# Properties

 Label 26.2.a.a.1.1 Level $26$ Weight $2$ Character 26.1 Self dual yes Analytic conductor $0.208$ Analytic rank $0$ 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] = [26,2,Mod(1,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("26.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 26.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: $$0.207611045255$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 26.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} +2.00000 q^{19} -3.00000 q^{20} -1.00000 q^{21} -6.00000 q^{22} -1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} +3.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +3.00000 q^{34} +3.00000 q^{35} -2.00000 q^{36} -7.00000 q^{37} -2.00000 q^{38} +1.00000 q^{39} +3.00000 q^{40} +1.00000 q^{42} -1.00000 q^{43} +6.00000 q^{44} +6.00000 q^{45} +3.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} +1.00000 q^{52} +5.00000 q^{54} -18.0000 q^{55} +1.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} -6.00000 q^{59} -3.00000 q^{60} +8.00000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -3.00000 q^{65} -6.00000 q^{66} +14.0000 q^{67} -3.00000 q^{68} -3.00000 q^{70} -3.00000 q^{71} +2.00000 q^{72} +2.00000 q^{73} +7.00000 q^{74} +4.00000 q^{75} +2.00000 q^{76} -6.00000 q^{77} -1.00000 q^{78} +8.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +12.0000 q^{83} -1.00000 q^{84} +9.00000 q^{85} +1.00000 q^{86} +6.00000 q^{87} -6.00000 q^{88} -6.00000 q^{89} -6.00000 q^{90} -1.00000 q^{91} -4.00000 q^{93} -3.00000 q^{94} -6.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} +6.00000 q^{98} -12.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 3.00000 0.948683
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 1.00000 0.277350
$$14$$ 1.00000 0.267261
$$15$$ −3.00000 −0.774597
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 2.00000 0.471405
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ −3.00000 −0.670820
$$21$$ −1.00000 −0.218218
$$22$$ −6.00000 −1.27920
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 4.00000 0.800000
$$26$$ −1.00000 −0.196116
$$27$$ −5.00000 −0.962250
$$28$$ −1.00000 −0.188982
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 3.00000 0.547723
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 6.00000 1.04447
$$34$$ 3.00000 0.514496
$$35$$ 3.00000 0.507093
$$36$$ −2.00000 −0.333333
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 1.00000 0.160128
$$40$$ 3.00000 0.474342
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 6.00000 0.894427
$$46$$ 0 0
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.00000 −0.857143
$$50$$ −4.00000 −0.565685
$$51$$ −3.00000 −0.420084
$$52$$ 1.00000 0.138675
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 5.00000 0.680414
$$55$$ −18.0000 −2.42712
$$56$$ 1.00000 0.133631
$$57$$ 2.00000 0.264906
$$58$$ −6.00000 −0.787839
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ −3.00000 −0.387298
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ −3.00000 −0.372104
$$66$$ −6.00000 −0.738549
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ −3.00000 −0.358569
$$71$$ −3.00000 −0.356034 −0.178017 0.984027i $$-0.556968\pi$$
−0.178017 + 0.984027i $$0.556968\pi$$
$$72$$ 2.00000 0.235702
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 4.00000 0.461880
$$76$$ 2.00000 0.229416
$$77$$ −6.00000 −0.683763
$$78$$ −1.00000 −0.113228
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 9.00000 0.976187
$$86$$ 1.00000 0.107833
$$87$$ 6.00000 0.643268
$$88$$ −6.00000 −0.639602
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ −6.00000 −0.632456
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ −3.00000 −0.309426
$$95$$ −6.00000 −0.615587
$$96$$ −1.00000 −0.102062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 6.00000 0.606092
$$99$$ −12.0000 −1.20605
$$100$$ 4.00000 0.400000
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 3.00000 0.297044
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 3.00000 0.292770
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 18.0000 1.71623
$$111$$ −7.00000 −0.664411
$$112$$ −1.00000 −0.0944911
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ −2.00000 −0.184900
$$118$$ 6.00000 0.552345
