# Properties

 Label 5610.2.a.s.1.1 Level $5610$ Weight $2$ Character 5610.1 Self dual yes Analytic conductor $44.796$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5610.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: $$44.7960755339$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 5610.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -5.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -5.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} -9.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} +5.00000 q^{38} +5.00000 q^{39} -1.00000 q^{40} -10.0000 q^{41} +1.00000 q^{42} -12.0000 q^{43} +1.00000 q^{44} +1.00000 q^{45} +5.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} +5.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +1.00000 q^{56} -5.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} +13.0000 q^{61} +9.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +5.00000 q^{65} -1.00000 q^{66} -5.00000 q^{67} +1.00000 q^{68} -5.00000 q^{69} +1.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} -12.0000 q^{73} +7.00000 q^{74} +1.00000 q^{75} -5.00000 q^{76} -1.00000 q^{77} -5.00000 q^{78} -2.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +15.0000 q^{83} -1.00000 q^{84} +1.00000 q^{85} +12.0000 q^{86} -2.00000 q^{87} -1.00000 q^{88} +2.00000 q^{89} -1.00000 q^{90} -5.00000 q^{91} -5.00000 q^{92} -9.00000 q^{93} +2.00000 q^{94} -5.00000 q^{95} -1.00000 q^{96} +13.0000 q^{97} +6.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## 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
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −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
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 1.00000 0.301511
$$12$$ 1.00000 0.288675
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536
$$18$$ −1.00000 −0.235702
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −1.00000 −0.218218
$$22$$ −1.00000 −0.213201
$$23$$ −5.00000 −1.04257 −0.521286 0.853382i $$-0.674548\pi$$
−0.521286 + 0.853382i $$0.674548\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ −5.00000 −0.980581
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −9.00000 −1.61645 −0.808224 0.588875i $$-0.799571\pi$$
−0.808224 + 0.588875i $$0.799571\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 1.00000 0.174078
$$34$$ −1.00000 −0.171499
$$35$$ −1.00000 −0.169031
$$36$$ 1.00000 0.166667
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 5.00000 0.811107
$$39$$ 5.00000 0.800641
$$40$$ −1.00000 −0.158114
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 1.00000 0.149071
$$46$$ 5.00000 0.737210
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.00000 −0.857143
$$50$$ −1.00000 −0.141421
$$51$$ 1.00000 0.140028
$$52$$ 5.00000 0.693375
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 1.00000 0.134840
$$56$$ 1.00000 0.133631
$$57$$ −5.00000 −0.662266
$$58$$ 2.00000 0.262613
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ 9.00000 1.14300
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 5.00000 0.620174
$$66$$ −1.00000 −0.123091
$$67$$ −5.00000 −0.610847 −0.305424 0.952217i $$-0.598798\pi$$
−0.305424 + 0.952217i $$0.598798\pi$$
$$68$$ 1.00000 0.121268
$$69$$ −5.00000 −0.601929
$$70$$ 1.00000 0.119523
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −12.0000 −1.40449 −0.702247 0.711934i $$-0.747820\pi$$
−0.702247 + 0.711934i $$0.747820\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 1.00000 0.115470
$$76$$ −5.00000 −0.573539
$$77$$ −1.00000 −0.113961
$$78$$ −5.00000 −0.566139
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ 15.0000 1.64646 0.823232 0.567705i $$-0.192169\pi$$
0.823232 + 0.567705i $$0.192169\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 1.00000 0.108465
$$86$$ 12.0000 1.29399
$$87$$ −2.00000 −0.214423
$$88$$ −1.00000 −0.106600
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ −5.00000 −0.524142
$$92$$ −5.00000 −0.521286
$$93$$ −9.00000 −0.933257
$$94$$ 2.00000 0.206284
$$95$$ −5.00000 −0.512989
$$96$$ −1.00000 −0.102062
$$97$$ 13.0000 1.31995 0.659975 0.751288i $$-0.270567\pi$$
0.659975 + 0.751288i $$0.270567\pi$$
$$98$$ 6.00000 0.606092
$$99$$ 1.00000 0.100504
$$100$$ 1.00000 0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ −1.00000 −0.0990148
$$103$$ −5.00000 −0.492665 −0.246332 0.969185i $$-0.579225\pi$$
