# Properties

 Label 666.2.a.b.1.1 Level $666$ Weight $2$ Character 666.1 Self dual yes Analytic conductor $5.318$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(1,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$666 = 2 \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 666.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: $$5.31803677462$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 222) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 666.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{7} -1.00000 q^{8} -1.00000 q^{11} +1.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +3.00000 q^{19} +1.00000 q^{22} +1.00000 q^{23} -5.00000 q^{25} -1.00000 q^{26} +3.00000 q^{28} +4.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} -1.00000 q^{37} -3.00000 q^{38} +10.0000 q^{41} +12.0000 q^{43} -1.00000 q^{44} -1.00000 q^{46} +6.00000 q^{47} +2.00000 q^{49} +5.00000 q^{50} +1.00000 q^{52} +1.00000 q^{53} -3.00000 q^{56} -4.00000 q^{58} +2.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +2.00000 q^{67} +3.00000 q^{68} -3.00000 q^{73} +1.00000 q^{74} +3.00000 q^{76} -3.00000 q^{77} +14.0000 q^{79} -10.0000 q^{82} -9.00000 q^{83} -12.0000 q^{86} +1.00000 q^{88} +3.00000 q^{89} +3.00000 q^{91} +1.00000 q^{92} -6.00000 q^{94} -10.0000 q^{97} -2.00000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ 3.00000 0.566947
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.00000 −0.164399
$$38$$ −3.00000 −0.486664
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 5.00000 0.707107
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3.00000 −0.400892
$$57$$ 0 0
$$58$$ −4.00000 −0.525226
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −3.00000 −0.351123 −0.175562 0.984468i $$-0.556174\pi$$
−0.175562 + 0.984468i $$0.556174\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 3.00000 0.344124
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −10.0000 −1.10432
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −12.0000 −1.29399
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 0 0
$$91$$ 3.00000 0.314485
$$92$$ 1.00000 0.104257
$$93$$ 0 0
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ −2.00000 −0.197066 −0.0985329 0.995134i $$-0.531415\pi$$
−0.0985329 + 0.995134i $$0.531415\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ 13.0000 1.25676 0.628379 0.777908i $$-0.283719\pi$$
0.628379 + 0.777908i $$0.283719\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 3.00000 0.283473
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 9.00000 0.825029
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −10.0000 −0.873704 −0.436852 0.899533i $$-0.643907\pi$$
−0.436852 + 0.899533i $$0.643907\pi$$
$$132$$ 0 0
$$133$$ 9.00000 0.780399
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ −20.0000 −1.70872 −0.854358 0.519685i $$-0.826049\pi$$
−0.854358 + 0.519685i $$0.826049\pi$$
$$138$$ 0 0
$$139$$ −10.0000 −0.848189 −0.424094 0.905618i $$-0.639408\pi$$
−0.424094 + 0.905618i $$0.639408\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −1.00000 −0.0836242
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3.00000 0.248282
$$147$$ 0 0
$$148$$ −1.00000 −0.0821995
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 1.00000 0.0813788 0.0406894 0.999172i $$-0.487045\pi$$
0.0406894 + 0.999172i $$0.487045\pi$$
$$152$$ −3.00000 −0.243332
$$153$$ 0 0
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ −14.0000 −1.11378
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ 17.0000 1.33154 0.665771 0.746156i $$-0.268103\pi$$
0.665771 + 0.746156i $$0.268103\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ 9.00000 0.698535
$$167$$ −21.0000 −1.62503 −0.812514 0.582941i $$-0.801902\pi$$
