# Properties

 Label 6525.2.a.cd.1.1 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $9$ 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: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10$$ x^9 - 2*x^8 - 12*x^7 + 21*x^6 + 48*x^5 - 68*x^4 - 73*x^3 + 66*x^2 + 40*x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.21081$$ 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)

 $$f(q)$$ $$=$$ $$q-2.21081 q^{2} +2.88766 q^{4} -1.31646 q^{7} -1.96245 q^{8} +O(q^{10})$$ $$q-2.21081 q^{2} +2.88766 q^{4} -1.31646 q^{7} -1.96245 q^{8} -4.29189 q^{11} -2.97543 q^{13} +2.91044 q^{14} -1.43674 q^{16} +0.642441 q^{17} -5.07923 q^{19} +9.48853 q^{22} +8.84038 q^{23} +6.57809 q^{26} -3.80150 q^{28} +1.00000 q^{29} -6.27016 q^{31} +7.10123 q^{32} -1.42031 q^{34} +0.934054 q^{37} +11.2292 q^{38} -11.0792 q^{41} +2.03981 q^{43} -12.3935 q^{44} -19.5444 q^{46} +9.57178 q^{47} -5.26692 q^{49} -8.59203 q^{52} -13.0891 q^{53} +2.58349 q^{56} -2.21081 q^{58} +2.55069 q^{59} +8.46575 q^{61} +13.8621 q^{62} -12.8260 q^{64} -12.9808 q^{67} +1.85515 q^{68} -5.10419 q^{71} -16.0007 q^{73} -2.06501 q^{74} -14.6671 q^{76} +5.65012 q^{77} -3.30518 q^{79} +24.4939 q^{82} +5.24640 q^{83} -4.50962 q^{86} +8.42260 q^{88} +5.05088 q^{89} +3.91704 q^{91} +25.5280 q^{92} -21.1614 q^{94} -2.04414 q^{97} +11.6441 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + 2 q^{2} + 10 q^{4} + q^{7} + 9 q^{8}+O(q^{10})$$ 9 * q + 2 * q^2 + 10 * q^4 + q^7 + 9 * q^8 $$9 q + 2 q^{2} + 10 q^{4} + q^{7} + 9 q^{8} + 2 q^{11} + q^{13} - 3 q^{14} + 4 q^{16} + 12 q^{17} - q^{19} + 3 q^{22} + 16 q^{23} + 6 q^{26} - 4 q^{28} + 9 q^{29} + 5 q^{31} + 20 q^{32} + 3 q^{34} + 30 q^{38} - 10 q^{41} + 3 q^{43} - 13 q^{44} + 4 q^{46} + 26 q^{47} - 8 q^{49} - 9 q^{52} + 22 q^{53} + 22 q^{56} + 2 q^{58} + 4 q^{59} + 7 q^{61} + 28 q^{62} + 9 q^{64} + 5 q^{67} + 39 q^{68} - 10 q^{73} - 34 q^{74} - 2 q^{76} + 34 q^{77} + 10 q^{79} - 8 q^{82} + 46 q^{83} + 28 q^{86} + 2 q^{88} + 4 q^{89} - 21 q^{91} + 20 q^{92} + 5 q^{94} + 7 q^{97} + 51 q^{98}+O(q^{100})$$ 9 * q + 2 * q^2 + 10 * q^4 + q^7 + 9 * q^8 + 2 * q^11 + q^13 - 3 * q^14 + 4 * q^16 + 12 * q^17 - q^19 + 3 * q^22 + 16 * q^23 + 6 * q^26 - 4 * q^28 + 9 * q^29 + 5 * q^31 + 20 * q^32 + 3 * q^34 + 30 * q^38 - 10 * q^41 + 3 * q^43 - 13 * q^44 + 4 * q^46 + 26 * q^47 - 8 * q^49 - 9 * q^52 + 22 * q^53 + 22 * q^56 + 2 * q^58 + 4 * q^59 + 7 * q^61 + 28 * q^62 + 9 * q^64 + 5 * q^67 + 39 * q^68 - 10 * q^73 - 34 * q^74 - 2 * q^76 + 34 * q^77 + 10 * q^79 - 8 * q^82 + 46 * q^83 + 28 * q^86 + 2 * q^88 + 4 * q^89 - 21 * q^91 + 20 * q^92 + 5 * q^94 + 7 * q^97 + 51 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.21081 −1.56328 −0.781638 0.623733i $$-0.785615\pi$$
−0.781638 + 0.623733i $$0.785615\pi$$
$$3$$ 0 0
$$4$$ 2.88766 1.44383
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.31646 −0.497576 −0.248788 0.968558i $$-0.580032\pi$$
−0.248788 + 0.968558i $$0.580032\pi$$
$$8$$ −1.96245 −0.693830
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.29189 −1.29405 −0.647027 0.762467i $$-0.723988\pi$$
−0.647027 + 0.762467i $$0.723988\pi$$
$$12$$ 0 0
$$13$$ −2.97543 −0.825235 −0.412617 0.910904i $$-0.635385\pi$$
−0.412617 + 0.910904i $$0.635385\pi$$
$$14$$ 2.91044 0.777849
$$15$$ 0 0
$$16$$ −1.43674 −0.359184
$$17$$ 0.642441 0.155815 0.0779074 0.996961i $$-0.475176\pi$$
0.0779074 + 0.996961i $$0.475176\pi$$
$$18$$ 0 0
$$19$$ −5.07923 −1.16525 −0.582627 0.812740i $$-0.697975\pi$$
−0.582627 + 0.812740i $$0.697975\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 9.48853 2.02296
$$23$$ 8.84038 1.84335 0.921674 0.387966i $$-0.126822\pi$$
0.921674 + 0.387966i $$0.126822\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.57809 1.29007
$$27$$ 0 0
$$28$$ −3.80150 −0.718416
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −6.27016 −1.12615 −0.563077 0.826404i $$-0.690383\pi$$
−0.563077 + 0.826404i $$0.690383\pi$$
$$32$$ 7.10123 1.25533
$$33$$ 0 0
$$34$$ −1.42031 −0.243582
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.934054 0.153558 0.0767788 0.997048i $$-0.475536\pi$$
0.0767788 + 0.997048i $$0.475536\pi$$
$$38$$ 11.2292 1.82161
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −11.0792 −1.73028 −0.865139 0.501533i $$-0.832770\pi$$
−0.865139 + 0.501533i $$0.832770\pi$$
$$42$$ 0 0
$$43$$ 2.03981 0.311068 0.155534 0.987831i $$-0.450290\pi$$
0.155534 + 0.987831i $$0.450290\pi$$
$$44$$ −12.3935 −1.86839
$$45$$ 0 0
$$46$$ −19.5444 −2.88166
$$47$$ 9.57178 1.39619 0.698094 0.716006i $$-0.254032\pi$$
0.698094 + 0.716006i $$0.254032\pi$$
$$48$$ 0 0
$$49$$ −5.26692 −0.752418
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −8.59203 −1.19150
