Properties

 Label 9680.2.a.dc.1.3 Level $9680$ Weight $2$ Character 9680.1 Self dual yes Analytic conductor $77.295$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.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: $$77.2951891566$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.25903625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5$$ x^6 - 3*x^5 - 7*x^4 + 17*x^3 + 16*x^2 - 20*x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$-0.220878$$ of defining polynomial Character $$\chi$$ $$=$$ 9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.220878 q^{3} -1.00000 q^{5} +2.08772 q^{7} -2.95121 q^{9} +O(q^{10})$$ $$q-0.220878 q^{3} -1.00000 q^{5} +2.08772 q^{7} -2.95121 q^{9} +1.87868 q^{13} +0.220878 q^{15} +5.28462 q^{17} +2.92388 q^{19} -0.461133 q^{21} -5.21521 q^{23} +1.00000 q^{25} +1.31449 q^{27} -3.81157 q^{29} -6.68385 q^{31} -2.08772 q^{35} -3.45754 q^{37} -0.414959 q^{39} +5.80287 q^{41} -0.884108 q^{43} +2.95121 q^{45} -7.69696 q^{47} -2.64141 q^{49} -1.16726 q^{51} -9.50720 q^{53} -0.645820 q^{57} +7.17454 q^{59} -7.88806 q^{61} -6.16132 q^{63} -1.87868 q^{65} +1.32232 q^{67} +1.15193 q^{69} +10.2325 q^{71} +8.17103 q^{73} -0.220878 q^{75} -5.23101 q^{79} +8.56330 q^{81} +9.63433 q^{83} -5.28462 q^{85} +0.841894 q^{87} -17.0929 q^{89} +3.92216 q^{91} +1.47632 q^{93} -2.92388 q^{95} -15.3296 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9}+O(q^{10})$$ 6 * q + 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 $$6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} - q^{13} - 3 q^{15} - 6 q^{17} - 7 q^{19} - 14 q^{21} + 9 q^{23} + 6 q^{25} + 21 q^{27} - 10 q^{29} - q^{31} + 7 q^{35} + 3 q^{37} - q^{39} + 6 q^{41} + 18 q^{43} - 5 q^{45} + 3 q^{47} + 17 q^{49} + 15 q^{51} - 23 q^{53} - 9 q^{57} + 2 q^{59} - 6 q^{61} - 49 q^{63} + q^{65} + 22 q^{67} + 2 q^{69} + 13 q^{71} - 10 q^{73} + 3 q^{75} - 22 q^{79} + 10 q^{81} + 10 q^{83} + 6 q^{85} - 3 q^{87} - 25 q^{89} - 12 q^{91} + 19 q^{93} + 7 q^{95} - 33 q^{97}+O(q^{100})$$ 6 * q + 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 - q^13 - 3 * q^15 - 6 * q^17 - 7 * q^19 - 14 * q^21 + 9 * q^23 + 6 * q^25 + 21 * q^27 - 10 * q^29 - q^31 + 7 * q^35 + 3 * q^37 - q^39 + 6 * q^41 + 18 * q^43 - 5 * q^45 + 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 - 9 * q^57 + 2 * q^59 - 6 * q^61 - 49 * q^63 + q^65 + 22 * q^67 + 2 * q^69 + 13 * q^71 - 10 * q^73 + 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 + 6 * q^85 - 3 * q^87 - 25 * q^89 - 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −0.220878 −0.127524 −0.0637621 0.997965i $$-0.520310\pi$$
−0.0637621 + 0.997965i $$0.520310\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 2.08772 0.789085 0.394543 0.918878i $$-0.370903\pi$$
0.394543 + 0.918878i $$0.370903\pi$$
$$8$$ 0 0
$$9$$ −2.95121 −0.983738
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 1.87868 0.521052 0.260526 0.965467i $$-0.416104\pi$$
0.260526 + 0.965467i $$0.416104\pi$$
$$14$$ 0 0
$$15$$ 0.220878 0.0570305
$$16$$ 0 0
$$17$$ 5.28462 1.28171 0.640854 0.767663i $$-0.278580\pi$$
0.640854 + 0.767663i $$0.278580\pi$$
$$18$$ 0 0
$$19$$ 2.92388 0.670783 0.335391 0.942079i $$-0.391131\pi$$
0.335391 + 0.942079i $$0.391131\pi$$
$$20$$ 0 0
$$21$$ −0.461133 −0.100627
$$22$$ 0 0
$$23$$ −5.21521 −1.08745 −0.543724 0.839264i $$-0.682986\pi$$
−0.543724 + 0.839264i $$0.682986\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.31449 0.252974
$$28$$ 0 0
$$29$$ −3.81157 −0.707792 −0.353896 0.935285i $$-0.615143\pi$$
−0.353896 + 0.935285i $$0.615143\pi$$
$$30$$ 0 0
$$31$$ −6.68385 −1.20045 −0.600227 0.799829i $$-0.704923\pi$$
−0.600227 + 0.799829i $$0.704923\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.08772 −0.352890
$$36$$ 0 0
$$37$$ −3.45754 −0.568416 −0.284208 0.958763i $$-0.591731\pi$$
−0.284208 + 0.958763i $$0.591731\pi$$
$$38$$ 0 0
$$39$$ −0.414959 −0.0664467
$$40$$ 0 0
$$41$$ 5.80287 0.906256 0.453128 0.891446i $$-0.350308\pi$$
0.453128 + 0.891446i $$0.350308\pi$$
$$42$$ 0 0
$$43$$ −0.884108 −0.134825 −0.0674126 0.997725i $$-0.521474\pi$$
−0.0674126 + 0.997725i $$0.521474\pi$$
$$44$$ 0 0
$$45$$ 2.95121 0.439941
$$46$$ 0 0
$$47$$ −7.69696 −1.12272 −0.561359 0.827573i $$-0.689721\pi$$
−0.561359 + 0.827573i $$0.689721\pi$$
$$48$$ 0 0
$$49$$ −2.64141 −0.377345
