# Properties

 Label 3800.2.d.n.3649.3 Level $3800$ Weight $2$ Character 3800.3649 Analytic conductor $30.343$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.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: $$30.3431527681$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ x^6 - 2*x^5 + 3*x^4 - 6*x^3 + 6*x^2 - 8*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 760) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3649.3 Root $$0.264658 + 1.38923i$$ of defining polynomial Character $$\chi$$ $$=$$ 3800.3649 Dual form 3800.2.d.n.3649.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.470683i q^{3} -2.71982i q^{7} +2.77846 q^{9} +O(q^{10})$$ $$q-0.470683i q^{3} -2.71982i q^{7} +2.77846 q^{9} -5.55691 q^{11} +2.02760i q^{13} +3.77846i q^{17} -1.00000 q^{19} -1.28018 q^{21} -5.77846i q^{23} -2.71982i q^{27} +5.66119 q^{29} +7.55691 q^{31} +2.61555i q^{33} +3.75086i q^{37} +0.954357 q^{39} -12.6155 q^{41} -9.43965i q^{43} -11.1138i q^{47} -0.397442 q^{49} +1.77846 q^{51} -8.85170i q^{53} +0.470683i q^{57} -11.4526 q^{59} -10.6155 q^{61} -7.55691i q^{63} +11.5845i q^{67} -2.71982 q^{69} -9.45264i q^{73} +15.1138i q^{77} -8.94137 q^{79} +7.05520 q^{81} -4.94137i q^{83} -2.66463i q^{87} -15.4948 q^{89} +5.51471 q^{91} -3.55691i q^{93} -10.8647i q^{97} -15.4396 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100})$$ 6 * q - 6 * q^19 - 26 * q^21 + 14 * q^29 + 12 * q^31 - 6 * q^39 - 44 * q^41 - 24 * q^49 - 6 * q^51 - 22 * q^59 - 32 * q^61 + 2 * q^69 - 52 * q^79 - 26 * q^81 + 12 * q^89 + 58 * q^91 - 56 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 0.470683i − 0.271749i −0.990726 0.135875i $$-0.956616\pi$$
0.990726 0.135875i $$-0.0433844\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.71982i − 1.02800i −0.857791 0.513998i $$-0.828164\pi$$
0.857791 0.513998i $$-0.171836\pi$$
$$8$$ 0 0
$$9$$ 2.77846 0.926152
$$10$$ 0 0
$$11$$ −5.55691 −1.67547 −0.837736 0.546075i $$-0.816121\pi$$
−0.837736 + 0.546075i $$0.816121\pi$$
$$12$$ 0 0
$$13$$ 2.02760i 0.562354i 0.959656 + 0.281177i $$0.0907249\pi$$
−0.959656 + 0.281177i $$0.909275\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.77846i 0.916410i 0.888846 + 0.458205i $$0.151508\pi$$
−0.888846 + 0.458205i $$0.848492\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.28018 −0.279357
$$22$$ 0 0
$$23$$ − 5.77846i − 1.20489i −0.798160 0.602446i $$-0.794193\pi$$
0.798160 0.602446i $$-0.205807\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 2.71982i − 0.523430i
$$28$$ 0 0
$$29$$ 5.66119 1.05126 0.525628 0.850714i $$-0.323830\pi$$
0.525628 + 0.850714i $$0.323830\pi$$
$$30$$ 0 0
$$31$$ 7.55691 1.35726 0.678631 0.734479i $$-0.262574\pi$$
0.678631 + 0.734479i $$0.262574\pi$$
$$32$$ 0 0
$$33$$ 2.61555i 0.455308i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.75086i 0.616637i 0.951283 + 0.308319i $$0.0997663\pi$$
−0.951283 + 0.308319i $$0.900234\pi$$
$$38$$ 0 0
$$39$$ 0.954357 0.152819
$$40$$ 0 0
$$41$$ −12.6155 −1.97022 −0.985109 0.171932i $$-0.944999\pi$$
−0.985109 + 0.171932i $$0.944999\pi$$
$$42$$ 0 0
$$43$$ − 9.43965i − 1.43953i −0.694216 0.719766i $$-0.744249\pi$$
0.694216 0.719766i $$-0.255751\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 11.1138i − 1.62112i −0.585657 0.810559i $$-0.699163\pi$$
0.585657 0.810559i $$-0.300837\pi$$
$$48$$ 0 0
$$49$$ −0.397442 −0.0567775
$$50$$ 0 0
$$51$$ 1.77846 0.249034
$$52$$ 0 0
$$53$$ − 8.85170i − 1.21587i −0.793985 0.607937i $$-0.791997\pi$$
0.793985 0.607937i $$-0.208003\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.470683i 0.0623435i
$$58$$ 0 0
$$59$$ −11.4526 −1.49101 −0.745503 0.666502i $$-0.767791\pi$$
−0.745503 + 0.666502i $$0.767791\pi$$
$$60$$ 0 0
$$61$$ −10.6155 −1.35918 −0.679591 0.733591i $$-0.737843\pi$$
−0.679591 + 0.733591i $$0.737843\pi$$
$$62$$ 0 0
$$63$$ − 7.55691i − 0.952082i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.5845i 1.41527i 0.706576 + 0.707637i $$0.250239\pi$$
−0.706576 + 0.707637i $$0.749761\pi$$
$$68$$ 0 0
$$69$$ −2.71982 −0.327428
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ − 9.45264i − 1.10635i −0.833066 0.553174i $$-0.813417\pi$$
