# Properties

 Label 7600.2.a.bg.1.1 Level $7600$ Weight $2$ Character 7600.1 Self dual yes Analytic conductor $60.686$ 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] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.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: $$60.6863055362$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 7600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.414214 q^{3} +4.41421 q^{7} -2.82843 q^{9} +O(q^{10})$$ $$q-0.414214 q^{3} +4.41421 q^{7} -2.82843 q^{9} +1.41421 q^{11} -5.82843 q^{13} -1.00000 q^{17} +1.00000 q^{19} -1.82843 q^{21} +0.757359 q^{23} +2.41421 q^{27} -0.171573 q^{29} -6.24264 q^{31} -0.585786 q^{33} +8.48528 q^{37} +2.41421 q^{39} -4.24264 q^{41} +1.75736 q^{43} +12.4853 q^{49} +0.414214 q^{51} -5.48528 q^{53} -0.414214 q^{57} -6.89949 q^{59} +14.2426 q^{61} -12.4853 q^{63} +4.75736 q^{67} -0.313708 q^{69} +13.4142 q^{71} -11.4853 q^{73} +6.24264 q^{77} +6.48528 q^{79} +7.48528 q^{81} +14.4853 q^{83} +0.0710678 q^{87} +7.07107 q^{89} -25.7279 q^{91} +2.58579 q^{93} +0.343146 q^{97} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 6 q^{7}+O(q^{10})$$ 2 * q + 2 * q^3 + 6 * q^7 $$2 q + 2 q^{3} + 6 q^{7} - 6 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{21} + 10 q^{23} + 2 q^{27} - 6 q^{29} - 4 q^{31} - 4 q^{33} + 2 q^{39} + 12 q^{43} + 8 q^{49} - 2 q^{51} + 6 q^{53} + 2 q^{57} + 6 q^{59} + 20 q^{61} - 8 q^{63} + 18 q^{67} + 22 q^{69} + 24 q^{71} - 6 q^{73} + 4 q^{77} - 4 q^{79} - 2 q^{81} + 12 q^{83} - 14 q^{87} - 26 q^{91} + 8 q^{93} + 12 q^{97} - 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 6 * q^7 - 6 * q^13 - 2 * q^17 + 2 * q^19 + 2 * q^21 + 10 * q^23 + 2 * q^27 - 6 * q^29 - 4 * q^31 - 4 * q^33 + 2 * q^39 + 12 * q^43 + 8 * q^49 - 2 * q^51 + 6 * q^53 + 2 * q^57 + 6 * q^59 + 20 * q^61 - 8 * q^63 + 18 * q^67 + 22 * q^69 + 24 * q^71 - 6 * q^73 + 4 * q^77 - 4 * q^79 - 2 * q^81 + 12 * q^83 - 14 * q^87 - 26 * q^91 + 8 * q^93 + 12 * q^97 - 8 * 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
$$3$$ −0.414214 −0.239146 −0.119573 0.992825i $$-0.538153\pi$$
−0.119573 + 0.992825i $$0.538153\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.41421 1.66842 0.834208 0.551450i $$-0.185925\pi$$
0.834208 + 0.551450i $$0.185925\pi$$
$$8$$ 0 0
$$9$$ −2.82843 −0.942809
$$10$$ 0 0
$$11$$ 1.41421 0.426401 0.213201 0.977008i $$-0.431611\pi$$
0.213201 + 0.977008i $$0.431611\pi$$
$$12$$ 0 0
$$13$$ −5.82843 −1.61651 −0.808257 0.588829i $$-0.799589\pi$$
−0.808257 + 0.588829i $$0.799589\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −1.82843 −0.398996
$$22$$ 0 0
$$23$$ 0.757359 0.157920 0.0789602 0.996878i $$-0.474840\pi$$
0.0789602 + 0.996878i $$0.474840\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 2.41421 0.464616
$$28$$ 0 0
$$29$$ −0.171573 −0.0318603 −0.0159301 0.999873i $$-0.505071\pi$$
−0.0159301 + 0.999873i $$0.505071\pi$$
$$30$$ 0 0
$$31$$ −6.24264 −1.12121 −0.560606 0.828083i $$-0.689432\pi$$
−0.560606 + 0.828083i $$0.689432\pi$$
$$32$$ 0 0
$$33$$ −0.585786 −0.101972
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.48528 1.39497 0.697486 0.716599i $$-0.254302\pi$$
0.697486 + 0.716599i $$0.254302\pi$$
$$38$$ 0 0
$$39$$ 2.41421 0.386584
$$40$$ 0 0
$$41$$ −4.24264 −0.662589 −0.331295 0.943527i $$-0.607485\pi$$
−0.331295 + 0.943527i $$0.607485\pi$$
$$42$$ 0 0
$$43$$ 1.75736 0.267995 0.133997 0.990982i $$-0.457219\pi$$
0.133997 + 0.990982i $$0.457219\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 12.4853 1.78361
$$50$$ 0 0
$$51$$ 0.414214 0.0580015
$$52$$ 0 0
$$53$$ −5.48528 −0.753461 −0.376731 0.926323i $$-0.622952\pi$$
−0.376731 + 0.926323i $$0.622952\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.414214 −0.0548639
$$58$$ 0 0
$$59$$ −6.89949 −0.898238 −0.449119 0.893472i $$-0.648262\pi$$
−0.449119 + 0.893472i $$0.648262\pi$$
$$60$$ 0 0
$$61$$ 14.2426 1.82358 0.911792 0.410653i $$-0.134699\pi$$
0.911792 + 0.410653i $$0.134699\pi$$
$$62$$ 0 0
$$63$$ −12.4853 −1.57300
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.75736 0.581204 0.290602 0.956844i $$-0.406144\pi$$