$$50$$ 0 0
$$51$$ −1.16726 −0.163449
$$52$$ 0 0
$$53$$ −9.50720 −1.30591 −0.652957 0.757395i $$-0.726472\pi$$
−0.652957 + 0.757395i $$0.726472\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.645820 −0.0855410
$$58$$ 0 0
$$59$$ 7.17454 0.934045 0.467023 0.884245i $$-0.345327\pi$$
0.467023 + 0.884245i $$0.345327\pi$$
$$60$$ 0 0
$$61$$ −7.88806 −1.00996 −0.504981 0.863130i $$-0.668501\pi$$
−0.504981 + 0.863130i $$0.668501\pi$$
$$62$$ 0 0
$$63$$ −6.16132 −0.776253
$$64$$ 0 0
$$65$$ −1.87868 −0.233022
$$66$$ 0 0
$$67$$ 1.32232 0.161547 0.0807733 0.996732i $$-0.474261\pi$$
0.0807733 + 0.996732i $$0.474261\pi$$
$$68$$ 0 0
$$69$$ 1.15193 0.138676
$$70$$ 0 0
$$71$$ 10.2325 1.21437 0.607186 0.794560i $$-0.292298\pi$$
0.607186 + 0.794560i $$0.292298\pi$$
$$72$$ 0 0
$$73$$ 8.17103 0.956346 0.478173 0.878266i $$-0.341299\pi$$
0.478173 + 0.878266i $$0.341299\pi$$
$$74$$ 0 0
$$75$$ −0.220878 −0.0255048
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −5.23101 −0.588534 −0.294267 0.955723i $$-0.595076\pi$$
−0.294267 + 0.955723i $$0.595076\pi$$
$$80$$ 0 0
$$81$$ 8.56330 0.951477
$$82$$ 0 0
$$83$$ 9.63433 1.05750 0.528752 0.848776i $$-0.322660\pi$$
0.528752 + 0.848776i $$0.322660\pi$$
$$84$$ 0 0
$$85$$ −5.28462 −0.573197
$$86$$ 0 0
$$87$$ 0.841894 0.0902605
$$88$$ 0 0
$$89$$ −17.0929 −1.81185 −0.905923 0.423443i $$-0.860821\pi$$
−0.905923 + 0.423443i $$0.860821\pi$$
$$90$$ 0 0
$$91$$ 3.92216 0.411154
$$92$$ 0 0
$$93$$ 1.47632 0.153087
$$94$$ 0 0
$$95$$ −2.92388 −0.299983
$$96$$ 0 0
$$97$$ −15.3296 −1.55648 −0.778242 0.627964i $$-0.783888\pi$$
−0.778242 + 0.627964i $$0.783888\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 15.5788 1.55015 0.775075 0.631869i $$-0.217712\pi$$
0.775075 + 0.631869i $$0.217712\pi$$
$$102$$ 0 0
$$103$$ 14.5038 1.42910 0.714549 0.699586i $$-0.246632\pi$$
0.714549 + 0.699586i $$0.246632\pi$$
$$104$$ 0 0
$$105$$ 0.461133 0.0450019
$$106$$ 0 0
$$107$$ −16.8702 −1.63090 −0.815450 0.578828i $$-0.803510\pi$$
−0.815450 + 0.578828i $$0.803510\pi$$
$$108$$ 0 0
$$109$$ −9.97408 −0.955344 −0.477672 0.878538i $$-0.658519\pi$$
−0.477672 + 0.878538i $$0.658519\pi$$
$$110$$ 0 0
$$111$$ 0.763696 0.0724868
$$112$$ 0 0
$$113$$ 0.00824325 0.000775459 0 0.000387730 1.00000i $$-0.499877\pi$$
0.000387730 1.00000i $$0.499877\pi$$
$$114$$ 0 0
$$115$$ 5.21521 0.486321
$$116$$ 0 0
$$117$$ −5.54438 −0.512578
$$118$$ 0 0
$$119$$ 11.0328 1.01138
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −1.28173 −0.115569
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 10.0723 0.893769 0.446885 0.894592i $$-0.352533\pi$$
0.446885 + 0.894592i $$0.352533\pi$$
$$128$$ 0 0
$$129$$ 0.195280 0.0171935
$$130$$ 0 0
$$131$$ 18.0249 1.57484 0.787422 0.616415i $$-0.211415\pi$$
0.787422 + 0.616415i $$0.211415\pi$$
$$132$$ 0 0
$$133$$ 6.10424 0.529305
$$134$$ 0 0
$$135$$ −1.31449 −0.113134
$$136$$ 0 0
$$137$$ 5.15806 0.440683 0.220341 0.975423i $$-0.429283\pi$$
0.220341 + 0.975423i $$0.429283\pi$$
$$138$$ 0 0
$$139$$ 1.61222 0.136747 0.0683735 0.997660i $$-0.478219\pi$$
0.0683735 + 0.997660i $$0.478219\pi$$
$$140$$ 0 0
$$141$$ 1.70009 0.143174
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 3.81157 0.316534
$$146$$ 0 0
$$147$$ 0.583431 0.0481205
$$148$$ 0 0
$$149$$ 3.00244 0.245970 0.122985 0.992409i $$-0.460753\pi$$
0.122985 + 0.992409i $$0.460753\pi$$
$$150$$ 0 0
$$151$$ −0.831273 −0.0676480 −0.0338240 0.999428i $$-0.510769\pi$$
−0.0338240 + 0.999428i $$0.510769\pi$$
$$152$$ 0 0
$$153$$ −15.5960 −1.26086
$$154$$ 0 0
$$155$$ 6.68385 0.536860
$$156$$ 0 0
$$157$$ −13.3102 −1.06227 −0.531133 0.847288i $$-0.678234\pi$$
−0.531133 + 0.847288i $$0.678234\pi$$
$$158$$ 0 0
$$159$$ 2.09993 0.166536
$$160$$ 0 0
$$161$$ −10.8879 −0.858088
$$162$$ 0 0
$$163$$ 6.36304 0.498392 0.249196 0.968453i $$-0.419834\pi$$
0.249196 + 0.968453i $$0.419834\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2.25595 0.174571 0.0872854 0.996183i $$-0.472181\pi$$
0.0872854 + 0.996183i $$0.472181\pi$$
$$168$$ 0 0
$$169$$ −9.47056 −0.728505
$$170$$ 0 0
$$171$$ −8.62898 −0.659874
$$172$$ 0 0
$$173$$ −10.8245 −0.822968 −0.411484 0.911417i $$-0.634989\pi$$
−0.411484 + 0.911417i $$0.634989\pi$$
$$174$$ 0 0
$$175$$ 2.08772 0.157817
$$176$$ 0 0
$$177$$ −1.58470 −0.119113
$$178$$ 0 0
