# Properties

 Label 8624.2.a.cb.1.1 Level $8624$ Weight $2$ Character 8624.1 Self dual yes Analytic conductor $68.863$ 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] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8624 = 2^{4} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8624.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: $$68.8629867032$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 8624.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.56155 q^{3} -3.56155 q^{5} -0.561553 q^{9} +O(q^{10})$$ $$q-1.56155 q^{3} -3.56155 q^{5} -0.561553 q^{9} +1.00000 q^{11} +5.12311 q^{13} +5.56155 q^{15} -2.00000 q^{17} -4.00000 q^{19} -2.43845 q^{23} +7.68466 q^{25} +5.56155 q^{27} -5.12311 q^{29} -5.56155 q^{31} -1.56155 q^{33} -7.56155 q^{37} -8.00000 q^{39} +1.12311 q^{41} +7.12311 q^{43} +2.00000 q^{45} +8.00000 q^{47} +3.12311 q^{51} +12.2462 q^{53} -3.56155 q^{55} +6.24621 q^{57} +7.80776 q^{59} -1.12311 q^{61} -18.2462 q^{65} -9.56155 q^{67} +3.80776 q^{69} +8.68466 q^{71} -5.12311 q^{73} -12.0000 q^{75} +11.1231 q^{79} -7.00000 q^{81} +0.876894 q^{83} +7.12311 q^{85} +8.00000 q^{87} -2.68466 q^{89} +8.68466 q^{93} +14.2462 q^{95} -15.5616 q^{97} -0.561553 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 - 3 * q^5 + 3 * q^9 $$2 q + q^{3} - 3 q^{5} + 3 q^{9} + 2 q^{11} + 2 q^{13} + 7 q^{15} - 4 q^{17} - 8 q^{19} - 9 q^{23} + 3 q^{25} + 7 q^{27} - 2 q^{29} - 7 q^{31} + q^{33} - 11 q^{37} - 16 q^{39} - 6 q^{41} + 6 q^{43} + 4 q^{45} + 16 q^{47} - 2 q^{51} + 8 q^{53} - 3 q^{55} - 4 q^{57} - 5 q^{59} + 6 q^{61} - 20 q^{65} - 15 q^{67} - 13 q^{69} + 5 q^{71} - 2 q^{73} - 24 q^{75} + 14 q^{79} - 14 q^{81} + 10 q^{83} + 6 q^{85} + 16 q^{87} + 7 q^{89} + 5 q^{93} + 12 q^{95} - 27 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q + q^3 - 3 * q^5 + 3 * q^9 + 2 * q^11 + 2 * q^13 + 7 * q^15 - 4 * q^17 - 8 * q^19 - 9 * q^23 + 3 * q^25 + 7 * q^27 - 2 * q^29 - 7 * q^31 + q^33 - 11 * q^37 - 16 * q^39 - 6 * q^41 + 6 * q^43 + 4 * q^45 + 16 * q^47 - 2 * q^51 + 8 * q^53 - 3 * q^55 - 4 * q^57 - 5 * q^59 + 6 * q^61 - 20 * q^65 - 15 * q^67 - 13 * q^69 + 5 * q^71 - 2 * q^73 - 24 * q^75 + 14 * q^79 - 14 * q^81 + 10 * q^83 + 6 * q^85 + 16 * q^87 + 7 * q^89 + 5 * q^93 + 12 * q^95 - 27 * q^97 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.56155 −0.901563 −0.450781 0.892634i $$-0.648855\pi$$
−0.450781 + 0.892634i $$0.648855\pi$$
$$4$$ 0 0
$$5$$ −3.56155 −1.59277 −0.796387 0.604787i $$-0.793258\pi$$
−0.796387 + 0.604787i $$0.793258\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 5.12311 1.42089 0.710447 0.703751i $$-0.248493\pi$$
0.710447 + 0.703751i $$0.248493\pi$$
$$14$$ 0 0
$$15$$ 5.56155 1.43599
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.43845 −0.508451 −0.254226 0.967145i $$-0.581821\pi$$
−0.254226 + 0.967145i $$0.581821\pi$$
$$24$$ 0 0
$$25$$ 7.68466 1.53693
$$26$$ 0 0
$$27$$ 5.56155 1.07032
$$28$$ 0 0
$$29$$ −5.12311 −0.951337 −0.475668 0.879625i $$-0.657794\pi$$
−0.475668 + 0.879625i $$0.657794\pi$$
$$30$$ 0 0
$$31$$ −5.56155 −0.998884 −0.499442 0.866347i $$-0.666462\pi$$
−0.499442 + 0.866347i $$0.666462\pi$$
$$32$$ 0 0
$$33$$ −1.56155 −0.271831
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.56155 −1.24311 −0.621556 0.783370i $$-0.713499\pi$$
−0.621556 + 0.783370i $$0.713499\pi$$
$$38$$ 0 0
$$39$$ −8.00000 −1.28103
$$40$$ 0 0
$$41$$ 1.12311 0.175400 0.0876998 0.996147i $$-0.472048\pi$$
0.0876998 + 0.996147i $$0.472048\pi$$
$$42$$ 0 0
$$43$$ 7.12311 1.08626 0.543132 0.839648i $$-0.317238\pi$$
0.543132 + 0.839648i $$0.317238\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 3.12311 0.437322
$$52$$ 0 0
$$53$$ 12.2462 1.68215 0.841073 0.540921i $$-0.181924\pi$$
0.841073 + 0.540921i $$0.181924\pi$$
$$54$$ 0 0
$$55$$ −3.56155 −0.480240
$$56$$ 0 0
$$57$$ 6.24621 0.827331
$$58$$ 0 0
$$59$$ 7.80776 1.01648 0.508242 0.861214i $$-0.330295\pi$$
0.508242 + 0.861214i $$0.330295\pi$$
$$60$$ 0 0
$$61$$ −1.12311 −0.143799 −0.0718995 0.997412i $$-0.522906\pi$$
−0.0718995 + 0.997412i $$0.522906\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −18.2462 −2.26316
$$66$$ 0 0
$$67$$ −9.56155 −1.16813 −0.584065 0.811707i $$-0.698539\pi$$
−0.584065 + 0.811707i $$0.698539\pi$$
$$68$$ 0 0
$$69$$ 3.80776 0.458401
