# Properties

 Label 338.2.a.f.1.1 Level $338$ Weight $2$ Character 338.1 Self dual yes Analytic conductor $2.699$ 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] = [338,2,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, 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("338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$2.69894358832$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 338.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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 $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 3.00000 0.948683
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 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 $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 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
$$26$$ 0 0
$$27$$ −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 $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 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 $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ 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 $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.00000 −0.857143
$$50$$ 4.00000 0.565685
$$51$$ −3.00000 −0.420084
$$52$$ 0 0
$$53$$ 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 $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 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
$$65$$ 0 0
$$66$$ −6.00000 −0.738549
$$67$$ −14.0000 −1.71037 −0.855186 0.518321i $$-0.826557\pi$$
−0.855186 + 0.518321i $$0.826557\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ 3.00000 0.358569
$$71$$ 3.00000 0.356034 0.178017 0.984027i $$-0.443032\pi$$
0.178017 + 0.984027i $$0.443032\pi$$
$$72$$ −2.00000 −0.235702
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 4.00000 0.461880
$$76$$ −2.00000 −0.229416
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ 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 $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 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 $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ −6.00000 −0.632456
$$91$$ 0 0
$$92$$ 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 $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −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 $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 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 $$-0.391183\pi$$
0.335239 + 0.942133i $$0.391183\pi$$
$$110$$ −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
$$117$$ 0 0
$$118$$ 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
$$130$$ 0 0
$$131$$ −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
$$143$$ 0 0
$$144$$ −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 $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 4.00000 0.326599
$$151$$ −17.0000 −1.38344 −0.691720 0.722166i $$-0.743147\pi$$
−0.691720 + 0.722166i $$0.743147\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 6.00000 0.485071
$$154$$ −6.00000 −0.483494
$$155$$ 12.0000 0.963863
$$156$$ 0 0
$$157$$ 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 $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 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
$$169$$ 0 0
$$170$$ −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 $$0.233262\pi$$
$$182$$ 0 0
$$183$$ 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 $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ 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
$$208$$ 0 0
$$209$$ 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
$$221$$ 0 0
$$222$$ 7.00000 0.469809
$$223$$ 19.0000 1.27233 0.636167 0.771551i $$-0.280519\pi$$
0.636167 + 0.771551i $$0.280519\pi$$
$$224$$ 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 $$-0.358679\pi$$
0.429532 + 0.903052i $$0.358679\pi$$
$$230$$ 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 $$0.845442\pi$$
$$234$$ 0 0
$$235$$ −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 $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 3.00000 0.193649
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 25.0000 1.60706
$$243$$ 16.0000 1.02640
$$244$$ 8.00000 0.512148
$$245$$ −18.0000 −1.14998
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 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
$$260$$ 0 0
$$261$$ −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 $$-0.608433\pi$$
−0.334101 + 0.942537i $$0.608433\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ 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 $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ −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
