# Properties

 Label 637.2.a.g.1.2 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ 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] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.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: $$5.08647060876$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.41421 q^{2} +1.41421 q^{3} -4.41421 q^{5} +2.00000 q^{6} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.41421 q^{2} +1.41421 q^{3} -4.41421 q^{5} +2.00000 q^{6} -2.82843 q^{8} -1.00000 q^{9} -6.24264 q^{10} -4.24264 q^{11} +1.00000 q^{13} -6.24264 q^{15} -4.00000 q^{16} +1.41421 q^{17} -1.41421 q^{18} -1.24264 q^{19} -6.00000 q^{22} -0.171573 q^{23} -4.00000 q^{24} +14.4853 q^{25} +1.41421 q^{26} -5.65685 q^{27} +5.82843 q^{29} -8.82843 q^{30} +5.24264 q^{31} -6.00000 q^{33} +2.00000 q^{34} -6.24264 q^{37} -1.75736 q^{38} +1.41421 q^{39} +12.4853 q^{40} -3.17157 q^{41} -5.00000 q^{43} +4.41421 q^{45} -0.242641 q^{46} -4.41421 q^{47} -5.65685 q^{48} +20.4853 q^{50} +2.00000 q^{51} -5.82843 q^{53} -8.00000 q^{54} +18.7279 q^{55} -1.75736 q^{57} +8.24264 q^{58} -11.6569 q^{59} -6.00000 q^{61} +7.41421 q^{62} +8.00000 q^{64} -4.41421 q^{65} -8.48528 q^{66} +2.48528 q^{67} -0.242641 q^{69} +1.07107 q^{71} +2.82843 q^{72} +0.757359 q^{73} -8.82843 q^{74} +20.4853 q^{75} +2.00000 q^{78} -1.48528 q^{79} +17.6569 q^{80} -5.00000 q^{81} -4.48528 q^{82} -4.75736 q^{83} -6.24264 q^{85} -7.07107 q^{86} +8.24264 q^{87} +12.0000 q^{88} -4.41421 q^{89} +6.24264 q^{90} +7.41421 q^{93} -6.24264 q^{94} +5.48528 q^{95} +13.7279 q^{97} +4.24264 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 6 * q^5 + 4 * q^6 - 2 * q^9 $$2 q - 6 q^{5} + 4 q^{6} - 2 q^{9} - 4 q^{10} + 2 q^{13} - 4 q^{15} - 8 q^{16} + 6 q^{19} - 12 q^{22} - 6 q^{23} - 8 q^{24} + 12 q^{25} + 6 q^{29} - 12 q^{30} + 2 q^{31} - 12 q^{33} + 4 q^{34} - 4 q^{37} - 12 q^{38} + 8 q^{40} - 12 q^{41} - 10 q^{43} + 6 q^{45} + 8 q^{46} - 6 q^{47} + 24 q^{50} + 4 q^{51} - 6 q^{53} - 16 q^{54} + 12 q^{55} - 12 q^{57} + 8 q^{58} - 12 q^{59} - 12 q^{61} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 12 q^{67} + 8 q^{69} - 12 q^{71} + 10 q^{73} - 12 q^{74} + 24 q^{75} + 4 q^{78} + 14 q^{79} + 24 q^{80} - 10 q^{81} + 8 q^{82} - 18 q^{83} - 4 q^{85} + 8 q^{87} + 24 q^{88} - 6 q^{89} + 4 q^{90} + 12 q^{93} - 4 q^{94} - 6 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q - 6 * q^5 + 4 * q^6 - 2 * q^9 - 4 * q^10 + 2 * q^13 - 4 * q^15 - 8 * q^16 + 6 * q^19 - 12 * q^22 - 6 * q^23 - 8 * q^24 + 12 * q^25 + 6 * q^29 - 12 * q^30 + 2 * q^31 - 12 * q^33 + 4 * q^34 - 4 * q^37 - 12 * q^38 + 8 * q^40 - 12 * q^41 - 10 * q^43 + 6 * q^45 + 8 * q^46 - 6 * q^47 + 24 * q^50 + 4 * q^51 - 6 * q^53 - 16 * q^54 + 12 * q^55 - 12 * q^57 + 8 * q^58 - 12 * q^59 - 12 * q^61 + 12 * q^62 + 16 * q^64 - 6 * q^65 - 12 * q^67 + 8 * q^69 - 12 * q^71 + 10 * q^73 - 12 * q^74 + 24 * q^75 + 4 * q^78 + 14 * q^79 + 24 * q^80 - 10 * q^81 + 8 * q^82 - 18 * q^83 - 4 * q^85 + 8 * q^87 + 24 * q^88 - 6 * q^89 + 4 * q^90 + 12 * q^93 - 4 * q^94 - 6 * q^95 + 2 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.41421 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$3$$ 1.41421 0.816497 0.408248 0.912871i $$-0.366140\pi$$
0.408248 + 0.912871i $$0.366140\pi$$
$$4$$ 0 0
$$5$$ −4.41421 −1.97410 −0.987048 0.160424i $$-0.948714\pi$$
−0.987048 + 0.160424i $$0.948714\pi$$
$$6$$ 2.00000 0.816497
$$7$$ 0 0
$$8$$ −2.82843 −1.00000
$$9$$ −1.00000 −0.333333
$$10$$ −6.24264 −1.97410
$$11$$ −4.24264 −1.27920 −0.639602 0.768706i $$-0.720901\pi$$
−0.639602 + 0.768706i $$0.720901\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −6.24264 −1.61184
$$16$$ −4.00000 −1.00000
$$17$$ 1.41421 0.342997 0.171499 0.985184i $$-0.445139\pi$$
0.171499 + 0.985184i $$0.445139\pi$$
$$18$$ −1.41421 −0.333333
$$19$$ −1.24264 −0.285081 −0.142541 0.989789i $$-0.545527\pi$$
−0.142541 + 0.989789i $$0.545527\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ −0.171573 −0.0357754 −0.0178877 0.999840i $$-0.505694\pi$$
−0.0178877 + 0.999840i $$0.505694\pi$$
$$24$$ −4.00000 −0.816497
$$25$$ 14.4853 2.89706
$$26$$ 1.41421 0.277350
$$27$$ −5.65685 −1.08866
$$28$$ 0 0
$$29$$ 5.82843 1.08231 0.541156 0.840922i $$-0.317987\pi$$
0.541156 + 0.840922i $$0.317987\pi$$
$$30$$ −8.82843 −1.61184
$$31$$ 5.24264 0.941606 0.470803 0.882238i $$-0.343964\pi$$
0.470803 + 0.882238i $$0.343964\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.24264 −1.02628 −0.513142 0.858304i $$-0.671519\pi$$
−0.513142 + 0.858304i $$0.671519\pi$$
$$38$$ −1.75736 −0.285081
$$39$$ 1.41421 0.226455
$$40$$ 12.4853 1.97410
$$41$$ −3.17157 −0.495316 −0.247658 0.968847i $$-0.579661\pi$$
−0.247658 + 0.968847i $$0.579661\pi$$
$$42$$ 0 0
$$43$$ −5.00000 −0.762493 −0.381246 0.924473i $$-0.624505\pi$$
−0.381246 + 0.924473i $$0.624505\pi$$
$$44$$ 0 0
$$45$$ 4.41421 0.658032
$$46$$ −0.242641 −0.0357754
