# Properties

 Label 7440.2.a.bg.1.2 Level $7440$ Weight $2$ Character 7440.1 Self dual yes Analytic conductor $59.409$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7440.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: $$59.4086991038$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 7440.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} -6.37228 q^{11} -2.00000 q^{13} -1.00000 q^{15} +6.74456 q^{17} +6.37228 q^{19} -2.37228 q^{21} +2.37228 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.74456 q^{29} +1.00000 q^{31} +6.37228 q^{33} +2.37228 q^{35} +10.7446 q^{37} +2.00000 q^{39} -10.7446 q^{41} -6.37228 q^{43} +1.00000 q^{45} -4.74456 q^{47} -1.37228 q^{49} -6.74456 q^{51} -4.37228 q^{53} -6.37228 q^{55} -6.37228 q^{57} +8.74456 q^{59} -11.4891 q^{61} +2.37228 q^{63} -2.00000 q^{65} +0.744563 q^{67} -2.37228 q^{69} +2.37228 q^{71} +9.11684 q^{73} -1.00000 q^{75} -15.1168 q^{77} +10.3723 q^{79} +1.00000 q^{81} +12.0000 q^{83} +6.74456 q^{85} -2.74456 q^{87} +4.37228 q^{89} -4.74456 q^{91} -1.00000 q^{93} +6.37228 q^{95} +2.00000 q^{97} -6.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 4 q^{13} - 2 q^{15} + 2 q^{17} + 7 q^{19} + q^{21} - q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 2 q^{31} + 7 q^{33} - q^{35} + 10 q^{37} + 4 q^{39} - 10 q^{41} - 7 q^{43} + 2 q^{45} + 2 q^{47} + 3 q^{49} - 2 q^{51} - 3 q^{53} - 7 q^{55} - 7 q^{57} + 6 q^{59} - q^{63} - 4 q^{65} - 10 q^{67} + q^{69} - q^{71} + q^{73} - 2 q^{75} - 13 q^{77} + 15 q^{79} + 2 q^{81} + 24 q^{83} + 2 q^{85} + 6 q^{87} + 3 q^{89} + 2 q^{91} - 2 q^{93} + 7 q^{95} + 4 q^{97} - 7 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 4 * q^13 - 2 * q^15 + 2 * q^17 + 7 * q^19 + q^21 - q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 + 2 * q^31 + 7 * q^33 - q^35 + 10 * q^37 + 4 * q^39 - 10 * q^41 - 7 * q^43 + 2 * q^45 + 2 * q^47 + 3 * q^49 - 2 * q^51 - 3 * q^53 - 7 * q^55 - 7 * q^57 + 6 * q^59 - q^63 - 4 * q^65 - 10 * q^67 + q^69 - q^71 + q^73 - 2 * q^75 - 13 * q^77 + 15 * q^79 + 2 * q^81 + 24 * q^83 + 2 * q^85 + 6 * q^87 + 3 * q^89 + 2 * q^91 - 2 * q^93 + 7 * q^95 + 4 * q^97 - 7 * 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
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.37228 0.896638 0.448319 0.893874i $$-0.352023\pi$$
0.448319 + 0.893874i $$0.352023\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −6.37228 −1.92132 −0.960658 0.277736i $$-0.910416\pi$$
−0.960658 + 0.277736i $$0.910416\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 6.74456 1.63580 0.817898 0.575363i $$-0.195139\pi$$
0.817898 + 0.575363i $$0.195139\pi$$
$$18$$ 0 0
$$19$$ 6.37228 1.46190 0.730951 0.682430i $$-0.239077\pi$$
0.730951 + 0.682430i $$0.239077\pi$$
$$20$$ 0 0
$$21$$ −2.37228 −0.517674
$$22$$ 0 0
$$23$$ 2.37228 0.494655 0.247327 0.968932i $$-0.420448\pi$$
0.247327 + 0.968932i $$0.420448\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 2.74456 0.509652 0.254826 0.966987i $$-0.417982\pi$$
0.254826 + 0.966987i $$0.417982\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ 0 0
$$33$$ 6.37228 1.10927
$$34$$ 0 0
$$35$$ 2.37228 0.400989
$$36$$ 0 0
$$37$$ 10.7446 1.76640 0.883198 0.469001i $$-0.155386\pi$$
0.883198 + 0.469001i $$0.155386\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −10.7446 −1.67802 −0.839009 0.544117i $$-0.816865\pi$$
−0.839009 + 0.544117i $$0.816865\pi$$
$$42$$ 0 0
$$43$$ −6.37228 −0.971764 −0.485882 0.874024i $$-0.661501\pi$$
−0.485882 + 0.874024i $$0.661501\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −4.74456 −0.692066 −0.346033 0.938222i $$-0.612471\pi$$
−0.346033 + 0.938222i $$0.612471\pi$$
$$48$$ 0 0
$$49$$ −1.37228 −0.196040
$$50$$ 0 0
$$51$$ −6.74456 −0.944428
$$52$$ 0 0
$$53$$ −4.37228 −0.600579 −0.300290 0.953848i $$-0.597083\pi$$
−0.300290 + 0.953848i $$0.597083\pi$$
$$54$$ 0 0
$$55$$ −6.37228 −0.859238
$$56$$ 0 0
$$57$$ −6.37228 −0.844029
$$58$$ 0 0
$$59$$ 8.74456 1.13845 0.569223 0.822183i $$-0.307244\pi$$
0.569223 + 0.822183i $$0.307244\pi$$
