# Properties

 Label 234.2.a.e.1.1 Level $234$ Weight $2$ Character 234.1 Self dual yes Analytic conductor $1.868$ 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] = [234,2,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 234.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: $$1.86849940730$$ 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 $$\chi$$ $$=$$ 234.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{10} -6.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{19} +3.00000 q^{20} -6.00000 q^{22} +4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} +2.00000 q^{38} +3.00000 q^{40} -1.00000 q^{43} -6.00000 q^{44} -3.00000 q^{47} -6.00000 q^{49} +4.00000 q^{50} +1.00000 q^{52} -18.0000 q^{55} -1.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} +8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +14.0000 q^{67} +3.00000 q^{68} -3.00000 q^{70} +3.00000 q^{71} +2.00000 q^{73} -7.00000 q^{74} +2.00000 q^{76} +6.00000 q^{77} +8.00000 q^{79} +3.00000 q^{80} -12.0000 q^{83} +9.00000 q^{85} -1.00000 q^{86} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} -3.00000 q^{94} +6.00000 q^{95} -10.0000 q^{97} -6.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 3.00000 0.948683
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 3.00000 0.670820
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 3.00000 0.514496
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 4.00000 0.565685
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −18.0000 −2.42712
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 3.00000 0.372104
$$66$$ 0 0
$$67$$ 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 $$-0.443032\pi$$
0.178017 + 0.984027i $$0.443032\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −7.00000 −0.813733
$$75$$ 0 0
$$76$$ 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
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 9.00000 0.976187
$$86$$ −1.00000 −0.107833
$$87$$ 0 0
$$88$$ −6.00000 −0.639602
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −3.00000 −0.309426
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −6.00000 −0.606092
$$99$$ 0 0
$$100$$ 4.00000 0.400000
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ −18.0000 −1.71623
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 6.00000 0.552345
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 8.00000 0.724286
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 3.00000 0.263117
$$131$$ 21.0000 1.83478 0.917389 0.397991i $$-0.130293\pi$$
0.917389 + 0.397991i $$0.130293\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ 14.0000 1.20942
$$135$$ 0 0
$$136$$ 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
$$141$$ 0 0
$$142$$ 3.00000 0.251754
$$143$$ −6.00000 −0.501745
$$144$$ 0 0
$$145$$ −18.0000 −1.49482
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ −7.00000 −0.575396
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 17.0000 1.38344 0.691720 0.722166i $$-0.256853\pi$$
0.691720 + 0.722166i $$0.256853\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 0 0
$$154$$ 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
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 9.00000 0.690268
$$171$$ 0 0
$$172$$ −1.00000 −0.0762493
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ −6.00000 −0.452267
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −3.00000 −0.224231 −0.112115 0.993695i $$-0.535763\pi$$
−0.112115 + 0.993695i $$0.535763\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −21.0000 −1.54395
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ −3.00000 −0.218797
$$189$$ 0 0
$$190$$ 6.00000 0.435286
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 0 0
$$202$$ 12.0000 0.844317
$$203$$ 6.00000 0.421117
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −3.00000 −0.204598
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ −7.00000 −0.474100
$$219$$ 0 0
$$220$$ −18.0000 −1.21356