$$53$$ −13.0891 −1.79792 −0.898960 0.438030i $$-0.855676\pi$$
−0.898960 + 0.438030i $$0.855676\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 2.58349 0.345233
$$57$$ 0 0
$$58$$ −2.21081 −0.290293
$$59$$ 2.55069 0.332071 0.166036 0.986120i $$-0.446903\pi$$
0.166036 + 0.986120i $$0.446903\pi$$
$$60$$ 0 0
$$61$$ 8.46575 1.08393 0.541964 0.840402i $$-0.317681\pi$$
0.541964 + 0.840402i $$0.317681\pi$$
$$62$$ 13.8621 1.76049
$$63$$ 0 0
$$64$$ −12.8260 −1.60325
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −12.9808 −1.58586 −0.792930 0.609313i $$-0.791445\pi$$
−0.792930 + 0.609313i $$0.791445\pi$$
$$68$$ 1.85515 0.224970
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −5.10419 −0.605756 −0.302878 0.953029i $$-0.597947\pi$$
−0.302878 + 0.953029i $$0.597947\pi$$
$$72$$ 0 0
$$73$$ −16.0007 −1.87274 −0.936372 0.351009i $$-0.885839\pi$$
−0.936372 + 0.351009i $$0.885839\pi$$
$$74$$ −2.06501 −0.240053
$$75$$ 0 0
$$76$$ −14.6671 −1.68243
$$77$$ 5.65012 0.643890
$$78$$ 0 0
$$79$$ −3.30518 −0.371861 −0.185931 0.982563i $$-0.559530\pi$$
−0.185931 + 0.982563i $$0.559530\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 24.4939 2.70490
$$83$$ 5.24640 0.575867 0.287934 0.957650i $$-0.407032\pi$$
0.287934 + 0.957650i $$0.407032\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.50962 −0.486284
$$87$$ 0 0
$$88$$ 8.42260 0.897853
$$89$$ 5.05088 0.535392 0.267696 0.963503i $$-0.413738\pi$$
0.267696 + 0.963503i $$0.413738\pi$$
$$90$$ 0 0
$$91$$ 3.91704 0.410617
$$92$$ 25.5280 2.66148
$$93$$ 0 0
$$94$$ −21.1614 −2.18263
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.04414 −0.207551 −0.103776 0.994601i $$-0.533092\pi$$
−0.103776 + 0.994601i $$0.533092\pi$$
$$98$$ 11.6441 1.17624
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.4871 1.24251 0.621255 0.783609i $$-0.286623\pi$$
0.621255 + 0.783609i $$0.286623\pi$$
$$102$$ 0 0
$$103$$ −10.6723 −1.05157 −0.525785 0.850617i $$-0.676228\pi$$
−0.525785 + 0.850617i $$0.676228\pi$$
$$104$$ 5.83912 0.572572
$$105$$ 0 0
$$106$$ 28.9374 2.81065
$$107$$ −4.85956 −0.469791 −0.234896 0.972021i $$-0.575475\pi$$
−0.234896 + 0.972021i $$0.575475\pi$$
$$108$$ 0 0
$$109$$ 15.3415 1.46945 0.734723 0.678367i $$-0.237312\pi$$
0.734723 + 0.678367i $$0.237312\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.89141 0.178721
$$113$$ 4.32180 0.406560 0.203280 0.979121i $$-0.434840\pi$$
0.203280 + 0.979121i $$0.434840\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.88766 0.268113
$$117$$ 0 0
$$118$$ −5.63908 −0.519119
$$119$$ −0.845750 −0.0775298
$$120$$ 0 0
$$121$$ 7.42032 0.674575
$$122$$ −18.7161 −1.69448
$$123$$ 0 0
$$124$$ −18.1061 −1.62598
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 21.5809 1.91500 0.957498 0.288440i $$-0.0931366\pi$$
0.957498 + 0.288440i $$0.0931366\pi$$
$$128$$ 14.1533 1.25098
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −9.25658 −0.808751 −0.404376 0.914593i $$-0.632511\pi$$
−0.404376 + 0.914593i $$0.632511\pi$$
$$132$$ 0 0
$$133$$ 6.68661 0.579803
$$134$$ 28.6981 2.47914
$$135$$ 0 0
$$136$$ −1.26076 −0.108109
$$137$$ −0.843767 −0.0720879 −0.0360439 0.999350i $$-0.511476\pi$$
−0.0360439 + 0.999350i $$0.511476\pi$$
$$138$$ 0 0
$$139$$ −13.9777 −1.18557 −0.592785 0.805361i $$-0.701972\pi$$
−0.592785 + 0.805361i $$0.701972\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 11.2844 0.946964
$$143$$ 12.7702 1.06790
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 35.3745 2.92762
$$147$$ 0 0
$$148$$ 2.69723 0.221711
$$149$$ −19.6274 −1.60794 −0.803971 0.594669i $$-0.797283\pi$$
−0.803971 + 0.594669i $$0.797283\pi$$
$$150$$ 0 0
$$151$$ −10.7559 −0.875301 −0.437650 0.899145i $$-0.644189\pi$$
−0.437650 + 0.899145i $$0.644189\pi$$
$$152$$ 9.96771 0.808488
$$153$$ 0 0
$$154$$ −12.4913 −1.00658
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 23.5513 1.87960 0.939800 0.341724i $$-0.111011\pi$$
0.939800 + 0.341724i $$0.111011\pi$$
$$158$$ 7.30710 0.581322
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −11.6380 −0.917206
$$162$$ 0 0
$$163$$ 9.50148 0.744213 0.372107 0.928190i $$-0.378636\pi$$
0.372107 + 0.928190i $$0.378636\pi$$
$$164$$ −31.9929 −2.49823
$$165$$ 0 0
$$166$$ −11.5988 −0.900239
$$167$$ −1.82786 −0.141444 −0.0707220 0.997496i $$-0.522530\pi$$
−0.0707220 + 0.997496i $$0.522530\pi$$
$$168$$ 0 0
$$169$$ −4.14683 −0.318987
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 5.89027 0.449129
$$173$$ 5.50734 0.418715 0.209358 0.977839i $$-0.432863\pi$$
0.209358 + 0.977839i $$0.432863\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6.16631 0.464803
$$177$$ 0 0
$$178$$ −11.1665 −0.836966
$$179$$ −9.36470 −0.699950 −0.349975 0.936759i $$-0.613810\pi$$
