# Properties

 Label 1520.2.a.k.1.2 Level $1520$ Weight $2$ Character 1520.1 Self dual yes Analytic conductor $12.137$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$1.73205$$ 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)

 $$f(q)$$ $$=$$ $$q+0.732051 q^{3} -1.00000 q^{5} -2.46410 q^{9} +O(q^{10})$$ $$q+0.732051 q^{3} -1.00000 q^{5} -2.46410 q^{9} -2.00000 q^{11} +2.73205 q^{13} -0.732051 q^{15} +0.535898 q^{17} -1.00000 q^{19} -5.46410 q^{23} +1.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} -4.00000 q^{31} -1.46410 q^{33} -9.66025 q^{37} +2.00000 q^{39} +7.46410 q^{41} -10.9282 q^{43} +2.46410 q^{45} -10.9282 q^{47} -7.00000 q^{49} +0.392305 q^{51} +5.66025 q^{53} +2.00000 q^{55} -0.732051 q^{57} +5.46410 q^{59} -13.4641 q^{61} -2.73205 q^{65} -6.19615 q^{67} -4.00000 q^{69} +2.92820 q^{71} +10.3923 q^{73} +0.732051 q^{75} -12.3923 q^{79} +4.46410 q^{81} -1.46410 q^{83} -0.535898 q^{85} +2.53590 q^{87} +3.46410 q^{89} -2.92820 q^{93} +1.00000 q^{95} +1.66025 q^{97} +4.92820 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 8 q^{17} - 2 q^{19} - 4 q^{23} + 2 q^{25} - 8 q^{27} - 8 q^{31} + 4 q^{33} - 2 q^{37} + 4 q^{39} + 8 q^{41} - 8 q^{43} - 2 q^{45} - 8 q^{47} - 14 q^{49} - 20 q^{51} - 6 q^{53} + 4 q^{55} + 2 q^{57} + 4 q^{59} - 20 q^{61} - 2 q^{65} - 2 q^{67} - 8 q^{69} - 8 q^{71} - 2 q^{75} - 4 q^{79} + 2 q^{81} + 4 q^{83} - 8 q^{85} + 12 q^{87} + 8 q^{93} + 2 q^{95} - 14 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 8 * q^17 - 2 * q^19 - 4 * q^23 + 2 * q^25 - 8 * q^27 - 8 * q^31 + 4 * q^33 - 2 * q^37 + 4 * q^39 + 8 * q^41 - 8 * q^43 - 2 * q^45 - 8 * q^47 - 14 * q^49 - 20 * q^51 - 6 * q^53 + 4 * q^55 + 2 * q^57 + 4 * q^59 - 20 * q^61 - 2 * q^65 - 2 * q^67 - 8 * q^69 - 8 * q^71 - 2 * q^75 - 4 * q^79 + 2 * q^81 + 4 * q^83 - 8 * q^85 + 12 * q^87 + 8 * q^93 + 2 * q^95 - 14 * 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
$$3$$ 0.732051 0.422650 0.211325 0.977416i $$-0.432222\pi$$
0.211325 + 0.977416i $$0.432222\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ −2.46410 −0.821367
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 2.73205 0.757735 0.378867 0.925451i $$-0.376314\pi$$
0.378867 + 0.925451i $$0.376314\pi$$
$$14$$ 0 0
$$15$$ −0.732051 −0.189015
$$16$$ 0 0
$$17$$ 0.535898 0.129974 0.0649872 0.997886i $$-0.479299\pi$$
0.0649872 + 0.997886i $$0.479299\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −5.46410 −1.13934 −0.569672 0.821872i $$-0.692930\pi$$
−0.569672 + 0.821872i $$0.692930\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ 3.46410 0.643268 0.321634 0.946864i $$-0.395768\pi$$
0.321634 + 0.946864i $$0.395768\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ −1.46410 −0.254867
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.66025 −1.58814 −0.794068 0.607829i $$-0.792041\pi$$
−0.794068 + 0.607829i $$0.792041\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 7.46410 1.16570 0.582848 0.812581i $$-0.301938\pi$$
0.582848 + 0.812581i $$0.301938\pi$$
$$42$$ 0 0
$$43$$ −10.9282 −1.66654 −0.833268 0.552870i $$-0.813533\pi$$
−0.833268 + 0.552870i $$0.813533\pi$$
$$44$$ 0 0
$$45$$ 2.46410 0.367327
$$46$$ 0 0
$$47$$ −10.9282 −1.59404 −0.797021 0.603951i $$-0.793592\pi$$
−0.797021 + 0.603951i $$0.793592\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 0.392305 0.0549337
$$52$$ 0 0
$$53$$ 5.66025 0.777496 0.388748 0.921344i $$-0.372908\pi$$
0.388748 + 0.921344i $$0.372908\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ −0.732051 −0.0969625
$$58$$ 0 0
$$59$$ 5.46410 0.711365 0.355683 0.934607i $$-0.384248\pi$$
0.355683 + 0.934607i $$0.384248\pi$$
$$60$$ 0 0
$$61$$ −13.4641 −1.72390 −0.861951 0.506992i $$-0.830757\pi$$
−0.861951 + 0.506992i $$0.830757\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.73205 −0.338869
$$66$$ 0 0
$$67$$ −6.19615 −0.756980 −0.378490 0.925605i $$-0.623557\pi$$
−0.378490 + 0.925605i $$0.623557\pi$$
$$68$$ 0 0
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 2.92820 0.347514 0.173757 0.984789i $$-0.444409\pi$$
