# Properties

 Label 1425.2.a.o.1.2 Level $1425$ Weight $2$ Character 1425.1 Self dual yes Analytic conductor $11.379$ 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] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.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: $$11.3786822880$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +2.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +2.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} +4.73205 q^{11} -1.00000 q^{12} -0.732051 q^{13} +4.73205 q^{14} -5.00000 q^{16} +1.73205 q^{18} +1.00000 q^{19} -2.73205 q^{21} +8.19615 q^{22} -3.46410 q^{23} +1.73205 q^{24} -1.26795 q^{26} -1.00000 q^{27} +2.73205 q^{28} +8.19615 q^{29} +8.92820 q^{31} -5.19615 q^{32} -4.73205 q^{33} +1.00000 q^{36} +6.19615 q^{37} +1.73205 q^{38} +0.732051 q^{39} +1.26795 q^{41} -4.73205 q^{42} -4.19615 q^{43} +4.73205 q^{44} -6.00000 q^{46} +3.46410 q^{47} +5.00000 q^{48} +0.464102 q^{49} -0.732051 q^{52} +9.46410 q^{53} -1.73205 q^{54} -4.73205 q^{56} -1.00000 q^{57} +14.1962 q^{58} +2.53590 q^{59} -6.53590 q^{61} +15.4641 q^{62} +2.73205 q^{63} +1.00000 q^{64} -8.19615 q^{66} -8.00000 q^{67} +3.46410 q^{69} -4.39230 q^{71} -1.73205 q^{72} +16.9282 q^{73} +10.7321 q^{74} +1.00000 q^{76} +12.9282 q^{77} +1.26795 q^{78} -10.9282 q^{79} +1.00000 q^{81} +2.19615 q^{82} +12.9282 q^{83} -2.73205 q^{84} -7.26795 q^{86} -8.19615 q^{87} -8.19615 q^{88} +10.7321 q^{89} -2.00000 q^{91} -3.46410 q^{92} -8.92820 q^{93} +6.00000 q^{94} +5.19615 q^{96} +6.19615 q^{97} +0.803848 q^{98} +4.73205 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} + 6 q^{14} - 10 q^{16} + 2 q^{19} - 2 q^{21} + 6 q^{22} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 6 q^{29} + 4 q^{31} - 6 q^{33} + 2 q^{36} + 2 q^{37} - 2 q^{39} + 6 q^{41} - 6 q^{42} + 2 q^{43} + 6 q^{44} - 12 q^{46} + 10 q^{48} - 6 q^{49} + 2 q^{52} + 12 q^{53} - 6 q^{56} - 2 q^{57} + 18 q^{58} + 12 q^{59} - 20 q^{61} + 24 q^{62} + 2 q^{63} + 2 q^{64} - 6 q^{66} - 16 q^{67} + 12 q^{71} + 20 q^{73} + 18 q^{74} + 2 q^{76} + 12 q^{77} + 6 q^{78} - 8 q^{79} + 2 q^{81} - 6 q^{82} + 12 q^{83} - 2 q^{84} - 18 q^{86} - 6 q^{87} - 6 q^{88} + 18 q^{89} - 4 q^{91} - 4 q^{93} + 12 q^{94} + 2 q^{97} + 12 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 2 * q^13 + 6 * q^14 - 10 * q^16 + 2 * q^19 - 2 * q^21 + 6 * q^22 - 6 * q^26 - 2 * q^27 + 2 * q^28 + 6 * q^29 + 4 * q^31 - 6 * q^33 + 2 * q^36 + 2 * q^37 - 2 * q^39 + 6 * q^41 - 6 * q^42 + 2 * q^43 + 6 * q^44 - 12 * q^46 + 10 * q^48 - 6 * q^49 + 2 * q^52 + 12 * q^53 - 6 * q^56 - 2 * q^57 + 18 * q^58 + 12 * q^59 - 20 * q^61 + 24 * q^62 + 2 * q^63 + 2 * q^64 - 6 * q^66 - 16 * q^67 + 12 * q^71 + 20 * q^73 + 18 * q^74 + 2 * q^76 + 12 * q^77 + 6 * q^78 - 8 * q^79 + 2 * q^81 - 6 * q^82 + 12 * q^83 - 2 * q^84 - 18 * q^86 - 6 * q^87 - 6 * q^88 + 18 * q^89 - 4 * q^91 - 4 * q^93 + 12 * q^94 + 2 * q^97 + 12 * q^98 + 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.73205 1.22474 0.612372 0.790569i $$-0.290215\pi$$
0.612372 + 0.790569i $$0.290215\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.73205 −0.707107
$$7$$ 2.73205 1.03262 0.516309 0.856402i $$-0.327306\pi$$
0.516309 + 0.856402i $$0.327306\pi$$
$$8$$ −1.73205 −0.612372
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.73205 1.42677 0.713384 0.700774i $$-0.247162\pi$$
0.713384 + 0.700774i $$0.247162\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −0.732051 −0.203034 −0.101517 0.994834i $$-0.532370\pi$$
−0.101517 + 0.994834i $$0.532370\pi$$
$$14$$ 4.73205 1.26469
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 1.73205 0.408248
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −2.73205 −0.596182
$$22$$ 8.19615 1.74743
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 1.73205 0.353553
$$25$$ 0 0
$$26$$ −1.26795 −0.248665
$$27$$ −1.00000 −0.192450
$$28$$ 2.73205 0.516309
$$29$$ 8.19615 1.52199 0.760994 0.648759i $$-0.224712\pi$$
0.760994 + 0.648759i $$0.224712\pi$$
$$30$$ 0 0
$$31$$ 8.92820 1.60355 0.801776 0.597624i $$-0.203889\pi$$
0.801776 + 0.597624i $$0.203889\pi$$
$$32$$ −5.19615 −0.918559
$$33$$ −4.73205 −0.823744
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.19615 1.01864 0.509321 0.860577i $$-0.329897\pi$$
0.509321 + 0.860577i $$0.329897\pi$$
$$38$$ 1.73205 0.280976
$$39$$ 0.732051 0.117222
$$40$$ 0 0
$$41$$ 1.26795 0.198020 0.0990102 0.995086i $$-0.468432\pi$$
0.0990102 + 0.995086i $$0.468432\pi$$
$$42$$ −4.73205 −0.730171
$$43$$ −4.19615 −0.639907 −0.319954 0.947433i $$-0.603667\pi$$
−0.319954 + 0.947433i $$0.603667\pi$$
$$44$$ 4.73205 0.713384
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ 3.46410 0.505291 0.252646 0.967559i $$-0.418699\pi$$
0.252646 + 0.967559i $$0.418699\pi$$
$$48$$ 5.00000 0.721688
$$49$$ 0.464102 0.0663002