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ −19.0000 −1.27233 −0.636167 0.771551i $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −13.0000 −0.859064 −0.429532 0.903052i $$-0.641321\pi$$
−0.429532 + 0.903052i $$0.641321\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 27.0000 1.76883 0.884414 0.466702i $$-0.154558\pi$$
0.884414 + 0.466702i $$0.154558\pi$$
$$234$$ 0 0
$$235$$ −9.00000 −0.587095
$$236$$ 6.00000 0.390567
$$237$$ 0 0
$$238$$ −3.00000 −0.194461
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 25.0000 1.60706
$$243$$ 0 0
$$244$$ 8.00000 0.512148
$$245$$ −18.0000 −1.14998
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ −3.00000 −0.189737
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 20.0000 1.25491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −9.00000 −0.561405 −0.280702 0.959795i $$-0.590567\pi$$
−0.280702 + 0.959795i $$0.590567\pi$$
$$258$$ 0 0
$$259$$ 7.00000 0.434959
$$260$$ 3.00000 0.186052
$$261$$ 0 0
$$262$$ 21.0000 1.29738
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ 0 0
$$268$$ 14.0000 0.855186
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 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
$$279$$ 0 0
$$280$$ −3.00000 −0.179284
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 3.00000 0.178017
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ −18.0000 −1.05700
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ −21.0000 −1.22683 −0.613417 0.789760i $$-0.710205\pi$$
−0.613417 + 0.789760i $$0.710205\pi$$
$$294$$ 0 0
$$295$$ 18.0000 1.04800
$$296$$ −7.00000 −0.406867
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 17.0000 0.978240
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 24.0000 1.37424
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 6.00000 0.341882
$$309$$ 0 0
$$310$$ −12.0000 −0.681554
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 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
$$315$$ 0 0
$$316$$ 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
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 6.00000 0.333849
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ −16.0000 −0.886158
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3.00000 0.165395
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 42.0000 2.29471
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ 9.00000 0.488094
$$341$$ 24.0000 1.29967
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −3.00000 −0.161048 −0.0805242 0.996753i $$-0.525659\pi$$
−0.0805242 + 0.996753i $$0.525659\pi$$
$$348$$ 0 0
$$349$$ −19.0000 −1.01705 −0.508523 0.861048i $$-0.669808\pi$$
−0.508523 + 0.861048i $$0.669808\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ −6.00000 −0.319801
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ 9.00000 0.477670
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ −3.00000 −0.158555
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 20.0000 1.05118
$$363$$ 0 0
$$364$$ −1.00000 −0.0524142
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ 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
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 0 0
$$376$$ −3.00000 −0.154713
$$377$$ −6.00000 −0.309016
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 0 0
$$382$$ 18.0000 0.920960
$$383$$ −21.0000 −1.07305 −0.536525 0.843884i $$-0.680263\pi$$
−0.536525 + 0.843884i $$0.680263\pi$$
$$384$$ 0 0
$$385$$ 18.0000 0.917365
$$386$$ −4.00000 −0.203595
$$387$$ 0 0
$$388$$ −10.0000 −0.507673
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −6.00000 −0.303046
$$393$$ 0 0
$$394$$ −3.00000 −0.151138
$$395$$ 24.0000 1.20757
$$396$$ 0 0
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ 2.00000 0.100251
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ −36.0000 −1.79775 −0.898877 0.438201i $$-0.855616\pi$$
−0.898877 + 0.438201i $$0.855616\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 42.0000 2.08186
$$408$$ 0 0
