Properties

 Label 6525.2.a.ca.1.4 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $1$ 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: $$1$$ 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 1.4 Root $$1.23642$$ 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)

 $$f(q)$$ $$=$$ $$q-1.23642 q^{2} -0.471260 q^{4} +3.27673 q^{7} +3.05552 q^{8} +O(q^{10})$$ $$q-1.23642 q^{2} -0.471260 q^{4} +3.27673 q^{7} +3.05552 q^{8} +2.30259 q^{11} -5.57932 q^{13} -4.05142 q^{14} -2.83539 q^{16} +1.94031 q^{17} -3.59158 q^{19} -2.84698 q^{22} +1.66263 q^{23} +6.89840 q^{26} -1.54419 q^{28} +1.00000 q^{29} +1.70877 q^{31} -2.60530 q^{32} -2.39904 q^{34} +9.16633 q^{37} +4.44071 q^{38} -8.54038 q^{41} +3.56165 q^{43} -1.08512 q^{44} -2.05571 q^{46} -11.5640 q^{47} +3.73698 q^{49} +2.62931 q^{52} -9.66312 q^{53} +10.0121 q^{56} -1.23642 q^{58} -9.83385 q^{59} +5.42714 q^{61} -2.11276 q^{62} +8.89204 q^{64} -5.20114 q^{67} -0.914391 q^{68} -6.02596 q^{71} -15.5143 q^{73} -11.3334 q^{74} +1.69257 q^{76} +7.54498 q^{77} +12.2465 q^{79} +10.5595 q^{82} -10.7179 q^{83} -4.40370 q^{86} +7.03562 q^{88} -2.53756 q^{89} -18.2820 q^{91} -0.783531 q^{92} +14.2980 q^{94} +5.89998 q^{97} -4.62048 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.23642 −0.874282 −0.437141 0.899393i $$-0.644009\pi$$
−0.437141 + 0.899393i $$0.644009\pi$$
$$3$$ 0 0
$$4$$ −0.471260 −0.235630
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.27673 1.23849 0.619244 0.785198i $$-0.287439\pi$$
0.619244 + 0.785198i $$0.287439\pi$$
$$8$$ 3.05552 1.08029
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.30259 0.694258 0.347129 0.937817i $$-0.387157\pi$$
0.347129 + 0.937817i $$0.387157\pi$$
$$12$$ 0 0
$$13$$ −5.57932 −1.54743 −0.773713 0.633536i $$-0.781603\pi$$
−0.773713 + 0.633536i $$0.781603\pi$$
$$14$$ −4.05142 −1.08279
$$15$$ 0 0
$$16$$ −2.83539 −0.708848
$$17$$ 1.94031 0.470594 0.235297 0.971923i $$-0.424394\pi$$
0.235297 + 0.971923i $$0.424394\pi$$
$$18$$ 0 0
$$19$$ −3.59158 −0.823965 −0.411982 0.911192i $$-0.635164\pi$$
−0.411982 + 0.911192i $$0.635164\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2.84698 −0.606977
$$23$$ 1.66263 0.346682 0.173341 0.984862i $$-0.444544\pi$$
0.173341 + 0.984862i $$0.444544\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.89840 1.35289
$$27$$ 0 0
$$28$$ −1.54419 −0.291825
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 1.70877 0.306903 0.153452 0.988156i $$-0.450961\pi$$
0.153452 + 0.988156i $$0.450961\pi$$
$$32$$ −2.60530 −0.460556
$$33$$ 0 0
$$34$$ −2.39904 −0.411432
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.16633 1.50693 0.753467 0.657485i $$-0.228380\pi$$
0.753467 + 0.657485i $$0.228380\pi$$
$$38$$ 4.44071 0.720378
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.54038 −1.33378 −0.666892 0.745154i $$-0.732376\pi$$
−0.666892 + 0.745154i $$0.732376\pi$$
$$42$$ 0 0
$$43$$ 3.56165 0.543146 0.271573 0.962418i $$-0.412456\pi$$
0.271573 + 0.962418i $$0.412456\pi$$
$$44$$ −1.08512 −0.163588
$$45$$ 0 0
$$46$$ −2.05571 −0.303098
$$47$$ −11.5640 −1.68678 −0.843391 0.537300i $$-0.819444\pi$$
−0.843391 + 0.537300i $$0.819444\pi$$
$$48$$ 0 0
$$49$$ 3.73698 0.533854
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.62931 0.364620
$$53$$ −9.66312 −1.32733 −0.663666 0.748029i $$-0.731000\pi$$
−0.663666 + 0.748029i $$0.731000\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 10.0121 1.33793
$$57$$ 0 0
$$58$$ −1.23642 −0.162350
$$59$$ −9.83385 −1.28026 −0.640129 0.768268i $$-0.721119\pi$$
−0.640129 + 0.768268i $$0.721119\pi$$
$$60$$ 0 0
$$61$$ 5.42714 0.694874 0.347437 0.937703i $$-0.387052\pi$$
0.347437 + 0.937703i $$0.387052\pi$$
$$62$$ −2.11276 −0.268320
$$63$$ 0 0
$$64$$ 8.89204 1.11150
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −5.20114 −0.635421 −0.317710 0.948188i $$-0.602914\pi$$
−0.317710 + 0.948188i $$0.602914\pi$$
$$68$$ −0.914391 −0.110886
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.02596 −0.715150 −0.357575 0.933884i $$-0.616396\pi$$
−0.357575 + 0.933884i $$0.616396\pi$$
$$72$$ 0 0
$$73$$ −15.5143 −1.81581 −0.907905 0.419177i $$-0.862319\pi$$
−0.907905 + 0.419177i $$0.862319\pi$$
$$74$$ −11.3334 −1.31749
$$75$$ 0 0
$$76$$ 1.69257 0.194151
$$77$$ 7.54498 0.859830
$$78$$ 0 0
$$79$$ 12.2465 1.37784 0.688922 0.724835i $$-0.258084\pi$$
0.688922 + 0.724835i $$0.258084\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 10.5595 1.16610
$$83$$ −10.7179 −1.17644 −0.588221 0.808700i $$-0.700172\pi$$