$$47$$ −4.41421 −0.643879 −0.321940 0.946760i $$-0.604335\pi$$
−0.321940 + 0.946760i $$0.604335\pi$$
$$48$$ −5.65685 −0.816497
$$49$$ 0 0
$$50$$ 20.4853 2.89706
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ −5.82843 −0.800596 −0.400298 0.916385i $$-0.631093\pi$$
−0.400298 + 0.916385i $$0.631093\pi$$
$$54$$ −8.00000 −1.08866
$$55$$ 18.7279 2.52527
$$56$$ 0 0
$$57$$ −1.75736 −0.232768
$$58$$ 8.24264 1.08231
$$59$$ −11.6569 −1.51759 −0.758797 0.651328i $$-0.774212\pi$$
−0.758797 + 0.651328i $$0.774212\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 7.41421 0.941606
$$63$$ 0 0
$$64$$ 8.00000 1.00000
$$65$$ −4.41421 −0.547516
$$66$$ −8.48528 −1.04447
$$67$$ 2.48528 0.303625 0.151813 0.988409i $$-0.451489\pi$$
0.151813 + 0.988409i $$0.451489\pi$$
$$68$$ 0 0
$$69$$ −0.242641 −0.0292105
$$70$$ 0 0
$$71$$ 1.07107 0.127112 0.0635562 0.997978i $$-0.479756\pi$$
0.0635562 + 0.997978i $$0.479756\pi$$
$$72$$ 2.82843 0.333333
$$73$$ 0.757359 0.0886422 0.0443211 0.999017i $$-0.485888\pi$$
0.0443211 + 0.999017i $$0.485888\pi$$
$$74$$ −8.82843 −1.02628
$$75$$ 20.4853 2.36544
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 2.00000 0.226455
$$79$$ −1.48528 −0.167107 −0.0835536 0.996503i $$-0.526627\pi$$
−0.0835536 + 0.996503i $$0.526627\pi$$
$$80$$ 17.6569 1.97410
$$81$$ −5.00000 −0.555556
$$82$$ −4.48528 −0.495316
$$83$$ −4.75736 −0.522188 −0.261094 0.965313i $$-0.584083\pi$$
−0.261094 + 0.965313i $$0.584083\pi$$
$$84$$ 0 0
$$85$$ −6.24264 −0.677109
$$86$$ −7.07107 −0.762493
$$87$$ 8.24264 0.883704
$$88$$ 12.0000 1.27920
$$89$$ −4.41421 −0.467906 −0.233953 0.972248i $$-0.575166\pi$$
−0.233953 + 0.972248i $$0.575166\pi$$
$$90$$ 6.24264 0.658032
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 7.41421 0.768818
$$94$$ −6.24264 −0.643879
$$95$$ 5.48528 0.562778
$$96$$ 0 0
$$97$$ 13.7279 1.39386 0.696930 0.717139i $$-0.254549\pi$$
0.696930 + 0.717139i $$0.254549\pi$$
$$98$$ 0 0
$$99$$ 4.24264 0.426401
$$100$$ 0 0
$$101$$ −1.75736 −0.174864 −0.0874319 0.996170i $$-0.527866\pi$$
−0.0874319 + 0.996170i $$0.527866\pi$$
$$102$$ 2.82843 0.280056
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −2.82843 −0.277350
$$105$$ 0 0
$$106$$ −8.24264 −0.800596
$$107$$ 8.14214 0.787130 0.393565 0.919297i $$-0.371242\pi$$
0.393565 + 0.919297i $$0.371242\pi$$
$$108$$ 0 0
$$109$$ −8.72792 −0.835983 −0.417992 0.908451i $$-0.637266\pi$$
−0.417992 + 0.908451i $$0.637266\pi$$
$$110$$ 26.4853 2.52527
$$111$$ −8.82843 −0.837957
$$112$$ 0 0
$$113$$ −20.3137 −1.91095 −0.955476 0.295067i $$-0.904658\pi$$
−0.955476 + 0.295067i $$0.904658\pi$$
$$114$$ −2.48528 −0.232768
$$115$$ 0.757359 0.0706241
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ −16.4853 −1.51759
$$119$$ 0 0
$$120$$ 17.6569 1.61184
$$121$$ 7.00000 0.636364
$$122$$ −8.48528 −0.768221
$$123$$ −4.48528 −0.404424
$$124$$ 0 0
$$125$$ −41.8701 −3.74497
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 11.3137 1.00000
$$129$$ −7.07107 −0.622573
$$130$$ −6.24264 −0.547516
$$131$$ −2.82843 −0.247121 −0.123560 0.992337i $$-0.539431\pi$$
−0.123560 + 0.992337i $$0.539431\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 3.51472 0.303625
$$135$$ 24.9706 2.14912
$$136$$ −4.00000 −0.342997
$$137$$ −7.41421 −0.633439 −0.316720 0.948519i $$-0.602581\pi$$
−0.316720 + 0.948519i $$0.602581\pi$$
$$138$$ −0.343146 −0.0292105
$$139$$ 2.24264 0.190218 0.0951092 0.995467i $$-0.469680\pi$$
0.0951092 + 0.995467i $$0.469680\pi$$
$$140$$ 0 0
$$141$$ −6.24264 −0.525725
$$142$$ 1.51472 0.127112
$$143$$ −4.24264 −0.354787
$$144$$ 4.00000 0.333333
$$145$$ −25.7279 −2.13659
$$146$$ 1.07107 0.0886422
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 7.75736 0.635508 0.317754 0.948173i $$-0.397071\pi$$
0.317754 + 0.948173i $$0.397071\pi$$
$$150$$ 28.9706 2.36544
$$151$$ −18.2426 −1.48457 −0.742283 0.670087i $$-0.766257\pi$$
−0.742283 + 0.670087i $$0.766257\pi$$
$$152$$ 3.51472 0.285081
$$153$$ −1.41421 −0.114332
$$154$$ 0 0
$$155$$ −23.1421 −1.85882
$$156$$ 0 0
$$157$$ 12.2426 0.977069 0.488535 0.872545i $$-0.337532\pi$$
0.488535 + 0.872545i $$0.337532\pi$$
$$158$$ −2.10051 −0.167107
$$159$$ −8.24264 −0.653684
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −7.07107 −0.555556
$$163$$ −8.48528 −0.664619 −0.332309 0.943170i $$-0.607828\pi$$
−0.332309 + 0.943170i $$0.607828\pi$$
$$164$$ 0 0
$$165$$ 26.4853 2.06188
$$166$$ −6.72792 −0.522188
$$167$$ 21.3848 1.65480 0.827402 0.561610i $$-0.189818\pi$$
0.827402 + 0.561610i $$0.189818\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −8.82843 −0.677109
$$171$$ 1.24264 0.0950271
$$172$$ 0 0
$$173$$ 0.727922 0.0553429 0.0276714 0.999617i $$-0.491191\pi$$
0.0276714 + 0.999617i $$0.491191\pi$$
$$174$$ 11.6569 0.883704
$$175$$ 0 0
$$176$$ 16.9706 1.27920
$$177$$ −16.4853 −1.23911
$$178$$ −6.24264 −0.467906
$$179$$ −9.00000 −0.672692 −0.336346 0.941739i $$-0.609191\pi$$
−0.336346 + 0.941739i $$0.609191\pi$$
