# Properties

 Label 7098.2.a.bu.1.2 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $0$ 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] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.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: $$56.6778153547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 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 1.2 Root $$4.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 7098.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.27492 q^{10} +2.27492 q^{11} -1.00000 q^{12} -1.00000 q^{14} -4.27492 q^{15} +1.00000 q^{16} +0.274917 q^{17} +1.00000 q^{18} +2.27492 q^{19} +4.27492 q^{20} +1.00000 q^{21} +2.27492 q^{22} +2.27492 q^{23} -1.00000 q^{24} +13.2749 q^{25} -1.00000 q^{27} -1.00000 q^{28} +8.27492 q^{29} -4.27492 q^{30} -8.00000 q^{31} +1.00000 q^{32} -2.27492 q^{33} +0.274917 q^{34} -4.27492 q^{35} +1.00000 q^{36} +4.27492 q^{37} +2.27492 q^{38} +4.27492 q^{40} -6.54983 q^{41} +1.00000 q^{42} -2.27492 q^{43} +2.27492 q^{44} +4.27492 q^{45} +2.27492 q^{46} -1.00000 q^{48} +1.00000 q^{49} +13.2749 q^{50} -0.274917 q^{51} +10.0000 q^{53} -1.00000 q^{54} +9.72508 q^{55} -1.00000 q^{56} -2.27492 q^{57} +8.27492 q^{58} -8.00000 q^{59} -4.27492 q^{60} -12.2749 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.27492 q^{66} -12.5498 q^{67} +0.274917 q^{68} -2.27492 q^{69} -4.27492 q^{70} +1.00000 q^{72} +12.8248 q^{73} +4.27492 q^{74} -13.2749 q^{75} +2.27492 q^{76} -2.27492 q^{77} +12.5498 q^{79} +4.27492 q^{80} +1.00000 q^{81} -6.54983 q^{82} +4.54983 q^{83} +1.00000 q^{84} +1.17525 q^{85} -2.27492 q^{86} -8.27492 q^{87} +2.27492 q^{88} +14.0000 q^{89} +4.27492 q^{90} +2.27492 q^{92} +8.00000 q^{93} +9.72508 q^{95} -1.00000 q^{96} -15.0997 q^{97} +1.00000 q^{98} +2.27492 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 3 q^{11} - 2 q^{12} - 2 q^{14} - q^{15} + 2 q^{16} - 7 q^{17} + 2 q^{18} - 3 q^{19} + q^{20} + 2 q^{21} - 3 q^{22} - 3 q^{23} - 2 q^{24} + 19 q^{25} - 2 q^{27} - 2 q^{28} + 9 q^{29} - q^{30} - 16 q^{31} + 2 q^{32} + 3 q^{33} - 7 q^{34} - q^{35} + 2 q^{36} + q^{37} - 3 q^{38} + q^{40} + 2 q^{41} + 2 q^{42} + 3 q^{43} - 3 q^{44} + q^{45} - 3 q^{46} - 2 q^{48} + 2 q^{49} + 19 q^{50} + 7 q^{51} + 20 q^{53} - 2 q^{54} + 27 q^{55} - 2 q^{56} + 3 q^{57} + 9 q^{58} - 16 q^{59} - q^{60} - 17 q^{61} - 16 q^{62} - 2 q^{63} + 2 q^{64} + 3 q^{66} - 10 q^{67} - 7 q^{68} + 3 q^{69} - q^{70} + 2 q^{72} + 3 q^{73} + q^{74} - 19 q^{75} - 3 q^{76} + 3 q^{77} + 10 q^{79} + q^{80} + 2 q^{81} + 2 q^{82} - 6 q^{83} + 2 q^{84} + 25 q^{85} + 3 q^{86} - 9 q^{87} - 3 q^{88} + 28 q^{89} + q^{90} - 3 q^{92} + 16 q^{93} + 27 q^{95} - 2 q^{96} + 2 q^{98} - 3 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + q^10 - 3 * q^11 - 2 * q^12 - 2 * q^14 - q^15 + 2 * q^16 - 7 * q^17 + 2 * q^18 - 3 * q^19 + q^20 + 2 * q^21 - 3 * q^22 - 3 * q^23 - 2 * q^24 + 19 * q^25 - 2 * q^27 - 2 * q^28 + 9 * q^29 - q^30 - 16 * q^31 + 2 * q^32 + 3 * q^33 - 7 * q^34 - q^35 + 2 * q^36 + q^37 - 3 * q^38 + q^40 + 2 * q^41 + 2 * q^42 + 3 * q^43 - 3 * q^44 + q^45 - 3 * q^46 - 2 * q^48 + 2 * q^49 + 19 * q^50 + 7 * q^51 + 20 * q^53 - 2 * q^54 + 27 * q^55 - 2 * q^56 + 3 * q^57 + 9 * q^58 - 16 * q^59 - q^60 - 17 * q^61 - 16 * q^62 - 2 * q^63 + 2 * q^64 + 3 * q^66 - 10 * q^67 - 7 * q^68 + 3 * q^69 - q^70 + 2 * q^72 + 3 * q^73 + q^74 - 19 * q^75 - 3 * q^76 + 3 * q^77 + 10 * q^79 + q^80 + 2 * q^81 + 2 * q^82 - 6 * q^83 + 2 * q^84 + 25 * q^85 + 3 * q^86 - 9 * q^87 - 3 * q^88 + 28 * q^89 + q^90 - 3 * q^92 + 16 * q^93 + 27 * q^95 - 2 * q^96 + 2 * q^98 - 3 * 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
$$5$$ 4.27492 1.91180 0.955901 0.293691i $$-0.0948835\pi$$
0.955901 + 0.293691i $$0.0948835\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 4.27492 1.35185
$$11$$ 2.27492 0.685913 0.342957 0.939351i $$-0.388572\pi$$
0.342957 + 0.939351i $$0.388572\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −1.00000 −0.267261
$$15$$ −4.27492 −1.10378
$$16$$ 1.00000 0.250000
$$17$$ 0.274917 0.0666772 0.0333386 0.999444i $$-0.489386\pi$$
0.0333386 + 0.999444i $$0.489386\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 2.27492 0.521902 0.260951 0.965352i $$-0.415964\pi$$
0.260951 + 0.965352i $$0.415964\pi$$
$$20$$ 4.27492 0.955901
$$21$$ 1.00000 0.218218
$$22$$ 2.27492 0.485014
$$23$$ 2.27492 0.474353 0.237177 0.971467i $$-0.423778\pi$$
0.237177 + 0.971467i $$0.423778\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 13.2749 2.65498
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −1.00000 −0.188982
$$29$$ 8.27492 1.53661 0.768307 0.640082i $$-0.221100\pi$$
0.768307 + 0.640082i $$0.221100\pi$$
$$30$$ −4.27492 −0.780490
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.27492 −0.396012
$$34$$ 0.274917 0.0471479
$$35$$ −4.27492 −0.722593
$$36$$ 1.00000 0.166667
$$37$$ 4.27492 0.702792 0.351396 0.936227i $$-0.385707\pi$$
0.351396 + 0.936227i $$0.385707\pi$$
$$38$$ 2.27492 0.369040
$$39$$ 0 0
$$40$$ 4.27492 0.675924
