LMFDB - Embedded newform 338.4.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, 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("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	338.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:	$19.9426455819$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6681389953.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 107x^{4} + 85x^{3} + 3703x^{2} - 1659x - 41951 $ x^6 - x^5 - 107x^4 + 85x^3 + 3703x^2 - 1659x - 41951
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.81752$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +8.95458 q^{3} +4.00000 q^{4} -17.3267 q^{5} -17.9092 q^{6} +16.7510 q^{7} -8.00000 q^{8} +53.1845 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +8.95458 q^{3} +4.00000 q^{4} -17.3267 q^{5} -17.9092 q^{6} +16.7510 q^{7} -8.00000 q^{8} +53.1845 q^{9} +34.6535 q^{10} +12.3000 q^{11} +35.8183 q^{12} -33.5020 q^{14} -155.154 q^{15} +16.0000 q^{16} -67.2070 q^{17} -106.369 q^{18} +112.331 q^{19} -69.3069 q^{20} +149.998 q^{21} -24.5999 q^{22} +173.127 q^{23} -71.6366 q^{24} +175.215 q^{25} +234.471 q^{27} +67.0040 q^{28} -30.9749 q^{29} +310.307 q^{30} +184.682 q^{31} -32.0000 q^{32} +110.141 q^{33} +134.414 q^{34} -290.240 q^{35} +212.738 q^{36} +154.964 q^{37} -224.662 q^{38} +138.614 q^{40} -123.827 q^{41} -299.996 q^{42} -201.562 q^{43} +49.1998 q^{44} -921.514 q^{45} -346.254 q^{46} +382.133 q^{47} +143.273 q^{48} -62.4041 q^{49} -350.431 q^{50} -601.810 q^{51} +394.866 q^{53} -468.943 q^{54} -213.118 q^{55} -134.008 q^{56} +1005.88 q^{57} +61.9498 q^{58} -110.728 q^{59} -620.614 q^{60} +747.271 q^{61} -369.364 q^{62} +890.894 q^{63} +64.0000 q^{64} -220.282 q^{66} +51.7433 q^{67} -268.828 q^{68} +1550.28 q^{69} +580.480 q^{70} -167.600 q^{71} -425.476 q^{72} -15.5647 q^{73} -309.927 q^{74} +1568.98 q^{75} +449.325 q^{76} +206.036 q^{77} -325.323 q^{79} -277.228 q^{80} +663.611 q^{81} +247.653 q^{82} -1158.87 q^{83} +599.993 q^{84} +1164.48 q^{85} +403.124 q^{86} -277.367 q^{87} -98.3996 q^{88} -1338.22 q^{89} +1843.03 q^{90} +692.509 q^{92} +1653.75 q^{93} -764.266 q^{94} -1946.33 q^{95} -286.547 q^{96} +1259.02 q^{97} +124.808 q^{98} +654.167 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10}) $ 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * 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
$3$	8.95458	1.72331	0.861655	−	0.507495i	$-0.169428\pi$
$3$	8.95458	1.72331	0.861655	+	0.507495i	$0.169428\pi$
$4$	4.00000	0.500000
$5$	−17.3267	−1.54975	−0.774875	−	0.632115i	$-0.782187\pi$
$5$	−17.3267	−1.54975	−0.774875	+	0.632115i	$0.782187\pi$
$6$	−17.9092	−1.21856
$7$	16.7510	0.904469	0.452234	−	0.891899i	$-0.350627\pi$
$7$	16.7510	0.904469	0.452234	+	0.891899i	$0.350627\pi$