−0.349975 + 0.936759i $$0.613810\pi$$
$$180$$ 0 0
$$181$$ 25.6019 1.90297 0.951487 0.307688i $$-0.0995552\pi$$
0.951487 + 0.307688i $$0.0995552\pi$$
$$182$$ −8.65981 −0.641908
$$183$$ 0 0
$$184$$ −17.3488 −1.27897
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −2.75729 −0.201633
$$188$$ 27.6401 2.01586
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 12.3658 0.894756 0.447378 0.894345i $$-0.352358\pi$$
0.447378 + 0.894345i $$0.352358\pi$$
$$192$$ 0 0
$$193$$ −15.7223 −1.13172 −0.565859 0.824502i $$-0.691455\pi$$
−0.565859 + 0.824502i $$0.691455\pi$$
$$194$$ 4.51920 0.324459
$$195$$ 0 0
$$196$$ −15.2091 −1.08636
$$197$$ 21.6728 1.54412 0.772061 0.635549i $$-0.219226\pi$$
0.772061 + 0.635549i $$0.219226\pi$$
$$198$$ 0 0
$$199$$ −1.23732 −0.0877116 −0.0438558 0.999038i $$-0.513964\pi$$
−0.0438558 + 0.999038i $$0.513964\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −27.6065 −1.94238
$$203$$ −1.31646 −0.0923976
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 23.5943 1.64389
$$207$$ 0 0
$$208$$ 4.27490 0.296411
$$209$$ 21.7995 1.50790
$$210$$ 0 0
$$211$$ −5.05346 −0.347895 −0.173947 0.984755i $$-0.555652\pi$$
−0.173947 + 0.984755i $$0.555652\pi$$
$$212$$ −37.7968 −2.59589
$$213$$ 0 0
$$214$$ 10.7435 0.734413
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 8.25444 0.560348
$$218$$ −33.9170 −2.29715
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1.91154 −0.128584
$$222$$ 0 0
$$223$$ −16.2248 −1.08649 −0.543245 0.839574i $$-0.682805\pi$$
−0.543245 + 0.839574i $$0.682805\pi$$
$$224$$ −9.34851 −0.624624
$$225$$ 0 0
$$226$$ −9.55465 −0.635566
$$227$$ 6.82690 0.453117 0.226559 0.973998i $$-0.427253\pi$$
0.226559 + 0.973998i $$0.427253\pi$$
$$228$$ 0 0
$$229$$ −5.87121 −0.387980 −0.193990 0.981003i $$-0.562143\pi$$
−0.193990 + 0.981003i $$0.562143\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1.96245 −0.128841
$$233$$ −14.1453 −0.926692 −0.463346 0.886178i $$-0.653351\pi$$
−0.463346 + 0.886178i $$0.653351\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 7.36553 0.479455
$$237$$ 0 0
$$238$$ 1.86979 0.121200
$$239$$ 20.6920 1.33845 0.669226 0.743059i $$-0.266626\pi$$
0.669226 + 0.743059i $$0.266626\pi$$
$$240$$ 0 0
$$241$$ −7.90845 −0.509428 −0.254714 0.967016i $$-0.581981\pi$$
−0.254714 + 0.967016i $$0.581981\pi$$
$$242$$ −16.4049 −1.05455
$$243$$ 0 0
$$244$$ 24.4462 1.56501
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 15.1129 0.961609
$$248$$ 12.3049 0.781359
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −11.8140 −0.745691 −0.372846 0.927893i $$-0.621618\pi$$
−0.372846 + 0.927893i $$0.621618\pi$$
$$252$$ 0 0
$$253$$ −37.9420 −2.38539
$$254$$ −47.7112 −2.99367
$$255$$ 0 0
$$256$$ −5.63818 −0.352386
$$257$$ −4.35440 −0.271620 −0.135810 0.990735i $$-0.543364\pi$$
−0.135810 + 0.990735i $$0.543364\pi$$
$$258$$ 0 0
$$259$$ −1.22965 −0.0764066
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 20.4645 1.26430
$$263$$ 27.6727 1.70637 0.853187 0.521605i $$-0.174666\pi$$
0.853187 + 0.521605i $$0.174666\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −14.7828 −0.906392
$$267$$ 0 0
$$268$$ −37.4842 −2.28971
$$269$$ 6.32037 0.385360 0.192680 0.981262i $$-0.438282\pi$$
0.192680 + 0.981262i $$0.438282\pi$$
$$270$$ 0 0
$$271$$ 4.37191 0.265575 0.132787 0.991145i $$-0.457607\pi$$
0.132787 + 0.991145i $$0.457607\pi$$
$$272$$ −0.923018 −0.0559662
$$273$$ 0 0
$$274$$ 1.86540 0.112693
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −4.01638 −0.241321 −0.120660 0.992694i $$-0.538501\pi$$
−0.120660 + 0.992694i $$0.538501\pi$$
$$278$$ 30.9019 1.85337
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 13.6990 0.817215 0.408607 0.912710i $$-0.366015\pi$$
0.408607 + 0.912710i $$0.366015\pi$$
$$282$$ 0 0
$$283$$ −14.8058 −0.880114 −0.440057 0.897970i $$-0.645042\pi$$
−0.440057 + 0.897970i $$0.645042\pi$$
$$284$$ −14.7392 −0.874609
$$285$$ 0 0
$$286$$ −28.2324 −1.66942
$$287$$ 14.5853 0.860945
$$288$$ 0 0
$$289$$ −16.5873 −0.975722
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −46.2047 −2.70393
$$293$$ −16.6890 −0.974983 −0.487492 0.873128i $$-0.662088\pi$$
−0.487492 + 0.873128i $$0.662088\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1.83303 −0.106543
$$297$$ 0 0
$$298$$ 43.3924 2.51366
$$299$$ −26.3039 −1.52119
$$300$$ 0 0
$$301$$ −2.68533 −0.154780
$$302$$ 23.7792 1.36834
$$303$$ 0 0
$$304$$ 7.29750 0.418540
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.97073 0.283695 0.141847 0.989889i $$-0.454696\pi$$
0.141847 + 0.989889i $$0.454696\pi$$
$$308$$ 16.3156 0.929669
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 21.0786 1.19526 0.597629 0.801773i $$-0.296110\pi$$
