# Properties

 Label 6897.2.a.f.1.1 Level $6897$ Weight $2$ Character 6897.1 Self dual yes Analytic conductor $55.073$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6897,2,Mod(1,6897)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6897, 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("6897.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6897 = 3 \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6897.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: $$55.0728222741$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 6897.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} +5.00000 q^{7} +1.00000 q^{9} -6.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} +10.0000 q^{14} +3.00000 q^{15} -4.00000 q^{16} +1.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} -6.00000 q^{20} -5.00000 q^{21} -4.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +10.0000 q^{28} +2.00000 q^{29} +6.00000 q^{30} -6.00000 q^{31} -8.00000 q^{32} +2.00000 q^{34} -15.0000 q^{35} +2.00000 q^{36} +2.00000 q^{38} +2.00000 q^{39} -10.0000 q^{42} +1.00000 q^{43} -3.00000 q^{45} -8.00000 q^{46} -9.00000 q^{47} +4.00000 q^{48} +18.0000 q^{49} +8.00000 q^{50} -1.00000 q^{51} -4.00000 q^{52} +10.0000 q^{53} -2.00000 q^{54} -1.00000 q^{57} +4.00000 q^{58} -8.00000 q^{59} +6.00000 q^{60} +1.00000 q^{61} -12.0000 q^{62} +5.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} +4.00000 q^{69} -30.0000 q^{70} -12.0000 q^{71} +11.0000 q^{73} -4.00000 q^{75} +2.00000 q^{76} +4.00000 q^{78} -16.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -10.0000 q^{84} -3.00000 q^{85} +2.00000 q^{86} -2.00000 q^{87} -6.00000 q^{89} -6.00000 q^{90} -10.0000 q^{91} -8.00000 q^{92} +6.00000 q^{93} -18.0000 q^{94} -3.00000 q^{95} +8.00000 q^{96} -10.0000 q^{97} +36.0000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 2.00000 1.00000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ −2.00000 −0.816497
$$7$$ 5.00000 1.88982 0.944911 0.327327i $$-0.106148\pi$$
0.944911 + 0.327327i $$0.106148\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ −6.00000 −1.89737
$$11$$ 0 0
$$12$$ −2.00000 −0.577350
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 10.0000 2.67261
$$15$$ 3.00000 0.774597
$$16$$ −4.00000 −1.00000
$$17$$ 1.00000 0.242536 0.121268 0.992620i $$-0.461304\pi$$
0.121268 + 0.992620i $$0.461304\pi$$
$$18$$ 2.00000 0.471405
$$19$$ 1.00000 0.229416
$$20$$ −6.00000 −1.34164
$$21$$ −5.00000 −1.09109
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ −4.00000 −0.784465
$$27$$ −1.00000 −0.192450
$$28$$ 10.0000 1.88982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 6.00000 1.09545
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ −15.0000 −2.53546
$$36$$ 2.00000 0.333333
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ −10.0000 −1.54303
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ −8.00000 −1.17954
$$47$$ −9.00000 −1.31278 −0.656392 0.754420i $$-0.727918\pi$$
−0.656392 + 0.754420i $$0.727918\pi$$
$$48$$ 4.00000 0.577350
$$49$$ 18.0000 2.57143
$$50$$ 8.00000 1.13137
$$51$$ −1.00000 −0.140028
$$52$$ −4.00000 −0.554700
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 4.00000 0.525226
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 6.00000 0.774597
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ −12.0000 −1.52400
$$63$$ 5.00000 0.629941
$$64$$ −8.00000 −1.00000
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 4.00000 0.481543
$$70$$ −30.0000 −3.58569
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 0 0
$$75$$ −4.00000 −0.461880
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ 4.00000 0.452911
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 12.0000 1.34164
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ −10.0000 −1.09109
$$85$$ −3.00000 −0.325396
$$86$$ 2.00000 0.215666
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ −6.00000 −0.632456
$$91$$ −10.0000 −1.04828
$$92$$ −8.00000 −0.834058
$$93$$ 6.00000 0.622171
$$94$$ −18.0000 −1.85656
$$95$$ −3.00000 −0.307794
$$96$$ 8.00000 0.816497
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 36.0000 3.63655
