# Properties

 Label 1520.2.a.m.1.2 Level $1520$ Weight $2$ Character 1520.1 Self dual yes Analytic conductor $12.137$ 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] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.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: $$12.1372611072$$ 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 760) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1520.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.41421 q^{3} -1.00000 q^{5} +2.82843 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.41421 q^{3} -1.00000 q^{5} +2.82843 q^{7} -1.00000 q^{9} +0.828427 q^{11} +3.41421 q^{13} -1.41421 q^{15} +4.82843 q^{17} +1.00000 q^{19} +4.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} -5.65685 q^{27} -0.828427 q^{29} +1.17157 q^{33} -2.82843 q^{35} +10.2426 q^{37} +4.82843 q^{39} -0.828427 q^{41} -2.82843 q^{43} +1.00000 q^{45} +8.48528 q^{47} +1.00000 q^{49} +6.82843 q^{51} +13.0711 q^{53} -0.828427 q^{55} +1.41421 q^{57} -2.82843 q^{59} -1.65685 q^{61} -2.82843 q^{63} -3.41421 q^{65} +9.41421 q^{67} -5.65685 q^{69} -15.3137 q^{71} +12.1421 q^{73} +1.41421 q^{75} +2.34315 q^{77} +9.17157 q^{79} -5.00000 q^{81} -8.00000 q^{83} -4.82843 q^{85} -1.17157 q^{87} -3.17157 q^{89} +9.65685 q^{91} -1.00000 q^{95} -2.24264 q^{97} -0.828427 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^9 $$2 q - 2 q^{5} - 2 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{17} + 2 q^{19} + 8 q^{21} - 8 q^{23} + 2 q^{25} + 4 q^{29} + 8 q^{33} + 12 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{45} + 2 q^{49} + 8 q^{51} + 12 q^{53} + 4 q^{55} + 8 q^{61} - 4 q^{65} + 16 q^{67} - 8 q^{71} - 4 q^{73} + 16 q^{77} + 24 q^{79} - 10 q^{81} - 16 q^{83} - 4 q^{85} - 8 q^{87} - 12 q^{89} + 8 q^{91} - 2 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^17 + 2 * q^19 + 8 * q^21 - 8 * q^23 + 2 * q^25 + 4 * q^29 + 8 * q^33 + 12 * q^37 + 4 * q^39 + 4 * q^41 + 2 * q^45 + 2 * q^49 + 8 * q^51 + 12 * q^53 + 4 * q^55 + 8 * q^61 - 4 * q^65 + 16 * q^67 - 8 * q^71 - 4 * q^73 + 16 * q^77 + 24 * q^79 - 10 * q^81 - 16 * q^83 - 4 * q^85 - 8 * q^87 - 12 * q^89 + 8 * q^91 - 2 * q^95 + 4 * q^97 + 4 * 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
$$3$$ 1.41421 0.816497 0.408248 0.912871i $$-0.366140\pi$$
0.408248 + 0.912871i $$0.366140\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 2.82843 1.06904 0.534522 0.845154i $$-0.320491\pi$$
0.534522 + 0.845154i $$0.320491\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0.828427 0.249780 0.124890 0.992171i $$-0.460142\pi$$
0.124890 + 0.992171i $$0.460142\pi$$
$$12$$ 0 0
$$13$$ 3.41421 0.946932 0.473466 0.880812i $$-0.343003\pi$$
0.473466 + 0.880812i $$0.343003\pi$$
$$14$$ 0 0
$$15$$ −1.41421 −0.365148
$$16$$ 0 0
$$17$$ 4.82843 1.17107 0.585533 0.810649i $$-0.300885\pi$$
0.585533 + 0.810649i $$0.300885\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.65685 −1.08866
$$28$$ 0 0
$$29$$ −0.828427 −0.153835 −0.0769175 0.997037i $$-0.524508\pi$$
−0.0769175 + 0.997037i $$0.524508\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 1.17157 0.203945
$$34$$ 0 0
$$35$$ −2.82843 −0.478091
$$36$$ 0 0
$$37$$ 10.2426 1.68388 0.841940 0.539571i $$-0.181414\pi$$
0.841940 + 0.539571i $$0.181414\pi$$
$$38$$ 0 0
$$39$$ 4.82843 0.773167
$$40$$ 0 0
$$41$$ −0.828427 −0.129379 −0.0646893 0.997905i $$-0.520606\pi$$
−0.0646893 + 0.997905i $$0.520606\pi$$
$$42$$ 0 0
$$43$$ −2.82843 −0.431331 −0.215666 0.976467i $$-0.569192\pi$$
−0.215666 + 0.976467i $$0.569192\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 8.48528 1.23771 0.618853 0.785507i $$-0.287598\pi$$
0.618853 + 0.785507i $$0.287598\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 6.82843 0.956171
$$52$$ 0 0
$$53$$ 13.0711 1.79545 0.897725 0.440557i $$-0.145219\pi$$
0.897725 + 0.440557i $$0.145219\pi$$
$$54$$ 0 0
$$55$$ −0.828427 −0.111705
$$56$$ 0 0
$$57$$ 1.41421 0.187317
$$58$$ 0 0
$$59$$ −2.82843 −0.368230 −0.184115 0.982905i $$-0.558942\pi$$
−0.184115 + 0.982905i $$0.558942\pi$$
$$60$$ 0 0
$$61$$ −1.65685 −0.212138 −0.106069 0.994359i $$-0.533827\pi$$
−0.106069 + 0.994359i $$0.533827\pi$$
$$62$$ 0 0
$$63$$ −2.82843 −0.356348
$$64$$ 0 0
