# Properties

 Label 18.3.b.a.17.2 Level $18$ Weight $3$ Character 18.17 Analytic conductor $0.490$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,3,Mod(17,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("18.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 18.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.490464475849$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 17.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 18.17 Dual form 18.3.b.a.17.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} -4.00000 q^{7} -2.82843i q^{8} +O(q^{10})$$ $$q+1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} -4.00000 q^{7} -2.82843i q^{8} +6.00000 q^{10} +16.9706i q^{11} +8.00000 q^{13} -5.65685i q^{14} +4.00000 q^{16} -12.7279i q^{17} -16.0000 q^{19} +8.48528i q^{20} -24.0000 q^{22} -16.9706i q^{23} +7.00000 q^{25} +11.3137i q^{26} +8.00000 q^{28} +4.24264i q^{29} +44.0000 q^{31} +5.65685i q^{32} +18.0000 q^{34} +16.9706i q^{35} -34.0000 q^{37} -22.6274i q^{38} -12.0000 q^{40} +46.6690i q^{41} -40.0000 q^{43} -33.9411i q^{44} +24.0000 q^{46} -84.8528i q^{47} -33.0000 q^{49} +9.89949i q^{50} -16.0000 q^{52} +38.1838i q^{53} +72.0000 q^{55} +11.3137i q^{56} -6.00000 q^{58} +33.9411i q^{59} +50.0000 q^{61} +62.2254i q^{62} -8.00000 q^{64} -33.9411i q^{65} +8.00000 q^{67} +25.4558i q^{68} -24.0000 q^{70} -50.9117i q^{71} -16.0000 q^{73} -48.0833i q^{74} +32.0000 q^{76} -67.8823i q^{77} -76.0000 q^{79} -16.9706i q^{80} -66.0000 q^{82} +118.794i q^{83} -54.0000 q^{85} -56.5685i q^{86} +48.0000 q^{88} +12.7279i q^{89} -32.0000 q^{91} +33.9411i q^{92} +120.000 q^{94} +67.8823i q^{95} +176.000 q^{97} -46.6690i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 8 q^{7}+O(q^{10})$$ 2 * q - 4 * q^4 - 8 * q^7 $$2 q - 4 q^{4} - 8 q^{7} + 12 q^{10} + 16 q^{13} + 8 q^{16} - 32 q^{19} - 48 q^{22} + 14 q^{25} + 16 q^{28} + 88 q^{31} + 36 q^{34} - 68 q^{37} - 24 q^{40} - 80 q^{43} + 48 q^{46} - 66 q^{49} - 32 q^{52} + 144 q^{55} - 12 q^{58} + 100 q^{61} - 16 q^{64} + 16 q^{67} - 48 q^{70} - 32 q^{73} + 64 q^{76} - 152 q^{79} - 132 q^{82} - 108 q^{85} + 96 q^{88} - 64 q^{91} + 240 q^{94} + 352 q^{97}+O(q^{100})$$ 2 * q - 4 * q^4 - 8 * q^7 + 12 * q^10 + 16 * q^13 + 8 * q^16 - 32 * q^19 - 48 * q^22 + 14 * q^25 + 16 * q^28 + 88 * q^31 + 36 * q^34 - 68 * q^37 - 24 * q^40 - 80 * q^43 + 48 * q^46 - 66 * q^49 - 32 * q^52 + 144 * q^55 - 12 * q^58 + 100 * q^61 - 16 * q^64 + 16 * q^67 - 48 * q^70 - 32 * q^73 + 64 * q^76 - 152 * q^79 - 132 * q^82 - 108 * q^85 + 96 * q^88 - 64 * q^91 + 240 * q^94 + 352 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$\chi(n)$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.41421i 0.707107i
$$3$$ 0 0
$$4$$ −2.00000 −0.500000
$$5$$ − 4.24264i − 0.848528i −0.905539 0.424264i $$-0.860533\pi$$
0.905539 0.424264i $$-0.139467\pi$$
$$6$$ 0 0
$$7$$ −4.00000 −0.571429 −0.285714 0.958315i $$-0.592231\pi$$
−0.285714 + 0.958315i $$0.592231\pi$$
$$8$$ − 2.82843i − 0.353553i
$$9$$ 0 0
$$10$$ 6.00000 0.600000
$$11$$ 16.9706i 1.54278i 0.636364 + 0.771389i $$0.280438\pi$$
−0.636364 + 0.771389i $$0.719562\pi$$
$$12$$ 0 0
$$13$$ 8.00000 0.615385 0.307692 0.951486i $$-0.400443\pi$$
0.307692 + 0.951486i $$0.400443\pi$$
$$14$$ − 5.65685i − 0.404061i
$$15$$ 0 0
$$16$$ 4.00000 0.250000
$$17$$ − 12.7279i − 0.748701i −0.927287 0.374351i $$-0.877866\pi$$
0.927287 0.374351i $$-0.122134\pi$$
$$18$$ 0 0
$$19$$ −16.0000 −0.842105 −0.421053 0.907036i $$-0.638339\pi$$
−0.421053 + 0.907036i $$0.638339\pi$$
$$20$$ 8.48528i 0.424264i
$$21$$ 0 0
$$22$$ −24.0000 −1.09091
$$23$$ − 16.9706i − 0.737851i −0.929459 0.368925i $$-0.879726\pi$$
0.929459 0.368925i $$-0.120274\pi$$
$$24$$ 0 0
$$25$$ 7.00000 0.280000
$$26$$ 11.3137i 0.435143i
$$27$$ 0 0
$$28$$ 8.00000 0.285714
$$29$$ 4.24264i 0.146298i 0.997321 + 0.0731490i $$0.0233049\pi$$
−0.997321 + 0.0731490i $$0.976695\pi$$
$$30$$ 0 0
$$31$$ 44.0000 1.41935 0.709677 0.704527i $$-0.248841\pi$$
0.709677 + 0.704527i $$0.248841\pi$$
$$32$$ 5.65685i 0.176777i
$$33$$ 0 0
$$34$$ 18.0000 0.529412
$$35$$ 16.9706i 0.484873i
$$36$$ 0 0
$$37$$ −34.0000 −0.918919 −0.459459 0.888199i $$-0.651957\pi$$
−0.459459 + 0.888199i $$0.651957\pi$$
$$38$$ − 22.6274i − 0.595458i
$$39$$ 0 0
$$40$$ −12.0000 −0.300000
$$41$$ 46.6690i 1.13827i 0.822244 + 0.569135i $$0.192722\pi$$
−0.822244 + 0.569135i $$0.807278\pi$$
$$42$$ 0 0
$$43$$ −40.0000 −0.930233 −0.465116 0.885250i $$-0.653987\pi$$
−0.465116 + 0.885250i $$0.653987\pi$$
$$44$$ − 33.9411i − 0.771389i
$$45$$ 0 0
$$46$$ 24.0000 0.521739
$$47$$ − 84.8528i − 1.80538i −0.430293 0.902690i $$-0.641590\pi$$
0.430293 0.902690i $$-0.358410\pi$$
$$48$$ 0 0
$$49$$ −33.0000 −0.673469
$$50$$ 9.89949i 0.197990i
$$51$$ 0 0
$$52$$ −16.0000 −0.307692
$$53$$ 38.1838i 0.720448i 0.932866 + 0.360224i $$0.117300\pi$$
−0.932866 + 0.360224i $$0.882700\pi$$
$$54$$ 0 0
$$55$$ 72.0000 1.30909
$$56$$ 11.3137i 0.202031i
$$57$$ 0 0
$$58$$ −6.00000 −0.103448
$$59$$ 33.9411i 0.575273i 0.957740 + 0.287637i $$0.0928695\pi$$
−0.957740 + 0.287637i $$0.907130\pi$$
$$60$$ 0 0
$$61$$ 50.0000 0.819672 0.409836 0.912159i $$-0.365586\pi$$
