# Properties

 Label 3800.2.d.f.3649.2 Level $3800$ Weight $2$ Character 3800.3649 Analytic conductor $30.343$ 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] = [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: $$2$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3649.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3800.3649 Dual form 3800.2.d.f.3649.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -3.00000i q^{7} +2.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -3.00000i q^{7} +2.00000 q^{9} +2.00000 q^{11} +1.00000i q^{13} +5.00000i q^{17} -1.00000 q^{19} +3.00000 q^{21} -1.00000i q^{23} +5.00000i q^{27} +3.00000 q^{29} +4.00000 q^{31} +2.00000i q^{33} -2.00000i q^{37} -1.00000 q^{39} -8.00000 q^{41} -8.00000i q^{43} +8.00000i q^{47} -2.00000 q^{49} -5.00000 q^{51} +9.00000i q^{53} -1.00000i q^{57} -1.00000 q^{59} +14.0000 q^{61} -6.00000i q^{63} -13.0000i q^{67} +1.00000 q^{69} +10.0000 q^{71} +9.00000i q^{73} -6.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +10.0000i q^{83} +3.00000i q^{87} +12.0000 q^{89} +3.00000 q^{91} +4.00000i q^{93} -14.0000i q^{97} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 4 q^{11} - 2 q^{19} + 6 q^{21} + 6 q^{29} + 8 q^{31} - 2 q^{39} - 16 q^{41} - 4 q^{49} - 10 q^{51} - 2 q^{59} + 28 q^{61} + 2 q^{69} + 20 q^{71} + 20 q^{79} + 2 q^{81} + 24 q^{89} + 6 q^{91} + 8 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 4 * q^11 - 2 * q^19 + 6 * q^21 + 6 * q^29 + 8 * q^31 - 2 * q^39 - 16 * q^41 - 4 * q^49 - 10 * q^51 - 2 * q^59 + 28 * q^61 + 2 * q^69 + 20 * q^71 + 20 * q^79 + 2 * q^81 + 24 * q^89 + 6 * q^91 + 8 * 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
$$3$$ 1.00000i 0.577350i 0.957427 + 0.288675i $$0.0932147\pi$$
−0.957427 + 0.288675i $$0.906785\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.00000i 1.21268i 0.795206 + 0.606339i $$0.207363\pi$$
−0.795206 + 0.606339i $$0.792637\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i −0.994550 0.104257i $$-0.966753\pi$$
0.994550 0.104257i $$-0.0332465\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.00000i 0.962250i
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 2.00000i 0.348155i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ −5.00000 −0.700140
$$52$$ 0 0
$$53$$ 9.00000i 1.23625i 0.786082 + 0.618123i $$0.212106\pi$$
−0.786082 + 0.618123i $$0.787894\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 1.00000i − 0.132453i
$$58$$ 0 0
$$59$$ −1.00000 −0.130189 −0.0650945 0.997879i $$-0.520735\pi$$
−0.0650945 + 0.997879i $$0.520735\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 0 0
$$63$$ − 6.00000i − 0.755929i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 13.0000i − 1.58820i −0.607785 0.794101i $$-0.707942\pi$$
0.607785 0.794101i $$-0.292058\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 10.0000 1.18678 0.593391 0.804914i $$-0.297789\pi$$
0.593391 + 0.804914i $$0.297789\pi$$
$$72$$ 0 0
$$73$$ 9.00000i 1.05337i 0.850060 + 0.526685i $$0.176565\pi$$
−0.850060 + 0.526685i $$0.823435\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 6.00000i − 0.683763i
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 10.0000i 1.09764i 0.835940 + 0.548821i $$0.184923\pi$$
−0.835940 + 0.548821i $$0.815077\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 3.00000i 0.321634i
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ 3.00000 0.314485
$$92$$ 0 0
$$93$$ 4.00000i 0.414781i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 6.00000i 0.591198i 0.955312 + 0.295599i $$0.0955191\pi$$
−0.955312 + 0.295599i $$0.904481\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 15.0000i − 1.45010i −0.688694 0.725052i $$-0.741816\pi$$
0.688694 0.725052i $$-0.258184\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ − 18.0000i − 1.69330i −0.532152 0.846649i $$-0.678617\pi$$