0.173757 + 0.984789i $$0.444409\pi$$
$$72$$ 0 0
$$73$$ 10.3923 1.21633 0.608164 0.793812i $$-0.291906\pi$$
0.608164 + 0.793812i $$0.291906\pi$$
$$74$$ 0 0
$$75$$ 0.732051 0.0845299
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −12.3923 −1.39424 −0.697122 0.716953i $$-0.745536\pi$$
−0.697122 + 0.716953i $$0.745536\pi$$
$$80$$ 0 0
$$81$$ 4.46410 0.496011
$$82$$ 0 0
$$83$$ −1.46410 −0.160706 −0.0803530 0.996766i $$-0.525605\pi$$
−0.0803530 + 0.996766i $$0.525605\pi$$
$$84$$ 0 0
$$85$$ −0.535898 −0.0581263
$$86$$ 0 0
$$87$$ 2.53590 0.271877
$$88$$ 0 0
$$89$$ 3.46410 0.367194 0.183597 0.983002i $$-0.441226\pi$$
0.183597 + 0.983002i $$0.441226\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −2.92820 −0.303641
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 1.66025 0.168573 0.0842866 0.996442i $$-0.473139\pi$$
0.0842866 + 0.996442i $$0.473139\pi$$
$$98$$ 0 0
$$99$$ 4.92820 0.495303
$$100$$ 0 0
$$101$$ −5.46410 −0.543698 −0.271849 0.962340i $$-0.587635\pi$$
−0.271849 + 0.962340i $$0.587635\pi$$
$$102$$ 0 0
$$103$$ 11.6603 1.14892 0.574459 0.818533i $$-0.305212\pi$$
0.574459 + 0.818533i $$0.305212\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 9.12436 0.882085 0.441042 0.897486i $$-0.354609\pi$$
0.441042 + 0.897486i $$0.354609\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −7.07180 −0.671225
$$112$$ 0 0
$$113$$ 13.2679 1.24814 0.624072 0.781367i $$-0.285477\pi$$
0.624072 + 0.781367i $$0.285477\pi$$
$$114$$ 0 0
$$115$$ 5.46410 0.509530
$$116$$ 0 0
$$117$$ −6.73205 −0.622378
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 5.46410 0.492681
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −0.339746 −0.0301476 −0.0150738 0.999886i $$-0.504798\pi$$
−0.0150738 + 0.999886i $$0.504798\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ 7.46410 0.637701 0.318851 0.947805i $$-0.396703\pi$$
0.318851 + 0.947805i $$0.396703\pi$$
$$138$$ 0 0
$$139$$ 19.8564 1.68420 0.842099 0.539323i $$-0.181320\pi$$
0.842099 + 0.539323i $$0.181320\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ −5.46410 −0.456931
$$144$$ 0 0
$$145$$ −3.46410 −0.287678
$$146$$ 0 0
$$147$$ −5.12436 −0.422650
$$148$$ 0 0
$$149$$ −19.3205 −1.58280 −0.791399 0.611300i $$-0.790647\pi$$
−0.791399 + 0.611300i $$0.790647\pi$$
$$150$$ 0 0
$$151$$ −1.46410 −0.119147 −0.0595734 0.998224i $$-0.518974\pi$$
−0.0595734 + 0.998224i $$0.518974\pi$$
$$152$$ 0 0
$$153$$ −1.32051 −0.106757
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −0.535898 −0.0427693 −0.0213847 0.999771i $$-0.506807\pi$$
−0.0213847 + 0.999771i $$0.506807\pi$$
$$158$$ 0 0
$$159$$ 4.14359 0.328608
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ 1.46410 0.113980
$$166$$ 0 0
$$167$$ 3.26795 0.252882 0.126441 0.991974i $$-0.459645\pi$$
0.126441 + 0.991974i $$0.459645\pi$$
$$168$$ 0 0
$$169$$ −5.53590 −0.425838
$$170$$ 0 0
$$171$$ 2.46410 0.188435
$$172$$ 0 0
$$173$$ 7.12436 0.541655 0.270827 0.962628i $$-0.412703\pi$$
0.270827 + 0.962628i $$0.412703\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 0 0
$$179$$ 2.53590 0.189542 0.0947710 0.995499i $$-0.469788\pi$$
0.0947710 + 0.995499i $$0.469788\pi$$
$$180$$ 0 0
$$181$$ 11.8564 0.881280 0.440640 0.897684i $$-0.354752\pi$$
0.440640 + 0.897684i $$0.354752\pi$$
$$182$$ 0 0
$$183$$ −9.85641 −0.728607
$$184$$ 0 0
$$185$$ 9.66025 0.710236
$$186$$ 0 0
$$187$$ −1.07180 −0.0783775
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2.92820 −0.211877 −0.105939 0.994373i $$-0.533785\pi$$
−0.105939 + 0.994373i $$0.533785\pi$$
$$192$$ 0 0
$$193$$ 3.80385 0.273807 0.136903 0.990584i $$-0.456285\pi$$
0.136903 + 0.990584i $$0.456285\pi$$
$$194$$ 0 0
$$195$$ −2.00000 −0.143223
$$196$$ 0 0
$$197$$ −15.8564 −1.12972 −0.564861 0.825186i $$-0.691070\pi$$
−0.564861 + 0.825186i $$0.691070\pi$$
$$198$$ 0 0
$$199$$ 5.07180 0.359530 0.179765 0.983710i $$-0.442466\pi$$
0.179765 + 0.983710i $$0.442466\pi$$
$$200$$ 0 0
$$201$$ −4.53590 −0.319938
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −7.46410 −0.521315