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −0.732051 −0.101517
$$53$$ 9.46410 1.29999 0.649997 0.759937i $$-0.274770\pi$$
0.649997 + 0.759937i $$0.274770\pi$$
$$54$$ −1.73205 −0.235702
$$55$$ 0 0
$$56$$ −4.73205 −0.632347
$$57$$ −1.00000 −0.132453
$$58$$ 14.1962 1.86405
$$59$$ 2.53590 0.330146 0.165073 0.986281i $$-0.447214\pi$$
0.165073 + 0.986281i $$0.447214\pi$$
$$60$$ 0 0
$$61$$ −6.53590 −0.836836 −0.418418 0.908255i $$-0.637415\pi$$
−0.418418 + 0.908255i $$0.637415\pi$$
$$62$$ 15.4641 1.96394
$$63$$ 2.73205 0.344206
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −8.19615 −1.00888
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 0 0
$$69$$ 3.46410 0.417029
$$70$$ 0 0
$$71$$ −4.39230 −0.521271 −0.260635 0.965437i $$-0.583932\pi$$
−0.260635 + 0.965437i $$0.583932\pi$$
$$72$$ −1.73205 −0.204124
$$73$$ 16.9282 1.98130 0.990648 0.136441i $$-0.0435665\pi$$
0.990648 + 0.136441i $$0.0435665\pi$$
$$74$$ 10.7321 1.24758
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 12.9282 1.47331
$$78$$ 1.26795 0.143567
$$79$$ −10.9282 −1.22952 −0.614759 0.788715i $$-0.710747\pi$$
−0.614759 + 0.788715i $$0.710747\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.19615 0.242524
$$83$$ 12.9282 1.41905 0.709527 0.704678i $$-0.248908\pi$$
0.709527 + 0.704678i $$0.248908\pi$$
$$84$$ −2.73205 −0.298091
$$85$$ 0 0
$$86$$ −7.26795 −0.783723
$$87$$ −8.19615 −0.878720
$$88$$ −8.19615 −0.873713
$$89$$ 10.7321 1.13760 0.568798 0.822478i $$-0.307409\pi$$
0.568798 + 0.822478i $$0.307409\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ −3.46410 −0.361158
$$93$$ −8.92820 −0.925812
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 5.19615 0.530330
$$97$$ 6.19615 0.629124 0.314562 0.949237i $$-0.398142\pi$$
0.314562 + 0.949237i $$0.398142\pi$$
$$98$$ 0.803848 0.0812009
$$99$$ 4.73205 0.475589
$$100$$ 0 0
$$101$$ −10.3923 −1.03407 −0.517036 0.855963i $$-0.672965\pi$$
−0.517036 + 0.855963i $$0.672965\pi$$
$$102$$ 0 0
$$103$$ −9.85641 −0.971181 −0.485590 0.874187i $$-0.661395\pi$$
−0.485590 + 0.874187i $$0.661395\pi$$
$$104$$ 1.26795 0.124333
$$105$$ 0 0
$$106$$ 16.3923 1.59216
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −14.3923 −1.37853 −0.689266 0.724508i $$-0.742067\pi$$
−0.689266 + 0.724508i $$0.742067\pi$$
$$110$$ 0 0
$$111$$ −6.19615 −0.588113
$$112$$ −13.6603 −1.29077
$$113$$ −18.9282 −1.78062 −0.890308 0.455359i $$-0.849511\pi$$
−0.890308 + 0.455359i $$0.849511\pi$$
$$114$$ −1.73205 −0.162221
$$115$$ 0 0
$$116$$ 8.19615 0.760994
$$117$$ −0.732051 −0.0676781
$$118$$ 4.39230 0.404344
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.3923 1.03566
$$122$$ −11.3205 −1.02491
$$123$$ −1.26795 −0.114327
$$124$$ 8.92820 0.801776
$$125$$ 0 0
$$126$$ 4.73205 0.421565
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 12.1244 1.07165
$$129$$ 4.19615 0.369451
$$130$$ 0 0
$$131$$ −9.12436 −0.797199 −0.398599 0.917125i $$-0.630504\pi$$
−0.398599 + 0.917125i $$0.630504\pi$$
$$132$$ −4.73205 −0.411872
$$133$$ 2.73205 0.236899
$$134$$ −13.8564 −1.19701
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −19.8564 −1.69645 −0.848224 0.529638i $$-0.822328\pi$$
−0.848224 + 0.529638i $$0.822328\pi$$
$$138$$ 6.00000 0.510754
$$139$$ −8.39230 −0.711826 −0.355913 0.934519i $$-0.615830\pi$$
−0.355913 + 0.934519i $$0.615830\pi$$
$$140$$ 0 0
$$141$$ −3.46410 −0.291730
$$142$$ −7.60770 −0.638424
$$143$$ −3.46410 −0.289683
$$144$$ −5.00000 −0.416667
$$145$$ 0 0
$$146$$ 29.3205 2.42658
$$147$$ −0.464102 −0.0382785
$$148$$ 6.19615 0.509321
$$149$$ −19.8564 −1.62670 −0.813350 0.581775i $$-0.802359\pi$$
−0.813350 + 0.581775i $$0.802359\pi$$
$$150$$ 0 0
$$151$$ 14.0000 1.13930 0.569652 0.821886i $$-0.307078\pi$$
0.569652 + 0.821886i $$0.307078\pi$$
$$152$$ −1.73205 −0.140488
$$153$$ 0 0
$$154$$ 22.3923 1.80442
$$155$$ 0 0
$$156$$ 0.732051 0.0586110
$$157$$ −6.39230 −0.510161 −0.255081 0.966920i $$-0.582102\pi$$
−0.255081 + 0.966920i $$0.582102\pi$$
$$158$$ −18.9282 −1.50585
$$159$$ −9.46410 −0.750552
$$160$$ 0 0
$$161$$ −9.46410 −0.745876
$$162$$ 1.73205 0.136083
$$163$$ −9.26795 −0.725922 −0.362961 0.931804i $$-0.618234\pi$$
−0.362961 + 0.931804i $$0.618234\pi$$
$$164$$ 1.26795 0.0990102
$$165$$ 0 0
$$166$$ 22.3923 1.73798
$$167$$ −3.46410 −0.268060 −0.134030 0.990977i $$-0.542792\pi$$
−0.134030 + 0.990977i $$0.542792\pi$$
$$168$$ 4.73205 0.365086
$$169$$ −12.4641 −0.958777
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ −4.19615 −0.319954
$$173$$ −6.92820 −0.526742 −0.263371 0.964695i $$-0.584834\pi$$
−0.263371 + 0.964695i $$0.584834\pi$$
$$174$$ −14.1962 −1.07621
$$175$$ 0 0
$$176$$ −23.6603 −1.78346
$$177$$ −2.53590 −0.190610
$$178$$ 18.5885 1.39326
$$179$$ 23.3205 1.74306 0.871528 0.490345i $$-0.163129\pi$$
0.871528 + 0.490345i $$0.163129\pi$$