$$180$$ 0 0
$$181$$ −6.72792 −0.500083 −0.250041 0.968235i $$-0.580444\pi$$
−0.250041 + 0.968235i $$0.580444\pi$$
$$182$$ 0 0
$$183$$ −8.48528 −0.627250
$$184$$ 0.485281 0.0357754
$$185$$ 27.5563 2.02598
$$186$$ 10.4853 0.768818
$$187$$ −6.00000 −0.438763
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 7.75736 0.562778
$$191$$ 20.8284 1.50709 0.753546 0.657395i $$-0.228342\pi$$
0.753546 + 0.657395i $$0.228342\pi$$
$$192$$ 11.3137 0.816497
$$193$$ 2.48528 0.178894 0.0894472 0.995992i $$-0.471490\pi$$
0.0894472 + 0.995992i $$0.471490\pi$$
$$194$$ 19.4142 1.39386
$$195$$ −6.24264 −0.447045
$$196$$ 0 0
$$197$$ 23.6569 1.68548 0.842741 0.538320i $$-0.180941\pi$$
0.842741 + 0.538320i $$0.180941\pi$$
$$198$$ 6.00000 0.426401
$$199$$ 12.2426 0.867858 0.433929 0.900947i $$-0.357127\pi$$
0.433929 + 0.900947i $$0.357127\pi$$
$$200$$ −40.9706 −2.89706
$$201$$ 3.51472 0.247909
$$202$$ −2.48528 −0.174864
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 14.0000 0.977802
$$206$$ 11.3137 0.788263
$$207$$ 0.171573 0.0119251
$$208$$ −4.00000 −0.277350
$$209$$ 5.27208 0.364677
$$210$$ 0 0
$$211$$ 17.9706 1.23714 0.618572 0.785728i $$-0.287711\pi$$
0.618572 + 0.785728i $$0.287711\pi$$
$$212$$ 0 0
$$213$$ 1.51472 0.103787
$$214$$ 11.5147 0.787130
$$215$$ 22.0711 1.50523
$$216$$ 16.0000 1.08866
$$217$$ 0 0
$$218$$ −12.3431 −0.835983
$$219$$ 1.07107 0.0723761
$$220$$ 0 0
$$221$$ 1.41421 0.0951303
$$222$$ −12.4853 −0.837957
$$223$$ −9.24264 −0.618933 −0.309466 0.950910i $$-0.600150\pi$$
−0.309466 + 0.950910i $$0.600150\pi$$
$$224$$ 0 0
$$225$$ −14.4853 −0.965685
$$226$$ −28.7279 −1.91095
$$227$$ −21.1716 −1.40521 −0.702603 0.711582i $$-0.747979\pi$$
−0.702603 + 0.711582i $$0.747979\pi$$
$$228$$ 0 0
$$229$$ 21.4558 1.41784 0.708921 0.705288i $$-0.249182\pi$$
0.708921 + 0.705288i $$0.249182\pi$$
$$230$$ 1.07107 0.0706241
$$231$$ 0 0
$$232$$ −16.4853 −1.08231
$$233$$ −3.34315 −0.219017 −0.109508 0.993986i $$-0.534928\pi$$
−0.109508 + 0.993986i $$0.534928\pi$$
$$234$$ −1.41421 −0.0924500
$$235$$ 19.4853 1.27108
$$236$$ 0 0
$$237$$ −2.10051 −0.136442
$$238$$ 0 0
$$239$$ 20.4853 1.32508 0.662541 0.749025i $$-0.269478\pi$$
0.662541 + 0.749025i $$0.269478\pi$$
$$240$$ 24.9706 1.61184
$$241$$ −22.2132 −1.43088 −0.715439 0.698675i $$-0.753773\pi$$
−0.715439 + 0.698675i $$0.753773\pi$$
$$242$$ 9.89949 0.636364
$$243$$ 9.89949 0.635053
$$244$$ 0 0
$$245$$ 0 0
$$246$$ −6.34315 −0.404424
$$247$$ −1.24264 −0.0790673
$$248$$ −14.8284 −0.941606
$$249$$ −6.72792 −0.426365
$$250$$ −59.2132 −3.74497
$$251$$ −19.4142 −1.22541 −0.612707 0.790310i $$-0.709919\pi$$
−0.612707 + 0.790310i $$0.709919\pi$$
$$252$$ 0 0
$$253$$ 0.727922 0.0457641
$$254$$ 2.82843 0.177471
$$255$$ −8.82843 −0.552858
$$256$$ 0 0
$$257$$ 16.5858 1.03459 0.517296 0.855806i $$-0.326938\pi$$
0.517296 + 0.855806i $$0.326938\pi$$
$$258$$ −10.0000 −0.622573
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −5.82843 −0.360771
$$262$$ −4.00000 −0.247121
$$263$$ −31.9706 −1.97139 −0.985695 0.168541i $$-0.946095\pi$$
−0.985695 + 0.168541i $$0.946095\pi$$
$$264$$ 16.9706 1.04447
$$265$$ 25.7279 1.58045
$$266$$ 0 0
$$267$$ −6.24264 −0.382043
$$268$$ 0 0
$$269$$ −14.8284 −0.904105 −0.452053 0.891991i $$-0.649308\pi$$
−0.452053 + 0.891991i $$0.649308\pi$$
$$270$$ 35.3137 2.14912
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ −5.65685 −0.342997
$$273$$ 0 0
$$274$$ −10.4853 −0.633439
$$275$$ −61.4558 −3.70593
$$276$$ 0 0
$$277$$ 9.48528 0.569915 0.284958 0.958540i $$-0.408020\pi$$
0.284958 + 0.958540i $$0.408020\pi$$
$$278$$ 3.17157 0.190218
$$279$$ −5.24264 −0.313869
$$280$$ 0 0
$$281$$ −15.5563 −0.928014 −0.464007 0.885832i $$-0.653589\pi$$
−0.464007 + 0.885832i $$0.653589\pi$$
$$282$$ −8.82843 −0.525725
$$283$$ −8.48528 −0.504398 −0.252199 0.967675i $$-0.581154\pi$$
−0.252199 + 0.967675i $$0.581154\pi$$
$$284$$ 0 0
$$285$$ 7.75736 0.459506
$$286$$ −6.00000 −0.354787
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −15.0000 −0.882353
$$290$$ −36.3848 −2.13659
$$291$$ 19.4142 1.13808
$$292$$ 0 0
$$293$$ 21.3848 1.24931 0.624656 0.780900i $$-0.285239\pi$$
0.624656 + 0.780900i $$0.285239\pi$$
$$294$$ 0 0
$$295$$ 51.4558 2.99588
$$296$$ 17.6569 1.02628
$$297$$ 24.0000 1.39262
$$298$$ 10.9706 0.635508
$$299$$ −0.171573 −0.00992232
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −25.7990 −1.48457
$$303$$ −2.48528 −0.142776
$$304$$ 4.97056 0.285081
$$305$$ 26.4853 1.51654
$$306$$ −2.00000 −0.114332
$$307$$ −4.75736 −0.271517 −0.135758 0.990742i $$-0.543347\pi$$
−0.135758 + 0.990742i $$0.543347\pi$$
$$308$$ 0 0
$$309$$ 11.3137 0.643614
$$310$$ −32.7279 −1.85882
$$311$$ 4.58579 0.260036 0.130018 0.991512i $$-0.458496\pi$$
0.130018 + 0.991512i $$0.458496\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ 19.2132 1.08599 0.542997 0.839734i $$-0.317289\pi$$
0.542997 + 0.839734i $$0.317289\pi$$
$$314$$ 17.3137 0.977069