$$179$$ 11.5996 0.866993 0.433496 0.901155i $$-0.357280\pi$$
0.433496 + 0.901155i $$0.357280\pi$$
$$180$$ 0 0
$$181$$ −3.77485 −0.280582 −0.140291 0.990110i $$-0.544804\pi$$
−0.140291 + 0.990110i $$0.544804\pi$$
$$182$$ 0 0
$$183$$ 1.74230 0.128795
$$184$$ 0 0
$$185$$ 3.45754 0.254203
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 2.74430 0.199618
$$190$$ 0 0
$$191$$ 21.5286 1.55775 0.778876 0.627178i $$-0.215790\pi$$
0.778876 + 0.627178i $$0.215790\pi$$
$$192$$ 0 0
$$193$$ −17.2964 −1.24503 −0.622513 0.782610i $$-0.713888\pi$$
−0.622513 + 0.782610i $$0.713888\pi$$
$$194$$ 0 0
$$195$$ 0.414959 0.0297159
$$196$$ 0 0
$$197$$ 8.27858 0.589824 0.294912 0.955524i $$-0.404710\pi$$
0.294912 + 0.955524i $$0.404710\pi$$
$$198$$ 0 0
$$199$$ −14.4762 −1.02619 −0.513096 0.858331i $$-0.671501\pi$$
−0.513096 + 0.858331i $$0.671501\pi$$
$$200$$ 0 0
$$201$$ −0.292071 −0.0206011
$$202$$ 0 0
$$203$$ −7.95751 −0.558508
$$204$$ 0 0
$$205$$ −5.80287 −0.405290
$$206$$ 0 0
$$207$$ 15.3912 1.06976
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.12519 −0.559361 −0.279681 0.960093i $$-0.590229\pi$$
−0.279681 + 0.960093i $$0.590229\pi$$
$$212$$ 0 0
$$213$$ −2.26013 −0.154862
$$214$$ 0 0
$$215$$ 0.884108 0.0602957
$$216$$ 0 0
$$217$$ −13.9540 −0.947261
$$218$$ 0 0
$$219$$ −1.80480 −0.121957
$$220$$ 0 0
$$221$$ 9.92810 0.667837
$$222$$ 0 0
$$223$$ −12.3446 −0.826658 −0.413329 0.910582i $$-0.635634\pi$$
−0.413329 + 0.910582i $$0.635634\pi$$
$$224$$ 0 0
$$225$$ −2.95121 −0.196748
$$226$$ 0 0
$$227$$ −19.0269 −1.26286 −0.631430 0.775432i $$-0.717532\pi$$
−0.631430 + 0.775432i $$0.717532\pi$$
$$228$$ 0 0
$$229$$ 3.14648 0.207926 0.103963 0.994581i $$-0.466848\pi$$
0.103963 + 0.994581i $$0.466848\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −15.8983 −1.04153 −0.520767 0.853699i $$-0.674354\pi$$
−0.520767 + 0.853699i $$0.674354\pi$$
$$234$$ 0 0
$$235$$ 7.69696 0.502095
$$236$$ 0 0
$$237$$ 1.15542 0.0750523
$$238$$ 0 0
$$239$$ −15.7958 −1.02174 −0.510872 0.859657i $$-0.670677\pi$$
−0.510872 + 0.859657i $$0.670677\pi$$
$$240$$ 0 0
$$241$$ −16.2927 −1.04950 −0.524752 0.851255i $$-0.675842\pi$$
−0.524752 + 0.851255i $$0.675842\pi$$
$$242$$ 0 0
$$243$$ −5.83493 −0.374311
$$244$$ 0 0
$$245$$ 2.64141 0.168754
$$246$$ 0 0
$$247$$ 5.49302 0.349513
$$248$$ 0 0
$$249$$ −2.12801 −0.134857
$$250$$ 0 0
$$251$$ −9.75176 −0.615525 −0.307763 0.951463i $$-0.599580\pi$$
−0.307763 + 0.951463i $$0.599580\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 1.16726 0.0730965
$$256$$ 0 0
$$257$$ −17.9647 −1.12061 −0.560303 0.828288i $$-0.689315\pi$$
−0.560303 + 0.828288i $$0.689315\pi$$
$$258$$ 0 0
$$259$$ −7.21839 −0.448529
$$260$$ 0 0
$$261$$ 11.2488 0.696281
$$262$$ 0 0
$$263$$ −11.6324 −0.717284 −0.358642 0.933475i $$-0.616760\pi$$
−0.358642 + 0.933475i $$0.616760\pi$$
$$264$$ 0 0
$$265$$ 9.50720 0.584023
$$266$$ 0 0
$$267$$ 3.77545 0.231054
$$268$$ 0 0
$$269$$ −11.8652 −0.723437 −0.361718 0.932287i $$-0.617810\pi$$
−0.361718 + 0.932287i $$0.617810\pi$$
$$270$$ 0 0
$$271$$ −12.5270 −0.760963 −0.380481 0.924789i $$-0.624242\pi$$
−0.380481 + 0.924789i $$0.624242\pi$$
$$272$$ 0 0
$$273$$ −0.866320 −0.0524321
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 27.4409 1.64876 0.824381 0.566035i $$-0.191523\pi$$
0.824381 + 0.566035i $$0.191523\pi$$
$$278$$ 0 0
$$279$$ 19.7255 1.18093
$$280$$ 0 0
$$281$$ 25.7658 1.53706 0.768530 0.639814i $$-0.220989\pi$$
0.768530 + 0.639814i $$0.220989\pi$$
$$282$$ 0 0
$$283$$ −10.2068 −0.606731 −0.303366 0.952874i $$-0.598110\pi$$
−0.303366 + 0.952874i $$0.598110\pi$$
$$284$$ 0 0
$$285$$ 0.645820 0.0382551
$$286$$ 0 0
$$287$$ 12.1148 0.715113
$$288$$ 0 0
$$289$$ 10.9272 0.642776
$$290$$ 0 0
$$291$$ 3.38597 0.198489
$$292$$ 0 0
$$293$$ −26.0333 −1.52088 −0.760441 0.649407i $$-0.775017\pi$$
−0.760441 + 0.649407i $$0.775017\pi$$
$$294$$ 0 0
$$295$$ −7.17454 −0.417718
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −9.79771 −0.566616
$$300$$ 0 0
$$301$$ −1.84577 −0.106389
$$302$$ 0 0
$$303$$ −3.44102 −0.197682
$$304$$ 0 0
$$305$$ 7.88806 0.451669
$$306$$ 0 0
$$307$$ 10.1591 0.579808 0.289904 0.957056i $$-0.406377\pi$$
0.289904 + 0.957056i $$0.406377\pi$$
$$308$$ 0 0
$$309$$ −3.20356 −0.182244
$$310$$ 0 0
$$311$$ −6.32891 −0.358879 −0.179440 0.983769i $$-0.557428\pi$$