$$65$$ −3.41421 −0.423481
$$66$$ 0 0
$$67$$ 9.41421 1.15013 0.575065 0.818108i $$-0.304977\pi$$
0.575065 + 0.818108i $$0.304977\pi$$
$$68$$ 0 0
$$69$$ −5.65685 −0.681005
$$70$$ 0 0
$$71$$ −15.3137 −1.81740 −0.908701 0.417447i $$-0.862925\pi$$
−0.908701 + 0.417447i $$0.862925\pi$$
$$72$$ 0 0
$$73$$ 12.1421 1.42113 0.710565 0.703632i $$-0.248440\pi$$
0.710565 + 0.703632i $$0.248440\pi$$
$$74$$ 0 0
$$75$$ 1.41421 0.163299
$$76$$ 0 0
$$77$$ 2.34315 0.267026
$$78$$ 0 0
$$79$$ 9.17157 1.03188 0.515941 0.856624i $$-0.327442\pi$$
0.515941 + 0.856624i $$0.327442\pi$$
$$80$$ 0 0
$$81$$ −5.00000 −0.555556
$$82$$ 0 0
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ −4.82843 −0.523716
$$86$$ 0 0
$$87$$ −1.17157 −0.125606
$$88$$ 0 0
$$89$$ −3.17157 −0.336186 −0.168093 0.985771i $$-0.553761\pi$$
−0.168093 + 0.985771i $$0.553761\pi$$
$$90$$ 0 0
$$91$$ 9.65685 1.01231
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −2.24264 −0.227706 −0.113853 0.993498i $$-0.536319\pi$$
−0.113853 + 0.993498i $$0.536319\pi$$
$$98$$ 0 0
$$99$$ −0.828427 −0.0832601
$$100$$ 0 0
$$101$$ 17.6569 1.75692 0.878461 0.477813i $$-0.158571\pi$$
0.878461 + 0.477813i $$0.158571\pi$$
$$102$$ 0 0
$$103$$ −8.24264 −0.812172 −0.406086 0.913835i $$-0.633107\pi$$
−0.406086 + 0.913835i $$0.633107\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ 0 0
$$107$$ −1.41421 −0.136717 −0.0683586 0.997661i $$-0.521776\pi$$
−0.0683586 + 0.997661i $$0.521776\pi$$
$$108$$ 0 0
$$109$$ 3.65685 0.350263 0.175132 0.984545i $$-0.443965\pi$$
0.175132 + 0.984545i $$0.443965\pi$$
$$110$$ 0 0
$$111$$ 14.4853 1.37488
$$112$$ 0 0
$$113$$ −14.7279 −1.38549 −0.692743 0.721184i $$-0.743598\pi$$
−0.692743 + 0.721184i $$0.743598\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 0 0
$$117$$ −3.41421 −0.315644
$$118$$ 0 0
$$119$$ 13.6569 1.25192
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 0 0
$$123$$ −1.17157 −0.105637
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 13.4142 1.19032 0.595159 0.803608i $$-0.297089\pi$$
0.595159 + 0.803608i $$0.297089\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −7.31371 −0.639002 −0.319501 0.947586i $$-0.603515\pi$$
−0.319501 + 0.947586i $$0.603515\pi$$
$$132$$ 0 0
$$133$$ 2.82843 0.245256
$$134$$ 0 0
$$135$$ 5.65685 0.486864
$$136$$ 0 0
$$137$$ −7.17157 −0.612709 −0.306354 0.951918i $$-0.599109\pi$$
−0.306354 + 0.951918i $$0.599109\pi$$
$$138$$ 0 0
$$139$$ 3.17157 0.269009 0.134505 0.990913i $$-0.457056\pi$$
0.134505 + 0.990913i $$0.457056\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 2.82843 0.236525
$$144$$ 0 0
$$145$$ 0.828427 0.0687971
$$146$$ 0 0
$$147$$ 1.41421 0.116642
$$148$$ 0 0
$$149$$ 1.65685 0.135735 0.0678674 0.997694i $$-0.478381\pi$$
0.0678674 + 0.997694i $$0.478381\pi$$
$$150$$ 0 0
$$151$$ −6.14214 −0.499840 −0.249920 0.968267i $$-0.580404\pi$$
−0.249920 + 0.968267i $$0.580404\pi$$
$$152$$ 0 0
$$153$$ −4.82843 −0.390355
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −10.4853 −0.836817 −0.418408 0.908259i $$-0.637412\pi$$
−0.418408 + 0.908259i $$0.637412\pi$$
$$158$$ 0 0
$$159$$ 18.4853 1.46598
$$160$$ 0 0
$$161$$ −11.3137 −0.891645
$$162$$ 0 0
$$163$$ −14.8284 −1.16145 −0.580726 0.814099i $$-0.697231\pi$$
−0.580726 + 0.814099i $$0.697231\pi$$
$$164$$ 0 0
$$165$$ −1.17157 −0.0912068
$$166$$ 0 0
$$167$$ −2.10051 −0.162542 −0.0812710 0.996692i $$-0.525898\pi$$
−0.0812710 + 0.996692i $$0.525898\pi$$
$$168$$ 0 0
$$169$$ −1.34315 −0.103319
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ −10.7279 −0.815629 −0.407814 0.913065i $$-0.633709\pi$$
−0.407814 + 0.913065i $$0.633709\pi$$
$$174$$ 0 0
$$175$$ 2.82843 0.213809
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ −8.48528 −0.634220 −0.317110 0.948389i $$-0.602712\pi$$
−0.317110 + 0.948389i $$0.602712\pi$$
$$180$$ 0 0
$$181$$ −5.31371 −0.394965 −0.197482 0.980306i $$-0.563277\pi$$
−0.197482 + 0.980306i $$0.563277\pi$$
$$182$$ 0 0
$$183$$ −2.34315 −0.173210
$$184$$ 0 0
$$185$$ −10.2426 −0.753054
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 0 0
$$189$$ −16.0000 −1.16383
$$190$$ 0 0
$$191$$ −3.31371 −0.239772 −0.119886 0.992788i $$-0.538253\pi$$
−0.119886 + 0.992788i $$0.538253\pi$$
