LMFDB - Embedded newform 38.4.a.c.1.1

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

Display 100 coefficients 10 coefficients

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0.227998	0.0438783	0.0219391	−	0.999759i	$-0.493016\pi$
$3$	0.227998	0.0438783	0.0219391	+	0.999759i	$0.493016\pi$
$4$	4.00000	0.500000
$5$	8.31601	0.743806	0.371903	−	0.928272i	$-0.378705\pi$
$5$	8.31601	0.743806	0.371903	+	0.928272i	$0.378705\pi$
$6$	0.455996	0.0310266
$7$	8.08801	0.436711	0.218356	−	0.975869i	$-0.429931\pi$
$7$	8.08801	0.436711	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	$-0.556006\pi$
$11$	−12.7720	−0.350082	−0.175041	+	0.984561i	$0.556006\pi$
$12$	0.911993	0.0219391
$13$	−47.0360	−1.00350	−0.501748	−	0.865014i	$-0.667309\pi$
$13$	−47.0360	−1.00350	−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	$-0.572038\pi$
$17$	−31.4560	−0.448776	−0.224388	+	0.974500i	$0.572038\pi$
$18$	−53.8960	−0.705745
$19$	19.0000	0.229416
$19$	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	$-0.527501\pi$
$23$	−19.0360	−0.172578	−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	$-0.405670\pi$
$29$	91.2120	0.584057	0.292028	+	0.956410i	$0.405670\pi$
$30$	3.79207	0.0230778
$31$	293.968	1.70317	0.851584	−	0.524218i	$-0.175642\pi$
$31$	293.968	1.70317	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	$-0.340994\pi$
$37$	215.616	0.958029	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	$-0.541169\pi$
$41$	−67.7200	−0.257953	−0.128977	+	0.991648i	$0.541169\pi$
$42$	3.68810	0.0135497
$43$	308.596	1.09443	0.547214	−	0.836992i	$-0.315688\pi$
$43$	308.596	1.09443	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	$-0.445995\pi$
$47$	108.812	0.337700	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	$-0.845114\pi$
$53$	−682.124	−1.76787	−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	$-0.589060\pi$
$59$	−250.300	−0.552310	−0.276155	+	0.961113i	$0.589060\pi$
$60$	7.58413	0.0163185
$61$	−317.692	−0.666825	−0.333412	−	0.942781i	$-0.608200\pi$
$61$	−317.692	−0.666825	−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	$-0.172069\pi$
$67$	940.444	1.71483	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	$-0.607247\pi$
$71$	−395.552	−0.661175	−0.330587	+	0.943775i	$0.607247\pi$
$72$	−215.584	−0.352873
$73$	975.048	1.56330	0.781649	−	0.623718i	$-0.214379\pi$
$73$	975.048	1.56330	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	$-0.271787\pi$
$79$	922.776	1.31418	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	$-0.779504\pi$
$83$	−1163.77	−1.53904	−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	$-0.366226\pi$
$89$	685.136	0.816003	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	$-0.464734\pi$
$97$	211.256	0.221132	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	$-0.816951\pi$
$101$	−1703.55	−1.67831	−0.839157	+	0.543889i	$0.816951\pi$
$102$	−14.3438	−0.0139240
$103$	−1393.52	−1.33308	−0.666542	−	0.745468i	$-0.732226\pi$
$103$	−1393.52	−1.33308	−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	$-0.365465\pi$
$107$	907.996	0.820367	0.410184	+	0.912003i	$0.365465\pi$
$108$	−49.2002	−0.0438360
$109$	862.077	0.757541	0.378770	−	0.925491i	$-0.376347\pi$
$109$	862.077	0.757541	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	$-0.284893\pi$
$113$	1502.72	1.25101	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	$-0.456577\pi$
$127$	389.280	0.271992	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	$-0.528519\pi$
$131$	−268.308	−0.178948	−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	$-0.189149\pi$
$137$	2657.33	1.65716	0.828580	+	0.559871i	$0.189149\pi$
$138$	−8.68036	−0.00535450
$139$	−2859.92	−1.74514	−0.872572	−	0.488486i	$-0.837549\pi$
$139$	−2859.92	−1.74514	−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	$-0.472681\pi$
$149$	311.812	0.171440	0.0857202	+	0.996319i	$0.472681\pi$
$150$	−25.4647	−0.0138612
$151$	−1462.32	−0.788093	−0.394046	−	0.919091i	$-0.628925\pi$
