# Properties

 Label 16.22.a.f.1.2 Level $16$ Weight $22$ Character 16.1 Self dual yes Analytic conductor $44.716$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,22,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$44.7163750859$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4963x + 96223$$ x^3 - x^2 - 4963*x + 96223 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{21}\cdot 3\cdot 5\cdot 7$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$57.9766$$ of defining polynomial Character $$\chi$$ $$=$$ 16.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+7983.67 q^{3} +3.09730e7 q^{5} -9.17385e7 q^{7} -1.03966e10 q^{9} +O(q^{10})$$ $$q+7983.67 q^{3} +3.09730e7 q^{5} -9.17385e7 q^{7} -1.03966e10 q^{9} -8.72158e10 q^{11} -2.36850e11 q^{13} +2.47278e11 q^{15} +7.42283e12 q^{17} -4.68680e9 q^{19} -7.32410e11 q^{21} -3.33703e14 q^{23} +4.82489e14 q^{25} -1.66515e14 q^{27} +3.23982e15 q^{29} -6.40473e15 q^{31} -6.96303e14 q^{33} -2.84142e15 q^{35} -1.61009e16 q^{37} -1.89093e15 q^{39} -5.77168e16 q^{41} +2.01468e17 q^{43} -3.22014e17 q^{45} -6.62056e17 q^{47} -5.50130e17 q^{49} +5.92615e16 q^{51} +4.62651e17 q^{53} -2.70133e18 q^{55} -3.74179e13 q^{57} -7.39502e18 q^{59} -5.50188e18 q^{61} +9.53770e17 q^{63} -7.33594e18 q^{65} -6.03520e18 q^{67} -2.66418e18 q^{69} +4.43147e19 q^{71} -2.48622e19 q^{73} +3.85204e18 q^{75} +8.00105e18 q^{77} -5.70948e19 q^{79} +1.07423e20 q^{81} +1.31820e20 q^{83} +2.29907e20 q^{85} +2.58657e19 q^{87} +3.97023e20 q^{89} +2.17282e19 q^{91} -5.11333e19 q^{93} -1.45164e17 q^{95} -9.80402e20 q^{97} +9.06749e20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 96764 q^{3} - 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9}+O(q^{10})$$ 3 * q - 96764 * q^3 - 24111774 * q^5 - 295988280 * q^7 + 18844697239 * q^9 $$3 q - 96764 q^{3} - 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9} + 40335108684 q^{11} + 133734425946 q^{13} + 1223136458200 q^{15} + 7797732274422 q^{17} - 35788199781996 q^{19} + 198539224853088 q^{21} - 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots + 94\!\cdots\!60 q^{99}+O(q^{100})$$ 3 * q - 96764 * q^3 - 24111774 * q^5 - 295988280 * q^7 + 18844697239 * q^9 + 40335108684 * q^11 + 133734425946 * q^13 + 1223136458200 * q^15 + 7797732274422 * q^17 - 35788199781996 * q^19 + 198539224853088 * q^21 - 193770761479080 * q^23 + 1127564438439501 * q^25 - 5282002293508952 * q^27 + 5607343422466122 * q^29 - 11246757871503072 * q^31 + 10014149026970384 * q^33 - 5274251425350096 * q^35 - 24272499791100606 * q^37 + 52925308377862264 * q^39 - 298159108991869602 * q^41 + 33333932139754860 * q^43 - 927411477977893478 * q^45 + 120874283547603888 * q^47 - 850403331975639477 * q^49 - 3954388789815661240 * q^51 - 1138443393004854222 * q^53 - 6957552484263571704 * q^55 + 681697547133656944 * q^57 - 9225624498709937412 * q^59 - 6554902294063924182 * q^61 - 21785214559631141976 * q^63 - 20714581819561144452 * q^65 + 15793054074531629124 * q^67 - 72988310309689939168 * q^69 + 41139582493467997704 * q^71 - 19422167949903851970 * q^73 + 75231657393126995900 * q^75 + 68380237365358617888 * q^77 + 131735321299806049488 * q^79 + 500425796062282339147 * q^81 - 64013993832679681068 * q^83 + 390258202763001297252 * q^85 - 101898953973185066568 * q^87 + 429891446897537246766 * q^89 + 297181701588021496176 * q^91 + 640035369009914700160 * q^93 + 1036372406649019824696 * q^95 - 324059336514148638042 * q^97 + 949982905688477352860 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 7983.67 0.0780602 0.0390301 0.999238i $$-0.487573\pi$$
0.0390301 + 0.999238i $$0.487573\pi$$
$$4$$ 0 0
$$5$$ 3.09730e7 1.41840 0.709199 0.705008i $$-0.249057\pi$$
0.709199 + 0.705008i $$0.249057\pi$$
$$6$$ 0 0
$$7$$ −9.17385e7 −0.122750 −0.0613751 0.998115i $$-0.519549\pi$$
−0.0613751 + 0.998115i $$0.519549\pi$$
$$8$$ 0 0
$$9$$ −1.03966e10 −0.993907
$$10$$ 0 0
$$11$$ −8.72158e10 −1.01385 −0.506923 0.861991i $$-0.669217\pi$$
−0.506923 + 0.861991i $$0.669217\pi$$
$$12$$ 0 0
$$13$$ −2.36850e11 −0.476505 −0.238253 0.971203i $$-0.576575\pi$$
−0.238253 + 0.971203i $$0.576575\pi$$
$$14$$ 0 0
$$15$$ 2.47278e11 0.110720
$$16$$ 0 0
$$17$$ 7.42283e12 0.893009 0.446505 0.894781i $$-0.352669\pi$$
0.446505 + 0.894781i $$0.352669\pi$$
$$18$$ 0 0
$$19$$ −4.68680e9 −0.000175373 0 −8.76867e−5 1.00000i $$-0.500028\pi$$
−8.76867e−5 1.00000i $$0.500028\pi$$
$$20$$ 0 0
$$21$$ −7.32410e11 −0.00958190
$$22$$ 0 0
$$23$$ −3.33703e14 −1.67965 −0.839823 0.542860i $$-0.817341\pi$$
−0.839823 + 0.542860i $$0.817341\pi$$
$$24$$ 0 0
$$25$$ 4.82489e14 1.01185
$$26$$ 0 0
$$27$$ −1.66515e14 −0.155645
$$28$$ 0 0