0.833066 0.553174i $$-0.186583\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 15.1138i 1.72238i
$$78$$ 0 0
$$79$$ −8.94137 −1.00598 −0.502991 0.864292i $$-0.667767\pi$$
−0.502991 + 0.864292i $$0.667767\pi$$
$$80$$ 0 0
$$81$$ 7.05520 0.783911
$$82$$ 0 0
$$83$$ − 4.94137i − 0.542385i −0.962525 0.271193i $$-0.912582\pi$$
0.962525 0.271193i $$-0.0874181\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 2.66463i − 0.285678i
$$88$$ 0 0
$$89$$ −15.4948 −1.64245 −0.821225 0.570604i $$-0.806709\pi$$
−0.821225 + 0.570604i $$0.806709\pi$$
$$90$$ 0 0
$$91$$ 5.51471 0.578099
$$92$$ 0 0
$$93$$ − 3.55691i − 0.368835i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 10.8647i − 1.10314i −0.834128 0.551571i $$-0.814029\pi$$
0.834128 0.551571i $$-0.185971\pi$$
$$98$$ 0 0
$$99$$ −15.4396 −1.55174
$$100$$ 0 0
$$101$$ 4.49828 0.447596 0.223798 0.974636i $$-0.428154\pi$$
0.223798 + 0.974636i $$0.428154\pi$$
$$102$$ 0 0
$$103$$ 4.36641i 0.430235i 0.976588 + 0.215117i $$0.0690134\pi$$
−0.976588 + 0.215117i $$0.930987\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.64658i 0.159181i 0.996828 + 0.0795906i $$0.0253613\pi$$
−0.996828 + 0.0795906i $$0.974639\pi$$
$$108$$ 0 0
$$109$$ 0.954357 0.0914108 0.0457054 0.998955i $$-0.485446\pi$$
0.0457054 + 0.998955i $$0.485446\pi$$
$$110$$ 0 0
$$111$$ 1.76547 0.167571
$$112$$ 0 0
$$113$$ 5.68879i 0.535156i 0.963536 + 0.267578i $$0.0862233\pi$$
−0.963536 + 0.267578i $$0.913777\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 5.63359i 0.520826i
$$118$$ 0 0
$$119$$ 10.2767 0.942067
$$120$$ 0 0
$$121$$ 19.8793 1.80721
$$122$$ 0 0
$$123$$ 5.93793i 0.535405i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.3043i 0.914362i 0.889374 + 0.457181i $$0.151141\pi$$
−0.889374 + 0.457181i $$0.848859\pi$$
$$128$$ 0 0
$$129$$ −4.44309 −0.391192
$$130$$ 0 0
$$131$$ 3.11383 0.272056 0.136028 0.990705i $$-0.456566\pi$$
0.136028 + 0.990705i $$0.456566\pi$$
$$132$$ 0 0
$$133$$ 2.71982i 0.235839i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 13.4526i − 1.14934i −0.818386 0.574668i $$-0.805131\pi$$
0.818386 0.574668i $$-0.194869\pi$$
$$138$$ 0 0
$$139$$ −9.55691 −0.810607 −0.405303 0.914182i $$-0.632834\pi$$
−0.405303 + 0.914182i $$0.632834\pi$$
$$140$$ 0 0
$$141$$ −5.23109 −0.440538
$$142$$ 0 0
$$143$$ − 11.2672i − 0.942209i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0.187070i 0.0154292i
$$148$$ 0 0
$$149$$ −19.6121 −1.60669 −0.803343 0.595516i $$-0.796948\pi$$
−0.803343 + 0.595516i $$0.796948\pi$$
$$150$$ 0 0
$$151$$ −17.0518 −1.38765 −0.693826 0.720143i $$-0.744076\pi$$
−0.693826 + 0.720143i $$0.744076\pi$$
$$152$$ 0 0
$$153$$ 10.4983i 0.848736i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.4983i 1.79556i 0.440446 + 0.897779i $$0.354820\pi$$
−0.440446 + 0.897779i $$0.645180\pi$$
$$158$$ 0 0
$$159$$ −4.16635 −0.330413
$$160$$ 0 0
$$161$$ −15.7164 −1.23862
$$162$$ 0 0
$$163$$ 22.4362i 1.75734i 0.477430 + 0.878670i $$0.341568\pi$$
−0.477430 + 0.878670i $$0.658432\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.69223i 0.517860i 0.965896 + 0.258930i $$0.0833699\pi$$
−0.965896 + 0.258930i $$0.916630\pi$$
$$168$$ 0 0
$$169$$ 8.88885 0.683758
$$170$$ 0 0
$$171$$ −2.77846 −0.212474
$$172$$ 0 0
$$173$$ − 2.86469i − 0.217798i −0.994053 0.108899i $$-0.965267\pi$$
0.994053 0.108899i $$-0.0347325\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 5.39057i 0.405180i
$$178$$ 0 0
$$179$$ −1.38445 −0.103479 −0.0517394 0.998661i $$-0.516477\pi$$
−0.0517394 + 0.998661i $$0.516477\pi$$
$$180$$ 0 0
$$181$$ −20.8793 −1.55195 −0.775973 0.630766i $$-0.782741\pi$$
−0.775973 + 0.630766i $$0.782741\pi$$
$$182$$ 0 0
$$183$$ 4.99656i 0.369357i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 20.9966i − 1.53542i
$$188$$ 0 0
$$189$$ −7.39744 −0.538085
$$190$$ 0 0
$$191$$ 19.2733 1.39457 0.697284 0.716795i $$-0.254392\pi$$
0.697284 + 0.716795i $$0.254392\pi$$
$$192$$ 0 0
$$193$$ − 8.01461i − 0.576904i −0.957494 0.288452i $$-0.906859\pi$$
0.957494 0.288452i $$-0.0931406\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 16.8793i − 1.20260i −0.799023 0.601300i $$-0.794650\pi$$
0.799023 0.601300i $$-0.205350\pi$$