$151$	−1462.32	−0.788093	−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	$-0.499645\pi$
$157$	4.38395	0.00222852	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	$-0.640487\pi$
$163$	−1777.89	−0.854325	−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	$-0.433660\pi$
$167$	893.064	0.413817	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	$-0.681166\pi$
$173$	−2452.56	−1.07783	−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	$-0.641872\pi$
$179$	−2064.81	−0.862185	−0.431092	+	0.902308i	$0.641872\pi$
$180$	−896.399	−0.371187
$181$	−2518.54	−1.03426	−0.517132	−	0.855906i	$-0.673000\pi$
$181$	−2518.54	−1.03426	−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	$-0.793455\pi$
$191$	−4206.38	−1.59352	−0.796761	+	0.604294i	$0.793455\pi$
$192$	14.5919	0.00548478
$193$	3245.82	1.21056	0.605282	−	0.796011i	$-0.293060\pi$
$193$	3245.82	1.21056	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	$-0.601565\pi$
$197$	−1734.71	−0.627377	−0.313688	+	0.949526i	$0.601565\pi$
$198$	688.360	0.247069
$199$	380.792	0.135646	0.0678232	−	0.997697i	$-0.478395\pi$
$199$	380.792	0.135646	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	$-0.447290\pi$
$211$	1010.44	0.329675	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	$-0.329534\pi$
$223$	3398.70	1.02060	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	$-0.181262\pi$
$227$	5760.80	1.68439	0.842197	+	0.539169i	$0.181262\pi$
$228$	17.3279	0.00503318
$229$	−2179.00	−0.628786	−0.314393	−	0.949293i	$-0.601801\pi$
$229$	−2179.00	−0.628786	−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	$-0.629196\pi$
$233$	−2808.49	−0.789659	−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	$-0.176239\pi$
$239$	6285.67	1.70120	0.850599	+	0.525815i	$0.176239\pi$
$240$	30.3365	0.00815923
$241$	1129.22	0.301825	0.150912	−	0.988547i	$-0.451779\pi$
$241$	1129.22	0.301825	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	$-0.617677\pi$
$251$	−2873.73	−0.722661	−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	$-0.648756\pi$
$257$	−3712.18	−0.901008	−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	$-0.452696\pi$
$263$	1263.04	0.296130	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	$-0.713493\pi$
$269$	−5484.39	−1.24308	−0.621541	+	0.783381i	$0.713493\pi$
$270$	−204.575	−0.0461111
$271$	−3217.66	−0.721251	−0.360625	−	0.932711i	$-0.617437\pi$
$271$	−3217.66	−0.721251	−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	$-0.187397\pi$
$277$	7668.13	1.66330	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	$-0.461836\pi$
$281$	1126.81	0.239216	0.119608	+	0.992821i	$0.461836\pi$
$282$	49.6179	0.0104777
$283$	−1502.63	−0.315625	−0.157813	−	0.987469i	$-0.550444\pi$
$283$	−1502.63	−0.315625	−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	$-0.514359\pi$
$293$	−452.324	−0.0901878	−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	$-0.569595\pi$
$307$	−2333.46	−0.433803	−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	$-0.0957959\pi$
$311$	10476.1	1.91011	0.955055	+	0.296429i	$0.0957959\pi$
$312$	−85.7930	−0.0155675
$313$	4160.33	0.751297	0.375648	−	0.926762i	$-0.377420\pi$
$313$	4160.33	0.751297	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	$-0.731644\pi$
$317$	−7508.56	−1.33036	−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	$-0.168960\pi$
$331$	10386.8	1.72480	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	$-0.349969\pi$
$337$	5618.29	0.908153	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	$-0.455180\pi$
$347$	1814.32	0.280686	0.140343	+	0.990103i	$0.455180\pi$
$348$	83.1847	0.0128137
$349$	−816.757	−0.125272	−0.0626361	−	0.998036i	$-0.519951\pi$
$349$	−816.757	−0.125272	−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	$-0.184833\pi$
$353$	11090.4	1.67219	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	$-0.575863\pi$
