# Properties

 Label 3822.2.a.bk.1.1 Level $3822$ Weight $2$ Character 3822.1 Self dual yes Analytic conductor $30.519$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.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: $$30.5188236525$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.64575$$ of defining polynomial Character $$\chi$$ $$=$$ 3822.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.64575 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.64575 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.64575 q^{10} +0.645751 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.64575 q^{15} +1.00000 q^{16} -6.64575 q^{17} -1.00000 q^{18} +5.00000 q^{19} -3.64575 q^{20} -0.645751 q^{22} -2.35425 q^{23} -1.00000 q^{24} +8.29150 q^{25} -1.00000 q^{26} +1.00000 q^{27} +4.29150 q^{29} +3.64575 q^{30} +3.29150 q^{31} -1.00000 q^{32} +0.645751 q^{33} +6.64575 q^{34} +1.00000 q^{36} +5.64575 q^{37} -5.00000 q^{38} +1.00000 q^{39} +3.64575 q^{40} -2.35425 q^{41} -5.29150 q^{43} +0.645751 q^{44} -3.64575 q^{45} +2.35425 q^{46} +3.00000 q^{47} +1.00000 q^{48} -8.29150 q^{50} -6.64575 q^{51} +1.00000 q^{52} -3.00000 q^{53} -1.00000 q^{54} -2.35425 q^{55} +5.00000 q^{57} -4.29150 q^{58} -7.93725 q^{59} -3.64575 q^{60} -11.9373 q^{61} -3.29150 q^{62} +1.00000 q^{64} -3.64575 q^{65} -0.645751 q^{66} +7.58301 q^{67} -6.64575 q^{68} -2.35425 q^{69} +16.2915 q^{71} -1.00000 q^{72} -13.6458 q^{73} -5.64575 q^{74} +8.29150 q^{75} +5.00000 q^{76} -1.00000 q^{78} -10.0000 q^{79} -3.64575 q^{80} +1.00000 q^{81} +2.35425 q^{82} -13.2915 q^{83} +24.2288 q^{85} +5.29150 q^{86} +4.29150 q^{87} -0.645751 q^{88} +16.9373 q^{89} +3.64575 q^{90} -2.35425 q^{92} +3.29150 q^{93} -3.00000 q^{94} -18.2288 q^{95} -1.00000 q^{96} +0.937254 q^{97} +0.645751 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 4 q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{15} + 2 q^{16} - 8 q^{17} - 2 q^{18} + 10 q^{19} - 2 q^{20} + 4 q^{22} - 10 q^{23} - 2 q^{24} + 6 q^{25} - 2 q^{26} + 2 q^{27} - 2 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} - 4 q^{33} + 8 q^{34} + 2 q^{36} + 6 q^{37} - 10 q^{38} + 2 q^{39} + 2 q^{40} - 10 q^{41} - 4 q^{44} - 2 q^{45} + 10 q^{46} + 6 q^{47} + 2 q^{48} - 6 q^{50} - 8 q^{51} + 2 q^{52} - 6 q^{53} - 2 q^{54} - 10 q^{55} + 10 q^{57} + 2 q^{58} - 2 q^{60} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 2 q^{65} + 4 q^{66} - 6 q^{67} - 8 q^{68} - 10 q^{69} + 22 q^{71} - 2 q^{72} - 22 q^{73} - 6 q^{74} + 6 q^{75} + 10 q^{76} - 2 q^{78} - 20 q^{79} - 2 q^{80} + 2 q^{81} + 10 q^{82} - 16 q^{83} + 22 q^{85} - 2 q^{87} + 4 q^{88} + 18 q^{89} + 2 q^{90} - 10 q^{92} - 4 q^{93} - 6 q^{94} - 10 q^{95} - 2 q^{96} - 14 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^10 - 4 * q^11 + 2 * q^12 + 2 * q^13 - 2 * q^15 + 2 * q^16 - 8 * q^17 - 2 * q^18 + 10 * q^19 - 2 * q^20 + 4 * q^22 - 10 * q^23 - 2 * q^24 + 6 * q^25 - 2 * q^26 + 2 * q^27 - 2 * q^29 + 2 * q^30 - 4 * q^31 - 2 * q^32 - 4 * q^33 + 8 * q^34 + 2 * q^36 + 6 * q^37 - 10 * q^38 + 2 * q^39 + 2 * q^40 - 10 * q^41 - 4 * q^44 - 2 * q^45 + 10 * q^46 + 6 * q^47 + 2 * q^48 - 6 * q^50 - 8 * q^51 + 2 * q^52 - 6 * q^53 - 2 * q^54 - 10 * q^55 + 10 * q^57 + 2 * q^58 - 2 * q^60 - 8 * q^61 + 4 * q^62 + 2 * q^64 - 2 * q^65 + 4 * q^66 - 6 * q^67 - 8 * q^68 - 10 * q^69 + 22 * q^71 - 2 * q^72 - 22 * q^73 - 6 * q^74 + 6 * q^75 + 10 * q^76 - 2 * q^78 - 20 * q^79 - 2 * q^80 + 2 * q^81 + 10 * q^82 - 16 * q^83 + 22 * q^85 - 2 * q^87 + 4 * q^88 + 18 * q^89 + 2 * q^90 - 10 * q^92 - 4 * q^93 - 6 * q^94 - 10 * q^95 - 2 * q^96 - 14 * q^97 - 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −3.64575 −1.63043 −0.815215 0.579159i $$-0.803381\pi$$
−0.815215 + 0.579159i $$0.803381\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 3.64575 1.15289
$$11$$ 0.645751 0.194701 0.0973507 0.995250i $$-0.468963\pi$$
0.0973507 + 0.995250i $$0.468963\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −3.64575 −0.941329
$$16$$ 1.00000 0.250000
$$17$$ −6.64575 −1.61183 −0.805916 0.592030i $$-0.798327\pi$$
−0.805916 + 0.592030i $$0.798327\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ −3.64575 −0.815215
$$21$$ 0 0
$$22$$ −0.645751 −0.137675
$$23$$ −2.35425 −0.490895 −0.245447 0.969410i $$-0.578935\pi$$
−0.245447 + 0.969410i $$0.578935\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 8.29150 1.65830
$$26$$ −1.00000 −0.196116
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 4.29150 0.796912 0.398456 0.917187i $$-0.369546\pi$$
0.398456 + 0.917187i $$0.369546\pi$$
$$30$$ 3.64575 0.665620
$$31$$ 3.29150 0.591171 0.295586 0.955316i $$-0.404485\pi$$
0.295586 + 0.955316i $$0.404485\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0.645751 0.112411
$$34$$ 6.64575 1.13974
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 5.64575 0.928156 0.464078 0.885794i $$-0.346386\pi$$
0.464078 + 0.885794i $$0.346386\pi$$
$$38$$ −5.00000 −0.811107
$$39$$ 1.00000 0.160128