−0.179440 + 0.983769i $$0.557428\pi$$
$$312$$ 0 0
$$313$$ −21.9366 −1.23993 −0.619964 0.784630i $$-0.712853\pi$$
−0.619964 + 0.784630i $$0.712853\pi$$
$$314$$ 0 0
$$315$$ 6.16132 0.347151
$$316$$ 0 0
$$317$$ −24.2225 −1.36047 −0.680237 0.732993i $$-0.738123\pi$$
−0.680237 + 0.732993i $$0.738123\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 3.72625 0.207979
$$322$$ 0 0
$$323$$ 15.4516 0.859748
$$324$$ 0 0
$$325$$ 1.87868 0.104210
$$326$$ 0 0
$$327$$ 2.20306 0.121829
$$328$$ 0 0
$$329$$ −16.0691 −0.885920
$$330$$ 0 0
$$331$$ −29.3673 −1.61417 −0.807086 0.590433i $$-0.798957\pi$$
−0.807086 + 0.590433i $$0.798957\pi$$
$$332$$ 0 0
$$333$$ 10.2039 0.559172
$$334$$ 0 0
$$335$$ −1.32232 −0.0722458
$$336$$ 0 0
$$337$$ −5.77943 −0.314825 −0.157413 0.987533i $$-0.550315\pi$$
−0.157413 + 0.987533i $$0.550315\pi$$
$$338$$ 0 0
$$339$$ −0.00182075 −9.88898e−5 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.1286 −1.08684
$$344$$ 0 0
$$345$$ −1.15193 −0.0620177
$$346$$ 0 0
$$347$$ 31.9559 1.71548 0.857742 0.514081i $$-0.171867\pi$$
0.857742 + 0.514081i $$0.171867\pi$$
$$348$$ 0 0
$$349$$ −10.9895 −0.588253 −0.294126 0.955767i $$-0.595029\pi$$
−0.294126 + 0.955767i $$0.595029\pi$$
$$350$$ 0 0
$$351$$ 2.46951 0.131813
$$352$$ 0 0
$$353$$ −19.8860 −1.05842 −0.529212 0.848489i $$-0.677513\pi$$
−0.529212 + 0.848489i $$0.677513\pi$$
$$354$$ 0 0
$$355$$ −10.2325 −0.543083
$$356$$ 0 0
$$357$$ −2.43691 −0.128975
$$358$$ 0 0
$$359$$ −33.1513 −1.74966 −0.874830 0.484429i $$-0.839027\pi$$
−0.874830 + 0.484429i $$0.839027\pi$$
$$360$$ 0 0
$$361$$ −10.4510 −0.550050
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8.17103 −0.427691
$$366$$ 0 0
$$367$$ −20.2195 −1.05545 −0.527723 0.849416i $$-0.676954\pi$$
−0.527723 + 0.849416i $$0.676954\pi$$
$$368$$ 0 0
$$369$$ −17.1255 −0.891518
$$370$$ 0 0
$$371$$ −19.8484 −1.03048
$$372$$ 0 0
$$373$$ −11.6099 −0.601136 −0.300568 0.953760i $$-0.597176\pi$$
−0.300568 + 0.953760i $$0.597176\pi$$
$$374$$ 0 0
$$375$$ 0.220878 0.0114061
$$376$$ 0 0
$$377$$ −7.16073 −0.368796
$$378$$ 0 0
$$379$$ 2.61724 0.134438 0.0672192 0.997738i $$-0.478587\pi$$
0.0672192 + 0.997738i $$0.478587\pi$$
$$380$$ 0 0
$$381$$ −2.22475 −0.113977
$$382$$ 0 0
$$383$$ −7.75206 −0.396112 −0.198056 0.980191i $$-0.563463\pi$$
−0.198056 + 0.980191i $$0.563463\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.60919 0.132633
$$388$$ 0 0
$$389$$ 19.8235 1.00509 0.502545 0.864551i $$-0.332397\pi$$
0.502545 + 0.864551i $$0.332397\pi$$
$$390$$ 0 0
$$391$$ −27.5604 −1.39379
$$392$$ 0 0
$$393$$ −3.98131 −0.200831
$$394$$ 0 0
$$395$$ 5.23101 0.263201
$$396$$ 0 0
$$397$$ −17.8853 −0.897636 −0.448818 0.893623i $$-0.648155\pi$$
−0.448818 + 0.893623i $$0.648155\pi$$
$$398$$ 0 0
$$399$$ −1.34829 −0.0674991
$$400$$ 0 0
$$401$$ 34.2258 1.70916 0.854578 0.519323i $$-0.173816\pi$$
0.854578 + 0.519323i $$0.173816\pi$$
$$402$$ 0 0
$$403$$ −12.5568 −0.625499
$$404$$ 0 0
$$405$$ −8.56330 −0.425514
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −2.21701 −0.109624 −0.0548121 0.998497i $$-0.517456\pi$$
−0.0548121 + 0.998497i $$0.517456\pi$$
$$410$$ 0 0
$$411$$ −1.13930 −0.0561977
$$412$$ 0 0
$$413$$ 14.9784 0.737041
$$414$$ 0 0
$$415$$ −9.63433 −0.472930
$$416$$ 0 0
$$417$$ −0.356105 −0.0174385
$$418$$ 0 0
$$419$$ 17.7908 0.869137 0.434569 0.900639i $$-0.356901\pi$$
0.434569 + 0.900639i $$0.356901\pi$$
$$420$$ 0 0
$$421$$ 35.7958 1.74458 0.872290 0.488989i $$-0.162634\pi$$
0.872290 + 0.488989i $$0.162634\pi$$
$$422$$ 0 0
$$423$$ 22.7154 1.10446
$$424$$ 0 0
$$425$$ 5.28462 0.256342
$$426$$ 0 0
$$427$$ −16.4681 −0.796946
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.10027 0.149335 0.0746673 0.997209i $$-0.476211\pi$$
0.0746673 + 0.997209i $$0.476211\pi$$
$$432$$ 0 0
$$433$$ 19.3592 0.930345 0.465172 0.885220i $$-0.345992\pi$$
0.465172 + 0.885220i $$0.345992\pi$$
$$434$$ 0 0
$$435$$ −0.841894 −0.0403657
$$436$$ 0 0
$$437$$ −15.2486 −0.729441
$$438$$ 0 0
$$439$$ −40.6115 −1.93828 −0.969142 0.246504i $$-0.920718\pi$$
−0.969142 + 0.246504i $$0.920718\pi$$
$$440$$ 0 0
$$441$$ 7.79537 0.371208
$$442$$ 0 0
$$443$$ 21.5805 1.02532 0.512661 0.858591i $$-0.328660\pi$$
0.512661 + 0.858591i $$0.328660\pi$$
$$444$$ 0 0
$$445$$ 17.0929 0.810282
$$446$$ 0 0
$$447$$ −0.663174 −0.0313671
$$448$$ 0 0