$$198$$ 0 0
$$199$$ −1.28018 −0.0907493 −0.0453746 0.998970i $$-0.514448\pi$$
−0.0453746 + 0.998970i $$0.514448\pi$$
$$200$$ 0 0
$$201$$ 5.45264 0.384599
$$202$$ 0 0
$$203$$ − 15.3974i − 1.08069i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 16.0552i − 1.11591i
$$208$$ 0 0
$$209$$ 5.55691 0.384380
$$210$$ 0 0
$$211$$ −1.54392 −0.106288 −0.0531441 0.998587i $$-0.516924\pi$$
−0.0531441 + 0.998587i $$0.516924\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 20.5535i − 1.39526i
$$218$$ 0 0
$$219$$ −4.44920 −0.300649
$$220$$ 0 0
$$221$$ −7.66119 −0.515347
$$222$$ 0 0
$$223$$ − 3.92332i − 0.262725i −0.991334 0.131363i $$-0.958065\pi$$
0.991334 0.131363i $$-0.0419352\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 10.9069i − 0.723916i −0.932194 0.361958i $$-0.882108\pi$$
0.932194 0.361958i $$-0.117892\pi$$
$$228$$ 0 0
$$229$$ −10.1725 −0.672215 −0.336108 0.941824i $$-0.609111\pi$$
−0.336108 + 0.941824i $$0.609111\pi$$
$$230$$ 0 0
$$231$$ 7.11383 0.468056
$$232$$ 0 0
$$233$$ − 9.11383i − 0.597067i −0.954399 0.298533i $$-0.903503\pi$$
0.954399 0.298533i $$-0.0964974\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 4.20855i 0.273375i
$$238$$ 0 0
$$239$$ −20.1595 −1.30401 −0.652004 0.758216i $$-0.726071\pi$$
−0.652004 + 0.758216i $$0.726071\pi$$
$$240$$ 0 0
$$241$$ −4.82410 −0.310748 −0.155374 0.987856i $$-0.549658\pi$$
−0.155374 + 0.987856i $$0.549658\pi$$
$$242$$ 0 0
$$243$$ − 11.4802i − 0.736457i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 2.02760i − 0.129013i
$$248$$ 0 0
$$249$$ −2.32582 −0.147393
$$250$$ 0 0
$$251$$ −18.2277 −1.15052 −0.575260 0.817971i $$-0.695099\pi$$
−0.575260 + 0.817971i $$0.695099\pi$$
$$252$$ 0 0
$$253$$ 32.1104i 2.01876i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 28.5941i − 1.78365i −0.452382 0.891824i $$-0.649426\pi$$
0.452382 0.891824i $$-0.350574\pi$$
$$258$$ 0 0
$$259$$ 10.2017 0.633901
$$260$$ 0 0
$$261$$ 15.7294 0.973624
$$262$$ 0 0
$$263$$ − 27.3776i − 1.68817i −0.536206 0.844087i $$-0.680143\pi$$
0.536206 0.844087i $$-0.319857\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 7.29317i 0.446334i
$$268$$ 0 0
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ 15.8337 0.961826 0.480913 0.876768i $$-0.340305\pi$$
0.480913 + 0.876768i $$0.340305\pi$$
$$272$$ 0 0
$$273$$ − 2.59568i − 0.157098i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6.70683i 0.402975i 0.979491 + 0.201487i $$0.0645775\pi$$
−0.979491 + 0.201487i $$0.935423\pi$$
$$278$$ 0 0
$$279$$ 20.9966 1.25703
$$280$$ 0 0
$$281$$ 8.38101 0.499969 0.249985 0.968250i $$-0.419574\pi$$
0.249985 + 0.968250i $$0.419574\pi$$
$$282$$ 0 0
$$283$$ − 20.2897i − 1.20610i −0.797704 0.603050i $$-0.793952\pi$$
0.797704 0.603050i $$-0.206048\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 34.3121i 2.02538i
$$288$$ 0 0
$$289$$ 2.72326 0.160192
$$290$$ 0 0
$$291$$ −5.11383 −0.299778
$$292$$ 0 0
$$293$$ − 6.76041i − 0.394947i −0.980308 0.197474i $$-0.936726\pi$$
0.980308 0.197474i $$-0.0632737\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 15.1138i 0.876993i
$$298$$ 0 0
$$299$$ 11.7164 0.677576
$$300$$ 0 0
$$301$$ −25.6742 −1.47984
$$302$$ 0 0
$$303$$ − 2.11727i − 0.121634i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 7.74398i − 0.441972i −0.975277 0.220986i $$-0.929072\pi$$
0.975277 0.220986i $$-0.0709276\pi$$
$$308$$ 0 0
$$309$$ 2.05520 0.116916
$$310$$ 0 0
$$311$$ −11.0456 −0.626341 −0.313170 0.949697i $$-0.601391\pi$$
−0.313170 + 0.949697i $$0.601391\pi$$
$$312$$ 0 0
$$313$$ 1.16291i 0.0657315i 0.999460 + 0.0328658i $$0.0104634\pi$$
−0.999460 + 0.0328658i $$0.989537\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 21.8742i 1.22858i 0.789080 + 0.614290i $$0.210557\pi$$
−0.789080 + 0.614290i $$0.789443\pi$$
$$318$$ 0 0
$$319$$ −31.4588 −1.76135
$$320$$ 0 0
$$321$$ 0.775019 0.0432574
$$322$$ 0 0
$$323$$ − 3.77846i − 0.210239i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 0.449200i − 0.0248408i
$$328$$ 0 0
$$329$$ −30.2277 −1.66650
$$330$$ 0 0
$$331$$ 14.6646 0.806041 0.403020 0.915191i $$-0.367960\pi$$
0.403020 + 0.915191i $$0.367960\pi$$
$$332$$ 0 0
