# Properties

 Label 3610.2.a.e.1.1 Level $3610$ Weight $2$ Character 3610.1 Self dual yes Analytic conductor $28.826$ 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] = [3610,2,Mod(1,3610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3610, 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("3610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3610 = 2 \cdot 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3610.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: $$28.8259951297$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3610.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +3.00000 q^{12} +1.00000 q^{13} +5.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -6.00000 q^{18} -1.00000 q^{20} -15.0000 q^{21} +4.00000 q^{22} +7.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} +9.00000 q^{27} -5.00000 q^{28} +3.00000 q^{29} +3.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -12.0000 q^{33} +3.00000 q^{34} +5.00000 q^{35} +6.00000 q^{36} +2.00000 q^{37} +3.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} +15.0000 q^{42} +6.00000 q^{43} -4.00000 q^{44} -6.00000 q^{45} -7.00000 q^{46} +3.00000 q^{48} +18.0000 q^{49} -1.00000 q^{50} -9.00000 q^{51} +1.00000 q^{52} +13.0000 q^{53} -9.00000 q^{54} +4.00000 q^{55} +5.00000 q^{56} -3.00000 q^{58} +9.00000 q^{59} -3.00000 q^{60} -12.0000 q^{61} -2.00000 q^{62} -30.0000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +12.0000 q^{66} +3.00000 q^{67} -3.00000 q^{68} +21.0000 q^{69} -5.00000 q^{70} -6.00000 q^{72} +11.0000 q^{73} -2.00000 q^{74} +3.00000 q^{75} +20.0000 q^{77} -3.00000 q^{78} +2.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} -10.0000 q^{83} -15.0000 q^{84} +3.00000 q^{85} -6.00000 q^{86} +9.00000 q^{87} +4.00000 q^{88} -2.00000 q^{89} +6.00000 q^{90} -5.00000 q^{91} +7.00000 q^{92} +6.00000 q^{93} -3.00000 q^{96} +2.00000 q^{97} -18.0000 q^{98} -24.0000 q^{99} +O(q^{100})$$

## 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
$$3$$ 3.00000 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −3.00000 −1.22474
$$7$$ −5.00000 −1.88982 −0.944911 0.327327i $$-0.893852\pi$$
−0.944911 + 0.327327i $$0.893852\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 6.00000 2.00000
$$10$$ 1.00000 0.316228
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 3.00000 0.866025
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 5.00000 1.33631
$$15$$ −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 $$0.618522\pi$$
$$18$$ −6.00000 −1.41421
$$19$$ 0 0
$$20$$ −1.00000 −0.223607
$$21$$ −15.0000 −3.27327
$$22$$ 4.00000 0.852803
$$23$$ 7.00000 1.45960 0.729800 0.683660i $$-0.239613\pi$$
0.729800 + 0.683660i $$0.239613\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 1.00000 0.200000
$$26$$ −1.00000 −0.196116
$$27$$ 9.00000 1.73205
$$28$$ −5.00000 −0.944911
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 3.00000 0.547723
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −12.0000 −2.08893
$$34$$ 3.00000 0.514496
$$35$$ 5.00000 0.845154
$$36$$ 6.00000 1.00000
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 3.00000 0.480384
$$40$$ 1.00000 0.158114
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 15.0000 2.31455
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ −6.00000 −0.894427
$$46$$ −7.00000 −1.03209
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 3.00000 0.433013
$$49$$ 18.0000 2.57143
$$50$$ −1.00000 −0.141421
$$51$$ −9.00000 −1.26025
$$52$$ 1.00000 0.138675
$$53$$ 13.0000 1.78569 0.892844 0.450367i $$-0.148707\pi$$
0.892844 + 0.450367i $$0.148707\pi$$
$$54$$ −9.00000 −1.22474
$$55$$ 4.00000 0.539360
$$56$$ 5.00000 0.668153
$$57$$ 0 0
$$58$$ −3.00000 −0.393919
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ −3.00000 −0.387298
$$61$$ −12.0000 −1.53644 −0.768221 0.640184i $$-0.778858\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ −30.0000 −3.77964
$$64$$ 1.00000 0.125000
$$65$$ −1.00000 −0.124035
$$66$$ 12.0000 1.47710
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 21.0000 2.52810
$$70$$ −5.00000 −0.597614
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −6.00000 −0.707107
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 3.00000 0.346410
$$76$$ 0 0
$$77$$ 20.0000 2.27921
$$78$$ −3.00000 −0.339683
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 9.00000 1.00000
$$82$$ −6.00000 −0.662589
$$83$$ −10.0000 −1.09764 −0.548821 0.835940i $$-0.684923\pi$$