$$60$$ 0 0
$$61$$ −11.4891 −1.47103 −0.735516 0.677507i $$-0.763060\pi$$
−0.735516 + 0.677507i $$0.763060\pi$$
$$62$$ 0 0
$$63$$ 2.37228 0.298879
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 0.744563 0.0909628 0.0454814 0.998965i $$-0.485518\pi$$
0.0454814 + 0.998965i $$0.485518\pi$$
$$68$$ 0 0
$$69$$ −2.37228 −0.285589
$$70$$ 0 0
$$71$$ 2.37228 0.281538 0.140769 0.990042i $$-0.455043\pi$$
0.140769 + 0.990042i $$0.455043\pi$$
$$72$$ 0 0
$$73$$ 9.11684 1.06705 0.533523 0.845786i $$-0.320868\pi$$
0.533523 + 0.845786i $$0.320868\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −15.1168 −1.72272
$$78$$ 0 0
$$79$$ 10.3723 1.16697 0.583486 0.812123i $$-0.301688\pi$$
0.583486 + 0.812123i $$0.301688\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 6.74456 0.731551
$$86$$ 0 0
$$87$$ −2.74456 −0.294248
$$88$$ 0 0
$$89$$ 4.37228 0.463461 0.231730 0.972780i $$-0.425561\pi$$
0.231730 + 0.972780i $$0.425561\pi$$
$$90$$ 0 0
$$91$$ −4.74456 −0.497365
$$92$$ 0 0
$$93$$ −1.00000 −0.103695
$$94$$ 0 0
$$95$$ 6.37228 0.653782
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ −6.37228 −0.640438
$$100$$ 0 0
$$101$$ −9.11684 −0.907160 −0.453580 0.891216i $$-0.649853\pi$$
−0.453580 + 0.891216i $$0.649853\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ −2.37228 −0.231511
$$106$$ 0 0
$$107$$ −6.37228 −0.616032 −0.308016 0.951381i $$-0.599665\pi$$
−0.308016 + 0.951381i $$0.599665\pi$$
$$108$$ 0 0
$$109$$ −6.74456 −0.646012 −0.323006 0.946397i $$-0.604693\pi$$
−0.323006 + 0.946397i $$0.604693\pi$$
$$110$$ 0 0
$$111$$ −10.7446 −1.01983
$$112$$ 0 0
$$113$$ −8.37228 −0.787598 −0.393799 0.919197i $$-0.628839\pi$$
−0.393799 + 0.919197i $$0.628839\pi$$
$$114$$ 0 0
$$115$$ 2.37228 0.221216
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 16.0000 1.46672
$$120$$ 0 0
$$121$$ 29.6060 2.69145
$$122$$ 0 0
$$123$$ 10.7446 0.968805
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 9.48913 0.842024 0.421012 0.907055i $$-0.361675\pi$$
0.421012 + 0.907055i $$0.361675\pi$$
$$128$$ 0 0
$$129$$ 6.37228 0.561048
$$130$$ 0 0
$$131$$ −18.2337 −1.59308 −0.796542 0.604583i $$-0.793340\pi$$
−0.796542 + 0.604583i $$0.793340\pi$$
$$132$$ 0 0
$$133$$ 15.1168 1.31080
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 19.4891 1.66507 0.832534 0.553974i $$-0.186889\pi$$
0.832534 + 0.553974i $$0.186889\pi$$
$$138$$ 0 0
$$139$$ −0.744563 −0.0631530 −0.0315765 0.999501i $$-0.510053\pi$$
−0.0315765 + 0.999501i $$0.510053\pi$$
$$140$$ 0 0
$$141$$ 4.74456 0.399564
$$142$$ 0 0
$$143$$ 12.7446 1.06575
$$144$$ 0 0
$$145$$ 2.74456 0.227924
$$146$$ 0 0
$$147$$ 1.37228 0.113184
$$148$$ 0 0
$$149$$ 5.11684 0.419188 0.209594 0.977788i $$-0.432786\pi$$
0.209594 + 0.977788i $$0.432786\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 6.74456 0.545266
$$154$$ 0 0
$$155$$ 1.00000 0.0803219
$$156$$ 0 0
$$157$$ 3.62772 0.289523 0.144762 0.989467i $$-0.453758\pi$$
0.144762 + 0.989467i $$0.453758\pi$$
$$158$$ 0 0
$$159$$ 4.37228 0.346744
$$160$$ 0 0
$$161$$ 5.62772 0.443526
$$162$$ 0 0
$$163$$ 10.2337 0.801564 0.400782 0.916173i $$-0.368738\pi$$
0.400782 + 0.916173i $$0.368738\pi$$
$$164$$ 0 0
$$165$$ 6.37228 0.496081
$$166$$ 0 0
$$167$$ 18.3723 1.42169 0.710845 0.703349i $$-0.248313\pi$$
0.710845 + 0.703349i $$0.248313\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 6.37228 0.487301
$$172$$ 0 0
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ 2.37228 0.179328
$$176$$ 0 0
$$177$$ −8.74456 −0.657282
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −13.8614 −1.03031 −0.515155 0.857097i $$-0.672266\pi$$
−0.515155 + 0.857097i $$0.672266\pi$$
$$182$$ 0 0
$$183$$ 11.4891 0.849301
$$184$$ 0 0
$$185$$ 10.7446 0.789956
$$186$$ 0 0
$$187$$ −42.9783 −3.14288
$$188$$ 0 0
$$189$$ −2.37228 −0.172558
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ −7.48913 −0.539079 −0.269540 0.962989i $$-0.586871\pi$$
−0.269540 + 0.962989i $$0.586871\pi$$
$$194$$ 0 0
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ −3.48913 −0.248590 −0.124295 0.992245i $$-0.539667\pi$$
