Properties

 Label 38.4.a.c.1.1 Level $38$ Weight $4$ Character 38.1 Self dual yes Analytic conductor $2.242$ 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] = [38,4,Mod(1,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 38.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: $$2.24207258022$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 18$$ x^2 - x - 18 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$4.77200$$ of defining polynomial Character $$\chi$$ $$=$$ 38.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +0.227998 q^{3} +4.00000 q^{4} +8.31601 q^{5} +0.455996 q^{6} +8.08801 q^{7} +8.00000 q^{8} -26.9480 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +0.227998 q^{3} +4.00000 q^{4} +8.31601 q^{5} +0.455996 q^{6} +8.08801 q^{7} +8.00000 q^{8} -26.9480 q^{9} +16.6320 q^{10} -12.7720 q^{11} +0.911993 q^{12} -47.0360 q^{13} +16.1760 q^{14} +1.89603 q^{15} +16.0000 q^{16} -31.4560 q^{17} -53.8960 q^{18} +19.0000 q^{19} +33.2640 q^{20} +1.84405 q^{21} -25.5440 q^{22} -19.0360 q^{23} +1.82399 q^{24} -55.8441 q^{25} -94.0720 q^{26} -12.3000 q^{27} +32.3520 q^{28} +91.2120 q^{29} +3.79207 q^{30} +293.968 q^{31} +32.0000 q^{32} -2.91199 q^{33} -62.9120 q^{34} +67.2599 q^{35} -107.792 q^{36} +215.616 q^{37} +38.0000 q^{38} -10.7241 q^{39} +66.5280 q^{40} -67.7200 q^{41} +3.68810 q^{42} +308.596 q^{43} -51.0880 q^{44} -224.100 q^{45} -38.0720 q^{46} +108.812 q^{47} +3.64797 q^{48} -277.584 q^{49} -111.688 q^{50} -7.17191 q^{51} -188.144 q^{52} -682.124 q^{53} -24.6001 q^{54} -106.212 q^{55} +64.7041 q^{56} +4.33196 q^{57} +182.424 q^{58} -250.300 q^{59} +7.58413 q^{60} -317.692 q^{61} +587.936 q^{62} -217.956 q^{63} +64.0000 q^{64} -391.152 q^{65} -5.82399 q^{66} +940.444 q^{67} -125.824 q^{68} -4.34018 q^{69} +134.520 q^{70} -395.552 q^{71} -215.584 q^{72} +975.048 q^{73} +431.232 q^{74} -12.7323 q^{75} +76.0000 q^{76} -103.300 q^{77} -21.4483 q^{78} +922.776 q^{79} +133.056 q^{80} +724.792 q^{81} -135.440 q^{82} -1163.77 q^{83} +7.37620 q^{84} -261.588 q^{85} +617.192 q^{86} +20.7962 q^{87} -102.176 q^{88} +685.136 q^{89} -448.200 q^{90} -380.428 q^{91} -76.1441 q^{92} +67.0242 q^{93} +217.624 q^{94} +158.004 q^{95} +7.29594 q^{96} +211.256 q^{97} -555.168 q^{98} +344.180 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 + 9 * q^3 + 8 * q^4 - 9 * q^5 + 18 * q^6 - 18 * q^7 + 16 * q^8 + 23 * q^9 $$2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9} - 18 q^{10} - 17 q^{11} + 36 q^{12} + 17 q^{13} - 36 q^{14} - 150 q^{15} + 32 q^{16} - 80 q^{17} + 46 q^{18} + 38 q^{19} - 36 q^{20} - 227 q^{21} - 34 q^{22} + 73 q^{23} + 72 q^{24} + 119 q^{25} + 34 q^{26} + 189 q^{27} - 72 q^{28} + 3 q^{29} - 300 q^{30} + 212 q^{31} + 64 q^{32} - 40 q^{33} - 160 q^{34} + 519 q^{35} + 92 q^{36} + 192 q^{37} + 76 q^{38} + 551 q^{39} - 72 q^{40} - 50 q^{41} - 454 q^{42} + 677 q^{43} - 68 q^{44} - 1089 q^{45} + 146 q^{46} - 389 q^{47} + 144 q^{48} + 60 q^{49} + 238 q^{50} - 433 q^{51} + 68 q^{52} - 1219 q^{53} + 378 q^{54} - 33 q^{55} - 144 q^{56} + 171 q^{57} + 6 q^{58} - 287 q^{59} - 600 q^{60} + 313 q^{61} + 424 q^{62} - 1521 q^{63} + 128 q^{64} - 1500 q^{65} - 80 q^{66} + 1223 q^{67} - 320 q^{68} + 803 q^{69} + 1038 q^{70} + 200 q^{71} + 184 q^{72} + 378 q^{73} + 384 q^{74} + 1521 q^{75} + 152 q^{76} + 7 q^{77} + 1102 q^{78} + 1350 q^{79} - 144 q^{80} + 1142 q^{81} - 100 q^{82} - 670 q^{83} - 908 q^{84} + 579 q^{85} + 1354 q^{86} - 753 q^{87} - 136 q^{88} - 236 q^{89} - 2178 q^{90} - 2051 q^{91} + 292 q^{92} - 652 q^{93} - 778 q^{94} - 171 q^{95} + 288 q^{96} + 1294 q^{97} + 120 q^{98} + 133 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 + 9 * q^3 + 8 * q^4 - 9 * q^5 + 18 * q^6 - 18 * q^7 + 16 * q^8 + 23 * q^9 - 18 * q^10 - 17 * q^11 + 36 * q^12 + 17 * q^13 - 36 * q^14 - 150 * q^15 + 32 * q^16 - 80 * q^17 + 46 * q^18 + 38 * q^19 - 36 * q^20 - 227 * q^21 - 34 * q^22 + 73 * q^23 + 72 * q^24 + 119 * q^25 + 34 * q^26 + 189 * q^27 - 72 * q^28 + 3 * q^29 - 300 * q^30 + 212 * q^31 + 64 * q^32 - 40 * q^33 - 160 * q^34 + 519 * q^35 + 92 * q^36 + 192 * q^37 + 76 * q^38 + 551 * q^39 - 72 * q^40 - 50 * q^41 - 454 * q^42 + 677 * q^43 - 68 * q^44 - 1089 * q^45 + 146 * q^46 - 389 * q^47 + 144 * q^48 + 60 * q^49 + 238 * q^50 - 433 * q^51 + 68 * q^52 - 1219 * q^53 + 378 * q^54 - 33 * q^55 - 144 * q^56 + 171 * q^57 + 6 * q^58 - 287 * q^59 - 600 * q^60 + 313 * q^61 + 424 * q^62 - 1521 * q^63 + 128 * q^64 - 1500 * q^65 - 80 * q^66 + 1223 * q^67 - 320 * q^68 + 803 * q^69 + 1038 * q^70 + 200 * q^71 + 184 * q^72 + 378 * q^73 + 384 * q^74 + 1521 * q^75 + 152 * q^76 + 7 * q^77 + 1102 * q^78 + 1350 * q^79 - 144 * q^80 + 1142 * q^81 - 100 * q^82 - 670 * q^83 - 908 * q^84 + 579 * q^85 + 1354 * q^86 - 753 * q^87 - 136 * q^88 - 236 * q^89 - 2178 * q^90 - 2051 * q^91 + 292 * q^92 - 652 * q^93 - 778 * q^94 - 171 * q^95 + 288 * q^96 + 1294 * q^97 + 120 * q^98 + 133 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.707107
$$3$$ 0.227998 0.0438783 0.0219391 0.999759i $$-0.493016\pi$$
0.0219391 + 0.999759i $$0.493016\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 8.31601 0.743806 0.371903 0.928272i $$-0.378705\pi$$
0.371903 + 0.928272i $$0.378705\pi$$
$$6$$ 0.455996 0.0310266
$$7$$ 8.08801 0.436711 0.218356 0.975869i $$-0.429931\pi$$
0.218356 + 0.975869i $$0.429931\pi$$
$$8$$ 8.00000 0.353553
$$9$$ −26.9480 −0.998075
$$10$$ 16.6320 0.525950
$$11$$ −12.7720 −0.350082 −0.175041 0.984561i $$-0.556006\pi$$
−0.175041 + 0.984561i $$0.556006\pi$$
$$12$$ 0.911993 0.0219391
$$13$$ −47.0360 −1.00350 −0.501748 0.865014i $$-0.667309\pi$$
