# Properties

 Label 507.4.b.k.337.16 Level $507$ Weight $4$ Character 507.337 Analytic conductor $29.914$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, not 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: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + 6703200 x^{4} + 137781 x^{2} + 729$$ x^18 + 112*x^16 + 5026*x^14 + 114847*x^12 + 1397921*x^10 + 8545747*x^8 + 21033277*x^6 + 6703200*x^4 + 137781*x^2 + 729 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$13^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.16 Root $$4.14324i$$ of defining polynomial Character $$\chi$$ $$=$$ 507.337 Dual form 507.4.b.k.337.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.69820i q^{2} +3.00000 q^{3} -14.0731 q^{4} +4.47249i q^{5} +14.0946i q^{6} -27.2096i q^{7} -28.5326i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+4.69820i q^{2} +3.00000 q^{3} -14.0731 q^{4} +4.47249i q^{5} +14.0946i q^{6} -27.2096i q^{7} -28.5326i q^{8} +9.00000 q^{9} -21.0127 q^{10} +5.99207i q^{11} -42.2193 q^{12} +127.836 q^{14} +13.4175i q^{15} +21.4672 q^{16} -105.037 q^{17} +42.2838i q^{18} -156.462i q^{19} -62.9418i q^{20} -81.6287i q^{21} -28.1519 q^{22} +175.423 q^{23} -85.5978i q^{24} +104.997 q^{25} +27.0000 q^{27} +382.923i q^{28} +204.886 q^{29} -63.0380 q^{30} +31.9570i q^{31} -127.404i q^{32} +17.9762i q^{33} -493.483i q^{34} +121.695 q^{35} -126.658 q^{36} -344.140i q^{37} +735.088 q^{38} +127.612 q^{40} +46.5921i q^{41} +383.508 q^{42} +173.286 q^{43} -84.3269i q^{44} +40.2524i q^{45} +824.175i q^{46} +265.613i q^{47} +64.4016 q^{48} -397.361 q^{49} +493.296i q^{50} -315.110 q^{51} +172.912 q^{53} +126.851i q^{54} -26.7995 q^{55} -776.360 q^{56} -469.385i q^{57} +962.596i q^{58} +137.566i q^{59} -188.825i q^{60} -58.9384 q^{61} -150.140 q^{62} -244.886i q^{63} +770.306 q^{64} -84.4558 q^{66} +211.668i q^{67} +1478.19 q^{68} +526.270 q^{69} +571.746i q^{70} -436.317i q^{71} -256.793i q^{72} -1159.11i q^{73} +1616.84 q^{74} +314.990 q^{75} +2201.90i q^{76} +163.042 q^{77} -1017.51 q^{79} +96.0118i q^{80} +81.0000 q^{81} -218.899 q^{82} -150.251i q^{83} +1148.77i q^{84} -469.775i q^{85} +814.131i q^{86} +614.659 q^{87} +170.969 q^{88} -565.984i q^{89} -189.114 q^{90} -2468.75 q^{92} +95.8709i q^{93} -1247.90 q^{94} +699.773 q^{95} -382.211i q^{96} -286.741i q^{97} -1866.88i q^{98} +53.9286i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10})$$ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 $$18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100})$$ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.69820i 1.66106i 0.556970 + 0.830532i $$0.311964\pi$$
−0.556970 + 0.830532i $$0.688036\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −14.0731 −1.75914
$$5$$ 4.47249i 0.400032i 0.979793 + 0.200016i $$0.0640994\pi$$
−0.979793 + 0.200016i $$0.935901\pi$$
$$6$$ 14.0946i 0.959016i
$$7$$ − 27.2096i − 1.46918i −0.678512 0.734589i $$-0.737375\pi$$
0.678512 0.734589i $$-0.262625\pi$$
$$8$$ − 28.5326i − 1.26098i
$$9$$ 9.00000 0.333333
$$10$$ −21.0127 −0.664479
$$11$$ 5.99207i 0.164243i 0.996622 + 0.0821216i $$0.0261696\pi$$
−0.996622 + 0.0821216i $$0.973830\pi$$
$$12$$ −42.2193 −1.01564
$$13$$ 0 0
$$14$$ 127.836 2.44040
$$15$$ 13.4175i 0.230958i
$$16$$ 21.4672 0.335425
$$17$$ −105.037 −1.49854 −0.749268 0.662267i $$-0.769594\pi$$
−0.749268 + 0.662267i $$0.769594\pi$$
$$18$$ 42.2838i 0.553688i
$$19$$ − 156.462i − 1.88920i −0.328227 0.944599i $$-0.606451\pi$$
0.328227 0.944599i $$-0.393549\pi$$
$$20$$ − 62.9418i − 0.703711i
$$21$$ − 81.6287i − 0.848231i
$$22$$ −28.1519 −0.272819
$$23$$ 175.423 1.59036 0.795181 0.606372i $$-0.207376\pi$$
0.795181 + 0.606372i $$0.207376\pi$$
$$24$$ − 85.5978i − 0.728024i
$$25$$ 104.997 0.839975
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 382.923i 2.58449i
$$29$$ 204.886 1.31195 0.655973 0.754785i $$-0.272259\pi$$
0.655973 + 0.754785i $$0.272259\pi$$
$$30$$ −63.0380 −0.383637
$$31$$ 31.9570i 0.185150i 0.995706 + 0.0925748i $$0.0295097\pi$$
−0.995706 + 0.0925748i $$0.970490\pi$$
$$32$$ − 127.404i − 0.703813i
$$33$$ 17.9762i 0.0948259i
$$34$$ − 493.483i − 2.48916i
$$35$$ 121.695 0.587718
$$36$$ −126.658 −0.586379
$$37$$ − 344.140i − 1.52909i −0.644571 0.764545i $$-0.722964\pi$$
0.644571 0.764545i $$-0.277036\pi$$
$$38$$ 735.088 3.13808
$$39$$ 0 0
$$40$$ 127.612 0.504430
$$41$$ 46.5921i 0.177475i 0.996055 + 0.0887373i $$0.0282832\pi$$
−0.996055 + 0.0887373i $$0.971717\pi$$
$$42$$ 383.508 1.40897
$$43$$ 173.286 0.614554 0.307277 0.951620i $$-0.400582\pi$$
0.307277 + 0.951620i $$0.400582\pi$$
$$44$$ − 84.3269i − 0.288926i
$$45$$ 40.2524i 0.133344i
$$46$$ 824.175i 2.64169i
$$47$$ 265.613i 0.824334i 0.911108 + 0.412167i $$0.135228\pi$$
−0.911108 + 0.412167i $$0.864772\pi$$
$$48$$ 64.4016 0.193658
$$49$$ −397.361 −1.15849
$$50$$ 493.296i 1.39525i
$$51$$ −315.110 −0.865180
$$52$$ 0 0
$$53$$ 172.912 0.448137 0.224068 0.974573i $$-0.428066\pi$$
0.224068 + 0.974573i $$0.428066\pi$$
$$54$$ 126.851i 0.319672i
$$55$$ −26.7995 −0.0657025
$$56$$ −776.360 −1.85260
$$57$$ − 469.385i − 1.09073i
$$58$$ 962.596i 2.17923i
$$59$$ 137.566i 0.303551i 0.988415 + 0.151775i $$0.0484991\pi$$
−0.988415 + 0.151775i $$0.951501\pi$$
$$60$$ − 188.825i − 0.406288i
