# Properties

 Label 2175.2.a.y.1.1 Level $2175$ Weight $2$ Character 2175.1 Self dual yes Analytic conductor $17.367$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.331312.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 7x^{3} - 4x^{2} + 7x + 4$$ x^5 - 7*x^3 - 4*x^2 + 7*x + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.70559$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.88448 q^{2} -1.00000 q^{3} +1.55125 q^{4} +1.88448 q^{6} +3.97545 q^{7} +0.845662 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.88448 q^{2} -1.00000 q^{3} +1.55125 q^{4} +1.88448 q^{6} +3.97545 q^{7} +0.845662 q^{8} +1.00000 q^{9} -5.05034 q^{11} -1.55125 q^{12} -1.97545 q^{13} -7.49164 q^{14} -4.69613 q^{16} +4.61461 q^{17} -1.88448 q^{18} +4.64986 q^{19} -3.97545 q^{21} +9.51724 q^{22} +1.20650 q^{23} -0.845662 q^{24} +3.72269 q^{26} -1.00000 q^{27} +6.16691 q^{28} -1.00000 q^{29} +6.45387 q^{31} +7.15841 q^{32} +5.05034 q^{33} -8.69613 q^{34} +1.55125 q^{36} +4.35033 q^{37} -8.76255 q^{38} +1.97545 q^{39} -11.2474 q^{41} +7.49164 q^{42} +7.53991 q^{43} -7.83433 q^{44} -2.27362 q^{46} +6.70001 q^{47} +4.69613 q^{48} +8.80420 q^{49} -4.61461 q^{51} -3.06441 q^{52} +3.70001 q^{53} +1.88448 q^{54} +3.36188 q^{56} -4.64986 q^{57} +1.88448 q^{58} -12.9218 q^{59} -7.28614 q^{61} -12.1622 q^{62} +3.97545 q^{63} -4.09760 q^{64} -9.51724 q^{66} -13.4019 q^{67} +7.15841 q^{68} -1.20650 q^{69} +6.06103 q^{71} +0.845662 q^{72} -6.53640 q^{73} -8.19809 q^{74} +7.21309 q^{76} -20.0774 q^{77} -3.72269 q^{78} +5.30104 q^{79} +1.00000 q^{81} +21.1954 q^{82} -7.97193 q^{83} -6.16691 q^{84} -14.2088 q^{86} +1.00000 q^{87} -4.27088 q^{88} +12.0927 q^{89} -7.85330 q^{91} +1.87158 q^{92} -6.45387 q^{93} -12.6260 q^{94} -7.15841 q^{96} +10.1052 q^{97} -16.5913 q^{98} -5.05034 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 2 q^{2} - 5 q^{3} + 8 q^{4} - 2 q^{6} + 4 q^{7} + 18 q^{8} + 5 q^{9}+O(q^{10})$$ 5 * q + 2 * q^2 - 5 * q^3 + 8 * q^4 - 2 * q^6 + 4 * q^7 + 18 * q^8 + 5 * q^9 $$5 q + 2 q^{2} - 5 q^{3} + 8 q^{4} - 2 q^{6} + 4 q^{7} + 18 q^{8} + 5 q^{9} - 5 q^{11} - 8 q^{12} + 6 q^{13} - 8 q^{14} + 14 q^{16} + 14 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{21} + 8 q^{22} + 13 q^{23} - 18 q^{24} + 12 q^{26} - 5 q^{27} - 4 q^{28} - 5 q^{29} + 2 q^{31} + 18 q^{32} + 5 q^{33} - 6 q^{34} + 8 q^{36} + 17 q^{37} + 14 q^{38} - 6 q^{39} - 19 q^{41} + 8 q^{42} + 7 q^{43} - 22 q^{44} + 30 q^{46} + 18 q^{47} - 14 q^{48} + 9 q^{49} - 14 q^{51} + 20 q^{52} + 3 q^{53} - 2 q^{54} - 16 q^{56} - 2 q^{57} - 2 q^{58} - 22 q^{59} - 8 q^{61} - 4 q^{62} + 4 q^{63} + 8 q^{64} - 8 q^{66} + 10 q^{67} + 18 q^{68} - 13 q^{69} - 18 q^{71} + 18 q^{72} + 19 q^{73} + 2 q^{74} + 52 q^{76} + 8 q^{77} - 12 q^{78} + 16 q^{79} + 5 q^{81} + 22 q^{82} - 3 q^{83} + 4 q^{84} + 5 q^{87} - 26 q^{88} + 2 q^{89} - 36 q^{91} + 48 q^{92} - 2 q^{93} + 2 q^{94} - 18 q^{96} + 5 q^{97} - 6 q^{98} - 5 q^{99}+O(q^{100})$$ 5 * q + 2 * q^2 - 5 * q^3 + 8 * q^4 - 2 * q^6 + 4 * q^7 + 18 * q^8 + 5 * q^9 - 5 * q^11 - 8 * q^12 + 6 * q^13 - 8 * q^14 + 14 * q^16 + 14 * q^17 + 2 * q^18 + 2 * q^19 - 4 * q^21 + 8 * q^22 + 13 * q^23 - 18 * q^24 + 12 * q^26 - 5 * q^27 - 4 * q^28 - 5 * q^29 + 2 * q^31 + 18 * q^32 + 5 * q^33 - 6 * q^34 + 8 * q^36 + 17 * q^37 + 14 * q^38 - 6 * q^39 - 19 * q^41 + 8 * q^42 + 7 * q^43 - 22 * q^44 + 30 * q^46 + 18 * q^47 - 14 * q^48 + 9 * q^49 - 14 * q^51 + 20 * q^52 + 3 * q^53 - 2 * q^54 - 16 * q^56 - 2 * q^57 - 2 * q^58 - 22 * q^59 - 8 * q^61 - 4 * q^62 + 4 * q^63 + 8 * q^64 - 8 * q^66 + 10 * q^67 + 18 * q^68 - 13 * q^69 - 18 * q^71 + 18 * q^72 + 19 * q^73 + 2 * q^74 + 52 * q^76 + 8 * q^77 - 12 * q^78 + 16 * q^79 + 5 * q^81 + 22 * q^82 - 3 * q^83 + 4 * q^84 + 5 * q^87 - 26 * q^88 + 2 * q^89 - 36 * q^91 + 48 * q^92 - 2 * q^93 + 2 * q^94 - 18 * q^96 + 5 * q^97 - 6 * q^98 - 5 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.88448 −1.33253 −0.666263 0.745717i $$-0.732107\pi$$
−0.666263 + 0.745717i $$0.732107\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 1.55125 0.775624
$$5$$ 0 0
$$6$$ 1.88448 0.769334
$$7$$ 3.97545 1.50258 0.751289 0.659973i $$-0.229432\pi$$
0.751289 + 0.659973i $$0.229432\pi$$
$$8$$ 0.845662 0.298987
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.05034 −1.52273 −0.761367 0.648321i $$-0.775471\pi$$
−0.761367 + 0.648321i $$0.775471\pi$$
$$12$$ −1.55125 −0.447807
$$13$$ −1.97545 −0.547891 −0.273946 0.961745i $$-0.588329\pi$$
−0.273946 + 0.961745i $$0.588329\pi$$
$$14$$ −7.49164 −2.00222
$$15$$ 0 0
$$16$$ −4.69613 −1.17403
$$17$$ 4.61461 1.11921 0.559604 0.828760i $$-0.310953\pi$$
0.559604 + 0.828760i $$0.310953\pi$$
$$18$$ −1.88448 −0.444175
$$19$$ 4.64986 1.06675 0.533376 0.845879i $$-0.320923\pi$$
0.533376 + 0.845879i $$0.320923\pi$$
$$20$$ 0 0
$$21$$ −3.97545 −0.867514
$$22$$ 9.51724 2.02908
$$23$$ 1.20650 0.251572 0.125786 0.992057i $$-0.459855\pi$$
0.125786 + 0.992057i $$0.459855\pi$$
$$24$$ −0.845662 −0.172620
$$25$$ 0 0
$$26$$ 3.72269 0.730079
