Properties

 Label 245.2.a.h.1.1 Level $245$ Weight $2$ Character 245.1 Self dual yes Analytic conductor $1.956$ 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] = [245,2,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("245.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.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: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ 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: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.414214 q^{2} +2.41421 q^{3} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.58579 q^{8} +2.82843 q^{9} +O(q^{10})$$ $$q-0.414214 q^{2} +2.41421 q^{3} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.58579 q^{8} +2.82843 q^{9} -0.414214 q^{10} +4.82843 q^{11} -4.41421 q^{12} +0.828427 q^{13} +2.41421 q^{15} +3.00000 q^{16} -0.828427 q^{17} -1.17157 q^{18} -2.82843 q^{19} -1.82843 q^{20} -2.00000 q^{22} -2.41421 q^{23} +3.82843 q^{24} +1.00000 q^{25} -0.343146 q^{26} -0.414214 q^{27} -1.00000 q^{29} -1.00000 q^{30} -6.00000 q^{31} -4.41421 q^{32} +11.6569 q^{33} +0.343146 q^{34} -5.17157 q^{36} +1.17157 q^{38} +2.00000 q^{39} +1.58579 q^{40} -2.17157 q^{41} +6.41421 q^{43} -8.82843 q^{44} +2.82843 q^{45} +1.00000 q^{46} +2.00000 q^{47} +7.24264 q^{48} -0.414214 q^{50} -2.00000 q^{51} -1.51472 q^{52} -6.82843 q^{53} +0.171573 q^{54} +4.82843 q^{55} -6.82843 q^{57} +0.414214 q^{58} -12.4853 q^{59} -4.41421 q^{60} -11.4853 q^{61} +2.48528 q^{62} -4.17157 q^{64} +0.828427 q^{65} -4.82843 q^{66} +12.4142 q^{67} +1.51472 q^{68} -5.82843 q^{69} -12.4853 q^{71} +4.48528 q^{72} +4.82843 q^{73} +2.41421 q^{75} +5.17157 q^{76} -0.828427 q^{78} +9.17157 q^{79} +3.00000 q^{80} -9.48528 q^{81} +0.899495 q^{82} -11.7279 q^{83} -0.828427 q^{85} -2.65685 q^{86} -2.41421 q^{87} +7.65685 q^{88} +2.65685 q^{89} -1.17157 q^{90} +4.41421 q^{92} -14.4853 q^{93} -0.828427 q^{94} -2.82843 q^{95} -10.6569 q^{96} +0.343146 q^{97} +13.6569 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8} + 2 q^{10} + 4 q^{11} - 6 q^{12} - 4 q^{13} + 2 q^{15} + 6 q^{16} + 4 q^{17} - 8 q^{18} + 2 q^{20} - 4 q^{22} - 2 q^{23} + 2 q^{24} + 2 q^{25} - 12 q^{26} + 2 q^{27} - 2 q^{29} - 2 q^{30} - 12 q^{31} - 6 q^{32} + 12 q^{33} + 12 q^{34} - 16 q^{36} + 8 q^{38} + 4 q^{39} + 6 q^{40} - 10 q^{41} + 10 q^{43} - 12 q^{44} + 2 q^{46} + 4 q^{47} + 6 q^{48} + 2 q^{50} - 4 q^{51} - 20 q^{52} - 8 q^{53} + 6 q^{54} + 4 q^{55} - 8 q^{57} - 2 q^{58} - 8 q^{59} - 6 q^{60} - 6 q^{61} - 12 q^{62} - 14 q^{64} - 4 q^{65} - 4 q^{66} + 22 q^{67} + 20 q^{68} - 6 q^{69} - 8 q^{71} - 8 q^{72} + 4 q^{73} + 2 q^{75} + 16 q^{76} + 4 q^{78} + 24 q^{79} + 6 q^{80} - 2 q^{81} - 18 q^{82} + 2 q^{83} + 4 q^{85} + 6 q^{86} - 2 q^{87} + 4 q^{88} - 6 q^{89} - 8 q^{90} + 6 q^{92} - 12 q^{93} + 4 q^{94} - 10 q^{96} + 12 q^{97} + 16 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 6 * q^8 + 2 * q^10 + 4 * q^11 - 6 * q^12 - 4 * q^13 + 2 * q^15 + 6 * q^16 + 4 * q^17 - 8 * q^18 + 2 * q^20 - 4 * q^22 - 2 * q^23 + 2 * q^24 + 2 * q^25 - 12 * q^26 + 2 * q^27 - 2 * q^29 - 2 * q^30 - 12 * q^31 - 6 * q^32 + 12 * q^33 + 12 * q^34 - 16 * q^36 + 8 * q^38 + 4 * q^39 + 6 * q^40 - 10 * q^41 + 10 * q^43 - 12 * q^44 + 2 * q^46 + 4 * q^47 + 6 * q^48 + 2 * q^50 - 4 * q^51 - 20 * q^52 - 8 * q^53 + 6 * q^54 + 4 * q^55 - 8 * q^57 - 2 * q^58 - 8 * q^59 - 6 * q^60 - 6 * q^61 - 12 * q^62 - 14 * q^64 - 4 * q^65 - 4 * q^66 + 22 * q^67 + 20 * q^68 - 6 * q^69 - 8 * q^71 - 8 * q^72 + 4 * q^73 + 2 * q^75 + 16 * q^76 + 4 * q^78 + 24 * q^79 + 6 * q^80 - 2 * q^81 - 18 * q^82 + 2 * q^83 + 4 * q^85 + 6 * q^86 - 2 * q^87 + 4 * q^88 - 6 * q^89 - 8 * q^90 + 6 * q^92 - 12 * q^93 + 4 * q^94 - 10 * q^96 + 12 * q^97 + 16 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.414214 −0.292893 −0.146447 0.989219i $$-0.546784\pi$$
−0.146447 + 0.989219i $$0.546784\pi$$
$$3$$ 2.41421 1.39385 0.696923 0.717146i $$-0.254552\pi$$
0.696923 + 0.717146i $$0.254552\pi$$
$$4$$ −1.82843 −0.914214
$$5$$ 1.00000 0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.58579 0.560660
$$9$$ 2.82843 0.942809
$$10$$ −0.414214 −0.130986
$$11$$ 4.82843 1.45583 0.727913 0.685670i $$-0.240491\pi$$
0.727913 + 0.685670i $$0.240491\pi$$
$$12$$ −4.41421 −1.27427
$$13$$ 0.828427 0.229764 0.114882 0.993379i $$-0.463351\pi$$
0.114882 + 0.993379i $$0.463351\pi$$
$$14$$ 0 0
$$15$$ 2.41421 0.623347
$$16$$ 3.00000 0.750000
$$17$$ −0.828427 −0.200923 −0.100462 0.994941i $$-0.532032\pi$$
−0.100462 + 0.994941i $$0.532032\pi$$
$$18$$ −1.17157 −0.276142
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ −1.82843 −0.408849
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ −2.41421 −0.503398 −0.251699 0.967806i $$-0.580989\pi$$
−0.251699 + 0.967806i $$0.580989\pi$$
$$24$$ 3.82843 0.781474
$$25$$ 1.00000 0.200000
$$26$$ −0.343146 −0.0672964
$$27$$ −0.414214 −0.0797154
$$28$$ 0 0
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ −4.41421 −0.780330
$$33$$ 11.6569 2.02920
$$34$$ 0.343146 0.0588490
$$35$$ 0 0
$$36$$ −5.17157 −0.861929
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 1.17157 0.190054
$$39$$ 2.00000 0.320256
$$40$$ 1.58579 0.250735
