# Properties

 Label 3192.2.a.u.1.1 Level $3192$ Weight $2$ Character 3192.1 Self dual yes Analytic conductor $25.488$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3192.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: $$25.4882483252$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.12489$$ of defining polynomial Character $$\chi$$ $$=$$ 3192.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -2.64002 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -2.64002 q^{5} -1.00000 q^{7} +1.00000 q^{9} -6.24977 q^{11} +4.00000 q^{13} +2.64002 q^{15} -2.64002 q^{17} -1.00000 q^{19} +1.00000 q^{21} -6.24977 q^{23} +1.96972 q^{25} -1.00000 q^{27} +3.60975 q^{29} -8.24977 q^{31} +6.24977 q^{33} +2.64002 q^{35} +2.00000 q^{37} -4.00000 q^{39} -5.21949 q^{41} +4.24977 q^{43} -2.64002 q^{45} +1.60975 q^{47} +1.00000 q^{49} +2.64002 q^{51} -12.8898 q^{53} +16.4995 q^{55} +1.00000 q^{57} +1.93945 q^{59} +0.0605522 q^{61} -1.00000 q^{63} -10.5601 q^{65} +6.24977 q^{67} +6.24977 q^{69} +0.329700 q^{71} -1.21949 q^{73} -1.96972 q^{75} +6.24977 q^{77} +0.969724 q^{79} +1.00000 q^{81} +3.67030 q^{83} +6.96972 q^{85} -3.60975 q^{87} +9.21949 q^{89} -4.00000 q^{91} +8.24977 q^{93} +2.64002 q^{95} +9.03028 q^{97} -6.24977 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 12 q^{13} - 3 q^{19} + 3 q^{21} - 2 q^{23} + 5 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} - 12 q^{39} + 2 q^{41} - 4 q^{43} - 4 q^{47} + 3 q^{49} - 14 q^{53} + 16 q^{55} + 3 q^{57} + 4 q^{59} + 2 q^{61} - 3 q^{63} + 2 q^{67} + 2 q^{69} + 8 q^{71} + 14 q^{73} - 5 q^{75} + 2 q^{77} + 2 q^{79} + 3 q^{81} + 4 q^{83} + 20 q^{85} - 2 q^{87} + 10 q^{89} - 12 q^{91} + 8 q^{93} + 28 q^{97} - 2 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 2 * q^11 + 12 * q^13 - 3 * q^19 + 3 * q^21 - 2 * q^23 + 5 * q^25 - 3 * q^27 + 2 * q^29 - 8 * q^31 + 2 * q^33 + 6 * q^37 - 12 * q^39 + 2 * q^41 - 4 * q^43 - 4 * q^47 + 3 * q^49 - 14 * q^53 + 16 * q^55 + 3 * q^57 + 4 * q^59 + 2 * q^61 - 3 * q^63 + 2 * q^67 + 2 * q^69 + 8 * q^71 + 14 * q^73 - 5 * q^75 + 2 * q^77 + 2 * q^79 + 3 * q^81 + 4 * q^83 + 20 * q^85 - 2 * q^87 + 10 * q^89 - 12 * q^91 + 8 * q^93 + 28 * q^97 - 2 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −2.64002 −1.18065 −0.590327 0.807164i $$-0.701001\pi$$
−0.590327 + 0.807164i $$0.701001\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −6.24977 −1.88438 −0.942188 0.335084i $$-0.891235\pi$$
−0.942188 + 0.335084i $$0.891235\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 2.64002 0.681651
$$16$$ 0 0
$$17$$ −2.64002 −0.640300 −0.320150 0.947367i $$-0.603733\pi$$
−0.320150 + 0.947367i $$0.603733\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −6.24977 −1.30317 −0.651584 0.758577i $$-0.725895\pi$$
−0.651584 + 0.758577i $$0.725895\pi$$
$$24$$ 0 0
$$25$$ 1.96972 0.393945
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 3.60975 0.670313 0.335157 0.942162i $$-0.391211\pi$$
0.335157 + 0.942162i $$0.391211\pi$$
$$30$$ 0 0
$$31$$ −8.24977 −1.48170 −0.740851 0.671669i $$-0.765578\pi$$
−0.740851 + 0.671669i $$0.765578\pi$$
$$32$$ 0 0
$$33$$ 6.24977 1.08795
$$34$$ 0 0
$$35$$ 2.64002 0.446245
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −5.21949 −0.815148 −0.407574 0.913172i $$-0.633625\pi$$
−0.407574 + 0.913172i $$0.633625\pi$$
$$42$$ 0 0
$$43$$ 4.24977 0.648084 0.324042 0.946043i $$-0.394958\pi$$
0.324042 + 0.946043i $$0.394958\pi$$
$$44$$ 0 0
$$45$$ −2.64002 −0.393551
$$46$$ 0 0
$$47$$ 1.60975 0.234806 0.117403 0.993084i $$-0.462543\pi$$
0.117403 + 0.993084i $$0.462543\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.64002 0.369677
$$52$$ 0 0
$$53$$ −12.8898 −1.77055 −0.885275 0.465068i $$-0.846030\pi$$
−0.885275 + 0.465068i $$0.846030\pi$$
$$54$$ 0 0
$$55$$ 16.4995 2.22480
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 1.93945 0.252495 0.126247 0.991999i $$-0.459707\pi$$
0.126247 + 0.991999i $$0.459707\pi$$
$$60$$ 0 0
$$61$$ 0.0605522 0.00775291 0.00387646 0.999992i $$-0.498766\pi$$
0.00387646 + 0.999992i $$0.498766\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ −10.5601 −1.30982
$$66$$ 0 0
$$67$$ 6.24977 0.763531 0.381766 0.924259i $$-0.375316\pi$$