0.409836 + 0.912159i $$0.365586\pi$$
$$62$$ 62.2254i 1.00364i
$$63$$ 0 0
$$64$$ −8.00000 −0.125000
$$65$$ − 33.9411i − 0.522171i
$$66$$ 0 0
$$67$$ 8.00000 0.119403 0.0597015 0.998216i $$-0.480985\pi$$
0.0597015 + 0.998216i $$0.480985\pi$$
$$68$$ 25.4558i 0.374351i
$$69$$ 0 0
$$70$$ −24.0000 −0.342857
$$71$$ − 50.9117i − 0.717066i −0.933517 0.358533i $$-0.883277\pi$$
0.933517 0.358533i $$-0.116723\pi$$
$$72$$ 0 0
$$73$$ −16.0000 −0.219178 −0.109589 0.993977i $$-0.534953\pi$$
−0.109589 + 0.993977i $$0.534953\pi$$
$$74$$ − 48.0833i − 0.649774i
$$75$$ 0 0
$$76$$ 32.0000 0.421053
$$77$$ − 67.8823i − 0.881588i
$$78$$ 0 0
$$79$$ −76.0000 −0.962025 −0.481013 0.876714i $$-0.659731\pi$$
−0.481013 + 0.876714i $$0.659731\pi$$
$$80$$ − 16.9706i − 0.212132i
$$81$$ 0 0
$$82$$ −66.0000 −0.804878
$$83$$ 118.794i 1.43125i 0.698484 + 0.715626i $$0.253859\pi$$
−0.698484 + 0.715626i $$0.746141\pi$$
$$84$$ 0 0
$$85$$ −54.0000 −0.635294
$$86$$ − 56.5685i − 0.657774i
$$87$$ 0 0
$$88$$ 48.0000 0.545455
$$89$$ 12.7279i 0.143010i 0.997440 + 0.0715052i $$0.0227802\pi$$
−0.997440 + 0.0715052i $$0.977220\pi$$
$$90$$ 0 0
$$91$$ −32.0000 −0.351648
$$92$$ 33.9411i 0.368925i
$$93$$ 0 0
$$94$$ 120.000 1.27660
$$95$$ 67.8823i 0.714550i
$$96$$ 0 0
$$97$$ 176.000 1.81443 0.907216 0.420664i $$-0.138203\pi$$
0.907216 + 0.420664i $$0.138203\pi$$
$$98$$ − 46.6690i − 0.476215i
$$99$$ 0 0
$$100$$ −14.0000 −0.140000
$$101$$ 29.6985i 0.294044i 0.989133 + 0.147022i $$0.0469689\pi$$
−0.989133 + 0.147022i $$0.953031\pi$$
$$102$$ 0 0
$$103$$ −28.0000 −0.271845 −0.135922 0.990719i $$-0.543400\pi$$
−0.135922 + 0.990719i $$0.543400\pi$$
$$104$$ − 22.6274i − 0.217571i
$$105$$ 0 0
$$106$$ −54.0000 −0.509434
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 56.0000 0.513761 0.256881 0.966443i $$-0.417305\pi$$
0.256881 + 0.966443i $$0.417305\pi$$
$$110$$ 101.823i 0.925667i
$$111$$ 0 0
$$112$$ −16.0000 −0.142857
$$113$$ − 156.978i − 1.38918i −0.719404 0.694592i $$-0.755585\pi$$
0.719404 0.694592i $$-0.244415\pi$$
$$114$$ 0 0
$$115$$ −72.0000 −0.626087
$$116$$ − 8.48528i − 0.0731490i
$$117$$ 0 0
$$118$$ −48.0000 −0.406780
$$119$$ 50.9117i 0.427829i
$$120$$ 0 0
$$121$$ −167.000 −1.38017
$$122$$ 70.7107i 0.579596i
$$123$$ 0 0
$$124$$ −88.0000 −0.709677
$$125$$ − 135.765i − 1.08612i
$$126$$ 0 0
$$127$$ 92.0000 0.724409 0.362205 0.932099i $$-0.382024\pi$$
0.362205 + 0.932099i $$0.382024\pi$$
$$128$$ − 11.3137i − 0.0883883i
$$129$$ 0 0
$$130$$ 48.0000 0.369231
$$131$$ − 169.706i − 1.29546i −0.761869 0.647731i $$-0.775718\pi$$
0.761869 0.647731i $$-0.224282\pi$$
$$132$$ 0 0
$$133$$ 64.0000 0.481203
$$134$$ 11.3137i 0.0844307i
$$135$$ 0 0
$$136$$ −36.0000 −0.264706
$$137$$ 156.978i 1.14582i 0.819617 + 0.572911i $$0.194186\pi$$
−0.819617 + 0.572911i $$0.805814\pi$$
$$138$$ 0 0
$$139$$ 152.000 1.09353 0.546763 0.837288i $$-0.315860\pi$$
0.546763 + 0.837288i $$0.315860\pi$$
$$140$$ − 33.9411i − 0.242437i
$$141$$ 0 0
$$142$$ 72.0000 0.507042
$$143$$ 135.765i 0.949402i
$$144$$ 0 0
$$145$$ 18.0000 0.124138
$$146$$ − 22.6274i − 0.154982i
$$147$$ 0 0
$$148$$ 68.0000 0.459459
$$149$$ 275.772i 1.85082i 0.378972 + 0.925408i $$0.376278\pi$$
−0.378972 + 0.925408i $$0.623722\pi$$
$$150$$ 0 0
$$151$$ −148.000 −0.980132 −0.490066 0.871685i $$-0.663027\pi$$
−0.490066 + 0.871685i $$0.663027\pi$$
$$152$$ 45.2548i 0.297729i
$$153$$ 0 0
$$154$$ 96.0000 0.623377
$$155$$ − 186.676i − 1.20436i
$$156$$ 0 0
$$157$$ −82.0000 −0.522293 −0.261146 0.965299i $$-0.584101\pi$$
−0.261146 + 0.965299i $$0.584101\pi$$
$$158$$ − 107.480i − 0.680255i
$$159$$ 0 0
$$160$$ 24.0000 0.150000
$$161$$ 67.8823i 0.421629i
$$162$$ 0 0
$$163$$ 56.0000 0.343558 0.171779 0.985135i $$-0.445048\pi$$
0.171779 + 0.985135i $$0.445048\pi$$
$$164$$ − 93.3381i − 0.569135i
$$165$$ 0 0
$$166$$ −168.000 −1.01205
$$167$$ 33.9411i 0.203240i 0.994823 + 0.101620i $$0.0324026\pi$$
−0.994823 + 0.101620i $$0.967597\pi$$
$$168$$ 0 0
$$169$$ −105.000 −0.621302
$$170$$ − 76.3675i − 0.449221i
$$171$$ 0 0
$$172$$ 80.0000 0.465116
$$173$$ − 173.948i − 1.00548i −0.864437 0.502741i $$-0.832325\pi$$
0.864437 0.502741i $$-0.167675\pi$$
$$174$$ 0 0
$$175$$ −28.0000 −0.160000
$$176$$ 67.8823i 0.385695i
$$177$$ 0 0
$$178$$ −18.0000 −0.101124
$$179$$ − 203.647i − 1.13769i −0.822444 0.568846i $$-0.807390\pi$$
0.822444 0.568846i $$-0.192610\pi$$
$$180$$ 0 0
$$181$$ −232.000 −1.28177 −0.640884 0.767638i $$-0.721432\pi$$
−0.640884 + 0.767638i $$0.721432\pi$$
$$182$$ − 45.2548i − 0.248653i
$$183$$ 0 0
$$184$$ −48.0000 −0.260870
$$185$$ 144.250i 0.779729i
$$186$$ 0 0
$$187$$ 216.000 1.15508
$$188$$ 169.706i 0.902690i
$$189$$ 0 0
$$190$$ −96.0000 −0.505263
$$191$$ − 33.9411i − 0.177702i −0.996045 0.0888511i $$-0.971680\pi$$
0.996045 0.0888511i $$-0.0283195\pi$$
$$192$$ 0 0
$$193$$ 206.000 1.06736 0.533679 0.845687i $$-0.320809\pi$$
0.533679 + 0.845687i $$0.320809\pi$$
$$194$$ 248.902i 1.28300i
$$195$$ 0 0
$$196$$ 66.0000 0.336735
$$197$$ 165.463i 0.839914i 0.907544 + 0.419957i $$0.137955\pi$$
−0.907544 + 0.419957i $$0.862045\pi$$
$$198$$ 0 0
$$199$$ 20.0000 0.100503 0.0502513 0.998737i $$-0.483998\pi$$