$$333$$ 10.4216i 0.571100i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 20.0958i 1.09469i 0.836908 + 0.547344i $$0.184361\pi$$
−0.836908 + 0.547344i $$0.815639\pi$$
$$338$$ 0 0
$$339$$ 2.67762 0.145428
$$340$$ 0 0
$$341$$ −41.9931 −2.27406
$$342$$ 0 0
$$343$$ − 17.9578i − 0.969630i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 11.0586i 0.593659i 0.954931 + 0.296829i $$0.0959292\pi$$
−0.954931 + 0.296829i $$0.904071\pi$$
$$348$$ 0 0
$$349$$ −8.11727 −0.434507 −0.217254 0.976115i $$-0.569710\pi$$
−0.217254 + 0.976115i $$0.569710\pi$$
$$350$$ 0 0
$$351$$ 5.51471 0.294353
$$352$$ 0 0
$$353$$ 12.0130i 0.639387i 0.947521 + 0.319693i $$0.103580\pi$$
−0.947521 + 0.319693i $$0.896420\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 4.83709i − 0.256006i
$$358$$ 0 0
$$359$$ 17.9509 0.947413 0.473707 0.880683i $$-0.342916\pi$$
0.473707 + 0.880683i $$0.342916\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ − 9.35685i − 0.491108i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3.11383i 0.162541i 0.996692 + 0.0812703i $$0.0258977\pi$$
−0.996692 + 0.0812703i $$0.974102\pi$$
$$368$$ 0 0
$$369$$ −35.0518 −1.82472
$$370$$ 0 0
$$371$$ −24.0751 −1.24991
$$372$$ 0 0
$$373$$ − 21.8742i − 1.13261i −0.824197 0.566303i $$-0.808373\pi$$
0.824197 0.566303i $$-0.191627\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 11.4786i 0.591179i
$$378$$ 0 0
$$379$$ 5.28018 0.271224 0.135612 0.990762i $$-0.456700\pi$$
0.135612 + 0.990762i $$0.456700\pi$$
$$380$$ 0 0
$$381$$ 4.85008 0.248477
$$382$$ 0 0
$$383$$ − 16.1319i − 0.824300i −0.911116 0.412150i $$-0.864778\pi$$
0.911116 0.412150i $$-0.135222\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 26.2277i − 1.33323i
$$388$$ 0 0
$$389$$ −3.43965 −0.174397 −0.0871985 0.996191i $$-0.527791\pi$$
−0.0871985 + 0.996191i $$0.527791\pi$$
$$390$$ 0 0
$$391$$ 21.8337 1.10418
$$392$$ 0 0
$$393$$ − 1.46563i − 0.0739311i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.29317i 0.0649021i 0.999473 + 0.0324511i $$0.0103313\pi$$
−0.999473 + 0.0324511i $$0.989669\pi$$
$$398$$ 0 0
$$399$$ 1.28018 0.0640890
$$400$$ 0 0
$$401$$ −35.1070 −1.75316 −0.876579 0.481258i $$-0.840180\pi$$
−0.876579 + 0.481258i $$0.840180\pi$$
$$402$$ 0 0
$$403$$ 15.3224i 0.763262i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 20.8432i − 1.03316i
$$408$$ 0 0
$$409$$ 24.8793 1.23020 0.615101 0.788448i $$-0.289115\pi$$
0.615101 + 0.788448i $$0.289115\pi$$
$$410$$ 0 0
$$411$$ −6.33193 −0.312331
$$412$$ 0 0
$$413$$ 31.1492i 1.53275i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.49828i 0.220282i
$$418$$ 0 0
$$419$$ 35.4328 1.73100 0.865502 0.500905i $$-0.167000\pi$$
0.865502 + 0.500905i $$0.167000\pi$$
$$420$$ 0 0
$$421$$ 14.2147 0.692780 0.346390 0.938091i $$-0.387407\pi$$
0.346390 + 0.938091i $$0.387407\pi$$
$$422$$ 0 0
$$423$$ − 30.8793i − 1.50140i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 28.8724i 1.39723i
$$428$$ 0 0
$$429$$ −5.30328 −0.256045
$$430$$ 0 0
$$431$$ −24.4983 −1.18004 −0.590020 0.807388i $$-0.700880\pi$$
−0.590020 + 0.807388i $$0.700880\pi$$
$$432$$ 0 0
$$433$$ 9.45426i 0.454343i 0.973855 + 0.227171i $$0.0729477\pi$$
−0.973855 + 0.227171i $$0.927052\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 5.77846i 0.276421i
$$438$$ 0 0
$$439$$ −9.70340 −0.463118 −0.231559 0.972821i $$-0.574383\pi$$
−0.231559 + 0.972821i $$0.574383\pi$$
$$440$$ 0 0
$$441$$ −1.10428 −0.0525846
$$442$$ 0 0
$$443$$ − 6.85008i − 0.325457i −0.986671 0.162729i $$-0.947971\pi$$
0.986671 0.162729i $$-0.0520295\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 9.23109i 0.436616i
$$448$$ 0 0
$$449$$ 25.7655 1.21595 0.607974 0.793957i $$-0.291983\pi$$
0.607974 + 0.793957i $$0.291983\pi$$
$$450$$ 0 0
$$451$$ 70.1035 3.30105
$$452$$ 0 0
$$453$$ 8.02598i 0.377093i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 0.400880i − 0.0187524i −0.999956 0.00937619i $$-0.997015\pi$$
0.999956 0.00937619i $$-0.00298458\pi$$
$$458$$ 0 0
$$459$$ 10.2767 0.479677
$$460$$ 0 0
$$461$$ 23.1070 1.07620 0.538099 0.842882i $$-0.319143\pi$$
0.538099 + 0.842882i $$0.319143\pi$$
$$462$$ 0 0
$$463$$ 4.96735i 0.230852i 0.993316 + 0.115426i $$0.0368233\pi$$
