# Properties

 Label 2200.2.a.u.1.2 Level $2200$ Weight $2$ Character 2200.1 Self dual yes Analytic conductor $17.567$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2200.a (trivial)

## 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: yes Analytic conductor: $$17.5670884447$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x + 6$$ x^3 - x^2 - 7*x + 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.841083$$ of defining polynomial Character $$\chi$$ $$=$$ 2200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.841083 q^{3} +3.29258 q^{7} -2.29258 q^{9} +O(q^{10})$$ $$q-0.841083 q^{3} +3.29258 q^{7} -2.29258 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.15892 q^{17} +1.31783 q^{19} -2.76933 q^{21} -3.84108 q^{23} +4.45150 q^{27} +6.61041 q^{29} +7.58516 q^{31} -0.841083 q^{33} -2.13366 q^{37} +0.841083 q^{39} +6.13366 q^{41} -8.81583 q^{43} -2.76933 q^{47} +3.84108 q^{49} -0.974745 q^{51} +10.1956 q^{53} -1.10841 q^{57} +2.76933 q^{59} -3.61041 q^{61} -7.54850 q^{63} -2.90299 q^{67} +3.23067 q^{69} +0.866337 q^{71} -5.87774 q^{73} +3.29258 q^{77} +11.7441 q^{79} +3.13366 q^{81} -8.92825 q^{83} -5.55991 q^{87} +14.2926 q^{89} -3.29258 q^{91} -6.37975 q^{93} +12.4262 q^{97} -2.29258 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q - q^3 - 3 * q^7 + 6 * q^9 $$3 q - q^{3} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 3 q^{13} + 5 q^{17} + 7 q^{19} - 10 q^{23} + 2 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 8 q^{37} + q^{39} + 4 q^{41} - 9 q^{43} + 10 q^{49} + 13 q^{51} - 5 q^{53} + 27 q^{57} - q^{61} - 34 q^{63} + 14 q^{67} + 18 q^{69} + 17 q^{71} + 21 q^{73} - 3 q^{77} + 11 q^{79} - 5 q^{81} - 20 q^{83} + 25 q^{87} + 30 q^{89} + 3 q^{91} - q^{93} + 10 q^{97} + 6 q^{99}+O(q^{100})$$ 3 * q - q^3 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 3 * q^13 + 5 * q^17 + 7 * q^19 - 10 * q^23 + 2 * q^27 + 10 * q^29 - 3 * q^31 - q^33 + 8 * q^37 + q^39 + 4 * q^41 - 9 * q^43 + 10 * q^49 + 13 * q^51 - 5 * q^53 + 27 * q^57 - q^61 - 34 * q^63 + 14 * q^67 + 18 * q^69 + 17 * q^71 + 21 * q^73 - 3 * q^77 + 11 * q^79 - 5 * q^81 - 20 * q^83 + 25 * q^87 + 30 * q^89 + 3 * q^91 - q^93 + 10 * q^97 + 6 * q^99

## 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
$$3$$ −0.841083 −0.485599 −0.242800 0.970076i $$-0.578066\pi$$
−0.242800 + 0.970076i $$0.578066\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.29258 1.24448 0.622239 0.782827i $$-0.286223\pi$$
0.622239 + 0.782827i $$0.286223\pi$$
$$8$$ 0 0
$$9$$ −2.29258 −0.764193
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.15892 0.281079 0.140539 0.990075i $$-0.455116\pi$$
0.140539 + 0.990075i $$0.455116\pi$$
$$18$$ 0 0
$$19$$ 1.31783 0.302332 0.151166 0.988508i $$-0.451697\pi$$
0.151166 + 0.988508i $$0.451697\pi$$
$$20$$ 0 0
$$21$$ −2.76933 −0.604318
$$22$$ 0 0
$$23$$ −3.84108 −0.800921 −0.400461 0.916314i $$-0.631150\pi$$
−0.400461 + 0.916314i $$0.631150\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.45150 0.856691
$$28$$ 0 0
$$29$$ 6.61041 1.22752 0.613762 0.789491i $$-0.289656\pi$$
0.613762 + 0.789491i $$0.289656\pi$$
$$30$$ 0 0
$$31$$ 7.58516 1.36233 0.681167 0.732128i $$-0.261473\pi$$
0.681167 + 0.732128i $$0.261473\pi$$
$$32$$ 0 0
$$33$$ −0.841083 −0.146414
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.13366 −0.350772 −0.175386 0.984500i $$-0.556117\pi$$
−0.175386 + 0.984500i $$0.556117\pi$$
$$38$$ 0 0
$$39$$ 0.841083 0.134681
$$40$$ 0 0
$$41$$ 6.13366 0.957917 0.478959 0.877838i $$-0.341014\pi$$
0.478959 + 0.877838i $$0.341014\pi$$
$$42$$ 0 0
$$43$$ −8.81583 −1.34440 −0.672201 0.740369i $$-0.734651\pi$$
−0.672201 + 0.740369i $$0.734651\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −2.76933 −0.403949 −0.201974 0.979391i $$-0.564736\pi$$
−0.201974 + 0.979391i $$0.564736\pi$$
$$48$$ 0 0
$$49$$ 3.84108 0.548726
$$50$$ 0 0
$$51$$ −0.974745 −0.136492
$$52$$ 0 0
$$53$$ 10.1956 1.40047 0.700235 0.713912i $$-0.253079\pi$$
0.700235 + 0.713912i $$0.253079\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.10841 −0.146812
$$58$$ 0 0
$$59$$ 2.76933 0.360536 0.180268 0.983618i $$-0.442303\pi$$
0.180268 + 0.983618i $$0.442303\pi$$
$$60$$ 0 0
