# Properties

 Label 704.2.a.p.1.1 Level $704$ Weight $2$ Character 704.1 Self dual yes Analytic conductor $5.621$ 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] = [704,2,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$704 = 2^{6} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 704.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$5.62146830230$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 704.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.56155 q^{3} -3.56155 q^{5} -3.12311 q^{7} -0.561553 q^{9} +O(q^{10})$$ $$q-1.56155 q^{3} -3.56155 q^{5} -3.12311 q^{7} -0.561553 q^{9} -1.00000 q^{11} +5.12311 q^{13} +5.56155 q^{15} +2.00000 q^{17} -4.00000 q^{19} +4.87689 q^{21} -2.43845 q^{23} +7.68466 q^{25} +5.56155 q^{27} +5.12311 q^{29} +5.56155 q^{31} +1.56155 q^{33} +11.1231 q^{35} +7.56155 q^{37} -8.00000 q^{39} -1.12311 q^{41} -7.12311 q^{43} +2.00000 q^{45} -8.00000 q^{47} +2.75379 q^{49} -3.12311 q^{51} -12.2462 q^{53} +3.56155 q^{55} +6.24621 q^{57} +7.80776 q^{59} -1.12311 q^{61} +1.75379 q^{63} -18.2462 q^{65} +9.56155 q^{67} +3.80776 q^{69} +8.68466 q^{71} +5.12311 q^{73} -12.0000 q^{75} +3.12311 q^{77} +11.1231 q^{79} -7.00000 q^{81} +0.876894 q^{83} -7.12311 q^{85} -8.00000 q^{87} +2.68466 q^{89} -16.0000 q^{91} -8.68466 q^{93} +14.2462 q^{95} +15.5616 q^{97} +0.561553 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 $$2 q + q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 7 q^{15} + 4 q^{17} - 8 q^{19} + 18 q^{21} - 9 q^{23} + 3 q^{25} + 7 q^{27} + 2 q^{29} + 7 q^{31} - q^{33} + 14 q^{35} + 11 q^{37} - 16 q^{39} + 6 q^{41} - 6 q^{43} + 4 q^{45} - 16 q^{47} + 22 q^{49} + 2 q^{51} - 8 q^{53} + 3 q^{55} - 4 q^{57} - 5 q^{59} + 6 q^{61} + 20 q^{63} - 20 q^{65} + 15 q^{67} - 13 q^{69} + 5 q^{71} + 2 q^{73} - 24 q^{75} - 2 q^{77} + 14 q^{79} - 14 q^{81} + 10 q^{83} - 6 q^{85} - 16 q^{87} - 7 q^{89} - 32 q^{91} - 5 q^{93} + 12 q^{95} + 27 q^{97} - 3 q^{99}+O(q^{100})$$ 2 * q + q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 7 * q^15 + 4 * q^17 - 8 * q^19 + 18 * q^21 - 9 * q^23 + 3 * q^25 + 7 * q^27 + 2 * q^29 + 7 * q^31 - q^33 + 14 * q^35 + 11 * q^37 - 16 * q^39 + 6 * q^41 - 6 * q^43 + 4 * q^45 - 16 * q^47 + 22 * q^49 + 2 * q^51 - 8 * q^53 + 3 * q^55 - 4 * q^57 - 5 * q^59 + 6 * q^61 + 20 * q^63 - 20 * q^65 + 15 * q^67 - 13 * q^69 + 5 * q^71 + 2 * q^73 - 24 * q^75 - 2 * q^77 + 14 * q^79 - 14 * q^81 + 10 * q^83 - 6 * q^85 - 16 * q^87 - 7 * q^89 - 32 * q^91 - 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
$$7$$ −3.12311 −1.18042 −0.590211 0.807249i $$-0.700956\pi$$
−0.590211 + 0.807249i $$0.700956\pi$$
$$8$$ 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 $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 4.87689 1.06423
$$22$$ 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 $$-0.342206\pi$$