$$192$$ 0 0
$$193$$ −6.92893 −0.498755 −0.249378 0.968406i $$-0.580226\pi$$
−0.249378 + 0.968406i $$0.580226\pi$$
$$194$$ 0 0
$$195$$ −4.82843 −0.345771
$$196$$ 0 0
$$197$$ 25.3137 1.80353 0.901764 0.432230i $$-0.142273\pi$$
0.901764 + 0.432230i $$0.142273\pi$$
$$198$$ 0 0
$$199$$ 5.65685 0.401004 0.200502 0.979693i $$-0.435743\pi$$
0.200502 + 0.979693i $$0.435743\pi$$
$$200$$ 0 0
$$201$$ 13.3137 0.939077
$$202$$ 0 0
$$203$$ −2.34315 −0.164457
$$204$$ 0 0
$$205$$ 0.828427 0.0578599
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 0.828427 0.0573035
$$210$$ 0 0
$$211$$ 16.4853 1.13489 0.567447 0.823410i $$-0.307931\pi$$
0.567447 + 0.823410i $$0.307931\pi$$
$$212$$ 0 0
$$213$$ −21.6569 −1.48390
$$214$$ 0 0
$$215$$ 2.82843 0.192897
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 17.1716 1.16035
$$220$$ 0 0
$$221$$ 16.4853 1.10892
$$222$$ 0 0
$$223$$ −13.4142 −0.898282 −0.449141 0.893461i $$-0.648270\pi$$
−0.449141 + 0.893461i $$0.648270\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ −24.0416 −1.59570 −0.797850 0.602857i $$-0.794029\pi$$
−0.797850 + 0.602857i $$0.794029\pi$$
$$228$$ 0 0
$$229$$ 11.3137 0.747631 0.373815 0.927503i $$-0.378049\pi$$
0.373815 + 0.927503i $$0.378049\pi$$
$$230$$ 0 0
$$231$$ 3.31371 0.218026
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ −8.48528 −0.553519
$$236$$ 0 0
$$237$$ 12.9706 0.842529
$$238$$ 0 0
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ 8.14214 0.524481 0.262241 0.965003i $$-0.415539\pi$$
0.262241 + 0.965003i $$0.415539\pi$$
$$242$$ 0 0
$$243$$ 9.89949 0.635053
$$244$$ 0 0
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ 3.41421 0.217241
$$248$$ 0 0
$$249$$ −11.3137 −0.716977
$$250$$ 0 0
$$251$$ −28.9706 −1.82861 −0.914303 0.405031i $$-0.867261\pi$$
−0.914303 + 0.405031i $$0.867261\pi$$
$$252$$ 0 0
$$253$$ −3.31371 −0.208331
$$254$$ 0 0
$$255$$ −6.82843 −0.427613
$$256$$ 0 0
$$257$$ −4.10051 −0.255782 −0.127891 0.991788i $$-0.540821\pi$$
−0.127891 + 0.991788i $$0.540821\pi$$
$$258$$ 0 0
$$259$$ 28.9706 1.80014
$$260$$ 0 0
$$261$$ 0.828427 0.0512784
$$262$$ 0 0
$$263$$ −27.3137 −1.68424 −0.842118 0.539294i $$-0.818691\pi$$
−0.842118 + 0.539294i $$0.818691\pi$$
$$264$$ 0 0
$$265$$ −13.0711 −0.802949
$$266$$ 0 0
$$267$$ −4.48528 −0.274495
$$268$$ 0 0
$$269$$ −29.3137 −1.78729 −0.893644 0.448776i $$-0.851860\pi$$
−0.893644 + 0.448776i $$0.851860\pi$$
$$270$$ 0 0
$$271$$ 9.51472 0.577978 0.288989 0.957332i $$-0.406681\pi$$
0.288989 + 0.957332i $$0.406681\pi$$
$$272$$ 0 0
$$273$$ 13.6569 0.826550
$$274$$ 0 0
$$275$$ 0.828427 0.0499560
$$276$$ 0 0
$$277$$ 19.1716 1.15191 0.575954 0.817482i $$-0.304631\pi$$
0.575954 + 0.817482i $$0.304631\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 11.1716 0.666440 0.333220 0.942849i $$-0.391865\pi$$
0.333220 + 0.942849i $$0.391865\pi$$
$$282$$ 0 0
$$283$$ −6.34315 −0.377061 −0.188530 0.982067i $$-0.560372\pi$$
−0.188530 + 0.982067i $$0.560372\pi$$
$$284$$ 0 0
$$285$$ −1.41421 −0.0837708
$$286$$ 0 0
$$287$$ −2.34315 −0.138312
$$288$$ 0 0
$$289$$ 6.31371 0.371395
$$290$$ 0 0
$$291$$ −3.17157 −0.185921
$$292$$ 0 0
$$293$$ 32.3848 1.89194 0.945969 0.324256i $$-0.105114\pi$$
0.945969 + 0.324256i $$0.105114\pi$$
$$294$$ 0 0
$$295$$ 2.82843 0.164677
$$296$$ 0 0
$$297$$ −4.68629 −0.271926
$$298$$ 0 0
$$299$$ −13.6569 −0.789796
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ 24.9706 1.43452
$$304$$ 0 0
$$305$$ 1.65685 0.0948712
$$306$$ 0 0
$$307$$ −1.41421 −0.0807134 −0.0403567 0.999185i $$-0.512849\pi$$
−0.0403567 + 0.999185i $$0.512849\pi$$
$$308$$ 0 0
$$309$$ −11.6569 −0.663135
$$310$$ 0 0
$$311$$ −19.1716 −1.08712 −0.543560 0.839370i $$-0.682924\pi$$
−0.543560 + 0.839370i $$0.682924\pi$$
$$312$$ 0 0
$$313$$ −5.31371 −0.300349 −0.150174 0.988660i $$-0.547983\pi$$
−0.150174 + 0.988660i $$0.547983\pi$$
$$314$$ 0 0
$$315$$ 2.82843 0.159364
$$316$$ 0 0
$$317$$ 0.100505 0.00564493 0.00282246 0.999996i $$-0.499102\pi$$
0.00282246 + 0.999996i $$0.499102\pi$$
$$318$$ 0 0
$$319$$ −0.686292 −0.0384249
$$320$$ 0 0
$$321$$ −2.00000 −0.111629
$$322$$ 0 0
$$323$$ 4.82843 0.268661
$$324$$ 0 0
$$325$$ 3.41421 0.189386
$$326$$ 0 0
$$327$$ 5.17157 0.285989
