# Properties

 Label 2535.2.a.ba.1.2 Level $2535$ Weight $2$ Character 2535.1 Self dual yes Analytic conductor $20.242$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2535,2,Mod(1,2535)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2535, 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("2535.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2535 = 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2535.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: $$20.2420769124$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.756.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 2$$ x^3 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.339877$$ of defining polynomial Character $$\chi$$ $$=$$ 2535.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.339877 q^{2} -1.00000 q^{3} -1.88448 q^{4} -1.00000 q^{5} -0.339877 q^{6} -0.660123 q^{7} -1.32025 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.339877 q^{2} -1.00000 q^{3} -1.88448 q^{4} -1.00000 q^{5} -0.339877 q^{6} -0.660123 q^{7} -1.32025 q^{8} +1.00000 q^{9} -0.339877 q^{10} -0.679754 q^{11} +1.88448 q^{12} -0.224361 q^{14} +1.00000 q^{15} +3.32025 q^{16} +7.42909 q^{17} +0.339877 q^{18} -0.115516 q^{19} +1.88448 q^{20} +0.660123 q^{21} -0.231033 q^{22} -7.76897 q^{23} +1.32025 q^{24} +1.00000 q^{25} -1.00000 q^{27} +1.24399 q^{28} +5.54461 q^{29} +0.339877 q^{30} +9.97370 q^{31} +3.76897 q^{32} +0.679754 q^{33} +2.52498 q^{34} +0.660123 q^{35} -1.88448 q^{36} -9.76897 q^{37} -0.0392613 q^{38} +1.32025 q^{40} -4.22436 q^{41} +0.224361 q^{42} -0.544607 q^{43} +1.28098 q^{44} -1.00000 q^{45} -2.64049 q^{46} +5.01963 q^{47} -3.32025 q^{48} -6.56424 q^{49} +0.339877 q^{50} -7.42909 q^{51} +0.679754 q^{53} -0.339877 q^{54} +0.679754 q^{55} +0.871525 q^{56} +0.115516 q^{57} +1.88448 q^{58} -2.22436 q^{59} -1.88448 q^{60} -4.20473 q^{61} +3.38983 q^{62} -0.660123 q^{63} -5.35951 q^{64} +0.231033 q^{66} +7.63382 q^{67} -14.0000 q^{68} +7.76897 q^{69} +0.224361 q^{70} -7.31357 q^{71} -1.32025 q^{72} -8.01963 q^{73} -3.32025 q^{74} -1.00000 q^{75} +0.217689 q^{76} +0.448721 q^{77} +9.97370 q^{79} -3.32025 q^{80} +1.00000 q^{81} -1.43576 q^{82} -1.76897 q^{83} -1.24399 q^{84} -7.42909 q^{85} -0.185099 q^{86} -5.54461 q^{87} +0.897442 q^{88} -13.5446 q^{89} -0.339877 q^{90} +14.6405 q^{92} -9.97370 q^{93} +1.70606 q^{94} +0.115516 q^{95} -3.76897 q^{96} -9.90411 q^{97} -2.23103 q^{98} -0.679754 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 6 q^{4} - 3 q^{5} - 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^5 - 3 * q^7 - 6 * q^8 + 3 * q^9 $$3 q - 3 q^{3} + 6 q^{4} - 3 q^{5} - 3 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{12} + 12 q^{14} + 3 q^{15} + 12 q^{16} - 12 q^{19} - 6 q^{20} + 3 q^{21} - 24 q^{22} + 6 q^{24} + 3 q^{25} - 3 q^{27} - 12 q^{28} + 6 q^{29} - 3 q^{31} - 12 q^{32} + 3 q^{35} + 6 q^{36} - 6 q^{37} + 6 q^{38} + 6 q^{40} - 12 q^{42} + 9 q^{43} + 12 q^{44} - 3 q^{45} - 12 q^{46} + 12 q^{47} - 12 q^{48} - 6 q^{49} + 30 q^{56} + 12 q^{57} - 6 q^{58} + 6 q^{59} + 6 q^{60} - 3 q^{61} - 6 q^{62} - 3 q^{63} - 12 q^{64} + 24 q^{66} - 9 q^{67} - 42 q^{68} - 12 q^{70} + 12 q^{71} - 6 q^{72} - 21 q^{73} - 12 q^{74} - 3 q^{75} - 48 q^{76} - 24 q^{77} - 3 q^{79} - 12 q^{80} + 3 q^{81} - 18 q^{82} + 18 q^{83} + 12 q^{84} + 6 q^{86} - 6 q^{87} - 48 q^{88} - 30 q^{89} + 48 q^{92} + 3 q^{93} + 36 q^{94} + 12 q^{95} + 12 q^{96} - 15 q^{97} - 30 q^{98}+O(q^{100})$$ 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^5 - 3 * q^7 - 6 * q^8 + 3 * q^9 - 6 * q^12 + 12 * q^14 + 3 * q^15 + 12 * q^16 - 12 * q^19 - 6 * q^20 + 3 * q^21 - 24 * q^22 + 6 * q^24 + 3 * q^25 - 3 * q^27 - 12 * q^28 + 6 * q^29 - 3 * q^31 - 12 * q^32 + 3 * q^35 + 6 * q^36 - 6 * q^37 + 6 * q^38 + 6 * q^40 - 12 * q^42 + 9 * q^43 + 12 * q^44 - 3 * q^45 - 12 * q^46 + 12 * q^47 - 12 * q^48 - 6 * q^49 + 30 * q^56 + 12 * q^57 - 6 * q^58 + 6 * q^59 + 6 * q^60 - 3 * q^61 - 6 * q^62 - 3 * q^63 - 12 * q^64 + 24 * q^66 - 9 * q^67 - 42 * q^68 - 12 * q^70 + 12 * q^71 - 6 * q^72 - 21 * q^73 - 12 * q^74 - 3 * q^75 - 48 * q^76 - 24 * q^77 - 3 * q^79 - 12 * q^80 + 3 * q^81 - 18 * q^82 + 18 * q^83 + 12 * q^84 + 6 * q^86 - 6 * q^87 - 48 * q^88 - 30 * q^89 + 48 * q^92 + 3 * q^93 + 36 * q^94 + 12 * q^95 + 12 * q^96 - 15 * q^97 - 30 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.339877 0.240329 0.120165 0.992754i $$-0.461658\pi$$
0.120165 + 0.992754i $$0.461658\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.88448 −0.942242
$$5$$ −1.00000 −0.447214
$$6$$ −0.339877 −0.138754
$$7$$ −0.660123 −0.249503 −0.124752 0.992188i $$-0.539813\pi$$
−0.124752 + 0.992188i $$0.539813\pi$$
$$8$$ −1.32025 −0.466778
$$9$$ 1.00000 0.333333
$$10$$ −0.339877 −0.107479
$$11$$ −0.679754 −0.204953 −0.102477 0.994735i $$-0.532677\pi$$
−0.102477 + 0.994735i $$0.532677\pi$$
$$12$$ 1.88448 0.544004
$$13$$ 0 0
$$14$$ −0.224361 −0.0599629
$$15$$ 1.00000 0.258199
$$16$$ 3.32025 0.830062
$$17$$ 7.42909 1.80182 0.900910 0.434007i $$-0.142901\pi$$
0.900910 + 0.434007i $$0.142901\pi$$
$$18$$ 0.339877 0.0801098
$$19$$ −0.115516 −0.0265013 −0.0132506 0.999912i $$-0.504218\pi$$
−0.0132506 + 0.999912i $$0.504218\pi$$
$$20$$ 1.88448 0.421383
$$21$$ 0.660123 0.144051
$$22$$ −0.231033 −0.0492563
$$23$$ −7.76897 −1.61994 −0.809971 0.586470i $$-0.800517\pi$$
−0.809971 + 0.586470i $$0.800517\pi$$
$$24$$ 1.32025 0.269494