$$206$$ 0 0
$$207$$ 13.4641 0.935820
$$208$$ 0 0
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ −4.39230 −0.302379 −0.151189 0.988505i $$-0.548310\pi$$
−0.151189 + 0.988505i $$0.548310\pi$$
$$212$$ 0 0
$$213$$ 2.14359 0.146877
$$214$$ 0 0
$$215$$ 10.9282 0.745297
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 7.60770 0.514080
$$220$$ 0 0
$$221$$ 1.46410 0.0984861
$$222$$ 0 0
$$223$$ 12.0526 0.807099 0.403550 0.914958i $$-0.367776\pi$$
0.403550 + 0.914958i $$0.367776\pi$$
$$224$$ 0 0
$$225$$ −2.46410 −0.164273
$$226$$ 0 0
$$227$$ −4.73205 −0.314077 −0.157039 0.987592i $$-0.550195\pi$$
−0.157039 + 0.987592i $$0.550195\pi$$
$$228$$ 0 0
$$229$$ −1.46410 −0.0967506 −0.0483753 0.998829i $$-0.515404\pi$$
−0.0483753 + 0.998829i $$0.515404\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ 10.9282 0.712877
$$236$$ 0 0
$$237$$ −9.07180 −0.589277
$$238$$ 0 0
$$239$$ 5.07180 0.328067 0.164034 0.986455i $$-0.447549\pi$$
0.164034 + 0.986455i $$0.447549\pi$$
$$240$$ 0 0
$$241$$ 10.3923 0.669427 0.334714 0.942320i $$-0.391360\pi$$
0.334714 + 0.942320i $$0.391360\pi$$
$$242$$ 0 0
$$243$$ 15.2679 0.979439
$$244$$ 0 0
$$245$$ 7.00000 0.447214
$$246$$ 0 0
$$247$$ −2.73205 −0.173836
$$248$$ 0 0
$$249$$ −1.07180 −0.0679224
$$250$$ 0 0
$$251$$ −25.8564 −1.63204 −0.816021 0.578022i $$-0.803825\pi$$
−0.816021 + 0.578022i $$0.803825\pi$$
$$252$$ 0 0
$$253$$ 10.9282 0.687050
$$254$$ 0 0
$$255$$ −0.392305 −0.0245671
$$256$$ 0 0
$$257$$ 21.2679 1.32666 0.663329 0.748328i $$-0.269143\pi$$
0.663329 + 0.748328i $$0.269143\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −8.53590 −0.528359
$$262$$ 0 0
$$263$$ 15.3205 0.944703 0.472351 0.881410i $$-0.343405\pi$$
0.472351 + 0.881410i $$0.343405\pi$$
$$264$$ 0 0
$$265$$ −5.66025 −0.347707
$$266$$ 0 0
$$267$$ 2.53590 0.155194
$$268$$ 0 0
$$269$$ 4.92820 0.300478 0.150239 0.988650i $$-0.451996\pi$$
0.150239 + 0.988650i $$0.451996\pi$$
$$270$$ 0 0
$$271$$ −14.0000 −0.850439 −0.425220 0.905090i $$-0.639803\pi$$
−0.425220 + 0.905090i $$0.639803\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ −27.4641 −1.65016 −0.825079 0.565017i $$-0.808869\pi$$
−0.825079 + 0.565017i $$0.808869\pi$$
$$278$$ 0 0
$$279$$ 9.85641 0.590088
$$280$$ 0 0
$$281$$ −1.60770 −0.0959071 −0.0479535 0.998850i $$-0.515270\pi$$
−0.0479535 + 0.998850i $$0.515270\pi$$
$$282$$ 0 0
$$283$$ 2.53590 0.150744 0.0753718 0.997156i $$-0.475986\pi$$
0.0753718 + 0.997156i $$0.475986\pi$$
$$284$$ 0 0
$$285$$ 0.732051 0.0433629
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.7128 −0.983107
$$290$$ 0 0
$$291$$ 1.21539 0.0712474
$$292$$ 0 0
$$293$$ 7.80385 0.455906 0.227953 0.973672i $$-0.426797\pi$$
0.227953 + 0.973672i $$0.426797\pi$$
$$294$$ 0 0
$$295$$ −5.46410 −0.318132
$$296$$ 0 0
$$297$$ 8.00000 0.464207
$$298$$ 0 0
$$299$$ −14.9282 −0.863320
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −4.00000 −0.229794
$$304$$ 0 0
$$305$$ 13.4641 0.770952
$$306$$ 0 0
$$307$$ −28.7321 −1.63982 −0.819912 0.572489i $$-0.805978\pi$$
−0.819912 + 0.572489i $$0.805978\pi$$
$$308$$ 0 0
$$309$$ 8.53590 0.485590
$$310$$ 0 0
$$311$$ 7.85641 0.445496 0.222748 0.974876i $$-0.428497\pi$$
0.222748 + 0.974876i $$0.428497\pi$$
$$312$$ 0 0
$$313$$ 11.8564 0.670164 0.335082 0.942189i $$-0.391236\pi$$
0.335082 + 0.942189i $$0.391236\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 24.5885 1.38103 0.690513 0.723320i $$-0.257385\pi$$
0.690513 + 0.723320i $$0.257385\pi$$
$$318$$ 0 0
$$319$$ −6.92820 −0.387905
$$320$$ 0 0
$$321$$ 6.67949 0.372813
$$322$$ 0 0
$$323$$ −0.535898 −0.0298182
$$324$$ 0 0
$$325$$ 2.73205 0.151547
$$326$$ 0 0
$$327$$ −10.2487 −0.566755
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 34.2487 1.88248 0.941240 0.337739i $$-0.109662\pi$$
0.941240 + 0.337739i $$0.109662\pi$$
$$332$$ 0 0
$$333$$ 23.8038 1.30444
$$334$$ 0 0
$$335$$ 6.19615 0.338532
$$336$$ 0 0
$$337$$ −5.66025 −0.308334 −0.154167 0.988045i $$-0.549269\pi$$
−0.154167 + 0.988045i $$0.549269\pi$$
$$338$$ 0 0