−0.246332 + 0.969185i $$0.579225\pi$$
$$104$$ −5.00000 −0.490290
$$105$$ −1.00000 −0.0975900
$$106$$ 10.0000 0.971286
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 15.0000 1.43674 0.718370 0.695662i $$-0.244889\pi$$
0.718370 + 0.695662i $$0.244889\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ −7.00000 −0.664411
$$112$$ −1.00000 −0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 5.00000 0.468293
$$115$$ −5.00000 −0.466252
$$116$$ −2.00000 −0.185695
$$117$$ 5.00000 0.462250
$$118$$ −4.00000 −0.368230
$$119$$ −1.00000 −0.0916698
$$120$$ −1.00000 −0.0912871
$$121$$ 1.00000 0.0909091
$$122$$ −13.0000 −1.17696
$$123$$ −10.0000 −0.901670
$$124$$ −9.00000 −0.808224
$$125$$ 1.00000 0.0894427
$$126$$ 1.00000 0.0890871
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −12.0000 −1.05654
$$130$$ −5.00000 −0.438529
$$131$$ 11.0000 0.961074 0.480537 0.876974i $$-0.340442\pi$$
0.480537 + 0.876974i $$0.340442\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 5.00000 0.433555
$$134$$ 5.00000 0.431934
$$135$$ 1.00000 0.0860663
$$136$$ −1.00000 −0.0857493
$$137$$ 1.00000 0.0854358 0.0427179 0.999087i $$-0.486398\pi$$
0.0427179 + 0.999087i $$0.486398\pi$$
$$138$$ 5.00000 0.425628
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ −2.00000 −0.168430
$$142$$ 2.00000 0.167836
$$143$$ 5.00000 0.418121
$$144$$ 1.00000 0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ 12.0000 0.993127
$$147$$ −6.00000 −0.494872
$$148$$ −7.00000 −0.575396
$$149$$ −7.00000 −0.573462 −0.286731 0.958011i $$-0.592569\pi$$
−0.286731 + 0.958011i $$0.592569\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −23.0000 −1.87171 −0.935857 0.352381i $$-0.885372\pi$$
−0.935857 + 0.352381i $$0.885372\pi$$
$$152$$ 5.00000 0.405554
$$153$$ 1.00000 0.0808452
$$154$$ 1.00000 0.0805823
$$155$$ −9.00000 −0.722897
$$156$$ 5.00000 0.400320
$$157$$ 24.0000 1.91541 0.957704 0.287754i $$-0.0929087\pi$$
0.957704 + 0.287754i $$0.0929087\pi$$
$$158$$ 2.00000 0.159111
$$159$$ −10.0000 −0.793052
$$160$$ −1.00000 −0.0790569
$$161$$ 5.00000 0.394055
$$162$$ −1.00000 −0.0785674
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 1.00000 0.0778499
$$166$$ −15.0000 −1.16423
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 12.0000 0.923077
$$170$$ −1.00000 −0.0766965
$$171$$ −5.00000 −0.382360
$$172$$ −12.0000 −0.914991
$$173$$ 3.00000 0.228086 0.114043 0.993476i $$-0.463620\pi$$
0.114043 + 0.993476i $$0.463620\pi$$
$$174$$ 2.00000 0.151620
$$175$$ −1.00000 −0.0755929
$$176$$ 1.00000 0.0753778
$$177$$ 4.00000 0.300658
$$178$$ −2.00000 −0.149906
$$179$$ −7.00000 −0.523205 −0.261602 0.965176i $$-0.584251\pi$$
−0.261602 + 0.965176i $$0.584251\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 5.00000 0.370625
$$183$$ 13.0000 0.960988
$$184$$ 5.00000 0.368605
$$185$$ −7.00000 −0.514650
$$186$$ 9.00000 0.659912
$$187$$ 1.00000 0.0731272
$$188$$ −2.00000 −0.145865
$$189$$ −1.00000 −0.0727393
$$190$$ 5.00000 0.362738
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ −13.0000 −0.933346
$$195$$ 5.00000 0.358057
$$196$$ −6.00000 −0.428571
$$197$$ 15.0000 1.06871 0.534353 0.845262i $$-0.320555\pi$$
0.534353 + 0.845262i $$0.320555\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ −17.0000 −1.20510 −0.602549 0.798082i $$-0.705848\pi$$
−0.602549 + 0.798082i $$0.705848\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −5.00000 −0.352673
$$202$$ −6.00000 −0.422159
$$203$$ 2.00000 0.140372
$$204$$ 1.00000 0.0700140
$$205$$ −10.0000 −0.698430
$$206$$ 5.00000 0.348367
$$207$$ −5.00000 −0.347524
$$208$$ 5.00000 0.346688
$$209$$ −5.00000 −0.345857
$$210$$ 1.00000 0.0690066
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ −2.00000 −0.137038
$$214$$ 2.00000 0.136717
$$215$$ −12.0000 −0.818393
$$216$$ −1.00000 −0.0680414
$$217$$ 9.00000 0.610960
$$218$$ −15.0000 −1.01593
$$219$$ −12.0000 −0.810885
$$220$$ 1.00000 0.0674200
$$221$$ 5.00000 0.336336
$$222$$ 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
$$225$$ 1.00000 0.0666667
$$226$$ −6.00000 −0.399114
$$227$$ 26.0000 1.72568 0.862840 0.505477i $$-0.168683\pi$$
0.862840 + 0.505477i $$0.168683\pi$$
$$228$$ −5.00000 −0.331133
$$229$$ −21.0000 −1.38772 −0.693860 0.720110i $$-0.744091\pi$$
−0.693860 + 0.720110i $$0.744091\pi$$
$$230$$ 5.00000 0.329690
$$231$$ −1.00000 −0.0657952
$$232$$ 2.00000 0.131306
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ −5.00000 −0.326860