0.381766 + 0.924259i $$0.375316\pi$$
$$68$$ 0 0
$$69$$ 6.24977 0.752384
$$70$$ 0 0
$$71$$ 0.329700 0.0391282 0.0195641 0.999809i $$-0.493772\pi$$
0.0195641 + 0.999809i $$0.493772\pi$$
$$72$$ 0 0
$$73$$ −1.21949 −0.142731 −0.0713655 0.997450i $$-0.522736\pi$$
−0.0713655 + 0.997450i $$0.522736\pi$$
$$74$$ 0 0
$$75$$ −1.96972 −0.227444
$$76$$ 0 0
$$77$$ 6.24977 0.712227
$$78$$ 0 0
$$79$$ 0.969724 0.109102 0.0545512 0.998511i $$-0.482627\pi$$
0.0545512 + 0.998511i $$0.482627\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 3.67030 0.402868 0.201434 0.979502i $$-0.435440\pi$$
0.201434 + 0.979502i $$0.435440\pi$$
$$84$$ 0 0
$$85$$ 6.96972 0.755973
$$86$$ 0 0
$$87$$ −3.60975 −0.387006
$$88$$ 0 0
$$89$$ 9.21949 0.977264 0.488632 0.872490i $$-0.337496\pi$$
0.488632 + 0.872490i $$0.337496\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ 8.24977 0.855461
$$94$$ 0 0
$$95$$ 2.64002 0.270861
$$96$$ 0 0
$$97$$ 9.03028 0.916886 0.458443 0.888724i $$-0.348407\pi$$
0.458443 + 0.888724i $$0.348407\pi$$
$$98$$ 0 0
$$99$$ −6.24977 −0.628126
$$100$$ 0 0
$$101$$ 2.64002 0.262692 0.131346 0.991337i $$-0.458070\pi$$
0.131346 + 0.991337i $$0.458070\pi$$
$$102$$ 0 0
$$103$$ −1.93945 −0.191099 −0.0955497 0.995425i $$-0.530461\pi$$
−0.0955497 + 0.995425i $$0.530461\pi$$
$$104$$ 0 0
$$105$$ −2.64002 −0.257640
$$106$$ 0 0
$$107$$ −8.82924 −0.853555 −0.426778 0.904357i $$-0.640351\pi$$
−0.426778 + 0.904357i $$0.640351\pi$$
$$108$$ 0 0
$$109$$ 7.28005 0.697302 0.348651 0.937253i $$-0.386640\pi$$
0.348651 + 0.937253i $$0.386640\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ −4.88979 −0.459993 −0.229997 0.973191i $$-0.573872\pi$$
−0.229997 + 0.973191i $$0.573872\pi$$
$$114$$ 0 0
$$115$$ 16.4995 1.53859
$$116$$ 0 0
$$117$$ 4.00000 0.369800
$$118$$ 0 0
$$119$$ 2.64002 0.242011
$$120$$ 0 0
$$121$$ 28.0596 2.55088
$$122$$ 0 0
$$123$$ 5.21949 0.470626
$$124$$ 0 0
$$125$$ 8.00000 0.715542
$$126$$ 0 0
$$127$$ −14.7493 −1.30879 −0.654395 0.756153i $$-0.727077\pi$$
−0.654395 + 0.756153i $$0.727077\pi$$
$$128$$ 0 0
$$129$$ −4.24977 −0.374171
$$130$$ 0 0
$$131$$ 20.9503 1.83044 0.915220 0.402954i $$-0.132017\pi$$
0.915220 + 0.402954i $$0.132017\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ 2.64002 0.227217
$$136$$ 0 0
$$137$$ 6.49954 0.555293 0.277647 0.960683i $$-0.410446\pi$$
0.277647 + 0.960683i $$0.410446\pi$$
$$138$$ 0 0
$$139$$ 7.21949 0.612350 0.306175 0.951975i $$-0.400951\pi$$
0.306175 + 0.951975i $$0.400951\pi$$
$$140$$ 0 0
$$141$$ −1.60975 −0.135565
$$142$$ 0 0
$$143$$ −24.9991 −2.09053
$$144$$ 0 0
$$145$$ −9.52982 −0.791408
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 21.8401 1.78921 0.894607 0.446854i $$-0.147456\pi$$
0.894607 + 0.446854i $$0.147456\pi$$
$$150$$ 0 0
$$151$$ −2.24977 −0.183084 −0.0915419 0.995801i $$-0.529180\pi$$
−0.0915419 + 0.995801i $$0.529180\pi$$
$$152$$ 0 0
$$153$$ −2.64002 −0.213433
$$154$$ 0 0
$$155$$ 21.7796 1.74938
$$156$$ 0 0
$$157$$ −6.49954 −0.518720 −0.259360 0.965781i $$-0.583512\pi$$
−0.259360 + 0.965781i $$0.583512\pi$$
$$158$$ 0 0
$$159$$ 12.8898 1.02223
$$160$$ 0 0
$$161$$ 6.24977 0.492551
$$162$$ 0 0
$$163$$ −14.8099 −1.16000 −0.579999 0.814617i $$-0.696947\pi$$
−0.579999 + 0.814617i $$0.696947\pi$$
$$164$$ 0 0
$$165$$ −16.4995 −1.28449
$$166$$ 0 0
$$167$$ 19.7190 1.52590 0.762952 0.646455i $$-0.223749\pi$$
0.762952 + 0.646455i $$0.223749\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ −3.28005 −0.249377 −0.124689 0.992196i $$-0.539793\pi$$
−0.124689 + 0.992196i $$0.539793\pi$$
$$174$$ 0 0
$$175$$ −1.96972 −0.148897
$$176$$ 0 0
$$177$$ −1.93945 −0.145778
$$178$$ 0 0
$$179$$ 19.3893 1.44923 0.724614 0.689155i $$-0.242018\pi$$
0.724614 + 0.689155i $$0.242018\pi$$
$$180$$ 0 0
$$181$$ 6.47018 0.480925 0.240462 0.970658i $$-0.422701\pi$$
0.240462 + 0.970658i $$0.422701\pi$$
$$182$$ 0 0
$$183$$ −0.0605522 −0.00447615
$$184$$ 0 0
$$185$$ −5.28005 −0.388197
$$186$$ 0 0
$$187$$ 16.4995 1.20657
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −8.31032 −0.601314 −0.300657 0.953732i $$-0.597206\pi$$
−0.300657 + 0.953732i $$0.597206\pi$$
$$192$$ 0 0
$$193$$ −8.56009 −0.616169 −0.308085 0.951359i $$-0.599688\pi$$