$$61$$ −3.61041 −0.462266 −0.231133 0.972922i $$-0.574243\pi$$
−0.231133 + 0.972922i $$0.574243\pi$$
$$62$$ 0 0
$$63$$ −7.54850 −0.951022
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.90299 −0.354657 −0.177329 0.984152i $$-0.556746\pi$$
−0.177329 + 0.984152i $$0.556746\pi$$
$$68$$ 0 0
$$69$$ 3.23067 0.388927
$$70$$ 0 0
$$71$$ 0.866337 0.102815 0.0514077 0.998678i $$-0.483629\pi$$
0.0514077 + 0.998678i $$0.483629\pi$$
$$72$$ 0 0
$$73$$ −5.87774 −0.687937 −0.343969 0.938981i $$-0.611771\pi$$
−0.343969 + 0.938981i $$0.611771\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.29258 0.375224
$$78$$ 0 0
$$79$$ 11.7441 1.32131 0.660656 0.750689i $$-0.270278\pi$$
0.660656 + 0.750689i $$0.270278\pi$$
$$80$$ 0 0
$$81$$ 3.13366 0.348185
$$82$$ 0 0
$$83$$ −8.92825 −0.980003 −0.490001 0.871722i $$-0.663004\pi$$
−0.490001 + 0.871722i $$0.663004\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −5.55991 −0.596084
$$88$$ 0 0
$$89$$ 14.2926 1.51501 0.757505 0.652829i $$-0.226418\pi$$
0.757505 + 0.652829i $$0.226418\pi$$
$$90$$ 0 0
$$91$$ −3.29258 −0.345156
$$92$$ 0 0
$$93$$ −6.37975 −0.661549
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 12.4262 1.26169 0.630847 0.775907i $$-0.282708\pi$$
0.630847 + 0.775907i $$0.282708\pi$$
$$98$$ 0 0
$$99$$ −2.29258 −0.230413
$$100$$ 0 0
$$101$$ 2.15892 0.214820 0.107410 0.994215i $$-0.465744\pi$$
0.107410 + 0.994215i $$0.465744\pi$$
$$102$$ 0 0
$$103$$ −3.07175 −0.302669 −0.151334 0.988483i $$-0.548357\pi$$
−0.151334 + 0.988483i $$0.548357\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 16.4882 1.59397 0.796985 0.603999i $$-0.206427\pi$$
0.796985 + 0.603999i $$0.206427\pi$$
$$108$$ 0 0
$$109$$ 10.2926 0.985850 0.492925 0.870072i $$-0.335928\pi$$
0.492925 + 0.870072i $$0.335928\pi$$
$$110$$ 0 0
$$111$$ 1.79459 0.170335
$$112$$ 0 0
$$113$$ 11.4980 1.08164 0.540820 0.841138i $$-0.318114\pi$$
0.540820 + 0.841138i $$0.318114\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.29258 0.211949
$$118$$ 0 0
$$119$$ 3.81583 0.349796
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −5.15892 −0.465164
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.37975 −0.388640 −0.194320 0.980938i $$-0.562250\pi$$
−0.194320 + 0.980938i $$0.562250\pi$$
$$128$$ 0 0
$$129$$ 7.41484 0.652840
$$130$$ 0 0
$$131$$ −5.01140 −0.437848 −0.218924 0.975742i $$-0.570255\pi$$
−0.218924 + 0.975742i $$0.570255\pi$$
$$132$$ 0 0
$$133$$ 4.33908 0.376246
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 16.2421 1.38765 0.693827 0.720142i $$-0.255923\pi$$
0.693827 + 0.720142i $$0.255923\pi$$
$$138$$ 0 0
$$139$$ −22.7555 −1.93009 −0.965047 0.262076i $$-0.915593\pi$$
−0.965047 + 0.262076i $$0.915593\pi$$
$$140$$ 0 0
$$141$$ 2.32924 0.196157
$$142$$ 0 0
$$143$$ −1.00000 −0.0836242
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −3.23067 −0.266461
$$148$$ 0 0
$$149$$ 18.2673 1.49652 0.748259 0.663407i $$-0.230890\pi$$
0.748259 + 0.663407i $$0.230890\pi$$
$$150$$ 0 0
$$151$$ 16.0832 1.30883 0.654414 0.756136i $$-0.272915\pi$$
0.654414 + 0.756136i $$0.272915\pi$$
$$152$$ 0 0
$$153$$ −2.65691 −0.214798
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 15.1703 1.21072 0.605362 0.795951i $$-0.293028\pi$$
0.605362 + 0.795951i $$0.293028\pi$$
$$158$$ 0 0
$$159$$ −8.57532 −0.680067
$$160$$ 0 0
$$161$$ −12.6471 −0.996729
$$162$$ 0 0
$$163$$ −6.24608 −0.489231 −0.244616 0.969620i $$-0.578662\pi$$
−0.244616 + 0.969620i $$0.578662\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0.594999 0.0460424 0.0230212 0.999735i $$-0.492671\pi$$
0.0230212 + 0.999735i $$0.492671\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −3.02124 −0.231040
$$172$$ 0 0
$$173$$ 1.31783 0.100193 0.0500966 0.998744i $$-0.484047\pi$$
0.0500966 + 0.998744i $$0.484047\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −2.32924 −0.175076
$$178$$ 0 0
$$179$$ −4.70742 −0.351849 −0.175925 0.984404i $$-0.556291\pi$$
−0.175925 + 0.984404i $$0.556291\pi$$
$$180$$ 0 0
$$181$$ −10.0212 −0.744873 −0.372437 0.928058i $$-0.621478\pi$$
−0.372437 + 0.928058i $$0.621478\pi$$
$$182$$ 0 0
$$183$$ 3.03666 0.224476
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.15892 0.0847484
