# Properties

 Label 950.2.b.a.799.1 Level $950$ Weight $2$ Character 950.799 Analytic conductor $7.586$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(799,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("950.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.799 Dual form 950.2.b.a.799.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} -4.00000 q^{11} +3.00000i q^{12} -1.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +6.00000i q^{18} -1.00000 q^{19} +15.0000 q^{21} +4.00000i q^{22} +7.00000i q^{23} +3.00000 q^{24} -1.00000 q^{26} +9.00000i q^{27} -5.00000i q^{28} +3.00000 q^{29} -2.00000 q^{31} -1.00000i q^{32} +12.0000i q^{33} +3.00000 q^{34} +6.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -3.00000 q^{39} -6.00000 q^{41} -15.0000i q^{42} +6.00000i q^{43} +4.00000 q^{44} +7.00000 q^{46} -3.00000i q^{48} -18.0000 q^{49} +9.00000 q^{51} +1.00000i q^{52} -13.0000i q^{53} +9.00000 q^{54} -5.00000 q^{56} +3.00000i q^{57} -3.00000i q^{58} +9.00000 q^{59} -12.0000 q^{61} +2.00000i q^{62} -30.0000i q^{63} -1.00000 q^{64} +12.0000 q^{66} +3.00000i q^{67} -3.00000i q^{68} +21.0000 q^{69} -6.00000i q^{72} +11.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} -20.0000i q^{77} +3.00000i q^{78} +2.00000 q^{79} +9.00000 q^{81} +6.00000i q^{82} -10.0000i q^{83} -15.0000 q^{84} +6.00000 q^{86} -9.00000i q^{87} -4.00000i q^{88} -2.00000 q^{89} +5.00000 q^{91} -7.00000i q^{92} +6.00000i q^{93} -3.00000 q^{96} +2.00000i q^{97} +18.0000i q^{98} +24.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 $$2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} - 8 q^{11} + 10 q^{14} + 2 q^{16} - 2 q^{19} + 30 q^{21} + 6 q^{24} - 2 q^{26} + 6 q^{29} - 4 q^{31} + 6 q^{34} + 12 q^{36} - 6 q^{39} - 12 q^{41} + 8 q^{44} + 14 q^{46} - 36 q^{49} + 18 q^{51} + 18 q^{54} - 10 q^{56} + 18 q^{59} - 24 q^{61} - 2 q^{64} + 24 q^{66} + 42 q^{69} + 4 q^{74} + 2 q^{76} + 4 q^{79} + 18 q^{81} - 30 q^{84} + 12 q^{86} - 4 q^{89} + 10 q^{91} - 6 q^{96} + 48 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 - 8 * q^11 + 10 * q^14 + 2 * q^16 - 2 * q^19 + 30 * q^21 + 6 * q^24 - 2 * q^26 + 6 * q^29 - 4 * q^31 + 6 * q^34 + 12 * q^36 - 6 * q^39 - 12 * q^41 + 8 * q^44 + 14 * q^46 - 36 * q^49 + 18 * q^51 + 18 * q^54 - 10 * q^56 + 18 * q^59 - 24 * q^61 - 2 * q^64 + 24 * q^66 + 42 * q^69 + 4 * q^74 + 2 * q^76 + 4 * q^79 + 18 * q^81 - 30 * q^84 + 12 * q^86 - 4 * q^89 + 10 * q^91 - 6 * q^96 + 48 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ − 3.00000i − 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −3.00000 −1.22474
$$7$$ 5.00000i 1.88982i 0.327327 + 0.944911i $$0.393852\pi$$
−0.327327 + 0.944911i $$0.606148\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −6.00000 −2.00000
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 3.00000i 0.866025i
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 5.00000 1.33631
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ 6.00000i 1.41421i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 15.0000 3.27327
$$22$$ 4.00000i 0.852803i
$$23$$ 7.00000i 1.45960i 0.683660 + 0.729800i $$0.260387\pi$$
−0.683660 + 0.729800i $$0.739613\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 9.00000i 1.73205i
$$28$$ − 5.00000i − 0.944911i
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 12.0000i 2.08893i
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ 6.00000 1.00000
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −3.00000 −0.480384
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ − 15.0000i − 2.31455i
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 7.00000 1.03209
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 3.00000i − 0.433013i
$$49$$ −18.0000 −2.57143
$$50$$ 0 0
$$51$$ 9.00000 1.26025
$$52$$ 1.00000i 0.138675i
$$53$$ − 13.0000i − 1.78569i −0.450367 0.892844i $$-0.648707\pi$$
0.450367 0.892844i $$-0.351293\pi$$
$$54$$ 9.00000 1.22474
$$55$$ 0 0
$$56$$ −5.00000 −0.668153
$$57$$ 3.00000i 0.397360i
$$58$$ − 3.00000i − 0.393919i
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ 0 0
$$61$$ −12.0000 −1.53644 −0.768221 0.640184i $$-0.778858\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ 2.00000i 0.254000i
$$63$$ − 30.0000i − 3.77964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 12.0000 1.47710
$$67$$ 3.00000i 0.366508i 0.983066 + 0.183254i $$0.0586631\pi$$
−0.983066 + 0.183254i $$0.941337\pi$$
$$68$$ − 3.00000i − 0.363803i
$$69$$ 21.0000 2.52810