$$61$$ −58.9384 −0.123710 −0.0618548 0.998085i $$-0.519702\pi$$
−0.0618548 + 0.998085i $$0.519702\pi$$
$$62$$ −150.140 −0.307546
$$63$$ − 244.886i − 0.489726i
$$64$$ 770.306 1.50450
$$65$$ 0 0
$$66$$ −84.4558 −0.157512
$$67$$ 211.668i 0.385961i 0.981203 + 0.192980i $$0.0618154\pi$$
−0.981203 + 0.192980i $$0.938185\pi$$
$$68$$ 1478.19 2.63613
$$69$$ 526.270 0.918196
$$70$$ 571.746i 0.976238i
$$71$$ − 436.317i − 0.729314i −0.931142 0.364657i $$-0.881186\pi$$
0.931142 0.364657i $$-0.118814\pi$$
$$72$$ − 256.793i − 0.420325i
$$73$$ − 1159.11i − 1.85840i −0.369579 0.929199i $$-0.620498\pi$$
0.369579 0.929199i $$-0.379502\pi$$
$$74$$ 1616.84 2.53992
$$75$$ 314.990 0.484960
$$76$$ 2201.90i 3.32336i
$$77$$ 163.042 0.241303
$$78$$ 0 0
$$79$$ −1017.51 −1.44910 −0.724548 0.689224i $$-0.757952\pi$$
−0.724548 + 0.689224i $$0.757952\pi$$
$$80$$ 96.0118i 0.134181i
$$81$$ 81.0000 0.111111
$$82$$ −218.899 −0.294797
$$83$$ − 150.251i − 0.198701i −0.995053 0.0993504i $$-0.968324\pi$$
0.995053 0.0993504i $$-0.0316765\pi$$
$$84$$ 1148.77i 1.49215i
$$85$$ − 469.775i − 0.599462i
$$86$$ 814.131i 1.02081i
$$87$$ 614.659 0.757452
$$88$$ 170.969 0.207107
$$89$$ − 565.984i − 0.674092i −0.941488 0.337046i $$-0.890572\pi$$
0.941488 0.337046i $$-0.109428\pi$$
$$90$$ −189.114 −0.221493
$$91$$ 0 0
$$92$$ −2468.75 −2.79766
$$93$$ 95.8709i 0.106896i
$$94$$ −1247.90 −1.36927
$$95$$ 699.773 0.755739
$$96$$ − 382.211i − 0.406346i
$$97$$ − 286.741i − 0.300145i −0.988675 0.150073i $$-0.952049\pi$$
0.988675 0.150073i $$-0.0479508\pi$$
$$98$$ − 1866.88i − 1.92432i
$$99$$ 53.9286i 0.0547478i
$$100$$ −1477.63 −1.47763
$$101$$ 1218.57 1.20052 0.600258 0.799807i $$-0.295065\pi$$
0.600258 + 0.799807i $$0.295065\pi$$
$$102$$ − 1480.45i − 1.43712i
$$103$$ −74.2485 −0.0710283 −0.0355142 0.999369i $$-0.511307\pi$$
−0.0355142 + 0.999369i $$0.511307\pi$$
$$104$$ 0 0
$$105$$ 365.084 0.339319
$$106$$ 812.374i 0.744384i
$$107$$ −1253.46 −1.13249 −0.566246 0.824237i $$-0.691605\pi$$
−0.566246 + 0.824237i $$0.691605\pi$$
$$108$$ −379.973 −0.338546
$$109$$ 722.643i 0.635015i 0.948256 + 0.317507i $$0.102846\pi$$
−0.948256 + 0.317507i $$0.897154\pi$$
$$110$$ − 125.909i − 0.109136i
$$111$$ − 1032.42i − 0.882820i
$$112$$ − 584.113i − 0.492799i
$$113$$ −855.913 −0.712544 −0.356272 0.934382i $$-0.615952\pi$$
−0.356272 + 0.934382i $$0.615952\pi$$
$$114$$ 2205.26 1.81177
$$115$$ 784.580i 0.636195i
$$116$$ −2883.38 −2.30789
$$117$$ 0 0
$$118$$ −646.310 −0.504218
$$119$$ 2858.00i 2.20162i
$$120$$ 382.836 0.291233
$$121$$ 1295.10 0.973024
$$122$$ − 276.905i − 0.205490i
$$123$$ 139.776i 0.102465i
$$124$$ − 449.733i − 0.325703i
$$125$$ 1028.66i 0.736048i
$$126$$ 1150.52 0.813467
$$127$$ −726.104 −0.507333 −0.253667 0.967292i $$-0.581637\pi$$
−0.253667 + 0.967292i $$0.581637\pi$$
$$128$$ 2599.82i 1.79526i
$$129$$ 519.857 0.354813
$$130$$ 0 0
$$131$$ −1456.91 −0.971685 −0.485843 0.874046i $$-0.661487\pi$$
−0.485843 + 0.874046i $$0.661487\pi$$
$$132$$ − 252.981i − 0.166812i
$$133$$ −4257.25 −2.77557
$$134$$ −994.460 −0.641106
$$135$$ 120.757i 0.0769862i
$$136$$ 2996.97i 1.88962i
$$137$$ − 1806.80i − 1.12675i −0.826200 0.563377i $$-0.809502\pi$$
0.826200 0.563377i $$-0.190498\pi$$
$$138$$ 2472.52i 1.52518i
$$139$$ 1229.28 0.750117 0.375059 0.927001i $$-0.377623\pi$$
0.375059 + 0.927001i $$0.377623\pi$$
$$140$$ −1712.62 −1.03388
$$141$$ 796.840i 0.475929i
$$142$$ 2049.90 1.21144
$$143$$ 0 0
$$144$$ 193.205 0.111808
$$145$$ 916.352i 0.524820i
$$146$$ 5445.71 3.08692
$$147$$ −1192.08 −0.668853
$$148$$ 4843.12i 2.68988i
$$149$$ 446.127i 0.245290i 0.992451 + 0.122645i $$0.0391376\pi$$
−0.992451 + 0.122645i $$0.960862\pi$$
$$150$$ 1479.89i 0.805549i
$$151$$ − 207.208i − 0.111671i −0.998440 0.0558355i $$-0.982218\pi$$
0.998440 0.0558355i $$-0.0177822\pi$$
$$152$$ −4464.26 −2.38223
$$153$$ −945.329 −0.499512
$$154$$ 766.002i 0.400819i
$$155$$ −142.927 −0.0740658
$$156$$ 0 0
$$157$$ 1096.60 0.557442 0.278721 0.960372i $$-0.410089\pi$$
0.278721 + 0.960372i $$0.410089\pi$$
$$158$$ − 4780.46i − 2.40704i
$$159$$ 518.735 0.258732
$$160$$ 569.812 0.281547
$$161$$ − 4773.20i − 2.33653i
$$162$$ 380.554i 0.184563i
$$163$$ 3154.25i 1.51571i 0.652425 + 0.757853i $$0.273752\pi$$
−0.652425 + 0.757853i $$0.726248\pi$$
$$164$$ − 655.694i − 0.312202i
$$165$$ −80.3984 −0.0379334
$$166$$ 705.908 0.330055
$$167$$ − 3679.65i − 1.70503i −0.522705 0.852514i $$-0.675077\pi$$
0.522705 0.852514i $$-0.324923\pi$$
$$168$$ −2329.08 −1.06960
$$169$$ 0 0
$$170$$ 2207.10 0.995745
$$171$$ − 1408.15i − 0.629733i
$$172$$ −2438.66 −1.08108
$$173$$ 3666.99 1.61154 0.805768 0.592231i $$-0.201753\pi$$
0.805768 + 0.592231i $$0.201753\pi$$
$$174$$ 2887.79i 1.25818i
$$175$$ − 2856.92i − 1.23407i
$$176$$ 128.633i 0.0550913i
$$177$$ 412.697i 0.175255i
$$178$$ 2659.11 1.11971
$$179$$ −4173.50 −1.74269 −0.871347 0.490666i $$-0.836753\pi$$
−0.871347 + 0.490666i $$0.836753\pi$$
$$180$$ − 566.476i − 0.234570i
$$181$$ 499.197 0.205000 0.102500 0.994733i $$-0.467316\pi$$
0.102500 + 0.994733i $$0.467316\pi$$
$$182$$ 0 0
$$183$$ −176.815 −0.0714238
$$184$$ − 5005.29i − 2.00541i
$$185$$ 1539.16 0.611685
$$186$$ −450.421 −0.177562
$$187$$ − 629.386i − 0.246124i
$$188$$ − 3738.00i − 1.45012i