$359$	−3211.68	−0.472161	−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	$-0.305210\pi$
$367$	8077.81	1.14893	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	$-0.614892\pi$
$373$	−5088.15	−0.706312	−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	$-0.444783\pi$
$379$	2547.00	0.345199	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	$-0.655996\pi$
$383$	−7056.11	−0.941384	−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	$-0.400294\pi$
$389$	4728.25	0.616277	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	$-0.485089\pi$
$397$	740.837	0.0936563	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	$-0.462661\pi$
$401$	1879.58	0.234070	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	$-0.533067\pi$
$409$	−1715.45	−0.207393	−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	$-0.453496\pi$
$419$	2497.15	0.291155	0.145578	+	0.989347i	$0.453496\pi$
$420$	61.3405	0.00712646
$421$	6582.52	0.762024	0.381012	−	0.924570i	$-0.375576\pi$
$421$	6582.52	0.762024	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	$-0.334812\pi$
$431$	8875.72	0.991946	0.495973	+	0.868338i	$0.334812\pi$
$432$	−196.801	−0.0219180
$433$	−3636.90	−0.403645	−0.201822	−	0.979422i	$-0.564686\pi$
$433$	−3636.90	−0.403645	−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	$-0.703573\pi$
$439$	−10979.4	−1.19366	−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	$-0.477789\pi$
$443$	1300.16	0.139442	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	$-0.814125\pi$
$449$	−15875.2	−1.66859	−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	$-0.449025\pi$
$457$	3115.66	0.318916	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	$-0.261578\pi$
$461$	13479.7	1.36185	0.680924	+	0.732354i	$0.261578\pi$
$462$	−47.1044	−0.00474350
$463$	7946.19	0.797604	0.398802	−	0.917037i	$-0.369426\pi$
$463$	7946.19	0.797604	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	$-0.649736\pi$
$467$	−9148.37	−0.906501	−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	$-0.380878\pi$
$479$	7664.64	0.731120	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	$-0.580027\pi$
$487$	−5347.21	−0.497547	−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	$-0.284210\pi$
$491$	13647.2	1.25436	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	$-0.834617\pi$
$499$	−19351.6	−1.73607	−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	$-0.825586\pi$
$503$	−19259.1	−1.70720	−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	$-0.449959\pi$
$509$	3595.77	0.313123	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	$-0.720877\pi$
$521$	−15211.0	−1.27909	−0.639544	+	0.768754i	$0.720877\pi$
$522$	−4915.97	−0.412195
$523$	18307.1	1.53062	0.765310	−	0.643662i	$-0.222586\pi$
$523$	18307.1	1.53062	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	$-0.382205\pi$
$541$	9102.17	0.723351	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	$-0.617325\pi$
$547$	−9218.75	−0.720595	−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	$-0.670728\pi$
$557$	−13435.1	−1.02202	−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	$-0.352507\pi$
$563$	11941.5	0.893916	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	$-0.575506\pi$
$569$	−6378.91	−0.469979	−0.234989	+	0.971998i	$0.575506\pi$
$570$	72.0493	0.00529441
$571$	24903.9	1.82521	0.912605	−	0.408843i	$-0.134068\pi$
$571$	24903.9	1.82521	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	$-0.635094\pi$
$577$	−11414.7	−0.823568	−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	$-0.240041\pi$
$587$	20732.1	1.45776	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	$-0.285665\pi$
$593$	18010.5	1.24722	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	$-0.0978942\pi$
$599$	27944.7	1.90616	0.953080	+	0.302719i	$0.0978942\pi$
$600$	−101.859	−0.00693061
$601$	−11598.1	−0.787179	−0.393590	−	0.919286i	$-0.628767\pi$
$601$	−11598.1	−0.787179	−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	$-0.264411\pi$