$$29$$ 3.23982e15 1.43002 0.715010 0.699114i $$-0.246422\pi$$
0.715010 + 0.699114i $$0.246422\pi$$
$$30$$ 0 0
$$31$$ −6.40473e15 −1.40347 −0.701735 0.712438i $$-0.747591\pi$$
−0.701735 + 0.712438i $$0.747591\pi$$
$$32$$ 0 0
$$33$$ −6.96303e14 −0.0791410
$$34$$ 0 0
$$35$$ −2.84142e15 −0.174109
$$36$$ 0 0
$$37$$ −1.61009e16 −0.550467 −0.275234 0.961377i $$-0.588755\pi$$
−0.275234 + 0.961377i $$0.588755\pi$$
$$38$$ 0 0
$$39$$ −1.89093e15 −0.0371961
$$40$$ 0 0
$$41$$ −5.77168e16 −0.671540 −0.335770 0.941944i $$-0.608997\pi$$
−0.335770 + 0.941944i $$0.608997\pi$$
$$42$$ 0 0
$$43$$ 2.01468e17 1.42163 0.710817 0.703377i $$-0.248325\pi$$
0.710817 + 0.703377i $$0.248325\pi$$
$$44$$ 0 0
$$45$$ −3.22014e17 −1.40976
$$46$$ 0 0
$$47$$ −6.62056e17 −1.83598 −0.917989 0.396607i $$-0.870188\pi$$
−0.917989 + 0.396607i $$0.870188\pi$$
$$48$$ 0 0
$$49$$ −5.50130e17 −0.984932
$$50$$ 0 0
$$51$$ 5.92615e16 0.0697085
$$52$$ 0 0
$$53$$ 4.62651e17 0.363376 0.181688 0.983356i $$-0.441844\pi$$
0.181688 + 0.983356i $$0.441844\pi$$
$$54$$ 0 0
$$55$$ −2.70133e18 −1.43804
$$56$$ 0 0
$$57$$ −3.74179e13 −1.36897e−5 0
$$58$$ 0 0
$$59$$ −7.39502e18 −1.88362 −0.941809 0.336148i $$-0.890876\pi$$
−0.941809 + 0.336148i $$0.890876\pi$$
$$60$$ 0 0
$$61$$ −5.50188e18 −0.987524 −0.493762 0.869597i $$-0.664379\pi$$
−0.493762 + 0.869597i $$0.664379\pi$$
$$62$$ 0 0
$$63$$ 9.53770e17 0.122002
$$64$$ 0 0
$$65$$ −7.33594e18 −0.675874
$$66$$ 0 0
$$67$$ −6.03520e18 −0.404489 −0.202244 0.979335i $$-0.564824\pi$$
−0.202244 + 0.979335i $$0.564824\pi$$
$$68$$ 0 0
$$69$$ −2.66418e18 −0.131113
$$70$$ 0 0
$$71$$ 4.43147e19 1.61561 0.807803 0.589453i $$-0.200657\pi$$
0.807803 + 0.589453i $$0.200657\pi$$
$$72$$ 0 0
$$73$$ −2.48622e19 −0.677095 −0.338548 0.940949i $$-0.609936\pi$$
−0.338548 + 0.940949i $$0.609936\pi$$
$$74$$ 0 0
$$75$$ 3.85204e18 0.0789855
$$76$$ 0 0
$$77$$ 8.00105e18 0.124450
$$78$$ 0 0
$$79$$ −5.70948e19 −0.678441 −0.339220 0.940707i $$-0.610163\pi$$
−0.339220 + 0.940707i $$0.610163\pi$$
$$80$$ 0 0
$$81$$ 1.07423e20 0.981757
$$82$$ 0 0
$$83$$ 1.31820e20 0.932528 0.466264 0.884646i $$-0.345600\pi$$
0.466264 + 0.884646i $$0.345600\pi$$
$$84$$ 0 0
$$85$$ 2.29907e20 1.26664
$$86$$ 0 0
$$87$$ 2.58657e19 0.111628
$$88$$ 0 0
$$89$$ 3.97023e20 1.34965 0.674823 0.737979i $$-0.264220\pi$$
0.674823 + 0.737979i $$0.264220\pi$$
$$90$$ 0 0
$$91$$ 2.17282e19 0.0584911
$$92$$ 0 0
$$93$$ −5.11333e19 −0.109555
$$94$$ 0 0
$$95$$ −1.45164e17 −0.000248749 0
$$96$$ 0 0
$$97$$ −9.80402e20 −1.34990 −0.674949 0.737864i $$-0.735835\pi$$
−0.674949 + 0.737864i $$0.735835\pi$$
$$98$$ 0 0
$$99$$ 9.06749e20 1.00767
$$100$$ 0 0
$$101$$ 6.46191e19 0.0582085 0.0291042 0.999576i $$-0.490735\pi$$
0.0291042 + 0.999576i $$0.490735\pi$$
$$102$$ 0 0
$$103$$ 7.39481e19 0.0542171 0.0271085 0.999632i $$-0.491370\pi$$
0.0271085 + 0.999632i $$0.491370\pi$$
$$104$$ 0 0
$$105$$ −2.26849e19 −0.0135910
$$106$$ 0 0
$$107$$ −7.69439e20 −0.378133 −0.189066 0.981964i $$-0.560546\pi$$
−0.189066 + 0.981964i $$0.560546\pi$$
$$108$$ 0 0
$$109$$ −3.27500e21 −1.32505 −0.662526 0.749039i $$-0.730516\pi$$
−0.662526 + 0.749039i $$0.730516\pi$$
$$110$$ 0 0
$$111$$ −1.28544e20 −0.0429696
$$112$$ 0 0
$$113$$ −2.53293e20 −0.0701938 −0.0350969 0.999384i $$-0.511174\pi$$
−0.0350969 + 0.999384i $$0.511174\pi$$
$$114$$ 0 0
$$115$$ −1.03358e22 −2.38241
$$116$$ 0 0
$$117$$ 2.46243e21 0.473602
$$118$$ 0 0
$$119$$ −6.80959e20 −0.109617
$$120$$ 0 0
$$121$$ 2.06345e20 0.0278836
$$122$$ 0 0
$$123$$ −4.60792e20 −0.0524206
$$124$$ 0 0
$$125$$ 1.75063e20 0.0168128
$$126$$ 0 0
$$127$$ 3.02098e21 0.245589 0.122795 0.992432i $$-0.460814\pi$$
0.122795 + 0.992432i $$0.460814\pi$$
$$128$$ 0 0
$$129$$ 1.60846e21 0.110973
$$130$$ 0 0
$$131$$ 1.70668e22 1.00185 0.500925 0.865491i $$-0.332993\pi$$
0.500925 + 0.865491i $$0.332993\pi$$
$$132$$ 0 0
$$133$$ 4.29960e17 2.15271e−5 0
$$134$$ 0 0
$$135$$ −5.15748e21 −0.220766
$$136$$ 0 0
$$137$$ −2.34811e22 −0.861296 −0.430648 0.902520i $$-0.641715\pi$$
−0.430648 + 0.902520i $$0.641715\pi$$
$$138$$ 0 0
$$139$$ 4.53132e22 1.42748 0.713738 0.700413i $$-0.247001\pi$$
0.713738 + 0.700413i $$0.247001\pi$$
$$140$$ 0 0
$$141$$ −5.28564e21 −0.143317
$$142$$ 0 0
$$143$$ 2.06570e22 0.483103
$$144$$ 0 0
$$145$$ 1.00347e23 2.02834
$$146$$ 0 0
$$147$$ −4.39206e21 −0.0768840
$$148$$ 0 0
$$149$$ 6.98352e22 1.06076 0.530381 0.847759i $$-0.322049\pi$$
0.530381 + 0.847759i $$0.322049\pi$$
$$150$$ 0 0
$$151$$ 4.73560e22 0.625341 0.312671 0.949862i $$-0.398776\pi$$
0.312671 + 0.949862i $$0.398776\pi$$
$$152$$ 0 0
$$153$$ −7.71723e22 −0.887568
$$154$$ 0 0
$$155$$ −1.98374e23 −1.99068
$$156$$ 0 0