$$41$$ −2.17157 −0.339143 −0.169571 0.985518i $$-0.554238\pi$$
−0.169571 + 0.985518i $$0.554238\pi$$
$$42$$ 0 0
$$43$$ 6.41421 0.978158 0.489079 0.872239i $$-0.337333\pi$$
0.489079 + 0.872239i $$0.337333\pi$$
$$44$$ −8.82843 −1.33094
$$45$$ 2.82843 0.421637
$$46$$ 1.00000 0.147442
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 7.24264 1.04539
$$49$$ 0 0
$$50$$ −0.414214 −0.0585786
$$51$$ −2.00000 −0.280056
$$52$$ −1.51472 −0.210054
$$53$$ −6.82843 −0.937957 −0.468978 0.883210i $$-0.655378\pi$$
−0.468978 + 0.883210i $$0.655378\pi$$
$$54$$ 0.171573 0.0233481
$$55$$ 4.82843 0.651065
$$56$$ 0 0
$$57$$ −6.82843 −0.904447
$$58$$ 0.414214 0.0543889
$$59$$ −12.4853 −1.62545 −0.812723 0.582651i $$-0.802016\pi$$
−0.812723 + 0.582651i $$0.802016\pi$$
$$60$$ −4.41421 −0.569873
$$61$$ −11.4853 −1.47054 −0.735270 0.677775i $$-0.762945\pi$$
−0.735270 + 0.677775i $$0.762945\pi$$
$$62$$ 2.48528 0.315631
$$63$$ 0 0
$$64$$ −4.17157 −0.521447
$$65$$ 0.828427 0.102754
$$66$$ −4.82843 −0.594338
$$67$$ 12.4142 1.51664 0.758319 0.651884i $$-0.226021\pi$$
0.758319 + 0.651884i $$0.226021\pi$$
$$68$$ 1.51472 0.183687
$$69$$ −5.82843 −0.701660
$$70$$ 0 0
$$71$$ −12.4853 −1.48173 −0.740865 0.671654i $$-0.765584\pi$$
−0.740865 + 0.671654i $$0.765584\pi$$
$$72$$ 4.48528 0.528595
$$73$$ 4.82843 0.565125 0.282562 0.959249i $$-0.408816\pi$$
0.282562 + 0.959249i $$0.408816\pi$$
$$74$$ 0 0
$$75$$ 2.41421 0.278769
$$76$$ 5.17157 0.593220
$$77$$ 0 0
$$78$$ −0.828427 −0.0938009
$$79$$ 9.17157 1.03188 0.515941 0.856624i $$-0.327442\pi$$
0.515941 + 0.856624i $$0.327442\pi$$
$$80$$ 3.00000 0.335410
$$81$$ −9.48528 −1.05392
$$82$$ 0.899495 0.0993326
$$83$$ −11.7279 −1.28731 −0.643653 0.765317i $$-0.722582\pi$$
−0.643653 + 0.765317i $$0.722582\pi$$
$$84$$ 0 0
$$85$$ −0.828427 −0.0898555
$$86$$ −2.65685 −0.286496
$$87$$ −2.41421 −0.258831
$$88$$ 7.65685 0.816223
$$89$$ 2.65685 0.281626 0.140813 0.990036i $$-0.455028\pi$$
0.140813 + 0.990036i $$0.455028\pi$$
$$90$$ −1.17157 −0.123495
$$91$$ 0 0
$$92$$ 4.41421 0.460214
$$93$$ −14.4853 −1.50205
$$94$$ −0.828427 −0.0854457
$$95$$ −2.82843 −0.290191
$$96$$ −10.6569 −1.08766
$$97$$ 0.343146 0.0348412 0.0174206 0.999848i $$-0.494455\pi$$
0.0174206 + 0.999848i $$0.494455\pi$$
$$98$$ 0 0
$$99$$ 13.6569 1.37257
$$100$$ −1.82843 −0.182843
$$101$$ 12.3137 1.22526 0.612630 0.790370i $$-0.290112\pi$$
0.612630 + 0.790370i $$0.290112\pi$$
$$102$$ 0.828427 0.0820265
$$103$$ 0.414214 0.0408137 0.0204068 0.999792i $$-0.493504\pi$$
0.0204068 + 0.999792i $$0.493504\pi$$
$$104$$ 1.31371 0.128820
$$105$$ 0 0
$$106$$ 2.82843 0.274721
$$107$$ 2.75736 0.266564 0.133282 0.991078i $$-0.457448\pi$$
0.133282 + 0.991078i $$0.457448\pi$$
$$108$$ 0.757359 0.0728769
$$109$$ 3.48528 0.333829 0.166915 0.985971i $$-0.446620\pi$$
0.166915 + 0.985971i $$0.446620\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.4853 1.17452 0.587258 0.809400i $$-0.300207\pi$$
0.587258 + 0.809400i $$0.300207\pi$$
$$114$$ 2.82843 0.264906
$$115$$ −2.41421 −0.225127
$$116$$ 1.82843 0.169765
$$117$$ 2.34315 0.216624
$$118$$ 5.17157 0.476082
$$119$$ 0 0
$$120$$ 3.82843 0.349486
$$121$$ 12.3137 1.11943
$$122$$ 4.75736 0.430711
$$123$$ −5.24264 −0.472713
$$124$$ 10.9706 0.985186
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 13.3137 1.18140 0.590700 0.806891i $$-0.298852\pi$$
0.590700 + 0.806891i $$0.298852\pi$$
$$128$$ 10.5563 0.933058
$$129$$ 15.4853 1.36340
$$130$$ −0.343146 −0.0300959
$$131$$ 3.31371 0.289520 0.144760 0.989467i $$-0.453759\pi$$
0.144760 + 0.989467i $$0.453759\pi$$
$$132$$ −21.3137 −1.85512
$$133$$ 0 0
$$134$$ −5.14214 −0.444213
$$135$$ −0.414214 −0.0356498
$$136$$ −1.31371 −0.112650
$$137$$ 1.65685 0.141555 0.0707773 0.997492i $$-0.477452\pi$$
0.0707773 + 0.997492i $$0.477452\pi$$
$$138$$ 2.41421 0.205512
$$139$$ 12.1421 1.02988 0.514941 0.857225i $$-0.327814\pi$$
0.514941 + 0.857225i $$0.327814\pi$$
$$140$$ 0 0
$$141$$ 4.82843 0.406627
$$142$$ 5.17157 0.433989
$$143$$ 4.00000 0.334497
$$144$$ 8.48528 0.707107
$$145$$ −1.00000 −0.0830455
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −7.82843 −0.641330 −0.320665 0.947193i $$-0.603906\pi$$
−0.320665 + 0.947193i $$0.603906\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 0.343146 0.0279248 0.0139624 0.999903i $$-0.495555\pi$$
0.0139624 + 0.999903i $$0.495555\pi$$
$$152$$ −4.48528 −0.363804
$$153$$ −2.34315 −0.189432
$$154$$ 0 0
$$155$$ −6.00000 −0.481932
$$156$$ −3.65685 −0.292783
$$157$$ −5.31371 −0.424080 −0.212040 0.977261i $$-0.568011\pi$$
−0.212040 + 0.977261i $$0.568011\pi$$
$$158$$ −3.79899 −0.302231
$$159$$ −16.4853 −1.30737
$$160$$ −4.41421 −0.348974
$$161$$ 0 0
$$162$$ 3.92893 0.308686
$$163$$ 23.6569 1.85295 0.926474 0.376359i $$-0.122824\pi$$
0.926474 + 0.376359i $$0.122824\pi$$
$$164$$ 3.97056 0.310049
$$165$$ 11.6569 0.907485
$$166$$ 4.85786 0.377043
$$167$$ 19.5858 1.51559 0.757797 0.652491i $$-0.226276\pi$$
0.757797 + 0.652491i $$0.226276\pi$$
$$168$$ 0 0
$$169$$ −12.3137 −0.947208
$$170$$ 0.343146 0.0263181
$$171$$ −8.00000 −0.611775
$$172$$ −11.7279 −0.894246
$$173$$ −19.3137 −1.46839 −0.734197 0.678936i $$-0.762441\pi$$
−0.734197 + 0.678936i $$0.762441\pi$$