−0.812514 + 0.582941i $$0.801902\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 12.0000 0.914991
$$173$$ −1.00000 −0.0760286 −0.0380143 0.999277i $$-0.512103\pi$$
−0.0380143 + 0.999277i $$0.512103\pi$$
$$174$$ 0 0
$$175$$ −15.0000 −1.13389
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ −3.00000 −0.224860
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ −3.00000 −0.222375
$$183$$ 0 0
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3.00000 −0.219382
$$188$$ 6.00000 0.437595
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3.00000 −0.217072 −0.108536 0.994092i $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ 0 0
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ −13.0000 −0.926212 −0.463106 0.886303i $$-0.653265\pi$$
−0.463106 + 0.886303i $$0.653265\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 0 0
$$202$$ −6.00000 −0.422159
$$203$$ 12.0000 0.842235
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2.00000 0.139347
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 1.00000 0.0686803
$$213$$ 0 0
$$214$$ −13.0000 −0.888662
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −18.0000 −1.22192
$$218$$ −11.0000 −0.745014
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −4.00000 −0.262613
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ −9.00000 −0.583383
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 10.0000 0.642824
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.00000 0.190885
$$248$$ 6.00000 0.381000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 0 0
$$253$$ −1.00000 −0.0628695
$$254$$ 7.00000 0.439219
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −29.0000 −1.80897 −0.904485 0.426505i $$-0.859745\pi$$
−0.904485 + 0.426505i $$0.859745\pi$$
$$258$$ 0 0
$$259$$ −3.00000 −0.186411
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 10.0000 0.617802
$$263$$ −26.0000 −1.60323 −0.801614 0.597841i $$-0.796025\pi$$
−0.801614 + 0.597841i $$0.796025\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −9.00000 −0.551825
$$267$$ 0 0
$$268$$ 2.00000 0.122169
$$269$$ 7.00000 0.426798 0.213399 0.976965i $$-0.431547\pi$$
0.213399 + 0.976965i $$0.431547\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 0 0
$$274$$ 20.0000 1.20824
$$275$$ 5.00000 0.301511
$$276$$ 0 0
$$277$$ −1.00000 −0.0600842 −0.0300421 0.999549i $$-0.509564\pi$$
−0.0300421 + 0.999549i $$0.509564\pi$$
$$278$$ 10.0000 0.599760
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 31.0000 1.84930 0.924652 0.380812i $$-0.124356\pi$$
0.924652 + 0.380812i $$0.124356\pi$$
$$282$$ 0 0
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 1.00000 0.0591312
$$287$$ 30.0000 1.77084
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −3.00000 −0.175562
$$293$$ −31.0000 −1.81104 −0.905520 0.424304i $$-0.860519\pi$$
−0.905520 + 0.424304i $$0.860519\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ 18.0000 1.04271
$$299$$ 1.00000 0.0578315
$$300$$ 0 0
$$301$$ 36.0000 2.07501
$$302$$ −1.00000 −0.0575435
$$303$$ 0 0
$$304$$ 3.00000 0.172062
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 30.0000 1.69570 0.847850 0.530236i $$-0.177897\pi$$
0.847850 + 0.530236i $$0.177897\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −3.00000 −0.167183
$$323$$ 9.00000 0.500773
$$324$$ 0 0
$$325$$ −5.00000 −0.277350
$$326$$ −17.0000 −0.941543
$$327$$ 0 0
$$328$$ −10.0000 −0.552158
$$329$$ 18.0000 0.992372
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ −9.00000 −0.493939
$$333$$ 0 0
$$334$$ 21.0000 1.14907
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 21.0000 1.14394 0.571971 0.820274i $$-0.306179\pi$$
0.571971 + 0.820274i $$0.306179\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 1.00000 0.0537603