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 1.24399 0.235092
$$29$$ 5.54461 1.02961 0.514804 0.857308i $$-0.327865\pi$$
0.514804 + 0.857308i $$0.327865\pi$$
$$30$$ 0.339877 0.0620527
$$31$$ 9.97370 1.79133 0.895664 0.444730i $$-0.146701\pi$$
0.895664 + 0.444730i $$0.146701\pi$$
$$32$$ 3.76897 0.666266
$$33$$ 0.679754 0.118330
$$34$$ 2.52498 0.433030
$$35$$ 0.660123 0.111581
$$36$$ −1.88448 −0.314081
$$37$$ −9.76897 −1.60601 −0.803004 0.595973i $$-0.796766\pi$$
−0.803004 + 0.595973i $$0.796766\pi$$
$$38$$ −0.0392613 −0.00636903
$$39$$ 0 0
$$40$$ 1.32025 0.208749
$$41$$ −4.22436 −0.659734 −0.329867 0.944027i $$-0.607004\pi$$
−0.329867 + 0.944027i $$0.607004\pi$$
$$42$$ 0.224361 0.0346196
$$43$$ −0.544607 −0.0830518 −0.0415259 0.999137i $$-0.513222\pi$$
−0.0415259 + 0.999137i $$0.513222\pi$$
$$44$$ 1.28098 0.193116
$$45$$ −1.00000 −0.149071
$$46$$ −2.64049 −0.389319
$$47$$ 5.01963 0.732188 0.366094 0.930578i $$-0.380695\pi$$
0.366094 + 0.930578i $$0.380695\pi$$
$$48$$ −3.32025 −0.479236
$$49$$ −6.56424 −0.937748
$$50$$ 0.339877 0.0480659
$$51$$ −7.42909 −1.04028
$$52$$ 0 0
$$53$$ 0.679754 0.0933714 0.0466857 0.998910i $$-0.485134\pi$$
0.0466857 + 0.998910i $$0.485134\pi$$
$$54$$ −0.339877 −0.0462514
$$55$$ 0.679754 0.0916580
$$56$$ 0.871525 0.116462
$$57$$ 0.115516 0.0153005
$$58$$ 1.88448 0.247445
$$59$$ −2.22436 −0.289587 −0.144794 0.989462i $$-0.546252\pi$$
−0.144794 + 0.989462i $$0.546252\pi$$
$$60$$ −1.88448 −0.243286
$$61$$ −4.20473 −0.538361 −0.269180 0.963090i $$-0.586753\pi$$
−0.269180 + 0.963090i $$0.586753\pi$$
$$62$$ 3.38983 0.430509
$$63$$ −0.660123 −0.0831677
$$64$$ −5.35951 −0.669938
$$65$$ 0 0
$$66$$ 0.231033 0.0284381
$$67$$ 7.63382 0.932620 0.466310 0.884621i $$-0.345583\pi$$
0.466310 + 0.884621i $$0.345583\pi$$
$$68$$ −14.0000 −1.69775
$$69$$ 7.76897 0.935274
$$70$$ 0.224361 0.0268162
$$71$$ −7.31357 −0.867962 −0.433981 0.900922i $$-0.642891\pi$$
−0.433981 + 0.900922i $$0.642891\pi$$
$$72$$ −1.32025 −0.155593
$$73$$ −8.01963 −0.938627 −0.469313 0.883032i $$-0.655499\pi$$
−0.469313 + 0.883032i $$0.655499\pi$$
$$74$$ −3.32025 −0.385971
$$75$$ −1.00000 −0.115470
$$76$$ 0.217689 0.0249706
$$77$$ 0.448721 0.0511365
$$78$$ 0 0
$$79$$ 9.97370 1.12213 0.561064 0.827772i $$-0.310392\pi$$
0.561064 + 0.827772i $$0.310392\pi$$
$$80$$ −3.32025 −0.371215
$$81$$ 1.00000 0.111111
$$82$$ −1.43576 −0.158553
$$83$$ −1.76897 −0.194169 −0.0970847 0.995276i $$-0.530952\pi$$
−0.0970847 + 0.995276i $$0.530952\pi$$
$$84$$ −1.24399 −0.135731
$$85$$ −7.42909 −0.805798
$$86$$ −0.185099 −0.0199598
$$87$$ −5.54461 −0.594444
$$88$$ 0.897442 0.0956677
$$89$$ −13.5446 −1.43573 −0.717863 0.696185i $$-0.754879\pi$$
−0.717863 + 0.696185i $$0.754879\pi$$
$$90$$ −0.339877 −0.0358262
$$91$$ 0 0
$$92$$ 14.6405 1.52638
$$93$$ −9.97370 −1.03422
$$94$$ 1.70606 0.175966
$$95$$ 0.115516 0.0118517
$$96$$ −3.76897 −0.384669
$$97$$ −9.90411 −1.00561 −0.502805 0.864400i $$-0.667699\pi$$
−0.502805 + 0.864400i $$0.667699\pi$$
$$98$$ −2.23103 −0.225368
$$99$$ −0.679754 −0.0683178
$$100$$ −1.88448 −0.188448
$$101$$ −9.35951 −0.931306 −0.465653 0.884967i $$-0.654180\pi$$
−0.465653 + 0.884967i $$0.654180\pi$$
$$102$$ −2.52498 −0.250010
$$103$$ 4.50535 0.443925 0.221962 0.975055i $$-0.428754\pi$$
0.221962 + 0.975055i $$0.428754\pi$$
$$104$$ 0 0
$$105$$ −0.660123 −0.0644214
$$106$$ 0.231033 0.0224399
$$107$$ 0.570909 0.0551919 0.0275960 0.999619i $$-0.491215\pi$$
0.0275960 + 0.999619i $$0.491215\pi$$
$$108$$ 1.88448 0.181335
$$109$$ −15.4095 −1.47596 −0.737979 0.674823i $$-0.764220\pi$$
−0.737979 + 0.674823i $$0.764220\pi$$
$$110$$ 0.231033 0.0220281
$$111$$ 9.76897 0.927229
$$112$$ −2.19177 −0.207103
$$113$$ −5.32025 −0.500487 −0.250243 0.968183i $$-0.580511\pi$$
−0.250243 + 0.968183i $$0.580511\pi$$
$$114$$ 0.0392613 0.00367716
$$115$$ 7.76897 0.724460
$$116$$ −10.4487 −0.970139
$$117$$ 0 0
$$118$$ −0.756009 −0.0695962
$$119$$ −4.90411 −0.449559
$$120$$ −1.32025 −0.120521
$$121$$ −10.5379 −0.957994
$$122$$ −1.42909 −0.129384
$$123$$ 4.22436 0.380898
$$124$$ −18.7953 −1.68787
$$125$$ −1.00000 −0.0894427
$$126$$ −0.224361 −0.0199876
$$127$$ −12.0196 −1.06657 −0.533285 0.845936i $$-0.679043\pi$$
−0.533285 + 0.845936i $$0.679043\pi$$
$$128$$ −9.35951 −0.827271
$$129$$ 0.544607 0.0479500
$$130$$ 0 0
$$131$$ 10.9041 0.952697 0.476348 0.879257i $$-0.341960\pi$$
0.476348 + 0.879257i $$0.341960\pi$$
$$132$$ −1.28098 −0.111495
$$133$$ 0.0762550 0.00661215
$$134$$ 2.59456 0.224136
$$135$$ 1.00000 0.0860663
$$136$$ −9.80823 −0.841049
$$137$$ −3.38983 −0.289613 −0.144806 0.989460i $$-0.546256\pi$$
−0.144806 + 0.989460i $$0.546256\pi$$
$$138$$ 2.64049 0.224774
$$139$$ 14.8082 1.25602 0.628009 0.778206i $$-0.283870\pi$$
0.628009 + 0.778206i $$0.283870\pi$$
$$140$$ −1.24399 −0.105136
$$141$$ −5.01963 −0.422729
$$142$$ −2.48571 −0.208597
$$143$$ 0 0
$$144$$ 3.32025 0.276687
$$145$$ −5.54461 −0.460455
$$146$$ −2.72569 −0.225579
$$147$$ 6.56424 0.541409
$$148$$ 18.4095 1.51325
$$149$$ 17.0892 1.40000 0.700001 0.714141i $$-0.253183\pi$$
0.700001 + 0.714141i $$0.253183\pi$$
$$150$$ −0.339877 −0.0277508
$$151$$ −13.0130 −1.05898 −0.529490 0.848316i $$-0.677617\pi$$
−0.529490 + 0.848316i $$0.677617\pi$$
$$152$$ 0.152510 0.0123702
$$153$$ 7.42909 0.600606
$$154$$ 0.152510 0.0122896
$$155$$ −9.97370 −0.801107
$$156$$ 0 0
$$157$$ −0.775639 −0.0619028 −0.0309514 0.999521i $$-0.509854\pi$$