$$40$$ 3.64575 0.576444
$$41$$ −2.35425 −0.367672 −0.183836 0.982957i $$-0.558852\pi$$
−0.183836 + 0.982957i $$0.558852\pi$$
$$42$$ 0 0
$$43$$ −5.29150 −0.806947 −0.403473 0.914991i $$-0.632197\pi$$
−0.403473 + 0.914991i $$0.632197\pi$$
$$44$$ 0.645751 0.0973507
$$45$$ −3.64575 −0.543477
$$46$$ 2.35425 0.347115
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ −8.29150 −1.17260
$$51$$ −6.64575 −0.930591
$$52$$ 1.00000 0.138675
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −2.35425 −0.317447
$$56$$ 0 0
$$57$$ 5.00000 0.662266
$$58$$ −4.29150 −0.563502
$$59$$ −7.93725 −1.03334 −0.516671 0.856184i $$-0.672829\pi$$
−0.516671 + 0.856184i $$0.672829\pi$$
$$60$$ −3.64575 −0.470664
$$61$$ −11.9373 −1.52841 −0.764204 0.644974i $$-0.776868\pi$$
−0.764204 + 0.644974i $$0.776868\pi$$
$$62$$ −3.29150 −0.418021
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −3.64575 −0.452200
$$66$$ −0.645751 −0.0794865
$$67$$ 7.58301 0.926412 0.463206 0.886251i $$-0.346699\pi$$
0.463206 + 0.886251i $$0.346699\pi$$
$$68$$ −6.64575 −0.805916
$$69$$ −2.35425 −0.283418
$$70$$ 0 0
$$71$$ 16.2915 1.93345 0.966723 0.255826i $$-0.0823474\pi$$
0.966723 + 0.255826i $$0.0823474\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −13.6458 −1.59711 −0.798557 0.601919i $$-0.794403\pi$$
−0.798557 + 0.601919i $$0.794403\pi$$
$$74$$ −5.64575 −0.656305
$$75$$ 8.29150 0.957420
$$76$$ 5.00000 0.573539
$$77$$ 0 0
$$78$$ −1.00000 −0.113228
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ −3.64575 −0.407607
$$81$$ 1.00000 0.111111
$$82$$ 2.35425 0.259983
$$83$$ −13.2915 −1.45893 −0.729466 0.684017i $$-0.760231\pi$$
−0.729466 + 0.684017i $$0.760231\pi$$
$$84$$ 0 0
$$85$$ 24.2288 2.62798
$$86$$ 5.29150 0.570597
$$87$$ 4.29150 0.460097
$$88$$ −0.645751 −0.0688373
$$89$$ 16.9373 1.79535 0.897673 0.440663i $$-0.145257\pi$$
0.897673 + 0.440663i $$0.145257\pi$$
$$90$$ 3.64575 0.384296
$$91$$ 0 0
$$92$$ −2.35425 −0.245447
$$93$$ 3.29150 0.341313
$$94$$ −3.00000 −0.309426
$$95$$ −18.2288 −1.87023
$$96$$ −1.00000 −0.102062
$$97$$ 0.937254 0.0951637 0.0475819 0.998867i $$-0.484849\pi$$
0.0475819 + 0.998867i $$0.484849\pi$$
$$98$$ 0 0
$$99$$ 0.645751 0.0649004
$$100$$ 8.29150 0.829150
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 6.64575 0.658027
$$103$$ −5.29150 −0.521387 −0.260694 0.965422i $$-0.583951\pi$$
−0.260694 + 0.965422i $$0.583951\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 3.00000 0.291386
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 2.35425 0.224469
$$111$$ 5.64575 0.535871
$$112$$ 0 0
$$113$$ 15.2288 1.43260 0.716300 0.697792i $$-0.245834\pi$$
0.716300 + 0.697792i $$0.245834\pi$$
$$114$$ −5.00000 −0.468293
$$115$$ 8.58301 0.800369
$$116$$ 4.29150 0.398456
$$117$$ 1.00000 0.0924500
$$118$$ 7.93725 0.730683
$$119$$ 0 0
$$120$$ 3.64575 0.332810
$$121$$ −10.5830 −0.962091
$$122$$ 11.9373 1.08075
$$123$$ −2.35425 −0.212275
$$124$$ 3.29150 0.295586
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −10.2288 −0.907655 −0.453828 0.891089i $$-0.649942\pi$$
−0.453828 + 0.891089i $$0.649942\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −5.29150 −0.465891
$$130$$ 3.64575 0.319754
$$131$$ 10.9373 0.955592 0.477796 0.878471i $$-0.341436\pi$$
0.477796 + 0.878471i $$0.341436\pi$$
$$132$$ 0.645751 0.0562054
$$133$$ 0 0
$$134$$ −7.58301 −0.655072
$$135$$ −3.64575 −0.313776
$$136$$ 6.64575 0.569868
$$137$$ −19.5203 −1.66773 −0.833864 0.551970i $$-0.813876\pi$$
−0.833864 + 0.551970i $$0.813876\pi$$
$$138$$ 2.35425 0.200407
$$139$$ −22.2288 −1.88542 −0.942709 0.333615i $$-0.891731\pi$$
−0.942709 + 0.333615i $$0.891731\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ −16.2915 −1.36715
$$143$$ 0.645751 0.0540004
$$144$$ 1.00000 0.0833333
$$145$$ −15.6458 −1.29931
$$146$$ 13.6458 1.12933
$$147$$ 0 0
$$148$$ 5.64575 0.464078
$$149$$ −7.06275 −0.578603 −0.289301 0.957238i $$-0.593423\pi$$
−0.289301 + 0.957238i $$0.593423\pi$$
$$150$$ −8.29150 −0.676998
$$151$$ 6.06275 0.493379 0.246690 0.969095i $$-0.420657\pi$$
0.246690 + 0.969095i $$0.420657\pi$$
$$152$$ −5.00000 −0.405554
$$153$$ −6.64575 −0.537277
$$154$$ 0 0
$$155$$ −12.0000 −0.963863
$$156$$ 1.00000 0.0800641
$$157$$ 2.64575 0.211154 0.105577 0.994411i $$-0.466331\pi$$
0.105577 + 0.994411i $$0.466331\pi$$
$$158$$ 10.0000 0.795557
$$159$$ −3.00000 −0.237915
$$160$$ 3.64575 0.288222
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 2.41699 0.189314 0.0946568 0.995510i $$-0.469825\pi$$
0.0946568 + 0.995510i $$0.469825\pi$$
$$164$$ −2.35425 −0.183836
$$165$$ −2.35425 −0.183278
$$166$$ 13.2915 1.03162
$$167$$ −6.87451 −0.531965 −0.265983 0.963978i $$-0.585696\pi$$
−0.265983 + 0.963978i $$0.585696\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −24.2288 −1.85826
$$171$$ 5.00000 0.382360
$$172$$ −5.29150 −0.403473
$$173$$ −22.2915 −1.69479 −0.847396 0.530962i $$-0.821831\pi$$
−0.847396 + 0.530962i $$0.821831\pi$$
$$174$$ −4.29150 −0.325338
$$175$$ 0 0