0.532152 0.846649i $$-0.321383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ 15.0000 1.37505
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ − 8.00000i − 0.721336i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 6.00000i 0.532414i 0.963916 + 0.266207i $$0.0857705\pi$$
−0.963916 + 0.266207i $$0.914230\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 3.00000i 0.260133i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 7.00000i − 0.598050i −0.954245 0.299025i $$-0.903339\pi$$
0.954245 0.299025i $$-0.0966615\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ 2.00000i 0.167248i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 2.00000i − 0.164957i
$$148$$ 0 0
$$149$$ 8.00000 0.655386 0.327693 0.944784i $$-0.393729\pi$$
0.327693 + 0.944784i $$0.393729\pi$$
$$150$$ 0 0
$$151$$ 22.0000 1.79033 0.895167 0.445730i $$-0.147056\pi$$
0.895167 + 0.445730i $$0.147056\pi$$
$$152$$ 0 0
$$153$$ 10.0000i 0.808452i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.0000i 1.75579i 0.478852 + 0.877896i $$0.341053\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 4.00000i − 0.309529i −0.987951 0.154765i $$-0.950538\pi$$
0.987951 0.154765i $$-0.0494619\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 22.0000i 1.67263i 0.548250 + 0.836315i $$0.315294\pi$$
−0.548250 + 0.836315i $$0.684706\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 1.00000i − 0.0751646i
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 14.0000i 1.03491i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 10.0000i 0.731272i
$$188$$ 0 0
$$189$$ 15.0000 1.09109
$$190$$ 0 0
$$191$$ 23.0000 1.66422 0.832111 0.554609i $$-0.187132\pi$$
0.832111 + 0.554609i $$0.187132\pi$$
$$192$$ 0 0
$$193$$ 6.00000i 0.431889i 0.976406 + 0.215945i $$0.0692831\pi$$
−0.976406 + 0.215945i $$0.930717\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 8.00000i 0.569976i 0.958531 + 0.284988i $$0.0919897\pi$$
−0.958531 + 0.284988i $$0.908010\pi$$
$$198$$ 0 0
$$199$$ 9.00000 0.637993 0.318997 0.947756i $$-0.396654\pi$$
0.318997 + 0.947756i $$0.396654\pi$$
$$200$$ 0 0
$$201$$ 13.0000 0.916949
$$202$$ 0 0
$$203$$ − 9.00000i − 0.631676i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 2.00000i − 0.139010i
$$208$$ 0 0
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ 21.0000 1.44570 0.722850 0.691005i $$-0.242832\pi$$
0.722850 + 0.691005i $$0.242832\pi$$
$$212$$ 0 0
$$213$$ 10.0000i 0.685189i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 12.0000i − 0.814613i
$$218$$ 0 0
$$219$$ −9.00000 −0.608164
$$220$$ 0 0
$$221$$ −5.00000 −0.336336
$$222$$ 0 0
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 9.00000i − 0.597351i −0.954355 0.298675i $$-0.903455\pi$$
0.954355 0.298675i $$-0.0965448\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ 26.0000i 1.70332i 0.524097 + 0.851658i $$0.324403\pi$$
−0.524097 + 0.851658i $$0.675597\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 10.0000i 0.649570i
$$238$$ 0 0
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 16.0000i 1.02640i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 1.00000i − 0.0636285i
$$248$$ 0 0
$$249$$ −10.0000 −0.633724
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ − 2.00000i − 0.125739i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4.00000i 0.249513i 0.992187 + 0.124757i $$0.0398150\pi$$
−0.992187 + 0.124757i $$0.960185\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 32.0000i 1.97320i 0.163144 + 0.986602i $$0.447836\pi$$
−0.163144 + 0.986602i $$0.552164\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.0000i 0.734388i
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −25.0000 −1.51864 −0.759321 0.650716i $$-0.774469\pi$$
−0.759321 + 0.650716i $$0.774469\pi$$
$$272$$ 0 0
$$273$$ 3.00000i 0.181568i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ − 6.00000i − 0.356663i −0.983970 0.178331i $$-0.942930\pi$$