$$174$$ 1.00000 0.0758098
$$175$$ 0 0
$$176$$ 14.4853 1.09187
$$177$$ −30.1421 −2.26562
$$178$$ −1.10051 −0.0824863
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ −5.17157 −0.385466
$$181$$ −8.65685 −0.643459 −0.321729 0.946832i $$-0.604264\pi$$
−0.321729 + 0.946832i $$0.604264\pi$$
$$182$$ 0 0
$$183$$ −27.7279 −2.04971
$$184$$ −3.82843 −0.282235
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ −4.00000 −0.292509
$$188$$ −3.65685 −0.266704
$$189$$ 0 0
$$190$$ 1.17157 0.0849948
$$191$$ −7.17157 −0.518917 −0.259458 0.965754i $$-0.583544\pi$$
−0.259458 + 0.965754i $$0.583544\pi$$
$$192$$ −10.0711 −0.726817
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −0.142136 −0.0102047
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ −23.6569 −1.68548 −0.842741 0.538320i $$-0.819059\pi$$
−0.842741 + 0.538320i $$0.819059\pi$$
$$198$$ −5.65685 −0.402015
$$199$$ 1.65685 0.117451 0.0587256 0.998274i $$-0.481296\pi$$
0.0587256 + 0.998274i $$0.481296\pi$$
$$200$$ 1.58579 0.112132
$$201$$ 29.9706 2.11396
$$202$$ −5.10051 −0.358870
$$203$$ 0 0
$$204$$ 3.65685 0.256031
$$205$$ −2.17157 −0.151669
$$206$$ −0.171573 −0.0119540
$$207$$ −6.82843 −0.474608
$$208$$ 2.48528 0.172323
$$209$$ −13.6569 −0.944664
$$210$$ 0 0
$$211$$ 3.51472 0.241963 0.120982 0.992655i $$-0.461396\pi$$
0.120982 + 0.992655i $$0.461396\pi$$
$$212$$ 12.4853 0.857493
$$213$$ −30.1421 −2.06531
$$214$$ −1.14214 −0.0780748
$$215$$ 6.41421 0.437446
$$216$$ −0.656854 −0.0446933
$$217$$ 0 0
$$218$$ −1.44365 −0.0977764
$$219$$ 11.6569 0.787697
$$220$$ −8.82843 −0.595212
$$221$$ −0.686292 −0.0461650
$$222$$ 0 0
$$223$$ 11.6569 0.780601 0.390300 0.920688i $$-0.372371\pi$$
0.390300 + 0.920688i $$0.372371\pi$$
$$224$$ 0 0
$$225$$ 2.82843 0.188562
$$226$$ −5.17157 −0.344008
$$227$$ 26.9706 1.79010 0.895050 0.445967i $$-0.147140\pi$$
0.895050 + 0.445967i $$0.147140\pi$$
$$228$$ 12.4853 0.826858
$$229$$ −0.343146 −0.0226757 −0.0113379 0.999936i $$-0.503609\pi$$
−0.0113379 + 0.999936i $$0.503609\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ 0 0
$$232$$ −1.58579 −0.104112
$$233$$ −11.1716 −0.731874 −0.365937 0.930640i $$-0.619251\pi$$
−0.365937 + 0.930640i $$0.619251\pi$$
$$234$$ −0.970563 −0.0634477
$$235$$ 2.00000 0.130466
$$236$$ 22.8284 1.48600
$$237$$ 22.1421 1.43829
$$238$$ 0 0
$$239$$ 1.31371 0.0849767 0.0424884 0.999097i $$-0.486471\pi$$
0.0424884 + 0.999097i $$0.486471\pi$$
$$240$$ 7.24264 0.467510
$$241$$ 16.3431 1.05275 0.526377 0.850251i $$-0.323550\pi$$
0.526377 + 0.850251i $$0.323550\pi$$
$$242$$ −5.10051 −0.327873
$$243$$ −21.6569 −1.38929
$$244$$ 21.0000 1.34439
$$245$$ 0 0
$$246$$ 2.17157 0.138454
$$247$$ −2.34315 −0.149091
$$248$$ −9.51472 −0.604185
$$249$$ −28.3137 −1.79431
$$250$$ −0.414214 −0.0261972
$$251$$ 13.3137 0.840354 0.420177 0.907442i $$-0.361968\pi$$
0.420177 + 0.907442i $$0.361968\pi$$
$$252$$ 0 0
$$253$$ −11.6569 −0.732860
$$254$$ −5.51472 −0.346024
$$255$$ −2.00000 −0.125245
$$256$$ 3.97056 0.248160
$$257$$ 17.6569 1.10140 0.550702 0.834702i $$-0.314360\pi$$
0.550702 + 0.834702i $$0.314360\pi$$
$$258$$ −6.41421 −0.399331
$$259$$ 0 0
$$260$$ −1.51472 −0.0939389
$$261$$ −2.82843 −0.175075
$$262$$ −1.37258 −0.0847985
$$263$$ 19.0416 1.17416 0.587079 0.809530i $$-0.300278\pi$$
0.587079 + 0.809530i $$0.300278\pi$$
$$264$$ 18.4853 1.13769
$$265$$ −6.82843 −0.419467
$$266$$ 0 0
$$267$$ 6.41421 0.392543
$$268$$ −22.6985 −1.38653
$$269$$ −30.4558 −1.85693 −0.928463 0.371425i $$-0.878869\pi$$
−0.928463 + 0.371425i $$0.878869\pi$$
$$270$$ 0.171573 0.0104416
$$271$$ −0.485281 −0.0294787 −0.0147394 0.999891i $$-0.504692\pi$$
−0.0147394 + 0.999891i $$0.504692\pi$$
$$272$$ −2.48528 −0.150692
$$273$$ 0 0
$$274$$ −0.686292 −0.0414604
$$275$$ 4.82843 0.291165
$$276$$ 10.6569 0.641467
$$277$$ −12.1421 −0.729550 −0.364775 0.931096i $$-0.618854\pi$$
−0.364775 + 0.931096i $$0.618854\pi$$
$$278$$ −5.02944 −0.301646
$$279$$ −16.9706 −1.01600
$$280$$ 0 0
$$281$$ 26.2843 1.56799 0.783994 0.620768i $$-0.213179\pi$$
0.783994 + 0.620768i $$0.213179\pi$$
$$282$$ −2.00000 −0.119098
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ 22.8284 1.35462
$$285$$ −6.82843 −0.404481
$$286$$ −1.65685 −0.0979718
$$287$$ 0 0
$$288$$ −12.4853 −0.735702
$$289$$ −16.3137 −0.959630
$$290$$ 0.414214 0.0243235
$$291$$ 0.828427 0.0485633
$$292$$ −8.82843 −0.516645
$$293$$ −16.0000 −0.934730 −0.467365 0.884064i $$-0.654797\pi$$
−0.467365 + 0.884064i $$0.654797\pi$$
$$294$$ 0 0
$$295$$ −12.4853 −0.726921
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 3.24264 0.187841
$$299$$ −2.00000 −0.115663
$$300$$ −4.41421 −0.254855
$$301$$ 0 0
$$302$$ −0.142136 −0.00817899
$$303$$ 29.7279 1.70782
$$304$$ −8.48528 −0.486664
$$305$$ −11.4853 −0.657645
$$306$$ 0.970563 0.0554834
$$307$$ −13.2426 −0.755797 −0.377899 0.925847i $$-0.623353\pi$$
−0.377899 + 0.925847i $$0.623353\pi$$
$$308$$ 0 0
$$309$$ 1.00000 0.0568880
$$310$$ 2.48528 0.141154
$$311$$ −18.8284 −1.06766 −0.533831 0.845591i $$-0.679248\pi$$
−0.533831 + 0.845591i $$0.679248\pi$$
$$312$$ 3.17157 0.179555
$$313$$ −17.6569 −0.998024 −0.499012 0.866595i $$-0.666304\pi$$
−0.499012 + 0.866595i $$0.666304\pi$$
$$314$$ 2.20101 0.124210