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 11.3137 0.635441 0.317721 0.948184i $$-0.397083\pi$$
0.317721 + 0.948184i $$0.397083\pi$$
$$318$$ −11.6569 −0.653684
$$319$$ −24.7279 −1.38450
$$320$$ −35.3137 −1.97410
$$321$$ 11.5147 0.642689
$$322$$ 0 0
$$323$$ −1.75736 −0.0977821
$$324$$ 0 0
$$325$$ 14.4853 0.803499
$$326$$ −12.0000 −0.664619
$$327$$ −12.3431 −0.682578
$$328$$ 8.97056 0.495316
$$329$$ 0 0
$$330$$ 37.4558 2.06188
$$331$$ −18.0000 −0.989369 −0.494685 0.869072i $$-0.664716\pi$$
−0.494685 + 0.869072i $$0.664716\pi$$
$$332$$ 0 0
$$333$$ 6.24264 0.342095
$$334$$ 30.2426 1.65480
$$335$$ −10.9706 −0.599386
$$336$$ 0 0
$$337$$ −33.0000 −1.79762 −0.898812 0.438334i $$-0.855569\pi$$
−0.898812 + 0.438334i $$0.855569\pi$$
$$338$$ 1.41421 0.0769231
$$339$$ −28.7279 −1.56029
$$340$$ 0 0
$$341$$ −22.2426 −1.20451
$$342$$ 1.75736 0.0950271
$$343$$ 0 0
$$344$$ 14.1421 0.762493
$$345$$ 1.07107 0.0576644
$$346$$ 1.02944 0.0553429
$$347$$ 5.65685 0.303676 0.151838 0.988405i $$-0.451481\pi$$
0.151838 + 0.988405i $$0.451481\pi$$
$$348$$ 0 0
$$349$$ 0.272078 0.0145640 0.00728200 0.999973i $$-0.497682\pi$$
0.00728200 + 0.999973i $$0.497682\pi$$
$$350$$ 0 0
$$351$$ −5.65685 −0.301941
$$352$$ 0 0
$$353$$ −8.48528 −0.451626 −0.225813 0.974171i $$-0.572504\pi$$
−0.225813 + 0.974171i $$0.572504\pi$$
$$354$$ −23.3137 −1.23911
$$355$$ −4.72792 −0.250932
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −12.7279 −0.672692
$$359$$ −8.10051 −0.427528 −0.213764 0.976885i $$-0.568572\pi$$
−0.213764 + 0.976885i $$0.568572\pi$$
$$360$$ −12.4853 −0.658032
$$361$$ −17.4558 −0.918729
$$362$$ −9.51472 −0.500083
$$363$$ 9.89949 0.519589
$$364$$ 0 0
$$365$$ −3.34315 −0.174988
$$366$$ −12.0000 −0.627250
$$367$$ −1.75736 −0.0917334 −0.0458667 0.998948i $$-0.514605\pi$$
−0.0458667 + 0.998948i $$0.514605\pi$$
$$368$$ 0.686292 0.0357754
$$369$$ 3.17157 0.165105
$$370$$ 38.9706 2.02598
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −8.48528 −0.439351 −0.219676 0.975573i $$-0.570500\pi$$
−0.219676 + 0.975573i $$0.570500\pi$$
$$374$$ −8.48528 −0.438763
$$375$$ −59.2132 −3.05776
$$376$$ 12.4853 0.643879
$$377$$ 5.82843 0.300179
$$378$$ 0 0
$$379$$ 32.2426 1.65619 0.828097 0.560585i $$-0.189424\pi$$
0.828097 + 0.560585i $$0.189424\pi$$
$$380$$ 0 0
$$381$$ 2.82843 0.144905
$$382$$ 29.4558 1.50709
$$383$$ 3.51472 0.179594 0.0897969 0.995960i $$-0.471378\pi$$
0.0897969 + 0.995960i $$0.471378\pi$$
$$384$$ 16.0000 0.816497
$$385$$ 0 0
$$386$$ 3.51472 0.178894
$$387$$ 5.00000 0.254164
$$388$$ 0 0
$$389$$ 18.3431 0.930034 0.465017 0.885302i $$-0.346048\pi$$
0.465017 + 0.885302i $$0.346048\pi$$
$$390$$ −8.82843 −0.447045
$$391$$ −0.242641 −0.0122709
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 33.4558 1.68548
$$395$$ 6.55635 0.329886
$$396$$ 0 0
$$397$$ −24.2132 −1.21523 −0.607613 0.794233i $$-0.707873\pi$$
−0.607613 + 0.794233i $$0.707873\pi$$
$$398$$ 17.3137 0.867858
$$399$$ 0 0
$$400$$ −57.9411 −2.89706
$$401$$ 17.6569 0.881741 0.440871 0.897571i $$-0.354670\pi$$
0.440871 + 0.897571i $$0.354670\pi$$
$$402$$ 4.97056 0.247909
$$403$$ 5.24264 0.261155
$$404$$ 0 0
$$405$$ 22.0711 1.09672
$$406$$ 0 0
$$407$$ 26.4853 1.31283
$$408$$ −5.65685 −0.280056
$$409$$ 5.24264 0.259232 0.129616 0.991564i $$-0.458626\pi$$
0.129616 + 0.991564i $$0.458626\pi$$
$$410$$ 19.7990 0.977802
$$411$$ −10.4853 −0.517201
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0.242641 0.0119251
$$415$$ 21.0000 1.03085
$$416$$ 0 0
$$417$$ 3.17157 0.155313
$$418$$ 7.45584 0.364677
$$419$$ −32.8701 −1.60581 −0.802904 0.596109i $$-0.796713\pi$$
−0.802904 + 0.596109i $$0.796713\pi$$
$$420$$ 0 0
$$421$$ −18.7279 −0.912743 −0.456372 0.889789i $$-0.650851\pi$$
−0.456372 + 0.889789i $$0.650851\pi$$
$$422$$ 25.4142 1.23714
$$423$$ 4.41421 0.214626
$$424$$ 16.4853 0.800596
$$425$$ 20.4853 0.993682
$$426$$ 2.14214 0.103787
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −6.00000 −0.289683
$$430$$ 31.2132 1.50523
$$431$$ −23.6569 −1.13951 −0.569755 0.821814i $$-0.692962\pi$$
−0.569755 + 0.821814i $$0.692962\pi$$
$$432$$ 22.6274 1.08866
$$433$$ −8.97056 −0.431098 −0.215549 0.976493i $$-0.569154\pi$$
−0.215549 + 0.976493i $$0.569154\pi$$
$$434$$ 0 0
$$435$$ −36.3848 −1.74452
$$436$$ 0 0
$$437$$ 0.213203 0.0101989
$$438$$ 1.51472 0.0723761
$$439$$ 17.5147 0.835932 0.417966 0.908463i $$-0.362743\pi$$
0.417966 + 0.908463i $$0.362743\pi$$
$$440$$ −52.9706 −2.52527
$$441$$ 0 0
$$442$$ 2.00000 0.0951303
$$443$$ −26.3137 −1.25020 −0.625101 0.780544i $$-0.714942\pi$$
−0.625101 + 0.780544i $$0.714942\pi$$
$$444$$ 0 0
$$445$$ 19.4853 0.923691
$$446$$ −13.0711 −0.618933
$$447$$ 10.9706 0.518890
$$448$$ 0 0
$$449$$ 26.8284 1.26611 0.633056 0.774106i $$-0.281800\pi$$
0.633056 + 0.774106i $$0.281800\pi$$
$$450$$ −20.4853 −0.965685
$$451$$ 13.4558 0.633611
$$452$$ 0 0
$$453$$ −25.7990 −1.21214
$$454$$ −29.9411 −1.40521
$$455$$ 0 0