$$99$$ 0 0
$$100$$ 8.00000 0.800000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ −2.00000 −0.197066 −0.0985329 0.995134i $$-0.531415\pi$$
−0.0985329 + 0.995134i $$0.531415\pi$$
$$104$$ 0 0
$$105$$ 15.0000 1.46385
$$106$$ 20.0000 1.94257
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ −2.00000 −0.192450
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −20.0000 −1.88982
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 12.0000 1.11901
$$116$$ 4.00000 0.371391
$$117$$ −2.00000 −0.184900
$$118$$ −16.0000 −1.47292
$$119$$ 5.00000 0.458349
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 2.00000 0.181071
$$123$$ 0 0
$$124$$ −12.0000 −1.07763
$$125$$ 3.00000 0.268328
$$126$$ 10.0000 0.890871
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ −1.00000 −0.0880451
$$130$$ 12.0000 1.05247
$$131$$ −7.00000 −0.611593 −0.305796 0.952097i $$-0.598923\pi$$
−0.305796 + 0.952097i $$0.598923\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ 16.0000 1.38219
$$135$$ 3.00000 0.258199
$$136$$ 0 0
$$137$$ −9.00000 −0.768922 −0.384461 0.923141i $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ 8.00000 0.681005
$$139$$ 13.0000 1.10265 0.551323 0.834292i $$-0.314123\pi$$
0.551323 + 0.834292i $$0.314123\pi$$
$$140$$ −30.0000 −2.53546
$$141$$ 9.00000 0.757937
$$142$$ −24.0000 −2.01404
$$143$$ 0 0
$$144$$ −4.00000 −0.333333
$$145$$ −6.00000 −0.498273
$$146$$ 22.0000 1.82073
$$147$$ −18.0000 −1.48461
$$148$$ 0 0
$$149$$ 21.0000 1.72039 0.860194 0.509968i $$-0.170343\pi$$
0.860194 + 0.509968i $$0.170343\pi$$
$$150$$ −8.00000 −0.653197
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 1.00000 0.0808452
$$154$$ 0 0
$$155$$ 18.0000 1.44579
$$156$$ 4.00000 0.320256
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ −32.0000 −2.54578
$$159$$ −10.0000 −0.793052
$$160$$ 24.0000 1.89737
$$161$$ −20.0000 −1.57622
$$162$$ 2.00000 0.157135
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −24.0000 −1.86276
$$167$$ −10.0000 −0.773823 −0.386912 0.922117i $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −6.00000 −0.460179
$$171$$ 1.00000 0.0764719
$$172$$ 2.00000 0.152499
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 20.0000 1.51186
$$176$$ 0 0
$$177$$ 8.00000 0.601317
$$178$$ −12.0000 −0.899438
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ −6.00000 −0.447214
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ −20.0000 −1.48250
$$183$$ −1.00000 −0.0739221
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 12.0000 0.879883
$$187$$ 0 0
$$188$$ −18.0000 −1.31278
$$189$$ −5.00000 −0.363696
$$190$$ −6.00000 −0.435286
$$191$$ 9.00000 0.651217 0.325609 0.945505i $$-0.394431\pi$$
0.325609 + 0.945505i $$0.394431\pi$$
$$192$$ 8.00000 0.577350
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −20.0000 −1.43592
$$195$$ −6.00000 −0.429669
$$196$$ 36.0000 2.57143
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ −21.0000 −1.48865 −0.744325 0.667817i $$-0.767229\pi$$
−0.744325 + 0.667817i $$0.767229\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ −4.00000 −0.281439
$$203$$ 10.0000 0.701862
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ −4.00000 −0.278019
$$208$$ 8.00000 0.554700
$$209$$ 0 0
$$210$$ 30.0000 2.07020
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 20.0000 1.37361
$$213$$ 12.0000 0.822226
$$214$$ −12.0000 −0.820303
$$215$$ −3.00000 −0.204598
$$216$$ 0 0
$$217$$ −30.0000 −2.03653
$$218$$ −8.00000 −0.541828
$$219$$ −11.0000 −0.743311
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ 12.0000 0.803579 0.401790 0.915732i $$-0.368388\pi$$
0.401790 + 0.915732i $$0.368388\pi$$
$$224$$ −40.0000 −2.67261
$$225$$ 4.00000 0.266667
$$226$$ 4.00000 0.266076
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ 25.0000 1.65205 0.826023 0.563636i $$-0.190598\pi$$
0.826023 + 0.563636i $$0.190598\pi$$
$$230$$ 24.0000 1.58251
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −9.00000 −0.589610 −0.294805 0.955557i $$-0.595255\pi$$
−0.294805 + 0.955557i $$0.595255\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 27.0000 1.76129
$$236$$ −16.0000 −1.04151