$$27$$ −1.00000 −0.192450
$$28$$ 6.16691 1.16544
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 6.45387 1.15915 0.579575 0.814919i $$-0.303219\pi$$
0.579575 + 0.814919i $$0.303219\pi$$
$$32$$ 7.15841 1.26544
$$33$$ 5.05034 0.879151
$$34$$ −8.69613 −1.49137
$$35$$ 0 0
$$36$$ 1.55125 0.258541
$$37$$ 4.35033 0.715189 0.357595 0.933877i $$-0.383597\pi$$
0.357595 + 0.933877i $$0.383597\pi$$
$$38$$ −8.76255 −1.42147
$$39$$ 1.97545 0.316325
$$40$$ 0 0
$$41$$ −11.2474 −1.75654 −0.878272 0.478161i $$-0.841304\pi$$
−0.878272 + 0.478161i $$0.841304\pi$$
$$42$$ 7.49164 1.15598
$$43$$ 7.53991 1.14983 0.574913 0.818215i $$-0.305036\pi$$
0.574913 + 0.818215i $$0.305036\pi$$
$$44$$ −7.83433 −1.18107
$$45$$ 0 0
$$46$$ −2.27362 −0.335226
$$47$$ 6.70001 0.977297 0.488648 0.872481i $$-0.337490\pi$$
0.488648 + 0.872481i $$0.337490\pi$$
$$48$$ 4.69613 0.677827
$$49$$ 8.80420 1.25774
$$50$$ 0 0
$$51$$ −4.61461 −0.646175
$$52$$ −3.06441 −0.424958
$$53$$ 3.70001 0.508235 0.254118 0.967173i $$-0.418215\pi$$
0.254118 + 0.967173i $$0.418215\pi$$
$$54$$ 1.88448 0.256445
$$55$$ 0 0
$$56$$ 3.36188 0.449251
$$57$$ −4.64986 −0.615889
$$58$$ 1.88448 0.247444
$$59$$ −12.9218 −1.68227 −0.841137 0.540823i $$-0.818113\pi$$
−0.841137 + 0.540823i $$0.818113\pi$$
$$60$$ 0 0
$$61$$ −7.28614 −0.932894 −0.466447 0.884549i $$-0.654466\pi$$
−0.466447 + 0.884549i $$0.654466\pi$$
$$62$$ −12.1622 −1.54460
$$63$$ 3.97545 0.500860
$$64$$ −4.09760 −0.512200
$$65$$ 0 0
$$66$$ −9.51724 −1.17149
$$67$$ −13.4019 −1.63730 −0.818651 0.574291i $$-0.805278\pi$$
−0.818651 + 0.574291i $$0.805278\pi$$
$$68$$ 7.15841 0.868085
$$69$$ −1.20650 −0.145245
$$70$$ 0 0
$$71$$ 6.06103 0.719312 0.359656 0.933085i $$-0.382894\pi$$
0.359656 + 0.933085i $$0.382894\pi$$
$$72$$ 0.845662 0.0996622
$$73$$ −6.53640 −0.765027 −0.382514 0.923950i $$-0.624942\pi$$
−0.382514 + 0.923950i $$0.624942\pi$$
$$74$$ −8.19809 −0.953008
$$75$$ 0 0
$$76$$ 7.21309 0.827398
$$77$$ −20.0774 −2.28803
$$78$$ −3.72269 −0.421511
$$79$$ 5.30104 0.596413 0.298207 0.954501i $$-0.403612\pi$$
0.298207 + 0.954501i $$0.403612\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 21.1954 2.34064
$$83$$ −7.97193 −0.875033 −0.437517 0.899210i $$-0.644142\pi$$
−0.437517 + 0.899210i $$0.644142\pi$$
$$84$$ −6.16691 −0.672865
$$85$$ 0 0
$$86$$ −14.2088 −1.53217
$$87$$ 1.00000 0.107211
$$88$$ −4.27088 −0.455277
$$89$$ 12.0927 1.28183 0.640913 0.767614i $$-0.278556\pi$$
0.640913 + 0.767614i $$0.278556\pi$$
$$90$$ 0 0
$$91$$ −7.85330 −0.823250
$$92$$ 1.87158 0.195126
$$93$$ −6.45387 −0.669235
$$94$$ −12.6260 −1.30227
$$95$$ 0 0
$$96$$ −7.15841 −0.730602
$$97$$ 10.1052 1.02603 0.513016 0.858379i $$-0.328528\pi$$
0.513016 + 0.858379i $$0.328528\pi$$
$$98$$ −16.5913 −1.67597
$$99$$ −5.05034 −0.507578
$$100$$ 0 0
$$101$$ −3.44029 −0.342321 −0.171161 0.985243i $$-0.554752\pi$$
−0.171161 + 0.985243i $$0.554752\pi$$
$$102$$ 8.69613 0.861045
$$103$$ 6.91278 0.681136 0.340568 0.940220i $$-0.389381\pi$$
0.340568 + 0.940220i $$0.389381\pi$$
$$104$$ −1.67056 −0.163812
$$105$$ 0 0
$$106$$ −6.97258 −0.677237
$$107$$ −9.11319 −0.881006 −0.440503 0.897751i $$-0.645200\pi$$
−0.440503 + 0.897751i $$0.645200\pi$$
$$108$$ −1.55125 −0.149269
$$109$$ 16.7429 1.60368 0.801839 0.597540i $$-0.203855\pi$$
0.801839 + 0.597540i $$0.203855\pi$$
$$110$$ 0 0
$$111$$ −4.35033 −0.412915
$$112$$ −18.6692 −1.76407
$$113$$ −7.15846 −0.673411 −0.336706 0.941610i $$-0.609313\pi$$
−0.336706 + 0.941610i $$0.609313\pi$$
$$114$$ 8.76255 0.820688
$$115$$ 0 0
$$116$$ −1.55125 −0.144030
$$117$$ −1.97545 −0.182630
$$118$$ 24.3508 2.24167
$$119$$ 18.3452 1.68170
$$120$$ 0 0
$$121$$ 14.5059 1.31872
$$122$$ 13.7305 1.24311
$$123$$ 11.2474 1.01414
$$124$$ 10.0116 0.899064
$$125$$ 0 0
$$126$$ −7.49164 −0.667408
$$127$$ 7.05015 0.625600 0.312800 0.949819i $$-0.398733\pi$$
0.312800 + 0.949819i $$0.398733\pi$$
$$128$$ −6.59500 −0.582921
$$129$$ −7.53991 −0.663852
$$130$$ 0 0
$$131$$ 15.5744 1.36074 0.680370 0.732869i $$-0.261819\pi$$
0.680370 + 0.732869i $$0.261819\pi$$
$$132$$ 7.83433 0.681891
$$133$$ 18.4853 1.60288
$$134$$ 25.2556 2.18175
$$135$$ 0 0
$$136$$ 3.90240 0.334628
$$137$$ 18.9602 1.61988 0.809941 0.586512i $$-0.199499\pi$$
0.809941 + 0.586512i $$0.199499\pi$$
$$138$$ 2.27362 0.193543
$$139$$ −5.78101 −0.490339 −0.245170 0.969480i $$-0.578844\pi$$
−0.245170 + 0.969480i $$0.578844\pi$$
$$140$$ 0 0
$$141$$ −6.70001 −0.564243
$$142$$ −11.4219 −0.958502
$$143$$ 9.97669 0.834292
$$144$$ −4.69613 −0.391344
$$145$$ 0 0
$$146$$ 12.3177 1.01942
$$147$$ −8.80420 −0.726158
$$148$$ 6.74844 0.554718
$$149$$ 0.928816 0.0760916 0.0380458 0.999276i $$-0.487887\pi$$
0.0380458 + 0.999276i $$0.487887\pi$$
$$150$$ 0 0
$$151$$ 1.48920 0.121189 0.0605947 0.998162i $$-0.480700\pi$$
0.0605947 + 0.998162i $$0.480700\pi$$
$$152$$ 3.93221 0.318944
$$153$$ 4.61461 0.373069
$$154$$ 37.8353 3.04885
$$155$$ 0 0
$$156$$ 3.06441 0.245349
$$157$$ 12.3479 0.985466 0.492733 0.870180i $$-0.335998\pi$$
0.492733 + 0.870180i $$0.335998\pi$$