$$157$$ −2.22945e21 −0.0195547 −0.00977733 0.999952i $$-0.503112\pi$$
−0.00977733 + 0.999952i $$0.503112\pi$$
$$158$$ 0 0
$$159$$ 3.69365e21 0.0283652
$$160$$ 0 0
$$161$$ 3.06134e22 0.206177
$$162$$ 0 0
$$163$$ −1.48202e23 −0.876768 −0.438384 0.898788i $$-0.644449\pi$$
−0.438384 + 0.898788i $$0.644449\pi$$
$$164$$ 0 0
$$165$$ −2.15666e22 −0.112253
$$166$$ 0 0
$$167$$ 2.25766e23 1.03547 0.517733 0.855542i $$-0.326776\pi$$
0.517733 + 0.855542i $$0.326776\pi$$
$$168$$ 0 0
$$169$$ −1.90967e23 −0.772943
$$170$$ 0 0
$$171$$ 4.87269e19 0.000174305 0
$$172$$ 0 0
$$173$$ −5.37523e23 −1.70182 −0.850909 0.525313i $$-0.823948\pi$$
−0.850909 + 0.525313i $$0.823948\pi$$
$$174$$ 0 0
$$175$$ −4.42628e22 −0.124205
$$176$$ 0 0
$$177$$ −5.90394e22 −0.147036
$$178$$ 0 0
$$179$$ −1.13222e23 −0.250597 −0.125299 0.992119i $$-0.539989\pi$$
−0.125299 + 0.992119i $$0.539989\pi$$
$$180$$ 0 0
$$181$$ 2.55968e23 0.504151 0.252075 0.967708i $$-0.418887\pi$$
0.252075 + 0.967708i $$0.418887\pi$$
$$182$$ 0 0
$$183$$ −4.39252e22 −0.0770863
$$184$$ 0 0
$$185$$ −4.98692e23 −0.780781
$$186$$ 0 0
$$187$$ −6.47388e23 −0.905374
$$188$$ 0 0
$$189$$ 1.52759e22 0.0191054
$$190$$ 0 0
$$191$$ 7.48387e23 0.838062 0.419031 0.907972i $$-0.362370\pi$$
0.419031 + 0.907972i $$0.362370\pi$$
$$192$$ 0 0
$$193$$ 1.77062e24 1.77735 0.888674 0.458539i $$-0.151627\pi$$
0.888674 + 0.458539i $$0.151627\pi$$
$$194$$ 0 0
$$195$$ −5.85678e22 −0.0527589
$$196$$ 0 0
$$197$$ 2.04387e24 1.65409 0.827044 0.562137i $$-0.190021\pi$$
0.827044 + 0.562137i $$0.190021\pi$$
$$198$$ 0 0
$$199$$ 8.13486e22 0.0592097 0.0296048 0.999562i $$-0.490575\pi$$
0.0296048 + 0.999562i $$0.490575\pi$$
$$200$$ 0 0
$$201$$ −4.81831e22 −0.0315745
$$202$$ 0 0
$$203$$ −2.97216e23 −0.175535
$$204$$ 0 0
$$205$$ −1.78766e24 −0.952511
$$206$$ 0 0
$$207$$ 3.46938e24 1.66941
$$208$$ 0 0
$$209$$ 4.08763e20 0.000177802 0
$$210$$ 0 0
$$211$$ −3.52722e24 −1.38825 −0.694123 0.719856i $$-0.744208\pi$$
−0.694123 + 0.719856i $$0.744208\pi$$
$$212$$ 0 0
$$213$$ 3.53794e23 0.126114
$$214$$ 0 0
$$215$$ 6.24008e24 2.01644
$$216$$ 0 0
$$217$$ 5.87560e23 0.172276
$$218$$ 0 0
$$219$$ −1.98492e23 −0.0528542
$$220$$ 0 0
$$221$$ −1.75809e24 −0.425523
$$222$$ 0 0
$$223$$ 2.37573e24 0.523114 0.261557 0.965188i $$-0.415764\pi$$
0.261557 + 0.965188i $$0.415764\pi$$
$$224$$ 0 0
$$225$$ −5.01625e24 −1.00569
$$226$$ 0 0
$$227$$ −6.71337e24 −1.22650 −0.613252 0.789887i $$-0.710139\pi$$
−0.613252 + 0.789887i $$0.710139\pi$$
$$228$$ 0 0
$$229$$ 3.10829e24 0.517903 0.258952 0.965890i $$-0.416623\pi$$
0.258952 + 0.965890i $$0.416623\pi$$
$$230$$ 0 0
$$231$$ 6.38777e22 0.00971457
$$232$$ 0 0
$$233$$ −3.90742e24 −0.542817 −0.271408 0.962464i $$-0.587489\pi$$
−0.271408 + 0.962464i $$0.587489\pi$$
$$234$$ 0 0
$$235$$ −2.05059e25 −2.60415
$$236$$ 0 0
$$237$$ −4.55826e23 −0.0529592
$$238$$ 0 0
$$239$$ −2.38784e23 −0.0253997 −0.0126998 0.999919i $$-0.504043\pi$$
−0.0126998 + 0.999919i $$0.504043\pi$$
$$240$$ 0 0
$$241$$ 1.14005e25 1.11107 0.555537 0.831492i $$-0.312513\pi$$
0.555537 + 0.831492i $$0.312513\pi$$
$$242$$ 0 0
$$243$$ 2.59944e24 0.232281
$$244$$ 0 0
$$245$$ −1.70392e25 −1.39703
$$246$$ 0 0
$$247$$ 1.11007e21 8.35663e−5 0
$$248$$ 0 0
$$249$$ 1.05241e24 0.0727934
$$250$$ 0 0
$$251$$ 5.80122e24 0.368931 0.184466 0.982839i $$-0.440945\pi$$
0.184466 + 0.982839i $$0.440945\pi$$
$$252$$ 0 0
$$253$$ 2.91042e25 1.70290
$$254$$ 0 0
$$255$$ 1.83551e24 0.0988744
$$256$$ 0 0
$$257$$ −2.21415e25 −1.09878 −0.549388 0.835567i $$-0.685139\pi$$
−0.549388 + 0.835567i $$0.685139\pi$$
$$258$$ 0 0
$$259$$ 1.47707e24 0.0675699
$$260$$ 0 0
$$261$$ −3.36832e25 −1.42131
$$262$$ 0 0
$$263$$ 3.74763e25 1.45956 0.729779 0.683683i $$-0.239623\pi$$
0.729779 + 0.683683i $$0.239623\pi$$
$$264$$ 0 0
$$265$$ 1.43297e25 0.515412
$$266$$ 0 0
$$267$$ 3.16970e24 0.105354
$$268$$ 0 0
$$269$$ 1.64824e25 0.506549 0.253274 0.967394i $$-0.418492\pi$$
0.253274 + 0.967394i $$0.418492\pi$$
$$270$$ 0 0
$$271$$ 2.86168e25 0.813660 0.406830 0.913504i $$-0.366634\pi$$
0.406830 + 0.913504i $$0.366634\pi$$
$$272$$ 0 0
$$273$$ 1.73471e23 0.00456583
$$274$$ 0 0
$$275$$ −4.20807e25 −1.02586
$$276$$ 0 0
$$277$$ 5.26085e25 1.18855 0.594277 0.804261i $$-0.297438\pi$$
0.594277 + 0.804261i $$0.297438\pi$$
$$278$$ 0 0
$$279$$ 6.65875e25 1.39492
$$280$$ 0 0
$$281$$ 6.45378e25 1.25429 0.627145 0.778903i $$-0.284223\pi$$
0.627145 + 0.778903i $$0.284223\pi$$
$$282$$ 0 0
$$283$$ −4.56042e25 −0.822711 −0.411355 0.911475i $$-0.634944\pi$$
−0.411355 + 0.911475i $$0.634944\pi$$
$$284$$ 0 0
$$285$$ −1.15894e21 −1.94174e−5 0
$$286$$ 0 0
$$287$$ 5.29485e24 0.0824317
$$288$$ 0 0
$$289$$ −1.39935e25 −0.202535
$$290$$ 0 0