0.597629 + 0.801773i $$0.296110\pi$$
$$312$$ 0 0
$$313$$ 24.3744 1.37772 0.688860 0.724894i $$-0.258111\pi$$
0.688860 + 0.724894i $$0.258111\pi$$
$$314$$ −52.0674 −2.93833
$$315$$ 0 0
$$316$$ −9.54423 −0.536905
$$317$$ −13.4102 −0.753194 −0.376597 0.926377i $$-0.622906\pi$$
−0.376597 + 0.926377i $$0.622906\pi$$
$$318$$ 0 0
$$319$$ −4.29189 −0.240300
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 25.7294 1.43385
$$323$$ −3.26310 −0.181564
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −21.0059 −1.16341
$$327$$ 0 0
$$328$$ 21.7423 1.20052
$$329$$ −12.6009 −0.694710
$$330$$ 0 0
$$331$$ 23.4187 1.28721 0.643603 0.765360i $$-0.277439\pi$$
0.643603 + 0.765360i $$0.277439\pi$$
$$332$$ 15.1498 0.831455
$$333$$ 0 0
$$334$$ 4.04104 0.221116
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.7865 1.24126 0.620631 0.784103i $$-0.286877\pi$$
0.620631 + 0.784103i $$0.286877\pi$$
$$338$$ 9.16784 0.498665
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 26.9108 1.45730
$$342$$ 0 0
$$343$$ 16.1490 0.871962
$$344$$ −4.00301 −0.215828
$$345$$ 0 0
$$346$$ −12.1757 −0.654567
$$347$$ 4.98513 0.267616 0.133808 0.991007i $$-0.457279\pi$$
0.133808 + 0.991007i $$0.457279\pi$$
$$348$$ 0 0
$$349$$ 25.6783 1.37453 0.687264 0.726407i $$-0.258811\pi$$
0.687264 + 0.726407i $$0.258811\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −30.4777 −1.62447
$$353$$ 20.3160 1.08131 0.540656 0.841244i $$-0.318176\pi$$
0.540656 + 0.841244i $$0.318176\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 14.5852 0.773016
$$357$$ 0 0
$$358$$ 20.7035 1.09422
$$359$$ 9.68907 0.511370 0.255685 0.966760i $$-0.417699\pi$$
0.255685 + 0.966760i $$0.417699\pi$$
$$360$$ 0 0
$$361$$ 6.79854 0.357818
$$362$$ −56.6008 −2.97487
$$363$$ 0 0
$$364$$ 11.3111 0.592862
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 9.16319 0.478315 0.239157 0.970981i $$-0.423129\pi$$
0.239157 + 0.970981i $$0.423129\pi$$
$$368$$ −12.7013 −0.662101
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 17.2313 0.894603
$$372$$ 0 0
$$373$$ −13.5773 −0.703007 −0.351504 0.936186i $$-0.614330\pi$$
−0.351504 + 0.936186i $$0.614330\pi$$
$$374$$ 6.09582 0.315208
$$375$$ 0 0
$$376$$ −18.7841 −0.968717
$$377$$ −2.97543 −0.153242
$$378$$ 0 0
$$379$$ 0.0601424 0.00308931 0.00154465 0.999999i $$-0.499508\pi$$
0.00154465 + 0.999999i $$0.499508\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −27.3383 −1.39875
$$383$$ −9.71642 −0.496486 −0.248243 0.968698i $$-0.579853\pi$$
−0.248243 + 0.968698i $$0.579853\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 34.7590 1.76919
$$387$$ 0 0
$$388$$ −5.90279 −0.299669
$$389$$ −5.68991 −0.288490 −0.144245 0.989542i $$-0.546075\pi$$
−0.144245 + 0.989542i $$0.546075\pi$$
$$390$$ 0 0
$$391$$ 5.67943 0.287221
$$392$$ 10.3361 0.522050
$$393$$ 0 0
$$394$$ −47.9143 −2.41389
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6.80729 0.341648 0.170824 0.985302i $$-0.445357\pi$$
0.170824 + 0.985302i $$0.445357\pi$$
$$398$$ 2.73548 0.137117
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −11.7255 −0.585543 −0.292771 0.956182i $$-0.594578\pi$$
−0.292771 + 0.956182i $$0.594578\pi$$
$$402$$ 0 0
$$403$$ 18.6564 0.929342
$$404$$ 36.0584 1.79397
$$405$$ 0 0
$$406$$ 2.91044 0.144443
$$407$$ −4.00886 −0.198712
$$408$$ 0 0
$$409$$ −22.2665 −1.10101 −0.550505 0.834832i $$-0.685565\pi$$
−0.550505 + 0.834832i $$0.685565\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −30.8179 −1.51829
$$413$$ −3.35789 −0.165231
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −21.1292 −1.03594
$$417$$ 0 0
$$418$$ −48.1944 −2.35727
$$419$$ 28.2149 1.37839 0.689194 0.724577i $$-0.257965\pi$$
0.689194 + 0.724577i $$0.257965\pi$$
$$420$$ 0 0
$$421$$ −34.5167 −1.68224 −0.841121 0.540847i $$-0.818104\pi$$
−0.841121 + 0.540847i $$0.818104\pi$$
$$422$$ 11.1722 0.543855
$$423$$ 0 0
$$424$$ 25.6866 1.24745
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −11.1449 −0.539337
$$428$$ −14.0328 −0.678299
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −23.0678 −1.11114 −0.555568 0.831471i $$-0.687499\pi$$
−0.555568 + 0.831471i $$0.687499\pi$$
$$432$$ 0 0
$$433$$ −17.8519 −0.857908 −0.428954 0.903326i $$-0.641118\pi$$
−0.428954 + 0.903326i $$0.641118\pi$$
$$434$$ −18.2490 −0.875978
$$435$$ 0 0
$$436$$ 44.3010 2.12163
$$437$$ −44.9023 −2.14797
$$438$$ 0 0
$$439$$ −3.31655 −0.158290 −0.0791452 0.996863i $$-0.525219\pi$$
−0.0791452 + 0.996863i $$0.525219\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 4.22604 0.201012
$$443$$ 29.8818 1.41973 0.709863 0.704340i $$-0.248757\pi$$
0.709863 + 0.704340i $$0.248757\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 35.8698 1.69848
$$447$$ 0 0
$$448$$ 16.8849 0.797738
