.95396 0.976981 0.213325i $$-0.0684295\pi$$
0.976981 + 0.213325i $$0.0684295\pi$$
$$240$$ 0 0
$$241$$ −13.5984 −0.875953 −0.437976 0.898986i $$-0.644305\pi$$
−0.437976 + 0.898986i $$0.644305\pi$$
$$242$$ 1.78099 0.114486
$$243$$ 5.64819 0.362332
$$244$$ −4.23126 −0.270879
$$245$$ 0 0
$$246$$ 0.111713 0.00712254
$$247$$ −1.11716 −0.0710835
$$248$$ 0.187272 0.0118918
$$249$$ −0.568025 −0.0359971
$$250$$ 0 0
$$251$$ 9.47667 0.598162 0.299081 0.954228i $$-0.403320\pi$$
0.299081 + 0.954228i $$0.403320\pi$$
$$252$$ −12.1504 −0.765406
$$253$$ −0.303981 −0.0191111
$$254$$ 1.79264 0.112480
$$255$$ 0 0
$$256$$ 20.3552 1.27220
$$257$$ 5.42346 0.338306 0.169153 0.985590i $$-0.445897\pi$$
0.169153 + 0.985590i $$0.445897\pi$$
$$258$$ 0.0687171 0.00427814
$$259$$ −17.6841 −1.09883
$$260$$ 0 0
$$261$$ 15.4764 0.957968
$$262$$ 14.8677 0.918531
$$263$$ −10.1945 −0.628618 −0.314309 0.949321i $$-0.601773\pi$$
−0.314309 + 0.949321i $$0.601773\pi$$
$$264$$ 0.314874 0.0193792
$$265$$ 0 0
$$266$$ 6.25008 0.383217
$$267$$ 0.0139362 0.000852884 0
$$268$$ −6.68948 −0.408625
$$269$$ −30.6046 −1.86600 −0.932999 0.359880i $$-0.882818\pi$$
−0.932999 + 0.359880i $$0.882818\pi$$
$$270$$ 0 0
$$271$$ −27.5296 −1.67230 −0.836152 0.548499i $$-0.815200\pi$$
−0.836152 + 0.548499i $$0.815200\pi$$
$$272$$ 16.3129 0.989115
$$273$$ −0.837033 −0.0506595
$$274$$ 26.5363 1.60311
$$275$$ 0 0
$$276$$ 0.0760574 0.00457811
$$277$$ −13.8247 −0.830643 −0.415321 0.909675i $$-0.636331\pi$$
−0.415321 + 0.909675i $$0.636331\pi$$
$$278$$ −5.40251 −0.324021
$$279$$ 0.375152 0.0224598
$$280$$ 0 0
$$281$$ 11.7359 0.700107 0.350054 0.936730i $$-0.386163\pi$$
0.350054 + 0.936730i $$0.386163\pi$$
$$282$$ −1.38742 −0.0826199
$$283$$ 2.10305 0.125013 0.0625067 0.998045i $$-0.480091\pi$$
0.0625067 + 0.998045i $$0.480091\pi$$
$$284$$ −12.8234 −0.760930
$$285$$ 0 0
$$286$$ 1.98966 0.117651
$$287$$ 1.03102 0.0608592
$$288$$ 17.4390 1.02760
$$289$$ −6.22861 −0.366389
$$290$$ 0 0
$$291$$ −3.73652 −0.219039
$$292$$ −14.5002 −0.848558
$$293$$ 6.28988 0.367458 0.183729 0.982977i $$-0.441183\pi$$
0.183729 + 0.982977i $$0.441183\pi$$
$$294$$ 2.02116 0.117876
$$295$$ 0 0
$$296$$ 7.43180 0.431964
$$297$$ 1.27127 0.0737668
$$298$$ −41.8701 −2.42547
$$299$$ −0.339596 −0.0196394
$$300$$ 0 0
$$301$$ 0.634204 0.0365549
$$302$$ −8.99801 −0.517777
$$303$$ 1.66890 0.0958757
$$304$$ −4.97045 −0.285075
$$305$$ 0 0
$$306$$ 17.2690 0.987206
$$307$$ 15.7997 0.901738 0.450869 0.892590i $$-0.351114\pi$$
0.450869 + 0.892590i $$0.351114\pi$$
$$308$$ −4.11264 −0.234339
$$309$$ 1.15730 0.0658363
$$310$$ 0 0
$$311$$ −7.80974 −0.442850 −0.221425 0.975177i $$-0.571071\pi$$
−0.221425 + 0.975177i $$0.571071\pi$$
$$312$$ 0.351766 0.0199148
$$313$$ −12.0946 −0.683629 −0.341814 0.939768i $$-0.611041\pi$$
−0.341814 + 0.939768i $$0.611041\pi$$
$$314$$ −24.9859 −1.41003
$$315$$ 0 0
$$316$$ −2.72508 −0.153298
$$317$$ 1.31596 0.0739116 0.0369558 0.999317i $$-0.488234\pi$$
0.0369558 + 0.999317i $$0.488234\pi$$
$$318$$ −0.445647 −0.0249907
$$319$$ 5.23841 0.293295
$$320$$ 0 0
$$321$$ 1.96920 0.109910
$$322$$ 1.89990 0.105878
$$323$$ −3.28198 −0.182614
$$324$$ 10.0689 0.559382
$$325$$ 0 0
$$326$$ 14.6486 0.811314
$$327$$ 0.823943 0.0455642
$$328$$ −0.433290 −0.0239244
$$329$$ −12.8048 −0.705953
$$330$$ 0 0
$$331$$ −4.16094 −0.228706 −0.114353 0.993440i $$-0.536480\pi$$
−0.114353 + 0.993440i $$0.536480\pi$$