−0.124295 + 0.992245i $$0.539667\pi$$
$$198$$ 0 0
$$199$$ 18.3723 1.30238 0.651188 0.758916i $$-0.274271\pi$$
0.651188 + 0.758916i $$0.274271\pi$$
$$200$$ 0 0
$$201$$ −0.744563 −0.0525174
$$202$$ 0 0
$$203$$ 6.51087 0.456974
$$204$$ 0 0
$$205$$ −10.7446 −0.750433
$$206$$ 0 0
$$207$$ 2.37228 0.164885
$$208$$ 0 0
$$209$$ −40.6060 −2.80877
$$210$$ 0 0
$$211$$ 6.37228 0.438686 0.219343 0.975648i $$-0.429609\pi$$
0.219343 + 0.975648i $$0.429609\pi$$
$$212$$ 0 0
$$213$$ −2.37228 −0.162546
$$214$$ 0 0
$$215$$ −6.37228 −0.434586
$$216$$ 0 0
$$217$$ 2.37228 0.161041
$$218$$ 0 0
$$219$$ −9.11684 −0.616059
$$220$$ 0 0
$$221$$ −13.4891 −0.907377
$$222$$ 0 0
$$223$$ −20.7446 −1.38916 −0.694579 0.719416i $$-0.744409\pi$$
−0.694579 + 0.719416i $$0.744409\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 11.1168 0.737851 0.368925 0.929459i $$-0.379726\pi$$
0.368925 + 0.929459i $$0.379726\pi$$
$$228$$ 0 0
$$229$$ 21.1168 1.39544 0.697720 0.716370i $$-0.254198\pi$$
0.697720 + 0.716370i $$0.254198\pi$$
$$230$$ 0 0
$$231$$ 15.1168 0.994615
$$232$$ 0 0
$$233$$ 13.8614 0.908091 0.454045 0.890979i $$-0.349980\pi$$
0.454045 + 0.890979i $$0.349980\pi$$
$$234$$ 0 0
$$235$$ −4.74456 −0.309501
$$236$$ 0 0
$$237$$ −10.3723 −0.673752
$$238$$ 0 0
$$239$$ 6.51087 0.421153 0.210577 0.977577i $$-0.432466\pi$$
0.210577 + 0.977577i $$0.432466\pi$$
$$240$$ 0 0
$$241$$ 27.4891 1.77073 0.885365 0.464896i $$-0.153908\pi$$
0.885365 + 0.464896i $$0.153908\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −1.37228 −0.0876718
$$246$$ 0 0
$$247$$ −12.7446 −0.810917
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ 0 0
$$253$$ −15.1168 −0.950388
$$254$$ 0 0
$$255$$ −6.74456 −0.422361
$$256$$ 0 0
$$257$$ 7.62772 0.475804 0.237902 0.971289i $$-0.423540\pi$$
0.237902 + 0.971289i $$0.423540\pi$$
$$258$$ 0 0
$$259$$ 25.4891 1.58382
$$260$$ 0 0
$$261$$ 2.74456 0.169884
$$262$$ 0 0
$$263$$ −26.9783 −1.66355 −0.831775 0.555113i $$-0.812675\pi$$
−0.831775 + 0.555113i $$0.812675\pi$$
$$264$$ 0 0
$$265$$ −4.37228 −0.268587
$$266$$ 0 0
$$267$$ −4.37228 −0.267579
$$268$$ 0 0
$$269$$ −2.00000 −0.121942 −0.0609711 0.998140i $$-0.519420\pi$$
−0.0609711 + 0.998140i $$0.519420\pi$$
$$270$$ 0 0
$$271$$ 31.1168 1.89021 0.945107 0.326762i $$-0.105957\pi$$
0.945107 + 0.326762i $$0.105957\pi$$
$$272$$ 0 0
$$273$$ 4.74456 0.287154
$$274$$ 0 0
$$275$$ −6.37228 −0.384263
$$276$$ 0 0
$$277$$ 26.7446 1.60693 0.803463 0.595355i $$-0.202989\pi$$
0.803463 + 0.595355i $$0.202989\pi$$
$$278$$ 0 0
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 5.25544 0.313513 0.156757 0.987637i $$-0.449896\pi$$
0.156757 + 0.987637i $$0.449896\pi$$
$$282$$ 0 0
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 0 0
$$285$$ −6.37228 −0.377461
$$286$$ 0 0
$$287$$ −25.4891 −1.50458
$$288$$ 0 0
$$289$$ 28.4891 1.67583
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 0 0
$$293$$ 15.4891 0.904884 0.452442 0.891794i $$-0.350553\pi$$
0.452442 + 0.891794i $$0.350553\pi$$
$$294$$ 0 0
$$295$$ 8.74456 0.509128
$$296$$ 0 0
$$297$$ 6.37228 0.369757
$$298$$ 0 0
$$299$$ −4.74456 −0.274385
$$300$$ 0 0
$$301$$ −15.1168 −0.871320
$$302$$ 0 0
$$303$$ 9.11684 0.523749
$$304$$ 0 0
$$305$$ −11.4891 −0.657865
$$306$$ 0 0
$$307$$ 14.9783 0.854854 0.427427 0.904050i $$-0.359420\pi$$
0.427427 + 0.904050i $$0.359420\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 0 0
$$315$$ 2.37228 0.133663
$$316$$ 0 0
$$317$$ −13.2554 −0.744500 −0.372250 0.928133i $$-0.621414\pi$$
−0.372250 + 0.928133i $$0.621414\pi$$
$$318$$ 0 0
$$319$$ −17.4891 −0.979203
$$320$$ 0 0
$$321$$ 6.37228 0.355666
$$322$$ 0 0
$$323$$ 42.9783 2.39137
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ 6.74456 0.372975
$$328$$ 0 0
$$329$$ −11.2554 −0.620532
$$330$$ 0 0
$$331$$ −21.4891 −1.18115 −0.590575 0.806983i $$-0.701099\pi$$
−0.590575 + 0.806983i $$0.701099\pi$$
$$332$$ 0 0
$$333$$ 10.7446 0.588798
$$334$$ 0 0
$$335$$ 0.744563 0.0406798
$$336$$ 0 0
$$337$$ −16.9783 −0.924864 −0.462432 0.886655i $$-0.653023\pi$$