0.475668 + 0.879625i $$0.342206\pi$$
$$30$$ 0 0
$$31$$ 5.56155 0.998884 0.499442 0.866347i $$-0.333538\pi$$
0.499442 + 0.866347i $$0.333538\pi$$
$$32$$ 0 0
$$33$$ 1.56155 0.271831
$$34$$ 0 0
$$35$$ 11.1231 1.88015
$$36$$ 0 0
$$37$$ 7.56155 1.24311 0.621556 0.783370i $$-0.286501\pi$$
0.621556 + 0.783370i $$0.286501\pi$$
$$38$$ 0 0
$$39$$ −8.00000 −1.28103
$$40$$ 0 0
$$41$$ −1.12311 −0.175400 −0.0876998 0.996147i $$-0.527952\pi$$
−0.0876998 + 0.996147i $$0.527952\pi$$
$$42$$ 0 0
$$43$$ −7.12311 −1.08626 −0.543132 0.839648i $$-0.682762\pi$$
−0.543132 + 0.839648i $$0.682762\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 2.75379 0.393398
$$50$$ 0 0
$$51$$ −3.12311 −0.437322
$$52$$ 0 0
$$53$$ −12.2462 −1.68215 −0.841073 0.540921i $$-0.818076\pi$$
−0.841073 + 0.540921i $$0.818076\pi$$
$$54$$ 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
$$63$$ 1.75379 0.220957
$$64$$ 0 0
$$65$$ −18.2462 −2.26316
$$66$$ 0 0
$$67$$ 9.56155 1.16813 0.584065 0.811707i $$-0.301461\pi$$
0.584065 + 0.811707i $$0.301461\pi$$
$$68$$ 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 $$-0.403078\pi$$
0.299807 + 0.954000i $$0.403078\pi$$
$$74$$ 0 0
$$75$$ −12.0000 −1.38564
$$76$$ 0 0
$$77$$ 3.12311 0.355911
$$78$$ 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 $$-0.454555\pi$$
0.142287 + 0.989825i $$0.454555\pi$$
$$90$$ 0 0
$$91$$ −16.0000 −1.67726
$$92$$ 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 $$-0.210071\pi$$
0.790018 + 0.613083i $$0.210071\pi$$
$$98$$ 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
$$105$$ −17.3693 −1.69507
$$106$$ 0 0
$$107$$ −13.3693 −1.29246 −0.646230 0.763142i $$-0.723655\pi$$
−0.646230 + 0.763142i $$0.723655\pi$$
$$108$$ 0 0
$$109$$ −12.2462 −1.17297 −0.586487 0.809959i $$-0.699490\pi$$
−0.586487 + 0.809959i $$0.699490\pi$$
$$110$$ 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
$$119$$ −6.24621 −0.572589
$$120$$ 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
$$133$$ 12.4924 1.08323
$$134$$ 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
$$147$$ −4.30019 −0.354673
$$148$$ 0 0
$$149$$ −4.24621 −0.347863 −0.173932 0.984758i $$-0.555647\pi$$
−0.173932 + 0.984758i $$0.555647\pi$$
$$150$$ 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
$$161$$ 7.61553 0.600188
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ −5.56155 −0.432966
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 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
$$175$$ −24.0000 −1.81423
$$176$$ 0 0
$$177$$ −12.1922 −0.916425
$$178$$ 0 0
$$179$$ −6.43845 −0.481232 −0.240616 0.970620i $$-0.577349\pi$$
−0.240616 + 0.970620i $$0.577349\pi$$
$$180$$ 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
$$189$$ −17.3693 −1.26343
$$190$$ 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 $$-0.327320\pi$$
0.516271 + 0.856425i $$0.327320\pi$$
$$198$$ 0 0