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ 17.4558 0.959460 0.479730 0.877416i $$-0.340735\pi$$
0.479730 + 0.877416i $$0.340735\pi$$
$$332$$ 0 0
$$333$$ −10.2426 −0.561293
$$334$$ 0 0
$$335$$ −9.41421 −0.514353
$$336$$ 0 0
$$337$$ −20.3848 −1.11043 −0.555215 0.831707i $$-0.687364\pi$$
−0.555215 + 0.831707i $$0.687364\pi$$
$$338$$ 0 0
$$339$$ −20.8284 −1.13124
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −16.9706 −0.916324
$$344$$ 0 0
$$345$$ 5.65685 0.304555
$$346$$ 0 0
$$347$$ 20.2843 1.08892 0.544458 0.838788i $$-0.316735\pi$$
0.544458 + 0.838788i $$0.316735\pi$$
$$348$$ 0 0
$$349$$ 12.3431 0.660713 0.330357 0.943856i $$-0.392831\pi$$
0.330357 + 0.943856i $$0.392831\pi$$
$$350$$ 0 0
$$351$$ −19.3137 −1.03089
$$352$$ 0 0
$$353$$ −12.1421 −0.646261 −0.323130 0.946354i $$-0.604735\pi$$
−0.323130 + 0.946354i $$0.604735\pi$$
$$354$$ 0 0
$$355$$ 15.3137 0.812767
$$356$$ 0 0
$$357$$ 19.3137 1.02219
$$358$$ 0 0
$$359$$ 19.4558 1.02684 0.513420 0.858137i $$-0.328378\pi$$
0.513420 + 0.858137i $$0.328378\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −14.5858 −0.765555
$$364$$ 0 0
$$365$$ −12.1421 −0.635548
$$366$$ 0 0
$$367$$ −31.1127 −1.62407 −0.812035 0.583609i $$-0.801640\pi$$
−0.812035 + 0.583609i $$0.801640\pi$$
$$368$$ 0 0
$$369$$ 0.828427 0.0431262
$$370$$ 0 0
$$371$$ 36.9706 1.91942
$$372$$ 0 0
$$373$$ −33.5563 −1.73748 −0.868741 0.495267i $$-0.835070\pi$$
−0.868741 + 0.495267i $$0.835070\pi$$
$$374$$ 0 0
$$375$$ −1.41421 −0.0730297
$$376$$ 0 0
$$377$$ −2.82843 −0.145671
$$378$$ 0 0
$$379$$ −9.65685 −0.496039 −0.248020 0.968755i $$-0.579780\pi$$
−0.248020 + 0.968755i $$0.579780\pi$$
$$380$$ 0 0
$$381$$ 18.9706 0.971891
$$382$$ 0 0
$$383$$ −32.7279 −1.67232 −0.836159 0.548487i $$-0.815204\pi$$
−0.836159 + 0.548487i $$0.815204\pi$$
$$384$$ 0 0
$$385$$ −2.34315 −0.119418
$$386$$ 0 0
$$387$$ 2.82843 0.143777
$$388$$ 0 0
$$389$$ 9.31371 0.472224 0.236112 0.971726i $$-0.424127\pi$$
0.236112 + 0.971726i $$0.424127\pi$$
$$390$$ 0 0
$$391$$ −19.3137 −0.976736
$$392$$ 0 0
$$393$$ −10.3431 −0.521743
$$394$$ 0 0
$$395$$ −9.17157 −0.461472
$$396$$ 0 0
$$397$$ 36.8284 1.84837 0.924183 0.381950i $$-0.124747\pi$$
0.924183 + 0.381950i $$0.124747\pi$$
$$398$$ 0 0
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ 14.9706 0.747594 0.373797 0.927510i $$-0.378056\pi$$
0.373797 + 0.927510i $$0.378056\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 5.00000 0.248452
$$406$$ 0 0
$$407$$ 8.48528 0.420600
$$408$$ 0 0
$$409$$ −30.2843 −1.49746 −0.748730 0.662875i $$-0.769336\pi$$
−0.748730 + 0.662875i $$0.769336\pi$$
$$410$$ 0 0
$$411$$ −10.1421 −0.500275
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 8.00000 0.392705
$$416$$ 0 0
$$417$$ 4.48528 0.219645
$$418$$ 0 0
$$419$$ −1.65685 −0.0809426 −0.0404713 0.999181i $$-0.512886\pi$$
−0.0404713 + 0.999181i $$0.512886\pi$$
$$420$$ 0 0
$$421$$ −20.8284 −1.01512 −0.507558 0.861618i $$-0.669452\pi$$
−0.507558 + 0.861618i $$0.669452\pi$$
$$422$$ 0 0
$$423$$ −8.48528 −0.412568
$$424$$ 0 0
$$425$$ 4.82843 0.234213
$$426$$ 0 0
$$427$$ −4.68629 −0.226786
$$428$$ 0 0
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ 14.8284 0.714260 0.357130 0.934055i $$-0.383755\pi$$
0.357130 + 0.934055i $$0.383755\pi$$
$$432$$ 0 0
$$433$$ −18.0416 −0.867025 −0.433513 0.901147i $$-0.642726\pi$$
−0.433513 + 0.901147i $$0.642726\pi$$
$$434$$ 0 0
$$435$$ 1.17157 0.0561726
$$436$$ 0 0
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ −14.1421 −0.674967 −0.337484 0.941331i $$-0.609576\pi$$
−0.337484 + 0.941331i $$0.609576\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ −0.686292 −0.0326067 −0.0163033 0.999867i $$-0.505190\pi$$
−0.0163033 + 0.999867i $$0.505190\pi$$
$$444$$ 0 0
$$445$$ 3.17157 0.150347
$$446$$ 0 0
$$447$$ 2.34315 0.110827
$$448$$ 0 0
$$449$$ 26.2843 1.24043 0.620216 0.784431i $$-0.287045\pi$$
0.620216 + 0.784431i $$0.287045\pi$$
$$450$$ 0 0
$$451$$ −0.686292 −0.0323162
$$452$$ 0 0
$$453$$ −8.68629 −0.408118
$$454$$ 0 0
$$455$$ −9.65685 −0.452720
$$456$$ 0 0
$$457$$ −34.2843 −1.60375 −0.801875 0.597491i $$-0.796164\pi$$
−0.801875 + 0.597491i $$0.796164\pi$$
$$458$$ 0 0
$$459$$ −27.3137 −1.27489
$$460$$ 0 0