−0.501748 + 0.865014i $$0.667309\pi$$
$$14$$ 16.1760 0.308802
$$15$$ 1.89603 0.0326369
$$16$$ 16.0000 0.250000
$$17$$ −31.4560 −0.448776 −0.224388 0.974500i $$-0.572038\pi$$
−0.224388 + 0.974500i $$0.572038\pi$$
$$18$$ −53.8960 −0.705745
$$19$$ 19.0000 0.229416
$$20$$ 33.2640 0.371903
$$21$$ 1.84405 0.0191621
$$22$$ −25.5440 −0.247545
$$23$$ −19.0360 −0.172578 −0.0862888 0.996270i $$-0.527501\pi$$
−0.0862888 + 0.996270i $$0.527501\pi$$
$$24$$ 1.82399 0.0155133
$$25$$ −55.8441 −0.446752
$$26$$ −94.0720 −0.709579
$$27$$ −12.3000 −0.0876720
$$28$$ 32.3520 0.218356
$$29$$ 91.2120 0.584057 0.292028 0.956410i $$-0.405670\pi$$
0.292028 + 0.956410i $$0.405670\pi$$
$$30$$ 3.79207 0.0230778
$$31$$ 293.968 1.70317 0.851584 0.524218i $$-0.175642\pi$$
0.851584 + 0.524218i $$0.175642\pi$$
$$32$$ 32.0000 0.176777
$$33$$ −2.91199 −0.0153610
$$34$$ −62.9120 −0.317333
$$35$$ 67.2599 0.324829
$$36$$ −107.792 −0.499037
$$37$$ 215.616 0.958029 0.479014 0.877807i $$-0.340994\pi$$
0.479014 + 0.877807i $$0.340994\pi$$
$$38$$ 38.0000 0.162221
$$39$$ −10.7241 −0.0440317
$$40$$ 66.5280 0.262975
$$41$$ −67.7200 −0.257953 −0.128977 0.991648i $$-0.541169\pi$$
−0.128977 + 0.991648i $$0.541169\pi$$
$$42$$ 3.68810 0.0135497
$$43$$ 308.596 1.09443 0.547214 0.836992i $$-0.315688\pi$$
0.547214 + 0.836992i $$0.315688\pi$$
$$44$$ −51.0880 −0.175041
$$45$$ −224.100 −0.742374
$$46$$ −38.0720 −0.122031
$$47$$ 108.812 0.337700 0.168850 0.985642i $$-0.445995\pi$$
0.168850 + 0.985642i $$0.445995\pi$$
$$48$$ 3.64797 0.0109696
$$49$$ −277.584 −0.809283
$$50$$ −111.688 −0.315902
$$51$$ −7.17191 −0.0196915
$$52$$ −188.144 −0.501748
$$53$$ −682.124 −1.76787 −0.883933 0.467613i $$-0.845114\pi$$
−0.883933 + 0.467613i $$0.845114\pi$$
$$54$$ −24.6001 −0.0619935
$$55$$ −106.212 −0.260393
$$56$$ 64.7041 0.154401
$$57$$ 4.33196 0.0100664
$$58$$ 182.424 0.412991
$$59$$ −250.300 −0.552310 −0.276155 0.961113i $$-0.589060\pi$$
−0.276155 + 0.961113i $$0.589060\pi$$
$$60$$ 7.58413 0.0163185
$$61$$ −317.692 −0.666825 −0.333412 0.942781i $$-0.608200\pi$$
−0.333412 + 0.942781i $$0.608200\pi$$
$$62$$ 587.936 1.20432
$$63$$ −217.956 −0.435871
$$64$$ 64.0000 0.125000
$$65$$ −391.152 −0.746406
$$66$$ −5.82399 −0.0108619
$$67$$ 940.444 1.71483 0.857414 0.514626i $$-0.172069\pi$$
0.857414 + 0.514626i $$0.172069\pi$$
$$68$$ −125.824 −0.224388
$$69$$ −4.34018 −0.00757241
$$70$$ 134.520 0.229689
$$71$$ −395.552 −0.661175 −0.330587 0.943775i $$-0.607247\pi$$
−0.330587 + 0.943775i $$0.607247\pi$$
$$72$$ −215.584 −0.352873
$$73$$ 975.048 1.56330 0.781649 0.623718i $$-0.214379\pi$$
0.781649 + 0.623718i $$0.214379\pi$$
$$74$$ 431.232 0.677429
$$75$$ −12.7323 −0.0196027
$$76$$ 76.0000 0.114708
$$77$$ −103.300 −0.152885
$$78$$ −21.4483 −0.0311351
$$79$$ 922.776 1.31418 0.657091 0.753811i $$-0.271787\pi$$
0.657091 + 0.753811i $$0.271787\pi$$
$$80$$ 133.056 0.185952
$$81$$ 724.792 0.994228
$$82$$ −135.440 −0.182401
$$83$$ −1163.77 −1.53904 −0.769519 0.638624i $$-0.779504\pi$$
−0.769519 + 0.638624i $$0.779504\pi$$
$$84$$ 7.37620 0.00958107
$$85$$ −261.588 −0.333803
$$86$$ 617.192 0.773878
$$87$$ 20.7962 0.0256274
$$88$$ −102.176 −0.123773
$$89$$ 685.136 0.816003 0.408002 0.912981i $$-0.366226\pi$$
0.408002 + 0.912981i $$0.366226\pi$$
$$90$$ −448.200 −0.524938
$$91$$ −380.428 −0.438238
$$92$$ −76.1441 −0.0862888
$$93$$ 67.0242 0.0747321
$$94$$ 217.624 0.238790
$$95$$ 158.004 0.170641
$$96$$ 7.29594 0.00775665
$$97$$ 211.256 0.221132 0.110566 0.993869i $$-0.464734\pi$$
0.110566 + 0.993869i $$0.464734\pi$$
$$98$$ −555.168 −0.572250
$$99$$ 344.180 0.349408
$$100$$ −223.376 −0.223376
$$101$$ −1703.55 −1.67831 −0.839157 0.543889i $$-0.816951\pi$$
−0.839157 + 0.543889i $$0.816951\pi$$
$$102$$ −14.3438 −0.0139240
$$103$$ −1393.52 −1.33308 −0.666542 0.745468i $$-0.732226\pi$$
−0.666542 + 0.745468i $$0.732226\pi$$
$$104$$ −376.288 −0.354789
$$105$$ 15.3351 0.0142529
$$106$$ −1364.25 −1.25007
$$107$$ 907.996 0.820367 0.410184 0.912003i $$-0.365465\pi$$
0.410184 + 0.912003i $$0.365465\pi$$
$$108$$ −49.2002 −0.0438360
$$109$$ 862.077 0.757541 0.378770 0.925491i $$-0.376347\pi$$
0.378770 + 0.925491i $$0.376347\pi$$
$$110$$ −212.424 −0.184126
$$111$$ 49.1601 0.0420366
$$112$$ 129.408 0.109178
$$113$$ 1502.72 1.25101 0.625505 0.780220i $$-0.284893\pi$$
0.625505 + 0.780220i $$0.284893\pi$$
$$114$$ 8.66393 0.00711799
$$115$$ −158.304 −0.128364
$$116$$ 364.848 0.292028
$$117$$ 1267.53 1.00156
$$118$$ −500.600 −0.390542
$$119$$ −254.416 −0.195986
$$120$$ 15.1683 0.0115389
$$121$$ −1167.88 −0.877443
$$122$$ −635.384 −0.471516
$$123$$ −15.4400 −0.0113185
$$124$$ 1175.87 0.851584
$$125$$ −1503.90 −1.07610
$$126$$ −435.912 −0.308207
$$127$$ 389.280 0.271992 0.135996 0.990709i $$-0.456577\pi$$
0.135996 + 0.990709i $$0.456577\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 70.3593 0.0480216
$$130$$ −782.304 −0.527789
$$131$$ −268.308 −0.178948 −0.0894739 0.995989i $$-0.528519\pi$$
−0.0894739 + 0.995989i $$0.528519\pi$$
$$132$$ −11.6480 −0.00768050
$$133$$ 153.672 0.100188
$$134$$ 1880.89 1.21257
$$135$$ −102.287 −0.0652110
$$136$$ −251.648 −0.158666
$$137$$ 2657.33 1.65716 0.828580 0.559871i $$-0.189149\pi$$
0.828580 + 0.559871i $$0.189149\pi$$
$$138$$ −8.68036 −0.00535450
$$139$$ −2859.92 −1.74514 −0.872572 0.488486i $$-0.837549\pi$$
−0.872572 + 0.488486i $$0.837549\pi$$
$$140$$ 269.040 0.162414
$$141$$ 24.8090 0.0148177
$$142$$ −791.104 −0.467521
$$143$$ 600.744 0.351306
$$144$$ −431.168 −0.249519
$$145$$ 758.520 0.434425
$$146$$ 1950.10 1.10542
$$147$$ −63.2887 −0.0355099