$$199$$ 12.4924 0.885564 0.442782 0.896629i $$-0.353991\pi$$
0.442782 + 0.896629i $$0.353991\pi$$
$$200$$ 0 0
$$201$$ −14.9309 −1.05314
$$202$$ 0 0
$$203$$ −16.0000 −1.12298
$$204$$ 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 $$-0.405572\pi$$
0.292321 + 0.956320i $$0.405572\pi$$
$$212$$ 0 0
$$213$$ −13.5616 −0.929222
$$214$$ 0 0
$$215$$ 25.3693 1.73017
$$216$$ 0 0
$$217$$ −17.3693 −1.17911
$$218$$ 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 $$-0.370622\pi$$
0.395353 + 0.918529i $$0.370622\pi$$
$$224$$ 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
$$231$$ −4.87689 −0.320876
$$232$$ 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 $$-0.112683\pi$$
0.937992 + 0.346657i $$0.112683\pi$$
$$242$$ 0 0
$$243$$ −5.75379 −0.369106
$$244$$ 0 0
$$245$$ −9.80776 −0.626595
$$246$$ 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 $$-0.380525\pi$$
0.366591 + 0.930382i $$0.380525\pi$$
$$258$$ 0 0
$$259$$ −23.6155 −1.46740
$$260$$ 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 $$-0.167067\pi$$
0.865396 + 0.501089i $$0.167067\pi$$
$$272$$ 0 0
$$273$$ 24.9848 1.51215
$$274$$ 0 0
$$275$$ −7.68466 −0.463402
$$276$$ 0 0
$$277$$ 18.0000 1.08152 0.540758 0.841178i $$-0.318138\pi$$
0.540758 + 0.841178i $$0.318138\pi$$
$$278$$ 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
$$287$$ 3.50758 0.207046
$$288$$ 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
$$301$$ 22.2462 1.28225
$$302$$ 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 $$-0.589189\pi$$
−0.276543 + 0.961001i $$0.589189\pi$$
$$312$$ 0 0
$$313$$ −9.80776 −0.554368 −0.277184 0.960817i $$-0.589401\pi$$
−0.277184 + 0.960817i $$0.589401\pi$$
$$314$$ 0 0
$$315$$ −6.24621 −0.351934
$$316$$ 0 0
$$317$$ 14.1922 0.797115 0.398558 0.917143i $$-0.369511\pi$$
0.398558 + 0.917143i $$0.369511\pi$$
$$318$$ 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
$$329$$ 24.9848 1.37746
$$330$$ 0 0
$$331$$ 34.9309 1.91997 0.959987 0.280044i $$-0.0903491\pi$$
0.959987 + 0.280044i $$0.0903491\pi$$
$$332$$ 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
$$343$$ 13.2614 0.716046
$$344$$ 0 0
$$345$$ −13.5616 −0.730129
$$346$$ 0 0
$$347$$ 22.7386 1.22067 0.610337 0.792142i $$-0.291034\pi$$
0.610337 + 0.792142i $$0.291034\pi$$
$$348$$ 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 $$-0.721120\pi$$
−0.640132 + 0.768265i $$0.721120\pi$$
$$354$$ 0 0
$$355$$ −30.9309 −1.64164
$$356$$ 0 0
$$357$$ 9.75379 0.516225
$$358$$ 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 $$-0.704233\pi$$
−0.598491 + 0.801130i $$0.704233\pi$$
$$368$$ 0 0
$$369$$ 0.630683 0.0328321
$$370$$ 0 0
$$371$$ 38.2462 1.98564
$$372$$ 0 0
$$373$$ 8.24621 0.426973 0.213486 0.976946i $$-0.431518\pi$$
0.213486 + 0.976946i $$0.431518\pi$$
$$374$$ 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 $$-0.498428\pi$$
0.00493725 + 0.999988i $$0.498428\pi$$