0.290602 + 0.956844i $$0.406144\pi$$
$$68$$ 0 0
$$69$$ −0.313708 −0.0377661
$$70$$ 0 0
$$71$$ 13.4142 1.59197 0.795987 0.605314i $$-0.206952\pi$$
0.795987 + 0.605314i $$0.206952\pi$$
$$72$$ 0 0
$$73$$ −11.4853 −1.34425 −0.672125 0.740437i $$-0.734618\pi$$
−0.672125 + 0.740437i $$0.734618\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.24264 0.711415
$$78$$ 0 0
$$79$$ 6.48528 0.729651 0.364826 0.931076i $$-0.381129\pi$$
0.364826 + 0.931076i $$0.381129\pi$$
$$80$$ 0 0
$$81$$ 7.48528 0.831698
$$82$$ 0 0
$$83$$ 14.4853 1.58997 0.794983 0.606632i $$-0.207480\pi$$
0.794983 + 0.606632i $$0.207480\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0.0710678 0.00761927
$$88$$ 0 0
$$89$$ 7.07107 0.749532 0.374766 0.927119i $$-0.377723\pi$$
0.374766 + 0.927119i $$0.377723\pi$$
$$90$$ 0 0
$$91$$ −25.7279 −2.69702
$$92$$ 0 0
$$93$$ 2.58579 0.268134
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0.343146 0.0348412 0.0174206 0.999848i $$-0.494455\pi$$
0.0174206 + 0.999848i $$0.494455\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 13.0711 1.30062 0.650310 0.759669i $$-0.274639\pi$$
0.650310 + 0.759669i $$0.274639\pi$$
$$102$$ 0 0
$$103$$ 4.24264 0.418040 0.209020 0.977911i $$-0.432973\pi$$
0.209020 + 0.977911i $$0.432973\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 19.7279 1.90717 0.953585 0.301124i $$-0.0973617\pi$$
0.953585 + 0.301124i $$0.0973617\pi$$
$$108$$ 0 0
$$109$$ −17.9706 −1.72127 −0.860634 0.509224i $$-0.829932\pi$$
−0.860634 + 0.509224i $$0.829932\pi$$
$$110$$ 0 0
$$111$$ −3.51472 −0.333602
$$112$$ 0 0
$$113$$ 10.2426 0.963547 0.481773 0.876296i $$-0.339993\pi$$
0.481773 + 0.876296i $$0.339993\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 16.4853 1.52406
$$118$$ 0 0
$$119$$ −4.41421 −0.404650
$$120$$ 0 0
$$121$$ −9.00000 −0.818182
$$122$$ 0 0
$$123$$ 1.75736 0.158456
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2.48528 −0.220533 −0.110267 0.993902i $$-0.535170\pi$$
−0.110267 + 0.993902i $$0.535170\pi$$
$$128$$ 0 0
$$129$$ −0.727922 −0.0640900
$$130$$ 0 0
$$131$$ 16.9706 1.48272 0.741362 0.671105i $$-0.234180\pi$$
0.741362 + 0.671105i $$0.234180\pi$$
$$132$$ 0 0
$$133$$ 4.41421 0.382761
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −13.0000 −1.11066 −0.555332 0.831628i $$-0.687409\pi$$
−0.555332 + 0.831628i $$0.687409\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.24264 −0.689284
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −5.17157 −0.426544
$$148$$ 0 0
$$149$$ 17.6569 1.44651 0.723253 0.690583i $$-0.242646\pi$$
0.723253 + 0.690583i $$0.242646\pi$$
$$150$$ 0 0
$$151$$ 10.4853 0.853280 0.426640 0.904422i $$-0.359697\pi$$
0.426640 + 0.904422i $$0.359697\pi$$
$$152$$ 0 0
$$153$$ 2.82843 0.228665
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −11.6569 −0.930318 −0.465159 0.885227i $$-0.654003\pi$$
−0.465159 + 0.885227i $$0.654003\pi$$
$$158$$ 0 0
$$159$$ 2.27208 0.180188
$$160$$ 0 0
$$161$$ 3.34315 0.263477
$$162$$ 0 0
$$163$$ −10.2426 −0.802266 −0.401133 0.916020i $$-0.631383\pi$$
−0.401133 + 0.916020i $$0.631383\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −18.2426 −1.41166 −0.705829 0.708382i $$-0.749425\pi$$
−0.705829 + 0.708382i $$0.749425\pi$$
$$168$$ 0 0
$$169$$ 20.9706 1.61312
$$170$$ 0 0
$$171$$ −2.82843 −0.216295
$$172$$ 0 0
$$173$$ −0.485281 −0.0368953 −0.0184476 0.999830i $$-0.505872\pi$$
−0.0184476 + 0.999830i $$0.505872\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2.85786 0.214810
$$178$$ 0 0
$$179$$ 11.6569 0.871274 0.435637 0.900122i $$-0.356523\pi$$
0.435637 + 0.900122i $$0.356523\pi$$
$$180$$ 0 0
$$181$$ −8.48528 −0.630706 −0.315353 0.948974i $$-0.602123\pi$$
−0.315353 + 0.948974i $$0.602123\pi$$
$$182$$ 0 0
$$183$$ −5.89949 −0.436103
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.41421 −0.103418
$$188$$ 0 0
$$189$$ 10.6569 0.775172
$$190$$ 0 0
$$191$$ −12.5563 −0.908546 −0.454273 0.890863i $$-0.650101\pi$$
−0.454273 + 0.890863i $$0.650101\pi$$
$$192$$ 0 0
$$193$$ 0.343146 0.0247002 0.0123501 0.999924i $$-0.496069\pi$$