$$332$$ −3.11790 −0.171117
$$333$$ 14.8877 0.815843
$$334$$ 7.18899 0.393364
$$335$$ 0 0
$$336$$ −3.72409 −0.203166
$$337$$ −28.5927 −1.55754 −0.778771 0.627308i $$-0.784157\pi$$
−0.778771 + 0.627308i $$0.784157\pi$$
$$338$$ −20.9301 −1.13844
$$339$$ 0.194234 0.0105493
$$340$$ 0 0
$$341$$ 0.126980 0.00687635
$$342$$ −5.26178 −0.284524
$$343$$ −5.91165 −0.319199
$$344$$ −0.266527 −0.0143702
$$345$$ 0 0
$$346$$ −5.37165 −0.288782
$$347$$ 9.07959 0.487418 0.243709 0.969848i $$-0.421636\pi$$
0.243709 + 0.969848i $$0.421636\pi$$
$$348$$ −1.31067 −0.0702595
$$349$$ 18.6493 0.998273 0.499136 0.866523i $$-0.333651\pi$$
0.499136 + 0.866523i $$0.333651\pi$$
$$350$$ 0 0
$$351$$ 1.42022 0.0758059
$$352$$ 5.90267 0.314613
$$353$$ 29.3091 1.55997 0.779984 0.625800i $$-0.215227\pi$$
0.779984 + 0.625800i $$0.215227\pi$$
$$354$$ 2.18017 0.115875
$$355$$ 0 0
$$356$$ 0.0764964 0.00405430
$$357$$ −2.45902 −0.130145
$$358$$ 14.6809 0.775909
$$359$$ −26.4877 −1.39797 −0.698984 0.715137i $$-0.746364\pi$$
−0.698984 + 0.715137i $$0.746364\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −32.4107 −1.70347
$$363$$ 0.213501 0.0112059
$$364$$ −4.59449 −0.240817
$$365$$ 0 0
$$366$$ −1.37289 −0.0717620
$$367$$ 27.3556 1.42795 0.713975 0.700171i $$-0.246893\pi$$
0.713975 + 0.700171i $$0.246893\pi$$
$$368$$ −1.51092 −0.0787621
$$369$$ −0.867988 −0.0451856
$$370$$ 0 0
$$371$$ −4.11297 −0.213535
$$372$$ −0.0317710 −0.00164725
$$373$$ 37.5358 1.94353 0.971766 0.235947i $$-0.0758192\pi$$
0.971766 + 0.235947i $$0.0758192\pi$$
$$374$$ 5.84516 0.302246
$$375$$ 0 0
$$376$$ 5.38128 0.277518
$$377$$ 5.85216 0.301402
$$378$$ −7.94557 −0.408676
$$379$$ −31.1607 −1.60062 −0.800308 0.599589i $$-0.795331\pi$$
−0.800308 + 0.599589i $$0.795331\pi$$
$$380$$ 0 0
$$381$$ 0.214899 0.0110096
$$382$$ 40.8554 2.09034
$$383$$ 22.2016 1.13445 0.567225 0.823563i $$-0.308017\pi$$
0.567225 + 0.823563i $$0.308017\pi$$
$$384$$ 2.30307 0.117528
$$385$$ 0 0
$$386$$ 8.06841 0.410671
$$387$$ −0.533919 −0.0271406
$$388$$ −20.5099 −1.04123
$$389$$ 29.7883 1.51033 0.755164 0.655535i $$-0.227557\pi$$
0.755164 + 0.655535i $$0.227557\pi$$
$$390$$ 0 0
$$391$$ −0.997659 −0.0504538
$$392$$ −7.83929 −0.395944
$$393$$ 1.78231 0.0899058
$$394$$ −24.0060 −1.20941
$$395$$ 0 0
$$396$$ 3.46232 0.173988
$$397$$ −31.8515 −1.59858 −0.799291 0.600944i $$-0.794791\pi$$
−0.799291 + 0.600944i $$0.794791\pi$$
$$398$$ −9.56415 −0.479408
$$399$$ 0.749247 0.0375093
$$400$$ 0 0
$$401$$ 26.7235 1.33451 0.667255 0.744830i $$-0.267469\pi$$
0.667255 + 0.744830i $$0.267469\pi$$
$$402$$ −2.17049 −0.108254
$$403$$ 0.141858 0.00706643
$$404$$ 9.16062 0.455758
$$405$$ 0 0
$$406$$ −32.7405 −1.62488
$$407$$ 5.03915 0.249781
$$408$$ 1.03341 0.0511614
$$409$$ 27.4840 1.35900 0.679498 0.733677i $$-0.262198\pi$$
0.679498 + 0.733677i $$0.262198\pi$$
$$410$$ 0 0
$$411$$ 3.18111 0.156913
$$412$$ 6.35243 0.312962
$$413$$ 20.1212 0.990101
$$414$$ −1.59948 −0.0786101
$$415$$ 0 0
$$416$$ 6.59426 0.323310
$$417$$ −0.647643 −0.0317152
$$418$$ −1.78099 −0.0871109
$$419$$ −18.4002 −0.898909 −0.449455 0.893303i $$-0.648382\pi$$
−0.449455 + 0.893303i $$0.648382\pi$$
$$420$$ 0 0
$$421$$ −36.7525 −1.79121 −0.895603 0.444855i $$-0.853255\pi$$
−0.895603 + 0.444855i $$0.853255\pi$$
$$422$$ 38.5910 1.87858
$$423$$ 10.7800 0.524143
$$424$$ 1.72849 0.0839430
$$425$$ 0 0
$$426$$ −4.16073 −0.201588
$$427$$ −12.6707 −0.613177