$$70$$ 0 0
$$71$$ 8.68466 1.03068 0.515340 0.856986i $$-0.327666\pi$$
0.515340 + 0.856986i $$0.327666\pi$$
$$72$$ 0 0
$$73$$ −5.12311 −0.599614 −0.299807 0.954000i $$-0.596922\pi$$
−0.299807 + 0.954000i $$0.596922\pi$$
$$74$$ 0 0
$$75$$ −12.0000 −1.38564
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 11.1231 1.25145 0.625724 0.780045i $$-0.284804\pi$$
0.625724 + 0.780045i $$0.284804\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 0.876894 0.0962517 0.0481258 0.998841i $$-0.484675\pi$$
0.0481258 + 0.998841i $$0.484675\pi$$
$$84$$ 0 0
$$85$$ 7.12311 0.772609
$$86$$ 0 0
$$87$$ 8.00000 0.857690
$$88$$ 0 0
$$89$$ −2.68466 −0.284573 −0.142287 0.989825i $$-0.545445\pi$$
−0.142287 + 0.989825i $$0.545445\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.68466 0.900557
$$94$$ 0 0
$$95$$ 14.2462 1.46163
$$96$$ 0 0
$$97$$ −15.5616 −1.58004 −0.790018 0.613083i $$-0.789929\pi$$
−0.790018 + 0.613083i $$0.789929\pi$$
$$98$$ 0 0
$$99$$ −0.561553 −0.0564382
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 13.3693 1.29246 0.646230 0.763142i $$-0.276345\pi$$
0.646230 + 0.763142i $$0.276345\pi$$
$$108$$ 0 0
$$109$$ 12.2462 1.17297 0.586487 0.809959i $$-0.300510\pi$$
0.586487 + 0.809959i $$0.300510\pi$$
$$110$$ 0 0
$$111$$ 11.8078 1.12074
$$112$$ 0 0
$$113$$ −0.438447 −0.0412456 −0.0206228 0.999787i $$-0.506565\pi$$
−0.0206228 + 0.999787i $$0.506565\pi$$
$$114$$ 0 0
$$115$$ 8.68466 0.809849
$$116$$ 0 0
$$117$$ −2.87689 −0.265969
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −1.75379 −0.158134
$$124$$ 0 0
$$125$$ −9.56155 −0.855211
$$126$$ 0 0
$$127$$ −6.24621 −0.554262 −0.277131 0.960832i $$-0.589384\pi$$
−0.277131 + 0.960832i $$0.589384\pi$$
$$128$$ 0 0
$$129$$ −11.1231 −0.979335
$$130$$ 0 0
$$131$$ 13.3693 1.16808 0.584041 0.811724i $$-0.301471\pi$$
0.584041 + 0.811724i $$0.301471\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −19.8078 −1.70478
$$136$$ 0 0
$$137$$ −8.43845 −0.720945 −0.360473 0.932770i $$-0.617385\pi$$
−0.360473 + 0.932770i $$0.617385\pi$$
$$138$$ 0 0
$$139$$ 15.1231 1.28273 0.641363 0.767238i $$-0.278369\pi$$
0.641363 + 0.767238i $$0.278369\pi$$
$$140$$ 0 0
$$141$$ −12.4924 −1.05205
$$142$$ 0 0
$$143$$ 5.12311 0.428416
$$144$$ 0 0
$$145$$ 18.2462 1.51527
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 4.24621 0.347863 0.173932 0.984758i $$-0.444353\pi$$
0.173932 + 0.984758i $$0.444353\pi$$
$$150$$ 0 0
$$151$$ 9.36932 0.762464 0.381232 0.924479i $$-0.375500\pi$$
0.381232 + 0.924479i $$0.375500\pi$$
$$152$$ 0 0
$$153$$ 1.12311 0.0907977
$$154$$ 0 0
$$155$$ 19.8078 1.59100
$$156$$ 0 0
$$157$$ 4.43845 0.354227 0.177113 0.984190i $$-0.443324\pi$$
0.177113 + 0.984190i $$0.443324\pi$$
$$158$$ 0 0
$$159$$ −19.1231 −1.51656
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ 5.56155 0.432966
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 13.2462 1.01894
$$170$$ 0 0
$$171$$ 2.24621 0.171772
$$172$$ 0 0
$$173$$ −12.2462 −0.931062 −0.465531 0.885032i $$-0.654137\pi$$
−0.465531 + 0.885032i $$0.654137\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −12.1922 −0.916425
$$178$$ 0 0
$$179$$ 6.43845 0.481232 0.240616 0.970620i $$-0.422651\pi$$
0.240616 + 0.970620i $$0.422651\pi$$
$$180$$ 0 0
$$181$$ 1.31534 0.0977686 0.0488843 0.998804i $$-0.484433\pi$$
0.0488843 + 0.998804i $$0.484433\pi$$
$$182$$ 0 0
$$183$$ 1.75379 0.129644
$$184$$ 0 0
$$185$$ 26.9309 1.98000
$$186$$ 0 0
$$187$$ −2.00000 −0.146254
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.4384 0.755300 0.377650 0.925949i $$-0.376732\pi$$
0.377650 + 0.925949i $$0.376732\pi$$
$$192$$ 0 0
$$193$$ −9.12311 −0.656696 −0.328348 0.944557i $$-0.606492\pi$$
−0.328348 + 0.944557i $$0.606492\pi$$
$$194$$ 0 0
$$195$$ 28.4924 2.04038
$$196$$ 0 0
$$197$$ −14.4924 −1.03254 −0.516271 0.856425i $$-0.672680\pi$$
−0.516271 + 0.856425i $$0.672680\pi$$
$$198$$ 0 0
$$199$$ −12.4924 −0.885564 −0.442782 0.896629i $$-0.646009\pi$$
−0.442782 + 0.896629i $$0.646009\pi$$
$$200$$ 0 0
$$201$$ 14.9309 1.05314
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −4.00000 −0.279372
$$206$$ 0 0
$$207$$ 1.36932 0.0951741
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −8.49242 −0.584642 −0.292321 0.956320i $$-0.594428\pi$$
−0.292321 + 0.956320i $$0.594428\pi$$
$$212$$ 0 0
