# Properties

 Label 4650.2.d.h.3349.1 Level $4650$ Weight $2$ Character 4650.3349 Analytic conductor $37.130$ 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] = [4650,2,Mod(3349,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4650.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.d (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: $$37.1304369399$$ 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 930) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3349.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.3349 Dual form 4650.2.d.h.3349.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +4.00000 q^{21} -2.00000i q^{22} +6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} +1.00000 q^{31} -1.00000i q^{32} -2.00000i q^{33} +1.00000 q^{36} -2.00000i q^{37} -2.00000 q^{39} -10.0000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -2.00000 q^{44} +6.00000 q^{46} +4.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} +4.00000 q^{59} -1.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} +4.00000i q^{67} +6.00000 q^{69} -16.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -2.00000 q^{74} +8.00000i q^{77} +2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} -8.00000i q^{83} -4.00000 q^{84} +4.00000 q^{86} +2.00000i q^{88} -6.00000 q^{89} +8.00000 q^{91} -6.00000i q^{92} -1.00000i q^{93} +4.00000 q^{94} -1.00000 q^{96} +14.0000i q^{97} +9.00000i q^{98} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{11} + 8 q^{14} + 2 q^{16} + 8 q^{21} + 2 q^{24} - 4 q^{26} + 2 q^{31} + 2 q^{36} - 4 q^{39} - 20 q^{41} - 4 q^{44} + 12 q^{46} - 18 q^{49} + 2 q^{54} - 8 q^{56} + 8 q^{59} - 2 q^{64} - 4 q^{66} + 12 q^{69} - 32 q^{71} - 4 q^{74} - 8 q^{79} + 2 q^{81} - 8 q^{84} + 8 q^{86} - 12 q^{89} + 16 q^{91} + 8 q^{94} - 2 q^{96} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^11 + 8 * q^14 + 2 * q^16 + 8 * q^21 + 2 * q^24 - 4 * q^26 + 2 * q^31 + 2 * q^36 - 4 * q^39 - 20 * q^41 - 4 * q^44 + 12 * q^46 - 18 * q^49 + 2 * q^54 - 8 * q^56 + 8 * q^59 - 2 * q^64 - 4 * q^66 + 12 * q^69 - 32 * q^71 - 4 * q^74 - 8 * q^79 + 2 * q^81 - 8 * q^84 + 8 * q^86 - 12 * q^89 + 16 * q^91 + 8 * q^94 - 2 * q^96 - 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ − 2.00000i − 0.426401i
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 2.00000i − 0.348155i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ − 1.00000i − 0.127000i
$$63$$ − 4.00000i − 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.00000i 0.911685i
$$78$$ 2.00000i 0.226455i
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000i 1.10432i
$$83$$ − 8.00000i − 0.878114i −0.898459 0.439057i $$-0.855313\pi$$
0.898459 0.439057i $$-0.144687\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 2.00000i 0.213201i
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ − 6.00000i − 0.625543i
$$93$$ − 1.00000i − 0.103695i
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 12.0000i 1.18240i 0.806527 + 0.591198i $$0.201345\pi$$
−0.806527 + 0.591198i $$0.798655\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 4.00000i 0.377964i
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ − 4.00000i − 0.368230i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 10.0000i 0.901670i
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ 18.0000i 1.59724i 0.601834 + 0.798621i $$0.294437\pi$$
−0.601834 + 0.798621i $$0.705563\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ − 6.00000i − 0.510754i
$$139$$ 6.00000 0.508913 0.254457 0.967084i $$-0.418103\pi$$
0.254457 + 0.967084i $$0.418103\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 16.0000i 1.34269i
$$143$$ − 4.00000i − 0.334497i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 9.00000i 0.742307i
$$148$$ 2.00000i 0.164399i