−0.308085 + 0.951359i $$0.599688\pi$$
$$194$$ 0 0
$$195$$ 10.5601 0.756224
$$196$$ 0 0
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ 0 0
$$199$$ −11.2195 −0.795329 −0.397664 0.917531i $$-0.630179\pi$$
−0.397664 + 0.917531i $$0.630179\pi$$
$$200$$ 0 0
$$201$$ −6.24977 −0.440825
$$202$$ 0 0
$$203$$ −3.60975 −0.253355
$$204$$ 0 0
$$205$$ 13.7796 0.962408
$$206$$ 0 0
$$207$$ −6.24977 −0.434389
$$208$$ 0 0
$$209$$ 6.24977 0.432306
$$210$$ 0 0
$$211$$ −18.0899 −1.24536 −0.622680 0.782476i $$-0.713956\pi$$
−0.622680 + 0.782476i $$0.713956\pi$$
$$212$$ 0 0
$$213$$ −0.329700 −0.0225907
$$214$$ 0 0
$$215$$ −11.2195 −0.765163
$$216$$ 0 0
$$217$$ 8.24977 0.560031
$$218$$ 0 0
$$219$$ 1.21949 0.0824058
$$220$$ 0 0
$$221$$ −10.5601 −0.710349
$$222$$ 0 0
$$223$$ 6.31032 0.422570 0.211285 0.977424i $$-0.432235\pi$$
0.211285 + 0.977424i $$0.432235\pi$$
$$224$$ 0 0
$$225$$ 1.96972 0.131315
$$226$$ 0 0
$$227$$ 7.21949 0.479175 0.239587 0.970875i $$-0.422988\pi$$
0.239587 + 0.970875i $$0.422988\pi$$
$$228$$ 0 0
$$229$$ 21.8401 1.44324 0.721619 0.692291i $$-0.243398\pi$$
0.721619 + 0.692291i $$0.243398\pi$$
$$230$$ 0 0
$$231$$ −6.24977 −0.411205
$$232$$ 0 0
$$233$$ −21.7190 −1.42286 −0.711431 0.702756i $$-0.751952\pi$$
−0.711431 + 0.702756i $$0.751952\pi$$
$$234$$ 0 0
$$235$$ −4.24977 −0.277224
$$236$$ 0 0
$$237$$ −0.969724 −0.0629903
$$238$$ 0 0
$$239$$ −28.0294 −1.81307 −0.906534 0.422132i $$-0.861282\pi$$
−0.906534 + 0.422132i $$0.861282\pi$$
$$240$$ 0 0
$$241$$ −5.52982 −0.356207 −0.178103 0.984012i $$-0.556996\pi$$
−0.178103 + 0.984012i $$0.556996\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −2.64002 −0.168665
$$246$$ 0 0
$$247$$ −4.00000 −0.254514
$$248$$ 0 0
$$249$$ −3.67030 −0.232596
$$250$$ 0 0
$$251$$ −12.9503 −0.817419 −0.408709 0.912665i $$-0.634021\pi$$
−0.408709 + 0.912665i $$0.634021\pi$$
$$252$$ 0 0
$$253$$ 39.0596 2.45566
$$254$$ 0 0
$$255$$ −6.96972 −0.436461
$$256$$ 0 0
$$257$$ −6.49954 −0.405430 −0.202715 0.979238i $$-0.564977\pi$$
−0.202715 + 0.979238i $$0.564977\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 3.60975 0.223438
$$262$$ 0 0
$$263$$ 28.5289 1.75917 0.879584 0.475744i $$-0.157821\pi$$
0.879584 + 0.475744i $$0.157821\pi$$
$$264$$ 0 0
$$265$$ 34.0294 2.09041
$$266$$ 0 0
$$267$$ −9.21949 −0.564224
$$268$$ 0 0
$$269$$ 24.9385 1.52053 0.760265 0.649614i $$-0.225069\pi$$
0.760265 + 0.649614i $$0.225069\pi$$
$$270$$ 0 0
$$271$$ 14.4390 0.877106 0.438553 0.898705i $$-0.355491\pi$$
0.438553 + 0.898705i $$0.355491\pi$$
$$272$$ 0 0
$$273$$ 4.00000 0.242091
$$274$$ 0 0
$$275$$ −12.3103 −0.742340
$$276$$ 0 0
$$277$$ 25.5904 1.53758 0.768788 0.639504i $$-0.220860\pi$$
0.768788 + 0.639504i $$0.220860\pi$$
$$278$$ 0 0
$$279$$ −8.24977 −0.493901
$$280$$ 0 0
$$281$$ −26.1698 −1.56116 −0.780581 0.625055i $$-0.785077\pi$$
−0.780581 + 0.625055i $$0.785077\pi$$
$$282$$ 0 0
$$283$$ 17.2800 1.02719 0.513596 0.858032i $$-0.328313\pi$$
0.513596 + 0.858032i $$0.328313\pi$$
$$284$$ 0 0
$$285$$ −2.64002 −0.156381
$$286$$ 0 0
$$287$$ 5.21949 0.308097
$$288$$ 0 0
$$289$$ −10.0303 −0.590016
$$290$$ 0 0
$$291$$ −9.03028 −0.529364
$$292$$ 0 0
$$293$$ 11.4012 0.666062 0.333031 0.942916i $$-0.391929\pi$$
0.333031 + 0.942916i $$0.391929\pi$$
$$294$$ 0 0
$$295$$ −5.12019 −0.298109
$$296$$ 0 0
$$297$$ 6.24977 0.362648
$$298$$ 0 0
$$299$$ −24.9991 −1.44573
$$300$$ 0 0
$$301$$ −4.24977 −0.244953
$$302$$ 0 0
$$303$$ −2.64002 −0.151665
$$304$$ 0 0
$$305$$ −0.159859 −0.00915351
$$306$$ 0 0
$$307$$ −12.7493 −0.727642 −0.363821 0.931469i $$-0.618528\pi$$
−0.363821 + 0.931469i $$0.618528\pi$$
$$308$$ 0 0
$$309$$ 1.93945 0.110331
$$310$$ 0 0
$$311$$ 18.8898 1.07114 0.535571 0.844490i $$-0.320096\pi$$
0.535571 + 0.844490i $$0.320096\pi$$
$$312$$ 0 0
$$313$$ 16.4390 0.929187 0.464593 0.885524i $$-0.346201\pi$$
0.464593 + 0.885524i $$0.346201\pi$$
$$314$$ 0 0
$$315$$ 2.64002 0.148748
$$316$$ 0 0
$$317$$ 4.23039 0.237603 0.118801 0.992918i $$-0.462095\pi$$
0.118801 + 0.992918i $$0.462095\pi$$
$$318$$ 0 0
$$319$$ −22.5601 −1.26312
$$320$$ 0 0
$$321$$ 8.82924 0.492800
$$322$$ 0 0
$$323$$ 2.64002 0.146895
$$324$$ 0 0
$$325$$ 7.87890 0.437042