$$213$$ −13.5616 −0.929222
$$214$$ 0 0
$$215$$ −25.3693 −1.73017
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 8.00000 0.540590
$$220$$ 0 0
$$221$$ −10.2462 −0.689235
$$222$$ 0 0
$$223$$ −11.8078 −0.790706 −0.395353 0.918529i $$-0.629378\pi$$
−0.395353 + 0.918529i $$0.629378\pi$$
$$224$$ 0 0
$$225$$ −4.31534 −0.287689
$$226$$ 0 0
$$227$$ 23.1231 1.53473 0.767367 0.641208i $$-0.221566\pi$$
0.767367 + 0.641208i $$0.221566\pi$$
$$228$$ 0 0
$$229$$ −14.6847 −0.970390 −0.485195 0.874406i $$-0.661251\pi$$
−0.485195 + 0.874406i $$0.661251\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −7.36932 −0.482780 −0.241390 0.970428i $$-0.577603\pi$$
−0.241390 + 0.970428i $$0.577603\pi$$
$$234$$ 0 0
$$235$$ −28.4924 −1.85864
$$236$$ 0 0
$$237$$ −17.3693 −1.12826
$$238$$ 0 0
$$239$$ 4.87689 0.315460 0.157730 0.987482i $$-0.449582\pi$$
0.157730 + 0.987482i $$0.449582\pi$$
$$240$$ 0 0
$$241$$ −29.1231 −1.87598 −0.937992 0.346657i $$-0.887317\pi$$
−0.937992 + 0.346657i $$0.887317\pi$$
$$242$$ 0 0
$$243$$ −5.75379 −0.369106
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −20.4924 −1.30390
$$248$$ 0 0
$$249$$ −1.36932 −0.0867769
$$250$$ 0 0
$$251$$ 1.56155 0.0985643 0.0492822 0.998785i $$-0.484307\pi$$
0.0492822 + 0.998785i $$0.484307\pi$$
$$252$$ 0 0
$$253$$ −2.43845 −0.153304
$$254$$ 0 0
$$255$$ −11.1231 −0.696556
$$256$$ 0 0
$$257$$ −11.7538 −0.733181 −0.366591 0.930382i $$-0.619475\pi$$
−0.366591 + 0.930382i $$0.619475\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.87689 0.178075
$$262$$ 0 0
$$263$$ −19.1231 −1.17918 −0.589591 0.807702i $$-0.700711\pi$$
−0.589591 + 0.807702i $$0.700711\pi$$
$$264$$ 0 0
$$265$$ −43.6155 −2.67928
$$266$$ 0 0
$$267$$ 4.19224 0.256561
$$268$$ 0 0
$$269$$ 20.7386 1.26446 0.632228 0.774782i $$-0.282140\pi$$
0.632228 + 0.774782i $$0.282140\pi$$
$$270$$ 0 0
$$271$$ −28.4924 −1.73079 −0.865396 0.501089i $$-0.832933\pi$$
−0.865396 + 0.501089i $$0.832933\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 7.68466 0.463402
$$276$$ 0 0
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 0 0
$$279$$ 3.12311 0.186975
$$280$$ 0 0
$$281$$ 16.2462 0.969168 0.484584 0.874745i $$-0.338971\pi$$
0.484584 + 0.874745i $$0.338971\pi$$
$$282$$ 0 0
$$283$$ 20.0000 1.18888 0.594438 0.804141i $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ 0 0
$$285$$ −22.2462 −1.31775
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 24.3002 1.42450
$$292$$ 0 0
$$293$$ 3.36932 0.196838 0.0984188 0.995145i $$-0.468622\pi$$
0.0984188 + 0.995145i $$0.468622\pi$$
$$294$$ 0 0
$$295$$ −27.8078 −1.61903
$$296$$ 0 0
$$297$$ 5.56155 0.322714
$$298$$ 0 0
$$299$$ −12.4924 −0.722455
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −3.12311 −0.179418
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ −32.4924 −1.85444 −0.927220 0.374516i $$-0.877809\pi$$
−0.927220 + 0.374516i $$0.877809\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 9.75379 0.553087 0.276543 0.961001i $$-0.410811\pi$$
0.276543 + 0.961001i $$0.410811\pi$$
$$312$$ 0 0
$$313$$ 9.80776 0.554368 0.277184 0.960817i $$-0.410599\pi$$
0.277184 + 0.960817i $$0.410599\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −14.1922 −0.797115 −0.398558 0.917143i $$-0.630489\pi$$
−0.398558 + 0.917143i $$0.630489\pi$$
$$318$$ 0 0
$$319$$ −5.12311 −0.286839
$$320$$ 0 0
$$321$$ −20.8769 −1.16523
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ 0 0
$$325$$ 39.3693 2.18382
$$326$$ 0 0
$$327$$ −19.1231 −1.05751
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −34.9309 −1.91997 −0.959987 0.280044i $$-0.909651\pi$$
−0.959987 + 0.280044i $$0.909651\pi$$
$$332$$ 0 0
$$333$$ 4.24621 0.232691
$$334$$ 0 0
$$335$$ 34.0540 1.86057
$$336$$ 0 0
$$337$$ −16.7386 −0.911811 −0.455906 0.890028i $$-0.650685\pi$$
−0.455906 + 0.890028i $$0.650685\pi$$
$$338$$ 0 0
$$339$$ 0.684658 0.0371855
$$340$$ 0 0
$$341$$ −5.56155 −0.301175
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −13.5616 −0.730129
$$346$$ 0 0
$$347$$ −22.7386 −1.22067 −0.610337 0.792142i $$-0.708966\pi$$
−0.610337 + 0.792142i $$0.708966\pi$$
$$348$$ 0 0
$$349$$ 32.2462 1.72610 0.863050 0.505118i $$-0.168551\pi$$
0.863050 + 0.505118i $$0.168551\pi$$
$$350$$ 0 0
$$351$$ 28.4924 1.52081
$$352$$ 0 0
$$353$$ 24.0540 1.28026 0.640132 0.768265i $$-0.278880\pi$$