$8$	−8.00000	−0.353553
$9$	53.1845	1.96980
$10$	34.6535	1.09584
$11$	12.3000	0.337143	0.168572	−	0.985689i	$-0.446085\pi$
$11$	12.3000	0.337143	0.168572	+	0.985689i	$0.446085\pi$
$12$	35.8183	0.861655
$13$	0	0
$13$	0	0
$14$	−33.5020	−0.639556
$15$	−155.154	−2.67070
$16$	16.0000	0.250000
$17$	−67.2070	−0.958829	−0.479414	−	0.877589i	$-0.659151\pi$
$17$	−67.2070	−0.958829	−0.479414	+	0.877589i	$0.659151\pi$
$18$	−106.369	−1.39286
$19$	112.331	1.35634	0.678172	−	0.734903i	$-0.262772\pi$
$19$	112.331	1.35634	0.678172	+	0.734903i	$0.262772\pi$
$20$	−69.3069	−0.774875
$21$	149.998	1.55868
$22$	−24.5999	−0.238396
$23$	173.127	1.56954	0.784772	−	0.619784i	$-0.212780\pi$
$23$	173.127	1.56954	0.784772	+	0.619784i	$0.212780\pi$
$24$	−71.6366	−0.609282
$25$	175.215	1.40172
$26$	0	0
$27$	234.471	1.67126
$28$	67.0040	0.452234
$29$	−30.9749	−0.198341	−0.0991705	−	0.995070i	$-0.531619\pi$
$29$	−30.9749	−0.198341	−0.0991705	+	0.995070i	$0.531619\pi$
$30$	310.307	1.88847
$31$	184.682	1.07000	0.534999	−	0.844853i	$-0.320312\pi$
$31$	184.682	1.07000	0.534999	+	0.844853i	$0.320312\pi$
$32$	−32.0000	−0.176777
$33$	110.141	0.581002
$34$	134.414	0.677994
$35$	−290.240	−1.40170
$36$	212.738	0.984898
$37$	154.964	0.688537	0.344269	−	0.938871i	$-0.388127\pi$
$37$	154.964	0.688537	0.344269	+	0.938871i	$0.388127\pi$
$38$	−224.662	−0.959080
$39$	0	0
$40$	138.614	0.547919
$41$	−123.827	−0.471670	−0.235835	−	0.971793i	$-0.575782\pi$
$41$	−123.827	−0.471670	−0.235835	+	0.971793i	$0.575782\pi$
$42$	−299.996	−1.10215
$43$	−201.562	−0.714836	−0.357418	−	0.933945i	$-0.616343\pi$
$43$	−201.562	−0.714836	−0.357418	+	0.933945i	$0.616343\pi$
$44$	49.1998	0.168572
$45$	−921.514	−3.05269
$46$	−346.254	−1.10984
$47$	382.133	1.18595	0.592977	−	0.805219i	$-0.297953\pi$
$47$	382.133	1.18595	0.592977	+	0.805219i	$0.297953\pi$
$48$	143.273	0.430827
$49$	−62.4041	−0.181936
$50$	−350.431	−0.991168
$51$	−601.810	−1.65236
$52$	0	0
$53$	394.866	1.02338	0.511689	−	0.859171i	$-0.329020\pi$
$53$	394.866	1.02338	0.511689	+	0.859171i	$0.329020\pi$
$54$	−468.943	−1.18176
$55$	−213.118	−0.522487
$56$	−134.008	−0.319778
$57$	1005.88	2.33740
$58$	61.9498	0.140248
$59$	−110.728	−0.244332	−0.122166	−	0.992510i	$-0.538984\pi$
$59$	−110.728	−0.244332	−0.122166	+	0.992510i	$0.538984\pi$
$60$	−620.614	−1.33535
$61$	747.271	1.56850	0.784248	−	0.620448i	$-0.213049\pi$
$61$	747.271	1.56850	0.784248	+	0.620448i	$0.213049\pi$
$62$	−369.364	−0.756602
$63$	890.894	1.78162
$64$	64.0000	0.125000
$65$	0	0
$66$	−220.282	−0.410831
$67$	51.7433	0.0943501	0.0471750	−	0.998887i	$-0.484978\pi$
$67$	51.7433	0.0943501	0.0471750	+	0.998887i	$0.484978\pi$
$68$	−268.828	−0.479414
$69$	1550.28	2.70481
$70$	580.480	0.991152
