# Properties

 Label 6525.2.a.bg.1.3 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 87) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.11491$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.47283 q^{2} +4.11491 q^{4} -1.64207 q^{7} +5.22982 q^{8} +O(q^{10})$$ $$q+2.47283 q^{2} +4.11491 q^{4} -1.64207 q^{7} +5.22982 q^{8} +2.35793 q^{11} +2.58774 q^{13} -4.06058 q^{14} +4.70265 q^{16} +5.87189 q^{17} +2.94567 q^{19} +5.83076 q^{22} -2.22982 q^{23} +6.39905 q^{26} -6.75698 q^{28} -1.00000 q^{29} -2.22982 q^{31} +1.16924 q^{32} +14.5202 q^{34} +0.945668 q^{37} +7.28415 q^{38} -0.568295 q^{41} +2.56829 q^{43} +9.70265 q^{44} -5.51396 q^{46} -4.92622 q^{47} -4.30359 q^{49} +10.6483 q^{52} +14.1212 q^{53} -8.58774 q^{56} -2.47283 q^{58} +7.28415 q^{59} +4.94567 q^{61} -5.51396 q^{62} -6.51396 q^{64} -0.926221 q^{67} +24.1623 q^{68} +16.1212 q^{71} -0.945668 q^{73} +2.33848 q^{74} +12.1212 q^{76} -3.87189 q^{77} -5.51396 q^{79} -1.40530 q^{82} +5.17548 q^{83} +6.35097 q^{86} +12.3315 q^{88} -8.47908 q^{89} -4.24926 q^{91} -9.17548 q^{92} -12.1817 q^{94} +1.66152 q^{97} -10.6421 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 6 q^{4} - 4 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^7 + 3 * q^8 $$3 q + 2 q^{2} + 6 q^{4} - 4 q^{7} + 3 q^{8} + 8 q^{11} - 4 q^{13} + 5 q^{14} - 4 q^{16} + 4 q^{17} - 2 q^{19} + 13 q^{22} + 6 q^{23} + 11 q^{26} - 13 q^{28} - 3 q^{29} + 6 q^{31} + 8 q^{32} + q^{34} - 8 q^{37} + 20 q^{38} + 2 q^{41} + 4 q^{43} + 11 q^{44} - 2 q^{46} - 12 q^{47} - 3 q^{49} + 3 q^{52} + 8 q^{53} - 14 q^{56} - 2 q^{58} + 20 q^{59} + 4 q^{61} - 2 q^{62} - 5 q^{64} + 29 q^{68} + 14 q^{71} + 8 q^{73} + 16 q^{74} + 2 q^{76} + 2 q^{77} - 2 q^{79} + 32 q^{82} - 8 q^{83} - 28 q^{86} - 2 q^{88} + 8 q^{89} + 8 q^{91} - 4 q^{92} + 15 q^{94} - 4 q^{97} - 31 q^{98}+O(q^{100})$$ 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^7 + 3 * q^8 + 8 * q^11 - 4 * q^13 + 5 * q^14 - 4 * q^16 + 4 * q^17 - 2 * q^19 + 13 * q^22 + 6 * q^23 + 11 * q^26 - 13 * q^28 - 3 * q^29 + 6 * q^31 + 8 * q^32 + q^34 - 8 * q^37 + 20 * q^38 + 2 * q^41 + 4 * q^43 + 11 * q^44 - 2 * q^46 - 12 * q^47 - 3 * q^49 + 3 * q^52 + 8 * q^53 - 14 * q^56 - 2 * q^58 + 20 * q^59 + 4 * q^61 - 2 * q^62 - 5 * q^64 + 29 * q^68 + 14 * q^71 + 8 * q^73 + 16 * q^74 + 2 * q^76 + 2 * q^77 - 2 * q^79 + 32 * q^82 - 8 * q^83 - 28 * q^86 - 2 * q^88 + 8 * q^89 + 8 * q^91 - 4 * q^92 + 15 * q^94 - 4 * q^97 - 31 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.47283 1.74856 0.874279 0.485424i $$-0.161335\pi$$
0.874279 + 0.485424i $$0.161335\pi$$
$$3$$ 0 0
$$4$$ 4.11491 2.05745
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.64207 −0.620645 −0.310323 0.950631i $$-0.600437\pi$$
−0.310323 + 0.950631i $$0.600437\pi$$
$$8$$ 5.22982 1.84902
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.35793 0.710942 0.355471 0.934687i $$-0.384321\pi$$
0.355471 + 0.934687i $$0.384321\pi$$
$$12$$ 0 0
$$13$$ 2.58774 0.717710 0.358855 0.933393i $$-0.383167\pi$$
0.358855 + 0.933393i $$0.383167\pi$$
$$14$$ −4.06058 −1.08523
$$15$$ 0 0
$$16$$ 4.70265 1.17566
$$17$$ 5.87189 1.42414 0.712071 0.702107i $$-0.247757\pi$$
0.712071 + 0.702107i $$0.247757\pi$$
$$18$$ 0 0
$$19$$ 2.94567 0.675783 0.337891 0.941185i $$-0.390286\pi$$
0.337891 + 0.941185i $$0.390286\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 5.83076 1.24312
$$23$$ −2.22982 −0.464949 −0.232474 0.972603i $$-0.574682\pi$$
−0.232474 + 0.972603i $$0.574682\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.39905 1.25496
$$27$$ 0 0
$$28$$ −6.75698 −1.27695
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −2.22982 −0.400487 −0.200243 0.979746i $$-0.564173\pi$$
−0.200243 + 0.979746i $$0.564173\pi$$
$$32$$ 1.16924 0.206694
$$33$$ 0 0
$$34$$ 14.5202 2.49019
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.945668 0.155467 0.0777334 0.996974i $$-0.475232\pi$$
0.0777334 + 0.996974i $$0.475232\pi$$
$$38$$ 7.28415 1.18164
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −0.568295 −0.0887527 −0.0443763 0.999015i $$-0.514130\pi$$
−0.0443763 + 0.999015i $$0.514130\pi$$
$$42$$ 0 0
$$43$$ 2.56829 0.391661 0.195831 0.980638i $$-0.437260\pi$$
0.195831 + 0.980638i $$0.437260\pi$$
$$44$$ 9.70265 1.46273
$$45$$ 0 0
$$46$$ −5.51396 −0.812989
$$47$$ −4.92622 −0.718563 −0.359282 0.933229i $$-0.616978\pi$$
−0.359282 + 0.933229i $$0.616978\pi$$
$$48$$ 0 0
$$49$$ −4.30359 −0.614799
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 10.6483 1.47666
$$53$$ 14.1212 1.93969 0.969845 0.243724i $$-0.0783691\pi$$
0.969845 + 0.243724i $$0.0783691\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −8.58774 −1.14759
$$57$$ 0 0
$$58$$ −2.47283 −0.324699