$$158$$ −9.98968 −0.794736
$$159$$ −3.70001 −0.293430
$$160$$ 0 0
$$161$$ 4.79637 0.378007
$$162$$ −1.88448 −0.148058
$$163$$ −13.9493 −1.09259 −0.546295 0.837593i $$-0.683962\pi$$
−0.546295 + 0.837593i $$0.683962\pi$$
$$164$$ −17.4475 −1.36242
$$165$$ 0 0
$$166$$ 15.0229 1.16600
$$167$$ −11.2844 −0.873217 −0.436608 0.899652i $$-0.643820\pi$$
−0.436608 + 0.899652i $$0.643820\pi$$
$$168$$ −3.36188 −0.259375
$$169$$ −9.09760 −0.699815
$$170$$ 0 0
$$171$$ 4.64986 0.355584
$$172$$ 11.6963 0.891833
$$173$$ 4.72255 0.359049 0.179524 0.983754i $$-0.442544\pi$$
0.179524 + 0.983754i $$0.442544\pi$$
$$174$$ −1.88448 −0.142862
$$175$$ 0 0
$$176$$ 23.7170 1.78774
$$177$$ 12.9218 0.971261
$$178$$ −22.7884 −1.70807
$$179$$ 9.36954 0.700312 0.350156 0.936691i $$-0.386129\pi$$
0.350156 + 0.936691i $$0.386129\pi$$
$$180$$ 0 0
$$181$$ 12.9621 0.963462 0.481731 0.876319i $$-0.340008\pi$$
0.481731 + 0.876319i $$0.340008\pi$$
$$182$$ 14.7994 1.09700
$$183$$ 7.28614 0.538607
$$184$$ 1.02029 0.0752167
$$185$$ 0 0
$$186$$ 12.1622 0.891773
$$187$$ −23.3053 −1.70426
$$188$$ 10.3934 0.758015
$$189$$ −3.97545 −0.289171
$$190$$ 0 0
$$191$$ 2.69237 0.194813 0.0974066 0.995245i $$-0.468945\pi$$
0.0974066 + 0.995245i $$0.468945\pi$$
$$192$$ 4.09760 0.295719
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ −19.0431 −1.36721
$$195$$ 0 0
$$196$$ 13.6575 0.975536
$$197$$ 10.2277 0.728695 0.364347 0.931263i $$-0.381292\pi$$
0.364347 + 0.931263i $$0.381292\pi$$
$$198$$ 9.51724 0.676361
$$199$$ −3.39810 −0.240885 −0.120442 0.992720i $$-0.538431\pi$$
−0.120442 + 0.992720i $$0.538431\pi$$
$$200$$ 0 0
$$201$$ 13.4019 0.945297
$$202$$ 6.48314 0.456152
$$203$$ −3.97545 −0.279022
$$204$$ −7.15841 −0.501189
$$205$$ 0 0
$$206$$ −13.0270 −0.907632
$$207$$ 1.20650 0.0838574
$$208$$ 9.27696 0.643241
$$209$$ −23.4834 −1.62438
$$210$$ 0 0
$$211$$ 16.5171 1.13708 0.568540 0.822655i $$-0.307508\pi$$
0.568540 + 0.822655i $$0.307508\pi$$
$$212$$ 5.73963 0.394200
$$213$$ −6.06103 −0.415295
$$214$$ 17.1736 1.17396
$$215$$ 0 0
$$216$$ −0.845662 −0.0575400
$$217$$ 25.6570 1.74171
$$218$$ −31.5516 −2.13694
$$219$$ 6.53640 0.441689
$$220$$ 0 0
$$221$$ −9.11593 −0.613204
$$222$$ 8.19809 0.550219
$$223$$ 3.48895 0.233637 0.116819 0.993153i $$-0.462730\pi$$
0.116819 + 0.993153i $$0.462730\pi$$
$$224$$ 28.4579 1.90142
$$225$$ 0 0
$$226$$ 13.4899 0.897338
$$227$$ 24.1163 1.60066 0.800328 0.599563i $$-0.204659\pi$$
0.800328 + 0.599563i $$0.204659\pi$$
$$228$$ −7.21309 −0.477698
$$229$$ −25.7198 −1.69961 −0.849805 0.527097i $$-0.823280\pi$$
−0.849805 + 0.527097i $$0.823280\pi$$
$$230$$ 0 0
$$231$$ 20.0774 1.32099
$$232$$ −0.845662 −0.0555204
$$233$$ −19.5181 −1.27867 −0.639337 0.768927i $$-0.720791\pi$$
−0.639337 + 0.768927i $$0.720791\pi$$
$$234$$ 3.72269 0.243360
$$235$$ 0 0
$$236$$ −20.0449 −1.30481
$$237$$ −5.30104 −0.344339
$$238$$ −34.5710 −2.24091
$$239$$ 4.35357 0.281609 0.140805 0.990037i $$-0.455031\pi$$
0.140805 + 0.990037i $$0.455031\pi$$
$$240$$ 0 0
$$241$$ 13.3619 0.860716 0.430358 0.902658i $$-0.358387\pi$$
0.430358 + 0.902658i $$0.358387\pi$$
$$242$$ −27.3360 −1.75723
$$243$$ −1.00000 −0.0641500
$$244$$ −11.3026 −0.723575
$$245$$ 0 0
$$246$$ −21.1954 −1.35137
$$247$$ −9.18556 −0.584463
$$248$$ 5.45779 0.346570
$$249$$ 7.97193 0.505201
$$250$$ 0 0
$$251$$ −5.34451 −0.337343 −0.168671 0.985672i $$-0.553948\pi$$
−0.168671 + 0.985672i $$0.553948\pi$$
$$252$$ 6.16691 0.388479
$$253$$ −6.09322 −0.383078
$$254$$ −13.2858 −0.833627
$$255$$ 0 0
$$256$$ 20.6233 1.28896
$$257$$ 11.5812 0.722418 0.361209 0.932485i $$-0.382364\pi$$
0.361209 + 0.932485i $$0.382364\pi$$
$$258$$ 14.2088 0.884600
$$259$$ 17.2945 1.07463
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ −29.3496 −1.81322
$$263$$ 15.2957 0.943173 0.471587 0.881820i $$-0.343681\pi$$
0.471587 + 0.881820i $$0.343681\pi$$
$$264$$ 4.27088 0.262854
$$265$$ 0 0
$$266$$ −34.8351 −2.13588
$$267$$ −12.0927 −0.740062
$$268$$ −20.7897 −1.26993
$$269$$ 6.70294 0.408685 0.204343 0.978899i $$-0.434494\pi$$
0.204343 + 0.978899i $$0.434494\pi$$
$$270$$ 0 0
$$271$$ 0.201885 0.0122637 0.00613183 0.999981i $$-0.498048\pi$$
0.00613183 + 0.999981i $$0.498048\pi$$
$$272$$ −21.6708 −1.31399
$$273$$ 7.85330 0.475303
$$274$$ −35.7301 −2.15853
$$275$$ 0 0
$$276$$ −1.87158 −0.112656
$$277$$ 13.4754 0.809659 0.404829 0.914392i $$-0.367331\pi$$
0.404829 + 0.914392i $$0.367331\pi$$
$$278$$ 10.8942 0.653390
$$279$$ 6.45387 0.386383
$$280$$ 0 0
$$281$$ −24.5512 −1.46460 −0.732301 0.680981i $$-0.761554\pi$$
−0.732301 + 0.680981i $$0.761554\pi$$
$$282$$ 12.6260 0.751868
$$283$$ −10.7258 −0.637582 −0.318791 0.947825i $$-0.603277\pi$$
−0.318791 + 0.947825i $$0.603277\pi$$
$$284$$ 9.40217 0.557916
$$285$$ 0 0
$$286$$ −18.8008 −1.11172
$$287$$ −44.7134 −2.63935
$$288$$ 7.15841 0.421813
$$289$$ 4.29465 0.252627
$$290$$ 0 0
$$291$$ −10.1052 −0.592380
$$292$$ −10.1396 −0.593374
$$293$$ 26.8072 1.56609 0.783046 0.621964i $$-0.213665\pi$$
0.783046 + 0.621964i $$0.213665\pi$$
$$294$$ 16.5913 0.967624