−0.993316 + 0.115426i $$0.963177\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 21.6190i − 1.00041i −0.865908 0.500204i $$-0.833258\pi$$
0.865908 0.500204i $$-0.166742\pi$$
$$468$$ 0 0
$$469$$ 31.5078 1.45490
$$470$$ 0 0
$$471$$ 10.5896 0.487942
$$472$$ 0 0
$$473$$ 52.4553i 2.41190i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 24.5941i − 1.12608i
$$478$$ 0 0
$$479$$ −2.20855 −0.100911 −0.0504557 0.998726i $$-0.516067\pi$$
−0.0504557 + 0.998726i $$0.516067\pi$$
$$480$$ 0 0
$$481$$ −7.60523 −0.346769
$$482$$ 0 0
$$483$$ 7.39744i 0.336595i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 11.4250i 0.517718i 0.965915 + 0.258859i $$0.0833465\pi$$
−0.965915 + 0.258859i $$0.916654\pi$$
$$488$$ 0 0
$$489$$ 10.5604 0.477556
$$490$$ 0 0
$$491$$ 23.6673 1.06809 0.534045 0.845456i $$-0.320671\pi$$
0.534045 + 0.845456i $$0.320671\pi$$
$$492$$ 0 0
$$493$$ 21.3906i 0.963383i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −3.99312 −0.178757 −0.0893784 0.995998i $$-0.528488\pi$$
−0.0893784 + 0.995998i $$0.528488\pi$$
$$500$$ 0 0
$$501$$ 3.14992 0.140728
$$502$$ 0 0
$$503$$ 0.338809i 0.0151068i 0.999971 + 0.00755338i $$0.00240434\pi$$
−0.999971 + 0.00755338i $$0.997596\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 4.18383i − 0.185811i
$$508$$ 0 0
$$509$$ 28.4914 1.26286 0.631430 0.775433i $$-0.282468\pi$$
0.631430 + 0.775433i $$0.282468\pi$$
$$510$$ 0 0
$$511$$ −25.7095 −1.13732
$$512$$ 0 0
$$513$$ 2.71982i 0.120083i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 61.7586i 2.71614i
$$518$$ 0 0
$$519$$ −1.34836 −0.0591865
$$520$$ 0 0
$$521$$ 41.4328 1.81520 0.907601 0.419833i $$-0.137911\pi$$
0.907601 + 0.419833i $$0.137911\pi$$
$$522$$ 0 0
$$523$$ − 22.1741i − 0.969605i −0.874624 0.484802i $$-0.838892\pi$$
0.874624 0.484802i $$-0.161108\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 28.5535i 1.24381i
$$528$$ 0 0
$$529$$ −10.3906 −0.451764
$$530$$ 0 0
$$531$$ −31.8207 −1.38090
$$532$$ 0 0
$$533$$ − 25.5793i − 1.10796i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0.651639i 0.0281203i
$$538$$ 0 0
$$539$$ 2.20855 0.0951291
$$540$$ 0 0
$$541$$ −7.61211 −0.327270 −0.163635 0.986521i $$-0.552322\pi$$
−0.163635 + 0.986521i $$0.552322\pi$$
$$542$$ 0 0
$$543$$ 9.82754i 0.421740i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 18.3303i 0.783748i 0.920019 + 0.391874i $$0.128173\pi$$
−0.920019 + 0.391874i $$0.871827\pi$$
$$548$$ 0 0
$$549$$ −29.4948 −1.25881
$$550$$ 0 0
$$551$$ −5.66119 −0.241175
$$552$$ 0 0
$$553$$ 24.3189i 1.03415i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 36.7000i − 1.55503i −0.628866 0.777514i $$-0.716481\pi$$
0.628866 0.777514i $$-0.283519\pi$$
$$558$$ 0 0
$$559$$ 19.1398 0.809528
$$560$$ 0 0
$$561$$ −9.88273 −0.417249
$$562$$ 0 0
$$563$$ − 16.1319i − 0.679877i −0.940448 0.339939i $$-0.889594\pi$$
0.940448 0.339939i $$-0.110406\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 19.1889i − 0.805858i
$$568$$ 0 0
$$569$$ 19.2051 0.805120 0.402560 0.915394i $$-0.368120\pi$$
0.402560 + 0.915394i $$0.368120\pi$$
$$570$$ 0 0
$$571$$ −2.46907 −0.103327 −0.0516636 0.998665i $$-0.516452\pi$$
−0.0516636 + 0.998665i $$0.516452\pi$$
$$572$$ 0 0
$$573$$ − 9.07162i − 0.378972i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 11.7233i − 0.488046i −0.969769 0.244023i $$-0.921533\pi$$
0.969769 0.244023i $$-0.0784672\pi$$
$$578$$ 0 0
$$579$$ −3.77234 −0.156773
$$580$$ 0 0
$$581$$ −13.4396 −0.557571
$$582$$ 0 0
$$583$$ 49.1881i 2.03716i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8.28973i 0.342154i 0.985258 + 0.171077i $$0.0547246\pi$$
−0.985258 + 0.171077i $$0.945275\pi$$
$$588$$ 0 0
$$589$$ −7.55691 −0.311377
$$590$$ 0 0
$$591$$ −7.94480 −0.326806
$$592$$ 0 0
$$593$$ − 31.5760i − 1.29667i −0.761354 0.648336i $$-0.775465\pi$$
0.761354 0.648336i $$-0.224535\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0.602558i 0.0246610i
$$598$$ 0 0
$$599$$ −1.28629 −0.0525564 −0.0262782 0.999655i $$-0.508366\pi$$
−0.0262782 + 0.999655i $$0.508366\pi$$
$$600$$ 0 0
$$601$$ 38.8432 1.58445 0.792224 0.610231i $$-0.208923\pi$$
0.792224 + 0.610231i $$0.208923\pi$$
$$602$$ 0 0