$$189$$ − 734.659i − 0.282744i
$$190$$ 3287.68i 1.25533i
$$191$$ −3086.68 −1.16934 −0.584671 0.811271i $$-0.698776\pi$$
−0.584671 + 0.811271i $$0.698776\pi$$
$$192$$ 2310.92 0.868625
$$193$$ − 1644.14i − 0.613201i −0.951838 0.306601i $$-0.900808\pi$$
0.951838 0.306601i $$-0.0991916\pi$$
$$194$$ 1347.17 0.498561
$$195$$ 0 0
$$196$$ 5592.10 2.03794
$$197$$ − 1371.21i − 0.495911i −0.968771 0.247956i $$-0.920241\pi$$
0.968771 0.247956i $$-0.0797587\pi$$
$$198$$ −253.367 −0.0909396
$$199$$ 4627.33 1.64836 0.824178 0.566332i $$-0.191638\pi$$
0.824178 + 0.566332i $$0.191638\pi$$
$$200$$ − 2995.83i − 1.05919i
$$201$$ 635.004i 0.222835i
$$202$$ 5725.08i 1.99413i
$$203$$ − 5574.87i − 1.92748i
$$204$$ 4434.57 1.52197
$$205$$ −208.383 −0.0709955
$$206$$ − 348.834i − 0.117983i
$$207$$ 1578.81 0.530121
$$208$$ 0 0
$$209$$ 937.528 0.310288
$$210$$ 1715.24i 0.563631i
$$211$$ 5088.31 1.66016 0.830080 0.557644i $$-0.188294\pi$$
0.830080 + 0.557644i $$0.188294\pi$$
$$212$$ −2433.40 −0.788334
$$213$$ − 1308.95i − 0.421069i
$$214$$ − 5889.01i − 1.88114i
$$215$$ 775.019i 0.245841i
$$216$$ − 770.380i − 0.242675i
$$217$$ 869.535 0.272018
$$218$$ −3395.12 −1.05480
$$219$$ − 3477.32i − 1.07295i
$$220$$ 377.151 0.115580
$$221$$ 0 0
$$222$$ 4850.52 1.46642
$$223$$ 4744.82i 1.42483i 0.701759 + 0.712415i $$0.252398\pi$$
−0.701759 + 0.712415i $$0.747602\pi$$
$$224$$ −3466.60 −1.03403
$$225$$ 944.971 0.279992
$$226$$ − 4021.25i − 1.18358i
$$227$$ 1145.52i 0.334937i 0.985877 + 0.167469i $$0.0535593\pi$$
−0.985877 + 0.167469i $$0.946441\pi$$
$$228$$ 6605.70i 1.91874i
$$229$$ − 1348.47i − 0.389123i −0.980890 0.194561i $$-0.937672\pi$$
0.980890 0.194561i $$-0.0623283\pi$$
$$230$$ −3686.12 −1.05676
$$231$$ 489.125 0.139316
$$232$$ − 5845.94i − 1.65433i
$$233$$ 952.002 0.267673 0.133836 0.991003i $$-0.457270\pi$$
0.133836 + 0.991003i $$0.457270\pi$$
$$234$$ 0 0
$$235$$ −1187.95 −0.329760
$$236$$ − 1935.97i − 0.533987i
$$237$$ −3052.52 −0.836636
$$238$$ −13427.5 −3.65703
$$239$$ − 3069.12i − 0.830647i −0.909674 0.415323i $$-0.863668\pi$$
0.909674 0.415323i $$-0.136332\pi$$
$$240$$ 288.036i 0.0774692i
$$241$$ − 2508.17i − 0.670396i −0.942148 0.335198i $$-0.891197\pi$$
0.942148 0.335198i $$-0.108803\pi$$
$$242$$ 6084.62i 1.61626i
$$243$$ 243.000 0.0641500
$$244$$ 829.446 0.217622
$$245$$ − 1777.19i − 0.463432i
$$246$$ −656.697 −0.170201
$$247$$ 0 0
$$248$$ 911.815 0.233469
$$249$$ − 450.752i − 0.114720i
$$250$$ −4832.85 −1.22262
$$251$$ −3405.91 −0.856490 −0.428245 0.903663i $$-0.640868\pi$$
−0.428245 + 0.903663i $$0.640868\pi$$
$$252$$ 3446.31i 0.861495i
$$253$$ 1051.15i 0.261206i
$$254$$ − 3411.38i − 0.842714i
$$255$$ − 1409.33i − 0.346100i
$$256$$ −6052.04 −1.47755
$$257$$ 4733.95 1.14901 0.574506 0.818501i $$-0.305194\pi$$
0.574506 + 0.818501i $$0.305194\pi$$
$$258$$ 2442.39i 0.589367i
$$259$$ −9363.91 −2.24651
$$260$$ 0 0
$$261$$ 1843.98 0.437315
$$262$$ − 6844.85i − 1.61403i
$$263$$ −1866.11 −0.437526 −0.218763 0.975778i $$-0.570202\pi$$
−0.218763 + 0.975778i $$0.570202\pi$$
$$264$$ 512.908 0.119573
$$265$$ 773.346i 0.179269i
$$266$$ − 20001.4i − 4.61040i
$$267$$ − 1697.95i − 0.389187i
$$268$$ − 2978.83i − 0.678958i
$$269$$ 1580.19 0.358164 0.179082 0.983834i $$-0.442687\pi$$
0.179082 + 0.983834i $$0.442687\pi$$
$$270$$ −567.342 −0.127879
$$271$$ − 4922.09i − 1.10330i −0.834074 0.551652i $$-0.813998\pi$$
0.834074 0.551652i $$-0.186002\pi$$
$$272$$ −2254.84 −0.502646
$$273$$ 0 0
$$274$$ 8488.71 1.87161
$$275$$ 629.148i 0.137960i
$$276$$ −7406.25 −1.61523
$$277$$ −2687.08 −0.582856 −0.291428 0.956593i $$-0.594130\pi$$
−0.291428 + 0.956593i $$0.594130\pi$$
$$278$$ 5775.41i 1.24599i
$$279$$ 287.613i 0.0617165i
$$280$$ − 3472.26i − 0.741098i
$$281$$ − 883.753i − 0.187617i −0.995590 0.0938083i $$-0.970096\pi$$
0.995590 0.0938083i $$-0.0299041\pi$$
$$282$$ −3743.71 −0.790550
$$283$$ 469.776 0.0986759 0.0493379 0.998782i $$-0.484289\pi$$
0.0493379 + 0.998782i $$0.484289\pi$$
$$284$$ 6140.33i 1.28296i
$$285$$ 2099.32 0.436326
$$286$$ 0 0
$$287$$ 1267.75 0.260742
$$288$$ − 1146.63i − 0.234604i
$$289$$ 6119.68 1.24561
$$290$$ −4305.20 −0.871760
$$291$$ − 860.222i − 0.173289i
$$292$$ 16312.2i 3.26918i
$$293$$ 3403.76i 0.678668i 0.940666 + 0.339334i $$0.110202\pi$$
−0.940666 + 0.339334i $$0.889798\pi$$
$$294$$ − 5600.65i − 1.11101i
$$295$$ −615.261 −0.121430
$$296$$ −9819.22 −1.92814
$$297$$ 161.786i 0.0316086i
$$298$$ −2095.99 −0.407442
$$299$$ 0 0
$$300$$ −4432.89 −0.853110
$$301$$ − 4715.03i − 0.902889i
$$302$$ 973.504 0.185493
$$303$$ 3655.70 0.693118
$$304$$ − 3358.79i − 0.633684i
$$305$$ − 263.602i − 0.0494878i
$$306$$ − 4441.35i − 0.829722i
$$307$$ 888.862i 0.165244i 0.996581 + 0.0826222i $$0.0263295\pi$$
−0.996581 + 0.0826222i $$0.973671\pi$$
$$308$$ −2294.50 −0.424484
$$309$$ −222.745 −0.0410082
$$310$$ − 671.501i − 0.123028i
$$311$$ 1218.36 0.222145 0.111072 0.993812i $$-0.464571\pi$$
0.111072 + 0.993812i $$0.464571\pi$$
$$312$$ 0 0
$$313$$ −4870.41 −0.879526 −0.439763 0.898114i $$-0.644938\pi$$
−0.439763 + 0.898114i $$0.644938\pi$$
$$314$$ 5152.06i 0.925948i
$$315$$ 1095.25 0.195906
$$316$$ 14319.5 2.54916
$$317$$ 4340.93i 0.769120i 0.923100 + 0.384560i $$0.125647\pi$$
−0.923100 + 0.384560i $$0.874353\pi$$