−0.588221 + 0.808700i $$0.700172\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.40370 −0.474863
$$87$$ 0 0
$$88$$ 7.03562 0.749999
$$89$$ −2.53756 −0.268981 −0.134491 0.990915i $$-0.542940\pi$$
−0.134491 + 0.990915i $$0.542940\pi$$
$$90$$ 0 0
$$91$$ −18.2820 −1.91647
$$92$$ −0.783531 −0.0816887
$$93$$ 0 0
$$94$$ 14.2980 1.47472
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.89998 0.599053 0.299526 0.954088i $$-0.403171\pi$$
0.299526 + 0.954088i $$0.403171\pi$$
$$98$$ −4.62048 −0.466739
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.33789 −0.431636 −0.215818 0.976434i $$-0.569242\pi$$
−0.215818 + 0.976434i $$0.569242\pi$$
$$102$$ 0 0
$$103$$ −7.62183 −0.751001 −0.375501 0.926822i $$-0.622529\pi$$
−0.375501 + 0.926822i $$0.622529\pi$$
$$104$$ −17.0477 −1.67167
$$105$$ 0 0
$$106$$ 11.9477 1.16046
$$107$$ −4.19703 −0.405742 −0.202871 0.979205i $$-0.565027\pi$$
−0.202871 + 0.979205i $$0.565027\pi$$
$$108$$ 0 0
$$109$$ 4.02677 0.385695 0.192847 0.981229i $$-0.438228\pi$$
0.192847 + 0.981229i $$0.438228\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −9.29082 −0.877900
$$113$$ −7.40527 −0.696629 −0.348314 0.937378i $$-0.613246\pi$$
−0.348314 + 0.937378i $$0.613246\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −0.471260 −0.0437554
$$117$$ 0 0
$$118$$ 12.1588 1.11931
$$119$$ 6.35788 0.582826
$$120$$ 0 0
$$121$$ −5.69807 −0.518006
$$122$$ −6.71024 −0.607516
$$123$$ 0 0
$$124$$ −0.805274 −0.0723157
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −12.9643 −1.15040 −0.575199 0.818014i $$-0.695075\pi$$
−0.575199 + 0.818014i $$0.695075\pi$$
$$128$$ −5.78371 −0.511213
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 17.6941 1.54594 0.772971 0.634442i $$-0.218770\pi$$
0.772971 + 0.634442i $$0.218770\pi$$
$$132$$ 0 0
$$133$$ −11.7686 −1.02047
$$134$$ 6.43081 0.555537
$$135$$ 0 0
$$136$$ 5.92866 0.508378
$$137$$ 22.5543 1.92694 0.963471 0.267813i $$-0.0863010\pi$$
0.963471 + 0.267813i $$0.0863010\pi$$
$$138$$ 0 0
$$139$$ −3.20075 −0.271484 −0.135742 0.990744i $$-0.543342\pi$$
−0.135742 + 0.990744i $$0.543342\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 7.45064 0.625243
$$143$$ −12.8469 −1.07431
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 19.1822 1.58753
$$147$$ 0 0
$$148$$ −4.31973 −0.355079
$$149$$ −0.612365 −0.0501669 −0.0250835 0.999685i $$-0.507985\pi$$
−0.0250835 + 0.999685i $$0.507985\pi$$
$$150$$ 0 0
$$151$$ 12.4363 1.01205 0.506026 0.862518i $$-0.331114\pi$$
0.506026 + 0.862518i $$0.331114\pi$$
$$152$$ −10.9741 −0.890121
$$153$$ 0 0
$$154$$ −9.32878 −0.751734
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.378638 0.0302186 0.0151093 0.999886i $$-0.495190\pi$$
0.0151093 + 0.999886i $$0.495190\pi$$
$$158$$ −15.1419 −1.20463
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 5.44799 0.429362
$$162$$ 0 0
$$163$$ −2.54446 −0.199297 −0.0996487 0.995023i $$-0.531772\pi$$
−0.0996487 + 0.995023i $$0.531772\pi$$
$$164$$ 4.02474 0.314280
$$165$$ 0 0
$$166$$ 13.2519 1.02854
$$167$$ 10.7799 0.834175 0.417088 0.908866i $$-0.363051\pi$$
0.417088 + 0.908866i $$0.363051\pi$$
$$168$$ 0 0
$$169$$ 18.1289 1.39453
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1.67846 −0.127982
$$173$$ −23.1221 −1.75794 −0.878970 0.476877i $$-0.841769\pi$$
−0.878970 + 0.476877i $$0.841769\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −6.52875 −0.492123
$$177$$ 0 0
$$178$$ 3.13750 0.235165
$$179$$ 23.7081 1.77203 0.886014 0.463659i $$-0.153464\pi$$
0.886014 + 0.463659i $$0.153464\pi$$
$$180$$ 0 0
$$181$$ 11.6809 0.868233 0.434116 0.900857i $$-0.357061\pi$$
0.434116 + 0.900857i $$0.357061\pi$$
$$182$$ 22.6042 1.67554
$$183$$ 0 0
$$184$$ 5.08020 0.374517
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.46774 0.326714
$$188$$ 5.44965 0.397457
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.68267 0.628256 0.314128 0.949381i $$-0.398288\pi$$
0.314128 + 0.949381i $$0.398288\pi$$
$$192$$ 0 0
$$193$$ 2.73727 0.197033 0.0985164 0.995135i $$-0.468590\pi$$
0.0985164 + 0.995135i $$0.468590\pi$$
$$194$$ −7.29487 −0.523741
$$195$$ 0 0
$$196$$ −1.76109 −0.125792
$$197$$ −26.2457 −1.86993 −0.934964 0.354741i $$-0.884569\pi$$
−0.934964 + 0.354741i $$0.884569\pi$$
$$198$$ 0 0
$$199$$ 9.94800 0.705195 0.352597 0.935775i $$-0.385299\pi$$
0.352597 + 0.935775i $$0.385299\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 5.36347 0.377372
$$203$$ 3.27673 0.229982
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 9.42380 0.656587
$$207$$ 0 0
$$208$$ 15.8196 1.09689
$$209$$ −8.26994 −0.572044