$$291$$ −7.82721e24 −0.105373
$$292$$ 0 0
$$293$$ −5.87863e25 −0.736489 −0.368244 0.929729i $$-0.620041\pi$$
−0.368244 + 0.929729i $$0.620041\pi$$
$$294$$ 0 0
$$295$$ −2.29046e26 −2.67172
$$296$$ 0 0
$$297$$ 1.45228e25 0.157800
$$298$$ 0 0
$$299$$ 7.90374e25 0.800360
$$300$$ 0 0
$$301$$ −1.84824e25 −0.174506
$$302$$ 0 0
$$303$$ 5.15898e23 0.00454377
$$304$$ 0 0
$$305$$ −1.70410e26 −1.40070
$$306$$ 0 0
$$307$$ 1.65443e26 1.26969 0.634843 0.772641i $$-0.281065\pi$$
0.634843 + 0.772641i $$0.281065\pi$$
$$308$$ 0 0
$$309$$ 5.90377e23 0.00423220
$$310$$ 0 0
$$311$$ 7.32700e24 0.0490842 0.0245421 0.999699i $$-0.492187\pi$$
0.0245421 + 0.999699i $$0.492187\pi$$
$$312$$ 0 0
$$313$$ −7.64901e25 −0.479059 −0.239530 0.970889i $$-0.576993\pi$$
−0.239530 + 0.970889i $$0.576993\pi$$
$$314$$ 0 0
$$315$$ 2.95411e25 0.173048
$$316$$ 0 0
$$317$$ −9.82997e25 −0.538803 −0.269402 0.963028i $$-0.586826\pi$$
−0.269402 + 0.963028i $$0.586826\pi$$
$$318$$ 0 0
$$319$$ −2.82564e26 −1.44982
$$320$$ 0 0
$$321$$ −6.14295e24 −0.0295171
$$322$$ 0 0
$$323$$ −3.47893e22 −0.000156610 0
$$324$$ 0 0
$$325$$ −1.14277e26 −0.482153
$$326$$ 0 0
$$327$$ −2.61465e25 −0.103434
$$328$$ 0 0
$$329$$ 6.07360e25 0.225367
$$330$$ 0 0
$$331$$ 2.46189e26 0.857187 0.428593 0.903498i $$-0.359009\pi$$
0.428593 + 0.903498i $$0.359009\pi$$
$$332$$ 0 0
$$333$$ 1.67395e26 0.547113
$$334$$ 0 0
$$335$$ −1.86928e26 −0.573726
$$336$$ 0 0
$$337$$ 4.89205e26 1.41051 0.705257 0.708952i $$-0.250832\pi$$
0.705257 + 0.708952i $$0.250832\pi$$
$$338$$ 0 0
$$339$$ −2.02221e24 −0.00547934
$$340$$ 0 0
$$341$$ 5.58594e26 1.42290
$$342$$ 0 0
$$343$$ 1.01708e26 0.243651
$$344$$ 0 0
$$345$$ −8.25175e25 −0.185971
$$346$$ 0 0
$$347$$ −1.00972e26 −0.214161 −0.107080 0.994250i $$-0.534150\pi$$
−0.107080 + 0.994250i $$0.534150\pi$$
$$348$$ 0 0
$$349$$ −4.95706e25 −0.0989822 −0.0494911 0.998775i $$-0.515760\pi$$
−0.0494911 + 0.998775i $$0.515760\pi$$
$$350$$ 0 0
$$351$$ 3.94391e25 0.0741655
$$352$$ 0 0
$$353$$ 1.88459e26 0.333873 0.166937 0.985968i $$-0.446613\pi$$
0.166937 + 0.985968i $$0.446613\pi$$
$$354$$ 0 0
$$355$$ 1.37256e27 2.29157
$$356$$ 0 0
$$357$$ −5.43656e24 −0.00855673
$$358$$ 0 0
$$359$$ −5.21557e26 −0.774123 −0.387062 0.922054i $$-0.626510\pi$$
−0.387062 + 0.922054i $$0.626510\pi$$
$$360$$ 0 0
$$361$$ −7.14209e26 −1.00000
$$362$$ 0 0
$$363$$ 1.64740e24 0.00217660
$$364$$ 0 0
$$365$$ −7.70057e26 −0.960390
$$366$$ 0 0
$$367$$ 4.38478e25 0.0516362 0.0258181 0.999667i $$-0.491781\pi$$
0.0258181 + 0.999667i $$0.491781\pi$$
$$368$$ 0 0
$$369$$ 6.00060e26 0.667448
$$370$$ 0 0
$$371$$ −4.24429e25 −0.0446045
$$372$$ 0 0
$$373$$ 1.92501e25 0.0191201 0.00956004 0.999954i $$-0.496957\pi$$
0.00956004 + 0.999954i $$0.496957\pi$$
$$374$$ 0 0
$$375$$ 1.39765e24 0.00131241
$$376$$ 0 0
$$377$$ −7.67351e26 −0.681412
$$378$$ 0 0
$$379$$ 2.04774e27 1.72014 0.860070 0.510175i $$-0.170419\pi$$
0.860070 + 0.510175i $$0.170419\pi$$
$$380$$ 0 0
$$381$$ 2.41186e25 0.0191707
$$382$$ 0 0
$$383$$ −1.75914e27 −1.32347 −0.661733 0.749740i $$-0.730179\pi$$
−0.661733 + 0.749740i $$0.730179\pi$$
$$384$$ 0 0
$$385$$ 2.47816e26 0.176519
$$386$$ 0 0
$$387$$ −2.09459e27 −1.41297
$$388$$ 0 0
$$389$$ −2.73236e26 −0.174609 −0.0873046 0.996182i $$-0.527825\pi$$
−0.0873046 + 0.996182i $$0.527825\pi$$
$$390$$ 0 0
$$391$$ −2.47702e27 −1.49994
$$392$$ 0 0
$$393$$ 1.36256e26 0.0782046
$$394$$ 0 0
$$395$$ −1.76840e27 −0.962299
$$396$$ 0 0
$$397$$ −1.21691e27 −0.627998 −0.313999 0.949423i $$-0.601669\pi$$
−0.313999 + 0.949423i $$0.601669\pi$$
$$398$$ 0 0
$$399$$ 3.43266e21 1.68041e−6 0
$$400$$ 0 0
$$401$$ −1.15061e27 −0.534455 −0.267228 0.963633i $$-0.586108\pi$$
−0.267228 + 0.963633i $$0.586108\pi$$
$$402$$ 0 0
$$403$$ 1.51696e27 0.668761
$$404$$ 0 0
$$405$$ 3.32721e27 1.39252
$$406$$ 0 0
$$407$$ 1.40425e27 0.558089
$$408$$ 0 0
$$409$$ −1.54628e26 −0.0583705 −0.0291852 0.999574i $$-0.509291\pi$$
−0.0291852 + 0.999574i $$0.509291\pi$$
$$410$$ 0 0
$$411$$ −1.87465e26 −0.0672330
$$412$$ 0 0
$$413$$ 6.78408e26 0.231215
$$414$$ 0 0
$$415$$ 4.08287e27 1.32270
$$416$$ 0 0
$$417$$ 3.61766e26 0.111429
$$418$$ 0 0
$$419$$ −4.10695e27 −1.20302 −0.601510 0.798866i $$-0.705434\pi$$
−0.601510 + 0.798866i $$0.705434\pi$$
$$420$$ 0 0
$$421$$ −4.80891e27 −1.33994 −0.669969 0.742389i $$-0.733693\pi$$
−0.669969 + 0.742389i $$0.733693\pi$$
$$422$$ 0 0
$$423$$ 6.88314e27 1.82479
$$424$$ 0 0
$$425$$ 3.58144e27 0.903594
$$426$$ 0 0
$$427$$ 5.04734e26 0.121219
$$428$$ 0 0
$$429$$ 1.64919e26 0.0377111
$$430$$ 0 0
$$431$$ 4.51446e27 0.983092 0.491546 0.870852i $$-0.336432\pi$$
0.491546 + 0.870852i $$0.336432\pi$$
$$432$$ 0 0