$$295$$ 0 0
$$296$$ 3.67890 0.213832
$$297$$ 5.05034 0.293050
$$298$$ −1.75033 −0.101394
$$299$$ −2.38338 −0.137834
$$300$$ 0 0
$$301$$ 29.9745 1.72770
$$302$$ −2.80636 −0.161488
$$303$$ 3.44029 0.197639
$$304$$ −21.8363 −1.25240
$$305$$ 0 0
$$306$$ −8.69613 −0.497124
$$307$$ 17.2029 0.981819 0.490909 0.871211i $$-0.336665\pi$$
0.490909 + 0.871211i $$0.336665\pi$$
$$308$$ −31.1450 −1.77465
$$309$$ −6.91278 −0.393254
$$310$$ 0 0
$$311$$ −5.23228 −0.296696 −0.148348 0.988935i $$-0.547396\pi$$
−0.148348 + 0.988935i $$0.547396\pi$$
$$312$$ 1.67056 0.0945769
$$313$$ 21.0916 1.19217 0.596083 0.802923i $$-0.296723\pi$$
0.596083 + 0.802923i $$0.296723\pi$$
$$314$$ −23.2692 −1.31316
$$315$$ 0 0
$$316$$ 8.22323 0.462593
$$317$$ 28.6493 1.60910 0.804551 0.593883i $$-0.202406\pi$$
0.804551 + 0.593883i $$0.202406\pi$$
$$318$$ 6.97258 0.391003
$$319$$ 5.05034 0.282765
$$320$$ 0 0
$$321$$ 9.11319 0.508649
$$322$$ −9.03865 −0.503704
$$323$$ 21.4573 1.19392
$$324$$ 1.55125 0.0861805
$$325$$ 0 0
$$326$$ 26.2870 1.45590
$$327$$ −16.7429 −0.925884
$$328$$ −9.51147 −0.525183
$$329$$ 26.6356 1.46847
$$330$$ 0 0
$$331$$ −30.3560 −1.66852 −0.834260 0.551372i $$-0.814105\pi$$
−0.834260 + 0.551372i $$0.814105\pi$$
$$332$$ −12.3664 −0.678697
$$333$$ 4.35033 0.238396
$$334$$ 21.2653 1.16358
$$335$$ 0 0
$$336$$ 18.6692 1.01849
$$337$$ 20.7033 1.12778 0.563890 0.825850i $$-0.309304\pi$$
0.563890 + 0.825850i $$0.309304\pi$$
$$338$$ 17.1442 0.932522
$$339$$ 7.15846 0.388794
$$340$$ 0 0
$$341$$ −32.5942 −1.76508
$$342$$ −8.76255 −0.473824
$$343$$ 7.17250 0.387279
$$344$$ 6.37621 0.343782
$$345$$ 0 0
$$346$$ −8.89953 −0.478442
$$347$$ 13.6308 0.731737 0.365869 0.930667i $$-0.380772\pi$$
0.365869 + 0.930667i $$0.380772\pi$$
$$348$$ 1.55125 0.0831556
$$349$$ 9.39303 0.502797 0.251399 0.967884i $$-0.419110\pi$$
0.251399 + 0.967884i $$0.419110\pi$$
$$350$$ 0 0
$$351$$ 1.97545 0.105442
$$352$$ −36.1524 −1.92693
$$353$$ −10.7225 −0.570701 −0.285351 0.958423i $$-0.592110\pi$$
−0.285351 + 0.958423i $$0.592110\pi$$
$$354$$ −24.3508 −1.29423
$$355$$ 0 0
$$356$$ 18.7588 0.994215
$$357$$ −18.3452 −0.970929
$$358$$ −17.6567 −0.933184
$$359$$ −5.62043 −0.296635 −0.148317 0.988940i $$-0.547386\pi$$
−0.148317 + 0.988940i $$0.547386\pi$$
$$360$$ 0 0
$$361$$ 2.62120 0.137958
$$362$$ −24.4267 −1.28384
$$363$$ −14.5059 −0.761362
$$364$$ −12.1824 −0.638532
$$365$$ 0 0
$$366$$ −13.7305 −0.717707
$$367$$ 12.4996 0.652476 0.326238 0.945288i $$-0.394219\pi$$
0.326238 + 0.945288i $$0.394219\pi$$
$$368$$ −5.66587 −0.295354
$$369$$ −11.2474 −0.585515
$$370$$ 0 0
$$371$$ 14.7092 0.763664
$$372$$ −10.0116 −0.519075
$$373$$ 20.5248 1.06274 0.531368 0.847141i $$-0.321678\pi$$
0.531368 + 0.847141i $$0.321678\pi$$
$$374$$ 43.9184 2.27096
$$375$$ 0 0
$$376$$ 5.66594 0.292199
$$377$$ 1.97545 0.101741
$$378$$ 7.49164 0.385328
$$379$$ 18.6409 0.957516 0.478758 0.877947i $$-0.341087\pi$$
0.478758 + 0.877947i $$0.341087\pi$$
$$380$$ 0 0
$$381$$ −7.05015 −0.361190
$$382$$ −5.07371 −0.259594
$$383$$ 16.9354 0.865356 0.432678 0.901548i $$-0.357569\pi$$
0.432678 + 0.901548i $$0.357569\pi$$
$$384$$ 6.59500 0.336549
$$385$$ 0 0
$$386$$ −26.3827 −1.34284
$$387$$ 7.53991 0.383275
$$388$$ 15.6757 0.795815
$$389$$ 23.2880 1.18075 0.590374 0.807130i $$-0.298980\pi$$
0.590374 + 0.807130i $$0.298980\pi$$
$$390$$ 0 0
$$391$$ 5.56752 0.281562
$$392$$ 7.44537 0.376048
$$393$$ −15.5744 −0.785624
$$394$$ −19.2739 −0.971005
$$395$$ 0 0
$$396$$ −7.83433 −0.393690
$$397$$ 8.20635 0.411865 0.205933 0.978566i $$-0.433977\pi$$
0.205933 + 0.978566i $$0.433977\pi$$
$$398$$ 6.40363 0.320985
$$399$$ −18.4853 −0.925422
$$400$$ 0 0
$$401$$ −20.0238 −0.999942 −0.499971 0.866042i $$-0.666656\pi$$
−0.499971 + 0.866042i $$0.666656\pi$$
$$402$$ −25.2556 −1.25963
$$403$$ −12.7493 −0.635088
$$404$$ −5.33674 −0.265513
$$405$$ 0 0
$$406$$ 7.49164 0.371804
$$407$$ −21.9706 −1.08904
$$408$$ −3.90240 −0.193198
$$409$$ 27.2928 1.34954 0.674771 0.738027i $$-0.264242\pi$$
0.674771 + 0.738027i $$0.264242\pi$$
$$410$$ 0 0
$$411$$ −18.9602 −0.935239
$$412$$ 10.7234 0.528306
$$413$$ −51.3699 −2.52775
$$414$$ −2.27362 −0.111742
$$415$$ 0 0
$$416$$ −14.1411 −0.693323
$$417$$ 5.78101 0.283098
$$418$$ 44.2538 2.16453
$$419$$ −34.5488 −1.68782 −0.843909 0.536486i $$-0.819751\pi$$
−0.843909 + 0.536486i $$0.819751\pi$$
$$420$$ 0 0
$$421$$ −26.1250 −1.27325 −0.636626 0.771173i $$-0.719671\pi$$
−0.636626 + 0.771173i $$0.719671\pi$$
$$422$$ −31.1260 −1.51519
$$423$$ 6.70001 0.325766
$$424$$ 3.12896 0.151956
$$425$$ 0 0
$$426$$ 11.4219 0.553391
$$427$$ −28.9657 −1.40175
$$428$$ −14.1368 −0.683329
$$429$$ −9.97669 −0.481679
$$430$$ 0 0
$$431$$ 30.4575 1.46709 0.733544 0.679642i $$-0.237865\pi$$
0.733544 + 0.679642i $$0.237865\pi$$
$$432$$ 4.69613 0.225942
$$433$$ 15.9802 0.767957 0.383979 0.923342i $$-0.374554\pi$$
0.383979 + 0.923342i $$0.374554\pi$$
$$434$$ −48.3501 −2.32088
$$435$$ 0 0
$$436$$ 25.9724 1.24385
$$437$$ 5.61005 0.268365
$$438$$ −12.3177 −0.588562