$$326$$ 0 0
$$327$$ −7.28005 −0.402588
$$328$$ 0 0
$$329$$ −1.60975 −0.0887482
$$330$$ 0 0
$$331$$ −20.1892 −1.10970 −0.554850 0.831950i $$-0.687224\pi$$
−0.554850 + 0.831950i $$0.687224\pi$$
$$332$$ 0 0
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ −16.4995 −0.901466
$$336$$ 0 0
$$337$$ 23.1589 1.26155 0.630774 0.775967i $$-0.282738\pi$$
0.630774 + 0.775967i $$0.282738\pi$$
$$338$$ 0 0
$$339$$ 4.88979 0.265577
$$340$$ 0 0
$$341$$ 51.5592 2.79209
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −16.4995 −0.888305
$$346$$ 0 0
$$347$$ −21.4693 −1.15253 −0.576265 0.817263i $$-0.695490\pi$$
−0.576265 + 0.817263i $$0.695490\pi$$
$$348$$ 0 0
$$349$$ −14.9991 −0.802883 −0.401441 0.915885i $$-0.631491\pi$$
−0.401441 + 0.915885i $$0.631491\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ −34.4802 −1.83519 −0.917597 0.397512i $$-0.869874\pi$$
−0.917597 + 0.397512i $$0.869874\pi$$
$$354$$ 0 0
$$355$$ −0.870417 −0.0461969
$$356$$ 0 0
$$357$$ −2.64002 −0.139725
$$358$$ 0 0
$$359$$ −8.68876 −0.458575 −0.229288 0.973359i $$-0.573640\pi$$
−0.229288 + 0.973359i $$0.573640\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −28.0596 −1.47275
$$364$$ 0 0
$$365$$ 3.21949 0.168516
$$366$$ 0 0
$$367$$ 27.7190 1.44692 0.723461 0.690365i $$-0.242550\pi$$
0.723461 + 0.690365i $$0.242550\pi$$
$$368$$ 0 0
$$369$$ −5.21949 −0.271716
$$370$$ 0 0
$$371$$ 12.8898 0.669205
$$372$$ 0 0
$$373$$ 16.5601 0.857449 0.428725 0.903435i $$-0.358963\pi$$
0.428725 + 0.903435i $$0.358963\pi$$
$$374$$ 0 0
$$375$$ −8.00000 −0.413118
$$376$$ 0 0
$$377$$ 14.4390 0.743646
$$378$$ 0 0
$$379$$ −8.80986 −0.452532 −0.226266 0.974066i $$-0.572652\pi$$
−0.226266 + 0.974066i $$0.572652\pi$$
$$380$$ 0 0
$$381$$ 14.7493 0.755630
$$382$$ 0 0
$$383$$ 30.2791 1.54719 0.773596 0.633680i $$-0.218456\pi$$
0.773596 + 0.633680i $$0.218456\pi$$
$$384$$ 0 0
$$385$$ −16.4995 −0.840895
$$386$$ 0 0
$$387$$ 4.24977 0.216028
$$388$$ 0 0
$$389$$ −34.2186 −1.73495 −0.867475 0.497480i $$-0.834259\pi$$
−0.867475 + 0.497480i $$0.834259\pi$$
$$390$$ 0 0
$$391$$ 16.4995 0.834418
$$392$$ 0 0
$$393$$ −20.9503 −1.05681
$$394$$ 0 0
$$395$$ −2.56009 −0.128812
$$396$$ 0 0
$$397$$ 12.0606 0.605302 0.302651 0.953101i $$-0.402128\pi$$
0.302651 + 0.953101i $$0.402128\pi$$
$$398$$ 0 0
$$399$$ −1.00000 −0.0500626
$$400$$ 0 0
$$401$$ −24.8898 −1.24294 −0.621469 0.783439i $$-0.713464\pi$$
−0.621469 + 0.783439i $$0.713464\pi$$
$$402$$ 0 0
$$403$$ −32.9991 −1.64380
$$404$$ 0 0
$$405$$ −2.64002 −0.131184
$$406$$ 0 0
$$407$$ −12.4995 −0.619579
$$408$$ 0 0
$$409$$ 18.5601 0.917738 0.458869 0.888504i $$-0.348255\pi$$
0.458869 + 0.888504i $$0.348255\pi$$
$$410$$ 0 0
$$411$$ −6.49954 −0.320599
$$412$$ 0 0
$$413$$ −1.93945 −0.0954340
$$414$$ 0 0
$$415$$ −9.68968 −0.475648
$$416$$ 0 0
$$417$$ −7.21949 −0.353540
$$418$$ 0 0
$$419$$ −13.6097 −0.664880 −0.332440 0.943124i $$-0.607872\pi$$
−0.332440 + 0.943124i $$0.607872\pi$$
$$420$$ 0 0
$$421$$ −11.6585 −0.568200 −0.284100 0.958795i $$-0.591695\pi$$
−0.284100 + 0.958795i $$0.591695\pi$$
$$422$$ 0 0
$$423$$ 1.60975 0.0782686
$$424$$ 0 0
$$425$$ −5.20012 −0.252243
$$426$$ 0 0
$$427$$ −0.0605522 −0.00293033
$$428$$ 0 0
$$429$$ 24.9991 1.20697
$$430$$ 0 0
$$431$$ −22.7687 −1.09673 −0.548365 0.836239i $$-0.684749\pi$$
−0.548365 + 0.836239i $$0.684749\pi$$
$$432$$ 0 0
$$433$$ −36.5895 −1.75838 −0.879188 0.476474i $$-0.841915\pi$$
−0.879188 + 0.476474i $$0.841915\pi$$
$$434$$ 0 0
$$435$$ 9.52982 0.456920
$$436$$ 0 0
$$437$$ 6.24977 0.298967
$$438$$ 0 0
$$439$$ −10.0681 −0.480525 −0.240262 0.970708i $$-0.577234\pi$$
−0.240262 + 0.970708i $$0.577234\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −23.6509 −1.12369 −0.561845 0.827243i $$-0.689908\pi$$
−0.561845 + 0.827243i $$0.689908\pi$$
$$444$$ 0 0
$$445$$ −24.3397 −1.15381
$$446$$ 0 0
$$447$$ −21.8401 −1.03300
$$448$$ 0 0
$$449$$ 10.2909 0.485660 0.242830 0.970069i $$-0.421924\pi$$
0.242830 + 0.970069i $$0.421924\pi$$
$$450$$ 0 0
$$451$$ 32.6206 1.53605
$$452$$ 0 0
$$453$$ 2.24977 0.105703
$$454$$ 0 0
$$455$$ 10.5601 0.495065
$$456$$ 0 0
$$457$$ 26.5895 1.24380 0.621901 0.783096i $$-0.286361\pi$$
0.621901 + 0.783096i $$0.286361\pi$$