$$339$$ 9.71281 0.527528
$$340$$ 0 0
$$341$$ 8.00000 0.433224
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 4.00000 0.215353
$$346$$ 0 0
$$347$$ −12.3923 −0.665254 −0.332627 0.943059i $$-0.607935\pi$$
−0.332627 + 0.943059i $$0.607935\pi$$
$$348$$ 0 0
$$349$$ 18.7846 1.00552 0.502759 0.864427i $$-0.332318\pi$$
0.502759 + 0.864427i $$0.332318\pi$$
$$350$$ 0 0
$$351$$ −10.9282 −0.583304
$$352$$ 0 0
$$353$$ −16.2487 −0.864832 −0.432416 0.901674i $$-0.642339\pi$$
−0.432416 + 0.901674i $$0.642339\pi$$
$$354$$ 0 0
$$355$$ −2.92820 −0.155413
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4.92820 0.260101 0.130050 0.991507i $$-0.458486\pi$$
0.130050 + 0.991507i $$0.458486\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −5.12436 −0.268959
$$364$$ 0 0
$$365$$ −10.3923 −0.543958
$$366$$ 0 0
$$367$$ 2.92820 0.152851 0.0764255 0.997075i $$-0.475649\pi$$
0.0764255 + 0.997075i $$0.475649\pi$$
$$368$$ 0 0
$$369$$ −18.3923 −0.957465
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −0.196152 −0.0101564 −0.00507819 0.999987i $$-0.501616\pi$$
−0.00507819 + 0.999987i $$0.501616\pi$$
$$374$$ 0 0
$$375$$ −0.732051 −0.0378029
$$376$$ 0 0
$$377$$ 9.46410 0.487426
$$378$$ 0 0
$$379$$ −34.6410 −1.77939 −0.889695 0.456556i $$-0.849083\pi$$
−0.889695 + 0.456556i $$0.849083\pi$$
$$380$$ 0 0
$$381$$ −0.248711 −0.0127419
$$382$$ 0 0
$$383$$ −21.8038 −1.11412 −0.557062 0.830471i $$-0.688072\pi$$
−0.557062 + 0.830471i $$0.688072\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 26.9282 1.36884
$$388$$ 0 0
$$389$$ 11.8564 0.601144 0.300572 0.953759i $$-0.402822\pi$$
0.300572 + 0.953759i $$0.402822\pi$$
$$390$$ 0 0
$$391$$ −2.92820 −0.148086
$$392$$ 0 0
$$393$$ −8.78461 −0.443125
$$394$$ 0 0
$$395$$ 12.3923 0.623525
$$396$$ 0 0
$$397$$ −12.5359 −0.629159 −0.314579 0.949231i $$-0.601863\pi$$
−0.314579 + 0.949231i $$0.601863\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ 0 0
$$403$$ −10.9282 −0.544373
$$404$$ 0 0
$$405$$ −4.46410 −0.221823
$$406$$ 0 0
$$407$$ 19.3205 0.957682
$$408$$ 0 0
$$409$$ 28.9282 1.43041 0.715204 0.698916i $$-0.246334\pi$$
0.715204 + 0.698916i $$0.246334\pi$$
$$410$$ 0 0
$$411$$ 5.46410 0.269524
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.46410 0.0718699
$$416$$ 0 0
$$417$$ 14.5359 0.711826
$$418$$ 0 0
$$419$$ 1.85641 0.0906914 0.0453457 0.998971i $$-0.485561\pi$$
0.0453457 + 0.998971i $$0.485561\pi$$
$$420$$ 0 0
$$421$$ −32.2487 −1.57171 −0.785853 0.618413i $$-0.787776\pi$$
−0.785853 + 0.618413i $$0.787776\pi$$
$$422$$ 0 0
$$423$$ 26.9282 1.30929
$$424$$ 0 0
$$425$$ 0.535898 0.0259949
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ −7.60770 −0.366450 −0.183225 0.983071i $$-0.558654\pi$$
−0.183225 + 0.983071i $$0.558654\pi$$
$$432$$ 0 0
$$433$$ 24.9808 1.20050 0.600249 0.799813i $$-0.295068\pi$$
0.600249 + 0.799813i $$0.295068\pi$$
$$434$$ 0 0
$$435$$ −2.53590 −0.121587
$$436$$ 0 0
$$437$$ 5.46410 0.261383
$$438$$ 0 0
$$439$$ 22.2487 1.06187 0.530937 0.847412i $$-0.321840\pi$$
0.530937 + 0.847412i $$0.321840\pi$$
$$440$$ 0 0
$$441$$ 17.2487 0.821367
$$442$$ 0 0
$$443$$ 26.5359 1.26076 0.630379 0.776287i $$-0.282899\pi$$
0.630379 + 0.776287i $$0.282899\pi$$
$$444$$ 0 0
$$445$$ −3.46410 −0.164214
$$446$$ 0 0
$$447$$ −14.1436 −0.668969
$$448$$ 0 0
$$449$$ 32.6410 1.54042 0.770212 0.637787i $$-0.220150\pi$$
0.770212 + 0.637787i $$0.220150\pi$$
$$450$$ 0 0
$$451$$ −14.9282 −0.702942
$$452$$ 0 0
$$453$$ −1.07180 −0.0503574
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 15.0718 0.705029 0.352514 0.935806i $$-0.385327\pi$$
0.352514 + 0.935806i $$0.385327\pi$$
$$458$$ 0 0
$$459$$ −2.14359 −0.100054
$$460$$ 0 0
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 0 0
$$463$$ 38.2487 1.77757 0.888784 0.458326i $$-0.151551\pi$$
0.888784 + 0.458326i $$0.151551\pi$$
$$464$$ 0 0
$$465$$ 2.92820 0.135792
$$466$$ 0 0
$$467$$ 0.679492 0.0314431 0.0157216 0.999876i $$-0.494995\pi$$
0.0157216 + 0.999876i $$0.494995\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −0.392305 −0.0180765