$$318$$ 2437.12i 0.429770i
$$319$$ 1227.69i 0.215478i
$$320$$ 3445.19i 0.601849i
$$321$$ −3760.38 −0.653844
$$322$$ 22425.4 3.88112
$$323$$ 16434.2i 2.83103i
$$324$$ −1139.92 −0.195460
$$325$$ 0 0
$$326$$ −14819.3 −2.51769
$$327$$ 2167.93i 0.366626i
$$328$$ 1329.39 0.223791
$$329$$ 7227.23 1.21109
$$330$$ − 377.728i − 0.0630098i
$$331$$ 907.289i 0.150662i 0.997159 + 0.0753310i $$0.0240013\pi$$
−0.997159 + 0.0753310i $$0.975999\pi$$
$$332$$ 2114.49i 0.349542i
$$333$$ − 3097.26i − 0.509697i
$$334$$ 17287.7 2.83216
$$335$$ −946.684 −0.154397
$$336$$ − 1752.34i − 0.284518i
$$337$$ −8660.88 −1.39997 −0.699983 0.714160i $$-0.746809\pi$$
−0.699983 + 0.714160i $$0.746809\pi$$
$$338$$ 0 0
$$339$$ −2567.74 −0.411388
$$340$$ 6611.19i 1.05454i
$$341$$ −191.488 −0.0304096
$$342$$ 6615.79 1.04603
$$343$$ 1479.14i 0.232846i
$$344$$ − 4944.29i − 0.774937i
$$345$$ 2353.74i 0.367308i
$$346$$ 17228.2i 2.67687i
$$347$$ −347.605 −0.0537763 −0.0268882 0.999638i $$-0.508560\pi$$
−0.0268882 + 0.999638i $$0.508560\pi$$
$$348$$ −8650.15 −1.33246
$$349$$ − 10970.2i − 1.68258i −0.540581 0.841292i $$-0.681795\pi$$
0.540581 0.841292i $$-0.318205\pi$$
$$350$$ 13422.4 2.04988
$$351$$ 0 0
$$352$$ 763.411 0.115596
$$353$$ − 10384.8i − 1.56580i −0.622146 0.782901i $$-0.713739\pi$$
0.622146 0.782901i $$-0.286261\pi$$
$$354$$ −1938.93 −0.291110
$$355$$ 1951.42 0.291749
$$356$$ 7965.15i 1.18582i
$$357$$ 8574.00i 1.27110i
$$358$$ − 19608.0i − 2.89473i
$$359$$ 8665.80i 1.27399i 0.770867 + 0.636996i $$0.219823\pi$$
−0.770867 + 0.636996i $$0.780177\pi$$
$$360$$ 1148.51 0.168143
$$361$$ −17621.2 −2.56907
$$362$$ 2345.33i 0.340518i
$$363$$ 3885.29 0.561776
$$364$$ 0 0
$$365$$ 5184.09 0.743419
$$366$$ − 830.714i − 0.118640i
$$367$$ 1234.22 0.175547 0.0877734 0.996140i $$-0.472025\pi$$
0.0877734 + 0.996140i $$0.472025\pi$$
$$368$$ 3765.85 0.533447
$$369$$ 419.329i 0.0591582i
$$370$$ 7231.31i 1.01605i
$$371$$ − 4704.85i − 0.658393i
$$372$$ − 1349.20i − 0.188045i
$$373$$ −427.483 −0.0593410 −0.0296705 0.999560i $$-0.509446\pi$$
−0.0296705 + 0.999560i $$0.509446\pi$$
$$374$$ 2956.98 0.408829
$$375$$ 3085.98i 0.424958i
$$376$$ 7578.64 1.03946
$$377$$ 0 0
$$378$$ 3451.57 0.469656
$$379$$ 124.241i 0.0168386i 0.999965 + 0.00841929i $$0.00267998\pi$$
−0.999965 + 0.00841929i $$0.997320\pi$$
$$380$$ −9847.98 −1.32945
$$381$$ −2178.31 −0.292909
$$382$$ − 14501.8i − 1.94235i
$$383$$ 9341.21i 1.24625i 0.782122 + 0.623125i $$0.214137\pi$$
−0.782122 + 0.623125i $$0.785863\pi$$
$$384$$ 7799.46i 1.03650i
$$385$$ 729.202i 0.0965288i
$$386$$ 7724.51 1.01857
$$387$$ 1559.57 0.204851
$$388$$ 4035.33i 0.527997i
$$389$$ −11368.9 −1.48182 −0.740908 0.671607i $$-0.765604\pi$$
−0.740908 + 0.671607i $$0.765604\pi$$
$$390$$ 0 0
$$391$$ −18425.9 −2.38321
$$392$$ 11337.7i 1.46082i
$$393$$ −4370.73 −0.561003
$$394$$ 6442.21 0.823741
$$395$$ − 4550.80i − 0.579685i
$$396$$ − 758.942i − 0.0963088i
$$397$$ 12077.8i 1.52687i 0.645883 + 0.763436i $$0.276489\pi$$
−0.645883 + 0.763436i $$0.723511\pi$$
$$398$$ 21740.1i 2.73802i
$$399$$ −12771.8 −1.60248
$$400$$ 2253.99 0.281748
$$401$$ 4856.74i 0.604823i 0.953178 + 0.302411i $$0.0977916\pi$$
−0.953178 + 0.302411i $$0.902208\pi$$
$$402$$ −2983.38 −0.370143
$$403$$ 0 0
$$404$$ −17149.0 −2.11187
$$405$$ 362.272i 0.0444480i
$$406$$ 26191.8 3.20167
$$407$$ 2062.11 0.251143
$$408$$ 8990.90i 1.09097i
$$409$$ 2981.80i 0.360490i 0.983622 + 0.180245i $$0.0576890\pi$$
−0.983622 + 0.180245i $$0.942311\pi$$
$$410$$ − 979.023i − 0.117928i
$$411$$ − 5420.40i − 0.650532i
$$412$$ 1044.91 0.124949
$$413$$ 3743.10 0.445971
$$414$$ 7417.57i 0.880565i
$$415$$ 671.995 0.0794866
$$416$$ 0 0
$$417$$ 3687.84 0.433080
$$418$$ 4404.70i 0.515409i
$$419$$ 7774.01 0.906408 0.453204 0.891407i $$-0.350281\pi$$
0.453204 + 0.891407i $$0.350281\pi$$
$$420$$ −5137.86 −0.596909
$$421$$ 3959.71i 0.458396i 0.973380 + 0.229198i $$0.0736103\pi$$
−0.973380 + 0.229198i $$0.926390\pi$$
$$422$$ 23905.9i 2.75763i
$$423$$ 2390.52i 0.274778i
$$424$$ − 4933.62i − 0.565089i
$$425$$ −11028.5 −1.25873
$$426$$ 6149.71 0.699424
$$427$$ 1603.69i 0.181752i
$$428$$ 17640.1 1.99221
$$429$$ 0 0
$$430$$ −3641.19 −0.408358
$$431$$ 4375.12i 0.488961i 0.969654 + 0.244480i $$0.0786174\pi$$
−0.969654 + 0.244480i $$0.921383\pi$$
$$432$$ 579.614 0.0645525
$$433$$ −8992.74 −0.998068 −0.499034 0.866582i $$-0.666312\pi$$
−0.499034 + 0.866582i $$0.666312\pi$$
$$434$$ 4085.25i 0.451839i
$$435$$ 2749.06i 0.303005i
$$436$$ − 10169.8i − 1.11708i
$$437$$ − 27447.0i − 3.00451i
$$438$$ 16337.1 1.78223
$$439$$ 195.465 0.0212506 0.0106253 0.999944i $$-0.496618\pi$$
0.0106253 + 0.999944i $$0.496618\pi$$
$$440$$ 764.659i 0.0828493i
$$441$$ −3576.25 −0.386162
$$442$$ 0 0
$$443$$ 7369.97 0.790424 0.395212 0.918590i $$-0.370671\pi$$
0.395212 + 0.918590i $$0.370671\pi$$
$$444$$ 14529.4i 1.55300i
$$445$$ 2531.36 0.269658
$$446$$ −22292.1 −2.36673
$$447$$ 1338.38i 0.141618i
$$448$$ − 20959.7i − 2.21038i
$$449$$ 164.281i 0.0172670i 0.999963 + 0.00863351i $$0.00274817\pi$$
−0.999963 + 0.00863351i $$0.997252\pi$$
$$450$$ 4439.67i 0.465084i
$$451$$ −279.183 −0.0291490
$$452$$ 12045.3 1.25346
$$453$$ − 621.623i − 0.0644733i
$$454$$ −5381.88 −0.556352
$$455$$ 0 0
$$456$$ −13392.8 −1.37538