$$176$$ 0.645751 0.0486753
$$177$$ −7.93725 −0.596601
$$178$$ −16.9373 −1.26950
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ −3.64575 −0.271738
$$181$$ 14.6458 1.08861 0.544305 0.838887i $$-0.316793\pi$$
0.544305 + 0.838887i $$0.316793\pi$$
$$182$$ 0 0
$$183$$ −11.9373 −0.882427
$$184$$ 2.35425 0.173558
$$185$$ −20.5830 −1.51329
$$186$$ −3.29150 −0.241345
$$187$$ −4.29150 −0.313826
$$188$$ 3.00000 0.218797
$$189$$ 0 0
$$190$$ 18.2288 1.32245
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 18.9373 1.36313 0.681567 0.731756i $$-0.261299\pi$$
0.681567 + 0.731756i $$0.261299\pi$$
$$194$$ −0.937254 −0.0672909
$$195$$ −3.64575 −0.261078
$$196$$ 0 0
$$197$$ −26.8118 −1.91026 −0.955129 0.296189i $$-0.904284\pi$$
−0.955129 + 0.296189i $$0.904284\pi$$
$$198$$ −0.645751 −0.0458915
$$199$$ −22.2288 −1.57575 −0.787877 0.615832i $$-0.788820\pi$$
−0.787877 + 0.615832i $$0.788820\pi$$
$$200$$ −8.29150 −0.586298
$$201$$ 7.58301 0.534864
$$202$$ 0 0
$$203$$ 0 0
$$204$$ −6.64575 −0.465296
$$205$$ 8.58301 0.599463
$$206$$ 5.29150 0.368676
$$207$$ −2.35425 −0.163632
$$208$$ 1.00000 0.0693375
$$209$$ 3.22876 0.223338
$$210$$ 0 0
$$211$$ −6.35425 −0.437445 −0.218722 0.975787i $$-0.570189\pi$$
−0.218722 + 0.975787i $$0.570189\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ 16.2915 1.11628
$$214$$ 12.0000 0.820303
$$215$$ 19.2915 1.31567
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 4.00000 0.270914
$$219$$ −13.6458 −0.922094
$$220$$ −2.35425 −0.158723
$$221$$ −6.64575 −0.447042
$$222$$ −5.64575 −0.378918
$$223$$ −20.5203 −1.37414 −0.687069 0.726592i $$-0.741103\pi$$
−0.687069 + 0.726592i $$0.741103\pi$$
$$224$$ 0 0
$$225$$ 8.29150 0.552767
$$226$$ −15.2288 −1.01300
$$227$$ −4.70850 −0.312514 −0.156257 0.987716i $$-0.549943\pi$$
−0.156257 + 0.987716i $$0.549943\pi$$
$$228$$ 5.00000 0.331133
$$229$$ 26.4575 1.74836 0.874181 0.485601i $$-0.161399\pi$$
0.874181 + 0.485601i $$0.161399\pi$$
$$230$$ −8.58301 −0.565947
$$231$$ 0 0
$$232$$ −4.29150 −0.281751
$$233$$ 0.645751 0.0423046 0.0211523 0.999776i $$-0.493267\pi$$
0.0211523 + 0.999776i $$0.493267\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ −10.9373 −0.713468
$$236$$ −7.93725 −0.516671
$$237$$ −10.0000 −0.649570
$$238$$ 0 0
$$239$$ 3.00000 0.194054 0.0970269 0.995282i $$-0.469067\pi$$
0.0970269 + 0.995282i $$0.469067\pi$$
$$240$$ −3.64575 −0.235332
$$241$$ −13.8745 −0.893736 −0.446868 0.894600i $$-0.647461\pi$$
−0.446868 + 0.894600i $$0.647461\pi$$
$$242$$ 10.5830 0.680301
$$243$$ 1.00000 0.0641500
$$244$$ −11.9373 −0.764204
$$245$$ 0 0
$$246$$ 2.35425 0.150101
$$247$$ 5.00000 0.318142
$$248$$ −3.29150 −0.209011
$$249$$ −13.2915 −0.842315
$$250$$ 12.0000 0.758947
$$251$$ −13.2915 −0.838952 −0.419476 0.907766i $$-0.637786\pi$$
−0.419476 + 0.907766i $$0.637786\pi$$
$$252$$ 0 0
$$253$$ −1.52026 −0.0955779
$$254$$ 10.2288 0.641809
$$255$$ 24.2288 1.51726
$$256$$ 1.00000 0.0625000
$$257$$ −15.8745 −0.990225 −0.495112 0.868829i $$-0.664873\pi$$
−0.495112 + 0.868829i $$0.664873\pi$$
$$258$$ 5.29150 0.329435
$$259$$ 0 0
$$260$$ −3.64575 −0.226100
$$261$$ 4.29150 0.265637
$$262$$ −10.9373 −0.675706
$$263$$ −3.64575 −0.224807 −0.112403 0.993663i $$-0.535855\pi$$
−0.112403 + 0.993663i $$0.535855\pi$$
$$264$$ −0.645751 −0.0397432
$$265$$ 10.9373 0.671870
$$266$$ 0 0
$$267$$ 16.9373 1.03654
$$268$$ 7.58301 0.463206
$$269$$ 6.87451 0.419146 0.209573 0.977793i $$-0.432793\pi$$
0.209573 + 0.977793i $$0.432793\pi$$
$$270$$ 3.64575 0.221873
$$271$$ −22.6458 −1.37563 −0.687816 0.725885i $$-0.741430\pi$$
−0.687816 + 0.725885i $$0.741430\pi$$
$$272$$ −6.64575 −0.402958
$$273$$ 0 0
$$274$$ 19.5203 1.17926
$$275$$ 5.35425 0.322873
$$276$$ −2.35425 −0.141709
$$277$$ 18.5203 1.11277 0.556387 0.830923i $$-0.312187\pi$$
0.556387 + 0.830923i $$0.312187\pi$$
$$278$$ 22.2288 1.33319
$$279$$ 3.29150 0.197057
$$280$$ 0 0
$$281$$ −2.58301 −0.154089 −0.0770446 0.997028i $$-0.524548\pi$$
−0.0770446 + 0.997028i $$0.524548\pi$$
$$282$$ −3.00000 −0.178647
$$283$$ −11.0627 −0.657612 −0.328806 0.944397i $$-0.606646\pi$$
−0.328806 + 0.944397i $$0.606646\pi$$
$$284$$ 16.2915 0.966723
$$285$$ −18.2288 −1.07978
$$286$$ −0.645751 −0.0381841
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 27.1660 1.59800
$$290$$ 15.6458 0.918750
$$291$$ 0.937254 0.0549428
$$292$$ −13.6458 −0.798557
$$293$$ 7.52026 0.439338 0.219669 0.975574i $$-0.429502\pi$$
0.219669 + 0.975574i $$0.429502\pi$$
$$294$$ 0 0
$$295$$ 28.9373 1.68479
$$296$$ −5.64575 −0.328153
$$297$$ 0.645751 0.0374703
$$298$$ 7.06275 0.409134
$$299$$ −2.35425 −0.136150
$$300$$ 8.29150 0.478710
$$301$$ 0 0
$$302$$ −6.06275 −0.348872
$$303$$ 0 0
$$304$$ 5.00000 0.286770
$$305$$ 43.5203 2.49196
$$306$$ 6.64575 0.379912
$$307$$ −9.58301 −0.546931 −0.273465 0.961882i $$-0.588170\pi$$
−0.273465 + 0.961882i $$0.588170\pi$$
$$308$$ 0 0
$$309$$ −5.29150 −0.301023
$$310$$ 12.0000 0.681554
$$311$$ −1.06275 −0.0602628 −0.0301314 0.999546i $$-0.509593\pi$$