0.0123501 + 0.999924i $$0.496069\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 11.7574 0.837677 0.418839 0.908061i $$-0.362437\pi$$
0.418839 + 0.908061i $$0.362437\pi$$
$$198$$ 0 0
$$199$$ −9.24264 −0.655193 −0.327597 0.944818i $$-0.606239\pi$$
−0.327597 + 0.944818i $$0.606239\pi$$
$$200$$ 0 0
$$201$$ −1.97056 −0.138993
$$202$$ 0 0
$$203$$ −0.757359 −0.0531562
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −2.14214 −0.148889
$$208$$ 0 0
$$209$$ 1.41421 0.0978232
$$210$$ 0 0
$$211$$ 19.7279 1.35813 0.679063 0.734080i $$-0.262386\pi$$
0.679063 + 0.734080i $$0.262386\pi$$
$$212$$ 0 0
$$213$$ −5.55635 −0.380715
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −27.5563 −1.87065
$$218$$ 0 0
$$219$$ 4.75736 0.321473
$$220$$ 0 0
$$221$$ 5.82843 0.392062
$$222$$ 0 0
$$223$$ −15.1716 −1.01596 −0.507982 0.861368i $$-0.669608\pi$$
−0.507982 + 0.861368i $$0.669608\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −16.7574 −1.11223 −0.556113 0.831107i $$-0.687708\pi$$
−0.556113 + 0.831107i $$0.687708\pi$$
$$228$$ 0 0
$$229$$ 14.9706 0.989283 0.494641 0.869097i $$-0.335299\pi$$
0.494641 + 0.869097i $$0.335299\pi$$
$$230$$ 0 0
$$231$$ −2.58579 −0.170132
$$232$$ 0 0
$$233$$ 24.9706 1.63588 0.817938 0.575306i $$-0.195117\pi$$
0.817938 + 0.575306i $$0.195117\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −2.68629 −0.174493
$$238$$ 0 0
$$239$$ 6.89949 0.446291 0.223146 0.974785i $$-0.428367\pi$$
0.223146 + 0.974785i $$0.428367\pi$$
$$240$$ 0 0
$$241$$ 8.97056 0.577845 0.288922 0.957353i $$-0.406703\pi$$
0.288922 + 0.957353i $$0.406703\pi$$
$$242$$ 0 0
$$243$$ −10.3431 −0.663513
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.82843 −0.370854
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ −3.55635 −0.224475 −0.112237 0.993681i $$-0.535802\pi$$
−0.112237 + 0.993681i $$0.535802\pi$$
$$252$$ 0 0
$$253$$ 1.07107 0.0673375
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −20.7279 −1.29297 −0.646486 0.762926i $$-0.723762\pi$$
−0.646486 + 0.762926i $$0.723762\pi$$
$$258$$ 0 0
$$259$$ 37.4558 2.32739
$$260$$ 0 0
$$261$$ 0.485281 0.0300382
$$262$$ 0 0
$$263$$ −26.9706 −1.66308 −0.831538 0.555468i $$-0.812539\pi$$
−0.831538 + 0.555468i $$0.812539\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −2.92893 −0.179248
$$268$$ 0 0
$$269$$ −16.6274 −1.01379 −0.506896 0.862007i $$-0.669207\pi$$
−0.506896 + 0.862007i $$0.669207\pi$$
$$270$$ 0 0
$$271$$ 27.2426 1.65487 0.827436 0.561560i $$-0.189798\pi$$
0.827436 + 0.561560i $$0.189798\pi$$
$$272$$ 0 0
$$273$$ 10.6569 0.644982
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 0 0
$$279$$ 17.6569 1.05709
$$280$$ 0 0
$$281$$ 4.24264 0.253095 0.126547 0.991961i $$-0.459610\pi$$
0.126547 + 0.991961i $$0.459610\pi$$
$$282$$ 0 0
$$283$$ 32.1421 1.91065 0.955326 0.295555i $$-0.0955045\pi$$
0.955326 + 0.295555i $$0.0955045\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −18.7279 −1.10547
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ −0.142136 −0.00833214
$$292$$ 0 0
$$293$$ 5.48528 0.320454 0.160227 0.987080i $$-0.448777\pi$$
0.160227 + 0.987080i $$0.448777\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.41421 0.198113
$$298$$ 0 0
$$299$$ −4.41421 −0.255281
$$300$$ 0 0
$$301$$ 7.75736 0.447127
$$302$$ 0 0
$$303$$ −5.41421 −0.311038
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −17.6569 −1.00773 −0.503865 0.863782i $$-0.668089\pi$$
−0.503865 + 0.863782i $$0.668089\pi$$
$$308$$ 0 0
$$309$$ −1.75736 −0.0999727
$$310$$ 0 0
$$311$$ 4.75736 0.269765 0.134883 0.990862i $$-0.456934\pi$$
0.134883 + 0.990862i $$0.456934\pi$$
$$312$$ 0 0
$$313$$ 7.97056 0.450523 0.225261 0.974298i $$-0.427676\pi$$
0.225261 + 0.974298i $$0.427676\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9.48528 0.532746 0.266373 0.963870i $$-0.414175\pi$$
0.266373 + 0.963870i $$0.414175\pi$$
$$318$$ 0 0
$$319$$ −0.242641 −0.0135853
$$320$$ 0 0
$$321$$ −8.17157 −0.456093
$$322$$ 0 0
$$323$$ −1.00000 −0.0556415
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 7.44365 0.411635