$$347$$ −22.0000 −1.18102 −0.590511 0.807030i $$-0.701074\pi$$
−0.590511 + 0.807030i $$0.701074\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 15.0000 0.801784
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 3.00000 0.159000
$$357$$ 0 0
$$358$$ −24.0000 −1.26844
$$359$$ −30.0000 −1.58334 −0.791670 0.610949i $$-0.790788\pi$$
−0.791670 + 0.610949i $$0.790788\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 20.0000 1.05118
$$363$$ 0 0
$$364$$ 3.00000 0.157243
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −21.0000 −1.09619 −0.548096 0.836416i $$-0.684647\pi$$
−0.548096 + 0.836416i $$0.684647\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3.00000 0.155752
$$372$$ 0 0
$$373$$ 20.0000 1.03556 0.517780 0.855514i $$-0.326758\pi$$
0.517780 + 0.855514i $$0.326758\pi$$
$$374$$ 3.00000 0.155126
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 3.00000 0.153493
$$383$$ −1.00000 −0.0510976 −0.0255488 0.999674i $$-0.508133\pi$$
−0.0255488 + 0.999674i $$0.508133\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 18.0000 0.916176
$$387$$ 0 0
$$388$$ −10.0000 −0.507673
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 3.00000 0.151717
$$392$$ −2.00000 −0.101015
$$393$$ 0 0
$$394$$ 13.0000 0.654931
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −12.0000 −0.602263 −0.301131 0.953583i $$-0.597364\pi$$
−0.301131 + 0.953583i $$0.597364\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ 11.0000 0.549314 0.274657 0.961542i $$-0.411436\pi$$
0.274657 + 0.961542i $$0.411436\pi$$
$$402$$ 0 0
$$403$$ −6.00000 −0.298881
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ −12.0000 −0.595550
$$407$$ 1.00000 0.0495682
$$408$$ 0 0
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −2.00000 −0.0985329
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ 3.00000 0.146735
$$419$$ 21.0000 1.02592 0.512959 0.858413i $$-0.328549\pi$$
0.512959 + 0.858413i $$0.328549\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 0 0
$$424$$ −1.00000 −0.0485643
$$425$$ −15.0000 −0.727607
$$426$$ 0 0
$$427$$ 6.00000 0.290360
$$428$$ 13.0000 0.628379
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5.00000 −0.240842 −0.120421 0.992723i $$-0.538424\pi$$
−0.120421 + 0.992723i $$0.538424\pi$$
$$432$$ 0 0
$$433$$ 7.00000 0.336399 0.168199 0.985753i $$-0.446205\pi$$
0.168199 + 0.985753i $$0.446205\pi$$
$$434$$ 18.0000 0.864028
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ 3.00000 0.143509
$$438$$ 0 0
$$439$$ 34.0000 1.62273 0.811366 0.584539i $$-0.198725\pi$$
0.811366 + 0.584539i $$0.198725\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3.00000 −0.142695
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 0 0
$$448$$ 3.00000 0.141737
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ −10.0000 −0.470882
$$452$$ −18.0000 −0.846649
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −20.0000 −0.935561 −0.467780 0.883845i $$-0.654946\pi$$
−0.467780 + 0.883845i $$0.654946\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ −26.0000 −1.20832 −0.604161 0.796862i $$-0.706492\pi$$
−0.604161 + 0.796862i $$0.706492\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 0 0
$$469$$ 6.00000 0.277054
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −12.0000 −0.551761
$$474$$ 0 0
$$475$$ −15.0000 −0.688247
$$476$$ 9.00000 0.412514
$$477$$ 0 0
$$478$$ −16.0000 −0.731823
$$479$$ 19.0000 0.868132 0.434066 0.900881i $$-0.357078\pi$$
0.434066 + 0.900881i $$0.357078\pi$$
$$480$$ 0 0
$$481$$ −1.00000 −0.0455961
$$482$$ −8.00000 −0.364390
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −6.00000 −0.271886 −0.135943 0.990717i $$-0.543406\pi$$