$$59$$ 7.28415 0.948315 0.474158 0.880440i $$-0.342753\pi$$
0.474158 + 0.880440i $$0.342753\pi$$
$$60$$ 0 0
$$61$$ 4.94567 0.633228 0.316614 0.948554i $$-0.397454\pi$$
0.316614 + 0.948554i $$0.397454\pi$$
$$62$$ −5.51396 −0.700274
$$63$$ 0 0
$$64$$ −6.51396 −0.814245
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −0.926221 −0.113156 −0.0565779 0.998398i $$-0.518019\pi$$
−0.0565779 + 0.998398i $$0.518019\pi$$
$$68$$ 24.1623 2.93011
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 16.1212 1.91323 0.956614 0.291357i $$-0.0941069\pi$$
0.956614 + 0.291357i $$0.0941069\pi$$
$$72$$ 0 0
$$73$$ −0.945668 −0.110682 −0.0553410 0.998468i $$-0.517625\pi$$
−0.0553410 + 0.998468i $$0.517625\pi$$
$$74$$ 2.33848 0.271843
$$75$$ 0 0
$$76$$ 12.1212 1.39039
$$77$$ −3.87189 −0.441243
$$78$$ 0 0
$$79$$ −5.51396 −0.620369 −0.310185 0.950676i $$-0.600391\pi$$
−0.310185 + 0.950676i $$0.600391\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −1.40530 −0.155189
$$83$$ 5.17548 0.568083 0.284042 0.958812i $$-0.408325\pi$$
0.284042 + 0.958812i $$0.408325\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 6.35097 0.684842
$$87$$ 0 0
$$88$$ 12.3315 1.31454
$$89$$ −8.47908 −0.898780 −0.449390 0.893336i $$-0.648359\pi$$
−0.449390 + 0.893336i $$0.648359\pi$$
$$90$$ 0 0
$$91$$ −4.24926 −0.445444
$$92$$ −9.17548 −0.956610
$$93$$ 0 0
$$94$$ −12.1817 −1.25645
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.66152 0.168702 0.0843509 0.996436i $$-0.473118\pi$$
0.0843509 + 0.996436i $$0.473118\pi$$
$$98$$ −10.6421 −1.07501
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 4.01945 0.399950 0.199975 0.979801i $$-0.435914\pi$$
0.199975 + 0.979801i $$0.435914\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 13.5334 1.32706
$$105$$ 0 0
$$106$$ 34.9193 3.39166
$$107$$ 13.9736 1.35088 0.675439 0.737416i $$-0.263954\pi$$
0.675439 + 0.737416i $$0.263954\pi$$
$$108$$ 0 0
$$109$$ −1.41226 −0.135270 −0.0676349 0.997710i $$-0.521545\pi$$
−0.0676349 + 0.997710i $$0.521545\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −7.72210 −0.729669
$$113$$ −3.04737 −0.286673 −0.143336 0.989674i $$-0.545783\pi$$
−0.143336 + 0.989674i $$0.545783\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −4.11491 −0.382060
$$117$$ 0 0
$$118$$ 18.0125 1.65818
$$119$$ −9.64207 −0.883887
$$120$$ 0 0
$$121$$ −5.44018 −0.494562
$$122$$ 12.2298 1.10724
$$123$$ 0 0
$$124$$ −9.17548 −0.823983
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 13.6351 1.20992 0.604960 0.796256i $$-0.293189\pi$$
0.604960 + 0.796256i $$0.293189\pi$$
$$128$$ −18.4464 −1.63045
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −7.99304 −0.698355 −0.349178 0.937057i $$-0.613539\pi$$
−0.349178 + 0.937057i $$0.613539\pi$$
$$132$$ 0 0
$$133$$ −4.83700 −0.419421
$$134$$ −2.29039 −0.197860
$$135$$ 0 0
$$136$$ 30.7089 2.63327
$$137$$ 5.54037 0.473346 0.236673 0.971589i $$-0.423943\pi$$
0.236673 + 0.971589i $$0.423943\pi$$
$$138$$ 0 0
$$139$$ 5.64207 0.478554 0.239277 0.970951i $$-0.423090\pi$$
0.239277 + 0.970951i $$0.423090\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 39.8649 3.34539
$$143$$ 6.10170 0.510250
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.33848 −0.193534
$$147$$ 0 0
$$148$$ 3.89134 0.319866
$$149$$ −11.6351 −0.953186 −0.476593 0.879124i $$-0.658128\pi$$
−0.476593 + 0.879124i $$0.658128\pi$$
$$150$$ 0 0
$$151$$ 16.9193 1.37687 0.688435 0.725298i $$-0.258298\pi$$
0.688435 + 0.725298i $$0.258298\pi$$
$$152$$ 15.4053 1.24953
$$153$$ 0 0
$$154$$ −9.57454 −0.771538
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −16.3510 −1.30495 −0.652475 0.757811i $$-0.726269\pi$$
−0.652475 + 0.757811i $$0.726269\pi$$
$$158$$ −13.6351 −1.08475
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.66152 0.288568
$$162$$ 0 0
$$163$$ 8.03889 0.629655 0.314827 0.949149i $$-0.398053\pi$$
0.314827 + 0.949149i $$0.398053\pi$$
$$164$$ −2.33848 −0.182605
$$165$$ 0 0
$$166$$ 12.7981 0.993326
$$167$$ 24.2423 1.87593 0.937963 0.346736i $$-0.112710\pi$$
0.937963 + 0.346736i $$0.112710\pi$$
$$168$$ 0 0
$$169$$ −6.30359 −0.484892
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 10.5683 0.805825
$$173$$ 15.1755 1.15377 0.576885 0.816825i $$-0.304268\pi$$
0.576885 + 0.816825i $$0.304268\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 11.0885 0.835827
$$177$$ 0 0
$$178$$ −20.9673 −1.57157
$$179$$ −18.6894 −1.39691 −0.698457 0.715652i $$-0.746130\pi$$
−0.698457 + 0.715652i $$0.746130\pi$$
$$180$$ 0 0
$$181$$ −15.0474 −1.11846 −0.559231 0.829012i $$-0.688904\pi$$
−0.559231 + 0.829012i $$0.688904\pi$$
$$182$$ −10.5077 −0.778884
$$183$$ 0 0
$$184$$ −11.6615 −0.859699
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 13.8455 1.01248