$$237$$ 16.0000 1.03931
$$238$$ 10.0000 0.648204
$$239$$ 3.00000 0.194054 0.0970269 0.995282i $$-0.469067\pi$$
0.0970269 + 0.995282i $$0.469067\pi$$
$$240$$ −12.0000 −0.774597
$$241$$ −20.0000 −1.28831 −0.644157 0.764894i $$-0.722792\pi$$
−0.644157 + 0.764894i $$0.722792\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ −54.0000 −3.44993
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 6.00000 0.379473
$$251$$ 7.00000 0.441836 0.220918 0.975292i $$-0.429095\pi$$
0.220918 + 0.975292i $$0.429095\pi$$
$$252$$ 10.0000 0.629941
$$253$$ 0 0
$$254$$ 4.00000 0.250982
$$255$$ 3.00000 0.187867
$$256$$ 16.0000 1.00000
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ −2.00000 −0.124515
$$259$$ 0 0
$$260$$ 12.0000 0.744208
$$261$$ 2.00000 0.123797
$$262$$ −14.0000 −0.864923
$$263$$ −23.0000 −1.41824 −0.709120 0.705087i $$-0.750908\pi$$
−0.709120 + 0.705087i $$0.750908\pi$$
$$264$$ 0 0
$$265$$ −30.0000 −1.84289
$$266$$ 10.0000 0.613139
$$267$$ 6.00000 0.367194
$$268$$ 16.0000 0.977356
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 6.00000 0.365148
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 10.0000 0.605228
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ 11.0000 0.660926 0.330463 0.943819i $$-0.392795\pi$$
0.330463 + 0.943819i $$0.392795\pi$$
$$278$$ 26.0000 1.55938
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 18.0000 1.07188
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ −24.0000 −1.42414
$$285$$ 3.00000 0.177705
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −8.00000 −0.471405
$$289$$ −16.0000 −0.941176
$$290$$ −12.0000 −0.704664
$$291$$ 10.0000 0.586210
$$292$$ 22.0000 1.28745
$$293$$ 28.0000 1.63578 0.817889 0.575376i $$-0.195144\pi$$
0.817889 + 0.575376i $$0.195144\pi$$
$$294$$ −36.0000 −2.09956
$$295$$ 24.0000 1.39733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 42.0000 2.43299
$$299$$ 8.00000 0.462652
$$300$$ −8.00000 −0.461880
$$301$$ 5.00000 0.288195
$$302$$ 0 0
$$303$$ 2.00000 0.114897
$$304$$ −4.00000 −0.229416
$$305$$ −3.00000 −0.171780
$$306$$ 2.00000 0.114332
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 2.00000 0.113776
$$310$$ 36.0000 2.04466
$$311$$ −21.0000 −1.19080 −0.595400 0.803429i $$-0.703007\pi$$
−0.595400 + 0.803429i $$0.703007\pi$$
$$312$$ 0 0
$$313$$ −2.00000 −0.113047 −0.0565233 0.998401i $$-0.518002\pi$$
−0.0565233 + 0.998401i $$0.518002\pi$$
$$314$$ −36.0000 −2.03160
$$315$$ −15.0000 −0.845154
$$316$$ −32.0000 −1.80014
$$317$$ −4.00000 −0.224662 −0.112331 0.993671i $$-0.535832\pi$$
−0.112331 + 0.993671i $$0.535832\pi$$
$$318$$ −20.0000 −1.12154
$$319$$ 0 0
$$320$$ 24.0000 1.34164
$$321$$ 6.00000 0.334887
$$322$$ −40.0000 −2.22911
$$323$$ 1.00000 0.0556415
$$324$$ 2.00000 0.111111
$$325$$ −8.00000 −0.443760
$$326$$ 0 0
$$327$$ 4.00000 0.221201
$$328$$ 0 0
$$329$$ −45.0000 −2.48093
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −24.0000 −1.31717
$$333$$ 0 0
$$334$$ −20.0000 −1.09435
$$335$$ −24.0000 −1.31126
$$336$$ 20.0000 1.09109
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −18.0000 −0.979071
$$339$$ −2.00000 −0.108625
$$340$$ −6.00000 −0.325396
$$341$$ 0 0
$$342$$ 2.00000 0.108148
$$343$$ 55.0000 2.96972
$$344$$ 0 0
$$345$$ −12.0000 −0.646058
$$346$$ −12.0000 −0.645124
$$347$$ 25.0000 1.34207 0.671035 0.741426i $$-0.265850\pi$$
0.671035 + 0.741426i $$0.265850\pi$$
$$348$$ −4.00000 −0.214423
$$349$$ −9.00000 −0.481759 −0.240879 0.970555i $$-0.577436\pi$$
−0.240879 + 0.970555i $$0.577436\pi$$
$$350$$ 40.0000 2.13809
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ 16.0000 0.850390
$$355$$ 36.0000 1.91068
$$356$$ −12.0000 −0.635999
$$357$$ −5.00000 −0.264628
$$358$$ −36.0000 −1.90266
$$359$$ −37.0000 −1.95279 −0.976393 0.216003i $$-0.930698\pi$$
−0.976393 + 0.216003i $$0.930698\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −28.0000 −1.47165
$$363$$ 0 0
$$364$$ −20.0000 −1.04828
$$365$$ −33.0000 −1.72730
$$366$$ −2.00000 −0.104542
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 16.0000 0.834058
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 50.0000 2.59587
$$372$$ 12.0000 0.622171