−0.0301314 + 0.999546i $$0.509593\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 13.1660 0.744187 0.372093 0.928195i $$-0.378640\pi$$
0.372093 + 0.928195i $$0.378640\pi$$
$$314$$ −2.64575 −0.149308
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −13.2915 −0.746525 −0.373263 0.927726i $$-0.621761\pi$$
−0.373263 + 0.927726i $$0.621761\pi$$
$$318$$ 3.00000 0.168232
$$319$$ 2.77124 0.155160
$$320$$ −3.64575 −0.203804
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ −33.2288 −1.84890
$$324$$ 1.00000 0.0555556
$$325$$ 8.29150 0.459930
$$326$$ −2.41699 −0.133865
$$327$$ −4.00000 −0.221201
$$328$$ 2.35425 0.129992
$$329$$ 0 0
$$330$$ 2.35425 0.129597
$$331$$ 11.4170 0.627535 0.313767 0.949500i $$-0.398409\pi$$
0.313767 + 0.949500i $$0.398409\pi$$
$$332$$ −13.2915 −0.729466
$$333$$ 5.64575 0.309385
$$334$$ 6.87451 0.376156
$$335$$ −27.6458 −1.51045
$$336$$ 0 0
$$337$$ −15.5830 −0.848860 −0.424430 0.905461i $$-0.639526\pi$$
−0.424430 + 0.905461i $$0.639526\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 15.2288 0.827113
$$340$$ 24.2288 1.31399
$$341$$ 2.12549 0.115102
$$342$$ −5.00000 −0.270369
$$343$$ 0 0
$$344$$ 5.29150 0.285299
$$345$$ 8.58301 0.462093
$$346$$ 22.2915 1.19840
$$347$$ 0.228757 0.0122803 0.00614015 0.999981i $$-0.498046\pi$$
0.00614015 + 0.999981i $$0.498046\pi$$
$$348$$ 4.29150 0.230049
$$349$$ 18.9373 1.01369 0.506844 0.862038i $$-0.330812\pi$$
0.506844 + 0.862038i $$0.330812\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ −0.645751 −0.0344187
$$353$$ −20.5830 −1.09552 −0.547761 0.836635i $$-0.684520\pi$$
−0.547761 + 0.836635i $$0.684520\pi$$
$$354$$ 7.93725 0.421860
$$355$$ −59.3948 −3.15235
$$356$$ 16.9373 0.897673
$$357$$ 0 0
$$358$$ 6.00000 0.317110
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 3.64575 0.192148
$$361$$ 6.00000 0.315789
$$362$$ −14.6458 −0.769764
$$363$$ −10.5830 −0.555464
$$364$$ 0 0
$$365$$ 49.7490 2.60398
$$366$$ 11.9373 0.623970
$$367$$ −0.583005 −0.0304326 −0.0152163 0.999884i $$-0.504844\pi$$
−0.0152163 + 0.999884i $$0.504844\pi$$
$$368$$ −2.35425 −0.122724
$$369$$ −2.35425 −0.122557
$$370$$ 20.5830 1.07006
$$371$$ 0 0
$$372$$ 3.29150 0.170656
$$373$$ 14.6458 0.758328 0.379164 0.925329i $$-0.376212\pi$$
0.379164 + 0.925329i $$0.376212\pi$$
$$374$$ 4.29150 0.221908
$$375$$ −12.0000 −0.619677
$$376$$ −3.00000 −0.154713
$$377$$ 4.29150 0.221024
$$378$$ 0 0
$$379$$ −9.16601 −0.470826 −0.235413 0.971895i $$-0.575644\pi$$
−0.235413 + 0.971895i $$0.575644\pi$$
$$380$$ −18.2288 −0.935115
$$381$$ −10.2288 −0.524035
$$382$$ 0 0
$$383$$ 25.7490 1.31571 0.657857 0.753143i $$-0.271463\pi$$
0.657857 + 0.753143i $$0.271463\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −18.9373 −0.963881
$$387$$ −5.29150 −0.268982
$$388$$ 0.937254 0.0475819
$$389$$ 7.70850 0.390836 0.195418 0.980720i $$-0.437394\pi$$
0.195418 + 0.980720i $$0.437394\pi$$
$$390$$ 3.64575 0.184610
$$391$$ 15.6458 0.791240
$$392$$ 0 0
$$393$$ 10.9373 0.551711
$$394$$ 26.8118 1.35076
$$395$$ 36.4575 1.83438
$$396$$ 0.645751 0.0324502
$$397$$ −14.9373 −0.749679 −0.374840 0.927090i $$-0.622302\pi$$
−0.374840 + 0.927090i $$0.622302\pi$$
$$398$$ 22.2288 1.11423
$$399$$ 0 0
$$400$$ 8.29150 0.414575
$$401$$ −18.2288 −0.910301 −0.455150 0.890415i $$-0.650414\pi$$
−0.455150 + 0.890415i $$0.650414\pi$$
$$402$$ −7.58301 −0.378206
$$403$$ 3.29150 0.163961
$$404$$ 0 0
$$405$$ −3.64575 −0.181159
$$406$$ 0 0
$$407$$ 3.64575 0.180713
$$408$$ 6.64575 0.329014
$$409$$ −26.9373 −1.33196 −0.665981 0.745969i $$-0.731987\pi$$
−0.665981 + 0.745969i $$0.731987\pi$$
$$410$$ −8.58301 −0.423884
$$411$$ −19.5203 −0.962863
$$412$$ −5.29150 −0.260694
$$413$$ 0 0
$$414$$ 2.35425 0.115705
$$415$$ 48.4575 2.37869
$$416$$ −1.00000 −0.0490290
$$417$$ −22.2288 −1.08855
$$418$$ −3.22876 −0.157924
$$419$$ 30.4575 1.48795 0.743973 0.668209i $$-0.232939\pi$$
0.743973 + 0.668209i $$0.232939\pi$$
$$420$$ 0 0
$$421$$ −24.3542 −1.18695 −0.593477 0.804851i $$-0.702245\pi$$
−0.593477 + 0.804851i $$0.702245\pi$$
$$422$$ 6.35425 0.309320
$$423$$ 3.00000 0.145865
$$424$$ 3.00000 0.145693
$$425$$ −55.1033 −2.67290
$$426$$ −16.2915 −0.789326
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 0.645751 0.0311772
$$430$$ −19.2915 −0.930319
$$431$$ 20.5830 0.991448 0.495724 0.868480i $$-0.334903\pi$$
0.495724 + 0.868480i $$0.334903\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −33.5830 −1.61390 −0.806948 0.590622i $$-0.798882\pi$$
−0.806948 + 0.590622i $$0.798882\pi$$
$$434$$ 0 0
$$435$$ −15.6458 −0.750156
$$436$$ −4.00000 −0.191565
$$437$$ −11.7712 −0.563095
$$438$$ 13.6458 0.652019
$$439$$ 23.6458 1.12855 0.564275 0.825587i $$-0.309156\pi$$
0.564275 + 0.825587i $$0.309156\pi$$
$$440$$ 2.35425 0.112234
$$441$$ 0 0
$$442$$ 6.64575 0.316106
$$443$$ 19.0627 0.905698 0.452849 0.891587i $$-0.350408\pi$$
0.452849 + 0.891587i $$0.350408\pi$$
$$444$$ 5.64575 0.267935
$$445$$ −61.7490 −2.92718
$$446$$ 20.5203 0.971662
$$447$$ −7.06275 −0.334056