−0.135943 + 0.990717i $$0.543406\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 11.0000 0.496423 0.248212 0.968706i $$-0.420157\pi$$
0.248212 + 0.968706i $$0.420157\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ −3.00000 −0.134976
$$495$$ 0 0
$$496$$ −6.00000 −0.269408
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −15.0000 −0.671492 −0.335746 0.941953i $$-0.608988\pi$$
−0.335746 + 0.941953i $$0.608988\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −18.0000 −0.803379
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1.00000 0.0444554
$$507$$ 0 0
$$508$$ −7.00000 −0.310575
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ −9.00000 −0.398137
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 29.0000 1.27914
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −6.00000 −0.263880
$$518$$ 3.00000 0.131812
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ 32.0000 1.39926 0.699631 0.714504i $$-0.253348\pi$$
0.699631 + 0.714504i $$0.253348\pi$$
$$524$$ −10.0000 −0.436852
$$525$$ 0 0
$$526$$ 26.0000 1.13365
$$527$$ −18.0000 −0.784092
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 9.00000 0.390199
$$533$$ 10.0000 0.433148
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −2.00000 −0.0863868
$$537$$ 0 0
$$538$$ −7.00000 −0.301791
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ −27.0000 −1.16082 −0.580410 0.814324i $$-0.697108\pi$$
−0.580410 + 0.814324i $$0.697108\pi$$
$$542$$ −32.0000 −1.37452
$$543$$ 0 0
$$544$$ −3.00000 −0.128624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −23.0000 −0.983409 −0.491704 0.870762i $$-0.663626\pi$$
−0.491704 + 0.870762i $$0.663626\pi$$
$$548$$ −20.0000 −0.854358
$$549$$ 0 0
$$550$$ −5.00000 −0.213201
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 42.0000 1.78602
$$554$$ 1.00000 0.0424859
$$555$$ 0 0
$$556$$ −10.0000 −0.424094
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −31.0000 −1.30766
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −13.0000 −0.546431
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −19.0000 −0.796521 −0.398261 0.917272i $$-0.630386\pi$$
−0.398261 + 0.917272i $$0.630386\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −1.00000 −0.0418121
$$573$$ 0 0
$$574$$ −30.0000 −1.25218
$$575$$ −5.00000 −0.208514
$$576$$ 0 0
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −27.0000 −1.12015
$$582$$ 0 0
$$583$$ −1.00000 −0.0414158
$$584$$ 3.00000 0.124141
$$585$$ 0 0
$$586$$ 31.0000 1.28060
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ 0 0
$$589$$ −18.0000 −0.741677
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −1.00000 −0.0410997
$$593$$ 16.0000 0.657041 0.328521 0.944497i $$-0.393450\pi$$
0.328521 + 0.944497i $$0.393450\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 0 0
$$598$$ −1.00000 −0.0408930
$$599$$ −34.0000 −1.38920 −0.694601 0.719395i $$-0.744419\pi$$
−0.694601 + 0.719395i $$0.744419\pi$$
$$600$$ 0 0
$$601$$ −1.00000 −0.0407909 −0.0203954 0.999792i $$-0.506493\pi$$
−0.0203954 + 0.999792i $$0.506493\pi$$
$$602$$ −36.0000 −1.46725
$$603$$ 0 0
$$604$$ 1.00000 0.0406894
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ −3.00000 −0.121666
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6.00000 0.242734
$$612$$ 0 0
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 3.00000 0.120873
$$617$$ 32.0000 1.28827 0.644136 0.764911i $$-0.277217\pi$$
0.644136 + 0.764911i $$0.277217\pi$$
$$618$$ 0 0
$$619$$ −18.0000 −0.723481 −0.361741 0.932279i $$-0.617817\pi$$
−0.361741 + 0.932279i $$0.617817\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 8.00000 0.320771
$$623$$ 9.00000 0.360577
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ −30.0000 −1.19904
$$627$$ 0 0
