# Properties

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +4.00000 q^{21} +2.00000i q^{22} -6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +1.00000 q^{31} +1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} +2.00000i q^{37} -2.00000 q^{39} -10.0000 q^{41} +4.00000i q^{42} -4.00000i q^{43} -2.00000 q^{44} +6.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} +4.00000 q^{59} +1.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -4.00000i q^{67} +6.00000 q^{69} -16.0000 q^{71} +1.00000i q^{72} +4.00000i q^{73} -2.00000 q^{74} -8.00000i q^{77} -2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +8.00000i q^{83} -4.00000 q^{84} +4.00000 q^{86} -2.00000i q^{88} -6.00000 q^{89} +8.00000 q^{91} +6.00000i q^{92} +1.00000i q^{93} +4.00000 q^{94} -1.00000 q^{96} -14.0000i q^{97} -9.00000i q^{98} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 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 $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ − 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 $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 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 $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ 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 $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 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 $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ − 4.00000i − 0.583460i −0.956501 0.291730i $$-0.905769\pi$$
0.956501 0.291730i $$-0.0942309\pi$$
$$48$$ 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 $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 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 $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 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 $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ −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 $$0.144687\pi$$
−0.898459 + 0.439057i $$0.855313\pi$$
$$84$$ −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 $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ − 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 $$-0.798655\pi$$
0.806527 0.591198i $$-0.201345\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 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 $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ 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 $$-0.705563\pi$$
0.601834 0.798621i $$-0.294437\pi$$
$$128$$ − 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 $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 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 $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ − 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 $$-0.844270\pi$$
0.882690 0.469956i $$-0.155730\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ − 14.0000i − 1.08335i −0.840587 0.541676i $$-0.817790\pi$$
0.840587 0.541676i $$-0.182210\pi$$
$$168$$ − 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 $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 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 $$-0.882808\pi$$
0.932988 0.359908i $$-0.117192\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ − 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 $$-0.891323\pi$$
0.942280 0.334825i $$-0.108677\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 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 $$-0.743851\pi$$
0.693316 0.720634i $$-0.256149\pi$$
$$234$$ 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 $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 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 $$0.0997645\pi$$
−0.951285 + 0.308313i $$0.900236\pi$$
$$264$$ 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 $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ 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 $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 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 $$-0.981394\pi$$
0.998292 0.0584206i $$-0.0186065\pi$$
$$294$$ 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 $$0.294649\pi$$
−0.601302 + 0.799022i $$0.705351\pi$$
$$308$$ 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 $$0.290609\pi$$
−0.611393 + 0.791327i $$0.709391\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ − 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 $$0.336901\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 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 $$-0.777193\pi$$
0.764862 0.644194i $$-0.222807\pi$$
$$348$$ 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 $$0.407467\pi$$
−0.286623 + 0.958043i $$0.592533\pi$$
$$354$$ −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 $$-0.762589\pi$$
0.734513 0.678594i $$-0.237411\pi$$
$$368$$ − 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 $$0.0496454\pi$$
−0.987862 + 0.155334i $$0.950355\pi$$
$$374$$ 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 $$0.0489878\pi$$
−0.988181 + 0.153293i $$0.951012\pi$$
$$384$$ 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 $$-0.850819\pi$$
0.892171 0.451697i $$-0.149181\pi$$
$$398$$ − 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 $$-0.840430\pi$$
0.876957 0.480569i $$-0.159570\pi$$
$$434$$ 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 $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 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 $$-0.940087\pi$$
0.982339 0.187112i $$-0.0599128\pi$$
$$458$$ 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 $$-0.862638\pi$$