$$148$$ 862.464 0.479014
$$149$$ 311.812 0.171440 0.0857202 0.996319i $$-0.472681\pi$$
0.0857202 + 0.996319i $$0.472681\pi$$
$$150$$ −25.4647 −0.0138612
$$151$$ −1462.32 −0.788093 −0.394046 0.919091i $$-0.628925\pi$$
−0.394046 + 0.919091i $$0.628925\pi$$
$$152$$ 152.000 0.0811107
$$153$$ 847.677 0.447912
$$154$$ −206.600 −0.108106
$$155$$ 2444.64 1.26683
$$156$$ −42.8965 −0.0220158
$$157$$ 4.38395 0.00222852 0.00111426 0.999999i $$-0.499645\pi$$
0.00111426 + 0.999999i $$0.499645\pi$$
$$158$$ 1845.55 0.929267
$$159$$ −155.523 −0.0775709
$$160$$ 266.112 0.131488
$$161$$ −153.964 −0.0753666
$$162$$ 1449.58 0.703025
$$163$$ −1777.89 −0.854325 −0.427162 0.904175i $$-0.640487\pi$$
−0.427162 + 0.904175i $$0.640487\pi$$
$$164$$ −270.880 −0.128977
$$165$$ −24.2161 −0.0114256
$$166$$ −2327.54 −1.08826
$$167$$ 893.064 0.413817 0.206908 0.978360i $$-0.433660\pi$$
0.206908 + 0.978360i $$0.433660\pi$$
$$168$$ 14.7524 0.00677484
$$169$$ 15.3876 0.00700391
$$170$$ −523.176 −0.236034
$$171$$ −512.012 −0.228974
$$172$$ 1234.38 0.547214
$$173$$ −2452.56 −1.07783 −0.538915 0.842360i $$-0.681166\pi$$
−0.538915 + 0.842360i $$0.681166\pi$$
$$174$$ 41.5923 0.0181213
$$175$$ −451.667 −0.195102
$$176$$ −204.352 −0.0875205
$$177$$ −57.0679 −0.0242344
$$178$$ 1370.27 0.577002
$$179$$ −2064.81 −0.862185 −0.431092 0.902308i $$-0.641872\pi$$
−0.431092 + 0.902308i $$0.641872\pi$$
$$180$$ −896.399 −0.371187
$$181$$ −2518.54 −1.03426 −0.517132 0.855906i $$-0.673000\pi$$
−0.517132 + 0.855906i $$0.673000\pi$$
$$182$$ −760.855 −0.309881
$$183$$ −72.4332 −0.0292591
$$184$$ −152.288 −0.0610154
$$185$$ 1793.06 0.712588
$$186$$ 134.048 0.0528436
$$187$$ 401.756 0.157109
$$188$$ 435.249 0.168850
$$189$$ −99.4829 −0.0382874
$$190$$ 316.008 0.120661
$$191$$ −4206.38 −1.59352 −0.796761 0.604294i $$-0.793455\pi$$
−0.796761 + 0.604294i $$0.793455\pi$$
$$192$$ 14.5919 0.00548478
$$193$$ 3245.82 1.21056 0.605282 0.796011i $$-0.293060\pi$$
0.605282 + 0.796011i $$0.293060\pi$$
$$194$$ 422.512 0.156364
$$195$$ −89.1819 −0.0327510
$$196$$ −1110.34 −0.404642
$$197$$ −1734.71 −0.627377 −0.313688 0.949526i $$-0.601565\pi$$
−0.313688 + 0.949526i $$0.601565\pi$$
$$198$$ 688.360 0.247069
$$199$$ 380.792 0.135646 0.0678232 0.997697i $$-0.478395\pi$$
0.0678232 + 0.997697i $$0.478395\pi$$
$$200$$ −446.752 −0.157951
$$201$$ 214.420 0.0752437
$$202$$ −3407.10 −1.18675
$$203$$ 737.724 0.255064
$$204$$ −28.6876 −0.00984576
$$205$$ −563.160 −0.191867
$$206$$ −2787.04 −0.942633
$$207$$ 512.983 0.172245
$$208$$ −752.576 −0.250874
$$209$$ −242.668 −0.0803143
$$210$$ 30.6703 0.0100783
$$211$$ 1010.44 0.329675 0.164837 0.986321i $$-0.447290\pi$$
0.164837 + 0.986321i $$0.447290\pi$$
$$212$$ −2728.50 −0.883933
$$213$$ −90.1852 −0.0290112
$$214$$ 1815.99 0.580087
$$215$$ 2566.29 0.814043
$$216$$ −98.4004 −0.0309967
$$217$$ 2377.62 0.743793
$$218$$ 1724.15 0.535662
$$219$$ 222.309 0.0685948
$$220$$ −424.848 −0.130197
$$221$$ 1479.57 0.450345
$$222$$ 98.3201 0.0297244
$$223$$ 3398.70 1.02060 0.510301 0.859996i $$-0.329534\pi$$
0.510301 + 0.859996i $$0.329534\pi$$
$$224$$ 258.816 0.0772004
$$225$$ 1504.89 0.445892
$$226$$ 3005.44 0.884597
$$227$$ 5760.80 1.68439 0.842197 0.539169i $$-0.181262\pi$$
0.842197 + 0.539169i $$0.181262\pi$$
$$228$$ 17.3279 0.00503318
$$229$$ −2179.00 −0.628786 −0.314393 0.949293i $$-0.601801\pi$$
−0.314393 + 0.949293i $$0.601801\pi$$
$$230$$ −316.607 −0.0907673
$$231$$ −23.5522 −0.00670832
$$232$$ 729.696 0.206495
$$233$$ −2808.49 −0.789659 −0.394830 0.918754i $$-0.629196\pi$$
−0.394830 + 0.918754i $$0.629196\pi$$
$$234$$ 2535.06 0.708213
$$235$$ 904.882 0.251183
$$236$$ −1001.20 −0.276155
$$237$$ 210.391 0.0576640
$$238$$ −508.833 −0.138583
$$239$$ 6285.67 1.70120 0.850599 0.525815i $$-0.176239\pi$$
0.850599 + 0.525815i $$0.176239\pi$$
$$240$$ 30.3365 0.00815923
$$241$$ 1129.22 0.301825 0.150912 0.988547i $$-0.451779\pi$$
0.150912 + 0.988547i $$0.451779\pi$$
$$242$$ −2335.75 −0.620446
$$243$$ 497.352 0.131297
$$244$$ −1270.77 −0.333412
$$245$$ −2308.39 −0.601950
$$246$$ −30.8801 −0.00800342
$$247$$ −893.684 −0.230218
$$248$$ 2351.74 0.602161
$$249$$ −265.337 −0.0675303
$$250$$ −3007.80 −0.760920
$$251$$ −2873.73 −0.722661 −0.361331 0.932438i $$-0.617677\pi$$
−0.361331 + 0.932438i $$0.617677\pi$$
$$252$$ −871.823 −0.217935
$$253$$ 243.128 0.0604163
$$254$$ 778.559 0.192327
$$255$$ −59.6416 −0.0146467
$$256$$ 256.000 0.0625000
$$257$$ −3712.18 −0.901008 −0.450504 0.892774i $$-0.648756\pi$$
−0.450504 + 0.892774i $$0.648756\pi$$
$$258$$ 140.719 0.0339564
$$259$$ 1743.90 0.418382
$$260$$ −1564.61 −0.373203
$$261$$ −2457.98 −0.582932
$$262$$ −536.616 −0.126535
$$263$$ 1263.04 0.296130 0.148065 0.988978i $$-0.452696\pi$$
0.148065 + 0.988978i $$0.452696\pi$$
$$264$$ −23.2959 −0.00543093
$$265$$ −5672.55 −1.31495
$$266$$ 307.344 0.0708439
$$267$$ 156.210 0.0358048
$$268$$ 3761.78 0.857414
$$269$$ −5484.39 −1.24308 −0.621541 0.783381i $$-0.713493\pi$$
−0.621541 + 0.783381i $$0.713493\pi$$
$$270$$ −204.575 −0.0461111
$$271$$ −3217.66 −0.721251 −0.360625 0.932711i $$-0.617437\pi$$
−0.360625 + 0.932711i $$0.617437\pi$$
$$272$$ −503.296 −0.112194
$$273$$ −86.7368 −0.0192291
$$274$$ 5314.66 1.17179
$$275$$ 713.240 0.156400
$$276$$ −17.3607 −0.00378620
$$277$$ 7668.13 1.66330 0.831649 0.555302i $$-0.187397\pi$$
0.831649 + 0.555302i $$0.187397\pi$$
$$278$$ −5719.83 −1.23400
$$279$$ −7921.86 −1.69989
$$280$$ 538.079 0.114844
$$281$$ 1126.81 0.239216 0.119608 0.992821i $$-0.461836\pi$$
0.119608 + 0.992821i $$0.461836\pi$$
$$282$$ 49.6179 0.0104777