0.640132 + 0.768265i $$0.278880\pi$$
$$354$$ 0 0
$$355$$ −30.9309 −1.64164
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4.49242 −0.237101 −0.118550 0.992948i $$-0.537825\pi$$
−0.118550 + 0.992948i $$0.537825\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ −1.56155 −0.0819603
$$364$$ 0 0
$$365$$ 18.2462 0.955050
$$366$$ 0 0
$$367$$ 22.9309 1.19698 0.598491 0.801130i $$-0.295767\pi$$
0.598491 + 0.801130i $$0.295767\pi$$
$$368$$ 0 0
$$369$$ −0.630683 −0.0328321
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −8.24621 −0.426973 −0.213486 0.976946i $$-0.568482\pi$$
−0.213486 + 0.976946i $$0.568482\pi$$
$$374$$ 0 0
$$375$$ 14.9309 0.771027
$$376$$ 0 0
$$377$$ −26.2462 −1.35175
$$378$$ 0 0
$$379$$ −0.192236 −0.00987450 −0.00493725 0.999988i $$-0.501572\pi$$
−0.00493725 + 0.999988i $$0.501572\pi$$
$$380$$ 0 0
$$381$$ 9.75379 0.499702
$$382$$ 0 0
$$383$$ −2.05398 −0.104953 −0.0524766 0.998622i $$-0.516712\pi$$
−0.0524766 + 0.998622i $$0.516712\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 3.56155 0.180578 0.0902889 0.995916i $$-0.471221\pi$$
0.0902889 + 0.995916i $$0.471221\pi$$
$$390$$ 0 0
$$391$$ 4.87689 0.246635
$$392$$ 0 0
$$393$$ −20.8769 −1.05310
$$394$$ 0 0
$$395$$ −39.6155 −1.99327
$$396$$ 0 0
$$397$$ −10.4924 −0.526600 −0.263300 0.964714i $$-0.584811\pi$$
−0.263300 + 0.964714i $$0.584811\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.4924 1.52272 0.761359 0.648330i $$-0.224532\pi$$
0.761359 + 0.648330i $$0.224532\pi$$
$$402$$ 0 0
$$403$$ −28.4924 −1.41931
$$404$$ 0 0
$$405$$ 24.9309 1.23882
$$406$$ 0 0
$$407$$ −7.56155 −0.374812
$$408$$ 0 0
$$409$$ −22.4924 −1.11218 −0.556089 0.831123i $$-0.687699\pi$$
−0.556089 + 0.831123i $$0.687699\pi$$
$$410$$ 0 0
$$411$$ 13.1771 0.649977
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −3.12311 −0.153307
$$416$$ 0 0
$$417$$ −23.6155 −1.15646
$$418$$ 0 0
$$419$$ −32.4924 −1.58736 −0.793679 0.608336i $$-0.791837\pi$$
−0.793679 + 0.608336i $$0.791837\pi$$
$$420$$ 0 0
$$421$$ 2.49242 0.121473 0.0607366 0.998154i $$-0.480655\pi$$
0.0607366 + 0.998154i $$0.480655\pi$$
$$422$$ 0 0
$$423$$ −4.49242 −0.218429
$$424$$ 0 0
$$425$$ −15.3693 −0.745521
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −8.00000 −0.386244
$$430$$ 0 0
$$431$$ −27.1231 −1.30647 −0.653237 0.757153i $$-0.726589\pi$$
−0.653237 + 0.757153i $$0.726589\pi$$
$$432$$ 0 0
$$433$$ 22.6847 1.09016 0.545078 0.838386i $$-0.316500\pi$$
0.545078 + 0.838386i $$0.316500\pi$$
$$434$$ 0 0
$$435$$ −28.4924 −1.36611
$$436$$ 0 0
$$437$$ 9.75379 0.466587
$$438$$ 0 0
$$439$$ 4.49242 0.214412 0.107206 0.994237i $$-0.465810\pi$$
0.107206 + 0.994237i $$0.465810\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −11.3153 −0.537608 −0.268804 0.963195i $$-0.586628\pi$$
−0.268804 + 0.963195i $$0.586628\pi$$
$$444$$ 0 0
$$445$$ 9.56155 0.453261
$$446$$ 0 0
$$447$$ −6.63068 −0.313621
$$448$$ 0 0
$$449$$ −36.5464 −1.72473 −0.862366 0.506286i $$-0.831018\pi$$
−0.862366 + 0.506286i $$0.831018\pi$$
$$450$$ 0 0
$$451$$ 1.12311 0.0528850
$$452$$ 0 0
$$453$$ −14.6307 −0.687409
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 23.8617 1.11621 0.558103 0.829772i $$-0.311530\pi$$
0.558103 + 0.829772i $$0.311530\pi$$
$$458$$ 0 0
$$459$$ −11.1231 −0.519182
$$460$$ 0 0
$$461$$ −1.12311 −0.0523082 −0.0261541 0.999658i $$-0.508326\pi$$
−0.0261541 + 0.999658i $$0.508326\pi$$
$$462$$ 0 0
$$463$$ −15.3153 −0.711764 −0.355882 0.934531i $$-0.615820\pi$$
−0.355882 + 0.934531i $$0.615820\pi$$
$$464$$ 0 0
$$465$$ −30.9309 −1.43438
$$466$$ 0 0
$$467$$ 28.3002 1.30958 0.654788 0.755812i $$-0.272758\pi$$
0.654788 + 0.755812i $$0.272758\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −6.93087 −0.319358
$$472$$ 0 0
$$473$$ 7.12311 0.327521
$$474$$ 0 0
$$475$$ −30.7386 −1.41039
$$476$$ 0 0
$$477$$ −6.87689 −0.314871
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −38.7386 −1.76633
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 55.4233 2.51664
$$486$$ 0 0
$$487$$ −14.9309 −0.676582 −0.338291 0.941041i $$-0.609849\pi$$
−0.338291 + 0.941041i $$0.609849\pi$$
$$488$$ 0 0
$$489$$ −6.24621 −0.282463
$$490$$ 0 0
$$491$$ −13.7538 −0.620700 −0.310350 0.950622i $$-0.600446\pi$$
−0.310350 + 0.950622i $$0.600446\pi$$