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 19.2426 1.05767 0.528836 0.848724i $$-0.322629\pi$$
0.528836 + 0.848724i $$0.322629\pi$$
$$332$$ 0 0
$$333$$ −24.0000 −1.31519
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −14.1005 −0.768103 −0.384052 0.923312i $$-0.625472\pi$$
−0.384052 + 0.923312i $$0.625472\pi$$
$$338$$ 0 0
$$339$$ −4.24264 −0.230429
$$340$$ 0 0
$$341$$ −8.82843 −0.478086
$$342$$ 0 0
$$343$$ 24.2132 1.30739
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −5.51472 −0.296046 −0.148023 0.988984i $$-0.547291\pi$$
−0.148023 + 0.988984i $$0.547291\pi$$
$$348$$ 0 0
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 0 0
$$351$$ −14.0711 −0.751058
$$352$$ 0 0
$$353$$ 2.51472 0.133845 0.0669225 0.997758i $$-0.478682\pi$$
0.0669225 + 0.997758i $$0.478682\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1.82843 0.0967706
$$358$$ 0 0
$$359$$ 22.7574 1.20109 0.600544 0.799592i $$-0.294951\pi$$
0.600544 + 0.799592i $$0.294951\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 3.72792 0.195665
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −25.4558 −1.32878 −0.664392 0.747384i $$-0.731309\pi$$
−0.664392 + 0.747384i $$0.731309\pi$$
$$368$$ 0 0
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ −24.2132 −1.25709
$$372$$ 0 0
$$373$$ −9.00000 −0.466002 −0.233001 0.972476i $$-0.574855\pi$$
−0.233001 + 0.972476i $$0.574855\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.00000 0.0515026
$$378$$ 0 0
$$379$$ −11.2426 −0.577496 −0.288748 0.957405i $$-0.593239\pi$$
−0.288748 + 0.957405i $$0.593239\pi$$
$$380$$ 0 0
$$381$$ 1.02944 0.0527397
$$382$$ 0 0
$$383$$ 12.2426 0.625570 0.312785 0.949824i $$-0.398738\pi$$
0.312785 + 0.949824i $$0.398738\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.97056 −0.252668
$$388$$ 0 0
$$389$$ 22.9289 1.16254 0.581272 0.813710i $$-0.302555\pi$$
0.581272 + 0.813710i $$0.302555\pi$$
$$390$$ 0 0
$$391$$ −0.757359 −0.0383013
$$392$$ 0 0
$$393$$ −7.02944 −0.354588
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −24.0000 −1.20453 −0.602263 0.798298i $$-0.705734\pi$$
−0.602263 + 0.798298i $$0.705734\pi$$
$$398$$ 0 0
$$399$$ −1.82843 −0.0915358
$$400$$ 0 0
$$401$$ −25.4142 −1.26913 −0.634563 0.772871i $$-0.718820\pi$$
−0.634563 + 0.772871i $$0.718820\pi$$
$$402$$ 0 0
$$403$$ 36.3848 1.81245
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 12.0000 0.594818
$$408$$ 0 0
$$409$$ 25.2132 1.24671 0.623356 0.781938i $$-0.285769\pi$$
0.623356 + 0.781938i $$0.285769\pi$$
$$410$$ 0 0
$$411$$ 5.38478 0.265611
$$412$$ 0 0
$$413$$ −30.4558 −1.49863
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −4.97056 −0.243410
$$418$$ 0 0
$$419$$ −19.4142 −0.948446 −0.474223 0.880405i $$-0.657271\pi$$
−0.474223 + 0.880405i $$0.657271\pi$$
$$420$$ 0 0
$$421$$ 19.4853 0.949655 0.474827 0.880079i $$-0.342511\pi$$
0.474827 + 0.880079i $$0.342511\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 62.8701 3.04250
$$428$$ 0 0
$$429$$ 3.41421 0.164840
$$430$$ 0 0
$$431$$ 6.38478 0.307544 0.153772 0.988106i $$-0.450858\pi$$
0.153772 + 0.988106i $$0.450858\pi$$
$$432$$ 0 0
$$433$$ 9.55635 0.459249 0.229624 0.973279i $$-0.426250\pi$$
0.229624 + 0.973279i $$0.426250\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0.757359 0.0362294
$$438$$ 0 0
$$439$$ 5.75736 0.274784 0.137392 0.990517i $$-0.456128\pi$$
0.137392 + 0.990517i $$0.456128\pi$$
$$440$$ 0 0
$$441$$ −35.3137 −1.68161
$$442$$ 0 0
$$443$$ 4.24264 0.201574 0.100787 0.994908i $$-0.467864\pi$$
0.100787 + 0.994908i $$0.467864\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −7.31371 −0.345927
$$448$$ 0 0
$$449$$ 17.3137 0.817084 0.408542 0.912739i $$-0.366037\pi$$
0.408542 + 0.912739i $$0.366037\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 0 0
$$453$$ −4.34315 −0.204059
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.00000 0.140334 0.0701670 0.997535i $$-0.477647\pi$$
0.0701670 + 0.997535i $$0.477647\pi$$
$$458$$ 0 0
$$459$$ −2.41421 −0.112686
$$460$$ 0 0
$$461$$ −3.55635 −0.165636 −0.0828178 0.996565i $$-0.526392\pi$$