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 6.00000i − 0.707107i
$$73$$ 11.0000i 1.28745i 0.765256 + 0.643726i $$0.222612\pi$$
−0.765256 + 0.643726i $$0.777388\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ − 20.0000i − 2.27921i
$$78$$ 3.00000i 0.339683i
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 6.00000i 0.662589i
$$83$$ − 10.0000i − 1.09764i −0.835940 0.548821i $$-0.815077\pi$$
0.835940 0.548821i $$-0.184923\pi$$
$$84$$ −15.0000 −1.63663
$$85$$ 0 0
$$86$$ 6.00000 0.646997
$$87$$ − 9.00000i − 0.964901i
$$88$$ − 4.00000i − 0.426401i
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 5.00000 0.524142
$$92$$ − 7.00000i − 0.729800i
$$93$$ 6.00000i 0.622171i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −3.00000 −0.306186
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 18.0000i 1.81827i
$$99$$ 24.0000 2.41209
$$100$$ 0 0
$$101$$ −8.00000 −0.796030 −0.398015 0.917379i $$-0.630301\pi$$
−0.398015 + 0.917379i $$0.630301\pi$$
$$102$$ − 9.00000i − 0.891133i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −13.0000 −1.26267
$$107$$ 13.0000i 1.25676i 0.777908 + 0.628379i $$0.216281\pi$$
−0.777908 + 0.628379i $$0.783719\pi$$
$$108$$ − 9.00000i − 0.866025i
$$109$$ −19.0000 −1.81987 −0.909935 0.414751i $$-0.863869\pi$$
−0.909935 + 0.414751i $$0.863869\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 5.00000i 0.472456i
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 3.00000 0.280976
$$115$$ 0 0
$$116$$ −3.00000 −0.278543
$$117$$ 6.00000i 0.554700i
$$118$$ − 9.00000i − 0.828517i
$$119$$ −15.0000 −1.37505
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 12.0000i 1.08643i
$$123$$ 18.0000i 1.62301i
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ −30.0000 −2.67261
$$127$$ 6.00000i 0.532414i 0.963916 + 0.266207i $$0.0857705\pi$$
−0.963916 + 0.266207i $$0.914230\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 18.0000 1.58481
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ − 12.0000i − 1.04447i
$$133$$ − 5.00000i − 0.433555i
$$134$$ 3.00000 0.259161
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ − 9.00000i − 0.768922i −0.923141 0.384461i $$-0.874387\pi$$
0.923141 0.384461i $$-0.125613\pi$$
$$138$$ − 21.0000i − 1.78764i
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.00000i 0.334497i
$$144$$ −6.00000 −0.500000
$$145$$ 0 0
$$146$$ 11.0000 0.910366
$$147$$ 54.0000i 4.45384i
$$148$$ − 2.00000i − 0.164399i
$$149$$ 4.00000 0.327693 0.163846 0.986486i $$-0.447610\pi$$
0.163846 + 0.986486i $$0.447610\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ − 18.0000i − 1.45521i
$$154$$ −20.0000 −1.61165
$$155$$ 0 0
$$156$$ 3.00000 0.240192
$$157$$ − 6.00000i − 0.478852i −0.970915 0.239426i $$-0.923041\pi$$
0.970915 0.239426i $$-0.0769593\pi$$
$$158$$ − 2.00000i − 0.159111i
$$159$$ −39.0000 −3.09290
$$160$$ 0 0
$$161$$ −35.0000 −2.75839
$$162$$ − 9.00000i − 0.707107i
$$163$$ 22.0000i 1.72317i 0.507611 + 0.861586i $$0.330529\pi$$
−0.507611 + 0.861586i $$0.669471\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −10.0000 −0.776151
$$167$$ 2.00000i 0.154765i 0.997001 + 0.0773823i $$0.0246562\pi$$
−0.997001 + 0.0773823i $$0.975344\pi$$
$$168$$ 15.0000i 1.15728i
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ − 6.00000i − 0.457496i
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ −9.00000 −0.682288
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ − 27.0000i − 2.02944i
$$178$$ 2.00000i 0.149906i
$$179$$ 8.00000 0.597948 0.298974 0.954261i $$-0.403356\pi$$
0.298974 + 0.954261i $$0.403356\pi$$
$$180$$ 0 0
$$181$$ 26.0000 1.93256 0.966282 0.257485i $$-0.0828937\pi$$
0.966282 + 0.257485i $$0.0828937\pi$$
$$182$$ − 5.00000i − 0.370625i
$$183$$ 36.0000i 2.66120i
$$184$$ −7.00000 −0.516047
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ − 12.0000i − 0.877527i
$$188$$ 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 $$0.394431\pi$$
$$192$$ 3.00000i 0.216506i
$$193$$ 10.0000i 0.719816i 0.932988 + 0.359908i $$0.117192\pi$$
−0.932988 + 0.359908i $$0.882808\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 18.0000 1.28571
$$197$$ 22.0000i 1.56744i 0.621117 + 0.783718i $$0.286679\pi$$
−0.621117 + 0.783718i $$0.713321\pi$$
$$198$$ − 24.0000i − 1.70561i
$$199$$ 15.0000 1.06332 0.531661 0.846957i $$-0.321568\pi$$
0.531661 + 0.846957i $$0.321568\pi$$
$$200$$ 0 0
$$201$$ 9.00000 0.634811
$$202$$ 8.00000i 0.562878i
$$203$$ 15.0000i 1.05279i
$$204$$ −9.00000 −0.630126