$$188$$ 0 0
$$189$$ 14.6569 1.06613
$$190$$ 0 0
$$191$$ 16.2421 1.17523 0.587617 0.809139i $$-0.300066\pi$$
0.587617 + 0.809139i $$0.300066\pi$$
$$192$$ 0 0
$$193$$ 15.3643 1.10595 0.552974 0.833198i $$-0.313493\pi$$
0.552974 + 0.833198i $$0.313493\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −7.52325 −0.536009 −0.268005 0.963418i $$-0.586364\pi$$
−0.268005 + 0.963418i $$0.586364\pi$$
$$198$$ 0 0
$$199$$ 24.5094 1.73742 0.868712 0.495317i $$-0.164948\pi$$
0.868712 + 0.495317i $$0.164948\pi$$
$$200$$ 0 0
$$201$$ 2.44166 0.172221
$$202$$ 0 0
$$203$$ 21.7653 1.52763
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 8.80599 0.612059
$$208$$ 0 0
$$209$$ 1.31783 0.0911565
$$210$$ 0 0
$$211$$ 5.35449 0.368618 0.184309 0.982868i $$-0.440995\pi$$
0.184309 + 0.982868i $$0.440995\pi$$
$$212$$ 0 0
$$213$$ −0.728661 −0.0499271
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 24.9747 1.69540
$$218$$ 0 0
$$219$$ 4.94367 0.334062
$$220$$ 0 0
$$221$$ −1.15892 −0.0779572
$$222$$ 0 0
$$223$$ −7.40500 −0.495876 −0.247938 0.968776i $$-0.579753\pi$$
−0.247938 + 0.968776i $$0.579753\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −16.2926 −1.08138 −0.540688 0.841223i $$-0.681836\pi$$
−0.540688 + 0.841223i $$0.681836\pi$$
$$228$$ 0 0
$$229$$ −19.0579 −1.25938 −0.629691 0.776846i $$-0.716818\pi$$
−0.629691 + 0.776846i $$0.716818\pi$$
$$230$$ 0 0
$$231$$ −2.76933 −0.182209
$$232$$ 0 0
$$233$$ 6.97475 0.456931 0.228465 0.973552i $$-0.426629\pi$$
0.228465 + 0.973552i $$0.426629\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −9.87774 −0.641628
$$238$$ 0 0
$$239$$ −0.513409 −0.0332097 −0.0166048 0.999862i $$-0.505286\pi$$
−0.0166048 + 0.999862i $$0.505286\pi$$
$$240$$ 0 0
$$241$$ 12.4727 0.803440 0.401720 0.915763i $$-0.368413\pi$$
0.401720 + 0.915763i $$0.368413\pi$$
$$242$$ 0 0
$$243$$ −15.9902 −1.02577
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.31783 −0.0838518
$$248$$ 0 0
$$249$$ 7.50940 0.475889
$$250$$ 0 0
$$251$$ 14.4727 0.913511 0.456756 0.889592i $$-0.349011\pi$$
0.456756 + 0.889592i $$0.349011\pi$$
$$252$$ 0 0
$$253$$ −3.84108 −0.241487
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −22.6218 −1.41111 −0.705555 0.708655i $$-0.749302\pi$$
−0.705555 + 0.708655i $$0.749302\pi$$
$$258$$ 0 0
$$259$$ −7.02525 −0.436528
$$260$$ 0 0
$$261$$ −15.1549 −0.938065
$$262$$ 0 0
$$263$$ −14.5445 −0.896852 −0.448426 0.893820i $$-0.648015\pi$$
−0.448426 + 0.893820i $$0.648015\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −12.0212 −0.735688
$$268$$ 0 0
$$269$$ 7.42624 0.452786 0.226393 0.974036i $$-0.427307\pi$$
0.226393 + 0.974036i $$0.427307\pi$$
$$270$$ 0 0
$$271$$ −1.40500 −0.0853477 −0.0426739 0.999089i $$-0.513588\pi$$
−0.0426739 + 0.999089i $$0.513588\pi$$
$$272$$ 0 0
$$273$$ 2.76933 0.167608
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −17.9397 −1.07789 −0.538945 0.842341i $$-0.681177\pi$$
−0.538945 + 0.842341i $$0.681177\pi$$
$$278$$ 0 0
$$279$$ −17.3896 −1.04109
$$280$$ 0 0
$$281$$ −5.49799 −0.327983 −0.163991 0.986462i $$-0.552437\pi$$
−0.163991 + 0.986462i $$0.552437\pi$$
$$282$$ 0 0
$$283$$ −8.25592 −0.490764 −0.245382 0.969427i $$-0.578913\pi$$
−0.245382 + 0.969427i $$0.578913\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 20.1956 1.19211
$$288$$ 0 0
$$289$$ −15.6569 −0.920995
$$290$$ 0 0
$$291$$ −10.4515 −0.612678
$$292$$ 0 0
$$293$$ 14.6357 0.855025 0.427512 0.904009i $$-0.359390\pi$$
0.427512 + 0.904009i $$0.359390\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 4.45150 0.258302
$$298$$ 0 0
$$299$$ 3.84108 0.222136
$$300$$ 0 0
$$301$$ −29.0268 −1.67308
$$302$$ 0 0
$$303$$ −1.81583 −0.104317
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −18.3292 −1.04610 −0.523052 0.852301i $$-0.675207\pi$$
−0.523052 + 0.852301i $$0.675207\pi$$
$$308$$ 0 0
$$309$$ 2.58360 0.146976
$$310$$ 0 0
$$311$$ 6.09701 0.345729 0.172865 0.984946i $$-0.444698\pi$$
0.172865 + 0.984946i $$0.444698\pi$$
$$312$$ 0 0
$$313$$ −0.317835 −0.0179651 −0.00898254 0.999960i $$-0.502859\pi$$
−0.00898254 + 0.999960i $$0.502859\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −24.3797 −1.36930 −0.684651 0.728871i $$-0.740046\pi$$