$$439$$ 19.2547 0.918976 0.459488 0.888184i $$-0.348033\pi$$
0.459488 + 0.888184i $$0.348033\pi$$
$$440$$ 0 0
$$441$$ 8.80420 0.419248
$$442$$ 17.1788 0.817110
$$443$$ 12.2199 0.580585 0.290293 0.956938i $$-0.406247\pi$$
0.290293 + 0.956938i $$0.406247\pi$$
$$444$$ −6.74844 −0.320267
$$445$$ 0 0
$$446$$ −6.57484 −0.311328
$$447$$ −0.928816 −0.0439315
$$448$$ −16.2898 −0.769621
$$449$$ −14.4104 −0.680069 −0.340035 0.940413i $$-0.610439\pi$$
−0.340035 + 0.940413i $$0.610439\pi$$
$$450$$ 0 0
$$451$$ 56.8030 2.67475
$$452$$ −11.1046 −0.522314
$$453$$ −1.48920 −0.0699688
$$454$$ −45.4466 −2.13291
$$455$$ 0 0
$$456$$ −3.93221 −0.184143
$$457$$ −17.5395 −0.820463 −0.410232 0.911981i $$-0.634552\pi$$
−0.410232 + 0.911981i $$0.634552\pi$$
$$458$$ 48.4683 2.26477
$$459$$ −4.61461 −0.215392
$$460$$ 0 0
$$461$$ 6.02126 0.280438 0.140219 0.990121i $$-0.455219\pi$$
0.140219 + 0.990121i $$0.455219\pi$$
$$462$$ −37.8353 −1.76026
$$463$$ −23.0161 −1.06965 −0.534824 0.844964i $$-0.679622\pi$$
−0.534824 + 0.844964i $$0.679622\pi$$
$$464$$ 4.69613 0.218012
$$465$$ 0 0
$$466$$ 36.7814 1.70386
$$467$$ 9.45324 0.437444 0.218722 0.975787i $$-0.429811\pi$$
0.218722 + 0.975787i $$0.429811\pi$$
$$468$$ −3.06441 −0.141653
$$469$$ −53.2786 −2.46018
$$470$$ 0 0
$$471$$ −12.3479 −0.568959
$$472$$ −10.9275 −0.502977
$$473$$ −38.0791 −1.75088
$$474$$ 9.98968 0.458841
$$475$$ 0 0
$$476$$ 28.4579 1.30437
$$477$$ 3.70001 0.169412
$$478$$ −8.20420 −0.375252
$$479$$ 40.2452 1.83885 0.919424 0.393268i $$-0.128656\pi$$
0.919424 + 0.393268i $$0.128656\pi$$
$$480$$ 0 0
$$481$$ −8.59385 −0.391846
$$482$$ −25.1802 −1.14693
$$483$$ −4.79637 −0.218243
$$484$$ 22.5023 1.02283
$$485$$ 0 0
$$486$$ 1.88448 0.0854815
$$487$$ −27.3621 −1.23989 −0.619947 0.784644i $$-0.712846\pi$$
−0.619947 + 0.784644i $$0.712846\pi$$
$$488$$ −6.16161 −0.278923
$$489$$ 13.9493 0.630807
$$490$$ 0 0
$$491$$ −16.1708 −0.729778 −0.364889 0.931051i $$-0.618893\pi$$
−0.364889 + 0.931051i $$0.618893\pi$$
$$492$$ 17.4475 0.786593
$$493$$ −4.61461 −0.207832
$$494$$ 17.3100 0.778812
$$495$$ 0 0
$$496$$ −30.3082 −1.36088
$$497$$ 24.0953 1.08082
$$498$$ −15.0229 −0.673193
$$499$$ −24.2809 −1.08696 −0.543482 0.839421i $$-0.682894\pi$$
−0.543482 + 0.839421i $$0.682894\pi$$
$$500$$ 0 0
$$501$$ 11.2844 0.504152
$$502$$ 10.0716 0.449518
$$503$$ −24.4131 −1.08853 −0.544263 0.838915i $$-0.683191\pi$$
−0.544263 + 0.838915i $$0.683191\pi$$
$$504$$ 3.36188 0.149750
$$505$$ 0 0
$$506$$ 11.4825 0.510461
$$507$$ 9.09760 0.404039
$$508$$ 10.9365 0.485230
$$509$$ 33.6547 1.49172 0.745859 0.666103i $$-0.232039\pi$$
0.745859 + 0.666103i $$0.232039\pi$$
$$510$$ 0 0
$$511$$ −25.9851 −1.14951
$$512$$ −25.6741 −1.13465
$$513$$ −4.64986 −0.205296
$$514$$ −21.8246 −0.962641
$$515$$ 0 0
$$516$$ −11.6963 −0.514900
$$517$$ −33.8373 −1.48816
$$518$$ −32.5911 −1.43197
$$519$$ −4.72255 −0.207297
$$520$$ 0 0
$$521$$ 8.83636 0.387128 0.193564 0.981088i $$-0.437995\pi$$
0.193564 + 0.981088i $$0.437995\pi$$
$$522$$ 1.88448 0.0824813
$$523$$ 9.71669 0.424881 0.212441 0.977174i $$-0.431859\pi$$
0.212441 + 0.977174i $$0.431859\pi$$
$$524$$ 24.1597 1.05542
$$525$$ 0 0
$$526$$ −28.8244 −1.25680
$$527$$ 29.7821 1.29733
$$528$$ −23.7170 −1.03215
$$529$$ −21.5444 −0.936711
$$530$$ 0 0
$$531$$ −12.9218 −0.560758
$$532$$ 28.6753 1.24323
$$533$$ 22.2186 0.962395
$$534$$ 22.7884 0.986152
$$535$$ 0 0
$$536$$ −11.3335 −0.489531
$$537$$ −9.36954 −0.404325
$$538$$ −12.6315 −0.544584
$$539$$ −44.4642 −1.91521
$$540$$ 0 0
$$541$$ −11.4439 −0.492012 −0.246006 0.969268i $$-0.579118\pi$$
−0.246006 + 0.969268i $$0.579118\pi$$
$$542$$ −0.380448 −0.0163416
$$543$$ −12.9621 −0.556255
$$544$$ 33.0333 1.41629
$$545$$ 0 0
$$546$$ −14.7994 −0.633354
$$547$$ −15.6211 −0.667908 −0.333954 0.942589i $$-0.608383\pi$$
−0.333954 + 0.942589i $$0.608383\pi$$
$$548$$ 29.4120 1.25642
$$549$$ −7.28614 −0.310965
$$550$$ 0 0
$$551$$ −4.64986 −0.198091
$$552$$ −1.02029 −0.0434264
$$553$$ 21.0740 0.896158
$$554$$ −25.3941 −1.07889
$$555$$ 0 0
$$556$$ −8.96779 −0.380319
$$557$$ −18.9932 −0.804766 −0.402383 0.915471i $$-0.631818\pi$$
−0.402383 + 0.915471i $$0.631818\pi$$
$$558$$ −12.1622 −0.514865
$$559$$ −14.8947 −0.629979
$$560$$ 0 0
$$561$$ 23.3053 0.983953
$$562$$ 46.2661 1.95162
$$563$$ 20.7203 0.873257 0.436629 0.899642i $$-0.356172\pi$$
0.436629 + 0.899642i $$0.356172\pi$$
$$564$$ −10.3934 −0.437640
$$565$$ 0 0
$$566$$ 20.2125 0.849595
$$567$$ 3.97545 0.166953
$$568$$ 5.12558 0.215065
$$569$$ −23.0163 −0.964892 −0.482446 0.875926i $$-0.660252\pi$$
−0.482446 + 0.875926i $$0.660252\pi$$
$$570$$ 0 0
$$571$$ −18.1927 −0.761341 −0.380670 0.924711i $$-0.624307\pi$$
−0.380670 + 0.924711i $$0.624307\pi$$
$$572$$ 15.4763 0.647097
$$573$$ −2.69237 −0.112475
$$574$$ 84.2612 3.51700
$$575$$ 0 0
$$576$$ −4.09760 −0.170733
$$577$$ −10.4435 −0.434771 −0.217385 0.976086i $$-0.569753\pi$$
−0.217385 + 0.976086i $$0.569753\pi$$
$$578$$ −8.09317 −0.336631
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ −31.6920 −1.31481