$$380$$ 0 0
$$381$$ 9.75379 0.499702
$$382$$ 0 0
$$383$$ 2.05398 0.104953 0.0524766 0.998622i $$-0.483288\pi$$
0.0524766 + 0.998622i $$0.483288\pi$$
$$384$$ 0 0
$$385$$ −11.1231 −0.566886
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ 0 0
$$389$$ −3.56155 −0.180578 −0.0902889 0.995916i $$-0.528779\pi$$
−0.0902889 + 0.995916i $$0.528779\pi$$
$$390$$ 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
$$399$$ −19.5076 −0.976600
$$400$$ 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 $$-0.312301\pi$$
0.556089 + 0.831123i $$0.312301\pi$$
$$410$$ 0 0
$$411$$ 13.1771 0.649977
$$412$$ 0 0
$$413$$ −24.3845 −1.19988
$$414$$ 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 $$-0.519345\pi$$
−0.0607366 + 0.998154i $$0.519345\pi$$
$$422$$ 0 0
$$423$$ 4.49242 0.218429
$$424$$ 0 0
$$425$$ 15.3693 0.745521
$$426$$ 0 0
$$427$$ 3.50758 0.169744
$$428$$ 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 $$-0.683500\pi$$
−0.545078 + 0.838386i $$0.683500\pi$$
$$434$$ 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 $$-0.534190\pi$$
−0.107206 + 0.994237i $$0.534190\pi$$
$$440$$ 0 0
$$441$$ −1.54640 −0.0736380
$$442$$ 0 0
$$443$$ 11.3153 0.537608 0.268804 0.963195i $$-0.413372\pi$$
0.268804 + 0.963195i $$0.413372\pi$$
$$444$$ 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
$$455$$ 56.9848 2.67149
$$456$$ 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
$$469$$ −29.8617 −1.37889
$$470$$ 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 $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 38.7386 1.76633
$$482$$ 0 0
$$483$$ −11.8920 −0.541107
$$484$$ 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 $$-0.399554\pi$$
0.310350 + 0.950622i $$0.399554\pi$$
$$492$$ 0 0
$$493$$ 10.2462 0.461466
$$494$$ 0 0
$$495$$ −2.00000 −0.0898933
$$496$$ 0 0
$$497$$ −27.1231 −1.21664
$$498$$ 0 0
$$499$$ −28.9848 −1.29754 −0.648770 0.760985i $$-0.724716\pi$$
−0.648770 + 0.760985i $$0.724716\pi$$
$$500$$ 0 0
$$501$$ 12.4924 0.558120
$$502$$ 0 0
$$503$$ 31.6155 1.40967 0.704833 0.709373i $$-0.251022\pi$$
0.704833 + 0.709373i $$0.251022\pi$$
$$504$$ 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
$$511$$ −16.0000 −0.707798
$$512$$ 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 $$-0.490827\pi$$
0.0288131 + 0.999585i $$0.490827\pi$$
$$522$$ 0 0
$$523$$ −12.0000 −0.524723 −0.262362 0.964970i $$-0.584501\pi$$
−0.262362 + 0.964970i $$0.584501\pi$$
$$524$$ 0 0
$$525$$ 37.4773 1.63564
$$526$$ 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
$$539$$ −2.75379 −0.118614
$$540$$ 0 0
$$541$$ 23.8617 1.02590 0.512948 0.858420i $$-0.328553\pi$$
0.512948 + 0.858420i $$0.328553\pi$$
$$542$$ 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 $$-0.858758\pi$$
−0.903159 + 0.429307i $$0.858758\pi$$
$$548$$ 0 0
$$549$$ 0.630683 0.0269169
$$550$$ 0 0
$$551$$ −20.4924 −0.873007