$$449$$ −34.4259 −1.62466 −0.812330 0.583198i $$-0.801801\pi$$
−0.812330 + 0.583198i $$0.801801\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0.183610 0.00862676
$$454$$ 0 0
$$455$$ −3.92216 −0.183874
$$456$$ 0 0
$$457$$ −12.4701 −0.583327 −0.291664 0.956521i $$-0.594209\pi$$
−0.291664 + 0.956521i $$0.594209\pi$$
$$458$$ 0 0
$$459$$ 6.94660 0.324239
$$460$$ 0 0
$$461$$ −23.5302 −1.09591 −0.547956 0.836507i $$-0.684594\pi$$
−0.547956 + 0.836507i $$0.684594\pi$$
$$462$$ 0 0
$$463$$ 29.5796 1.37468 0.687340 0.726336i $$-0.258778\pi$$
0.687340 + 0.726336i $$0.258778\pi$$
$$464$$ 0 0
$$465$$ −1.47632 −0.0684626
$$466$$ 0 0
$$467$$ 21.8760 1.01230 0.506150 0.862446i $$-0.331068\pi$$
0.506150 + 0.862446i $$0.331068\pi$$
$$468$$ 0 0
$$469$$ 2.76063 0.127474
$$470$$ 0 0
$$471$$ 2.93992 0.135465
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 2.92388 0.134157
$$476$$ 0 0
$$477$$ 28.0578 1.28468
$$478$$ 0 0
$$479$$ 30.9759 1.41533 0.707664 0.706550i $$-0.249749\pi$$
0.707664 + 0.706550i $$0.249749\pi$$
$$480$$ 0 0
$$481$$ −6.49561 −0.296174
$$482$$ 0 0
$$483$$ 2.40491 0.109427
$$484$$ 0 0
$$485$$ 15.3296 0.696081
$$486$$ 0 0
$$487$$ 9.02364 0.408900 0.204450 0.978877i $$-0.434459\pi$$
0.204450 + 0.978877i $$0.434459\pi$$
$$488$$ 0 0
$$489$$ −1.40546 −0.0635570
$$490$$ 0 0
$$491$$ 29.4524 1.32917 0.664583 0.747214i $$-0.268609\pi$$
0.664583 + 0.747214i $$0.268609\pi$$
$$492$$ 0 0
$$493$$ −20.1427 −0.907182
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 21.3626 0.958243
$$498$$ 0 0
$$499$$ −13.2183 −0.591732 −0.295866 0.955229i $$-0.595608\pi$$
−0.295866 + 0.955229i $$0.595608\pi$$
$$500$$ 0 0
$$501$$ −0.498291 −0.0222620
$$502$$ 0 0
$$503$$ 8.17945 0.364704 0.182352 0.983233i $$-0.441629\pi$$
0.182352 + 0.983233i $$0.441629\pi$$
$$504$$ 0 0
$$505$$ −15.5788 −0.693248
$$506$$ 0 0
$$507$$ 2.09184 0.0929019
$$508$$ 0 0
$$509$$ −7.71064 −0.341768 −0.170884 0.985291i $$-0.554662\pi$$
−0.170884 + 0.985291i $$0.554662\pi$$
$$510$$ 0 0
$$511$$ 17.0588 0.754639
$$512$$ 0 0
$$513$$ 3.84342 0.169691
$$514$$ 0 0
$$515$$ −14.5038 −0.639112
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 2.39089 0.104948
$$520$$ 0 0
$$521$$ −22.4464 −0.983396 −0.491698 0.870766i $$-0.663624\pi$$
−0.491698 + 0.870766i $$0.663624\pi$$
$$522$$ 0 0
$$523$$ 4.50517 0.196997 0.0984987 0.995137i $$-0.468596\pi$$
0.0984987 + 0.995137i $$0.468596\pi$$
$$524$$ 0 0
$$525$$ −0.461133 −0.0201255
$$526$$ 0 0
$$527$$ −35.3216 −1.53863
$$528$$ 0 0
$$529$$ 4.19845 0.182541
$$530$$ 0 0
$$531$$ −21.1736 −0.918855
$$532$$ 0 0
$$533$$ 10.9017 0.472206
$$534$$ 0 0
$$535$$ 16.8702 0.729360
$$536$$ 0 0
$$537$$ −2.56209 −0.110562
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −24.9692 −1.07351 −0.536755 0.843738i $$-0.680350\pi$$
−0.536755 + 0.843738i $$0.680350\pi$$
$$542$$ 0 0
$$543$$ 0.833782 0.0357810
$$544$$ 0 0
$$545$$ 9.97408 0.427243
$$546$$ 0 0
$$547$$ −9.75488 −0.417089 −0.208544 0.978013i $$-0.566873\pi$$
−0.208544 + 0.978013i $$0.566873\pi$$
$$548$$ 0 0
$$549$$ 23.2793 0.993538
$$550$$ 0 0
$$551$$ −11.1446 −0.474775
$$552$$ 0 0
$$553$$ −10.9209 −0.464404
$$554$$ 0 0
$$555$$ −0.763696 −0.0324171
$$556$$ 0 0
$$557$$ 35.0288 1.48422 0.742110 0.670278i $$-0.233825\pi$$
0.742110 + 0.670278i $$0.233825\pi$$
$$558$$ 0 0
$$559$$ −1.66096 −0.0702509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −1.82310 −0.0768347 −0.0384173 0.999262i $$-0.512232\pi$$
−0.0384173 + 0.999262i $$0.512232\pi$$
$$564$$ 0 0
$$565$$ −0.00824325 −0.000346796 0
$$566$$ 0 0
$$567$$ 17.8778 0.750797
$$568$$ 0 0
$$569$$ −12.9416 −0.542542 −0.271271 0.962503i $$-0.587444\pi$$
−0.271271 + 0.962503i $$0.587444\pi$$
$$570$$ 0 0
$$571$$ 8.07735 0.338027 0.169013 0.985614i $$-0.445942\pi$$
0.169013 + 0.985614i $$0.445942\pi$$
$$572$$ 0 0
$$573$$ −4.75519 −0.198651
$$574$$ 0 0
$$575$$ −5.21521 −0.217489
$$576$$ 0 0
$$577$$ 11.8678 0.494065 0.247032 0.969007i $$-0.420545\pi$$
0.247032 + 0.969007i $$0.420545\pi$$
$$578$$ 0 0
$$579$$ 3.82041 0.158771
$$580$$ 0 0
$$581$$ 20.1138 0.834461
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 5.54438 0.229232
$$586$$ 0 0
$$587$$ 43.5701 1.79833 0.899165 0.437611i $$-0.144175\pi$$
0.899165 + 0.437611i $$0.144175\pi$$
$$588$$ 0 0
$$589$$ −19.5427 −0.805244
$$590$$ 0 0
$$591$$ −1.82856 −0.0752168
$$592$$ 0 0