$$628$$ −4.00000 −0.159617
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −14.0000 −0.556890
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.00000 0.0792429
$$638$$ 4.00000 0.158362
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 32.0000 1.26392 0.631962 0.774999i $$-0.282250\pi$$
0.631962 + 0.774999i $$0.282250\pi$$
$$642$$ 0 0
$$643$$ 31.0000 1.22252 0.611260 0.791430i $$-0.290663\pi$$
0.611260 + 0.791430i $$0.290663\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 0 0
$$646$$ −9.00000 −0.354100
$$647$$ −31.0000 −1.21874 −0.609368 0.792888i $$-0.708577\pi$$
−0.609368 + 0.792888i $$0.708577\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 5.00000 0.196116
$$651$$ 0 0
$$652$$ 17.0000 0.665771
$$653$$ 8.00000 0.313064 0.156532 0.987673i $$-0.449969\pi$$
0.156532 + 0.987673i $$0.449969\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 0 0
$$658$$ −18.0000 −0.701713
$$659$$ 32.0000 1.24654 0.623272 0.782006i $$-0.285803\pi$$
0.623272 + 0.782006i $$0.285803\pi$$
$$660$$ 0 0
$$661$$ 1.00000 0.0388955 0.0194477 0.999811i $$-0.493809\pi$$
0.0194477 + 0.999811i $$0.493809\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 0 0
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.00000 0.154881
$$668$$ −21.0000 −0.812514
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −2.00000 −0.0772091
$$672$$ 0 0
$$673$$ −51.0000 −1.96591 −0.982953 0.183858i $$-0.941141\pi$$
−0.982953 + 0.183858i $$0.941141\pi$$
$$674$$ −21.0000 −0.808890
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ −13.0000 −0.499631 −0.249815 0.968294i $$-0.580370\pi$$
−0.249815 + 0.968294i $$0.580370\pi$$
$$678$$ 0 0
$$679$$ −30.0000 −1.15129
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −6.00000 −0.229752
$$683$$ 2.00000 0.0765279 0.0382639 0.999268i $$-0.487817\pi$$
0.0382639 + 0.999268i $$0.487817\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 15.0000 0.572703
$$687$$ 0 0
$$688$$ 12.0000 0.457496
$$689$$ 1.00000 0.0380970
$$690$$ 0 0
$$691$$ 24.0000 0.913003 0.456502 0.889723i $$-0.349102\pi$$
0.456502 + 0.889723i $$0.349102\pi$$
$$692$$ −1.00000 −0.0380143
$$693$$ 0 0
$$694$$ 22.0000 0.835109
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 30.0000 1.13633
$$698$$ −10.0000 −0.378506
$$699$$ 0 0
$$700$$ −15.0000 −0.566947
$$701$$ 16.0000 0.604312 0.302156 0.953259i $$-0.402294\pi$$
0.302156 + 0.953259i $$0.402294\pi$$
$$702$$ 0 0
$$703$$ −3.00000 −0.113147
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 2.00000 0.0752710
$$707$$ 18.0000 0.676960
$$708$$ 0 0
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −3.00000 −0.112430
$$713$$ −6.00000 −0.224702
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 0 0
$$718$$ 30.0000 1.11959
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 0 0
$$721$$ −6.00000 −0.223452
$$722$$ 10.0000 0.372161
$$723$$ 0 0
$$724$$ −20.0000 −0.743294
$$725$$ −20.0000 −0.742781
$$726$$ 0 0
$$727$$ −4.00000 −0.148352 −0.0741759 0.997245i $$-0.523633\pi$$
−0.0741759 + 0.997245i $$0.523633\pi$$
$$728$$ −3.00000 −0.111187
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 36.0000 1.33151
$$732$$ 0 0
$$733$$ 18.0000 0.664845 0.332423 0.943131i $$-0.392134\pi$$
0.332423 + 0.943131i $$0.392134\pi$$
$$734$$ 21.0000 0.775124
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ −2.00000 −0.0736709
$$738$$ 0 0
$$739$$ 34.0000 1.25071 0.625355 0.780340i $$-0.284954\pi$$
0.625355 + 0.780340i $$0.284954\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −3.00000 −0.110133
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −20.0000 −0.732252
$$747$$ 0 0
$$748$$ −3.00000 −0.109691
$$749$$ 39.0000 1.42503
$$750$$ 0 0
$$751$$ −48.0000 −1.75154 −0.875772 0.482724i $$-0.839647\pi$$