$$603$$ 32.1871i 1.31076i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 43.6888i − 1.77327i −0.462467 0.886637i $$-0.653036\pi$$
0.462467 0.886637i $$-0.346964\pi$$
$$608$$ 0 0
$$609$$ −7.24732 −0.293676
$$610$$ 0 0
$$611$$ 22.5344 0.911643
$$612$$ 0 0
$$613$$ − 30.8172i − 1.24470i −0.782741 0.622348i $$-0.786179\pi$$
0.782741 0.622348i $$-0.213821\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 10.4983i − 0.422645i −0.977416 0.211322i $$-0.932223\pi$$
0.977416 0.211322i $$-0.0677770\pi$$
$$618$$ 0 0
$$619$$ −13.3224 −0.535472 −0.267736 0.963492i $$-0.586275\pi$$
−0.267736 + 0.963492i $$0.586275\pi$$
$$620$$ 0 0
$$621$$ −15.7164 −0.630677
$$622$$ 0 0
$$623$$ 42.1432i 1.68843i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 2.61555i − 0.104455i
$$628$$ 0 0
$$629$$ −14.1725 −0.565093
$$630$$ 0 0
$$631$$ 1.21199 0.0482486 0.0241243 0.999709i $$-0.492320\pi$$
0.0241243 + 0.999709i $$0.492320\pi$$
$$632$$ 0 0
$$633$$ 0.726700i 0.0288837i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 0.805853i − 0.0319291i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −49.6965 −1.96289 −0.981447 0.191732i $$-0.938589\pi$$
−0.981447 + 0.191732i $$0.938589\pi$$
$$642$$ 0 0
$$643$$ − 39.0449i − 1.53978i −0.638177 0.769890i $$-0.720311\pi$$
0.638177 0.769890i $$-0.279689\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.4232i 1.11743i 0.829359 + 0.558716i $$0.188706\pi$$
−0.829359 + 0.558716i $$0.811294\pi$$
$$648$$ 0 0
$$649$$ 63.6413 2.49814
$$650$$ 0 0
$$651$$ −9.67418 −0.379161
$$652$$ 0 0
$$653$$ 29.0586i 1.13715i 0.822631 + 0.568576i $$0.192506\pi$$
−0.822631 + 0.568576i $$0.807494\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 26.2637i − 1.02465i
$$658$$ 0 0
$$659$$ −11.8957 −0.463392 −0.231696 0.972788i $$-0.574427\pi$$
−0.231696 + 0.972788i $$0.574427\pi$$
$$660$$ 0 0
$$661$$ 30.8923 1.20157 0.600785 0.799410i $$-0.294855\pi$$
0.600785 + 0.799410i $$0.294855\pi$$
$$662$$ 0 0
$$663$$ 3.60600i 0.140045i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 32.7129i − 1.26665i
$$668$$ 0 0
$$669$$ −1.84664 −0.0713953
$$670$$ 0 0
$$671$$ 58.9897 2.27727
$$672$$ 0 0
$$673$$ 37.5354i 1.44688i 0.690385 + 0.723442i $$0.257441\pi$$
−0.690385 + 0.723442i $$0.742559\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 31.6466i 1.21628i 0.793831 + 0.608138i $$0.208083\pi$$
−0.793831 + 0.608138i $$0.791917\pi$$
$$678$$ 0 0
$$679$$ −29.5500 −1.13403
$$680$$ 0 0
$$681$$ −5.13369 −0.196724
$$682$$ 0 0
$$683$$ 22.6233i 0.865656i 0.901477 + 0.432828i $$0.142484\pi$$
−0.901477 + 0.432828i $$0.857516\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 4.78801i 0.182674i
$$688$$ 0 0
$$689$$ 17.9477 0.683752
$$690$$ 0 0
$$691$$ 17.7655 0.675830 0.337915 0.941177i $$-0.390278\pi$$
0.337915 + 0.941177i $$0.390278\pi$$
$$692$$ 0 0
$$693$$ 41.9931i 1.59519i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 47.6673i − 1.80553i
$$698$$ 0 0
$$699$$ −4.28973 −0.162252
$$700$$ 0 0
$$701$$ 25.6052 0.967096 0.483548 0.875318i $$-0.339348\pi$$
0.483548 + 0.875318i $$0.339348\pi$$
$$702$$ 0 0
$$703$$ − 3.75086i − 0.141466i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 12.2345i − 0.460127i
$$708$$ 0 0
$$709$$ −23.2311 −0.872462 −0.436231 0.899835i $$-0.643687\pi$$
−0.436231 + 0.899835i $$0.643687\pi$$
$$710$$ 0 0
$$711$$ −24.8432 −0.931693
$$712$$ 0 0
$$713$$ − 43.6673i − 1.63535i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 9.48873i 0.354363i
$$718$$ 0 0
$$719$$ −24.9544 −0.930640 −0.465320 0.885142i $$-0.654061\pi$$
−0.465320 + 0.885142i $$0.654061\pi$$
$$720$$ 0 0
$$721$$ 11.8759 0.442280
$$722$$ 0 0
$$723$$ 2.27062i 0.0844454i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 3.39744i 0.126004i 0.998013 + 0.0630021i $$0.0200675\pi$$
−0.998013 + 0.0630021i $$0.979933\pi$$
$$728$$ 0 0
$$729$$ 15.7620 0.583779
$$730$$ 0 0
$$731$$ 35.6673 1.31920
$$732$$ 0 0
$$733$$ − 9.66730i − 0.357070i −0.983934 0.178535i $$-0.942864\pi$$
0.983934 0.178535i $$-0.0571358\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 64.3741i − 2.37125i
$$738$$ 0 0
$$739$$ −17.0225 −0.626184 −0.313092 0.949723i $$-0.601365\pi$$
−0.313092 + 0.949723i $$0.601365\pi$$
$$740$$ 0 0