$$119$$ 3.00000 0.275010
$$120$$ 3.00000 0.273861
$$121$$ 25.0000 2.27273
$$122$$ −8.00000 −0.724286
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 3.00000 0.268328
$$126$$ −2.00000 −0.178174
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −1.00000 −0.0880451
$$130$$ 3.00000 0.263117
$$131$$ −21.0000 −1.83478 −0.917389 0.397991i $$-0.869707\pi$$
−0.917389 + 0.397991i $$0.869707\pi$$
$$132$$ 6.00000 0.522233
$$133$$ −2.00000 −0.173422
$$134$$ −14.0000 −1.20942
$$135$$ 15.0000 1.29099
$$136$$ 3.00000 0.257248
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ −13.0000 −1.10265 −0.551323 0.834292i $$-0.685877\pi$$
−0.551323 + 0.834292i $$0.685877\pi$$
$$140$$ 3.00000 0.253546
$$141$$ 3.00000 0.252646
$$142$$ 3.00000 0.251754
$$143$$ 6.00000 0.501745
$$144$$ −2.00000 −0.166667
$$145$$ −18.0000 −1.49482
$$146$$ −2.00000 −0.165521
$$147$$ −6.00000 −0.494872
$$148$$ −7.00000 −0.575396
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ −4.00000 −0.326599
$$151$$ 17.0000 1.38344 0.691720 0.722166i $$-0.256853\pi$$
0.691720 + 0.722166i $$0.256853\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 6.00000 0.485071
$$154$$ 6.00000 0.483494
$$155$$ 12.0000 0.963863
$$156$$ 1.00000 0.0800641
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 0 0
$$160$$ 3.00000 0.237171
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ −18.0000 −1.40130
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 1.00000 0.0769231
$$170$$ −9.00000 −0.690268
$$171$$ −4.00000 −0.305888
$$172$$ −1.00000 −0.0762493
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ −4.00000 −0.302372
$$176$$ 6.00000 0.452267
$$177$$ −6.00000 −0.450988
$$178$$ 6.00000 0.449719
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 6.00000 0.447214
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ 1.00000 0.0741249
$$183$$ 8.00000 0.591377
$$184$$ 0 0
$$185$$ 21.0000 1.54395
$$186$$ 4.00000 0.293294
$$187$$ −18.0000 −1.31629
$$188$$ 3.00000 0.218797
$$189$$ 5.00000 0.363696
$$190$$ 6.00000 0.435286
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 10.0000 0.717958
$$195$$ −3.00000 −0.214834
$$196$$ −6.00000 −0.428571
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 12.0000 0.852803
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ 14.0000 0.987484
$$202$$ 12.0000 0.844317
$$203$$ −6.00000 −0.421117
$$204$$ −3.00000 −0.210042
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ 12.0000 0.830057
$$210$$ −3.00000 −0.207020
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 0 0
$$213$$ −3.00000 −0.205557
$$214$$ −12.0000 −0.820303
$$215$$ 3.00000 0.204598
$$216$$ 5.00000 0.340207
$$217$$ 4.00000 0.271538
$$218$$ 7.00000 0.474100
$$219$$ 2.00000 0.135147
$$220$$ −18.0000 −1.21356
$$221$$ −3.00000 −0.201802
$$222$$ 7.00000 0.469809
$$223$$ −19.0000 −1.27233 −0.636167 0.771551i $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −8.00000 −0.533333
$$226$$ 6.00000 0.399114
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 2.00000 0.132453
$$229$$ −13.0000 −0.859064 −0.429532 0.903052i $$-0.641321\pi$$
−0.429532 + 0.903052i $$0.641321\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ −6.00000 −0.393919
$$233$$ −27.0000 −1.76883 −0.884414 0.466702i $$-0.845442\pi$$
−0.884414 + 0.466702i $$0.845442\pi$$
$$234$$ 2.00000 0.130744
$$235$$ −9.00000 −0.587095
$$236$$ −6.00000 −0.390567
$$237$$ 8.00000 0.519656
$$238$$ −3.00000 −0.194461
$$239$$ 15.0000 0.970269 0.485135 0.874439i $$-0.338771\pi$$
0.485135 + 0.874439i $$0.338771\pi$$
$$240$$ −3.00000 −0.193649
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ −25.0000 −1.60706
$$243$$ 16.0000 1.02640
$$244$$ 8.00000 0.512148
$$245$$ 18.0000 1.14998
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 4.00000 0.254000
$$249$$ 12.0000 0.760469
$$250$$ −3.00000 −0.189737
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 0 0
$$254$$ −20.0000 −1.25491
$$255$$ 9.00000 0.563602
$$256$$ 1.00000 0.0625000
$$257$$ 9.00000 0.561405 0.280702 0.959795i $$-0.409433\pi$$
0.280702 + 0.959795i $$0.409433\pi$$
$$258$$ 1.00000 0.0622573
$$259$$ 7.00000 0.434959
$$260$$ −3.00000 −0.186052
$$261$$ −12.0000 −0.742781
$$262$$ 21.0000 1.29738
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ −6.00000 −0.367194
$$268$$ 14.0000 0.855186