−0.875772 + 0.482724i $$0.839647\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3.00000 0.109037 0.0545184 0.998513i $$-0.482638\pi$$
0.0545184 + 0.998513i $$0.482638\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ 33.0000 1.19468
$$764$$ −3.00000 −0.108536
$$765$$ 0 0
$$766$$ 1.00000 0.0361315
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −4.00000 −0.144244 −0.0721218 0.997396i $$-0.522977\pi$$
−0.0721218 + 0.997396i $$0.522977\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −18.0000 −0.647834
$$773$$ 35.0000 1.25886 0.629431 0.777056i $$-0.283288\pi$$
0.629431 + 0.777056i $$0.283288\pi$$
$$774$$ 0 0
$$775$$ 30.0000 1.07763
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ 30.0000 1.07486
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −3.00000 −0.107280
$$783$$ 0 0
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −38.0000 −1.35455 −0.677277 0.735728i $$-0.736840\pi$$
−0.677277 + 0.735728i $$0.736840\pi$$
$$788$$ −13.0000 −0.463106
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −54.0000 −1.92002
$$792$$ 0 0
$$793$$ 2.00000 0.0710221
$$794$$ 12.0000 0.425864
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ −16.0000 −0.566749 −0.283375 0.959009i $$-0.591454\pi$$
−0.283375 + 0.959009i $$0.591454\pi$$
$$798$$ 0 0
$$799$$ 18.0000 0.636794
$$800$$ 5.00000 0.176777
$$801$$ 0 0
$$802$$ −11.0000 −0.388424
$$803$$ 3.00000 0.105868
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 6.00000 0.211341
$$807$$ 0 0
$$808$$ −6.00000 −0.211079
$$809$$ −21.0000 −0.738321 −0.369160 0.929366i $$-0.620355\pi$$
−0.369160 + 0.929366i $$0.620355\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 12.0000 0.421117
$$813$$ 0 0
$$814$$ −1.00000 −0.0350500
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 36.0000 1.25948
$$818$$ 6.00000 0.209785
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 45.0000 1.57051 0.785255 0.619172i $$-0.212532\pi$$
0.785255 + 0.619172i $$0.212532\pi$$
$$822$$ 0 0
$$823$$ 29.0000 1.01088 0.505438 0.862863i $$-0.331331\pi$$
0.505438 + 0.862863i $$0.331331\pi$$
$$824$$ 2.00000 0.0696733
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −2.00000 −0.0695468 −0.0347734 0.999395i $$-0.511071\pi$$
−0.0347734 + 0.999395i $$0.511071\pi$$
$$828$$ 0 0
$$829$$ −17.0000 −0.590434 −0.295217 0.955430i $$-0.595392\pi$$
−0.295217 + 0.955430i $$0.595392\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −3.00000 −0.103757
$$837$$ 0 0
$$838$$ −21.0000 −0.725433
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ −26.0000 −0.896019
$$843$$ 0 0
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −30.0000 −1.03081
$$848$$ 1.00000 0.0343401
$$849$$ 0 0
$$850$$ 15.0000 0.514496
$$851$$ −1.00000 −0.0342796
$$852$$ 0 0
$$853$$ −35.0000 −1.19838 −0.599189 0.800608i $$-0.704510\pi$$
−0.599189 + 0.800608i $$0.704510\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ −13.0000 −0.444331
$$857$$ 45.0000 1.53717 0.768585 0.639747i $$-0.220961\pi$$
0.768585 + 0.639747i $$0.220961\pi$$
$$858$$ 0 0
$$859$$ 49.0000 1.67186 0.835929 0.548837i $$-0.184929\pi$$
0.835929 + 0.548837i $$0.184929\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 5.00000 0.170301
$$863$$ 18.0000 0.612727 0.306364 0.951915i $$-0.400888\pi$$
0.306364 + 0.951915i $$0.400888\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −7.00000 −0.237870
$$867$$ 0 0
$$868$$ −18.0000 −0.610960
$$869$$ −14.0000 −0.474917
$$870$$ 0 0
$$871$$ 2.00000 0.0677674
$$872$$ −11.0000 −0.372507
$$873$$ 0 0
$$874$$ −3.00000 −0.101477
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −30.0000 −1.01303 −0.506514 0.862232i $$-0.669066\pi$$
−0.506514 + 0.862232i $$0.669066\pi$$
$$878$$ −34.0000 −1.14744