$$741$$ −0.954357 −0.0350592
$$742$$ 0 0
$$743$$ 8.57496i 0.314585i 0.987552 + 0.157292i $$0.0502765\pi$$
−0.987552 + 0.157292i $$0.949723\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 13.7294i − 0.502332i
$$748$$ 0 0
$$749$$ 4.47842 0.163638
$$750$$ 0 0
$$751$$ 12.8310 0.468209 0.234104 0.972211i $$-0.424784\pi$$
0.234104 + 0.972211i $$0.424784\pi$$
$$752$$ 0 0
$$753$$ 8.57946i 0.312653i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 35.8827i 1.30418i 0.758142 + 0.652090i $$0.226108\pi$$
−0.758142 + 0.652090i $$0.773892\pi$$
$$758$$ 0 0
$$759$$ 15.1138 0.548597
$$760$$ 0 0
$$761$$ −30.1234 −1.09197 −0.545986 0.837794i $$-0.683845\pi$$
−0.545986 + 0.837794i $$0.683845\pi$$
$$762$$ 0 0
$$763$$ − 2.59568i − 0.0939700i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 23.2213i − 0.838474i
$$768$$ 0 0
$$769$$ 6.22154 0.224355 0.112177 0.993688i $$-0.464218\pi$$
0.112177 + 0.993688i $$0.464218\pi$$
$$770$$ 0 0
$$771$$ −13.4588 −0.484705
$$772$$ 0 0
$$773$$ 10.2362i 0.368169i 0.982910 + 0.184084i $$0.0589320\pi$$
−0.982910 + 0.184084i $$0.941068\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 4.80176i − 0.172262i
$$778$$ 0 0
$$779$$ 12.6155 0.451999
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 15.3974i − 0.550260i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4.34654i 0.154937i 0.996995 + 0.0774687i $$0.0246838\pi$$
−0.996995 + 0.0774687i $$0.975316\pi$$
$$788$$ 0 0
$$789$$ −12.8862 −0.458760
$$790$$ 0 0
$$791$$ 15.4725 0.550139
$$792$$ 0 0
$$793$$ − 21.5241i − 0.764342i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 30.1932i 1.06950i 0.845011 + 0.534749i $$0.179594\pi$$
−0.845011 + 0.534749i $$0.820406\pi$$
$$798$$ 0 0
$$799$$ 41.9931 1.48561
$$800$$ 0 0
$$801$$ −43.0518 −1.52116
$$802$$ 0 0
$$803$$ 52.5275i 1.85366i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 6.58957i − 0.231964i
$$808$$ 0 0
$$809$$ 11.8077 0.415136 0.207568 0.978221i $$-0.433445\pi$$
0.207568 + 0.978221i $$0.433445\pi$$
$$810$$ 0 0
$$811$$ 0.811111 0.0284820 0.0142410 0.999899i $$-0.495467\pi$$
0.0142410 + 0.999899i $$0.495467\pi$$
$$812$$ 0 0
$$813$$ − 7.45264i − 0.261375i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 9.43965i 0.330251i
$$818$$ 0 0
$$819$$ 15.3224 0.535407
$$820$$ 0 0
$$821$$ 40.6707 1.41942 0.709709 0.704495i $$-0.248826\pi$$
0.709709 + 0.704495i $$0.248826\pi$$
$$822$$ 0 0
$$823$$ − 7.48185i − 0.260801i −0.991461 0.130401i $$-0.958374\pi$$
0.991461 0.130401i $$-0.0416263\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 46.4346i 1.61469i 0.590080 + 0.807344i $$0.299096\pi$$
−0.590080 + 0.807344i $$0.700904\pi$$
$$828$$ 0 0
$$829$$ 10.7267 0.372554 0.186277 0.982497i $$-0.440358\pi$$
0.186277 + 0.982497i $$0.440358\pi$$
$$830$$ 0 0
$$831$$ 3.15680 0.109508
$$832$$ 0 0
$$833$$ − 1.50172i − 0.0520315i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 20.5535i − 0.710432i
$$838$$ 0 0
$$839$$ −10.1465 −0.350295 −0.175148 0.984542i $$-0.556040\pi$$
−0.175148 + 0.984542i $$0.556040\pi$$
$$840$$ 0 0
$$841$$ 3.04908 0.105141
$$842$$ 0 0
$$843$$ − 3.94480i − 0.135866i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 54.0682i − 1.85780i
$$848$$ 0 0
$$849$$ −9.55004 −0.327756
$$850$$ 0 0
$$851$$ 21.6742 0.742981
$$852$$ 0 0
$$853$$ − 26.1104i − 0.894003i −0.894533 0.447001i $$-0.852492\pi$$
0.894533 0.447001i $$-0.147508\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 20.6922i − 0.706833i −0.935466 0.353416i $$-0.885020\pi$$
0.935466 0.353416i $$-0.114980\pi$$
$$858$$ 0 0
$$859$$ −3.53093 −0.120474 −0.0602370 0.998184i $$-0.519186\pi$$
−0.0602370 + 0.998184i $$0.519186\pi$$
$$860$$ 0 0
$$861$$ 16.1501 0.550395
$$862$$ 0 0
$$863$$ − 3.39906i − 0.115705i −0.998325 0.0578527i $$-0.981575\pi$$
0.998325 0.0578527i $$-0.0184254\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 1.28179i − 0.0435320i
$$868$$ 0 0
$$869$$ 49.6864 1.68550
$$870$$ 0 0
$$871$$ −23.4887 −0.795885
$$872$$ 0 0
$$873$$ − 30.1871i − 1.02168i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0.422364i 0.0142622i 0.999975 + 0.00713111i $$0.00226992\pi$$
−0.999975 + 0.00713111i $$0.997730\pi$$
$$878$$ 0 0