0.0502513 + 0.998737i $$0.483998\pi$$
$$200$$ − 19.7990i − 0.0989949i
$$201$$ 0 0
$$202$$ −42.0000 −0.207921
$$203$$ − 16.9706i − 0.0835988i
$$204$$ 0 0
$$205$$ 198.000 0.965854
$$206$$ − 39.5980i − 0.192223i
$$207$$ 0 0
$$208$$ 32.0000 0.153846
$$209$$ − 271.529i − 1.29918i
$$210$$ 0 0
$$211$$ 296.000 1.40284 0.701422 0.712746i $$-0.252549\pi$$
0.701422 + 0.712746i $$0.252549\pi$$
$$212$$ − 76.3675i − 0.360224i
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 169.706i 0.789328i
$$216$$ 0 0
$$217$$ −176.000 −0.811060
$$218$$ 79.1960i 0.363284i
$$219$$ 0 0
$$220$$ −144.000 −0.654545
$$221$$ − 101.823i − 0.460739i
$$222$$ 0 0
$$223$$ −436.000 −1.95516 −0.977578 0.210571i $$-0.932468\pi$$
−0.977578 + 0.210571i $$0.932468\pi$$
$$224$$ − 22.6274i − 0.101015i
$$225$$ 0 0
$$226$$ 222.000 0.982301
$$227$$ 16.9706i 0.0747602i 0.999301 + 0.0373801i $$0.0119012\pi$$
−0.999301 + 0.0373801i $$0.988099\pi$$
$$228$$ 0 0
$$229$$ 8.00000 0.0349345 0.0174672 0.999847i $$-0.494440\pi$$
0.0174672 + 0.999847i $$0.494440\pi$$
$$230$$ − 101.823i − 0.442710i
$$231$$ 0 0
$$232$$ 12.0000 0.0517241
$$233$$ − 12.7279i − 0.0546263i −0.999627 0.0273131i $$-0.991305\pi$$
0.999627 0.0273131i $$-0.00869512\pi$$
$$234$$ 0 0
$$235$$ −360.000 −1.53191
$$236$$ − 67.8823i − 0.287637i
$$237$$ 0 0
$$238$$ −72.0000 −0.302521
$$239$$ 135.765i 0.568052i 0.958817 + 0.284026i $$0.0916703\pi$$
−0.958817 + 0.284026i $$0.908330\pi$$
$$240$$ 0 0
$$241$$ 32.0000 0.132780 0.0663900 0.997794i $$-0.478852\pi$$
0.0663900 + 0.997794i $$0.478852\pi$$
$$242$$ − 236.174i − 0.975924i
$$243$$ 0 0
$$244$$ −100.000 −0.409836
$$245$$ 140.007i 0.571458i
$$246$$ 0 0
$$247$$ −128.000 −0.518219
$$248$$ − 124.451i − 0.501818i
$$249$$ 0 0
$$250$$ 192.000 0.768000
$$251$$ 50.9117i 0.202835i 0.994844 + 0.101418i $$0.0323379\pi$$
−0.994844 + 0.101418i $$0.967662\pi$$
$$252$$ 0 0
$$253$$ 288.000 1.13834
$$254$$ 130.108i 0.512235i
$$255$$ 0 0
$$256$$ 16.0000 0.0625000
$$257$$ − 182.434i − 0.709858i −0.934893 0.354929i $$-0.884505\pi$$
0.934893 0.354929i $$-0.115495\pi$$
$$258$$ 0 0
$$259$$ 136.000 0.525097
$$260$$ 67.8823i 0.261086i
$$261$$ 0 0
$$262$$ 240.000 0.916031
$$263$$ 373.352i 1.41959i 0.704408 + 0.709795i $$0.251213\pi$$
−0.704408 + 0.709795i $$0.748787\pi$$
$$264$$ 0 0
$$265$$ 162.000 0.611321
$$266$$ 90.5097i 0.340262i
$$267$$ 0 0
$$268$$ −16.0000 −0.0597015
$$269$$ − 343.654i − 1.27752i −0.769404 0.638762i $$-0.779447\pi$$
0.769404 0.638762i $$-0.220553\pi$$
$$270$$ 0 0
$$271$$ 380.000 1.40221 0.701107 0.713056i $$-0.252690\pi$$
0.701107 + 0.713056i $$0.252690\pi$$
$$272$$ − 50.9117i − 0.187175i
$$273$$ 0 0
$$274$$ −222.000 −0.810219
$$275$$ 118.794i 0.431978i
$$276$$ 0 0
$$277$$ −328.000 −1.18412 −0.592058 0.805896i $$-0.701684\pi$$
−0.592058 + 0.805896i $$0.701684\pi$$
$$278$$ 214.960i 0.773239i
$$279$$ 0 0
$$280$$ 48.0000 0.171429
$$281$$ 284.257i 1.01159i 0.862654 + 0.505795i $$0.168801\pi$$
−0.862654 + 0.505795i $$0.831199\pi$$
$$282$$ 0 0
$$283$$ −208.000 −0.734982 −0.367491 0.930027i $$-0.619783\pi$$
−0.367491 + 0.930027i $$0.619783\pi$$
$$284$$ 101.823i 0.358533i
$$285$$ 0 0
$$286$$ −192.000 −0.671329
$$287$$ − 186.676i − 0.650440i
$$288$$ 0 0
$$289$$ 127.000 0.439446
$$290$$ 25.4558i 0.0877788i
$$291$$ 0 0
$$292$$ 32.0000 0.109589
$$293$$ − 436.992i − 1.49144i −0.666259 0.745720i $$-0.732106\pi$$
0.666259 0.745720i $$-0.267894\pi$$
$$294$$ 0 0
$$295$$ 144.000 0.488136
$$296$$ 96.1665i 0.324887i
$$297$$ 0 0
$$298$$ −390.000 −1.30872
$$299$$ − 135.765i − 0.454062i
$$300$$ 0 0
$$301$$ 160.000 0.531561
$$302$$ − 209.304i − 0.693058i
$$303$$ 0 0
$$304$$ −64.0000 −0.210526
$$305$$ − 212.132i − 0.695515i
$$306$$ 0 0
$$307$$ −520.000 −1.69381 −0.846906 0.531743i $$-0.821537\pi$$
−0.846906 + 0.531743i $$0.821537\pi$$
$$308$$ 135.765i 0.440794i
$$309$$ 0 0
$$310$$ 264.000 0.851613
$$311$$ − 373.352i − 1.20049i −0.799816 0.600245i $$-0.795070\pi$$
0.799816 0.600245i $$-0.204930\pi$$
$$312$$ 0 0
$$313$$ −94.0000 −0.300319 −0.150160 0.988662i $$-0.547979\pi$$
−0.150160 + 0.988662i $$0.547979\pi$$
$$314$$ − 115.966i − 0.369317i
$$315$$ 0 0
$$316$$ 152.000 0.481013
$$317$$ 335.169i 1.05731i 0.848835 + 0.528657i $$0.177304\pi$$
−0.848835 + 0.528657i $$0.822696\pi$$
$$318$$ 0 0
$$319$$ −72.0000 −0.225705
$$320$$ 33.9411i 0.106066i
$$321$$ 0 0
$$322$$ −96.0000 −0.298137
$$323$$ 203.647i 0.630485i
$$324$$ 0 0
$$325$$ 56.0000 0.172308
$$326$$ 79.1960i 0.242932i
$$327$$ 0 0
$$328$$ 132.000 0.402439
$$329$$ 339.411i 1.03165i
$$330$$ 0 0
$$331$$ 536.000 1.61934 0.809668 0.586889i $$-0.199647\pi$$
0.809668 + 0.586889i $$0.199647\pi$$
$$332$$ − 237.588i − 0.715626i
$$333$$ 0 0
$$334$$ −48.0000 −0.143713
$$335$$ − 33.9411i − 0.101317i
$$336$$ 0 0
$$337$$ −208.000 −0.617211 −0.308605 0.951190i $$-0.599862\pi$$
−0.308605 + 0.951190i $$0.599862\pi$$
$$338$$ − 148.492i − 0.439327i
$$339$$ 0 0
$$340$$ 108.000 0.317647
$$341$$ 746.705i 2.18975i
$$342$$ 0 0
$$343$$ 328.000 0.956268
$$344$$ 113.137i 0.328887i
$$345$$ 0 0
$$346$$ 246.000 0.710983
$$347$$ 288.500i 0.831411i 0.909499 + 0.415705i $$0.136465\pi$$
−0.909499 + 0.415705i $$0.863535\pi$$