$$428$$ 10.8090 0.522473
$$429$$ 0.238516 0.0115157
$$430$$ 0 0
$$431$$ −5.14958 −0.248046 −0.124023 0.992279i $$-0.539580\pi$$
−0.124023 + 0.992279i $$0.539580\pi$$
$$432$$ 6.31880 0.304014
$$433$$ −23.7703 −1.14233 −0.571164 0.820836i $$-0.693508\pi$$
−0.571164 + 0.820836i $$0.693508\pi$$
$$434$$ −0.793635 −0.0380957
$$435$$ 0 0
$$436$$ 4.52264 0.216595
$$437$$ 0.303981 0.0145414
$$438$$ −4.70477 −0.224803
$$439$$ 0.861025 0.0410945 0.0205472 0.999789i $$-0.493459\pi$$
0.0205472 + 0.999789i $$0.493459\pi$$
$$440$$ 0 0
$$441$$ −15.7041 −0.747812
$$442$$ 6.53001 0.310601
$$443$$ 25.1374 1.19432 0.597158 0.802124i $$-0.296297\pi$$
0.597158 + 0.802124i $$0.296297\pi$$
$$444$$ −1.26082 −0.0598358
$$445$$ 0 0
$$446$$ 49.5335 2.34548
$$447$$ −5.01930 −0.237405
$$448$$ −2.00623 −0.0947856
$$449$$ −7.27114 −0.343146 −0.171573 0.985171i $$-0.554885\pi$$
−0.171573 + 0.985171i $$0.554885\pi$$
$$450$$ 0 0
$$451$$ −0.293793 −0.0138342
$$452$$ 1.06616 0.0501477
$$453$$ −1.07866 −0.0506800
$$454$$ 13.5654 0.636658
$$455$$ 0 0
$$456$$ −0.314874 −0.0147453
$$457$$ 19.7002 0.921536 0.460768 0.887521i $$-0.347574\pi$$
0.460768 + 0.887521i $$0.347574\pi$$
$$458$$ −29.3742 −1.37257
$$459$$ 4.17230 0.194746
$$460$$ 0 0
$$461$$ −11.4754 −0.534461 −0.267231 0.963633i $$-0.586109\pi$$
−0.267231 + 0.963633i $$0.586109\pi$$
$$462$$ −1.33440 −0.0620819
$$463$$ 17.6025 0.818058 0.409029 0.912521i $$-0.365868\pi$$
0.409029 + 0.912521i $$0.365868\pi$$
$$464$$ 26.0372 1.20875
$$465$$ 0 0
$$466$$ −36.8635 −1.70767
$$467$$ 5.49601 0.254325 0.127163 0.991882i $$-0.459413\pi$$
0.127163 + 0.991882i $$0.459413\pi$$
$$468$$ 3.86798 0.178797
$$469$$ −20.0319 −0.924988
$$470$$ 0 0
$$471$$ −2.99526 −0.138014
$$472$$ −8.45602 −0.389220
$$473$$ −0.180719 −0.00830947
$$474$$ −0.884189 −0.0406121
$$475$$ 0 0
$$476$$ −13.4976 −0.618661
$$477$$ 3.46260 0.158542
$$478$$ 53.7992 2.46072
$$479$$ −11.8626 −0.542018 −0.271009 0.962577i $$-0.587357\pi$$
−0.271009 + 0.962577i $$0.587357\pi$$
$$480$$ 0 0
$$481$$ 5.62956 0.256686
$$482$$ −24.2186 −1.10313
$$483$$ 0.227757 0.0103633
$$484$$ 1.17191 0.0532687
$$485$$ 0 0
$$486$$ 10.0593 0.456301
$$487$$ 33.1690 1.50303 0.751516 0.659714i $$-0.229323\pi$$
0.751516 + 0.659714i $$0.229323\pi$$
$$488$$ 5.32490 0.241047
$$489$$ 1.75605 0.0794114
$$490$$ 0 0
$$491$$ 8.69377 0.392344 0.196172 0.980569i $$-0.437149\pi$$
0.196172 + 0.980569i $$0.437149\pi$$
$$492$$ 0.0735084 0.00331401
$$493$$ 17.1923 0.774304
$$494$$ −1.98966 −0.0895188
$$495$$ 0 0
$$496$$ 0.631147 0.0283393
$$497$$ −38.4002 −1.72249
$$498$$ −1.01164 −0.0453329
$$499$$ 2.21585 0.0991951 0.0495975 0.998769i $$-0.484206\pi$$
0.0495975 + 0.998769i $$0.484206\pi$$
$$500$$ 0 0
$$501$$ 0.861802 0.0385025
$$502$$ 16.8778 0.753294
$$503$$ 25.4694 1.13562 0.567812 0.823159i $$-0.307790\pi$$
0.567812 + 0.823159i $$0.307790\pi$$
$$504$$ 15.2909 0.681112
$$505$$ 0 0
$$506$$ −0.541385 −0.0240675
$$507$$ −2.50905 −0.111431
$$508$$ 1.17958 0.0523355
$$509$$ 31.0920 1.37813 0.689063 0.724701i $$-0.258022\pi$$
0.689063 + 0.724701i $$0.258022\pi$$
$$510$$ 0 0
$$511$$ −43.4213 −1.92085
$$512$$ 14.6780 0.648680
$$513$$ −1.27127 −0.0561281
$$514$$ 9.65910 0.426045
$$515$$ 0 0
$$516$$ 0.0452167 0.00199056
$$517$$ 3.64879 0.160473
$$518$$ −31.4951 −1.38381
$$519$$ −0.643943 −0.0282660
$$520$$ 0 0
$$521$$ −34.0870 −1.49338 −0.746688 0.665174i $$-0.768357\pi$$