−0.548821 + 0.835940i $$0.684923\pi$$
$$84$$ −15.0000 −1.63663
$$85$$ 3.00000 0.325396
$$86$$ −6.00000 −0.646997
$$87$$ 9.00000 0.964901
$$88$$ 4.00000 0.426401
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 6.00000 0.632456
$$91$$ −5.00000 −0.524142
$$92$$ 7.00000 0.729800
$$93$$ 6.00000 0.622171
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −3.00000 −0.306186
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −18.0000 −1.81827
$$99$$ −24.0000 −2.41209
$$100$$ 1.00000 0.100000
$$101$$ −8.00000 −0.796030 −0.398015 0.917379i $$-0.630301\pi$$
−0.398015 + 0.917379i $$0.630301\pi$$
$$102$$ 9.00000 0.891133
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 15.0000 1.46385
$$106$$ −13.0000 −1.26267
$$107$$ 13.0000 1.25676 0.628379 0.777908i $$-0.283719\pi$$
0.628379 + 0.777908i $$0.283719\pi$$
$$108$$ 9.00000 0.866025
$$109$$ −19.0000 −1.81987 −0.909935 0.414751i $$-0.863869\pi$$
−0.909935 + 0.414751i $$0.863869\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 6.00000 0.569495
$$112$$ −5.00000 −0.472456
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ −7.00000 −0.652753
$$116$$ 3.00000 0.278543
$$117$$ 6.00000 0.554700
$$118$$ −9.00000 −0.828517
$$119$$ 15.0000 1.37505
$$120$$ 3.00000 0.273861
$$121$$ 5.00000 0.454545
$$122$$ 12.0000 1.08643
$$123$$ 18.0000 1.62301
$$124$$ 2.00000 0.179605
$$125$$ −1.00000 −0.0894427
$$126$$ 30.0000 2.67261
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 18.0000 1.58481
$$130$$ 1.00000 0.0877058
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ −12.0000 −1.04447
$$133$$ 0 0
$$134$$ −3.00000 −0.259161
$$135$$ −9.00000 −0.774597
$$136$$ 3.00000 0.257248
$$137$$ 9.00000 0.768922 0.384461 0.923141i $$-0.374387\pi$$
0.384461 + 0.923141i $$0.374387\pi$$
$$138$$ −21.0000 −1.78764
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 5.00000 0.422577
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −4.00000 −0.334497
$$144$$ 6.00000 0.500000
$$145$$ −3.00000 −0.249136
$$146$$ −11.0000 −0.910366
$$147$$ 54.0000 4.45384
$$148$$ 2.00000 0.164399
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ −3.00000 −0.244949
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 0 0
$$153$$ −18.0000 −1.45521
$$154$$ −20.0000 −1.61165
$$155$$ −2.00000 −0.160644
$$156$$ 3.00000 0.240192
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ −2.00000 −0.159111
$$159$$ 39.0000 3.09290
$$160$$ 1.00000 0.0790569
$$161$$ −35.0000 −2.75839
$$162$$ −9.00000 −0.707107
$$163$$ 22.0000 1.72317 0.861586 0.507611i $$-0.169471\pi$$
0.861586 + 0.507611i $$0.169471\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 12.0000 0.934199
$$166$$ 10.0000 0.776151
$$167$$ 2.00000 0.154765 0.0773823 0.997001i $$-0.475344\pi$$
0.0773823 + 0.997001i $$0.475344\pi$$
$$168$$ 15.0000 1.15728
$$169$$ −12.0000 −0.923077
$$170$$ −3.00000 −0.230089
$$171$$ 0 0
$$172$$ 6.00000 0.457496
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ −9.00000 −0.682288
$$175$$ −5.00000 −0.377964
$$176$$ −4.00000 −0.301511
$$177$$ 27.0000 2.02944
$$178$$ 2.00000 0.149906
$$179$$ 8.00000 0.597948 0.298974 0.954261i $$-0.403356\pi$$
0.298974 + 0.954261i $$0.403356\pi$$
$$180$$ −6.00000 −0.447214
$$181$$ −26.0000 −1.93256 −0.966282 0.257485i $$-0.917106\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ 5.00000 0.370625
$$183$$ −36.0000 −2.66120
$$184$$ −7.00000 −0.516047
$$185$$ −2.00000 −0.147043
$$186$$ −6.00000 −0.439941
$$187$$ 12.0000 0.877527
$$188$$ 0 0
$$189$$ −45.0000 −3.27327
$$190$$ 0 0
$$191$$ 9.00000 0.651217 0.325609 0.945505i $$-0.394431\pi$$
0.325609 + 0.945505i $$0.394431\pi$$
$$192$$ 3.00000 0.216506
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ −3.00000 −0.214834
$$196$$ 18.0000 1.28571
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 24.0000 1.70561
$$199$$ −15.0000 −1.06332 −0.531661 0.846957i $$-0.678432\pi$$
−0.531661 + 0.846957i $$0.678432\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 9.00000 0.634811
$$202$$ 8.00000 0.562878
$$203$$ −15.0000 −1.05279
$$204$$ −9.00000 −0.630126
$$205$$ −6.00000 −0.419058
$$206$$ 4.00000 0.278693
$$207$$ 42.0000 2.91920
$$208$$ 1.00000 0.0693375
$$209$$ 0 0
$$210$$ −15.0000 −1.03510
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ 13.0000 0.892844
$$213$$ 0 0
$$214$$ −13.0000 −0.888662
$$215$$ −6.00000 −0.409197
$$216$$ −9.00000 −0.612372
$$217$$ −10.0000 −0.678844
$$218$$ 19.0000 1.28684
$$219$$ 33.0000 2.22993