$71$	−167.600	−0.280147	−0.140073	−	0.990141i	$-0.544734\pi$
$71$	−167.600	−0.280147	−0.140073	+	0.990141i	$0.544734\pi$
$72$	−425.476	−0.696428
$73$	−15.5647	−0.0249549	−0.0124775	−	0.999922i	$-0.503972\pi$
$73$	−15.5647	−0.0249549	−0.0124775	+	0.999922i	$0.503972\pi$
$74$	−309.927	−0.486869
$75$	1568.98	2.41560
$76$	449.325	0.678172
$77$	206.036	0.304935
$78$	0	0
$79$	−325.323	−0.463313	−0.231657	−	0.972798i	$-0.574415\pi$
$79$	−325.323	−0.463313	−0.231657	+	0.972798i	$0.574415\pi$
$80$	−277.228	−0.387437
$81$	663.611	0.910303
$82$	247.653	0.333521
$83$	−1158.87	−1.53257	−0.766283	−	0.642503i	$-0.777896\pi$
$83$	−1158.87	−1.53257	−0.766283	+	0.642503i	$0.777896\pi$
$84$	599.993	0.779340
$85$	1164.48	1.48594
$86$	403.124	0.505465
$87$	−277.367	−0.341803
$88$	−98.3996	−0.119198
$89$	−1338.22	−1.59383	−0.796915	−	0.604091i	$-0.793536\pi$
$89$	−1338.22	−1.59383	−0.796915	+	0.604091i	$0.793536\pi$
$90$	1843.03	2.15858
$91$	0	0
$92$	692.509	0.784772
$93$	1653.75	1.84394
$94$	−764.266	−0.838596
$95$	−1946.33	−2.10199
$96$	−286.547	−0.304641
$97$	1259.02	1.31788	0.658940	−	0.752195i	$-0.271005\pi$
$97$	1259.02	1.31788	0.658940	+	0.752195i	$0.271005\pi$
$98$	124.808	0.128648
$99$	654.167	0.664104
$100$	700.862	0.700862
$101$	883.670	0.870579	0.435289	−	0.900291i	$-0.356646\pi$
$101$	883.670	0.870579	0.435289	+	0.900291i	$0.356646\pi$
$102$	1203.62	1.16839
$103$	411.334	0.393495	0.196747	−	0.980454i	$-0.436962\pi$
$103$	411.334	0.393495	0.196747	+	0.980454i	$0.436962\pi$
$104$	0	0
$105$	−2598.98	−2.41556
$106$	−789.732	−0.723638
$107$	−684.140	−0.618115	−0.309058	−	0.951043i	$-0.600014\pi$
$107$	−684.140	−0.618115	−0.309058	+	0.951043i	$0.600014\pi$
$108$	937.885	0.835630
$109$	922.489	0.810628	0.405314	−	0.914178i	$-0.367162\pi$
$109$	922.489	0.810628	0.405314	+	0.914178i	$0.367162\pi$
$110$	426.236	0.369454
$111$	1387.64	1.18656
$112$	268.016	0.226117
$113$	1762.33	1.46713	0.733566	−	0.679618i	$-0.237854\pi$
$113$	1762.33	1.46713	0.733566	+	0.679618i	$0.237854\pi$
$114$	−2011.76	−1.65279
$115$	−2999.73	−2.43240
$116$	−123.900	−0.0991705
$117$	0	0
$118$	221.457	0.172769
$119$	−1125.78	−0.867231
$120$	1241.23	0.944235
$121$	−1179.71	−0.886334
$122$	−1494.54	−1.10909
$123$	−1108.82	−0.812834
$124$	738.729	0.534999
$125$	−870.069	−0.622571
$126$	−1781.79	−1.25980
$127$	−1127.58	−0.787850	−0.393925	−	0.919143i	$-0.628883\pi$
$127$	−1127.58	−0.787850	−0.393925	+	0.919143i	$0.628883\pi$
$128$	−128.000	−0.0883883
$129$	−1804.90	−1.23188
$130$	0	0
$131$	1988.93	1.32651	0.663257	−	0.748392i	$-0.269174\pi$
$131$	1988.93	1.32651	0.663257	+	0.748392i	$0.269174\pi$
$132$	440.564	0.290501
$133$	1881.66	1.22677
$134$	−103.487	−0.0667156
$135$	−4062.62	−2.59004
$136$	537.656	0.338997