$$373$$ −16.0000 −0.828449 −0.414224 0.910175i $$-0.635947\pi$$
−0.414224 + 0.910175i $$0.635947\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ −10.0000 −0.514344
$$379$$ 34.0000 1.74646 0.873231 0.487306i $$-0.162020\pi$$
0.873231 + 0.487306i $$0.162020\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ −2.00000 −0.102463
$$382$$ 18.0000 0.920960
$$383$$ −34.0000 −1.73732 −0.868659 0.495410i $$-0.835018\pi$$
−0.868659 + 0.495410i $$0.835018\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ 1.00000 0.0508329
$$388$$ −20.0000 −1.01535
$$389$$ −27.0000 −1.36895 −0.684477 0.729034i $$-0.739969\pi$$
−0.684477 + 0.729034i $$0.739969\pi$$
$$390$$ −12.0000 −0.607644
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ 7.00000 0.353103
$$394$$ 4.00000 0.201517
$$395$$ 48.0000 2.41514
$$396$$ 0 0
$$397$$ 25.0000 1.25471 0.627357 0.778732i $$-0.284137\pi$$
0.627357 + 0.778732i $$0.284137\pi$$
$$398$$ −42.0000 −2.10527
$$399$$ −5.00000 −0.250313
$$400$$ −16.0000 −0.800000
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ 12.0000 0.597763
$$404$$ −4.00000 −0.199007
$$405$$ −3.00000 −0.149071
$$406$$ 20.0000 0.992583
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 9.00000 0.443937
$$412$$ −4.00000 −0.197066
$$413$$ −40.0000 −1.96827
$$414$$ −8.00000 −0.393179
$$415$$ 36.0000 1.76717
$$416$$ 16.0000 0.784465
$$417$$ −13.0000 −0.636613
$$418$$ 0 0
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 30.0000 1.46385
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ −24.0000 −1.16830
$$423$$ −9.00000 −0.437595
$$424$$ 0 0
$$425$$ 4.00000 0.194029
$$426$$ 24.0000 1.16280
$$427$$ 5.00000 0.241967
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −6.00000 −0.289346
$$431$$ 34.0000 1.63772 0.818861 0.573992i $$-0.194606\pi$$
0.818861 + 0.573992i $$0.194606\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 6.00000 0.288342 0.144171 0.989553i $$-0.453949\pi$$
0.144171 + 0.989553i $$0.453949\pi$$
$$434$$ −60.0000 −2.88009
$$435$$ 6.00000 0.287678
$$436$$ −8.00000 −0.383131
$$437$$ −4.00000 −0.191346
$$438$$ −22.0000 −1.05120
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ −4.00000 −0.190261
$$443$$ −5.00000 −0.237557 −0.118779 0.992921i $$-0.537898\pi$$
−0.118779 + 0.992921i $$0.537898\pi$$
$$444$$ 0 0
$$445$$ 18.0000 0.853282
$$446$$ 24.0000 1.13643
$$447$$ −21.0000 −0.993266
$$448$$ −40.0000 −1.88982
$$449$$ −36.0000 −1.69895 −0.849473 0.527633i $$-0.823080\pi$$
−0.849473 + 0.527633i $$0.823080\pi$$
$$450$$ 8.00000 0.377124
$$451$$ 0 0
$$452$$ 4.00000 0.188144
$$453$$ 0 0
$$454$$ −36.0000 −1.68956
$$455$$ 30.0000 1.40642
$$456$$ 0 0
$$457$$ 29.0000 1.35656 0.678281 0.734802i $$-0.262725\pi$$
0.678281 + 0.734802i $$0.262725\pi$$
$$458$$ 50.0000 2.33635
$$459$$ −1.00000 −0.0466760
$$460$$ 24.0000 1.11901
$$461$$ −27.0000 −1.25752 −0.628758 0.777601i $$-0.716436\pi$$
−0.628758 + 0.777601i $$0.716436\pi$$
$$462$$ 0 0
$$463$$ 17.0000 0.790057 0.395029 0.918669i $$-0.370735\pi$$
0.395029 + 0.918669i $$0.370735\pi$$
$$464$$ −8.00000 −0.371391
$$465$$ −18.0000 −0.834730
$$466$$ −18.0000 −0.833834
$$467$$ −5.00000 −0.231372 −0.115686 0.993286i $$-0.536907\pi$$
−0.115686 + 0.993286i $$0.536907\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ 40.0000 1.84703
$$470$$ 54.0000 2.49083
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 32.0000 1.46981
$$475$$ 4.00000 0.183533
$$476$$ 10.0000 0.458349
$$477$$ 10.0000 0.457869
$$478$$ 6.00000 0.274434
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ −24.0000 −1.09545
$$481$$ 0 0
$$482$$ −40.0000 −1.82195
$$483$$ 20.0000 0.910032
$$484$$ 0 0
$$485$$ 30.0000 1.36223
$$486$$ −2.00000 −0.0907218
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ −108.000 −4.87894
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 2.00000 0.0900755
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 24.0000 1.07763
$$497$$ −60.0000 −2.69137
$$498$$ 24.0000 1.07547
$$499$$ 5.00000 0.223831 0.111915 0.993718i $$-0.464301\pi$$
0.111915 + 0.993718i $$0.464301\pi$$
$$500$$ 6.00000 0.268328
$$501$$ 10.0000 0.446767
$$502$$ 14.0000 0.624851