$$457$$ 11687.0i 1.19627i 0.801397 + 0.598133i $$0.204091\pi$$
−0.801397 + 0.598133i $$0.795909\pi$$
$$458$$ 6335.36 0.646358
$$459$$ −2835.99 −0.288393
$$460$$ − 11041.5i − 1.11915i
$$461$$ 1057.53i 0.106842i 0.998572 + 0.0534211i $$0.0170126\pi$$
−0.998572 + 0.0534211i $$0.982987\pi$$
$$462$$ 2298.01i 0.231413i
$$463$$ − 8554.74i − 0.858688i −0.903141 0.429344i $$-0.858745\pi$$
0.903141 0.429344i $$-0.141255\pi$$
$$464$$ 4398.33 0.440059
$$465$$ −428.782 −0.0427619
$$466$$ 4472.70i 0.444622i
$$467$$ 7705.26 0.763506 0.381753 0.924264i $$-0.375321\pi$$
0.381753 + 0.924264i $$0.375321\pi$$
$$468$$ 0 0
$$469$$ 5759.40 0.567046
$$470$$ − 5581.24i − 0.547752i
$$471$$ 3289.81 0.321839
$$472$$ 3925.10 0.382770
$$473$$ 1038.34i 0.100936i
$$474$$ − 14341.4i − 1.38971i
$$475$$ − 16428.0i − 1.58688i
$$476$$ − 40220.9i − 3.87295i
$$477$$ 1556.21 0.149379
$$478$$ 14419.3 1.37976
$$479$$ 4508.59i 0.430069i 0.976606 + 0.215034i $$0.0689863\pi$$
−0.976606 + 0.215034i $$0.931014\pi$$
$$480$$ 1709.44 0.162551
$$481$$ 0 0
$$482$$ 11783.9 1.11357
$$483$$ − 14319.6i − 1.34899i
$$484$$ −18226.0 −1.71168
$$485$$ 1282.45 0.120068
$$486$$ 1141.66i 0.106557i
$$487$$ 6725.73i 0.625815i 0.949784 + 0.312908i $$0.101303\pi$$
−0.949784 + 0.312908i $$0.898697\pi$$
$$488$$ 1681.67i 0.155995i
$$489$$ 9462.76i 0.875094i
$$490$$ 8349.61 0.769790
$$491$$ −11517.5 −1.05861 −0.529303 0.848433i $$-0.677546\pi$$
−0.529303 + 0.848433i $$0.677546\pi$$
$$492$$ − 1967.08i − 0.180250i
$$493$$ −21520.5 −1.96600
$$494$$ 0 0
$$495$$ −241.195 −0.0219008
$$496$$ 686.026i 0.0621038i
$$497$$ −11872.0 −1.07149
$$498$$ 2117.72 0.190557
$$499$$ 19907.9i 1.78598i 0.450080 + 0.892988i $$0.351395\pi$$
−0.450080 + 0.892988i $$0.648605\pi$$
$$500$$ − 14476.4i − 1.29481i
$$501$$ − 11038.9i − 0.984398i
$$502$$ − 16001.6i − 1.42269i
$$503$$ −5735.48 −0.508415 −0.254207 0.967150i $$-0.581815\pi$$
−0.254207 + 0.967150i $$0.581815\pi$$
$$504$$ −6987.24 −0.617533
$$505$$ 5450.04i 0.480244i
$$506$$ −4938.51 −0.433881
$$507$$ 0 0
$$508$$ 10218.5 0.892469
$$509$$ 9253.84i 0.805834i 0.915237 + 0.402917i $$0.132004\pi$$
−0.915237 + 0.402917i $$0.867996\pi$$
$$510$$ 6621.29 0.574894
$$511$$ −31538.8 −2.73032
$$512$$ − 7635.12i − 0.659039i
$$513$$ − 4224.46i − 0.363576i
$$514$$ 22241.1i 1.90858i
$$515$$ − 332.076i − 0.0284136i
$$516$$ −7315.99 −0.624164
$$517$$ −1591.57 −0.135391
$$518$$ − 43993.5i − 3.73159i
$$519$$ 11001.0 0.930421
$$520$$ 0 0
$$521$$ 3887.42 0.326892 0.163446 0.986552i $$-0.447739\pi$$
0.163446 + 0.986552i $$0.447739\pi$$
$$522$$ 8663.37i 0.726409i
$$523$$ 4782.27 0.399836 0.199918 0.979813i $$-0.435932\pi$$
0.199918 + 0.979813i $$0.435932\pi$$
$$524$$ 20503.2 1.70933
$$525$$ − 8570.76i − 0.712492i
$$526$$ − 8767.37i − 0.726759i
$$527$$ − 3356.65i − 0.277453i
$$528$$ 385.898i 0.0318070i
$$529$$ 18606.4 1.52925
$$530$$ −3633.34 −0.297777
$$531$$ 1238.09i 0.101184i
$$532$$ 59912.7 4.88261
$$533$$ 0 0
$$534$$ 7977.32 0.646465
$$535$$ − 5606.09i − 0.453033i
$$536$$ 6039.44 0.486687
$$537$$ −12520.5 −1.00615
$$538$$ 7424.06i 0.594933i
$$539$$ − 2381.01i − 0.190274i
$$540$$ − 1699.43i − 0.135429i
$$541$$ 14872.6i 1.18192i 0.806699 + 0.590962i $$0.201252\pi$$
−0.806699 + 0.590962i $$0.798748\pi$$
$$542$$ 23125.0 1.83266
$$543$$ 1497.59 0.118357
$$544$$ 13382.0i 1.05469i
$$545$$ −3232.01 −0.254026
$$546$$ 0 0
$$547$$ 16965.3 1.32611 0.663055 0.748570i $$-0.269259\pi$$
0.663055 + 0.748570i $$0.269259\pi$$
$$548$$ 25427.3i 1.98212i
$$549$$ −530.446 −0.0412366
$$550$$ −2955.86 −0.229161
$$551$$ − 32056.8i − 2.47852i
$$552$$ − 15015.9i − 1.15782i
$$553$$ 27686.0i 2.12898i
$$554$$ − 12624.5i − 0.968162i
$$555$$ 4617.49 0.353156
$$556$$ −17299.8 −1.31956
$$557$$ − 1934.05i − 0.147125i −0.997291 0.0735623i $$-0.976563\pi$$
0.997291 0.0735623i $$-0.0234368\pi$$
$$558$$ −1351.26 −0.102515
$$559$$ 0 0
$$560$$ 2612.44 0.197135
$$561$$ − 1888.16i − 0.142100i
$$562$$ 4152.05 0.311643
$$563$$ −6592.13 −0.493473 −0.246736 0.969083i $$-0.579358\pi$$
−0.246736 + 0.969083i $$0.579358\pi$$
$$564$$ − 11214.0i − 0.837225i
$$565$$ − 3828.06i − 0.285040i
$$566$$ 2207.10i 0.163907i
$$567$$ − 2203.98i − 0.163242i
$$568$$ −12449.3 −0.919646
$$569$$ 14477.7 1.06667 0.533336 0.845903i $$-0.320938\pi$$
0.533336 + 0.845903i $$0.320938\pi$$
$$570$$ 9863.03i 0.724766i
$$571$$ 4998.21 0.366320 0.183160 0.983083i $$-0.441367\pi$$
0.183160 + 0.983083i $$0.441367\pi$$
$$572$$ 0 0
$$573$$ −9260.04 −0.675120
$$574$$ 5956.15i 0.433109i
$$575$$ 18418.9 1.33586
$$576$$ 6932.75 0.501501
$$577$$ 9291.73i 0.670398i 0.942147 + 0.335199i $$0.108804\pi$$
−0.942147 + 0.335199i $$0.891196\pi$$
$$578$$ 28751.5i 2.06904i
$$579$$ − 4932.42i − 0.354032i
$$580$$ − 12895.9i − 0.923230i
$$581$$ −4088.26 −0.291927
$$582$$ 4041.50 0.287844
$$583$$ 1036.10i 0.0736034i
$$584$$ −33072.3 −2.34339
$$585$$ 0 0
$$586$$ −15991.5 −1.12731
$$587$$ − 5602.64i − 0.393945i −0.980409 0.196972i $$-0.936889\pi$$
0.980409 0.196972i $$-0.0631109\pi$$
$$588$$ 16776.3 1.17660
$$589$$ 5000.04 0.349784
$$590$$ − 2890.62i − 0.201703i
$$591$$ − 4113.62i − 0.286314i
$$592$$ − 7387.73i − 0.512895i
$$593$$ 10885.8i 0.753839i 0.926246 + 0.376919i $$0.123017\pi$$
−0.926246 + 0.376919i $$0.876983\pi$$
$$594$$ −760.102 −0.0525040