$$283$$ −1502.63 −0.315625 −0.157813 0.987469i $$-0.550444\pi$$
−0.157813 + 0.987469i $$0.550444\pi$$
$$284$$ −1582.21 −0.330587
$$285$$ 36.0246 0.00748742
$$286$$ 1201.49 0.248411
$$287$$ −547.720 −0.112651
$$288$$ −862.337 −0.176436
$$289$$ −3923.52 −0.798600
$$290$$ 1517.04 0.307185
$$291$$ 48.1659 0.00970288
$$292$$ 3900.19 0.781649
$$293$$ −452.324 −0.0901878 −0.0450939 0.998983i $$-0.514359\pi$$
−0.0450939 + 0.998983i $$0.514359\pi$$
$$294$$ −126.577 −0.0251093
$$295$$ −2081.50 −0.410812
$$296$$ 1724.93 0.338714
$$297$$ 157.096 0.0306924
$$298$$ 623.623 0.121227
$$299$$ 895.379 0.173181
$$300$$ −50.9294 −0.00980136
$$301$$ 2495.93 0.477950
$$302$$ −2924.64 −0.557266
$$303$$ −388.407 −0.0736415
$$304$$ 304.000 0.0573539
$$305$$ −2641.93 −0.495988
$$306$$ 1695.35 0.316722
$$307$$ −2333.46 −0.433803 −0.216901 0.976194i $$-0.569595\pi$$
−0.216901 + 0.976194i $$0.569595\pi$$
$$308$$ −413.200 −0.0764424
$$309$$ −317.720 −0.0584934
$$310$$ 4889.28 0.895782
$$311$$ 10476.1 1.91011 0.955055 0.296429i $$-0.0957959\pi$$
0.955055 + 0.296429i $$0.0957959\pi$$
$$312$$ −85.7930 −0.0155675
$$313$$ 4160.33 0.751297 0.375648 0.926762i $$-0.377420\pi$$
0.375648 + 0.926762i $$0.377420\pi$$
$$314$$ 8.76790 0.00157580
$$315$$ −1812.52 −0.324203
$$316$$ 3691.10 0.657091
$$317$$ −7508.56 −1.33036 −0.665178 0.746685i $$-0.731644\pi$$
−0.665178 + 0.746685i $$0.731644\pi$$
$$318$$ −311.046 −0.0548509
$$319$$ −1164.96 −0.204468
$$320$$ 532.224 0.0929758
$$321$$ 207.021 0.0359963
$$322$$ −307.927 −0.0532922
$$323$$ −597.664 −0.102956
$$324$$ 2899.17 0.497114
$$325$$ 2626.68 0.448314
$$326$$ −3555.78 −0.604099
$$327$$ 196.552 0.0332396
$$328$$ −541.760 −0.0912003
$$329$$ 880.073 0.147477
$$330$$ −48.4323 −0.00807912
$$331$$ 10386.8 1.72480 0.862400 0.506227i $$-0.168960\pi$$
0.862400 + 0.506227i $$0.168960\pi$$
$$332$$ −4655.07 −0.769519
$$333$$ −5810.43 −0.956184
$$334$$ 1786.13 0.292613
$$335$$ 7820.74 1.27550
$$336$$ 29.5048 0.00479053
$$337$$ 5618.29 0.908153 0.454077 0.890963i $$-0.349969\pi$$
0.454077 + 0.890963i $$0.349969\pi$$
$$338$$ 30.7752 0.00495251
$$339$$ 342.617 0.0548921
$$340$$ −1046.35 −0.166901
$$341$$ −3754.56 −0.596249
$$342$$ −1024.02 −0.161909
$$343$$ −5019.29 −0.790135
$$344$$ 2468.77 0.386939
$$345$$ −36.0929 −0.00563240
$$346$$ −4905.12 −0.762142
$$347$$ 1814.32 0.280686 0.140343 0.990103i $$-0.455180\pi$$
0.140343 + 0.990103i $$0.455180\pi$$
$$348$$ 83.1847 0.0128137
$$349$$ −816.757 −0.125272 −0.0626361 0.998036i $$-0.519951\pi$$
−0.0626361 + 0.998036i $$0.519951\pi$$
$$350$$ −903.334 −0.137958
$$351$$ 578.545 0.0879785
$$352$$ −408.704 −0.0618864
$$353$$ 11090.4 1.67219 0.836095 0.548585i $$-0.184833\pi$$
0.836095 + 0.548585i $$0.184833\pi$$
$$354$$ −114.136 −0.0171363
$$355$$ −3289.41 −0.491786
$$356$$ 2740.55 0.408002
$$357$$ −58.0064 −0.00859951
$$358$$ −4129.62 −0.609657
$$359$$ −3211.68 −0.472161 −0.236081 0.971733i $$-0.575863\pi$$
−0.236081 + 0.971733i $$0.575863\pi$$
$$360$$ −1792.80 −0.262469
$$361$$ 361.000 0.0526316
$$362$$ −5037.09 −0.731335
$$363$$ −266.274 −0.0385007
$$364$$ −1521.71 −0.219119
$$365$$ 8108.51 1.16279
$$366$$ −144.866 −0.0206893
$$367$$ 8077.81 1.14893 0.574466 0.818528i $$-0.305210\pi$$
0.574466 + 0.818528i $$0.305210\pi$$
$$368$$ −304.576 −0.0431444
$$369$$ 1824.92 0.257457
$$370$$ 3586.13 0.503876
$$371$$ −5517.02 −0.772048
$$372$$ 268.097 0.0373660
$$373$$ −5088.15 −0.706312 −0.353156 0.935564i $$-0.614892\pi$$
−0.353156 + 0.935564i $$0.614892\pi$$
$$374$$ 803.512 0.111093
$$375$$ −342.886 −0.0472175
$$376$$ 870.497 0.119395
$$377$$ −4290.25 −0.586099
$$378$$ −198.966 −0.0270733
$$379$$ 2547.00 0.345199 0.172600 0.984992i $$-0.444783\pi$$
0.172600 + 0.984992i $$0.444783\pi$$
$$380$$ 632.016 0.0853204
$$381$$ 88.7550 0.0119345
$$382$$ −8412.76 −1.12679
$$383$$ −7056.11 −0.941384 −0.470692 0.882297i $$-0.655996\pi$$
−0.470692 + 0.882297i $$0.655996\pi$$
$$384$$ 29.1838 0.00387833
$$385$$ −859.044 −0.113717
$$386$$ 6491.63 0.855999
$$387$$ −8316.05 −1.09232
$$388$$ 845.023 0.110566
$$389$$ 4728.25 0.616277 0.308138 0.951342i $$-0.400294\pi$$
0.308138 + 0.951342i $$0.400294\pi$$
$$390$$ −178.364 −0.0231585
$$391$$ 598.797 0.0774488
$$392$$ −2220.67 −0.286125
$$393$$ −61.1737 −0.00785192
$$394$$ −3469.43 −0.443622
$$395$$ 7673.81 0.977497
$$396$$ 1376.72 0.174704
$$397$$ 740.837 0.0936563 0.0468281 0.998903i $$-0.485089\pi$$
0.0468281 + 0.998903i $$0.485089\pi$$
$$398$$ 761.584 0.0959165
$$399$$ 35.0370 0.00439610
$$400$$ −893.505 −0.111688
$$401$$ 1879.58 0.234070 0.117035 0.993128i $$-0.462661\pi$$
0.117035 + 0.993128i $$0.462661\pi$$
$$402$$ 428.839 0.0532053
$$403$$ −13827.1 −1.70912
$$404$$ −6814.21 −0.839157
$$405$$ 6027.37 0.739513
$$406$$ 1475.45 0.180358
$$407$$ −2753.85 −0.335389
$$408$$ −57.3753 −0.00696201
$$409$$ −1715.45 −0.207393 −0.103697 0.994609i $$-0.533067\pi$$
−0.103697 + 0.994609i $$0.533067\pi$$
$$410$$ −1126.32 −0.135671
$$411$$ 605.866 0.0727133
$$412$$ −5574.08 −0.666542
$$413$$ −2024.43 −0.241200
$$414$$ 1025.97 0.121796
$$415$$ −9677.90 −1.14475
$$416$$ −1505.15 −0.177395
$$417$$ −652.056 −0.0765739
$$418$$ −485.336 −0.0567908
$$419$$ 2497.15 0.291155 0.145578 0.989347i $$-0.453496\pi$$
0.145578 + 0.989347i $$0.453496\pi$$
$$420$$ 61.3405 0.00712646
$$421$$ 6582.52 0.762024 0.381012 0.924570i $$-0.375576\pi$$
0.381012 + 0.924570i $$0.375576\pi$$
$$422$$ 2020.87 0.233115
$$423$$ −2932.27 −0.337049
$$424$$ −5456.99 −0.625035
$$425$$ 1756.63 0.200492
$$426$$ −180.370 −0.0205140
$$427$$ −2569.50 −0.291210
$$428$$ 3631.99 0.410184
$$429$$ 136.969 0.0154147