$$461$$ −8.62742 −0.401819 −0.200909 0.979610i $$-0.564390\pi$$
−0.200909 + 0.979610i $$0.564390\pi$$
$$462$$ 0 0
$$463$$ −26.6274 −1.23748 −0.618741 0.785595i $$-0.712357\pi$$
−0.618741 + 0.785595i $$0.712357\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −24.0000 −1.11059 −0.555294 0.831654i $$-0.687394\pi$$
−0.555294 + 0.831654i $$0.687394\pi$$
$$468$$ 0 0
$$469$$ 26.6274 1.22954
$$470$$ 0 0
$$471$$ −14.8284 −0.683258
$$472$$ 0 0
$$473$$ −2.34315 −0.107738
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −13.0711 −0.598483
$$478$$ 0 0
$$479$$ −23.4558 −1.07172 −0.535862 0.844305i $$-0.680013\pi$$
−0.535862 + 0.844305i $$0.680013\pi$$
$$480$$ 0 0
$$481$$ 34.9706 1.59452
$$482$$ 0 0
$$483$$ −16.0000 −0.728025
$$484$$ 0 0
$$485$$ 2.24264 0.101833
$$486$$ 0 0
$$487$$ −27.5563 −1.24870 −0.624349 0.781146i $$-0.714636\pi$$
−0.624349 + 0.781146i $$0.714636\pi$$
$$488$$ 0 0
$$489$$ −20.9706 −0.948322
$$490$$ 0 0
$$491$$ 12.9706 0.585353 0.292677 0.956211i $$-0.405454\pi$$
0.292677 + 0.956211i $$0.405454\pi$$
$$492$$ 0 0
$$493$$ −4.00000 −0.180151
$$494$$ 0 0
$$495$$ 0.828427 0.0372350
$$496$$ 0 0
$$497$$ −43.3137 −1.94289
$$498$$ 0 0
$$499$$ −4.14214 −0.185427 −0.0927137 0.995693i $$-0.529554\pi$$
−0.0927137 + 0.995693i $$0.529554\pi$$
$$500$$ 0 0
$$501$$ −2.97056 −0.132715
$$502$$ 0 0
$$503$$ 1.65685 0.0738755 0.0369377 0.999318i $$-0.488240\pi$$
0.0369377 + 0.999318i $$0.488240\pi$$
$$504$$ 0 0
$$505$$ −17.6569 −0.785720
$$506$$ 0 0
$$507$$ −1.89949 −0.0843595
$$508$$ 0 0
$$509$$ −15.4558 −0.685068 −0.342534 0.939505i $$-0.611285\pi$$
−0.342534 + 0.939505i $$0.611285\pi$$
$$510$$ 0 0
$$511$$ 34.3431 1.51925
$$512$$ 0 0
$$513$$ −5.65685 −0.249756
$$514$$ 0 0
$$515$$ 8.24264 0.363214
$$516$$ 0 0
$$517$$ 7.02944 0.309154
$$518$$ 0 0
$$519$$ −15.1716 −0.665958
$$520$$ 0 0
$$521$$ 14.6863 0.643418 0.321709 0.946839i $$-0.395743\pi$$
0.321709 + 0.946839i $$0.395743\pi$$
$$522$$ 0 0
$$523$$ 36.2426 1.58478 0.792390 0.610015i $$-0.208837\pi$$
0.792390 + 0.610015i $$0.208837\pi$$
$$524$$ 0 0
$$525$$ 4.00000 0.174574
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 2.82843 0.122743
$$532$$ 0 0
$$533$$ −2.82843 −0.122513
$$534$$ 0 0
$$535$$ 1.41421 0.0611418
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 0.828427 0.0356829
$$540$$ 0 0
$$541$$ 36.2843 1.55998 0.779991 0.625790i $$-0.215223\pi$$
0.779991 + 0.625790i $$0.215223\pi$$
$$542$$ 0 0
$$543$$ −7.51472 −0.322487
$$544$$ 0 0
$$545$$ −3.65685 −0.156642
$$546$$ 0 0
$$547$$ 39.0711 1.67056 0.835279 0.549826i $$-0.185306\pi$$
0.835279 + 0.549826i $$0.185306\pi$$
$$548$$ 0 0
$$549$$ 1.65685 0.0707128
$$550$$ 0 0
$$551$$ −0.828427 −0.0352922
$$552$$ 0 0
$$553$$ 25.9411 1.10313
$$554$$ 0 0
$$555$$ −14.4853 −0.614866
$$556$$ 0 0
$$557$$ −5.79899 −0.245711 −0.122856 0.992425i $$-0.539205\pi$$
−0.122856 + 0.992425i $$0.539205\pi$$
$$558$$ 0 0
$$559$$ −9.65685 −0.408441
$$560$$ 0 0
$$561$$ 5.65685 0.238833
$$562$$ 0 0
$$563$$ 4.24264 0.178806 0.0894030 0.995996i $$-0.471504\pi$$
0.0894030 + 0.995996i $$0.471504\pi$$
$$564$$ 0 0
$$565$$ 14.7279 0.619608
$$566$$ 0 0
$$567$$ −14.1421 −0.593914
$$568$$ 0 0
$$569$$ −31.9411 −1.33904 −0.669521 0.742793i $$-0.733501\pi$$
−0.669521 + 0.742793i $$0.733501\pi$$
$$570$$ 0 0
$$571$$ 30.4853 1.27577 0.637885 0.770132i $$-0.279810\pi$$
0.637885 + 0.770132i $$0.279810\pi$$
$$572$$ 0 0
$$573$$ −4.68629 −0.195773
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ 44.6274 1.85786 0.928932 0.370251i $$-0.120728\pi$$
0.928932 + 0.370251i $$0.120728\pi$$
$$578$$ 0 0
$$579$$ −9.79899 −0.407232
$$580$$ 0 0
$$581$$ −22.6274 −0.938743
$$582$$ 0 0
$$583$$ 10.8284 0.448468
$$584$$ 0 0
$$585$$ 3.41421 0.141160
$$586$$ 0 0
$$587$$ 5.65685 0.233483 0.116742 0.993162i $$-0.462755\pi$$
0.116742 + 0.993162i $$0.462755\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 35.7990 1.47257
$$592$$ 0 0
$$593$$ 16.6274 0.682806 0.341403 0.939917i $$-0.389098\pi$$
0.341403 + 0.939917i $$0.389098\pi$$
$$594$$ 0 0
$$595$$ −13.6569 −0.559876
$$596$$ 0 0
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ −6.14214 −0.250961 −0.125480 0.992096i $$-0.540047\pi$$
−0.125480 + 0.992096i $$0.540047\pi$$
$$600$$ 0 0