−0.0828178 + 0.996565i $$0.526392\pi$$
$$462$$ 0 0
$$463$$ 14.1421 0.657241 0.328620 0.944462i $$-0.393416\pi$$
0.328620 + 0.944462i $$0.393416\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0.727922 0.0336842 0.0168421 0.999858i $$-0.494639\pi$$
0.0168421 + 0.999858i $$0.494639\pi$$
$$468$$ 0 0
$$469$$ 21.0000 0.969690
$$470$$ 0 0
$$471$$ 4.82843 0.222482
$$472$$ 0 0
$$473$$ 2.48528 0.114273
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 15.5147 0.710370
$$478$$ 0 0
$$479$$ 31.1127 1.42158 0.710788 0.703407i $$-0.248339\pi$$
0.710788 + 0.703407i $$0.248339\pi$$
$$480$$ 0 0
$$481$$ −49.4558 −2.25499
$$482$$ 0 0
$$483$$ −1.38478 −0.0630095
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 37.7990 1.71284 0.856418 0.516283i $$-0.172685\pi$$
0.856418 + 0.516283i $$0.172685\pi$$
$$488$$ 0 0
$$489$$ 4.24264 0.191859
$$490$$ 0 0
$$491$$ 33.5563 1.51438 0.757188 0.653197i $$-0.226572\pi$$
0.757188 + 0.653197i $$0.226572\pi$$
$$492$$ 0 0
$$493$$ 0.171573 0.00772725
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 59.2132 2.65608
$$498$$ 0 0
$$499$$ 25.7574 1.15306 0.576529 0.817077i $$-0.304407\pi$$
0.576529 + 0.817077i $$0.304407\pi$$
$$500$$ 0 0
$$501$$ 7.55635 0.337593
$$502$$ 0 0
$$503$$ 14.2721 0.636361 0.318180 0.948030i $$-0.396928\pi$$
0.318180 + 0.948030i $$0.396928\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −8.68629 −0.385772
$$508$$ 0 0
$$509$$ 28.9706 1.28410 0.642049 0.766664i $$-0.278085\pi$$
0.642049 + 0.766664i $$0.278085\pi$$
$$510$$ 0 0
$$511$$ −50.6985 −2.24277
$$512$$ 0 0
$$513$$ 2.41421 0.106590
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0.201010 0.00882337
$$520$$ 0 0
$$521$$ 23.3137 1.02139 0.510696 0.859761i $$-0.329388\pi$$
0.510696 + 0.859761i $$0.329388\pi$$
$$522$$ 0 0
$$523$$ −2.27208 −0.0993510 −0.0496755 0.998765i $$-0.515819\pi$$
−0.0496755 + 0.998765i $$0.515819\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.24264 0.271934
$$528$$ 0 0
$$529$$ −22.4264 −0.975061
$$530$$ 0 0
$$531$$ 19.5147 0.846867
$$532$$ 0 0
$$533$$ 24.7279 1.07109
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −4.82843 −0.208362
$$538$$ 0 0
$$539$$ 17.6569 0.760535
$$540$$ 0 0
$$541$$ 9.75736 0.419502 0.209751 0.977755i $$-0.432735\pi$$
0.209751 + 0.977755i $$0.432735\pi$$
$$542$$ 0 0
$$543$$ 3.51472 0.150831
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −17.3137 −0.740281 −0.370140 0.928976i $$-0.620690\pi$$
−0.370140 + 0.928976i $$0.620690\pi$$
$$548$$ 0 0
$$549$$ −40.2843 −1.71929
$$550$$ 0 0
$$551$$ −0.171573 −0.00730925
$$552$$ 0 0
$$553$$ 28.6274 1.21736
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −16.0000 −0.677942 −0.338971 0.940797i $$-0.610079\pi$$
−0.338971 + 0.940797i $$0.610079\pi$$
$$558$$ 0 0
$$559$$ −10.2426 −0.433218
$$560$$ 0 0
$$561$$ 0.585786 0.0247319
$$562$$ 0 0
$$563$$ 4.97056 0.209484 0.104742 0.994499i $$-0.466598\pi$$
0.104742 + 0.994499i $$0.466598\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 33.0416 1.38762
$$568$$ 0 0
$$569$$ −28.2843 −1.18574 −0.592869 0.805299i $$-0.702005\pi$$
−0.592869 + 0.805299i $$0.702005\pi$$
$$570$$ 0 0
$$571$$ 2.24264 0.0938516 0.0469258 0.998898i $$-0.485058\pi$$
0.0469258 + 0.998898i $$0.485058\pi$$
$$572$$ 0 0
$$573$$ 5.20101 0.217275
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 20.3137 0.845671 0.422835 0.906207i $$-0.361035\pi$$
0.422835 + 0.906207i $$0.361035\pi$$
$$578$$ 0 0
$$579$$ −0.142136 −0.00590695
$$580$$ 0 0
$$581$$ 63.9411 2.65272
$$582$$ 0 0
$$583$$ −7.75736 −0.321277
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 30.2426 1.24825 0.624124 0.781326i $$-0.285456\pi$$
0.624124 + 0.781326i $$0.285456\pi$$
$$588$$ 0 0
$$589$$ −6.24264 −0.257224
$$590$$ 0 0
$$591$$ −4.87006 −0.200327
$$592$$ 0 0
$$593$$ −22.0000 −0.903432 −0.451716 0.892162i $$-0.649188\pi$$
−0.451716 + 0.892162i $$0.649188\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 3.82843 0.156687
$$598$$ 0 0
$$599$$ 15.2132 0.621595 0.310797 0.950476i $$-0.399404\pi$$
0.310797 + 0.950476i $$0.399404\pi$$
$$600$$ 0 0