$$448$$ 0 0
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ −8.29150 −0.390865
$$451$$ −1.52026 −0.0715862
$$452$$ 15.2288 0.716300
$$453$$ 6.06275 0.284853
$$454$$ 4.70850 0.220981
$$455$$ 0 0
$$456$$ −5.00000 −0.234146
$$457$$ 32.2288 1.50760 0.753799 0.657105i $$-0.228219\pi$$
0.753799 + 0.657105i $$0.228219\pi$$
$$458$$ −26.4575 −1.23628
$$459$$ −6.64575 −0.310197
$$460$$ 8.58301 0.400185
$$461$$ −36.4575 −1.69800 −0.848998 0.528396i $$-0.822794\pi$$
−0.848998 + 0.528396i $$0.822794\pi$$
$$462$$ 0 0
$$463$$ 25.1660 1.16956 0.584782 0.811191i $$-0.301180\pi$$
0.584782 + 0.811191i $$0.301180\pi$$
$$464$$ 4.29150 0.199228
$$465$$ −12.0000 −0.556487
$$466$$ −0.645751 −0.0299139
$$467$$ −37.5203 −1.73623 −0.868115 0.496363i $$-0.834669\pi$$
−0.868115 + 0.496363i $$0.834669\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 0 0
$$470$$ 10.9373 0.504498
$$471$$ 2.64575 0.121910
$$472$$ 7.93725 0.365342
$$473$$ −3.41699 −0.157114
$$474$$ 10.0000 0.459315
$$475$$ 41.4575 1.90220
$$476$$ 0 0
$$477$$ −3.00000 −0.137361
$$478$$ −3.00000 −0.137217
$$479$$ −34.7490 −1.58772 −0.793862 0.608099i $$-0.791933\pi$$
−0.793862 + 0.608099i $$0.791933\pi$$
$$480$$ 3.64575 0.166405
$$481$$ 5.64575 0.257424
$$482$$ 13.8745 0.631967
$$483$$ 0 0
$$484$$ −10.5830 −0.481046
$$485$$ −3.41699 −0.155158
$$486$$ −1.00000 −0.0453609
$$487$$ −23.9373 −1.08470 −0.542350 0.840152i $$-0.682465\pi$$
−0.542350 + 0.840152i $$0.682465\pi$$
$$488$$ 11.9373 0.540374
$$489$$ 2.41699 0.109300
$$490$$ 0 0
$$491$$ 11.1660 0.503915 0.251957 0.967738i $$-0.418926\pi$$
0.251957 + 0.967738i $$0.418926\pi$$
$$492$$ −2.35425 −0.106138
$$493$$ −28.5203 −1.28449
$$494$$ −5.00000 −0.224961
$$495$$ −2.35425 −0.105816
$$496$$ 3.29150 0.147793
$$497$$ 0 0
$$498$$ 13.2915 0.595606
$$499$$ −29.2915 −1.31127 −0.655634 0.755079i $$-0.727599\pi$$
−0.655634 + 0.755079i $$0.727599\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ −6.87451 −0.307130
$$502$$ 13.2915 0.593229
$$503$$ 36.4575 1.62556 0.812780 0.582571i $$-0.197953\pi$$
0.812780 + 0.582571i $$0.197953\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1.52026 0.0675838
$$507$$ 1.00000 0.0444116
$$508$$ −10.2288 −0.453828
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ −24.2288 −1.07287
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 5.00000 0.220755
$$514$$ 15.8745 0.700195
$$515$$ 19.2915 0.850085
$$516$$ −5.29150 −0.232945
$$517$$ 1.93725 0.0852003
$$518$$ 0 0
$$519$$ −22.2915 −0.978488
$$520$$ 3.64575 0.159877
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ −4.29150 −0.187834
$$523$$ −26.7085 −1.16788 −0.583941 0.811796i $$-0.698490\pi$$
−0.583941 + 0.811796i $$0.698490\pi$$
$$524$$ 10.9373 0.477796
$$525$$ 0 0
$$526$$ 3.64575 0.158962
$$527$$ −21.8745 −0.952869
$$528$$ 0.645751 0.0281027
$$529$$ −17.4575 −0.759022
$$530$$ −10.9373 −0.475084
$$531$$ −7.93725 −0.344447
$$532$$ 0 0
$$533$$ −2.35425 −0.101974
$$534$$ −16.9373 −0.732947
$$535$$ 43.7490 1.89143
$$536$$ −7.58301 −0.327536
$$537$$ −6.00000 −0.258919
$$538$$ −6.87451 −0.296381
$$539$$ 0 0
$$540$$ −3.64575 −0.156888
$$541$$ 41.6458 1.79049 0.895245 0.445574i $$-0.147000\pi$$
0.895245 + 0.445574i $$0.147000\pi$$
$$542$$ 22.6458 0.972718
$$543$$ 14.6458 0.628509
$$544$$ 6.64575 0.284934
$$545$$ 14.5830 0.624667
$$546$$ 0 0
$$547$$ −20.9373 −0.895212 −0.447606 0.894231i $$-0.647723\pi$$
−0.447606 + 0.894231i $$0.647723\pi$$
$$548$$ −19.5203 −0.833864
$$549$$ −11.9373 −0.509470
$$550$$ −5.35425 −0.228306
$$551$$ 21.4575 0.914121
$$552$$ 2.35425 0.100203
$$553$$ 0 0
$$554$$ −18.5203 −0.786850
$$555$$ −20.5830 −0.873700
$$556$$ −22.2288 −0.942709
$$557$$ 11.1660 0.473119 0.236560 0.971617i $$-0.423980\pi$$
0.236560 + 0.971617i $$0.423980\pi$$
$$558$$ −3.29150 −0.139340
$$559$$ −5.29150 −0.223807
$$560$$ 0 0
$$561$$ −4.29150 −0.181187
$$562$$ 2.58301 0.108958
$$563$$ −27.0405 −1.13962 −0.569811 0.821776i $$-0.692984\pi$$
−0.569811 + 0.821776i $$0.692984\pi$$
$$564$$ 3.00000 0.126323
$$565$$ −55.5203 −2.33575
$$566$$ 11.0627 0.465002
$$567$$ 0 0
$$568$$ −16.2915 −0.683576
$$569$$ −33.2288 −1.39302 −0.696511 0.717546i $$-0.745265\pi$$
−0.696511 + 0.717546i $$0.745265\pi$$
$$570$$ 18.2288 0.763519
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 0.645751 0.0270002
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −19.5203 −0.814051
$$576$$ 1.00000 0.0416667
$$577$$ −12.5830 −0.523837 −0.261919 0.965090i $$-0.584355\pi$$
−0.261919 + 0.965090i $$0.584355\pi$$
$$578$$ −27.1660 −1.12996
$$579$$ 18.9373 0.787005
$$580$$ −15.6458 −0.649654
$$581$$ 0 0
$$582$$ −0.937254 −0.0388504
$$583$$ −1.93725 −0.0802329
$$584$$ 13.6458 0.564665
$$585$$ −3.64575 −0.150733
$$586$$ −7.52026 −0.310659
$$587$$ 31.9373 1.31819 0.659096 0.752059i $$-0.270939\pi$$
0.659096 + 0.752059i $$0.270939\pi$$
$$588$$ 0 0
$$589$$ 16.4575 0.678120
$$590$$ −28.9373 −1.19133
$$591$$ −26.8118 −1.10289
$$592$$ 5.64575 0.232039