$137$	−1643.36	−1.02483	−0.512416	−	0.858737i	$-0.671249\pi$
$137$	−1643.36	−1.02483	−0.512416	+	0.858737i	$0.671249\pi$
$138$	−3100.56	−1.91259
$139$	−2305.50	−1.40683	−0.703416	−	0.710778i	$-0.748343\pi$
$139$	−2305.50	−1.40683	−0.703416	+	0.710778i	$0.748343\pi$
$140$	−1160.96	−0.700850
$141$	3421.84	2.04377
$142$	335.199	0.198094
$143$	0	0
$144$	850.952	0.492449
$145$	536.693	0.307379
$146$	31.1294	0.0176458
$147$	−558.803	−0.313532
$148$	619.855	0.344269
$149$	−810.915	−0.445857	−0.222929	−	0.974835i	$-0.571562\pi$
$149$	−810.915	−0.445857	−0.222929	+	0.974835i	$0.571562\pi$
$150$	−3137.96	−1.70809
$151$	338.882	0.182634	0.0913172	−	0.995822i	$-0.470892\pi$
$151$	338.882	0.182634	0.0913172	+	0.995822i	$0.470892\pi$
$152$	−898.649	−0.479540
$153$	−3574.37	−1.88870
$154$	−412.073	−0.215622
$155$	−3199.94	−1.65823
$156$	0	0
$157$	−3838.59	−1.95129	−0.975647	−	0.219346i	$-0.929608\pi$
$157$	−3838.59	−1.95129	−0.975647	+	0.219346i	$0.929608\pi$
$158$	650.647	0.327612
$159$	3535.86	1.76360
$160$	554.455	0.273960
$161$	2900.05	1.41960
$162$	−1327.22	−0.643681
$163$	−2083.98	−1.00141	−0.500706	−	0.865617i	$-0.666926\pi$
$163$	−2083.98	−1.00141	−0.500706	+	0.865617i	$0.666926\pi$
$164$	−495.307	−0.235835
$165$	−1908.38	−0.900408
$166$	2317.75	1.08369
$167$	823.438	0.381554	0.190777	−	0.981633i	$-0.438899\pi$
$167$	823.438	0.381554	0.190777	+	0.981633i	$0.438899\pi$
$168$	−1199.99	−0.551077
$169$	0	0
$170$	−2328.95	−1.05072
$171$	5974.28	2.67172
$172$	−806.248	−0.357418
$173$	−3111.64	−1.36748	−0.683739	−	0.729727i	$-0.739647\pi$
$173$	−3111.64	−1.36748	−0.683739	+	0.729727i	$0.739647\pi$
$174$	554.734	0.241691
$175$	2935.03	1.26782
$176$	196.799	0.0842858
$177$	−991.527	−0.421061
$178$	2676.44	1.12701
$179$	−3113.71	−1.30017	−0.650083	−	0.759863i	$-0.725266\pi$
$179$	−3113.71	−1.30017	−0.650083	+	0.759863i	$0.725266\pi$
$180$	−3686.05	−1.52635
$181$	−307.468	−0.126265	−0.0631323	−	0.998005i	$-0.520109\pi$
$181$	−307.468	−0.126265	−0.0631323	+	0.998005i	$0.520109\pi$
$182$	0	0
$183$	6691.50	2.70300
$184$	−1385.02	−0.554918
$185$	−2685.01	−1.06706
$186$	−3307.50	−1.30386
$187$	−826.643	−0.323263
$188$	1528.53	0.592977
$189$	3927.63	1.51160
$190$	3892.66	1.48633
$191$	−1061.40	−0.402094	−0.201047	−	0.979582i	$-0.564434\pi$
$191$	−1061.40	−0.402094	−0.201047	+	0.979582i	$0.564434\pi$
$192$	573.093	0.215414
$193$	762.766	0.284482	0.142241	−	0.989832i	$-0.454569\pi$
$193$	762.766	0.284482	0.142241	+	0.989832i	$0.454569\pi$
$194$	−2518.05	−0.931882
$195$	0	0
$196$	−249.616	−0.0909681
$197$	−1145.77	−0.414378	−0.207189	−	0.978301i	$-0.566432\pi$
$197$	−1145.77	−0.414378	−0.207189	+	0.978301i	$0.566432\pi$
$198$	−1308.33	−0.469592
$199$	−1522.31	−0.542279	−0.271140	−	0.962540i	$-0.587401\pi$