0.983970 0.178331i $$-0.0570699\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000i 1.41668i
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 0 0
$$293$$ 7.00000i 0.408944i 0.978872 + 0.204472i $$0.0655478\pi$$
−0.978872 + 0.204472i $$0.934452\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 10.0000i 0.580259i
$$298$$ 0 0
$$299$$ 1.00000 0.0578315
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ 0 0
$$303$$ − 14.0000i − 0.804279i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ −6.00000 −0.341328
$$310$$ 0 0
$$311$$ −9.00000 −0.510343 −0.255172 0.966896i $$-0.582132\pi$$
−0.255172 + 0.966896i $$0.582132\pi$$
$$312$$ 0 0
$$313$$ 13.0000i 0.734803i 0.930062 + 0.367402i $$0.119753\pi$$
−0.930062 + 0.367402i $$0.880247\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 3.00000i − 0.168497i −0.996445 0.0842484i $$-0.973151\pi$$
0.996445 0.0842484i $$-0.0268489\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 15.0000 0.837218
$$322$$ 0 0
$$323$$ − 5.00000i − 0.278207i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 7.00000i − 0.387101i
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ 0 0
$$333$$ − 4.00000i − 0.219199i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 12.0000i − 0.653682i −0.945079 0.326841i $$-0.894016\pi$$
0.945079 0.326841i $$-0.105984\pi$$
$$338$$ 0 0
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 8.00000 0.433224
$$342$$ 0 0
$$343$$ − 15.0000i − 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 2.00000i − 0.107366i −0.998558 0.0536828i $$-0.982904\pi$$
0.998558 0.0536828i $$-0.0170960\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 0 0
$$353$$ − 7.00000i − 0.372572i −0.982496 0.186286i $$-0.940355\pi$$
0.982496 0.186286i $$-0.0596452\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 15.0000i 0.793884i
$$358$$ 0 0
$$359$$ −17.0000 −0.897226 −0.448613 0.893726i $$-0.648082\pi$$
−0.448613 + 0.893726i $$0.648082\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ − 7.00000i − 0.367405i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 28.0000i − 1.46159i −0.682598 0.730794i $$-0.739150\pi$$
0.682598 0.730794i $$-0.260850\pi$$
$$368$$ 0 0
$$369$$ −16.0000 −0.832927
$$370$$ 0 0
$$371$$ 27.0000 1.40177
$$372$$ 0 0
$$373$$ − 13.0000i − 0.673114i −0.941663 0.336557i $$-0.890737\pi$$
0.941663 0.336557i $$-0.109263\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.00000i 0.154508i
$$378$$ 0 0
$$379$$ −9.00000 −0.462299 −0.231149 0.972918i $$-0.574249\pi$$
−0.231149 + 0.972918i $$0.574249\pi$$
$$380$$ 0 0
$$381$$ −6.00000 −0.307389
$$382$$ 0 0
$$383$$ − 6.00000i − 0.306586i −0.988181 0.153293i $$-0.951012\pi$$
0.988181 0.153293i $$-0.0489878\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 16.0000i − 0.813326i
$$388$$ 0 0
$$389$$ −34.0000 −1.72387 −0.861934 0.507020i $$-0.830747\pi$$
−0.861934 + 0.507020i $$0.830747\pi$$
$$390$$ 0 0
$$391$$ 5.00000 0.252861
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4.00000i 0.200754i 0.994949 + 0.100377i $$0.0320049\pi$$
−0.994949 + 0.100377i $$0.967995\pi$$
$$398$$ 0 0
$$399$$ −3.00000 −0.150188
$$400$$ 0 0
$$401$$ −16.0000 −0.799002 −0.399501 0.916733i $$-0.630817\pi$$
−0.399501 + 0.916733i $$0.630817\pi$$
$$402$$ 0 0
$$403$$ 4.00000i 0.199254i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 4.00000i − 0.198273i
$$408$$ 0 0
$$409$$ 8.00000 0.395575 0.197787 0.980245i $$-0.436624\pi$$
0.197787 + 0.980245i $$0.436624\pi$$
$$410$$ 0 0
$$411$$ 7.00000 0.345285
$$412$$ 0 0
$$413$$ 3.00000i 0.147620i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 12.0000i 0.587643i
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −35.0000 −1.70580 −0.852898 0.522078i $$-0.825157\pi$$
−0.852898 + 0.522078i $$0.825157\pi$$
$$422$$ 0 0
$$423$$ 16.0000i 0.777947i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 42.0000i − 2.03252i