$$235$$ −2.00000 −0.130466
$$236$$ 4.00000 0.260378
$$237$$ −2.00000 −0.129914
$$238$$ 1.00000 0.0648204
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −17.0000 −1.09507 −0.547533 0.836784i $$-0.684433\pi$$
−0.547533 + 0.836784i $$0.684433\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 1.00000 0.0641500
$$244$$ 13.0000 0.832240
$$245$$ −6.00000 −0.383326
$$246$$ 10.0000 0.637577
$$247$$ −25.0000 −1.59071
$$248$$ 9.00000 0.571501
$$249$$ 15.0000 0.950586
$$250$$ −1.00000 −0.0632456
$$251$$ −27.0000 −1.70422 −0.852112 0.523359i $$-0.824679\pi$$
−0.852112 + 0.523359i $$0.824679\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ −5.00000 −0.314347
$$254$$ −12.0000 −0.752947
$$255$$ 1.00000 0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 7.00000 0.434959
$$260$$ 5.00000 0.310087
$$261$$ −2.00000 −0.123797
$$262$$ −11.0000 −0.679582
$$263$$ 3.00000 0.184988 0.0924940 0.995713i $$-0.470516\pi$$
0.0924940 + 0.995713i $$0.470516\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ −10.0000 −0.614295
$$266$$ −5.00000 −0.306570
$$267$$ 2.00000 0.122398
$$268$$ −5.00000 −0.305424
$$269$$ −7.00000 −0.426798 −0.213399 0.976965i $$-0.568453\pi$$
−0.213399 + 0.976965i $$0.568453\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ −5.00000 −0.302614
$$274$$ −1.00000 −0.0604122
$$275$$ 1.00000 0.0603023
$$276$$ −5.00000 −0.300965
$$277$$ 4.00000 0.240337 0.120168 0.992754i $$-0.461657\pi$$
0.120168 + 0.992754i $$0.461657\pi$$
$$278$$ 14.0000 0.839664
$$279$$ −9.00000 −0.538816
$$280$$ 1.00000 0.0597614
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 2.00000 0.119098
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ −5.00000 −0.296174
$$286$$ −5.00000 −0.295656
$$287$$ 10.0000 0.590281
$$288$$ −1.00000 −0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 2.00000 0.117444
$$291$$ 13.0000 0.762073
$$292$$ −12.0000 −0.702247
$$293$$ −24.0000 −1.40209 −0.701047 0.713115i $$-0.747284\pi$$
−0.701047 + 0.713115i $$0.747284\pi$$
$$294$$ 6.00000 0.349927
$$295$$ 4.00000 0.232889
$$296$$ 7.00000 0.406867
$$297$$ 1.00000 0.0580259
$$298$$ 7.00000 0.405499
$$299$$ −25.0000 −1.44579
$$300$$ 1.00000 0.0577350
$$301$$ 12.0000 0.691669
$$302$$ 23.0000 1.32350
$$303$$ 6.00000 0.344691
$$304$$ −5.00000 −0.286770
$$305$$ 13.0000 0.744378
$$306$$ −1.00000 −0.0571662
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ −1.00000 −0.0569803
$$309$$ −5.00000 −0.284440
$$310$$ 9.00000 0.511166
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ −5.00000 −0.283069
$$313$$ −21.0000 −1.18699 −0.593495 0.804838i $$-0.702252\pi$$
−0.593495 + 0.804838i $$0.702252\pi$$
$$314$$ −24.0000 −1.35440
$$315$$ −1.00000 −0.0563436
$$316$$ −2.00000 −0.112509
$$317$$ 26.0000 1.46031 0.730153 0.683284i $$-0.239449\pi$$
0.730153 + 0.683284i $$0.239449\pi$$
$$318$$ 10.0000 0.560772
$$319$$ −2.00000 −0.111979
$$320$$ 1.00000 0.0559017
$$321$$ −2.00000 −0.111629
$$322$$ −5.00000 −0.278639
$$323$$ −5.00000 −0.278207
$$324$$ 1.00000 0.0555556
$$325$$ 5.00000 0.277350
$$326$$ 12.0000 0.664619
$$327$$ 15.0000 0.829502
$$328$$ 10.0000 0.552158
$$329$$ 2.00000 0.110264
$$330$$ −1.00000 −0.0550482
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 15.0000 0.823232
$$333$$ −7.00000 −0.383598
$$334$$ 16.0000 0.875481
$$335$$ −5.00000 −0.273179
$$336$$ −1.00000 −0.0545545
$$337$$ 28.0000 1.52526 0.762629 0.646837i $$-0.223908\pi$$
0.762629 + 0.646837i $$0.223908\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 6.00000 0.325875
$$340$$ 1.00000 0.0542326
$$341$$ −9.00000 −0.487377
$$342$$ 5.00000 0.270369
$$343$$ 13.0000 0.701934
$$344$$ 12.0000 0.646997
$$345$$ −5.00000 −0.269191
$$346$$ −3.00000 −0.161281
$$347$$ 30.0000 1.61048 0.805242 0.592946i $$-0.202035\pi$$
0.805242 + 0.592946i $$0.202035\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 5.00000 0.266880
$$352$$ −1.00000 −0.0533002
$$353$$ −1.00000 −0.0532246 −0.0266123 0.999646i $$-0.508472\pi$$
−0.0266123 + 0.999646i $$0.508472\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ −2.00000 −0.106149
$$356$$ 2.00000 0.106000
$$357$$ −1.00000 −0.0529256
$$358$$ 7.00000 0.369961
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 6.00000 0.315789
$$362$$ 16.0000 0.840941
$$363$$ 1.00000 0.0524864
$$364$$ −5.00000 −0.262071
$$365$$ −12.0000 −0.628109
$$366$$ −13.0000 −0.679521