$$210$$ 0 0
$$211$$ 7.76180 0.534344 0.267172 0.963649i $$-0.413911\pi$$
0.267172 + 0.963649i $$0.413911\pi$$
$$212$$ 4.55384 0.312759
$$213$$ 0 0
$$214$$ 5.18930 0.354733
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 5.59917 0.380096
$$218$$ −4.97879 −0.337206
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −10.8256 −0.728210
$$222$$ 0 0
$$223$$ 9.63976 0.645526 0.322763 0.946480i $$-0.395388\pi$$
0.322763 + 0.946480i $$0.395388\pi$$
$$224$$ −8.53687 −0.570394
$$225$$ 0 0
$$226$$ 9.15603 0.609050
$$227$$ −9.21807 −0.611825 −0.305913 0.952060i $$-0.598962\pi$$
−0.305913 + 0.952060i $$0.598962\pi$$
$$228$$ 0 0
$$229$$ −6.89586 −0.455691 −0.227846 0.973697i $$-0.573168\pi$$
−0.227846 + 0.973697i $$0.573168\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.05552 0.200605
$$233$$ 13.6229 0.892466 0.446233 0.894917i $$-0.352765\pi$$
0.446233 + 0.894917i $$0.352765\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 4.63430 0.301667
$$237$$ 0 0
$$238$$ −7.86102 −0.509554
$$239$$ −0.749144 −0.0484581 −0.0242290 0.999706i $$-0.507713\pi$$
−0.0242290 + 0.999706i $$0.507713\pi$$
$$240$$ 0 0
$$241$$ −12.5380 −0.807645 −0.403822 0.914837i $$-0.632319\pi$$
−0.403822 + 0.914837i $$0.632319\pi$$
$$242$$ 7.04522 0.452884
$$243$$ 0 0
$$244$$ −2.55760 −0.163733
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 20.0386 1.27503
$$248$$ 5.22117 0.331545
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 11.7441 0.741281 0.370640 0.928776i $$-0.379138\pi$$
0.370640 + 0.928776i $$0.379138\pi$$
$$252$$ 0 0
$$253$$ 3.82836 0.240687
$$254$$ 16.0294 1.00577
$$255$$ 0 0
$$256$$ −10.6330 −0.664560
$$257$$ −13.9304 −0.868955 −0.434477 0.900683i $$-0.643067\pi$$
−0.434477 + 0.900683i $$0.643067\pi$$
$$258$$ 0 0
$$259$$ 30.0356 1.86632
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −21.8774 −1.35159
$$263$$ −25.8059 −1.59126 −0.795631 0.605781i $$-0.792861\pi$$
−0.795631 + 0.605781i $$0.792861\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 14.5510 0.892180
$$267$$ 0 0
$$268$$ 2.45109 0.149724
$$269$$ −11.4590 −0.698667 −0.349333 0.936998i $$-0.613592\pi$$
−0.349333 + 0.936998i $$0.613592\pi$$
$$270$$ 0 0
$$271$$ −21.4783 −1.30472 −0.652358 0.757911i $$-0.726220\pi$$
−0.652358 + 0.757911i $$0.726220\pi$$
$$272$$ −5.50154 −0.333580
$$273$$ 0 0
$$274$$ −27.8866 −1.68469
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7.16907 0.430747 0.215374 0.976532i $$-0.430903\pi$$
0.215374 + 0.976532i $$0.430903\pi$$
$$278$$ 3.95748 0.237354
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 14.0042 0.835420 0.417710 0.908580i $$-0.362833\pi$$
0.417710 + 0.908580i $$0.362833\pi$$
$$282$$ 0 0
$$283$$ 10.3513 0.615324 0.307662 0.951496i $$-0.400453\pi$$
0.307662 + 0.951496i $$0.400453\pi$$
$$284$$ 2.83980 0.168511
$$285$$ 0 0
$$286$$ 15.8842 0.939253
$$287$$ −27.9846 −1.65188
$$288$$ 0 0
$$289$$ −13.2352 −0.778541
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 7.31127 0.427859
$$293$$ −26.6063 −1.55435 −0.777177 0.629282i $$-0.783349\pi$$
−0.777177 + 0.629282i $$0.783349\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 28.0079 1.62793
$$297$$ 0 0
$$298$$ 0.757142 0.0438601
$$299$$ −9.27634 −0.536465
$$300$$ 0 0
$$301$$ 11.6706 0.672680
$$302$$ −15.3765 −0.884819
$$303$$ 0 0
$$304$$ 10.1835 0.584066
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 29.2808 1.67114 0.835571 0.549383i $$-0.185137\pi$$
0.835571 + 0.549383i $$0.185137\pi$$
$$308$$ −3.55565 −0.202602
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −5.03104 −0.285284 −0.142642 0.989774i $$-0.545560\pi$$
−0.142642 + 0.989774i $$0.545560\pi$$
$$312$$ 0 0
$$313$$ −27.5875 −1.55934 −0.779670 0.626191i $$-0.784613\pi$$
−0.779670 + 0.626191i $$0.784613\pi$$
$$314$$ −0.468157 −0.0264196
$$315$$ 0 0
$$316$$ −5.77131 −0.324662
$$317$$ 7.70663 0.432848 0.216424 0.976300i $$-0.430561\pi$$
0.216424 + 0.976300i $$0.430561\pi$$
$$318$$ 0 0
$$319$$ 2.30259 0.128920
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −6.73601 −0.375383
$$323$$ −6.96878 −0.387753
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 3.14602 0.174242
$$327$$ 0 0
$$328$$ −26.0953 −1.44087
$$329$$ −37.8921 −2.08906
$$330$$ 0 0
$$331$$ 18.9943 1.04402 0.522010 0.852940i $$-0.325182\pi$$
0.522010 + 0.852940i $$0.325182\pi$$
$$332$$ 5.05092 0.277205
$$333$$ 0 0
$$334$$ −13.3285 −0.729305
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −20.1983 −1.10027 −0.550136 0.835075i $$-0.685424\pi$$
−0.550136 + 0.835075i $$0.685424\pi$$
$$338$$ −22.4149 −1.21921