$607$	20170.5	1.34876	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	$-0.340061\pi$
$613$	14618.3	0.963174	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	$-0.693899\pi$
$617$	−17538.1	−1.14434	−0.572171	+	0.820134i	$0.693899\pi$
$618$	−635.440	−0.0413611
$619$	−8815.75	−0.572431	−0.286216	−	0.958165i	$-0.592397\pi$
$619$	−8815.75	−0.572431	−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	$-0.746536\pi$
$631$	−22170.8	−1.39874	−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	$-0.737973\pi$
$641$	−22067.7	−1.35978	−0.679891	+	0.733313i	$0.737973\pi$
$642$	414.043	0.0254532
$643$	−11795.4	−0.723428	−0.361714	−	0.932289i	$-0.617808\pi$
$643$	−11795.4	−0.723428	−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	$-0.595383\pi$
$647$	−9716.04	−0.590382	−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	$-0.599989\pi$
$653$	−10311.9	−0.617969	−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	$-0.462093\pi$
$659$	4019.80	0.237616	0.118808	+	0.992917i	$0.462093\pi$
$660$	−96.8646	−0.00571280
$661$	−22702.6	−1.33590	−0.667951	−	0.744206i	$-0.732828\pi$
$661$	−22702.6	−1.33590	−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	$-0.396711\pi$
$673$	11132.8	0.637652	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	$-0.629759\pi$
$677$	−13967.0	−0.792903	−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	$-0.489531\pi$
$683$	1173.88	0.0657648	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	$-0.422902\pi$
$691$	8713.33	0.479697	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	$-0.814661\pi$
$701$	−31003.4	−1.67045	−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	$-0.604242\pi$
$709$	−12145.1	−0.643328	−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	$-0.277748\pi$
$719$	24787.8	1.28572	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	$-0.335826\pi$
$727$	19335.6	0.986409	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	$-0.329996\pi$
$733$	20204.5	1.01810	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	$-0.372699\pi$
$739$	15643.7	0.778706	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	$-0.535438\pi$
$743$	−4500.20	−0.222202	−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	$-0.175231\pi$
$751$	35080.2	1.70452	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	$-0.580256\pi$
$757$	−10391.8	−0.498938	−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	$-0.409236\pi$
$761$	11810.5	0.562590	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	$-0.191977\pi$
$769$	35125.5	1.64715	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	$-0.345935\pi$
$773$	20001.5	0.930665	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	$-0.400392\pi$
$787$	13593.3	0.615690	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	$-0.549335\pi$
$797$	−6946.75	−0.308741	−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	$-0.317305\pi$
$809$	24987.2	1.08591	0.542955	+	0.839762i	$0.317305\pi$
$810$	12054.7	0.522914
$811$	−23172.5	−1.00332	−0.501662	−	0.865064i	$-0.667278\pi$
$811$	−23172.5	−1.00332	−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	$-0.273678\pi$
$821$	30703.8	1.30520	0.652600	+	0.757703i	$0.273678\pi$
$822$	1211.73	0.0514161
$823$	−15940.1	−0.675135	−0.337568	−	0.941301i	$-0.609604\pi$
$823$	−15940.1	−0.675135	−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	$-0.544731\pi$
$827$	−6662.20	−0.280130	−0.140065	+	0.990142i	$0.544731\pi$
$828$	2051.93	0.0861227
$829$	−20606.0	−0.863299	−0.431649	−	0.902041i	$-0.642068\pi$
$829$	−20606.0	−0.863299	−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	$-0.889753\pi$
$839$	−45717.9	−1.88124	−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	$-0.387603\pi$
$853$	17230.4	0.691626	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	$-0.153090\pi$
$857$	44484.4	1.77311	0.886557	+	0.462619i	$0.153090\pi$
$858$	273.937	0.0108998
$859$	−23213.4	−0.922039	−0.461019	−	0.887390i	$-0.652516\pi$