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ −5.00000 −0.260643
$$369$$ −10.0000 −0.520579
$$370$$ 7.00000 0.363913
$$371$$ 10.0000 0.519174
$$372$$ −9.00000 −0.466628
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ −1.00000 −0.0517088
$$375$$ 1.00000 0.0516398
$$376$$ 2.00000 0.103142
$$377$$ −10.0000 −0.515026
$$378$$ 1.00000 0.0514344
$$379$$ 7.00000 0.359566 0.179783 0.983706i $$-0.442460\pi$$
0.179783 + 0.983706i $$0.442460\pi$$
$$380$$ −5.00000 −0.256495
$$381$$ 12.0000 0.614779
$$382$$ 0 0
$$383$$ 26.0000 1.32854 0.664269 0.747494i $$-0.268743\pi$$
0.664269 + 0.747494i $$0.268743\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −1.00000 −0.0509647
$$386$$ 0 0
$$387$$ −12.0000 −0.609994
$$388$$ 13.0000 0.659975
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ −5.00000 −0.253185
$$391$$ −5.00000 −0.252861
$$392$$ 6.00000 0.303046
$$393$$ 11.0000 0.554877
$$394$$ −15.0000 −0.755689
$$395$$ −2.00000 −0.100631
$$396$$ 1.00000 0.0502519
$$397$$ 10.0000 0.501886 0.250943 0.968002i $$-0.419259\pi$$
0.250943 + 0.968002i $$0.419259\pi$$
$$398$$ 17.0000 0.852133
$$399$$ 5.00000 0.250313
$$400$$ 1.00000 0.0500000
$$401$$ 19.0000 0.948815 0.474407 0.880305i $$-0.342662\pi$$
0.474407 + 0.880305i $$0.342662\pi$$
$$402$$ 5.00000 0.249377
$$403$$ −45.0000 −2.24161
$$404$$ 6.00000 0.298511
$$405$$ 1.00000 0.0496904
$$406$$ −2.00000 −0.0992583
$$407$$ −7.00000 −0.346977
$$408$$ −1.00000 −0.0495074
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 10.0000 0.493865
$$411$$ 1.00000 0.0493264
$$412$$ −5.00000 −0.246332
$$413$$ −4.00000 −0.196827
$$414$$ 5.00000 0.245737
$$415$$ 15.0000 0.736321
$$416$$ −5.00000 −0.245145
$$417$$ −14.0000 −0.685583
$$418$$ 5.00000 0.244558
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ −1.00000 −0.0487950
$$421$$ −29.0000 −1.41337 −0.706687 0.707527i $$-0.749811\pi$$
−0.706687 + 0.707527i $$0.749811\pi$$
$$422$$ −14.0000 −0.681509
$$423$$ −2.00000 −0.0972433
$$424$$ 10.0000 0.485643
$$425$$ 1.00000 0.0485071
$$426$$ 2.00000 0.0969003
$$427$$ −13.0000 −0.629114
$$428$$ −2.00000 −0.0966736
$$429$$ 5.00000 0.241402
$$430$$ 12.0000 0.578691
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ −9.00000 −0.432014
$$435$$ −2.00000 −0.0958927
$$436$$ 15.0000 0.718370
$$437$$ 25.0000 1.19591
$$438$$ 12.0000 0.573382
$$439$$ −14.0000 −0.668184 −0.334092 0.942541i $$-0.608430\pi$$
−0.334092 + 0.942541i $$0.608430\pi$$
$$440$$ −1.00000 −0.0476731
$$441$$ −6.00000 −0.285714
$$442$$ −5.00000 −0.237826
$$443$$ 18.0000 0.855206 0.427603 0.903967i $$-0.359358\pi$$
0.427603 + 0.903967i $$0.359358\pi$$
$$444$$ −7.00000 −0.332205
$$445$$ 2.00000 0.0948091
$$446$$ 19.0000 0.899676
$$447$$ −7.00000 −0.331089
$$448$$ −1.00000 −0.0472456
$$449$$ 25.0000 1.17982 0.589911 0.807468i $$-0.299163\pi$$
0.589911 + 0.807468i $$0.299163\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −10.0000 −0.470882
$$452$$ 6.00000 0.282216
$$453$$ −23.0000 −1.08063
$$454$$ −26.0000 −1.22024
$$455$$ −5.00000 −0.234404
$$456$$ 5.00000 0.234146
$$457$$ −29.0000 −1.35656 −0.678281 0.734802i $$-0.737275\pi$$
−0.678281 + 0.734802i $$0.737275\pi$$
$$458$$ 21.0000 0.981266
$$459$$ 1.00000 0.0466760
$$460$$ −5.00000 −0.233126
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 1.00000 0.0465242
$$463$$ −23.0000 −1.06890 −0.534450 0.845200i $$-0.679481\pi$$
−0.534450 + 0.845200i $$0.679481\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ −9.00000 −0.417365
$$466$$ −22.0000 −1.01913
$$467$$ 32.0000 1.48078 0.740392 0.672176i $$-0.234640\pi$$
0.740392 + 0.672176i $$0.234640\pi$$
$$468$$ 5.00000 0.231125
$$469$$ 5.00000 0.230879
$$470$$ 2.00000 0.0922531
$$471$$ 24.0000 1.10586
$$472$$ −4.00000 −0.184115
$$473$$ −12.0000 −0.551761
$$474$$ 2.00000 0.0918630
$$475$$ −5.00000 −0.229416
$$476$$ −1.00000 −0.0458349
$$477$$ −10.0000 −0.457869
$$478$$ −6.00000 −0.274434
$$479$$ −15.0000 −0.685367 −0.342684 0.939451i $$-0.611336\pi$$
−0.342684 + 0.939451i $$0.611336\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ −35.0000 −1.59586
$$482$$ 17.0000 0.774329
$$483$$ 5.00000 0.227508
$$484$$ 1.00000 0.0454545
$$485$$ 13.0000 0.590300
$$486$$ −1.00000 −0.0453609
$$487$$ 40.0000 1.81257 0.906287 0.422664i $$-0.138905\pi$$
0.906287 + 0.422664i $$0.138905\pi$$
$$488$$ −13.0000 −0.588482
$$489$$ −12.0000 −0.542659
$$490$$ 6.00000 0.271052