−0.0309514 + 0.999521i $$0.509854\pi$$
$$158$$ 3.38983 0.269680
$$159$$ −0.679754 −0.0539080
$$160$$ −3.76897 −0.297963
$$161$$ 5.12847 0.404180
$$162$$ 0.339877 0.0267033
$$163$$ −12.1981 −0.955426 −0.477713 0.878516i $$-0.658534\pi$$
−0.477713 + 0.878516i $$0.658534\pi$$
$$164$$ 7.96074 0.621629
$$165$$ −0.679754 −0.0529188
$$166$$ −0.601231 −0.0466646
$$167$$ 1.59054 0.123080 0.0615398 0.998105i $$-0.480399\pi$$
0.0615398 + 0.998105i $$0.480399\pi$$
$$168$$ −0.871525 −0.0672396
$$169$$ 0 0
$$170$$ −2.52498 −0.193657
$$171$$ −0.115516 −0.00883375
$$172$$ 1.02630 0.0782548
$$173$$ 12.7493 0.969314 0.484657 0.874704i $$-0.338944\pi$$
0.484657 + 0.874704i $$0.338944\pi$$
$$174$$ −1.88448 −0.142862
$$175$$ −0.660123 −0.0499006
$$176$$ −2.25695 −0.170124
$$177$$ 2.22436 0.167193
$$178$$ −4.60350 −0.345047
$$179$$ 17.7623 1.32762 0.663808 0.747903i $$-0.268939\pi$$
0.663808 + 0.747903i $$0.268939\pi$$
$$180$$ 1.88448 0.140461
$$181$$ 7.02630 0.522261 0.261130 0.965304i $$-0.415905\pi$$
0.261130 + 0.965304i $$0.415905\pi$$
$$182$$ 0 0
$$183$$ 4.20473 0.310823
$$184$$ 10.2569 0.756152
$$185$$ 9.76897 0.718229
$$186$$ −3.38983 −0.248554
$$187$$ −5.04995 −0.369289
$$188$$ −9.45941 −0.689899
$$189$$ 0.660123 0.0480169
$$190$$ 0.0392613 0.00284832
$$191$$ −13.9541 −1.00968 −0.504840 0.863213i $$-0.668449\pi$$
−0.504840 + 0.863213i $$0.668449\pi$$
$$192$$ 5.35951 0.386789
$$193$$ −23.7493 −1.70951 −0.854757 0.519028i $$-0.826294\pi$$
−0.854757 + 0.519028i $$0.826294\pi$$
$$194$$ −3.36618 −0.241678
$$195$$ 0 0
$$196$$ 12.3702 0.883586
$$197$$ −15.8082 −1.12629 −0.563145 0.826358i $$-0.690409\pi$$
−0.563145 + 0.826358i $$0.690409\pi$$
$$198$$ −0.231033 −0.0164188
$$199$$ 8.76897 0.621616 0.310808 0.950473i $$-0.399400\pi$$
0.310808 + 0.950473i $$0.399400\pi$$
$$200$$ −1.32025 −0.0933555
$$201$$ −7.63382 −0.538448
$$202$$ −3.18108 −0.223820
$$203$$ −3.66012 −0.256890
$$204$$ 14.0000 0.980196
$$205$$ 4.22436 0.295042
$$206$$ 1.53126 0.106688
$$207$$ −7.76897 −0.539981
$$208$$ 0 0
$$209$$ 0.0785226 0.00543152
$$210$$ −0.224361 −0.0154824
$$211$$ 8.80823 0.606383 0.303192 0.952930i $$-0.401948\pi$$
0.303192 + 0.952930i $$0.401948\pi$$
$$212$$ −1.28098 −0.0879784
$$213$$ 7.31357 0.501118
$$214$$ 0.194039 0.0132642
$$215$$ 0.544607 0.0371419
$$216$$ 1.32025 0.0898314
$$217$$ −6.58387 −0.446942
$$218$$ −5.23732 −0.354716
$$219$$ 8.01963 0.541916
$$220$$ −1.28098 −0.0863640
$$221$$ 0 0
$$222$$ 3.32025 0.222840
$$223$$ −10.0393 −0.672279 −0.336139 0.941812i $$-0.609121\pi$$
−0.336139 + 0.941812i $$0.609121\pi$$
$$224$$ −2.48798 −0.166235
$$225$$ 1.00000 0.0666667
$$226$$ −1.80823 −0.120282
$$227$$ −1.21140 −0.0804036 −0.0402018 0.999192i $$-0.512800\pi$$
−0.0402018 + 0.999192i $$0.512800\pi$$
$$228$$ −0.217689 −0.0144168
$$229$$ −19.2440 −1.27168 −0.635839 0.771821i $$-0.719346\pi$$
−0.635839 + 0.771821i $$0.719346\pi$$
$$230$$ 2.64049 0.174109
$$231$$ −0.448721 −0.0295237
$$232$$ −7.32025 −0.480598
$$233$$ −23.9081 −1.56627 −0.783137 0.621849i $$-0.786382\pi$$
−0.783137 + 0.621849i $$0.786382\pi$$
$$234$$ 0 0
$$235$$ −5.01963 −0.327445
$$236$$ 4.19177 0.272861
$$237$$ −9.97370 −0.647861
$$238$$ −1.66680 −0.108042
$$239$$ −18.6798 −1.20829 −0.604146 0.796873i $$-0.706486\pi$$
−0.604146 + 0.796873i $$0.706486\pi$$
$$240$$ 3.32025 0.214321
$$241$$ −6.11552 −0.393935 −0.196968 0.980410i $$-0.563109\pi$$
−0.196968 + 0.980410i $$0.563109\pi$$
$$242$$ −3.58160 −0.230234
$$243$$ −1.00000 −0.0641500
$$244$$ 7.92375 0.507266
$$245$$ 6.56424 0.419374
$$246$$ 1.43576 0.0915409
$$247$$ 0 0
$$248$$ −13.1677 −0.836152
$$249$$ 1.76897 0.112104
$$250$$ −0.339877 −0.0214957
$$251$$ 3.35951 0.212050 0.106025 0.994363i $$-0.466188\pi$$
0.106025 + 0.994363i $$0.466188\pi$$
$$252$$ 1.24399 0.0783641
$$253$$ 5.28098 0.332013
$$254$$ −4.08519 −0.256328
$$255$$ 7.42909 0.465228
$$256$$ 7.53793 0.471121
$$257$$ −10.1088 −0.630572 −0.315286 0.948997i $$-0.602101\pi$$
−0.315286 + 0.948997i $$0.602101\pi$$
$$258$$ 0.185099 0.0115238
$$259$$ 6.44872 0.400704
$$260$$ 0 0
$$261$$ 5.54461 0.343203
$$262$$ 3.70606 0.228961
$$263$$ −22.5183 −1.38854 −0.694269 0.719716i $$-0.744272\pi$$
−0.694269 + 0.719716i $$0.744272\pi$$
$$264$$ −0.897442 −0.0552338
$$265$$ −0.679754 −0.0417569
$$266$$ 0.0259173 0.00158909
$$267$$ 13.5446 0.828916
$$268$$ −14.3858 −0.878753
$$269$$ 8.49465 0.517928 0.258964 0.965887i $$-0.416619\pi$$
0.258964 + 0.965887i $$0.416619\pi$$
$$270$$ 0.339877 0.0206842
$$271$$ −9.37020 −0.569199 −0.284600 0.958647i $$-0.591861\pi$$
−0.284600 + 0.958647i $$0.591861\pi$$
$$272$$ 24.6664 1.49562
$$273$$ 0 0
$$274$$ −1.15212 −0.0696024
$$275$$ −0.679754 −0.0409907
$$276$$ −14.6405 −0.881254
$$277$$ 0.719015 0.0432014 0.0216007 0.999767i $$-0.493124\pi$$
0.0216007 + 0.999767i $$0.493124\pi$$
$$278$$ 5.03297 0.301858
$$279$$ 9.97370 0.597110
$$280$$ −0.871525 −0.0520836
$$281$$ −1.54461 −0.0921435 −0.0460718 0.998938i $$-0.514670\pi$$
−0.0460718 + 0.998938i $$0.514670\pi$$
$$282$$ −1.70606 −0.101594
$$283$$ −18.1195 −1.07709 −0.538547 0.842595i $$-0.681027\pi$$
−0.538547 + 0.842595i $$0.681027\pi$$
$$284$$ 13.7823 0.817830
$$285$$ −0.115516 −0.00684259
$$286$$ 0 0
$$287$$ 2.78860 0.164606
$$288$$ 3.76897 0.222089
$$289$$ 38.1914 2.24655
$$290$$ −1.88448 −0.110661
$$291$$ 9.90411 0.580589
$$292$$ 15.1129 0.884413
$$293$$ 30.5050 1.78212 0.891059 0.453887i $$-0.149963\pi$$
0.891059 + 0.453887i $$0.149963\pi$$