$$456$$ 4.97056 0.232768
$$457$$ 35.2132 1.64720 0.823602 0.567168i $$-0.191961\pi$$
0.823602 + 0.567168i $$0.191961\pi$$
$$458$$ 30.3431 1.41784
$$459$$ −8.00000 −0.373408
$$460$$ 0 0
$$461$$ −3.17157 −0.147715 −0.0738574 0.997269i $$-0.523531\pi$$
−0.0738574 + 0.997269i $$0.523531\pi$$
$$462$$ 0 0
$$463$$ 4.24264 0.197172 0.0985861 0.995129i $$-0.468568\pi$$
0.0985861 + 0.995129i $$0.468568\pi$$
$$464$$ −23.3137 −1.08231
$$465$$ −32.7279 −1.51772
$$466$$ −4.72792 −0.219017
$$467$$ −15.8995 −0.735741 −0.367870 0.929877i $$-0.619913\pi$$
−0.367870 + 0.929877i $$0.619913\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 27.5563 1.27108
$$471$$ 17.3137 0.797774
$$472$$ 32.9706 1.51759
$$473$$ 21.2132 0.975384
$$474$$ −2.97056 −0.136442
$$475$$ −18.0000 −0.825897
$$476$$ 0 0
$$477$$ 5.82843 0.266865
$$478$$ 28.9706 1.32508
$$479$$ −36.2132 −1.65462 −0.827312 0.561743i $$-0.810131\pi$$
−0.827312 + 0.561743i $$0.810131\pi$$
$$480$$ 0 0
$$481$$ −6.24264 −0.284640
$$482$$ −31.4142 −1.43088
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −60.5980 −2.75161
$$486$$ 14.0000 0.635053
$$487$$ 39.4558 1.78791 0.893957 0.448152i $$-0.147918\pi$$
0.893957 + 0.448152i $$0.147918\pi$$
$$488$$ 16.9706 0.768221
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ −28.6274 −1.29194 −0.645969 0.763364i $$-0.723546\pi$$
−0.645969 + 0.763364i $$0.723546\pi$$
$$492$$ 0 0
$$493$$ 8.24264 0.371230
$$494$$ −1.75736 −0.0790673
$$495$$ −18.7279 −0.841757
$$496$$ −20.9706 −0.941606
$$497$$ 0 0
$$498$$ −9.51472 −0.426365
$$499$$ −38.7279 −1.73370 −0.866850 0.498569i $$-0.833859\pi$$
−0.866850 + 0.498569i $$0.833859\pi$$
$$500$$ 0 0
$$501$$ 30.2426 1.35114
$$502$$ −27.4558 −1.22541
$$503$$ −16.6274 −0.741380 −0.370690 0.928757i $$-0.620879\pi$$
−0.370690 + 0.928757i $$0.620879\pi$$
$$504$$ 0 0
$$505$$ 7.75736 0.345198
$$506$$ 1.02944 0.0457641
$$507$$ 1.41421 0.0628074
$$508$$ 0 0
$$509$$ 24.8995 1.10365 0.551825 0.833960i $$-0.313931\pi$$
0.551825 + 0.833960i $$0.313931\pi$$
$$510$$ −12.4853 −0.552858
$$511$$ 0 0
$$512$$ −22.6274 −1.00000
$$513$$ 7.02944 0.310357
$$514$$ 23.4558 1.03459
$$515$$ −35.3137 −1.55611
$$516$$ 0 0
$$517$$ 18.7279 0.823653
$$518$$ 0 0
$$519$$ 1.02944 0.0451873
$$520$$ 12.4853 0.547516
$$521$$ 17.6569 0.773561 0.386780 0.922172i $$-0.373587\pi$$
0.386780 + 0.922172i $$0.373587\pi$$
$$522$$ −8.24264 −0.360771
$$523$$ 2.97056 0.129894 0.0649468 0.997889i $$-0.479312\pi$$
0.0649468 + 0.997889i $$0.479312\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −45.2132 −1.97139
$$527$$ 7.41421 0.322968
$$528$$ 24.0000 1.04447
$$529$$ −22.9706 −0.998720
$$530$$ 36.3848 1.58045
$$531$$ 11.6569 0.505864
$$532$$ 0 0
$$533$$ −3.17157 −0.137376
$$534$$ −8.82843 −0.382043
$$535$$ −35.9411 −1.55387
$$536$$ −7.02944 −0.303625
$$537$$ −12.7279 −0.549250
$$538$$ −20.9706 −0.904105
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 35.2132 1.51393 0.756967 0.653453i $$-0.226680\pi$$
0.756967 + 0.653453i $$0.226680\pi$$
$$542$$ −28.2843 −1.21491
$$543$$ −9.51472 −0.408316
$$544$$ 0 0
$$545$$ 38.5269 1.65031
$$546$$ 0 0
$$547$$ 18.5147 0.791632 0.395816 0.918330i $$-0.370462\pi$$
0.395816 + 0.918330i $$0.370462\pi$$
$$548$$ 0 0
$$549$$ 6.00000 0.256074
$$550$$ −86.9117 −3.70593
$$551$$ −7.24264 −0.308547
$$552$$ 0.686292 0.0292105
$$553$$ 0 0
$$554$$ 13.4142 0.569915
$$555$$ 38.9706 1.65421
$$556$$ 0 0
$$557$$ 12.0000 0.508456 0.254228 0.967144i $$-0.418179\pi$$
0.254228 + 0.967144i $$0.418179\pi$$
$$558$$ −7.41421 −0.313869
$$559$$ −5.00000 −0.211477
$$560$$ 0 0
$$561$$ −8.48528 −0.358249
$$562$$ −22.0000 −0.928014
$$563$$ 17.6569 0.744148 0.372074 0.928203i $$-0.378647\pi$$
0.372074 + 0.928203i $$0.378647\pi$$
$$564$$ 0 0
$$565$$ 89.6690 3.77241
$$566$$ −12.0000 −0.504398
$$567$$ 0 0
$$568$$ −3.02944 −0.127112
$$569$$ −23.1421 −0.970169 −0.485084 0.874467i $$-0.661211\pi$$
−0.485084 + 0.874467i $$0.661211\pi$$
$$570$$ 10.9706 0.459506
$$571$$ −8.45584 −0.353866 −0.176933 0.984223i $$-0.556618\pi$$
−0.176933 + 0.984223i $$0.556618\pi$$
$$572$$ 0 0
$$573$$ 29.4558 1.23054
$$574$$ 0 0
$$575$$ −2.48528 −0.103643
$$576$$ −8.00000 −0.333333
$$577$$ −26.9706 −1.12280 −0.561400 0.827545i $$-0.689737\pi$$
−0.561400 + 0.827545i $$0.689737\pi$$
$$578$$ −21.2132 −0.882353
$$579$$ 3.51472 0.146067
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 27.4558 1.13808
$$583$$ 24.7279 1.02413
$$584$$ −2.14214 −0.0886422
$$585$$ 4.41421 0.182505
$$586$$ 30.2426 1.24931
$$587$$ −24.5563 −1.01355 −0.506775 0.862079i $$-0.669162\pi$$
−0.506775 + 0.862079i $$0.669162\pi$$
$$588$$ 0 0
$$589$$ −6.51472 −0.268434
$$590$$ 72.7696 2.99588
$$591$$ 33.4558 1.37619
$$592$$ 24.9706 1.02628
$$593$$ −30.5563 −1.25480 −0.627399 0.778698i $$-0.715881\pi$$
−0.627399 + 0.778698i $$0.715881\pi$$
$$594$$ 33.9411 1.39262
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 17.3137 0.708603
$$598$$ −0.242641 −0.00992232