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ −2.00000 −0.0900755
$$494$$ 25.0000 1.12480
$$495$$ 1.00000 0.0449467
$$496$$ −9.00000 −0.404112
$$497$$ 2.00000 0.0897123
$$498$$ −15.0000 −0.672166
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −16.0000 −0.714827
$$502$$ 27.0000 1.20507
$$503$$ 4.00000 0.178351 0.0891756 0.996016i $$-0.471577\pi$$
0.0891756 + 0.996016i $$0.471577\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 6.00000 0.266996
$$506$$ 5.00000 0.222277
$$507$$ 12.0000 0.532939
$$508$$ 12.0000 0.532414
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ −1.00000 −0.0442807
$$511$$ 12.0000 0.530849
$$512$$ −1.00000 −0.0441942
$$513$$ −5.00000 −0.220755
$$514$$ 6.00000 0.264649
$$515$$ −5.00000 −0.220326
$$516$$ −12.0000 −0.528271
$$517$$ −2.00000 −0.0879599
$$518$$ −7.00000 −0.307562
$$519$$ 3.00000 0.131685
$$520$$ −5.00000 −0.219265
$$521$$ −29.0000 −1.27051 −0.635257 0.772301i $$-0.719106\pi$$
−0.635257 + 0.772301i $$0.719106\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ −32.0000 −1.39926 −0.699631 0.714504i $$-0.746652\pi$$
−0.699631 + 0.714504i $$0.746652\pi$$
$$524$$ 11.0000 0.480537
$$525$$ −1.00000 −0.0436436
$$526$$ −3.00000 −0.130806
$$527$$ −9.00000 −0.392046
$$528$$ 1.00000 0.0435194
$$529$$ 2.00000 0.0869565
$$530$$ 10.0000 0.434372
$$531$$ 4.00000 0.173585
$$532$$ 5.00000 0.216777
$$533$$ −50.0000 −2.16574
$$534$$ −2.00000 −0.0865485
$$535$$ −2.00000 −0.0864675
$$536$$ 5.00000 0.215967
$$537$$ −7.00000 −0.302072
$$538$$ 7.00000 0.301791
$$539$$ −6.00000 −0.258438
$$540$$ 1.00000 0.0430331
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ −16.0000 −0.686626
$$544$$ −1.00000 −0.0428746
$$545$$ 15.0000 0.642529
$$546$$ 5.00000 0.213980
$$547$$ −27.0000 −1.15444 −0.577218 0.816590i $$-0.695862\pi$$
−0.577218 + 0.816590i $$0.695862\pi$$
$$548$$ 1.00000 0.0427179
$$549$$ 13.0000 0.554826
$$550$$ −1.00000 −0.0426401
$$551$$ 10.0000 0.426014
$$552$$ 5.00000 0.212814
$$553$$ 2.00000 0.0850487
$$554$$ −4.00000 −0.169944
$$555$$ −7.00000 −0.297133
$$556$$ −14.0000 −0.593732
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 9.00000 0.381000
$$559$$ −60.0000 −2.53773
$$560$$ −1.00000 −0.0422577
$$561$$ 1.00000 0.0422200
$$562$$ 18.0000 0.759284
$$563$$ 21.0000 0.885044 0.442522 0.896758i $$-0.354084\pi$$
0.442522 + 0.896758i $$0.354084\pi$$
$$564$$ −2.00000 −0.0842152
$$565$$ 6.00000 0.252422
$$566$$ −4.00000 −0.168133
$$567$$ −1.00000 −0.0419961
$$568$$ 2.00000 0.0839181
$$569$$ 7.00000 0.293455 0.146728 0.989177i $$-0.453126\pi$$
0.146728 + 0.989177i $$0.453126\pi$$
$$570$$ 5.00000 0.209427
$$571$$ 42.0000 1.75765 0.878823 0.477149i $$-0.158330\pi$$
0.878823 + 0.477149i $$0.158330\pi$$
$$572$$ 5.00000 0.209061
$$573$$ 0 0
$$574$$ −10.0000 −0.417392
$$575$$ −5.00000 −0.208514
$$576$$ 1.00000 0.0416667
$$577$$ −18.0000 −0.749350 −0.374675 0.927156i $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ −2.00000 −0.0830455
$$581$$ −15.0000 −0.622305
$$582$$ −13.0000 −0.538867
$$583$$ −10.0000 −0.414158
$$584$$ 12.0000 0.496564
$$585$$ 5.00000 0.206725
$$586$$ 24.0000 0.991431
$$587$$ −2.00000 −0.0825488 −0.0412744 0.999148i $$-0.513142\pi$$
−0.0412744 + 0.999148i $$0.513142\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ 45.0000 1.85419
$$590$$ −4.00000 −0.164677
$$591$$ 15.0000 0.617018
$$592$$ −7.00000 −0.287698
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ −1.00000 −0.0409960
$$596$$ −7.00000 −0.286731
$$597$$ −17.0000 −0.695764
$$598$$ 25.0000 1.02233
$$599$$ −21.0000 −0.858037 −0.429018 0.903296i $$-0.641140\pi$$
−0.429018 + 0.903296i $$0.641140\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −17.0000 −0.693444 −0.346722 0.937968i $$-0.612705\pi$$
−0.346722 + 0.937968i $$0.612705\pi$$
$$602$$ −12.0000 −0.489083
$$603$$ −5.00000 −0.203616
$$604$$ −23.0000 −0.935857
$$605$$ 1.00000 0.0406558
$$606$$ −6.00000 −0.243733
$$607$$ 7.00000 0.284121 0.142061 0.989858i $$-0.454627\pi$$
0.142061 + 0.989858i $$0.454627\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 2.00000 0.0810441
$$610$$ −13.0000 −0.526355
$$611$$ −10.0000 −0.404557
$$612$$ 1.00000 0.0404226
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 20.0000 0.807134
$$615$$ −10.0000 −0.403239
$$616$$ 1.00000 0.0402911
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 5.00000 0.201129
$$619$$ 17.0000 0.683288 0.341644 0.939829i $$-0.389016\pi$$