$$552$$ 0 0
$$553$$ −34.7386 −1.47724
$$554$$ 0 0
$$555$$ 42.0540 1.78509
$$556$$ 0 0
$$557$$ 3.75379 0.159053 0.0795266 0.996833i $$-0.474659\pi$$
0.0795266 + 0.996833i $$0.474659\pi$$
$$558$$ 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
$$567$$ 21.8617 0.918107
$$568$$ 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 $$-0.387847\pi$$
0.345093 + 0.938568i $$0.387847\pi$$
$$572$$ 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 $$-0.395000\pi$$
0.323918 + 0.946085i $$0.395000\pi$$
$$578$$ 0 0
$$579$$ 14.2462 0.592052
$$580$$ 0 0
$$581$$ −2.73863 −0.113618
$$582$$ 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 $$-0.477962\pi$$
0.0691806 + 0.997604i $$0.477962\pi$$
$$594$$ 0 0
$$595$$ 22.2462 0.912006
$$596$$ 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 $$-0.475606\pi$$
0.0765601 + 0.997065i $$0.475606\pi$$
$$602$$ 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 $$-0.119168\pi$$
0.930735 + 0.365694i $$0.119168\pi$$
$$608$$ 0 0
$$609$$ 24.9848 1.01244
$$610$$ 0 0
$$611$$ −40.9848 −1.65807
$$612$$ 0 0
$$613$$ −11.8617 −0.479091 −0.239546 0.970885i $$-0.576998\pi$$
−0.239546 + 0.970885i $$0.576998\pi$$
$$614$$ 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
$$623$$ −8.38447 −0.335917
$$624$$ 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
$$637$$ 14.1080 0.558977
$$638$$ 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 $$-0.585894\pi$$
−0.266580 + 0.963813i $$0.585894\pi$$
$$648$$ 0 0
$$649$$ −7.80776 −0.306482
$$650$$ 0 0
$$651$$ 27.1231 1.06304
$$652$$ 0 0
$$653$$ −35.1771 −1.37659 −0.688293 0.725433i $$-0.741640\pi$$
−0.688293 + 0.725433i $$0.741640\pi$$
$$654$$ 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 $$-0.572643\pi$$
−0.226238 + 0.974072i $$0.572643\pi$$
$$660$$ 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
$$665$$ −44.4924 −1.72534
$$666$$ 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
$$679$$ −48.6004 −1.86511
$$680$$ 0 0
$$681$$ −36.1080 −1.38366
$$682$$ 0 0
$$683$$ 6.73863 0.257847 0.128923 0.991655i $$-0.458848\pi$$
0.128923 + 0.991655i $$0.458848\pi$$
$$684$$ 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
$$693$$ −1.75379 −0.0666209
$$694$$ 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 $$-0.902587\pi$$
−0.953536 + 0.301278i $$0.902587\pi$$
$$702$$ 0 0
$$703$$ −30.2462 −1.14076
$$704$$ 0 0
$$705$$ −44.4924 −1.67568
$$706$$ 0 0
$$707$$ −6.24621 −0.234913
$$708$$ 0 0
$$709$$ −2.19224 −0.0823311 −0.0411656 0.999152i $$-0.513107\pi$$
−0.0411656 + 0.999152i $$0.513107\pi$$
$$710$$ 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 $$-0.729670\pi$$
−0.660533 + 0.750797i $$0.729670\pi$$
$$720$$ 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 $$-0.357681\pi$$
0.432359 + 0.901702i $$0.357681\pi$$
$$728$$ 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
$$735$$ 15.3153 0.564915