$$286$$ 0 0
$$287$$ 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 $$-0.710205\pi$$
−0.613417 + 0.789760i $$0.710205\pi$$
$$294$$ −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 $$-0.518177\pi$$
−0.0570730 + 0.998370i $$0.518177\pi$$
$$308$$ −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 $$0.823744\pi$$
$$312$$ 0 0
$$313$$ −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 $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 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
$$325$$ 0 0
$$326$$ 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 $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ −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 $$0.284509\pi$$
$$338$$ 0 0
$$339$$ −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 $$-0.330192\pi$$
0.508523 + 0.861048i $$0.330192\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 0 0
$$352$$ −6.00000 −0.319801
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 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
$$364$$ 0 0
$$365$$ −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
$$377$$ 0 0
$$378$$ −5.00000 −0.257172
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ 20.0000 1.02463
$$382$$ −18.0000 −0.920960
$$383$$ −21.0000 −1.07305 −0.536525 0.843884i $$-0.680263\pi$$
−0.536525 + 0.843884i $$0.680263\pi$$
$$384$$ 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 $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 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 $$-0.174655\pi$$
0.853206 + 0.521575i $$0.174655\pi$$
$$398$$ 2.00000 0.100251
$$399$$ −2.00000 −0.100125
$$400$$ 4.00000 0.200000
$$401$$ −36.0000 −1.79775 −0.898877 0.438201i $$-0.855616\pi$$
−0.898877 + 0.438201i $$0.855616\pi$$
$$402$$ −14.0000 −0.698257
$$403$$ 0 0
$$404$$ −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 $$-0.790517\pi$$
−0.791149 + 0.611623i $$0.790517\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ 6.00000 0.295241
$$414$$ 0 0
$$415$$ −36.0000 −1.76717
$$416$$ 0 0
$$417$$ −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 $$-0.635961\pi$$
−0.414265 + 0.910156i $$0.635961\pi$$
$$422$$ −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
$$429$$ 0 0
$$430$$ −3.00000 −0.144673
$$431$$ 33.0000 1.58955 0.794777 0.606902i $$-0.207588\pi$$
0.794777 + 0.606902i $$0.207588\pi$$
$$432$$ −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
$$442$$ 0 0
$$443$$ 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 $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ −8.00000 −0.377124
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ −17.0000 −0.798730
$$454$$ 0 0
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 13.0000 0.607450
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ −9.00000 −0.419172 −0.209586 0.977790i $$-0.567212\pi$$
−0.209586 + 0.977790i $$0.567212\pi$$
$$462$$ −6.00000 −0.279145
$$463$$ 40.0000 1.85896 0.929479 0.368875i $$-0.120257\pi$$
0.929479 + 0.368875i $$0.120257\pi$$
$$464$$ 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 $$0.186655\pi$$
$$468$$ 0 0
$$469$$ −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 $$-0.340725\pi$$
0.479757 + 0.877401i $$0.340725\pi$$
$$480$$ 3.00000 0.136931
$$481$$ 0 0
$$482$$ 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 $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 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
$$494$$ 0 0
$$495$$ 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 $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ −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
$$507$$ 0 0
$$508$$ 20.0000 0.887357
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ −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
$$520$$ 0 0
$$521$$ −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 $$-0.575988\pi$$
−0.236463 + 0.971640i $$0.575988\pi$$
$$542$$ −11.0000 −0.472490
$$543$$ 20.0000 0.858282
$$544$$ −3.00000 −0.128624
$$545$$ 21.0000 0.899541
$$546$$ 0 0
$$547$$ 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 $$-0.520244\pi$$
−0.0635570 + 0.997978i $$0.520244\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 0 0
$$560$$ 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 $$0.466637\pi$$
$$572$$ 0 0
$$573$$ −18.0000 −0.751961
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ −38.0000 −1.58196 −0.790980 0.611842i $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ −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