$$348$$ 0 0
$$349$$ −238.000 −0.681948 −0.340974 0.940073i $$-0.610757\pi$$
−0.340974 + 0.940073i $$0.610757\pi$$
$$350$$ − 39.5980i − 0.113137i
$$351$$ 0 0
$$352$$ −96.0000 −0.272727
$$353$$ − 224.860i − 0.636997i −0.947923 0.318499i $$-0.896821\pi$$
0.947923 0.318499i $$-0.103179\pi$$
$$354$$ 0 0
$$355$$ −216.000 −0.608451
$$356$$ − 25.4558i − 0.0715052i
$$357$$ 0 0
$$358$$ 288.000 0.804469
$$359$$ − 560.029i − 1.55997i −0.625799 0.779984i $$-0.715227\pi$$
0.625799 0.779984i $$-0.284773\pi$$
$$360$$ 0 0
$$361$$ −105.000 −0.290859
$$362$$ − 328.098i − 0.906347i
$$363$$ 0 0
$$364$$ 64.0000 0.175824
$$365$$ 67.8823i 0.185979i
$$366$$ 0 0
$$367$$ 284.000 0.773842 0.386921 0.922113i $$-0.373539\pi$$
0.386921 + 0.922113i $$0.373539\pi$$
$$368$$ − 67.8823i − 0.184463i
$$369$$ 0 0
$$370$$ −204.000 −0.551351
$$371$$ − 152.735i − 0.411685i
$$372$$ 0 0
$$373$$ −190.000 −0.509383 −0.254692 0.967022i $$-0.581974\pi$$
−0.254692 + 0.967022i $$0.581974\pi$$
$$374$$ 305.470i 0.816765i
$$375$$ 0 0
$$376$$ −240.000 −0.638298
$$377$$ 33.9411i 0.0900295i
$$378$$ 0 0
$$379$$ −160.000 −0.422164 −0.211082 0.977468i $$-0.567699\pi$$
−0.211082 + 0.977468i $$0.567699\pi$$
$$380$$ − 135.765i − 0.357275i
$$381$$ 0 0
$$382$$ 48.0000 0.125654
$$383$$ − 271.529i − 0.708953i −0.935065 0.354477i $$-0.884659\pi$$
0.935065 0.354477i $$-0.115341\pi$$
$$384$$ 0 0
$$385$$ −288.000 −0.748052
$$386$$ 291.328i 0.754736i
$$387$$ 0 0
$$388$$ −352.000 −0.907216
$$389$$ − 403.051i − 1.03612i −0.855344 0.518060i $$-0.826654\pi$$
0.855344 0.518060i $$-0.173346\pi$$
$$390$$ 0 0
$$391$$ −216.000 −0.552430
$$392$$ 93.3381i 0.238107i
$$393$$ 0 0
$$394$$ −234.000 −0.593909
$$395$$ 322.441i 0.816306i
$$396$$ 0 0
$$397$$ 146.000 0.367758 0.183879 0.982949i $$-0.441135\pi$$
0.183879 + 0.982949i $$0.441135\pi$$
$$398$$ 28.2843i 0.0710660i
$$399$$ 0 0
$$400$$ 28.0000 0.0700000
$$401$$ 326.683i 0.814672i 0.913278 + 0.407336i $$0.133542\pi$$
−0.913278 + 0.407336i $$0.866458\pi$$
$$402$$ 0 0
$$403$$ 352.000 0.873449
$$404$$ − 59.3970i − 0.147022i
$$405$$ 0 0
$$406$$ 24.0000 0.0591133
$$407$$ − 576.999i − 1.41769i
$$408$$ 0 0
$$409$$ 368.000 0.899756 0.449878 0.893090i $$-0.351468\pi$$
0.449878 + 0.893090i $$0.351468\pi$$
$$410$$ 280.014i 0.682962i
$$411$$ 0 0
$$412$$ 56.0000 0.135922
$$413$$ − 135.765i − 0.328728i
$$414$$ 0 0
$$415$$ 504.000 1.21446
$$416$$ 45.2548i 0.108786i
$$417$$ 0 0
$$418$$ 384.000 0.918660
$$419$$ 390.323i 0.931558i 0.884901 + 0.465779i $$0.154226\pi$$
−0.884901 + 0.465779i $$0.845774\pi$$
$$420$$ 0 0
$$421$$ −40.0000 −0.0950119 −0.0475059 0.998871i $$-0.515127\pi$$
−0.0475059 + 0.998871i $$0.515127\pi$$
$$422$$ 418.607i 0.991960i
$$423$$ 0 0
$$424$$ 108.000 0.254717
$$425$$ − 89.0955i − 0.209636i
$$426$$ 0 0
$$427$$ −200.000 −0.468384
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −240.000 −0.558140
$$431$$ − 152.735i − 0.354374i −0.984177 0.177187i $$-0.943300\pi$$
0.984177 0.177187i $$-0.0566997\pi$$
$$432$$ 0 0
$$433$$ 542.000 1.25173 0.625866 0.779931i $$-0.284746\pi$$
0.625866 + 0.779931i $$0.284746\pi$$
$$434$$ − 248.902i − 0.573506i
$$435$$ 0 0
$$436$$ −112.000 −0.256881
$$437$$ 271.529i 0.621348i
$$438$$ 0 0
$$439$$ −4.00000 −0.00911162 −0.00455581 0.999990i $$-0.501450\pi$$
−0.00455581 + 0.999990i $$0.501450\pi$$
$$440$$ − 203.647i − 0.462834i
$$441$$ 0 0
$$442$$ 144.000 0.325792
$$443$$ 322.441i 0.727857i 0.931427 + 0.363929i $$0.118565\pi$$
−0.931427 + 0.363929i $$0.881435\pi$$
$$444$$ 0 0
$$445$$ 54.0000 0.121348
$$446$$ − 616.597i − 1.38250i
$$447$$ 0 0
$$448$$ 32.0000 0.0714286
$$449$$ 216.375i 0.481904i 0.970537 + 0.240952i $$0.0774596\pi$$
−0.970537 + 0.240952i $$0.922540\pi$$
$$450$$ 0 0
$$451$$ −792.000 −1.75610
$$452$$ 313.955i 0.694592i
$$453$$ 0 0
$$454$$ −24.0000 −0.0528634
$$455$$ 135.765i 0.298384i
$$456$$ 0 0
$$457$$ −400.000 −0.875274 −0.437637 0.899152i $$-0.644184\pi$$
−0.437637 + 0.899152i $$0.644184\pi$$
$$458$$ 11.3137i 0.0247024i
$$459$$ 0 0
$$460$$ 144.000 0.313043
$$461$$ − 301.227i − 0.653422i −0.945124 0.326711i $$-0.894060\pi$$
0.945124 0.326711i $$-0.105940\pi$$
$$462$$ 0 0
$$463$$ −604.000 −1.30454 −0.652268 0.757989i $$-0.726182\pi$$
−0.652268 + 0.757989i $$0.726182\pi$$
$$464$$ 16.9706i 0.0365745i
$$465$$ 0 0
$$466$$ 18.0000 0.0386266
$$467$$ 356.382i 0.763130i 0.924342 + 0.381565i $$0.124615\pi$$
−0.924342 + 0.381565i $$0.875385\pi$$
$$468$$ 0 0
$$469$$ −32.0000 −0.0682303
$$470$$ − 509.117i − 1.08323i
$$471$$ 0 0
$$472$$ 96.0000 0.203390
$$473$$ − 678.823i − 1.43514i
$$474$$ 0 0
$$475$$ −112.000 −0.235789
$$476$$ − 101.823i − 0.213915i
$$477$$ 0 0
$$478$$ −192.000 −0.401674
$$479$$ 526.087i 1.09830i 0.835723 + 0.549152i $$0.185049\pi$$
−0.835723 + 0.549152i $$0.814951\pi$$
$$480$$ 0 0
$$481$$ −272.000 −0.565489
$$482$$ 45.2548i 0.0938897i
$$483$$ 0 0
$$484$$ 334.000 0.690083
$$485$$ − 746.705i − 1.53960i
$$486$$ 0 0
$$487$$ 596.000 1.22382 0.611910 0.790928i $$-0.290402\pi$$
0.611910 + 0.790928i $$0.290402\pi$$
$$488$$ − 141.421i − 0.289798i
$$489$$ 0 0
$$490$$ −198.000 −0.404082
$$491$$ − 271.529i − 0.553012i −0.961012 0.276506i $$-0.910823\pi$$