$$736$$ 0 0
$$737$$ −9.56155 −0.352204
$$738$$ 0 0
$$739$$ −2.63068 −0.0967712 −0.0483856 0.998829i $$-0.515408\pi$$
−0.0483856 + 0.998829i $$0.515408\pi$$
$$740$$ 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
$$749$$ 41.7538 1.52565
$$750$$ 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 $$-0.592422\pi$$
−0.286291 + 0.958143i $$0.592422\pi$$
$$758$$ 0 0
$$759$$ −3.80776 −0.138213
$$760$$ 0 0
$$761$$ 5.12311 0.185712 0.0928562 0.995680i $$-0.470400\pi$$
0.0928562 + 0.995680i $$0.470400\pi$$
$$762$$ 0 0
$$763$$ 38.2462 1.38461
$$764$$ 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 $$-0.347182\pi$$
0.461860 + 0.886953i $$0.347182\pi$$
$$770$$ 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
$$777$$ 36.8769 1.32295
$$778$$ 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
$$791$$ 1.36932 0.0486873
$$792$$ 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
$$805$$ −27.1231 −0.955964
$$806$$ 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
$$819$$ 8.98485 0.313956
$$820$$ 0 0
$$821$$ 42.9848 1.50018 0.750091 0.661335i $$-0.230010\pi$$
0.750091 + 0.661335i $$0.230010\pi$$
$$822$$ 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 $$-0.264774\pi$$
0.673537 + 0.739153i $$0.264774\pi$$
$$828$$ 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
$$833$$ 5.50758 0.190826
$$834$$ 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 $$-0.388923\pi$$
0.341920 + 0.939729i $$0.388923\pi$$
$$840$$ 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
$$847$$ −3.12311 −0.107311
$$848$$ 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 $$-0.328076\pi$$
0.514234 + 0.857650i $$0.328076\pi$$
$$858$$ 0 0
$$859$$ 30.0540 1.02543 0.512714 0.858559i $$-0.328640\pi$$
0.512714 + 0.858559i $$0.328640\pi$$
$$860$$ 0 0
$$861$$ −5.47727 −0.186665
$$862$$ 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
$$875$$ 29.8617 1.00951
$$876$$ 0 0
$$877$$ −55.3693 −1.86969 −0.934844 0.355057i $$-0.884461\pi$$
−0.934844 + 0.355057i $$0.884461\pi$$
$$878$$ 0 0
$$879$$ −5.26137 −0.177461
$$880$$ 0 0
$$881$$ 34.3002 1.15560 0.577801 0.816177i $$-0.303911\pi$$
0.577801 + 0.816177i $$0.303911\pi$$
$$882$$ 0 0
$$883$$ 8.49242 0.285793 0.142896 0.989738i $$-0.454358\pi$$
0.142896 + 0.989738i $$0.454358\pi$$
$$884$$ 0 0
$$885$$ 43.4233 1.45966
$$886$$ 0 0
$$887$$ −31.6155 −1.06155 −0.530773 0.847514i $$-0.678098\pi$$
−0.530773 + 0.847514i $$0.678098\pi$$
$$888$$ 0 0
$$889$$ 19.5076 0.654263
$$890$$ 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
$$903$$ −34.7386 −1.15603
$$904$$ 0 0
$$905$$ −4.68466 −0.155723
$$906$$ 0 0
$$907$$ 16.4924 0.547622 0.273811 0.961784i $$-0.411716\pi$$
0.273811 + 0.961784i $$0.411716\pi$$
$$908$$ 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
$$917$$ −41.7538 −1.37883
$$918$$ 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 $$-0.223881\pi$$