$$41$$ −6.54983 −1.02291 −0.511456 0.859309i $$-0.670894\pi$$
−0.511456 + 0.859309i $$0.670894\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −2.27492 −0.346922 −0.173461 0.984841i $$-0.555495\pi$$
−0.173461 + 0.984841i $$0.555495\pi$$
$$44$$ 2.27492 0.342957
$$45$$ 4.27492 0.637267
$$46$$ 2.27492 0.335418
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 13.2749 1.87736
$$51$$ −0.274917 −0.0384961
$$52$$ 0 0
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 9.72508 1.31133
$$56$$ −1.00000 −0.133631
$$57$$ −2.27492 −0.301320
$$58$$ 8.27492 1.08655
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ −4.27492 −0.551889
$$61$$ −12.2749 −1.57164 −0.785821 0.618454i $$-0.787759\pi$$
−0.785821 + 0.618454i $$0.787759\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.27492 −0.280023
$$67$$ −12.5498 −1.53321 −0.766603 0.642121i $$-0.778055\pi$$
−0.766603 + 0.642121i $$0.778055\pi$$
$$68$$ 0.274917 0.0333386
$$69$$ −2.27492 −0.273868
$$70$$ −4.27492 −0.510950
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 12.8248 1.50102 0.750512 0.660857i $$-0.229807\pi$$
0.750512 + 0.660857i $$0.229807\pi$$
$$74$$ 4.27492 0.496949
$$75$$ −13.2749 −1.53286
$$76$$ 2.27492 0.260951
$$77$$ −2.27492 −0.259251
$$78$$ 0 0
$$79$$ 12.5498 1.41197 0.705983 0.708228i $$-0.250505\pi$$
0.705983 + 0.708228i $$0.250505\pi$$
$$80$$ 4.27492 0.477950
$$81$$ 1.00000 0.111111
$$82$$ −6.54983 −0.723308
$$83$$ 4.54983 0.499409 0.249705 0.968322i $$-0.419666\pi$$
0.249705 + 0.968322i $$0.419666\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 1.17525 0.127474
$$86$$ −2.27492 −0.245311
$$87$$ −8.27492 −0.887164
$$88$$ 2.27492 0.242507
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 4.27492 0.450616
$$91$$ 0 0
$$92$$ 2.27492 0.237177
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 9.72508 0.997772
$$96$$ −1.00000 −0.102062
$$97$$ −15.0997 −1.53314 −0.766570 0.642161i $$-0.778038\pi$$
−0.766570 + 0.642161i $$0.778038\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 2.27492 0.228638
$$100$$ 13.2749 1.32749
$$101$$ 2.54983 0.253718 0.126859 0.991921i $$-0.459510\pi$$
0.126859 + 0.991921i $$0.459510\pi$$
$$102$$ −0.274917 −0.0272209
$$103$$ 10.2749 1.01242 0.506209 0.862411i $$-0.331046\pi$$
0.506209 + 0.862411i $$0.331046\pi$$
$$104$$ 0 0
$$105$$ 4.27492 0.417189
$$106$$ 10.0000 0.971286
$$107$$ 0.549834 0.0531545 0.0265773 0.999647i $$-0.491539\pi$$
0.0265773 + 0.999647i $$0.491539\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −0.274917 −0.0263323 −0.0131661 0.999913i $$-0.504191\pi$$
−0.0131661 + 0.999913i $$0.504191\pi$$
$$110$$ 9.72508 0.927250
$$111$$ −4.27492 −0.405757
$$112$$ −1.00000 −0.0944911
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −2.27492 −0.213066
$$115$$ 9.72508 0.906869
$$116$$ 8.27492 0.768307
$$117$$ 0 0
$$118$$ −8.00000 −0.736460
$$119$$ −0.274917 −0.0252016
$$120$$ −4.27492 −0.390245
$$121$$ −5.82475 −0.529523
$$122$$ −12.2749 −1.11132
$$123$$ 6.54983 0.590579
$$124$$ −8.00000 −0.718421
$$125$$ 35.3746 3.16400
$$126$$ −1.00000 −0.0890871
$$127$$ −20.5498 −1.82350 −0.911751 0.410742i $$-0.865270\pi$$
−0.911751 + 0.410742i $$0.865270\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 2.27492 0.200295
$$130$$ 0 0
$$131$$ 6.82475 0.596281 0.298141 0.954522i $$-0.403634\pi$$
0.298141 + 0.954522i $$0.403634\pi$$
$$132$$ −2.27492 −0.198006
$$133$$ −2.27492 −0.197260
$$134$$ −12.5498 −1.08414
$$135$$ −4.27492 −0.367926
$$136$$ 0.274917 0.0235740
$$137$$ 17.3746 1.48441 0.742206 0.670172i $$-0.233780\pi$$
0.742206 + 0.670172i $$0.233780\pi$$
$$138$$ −2.27492 −0.193654
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ −4.27492 −0.361296
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 35.3746 2.93770
$$146$$ 12.8248 1.06138
$$147$$ −1.00000 −0.0824786
$$148$$ 4.27492 0.351396
$$149$$ −22.5498 −1.84735 −0.923677 0.383172i $$-0.874832\pi$$
−0.923677 + 0.383172i $$0.874832\pi$$
$$150$$ −13.2749 −1.08389
$$151$$ −6.27492 −0.510646 −0.255323 0.966856i $$-0.582182\pi$$
−0.255323 + 0.966856i $$0.582182\pi$$
$$152$$ 2.27492 0.184520
$$153$$ 0.274917 0.0222257
$$154$$ −2.27492 −0.183318
$$155$$ −34.1993 −2.74696
$$156$$ 0 0
$$157$$ −13.3746 −1.06741 −0.533704 0.845671i $$-0.679200\pi$$
−0.533704 + 0.845671i $$0.679200\pi$$
$$158$$ 12.5498 0.998411
$$159$$ −10.0000 −0.793052
$$160$$ 4.27492 0.337962
$$161$$ −2.27492 −0.179289
$$162$$ 1.00000 0.0785674
$$163$$ −12.5498 −0.982979 −0.491489 0.870884i $$-0.663547\pi$$
−0.491489 + 0.870884i $$0.663547\pi$$
$$164$$ −6.54983 −0.511456
$$165$$ −9.72508 −0.757097
$$166$$ 4.54983 0.353136
$$167$$ 1.72508 0.133491 0.0667455 0.997770i $$-0.478738\pi$$
0.0667455 + 0.997770i $$0.478738\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 0 0
$$170$$ 1.17525 0.0901374
$$171$$ 2.27492 0.173967
$$172$$ −2.27492 −0.173461
$$173$$ 7.09967 0.539778 0.269889 0.962891i $$-0.413013\pi$$
0.269889 + 0.962891i $$0.413013\pi$$
$$174$$ −8.27492 −0.627320
$$175$$ −13.2749 −1.00349
$$176$$ 2.27492 0.171478