$$220$$ 4.00000 0.269680
$$221$$ −3.00000 −0.201802
$$222$$ −6.00000 −0.402694
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 6.00000 0.400000
$$226$$ 0 0
$$227$$ −5.00000 −0.331862 −0.165931 0.986137i $$-0.553063\pi$$
−0.165931 + 0.986137i $$0.553063\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 7.00000 0.461566
$$231$$ 60.0000 3.94771
$$232$$ −3.00000 −0.196960
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 9.00000 0.585850
$$237$$ 6.00000 0.389742
$$238$$ −15.0000 −0.972306
$$239$$ −11.0000 −0.711531 −0.355765 0.934575i $$-0.615780\pi$$
−0.355765 + 0.934575i $$0.615780\pi$$
$$240$$ −3.00000 −0.193649
$$241$$ 12.0000 0.772988 0.386494 0.922292i $$-0.373686\pi$$
0.386494 + 0.922292i $$0.373686\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ −12.0000 −0.768221
$$245$$ −18.0000 −1.14998
$$246$$ −18.0000 −1.14764
$$247$$ 0 0
$$248$$ −2.00000 −0.127000
$$249$$ −30.0000 −1.90117
$$250$$ 1.00000 0.0632456
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ −30.0000 −1.88982
$$253$$ −28.0000 −1.76034
$$254$$ −6.00000 −0.376473
$$255$$ 9.00000 0.563602
$$256$$ 1.00000 0.0625000
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ −18.0000 −1.12063
$$259$$ −10.0000 −0.621370
$$260$$ −1.00000 −0.0620174
$$261$$ 18.0000 1.11417
$$262$$ −16.0000 −0.988483
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 12.0000 0.738549
$$265$$ −13.0000 −0.798584
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 3.00000 0.183254
$$269$$ −2.00000 −0.121942 −0.0609711 0.998140i $$-0.519420\pi$$
−0.0609711 + 0.998140i $$0.519420\pi$$
$$270$$ 9.00000 0.547723
$$271$$ −27.0000 −1.64013 −0.820067 0.572268i $$-0.806064\pi$$
−0.820067 + 0.572268i $$0.806064\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ −15.0000 −0.907841
$$274$$ −9.00000 −0.543710
$$275$$ −4.00000 −0.241209
$$276$$ 21.0000 1.26405
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ 12.0000 0.718421
$$280$$ −5.00000 −0.298807
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ −30.0000 −1.77084
$$288$$ −6.00000 −0.353553
$$289$$ −8.00000 −0.470588
$$290$$ 3.00000 0.176166
$$291$$ 6.00000 0.351726
$$292$$ 11.0000 0.643726
$$293$$ 27.0000 1.57736 0.788678 0.614806i $$-0.210766\pi$$
0.788678 + 0.614806i $$0.210766\pi$$
$$294$$ −54.0000 −3.14934
$$295$$ −9.00000 −0.524000
$$296$$ −2.00000 −0.116248
$$297$$ −36.0000 −2.08893
$$298$$ 4.00000 0.231714
$$299$$ 7.00000 0.404820
$$300$$ 3.00000 0.173205
$$301$$ −30.0000 −1.72917
$$302$$ −10.0000 −0.575435
$$303$$ −24.0000 −1.37876
$$304$$ 0 0
$$305$$ 12.0000 0.687118
$$306$$ 18.0000 1.02899
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 20.0000 1.13961
$$309$$ −12.0000 −0.682656
$$310$$ 2.00000 0.113592
$$311$$ 25.0000 1.41762 0.708810 0.705399i $$-0.249232\pi$$
0.708810 + 0.705399i $$0.249232\pi$$
$$312$$ −3.00000 −0.169842
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 30.0000 1.69031
$$316$$ 2.00000 0.112509
$$317$$ −9.00000 −0.505490 −0.252745 0.967533i $$-0.581333\pi$$
−0.252745 + 0.967533i $$0.581333\pi$$
$$318$$ −39.0000 −2.18701
$$319$$ −12.0000 −0.671871
$$320$$ −1.00000 −0.0559017
$$321$$ 39.0000 2.17677
$$322$$ 35.0000 1.95047
$$323$$ 0 0
$$324$$ 9.00000 0.500000
$$325$$ 1.00000 0.0554700
$$326$$ −22.0000 −1.21847
$$327$$ −57.0000 −3.15211
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ −12.0000 −0.660578
$$331$$ 7.00000 0.384755 0.192377 0.981321i $$-0.438380\pi$$
0.192377 + 0.981321i $$0.438380\pi$$
$$332$$ −10.0000 −0.548821
$$333$$ 12.0000 0.657596
$$334$$ −2.00000 −0.109435
$$335$$ −3.00000 −0.163908
$$336$$ −15.0000 −0.818317
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 0 0
$$340$$ 3.00000 0.162698
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ −55.0000 −2.96972
$$344$$ −6.00000 −0.323498
$$345$$ −21.0000 −1.13060
$$346$$ −14.0000 −0.752645
$$347$$ −6.00000 −0.322097 −0.161048 0.986947i $$-0.551488\pi$$
−0.161048 + 0.986947i $$0.551488\pi$$
$$348$$ 9.00000 0.482451
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 5.00000 0.267261
$$351$$ 9.00000 0.480384
$$352$$ 4.00000 0.213201
$$353$$ 7.00000 0.372572 0.186286 0.982496i $$-0.440355\pi$$
0.186286 + 0.982496i $$0.440355\pi$$
$$354$$ −27.0000 −1.43503
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 45.0000 2.38165
$$358$$ −8.00000 −0.422813
$$359$$ −5.00000 −0.263890 −0.131945 0.991257i $$-0.542122\pi$$
−0.131945 + 0.991257i $$0.542122\pi$$
$$360$$ 6.00000 0.316228
$$361$$ 0 0