−0.746688 + 0.665174i $$0.768357\pi$$
$$522$$ 27.5633 1.20641
$$523$$ 26.0002 1.13691 0.568456 0.822714i $$-0.307541\pi$$
0.568456 + 0.822714i $$0.307541\pi$$
$$524$$ 9.78315 0.427379
$$525$$ 0 0
$$526$$ −18.1562 −0.791649
$$527$$ 0.416746 0.0181537
$$528$$ 1.06120 0.0461826
$$529$$ −22.9076 −0.995982
$$530$$ 0 0
$$531$$ −16.9395 −0.735113
$$532$$ 4.11264 0.178305
$$533$$ −0.328215 −0.0142166
$$534$$ 0.0248203 0.00107408
$$535$$ 0 0
$$536$$ 8.41849 0.363623
$$537$$ 1.75992 0.0759460
$$538$$ −54.5064 −2.34994
$$539$$ −5.31545 −0.228953
$$540$$ 0 0
$$541$$ −5.06892 −0.217930 −0.108965 0.994046i $$-0.534754\pi$$
−0.108965 + 0.994046i $$0.534754\pi$$
$$542$$ −49.0298 −2.10601
$$543$$ −3.88534 −0.166736
$$544$$ 19.3724 0.830587
$$545$$ 0 0
$$546$$ −1.49074 −0.0637980
$$547$$ −10.9056 −0.466289 −0.233145 0.972442i $$-0.574902\pi$$
−0.233145 + 0.972442i $$0.574902\pi$$
$$548$$ 17.4612 0.745906
$$549$$ 10.6671 0.455261
$$550$$ 0 0
$$551$$ −5.23841 −0.223164
$$552$$ −0.0957157 −0.00407393
$$553$$ −8.16036 −0.347014
$$554$$ −24.6215 −1.04607
$$555$$ 0 0
$$556$$ −3.55493 −0.150762
$$557$$ −16.6435 −0.705209 −0.352605 0.935772i $$-0.614704\pi$$
−0.352605 + 0.935772i $$0.614704\pi$$
$$558$$ 0.668140 0.0282846
$$559$$ −0.201893 −0.00853916
$$560$$ 0 0
$$561$$ 0.700706 0.0295838
$$562$$ 20.9015 0.881678
$$563$$ −32.9233 −1.38755 −0.693776 0.720190i $$-0.744054\pi$$
−0.693776 + 0.720190i $$0.744054\pi$$
$$564$$ −0.912943 −0.0384418
$$565$$ 0 0
$$566$$ 3.74550 0.157435
$$567$$ 30.1516 1.26625
$$568$$ 16.1378 0.677129
$$569$$ 5.35168 0.224354 0.112177 0.993688i $$-0.464218\pi$$
0.112177 + 0.993688i $$0.464218\pi$$
$$570$$ 0 0
$$571$$ −32.1025 −1.34345 −0.671724 0.740802i $$-0.734446\pi$$
−0.671724 + 0.740802i $$0.734446\pi$$
$$572$$ 1.30922 0.0547412
$$573$$ 4.89766 0.204603
$$574$$ 1.83623 0.0766428
$$575$$ 0 0
$$576$$ 1.68899 0.0703748
$$577$$ 26.5229 1.10416 0.552082 0.833790i $$-0.313834\pi$$
0.552082 + 0.833790i $$0.313834\pi$$
$$578$$ −11.0931 −0.461411
$$579$$ 0.967224 0.0401965
$$580$$ 0 0
$$581$$ −9.33668 −0.387351
$$582$$ −6.65470 −0.275846
$$583$$ 1.17201 0.0485396
$$584$$ 18.2480 0.755107
$$585$$ 0 0
$$586$$ 11.2022 0.462758
$$587$$ 38.3527 1.58298 0.791492 0.611180i $$-0.209305\pi$$
0.791492 + 0.611180i $$0.209305\pi$$
$$588$$ 1.32995 0.0548462
$$589$$ −0.126980 −0.00523212
$$590$$ 0 0
$$591$$ −2.87780 −0.118377
$$592$$ 25.0468 1.02942
$$593$$ −10.8794 −0.446763 −0.223381 0.974731i $$-0.571709\pi$$
−0.223381 + 0.974731i $$0.571709\pi$$
$$594$$ 2.26412 0.0928981
$$595$$ 0 0
$$596$$ −27.5510 −1.12853
$$597$$ −1.14653 −0.0469244
$$598$$ −0.604817 −0.0247328
$$599$$ 25.8612 1.05666 0.528329 0.849040i $$-0.322819\pi$$
0.528329 + 0.849040i $$0.322819\pi$$
$$600$$ 0 0
$$601$$ −41.9127 −1.70965 −0.854827 0.518914i $$-0.826337\pi$$
−0.854827 + 0.518914i $$0.826337\pi$$
$$602$$ 1.12951 0.0460353
$$603$$ 16.8643 0.686769
$$604$$ −5.92081 −0.240914
$$605$$ 0 0
$$606$$ 2.97228 0.120741
$$607$$ −41.4401 −1.68200 −0.841002 0.541032i $$-0.818034\pi$$
−0.841002 + 0.541032i $$0.818034\pi$$
$$608$$ −5.90267 −0.239385
$$609$$ −3.92486 −0.159043
$$610$$ 0 0
$$611$$ 4.07629 0.164909
$$612$$ 11.3633 0.459333
$$613$$ −5.66031 −0.228618 −0.114309 0.993445i $$-0.536465\pi$$
−0.114309 + 0.993445i $$0.536465\pi$$
$$614$$ 28.1391 1.13560
$$615$$ 0 0
$$616$$ 5.17562 0.208532
$$617$$ 21.2337 0.854837 0.427419 0.904054i $$-0.359423\pi$$