$$177$$ 8.00000 0.601317
$$178$$ 14.0000 1.04934
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 4.27492 0.318634
$$181$$ 19.0997 1.41967 0.709834 0.704369i $$-0.248770\pi$$
0.709834 + 0.704369i $$0.248770\pi$$
$$182$$ 0 0
$$183$$ 12.2749 0.907388
$$184$$ 2.27492 0.167709
$$185$$ 18.2749 1.34360
$$186$$ 8.00000 0.586588
$$187$$ 0.625414 0.0457348
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 9.72508 0.705532
$$191$$ 2.27492 0.164607 0.0823036 0.996607i $$-0.473772\pi$$
0.0823036 + 0.996607i $$0.473772\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 18.5498 1.33525 0.667623 0.744499i $$-0.267312\pi$$
0.667623 + 0.744499i $$0.267312\pi$$
$$194$$ −15.0997 −1.08409
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 2.54983 0.181668 0.0908341 0.995866i $$-0.471047\pi$$
0.0908341 + 0.995866i $$0.471047\pi$$
$$198$$ 2.27492 0.161671
$$199$$ 6.82475 0.483794 0.241897 0.970302i $$-0.422230\pi$$
0.241897 + 0.970302i $$0.422230\pi$$
$$200$$ 13.2749 0.938678
$$201$$ 12.5498 0.885197
$$202$$ 2.54983 0.179406
$$203$$ −8.27492 −0.580785
$$204$$ −0.274917 −0.0192481
$$205$$ −28.0000 −1.95560
$$206$$ 10.2749 0.715887
$$207$$ 2.27492 0.158118
$$208$$ 0 0
$$209$$ 5.17525 0.357979
$$210$$ 4.27492 0.294997
$$211$$ 1.17525 0.0809074 0.0404537 0.999181i $$-0.487120\pi$$
0.0404537 + 0.999181i $$0.487120\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 0 0
$$214$$ 0.549834 0.0375859
$$215$$ −9.72508 −0.663245
$$216$$ −1.00000 −0.0680414
$$217$$ 8.00000 0.543075
$$218$$ −0.274917 −0.0186197
$$219$$ −12.8248 −0.866616
$$220$$ 9.72508 0.655665
$$221$$ 0 0
$$222$$ −4.27492 −0.286914
$$223$$ −21.6495 −1.44976 −0.724879 0.688876i $$-0.758104\pi$$
−0.724879 + 0.688876i $$0.758104\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 13.2749 0.884994
$$226$$ −6.00000 −0.399114
$$227$$ 20.5498 1.36394 0.681970 0.731380i $$-0.261123\pi$$
0.681970 + 0.731380i $$0.261123\pi$$
$$228$$ −2.27492 −0.150660
$$229$$ −19.0997 −1.26214 −0.631071 0.775725i $$-0.717384\pi$$
−0.631071 + 0.775725i $$0.717384\pi$$
$$230$$ 9.72508 0.641253
$$231$$ 2.27492 0.149679
$$232$$ 8.27492 0.543275
$$233$$ 5.45017 0.357052 0.178526 0.983935i $$-0.442867\pi$$
0.178526 + 0.983935i $$0.442867\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −8.00000 −0.520756
$$237$$ −12.5498 −0.815199
$$238$$ −0.274917 −0.0178202
$$239$$ −25.0997 −1.62356 −0.811781 0.583962i $$-0.801502\pi$$
−0.811781 + 0.583962i $$0.801502\pi$$
$$240$$ −4.27492 −0.275945
$$241$$ −15.0997 −0.972655 −0.486328 0.873777i $$-0.661664\pi$$
−0.486328 + 0.873777i $$0.661664\pi$$
$$242$$ −5.82475 −0.374429
$$243$$ −1.00000 −0.0641500
$$244$$ −12.2749 −0.785821
$$245$$ 4.27492 0.273114
$$246$$ 6.54983 0.417602
$$247$$ 0 0
$$248$$ −8.00000 −0.508001
$$249$$ −4.54983 −0.288334
$$250$$ 35.3746 2.23729
$$251$$ 23.9244 1.51010 0.755048 0.655669i $$-0.227614\pi$$
0.755048 + 0.655669i $$0.227614\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 5.17525 0.325365
$$254$$ −20.5498 −1.28941
$$255$$ −1.17525 −0.0735969
$$256$$ 1.00000 0.0625000
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 2.27492 0.141630
$$259$$ −4.27492 −0.265630
$$260$$ 0 0
$$261$$ 8.27492 0.512205
$$262$$ 6.82475 0.421635
$$263$$ 28.0000 1.72655 0.863277 0.504730i $$-0.168408\pi$$
0.863277 + 0.504730i $$0.168408\pi$$
$$264$$ −2.27492 −0.140011
$$265$$ 42.7492 2.62606
$$266$$ −2.27492 −0.139484
$$267$$ −14.0000 −0.856786
$$268$$ −12.5498 −0.766603
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ −4.27492 −0.260163
$$271$$ −28.5498 −1.73428 −0.867139 0.498065i $$-0.834044\pi$$
−0.867139 + 0.498065i $$0.834044\pi$$
$$272$$ 0.274917 0.0166693
$$273$$ 0 0
$$274$$ 17.3746 1.04964
$$275$$ 30.1993 1.82109
$$276$$ −2.27492 −0.136934
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 4.00000 0.239904
$$279$$ −8.00000 −0.478947
$$280$$ −4.27492 −0.255475
$$281$$ −15.0997 −0.900771 −0.450385 0.892834i $$-0.648713\pi$$
−0.450385 + 0.892834i $$0.648713\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ −9.72508 −0.576064
$$286$$ 0 0
$$287$$ 6.54983 0.386625
$$288$$ 1.00000 0.0589256
$$289$$ −16.9244 −0.995554
$$290$$ 35.3746 2.07727
$$291$$ 15.0997 0.885158
$$292$$ 12.8248 0.750512
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ −34.1993 −1.99116
$$296$$ 4.27492 0.248475
$$297$$ −2.27492 −0.132004
$$298$$ −22.5498 −1.30628
$$299$$ 0 0
$$300$$ −13.2749 −0.766428
$$301$$ 2.27492 0.131124
$$302$$ −6.27492 −0.361081
$$303$$ −2.54983 −0.146484
$$304$$ 2.27492 0.130475
$$305$$ −52.4743 −3.00467
$$306$$ 0.274917 0.0157160
$$307$$ −21.0997 −1.20422 −0.602111 0.798412i $$-0.705673\pi$$
−0.602111 + 0.798412i $$0.705673\pi$$
$$308$$ −2.27492 −0.129625
$$309$$ −10.2749 −0.584520
$$310$$ −34.1993 −1.94239
$$311$$ 3.45017 0.195641 0.0978205 0.995204i $$-0.468813\pi$$
0.0978205 + 0.995204i $$0.468813\pi$$
$$312$$ 0 0
$$313$$ −27.6495 −1.56284 −0.781421 0.624004i $$-0.785505\pi$$
−0.781421 + 0.624004i $$0.785505\pi$$
$$314$$ −13.3746 −0.754772
$$315$$ −4.27492 −0.240864
$$316$$ 12.5498 0.705983