$$879$$ −3.18201 −0.107327
$$880$$ 0 0
$$881$$ 16.5243 0.556716 0.278358 0.960477i $$-0.410210\pi$$
0.278358 + 0.960477i $$0.410210\pi$$
$$882$$ 0 0
$$883$$ 24.5535i 0.826290i 0.910665 + 0.413145i $$0.135570\pi$$
−0.910665 + 0.413145i $$0.864430\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 41.5975i 1.39671i 0.715753 + 0.698354i $$0.246084\pi$$
−0.715753 + 0.698354i $$0.753916\pi$$
$$888$$ 0 0
$$889$$ 28.0260 0.939961
$$890$$ 0 0
$$891$$ −39.2051 −1.31342
$$892$$ 0 0
$$893$$ 11.1138i 0.371910i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 5.51471i − 0.184131i
$$898$$ 0 0
$$899$$ 42.7811 1.42683
$$900$$ 0 0
$$901$$ 33.4458 1.11424
$$902$$ 0 0
$$903$$ 12.0844i 0.402144i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 31.9294i − 1.06020i −0.847935 0.530100i $$-0.822154\pi$$
0.847935 0.530100i $$-0.177846\pi$$
$$908$$ 0 0
$$909$$ 12.4983 0.414542
$$910$$ 0 0
$$911$$ 26.9605 0.893240 0.446620 0.894724i $$-0.352628\pi$$
0.446620 + 0.894724i $$0.352628\pi$$
$$912$$ 0 0
$$913$$ 27.4588i 0.908752i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 8.46907i − 0.279673i
$$918$$ 0 0
$$919$$ 0.394005 0.0129970 0.00649850 0.999979i $$-0.497931\pi$$
0.00649850 + 0.999979i $$0.497931\pi$$
$$920$$ 0 0
$$921$$ −3.64496 −0.120106
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 12.1319i 0.398463i
$$928$$ 0 0
$$929$$ 32.9284 1.08035 0.540173 0.841554i $$-0.318359\pi$$
0.540173 + 0.841554i $$0.318359\pi$$
$$930$$ 0 0
$$931$$ 0.397442 0.0130256
$$932$$ 0 0
$$933$$ 5.19900i 0.170208i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.3388i 0.860451i 0.902721 + 0.430226i $$0.141566\pi$$
−0.902721 + 0.430226i $$0.858434\pi$$
$$938$$ 0 0
$$939$$ 0.547362 0.0178625
$$940$$ 0 0
$$941$$ 4.71982 0.153862 0.0769309 0.997036i $$-0.475488\pi$$
0.0769309 + 0.997036i $$0.475488\pi$$
$$942$$ 0 0
$$943$$ 72.8984i 2.37390i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 53.9639i − 1.75359i −0.480863 0.876796i $$-0.659677\pi$$
0.480863 0.876796i $$-0.340323\pi$$
$$948$$ 0 0
$$949$$ 19.1661 0.622159
$$950$$ 0 0
$$951$$ 10.2958 0.333866
$$952$$ 0 0
$$953$$ − 13.0801i − 0.423707i −0.977301 0.211853i $$-0.932050\pi$$
0.977301 0.211853i $$-0.0679499\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 14.8071i 0.478646i
$$958$$ 0 0
$$959$$ −36.5888 −1.18151
$$960$$ 0 0
$$961$$ 26.1070 0.842160
$$962$$ 0 0
$$963$$ 4.57496i 0.147426i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 10.4914i − 0.337381i −0.985669 0.168690i $$-0.946046\pi$$
0.985669 0.168690i $$-0.0539538\pi$$
$$968$$ 0 0
$$969$$ −1.77846 −0.0571323
$$970$$ 0 0
$$971$$ −12.4691 −0.400151 −0.200076 0.979780i $$-0.564119\pi$$
−0.200076 + 0.979780i $$0.564119\pi$$
$$972$$ 0 0
$$973$$ 25.9931i 0.833301i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 52.4699i 1.67866i 0.543621 + 0.839331i $$0.317053\pi$$
−0.543621 + 0.839331i $$0.682947\pi$$
$$978$$ 0 0
$$979$$ 86.1035 2.75188
$$980$$ 0 0
$$981$$ 2.65164 0.0846603
$$982$$ 0 0
$$983$$ 14.1939i 0.452717i 0.974044 + 0.226358i $$0.0726820\pi$$
−0.974044 + 0.226358i $$0.927318\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 14.2277i 0.452871i
$$988$$ 0 0
$$989$$ −54.5466 −1.73448
$$990$$ 0 0
$$991$$ 19.4036 0.616374 0.308187 0.951326i $$-0.400278\pi$$
0.308187 + 0.951326i $$0.400278\pi$$
$$992$$ 0 0
$$993$$ − 6.90240i − 0.219041i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 8.14648i 0.258002i 0.991644 + 0.129001i $$0.0411770\pi$$
−0.991644 + 0.129001i $$0.958823\pi$$
$$998$$ 0 0
$$999$$ 10.2017 0.322767
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 3800.2.d.n.3649.3 6
5.2 odd 4 760.2.a.i.1.2 3
5.3 odd 4 3800.2.a.w.1.2 3
5.4 even 2 inner 3800.2.d.n.3649.4 6
15.2 even 4 6840.2.a.bm.1.3 3
20.3 even 4 7600.2.a.bp.1.2 3
20.7 even 4 1520.2.a.q.1.2 3
40.27 even 4 6080.2.a.br.1.2 3
40.37 odd 4 6080.2.a.bx.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.2 3 5.2 odd 4
1520.2.a.q.1.2 3 20.7 even 4
3800.2.a.w.1.2 3 5.3 odd 4
3800.2.d.n.3649.3 6 1.1 even 1 trivial
3800.2.d.n.3649.4 6 5.4 even 2 inner
6080.2.a.br.1.2 3 40.27 even 4
6080.2.a.bx.1.2 3 40.37 odd 4
6840.2.a.bm.1.3 3 15.2 even 4
7600.2.a.bp.1.2 3 20.3 even 4