$199$	−1522.31	−0.542279	−0.271140	+	0.962540i	$0.587401\pi$
$200$	−1401.72	−0.495584
$201$	463.340	0.162594
$202$	−1767.34	−0.615592
$203$	−518.860	−0.179393
$204$	−2407.24	−0.826179
$205$	2145.51	0.730970
$206$	−822.669	−0.278243
$207$	9207.69	3.09168
$208$	0	0
$209$	1381.67	0.457282
$210$	5197.95	1.70806
$211$	1775.03	0.579138	0.289569	−	0.957157i	$-0.406488\pi$
$211$	1775.03	0.579138	0.289569	+	0.957157i	$0.406488\pi$
$212$	1579.46	0.511689
$213$	−1500.78	−0.482779
$214$	1368.28	0.437073
$215$	3492.41	1.10782
$216$	−1875.77	−0.590880
$217$	3093.61	0.967779
$218$	−1844.98	−0.573200
$219$	−139.375	−0.0430050
$220$	−852.472	−0.261244
$221$	0	0
$222$	−2775.27	−0.839027
$223$	−3650.62	−1.09625	−0.548125	−	0.836396i	$-0.684658\pi$
$223$	−3650.62	−1.09625	−0.548125	+	0.836396i	$0.684658\pi$
$224$	−536.032	−0.159889
$225$	9318.75	2.76111
$226$	−3524.66	−1.03742
$227$	−2970.87	−0.868650	−0.434325	−	0.900756i	$-0.643013\pi$
$227$	−2970.87	−0.868650	−0.434325	+	0.900756i	$0.643013\pi$
$228$	4023.51	1.16870
$229$	−5943.74	−1.71517	−0.857584	−	0.514343i	$-0.828036\pi$
$229$	−5943.74	−1.71517	−0.857584	+	0.514343i	$0.828036\pi$
$230$	5999.46	1.71997
$231$	1844.97	0.525498
$232$	247.799	0.0701242
$233$	3771.83	1.06052	0.530260	−	0.847835i	$-0.322094\pi$
$233$	3771.83	1.06052	0.530260	+	0.847835i	$0.322094\pi$
$234$	0	0
$235$	−6621.11	−1.83793
$236$	−442.914	−0.122166
$237$	−2913.14	−0.798432
$238$	2251.57	0.613225
$239$	6057.29	1.63939	0.819694	−	0.572802i	$-0.194144\pi$
$239$	6057.29	1.63939	0.819694	+	0.572802i	$0.194144\pi$
$240$	−2482.46	−0.667675
$241$	1541.49	0.412016	0.206008	−	0.978550i	$-0.433953\pi$
$241$	1541.49	0.412016	0.206008	+	0.978550i	$0.433953\pi$
$242$	2359.42	0.626733
$243$	−388.370	−0.102526
$244$	2989.08	0.784248
$245$	1081.26	0.281955
$246$	2217.63	0.574760
$247$	0	0
$248$	−1477.46	−0.378301
$249$	−10377.2	−2.64109
$250$	1740.14	0.440224
$251$	2100.31	0.528169	0.264085	−	0.964499i	$-0.414930\pi$
$251$	2100.31	0.528169	0.264085	+	0.964499i	$0.414930\pi$
$252$	3563.57	0.890810
$253$	2129.46	0.529161
$254$	2255.17	0.557094
$255$	10427.4	2.56074
$256$	256.000	0.0625000
$257$	−2120.74	−0.514741	−0.257370	−	0.966313i	$-0.582856\pi$
$257$	−2120.74	−0.514741	−0.257370	+	0.966313i	$0.582856\pi$
$258$	3609.81	0.871073
$259$	2595.80	0.622761
$260$	0	0
$261$	−1647.38	−0.390692
$262$	−3977.85	−0.937987
$263$	−3856.54	−0.904199	−0.452100	−	0.891967i	$-0.649325\pi$
$263$	−3856.54	−0.904199	−0.452100	+	0.891967i	$0.649325\pi$
$264$	−881.127	−0.205415
$265$	−6841.74	−1.58598
$266$	−3763.32	−0.867458
$267$	−11983.2	−2.74666
$268$	206.973	0.0471750
$269$	6243.14	1.41506	0.707529	−	0.706684i	$-0.249810\pi$
$269$	6243.14	1.41506	0.707529	+	0.706684i	$0.249810\pi$