$$188$$ −20.2709 −1.47841
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 26.3510 1.90669 0.953344 0.301885i $$-0.0976157\pi$$
0.953344 + 0.301885i $$0.0976157\pi$$
$$192$$ 0 0
$$193$$ −9.32304 −0.671087 −0.335544 0.942025i $$-0.608920\pi$$
−0.335544 + 0.942025i $$0.608920\pi$$
$$194$$ 4.10866 0.294985
$$195$$ 0 0
$$196$$ −17.7089 −1.26492
$$197$$ 3.09323 0.220383 0.110192 0.993910i $$-0.464854\pi$$
0.110192 + 0.993910i $$0.464854\pi$$
$$198$$ 0 0
$$199$$ −6.10170 −0.432538 −0.216269 0.976334i $$-0.569389\pi$$
−0.216269 + 0.976334i $$0.569389\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 9.93942 0.699335
$$203$$ 1.64207 0.115251
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −19.7827 −1.37832
$$207$$ 0 0
$$208$$ 12.1692 0.843785
$$209$$ 6.94567 0.480442
$$210$$ 0 0
$$211$$ −5.17548 −0.356295 −0.178147 0.984004i $$-0.557010\pi$$
−0.178147 + 0.984004i $$0.557010\pi$$
$$212$$ 58.1072 3.99082
$$213$$ 0 0
$$214$$ 34.5544 2.36209
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.66152 0.248560
$$218$$ −3.49228 −0.236527
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 15.1949 1.02212
$$222$$ 0 0
$$223$$ 14.1017 0.944320 0.472160 0.881513i $$-0.343474\pi$$
0.472160 + 0.881513i $$0.343474\pi$$
$$224$$ −1.91998 −0.128284
$$225$$ 0 0
$$226$$ −7.53564 −0.501264
$$227$$ −13.1755 −0.874488 −0.437244 0.899343i $$-0.644045\pi$$
−0.437244 + 0.899343i $$0.644045\pi$$
$$228$$ 0 0
$$229$$ −22.5808 −1.49218 −0.746090 0.665845i $$-0.768071\pi$$
−0.746090 + 0.665845i $$0.768071\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −5.22982 −0.343354
$$233$$ −3.89134 −0.254930 −0.127465 0.991843i $$-0.540684\pi$$
−0.127465 + 0.991843i $$0.540684\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 29.9736 1.95111
$$237$$ 0 0
$$238$$ −23.8432 −1.54553
$$239$$ −9.05433 −0.585676 −0.292838 0.956162i $$-0.594600\pi$$
−0.292838 + 0.956162i $$0.594600\pi$$
$$240$$ 0 0
$$241$$ 11.5070 0.741231 0.370616 0.928786i $$-0.379147\pi$$
0.370616 + 0.928786i $$0.379147\pi$$
$$242$$ −13.4527 −0.864770
$$243$$ 0 0
$$244$$ 20.3510 1.30284
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 7.62263 0.485016
$$248$$ −11.6615 −0.740507
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.7089 1.30713 0.653567 0.756869i $$-0.273272\pi$$
0.653567 + 0.756869i $$0.273272\pi$$
$$252$$ 0 0
$$253$$ −5.25774 −0.330551
$$254$$ 33.7174 2.11562
$$255$$ 0 0
$$256$$ −32.5870 −2.03669
$$257$$ 21.2841 1.32767 0.663834 0.747880i $$-0.268928\pi$$
0.663834 + 0.747880i $$0.268928\pi$$
$$258$$ 0 0
$$259$$ −1.55286 −0.0964898
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −19.7655 −1.22111
$$263$$ −11.7827 −0.726551 −0.363275 0.931682i $$-0.618342\pi$$
−0.363275 + 0.931682i $$0.618342\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −11.9611 −0.733382
$$267$$ 0 0
$$268$$ −3.81131 −0.232813
$$269$$ 7.55982 0.460930 0.230465 0.973081i $$-0.425975\pi$$
0.230465 + 0.973081i $$0.425975\pi$$
$$270$$ 0 0
$$271$$ −18.6072 −1.13031 −0.565153 0.824986i $$-0.691183\pi$$
−0.565153 + 0.824986i $$0.691183\pi$$
$$272$$ 27.6134 1.67431
$$273$$ 0 0
$$274$$ 13.7004 0.827672
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −29.3983 −1.76637 −0.883187 0.469020i $$-0.844607\pi$$
−0.883187 + 0.469020i $$0.844607\pi$$
$$278$$ 13.9519 0.836780
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 7.59622 0.453152 0.226576 0.973993i $$-0.427247\pi$$
0.226576 + 0.973993i $$0.427247\pi$$
$$282$$ 0 0
$$283$$ −25.8913 −1.53908 −0.769540 0.638599i $$-0.779514\pi$$
−0.769540 + 0.638599i $$0.779514\pi$$
$$284$$ 66.3370 3.93638
$$285$$ 0 0
$$286$$ 15.0885 0.892202
$$287$$ 0.933181 0.0550840
$$288$$ 0 0
$$289$$ 17.4791 1.02818
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −3.89134 −0.227723
$$293$$ −23.3036 −1.36141 −0.680705 0.732557i $$-0.738327\pi$$
−0.680705 + 0.732557i $$0.738327\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 4.94567 0.287461
$$297$$ 0 0
$$298$$ −28.7717 −1.66670
$$299$$ −5.77018 −0.333698
$$300$$ 0 0
$$301$$ −4.21733 −0.243083
$$302$$ 41.8385 2.40754
$$303$$ 0 0
$$304$$ 13.8524 0.794492
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −18.0125 −1.02803 −0.514013 0.857782i $$-0.671842\pi$$
−0.514013 + 0.857782i $$0.671842\pi$$
$$308$$ −15.9325 −0.907836
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 12.9262 0.732979 0.366489 0.930422i $$-0.380560\pi$$
0.366489 + 0.930422i $$0.380560\pi$$
$$312$$ 0 0
$$313$$ −4.44018 −0.250974 −0.125487 0.992095i $$-0.540049\pi$$
−0.125487 + 0.992095i $$0.540049\pi$$
$$314$$ −40.4332 −2.28178
$$315$$ 0 0
$$316$$ −22.6894 −1.27638
$$317$$ −19.7632 −1.11001 −0.555007 0.831846i $$-0.687284\pi$$
−0.555007 + 0.831846i $$0.687284\pi$$