$$433$$ −4.04253e27 −0.838552 −0.419276 0.907859i $$-0.637716\pi$$
−0.419276 + 0.907859i $$0.637716\pi$$
$$434$$ 0 0
$$435$$ 8.01138e26 0.158332
$$436$$ 0 0
$$437$$ 1.56400e24 0.000294565 0
$$438$$ 0 0
$$439$$ −3.73290e26 −0.0670145 −0.0335072 0.999438i $$-0.510668\pi$$
−0.0335072 + 0.999438i $$0.510668\pi$$
$$440$$ 0 0
$$441$$ 5.71949e27 0.978931
$$442$$ 0 0
$$443$$ −8.67248e27 −1.41548 −0.707741 0.706472i $$-0.750286\pi$$
−0.707741 + 0.706472i $$0.750286\pi$$
$$444$$ 0 0
$$445$$ 1.22970e28 1.91434
$$446$$ 0 0
$$447$$ 5.57541e26 0.0828034
$$448$$ 0 0
$$449$$ −4.22534e27 −0.598790 −0.299395 0.954129i $$-0.596785\pi$$
−0.299395 + 0.954129i $$0.596785\pi$$
$$450$$ 0 0
$$451$$ 5.03382e27 0.680838
$$452$$ 0 0
$$453$$ 3.78075e26 0.0488143
$$454$$ 0 0
$$455$$ 6.72988e26 0.0829637
$$456$$ 0 0
$$457$$ 8.06069e27 0.948969 0.474484 0.880264i $$-0.342635\pi$$
0.474484 + 0.880264i $$0.342635\pi$$
$$458$$ 0 0
$$459$$ −1.23601e27 −0.138992
$$460$$ 0 0
$$461$$ −9.88705e27 −1.06220 −0.531101 0.847309i $$-0.678221\pi$$
−0.531101 + 0.847309i $$0.678221\pi$$
$$462$$ 0 0
$$463$$ −1.00097e28 −1.02759 −0.513794 0.857913i $$-0.671761\pi$$
−0.513794 + 0.857913i $$0.671761\pi$$
$$464$$ 0 0
$$465$$ −1.58375e27 −0.155393
$$466$$ 0 0
$$467$$ −1.60212e28 −1.50268 −0.751342 0.659913i $$-0.770593\pi$$
−0.751342 + 0.659913i $$0.770593\pi$$
$$468$$ 0 0
$$469$$ 5.53660e26 0.0496511
$$470$$ 0 0
$$471$$ −1.77992e25 −0.00152644
$$472$$ 0 0
$$473$$ −1.75712e28 −1.44132
$$474$$ 0 0
$$475$$ −2.26133e24 −0.000177452 0
$$476$$ 0 0
$$477$$ −4.81000e27 −0.361162
$$478$$ 0 0
$$479$$ 9.14679e27 0.657273 0.328636 0.944457i $$-0.393411\pi$$
0.328636 + 0.944457i $$0.393411\pi$$
$$480$$ 0 0
$$481$$ 3.81349e27 0.262300
$$482$$ 0 0
$$483$$ 2.44407e26 0.0160942
$$484$$ 0 0
$$485$$ −3.03660e28 −1.91469
$$486$$ 0 0
$$487$$ 1.63338e28 0.986357 0.493179 0.869928i $$-0.335835\pi$$
0.493179 + 0.869928i $$0.335835\pi$$
$$488$$ 0 0
$$489$$ −1.18320e27 −0.0684407
$$490$$ 0 0
$$491$$ −1.10020e28 −0.609702 −0.304851 0.952400i $$-0.598607\pi$$
−0.304851 + 0.952400i $$0.598607\pi$$
$$492$$ 0 0
$$493$$ 2.40487e28 1.27702
$$494$$ 0 0
$$495$$ 2.80847e28 1.42927
$$496$$ 0 0
$$497$$ −4.06537e27 −0.198316
$$498$$ 0 0
$$499$$ −2.04342e28 −0.955657 −0.477829 0.878453i $$-0.658576\pi$$
−0.477829 + 0.878453i $$0.658576\pi$$
$$500$$ 0 0
$$501$$ 1.80244e27 0.0808288
$$502$$ 0 0
$$503$$ −1.67038e28 −0.718375 −0.359188 0.933265i $$-0.616946\pi$$
−0.359188 + 0.933265i $$0.616946\pi$$
$$504$$ 0 0
$$505$$ 2.00145e27 0.0825628
$$506$$ 0 0
$$507$$ −1.52462e27 −0.0603361
$$508$$ 0 0
$$509$$ −5.42146e27 −0.205864 −0.102932 0.994688i $$-0.532822\pi$$
−0.102932 + 0.994688i $$0.532822\pi$$
$$510$$ 0 0
$$511$$ 2.28082e27 0.0831136
$$512$$ 0 0
$$513$$ 7.80424e23 2.72960e−5 0
$$514$$ 0 0
$$515$$ 2.29039e27 0.0769014
$$516$$ 0 0
$$517$$ 5.77418e28 1.86140
$$518$$ 0 0
$$519$$ −4.29141e27 −0.132844
$$520$$ 0 0
$$521$$ 1.37414e28 0.408541 0.204271 0.978914i $$-0.434518\pi$$
0.204271 + 0.978914i $$0.434518\pi$$
$$522$$ 0 0
$$523$$ 3.33678e28 0.952927 0.476463 0.879194i $$-0.341919\pi$$
0.476463 + 0.879194i $$0.341919\pi$$
$$524$$ 0 0
$$525$$ −3.53380e26 −0.00969548
$$526$$ 0 0
$$527$$ −4.75412e28 −1.25331
$$528$$ 0 0
$$529$$ 7.18860e28 1.82121
$$530$$ 0 0
$$531$$ 7.68831e28 1.87214
$$532$$ 0 0
$$533$$ 1.36702e28 0.319992
$$534$$ 0 0
$$535$$ −2.38318e28 −0.536343
$$536$$ 0 0
$$537$$ −9.03932e26 −0.0195617
$$538$$ 0 0
$$539$$ 4.79800e28 0.998570
$$540$$ 0 0
$$541$$ −9.34113e28 −1.86994 −0.934971 0.354723i $$-0.884575\pi$$
−0.934971 + 0.354723i $$0.884575\pi$$
$$542$$ 0 0
$$543$$ 2.04356e27 0.0393541
$$544$$ 0 0
$$545$$ −1.01437e29 −1.87945
$$546$$ 0 0
$$547$$ −1.90655e28 −0.339923 −0.169961 0.985451i $$-0.554364\pi$$
−0.169961 + 0.985451i $$0.554364\pi$$
$$548$$ 0 0
$$549$$ 5.72009e28 0.981507
$$550$$ 0 0
$$551$$ −1.51844e25 −0.000250788 0
$$552$$ 0 0
$$553$$ 5.23779e27 0.0832787
$$554$$ 0 0
$$555$$ −3.98140e27 −0.0609480
$$556$$ 0 0
$$557$$ −4.85895e28 −0.716247 −0.358124 0.933674i $$-0.616583\pi$$
−0.358124 + 0.933674i $$0.616583\pi$$
$$558$$ 0 0
$$559$$ −4.77177e28 −0.677416
$$560$$ 0 0
$$561$$ −5.16854e27 −0.0706736
$$562$$ 0 0
$$563$$ 4.82251e28 0.635236 0.317618 0.948219i $$-0.397117\pi$$
0.317618 + 0.948219i $$0.397117\pi$$
$$564$$ 0 0
$$565$$ −7.84523e27 −0.0995628
$$566$$ 0 0
$$567$$ −9.85481e27 −0.120511
$$568$$ 0 0
$$569$$ −8.25601e28 −0.972952 −0.486476 0.873694i $$-0.661718\pi$$
−0.486476 + 0.873694i $$0.661718\pi$$
$$570$$ 0 0
$$571$$ 3.00348e28 0.341150 0.170575 0.985345i $$-0.445437\pi$$
0.170575 + 0.985345i $$0.445437\pi$$
$$572$$ 0 0
$$573$$ 5.97488e27 0.0654193
$$574$$ 0 0
$$575$$ −1.61008e29 −1.69956