$$582$$ 19.0431 0.789361
$$583$$ −18.6863 −0.773907
$$584$$ −5.52758 −0.228733
$$585$$ 0 0
$$586$$ −50.5175 −2.08686
$$587$$ 20.7562 0.856700 0.428350 0.903613i $$-0.359095\pi$$
0.428350 + 0.903613i $$0.359095\pi$$
$$588$$ −13.6575 −0.563226
$$589$$ 30.0096 1.23652
$$590$$ 0 0
$$591$$ −10.2277 −0.420712
$$592$$ −20.4297 −0.839655
$$593$$ 2.43875 0.100147 0.0500737 0.998746i $$-0.484054\pi$$
0.0500737 + 0.998746i $$0.484054\pi$$
$$594$$ −9.51724 −0.390497
$$595$$ 0 0
$$596$$ 1.44082 0.0590185
$$597$$ 3.39810 0.139075
$$598$$ 4.49141 0.183668
$$599$$ −10.9851 −0.448840 −0.224420 0.974493i $$-0.572049\pi$$
−0.224420 + 0.974493i $$0.572049\pi$$
$$600$$ 0 0
$$601$$ 16.7056 0.681435 0.340717 0.940166i $$-0.389330\pi$$
0.340717 + 0.940166i $$0.389330\pi$$
$$602$$ −56.4863 −2.30221
$$603$$ −13.4019 −0.545768
$$604$$ 2.31012 0.0939975
$$605$$ 0 0
$$606$$ −6.48314 −0.263360
$$607$$ −47.0785 −1.91086 −0.955429 0.295222i $$-0.904606\pi$$
−0.955429 + 0.295222i $$0.904606\pi$$
$$608$$ 33.2856 1.34991
$$609$$ 3.97545 0.161093
$$610$$ 0 0
$$611$$ −13.2355 −0.535452
$$612$$ 7.15841 0.289362
$$613$$ −0.285402 −0.0115273 −0.00576364 0.999983i $$-0.501835\pi$$
−0.00576364 + 0.999983i $$0.501835\pi$$
$$614$$ −32.4184 −1.30830
$$615$$ 0 0
$$616$$ −16.9787 −0.684089
$$617$$ 45.4926 1.83146 0.915731 0.401792i $$-0.131613\pi$$
0.915731 + 0.401792i $$0.131613\pi$$
$$618$$ 13.0270 0.524021
$$619$$ −37.4525 −1.50534 −0.752672 0.658396i $$-0.771235\pi$$
−0.752672 + 0.658396i $$0.771235\pi$$
$$620$$ 0 0
$$621$$ −1.20650 −0.0484151
$$622$$ 9.86011 0.395354
$$623$$ 48.0740 1.92604
$$624$$ −9.27696 −0.371376
$$625$$ 0 0
$$626$$ −39.7466 −1.58859
$$627$$ 23.4834 0.937835
$$628$$ 19.1546 0.764352
$$629$$ 20.0751 0.800446
$$630$$ 0 0
$$631$$ −12.1278 −0.482799 −0.241400 0.970426i $$-0.577606\pi$$
−0.241400 + 0.970426i $$0.577606\pi$$
$$632$$ 4.48288 0.178320
$$633$$ −16.5171 −0.656494
$$634$$ −53.9888 −2.14417
$$635$$ 0 0
$$636$$ −5.73963 −0.227591
$$637$$ −17.3923 −0.689106
$$638$$ −9.51724 −0.376791
$$639$$ 6.06103 0.239771
$$640$$ 0 0
$$641$$ 1.98414 0.0783687 0.0391843 0.999232i $$-0.487524\pi$$
0.0391843 + 0.999232i $$0.487524\pi$$
$$642$$ −17.1736 −0.677788
$$643$$ −38.6749 −1.52519 −0.762595 0.646876i $$-0.776075\pi$$
−0.762595 + 0.646876i $$0.776075\pi$$
$$644$$ 7.44037 0.293191
$$645$$ 0 0
$$646$$ −40.4358 −1.59092
$$647$$ 27.6995 1.08898 0.544489 0.838768i $$-0.316723\pi$$
0.544489 + 0.838768i $$0.316723\pi$$
$$648$$ 0.845662 0.0332207
$$649$$ 65.2594 2.56165
$$650$$ 0 0
$$651$$ −25.6570 −1.00558
$$652$$ −21.6388 −0.847440
$$653$$ 34.8231 1.36273 0.681366 0.731942i $$-0.261386\pi$$
0.681366 + 0.731942i $$0.261386\pi$$
$$654$$ 31.5516 1.23376
$$655$$ 0 0
$$656$$ 52.8191 2.06224
$$657$$ −6.53640 −0.255009
$$658$$ −50.1940 −1.95677
$$659$$ 7.37091 0.287130 0.143565 0.989641i $$-0.454143\pi$$
0.143565 + 0.989641i $$0.454143\pi$$
$$660$$ 0 0
$$661$$ 25.7621 1.00203 0.501015 0.865439i $$-0.332960\pi$$
0.501015 + 0.865439i $$0.332960\pi$$
$$662$$ 57.2052 2.22334
$$663$$ 9.11593 0.354034
$$664$$ −6.74156 −0.261623
$$665$$ 0 0
$$666$$ −8.19809 −0.317669
$$667$$ −1.20650 −0.0467158
$$668$$ −17.5050 −0.677288
$$669$$ −3.48895 −0.134891
$$670$$ 0 0
$$671$$ 36.7974 1.42055
$$672$$ −28.4579 −1.09779
$$673$$ −39.0838 −1.50657 −0.753285 0.657694i $$-0.771532\pi$$
−0.753285 + 0.657694i $$0.771532\pi$$
$$674$$ −39.0149 −1.50280
$$675$$ 0 0
$$676$$ −14.1126 −0.542794
$$677$$ −40.1309 −1.54236 −0.771178 0.636619i $$-0.780332\pi$$
−0.771178 + 0.636619i $$0.780332\pi$$
$$678$$ −13.4899 −0.518078
$$679$$ 40.1729 1.54169
$$680$$ 0 0
$$681$$ −24.1163 −0.924139
$$682$$ 61.4230 2.35201
$$683$$ −21.7747 −0.833185 −0.416592 0.909093i $$-0.636776\pi$$
−0.416592 + 0.909093i $$0.636776\pi$$
$$684$$ 7.21309 0.275799
$$685$$ 0 0
$$686$$ −13.5164 −0.516059
$$687$$ 25.7198 0.981270
$$688$$ −35.4084 −1.34993
$$689$$ −7.30918 −0.278458
$$690$$ 0 0
$$691$$ 27.2866 1.03803 0.519016 0.854765i $$-0.326299\pi$$
0.519016 + 0.854765i $$0.326299\pi$$
$$692$$ 7.32585 0.278487
$$693$$ −20.0774 −0.762676
$$694$$ −25.6868 −0.975058
$$695$$ 0 0
$$696$$ 0.845662 0.0320547
$$697$$ −51.9023 −1.96594
$$698$$ −17.7009 −0.669990
$$699$$ 19.5181 0.738242
$$700$$ 0 0
$$701$$ −9.80694 −0.370403 −0.185202 0.982701i $$-0.559294\pi$$
−0.185202 + 0.982701i $$0.559294\pi$$
$$702$$ −3.72269 −0.140504
$$703$$ 20.2284 0.762929
$$704$$ 20.6943 0.779944
$$705$$ 0 0
$$706$$ 20.2063 0.760474
$$707$$ −13.6767 −0.514365
$$708$$ 20.0449 0.753333
$$709$$ 34.6952 1.30301 0.651503 0.758646i $$-0.274139\pi$$
0.651503 + 0.758646i $$0.274139\pi$$
$$710$$ 0 0
$$711$$ 5.30104 0.198804
$$712$$ 10.2263 0.383249
$$713$$ 7.78659 0.291610
$$714$$ 34.5710 1.29379
$$715$$ 0 0
$$716$$ 14.5345 0.543179
$$717$$ −4.35357 −0.162587
$$718$$ 10.5916 0.395273
$$719$$ −0.619226 −0.0230932 −0.0115466 0.999933i $$-0.503675\pi$$
−0.0115466 + 0.999933i $$0.503675\pi$$
$$720$$ 0 0
$$721$$ 27.4814 1.02346
$$722$$ −4.93959 −0.183833
$$723$$ −13.3619 −0.496934
$$724$$ 20.1074 0.747284