$$593$$ 20.0344 0.822715 0.411358 0.911474i $$-0.365055\pi$$
0.411358 + 0.911474i $$0.365055\pi$$
$$594$$ 0 0
$$595$$ −11.0328 −0.452302
$$596$$ 0 0
$$597$$ 3.19748 0.130864
$$598$$ 0 0
$$599$$ −14.1406 −0.577768 −0.288884 0.957364i $$-0.593284\pi$$
−0.288884 + 0.957364i $$0.593284\pi$$
$$600$$ 0 0
$$601$$ −38.5427 −1.57219 −0.786094 0.618106i $$-0.787900\pi$$
−0.786094 + 0.618106i $$0.787900\pi$$
$$602$$ 0 0
$$603$$ −3.90244 −0.158919
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 15.8199 0.642110 0.321055 0.947061i $$-0.395963\pi$$
0.321055 + 0.947061i $$0.395963\pi$$
$$608$$ 0 0
$$609$$ 1.75764 0.0712232
$$610$$ 0 0
$$611$$ −14.4601 −0.584994
$$612$$ 0 0
$$613$$ 4.94110 0.199569 0.0997847 0.995009i $$-0.468185\pi$$
0.0997847 + 0.995009i $$0.468185\pi$$
$$614$$ 0 0
$$615$$ 1.28173 0.0516842
$$616$$ 0 0
$$617$$ 8.96192 0.360793 0.180397 0.983594i $$-0.442262\pi$$
0.180397 + 0.983594i $$0.442262\pi$$
$$618$$ 0 0
$$619$$ 22.7324 0.913693 0.456846 0.889546i $$-0.348979\pi$$
0.456846 + 0.889546i $$0.348979\pi$$
$$620$$ 0 0
$$621$$ −6.85536 −0.275096
$$622$$ 0 0
$$623$$ −35.6853 −1.42970
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −18.2718 −0.728544
$$630$$ 0 0
$$631$$ 12.7142 0.506144 0.253072 0.967447i $$-0.418559\pi$$
0.253072 + 0.967447i $$0.418559\pi$$
$$632$$ 0 0
$$633$$ 1.79468 0.0713320
$$634$$ 0 0
$$635$$ −10.0723 −0.399706
$$636$$ 0 0
$$637$$ −4.96237 −0.196616
$$638$$ 0 0
$$639$$ −30.1982 −1.19462
$$640$$ 0 0
$$641$$ −13.5734 −0.536118 −0.268059 0.963402i $$-0.586382\pi$$
−0.268059 + 0.963402i $$0.586382\pi$$
$$642$$ 0 0
$$643$$ −22.7780 −0.898275 −0.449138 0.893463i $$-0.648269\pi$$
−0.449138 + 0.893463i $$0.648269\pi$$
$$644$$ 0 0
$$645$$ −0.195280 −0.00768915
$$646$$ 0 0
$$647$$ −12.7235 −0.500212 −0.250106 0.968218i $$-0.580465\pi$$
−0.250106 + 0.968218i $$0.580465\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 3.08214 0.120799
$$652$$ 0 0
$$653$$ −15.3118 −0.599197 −0.299598 0.954065i $$-0.596853\pi$$
−0.299598 + 0.954065i $$0.596853\pi$$
$$654$$ 0 0
$$655$$ −18.0249 −0.704291
$$656$$ 0 0
$$657$$ −24.1144 −0.940794
$$658$$ 0 0
$$659$$ 41.8861 1.63165 0.815826 0.578297i $$-0.196282\pi$$
0.815826 + 0.578297i $$0.196282\pi$$
$$660$$ 0 0
$$661$$ 3.83538 0.149179 0.0745895 0.997214i $$-0.476235\pi$$
0.0745895 + 0.997214i $$0.476235\pi$$
$$662$$ 0 0
$$663$$ −2.19290 −0.0851653
$$664$$ 0 0
$$665$$ −6.10424 −0.236712
$$666$$ 0 0
$$667$$ 19.8782 0.769686
$$668$$ 0 0
$$669$$ 2.72666 0.105419
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 28.5708 1.10132 0.550662 0.834728i $$-0.314375\pi$$
0.550662 + 0.834728i $$0.314375\pi$$
$$674$$ 0 0
$$675$$ 1.31449 0.0505949
$$676$$ 0 0
$$677$$ −18.9267 −0.727413 −0.363706 0.931514i $$-0.618489\pi$$
−0.363706 + 0.931514i $$0.618489\pi$$
$$678$$ 0 0
$$679$$ −32.0039 −1.22820
$$680$$ 0 0
$$681$$ 4.20263 0.161045
$$682$$ 0 0
$$683$$ −34.3479 −1.31429 −0.657143 0.753766i $$-0.728235\pi$$
−0.657143 + 0.753766i $$0.728235\pi$$
$$684$$ 0 0
$$685$$ −5.15806 −0.197079
$$686$$ 0 0
$$687$$ −0.694990 −0.0265155
$$688$$ 0 0
$$689$$ −17.8610 −0.680449
$$690$$ 0 0
$$691$$ −48.9701 −1.86291 −0.931456 0.363854i $$-0.881461\pi$$
−0.931456 + 0.363854i $$0.881461\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1.61222 −0.0611551
$$696$$ 0 0
$$697$$ 30.6659 1.16156
$$698$$ 0 0
$$699$$ 3.51159 0.132821
$$700$$ 0 0
$$701$$ −17.1191 −0.646581 −0.323291 0.946300i $$-0.604789\pi$$
−0.323291 + 0.946300i $$0.604789\pi$$
$$702$$ 0 0
$$703$$ −10.1094 −0.381284
$$704$$ 0 0
$$705$$ −1.70009 −0.0640292
$$706$$ 0 0
$$707$$ 32.5243 1.22320
$$708$$ 0 0
$$709$$ 40.2510 1.51166 0.755829 0.654769i $$-0.227234\pi$$
0.755829 + 0.654769i $$0.227234\pi$$
$$710$$ 0 0
$$711$$ 15.4378 0.578963
$$712$$ 0 0
$$713$$ 34.8577 1.30543
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 3.48894 0.130297
$$718$$ 0 0
$$719$$ −31.1437 −1.16146 −0.580732 0.814094i $$-0.697234\pi$$
−0.580732 + 0.814094i $$0.697234\pi$$
$$720$$ 0 0
$$721$$ 30.2798 1.12768
$$722$$ 0 0
$$723$$ 3.59870 0.133837
$$724$$ 0 0
$$725$$ −3.81157 −0.141558
$$726$$ 0 0
$$727$$ 29.3135 1.08718 0.543589 0.839351i $$-0.317065\pi$$
0.543589 + 0.839351i $$0.317065\pi$$
$$728$$ 0 0
$$729$$ −24.4011 −0.903744
$$730$$ 0 0
$$731$$ −4.67217 −0.172807
$$732$$ 0 0