$$458$$ 0 0
$$459$$ 2.64002 0.123226
$$460$$ 0 0
$$461$$ 13.2001 0.614791 0.307395 0.951582i $$-0.400543\pi$$
0.307395 + 0.951582i $$0.400543\pi$$
$$462$$ 0 0
$$463$$ 36.8780 1.71387 0.856933 0.515429i $$-0.172367\pi$$
0.856933 + 0.515429i $$0.172367\pi$$
$$464$$ 0 0
$$465$$ −21.7796 −1.01000
$$466$$ 0 0
$$467$$ 18.7687 0.868511 0.434256 0.900790i $$-0.357011\pi$$
0.434256 + 0.900790i $$0.357011\pi$$
$$468$$ 0 0
$$469$$ −6.24977 −0.288588
$$470$$ 0 0
$$471$$ 6.49954 0.299483
$$472$$ 0 0
$$473$$ −26.5601 −1.22123
$$474$$ 0 0
$$475$$ −1.96972 −0.0903771
$$476$$ 0 0
$$477$$ −12.8898 −0.590183
$$478$$ 0 0
$$479$$ 9.32878 0.426243 0.213122 0.977026i $$-0.431637\pi$$
0.213122 + 0.977026i $$0.431637\pi$$
$$480$$ 0 0
$$481$$ 8.00000 0.364769
$$482$$ 0 0
$$483$$ −6.24977 −0.284374
$$484$$ 0 0
$$485$$ −23.8401 −1.08253
$$486$$ 0 0
$$487$$ −25.4693 −1.15412 −0.577061 0.816701i $$-0.695801\pi$$
−0.577061 + 0.816701i $$0.695801\pi$$
$$488$$ 0 0
$$489$$ 14.8099 0.669725
$$490$$ 0 0
$$491$$ 7.65092 0.345281 0.172641 0.984985i $$-0.444770\pi$$
0.172641 + 0.984985i $$0.444770\pi$$
$$492$$ 0 0
$$493$$ −9.52982 −0.429201
$$494$$ 0 0
$$495$$ 16.4995 0.741599
$$496$$ 0 0
$$497$$ −0.329700 −0.0147891
$$498$$ 0 0
$$499$$ −36.6206 −1.63937 −0.819683 0.572818i $$-0.805850\pi$$
−0.819683 + 0.572818i $$0.805850\pi$$
$$500$$ 0 0
$$501$$ −19.7190 −0.880982
$$502$$ 0 0
$$503$$ −15.6703 −0.698704 −0.349352 0.936992i $$-0.613598\pi$$
−0.349352 + 0.936992i $$0.613598\pi$$
$$504$$ 0 0
$$505$$ −6.96972 −0.310149
$$506$$ 0 0
$$507$$ −3.00000 −0.133235
$$508$$ 0 0
$$509$$ −10.6206 −0.470752 −0.235376 0.971904i $$-0.575632\pi$$
−0.235376 + 0.971904i $$0.575632\pi$$
$$510$$ 0 0
$$511$$ 1.21949 0.0539473
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ 5.12019 0.225622
$$516$$ 0 0
$$517$$ −10.0606 −0.442463
$$518$$ 0 0
$$519$$ 3.28005 0.143978
$$520$$ 0 0
$$521$$ −34.0000 −1.48957 −0.744784 0.667306i $$-0.767447\pi$$
−0.744784 + 0.667306i $$0.767447\pi$$
$$522$$ 0 0
$$523$$ −0.870417 −0.0380607 −0.0190303 0.999819i $$-0.506058\pi$$
−0.0190303 + 0.999819i $$0.506058\pi$$
$$524$$ 0 0
$$525$$ 1.96972 0.0859658
$$526$$ 0 0
$$527$$ 21.7796 0.948734
$$528$$ 0 0
$$529$$ 16.0596 0.698245
$$530$$ 0 0
$$531$$ 1.93945 0.0841649
$$532$$ 0 0
$$533$$ −20.8780 −0.904326
$$534$$ 0 0
$$535$$ 23.3094 1.00775
$$536$$ 0 0
$$537$$ −19.3893 −0.836712
$$538$$ 0 0
$$539$$ −6.24977 −0.269197
$$540$$ 0 0
$$541$$ −2.12110 −0.0911934 −0.0455967 0.998960i $$-0.514519\pi$$
−0.0455967 + 0.998960i $$0.514519\pi$$
$$542$$ 0 0
$$543$$ −6.47018 −0.277662
$$544$$ 0 0
$$545$$ −19.2195 −0.823273
$$546$$ 0 0
$$547$$ −25.8477 −1.10517 −0.552584 0.833457i $$-0.686358\pi$$
−0.552584 + 0.833457i $$0.686358\pi$$
$$548$$ 0 0
$$549$$ 0.0605522 0.00258430
$$550$$ 0 0
$$551$$ −3.60975 −0.153780
$$552$$ 0 0
$$553$$ −0.969724 −0.0412369
$$554$$ 0 0
$$555$$ 5.28005 0.224126
$$556$$ 0 0
$$557$$ 26.2186 1.11092 0.555458 0.831544i $$-0.312543\pi$$
0.555458 + 0.831544i $$0.312543\pi$$
$$558$$ 0 0
$$559$$ 16.9991 0.718985
$$560$$ 0 0
$$561$$ −16.4995 −0.696611
$$562$$ 0 0
$$563$$ 35.8401 1.51048 0.755241 0.655447i $$-0.227520\pi$$
0.755241 + 0.655447i $$0.227520\pi$$
$$564$$ 0 0
$$565$$ 12.9092 0.543093
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 24.4490 1.02495 0.512477 0.858701i $$-0.328728\pi$$
0.512477 + 0.858701i $$0.328728\pi$$
$$570$$ 0 0
$$571$$ −22.5601 −0.944111 −0.472055 0.881569i $$-0.656488\pi$$
−0.472055 + 0.881569i $$0.656488\pi$$
$$572$$ 0 0
$$573$$ 8.31032 0.347169
$$574$$ 0 0
$$575$$ −12.3103 −0.513376
$$576$$ 0 0
$$577$$ −24.5601 −1.02245 −0.511225 0.859447i $$-0.670808\pi$$
−0.511225 + 0.859447i $$0.670808\pi$$
$$578$$ 0 0
$$579$$ 8.56009 0.355745
$$580$$ 0 0
$$581$$ −3.67030 −0.152270
$$582$$ 0 0
$$583$$ 80.5583 3.33638
$$584$$ 0 0
$$585$$ −10.5601 −0.436606
$$586$$ 0 0
$$587$$ −35.7678 −1.47629 −0.738147 0.674640i $$-0.764299\pi$$
−0.738147 + 0.674640i $$0.764299\pi$$
$$588$$ 0 0
$$589$$ 8.24977 0.339926
$$590$$ 0 0
$$591$$ −14.0000 −0.575883
$$592$$ 0 0
$$593$$ 0.0411748 0.00169085 0.000845423 1.00000i $$-0.499731\pi$$
0.000845423 1.00000i $$0.499731\pi$$
$$594$$ 0 0
$$595$$ −6.96972 −0.285731
$$596$$ 0 0