$$472$$ 0 0
$$473$$ 21.8564 1.00496
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ −13.9474 −0.638609
$$478$$ 0 0
$$479$$ 7.85641 0.358968 0.179484 0.983761i $$-0.442557\pi$$
0.179484 + 0.983761i $$0.442557\pi$$
$$480$$ 0 0
$$481$$ −26.3923 −1.20339
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1.66025 −0.0753883
$$486$$ 0 0
$$487$$ −39.2679 −1.77940 −0.889700 0.456545i $$-0.849087\pi$$
−0.889700 + 0.456545i $$0.849087\pi$$
$$488$$ 0 0
$$489$$ 2.92820 0.132418
$$490$$ 0 0
$$491$$ −25.8564 −1.16688 −0.583442 0.812155i $$-0.698294\pi$$
−0.583442 + 0.812155i $$0.698294\pi$$
$$492$$ 0 0
$$493$$ 1.85641 0.0836083
$$494$$ 0 0
$$495$$ −4.92820 −0.221506
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 10.7846 0.482785 0.241393 0.970428i $$-0.422396\pi$$
0.241393 + 0.970428i $$0.422396\pi$$
$$500$$ 0 0
$$501$$ 2.39230 0.106880
$$502$$ 0 0
$$503$$ 11.3205 0.504757 0.252378 0.967629i $$-0.418787\pi$$
0.252378 + 0.967629i $$0.418787\pi$$
$$504$$ 0 0
$$505$$ 5.46410 0.243149
$$506$$ 0 0
$$507$$ −4.05256 −0.179980
$$508$$ 0 0
$$509$$ −18.3923 −0.815225 −0.407612 0.913155i $$-0.633639\pi$$
−0.407612 + 0.913155i $$0.633639\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ −11.6603 −0.513812
$$516$$ 0 0
$$517$$ 21.8564 0.961244
$$518$$ 0 0
$$519$$ 5.21539 0.228930
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 15.2679 0.667621 0.333810 0.942640i $$-0.391665\pi$$
0.333810 + 0.942640i $$0.391665\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.14359 −0.0933764
$$528$$ 0 0
$$529$$ 6.85641 0.298105
$$530$$ 0 0
$$531$$ −13.4641 −0.584292
$$532$$ 0 0
$$533$$ 20.3923 0.883289
$$534$$ 0 0
$$535$$ −9.12436 −0.394480
$$536$$ 0 0
$$537$$ 1.85641 0.0801099
$$538$$ 0 0
$$539$$ 14.0000 0.603023
$$540$$ 0 0
$$541$$ −44.3923 −1.90857 −0.954287 0.298891i $$-0.903383\pi$$
−0.954287 + 0.298891i $$0.903383\pi$$
$$542$$ 0 0
$$543$$ 8.67949 0.372473
$$544$$ 0 0
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ 14.5885 0.623757 0.311879 0.950122i $$-0.399042\pi$$
0.311879 + 0.950122i $$0.399042\pi$$
$$548$$ 0 0
$$549$$ 33.1769 1.41596
$$550$$ 0 0
$$551$$ −3.46410 −0.147576
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 7.07180 0.300181
$$556$$ 0 0
$$557$$ −36.2487 −1.53591 −0.767954 0.640505i $$-0.778725\pi$$
−0.767954 + 0.640505i $$0.778725\pi$$
$$558$$ 0 0
$$559$$ −29.8564 −1.26279
$$560$$ 0 0
$$561$$ −0.784610 −0.0331262
$$562$$ 0 0
$$563$$ −22.5885 −0.951990 −0.475995 0.879448i $$-0.657912\pi$$
−0.475995 + 0.879448i $$0.657912\pi$$
$$564$$ 0 0
$$565$$ −13.2679 −0.558187
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −26.7846 −1.12287 −0.561435 0.827521i $$-0.689750\pi$$
−0.561435 + 0.827521i $$0.689750\pi$$
$$570$$ 0 0
$$571$$ −10.7846 −0.451322 −0.225661 0.974206i $$-0.572454\pi$$
−0.225661 + 0.974206i $$0.572454\pi$$
$$572$$ 0 0
$$573$$ −2.14359 −0.0895499
$$574$$ 0 0
$$575$$ −5.46410 −0.227869
$$576$$ 0 0
$$577$$ 8.14359 0.339022 0.169511 0.985528i $$-0.445781\pi$$
0.169511 + 0.985528i $$0.445781\pi$$
$$578$$ 0 0
$$579$$ 2.78461 0.115724
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −11.3205 −0.468848
$$584$$ 0 0
$$585$$ 6.73205 0.278336
$$586$$ 0 0
$$587$$ 40.1051 1.65532 0.827658 0.561233i $$-0.189673\pi$$
0.827658 + 0.561233i $$0.189673\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ −11.6077 −0.477477
$$592$$ 0 0
$$593$$ −8.14359 −0.334417 −0.167209 0.985922i $$-0.553475\pi$$
−0.167209 + 0.985922i $$0.553475\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 3.71281 0.151955
$$598$$ 0 0
$$599$$ −47.0333 −1.92173 −0.960865 0.277018i $$-0.910654\pi$$
−0.960865 + 0.277018i $$0.910654\pi$$
$$600$$ 0 0
$$601$$ −43.4641 −1.77294 −0.886469 0.462788i $$-0.846849\pi$$
−0.886469 + 0.462788i $$0.846849\pi$$
$$602$$ 0 0
$$603$$ 15.2679 0.621759
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 0 0
$$607$$ 20.4449 0.829831 0.414916 0.909860i $$-0.363811\pi$$
0.414916 + 0.909860i $$0.363811\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −29.8564 −1.20786