$$362$$ 26.0000 1.36653
$$363$$ 15.0000 0.787296
$$364$$ −5.00000 −0.262071
$$365$$ −11.0000 −0.575766
$$366$$ 36.0000 1.88175
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 7.00000 0.364900
$$369$$ 36.0000 1.87409
$$370$$ 2.00000 0.103975
$$371$$ −65.0000 −3.37463
$$372$$ 6.00000 0.311086
$$373$$ 23.0000 1.19089 0.595447 0.803394i $$-0.296975\pi$$
0.595447 + 0.803394i $$0.296975\pi$$
$$374$$ −12.0000 −0.620505
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ 3.00000 0.154508
$$378$$ 45.0000 2.31455
$$379$$ 33.0000 1.69510 0.847548 0.530719i $$-0.178078\pi$$
0.847548 + 0.530719i $$0.178078\pi$$
$$380$$ 0 0
$$381$$ 18.0000 0.922168
$$382$$ −9.00000 −0.460480
$$383$$ 4.00000 0.204390 0.102195 0.994764i $$-0.467413\pi$$
0.102195 + 0.994764i $$0.467413\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ −20.0000 −1.01929
$$386$$ 10.0000 0.508987
$$387$$ 36.0000 1.82998
$$388$$ 2.00000 0.101535
$$389$$ −4.00000 −0.202808 −0.101404 0.994845i $$-0.532333\pi$$
−0.101404 + 0.994845i $$0.532333\pi$$
$$390$$ 3.00000 0.151911
$$391$$ −21.0000 −1.06202
$$392$$ −18.0000 −0.909137
$$393$$ 48.0000 2.42128
$$394$$ 22.0000 1.10834
$$395$$ −2.00000 −0.100631
$$396$$ −24.0000 −1.20605
$$397$$ −16.0000 −0.803017 −0.401508 0.915855i $$-0.631514\pi$$
−0.401508 + 0.915855i $$0.631514\pi$$
$$398$$ 15.0000 0.751882
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ −9.00000 −0.448879
$$403$$ 2.00000 0.0996271
$$404$$ −8.00000 −0.398015
$$405$$ −9.00000 −0.447214
$$406$$ 15.0000 0.744438
$$407$$ −8.00000 −0.396545
$$408$$ 9.00000 0.445566
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 6.00000 0.296319
$$411$$ 27.0000 1.33181
$$412$$ −4.00000 −0.197066
$$413$$ −45.0000 −2.21431
$$414$$ −42.0000 −2.06419
$$415$$ 10.0000 0.490881
$$416$$ −1.00000 −0.0490290
$$417$$ 48.0000 2.35057
$$418$$ 0 0
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 15.0000 0.731925
$$421$$ −1.00000 −0.0487370 −0.0243685 0.999703i $$-0.507758\pi$$
−0.0243685 + 0.999703i $$0.507758\pi$$
$$422$$ −5.00000 −0.243396
$$423$$ 0 0
$$424$$ −13.0000 −0.631336
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ 60.0000 2.90360
$$428$$ 13.0000 0.628379
$$429$$ −12.0000 −0.579365
$$430$$ 6.00000 0.289346
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 9.00000 0.433013
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ 10.0000 0.480015
$$435$$ −9.00000 −0.431517
$$436$$ −19.0000 −0.909935
$$437$$ 0 0
$$438$$ −33.0000 −1.57680
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ 108.000 5.14286
$$442$$ 3.00000 0.142695
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 2.00000 0.0948091
$$446$$ −2.00000 −0.0947027
$$447$$ −12.0000 −0.567581
$$448$$ −5.00000 −0.236228
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ −6.00000 −0.282843
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ 30.0000 1.40952
$$454$$ 5.00000 0.234662
$$455$$ 5.00000 0.234404
$$456$$ 0 0
$$457$$ −29.0000 −1.35656 −0.678281 0.734802i $$-0.737275\pi$$
−0.678281 + 0.734802i $$0.737275\pi$$
$$458$$ 6.00000 0.280362
$$459$$ −27.0000 −1.26025
$$460$$ −7.00000 −0.326377
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ −60.0000 −2.79145
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 3.00000 0.139272
$$465$$ −6.00000 −0.278243
$$466$$ −10.0000 −0.463241
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 6.00000 0.277350
$$469$$ −15.0000 −0.692636
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ −9.00000 −0.414259
$$473$$ −24.0000 −1.10352
$$474$$ −6.00000 −0.275589
$$475$$ 0 0
$$476$$ 15.0000 0.687524
$$477$$ 78.0000 3.57137
$$478$$ 11.0000 0.503128
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 3.00000 0.136931
$$481$$ 2.00000 0.0911922
$$482$$ −12.0000 −0.546585
$$483$$ −105.000 −4.77767
$$484$$ 5.00000 0.227273
$$485$$ −2.00000 −0.0908153
$$486$$ 0 0
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ 12.0000 0.543214
$$489$$ 66.0000 2.98462
$$490$$ 18.0000 0.813157
$$491$$ 18.0000 0.812329 0.406164 0.913800i $$-0.366866\pi$$
0.406164 + 0.913800i $$0.366866\pi$$
$$492$$ 18.0000 0.811503
$$493$$ −9.00000 −0.405340
$$494$$ 0 0
$$495$$ 24.0000 1.07872
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ 30.0000 1.34433
$$499$$ 42.0000 1.88018 0.940089 0.340929i $$-0.110742\pi$$
0.940089 + 0.340929i $$0.110742\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 6.00000 0.268060
$$502$$ 12.0000 0.535586
$$503$$ −21.0000 −0.936344 −0.468172 0.883637i $$-0.655087\pi$$