$$430$$ 5132.57 0.575615
$$431$$ 8875.72 0.991946 0.495973 0.868338i $$-0.334812\pi$$
0.495973 + 0.868338i $$0.334812\pi$$
$$432$$ −196.801 −0.0219180
$$433$$ −3636.90 −0.403645 −0.201822 0.979422i $$-0.564686\pi$$
−0.201822 + 0.979422i $$0.564686\pi$$
$$434$$ 4755.23 0.525941
$$435$$ 172.941 0.0190618
$$436$$ 3448.31 0.378770
$$437$$ −361.684 −0.0395920
$$438$$ 444.618 0.0485039
$$439$$ −10979.4 −1.19366 −0.596829 0.802368i $$-0.703573\pi$$
−0.596829 + 0.802368i $$0.703573\pi$$
$$440$$ −849.696 −0.0920629
$$441$$ 7480.34 0.807725
$$442$$ 2959.13 0.318442
$$443$$ 1300.16 0.139442 0.0697208 0.997567i $$-0.477789\pi$$
0.0697208 + 0.997567i $$0.477789\pi$$
$$444$$ 196.640 0.0210183
$$445$$ 5697.60 0.606948
$$446$$ 6797.41 0.721674
$$447$$ 71.0925 0.00752250
$$448$$ 517.632 0.0545889
$$449$$ −15875.2 −1.66859 −0.834296 0.551317i $$-0.814125\pi$$
−0.834296 + 0.551317i $$0.814125\pi$$
$$450$$ 3009.77 0.315293
$$451$$ 864.920 0.0903049
$$452$$ 6010.88 0.625505
$$453$$ −333.406 −0.0345801
$$454$$ 11521.6 1.19105
$$455$$ −3163.64 −0.325964
$$456$$ 34.6557 0.00355900
$$457$$ 3115.66 0.318916 0.159458 0.987205i $$-0.449025\pi$$
0.159458 + 0.987205i $$0.449025\pi$$
$$458$$ −4357.99 −0.444619
$$459$$ 386.910 0.0393451
$$460$$ −633.215 −0.0641822
$$461$$ 13479.7 1.36185 0.680924 0.732354i $$-0.261578\pi$$
0.680924 + 0.732354i $$0.261578\pi$$
$$462$$ −47.1044 −0.00474350
$$463$$ 7946.19 0.797604 0.398802 0.917037i $$-0.369426\pi$$
0.398802 + 0.917037i $$0.369426\pi$$
$$464$$ 1459.39 0.146014
$$465$$ 557.373 0.0555862
$$466$$ −5616.99 −0.558373
$$467$$ −9148.37 −0.906501 −0.453250 0.891383i $$-0.649736\pi$$
−0.453250 + 0.891383i $$0.649736\pi$$
$$468$$ 5070.11 0.500782
$$469$$ 7606.32 0.748885
$$470$$ 1809.76 0.177613
$$471$$ 0.999532 9.77834e−5 0
$$472$$ −2002.40 −0.195271
$$473$$ −3941.39 −0.383140
$$474$$ 420.782 0.0407746
$$475$$ −1061.04 −0.102492
$$476$$ −1017.67 −0.0979929
$$477$$ 18381.9 1.76446
$$478$$ 12571.3 1.20293
$$479$$ 7664.64 0.731120 0.365560 0.930788i $$-0.380878\pi$$
0.365560 + 0.930788i $$0.380878\pi$$
$$480$$ 60.6731 0.00576945
$$481$$ −10141.7 −0.961378
$$482$$ 2258.45 0.213422
$$483$$ −35.1034 −0.00330696
$$484$$ −4671.50 −0.438721
$$485$$ 1756.80 0.164479
$$486$$ 994.705 0.0928410
$$487$$ −5347.21 −0.497547 −0.248774 0.968562i $$-0.580027\pi$$
−0.248774 + 0.968562i $$0.580027\pi$$
$$488$$ −2541.54 −0.235758
$$489$$ −405.355 −0.0374863
$$490$$ −4616.78 −0.425643
$$491$$ 13647.2 1.25436 0.627178 0.778876i $$-0.284210\pi$$
0.627178 + 0.778876i $$0.284210\pi$$
$$492$$ −61.7601 −0.00565927
$$493$$ −2869.17 −0.262111
$$494$$ −1787.37 −0.162789
$$495$$ 2862.20 0.259892
$$496$$ 4703.49 0.425792
$$497$$ −3199.23 −0.288743
$$498$$ −530.674 −0.0477511
$$499$$ −19351.6 −1.73607 −0.868034 0.496504i $$-0.834617\pi$$
−0.868034 + 0.496504i $$0.834617\pi$$
$$500$$ −6015.60 −0.538052
$$501$$ 203.617 0.0181576
$$502$$ −5747.45 −0.510999
$$503$$ −19259.1 −1.70720 −0.853600 0.520929i $$-0.825586\pi$$
−0.853600 + 0.520929i $$0.825586\pi$$
$$504$$ −1743.65 −0.154104
$$505$$ −14166.7 −1.24834
$$506$$ 486.256 0.0427208
$$507$$ 3.50834 0.000307319 0
$$508$$ 1557.12 0.135996
$$509$$ 3595.77 0.313123 0.156561 0.987668i $$-0.449959\pi$$
0.156561 + 0.987668i $$0.449959\pi$$
$$510$$ −119.283 −0.0103568
$$511$$ 7886.20 0.682710
$$512$$ 512.000 0.0441942
$$513$$ −233.701 −0.0201133
$$514$$ −7424.35 −0.637109
$$515$$ −11588.5 −0.991556
$$516$$ 281.437 0.0240108
$$517$$ −1389.75 −0.118223
$$518$$ 3487.81 0.295841
$$519$$ −559.179 −0.0472933
$$520$$ −3129.21 −0.263895
$$521$$ −15211.0 −1.27909 −0.639544 0.768754i $$-0.720877\pi$$
−0.639544 + 0.768754i $$0.720877\pi$$
$$522$$ −4915.97 −0.412195
$$523$$ 18307.1 1.53062 0.765310 0.643662i $$-0.222586\pi$$
0.765310 + 0.643662i $$0.222586\pi$$
$$524$$ −1073.23 −0.0894739
$$525$$ −102.979 −0.00856073
$$526$$ 2526.07 0.209395
$$527$$ −9247.06 −0.764342
$$528$$ −46.5919 −0.00384025
$$529$$ −11804.6 −0.970217
$$530$$ −11345.1 −0.929810
$$531$$ 6745.09 0.551247
$$532$$ 614.689 0.0500942
$$533$$ 3185.28 0.258855
$$534$$ 312.420 0.0253178
$$535$$ 7550.90 0.610194
$$536$$ 7523.55 0.606284
$$537$$ −470.773 −0.0378312
$$538$$ −10968.8 −0.878992
$$539$$ 3545.31 0.283316
$$540$$ −409.149 −0.0326055
$$541$$ 9102.17 0.723351 0.361676 0.932304i $$-0.382205\pi$$
0.361676 + 0.932304i $$0.382205\pi$$
$$542$$ −6435.32 −0.510001
$$543$$ −574.223 −0.0453817
$$544$$ −1006.59 −0.0793332
$$545$$ 7169.03 0.563464
$$546$$ −173.474 −0.0135970
$$547$$ −9218.75 −0.720595 −0.360297 0.932837i $$-0.617325\pi$$
−0.360297 + 0.932837i $$0.617325\pi$$
$$548$$ 10629.3 0.828580
$$549$$ 8561.17 0.665541
$$550$$ 1426.48 0.110592
$$551$$ 1733.03 0.133992
$$552$$ −34.7214 −0.00267725
$$553$$ 7463.42 0.573918
$$554$$ 15336.3 1.17613
$$555$$ 408.815 0.0312671
$$556$$ −11439.7 −0.872572
$$557$$ −13435.1 −1.02202 −0.511010 0.859575i $$-0.670728\pi$$
−0.511010 + 0.859575i $$0.670728\pi$$
$$558$$ −15843.7 −1.20200
$$559$$ −14515.1 −1.09825
$$560$$ 1076.16 0.0812071
$$561$$ 91.5996 0.00689365
$$562$$ 2253.62 0.169151
$$563$$ 11941.5 0.893916 0.446958 0.894555i $$-0.352507\pi$$
0.446958 + 0.894555i $$0.352507\pi$$
$$564$$ 99.2359 0.00740884
$$565$$ 12496.6 0.930508
$$566$$ −3005.26 −0.223181
$$567$$ 5862.12 0.434191
$$568$$ −3164.42 −0.233761
$$569$$ −6378.91 −0.469979 −0.234989 0.971998i $$-0.575506\pi$$
−0.234989 + 0.971998i $$0.575506\pi$$
$$570$$ 72.0493 0.00529441
$$571$$ 24903.9 1.82521 0.912605 0.408843i $$-0.134068\pi$$
0.912605 + 0.408843i $$0.134068\pi$$
$$572$$ 2402.98 0.175653
$$573$$ −959.046 −0.0699210
$$574$$ −1095.44 −0.0796564