−0.462432 + 0.886655i $$0.653023\pi$$
$$338$$ 0 0
$$339$$ 8.37228 0.454720
$$340$$ 0 0
$$341$$ −6.37228 −0.345078
$$342$$ 0 0
$$343$$ −19.8614 −1.07242
$$344$$ 0 0
$$345$$ −2.37228 −0.127719
$$346$$ 0 0
$$347$$ 32.4674 1.74294 0.871470 0.490449i $$-0.163167\pi$$
0.871470 + 0.490449i $$0.163167\pi$$
$$348$$ 0 0
$$349$$ 18.7446 1.00337 0.501687 0.865049i $$-0.332713\pi$$
0.501687 + 0.865049i $$0.332713\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −7.48913 −0.398606 −0.199303 0.979938i $$-0.563868\pi$$
−0.199303 + 0.979938i $$0.563868\pi$$
$$354$$ 0 0
$$355$$ 2.37228 0.125908
$$356$$ 0 0
$$357$$ −16.0000 −0.846810
$$358$$ 0 0
$$359$$ 2.37228 0.125204 0.0626021 0.998039i $$-0.480060\pi$$
0.0626021 + 0.998039i $$0.480060\pi$$
$$360$$ 0 0
$$361$$ 21.6060 1.13716
$$362$$ 0 0
$$363$$ −29.6060 −1.55391
$$364$$ 0 0
$$365$$ 9.11684 0.477197
$$366$$ 0 0
$$367$$ −14.2337 −0.742992 −0.371496 0.928434i $$-0.621155\pi$$
−0.371496 + 0.928434i $$0.621155\pi$$
$$368$$ 0 0
$$369$$ −10.7446 −0.559340
$$370$$ 0 0
$$371$$ −10.3723 −0.538502
$$372$$ 0 0
$$373$$ −13.8614 −0.717716 −0.358858 0.933392i $$-0.616834\pi$$
−0.358858 + 0.933392i $$0.616834\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −5.48913 −0.282704
$$378$$ 0 0
$$379$$ 0.138593 0.00711906 0.00355953 0.999994i $$-0.498867\pi$$
0.00355953 + 0.999994i $$0.498867\pi$$
$$380$$ 0 0
$$381$$ −9.48913 −0.486143
$$382$$ 0 0
$$383$$ 9.48913 0.484872 0.242436 0.970167i $$-0.422054\pi$$
0.242436 + 0.970167i $$0.422054\pi$$
$$384$$ 0 0
$$385$$ −15.1168 −0.770426
$$386$$ 0 0
$$387$$ −6.37228 −0.323921
$$388$$ 0 0
$$389$$ −28.9783 −1.46926 −0.734628 0.678470i $$-0.762643\pi$$
−0.734628 + 0.678470i $$0.762643\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 18.2337 0.919768
$$394$$ 0 0
$$395$$ 10.3723 0.521886
$$396$$ 0 0
$$397$$ 38.6060 1.93758 0.968789 0.247887i $$-0.0797361\pi$$
0.968789 + 0.247887i $$0.0797361\pi$$
$$398$$ 0 0
$$399$$ −15.1168 −0.756789
$$400$$ 0 0
$$401$$ 28.3723 1.41684 0.708422 0.705789i $$-0.249407\pi$$
0.708422 + 0.705789i $$0.249407\pi$$
$$402$$ 0 0
$$403$$ −2.00000 −0.0996271
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −68.4674 −3.39380
$$408$$ 0 0
$$409$$ 24.2337 1.19828 0.599139 0.800645i $$-0.295510\pi$$
0.599139 + 0.800645i $$0.295510\pi$$
$$410$$ 0 0
$$411$$ −19.4891 −0.961328
$$412$$ 0 0
$$413$$ 20.7446 1.02077
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 0.744563 0.0364614
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −3.48913 −0.170050 −0.0850248 0.996379i $$-0.527097\pi$$
−0.0850248 + 0.996379i $$0.527097\pi$$
$$422$$ 0 0
$$423$$ −4.74456 −0.230689
$$424$$ 0 0
$$425$$ 6.74456 0.327159
$$426$$ 0 0
$$427$$ −27.2554 −1.31898
$$428$$ 0 0
$$429$$ −12.7446 −0.615313
$$430$$ 0 0
$$431$$ −18.9783 −0.914150 −0.457075 0.889428i $$-0.651103\pi$$
−0.457075 + 0.889428i $$0.651103\pi$$
$$432$$ 0 0
$$433$$ 18.8832 0.907467 0.453733 0.891138i $$-0.350092\pi$$
0.453733 + 0.891138i $$0.350092\pi$$
$$434$$ 0 0
$$435$$ −2.74456 −0.131592
$$436$$ 0 0
$$437$$ 15.1168 0.723137
$$438$$ 0 0
$$439$$ −3.25544 −0.155374 −0.0776868 0.996978i $$-0.524753\pi$$
−0.0776868 + 0.996978i $$0.524753\pi$$
$$440$$ 0 0
$$441$$ −1.37228 −0.0653467
$$442$$ 0 0
$$443$$ 17.3505 0.824349 0.412174 0.911105i $$-0.364769\pi$$
0.412174 + 0.911105i $$0.364769\pi$$
$$444$$ 0 0
$$445$$ 4.37228 0.207266
$$446$$ 0 0
$$447$$ −5.11684 −0.242018
$$448$$ 0 0
$$449$$ 36.9783 1.74511 0.872556 0.488515i $$-0.162461\pi$$
0.872556 + 0.488515i $$0.162461\pi$$
$$450$$ 0 0
$$451$$ 68.4674 3.22400
$$452$$ 0 0
$$453$$ −8.00000 −0.375873
$$454$$ 0 0
$$455$$ −4.74456 −0.222429
$$456$$ 0 0
$$457$$ −34.4674 −1.61232 −0.806158 0.591700i $$-0.798457\pi$$
−0.806158 + 0.591700i $$0.798457\pi$$
$$458$$ 0 0
$$459$$ −6.74456 −0.314809
$$460$$ 0 0
$$461$$ 37.7228 1.75693 0.878463 0.477810i $$-0.158569\pi$$
0.878463 + 0.477810i $$0.158569\pi$$
$$462$$ 0 0
$$463$$ −25.4891 −1.18458 −0.592290 0.805725i $$-0.701776\pi$$
−0.592290 + 0.805725i $$0.701776\pi$$