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3.93459 0.213070
$$342$$ 0 0
$$343$$ −10.6921 −0.577317
$$344$$ 10.8827 0.586755
$$345$$ 0 0
$$346$$ 28.5887 1.53694
$$347$$ −23.7995 −1.27763 −0.638813 0.769362i $$-0.720574\pi$$
−0.638813 + 0.769362i $$0.720574\pi$$
$$348$$ 0 0
$$349$$ −22.8554 −1.22342 −0.611710 0.791082i $$-0.709518\pi$$
−0.611710 + 0.791082i $$0.709518\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5.99894 −0.319745
$$353$$ 17.5338 0.933231 0.466615 0.884460i $$-0.345473\pi$$
0.466615 + 0.884460i $$0.345473\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 1.19585 0.0633801
$$357$$ 0 0
$$358$$ −29.3132 −1.54925
$$359$$ 26.6594 1.40703 0.703513 0.710682i $$-0.251613\pi$$
0.703513 + 0.710682i $$0.251613\pi$$
$$360$$ 0 0
$$361$$ −6.10055 −0.321082
$$362$$ −14.4425 −0.759081
$$363$$ 0 0
$$364$$ 8.61556 0.451578
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −22.5869 −1.17902 −0.589512 0.807760i $$-0.700680\pi$$
−0.589512 + 0.807760i $$0.700680\pi$$
$$368$$ −4.71420 −0.245745
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −31.6634 −1.64388
$$372$$ 0 0
$$373$$ 11.0978 0.574623 0.287311 0.957837i $$-0.407239\pi$$
0.287311 + 0.957837i $$0.407239\pi$$
$$374$$ −5.52402 −0.285640
$$375$$ 0 0
$$376$$ −35.3340 −1.82221
$$377$$ −5.57932 −0.287350
$$378$$ 0 0
$$379$$ −28.2474 −1.45097 −0.725485 0.688238i $$-0.758384\pi$$
−0.725485 + 0.688238i $$0.758384\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −10.7354 −0.549273
$$383$$ −7.24537 −0.370221 −0.185110 0.982718i $$-0.559264\pi$$
−0.185110 + 0.982718i $$0.559264\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3.38442 −0.172262
$$387$$ 0 0
$$388$$ −2.78043 −0.141155
$$389$$ −34.4482 −1.74659 −0.873296 0.487190i $$-0.838022\pi$$
−0.873296 + 0.487190i $$0.838022\pi$$
$$390$$ 0 0
$$391$$ 3.22601 0.163147
$$392$$ 11.4184 0.576717
$$393$$ 0 0
$$394$$ 32.4508 1.63485
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −13.6412 −0.684632 −0.342316 0.939585i $$-0.611211\pi$$
−0.342316 + 0.939585i $$0.611211\pi$$
$$398$$ −12.2999 −0.616540
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −10.3741 −0.518059 −0.259029 0.965869i $$-0.583403\pi$$
−0.259029 + 0.965869i $$0.583403\pi$$
$$402$$ 0 0
$$403$$ −9.53376 −0.474910
$$404$$ 2.04428 0.101707
$$405$$ 0 0
$$406$$ −4.05142 −0.201069
$$407$$ 21.1063 1.04620
$$408$$ 0 0
$$409$$ 38.5394 1.90565 0.952825 0.303520i $$-0.0981620\pi$$
0.952825 + 0.303520i $$0.0981620\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 3.59187 0.176959
$$413$$ −32.2229 −1.58558
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 14.5358 0.712677
$$417$$ 0 0
$$418$$ 10.2251 0.500128
$$419$$ −9.87578 −0.482464 −0.241232 0.970468i $$-0.577551\pi$$
−0.241232 + 0.970468i $$0.577551\pi$$
$$420$$ 0 0
$$421$$ −10.2857 −0.501292 −0.250646 0.968079i $$-0.580643\pi$$
−0.250646 + 0.968079i $$0.580643\pi$$
$$422$$ −9.59686 −0.467168
$$423$$ 0 0
$$424$$ −29.5259 −1.43390
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 17.7833 0.860594
$$428$$ 1.97789 0.0956051
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.2705 0.735553 0.367776 0.929914i $$-0.380119\pi$$
0.367776 + 0.929914i $$0.380119\pi$$
$$432$$ 0 0
$$433$$ −13.8370 −0.664966 −0.332483 0.943109i $$-0.607886\pi$$
−0.332483 + 0.943109i $$0.607886\pi$$
$$434$$ −6.92294 −0.332312
$$435$$ 0 0
$$436$$ −1.89766 −0.0908813
$$437$$ −5.97146 −0.285654
$$438$$ 0 0
$$439$$ −6.48836 −0.309673 −0.154836 0.987940i $$-0.549485\pi$$
−0.154836 + 0.987940i $$0.549485\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 13.3850 0.636661
$$443$$ −22.1026 −1.05012 −0.525062 0.851064i $$-0.675958\pi$$
−0.525062 + 0.851064i $$0.675958\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −11.9188 −0.564372
$$447$$ 0 0
$$448$$ 29.1368 1.37659
$$449$$ −0.247203 −0.0116662 −0.00583311 0.999983i $$-0.501857\pi$$
−0.00583311 + 0.999983i $$0.501857\pi$$
$$450$$ 0 0
$$451$$ −19.6650 −0.925990
$$452$$ 3.48981 0.164147
$$453$$ 0 0
$$454$$ 11.3974 0.534908
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 35.4153 1.65666 0.828328 0.560243i $$-0.189292\pi$$
0.828328 + 0.560243i $$0.189292\pi$$
$$458$$ 8.52619 0.398403
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −13.5690 −0.631969 −0.315985 0.948764i $$-0.602335\pi$$
−0.315985 + 0.948764i $$0.602335\pi$$
$$462$$ 0 0
$$463$$ 7.88217 0.366316 0.183158 0.983084i $$-0.441368\pi$$
0.183158 + 0.983084i $$0.441368\pi$$
$$464$$ −2.83539 −0.131630
$$465$$ 0 0
$$466$$ −16.8437 −0.780268