−0.468172 + 0.883637i $$0.655087\pi$$
$$504$$ 30.0000 1.33631
$$505$$ 8.00000 0.355995
$$506$$ 28.0000 1.24475
$$507$$ −36.0000 −1.59882
$$508$$ 6.00000 0.266207
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ −9.00000 −0.398527
$$511$$ −55.0000 −2.43306
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ 4.00000 0.176261
$$516$$ 18.0000 0.792406
$$517$$ 0 0
$$518$$ 10.0000 0.439375
$$519$$ 42.0000 1.84360
$$520$$ 1.00000 0.0438529
$$521$$ −24.0000 −1.05146 −0.525730 0.850652i $$-0.676208\pi$$
−0.525730 + 0.850652i $$0.676208\pi$$
$$522$$ −18.0000 −0.787839
$$523$$ 9.00000 0.393543 0.196771 0.980449i $$-0.436954\pi$$
0.196771 + 0.980449i $$0.436954\pi$$
$$524$$ 16.0000 0.698963
$$525$$ −15.0000 −0.654654
$$526$$ −8.00000 −0.348817
$$527$$ −6.00000 −0.261364
$$528$$ −12.0000 −0.522233
$$529$$ 26.0000 1.13043
$$530$$ 13.0000 0.564684
$$531$$ 54.0000 2.34340
$$532$$ 0 0
$$533$$ 6.00000 0.259889
$$534$$ 6.00000 0.259645
$$535$$ −13.0000 −0.562039
$$536$$ −3.00000 −0.129580
$$537$$ 24.0000 1.03568
$$538$$ 2.00000 0.0862261
$$539$$ −72.0000 −3.10126
$$540$$ −9.00000 −0.387298
$$541$$ 16.0000 0.687894 0.343947 0.938989i $$-0.388236\pi$$
0.343947 + 0.938989i $$0.388236\pi$$
$$542$$ 27.0000 1.15975
$$543$$ −78.0000 −3.34730
$$544$$ 3.00000 0.128624
$$545$$ 19.0000 0.813871
$$546$$ 15.0000 0.641941
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 9.00000 0.384461
$$549$$ −72.0000 −3.07289
$$550$$ 4.00000 0.170561
$$551$$ 0 0
$$552$$ −21.0000 −0.893819
$$553$$ −10.0000 −0.425243
$$554$$ 8.00000 0.339887
$$555$$ −6.00000 −0.254686
$$556$$ 16.0000 0.678551
$$557$$ 12.0000 0.508456 0.254228 0.967144i $$-0.418179\pi$$
0.254228 + 0.967144i $$0.418179\pi$$
$$558$$ −12.0000 −0.508001
$$559$$ 6.00000 0.253773
$$560$$ 5.00000 0.211289
$$561$$ 36.0000 1.51992
$$562$$ −18.0000 −0.759284
$$563$$ −20.0000 −0.842900 −0.421450 0.906852i $$-0.638479\pi$$
−0.421450 + 0.906852i $$0.638479\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 2.00000 0.0840663
$$567$$ −45.0000 −1.88982
$$568$$ 0 0
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ 6.00000 0.251092 0.125546 0.992088i $$-0.459932\pi$$
0.125546 + 0.992088i $$0.459932\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 27.0000 1.12794
$$574$$ 30.0000 1.25218
$$575$$ 7.00000 0.291920
$$576$$ 6.00000 0.250000
$$577$$ −7.00000 −0.291414 −0.145707 0.989328i $$-0.546546\pi$$
−0.145707 + 0.989328i $$0.546546\pi$$
$$578$$ 8.00000 0.332756
$$579$$ −30.0000 −1.24676
$$580$$ −3.00000 −0.124568
$$581$$ 50.0000 2.07435
$$582$$ −6.00000 −0.248708
$$583$$ −52.0000 −2.15362
$$584$$ −11.0000 −0.455183
$$585$$ −6.00000 −0.248069
$$586$$ −27.0000 −1.11536
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 54.0000 2.22692
$$589$$ 0 0
$$590$$ 9.00000 0.370524
$$591$$ −66.0000 −2.71488
$$592$$ 2.00000 0.0821995
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 36.0000 1.47710
$$595$$ −15.0000 −0.614940
$$596$$ −4.00000 −0.163846
$$597$$ −45.0000 −1.84173
$$598$$ −7.00000 −0.286251
$$599$$ 26.0000 1.06233 0.531166 0.847268i $$-0.321754\pi$$
0.531166 + 0.847268i $$0.321754\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ 30.0000 1.22271
$$603$$ 18.0000 0.733017
$$604$$ 10.0000 0.406894
$$605$$ −5.00000 −0.203279
$$606$$ 24.0000 0.974933
$$607$$ 26.0000 1.05531 0.527654 0.849460i $$-0.323072\pi$$
0.527654 + 0.849460i $$0.323072\pi$$
$$608$$ 0 0
$$609$$ −45.0000 −1.82349
$$610$$ −12.0000 −0.485866
$$611$$ 0 0
$$612$$ −18.0000 −0.727607
$$613$$ −20.0000 −0.807792 −0.403896 0.914805i $$-0.632344\pi$$
−0.403896 + 0.914805i $$0.632344\pi$$
$$614$$ 4.00000 0.161427
$$615$$ −18.0000 −0.725830
$$616$$ −20.0000 −0.805823
$$617$$ 14.0000 0.563619 0.281809 0.959470i $$-0.409065\pi$$
0.281809 + 0.959470i $$0.409065\pi$$
$$618$$ 12.0000 0.482711
$$619$$ −24.0000 −0.964641 −0.482321 0.875995i $$-0.660206\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ −2.00000 −0.0803219
$$621$$ 63.0000 2.52810
$$622$$ −25.0000 −1.00241
$$623$$ 10.0000 0.400642
$$624$$ 3.00000 0.120096
$$625$$ 1.00000 0.0400000
$$626$$ 1.00000 0.0399680
$$627$$ 0 0
$$628$$ 6.00000 0.239426
$$629$$ −6.00000 −0.239236
$$630$$ −30.0000 −1.19523
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ −2.00000 −0.0795557
$$633$$ 15.0000 0.596196
$$634$$ 9.00000 0.357436
$$635$$ −6.00000 −0.238103
$$636$$ 39.0000 1.54645
$$637$$ 18.0000 0.713186
$$638$$ 12.0000 0.475085