$$294$$ 2.23103 0.130116
$$295$$ 2.22436 0.129507
$$296$$ 12.8974 0.749649
$$297$$ 0.679754 0.0394433
$$298$$ 5.80823 0.336462
$$299$$ 0 0
$$300$$ 1.88448 0.108801
$$301$$ 0.359508 0.0207217
$$302$$ −4.42280 −0.254504
$$303$$ 9.35951 0.537690
$$304$$ −0.383543 −0.0219977
$$305$$ 4.20473 0.240762
$$306$$ 2.52498 0.144343
$$307$$ 4.77564 0.272560 0.136280 0.990670i $$-0.456485\pi$$
0.136280 + 0.990670i $$0.456485\pi$$
$$308$$ −0.845608 −0.0481830
$$309$$ −4.50535 −0.256300
$$310$$ −3.38983 −0.192529
$$311$$ 30.5812 1.73410 0.867051 0.498220i $$-0.166013\pi$$
0.867051 + 0.498220i $$0.166013\pi$$
$$312$$ 0 0
$$313$$ −26.7230 −1.51048 −0.755238 0.655451i $$-0.772479\pi$$
−0.755238 + 0.655451i $$0.772479\pi$$
$$314$$ −0.263622 −0.0148770
$$315$$ 0.660123 0.0371937
$$316$$ −18.7953 −1.05732
$$317$$ 20.6271 1.15854 0.579268 0.815137i $$-0.303338\pi$$
0.579268 + 0.815137i $$0.303338\pi$$
$$318$$ −0.231033 −0.0129557
$$319$$ −3.76897 −0.211022
$$320$$ 5.35951 0.299606
$$321$$ −0.570909 −0.0318651
$$322$$ 1.74305 0.0971364
$$323$$ −0.858181 −0.0477505
$$324$$ −1.88448 −0.104694
$$325$$ 0 0
$$326$$ −4.14584 −0.229617
$$327$$ 15.4095 0.852145
$$328$$ 5.57720 0.307949
$$329$$ −3.31357 −0.182683
$$330$$ −0.231033 −0.0127179
$$331$$ −5.37020 −0.295173 −0.147586 0.989049i $$-0.547150\pi$$
−0.147586 + 0.989049i $$0.547150\pi$$
$$332$$ 3.33359 0.182955
$$333$$ −9.76897 −0.535336
$$334$$ 0.540588 0.0295797
$$335$$ −7.63382 −0.417080
$$336$$ 2.19177 0.119571
$$337$$ 28.7623 1.56678 0.783391 0.621529i $$-0.213488\pi$$
0.783391 + 0.621529i $$0.213488\pi$$
$$338$$ 0 0
$$339$$ 5.32025 0.288956
$$340$$ 14.0000 0.759257
$$341$$ −6.77966 −0.367139
$$342$$ −0.0392613 −0.00212301
$$343$$ 8.95407 0.483474
$$344$$ 0.719015 0.0387667
$$345$$ −7.76897 −0.418267
$$346$$ 4.33320 0.232955
$$347$$ 22.3961 1.20229 0.601143 0.799141i $$-0.294712\pi$$
0.601143 + 0.799141i $$0.294712\pi$$
$$348$$ 10.4487 0.560110
$$349$$ 11.9737 0.640937 0.320469 0.947259i $$-0.396160\pi$$
0.320469 + 0.947259i $$0.396160\pi$$
$$350$$ −0.224361 −0.0119926
$$351$$ 0 0
$$352$$ −2.56197 −0.136553
$$353$$ 17.0366 0.906767 0.453384 0.891316i $$-0.350217\pi$$
0.453384 + 0.891316i $$0.350217\pi$$
$$354$$ 0.756009 0.0401814
$$355$$ 7.31357 0.388164
$$356$$ 25.5246 1.35280
$$357$$ 4.90411 0.259553
$$358$$ 6.03699 0.319065
$$359$$ 35.0825 1.85159 0.925793 0.378031i $$-0.123399\pi$$
0.925793 + 0.378031i $$0.123399\pi$$
$$360$$ 1.32025 0.0695831
$$361$$ −18.9867 −0.999298
$$362$$ 2.38808 0.125515
$$363$$ 10.5379 0.553098
$$364$$ 0 0
$$365$$ 8.01963 0.419767
$$366$$ 1.42909 0.0746998
$$367$$ 20.0825 1.04830 0.524150 0.851626i $$-0.324383\pi$$
0.524150 + 0.851626i $$0.324383\pi$$
$$368$$ −25.7949 −1.34465
$$369$$ −4.22436 −0.219911
$$370$$ 3.32025 0.172611
$$371$$ −0.448721 −0.0232964
$$372$$ 18.7953 0.974489
$$373$$ −23.4790 −1.21570 −0.607849 0.794052i $$-0.707968\pi$$
−0.607849 + 0.794052i $$0.707968\pi$$
$$374$$ −1.71636 −0.0887510
$$375$$ 1.00000 0.0516398
$$376$$ −6.62715 −0.341769
$$377$$ 0 0
$$378$$ 0.224361 0.0115399
$$379$$ −21.1414 −1.08596 −0.542981 0.839745i $$-0.682705\pi$$
−0.542981 + 0.839745i $$0.682705\pi$$
$$380$$ −0.217689 −0.0111672
$$381$$ 12.0196 0.615784
$$382$$ −4.74266 −0.242656
$$383$$ −1.73865 −0.0888406 −0.0444203 0.999013i $$-0.514144\pi$$
−0.0444203 + 0.999013i $$0.514144\pi$$
$$384$$ 9.35951 0.477625
$$385$$ −0.448721 −0.0228689
$$386$$ −8.07185 −0.410846
$$387$$ −0.544607 −0.0276839
$$388$$ 18.6641 0.947528
$$389$$ −26.6664 −1.35204 −0.676020 0.736883i $$-0.736297\pi$$
−0.676020 + 0.736883i $$0.736297\pi$$
$$390$$ 0 0
$$391$$ −57.7164 −2.91884
$$392$$ 8.66641 0.437720
$$393$$ −10.9041 −0.550040
$$394$$ −5.37285 −0.270680
$$395$$ −9.97370 −0.501831
$$396$$ 1.28098 0.0643719
$$397$$ −14.5446 −0.729973 −0.364986 0.931013i $$-0.618926\pi$$
−0.364986 + 0.931013i $$0.618926\pi$$
$$398$$ 2.98037 0.149392
$$399$$ −0.0762550 −0.00381752
$$400$$ 3.32025 0.166012
$$401$$ 8.37020 0.417988 0.208994 0.977917i $$-0.432981\pi$$
0.208994 + 0.977917i $$0.432981\pi$$
$$402$$ −2.59456 −0.129405
$$403$$ 0 0
$$404$$ 17.6378 0.877515
$$405$$ −1.00000 −0.0496904
$$406$$ −1.24399 −0.0617382
$$407$$ 6.64049 0.329157
$$408$$ 9.80823 0.485580
$$409$$ −17.2547 −0.853189 −0.426595 0.904443i $$-0.640287\pi$$
−0.426595 + 0.904443i $$0.640287\pi$$
$$410$$ 1.43576 0.0709073
$$411$$ 3.38983 0.167208
$$412$$ −8.49025 −0.418285
$$413$$ 1.46835 0.0722529
$$414$$ −2.64049 −0.129773
$$415$$ 1.76897 0.0868352
$$416$$ 0 0
$$417$$ −14.8082 −0.725162
$$418$$ 0.0266880 0.00130535
$$419$$ 12.8649 0.628489 0.314245 0.949342i $$-0.398249\pi$$
0.314245 + 0.949342i $$0.398249\pi$$
$$420$$ 1.24399 0.0607006
$$421$$ −24.5616 −1.19706 −0.598529 0.801101i $$-0.704248\pi$$
−0.598529 + 0.801101i $$0.704248\pi$$
$$422$$ 2.99371 0.145732
$$423$$ 5.01963 0.244063
$$424$$ −0.897442 −0.0435837
$$425$$ 7.42909 0.360364
$$426$$ 2.48571 0.120433
$$427$$ 2.77564 0.134323
$$428$$ −1.07587 −0.0520041
$$429$$ 0 0
$$430$$ 0.185099 0.00892628
$$431$$ −24.7797 −1.19359 −0.596797 0.802392i $$-0.703560\pi$$
−0.596797 + 0.802392i $$0.703560\pi$$
$$432$$ −3.32025 −0.159745
$$433$$ −14.0589 −0.675627 −0.337814 0.941213i $$-0.609687\pi$$
−0.337814 + 0.941213i $$0.609687\pi$$
$$434$$ −2.23770 −0.107413
$$435$$ 5.54461 0.265844
$$436$$ 29.0389 1.39071
$$437$$ 0.897442 0.0429305
$$438$$ 2.72569 0.130238
$$439$$ 20.8452 0.994888 0.497444 0.867496i $$-0.334272\pi$$