$$601$$ −20.2426 −0.825715 −0.412857 0.910796i $$-0.635469\pi$$
−0.412857 + 0.910796i $$0.635469\pi$$
$$602$$ 0 0
$$603$$ −13.4558 −0.547964
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 3.17157 0.128730 0.0643651 0.997926i $$-0.479498\pi$$
0.0643651 + 0.997926i $$0.479498\pi$$
$$608$$ 0 0
$$609$$ 0.313708 0.0127121
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 11.6985 0.472497 0.236249 0.971693i $$-0.424082\pi$$
0.236249 + 0.971693i $$0.424082\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4.48528 0.180571 0.0902853 0.995916i $$-0.471222\pi$$
0.0902853 + 0.995916i $$0.471222\pi$$
$$618$$ 0 0
$$619$$ −7.75736 −0.311795 −0.155897 0.987773i $$-0.549827\pi$$
−0.155897 + 0.987773i $$0.549827\pi$$
$$620$$ 0 0
$$621$$ 1.82843 0.0733723
$$622$$ 0 0
$$623$$ 31.2132 1.25053
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −0.585786 −0.0233941
$$628$$ 0 0
$$629$$ −8.48528 −0.338330
$$630$$ 0 0
$$631$$ 38.9706 1.55139 0.775697 0.631106i $$-0.217399\pi$$
0.775697 + 0.631106i $$0.217399\pi$$
$$632$$ 0 0
$$633$$ −8.17157 −0.324791
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −72.7696 −2.88323
$$638$$ 0 0
$$639$$ −37.9411 −1.50093
$$640$$ 0 0
$$641$$ −18.0416 −0.712602 −0.356301 0.934371i $$-0.615962\pi$$
−0.356301 + 0.934371i $$0.615962\pi$$
$$642$$ 0 0
$$643$$ −14.4853 −0.571244 −0.285622 0.958342i $$-0.592200\pi$$
−0.285622 + 0.958342i $$0.592200\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −18.7574 −0.737428 −0.368714 0.929543i $$-0.620202\pi$$
−0.368714 + 0.929543i $$0.620202\pi$$
$$648$$ 0 0
$$649$$ −9.75736 −0.383010
$$650$$ 0 0
$$651$$ 11.4142 0.447358
$$652$$ 0 0
$$653$$ 28.9706 1.13371 0.566853 0.823819i $$-0.308161\pi$$
0.566853 + 0.823819i $$0.308161\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 32.4853 1.26737
$$658$$ 0 0
$$659$$ −18.8995 −0.736220 −0.368110 0.929782i $$-0.619995\pi$$
−0.368110 + 0.929782i $$0.619995\pi$$
$$660$$ 0 0
$$661$$ −18.4558 −0.717849 −0.358925 0.933367i $$-0.616856\pi$$
−0.358925 + 0.933367i $$0.616856\pi$$
$$662$$ 0 0
$$663$$ −2.41421 −0.0937603
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −0.129942 −0.00503139
$$668$$ 0 0
$$669$$ 6.28427 0.242964
$$670$$ 0 0
$$671$$ 20.1421 0.777579
$$672$$ 0 0
$$673$$ 12.0000 0.462566 0.231283 0.972887i $$-0.425708\pi$$
0.231283 + 0.972887i $$0.425708\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 40.9411 1.57350 0.786748 0.617275i $$-0.211763\pi$$
0.786748 + 0.617275i $$0.211763\pi$$
$$678$$ 0 0
$$679$$ 1.51472 0.0581296
$$680$$ 0 0
$$681$$ 6.94113 0.265985
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −6.20101 −0.236583
$$688$$ 0 0
$$689$$ 31.9706 1.21798
$$690$$ 0 0
$$691$$ −22.4853 −0.855380 −0.427690 0.903925i $$-0.640673\pi$$
−0.427690 + 0.903925i $$0.640673\pi$$
$$692$$ 0 0
$$693$$ −17.6569 −0.670728
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4.24264 0.160701
$$698$$ 0 0
$$699$$ −10.3431 −0.391214
$$700$$ 0 0
$$701$$ 16.9706 0.640969 0.320485 0.947254i $$-0.396154\pi$$
0.320485 + 0.947254i $$0.396154\pi$$
$$702$$ 0 0
$$703$$ 8.48528 0.320028
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 57.6985 2.16997
$$708$$ 0 0
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ 0 0
$$711$$ −18.3431 −0.687922
$$712$$ 0 0
$$713$$ −4.72792 −0.177062
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −2.85786 −0.106729
$$718$$ 0 0
$$719$$ 5.10051 0.190217 0.0951084 0.995467i $$-0.469680\pi$$
0.0951084 + 0.995467i $$0.469680\pi$$
$$720$$ 0 0
$$721$$ 18.7279 0.697464
$$722$$ 0 0
$$723$$ −3.71573 −0.138189
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 9.72792 0.360789 0.180394 0.983594i $$-0.442263\pi$$
0.180394 + 0.983594i $$0.442263\pi$$
$$728$$ 0 0
$$729$$ −18.1716 −0.673021
$$730$$ 0 0
$$731$$ −1.75736 −0.0649983
$$732$$ 0 0
$$733$$ −12.0000 −0.443230 −0.221615 0.975134i $$-0.571133\pi$$
−0.221615 + 0.975134i $$0.571133\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.72792 0.247826
$$738$$ 0 0
$$739$$ −1.27208 −0.0467941 −0.0233971 0.999726i $$-0.507448\pi$$
−0.0233971 + 0.999726i $$0.507448\pi$$