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ −8.00000 −0.315981 −0.157991 0.987441i $$-0.550502\pi$$
−0.157991 + 0.987441i $$0.550502\pi$$
$$642$$ −39.0000 −1.53921
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ −35.0000 −1.37919
$$645$$ −18.0000 −0.708749
$$646$$ 0 0
$$647$$ −21.0000 −0.825595 −0.412798 0.910823i $$-0.635448\pi$$
−0.412798 + 0.910823i $$0.635448\pi$$
$$648$$ −9.00000 −0.353553
$$649$$ −36.0000 −1.41312
$$650$$ −1.00000 −0.0392232
$$651$$ −30.0000 −1.17579
$$652$$ 22.0000 0.861586
$$653$$ −16.0000 −0.626128 −0.313064 0.949732i $$-0.601356\pi$$
−0.313064 + 0.949732i $$0.601356\pi$$
$$654$$ 57.0000 2.22888
$$655$$ −16.0000 −0.625172
$$656$$ 6.00000 0.234261
$$657$$ 66.0000 2.57491
$$658$$ 0 0
$$659$$ −33.0000 −1.28550 −0.642749 0.766077i $$-0.722206\pi$$
−0.642749 + 0.766077i $$0.722206\pi$$
$$660$$ 12.0000 0.467099
$$661$$ −15.0000 −0.583432 −0.291716 0.956505i $$-0.594226\pi$$
−0.291716 + 0.956505i $$0.594226\pi$$
$$662$$ −7.00000 −0.272063
$$663$$ −9.00000 −0.349531
$$664$$ 10.0000 0.388075
$$665$$ 0 0
$$666$$ −12.0000 −0.464991
$$667$$ 21.0000 0.813123
$$668$$ 2.00000 0.0773823
$$669$$ 6.00000 0.231973
$$670$$ 3.00000 0.115900
$$671$$ 48.0000 1.85302
$$672$$ 15.0000 0.578638
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 9.00000 0.346410
$$676$$ −12.0000 −0.461538
$$677$$ 39.0000 1.49889 0.749446 0.662066i $$-0.230320\pi$$
0.749446 + 0.662066i $$0.230320\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ −3.00000 −0.115045
$$681$$ −15.0000 −0.574801
$$682$$ 8.00000 0.306336
$$683$$ 44.0000 1.68361 0.841807 0.539779i $$-0.181492\pi$$
0.841807 + 0.539779i $$0.181492\pi$$
$$684$$ 0 0
$$685$$ −9.00000 −0.343872
$$686$$ 55.0000 2.09991
$$687$$ −18.0000 −0.686743
$$688$$ 6.00000 0.228748
$$689$$ 13.0000 0.495261
$$690$$ 21.0000 0.799456
$$691$$ −42.0000 −1.59776 −0.798878 0.601494i $$-0.794573\pi$$
−0.798878 + 0.601494i $$0.794573\pi$$
$$692$$ 14.0000 0.532200
$$693$$ 120.000 4.55842
$$694$$ 6.00000 0.227757
$$695$$ −16.0000 −0.606915
$$696$$ −9.00000 −0.341144
$$697$$ −18.0000 −0.681799
$$698$$ −14.0000 −0.529908
$$699$$ 30.0000 1.13470
$$700$$ −5.00000 −0.188982
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ −9.00000 −0.339683
$$703$$ 0 0
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −7.00000 −0.263448
$$707$$ 40.0000 1.50435
$$708$$ 27.0000 1.01472
$$709$$ 2.00000 0.0751116 0.0375558 0.999295i $$-0.488043\pi$$
0.0375558 + 0.999295i $$0.488043\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 2.00000 0.0749532
$$713$$ 14.0000 0.524304
$$714$$ −45.0000 −1.68408
$$715$$ 4.00000 0.149592
$$716$$ 8.00000 0.298974
$$717$$ −33.0000 −1.23241
$$718$$ 5.00000 0.186598
$$719$$ −27.0000 −1.00693 −0.503465 0.864016i $$-0.667942\pi$$
−0.503465 + 0.864016i $$0.667942\pi$$
$$720$$ −6.00000 −0.223607
$$721$$ 20.0000 0.744839
$$722$$ 0 0
$$723$$ 36.0000 1.33885
$$724$$ −26.0000 −0.966282
$$725$$ 3.00000 0.111417
$$726$$ −15.0000 −0.556702
$$727$$ 23.0000 0.853023 0.426511 0.904482i $$-0.359742\pi$$
0.426511 + 0.904482i $$0.359742\pi$$
$$728$$ 5.00000 0.185312
$$729$$ −27.0000 −1.00000
$$730$$ 11.0000 0.407128
$$731$$ −18.0000 −0.665754
$$732$$ −36.0000 −1.33060
$$733$$ 36.0000 1.32969 0.664845 0.746981i $$-0.268498\pi$$
0.664845 + 0.746981i $$0.268498\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ −54.0000 −1.99182
$$736$$ −7.00000 −0.258023
$$737$$ −12.0000 −0.442026
$$738$$ −36.0000 −1.32518
$$739$$ −10.0000 −0.367856 −0.183928 0.982940i $$-0.558881\pi$$
−0.183928 + 0.982940i $$0.558881\pi$$
$$740$$ −2.00000 −0.0735215
$$741$$ 0 0
$$742$$ 65.0000 2.38623
$$743$$ 18.0000 0.660356 0.330178 0.943919i $$-0.392891\pi$$
0.330178 + 0.943919i $$0.392891\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 4.00000 0.146549
$$746$$ −23.0000 −0.842090
$$747$$ −60.0000 −2.19529
$$748$$ 12.0000 0.438763
$$749$$ −65.0000 −2.37505
$$750$$ 3.00000 0.109545
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ 0 0
$$753$$ −36.0000 −1.31191
$$754$$ −3.00000 −0.109254
$$755$$ −10.0000 −0.363937
$$756$$ −45.0000 −1.63663
$$757$$ −6.00000 −0.218074 −0.109037 0.994038i $$-0.534777\pi$$
−0.109037 + 0.994038i $$0.534777\pi$$
$$758$$ −33.0000 −1.19861
$$759$$ −84.0000 −3.04901
$$760$$ 0 0
$$761$$ 11.0000 0.398750 0.199375 0.979923i $$-0.436109\pi$$
0.199375 + 0.979923i $$0.436109\pi$$
$$762$$ −18.0000 −0.652071
$$763$$ 95.0000 3.43923
$$764$$ 9.00000 0.325609
$$765$$ 18.0000 0.650791
$$766$$ −4.00000 −0.144526