$$576$$ 0 0
$$577$$ 7.52665e28 0.766047 0.383024 0.923739i $$-0.374883\pi$$
0.383024 + 0.923739i $$0.374883\pi$$
$$578$$ 0 0
$$579$$ 1.41360e28 0.138740
$$580$$ 0 0
$$581$$ −1.20930e28 −0.114468
$$582$$ 0 0
$$583$$ −4.03504e28 −0.368408
$$584$$ 0 0
$$585$$ 7.62690e28 0.671756
$$586$$ 0 0
$$587$$ −1.52918e29 −1.29945 −0.649723 0.760171i $$-0.725115\pi$$
−0.649723 + 0.760171i $$0.725115\pi$$
$$588$$ 0 0
$$589$$ 3.00177e25 0.000246131 0
$$590$$ 0 0
$$591$$ 1.63176e28 0.129118
$$592$$ 0 0
$$593$$ 3.89818e28 0.297706 0.148853 0.988859i $$-0.452442\pi$$
0.148853 + 0.988859i $$0.452442\pi$$
$$594$$ 0 0
$$595$$ −2.10913e28 −0.155481
$$596$$ 0 0
$$597$$ 6.49461e26 0.00462192
$$598$$ 0 0
$$599$$ −1.28684e29 −0.884186 −0.442093 0.896969i $$-0.645764\pi$$
−0.442093 + 0.896969i $$0.645764\pi$$
$$600$$ 0 0
$$601$$ −1.81454e29 −1.20388 −0.601942 0.798540i $$-0.705606\pi$$
−0.601942 + 0.798540i $$0.705606\pi$$
$$602$$ 0 0
$$603$$ 6.27457e28 0.402024
$$604$$ 0 0
$$605$$ 6.39114e27 0.0395500
$$606$$ 0 0
$$607$$ 2.81972e29 1.68549 0.842743 0.538316i $$-0.180939\pi$$
0.842743 + 0.538316i $$0.180939\pi$$
$$608$$ 0 0
$$609$$ −2.37288e27 −0.0137023
$$610$$ 0 0
$$611$$ 1.56808e29 0.874853
$$612$$ 0 0
$$613$$ 8.09843e28 0.436582 0.218291 0.975884i $$-0.429952\pi$$
0.218291 + 0.975884i $$0.429952\pi$$
$$614$$ 0 0
$$615$$ −1.42721e28 −0.0743532
$$616$$ 0 0
$$617$$ 3.47454e29 1.74946 0.874728 0.484614i $$-0.161040\pi$$
0.874728 + 0.484614i $$0.161040\pi$$
$$618$$ 0 0
$$619$$ 2.90076e29 1.41176 0.705878 0.708333i $$-0.250553\pi$$
0.705878 + 0.708333i $$0.250553\pi$$
$$620$$ 0 0
$$621$$ 5.55666e28 0.261428
$$622$$ 0 0
$$623$$ −3.64223e28 −0.165669
$$624$$ 0 0
$$625$$ −2.24647e29 −0.988006
$$626$$ 0 0
$$627$$ 3.26343e24 1.38792e−5 0
$$628$$ 0 0
$$629$$ −1.19514e29 −0.491572
$$630$$ 0 0
$$631$$ 4.23594e29 1.68516 0.842581 0.538570i $$-0.181035\pi$$
0.842581 + 0.538570i $$0.181035\pi$$
$$632$$ 0 0
$$633$$ −2.81602e28 −0.108367
$$634$$ 0 0
$$635$$ 9.35689e28 0.348343
$$636$$ 0 0
$$637$$ 1.30298e29 0.469325
$$638$$ 0 0
$$639$$ −4.60723e29 −1.60576
$$640$$ 0 0
$$641$$ −3.43690e29 −1.15920 −0.579598 0.814902i $$-0.696791\pi$$
−0.579598 + 0.814902i $$0.696791\pi$$
$$642$$ 0 0
$$643$$ −1.89646e29 −0.619055 −0.309527 0.950891i $$-0.600171\pi$$
−0.309527 + 0.950891i $$0.600171\pi$$
$$644$$ 0 0
$$645$$ 4.98187e28 0.157404
$$646$$ 0 0
$$647$$ −3.06810e29 −0.938370 −0.469185 0.883100i $$-0.655452\pi$$
−0.469185 + 0.883100i $$0.655452\pi$$
$$648$$ 0 0
$$649$$ 6.44962e29 1.90970
$$650$$ 0 0
$$651$$ 4.69089e27 0.0134479
$$652$$ 0 0
$$653$$ 1.17693e29 0.326710 0.163355 0.986567i $$-0.447768\pi$$
0.163355 + 0.986567i $$0.447768\pi$$
$$654$$ 0 0
$$655$$ 5.28609e29 1.42102
$$656$$ 0 0
$$657$$ 2.58483e29 0.672969
$$658$$ 0 0
$$659$$ −3.42901e29 −0.864712 −0.432356 0.901703i $$-0.642318\pi$$
−0.432356 + 0.901703i $$0.642318\pi$$
$$660$$ 0 0
$$661$$ −3.95818e29 −0.966897 −0.483448 0.875373i $$-0.660616\pi$$
−0.483448 + 0.875373i $$0.660616\pi$$
$$662$$ 0 0
$$663$$ −1.40361e28 −0.0332164
$$664$$ 0 0
$$665$$ 1.33172e25 3.05340e−5 0
$$666$$ 0 0
$$667$$ −1.08114e30 −2.40193
$$668$$ 0 0
$$669$$ 1.89671e28 0.0408343
$$670$$ 0 0
$$671$$ 4.79851e29 1.00120
$$672$$ 0 0
$$673$$ −9.58339e29 −1.93803 −0.969017 0.246996i $$-0.920557\pi$$
−0.969017 + 0.246996i $$0.920557\pi$$
$$674$$ 0 0
$$675$$ −8.03418e28 −0.157490
$$676$$ 0 0
$$677$$ 5.12797e29 0.974461 0.487230 0.873273i $$-0.338007\pi$$
0.487230 + 0.873273i $$0.338007\pi$$
$$678$$ 0 0
$$679$$ 8.99406e28 0.165700
$$680$$ 0 0
$$681$$ −5.35973e28 −0.0957411
$$682$$ 0 0
$$683$$ −5.75298e29 −0.996495 −0.498248 0.867035i $$-0.666023\pi$$
−0.498248 + 0.867035i $$0.666023\pi$$
$$684$$ 0 0
$$685$$ −7.27280e29 −1.22166
$$686$$ 0 0
$$687$$ 2.48156e28 0.0404276
$$688$$ 0 0
$$689$$ −1.09579e29 −0.173151
$$690$$ 0 0
$$691$$ −2.46471e29 −0.377787 −0.188893 0.981998i $$-0.560490\pi$$
−0.188893 + 0.981998i $$0.560490\pi$$
$$692$$ 0 0
$$693$$ −8.31838e28 −0.123691
$$694$$ 0 0
$$695$$ 1.40348e30 2.02473
$$696$$ 0 0
$$697$$ −4.28422e29 −0.599691
$$698$$ 0 0
$$699$$ −3.11956e28 −0.0423724
$$700$$ 0 0
$$701$$ −8.94607e29 −1.17922 −0.589608 0.807690i $$-0.700718\pi$$
−0.589608 + 0.807690i $$0.700718\pi$$
$$702$$ 0 0
$$703$$ 7.54616e25 9.65373e−5 0
$$704$$ 0 0
$$705$$ −1.63712e29 −0.203280
$$706$$ 0 0
$$707$$ −5.92806e27 −0.00714510
$$708$$ 0 0
$$709$$ 7.52975e29 0.881039 0.440520 0.897743i $$-0.354794\pi$$
0.440520 + 0.897743i $$0.354794\pi$$
$$710$$ 0 0
$$711$$ 5.93592e29 0.674307
$$712$$ 0 0
$$713$$ 2.13728e30 2.35733
$$714$$ 0 0
$$715$$ 6.39810e29 0.685232
$$716$$ 0 0
$$717$$ −1.90638e27 −0.00198270
$$718$$ 0 0