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ −15.0000 −0.912871
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ −1.00000 −0.0605228
$$274$$ 0 0
$$275$$ 24.0000 1.44725
$$276$$ 0 0
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 13.0000 0.779688
$$279$$ 8.00000 0.478947
$$280$$ −3.00000 −0.179284
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ −3.00000 −0.178647
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −3.00000 −0.178017
$$285$$ −6.00000 −0.355409
$$286$$ −6.00000 −0.354787
$$287$$ 0 0
$$288$$ 2.00000 0.117851
$$289$$ −8.00000 −0.470588
$$290$$ 18.0000 1.05700
$$291$$ −10.0000 −0.586210
$$292$$ 2.00000 0.117041
$$293$$ 21.0000 1.22683 0.613417 0.789760i $$-0.289795\pi$$
0.613417 + 0.789760i $$0.289795\pi$$
$$294$$ 6.00000 0.349927
$$295$$ 18.0000 1.04800
$$296$$ 7.00000 0.406867
$$297$$ −30.0000 −1.74078
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ 4.00000 0.230940
$$301$$ 1.00000 0.0576390
$$302$$ −17.0000 −0.978240
$$303$$ −12.0000 −0.689382
$$304$$ 2.00000 0.114708
$$305$$ −24.0000 −1.37424
$$306$$ −6.00000 −0.342997
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ −6.00000 −0.341882
$$309$$ −4.00000 −0.227552
$$310$$ −12.0000 −0.681554
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ −6.00000 −0.338062
$$316$$ 8.00000 0.450035
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 36.0000 2.01561
$$320$$ −3.00000 −0.167705
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ −6.00000 −0.333849
$$324$$ 1.00000 0.0555556
$$325$$ 4.00000 0.221880
$$326$$ 16.0000 0.886158
$$327$$ −7.00000 −0.387101
$$328$$ 0 0
$$329$$ −3.00000 −0.165395
$$330$$ 18.0000 0.990867
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 14.0000 0.767195
$$334$$ 0 0
$$335$$ −42.0000 −2.29471
$$336$$ −1.00000 −0.0545545
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ −6.00000 −0.325875
$$340$$ 9.00000 0.488094
$$341$$ −24.0000 −1.29967
$$342$$ 4.00000 0.216295
$$343$$ 13.0000 0.701934
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 3.00000 0.161048 0.0805242 0.996753i $$-0.474341\pi$$
0.0805242 + 0.996753i $$0.474341\pi$$
$$348$$ 6.00000 0.321634
$$349$$ −19.0000 −1.01705 −0.508523 0.861048i $$-0.669808\pi$$
−0.508523 + 0.861048i $$0.669808\pi$$
$$350$$ 4.00000 0.213809
$$351$$ −5.00000 −0.266880
$$352$$ −6.00000 −0.319801
$$353$$ 24.0000 1.27739 0.638696 0.769460i $$-0.279474\pi$$
0.638696 + 0.769460i $$0.279474\pi$$
$$354$$ 6.00000 0.318896
$$355$$ 9.00000 0.477670
$$356$$ −6.00000 −0.317999
$$357$$ 3.00000 0.158777
$$358$$ −3.00000 −0.158555
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ −6.00000 −0.316228
$$361$$ −15.0000 −0.789474
$$362$$ −20.0000 −1.05118
$$363$$ 25.0000 1.31216
$$364$$ −1.00000 −0.0524142
$$365$$ −6.00000 −0.314054
$$366$$ −8.00000 −0.418167
$$367$$ 26.0000 1.35719 0.678594 0.734513i $$-0.262589\pi$$
0.678594 + 0.734513i $$0.262589\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ −21.0000 −1.09174
$$371$$ 0 0
$$372$$ −4.00000 −0.207390
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 18.0000 0.930758
$$375$$ 3.00000 0.154919
$$376$$ −3.00000 −0.154713
$$377$$ 6.00000 0.309016
$$378$$ −5.00000 −0.257172
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ 20.0000 1.02463
$$382$$ 18.0000 0.920960
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 18.0000 0.917365
$$386$$ 4.00000 0.203595
$$387$$ 2.00000 0.101666
$$388$$ −10.0000 −0.507673
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 3.00000 0.151911
$$391$$ 0 0
$$392$$ 6.00000 0.303046
$$393$$ −21.0000 −1.05931
$$394$$ −3.00000 −0.151138
$$395$$ −24.0000 −1.20757
$$396$$ −12.0000 −0.603023
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ −2.00000 −0.100251
$$399$$ −2.00000 −0.100125
$$400$$ 4.00000 0.200000
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ −14.0000 −0.698257
$$403$$ −4.00000 −0.199254
$$404$$ −12.0000 −0.597022
$$405$$ −3.00000 −0.149071
$$406$$ 6.00000 0.297775
$$407$$ −42.0000 −2.08186
$$408$$ 3.00000 0.148522
$$409$$ 32.0000 1.58230 0.791149 0.611623i $$-0.209483\pi$$
0.791149 + 0.611623i $$0.209483\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ 6.00000 0.295241
$$414$$ 0 0
$$415$$ −36.0000 −1.76717
$$416$$ −1.00000 −0.0490290
$$417$$ −13.0000 −0.636613
$$418$$ −12.0000 −0.586939