$$595$$ −12782.4 −0.880717
$$596$$ − 6278.38i − 0.431498i
$$597$$ 13882.0 0.951678
$$598$$ 0 0
$$599$$ 20403.2 1.39174 0.695872 0.718166i $$-0.255018\pi$$
0.695872 + 0.718166i $$0.255018\pi$$
$$600$$ − 8987.50i − 0.611522i
$$601$$ −6312.19 −0.428419 −0.214209 0.976788i $$-0.568717\pi$$
−0.214209 + 0.976788i $$0.568717\pi$$
$$602$$ 22152.2 1.49976
$$603$$ 1905.01i 0.128654i
$$604$$ 2916.05i 0.196445i
$$605$$ 5792.30i 0.389241i
$$606$$ 17175.2i 1.15131i
$$607$$ −21848.6 −1.46097 −0.730484 0.682930i $$-0.760705\pi$$
−0.730484 + 0.682930i $$0.760705\pi$$
$$608$$ −19933.8 −1.32964
$$609$$ − 16724.6i − 1.11283i
$$610$$ 1238.45 0.0822025
$$611$$ 0 0
$$612$$ 13303.7 0.878710
$$613$$ − 1335.14i − 0.0879704i −0.999032 0.0439852i $$-0.985995\pi$$
0.999032 0.0439852i $$-0.0140054\pi$$
$$614$$ −4176.05 −0.274482
$$615$$ −625.148 −0.0409893
$$616$$ − 4652.00i − 0.304277i
$$617$$ 18908.3i 1.23374i 0.787064 + 0.616871i $$0.211600\pi$$
−0.787064 + 0.616871i $$0.788400\pi$$
$$618$$ − 1046.50i − 0.0681173i
$$619$$ − 6722.49i − 0.436510i −0.975892 0.218255i $$-0.929964\pi$$
0.975892 0.218255i $$-0.0700365\pi$$
$$620$$ 2011.43 0.130292
$$621$$ 4736.43 0.306065
$$622$$ 5724.11i 0.368997i
$$623$$ −15400.2 −0.990362
$$624$$ 0 0
$$625$$ 8523.93 0.545532
$$626$$ − 22882.2i − 1.46095i
$$627$$ 2812.59 0.179145
$$628$$ −15432.6 −0.980617
$$629$$ 36147.3i 2.29140i
$$630$$ 5145.71i 0.325413i
$$631$$ 8098.44i 0.510925i 0.966819 + 0.255463i $$0.0822278\pi$$
−0.966819 + 0.255463i $$0.917772\pi$$
$$632$$ 29032.2i 1.82727i
$$633$$ 15264.9 0.958494
$$634$$ −20394.6 −1.27756
$$635$$ − 3247.50i − 0.202950i
$$636$$ −7300.21 −0.455145
$$637$$ 0 0
$$638$$ −5767.94 −0.357923
$$639$$ − 3926.85i − 0.243105i
$$640$$ −11627.7 −0.718163
$$641$$ 10955.9 0.675090 0.337545 0.941309i $$-0.390403\pi$$
0.337545 + 0.941309i $$0.390403\pi$$
$$642$$ − 17667.0i − 1.08608i
$$643$$ − 28125.0i − 1.72495i −0.506104 0.862473i $$-0.668915\pi$$
0.506104 0.862473i $$-0.331085\pi$$
$$644$$ 67173.7i 4.11027i
$$645$$ 2325.06i 0.141936i
$$646$$ −77211.1 −4.70253
$$647$$ 29001.4 1.76223 0.881115 0.472901i $$-0.156793\pi$$
0.881115 + 0.472901i $$0.156793\pi$$
$$648$$ − 2311.14i − 0.140108i
$$649$$ −824.302 −0.0498562
$$650$$ 0 0
$$651$$ 2608.61 0.157050
$$652$$ − 44390.1i − 2.66634i
$$653$$ 19506.3 1.16898 0.584488 0.811402i $$-0.301295\pi$$
0.584488 + 0.811402i $$0.301295\pi$$
$$654$$ −10185.4 −0.608989
$$655$$ − 6516.01i − 0.388705i
$$656$$ 1000.20i 0.0595294i
$$657$$ − 10432.0i − 0.619466i
$$658$$ 33955.0i 2.01171i
$$659$$ 5985.86 0.353833 0.176917 0.984226i $$-0.443388\pi$$
0.176917 + 0.984226i $$0.443388\pi$$
$$660$$ 1131.45 0.0667300
$$661$$ 280.836i 0.0165254i 0.999966 + 0.00826268i $$0.00263012\pi$$
−0.999966 + 0.00826268i $$0.997370\pi$$
$$662$$ −4262.63 −0.250259
$$663$$ 0 0
$$664$$ −4287.04 −0.250557
$$665$$ − 19040.5i − 1.11032i
$$666$$ 14551.6 0.846639
$$667$$ 35941.8 2.08647
$$668$$ 51784.0i 2.99938i
$$669$$ 14234.5i 0.822626i
$$670$$ − 4447.71i − 0.256463i
$$671$$ − 353.163i − 0.0203185i
$$672$$ −10399.8 −0.596996
$$673$$ 14868.9 0.851640 0.425820 0.904808i $$-0.359986\pi$$
0.425820 + 0.904808i $$0.359986\pi$$
$$674$$ − 40690.6i − 2.32543i
$$675$$ 2834.91 0.161653
$$676$$ 0 0
$$677$$ −20769.5 −1.17908 −0.589540 0.807739i $$-0.700691\pi$$
−0.589540 + 0.807739i $$0.700691\pi$$
$$678$$ − 12063.7i − 0.683341i
$$679$$ −7802.09 −0.440967
$$680$$ −13403.9 −0.755907
$$681$$ 3436.56i 0.193376i
$$682$$ − 899.650i − 0.0505123i
$$683$$ − 14980.8i − 0.839275i −0.907692 0.419638i $$-0.862157\pi$$
0.907692 0.419638i $$-0.137843\pi$$
$$684$$ 19817.1i 1.10779i
$$685$$ 8080.90 0.450738
$$686$$ −6949.30 −0.386772
$$687$$ − 4045.40i − 0.224660i
$$688$$ 3719.96 0.206137
$$689$$ 0 0
$$690$$ −11058.3 −0.610122
$$691$$ 9472.45i 0.521489i 0.965408 + 0.260745i $$0.0839680\pi$$
−0.965408 + 0.260745i $$0.916032\pi$$
$$692$$ −51605.8 −2.83491
$$693$$ 1467.37 0.0804342
$$694$$ − 1633.12i − 0.0893260i
$$695$$ 5497.95i 0.300071i
$$696$$ − 17537.8i − 0.955128i
$$697$$ − 4893.87i − 0.265952i
$$698$$ 51540.3 2.79488
$$699$$ 2856.01 0.154541
$$700$$ 40205.7i 2.17090i
$$701$$ −1035.34 −0.0557834 −0.0278917 0.999611i $$-0.508879\pi$$
−0.0278917 + 0.999611i $$0.508879\pi$$
$$702$$ 0 0
$$703$$ −53844.8 −2.88875
$$704$$ 4615.72i 0.247105i
$$705$$ −3563.86 −0.190387
$$706$$ 48790.0 2.60090
$$707$$ − 33156.7i − 1.76377i
$$708$$ − 5807.92i − 0.308298i
$$709$$ − 16800.4i − 0.889917i −0.895551 0.444958i $$-0.853219\pi$$
0.895551 0.444958i $$-0.146781\pi$$
$$710$$ 9168.18i 0.484614i
$$711$$ −9157.57 −0.483032
$$712$$ −16149.0 −0.850013
$$713$$ 5606.00i 0.294455i
$$714$$ −40282.4 −2.11139
$$715$$ 0 0
$$716$$ 58734.1 3.06564
$$717$$ − 9207.35i − 0.479574i
$$718$$ −40713.7 −2.11618
$$719$$ −14873.1 −0.771449 −0.385724 0.922614i $$-0.626048\pi$$
−0.385724 + 0.922614i $$0.626048\pi$$
$$720$$ 864.107i 0.0447269i
$$721$$ 2020.27i 0.104353i
$$722$$ − 82788.1i − 4.26739i
$$723$$ − 7524.51i − 0.387053i
$$724$$ −7025.24 −0.360623
$$725$$ 21512.4 1.10200
$$726$$ 18253.9i 0.933146i
$$727$$ 2318.92 0.118300 0.0591500 0.998249i $$-0.481161\pi$$
0.0591500 + 0.998249i $$0.481161\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 24355.9i 1.23487i
$$731$$ −18201.3 −0.920931
$$732$$ 2488.34 0.125644