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 4.00000 0.177471
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 6.00000 0.265684
$$511$$ 55.0000 2.43306
$$512$$ 32.0000 1.41421
$$513$$ −1.00000 −0.0441511
$$514$$ −16.0000 −0.705730
$$515$$ 6.00000 0.264392
$$516$$ −2.00000 −0.0880451
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ 4.00000 0.175075
$$523$$ 34.0000 1.48672 0.743358 0.668894i $$-0.233232\pi$$
0.743358 + 0.668894i $$0.233232\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ −20.0000 −0.872872
$$526$$ −46.0000 −2.00570
$$527$$ −6.00000 −0.261364
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −60.0000 −2.60623
$$531$$ −8.00000 −0.347170
$$532$$ 10.0000 0.433555
$$533$$ 0 0
$$534$$ 12.0000 0.519291
$$535$$ 18.0000 0.778208
$$536$$ 0 0
$$537$$ 18.0000 0.776757
$$538$$ −28.0000 −1.20717
$$539$$ 0 0
$$540$$ 6.00000 0.258199
$$541$$ −3.00000 −0.128980 −0.0644900 0.997918i $$-0.520542\pi$$
−0.0644900 + 0.997918i $$0.520542\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ 14.0000 0.600798
$$544$$ −8.00000 −0.342997
$$545$$ 12.0000 0.514024
$$546$$ 20.0000 0.855921
$$547$$ 26.0000 1.11168 0.555840 0.831289i $$-0.312397\pi$$
0.555840 + 0.831289i $$0.312397\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 1.00000 0.0426790
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ −80.0000 −3.40195
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ 26.0000 1.10265
$$557$$ 41.0000 1.73723 0.868613 0.495491i $$-0.165012\pi$$
0.868613 + 0.495491i $$0.165012\pi$$
$$558$$ −12.0000 −0.508001
$$559$$ −2.00000 −0.0845910
$$560$$ 60.0000 2.53546
$$561$$ 0 0
$$562$$ −20.0000 −0.843649
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 18.0000 0.757937
$$565$$ −6.00000 −0.252422
$$566$$ 26.0000 1.09286
$$567$$ 5.00000 0.209980
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 6.00000 0.251312
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ −9.00000 −0.375980
$$574$$ 0 0
$$575$$ −16.0000 −0.667246
$$576$$ −8.00000 −0.333333
$$577$$ 27.0000 1.12402 0.562012 0.827129i $$-0.310027\pi$$
0.562012 + 0.827129i $$0.310027\pi$$
$$578$$ −32.0000 −1.33102
$$579$$ 4.00000 0.166234
$$580$$ −12.0000 −0.498273
$$581$$ −60.0000 −2.48922
$$582$$ 20.0000 0.829027
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 6.00000 0.248069
$$586$$ 56.0000 2.31334
$$587$$ 7.00000 0.288921 0.144460 0.989511i $$-0.453855\pi$$
0.144460 + 0.989511i $$0.453855\pi$$
$$588$$ −36.0000 −1.48461
$$589$$ −6.00000 −0.247226
$$590$$ 48.0000 1.97613
$$591$$ −2.00000 −0.0822690
$$592$$ 0 0
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ −15.0000 −0.614940
$$596$$ 42.0000 1.72039
$$597$$ 21.0000 0.859473
$$598$$ 16.0000 0.654289
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 10.0000 0.407570
$$603$$ 8.00000 0.325785
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 4.00000 0.162489
$$607$$ −26.0000 −1.05531 −0.527654 0.849460i $$-0.676928\pi$$
−0.527654 + 0.849460i $$0.676928\pi$$
$$608$$ −8.00000 −0.324443
$$609$$ −10.0000 −0.405220
$$610$$ −6.00000 −0.242933
$$611$$ 18.0000 0.728202
$$612$$ 2.00000 0.0808452
$$613$$ −33.0000 −1.33286 −0.666429 0.745569i $$-0.732178\pi$$
−0.666429 + 0.745569i $$0.732178\pi$$
$$614$$ 24.0000 0.968561
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 27.0000 1.08698 0.543490 0.839416i $$-0.317103\pi$$
0.543490 + 0.839416i $$0.317103\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 36.0000 1.44579
$$621$$ 4.00000 0.160514
$$622$$ −42.0000 −1.68405
$$623$$ −30.0000 −1.20192
$$624$$ −8.00000 −0.320256
$$625$$ −29.0000 −1.16000
$$626$$ −4.00000 −0.159872
$$627$$ 0 0
$$628$$ −36.0000 −1.43656
$$629$$ 0 0
$$630$$ −30.0000 −1.19523
$$631$$ 15.0000 0.597141 0.298570 0.954388i $$-0.403490\pi$$
0.298570 + 0.954388i $$0.403490\pi$$
$$632$$ 0 0
$$633$$ 12.0000 0.476957
$$634$$ −8.00000 −0.317721
$$635$$ −6.00000 −0.238103
$$636$$ −20.0000 −0.793052
$$637$$ −36.0000 −1.42637
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −1.00000 −0.0394362 −0.0197181 0.999806i $$-0.506277\pi$$
−0.0197181 + 0.999806i $$0.506277\pi$$
$$644$$ −40.0000 −1.57622
$$645$$ 3.00000 0.118125