$$419$$ 9.00000 0.439679 0.219839 0.975536i $$-0.429447\pi$$
0.219839 + 0.975536i $$0.429447\pi$$
$$420$$ 3.00000 0.146385
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ 13.0000 0.632830
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ −12.0000 −0.582086
$$426$$ 3.00000 0.145350
$$427$$ −8.00000 −0.387147
$$428$$ 12.0000 0.580042
$$429$$ 6.00000 0.289683
$$430$$ −3.00000 −0.144673
$$431$$ −33.0000 −1.58955 −0.794777 0.606902i $$-0.792412\pi$$
−0.794777 + 0.606902i $$0.792412\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ −25.0000 −1.20142 −0.600712 0.799466i $$-0.705116\pi$$
−0.600712 + 0.799466i $$0.705116\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ −18.0000 −0.863034
$$436$$ −7.00000 −0.335239
$$437$$ 0 0
$$438$$ −2.00000 −0.0955637
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 18.0000 0.858116
$$441$$ 12.0000 0.571429
$$442$$ 3.00000 0.142695
$$443$$ 21.0000 0.997740 0.498870 0.866677i $$-0.333748\pi$$
0.498870 + 0.866677i $$0.333748\pi$$
$$444$$ −7.00000 −0.332205
$$445$$ 18.0000 0.853282
$$446$$ 19.0000 0.899676
$$447$$ −6.00000 −0.283790
$$448$$ −1.00000 −0.0472456
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 8.00000 0.377124
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ 17.0000 0.798730
$$454$$ 0 0
$$455$$ 3.00000 0.140642
$$456$$ −2.00000 −0.0936586
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 13.0000 0.607450
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ 9.00000 0.419172 0.209586 0.977790i $$-0.432788\pi$$
0.209586 + 0.977790i $$0.432788\pi$$
$$462$$ 6.00000 0.279145
$$463$$ −40.0000 −1.85896 −0.929479 0.368875i $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 12.0000 0.556487
$$466$$ 27.0000 1.25075
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ −14.0000 −0.646460
$$470$$ 9.00000 0.415139
$$471$$ 14.0000 0.645086
$$472$$ 6.00000 0.276172
$$473$$ −6.00000 −0.275880
$$474$$ −8.00000 −0.367452
$$475$$ 8.00000 0.367065
$$476$$ 3.00000 0.137505
$$477$$ 0 0
$$478$$ −15.0000 −0.686084
$$479$$ −21.0000 −0.959514 −0.479757 0.877401i $$-0.659275\pi$$
−0.479757 + 0.877401i $$0.659275\pi$$
$$480$$ 3.00000 0.136931
$$481$$ −7.00000 −0.319173
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ 25.0000 1.13636
$$485$$ 30.0000 1.36223
$$486$$ −16.0000 −0.725775
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ −16.0000 −0.723545
$$490$$ −18.0000 −0.813157
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 0 0
$$493$$ −18.0000 −0.810679
$$494$$ −2.00000 −0.0899843
$$495$$ 36.0000 1.61808
$$496$$ −4.00000 −0.179605
$$497$$ 3.00000 0.134568
$$498$$ −12.0000 −0.537733
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ 3.00000 0.134164
$$501$$ 0 0
$$502$$ −24.0000 −1.07117
$$503$$ −30.0000 −1.33763 −0.668817 0.743427i $$-0.733199\pi$$
−0.668817 + 0.743427i $$0.733199\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 36.0000 1.60198
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 20.0000 0.887357
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ −9.00000 −0.398527
$$511$$ −2.00000 −0.0884748
$$512$$ −1.00000 −0.0441942
$$513$$ −10.0000 −0.441511
$$514$$ −9.00000 −0.396973
$$515$$ 12.0000 0.528783
$$516$$ −1.00000 −0.0440225
$$517$$ 18.0000 0.791639
$$518$$ −7.00000 −0.307562
$$519$$ 0 0
$$520$$ 3.00000 0.131559
$$521$$ −9.00000 −0.394297 −0.197149 0.980374i $$-0.563168\pi$$
−0.197149 + 0.980374i $$0.563168\pi$$
$$522$$ 12.0000 0.525226
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −21.0000 −0.917389
$$525$$ −4.00000 −0.174574
$$526$$ 12.0000 0.523225
$$527$$ 12.0000 0.522728
$$528$$ 6.00000 0.261116
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ −2.00000 −0.0867110
$$533$$ 0 0
$$534$$ 6.00000 0.259645
$$535$$ −36.0000 −1.55642
$$536$$ −14.0000 −0.604708
$$537$$ 3.00000 0.129460
$$538$$ −24.0000 −1.03471
$$539$$ −36.0000 −1.55063
$$540$$ 15.0000 0.645497
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ −11.0000 −0.472490
$$543$$ 20.0000 0.858282
$$544$$ 3.00000 0.128624
$$545$$ 21.0000 0.899541
$$546$$ 1.00000 0.0427960
$$547$$ 17.0000 0.726868 0.363434 0.931620i $$-0.381604\pi$$
0.363434 + 0.931620i $$0.381604\pi$$
$$548$$ 0 0
$$549$$ −16.0000 −0.682863
$$550$$ −24.0000 −1.02336
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 28.0000 1.18961