0.497444 + 0.867496i $$0.334272\pi$$
$$440$$ −0.897442 −0.0427839
$$441$$ −6.56424 −0.312583
$$442$$ 0 0
$$443$$ −11.4291 −0.543012 −0.271506 0.962437i $$-0.587522\pi$$
−0.271506 + 0.962437i $$0.587522\pi$$
$$444$$ −18.4095 −0.873674
$$445$$ 13.5446 0.642076
$$446$$ −3.41211 −0.161568
$$447$$ −17.0892 −0.808292
$$448$$ 3.53793 0.167152
$$449$$ −25.9081 −1.22268 −0.611340 0.791368i $$-0.709369\pi$$
−0.611340 + 0.791368i $$0.709369\pi$$
$$450$$ 0.339877 0.0160220
$$451$$ 2.87153 0.135215
$$452$$ 10.0259 0.471579
$$453$$ 13.0130 0.611402
$$454$$ −0.411728 −0.0193233
$$455$$ 0 0
$$456$$ −0.152510 −0.00714193
$$457$$ −5.90411 −0.276183 −0.138091 0.990419i $$-0.544097\pi$$
−0.138091 + 0.990419i $$0.544097\pi$$
$$458$$ −6.54059 −0.305622
$$459$$ −7.42909 −0.346760
$$460$$ −14.6405 −0.682616
$$461$$ 15.4920 0.721534 0.360767 0.932656i $$-0.382515\pi$$
0.360767 + 0.932656i $$0.382515\pi$$
$$462$$ −0.152510 −0.00709541
$$463$$ −17.4420 −0.810601 −0.405300 0.914184i $$-0.632833\pi$$
−0.405300 + 0.914184i $$0.632833\pi$$
$$464$$ 18.4095 0.854638
$$465$$ 9.97370 0.462519
$$466$$ −8.12582 −0.376421
$$467$$ −20.4790 −0.947657 −0.473829 0.880617i $$-0.657128\pi$$
−0.473829 + 0.880617i $$0.657128\pi$$
$$468$$ 0 0
$$469$$ −5.03926 −0.232691
$$470$$ −1.70606 −0.0786945
$$471$$ 0.775639 0.0357396
$$472$$ 2.93670 0.135173
$$473$$ 0.370199 0.0170217
$$474$$ −3.38983 −0.155700
$$475$$ −0.115516 −0.00530025
$$476$$ 9.24172 0.423594
$$477$$ 0.679754 0.0311238
$$478$$ −6.34882 −0.290388
$$479$$ 40.9907 1.87291 0.936456 0.350785i $$-0.114085\pi$$
0.936456 + 0.350785i $$0.114085\pi$$
$$480$$ 3.76897 0.172029
$$481$$ 0 0
$$482$$ −2.07852 −0.0946741
$$483$$ −5.12847 −0.233354
$$484$$ 19.8586 0.902662
$$485$$ 9.90411 0.449723
$$486$$ −0.339877 −0.0154171
$$487$$ −24.0629 −1.09039 −0.545197 0.838308i $$-0.683545\pi$$
−0.545197 + 0.838308i $$0.683545\pi$$
$$488$$ 5.55128 0.251295
$$489$$ 12.1981 0.551615
$$490$$ 2.23103 0.100788
$$491$$ 36.7730 1.65954 0.829771 0.558104i $$-0.188471\pi$$
0.829771 + 0.558104i $$0.188471\pi$$
$$492$$ −7.96074 −0.358898
$$493$$ 41.1914 1.85517
$$494$$ 0 0
$$495$$ 0.679754 0.0305527
$$496$$ 33.1151 1.48691
$$497$$ 4.82786 0.216559
$$498$$ 0.601231 0.0269418
$$499$$ −34.8212 −1.55881 −0.779405 0.626520i $$-0.784479\pi$$
−0.779405 + 0.626520i $$0.784479\pi$$
$$500$$ 1.88448 0.0842767
$$501$$ −1.59054 −0.0710601
$$502$$ 1.14182 0.0509619
$$503$$ −32.0259 −1.42797 −0.713983 0.700164i $$-0.753110\pi$$
−0.713983 + 0.700164i $$0.753110\pi$$
$$504$$ 0.871525 0.0388208
$$505$$ 9.35951 0.416493
$$506$$ 1.79488 0.0797924
$$507$$ 0 0
$$508$$ 22.6508 1.00497
$$509$$ 41.3462 1.83264 0.916318 0.400451i $$-0.131146\pi$$
0.916318 + 0.400451i $$0.131146\pi$$
$$510$$ 2.52498 0.111808
$$511$$ 5.29394 0.234190
$$512$$ 21.2810 0.940496
$$513$$ 0.115516 0.00510017
$$514$$ −3.43576 −0.151545
$$515$$ −4.50535 −0.198529
$$516$$ −1.02630 −0.0451805
$$517$$ −3.41211 −0.150065
$$518$$ 2.19177 0.0963009
$$519$$ −12.7493 −0.559634
$$520$$ 0 0
$$521$$ 4.40279 0.192890 0.0964448 0.995338i $$-0.469253\pi$$
0.0964448 + 0.995338i $$0.469253\pi$$
$$522$$ 1.88448 0.0824816
$$523$$ −32.4487 −1.41888 −0.709442 0.704764i $$-0.751053\pi$$
−0.709442 + 0.704764i $$0.751053\pi$$
$$524$$ −20.5486 −0.897671
$$525$$ 0.660123 0.0288101
$$526$$ −7.65345 −0.333706
$$527$$ 74.0955 3.22765
$$528$$ 2.25695 0.0982211
$$529$$ 37.3569 1.62421
$$530$$ −0.231033 −0.0100354
$$531$$ −2.22436 −0.0965290
$$532$$ −0.143701 −0.00623024
$$533$$ 0 0
$$534$$ 4.60350 0.199213
$$535$$ −0.570909 −0.0246826
$$536$$ −10.0785 −0.435326
$$537$$ −17.7623 −0.766500
$$538$$ 2.88714 0.124473
$$539$$ 4.46207 0.192195
$$540$$ −1.88448 −0.0810953
$$541$$ 8.57720 0.368762 0.184381 0.982855i $$-0.440972\pi$$
0.184381 + 0.982855i $$0.440972\pi$$
$$542$$ −3.18471 −0.136795
$$543$$ −7.02630 −0.301528
$$544$$ 28.0000 1.20049
$$545$$ 15.4095 0.660069
$$546$$ 0 0
$$547$$ −32.4920 −1.38926 −0.694629 0.719368i $$-0.744431\pi$$
−0.694629 + 0.719368i $$0.744431\pi$$
$$548$$ 6.38808 0.272885
$$549$$ −4.20473 −0.179454
$$550$$ −0.231033 −0.00985126
$$551$$ −0.640492 −0.0272859
$$552$$ −10.2569 −0.436565
$$553$$ −6.58387 −0.279975
$$554$$ 0.244377 0.0103826
$$555$$ −9.76897 −0.414670
$$556$$ −27.9059 −1.18347
$$557$$ 41.2150 1.74634 0.873169 0.487418i $$-0.162061\pi$$
0.873169 + 0.487418i $$0.162061\pi$$
$$558$$ 3.38983 0.143503
$$559$$ 0 0
$$560$$ 2.19177 0.0926192
$$561$$ 5.04995 0.213209
$$562$$ −0.524976 −0.0221448
$$563$$ −12.1784 −0.513260 −0.256630 0.966510i $$-0.582612\pi$$
−0.256630 + 0.966510i $$0.582612\pi$$
$$564$$ 9.45941 0.398313
$$565$$ 5.32025 0.223824
$$566$$ −6.15841 −0.258857
$$567$$ −0.660123 −0.0277226
$$568$$ 9.65572 0.405145
$$569$$ 6.94338 0.291081 0.145541 0.989352i $$-0.453508\pi$$
0.145541 + 0.989352i $$0.453508\pi$$
$$570$$ −0.0392613 −0.00164448
$$571$$ 3.51429 0.147068 0.0735341 0.997293i $$-0.476572\pi$$
0.0735341 + 0.997293i $$0.476572\pi$$
$$572$$ 0 0
$$573$$ 13.9541 0.582939
$$574$$ 0.947780 0.0395596
$$575$$ −7.76897 −0.323988
$$576$$ −5.35951 −0.223313
$$577$$ −3.14182 −0.130796 −0.0653978 0.997859i $$-0.520832\pi$$
−0.0653978 + 0.997859i $$0.520832\pi$$
$$578$$ 12.9804 0.539912
$$579$$ 23.7493 0.986989
$$580$$ 10.4487 0.433860
$$581$$ 1.16774 0.0484459
$$582$$ 3.36618 0.139533
$$583$$ −0.462065 −0.0191368
$$584$$ 10.5879 0.438130
$$585$$ 0 0
$$586$$ 10.3679 0.428295