$$492$$ 0 0
$$493$$ 10.2462 0.461466
$$494$$ 0 0
$$495$$ 2.00000 0.0898933
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 28.9848 1.29754 0.648770 0.760985i $$-0.275284\pi$$
0.648770 + 0.760985i $$0.275284\pi$$
$$500$$ 0 0
$$501$$ −12.4924 −0.558120
$$502$$ 0 0
$$503$$ −31.6155 −1.40967 −0.704833 0.709373i $$-0.748978\pi$$
−0.704833 + 0.709373i $$0.748978\pi$$
$$504$$ 0 0
$$505$$ −7.12311 −0.316974
$$506$$ 0 0
$$507$$ −20.6847 −0.918638
$$508$$ 0 0
$$509$$ 18.3002 0.811142 0.405571 0.914064i $$-0.367073\pi$$
0.405571 + 0.914064i $$0.367073\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −22.2462 −0.982194
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 8.00000 0.351840
$$518$$ 0 0
$$519$$ 19.1231 0.839411
$$520$$ 0 0
$$521$$ −1.31534 −0.0576262 −0.0288131 0.999585i $$-0.509173\pi$$
−0.0288131 + 0.999585i $$0.509173\pi$$
$$522$$ 0 0
$$523$$ −12.0000 −0.524723 −0.262362 0.964970i $$-0.584501\pi$$
−0.262362 + 0.964970i $$0.584501\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 11.1231 0.484530
$$528$$ 0 0
$$529$$ −17.0540 −0.741477
$$530$$ 0 0
$$531$$ −4.38447 −0.190270
$$532$$ 0 0
$$533$$ 5.75379 0.249224
$$534$$ 0 0
$$535$$ −47.6155 −2.05860
$$536$$ 0 0
$$537$$ −10.0540 −0.433861
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −23.8617 −1.02590 −0.512948 0.858420i $$-0.671447\pi$$
−0.512948 + 0.858420i $$0.671447\pi$$
$$542$$ 0 0
$$543$$ −2.05398 −0.0881445
$$544$$ 0 0
$$545$$ −43.6155 −1.86828
$$546$$ 0 0
$$547$$ 42.2462 1.80632 0.903159 0.429307i $$-0.141242\pi$$
0.903159 + 0.429307i $$0.141242\pi$$
$$548$$ 0 0
$$549$$ 0.630683 0.0269169
$$550$$ 0 0
$$551$$ 20.4924 0.873007
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −42.0540 −1.78509
$$556$$ 0 0
$$557$$ −3.75379 −0.159053 −0.0795266 0.996833i $$-0.525341\pi$$
−0.0795266 + 0.996833i $$0.525341\pi$$
$$558$$ 0 0
$$559$$ 36.4924 1.54347
$$560$$ 0 0
$$561$$ 3.12311 0.131858
$$562$$ 0 0
$$563$$ 24.4924 1.03223 0.516116 0.856519i $$-0.327377\pi$$
0.516116 + 0.856519i $$0.327377\pi$$
$$564$$ 0 0
$$565$$ 1.56155 0.0656950
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −26.8769 −1.12674 −0.563369 0.826205i $$-0.690495\pi$$
−0.563369 + 0.826205i $$0.690495\pi$$
$$570$$ 0 0
$$571$$ −16.4924 −0.690186 −0.345093 0.938568i $$-0.612153\pi$$
−0.345093 + 0.938568i $$0.612153\pi$$
$$572$$ 0 0
$$573$$ −16.3002 −0.680950
$$574$$ 0 0
$$575$$ −18.7386 −0.781455
$$576$$ 0 0
$$577$$ −15.5616 −0.647836 −0.323918 0.946085i $$-0.605000\pi$$
−0.323918 + 0.946085i $$0.605000\pi$$
$$578$$ 0 0
$$579$$ 14.2462 0.592052
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 12.2462 0.507186
$$584$$ 0 0
$$585$$ 10.2462 0.423629
$$586$$ 0 0
$$587$$ −24.4924 −1.01091 −0.505455 0.862853i $$-0.668675\pi$$
−0.505455 + 0.862853i $$0.668675\pi$$
$$588$$ 0 0
$$589$$ 22.2462 0.916639
$$590$$ 0 0
$$591$$ 22.6307 0.930902
$$592$$ 0 0
$$593$$ −3.36932 −0.138361 −0.0691806 0.997604i $$-0.522038\pi$$
−0.0691806 + 0.997604i $$0.522038\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 19.5076 0.798392
$$598$$ 0 0
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −3.75379 −0.153120 −0.0765601 0.997065i $$-0.524394\pi$$
−0.0765601 + 0.997065i $$0.524394\pi$$
$$602$$ 0 0
$$603$$ 5.36932 0.218655
$$604$$ 0 0
$$605$$ −3.56155 −0.144798
$$606$$ 0 0
$$607$$ −45.8617 −1.86147 −0.930735 0.365694i $$-0.880832\pi$$
−0.930735 + 0.365694i $$0.880832\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 40.9848 1.65807
$$612$$ 0 0
$$613$$ 11.8617 0.479091 0.239546 0.970885i $$-0.423002\pi$$
0.239546 + 0.970885i $$0.423002\pi$$
$$614$$ 0 0
$$615$$ 6.24621 0.251872
$$616$$ 0 0
$$617$$ −2.49242 −0.100341 −0.0501706 0.998741i $$-0.515976\pi$$
−0.0501706 + 0.998741i $$0.515976\pi$$
$$618$$ 0 0
$$619$$ 18.9309 0.760896 0.380448 0.924802i $$-0.375770\pi$$
0.380448 + 0.924802i $$0.375770\pi$$
$$620$$ 0 0
$$621$$ −13.5616 −0.544206
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −4.36932 −0.174773
$$626$$ 0 0
$$627$$ 6.24621 0.249450
$$628$$ 0 0
$$629$$ 15.1231 0.602998
$$630$$ 0 0
$$631$$ 42.0540 1.67414 0.837071 0.547094i $$-0.184266\pi$$
0.837071 + 0.547094i $$0.184266\pi$$
$$632$$ 0 0
$$633$$ 13.2614 0.527092
$$634$$ 0 0
$$635$$ 22.2462 0.882814
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −4.87689 −0.192927