$$315$$ 0 0
$$316$$ −16.7696 −0.943361
$$317$$ 25.7990 1.44902 0.724508 0.689267i $$-0.242067\pi$$
0.724508 + 0.689267i $$0.242067\pi$$
$$318$$ 6.82843 0.382919
$$319$$ −4.82843 −0.270340
$$320$$ −4.17157 −0.233198
$$321$$ 6.65685 0.371549
$$322$$ 0 0
$$323$$ 2.34315 0.130376
$$324$$ 17.3431 0.963508
$$325$$ 0.828427 0.0459529
$$326$$ −9.79899 −0.542716
$$327$$ 8.41421 0.465307
$$328$$ −3.44365 −0.190144
$$329$$ 0 0
$$330$$ −4.82843 −0.265796
$$331$$ −10.9706 −0.602997 −0.301498 0.953467i $$-0.597487\pi$$
−0.301498 + 0.953467i $$0.597487\pi$$
$$332$$ 21.4437 1.17687
$$333$$ 0 0
$$334$$ −8.11270 −0.443907
$$335$$ 12.4142 0.678261
$$336$$ 0 0
$$337$$ 14.8284 0.807756 0.403878 0.914813i $$-0.367662\pi$$
0.403878 + 0.914813i $$0.367662\pi$$
$$338$$ 5.10051 0.277431
$$339$$ 30.1421 1.63710
$$340$$ 1.51472 0.0821472
$$341$$ −28.9706 −1.56884
$$342$$ 3.31371 0.179185
$$343$$ 0 0
$$344$$ 10.1716 0.548414
$$345$$ −5.82843 −0.313792
$$346$$ 8.00000 0.430083
$$347$$ 22.0711 1.18484 0.592418 0.805630i $$-0.298173\pi$$
0.592418 + 0.805630i $$0.298173\pi$$
$$348$$ 4.41421 0.236627
$$349$$ −26.6569 −1.42691 −0.713454 0.700702i $$-0.752870\pi$$
−0.713454 + 0.700702i $$0.752870\pi$$
$$350$$ 0 0
$$351$$ −0.343146 −0.0183158
$$352$$ −21.3137 −1.13602
$$353$$ −21.1716 −1.12685 −0.563425 0.826168i $$-0.690516\pi$$
−0.563425 + 0.826168i $$0.690516\pi$$
$$354$$ 12.4853 0.663585
$$355$$ −12.4853 −0.662650
$$356$$ −4.85786 −0.257466
$$357$$ 0 0
$$358$$ 4.14214 0.218919
$$359$$ −10.0000 −0.527780 −0.263890 0.964553i $$-0.585006\pi$$
−0.263890 + 0.964553i $$0.585006\pi$$
$$360$$ 4.48528 0.236395
$$361$$ −11.0000 −0.578947
$$362$$ 3.58579 0.188465
$$363$$ 29.7279 1.56031
$$364$$ 0 0
$$365$$ 4.82843 0.252731
$$366$$ 11.4853 0.600345
$$367$$ −11.2426 −0.586861 −0.293431 0.955980i $$-0.594797\pi$$
−0.293431 + 0.955980i $$0.594797\pi$$
$$368$$ −7.24264 −0.377549
$$369$$ −6.14214 −0.319747
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 26.4853 1.37320
$$373$$ 12.9706 0.671590 0.335795 0.941935i $$-0.390995\pi$$
0.335795 + 0.941935i $$0.390995\pi$$
$$374$$ 1.65685 0.0856739
$$375$$ 2.41421 0.124669
$$376$$ 3.17157 0.163561
$$377$$ −0.828427 −0.0426662
$$378$$ 0 0
$$379$$ 21.1716 1.08751 0.543755 0.839244i $$-0.317002\pi$$
0.543755 + 0.839244i $$0.317002\pi$$
$$380$$ 5.17157 0.265296
$$381$$ 32.1421 1.64669
$$382$$ 2.97056 0.151987
$$383$$ 16.8995 0.863524 0.431762 0.901988i $$-0.357892\pi$$
0.431762 + 0.901988i $$0.357892\pi$$
$$384$$ 25.4853 1.30054
$$385$$ 0 0
$$386$$ 0.828427 0.0421658
$$387$$ 18.1421 0.922217
$$388$$ −0.627417 −0.0318523
$$389$$ 12.3431 0.625822 0.312911 0.949782i $$-0.398696\pi$$
0.312911 + 0.949782i $$0.398696\pi$$
$$390$$ −0.828427 −0.0419490
$$391$$ 2.00000 0.101144
$$392$$ 0 0
$$393$$ 8.00000 0.403547
$$394$$ 9.79899 0.493666
$$395$$ 9.17157 0.461472
$$396$$ −24.9706 −1.25482
$$397$$ 28.6274 1.43677 0.718384 0.695646i $$-0.244882\pi$$
0.718384 + 0.695646i $$0.244882\pi$$
$$398$$ −0.686292 −0.0344007
$$399$$ 0 0
$$400$$ 3.00000 0.150000
$$401$$ 7.68629 0.383835 0.191918 0.981411i $$-0.438529\pi$$
0.191918 + 0.981411i $$0.438529\pi$$
$$402$$ −12.4142 −0.619165
$$403$$ −4.97056 −0.247601
$$404$$ −22.5147 −1.12015
$$405$$ −9.48528 −0.471327
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −3.17157 −0.157016
$$409$$ 24.7990 1.22623 0.613116 0.789993i $$-0.289916\pi$$
0.613116 + 0.789993i $$0.289916\pi$$
$$410$$ 0.899495 0.0444229
$$411$$ 4.00000 0.197305
$$412$$ −0.757359 −0.0373124
$$413$$ 0 0
$$414$$ 2.82843 0.139010
$$415$$ −11.7279 −0.575701
$$416$$ −3.65685 −0.179292
$$417$$ 29.3137 1.43550
$$418$$ 5.65685 0.276686
$$419$$ −23.3137 −1.13895 −0.569475 0.822009i $$-0.692853\pi$$
−0.569475 + 0.822009i $$0.692853\pi$$
$$420$$ 0 0
$$421$$ −3.48528 −0.169862 −0.0849311 0.996387i $$-0.527067\pi$$
−0.0849311 + 0.996387i $$0.527067\pi$$
$$422$$ −1.45584 −0.0708694
$$423$$ 5.65685 0.275046
$$424$$ −10.8284 −0.525875
$$425$$ −0.828427 −0.0401846
$$426$$ 12.4853 0.604914
$$427$$ 0 0
$$428$$ −5.04163 −0.243696
$$429$$ 9.65685 0.466237
$$430$$ −2.65685 −0.128125
$$431$$ −21.7990 −1.05002 −0.525010 0.851096i $$-0.675938\pi$$
−0.525010 + 0.851096i $$0.675938\pi$$
$$432$$ −1.24264 −0.0597866
$$433$$ −31.7990 −1.52816 −0.764081 0.645120i $$-0.776807\pi$$
−0.764081 + 0.645120i $$0.776807\pi$$
$$434$$ 0 0
$$435$$ −2.41421 −0.115753
$$436$$ −6.37258 −0.305191
$$437$$ 6.82843 0.326648
$$438$$ −4.82843 −0.230711
$$439$$ 33.9411 1.61992 0.809961 0.586484i $$-0.199488\pi$$
0.809961 + 0.586484i $$0.199488\pi$$
$$440$$ 7.65685 0.365026
$$441$$ 0 0
$$442$$ 0.284271 0.0135214
$$443$$ −12.2132 −0.580267 −0.290133 0.956986i $$-0.593700\pi$$
−0.290133 + 0.956986i $$0.593700\pi$$
$$444$$ 0 0
$$445$$ 2.65685 0.125947
$$446$$ −4.82843 −0.228633
$$447$$ −18.8995 −0.893915
$$448$$ 0 0
$$449$$ −1.82843 −0.0862888 −0.0431444 0.999069i $$-0.513738\pi$$
−0.0431444 + 0.999069i $$0.513738\pi$$
$$450$$ −1.17157 −0.0552285
$$451$$ −10.4853 −0.493733
$$452$$ −22.8284 −1.07376
$$453$$ 0.828427 0.0389229
$$454$$ −11.1716 −0.524308
$$455$$ 0 0
$$456$$ −10.8284 −0.507088
$$457$$ −32.2843 −1.51019 −0.755097 0.655613i $$-0.772410\pi$$
−0.755097 + 0.655613i $$0.772410\pi$$