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ − 42.0000i − 2.91920i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ 13.0000i 0.892844i
$$213$$ 0 0
$$214$$ 13.0000 0.888662
$$215$$ 0 0
$$216$$ −9.00000 −0.612372
$$217$$ − 10.0000i − 0.678844i
$$218$$ 19.0000i 1.28684i
$$219$$ 33.0000 2.22993
$$220$$ 0 0
$$221$$ 3.00000 0.201802
$$222$$ − 6.00000i − 0.402694i
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 5.00000i − 0.331862i −0.986137 0.165931i $$-0.946937\pi$$
0.986137 0.165931i $$-0.0530628\pi$$
$$228$$ − 3.00000i − 0.198680i
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ −60.0000 −3.94771
$$232$$ 3.00000i 0.196960i
$$233$$ 10.0000i 0.655122i 0.944830 + 0.327561i $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ −9.00000 −0.585850
$$237$$ − 6.00000i − 0.389742i
$$238$$ 15.0000i 0.972306i
$$239$$ 11.0000 0.711531 0.355765 0.934575i $$-0.384220\pi$$
0.355765 + 0.934575i $$0.384220\pi$$
$$240$$ 0 0
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 0 0
$$244$$ 12.0000 0.768221
$$245$$ 0 0
$$246$$ 18.0000 1.14764
$$247$$ 1.00000i 0.0636285i
$$248$$ − 2.00000i − 0.127000i
$$249$$ −30.0000 −1.90117
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 30.0000i 1.88982i
$$253$$ − 28.0000i − 1.76034i
$$254$$ 6.00000 0.376473
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 22.0000i − 1.37232i −0.727450 0.686161i $$-0.759294\pi$$
0.727450 0.686161i $$-0.240706\pi$$
$$258$$ − 18.0000i − 1.12063i
$$259$$ −10.0000 −0.621370
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ − 16.0000i − 0.988483i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ −12.0000 −0.738549
$$265$$ 0 0
$$266$$ −5.00000 −0.306570
$$267$$ 6.00000i 0.367194i
$$268$$ − 3.00000i − 0.183254i
$$269$$ −2.00000 −0.121942 −0.0609711 0.998140i $$-0.519420\pi$$
−0.0609711 + 0.998140i $$0.519420\pi$$
$$270$$ 0 0
$$271$$ −27.0000 −1.64013 −0.820067 0.572268i $$-0.806064\pi$$
−0.820067 + 0.572268i $$0.806064\pi$$
$$272$$ 3.00000i 0.181902i
$$273$$ − 15.0000i − 0.907841i
$$274$$ −9.00000 −0.543710
$$275$$ 0 0
$$276$$ −21.0000 −1.26405
$$277$$ 8.00000i 0.480673i 0.970690 + 0.240337i $$0.0772579\pi$$
−0.970690 + 0.240337i $$0.922742\pi$$
$$278$$ 16.0000i 0.959616i
$$279$$ 12.0000 0.718421
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ − 2.00000i − 0.118888i −0.998232 0.0594438i $$-0.981067\pi$$
0.998232 0.0594438i $$-0.0189327\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ − 30.0000i − 1.77084i
$$288$$ 6.00000i 0.353553i
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ − 11.0000i − 0.643726i
$$293$$ − 27.0000i − 1.57736i −0.614806 0.788678i $$-0.710766\pi$$
0.614806 0.788678i $$-0.289234\pi$$
$$294$$ 54.0000 3.14934
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ − 36.0000i − 2.08893i
$$298$$ − 4.00000i − 0.231714i
$$299$$ 7.00000 0.404820
$$300$$ 0 0
$$301$$ −30.0000 −1.72917
$$302$$ 10.0000i 0.575435i
$$303$$ 24.0000i 1.37876i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ −18.0000 −1.02899
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 20.0000i 1.13961i
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 25.0000 1.41762 0.708810 0.705399i $$-0.249232\pi$$
0.708810 + 0.705399i $$0.249232\pi$$
$$312$$ − 3.00000i − 0.169842i
$$313$$ − 1.00000i − 0.0565233i −0.999601 0.0282617i $$-0.991003\pi$$
0.999601 0.0282617i $$-0.00899717\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ − 9.00000i − 0.505490i −0.967533 0.252745i $$-0.918667\pi$$
0.967533 0.252745i $$-0.0813334\pi$$
$$318$$ 39.0000i 2.18701i
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ 39.0000 2.17677
$$322$$ 35.0000i 1.95047i
$$323$$ − 3.00000i − 0.166924i
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ 22.0000 1.21847
$$327$$ 57.0000i 3.15211i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ 10.0000i 0.548821i
$$333$$ − 12.0000i − 0.657596i
$$334$$ 2.00000 0.109435
$$335$$ 0 0
$$336$$ 15.0000 0.818317
$$337$$ − 6.00000i − 0.326841i −0.986557 0.163420i $$-0.947747\pi$$
0.986557 0.163420i $$-0.0522527\pi$$
$$338$$ − 12.0000i − 0.652714i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 8.00000 0.433224
$$342$$ − 6.00000i − 0.324443i
$$343$$ − 55.0000i − 2.96972i
$$344$$ −6.00000 −0.323498
$$345$$ 0 0
$$346$$ −14.0000 −0.752645
$$347$$ 6.00000i 0.322097i 0.986947 + 0.161048i $$0.0514875\pi$$
−0.986947 + 0.161048i $$0.948512\pi$$
$$348$$ 9.00000i 0.482451i
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 9.00000 0.480384
$$352$$ 4.00000i 0.213201i