$$180$$ 0 0
$$181$$ −2.39230 −0.177819 −0.0889093 0.996040i $$-0.528338\pi$$
−0.0889093 + 0.996040i $$0.528338\pi$$
$$182$$ −3.46410 −0.256776
$$183$$ 6.53590 0.483148
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ −15.4641 −1.13388
$$187$$ 0 0
$$188$$ 3.46410 0.252646
$$189$$ −2.73205 −0.198727
$$190$$ 0 0
$$191$$ 0.339746 0.0245832 0.0122916 0.999924i $$-0.496087\pi$$
0.0122916 + 0.999924i $$0.496087\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −17.1244 −1.23264 −0.616319 0.787497i $$-0.711377\pi$$
−0.616319 + 0.787497i $$0.711377\pi$$
$$194$$ 10.7321 0.770516
$$195$$ 0 0
$$196$$ 0.464102 0.0331501
$$197$$ 24.0000 1.70993 0.854965 0.518686i $$-0.173579\pi$$
0.854965 + 0.518686i $$0.173579\pi$$
$$198$$ 8.19615 0.582475
$$199$$ −15.3205 −1.08604 −0.543021 0.839719i $$-0.682720\pi$$
−0.543021 + 0.839719i $$0.682720\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ −18.0000 −1.26648
$$203$$ 22.3923 1.57163
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −17.0718 −1.18945
$$207$$ −3.46410 −0.240772
$$208$$ 3.66025 0.253793
$$209$$ 4.73205 0.327323
$$210$$ 0 0
$$211$$ 1.07180 0.0737855 0.0368928 0.999319i $$-0.488254\pi$$
0.0368928 + 0.999319i $$0.488254\pi$$
$$212$$ 9.46410 0.649997
$$213$$ 4.39230 0.300956
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 1.73205 0.117851
$$217$$ 24.3923 1.65586
$$218$$ −24.9282 −1.68835
$$219$$ −16.9282 −1.14390
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −10.7321 −0.720288
$$223$$ 17.8564 1.19575 0.597877 0.801588i $$-0.296011\pi$$
0.597877 + 0.801588i $$0.296011\pi$$
$$224$$ −14.1962 −0.948520
$$225$$ 0 0
$$226$$ −32.7846 −2.18080
$$227$$ −10.3923 −0.689761 −0.344881 0.938647i $$-0.612081\pi$$
−0.344881 + 0.938647i $$0.612081\pi$$
$$228$$ −1.00000 −0.0662266
$$229$$ −18.5359 −1.22489 −0.612443 0.790515i $$-0.709813\pi$$
−0.612443 + 0.790515i $$0.709813\pi$$
$$230$$ 0 0
$$231$$ −12.9282 −0.850613
$$232$$ −14.1962 −0.932023
$$233$$ 7.85641 0.514690 0.257345 0.966320i $$-0.417152\pi$$
0.257345 + 0.966320i $$0.417152\pi$$
$$234$$ −1.26795 −0.0828884
$$235$$ 0 0
$$236$$ 2.53590 0.165073
$$237$$ 10.9282 0.709863
$$238$$ 0 0
$$239$$ −9.80385 −0.634158 −0.317079 0.948399i $$-0.602702\pi$$
−0.317079 + 0.948399i $$0.602702\pi$$
$$240$$ 0 0
$$241$$ −3.07180 −0.197872 −0.0989359 0.995094i $$-0.531544\pi$$
−0.0989359 + 0.995094i $$0.531544\pi$$
$$242$$ 19.7321 1.26842
$$243$$ −1.00000 −0.0641500
$$244$$ −6.53590 −0.418418
$$245$$ 0 0
$$246$$ −2.19615 −0.140022
$$247$$ −0.732051 −0.0465793
$$248$$ −15.4641 −0.981971
$$249$$ −12.9282 −0.819292
$$250$$ 0 0
$$251$$ 28.0526 1.77066 0.885331 0.464961i $$-0.153932\pi$$
0.885331 + 0.464961i $$0.153932\pi$$
$$252$$ 2.73205 0.172103
$$253$$ −16.3923 −1.03058
$$254$$ 6.92820 0.434714
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ 24.0000 1.49708 0.748539 0.663090i $$-0.230755\pi$$
0.748539 + 0.663090i $$0.230755\pi$$
$$258$$ 7.26795 0.452483
$$259$$ 16.9282 1.05187
$$260$$ 0 0
$$261$$ 8.19615 0.507329
$$262$$ −15.8038 −0.976365
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 8.19615 0.504438
$$265$$ 0 0
$$266$$ 4.73205 0.290141
$$267$$ −10.7321 −0.656791
$$268$$ −8.00000 −0.488678
$$269$$ 0.588457 0.0358789 0.0179394 0.999839i $$-0.494289\pi$$
0.0179394 + 0.999839i $$0.494289\pi$$
$$270$$ 0 0
$$271$$ 0.392305 0.0238308 0.0119154 0.999929i $$-0.496207\pi$$
0.0119154 + 0.999929i $$0.496207\pi$$
$$272$$ 0 0
$$273$$ 2.00000 0.121046
$$274$$ −34.3923 −2.07772
$$275$$ 0 0
$$276$$ 3.46410 0.208514
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −14.5359 −0.871805
$$279$$ 8.92820 0.534518
$$280$$ 0 0
$$281$$ 1.26795 0.0756395 0.0378198 0.999285i $$-0.487959\pi$$
0.0378198 + 0.999285i $$0.487959\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ −24.9808 −1.48495 −0.742476 0.669873i $$-0.766349\pi$$
−0.742476 + 0.669873i $$0.766349\pi$$
$$284$$ −4.39230 −0.260635
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ 3.46410 0.204479
$$288$$ −5.19615 −0.306186
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −6.19615 −0.363225
$$292$$ 16.9282 0.990648
$$293$$ −27.7128 −1.61900 −0.809500 0.587120i $$-0.800262\pi$$
−0.809500 + 0.587120i $$0.800262\pi$$
$$294$$ −0.803848 −0.0468813
$$295$$ 0 0
$$296$$ −10.7321 −0.623788
$$297$$ −4.73205 −0.274581
$$298$$ −34.3923 −1.99229
$$299$$ 2.53590 0.146655
$$300$$ 0 0
$$301$$ −11.4641 −0.660780
$$302$$ 24.2487 1.39536
$$303$$ 10.3923 0.597022
$$304$$ −5.00000 −0.286770
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 32.3923 1.84873 0.924363 0.381514i $$-0.124597\pi$$
0.924363 + 0.381514i $$0.124597\pi$$
$$308$$ 12.9282 0.736653
$$309$$ 9.85641 0.560711
$$310$$ 0 0
$$311$$ 32.4449 1.83978 0.919890 0.392177i $$-0.128278\pi$$
0.919890 + 0.392177i $$0.128278\pi$$