0.961012 0.276506i $$-0.0891766\pi$$
$$492$$ 0 0
$$493$$ 54.0000 0.109533
$$494$$ − 181.019i − 0.366436i
$$495$$ 0 0
$$496$$ 176.000 0.354839
$$497$$ 203.647i 0.409752i
$$498$$ 0 0
$$499$$ 224.000 0.448898 0.224449 0.974486i $$-0.427942\pi$$
0.224449 + 0.974486i $$0.427942\pi$$
$$500$$ 271.529i 0.543058i
$$501$$ 0 0
$$502$$ −72.0000 −0.143426
$$503$$ 865.499i 1.72067i 0.509726 + 0.860337i $$0.329747\pi$$
−0.509726 + 0.860337i $$0.670253\pi$$
$$504$$ 0 0
$$505$$ 126.000 0.249505
$$506$$ 407.294i 0.804928i
$$507$$ 0 0
$$508$$ −184.000 −0.362205
$$509$$ 479.418i 0.941883i 0.882164 + 0.470941i $$0.156086\pi$$
−0.882164 + 0.470941i $$0.843914\pi$$
$$510$$ 0 0
$$511$$ 64.0000 0.125245
$$512$$ 22.6274i 0.0441942i
$$513$$ 0 0
$$514$$ 258.000 0.501946
$$515$$ 118.794i 0.230668i
$$516$$ 0 0
$$517$$ 1440.00 2.78530
$$518$$ 192.333i 0.371299i
$$519$$ 0 0
$$520$$ −96.0000 −0.184615
$$521$$ − 521.845i − 1.00162i −0.865557 0.500811i $$-0.833035\pi$$
0.865557 0.500811i $$-0.166965\pi$$
$$522$$ 0 0
$$523$$ −736.000 −1.40727 −0.703633 0.710564i $$-0.748440\pi$$
−0.703633 + 0.710564i $$0.748440\pi$$
$$524$$ 339.411i 0.647731i
$$525$$ 0 0
$$526$$ −528.000 −1.00380
$$527$$ − 560.029i − 1.06267i
$$528$$ 0 0
$$529$$ 241.000 0.455577
$$530$$ 229.103i 0.432269i
$$531$$ 0 0
$$532$$ −128.000 −0.240602
$$533$$ 373.352i 0.700474i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ − 22.6274i − 0.0422153i
$$537$$ 0 0
$$538$$ 486.000 0.903346
$$539$$ − 560.029i − 1.03901i
$$540$$ 0 0
$$541$$ −808.000 −1.49353 −0.746765 0.665088i $$-0.768394\pi$$
−0.746765 + 0.665088i $$0.768394\pi$$
$$542$$ 537.401i 0.991515i
$$543$$ 0 0
$$544$$ 72.0000 0.132353
$$545$$ − 237.588i − 0.435941i
$$546$$ 0 0
$$547$$ 536.000 0.979890 0.489945 0.871753i $$-0.337017\pi$$
0.489945 + 0.871753i $$0.337017\pi$$
$$548$$ − 313.955i − 0.572911i
$$549$$ 0 0
$$550$$ −168.000 −0.305455
$$551$$ − 67.8823i − 0.123198i
$$552$$ 0 0
$$553$$ 304.000 0.549729
$$554$$ − 463.862i − 0.837296i
$$555$$ 0 0
$$556$$ −304.000 −0.546763
$$557$$ − 165.463i − 0.297061i −0.988908 0.148531i $$-0.952546\pi$$
0.988908 0.148531i $$-0.0474543\pi$$
$$558$$ 0 0
$$559$$ −320.000 −0.572451
$$560$$ 67.8823i 0.121218i
$$561$$ 0 0
$$562$$ −402.000 −0.715302
$$563$$ − 322.441i − 0.572719i −0.958122 0.286359i $$-0.907555\pi$$
0.958122 0.286359i $$-0.0924451\pi$$
$$564$$ 0 0
$$565$$ −666.000 −1.17876
$$566$$ − 294.156i − 0.519711i
$$567$$ 0 0
$$568$$ −144.000 −0.253521
$$569$$ 156.978i 0.275883i 0.990440 + 0.137942i $$0.0440487\pi$$
−0.990440 + 0.137942i $$0.955951\pi$$
$$570$$ 0 0
$$571$$ 368.000 0.644483 0.322242 0.946657i $$-0.395564\pi$$
0.322242 + 0.946657i $$0.395564\pi$$
$$572$$ − 271.529i − 0.474701i
$$573$$ 0 0
$$574$$ 264.000 0.459930
$$575$$ − 118.794i − 0.206598i
$$576$$ 0 0
$$577$$ −142.000 −0.246101 −0.123050 0.992400i $$-0.539268\pi$$
−0.123050 + 0.992400i $$0.539268\pi$$
$$578$$ 179.605i 0.310736i
$$579$$ 0 0
$$580$$ −36.0000 −0.0620690
$$581$$ − 475.176i − 0.817858i
$$582$$ 0 0
$$583$$ −648.000 −1.11149
$$584$$ 45.2548i 0.0774912i
$$585$$ 0 0
$$586$$ 618.000 1.05461
$$587$$ 373.352i 0.636035i 0.948085 + 0.318017i $$0.103017\pi$$
−0.948085 + 0.318017i $$0.896983\pi$$
$$588$$ 0 0
$$589$$ −704.000 −1.19525
$$590$$ 203.647i 0.345164i
$$591$$ 0 0
$$592$$ −136.000 −0.229730
$$593$$ 1107.33i 1.86733i 0.358142 + 0.933667i $$0.383410\pi$$
−0.358142 + 0.933667i $$0.616590\pi$$
$$594$$ 0 0
$$595$$ 216.000 0.363025
$$596$$ − 551.543i − 0.925408i
$$597$$ 0 0
$$598$$ 192.000 0.321070
$$599$$ 797.616i 1.33158i 0.746139 + 0.665790i $$0.231905\pi$$
−0.746139 + 0.665790i $$0.768095\pi$$
$$600$$ 0 0
$$601$$ 158.000 0.262895 0.131448 0.991323i $$-0.458037\pi$$
0.131448 + 0.991323i $$0.458037\pi$$
$$602$$ 226.274i 0.375871i
$$603$$ 0 0
$$604$$ 296.000 0.490066
$$605$$ 708.521i 1.17111i
$$606$$ 0 0
$$607$$ 332.000 0.546952 0.273476 0.961879i $$-0.411827\pi$$
0.273476 + 0.961879i $$0.411827\pi$$
$$608$$ − 90.5097i − 0.148865i
$$609$$ 0 0
$$610$$ 300.000 0.491803
$$611$$ − 678.823i − 1.11100i
$$612$$ 0 0
$$613$$ 578.000 0.942904 0.471452 0.881892i $$-0.343730\pi$$
0.471452 + 0.881892i $$0.343730\pi$$
$$614$$ − 735.391i − 1.19771i
$$615$$ 0 0
$$616$$ −192.000 −0.311688
$$617$$ − 55.1543i − 0.0893911i −0.999001 0.0446956i $$-0.985768\pi$$
0.999001 0.0446956i $$-0.0142318\pi$$
$$618$$ 0 0
$$619$$ 896.000 1.44750 0.723748 0.690064i $$-0.242418\pi$$
0.723748 + 0.690064i $$0.242418\pi$$
$$620$$ 373.352i 0.602181i
$$621$$ 0 0
$$622$$ 528.000 0.848875
$$623$$ − 50.9117i − 0.0817202i
$$624$$ 0 0
$$625$$ −401.000 −0.641600
$$626$$ − 132.936i − 0.212358i
$$627$$ 0 0
$$628$$ 164.000 0.261146
$$629$$ 432.749i 0.687996i
$$630$$ 0 0
$$631$$ 20.0000 0.0316957 0.0158479 0.999874i $$-0.494955\pi$$
0.0158479 + 0.999874i $$0.494955\pi$$
$$632$$ 214.960i 0.340127i
$$633$$ 0 0
$$634$$ −474.000 −0.747634
$$635$$ − 390.323i − 0.614682i
$$636$$ 0 0
$$637$$ −264.000 −0.414443
$$638$$ − 101.823i − 0.159598i
$$639$$ 0 0
$$640$$ −48.0000 −0.0750000
$$641$$ 258.801i 0.403746i 0.979412 + 0.201873i $$0.0647028\pi$$
−0.979412 + 0.201873i $$0.935297\pi$$
$$642$$ 0 0
$$643$$ 728.000 1.13219 0.566096 0.824339i $$-0.308453\pi$$