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 8.00000 0.644658
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ 4.00000i 0.318223i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −24.0000 −1.89146
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 14.0000i 1.08335i 0.840587 + 0.541676i $$0.182210\pi$$
−0.840587 + 0.541676i $$0.817790\pi$$
$$168$$ 4.00000i 0.308607i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 4.00000i − 0.304997i
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ − 4.00000i − 0.300658i
$$178$$ 6.00000i 0.449719i
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ − 8.00000i − 0.592999i
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ −1.00000 −0.0733236
$$187$$ 0 0
$$188$$ − 4.00000i − 0.291730i
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 10.0000i 0.719816i 0.932988 + 0.359908i $$0.117192\pi$$
−0.932988 + 0.359908i $$0.882808\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 2.00000i 0.142134i
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 10.0000i 0.703598i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 12.0000 0.836080
$$207$$ − 6.00000i − 0.417029i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 16.0000i 1.09630i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 4.00000i 0.271538i
$$218$$ − 2.00000i − 0.135457i
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 2.00000i 0.134231i
$$223$$ 10.0000i 0.669650i 0.942280 + 0.334825i $$0.108677\pi$$
−0.942280 + 0.334825i $$0.891323\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 0 0
$$233$$ 22.0000i 1.44127i 0.693316 + 0.720634i $$0.256149\pi$$
−0.693316 + 0.720634i $$0.743851\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 4.00000i 0.259828i
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 7.00000i 0.449977i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ 0 0
$$248$$ 1.00000i 0.0635001i
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 12.0000i 0.754434i
$$254$$ 18.0000 1.12942
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 4.00000i − 0.247121i
$$263$$ − 10.0000i − 0.616626i −0.951285 0.308313i $$-0.900236\pi$$
0.951285 0.308313i $$-0.0997645\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ − 4.00000i − 0.244339i
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ − 8.00000i − 0.484182i
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ − 6.00000i − 0.359856i
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ − 40.0000i − 2.36113i
$$288$$ 1.00000i 0.0589256i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 4.00000i 0.234082i
$$293$$ 2.00000i 0.116841i 0.998292 + 0.0584206i $$0.0186065\pi$$
−0.998292 + 0.0584206i $$0.981394\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 2.00000i 0.116052i
$$298$$ − 2.00000i − 0.115857i
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ − 4.00000i − 0.230174i
$$303$$ 10.0000i 0.574485i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 28.0000i − 1.59804i −0.601302 0.799022i $$-0.705351\pi$$
0.601302 0.799022i $$-0.294649\pi$$
$$308$$ − 8.00000i − 0.455842i
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ − 28.0000i − 1.58265i −0.611393 0.791327i $$-0.709391\pi$$
0.611393 0.791327i $$-0.290609\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 24.0000i 1.33747i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ − 2.00000i − 0.110600i
$$328$$ − 10.0000i − 0.552158i
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 18.0000 0.989369 0.494685 0.869072i $$-0.335284\pi$$
0.494685 + 0.869072i $$0.335284\pi$$
$$332$$ 8.00000i 0.439057i
$$333$$ 2.00000i 0.109599i
$$334$$ 14.0000 0.766046
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ − 32.0000i − 1.74315i −0.490261 0.871576i $$-0.663099\pi$$
0.490261 0.871576i $$-0.336901\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 2.00000 0.108306