$$719$$ −4.85535e29 −0.490419 −0.245210 0.969470i $$-0.578857\pi$$
−0.245210 + 0.969470i $$0.578857\pi$$
$$720$$ 0 0
$$721$$ −6.78389e27 −0.00665516
$$722$$ 0 0
$$723$$ 9.10175e28 0.0867307
$$724$$ 0 0
$$725$$ 1.56318e30 1.44697
$$726$$ 0 0
$$727$$ −1.86351e30 −1.67579 −0.837895 0.545831i $$-0.816214\pi$$
−0.837895 + 0.545831i $$0.816214\pi$$
$$728$$ 0 0
$$729$$ −1.10293e30 −0.963625
$$730$$ 0 0
$$731$$ 1.49546e30 1.26953
$$732$$ 0 0
$$733$$ −3.73906e29 −0.308440 −0.154220 0.988037i $$-0.549286\pi$$
−0.154220 + 0.988037i $$0.549286\pi$$
$$734$$ 0 0
$$735$$ −1.36035e29 −0.109052
$$736$$ 0 0
$$737$$ 5.26365e29 0.410089
$$738$$ 0 0
$$739$$ −8.38257e29 −0.634762 −0.317381 0.948298i $$-0.602803\pi$$
−0.317381 + 0.948298i $$0.602803\pi$$
$$740$$ 0 0
$$741$$ 8.86242e24 6.52321e−6 0
$$742$$ 0 0
$$743$$ 7.60139e29 0.543889 0.271945 0.962313i $$-0.412333\pi$$
0.271945 + 0.962313i $$0.412333\pi$$
$$744$$ 0 0
$$745$$ 2.16300e30 1.50458
$$746$$ 0 0
$$747$$ −1.37048e30 −0.926846
$$748$$ 0 0
$$749$$ 7.05871e28 0.0464159
$$750$$ 0 0
$$751$$ 1.08974e30 0.696794 0.348397 0.937347i $$-0.386726\pi$$
0.348397 + 0.937347i $$0.386726\pi$$
$$752$$ 0 0
$$753$$ 4.63151e28 0.0287988
$$754$$ 0 0
$$755$$ 1.46676e30 0.886983
$$756$$ 0 0
$$757$$ 3.06224e30 1.80108 0.900540 0.434774i $$-0.143172\pi$$
0.900540 + 0.434774i $$0.143172\pi$$
$$758$$ 0 0
$$759$$ 2.32358e29 0.132929
$$760$$ 0 0
$$761$$ −3.46912e30 −1.93055 −0.965274 0.261241i $$-0.915868\pi$$
−0.965274 + 0.261241i $$0.915868\pi$$
$$762$$ 0 0
$$763$$ 3.00444e29 0.162650
$$764$$ 0 0
$$765$$ −2.39026e30 −1.25892
$$766$$ 0 0
$$767$$ 1.75151e30 0.897554
$$768$$ 0 0
$$769$$ −7.20768e29 −0.359392 −0.179696 0.983722i $$-0.557511\pi$$
−0.179696 + 0.983722i $$0.557511\pi$$
$$770$$ 0 0
$$771$$ −1.76770e29 −0.0857707
$$772$$ 0 0
$$773$$ −1.63916e30 −0.773990 −0.386995 0.922082i $$-0.626487\pi$$
−0.386995 + 0.922082i $$0.626487\pi$$
$$774$$ 0 0
$$775$$ −3.09021e30 −1.42011
$$776$$ 0 0
$$777$$ 1.17924e28 0.00527452
$$778$$ 0 0
$$779$$ 2.70507e26 0.000117770 0
$$780$$ 0 0
$$781$$ −3.86494e30 −1.63798
$$782$$ 0 0
$$783$$ −5.39480e29 −0.222575
$$784$$ 0 0
$$785$$ −6.90526e28 −0.0277363
$$786$$ 0 0
$$787$$ −1.22284e30 −0.478227 −0.239114 0.970992i $$-0.576857\pi$$
−0.239114 + 0.970992i $$0.576857\pi$$
$$788$$ 0 0
$$789$$ 2.99199e29 0.113933
$$790$$ 0 0
$$791$$ 2.32367e28 0.00861630
$$792$$ 0 0
$$793$$ 1.30312e30 0.470560
$$794$$ 0 0
$$795$$ 1.14403e29 0.0402332
$$796$$ 0 0
$$797$$ 1.32063e30 0.452345 0.226173 0.974087i $$-0.427379\pi$$
0.226173 + 0.974087i $$0.427379\pi$$
$$798$$ 0 0
$$799$$ −4.91433e30 −1.63954
$$800$$ 0 0
$$801$$ −4.12769e30 −1.34142
$$802$$ 0 0
$$803$$ 2.16838e30 0.686470
$$804$$ 0 0
$$805$$ 9.48189e29 0.292441
$$806$$ 0 0
$$807$$ 1.31590e29 0.0395413
$$808$$ 0 0
$$809$$ 5.54086e30 1.62225 0.811124 0.584874i $$-0.198856\pi$$
0.811124 + 0.584874i $$0.198856\pi$$
$$810$$ 0 0
$$811$$ 4.51703e30 1.28865 0.644323 0.764753i $$-0.277139\pi$$
0.644323 + 0.764753i $$0.277139\pi$$
$$812$$ 0 0
$$813$$ 2.28467e29 0.0635145
$$814$$ 0 0
$$815$$ −4.59026e30 −1.24361
$$816$$ 0 0
$$817$$ −9.44242e26 −0.000249317 0
$$818$$ 0 0
$$819$$ −2.25900e29 −0.0581347
$$820$$ 0 0
$$821$$ 6.49500e30 1.62921 0.814603 0.580019i $$-0.196955\pi$$
0.814603 + 0.580019i $$0.196955\pi$$
$$822$$ 0 0
$$823$$ −6.58977e30 −1.61128 −0.805642 0.592402i $$-0.798180\pi$$
−0.805642 + 0.592402i $$0.798180\pi$$
$$824$$ 0 0
$$825$$ −3.35959e29 −0.0800791
$$826$$ 0 0
$$827$$ 2.99217e29 0.0695309 0.0347655 0.999395i $$-0.488932\pi$$
0.0347655 + 0.999395i $$0.488932\pi$$
$$828$$ 0 0
$$829$$ −3.02375e30 −0.685053 −0.342526 0.939508i $$-0.611283\pi$$
−0.342526 + 0.939508i $$0.611283\pi$$
$$830$$ 0 0
$$831$$ 4.20009e29 0.0927787
$$832$$ 0 0
$$833$$ −4.08352e30 −0.879554
$$834$$ 0 0
$$835$$ 6.99266e30 1.46870
$$836$$ 0 0
$$837$$ 1.06649e30 0.218443
$$838$$ 0 0
$$839$$ 6.48734e30 1.29588 0.647942 0.761690i $$-0.275630\pi$$
0.647942 + 0.761690i $$0.275630\pi$$
$$840$$ 0 0
$$841$$ 5.36360e30 1.04496
$$842$$ 0 0
$$843$$ 5.15249e29 0.0979101
$$844$$ 0 0
$$845$$ −5.91481e30 −1.09634
$$846$$ 0 0
$$847$$ −1.89298e28 −0.00342272
$$848$$ 0 0
$$849$$ −3.64089e29 −0.0642210
$$850$$ 0 0
$$851$$ 5.37291e30 0.924590
$$852$$ 0 0
$$853$$ 1.83172e30 0.307535 0.153767 0.988107i $$-0.450859\pi$$
0.153767 + 0.988107i $$0.450859\pi$$
$$854$$ 0 0
$$855$$ 1.50922e27 0.000247234 0
$$856$$ 0 0
$$857$$ 1.04074e31 1.66358 0.831792 0.555088i $$-0.187315\pi$$
0.831792 + 0.555088i $$0.187315\pi$$
$$858$$ 0 0
$$859$$ 8.40237e29 0.131061 0.0655306 0.997851i $$-0.479126\pi$$
0.0655306 + 0.997851i $$0.479126\pi$$
$$860$$ 0 0
$$861$$ 4.22724e28 0.00643463
$$862$$ 0 0