$$733$$ −27.1541 −1.00296 −0.501479 0.865170i $$-0.667211\pi$$
−0.501479 + 0.865170i $$0.667211\pi$$
$$734$$ 0 0
$$735$$ −0.583431 −0.0215202
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 10.6933 0.393361 0.196680 0.980468i $$-0.436984\pi$$
0.196680 + 0.980468i $$0.436984\pi$$
$$740$$ 0 0
$$741$$ −1.21329 −0.0445713
$$742$$ 0 0
$$743$$ −53.4828 −1.96209 −0.981046 0.193775i $$-0.937927\pi$$
−0.981046 + 0.193775i $$0.937927\pi$$
$$744$$ 0 0
$$745$$ −3.00244 −0.110001
$$746$$ 0 0
$$747$$ −28.4329 −1.04031
$$748$$ 0 0
$$749$$ −35.2202 −1.28692
$$750$$ 0 0
$$751$$ −49.3066 −1.79922 −0.899612 0.436690i $$-0.856151\pi$$
−0.899612 + 0.436690i $$0.856151\pi$$
$$752$$ 0 0
$$753$$ 2.15395 0.0784943
$$754$$ 0 0
$$755$$ 0.831273 0.0302531
$$756$$ 0 0
$$757$$ −40.7211 −1.48003 −0.740016 0.672589i $$-0.765182\pi$$
−0.740016 + 0.672589i $$0.765182\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −33.3345 −1.20838 −0.604188 0.796842i $$-0.706502\pi$$
−0.604188 + 0.796842i $$0.706502\pi$$
$$762$$ 0 0
$$763$$ −20.8231 −0.753848
$$764$$ 0 0
$$765$$ 15.5960 0.563876
$$766$$ 0 0
$$767$$ 13.4787 0.486686
$$768$$ 0 0
$$769$$ 43.5317 1.56979 0.784896 0.619627i $$-0.212716\pi$$
0.784896 + 0.619627i $$0.212716\pi$$
$$770$$ 0 0
$$771$$ 3.96801 0.142904
$$772$$ 0 0
$$773$$ −36.7061 −1.32023 −0.660113 0.751166i $$-0.729492\pi$$
−0.660113 + 0.751166i $$0.729492\pi$$
$$774$$ 0 0
$$775$$ −6.68385 −0.240091
$$776$$ 0 0
$$777$$ 1.59439 0.0571982
$$778$$ 0 0
$$779$$ 16.9669 0.607901
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −5.01029 −0.179053
$$784$$ 0 0
$$785$$ 13.3102 0.475060
$$786$$ 0 0
$$787$$ 26.2255 0.934837 0.467418 0.884036i $$-0.345184\pi$$
0.467418 + 0.884036i $$0.345184\pi$$
$$788$$ 0 0
$$789$$ 2.56934 0.0914710
$$790$$ 0 0
$$791$$ 0.0172096 0.000611903 0
$$792$$ 0 0
$$793$$ −14.8191 −0.526243
$$794$$ 0 0
$$795$$ −2.09993 −0.0744770
$$796$$ 0 0
$$797$$ −7.79065 −0.275959 −0.137980 0.990435i $$-0.544061\pi$$
−0.137980 + 0.990435i $$0.544061\pi$$
$$798$$ 0 0
$$799$$ −40.6755 −1.43900
$$800$$ 0 0
$$801$$ 50.4448 1.78238
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 10.8879 0.383749
$$806$$ 0 0
$$807$$ 2.62077 0.0922556
$$808$$ 0 0
$$809$$ −23.6845 −0.832704 −0.416352 0.909203i $$-0.636692\pi$$
−0.416352 + 0.909203i $$0.636692\pi$$
$$810$$ 0 0
$$811$$ −4.53483 −0.159239 −0.0796197 0.996825i $$-0.525371\pi$$
−0.0796197 + 0.996825i $$0.525371\pi$$
$$812$$ 0 0
$$813$$ 2.76695 0.0970411
$$814$$ 0 0
$$815$$ −6.36304 −0.222888
$$816$$ 0 0
$$817$$ −2.58502 −0.0904385
$$818$$ 0 0
$$819$$ −11.5751 −0.404468
$$820$$ 0 0
$$821$$ 20.1204 0.702207 0.351104 0.936337i $$-0.385806\pi$$
0.351104 + 0.936337i $$0.385806\pi$$
$$822$$ 0 0
$$823$$ 10.7823 0.375849 0.187924 0.982184i $$-0.439824\pi$$
0.187924 + 0.982184i $$0.439824\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 49.7772 1.73092 0.865462 0.500975i $$-0.167025\pi$$
0.865462 + 0.500975i $$0.167025\pi$$
$$828$$ 0 0
$$829$$ 15.8842 0.551682 0.275841 0.961203i $$-0.411044\pi$$
0.275841 + 0.961203i $$0.411044\pi$$
$$830$$ 0 0
$$831$$ −6.06109 −0.210257
$$832$$ 0 0
$$833$$ −13.9589 −0.483646
$$834$$ 0 0
$$835$$ −2.25595 −0.0780704
$$836$$ 0 0
$$837$$ −8.78588 −0.303684
$$838$$ 0 0
$$839$$ −5.99554 −0.206989 −0.103494 0.994630i $$-0.533002\pi$$
−0.103494 + 0.994630i $$0.533002\pi$$
$$840$$ 0 0
$$841$$ −14.4719 −0.499031
$$842$$ 0 0
$$843$$ −5.69111 −0.196012
$$844$$ 0 0
$$845$$ 9.47056 0.325797
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 2.25446 0.0773729
$$850$$ 0 0
$$851$$ 18.0318 0.618123
$$852$$ 0 0
$$853$$ 18.9322 0.648226 0.324113 0.946018i $$-0.394934\pi$$
0.324113 + 0.946018i $$0.394934\pi$$
$$854$$ 0 0
$$855$$ 8.62898 0.295105
$$856$$ 0 0
$$857$$ 33.4506 1.14265 0.571325 0.820724i $$-0.306429\pi$$
0.571325 + 0.820724i $$0.306429\pi$$
$$858$$ 0 0
$$859$$ 0.368811 0.0125837 0.00629184 0.999980i $$-0.497997\pi$$
0.00629184 + 0.999980i $$0.497997\pi$$
$$860$$ 0 0
$$861$$ −2.67589 −0.0911941
$$862$$ 0 0
$$863$$ 5.94784 0.202467 0.101233 0.994863i $$-0.467721\pi$$
0.101233 + 0.994863i $$0.467721\pi$$
$$864$$ 0 0
$$865$$ 10.8245 0.368042
$$866$$ 0 0
$$867$$ −2.41358 −0.0819695
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 2.48421 0.0841742
$$872$$ 0 0
$$873$$ 45.2409 1.53117
$$874$$ 0 0
$$875$$ −2.08772 −0.0705779