$$318$$ 0 0
$$319$$ −2.35793 −0.132019
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 9.05433 0.504578
$$323$$ 17.2966 0.962410
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 19.8788 1.10099
$$327$$ 0 0
$$328$$ −2.97208 −0.164105
$$329$$ 8.08922 0.445973
$$330$$ 0 0
$$331$$ −18.4332 −1.01318 −0.506591 0.862187i $$-0.669094\pi$$
−0.506591 + 0.862187i $$0.669094\pi$$
$$332$$ 21.2966 1.16880
$$333$$ 0 0
$$334$$ 59.9472 3.28016
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −23.2577 −1.26693 −0.633465 0.773771i $$-0.718368\pi$$
−0.633465 + 0.773771i $$0.718368\pi$$
$$338$$ −15.5877 −0.847861
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −5.25774 −0.284723
$$342$$ 0 0
$$343$$ 18.5613 1.00222
$$344$$ 13.4317 0.724189
$$345$$ 0 0
$$346$$ 37.5264 2.01743
$$347$$ −16.5808 −0.890103 −0.445051 0.895505i $$-0.646815\pi$$
−0.445051 + 0.895505i $$0.646815\pi$$
$$348$$ 0 0
$$349$$ −29.0279 −1.55383 −0.776915 0.629606i $$-0.783216\pi$$
−0.776915 + 0.629606i $$0.783216\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.75698 0.146948
$$353$$ −3.25774 −0.173392 −0.0866960 0.996235i $$-0.527631\pi$$
−0.0866960 + 0.996235i $$0.527631\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −34.8906 −1.84920
$$357$$ 0 0
$$358$$ −46.2159 −2.44259
$$359$$ −29.8913 −1.57760 −0.788802 0.614647i $$-0.789298\pi$$
−0.788802 + 0.614647i $$0.789298\pi$$
$$360$$ 0 0
$$361$$ −10.3230 −0.543318
$$362$$ −37.2097 −1.95570
$$363$$ 0 0
$$364$$ −17.4853 −0.916480
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 21.8913 1.14272 0.571359 0.820700i $$-0.306416\pi$$
0.571359 + 0.820700i $$0.306416\pi$$
$$368$$ −10.4860 −0.546622
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −23.1880 −1.20386
$$372$$ 0 0
$$373$$ −8.10866 −0.419851 −0.209925 0.977717i $$-0.567322\pi$$
−0.209925 + 0.977717i $$0.567322\pi$$
$$374$$ 34.2376 1.77038
$$375$$ 0 0
$$376$$ −25.7632 −1.32864
$$377$$ −2.58774 −0.133275
$$378$$ 0 0
$$379$$ −25.7563 −1.32301 −0.661505 0.749941i $$-0.730082\pi$$
−0.661505 + 0.749941i $$0.730082\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 65.1616 3.33396
$$383$$ −24.5419 −1.25403 −0.627016 0.779006i $$-0.715724\pi$$
−0.627016 + 0.779006i $$0.715724\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −23.0543 −1.17343
$$387$$ 0 0
$$388$$ 6.83700 0.347096
$$389$$ −22.5877 −1.14524 −0.572622 0.819820i $$-0.694074\pi$$
−0.572622 + 0.819820i $$0.694074\pi$$
$$390$$ 0 0
$$391$$ −13.0932 −0.662153
$$392$$ −22.5070 −1.13678
$$393$$ 0 0
$$394$$ 7.64903 0.385353
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 11.4317 0.573741 0.286870 0.957969i $$-0.407385\pi$$
0.286870 + 0.957969i $$0.407385\pi$$
$$398$$ −15.0885 −0.756318
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −16.6894 −0.833431 −0.416716 0.909037i $$-0.636819\pi$$
−0.416716 + 0.909037i $$0.636819\pi$$
$$402$$ 0 0
$$403$$ −5.77018 −0.287433
$$404$$ 16.5397 0.822878
$$405$$ 0 0
$$406$$ 4.06058 0.201523
$$407$$ 2.22982 0.110528
$$408$$ 0 0
$$409$$ 5.54037 0.273954 0.136977 0.990574i $$-0.456261\pi$$
0.136977 + 0.990574i $$0.456261\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −32.9193 −1.62182
$$413$$ −11.9611 −0.588568
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 3.02569 0.148347
$$417$$ 0 0
$$418$$ 17.1755 0.840080
$$419$$ 5.05433 0.246920 0.123460 0.992350i $$-0.460601\pi$$
0.123460 + 0.992350i $$0.460601\pi$$
$$420$$ 0 0
$$421$$ 8.86341 0.431976 0.215988 0.976396i $$-0.430703\pi$$
0.215988 + 0.976396i $$0.430703\pi$$
$$422$$ −12.7981 −0.623002
$$423$$ 0 0
$$424$$ 73.8510 3.58652
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −8.12115 −0.393010
$$428$$ 57.5000 2.77937
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −6.56829 −0.316384 −0.158192 0.987408i $$-0.550566\pi$$
−0.158192 + 0.987408i $$0.550566\pi$$
$$432$$ 0 0
$$433$$ 12.6894 0.609816 0.304908 0.952382i $$-0.401374\pi$$
0.304908 + 0.952382i $$0.401374\pi$$
$$434$$ 9.05433 0.434622
$$435$$ 0 0
$$436$$ −5.81131 −0.278311
$$437$$ −6.56829 −0.314204
$$438$$ 0 0
$$439$$ 34.8176 1.66175 0.830876 0.556458i $$-0.187840\pi$$
0.830876 + 0.556458i $$0.187840\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 37.5745 1.78724
$$443$$ 16.9262 0.804189 0.402095 0.915598i $$-0.368282\pi$$
0.402095 + 0.915598i $$0.368282\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 34.8712 1.65120
$$447$$ 0 0
$$448$$ 10.6964 0.505358
$$449$$ 5.15604 0.243328 0.121664 0.992571i $$-0.461177\pi$$
0.121664 + 0.992571i $$0.461177\pi$$
$$450$$ 0 0
$$451$$ −1.34000 −0.0630980
$$452$$ −12.5397 −0.589816
$$453$$ 0 0
$$454$$ −32.5808 −1.52909
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 15.5070 0.725387 0.362693 0.931908i $$-0.381857\pi$$