$$428$$ 0 0
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 0 0
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.00000i 0.0478365i
$$438$$ 0 0
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 0 0
$$443$$ 6.00000i 0.285069i 0.989790 + 0.142534i $$0.0455251\pi$$
−0.989790 + 0.142534i $$0.954475\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 8.00000i 0.378387i
$$448$$ 0 0
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ 0 0
$$453$$ 22.0000i 1.03365i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 1.00000i − 0.0467780i −0.999726 0.0233890i $$-0.992554\pi$$
0.999726 0.0233890i $$-0.00744563\pi$$
$$458$$ 0 0
$$459$$ −25.0000 −1.16690
$$460$$ 0 0
$$461$$ 20.0000 0.931493 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$462$$ 0 0
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.00000i 0.277647i 0.990317 + 0.138823i $$0.0443321\pi$$
−0.990317 + 0.138823i $$0.955668\pi$$
$$468$$ 0 0
$$469$$ −39.0000 −1.80085
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ 0 0
$$473$$ − 16.0000i − 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 18.0000i 0.824163i
$$478$$ 0 0
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 0 0
$$483$$ − 3.00000i − 0.136505i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 26.0000i − 1.17817i −0.808070 0.589086i $$-0.799488\pi$$
0.808070 0.589086i $$-0.200512\pi$$
$$488$$ 0 0
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 15.0000i 0.675566i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 30.0000i − 1.34568i
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 4.00000 0.178707
$$502$$ 0 0
$$503$$ − 33.0000i − 1.47140i −0.677309 0.735699i $$-0.736854\pi$$
0.677309 0.735699i $$-0.263146\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 12.0000i 0.532939i
$$508$$ 0 0
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ 27.0000 1.19441
$$512$$ 0 0
$$513$$ − 5.00000i − 0.220755i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 16.0000i 0.703679i
$$518$$ 0 0
$$519$$ −22.0000 −0.965693
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 0 0
$$523$$ − 5.00000i − 0.218635i −0.994007 0.109317i $$-0.965134\pi$$
0.994007 0.109317i $$-0.0348665\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 20.0000i 0.871214i
$$528$$ 0 0
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ −2.00000 −0.0867926
$$532$$ 0 0
$$533$$ − 8.00000i − 0.346518i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ 0 0
$$543$$ − 14.0000i − 0.600798i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 28.0000i − 1.19719i −0.801050 0.598597i $$-0.795725\pi$$
0.801050 0.598597i $$-0.204275\pi$$
$$548$$ 0 0
$$549$$ 28.0000 1.19501
$$550$$ 0 0
$$551$$ −3.00000 −0.127804
$$552$$ 0 0
$$553$$ − 30.0000i − 1.27573i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 24.0000i − 1.01691i −0.861088 0.508456i $$-0.830216\pi$$
0.861088 0.508456i $$-0.169784\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −10.0000 −0.422200
$$562$$ 0 0
$$563$$ − 44.0000i − 1.85438i −0.374593 0.927189i $$-0.622217\pi$$
0.374593 0.927189i $$-0.377783\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 3.00000i − 0.125988i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 23.0000i 0.960839i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 29.0000i 1.20729i 0.797255 + 0.603643i $$0.206285\pi$$
−0.797255 + 0.603643i $$0.793715\pi$$
$$578$$ 0 0
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ 30.0000 1.24461
$$582$$ 0 0
$$583$$ 18.0000i 0.745484i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 44.0000i 1.81607i 0.418890 + 0.908037i $$0.362419\pi$$
−0.418890 + 0.908037i $$0.637581\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ −8.00000 −0.329076
$$592$$ 0 0
$$593$$ 2.00000i 0.0821302i 0.999156 + 0.0410651i $$0.0130751\pi$$