$$646$$ 2.00000 0.0786889
$$647$$ −39.0000 −1.53325 −0.766624 0.642096i $$-0.778065\pi$$
−0.766624 + 0.642096i $$0.778065\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −16.0000 −0.627572
$$651$$ 30.0000 1.17579
$$652$$ 0 0
$$653$$ 3.00000 0.117399 0.0586995 0.998276i $$-0.481305\pi$$
0.0586995 + 0.998276i $$0.481305\pi$$
$$654$$ 8.00000 0.312825
$$655$$ 21.0000 0.820538
$$656$$ 0 0
$$657$$ 11.0000 0.429151
$$658$$ −90.0000 −3.50857
$$659$$ 14.0000 0.545363 0.272681 0.962104i $$-0.412090\pi$$
0.272681 + 0.962104i $$0.412090\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ 2.00000 0.0776736
$$664$$ 0 0
$$665$$ −15.0000 −0.581675
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ −20.0000 −0.773823
$$669$$ −12.0000 −0.463947
$$670$$ −48.0000 −1.85440
$$671$$ 0 0
$$672$$ 40.0000 1.54303
$$673$$ 24.0000 0.925132 0.462566 0.886585i $$-0.346929\pi$$
0.462566 + 0.886585i $$0.346929\pi$$
$$674$$ 28.0000 1.07852
$$675$$ −4.00000 −0.153960
$$676$$ −18.0000 −0.692308
$$677$$ −34.0000 −1.30673 −0.653363 0.757045i $$-0.726642\pi$$
−0.653363 + 0.757045i $$0.726642\pi$$
$$678$$ −4.00000 −0.153619
$$679$$ −50.0000 −1.91882
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 0 0
$$683$$ −6.00000 −0.229584 −0.114792 0.993390i $$-0.536620\pi$$
−0.114792 + 0.993390i $$0.536620\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 27.0000 1.03162
$$686$$ 110.000 4.19982
$$687$$ −25.0000 −0.953809
$$688$$ −4.00000 −0.152499
$$689$$ −20.0000 −0.761939
$$690$$ −24.0000 −0.913664
$$691$$ −31.0000 −1.17930 −0.589648 0.807661i $$-0.700733\pi$$
−0.589648 + 0.807661i $$0.700733\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 50.0000 1.89797
$$695$$ −39.0000 −1.47935
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −18.0000 −0.681310
$$699$$ 9.00000 0.340411
$$700$$ 40.0000 1.51186
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 4.00000 0.150970
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −27.0000 −1.01688
$$706$$ −4.00000 −0.150542
$$707$$ −10.0000 −0.376089
$$708$$ 16.0000 0.601317
$$709$$ −42.0000 −1.57734 −0.788672 0.614815i $$-0.789231\pi$$
−0.788672 + 0.614815i $$0.789231\pi$$
$$710$$ 72.0000 2.70211
$$711$$ −16.0000 −0.600047
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ −10.0000 −0.374241
$$715$$ 0 0
$$716$$ −36.0000 −1.34538
$$717$$ −3.00000 −0.112037
$$718$$ −74.0000 −2.76166
$$719$$ 33.0000 1.23069 0.615346 0.788257i $$-0.289016\pi$$
0.615346 + 0.788257i $$0.289016\pi$$
$$720$$ 12.0000 0.447214
$$721$$ −10.0000 −0.372419
$$722$$ 2.00000 0.0744323
$$723$$ 20.0000 0.743808
$$724$$ −28.0000 −1.04061
$$725$$ 8.00000 0.297113
$$726$$ 0 0
$$727$$ −23.0000 −0.853023 −0.426511 0.904482i $$-0.640258\pi$$
−0.426511 + 0.904482i $$0.640258\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −66.0000 −2.44277
$$731$$ 1.00000 0.0369863
$$732$$ −2.00000 −0.0739221
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 54.0000 1.99182
$$736$$ 32.0000 1.17954
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 5.00000 0.183928 0.0919640 0.995762i $$-0.470686\pi$$
0.0919640 + 0.995762i $$0.470686\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 100.000 3.67112
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 0 0
$$745$$ −63.0000 −2.30814
$$746$$ −32.0000 −1.17160
$$747$$ −12.0000 −0.439057
$$748$$ 0 0
$$749$$ −30.0000 −1.09618
$$750$$ −6.00000 −0.219089
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 36.0000 1.31278
$$753$$ −7.00000 −0.255094
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ −10.0000 −0.363696
$$757$$ −17.0000 −0.617876 −0.308938 0.951082i $$-0.599973\pi$$
−0.308938 + 0.951082i $$0.599973\pi$$
$$758$$ 68.0000 2.46987
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −15.0000 −0.543750 −0.271875 0.962333i $$-0.587644\pi$$
−0.271875 + 0.962333i $$0.587644\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ −20.0000 −0.724049
$$764$$ 18.0000 0.651217
$$765$$ −3.00000 −0.108465
$$766$$ −68.0000 −2.45694
$$767$$ 16.0000 0.577727
$$768$$ −16.0000 −0.577350
$$769$$ −11.0000 −0.396670 −0.198335 0.980134i $$-0.563553\pi$$
−0.198335 + 0.980134i $$0.563553\pi$$