$$597$$ 11.2195 0.459183
$$598$$ 0 0
$$599$$ 35.8889 1.46638 0.733190 0.680024i $$-0.238031\pi$$
0.733190 + 0.680024i $$0.238031\pi$$
$$600$$ 0 0
$$601$$ −9.52982 −0.388729 −0.194365 0.980929i $$-0.562265\pi$$
−0.194365 + 0.980929i $$0.562265\pi$$
$$602$$ 0 0
$$603$$ 6.24977 0.254510
$$604$$ 0 0
$$605$$ −74.0781 −3.01170
$$606$$ 0 0
$$607$$ 5.12019 0.207822 0.103911 0.994587i $$-0.466864\pi$$
0.103911 + 0.994587i $$0.466864\pi$$
$$608$$ 0 0
$$609$$ 3.60975 0.146274
$$610$$ 0 0
$$611$$ 6.43899 0.260494
$$612$$ 0 0
$$613$$ −1.21193 −0.0489495 −0.0244747 0.999700i $$-0.507791\pi$$
−0.0244747 + 0.999700i $$0.507791\pi$$
$$614$$ 0 0
$$615$$ −13.7796 −0.555647
$$616$$ 0 0
$$617$$ 30.9991 1.24798 0.623988 0.781434i $$-0.285511\pi$$
0.623988 + 0.781434i $$0.285511\pi$$
$$618$$ 0 0
$$619$$ 3.34060 0.134270 0.0671350 0.997744i $$-0.478614\pi$$
0.0671350 + 0.997744i $$0.478614\pi$$
$$620$$ 0 0
$$621$$ 6.24977 0.250795
$$622$$ 0 0
$$623$$ −9.21949 −0.369371
$$624$$ 0 0
$$625$$ −30.9688 −1.23875
$$626$$ 0 0
$$627$$ −6.24977 −0.249592
$$628$$ 0 0
$$629$$ −5.28005 −0.210529
$$630$$ 0 0
$$631$$ 35.8089 1.42553 0.712766 0.701402i $$-0.247442\pi$$
0.712766 + 0.701402i $$0.247442\pi$$
$$632$$ 0 0
$$633$$ 18.0899 0.719009
$$634$$ 0 0
$$635$$ 38.9385 1.54523
$$636$$ 0 0
$$637$$ 4.00000 0.158486
$$638$$ 0 0
$$639$$ 0.329700 0.0130427
$$640$$ 0 0
$$641$$ −4.55011 −0.179719 −0.0898593 0.995954i $$-0.528642\pi$$
−0.0898593 + 0.995954i $$0.528642\pi$$
$$642$$ 0 0
$$643$$ 44.8392 1.76829 0.884143 0.467216i $$-0.154743\pi$$
0.884143 + 0.467216i $$0.154743\pi$$
$$644$$ 0 0
$$645$$ 11.2195 0.441767
$$646$$ 0 0
$$647$$ −9.32878 −0.366752 −0.183376 0.983043i $$-0.558703\pi$$
−0.183376 + 0.983043i $$0.558703\pi$$
$$648$$ 0 0
$$649$$ −12.1211 −0.475795
$$650$$ 0 0
$$651$$ −8.24977 −0.323334
$$652$$ 0 0
$$653$$ −13.1807 −0.515802 −0.257901 0.966171i $$-0.583031\pi$$
−0.257901 + 0.966171i $$0.583031\pi$$
$$654$$ 0 0
$$655$$ −55.3094 −2.16112
$$656$$ 0 0
$$657$$ −1.21949 −0.0475770
$$658$$ 0 0
$$659$$ 29.2900 1.14098 0.570489 0.821305i $$-0.306754\pi$$
0.570489 + 0.821305i $$0.306754\pi$$
$$660$$ 0 0
$$661$$ 31.4381 1.22280 0.611400 0.791322i $$-0.290607\pi$$
0.611400 + 0.791322i $$0.290607\pi$$
$$662$$ 0 0
$$663$$ 10.5601 0.410120
$$664$$ 0 0
$$665$$ −2.64002 −0.102376
$$666$$ 0 0
$$667$$ −22.5601 −0.873530
$$668$$ 0 0
$$669$$ −6.31032 −0.243971
$$670$$ 0 0
$$671$$ −0.378437 −0.0146094
$$672$$ 0 0
$$673$$ 16.0606 0.619089 0.309544 0.950885i $$-0.399823\pi$$
0.309544 + 0.950885i $$0.399823\pi$$
$$674$$ 0 0
$$675$$ −1.96972 −0.0758147
$$676$$ 0 0
$$677$$ 30.4995 1.17219 0.586096 0.810241i $$-0.300664\pi$$
0.586096 + 0.810241i $$0.300664\pi$$
$$678$$ 0 0
$$679$$ −9.03028 −0.346550
$$680$$ 0 0
$$681$$ −7.21949 −0.276652
$$682$$ 0 0
$$683$$ −18.1093 −0.692933 −0.346466 0.938062i $$-0.612619\pi$$
−0.346466 + 0.938062i $$0.612619\pi$$
$$684$$ 0 0
$$685$$ −17.1589 −0.655609
$$686$$ 0 0
$$687$$ −21.8401 −0.833253
$$688$$ 0 0
$$689$$ −51.5592 −1.96425
$$690$$ 0 0
$$691$$ −6.31789 −0.240344 −0.120172 0.992753i $$-0.538345\pi$$
−0.120172 + 0.992753i $$0.538345\pi$$
$$692$$ 0 0
$$693$$ 6.24977 0.237409
$$694$$ 0 0
$$695$$ −19.0596 −0.722973
$$696$$ 0 0
$$697$$ 13.7796 0.521939
$$698$$ 0 0
$$699$$ 21.7190 0.821489
$$700$$ 0 0
$$701$$ −19.4399 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$702$$ 0 0
$$703$$ −2.00000 −0.0754314
$$704$$ 0 0
$$705$$ 4.24977 0.160056
$$706$$ 0 0
$$707$$ −2.64002 −0.0992883
$$708$$ 0 0
$$709$$ −25.8477 −0.970731 −0.485365 0.874311i $$-0.661313\pi$$
−0.485365 + 0.874311i $$0.661313\pi$$
$$710$$ 0 0
$$711$$ 0.969724 0.0363675
$$712$$ 0 0
$$713$$ 51.5592 1.93091
$$714$$ 0 0
$$715$$ 65.9982 2.46819
$$716$$ 0 0
$$717$$ 28.0294 1.04678
$$718$$ 0 0
$$719$$ −8.32970 −0.310645 −0.155323 0.987864i $$-0.549642\pi$$
−0.155323 + 0.987864i $$0.549642\pi$$
$$720$$ 0 0
$$721$$ 1.93945 0.0722288
$$722$$ 0 0
$$723$$ 5.52982 0.205656
$$724$$ 0 0
$$725$$ 7.11021 0.264066
$$726$$ 0 0
$$727$$ −29.9394 −1.11039 −0.555196 0.831719i $$-0.687357\pi$$
−0.555196 + 0.831719i $$0.687357\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −11.2195 −0.414968
$$732$$ 0 0