$$464$$ 0 0
$$465$$ −1.00000 −0.0463739
$$466$$ 0 0
$$467$$ −29.4891 −1.36459 −0.682297 0.731075i $$-0.739019\pi$$
−0.682297 + 0.731075i $$0.739019\pi$$
$$468$$ 0 0
$$469$$ 1.76631 0.0815607
$$470$$ 0 0
$$471$$ −3.62772 −0.167156
$$472$$ 0 0
$$473$$ 40.6060 1.86706
$$474$$ 0 0
$$475$$ 6.37228 0.292380
$$476$$ 0 0
$$477$$ −4.37228 −0.200193
$$478$$ 0 0
$$479$$ 19.8614 0.907491 0.453745 0.891131i $$-0.350088\pi$$
0.453745 + 0.891131i $$0.350088\pi$$
$$480$$ 0 0
$$481$$ −21.4891 −0.979820
$$482$$ 0 0
$$483$$ −5.62772 −0.256070
$$484$$ 0 0
$$485$$ 2.00000 0.0908153
$$486$$ 0 0
$$487$$ 14.5109 0.657550 0.328775 0.944408i $$-0.393364\pi$$
0.328775 + 0.944408i $$0.393364\pi$$
$$488$$ 0 0
$$489$$ −10.2337 −0.462783
$$490$$ 0 0
$$491$$ 12.6060 0.568899 0.284450 0.958691i $$-0.408189\pi$$
0.284450 + 0.958691i $$0.408189\pi$$
$$492$$ 0 0
$$493$$ 18.5109 0.833688
$$494$$ 0 0
$$495$$ −6.37228 −0.286413
$$496$$ 0 0
$$497$$ 5.62772 0.252438
$$498$$ 0 0
$$499$$ −10.5109 −0.470531 −0.235266 0.971931i $$-0.575596\pi$$
−0.235266 + 0.971931i $$0.575596\pi$$
$$500$$ 0 0
$$501$$ −18.3723 −0.820813
$$502$$ 0 0
$$503$$ 17.4891 0.779802 0.389901 0.920857i $$-0.372509\pi$$
0.389901 + 0.920857i $$0.372509\pi$$
$$504$$ 0 0
$$505$$ −9.11684 −0.405694
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 0 0
$$509$$ −38.7446 −1.71732 −0.858661 0.512543i $$-0.828703\pi$$
−0.858661 + 0.512543i $$0.828703\pi$$
$$510$$ 0 0
$$511$$ 21.6277 0.956754
$$512$$ 0 0
$$513$$ −6.37228 −0.281343
$$514$$ 0 0
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ 30.2337 1.32968
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −8.97825 −0.393344 −0.196672 0.980469i $$-0.563013\pi$$
−0.196672 + 0.980469i $$0.563013\pi$$
$$522$$ 0 0
$$523$$ −15.8614 −0.693571 −0.346785 0.937944i $$-0.612727\pi$$
−0.346785 + 0.937944i $$0.612727\pi$$
$$524$$ 0 0
$$525$$ −2.37228 −0.103535
$$526$$ 0 0
$$527$$ 6.74456 0.293798
$$528$$ 0 0
$$529$$ −17.3723 −0.755317
$$530$$ 0 0
$$531$$ 8.74456 0.379482
$$532$$ 0 0
$$533$$ 21.4891 0.930797
$$534$$ 0 0
$$535$$ −6.37228 −0.275498
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 8.74456 0.376655
$$540$$ 0 0
$$541$$ 7.48913 0.321983 0.160991 0.986956i $$-0.448531\pi$$
0.160991 + 0.986956i $$0.448531\pi$$
$$542$$ 0 0
$$543$$ 13.8614 0.594850
$$544$$ 0 0
$$545$$ −6.74456 −0.288905
$$546$$ 0 0
$$547$$ −8.74456 −0.373890 −0.186945 0.982370i $$-0.559859\pi$$
−0.186945 + 0.982370i $$0.559859\pi$$
$$548$$ 0 0
$$549$$ −11.4891 −0.490344
$$550$$ 0 0
$$551$$ 17.4891 0.745062
$$552$$ 0 0
$$553$$ 24.6060 1.04635
$$554$$ 0 0
$$555$$ −10.7446 −0.456081
$$556$$ 0 0
$$557$$ 29.1168 1.23372 0.616860 0.787073i $$-0.288404\pi$$
0.616860 + 0.787073i $$0.288404\pi$$
$$558$$ 0 0
$$559$$ 12.7446 0.539038
$$560$$ 0 0
$$561$$ 42.9783 1.81454
$$562$$ 0 0
$$563$$ −38.9783 −1.64274 −0.821369 0.570398i $$-0.806789\pi$$
−0.821369 + 0.570398i $$0.806789\pi$$
$$564$$ 0 0
$$565$$ −8.37228 −0.352225
$$566$$ 0 0
$$567$$ 2.37228 0.0996265
$$568$$ 0 0
$$569$$ 4.37228 0.183296 0.0916478 0.995791i $$-0.470787\pi$$
0.0916478 + 0.995791i $$0.470787\pi$$
$$570$$ 0 0
$$571$$ 15.2554 0.638420 0.319210 0.947684i $$-0.396582\pi$$
0.319210 + 0.947684i $$0.396582\pi$$
$$572$$ 0 0
$$573$$ 16.0000 0.668410
$$574$$ 0 0
$$575$$ 2.37228 0.0989310
$$576$$ 0 0
$$577$$ 32.2337 1.34191 0.670953 0.741500i $$-0.265885\pi$$
0.670953 + 0.741500i $$0.265885\pi$$
$$578$$ 0 0
$$579$$ 7.48913 0.311237
$$580$$ 0 0
$$581$$ 28.4674 1.18103
$$582$$ 0 0
$$583$$ 27.8614 1.15390
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 6.37228 0.262565
$$590$$ 0 0
$$591$$ 3.48913 0.143523
$$592$$ 0 0
$$593$$ −0.978251 −0.0401719 −0.0200860 0.999798i $$-0.506394\pi$$
−0.0200860 + 0.999798i $$0.506394\pi$$
$$594$$ 0 0
$$595$$ 16.0000 0.655936
$$596$$ 0 0
$$597$$ −18.3723 −0.751927
$$598$$ 0 0
$$599$$ 7.11684 0.290786 0.145393 0.989374i $$-0.453555\pi$$
0.145393 + 0.989374i $$0.453555\pi$$
$$600$$ 0 0
$$601$$ 35.4891 1.44763 0.723816 0.689993i $$-0.242387\pi$$
0.723816 + 0.689993i $$0.242387\pi$$