$$555$$ 21.0000 0.891400
$$556$$ −13.0000 −0.551323
$$557$$ 3.00000 0.127114 0.0635570 0.997978i $$-0.479756\pi$$
0.0635570 + 0.997978i $$0.479756\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ −1.00000 −0.0422955
$$560$$ 3.00000 0.126773
$$561$$ −18.0000 −0.759961
$$562$$ 6.00000 0.253095
$$563$$ 39.0000 1.64365 0.821827 0.569737i $$-0.192955\pi$$
0.821827 + 0.569737i $$0.192955\pi$$
$$564$$ 3.00000 0.126323
$$565$$ 18.0000 0.757266
$$566$$ 4.00000 0.168133
$$567$$ −1.00000 −0.0419961
$$568$$ 3.00000 0.125877
$$569$$ 15.0000 0.628833 0.314416 0.949285i $$-0.398191\pi$$
0.314416 + 0.949285i $$0.398191\pi$$
$$570$$ 6.00000 0.251312
$$571$$ 5.00000 0.209243 0.104622 0.994512i $$-0.466637\pi$$
0.104622 + 0.994512i $$0.466637\pi$$
$$572$$ 6.00000 0.250873
$$573$$ −18.0000 −0.751961
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ 8.00000 0.332756
$$579$$ −4.00000 −0.166234
$$580$$ −18.0000 −0.747409
$$581$$ −12.0000 −0.497844
$$582$$ 10.0000 0.414513
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 6.00000 0.248069
$$586$$ −21.0000 −0.867502
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ −8.00000 −0.329634
$$590$$ −18.0000 −0.741048
$$591$$ 3.00000 0.123404
$$592$$ −7.00000 −0.287698
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 30.0000 1.23091
$$595$$ −9.00000 −0.368964
$$596$$ −6.00000 −0.245770
$$597$$ 2.00000 0.0818546
$$598$$ 0 0
$$599$$ 6.00000 0.245153 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$600$$ −4.00000 −0.163299
$$601$$ −19.0000 −0.775026 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$602$$ −1.00000 −0.0407570
$$603$$ −28.0000 −1.14025
$$604$$ 17.0000 0.691720
$$605$$ −75.0000 −3.04918
$$606$$ 12.0000 0.487467
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ −6.00000 −0.243132
$$610$$ 24.0000 0.971732
$$611$$ 3.00000 0.121367
$$612$$ 6.00000 0.242536
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 6.00000 0.241747
$$617$$ −24.0000 −0.966204 −0.483102 0.875564i $$-0.660490\pi$$
−0.483102 + 0.875564i $$0.660490\pi$$
$$618$$ 4.00000 0.160904
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 12.0000 0.481932
$$621$$ 0 0
$$622$$ 30.0000 1.20289
$$623$$ 6.00000 0.240385
$$624$$ 1.00000 0.0400320
$$625$$ −29.0000 −1.16000
$$626$$ 1.00000 0.0399680
$$627$$ 12.0000 0.479234
$$628$$ 14.0000 0.558661
$$629$$ 21.0000 0.837325
$$630$$ 6.00000 0.239046
$$631$$ 29.0000 1.15447 0.577236 0.816577i $$-0.304131\pi$$
0.577236 + 0.816577i $$0.304131\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ −13.0000 −0.516704
$$634$$ 6.00000 0.238290
$$635$$ −60.0000 −2.38103
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ −36.0000 −1.42525
$$639$$ 6.00000 0.237356
$$640$$ 3.00000 0.118585
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 0 0
$$645$$ 3.00000 0.118125
$$646$$ 6.00000 0.236067
$$647$$ −6.00000 −0.235884 −0.117942 0.993020i $$-0.537630\pi$$
−0.117942 + 0.993020i $$0.537630\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −36.0000 −1.41312
$$650$$ −4.00000 −0.156893
$$651$$ 4.00000 0.156772
$$652$$ −16.0000 −0.626608
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 7.00000 0.273722
$$655$$ 63.0000 2.46161
$$656$$ 0 0
$$657$$ −4.00000 −0.156055
$$658$$ 3.00000 0.116952
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ −18.0000 −0.700649
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ −3.00000 −0.116510
$$664$$ −12.0000 −0.465690
$$665$$ 6.00000 0.232670
$$666$$ −14.0000 −0.542489
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −19.0000 −0.734582
$$670$$ 42.0000 1.62260
$$671$$ 48.0000 1.85302
$$672$$ 1.00000 0.0385758
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ −23.0000 −0.885927
$$675$$ −20.0000 −0.769800
$$676$$ 1.00000 0.0384615
$$677$$ 48.0000 1.84479 0.922395 0.386248i $$-0.126229\pi$$
0.922395 + 0.386248i $$0.126229\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 10.0000 0.383765
$$680$$ −9.00000 −0.345134
$$681$$ 0 0
$$682$$ 24.0000 0.919007
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ −13.0000 −0.495981
$$688$$ −1.00000 −0.0381246
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ 12.0000 0.455842
$$694$$ −3.00000 −0.113878
$$695$$ 39.0000 1.47935
$$696$$ −6.00000 −0.227429
$$697$$ 0 0
$$698$$ 19.0000 0.719161