$$409$$ 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
$$417$$ 0 0
$$418$$ −12.0000 −0.586939
$$419$$ −9.00000 −0.439679 −0.219839 0.975536i $$-0.570553\pi$$
−0.219839 + 0.975536i $$0.570553\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ −13.0000 −0.632830
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 12.0000 0.582086
$$426$$ 0 0
$$427$$ −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 $$0.207588\pi$$
$$432$$ 0 0
$$433$$ −25.0000 −1.20142 −0.600712 0.799466i $$-0.705116\pi$$
−0.600712 + 0.799466i $$0.705116\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ −7.00000 −0.335239
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ −18.0000 −0.858116
$$441$$ 0 0
$$442$$ 3.00000 0.142695
$$443$$ −21.0000 −0.997740 −0.498870 0.866677i $$-0.666252\pi$$
−0.498870 + 0.866677i $$0.666252\pi$$
$$444$$ 0 0
$$445$$ 18.0000 0.853282
$$446$$ −19.0000 −0.899676
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −3.00000 −0.140642
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ −13.0000 −0.607450
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −9.00000 −0.419172 −0.209586 0.977790i $$-0.567212\pi$$
−0.209586 + 0.977790i $$0.567212\pi$$
$$462$$ 0 0
$$463$$ −40.0000 −1.85896 −0.929479 0.368875i $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 27.0000 1.25075
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 0 0
$$469$$ −14.0000 −0.646460
$$470$$ −9.00000 −0.415139
$$471$$ 0 0
$$472$$ 6.00000 0.276172
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ 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 $$0.340725\pi$$
$$480$$ 0 0
$$481$$ −7.00000 −0.319173
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ 25.0000 1.13636
$$485$$ −30.0000 −1.36223
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 8.00000 0.362143
$$489$$ 0 0
$$490$$ −18.0000 −0.813157
$$491$$ 9.00000 0.406164 0.203082 0.979162i $$-0.434904\pi$$
0.203082 + 0.979162i $$0.434904\pi$$
$$492$$ 0 0
$$493$$ −18.0000 −0.810679
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ −3.00000 −0.134568
$$498$$ 0 0
$$499$$ −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 $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$504$$ 0 0
$$505$$ 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 $$0.369386\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −9.00000 −0.396973
$$515$$ −12.0000 −0.528783
$$516$$ 0 0
$$517$$ 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 $$-0.436832\pi$$
0.197149 + 0.980374i $$0.436832\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 21.0000 0.917389
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −2.00000 −0.0867110
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −36.0000 −1.55642
$$536$$ 14.0000 0.604708
$$537$$ 0 0
$$538$$ −24.0000 −1.03471
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 11.0000 0.472490
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ −24.0000 −1.02336
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ −28.0000 −1.18961
$$555$$ 0 0
$$556$$ −13.0000 −0.551323
$$557$$ −3.00000 −0.127114 −0.0635570 0.997978i $$-0.520244\pi$$
−0.0635570 + 0.997978i $$0.520244\pi$$
$$558$$ 0 0
$$559$$ −1.00000 −0.0422955
$$560$$ −3.00000 −0.126773
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ −39.0000 −1.64365 −0.821827 0.569737i $$-0.807045\pi$$
−0.821827 + 0.569737i $$0.807045\pi$$
$$564$$ 0 0
$$565$$ 18.0000 0.757266
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ 3.00000 0.125877
$$569$$ −15.0000 −0.628833 −0.314416 0.949285i $$-0.601809\pi$$
−0.314416 + 0.949285i $$0.601809\pi$$
$$570$$ 0 0
$$571$$ 5.00000 0.209243 0.104622 0.994512i $$-0.466637\pi$$
0.104622 + 0.994512i $$0.466637\pi$$
$$572$$ −6.00000 −0.250873
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 0 0
$$580$$ −18.0000 −0.747409
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ 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 $$0.664939\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 18.0000 0.741048