0.566096 + 0.824339i $$0.308453\pi$$
$$644$$ − 135.765i − 0.210814i
$$645$$ 0 0
$$646$$ −288.000 −0.445820
$$647$$ − 458.205i − 0.708200i −0.935208 0.354100i $$-0.884787\pi$$
0.935208 0.354100i $$-0.115213\pi$$
$$648$$ 0 0
$$649$$ −576.000 −0.887519
$$650$$ 79.1960i 0.121840i
$$651$$ 0 0
$$652$$ −112.000 −0.171779
$$653$$ 301.227i 0.461298i 0.973037 + 0.230649i $$0.0740849\pi$$
−0.973037 + 0.230649i $$0.925915\pi$$
$$654$$ 0 0
$$655$$ −720.000 −1.09924
$$656$$ 186.676i 0.284567i
$$657$$ 0 0
$$658$$ −480.000 −0.729483
$$659$$ − 1052.17i − 1.59662i −0.602244 0.798312i $$-0.705727\pi$$
0.602244 0.798312i $$-0.294273\pi$$
$$660$$ 0 0
$$661$$ 62.0000 0.0937973 0.0468986 0.998900i $$-0.485066\pi$$
0.0468986 + 0.998900i $$0.485066\pi$$
$$662$$ 758.018i 1.14504i
$$663$$ 0 0
$$664$$ 336.000 0.506024
$$665$$ − 271.529i − 0.408314i
$$666$$ 0 0
$$667$$ 72.0000 0.107946
$$668$$ − 67.8823i − 0.101620i
$$669$$ 0 0
$$670$$ 48.0000 0.0716418
$$671$$ 848.528i 1.26457i
$$672$$ 0 0
$$673$$ −670.000 −0.995542 −0.497771 0.867308i $$-0.665848\pi$$
−0.497771 + 0.867308i $$0.665848\pi$$
$$674$$ − 294.156i − 0.436434i
$$675$$ 0 0
$$676$$ 210.000 0.310651
$$677$$ − 1294.01i − 1.91138i −0.294372 0.955691i $$-0.595111\pi$$
0.294372 0.955691i $$-0.404889\pi$$
$$678$$ 0 0
$$679$$ −704.000 −1.03682
$$680$$ 152.735i 0.224610i
$$681$$ 0 0
$$682$$ −1056.00 −1.54839
$$683$$ − 560.029i − 0.819954i −0.912096 0.409977i $$-0.865537\pi$$
0.912096 0.409977i $$-0.134463\pi$$
$$684$$ 0 0
$$685$$ 666.000 0.972263
$$686$$ 463.862i 0.676184i
$$687$$ 0 0
$$688$$ −160.000 −0.232558
$$689$$ 305.470i 0.443353i
$$690$$ 0 0
$$691$$ −40.0000 −0.0578871 −0.0289436 0.999581i $$-0.509214\pi$$
−0.0289436 + 0.999581i $$0.509214\pi$$
$$692$$ 347.897i 0.502741i
$$693$$ 0 0
$$694$$ −408.000 −0.587896
$$695$$ − 644.881i − 0.927887i
$$696$$ 0 0
$$697$$ 594.000 0.852224
$$698$$ − 336.583i − 0.482210i
$$699$$ 0 0
$$700$$ 56.0000 0.0800000
$$701$$ 954.594i 1.36176i 0.732395 + 0.680880i $$0.238403\pi$$
−0.732395 + 0.680880i $$0.761597\pi$$
$$702$$ 0 0
$$703$$ 544.000 0.773826
$$704$$ − 135.765i − 0.192847i
$$705$$ 0 0
$$706$$ 318.000 0.450425
$$707$$ − 118.794i − 0.168025i
$$708$$ 0 0
$$709$$ 968.000 1.36530 0.682652 0.730744i $$-0.260827\pi$$
0.682652 + 0.730744i $$0.260827\pi$$
$$710$$ − 305.470i − 0.430240i
$$711$$ 0 0
$$712$$ 36.0000 0.0505618
$$713$$ − 746.705i − 1.04727i
$$714$$ 0 0
$$715$$ 576.000 0.805594
$$716$$ 407.294i 0.568846i
$$717$$ 0 0
$$718$$ 792.000 1.10306
$$719$$ 1170.97i 1.62861i 0.580439 + 0.814304i $$0.302881\pi$$
−0.580439 + 0.814304i $$0.697119\pi$$
$$720$$ 0 0
$$721$$ 112.000 0.155340
$$722$$ − 148.492i − 0.205668i
$$723$$ 0 0
$$724$$ 464.000 0.640884
$$725$$ 29.6985i 0.0409634i
$$726$$ 0 0
$$727$$ −508.000 −0.698762 −0.349381 0.936981i $$-0.613608\pi$$
−0.349381 + 0.936981i $$0.613608\pi$$
$$728$$ 90.5097i 0.124326i
$$729$$ 0 0
$$730$$ −96.0000 −0.131507
$$731$$ 509.117i 0.696466i
$$732$$ 0 0
$$733$$ −1144.00 −1.56071 −0.780355 0.625337i $$-0.784961\pi$$
−0.780355 + 0.625337i $$0.784961\pi$$
$$734$$ 401.637i 0.547189i
$$735$$ 0 0
$$736$$ 96.0000 0.130435
$$737$$ 135.765i 0.184212i
$$738$$ 0 0
$$739$$ −304.000 −0.411367 −0.205683 0.978619i $$-0.565942\pi$$
−0.205683 + 0.978619i $$0.565942\pi$$
$$740$$ − 288.500i − 0.389864i
$$741$$ 0 0
$$742$$ 216.000 0.291105
$$743$$ 848.528i 1.14203i 0.820940 + 0.571015i $$0.193450\pi$$
−0.820940 + 0.571015i $$0.806550\pi$$
$$744$$ 0 0
$$745$$ 1170.00 1.57047
$$746$$ − 268.701i − 0.360188i
$$747$$ 0 0
$$748$$ −432.000 −0.577540
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 188.000 0.250333 0.125166 0.992136i $$-0.460054\pi$$
0.125166 + 0.992136i $$0.460054\pi$$
$$752$$ − 339.411i − 0.451345i
$$753$$ 0 0
$$754$$ −48.0000 −0.0636605
$$755$$ 627.911i 0.831670i
$$756$$ 0 0
$$757$$ −1240.00 −1.63804 −0.819022 0.573761i $$-0.805484\pi$$
−0.819022 + 0.573761i $$0.805484\pi$$
$$758$$ − 226.274i − 0.298515i
$$759$$ 0 0
$$760$$ 192.000 0.252632
$$761$$ − 156.978i − 0.206278i −0.994667 0.103139i $$-0.967111\pi$$
0.994667 0.103139i $$-0.0328887\pi$$
$$762$$ 0 0
$$763$$ −224.000 −0.293578
$$764$$ 67.8823i 0.0888511i
$$765$$ 0 0
$$766$$ 384.000 0.501305
$$767$$ 271.529i 0.354014i
$$768$$ 0 0
$$769$$ −910.000 −1.18336 −0.591678 0.806175i $$-0.701534\pi$$
−0.591678 + 0.806175i $$0.701534\pi$$
$$770$$ − 407.294i − 0.528953i
$$771$$ 0 0
$$772$$ −412.000 −0.533679
$$773$$ − 1387.34i − 1.79475i −0.441266 0.897376i $$-0.645471\pi$$
0.441266 0.897376i $$-0.354529\pi$$
$$774$$ 0 0
$$775$$ 308.000 0.397419
$$776$$ − 497.803i − 0.641499i
$$777$$ 0 0
$$778$$ 570.000 0.732648
$$779$$ − 746.705i − 0.958543i
$$780$$ 0 0
$$781$$ 864.000 1.10627
$$782$$ − 305.470i − 0.390627i
$$783$$ 0 0
$$784$$ −132.000 −0.168367
$$785$$ 347.897i 0.443180i
$$786$$ 0 0
$$787$$ −1360.00 −1.72808 −0.864041 0.503422i $$-0.832074\pi$$
−0.864041 + 0.503422i $$0.832074\pi$$
$$788$$ − 330.926i − 0.419957i
$$789$$ 0 0
$$790$$ −456.000 −0.577215
$$791$$ 627.911i 0.793819i
$$792$$ 0 0
$$793$$ 400.000 0.504414
$$794$$ 206.475i 0.260044i
$$795$$ 0 0
$$796$$ −40.0000 −0.0502513