$$317$$ 1.45017 0.0814494 0.0407247 0.999170i $$-0.487033\pi$$
0.0407247 + 0.999170i $$0.487033\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ 18.8248 1.05398
$$320$$ 4.27492 0.238975
$$321$$ −0.549834 −0.0306888
$$322$$ −2.27492 −0.126776
$$323$$ 0.625414 0.0347990
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −12.5498 −0.695071
$$327$$ 0.274917 0.0152030
$$328$$ −6.54983 −0.361654
$$329$$ 0 0
$$330$$ −9.72508 −0.535348
$$331$$ 29.6495 1.62968 0.814842 0.579683i $$-0.196824\pi$$
0.814842 + 0.579683i $$0.196824\pi$$
$$332$$ 4.54983 0.249705
$$333$$ 4.27492 0.234264
$$334$$ 1.72508 0.0943923
$$335$$ −53.6495 −2.93119
$$336$$ 1.00000 0.0545545
$$337$$ 9.37459 0.510666 0.255333 0.966853i $$-0.417815\pi$$
0.255333 + 0.966853i $$0.417815\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 1.17525 0.0637368
$$341$$ −18.1993 −0.985549
$$342$$ 2.27492 0.123013
$$343$$ −1.00000 −0.0539949
$$344$$ −2.27492 −0.122655
$$345$$ −9.72508 −0.523581
$$346$$ 7.09967 0.381681
$$347$$ 5.09967 0.273765 0.136882 0.990587i $$-0.456292\pi$$
0.136882 + 0.990587i $$0.456292\pi$$
$$348$$ −8.27492 −0.443582
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ −13.2749 −0.709574
$$351$$ 0 0
$$352$$ 2.27492 0.121253
$$353$$ −27.0997 −1.44237 −0.721185 0.692743i $$-0.756402\pi$$
−0.721185 + 0.692743i $$0.756402\pi$$
$$354$$ 8.00000 0.425195
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ 0.274917 0.0145502
$$358$$ 12.0000 0.634220
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 4.27492 0.225308
$$361$$ −13.8248 −0.727619
$$362$$ 19.0997 1.00386
$$363$$ 5.82475 0.305720
$$364$$ 0 0
$$365$$ 54.8248 2.86966
$$366$$ 12.2749 0.641620
$$367$$ −2.90033 −0.151396 −0.0756980 0.997131i $$-0.524119\pi$$
−0.0756980 + 0.997131i $$0.524119\pi$$
$$368$$ 2.27492 0.118588
$$369$$ −6.54983 −0.340971
$$370$$ 18.2749 0.950068
$$371$$ −10.0000 −0.519174
$$372$$ 8.00000 0.414781
$$373$$ 26.5498 1.37470 0.687349 0.726327i $$-0.258774\pi$$
0.687349 + 0.726327i $$0.258774\pi$$
$$374$$ 0.625414 0.0323394
$$375$$ −35.3746 −1.82674
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 1.00000 0.0514344
$$379$$ 33.0997 1.70022 0.850108 0.526609i $$-0.176537\pi$$
0.850108 + 0.526609i $$0.176537\pi$$
$$380$$ 9.72508 0.498886
$$381$$ 20.5498 1.05280
$$382$$ 2.27492 0.116395
$$383$$ −30.2749 −1.54698 −0.773488 0.633811i $$-0.781490\pi$$
−0.773488 + 0.633811i $$0.781490\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −9.72508 −0.495636
$$386$$ 18.5498 0.944162
$$387$$ −2.27492 −0.115641
$$388$$ −15.0997 −0.766570
$$389$$ 28.1993 1.42976 0.714882 0.699246i $$-0.246481\pi$$
0.714882 + 0.699246i $$0.246481\pi$$
$$390$$ 0 0
$$391$$ 0.625414 0.0316285
$$392$$ 1.00000 0.0505076
$$393$$ −6.82475 −0.344263
$$394$$ 2.54983 0.128459
$$395$$ 53.6495 2.69940
$$396$$ 2.27492 0.114319
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 6.82475 0.342094
$$399$$ 2.27492 0.113888
$$400$$ 13.2749 0.663746
$$401$$ 3.09967 0.154790 0.0773950 0.997001i $$-0.475340\pi$$
0.0773950 + 0.997001i $$0.475340\pi$$
$$402$$ 12.5498 0.625929
$$403$$ 0 0
$$404$$ 2.54983 0.126859
$$405$$ 4.27492 0.212422
$$406$$ −8.27492 −0.410677
$$407$$ 9.72508 0.482054
$$408$$ −0.274917 −0.0136104
$$409$$ 7.17525 0.354793 0.177397 0.984139i $$-0.443232\pi$$
0.177397 + 0.984139i $$0.443232\pi$$
$$410$$ −28.0000 −1.38282
$$411$$ −17.3746 −0.857025
$$412$$ 10.2749 0.506209
$$413$$ 8.00000 0.393654
$$414$$ 2.27492 0.111806
$$415$$ 19.4502 0.954771
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 5.17525 0.253130
$$419$$ 2.27492 0.111137 0.0555685 0.998455i $$-0.482303\pi$$
0.0555685 + 0.998455i $$0.482303\pi$$
$$420$$ 4.27492 0.208595
$$421$$ −3.09967 −0.151069 −0.0755343 0.997143i $$-0.524066\pi$$
−0.0755343 + 0.997143i $$0.524066\pi$$
$$422$$ 1.17525 0.0572102
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 3.64950 0.177027
$$426$$ 0 0
$$427$$ 12.2749 0.594025
$$428$$ 0.549834 0.0265773
$$429$$ 0 0
$$430$$ −9.72508 −0.468985
$$431$$ 11.4502 0.551535 0.275768 0.961224i $$-0.411068\pi$$
0.275768 + 0.961224i $$0.411068\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 23.6495 1.13652 0.568261 0.822848i $$-0.307616\pi$$
0.568261 + 0.822848i $$0.307616\pi$$
$$434$$ 8.00000 0.384012
$$435$$ −35.3746 −1.69608
$$436$$ −0.274917 −0.0131661
$$437$$ 5.17525 0.247566
$$438$$ −12.8248 −0.612790
$$439$$ −14.8248 −0.707547 −0.353773 0.935331i $$-0.615102\pi$$
−0.353773 + 0.935331i $$0.615102\pi$$
$$440$$ 9.72508 0.463625
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 7.45017 0.353968 0.176984 0.984214i $$-0.443366\pi$$
0.176984 + 0.984214i $$0.443366\pi$$
$$444$$ −4.27492 −0.202879
$$445$$ 59.8488 2.83711
$$446$$ −21.6495 −1.02513
$$447$$ 22.5498 1.06657
$$448$$ −1.00000 −0.0472456
$$449$$ 0.274917 0.0129741 0.00648707 0.999979i $$-0.497935\pi$$
0.00648707 + 0.999979i $$0.497935\pi$$
$$450$$ 13.2749 0.625786
$$451$$ −14.9003 −0.701629
$$452$$ −6.00000 −0.282216
$$453$$ 6.27492 0.294821
$$454$$ 20.5498 0.964452
$$455$$ 0 0