$$312$$ −1.26795 −0.0717835
$$313$$ −6.39230 −0.361314 −0.180657 0.983546i $$-0.557822\pi$$
−0.180657 + 0.983546i $$0.557822\pi$$
$$314$$ −11.0718 −0.624818
$$315$$ 0 0
$$316$$ −10.9282 −0.614759
$$317$$ 11.3205 0.635823 0.317912 0.948120i $$-0.397018\pi$$
0.317912 + 0.948120i $$0.397018\pi$$
$$318$$ −16.3923 −0.919235
$$319$$ 38.7846 2.17152
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −16.3923 −0.913507
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −16.0526 −0.889069
$$327$$ 14.3923 0.795896
$$328$$ −2.19615 −0.121262
$$329$$ 9.46410 0.521773
$$330$$ 0 0
$$331$$ −25.7128 −1.41330 −0.706652 0.707561i $$-0.749795\pi$$
−0.706652 + 0.707561i $$0.749795\pi$$
$$332$$ 12.9282 0.709527
$$333$$ 6.19615 0.339547
$$334$$ −6.00000 −0.328305
$$335$$ 0 0
$$336$$ 13.6603 0.745228
$$337$$ −5.12436 −0.279141 −0.139571 0.990212i $$-0.544572\pi$$
−0.139571 + 0.990212i $$0.544572\pi$$
$$338$$ −21.5885 −1.17426
$$339$$ 18.9282 1.02804
$$340$$ 0 0
$$341$$ 42.2487 2.28790
$$342$$ 1.73205 0.0936586
$$343$$ −17.8564 −0.964155
$$344$$ 7.26795 0.391862
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ 0.928203 0.0498286 0.0249143 0.999690i $$-0.492069\pi$$
0.0249143 + 0.999690i $$0.492069\pi$$
$$348$$ −8.19615 −0.439360
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ 0.732051 0.0390740
$$352$$ −24.5885 −1.31057
$$353$$ 14.7846 0.786905 0.393453 0.919345i $$-0.371281\pi$$
0.393453 + 0.919345i $$0.371281\pi$$
$$354$$ −4.39230 −0.233448
$$355$$ 0 0
$$356$$ 10.7321 0.568798
$$357$$ 0 0
$$358$$ 40.3923 2.13480
$$359$$ 0.339746 0.0179311 0.00896555 0.999960i $$-0.497146\pi$$
0.00896555 + 0.999960i $$0.497146\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −4.14359 −0.217782
$$363$$ −11.3923 −0.597941
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 11.3205 0.591732
$$367$$ −16.1962 −0.845432 −0.422716 0.906262i $$-0.638923\pi$$
−0.422716 + 0.906262i $$0.638923\pi$$
$$368$$ 17.3205 0.902894
$$369$$ 1.26795 0.0660068
$$370$$ 0 0
$$371$$ 25.8564 1.34240
$$372$$ −8.92820 −0.462906
$$373$$ 6.19615 0.320825 0.160412 0.987050i $$-0.448718\pi$$
0.160412 + 0.987050i $$0.448718\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ −6.00000 −0.309016
$$378$$ −4.73205 −0.243390
$$379$$ 20.9282 1.07501 0.537505 0.843261i $$-0.319367\pi$$
0.537505 + 0.843261i $$0.319367\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0.588457 0.0301081
$$383$$ 17.0718 0.872328 0.436164 0.899867i $$-0.356337\pi$$
0.436164 + 0.899867i $$0.356337\pi$$
$$384$$ −12.1244 −0.618718
$$385$$ 0 0
$$386$$ −29.6603 −1.50967
$$387$$ −4.19615 −0.213302
$$388$$ 6.19615 0.314562
$$389$$ 7.85641 0.398336 0.199168 0.979965i $$-0.436176\pi$$
0.199168 + 0.979965i $$0.436176\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −0.803848 −0.0406004
$$393$$ 9.12436 0.460263
$$394$$ 41.5692 2.09423
$$395$$ 0 0
$$396$$ 4.73205 0.237795
$$397$$ −8.92820 −0.448094 −0.224047 0.974578i $$-0.571927\pi$$
−0.224047 + 0.974578i $$0.571927\pi$$
$$398$$ −26.5359 −1.33012
$$399$$ −2.73205 −0.136774
$$400$$ 0 0
$$401$$ −34.0526 −1.70050 −0.850252 0.526376i $$-0.823550\pi$$
−0.850252 + 0.526376i $$0.823550\pi$$
$$402$$ 13.8564 0.691095
$$403$$ −6.53590 −0.325576
$$404$$ −10.3923 −0.517036
$$405$$ 0 0
$$406$$ 38.7846 1.92485
$$407$$ 29.3205 1.45336
$$408$$ 0 0
$$409$$ −26.3923 −1.30502 −0.652508 0.757782i $$-0.726283\pi$$
−0.652508 + 0.757782i $$0.726283\pi$$
$$410$$ 0 0
$$411$$ 19.8564 0.979444
$$412$$ −9.85641 −0.485590
$$413$$ 6.92820 0.340915
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ 3.80385 0.186499
$$417$$ 8.39230 0.410973
$$418$$ 8.19615 0.400887
$$419$$ −28.0526 −1.37046 −0.685229 0.728328i $$-0.740298\pi$$
−0.685229 + 0.728328i $$0.740298\pi$$
$$420$$ 0 0
$$421$$ −18.7846 −0.915506 −0.457753 0.889079i $$-0.651346\pi$$
−0.457753 + 0.889079i $$0.651346\pi$$
$$422$$ 1.85641 0.0903685
$$423$$ 3.46410 0.168430
$$424$$ −16.3923 −0.796081
$$425$$ 0 0
$$426$$ 7.60770 0.368594
$$427$$ −17.8564 −0.864132
$$428$$ 0 0
$$429$$ 3.46410 0.167248
$$430$$ 0 0
$$431$$ −11.3205 −0.545290 −0.272645 0.962115i $$-0.587898\pi$$
−0.272645 + 0.962115i $$0.587898\pi$$
$$432$$ 5.00000 0.240563
$$433$$ 10.5885 0.508849 0.254424 0.967093i $$-0.418114\pi$$
0.254424 + 0.967093i $$0.418114\pi$$
$$434$$ 42.2487 2.02800
$$435$$ 0 0
$$436$$ −14.3923 −0.689266
$$437$$ −3.46410 −0.165710
$$438$$ −29.3205 −1.40099
$$439$$ 26.9282 1.28521 0.642607 0.766196i $$-0.277853\pi$$
0.642607 + 0.766196i $$0.277853\pi$$
$$440$$ 0 0
$$441$$ 0.464102 0.0221001
$$442$$ 0 0
$$443$$ 5.32051 0.252785 0.126392 0.991980i $$-0.459660\pi$$
0.126392 + 0.991980i $$0.459660\pi$$
$$444$$ −6.19615 −0.294056
$$445$$ 0 0
$$446$$ 30.9282 1.46449
$$447$$ 19.8564 0.939176
$$448$$ 2.73205 0.129077
$$449$$ −5.66025 −0.267124 −0.133562 0.991040i $$-0.542642\pi$$