$$725$$ 0 0
$$726$$ 27.3360 1.01453
$$727$$ 5.82966 0.216210 0.108105 0.994139i $$-0.465522\pi$$
0.108105 + 0.994139i $$0.465522\pi$$
$$728$$ −6.64123 −0.246141
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 34.7938 1.28689
$$732$$ 11.3026 0.417756
$$733$$ 4.63787 0.171304 0.0856519 0.996325i $$-0.472703\pi$$
0.0856519 + 0.996325i $$0.472703\pi$$
$$734$$ −23.5553 −0.869441
$$735$$ 0 0
$$736$$ 8.63661 0.318350
$$737$$ 67.6841 2.49318
$$738$$ 21.1954 0.780214
$$739$$ 21.0686 0.775022 0.387511 0.921865i $$-0.373335\pi$$
0.387511 + 0.921865i $$0.373335\pi$$
$$740$$ 0 0
$$741$$ 9.18556 0.337440
$$742$$ −27.7191 −1.01760
$$743$$ 38.5779 1.41529 0.707643 0.706571i $$-0.249759\pi$$
0.707643 + 0.706571i $$0.249759\pi$$
$$744$$ −5.45779 −0.200092
$$745$$ 0 0
$$746$$ −38.6785 −1.41612
$$747$$ −7.97193 −0.291678
$$748$$ −36.1524 −1.32186
$$749$$ −36.2290 −1.32378
$$750$$ 0 0
$$751$$ −41.4212 −1.51148 −0.755740 0.654872i $$-0.772722\pi$$
−0.755740 + 0.654872i $$0.772722\pi$$
$$752$$ −31.4641 −1.14738
$$753$$ 5.34451 0.194765
$$754$$ −3.72269 −0.135572
$$755$$ 0 0
$$756$$ −6.16691 −0.224288
$$757$$ −15.7076 −0.570901 −0.285450 0.958393i $$-0.592143\pi$$
−0.285450 + 0.958393i $$0.592143\pi$$
$$758$$ −35.1282 −1.27592
$$759$$ 6.09322 0.221170
$$760$$ 0 0
$$761$$ 15.1038 0.547513 0.273756 0.961799i $$-0.411734\pi$$
0.273756 + 0.961799i $$0.411734\pi$$
$$762$$ 13.2858 0.481295
$$763$$ 66.5605 2.40965
$$764$$ 4.17654 0.151102
$$765$$ 0 0
$$766$$ −31.9143 −1.15311
$$767$$ 25.5263 0.921702
$$768$$ −20.6233 −0.744179
$$769$$ −10.4395 −0.376457 −0.188228 0.982125i $$-0.560275\pi$$
−0.188228 + 0.982125i $$0.560275\pi$$
$$770$$ 0 0
$$771$$ −11.5812 −0.417088
$$772$$ 21.7175 0.781629
$$773$$ 23.0481 0.828982 0.414491 0.910053i $$-0.363960\pi$$
0.414491 + 0.910053i $$0.363960\pi$$
$$774$$ −14.2088 −0.510724
$$775$$ 0 0
$$776$$ 8.54561 0.306770
$$777$$ −17.2945 −0.620437
$$778$$ −43.8856 −1.57338
$$779$$ −52.2987 −1.87380
$$780$$ 0 0
$$781$$ −30.6103 −1.09532
$$782$$ −10.4919 −0.375188
$$783$$ 1.00000 0.0357371
$$784$$ −41.3456 −1.47663
$$785$$ 0 0
$$786$$ 29.3496 1.04686
$$787$$ −0.591462 −0.0210833 −0.0105417 0.999944i $$-0.503356\pi$$
−0.0105417 + 0.999944i $$0.503356\pi$$
$$788$$ 15.8657 0.565193
$$789$$ −15.2957 −0.544541
$$790$$ 0 0
$$791$$ −28.4581 −1.01185
$$792$$ −4.27088 −0.151759
$$793$$ 14.3934 0.511124
$$794$$ −15.4647 −0.548821
$$795$$ 0 0
$$796$$ −5.27129 −0.186836
$$797$$ −25.4764 −0.902419 −0.451210 0.892418i $$-0.649007\pi$$
−0.451210 + 0.892418i $$0.649007\pi$$
$$798$$ 34.8351 1.23315
$$799$$ 30.9180 1.09380
$$800$$ 0 0
$$801$$ 12.0927 0.427275
$$802$$ 37.7344 1.33245
$$803$$ 33.0110 1.16493
$$804$$ 20.7897 0.733195
$$805$$ 0 0
$$806$$ 24.0257 0.846271
$$807$$ −6.70294 −0.235955
$$808$$ −2.90932 −0.102349
$$809$$ −8.55103 −0.300638 −0.150319 0.988638i $$-0.548030\pi$$
−0.150319 + 0.988638i $$0.548030\pi$$
$$810$$ 0 0
$$811$$ −19.9128 −0.699232 −0.349616 0.936893i $$-0.613688\pi$$
−0.349616 + 0.936893i $$0.613688\pi$$
$$812$$ −6.16691 −0.216416
$$813$$ −0.201885 −0.00708043
$$814$$ 41.4031 1.45118
$$815$$ 0 0
$$816$$ 21.6708 0.758630
$$817$$ 35.0595 1.22658
$$818$$ −51.4326 −1.79830
$$819$$ −7.85330 −0.274417
$$820$$ 0 0
$$821$$ 38.1575 1.33170 0.665852 0.746084i $$-0.268068\pi$$
0.665852 + 0.746084i $$0.268068\pi$$
$$822$$ 35.7301 1.24623
$$823$$ −27.8681 −0.971421 −0.485710 0.874120i $$-0.661439\pi$$
−0.485710 + 0.874120i $$0.661439\pi$$
$$824$$ 5.84587 0.203651
$$825$$ 0 0
$$826$$ 96.8053 3.36829
$$827$$ −19.0910 −0.663858 −0.331929 0.943304i $$-0.607700\pi$$
−0.331929 + 0.943304i $$0.607700\pi$$
$$828$$ 1.87158 0.0650418
$$829$$ −20.6808 −0.718274 −0.359137 0.933285i $$-0.616929\pi$$
−0.359137 + 0.933285i $$0.616929\pi$$
$$830$$ 0 0
$$831$$ −13.4754 −0.467457
$$832$$ 8.09460 0.280630
$$833$$ 40.6280 1.40768
$$834$$ −10.8942 −0.377235
$$835$$ 0 0
$$836$$ −36.4285 −1.25991
$$837$$ −6.45387 −0.223078
$$838$$ 65.1063 2.24906
$$839$$ −18.3865 −0.634772 −0.317386 0.948296i $$-0.602805\pi$$
−0.317386 + 0.948296i $$0.602805\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 49.2318 1.69664
$$843$$ 24.5512 0.845588
$$844$$ 25.6221 0.881947
$$845$$ 0 0
$$846$$ −12.6260 −0.434091
$$847$$ 57.6675 1.98148
$$848$$ −17.3757 −0.596684
$$849$$ 10.7258 0.368108
$$850$$ 0 0
$$851$$ 5.24866 0.179922
$$852$$ −9.40217 −0.322113
$$853$$ −28.4263 −0.973297 −0.486648 0.873598i $$-0.661781\pi$$
−0.486648 + 0.873598i $$0.661781\pi$$
$$854$$ 54.5851 1.86786
$$855$$ 0 0
$$856$$ −7.70668 −0.263409
$$857$$ −22.6832 −0.774842 −0.387421 0.921903i $$-0.626634\pi$$
−0.387421 + 0.921903i $$0.626634\pi$$
$$858$$ 18.8008 0.641849
$$859$$ −17.3179 −0.590878 −0.295439 0.955362i $$-0.595466\pi$$
−0.295439 + 0.955362i $$0.595466\pi$$
$$860$$ 0 0
$$861$$ 44.7134 1.52383
$$862$$ −57.3965 −1.95493
$$863$$ 25.9340 0.882805 0.441403 0.897309i $$-0.354481\pi$$
0.441403 + 0.897309i $$0.354481\pi$$
$$864$$ −7.15841 −0.243534
$$865$$ 0 0
$$866$$ −30.1142 −1.02332
$$867$$ −4.29465 −0.145854