$$612$$ 0 0
$$613$$ −4.24871 −0.171604 −0.0858019 0.996312i $$-0.527345\pi$$
−0.0858019 + 0.996312i $$0.527345\pi$$
$$614$$ 0 0
$$615$$ −5.46410 −0.220334
$$616$$ 0 0
$$617$$ −3.46410 −0.139459 −0.0697297 0.997566i $$-0.522214\pi$$
−0.0697297 + 0.997566i $$0.522214\pi$$
$$618$$ 0 0
$$619$$ 14.0000 0.562708 0.281354 0.959604i $$-0.409217\pi$$
0.281354 + 0.959604i $$0.409217\pi$$
$$620$$ 0 0
$$621$$ 21.8564 0.877067
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 1.46410 0.0584706
$$628$$ 0 0
$$629$$ −5.17691 −0.206417
$$630$$ 0 0
$$631$$ 48.6410 1.93637 0.968184 0.250239i $$-0.0805091\pi$$
0.968184 + 0.250239i $$0.0805091\pi$$
$$632$$ 0 0
$$633$$ −3.21539 −0.127800
$$634$$ 0 0
$$635$$ 0.339746 0.0134824
$$636$$ 0 0
$$637$$ −19.1244 −0.757735
$$638$$ 0 0
$$639$$ −7.21539 −0.285436
$$640$$ 0 0
$$641$$ −45.0333 −1.77871 −0.889355 0.457218i $$-0.848846\pi$$
−0.889355 + 0.457218i $$0.848846\pi$$
$$642$$ 0 0
$$643$$ −50.2487 −1.98162 −0.990808 0.135277i $$-0.956808\pi$$
−0.990808 + 0.135277i $$0.956808\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ 0 0
$$647$$ 12.6795 0.498482 0.249241 0.968441i $$-0.419819\pi$$
0.249241 + 0.968441i $$0.419819\pi$$
$$648$$ 0 0
$$649$$ −10.9282 −0.428969
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −13.6077 −0.532510 −0.266255 0.963903i $$-0.585786\pi$$
−0.266255 + 0.963903i $$0.585786\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ 0 0
$$657$$ −25.6077 −0.999051
$$658$$ 0 0
$$659$$ −11.6077 −0.452172 −0.226086 0.974107i $$-0.572593\pi$$
−0.226086 + 0.974107i $$0.572593\pi$$
$$660$$ 0 0
$$661$$ −23.4641 −0.912648 −0.456324 0.889814i $$-0.650834\pi$$
−0.456324 + 0.889814i $$0.650834\pi$$
$$662$$ 0 0
$$663$$ 1.07180 0.0416251
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −18.9282 −0.732903
$$668$$ 0 0
$$669$$ 8.82309 0.341120
$$670$$ 0 0
$$671$$ 26.9282 1.03955
$$672$$ 0 0
$$673$$ −44.1962 −1.70364 −0.851818 0.523837i $$-0.824500\pi$$
−0.851818 + 0.523837i $$0.824500\pi$$
$$674$$ 0 0
$$675$$ −4.00000 −0.153960
$$676$$ 0 0
$$677$$ −35.9090 −1.38009 −0.690047 0.723765i $$-0.742410\pi$$
−0.690047 + 0.723765i $$0.742410\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −3.46410 −0.132745
$$682$$ 0 0
$$683$$ 26.1962 1.00237 0.501184 0.865341i $$-0.332898\pi$$
0.501184 + 0.865341i $$0.332898\pi$$
$$684$$ 0 0
$$685$$ −7.46410 −0.285189
$$686$$ 0 0
$$687$$ −1.07180 −0.0408916
$$688$$ 0 0
$$689$$ 15.4641 0.589135
$$690$$ 0 0
$$691$$ −2.78461 −0.105932 −0.0529658 0.998596i $$-0.516867\pi$$
−0.0529658 + 0.998596i $$0.516867\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −19.8564 −0.753196
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 0 0
$$699$$ 10.2487 0.387642
$$700$$ 0 0
$$701$$ 26.2487 0.991400 0.495700 0.868494i $$-0.334912\pi$$
0.495700 + 0.868494i $$0.334912\pi$$
$$702$$ 0 0
$$703$$ 9.66025 0.364343
$$704$$ 0 0
$$705$$ 8.00000 0.301297
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −22.7846 −0.855694 −0.427847 0.903851i $$-0.640728\pi$$
−0.427847 + 0.903851i $$0.640728\pi$$
$$710$$ 0 0
$$711$$ 30.5359 1.14519
$$712$$ 0 0
$$713$$ 21.8564 0.818529
$$714$$ 0 0
$$715$$ 5.46410 0.204346
$$716$$ 0 0
$$717$$ 3.71281 0.138658
$$718$$ 0 0
$$719$$ −35.5692 −1.32651 −0.663254 0.748394i $$-0.730825\pi$$
−0.663254 + 0.748394i $$0.730825\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 7.60770 0.282933
$$724$$ 0 0
$$725$$ 3.46410 0.128654
$$726$$ 0 0
$$727$$ −34.9282 −1.29542 −0.647708 0.761889i $$-0.724272\pi$$
−0.647708 + 0.761889i $$0.724272\pi$$
$$728$$ 0 0
$$729$$ −2.21539 −0.0820515
$$730$$ 0 0
$$731$$ −5.85641 −0.216607
$$732$$ 0 0
$$733$$ −43.8564 −1.61987 −0.809937 0.586517i $$-0.800499\pi$$
−0.809937 + 0.586517i $$0.800499\pi$$
$$734$$ 0 0
$$735$$ 5.12436 0.189015
$$736$$ 0 0
$$737$$ 12.3923 0.456476
$$738$$ 0 0
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 0 0
$$743$$ 43.3731 1.59120 0.795602 0.605820i $$-0.207155\pi$$
0.795602 + 0.605820i $$0.207155\pi$$
$$744$$ 0 0
$$745$$ 19.3205 0.707849
$$746$$ 0 0