$$449$$ −8.89198 −0.419639 −0.209819 0.977740i $$-0.567288\pi$$
−0.209819 + 0.977740i $$0.567288\pi$$
$$450$$ 0 0
$$451$$ 47.5506 2.23907
$$452$$ 12.4799 0.587004
$$453$$ 0 0
$$454$$ −15.0929 −0.708347
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −19.8853 −0.930194 −0.465097 0.885260i $$-0.653981\pi$$
−0.465097 + 0.885260i $$0.653981\pi$$
$$458$$ 12.9801 0.606520
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 13.5776 0.632372 0.316186 0.948697i $$-0.397598\pi$$
0.316186 + 0.948697i $$0.397598\pi$$
$$462$$ 0 0
$$463$$ 29.7645 1.38328 0.691638 0.722244i $$-0.256889\pi$$
0.691638 + 0.722244i $$0.256889\pi$$
$$464$$ −1.43674 −0.0666988
$$465$$ 0 0
$$466$$ 31.2726 1.44867
$$467$$ 6.56767 0.303915 0.151958 0.988387i $$-0.451442\pi$$
0.151958 + 0.988387i $$0.451442\pi$$
$$468$$ 0 0
$$469$$ 17.0888 0.789086
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −5.00559 −0.230401
$$473$$ −8.75463 −0.402538
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.44224 −0.111940
$$477$$ 0 0
$$478$$ −45.7459 −2.09237
$$479$$ −36.9147 −1.68668 −0.843338 0.537383i $$-0.819413\pi$$
−0.843338 + 0.537383i $$0.819413\pi$$
$$480$$ 0 0
$$481$$ −2.77921 −0.126721
$$482$$ 17.4840 0.796376
$$483$$ 0 0
$$484$$ 21.4274 0.973972
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 24.4257 1.10684 0.553418 0.832904i $$-0.313323\pi$$
0.553418 + 0.832904i $$0.313323\pi$$
$$488$$ −16.6136 −0.752062
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0.828059 0.0373698 0.0186849 0.999825i $$-0.494052\pi$$
0.0186849 + 0.999825i $$0.494052\pi$$
$$492$$ 0 0
$$493$$ 0.642441 0.0289341
$$494$$ −33.4116 −1.50326
$$495$$ 0 0
$$496$$ 9.00856 0.404496
$$497$$ 6.71948 0.301410
$$498$$ 0 0
$$499$$ 32.7413 1.46570 0.732850 0.680390i $$-0.238189\pi$$
0.732850 + 0.680390i $$0.238189\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 26.1184 1.16572
$$503$$ 3.04972 0.135980 0.0679901 0.997686i $$-0.478341\pi$$
0.0679901 + 0.997686i $$0.478341\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 83.8823 3.72902
$$507$$ 0 0
$$508$$ 62.3183 2.76493
$$509$$ −11.2666 −0.499384 −0.249692 0.968325i $$-0.580329\pi$$
−0.249692 + 0.968325i $$0.580329\pi$$
$$510$$ 0 0
$$511$$ 21.0644 0.931833
$$512$$ −15.8416 −0.700107
$$513$$ 0 0
$$514$$ 9.62673 0.424617
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −41.0810 −1.80674
$$518$$ 2.71851 0.119445
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 1.39366 0.0610573 0.0305287 0.999534i $$-0.490281\pi$$
0.0305287 + 0.999534i $$0.490281\pi$$
$$522$$ 0 0
$$523$$ −10.6212 −0.464434 −0.232217 0.972664i $$-0.574598\pi$$
−0.232217 + 0.972664i $$0.574598\pi$$
$$524$$ −26.7299 −1.16770
$$525$$ 0 0
$$526$$ −61.1791 −2.66753
$$527$$ −4.02821 −0.175472
$$528$$ 0 0
$$529$$ 55.1524 2.39793
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 19.3087 0.837137
$$533$$ 32.9653 1.42789
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 25.4742 1.10032
$$537$$ 0 0
$$538$$ −13.9731 −0.602424
$$539$$ 22.6051 0.973669
$$540$$ 0 0
$$541$$ 31.1071 1.33740 0.668698 0.743534i $$-0.266852\pi$$
0.668698 + 0.743534i $$0.266852\pi$$
$$542$$ −9.66544 −0.415166
$$543$$ 0 0
$$544$$ 4.56212 0.195599
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −17.0828 −0.730408 −0.365204 0.930927i $$-0.619001\pi$$
−0.365204 + 0.930927i $$0.619001\pi$$
$$548$$ −2.43651 −0.104083
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5.07923 −0.216382
$$552$$ 0 0
$$553$$ 4.35114 0.185029
$$554$$ 8.87942 0.377251
$$555$$ 0 0
$$556$$ −40.3628 −1.71176
$$557$$ −15.1831 −0.643329 −0.321665 0.946854i $$-0.604242\pi$$
−0.321665 + 0.946854i $$0.604242\pi$$
$$558$$ 0 0
$$559$$ −6.06930 −0.256704
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −30.2859 −1.27753
$$563$$ −34.9344 −1.47231 −0.736154 0.676814i $$-0.763360\pi$$
−0.736154 + 0.676814i $$0.763360\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 32.7328 1.37586
$$567$$ 0 0
$$568$$ 10.0167 0.420291
$$569$$ 2.51836 0.105575 0.0527875 0.998606i $$-0.483189\pi$$
0.0527875 + 0.998606i $$0.483189\pi$$
$$570$$ 0 0
$$571$$ 18.9130 0.791485 0.395742 0.918362i $$-0.370487\pi$$
0.395742 + 0.918362i $$0.370487\pi$$
$$572$$ 36.8760 1.54186
$$573$$ 0 0
$$574$$ −32.2453 −1.34589
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −4.64049 −0.193186 −0.0965930 0.995324i $$-0.530795\pi$$
−0.0965930 + 0.995324i $$0.530795\pi$$
$$578$$ 36.6712 1.52532
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −6.90669 −0.286538
$$582$$ 0 0
$$583$$ 56.1768 2.32661
$$584$$ 31.4006 1.29937
$$585$$ 0 0
$$586$$ 36.8962 1.52417
$$587$$ 25.7669 1.06351 0.531756 0.846897i $$-0.321532\pi$$
0.531756 + 0.846897i $$0.321532\pi$$
$$588$$ 0 0