$$770$$ 0 0
$$771$$ 8.00000 0.288113
$$772$$ −8.00000 −0.287926
$$773$$ 20.0000 0.719350 0.359675 0.933078i $$-0.382888\pi$$
0.359675 + 0.933078i $$0.382888\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ −24.0000 −0.862105
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −54.0000 −1.93599
$$779$$ 0 0
$$780$$ −12.0000 −0.429669
$$781$$ 0 0
$$782$$ −8.00000 −0.286079
$$783$$ −2.00000 −0.0714742
$$784$$ −72.0000 −2.57143
$$785$$ 54.0000 1.92734
$$786$$ 14.0000 0.499363
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ 4.00000 0.142494
$$789$$ 23.0000 0.818822
$$790$$ 96.0000 3.41553
$$791$$ 10.0000 0.355559
$$792$$ 0 0
$$793$$ −2.00000 −0.0710221
$$794$$ 50.0000 1.77443
$$795$$ 30.0000 1.06399
$$796$$ −42.0000 −1.48865
$$797$$ −44.0000 −1.55856 −0.779280 0.626676i $$-0.784415\pi$$
−0.779280 + 0.626676i $$0.784415\pi$$
$$798$$ −10.0000 −0.353996
$$799$$ −9.00000 −0.318397
$$800$$ −32.0000 −1.13137
$$801$$ −6.00000 −0.212000
$$802$$ 72.0000 2.54241
$$803$$ 0 0
$$804$$ −16.0000 −0.564276
$$805$$ 60.0000 2.11472
$$806$$ 24.0000 0.845364
$$807$$ 14.0000 0.492823
$$808$$ 0 0
$$809$$ 55.0000 1.93370 0.966849 0.255351i $$-0.0821909\pi$$
0.966849 + 0.255351i $$0.0821909\pi$$
$$810$$ −6.00000 −0.210819
$$811$$ 38.0000 1.33436 0.667180 0.744896i $$-0.267501\pi$$
0.667180 + 0.744896i $$0.267501\pi$$
$$812$$ 20.0000 0.701862
$$813$$ 12.0000 0.420858
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ 1.00000 0.0349856
$$818$$ 28.0000 0.978997
$$819$$ −10.0000 −0.349428
$$820$$ 0 0
$$821$$ 45.0000 1.57051 0.785255 0.619172i $$-0.212532\pi$$
0.785255 + 0.619172i $$0.212532\pi$$
$$822$$ 18.0000 0.627822
$$823$$ 43.0000 1.49889 0.749443 0.662069i $$-0.230321\pi$$
0.749443 + 0.662069i $$0.230321\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −80.0000 −2.78356
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ −8.00000 −0.278019
$$829$$ 52.0000 1.80603 0.903017 0.429604i $$-0.141347\pi$$
0.903017 + 0.429604i $$0.141347\pi$$
$$830$$ 72.0000 2.49916
$$831$$ −11.0000 −0.381586
$$832$$ 16.0000 0.554700
$$833$$ 18.0000 0.623663
$$834$$ −26.0000 −0.900306
$$835$$ 30.0000 1.03819
$$836$$ 0 0
$$837$$ 6.00000 0.207390
$$838$$ 56.0000 1.93449
$$839$$ 54.0000 1.86429 0.932144 0.362089i $$-0.117936\pi$$
0.932144 + 0.362089i $$0.117936\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 52.0000 1.79204
$$843$$ 10.0000 0.344418
$$844$$ −24.0000 −0.826114
$$845$$ 27.0000 0.928828
$$846$$ −18.0000 −0.618853
$$847$$ 0 0
$$848$$ −40.0000 −1.37361
$$849$$ −13.0000 −0.446159
$$850$$ 8.00000 0.274398
$$851$$ 0 0
$$852$$ 24.0000 0.822226
$$853$$ 14.0000 0.479351 0.239675 0.970853i $$-0.422959\pi$$
0.239675 + 0.970853i $$0.422959\pi$$
$$854$$ 10.0000 0.342193
$$855$$ −3.00000 −0.102598
$$856$$ 0 0
$$857$$ 8.00000 0.273275 0.136637 0.990621i $$-0.456370\pi$$
0.136637 + 0.990621i $$0.456370\pi$$
$$858$$ 0 0
$$859$$ 27.0000 0.921228 0.460614 0.887601i $$-0.347629\pi$$
0.460614 + 0.887601i $$0.347629\pi$$
$$860$$ −6.00000 −0.204598
$$861$$ 0 0
$$862$$ 68.0000 2.31609
$$863$$ −44.0000 −1.49778 −0.748889 0.662696i $$-0.769412\pi$$
−0.748889 + 0.662696i $$0.769412\pi$$
$$864$$ 8.00000 0.272166
$$865$$ 18.0000 0.612018
$$866$$ 12.0000 0.407777
$$867$$ 16.0000 0.543388
$$868$$ −60.0000 −2.03653
$$869$$ 0 0
$$870$$ 12.0000 0.406838
$$871$$ −16.0000 −0.542139
$$872$$ 0 0
$$873$$ −10.0000 −0.338449
$$874$$ −8.00000 −0.270604
$$875$$ 15.0000 0.507093
$$876$$ −22.0000 −0.743311
$$877$$ 6.00000 0.202606 0.101303 0.994856i $$-0.467699\pi$$
0.101303 + 0.994856i $$0.467699\pi$$
$$878$$ −52.0000 −1.75491
$$879$$ −28.0000 −0.944417
$$880$$ 0 0
$$881$$ −37.0000 −1.24656 −0.623281 0.781998i $$-0.714201\pi$$
−0.623281 + 0.781998i $$0.714201\pi$$
$$882$$ 36.0000 1.21218
$$883$$ 35.0000 1.17784 0.588922 0.808190i $$-0.299553\pi$$
0.588922 + 0.808190i $$0.299553\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ −24.0000 −0.806751
$$886$$ −10.0000 −0.335957
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 10.0000 0.335389
$$890$$ 36.0000 1.20672
$$891$$ 0 0
$$892$$ 24.0000 0.803579
$$893$$ −9.00000 −0.301174
$$894$$ −42.0000 −1.40469