$$458$$ 0.142136 0.00664156
$$459$$ 0.343146 0.0160167
$$460$$ 4.41421 0.205814
$$461$$ 18.6863 0.870307 0.435154 0.900356i $$-0.356694\pi$$
0.435154 + 0.900356i $$0.356694\pi$$
$$462$$ 0 0
$$463$$ 11.0416 0.513148 0.256574 0.966525i $$-0.417406\pi$$
0.256574 + 0.966525i $$0.417406\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ −14.4853 −0.671739
$$466$$ 4.62742 0.214361
$$467$$ −22.8995 −1.05966 −0.529831 0.848103i $$-0.677745\pi$$
−0.529831 + 0.848103i $$0.677745\pi$$
$$468$$ −4.28427 −0.198041
$$469$$ 0 0
$$470$$ −0.828427 −0.0382125
$$471$$ −12.8284 −0.591103
$$472$$ −19.7990 −0.911322
$$473$$ 30.9706 1.42403
$$474$$ −9.17157 −0.421264
$$475$$ −2.82843 −0.129777
$$476$$ 0 0
$$477$$ −19.3137 −0.884314
$$478$$ −0.544156 −0.0248891
$$479$$ −24.3431 −1.11227 −0.556133 0.831093i $$-0.687716\pi$$
−0.556133 + 0.831093i $$0.687716\pi$$
$$480$$ −10.6569 −0.486417
$$481$$ 0 0
$$482$$ −6.76955 −0.308345
$$483$$ 0 0
$$484$$ −22.5147 −1.02340
$$485$$ 0.343146 0.0155814
$$486$$ 8.97056 0.406913
$$487$$ 15.6569 0.709480 0.354740 0.934965i $$-0.384569\pi$$
0.354740 + 0.934965i $$0.384569\pi$$
$$488$$ −18.2132 −0.824473
$$489$$ 57.1127 2.58273
$$490$$ 0 0
$$491$$ −13.3137 −0.600839 −0.300420 0.953807i $$-0.597127\pi$$
−0.300420 + 0.953807i $$0.597127\pi$$
$$492$$ 9.58579 0.432161
$$493$$ 0.828427 0.0373105
$$494$$ 0.970563 0.0436677
$$495$$ 13.6569 0.613830
$$496$$ −18.0000 −0.808224
$$497$$ 0 0
$$498$$ 11.7279 0.525541
$$499$$ 4.82843 0.216150 0.108075 0.994143i $$-0.465531\pi$$
0.108075 + 0.994143i $$0.465531\pi$$
$$500$$ −1.82843 −0.0817697
$$501$$ 47.2843 2.11251
$$502$$ −5.51472 −0.246134
$$503$$ 37.8701 1.68854 0.844271 0.535916i $$-0.180034\pi$$
0.844271 + 0.535916i $$0.180034\pi$$
$$504$$ 0 0
$$505$$ 12.3137 0.547953
$$506$$ 4.82843 0.214650
$$507$$ −29.7279 −1.32026
$$508$$ −24.3431 −1.08005
$$509$$ 24.6569 1.09290 0.546448 0.837493i $$-0.315980\pi$$
0.546448 + 0.837493i $$0.315980\pi$$
$$510$$ 0.828427 0.0366834
$$511$$ 0 0
$$512$$ −22.7574 −1.00574
$$513$$ 1.17157 0.0517262
$$514$$ −7.31371 −0.322594
$$515$$ 0.414214 0.0182524
$$516$$ −28.3137 −1.24644
$$517$$ 9.65685 0.424708
$$518$$ 0 0
$$519$$ −46.6274 −2.04672
$$520$$ 1.31371 0.0576099
$$521$$ −18.9706 −0.831115 −0.415558 0.909567i $$-0.636414\pi$$
−0.415558 + 0.909567i $$0.636414\pi$$
$$522$$ 1.17157 0.0512784
$$523$$ 24.3431 1.06445 0.532226 0.846602i $$-0.321356\pi$$
0.532226 + 0.846602i $$0.321356\pi$$
$$524$$ −6.05887 −0.264683
$$525$$ 0 0
$$526$$ −7.88730 −0.343903
$$527$$ 4.97056 0.216521
$$528$$ 34.9706 1.52190
$$529$$ −17.1716 −0.746590
$$530$$ 2.82843 0.122859
$$531$$ −35.3137 −1.53248
$$532$$ 0 0
$$533$$ −1.79899 −0.0779229
$$534$$ −2.65685 −0.114973
$$535$$ 2.75736 0.119211
$$536$$ 19.6863 0.850318
$$537$$ −24.1421 −1.04181
$$538$$ 12.6152 0.543881
$$539$$ 0 0
$$540$$ 0.757359 0.0325916
$$541$$ 18.6569 0.802121 0.401060 0.916052i $$-0.368642\pi$$
0.401060 + 0.916052i $$0.368642\pi$$
$$542$$ 0.201010 0.00863412
$$543$$ −20.8995 −0.896883
$$544$$ 3.65685 0.156786
$$545$$ 3.48528 0.149293
$$546$$ 0 0
$$547$$ 5.10051 0.218082 0.109041 0.994037i $$-0.465222\pi$$
0.109041 + 0.994037i $$0.465222\pi$$
$$548$$ −3.02944 −0.129411
$$549$$ −32.4853 −1.38644
$$550$$ −2.00000 −0.0852803
$$551$$ 2.82843 0.120495
$$552$$ −9.24264 −0.393393
$$553$$ 0 0
$$554$$ 5.02944 0.213680
$$555$$ 0 0
$$556$$ −22.2010 −0.941533
$$557$$ −34.2843 −1.45267 −0.726336 0.687340i $$-0.758778\pi$$
−0.726336 + 0.687340i $$0.758778\pi$$
$$558$$ 7.02944 0.297580
$$559$$ 5.31371 0.224746
$$560$$ 0 0
$$561$$ −9.65685 −0.407713
$$562$$ −10.8873 −0.459253
$$563$$ 16.2721 0.685786 0.342893 0.939374i $$-0.388593\pi$$
0.342893 + 0.939374i $$0.388593\pi$$
$$564$$ −8.82843 −0.371744
$$565$$ 12.4853 0.525260
$$566$$ −5.79899 −0.243750
$$567$$ 0 0
$$568$$ −19.7990 −0.830747
$$569$$ −3.65685 −0.153303 −0.0766517 0.997058i $$-0.524423\pi$$
−0.0766517 + 0.997058i $$0.524423\pi$$
$$570$$ 2.82843 0.118470
$$571$$ 14.8284 0.620550 0.310275 0.950647i $$-0.399579\pi$$
0.310275 + 0.950647i $$0.399579\pi$$
$$572$$ −7.31371 −0.305802
$$573$$ −17.3137 −0.723291
$$574$$ 0 0
$$575$$ −2.41421 −0.100680
$$576$$ −11.7990 −0.491625
$$577$$ 23.9411 0.996682 0.498341 0.866981i $$-0.333943\pi$$
0.498341 + 0.866981i $$0.333943\pi$$
$$578$$ 6.75736 0.281069
$$579$$ −4.82843 −0.200663
$$580$$ 1.82843 0.0759213
$$581$$ 0 0
$$582$$ −0.343146 −0.0142238
$$583$$ −32.9706 −1.36550
$$584$$ 7.65685 0.316843
$$585$$ 2.34315 0.0968772
$$586$$ 6.62742 0.273776
$$587$$ 22.2843 0.919770 0.459885 0.887978i $$-0.347891\pi$$
0.459885 + 0.887978i $$0.347891\pi$$
$$588$$ 0 0
$$589$$ 16.9706 0.699260
$$590$$ 5.17157 0.212910
$$591$$ −57.1127 −2.34930
$$592$$ 0 0
$$593$$ 43.7990 1.79861 0.899304 0.437323i $$-0.144073\pi$$
0.899304 + 0.437323i $$0.144073\pi$$
$$594$$ 0.828427 0.0339908
$$595$$ 0 0
$$596$$ 14.3137 0.586312
$$597$$ 4.00000 0.163709
$$598$$ 0.828427 0.0338769
$$599$$ 17.6569 0.721440 0.360720 0.932674i $$-0.382531\pi$$
0.360720 + 0.932674i $$0.382531\pi$$
$$600$$ 3.82843 0.156295
$$601$$ 8.34315 0.340324 0.170162 0.985416i $$-0.445571\pi$$
0.170162 + 0.985416i $$0.445571\pi$$
$$602$$ 0 0
$$603$$ 35.1127 1.42990