$$342$$ 0 0
$$343$$ − 8.00000i − 0.431959i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 24.0000i 1.28839i 0.764862 + 0.644194i $$0.222807\pi$$
−0.764862 + 0.644194i $$0.777193\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ − 2.00000i − 0.106600i
$$353$$ − 36.0000i − 1.91609i −0.286623 0.958043i $$-0.592533\pi$$
0.286623 0.958043i $$-0.407467\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ − 2.00000i − 0.105703i
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 20.0000i 1.05118i
$$363$$ 7.00000i 0.367405i
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 26.0000i 1.35719i 0.734513 + 0.678594i $$0.237411\pi$$
−0.734513 + 0.678594i $$0.762589\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 24.0000 1.24602
$$372$$ 1.00000i 0.0518476i
$$373$$ − 6.00000i − 0.310668i −0.987862 0.155334i $$-0.950355\pi$$
0.987862 0.155334i $$-0.0496454\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 0 0
$$378$$ 4.00000i 0.205738i
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 18.0000 0.922168
$$382$$ 0 0
$$383$$ − 6.00000i − 0.306586i −0.988181 0.153293i $$-0.951012\pi$$
0.988181 0.153293i $$-0.0489878\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ − 4.00000i − 0.203331i
$$388$$ − 14.0000i − 0.710742i
$$389$$ −20.0000 −1.01404 −0.507020 0.861934i $$-0.669253\pi$$
−0.507020 + 0.861934i $$0.669253\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 9.00000i − 0.454569i
$$393$$ − 4.00000i − 0.201773i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 18.0000i 0.903394i 0.892171 + 0.451697i $$0.149181\pi$$
−0.892171 + 0.451697i $$0.850819\pi$$
$$398$$ 20.0000i 1.00251i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ − 4.00000i − 0.199502i
$$403$$ − 2.00000i − 0.0996271i
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 4.00000i − 0.198273i
$$408$$ 0 0
$$409$$ 18.0000 0.890043 0.445021 0.895520i $$-0.353196\pi$$
0.445021 + 0.895520i $$0.353196\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ − 12.0000i − 0.591198i
$$413$$ 16.0000i 0.787309i
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 6.00000i − 0.293821i
$$418$$ 0 0
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ − 4.00000i − 0.194487i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 16.0000 0.775203
$$427$$ 0 0
$$428$$ − 12.0000i − 0.580042i
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ 40.0000 1.92673 0.963366 0.268190i $$-0.0864254\pi$$
0.963366 + 0.268190i $$0.0864254\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 20.0000i 0.961139i 0.876957 + 0.480569i $$0.159570\pi$$
−0.876957 + 0.480569i $$0.840430\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 0 0
$$438$$ 4.00000i 0.191127i
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ 10.0000 0.473514
$$447$$ − 2.00000i − 0.0945968i
$$448$$ − 4.00000i − 0.188982i
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ −20.0000 −0.941763
$$452$$ 2.00000i 0.0940721i
$$453$$ − 4.00000i − 0.187936i
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.00000i 0.374224i 0.982339 + 0.187112i $$0.0599128\pi$$
−0.982339 + 0.187112i $$0.940087\pi$$
$$458$$ − 20.0000i − 0.934539i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ − 8.00000i − 0.372194i
$$463$$ 18.0000i 0.836531i 0.908325 + 0.418265i $$0.137362\pi$$
−0.908325 + 0.418265i $$0.862638\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ − 20.0000i − 0.925490i −0.886492 0.462745i $$-0.846865\pi$$
0.886492 0.462745i $$-0.153135\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 4.00000i 0.184115i
$$473$$ 8.00000i 0.367840i
$$474$$ 4.00000 0.183726
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ − 8.00000i − 0.365911i
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 22.0000i 1.00207i
$$483$$ 24.0000i 1.09204i