0.362693 + 0.931908i $$0.381857\pi$$
$$458$$ −55.8385 −2.60916
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ −22.6002 −1.05032 −0.525161 0.851003i $$-0.675995\pi$$
−0.525161 + 0.851003i $$0.675995\pi$$
$$464$$ −4.70265 −0.218315
$$465$$ 0 0
$$466$$ −9.62263 −0.445760
$$467$$ −28.9193 −1.33822 −0.669112 0.743162i $$-0.733325\pi$$
−0.669112 + 0.743162i $$0.733325\pi$$
$$468$$ 0 0
$$469$$ 1.52092 0.0702297
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 38.0947 1.75345
$$473$$ 6.05585 0.278448
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −39.6762 −1.81856
$$477$$ 0 0
$$478$$ −22.3899 −1.02409
$$479$$ −14.8106 −0.676713 −0.338357 0.941018i $$-0.609871\pi$$
−0.338357 + 0.941018i $$0.609871\pi$$
$$480$$ 0 0
$$481$$ 2.44714 0.111580
$$482$$ 28.4549 1.29609
$$483$$ 0 0
$$484$$ −22.3859 −1.01754
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −21.1366 −0.957790 −0.478895 0.877872i $$-0.658963\pi$$
−0.478895 + 0.877872i $$0.658963\pi$$
$$488$$ 25.8649 1.17085
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 27.4876 1.24050 0.620248 0.784406i $$-0.287032\pi$$
0.620248 + 0.784406i $$0.287032\pi$$
$$492$$ 0 0
$$493$$ −5.87189 −0.264457
$$494$$ 18.8495 0.848079
$$495$$ 0 0
$$496$$ −10.4860 −0.470837
$$497$$ −26.4721 −1.18744
$$498$$ 0 0
$$499$$ 17.3470 0.776556 0.388278 0.921542i $$-0.373070\pi$$
0.388278 + 0.921542i $$0.373070\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 51.2097 2.28560
$$503$$ 16.2104 0.722785 0.361392 0.932414i $$-0.382302\pi$$
0.361392 + 0.932414i $$0.382302\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −13.0015 −0.577988
$$507$$ 0 0
$$508$$ 56.1072 2.48936
$$509$$ 30.6241 1.35739 0.678696 0.734420i $$-0.262546\pi$$
0.678696 + 0.734420i $$0.262546\pi$$
$$510$$ 0 0
$$511$$ 1.55286 0.0686943
$$512$$ −43.6894 −1.93082
$$513$$ 0 0
$$514$$ 52.6322 2.32151
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −11.6157 −0.510856
$$518$$ −3.83996 −0.168718
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 5.06682 0.221981 0.110991 0.993821i $$-0.464598\pi$$
0.110991 + 0.993821i $$0.464598\pi$$
$$522$$ 0 0
$$523$$ −14.5613 −0.636723 −0.318361 0.947969i $$-0.603133\pi$$
−0.318361 + 0.947969i $$0.603133\pi$$
$$524$$ −32.8906 −1.43683
$$525$$ 0 0
$$526$$ −29.1366 −1.27042
$$527$$ −13.0932 −0.570350
$$528$$ 0 0
$$529$$ −18.0279 −0.783823
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −19.9038 −0.862940
$$533$$ −1.47060 −0.0636987
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −4.84396 −0.209227
$$537$$ 0 0
$$538$$ 18.6942 0.805963
$$539$$ −10.1476 −0.437086
$$540$$ 0 0
$$541$$ −10.1212 −0.435142 −0.217571 0.976044i $$-0.569813\pi$$
−0.217571 + 0.976044i $$0.569813\pi$$
$$542$$ −46.0125 −1.97641
$$543$$ 0 0
$$544$$ 6.86565 0.294362
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 22.8176 0.975608 0.487804 0.872953i $$-0.337798\pi$$
0.487804 + 0.872953i $$0.337798\pi$$
$$548$$ 22.7981 0.973887
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2.94567 −0.125490
$$552$$ 0 0
$$553$$ 9.05433 0.385029
$$554$$ −72.6972 −3.08861
$$555$$ 0 0
$$556$$ 23.2166 0.984604
$$557$$ 6.67696 0.282912 0.141456 0.989945i $$-0.454822\pi$$
0.141456 + 0.989945i $$0.454822\pi$$
$$558$$ 0 0
$$559$$ 6.64608 0.281099
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 18.7842 0.792363
$$563$$ −39.3161 −1.65698 −0.828488 0.560007i $$-0.810798\pi$$
−0.828488 + 0.560007i $$0.810798\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −64.0250 −2.69117
$$567$$ 0 0
$$568$$ 84.3106 3.53760
$$569$$ −33.1560 −1.38997 −0.694987 0.719023i $$-0.744590\pi$$
−0.694987 + 0.719023i $$0.744590\pi$$
$$570$$ 0 0
$$571$$ 23.0279 0.963689 0.481844 0.876257i $$-0.339967\pi$$
0.481844 + 0.876257i $$0.339967\pi$$
$$572$$ 25.1079 1.04982
$$573$$ 0 0
$$574$$ 2.30760 0.0963175
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −37.5264 −1.56225 −0.781123 0.624377i $$-0.785353\pi$$
−0.781123 + 0.624377i $$0.785353\pi$$
$$578$$ 43.2229 1.79783
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −8.49852 −0.352578
$$582$$ 0 0
$$583$$ 33.2966 1.37901
$$584$$ −4.94567 −0.204653
$$585$$ 0 0
$$586$$ −57.6259 −2.38050
$$587$$ 2.98903 0.123371 0.0616853 0.998096i $$-0.480353\pi$$
0.0616853 + 0.998096i $$0.480353\pi$$
$$588$$ 0 0
$$589$$ −6.56829 −0.270642
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 4.44714 0.182776
$$593$$ 3.55286 0.145898 0.0729492 0.997336i $$-0.476759\pi$$
0.0729492 + 0.997336i $$0.476759\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −47.8774 −1.96114
$$597$$ 0 0
$$598$$ −14.2687 −0.583491
$$599$$ −19.8285 −0.810172 −0.405086 0.914279i $$-0.632758\pi$$
−0.405086 + 0.914279i $$0.632758\pi$$
$$600$$ 0 0