$$575$$ 1063.05 0.0770995
$$576$$ −1724.67 −0.124759
$$577$$ −11414.7 −0.823568 −0.411784 0.911281i $$-0.635094\pi$$
−0.411784 + 0.911281i $$0.635094\pi$$
$$578$$ −7847.04 −0.564695
$$579$$ 740.040 0.0531175
$$580$$ 3034.08 0.217213
$$581$$ −9412.57 −0.672115
$$582$$ 96.3319 0.00686097
$$583$$ 8712.09 0.618899
$$584$$ 7800.39 0.552709
$$585$$ 10540.8 0.744969
$$586$$ −904.648 −0.0637724
$$587$$ 20732.1 1.45776 0.728881 0.684641i $$-0.240041\pi$$
0.728881 + 0.684641i $$0.240041\pi$$
$$588$$ −253.155 −0.0177550
$$589$$ 5585.39 0.390734
$$590$$ −4162.99 −0.290488
$$591$$ −395.511 −0.0275282
$$592$$ 3449.86 0.239507
$$593$$ 18010.5 1.24722 0.623611 0.781735i $$-0.285665\pi$$
0.623611 + 0.781735i $$0.285665\pi$$
$$594$$ 314.192 0.0217028
$$595$$ −2115.73 −0.145775
$$596$$ 1247.25 0.0857202
$$597$$ 86.8199 0.00595193
$$598$$ 1790.76 0.122457
$$599$$ 27944.7 1.90616 0.953080 0.302719i $$-0.0978942\pi$$
0.953080 + 0.302719i $$0.0978942\pi$$
$$600$$ −101.859 −0.00693061
$$601$$ −11598.1 −0.787179 −0.393590 0.919286i $$-0.628767\pi$$
−0.393590 + 0.919286i $$0.628767\pi$$
$$602$$ 4991.85 0.337961
$$603$$ −25343.1 −1.71153
$$604$$ −5849.28 −0.394046
$$605$$ −9712.06 −0.652647
$$606$$ −776.813 −0.0520724
$$607$$ 20170.5 1.34876 0.674379 0.738385i $$-0.264411\pi$$
0.674379 + 0.738385i $$0.264411\pi$$
$$608$$ 608.000 0.0405554
$$609$$ 168.200 0.0111918
$$610$$ −5283.86 −0.350717
$$611$$ −5118.09 −0.338880
$$612$$ 3390.71 0.223956
$$613$$ 14618.3 0.963174 0.481587 0.876398i $$-0.340061\pi$$
0.481587 + 0.876398i $$0.340061\pi$$
$$614$$ −4666.91 −0.306745
$$615$$ −128.399 −0.00841880
$$616$$ −826.400 −0.0540530
$$617$$ −17538.1 −1.14434 −0.572171 0.820134i $$-0.693899\pi$$
−0.572171 + 0.820134i $$0.693899\pi$$
$$618$$ −635.440 −0.0413611
$$619$$ −8815.75 −0.572431 −0.286216 0.958165i $$-0.592397\pi$$
−0.286216 + 0.958165i $$0.592397\pi$$
$$620$$ 9778.56 0.633414
$$621$$ 234.144 0.0151302
$$622$$ 20952.2 1.35065
$$623$$ 5541.39 0.356358
$$624$$ −171.586 −0.0110079
$$625$$ −5525.94 −0.353660
$$626$$ 8320.66 0.531247
$$627$$ −55.3279 −0.00352405
$$628$$ 17.5358 0.00111426
$$629$$ −6782.42 −0.429941
$$630$$ −3625.04 −0.229246
$$631$$ −22170.8 −1.39874 −0.699370 0.714759i $$-0.746536\pi$$
−0.699370 + 0.714759i $$0.746536\pi$$
$$632$$ 7382.21 0.464634
$$633$$ 230.378 0.0144655
$$634$$ −15017.1 −0.940703
$$635$$ 3237.25 0.202309
$$636$$ −622.092 −0.0387855
$$637$$ 13056.5 0.812112
$$638$$ −2329.92 −0.144581
$$639$$ 10659.3 0.659902
$$640$$ 1064.45 0.0657438
$$641$$ −22067.7 −1.35978 −0.679891 0.733313i $$-0.737973\pi$$
−0.679891 + 0.733313i $$0.737973\pi$$
$$642$$ 414.043 0.0254532
$$643$$ −11795.4 −0.723428 −0.361714 0.932289i $$-0.617808\pi$$
−0.361714 + 0.932289i $$0.617808\pi$$
$$644$$ −615.854 −0.0376833
$$645$$ 585.108 0.0357188
$$646$$ −1195.33 −0.0728012
$$647$$ −9716.04 −0.590382 −0.295191 0.955438i $$-0.595383\pi$$
−0.295191 + 0.955438i $$0.595383\pi$$
$$648$$ 5798.34 0.351513
$$649$$ 3196.83 0.193354
$$650$$ 5253.36 0.317006
$$651$$ 542.092 0.0326363
$$652$$ −7111.55 −0.427162
$$653$$ −10311.9 −0.617969 −0.308985 0.951067i $$-0.599989\pi$$
−0.308985 + 0.951067i $$0.599989\pi$$
$$654$$ 393.104 0.0235039
$$655$$ −2231.25 −0.133102
$$656$$ −1083.52 −0.0644884
$$657$$ −26275.6 −1.56029
$$658$$ 1760.15 0.104282
$$659$$ 4019.80 0.237616 0.118808 0.992917i $$-0.462093\pi$$
0.118808 + 0.992917i $$0.462093\pi$$
$$660$$ −96.8646 −0.00571280
$$661$$ −22702.6 −1.33590 −0.667951 0.744206i $$-0.732828\pi$$
−0.667951 + 0.744206i $$0.732828\pi$$
$$662$$ 20773.6 1.21962
$$663$$ 337.338 0.0197604
$$664$$ −9310.15 −0.544132
$$665$$ 1277.94 0.0745208
$$666$$ −11620.9 −0.676124
$$667$$ −1736.31 −0.100795
$$668$$ 3572.26 0.206908
$$669$$ 774.898 0.0447822
$$670$$ 15641.5 0.901915
$$671$$ 4057.57 0.233443
$$672$$ 59.0096 0.00338742
$$673$$ 11132.8 0.637652 0.318826 0.947813i $$-0.396711\pi$$
0.318826 + 0.947813i $$0.396711\pi$$
$$674$$ 11236.6 0.642161
$$675$$ 686.884 0.0391677
$$676$$ 61.5503 0.00350195
$$677$$ −13967.0 −0.792903 −0.396452 0.918056i $$-0.629759\pi$$
−0.396452 + 0.918056i $$0.629759\pi$$
$$678$$ 685.235 0.0388146
$$679$$ 1708.64 0.0965707
$$680$$ −2092.71 −0.118017
$$681$$ 1313.45 0.0739083
$$682$$ −7509.12 −0.421612
$$683$$ 1173.88 0.0657648 0.0328824 0.999459i $$-0.489531\pi$$
0.0328824 + 0.999459i $$0.489531\pi$$
$$684$$ −2048.05 −0.114487
$$685$$ 22098.4 1.23261
$$686$$ −10038.6 −0.558709
$$687$$ −496.807 −0.0275901
$$688$$ 4937.54 0.273607
$$689$$ 32084.4 1.77405
$$690$$ −72.1859 −0.00398271
$$691$$ 8713.33 0.479697 0.239849 0.970810i $$-0.422902\pi$$
0.239849 + 0.970810i $$0.422902\pi$$
$$692$$ −9810.24 −0.538915
$$693$$ 2783.73 0.152590
$$694$$ 3628.64 0.198475
$$695$$ −23783.1 −1.29805
$$696$$ 166.369 0.00906065
$$697$$ 2130.20 0.115763
$$698$$ −1633.51 −0.0885809
$$699$$ −640.331 −0.0346489
$$700$$ −1806.67 −0.0975509
$$701$$ −31003.4 −1.67045 −0.835223 0.549912i $$-0.814661\pi$$
−0.835223 + 0.549912i $$0.814661\pi$$
$$702$$ 1157.09 0.0622102
$$703$$ 4096.70 0.219787
$$704$$ −817.408 −0.0437603
$$705$$ 206.311 0.0110215
$$706$$ 22180.8 1.18242
$$707$$ −13778.3 −0.732939
$$708$$ −228.272 −0.0121172
$$709$$ −12145.1 −0.643328 −0.321664 0.946854i $$-0.604242\pi$$
−0.321664 + 0.946854i $$0.604242\pi$$
$$710$$ −6578.83 −0.347745
$$711$$ −24867.0 −1.31165
$$712$$ 5481.09 0.288501
$$713$$ −5595.98 −0.293929
$$714$$ −116.013 −0.00608078
$$715$$ 4995.79 0.261304
$$716$$ −8259.24 −0.431092
$$717$$ 1433.12 0.0746456
$$718$$ −6423.36 −0.333868
$$719$$ 24787.8 1.28572 0.642858 0.765985i $$-0.277748\pi$$