$$740$$ 0 0
$$741$$ 2.41421 0.0886884
$$742$$ 0 0
$$743$$ 6.72792 0.246824 0.123412 0.992356i $$-0.460616\pi$$
0.123412 + 0.992356i $$0.460616\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −40.9706 −1.49903
$$748$$ 0 0
$$749$$ 87.0833 3.18195
$$750$$ 0 0
$$751$$ 26.7279 0.975316 0.487658 0.873035i $$-0.337851\pi$$
0.487658 + 0.873035i $$0.337851\pi$$
$$752$$ 0 0
$$753$$ 1.47309 0.0536823
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −12.3431 −0.448619 −0.224310 0.974518i $$-0.572013\pi$$
−0.224310 + 0.974518i $$0.572013\pi$$
$$758$$ 0 0
$$759$$ −0.443651 −0.0161035
$$760$$ 0 0
$$761$$ 43.9706 1.59393 0.796966 0.604024i $$-0.206437\pi$$
0.796966 + 0.604024i $$0.206437\pi$$
$$762$$ 0 0
$$763$$ −79.3259 −2.87179
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 40.2132 1.45201
$$768$$ 0 0
$$769$$ −36.4558 −1.31463 −0.657316 0.753615i $$-0.728308\pi$$
−0.657316 + 0.753615i $$0.728308\pi$$
$$770$$ 0 0
$$771$$ 8.58579 0.309210
$$772$$ 0 0
$$773$$ −13.9706 −0.502486 −0.251243 0.967924i $$-0.580839\pi$$
−0.251243 + 0.967924i $$0.580839\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −15.5147 −0.556587
$$778$$ 0 0
$$779$$ −4.24264 −0.152008
$$780$$ 0 0
$$781$$ 18.9706 0.678820
$$782$$ 0 0
$$783$$ −0.414214 −0.0148028
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −41.1838 −1.46804 −0.734021 0.679126i $$-0.762359\pi$$
−0.734021 + 0.679126i $$0.762359\pi$$
$$788$$ 0 0
$$789$$ 11.1716 0.397719
$$790$$ 0 0
$$791$$ 45.2132 1.60760
$$792$$ 0 0
$$793$$ −83.0122 −2.94785
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 47.4853 1.68201 0.841007 0.541023i $$-0.181963\pi$$
0.841007 + 0.541023i $$0.181963\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −20.0000 −0.706665
$$802$$ 0 0
$$803$$ −16.2426 −0.573190
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.88730 0.242444
$$808$$ 0 0
$$809$$ −28.7990 −1.01252 −0.506259 0.862381i $$-0.668972\pi$$
−0.506259 + 0.862381i $$0.668972\pi$$
$$810$$ 0 0
$$811$$ −52.6985 −1.85049 −0.925247 0.379365i $$-0.876142\pi$$
−0.925247 + 0.379365i $$0.876142\pi$$
$$812$$ 0 0
$$813$$ −11.2843 −0.395757
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.75736 0.0614822
$$818$$ 0 0
$$819$$ 72.7696 2.54277
$$820$$ 0 0
$$821$$ 6.34315 0.221377 0.110689 0.993855i $$-0.464694\pi$$
0.110689 + 0.993855i $$0.464694\pi$$
$$822$$ 0 0
$$823$$ −15.7279 −0.548241 −0.274120 0.961695i $$-0.588387\pi$$
−0.274120 + 0.961695i $$0.588387\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −20.6985 −0.719757 −0.359878 0.932999i $$-0.617182\pi$$
−0.359878 + 0.932999i $$0.617182\pi$$
$$828$$ 0 0
$$829$$ 10.4558 0.363146 0.181573 0.983377i $$-0.441881\pi$$
0.181573 + 0.983377i $$0.441881\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −12.4853 −0.432589
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −15.0711 −0.520932
$$838$$ 0 0
$$839$$ −22.6274 −0.781185 −0.390593 0.920564i $$-0.627730\pi$$
−0.390593 + 0.920564i $$0.627730\pi$$
$$840$$ 0 0
$$841$$ −28.9706 −0.998985
$$842$$ 0 0
$$843$$ −1.75736 −0.0605267
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −39.7279 −1.36507
$$848$$ 0 0
$$849$$ −13.3137 −0.456925
$$850$$ 0 0
$$851$$ 6.42641 0.220294
$$852$$ 0 0
$$853$$ 27.9411 0.956686 0.478343 0.878173i $$-0.341238\pi$$
0.478343 + 0.878173i $$0.341238\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ −4.72792 −0.161315 −0.0806573 0.996742i $$-0.525702\pi$$
−0.0806573 + 0.996742i $$0.525702\pi$$
$$860$$ 0 0
$$861$$ 7.75736 0.264370
$$862$$ 0 0
$$863$$ 0.727922 0.0247788 0.0123894 0.999923i $$-0.496056\pi$$
0.0123894 + 0.999923i $$0.496056\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 6.62742 0.225079
$$868$$ 0 0
$$869$$ 9.17157 0.311124
$$870$$ 0 0
$$871$$ −27.7279 −0.939525
$$872$$ 0 0
$$873$$ −0.970563 −0.0328486
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 18.8579 0.636785 0.318392 0.947959i $$-0.396857\pi$$
0.318392 + 0.947959i $$0.396857\pi$$
$$878$$ 0 0
$$879$$ −2.27208 −0.0766353
$$880$$ 0 0
$$881$$ −39.5980 −1.33409 −0.667045 0.745018i $$-0.732441\pi$$