$$733$$ 12.7200 0.469822 0.234911 0.972017i $$-0.424520\pi$$
0.234911 + 0.972017i $$0.424520\pi$$
$$734$$ 0 0
$$735$$ 2.64002 0.0973787
$$736$$ 0 0
$$737$$ −39.0596 −1.43878
$$738$$ 0 0
$$739$$ 27.8089 1.02297 0.511484 0.859293i $$-0.329096\pi$$
0.511484 + 0.859293i $$0.329096\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ 0 0
$$743$$ 1.60975 0.0590559 0.0295280 0.999564i $$-0.490600\pi$$
0.0295280 + 0.999564i $$0.490600\pi$$
$$744$$ 0 0
$$745$$ −57.6585 −2.11244
$$746$$ 0 0
$$747$$ 3.67030 0.134289
$$748$$ 0 0
$$749$$ 8.82924 0.322613
$$750$$ 0 0
$$751$$ −39.7484 −1.45044 −0.725220 0.688517i $$-0.758262\pi$$
−0.725220 + 0.688517i $$0.758262\pi$$
$$752$$ 0 0
$$753$$ 12.9503 0.471937
$$754$$ 0 0
$$755$$ 5.93945 0.216159
$$756$$ 0 0
$$757$$ −11.4986 −0.417925 −0.208962 0.977924i $$-0.567009\pi$$
−0.208962 + 0.977924i $$0.567009\pi$$
$$758$$ 0 0
$$759$$ −39.0596 −1.41777
$$760$$ 0 0
$$761$$ −19.7602 −0.716307 −0.358154 0.933663i $$-0.616594\pi$$
−0.358154 + 0.933663i $$0.616594\pi$$
$$762$$ 0 0
$$763$$ −7.28005 −0.263555
$$764$$ 0 0
$$765$$ 6.96972 0.251991
$$766$$ 0 0
$$767$$ 7.75779 0.280118
$$768$$ 0 0
$$769$$ 33.4381 1.20581 0.602904 0.797814i $$-0.294010\pi$$
0.602904 + 0.797814i $$0.294010\pi$$
$$770$$ 0 0
$$771$$ 6.49954 0.234075
$$772$$ 0 0
$$773$$ −12.8411 −0.461861 −0.230930 0.972970i $$-0.574177\pi$$
−0.230930 + 0.972970i $$0.574177\pi$$
$$774$$ 0 0
$$775$$ −16.2498 −0.583709
$$776$$ 0 0
$$777$$ 2.00000 0.0717496
$$778$$ 0 0
$$779$$ 5.21949 0.187008
$$780$$ 0 0
$$781$$ −2.06055 −0.0737324
$$782$$ 0 0
$$783$$ −3.60975 −0.129002
$$784$$ 0 0
$$785$$ 17.1589 0.612429
$$786$$ 0 0
$$787$$ 0.870417 0.0310270 0.0155135 0.999880i $$-0.495062\pi$$
0.0155135 + 0.999880i $$0.495062\pi$$
$$788$$ 0 0
$$789$$ −28.5289 −1.01566
$$790$$ 0 0
$$791$$ 4.88979 0.173861
$$792$$ 0 0
$$793$$ 0.242209 0.00860109
$$794$$ 0 0
$$795$$ −34.0294 −1.20690
$$796$$ 0 0
$$797$$ 31.7796 1.12569 0.562845 0.826562i $$-0.309707\pi$$
0.562845 + 0.826562i $$0.309707\pi$$
$$798$$ 0 0
$$799$$ −4.24977 −0.150346
$$800$$ 0 0
$$801$$ 9.21949 0.325755
$$802$$ 0 0
$$803$$ 7.62156 0.268959
$$804$$ 0 0
$$805$$ −16.4995 −0.581532
$$806$$ 0 0
$$807$$ −24.9385 −0.877878
$$808$$ 0 0
$$809$$ −43.9394 −1.54483 −0.772414 0.635119i $$-0.780951\pi$$
−0.772414 + 0.635119i $$0.780951\pi$$
$$810$$ 0 0
$$811$$ 3.37935 0.118665 0.0593326 0.998238i $$-0.481103\pi$$
0.0593326 + 0.998238i $$0.481103\pi$$
$$812$$ 0 0
$$813$$ −14.4390 −0.506397
$$814$$ 0 0
$$815$$ 39.0984 1.36956
$$816$$ 0 0
$$817$$ −4.24977 −0.148681
$$818$$ 0 0
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ −8.59885 −0.300102 −0.150051 0.988678i $$-0.547944\pi$$
−0.150051 + 0.988678i $$0.547944\pi$$
$$822$$ 0 0
$$823$$ −12.0076 −0.418557 −0.209279 0.977856i $$-0.567112\pi$$
−0.209279 + 0.977856i $$0.567112\pi$$
$$824$$ 0 0
$$825$$ 12.3103 0.428590
$$826$$ 0 0
$$827$$ 27.3893 0.952421 0.476210 0.879331i $$-0.342010\pi$$
0.476210 + 0.879331i $$0.342010\pi$$
$$828$$ 0 0
$$829$$ 0.909172 0.0315768 0.0157884 0.999875i $$-0.494974\pi$$
0.0157884 + 0.999875i $$0.494974\pi$$
$$830$$ 0 0
$$831$$ −25.5904 −0.887720
$$832$$ 0 0
$$833$$ −2.64002 −0.0914714
$$834$$ 0 0
$$835$$ −52.0587 −1.80157
$$836$$ 0 0
$$837$$ 8.24977 0.285154
$$838$$ 0 0
$$839$$ −40.1798 −1.38716 −0.693581 0.720379i $$-0.743968\pi$$
−0.693581 + 0.720379i $$0.743968\pi$$
$$840$$ 0 0
$$841$$ −15.9697 −0.550680
$$842$$ 0 0
$$843$$ 26.1698 0.901337
$$844$$ 0 0
$$845$$ −7.92007 −0.272459
$$846$$ 0 0
$$847$$ −28.0596 −0.964140
$$848$$ 0 0
$$849$$ −17.2800 −0.593050
$$850$$ 0 0
$$851$$ −12.4995 −0.428479
$$852$$ 0 0
$$853$$ 2.37844 0.0814361 0.0407181 0.999171i $$-0.487035\pi$$
0.0407181 + 0.999171i $$0.487035\pi$$
$$854$$ 0 0
$$855$$ 2.64002 0.0902869
$$856$$ 0 0
$$857$$ −39.2800 −1.34178 −0.670890 0.741556i $$-0.734088\pi$$
−0.670890 + 0.741556i $$0.734088\pi$$
$$858$$ 0 0
$$859$$ −45.7408 −1.56066 −0.780329 0.625370i $$-0.784948\pi$$
−0.780329 + 0.625370i $$0.784948\pi$$
$$860$$ 0 0
$$861$$ −5.21949 −0.177880
$$862$$ 0 0
$$863$$ −1.60975 −0.0547964 −0.0273982 0.999625i $$-0.508722\pi$$
−0.0273982 + 0.999625i $$0.508722\pi$$
$$864$$ 0 0
$$865$$ 8.65940 0.294428
$$866$$ 0 0