0.427419 + 0.904054i $$0.359423\pi$$
$$618$$ 2.06113 0.0829108
$$619$$ 15.5489 0.624962 0.312481 0.949924i $$-0.398840\pi$$
0.312481 + 0.949924i $$0.398840\pi$$
$$620$$ 0 0
$$621$$ −0.386443 −0.0155074
$$622$$ −13.9090 −0.557701
$$623$$ 0.229071 0.00917755
$$624$$ 1.18553 0.0474592
$$625$$ 0 0
$$626$$ −21.5404 −0.860926
$$627$$ −0.213501 −0.00852641
$$628$$ −16.4410 −0.656069
$$629$$ 16.5384 0.659428
$$630$$ 0 0
$$631$$ −36.0897 −1.43671 −0.718354 0.695678i $$-0.755104\pi$$
−0.718354 + 0.695678i $$0.755104\pi$$
$$632$$ 3.42942 0.136415
$$633$$ 4.62622 0.183876
$$634$$ 2.34370 0.0930804
$$635$$ 0 0
$$636$$ −0.293242 −0.0116278
$$637$$ −5.93823 −0.235281
$$638$$ 9.32953 0.369360
$$639$$ 32.3281 1.27888
$$640$$ 0 0
$$641$$ −3.17128 −0.125258 −0.0626290 0.998037i $$-0.519948\pi$$
−0.0626290 + 0.998037i $$0.519948\pi$$
$$642$$ 3.50712 0.138415
$$643$$ −17.2355 −0.679702 −0.339851 0.940479i $$-0.610377\pi$$
−0.339851 + 0.940479i $$0.610377\pi$$
$$644$$ 1.25016 0.0492633
$$645$$ 0 0
$$646$$ −5.84516 −0.229975
$$647$$ 21.6081 0.849504 0.424752 0.905310i $$-0.360361\pi$$
0.424752 + 0.905310i $$0.360361\pi$$
$$648$$ −12.6713 −0.497777
$$649$$ −5.73363 −0.225064
$$650$$ 0 0
$$651$$ −0.0951394 −0.00372881
$$652$$ 9.63900 0.377492
$$653$$ −1.82065 −0.0712477 −0.0356239 0.999365i $$-0.511342\pi$$
−0.0356239 + 0.999365i $$0.511342\pi$$
$$654$$ 1.46743 0.0573811
$$655$$ 0 0
$$656$$ −1.46028 −0.0570145
$$657$$ 36.5553 1.42616
$$658$$ −22.8052 −0.889040
$$659$$ −7.52543 −0.293149 −0.146575 0.989200i $$-0.546825\pi$$
−0.146575 + 0.989200i $$0.546825\pi$$
$$660$$ 0 0
$$661$$ −4.05494 −0.157719 −0.0788595 0.996886i $$-0.525128\pi$$
−0.0788595 + 0.996886i $$0.525128\pi$$
$$662$$ −7.41058 −0.288020
$$663$$ 0.782804 0.0304016
$$664$$ 3.92378 0.152272
$$665$$ 0 0
$$666$$ 26.5149 1.02743
$$667$$ −1.59237 −0.0616570
$$668$$ 4.73045 0.183027
$$669$$ 5.93797 0.229575
$$670$$ 0 0
$$671$$ 3.61056 0.139384
$$672$$ −4.42256 −0.170604
$$673$$ 5.99814 0.231211 0.115606 0.993295i $$-0.463119\pi$$
0.115606 + 0.993295i $$0.463119\pi$$
$$674$$ −50.9232 −1.96149
$$675$$ 0 0
$$676$$ −13.7722 −0.529702
$$677$$ −4.64183 −0.178400 −0.0891999 0.996014i $$-0.528431\pi$$
−0.0891999 + 0.996014i $$0.528431\pi$$
$$678$$ 0.345928 0.0132853
$$679$$ −61.4176 −2.35699
$$680$$ 0 0
$$681$$ 1.62620 0.0623161
$$682$$ 0.226150 0.00865972
$$683$$ 9.58079 0.366599 0.183299 0.983057i $$-0.441322\pi$$
0.183299 + 0.983057i $$0.441322\pi$$
$$684$$ −3.46232 −0.132385
$$685$$ 0 0
$$686$$ −10.5286 −0.401983
$$687$$ −3.52132 −0.134347
$$688$$ −0.898254 −0.0342456
$$689$$ 1.30933 0.0498814
$$690$$ 0 0
$$691$$ −22.4231 −0.853014 −0.426507 0.904484i $$-0.640256\pi$$
−0.426507 + 0.904484i $$0.640256\pi$$
$$692$$ −3.53462 −0.134366
$$693$$ 10.3680 0.393850
$$694$$ 16.1706 0.613828
$$695$$ 0 0
$$696$$ 1.64944 0.0625218
$$697$$ −0.964223 −0.0365226
$$698$$ 33.2141 1.25717
$$699$$ −4.41912 −0.167147
$$700$$ 0 0
$$701$$ 5.29648 0.200045 0.100023 0.994985i $$-0.468108\pi$$
0.100023 + 0.994985i $$0.468108\pi$$
$$702$$ 2.52940 0.0954660
$$703$$ −5.03915 −0.190055
$$704$$ 0.571685 0.0215462
$$705$$ 0 0
$$706$$ 52.1992 1.96454
$$707$$ 27.4318 1.03168
$$708$$ 1.43458 0.0539148
$$709$$ −32.7803 −1.23109 −0.615545 0.788102i $$-0.711064\pi$$
−0.615545 + 0.788102i $$0.711064\pi$$
$$710$$ 0 0
$$711$$ 6.86999 0.257645
$$712$$ −0.0962681 −0.00360780
$$713$$ −0.0385995 −0.00144556