0.341644 + 0.939829i $$0.389016\pi$$
$$620$$ −9.00000 −0.361449
$$621$$ −5.00000 −0.200643
$$622$$ 6.00000 0.240578
$$623$$ −2.00000 −0.0801283
$$624$$ 5.00000 0.200160
$$625$$ 1.00000 0.0400000
$$626$$ 21.0000 0.839329
$$627$$ −5.00000 −0.199681
$$628$$ 24.0000 0.957704
$$629$$ −7.00000 −0.279108
$$630$$ 1.00000 0.0398410
$$631$$ −10.0000 −0.398094 −0.199047 0.979990i $$-0.563785\pi$$
−0.199047 + 0.979990i $$0.563785\pi$$
$$632$$ 2.00000 0.0795557
$$633$$ 14.0000 0.556450
$$634$$ −26.0000 −1.03259
$$635$$ 12.0000 0.476205
$$636$$ −10.0000 −0.396526
$$637$$ −30.0000 −1.18864
$$638$$ 2.00000 0.0791808
$$639$$ −2.00000 −0.0791188
$$640$$ −1.00000 −0.0395285
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 2.00000 0.0789337
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 5.00000 0.197028
$$645$$ −12.0000 −0.472500
$$646$$ 5.00000 0.196722
$$647$$ −6.00000 −0.235884 −0.117942 0.993020i $$-0.537630\pi$$
−0.117942 + 0.993020i $$0.537630\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 4.00000 0.157014
$$650$$ −5.00000 −0.196116
$$651$$ 9.00000 0.352738
$$652$$ −12.0000 −0.469956
$$653$$ −36.0000 −1.40879 −0.704394 0.709809i $$-0.748781\pi$$
−0.704394 + 0.709809i $$0.748781\pi$$
$$654$$ −15.0000 −0.586546
$$655$$ 11.0000 0.429806
$$656$$ −10.0000 −0.390434
$$657$$ −12.0000 −0.468165
$$658$$ −2.00000 −0.0779681
$$659$$ −42.0000 −1.63609 −0.818044 0.575156i $$-0.804941\pi$$
−0.818044 + 0.575156i $$0.804941\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ −11.0000 −0.427850 −0.213925 0.976850i $$-0.568625\pi$$
−0.213925 + 0.976850i $$0.568625\pi$$
$$662$$ 12.0000 0.466393
$$663$$ 5.00000 0.194184
$$664$$ −15.0000 −0.582113
$$665$$ 5.00000 0.193892
$$666$$ 7.00000 0.271244
$$667$$ 10.0000 0.387202
$$668$$ −16.0000 −0.619059
$$669$$ −19.0000 −0.734582
$$670$$ 5.00000 0.193167
$$671$$ 13.0000 0.501859
$$672$$ 1.00000 0.0385758
$$673$$ −28.0000 −1.07932 −0.539660 0.841883i $$-0.681447\pi$$
−0.539660 + 0.841883i $$0.681447\pi$$
$$674$$ −28.0000 −1.07852
$$675$$ 1.00000 0.0384900
$$676$$ 12.0000 0.461538
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ −13.0000 −0.498894
$$680$$ −1.00000 −0.0383482
$$681$$ 26.0000 0.996322
$$682$$ 9.00000 0.344628
$$683$$ 39.0000 1.49229 0.746147 0.665782i $$-0.231902\pi$$
0.746147 + 0.665782i $$0.231902\pi$$
$$684$$ −5.00000 −0.191180
$$685$$ 1.00000 0.0382080
$$686$$ −13.0000 −0.496342
$$687$$ −21.0000 −0.801200
$$688$$ −12.0000 −0.457496
$$689$$ −50.0000 −1.90485
$$690$$ 5.00000 0.190347
$$691$$ −27.0000 −1.02713 −0.513564 0.858051i $$-0.671675\pi$$
−0.513564 + 0.858051i $$0.671675\pi$$
$$692$$ 3.00000 0.114043
$$693$$ −1.00000 −0.0379869
$$694$$ −30.0000 −1.13878
$$695$$ −14.0000 −0.531050
$$696$$ 2.00000 0.0758098
$$697$$ −10.0000 −0.378777
$$698$$ −10.0000 −0.378506
$$699$$ 22.0000 0.832116
$$700$$ −1.00000 −0.0377964
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ −5.00000 −0.188713
$$703$$ 35.0000 1.32005
$$704$$ 1.00000 0.0376889
$$705$$ −2.00000 −0.0753244
$$706$$ 1.00000 0.0376355
$$707$$ −6.00000 −0.225653
$$708$$ 4.00000 0.150329
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 2.00000 0.0750587
$$711$$ −2.00000 −0.0750059
$$712$$ −2.00000 −0.0749532
$$713$$ 45.0000 1.68526
$$714$$ 1.00000 0.0374241
$$715$$ 5.00000 0.186989
$$716$$ −7.00000 −0.261602
$$717$$ 6.00000 0.224074
$$718$$ −10.0000 −0.373197
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 5.00000 0.186210
$$722$$ −6.00000 −0.223297
$$723$$ −17.0000 −0.632237
$$724$$ −16.0000 −0.594635
$$725$$ −2.00000 −0.0742781
$$726$$ −1.00000 −0.0371135
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 5.00000 0.185312
$$729$$ 1.00000 0.0370370
$$730$$ 12.0000 0.444140
$$731$$ −12.0000 −0.443836
$$732$$ 13.0000 0.480494
$$733$$ 15.0000 0.554038 0.277019 0.960864i $$-0.410654\pi$$
0.277019 + 0.960864i $$0.410654\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ −6.00000 −0.221313
$$736$$ 5.00000 0.184302
$$737$$ −5.00000 −0.184177
$$738$$ 10.0000 0.368105
$$739$$ −43.0000 −1.58178 −0.790890 0.611958i $$-0.790382\pi$$
−0.790890 + 0.611958i $$0.790382\pi$$
$$740$$ −7.00000 −0.257325
$$741$$ −25.0000 −0.918398
$$742$$ −10.0000 −0.367112
$$743$$ 12.0000 0.440237 0.220119 0.975473i $$-0.429356\pi$$
0.220119 + 0.975473i $$0.429356\pi$$
$$744$$ 9.00000 0.329956
$$745$$ −7.00000 −0.256460
$$746$$ −6.00000 −0.219676
$$747$$ 15.0000 0.548821