$$747$$ 3.60770 0.131999
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −18.5359 −0.676385 −0.338192 0.941077i $$-0.609815\pi$$
−0.338192 + 0.941077i $$0.609815\pi$$
$$752$$ 0 0
$$753$$ −18.9282 −0.689782
$$754$$ 0 0
$$755$$ 1.46410 0.0532841
$$756$$ 0 0
$$757$$ −42.7846 −1.55503 −0.777517 0.628862i $$-0.783521\pi$$
−0.777517 + 0.628862i $$0.783521\pi$$
$$758$$ 0 0
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ 3.60770 0.130779 0.0653894 0.997860i $$-0.479171\pi$$
0.0653894 + 0.997860i $$0.479171\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 1.32051 0.0477431
$$766$$ 0 0
$$767$$ 14.9282 0.539026
$$768$$ 0 0
$$769$$ 3.60770 0.130097 0.0650484 0.997882i $$-0.479280\pi$$
0.0650484 + 0.997882i $$0.479280\pi$$
$$770$$ 0 0
$$771$$ 15.5692 0.560712
$$772$$ 0 0
$$773$$ −0.196152 −0.00705511 −0.00352756 0.999994i $$-0.501123\pi$$
−0.00352756 + 0.999994i $$0.501123\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −7.46410 −0.267429
$$780$$ 0 0
$$781$$ −5.85641 −0.209559
$$782$$ 0 0
$$783$$ −13.8564 −0.495188
$$784$$ 0 0
$$785$$ 0.535898 0.0191270
$$786$$ 0 0
$$787$$ −10.1962 −0.363454 −0.181727 0.983349i $$-0.558169\pi$$
−0.181727 + 0.983349i $$0.558169\pi$$
$$788$$ 0 0
$$789$$ 11.2154 0.399278
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −36.7846 −1.30626
$$794$$ 0 0
$$795$$ −4.14359 −0.146958
$$796$$ 0 0
$$797$$ −22.0526 −0.781142 −0.390571 0.920573i $$-0.627722\pi$$
−0.390571 + 0.920573i $$0.627722\pi$$
$$798$$ 0 0
$$799$$ −5.85641 −0.207185
$$800$$ 0 0
$$801$$ −8.53590 −0.301601
$$802$$ 0 0
$$803$$ −20.7846 −0.733473
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 3.60770 0.126997
$$808$$ 0 0
$$809$$ −45.7128 −1.60718 −0.803588 0.595185i $$-0.797079\pi$$
−0.803588 + 0.595185i $$0.797079\pi$$
$$810$$ 0 0
$$811$$ 13.8564 0.486564 0.243282 0.969956i $$-0.421776\pi$$
0.243282 + 0.969956i $$0.421776\pi$$
$$812$$ 0 0
$$813$$ −10.2487 −0.359438
$$814$$ 0 0
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 10.9282 0.382329
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.7846 −0.655587 −0.327794 0.944749i $$-0.606305\pi$$
−0.327794 + 0.944749i $$0.606305\pi$$
$$822$$ 0 0
$$823$$ 13.0718 0.455654 0.227827 0.973702i $$-0.426838\pi$$
0.227827 + 0.973702i $$0.426838\pi$$
$$824$$ 0 0
$$825$$ −1.46410 −0.0509735
$$826$$ 0 0
$$827$$ 29.9090 1.04004 0.520018 0.854155i $$-0.325925\pi$$
0.520018 + 0.854155i $$0.325925\pi$$
$$828$$ 0 0
$$829$$ 21.7128 0.754117 0.377059 0.926189i $$-0.376936\pi$$
0.377059 + 0.926189i $$0.376936\pi$$
$$830$$ 0 0
$$831$$ −20.1051 −0.697439
$$832$$ 0 0
$$833$$ −3.75129 −0.129974
$$834$$ 0 0
$$835$$ −3.26795 −0.113092
$$836$$ 0 0
$$837$$ 16.0000 0.553041
$$838$$ 0 0
$$839$$ −44.3923 −1.53259 −0.766296 0.642487i $$-0.777903\pi$$
−0.766296 + 0.642487i $$0.777903\pi$$
$$840$$ 0 0
$$841$$ −17.0000 −0.586207
$$842$$ 0 0
$$843$$ −1.17691 −0.0405351
$$844$$ 0 0
$$845$$ 5.53590 0.190441
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 1.85641 0.0637117
$$850$$ 0 0
$$851$$ 52.7846 1.80943
$$852$$ 0 0
$$853$$ −26.7846 −0.917088 −0.458544 0.888672i $$-0.651629\pi$$
−0.458544 + 0.888672i $$0.651629\pi$$
$$854$$ 0 0
$$855$$ −2.46410 −0.0842705
$$856$$ 0 0
$$857$$ −4.87564 −0.166549 −0.0832744 0.996527i $$-0.526538\pi$$
−0.0832744 + 0.996527i $$0.526538\pi$$
$$858$$ 0 0
$$859$$ −57.8564 −1.97404 −0.987018 0.160612i $$-0.948653\pi$$
−0.987018 + 0.160612i $$0.948653\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 28.7321 0.978050 0.489025 0.872270i $$-0.337353\pi$$
0.489025 + 0.872270i $$0.337353\pi$$
$$864$$ 0 0
$$865$$ −7.12436 −0.242235
$$866$$ 0 0
$$867$$ −12.2346 −0.415510
$$868$$ 0 0
$$869$$ 24.7846 0.840760
$$870$$ 0 0
$$871$$ −16.9282 −0.573590
$$872$$ 0 0
$$873$$ −4.09103 −0.138461
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 46.4449 1.56833 0.784166 0.620551i $$-0.213091\pi$$
0.784166 + 0.620551i $$0.213091\pi$$
$$878$$ 0 0
$$879$$ 5.71281 0.192688
$$880$$ 0 0
$$881$$ −8.39230 −0.282744 −0.141372 0.989957i $$-0.545151\pi$$