$$604$$ −0.627417 −0.0255292
$$605$$ 12.3137 0.500623
$$606$$ −12.3137 −0.500210
$$607$$ −4.21320 −0.171009 −0.0855043 0.996338i $$-0.527250\pi$$
−0.0855043 + 0.996338i $$0.527250\pi$$
$$608$$ 12.4853 0.506345
$$609$$ 0 0
$$610$$ 4.75736 0.192620
$$611$$ 1.65685 0.0670291
$$612$$ 4.28427 0.173181
$$613$$ −15.4558 −0.624256 −0.312128 0.950040i $$-0.601042\pi$$
−0.312128 + 0.950040i $$0.601042\pi$$
$$614$$ 5.48528 0.221368
$$615$$ −5.24264 −0.211404
$$616$$ 0 0
$$617$$ −11.3137 −0.455473 −0.227736 0.973723i $$-0.573132\pi$$
−0.227736 + 0.973723i $$0.573132\pi$$
$$618$$ −0.414214 −0.0166621
$$619$$ 42.4853 1.70763 0.853814 0.520578i $$-0.174284\pi$$
0.853814 + 0.520578i $$0.174284\pi$$
$$620$$ 10.9706 0.440588
$$621$$ 1.00000 0.0401286
$$622$$ 7.79899 0.312711
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ 1.00000 0.0400000
$$626$$ 7.31371 0.292315
$$627$$ −32.9706 −1.31672
$$628$$ 9.71573 0.387700
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 8.14214 0.324133 0.162067 0.986780i $$-0.448184\pi$$
0.162067 + 0.986780i $$0.448184\pi$$
$$632$$ 14.5442 0.578535
$$633$$ 8.48528 0.337260
$$634$$ −10.6863 −0.424407
$$635$$ 13.3137 0.528338
$$636$$ 30.1421 1.19521
$$637$$ 0 0
$$638$$ 2.00000 0.0791808
$$639$$ −35.3137 −1.39699
$$640$$ 10.5563 0.417276
$$641$$ 14.5147 0.573297 0.286648 0.958036i $$-0.407459\pi$$
0.286648 + 0.958036i $$0.407459\pi$$
$$642$$ −2.75736 −0.108824
$$643$$ −30.2843 −1.19430 −0.597148 0.802131i $$-0.703699\pi$$
−0.597148 + 0.802131i $$0.703699\pi$$
$$644$$ 0 0
$$645$$ 15.4853 0.609732
$$646$$ −0.970563 −0.0381863
$$647$$ 17.0416 0.669976 0.334988 0.942222i $$-0.391268\pi$$
0.334988 + 0.942222i $$0.391268\pi$$
$$648$$ −15.0416 −0.590891
$$649$$ −60.2843 −2.36636
$$650$$ −0.343146 −0.0134593
$$651$$ 0 0
$$652$$ −43.2548 −1.69399
$$653$$ −24.8284 −0.971611 −0.485806 0.874067i $$-0.661474\pi$$
−0.485806 + 0.874067i $$0.661474\pi$$
$$654$$ −3.48528 −0.136285
$$655$$ 3.31371 0.129477
$$656$$ −6.51472 −0.254357
$$657$$ 13.6569 0.532805
$$658$$ 0 0
$$659$$ 26.8284 1.04509 0.522544 0.852613i $$-0.324983\pi$$
0.522544 + 0.852613i $$0.324983\pi$$
$$660$$ −21.3137 −0.829635
$$661$$ 26.1716 1.01796 0.508978 0.860779i $$-0.330023\pi$$
0.508978 + 0.860779i $$0.330023\pi$$
$$662$$ 4.54416 0.176614
$$663$$ −1.65685 −0.0643469
$$664$$ −18.5980 −0.721742
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.41421 0.0934787
$$668$$ −35.8112 −1.38558
$$669$$ 28.1421 1.08804
$$670$$ −5.14214 −0.198658
$$671$$ −55.4558 −2.14085
$$672$$ 0 0
$$673$$ 18.3431 0.707076 0.353538 0.935420i $$-0.384978\pi$$
0.353538 + 0.935420i $$0.384978\pi$$
$$674$$ −6.14214 −0.236586
$$675$$ −0.414214 −0.0159431
$$676$$ 22.5147 0.865951
$$677$$ 0.142136 0.00546272 0.00273136 0.999996i $$-0.499131\pi$$
0.00273136 + 0.999996i $$0.499131\pi$$
$$678$$ −12.4853 −0.479494
$$679$$ 0 0
$$680$$ −1.31371 −0.0503784
$$681$$ 65.1127 2.49512
$$682$$ 12.0000 0.459504
$$683$$ −43.2426 −1.65463 −0.827317 0.561736i $$-0.810134\pi$$
−0.827317 + 0.561736i $$0.810134\pi$$
$$684$$ 14.6274 0.559293
$$685$$ 1.65685 0.0633051
$$686$$ 0 0
$$687$$ −0.828427 −0.0316065
$$688$$ 19.2426 0.733619
$$689$$ −5.65685 −0.215509
$$690$$ 2.41421 0.0919075
$$691$$ −4.82843 −0.183682 −0.0918410 0.995774i $$-0.529275\pi$$
−0.0918410 + 0.995774i $$0.529275\pi$$
$$692$$ 35.3137 1.34243
$$693$$ 0 0
$$694$$ −9.14214 −0.347031
$$695$$ 12.1421 0.460577
$$696$$ −3.82843 −0.145116
$$697$$ 1.79899 0.0681416
$$698$$ 11.0416 0.417932
$$699$$ −26.9706 −1.02012
$$700$$ 0 0
$$701$$ −42.7990 −1.61650 −0.808248 0.588843i $$-0.799584\pi$$
−0.808248 + 0.588843i $$0.799584\pi$$
$$702$$ 0.142136 0.00536456
$$703$$ 0 0
$$704$$ −20.1421 −0.759135
$$705$$ 4.82843 0.181849
$$706$$ 8.76955 0.330046
$$707$$ 0 0
$$708$$ 55.1127 2.07126
$$709$$ 38.3137 1.43890 0.719451 0.694543i $$-0.244394\pi$$
0.719451 + 0.694543i $$0.244394\pi$$
$$710$$ 5.17157 0.194086
$$711$$ 25.9411 0.972868
$$712$$ 4.21320 0.157896
$$713$$ 14.4853 0.542478
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ 18.2843 0.683315
$$717$$ 3.17157 0.118445
$$718$$ 4.14214 0.154583
$$719$$ −41.1127 −1.53324 −0.766622 0.642098i $$-0.778064\pi$$
−0.766622 + 0.642098i $$0.778064\pi$$
$$720$$ 8.48528 0.316228
$$721$$ 0 0
$$722$$ 4.55635 0.169570
$$723$$ 39.4558 1.46738
$$724$$ 15.8284 0.588259
$$725$$ −1.00000 −0.0371391
$$726$$ −12.3137 −0.457005
$$727$$ −40.4142 −1.49888 −0.749440 0.662072i $$-0.769677\pi$$
−0.749440 + 0.662072i $$0.769677\pi$$
$$728$$ 0 0
$$729$$ −23.8284 −0.882534
$$730$$ −2.00000 −0.0740233
$$731$$ −5.31371 −0.196535
$$732$$ 50.6985 1.87387
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ 4.65685 0.171888
$$735$$ 0 0
$$736$$ 10.6569 0.392817
$$737$$ 59.9411 2.20796
$$738$$ 2.54416 0.0936517
$$739$$ 41.1127 1.51236 0.756178 0.654367i $$-0.227065\pi$$
0.756178 + 0.654367i $$0.227065\pi$$
$$740$$ 0 0
$$741$$ −5.65685 −0.207810
$$742$$ 0 0
$$743$$ −1.92893 −0.0707657 −0.0353828 0.999374i $$-0.511265\pi$$
−0.0353828 + 0.999374i $$0.511265\pi$$
$$744$$ −22.9706 −0.842142
$$745$$ −7.82843 −0.286811
$$746$$ −5.37258 −0.196704
$$747$$ −33.1716 −1.21368
$$748$$ 7.31371 0.267416