−0.999156 + 0.0410651i $$0.986925\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 9.00000i 0.368345i
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ −4.00000 −0.163163 −0.0815817 0.996667i $$-0.525997\pi$$
−0.0815817 + 0.996667i $$0.525997\pi$$
$$602$$ 0 0
$$603$$ − 26.0000i − 1.05880i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 42.0000i − 1.70473i −0.522949 0.852364i $$-0.675168\pi$$
0.522949 0.852364i $$-0.324832\pi$$
$$608$$ 0 0
$$609$$ 9.00000 0.364698
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ 26.0000i 1.05013i 0.851062 + 0.525065i $$0.175959\pi$$
−0.851062 + 0.525065i $$0.824041\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 0 0
$$619$$ −22.0000 −0.884255 −0.442127 0.896952i $$-0.645776\pi$$
−0.442127 + 0.896952i $$0.645776\pi$$
$$620$$ 0 0
$$621$$ 5.00000 0.200643
$$622$$ 0 0
$$623$$ − 36.0000i − 1.44231i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 2.00000i − 0.0798723i
$$628$$ 0 0
$$629$$ 10.0000 0.398726
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ 21.0000i 0.834675i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 0 0
$$639$$ 20.0000 0.791188
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ − 46.0000i − 1.81406i −0.421063 0.907031i $$-0.638343\pi$$
0.421063 0.907031i $$-0.361657\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.00000i 0.353827i 0.984226 + 0.176913i $$0.0566112\pi$$
−0.984226 + 0.176913i $$0.943389\pi$$
$$648$$ 0 0
$$649$$ −2.00000 −0.0785069
$$650$$ 0 0
$$651$$ 12.0000 0.470317
$$652$$ 0 0
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 18.0000i 0.702247i
$$658$$ 0 0
$$659$$ −19.0000 −0.740135 −0.370067 0.929005i $$-0.620665\pi$$
−0.370067 + 0.929005i $$0.620665\pi$$
$$660$$ 0 0
$$661$$ −17.0000 −0.661223 −0.330612 0.943767i $$-0.607255\pi$$
−0.330612 + 0.943767i $$0.607255\pi$$
$$662$$ 0 0
$$663$$ − 5.00000i − 0.194184i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 3.00000i − 0.116160i
$$668$$ 0 0
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ 28.0000 1.08093
$$672$$ 0 0
$$673$$ − 12.0000i − 0.462566i −0.972887 0.231283i $$-0.925708\pi$$
0.972887 0.231283i $$-0.0742923\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 27.0000i − 1.03769i −0.854867 0.518847i $$-0.826361\pi$$
0.854867 0.518847i $$-0.173639\pi$$
$$678$$ 0 0
$$679$$ −42.0000 −1.61181
$$680$$ 0 0
$$681$$ 9.00000 0.344881
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 2.00000i − 0.0763048i
$$688$$ 0 0
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ −42.0000 −1.59776 −0.798878 0.601494i $$-0.794573\pi$$
−0.798878 + 0.601494i $$0.794573\pi$$
$$692$$ 0 0
$$693$$ − 12.0000i − 0.455842i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 40.0000i − 1.51511i
$$698$$ 0 0
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ 44.0000 1.66186 0.830929 0.556379i $$-0.187810\pi$$
0.830929 + 0.556379i $$0.187810\pi$$
$$702$$ 0 0
$$703$$ 2.00000i 0.0754314i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 42.0000i 1.57957i
$$708$$ 0 0
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ 20.0000 0.750059
$$712$$ 0 0
$$713$$ − 4.00000i − 0.149801i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 9.00000i 0.336111i
$$718$$ 0 0
$$719$$ −27.0000 −1.00693 −0.503465 0.864016i $$-0.667942\pi$$
−0.503465 + 0.864016i $$0.667942\pi$$
$$720$$ 0 0
$$721$$ 18.0000 0.670355
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 7.00000i − 0.259616i −0.991539 0.129808i $$-0.958564\pi$$
0.991539 0.129808i $$-0.0414360\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 40.0000 1.47945
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 26.0000i − 0.957722i
$$738$$ 0 0
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 1.00000 0.0367359
$$742$$ 0 0
$$743$$ − 28.0000i − 1.02722i −0.858024 0.513610i $$-0.828308\pi$$
0.858024 0.513610i $$-0.171692\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 20.0000i 0.731762i