0.762683 + 0.646772i $$0.223881\pi$$
$$930$$ 0 0
$$931$$ −11.0152 −0.361007
$$932$$ 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 $$-0.741425\pi$$
−0.687803 + 0.725897i $$0.741425\pi$$
$$938$$ 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
$$945$$ 61.8617 2.01236
$$946$$ 0 0
$$947$$ −12.6847 −0.412196 −0.206098 0.978531i $$-0.566077\pi$$
−0.206098 + 0.978531i $$0.566077\pi$$
$$948$$ 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
$$959$$ 26.3542 0.851020
$$960$$ 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
$$973$$ −47.2311 −1.51416
$$974$$ 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 $$-0.664199\pi$$
−0.493271 + 0.869876i $$0.664199\pi$$
$$984$$ 0 0
$$985$$ −51.6155 −1.64461
$$986$$ 0 0
$$987$$ −39.0152 −1.24187
$$988$$ 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 704.2.a.p.1.1 2
3.2 odd 2 6336.2.a.cx.1.2 2
4.3 odd 2 704.2.a.m.1.2 2
8.3 odd 2 88.2.a.b.1.1 2
8.5 even 2 176.2.a.d.1.2 2
11.10 odd 2 7744.2.a.cl.1.1 2
12.11 even 2 6336.2.a.cu.1.2 2
16.3 odd 4 2816.2.c.w.1409.3 4
16.5 even 4 2816.2.c.p.1409.3 4
16.11 odd 4 2816.2.c.w.1409.2 4
16.13 even 4 2816.2.c.p.1409.2 4
24.5 odd 2 1584.2.a.t.1.1 2
24.11 even 2 792.2.a.h.1.1 2
40.3 even 4 2200.2.b.g.1849.2 4
40.13 odd 4 4400.2.b.v.4049.3 4
40.19 odd 2 2200.2.a.o.1.2 2
40.27 even 4 2200.2.b.g.1849.3 4
40.29 even 2 4400.2.a.bp.1.1 2
40.37 odd 4 4400.2.b.v.4049.2 4
44.43 even 2 7744.2.a.by.1.2 2
56.13 odd 2 8624.2.a.cb.1.1 2
56.27 even 2 4312.2.a.n.1.2 2
88.3 odd 10 968.2.i.r.9.2 8
88.19 even 10 968.2.i.q.9.2 8
88.21 odd 2 1936.2.a.r.1.2 2
88.27 odd 10 968.2.i.r.729.1 8
88.35 even 10 968.2.i.q.81.1 8
88.43 even 2 968.2.a.j.1.1 2
88.51 even 10 968.2.i.q.753.2 8
88.59 odd 10 968.2.i.r.753.2 8
88.75 odd 10 968.2.i.r.81.1 8
88.83 even 10 968.2.i.q.729.1 8
264.131 odd 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 8.3 odd 2
176.2.a.d.1.2 2 8.5 even 2
704.2.a.m.1.2 2 4.3 odd 2
704.2.a.p.1.1 2 1.1 even 1 trivial
792.2.a.h.1.1 2 24.11 even 2
968.2.a.j.1.1 2 88.43 even 2
968.2.i.q.9.2 8 88.19 even 10
968.2.i.q.81.1 8 88.35 even 10
968.2.i.q.729.1 8 88.83 even 10
968.2.i.q.753.2 8 88.51 even 10
968.2.i.r.9.2 8 88.3 odd 10
968.2.i.r.81.1 8 88.75 odd 10
968.2.i.r.729.1 8 88.27 odd 10
968.2.i.r.753.2 8 88.59 odd 10
1584.2.a.t.1.1 2 24.5 odd 2
1936.2.a.r.1.2 2 88.21 odd 2
2200.2.a.o.1.2 2 40.19 odd 2
2200.2.b.g.1849.2 4 40.3 even 4
2200.2.b.g.1849.3 4 40.27 even 4
2816.2.c.p.1409.2 4 16.13 even 4
2816.2.c.p.1409.3 4 16.5 even 4
2816.2.c.w.1409.2 4 16.11 odd 4
2816.2.c.w.1409.3 4 16.3 odd 4
4312.2.a.n.1.2 2 56.27 even 2
4400.2.a.bp.1.1 2 40.29 even 2
4400.2.b.v.4049.2 4 40.37 odd 4
4400.2.b.v.4049.3 4 40.13 odd 4
6336.2.a.cu.1.2 2 12.11 even 2
6336.2.a.cx.1.2 2 3.2 odd 2
7744.2.a.by.1.2 2 44.43 even 2
7744.2.a.cl.1.1 2 11.10 odd 2
8624.2.a.cb.1.1 2 56.13 odd 2
8712.2.a.bb.1.1 2 264.131 odd 2