$$733$$ − 3350.95i − 0.168854i −0.996430 0.0844270i $$-0.973094\pi$$
0.996430 0.0844270i $$-0.0269060\pi$$
$$734$$ 5798.61i 0.291595i
$$735$$ − 5331.58i − 0.267562i
$$736$$ − 22349.6i − 1.11932i
$$737$$ −1268.33 −0.0633915
$$738$$ −1970.09 −0.0982656
$$739$$ 29348.1i 1.46088i 0.682979 + 0.730438i $$0.260684\pi$$
−0.682979 + 0.730438i $$0.739316\pi$$
$$740$$ −21660.8 −1.07604
$$741$$ 0 0
$$742$$ 22104.3 1.09363
$$743$$ 23622.2i 1.16637i 0.812338 + 0.583187i $$0.198195\pi$$
−0.812338 + 0.583187i $$0.801805\pi$$
$$744$$ 2735.45 0.134793
$$745$$ −1995.30 −0.0981236
$$746$$ − 2008.40i − 0.0985693i
$$747$$ − 1352.26i − 0.0662336i
$$748$$ 8857.41i 0.432966i
$$749$$ 34106.1i 1.66383i
$$750$$ −14498.5 −0.705882
$$751$$ −29751.4 −1.44560 −0.722798 0.691059i $$-0.757144\pi$$
−0.722798 + 0.691059i $$0.757144\pi$$
$$752$$ 5701.97i 0.276502i
$$753$$ −10217.7 −0.494495
$$754$$ 0 0
$$755$$ 926.735 0.0446720
$$756$$ 10338.9i 0.497385i
$$757$$ −36907.2 −1.77201 −0.886006 0.463673i $$-0.846531\pi$$
−0.886006 + 0.463673i $$0.846531\pi$$
$$758$$ −583.709 −0.0279700
$$759$$ 3153.45i 0.150807i
$$760$$ − 19966.4i − 0.952968i
$$761$$ 20208.2i 0.962613i 0.876552 + 0.481306i $$0.159838\pi$$
−0.876552 + 0.481306i $$0.840162\pi$$
$$762$$ − 10234.2i − 0.486541i
$$763$$ 19662.8 0.932950
$$764$$ 43439.1 2.05703
$$765$$ − 4227.98i − 0.199821i
$$766$$ −43886.9 −2.07010
$$767$$ 0 0
$$768$$ −18156.1 −0.853063
$$769$$ 24751.0i 1.16066i 0.814382 + 0.580329i $$0.197076\pi$$
−0.814382 + 0.580329i $$0.802924\pi$$
$$770$$ −3425.94 −0.160341
$$771$$ 14201.9 0.663382
$$772$$ 23138.2i 1.07871i
$$773$$ − 30673.4i − 1.42722i −0.700541 0.713612i $$-0.747058\pi$$
0.700541 0.713612i $$-0.252942\pi$$
$$774$$ 7327.18i 0.340271i
$$775$$ 3355.38i 0.155521i
$$776$$ −8181.46 −0.378476
$$777$$ −28091.7 −1.29702
$$778$$ − 53413.4i − 2.46139i
$$779$$ 7289.87 0.335285
$$780$$ 0 0
$$781$$ 2614.44 0.119785
$$782$$ − 86568.5i − 3.95867i
$$783$$ 5531.93 0.252484
$$784$$ −8530.23 −0.388585
$$785$$ 4904.55i 0.222995i
$$786$$ − 20534.6i − 0.931862i
$$787$$ − 29009.9i − 1.31397i −0.753905 0.656983i $$-0.771832\pi$$
0.753905 0.656983i $$-0.228168\pi$$
$$788$$ 19297.1i 0.872375i
$$789$$ −5598.33 −0.252606
$$790$$ 21380.6 0.962894
$$791$$ 23289.0i 1.04685i
$$792$$ 1538.72 0.0690355
$$793$$ 0 0
$$794$$ −56744.0 −2.53623
$$795$$ 2320.04i 0.103501i
$$796$$ −65120.8 −2.89968
$$797$$ −6778.24 −0.301252 −0.150626 0.988591i $$-0.548129\pi$$
−0.150626 + 0.988591i $$0.548129\pi$$
$$798$$ − 60004.3i − 2.66182i
$$799$$ − 27899.1i − 1.23529i
$$800$$ − 13377.0i − 0.591185i
$$801$$ − 5093.86i − 0.224697i
$$802$$ −22817.9 −1.00465
$$803$$ 6945.44 0.305229
$$804$$ − 8936.48i − 0.391997i
$$805$$ 21348.1 0.934685
$$806$$ 0 0
$$807$$ 4740.58 0.206786
$$808$$ − 34768.9i − 1.51382i
$$809$$ −34862.9 −1.51510 −0.757549 0.652778i $$-0.773603\pi$$
−0.757549 + 0.652778i $$0.773603\pi$$
$$810$$ −1702.03 −0.0738310
$$811$$ 22665.4i 0.981370i 0.871337 + 0.490685i $$0.163253\pi$$
−0.871337 + 0.490685i $$0.836747\pi$$
$$812$$ 78455.6i 3.39070i
$$813$$ − 14766.3i − 0.636993i
$$814$$ 9688.21i 0.417164i
$$815$$ −14107.4 −0.606331
$$816$$ −6764.52 −0.290203
$$817$$ − 27112.6i − 1.16101i
$$818$$ −14009.1 −0.598797
$$819$$ 0 0
$$820$$ 2932.59 0.124891
$$821$$ 20748.7i 0.882016i 0.897503 + 0.441008i $$0.145379\pi$$
−0.897503 + 0.441008i $$0.854621\pi$$
$$822$$ 25466.1 1.08058
$$823$$ −6141.41 −0.260117 −0.130058 0.991506i $$-0.541516\pi$$
−0.130058 + 0.991506i $$0.541516\pi$$
$$824$$ 2118.50i 0.0895650i
$$825$$ 1887.44i 0.0796513i
$$826$$ 17585.8i 0.740786i
$$827$$ 28383.5i 1.19346i 0.802442 + 0.596730i $$0.203533\pi$$
−0.802442 + 0.596730i $$0.796467\pi$$
$$828$$ −22218.8 −0.932555
$$829$$ 908.734 0.0380720 0.0190360 0.999819i $$-0.493940\pi$$
0.0190360 + 0.999819i $$0.493940\pi$$
$$830$$ 3157.17i 0.132032i
$$831$$ −8061.25 −0.336512
$$832$$ 0 0
$$833$$ 41737.4 1.73603
$$834$$ 17326.2i 0.719375i
$$835$$ 16457.2 0.682065
$$836$$ −13193.9 −0.545839
$$837$$ 862.838i 0.0356321i
$$838$$ 36523.8i 1.50560i
$$839$$ 27820.3i 1.14477i 0.819985 + 0.572385i $$0.193982\pi$$
−0.819985 + 0.572385i $$0.806018\pi$$
$$840$$ − 10416.8i − 0.427873i
$$841$$ 17589.3 0.721200
$$842$$ −18603.5 −0.761425
$$843$$ − 2651.26i − 0.108321i
$$844$$ −71608.3 −2.92045
$$845$$ 0 0
$$846$$ −11231.1 −0.456424
$$847$$ − 35239.0i − 1.42955i
$$848$$ 3711.93 0.150316
$$849$$ 1409.33 0.0569706
$$850$$ − 51814.1i − 2.09084i
$$851$$ − 60370.3i − 2.43181i
$$852$$ 18421.0i 0.740719i
$$853$$ − 5802.11i − 0.232896i −0.993197 0.116448i $$-0.962849\pi$$
0.993197 0.116448i $$-0.0371509\pi$$
$$854$$ −7534.45 −0.301901
$$855$$ 6297.96 0.251913
$$856$$ 35764.5i 1.42804i
$$857$$ 43311.1 1.72635 0.863173 0.504909i $$-0.168474\pi$$
0.863173 + 0.504909i $$0.168474\pi$$
$$858$$ 0 0
$$859$$ −16698.2 −0.663254 −0.331627 0.943411i $$-0.607598\pi$$
−0.331627 + 0.943411i $$0.607598\pi$$
$$860$$ − 10906.9i − 0.432468i
$$861$$ 3803.25 0.150539
$$862$$ −20555.2 −0.812196
$$863$$ 19429.2i 0.766369i 0.923672 + 0.383185i $$0.125173\pi$$
−0.923672 + 0.383185i $$0.874827\pi$$
$$864$$ − 3439.90i − 0.135449i
$$865$$ 16400.6i 0.644666i
$$866$$ − 42249.7i − 1.65786i
$$867$$ 18359.0 0.719153
$$868$$ −12237.0 −0.478517
$$869$$ − 6096.98i − 0.238004i