$$585$$ 0 0
$$586$$ −21.0000 −0.867502
$$587$$ −24.0000 −0.990586 −0.495293 0.868726i $$-0.664939\pi$$
−0.495293 + 0.868726i $$0.664939\pi$$
$$588$$ −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 $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 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
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ −38.0000 −1.53481 −0.767403 0.641165i $$-0.778451\pi$$
−0.767403 + 0.641165i $$0.778451\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ 24.0000 0.966204 0.483102 0.875564i $$-0.339510\pi$$
0.483102 + 0.875564i $$0.339510\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 12.0000 0.481932
$$621$$ 0 0
$$622$$ −30.0000 −1.20289
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ −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 $$-0.695869\pi$$
−0.577236 + 0.816577i $$0.695869\pi$$
$$632$$ 8.00000 0.318223
$$633$$ −13.0000 −0.516704
$$634$$ 6.00000 0.238290
$$635$$ 60.0000 2.38103
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −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 $$-0.589027\pi$$
−0.276053 + 0.961142i $$0.589027\pi$$
$$644$$ 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
$$650$$ 0 0
$$651$$ 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 $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ 0 0
$$664$$ −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
$$676$$ 0 0
$$677$$ 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 $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ 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 $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 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 $$0.5\pi$$
$$702$$ 0 0
$$703$$ −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 $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 9.00000 0.337764
$$711$$ −16.0000 −0.600047
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ −3.00000 −0.112272
$$715$$ 0 0
$$716$$ 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 $$0.559371\pi$$
$$728$$ 0 0
$$729$$ 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 $$-0.639642\pi$$
−0.424762 + 0.905305i $$0.639642\pi$$
$$734$$ 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 $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 21.0000 0.771975
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.00000 0.330178 0.165089 0.986279i $$-0.447209\pi$$
0.165089 + 0.986279i $$0.447209\pi$$
$$744$$ 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
$$754$$ 0 0
$$755$$ −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 $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 20.0000 0.724524
$$763$$ 7.00000 0.253417
$$764$$ −18.0000 −0.651217
$$765$$ 18.0000 0.650791
$$766$$ −21.0000 −0.758761
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −32.0000 −1.15395 −0.576975 0.816762i $$-0.695767\pi$$
−0.576975 + 0.816762i $$0.695767\pi$$
$$770$$ −18.0000 −0.648675
$$771$$ 9.00000 0.324127
$$772$$ 4.00000 0.143963
$$773$$ 39.0000 1.40273 0.701366 0.712801i $$-0.252574\pi$$
0.701366 + 0.712801i $$0.252574\pi$$
$$774$$ 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
$$780$$ 0 0
$$781$$ −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 $$-0.247371\pi$$
0.712923 + 0.701242i $$0.247371\pi$$
$$788$$ −3.00000 −0.106871
$$789$$ −12.0000 −0.427211
$$790$$ 24.0000 0.853882
$$791$$ −6.00000 −0.213335
$$792$$ 12.0000 0.426401
$$793$$ 0 0
$$794$$ 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
$$806$$ 0 0
$$807$$ 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 $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 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
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3.00000 0.104701 0.0523504 0.998629i $$-0.483329\pi$$
0.0523504 + 0.998629i $$0.483329\pi$$
$$822$$ 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 $$-0.601321\pi$$
−0.312961 + 0.949766i $$0.601321\pi$$
$$828$$ 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
$$832$$ 0 0
$$833$$ 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
$$845$$ 0 0
$$846$$ 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 $$-0.281647\pi$$
0.633428 + 0.773802i $$0.281647\pi$$
$$854$$ 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 $$0.754643\pi$$
$$858$$ 0 0
$$859$$ −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 $$-0.222289\pi$$
0.765909 + 0.642949i $$0.222289\pi$$
$$864$$ −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
$$871$$ 0 0