$$602$$ 0 0
$$603$$ 0.744563 0.0303209
$$604$$ 0 0
$$605$$ 29.6060 1.20365
$$606$$ 0 0
$$607$$ 10.3723 0.420998 0.210499 0.977594i $$-0.432491\pi$$
0.210499 + 0.977594i $$0.432491\pi$$
$$608$$ 0 0
$$609$$ −6.51087 −0.263834
$$610$$ 0 0
$$611$$ 9.48913 0.383889
$$612$$ 0 0
$$613$$ 12.5109 0.505309 0.252655 0.967557i $$-0.418696\pi$$
0.252655 + 0.967557i $$0.418696\pi$$
$$614$$ 0 0
$$615$$ 10.7446 0.433263
$$616$$ 0 0
$$617$$ 25.1168 1.01117 0.505583 0.862778i $$-0.331277\pi$$
0.505583 + 0.862778i $$0.331277\pi$$
$$618$$ 0 0
$$619$$ 18.2337 0.732874 0.366437 0.930443i $$-0.380578\pi$$
0.366437 + 0.930443i $$0.380578\pi$$
$$620$$ 0 0
$$621$$ −2.37228 −0.0951964
$$622$$ 0 0
$$623$$ 10.3723 0.415557
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 40.6060 1.62165
$$628$$ 0 0
$$629$$ 72.4674 2.88946
$$630$$ 0 0
$$631$$ −5.35053 −0.213001 −0.106501 0.994313i $$-0.533965\pi$$
−0.106501 + 0.994313i $$0.533965\pi$$
$$632$$ 0 0
$$633$$ −6.37228 −0.253275
$$634$$ 0 0
$$635$$ 9.48913 0.376564
$$636$$ 0 0
$$637$$ 2.74456 0.108744
$$638$$ 0 0
$$639$$ 2.37228 0.0938460
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ −38.0951 −1.50232 −0.751162 0.660118i $$-0.770506\pi$$
−0.751162 + 0.660118i $$0.770506\pi$$
$$644$$ 0 0
$$645$$ 6.37228 0.250908
$$646$$ 0 0
$$647$$ 35.5842 1.39896 0.699480 0.714652i $$-0.253415\pi$$
0.699480 + 0.714652i $$0.253415\pi$$
$$648$$ 0 0
$$649$$ −55.7228 −2.18731
$$650$$ 0 0
$$651$$ −2.37228 −0.0929770
$$652$$ 0 0
$$653$$ −4.97825 −0.194814 −0.0974070 0.995245i $$-0.531055\pi$$
−0.0974070 + 0.995245i $$0.531055\pi$$
$$654$$ 0 0
$$655$$ −18.2337 −0.712449
$$656$$ 0 0
$$657$$ 9.11684 0.355682
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 40.9783 1.59387 0.796935 0.604066i $$-0.206454\pi$$
0.796935 + 0.604066i $$0.206454\pi$$
$$662$$ 0 0
$$663$$ 13.4891 0.523874
$$664$$ 0 0
$$665$$ 15.1168 0.586206
$$666$$ 0 0
$$667$$ 6.51087 0.252102
$$668$$ 0 0
$$669$$ 20.7446 0.802031
$$670$$ 0 0
$$671$$ 73.2119 2.82632
$$672$$ 0 0
$$673$$ −23.4891 −0.905439 −0.452720 0.891653i $$-0.649546\pi$$
−0.452720 + 0.891653i $$0.649546\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −1.39403 −0.0535770 −0.0267885 0.999641i $$-0.508528\pi$$
−0.0267885 + 0.999641i $$0.508528\pi$$
$$678$$ 0 0
$$679$$ 4.74456 0.182080
$$680$$ 0 0
$$681$$ −11.1168 −0.425998
$$682$$ 0 0
$$683$$ −15.8614 −0.606920 −0.303460 0.952844i $$-0.598142\pi$$
−0.303460 + 0.952844i $$0.598142\pi$$
$$684$$ 0 0
$$685$$ 19.4891 0.744641
$$686$$ 0 0
$$687$$ −21.1168 −0.805658
$$688$$ 0 0
$$689$$ 8.74456 0.333141
$$690$$ 0 0
$$691$$ 47.8614 1.82073 0.910367 0.413802i $$-0.135799\pi$$
0.910367 + 0.413802i $$0.135799\pi$$
$$692$$ 0 0
$$693$$ −15.1168 −0.574241
$$694$$ 0 0
$$695$$ −0.744563 −0.0282429
$$696$$ 0 0
$$697$$ −72.4674 −2.74490
$$698$$ 0 0
$$699$$ −13.8614 −0.524287
$$700$$ 0 0
$$701$$ 5.39403 0.203730 0.101865 0.994798i $$-0.467519\pi$$
0.101865 + 0.994798i $$0.467519\pi$$
$$702$$ 0 0
$$703$$ 68.4674 2.58230
$$704$$ 0 0
$$705$$ 4.74456 0.178691
$$706$$ 0 0
$$707$$ −21.6277 −0.813394
$$708$$ 0 0
$$709$$ 22.8832 0.859395 0.429697 0.902973i $$-0.358620\pi$$
0.429697 + 0.902973i $$0.358620\pi$$
$$710$$ 0 0
$$711$$ 10.3723 0.388991
$$712$$ 0 0
$$713$$ 2.37228 0.0888426
$$714$$ 0 0
$$715$$ 12.7446 0.476620
$$716$$ 0 0
$$717$$ −6.51087 −0.243153
$$718$$ 0 0
$$719$$ −27.2554 −1.01646 −0.508228 0.861222i $$-0.669699\pi$$
−0.508228 + 0.861222i $$0.669699\pi$$
$$720$$ 0 0
$$721$$ 18.9783 0.706787
$$722$$ 0 0
$$723$$ −27.4891 −1.02233
$$724$$ 0 0
$$725$$ 2.74456 0.101930
$$726$$ 0 0
$$727$$ −13.6277 −0.505424 −0.252712 0.967542i $$-0.581323\pi$$
−0.252712 + 0.967542i $$0.581323\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −42.9783 −1.58961
$$732$$ 0 0
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ 0 0
$$735$$ 1.37228 0.0506174
$$736$$ 0 0
$$737$$ −4.74456 −0.174768
$$738$$ 0 0
$$739$$ −22.9783 −0.845269 −0.422634 0.906300i $$-0.638895\pi$$
−0.422634 + 0.906300i $$0.638895\pi$$
$$740$$ 0 0
$$741$$ 12.7446 0.468183