$$767$$ 9.00000 0.324971
$$768$$ 3.00000 0.108253
$$769$$ −47.0000 −1.69486 −0.847432 0.530904i $$-0.821852\pi$$
−0.847432 + 0.530904i $$0.821852\pi$$
$$770$$ 20.0000 0.720750
$$771$$ −66.0000 −2.37693
$$772$$ −10.0000 −0.359908
$$773$$ 51.0000 1.83434 0.917171 0.398493i $$-0.130467\pi$$
0.917171 + 0.398493i $$0.130467\pi$$
$$774$$ −36.0000 −1.29399
$$775$$ 2.00000 0.0718421
$$776$$ −2.00000 −0.0717958
$$777$$ −30.0000 −1.07624
$$778$$ 4.00000 0.143407
$$779$$ 0 0
$$780$$ −3.00000 −0.107417
$$781$$ 0 0
$$782$$ 21.0000 0.750958
$$783$$ 27.0000 0.964901
$$784$$ 18.0000 0.642857
$$785$$ −6.00000 −0.214149
$$786$$ −48.0000 −1.71210
$$787$$ 39.0000 1.39020 0.695100 0.718913i $$-0.255360\pi$$
0.695100 + 0.718913i $$0.255360\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 24.0000 0.854423
$$790$$ 2.00000 0.0711568
$$791$$ 0 0
$$792$$ 24.0000 0.852803
$$793$$ −12.0000 −0.426132
$$794$$ 16.0000 0.567819
$$795$$ −39.0000 −1.38319
$$796$$ −15.0000 −0.531661
$$797$$ −31.0000 −1.09808 −0.549038 0.835797i $$-0.685006\pi$$
−0.549038 + 0.835797i $$0.685006\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ −12.0000 −0.423999
$$802$$ −6.00000 −0.211867
$$803$$ −44.0000 −1.55273
$$804$$ 9.00000 0.317406
$$805$$ 35.0000 1.23359
$$806$$ −2.00000 −0.0704470
$$807$$ −6.00000 −0.211210
$$808$$ 8.00000 0.281439
$$809$$ 25.0000 0.878953 0.439477 0.898254i $$-0.355164\pi$$
0.439477 + 0.898254i $$0.355164\pi$$
$$810$$ 9.00000 0.316228
$$811$$ −37.0000 −1.29925 −0.649623 0.760257i $$-0.725073\pi$$
−0.649623 + 0.760257i $$0.725073\pi$$
$$812$$ −15.0000 −0.526397
$$813$$ −81.0000 −2.84079
$$814$$ 8.00000 0.280400
$$815$$ −22.0000 −0.770626
$$816$$ −9.00000 −0.315063
$$817$$ 0 0
$$818$$ 22.0000 0.769212
$$819$$ −30.0000 −1.04828
$$820$$ −6.00000 −0.209529
$$821$$ −52.0000 −1.81481 −0.907406 0.420255i $$-0.861941\pi$$
−0.907406 + 0.420255i $$0.861941\pi$$
$$822$$ −27.0000 −0.941733
$$823$$ −43.0000 −1.49889 −0.749443 0.662069i $$-0.769679\pi$$
−0.749443 + 0.662069i $$0.769679\pi$$
$$824$$ 4.00000 0.139347
$$825$$ −12.0000 −0.417786
$$826$$ 45.0000 1.56575
$$827$$ 3.00000 0.104320 0.0521601 0.998639i $$-0.483389\pi$$
0.0521601 + 0.998639i $$0.483389\pi$$
$$828$$ 42.0000 1.45960
$$829$$ −35.0000 −1.21560 −0.607800 0.794090i $$-0.707948\pi$$
−0.607800 + 0.794090i $$0.707948\pi$$
$$830$$ −10.0000 −0.347105
$$831$$ −24.0000 −0.832551
$$832$$ 1.00000 0.0346688
$$833$$ −54.0000 −1.87099
$$834$$ −48.0000 −1.66210
$$835$$ −2.00000 −0.0692129
$$836$$ 0 0
$$837$$ 18.0000 0.622171
$$838$$ −14.0000 −0.483622
$$839$$ 4.00000 0.138095 0.0690477 0.997613i $$-0.478004\pi$$
0.0690477 + 0.997613i $$0.478004\pi$$
$$840$$ −15.0000 −0.517549
$$841$$ −20.0000 −0.689655
$$842$$ 1.00000 0.0344623
$$843$$ 54.0000 1.85986
$$844$$ 5.00000 0.172107
$$845$$ 12.0000 0.412813
$$846$$ 0 0
$$847$$ −25.0000 −0.859010
$$848$$ 13.0000 0.446422
$$849$$ −6.00000 −0.205919
$$850$$ 3.00000 0.102899
$$851$$ 14.0000 0.479914
$$852$$ 0 0
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ −60.0000 −2.05316
$$855$$ 0 0
$$856$$ −13.0000 −0.444331
$$857$$ 40.0000 1.36637 0.683187 0.730243i $$-0.260593\pi$$
0.683187 + 0.730243i $$0.260593\pi$$
$$858$$ 12.0000 0.409673
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ −6.00000 −0.204598
$$861$$ −90.0000 −3.06719
$$862$$ −36.0000 −1.22616
$$863$$ −56.0000 −1.90626 −0.953131 0.302558i $$-0.902160\pi$$
−0.953131 + 0.302558i $$0.902160\pi$$
$$864$$ −9.00000 −0.306186
$$865$$ −14.0000 −0.476014
$$866$$ −16.0000 −0.543702
$$867$$ −24.0000 −0.815083
$$868$$ −10.0000 −0.339422
$$869$$ −8.00000 −0.271381
$$870$$ 9.00000 0.305129
$$871$$ 3.00000 0.101651
$$872$$ 19.0000 0.643421
$$873$$ 12.0000 0.406138
$$874$$ 0 0
$$875$$ 5.00000 0.169031
$$876$$ 33.0000 1.11497
$$877$$ −33.0000 −1.11433 −0.557165 0.830402i $$-0.688111\pi$$
−0.557165 + 0.830402i $$0.688111\pi$$
$$878$$ 26.0000 0.877457
$$879$$ 81.0000 2.73206
$$880$$ 4.00000 0.134840
$$881$$ 10.0000 0.336909 0.168454 0.985709i $$-0.446122\pi$$
0.168454 + 0.985709i $$0.446122\pi$$
$$882$$ −108.000 −3.63655
$$883$$ 30.0000 1.00958 0.504790 0.863242i $$-0.331570\pi$$
0.504790 + 0.863242i $$0.331570\pi$$
$$884$$ −3.00000 −0.100901
$$885$$ −27.0000 −0.907595
$$886$$ −36.0000 −1.20944
$$887$$ 28.0000 0.940148 0.470074 0.882627i $$-0.344227\pi$$
0.470074 + 0.882627i $$0.344227\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ −30.0000 −1.00617
$$890$$ −2.00000 −0.0670402
$$891$$ −36.0000 −1.20605