−0.684651 + 0.728871i $$0.740046\pi$$
$$318$$ 0 0
$$319$$ 6.61041 0.370112
$$320$$ 0 0
$$321$$ −13.8679 −0.774031
$$322$$ 0 0
$$323$$ 1.52726 0.0849791
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −8.65691 −0.478728
$$328$$ 0 0
$$329$$ −9.11825 −0.502705
$$330$$ 0 0
$$331$$ −3.54850 −0.195043 −0.0975217 0.995233i $$-0.531092\pi$$
−0.0975217 + 0.995233i $$0.531092\pi$$
$$332$$ 0 0
$$333$$ 4.89159 0.268058
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ 0 0
$$339$$ −9.67076 −0.525244
$$340$$ 0 0
$$341$$ 7.58516 0.410759
$$342$$ 0 0
$$343$$ −10.4010 −0.561601
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −8.70742 −0.467439 −0.233719 0.972304i $$-0.575090\pi$$
−0.233719 + 0.972304i $$0.575090\pi$$
$$348$$ 0 0
$$349$$ 16.8565 0.902308 0.451154 0.892446i $$-0.351013\pi$$
0.451154 + 0.892446i $$0.351013\pi$$
$$350$$ 0 0
$$351$$ −4.45150 −0.237603
$$352$$ 0 0
$$353$$ 10.3643 0.551638 0.275819 0.961210i $$-0.411051\pi$$
0.275819 + 0.961210i $$0.411051\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −3.20943 −0.169861
$$358$$ 0 0
$$359$$ −24.9030 −1.31433 −0.657165 0.753747i $$-0.728244\pi$$
−0.657165 + 0.753747i $$0.728244\pi$$
$$360$$ 0 0
$$361$$ −17.2633 −0.908595
$$362$$ 0 0
$$363$$ −0.841083 −0.0441454
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −2.98860 −0.156004 −0.0780018 0.996953i $$-0.524854\pi$$
−0.0780018 + 0.996953i $$0.524854\pi$$
$$368$$ 0 0
$$369$$ −14.0619 −0.732034
$$370$$ 0 0
$$371$$ 33.5697 1.74285
$$372$$ 0 0
$$373$$ −21.4475 −1.11051 −0.555254 0.831681i $$-0.687379\pi$$
−0.555254 + 0.831681i $$0.687379\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.61041 −0.340454
$$378$$ 0 0
$$379$$ 6.45150 0.331391 0.165696 0.986177i $$-0.447013\pi$$
0.165696 + 0.986177i $$0.447013\pi$$
$$380$$ 0 0
$$381$$ 3.68373 0.188723
$$382$$ 0 0
$$383$$ 22.0872 1.12860 0.564301 0.825569i $$-0.309146\pi$$
0.564301 + 0.825569i $$0.309146\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 20.2110 1.02738
$$388$$ 0 0
$$389$$ −31.2633 −1.58511 −0.792556 0.609799i $$-0.791250\pi$$
−0.792556 + 0.609799i $$0.791250\pi$$
$$390$$ 0 0
$$391$$ −4.45150 −0.225122
$$392$$ 0 0
$$393$$ 4.21500 0.212619
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 5.74408 0.288287 0.144143 0.989557i $$-0.453957\pi$$
0.144143 + 0.989557i $$0.453957\pi$$
$$398$$ 0 0
$$399$$ −3.64952 −0.182705
$$400$$ 0 0
$$401$$ 34.1605 1.70589 0.852947 0.521998i $$-0.174813\pi$$
0.852947 + 0.521998i $$0.174813\pi$$
$$402$$ 0 0
$$403$$ −7.58516 −0.377844
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.13366 −0.105762
$$408$$ 0 0
$$409$$ −20.7188 −1.02448 −0.512240 0.858842i $$-0.671184\pi$$
−0.512240 + 0.858842i $$0.671184\pi$$
$$410$$ 0 0
$$411$$ −13.6609 −0.673844
$$412$$ 0 0
$$413$$ 9.11825 0.448680
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 19.1392 0.937253
$$418$$ 0 0
$$419$$ 6.81984 0.333171 0.166586 0.986027i $$-0.446726\pi$$
0.166586 + 0.986027i $$0.446726\pi$$
$$420$$ 0 0
$$421$$ −36.5134 −1.77955 −0.889777 0.456395i $$-0.849140\pi$$
−0.889777 + 0.456395i $$0.849140\pi$$
$$422$$ 0 0
$$423$$ 6.34891 0.308695
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −11.8876 −0.575280
$$428$$ 0 0
$$429$$ 0.841083 0.0406079
$$430$$ 0 0
$$431$$ −10.4010 −0.500998 −0.250499 0.968117i $$-0.580595\pi$$
−0.250499 + 0.968117i $$0.580595\pi$$
$$432$$ 0 0
$$433$$ 29.7090 1.42772 0.713861 0.700287i $$-0.246945\pi$$
0.713861 + 0.700287i $$0.246945\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −5.06191 −0.242144
$$438$$ 0 0
$$439$$ −30.3292 −1.44754 −0.723768 0.690044i $$-0.757591\pi$$
−0.723768 + 0.690044i $$0.757591\pi$$
$$440$$ 0 0
$$441$$ −8.80599 −0.419333
$$442$$ 0 0
$$443$$ 25.0465 1.18999 0.594997 0.803728i $$-0.297153\pi$$
0.594997 + 0.803728i $$0.297153\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −15.3643 −0.726708
$$448$$ 0 0
$$449$$ −38.2322 −1.80429 −0.902145 0.431432i $$-0.858008\pi$$
−0.902145 + 0.431432i $$0.858008\pi$$
$$450$$ 0 0
$$451$$ 6.13366 0.288823
$$452$$ 0 0
$$453$$ −13.5273 −0.635566
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 32.7302 1.53106 0.765528 0.643403i $$-0.222478\pi$$