$$601$$ −14.4853 −0.590867 −0.295433 0.955363i $$-0.595464\pi$$
−0.295433 + 0.955363i $$0.595464\pi$$
$$602$$ 0 0
$$603$$ −9.41421 −0.383376
$$604$$ 0 0
$$605$$ 10.3137 0.419312
$$606$$ 0 0
$$607$$ 7.75736 0.314862 0.157431 0.987530i $$-0.449679\pi$$
0.157431 + 0.987530i $$0.449679\pi$$
$$608$$ 0 0
$$609$$ −3.31371 −0.134278
$$610$$ 0 0
$$611$$ 28.9706 1.17202
$$612$$ 0 0
$$613$$ 17.7990 0.718894 0.359447 0.933165i $$-0.382965\pi$$
0.359447 + 0.933165i $$0.382965\pi$$
$$614$$ 0 0
$$615$$ 1.17157 0.0472424
$$616$$ 0 0
$$617$$ −37.7990 −1.52173 −0.760865 0.648910i $$-0.775225\pi$$
−0.760865 + 0.648910i $$0.775225\pi$$
$$618$$ 0 0
$$619$$ −40.1421 −1.61345 −0.806724 0.590928i $$-0.798762\pi$$
−0.806724 + 0.590928i $$0.798762\pi$$
$$620$$ 0 0
$$621$$ 22.6274 0.908007
$$622$$ 0 0
$$623$$ −8.97056 −0.359398
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 1.17157 0.0467881
$$628$$ 0 0
$$629$$ 49.4558 1.97193
$$630$$ 0 0
$$631$$ 34.4853 1.37284 0.686419 0.727207i $$-0.259182\pi$$
0.686419 + 0.727207i $$0.259182\pi$$
$$632$$ 0 0
$$633$$ 23.3137 0.926637
$$634$$ 0 0
$$635$$ −13.4142 −0.532327
$$636$$ 0 0
$$637$$ 3.41421 0.135276
$$638$$ 0 0
$$639$$ 15.3137 0.605801
$$640$$ 0 0
$$641$$ 27.4558 1.08444 0.542220 0.840236i $$-0.317584\pi$$
0.542220 + 0.840236i $$0.317584\pi$$
$$642$$ 0 0
$$643$$ 35.3137 1.39264 0.696318 0.717733i $$-0.254820\pi$$
0.696318 + 0.717733i $$0.254820\pi$$
$$644$$ 0 0
$$645$$ 4.00000 0.157500
$$646$$ 0 0
$$647$$ 16.6863 0.656006 0.328003 0.944677i $$-0.393624\pi$$
0.328003 + 0.944677i $$0.393624\pi$$
$$648$$ 0 0
$$649$$ −2.34315 −0.0919765
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 12.1421 0.475158 0.237579 0.971368i $$-0.423646\pi$$
0.237579 + 0.971368i $$0.423646\pi$$
$$654$$ 0 0
$$655$$ 7.31371 0.285770
$$656$$ 0 0
$$657$$ −12.1421 −0.473710
$$658$$ 0 0
$$659$$ 28.4853 1.10963 0.554815 0.831974i $$-0.312789\pi$$
0.554815 + 0.831974i $$0.312789\pi$$
$$660$$ 0 0
$$661$$ 8.82843 0.343386 0.171693 0.985151i $$-0.445076\pi$$
0.171693 + 0.985151i $$0.445076\pi$$
$$662$$ 0 0
$$663$$ 23.3137 0.905429
$$664$$ 0 0
$$665$$ −2.82843 −0.109682
$$666$$ 0 0
$$667$$ 3.31371 0.128307
$$668$$ 0 0
$$669$$ −18.9706 −0.733444
$$670$$ 0 0
$$671$$ −1.37258 −0.0529880
$$672$$ 0 0
$$673$$ 8.38478 0.323209 0.161605 0.986856i $$-0.448333\pi$$
0.161605 + 0.986856i $$0.448333\pi$$
$$674$$ 0 0
$$675$$ −5.65685 −0.217732
$$676$$ 0 0
$$677$$ −7.61522 −0.292677 −0.146338 0.989235i $$-0.546749\pi$$
−0.146338 + 0.989235i $$0.546749\pi$$
$$678$$ 0 0
$$679$$ −6.34315 −0.243428
$$680$$ 0 0
$$681$$ −34.0000 −1.30288
$$682$$ 0 0
$$683$$ 9.89949 0.378794 0.189397 0.981901i $$-0.439347\pi$$
0.189397 + 0.981901i $$0.439347\pi$$
$$684$$ 0 0
$$685$$ 7.17157 0.274012
$$686$$ 0 0
$$687$$ 16.0000 0.610438
$$688$$ 0 0
$$689$$ 44.6274 1.70017
$$690$$ 0 0
$$691$$ 35.1716 1.33799 0.668995 0.743267i $$-0.266725\pi$$
0.668995 + 0.743267i $$0.266725\pi$$
$$692$$ 0 0
$$693$$ −2.34315 −0.0890087
$$694$$ 0 0
$$695$$ −3.17157 −0.120305
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 0 0
$$699$$ −14.1421 −0.534905
$$700$$ 0 0
$$701$$ −24.0000 −0.906467 −0.453234 0.891392i $$-0.649730\pi$$
−0.453234 + 0.891392i $$0.649730\pi$$
$$702$$ 0 0
$$703$$ 10.2426 0.386309
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ 0 0
$$707$$ 49.9411 1.87823
$$708$$ 0 0
$$709$$ −18.2843 −0.686680 −0.343340 0.939211i $$-0.611558\pi$$
−0.343340 + 0.939211i $$0.611558\pi$$
$$710$$ 0 0
$$711$$ −9.17157 −0.343961
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −2.82843 −0.105777
$$716$$ 0 0
$$717$$ −11.3137 −0.422518
$$718$$ 0 0
$$719$$ 31.4558 1.17311 0.586553 0.809911i $$-0.300485\pi$$
0.586553 + 0.809911i $$0.300485\pi$$
$$720$$ 0 0
$$721$$ −23.3137 −0.868248
$$722$$ 0 0
$$723$$ 11.5147 0.428237
$$724$$ 0 0
$$725$$ −0.828427 −0.0307670
$$726$$ 0 0
$$727$$ 41.4558 1.53751 0.768756 0.639542i $$-0.220876\pi$$
0.768756 + 0.639542i $$0.220876\pi$$
$$728$$ 0 0
$$729$$ 29.0000 1.07407
$$730$$ 0 0
$$731$$ −13.6569 −0.505117
$$732$$ 0 0
$$733$$ 4.34315 0.160418 0.0802089 0.996778i $$-0.474441\pi$$
0.0802089 + 0.996778i $$0.474441\pi$$
$$734$$ 0 0
$$735$$ −1.41421 −0.0521641
$$736$$ 0 0
$$737$$ 7.79899 0.287279
$$738$$ 0 0