0.642858 + 0.765985i $$0.277748\pi$$
$$720$$ −3585.60 −0.185594
$$721$$ −11270.8 −0.582173
$$722$$ 722.000 0.0372161
$$723$$ 257.461 0.0132435
$$724$$ −10074.2 −0.517132
$$725$$ −5093.65 −0.260929
$$726$$ −532.547 −0.0272241
$$727$$ 19335.6 0.986409 0.493204 0.869914i $$-0.335826\pi$$
0.493204 + 0.869914i $$0.335826\pi$$
$$728$$ −3043.42 −0.154941
$$729$$ −19456.0 −0.988467
$$730$$ 16217.0 0.822217
$$731$$ −9707.19 −0.491154
$$732$$ −289.733 −0.0146296
$$733$$ 20204.5 1.01810 0.509052 0.860735i $$-0.329996\pi$$
0.509052 + 0.860735i $$0.329996\pi$$
$$734$$ 16155.6 0.812418
$$735$$ −526.309 −0.0264125
$$736$$ −609.153 −0.0305077
$$737$$ −12011.4 −0.600331
$$738$$ 3649.84 0.182049
$$739$$ 15643.7 0.778706 0.389353 0.921089i $$-0.372699\pi$$
0.389353 + 0.921089i $$0.372699\pi$$
$$740$$ 7172.26 0.356294
$$741$$ −203.758 −0.0101016
$$742$$ −11034.0 −0.545920
$$743$$ −4500.20 −0.222202 −0.111101 0.993809i $$-0.535438\pi$$
−0.111101 + 0.993809i $$0.535438\pi$$
$$744$$ 536.193 0.0264218
$$745$$ 2593.03 0.127518
$$746$$ −10176.3 −0.499438
$$747$$ 31361.2 1.53608
$$748$$ 1607.02 0.0785543
$$749$$ 7343.88 0.358264
$$750$$ −685.773 −0.0333878
$$751$$ 35080.2 1.70452 0.852261 0.523117i $$-0.175231\pi$$
0.852261 + 0.523117i $$0.175231\pi$$
$$752$$ 1740.99 0.0844249
$$753$$ −655.204 −0.0317091
$$754$$ −8580.50 −0.414434
$$755$$ −12160.7 −0.586188
$$756$$ −397.931 −0.0191437
$$757$$ −10391.8 −0.498938 −0.249469 0.968383i $$-0.580256\pi$$
−0.249469 + 0.968383i $$0.580256\pi$$
$$758$$ 5094.00 0.244093
$$759$$ 55.4328 0.00265096
$$760$$ 1264.03 0.0603306
$$761$$ 11810.5 0.562590 0.281295 0.959621i $$-0.409236\pi$$
0.281295 + 0.959621i $$0.409236\pi$$
$$762$$ 177.510 0.00843899
$$763$$ 6972.48 0.330827
$$764$$ −16825.5 −0.796761
$$765$$ 7049.28 0.333160
$$766$$ −14112.2 −0.665659
$$767$$ 11773.1 0.554241
$$768$$ 58.3675 0.00274239
$$769$$ 35125.5 1.64715 0.823574 0.567209i $$-0.191977\pi$$
0.823574 + 0.567209i $$0.191977\pi$$
$$770$$ −1718.09 −0.0804098
$$771$$ −846.369 −0.0395347
$$772$$ 12983.3 0.605282
$$773$$ 20001.5 0.930665 0.465332 0.885136i $$-0.345935\pi$$
0.465332 + 0.885136i $$0.345935\pi$$
$$774$$ −16632.1 −0.772388
$$775$$ −16416.4 −0.760895
$$776$$ 1690.05 0.0781819
$$777$$ 397.607 0.0183579
$$778$$ 9456.49 0.435773
$$779$$ −1286.68 −0.0591786
$$780$$ −356.728 −0.0163755
$$781$$ 5051.99 0.231465
$$782$$ 1197.59 0.0547646
$$783$$ −1121.91 −0.0512055
$$784$$ −4441.35 −0.202321
$$785$$ 36.4569 0.00165758
$$786$$ −122.347 −0.00555214
$$787$$ 13593.3 0.615690 0.307845 0.951437i $$-0.400392\pi$$
0.307845 + 0.951437i $$0.400392\pi$$
$$788$$ −6938.85 −0.313688
$$789$$ 287.970 0.0129937
$$790$$ 15347.6 0.691195
$$791$$ 12154.0 0.546330
$$792$$ 2753.44 0.123534
$$793$$ 14943.0 0.669156
$$794$$ 1481.67 0.0662250
$$795$$ −1293.33 −0.0576977
$$796$$ 1523.17 0.0678232
$$797$$ −6946.75 −0.308741 −0.154370 0.988013i $$-0.549335\pi$$
−0.154370 + 0.988013i $$0.549335\pi$$
$$798$$ 70.0739 0.00310851
$$799$$ −3422.79 −0.151552
$$800$$ −1787.01 −0.0789754
$$801$$ −18463.1 −0.814432
$$802$$ 3759.17 0.165512
$$803$$ −12453.3 −0.547283
$$804$$ 857.678 0.0376219
$$805$$ −1280.36 −0.0560581
$$806$$ −27654.2 −1.20853
$$807$$ −1250.43 −0.0545443
$$808$$ −13628.4 −0.593374
$$809$$ 24987.2 1.08591 0.542955 0.839762i $$-0.317305\pi$$
0.542955 + 0.839762i $$0.317305\pi$$
$$810$$ 12054.7 0.522914
$$811$$ −23172.5 −1.00332 −0.501662 0.865064i $$-0.667278\pi$$
−0.501662 + 0.865064i $$0.667278\pi$$
$$812$$ 2950.89 0.127532
$$813$$ −733.621 −0.0316472
$$814$$ −5507.70 −0.237156
$$815$$ −14784.9 −0.635452
$$816$$ −114.751 −0.00492288
$$817$$ 5863.32 0.251079
$$818$$ −3430.91 −0.146649
$$819$$ 10251.8 0.437394
$$820$$ −2252.64 −0.0959337
$$821$$ 30703.8 1.30520 0.652600 0.757703i $$-0.273678\pi$$
0.652600 + 0.757703i $$0.273678\pi$$
$$822$$ 1211.73 0.0514161
$$823$$ −15940.1 −0.675135 −0.337568 0.941301i $$-0.609604\pi$$
−0.337568 + 0.941301i $$0.609604\pi$$
$$824$$ −11148.2 −0.471316
$$825$$ 162.617 0.00686256
$$826$$ −4048.86 −0.170554
$$827$$ −6662.20 −0.280130 −0.140065 0.990142i $$-0.544731\pi$$
−0.140065 + 0.990142i $$0.544731\pi$$
$$828$$ 2051.93 0.0861227
$$829$$ −20606.0 −0.863299 −0.431649 0.902041i $$-0.642068\pi$$
−0.431649 + 0.902041i $$0.642068\pi$$
$$830$$ −19355.8 −0.809458
$$831$$ 1748.32 0.0729826
$$832$$ −3010.31 −0.125437
$$833$$ 8731.69 0.363187
$$834$$ −1304.11 −0.0541459
$$835$$ 7426.73 0.307799
$$836$$ −970.672 −0.0401572
$$837$$ −3615.82 −0.149320
$$838$$ 4994.31 0.205878
$$839$$ −45717.9 −1.88124 −0.940618 0.339468i $$-0.889753\pi$$
−0.940618 + 0.339468i $$0.889753\pi$$
$$840$$ 122.681 0.00503917
$$841$$ −16069.4 −0.658878
$$842$$ 13165.0 0.538833
$$843$$ 256.910 0.0104964
$$844$$ 4041.75 0.164837
$$845$$ 127.963 0.00520955
$$846$$ −5864.54 −0.238330
$$847$$ −9445.79 −0.383189
$$848$$ −10914.0 −0.441967
$$849$$ −342.597 −0.0138491
$$850$$ 3513.26 0.141769
$$851$$ −4104.47 −0.165334
$$852$$ −360.741 −0.0145056
$$853$$ 17230.4 0.691626 0.345813 0.938303i $$-0.387603\pi$$
0.345813 + 0.938303i $$0.387603\pi$$
$$854$$ −5138.99 −0.205917
$$855$$ −4257.90 −0.170312
$$856$$ 7263.97 0.290044
$$857$$ 44484.4 1.77311 0.886557 0.462619i $$-0.153090\pi$$
0.886557 + 0.462619i $$0.153090\pi$$
$$858$$ 273.937 0.0108998
$$859$$ −23213.4 −0.922039 −0.461019 0.887390i $$-0.652516\pi$$
−0.461019 + 0.887390i $$0.652516\pi$$
$$860$$ 10265.1 0.407022
$$861$$ −124.879 −0.00494294
$$862$$ 17751.4 0.701411
$$863$$ −9640.68 −0.380270 −0.190135 0.981758i $$-0.560893\pi$$
−0.190135 + 0.981758i $$0.560893\pi$$
$$864$$ −393.601 −0.0154984
$$865$$ −20395.5 −0.801697