$$640$$ 0 0
$$641$$ −46.3002 −1.82875 −0.914374 0.404871i $$-0.867316\pi$$
−0.914374 + 0.404871i $$0.867316\pi$$
$$642$$ 0 0
$$643$$ −9.17708 −0.361909 −0.180954 0.983491i $$-0.557919\pi$$
−0.180954 + 0.983491i $$0.557919\pi$$
$$644$$ 0 0
$$645$$ 39.6155 1.55986
$$646$$ 0 0
$$647$$ 13.5616 0.533160 0.266580 0.963813i $$-0.414106\pi$$
0.266580 + 0.963813i $$0.414106\pi$$
$$648$$ 0 0
$$649$$ 7.80776 0.306482
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 35.1771 1.37659 0.688293 0.725433i $$-0.258360\pi$$
0.688293 + 0.725433i $$0.258360\pi$$
$$654$$ 0 0
$$655$$ −47.6155 −1.86049
$$656$$ 0 0
$$657$$ 2.87689 0.112238
$$658$$ 0 0
$$659$$ 11.6155 0.452477 0.226238 0.974072i $$-0.427357\pi$$
0.226238 + 0.974072i $$0.427357\pi$$
$$660$$ 0 0
$$661$$ −41.8078 −1.62613 −0.813067 0.582170i $$-0.802204\pi$$
−0.813067 + 0.582170i $$0.802204\pi$$
$$662$$ 0 0
$$663$$ 16.0000 0.621389
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12.4924 0.483709
$$668$$ 0 0
$$669$$ 18.4384 0.712872
$$670$$ 0 0
$$671$$ −1.12311 −0.0433570
$$672$$ 0 0
$$673$$ 33.2311 1.28096 0.640482 0.767974i $$-0.278735\pi$$
0.640482 + 0.767974i $$0.278735\pi$$
$$674$$ 0 0
$$675$$ 42.7386 1.64501
$$676$$ 0 0
$$677$$ 20.7386 0.797050 0.398525 0.917157i $$-0.369522\pi$$
0.398525 + 0.917157i $$0.369522\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −36.1080 −1.38366
$$682$$ 0 0
$$683$$ −6.73863 −0.257847 −0.128923 0.991655i $$-0.541152\pi$$
−0.128923 + 0.991655i $$0.541152\pi$$
$$684$$ 0 0
$$685$$ 30.0540 1.14830
$$686$$ 0 0
$$687$$ 22.9309 0.874867
$$688$$ 0 0
$$689$$ 62.7386 2.39015
$$690$$ 0 0
$$691$$ −9.94602 −0.378365 −0.189182 0.981942i $$-0.560584\pi$$
−0.189182 + 0.981942i $$0.560584\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −53.8617 −2.04309
$$696$$ 0 0
$$697$$ −2.24621 −0.0850813
$$698$$ 0 0
$$699$$ 11.5076 0.435257
$$700$$ 0 0
$$701$$ 50.4924 1.90707 0.953536 0.301278i $$-0.0974133\pi$$
0.953536 + 0.301278i $$0.0974133\pi$$
$$702$$ 0 0
$$703$$ 30.2462 1.14076
$$704$$ 0 0
$$705$$ 44.4924 1.67568
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.19224 0.0823311 0.0411656 0.999152i $$-0.486893\pi$$
0.0411656 + 0.999152i $$0.486893\pi$$
$$710$$ 0 0
$$711$$ −6.24621 −0.234251
$$712$$ 0 0
$$713$$ 13.5616 0.507884
$$714$$ 0 0
$$715$$ −18.2462 −0.682370
$$716$$ 0 0
$$717$$ −7.61553 −0.284407
$$718$$ 0 0
$$719$$ 35.4233 1.32107 0.660533 0.750797i $$-0.270330\pi$$
0.660533 + 0.750797i $$0.270330\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 45.4773 1.69132
$$724$$ 0 0
$$725$$ −39.3693 −1.46214
$$726$$ 0 0
$$727$$ −23.3153 −0.864718 −0.432359 0.901702i $$-0.642319\pi$$
−0.432359 + 0.901702i $$0.642319\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ −14.2462 −0.526915
$$732$$ 0 0
$$733$$ −1.12311 −0.0414829 −0.0207414 0.999785i $$-0.506603\pi$$
−0.0207414 + 0.999785i $$0.506603\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −9.56155 −0.352204
$$738$$ 0 0
$$739$$ 2.63068 0.0967712 0.0483856 0.998829i $$-0.484592\pi$$
0.0483856 + 0.998829i $$0.484592\pi$$
$$740$$ 0 0
$$741$$ 32.0000 1.17555
$$742$$ 0 0
$$743$$ −10.7386 −0.393962 −0.196981 0.980407i $$-0.563114\pi$$
−0.196981 + 0.980407i $$0.563114\pi$$
$$744$$ 0 0
$$745$$ −15.1231 −0.554068
$$746$$ 0 0
$$747$$ −0.492423 −0.0180168
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −5.56155 −0.202944 −0.101472 0.994838i $$-0.532355\pi$$
−0.101472 + 0.994838i $$0.532355\pi$$
$$752$$ 0 0
$$753$$ −2.43845 −0.0888620
$$754$$ 0 0
$$755$$ −33.3693 −1.21443
$$756$$ 0 0
$$757$$ 15.7538 0.572581 0.286291 0.958143i $$-0.407578\pi$$
0.286291 + 0.958143i $$0.407578\pi$$
$$758$$ 0 0
$$759$$ 3.80776 0.138213
$$760$$ 0 0
$$761$$ −5.12311 −0.185712 −0.0928562 0.995680i $$-0.529600\pi$$
−0.0928562 + 0.995680i $$0.529600\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −4.00000 −0.144620
$$766$$ 0 0
$$767$$ 40.0000 1.44432
$$768$$ 0 0
$$769$$ −25.6155 −0.923720 −0.461860 0.886953i $$-0.652818\pi$$
−0.461860 + 0.886953i $$0.652818\pi$$
$$770$$ 0 0
$$771$$ 18.3542 0.661009
$$772$$ 0 0
$$773$$ −40.7386 −1.46527 −0.732633 0.680623i $$-0.761709\pi$$
−0.732633 + 0.680623i $$0.761709\pi$$
$$774$$ 0 0
$$775$$ −42.7386 −1.53522
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −4.49242 −0.160958
$$780$$ 0 0