$$748$$ 1.00000 0.0365636
$$749$$ 2.00000 0.0730784
$$750$$ −1.00000 −0.0365148
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ −2.00000 −0.0729325
$$753$$ −27.0000 −0.983935
$$754$$ 10.0000 0.364179
$$755$$ −23.0000 −0.837056
$$756$$ −1.00000 −0.0363696
$$757$$ −6.00000 −0.218074 −0.109037 0.994038i $$-0.534777\pi$$
−0.109037 + 0.994038i $$0.534777\pi$$
$$758$$ −7.00000 −0.254251
$$759$$ −5.00000 −0.181489
$$760$$ 5.00000 0.181369
$$761$$ 31.0000 1.12375 0.561875 0.827222i $$-0.310080\pi$$
0.561875 + 0.827222i $$0.310080\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ −15.0000 −0.543036
$$764$$ 0 0
$$765$$ 1.00000 0.0361551
$$766$$ −26.0000 −0.939418
$$767$$ 20.0000 0.722158
$$768$$ 1.00000 0.0360844
$$769$$ 4.00000 0.144244 0.0721218 0.997396i $$-0.477023\pi$$
0.0721218 + 0.997396i $$0.477023\pi$$
$$770$$ 1.00000 0.0360375
$$771$$ −6.00000 −0.216085
$$772$$ 0 0
$$773$$ 27.0000 0.971123 0.485561 0.874203i $$-0.338615\pi$$
0.485561 + 0.874203i $$0.338615\pi$$
$$774$$ 12.0000 0.431331
$$775$$ −9.00000 −0.323290
$$776$$ −13.0000 −0.466673
$$777$$ 7.00000 0.251124
$$778$$ −12.0000 −0.430221
$$779$$ 50.0000 1.79144
$$780$$ 5.00000 0.179029
$$781$$ −2.00000 −0.0715656
$$782$$ 5.00000 0.178800
$$783$$ −2.00000 −0.0714742
$$784$$ −6.00000 −0.214286
$$785$$ 24.0000 0.856597
$$786$$ −11.0000 −0.392357
$$787$$ 7.00000 0.249523 0.124762 0.992187i $$-0.460183\pi$$
0.124762 + 0.992187i $$0.460183\pi$$
$$788$$ 15.0000 0.534353
$$789$$ 3.00000 0.106803
$$790$$ 2.00000 0.0711568
$$791$$ −6.00000 −0.213335
$$792$$ −1.00000 −0.0355335
$$793$$ 65.0000 2.30822
$$794$$ −10.0000 −0.354887
$$795$$ −10.0000 −0.354663
$$796$$ −17.0000 −0.602549
$$797$$ 33.0000 1.16892 0.584460 0.811423i $$-0.301306\pi$$
0.584460 + 0.811423i $$0.301306\pi$$
$$798$$ −5.00000 −0.176998
$$799$$ −2.00000 −0.0707549
$$800$$ −1.00000 −0.0353553
$$801$$ 2.00000 0.0706665
$$802$$ −19.0000 −0.670913
$$803$$ −12.0000 −0.423471
$$804$$ −5.00000 −0.176336
$$805$$ 5.00000 0.176227
$$806$$ 45.0000 1.58506
$$807$$ −7.00000 −0.246412
$$808$$ −6.00000 −0.211079
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 46.0000 1.61528 0.807639 0.589677i $$-0.200745\pi$$
0.807639 + 0.589677i $$0.200745\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ 20.0000 0.701431
$$814$$ 7.00000 0.245350
$$815$$ −12.0000 −0.420342
$$816$$ 1.00000 0.0350070
$$817$$ 60.0000 2.09913
$$818$$ 2.00000 0.0699284
$$819$$ −5.00000 −0.174714
$$820$$ −10.0000 −0.349215
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ −1.00000 −0.0348790
$$823$$ 14.0000 0.488009 0.244005 0.969774i $$-0.421539\pi$$
0.244005 + 0.969774i $$0.421539\pi$$
$$824$$ 5.00000 0.174183
$$825$$ 1.00000 0.0348155
$$826$$ 4.00000 0.139178
$$827$$ −2.00000 −0.0695468 −0.0347734 0.999395i $$-0.511071\pi$$
−0.0347734 + 0.999395i $$0.511071\pi$$
$$828$$ −5.00000 −0.173762
$$829$$ −13.0000 −0.451509 −0.225754 0.974184i $$-0.572485\pi$$
−0.225754 + 0.974184i $$0.572485\pi$$
$$830$$ −15.0000 −0.520658
$$831$$ 4.00000 0.138758
$$832$$ 5.00000 0.173344
$$833$$ −6.00000 −0.207888
$$834$$ 14.0000 0.484780
$$835$$ −16.0000 −0.553703
$$836$$ −5.00000 −0.172929
$$837$$ −9.00000 −0.311086
$$838$$ −20.0000 −0.690889
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 1.00000 0.0345033
$$841$$ −25.0000 −0.862069
$$842$$ 29.0000 0.999406
$$843$$ −18.0000 −0.619953
$$844$$ 14.0000 0.481900
$$845$$ 12.0000 0.412813
$$846$$ 2.00000 0.0687614
$$847$$ −1.00000 −0.0343604
$$848$$ −10.0000 −0.343401
$$849$$ 4.00000 0.137280
$$850$$ −1.00000 −0.0342997
$$851$$ 35.0000 1.19978
$$852$$ −2.00000 −0.0685189
$$853$$ −28.0000 −0.958702 −0.479351 0.877623i $$-0.659128\pi$$
−0.479351 + 0.877623i $$0.659128\pi$$
$$854$$ 13.0000 0.444851
$$855$$ −5.00000 −0.170996
$$856$$ 2.00000 0.0683586
$$857$$ 11.0000 0.375753 0.187876 0.982193i $$-0.439840\pi$$
0.187876 + 0.982193i $$0.439840\pi$$
$$858$$ −5.00000 −0.170697
$$859$$ −8.00000 −0.272956 −0.136478 0.990643i $$-0.543578\pi$$
−0.136478 + 0.990643i $$0.543578\pi$$
$$860$$ −12.0000 −0.409197
$$861$$ 10.0000 0.340799
$$862$$ 32.0000 1.08992
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 3.00000 0.102003
$$866$$ 30.0000 1.01944
$$867$$ 1.00000 0.0339618
$$868$$ 9.00000 0.305480
$$869$$ −2.00000 −0.0678454
$$870$$ 2.00000 0.0678064
$$871$$ −25.0000 −0.847093
$$872$$ −15.0000 −0.507964
$$873$$ 13.0000 0.439983
$$874$$ −25.0000 −0.845638