0.765528 + 0.643403i $$0.222478\pi$$
$$458$$ 0 0
$$459$$ 5.15892 0.240798
$$460$$ 0 0
$$461$$ 20.8118 0.969303 0.484651 0.874707i $$-0.338946\pi$$
0.484651 + 0.874707i $$0.338946\pi$$
$$462$$ 0 0
$$463$$ −1.51185 −0.0702614 −0.0351307 0.999383i $$-0.511185\pi$$
−0.0351307 + 0.999383i $$0.511185\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −23.4783 −1.08645 −0.543223 0.839588i $$-0.682796\pi$$
−0.543223 + 0.839588i $$0.682796\pi$$
$$468$$ 0 0
$$469$$ −9.55834 −0.441363
$$470$$ 0 0
$$471$$ −12.7595 −0.587926
$$472$$ 0 0
$$473$$ −8.81583 −0.405352
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −23.3742 −1.07023
$$478$$ 0 0
$$479$$ −14.7188 −0.672520 −0.336260 0.941769i $$-0.609162\pi$$
−0.336260 + 0.941769i $$0.609162\pi$$
$$480$$ 0 0
$$481$$ 2.13366 0.0972866
$$482$$ 0 0
$$483$$ 10.6372 0.484011
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −23.7595 −1.07665 −0.538323 0.842739i $$-0.680942\pi$$
−0.538323 + 0.842739i $$0.680942\pi$$
$$488$$ 0 0
$$489$$ 5.25347 0.237570
$$490$$ 0 0
$$491$$ 18.7555 0.846423 0.423211 0.906031i $$-0.360903\pi$$
0.423211 + 0.906031i $$0.360903\pi$$
$$492$$ 0 0
$$493$$ 7.66092 0.345031
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.85249 0.127951
$$498$$ 0 0
$$499$$ −26.9747 −1.20756 −0.603778 0.797153i $$-0.706339\pi$$
−0.603778 + 0.797153i $$0.706339\pi$$
$$500$$ 0 0
$$501$$ −0.500443 −0.0223582
$$502$$ 0 0
$$503$$ −43.2941 −1.93039 −0.965195 0.261530i $$-0.915773\pi$$
−0.965195 + 0.261530i $$0.915773\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 10.0930 0.448246
$$508$$ 0 0
$$509$$ −10.8312 −0.480086 −0.240043 0.970762i $$-0.577162\pi$$
−0.240043 + 0.970762i $$0.577162\pi$$
$$510$$ 0 0
$$511$$ −19.3529 −0.856123
$$512$$ 0 0
$$513$$ 5.86634 0.259005
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −2.76933 −0.121795
$$518$$ 0 0
$$519$$ −1.10841 −0.0486537
$$520$$ 0 0
$$521$$ 21.1337 0.925883 0.462941 0.886389i $$-0.346794\pi$$
0.462941 + 0.886389i $$0.346794\pi$$
$$522$$ 0 0
$$523$$ 31.7921 1.39017 0.695087 0.718926i $$-0.255366\pi$$
0.695087 + 0.718926i $$0.255366\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.79057 0.382923
$$528$$ 0 0
$$529$$ −8.24608 −0.358525
$$530$$ 0 0
$$531$$ −6.34891 −0.275519
$$532$$ 0 0
$$533$$ −6.13366 −0.265678
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 3.95933 0.170858
$$538$$ 0 0
$$539$$ 3.84108 0.165447
$$540$$ 0 0
$$541$$ 30.3757 1.30595 0.652977 0.757377i $$-0.273520\pi$$
0.652977 + 0.757377i $$0.273520\pi$$
$$542$$ 0 0
$$543$$ 8.42869 0.361710
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −4.83707 −0.206818 −0.103409 0.994639i $$-0.532975\pi$$
−0.103409 + 0.994639i $$0.532975\pi$$
$$548$$ 0 0
$$549$$ 8.27716 0.353261
$$550$$ 0 0
$$551$$ 8.71143 0.371120
$$552$$ 0 0
$$553$$ 38.6683 1.64434
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −30.6317 −1.29790 −0.648952 0.760829i $$-0.724793\pi$$
−0.648952 + 0.760829i $$0.724793\pi$$
$$558$$ 0 0
$$559$$ 8.81583 0.372870
$$560$$ 0 0
$$561$$ −0.974745 −0.0411538
$$562$$ 0 0
$$563$$ 14.2461 0.600401 0.300200 0.953876i $$-0.402946\pi$$
0.300200 + 0.953876i $$0.402946\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 10.3178 0.433308
$$568$$ 0 0
$$569$$ 40.3194 1.69028 0.845139 0.534547i $$-0.179518\pi$$
0.845139 + 0.534547i $$0.179518\pi$$
$$570$$ 0 0
$$571$$ −8.98860 −0.376161 −0.188081 0.982154i $$-0.560227\pi$$
−0.188081 + 0.982154i $$0.560227\pi$$
$$572$$ 0 0
$$573$$ −13.6609 −0.570693
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −10.2787 −0.427909 −0.213955 0.976844i $$-0.568634\pi$$
−0.213955 + 0.976844i $$0.568634\pi$$
$$578$$ 0 0
$$579$$ −12.9227 −0.537048
$$580$$ 0 0
$$581$$ −29.3970 −1.21959
$$582$$ 0 0
$$583$$ 10.1956 0.422258
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −4.52325 −0.186694 −0.0933472 0.995634i $$-0.529757\pi$$
−0.0933472 + 0.995634i $$0.529757\pi$$
$$588$$ 0 0
$$589$$ 9.99599 0.411877
$$590$$ 0 0
$$591$$ 6.32767 0.260286
$$592$$ 0 0
$$593$$ 28.0326 1.15116 0.575581 0.817745i $$-0.304776\pi$$
0.575581 + 0.817745i $$0.304776\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −20.6144 −0.843692
$$598$$ 0 0
$$599$$ 15.8876 0.649149 0.324574 0.945860i $$-0.394779\pi$$