$$353$$ 7.00000i 0.372572i 0.982496 + 0.186286i $$0.0596452\pi$$
−0.982496 + 0.186286i $$0.940355\pi$$
$$354$$ −27.0000 −1.43503
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 45.0000i 2.38165i
$$358$$ − 8.00000i − 0.422813i
$$359$$ 5.00000 0.263890 0.131945 0.991257i $$-0.457878\pi$$
0.131945 + 0.991257i $$0.457878\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 26.0000i − 1.36653i
$$363$$ − 15.0000i − 0.787296i
$$364$$ −5.00000 −0.262071
$$365$$ 0 0
$$366$$ 36.0000 1.88175
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 7.00000i 0.364900i
$$369$$ 36.0000 1.87409
$$370$$ 0 0
$$371$$ 65.0000 3.37463
$$372$$ − 6.00000i − 0.311086i
$$373$$ − 23.0000i − 1.19089i −0.803394 0.595447i $$-0.796975\pi$$
0.803394 0.595447i $$-0.203025\pi$$
$$374$$ −12.0000 −0.620505
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 3.00000i − 0.154508i
$$378$$ 45.0000i 2.31455i
$$379$$ 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
$$382$$ − 9.00000i − 0.460480i
$$383$$ − 4.00000i − 0.204390i −0.994764 0.102195i $$-0.967413\pi$$
0.994764 0.102195i $$-0.0325866\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ − 36.0000i − 1.82998i
$$388$$ − 2.00000i − 0.101535i
$$389$$ 4.00000 0.202808 0.101404 0.994845i $$-0.467667\pi$$
0.101404 + 0.994845i $$0.467667\pi$$
$$390$$ 0 0
$$391$$ −21.0000 −1.06202
$$392$$ − 18.0000i − 0.909137i
$$393$$ − 48.0000i − 2.42128i
$$394$$ 22.0000 1.10834
$$395$$ 0 0
$$396$$ −24.0000 −1.20605
$$397$$ 16.0000i 0.803017i 0.915855 + 0.401508i $$0.131514\pi$$
−0.915855 + 0.401508i $$0.868486\pi$$
$$398$$ − 15.0000i − 0.751882i
$$399$$ −15.0000 −0.750939
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ − 9.00000i − 0.448879i
$$403$$ 2.00000i 0.0996271i
$$404$$ 8.00000 0.398015
$$405$$ 0 0
$$406$$ 15.0000 0.744438
$$407$$ − 8.00000i − 0.396545i
$$408$$ 9.00000i 0.445566i
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ −27.0000 −1.33181
$$412$$ − 4.00000i − 0.197066i
$$413$$ 45.0000i 2.21431i
$$414$$ −42.0000 −2.06419
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 48.0000i 2.35057i
$$418$$ − 4.00000i − 0.195646i
$$419$$ −14.0000 −0.683945 −0.341972 0.939710i $$-0.611095\pi$$
−0.341972 + 0.939710i $$0.611095\pi$$
$$420$$ 0 0
$$421$$ 1.00000 0.0487370 0.0243685 0.999703i $$-0.492242\pi$$
0.0243685 + 0.999703i $$0.492242\pi$$
$$422$$ 5.00000i 0.243396i
$$423$$ 0 0
$$424$$ 13.0000 0.631336
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 60.0000i − 2.90360i
$$428$$ − 13.0000i − 0.628379i
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 9.00000i 0.433013i
$$433$$ − 16.0000i − 0.768911i −0.923144 0.384455i $$-0.874389\pi$$
0.923144 0.384455i $$-0.125611\pi$$
$$434$$ −10.0000 −0.480015
$$435$$ 0 0
$$436$$ 19.0000 0.909935
$$437$$ − 7.00000i − 0.334855i
$$438$$ − 33.0000i − 1.57680i
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ 0 0
$$441$$ 108.000 5.14286
$$442$$ − 3.00000i − 0.142695i
$$443$$ 36.0000i 1.71041i 0.518289 + 0.855206i $$0.326569\pi$$
−0.518289 + 0.855206i $$0.673431\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ −2.00000 −0.0947027
$$447$$ − 12.0000i − 0.567581i
$$448$$ − 5.00000i − 0.236228i
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 0 0
$$453$$ 30.0000i 1.40952i
$$454$$ −5.00000 −0.234662
$$455$$ 0 0
$$456$$ −3.00000 −0.140488
$$457$$ 29.0000i 1.35656i 0.734802 + 0.678281i $$0.237275\pi$$
−0.734802 + 0.678281i $$0.762725\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ −27.0000 −1.26025
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 60.0000i 2.79145i
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ 8.00000i 0.370196i 0.982720 + 0.185098i $$0.0592602\pi$$
−0.982720 + 0.185098i $$0.940740\pi$$
$$468$$ − 6.00000i − 0.277350i
$$469$$ −15.0000 −0.692636
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 9.00000i 0.414259i
$$473$$ − 24.0000i − 1.10352i
$$474$$ −6.00000 −0.275589
$$475$$ 0 0
$$476$$ 15.0000 0.687524
$$477$$ 78.0000i 3.57137i
$$478$$ − 11.0000i − 0.503128i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 12.0000i 0.546585i
$$483$$ 105.000i 4.77767i
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 38.0000i 1.72194i 0.508652 + 0.860972i $$0.330144\pi$$
−0.508652 + 0.860972i $$0.669856\pi$$
$$488$$ − 12.0000i − 0.543214i
$$489$$ 66.0000 2.98462
$$490$$ 0 0
$$491$$ 18.0000 0.812329 0.406164 0.913800i $$-0.366866\pi$$
0.406164 + 0.913800i $$0.366866\pi$$
$$492$$ − 18.0000i − 0.811503i
$$493$$ 9.00000i 0.405340i
$$494$$ 1.00000 0.0449921