$$797$$ − 106.066i − 0.133082i −0.997784 0.0665408i $$-0.978804\pi$$
0.997784 0.0665408i $$-0.0211963\pi$$
$$798$$ 0 0
$$799$$ −1080.00 −1.35169
$$800$$ 39.5980i 0.0494975i
$$801$$ 0 0
$$802$$ −462.000 −0.576060
$$803$$ − 271.529i − 0.338143i
$$804$$ 0 0
$$805$$ 288.000 0.357764
$$806$$ 497.803i 0.617622i
$$807$$ 0 0
$$808$$ 84.0000 0.103960
$$809$$ − 1107.33i − 1.36876i −0.729124 0.684381i $$-0.760072\pi$$
0.729124 0.684381i $$-0.239928\pi$$
$$810$$ 0 0
$$811$$ −160.000 −0.197287 −0.0986436 0.995123i $$-0.531450\pi$$
−0.0986436 + 0.995123i $$0.531450\pi$$
$$812$$ 33.9411i 0.0417994i
$$813$$ 0 0
$$814$$ 816.000 1.00246
$$815$$ − 237.588i − 0.291519i
$$816$$ 0 0
$$817$$ 640.000 0.783354
$$818$$ 520.431i 0.636223i
$$819$$ 0 0
$$820$$ −396.000 −0.482927
$$821$$ 436.992i 0.532268i 0.963936 + 0.266134i $$0.0857464\pi$$
−0.963936 + 0.266134i $$0.914254\pi$$
$$822$$ 0 0
$$823$$ 332.000 0.403402 0.201701 0.979447i $$-0.435353\pi$$
0.201701 + 0.979447i $$0.435353\pi$$
$$824$$ 79.1960i 0.0961116i
$$825$$ 0 0
$$826$$ 192.000 0.232446
$$827$$ 101.823i 0.123124i 0.998103 + 0.0615619i $$0.0196082\pi$$
−0.998103 + 0.0615619i $$0.980392\pi$$
$$828$$ 0 0
$$829$$ 632.000 0.762364 0.381182 0.924500i $$-0.375517\pi$$
0.381182 + 0.924500i $$0.375517\pi$$
$$830$$ 712.764i 0.858751i
$$831$$ 0 0
$$832$$ −64.0000 −0.0769231
$$833$$ 420.021i 0.504227i
$$834$$ 0 0
$$835$$ 144.000 0.172455
$$836$$ 543.058i 0.649591i
$$837$$ 0 0
$$838$$ −552.000 −0.658711
$$839$$ 729.734i 0.869767i 0.900487 + 0.434883i $$0.143210\pi$$
−0.900487 + 0.434883i $$0.856790\pi$$
$$840$$ 0 0
$$841$$ 823.000 0.978597
$$842$$ − 56.5685i − 0.0671835i
$$843$$ 0 0
$$844$$ −592.000 −0.701422
$$845$$ 445.477i 0.527192i
$$846$$ 0 0
$$847$$ 668.000 0.788666
$$848$$ 152.735i 0.180112i
$$849$$ 0 0
$$850$$ 126.000 0.148235
$$851$$ 576.999i 0.678025i
$$852$$ 0 0
$$853$$ 446.000 0.522860 0.261430 0.965222i $$-0.415806\pi$$
0.261430 + 0.965222i $$0.415806\pi$$
$$854$$ − 282.843i − 0.331198i
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 428.507i − 0.500008i −0.968245 0.250004i $$-0.919568\pi$$
0.968245 0.250004i $$-0.0804319\pi$$
$$858$$ 0 0
$$859$$ 728.000 0.847497 0.423749 0.905780i $$-0.360714\pi$$
0.423749 + 0.905780i $$0.360714\pi$$
$$860$$ − 339.411i − 0.394664i
$$861$$ 0 0
$$862$$ 216.000 0.250580
$$863$$ − 916.410i − 1.06189i −0.847407 0.530945i $$-0.821837\pi$$
0.847407 0.530945i $$-0.178163\pi$$
$$864$$ 0 0
$$865$$ −738.000 −0.853179
$$866$$ 766.504i 0.885108i
$$867$$ 0 0
$$868$$ 352.000 0.405530
$$869$$ − 1289.76i − 1.48419i
$$870$$ 0 0
$$871$$ 64.0000 0.0734788
$$872$$ − 158.392i − 0.181642i
$$873$$ 0 0
$$874$$ −384.000 −0.439359
$$875$$ 543.058i 0.620638i
$$876$$ 0 0
$$877$$ −910.000 −1.03763 −0.518814 0.854887i $$-0.673626\pi$$
−0.518814 + 0.854887i $$0.673626\pi$$
$$878$$ − 5.65685i − 0.00644289i
$$879$$ 0 0
$$880$$ 288.000 0.327273
$$881$$ − 929.138i − 1.05464i −0.849667 0.527320i $$-0.823197\pi$$
0.849667 0.527320i $$-0.176803\pi$$
$$882$$ 0 0
$$883$$ 1064.00 1.20498 0.602492 0.798125i $$-0.294175\pi$$
0.602492 + 0.798125i $$0.294175\pi$$
$$884$$ 203.647i 0.230370i
$$885$$ 0 0
$$886$$ −456.000 −0.514673
$$887$$ − 1391.59i − 1.56887i −0.620212 0.784434i $$-0.712953\pi$$
0.620212 0.784434i $$-0.287047\pi$$
$$888$$ 0 0
$$889$$ −368.000 −0.413948
$$890$$ 76.3675i 0.0858062i
$$891$$ 0 0
$$892$$ 872.000 0.977578
$$893$$ 1357.65i 1.52032i
$$894$$ 0 0
$$895$$ −864.000 −0.965363
$$896$$ 45.2548i 0.0505076i
$$897$$ 0 0
$$898$$ −306.000 −0.340757
$$899$$ 186.676i 0.207649i
$$900$$ 0 0
$$901$$ 486.000 0.539401
$$902$$ − 1120.06i − 1.24175i
$$903$$ 0 0
$$904$$ −444.000 −0.491150
$$905$$ 984.293i 1.08762i
$$906$$ 0 0
$$907$$ −1768.00 −1.94928 −0.974642 0.223771i $$-0.928163\pi$$
−0.974642 + 0.223771i $$0.928163\pi$$
$$908$$ − 33.9411i − 0.0373801i
$$909$$ 0 0
$$910$$ −192.000 −0.210989
$$911$$ − 237.588i − 0.260799i −0.991462 0.130399i $$-0.958374\pi$$
0.991462 0.130399i $$-0.0416260\pi$$
$$912$$ 0 0
$$913$$ −2016.00 −2.20811
$$914$$ − 565.685i − 0.618912i
$$915$$ 0 0
$$916$$ −16.0000 −0.0174672
$$917$$ 678.823i 0.740264i
$$918$$ 0 0
$$919$$ 380.000 0.413493 0.206746 0.978395i $$-0.433712\pi$$
0.206746 + 0.978395i $$0.433712\pi$$
$$920$$ 203.647i 0.221355i
$$921$$ 0 0
$$922$$ 426.000 0.462039
$$923$$ − 407.294i − 0.441271i
$$924$$ 0 0
$$925$$ −238.000 −0.257297
$$926$$ − 854.185i − 0.922446i
$$927$$ 0 0
$$928$$ −24.0000 −0.0258621
$$929$$ 666.095i 0.717002i 0.933529 + 0.358501i $$0.116712\pi$$
−0.933529 + 0.358501i $$0.883288\pi$$
$$930$$ 0 0
$$931$$ 528.000 0.567132
$$932$$ 25.4558i 0.0273131i
$$933$$ 0 0
$$934$$ −504.000 −0.539615
$$935$$ − 916.410i − 0.980118i
$$936$$ 0 0
$$937$$ −178.000 −0.189968 −0.0949840 0.995479i $$-0.530280\pi$$
−0.0949840 + 0.995479i $$0.530280\pi$$
$$938$$ − 45.2548i − 0.0482461i
$$939$$ 0 0
$$940$$ 720.000 0.765957
$$941$$ − 436.992i − 0.464391i −0.972669 0.232196i $$-0.925409\pi$$
0.972669 0.232196i $$-0.0745909\pi$$
$$942$$ 0 0
$$943$$ 792.000 0.839873
$$944$$ 135.765i 0.143818i
$$945$$ 0 0
$$946$$ 960.000 1.01480
$$947$$ 1798.88i 1.89956i 0.312924 + 0.949778i $$0.398691\pi$$
−0.312924 + 0.949778i $$0.601309\pi$$