0.908325 0.418265i $$-0.137362\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ 20.0000i 0.925490i 0.886492 + 0.462745i $$0.153135\pi$$
−0.886492 + 0.462745i $$0.846865\pi$$
$$468$$ 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 $$-0.927248\pi$$
0.973995 0.226572i $$-0.0727517\pi$$
$$488$$ 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 $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 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 $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ −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 $$-0.720443\pi$$
0.638497 0.769624i $$-0.279557\pi$$
$$548$$ − 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 $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ − 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 $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ −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 $$0.0398583\pi$$
−0.992170 + 0.124892i $$0.960142\pi$$
$$578$$ 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 $$0.164939\pi$$
−0.868726 + 0.495293i $$0.835061\pi$$
$$588$$ 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 $$-0.754026\pi$$
0.715994 0.698106i $$-0.245974\pi$$
$$594$$ 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 $$0.105267\pi$$
−0.945814 + 0.324710i $$0.894733\pi$$
$$608$$ 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 $$-0.677697\pi$$
0.529705 0.848182i $$-0.322303\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ −8.00000 −0.322329
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 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 $$0.334335\pi$$
−0.497271 + 0.867595i $$0.665665\pi$$
$$644$$ 24.0000 0.945732
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 18.0000i 0.707653i 0.935311 + 0.353827i $$0.115120\pi$$
−0.935311 + 0.353827i $$0.884880\pi$$
$$648$$ − 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 $$0.199689\pi$$
−0.809590 + 0.586995i $$0.800311\pi$$
$$654$$ −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 $$-0.719785\pi$$
0.636904 0.770943i $$-0.280215\pi$$
$$674$$ −32.0000 −1.23259
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 42.0000i − 1.61419i −0.590421 0.807096i $$-0.701038\pi$$
0.590421 0.807096i $$-0.298962\pi$$
$$678$$ − 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 $$-0.875015\pi$$
0.923898 0.382639i $$-0.124985\pi$$
$$684$$ 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 $$-0.650627\pi$$
0.455744 0.890111i $$-0.349373\pi$$
$$728$$ − 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 $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ 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 $$0.158248\pi$$
−0.878945 + 0.476924i $$0.841752\pi$$
$$744$$ 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 $$0.106075\pi$$
−0.944986 + 0.327111i $$0.893925\pi$$
$$758$$ 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 $$-0.988549\pi$$
0.999353 0.0359675i $$-0.0114513\pi$$
$$774$$ −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 $$0.252629\pi$$
−0.701242 + 0.712923i $$0.747371\pi$$
$$788$$ − 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 $$-0.764999\pi$$
0.739629 0.673015i $$-0.235001\pi$$
$$798$$ 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 $$0.101575\pi$$
−0.949515 + 0.313720i $$0.898425\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ 16.0000 0.556711
$$827$$ 56.0000i 1.94731i 0.228024 + 0.973655i $$0.426773\pi$$
−0.228024 + 0.973655i $$0.573227\pi$$
$$828$$ − 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 $$0.0327539\pi$$
−0.994711 + 0.102718i $$0.967246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 46.0000i 1.57133i 0.618652 + 0.785665i $$0.287679\pi$$
−0.618652 + 0.785665i $$0.712321\pi$$
$$858$$ − 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 $$0.170579\pi$$
−0.859815 + 0.510606i $$0.829421\pi$$
$$864$$ 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 $$-0.901708\pi$$
0.952701 0.303908i $$-0.0982917\pi$$
$$878$$ − 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 $$-0.792874\pi$$
0.795656 0.605748i $$-0.207126\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 28.0000i 0.940148i 0.882627 + 0.470074i $$0.155773\pi$$
−0.882627 + 0.470074i $$0.844227\pi$$
$$888$$ 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 $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ − 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 $$-0.836986\pi$$
0.871706 0.490029i $$-0.163014\pi$$
$$938$$ − 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 $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 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 $$-0.937737\pi$$
0.980930 0.194359i $$-0.0622627\pi$$
$$954$$ 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 $$-0.862713\pi$$
0.908423 0.418052i $$-0.137287\pi$$
$$968$$ 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 $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ 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 $$0.0509802\pi$$
−0.987202 + 0.159475i $$0.949020\pi$$
$$984$$ −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 $$-0.969711\pi$$
0.995476 0.0950110i $$-0.0302886\pi$$
$$998$$ − 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.3349.2 2
5.2 odd 4 930.2.a.j.1.1 1
5.3 odd 4 4650.2.a.x.1.1 1
5.4 even 2 inner 4650.2.d.h.3349.1 2
15.2 even 4 2790.2.a.v.1.1 1
20.7 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 5.2 odd 4
2790.2.a.v.1.1 1 15.2 even 4
4650.2.a.x.1.1 1 5.3 odd 4
4650.2.d.h.3349.1 2 5.4 even 2 inner
4650.2.d.h.3349.2 2 1.1 even 1 trivial
7440.2.a.g.1.1 1 20.7 even 4