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 0 0
$$498$$ 30.0000i 1.34433i
$$499$$ −42.0000 −1.88018 −0.940089 0.340929i $$-0.889258\pi$$
−0.940089 + 0.340929i $$0.889258\pi$$
$$500$$ 0 0
$$501$$ 6.00000 0.268060
$$502$$ 12.0000i 0.535586i
$$503$$ − 21.0000i − 0.936344i −0.883637 0.468172i $$-0.844913\pi$$
0.883637 0.468172i $$-0.155087\pi$$
$$504$$ 30.0000 1.33631
$$505$$ 0 0
$$506$$ −28.0000 −1.24475
$$507$$ − 36.0000i − 1.59882i
$$508$$ − 6.00000i − 0.266207i
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ −55.0000 −2.43306
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 9.00000i − 0.397360i
$$514$$ −22.0000 −0.970378
$$515$$ 0 0
$$516$$ −18.0000 −0.792406
$$517$$ 0 0
$$518$$ 10.0000i 0.439375i
$$519$$ −42.0000 −1.84360
$$520$$ 0 0
$$521$$ 24.0000 1.05146 0.525730 0.850652i $$-0.323792\pi$$
0.525730 + 0.850652i $$0.323792\pi$$
$$522$$ 18.0000i 0.787839i
$$523$$ − 9.00000i − 0.393543i −0.980449 0.196771i $$-0.936954\pi$$
0.980449 0.196771i $$-0.0630456\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ − 6.00000i − 0.261364i
$$528$$ 12.0000i 0.522233i
$$529$$ −26.0000 −1.13043
$$530$$ 0 0
$$531$$ −54.0000 −2.34340
$$532$$ 5.00000i 0.216777i
$$533$$ 6.00000i 0.259889i
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ −3.00000 −0.129580
$$537$$ − 24.0000i − 1.03568i
$$538$$ 2.00000i 0.0862261i
$$539$$ 72.0000 3.10126
$$540$$ 0 0
$$541$$ 16.0000 0.687894 0.343947 0.938989i $$-0.388236\pi$$
0.343947 + 0.938989i $$0.388236\pi$$
$$542$$ 27.0000i 1.15975i
$$543$$ − 78.0000i − 3.34730i
$$544$$ 3.00000 0.128624
$$545$$ 0 0
$$546$$ −15.0000 −0.641941
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ 9.00000i 0.384461i
$$549$$ 72.0000 3.07289
$$550$$ 0 0
$$551$$ −3.00000 −0.127804
$$552$$ 21.0000i 0.893819i
$$553$$ 10.0000i 0.425243i
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ − 12.0000i − 0.508456i −0.967144 0.254228i $$-0.918179\pi$$
0.967144 0.254228i $$-0.0818214\pi$$
$$558$$ − 12.0000i − 0.508001i
$$559$$ 6.00000 0.253773
$$560$$ 0 0
$$561$$ −36.0000 −1.51992
$$562$$ 18.0000i 0.759284i
$$563$$ 20.0000i 0.842900i 0.906852 + 0.421450i $$0.138479\pi$$
−0.906852 + 0.421450i $$0.861521\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −2.00000 −0.0840663
$$567$$ 45.0000i 1.88982i
$$568$$ 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 $$0.459932\pi$$
$$572$$ − 4.00000i − 0.167248i
$$573$$ − 27.0000i − 1.12794i
$$574$$ −30.0000 −1.25218
$$575$$ 0 0
$$576$$ 6.00000 0.250000
$$577$$ 7.00000i 0.291414i 0.989328 + 0.145707i $$0.0465456\pi$$
−0.989328 + 0.145707i $$0.953454\pi$$
$$578$$ − 8.00000i − 0.332756i
$$579$$ 30.0000 1.24676
$$580$$ 0 0
$$581$$ 50.0000 2.07435
$$582$$ − 6.00000i − 0.248708i
$$583$$ 52.0000i 2.15362i
$$584$$ −11.0000 −0.455183
$$585$$ 0 0
$$586$$ −27.0000 −1.11536
$$587$$ − 18.0000i − 0.742940i −0.928445 0.371470i $$-0.878854\pi$$
0.928445 0.371470i $$-0.121146\pi$$
$$588$$ − 54.0000i − 2.22692i
$$589$$ 2.00000 0.0824086
$$590$$ 0 0
$$591$$ 66.0000 2.71488
$$592$$ 2.00000i 0.0821995i
$$593$$ 30.0000i 1.23195i 0.787765 + 0.615976i $$0.211238\pi$$
−0.787765 + 0.615976i $$0.788762\pi$$
$$594$$ −36.0000 −1.47710
$$595$$ 0 0
$$596$$ −4.00000 −0.163846
$$597$$ − 45.0000i − 1.84173i
$$598$$ − 7.00000i − 0.286251i
$$599$$ 26.0000 1.06233 0.531166 0.847268i $$-0.321754\pi$$
0.531166 + 0.847268i $$0.321754\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ 30.0000i 1.22271i
$$603$$ − 18.0000i − 0.733017i
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 24.0000 0.974933
$$607$$ 26.0000i 1.05531i 0.849460 + 0.527654i $$0.176928\pi$$
−0.849460 + 0.527654i $$0.823072\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ 45.0000 1.82349
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 18.0000i 0.727607i
$$613$$ − 20.0000i − 0.807792i −0.914805 0.403896i $$-0.867656\pi$$
0.914805 0.403896i $$-0.132344\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 20.0000 0.805823
$$617$$ − 14.0000i − 0.563619i −0.959470 0.281809i $$-0.909065\pi$$
0.959470 0.281809i $$-0.0909346\pi$$
$$618$$ − 12.0000i − 0.482711i
$$619$$ 24.0000 0.964641 0.482321 0.875995i $$-0.339794\pi$$
0.482321 + 0.875995i $$0.339794\pi$$
$$620$$ 0 0
$$621$$ −63.0000 −2.52810
$$622$$ − 25.0000i − 1.00241i
$$623$$ − 10.0000i − 0.400642i
$$624$$ −3.00000 −0.120096
$$625$$ 0 0
$$626$$ −1.00000 −0.0399680
$$627$$ − 12.0000i − 0.479234i
$$628$$ 6.00000i 0.239426i
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 2.00000i 0.0795557i
$$633$$ 15.0000i 0.596196i