$$589$$ 31.8476 1.31226
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −1.34199 −0.0551554
$$593$$ −3.02524 −0.124232 −0.0621158 0.998069i $$-0.519785\pi$$
−0.0621158 + 0.998069i $$0.519785\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −56.6774 −2.32160
$$597$$ 0 0
$$598$$ 58.1529 2.37805
$$599$$ −0.521774 −0.0213191 −0.0106596 0.999943i $$-0.503393\pi$$
−0.0106596 + 0.999943i $$0.503393\pi$$
$$600$$ 0 0
$$601$$ −34.7237 −1.41641 −0.708204 0.706008i $$-0.750494\pi$$
−0.708204 + 0.706008i $$0.750494\pi$$
$$602$$ 5.93674 0.241964
$$603$$ 0 0
$$604$$ −31.0593 −1.26379
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14.0480 0.570190 0.285095 0.958499i $$-0.407975\pi$$
0.285095 + 0.958499i $$0.407975\pi$$
$$608$$ −36.0688 −1.46278
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −28.4801 −1.15218
$$612$$ 0 0
$$613$$ −15.3819 −0.621269 −0.310635 0.950529i $$-0.600542\pi$$
−0.310635 + 0.950529i $$0.600542\pi$$
$$614$$ −10.9893 −0.443493
$$615$$ 0 0
$$616$$ −11.0880 −0.446750
$$617$$ 15.6734 0.630988 0.315494 0.948928i $$-0.397830\pi$$
0.315494 + 0.948928i $$0.397830\pi$$
$$618$$ 0 0
$$619$$ 36.1380 1.45251 0.726254 0.687427i $$-0.241260\pi$$
0.726254 + 0.687427i $$0.241260\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −46.6007 −1.86852
$$623$$ −6.64930 −0.266399
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −53.8870 −2.15376
$$627$$ 0 0
$$628$$ 68.0083 2.71383
$$629$$ 0.600075 0.0239265
$$630$$ 0 0
$$631$$ −5.20523 −0.207217 −0.103608 0.994618i $$-0.533039\pi$$
−0.103608 + 0.994618i $$0.533039\pi$$
$$632$$ 6.48623 0.258008
$$633$$ 0 0
$$634$$ 29.6474 1.17745
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 15.6714 0.620921
$$638$$ 9.48853 0.375655
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −28.6342 −1.13098 −0.565492 0.824754i $$-0.691314\pi$$
−0.565492 + 0.824754i $$0.691314\pi$$
$$642$$ 0 0
$$643$$ 35.6717 1.40676 0.703378 0.710816i $$-0.251674\pi$$
0.703378 + 0.710816i $$0.251674\pi$$
$$644$$ −33.6067 −1.32429
$$645$$ 0 0
$$646$$ 7.21409 0.283834
$$647$$ 6.14445 0.241563 0.120782 0.992679i $$-0.461460\pi$$
0.120782 + 0.992679i $$0.461460\pi$$
$$648$$ 0 0
$$649$$ −10.9473 −0.429718
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 27.4371 1.07452
$$653$$ 8.62982 0.337711 0.168855 0.985641i $$-0.445993\pi$$
0.168855 + 0.985641i $$0.445993\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 15.9178 0.621488
$$657$$ 0 0
$$658$$ 27.8581 1.08602
$$659$$ 35.4336 1.38030 0.690148 0.723668i $$-0.257545\pi$$
0.690148 + 0.723668i $$0.257545\pi$$
$$660$$ 0 0
$$661$$ 12.7938 0.497622 0.248811 0.968552i $$-0.419960\pi$$
0.248811 + 0.968552i $$0.419960\pi$$
$$662$$ −51.7741 −2.01226
$$663$$ 0 0
$$664$$ −10.2958 −0.399554
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.84038 0.342301
$$668$$ −5.27824 −0.204221
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −36.3341 −1.40266
$$672$$ 0 0
$$673$$ −39.0602 −1.50566 −0.752830 0.658215i $$-0.771312\pi$$
−0.752830 + 0.658215i $$0.771312\pi$$
$$674$$ −50.3766 −1.94043
$$675$$ 0 0
$$676$$ −11.9747 −0.460564
$$677$$ 51.8790 1.99387 0.996936 0.0782187i $$-0.0249233\pi$$
0.996936 + 0.0782187i $$0.0249233\pi$$
$$678$$ 0 0
$$679$$ 2.69104 0.103272
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −59.4946 −2.27817
$$683$$ 27.2220 1.04162 0.520811 0.853672i $$-0.325630\pi$$
0.520811 + 0.853672i $$0.325630\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −35.7022 −1.36312
$$687$$ 0 0
$$688$$ −2.93066 −0.111730
$$689$$ 38.9455 1.48371
$$690$$ 0 0
$$691$$ 21.7474 0.827309 0.413655 0.910434i $$-0.364252\pi$$
0.413655 + 0.910434i $$0.364252\pi$$
$$692$$ 15.9033 0.604554
$$693$$ 0 0
$$694$$ −11.0212 −0.418357
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −7.11772 −0.269603
$$698$$ −56.7698 −2.14877
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −35.4994 −1.34079 −0.670397 0.742003i $$-0.733876\pi$$
−0.670397 + 0.742003i $$0.733876\pi$$
$$702$$ 0 0
$$703$$ −4.74427 −0.178934
$$704$$ 55.0477 2.07469
$$705$$ 0 0
$$706$$ −44.9148 −1.69039
$$707$$ −16.4388 −0.618243
$$708$$ 0 0
$$709$$ 33.4325 1.25558 0.627792 0.778381i $$-0.283959\pi$$
0.627792 + 0.778381i $$0.283959\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −9.91208 −0.371471
$$713$$ −55.4306 −2.07589
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −27.0421 −1.01061
$$717$$ 0 0
$$718$$ −21.4207 −0.799412
$$719$$ −44.5343 −1.66085 −0.830425 0.557130i $$-0.811902\pi$$
−0.830425 + 0.557130i $$0.811902\pi$$
$$720$$ 0 0
$$721$$ 14.0497 0.523237
$$722$$ −15.0302 −0.559368
$$723$$ 0 0
$$724$$ 73.9296 2.74757
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 15.6700 0.581170 0.290585 0.956849i $$-0.406150\pi$$