$$587$$ −37.9777 −1.56751 −0.783754 0.621071i $$-0.786698\pi$$
−0.783754 + 0.621071i $$0.786698\pi$$
$$588$$ −12.3702 −0.510138
$$589$$ −1.15212 −0.0474725
$$590$$ 0.756009 0.0311244
$$591$$ 15.8082 0.650264
$$592$$ −32.4354 −1.33309
$$593$$ −10.4487 −0.429078 −0.214539 0.976715i $$-0.568825\pi$$
−0.214539 + 0.976715i $$0.568825\pi$$
$$594$$ 0.231033 0.00947938
$$595$$ 4.90411 0.201049
$$596$$ −32.2043 −1.31914
$$597$$ −8.76897 −0.358890
$$598$$ 0 0
$$599$$ −45.5705 −1.86196 −0.930981 0.365069i $$-0.881045\pi$$
−0.930981 + 0.365069i $$0.881045\pi$$
$$600$$ 1.32025 0.0538988
$$601$$ −14.7819 −0.602967 −0.301484 0.953471i $$-0.597482\pi$$
−0.301484 + 0.953471i $$0.597482\pi$$
$$602$$ 0.122188 0.00498002
$$603$$ 7.63382 0.310873
$$604$$ 24.5227 0.997815
$$605$$ 10.5379 0.428428
$$606$$ 3.18108 0.129223
$$607$$ −18.9344 −0.768525 −0.384263 0.923224i $$-0.625544\pi$$
−0.384263 + 0.923224i $$0.625544\pi$$
$$608$$ −0.435377 −0.0176569
$$609$$ 3.66012 0.148316
$$610$$ 1.42909 0.0578622
$$611$$ 0 0
$$612$$ −14.0000 −0.565916
$$613$$ 2.58387 0.104361 0.0521807 0.998638i $$-0.483383\pi$$
0.0521807 + 0.998638i $$0.483383\pi$$
$$614$$ 1.62313 0.0655042
$$615$$ −4.22436 −0.170343
$$616$$ −0.592422 −0.0238694
$$617$$ −14.0393 −0.565199 −0.282600 0.959238i $$-0.591197\pi$$
−0.282600 + 0.959238i $$0.591197\pi$$
$$618$$ −1.53126 −0.0615964
$$619$$ 17.8582 0.717781 0.358890 0.933380i $$-0.383155\pi$$
0.358890 + 0.933380i $$0.383155\pi$$
$$620$$ 18.7953 0.754836
$$621$$ 7.76897 0.311758
$$622$$ 10.3938 0.416755
$$623$$ 8.94111 0.358218
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −9.08254 −0.363011
$$627$$ −0.0785226 −0.00313589
$$628$$ 1.46168 0.0583274
$$629$$ −72.5745 −2.89374
$$630$$ 0.224361 0.00893874
$$631$$ 24.9630 0.993762 0.496881 0.867819i $$-0.334479\pi$$
0.496881 + 0.867819i $$0.334479\pi$$
$$632$$ −13.1677 −0.523784
$$633$$ −8.80823 −0.350096
$$634$$ 7.01069 0.278430
$$635$$ 12.0196 0.476984
$$636$$ 1.28098 0.0507944
$$637$$ 0 0
$$638$$ −1.28098 −0.0507147
$$639$$ −7.31357 −0.289321
$$640$$ 9.35951 0.369967
$$641$$ 23.3069 0.920567 0.460284 0.887772i $$-0.347748\pi$$
0.460284 + 0.887772i $$0.347748\pi$$
$$642$$ −0.194039 −0.00765811
$$643$$ −45.4790 −1.79352 −0.896759 0.442519i $$-0.854085\pi$$
−0.896759 + 0.442519i $$0.854085\pi$$
$$644$$ −9.66453 −0.380836
$$645$$ −0.544607 −0.0214439
$$646$$ −0.291676 −0.0114758
$$647$$ 6.40946 0.251982 0.125991 0.992031i $$-0.459789\pi$$
0.125991 + 0.992031i $$0.459789\pi$$
$$648$$ −1.32025 −0.0518642
$$649$$ 1.51202 0.0593519
$$650$$ 0 0
$$651$$ 6.58387 0.258042
$$652$$ 22.9870 0.900242
$$653$$ 1.48170 0.0579832 0.0289916 0.999580i $$-0.490770\pi$$
0.0289916 + 0.999580i $$0.490770\pi$$
$$654$$ 5.23732 0.204795
$$655$$ −10.9041 −0.426059
$$656$$ −14.0259 −0.547620
$$657$$ −8.01963 −0.312876
$$658$$ −1.12621 −0.0439041
$$659$$ −7.08921 −0.276157 −0.138078 0.990421i $$-0.544093\pi$$
−0.138078 + 0.990421i $$0.544093\pi$$
$$660$$ 1.28098 0.0498623
$$661$$ −1.17843 −0.0458355 −0.0229178 0.999737i $$-0.507296\pi$$
−0.0229178 + 0.999737i $$0.507296\pi$$
$$662$$ −1.82521 −0.0709387
$$663$$ 0 0
$$664$$ 2.33547 0.0906339
$$665$$ −0.0762550 −0.00295704
$$666$$ −3.32025 −0.128657
$$667$$ −43.0759 −1.66790
$$668$$ −2.99735 −0.115971
$$669$$ 10.0393 0.388140
$$670$$ −2.59456 −0.100237
$$671$$ 2.85818 0.110339
$$672$$ 2.48798 0.0959760
$$673$$ 13.5576 0.522606 0.261303 0.965257i $$-0.415848\pi$$
0.261303 + 0.965257i $$0.415848\pi$$
$$674$$ 9.77564 0.376544
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 9.53793 0.366573 0.183286 0.983060i $$-0.441326\pi$$
0.183286 + 0.983060i $$0.441326\pi$$
$$678$$ 1.80823 0.0694446
$$679$$ 6.53793 0.250903
$$680$$ 9.80823 0.376128
$$681$$ 1.21140 0.0464210
$$682$$ −2.30425 −0.0882343
$$683$$ 44.5183 1.70345 0.851723 0.523993i $$-0.175558\pi$$
0.851723 + 0.523993i $$0.175558\pi$$
$$684$$ 0.217689 0.00832353
$$685$$ 3.38983 0.129519
$$686$$ 3.04328 0.116193
$$687$$ 19.2440 0.734204
$$688$$ −1.80823 −0.0689381
$$689$$ 0 0
$$690$$ −2.64049 −0.100522
$$691$$ 5.28060 0.200883 0.100442 0.994943i $$-0.467974\pi$$
0.100442 + 0.994943i $$0.467974\pi$$
$$692$$ −24.0259 −0.913328
$$693$$ 0.448721 0.0170455
$$694$$ 7.61192 0.288945
$$695$$ −14.8082 −0.561708
$$696$$ 7.32025 0.277473
$$697$$ −31.3832 −1.18872
$$698$$ 4.06958 0.154036
$$699$$ 23.9081 0.904289
$$700$$ 1.24399 0.0470184
$$701$$ 20.1392 0.760646 0.380323 0.924854i $$-0.375813\pi$$
0.380323 + 0.924854i $$0.375813\pi$$
$$702$$ 0 0
$$703$$ 1.12847 0.0425612
$$704$$ 3.64315 0.137306
$$705$$ 5.01963 0.189050
$$706$$ 5.79035 0.217923
$$707$$ 6.17843 0.232364
$$708$$ −4.19177 −0.157536
$$709$$ −1.54059 −0.0578580 −0.0289290 0.999581i $$-0.509210\pi$$
−0.0289290 + 0.999581i $$0.509210\pi$$
$$710$$ 2.48571 0.0932872
$$711$$ 9.97370 0.374043
$$712$$ 17.8822 0.670164
$$713$$ −77.4853 −2.90185
$$714$$ 1.66680 0.0623782
$$715$$ 0 0
$$716$$ −33.4728 −1.25094
$$717$$ 18.6798 0.697608
$$718$$ 11.9237 0.444990
$$719$$ 37.0040 1.38002 0.690009 0.723801i $$-0.257607\pi$$
0.690009 + 0.723801i $$0.257607\pi$$
$$720$$ −3.32025 −0.123738
$$721$$ −2.97408 −0.110761
$$722$$ −6.45313 −0.240160
$$723$$ 6.11552 0.227438
$$724$$ −13.2410 −0.492096
$$725$$ 5.54461 0.205922
$$726$$ 3.58160 0.132926
$$727$$ −44.9015 −1.66530 −0.832652 0.553797i $$-0.813178\pi$$
−0.832652 + 0.553797i $$0.813178\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.72569 0.100882