$270$	8125.24	1.83143
$271$	−2225.82	−0.498926	−0.249463	−	0.968384i	$-0.580254\pi$
$271$	−2225.82	−0.498926	−0.249463	+	0.968384i	$0.580254\pi$
$272$	−1075.31	−0.239707
$273$	0	0
$274$	3286.73	0.724666
$275$	2155.14	0.472581
$276$	6201.13	1.35241
$277$	−4627.44	−1.00374	−0.501870	−	0.864943i	$-0.667354\pi$
$277$	−4627.44	−1.00374	−0.501870	+	0.864943i	$0.667354\pi$
$278$	4610.99	0.994781
$279$	9822.24	2.10768
$280$	2321.92	0.495576
$281$	−5416.18	−1.14983	−0.574915	−	0.818213i	$-0.694965\pi$
$281$	−5416.18	−1.14983	−0.574915	+	0.818213i	$0.694965\pi$
$282$	−6843.68	−1.44516
$283$	6379.35	1.33997	0.669987	−	0.742373i	$-0.266300\pi$
$283$	6379.35	1.33997	0.669987	+	0.742373i	$0.266300\pi$
$284$	−670.398	−0.140073
$285$	−17428.6	−3.62239
$286$	0	0
$287$	−2074.22	−0.426611
$288$	−1701.90	−0.348214
$289$	−396.222	−0.0806476
$290$	−1073.39	−0.217350
$291$	11274.0	2.27112
$292$	−62.2587	−0.0124775
$293$	−525.025	−0.104684	−0.0523418	−	0.998629i	$-0.516669\pi$
$293$	−525.025	−0.104684	−0.0523418	+	0.998629i	$0.516669\pi$
$294$	1117.61	0.221701
$295$	1918.56	0.378654
$296$	−1239.71	−0.243435
$297$	2883.99	0.563454
$298$	1621.83	0.315269
$299$	0	0
$300$	6275.92	1.20780
$301$	−3376.37	−0.646547
$302$	−677.764	−0.129142
$303$	7912.90	1.50028
$304$	1797.30	0.339086
$305$	−12947.8	−2.43078
$306$	7148.74	1.33551
$307$	3597.21	0.668741	0.334370	−	0.942442i	$-0.391476\pi$
$307$	3597.21	0.668741	0.334370	+	0.942442i	$0.391476\pi$
$308$	824.146	0.152468
$309$	3683.33	0.678114
$310$	6399.88	1.17254
$311$	2168.56	0.395395	0.197698	−	0.980263i	$-0.436654\pi$
$311$	2168.56	0.395395	0.197698	+	0.980263i	$0.436654\pi$
$312$	0	0
$313$	9414.34	1.70010	0.850048	−	0.526706i	$-0.176573\pi$
$313$	9414.34	1.70010	0.850048	+	0.526706i	$0.176573\pi$
$314$	7677.19	1.37977
$315$	−15436.3	−2.76106
$316$	−1301.29	−0.231657
$317$	850.476	0.150686	0.0753431	−	0.997158i	$-0.475995\pi$
$317$	850.476	0.150686	0.0753431	+	0.997158i	$0.475995\pi$
$318$	−7071.72	−1.24705
$319$	−380.990	−0.0668693
$320$	−1108.91	−0.193719
$321$	−6126.19	−1.06520
$322$	−5800.11	−1.00381
$323$	−7549.44	−1.30050
$324$	2654.44	0.455151
$325$	0	0
$326$	4167.97	0.708105
$327$	8260.50	1.39696
$328$	990.613	0.166761
$329$	6401.11	1.07266
$330$	3816.76	0.636684
$331$	−1736.28	−0.288322	−0.144161	−	0.989554i	$-0.546048\pi$
$331$	−1736.28	−0.288322	−0.144161	+	0.989554i	$0.546048\pi$
$332$	−4635.50	−0.766283
$333$	8241.67	1.35628
$334$	−1646.88	−0.269800
$335$	−896.543	−0.146219
$336$	2399.97	0.389670
$337$	11495.0	1.85808	0.929042	−	0.369973i	$-0.120633\pi$
$337$	11495.0	1.85808	0.929042	+	0.369973i	$0.120633\pi$
$338$	0	0
$339$	15780.9	2.52832
$340$	4657.91	0.742972
$341$	2271.58	0.360742
$342$	−11948.6	−1.88919
$343$	−6790.92	−1.06902