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ −41.6569 −1.52008 −0.760040 0.649876i $$-0.774821\pi$$
−0.760040 + 0.649876i $$0.774821\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 32.1421 1.17132
$$754$$ 0.343146 0.0124966
$$755$$ 0.343146 0.0124884
$$756$$ 0 0
$$757$$ 19.4558 0.707135 0.353567 0.935409i $$-0.384969\pi$$
0.353567 + 0.935409i $$0.384969\pi$$
$$758$$ −8.76955 −0.318524
$$759$$ −28.1421 −1.02149
$$760$$ −4.48528 −0.162698
$$761$$ −13.3137 −0.482622 −0.241311 0.970448i $$-0.577577\pi$$
−0.241311 + 0.970448i $$0.577577\pi$$
$$762$$ −13.3137 −0.482305
$$763$$ 0 0
$$764$$ 13.1127 0.474401
$$765$$ −2.34315 −0.0847166
$$766$$ −7.00000 −0.252920
$$767$$ −10.3431 −0.373469
$$768$$ 9.58579 0.345897
$$769$$ 44.6274 1.60931 0.804653 0.593745i $$-0.202351\pi$$
0.804653 + 0.593745i $$0.202351\pi$$
$$770$$ 0 0
$$771$$ 42.6274 1.53519
$$772$$ 3.65685 0.131613
$$773$$ 25.1127 0.903241 0.451620 0.892210i $$-0.350846\pi$$
0.451620 + 0.892210i $$0.350846\pi$$
$$774$$ −7.51472 −0.270111
$$775$$ −6.00000 −0.215526
$$776$$ 0.544156 0.0195341
$$777$$ 0 0
$$778$$ −5.11270 −0.183299
$$779$$ 6.14214 0.220065
$$780$$ −3.65685 −0.130936
$$781$$ −60.2843 −2.15714
$$782$$ −0.828427 −0.0296245
$$783$$ 0.414214 0.0148028
$$784$$ 0 0
$$785$$ −5.31371 −0.189654
$$786$$ −3.31371 −0.118196
$$787$$ −28.5563 −1.01792 −0.508962 0.860789i $$-0.669971\pi$$
−0.508962 + 0.860789i $$0.669971\pi$$
$$788$$ 43.2548 1.54089
$$789$$ 45.9706 1.63660
$$790$$ −3.79899 −0.135162
$$791$$ 0 0
$$792$$ 21.6569 0.769543
$$793$$ −9.51472 −0.337878
$$794$$ −11.8579 −0.420820
$$795$$ −16.4853 −0.584673
$$796$$ −3.02944 −0.107376
$$797$$ −8.00000 −0.283375 −0.141687 0.989911i $$-0.545253\pi$$
−0.141687 + 0.989911i $$0.545253\pi$$
$$798$$ 0 0
$$799$$ −1.65685 −0.0586153
$$800$$ −4.41421 −0.156066
$$801$$ 7.51472 0.265520
$$802$$ −3.18377 −0.112423
$$803$$ 23.3137 0.822723
$$804$$ −54.7990 −1.93261
$$805$$ 0 0
$$806$$ 2.05887 0.0725208
$$807$$ −73.5269 −2.58827
$$808$$ 19.5269 0.686954
$$809$$ −9.62742 −0.338482 −0.169241 0.985575i $$-0.554132\pi$$
−0.169241 + 0.985575i $$0.554132\pi$$
$$810$$ 3.92893 0.138049
$$811$$ 24.6274 0.864786 0.432393 0.901685i $$-0.357669\pi$$
0.432393 + 0.901685i $$0.357669\pi$$
$$812$$ 0 0
$$813$$ −1.17157 −0.0410889
$$814$$ 0 0
$$815$$ 23.6569 0.828663
$$816$$ −6.00000 −0.210042
$$817$$ −18.1421 −0.634713
$$818$$ −10.2721 −0.359155
$$819$$ 0 0
$$820$$ 3.97056 0.138658
$$821$$ −19.9411 −0.695950 −0.347975 0.937504i $$-0.613131\pi$$
−0.347975 + 0.937504i $$0.613131\pi$$
$$822$$ −1.65685 −0.0577894
$$823$$ −12.0711 −0.420771 −0.210385 0.977619i $$-0.567472\pi$$
−0.210385 + 0.977619i $$0.567472\pi$$
$$824$$ 0.656854 0.0228826
$$825$$ 11.6569 0.405840
$$826$$ 0 0
$$827$$ 16.2132 0.563788 0.281894 0.959446i $$-0.409037\pi$$
0.281894 + 0.959446i $$0.409037\pi$$
$$828$$ 12.4853 0.433894
$$829$$ 6.68629 0.232225 0.116112 0.993236i $$-0.462957\pi$$
0.116112 + 0.993236i $$0.462957\pi$$
$$830$$ 4.85786 0.168619
$$831$$ −29.3137 −1.01688
$$832$$ −3.45584 −0.119810
$$833$$ 0 0
$$834$$ −12.1421 −0.420448
$$835$$ 19.5858 0.677794
$$836$$ 24.9706 0.863625
$$837$$ 2.48528 0.0859039
$$838$$ 9.65685 0.333590
$$839$$ 20.8284 0.719077 0.359539 0.933130i $$-0.382934\pi$$
0.359539 + 0.933130i $$0.382934\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 1.44365 0.0497515
$$843$$ 63.4558 2.18554
$$844$$ −6.42641 −0.221206
$$845$$ −12.3137 −0.423604
$$846$$ −2.34315 −0.0805590
$$847$$ 0 0
$$848$$ −20.4853 −0.703467
$$849$$ 33.7990 1.15998
$$850$$ 0.343146 0.0117698
$$851$$ 0 0
$$852$$ 55.1127 1.88813
$$853$$ 53.4558 1.83029 0.915147 0.403121i $$-0.132075\pi$$
0.915147 + 0.403121i $$0.132075\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ 4.37258 0.149452
$$857$$ −22.2843 −0.761216 −0.380608 0.924736i $$-0.624285\pi$$
−0.380608 + 0.924736i $$0.624285\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ 46.6274 1.59091 0.795453 0.606015i $$-0.207233\pi$$
0.795453 + 0.606015i $$0.207233\pi$$
$$860$$ −11.7279 −0.399919
$$861$$ 0 0
$$862$$ 9.02944 0.307544
$$863$$ −16.5563 −0.563585 −0.281792 0.959475i $$-0.590929\pi$$
−0.281792 + 0.959475i $$0.590929\pi$$
$$864$$ 1.82843 0.0622044
$$865$$ −19.3137 −0.656686
$$866$$ 13.1716 0.447588
$$867$$ −39.3848 −1.33758
$$868$$ 0 0
$$869$$ 44.2843 1.50224
$$870$$ 1.00000 0.0339032
$$871$$ 10.2843 0.348469
$$872$$ 5.52691 0.187165
$$873$$ 0.970563 0.0328486
$$874$$ −2.82843 −0.0956730
$$875$$ 0 0
$$876$$ −21.3137 −0.720123
$$877$$ 30.8284 1.04100 0.520501 0.853861i $$-0.325745\pi$$
0.520501 + 0.853861i $$0.325745\pi$$
$$878$$ −14.0589 −0.474464
$$879$$ −38.6274 −1.30287
$$880$$ 14.4853 0.488299
$$881$$ −3.82843 −0.128983 −0.0644915 0.997918i $$-0.520543\pi$$
−0.0644915 + 0.997918i $$0.520543\pi$$
$$882$$ 0 0
$$883$$ −38.2843 −1.28837 −0.644184 0.764870i $$-0.722803\pi$$
−0.644184 + 0.764870i $$0.722803\pi$$
$$884$$ 1.25483 0.0422046
$$885$$ −30.1421 −1.01322
$$886$$ 5.05887 0.169956
$$887$$ 44.0711 1.47976 0.739881 0.672738i $$-0.234882\pi$$
0.739881 + 0.672738i $$0.234882\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −1.10051 −0.0368890
$$891$$ −45.7990 −1.53432
$$892$$ −21.3137 −0.713636
$$893$$ −5.65685 −0.189299
$$894$$ 7.82843 0.261822