$$866$$ −7273.79 −0.285420
$$867$$ −894.555 −0.0350412
$$868$$ 9510.46 0.371897
$$869$$ −11785.7 −0.460072
$$870$$ 345.882 0.0134787
$$871$$ −44234.8 −1.72082
$$872$$ 6896.61 0.267831
$$873$$ −5692.93 −0.220706
$$874$$ −723.369 −0.0279958
$$875$$ −12163.6 −0.469947
$$876$$ 889.237 0.0342974
$$877$$ −9499.62 −0.365769 −0.182885 0.983134i $$-0.558543\pi$$
−0.182885 + 0.983134i $$0.558543\pi$$
$$878$$ −21958.7 −0.844044
$$879$$ −103.129 −0.00395729
$$880$$ −1699.39 −0.0650983
$$881$$ −8252.54 −0.315590 −0.157795 0.987472i $$-0.550439\pi$$
−0.157795 + 0.987472i $$0.550439\pi$$
$$882$$ 14960.7 0.571148
$$883$$ −34768.9 −1.32510 −0.662552 0.749016i $$-0.730526\pi$$
−0.662552 + 0.749016i $$0.730526\pi$$
$$884$$ 5918.26 0.225173
$$885$$ −474.577 −0.0180257
$$886$$ 2600.33 0.0986000
$$887$$ −3288.58 −0.124487 −0.0622433 0.998061i $$-0.519825\pi$$
−0.0622433 + 0.998061i $$0.519825\pi$$
$$888$$ 393.280 0.0148622
$$889$$ 3148.50 0.118782
$$890$$ 11395.2 0.429177
$$891$$ −9257.05 −0.348061
$$892$$ 13594.8 0.510301
$$893$$ 2067.43 0.0774736
$$894$$ 142.185 0.00531921
$$895$$ −17171.0 −0.641298
$$896$$ 1035.26 0.0386002
$$897$$ 204.145 0.00759888
$$898$$ −31750.4 −1.17987
$$899$$ 26813.4 0.994747
$$900$$ 6019.55 0.222946
$$901$$ 21456.9 0.793377
$$902$$ 1729.84 0.0638552
$$903$$ 569.067 0.0209716
$$904$$ 12021.8 0.442299
$$905$$ −20944.2 −0.769292
$$906$$ −666.813 −0.0244518
$$907$$ 37686.1 1.37966 0.689828 0.723973i $$-0.257686\pi$$
0.689828 + 0.723973i $$0.257686\pi$$
$$908$$ 23043.2 0.842197
$$909$$ 45907.4 1.67508
$$910$$ −6327.28 −0.230491
$$911$$ 15090.9 0.548828 0.274414 0.961612i $$-0.411516\pi$$
0.274414 + 0.961612i $$0.411516\pi$$
$$912$$ 69.3114 0.00251659
$$913$$ 14863.7 0.538790
$$914$$ 6231.32 0.225508
$$915$$ −602.355 −0.0217631
$$916$$ −8715.98 −0.314393
$$917$$ −2170.08 −0.0781485
$$918$$ 773.820 0.0278212
$$919$$ 43617.4 1.56562 0.782810 0.622261i $$-0.213786\pi$$
0.782810 + 0.622261i $$0.213786\pi$$
$$920$$ −1266.43 −0.0453836
$$921$$ −532.024 −0.0190345
$$922$$ 26959.4 0.962972
$$923$$ 18605.2 0.663486
$$924$$ −94.2089 −0.00335416
$$925$$ −12040.9 −0.428002
$$926$$ 15892.4 0.563991
$$927$$ 37552.6 1.33052
$$928$$ 2918.79 0.103248
$$929$$ 32446.2 1.14588 0.572942 0.819596i $$-0.305802\pi$$
0.572942 + 0.819596i $$0.305802\pi$$
$$930$$ 1114.75 0.0393054
$$931$$ −5274.10 −0.185662
$$932$$ −11234.0 −0.394830
$$933$$ 2388.53 0.0838123
$$934$$ −18296.7 −0.640993
$$935$$ 3341.01 0.116858
$$936$$ 10140.2 0.354106
$$937$$ −28355.4 −0.988614 −0.494307 0.869287i $$-0.664578\pi$$
−0.494307 + 0.869287i $$0.664578\pi$$
$$938$$ 15212.6 0.529542
$$939$$ 948.548 0.0329656
$$940$$ 3619.53 0.125592
$$941$$ −48970.8 −1.69650 −0.848248 0.529599i $$-0.822342\pi$$
−0.848248 + 0.529599i $$0.822342\pi$$
$$942$$ 1.99906 6.91433e−5 0
$$943$$ 1289.12 0.0445170
$$944$$ −4004.80 −0.138078
$$945$$ −827.300 −0.0284784
$$946$$ −7882.78 −0.270921
$$947$$ −9198.84 −0.315652 −0.157826 0.987467i $$-0.550448\pi$$
−0.157826 + 0.987467i $$0.550448\pi$$
$$948$$ 841.565 0.0288320
$$949$$ −45862.4 −1.56876
$$950$$ −2122.07 −0.0724728
$$951$$ −1711.94 −0.0583737
$$952$$ −2035.33 −0.0692914
$$953$$ 28428.9 0.966321 0.483160 0.875532i $$-0.339489\pi$$
0.483160 + 0.875532i $$0.339489\pi$$
$$954$$ 36763.8 1.24766
$$955$$ −34980.3 −1.18527
$$956$$ 25142.7 0.850599
$$957$$ −265.609 −0.00897170
$$958$$ 15329.3 0.516980
$$959$$ 21492.5 0.723700
$$960$$ 121.346 0.00407961
$$961$$ 56626.2 1.90078
$$962$$ −20283.4 −0.679797
$$963$$ −24468.7 −0.818788
$$964$$ 4516.90 0.150912
$$965$$ 26992.2 0.900426
$$966$$ −70.2068 −0.00233837
$$967$$ −22315.5 −0.742109 −0.371054 0.928611i $$-0.621004\pi$$
−0.371054 + 0.928611i $$0.621004\pi$$
$$968$$ −9343.01 −0.310223
$$969$$ −136.266 −0.00451755
$$970$$ 3513.61 0.116304
$$971$$ 208.410 0.00688795 0.00344398 0.999994i $$-0.498904\pi$$
0.00344398 + 0.999994i $$0.498904\pi$$
$$972$$ 1989.41 0.0656485
$$973$$ −23131.0 −0.762124
$$974$$ −10694.4 −0.351819
$$975$$ 598.879 0.0196712
$$976$$ −5083.08 −0.166706
$$977$$ 35744.3 1.17048 0.585241 0.810860i $$-0.301000\pi$$
0.585241 + 0.810860i $$0.301000\pi$$
$$978$$ −810.710 −0.0265068
$$979$$ −8750.56 −0.285668
$$980$$ −9233.56 −0.300975
$$981$$ −23231.3 −0.756082
$$982$$ 27294.3 0.886963
$$983$$ −36175.5 −1.17377 −0.586887 0.809669i $$-0.699647\pi$$
−0.586887 + 0.809669i $$0.699647\pi$$
$$984$$ −123.520 −0.00400171
$$985$$ −14425.9 −0.466647
$$986$$ −5738.33 −0.185340
$$987$$ 200.655 0.00647104
$$988$$ −3574.74 −0.115109
$$989$$ −5874.44 −0.188874
$$990$$ 5724.41 0.183771
$$991$$ 34654.3 1.11083 0.555413 0.831575i $$-0.312560\pi$$
0.555413 + 0.831575i $$0.312560\pi$$
$$992$$ 9406.98 0.301081
$$993$$ 2368.17 0.0756812
$$994$$ −6398.46 −0.204172
$$995$$ 3166.67 0.100895
$$996$$ −1061.35 −0.0337652
$$997$$ 4756.72 0.151100 0.0755501 0.997142i $$-0.475929\pi$$
0.0755501 + 0.997142i $$0.475929\pi$$
$$998$$ −38703.3 −1.22759
$$999$$ −2652.09 −0.0839923
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 38.4.a.c.1.1 2
3.2 odd 2 342.4.a.h.1.1 2
4.3 odd 2 304.4.a.c.1.2 2
5.2 odd 4 950.4.b.i.799.4 4
5.3 odd 4 950.4.b.i.799.1 4
5.4 even 2 950.4.a.e.1.2 2
7.6 odd 2 1862.4.a.e.1.2 2
8.3 odd 2 1216.4.a.p.1.1 2
8.5 even 2 1216.4.a.g.1.2 2
19.18 odd 2 722.4.a.f.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
38.4.a.c.1.1 2 1.1 even 1 trivial
304.4.a.c.1.2 2 4.3 odd 2
342.4.a.h.1.1 2 3.2 odd 2
722.4.a.f.1.2 2 19.18 odd 2
950.4.a.e.1.2 2 5.4 even 2
950.4.b.i.799.1 4 5.3 odd 4
950.4.b.i.799.4 4 5.2 odd 4
1216.4.a.g.1.2 2 8.5 even 2
1216.4.a.p.1.1 2 8.3 odd 2
1862.4.a.e.1.2 2 7.6 odd 2