$$875$$ −1.00000 −0.0338062
$$876$$ −12.0000 −0.405442
$$877$$ 12.0000 0.405211 0.202606 0.979260i $$-0.435059\pi$$
0.202606 + 0.979260i $$0.435059\pi$$
$$878$$ 14.0000 0.472477
$$879$$ −24.0000 −0.809500
$$880$$ 1.00000 0.0337100
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 6.00000 0.202031
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 5.00000 0.168168
$$885$$ 4.00000 0.134459
$$886$$ −18.0000 −0.604722
$$887$$ −20.0000 −0.671534 −0.335767 0.941945i $$-0.608996\pi$$
−0.335767 + 0.941945i $$0.608996\pi$$
$$888$$ 7.00000 0.234905
$$889$$ −12.0000 −0.402467
$$890$$ −2.00000 −0.0670402
$$891$$ 1.00000 0.0335013
$$892$$ −19.0000 −0.636167
$$893$$ 10.0000 0.334637
$$894$$ 7.00000 0.234115
$$895$$ −7.00000 −0.233984
$$896$$ 1.00000 0.0334077
$$897$$ −25.0000 −0.834726
$$898$$ −25.0000 −0.834261
$$899$$ 18.0000 0.600334
$$900$$ 1.00000 0.0333333
$$901$$ −10.0000 −0.333148
$$902$$ 10.0000 0.332964
$$903$$ 12.0000 0.399335
$$904$$ −6.00000 −0.199557
$$905$$ −16.0000 −0.531858
$$906$$ 23.0000 0.764124
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ 26.0000 0.862840
$$909$$ 6.00000 0.199007
$$910$$ 5.00000 0.165748
$$911$$ −28.0000 −0.927681 −0.463841 0.885919i $$-0.653529\pi$$
−0.463841 + 0.885919i $$0.653529\pi$$
$$912$$ −5.00000 −0.165567
$$913$$ 15.0000 0.496428
$$914$$ 29.0000 0.959235
$$915$$ 13.0000 0.429767
$$916$$ −21.0000 −0.693860
$$917$$ −11.0000 −0.363252
$$918$$ −1.00000 −0.0330049
$$919$$ −11.0000 −0.362857 −0.181428 0.983404i $$-0.558072\pi$$
−0.181428 + 0.983404i $$0.558072\pi$$
$$920$$ 5.00000 0.164845
$$921$$ −20.0000 −0.659022
$$922$$ −30.0000 −0.987997
$$923$$ −10.0000 −0.329154
$$924$$ −1.00000 −0.0328976
$$925$$ −7.00000 −0.230159
$$926$$ 23.0000 0.755827
$$927$$ −5.00000 −0.164222
$$928$$ 2.00000 0.0656532
$$929$$ −19.0000 −0.623370 −0.311685 0.950186i $$-0.600893\pi$$
−0.311685 + 0.950186i $$0.600893\pi$$
$$930$$ 9.00000 0.295122
$$931$$ 30.0000 0.983210
$$932$$ 22.0000 0.720634
$$933$$ −6.00000 −0.196431
$$934$$ −32.0000 −1.04707
$$935$$ 1.00000 0.0327035
$$936$$ −5.00000 −0.163430
$$937$$ −29.0000 −0.947389 −0.473694 0.880689i $$-0.657080\pi$$
−0.473694 + 0.880689i $$0.657080\pi$$
$$938$$ −5.00000 −0.163256
$$939$$ −21.0000 −0.685309
$$940$$ −2.00000 −0.0652328
$$941$$ 28.0000 0.912774 0.456387 0.889781i $$-0.349143\pi$$
0.456387 + 0.889781i $$0.349143\pi$$
$$942$$ −24.0000 −0.781962
$$943$$ 50.0000 1.62822
$$944$$ 4.00000 0.130189
$$945$$ −1.00000 −0.0325300
$$946$$ 12.0000 0.390154
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ −2.00000 −0.0649570
$$949$$ −60.0000 −1.94768
$$950$$ 5.00000 0.162221
$$951$$ 26.0000 0.843108
$$952$$ 1.00000 0.0324102
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ −2.00000 −0.0646508
$$958$$ 15.0000 0.484628
$$959$$ −1.00000 −0.0322917
$$960$$ 1.00000 0.0322749
$$961$$ 50.0000 1.61290
$$962$$ 35.0000 1.12845
$$963$$ −2.00000 −0.0644491
$$964$$ −17.0000 −0.547533
$$965$$ 0 0
$$966$$ −5.00000 −0.160872
$$967$$ 14.0000 0.450210 0.225105 0.974335i $$-0.427728\pi$$
0.225105 + 0.974335i $$0.427728\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ −5.00000 −0.160623
$$970$$ −13.0000 −0.417405
$$971$$ −61.0000 −1.95758 −0.978792 0.204859i $$-0.934327\pi$$
−0.978792 + 0.204859i $$0.934327\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 14.0000 0.448819
$$974$$ −40.0000 −1.28168
$$975$$ 5.00000 0.160128
$$976$$ 13.0000 0.416120
$$977$$ −53.0000 −1.69562 −0.847810 0.530300i $$-0.822079\pi$$
−0.847810 + 0.530300i $$0.822079\pi$$
$$978$$ 12.0000 0.383718
$$979$$ 2.00000 0.0639203
$$980$$ −6.00000 −0.191663
$$981$$ 15.0000 0.478913
$$982$$ −6.00000 −0.191468
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 15.0000 0.477940
$$986$$ 2.00000 0.0636930
$$987$$ 2.00000 0.0636607
$$988$$ −25.0000 −0.795356
$$989$$ 60.0000 1.90789
$$990$$ −1.00000 −0.0317821
$$991$$ 21.0000 0.667087 0.333543 0.942735i $$-0.391756\pi$$
0.333543 + 0.942735i $$0.391756\pi$$
$$992$$ 9.00000 0.285750
$$993$$ −12.0000 −0.380808
$$994$$ −2.00000 −0.0634361
$$995$$ −17.0000 −0.538936
$$996$$ 15.0000 0.475293
$$997$$ 42.0000 1.33015 0.665077 0.746775i $$-0.268399\pi$$
0.665077 + 0.746775i $$0.268399\pi$$
$$998$$ 28.0000 0.886325
$$999$$ −7.00000 −0.221470
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 5610.2.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.s.1.1 1 1.1 even 1 trivial