$$591$$ 0 0
$$592$$ −7.00000 −0.287698
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ −9.00000 −0.368964
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −6.00000 −0.245153 −0.122577 0.992459i $$-0.539116\pi$$
−0.122577 + 0.992459i $$0.539116\pi$$
$$600$$ 0 0
$$601$$ −19.0000 −0.775026 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$602$$ 1.00000 0.0407570
$$603$$ 0 0
$$604$$ 17.0000 0.691720
$$605$$ 75.0000 3.04918
$$606$$ 0 0
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ 24.0000 0.971732
$$611$$ −3.00000 −0.121367
$$612$$ 0 0
$$613$$ 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 $$-0.339510\pi$$
0.483102 + 0.875564i $$0.339510\pi$$
$$618$$ 0 0
$$619$$ −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
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ −1.00000 −0.0399680
$$627$$ 0 0
$$628$$ 14.0000 0.558661
$$629$$ −21.0000 −0.837325
$$630$$ 0 0
$$631$$ 29.0000 1.15447 0.577236 0.816577i $$-0.304131\pi$$
0.577236 + 0.816577i $$0.304131\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ 60.0000 2.38103
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 36.0000 1.42525
$$639$$ 0 0
$$640$$ 3.00000 0.118585
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 4.00000 0.156893
$$651$$ 0 0
$$652$$ −16.0000 −0.626608
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 63.0000 2.46161
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 3.00000 0.116952
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 8.00000 0.310929
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ −6.00000 −0.232670
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 42.0000 1.62260
$$671$$ −48.0000 −1.85302
$$672$$ 0 0
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ 23.0000 0.885927
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −48.0000 −1.84479 −0.922395 0.386248i $$-0.873771\pi$$
−0.922395 + 0.386248i $$0.873771\pi$$
$$678$$ 0 0
$$679$$ 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 $$0.651852\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 13.0000 0.496342
$$687$$ 0 0
$$688$$ −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
$$693$$ 0 0
$$694$$ −3.00000 −0.113878
$$695$$ −39.0000 −1.47935
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −19.0000 −0.719161
$$699$$ 0 0
$$700$$ −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
$$705$$ 0 0
$$706$$ −24.0000 −0.903252
$$707$$ −12.0000 −0.451306
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 9.00000 0.337764
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −18.0000 −0.673162
$$716$$ −3.00000 −0.112115
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ −15.0000 −0.558242
$$723$$ 0 0
$$724$$ 20.0000 0.743294
$$725$$ −24.0000 −0.891338
$$726$$ 0 0
$$727$$ −10.0000 −0.370879 −0.185440 0.982656i $$-0.559371\pi$$
−0.185440 + 0.982656i $$0.559371\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ 0 0
$$730$$ 6.00000 0.222070
$$731$$ −3.00000 −0.110959
$$732$$ 0 0
$$733$$ 23.0000 0.849524 0.424762 0.905305i $$-0.360358\pi$$
0.424762 + 0.905305i $$0.360358\pi$$
$$734$$ 26.0000 0.959678
$$735$$ 0 0
$$736$$ 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
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.00000 0.330178 0.165089 0.986279i $$-0.447209\pi$$
0.165089 + 0.986279i $$0.447209\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ −4.00000 −0.146450
$$747$$ 0 0
$$748$$ −18.0000 −0.658145
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ 0 0
$$754$$ −6.00000 −0.218507
$$755$$ 51.0000 1.85608
$$756$$ 0 0
$$757$$ −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 $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 7.00000 0.253417
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ −21.0000 −0.758761
$$767$$ 6.00000 0.216647
$$768$$ 0 0
$$769$$ 32.0000 1.15395 0.576975 0.816762i $$-0.304233\pi$$
0.576975 + 0.816762i $$0.304233\pi$$
$$770$$ 18.0000 0.648675
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ 39.0000 1.40273 0.701366 0.712801i $$-0.252574\pi$$