$$467$$ −20.8411 −0.964411 −0.482206 0.876058i $$-0.660164\pi$$
−0.482206 + 0.876058i $$0.660164\pi$$
$$468$$ 0 0
$$469$$ −17.0428 −0.786961
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −30.0475 −1.38305
$$473$$ 8.20102 0.377083
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.99622 −0.137331
$$477$$ 0 0
$$478$$ 0.926258 0.0423661
$$479$$ −28.9598 −1.32321 −0.661604 0.749853i $$-0.730124\pi$$
−0.661604 + 0.749853i $$0.730124\pi$$
$$480$$ 0 0
$$481$$ −51.1419 −2.33187
$$482$$ 15.5023 0.706110
$$483$$ 0 0
$$484$$ 2.68527 0.122058
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 23.3354 1.05743 0.528714 0.848800i $$-0.322674\pi$$
0.528714 + 0.848800i $$0.322674\pi$$
$$488$$ 16.5827 0.750665
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 25.9951 1.17314 0.586570 0.809898i $$-0.300478\pi$$
0.586570 + 0.809898i $$0.300478\pi$$
$$492$$ 0 0
$$493$$ 1.94031 0.0873872
$$494$$ −24.7762 −1.11473
$$495$$ 0 0
$$496$$ −4.84502 −0.217548
$$497$$ −19.7455 −0.885706
$$498$$ 0 0
$$499$$ −31.9575 −1.43061 −0.715307 0.698810i $$-0.753713\pi$$
−0.715307 + 0.698810i $$0.753713\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −14.5207 −0.648089
$$503$$ −4.75172 −0.211869 −0.105934 0.994373i $$-0.533783\pi$$
−0.105934 + 0.994373i $$0.533783\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −4.73346 −0.210428
$$507$$ 0 0
$$508$$ 6.10957 0.271068
$$509$$ −10.8427 −0.480595 −0.240297 0.970699i $$-0.577245\pi$$
−0.240297 + 0.970699i $$0.577245\pi$$
$$510$$ 0 0
$$511$$ −50.8361 −2.24886
$$512$$ 24.7143 1.09223
$$513$$ 0 0
$$514$$ 17.2239 0.759712
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −26.6272 −1.17106
$$518$$ −37.1367 −1.63169
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −33.8929 −1.48487 −0.742437 0.669916i $$-0.766330\pi$$
−0.742437 + 0.669916i $$0.766330\pi$$
$$522$$ 0 0
$$523$$ 25.7652 1.12664 0.563318 0.826240i $$-0.309525\pi$$
0.563318 + 0.826240i $$0.309525\pi$$
$$524$$ −8.33853 −0.364270
$$525$$ 0 0
$$526$$ 31.9070 1.39121
$$527$$ 3.31554 0.144427
$$528$$ 0 0
$$529$$ −20.2357 −0.879812
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 5.54610 0.240454
$$533$$ 47.6496 2.06393
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −15.8922 −0.686438
$$537$$ 0 0
$$538$$ 14.1681 0.610832
$$539$$ 8.60473 0.370632
$$540$$ 0 0
$$541$$ 4.96888 0.213629 0.106814 0.994279i $$-0.465935\pi$$
0.106814 + 0.994279i $$0.465935\pi$$
$$542$$ 26.5563 1.14069
$$543$$ 0 0
$$544$$ −5.05509 −0.216735
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −9.03353 −0.386246 −0.193123 0.981175i $$-0.561862\pi$$
−0.193123 + 0.981175i $$0.561862\pi$$
$$548$$ −10.6289 −0.454046
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.59158 −0.153006
$$552$$ 0 0
$$553$$ 40.1287 1.70644
$$554$$ −8.86399 −0.376595
$$555$$ 0 0
$$556$$ 1.50839 0.0639699
$$557$$ −22.5265 −0.954477 −0.477238 0.878774i $$-0.658362\pi$$
−0.477238 + 0.878774i $$0.658362\pi$$
$$558$$ 0 0
$$559$$ −19.8716 −0.840479
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −17.3151 −0.730393
$$563$$ −14.0601 −0.592561 −0.296280 0.955101i $$-0.595746\pi$$
−0.296280 + 0.955101i $$0.595746\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −12.7986 −0.537967
$$567$$ 0 0
$$568$$ −18.4125 −0.772570
$$569$$ 5.59236 0.234444 0.117222 0.993106i $$-0.462601\pi$$
0.117222 + 0.993106i $$0.462601\pi$$
$$570$$ 0 0
$$571$$ −20.9909 −0.878443 −0.439221 0.898379i $$-0.644746\pi$$
−0.439221 + 0.898379i $$0.644746\pi$$
$$572$$ 6.05424 0.253140
$$573$$ 0 0
$$574$$ 34.6007 1.44421
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 25.7139 1.07048 0.535242 0.844699i $$-0.320221\pi$$
0.535242 + 0.844699i $$0.320221\pi$$
$$578$$ 16.3643 0.680665
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −35.1197 −1.45701
$$582$$ 0 0
$$583$$ −22.2502 −0.921510
$$584$$ −47.4042 −1.96160
$$585$$ 0 0
$$586$$ 32.8966 1.35894
$$587$$ 9.26970 0.382602 0.191301 0.981531i $$-0.438729\pi$$
0.191301 + 0.981531i $$0.438729\pi$$
$$588$$ 0 0
$$589$$ −6.13717 −0.252878
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −25.9901 −1.06819
$$593$$ 31.5912 1.29729 0.648647 0.761089i $$-0.275335\pi$$
0.648647 + 0.761089i $$0.275335\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0.288584 0.0118208
$$597$$ 0 0
$$598$$ 11.4695 0.469022
$$599$$ 13.1545 0.537480 0.268740 0.963213i $$-0.413393\pi$$
0.268740 + 0.963213i $$0.413393\pi$$
$$600$$ 0 0
$$601$$ 34.2150 1.39566 0.697829 0.716265i $$-0.254150\pi$$
0.697829 + 0.716265i $$0.254150\pi$$
$$602$$ −14.4297 −0.588113