$$599$$ 10.7990 0.441235 0.220617 0.975360i $$-0.429193\pi$$
0.220617 + 0.975360i $$0.429193\pi$$
$$600$$ −57.9411 −2.36544
$$601$$ 5.02944 0.205155 0.102578 0.994725i $$-0.467291\pi$$
0.102578 + 0.994725i $$0.467291\pi$$
$$602$$ 0 0
$$603$$ −2.48528 −0.101208
$$604$$ 0 0
$$605$$ −30.8995 −1.25624
$$606$$ −3.51472 −0.142776
$$607$$ 1.27208 0.0516321 0.0258160 0.999667i $$-0.491782\pi$$
0.0258160 + 0.999667i $$0.491782\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 37.4558 1.51654
$$611$$ −4.41421 −0.178580
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ −6.72792 −0.271517
$$615$$ 19.7990 0.798372
$$616$$ 0 0
$$617$$ 23.6569 0.952389 0.476195 0.879340i $$-0.342016\pi$$
0.476195 + 0.879340i $$0.342016\pi$$
$$618$$ 16.0000 0.643614
$$619$$ 32.9706 1.32520 0.662599 0.748974i $$-0.269453\pi$$
0.662599 + 0.748974i $$0.269453\pi$$
$$620$$ 0 0
$$621$$ 0.970563 0.0389473
$$622$$ 6.48528 0.260036
$$623$$ 0 0
$$624$$ −5.65685 −0.226455
$$625$$ 112.397 4.49588
$$626$$ 27.1716 1.08599
$$627$$ 7.45584 0.297758
$$628$$ 0 0
$$629$$ −8.82843 −0.352012
$$630$$ 0 0
$$631$$ 2.00000 0.0796187 0.0398094 0.999207i $$-0.487325\pi$$
0.0398094 + 0.999207i $$0.487325\pi$$
$$632$$ 4.20101 0.167107
$$633$$ 25.4142 1.01012
$$634$$ 16.0000 0.635441
$$635$$ −8.82843 −0.350345
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −34.9706 −1.38450
$$639$$ −1.07107 −0.0423708
$$640$$ −49.9411 −1.97410
$$641$$ 26.6569 1.05288 0.526441 0.850212i $$-0.323526\pi$$
0.526441 + 0.850212i $$0.323526\pi$$
$$642$$ 16.2843 0.642689
$$643$$ −4.48528 −0.176882 −0.0884411 0.996081i $$-0.528189\pi$$
−0.0884411 + 0.996081i $$0.528189\pi$$
$$644$$ 0 0
$$645$$ 31.2132 1.22902
$$646$$ −2.48528 −0.0977821
$$647$$ 50.5269 1.98642 0.993209 0.116344i $$-0.0371176\pi$$
0.993209 + 0.116344i $$0.0371176\pi$$
$$648$$ 14.1421 0.555556
$$649$$ 49.4558 1.94131
$$650$$ 20.4853 0.803499
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −5.31371 −0.207941 −0.103971 0.994580i $$-0.533155\pi$$
−0.103971 + 0.994580i $$0.533155\pi$$
$$654$$ −17.4558 −0.682578
$$655$$ 12.4853 0.487840
$$656$$ 12.6863 0.495316
$$657$$ −0.757359 −0.0295474
$$658$$ 0 0
$$659$$ −26.6569 −1.03840 −0.519202 0.854652i $$-0.673771\pi$$
−0.519202 + 0.854652i $$0.673771\pi$$
$$660$$ 0 0
$$661$$ 19.2426 0.748452 0.374226 0.927338i $$-0.377908\pi$$
0.374226 + 0.927338i $$0.377908\pi$$
$$662$$ −25.4558 −0.989369
$$663$$ 2.00000 0.0776736
$$664$$ 13.4558 0.522188
$$665$$ 0 0
$$666$$ 8.82843 0.342095
$$667$$ −1.00000 −0.0387202
$$668$$ 0 0
$$669$$ −13.0711 −0.505357
$$670$$ −15.5147 −0.599386
$$671$$ 25.4558 0.982712
$$672$$ 0 0
$$673$$ 40.9411 1.57816 0.789082 0.614288i $$-0.210557\pi$$
0.789082 + 0.614288i $$0.210557\pi$$
$$674$$ −46.6690 −1.79762
$$675$$ −81.9411 −3.15392
$$676$$ 0 0
$$677$$ −29.3553 −1.12822 −0.564109 0.825701i $$-0.690780\pi$$
−0.564109 + 0.825701i $$0.690780\pi$$
$$678$$ −40.6274 −1.56029
$$679$$ 0 0
$$680$$ 17.6569 0.677109
$$681$$ −29.9411 −1.14735
$$682$$ −31.4558 −1.20451
$$683$$ −26.8284 −1.02656 −0.513281 0.858221i $$-0.671570\pi$$
−0.513281 + 0.858221i $$0.671570\pi$$
$$684$$ 0 0
$$685$$ 32.7279 1.25047
$$686$$ 0 0
$$687$$ 30.3431 1.15766
$$688$$ 20.0000 0.762493
$$689$$ −5.82843 −0.222045
$$690$$ 1.51472 0.0576644
$$691$$ 30.6985 1.16783 0.583913 0.811816i $$-0.301521\pi$$
0.583913 + 0.811816i $$0.301521\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 8.00000 0.303676
$$695$$ −9.89949 −0.375509
$$696$$ −23.3137 −0.883704
$$697$$ −4.48528 −0.169892
$$698$$ 0.384776 0.0145640
$$699$$ −4.72792 −0.178826
$$700$$ 0 0
$$701$$ 28.7990 1.08772 0.543861 0.839175i $$-0.316962\pi$$
0.543861 + 0.839175i $$0.316962\pi$$
$$702$$ −8.00000 −0.301941
$$703$$ 7.75736 0.292574
$$704$$ −33.9411 −1.27920
$$705$$ 27.5563 1.03783
$$706$$ −12.0000 −0.451626
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −24.7279 −0.928677 −0.464338 0.885658i $$-0.653708\pi$$
−0.464338 + 0.885658i $$0.653708\pi$$
$$710$$ −6.68629 −0.250932
$$711$$ 1.48528 0.0557024
$$712$$ 12.4853 0.467906
$$713$$ −0.899495 −0.0336864
$$714$$ 0 0
$$715$$ 18.7279 0.700385
$$716$$ 0 0
$$717$$ 28.9706 1.08193
$$718$$ −11.4558 −0.427528
$$719$$ 10.2426 0.381986 0.190993 0.981591i $$-0.438829\pi$$
0.190993 + 0.981591i $$0.438829\pi$$
$$720$$ −17.6569 −0.658032
$$721$$ 0 0
$$722$$ −24.6863 −0.918729
$$723$$ −31.4142 −1.16831
$$724$$ 0 0
$$725$$ 84.4264 3.13552
$$726$$ 14.0000 0.519589
$$727$$ −42.0000 −1.55769 −0.778847 0.627214i $$-0.784195\pi$$
−0.778847 + 0.627214i $$0.784195\pi$$
$$728$$ 0 0
$$729$$ 29.0000 1.07407
$$730$$ −4.72792 −0.174988
$$731$$ −7.07107 −0.261533
$$732$$ 0 0
$$733$$ −42.6985 −1.57710 −0.788552 0.614968i $$-0.789169\pi$$
−0.788552 + 0.614968i $$0.789169\pi$$
$$734$$ −2.48528 −0.0917334
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10.5442 −0.388399
$$738$$ 4.48528 0.165105
$$739$$ 41.6985 1.53390 0.766952 0.641705i $$-0.221773\pi$$