$$634$$ −9.00000 −0.357436
$$635$$ 0 0
$$636$$ 39.0000 1.54645
$$637$$ 18.0000i 0.713186i
$$638$$ 12.0000i 0.475085i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 8.00000 0.315981 0.157991 0.987441i $$-0.449498\pi$$
0.157991 + 0.987441i $$0.449498\pi$$
$$642$$ − 39.0000i − 1.53921i
$$643$$ 26.0000i 1.02534i 0.858586 + 0.512670i $$0.171344\pi$$
−0.858586 + 0.512670i $$0.828656\pi$$
$$644$$ 35.0000 1.37919
$$645$$ 0 0
$$646$$ −3.00000 −0.118033
$$647$$ 21.0000i 0.825595i 0.910823 + 0.412798i $$0.135448\pi$$
−0.910823 + 0.412798i $$0.864552\pi$$
$$648$$ 9.00000i 0.353553i
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ −30.0000 −1.17579
$$652$$ − 22.0000i − 0.861586i
$$653$$ − 16.0000i − 0.626128i −0.949732 0.313064i $$-0.898644\pi$$
0.949732 0.313064i $$-0.101356\pi$$
$$654$$ 57.0000 2.22888
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ − 66.0000i − 2.57491i
$$658$$ 0 0
$$659$$ −33.0000 −1.28550 −0.642749 0.766077i $$-0.722206\pi$$
−0.642749 + 0.766077i $$0.722206\pi$$
$$660$$ 0 0
$$661$$ 15.0000 0.583432 0.291716 0.956505i $$-0.405774\pi$$
0.291716 + 0.956505i $$0.405774\pi$$
$$662$$ 7.00000i 0.272063i
$$663$$ − 9.00000i − 0.349531i
$$664$$ 10.0000 0.388075
$$665$$ 0 0
$$666$$ −12.0000 −0.464991
$$667$$ 21.0000i 0.813123i
$$668$$ − 2.00000i − 0.0773823i
$$669$$ −6.00000 −0.231973
$$670$$ 0 0
$$671$$ 48.0000 1.85302
$$672$$ − 15.0000i − 0.578638i
$$673$$ − 44.0000i − 1.69608i −0.529936 0.848038i $$-0.677784\pi$$
0.529936 0.848038i $$-0.322216\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ 39.0000i 1.49889i 0.662066 + 0.749446i $$0.269680\pi$$
−0.662066 + 0.749446i $$0.730320\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ −15.0000 −0.574801
$$682$$ − 8.00000i − 0.306336i
$$683$$ − 44.0000i − 1.68361i −0.539779 0.841807i $$-0.681492\pi$$
0.539779 0.841807i $$-0.318508\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ 0 0
$$686$$ −55.0000 −2.09991
$$687$$ − 18.0000i − 0.686743i
$$688$$ 6.00000i 0.228748i
$$689$$ −13.0000 −0.495261
$$690$$ 0 0
$$691$$ −42.0000 −1.59776 −0.798878 0.601494i $$-0.794573\pi$$
−0.798878 + 0.601494i $$0.794573\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ 120.000i 4.55842i
$$694$$ 6.00000 0.227757
$$695$$ 0 0
$$696$$ 9.00000 0.341144
$$697$$ − 18.0000i − 0.681799i
$$698$$ 14.0000i 0.529908i
$$699$$ 30.0000 1.13470
$$700$$ 0 0
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ − 9.00000i − 0.339683i
$$703$$ − 2.00000i − 0.0754314i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 7.00000 0.263448
$$707$$ − 40.0000i − 1.50435i
$$708$$ 27.0000i 1.01472i
$$709$$ −2.00000 −0.0751116 −0.0375558 0.999295i $$-0.511957\pi$$
−0.0375558 + 0.999295i $$0.511957\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ − 2.00000i − 0.0749532i
$$713$$ − 14.0000i − 0.524304i
$$714$$ 45.0000 1.68408
$$715$$ 0 0
$$716$$ −8.00000 −0.298974
$$717$$ − 33.0000i − 1.23241i
$$718$$ − 5.00000i − 0.186598i
$$719$$ 27.0000 1.00693 0.503465 0.864016i $$-0.332058\pi$$
0.503465 + 0.864016i $$0.332058\pi$$
$$720$$ 0 0
$$721$$ −20.0000 −0.744839
$$722$$ − 1.00000i − 0.0372161i
$$723$$ 36.0000i 1.33885i
$$724$$ −26.0000 −0.966282
$$725$$ 0 0
$$726$$ −15.0000 −0.556702
$$727$$ − 23.0000i − 0.853023i −0.904482 0.426511i $$-0.859742\pi$$
0.904482 0.426511i $$-0.140258\pi$$
$$728$$ 5.00000i 0.185312i
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ −18.0000 −0.665754
$$732$$ − 36.0000i − 1.33060i
$$733$$ 36.0000i 1.32969i 0.746981 + 0.664845i $$0.231502\pi$$
−0.746981 + 0.664845i $$0.768498\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 7.00000 0.258023
$$737$$ − 12.0000i − 0.442026i
$$738$$ − 36.0000i − 1.32518i
$$739$$ 10.0000 0.367856 0.183928 0.982940i $$-0.441119\pi$$
0.183928 + 0.982940i $$0.441119\pi$$
$$740$$ 0 0
$$741$$ 3.00000 0.110208
$$742$$ − 65.0000i − 2.38623i
$$743$$ − 18.0000i − 0.660356i −0.943919 0.330178i $$-0.892891\pi$$
0.943919 0.330178i $$-0.107109\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 0 0
$$746$$ −23.0000 −0.842090
$$747$$ 60.0000i 2.19529i
$$748$$ 12.0000i 0.438763i
$$749$$ −65.0000 −2.37505
$$750$$ 0 0
$$751$$ −26.0000 −0.948753 −0.474377 0.880322i $$-0.657327\pi$$
−0.474377 + 0.880322i $$0.657327\pi$$
$$752$$ 0 0
$$753$$ 36.0000i 1.31191i
$$754$$ −3.00000 −0.109254
$$755$$ 0 0
$$756$$ 45.0000 1.63663
$$757$$ 6.00000i 0.218074i 0.994038 + 0.109037i $$0.0347767\pi$$
−0.994038 + 0.109037i $$0.965223\pi$$
$$758$$ − 33.0000i − 1.19861i
$$759$$ −84.0000 −3.04901