$$714$$ −4.37947 −0.163898
$$715$$ 0 0
$$716$$ 9.66022 0.361019
$$717$$ 6.44934 0.240855
$$718$$ −47.1743 −1.76053
$$719$$ −3.60564 −0.134468 −0.0672338 0.997737i $$-0.521417\pi$$
−0.0672338 + 0.997737i $$0.521417\pi$$
$$720$$ 0 0
$$721$$ 19.0226 0.708438
$$722$$ 1.78099 0.0662814
$$723$$ −2.90328 −0.107974
$$724$$ −21.3267 −0.792600
$$725$$ 0 0
$$726$$ 0.380243 0.0141121
$$727$$ 18.4897 0.685744 0.342872 0.939382i $$-0.388600\pi$$
0.342872 + 0.939382i $$0.388600\pi$$
$$728$$ 5.78202 0.214296
$$729$$ −24.5696 −0.909985
$$730$$ 0 0
$$731$$ −0.593116 −0.0219372
$$732$$ −0.903378 −0.0333898
$$733$$ 29.3667 1.08468 0.542341 0.840158i $$-0.317538\pi$$
0.542341 + 0.840158i $$0.317538\pi$$
$$734$$ 48.7199 1.79829
$$735$$ 0 0
$$736$$ −1.79430 −0.0661387
$$737$$ 5.70818 0.210263
$$738$$ −1.54587 −0.0569044
$$739$$ 22.8091 0.839048 0.419524 0.907744i $$-0.362197\pi$$
0.419524 + 0.907744i $$0.362197\pi$$
$$740$$ 0 0
$$741$$ −0.238516 −0.00876210
$$742$$ −7.32515 −0.268915
$$743$$ −40.2104 −1.47518 −0.737589 0.675250i $$-0.764036\pi$$
−0.737589 + 0.675250i $$0.764036\pi$$
$$744$$ 0.0399827 0.00146584
$$745$$ 0 0
$$746$$ 66.8508 2.44758
$$747$$ 7.86030 0.287593
$$748$$ 3.84619 0.140631
$$749$$ 32.3680 1.18270
$$750$$ 0 0
$$751$$ 45.2958 1.65287 0.826434 0.563034i $$-0.190366\pi$$
0.826434 + 0.563034i $$0.190366\pi$$
$$752$$ 18.1361 0.661355
$$753$$ 2.02328 0.0737324
$$754$$ 10.4226 0.379570
$$755$$ 0 0
$$756$$ −5.22829 −0.190151
$$757$$ 38.7580 1.40868 0.704341 0.709862i $$-0.251243\pi$$
0.704341 + 0.709862i $$0.251243\pi$$
$$758$$ −55.4967 −2.01573
$$759$$ −0.0649002 −0.00235573
$$760$$ 0 0
$$761$$ 29.0483 1.05300 0.526500 0.850175i $$-0.323504\pi$$
0.526500 + 0.850175i $$0.323504\pi$$
$$762$$ 0.382731 0.0138649
$$763$$ 13.5432 0.490298
$$764$$ 26.8834 0.972607
$$765$$ 0 0
$$766$$ 39.5408 1.42867
$$767$$ −6.40540 −0.231286
$$768$$ 4.34586 0.156818
$$769$$ −31.9246 −1.15123 −0.575616 0.817720i $$-0.695238\pi$$
−0.575616 + 0.817720i $$0.695238\pi$$
$$770$$ 0 0
$$771$$ 1.15791 0.0417013
$$772$$ 5.30912 0.191079
$$773$$ −4.83464 −0.173890 −0.0869449 0.996213i $$-0.527710\pi$$
−0.0869449 + 0.996213i $$0.527710\pi$$
$$774$$ −0.950903 −0.0341795
$$775$$ 0 0
$$776$$ 25.8110 0.926560
$$777$$ −3.77557 −0.135448
$$778$$ 53.0526 1.90203
$$779$$ 0.293793 0.0105262
$$780$$ 0 0
$$781$$ 10.9423 0.391546
$$782$$ −1.77682 −0.0635388
$$783$$ 6.65945 0.237989
$$784$$ −26.4202 −0.943577
$$785$$ 0 0
$$786$$ 3.17427 0.113223
$$787$$ 34.5482 1.23151 0.615755 0.787938i $$-0.288851\pi$$
0.615755 + 0.787938i $$0.288851\pi$$
$$788$$ −15.7963 −0.562720
$$789$$ −2.17653 −0.0774866
$$790$$ 0 0
$$791$$ 3.19264 0.113517
$$792$$ −4.35721 −0.154827
$$793$$ 4.03359 0.143237
$$794$$ −56.7271 −2.01317
$$795$$ 0 0
$$796$$ −6.29334 −0.223061
$$797$$ 49.3536 1.74819 0.874097 0.485751i $$-0.161454\pi$$
0.874097 + 0.485751i $$0.161454\pi$$
$$798$$ 1.33440 0.0472372
$$799$$ 11.9752 0.423654
$$800$$ 0 0
$$801$$ −0.192849 −0.00681399
$$802$$ 47.5942 1.68061
$$803$$ 12.3731 0.436637
$$804$$ −1.42821 −0.0503691
$$805$$ 0 0
$$806$$ 0.252646 0.00889909
$$807$$ −6.53412 −0.230012
$$808$$ −11.5283 −0.405565
$$809$$ 22.5147 0.791575 0.395787 0.918342i $$-0.370472\pi$$
0.395787 + 0.918342i $$0.370472\pi$$
$$810$$ 0 0
$$811$$ −29.8110 −1.04680 −0.523402 0.852086i $$-0.675337\pi$$
−0.523402 + 0.852086i $$0.675337\pi$$
$$812$$ −21.5437 −0.756034
$$813$$ −5.87760 −0.206136