0.766952 + 0.641705i $$0.221773\pi$$
$$740$$ 0 0
$$741$$ −1.75736 −0.0645582
$$742$$ 0 0
$$743$$ −18.3431 −0.672945 −0.336472 0.941693i $$-0.609234\pi$$
−0.336472 + 0.941693i $$0.609234\pi$$
$$744$$ −20.9706 −0.768818
$$745$$ −34.2426 −1.25455
$$746$$ −12.0000 −0.439351
$$747$$ 4.75736 0.174063
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −83.7401 −3.05776
$$751$$ −1.48528 −0.0541987 −0.0270993 0.999633i $$-0.508627\pi$$
−0.0270993 + 0.999633i $$0.508627\pi$$
$$752$$ 17.6569 0.643879
$$753$$ −27.4558 −1.00055
$$754$$ 8.24264 0.300179
$$755$$ 80.5269 2.93067
$$756$$ 0 0
$$757$$ 4.51472 0.164090 0.0820451 0.996629i $$-0.473855\pi$$
0.0820451 + 0.996629i $$0.473855\pi$$
$$758$$ 45.5980 1.65619
$$759$$ 1.02944 0.0373662
$$760$$ −15.5147 −0.562778
$$761$$ −43.2426 −1.56754 −0.783772 0.621048i $$-0.786707\pi$$
−0.783772 + 0.621048i $$0.786707\pi$$
$$762$$ 4.00000 0.144905
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 6.24264 0.225703
$$766$$ 4.97056 0.179594
$$767$$ −11.6569 −0.420905
$$768$$ 0 0
$$769$$ −9.78680 −0.352921 −0.176460 0.984308i $$-0.556465\pi$$
−0.176460 + 0.984308i $$0.556465\pi$$
$$770$$ 0 0
$$771$$ 23.4558 0.844742
$$772$$ 0 0
$$773$$ 27.1716 0.977294 0.488647 0.872482i $$-0.337491\pi$$
0.488647 + 0.872482i $$0.337491\pi$$
$$774$$ 7.07107 0.254164
$$775$$ 75.9411 2.72789
$$776$$ −38.8284 −1.39386
$$777$$ 0 0
$$778$$ 25.9411 0.930034
$$779$$ 3.94113 0.141205
$$780$$ 0 0
$$781$$ −4.54416 −0.162603
$$782$$ −0.343146 −0.0122709
$$783$$ −32.9706 −1.17827
$$784$$ 0 0
$$785$$ −54.0416 −1.92883
$$786$$ −5.65685 −0.201773
$$787$$ 33.2426 1.18497 0.592486 0.805581i $$-0.298147\pi$$
0.592486 + 0.805581i $$0.298147\pi$$
$$788$$ 0 0
$$789$$ −45.2132 −1.60963
$$790$$ 9.27208 0.329886
$$791$$ 0 0
$$792$$ −12.0000 −0.426401
$$793$$ −6.00000 −0.213066
$$794$$ −34.2426 −1.21523
$$795$$ 36.3848 1.29044
$$796$$ 0 0
$$797$$ 35.6569 1.26303 0.631515 0.775363i $$-0.282433\pi$$
0.631515 + 0.775363i $$0.282433\pi$$
$$798$$ 0 0
$$799$$ −6.24264 −0.220849
$$800$$ 0 0
$$801$$ 4.41421 0.155969
$$802$$ 24.9706 0.881741
$$803$$ −3.21320 −0.113391
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 7.41421 0.261155
$$807$$ −20.9706 −0.738199
$$808$$ 4.97056 0.174864
$$809$$ 0.514719 0.0180965 0.00904827 0.999959i $$-0.497120\pi$$
0.00904827 + 0.999959i $$0.497120\pi$$
$$810$$ 31.2132 1.09672
$$811$$ −45.9411 −1.61321 −0.806606 0.591090i $$-0.798698\pi$$
−0.806606 + 0.591090i $$0.798698\pi$$
$$812$$ 0 0
$$813$$ −28.2843 −0.991973
$$814$$ 37.4558 1.31283
$$815$$ 37.4558 1.31202
$$816$$ −8.00000 −0.280056
$$817$$ 6.21320 0.217372
$$818$$ 7.41421 0.259232
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8.14214 0.284162 0.142081 0.989855i $$-0.454621\pi$$
0.142081 + 0.989855i $$0.454621\pi$$
$$822$$ −14.8284 −0.517201
$$823$$ −6.48528 −0.226063 −0.113031 0.993591i $$-0.536056\pi$$
−0.113031 + 0.993591i $$0.536056\pi$$
$$824$$ −22.6274 −0.788263
$$825$$ −86.9117 −3.02588
$$826$$ 0 0
$$827$$ 1.45584 0.0506247 0.0253123 0.999680i $$-0.491942\pi$$
0.0253123 + 0.999680i $$0.491942\pi$$
$$828$$ 0 0
$$829$$ −25.2721 −0.877736 −0.438868 0.898552i $$-0.644620\pi$$
−0.438868 + 0.898552i $$0.644620\pi$$
$$830$$ 29.6985 1.03085
$$831$$ 13.4142 0.465334
$$832$$ 8.00000 0.277350
$$833$$ 0 0
$$834$$ 4.48528 0.155313
$$835$$ −94.3970 −3.26674
$$836$$ 0 0
$$837$$ −29.6569 −1.02509
$$838$$ −46.4853 −1.60581
$$839$$ 38.8284 1.34051 0.670253 0.742133i $$-0.266186\pi$$
0.670253 + 0.742133i $$0.266186\pi$$
$$840$$ 0 0
$$841$$ 4.97056 0.171399
$$842$$ −26.4853 −0.912743
$$843$$ −22.0000 −0.757720
$$844$$ 0 0
$$845$$ −4.41421 −0.151854
$$846$$ 6.24264 0.214626
$$847$$ 0 0
$$848$$ 23.3137 0.800596
$$849$$ −12.0000 −0.411839
$$850$$ 28.9706 0.993682
$$851$$ 1.07107 0.0367157
$$852$$ 0 0
$$853$$ −14.2721 −0.488667 −0.244333 0.969691i $$-0.578569\pi$$
−0.244333 + 0.969691i $$0.578569\pi$$
$$854$$ 0 0
$$855$$ −5.48528 −0.187593
$$856$$ −23.0294 −0.787130
$$857$$ −18.7696 −0.641156 −0.320578 0.947222i $$-0.603877\pi$$
−0.320578 + 0.947222i $$0.603877\pi$$
$$858$$ −8.48528 −0.289683
$$859$$ 2.97056 0.101354 0.0506771 0.998715i $$-0.483862\pi$$
0.0506771 + 0.998715i $$0.483862\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −33.4558 −1.13951
$$863$$ −28.2843 −0.962808 −0.481404 0.876499i $$-0.659873\pi$$
−0.481404 + 0.876499i $$0.659873\pi$$
$$864$$ 0 0
$$865$$ −3.21320 −0.109252
$$866$$ −12.6863 −0.431098
$$867$$ −21.2132 −0.720438
$$868$$ 0 0
$$869$$ 6.30152 0.213764
$$870$$ −51.4558 −1.74452
$$871$$ 2.48528 0.0842105
$$872$$ 24.6863 0.835983
$$873$$ −13.7279 −0.464620
$$874$$ 0.301515 0.0101989
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.75736 −0.0593418 −0.0296709 0.999560i $$-0.509446\pi$$
−0.0296709 + 0.999560i $$0.509446\pi$$
$$878$$ 24.7696 0.835932
$$879$$ 30.2426 1.02006
$$880$$ −74.9117 −2.52527
$$881$$ 49.1127 1.65465 0.827324 0.561724i $$-0.189862\pi$$