0.290585 + 0.956849i $$0.406150\pi$$
$$728$$ −7.68698 −0.284898
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1.31046 0.0484689
$$732$$ 0 0
$$733$$ 29.7350 1.09829 0.549143 0.835728i $$-0.314954\pi$$
0.549143 + 0.835728i $$0.314954\pi$$
$$734$$ −20.2580 −0.747738
$$735$$ 0 0
$$736$$ 62.7776 2.31401
$$737$$ 55.7123 2.05219
$$738$$ 0 0
$$739$$ 42.9178 1.57876 0.789379 0.613906i $$-0.210403\pi$$
0.789379 + 0.613906i $$0.210403\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −38.0950 −1.39851
$$743$$ 10.9064 0.400117 0.200058 0.979784i $$-0.435887\pi$$
0.200058 + 0.979784i $$0.435887\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 30.0168 1.09899
$$747$$ 0 0
$$748$$ −7.96211 −0.291124
$$749$$ 6.39743 0.233757
$$750$$ 0 0
$$751$$ 47.2977 1.72592 0.862960 0.505273i $$-0.168608\pi$$
0.862960 + 0.505273i $$0.168608\pi$$
$$752$$ −13.7521 −0.501488
$$753$$ 0 0
$$754$$ 6.57809 0.239560
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 33.3406 1.21179 0.605893 0.795546i $$-0.292816\pi$$
0.605893 + 0.795546i $$0.292816\pi$$
$$758$$ −0.132963 −0.00482944
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 45.9707 1.66644 0.833219 0.552944i $$-0.186496\pi$$
0.833219 + 0.552944i $$0.186496\pi$$
$$762$$ 0 0
$$763$$ −20.1965 −0.731162
$$764$$ 35.7082 1.29188
$$765$$ 0 0
$$766$$ 21.4811 0.776144
$$767$$ −7.58939 −0.274037
$$768$$ 0 0
$$769$$ −9.21048 −0.332138 −0.166069 0.986114i $$-0.553108\pi$$
−0.166069 + 0.986114i $$0.553108\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −45.4007 −1.63401
$$773$$ 28.6595 1.03081 0.515405 0.856947i $$-0.327642\pi$$
0.515405 + 0.856947i $$0.327642\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 4.01152 0.144005
$$777$$ 0 0
$$778$$ 12.5793 0.450989
$$779$$ 56.2737 2.01621
$$780$$ 0 0
$$781$$ 21.9066 0.783881
$$782$$ −12.5561 −0.449005
$$783$$ 0 0
$$784$$ 7.56718 0.270256
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 20.8195 0.742136 0.371068 0.928606i $$-0.378992\pi$$
0.371068 + 0.928606i $$0.378992\pi$$
$$788$$ 62.5837 2.22945
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −5.68948 −0.202295
$$792$$ 0 0
$$793$$ −25.1892 −0.894496
$$794$$ −15.0496 −0.534090
$$795$$ 0 0
$$796$$ −3.57298 −0.126641
$$797$$ 44.8656 1.58922 0.794610 0.607120i $$-0.207675\pi$$
0.794610 + 0.607120i $$0.207675\pi$$
$$798$$ 0 0
$$799$$ 6.14931 0.217547
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 25.9228 0.915365
$$803$$ 68.6734 2.42343
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −41.2457 −1.45282
$$807$$ 0 0
$$808$$ −24.5052 −0.862090
$$809$$ 26.6975 0.938633 0.469317 0.883030i $$-0.344500\pi$$
0.469317 + 0.883030i $$0.344500\pi$$
$$810$$ 0 0
$$811$$ −21.6414 −0.759934 −0.379967 0.925000i $$-0.624065\pi$$
−0.379967 + 0.925000i $$0.624065\pi$$
$$812$$ −3.80150 −0.133406
$$813$$ 0 0
$$814$$ 8.86281 0.310641
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −10.3606 −0.362473
$$818$$ 49.2270 1.72118
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −36.0374 −1.25771 −0.628857 0.777521i $$-0.716477\pi$$
−0.628857 + 0.777521i $$0.716477\pi$$
$$822$$ 0 0
$$823$$ −23.4577 −0.817684 −0.408842 0.912605i $$-0.634067\pi$$
−0.408842 + 0.912605i $$0.634067\pi$$
$$824$$ 20.9438 0.729611
$$825$$ 0 0
$$826$$ 7.42364 0.258301
$$827$$ −14.7623 −0.513334 −0.256667 0.966500i $$-0.582624\pi$$
−0.256667 + 0.966500i $$0.582624\pi$$
$$828$$ 0 0
$$829$$ 5.63801 0.195816 0.0979081 0.995195i $$-0.468785\pi$$
0.0979081 + 0.995195i $$0.468785\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 38.1628 1.32306
$$833$$ −3.38369 −0.117238
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 62.9495 2.17715
$$837$$ 0 0
$$838$$ −62.3776 −2.15480
$$839$$ 56.4632 1.94932 0.974662 0.223681i $$-0.0718073\pi$$
0.974662 + 0.223681i $$0.0718073\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 76.3098 2.62981
$$843$$ 0 0
$$844$$ −14.5927 −0.502301
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −9.76858 −0.335652
$$848$$ 18.8055 0.645784
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.25740 0.283060
$$852$$ 0 0
$$853$$ 38.2731 1.31045 0.655223 0.755436i $$-0.272575\pi$$
0.655223 + 0.755436i $$0.272575\pi$$
$$854$$ 24.6391 0.843133
$$855$$ 0 0
$$856$$ 9.53663 0.325955
$$857$$ 47.5891 1.62561 0.812806 0.582534i $$-0.197939\pi$$
0.812806 + 0.582534i $$0.197939\pi$$
$$858$$ 0 0
$$859$$ −19.4348 −0.663108 −0.331554 0.943436i $$-0.607573\pi$$
−0.331554 + 0.943436i $$0.607573\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 50.9984 1.73701
$$863$$ 24.3695 0.829546 0.414773 0.909925i $$-0.363861\pi$$
0.414773 + 0.909925i $$0.363861\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 39.4671 1.34115