$$456$$ −2.27492 −0.106533
$$457$$ 18.5498 0.867725 0.433862 0.900979i $$-0.357150\pi$$
0.433862 + 0.900979i $$0.357150\pi$$
$$458$$ −19.0997 −0.892469
$$459$$ −0.274917 −0.0128320
$$460$$ 9.72508 0.453434
$$461$$ 17.9244 0.834823 0.417412 0.908717i $$-0.362937\pi$$
0.417412 + 0.908717i $$0.362937\pi$$
$$462$$ 2.27492 0.105839
$$463$$ −30.2749 −1.40699 −0.703497 0.710698i $$-0.748379\pi$$
−0.703497 + 0.710698i $$0.748379\pi$$
$$464$$ 8.27492 0.384153
$$465$$ 34.1993 1.58596
$$466$$ 5.45017 0.252474
$$467$$ −26.2749 −1.21586 −0.607929 0.793991i $$-0.708000\pi$$
−0.607929 + 0.793991i $$0.708000\pi$$
$$468$$ 0 0
$$469$$ 12.5498 0.579498
$$470$$ 0 0
$$471$$ 13.3746 0.616268
$$472$$ −8.00000 −0.368230
$$473$$ −5.17525 −0.237958
$$474$$ −12.5498 −0.576433
$$475$$ 30.1993 1.38564
$$476$$ −0.274917 −0.0126008
$$477$$ 10.0000 0.457869
$$478$$ −25.0997 −1.14803
$$479$$ 23.3746 1.06801 0.534006 0.845481i $$-0.320686\pi$$
0.534006 + 0.845481i $$0.320686\pi$$
$$480$$ −4.27492 −0.195122
$$481$$ 0 0
$$482$$ −15.0997 −0.687771
$$483$$ 2.27492 0.103512
$$484$$ −5.82475 −0.264761
$$485$$ −64.5498 −2.93106
$$486$$ −1.00000 −0.0453609
$$487$$ 18.1993 0.824691 0.412345 0.911028i $$-0.364710\pi$$
0.412345 + 0.911028i $$0.364710\pi$$
$$488$$ −12.2749 −0.555659
$$489$$ 12.5498 0.567523
$$490$$ 4.27492 0.193121
$$491$$ 8.54983 0.385849 0.192924 0.981214i $$-0.438203\pi$$
0.192924 + 0.981214i $$0.438203\pi$$
$$492$$ 6.54983 0.295289
$$493$$ 2.27492 0.102457
$$494$$ 0 0
$$495$$ 9.72508 0.437110
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ −4.54983 −0.203883
$$499$$ −25.0997 −1.12362 −0.561808 0.827268i $$-0.689894\pi$$
−0.561808 + 0.827268i $$0.689894\pi$$
$$500$$ 35.3746 1.58200
$$501$$ −1.72508 −0.0770710
$$502$$ 23.9244 1.06780
$$503$$ −3.45017 −0.153835 −0.0769176 0.997037i $$-0.524508\pi$$
−0.0769176 + 0.997037i $$0.524508\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 10.9003 0.485058
$$506$$ 5.17525 0.230068
$$507$$ 0 0
$$508$$ −20.5498 −0.911751
$$509$$ 9.92442 0.439892 0.219946 0.975512i $$-0.429412\pi$$
0.219946 + 0.975512i $$0.429412\pi$$
$$510$$ −1.17525 −0.0520409
$$511$$ −12.8248 −0.567334
$$512$$ 1.00000 0.0441942
$$513$$ −2.27492 −0.100440
$$514$$ −14.0000 −0.617514
$$515$$ 43.9244 1.93554
$$516$$ 2.27492 0.100148
$$517$$ 0 0
$$518$$ −4.27492 −0.187829
$$519$$ −7.09967 −0.311641
$$520$$ 0 0
$$521$$ −4.27492 −0.187288 −0.0936438 0.995606i $$-0.529851\pi$$
−0.0936438 + 0.995606i $$0.529851\pi$$
$$522$$ 8.27492 0.362183
$$523$$ 36.0000 1.57417 0.787085 0.616844i $$-0.211589\pi$$
0.787085 + 0.616844i $$0.211589\pi$$
$$524$$ 6.82475 0.298141
$$525$$ 13.2749 0.579365
$$526$$ 28.0000 1.22086
$$527$$ −2.19934 −0.0958047
$$528$$ −2.27492 −0.0990031
$$529$$ −17.8248 −0.774989
$$530$$ 42.7492 1.85691
$$531$$ −8.00000 −0.347170
$$532$$ −2.27492 −0.0986302
$$533$$ 0 0
$$534$$ −14.0000 −0.605839
$$535$$ 2.35050 0.101621
$$536$$ −12.5498 −0.542070
$$537$$ −12.0000 −0.517838
$$538$$ 6.00000 0.258678
$$539$$ 2.27492 0.0979876
$$540$$ −4.27492 −0.183963
$$541$$ −42.4743 −1.82611 −0.913055 0.407835i $$-0.866284\pi$$
−0.913055 + 0.407835i $$0.866284\pi$$
$$542$$ −28.5498 −1.22632
$$543$$ −19.0997 −0.819645
$$544$$ 0.274917 0.0117870
$$545$$ −1.17525 −0.0503421
$$546$$ 0 0
$$547$$ −4.00000 −0.171028 −0.0855138 0.996337i $$-0.527253\pi$$
−0.0855138 + 0.996337i $$0.527253\pi$$
$$548$$ 17.3746 0.742206
$$549$$ −12.2749 −0.523881
$$550$$ 30.1993 1.28770
$$551$$ 18.8248 0.801961
$$552$$ −2.27492 −0.0968269
$$553$$ −12.5498 −0.533673
$$554$$ −10.0000 −0.424859
$$555$$ −18.2749 −0.775727
$$556$$ 4.00000 0.169638
$$557$$ 44.7492 1.89608 0.948042 0.318146i $$-0.103060\pi$$
0.948042 + 0.318146i $$0.103060\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 0 0
$$560$$ −4.27492 −0.180648
$$561$$ −0.625414 −0.0264050
$$562$$ −15.0997 −0.636941
$$563$$ −35.3746 −1.49086 −0.745431 0.666583i $$-0.767756\pi$$
−0.745431 + 0.666583i $$0.767756\pi$$
$$564$$ 0 0
$$565$$ −25.6495 −1.07908
$$566$$ −4.00000 −0.168133
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 3.09967 0.129945 0.0649724 0.997887i $$-0.479304\pi$$
0.0649724 + 0.997887i $$0.479304\pi$$
$$570$$ −9.72508 −0.407339
$$571$$ −37.0997 −1.55257 −0.776286 0.630380i $$-0.782899\pi$$
−0.776286 + 0.630380i $$0.782899\pi$$
$$572$$ 0 0
$$573$$ −2.27492 −0.0950360
$$574$$ 6.54983 0.273385
$$575$$ 30.1993 1.25940
$$576$$ 1.00000 0.0416667
$$577$$ 8.90033 0.370526 0.185263 0.982689i $$-0.440686\pi$$
0.185263 + 0.982689i $$0.440686\pi$$
$$578$$ −16.9244 −0.703963
$$579$$ −18.5498 −0.770905
$$580$$ 35.3746 1.46885
$$581$$ −4.54983 −0.188759
$$582$$ 15.0997 0.625901
$$583$$ 22.7492 0.942174
$$584$$ 12.8248 0.530692
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ −46.7492 −1.92954 −0.964772 0.263086i $$-0.915260\pi$$
−0.964772 + 0.263086i $$0.915260\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ −18.1993 −0.749891
$$590$$ −34.1993 −1.40796
$$591$$ −2.54983 −0.104886
$$592$$ 4.27492 0.175698
$$593$$ 7.09967 0.291548 0.145774 0.989318i $$-0.453433\pi$$