0.827324 + 0.561724i $$0.189862\pi$$
$$882$$ 0 0
$$883$$ 2.00000 0.0673054 0.0336527 0.999434i $$-0.489286\pi$$
0.0336527 + 0.999434i $$0.489286\pi$$
$$884$$ 0 0
$$885$$ 72.7696 2.44612
$$886$$ −37.2132 −1.25020
$$887$$ 43.1127 1.44758 0.723791 0.690019i $$-0.242398\pi$$
0.723791 + 0.690019i $$0.242398\pi$$
$$888$$ 24.9706 0.837957
$$889$$ 0 0
$$890$$ 27.5563 0.923691
$$891$$ 21.2132 0.710669
$$892$$ 0 0
$$893$$ 5.48528 0.183558
$$894$$ 15.5147 0.518890
$$895$$ 39.7279 1.32796
$$896$$ 0 0
$$897$$ −0.242641 −0.00810154
$$898$$ 37.9411 1.26611
$$899$$ 30.5563 1.01911
$$900$$ 0 0
$$901$$ −8.24264 −0.274602
$$902$$ 19.0294 0.633611
$$903$$ 0 0
$$904$$ 57.4558 1.91095
$$905$$ 29.6985 0.987211
$$906$$ −36.4853 −1.21214
$$907$$ −31.9706 −1.06157 −0.530783 0.847508i $$-0.678102\pi$$
−0.530783 + 0.847508i $$0.678102\pi$$
$$908$$ 0 0
$$909$$ 1.75736 0.0582879
$$910$$ 0 0
$$911$$ −10.0294 −0.332290 −0.166145 0.986101i $$-0.553132\pi$$
−0.166145 + 0.986101i $$0.553132\pi$$
$$912$$ 7.02944 0.232768
$$913$$ 20.1838 0.667985
$$914$$ 49.7990 1.64720
$$915$$ 37.4558 1.23825
$$916$$ 0 0
$$917$$ 0 0
$$918$$ −11.3137 −0.373408
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ −2.14214 −0.0706241
$$921$$ −6.72792 −0.221693
$$922$$ −4.48528 −0.147715
$$923$$ 1.07107 0.0352546
$$924$$ 0 0
$$925$$ −90.4264 −2.97320
$$926$$ 6.00000 0.197172
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ −17.1005 −0.561049 −0.280525 0.959847i $$-0.590508\pi$$
−0.280525 + 0.959847i $$0.590508\pi$$
$$930$$ −46.2843 −1.51772
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 6.48528 0.212319
$$934$$ −22.4853 −0.735741
$$935$$ 26.4853 0.866161
$$936$$ 2.82843 0.0924500
$$937$$ −2.78680 −0.0910407 −0.0455203 0.998963i $$-0.514495\pi$$
−0.0455203 + 0.998963i $$0.514495\pi$$
$$938$$ 0 0
$$939$$ 27.1716 0.886711
$$940$$ 0 0
$$941$$ −25.9289 −0.845259 −0.422630 0.906303i $$-0.638893\pi$$
−0.422630 + 0.906303i $$0.638893\pi$$
$$942$$ 24.4853 0.797774
$$943$$ 0.544156 0.0177202
$$944$$ 46.6274 1.51759
$$945$$ 0 0
$$946$$ 30.0000 0.975384
$$947$$ 20.8284 0.676833 0.338416 0.940996i $$-0.390109\pi$$
0.338416 + 0.940996i $$0.390109\pi$$
$$948$$ 0 0
$$949$$ 0.757359 0.0245849
$$950$$ −25.4558 −0.825897
$$951$$ 16.0000 0.518836
$$952$$ 0 0
$$953$$ −17.1421 −0.555288 −0.277644 0.960684i $$-0.589554\pi$$
−0.277644 + 0.960684i $$0.589554\pi$$
$$954$$ 8.24264 0.266865
$$955$$ −91.9411 −2.97514
$$956$$ 0 0
$$957$$ −34.9706 −1.13044
$$958$$ −51.2132 −1.65462
$$959$$ 0 0
$$960$$ −49.9411 −1.61184
$$961$$ −3.51472 −0.113378
$$962$$ −8.82843 −0.284640
$$963$$ −8.14214 −0.262377
$$964$$ 0 0
$$965$$ −10.9706 −0.353155
$$966$$ 0 0
$$967$$ −41.6985 −1.34093 −0.670466 0.741940i $$-0.733906\pi$$
−0.670466 + 0.741940i $$0.733906\pi$$
$$968$$ −19.7990 −0.636364
$$969$$ −2.48528 −0.0798387
$$970$$ −85.6985 −2.75161
$$971$$ 7.45584 0.239269 0.119635 0.992818i $$-0.461828\pi$$
0.119635 + 0.992818i $$0.461828\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 55.7990 1.78791
$$975$$ 20.4853 0.656054
$$976$$ 24.0000 0.768221
$$977$$ −24.0416 −0.769160 −0.384580 0.923092i $$-0.625654\pi$$
−0.384580 + 0.923092i $$0.625654\pi$$
$$978$$ −16.9706 −0.542659
$$979$$ 18.7279 0.598547
$$980$$ 0 0
$$981$$ 8.72792 0.278661
$$982$$ −40.4853 −1.29194
$$983$$ −33.0416 −1.05386 −0.526932 0.849907i $$-0.676658\pi$$
−0.526932 + 0.849907i $$0.676658\pi$$
$$984$$ 12.6863 0.404424
$$985$$ −104.426 −3.32730
$$986$$ 11.6569 0.371230
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0.857864 0.0272785
$$990$$ −26.4853 −0.841757
$$991$$ 34.9706 1.11088 0.555438 0.831558i $$-0.312551\pi$$
0.555438 + 0.831558i $$0.312551\pi$$
$$992$$ 0 0
$$993$$ −25.4558 −0.807817
$$994$$ 0 0
$$995$$ −54.0416 −1.71323
$$996$$ 0 0
$$997$$ 28.4853 0.902138 0.451069 0.892489i $$-0.351043\pi$$
0.451069 + 0.892489i $$0.351043\pi$$
$$998$$ −54.7696 −1.73370
$$999$$ 35.3137 1.11728
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 637.2.a.g.1.2 2
3.2 odd 2 5733.2.a.s.1.1 2
7.2 even 3 637.2.e.g.508.1 4
7.3 odd 6 637.2.e.f.79.1 4
7.4 even 3 637.2.e.g.79.1 4
7.5 odd 6 637.2.e.f.508.1 4
7.6 odd 2 91.2.a.c.1.2 2
13.12 even 2 8281.2.a.v.1.1 2
21.20 even 2 819.2.a.h.1.1 2
28.27 even 2 1456.2.a.q.1.2 2
35.34 odd 2 2275.2.a.j.1.1 2
56.13 odd 2 5824.2.a.bl.1.2 2
56.27 even 2 5824.2.a.bk.1.1 2
91.34 even 4 1183.2.c.d.337.3 4
91.83 even 4 1183.2.c.d.337.1 4
91.90 odd 2 1183.2.a.d.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.c.1.2 2 7.6 odd 2
637.2.a.g.1.2 2 1.1 even 1 trivial
637.2.e.f.79.1 4 7.3 odd 6
637.2.e.f.508.1 4 7.5 odd 6
637.2.e.g.79.1 4 7.4 even 3
637.2.e.g.508.1 4 7.2 even 3
819.2.a.h.1.1 2 21.20 even 2
1183.2.a.d.1.1 2 91.90 odd 2
1183.2.c.d.337.1 4 91.83 even 4
1183.2.c.d.337.3 4 91.34 even 4
1456.2.a.q.1.2 2 28.27 even 2
2275.2.a.j.1.1 2 35.34 odd 2
5733.2.a.s.1.1 2 3.2 odd 2
5824.2.a.bk.1.1 2 56.27 even 2
5824.2.a.bl.1.2 2 56.13 odd 2
8281.2.a.v.1.1 2 13.12 even 2