−0.133562 + 0.991040i $$0.542642\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ −18.9282 −0.890308
$$453$$ −14.0000 −0.657777
$$454$$ −18.0000 −0.844782
$$455$$ 0 0
$$456$$ 1.73205 0.0811107
$$457$$ −4.53590 −0.212180 −0.106090 0.994357i $$-0.533833\pi$$
−0.106090 + 0.994357i $$0.533833\pi$$
$$458$$ −32.1051 −1.50017
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ −22.3923 −1.04178
$$463$$ 35.5167 1.65060 0.825300 0.564695i $$-0.191006\pi$$
0.825300 + 0.564695i $$0.191006\pi$$
$$464$$ −40.9808 −1.90248
$$465$$ 0 0
$$466$$ 13.6077 0.630364
$$467$$ −20.5359 −0.950288 −0.475144 0.879908i $$-0.657604\pi$$
−0.475144 + 0.879908i $$0.657604\pi$$
$$468$$ −0.732051 −0.0338391
$$469$$ −21.8564 −1.00924
$$470$$ 0 0
$$471$$ 6.39230 0.294542
$$472$$ −4.39230 −0.202172
$$473$$ −19.8564 −0.912999
$$474$$ 18.9282 0.869401
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 9.46410 0.433331
$$478$$ −16.9808 −0.776682
$$479$$ 25.5167 1.16589 0.582943 0.812513i $$-0.301901\pi$$
0.582943 + 0.812513i $$0.301901\pi$$
$$480$$ 0 0
$$481$$ −4.53590 −0.206819
$$482$$ −5.32051 −0.242343
$$483$$ 9.46410 0.430632
$$484$$ 11.3923 0.517832
$$485$$ 0 0
$$486$$ −1.73205 −0.0785674
$$487$$ 32.3923 1.46784 0.733918 0.679238i $$-0.237690\pi$$
0.733918 + 0.679238i $$0.237690\pi$$
$$488$$ 11.3205 0.512455
$$489$$ 9.26795 0.419111
$$490$$ 0 0
$$491$$ 16.0526 0.724442 0.362221 0.932092i $$-0.382019\pi$$
0.362221 + 0.932092i $$0.382019\pi$$
$$492$$ −1.26795 −0.0571636
$$493$$ 0 0
$$494$$ −1.26795 −0.0570477
$$495$$ 0 0
$$496$$ −44.6410 −2.00444
$$497$$ −12.0000 −0.538274
$$498$$ −22.3923 −1.00342
$$499$$ 10.5359 0.471652 0.235826 0.971795i $$-0.424221\pi$$
0.235826 + 0.971795i $$0.424221\pi$$
$$500$$ 0 0
$$501$$ 3.46410 0.154765
$$502$$ 48.5885 2.16861
$$503$$ 23.0718 1.02872 0.514360 0.857574i $$-0.328029\pi$$
0.514360 + 0.857574i $$0.328029\pi$$
$$504$$ −4.73205 −0.210782
$$505$$ 0 0
$$506$$ −28.3923 −1.26219
$$507$$ 12.4641 0.553550
$$508$$ 4.00000 0.177471
$$509$$ −10.0526 −0.445572 −0.222786 0.974867i $$-0.571515\pi$$
−0.222786 + 0.974867i $$0.571515\pi$$
$$510$$ 0 0
$$511$$ 46.2487 2.04592
$$512$$ 8.66025 0.382733
$$513$$ −1.00000 −0.0441511
$$514$$ 41.5692 1.83354
$$515$$ 0 0
$$516$$ 4.19615 0.184725
$$517$$ 16.3923 0.720933
$$518$$ 29.3205 1.28827
$$519$$ 6.92820 0.304114
$$520$$ 0 0
$$521$$ 37.2679 1.63274 0.816369 0.577530i $$-0.195983\pi$$
0.816369 + 0.577530i $$0.195983\pi$$
$$522$$ 14.1962 0.621349
$$523$$ −8.67949 −0.379528 −0.189764 0.981830i $$-0.560772\pi$$
−0.189764 + 0.981830i $$0.560772\pi$$
$$524$$ −9.12436 −0.398599
$$525$$ 0 0
$$526$$ −10.3923 −0.453126
$$527$$ 0 0
$$528$$ 23.6603 1.02968
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 2.53590 0.110049
$$532$$ 2.73205 0.118449
$$533$$ −0.928203 −0.0402049
$$534$$ −18.5885 −0.804401
$$535$$ 0 0
$$536$$ 13.8564 0.598506
$$537$$ −23.3205 −1.00635
$$538$$ 1.01924 0.0439425
$$539$$ 2.19615 0.0945950
$$540$$ 0 0
$$541$$ 41.7128 1.79337 0.896687 0.442665i $$-0.145967\pi$$
0.896687 + 0.442665i $$0.145967\pi$$
$$542$$ 0.679492 0.0291867
$$543$$ 2.39230 0.102664
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 3.46410 0.148250
$$547$$ −43.3205 −1.85225 −0.926126 0.377215i $$-0.876882\pi$$
−0.926126 + 0.377215i $$0.876882\pi$$
$$548$$ −19.8564 −0.848224
$$549$$ −6.53590 −0.278945
$$550$$ 0 0
$$551$$ 8.19615 0.349168
$$552$$ −6.00000 −0.255377
$$553$$ −29.8564 −1.26962
$$554$$ −3.46410 −0.147176
$$555$$ 0 0
$$556$$ −8.39230 −0.355913
$$557$$ −0.928203 −0.0393292 −0.0196646 0.999807i $$-0.506260\pi$$
−0.0196646 + 0.999807i $$0.506260\pi$$
$$558$$ 15.4641 0.654648
$$559$$ 3.07180 0.129923
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2.19615 0.0926391
$$563$$ 27.4641 1.15747 0.578737 0.815514i $$-0.303546\pi$$
0.578737 + 0.815514i $$0.303546\pi$$
$$564$$ −3.46410 −0.145865
$$565$$ 0 0
$$566$$ −43.2679 −1.81869
$$567$$ 2.73205 0.114735
$$568$$ 7.60770 0.319212
$$569$$ 22.0526 0.924491 0.462246 0.886752i $$-0.347044\pi$$
0.462246 + 0.886752i $$0.347044\pi$$
$$570$$ 0 0
$$571$$ −34.2487 −1.43326 −0.716632 0.697452i $$-0.754317\pi$$
−0.716632 + 0.697452i $$0.754317\pi$$
$$572$$ −3.46410 −0.144841
$$573$$ −0.339746 −0.0141931
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −15.1769 −0.631823 −0.315912 0.948789i $$-0.602310\pi$$
−0.315912 + 0.948789i $$0.602310\pi$$
$$578$$ −29.4449 −1.22474
$$579$$ 17.1244 0.711664
$$580$$ 0 0
$$581$$ 35.3205 1.46534
$$582$$ −10.7321 −0.444858
$$583$$ 44.7846 1.85479
$$584$$ −29.3205 −1.21329
$$585$$ 0 0
$$586$$ −48.0000 −1.98286
$$587$$ 3.46410 0.142979 0.0714894 0.997441i $$-0.477225\pi$$
0.0714894 + 0.997441i $$0.477225\pi$$
$$588$$ −0.464102 −0.0191392
$$589$$ 8.92820 0.367880
$$590$$ 0 0
$$591$$ −24.0000 −0.987228
$$592$$ −30.9808 −1.27330