$$739$$ −0.686292 −0.0252456 −0.0126228 0.999920i $$-0.504018\pi$$
−0.0126228 + 0.999920i $$0.504018\pi$$
$$740$$ 0 0
$$741$$ 4.82843 0.177377
$$742$$ 0 0
$$743$$ 15.7574 0.578081 0.289041 0.957317i $$-0.406664\pi$$
0.289041 + 0.957317i $$0.406664\pi$$
$$744$$ 0 0
$$745$$ −1.65685 −0.0607024
$$746$$ 0 0
$$747$$ 8.00000 0.292705
$$748$$ 0 0
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ 44.4853 1.62329 0.811645 0.584150i $$-0.198572\pi$$
0.811645 + 0.584150i $$0.198572\pi$$
$$752$$ 0 0
$$753$$ −40.9706 −1.49305
$$754$$ 0 0
$$755$$ 6.14214 0.223535
$$756$$ 0 0
$$757$$ −22.2843 −0.809936 −0.404968 0.914331i $$-0.632717\pi$$
−0.404968 + 0.914331i $$0.632717\pi$$
$$758$$ 0 0
$$759$$ −4.68629 −0.170102
$$760$$ 0 0
$$761$$ 1.37258 0.0497561 0.0248780 0.999690i $$-0.492080\pi$$
0.0248780 + 0.999690i $$0.492080\pi$$
$$762$$ 0 0
$$763$$ 10.3431 0.374447
$$764$$ 0 0
$$765$$ 4.82843 0.174572
$$766$$ 0 0
$$767$$ −9.65685 −0.348689
$$768$$ 0 0
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ −5.79899 −0.208846
$$772$$ 0 0
$$773$$ −48.8701 −1.75773 −0.878867 0.477067i $$-0.841700\pi$$
−0.878867 + 0.477067i $$0.841700\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 40.9706 1.46981
$$778$$ 0 0
$$779$$ −0.828427 −0.0296815
$$780$$ 0 0
$$781$$ −12.6863 −0.453951
$$782$$ 0 0
$$783$$ 4.68629 0.167474
$$784$$ 0 0
$$785$$ 10.4853 0.374236
$$786$$ 0 0
$$787$$ 1.41421 0.0504113 0.0252056 0.999682i $$-0.491976\pi$$
0.0252056 + 0.999682i $$0.491976\pi$$
$$788$$ 0 0
$$789$$ −38.6274 −1.37517
$$790$$ 0 0
$$791$$ −41.6569 −1.48115
$$792$$ 0 0
$$793$$ −5.65685 −0.200881
$$794$$ 0 0
$$795$$ −18.4853 −0.655605
$$796$$ 0 0
$$797$$ 27.6985 0.981131 0.490565 0.871404i $$-0.336790\pi$$
0.490565 + 0.871404i $$0.336790\pi$$
$$798$$ 0 0
$$799$$ 40.9706 1.44943
$$800$$ 0 0
$$801$$ 3.17157 0.112062
$$802$$ 0 0
$$803$$ 10.0589 0.354970
$$804$$ 0 0
$$805$$ 11.3137 0.398756
$$806$$ 0 0
$$807$$ −41.4558 −1.45931
$$808$$ 0 0
$$809$$ 38.0000 1.33601 0.668004 0.744157i $$-0.267149\pi$$
0.668004 + 0.744157i $$0.267149\pi$$
$$810$$ 0 0
$$811$$ 0.686292 0.0240990 0.0120495 0.999927i $$-0.496164\pi$$
0.0120495 + 0.999927i $$0.496164\pi$$
$$812$$ 0 0
$$813$$ 13.4558 0.471917
$$814$$ 0 0
$$815$$ 14.8284 0.519417
$$816$$ 0 0
$$817$$ −2.82843 −0.0989541
$$818$$ 0 0
$$819$$ −9.65685 −0.337438
$$820$$ 0 0
$$821$$ −49.5980 −1.73098 −0.865491 0.500925i $$-0.832993\pi$$
−0.865491 + 0.500925i $$0.832993\pi$$
$$822$$ 0 0
$$823$$ 16.4853 0.574641 0.287320 0.957835i $$-0.407236\pi$$
0.287320 + 0.957835i $$0.407236\pi$$
$$824$$ 0 0
$$825$$ 1.17157 0.0407889
$$826$$ 0 0
$$827$$ −13.6985 −0.476343 −0.238171 0.971223i $$-0.576548\pi$$
−0.238171 + 0.971223i $$0.576548\pi$$
$$828$$ 0 0
$$829$$ −49.5980 −1.72261 −0.861305 0.508089i $$-0.830352\pi$$
−0.861305 + 0.508089i $$0.830352\pi$$
$$830$$ 0 0
$$831$$ 27.1127 0.940529
$$832$$ 0 0
$$833$$ 4.82843 0.167295
$$834$$ 0 0
$$835$$ 2.10051 0.0726910
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −0.201010 −0.00693964 −0.00346982 0.999994i $$-0.501104\pi$$
−0.00346982 + 0.999994i $$0.501104\pi$$
$$840$$ 0 0
$$841$$ −28.3137 −0.976335
$$842$$ 0 0
$$843$$ 15.7990 0.544146
$$844$$ 0 0
$$845$$ 1.34315 0.0462056
$$846$$ 0 0
$$847$$ −29.1716 −1.00235
$$848$$ 0 0
$$849$$ −8.97056 −0.307869
$$850$$ 0 0
$$851$$ −40.9706 −1.40445
$$852$$ 0 0
$$853$$ −27.9411 −0.956686 −0.478343 0.878173i $$-0.658762\pi$$
−0.478343 + 0.878173i $$0.658762\pi$$
$$854$$ 0 0
$$855$$ 1.00000 0.0341993
$$856$$ 0 0
$$857$$ 31.2132 1.06622 0.533111 0.846045i $$-0.321023\pi$$
0.533111 + 0.846045i $$0.321023\pi$$
$$858$$ 0 0
$$859$$ −6.34315 −0.216425 −0.108213 0.994128i $$-0.534513\pi$$
−0.108213 + 0.994128i $$0.534513\pi$$
$$860$$ 0 0
$$861$$ −3.31371 −0.112931
$$862$$ 0 0
$$863$$ 44.0416 1.49919 0.749597 0.661894i $$-0.230247\pi$$
0.749597 + 0.661894i $$0.230247\pi$$
$$864$$ 0 0
$$865$$ 10.7279 0.364760
$$866$$ 0 0
$$867$$ 8.92893 0.303242
$$868$$ 0 0
$$869$$ 7.59798 0.257744
$$870$$ 0 0
$$871$$ 32.1421 1.08909
$$872$$ 0 0
$$873$$ 2.24264 0.0759019
$$874$$ 0 0
$$875$$ −2.82843 −0.0956183
$$876$$ 0 0
$$877$$ 18.7279 0.632397 0.316198 0.948693i $$-0.397593\pi$$
0.316198 + 0.948693i $$0.397593\pi$$
$$878$$ 0 0