$$868$$ 39.8004 1.35092
$$869$$ −26.7720 −0.908179
$$870$$ 0 0
$$871$$ 26.4748 0.897064
$$872$$ 14.1588 0.479478
$$873$$ 10.1052 0.342010
$$874$$ −10.5720 −0.357603
$$875$$ 0 0
$$876$$ 10.1396 0.342585
$$877$$ 8.52354 0.287819 0.143910 0.989591i $$-0.454033\pi$$
0.143910 + 0.989591i $$0.454033\pi$$
$$878$$ −36.2850 −1.22456
$$879$$ −26.8072 −0.904184
$$880$$ 0 0
$$881$$ 48.4352 1.63182 0.815912 0.578177i $$-0.196236\pi$$
0.815912 + 0.578177i $$0.196236\pi$$
$$882$$ −16.5913 −0.558658
$$883$$ 5.87273 0.197633 0.0988166 0.995106i $$-0.468494\pi$$
0.0988166 + 0.995106i $$0.468494\pi$$
$$884$$ −14.1411 −0.475616
$$885$$ 0 0
$$886$$ −23.0281 −0.773644
$$887$$ −29.9614 −1.00601 −0.503003 0.864285i $$-0.667771\pi$$
−0.503003 + 0.864285i $$0.667771\pi$$
$$888$$ −3.67890 −0.123456
$$889$$ 28.0275 0.940013
$$890$$ 0 0
$$891$$ −5.05034 −0.169193
$$892$$ 5.41222 0.181215
$$893$$ 31.1541 1.04253
$$894$$ 1.75033 0.0585398
$$895$$ 0 0
$$896$$ −26.2181 −0.875884
$$897$$ 2.38338 0.0795786
$$898$$ 27.1561 0.906210
$$899$$ −6.45387 −0.215249
$$900$$ 0 0
$$901$$ 17.0741 0.568821
$$902$$ −107.044 −3.56417
$$903$$ −29.9745 −0.997490
$$904$$ −6.05364 −0.201341
$$905$$ 0 0
$$906$$ 2.80636 0.0932352
$$907$$ −41.3847 −1.37416 −0.687078 0.726584i $$-0.741107\pi$$
−0.687078 + 0.726584i $$0.741107\pi$$
$$908$$ 37.4104 1.24151
$$909$$ −3.44029 −0.114107
$$910$$ 0 0
$$911$$ 15.1439 0.501739 0.250870 0.968021i $$-0.419283\pi$$
0.250870 + 0.968021i $$0.419283\pi$$
$$912$$ 21.8363 0.723073
$$913$$ 40.2610 1.33244
$$914$$ 33.0528 1.09329
$$915$$ 0 0
$$916$$ −39.8977 −1.31826
$$917$$ 61.9152 2.04462
$$918$$ 8.69613 0.287015
$$919$$ −1.12139 −0.0369911 −0.0184956 0.999829i $$-0.505888\pi$$
−0.0184956 + 0.999829i $$0.505888\pi$$
$$920$$ 0 0
$$921$$ −17.2029 −0.566853
$$922$$ −11.3469 −0.373691
$$923$$ −11.9733 −0.394105
$$924$$ 31.1450 1.02459
$$925$$ 0 0
$$926$$ 43.3732 1.42533
$$927$$ 6.91278 0.227045
$$928$$ −7.15841 −0.234986
$$929$$ 3.38813 0.111161 0.0555804 0.998454i $$-0.482299\pi$$
0.0555804 + 0.998454i $$0.482299\pi$$
$$930$$ 0 0
$$931$$ 40.9383 1.34170
$$932$$ −30.2774 −0.991770
$$933$$ 5.23228 0.171297
$$934$$ −17.8144 −0.582905
$$935$$ 0 0
$$936$$ −1.67056 −0.0546040
$$937$$ 17.9125 0.585175 0.292587 0.956239i $$-0.405484\pi$$
0.292587 + 0.956239i $$0.405484\pi$$
$$938$$ 100.402 3.27825
$$939$$ −21.0916 −0.688297
$$940$$ 0 0
$$941$$ −26.2459 −0.855592 −0.427796 0.903875i $$-0.640710\pi$$
−0.427796 + 0.903875i $$0.640710\pi$$
$$942$$ 23.2692 0.758153
$$943$$ −13.5699 −0.441898
$$944$$ 60.6823 1.97504
$$945$$ 0 0
$$946$$ 71.7591 2.33309
$$947$$ 24.9518 0.810825 0.405413 0.914134i $$-0.367128\pi$$
0.405413 + 0.914134i $$0.367128\pi$$
$$948$$ −8.22323 −0.267078
$$949$$ 12.9123 0.419152
$$950$$ 0 0
$$951$$ −28.6493 −0.929016
$$952$$ 15.5138 0.502805
$$953$$ 34.5021 1.11763 0.558817 0.829291i $$-0.311255\pi$$
0.558817 + 0.829291i $$0.311255\pi$$
$$954$$ −6.97258 −0.225746
$$955$$ 0 0
$$956$$ 6.75347 0.218423
$$957$$ −5.05034 −0.163254
$$958$$ −75.8410 −2.45031
$$959$$ 75.3754 2.43400
$$960$$ 0 0
$$961$$ 10.6525 0.343628
$$962$$ 16.1949 0.522145
$$963$$ −9.11319 −0.293669
$$964$$ 20.7276 0.667592
$$965$$ 0 0
$$966$$ 9.03865 0.290814
$$967$$ −34.8694 −1.12132 −0.560661 0.828045i $$-0.689453\pi$$
−0.560661 + 0.828045i $$0.689453\pi$$
$$968$$ 12.2671 0.394279
$$969$$ −21.4573 −0.689308
$$970$$ 0 0
$$971$$ −50.6681 −1.62602 −0.813008 0.582252i $$-0.802172\pi$$
−0.813008 + 0.582252i $$0.802172\pi$$
$$972$$ −1.55125 −0.0497563
$$973$$ −22.9821 −0.736773
$$974$$ 51.5632 1.65219
$$975$$ 0 0
$$976$$ 34.2166 1.09525
$$977$$ 20.0017 0.639911 0.319956 0.947433i $$-0.396332\pi$$
0.319956 + 0.947433i $$0.396332\pi$$
$$978$$ −26.2870 −0.840567
$$979$$ −61.0723 −1.95188
$$980$$ 0 0
$$981$$ 16.7429 0.534560
$$982$$ 30.4735 0.972447
$$983$$ 8.51776 0.271674 0.135837 0.990731i $$-0.456628\pi$$
0.135837 + 0.990731i $$0.456628\pi$$
$$984$$ 9.51147 0.303215
$$985$$ 0 0
$$986$$ 8.69613 0.276941
$$987$$ −26.6356 −0.847819
$$988$$ −14.2491 −0.453324
$$989$$ 9.09689 0.289264
$$990$$ 0 0
$$991$$ 41.0770 1.30485 0.652427 0.757852i $$-0.273751\pi$$
0.652427 + 0.757852i $$0.273751\pi$$
$$992$$ 46.1995 1.46683
$$993$$ 30.3560 0.963320
$$994$$ −45.4071 −1.44022
$$995$$ 0 0
$$996$$ 12.3664 0.391846
$$997$$ 28.8974 0.915189 0.457594 0.889161i $$-0.348711\pi$$
0.457594 + 0.889161i $$0.348711\pi$$
$$998$$ 45.7568 1.44841
$$999$$ −4.35033 −0.137638
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 2175.2.a.y.1.1 5
3.2 odd 2 6525.2.a.bn.1.5 5
5.2 odd 4 435.2.c.d.349.3 10
5.3 odd 4 435.2.c.d.349.8 yes 10
5.4 even 2 2175.2.a.x.1.5 5
15.2 even 4 1305.2.c.i.784.8 10
15.8 even 4 1305.2.c.i.784.3 10
15.14 odd 2 6525.2.a.br.1.1 5

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.d.349.3 10 5.2 odd 4
435.2.c.d.349.8 yes 10 5.3 odd 4
1305.2.c.i.784.3 10 15.8 even 4
1305.2.c.i.784.8 10 15.2 even 4
2175.2.a.x.1.5 5 5.4 even 2
2175.2.a.y.1.1 5 1.1 even 1 trivial
6525.2.a.bn.1.5 5 3.2 odd 2
6525.2.a.br.1.1 5 15.14 odd 2