$$601$$ 43.2827 1.76554 0.882769 0.469807i $$-0.155676\pi$$
0.882769 + 0.469807i $$0.155676\pi$$
$$602$$ −10.4288 −0.425044
$$603$$ 0 0
$$604$$ 69.6212 2.83285
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 38.5544 1.56487 0.782437 0.622730i $$-0.213976\pi$$
0.782437 + 0.622730i $$0.213976\pi$$
$$608$$ 3.44419 0.139680
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12.7478 −0.515720
$$612$$ 0 0
$$613$$ −26.1142 −1.05474 −0.527371 0.849635i $$-0.676822\pi$$
−0.527371 + 0.849635i $$0.676822\pi$$
$$614$$ −44.5419 −1.79756
$$615$$ 0 0
$$616$$ −20.2493 −0.815866
$$617$$ 25.7299 1.03585 0.517923 0.855428i $$-0.326706\pi$$
0.517923 + 0.855428i $$0.326706\pi$$
$$618$$ 0 0
$$619$$ −2.94567 −0.118396 −0.0591982 0.998246i $$-0.518854\pi$$
−0.0591982 + 0.998246i $$0.518854\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 31.9644 1.28166
$$623$$ 13.9233 0.557824
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −10.9798 −0.438842
$$627$$ 0 0
$$628$$ −67.2827 −2.68487
$$629$$ 5.55286 0.221407
$$630$$ 0 0
$$631$$ 15.9541 0.635125 0.317562 0.948237i $$-0.397136\pi$$
0.317562 + 0.948237i $$0.397136\pi$$
$$632$$ −28.8370 −1.14707
$$633$$ 0 0
$$634$$ −48.8712 −1.94092
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −11.1366 −0.441248
$$638$$ −5.83076 −0.230842
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 3.98055 0.157222 0.0786112 0.996905i $$-0.474951\pi$$
0.0786112 + 0.996905i $$0.474951\pi$$
$$642$$ 0 0
$$643$$ 9.85940 0.388817 0.194408 0.980921i $$-0.437721\pi$$
0.194408 + 0.980921i $$0.437721\pi$$
$$644$$ 15.0668 0.593716
$$645$$ 0 0
$$646$$ 42.7717 1.68283
$$647$$ 13.2577 0.521216 0.260608 0.965445i $$-0.416077\pi$$
0.260608 + 0.965445i $$0.416077\pi$$
$$648$$ 0 0
$$649$$ 17.1755 0.674197
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 33.0793 1.29549
$$653$$ −38.8689 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.67249 −0.104343
$$657$$ 0 0
$$658$$ 20.0033 0.779809
$$659$$ 36.4527 1.41999 0.709997 0.704204i $$-0.248696\pi$$
0.709997 + 0.704204i $$0.248696\pi$$
$$660$$ 0 0
$$661$$ −26.3176 −1.02364 −0.511818 0.859094i $$-0.671028\pi$$
−0.511818 + 0.859094i $$0.671028\pi$$
$$662$$ −45.5823 −1.77161
$$663$$ 0 0
$$664$$ 27.0668 1.05040
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.22982 0.0863388
$$668$$ 99.7548 3.85963
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 11.6615 0.450188
$$672$$ 0 0
$$673$$ −10.8440 −0.418004 −0.209002 0.977915i $$-0.567022\pi$$
−0.209002 + 0.977915i $$0.567022\pi$$
$$674$$ −57.5125 −2.21530
$$675$$ 0 0
$$676$$ −25.9387 −0.997643
$$677$$ −10.5877 −0.406920 −0.203460 0.979083i $$-0.565219\pi$$
−0.203460 + 0.979083i $$0.565219\pi$$
$$678$$ 0 0
$$679$$ −2.72834 −0.104704
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −13.0015 −0.497854
$$683$$ −18.6461 −0.713473 −0.356736 0.934205i $$-0.616111\pi$$
−0.356736 + 0.934205i $$0.616111\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 45.8991 1.75244
$$687$$ 0 0
$$688$$ 12.0778 0.460461
$$689$$ 36.5419 1.39214
$$690$$ 0 0
$$691$$ −4.04585 −0.153912 −0.0769558 0.997035i $$-0.524520\pi$$
−0.0769558 + 0.997035i $$0.524520\pi$$
$$692$$ 62.4457 2.37383
$$693$$ 0 0
$$694$$ −41.0015 −1.55640
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −3.33696 −0.126396
$$698$$ −71.7812 −2.71696
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −16.7283 −0.631821 −0.315910 0.948789i $$-0.602310\pi$$
−0.315910 + 0.948789i $$0.602310\pi$$
$$702$$ 0 0
$$703$$ 2.78562 0.105062
$$704$$ −15.3594 −0.578881
$$705$$ 0 0
$$706$$ −8.05585 −0.303186
$$707$$ −6.60023 −0.248227
$$708$$ 0 0
$$709$$ 25.7049 0.965367 0.482684 0.875795i $$-0.339662\pi$$
0.482684 + 0.875795i $$0.339662\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −44.3440 −1.66186
$$713$$ 4.97208 0.186206
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −76.9053 −2.87409
$$717$$ 0 0
$$718$$ −73.9163 −2.75853
$$719$$ −42.5155 −1.58556 −0.792780 0.609508i $$-0.791367\pi$$
−0.792780 + 0.609508i $$0.791367\pi$$
$$720$$ 0 0
$$721$$ 13.1366 0.489232
$$722$$ −25.5272 −0.950023
$$723$$ 0 0
$$724$$ −61.9185 −2.30118
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −18.3076 −0.678991 −0.339496 0.940608i $$-0.610256\pi$$
−0.339496 + 0.940608i $$0.610256\pi$$
$$728$$ −22.2229 −0.823634
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 15.0807 0.557781
$$732$$ 0 0
$$733$$ 2.93318 0.108340 0.0541698 0.998532i $$-0.482749\pi$$
0.0541698 + 0.998532i $$0.482749\pi$$
$$734$$ 54.1336 1.99811
$$735$$ 0 0
$$736$$ −2.60719 −0.0961022
$$737$$ −2.18396 −0.0804472
$$738$$ 0 0
$$739$$ 45.2827 1.66575 0.832876 0.553460i $$-0.186693\pi$$
0.832876 + 0.553460i $$0.186693\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −57.3400 −2.10502