0.701366 + 0.712801i $$0.252574\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ 6.00000 0.215110
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −18.0000 −0.644091
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −6.00000 −0.214286
$$785$$ 42.0000 1.49904
$$786$$ 0 0
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ −3.00000 −0.106871
$$789$$ 0 0
$$790$$ 24.0000 0.853882
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ 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 $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ −9.00000 −0.318397
$$800$$ 4.00000 0.141421
$$801$$ 0 0
$$802$$ −36.0000 −1.27120
$$803$$ −12.0000 −0.423471
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 12.0000 0.422159
$$809$$ 33.0000 1.16022 0.580109 0.814539i $$-0.303010\pi$$
0.580109 + 0.814539i $$0.303010\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 6.00000 0.210559
$$813$$ 0 0
$$814$$ 42.0000 1.47210
$$815$$ −48.0000 −1.68137
$$816$$ 0 0
$$817$$ −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
$$825$$ 0 0
$$826$$ −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
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −12.0000 −0.415029
$$837$$ 0 0
$$838$$ −9.00000 −0.310900
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 17.0000 0.585859
$$843$$ 0 0
$$844$$ −13.0000 −0.447478
$$845$$ 3.00000 0.103203
$$846$$ 0 0
$$847$$ −25.0000 −0.859010
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 12.0000 0.411597
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −37.0000 −1.26686 −0.633428 0.773802i $$-0.718353\pi$$
−0.633428 + 0.773802i $$0.718353\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 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 $$0.222289\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −25.0000 −0.849535
$$867$$ 0 0
$$868$$ 4.00000 0.135769
$$869$$ −48.0000 −1.62829
$$870$$ 0 0
$$871$$ 14.0000 0.474372
$$872$$ −7.00000 −0.237050
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ 26.0000 0.877457
$$879$$ 0 0
$$880$$ −18.0000 −0.606780
$$881$$ −21.0000 −0.707508 −0.353754 0.935339i $$-0.615095\pi$$
−0.353754 + 0.935339i $$0.615095\pi$$
$$882$$ 0 0
$$883$$ 29.0000 0.975928 0.487964 0.872864i $$-0.337740\pi$$
0.487964 + 0.872864i $$0.337740\pi$$
$$884$$ 3.00000 0.100901
$$885$$ 0 0
$$886$$ −21.0000 −0.705509
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ −20.0000 −0.670778
$$890$$ 18.0000 0.603361
$$891$$ 0 0
$$892$$ −19.0000 −0.636167
$$893$$ −6.00000 −0.200782
$$894$$ 0 0
$$895$$ −9.00000 −0.300837
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 60.0000 1.99447
$$906$$ 0 0
$$907$$ −37.0000 −1.22856 −0.614282 0.789086i $$-0.710554\pi$$
−0.614282 + 0.789086i $$0.710554\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ −3.00000 −0.0994490
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ 0 0
$$913$$ 72.0000 2.38285
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −13.0000 −0.429532
$$917$$ −21.0000 −0.693481
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −9.00000 −0.296399
$$923$$ 3.00000 0.0987462
$$924$$ 0 0
$$925$$ −28.0000 −0.920634
$$926$$ −40.0000 −1.31448
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ −36.0000 −1.18112 −0.590561 0.806993i $$-0.701093\pi$$
−0.590561 + 0.806993i $$0.701093\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 27.0000 0.884414
$$933$$ 0 0
$$934$$ −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
$$939$$ 0 0
$$940$$ −9.00000 −0.293548
$$941$$ 21.0000 0.684580 0.342290 0.939594i $$-0.388797\pi$$
0.342290 + 0.939594i $$0.388797\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ 6.00000 0.195077
$$947$$ −6.00000 −0.194974 −0.0974869 0.995237i $$-0.531080\pi$$
−0.0974869 + 0.995237i $$0.531080\pi$$
$$948$$ 0 0
$$949$$ 2.00000 0.0649227
$$950$$ 8.00000 0.259554
$$951$$ 0 0
$$952$$ −3.00000 −0.0972306
$$953$$ −15.0000 −0.485898 −0.242949 0.970039i $$-0.578115\pi$$