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 10.0000i 0.453143i 0.973995 + 0.226572i $$0.0727517\pi$$
−0.973995 + 0.226572i $$0.927248\pi$$
$$488$$ 0 0
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ − 10.0000i − 0.450835i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ − 64.0000i − 2.87079i
$$498$$ 8.00000i 0.358489i
$$499$$ −22.0000 −0.984855 −0.492428 0.870353i $$-0.663890\pi$$
−0.492428 + 0.870353i $$0.663890\pi$$
$$500$$ 0 0
$$501$$ 14.0000 0.625474
$$502$$ − 2.00000i − 0.0892644i
$$503$$ − 16.0000i − 0.713405i −0.934218 0.356702i $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 12.0000 0.533465
$$507$$ − 9.00000i − 0.399704i
$$508$$ − 18.0000i − 0.798621i
$$509$$ 4.00000 0.177297 0.0886484 0.996063i $$-0.471745\pi$$
0.0886484 + 0.996063i $$0.471745\pi$$
$$510$$ 0 0
$$511$$ 16.0000 0.707798
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 8.00000i 0.351840i
$$518$$ − 8.00000i − 0.351500i
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ −42.0000 −1.84005 −0.920027 0.391856i $$-0.871833\pi$$
−0.920027 + 0.391856i $$0.871833\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ −10.0000 −0.436021
$$527$$ 0 0
$$528$$ − 2.00000i − 0.0870388i
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 20.0000i 0.866296i
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ − 2.00000i − 0.0863064i
$$538$$ − 24.0000i − 1.03471i
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 6.00000 0.257960 0.128980 0.991647i $$-0.458830\pi$$
0.128980 + 0.991647i $$0.458830\pi$$
$$542$$ 0 0
$$543$$ 20.0000i 0.858282i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 36.0000i 1.53925i 0.638497 + 0.769624i $$0.279557\pi$$
−0.638497 + 0.769624i $$0.720443\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 6.00000i 0.255377i
$$553$$ − 16.0000i − 0.680389i
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −6.00000 −0.254457
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 1.00000i 0.0423334i
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 6.00000i − 0.253095i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 4.00000i 0.167984i
$$568$$ − 16.0000i − 0.671345i
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ −26.0000 −1.08807 −0.544033 0.839064i $$-0.683103\pi$$
−0.544033 + 0.839064i $$0.683103\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 0 0
$$574$$ −40.0000 −1.66957
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 6.00000i − 0.249783i −0.992170 0.124892i $$-0.960142\pi$$
0.992170 0.124892i $$-0.0398583\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ 10.0000 0.415586
$$580$$ 0 0
$$581$$ 32.0000 1.32758
$$582$$ − 14.0000i − 0.580319i
$$583$$ − 12.0000i − 0.496989i
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 2.00000 0.0826192
$$587$$ − 24.0000i − 0.990586i −0.868726 0.495293i $$-0.835061\pi$$
0.868726 0.495293i $$-0.164939\pi$$
$$588$$ − 9.00000i − 0.371154i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ − 2.00000i − 0.0821995i
$$593$$ 34.0000i 1.39621i 0.715994 + 0.698106i $$0.245974\pi$$
−0.715994 + 0.698106i $$0.754026\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ −2.00000 −0.0819232
$$597$$ 20.0000i 0.818546i
$$598$$ − 12.0000i − 0.490716i
$$599$$ 48.0000 1.96123 0.980613 0.195952i $$-0.0627798\pi$$
0.980613 + 0.195952i $$0.0627798\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ − 4.00000i − 0.162893i
$$604$$ −4.00000 −0.162758
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ − 16.0000i − 0.649420i −0.945814 0.324710i $$-0.894733\pi$$
0.945814 0.324710i $$-0.105267\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 42.0000i 1.69636i 0.529705 + 0.848182i $$0.322303\pi$$
−0.529705 + 0.848182i $$0.677697\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ −8.00000 −0.322329