$$593$$ 38.7846 1.59269 0.796347 0.604841i $$-0.206763\pi$$
0.796347 + 0.604841i $$0.206763\pi$$
$$594$$ −8.19615 −0.336292
$$595$$ 0 0
$$596$$ −19.8564 −0.813350
$$597$$ 15.3205 0.627027
$$598$$ 4.39230 0.179615
$$599$$ −13.8564 −0.566157 −0.283079 0.959097i $$-0.591356\pi$$
−0.283079 + 0.959097i $$0.591356\pi$$
$$600$$ 0 0
$$601$$ −47.1769 −1.92439 −0.962193 0.272368i $$-0.912193\pi$$
−0.962193 + 0.272368i $$0.912193\pi$$
$$602$$ −19.8564 −0.809287
$$603$$ −8.00000 −0.325785
$$604$$ 14.0000 0.569652
$$605$$ 0 0
$$606$$ 18.0000 0.731200
$$607$$ 11.6077 0.471142 0.235571 0.971857i $$-0.424304\pi$$
0.235571 + 0.971857i $$0.424304\pi$$
$$608$$ −5.19615 −0.210732
$$609$$ −22.3923 −0.907382
$$610$$ 0 0
$$611$$ −2.53590 −0.102591
$$612$$ 0 0
$$613$$ −42.3923 −1.71221 −0.856105 0.516803i $$-0.827122\pi$$
−0.856105 + 0.516803i $$0.827122\pi$$
$$614$$ 56.1051 2.26422
$$615$$ 0 0
$$616$$ −22.3923 −0.902212
$$617$$ −27.7128 −1.11568 −0.557838 0.829950i $$-0.688369\pi$$
−0.557838 + 0.829950i $$0.688369\pi$$
$$618$$ 17.0718 0.686728
$$619$$ −15.3205 −0.615783 −0.307892 0.951421i $$-0.599623\pi$$
−0.307892 + 0.951421i $$0.599623\pi$$
$$620$$ 0 0
$$621$$ 3.46410 0.139010
$$622$$ 56.1962 2.25326
$$623$$ 29.3205 1.17470
$$624$$ −3.66025 −0.146527
$$625$$ 0 0
$$626$$ −11.0718 −0.442518
$$627$$ −4.73205 −0.188980
$$628$$ −6.39230 −0.255081
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −34.9282 −1.39047 −0.695235 0.718783i $$-0.744700\pi$$
−0.695235 + 0.718783i $$0.744700\pi$$
$$632$$ 18.9282 0.752923
$$633$$ −1.07180 −0.0426001
$$634$$ 19.6077 0.778721
$$635$$ 0 0
$$636$$ −9.46410 −0.375276
$$637$$ −0.339746 −0.0134612
$$638$$ 67.1769 2.65956
$$639$$ −4.39230 −0.173757
$$640$$ 0 0
$$641$$ 48.5885 1.91913 0.959564 0.281489i $$-0.0908284\pi$$
0.959564 + 0.281489i $$0.0908284\pi$$
$$642$$ 0 0
$$643$$ 12.1962 0.480969 0.240485 0.970653i $$-0.422694\pi$$
0.240485 + 0.970653i $$0.422694\pi$$
$$644$$ −9.46410 −0.372938
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 4.14359 0.162901 0.0814507 0.996677i $$-0.474045\pi$$
0.0814507 + 0.996677i $$0.474045\pi$$
$$648$$ −1.73205 −0.0680414
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ −24.3923 −0.956010
$$652$$ −9.26795 −0.362961
$$653$$ −17.0718 −0.668071 −0.334036 0.942560i $$-0.608411\pi$$
−0.334036 + 0.942560i $$0.608411\pi$$
$$654$$ 24.9282 0.974770
$$655$$ 0 0
$$656$$ −6.33975 −0.247525
$$657$$ 16.9282 0.660432
$$658$$ 16.3923 0.639039
$$659$$ 5.07180 0.197569 0.0987846 0.995109i $$-0.468505\pi$$
0.0987846 + 0.995109i $$0.468505\pi$$
$$660$$ 0 0
$$661$$ 39.1769 1.52381 0.761903 0.647692i $$-0.224265\pi$$
0.761903 + 0.647692i $$0.224265\pi$$
$$662$$ −44.5359 −1.73094
$$663$$ 0 0
$$664$$ −22.3923 −0.868990
$$665$$ 0 0
$$666$$ 10.7321 0.415859
$$667$$ −28.3923 −1.09935
$$668$$ −3.46410 −0.134030
$$669$$ −17.8564 −0.690369
$$670$$ 0 0
$$671$$ −30.9282 −1.19397
$$672$$ 14.1962 0.547628
$$673$$ −17.1244 −0.660095 −0.330048 0.943964i $$-0.607065\pi$$
−0.330048 + 0.943964i $$0.607065\pi$$
$$674$$ −8.87564 −0.341877
$$675$$ 0 0
$$676$$ −12.4641 −0.479389
$$677$$ −0.679492 −0.0261150 −0.0130575 0.999915i $$-0.504156\pi$$
−0.0130575 + 0.999915i $$0.504156\pi$$
$$678$$ 32.7846 1.25909
$$679$$ 16.9282 0.649645
$$680$$ 0 0
$$681$$ 10.3923 0.398234
$$682$$ 73.1769 2.80209
$$683$$ 5.07180 0.194067 0.0970335 0.995281i $$-0.469065\pi$$
0.0970335 + 0.995281i $$0.469065\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ −30.9282 −1.18084
$$687$$ 18.5359 0.707189
$$688$$ 20.9808 0.799884
$$689$$ −6.92820 −0.263944
$$690$$ 0 0
$$691$$ 12.3923 0.471425 0.235713 0.971823i $$-0.424258\pi$$
0.235713 + 0.971823i $$0.424258\pi$$
$$692$$ −6.92820 −0.263371
$$693$$ 12.9282 0.491102
$$694$$ 1.60770 0.0610273
$$695$$ 0 0
$$696$$ 14.1962 0.538104
$$697$$ 0 0
$$698$$ −38.1051 −1.44230
$$699$$ −7.85641 −0.297157
$$700$$ 0 0
$$701$$ −33.7128 −1.27332 −0.636658 0.771147i $$-0.719684\pi$$
−0.636658 + 0.771147i $$0.719684\pi$$
$$702$$ 1.26795 0.0478557
$$703$$ 6.19615 0.233692
$$704$$ 4.73205 0.178346
$$705$$ 0 0
$$706$$ 25.6077 0.963758
$$707$$ −28.3923 −1.06780
$$708$$ −2.53590 −0.0953049
$$709$$ −29.1769 −1.09576 −0.547881 0.836556i $$-0.684565\pi$$
−0.547881 + 0.836556i $$0.684565\pi$$
$$710$$ 0 0
$$711$$ −10.9282 −0.409840
$$712$$ −18.5885 −0.696632
$$713$$ −30.9282 −1.15827
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 23.3205 0.871528
$$717$$ 9.80385 0.366131
$$718$$ 0.588457 0.0219610
$$719$$ −11.6603 −0.434854 −0.217427 0.976077i $$-0.569766\pi$$
−0.217427 + 0.976077i $$0.569766\pi$$
$$720$$ 0 0
$$721$$ −26.9282 −1.00286
$$722$$ 1.73205 0.0644603
$$723$$ 3.07180 0.114241
$$724$$ −2.39230 −0.0889093
$$725$$ 0 0
$$726$$ −19.7321 −0.732325
$$727$$ −25.6603 −0.951686 −0.475843 0.879530i $$-0.657857\pi$$
−0.475843 + 0.879530i $$0.657857\pi$$