$$814$$ 8.97465 0.314561
$$815$$ 0 0
$$816$$ 3.48282 0.121923
$$817$$ 0.180719 0.00632256
$$818$$ 48.9486 1.71145
$$819$$ 11.5828 0.404736
$$820$$ 0 0
$$821$$ 23.5479 0.821828 0.410914 0.911674i $$-0.365210\pi$$
0.410914 + 0.911674i $$0.365210\pi$$
$$822$$ 5.66552 0.197608
$$823$$ 5.77193 0.201197 0.100598 0.994927i $$-0.467924\pi$$
0.100598 + 0.994927i $$0.467924\pi$$
$$824$$ −7.99432 −0.278495
$$825$$ 0 0
$$826$$ 35.8356 1.24688
$$827$$ 34.9143 1.21409 0.607045 0.794668i $$-0.292355\pi$$
0.607045 + 0.794668i $$0.292355\pi$$
$$828$$ −1.05248 −0.0365761
$$829$$ 12.1015 0.420303 0.210152 0.977669i $$-0.432604\pi$$
0.210152 + 0.977669i $$0.432604\pi$$
$$830$$ 0 0
$$831$$ −2.95158 −0.102389
$$832$$ 0.638666 0.0221418
$$833$$ −17.4452 −0.604440
$$834$$ −1.15344 −0.0399405
$$835$$ 0 0
$$836$$ −1.17191 −0.0405314
$$837$$ 0.161426 0.00557971
$$838$$ −32.7705 −1.13204
$$839$$ 18.3333 0.632937 0.316469 0.948603i $$-0.397503\pi$$
0.316469 + 0.948603i $$0.397503\pi$$
$$840$$ 0 0
$$841$$ −1.55909 −0.0537616
$$842$$ −65.4556 −2.25575
$$843$$ 2.50564 0.0862987
$$844$$ 25.3934 0.874077
$$845$$ 0 0
$$846$$ 19.1991 0.660079
$$847$$ 3.50934 0.120582
$$848$$ 5.82540 0.200045
$$849$$ 0.449003 0.0154098
$$850$$ 0 0
$$851$$ −1.53180 −0.0525095
$$852$$ −2.73781 −0.0937959
$$853$$ 26.7354 0.915402 0.457701 0.889106i $$-0.348673\pi$$
0.457701 + 0.889106i $$0.348673\pi$$
$$854$$ −22.5663 −0.772203
$$855$$ 0 0
$$856$$ −13.6028 −0.464933
$$857$$ −16.8931 −0.577057 −0.288529 0.957471i $$-0.593166\pi$$
−0.288529 + 0.957471i $$0.593166\pi$$
$$858$$ 0.424794 0.0145022
$$859$$ 5.29172 0.180551 0.0902755 0.995917i $$-0.471225\pi$$
0.0902755 + 0.995917i $$0.471225\pi$$
$$860$$ 0 0
$$861$$ 0.220124 0.00750180
$$862$$ −9.17133 −0.312377
$$863$$ −26.4397 −0.900018 −0.450009 0.893024i $$-0.648579\pi$$
−0.450009 + 0.893024i $$0.648579\pi$$
$$864$$ 7.50392 0.255288
$$865$$ 0 0
$$866$$ −42.3346 −1.43859
$$867$$ −1.32981 −0.0451629
$$868$$ −0.522222 −0.0177254
$$869$$ 2.32533 0.0788814
$$870$$ 0 0
$$871$$ 6.37697 0.216075
$$872$$ −5.69159 −0.192742
$$873$$ 51.7058 1.74998
$$874$$ 0.541385 0.0183126
$$875$$ 0 0
$$876$$ −3.09580 −0.104597
$$877$$ 5.61130 0.189480 0.0947401 0.995502i $$-0.469798\pi$$
0.0947401 + 0.995502i $$0.469798\pi$$
$$878$$ 1.53347 0.0517522
$$879$$ 1.34290 0.0452947
$$880$$ 0 0
$$881$$ −7.21640 −0.243127 −0.121563 0.992584i $$-0.538791\pi$$
−0.121563 + 0.992584i $$0.538791\pi$$
$$882$$ −27.9687 −0.941756
$$883$$ 36.5678 1.23061 0.615303 0.788291i $$-0.289034\pi$$
0.615303 + 0.788291i $$0.289034\pi$$
$$884$$ 4.29683 0.144518
$$885$$ 0 0
$$886$$ 44.7694 1.50406
$$887$$ −47.3649 −1.59036 −0.795179 0.606374i $$-0.792623\pi$$
−0.795179 + 0.606374i $$0.792623\pi$$
$$888$$ 1.58670 0.0532461
$$889$$ 3.53231 0.118470
$$890$$ 0 0
$$891$$ −8.59183 −0.287837
$$892$$ 32.5937 1.09132
$$893$$ −3.64879 −0.122102
$$894$$ −8.93930 −0.298975
$$895$$ 0 0
$$896$$ 37.8559 1.26468
$$897$$ −0.0725042 −0.00242085
$$898$$ −12.9498 −0.432140
$$899$$ 0.665173 0.0221848
$$900$$ 0 0
$$901$$ 3.84651 0.128146
$$902$$ −0.523242 −0.0174220
$$903$$ 0.135403 0.00450594
$$904$$ −1.34172 −0.0446250
$$905$$ 0 0
$$906$$ −1.92109 −0.0638238
$$907$$ −48.1508 −1.59882 −0.799411 0.600784i $$-0.794855\pi$$
−0.799411 + 0.600784i $$0.794855\pi$$
$$908$$ 8.92624 0.296228
$$909$$ −23.0941 −0.765984
$$910$$ 0 0
$$911$$ −20.7792 −0.688446 −0.344223 0.938888i $$-0.611858\pi$$