$$948$$ 0 0
$$949$$ −128.000 −0.134879
$$950$$ − 158.392i − 0.166728i
$$951$$ 0 0
$$952$$ 144.000 0.151261
$$953$$ 1310.98i 1.37563i 0.725886 + 0.687815i $$0.241430\pi$$
−0.725886 + 0.687815i $$0.758570\pi$$
$$954$$ 0 0
$$955$$ −144.000 −0.150785
$$956$$ − 271.529i − 0.284026i
$$957$$ 0 0
$$958$$ −744.000 −0.776618
$$959$$ − 627.911i − 0.654756i
$$960$$ 0 0
$$961$$ 975.000 1.01457
$$962$$ − 384.666i − 0.399861i
$$963$$ 0 0
$$964$$ −64.0000 −0.0663900
$$965$$ − 873.984i − 0.905683i
$$966$$ 0 0
$$967$$ 1700.00 1.75801 0.879007 0.476808i $$-0.158206\pi$$
0.879007 + 0.476808i $$0.158206\pi$$
$$968$$ 472.347i 0.487962i
$$969$$ 0 0
$$970$$ 1056.00 1.08866
$$971$$ 458.205i 0.471890i 0.971766 + 0.235945i $$0.0758185\pi$$
−0.971766 + 0.235945i $$0.924181\pi$$
$$972$$ 0 0
$$973$$ −608.000 −0.624872
$$974$$ 842.871i 0.865371i
$$975$$ 0 0
$$976$$ 200.000 0.204918
$$977$$ 759.433i 0.777311i 0.921383 + 0.388655i $$0.127060\pi$$
−0.921383 + 0.388655i $$0.872940\pi$$
$$978$$ 0 0
$$979$$ −216.000 −0.220633
$$980$$ − 280.014i − 0.285729i
$$981$$ 0 0
$$982$$ 384.000 0.391039
$$983$$ − 1052.17i − 1.07037i −0.844734 0.535186i $$-0.820242\pi$$
0.844734 0.535186i $$-0.179758\pi$$
$$984$$ 0 0
$$985$$ 702.000 0.712690
$$986$$ 76.3675i 0.0774519i
$$987$$ 0 0
$$988$$ 256.000 0.259109
$$989$$ 678.823i 0.686373i
$$990$$ 0 0
$$991$$ −772.000 −0.779011 −0.389506 0.921024i $$-0.627354\pi$$
−0.389506 + 0.921024i $$0.627354\pi$$
$$992$$ 248.902i 0.250909i
$$993$$ 0 0
$$994$$ −288.000 −0.289738
$$995$$ − 84.8528i − 0.0852792i
$$996$$ 0 0
$$997$$ 194.000 0.194584 0.0972919 0.995256i $$-0.468982\pi$$
0.0972919 + 0.995256i $$0.468982\pi$$
$$998$$ 316.784i 0.317419i
$$999$$ 0 0
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 18.3.b.a.17.2 yes 2
3.2 odd 2 inner 18.3.b.a.17.1 2
4.3 odd 2 144.3.e.b.17.1 2
5.2 odd 4 450.3.b.b.449.1 4
5.3 odd 4 450.3.b.b.449.4 4
5.4 even 2 450.3.d.f.251.1 2
7.2 even 3 882.3.s.b.557.2 4
7.3 odd 6 882.3.s.d.863.1 4
7.4 even 3 882.3.s.b.863.1 4
7.5 odd 6 882.3.s.d.557.2 4
7.6 odd 2 882.3.b.a.197.2 2
8.3 odd 2 576.3.e.f.449.2 2
8.5 even 2 576.3.e.c.449.2 2
9.2 odd 6 162.3.d.b.53.1 4
9.4 even 3 162.3.d.b.107.1 4
9.5 odd 6 162.3.d.b.107.2 4
9.7 even 3 162.3.d.b.53.2 4
11.10 odd 2 2178.3.c.d.485.1 2
12.11 even 2 144.3.e.b.17.2 2
13.5 odd 4 3042.3.d.a.3041.3 4
13.8 odd 4 3042.3.d.a.3041.2 4
13.12 even 2 3042.3.c.e.1691.1 2
15.2 even 4 450.3.b.b.449.3 4
15.8 even 4 450.3.b.b.449.2 4
15.14 odd 2 450.3.d.f.251.2 2
16.3 odd 4 2304.3.h.c.2177.2 4
16.5 even 4 2304.3.h.f.2177.3 4
16.11 odd 4 2304.3.h.c.2177.3 4
16.13 even 4 2304.3.h.f.2177.2 4
20.3 even 4 3600.3.c.b.449.1 4
20.7 even 4 3600.3.c.b.449.3 4
20.19 odd 2 3600.3.l.d.1601.1 2
21.2 odd 6 882.3.s.b.557.1 4
21.5 even 6 882.3.s.d.557.1 4
21.11 odd 6 882.3.s.b.863.2 4
21.17 even 6 882.3.s.d.863.2 4
21.20 even 2 882.3.b.a.197.1 2
24.5 odd 2 576.3.e.c.449.1 2
24.11 even 2 576.3.e.f.449.1 2
33.32 even 2 2178.3.c.d.485.2 2
36.7 odd 6 1296.3.q.f.1025.2 4
36.11 even 6 1296.3.q.f.1025.1 4
36.23 even 6 1296.3.q.f.593.2 4
36.31 odd 6 1296.3.q.f.593.1 4
39.5 even 4 3042.3.d.a.3041.1 4
39.8 even 4 3042.3.d.a.3041.4 4
39.38 odd 2 3042.3.c.e.1691.2 2
48.5 odd 4 2304.3.h.f.2177.1 4
48.11 even 4 2304.3.h.c.2177.1 4
48.29 odd 4 2304.3.h.f.2177.4 4
48.35 even 4 2304.3.h.c.2177.4 4
60.23 odd 4 3600.3.c.b.449.2 4
60.47 odd 4 3600.3.c.b.449.4 4
60.59 even 2 3600.3.l.d.1601.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.b.a.17.1 2 3.2 odd 2 inner
18.3.b.a.17.2 yes 2 1.1 even 1 trivial
144.3.e.b.17.1 2 4.3 odd 2
144.3.e.b.17.2 2 12.11 even 2
162.3.d.b.53.1 4 9.2 odd 6
162.3.d.b.53.2 4 9.7 even 3
162.3.d.b.107.1 4 9.4 even 3
162.3.d.b.107.2 4 9.5 odd 6
450.3.b.b.449.1 4 5.2 odd 4
450.3.b.b.449.2 4 15.8 even 4
450.3.b.b.449.3 4 15.2 even 4
450.3.b.b.449.4 4 5.3 odd 4
450.3.d.f.251.1 2 5.4 even 2
450.3.d.f.251.2 2 15.14 odd 2
576.3.e.c.449.1 2 24.5 odd 2
576.3.e.c.449.2 2 8.5 even 2
576.3.e.f.449.1 2 24.11 even 2
576.3.e.f.449.2 2 8.3 odd 2
882.3.b.a.197.1 2 21.20 even 2
882.3.b.a.197.2 2 7.6 odd 2
882.3.s.b.557.1 4 21.2 odd 6
882.3.s.b.557.2 4 7.2 even 3
882.3.s.b.863.1 4 7.4 even 3
882.3.s.b.863.2 4 21.11 odd 6
882.3.s.d.557.1 4 21.5 even 6
882.3.s.d.557.2 4 7.5 odd 6
882.3.s.d.863.1 4 7.3 odd 6
882.3.s.d.863.2 4 21.17 even 6
1296.3.q.f.593.1 4 36.31 odd 6
1296.3.q.f.593.2 4 36.23 even 6
1296.3.q.f.1025.1 4 36.11 even 6
1296.3.q.f.1025.2 4 36.7 odd 6
2178.3.c.d.485.1 2 11.10 odd 2
2178.3.c.d.485.2 2 33.32 even 2
2304.3.h.c.2177.1 4 48.11 even 4
2304.3.h.c.2177.2 4 16.3 odd 4
2304.3.h.c.2177.3 4 16.11 odd 4
2304.3.h.c.2177.4 4 48.35 even 4
2304.3.h.f.2177.1 4 48.5 odd 4
2304.3.h.f.2177.2 4 16.13 even 4
2304.3.h.f.2177.3 4 16.5 even 4
2304.3.h.f.2177.4 4 48.29 odd 4
3042.3.c.e.1691.1 2 13.12 even 2
3042.3.c.e.1691.2 2 39.38 odd 2
3042.3.d.a.3041.1 4 39.5 even 4
3042.3.d.a.3041.2 4 13.8 odd 4
3042.3.d.a.3041.3 4 13.5 odd 4
3042.3.d.a.3041.4 4 39.8 even 4
3600.3.c.b.449.1 4 20.3 even 4
3600.3.c.b.449.2 4 60.23 odd 4
3600.3.c.b.449.3 4 20.7 even 4
3600.3.c.b.449.4 4 60.47 odd 4
3600.3.l.d.1601.1 2 20.19 odd 2
3600.3.l.d.1601.2 2 60.59 even 2