$$867$$ 0 0
$$868$$ 23.8360 0.809047
$$869$$ 14.1855 0.481209
$$870$$ 0 0
$$871$$ 38.6235 1.30871
$$872$$ −30.1068 −1.01955
$$873$$ 0 0
$$874$$ 99.2703 3.35787
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 23.4009 0.790193 0.395097 0.918640i $$-0.370711\pi$$
0.395097 + 0.918640i $$0.370711\pi$$
$$878$$ 7.33225 0.247451
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −49.1434 −1.65568 −0.827841 0.560963i $$-0.810431\pi$$
−0.827841 + 0.560963i $$0.810431\pi$$
$$882$$ 0 0
$$883$$ −42.0245 −1.41424 −0.707118 0.707095i $$-0.750005\pi$$
−0.707118 + 0.707095i $$0.750005\pi$$
$$884$$ −5.51987 −0.185653
$$885$$ 0 0
$$886$$ −66.0628 −2.21942
$$887$$ 38.7280 1.30036 0.650179 0.759781i $$-0.274694\pi$$
0.650179 + 0.759781i $$0.274694\pi$$
$$888$$ 0 0
$$889$$ −28.4105 −0.952857
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −46.8516 −1.56871
$$893$$ −48.6173 −1.62691
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −18.6323 −0.622460
$$897$$ 0 0
$$898$$ 19.6584 0.656011
$$899$$ −6.27016 −0.209122
$$900$$ 0 0
$$901$$ −8.40895 −0.280143
$$902$$ −105.125 −3.50029
$$903$$ 0 0
$$904$$ −8.48129 −0.282084
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 22.7366 0.754956 0.377478 0.926019i $$-0.376791\pi$$
0.377478 + 0.926019i $$0.376791\pi$$
$$908$$ 19.7138 0.654224
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −5.74791 −0.190437 −0.0952183 0.995456i $$-0.530355\pi$$
−0.0952183 + 0.995456i $$0.530355\pi$$
$$912$$ 0 0
$$913$$ −22.5170 −0.745203
$$914$$ 43.9625 1.45415
$$915$$ 0 0
$$916$$ −16.9541 −0.560178
$$917$$ 12.1859 0.402415
$$918$$ 0 0
$$919$$ 52.1835 1.72137 0.860687 0.509135i $$-0.170035\pi$$
0.860687 + 0.509135i $$0.170035\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −30.0175 −0.988572
$$923$$ 15.1871 0.499891
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −65.8036 −2.16244
$$927$$ 0 0
$$928$$ 7.10123 0.233109
$$929$$ 4.61899 0.151544 0.0757721 0.997125i $$-0.475858\pi$$
0.0757721 + 0.997125i $$0.475858\pi$$
$$930$$ 0 0
$$931$$ 26.7519 0.876758
$$932$$ −40.8469 −1.33799
$$933$$ 0 0
$$934$$ −14.5198 −0.475104
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −22.8991 −0.748082 −0.374041 0.927412i $$-0.622028\pi$$
−0.374041 + 0.927412i $$0.622028\pi$$
$$938$$ −37.7800 −1.23356
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −24.1297 −0.786604 −0.393302 0.919409i $$-0.628667\pi$$
−0.393302 + 0.919409i $$0.628667\pi$$
$$942$$ 0 0
$$943$$ −97.9442 −3.18950
$$944$$ −3.66466 −0.119275
$$945$$ 0 0
$$946$$ 19.3548 0.629278
$$947$$ −13.6427 −0.443329 −0.221665 0.975123i $$-0.571149\pi$$
−0.221665 + 0.975123i $$0.571149\pi$$
$$948$$ 0 0
$$949$$ 47.6090 1.54545
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 1.65974 0.0537925
$$953$$ −45.3303 −1.46839 −0.734196 0.678937i $$-0.762441\pi$$
−0.734196 + 0.678937i $$0.762441\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 59.7514 1.93250
$$957$$ 0 0
$$958$$ 81.6113 2.63674
$$959$$ 1.11079 0.0358692
$$960$$ 0 0
$$961$$ 8.31492 0.268223
$$962$$ 6.14429 0.198100
$$963$$ 0 0
$$964$$ −22.8369 −0.735528
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −45.2638 −1.45558 −0.727792 0.685798i $$-0.759453\pi$$
−0.727792 + 0.685798i $$0.759453\pi$$
$$968$$ −14.5620 −0.468040
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 47.1089 1.51180 0.755899 0.654688i $$-0.227200\pi$$
0.755899 + 0.654688i $$0.227200\pi$$
$$972$$ 0 0
$$973$$ 18.4011 0.589912
$$974$$ −54.0006 −1.73029
$$975$$ 0 0
$$976$$ −12.1630 −0.389330
$$977$$ 2.89922 0.0927542 0.0463771 0.998924i $$-0.485232\pi$$
0.0463771 + 0.998924i $$0.485232\pi$$
$$978$$ 0 0
$$979$$ −21.6778 −0.692826
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −1.83068 −0.0584193
$$983$$ −37.0610 −1.18206 −0.591031 0.806649i $$-0.701279\pi$$
−0.591031 + 0.806649i $$0.701279\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −1.42031 −0.0452320
$$987$$ 0 0
$$988$$ 43.6408 1.38840
$$989$$ 18.0327 0.573406
$$990$$ 0 0
$$991$$ 17.4669 0.554854 0.277427 0.960747i $$-0.410518\pi$$
0.277427 + 0.960747i $$0.410518\pi$$
$$992$$ −44.5259 −1.41370
$$993$$ 0 0
$$994$$ −14.8555 −0.471187
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −19.4726 −0.616703 −0.308352 0.951272i $$-0.599777\pi$$
−0.308352 + 0.951272i $$0.599777\pi$$
$$998$$ −72.3846 −2.29129
$$999$$ 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.cd.1.1 yes 9
3.2 odd 2 6525.2.a.cb.1.9 yes 9
5.4 even 2 6525.2.a.ca.1.9 9
15.14 odd 2 6525.2.a.cc.1.1 yes 9

By twisted newform
Twist Min Dim Char Parity Ord Type
6525.2.a.ca.1.9 9 5.4 even 2
6525.2.a.cb.1.9 yes 9 3.2 odd 2
6525.2.a.cc.1.1 yes 9 15.14 odd 2
6525.2.a.cd.1.1 yes 9 1.1 even 1 trivial