$$731$$ −4.04593 −0.149644
$$732$$ −7.92375 −0.292870
$$733$$ 4.94072 0.182490 0.0912449 0.995828i $$-0.470915\pi$$
0.0912449 + 0.995828i $$0.470915\pi$$
$$734$$ 6.82559 0.251937
$$735$$ −6.56424 −0.242126
$$736$$ −29.2810 −1.07931
$$737$$ −5.18912 −0.191144
$$738$$ −1.43576 −0.0528511
$$739$$ −6.03926 −0.222158 −0.111079 0.993812i $$-0.535431\pi$$
−0.111079 + 0.993812i $$0.535431\pi$$
$$740$$ −18.4095 −0.676745
$$741$$ 0 0
$$742$$ −0.152510 −0.00559882
$$743$$ 13.0589 0.479084 0.239542 0.970886i $$-0.423003\pi$$
0.239542 + 0.970886i $$0.423003\pi$$
$$744$$ 13.1677 0.482753
$$745$$ −17.0892 −0.626100
$$746$$ −7.97998 −0.292168
$$747$$ −1.76897 −0.0647231
$$748$$ 9.51655 0.347960
$$749$$ −0.376871 −0.0137706
$$750$$ 0.339877 0.0124105
$$751$$ 48.0236 1.75241 0.876204 0.481941i $$-0.160068\pi$$
0.876204 + 0.481941i $$0.160068\pi$$
$$752$$ 16.6664 0.607761
$$753$$ −3.35951 −0.122427
$$754$$ 0 0
$$755$$ 13.0130 0.473590
$$756$$ −1.24399 −0.0452435
$$757$$ −4.56424 −0.165890 −0.0829450 0.996554i $$-0.526433\pi$$
−0.0829450 + 0.996554i $$0.526433\pi$$
$$758$$ −7.18548 −0.260989
$$759$$ −5.28098 −0.191688
$$760$$ −0.152510 −0.00553212
$$761$$ −21.0892 −0.764483 −0.382242 0.924062i $$-0.624848\pi$$
−0.382242 + 0.924062i $$0.624848\pi$$
$$762$$ 4.08519 0.147991
$$763$$ 10.1721 0.368256
$$764$$ 26.2962 0.951364
$$765$$ −7.42909 −0.268599
$$766$$ −0.590926 −0.0213510
$$767$$ 0 0
$$768$$ −7.53793 −0.272002
$$769$$ −42.3332 −1.52657 −0.763287 0.646059i $$-0.776416\pi$$
−0.763287 + 0.646059i $$0.776416\pi$$
$$770$$ −0.152510 −0.00549608
$$771$$ 10.1088 0.364061
$$772$$ 44.7552 1.61078
$$773$$ −50.2436 −1.80714 −0.903568 0.428444i $$-0.859062\pi$$
−0.903568 + 0.428444i $$0.859062\pi$$
$$774$$ −0.185099 −0.00665326
$$775$$ 9.97370 0.358266
$$776$$ 13.0759 0.469396
$$777$$ −6.44872 −0.231347
$$778$$ −9.06330 −0.324935
$$779$$ 0.487982 0.0174838
$$780$$ 0 0
$$781$$ 4.97143 0.177892
$$782$$ −19.6165 −0.701483
$$783$$ −5.54461 −0.198148
$$784$$ −21.7949 −0.778389
$$785$$ 0.775639 0.0276838
$$786$$ −3.70606 −0.132191
$$787$$ 10.5816 0.377193 0.188597 0.982055i $$-0.439606\pi$$
0.188597 + 0.982055i $$0.439606\pi$$
$$788$$ 29.7903 1.06124
$$789$$ 22.5183 0.801673
$$790$$ −3.38983 −0.120605
$$791$$ 3.51202 0.124873
$$792$$ 0.897442 0.0318892
$$793$$ 0 0
$$794$$ −4.94338 −0.175434
$$795$$ 0.679754 0.0241084
$$796$$ −16.5250 −0.585712
$$797$$ −20.1481 −0.713683 −0.356841 0.934165i $$-0.616146\pi$$
−0.356841 + 0.934165i $$0.616146\pi$$
$$798$$ −0.0259173 −0.000917463 0
$$799$$ 37.2913 1.31927
$$800$$ 3.76897 0.133253
$$801$$ −13.5446 −0.478575
$$802$$ 2.84484 0.100455
$$803$$ 5.45137 0.192375
$$804$$ 14.3858 0.507348
$$805$$ −5.12847 −0.180755
$$806$$ 0 0
$$807$$ −8.49465 −0.299026
$$808$$ 12.3569 0.434713
$$809$$ 12.9108 0.453919 0.226960 0.973904i $$-0.427121\pi$$
0.226960 + 0.973904i $$0.427121\pi$$
$$810$$ −0.339877 −0.0119421
$$811$$ 15.8974 0.558235 0.279117 0.960257i $$-0.409958\pi$$
0.279117 + 0.960257i $$0.409958\pi$$
$$812$$ 6.89744 0.242053
$$813$$ 9.37020 0.328627
$$814$$ 2.25695 0.0791061
$$815$$ 12.1981 0.427279
$$816$$ −24.6664 −0.863497
$$817$$ 0.0629110 0.00220098
$$818$$ −5.86447 −0.205046
$$819$$ 0 0
$$820$$ −7.96074 −0.278001
$$821$$ 37.2284 1.29928 0.649640 0.760242i $$-0.274920\pi$$
0.649640 + 0.760242i $$0.274920\pi$$
$$822$$ 1.15212 0.0401850
$$823$$ −4.89744 −0.170714 −0.0853571 0.996350i $$-0.527203\pi$$
−0.0853571 + 0.996350i $$0.527203\pi$$
$$824$$ −5.94817 −0.207214
$$825$$ 0.679754 0.0236660
$$826$$ 0.499059 0.0173645
$$827$$ −37.6334 −1.30864 −0.654321 0.756217i $$-0.727046\pi$$
−0.654321 + 0.756217i $$0.727046\pi$$
$$828$$ 14.6405 0.508792
$$829$$ −48.9104 −1.69873 −0.849364 0.527807i $$-0.823014\pi$$
−0.849364 + 0.527807i $$0.823014\pi$$
$$830$$ 0.601231 0.0208690
$$831$$ −0.719015 −0.0249424
$$832$$ 0 0
$$833$$ −48.7663 −1.68965
$$834$$ −5.03297 −0.174278
$$835$$ −1.59054 −0.0550429
$$836$$ −0.147975 −0.00511781
$$837$$ −9.97370 −0.344741
$$838$$ 4.37247 0.151044
$$839$$ −48.2783 −1.66675 −0.833377 0.552706i $$-0.813595\pi$$
−0.833377 + 0.552706i $$0.813595\pi$$
$$840$$ 0.871525 0.0300705
$$841$$ 1.74266 0.0600919
$$842$$ −8.34791 −0.287688
$$843$$ 1.54461 0.0531991
$$844$$ −16.5990 −0.571360
$$845$$ 0 0
$$846$$ 1.70606 0.0586554
$$847$$ 6.95633 0.239022
$$848$$ 2.25695 0.0775040
$$849$$ 18.1195 0.621861
$$850$$ 2.52498 0.0866060
$$851$$ 75.8948 2.60164
$$852$$ −13.7823 −0.472174
$$853$$ 14.4291 0.494043 0.247021 0.969010i $$-0.420548\pi$$
0.247021 + 0.969010i $$0.420548\pi$$
$$854$$ 0.943376 0.0322817
$$855$$ 0.115516 0.00395057
$$856$$ −0.753741 −0.0257623
$$857$$ 3.64678 0.124572 0.0622858 0.998058i $$-0.480161\pi$$
0.0622858 + 0.998058i $$0.480161\pi$$
$$858$$ 0 0
$$859$$ 21.7427 0.741850 0.370925 0.928663i $$-0.379041\pi$$
0.370925 + 0.928663i $$0.379041\pi$$
$$860$$ −1.02630 −0.0349966
$$861$$ −2.78860 −0.0950352
$$862$$ −8.42203 −0.286856
$$863$$ 17.0892 0.581724 0.290862 0.956765i $$-0.406058\pi$$
0.290862 + 0.956765i $$0.406058\pi$$
$$864$$ −3.76897 −0.128223
$$865$$ −12.7493 −0.433490
$$866$$ −4.77829 −0.162373
$$867$$ −38.1914 −1.29705
$$868$$ 12.4072 0.421128
$$869$$ −6.77966 −0.229984
$$870$$ 1.88448 0.0638900
$$871$$ 0 0
$$872$$ 20.3443 0.688944
$$873$$ −9.90411 −0.335203
$$874$$ 0.305020 0.0103175
$$875$$ 0.660123 0.0223162
$$876$$ −15.1129 −0.510616
$$877$$ 4.85818 0.164049 0.0820246 0.996630i $$-0.473861\pi$$