0.145774 + 0.989318i $$0.453433\pi$$
$$594$$ −2.27492 −0.0933410
$$595$$ −1.17525 −0.0481805
$$596$$ −22.5498 −0.923677
$$597$$ −6.82475 −0.279318
$$598$$ 0 0
$$599$$ 21.7251 0.887663 0.443831 0.896110i $$-0.353619\pi$$
0.443831 + 0.896110i $$0.353619\pi$$
$$600$$ −13.2749 −0.541946
$$601$$ 23.6495 0.964683 0.482342 0.875983i $$-0.339786\pi$$
0.482342 + 0.875983i $$0.339786\pi$$
$$602$$ 2.27492 0.0927187
$$603$$ −12.5498 −0.511069
$$604$$ −6.27492 −0.255323
$$605$$ −24.9003 −1.01234
$$606$$ −2.54983 −0.103580
$$607$$ 9.17525 0.372412 0.186206 0.982511i $$-0.440381\pi$$
0.186206 + 0.982511i $$0.440381\pi$$
$$608$$ 2.27492 0.0922601
$$609$$ 8.27492 0.335317
$$610$$ −52.4743 −2.12462
$$611$$ 0 0
$$612$$ 0.274917 0.0111129
$$613$$ −16.2749 −0.657338 −0.328669 0.944445i $$-0.606600\pi$$
−0.328669 + 0.944445i $$0.606600\pi$$
$$614$$ −21.0997 −0.851513
$$615$$ 28.0000 1.12907
$$616$$ −2.27492 −0.0916590
$$617$$ 20.8248 0.838373 0.419186 0.907900i $$-0.362315\pi$$
0.419186 + 0.907900i $$0.362315\pi$$
$$618$$ −10.2749 −0.413318
$$619$$ −29.7251 −1.19475 −0.597376 0.801961i $$-0.703790\pi$$
−0.597376 + 0.801961i $$0.703790\pi$$
$$620$$ −34.1993 −1.37348
$$621$$ −2.27492 −0.0912893
$$622$$ 3.45017 0.138339
$$623$$ −14.0000 −0.560898
$$624$$ 0 0
$$625$$ 84.8488 3.39395
$$626$$ −27.6495 −1.10510
$$627$$ −5.17525 −0.206680
$$628$$ −13.3746 −0.533704
$$629$$ 1.17525 0.0468602
$$630$$ −4.27492 −0.170317
$$631$$ 10.8248 0.430927 0.215463 0.976512i $$-0.430874\pi$$
0.215463 + 0.976512i $$0.430874\pi$$
$$632$$ 12.5498 0.499206
$$633$$ −1.17525 −0.0467119
$$634$$ 1.45017 0.0575934
$$635$$ −87.8488 −3.48617
$$636$$ −10.0000 −0.396526
$$637$$ 0 0
$$638$$ 18.8248 0.745279
$$639$$ 0 0
$$640$$ 4.27492 0.168981
$$641$$ −15.0997 −0.596401 −0.298201 0.954503i $$-0.596386\pi$$
−0.298201 + 0.954503i $$0.596386\pi$$
$$642$$ −0.549834 −0.0217002
$$643$$ 23.9244 0.943487 0.471744 0.881736i $$-0.343625\pi$$
0.471744 + 0.881736i $$0.343625\pi$$
$$644$$ −2.27492 −0.0896443
$$645$$ 9.72508 0.382925
$$646$$ 0.625414 0.0246066
$$647$$ −18.1993 −0.715490 −0.357745 0.933819i $$-0.616454\pi$$
−0.357745 + 0.933819i $$0.616454\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −18.1993 −0.714386
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ −12.5498 −0.491489
$$653$$ −5.37459 −0.210324 −0.105162 0.994455i $$-0.533536\pi$$
−0.105162 + 0.994455i $$0.533536\pi$$
$$654$$ 0.274917 0.0107501
$$655$$ 29.1752 1.13997
$$656$$ −6.54983 −0.255728
$$657$$ 12.8248 0.500341
$$658$$ 0 0
$$659$$ −7.45017 −0.290217 −0.145109 0.989416i $$-0.546353\pi$$
−0.145109 + 0.989416i $$0.546353\pi$$
$$660$$ −9.72508 −0.378548
$$661$$ 40.1993 1.56357 0.781787 0.623546i $$-0.214309\pi$$
0.781787 + 0.623546i $$0.214309\pi$$
$$662$$ 29.6495 1.15236
$$663$$ 0 0
$$664$$ 4.54983 0.176568
$$665$$ −9.72508 −0.377123
$$666$$ 4.27492 0.165650
$$667$$ 18.8248 0.728897
$$668$$ 1.72508 0.0667455
$$669$$ 21.6495 0.837018
$$670$$ −53.6495 −2.07266
$$671$$ −27.9244 −1.07801
$$672$$ 1.00000 0.0385758
$$673$$ 16.2749 0.627352 0.313676 0.949530i $$-0.398439\pi$$
0.313676 + 0.949530i $$0.398439\pi$$
$$674$$ 9.37459 0.361096
$$675$$ −13.2749 −0.510952
$$676$$ 0 0
$$677$$ 16.1993 0.622591 0.311296 0.950313i $$-0.399237\pi$$
0.311296 + 0.950313i $$0.399237\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 15.0997 0.579472
$$680$$ 1.17525 0.0450687
$$681$$ −20.5498 −0.787471
$$682$$ −18.1993 −0.696889
$$683$$ −3.37459 −0.129125 −0.0645625 0.997914i $$-0.520565\pi$$
−0.0645625 + 0.997914i $$0.520565\pi$$
$$684$$ 2.27492 0.0869836
$$685$$ 74.2749 2.83790
$$686$$ −1.00000 −0.0381802
$$687$$ 19.0997 0.728698
$$688$$ −2.27492 −0.0867304
$$689$$ 0 0
$$690$$ −9.72508 −0.370228
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 7.09967 0.269889
$$693$$ −2.27492 −0.0864170
$$694$$ 5.09967 0.193581
$$695$$ 17.0997 0.648627
$$696$$ −8.27492 −0.313660
$$697$$ −1.80066 −0.0682049
$$698$$ −2.00000 −0.0757011
$$699$$ −5.45017 −0.206144
$$700$$ −13.2749 −0.501745
$$701$$ −15.0997 −0.570307 −0.285153 0.958482i $$-0.592045\pi$$
−0.285153 + 0.958482i $$0.592045\pi$$
$$702$$ 0 0
$$703$$ 9.72508 0.366788
$$704$$ 2.27492 0.0857392
$$705$$ 0 0
$$706$$ −27.0997 −1.01991
$$707$$ −2.54983 −0.0958964
$$708$$ 8.00000 0.300658
$$709$$ 23.0997 0.867526 0.433763 0.901027i $$-0.357185\pi$$
0.433763 + 0.901027i $$0.357185\pi$$
$$710$$ 0 0
$$711$$ 12.5498 0.470656
$$712$$ 14.0000 0.524672
$$713$$ −18.1993 −0.681571
$$714$$ 0.274917 0.0102885
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 25.0997 0.937364
$$718$$ 32.0000 1.19423
$$719$$ −29.6495 −1.10574 −0.552870 0.833268i $$-0.686467\pi$$
−0.552870 + 0.833268i $$0.686467\pi$$
$$720$$ 4.27492 0.159317
$$721$$ −10.2749 −0.382658
$$722$$ −13.8248 −0.514504
$$723$$ 15.0997 0.561563
$$724$$ 19.0997 0.709834
$$725$$ 109.849 4.07968
$$726$$ 5.82475 0.216177
$$727$$ 35.3746 1.31197 0.655985 0.754774i $$-0.272253\pi$$
0.655985 + 0.754774i $$0.272253\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 54.8248 2.02916