$$617$$ 30.0000i 1.20775i 0.797077 + 0.603877i $$0.206378\pi$$
−0.797077 + 0.603877i $$0.793622\pi$$
$$618$$ − 12.0000i − 0.482711i
$$619$$ −34.0000 −1.36658 −0.683288 0.730149i $$-0.739451\pi$$
−0.683288 + 0.730149i $$0.739451\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ − 24.0000i − 0.961540i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −28.0000 −1.11911
$$627$$ 0 0
$$628$$ − 18.0000i − 0.718278i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ − 4.00000i − 0.159111i
$$633$$ 4.00000i 0.158986i
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 18.0000i 0.713186i
$$638$$ 0 0
$$639$$ 16.0000 0.632950
$$640$$ 0 0
$$641$$ 10.0000 0.394976 0.197488 0.980305i $$-0.436722\pi$$
0.197488 + 0.980305i $$0.436722\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ − 44.0000i − 1.73519i −0.497271 0.867595i $$-0.665665\pi$$
0.497271 0.867595i $$-0.334335\pi$$
$$644$$ 24.0000 0.945732
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 18.0000i − 0.707653i −0.935311 0.353827i $$-0.884880\pi$$
0.935311 0.353827i $$-0.115120\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ − 12.0000i − 0.469956i
$$653$$ − 30.0000i − 1.17399i −0.809590 0.586995i $$-0.800311\pi$$
0.809590 0.586995i $$-0.199689\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ 4.00000i 0.156055i
$$658$$ 16.0000i 0.623745i
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ − 18.0000i − 0.699590i
$$663$$ 0 0
$$664$$ 8.00000 0.310460
$$665$$ 0 0
$$666$$ 2.00000 0.0774984
$$667$$ 0 0
$$668$$ − 14.0000i − 0.541676i
$$669$$ 10.0000 0.386622
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 4.00000i − 0.154303i
$$673$$ 40.0000i 1.54189i 0.636904 + 0.770943i $$0.280215\pi$$
−0.636904 + 0.770943i $$0.719785\pi$$
$$674$$ −32.0000 −1.23259
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 42.0000i 1.61419i 0.590421 + 0.807096i $$0.298962\pi$$
−0.590421 + 0.807096i $$0.701038\pi$$
$$678$$ 2.00000i 0.0768095i
$$679$$ −56.0000 −2.14908
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ − 2.00000i − 0.0765840i
$$683$$ 20.0000i 0.765279i 0.923898 + 0.382639i $$0.124985\pi$$
−0.923898 + 0.382639i $$0.875015\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ − 20.0000i − 0.763048i
$$688$$ 4.00000i 0.152499i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ − 14.0000i − 0.532200i
$$693$$ − 8.00000i − 0.303895i
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ − 14.0000i − 0.529908i
$$699$$ 22.0000 0.832116
$$700$$ 0 0
$$701$$ −38.0000 −1.43524 −0.717620 0.696435i $$-0.754769\pi$$
−0.717620 + 0.696435i $$0.754769\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ 0 0
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ −36.0000 −1.35488
$$707$$ − 40.0000i − 1.50435i
$$708$$ 4.00000i 0.150329i
$$709$$ 32.0000 1.20179 0.600893 0.799330i $$-0.294812\pi$$
0.600893 + 0.799330i $$0.294812\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ − 6.00000i − 0.224860i
$$713$$ 6.00000i 0.224702i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −2.00000 −0.0747435
$$717$$ − 8.00000i − 0.298765i
$$718$$ 24.0000i 0.895672i
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ 0 0
$$721$$ −48.0000 −1.78761
$$722$$ 19.0000i 0.707107i
$$723$$ 22.0000i 0.818189i
$$724$$ 20.0000 0.743294
$$725$$ 0 0
$$726$$ 7.00000 0.259794
$$727$$ 48.0000i 1.78022i 0.455744 + 0.890111i $$0.349373\pi$$
−0.455744 + 0.890111i $$0.650627\pi$$
$$728$$ 8.00000i 0.296500i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 26.0000 0.959678
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ 8.00000i 0.294684i
$$738$$ − 10.0000i − 0.368105i
$$739$$ −2.00000 −0.0735712 −0.0367856 0.999323i $$-0.511712\pi$$
−0.0367856 + 0.999323i $$0.511712\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 24.0000i − 0.881068i
$$743$$ − 26.0000i − 0.953847i −0.878945 0.476924i $$-0.841752\pi$$