0.324574 + 0.945860i $$0.394779\pi$$
$$600$$ 0 0
$$601$$ 4.83124 0.197071 0.0985353 0.995134i $$-0.468584\pi$$
0.0985353 + 0.995134i $$0.468584\pi$$
$$602$$ 0 0
$$603$$ 6.65535 0.271027
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 37.2633 1.51247 0.756236 0.654299i $$-0.227036\pi$$
0.756236 + 0.654299i $$0.227036\pi$$
$$608$$ 0 0
$$609$$ −18.3064 −0.741814
$$610$$ 0 0
$$611$$ 2.76933 0.112035
$$612$$ 0 0
$$613$$ −44.1857 −1.78465 −0.892323 0.451398i $$-0.850925\pi$$
−0.892323 + 0.451398i $$0.850925\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 13.3080 0.535760 0.267880 0.963452i $$-0.413677\pi$$
0.267880 + 0.963452i $$0.413677\pi$$
$$618$$ 0 0
$$619$$ −33.4783 −1.34561 −0.672804 0.739821i $$-0.734910\pi$$
−0.672804 + 0.739821i $$0.734910\pi$$
$$620$$ 0 0
$$621$$ −17.0986 −0.686142
$$622$$ 0 0
$$623$$ 47.0595 1.88540
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −1.10841 −0.0442655
$$628$$ 0 0
$$629$$ −2.47274 −0.0985945
$$630$$ 0 0
$$631$$ −49.1411 −1.95627 −0.978137 0.207961i $$-0.933317\pi$$
−0.978137 + 0.207961i $$0.933317\pi$$
$$632$$ 0 0
$$633$$ −4.50357 −0.179001
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3.84108 −0.152189
$$638$$ 0 0
$$639$$ −1.98615 −0.0785708
$$640$$ 0 0
$$641$$ 8.36433 0.330371 0.165186 0.986262i $$-0.447178\pi$$
0.165186 + 0.986262i $$0.447178\pi$$
$$642$$ 0 0
$$643$$ 40.8020 1.60907 0.804536 0.593903i $$-0.202414\pi$$
0.804536 + 0.593903i $$0.202414\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −40.0832 −1.57583 −0.787916 0.615783i $$-0.788840\pi$$
−0.787916 + 0.615783i $$0.788840\pi$$
$$648$$ 0 0
$$649$$ 2.76933 0.108706
$$650$$ 0 0
$$651$$ −21.0058 −0.823283
$$652$$ 0 0
$$653$$ −14.1352 −0.553154 −0.276577 0.960992i $$-0.589200\pi$$
−0.276577 + 0.960992i $$0.589200\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 13.4752 0.525717
$$658$$ 0 0
$$659$$ −1.51341 −0.0589541 −0.0294770 0.999565i $$-0.509384\pi$$
−0.0294770 + 0.999565i $$0.509384\pi$$
$$660$$ 0 0
$$661$$ 15.4050 0.599185 0.299593 0.954067i $$-0.403149\pi$$
0.299593 + 0.954067i $$0.403149\pi$$
$$662$$ 0 0
$$663$$ 0.974745 0.0378560
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −25.3911 −0.983149
$$668$$ 0 0
$$669$$ 6.22822 0.240797
$$670$$ 0 0
$$671$$ −3.61041 −0.139379
$$672$$ 0 0
$$673$$ −36.8426 −1.42018 −0.710090 0.704111i $$-0.751346\pi$$
−0.710090 + 0.704111i $$0.751346\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −28.0367 −1.07754 −0.538768 0.842454i $$-0.681110\pi$$
−0.538768 + 0.842454i $$0.681110\pi$$
$$678$$ 0 0
$$679$$ 40.9144 1.57015
$$680$$ 0 0
$$681$$ 13.7034 0.525116
$$682$$ 0 0
$$683$$ −28.7416 −1.09977 −0.549884 0.835241i $$-0.685328\pi$$
−0.549884 + 0.835241i $$0.685328\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 16.0293 0.611555
$$688$$ 0 0
$$689$$ −10.1956 −0.388420
$$690$$ 0 0
$$691$$ −23.2729 −0.885343 −0.442671 0.896684i $$-0.645969\pi$$
−0.442671 + 0.896684i $$0.645969\pi$$
$$692$$ 0 0
$$693$$ −7.54850 −0.286744
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 7.10841 0.269250
$$698$$ 0 0
$$699$$ −5.86634 −0.221885
$$700$$ 0 0
$$701$$ −26.8891 −1.01559 −0.507794 0.861478i $$-0.669539\pi$$
−0.507794 + 0.861478i $$0.669539\pi$$
$$702$$ 0 0
$$703$$ −2.81181 −0.106050
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 7.10841 0.267339
$$708$$ 0 0
$$709$$ −16.6357 −0.624766 −0.312383 0.949956i $$-0.601127\pi$$
−0.312383 + 0.949956i $$0.601127\pi$$
$$710$$ 0 0
$$711$$ −26.9242 −1.00974
$$712$$ 0 0
$$713$$ −29.1352 −1.09112
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0.431820 0.0161266
$$718$$ 0 0
$$719$$ −11.8158 −0.440656 −0.220328 0.975426i $$-0.570713\pi$$
−0.220328 + 0.975426i $$0.570713\pi$$
$$720$$ 0 0
$$721$$ −10.1140 −0.376664
$$722$$ 0 0
$$723$$ −10.4906 −0.390150
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 30.7767 1.14145 0.570723 0.821143i $$-0.306663\pi$$
0.570723 + 0.821143i $$0.306663\pi$$
$$728$$ 0 0
$$729$$ 4.04806 0.149928
$$730$$ 0 0
$$731$$ −10.2168 −0.377883
$$732$$ 0 0
$$733$$ 23.6782 0.874572 0.437286 0.899322i $$-0.355940\pi$$
0.437286 + 0.899322i $$0.355940\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −2.90299 −0.106933