0.0820246 + 0.996630i $$0.473861\pi$$
$$878$$ 7.08481 0.239101
$$879$$ −30.5050 −1.02891
$$880$$ 2.25695 0.0760818
$$881$$ −23.3202 −0.785679 −0.392840 0.919607i $$-0.628507\pi$$
−0.392840 + 0.919607i $$0.628507\pi$$
$$882$$ −2.23103 −0.0751228
$$883$$ −12.8934 −0.433898 −0.216949 0.976183i $$-0.569611\pi$$
−0.216949 + 0.976183i $$0.569611\pi$$
$$884$$ 0 0
$$885$$ −2.22436 −0.0747711
$$886$$ −3.88448 −0.130502
$$887$$ 30.9278 1.03845 0.519226 0.854637i $$-0.326220\pi$$
0.519226 + 0.854637i $$0.326220\pi$$
$$888$$ −12.8974 −0.432810
$$889$$ 7.93444 0.266112
$$890$$ 4.60350 0.154310
$$891$$ −0.679754 −0.0227726
$$892$$ 18.9188 0.633449
$$893$$ −0.579849 −0.0194039
$$894$$ −5.80823 −0.194256
$$895$$ −17.7623 −0.593728
$$896$$ 6.17843 0.206407
$$897$$ 0 0
$$898$$ −8.80558 −0.293846
$$899$$ 55.3002 1.84437
$$900$$ −1.88448 −0.0628161
$$901$$ 5.04995 0.168238
$$902$$ 0.975965 0.0324961
$$903$$ −0.359508 −0.0119637
$$904$$ 7.02404 0.233616
$$905$$ −7.02630 −0.233562
$$906$$ 4.42280 0.146938
$$907$$ −16.9604 −0.563159 −0.281580 0.959538i $$-0.590858\pi$$
−0.281580 + 0.959538i $$0.590858\pi$$
$$908$$ 2.28287 0.0757596
$$909$$ −9.35951 −0.310435
$$910$$ 0 0
$$911$$ 37.7297 1.25004 0.625020 0.780608i $$-0.285091\pi$$
0.625020 + 0.780608i $$0.285091\pi$$
$$912$$ 0.383543 0.0127004
$$913$$ 1.20246 0.0397957
$$914$$ −2.00667 −0.0663748
$$915$$ −4.20473 −0.139004
$$916$$ 36.2650 1.19823
$$917$$ −7.19806 −0.237701
$$918$$ −2.52498 −0.0833366
$$919$$ 40.1628 1.32485 0.662425 0.749129i $$-0.269528\pi$$
0.662425 + 0.749129i $$0.269528\pi$$
$$920$$ −10.2569 −0.338162
$$921$$ −4.77564 −0.157363
$$922$$ 5.26537 0.173406
$$923$$ 0 0
$$924$$ 0.845608 0.0278185
$$925$$ −9.76897 −0.321202
$$926$$ −5.92815 −0.194811
$$927$$ 4.50535 0.147975
$$928$$ 20.8974 0.685992
$$929$$ −23.7690 −0.779835 −0.389917 0.920850i $$-0.627496\pi$$
−0.389917 + 0.920850i $$0.627496\pi$$
$$930$$ 3.38983 0.111157
$$931$$ 0.758276 0.0248515
$$932$$ 45.0545 1.47581
$$933$$ −30.5812 −1.00118
$$934$$ −6.96035 −0.227750
$$935$$ 5.04995 0.165151
$$936$$ 0 0
$$937$$ −7.43803 −0.242990 −0.121495 0.992592i $$-0.538769\pi$$
−0.121495 + 0.992592i $$0.538769\pi$$
$$938$$ −1.71273 −0.0559226
$$939$$ 26.7230 0.872073
$$940$$ 9.45941 0.308532
$$941$$ 19.3528 0.630884 0.315442 0.948945i $$-0.397847\pi$$
0.315442 + 0.948945i $$0.397847\pi$$
$$942$$ 0.263622 0.00858927
$$943$$ 32.8189 1.06873
$$944$$ −7.38542 −0.240375
$$945$$ −0.660123 −0.0214738
$$946$$ 0.125822 0.00409082
$$947$$ −13.9171 −0.452244 −0.226122 0.974099i $$-0.572605\pi$$
−0.226122 + 0.974099i $$0.572605\pi$$
$$948$$ 18.7953 0.610442
$$949$$ 0 0
$$950$$ −0.0392613 −0.00127381
$$951$$ −20.6271 −0.668881
$$952$$ 6.47464 0.209844
$$953$$ −10.6271 −0.344247 −0.172124 0.985075i $$-0.555063\pi$$
−0.172124 + 0.985075i $$0.555063\pi$$
$$954$$ 0.231033 0.00747996
$$955$$ 13.9541 0.451543
$$956$$ 35.2017 1.13850
$$957$$ 3.76897 0.121833
$$958$$ 13.9318 0.450115
$$959$$ 2.23770 0.0722593
$$960$$ −5.35951 −0.172977
$$961$$ 68.4746 2.20886
$$962$$ 0 0
$$963$$ 0.570909 0.0183973
$$964$$ 11.5246 0.371182
$$965$$ 23.7493 0.764518
$$966$$ −1.74305 −0.0560817
$$967$$ −51.6771 −1.66182 −0.830912 0.556404i $$-0.812181\pi$$
−0.830912 + 0.556404i $$0.812181\pi$$
$$968$$ 13.9127 0.447170
$$969$$ 0.858181 0.0275687
$$970$$ 3.36618 0.108082
$$971$$ −42.8974 −1.37664 −0.688322 0.725405i $$-0.741652\pi$$
−0.688322 + 0.725405i $$0.741652\pi$$
$$972$$ 1.88448 0.0604448
$$973$$ −9.77525 −0.313380
$$974$$ −8.17843 −0.262054
$$975$$ 0 0
$$976$$ −13.9607 −0.446872
$$977$$ −18.1651 −0.581153 −0.290576 0.956852i $$-0.593847\pi$$
−0.290576 + 0.956852i $$0.593847\pi$$
$$978$$ 4.14584 0.132569
$$979$$ 9.20700 0.294257
$$980$$ −12.3702 −0.395151
$$981$$ −15.4095 −0.491986
$$982$$ 12.4983 0.398836
$$983$$ 10.8582 0.346322 0.173161 0.984894i $$-0.444602\pi$$
0.173161 + 0.984894i $$0.444602\pi$$
$$984$$ −5.57720 −0.177795
$$985$$ 15.8082 0.503692
$$986$$ 14.0000 0.445851
$$987$$ 3.31357 0.105472
$$988$$ 0 0
$$989$$ 4.23103 0.134539
$$990$$ 0.231033 0.00734270
$$991$$ −22.3725 −0.710685 −0.355342 0.934736i $$-0.615636\pi$$
−0.355342 + 0.934736i $$0.615636\pi$$
$$992$$ 37.5905 1.19350
$$993$$ 5.37020 0.170418
$$994$$ 1.64088 0.0520455
$$995$$ −8.76897 −0.277995
$$996$$ −3.33359 −0.105629
$$997$$ −24.2373 −0.767604 −0.383802 0.923415i $$-0.625385\pi$$
−0.383802 + 0.923415i $$0.625385\pi$$
$$998$$ −11.8349 −0.374628
$$999$$ 9.76897 0.309076
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 2535.2.a.ba.1.2 3
3.2 odd 2 7605.2.a.bw.1.2 3
13.4 even 6 195.2.i.d.16.2 6
13.10 even 6 195.2.i.d.61.2 yes 6
13.12 even 2 2535.2.a.bb.1.2 3
39.17 odd 6 585.2.j.f.406.2 6
39.23 odd 6 585.2.j.f.451.2 6
39.38 odd 2 7605.2.a.bv.1.2 3
65.4 even 6 975.2.i.l.601.2 6
65.17 odd 12 975.2.bb.k.874.3 12
65.23 odd 12 975.2.bb.k.724.3 12
65.43 odd 12 975.2.bb.k.874.4 12
65.49 even 6 975.2.i.l.451.2 6
65.62 odd 12 975.2.bb.k.724.4 12

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.i.d.16.2 6 13.4 even 6
195.2.i.d.61.2 yes 6 13.10 even 6
585.2.j.f.406.2 6 39.17 odd 6
585.2.j.f.451.2 6 39.23 odd 6
975.2.i.l.451.2 6 65.49 even 6
975.2.i.l.601.2 6 65.4 even 6
975.2.bb.k.724.3 12 65.23 odd 12
975.2.bb.k.724.4 12 65.62 odd 12
975.2.bb.k.874.3 12 65.17 odd 12
975.2.bb.k.874.4 12 65.43 odd 12
2535.2.a.ba.1.2 3 1.1 even 1 trivial
2535.2.a.bb.1.2 3 13.12 even 2
7605.2.a.bv.1.2 3 39.38 odd 2
7605.2.a.bw.1.2 3 3.2 odd 2