$$731$$ −0.625414 −0.0231318
$$732$$ 12.2749 0.453694
$$733$$ −14.5498 −0.537410 −0.268705 0.963222i $$-0.586596\pi$$
−0.268705 + 0.963222i $$0.586596\pi$$
$$734$$ −2.90033 −0.107053
$$735$$ −4.27492 −0.157683
$$736$$ 2.27492 0.0838546
$$737$$ −28.5498 −1.05165
$$738$$ −6.54983 −0.241103
$$739$$ −37.6495 −1.38496 −0.692480 0.721437i $$-0.743482\pi$$
−0.692480 + 0.721437i $$0.743482\pi$$
$$740$$ 18.2749 0.671799
$$741$$ 0 0
$$742$$ −10.0000 −0.367112
$$743$$ −11.4502 −0.420066 −0.210033 0.977694i $$-0.567357\pi$$
−0.210033 + 0.977694i $$0.567357\pi$$
$$744$$ 8.00000 0.293294
$$745$$ −96.3987 −3.53177
$$746$$ 26.5498 0.972059
$$747$$ 4.54983 0.166470
$$748$$ 0.625414 0.0228674
$$749$$ −0.549834 −0.0200905
$$750$$ −35.3746 −1.29170
$$751$$ 10.1993 0.372179 0.186090 0.982533i $$-0.440419\pi$$
0.186090 + 0.982533i $$0.440419\pi$$
$$752$$ 0 0
$$753$$ −23.9244 −0.871854
$$754$$ 0 0
$$755$$ −26.8248 −0.976253
$$756$$ 1.00000 0.0363696
$$757$$ −35.0997 −1.27572 −0.637860 0.770153i $$-0.720180\pi$$
−0.637860 + 0.770153i $$0.720180\pi$$
$$758$$ 33.0997 1.20223
$$759$$ −5.17525 −0.187850
$$760$$ 9.72508 0.352766
$$761$$ −38.5498 −1.39743 −0.698715 0.715400i $$-0.746245\pi$$
−0.698715 + 0.715400i $$0.746245\pi$$
$$762$$ 20.5498 0.744442
$$763$$ 0.274917 0.00995267
$$764$$ 2.27492 0.0823036
$$765$$ 1.17525 0.0424912
$$766$$ −30.2749 −1.09388
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −32.8248 −1.18369 −0.591845 0.806051i $$-0.701600\pi$$
−0.591845 + 0.806051i $$0.701600\pi$$
$$770$$ −9.72508 −0.350468
$$771$$ 14.0000 0.504198
$$772$$ 18.5498 0.667623
$$773$$ 9.92442 0.356957 0.178478 0.983944i $$-0.442883\pi$$
0.178478 + 0.983944i $$0.442883\pi$$
$$774$$ −2.27492 −0.0817702
$$775$$ −106.199 −3.81479
$$776$$ −15.0997 −0.542047
$$777$$ 4.27492 0.153362
$$778$$ 28.1993 1.01100
$$779$$ −14.9003 −0.533860
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0.625414 0.0223648
$$783$$ −8.27492 −0.295721
$$784$$ 1.00000 0.0357143
$$785$$ −57.1752 −2.04067
$$786$$ −6.82475 −0.243431
$$787$$ 10.2749 0.366261 0.183131 0.983089i $$-0.441377\pi$$
0.183131 + 0.983089i $$0.441377\pi$$
$$788$$ 2.54983 0.0908341
$$789$$ −28.0000 −0.996826
$$790$$ 53.6495 1.90876
$$791$$ 6.00000 0.213335
$$792$$ 2.27492 0.0808357
$$793$$ 0 0
$$794$$ 14.0000 0.496841
$$795$$ −42.7492 −1.51616
$$796$$ 6.82475 0.241897
$$797$$ −15.6495 −0.554334 −0.277167 0.960822i $$-0.589396\pi$$
−0.277167 + 0.960822i $$0.589396\pi$$
$$798$$ 2.27492 0.0805312
$$799$$ 0 0
$$800$$ 13.2749 0.469339
$$801$$ 14.0000 0.494666
$$802$$ 3.09967 0.109453
$$803$$ 29.1752 1.02957
$$804$$ 12.5498 0.442599
$$805$$ −9.72508 −0.342764
$$806$$ 0 0
$$807$$ −6.00000 −0.211210
$$808$$ 2.54983 0.0897029
$$809$$ −56.1993 −1.97586 −0.987932 0.154890i $$-0.950498\pi$$
−0.987932 + 0.154890i $$0.950498\pi$$
$$810$$ 4.27492 0.150205
$$811$$ 11.3746 0.399416 0.199708 0.979855i $$-0.436001\pi$$
0.199708 + 0.979855i $$0.436001\pi$$
$$812$$ −8.27492 −0.290393
$$813$$ 28.5498 1.00129
$$814$$ 9.72508 0.340864
$$815$$ −53.6495 −1.87926
$$816$$ −0.274917 −0.00962403
$$817$$ −5.17525 −0.181059
$$818$$ 7.17525 0.250877
$$819$$ 0 0
$$820$$ −28.0000 −0.977802
$$821$$ −31.6495 −1.10458 −0.552288 0.833654i $$-0.686245\pi$$
−0.552288 + 0.833654i $$0.686245\pi$$
$$822$$ −17.3746 −0.606008
$$823$$ −25.0997 −0.874919 −0.437460 0.899238i $$-0.644122\pi$$
−0.437460 + 0.899238i $$0.644122\pi$$
$$824$$ 10.2749 0.357944
$$825$$ −30.1993 −1.05141
$$826$$ 8.00000 0.278356
$$827$$ 26.2749 0.913668 0.456834 0.889552i $$-0.348983\pi$$
0.456834 + 0.889552i $$0.348983\pi$$
$$828$$ 2.27492 0.0790588
$$829$$ 11.7251 0.407229 0.203614 0.979051i $$-0.434731\pi$$
0.203614 + 0.979051i $$0.434731\pi$$
$$830$$ 19.4502 0.675125
$$831$$ 10.0000 0.346896
$$832$$ 0 0
$$833$$ 0.274917 0.00952532
$$834$$ −4.00000 −0.138509
$$835$$ 7.37459 0.255208
$$836$$ 5.17525 0.178990
$$837$$ 8.00000 0.276520
$$838$$ 2.27492 0.0785857
$$839$$ −22.9003 −0.790607 −0.395304 0.918551i $$-0.629361\pi$$
−0.395304 + 0.918551i $$0.629361\pi$$
$$840$$ 4.27492 0.147499
$$841$$ 39.4743 1.36118
$$842$$ −3.09967 −0.106822
$$843$$ 15.0997 0.520060
$$844$$ 1.17525 0.0404537
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.82475 0.200141
$$848$$ 10.0000 0.343401
$$849$$ 4.00000 0.137280
$$850$$ 3.64950 0.125177
$$851$$ 9.72508 0.333372
$$852$$ 0 0
$$853$$ 27.6495 0.946701 0.473350 0.880874i $$-0.343044\pi$$
0.473350 + 0.880874i $$0.343044\pi$$
$$854$$ 12.2749 0.420039
$$855$$ 9.72508 0.332591
$$856$$ 0.549834 0.0187930
$$857$$ 19.0997 0.652432 0.326216 0.945295i $$-0.394226\pi$$
0.326216 + 0.945295i $$0.394226\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ −9.72508 −0.331623
$$861$$ −6.54983 −0.223218
$$862$$ 11.4502 0.389994
$$863$$ −29.6495 −1.00928 −0.504640 0.863330i $$-0.668375\pi$$
−0.504640 + 0.863330i $$0.668375\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 30.3505 1.03195
$$866$$ 23.6495 0.803643
$$867$$ 16.9244 0.574783
$$868$$ 8.00000 0.271538
$$869$$ 28.5498 0.968487