$$699$$ −27.0000 −1.02123
$$700$$ −4.00000 −0.151186
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 5.00000 0.188713
$$703$$ −14.0000 −0.528020
$$704$$ 6.00000 0.226134
$$705$$ −9.00000 −0.338960
$$706$$ −24.0000 −0.903252
$$707$$ 12.0000 0.451306
$$708$$ −6.00000 −0.225494
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ −9.00000 −0.337764
$$711$$ −16.0000 −0.600047
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ −3.00000 −0.112272
$$715$$ −18.0000 −0.673162
$$716$$ 3.00000 0.112115
$$717$$ 15.0000 0.560185
$$718$$ 0 0
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 6.00000 0.223607
$$721$$ 4.00000 0.148968
$$722$$ 15.0000 0.558242
$$723$$ −10.0000 −0.371904
$$724$$ 20.0000 0.743294
$$725$$ 24.0000 0.891338
$$726$$ −25.0000 −0.927837
$$727$$ −10.0000 −0.370879 −0.185440 0.982656i $$-0.559371\pi$$
−0.185440 + 0.982656i $$0.559371\pi$$
$$728$$ 1.00000 0.0370625
$$729$$ 13.0000 0.481481
$$730$$ 6.00000 0.222070
$$731$$ 3.00000 0.110959
$$732$$ 8.00000 0.295689
$$733$$ 23.0000 0.849524 0.424762 0.905305i $$-0.360358\pi$$
0.424762 + 0.905305i $$0.360358\pi$$
$$734$$ −26.0000 −0.959678
$$735$$ 18.0000 0.663940
$$736$$ 0 0
$$737$$ 84.0000 3.09418
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 21.0000 0.771975
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ −9.00000 −0.330178 −0.165089 0.986279i $$-0.552791\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 18.0000 0.659469
$$746$$ 4.00000 0.146450
$$747$$ −24.0000 −0.878114
$$748$$ −18.0000 −0.658145
$$749$$ −12.0000 −0.438470
$$750$$ −3.00000 −0.109545
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 3.00000 0.109399
$$753$$ 24.0000 0.874609
$$754$$ −6.00000 −0.218507
$$755$$ −51.0000 −1.85608
$$756$$ 5.00000 0.181848
$$757$$ −16.0000 −0.581530 −0.290765 0.956795i $$-0.593910\pi$$
−0.290765 + 0.956795i $$0.593910\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 6.00000 0.217643
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ −20.0000 −0.724524
$$763$$ 7.00000 0.253417
$$764$$ −18.0000 −0.651217
$$765$$ −18.0000 −0.650791
$$766$$ −21.0000 −0.758761
$$767$$ −6.00000 −0.216647
$$768$$ 1.00000 0.0360844
$$769$$ 32.0000 1.15395 0.576975 0.816762i $$-0.304233\pi$$
0.576975 + 0.816762i $$0.304233\pi$$
$$770$$ −18.0000 −0.648675
$$771$$ 9.00000 0.324127
$$772$$ −4.00000 −0.143963
$$773$$ −39.0000 −1.40273 −0.701366 0.712801i $$-0.747426\pi$$
−0.701366 + 0.712801i $$0.747426\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ −16.0000 −0.574737
$$776$$ 10.0000 0.358979
$$777$$ 7.00000 0.251124
$$778$$ 6.00000 0.215110
$$779$$ 0 0
$$780$$ −3.00000 −0.107417
$$781$$ −18.0000 −0.644091
$$782$$ 0 0
$$783$$ −30.0000 −1.07211
$$784$$ −6.00000 −0.214286
$$785$$ −42.0000 −1.49904
$$786$$ 21.0000 0.749045
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ 3.00000 0.106871
$$789$$ −12.0000 −0.427211
$$790$$ 24.0000 0.853882
$$791$$ 6.00000 0.213335
$$792$$ 12.0000 0.426401
$$793$$ 8.00000 0.284088
$$794$$ 34.0000 1.20661
$$795$$ 0 0
$$796$$ 2.00000 0.0708881
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 2.00000 0.0707992
$$799$$ −9.00000 −0.318397
$$800$$ −4.00000 −0.141421
$$801$$ 12.0000 0.423999
$$802$$ −36.0000 −1.27120
$$803$$ 12.0000 0.423471
$$804$$ 14.0000 0.493742
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 24.0000 0.844840
$$808$$ 12.0000 0.422159
$$809$$ −33.0000 −1.16022 −0.580109 0.814539i $$-0.696990\pi$$
−0.580109 + 0.814539i $$0.696990\pi$$
$$810$$ 3.00000 0.105409
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ 11.0000 0.385787
$$814$$ 42.0000 1.47210
$$815$$ 48.0000 1.68137
$$816$$ −3.00000 −0.105021
$$817$$ −2.00000 −0.0699711
$$818$$ −32.0000 −1.11885
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −3.00000 −0.104701 −0.0523504 0.998629i $$-0.516671\pi$$
−0.0523504 + 0.998629i $$0.516671\pi$$
$$822$$ 0 0
$$823$$ 14.0000 0.488009 0.244005 0.969774i $$-0.421539\pi$$
0.244005 + 0.969774i $$0.421539\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 24.0000 0.835573
$$826$$ −6.00000 −0.208767
$$827$$ 18.0000 0.625921 0.312961 0.949766i $$-0.398679\pi$$
0.312961 + 0.949766i $$0.398679\pi$$
$$828$$ 0 0
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 36.0000 1.24958
$$831$$ −28.0000 −0.971309