$$879$$ 45.7990 1.54476
$$880$$ 0 0
$$881$$ −1.65685 −0.0558208 −0.0279104 0.999610i $$-0.508885\pi$$
−0.0279104 + 0.999610i $$0.508885\pi$$
$$882$$ 0 0
$$883$$ 10.8284 0.364406 0.182203 0.983261i $$-0.441677\pi$$
0.182203 + 0.983261i $$0.441677\pi$$
$$884$$ 0 0
$$885$$ 4.00000 0.134459
$$886$$ 0 0
$$887$$ 15.2721 0.512786 0.256393 0.966573i $$-0.417466\pi$$
0.256393 + 0.966573i $$0.417466\pi$$
$$888$$ 0 0
$$889$$ 37.9411 1.27250
$$890$$ 0 0
$$891$$ −4.14214 −0.138767
$$892$$ 0 0
$$893$$ 8.48528 0.283949
$$894$$ 0 0
$$895$$ 8.48528 0.283632
$$896$$ 0 0
$$897$$ −19.3137 −0.644866
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 63.1127 2.10259
$$902$$ 0 0
$$903$$ −11.3137 −0.376497
$$904$$ 0 0
$$905$$ 5.31371 0.176634
$$906$$ 0 0
$$907$$ −48.0416 −1.59520 −0.797598 0.603189i $$-0.793896\pi$$
−0.797598 + 0.603189i $$0.793896\pi$$
$$908$$ 0 0
$$909$$ −17.6569 −0.585641
$$910$$ 0 0
$$911$$ 9.17157 0.303868 0.151934 0.988391i $$-0.451450\pi$$
0.151934 + 0.988391i $$0.451450\pi$$
$$912$$ 0 0
$$913$$ −6.62742 −0.219335
$$914$$ 0 0
$$915$$ 2.34315 0.0774620
$$916$$ 0 0
$$917$$ −20.6863 −0.683122
$$918$$ 0 0
$$919$$ 17.9411 0.591823 0.295912 0.955215i $$-0.404377\pi$$
0.295912 + 0.955215i $$0.404377\pi$$
$$920$$ 0 0
$$921$$ −2.00000 −0.0659022
$$922$$ 0 0
$$923$$ −52.2843 −1.72096
$$924$$ 0 0
$$925$$ 10.2426 0.336776
$$926$$ 0 0
$$927$$ 8.24264 0.270724
$$928$$ 0 0
$$929$$ 41.3137 1.35546 0.677729 0.735311i $$-0.262964\pi$$
0.677729 + 0.735311i $$0.262964\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ −27.1127 −0.887630
$$934$$ 0 0
$$935$$ −4.00000 −0.130814
$$936$$ 0 0
$$937$$ 55.4558 1.81166 0.905832 0.423638i $$-0.139247\pi$$
0.905832 + 0.423638i $$0.139247\pi$$
$$938$$ 0 0
$$939$$ −7.51472 −0.245234
$$940$$ 0 0
$$941$$ 21.5980 0.704074 0.352037 0.935986i $$-0.385489\pi$$
0.352037 + 0.935986i $$0.385489\pi$$
$$942$$ 0 0
$$943$$ 3.31371 0.107909
$$944$$ 0 0
$$945$$ 16.0000 0.520480
$$946$$ 0 0
$$947$$ −47.3137 −1.53749 −0.768744 0.639556i $$-0.779118\pi$$
−0.768744 + 0.639556i $$0.779118\pi$$
$$948$$ 0 0
$$949$$ 41.4558 1.34571
$$950$$ 0 0
$$951$$ 0.142136 0.00460906
$$952$$ 0 0
$$953$$ −34.7279 −1.12495 −0.562474 0.826815i $$-0.690150\pi$$
−0.562474 + 0.826815i $$0.690150\pi$$
$$954$$ 0 0
$$955$$ 3.31371 0.107229
$$956$$ 0 0
$$957$$ −0.970563 −0.0313738
$$958$$ 0 0
$$959$$ −20.2843 −0.655013
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 1.41421 0.0455724
$$964$$ 0 0
$$965$$ 6.92893 0.223050
$$966$$ 0 0
$$967$$ −37.6569 −1.21096 −0.605481 0.795859i $$-0.707019\pi$$
−0.605481 + 0.795859i $$0.707019\pi$$
$$968$$ 0 0
$$969$$ 6.82843 0.219361
$$970$$ 0 0
$$971$$ −25.9411 −0.832490 −0.416245 0.909252i $$-0.636654\pi$$
−0.416245 + 0.909252i $$0.636654\pi$$
$$972$$ 0 0
$$973$$ 8.97056 0.287583
$$974$$ 0 0
$$975$$ 4.82843 0.154633
$$976$$ 0 0
$$977$$ −53.8406 −1.72251 −0.861257 0.508170i $$-0.830322\pi$$
−0.861257 + 0.508170i $$0.830322\pi$$
$$978$$ 0 0
$$979$$ −2.62742 −0.0839726
$$980$$ 0 0
$$981$$ −3.65685 −0.116754
$$982$$ 0 0
$$983$$ −4.04163 −0.128908 −0.0644540 0.997921i $$-0.520531\pi$$
−0.0644540 + 0.997921i $$0.520531\pi$$
$$984$$ 0 0
$$985$$ −25.3137 −0.806562
$$986$$ 0 0
$$987$$ 33.9411 1.08036
$$988$$ 0 0
$$989$$ 11.3137 0.359755
$$990$$ 0 0
$$991$$ −44.7696 −1.42215 −0.711076 0.703115i $$-0.751792\pi$$
−0.711076 + 0.703115i $$0.751792\pi$$
$$992$$ 0 0
$$993$$ 24.6863 0.783396
$$994$$ 0 0
$$995$$ −5.65685 −0.179334
$$996$$ 0 0
$$997$$ 12.8284 0.406280 0.203140 0.979150i $$-0.434885\pi$$
0.203140 + 0.979150i $$0.434885\pi$$
$$998$$ 0 0
$$999$$ −57.9411 −1.83318
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 1520.2.a.m.1.2 2
4.3 odd 2 760.2.a.f.1.1 2
5.4 even 2 7600.2.a.ba.1.1 2
8.3 odd 2 6080.2.a.bf.1.2 2
8.5 even 2 6080.2.a.bg.1.1 2
12.11 even 2 6840.2.a.z.1.1 2
20.3 even 4 3800.2.d.i.3649.1 4
20.7 even 4 3800.2.d.i.3649.3 4
20.19 odd 2 3800.2.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.f.1.1 2 4.3 odd 2
1520.2.a.m.1.2 2 1.1 even 1 trivial
3800.2.a.n.1.2 2 20.19 odd 2
3800.2.d.i.3649.1 4 20.3 even 4
3800.2.d.i.3649.3 4 20.7 even 4
6080.2.a.bf.1.2 2 8.3 odd 2
6080.2.a.bg.1.1 2 8.5 even 2
6840.2.a.z.1.1 2 12.11 even 2
7600.2.a.ba.1.1 2 5.4 even 2