$$593$$ 43.5203 1.78716 0.893581 0.448901i $$-0.148184\pi$$
0.893581 + 0.448901i $$0.148184\pi$$
$$594$$ −0.645751 −0.0264955
$$595$$ 0 0
$$596$$ −7.06275 −0.289301
$$597$$ −22.2288 −0.909762
$$598$$ 2.35425 0.0962724
$$599$$ −32.8118 −1.34065 −0.670326 0.742067i $$-0.733846\pi$$
−0.670326 + 0.742067i $$0.733846\pi$$
$$600$$ −8.29150 −0.338499
$$601$$ 14.8745 0.606744 0.303372 0.952872i $$-0.401888\pi$$
0.303372 + 0.952872i $$0.401888\pi$$
$$602$$ 0 0
$$603$$ 7.58301 0.308804
$$604$$ 6.06275 0.246690
$$605$$ 38.5830 1.56862
$$606$$ 0 0
$$607$$ 16.8118 0.682368 0.341184 0.939996i $$-0.389172\pi$$
0.341184 + 0.939996i $$0.389172\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ 0 0
$$610$$ −43.5203 −1.76208
$$611$$ 3.00000 0.121367
$$612$$ −6.64575 −0.268639
$$613$$ 41.8745 1.69130 0.845648 0.533741i $$-0.179214\pi$$
0.845648 + 0.533741i $$0.179214\pi$$
$$614$$ 9.58301 0.386739
$$615$$ 8.58301 0.346100
$$616$$ 0 0
$$617$$ −10.7085 −0.431108 −0.215554 0.976492i $$-0.569156\pi$$
−0.215554 + 0.976492i $$0.569156\pi$$
$$618$$ 5.29150 0.212855
$$619$$ −41.7490 −1.67803 −0.839017 0.544105i $$-0.816869\pi$$
−0.839017 + 0.544105i $$0.816869\pi$$
$$620$$ −12.0000 −0.481932
$$621$$ −2.35425 −0.0944727
$$622$$ 1.06275 0.0426122
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 2.29150 0.0916601
$$626$$ −13.1660 −0.526220
$$627$$ 3.22876 0.128944
$$628$$ 2.64575 0.105577
$$629$$ −37.5203 −1.49603
$$630$$ 0 0
$$631$$ 14.4575 0.575545 0.287772 0.957699i $$-0.407085\pi$$
0.287772 + 0.957699i $$0.407085\pi$$
$$632$$ 10.0000 0.397779
$$633$$ −6.35425 −0.252559
$$634$$ 13.2915 0.527873
$$635$$ 37.2915 1.47987
$$636$$ −3.00000 −0.118958
$$637$$ 0 0
$$638$$ −2.77124 −0.109715
$$639$$ 16.2915 0.644482
$$640$$ 3.64575 0.144111
$$641$$ −21.4170 −0.845920 −0.422960 0.906148i $$-0.639009\pi$$
−0.422960 + 0.906148i $$0.639009\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −10.8745 −0.428849 −0.214424 0.976741i $$-0.568788\pi$$
−0.214424 + 0.976741i $$0.568788\pi$$
$$644$$ 0 0
$$645$$ 19.2915 0.759602
$$646$$ 33.2288 1.30737
$$647$$ 25.7490 1.01230 0.506149 0.862446i $$-0.331069\pi$$
0.506149 + 0.862446i $$0.331069\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −5.12549 −0.201193
$$650$$ −8.29150 −0.325219
$$651$$ 0 0
$$652$$ 2.41699 0.0946568
$$653$$ 12.0000 0.469596 0.234798 0.972044i $$-0.424557\pi$$
0.234798 + 0.972044i $$0.424557\pi$$
$$654$$ 4.00000 0.156412
$$655$$ −39.8745 −1.55803
$$656$$ −2.35425 −0.0919180
$$657$$ −13.6458 −0.532371
$$658$$ 0 0
$$659$$ −16.9373 −0.659782 −0.329891 0.944019i $$-0.607012\pi$$
−0.329891 + 0.944019i $$0.607012\pi$$
$$660$$ −2.35425 −0.0916390
$$661$$ 15.5203 0.603668 0.301834 0.953360i $$-0.402401\pi$$
0.301834 + 0.953360i $$0.402401\pi$$
$$662$$ −11.4170 −0.443734
$$663$$ −6.64575 −0.258100
$$664$$ 13.2915 0.515810
$$665$$ 0 0
$$666$$ −5.64575 −0.218768
$$667$$ −10.1033 −0.391200
$$668$$ −6.87451 −0.265983
$$669$$ −20.5203 −0.793359
$$670$$ 27.6458 1.06805
$$671$$ −7.70850 −0.297583
$$672$$ 0 0
$$673$$ −6.58301 −0.253756 −0.126878 0.991918i $$-0.540496\pi$$
−0.126878 + 0.991918i $$0.540496\pi$$
$$674$$ 15.5830 0.600235
$$675$$ 8.29150 0.319140
$$676$$ 1.00000 0.0384615
$$677$$ 20.1660 0.775043 0.387521 0.921861i $$-0.373331\pi$$
0.387521 + 0.921861i $$0.373331\pi$$
$$678$$ −15.2288 −0.584857
$$679$$ 0 0
$$680$$ −24.2288 −0.929130
$$681$$ −4.70850 −0.180430
$$682$$ −2.12549 −0.0813893
$$683$$ 19.2915 0.738169 0.369084 0.929396i $$-0.379671\pi$$
0.369084 + 0.929396i $$0.379671\pi$$
$$684$$ 5.00000 0.191180
$$685$$ 71.1660 2.71911
$$686$$ 0 0
$$687$$ 26.4575 1.00942
$$688$$ −5.29150 −0.201737
$$689$$ −3.00000 −0.114291
$$690$$ −8.58301 −0.326749
$$691$$ −40.0405 −1.52321 −0.761607 0.648040i $$-0.775589\pi$$
−0.761607 + 0.648040i $$0.775589\pi$$
$$692$$ −22.2915 −0.847396
$$693$$ 0 0
$$694$$ −0.228757 −0.00868348
$$695$$ 81.0405 3.07404
$$696$$ −4.29150 −0.162669
$$697$$ 15.6458 0.592625
$$698$$ −18.9373 −0.716786
$$699$$ 0.645751 0.0244246
$$700$$ 0 0
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ 28.2288 1.06467
$$704$$ 0.645751 0.0243377
$$705$$ −10.9373 −0.411921
$$706$$ 20.5830 0.774652
$$707$$ 0 0
$$708$$ −7.93725 −0.298300
$$709$$ 6.47974 0.243352 0.121676 0.992570i $$-0.461173\pi$$
0.121676 + 0.992570i $$0.461173\pi$$
$$710$$ 59.3948 2.22905
$$711$$ −10.0000 −0.375029
$$712$$ −16.9373 −0.634750
$$713$$ −7.74902 −0.290203
$$714$$ 0 0
$$715$$ −2.35425 −0.0880439
$$716$$ −6.00000 −0.224231
$$717$$ 3.00000 0.112037
$$718$$ 6.00000 0.223918
$$719$$ 44.5830 1.66267 0.831333 0.555775i $$-0.187578\pi$$
0.831333 + 0.555775i $$0.187578\pi$$
$$720$$ −3.64575 −0.135869
$$721$$ 0 0
$$722$$ −6.00000 −0.223297
$$723$$ −13.8745 −0.515998
$$724$$ 14.6458 0.544305
$$725$$ 35.5830 1.32152
$$726$$ 10.5830 0.392772
$$727$$ −16.4575 −0.610375 −0.305188 0.952292i $$-0.598719\pi$$
−0.305188 + 0.952292i $$0.598719\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −49.7490 −1.84129
$$731$$ 35.1660 1.30066