$$832$$ 1.00000 0.0346688
$$833$$ 18.0000 0.623663
$$834$$ 13.0000 0.450153
$$835$$ 0 0
$$836$$ 12.0000 0.415029
$$837$$ 20.0000 0.691301
$$838$$ −9.00000 −0.310900
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ −3.00000 −0.103510
$$841$$ 7.00000 0.241379
$$842$$ −17.0000 −0.585859
$$843$$ −6.00000 −0.206651
$$844$$ −13.0000 −0.447478
$$845$$ −3.00000 −0.103203
$$846$$ 6.00000 0.206284
$$847$$ −25.0000 −0.859010
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 12.0000 0.411597
$$851$$ 0 0
$$852$$ −3.00000 −0.102778
$$853$$ −37.0000 −1.26686 −0.633428 0.773802i $$-0.718353\pi$$
−0.633428 + 0.773802i $$0.718353\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 12.0000 0.410391
$$856$$ −12.0000 −0.410152
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ −6.00000 −0.204837
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 3.00000 0.102299
$$861$$ 0 0
$$862$$ 33.0000 1.12398
$$863$$ −45.0000 −1.53182 −0.765909 0.642949i $$-0.777711\pi$$
−0.765909 + 0.642949i $$0.777711\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ 25.0000 0.849535
$$867$$ −8.00000 −0.271694
$$868$$ 4.00000 0.135769
$$869$$ 48.0000 1.62829
$$870$$ 18.0000 0.610257
$$871$$ 14.0000 0.474372
$$872$$ 7.00000 0.237050
$$873$$ 20.0000 0.676897
$$874$$ 0 0
$$875$$ −3.00000 −0.101419
$$876$$ 2.00000 0.0675737
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ −26.0000 −0.877457
$$879$$ 21.0000 0.708312
$$880$$ −18.0000 −0.606780
$$881$$ 21.0000 0.707508 0.353754 0.935339i $$-0.384905\pi$$
0.353754 + 0.935339i $$0.384905\pi$$
$$882$$ −12.0000 −0.404061
$$883$$ 29.0000 0.975928 0.487964 0.872864i $$-0.337740\pi$$
0.487964 + 0.872864i $$0.337740\pi$$
$$884$$ −3.00000 −0.100901
$$885$$ 18.0000 0.605063
$$886$$ −21.0000 −0.705509
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 7.00000 0.234905
$$889$$ −20.0000 −0.670778
$$890$$ −18.0000 −0.603361
$$891$$ 6.00000 0.201008
$$892$$ −19.0000 −0.636167
$$893$$ 6.00000 0.200782
$$894$$ 6.00000 0.200670
$$895$$ −9.00000 −0.300837
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ −24.0000 −0.800445
$$900$$ −8.00000 −0.266667
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 1.00000 0.0332779
$$904$$ 6.00000 0.199557
$$905$$ −60.0000 −1.99447
$$906$$ −17.0000 −0.564787
$$907$$ −37.0000 −1.22856 −0.614282 0.789086i $$-0.710554\pi$$
−0.614282 + 0.789086i $$0.710554\pi$$
$$908$$ 0 0
$$909$$ 24.0000 0.796030
$$910$$ −3.00000 −0.0994490
$$911$$ 30.0000 0.993944 0.496972 0.867766i $$-0.334445\pi$$
0.496972 + 0.867766i $$0.334445\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ 72.0000 2.38285
$$914$$ 10.0000 0.330771
$$915$$ −24.0000 −0.793416
$$916$$ −13.0000 −0.429532
$$917$$ 21.0000 0.693481
$$918$$ −15.0000 −0.495074
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 2.00000 0.0659022
$$922$$ −9.00000 −0.296399
$$923$$ −3.00000 −0.0987462
$$924$$ −6.00000 −0.197386
$$925$$ −28.0000 −0.920634
$$926$$ 40.0000 1.31448
$$927$$ 8.00000 0.262754
$$928$$ −6.00000 −0.196960
$$929$$ 36.0000 1.18112 0.590561 0.806993i $$-0.298907\pi$$
0.590561 + 0.806993i $$0.298907\pi$$
$$930$$ −12.0000 −0.393496
$$931$$ −12.0000 −0.393284
$$932$$ −27.0000 −0.884414
$$933$$ −30.0000 −0.982156
$$934$$ −36.0000 −1.17796
$$935$$ 54.0000 1.76599
$$936$$ 2.00000 0.0653720
$$937$$ −34.0000 −1.11073 −0.555366 0.831606i $$-0.687422\pi$$
−0.555366 + 0.831606i $$0.687422\pi$$
$$938$$ 14.0000 0.457116
$$939$$ −1.00000 −0.0326338
$$940$$ −9.00000 −0.293548
$$941$$ −21.0000 −0.684580 −0.342290 0.939594i $$-0.611203\pi$$
−0.342290 + 0.939594i $$0.611203\pi$$
$$942$$ −14.0000 −0.456145
$$943$$ 0 0
$$944$$ −6.00000 −0.195283
$$945$$ −15.0000 −0.487950
$$946$$ 6.00000 0.195077
$$947$$ 6.00000 0.194974 0.0974869 0.995237i $$-0.468920\pi$$
0.0974869 + 0.995237i $$0.468920\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 2.00000 0.0649227
$$950$$ −8.00000 −0.259554
$$951$$ −6.00000 −0.194563
$$952$$ −3.00000 −0.0972306
$$953$$ 15.0000 0.485898 0.242949 0.970039i $$-0.421885\pi$$
0.242949 + 0.970039i $$0.421885\pi$$
$$954$$ 0 0
$$955$$ 54.0000 1.74740
$$956$$ 15.0000 0.485135
$$957$$ 36.0000 1.16371
$$958$$ 21.0000 0.678479
$$959$$ 0 0
$$960$$ −3.00000 −0.0968246
$$961$$ −15.0000 −0.483871
$$962$$ 7.00000 0.225689
$$963$$ −24.0000 −0.773389
$$964$$ −10.0000 −0.322078