$$728$$ 3.46410 0.128388
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 6.53590 0.241574
$$733$$ 18.7846 0.693825 0.346913 0.937897i $$-0.387230\pi$$
0.346913 + 0.937897i $$0.387230\pi$$
$$734$$ −28.0526 −1.03544
$$735$$ 0 0
$$736$$ 18.0000 0.663489
$$737$$ −37.8564 −1.39446
$$738$$ 2.19615 0.0808415
$$739$$ 6.14359 0.225996 0.112998 0.993595i $$-0.463955\pi$$
0.112998 + 0.993595i $$0.463955\pi$$
$$740$$ 0 0
$$741$$ 0.732051 0.0268926
$$742$$ 44.7846 1.64409
$$743$$ 3.21539 0.117961 0.0589806 0.998259i $$-0.481215\pi$$
0.0589806 + 0.998259i $$0.481215\pi$$
$$744$$ 15.4641 0.566941
$$745$$ 0 0
$$746$$ 10.7321 0.392928
$$747$$ 12.9282 0.473018
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ −17.3205 −0.631614
$$753$$ −28.0526 −1.02229
$$754$$ −10.3923 −0.378465
$$755$$ 0 0
$$756$$ −2.73205 −0.0993637
$$757$$ −32.2487 −1.17210 −0.586050 0.810275i $$-0.699318\pi$$
−0.586050 + 0.810275i $$0.699318\pi$$
$$758$$ 36.2487 1.31661
$$759$$ 16.3923 0.595003
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ −6.92820 −0.250982
$$763$$ −39.3205 −1.42350
$$764$$ 0.339746 0.0122916
$$765$$ 0 0
$$766$$ 29.5692 1.06838
$$767$$ −1.85641 −0.0670310
$$768$$ −19.0000 −0.685603
$$769$$ −20.6410 −0.744334 −0.372167 0.928166i $$-0.621385\pi$$
−0.372167 + 0.928166i $$0.621385\pi$$
$$770$$ 0 0
$$771$$ −24.0000 −0.864339
$$772$$ −17.1244 −0.616319
$$773$$ 25.1769 0.905551 0.452775 0.891625i $$-0.350434\pi$$
0.452775 + 0.891625i $$0.350434\pi$$
$$774$$ −7.26795 −0.261241
$$775$$ 0 0
$$776$$ −10.7321 −0.385258
$$777$$ −16.9282 −0.607296
$$778$$ 13.6077 0.487860
$$779$$ 1.26795 0.0454290
$$780$$ 0 0
$$781$$ −20.7846 −0.743732
$$782$$ 0 0
$$783$$ −8.19615 −0.292907
$$784$$ −2.32051 −0.0828753
$$785$$ 0 0
$$786$$ 15.8038 0.563705
$$787$$ −8.67949 −0.309390 −0.154695 0.987962i $$-0.549440\pi$$
−0.154695 + 0.987962i $$0.549440\pi$$
$$788$$ 24.0000 0.854965
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ −51.7128 −1.83870
$$792$$ −8.19615 −0.291238
$$793$$ 4.78461 0.169906
$$794$$ −15.4641 −0.548800
$$795$$ 0 0
$$796$$ −15.3205 −0.543021
$$797$$ −44.7846 −1.58635 −0.793176 0.608992i $$-0.791574\pi$$
−0.793176 + 0.608992i $$0.791574\pi$$
$$798$$ −4.73205 −0.167513
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 10.7321 0.379198
$$802$$ −58.9808 −2.08268
$$803$$ 80.1051 2.82685
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −11.3205 −0.398748
$$807$$ −0.588457 −0.0207147
$$808$$ 18.0000 0.633238
$$809$$ 14.7846 0.519799 0.259900 0.965636i $$-0.416311\pi$$
0.259900 + 0.965636i $$0.416311\pi$$
$$810$$ 0 0
$$811$$ 37.5692 1.31923 0.659617 0.751602i $$-0.270719\pi$$
0.659617 + 0.751602i $$0.270719\pi$$
$$812$$ 22.3923 0.785816
$$813$$ −0.392305 −0.0137587
$$814$$ 50.7846 1.78000
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −4.19615 −0.146805
$$818$$ −45.7128 −1.59831
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ −32.5359 −1.13551 −0.567755 0.823197i $$-0.692188\pi$$
−0.567755 + 0.823197i $$0.692188\pi$$
$$822$$ 34.3923 1.19957
$$823$$ −12.9808 −0.452481 −0.226240 0.974071i $$-0.572644\pi$$
−0.226240 + 0.974071i $$0.572644\pi$$
$$824$$ 17.0718 0.594724
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 5.32051 0.185012 0.0925061 0.995712i $$-0.470512\pi$$
0.0925061 + 0.995712i $$0.470512\pi$$
$$828$$ −3.46410 −0.120386
$$829$$ 34.1051 1.18452 0.592260 0.805747i $$-0.298236\pi$$
0.592260 + 0.805747i $$0.298236\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ −0.732051 −0.0253793
$$833$$ 0 0
$$834$$ 14.5359 0.503337
$$835$$ 0 0
$$836$$ 4.73205 0.163661
$$837$$ −8.92820 −0.308604
$$838$$ −48.5885 −1.67846
$$839$$ −19.6077 −0.676933 −0.338466 0.940978i $$-0.609908\pi$$
−0.338466 + 0.940978i $$0.609908\pi$$
$$840$$ 0 0
$$841$$ 38.1769 1.31645
$$842$$ −32.5359 −1.12126
$$843$$ −1.26795 −0.0436705
$$844$$ 1.07180 0.0368928
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ 31.1244 1.06945
$$848$$ −47.3205 −1.62499
$$849$$ 24.9808 0.857338
$$850$$ 0 0
$$851$$ −21.4641 −0.735780
$$852$$ 4.39230 0.150478
$$853$$ −27.1769 −0.930520 −0.465260 0.885174i $$-0.654039\pi$$
−0.465260 + 0.885174i $$0.654039\pi$$
$$854$$ −30.9282 −1.05834
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 42.2487 1.44319 0.721594 0.692316i $$-0.243410\pi$$
0.721594 + 0.692316i $$0.243410\pi$$
$$858$$ 6.00000 0.204837
$$859$$ 32.0000 1.09183 0.545913 0.837842i $$-0.316183\pi$$
0.545913 + 0.837842i $$0.316183\pi$$
$$860$$ 0 0
$$861$$ −3.46410 −0.118056
$$862$$ −19.6077 −0.667841
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 5.19615 0.176777
$$865$$ 0 0
$$866$$ 18.3397 0.623210
$$867$$ 17.0000 0.577350
$$868$$ 24.3923 0.827929
$$869$$ −51.7128 −1.75424
$$870$$ 0 0
$$871$$ 5.85641 0.198437
$$872$$ 24.9282 0.844175
$$873$$ 6.19615 0.209708