$$742$$ 0 0
$$743$$ 29.6277 1.08694 0.543468 0.839430i $$-0.317111\pi$$
0.543468 + 0.839430i $$0.317111\pi$$
$$744$$ 0 0
$$745$$ 5.11684 0.187467
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ −15.1168 −0.552357
$$750$$ 0 0
$$751$$ 27.2554 0.994565 0.497283 0.867589i $$-0.334331\pi$$
0.497283 + 0.867589i $$0.334331\pi$$
$$752$$ 0 0
$$753$$ −4.00000 −0.145768
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 0 0
$$759$$ 15.1168 0.548707
$$760$$ 0 0
$$761$$ −18.1386 −0.657523 −0.328762 0.944413i $$-0.606631\pi$$
−0.328762 + 0.944413i $$0.606631\pi$$
$$762$$ 0 0
$$763$$ −16.0000 −0.579239
$$764$$ 0 0
$$765$$ 6.74456 0.243850
$$766$$ 0 0
$$767$$ −17.4891 −0.631496
$$768$$ 0 0
$$769$$ −3.62772 −0.130819 −0.0654094 0.997859i $$-0.520835\pi$$
−0.0654094 + 0.997859i $$0.520835\pi$$
$$770$$ 0 0
$$771$$ −7.62772 −0.274706
$$772$$ 0 0
$$773$$ −18.6060 −0.669210 −0.334605 0.942358i $$-0.608603\pi$$
−0.334605 + 0.942358i $$0.608603\pi$$
$$774$$ 0 0
$$775$$ 1.00000 0.0359211
$$776$$ 0 0
$$777$$ −25.4891 −0.914417
$$778$$ 0 0
$$779$$ −68.4674 −2.45310
$$780$$ 0 0
$$781$$ −15.1168 −0.540923
$$782$$ 0 0
$$783$$ −2.74456 −0.0980827
$$784$$ 0 0
$$785$$ 3.62772 0.129479
$$786$$ 0 0
$$787$$ 27.1168 0.966611 0.483306 0.875452i $$-0.339436\pi$$
0.483306 + 0.875452i $$0.339436\pi$$
$$788$$ 0 0
$$789$$ 26.9783 0.960451
$$790$$ 0 0
$$791$$ −19.8614 −0.706190
$$792$$ 0 0
$$793$$ 22.9783 0.815982
$$794$$ 0 0
$$795$$ 4.37228 0.155069
$$796$$ 0 0
$$797$$ −43.4891 −1.54046 −0.770232 0.637764i $$-0.779860\pi$$
−0.770232 + 0.637764i $$0.779860\pi$$
$$798$$ 0 0
$$799$$ −32.0000 −1.13208
$$800$$ 0 0
$$801$$ 4.37228 0.154487
$$802$$ 0 0
$$803$$ −58.0951 −2.05013
$$804$$ 0 0
$$805$$ 5.62772 0.198351
$$806$$ 0 0
$$807$$ 2.00000 0.0704033
$$808$$ 0 0
$$809$$ −30.6060 −1.07605 −0.538024 0.842929i $$-0.680829\pi$$
−0.538024 + 0.842929i $$0.680829\pi$$
$$810$$ 0 0
$$811$$ 46.3723 1.62835 0.814176 0.580619i $$-0.197189\pi$$
0.814176 + 0.580619i $$0.197189\pi$$
$$812$$ 0 0
$$813$$ −31.1168 −1.09132
$$814$$ 0 0
$$815$$ 10.2337 0.358470
$$816$$ 0 0
$$817$$ −40.6060 −1.42062
$$818$$ 0 0
$$819$$ −4.74456 −0.165788
$$820$$ 0 0
$$821$$ 26.7446 0.933392 0.466696 0.884418i $$-0.345444\pi$$
0.466696 + 0.884418i $$0.345444\pi$$
$$822$$ 0 0
$$823$$ 28.7446 1.00197 0.500986 0.865455i $$-0.332971\pi$$
0.500986 + 0.865455i $$0.332971\pi$$
$$824$$ 0 0
$$825$$ 6.37228 0.221854
$$826$$ 0 0
$$827$$ −5.48913 −0.190876 −0.0954378 0.995435i $$-0.530425\pi$$
−0.0954378 + 0.995435i $$0.530425\pi$$
$$828$$ 0 0
$$829$$ 32.0951 1.11471 0.557354 0.830275i $$-0.311816\pi$$
0.557354 + 0.830275i $$0.311816\pi$$
$$830$$ 0 0
$$831$$ −26.7446 −0.927759
$$832$$ 0 0
$$833$$ −9.25544 −0.320682
$$834$$ 0 0
$$835$$ 18.3723 0.635799
$$836$$ 0 0
$$837$$ −1.00000 −0.0345651
$$838$$ 0 0
$$839$$ 11.8614 0.409501 0.204751 0.978814i $$-0.434362\pi$$
0.204751 + 0.978814i $$0.434362\pi$$
$$840$$ 0 0
$$841$$ −21.4674 −0.740254
$$842$$ 0 0
$$843$$ −5.25544 −0.181007
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 70.2337 2.41326
$$848$$ 0 0
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ 25.4891 0.873756
$$852$$ 0 0
$$853$$ 24.0951 0.825000 0.412500 0.910958i $$-0.364656\pi$$
0.412500 + 0.910958i $$0.364656\pi$$
$$854$$ 0 0
$$855$$ 6.37228 0.217927
$$856$$ 0 0
$$857$$ −15.4891 −0.529098 −0.264549 0.964372i $$-0.585223\pi$$
−0.264549 + 0.964372i $$0.585223\pi$$
$$858$$ 0 0
$$859$$ −5.48913 −0.187287 −0.0936433 0.995606i $$-0.529851\pi$$
−0.0936433 + 0.995606i $$0.529851\pi$$
$$860$$ 0 0
$$861$$ 25.4891 0.868667
$$862$$ 0 0
$$863$$ −45.3505 −1.54375 −0.771875 0.635774i $$-0.780681\pi$$
−0.771875 + 0.635774i $$0.780681\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ −28.4891 −0.967541
$$868$$ 0 0
$$869$$ −66.0951 −2.24212
$$870$$ 0 0
$$871$$ −1.48913 −0.0504571
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 2.37228 0.0801977
$$876$$ 0 0
$$877$$ 0.978251 0.0330332 0.0165166 0.999864i $$-0.494742\pi$$
0.0165166 + 0.999864i $$0.494742\pi$$
$$878$$ 0 0