$$781$$ 8.68466 0.310762
$$782$$ 0 0
$$783$$ −28.4924 −1.01824
$$784$$ 0 0
$$785$$ −15.8078 −0.564203
$$786$$ 0 0
$$787$$ −29.7538 −1.06061 −0.530304 0.847808i $$-0.677922\pi$$
−0.530304 + 0.847808i $$0.677922\pi$$
$$788$$ 0 0
$$789$$ 29.8617 1.06311
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −5.75379 −0.204323
$$794$$ 0 0
$$795$$ 68.1080 2.41554
$$796$$ 0 0
$$797$$ 14.1922 0.502715 0.251357 0.967894i $$-0.419123\pi$$
0.251357 + 0.967894i $$0.419123\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ 1.50758 0.0532676
$$802$$ 0 0
$$803$$ −5.12311 −0.180790
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −32.3845 −1.13999
$$808$$ 0 0
$$809$$ −45.6155 −1.60376 −0.801878 0.597487i $$-0.796166\pi$$
−0.801878 + 0.597487i $$0.796166\pi$$
$$810$$ 0 0
$$811$$ −7.12311 −0.250126 −0.125063 0.992149i $$-0.539913\pi$$
−0.125063 + 0.992149i $$0.539913\pi$$
$$812$$ 0 0
$$813$$ 44.4924 1.56042
$$814$$ 0 0
$$815$$ −14.2462 −0.499023
$$816$$ 0 0
$$817$$ −28.4924 −0.996824
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −42.9848 −1.50018 −0.750091 0.661335i $$-0.769990\pi$$
−0.750091 + 0.661335i $$0.769990\pi$$
$$822$$ 0 0
$$823$$ 54.5464 1.90137 0.950684 0.310161i $$-0.100383\pi$$
0.950684 + 0.310161i $$0.100383\pi$$
$$824$$ 0 0
$$825$$ −12.0000 −0.417786
$$826$$ 0 0
$$827$$ −38.7386 −1.34707 −0.673537 0.739153i $$-0.735226\pi$$
−0.673537 + 0.739153i $$0.735226\pi$$
$$828$$ 0 0
$$829$$ −15.0691 −0.523373 −0.261686 0.965153i $$-0.584279\pi$$
−0.261686 + 0.965153i $$0.584279\pi$$
$$830$$ 0 0
$$831$$ 28.1080 0.975054
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −28.4924 −0.986021
$$836$$ 0 0
$$837$$ −30.9309 −1.06913
$$838$$ 0 0
$$839$$ −19.8078 −0.683840 −0.341920 0.939729i $$-0.611077\pi$$
−0.341920 + 0.939729i $$0.611077\pi$$
$$840$$ 0 0
$$841$$ −2.75379 −0.0949582
$$842$$ 0 0
$$843$$ −25.3693 −0.873766
$$844$$ 0 0
$$845$$ −47.1771 −1.62294
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −31.2311 −1.07185
$$850$$ 0 0
$$851$$ 18.4384 0.632062
$$852$$ 0 0
$$853$$ 46.4924 1.59187 0.795935 0.605382i $$-0.206980\pi$$
0.795935 + 0.605382i $$0.206980\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ 0 0
$$857$$ −30.1080 −1.02847 −0.514234 0.857650i $$-0.671924\pi$$
−0.514234 + 0.857650i $$0.671924\pi$$
$$858$$ 0 0
$$859$$ 30.0540 1.02543 0.512714 0.858559i $$-0.328640\pi$$
0.512714 + 0.858559i $$0.328640\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 36.4924 1.24222 0.621108 0.783725i $$-0.286683\pi$$
0.621108 + 0.783725i $$0.286683\pi$$
$$864$$ 0 0
$$865$$ 43.6155 1.48297
$$866$$ 0 0
$$867$$ 20.3002 0.689430
$$868$$ 0 0
$$869$$ 11.1231 0.377326
$$870$$ 0 0
$$871$$ −48.9848 −1.65979
$$872$$ 0 0
$$873$$ 8.73863 0.295758
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 55.3693 1.86969 0.934844 0.355057i $$-0.115539\pi$$
0.934844 + 0.355057i $$0.115539\pi$$
$$878$$ 0 0
$$879$$ −5.26137 −0.177461
$$880$$ 0 0
$$881$$ −34.3002 −1.15560 −0.577801 0.816177i $$-0.696089\pi$$
−0.577801 + 0.816177i $$0.696089\pi$$
$$882$$ 0 0
$$883$$ −8.49242 −0.285793 −0.142896 0.989738i $$-0.545642\pi$$
−0.142896 + 0.989738i $$0.545642\pi$$
$$884$$ 0 0
$$885$$ 43.4233 1.45966
$$886$$ 0 0
$$887$$ 31.6155 1.06155 0.530773 0.847514i $$-0.321902\pi$$
0.530773 + 0.847514i $$0.321902\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −7.00000 −0.234509
$$892$$ 0 0
$$893$$ −32.0000 −1.07084
$$894$$ 0 0
$$895$$ −22.9309 −0.766494
$$896$$ 0 0
$$897$$ 19.5076 0.651339
$$898$$ 0 0
$$899$$ 28.4924 0.950275
$$900$$ 0 0
$$901$$ −24.4924 −0.815961
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4.68466 −0.155723
$$906$$ 0 0
$$907$$ −16.4924 −0.547622 −0.273811 0.961784i $$-0.588284\pi$$
−0.273811 + 0.961784i $$0.588284\pi$$
$$908$$ 0 0
$$909$$ −1.12311 −0.0372511
$$910$$ 0 0
$$911$$ 26.7386 0.885890 0.442945 0.896549i $$-0.353934\pi$$
0.442945 + 0.896549i $$0.353934\pi$$
$$912$$ 0 0
$$913$$ 0.876894 0.0290210
$$914$$ 0 0
$$915$$ −6.24621 −0.206493
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −6.63068 −0.218726 −0.109363 0.994002i $$-0.534881\pi$$
−0.109363 + 0.994002i $$0.534881\pi$$
$$920$$ 0 0
$$921$$ 50.7386 1.67189
$$922$$ 0 0
$$923$$ 44.4924 1.46449
$$924$$ 0 0
$$925$$ −58.1080 −1.91058