$$870$$ −35.3746 −1.19931
$$871$$ 0 0
$$872$$ −0.274917 −0.00930987
$$873$$ −15.0997 −0.511046
$$874$$ 5.17525 0.175055
$$875$$ −35.3746 −1.19588
$$876$$ −12.8248 −0.433308
$$877$$ −51.0997 −1.72551 −0.862757 0.505619i $$-0.831264\pi$$
−0.862757 + 0.505619i $$0.831264\pi$$
$$878$$ −14.8248 −0.500311
$$879$$ −6.00000 −0.202375
$$880$$ 9.72508 0.327832
$$881$$ −31.7251 −1.06885 −0.534423 0.845217i $$-0.679471\pi$$
−0.534423 + 0.845217i $$0.679471\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 31.9244 1.07434 0.537171 0.843473i $$-0.319493\pi$$
0.537171 + 0.843473i $$0.319493\pi$$
$$884$$ 0 0
$$885$$ 34.1993 1.14960
$$886$$ 7.45017 0.250293
$$887$$ −33.0997 −1.11138 −0.555689 0.831390i $$-0.687545\pi$$
−0.555689 + 0.831390i $$0.687545\pi$$
$$888$$ −4.27492 −0.143457
$$889$$ 20.5498 0.689219
$$890$$ 59.8488 2.00614
$$891$$ 2.27492 0.0762126
$$892$$ −21.6495 −0.724879
$$893$$ 0 0
$$894$$ 22.5498 0.754179
$$895$$ 51.2990 1.71474
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 0.274917 0.00917411
$$899$$ −66.1993 −2.20787
$$900$$ 13.2749 0.442497
$$901$$ 2.74917 0.0915882
$$902$$ −14.9003 −0.496127
$$903$$ −2.27492 −0.0757045
$$904$$ −6.00000 −0.199557
$$905$$ 81.6495 2.71412
$$906$$ 6.27492 0.208470
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 20.5498 0.681970
$$909$$ 2.54983 0.0845727
$$910$$ 0 0
$$911$$ −52.4743 −1.73855 −0.869275 0.494329i $$-0.835414\pi$$
−0.869275 + 0.494329i $$0.835414\pi$$
$$912$$ −2.27492 −0.0753300
$$913$$ 10.3505 0.342551
$$914$$ 18.5498 0.613574
$$915$$ 52.4743 1.73475
$$916$$ −19.0997 −0.631071
$$917$$ −6.82475 −0.225373
$$918$$ −0.274917 −0.00907362
$$919$$ −12.5498 −0.413981 −0.206990 0.978343i $$-0.566367\pi$$
−0.206990 + 0.978343i $$0.566367\pi$$
$$920$$ 9.72508 0.320626
$$921$$ 21.0997 0.695258
$$922$$ 17.9244 0.590309
$$923$$ 0 0
$$924$$ 2.27492 0.0748393
$$925$$ 56.7492 1.86590
$$926$$ −30.2749 −0.994896
$$927$$ 10.2749 0.337473
$$928$$ 8.27492 0.271637
$$929$$ −14.5498 −0.477365 −0.238682 0.971098i $$-0.576715\pi$$
−0.238682 + 0.971098i $$0.576715\pi$$
$$930$$ 34.1993 1.12144
$$931$$ 2.27492 0.0745574
$$932$$ 5.45017 0.178526
$$933$$ −3.45017 −0.112953
$$934$$ −26.2749 −0.859742
$$935$$ 2.67359 0.0874358
$$936$$ 0 0
$$937$$ 16.9003 0.552110 0.276055 0.961142i $$-0.410973\pi$$
0.276055 + 0.961142i $$0.410973\pi$$
$$938$$ 12.5498 0.409767
$$939$$ 27.6495 0.902307
$$940$$ 0 0
$$941$$ −51.0997 −1.66580 −0.832901 0.553422i $$-0.813322\pi$$
−0.832901 + 0.553422i $$0.813322\pi$$
$$942$$ 13.3746 0.435768
$$943$$ −14.9003 −0.485222
$$944$$ −8.00000 −0.260378
$$945$$ 4.27492 0.139063
$$946$$ −5.17525 −0.168262
$$947$$ 17.1752 0.558121 0.279060 0.960274i $$-0.409977\pi$$
0.279060 + 0.960274i $$0.409977\pi$$
$$948$$ −12.5498 −0.407600
$$949$$ 0 0
$$950$$ 30.1993 0.979796
$$951$$ −1.45017 −0.0470248
$$952$$ −0.274917 −0.00891012
$$953$$ 31.6495 1.02523 0.512614 0.858619i $$-0.328677\pi$$
0.512614 + 0.858619i $$0.328677\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 9.72508 0.314696
$$956$$ −25.0997 −0.811781
$$957$$ −18.8248 −0.608518
$$958$$ 23.3746 0.755199
$$959$$ −17.3746 −0.561055
$$960$$ −4.27492 −0.137972
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 0.549834 0.0177182
$$964$$ −15.0997 −0.486328
$$965$$ 79.2990 2.55273
$$966$$ 2.27492 0.0731943
$$967$$ −42.8248 −1.37715 −0.688576 0.725165i $$-0.741764\pi$$
−0.688576 + 0.725165i $$0.741764\pi$$
$$968$$ −5.82475 −0.187215
$$969$$ −0.625414 −0.0200912
$$970$$ −64.5498 −2.07257
$$971$$ −29.0997 −0.933853 −0.466926 0.884296i $$-0.654639\pi$$
−0.466926 + 0.884296i $$0.654639\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −4.00000 −0.128234
$$974$$ 18.1993 0.583144
$$975$$ 0 0
$$976$$ −12.2749 −0.392911
$$977$$ 13.9244 0.445482 0.222741 0.974878i $$-0.428500\pi$$
0.222741 + 0.974878i $$0.428500\pi$$
$$978$$ 12.5498 0.401299
$$979$$ 31.8488 1.01789
$$980$$ 4.27492 0.136557
$$981$$ −0.274917 −0.00877743
$$982$$ 8.54983 0.272836
$$983$$ −61.0241 −1.94637 −0.973183 0.230032i $$-0.926117\pi$$
−0.973183 + 0.230032i $$0.926117\pi$$
$$984$$ 6.54983 0.208801
$$985$$ 10.9003 0.347313
$$986$$ 2.27492 0.0724481
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −5.17525 −0.164563
$$990$$ 9.72508 0.309083
$$991$$ −46.7492 −1.48504 −0.742518 0.669826i $$-0.766369\pi$$
−0.742518 + 0.669826i $$0.766369\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ −29.6495 −0.940899
$$994$$ 0 0
$$995$$ 29.1752 0.924918
$$996$$ −4.54983 −0.144167
$$997$$ −12.9003 −0.408558 −0.204279 0.978913i $$-0.565485\pi$$
−0.204279 + 0.978913i $$0.565485\pi$$
$$998$$ −25.0997 −0.794516
$$999$$ −4.27492 −0.135252
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 7098.2.a.bu.1.2 2
13.12 even 2 546.2.a.h.1.1 2
39.38 odd 2 1638.2.a.y.1.2 2
52.51 odd 2 4368.2.a.bh.1.1 2
91.90 odd 2 3822.2.a.bm.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.h.1.1 2 13.12 even 2
1638.2.a.y.1.2 2 39.38 odd 2
3822.2.a.bm.1.2 2 91.90 odd 2
4368.2.a.bh.1.1 2 52.51 odd 2
7098.2.a.bu.1.2 2 1.1 even 1 trivial