$$870$$ −12915.6 −0.503311
$$871$$ 0 0
$$872$$ 20618.9 0.800738
$$873$$ − 2580.67i − 0.100048i
$$874$$ 128952. 4.99068
$$875$$ 27989.4 1.08139
$$876$$ 48936.6i 1.88746i
$$877$$ 11619.7i 0.447400i 0.974658 + 0.223700i $$0.0718137\pi$$
−0.974658 + 0.223700i $$0.928186\pi$$
$$878$$ 918.332i 0.0352986i
$$879$$ 10211.3i 0.391829i
$$880$$ −575.309 −0.0220383
$$881$$ 51102.0 1.95422 0.977112 0.212726i $$-0.0682341\pi$$
0.977112 + 0.212726i $$0.0682341\pi$$
$$882$$ − 16801.9i − 0.641441i
$$883$$ 37838.5 1.44209 0.721046 0.692888i $$-0.243662\pi$$
0.721046 + 0.692888i $$0.243662\pi$$
$$884$$ 0 0
$$885$$ −1845.78 −0.0701077
$$886$$ 34625.6i 1.31295i
$$887$$ −9626.92 −0.364420 −0.182210 0.983260i $$-0.558325\pi$$
−0.182210 + 0.983260i $$0.558325\pi$$
$$888$$ −29457.7 −1.11321
$$889$$ 19757.0i 0.745364i
$$890$$ 11892.8i 0.447920i
$$891$$ 485.357i 0.0182493i
$$892$$ − 66774.3i − 2.50647i
$$893$$ 41558.3 1.55733
$$894$$ −6287.98 −0.235237
$$895$$ − 18666.0i − 0.697133i
$$896$$ 70740.0 2.63757
$$897$$ 0 0
$$898$$ −771.825 −0.0286816
$$899$$ 6547.54i 0.242906i
$$900$$ −13298.7 −0.492543
$$901$$ −18162.0 −0.671549
$$902$$ − 1311.66i − 0.0484184i
$$903$$ − 14145.1i − 0.521283i
$$904$$ 24421.4i 0.898500i
$$905$$ 2232.65i 0.0820065i
$$906$$ 2920.51 0.107094
$$907$$ −17066.0 −0.624772 −0.312386 0.949955i $$-0.601128\pi$$
−0.312386 + 0.949955i $$0.601128\pi$$
$$908$$ − 16121.0i − 0.589200i
$$909$$ 10967.1 0.400172
$$910$$ 0 0
$$911$$ −37423.9 −1.36104 −0.680521 0.732729i $$-0.738246\pi$$
−0.680521 + 0.732729i $$0.738246\pi$$
$$912$$ − 10076.4i − 0.365858i
$$913$$ 900.312 0.0326353
$$914$$ −54907.7 −1.98708
$$915$$ − 790.805i − 0.0285718i
$$916$$ 18977.1i 0.684520i
$$917$$ 39641.9i 1.42758i
$$918$$ − 13324.0i − 0.479040i
$$919$$ −2783.88 −0.0999256 −0.0499628 0.998751i $$-0.515910\pi$$
−0.0499628 + 0.998751i $$0.515910\pi$$
$$920$$ 22386.1 0.802227
$$921$$ 2666.59i 0.0954039i
$$922$$ −4968.50 −0.177472
$$923$$ 0 0
$$924$$ −6883.50 −0.245076
$$925$$ − 36133.6i − 1.28440i
$$926$$ 40191.9 1.42634
$$927$$ −668.236 −0.0236761
$$928$$ − 26103.3i − 0.923363i
$$929$$ 26585.2i 0.938894i 0.882961 + 0.469447i $$0.155547\pi$$
−0.882961 + 0.469447i $$0.844453\pi$$
$$930$$ − 2014.50i − 0.0710303i
$$931$$ 62171.8i 2.18861i
$$932$$ −13397.6 −0.470873
$$933$$ 3655.09 0.128255
$$934$$ 36200.9i 1.26823i
$$935$$ 2814.92 0.0984576
$$936$$ 0 0
$$937$$ 34474.0 1.20194 0.600970 0.799272i $$-0.294781\pi$$
0.600970 + 0.799272i $$0.294781\pi$$
$$938$$ 27058.8i 0.941900i
$$939$$ −14611.2 −0.507795
$$940$$ 16718.2 0.580092
$$941$$ − 41994.3i − 1.45481i −0.686209 0.727404i $$-0.740726\pi$$
0.686209 0.727404i $$-0.259274\pi$$
$$942$$ 15456.2i 0.534596i
$$943$$ 8173.34i 0.282249i
$$944$$ 2953.15i 0.101819i
$$945$$ 3285.75 0.113106
$$946$$ −4878.32 −0.167662
$$947$$ 49352.0i 1.69348i 0.532008 + 0.846739i $$0.321438\pi$$
−0.532008 + 0.846739i $$0.678562\pi$$
$$948$$ 42958.5 1.47176
$$949$$ 0 0
$$950$$ 77181.9 2.63591
$$951$$ 13022.8i 0.444052i
$$952$$ 81546.2 2.77618
$$953$$ 51144.5 1.73844 0.869220 0.494425i $$-0.164621\pi$$
0.869220 + 0.494425i $$0.164621\pi$$
$$954$$ 7311.36i 0.248128i
$$955$$ − 13805.2i − 0.467774i
$$956$$ 43192.0i 1.46122i
$$957$$ 3683.07i 0.124406i
$$958$$ −21182.3 −0.714372
$$959$$ −49162.3 −1.65540
$$960$$ 10335.6i 0.347478i
$$961$$ 28769.8 0.965720
$$962$$ 0 0
$$963$$ −11281.1 −0.377497
$$964$$ 35297.7i 1.17932i
$$965$$ 7353.41 0.245300
$$966$$ 67276.3 2.24077
$$967$$ − 24895.1i − 0.827892i −0.910301 0.413946i $$-0.864150\pi$$
0.910301 0.413946i $$-0.135850\pi$$
$$968$$ − 36952.4i − 1.22696i
$$969$$ 49302.6i 1.63450i
$$970$$ 6025.19i 0.199440i
$$971$$ 42942.9 1.41926 0.709630 0.704574i $$-0.248862\pi$$
0.709630 + 0.704574i $$0.248862\pi$$
$$972$$ −3419.76 −0.112849
$$973$$ − 33448.2i − 1.10206i
$$974$$ −31598.8 −1.03952
$$975$$ 0 0
$$976$$ −1265.24 −0.0414953
$$977$$ 42555.7i 1.39353i 0.717301 + 0.696764i $$0.245377\pi$$
−0.717301 + 0.696764i $$0.754623\pi$$
$$978$$ −44458.0 −1.45359
$$979$$ 3391.41 0.110715
$$980$$ 25010.6i 0.815240i
$$981$$ 6503.78i 0.211672i
$$982$$ − 54111.3i − 1.75841i
$$983$$ 4345.38i 0.140993i 0.997512 + 0.0704965i $$0.0224584\pi$$
−0.997512 + 0.0704965i $$0.977542\pi$$
$$984$$ 3988.18 0.129206
$$985$$ 6132.71 0.198380
$$986$$ − 101108.i − 3.26565i
$$987$$ 21681.7 0.699225
$$988$$ 0 0
$$989$$ 30398.4 0.977363
$$990$$ − 1133.18i − 0.0363787i
$$991$$ 7934.89 0.254349 0.127175 0.991880i $$-0.459409\pi$$
0.127175 + 0.991880i $$0.459409\pi$$
$$992$$ 4071.43 0.130311
$$993$$ 2721.87i 0.0869848i
$$994$$ − 55777.0i − 1.77982i
$$995$$ 20695.7i 0.659395i
$$996$$ 6343.48i 0.201808i
$$997$$ 2725.62 0.0865810 0.0432905 0.999063i $$-0.486216\pi$$
0.0432905 + 0.999063i $$0.486216\pi$$
$$998$$ −93531.5 −2.96662
$$999$$ − 9291.79i − 0.294273i
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 507.4.b.k.337.16 18
13.5 odd 4 507.4.a.p.1.8 yes 9
13.8 odd 4 507.4.a.o.1.2 9
13.12 even 2 inner 507.4.b.k.337.3 18
39.5 even 4 1521.4.a.bf.1.2 9
39.8 even 4 1521.4.a.bi.1.8 9

By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.2 9 13.8 odd 4
507.4.a.p.1.8 yes 9 13.5 odd 4
507.4.b.k.337.3 18 13.12 even 2 inner
507.4.b.k.337.16 18 1.1 even 1 trivial
1521.4.a.bf.1.2 9 39.5 even 4
1521.4.a.bi.1.8 9 39.8 even 4