−0.242949 + 0.970039i $$0.578115\pi$$
$$954$$ 0 0
$$955$$ 54.0000 1.74740
$$956$$ −15.0000 −0.485135
$$957$$ 0 0
$$958$$ 21.0000 0.678479
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −7.00000 −0.225689
$$963$$ 0 0
$$964$$ −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
$$969$$ 0 0
$$970$$ −30.0000 −0.963242
$$971$$ 3.00000 0.0962746 0.0481373 0.998841i $$-0.484672\pi$$
0.0481373 + 0.998841i $$0.484672\pi$$
$$972$$ 0 0
$$973$$ 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 $$0.168074\pi$$
$$978$$ 0 0
$$979$$ −36.0000 −1.15056
$$980$$ −18.0000 −0.574989
$$981$$ 0 0
$$982$$ 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
$$987$$ 0 0
$$988$$ 2.00000 0.0636285
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ −3.00000 −0.0951542
$$995$$ 6.00000 0.190213
$$996$$ 0 0
$$997$$ −46.0000 −1.45683 −0.728417 0.685134i $$-0.759744\pi$$
−0.728417 + 0.685134i $$0.759744\pi$$
$$998$$ −40.0000 −1.26618
$$999$$ 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 234.2.a.e.1.1 1
3.2 odd 2 26.2.a.a.1.1 1
4.3 odd 2 1872.2.a.q.1.1 1
5.2 odd 4 5850.2.e.a.5149.2 2
5.3 odd 4 5850.2.e.a.5149.1 2
5.4 even 2 5850.2.a.p.1.1 1
8.3 odd 2 7488.2.a.h.1.1 1
8.5 even 2 7488.2.a.g.1.1 1
9.2 odd 6 2106.2.e.ba.1405.1 2
9.4 even 3 2106.2.e.b.703.1 2
9.5 odd 6 2106.2.e.ba.703.1 2
9.7 even 3 2106.2.e.b.1405.1 2
12.11 even 2 208.2.a.a.1.1 1
13.5 odd 4 3042.2.b.a.1351.1 2
13.8 odd 4 3042.2.b.a.1351.2 2
13.12 even 2 3042.2.a.a.1.1 1
15.2 even 4 650.2.b.d.599.1 2
15.8 even 4 650.2.b.d.599.2 2
15.14 odd 2 650.2.a.j.1.1 1
21.2 odd 6 1274.2.f.p.1145.1 2
21.5 even 6 1274.2.f.r.1145.1 2
21.11 odd 6 1274.2.f.p.79.1 2
21.17 even 6 1274.2.f.r.79.1 2
21.20 even 2 1274.2.a.d.1.1 1
24.5 odd 2 832.2.a.d.1.1 1
24.11 even 2 832.2.a.i.1.1 1
33.32 even 2 3146.2.a.n.1.1 1
39.2 even 12 338.2.e.a.147.1 4
39.5 even 4 338.2.b.c.337.2 2
39.8 even 4 338.2.b.c.337.1 2
39.11 even 12 338.2.e.a.147.2 4
39.17 odd 6 338.2.c.a.315.1 2
39.20 even 12 338.2.e.a.23.1 4
39.23 odd 6 338.2.c.a.191.1 2
39.29 odd 6 338.2.c.d.191.1 2
39.32 even 12 338.2.e.a.23.2 4
39.35 odd 6 338.2.c.d.315.1 2
39.38 odd 2 338.2.a.f.1.1 1
48.5 odd 4 3328.2.b.m.1665.1 2
48.11 even 4 3328.2.b.j.1665.2 2
48.29 odd 4 3328.2.b.m.1665.2 2
48.35 even 4 3328.2.b.j.1665.1 2
51.50 odd 2 7514.2.a.c.1.1 1
57.56 even 2 9386.2.a.j.1.1 1
60.59 even 2 5200.2.a.x.1.1 1
156.47 odd 4 2704.2.f.d.337.1 2
156.83 odd 4 2704.2.f.d.337.2 2
156.155 even 2 2704.2.a.f.1.1 1
195.194 odd 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 3.2 odd 2
208.2.a.a.1.1 1 12.11 even 2
234.2.a.e.1.1 1 1.1 even 1 trivial
338.2.a.f.1.1 1 39.38 odd 2
338.2.b.c.337.1 2 39.8 even 4
338.2.b.c.337.2 2 39.5 even 4
338.2.c.a.191.1 2 39.23 odd 6
338.2.c.a.315.1 2 39.17 odd 6
338.2.c.d.191.1 2 39.29 odd 6
338.2.c.d.315.1 2 39.35 odd 6
338.2.e.a.23.1 4 39.20 even 12
338.2.e.a.23.2 4 39.32 even 12
338.2.e.a.147.1 4 39.2 even 12
338.2.e.a.147.2 4 39.11 even 12
650.2.a.j.1.1 1 15.14 odd 2
650.2.b.d.599.1 2 15.2 even 4
650.2.b.d.599.2 2 15.8 even 4
832.2.a.d.1.1 1 24.5 odd 2
832.2.a.i.1.1 1 24.11 even 2
1274.2.a.d.1.1 1 21.20 even 2
1274.2.f.p.79.1 2 21.11 odd 6
1274.2.f.p.1145.1 2 21.2 odd 6
1274.2.f.r.79.1 2 21.17 even 6
1274.2.f.r.1145.1 2 21.5 even 6
1872.2.a.q.1.1 1 4.3 odd 2
2106.2.e.b.703.1 2 9.4 even 3
2106.2.e.b.1405.1 2 9.7 even 3
2106.2.e.ba.703.1 2 9.5 odd 6
2106.2.e.ba.1405.1 2 9.2 odd 6
2704.2.a.f.1.1 1 156.155 even 2
2704.2.f.d.337.1 2 156.47 odd 4
2704.2.f.d.337.2 2 156.83 odd 4
3042.2.a.a.1.1 1 13.12 even 2
3042.2.b.a.1351.1 2 13.5 odd 4
3042.2.b.a.1351.2 2 13.8 odd 4
3146.2.a.n.1.1 1 33.32 even 2
3328.2.b.j.1665.1 2 48.35 even 4
3328.2.b.j.1665.2 2 48.11 even 4
3328.2.b.m.1665.1 2 48.5 odd 4
3328.2.b.m.1665.2 2 48.29 odd 4
5200.2.a.x.1.1 1 60.59 even 2
5850.2.a.p.1.1 1 5.4 even 2
5850.2.e.a.5149.1 2 5.3 odd 4
5850.2.e.a.5149.2 2 5.2 odd 4
7488.2.a.g.1.1 1 8.5 even 2
7488.2.a.h.1.1 1 8.3 odd 2
7514.2.a.c.1.1 1 51.50 odd 2
8450.2.a.c.1.1 1 195.194 odd 2
9386.2.a.j.1.1 1 57.56 even 2