−0.667045 + 0.745018i $$0.732441\pi$$
$$882$$ 0 0
$$883$$ 14.4853 0.487469 0.243734 0.969842i $$-0.421628\pi$$
0.243734 + 0.969842i $$0.421628\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −45.2132 −1.51811 −0.759055 0.651026i $$-0.774339\pi$$
−0.759055 + 0.651026i $$0.774339\pi$$
$$888$$ 0 0
$$889$$ −10.9706 −0.367941
$$890$$ 0 0
$$891$$ 10.5858 0.354637
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.82843 0.0610494
$$898$$ 0 0
$$899$$ 1.07107 0.0357221
$$900$$ 0 0
$$901$$ 5.48528 0.182741
$$902$$ 0 0
$$903$$ −3.21320 −0.106929
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 38.6985 1.28496 0.642481 0.766302i $$-0.277905\pi$$
0.642481 + 0.766302i $$0.277905\pi$$
$$908$$ 0 0
$$909$$ −36.9706 −1.22624
$$910$$ 0 0
$$911$$ −40.2843 −1.33468 −0.667339 0.744754i $$-0.732567\pi$$
−0.667339 + 0.744754i $$0.732567\pi$$
$$912$$ 0 0
$$913$$ 20.4853 0.677964
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 74.9117 2.47380
$$918$$ 0 0
$$919$$ 6.21320 0.204955 0.102477 0.994735i $$-0.467323\pi$$
0.102477 + 0.994735i $$0.467323\pi$$
$$920$$ 0 0
$$921$$ 7.31371 0.240995
$$922$$ 0 0
$$923$$ −78.1838 −2.57345
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −12.0000 −0.394132
$$928$$ 0 0
$$929$$ −12.1716 −0.399336 −0.199668 0.979864i $$-0.563986\pi$$
−0.199668 + 0.979864i $$0.563986\pi$$
$$930$$ 0 0
$$931$$ 12.4853 0.409189
$$932$$ 0 0
$$933$$ −1.97056 −0.0645133
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −27.0000 −0.882052 −0.441026 0.897494i $$-0.645385\pi$$
−0.441026 + 0.897494i $$0.645385\pi$$
$$938$$ 0 0
$$939$$ −3.30152 −0.107741
$$940$$ 0 0
$$941$$ 53.1421 1.73238 0.866192 0.499711i $$-0.166561\pi$$
0.866192 + 0.499711i $$0.166561\pi$$
$$942$$ 0 0
$$943$$ −3.21320 −0.104636
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ 66.9411 2.17300
$$950$$ 0 0
$$951$$ −3.92893 −0.127404
$$952$$ 0 0
$$953$$ −18.7279 −0.606657 −0.303328 0.952886i $$-0.598098\pi$$
−0.303328 + 0.952886i $$0.598098\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0.100505 0.00324887
$$958$$ 0 0
$$959$$ −57.3848 −1.85305
$$960$$ 0 0
$$961$$ 7.97056 0.257115
$$962$$ 0 0
$$963$$ −55.7990 −1.79810
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 43.1127 1.38641 0.693205 0.720740i $$-0.256198\pi$$
0.693205 + 0.720740i $$0.256198\pi$$
$$968$$ 0 0
$$969$$ 0.414214 0.0133065
$$970$$ 0 0
$$971$$ −41.6569 −1.33683 −0.668416 0.743788i $$-0.733027\pi$$
−0.668416 + 0.743788i $$0.733027\pi$$
$$972$$ 0 0
$$973$$ 52.9706 1.69816
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −0.242641 −0.00776276 −0.00388138 0.999992i $$-0.501235\pi$$
−0.00388138 + 0.999992i $$0.501235\pi$$
$$978$$ 0 0
$$979$$ 10.0000 0.319601
$$980$$ 0 0
$$981$$ 50.8284 1.62283
$$982$$ 0 0
$$983$$ −51.4558 −1.64119 −0.820593 0.571513i $$-0.806357\pi$$
−0.820593 + 0.571513i $$0.806357\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.33095 0.0423218
$$990$$ 0 0
$$991$$ −13.7574 −0.437017 −0.218508 0.975835i $$-0.570119\pi$$
−0.218508 + 0.975835i $$0.570119\pi$$
$$992$$ 0 0
$$993$$ −7.97056 −0.252938
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −48.7279 −1.54323 −0.771614 0.636091i $$-0.780550\pi$$
−0.771614 + 0.636091i $$0.780550\pi$$
$$998$$ 0 0
$$999$$ 20.4853 0.648126
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 7600.2.a.bg.1.1 2
4.3 odd 2 950.2.a.g.1.2 2
5.2 odd 4 1520.2.d.e.609.3 4
5.3 odd 4 1520.2.d.e.609.2 4
5.4 even 2 7600.2.a.v.1.2 2
12.11 even 2 8550.2.a.bn.1.1 2
20.3 even 4 190.2.b.a.39.2 4
20.7 even 4 190.2.b.a.39.3 yes 4
20.19 odd 2 950.2.a.f.1.1 2
60.23 odd 4 1710.2.d.c.1369.3 4
60.47 odd 4 1710.2.d.c.1369.1 4
60.59 even 2 8550.2.a.cb.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.a.39.2 4 20.3 even 4
190.2.b.a.39.3 yes 4 20.7 even 4
950.2.a.f.1.1 2 20.19 odd 2
950.2.a.g.1.2 2 4.3 odd 2
1520.2.d.e.609.2 4 5.3 odd 4
1520.2.d.e.609.3 4 5.2 odd 4
1710.2.d.c.1369.1 4 60.47 odd 4
1710.2.d.c.1369.3 4 60.23 odd 4
7600.2.a.v.1.2 2 5.4 even 2
7600.2.a.bg.1.1 2 1.1 even 1 trivial
8550.2.a.bn.1.1 2 12.11 even 2
8550.2.a.cb.1.2 2 60.59 even 2