$859$	−23213.4	−0.922039	−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	$-0.560893\pi$
$863$	−9640.68	−0.380270	−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	$-0.558543\pi$
$877$	−9499.62	−0.365769	−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	$-0.550439\pi$
$881$	−8252.54	−0.315590	−0.157795	+	0.987472i	$0.550439\pi$
$882$	14960.7	0.571148
$883$	−34768.9	−1.32510	−0.662552	−	0.749016i	$-0.730526\pi$
$883$	−34768.9	−1.32510	−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	$-0.519825\pi$
$887$	−3288.58	−0.124487	−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	$-0.257686\pi$
$907$	37686.1	1.37966	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	$-0.411516\pi$
$911$	15090.9	0.548828	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	$-0.213786\pi$
$919$	43617.4	1.56562	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	$-0.305802\pi$
$929$	32446.2	1.14588	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	$-0.664578\pi$
$937$	−28355.4	−0.988614	−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	$-0.822342\pi$
$941$	−48970.8	−1.69650	−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	$-0.550448\pi$
$947$	−9198.84	−0.315652	−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	$-0.339489\pi$
$953$	28428.9	0.966321	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	$-0.621004\pi$
$967$	−22315.5	−0.742109	−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	$-0.498904\pi$
$971$	208.410	0.00688795	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	$-0.301000\pi$
$977$	35744.3	1.17048	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	$-0.699647\pi$
$983$	−36175.5	−1.17377	−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	$-0.312560\pi$
$991$	34654.3	1.11083	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	$-0.475929\pi$
$997$	4756.72	0.151100	0.0755501	+	0.997142i	$0.475929\pi$
$998$	−38703.3	−1.22759
$999$	−2652.09	−0.0839923

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +0.227998 q^{3} +4.00000 q^{4} +8.31601 q^{5} +0.455996 q^{6} +8.08801 q^{7} +8.00000 q^{8} -26.9480 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +0.227998 q^{3} +4.00000 q^{4} +8.31601 q^{5} +0.455996 q^{6} +8.08801 q^{7} +8.00000 q^{8} -26.9480 q^{9} +16.6320 q^{10} -12.7720 q^{11} +0.911993 q^{12} -47.0360 q^{13} +16.1760 q^{14} +1.89603 q^{15} +16.0000 q^{16} -31.4560 q^{17} -53.8960 q^{18} +19.0000 q^{19} +33.2640 q^{20} +1.84405 q^{21} -25.5440 q^{22} -19.0360 q^{23} +1.82399 q^{24} -55.8441 q^{25} -94.0720 q^{26} -12.3000 q^{27} +32.3520 q^{28} +91.2120 q^{29} +3.79207 q^{30} +293.968 q^{31} +32.0000 q^{32} -2.91199 q^{33} -62.9120 q^{34} +67.2599 q^{35} -107.792 q^{36} +215.616 q^{37} +38.0000 q^{38} -10.7241 q^{39} +66.5280 q^{40} -67.7200 q^{41} +3.68810 q^{42} +308.596 q^{43} -51.0880 q^{44} -224.100 q^{45} -38.0720 q^{46} +108.812 q^{47} +3.64797 q^{48} -277.584 q^{49} -111.688 q^{50} -7.17191 q^{51} -188.144 q^{52} -682.124 q^{53} -24.6001 q^{54} -106.212 q^{55} +64.7041 q^{56} +4.33196 q^{57} +182.424 q^{58} -250.300 q^{59} +7.58413 q^{60} -317.692 q^{61} +587.936 q^{62} -217.956 q^{63} +64.0000 q^{64} -391.152 q^{65} -5.82399 q^{66} +940.444 q^{67} -125.824 q^{68} -4.34018 q^{69} +134.520 q^{70} -395.552 q^{71} -215.584 q^{72} +975.048 q^{73} +431.232 q^{74} -12.7323 q^{75} +76.0000 q^{76} -103.300 q^{77} -21.4483 q^{78} +922.776 q^{79} +133.056 q^{80} +724.792 q^{81} -135.440 q^{82} -1163.77 q^{83} +7.37620 q^{84} -261.588 q^{85} +617.192 q^{86} +20.7962 q^{87} -102.176 q^{88} +685.136 q^{89} -448.200 q^{90} -380.428 q^{91} -76.1441 q^{92} +67.0242 q^{93} +217.624 q^{94} +158.004 q^{95} +7.29594 q^{96} +211.256 q^{97} -555.168 q^{98} +344.180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9}+O(q^{10}) \) 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

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