$$872$$ 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 $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ 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 $$0.337740\pi$$
$$884$$ 0 0
$$885$$ 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
$$910$$ 0 0
$$911$$ 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
$$923$$ 0 0
$$924$$ −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 $$-0.701093\pi$$
−0.590561 + 0.806993i $$0.701093\pi$$
$$930$$ 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
$$936$$ 0 0
$$937$$ −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 $$-0.388797\pi$$
0.342290 + 0.939594i $$0.388797\pi$$
$$942$$ 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 $$-0.531080\pi$$
−0.0974869 + 0.995237i $$0.531080\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 0 0
$$950$$ −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
$$962$$ 0 0
$$963$$ −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 $$-0.333904\pi$$
0.498446 + 0.866921i $$0.333904\pi$$
$$968$$ 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
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ 54.0000 1.72761 0.863807 0.503824i $$-0.168074\pi$$
0.863807 + 0.503824i $$0.168074\pi$$
$$978$$ 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 $$-0.713661\pi$$
−0.621953 + 0.783054i $$0.713661\pi$$
$$984$$ 0 0
$$985$$ −9.00000 −0.286764
$$986$$ −18.0000 −0.573237
$$987$$ −3.00000 −0.0954911
$$988$$ 0 0
$$989$$ 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 338.2.a.f.1.1 1
3.2 odd 2 3042.2.a.a.1.1 1
4.3 odd 2 2704.2.a.f.1.1 1
5.4 even 2 8450.2.a.c.1.1 1
13.2 odd 12 338.2.e.a.147.2 4
13.3 even 3 338.2.c.a.191.1 2
13.4 even 6 338.2.c.d.315.1 2
13.5 odd 4 338.2.b.c.337.1 2
13.6 odd 12 338.2.e.a.23.1 4
13.7 odd 12 338.2.e.a.23.2 4
13.8 odd 4 338.2.b.c.337.2 2
13.9 even 3 338.2.c.a.315.1 2
13.10 even 6 338.2.c.d.191.1 2
13.11 odd 12 338.2.e.a.147.1 4
13.12 even 2 26.2.a.a.1.1 1
39.5 even 4 3042.2.b.a.1351.2 2
39.8 even 4 3042.2.b.a.1351.1 2
39.38 odd 2 234.2.a.e.1.1 1
52.31 even 4 2704.2.f.d.337.1 2
52.47 even 4 2704.2.f.d.337.2 2
52.51 odd 2 208.2.a.a.1.1 1
65.12 odd 4 650.2.b.d.599.1 2
65.38 odd 4 650.2.b.d.599.2 2
65.64 even 2 650.2.a.j.1.1 1
91.12 odd 6 1274.2.f.r.1145.1 2
91.25 even 6 1274.2.f.p.79.1 2
91.38 odd 6 1274.2.f.r.79.1 2
91.51 even 6 1274.2.f.p.1145.1 2
91.90 odd 2 1274.2.a.d.1.1 1
104.51 odd 2 832.2.a.i.1.1 1
104.77 even 2 832.2.a.d.1.1 1
117.25 even 6 2106.2.e.ba.1405.1 2
117.38 odd 6 2106.2.e.b.1405.1 2
117.77 odd 6 2106.2.e.b.703.1 2
117.103 even 6 2106.2.e.ba.703.1 2
143.142 odd 2 3146.2.a.n.1.1 1
156.155 even 2 1872.2.a.q.1.1 1
195.38 even 4 5850.2.e.a.5149.1 2
195.77 even 4 5850.2.e.a.5149.2 2
195.194 odd 2 5850.2.a.p.1.1 1
208.51 odd 4 3328.2.b.j.1665.1 2
208.77 even 4 3328.2.b.m.1665.2 2
208.155 odd 4 3328.2.b.j.1665.2 2
208.181 even 4 3328.2.b.m.1665.1 2
221.220 even 2 7514.2.a.c.1.1 1
247.246 odd 2 9386.2.a.j.1.1 1
260.259 odd 2 5200.2.a.x.1.1 1
312.77 odd 2 7488.2.a.g.1.1 1
312.155 even 2 7488.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.a.1.1 1 13.12 even 2
208.2.a.a.1.1 1 52.51 odd 2
234.2.a.e.1.1 1 39.38 odd 2
338.2.a.f.1.1 1 1.1 even 1 trivial
338.2.b.c.337.1 2 13.5 odd 4
338.2.b.c.337.2 2 13.8 odd 4
338.2.c.a.191.1 2 13.3 even 3
338.2.c.a.315.1 2 13.9 even 3
338.2.c.d.191.1 2 13.10 even 6
338.2.c.d.315.1 2 13.4 even 6
338.2.e.a.23.1 4 13.6 odd 12
338.2.e.a.23.2 4 13.7 odd 12
338.2.e.a.147.1 4 13.11 odd 12
338.2.e.a.147.2 4 13.2 odd 12
650.2.a.j.1.1 1 65.64 even 2
650.2.b.d.599.1 2 65.12 odd 4
650.2.b.d.599.2 2 65.38 odd 4
832.2.a.d.1.1 1 104.77 even 2
832.2.a.i.1.1 1 104.51 odd 2
1274.2.a.d.1.1 1 91.90 odd 2
1274.2.f.p.79.1 2 91.25 even 6
1274.2.f.p.1145.1 2 91.51 even 6
1274.2.f.r.79.1 2 91.38 odd 6
1274.2.f.r.1145.1 2 91.12 odd 6
1872.2.a.q.1.1 1 156.155 even 2
2106.2.e.b.703.1 2 117.77 odd 6
2106.2.e.b.1405.1 2 117.38 odd 6
2106.2.e.ba.703.1 2 117.103 even 6
2106.2.e.ba.1405.1 2 117.25 even 6
2704.2.a.f.1.1 1 4.3 odd 2
2704.2.f.d.337.1 2 52.31 even 4
2704.2.f.d.337.2 2 52.47 even 4
3042.2.a.a.1.1 1 3.2 odd 2
3042.2.b.a.1351.1 2 39.8 even 4
3042.2.b.a.1351.2 2 39.5 even 4
3146.2.a.n.1.1 1 143.142 odd 2
3328.2.b.j.1665.1 2 208.51 odd 4
3328.2.b.j.1665.2 2 208.155 odd 4
3328.2.b.m.1665.1 2 208.181 even 4
3328.2.b.m.1665.2 2 208.77 even 4
5200.2.a.x.1.1 1 260.259 odd 2
5850.2.a.p.1.1 1 195.194 odd 2
5850.2.e.a.5149.1 2 195.38 even 4
5850.2.e.a.5149.2 2 195.77 even 4
7488.2.a.g.1.1 1 312.77 odd 2
7488.2.a.h.1.1 1 312.155 even 2
7514.2.a.c.1.1 1 221.220 even 2
8450.2.a.c.1.1 1 5.4 even 2
9386.2.a.j.1.1 1 247.246 odd 2