$$603$$ 0 0
$$604$$ −5.86073 −0.238470
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −3.51939 −0.142848 −0.0714238 0.997446i $$-0.522754\pi$$
−0.0714238 + 0.997446i $$0.522754\pi$$
$$608$$ 9.35714 0.379482
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 64.5193 2.61017
$$612$$ 0 0
$$613$$ 5.90592 0.238538 0.119269 0.992862i $$-0.461945\pi$$
0.119269 + 0.992862i $$0.461945\pi$$
$$614$$ −36.2034 −1.46105
$$615$$ 0 0
$$616$$ 23.0538 0.928866
$$617$$ −30.3173 −1.22053 −0.610264 0.792198i $$-0.708937\pi$$
−0.610264 + 0.792198i $$0.708937\pi$$
$$618$$ 0 0
$$619$$ −23.1044 −0.928646 −0.464323 0.885666i $$-0.653702\pi$$
−0.464323 + 0.885666i $$0.653702\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 6.22049 0.249419
$$623$$ −8.31491 −0.333130
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 34.1098 1.36330
$$627$$ 0 0
$$628$$ −0.178437 −0.00712042
$$629$$ 17.7855 0.709155
$$630$$ 0 0
$$631$$ −27.6228 −1.09965 −0.549824 0.835280i $$-0.685305\pi$$
−0.549824 + 0.835280i $$0.685305\pi$$
$$632$$ 37.4196 1.48847
$$633$$ 0 0
$$634$$ −9.52865 −0.378431
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −20.8498 −0.826099
$$638$$ −2.84698 −0.112713
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −10.7286 −0.423756 −0.211878 0.977296i $$-0.567958\pi$$
−0.211878 + 0.977296i $$0.567958\pi$$
$$642$$ 0 0
$$643$$ 18.0482 0.711751 0.355876 0.934533i $$-0.384183\pi$$
0.355876 + 0.934533i $$0.384183\pi$$
$$644$$ −2.56742 −0.101171
$$645$$ 0 0
$$646$$ 8.61635 0.339006
$$647$$ −26.9395 −1.05910 −0.529551 0.848278i $$-0.677640\pi$$
−0.529551 + 0.848278i $$0.677640\pi$$
$$648$$ 0 0
$$649$$ −22.6433 −0.888829
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 1.19910 0.0469605
$$653$$ −13.4857 −0.527737 −0.263868 0.964559i $$-0.584998\pi$$
−0.263868 + 0.964559i $$0.584998\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 24.2153 0.945450
$$657$$ 0 0
$$658$$ 46.8507 1.82643
$$659$$ 27.5560 1.07343 0.536716 0.843763i $$-0.319665\pi$$
0.536716 + 0.843763i $$0.319665\pi$$
$$660$$ 0 0
$$661$$ 18.1148 0.704583 0.352292 0.935890i $$-0.385403\pi$$
0.352292 + 0.935890i $$0.385403\pi$$
$$662$$ −23.4849 −0.912768
$$663$$ 0 0
$$664$$ −32.7488 −1.27090
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.66263 0.0643772
$$668$$ −5.08015 −0.196557
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 12.4965 0.482422
$$672$$ 0 0
$$673$$ 45.5751 1.75679 0.878396 0.477933i $$-0.158614\pi$$
0.878396 + 0.477933i $$0.158614\pi$$
$$674$$ 24.9736 0.961948
$$675$$ 0 0
$$676$$ −8.54342 −0.328593
$$677$$ −10.2385 −0.393497 −0.196748 0.980454i $$-0.563038\pi$$
−0.196748 + 0.980454i $$0.563038\pi$$
$$678$$ 0 0
$$679$$ 19.3327 0.741920
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −4.86482 −0.186283
$$683$$ 41.9562 1.60541 0.802705 0.596376i $$-0.203393\pi$$
0.802705 + 0.596376i $$0.203393\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 13.2199 0.504738
$$687$$ 0 0
$$688$$ −10.0987 −0.385008
$$689$$ 53.9137 2.05395
$$690$$ 0 0
$$691$$ 50.6633 1.92732 0.963662 0.267126i $$-0.0860739\pi$$
0.963662 + 0.267126i $$0.0860739\pi$$
$$692$$ 10.8965 0.414224
$$693$$ 0 0
$$694$$ 29.4263 1.11701
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −16.5710 −0.627671
$$698$$ 28.2589 1.06961
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −17.2666 −0.652151 −0.326076 0.945344i $$-0.605726\pi$$
−0.326076 + 0.945344i $$0.605726\pi$$
$$702$$ 0 0
$$703$$ −32.9216 −1.24166
$$704$$ 20.4747 0.771671
$$705$$ 0 0
$$706$$ −21.6792 −0.815907
$$707$$ −14.2141 −0.534577
$$708$$ 0 0
$$709$$ −18.3307 −0.688425 −0.344213 0.938892i $$-0.611854\pi$$
−0.344213 + 0.938892i $$0.611854\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −7.75357 −0.290577
$$713$$ 2.84104 0.106398
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −11.1727 −0.417543
$$717$$ 0 0
$$718$$ −32.9622 −1.23014
$$719$$ 10.6931 0.398785 0.199392 0.979920i $$-0.436103\pi$$
0.199392 + 0.979920i $$0.436103\pi$$
$$720$$ 0 0
$$721$$ −24.9747 −0.930107
$$722$$ 7.54286 0.280716
$$723$$ 0 0
$$724$$ −5.50474 −0.204582
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 5.74209 0.212962 0.106481 0.994315i $$-0.466042\pi$$
0.106481 + 0.994315i $$0.466042\pi$$
$$728$$ −55.8609 −2.07034
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 6.91070 0.255601
$$732$$ 0 0
$$733$$ −6.03079 −0.222752 −0.111376 0.993778i $$-0.535526\pi$$
−0.111376 + 0.993778i $$0.535526\pi$$
$$734$$ 27.9269 1.03080
$$735$$ 0 0
$$736$$ −4.33164 −0.159667
$$737$$ −11.9761 −0.441146
$$738$$ 0 0