$$965$$ 12.0000 0.386294
$$966$$ 0 0
$$967$$ −31.0000 −0.996893 −0.498446 0.866921i $$-0.666096\pi$$
−0.498446 + 0.866921i $$0.666096\pi$$
$$968$$ −25.0000 −0.803530
$$969$$ −6.00000 −0.192748
$$970$$ −30.0000 −0.963242
$$971$$ −3.00000 −0.0962746 −0.0481373 0.998841i $$-0.515328\pi$$
−0.0481373 + 0.998841i $$0.515328\pi$$
$$972$$ 16.0000 0.513200
$$973$$ 13.0000 0.416761
$$974$$ 16.0000 0.512673
$$975$$ 4.00000 0.128103
$$976$$ 8.00000 0.256074
$$977$$ −54.0000 −1.72761 −0.863807 0.503824i $$-0.831926\pi$$
−0.863807 + 0.503824i $$0.831926\pi$$
$$978$$ 16.0000 0.511624
$$979$$ −36.0000 −1.15056
$$980$$ 18.0000 0.574989
$$981$$ 14.0000 0.446986
$$982$$ 9.00000 0.287202
$$983$$ 39.0000 1.24391 0.621953 0.783054i $$-0.286339\pi$$
0.621953 + 0.783054i $$0.286339\pi$$
$$984$$ 0 0
$$985$$ −9.00000 −0.286764
$$986$$ 18.0000 0.573237
$$987$$ −3.00000 −0.0954911
$$988$$ 2.00000 0.0636285
$$989$$ 0 0
$$990$$ −36.0000 −1.14416
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 8.00000 0.253872
$$994$$ −3.00000 −0.0951542
$$995$$ −6.00000 −0.190213
$$996$$ 12.0000 0.380235
$$997$$ −46.0000 −1.45683 −0.728417 0.685134i $$-0.759744\pi$$
−0.728417 + 0.685134i $$0.759744\pi$$
$$998$$ 40.0000 1.26618
$$999$$ 35.0000 1.10735
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 26.2.a.a.1.1 1
3.2 odd 2 234.2.a.e.1.1 1
4.3 odd 2 208.2.a.a.1.1 1
5.2 odd 4 650.2.b.d.599.1 2
5.3 odd 4 650.2.b.d.599.2 2
5.4 even 2 650.2.a.j.1.1 1
7.2 even 3 1274.2.f.p.1145.1 2
7.3 odd 6 1274.2.f.r.79.1 2
7.4 even 3 1274.2.f.p.79.1 2
7.5 odd 6 1274.2.f.r.1145.1 2
7.6 odd 2 1274.2.a.d.1.1 1
8.3 odd 2 832.2.a.i.1.1 1
8.5 even 2 832.2.a.d.1.1 1
9.2 odd 6 2106.2.e.b.1405.1 2
9.4 even 3 2106.2.e.ba.703.1 2
9.5 odd 6 2106.2.e.b.703.1 2
9.7 even 3 2106.2.e.ba.1405.1 2
11.10 odd 2 3146.2.a.n.1.1 1
12.11 even 2 1872.2.a.q.1.1 1
13.2 odd 12 338.2.e.a.147.1 4
13.3 even 3 338.2.c.d.191.1 2
13.4 even 6 338.2.c.a.315.1 2
13.5 odd 4 338.2.b.c.337.2 2
13.6 odd 12 338.2.e.a.23.2 4
13.7 odd 12 338.2.e.a.23.1 4
13.8 odd 4 338.2.b.c.337.1 2
13.9 even 3 338.2.c.d.315.1 2
13.10 even 6 338.2.c.a.191.1 2
13.11 odd 12 338.2.e.a.147.2 4
13.12 even 2 338.2.a.f.1.1 1
15.2 even 4 5850.2.e.a.5149.2 2
15.8 even 4 5850.2.e.a.5149.1 2
15.14 odd 2 5850.2.a.p.1.1 1
16.3 odd 4 3328.2.b.j.1665.1 2
16.5 even 4 3328.2.b.m.1665.1 2
16.11 odd 4 3328.2.b.j.1665.2 2
16.13 even 4 3328.2.b.m.1665.2 2
17.16 even 2 7514.2.a.c.1.1 1
19.18 odd 2 9386.2.a.j.1.1 1
20.19 odd 2 5200.2.a.x.1.1 1
24.5 odd 2 7488.2.a.g.1.1 1
24.11 even 2 7488.2.a.h.1.1 1
39.5 even 4 3042.2.b.a.1351.1 2
39.8 even 4 3042.2.b.a.1351.2 2
39.38 odd 2 3042.2.a.a.1.1 1
52.31 even 4 2704.2.f.d.337.2 2
52.47 even 4 2704.2.f.d.337.1 2
52.51 odd 2 2704.2.a.f.1.1 1
65.64 even 2 8450.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.a.1.1 1 1.1 even 1 trivial
208.2.a.a.1.1 1 4.3 odd 2
234.2.a.e.1.1 1 3.2 odd 2
338.2.a.f.1.1 1 13.12 even 2
338.2.b.c.337.1 2 13.8 odd 4
338.2.b.c.337.2 2 13.5 odd 4
338.2.c.a.191.1 2 13.10 even 6
338.2.c.a.315.1 2 13.4 even 6
338.2.c.d.191.1 2 13.3 even 3
338.2.c.d.315.1 2 13.9 even 3
338.2.e.a.23.1 4 13.7 odd 12
338.2.e.a.23.2 4 13.6 odd 12
338.2.e.a.147.1 4 13.2 odd 12
338.2.e.a.147.2 4 13.11 odd 12
650.2.a.j.1.1 1 5.4 even 2
650.2.b.d.599.1 2 5.2 odd 4
650.2.b.d.599.2 2 5.3 odd 4
832.2.a.d.1.1 1 8.5 even 2
832.2.a.i.1.1 1 8.3 odd 2
1274.2.a.d.1.1 1 7.6 odd 2
1274.2.f.p.79.1 2 7.4 even 3
1274.2.f.p.1145.1 2 7.2 even 3
1274.2.f.r.79.1 2 7.3 odd 6
1274.2.f.r.1145.1 2 7.5 odd 6
1872.2.a.q.1.1 1 12.11 even 2
2106.2.e.b.703.1 2 9.5 odd 6
2106.2.e.b.1405.1 2 9.2 odd 6
2106.2.e.ba.703.1 2 9.4 even 3
2106.2.e.ba.1405.1 2 9.7 even 3
2704.2.a.f.1.1 1 52.51 odd 2
2704.2.f.d.337.1 2 52.47 even 4
2704.2.f.d.337.2 2 52.31 even 4
3042.2.a.a.1.1 1 39.38 odd 2
3042.2.b.a.1351.1 2 39.5 even 4
3042.2.b.a.1351.2 2 39.8 even 4
3146.2.a.n.1.1 1 11.10 odd 2
3328.2.b.j.1665.1 2 16.3 odd 4
3328.2.b.j.1665.2 2 16.11 odd 4
3328.2.b.m.1665.1 2 16.5 even 4
3328.2.b.m.1665.2 2 16.13 even 4
5200.2.a.x.1.1 1 20.19 odd 2
5850.2.a.p.1.1 1 15.14 odd 2
5850.2.e.a.5149.1 2 15.8 even 4
5850.2.e.a.5149.2 2 15.2 even 4
7488.2.a.g.1.1 1 24.5 odd 2
7488.2.a.h.1.1 1 24.11 even 2
7514.2.a.c.1.1 1 17.16 even 2
8450.2.a.c.1.1 1 65.64 even 2
9386.2.a.j.1.1 1 19.18 odd 2