$$738$$ 0 0
$$739$$ 30.6569 1.12773 0.563866 0.825866i $$-0.309313\pi$$
0.563866 + 0.825866i $$0.309313\pi$$
$$740$$ 0 0
$$741$$ 1.10841 0.0407184
$$742$$ 0 0
$$743$$ −17.4164 −0.638946 −0.319473 0.947595i $$-0.603506\pi$$
−0.319473 + 0.947595i $$0.603506\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 20.4687 0.748911
$$748$$ 0 0
$$749$$ 54.2886 1.98366
$$750$$ 0 0
$$751$$ −24.2926 −0.886449 −0.443224 0.896411i $$-0.646166\pi$$
−0.443224 + 0.896411i $$0.646166\pi$$
$$752$$ 0 0
$$753$$ −12.1728 −0.443600
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −34.1565 −1.24144 −0.620719 0.784033i $$-0.713159\pi$$
−0.620719 + 0.784033i $$0.713159\pi$$
$$758$$ 0 0
$$759$$ 3.23067 0.117266
$$760$$ 0 0
$$761$$ 52.5347 1.90438 0.952190 0.305507i $$-0.0988260\pi$$
0.952190 + 0.305507i $$0.0988260\pi$$
$$762$$ 0 0
$$763$$ 33.8891 1.22687
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −2.76933 −0.0999948
$$768$$ 0 0
$$769$$ −24.4010 −0.879922 −0.439961 0.898017i $$-0.645008\pi$$
−0.439961 + 0.898017i $$0.645008\pi$$
$$770$$ 0 0
$$771$$ 19.0268 0.685234
$$772$$ 0 0
$$773$$ 12.2966 0.442278 0.221139 0.975242i $$-0.429023\pi$$
0.221139 + 0.975242i $$0.429023\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 5.90882 0.211978
$$778$$ 0 0
$$779$$ 8.08315 0.289609
$$780$$ 0 0
$$781$$ 0.866337 0.0310000
$$782$$ 0 0
$$783$$ 29.4262 1.05161
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 2.66831 0.0951151 0.0475575 0.998868i $$-0.484856\pi$$
0.0475575 + 0.998868i $$0.484856\pi$$
$$788$$ 0 0
$$789$$ 12.2331 0.435511
$$790$$ 0 0
$$791$$ 37.8581 1.34608
$$792$$ 0 0
$$793$$ 3.61041 0.128210
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −26.2054 −0.928243 −0.464122 0.885771i $$-0.653630\pi$$
−0.464122 + 0.885771i $$0.653630\pi$$
$$798$$ 0 0
$$799$$ −3.20943 −0.113541
$$800$$ 0 0
$$801$$ −32.7669 −1.15776
$$802$$ 0 0
$$803$$ −5.87774 −0.207421
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.24608 −0.219873
$$808$$ 0 0
$$809$$ −20.0733 −0.705740 −0.352870 0.935672i $$-0.614794\pi$$
−0.352870 + 0.935672i $$0.614794\pi$$
$$810$$ 0 0
$$811$$ −14.2575 −0.500648 −0.250324 0.968162i $$-0.580537\pi$$
−0.250324 + 0.968162i $$0.580537\pi$$
$$812$$ 0 0
$$813$$ 1.18172 0.0414448
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −11.6178 −0.406455
$$818$$ 0 0
$$819$$ 7.54850 0.263766
$$820$$ 0 0
$$821$$ −16.0872 −0.561446 −0.280723 0.959789i $$-0.590574\pi$$
−0.280723 + 0.959789i $$0.590574\pi$$
$$822$$ 0 0
$$823$$ −27.6120 −0.962493 −0.481247 0.876585i $$-0.659816\pi$$
−0.481247 + 0.876585i $$0.659816\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 20.4980 0.712785 0.356393 0.934336i $$-0.384007\pi$$
0.356393 + 0.934336i $$0.384007\pi$$
$$828$$ 0 0
$$829$$ −30.5232 −1.06012 −0.530058 0.847961i $$-0.677830\pi$$
−0.530058 + 0.847961i $$0.677830\pi$$
$$830$$ 0 0
$$831$$ 15.0887 0.523422
$$832$$ 0 0
$$833$$ 4.45150 0.154235
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 33.7653 1.16710
$$838$$ 0 0
$$839$$ −28.8737 −0.996832 −0.498416 0.866938i $$-0.666085\pi$$
−0.498416 + 0.866938i $$0.666085\pi$$
$$840$$ 0 0
$$841$$ 14.6976 0.506813
$$842$$ 0 0
$$843$$ 4.62427 0.159268
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 3.29258 0.113134
$$848$$ 0 0
$$849$$ 6.94391 0.238314
$$850$$ 0 0
$$851$$ 8.19557 0.280941
$$852$$ 0 0
$$853$$ 47.5729 1.62886 0.814432 0.580259i $$-0.197049\pi$$
0.814432 + 0.580259i $$0.197049\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −19.7148 −0.673445 −0.336723 0.941604i $$-0.609318\pi$$
−0.336723 + 0.941604i $$0.609318\pi$$
$$858$$ 0 0
$$859$$ 47.6822 1.62689 0.813447 0.581639i $$-0.197588\pi$$
0.813447 + 0.581639i $$0.197588\pi$$
$$860$$ 0 0
$$861$$ −16.9861 −0.578886
$$862$$ 0 0
$$863$$ 15.8297 0.538849 0.269424 0.963022i $$-0.413167\pi$$
0.269424 + 0.963022i $$0.413167\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.1688 0.447234
$$868$$ 0 0
$$869$$ 11.7441 0.398391
$$870$$ 0 0
$$871$$ 2.90299 0.0983642
$$872$$ 0 0
$$873$$ −28.4882 −0.964178
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 43.5248 1.46973 0.734864 0.678214i $$-0.237246\pi$$