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −46.4924 −1.52537 −0.762683 0.646772i $$-0.776119\pi$$
−0.762683 + 0.646772i $$0.776119\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −15.2311 −0.498642
$$934$$ 0 0
$$935$$ 7.12311 0.232950
$$936$$ 0 0
$$937$$ 42.1080 1.37561 0.687803 0.725897i $$-0.258575\pi$$
0.687803 + 0.725897i $$0.258575\pi$$
$$938$$ 0 0
$$939$$ −15.3153 −0.499797
$$940$$ 0 0
$$941$$ 32.2462 1.05120 0.525598 0.850733i $$-0.323842\pi$$
0.525598 + 0.850733i $$0.323842\pi$$
$$942$$ 0 0
$$943$$ −2.73863 −0.0891822
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.6847 0.412196 0.206098 0.978531i $$-0.433923\pi$$
0.206098 + 0.978531i $$0.433923\pi$$
$$948$$ 0 0
$$949$$ −26.2462 −0.851988
$$950$$ 0 0
$$951$$ 22.1619 0.718650
$$952$$ 0 0
$$953$$ 0.246211 0.00797556 0.00398778 0.999992i $$-0.498731\pi$$
0.00398778 + 0.999992i $$0.498731\pi$$
$$954$$ 0 0
$$955$$ −37.1771 −1.20302
$$956$$ 0 0
$$957$$ 8.00000 0.258603
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −0.0691303 −0.00223001
$$962$$ 0 0
$$963$$ −7.50758 −0.241928
$$964$$ 0 0
$$965$$ 32.4924 1.04597
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ −12.4924 −0.401314
$$970$$ 0 0
$$971$$ −34.5464 −1.10865 −0.554323 0.832301i $$-0.687023\pi$$
−0.554323 + 0.832301i $$0.687023\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −61.4773 −1.96885
$$976$$ 0 0
$$977$$ 53.8078 1.72146 0.860731 0.509059i $$-0.170007\pi$$
0.860731 + 0.509059i $$0.170007\pi$$
$$978$$ 0 0
$$979$$ −2.68466 −0.0858021
$$980$$ 0 0
$$981$$ −6.87689 −0.219562
$$982$$ 0 0
$$983$$ 30.9309 0.986542 0.493271 0.869876i $$-0.335801\pi$$
0.493271 + 0.869876i $$0.335801\pi$$
$$984$$ 0 0
$$985$$ 51.6155 1.64461
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −17.3693 −0.552312
$$990$$ 0 0
$$991$$ 4.49242 0.142707 0.0713533 0.997451i $$-0.477268\pi$$
0.0713533 + 0.997451i $$0.477268\pi$$
$$992$$ 0 0
$$993$$ 54.5464 1.73098
$$994$$ 0 0
$$995$$ 44.4924 1.41050
$$996$$ 0 0
$$997$$ −52.2462 −1.65465 −0.827327 0.561721i $$-0.810140\pi$$
−0.827327 + 0.561721i $$0.810140\pi$$
$$998$$ 0 0
$$999$$ −42.0540 −1.33053
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 8624.2.a.cb.1.1 2
4.3 odd 2 4312.2.a.n.1.2 2
7.6 odd 2 176.2.a.d.1.2 2
21.20 even 2 1584.2.a.t.1.1 2
28.27 even 2 88.2.a.b.1.1 2
35.13 even 4 4400.2.b.v.4049.3 4
35.27 even 4 4400.2.b.v.4049.2 4
35.34 odd 2 4400.2.a.bp.1.1 2
56.13 odd 2 704.2.a.p.1.1 2
56.27 even 2 704.2.a.m.1.2 2
77.76 even 2 1936.2.a.r.1.2 2
84.83 odd 2 792.2.a.h.1.1 2
112.13 odd 4 2816.2.c.p.1409.3 4
112.27 even 4 2816.2.c.w.1409.3 4
112.69 odd 4 2816.2.c.p.1409.2 4
112.83 even 4 2816.2.c.w.1409.2 4
140.27 odd 4 2200.2.b.g.1849.3 4
140.83 odd 4 2200.2.b.g.1849.2 4
140.139 even 2 2200.2.a.o.1.2 2
168.83 odd 2 6336.2.a.cu.1.2 2
168.125 even 2 6336.2.a.cx.1.2 2
308.27 even 10 968.2.i.r.729.1 8
308.83 odd 10 968.2.i.q.729.1 8
308.139 odd 10 968.2.i.q.753.2 8
308.167 odd 10 968.2.i.q.81.1 8
308.195 odd 10 968.2.i.q.9.2 8
308.223 even 10 968.2.i.r.9.2 8
308.251 even 10 968.2.i.r.81.1 8
308.279 even 10 968.2.i.r.753.2 8
308.307 odd 2 968.2.a.j.1.1 2
616.307 odd 2 7744.2.a.by.1.2 2
616.461 even 2 7744.2.a.cl.1.1 2
924.923 even 2 8712.2.a.bb.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.a.b.1.1 2 28.27 even 2
176.2.a.d.1.2 2 7.6 odd 2
704.2.a.m.1.2 2 56.27 even 2
704.2.a.p.1.1 2 56.13 odd 2
792.2.a.h.1.1 2 84.83 odd 2
968.2.a.j.1.1 2 308.307 odd 2
968.2.i.q.9.2 8 308.195 odd 10
968.2.i.q.81.1 8 308.167 odd 10
968.2.i.q.729.1 8 308.83 odd 10
968.2.i.q.753.2 8 308.139 odd 10
968.2.i.r.9.2 8 308.223 even 10
968.2.i.r.81.1 8 308.251 even 10
968.2.i.r.729.1 8 308.27 even 10
968.2.i.r.753.2 8 308.279 even 10
1584.2.a.t.1.1 2 21.20 even 2
1936.2.a.r.1.2 2 77.76 even 2
2200.2.a.o.1.2 2 140.139 even 2
2200.2.b.g.1849.2 4 140.83 odd 4
2200.2.b.g.1849.3 4 140.27 odd 4
2816.2.c.p.1409.2 4 112.69 odd 4
2816.2.c.p.1409.3 4 112.13 odd 4
2816.2.c.w.1409.2 4 112.83 even 4
2816.2.c.w.1409.3 4 112.27 even 4
4312.2.a.n.1.2 2 4.3 odd 2
4400.2.a.bp.1.1 2 35.34 odd 2
4400.2.b.v.4049.2 4 35.27 even 4
4400.2.b.v.4049.3 4 35.13 even 4
6336.2.a.cu.1.2 2 168.83 odd 2
6336.2.a.cx.1.2 2 168.125 even 2
7744.2.a.by.1.2 2 616.307 odd 2
7744.2.a.cl.1.1 2 616.461 even 2
8624.2.a.cb.1.1 2 1.1 even 1 trivial
8712.2.a.bb.1.1 2 924.923 even 2