$344$	1612.50	0.252733
$345$	−26861.3	−4.19178
$346$	6223.28	0.966953
$347$	−8299.19	−1.28393	−0.641965	−	0.766734i	$-0.721881\pi$
$347$	−8299.19	−1.28393	−0.641965	+	0.766734i	$0.721881\pi$
$348$	−1109.47	−0.170902
$349$	−2539.56	−0.389511	−0.194756	−	0.980852i	$-0.562391\pi$
$349$	−2539.56	−0.389511	−0.194756	+	0.980852i	$0.562391\pi$
$350$	−5870.07	−0.896481
$351$	0	0
$352$	−393.598	−0.0595991
$353$	−4207.80	−0.634443	−0.317222	−	0.948351i	$-0.602750\pi$
$353$	−4207.80	−0.634443	−0.317222	+	0.948351i	$0.602750\pi$
$354$	1983.05	0.297735
$355$	2903.95	0.434157
$356$	−5352.88	−0.796915
$357$	−10080.9	−1.49451
$358$	6227.43	0.919357
$359$	−10588.4	−1.55664	−0.778320	−	0.627868i	$-0.783928\pi$
$359$	−10588.4	−1.55664	−0.778320	+	0.627868i	$0.783928\pi$
$360$	7372.11	1.07929
$361$	5759.29	0.839669
$362$	614.936	0.0892826
$363$	−10563.8	−1.52743
$364$	0	0
$365$	269.685	0.0386739
$366$	−13383.0	−1.91131
$367$	−3941.65	−0.560633	−0.280316	−	0.959908i	$-0.590439\pi$
$367$	−3941.65	−0.560633	−0.280316	+	0.959908i	$0.590439\pi$
$368$	2770.04	0.392386
$369$	−6585.66	−0.929094
$370$	5370.03	0.754526
$371$	6614.40	0.925614
$372$	6615.01	0.921968
$373$	−4999.19	−0.693962	−0.346981	−	0.937872i	$-0.612793\pi$
$373$	−4999.19	−0.693962	−0.346981	+	0.937872i	$0.612793\pi$
$374$	1653.29	0.228581
$375$	−7791.10	−1.07288
$376$	−3057.06	−0.419298
$377$	0	0
$378$	−7855.26	−1.06886
$379$	−1407.01	−0.190695	−0.0953474	−	0.995444i	$-0.530396\pi$
$379$	−1407.01	−0.190695	−0.0953474	+	0.995444i	$0.530396\pi$
$380$	−7785.32	−1.05100
$381$	−10097.0	−1.35771
$382$	2122.79	0.284323
$383$	7504.62	1.00122	0.500611	−	0.865672i	$-0.333109\pi$
$383$	7504.62	1.00122	0.500611	+	0.865672i	$0.333109\pi$
$384$	−1146.19	−0.152321
$385$	−3569.94	−0.472574
$386$	−1525.53	−0.201159
$387$	−10720.0	−1.40808
$388$	5036.09	0.658940
$389$	−3780.08	−0.492693	−0.246347	−	0.969182i	$-0.579230\pi$
$389$	−3780.08	−0.492693	−0.246347	+	0.969182i	$0.579230\pi$
$390$	0	0
$391$	−11635.4	−1.50492
$392$	499.233	0.0643241
$393$	17810.0	2.28599
$394$	2291.53	0.293009
$395$	5636.79	0.718019
$396$	2616.67	0.332052
$397$	9352.68	1.18236	0.591181	−	0.806539i	$-0.298662\pi$
$397$	9352.68	1.18236	0.591181	+	0.806539i	$0.298662\pi$
$398$	3044.62	0.383449
$399$	16849.5	2.11411
$400$	2803.45	0.350431
$401$	8547.54	1.06445	0.532224	−	0.846604i	$-0.321356\pi$
$401$	8547.54	1.06445	0.532224	+	0.846604i	$0.321356\pi$
$402$	−926.680	−0.114972
$403$	0	0
$404$	3534.68	0.435289
$405$	−11498.2	−1.41074
$406$	1037.72	0.126850
$407$	1906.05	0.232136
$408$	4814.48	0.584197
$409$	13563.2	1.63975	0.819876	−	0.572542i	$-0.194043\pi$
$409$	13563.2	1.63975	0.819876	+	0.572542i	$0.194043\pi$
$410$	−4291.02	−0.516874
$411$	−14715.6	−1.76610
$412$	1645.34	0.196747