0.878945 0.476924i $$-0.158248\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ −6.00000 −0.219676
$$747$$ 8.00000i 0.292705i
$$748$$ 0 0
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 4.00000i 0.145865i
$$753$$ − 2.00000i − 0.0728841i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ − 18.0000i − 0.654221i −0.944986 0.327111i $$-0.893925\pi$$
0.944986 0.327111i $$-0.106075\pi$$
$$758$$ − 16.0000i − 0.581146i
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ −26.0000 −0.942499 −0.471250 0.882000i $$-0.656197\pi$$
−0.471250 + 0.882000i $$0.656197\pi$$
$$762$$ − 18.0000i − 0.652071i
$$763$$ 8.00000i 0.289619i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −6.00000 −0.216789
$$767$$ − 8.00000i − 0.288863i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 10.0000i − 0.359908i
$$773$$ 2.00000i 0.0719350i 0.999353 + 0.0359675i $$0.0114513\pi$$
−0.999353 + 0.0359675i $$0.988549\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ − 8.00000i − 0.286998i
$$778$$ 20.0000i 0.717035i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ − 40.0000i − 1.42585i −0.701242 0.712923i $$-0.747371\pi$$
0.701242 0.712923i $$-0.252629\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ −10.0000 −0.356009
$$790$$ 0 0
$$791$$ 8.00000 0.284447
$$792$$ − 2.00000i − 0.0710669i
$$793$$ 0 0
$$794$$ 18.0000 0.638796
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 38.0000i 1.34603i 0.739629 + 0.673015i $$0.235001\pi$$
−0.739629 + 0.673015i $$0.764999\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ − 30.0000i − 1.05934i
$$803$$ − 8.00000i − 0.282314i
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −2.00000 −0.0704470
$$807$$ − 24.0000i − 0.844840i
$$808$$ − 10.0000i − 0.351799i
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 32.0000 1.12367 0.561836 0.827249i $$-0.310095\pi$$
0.561836 + 0.827249i $$0.310095\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −4.00000 −0.140200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ − 18.0000i − 0.629355i
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ − 18.0000i − 0.627441i −0.949515 0.313720i $$-0.898425\pi$$
0.949515 0.313720i $$-0.101575\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ 16.0000 0.556711
$$827$$ − 56.0000i − 1.94731i −0.228024 0.973655i $$-0.573227\pi$$
0.228024 0.973655i $$-0.426773\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 2.00000i 0.0693375i
$$833$$ 0 0
$$834$$ −6.00000 −0.207763
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1.00000i 0.0345651i
$$838$$ 36.0000i 1.24360i
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 30.0000i 1.03387i
$$843$$ − 6.00000i − 0.206651i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ − 28.0000i − 0.962091i
$$848$$ − 6.00000i − 0.206041i
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ − 16.0000i − 0.548151i
$$853$$ − 6.00000i − 0.205436i −0.994711 0.102718i $$-0.967246\pi$$
0.994711 0.102718i $$-0.0327539\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ − 46.0000i − 1.57133i −0.618652 0.785665i $$-0.712321\pi$$
0.618652 0.785665i $$-0.287679\pi$$
$$858$$ 4.00000i 0.136558i
$$859$$ 2.00000 0.0682391 0.0341196 0.999418i $$-0.489137\pi$$
0.0341196 + 0.999418i $$0.489137\pi$$
$$860$$ 0 0
$$861$$ −40.0000 −1.36320
$$862$$ − 40.0000i − 1.36241i
$$863$$ − 30.0000i − 1.02121i −0.859815 0.510606i $$-0.829421\pi$$
0.859815 0.510606i $$-0.170579\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 20.0000 0.679628
$$867$$ − 17.0000i − 0.577350i
$$868$$ − 4.00000i − 0.135769i
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 2.00000i 0.0677285i
$$873$$ − 14.0000i − 0.473828i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 4.00000 0.135147
$$877$$ 18.0000i 0.607817i 0.952701 + 0.303908i $$0.0982917\pi$$