$$748$$ 0 0
$$749$$ −45.0000 −1.64426
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ − 18.0000i − 0.655956i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22.0000i 0.799604i 0.916602 + 0.399802i $$0.130921\pi$$
−0.916602 + 0.399802i $$0.869079\pi$$
$$758$$ 0 0
$$759$$ 2.00000 0.0725954
$$760$$ 0 0
$$761$$ −37.0000 −1.34125 −0.670624 0.741797i $$-0.733974\pi$$
−0.670624 + 0.741797i $$0.733974\pi$$
$$762$$ 0 0
$$763$$ 21.0000i 0.760251i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 1.00000i − 0.0361079i
$$768$$ 0 0
$$769$$ −5.00000 −0.180305 −0.0901523 0.995928i $$-0.528735\pi$$
−0.0901523 + 0.995928i $$0.528735\pi$$
$$770$$ 0 0
$$771$$ −4.00000 −0.144056
$$772$$ 0 0
$$773$$ − 49.0000i − 1.76241i −0.472737 0.881204i $$-0.656734\pi$$
0.472737 0.881204i $$-0.343266\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 6.00000i − 0.215249i
$$778$$ 0 0
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ 20.0000 0.715656
$$782$$ 0 0
$$783$$ 15.0000i 0.536056i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 7.00000i 0.249523i 0.992187 + 0.124762i $$0.0398166\pi$$
−0.992187 + 0.124762i $$0.960183\pi$$
$$788$$ 0 0
$$789$$ −32.0000 −1.13923
$$790$$ 0 0
$$791$$ −54.0000 −1.92002
$$792$$ 0 0
$$793$$ 14.0000i 0.497155i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 3.00000i 0.106265i 0.998587 + 0.0531327i $$0.0169206\pi$$
−0.998587 + 0.0531327i $$0.983079\pi$$
$$798$$ 0 0
$$799$$ −40.0000 −1.41510
$$800$$ 0 0
$$801$$ 24.0000 0.847998
$$802$$ 0 0
$$803$$ 18.0000i 0.635206i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 10.0000i − 0.352017i
$$808$$ 0 0
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ 27.0000 0.948098 0.474049 0.880498i $$-0.342792\pi$$
0.474049 + 0.880498i $$0.342792\pi$$
$$812$$ 0 0
$$813$$ − 25.0000i − 0.876788i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.00000i 0.279885i
$$818$$ 0 0
$$819$$ 6.00000 0.209657
$$820$$ 0 0
$$821$$ 8.00000 0.279202 0.139601 0.990208i $$-0.455418\pi$$
0.139601 + 0.990208i $$0.455418\pi$$
$$822$$ 0 0
$$823$$ − 11.0000i − 0.383436i −0.981450 0.191718i $$-0.938594\pi$$
0.981450 0.191718i $$-0.0614059\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 33.0000i − 1.14752i −0.819023 0.573761i $$-0.805484\pi$$
0.819023 0.573761i $$-0.194516\pi$$
$$828$$ 0 0
$$829$$ −39.0000 −1.35453 −0.677263 0.735741i $$-0.736834\pi$$
−0.677263 + 0.735741i $$0.736834\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 10.0000i − 0.346479i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 20.0000i 0.691301i
$$838$$ 0 0
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 21.0000i 0.721569i
$$848$$ 0 0
$$849$$ 6.00000 0.205919
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 0 0
$$853$$ − 6.00000i − 0.205436i −0.994711 0.102718i $$-0.967246\pi$$
0.994711 0.102718i $$-0.0327539\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 44.0000i 1.50301i 0.659727 + 0.751506i $$0.270672\pi$$
−0.659727 + 0.751506i $$0.729328\pi$$
$$858$$ 0 0
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ 0 0
$$863$$ 54.0000i 1.83818i 0.394046 + 0.919091i $$0.371075\pi$$
−0.394046 + 0.919091i $$0.628925\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 8.00000i − 0.271694i
$$868$$ 0 0
$$869$$ 20.0000 0.678454
$$870$$ 0 0
$$871$$ 13.0000 0.440488
$$872$$ 0 0
$$873$$ − 28.0000i − 0.947656i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 37.0000i 1.24940i 0.780864 + 0.624701i $$0.214779\pi$$
−0.780864 + 0.624701i $$0.785221\pi$$
$$878$$ 0 0
$$879$$ −7.00000 −0.236104
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ 46.0000i 1.54802i 0.633171 + 0.774012i $$0.281753\pi$$
−0.633171 + 0.774012i $$0.718247\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 18.0000i 0.604381i 0.953248 + 0.302190i $$0.0977178\pi$$
−0.953248 + 0.302190i $$0.902282\pi$$
$$888$$ 0 0
$$889$$ 18.0000 0.603701