$$879$$ −15.4891 −0.522435
$$880$$ 0 0
$$881$$ −48.9783 −1.65012 −0.825060 0.565046i $$-0.808859\pi$$
−0.825060 + 0.565046i $$0.808859\pi$$
$$882$$ 0 0
$$883$$ −16.1386 −0.543107 −0.271553 0.962423i $$-0.587537\pi$$
−0.271553 + 0.962423i $$0.587537\pi$$
$$884$$ 0 0
$$885$$ −8.74456 −0.293945
$$886$$ 0 0
$$887$$ 19.2554 0.646534 0.323267 0.946308i $$-0.395219\pi$$
0.323267 + 0.946308i $$0.395219\pi$$
$$888$$ 0 0
$$889$$ 22.5109 0.754991
$$890$$ 0 0
$$891$$ −6.37228 −0.213479
$$892$$ 0 0
$$893$$ −30.2337 −1.01173
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ 0 0
$$897$$ 4.74456 0.158416
$$898$$ 0 0
$$899$$ 2.74456 0.0915363
$$900$$ 0 0
$$901$$ −29.4891 −0.982425
$$902$$ 0 0
$$903$$ 15.1168 0.503057
$$904$$ 0 0
$$905$$ −13.8614 −0.460769
$$906$$ 0 0
$$907$$ −5.48913 −0.182263 −0.0911317 0.995839i $$-0.529048\pi$$
−0.0911317 + 0.995839i $$0.529048\pi$$
$$908$$ 0 0
$$909$$ −9.11684 −0.302387
$$910$$ 0 0
$$911$$ 25.4891 0.844492 0.422246 0.906481i $$-0.361242\pi$$
0.422246 + 0.906481i $$0.361242\pi$$
$$912$$ 0 0
$$913$$ −76.4674 −2.53070
$$914$$ 0 0
$$915$$ 11.4891 0.379819
$$916$$ 0 0
$$917$$ −43.2554 −1.42842
$$918$$ 0 0
$$919$$ −22.2337 −0.733422 −0.366711 0.930335i $$-0.619516\pi$$
−0.366711 + 0.930335i $$0.619516\pi$$
$$920$$ 0 0
$$921$$ −14.9783 −0.493550
$$922$$ 0 0
$$923$$ −4.74456 −0.156169
$$924$$ 0 0
$$925$$ 10.7446 0.353279
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 1.11684 0.0366425 0.0183212 0.999832i $$-0.494168\pi$$
0.0183212 + 0.999832i $$0.494168\pi$$
$$930$$ 0 0
$$931$$ −8.74456 −0.286591
$$932$$ 0 0
$$933$$ 8.00000 0.261908
$$934$$ 0 0
$$935$$ −42.9783 −1.40554
$$936$$ 0 0
$$937$$ −42.7446 −1.39640 −0.698202 0.715901i $$-0.746016\pi$$
−0.698202 + 0.715901i $$0.746016\pi$$
$$938$$ 0 0
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 37.7228 1.22973 0.614864 0.788633i $$-0.289211\pi$$
0.614864 + 0.788633i $$0.289211\pi$$
$$942$$ 0 0
$$943$$ −25.4891 −0.830040
$$944$$ 0 0
$$945$$ −2.37228 −0.0771703
$$946$$ 0 0
$$947$$ −8.74456 −0.284160 −0.142080 0.989855i $$-0.545379\pi$$
−0.142080 + 0.989855i $$0.545379\pi$$
$$948$$ 0 0
$$949$$ −18.2337 −0.591891
$$950$$ 0 0
$$951$$ 13.2554 0.429837
$$952$$ 0 0
$$953$$ −45.7228 −1.48111 −0.740554 0.671997i $$-0.765437\pi$$
−0.740554 + 0.671997i $$0.765437\pi$$
$$954$$ 0 0
$$955$$ −16.0000 −0.517748
$$956$$ 0 0
$$957$$ 17.4891 0.565343
$$958$$ 0 0
$$959$$ 46.2337 1.49296
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ −6.37228 −0.205344
$$964$$ 0 0
$$965$$ −7.48913 −0.241083
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ −42.9783 −1.38066
$$970$$ 0 0
$$971$$ −54.7011 −1.75544 −0.877720 0.479173i $$-0.840937\pi$$
−0.877720 + 0.479173i $$0.840937\pi$$
$$972$$ 0 0
$$973$$ −1.76631 −0.0566254
$$974$$ 0 0
$$975$$ 2.00000 0.0640513
$$976$$ 0 0
$$977$$ −46.0000 −1.47167 −0.735835 0.677161i $$-0.763210\pi$$
−0.735835 + 0.677161i $$0.763210\pi$$
$$978$$ 0 0
$$979$$ −27.8614 −0.890454
$$980$$ 0 0
$$981$$ −6.74456 −0.215337
$$982$$ 0 0
$$983$$ −42.9783 −1.37079 −0.685397 0.728170i $$-0.740371\pi$$
−0.685397 + 0.728170i $$0.740371\pi$$
$$984$$ 0 0
$$985$$ −3.48913 −0.111173
$$986$$ 0 0
$$987$$ 11.2554 0.358265
$$988$$ 0 0
$$989$$ −15.1168 −0.480688
$$990$$ 0 0
$$991$$ 7.39403 0.234879 0.117440 0.993080i $$-0.462531\pi$$
0.117440 + 0.993080i $$0.462531\pi$$
$$992$$ 0 0
$$993$$ 21.4891 0.681937
$$994$$ 0 0
$$995$$ 18.3723 0.582440
$$996$$ 0 0
$$997$$ 6.00000 0.190022 0.0950110 0.995476i $$-0.469711\pi$$
0.0950110 + 0.995476i $$0.469711\pi$$
$$998$$ 0 0
$$999$$ −10.7446 −0.339943
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 7440.2.a.bg.1.2 2
4.3 odd 2 930.2.a.r.1.1 2
12.11 even 2 2790.2.a.bd.1.1 2
20.3 even 4 4650.2.d.bh.3349.2 4
20.7 even 4 4650.2.d.bh.3349.3 4
20.19 odd 2 4650.2.a.by.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.r.1.1 2 4.3 odd 2
2790.2.a.bd.1.1 2 12.11 even 2
4650.2.a.by.1.2 2 20.19 odd 2
4650.2.d.bh.3349.2 4 20.3 even 4
4650.2.d.bh.3349.3 4 20.7 even 4
7440.2.a.bg.1.2 2 1.1 even 1 trivial