$$874$$ −6.00000 −0.202953
$$875$$ 0 0
$$876$$ −16.9282 −0.571951
$$877$$ −53.1244 −1.79388 −0.896941 0.442150i $$-0.854216\pi$$
−0.896941 + 0.442150i $$0.854216\pi$$
$$878$$ 46.6410 1.57406
$$879$$ 27.7128 0.934730
$$880$$ 0 0
$$881$$ 8.53590 0.287582 0.143791 0.989608i $$-0.454071\pi$$
0.143791 + 0.989608i $$0.454071\pi$$
$$882$$ 0.803848 0.0270670
$$883$$ −36.9808 −1.24450 −0.622251 0.782818i $$-0.713782\pi$$
−0.622251 + 0.782818i $$0.713782\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 9.21539 0.309597
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 10.7321 0.360144
$$889$$ 10.9282 0.366520
$$890$$ 0 0
$$891$$ 4.73205 0.158530
$$892$$ 17.8564 0.597877
$$893$$ 3.46410 0.115922
$$894$$ 34.3923 1.15025
$$895$$ 0 0
$$896$$ 33.1244 1.10661
$$897$$ −2.53590 −0.0846712
$$898$$ −9.80385 −0.327159
$$899$$ 73.1769 2.44059
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 10.3923 0.346026
$$903$$ 11.4641 0.381501
$$904$$ 32.7846 1.09040
$$905$$ 0 0
$$906$$ −24.2487 −0.805609
$$907$$ 32.3923 1.07557 0.537784 0.843082i $$-0.319261\pi$$
0.537784 + 0.843082i $$0.319261\pi$$
$$908$$ −10.3923 −0.344881
$$909$$ −10.3923 −0.344691
$$910$$ 0 0
$$911$$ −54.9282 −1.81985 −0.909926 0.414770i $$-0.863862\pi$$
−0.909926 + 0.414770i $$0.863862\pi$$
$$912$$ 5.00000 0.165567
$$913$$ 61.1769 2.02466
$$914$$ −7.85641 −0.259867
$$915$$ 0 0
$$916$$ −18.5359 −0.612443
$$917$$ −24.9282 −0.823202
$$918$$ 0 0
$$919$$ 51.4256 1.69637 0.848187 0.529696i $$-0.177694\pi$$
0.848187 + 0.529696i $$0.177694\pi$$
$$920$$ 0 0
$$921$$ −32.3923 −1.06736
$$922$$ 10.3923 0.342252
$$923$$ 3.21539 0.105836
$$924$$ −12.9282 −0.425307
$$925$$ 0 0
$$926$$ 61.5167 2.02156
$$927$$ −9.85641 −0.323727
$$928$$ −42.5885 −1.39803
$$929$$ −1.60770 −0.0527468 −0.0263734 0.999652i $$-0.508396\pi$$
−0.0263734 + 0.999652i $$0.508396\pi$$
$$930$$ 0 0
$$931$$ 0.464102 0.0152103
$$932$$ 7.85641 0.257345
$$933$$ −32.4449 −1.06220
$$934$$ −35.5692 −1.16386
$$935$$ 0 0
$$936$$ 1.26795 0.0414442
$$937$$ 16.2487 0.530822 0.265411 0.964135i $$-0.414492\pi$$
0.265411 + 0.964135i $$0.414492\pi$$
$$938$$ −37.8564 −1.23606
$$939$$ 6.39230 0.208605
$$940$$ 0 0
$$941$$ 0.588457 0.0191832 0.00959158 0.999954i $$-0.496947\pi$$
0.00959158 + 0.999954i $$0.496947\pi$$
$$942$$ 11.0718 0.360739
$$943$$ −4.39230 −0.143033
$$944$$ −12.6795 −0.412682
$$945$$ 0 0
$$946$$ −34.3923 −1.11819
$$947$$ −28.1436 −0.914544 −0.457272 0.889327i $$-0.651173\pi$$
−0.457272 + 0.889327i $$0.651173\pi$$
$$948$$ 10.9282 0.354932
$$949$$ −12.3923 −0.402271
$$950$$ 0 0
$$951$$ −11.3205 −0.367093
$$952$$ 0 0
$$953$$ −37.8564 −1.22629 −0.613145 0.789971i $$-0.710096\pi$$
−0.613145 + 0.789971i $$0.710096\pi$$
$$954$$ 16.3923 0.530720
$$955$$ 0 0
$$956$$ −9.80385 −0.317079
$$957$$ −38.7846 −1.25373
$$958$$ 44.1962 1.42791
$$959$$ −54.2487 −1.75178
$$960$$ 0 0
$$961$$ 48.7128 1.57138
$$962$$ −7.85641 −0.253301
$$963$$ 0 0
$$964$$ −3.07180 −0.0989359
$$965$$ 0 0
$$966$$ 16.3923 0.527414
$$967$$ −4.87564 −0.156790 −0.0783951 0.996922i $$-0.524980\pi$$
−0.0783951 + 0.996922i $$0.524980\pi$$
$$968$$ −19.7321 −0.634212
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −27.7128 −0.889346 −0.444673 0.895693i $$-0.646680\pi$$
−0.444673 + 0.895693i $$0.646680\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −22.9282 −0.735044
$$974$$ 56.1051 1.79772
$$975$$ 0 0
$$976$$ 32.6795 1.04605
$$977$$ 39.0333 1.24879 0.624393 0.781110i $$-0.285346\pi$$
0.624393 + 0.781110i $$0.285346\pi$$
$$978$$ 16.0526 0.513304
$$979$$ 50.7846 1.62308
$$980$$ 0 0
$$981$$ −14.3923 −0.459511
$$982$$ 27.8038 0.887256
$$983$$ 41.3205 1.31792 0.658960 0.752178i $$-0.270997\pi$$
0.658960 + 0.752178i $$0.270997\pi$$
$$984$$ 2.19615 0.0700108
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −9.46410 −0.301246
$$988$$ −0.732051 −0.0232896
$$989$$ 14.5359 0.462215
$$990$$ 0 0
$$991$$ 13.0718 0.415239 0.207620 0.978210i $$-0.433428\pi$$
0.207620 + 0.978210i $$0.433428\pi$$
$$992$$ −46.3923 −1.47296
$$993$$ 25.7128 0.815971
$$994$$ −20.7846 −0.659248
$$995$$ 0 0
$$996$$ −12.9282 −0.409646
$$997$$ 17.6077 0.557641 0.278821 0.960343i $$-0.410056\pi$$
0.278821 + 0.960343i $$0.410056\pi$$
$$998$$ 18.2487 0.577653
$$999$$ −6.19615 −0.196038
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 1425.2.a.o.1.2 2
3.2 odd 2 4275.2.a.t.1.1 2
5.2 odd 4 1425.2.c.k.799.4 4
5.3 odd 4 1425.2.c.k.799.1 4
5.4 even 2 285.2.a.e.1.1 2
15.14 odd 2 855.2.a.f.1.2 2
20.19 odd 2 4560.2.a.bh.1.2 2
95.94 odd 2 5415.2.a.r.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.1 2 5.4 even 2
855.2.a.f.1.2 2 15.14 odd 2
1425.2.a.o.1.2 2 1.1 even 1 trivial
1425.2.c.k.799.1 4 5.3 odd 4
1425.2.c.k.799.4 4 5.2 odd 4
4275.2.a.t.1.1 2 3.2 odd 2
4560.2.a.bh.1.2 2 20.19 odd 2
5415.2.a.r.1.2 2 95.94 odd 2