$$895$$ −10.0000 −0.334263
$$896$$ 0 0
$$897$$ −4.82843 −0.161216
$$898$$ 0.757359 0.0252734
$$899$$ 6.00000 0.200111
$$900$$ −5.17157 −0.172386
$$901$$ 5.65685 0.188457
$$902$$ 4.34315 0.144611
$$903$$ 0 0
$$904$$ 19.7990 0.658505
$$905$$ −8.65685 −0.287764
$$906$$ −0.343146 −0.0114003
$$907$$ 28.2132 0.936804 0.468402 0.883515i $$-0.344830\pi$$
0.468402 + 0.883515i $$0.344830\pi$$
$$908$$ −49.3137 −1.63653
$$909$$ 34.8284 1.15519
$$910$$ 0 0
$$911$$ −49.7990 −1.64991 −0.824957 0.565195i $$-0.808801\pi$$
−0.824957 + 0.565195i $$0.808801\pi$$
$$912$$ −20.4853 −0.678335
$$913$$ −56.6274 −1.87409
$$914$$ 13.3726 0.442326
$$915$$ −27.7279 −0.916657
$$916$$ 0.627417 0.0207304
$$917$$ 0 0
$$918$$ −0.142136 −0.00469117
$$919$$ 19.1127 0.630470 0.315235 0.949014i $$-0.397917\pi$$
0.315235 + 0.949014i $$0.397917\pi$$
$$920$$ −3.82843 −0.126220
$$921$$ −31.9706 −1.05347
$$922$$ −7.74012 −0.254907
$$923$$ −10.3431 −0.340449
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.57359 −0.150298
$$927$$ 1.17157 0.0384795
$$928$$ 4.41421 0.144904
$$929$$ −11.4853 −0.376820 −0.188410 0.982090i $$-0.560333\pi$$
−0.188410 + 0.982090i $$0.560333\pi$$
$$930$$ 6.00000 0.196748
$$931$$ 0 0
$$932$$ 20.4264 0.669089
$$933$$ −45.4558 −1.48816
$$934$$ 9.48528 0.310368
$$935$$ −4.00000 −0.130814
$$936$$ 3.71573 0.121452
$$937$$ −10.6274 −0.347183 −0.173591 0.984818i $$-0.555537\pi$$
−0.173591 + 0.984818i $$0.555537\pi$$
$$938$$ 0 0
$$939$$ −42.6274 −1.39109
$$940$$ −3.65685 −0.119273
$$941$$ 10.2843 0.335258 0.167629 0.985850i $$-0.446389\pi$$
0.167629 + 0.985850i $$0.446389\pi$$
$$942$$ 5.31371 0.173130
$$943$$ 5.24264 0.170724
$$944$$ −37.4558 −1.21908
$$945$$ 0 0
$$946$$ −12.8284 −0.417088
$$947$$ −43.1838 −1.40328 −0.701642 0.712530i $$-0.747549\pi$$
−0.701642 + 0.712530i $$0.747549\pi$$
$$948$$ −40.4853 −1.31490
$$949$$ 4.00000 0.129845
$$950$$ 1.17157 0.0380108
$$951$$ 62.2843 2.01971
$$952$$ 0 0
$$953$$ −2.34315 −0.0759019 −0.0379510 0.999280i $$-0.512083\pi$$
−0.0379510 + 0.999280i $$0.512083\pi$$
$$954$$ 8.00000 0.259010
$$955$$ −7.17157 −0.232067
$$956$$ −2.40202 −0.0776869
$$957$$ −11.6569 −0.376813
$$958$$ 10.0833 0.325775
$$959$$ 0 0
$$960$$ −10.0711 −0.325042
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 7.79899 0.251319
$$964$$ −29.8823 −0.962442
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ −27.5269 −0.885206 −0.442603 0.896718i $$-0.645945\pi$$
−0.442603 + 0.896718i $$0.645945\pi$$
$$968$$ 19.5269 0.627619
$$969$$ 5.65685 0.181724
$$970$$ −0.142136 −0.00456370
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 39.5980 1.27011
$$973$$ 0 0
$$974$$ −6.48528 −0.207802
$$975$$ 2.00000 0.0640513
$$976$$ −34.4558 −1.10290
$$977$$ 21.3137 0.681886 0.340943 0.940084i $$-0.389254\pi$$
0.340943 + 0.940084i $$0.389254\pi$$
$$978$$ −23.6569 −0.756463
$$979$$ 12.8284 0.409998
$$980$$ 0 0
$$981$$ 9.85786 0.314737
$$982$$ 5.51472 0.175982
$$983$$ −14.2132 −0.453331 −0.226665 0.973973i $$-0.572782\pi$$
−0.226665 + 0.973973i $$0.572782\pi$$
$$984$$ −8.31371 −0.265031
$$985$$ −23.6569 −0.753770
$$986$$ −0.343146 −0.0109280
$$987$$ 0 0
$$988$$ 4.28427 0.136301
$$989$$ −15.4853 −0.492403
$$990$$ −5.65685 −0.179787
$$991$$ −15.6569 −0.497356 −0.248678 0.968586i $$-0.579996\pi$$
−0.248678 + 0.968586i $$0.579996\pi$$
$$992$$ 26.4853 0.840909
$$993$$ −26.4853 −0.840485
$$994$$ 0 0
$$995$$ 1.65685 0.0525258
$$996$$ 51.7696 1.64038
$$997$$ −17.4558 −0.552832 −0.276416 0.961038i $$-0.589147\pi$$
−0.276416 + 0.961038i $$0.589147\pi$$
$$998$$ −2.00000 −0.0633089
$$999$$ 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 245.2.a.h.1.1 2
3.2 odd 2 2205.2.a.n.1.2 2
4.3 odd 2 3920.2.a.bq.1.1 2
5.2 odd 4 1225.2.b.g.99.2 4
5.3 odd 4 1225.2.b.g.99.3 4
5.4 even 2 1225.2.a.k.1.2 2
7.2 even 3 35.2.e.a.11.2 4
7.3 odd 6 245.2.e.e.226.2 4
7.4 even 3 35.2.e.a.16.2 yes 4
7.5 odd 6 245.2.e.e.116.2 4
7.6 odd 2 245.2.a.g.1.1 2
21.2 odd 6 315.2.j.e.46.1 4
21.11 odd 6 315.2.j.e.226.1 4
21.20 even 2 2205.2.a.q.1.2 2
28.11 odd 6 560.2.q.k.401.2 4
28.23 odd 6 560.2.q.k.81.2 4
28.27 even 2 3920.2.a.bv.1.2 2
35.2 odd 12 175.2.k.a.74.2 8
35.4 even 6 175.2.e.c.51.1 4
35.9 even 6 175.2.e.c.151.1 4
35.13 even 4 1225.2.b.h.99.3 4
35.18 odd 12 175.2.k.a.149.2 8
35.23 odd 12 175.2.k.a.74.3 8
35.27 even 4 1225.2.b.h.99.2 4
35.32 odd 12 175.2.k.a.149.3 8
35.34 odd 2 1225.2.a.m.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.e.a.11.2 4 7.2 even 3
35.2.e.a.16.2 yes 4 7.4 even 3
175.2.e.c.51.1 4 35.4 even 6
175.2.e.c.151.1 4 35.9 even 6
175.2.k.a.74.2 8 35.2 odd 12
175.2.k.a.74.3 8 35.23 odd 12
175.2.k.a.149.2 8 35.18 odd 12
175.2.k.a.149.3 8 35.32 odd 12
245.2.a.g.1.1 2 7.6 odd 2
245.2.a.h.1.1 2 1.1 even 1 trivial
245.2.e.e.116.2 4 7.5 odd 6
245.2.e.e.226.2 4 7.3 odd 6
315.2.j.e.46.1 4 21.2 odd 6
315.2.j.e.226.1 4 21.11 odd 6
560.2.q.k.81.2 4 28.23 odd 6
560.2.q.k.401.2 4 28.11 odd 6
1225.2.a.k.1.2 2 5.4 even 2
1225.2.a.m.1.2 2 35.34 odd 2
1225.2.b.g.99.2 4 5.2 odd 4
1225.2.b.g.99.3 4 5.3 odd 4
1225.2.b.h.99.2 4 35.27 even 4
1225.2.b.h.99.3 4 35.13 even 4
2205.2.a.n.1.2 2 3.2 odd 2
2205.2.a.q.1.2 2 21.20 even 2
3920.2.a.bq.1.1 2 4.3 odd 2
3920.2.a.bv.1.2 2 28.27 even 2