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 12.0000 0.404290 0.202145 0.979356i $$-0.435209\pi$$
0.202145 + 0.979356i $$0.435209\pi$$
$$882$$ 0 0
$$883$$ 21.0000 0.706706 0.353353 0.935490i $$-0.385041\pi$$
0.353353 + 0.935490i $$0.385041\pi$$
$$884$$ 3.00000 0.100901
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 22.0000 0.738688 0.369344 0.929293i $$-0.379582\pi$$
0.369344 + 0.929293i $$0.379582\pi$$
$$888$$ 0 0
$$889$$ −21.0000 −0.704317
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 16.0000 0.535720
$$893$$ 18.0000 0.602347
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −3.00000 −0.100223
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 3.00000 0.0999445
$$902$$ 10.0000 0.332964
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 5.00000 0.166022 0.0830111 0.996549i $$-0.473546\pi$$
0.0830111 + 0.996549i $$0.473546\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 0 0
$$913$$ 9.00000 0.297857
$$914$$ 20.0000 0.661541
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −30.0000 −0.990687
$$918$$ 0 0
$$919$$ 22.0000 0.725713 0.362857 0.931845i $$-0.381802\pi$$
0.362857 + 0.931845i $$0.381802\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 5.00000 0.164399
$$926$$ 26.0000 0.854413
$$927$$ 0 0
$$928$$ −4.00000 −0.131306
$$929$$ 10.0000 0.328089 0.164045 0.986453i $$-0.447546\pi$$
0.164045 + 0.986453i $$0.447546\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ −6.00000 −0.196326
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ −6.00000 −0.195907
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ 0 0
$$943$$ 10.0000 0.325645
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ −10.0000 −0.324956 −0.162478 0.986712i $$-0.551949\pi$$
−0.162478 + 0.986712i $$0.551949\pi$$
$$948$$ 0 0
$$949$$ −3.00000 −0.0973841
$$950$$ 15.0000 0.486664
$$951$$ 0 0
$$952$$ −9.00000 −0.291692
$$953$$ 28.0000 0.907009 0.453504 0.891254i $$-0.350174\pi$$
0.453504 + 0.891254i $$0.350174\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ −19.0000 −0.613862
$$959$$ −60.0000 −1.93750
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 1.00000 0.0322413
$$963$$ 0 0
$$964$$ 8.00000 0.257663
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ 10.0000 0.321412
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −4.00000 −0.128366 −0.0641831 0.997938i $$-0.520444\pi$$
−0.0641831 + 0.997938i $$0.520444\pi$$
$$972$$ 0 0
$$973$$ −30.0000 −0.961756
$$974$$ 6.00000 0.192252
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 43.0000 1.37569 0.687846 0.725857i $$-0.258556\pi$$
0.687846 + 0.725857i $$0.258556\pi$$
$$978$$ 0 0
$$979$$ −3.00000 −0.0958804
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −11.0000 −0.351024
$$983$$ −46.0000 −1.46717 −0.733586 0.679597i $$-0.762155\pi$$
−0.733586 + 0.679597i $$0.762155\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ 3.00000 0.0954427
$$989$$ 12.0000 0.381578
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 6.00000 0.190500
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 27.0000 0.855099 0.427549 0.903992i $$-0.359377\pi$$
0.427549 + 0.903992i $$0.359377\pi$$
$$998$$ 15.0000 0.474817
$$999$$ 0 0
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 666.2.a.b.1.1 1
3.2 odd 2 222.2.a.d.1.1 1
4.3 odd 2 5328.2.a.h.1.1 1
12.11 even 2 1776.2.a.h.1.1 1
15.14 odd 2 5550.2.a.n.1.1 1
24.5 odd 2 7104.2.a.v.1.1 1
24.11 even 2 7104.2.a.f.1.1 1
111.110 odd 2 8214.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
222.2.a.d.1.1 1 3.2 odd 2
666.2.a.b.1.1 1 1.1 even 1 trivial
1776.2.a.h.1.1 1 12.11 even 2
5328.2.a.h.1.1 1 4.3 odd 2
5550.2.a.n.1.1 1 15.14 odd 2
7104.2.a.f.1.1 1 24.11 even 2
7104.2.a.v.1.1 1 24.5 odd 2
8214.2.a.b.1.1 1 111.110 odd 2