$$743$$ 2.56133 0.0939662 0.0469831 0.998896i $$-0.485039\pi$$
0.0469831 + 0.998896i $$0.485039\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −20.0514 −0.734133
$$747$$ 0 0
$$748$$ 56.9729 2.08313
$$749$$ −22.9457 −0.838416
$$750$$ 0 0
$$751$$ −5.13659 −0.187437 −0.0937184 0.995599i $$-0.529875\pi$$
−0.0937184 + 0.995599i $$0.529875\pi$$
$$752$$ −23.1663 −0.844788
$$753$$ 0 0
$$754$$ −6.39905 −0.233040
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −41.5264 −1.50930 −0.754652 0.656125i $$-0.772194\pi$$
−0.754652 + 0.656125i $$0.772194\pi$$
$$758$$ −63.6910 −2.31336
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.8759 1.11925 0.559625 0.828746i $$-0.310945\pi$$
0.559625 + 0.828746i $$0.310945\pi$$
$$762$$ 0 0
$$763$$ 2.31903 0.0839546
$$764$$ 108.432 3.92292
$$765$$ 0 0
$$766$$ −60.6880 −2.19275
$$767$$ 18.8495 0.680616
$$768$$ 0 0
$$769$$ 28.8929 1.04190 0.520951 0.853586i $$-0.325577\pi$$
0.520951 + 0.853586i $$0.325577\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −38.3635 −1.38073
$$773$$ −19.1616 −0.689193 −0.344597 0.938751i $$-0.611984\pi$$
−0.344597 + 0.938751i $$0.611984\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 8.68945 0.311933
$$777$$ 0 0
$$778$$ −55.8557 −2.00253
$$779$$ −1.67401 −0.0599775
$$780$$ 0 0
$$781$$ 38.0125 1.36019
$$782$$ −32.3774 −1.15781
$$783$$ 0 0
$$784$$ −20.2383 −0.722796
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −20.2423 −0.721560 −0.360780 0.932651i $$-0.617489\pi$$
−0.360780 + 0.932651i $$0.617489\pi$$
$$788$$ 12.7283 0.453428
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5.00401 0.177922
$$792$$ 0 0
$$793$$ 12.7981 0.454474
$$794$$ 28.2687 1.00322
$$795$$ 0 0
$$796$$ −25.1079 −0.889928
$$797$$ 4.56829 0.161817 0.0809086 0.996722i $$-0.474218\pi$$
0.0809086 + 0.996722i $$0.474218\pi$$
$$798$$ 0 0
$$799$$ −28.9262 −1.02334
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −41.2702 −1.45730
$$803$$ −2.22982 −0.0786885
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −14.2687 −0.502594
$$807$$ 0 0
$$808$$ 21.0210 0.739515
$$809$$ 9.45115 0.332285 0.166142 0.986102i $$-0.446869\pi$$
0.166142 + 0.986102i $$0.446869\pi$$
$$810$$ 0 0
$$811$$ 38.3051 1.34507 0.672537 0.740063i $$-0.265205\pi$$
0.672537 + 0.740063i $$0.265205\pi$$
$$812$$ 6.75698 0.237124
$$813$$ 0 0
$$814$$ 5.51396 0.193264
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.56534 0.264678
$$818$$ 13.7004 0.479024
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 5.02792 0.175476 0.0877379 0.996144i $$-0.472036\pi$$
0.0877379 + 0.996144i $$0.472036\pi$$
$$822$$ 0 0
$$823$$ 35.1924 1.22673 0.613366 0.789799i $$-0.289815\pi$$
0.613366 + 0.789799i $$0.289815\pi$$
$$824$$ −41.8385 −1.45751
$$825$$ 0 0
$$826$$ −29.5778 −1.02914
$$827$$ 47.2702 1.64375 0.821873 0.569670i $$-0.192929\pi$$
0.821873 + 0.569670i $$0.192929\pi$$
$$828$$ 0 0
$$829$$ 10.2034 0.354379 0.177189 0.984177i $$-0.443299\pi$$
0.177189 + 0.984177i $$0.443299\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −16.8565 −0.584392
$$833$$ −25.2702 −0.875561
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 28.5808 0.988487
$$837$$ 0 0
$$838$$ 12.4985 0.431754
$$839$$ −50.8036 −1.75394 −0.876968 0.480549i $$-0.840438\pi$$
−0.876968 + 0.480549i $$0.840438\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 21.9177 0.755335
$$843$$ 0 0
$$844$$ −21.2966 −0.733060
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 8.93318 0.306948
$$848$$ 66.4068 2.28042
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2.10866 −0.0722841
$$852$$ 0 0
$$853$$ 30.2812 1.03681 0.518404 0.855136i $$-0.326526\pi$$
0.518404 + 0.855136i $$0.326526\pi$$
$$854$$ −20.0823 −0.687201
$$855$$ 0 0
$$856$$ 73.0793 2.49780
$$857$$ −9.66152 −0.330031 −0.165016 0.986291i $$-0.552767\pi$$
−0.165016 + 0.986291i $$0.552767\pi$$
$$858$$ 0 0
$$859$$ 43.7438 1.49252 0.746259 0.665655i $$-0.231848\pi$$
0.746259 + 0.665655i $$0.231848\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −16.2423 −0.553215
$$863$$ −33.7952 −1.15040 −0.575200 0.818013i $$-0.695076\pi$$
−0.575200 + 0.818013i $$0.695076\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 31.3789 1.06630
$$867$$ 0 0
$$868$$ 15.0668 0.511401
$$869$$ −13.0015 −0.441046
$$870$$ 0 0
$$871$$ −2.39682 −0.0812132
$$872$$ −7.38585 −0.250116
$$873$$ 0 0
$$874$$ −16.2423 −0.549404
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 15.3789 0.519308 0.259654 0.965702i $$-0.416391\pi$$
0.259654 + 0.965702i $$0.416391\pi$$
$$878$$ 86.0980 2.90567
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 35.4681 1.19495 0.597475 0.801887i $$-0.296171\pi$$
0.597475 + 0.801887i $$0.296171\pi$$
$$882$$ 0 0