−0.344223 + 0.938888i $$0.611858\pi$$
$$912$$ −1.06120 −0.0351397
$$913$$ 2.66052 0.0880505
$$914$$ 35.0858 1.16053
$$915$$ 0 0
$$916$$ −19.3286 −0.638636
$$917$$ 29.2960 0.967440
$$918$$ 7.43081 0.245253
$$919$$ 9.76309 0.322055 0.161027 0.986950i $$-0.448519\pi$$
0.161027 + 0.986950i $$0.448519\pi$$
$$920$$ 0 0
$$921$$ 3.37326 0.111153
$$922$$ −20.4375 −0.673073
$$923$$ 12.2244 0.402369
$$924$$ −0.878052 −0.0288858
$$925$$ 0 0
$$926$$ 31.3498 1.03022
$$927$$ −16.0146 −0.525989
$$928$$ 30.9206 1.01502
$$929$$ 47.0624 1.54407 0.772034 0.635582i $$-0.219240\pi$$
0.772034 + 0.635582i $$0.219240\pi$$
$$930$$ 0 0
$$931$$ 5.31545 0.174207
$$932$$ −24.2567 −0.794553
$$933$$ −1.66739 −0.0545878
$$934$$ 9.78832 0.320284
$$935$$ 0 0
$$936$$ −4.86772 −0.159107
$$937$$ −37.3958 −1.22167 −0.610833 0.791759i $$-0.709165\pi$$
−0.610833 + 0.791759i $$0.709165\pi$$
$$938$$ −35.6766 −1.16488
$$939$$ −2.58222 −0.0842674
$$940$$ 0 0
$$941$$ 16.4787 0.537191 0.268595 0.963253i $$-0.413441\pi$$
0.268595 + 0.963253i $$0.413441\pi$$
$$942$$ −5.33451 −0.173808
$$943$$ 0.0893075 0.00290825
$$944$$ −28.4987 −0.927553
$$945$$ 0 0
$$946$$ −0.321858 −0.0104645
$$947$$ 15.8450 0.514893 0.257447 0.966292i $$-0.417119\pi$$
0.257447 + 0.966292i $$0.417119\pi$$
$$948$$ −0.581808 −0.0188962
$$949$$ 13.8228 0.448706
$$950$$ 0 0
$$951$$ 0.280959 0.00911071
$$952$$ 16.9863 0.550528
$$953$$ −29.0991 −0.942612 −0.471306 0.881970i $$-0.656217\pi$$
−0.471306 + 0.881970i $$0.656217\pi$$
$$954$$ 6.16685 0.199659
$$955$$ 0 0
$$956$$ 35.4006 1.14494
$$957$$ 1.11841 0.0361529
$$958$$ −21.1272 −0.682589
$$959$$ 52.2883 1.68848
$$960$$ 0 0
$$961$$ −30.9839 −0.999480
$$962$$ 10.0262 0.323256
$$963$$ −27.2497 −0.878110
$$964$$ −15.9362 −0.513270
$$965$$ 0 0
$$966$$ 0.405632 0.0130510
$$967$$ 27.6570 0.889390 0.444695 0.895682i $$-0.353312\pi$$
0.444695 + 0.895682i $$0.353312\pi$$
$$968$$ −1.47481 −0.0474023
$$969$$ −0.700706 −0.0225099
$$970$$ 0 0
$$971$$ 29.9538 0.961264 0.480632 0.876922i $$-0.340407\pi$$
0.480632 + 0.876922i $$0.340407\pi$$
$$972$$ 6.61918 0.212310
$$973$$ −10.6454 −0.341275
$$974$$ 59.0736 1.89284
$$975$$ 0 0
$$976$$ 17.9461 0.574440
$$977$$ −11.1701 −0.357364 −0.178682 0.983907i $$-0.557183\pi$$
−0.178682 + 0.983907i $$0.557183\pi$$
$$978$$ 3.12750 0.100006
$$979$$ −0.0652748 −0.00208619
$$980$$ 0 0
$$981$$ −11.4017 −0.364028
$$982$$ 15.4835 0.494098
$$983$$ 22.5894 0.720491 0.360245 0.932858i $$-0.382693\pi$$
0.360245 + 0.932858i $$0.382693\pi$$
$$984$$ −0.0925079 −0.00294904
$$985$$ 0 0
$$986$$ 30.6193 0.975118
$$987$$ −2.73384 −0.0870192
$$988$$ −1.30922 −0.0416518
$$989$$ 0.0549351 0.00174683
$$990$$ 0 0
$$991$$ −26.8461 −0.852796 −0.426398 0.904536i $$-0.640218\pi$$
−0.426398 + 0.904536i $$0.640218\pi$$
$$992$$ 0.749521 0.0237973
$$993$$ −0.888366 −0.0281914
$$994$$ −68.3903 −2.16921
$$995$$ 0 0
$$996$$ −0.665675 −0.0210927
$$997$$ −31.2387 −0.989339 −0.494669 0.869081i $$-0.664711\pi$$
−0.494669 + 0.869081i $$0.664711\pi$$
$$998$$ 3.94640 0.124921
$$999$$ 6.40614 0.202681
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.1.13 16
5.2 odd 4 1045.2.b.b.419.13 yes 16
5.3 odd 4 1045.2.b.b.419.4 16
5.4 even 2 inner 5225.2.a.z.1.4 16

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.b.419.4 16 5.3 odd 4
1045.2.b.b.419.13 yes 16 5.2 odd 4
5225.2.a.z.1.4 16 5.4 even 2 inner
5225.2.a.z.1.13 16 1.1 even 1 trivial