0.734864 + 0.678214i $$0.237246\pi$$
$$878$$ 0 0
$$879$$ −12.3098 −0.415200
$$880$$ 0 0
$$881$$ 53.5461 1.80401 0.902006 0.431723i $$-0.142094\pi$$
0.902006 + 0.431723i $$0.142094\pi$$
$$882$$ 0 0
$$883$$ −45.2535 −1.52290 −0.761450 0.648223i $$-0.775512\pi$$
−0.761450 + 0.648223i $$0.775512\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1.54850 0.0519936 0.0259968 0.999662i $$-0.491724\pi$$
0.0259968 + 0.999662i $$0.491724\pi$$
$$888$$ 0 0
$$889$$ −14.4207 −0.483654
$$890$$ 0 0
$$891$$ 3.13366 0.104982
$$892$$ 0 0
$$893$$ −3.64952 −0.122127
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −3.23067 −0.107869
$$898$$ 0 0
$$899$$ 50.1411 1.67230
$$900$$ 0 0
$$901$$ 11.8158 0.393642
$$902$$ 0 0
$$903$$ 24.4140 0.812446
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 40.7090 1.35172 0.675860 0.737030i $$-0.263772\pi$$
0.675860 + 0.737030i $$0.263772\pi$$
$$908$$ 0 0
$$909$$ −4.94949 −0.164164
$$910$$ 0 0
$$911$$ −24.9030 −0.825073 −0.412537 0.910941i $$-0.635357\pi$$
−0.412537 + 0.910941i $$0.635357\pi$$
$$912$$ 0 0
$$913$$ −8.92825 −0.295482
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −16.5004 −0.544893
$$918$$ 0 0
$$919$$ −2.09299 −0.0690414 −0.0345207 0.999404i $$-0.510990\pi$$
−0.0345207 + 0.999404i $$0.510990\pi$$
$$920$$ 0 0
$$921$$ 15.4164 0.507988
$$922$$ 0 0
$$923$$ −0.866337 −0.0285158
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 7.04223 0.231297
$$928$$ 0 0
$$929$$ −36.7555 −1.20591 −0.602954 0.797776i $$-0.706010\pi$$
−0.602954 + 0.797776i $$0.706010\pi$$
$$930$$ 0 0
$$931$$ 5.06191 0.165897
$$932$$ 0 0
$$933$$ −5.12809 −0.167886
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 50.2673 1.64216 0.821081 0.570812i $$-0.193371\pi$$
0.821081 + 0.570812i $$0.193371\pi$$
$$938$$ 0 0
$$939$$ 0.267325 0.00872383
$$940$$ 0 0
$$941$$ 40.3000 1.31374 0.656871 0.754003i $$-0.271880\pi$$
0.656871 + 0.754003i $$0.271880\pi$$
$$942$$ 0 0
$$943$$ −23.5599 −0.767216
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −0.133663 −0.00434345 −0.00217173 0.999998i $$-0.500691\pi$$
−0.00217173 + 0.999998i $$0.500691\pi$$
$$948$$ 0 0
$$949$$ 5.87774 0.190800
$$950$$ 0 0
$$951$$ 20.5054 0.664933
$$952$$ 0 0
$$953$$ −0.728661 −0.0236037 −0.0118018 0.999930i $$-0.503757\pi$$
−0.0118018 + 0.999930i $$0.503757\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −5.55991 −0.179726
$$958$$ 0 0
$$959$$ 53.4783 1.72690
$$960$$ 0 0
$$961$$ 26.5347 0.855956
$$962$$ 0 0
$$963$$ −37.8004 −1.21810
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 56.5769 1.81939 0.909695 0.415277i $$-0.136315\pi$$
0.909695 + 0.415277i $$0.136315\pi$$
$$968$$ 0 0
$$969$$ −1.28455 −0.0412658
$$970$$ 0 0
$$971$$ 48.8232 1.56681 0.783406 0.621511i $$-0.213481\pi$$
0.783406 + 0.621511i $$0.213481\pi$$
$$972$$ 0 0
$$973$$ −74.9242 −2.40196
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 7.21099 0.230700 0.115350 0.993325i $$-0.463201\pi$$
0.115350 + 0.993325i $$0.463201\pi$$
$$978$$ 0 0
$$979$$ 14.2926 0.456793
$$980$$ 0 0
$$981$$ −23.5966 −0.753380
$$982$$ 0 0
$$983$$ 58.4278 1.86356 0.931779 0.363027i $$-0.118257\pi$$
0.931779 + 0.363027i $$0.118257\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 7.66920 0.244113
$$988$$ 0 0
$$989$$ 33.8623 1.07676
$$990$$ 0 0
$$991$$ −24.0114 −0.762747 −0.381374 0.924421i $$-0.624549\pi$$
−0.381374 + 0.924421i $$0.624549\pi$$
$$992$$ 0 0
$$993$$ 2.98458 0.0947129
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −38.4222 −1.21684 −0.608422 0.793614i $$-0.708197\pi$$
−0.608422 + 0.793614i $$0.708197\pi$$
$$998$$ 0 0
$$999$$ −9.49799 −0.300503
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 2200.2.a.u.1.2 3
4.3 odd 2 4400.2.a.bz.1.2 3
5.2 odd 4 2200.2.b.m.1849.4 6
5.3 odd 4 2200.2.b.m.1849.3 6
5.4 even 2 2200.2.a.v.1.2 yes 3
20.3 even 4 4400.2.b.bb.4049.4 6
20.7 even 4 4400.2.b.bb.4049.3 6
20.19 odd 2 4400.2.a.by.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
2200.2.a.u.1.2 3 1.1 even 1 trivial
2200.2.a.v.1.2 yes 3 5.4 even 2
2200.2.b.m.1849.3 6 5.3 odd 4
2200.2.b.m.1849.4 6 5.2 odd 4
4400.2.a.by.1.2 3 20.19 odd 2
4400.2.a.bz.1.2 3 4.3 odd 2
4400.2.b.bb.4049.3 6 20.7 even 4
4400.2.b.bb.4049.4 6 20.3 even 4