$$876$$ 0 0
$$877$$ −5.80988 −0.196186 −0.0980929 0.995177i $$-0.531274\pi$$
−0.0980929 + 0.995177i $$0.531274\pi$$
$$878$$ 0 0
$$879$$ 5.75019 0.193949
$$880$$ 0 0
$$881$$ 27.3096 0.920083 0.460042 0.887897i $$-0.347834\pi$$
0.460042 + 0.887897i $$0.347834\pi$$
$$882$$ 0 0
$$883$$ −14.9649 −0.503609 −0.251804 0.967778i $$-0.581024\pi$$
−0.251804 + 0.967778i $$0.581024\pi$$
$$884$$ 0 0
$$885$$ 1.58470 0.0532691
$$886$$ 0 0
$$887$$ 9.81551 0.329573 0.164786 0.986329i $$-0.447307\pi$$
0.164786 + 0.986329i $$0.447307\pi$$
$$888$$ 0 0
$$889$$ 21.0281 0.705260
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −22.5050 −0.753100
$$894$$ 0 0
$$895$$ −11.5996 −0.387731
$$896$$ 0 0
$$897$$ 2.16410 0.0722573
$$898$$ 0 0
$$899$$ 25.4760 0.849672
$$900$$ 0 0
$$901$$ −50.2419 −1.67380
$$902$$ 0 0
$$903$$ 0.407691 0.0135671
$$904$$ 0 0
$$905$$ 3.77485 0.125480
$$906$$ 0 0
$$907$$ −54.5458 −1.81116 −0.905582 0.424172i $$-0.860565\pi$$
−0.905582 + 0.424172i $$0.860565\pi$$
$$908$$ 0 0
$$909$$ −45.9764 −1.52494
$$910$$ 0 0
$$911$$ −38.0190 −1.25962 −0.629812 0.776747i $$-0.716868\pi$$
−0.629812 + 0.776747i $$0.716868\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −1.74230 −0.0575987
$$916$$ 0 0
$$917$$ 37.6310 1.24269
$$918$$ 0 0
$$919$$ −55.3414 −1.82555 −0.912773 0.408467i $$-0.866063\pi$$
−0.912773 + 0.408467i $$0.866063\pi$$
$$920$$ 0 0
$$921$$ −2.24392 −0.0739396
$$922$$ 0 0
$$923$$ 19.2235 0.632751
$$924$$ 0 0
$$925$$ −3.45754 −0.113683
$$926$$ 0 0
$$927$$ −42.8037 −1.40586
$$928$$ 0 0
$$929$$ −0.719940 −0.0236205 −0.0118102 0.999930i $$-0.503759\pi$$
−0.0118102 + 0.999930i $$0.503759\pi$$
$$930$$ 0 0
$$931$$ −7.72316 −0.253116
$$932$$ 0 0
$$933$$ 1.39792 0.0457658
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −36.9846 −1.20823 −0.604117 0.796896i $$-0.706474\pi$$
−0.604117 + 0.796896i $$0.706474\pi$$
$$938$$ 0 0
$$939$$ 4.84531 0.158121
$$940$$ 0 0
$$941$$ 11.6784 0.380704 0.190352 0.981716i $$-0.439037\pi$$
0.190352 + 0.981716i $$0.439037\pi$$
$$942$$ 0 0
$$943$$ −30.2632 −0.985505
$$944$$ 0 0
$$945$$ −2.74430 −0.0892720
$$946$$ 0 0
$$947$$ 60.3234 1.96025 0.980123 0.198390i $$-0.0635713\pi$$
0.980123 + 0.198390i $$0.0635713\pi$$
$$948$$ 0 0
$$949$$ 15.3507 0.498306
$$950$$ 0 0
$$951$$ 5.35023 0.173493
$$952$$ 0 0
$$953$$ 2.02655 0.0656464 0.0328232 0.999461i $$-0.489550\pi$$
0.0328232 + 0.999461i $$0.489550\pi$$
$$954$$ 0 0
$$955$$ −21.5286 −0.696648
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 10.7686 0.347736
$$960$$ 0 0
$$961$$ 13.6738 0.441091
$$962$$ 0 0
$$963$$ 49.7874 1.60438
$$964$$ 0 0
$$965$$ 17.2964 0.556792
$$966$$ 0 0
$$967$$ −47.8068 −1.53736 −0.768682 0.639631i $$-0.779087\pi$$
−0.768682 + 0.639631i $$0.779087\pi$$
$$968$$ 0 0
$$969$$ −3.41292 −0.109639
$$970$$ 0 0
$$971$$ 7.71139 0.247470 0.123735 0.992315i $$-0.460513\pi$$
0.123735 + 0.992315i $$0.460513\pi$$
$$972$$ 0 0
$$973$$ 3.36588 0.107905
$$974$$ 0 0
$$975$$ −0.414959 −0.0132893
$$976$$ 0 0
$$977$$ −51.2365 −1.63920 −0.819600 0.572936i $$-0.805804\pi$$
−0.819600 + 0.572936i $$0.805804\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 29.4356 0.939808
$$982$$ 0 0
$$983$$ −33.1192 −1.05634 −0.528170 0.849139i $$-0.677121\pi$$
−0.528170 + 0.849139i $$0.677121\pi$$
$$984$$ 0 0
$$985$$ −8.27858 −0.263778
$$986$$ 0 0
$$987$$ 3.54932 0.112976
$$988$$ 0 0
$$989$$ 4.61081 0.146615
$$990$$ 0 0
$$991$$ −36.3110 −1.15346 −0.576729 0.816936i $$-0.695671\pi$$
−0.576729 + 0.816936i $$0.695671\pi$$
$$992$$ 0 0
$$993$$ 6.48660 0.205846
$$994$$ 0 0
$$995$$ 14.4762 0.458927
$$996$$ 0 0
$$997$$ 57.1807 1.81093 0.905466 0.424419i $$-0.139522\pi$$
0.905466 + 0.424419i $$0.139522\pi$$
$$998$$ 0 0
$$999$$ −4.54492 −0.143795
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 9680.2.a.dc.1.3 6
4.3 odd 2 4840.2.a.bb.1.4 6
11.3 even 5 880.2.bo.i.801.2 12
11.4 even 5 880.2.bo.i.401.2 12
11.10 odd 2 9680.2.a.dd.1.3 6
44.3 odd 10 440.2.y.c.361.2 12
44.15 odd 10 440.2.y.c.401.2 yes 12
44.43 even 2 4840.2.a.ba.1.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.361.2 12 44.3 odd 10
440.2.y.c.401.2 yes 12 44.15 odd 10
880.2.bo.i.401.2 12 11.4 even 5
880.2.bo.i.801.2 12 11.3 even 5
4840.2.a.ba.1.4 6 44.43 even 2
4840.2.a.bb.1.4 6 4.3 odd 2
9680.2.a.dc.1.3 6 1.1 even 1 trivial
9680.2.a.dd.1.3 6 11.10 odd 2