$$867$$ 10.0303 0.340646
$$868$$ 0 0
$$869$$ −6.06055 −0.205590
$$870$$ 0 0
$$871$$ 24.9991 0.847062
$$872$$ 0 0
$$873$$ 9.03028 0.305629
$$874$$ 0 0
$$875$$ −8.00000 −0.270449
$$876$$ 0 0
$$877$$ −50.0587 −1.69036 −0.845181 0.534480i $$-0.820508\pi$$
−0.845181 + 0.534480i $$0.820508\pi$$
$$878$$ 0 0
$$879$$ −11.4012 −0.384551
$$880$$ 0 0
$$881$$ 17.0790 0.575407 0.287703 0.957720i $$-0.407108\pi$$
0.287703 + 0.957720i $$0.407108\pi$$
$$882$$ 0 0
$$883$$ 1.06147 0.0357213 0.0178606 0.999840i $$-0.494314\pi$$
0.0178606 + 0.999840i $$0.494314\pi$$
$$884$$ 0 0
$$885$$ 5.12019 0.172113
$$886$$ 0 0
$$887$$ 21.7796 0.731287 0.365644 0.930755i $$-0.380849\pi$$
0.365644 + 0.930755i $$0.380849\pi$$
$$888$$ 0 0
$$889$$ 14.7493 0.494676
$$890$$ 0 0
$$891$$ −6.24977 −0.209375
$$892$$ 0 0
$$893$$ −1.60975 −0.0538681
$$894$$ 0 0
$$895$$ −51.1883 −1.71104
$$896$$ 0 0
$$897$$ 24.9991 0.834695
$$898$$ 0 0
$$899$$ −29.7796 −0.993205
$$900$$ 0 0
$$901$$ 34.0294 1.13368
$$902$$ 0 0
$$903$$ 4.24977 0.141424
$$904$$ 0 0
$$905$$ −17.0814 −0.567806
$$906$$ 0 0
$$907$$ 44.9073 1.49112 0.745562 0.666436i $$-0.232181\pi$$
0.745562 + 0.666436i $$0.232181\pi$$
$$908$$ 0 0
$$909$$ 2.64002 0.0875641
$$910$$ 0 0
$$911$$ −19.8889 −0.658948 −0.329474 0.944165i $$-0.606871\pi$$
−0.329474 + 0.944165i $$0.606871\pi$$
$$912$$ 0 0
$$913$$ −22.9385 −0.759155
$$914$$ 0 0
$$915$$ 0.159859 0.00528478
$$916$$ 0 0
$$917$$ −20.9503 −0.691841
$$918$$ 0 0
$$919$$ 32.9991 1.08854 0.544270 0.838910i $$-0.316807\pi$$
0.544270 + 0.838910i $$0.316807\pi$$
$$920$$ 0 0
$$921$$ 12.7493 0.420104
$$922$$ 0 0
$$923$$ 1.31880 0.0434089
$$924$$ 0 0
$$925$$ 3.93945 0.129528
$$926$$ 0 0
$$927$$ −1.93945 −0.0636998
$$928$$ 0 0
$$929$$ −3.95883 −0.129885 −0.0649424 0.997889i $$-0.520686\pi$$
−0.0649424 + 0.997889i $$0.520686\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 0 0
$$933$$ −18.8898 −0.618424
$$934$$ 0 0
$$935$$ −43.5592 −1.42454
$$936$$ 0 0
$$937$$ −35.4986 −1.15969 −0.579845 0.814727i $$-0.696887\pi$$
−0.579845 + 0.814727i $$0.696887\pi$$
$$938$$ 0 0
$$939$$ −16.4390 −0.536466
$$940$$ 0 0
$$941$$ 5.55918 0.181224 0.0906120 0.995886i $$-0.471118\pi$$
0.0906120 + 0.995886i $$0.471118\pi$$
$$942$$ 0 0
$$943$$ 32.6206 1.06227
$$944$$ 0 0
$$945$$ −2.64002 −0.0858800
$$946$$ 0 0
$$947$$ −19.8108 −0.643764 −0.321882 0.946780i $$-0.604315\pi$$
−0.321882 + 0.946780i $$0.604315\pi$$
$$948$$ 0 0
$$949$$ −4.87798 −0.158346
$$950$$ 0 0
$$951$$ −4.23039 −0.137180
$$952$$ 0 0
$$953$$ 57.7290 1.87003 0.935013 0.354613i $$-0.115387\pi$$
0.935013 + 0.354613i $$0.115387\pi$$
$$954$$ 0 0
$$955$$ 21.9394 0.709944
$$956$$ 0 0
$$957$$ 22.5601 0.729264
$$958$$ 0 0
$$959$$ −6.49954 −0.209881
$$960$$ 0 0
$$961$$ 37.0587 1.19544
$$962$$ 0 0
$$963$$ −8.82924 −0.284518
$$964$$ 0 0
$$965$$ 22.5988 0.727483
$$966$$ 0 0
$$967$$ 51.3094 1.65000 0.825000 0.565133i $$-0.191175\pi$$
0.825000 + 0.565133i $$0.191175\pi$$
$$968$$ 0 0
$$969$$ −2.64002 −0.0848098
$$970$$ 0 0
$$971$$ −16.6206 −0.533382 −0.266691 0.963782i $$-0.585930\pi$$
−0.266691 + 0.963782i $$0.585930\pi$$
$$972$$ 0 0
$$973$$ −7.21949 −0.231446
$$974$$ 0 0
$$975$$ −7.87890 −0.252327
$$976$$ 0 0
$$977$$ 34.6694 1.10917 0.554586 0.832126i $$-0.312877\pi$$
0.554586 + 0.832126i $$0.312877\pi$$
$$978$$ 0 0
$$979$$ −57.6197 −1.84153
$$980$$ 0 0
$$981$$ 7.28005 0.232434
$$982$$ 0 0
$$983$$ −22.1193 −0.705495 −0.352748 0.935719i $$-0.614753\pi$$
−0.352748 + 0.935719i $$0.614753\pi$$
$$984$$ 0 0
$$985$$ −36.9603 −1.17765
$$986$$ 0 0
$$987$$ 1.60975 0.0512388
$$988$$ 0 0
$$989$$ −26.5601 −0.844562
$$990$$ 0 0
$$991$$ −28.9697 −0.920254 −0.460127 0.887853i $$-0.652196\pi$$
−0.460127 + 0.887853i $$0.652196\pi$$
$$992$$ 0 0
$$993$$ 20.1892 0.640685
$$994$$ 0 0
$$995$$ 29.6197 0.939009
$$996$$ 0 0
$$997$$ −43.3775 −1.37378 −0.686890 0.726761i $$-0.741025\pi$$
−0.686890 + 0.726761i $$0.741025\pi$$
$$998$$ 0 0
$$999$$ −2.00000 −0.0632772
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 3192.2.a.u.1.1 3
3.2 odd 2 9576.2.a.cb.1.3 3
4.3 odd 2 6384.2.a.bw.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.u.1.1 3 1.1 even 1 trivial
6384.2.a.bw.1.1 3 4.3 odd 2
9576.2.a.cb.1.3 3 3.2 odd 2