−0.141372 + 0.989957i $$0.545151\pi$$
$$882$$ 0 0
$$883$$ 30.6410 1.03115 0.515576 0.856844i $$-0.327578\pi$$
0.515576 + 0.856844i $$0.327578\pi$$
$$884$$ 0 0
$$885$$ −4.00000 −0.134459
$$886$$ 0 0
$$887$$ 45.1244 1.51513 0.757564 0.652761i $$-0.226389\pi$$
0.757564 + 0.652761i $$0.226389\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −8.92820 −0.299106
$$892$$ 0 0
$$893$$ 10.9282 0.365698
$$894$$ 0 0
$$895$$ −2.53590 −0.0847657
$$896$$ 0 0
$$897$$ −10.9282 −0.364882
$$898$$ 0 0
$$899$$ −13.8564 −0.462137
$$900$$ 0 0
$$901$$ 3.03332 0.101055
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −11.8564 −0.394120
$$906$$ 0 0
$$907$$ −24.4449 −0.811678 −0.405839 0.913945i $$-0.633021\pi$$
−0.405839 + 0.913945i $$0.633021\pi$$
$$908$$ 0 0
$$909$$ 13.4641 0.446576
$$910$$ 0 0
$$911$$ −42.5359 −1.40928 −0.704639 0.709566i $$-0.748891\pi$$
−0.704639 + 0.709566i $$0.748891\pi$$
$$912$$ 0 0
$$913$$ 2.92820 0.0969094
$$914$$ 0 0
$$915$$ 9.85641 0.325843
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 10.9282 0.360488 0.180244 0.983622i $$-0.442311\pi$$
0.180244 + 0.983622i $$0.442311\pi$$
$$920$$ 0 0
$$921$$ −21.0333 −0.693071
$$922$$ 0 0
$$923$$ 8.00000 0.263323
$$924$$ 0 0
$$925$$ −9.66025 −0.317627
$$926$$ 0 0
$$927$$ −28.7321 −0.943684
$$928$$ 0 0
$$929$$ −50.0000 −1.64045 −0.820223 0.572043i $$-0.806151\pi$$
−0.820223 + 0.572043i $$0.806151\pi$$
$$930$$ 0 0
$$931$$ 7.00000 0.229416
$$932$$ 0 0
$$933$$ 5.75129 0.188289
$$934$$ 0 0
$$935$$ 1.07180 0.0350515
$$936$$ 0 0
$$937$$ 50.3923 1.64624 0.823122 0.567864i $$-0.192230\pi$$
0.823122 + 0.567864i $$0.192230\pi$$
$$938$$ 0 0
$$939$$ 8.67949 0.283245
$$940$$ 0 0
$$941$$ 48.9282 1.59501 0.797507 0.603310i $$-0.206152\pi$$
0.797507 + 0.603310i $$0.206152\pi$$
$$942$$ 0 0
$$943$$ −40.7846 −1.32813
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.75129 0.0569092 0.0284546 0.999595i $$-0.490941\pi$$
0.0284546 + 0.999595i $$0.490941\pi$$
$$948$$ 0 0
$$949$$ 28.3923 0.921653
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ −48.5885 −1.57393 −0.786967 0.616995i $$-0.788350\pi$$
−0.786967 + 0.616995i $$0.788350\pi$$
$$954$$ 0 0
$$955$$ 2.92820 0.0947544
$$956$$ 0 0
$$957$$ −5.07180 −0.163948
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −22.4833 −0.724515
$$964$$ 0 0
$$965$$ −3.80385 −0.122450
$$966$$ 0 0
$$967$$ −5.17691 −0.166478 −0.0832392 0.996530i $$-0.526527\pi$$
−0.0832392 + 0.996530i $$0.526527\pi$$
$$968$$ 0 0
$$969$$ −0.392305 −0.0126026
$$970$$ 0 0
$$971$$ 6.92820 0.222337 0.111168 0.993802i $$-0.464541\pi$$
0.111168 + 0.993802i $$0.464541\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 2.00000 0.0640513
$$976$$ 0 0
$$977$$ 50.4449 1.61387 0.806937 0.590637i $$-0.201124\pi$$
0.806937 + 0.590637i $$0.201124\pi$$
$$978$$ 0 0
$$979$$ −6.92820 −0.221426
$$980$$ 0 0
$$981$$ 34.4974 1.10142
$$982$$ 0 0
$$983$$ −41.5167 −1.32418 −0.662088 0.749426i $$-0.730329\pi$$
−0.662088 + 0.749426i $$0.730329\pi$$
$$984$$ 0 0
$$985$$ 15.8564 0.505227
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 59.7128 1.89876
$$990$$ 0 0
$$991$$ −31.3205 −0.994929 −0.497464 0.867484i $$-0.665735\pi$$
−0.497464 + 0.867484i $$0.665735\pi$$
$$992$$ 0 0
$$993$$ 25.0718 0.795629
$$994$$ 0 0
$$995$$ −5.07180 −0.160787
$$996$$ 0 0
$$997$$ −17.6077 −0.557641 −0.278821 0.960343i $$-0.589944\pi$$
−0.278821 + 0.960343i $$0.589944\pi$$
$$998$$ 0 0
$$999$$ 38.6410 1.22255
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.k.1.2 2
4.3 odd 2 760.2.a.g.1.1 2
5.4 even 2 7600.2.a.bd.1.1 2
8.3 odd 2 6080.2.a.ba.1.2 2
8.5 even 2 6080.2.a.bk.1.1 2
12.11 even 2 6840.2.a.y.1.2 2
20.3 even 4 3800.2.d.h.3649.2 4
20.7 even 4 3800.2.d.h.3649.3 4
20.19 odd 2 3800.2.a.j.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.g.1.1 2 4.3 odd 2
1520.2.a.k.1.2 2 1.1 even 1 trivial
3800.2.a.j.1.2 2 20.19 odd 2
3800.2.d.h.3649.2 4 20.3 even 4
3800.2.d.h.3649.3 4 20.7 even 4
6080.2.a.ba.1.2 2 8.3 odd 2
6080.2.a.bk.1.1 2 8.5 even 2
6840.2.a.y.1.2 2 12.11 even 2
7600.2.a.bd.1.1 2 5.4 even 2