$$892$$ 2.00000 0.0669650
$$893$$ 0 0
$$894$$ 12.0000 0.401340
$$895$$ −8.00000 −0.267411
$$896$$ 5.00000 0.167038
$$897$$ 21.0000 0.701170
$$898$$ −22.0000 −0.734150
$$899$$ 6.00000 0.200111
$$900$$ 6.00000 0.200000
$$901$$ −39.0000 −1.29928
$$902$$ 24.0000 0.799113
$$903$$ −90.0000 −2.99501
$$904$$ 0 0
$$905$$ 26.0000 0.864269
$$906$$ −30.0000 −0.996683
$$907$$ 1.00000 0.0332045 0.0166022 0.999862i $$-0.494715\pi$$
0.0166022 + 0.999862i $$0.494715\pi$$
$$908$$ −5.00000 −0.165931
$$909$$ −48.0000 −1.59206
$$910$$ −5.00000 −0.165748
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ 40.0000 1.32381
$$914$$ 29.0000 0.959235
$$915$$ 36.0000 1.19012
$$916$$ −6.00000 −0.198246
$$917$$ −80.0000 −2.64183
$$918$$ 27.0000 0.891133
$$919$$ −5.00000 −0.164935 −0.0824674 0.996594i $$-0.526280\pi$$
−0.0824674 + 0.996594i $$0.526280\pi$$
$$920$$ 7.00000 0.230783
$$921$$ −12.0000 −0.395413
$$922$$ −18.0000 −0.592798
$$923$$ 0 0
$$924$$ 60.0000 1.97386
$$925$$ 2.00000 0.0657596
$$926$$ 8.00000 0.262896
$$927$$ −24.0000 −0.788263
$$928$$ −3.00000 −0.0984798
$$929$$ −3.00000 −0.0984268 −0.0492134 0.998788i $$-0.515671\pi$$
−0.0492134 + 0.998788i $$0.515671\pi$$
$$930$$ 6.00000 0.196748
$$931$$ 0 0
$$932$$ 10.0000 0.327561
$$933$$ 75.0000 2.45539
$$934$$ 8.00000 0.261768
$$935$$ −12.0000 −0.392442
$$936$$ −6.00000 −0.196116
$$937$$ 47.0000 1.53542 0.767712 0.640796i $$-0.221395\pi$$
0.767712 + 0.640796i $$0.221395\pi$$
$$938$$ 15.0000 0.489767
$$939$$ −3.00000 −0.0979013
$$940$$ 0 0
$$941$$ 51.0000 1.66255 0.831276 0.555860i $$-0.187611\pi$$
0.831276 + 0.555860i $$0.187611\pi$$
$$942$$ −18.0000 −0.586472
$$943$$ 42.0000 1.36771
$$944$$ 9.00000 0.292925
$$945$$ 45.0000 1.46385
$$946$$ 24.0000 0.780307
$$947$$ −24.0000 −0.779895 −0.389948 0.920837i $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$948$$ 6.00000 0.194871
$$949$$ 11.0000 0.357075
$$950$$ 0 0
$$951$$ −27.0000 −0.875535
$$952$$ −15.0000 −0.486153
$$953$$ 24.0000 0.777436 0.388718 0.921357i $$-0.372918\pi$$
0.388718 + 0.921357i $$0.372918\pi$$
$$954$$ −78.0000 −2.52534
$$955$$ −9.00000 −0.291233
$$956$$ −11.0000 −0.355765
$$957$$ −36.0000 −1.16371
$$958$$ 0 0
$$959$$ −45.0000 −1.45313
$$960$$ −3.00000 −0.0968246
$$961$$ −27.0000 −0.870968
$$962$$ −2.00000 −0.0644826
$$963$$ 78.0000 2.51351
$$964$$ 12.0000 0.386494
$$965$$ 10.0000 0.321911
$$966$$ 105.000 3.37832
$$967$$ −44.0000 −1.41494 −0.707472 0.706741i $$-0.750165\pi$$
−0.707472 + 0.706741i $$0.750165\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 0 0
$$970$$ 2.00000 0.0642161
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ −80.0000 −2.56468
$$974$$ −38.0000 −1.21760
$$975$$ 3.00000 0.0960769
$$976$$ −12.0000 −0.384111
$$977$$ 62.0000 1.98356 0.991778 0.127971i $$-0.0408466\pi$$
0.991778 + 0.127971i $$0.0408466\pi$$
$$978$$ −66.0000 −2.11045
$$979$$ 8.00000 0.255681
$$980$$ −18.0000 −0.574989
$$981$$ −114.000 −3.63974
$$982$$ −18.0000 −0.574403
$$983$$ 42.0000 1.33959 0.669796 0.742545i $$-0.266382\pi$$
0.669796 + 0.742545i $$0.266382\pi$$
$$984$$ −18.0000 −0.573819
$$985$$ 22.0000 0.700978
$$986$$ 9.00000 0.286618
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 42.0000 1.33552
$$990$$ −24.0000 −0.762770
$$991$$ −30.0000 −0.952981 −0.476491 0.879180i $$-0.658091\pi$$
−0.476491 + 0.879180i $$0.658091\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 21.0000 0.666415
$$994$$ 0 0
$$995$$ 15.0000 0.475532
$$996$$ −30.0000 −0.950586
$$997$$ −50.0000 −1.58352 −0.791758 0.610835i $$-0.790834\pi$$
−0.791758 + 0.610835i $$0.790834\pi$$
$$998$$ −42.0000 −1.32949
$$999$$ 18.0000 0.569495
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 3610.2.a.e.1.1 1
19.18 odd 2 190.2.a.b.1.1 1
57.56 even 2 1710.2.a.g.1.1 1
76.75 even 2 1520.2.a.j.1.1 1
95.18 even 4 950.2.b.a.799.1 2
95.37 even 4 950.2.b.a.799.2 2
95.94 odd 2 950.2.a.c.1.1 1
133.132 even 2 9310.2.a.u.1.1 1
152.37 odd 2 6080.2.a.x.1.1 1
152.75 even 2 6080.2.a.b.1.1 1
285.284 even 2 8550.2.a.bm.1.1 1
380.379 even 2 7600.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.b.1.1 1 19.18 odd 2
950.2.a.c.1.1 1 95.94 odd 2
950.2.b.a.799.1 2 95.18 even 4
950.2.b.a.799.2 2 95.37 even 4
1520.2.a.j.1.1 1 76.75 even 2
1710.2.a.g.1.1 1 57.56 even 2
3610.2.a.e.1.1 1 1.1 even 1 trivial
6080.2.a.b.1.1 1 152.75 even 2
6080.2.a.x.1.1 1 152.37 odd 2
7600.2.a.a.1.1 1 380.379 even 2
8550.2.a.bm.1.1 1 285.284 even 2
9310.2.a.u.1.1 1 133.132 even 2