$$739$$ 23.7939 0.875273 0.437637 0.899152i $$-0.355816\pi$$
0.437637 + 0.899152i $$0.355816\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 39.1494 1.43722
$$743$$ −24.4388 −0.896574 −0.448287 0.893890i $$-0.647966\pi$$
−0.448287 + 0.893890i $$0.647966\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −13.7216 −0.502383
$$747$$ 0 0
$$748$$ −2.10547 −0.0769836
$$749$$ −13.7525 −0.502507
$$750$$ 0 0
$$751$$ 38.9811 1.42244 0.711220 0.702970i $$-0.248143\pi$$
0.711220 + 0.702970i $$0.248143\pi$$
$$752$$ 32.7885 1.19567
$$753$$ 0 0
$$754$$ 6.89840 0.251225
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 6.78690 0.246674 0.123337 0.992365i $$-0.460640\pi$$
0.123337 + 0.992365i $$0.460640\pi$$
$$758$$ 34.9257 1.26856
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −8.25388 −0.299203 −0.149601 0.988746i $$-0.547799\pi$$
−0.149601 + 0.988746i $$0.547799\pi$$
$$762$$ 0 0
$$763$$ 13.1946 0.477678
$$764$$ −4.09180 −0.148036
$$765$$ 0 0
$$766$$ 8.95833 0.323678
$$767$$ 54.8662 1.98110
$$768$$ 0 0
$$769$$ −12.3938 −0.446930 −0.223465 0.974712i $$-0.571737\pi$$
−0.223465 + 0.974712i $$0.571737\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −1.28997 −0.0464269
$$773$$ 38.1311 1.37148 0.685740 0.727846i $$-0.259479\pi$$
0.685740 + 0.727846i $$0.259479\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 18.0275 0.647150
$$777$$ 0 0
$$778$$ 42.5925 1.52701
$$779$$ 30.6735 1.09899
$$780$$ 0 0
$$781$$ −13.8753 −0.496499
$$782$$ −3.98872 −0.142636
$$783$$ 0 0
$$784$$ −10.5958 −0.378421
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −37.5823 −1.33967 −0.669833 0.742512i $$-0.733634\pi$$
−0.669833 + 0.742512i $$0.733634\pi$$
$$788$$ 12.3686 0.440612
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −24.2651 −0.862767
$$792$$ 0 0
$$793$$ −30.2798 −1.07527
$$794$$ 16.8663 0.598561
$$795$$ 0 0
$$796$$ −4.68810 −0.166165
$$797$$ 14.9045 0.527945 0.263972 0.964530i $$-0.414967\pi$$
0.263972 + 0.964530i $$0.414967\pi$$
$$798$$ 0 0
$$799$$ −22.4377 −0.793790
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 12.8268 0.452930
$$803$$ −35.7231 −1.26064
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 11.7878 0.415206
$$807$$ 0 0
$$808$$ −13.2545 −0.466292
$$809$$ −12.5838 −0.442424 −0.221212 0.975226i $$-0.571001\pi$$
−0.221212 + 0.975226i $$0.571001\pi$$
$$810$$ 0 0
$$811$$ −27.9894 −0.982841 −0.491420 0.870922i $$-0.663522\pi$$
−0.491420 + 0.870922i $$0.663522\pi$$
$$812$$ −1.54419 −0.0541906
$$813$$ 0 0
$$814$$ −26.0963 −0.914675
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −12.7919 −0.447533
$$818$$ −47.6509 −1.66608
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1.20609 −0.0420929 −0.0210464 0.999778i $$-0.506700\pi$$
−0.0210464 + 0.999778i $$0.506700\pi$$
$$822$$ 0 0
$$823$$ 25.7664 0.898162 0.449081 0.893491i $$-0.351751\pi$$
0.449081 + 0.893491i $$0.351751\pi$$
$$824$$ −23.2887 −0.811299
$$825$$ 0 0
$$826$$ 39.8411 1.38625
$$827$$ −56.1443 −1.95233 −0.976165 0.217029i $$-0.930363\pi$$
−0.976165 + 0.217029i $$0.930363\pi$$
$$828$$ 0 0
$$829$$ −14.6534 −0.508935 −0.254467 0.967081i $$-0.581900\pi$$
−0.254467 + 0.967081i $$0.581900\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −49.6116 −1.71997
$$833$$ 7.25089 0.251229
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 3.89730 0.134791
$$837$$ 0 0
$$838$$ 12.2106 0.421809
$$839$$ 8.72873 0.301349 0.150675 0.988583i $$-0.451855\pi$$
0.150675 + 0.988583i $$0.451855\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 12.7174 0.438271
$$843$$ 0 0
$$844$$ −3.65783 −0.125908
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −18.6710 −0.641545
$$848$$ 27.3987 0.940876
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 15.2402 0.522427
$$852$$ 0 0
$$853$$ 42.3533 1.45015 0.725075 0.688670i $$-0.241805\pi$$
0.725075 + 0.688670i $$0.241805\pi$$
$$854$$ −21.9876 −0.752402
$$855$$ 0 0
$$856$$ −12.8241 −0.438319
$$857$$ −49.8924 −1.70429 −0.852145 0.523305i $$-0.824699\pi$$
−0.852145 + 0.523305i $$0.824699\pi$$
$$858$$ 0 0
$$859$$ −33.4585 −1.14159 −0.570795 0.821092i $$-0.693365\pi$$
−0.570795 + 0.821092i $$0.693365\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −18.8807 −0.643081
$$863$$ −36.8016 −1.25274 −0.626371 0.779525i $$-0.715460\pi$$
−0.626371 + 0.779525i $$0.715460\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 17.1084 0.581368
$$867$$ 0 0
$$868$$ −2.63867 −0.0895622
$$869$$ 28.1988 0.956579
$$870$$ 0 0
$$871$$ 29.0189 0.983267
$$872$$ 12.3039 0.416662
$$873$$ 0 0