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 0 0
$$893$$ − 8.00000i − 0.267710i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.00000i 0.0333890i
$$898$$ 0 0
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ −45.0000 −1.49917
$$902$$ 0 0
$$903$$ − 24.0000i − 0.798670i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 5.00000i 0.166022i 0.996549 + 0.0830111i $$0.0264537\pi$$
−0.996549 + 0.0830111i $$0.973546\pi$$
$$908$$ 0 0
$$909$$ −28.0000 −0.928701
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ 0 0
$$913$$ 20.0000i 0.661903i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 43.0000 1.41844 0.709220 0.704988i $$-0.249047\pi$$
0.709220 + 0.704988i $$0.249047\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 0 0
$$923$$ 10.0000i 0.329154i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 12.0000i 0.394132i
$$928$$ 0 0
$$929$$ −1.00000 −0.0328089 −0.0164045 0.999865i $$-0.505222\pi$$
−0.0164045 + 0.999865i $$0.505222\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 0 0
$$933$$ − 9.00000i − 0.294647i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 49.0000i − 1.60076i −0.599493 0.800380i $$-0.704631\pi$$
0.599493 0.800380i $$-0.295369\pi$$
$$938$$ 0 0
$$939$$ −13.0000 −0.424239
$$940$$ 0 0
$$941$$ −15.0000 −0.488986 −0.244493 0.969651i $$-0.578622\pi$$
−0.244493 + 0.969651i $$0.578622\pi$$
$$942$$ 0 0
$$943$$ 8.00000i 0.260516i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 32.0000i 1.03986i 0.854209 + 0.519930i $$0.174042\pi$$
−0.854209 + 0.519930i $$0.825958\pi$$
$$948$$ 0 0
$$949$$ −9.00000 −0.292152
$$950$$ 0 0
$$951$$ 3.00000 0.0972817
$$952$$ 0 0
$$953$$ − 18.0000i − 0.583077i −0.956559 0.291539i $$-0.905833\pi$$
0.956559 0.291539i $$-0.0941672\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6.00000i 0.193952i
$$958$$ 0 0
$$959$$ −21.0000 −0.678125
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ − 30.0000i − 0.966736i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 40.0000i − 1.28631i −0.765735 0.643157i $$-0.777624\pi$$
0.765735 0.643157i $$-0.222376\pi$$
$$968$$ 0 0
$$969$$ 5.00000 0.160623
$$970$$ 0 0
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ 0 0
$$973$$ − 36.0000i − 1.15411i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 12.0000i − 0.383914i −0.981403 0.191957i $$-0.938517\pi$$
0.981403 0.191957i $$-0.0614834\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 0 0
$$983$$ 10.0000i 0.318950i 0.987202 + 0.159475i $$0.0509802\pi$$
−0.987202 + 0.159475i $$0.949020\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 24.0000i 0.763928i
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ −28.0000 −0.889449 −0.444725 0.895667i $$-0.646698\pi$$
−0.444725 + 0.895667i $$0.646698\pi$$
$$992$$ 0 0
$$993$$ − 25.0000i − 0.793351i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 16.0000i 0.506725i 0.967371 + 0.253363i $$0.0815366\pi$$
−0.967371 + 0.253363i $$0.918463\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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.f.3649.2 2
5.2 odd 4 152.2.a.b.1.1 1
5.3 odd 4 3800.2.a.d.1.1 1
5.4 even 2 inner 3800.2.d.f.3649.1 2
15.2 even 4 1368.2.a.g.1.1 1
20.3 even 4 7600.2.a.o.1.1 1
20.7 even 4 304.2.a.b.1.1 1
35.27 even 4 7448.2.a.g.1.1 1
40.27 even 4 1216.2.a.l.1.1 1
40.37 odd 4 1216.2.a.f.1.1 1
60.47 odd 4 2736.2.a.k.1.1 1
95.37 even 4 2888.2.a.b.1.1 1
380.227 odd 4 5776.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.b.1.1 1 5.2 odd 4
304.2.a.b.1.1 1 20.7 even 4
1216.2.a.f.1.1 1 40.37 odd 4
1216.2.a.l.1.1 1 40.27 even 4
1368.2.a.g.1.1 1 15.2 even 4
2736.2.a.k.1.1 1 60.47 odd 4
2888.2.a.b.1.1 1 95.37 even 4
3800.2.a.d.1.1 1 5.3 odd 4
3800.2.d.f.3649.1 2 5.4 even 2 inner
3800.2.d.f.3649.2 2 1.1 even 1 trivial
5776.2.a.l.1.1 1 380.227 odd 4
7448.2.a.g.1.1 1 35.27 even 4
7600.2.a.o.1.1 1 20.3 even 4