$$760$$ 0 0
$$761$$ 11.0000 0.398750 0.199375 0.979923i $$-0.436109\pi$$
0.199375 + 0.979923i $$0.436109\pi$$
$$762$$ − 18.0000i − 0.652071i
$$763$$ − 95.0000i − 3.43923i
$$764$$ −9.00000 −0.325609
$$765$$ 0 0
$$766$$ −4.00000 −0.144526
$$767$$ − 9.00000i − 0.324971i
$$768$$ − 3.00000i − 0.108253i
$$769$$ 47.0000 1.69486 0.847432 0.530904i $$-0.178148\pi$$
0.847432 + 0.530904i $$0.178148\pi$$
$$770$$ 0 0
$$771$$ −66.0000 −2.37693
$$772$$ − 10.0000i − 0.359908i
$$773$$ − 51.0000i − 1.83434i −0.398493 0.917171i $$-0.630467\pi$$
0.398493 0.917171i $$-0.369533\pi$$
$$774$$ −36.0000 −1.29399
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 30.0000i 1.07624i
$$778$$ − 4.00000i − 0.143407i
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 21.0000i 0.750958i
$$783$$ 27.0000i 0.964901i
$$784$$ −18.0000 −0.642857
$$785$$ 0 0
$$786$$ −48.0000 −1.71210
$$787$$ 39.0000i 1.39020i 0.718913 + 0.695100i $$0.244640\pi$$
−0.718913 + 0.695100i $$0.755360\pi$$
$$788$$ − 22.0000i − 0.783718i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 24.0000i 0.852803i
$$793$$ 12.0000i 0.426132i
$$794$$ 16.0000 0.567819
$$795$$ 0 0
$$796$$ −15.0000 −0.531661
$$797$$ − 31.0000i − 1.09808i −0.835797 0.549038i $$-0.814994\pi$$
0.835797 0.549038i $$-0.185006\pi$$
$$798$$ 15.0000i 0.530994i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 12.0000 0.423999
$$802$$ 6.00000i 0.211867i
$$803$$ − 44.0000i − 1.55273i
$$804$$ −9.00000 −0.317406
$$805$$ 0 0
$$806$$ 2.00000 0.0704470
$$807$$ 6.00000i 0.211210i
$$808$$ − 8.00000i − 0.281439i
$$809$$ −25.0000 −0.878953 −0.439477 0.898254i $$-0.644836\pi$$
−0.439477 + 0.898254i $$0.644836\pi$$
$$810$$ 0 0
$$811$$ 37.0000 1.29925 0.649623 0.760257i $$-0.274927\pi$$
0.649623 + 0.760257i $$0.274927\pi$$
$$812$$ − 15.0000i − 0.526397i
$$813$$ 81.0000i 2.84079i
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 9.00000 0.315063
$$817$$ − 6.00000i − 0.209913i
$$818$$ 22.0000i 0.769212i
$$819$$ −30.0000 −1.04828
$$820$$ 0 0
$$821$$ −52.0000 −1.81481 −0.907406 0.420255i $$-0.861941\pi$$
−0.907406 + 0.420255i $$0.861941\pi$$
$$822$$ 27.0000i 0.941733i
$$823$$ − 43.0000i − 1.49889i −0.662069 0.749443i $$-0.730321\pi$$
0.662069 0.749443i $$-0.269679\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 45.0000 1.56575
$$827$$ 3.00000i 0.104320i 0.998639 + 0.0521601i $$0.0166106\pi$$
−0.998639 + 0.0521601i $$0.983389\pi$$
$$828$$ 42.0000i 1.45960i
$$829$$ −35.0000 −1.21560 −0.607800 0.794090i $$-0.707948\pi$$
−0.607800 + 0.794090i $$0.707948\pi$$
$$830$$ 0 0
$$831$$ 24.0000 0.832551
$$832$$ 1.00000i 0.0346688i
$$833$$ − 54.0000i − 1.87099i
$$834$$ 48.0000 1.66210
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ − 18.0000i − 0.622171i
$$838$$ 14.0000i 0.483622i
$$839$$ 4.00000 0.138095 0.0690477 0.997613i $$-0.478004\pi$$
0.0690477 + 0.997613i $$0.478004\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ − 1.00000i − 0.0344623i
$$843$$ 54.0000i 1.85986i
$$844$$ 5.00000 0.172107
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 25.0000i 0.859010i
$$848$$ − 13.0000i − 0.446422i
$$849$$ −6.00000 −0.205919
$$850$$ 0 0
$$851$$ −14.0000 −0.479914
$$852$$ 0 0
$$853$$ 42.0000i 1.43805i 0.694983 + 0.719026i $$0.255412\pi$$
−0.694983 + 0.719026i $$0.744588\pi$$
$$854$$ −60.0000 −2.05316
$$855$$ 0 0
$$856$$ −13.0000 −0.444331
$$857$$ 40.0000i 1.36637i 0.730243 + 0.683187i $$0.239407\pi$$
−0.730243 + 0.683187i $$0.760593\pi$$
$$858$$ − 12.0000i − 0.409673i
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ −90.0000 −3.06719
$$862$$ 36.0000i 1.22616i
$$863$$ 56.0000i 1.90626i 0.302558 + 0.953131i $$0.402160\pi$$
−0.302558 + 0.953131i $$0.597840\pi$$
$$864$$ 9.00000 0.306186
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ − 24.0000i − 0.815083i
$$868$$ 10.0000i 0.339422i
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ − 19.0000i − 0.643421i
$$873$$ − 12.0000i − 0.406138i
$$874$$ −7.00000 −0.236779
$$875$$ 0 0
$$876$$ −33.0000 −1.11497
$$877$$ − 33.0000i − 1.11433i −0.830402 0.557165i $$-0.811889\pi$$
0.830402 0.557165i $$-0.188111\pi$$
$$878$$ 26.0000i 0.877457i
$$879$$ −81.0000 −2.73206
$$880$$ 0 0
$$881$$ 10.0000 0.336909 0.168454 0.985709i $$-0.446122\pi$$
0.168454 + 0.985709i $$0.446122\pi$$
$$882$$ − 108.000i − 3.63655i
$$883$$ 30.0000i 1.00958i 0.863242 + 0.504790i $$0.168430\pi$$
−0.863242 + 0.504790i $$0.831570\pi$$
$$884$$ −3.00000 −0.100901
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ 28.0000i 0.940148i 0.882627 + 0.470074i $$0.155773\pi$$