$$732$$ −11.9373 −0.441214
$$733$$ 14.2288 0.525551 0.262776 0.964857i $$-0.415362\pi$$
0.262776 + 0.964857i $$0.415362\pi$$
$$734$$ 0.583005 0.0215191
$$735$$ 0 0
$$736$$ 2.35425 0.0867788
$$737$$ 4.89674 0.180374
$$738$$ 2.35425 0.0866611
$$739$$ −17.2915 −0.636078 −0.318039 0.948078i $$-0.603024\pi$$
−0.318039 + 0.948078i $$0.603024\pi$$
$$740$$ −20.5830 −0.756646
$$741$$ 5.00000 0.183680
$$742$$ 0 0
$$743$$ −51.4575 −1.88779 −0.943897 0.330241i $$-0.892870\pi$$
−0.943897 + 0.330241i $$0.892870\pi$$
$$744$$ −3.29150 −0.120672
$$745$$ 25.7490 0.943371
$$746$$ −14.6458 −0.536219
$$747$$ −13.2915 −0.486311
$$748$$ −4.29150 −0.156913
$$749$$ 0 0
$$750$$ 12.0000 0.438178
$$751$$ −33.1660 −1.21024 −0.605122 0.796132i $$-0.706876\pi$$
−0.605122 + 0.796132i $$0.706876\pi$$
$$752$$ 3.00000 0.109399
$$753$$ −13.2915 −0.484369
$$754$$ −4.29150 −0.156287
$$755$$ −22.1033 −0.804420
$$756$$ 0 0
$$757$$ 26.6458 0.968456 0.484228 0.874942i $$-0.339100\pi$$
0.484228 + 0.874942i $$0.339100\pi$$
$$758$$ 9.16601 0.332924
$$759$$ −1.52026 −0.0551819
$$760$$ 18.2288 0.661226
$$761$$ 7.29150 0.264317 0.132158 0.991229i $$-0.457809\pi$$
0.132158 + 0.991229i $$0.457809\pi$$
$$762$$ 10.2288 0.370549
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 24.2288 0.875993
$$766$$ −25.7490 −0.930350
$$767$$ −7.93725 −0.286598
$$768$$ 1.00000 0.0360844
$$769$$ 5.64575 0.203591 0.101795 0.994805i $$-0.467541\pi$$
0.101795 + 0.994805i $$0.467541\pi$$
$$770$$ 0 0
$$771$$ −15.8745 −0.571706
$$772$$ 18.9373 0.681567
$$773$$ 40.3320 1.45064 0.725321 0.688411i $$-0.241691\pi$$
0.725321 + 0.688411i $$0.241691\pi$$
$$774$$ 5.29150 0.190199
$$775$$ 27.2915 0.980340
$$776$$ −0.937254 −0.0336455
$$777$$ 0 0
$$778$$ −7.70850 −0.276363
$$779$$ −11.7712 −0.421749
$$780$$ −3.64575 −0.130539
$$781$$ 10.5203 0.376444
$$782$$ −15.6458 −0.559491
$$783$$ 4.29150 0.153366
$$784$$ 0 0
$$785$$ −9.64575 −0.344272
$$786$$ −10.9373 −0.390119
$$787$$ 52.6235 1.87583 0.937913 0.346871i $$-0.112756\pi$$
0.937913 + 0.346871i $$0.112756\pi$$
$$788$$ −26.8118 −0.955129
$$789$$ −3.64575 −0.129792
$$790$$ −36.4575 −1.29710
$$791$$ 0 0
$$792$$ −0.645751 −0.0229458
$$793$$ −11.9373 −0.423904
$$794$$ 14.9373 0.530103
$$795$$ 10.9373 0.387904
$$796$$ −22.2288 −0.787877
$$797$$ −45.8745 −1.62496 −0.812479 0.582990i $$-0.801883\pi$$
−0.812479 + 0.582990i $$0.801883\pi$$
$$798$$ 0 0
$$799$$ −19.9373 −0.705329
$$800$$ −8.29150 −0.293149
$$801$$ 16.9373 0.598448
$$802$$ 18.2288 0.643680
$$803$$ −8.81176 −0.310960
$$804$$ 7.58301 0.267432
$$805$$ 0 0
$$806$$ −3.29150 −0.115938
$$807$$ 6.87451 0.241994
$$808$$ 0 0
$$809$$ 23.8118 0.837177 0.418588 0.908176i $$-0.362525\pi$$
0.418588 + 0.908176i $$0.362525\pi$$
$$810$$ 3.64575 0.128099
$$811$$ 38.0000 1.33436 0.667180 0.744896i $$-0.267501\pi$$
0.667180 + 0.744896i $$0.267501\pi$$
$$812$$ 0 0
$$813$$ −22.6458 −0.794221
$$814$$ −3.64575 −0.127784
$$815$$ −8.81176 −0.308663
$$816$$ −6.64575 −0.232648
$$817$$ −26.4575 −0.925631
$$818$$ 26.9373 0.941839
$$819$$ 0 0
$$820$$ 8.58301 0.299732
$$821$$ −5.77124 −0.201418 −0.100709 0.994916i $$-0.532111\pi$$
−0.100709 + 0.994916i $$0.532111\pi$$
$$822$$ 19.5203 0.680847
$$823$$ 15.5203 0.541002 0.270501 0.962720i $$-0.412811\pi$$
0.270501 + 0.962720i $$0.412811\pi$$
$$824$$ 5.29150 0.184338
$$825$$ 5.35425 0.186411
$$826$$ 0 0
$$827$$ 35.3542 1.22939 0.614694 0.788766i $$-0.289280\pi$$
0.614694 + 0.788766i $$0.289280\pi$$
$$828$$ −2.35425 −0.0818158
$$829$$ 47.2288 1.64032 0.820161 0.572132i $$-0.193884\pi$$
0.820161 + 0.572132i $$0.193884\pi$$
$$830$$ −48.4575 −1.68198
$$831$$ 18.5203 0.642461
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ 22.2288 0.769719
$$835$$ 25.0627 0.867332
$$836$$ 3.22876 0.111669
$$837$$ 3.29150 0.113771
$$838$$ −30.4575 −1.05214
$$839$$ 21.0000 0.725001 0.362500 0.931984i $$-0.381923\pi$$
0.362500 + 0.931984i $$0.381923\pi$$
$$840$$ 0 0
$$841$$ −10.5830 −0.364931
$$842$$ 24.3542 0.839303
$$843$$ −2.58301 −0.0889634
$$844$$ −6.35425 −0.218722
$$845$$ −3.64575 −0.125418
$$846$$ −3.00000 −0.103142
$$847$$ 0 0
$$848$$ −3.00000 −0.103020
$$849$$ −11.0627 −0.379672
$$850$$ 55.1033 1.89003
$$851$$ −13.2915 −0.455627
$$852$$ 16.2915 0.558138
$$853$$ −20.1033 −0.688323 −0.344161 0.938911i $$-0.611837\pi$$
−0.344161 + 0.938911i $$0.611837\pi$$
$$854$$ 0 0
$$855$$ −18.2288 −0.623410
$$856$$ 12.0000 0.410152
$$857$$ 13.9373 0.476088 0.238044 0.971254i $$-0.423494\pi$$
0.238044 + 0.971254i $$0.423494\pi$$
$$858$$ −0.645751 −0.0220456
$$859$$ 44.4575 1.51687 0.758435 0.651748i $$-0.225964\pi$$
0.758435 + 0.651748i $$0.225964\pi$$
$$860$$ 19.2915 0.657835
$$861$$ 0 0
$$862$$ −20.5830 −0.701060
$$863$$ 31.7490 1.08075 0.540375 0.841425i $$-0.318283\pi$$
0.540375 + 0.841425i $$0.318283\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 81.2693 2.76324
$$866$$ 33.5830 1.14120
$$867$$ 27.1660 0.922606
$$868$$ 0 0
$$869$$ −6.45751 −0.219056
$$870$$ 15.6458 0.530441
$$871$$ 7.58301 0.256940