$$863$$ 6.66021e30 0.989406 0.494703 0.869062i $$-0.335277\pi$$
0.494703 + 0.869062i $$0.335277\pi$$
$$864$$ 0 0
$$865$$ −1.66487e31 −2.41386
$$866$$ 0 0
$$867$$ −1.11720e29 −0.0158099
$$868$$ 0 0
$$869$$ 4.97957e30 0.687835
$$870$$ 0 0
$$871$$ 1.42944e30 0.192741
$$872$$ 0 0
$$873$$ 1.01929e31 1.34167
$$874$$ 0 0
$$875$$ −1.60600e28 −0.00206377
$$876$$ 0 0
$$877$$ −1.99416e30 −0.250187 −0.125093 0.992145i $$-0.539923\pi$$
−0.125093 + 0.992145i $$0.539923\pi$$
$$878$$ 0 0
$$879$$ −4.69331e29 −0.0574905
$$880$$ 0 0
$$881$$ −7.25457e30 −0.867690 −0.433845 0.900987i $$-0.642843\pi$$
−0.433845 + 0.900987i $$0.642843\pi$$
$$882$$ 0 0
$$883$$ −7.43622e30 −0.868491 −0.434245 0.900795i $$-0.642985\pi$$
−0.434245 + 0.900795i $$0.642985\pi$$
$$884$$ 0 0
$$885$$ −1.82863e30 −0.208555
$$886$$ 0 0
$$887$$ −5.32005e30 −0.592540 −0.296270 0.955104i $$-0.595743\pi$$
−0.296270 + 0.955104i $$0.595743\pi$$
$$888$$ 0 0
$$889$$ −2.77140e29 −0.0301461
$$890$$ 0 0
$$891$$ −9.36897e30 −0.995350
$$892$$ 0 0
$$893$$ 3.10293e27 0.000321982 0
$$894$$ 0 0
$$895$$ −3.50684e30 −0.355446
$$896$$ 0 0
$$897$$ 6.31009e29 0.0624762
$$898$$ 0 0
$$899$$ −2.07502e31 −2.00699
$$900$$ 0 0
$$901$$ 3.43418e30 0.324498
$$902$$ 0 0
$$903$$ −1.47557e29 −0.0136220
$$904$$ 0 0
$$905$$ 7.92809e30 0.715086
$$906$$ 0 0
$$907$$ 3.82478e30 0.337077 0.168539 0.985695i $$-0.446095\pi$$
0.168539 + 0.985695i $$0.446095\pi$$
$$908$$ 0 0
$$909$$ −6.71820e29 −0.0578538
$$910$$ 0 0
$$911$$ 3.98935e30 0.335706 0.167853 0.985812i $$-0.446317\pi$$
0.167853 + 0.985812i $$0.446317\pi$$
$$912$$ 0 0
$$913$$ −1.14968e31 −0.945440
$$914$$ 0 0
$$915$$ −1.36050e30 −0.109339
$$916$$ 0 0
$$917$$ −1.56568e30 −0.122977
$$918$$ 0 0
$$919$$ 2.24335e31 1.72220 0.861102 0.508432i $$-0.169775\pi$$
0.861102 + 0.508432i $$0.169775\pi$$
$$920$$ 0 0
$$921$$ 1.32085e30 0.0991119
$$922$$ 0 0
$$923$$ −1.04959e31 −0.769844
$$924$$ 0 0
$$925$$ −7.76850e30 −0.556992
$$926$$ 0 0
$$927$$ −7.68810e29 −0.0538867
$$928$$ 0 0
$$929$$ −5.26649e30 −0.360875 −0.180437 0.983586i $$-0.557751\pi$$
−0.180437 + 0.983586i $$0.557751\pi$$
$$930$$ 0 0
$$931$$ 2.57835e27 0.000172731 0
$$932$$ 0 0
$$933$$ 5.84963e28 0.00383152
$$934$$ 0 0
$$935$$ −2.00515e31 −1.28418
$$936$$ 0 0
$$937$$ −3.29722e30 −0.206482 −0.103241 0.994656i $$-0.532921\pi$$
−0.103241 + 0.994656i $$0.532921\pi$$
$$938$$ 0 0
$$939$$ −6.10672e29 −0.0373955
$$940$$ 0 0
$$941$$ −2.14709e31 −1.28576 −0.642879 0.765968i $$-0.722260\pi$$
−0.642879 + 0.765968i $$0.722260\pi$$
$$942$$ 0 0
$$943$$ 1.92603e31 1.12795
$$944$$ 0 0
$$945$$ 4.73139e29 0.0270991
$$946$$ 0 0
$$947$$ 1.56465e31 0.876483 0.438242 0.898857i $$-0.355601\pi$$
0.438242 + 0.898857i $$0.355601\pi$$
$$948$$ 0 0
$$949$$ 5.88861e30 0.322639
$$950$$ 0 0
$$951$$ −7.84793e29 −0.0420591
$$952$$ 0 0
$$953$$ −4.71750e30 −0.247307 −0.123654 0.992325i $$-0.539461\pi$$
−0.123654 + 0.992325i $$0.539461\pi$$
$$954$$ 0 0
$$955$$ 2.31798e31 1.18871
$$956$$ 0 0
$$957$$ −2.25590e30 −0.113173
$$958$$ 0 0
$$959$$ 2.15412e30 0.105724
$$960$$ 0 0
$$961$$ 2.01951e31 0.969728
$$962$$ 0 0
$$963$$ 7.99956e30 0.375829
$$964$$ 0 0
$$965$$ 5.48413e31 2.52099
$$966$$ 0 0
$$967$$ 3.43670e31 1.54584 0.772919 0.634505i $$-0.218796\pi$$
0.772919 + 0.634505i $$0.218796\pi$$
$$968$$ 0 0
$$969$$ −2.77747e26 −1.22250e−5 0
$$970$$ 0 0
$$971$$ −4.39842e31 −1.89450 −0.947249 0.320497i $$-0.896150\pi$$
−0.947249 + 0.320497i $$0.896150\pi$$
$$972$$ 0 0
$$973$$ −4.15696e30 −0.175223
$$974$$ 0 0
$$975$$ −9.12354e29 −0.0376370
$$976$$ 0 0
$$977$$ −1.85934e31 −0.750700 −0.375350 0.926883i $$-0.622477\pi$$
−0.375350 + 0.926883i $$0.622477\pi$$
$$978$$ 0 0
$$979$$ −3.46266e31 −1.36833
$$980$$ 0 0
$$981$$ 3.40489e31 1.31698
$$982$$ 0 0
$$983$$ 2.18619e31 0.827704 0.413852 0.910344i $$-0.364183\pi$$
0.413852 + 0.910344i $$0.364183\pi$$
$$984$$ 0 0
$$985$$ 6.33049e31 2.34616
$$986$$ 0 0
$$987$$ 4.84897e29 0.0175922
$$988$$ 0 0
$$989$$ −6.72306e31 −2.38784
$$990$$ 0 0
$$991$$ 8.68993e30 0.302164 0.151082 0.988521i $$-0.451724\pi$$
0.151082 + 0.988521i $$0.451724\pi$$
$$992$$ 0 0
$$993$$ 1.96550e30 0.0669122
$$994$$ 0 0
$$995$$ 2.51961e30 0.0839829
$$996$$ 0 0
$$997$$ 1.31572e31 0.429403 0.214701 0.976680i $$-0.431122\pi$$
0.214701 + 0.976680i $$0.431122\pi$$
$$998$$ 0 0
$$999$$ 2.68104e30 0.0856773
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 16.22.a.f.1.2 3
4.3 odd 2 8.22.a.b.1.2 3
8.3 odd 2 64.22.a.l.1.2 3
8.5 even 2 64.22.a.m.1.2 3
12.11 even 2 72.22.a.f.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.2 3 4.3 odd 2
16.22.a.f.1.2 3 1.1 even 1 trivial
64.22.a.l.1.2 3 8.3 odd 2
64.22.a.m.1.2 3 8.5 even 2
72.22.a.f.1.1 3 12.11 even 2