−0.952701 + 0.303908i $$0.901708\pi$$
$$878$$ 24.0000i 0.809961i
$$879$$ 2.00000 0.0674583
$$880$$ 0 0
$$881$$ 22.0000 0.741199 0.370599 0.928793i $$-0.379152\pi$$
0.370599 + 0.928793i $$0.379152\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ − 28.0000i − 0.940148i −0.882627 0.470074i $$-0.844227\pi$$
0.882627 0.470074i $$-0.155773\pi$$
$$888$$ − 2.00000i − 0.0671156i
$$889$$ −72.0000 −2.41480
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ − 10.0000i − 0.334825i
$$893$$ 0 0
$$894$$ −2.00000 −0.0668900
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ − 12.0000i − 0.400668i
$$898$$ 6.00000i 0.200223i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 20.0000i 0.665927i
$$903$$ 16.0000i 0.532447i
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ −4.00000 −0.132891
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ 20.0000i 0.663723i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 0 0
$$913$$ − 16.0000i − 0.529523i
$$914$$ 8.00000 0.264616
$$915$$ 0 0
$$916$$ −20.0000 −0.660819
$$917$$ 16.0000i 0.528367i
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 0 0
$$923$$ 32.0000i 1.05329i
$$924$$ −8.00000 −0.263181
$$925$$ 0 0
$$926$$ 18.0000 0.591517
$$927$$ − 12.0000i − 0.394132i
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 22.0000i − 0.720634i
$$933$$ 0 0
$$934$$ −20.0000 −0.654420
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 30.0000i 0.980057i 0.871706 + 0.490029i $$0.163014\pi$$
−0.871706 + 0.490029i $$0.836986\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ −28.0000 −0.913745
$$940$$ 0 0
$$941$$ 28.0000 0.912774 0.456387 0.889781i $$-0.349143\pi$$
0.456387 + 0.889781i $$0.349143\pi$$
$$942$$ − 18.0000i − 0.586472i
$$943$$ − 60.0000i − 1.95387i
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ − 36.0000i − 1.16984i −0.811090 0.584921i $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ − 4.00000i − 0.129914i
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ 12.0000i 0.388718i 0.980930 + 0.194359i $$0.0622627\pi$$
−0.980930 + 0.194359i $$0.937737\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ 16.0000i 0.516937i
$$959$$ 48.0000 1.55000
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 4.00000i 0.128965i
$$963$$ − 12.0000i − 0.386695i
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ 24.0000 0.772187
$$967$$ 26.0000i 0.836104i 0.908423 + 0.418052i $$0.137287\pi$$
−0.908423 + 0.418052i $$0.862713\pi$$
$$968$$ − 7.00000i − 0.224989i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 24.0000i 0.769405i
$$974$$ 10.0000 0.320421
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ − 12.0000i − 0.383718i
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ − 6.00000i − 0.191468i
$$983$$ − 10.0000i − 0.318950i −0.987202 0.159475i $$-0.949020\pi$$
0.987202 0.159475i $$-0.0509802\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 16.0000i 0.509286i
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 28.0000 0.889449 0.444725 0.895667i $$-0.353302\pi$$
0.444725 + 0.895667i $$0.353302\pi$$
$$992$$ − 1.00000i − 0.0317500i
$$993$$ − 18.0000i − 0.571213i
$$994$$ −64.0000 −2.02996
$$995$$ 0 0
$$996$$ 8.00000 0.253490
$$997$$ 6.00000i 0.190022i 0.995476 + 0.0950110i $$0.0302886\pi$$
−0.995476 + 0.0950110i $$0.969711\pi$$
$$998$$ 22.0000i 0.696398i
$$999$$ 2.00000 0.0632772
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 4650.2.d.h.3349.1 2
5.2 odd 4 4650.2.a.x.1.1 1
5.3 odd 4 930.2.a.j.1.1 1
5.4 even 2 inner 4650.2.d.h.3349.2 2
15.8 even 4 2790.2.a.v.1.1 1
20.3 even 4 7440.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.j.1.1 1 5.3 odd 4
2790.2.a.v.1.1 1 15.8 even 4
4650.2.a.x.1.1 1 5.2 odd 4
4650.2.d.h.3349.1 2 1.1 even 1 trivial
4650.2.d.h.3349.2 2 5.4 even 2 inner
7440.2.a.g.1.1 1 20.3 even 4