$$872$$ 4.00000 0.135457
$$873$$ 0.937254 0.0317212
$$874$$ 11.7712 0.398168
$$875$$ 0 0
$$876$$ −13.6458 −0.461047
$$877$$ −54.8118 −1.85086 −0.925431 0.378917i $$-0.876297\pi$$
−0.925431 + 0.378917i $$0.876297\pi$$
$$878$$ −23.6458 −0.798005
$$879$$ 7.52026 0.253652
$$880$$ −2.35425 −0.0793617
$$881$$ −10.7085 −0.360778 −0.180389 0.983595i $$-0.557736\pi$$
−0.180389 + 0.983595i $$0.557736\pi$$
$$882$$ 0 0
$$883$$ −29.0627 −0.978039 −0.489020 0.872273i $$-0.662645\pi$$
−0.489020 + 0.872273i $$0.662645\pi$$
$$884$$ −6.64575 −0.223521
$$885$$ 28.9373 0.972715
$$886$$ −19.0627 −0.640425
$$887$$ 20.5830 0.691110 0.345555 0.938399i $$-0.387691\pi$$
0.345555 + 0.938399i $$0.387691\pi$$
$$888$$ −5.64575 −0.189459
$$889$$ 0 0
$$890$$ 61.7490 2.06983
$$891$$ 0.645751 0.0216335
$$892$$ −20.5203 −0.687069
$$893$$ 15.0000 0.501956
$$894$$ 7.06275 0.236214
$$895$$ 21.8745 0.731184
$$896$$ 0 0
$$897$$ −2.35425 −0.0786061
$$898$$ −12.0000 −0.400445
$$899$$ 14.1255 0.471112
$$900$$ 8.29150 0.276383
$$901$$ 19.9373 0.664206
$$902$$ 1.52026 0.0506191
$$903$$ 0 0
$$904$$ −15.2288 −0.506501
$$905$$ −53.3948 −1.77490
$$906$$ −6.06275 −0.201421
$$907$$ −3.77124 −0.125222 −0.0626110 0.998038i $$-0.519943\pi$$
−0.0626110 + 0.998038i $$0.519943\pi$$
$$908$$ −4.70850 −0.156257
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −37.7490 −1.25068 −0.625340 0.780352i $$-0.715040\pi$$
−0.625340 + 0.780352i $$0.715040\pi$$
$$912$$ 5.00000 0.165567
$$913$$ −8.58301 −0.284056
$$914$$ −32.2288 −1.06603
$$915$$ 43.5203 1.43874
$$916$$ 26.4575 0.874181
$$917$$ 0 0
$$918$$ 6.64575 0.219342
$$919$$ 47.8745 1.57923 0.789617 0.613600i $$-0.210279\pi$$
0.789617 + 0.613600i $$0.210279\pi$$
$$920$$ −8.58301 −0.282973
$$921$$ −9.58301 −0.315771
$$922$$ 36.4575 1.20066
$$923$$ 16.2915 0.536241
$$924$$ 0 0
$$925$$ 46.8118 1.53916
$$926$$ −25.1660 −0.827006
$$927$$ −5.29150 −0.173796
$$928$$ −4.29150 −0.140875
$$929$$ −13.7490 −0.451091 −0.225545 0.974233i $$-0.572416\pi$$
−0.225545 + 0.974233i $$0.572416\pi$$
$$930$$ 12.0000 0.393496
$$931$$ 0 0
$$932$$ 0.645751 0.0211523
$$933$$ −1.06275 −0.0347927
$$934$$ 37.5203 1.22770
$$935$$ 15.6458 0.511671
$$936$$ −1.00000 −0.0326860
$$937$$ −31.4575 −1.02767 −0.513836 0.857888i $$-0.671776\pi$$
−0.513836 + 0.857888i $$0.671776\pi$$
$$938$$ 0 0
$$939$$ 13.1660 0.429657
$$940$$ −10.9373 −0.356734
$$941$$ −17.1660 −0.559596 −0.279798 0.960059i $$-0.590267\pi$$
−0.279798 + 0.960059i $$0.590267\pi$$
$$942$$ −2.64575 −0.0862032
$$943$$ 5.54249 0.180488
$$944$$ −7.93725 −0.258336
$$945$$ 0 0
$$946$$ 3.41699 0.111096
$$947$$ −10.5203 −0.341862 −0.170931 0.985283i $$-0.554678\pi$$
−0.170931 + 0.985283i $$0.554678\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ −13.6458 −0.442960
$$950$$ −41.4575 −1.34506
$$951$$ −13.2915 −0.431007
$$952$$ 0 0
$$953$$ −25.1033 −0.813174 −0.406587 0.913612i $$-0.633281\pi$$
−0.406587 + 0.913612i $$0.633281\pi$$
$$954$$ 3.00000 0.0971286
$$955$$ 0 0
$$956$$ 3.00000 0.0970269
$$957$$ 2.77124 0.0895816
$$958$$ 34.7490 1.12269
$$959$$ 0 0
$$960$$ −3.64575 −0.117666
$$961$$ −20.1660 −0.650516
$$962$$ −5.64575 −0.182026
$$963$$ −12.0000 −0.386695
$$964$$ −13.8745 −0.446868
$$965$$ −69.0405 −2.22249
$$966$$ 0 0
$$967$$ 9.10326 0.292741 0.146371 0.989230i $$-0.453241\pi$$
0.146371 + 0.989230i $$0.453241\pi$$
$$968$$ 10.5830 0.340151
$$969$$ −33.2288 −1.06746
$$970$$ 3.41699 0.109713
$$971$$ 16.9373 0.543542 0.271771 0.962362i $$-0.412391\pi$$
0.271771 + 0.962362i $$0.412391\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 23.9373 0.766999
$$975$$ 8.29150 0.265541
$$976$$ −11.9373 −0.382102
$$977$$ 20.1255 0.643872 0.321936 0.946762i $$-0.395666\pi$$
0.321936 + 0.946762i $$0.395666\pi$$
$$978$$ −2.41699 −0.0772870
$$979$$ 10.9373 0.349556
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ −11.1660 −0.356322
$$983$$ −30.0405 −0.958144 −0.479072 0.877776i $$-0.659027\pi$$
−0.479072 + 0.877776i $$0.659027\pi$$
$$984$$ 2.35425 0.0750507
$$985$$ 97.7490 3.11454
$$986$$ 28.5203 0.908270
$$987$$ 0 0
$$988$$ 5.00000 0.159071
$$989$$ 12.4575 0.396126
$$990$$ 2.35425 0.0748229
$$991$$ 5.41699 0.172077 0.0860383 0.996292i $$-0.472579\pi$$
0.0860383 + 0.996292i $$0.472579\pi$$
$$992$$ −3.29150 −0.104505
$$993$$ 11.4170 0.362307
$$994$$ 0 0
$$995$$ 81.0405 2.56916
$$996$$ −13.2915 −0.421157
$$997$$ −1.22876 −0.0389151 −0.0194576 0.999811i $$-0.506194\pi$$
−0.0194576 + 0.999811i $$0.506194\pi$$
$$998$$ 29.2915 0.927206
$$999$$ 5.64575 0.178624
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 3822.2.a.bk.1.1 2
7.2 even 3 546.2.i.j.235.2 yes 4
7.4 even 3 546.2.i.j.79.2 4
7.6 odd 2 3822.2.a.bi.1.2 2
21.2 odd 6 1638.2.j.k.235.1 4
21.11 odd 6 1638.2.j.k.1171.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.j.79.2 4 7.4 even 3
546.2.i.j.235.2 yes 4 7.2 even 3
1638.2.j.k.235.1 4 21.2 odd 6
1638.2.j.k.1171.1 4 21.11 odd 6
3822.2.a.bi.1.2 2 7.6 odd 2
3822.2.a.bk.1.1 2 1.1 even 1 trivial