$$883$$ 31.1057 1.04679 0.523395 0.852090i $$-0.324665\pi$$
0.523395 + 0.852090i $$0.324665\pi$$
$$884$$ 62.5257 2.10297
$$885$$ 0 0
$$886$$ 41.8557 1.40617
$$887$$ −3.31608 −0.111343 −0.0556715 0.998449i $$-0.517730\pi$$
−0.0556715 + 0.998449i $$0.517730\pi$$
$$888$$ 0 0
$$889$$ −22.3899 −0.750932
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 58.0272 1.94289
$$893$$ −14.5110 −0.485592
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 30.2904 1.01193
$$897$$ 0 0
$$898$$ 12.7500 0.425474
$$899$$ 2.22982 0.0743685
$$900$$ 0 0
$$901$$ 82.9178 2.76239
$$902$$ −3.31359 −0.110330
$$903$$ 0 0
$$904$$ −15.9372 −0.530063
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −55.5264 −1.84373 −0.921863 0.387517i $$-0.873333\pi$$
−0.921863 + 0.387517i $$0.873333\pi$$
$$908$$ −54.2159 −1.79922
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −31.7757 −1.05278 −0.526388 0.850244i $$-0.676454\pi$$
−0.526388 + 0.850244i $$0.676454\pi$$
$$912$$ 0 0
$$913$$ 12.2034 0.403874
$$914$$ 38.3462 1.26838
$$915$$ 0 0
$$916$$ −92.9178 −3.07009
$$917$$ 13.1252 0.433431
$$918$$ 0 0
$$919$$ −46.5085 −1.53417 −0.767087 0.641543i $$-0.778295\pi$$
−0.767087 + 0.641543i $$0.778295\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −34.6197 −1.14014
$$923$$ 41.7174 1.37314
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −55.8866 −1.83655
$$927$$ 0 0
$$928$$ −1.16924 −0.0383822
$$929$$ 2.97208 0.0975106 0.0487553 0.998811i $$-0.484475\pi$$
0.0487553 + 0.998811i $$0.484475\pi$$
$$930$$ 0 0
$$931$$ −12.6770 −0.415471
$$932$$ −16.0125 −0.524506
$$933$$ 0 0
$$934$$ −71.5125 −2.33996
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −28.9387 −0.945386 −0.472693 0.881227i $$-0.656718\pi$$
−0.472693 + 0.881227i $$0.656718\pi$$
$$938$$ 3.76099 0.122801
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 6.88037 0.224294 0.112147 0.993692i $$-0.464227\pi$$
0.112147 + 0.993692i $$0.464227\pi$$
$$942$$ 0 0
$$943$$ 1.26719 0.0412654
$$944$$ 34.2548 1.11490
$$945$$ 0 0
$$946$$ 14.9751 0.486883
$$947$$ −39.4945 −1.28340 −0.641700 0.766956i $$-0.721770\pi$$
−0.641700 + 0.766956i $$0.721770\pi$$
$$948$$ 0 0
$$949$$ −2.44714 −0.0794376
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −50.4263 −1.63432
$$953$$ 12.5294 0.405867 0.202934 0.979193i $$-0.434952\pi$$
0.202934 + 0.979193i $$0.434952\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −37.2577 −1.20500
$$957$$ 0 0
$$958$$ −36.6241 −1.18327
$$959$$ −9.09770 −0.293780
$$960$$ 0 0
$$961$$ −26.0279 −0.839610
$$962$$ 6.05138 0.195104
$$963$$ 0 0
$$964$$ 47.3502 1.52505
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −6.43322 −0.206878 −0.103439 0.994636i $$-0.532985\pi$$
−0.103439 + 0.994636i $$0.532985\pi$$
$$968$$ −28.4512 −0.914455
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 47.4347 1.52225 0.761127 0.648603i $$-0.224647\pi$$
0.761127 + 0.648603i $$0.224647\pi$$
$$972$$ 0 0
$$973$$ −9.26470 −0.297013
$$974$$ −52.2673 −1.67475
$$975$$ 0 0
$$976$$ 23.2577 0.744462
$$977$$ −46.7064 −1.49427 −0.747135 0.664672i $$-0.768571\pi$$
−0.747135 + 0.664672i $$0.768571\pi$$
$$978$$ 0 0
$$979$$ −19.9930 −0.638980
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 67.9722 2.16908
$$983$$ 20.2173 0.644833 0.322416 0.946598i $$-0.395505\pi$$
0.322416 + 0.946598i $$0.395505\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −14.5202 −0.462418
$$987$$ 0 0
$$988$$ 31.3664 0.997898
$$989$$ −5.72682 −0.182102
$$990$$ 0 0
$$991$$ −23.1127 −0.734198 −0.367099 0.930182i $$-0.619649\pi$$
−0.367099 + 0.930182i $$0.619649\pi$$
$$992$$ −2.60719 −0.0827783
$$993$$ 0 0
$$994$$ −65.4611 −2.07630
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 3.85244 0.122008 0.0610040 0.998138i $$-0.480570\pi$$
0.0610040 + 0.998138i $$0.480570\pi$$
$$998$$ 42.8961 1.35785
$$999$$ 0 0
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 6525.2.a.bg.1.3 3
3.2 odd 2 2175.2.a.t.1.1 3
5.4 even 2 261.2.a.e.1.1 3
15.2 even 4 2175.2.c.l.349.1 6
15.8 even 4 2175.2.c.l.349.6 6
15.14 odd 2 87.2.a.b.1.3 3
20.19 odd 2 4176.2.a.bx.1.3 3
60.59 even 2 1392.2.a.u.1.1 3
105.104 even 2 4263.2.a.m.1.3 3
120.29 odd 2 5568.2.a.cb.1.3 3
120.59 even 2 5568.2.a.bx.1.3 3
145.144 even 2 7569.2.a.t.1.3 3
435.434 odd 2 2523.2.a.h.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.a.b.1.3 3 15.14 odd 2
261.2.a.e.1.1 3 5.4 even 2
1392.2.a.u.1.1 3 60.59 even 2
2175.2.a.t.1.1 3 3.2 odd 2
2175.2.c.l.349.1 6 15.2 even 4
2175.2.c.l.349.6 6 15.8 even 4
2523.2.a.h.1.1 3 435.434 odd 2
4176.2.a.bx.1.3 3 20.19 odd 2
4263.2.a.m.1.3 3 105.104 even 2
5568.2.a.bx.1.3 3 120.59 even 2
5568.2.a.cb.1.3 3 120.29 odd 2
6525.2.a.bg.1.3 3 1.1 even 1 trivial
7569.2.a.t.1.3 3 145.144 even 2