$$895$$ 54.0000 1.80502
$$896$$ 0 0
$$897$$ −8.00000 −0.267112
$$898$$ −72.0000 −2.40267
$$899$$ −12.0000 −0.400222
$$900$$ 8.00000 0.266667
$$901$$ 10.0000 0.333148
$$902$$ 0 0
$$903$$ −5.00000 −0.166390
$$904$$ 0 0
$$905$$ 42.0000 1.39613
$$906$$ 0 0
$$907$$ −26.0000 −0.863316 −0.431658 0.902037i $$-0.642071\pi$$
−0.431658 + 0.902037i $$0.642071\pi$$
$$908$$ −36.0000 −1.19470
$$909$$ −2.00000 −0.0663358
$$910$$ 60.0000 1.98898
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 0 0
$$914$$ 58.0000 1.91847
$$915$$ 3.00000 0.0991769
$$916$$ 50.0000 1.65205
$$917$$ −35.0000 −1.15580
$$918$$ −2.00000 −0.0660098
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ −54.0000 −1.77840
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 34.0000 1.11731
$$927$$ −2.00000 −0.0656886
$$928$$ −16.0000 −0.525226
$$929$$ −2.00000 −0.0656179 −0.0328089 0.999462i $$-0.510445\pi$$
−0.0328089 + 0.999462i $$0.510445\pi$$
$$930$$ −36.0000 −1.18049
$$931$$ 18.0000 0.589926
$$932$$ −18.0000 −0.589610
$$933$$ 21.0000 0.687509
$$934$$ −10.0000 −0.327210
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −21.0000 −0.686040 −0.343020 0.939328i $$-0.611450\pi$$
−0.343020 + 0.939328i $$0.611450\pi$$
$$938$$ 80.0000 2.61209
$$939$$ 2.00000 0.0652675
$$940$$ 54.0000 1.76129
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ 36.0000 1.17294
$$943$$ 0 0
$$944$$ 32.0000 1.04151
$$945$$ 15.0000 0.487950
$$946$$ 0 0
$$947$$ −28.0000 −0.909878 −0.454939 0.890523i $$-0.650339\pi$$
−0.454939 + 0.890523i $$0.650339\pi$$
$$948$$ 32.0000 1.03931
$$949$$ −22.0000 −0.714150
$$950$$ 8.00000 0.259554
$$951$$ 4.00000 0.129709
$$952$$ 0 0
$$953$$ 32.0000 1.03658 0.518291 0.855204i $$-0.326568\pi$$
0.518291 + 0.855204i $$0.326568\pi$$
$$954$$ 20.0000 0.647524
$$955$$ −27.0000 −0.873699
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ 32.0000 1.03387
$$959$$ −45.0000 −1.45313
$$960$$ −24.0000 −0.774597
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ −40.0000 −1.28831
$$965$$ 12.0000 0.386294
$$966$$ 40.0000 1.28698
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 0 0
$$969$$ −1.00000 −0.0321246
$$970$$ 60.0000 1.92648
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ −2.00000 −0.0641500
$$973$$ 65.0000 2.08380
$$974$$ −32.0000 −1.02535
$$975$$ 8.00000 0.256205
$$976$$ −4.00000 −0.128037
$$977$$ 54.0000 1.72761 0.863807 0.503824i $$-0.168074\pi$$
0.863807 + 0.503824i $$0.168074\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −108.000 −3.44993
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ −44.0000 −1.40338 −0.701691 0.712481i $$-0.747571\pi$$
−0.701691 + 0.712481i $$0.747571\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 4.00000 0.127386
$$987$$ 45.0000 1.43237
$$988$$ −4.00000 −0.127257
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 48.0000 1.52400
$$993$$ 4.00000 0.126936
$$994$$ −120.000 −3.80617
$$995$$ 63.0000 1.99723
$$996$$ 24.0000 0.760469
$$997$$ 47.0000 1.48850 0.744252 0.667898i $$-0.232806\pi$$
0.744252 + 0.667898i $$0.232806\pi$$
$$998$$ 10.0000 0.316544
$$999$$ 0 0
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 6897.2.a.f.1.1 1
11.10 odd 2 57.2.a.a.1.1 1
33.32 even 2 171.2.a.d.1.1 1
44.43 even 2 912.2.a.g.1.1 1
55.32 even 4 1425.2.c.b.799.1 2
55.43 even 4 1425.2.c.b.799.2 2
55.54 odd 2 1425.2.a.j.1.1 1
77.76 even 2 2793.2.a.b.1.1 1
88.21 odd 2 3648.2.a.bh.1.1 1
88.43 even 2 3648.2.a.r.1.1 1
132.131 odd 2 2736.2.a.v.1.1 1
143.142 odd 2 9633.2.a.o.1.1 1
165.164 even 2 4275.2.a.b.1.1 1
209.208 even 2 1083.2.a.e.1.1 1
231.230 odd 2 8379.2.a.p.1.1 1
627.626 odd 2 3249.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.a.1.1 1 11.10 odd 2
171.2.a.d.1.1 1 33.32 even 2
912.2.a.g.1.1 1 44.43 even 2
1083.2.a.e.1.1 1 209.208 even 2
1425.2.a.j.1.1 1 55.54 odd 2
1425.2.c.b.799.1 2 55.32 even 4
1425.2.c.b.799.2 2 55.43 even 4
2736.2.a.v.1.1 1 132.131 odd 2
2793.2.a.b.1.1 1 77.76 even 2
3249.2.a.b.1.1 1 627.626 odd 2
3648.2.a.r.1.1 1 88.43 even 2
3648.2.a.bh.1.1 1 88.21 odd 2
4275.2.a.b.1.1 1 165.164 even 2
6897.2.a.f.1.1 1 1.1 even 1 trivial
8379.2.a.p.1.1 1 231.230 odd 2
9633.2.a.o.1.1 1 143.142 odd 2