$$874$$ 7.38325 0.249742
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 19.7336 0.666358 0.333179 0.942864i $$-0.391879\pi$$
0.333179 + 0.942864i $$0.391879\pi$$
$$878$$ 8.02235 0.270741
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 38.2999 1.29036 0.645179 0.764032i $$-0.276783\pi$$
0.645179 + 0.764032i $$0.276783\pi$$
$$882$$ 0 0
$$883$$ −32.9689 −1.10949 −0.554745 0.832020i $$-0.687184\pi$$
−0.554745 + 0.832020i $$0.687184\pi$$
$$884$$ 5.10169 0.171588
$$885$$ 0 0
$$886$$ 27.3281 0.918105
$$887$$ 15.0022 0.503724 0.251862 0.967763i $$-0.418957\pi$$
0.251862 + 0.967763i $$0.418957\pi$$
$$888$$ 0 0
$$889$$ −42.4806 −1.42475
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −4.54284 −0.152105
$$893$$ 41.5330 1.38985
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −18.9517 −0.633131
$$897$$ 0 0
$$898$$ 0.305647 0.0101996
$$899$$ 1.70877 0.0569905
$$900$$ 0 0
$$901$$ −18.7494 −0.624634
$$902$$ 24.3143 0.809577
$$903$$ 0 0
$$904$$ −22.6269 −0.752561
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 45.4276 1.50840 0.754199 0.656646i $$-0.228026\pi$$
0.754199 + 0.656646i $$0.228026\pi$$
$$908$$ 4.34411 0.144164
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 59.7103 1.97829 0.989146 0.146937i $$-0.0469415\pi$$
0.989146 + 0.146937i $$0.0469415\pi$$
$$912$$ 0 0
$$913$$ −24.6790 −0.816755
$$914$$ −43.7882 −1.44839
$$915$$ 0 0
$$916$$ 3.24975 0.107375
$$917$$ 57.9789 1.91463
$$918$$ 0 0
$$919$$ −1.59828 −0.0527224 −0.0263612 0.999652i $$-0.508392\pi$$
−0.0263612 + 0.999652i $$0.508392\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 16.7770 0.552520
$$923$$ 33.6208 1.10664
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −9.74569 −0.320263
$$927$$ 0 0
$$928$$ −2.60530 −0.0855231
$$929$$ 35.5116 1.16510 0.582549 0.812795i $$-0.302055\pi$$
0.582549 + 0.812795i $$0.302055\pi$$
$$930$$ 0 0
$$931$$ −13.4216 −0.439877
$$932$$ −6.41994 −0.210292
$$933$$ 0 0
$$934$$ 25.7684 0.843168
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −52.7492 −1.72324 −0.861621 0.507552i $$-0.830551\pi$$
−0.861621 + 0.507552i $$0.830551\pi$$
$$938$$ 21.0720 0.688026
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −27.2935 −0.889743 −0.444872 0.895594i $$-0.646751\pi$$
−0.444872 + 0.895594i $$0.646751\pi$$
$$942$$ 0 0
$$943$$ −14.1995 −0.462399
$$944$$ 27.8828 0.907508
$$945$$ 0 0
$$946$$ −10.1399 −0.329677
$$947$$ −21.1187 −0.686266 −0.343133 0.939287i $$-0.611488\pi$$
−0.343133 + 0.939287i $$0.611488\pi$$
$$948$$ 0 0
$$949$$ 86.5592 2.80983
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 19.4266 0.629621
$$953$$ −3.73049 −0.120842 −0.0604211 0.998173i $$-0.519244\pi$$
−0.0604211 + 0.998173i $$0.519244\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0.353042 0.0114182
$$957$$ 0 0
$$958$$ 35.8066 1.15686
$$959$$ 73.9043 2.38650
$$960$$ 0 0
$$961$$ −28.0801 −0.905810
$$962$$ 63.2330 2.03871
$$963$$ 0 0
$$964$$ 5.90867 0.190305
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −22.8453 −0.734654 −0.367327 0.930092i $$-0.619727\pi$$
−0.367327 + 0.930092i $$0.619727\pi$$
$$968$$ −17.4106 −0.559597
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −42.3917 −1.36042 −0.680208 0.733019i $$-0.738110\pi$$
−0.680208 + 0.733019i $$0.738110\pi$$
$$972$$ 0 0
$$973$$ −10.4880 −0.336230
$$974$$ −28.8524 −0.924492
$$975$$ 0 0
$$976$$ −15.3881 −0.492560
$$977$$ 0.0842590 0.00269568 0.00134784 0.999999i $$-0.499571\pi$$
0.00134784 + 0.999999i $$0.499571\pi$$
$$978$$ 0 0
$$979$$ −5.84297 −0.186742
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −32.1409 −1.02566
$$983$$ −15.0122 −0.478813 −0.239407 0.970919i $$-0.576953\pi$$
−0.239407 + 0.970919i $$0.576953\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2.39904 −0.0764011
$$987$$ 0 0
$$988$$ −9.44339 −0.300434
$$989$$ 5.92170 0.188299
$$990$$ 0 0
$$991$$ 3.36536 0.106904 0.0534521 0.998570i $$-0.482978\pi$$
0.0534521 + 0.998570i $$0.482978\pi$$
$$992$$ −4.45185 −0.141346
$$993$$ 0 0
$$994$$ 24.4137 0.774357
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.94573 0.0616220 0.0308110 0.999525i $$-0.490191\pi$$
0.0308110 + 0.999525i $$0.490191\pi$$
$$998$$ 39.5130 1.25076
$$999$$ 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.ca.1.4 9
3.2 odd 2 6525.2.a.cc.1.6 yes 9
5.4 even 2 6525.2.a.cd.1.6 yes 9
15.14 odd 2 6525.2.a.cb.1.4 yes 9

By twisted newform
Twist Min Dim Char Parity Ord Type
6525.2.a.ca.1.4 9 1.1 even 1 trivial
6525.2.a.cb.1.4 yes 9 15.14 odd 2
6525.2.a.cc.1.6 yes 9 3.2 odd 2
6525.2.a.cd.1.6 yes 9 5.4 even 2