−0.882627 + 0.470074i $$0.844227\pi$$
$$888$$ 6.00000i 0.201347i
$$889$$ −30.0000 −1.00617
$$890$$ 0 0
$$891$$ −36.0000 −1.20605
$$892$$ 2.00000i 0.0669650i
$$893$$ 0 0
$$894$$ −12.0000 −0.401340
$$895$$ 0 0
$$896$$ −5.00000 −0.167038
$$897$$ − 21.0000i − 0.701170i
$$898$$ − 22.0000i − 0.734150i
$$899$$ −6.00000 −0.200111
$$900$$ 0 0
$$901$$ 39.0000 1.29928
$$902$$ − 24.0000i − 0.799113i
$$903$$ 90.0000i 2.99501i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 30.0000 0.996683
$$907$$ 1.00000i 0.0332045i 0.999862 + 0.0166022i $$0.00528490\pi$$
−0.999862 + 0.0166022i $$0.994715\pi$$
$$908$$ 5.00000i 0.165931i
$$909$$ 48.0000 1.59206
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 3.00000i 0.0993399i
$$913$$ 40.0000i 1.32381i
$$914$$ 29.0000 0.959235
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 80.0000i 2.64183i
$$918$$ 27.0000i 0.891133i
$$919$$ 5.00000 0.164935 0.0824674 0.996594i $$-0.473720\pi$$
0.0824674 + 0.996594i $$0.473720\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ − 18.0000i − 0.592798i
$$923$$ 0 0
$$924$$ 60.0000 1.97386
$$925$$ 0 0
$$926$$ −8.00000 −0.262896
$$927$$ − 24.0000i − 0.788263i
$$928$$ − 3.00000i − 0.0984798i
$$929$$ 3.00000 0.0984268 0.0492134 0.998788i $$-0.484329\pi$$
0.0492134 + 0.998788i $$0.484329\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ − 10.0000i − 0.327561i
$$933$$ − 75.0000i − 2.45539i
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ − 47.0000i − 1.53542i −0.640796 0.767712i $$-0.721395\pi$$
0.640796 0.767712i $$-0.278605\pi$$
$$938$$ 15.0000i 0.489767i
$$939$$ −3.00000 −0.0979013
$$940$$ 0 0
$$941$$ −51.0000 −1.66255 −0.831276 0.555860i $$-0.812389\pi$$
−0.831276 + 0.555860i $$0.812389\pi$$
$$942$$ 18.0000i 0.586472i
$$943$$ − 42.0000i − 1.36771i
$$944$$ 9.00000 0.292925
$$945$$ 0 0
$$946$$ −24.0000 −0.780307
$$947$$ 24.0000i 0.779895i 0.920837 + 0.389948i $$0.127507\pi$$
−0.920837 + 0.389948i $$0.872493\pi$$
$$948$$ 6.00000i 0.194871i
$$949$$ 11.0000 0.357075
$$950$$ 0 0
$$951$$ −27.0000 −0.875535
$$952$$ − 15.0000i − 0.486153i
$$953$$ − 24.0000i − 0.777436i −0.921357 0.388718i $$-0.872918\pi$$
0.921357 0.388718i $$-0.127082\pi$$
$$954$$ 78.0000 2.52534
$$955$$ 0 0
$$956$$ −11.0000 −0.355765
$$957$$ 36.0000i 1.16371i
$$958$$ 0 0
$$959$$ 45.0000 1.45313
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ − 2.00000i − 0.0644826i
$$963$$ − 78.0000i − 2.51351i
$$964$$ 12.0000 0.386494
$$965$$ 0 0
$$966$$ 105.000 3.37832
$$967$$ 44.0000i 1.41494i 0.706741 + 0.707472i $$0.250165\pi$$
−0.706741 + 0.707472i $$0.749835\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ −9.00000 −0.289122
$$970$$ 0 0
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 0 0
$$973$$ − 80.0000i − 2.56468i
$$974$$ 38.0000 1.21760
$$975$$ 0 0
$$976$$ −12.0000 −0.384111
$$977$$ 62.0000i 1.98356i 0.127971 + 0.991778i $$0.459153\pi$$
−0.127971 + 0.991778i $$0.540847\pi$$
$$978$$ − 66.0000i − 2.11045i
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ 114.000 3.63974
$$982$$ − 18.0000i − 0.574403i
$$983$$ − 42.0000i − 1.33959i −0.742545 0.669796i $$-0.766382\pi$$
0.742545 0.669796i $$-0.233618\pi$$
$$984$$ −18.0000 −0.573819
$$985$$ 0 0
$$986$$ 9.00000 0.286618
$$987$$ 0 0
$$988$$ − 1.00000i − 0.0318142i
$$989$$ −42.0000 −1.33552
$$990$$ 0 0
$$991$$ 30.0000 0.952981 0.476491 0.879180i $$-0.341909\pi$$
0.476491 + 0.879180i $$0.341909\pi$$
$$992$$ 2.00000i 0.0635001i
$$993$$ 21.0000i 0.666415i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 30.0000 0.950586
$$997$$ 50.0000i 1.58352i 0.610835 + 0.791758i $$0.290834\pi$$
−0.610835 + 0.791758i $$0.709166\pi$$
$$998$$ 42.0000i 1.32949i
$$999$$ −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 950.2.b.a.799.1 2
5.2 odd 4 190.2.a.b.1.1 1
5.3 odd 4 950.2.a.c.1.1 1
5.4 even 2 inner 950.2.b.a.799.2 2
15.2 even 4 1710.2.a.g.1.1 1
15.8 even 4 8550.2.a.bm.1.1 1
20.3 even 4 7600.2.a.a.1.1 1
20.7 even 4 1520.2.a.j.1.1 1
35.27 even 4 9310.2.a.u.1.1 1
40.27 even 4 6080.2.a.b.1.1 1
40.37 odd 4 6080.2.a.x.1.1 1
95.37 even 4 3610.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.b.1.1 1 5.2 odd 4
950.2.a.c.1.1 1 5.3 odd 4
950.2.b.a.799.1 2 1.1 even 1 trivial
950.2.b.a.799.2 2 5.4 even 2 inner
1520.2.a.j.1.1 1 20.7 even 4
1710.2.a.g.1.1 1 15.2 even 4
3610.2.a.e.1.1 1 95.37 even 4
6080.2.a.b.1.1 1 40.27 even 4
6080.2.a.x.1.1 1 40.37 odd 4
7600.2.a.a.1.1 1 20.3 even 4
8550.2.a.bm.1.1 1 15.8 even 4
9310.2.a.u.1.1 1 35.27 even 4