$413$	−1854.81	−0.220991
$414$	−18415.4	−2.18615
$415$	20079.5	2.37509
$416$	0	0
$417$	−20644.8	−2.42441
$418$	−2763.34	−0.323347
$419$	6396.55	0.745804	0.372902	−	0.927871i	$-0.378363\pi$
$419$	6396.55	0.745804	0.372902	+	0.927871i	$0.378363\pi$
$420$	−10395.9	−1.20778
$421$	−11234.3	−1.30054	−0.650268	−	0.759705i	$-0.725343\pi$
$421$	−11234.3	−1.30054	−0.650268	+	0.759705i	$0.725343\pi$
$422$	−3550.06	−0.409512
$423$	20323.6	2.33609
$424$	−3158.93	−0.361819
$425$	−11775.7	−1.34401
$426$	3001.57	0.341377
$427$	12517.5	1.41866
$428$	−2736.56	−0.309058
$429$	0	0
$430$	−6984.82	−0.783344
$431$	9395.92	1.05008	0.525041	−	0.851077i	$-0.324050\pi$
$431$	9395.92	1.05008	0.525041	+	0.851077i	$0.324050\pi$
$432$	3751.54	0.417815
$433$	16110.3	1.78802	0.894011	−	0.448044i	$-0.147879\pi$
$433$	16110.3	1.78802	0.894011	+	0.448044i	$0.147879\pi$
$434$	−6187.22	−0.684323
$435$	4805.86	0.529709
$436$	3689.96	0.405314
$437$	19447.6	2.12884
$438$	278.750	0.0304092
$439$	11954.8	1.29971	0.649854	−	0.760059i	$-0.274830\pi$
$439$	11954.8	1.29971	0.649854	+	0.760059i	$0.274830\pi$
$440$	1704.94	0.184727
$441$	−3318.93	−0.358377
$442$	0	0
$443$	8624.03	0.924921	0.462461	−	0.886640i	$-0.346967\pi$
$443$	8624.03	0.924921	0.462461	+	0.886640i	$0.346967\pi$
$444$	5550.54	0.593282
$445$	23187.0	2.47004
$446$	7301.25	0.775166
$447$	−7261.40	−0.768350
$448$	1072.06	0.113059
$449$	−10811.0	−1.13631	−0.568157	−	0.822920i	$-0.692343\pi$
$449$	−10811.0	−1.13631	−0.568157	+	0.822920i	$0.692343\pi$
$450$	−18637.5	−1.95240
$451$	−1523.06	−0.159020
$452$	7049.32	0.733566
$453$	3034.54	0.314736
$454$	5941.74	0.614229
$455$	0	0
$456$	−8047.03	−0.826396
$457$	−8551.05	−0.875276	−0.437638	−	0.899151i	$-0.644185\pi$
$457$	−8551.05	−0.875276	−0.437638	+	0.899151i	$0.644185\pi$
$458$	11887.5	1.21281
$459$	−15758.1	−1.60245
$460$	−11998.9	−1.21620
$461$	−2442.47	−0.246761	−0.123381	−	0.992359i	$-0.539374\pi$
$461$	−2442.47	−0.246761	−0.123381	+	0.992359i	$0.539374\pi$
$462$	−3689.94	−0.371583
$463$	11308.5	1.13510	0.567551	−	0.823338i	$-0.307891\pi$
$463$	11308.5	1.13510	0.567551	+	0.823338i	$0.307891\pi$
$464$	−495.598	−0.0495853
$465$	−28654.1	−2.85764
$466$	−7543.67	−0.749901
$467$	16223.3	1.60755	0.803776	−	0.594933i	$-0.202821\pi$
$467$	16223.3	1.60755	0.803776	+	0.594933i	$0.202821\pi$
$468$	0	0
$469$	866.752	0.0853367
$470$	13242.2	1.29961
$471$	−34373.0	−3.36268
$472$	885.828	0.0863846
$473$	−2479.20	−0.241002
$474$	5826.27	0.564577
$475$	19682.2	1.90122
$476$	−4503.14	−0.433615
$477$	21000.8	2.01585
$478$	−12114.6	−1.15922
$479$	6820.18	0.650568	0.325284	−	0.945616i	$-0.394540\pi$
$479$	6820.18	0.650568	0.325284	+	0.945616i	$0.394540\pi$
$480$	4964.91	0.472117
$481$	0	0
$482$	−3082.97	−0.291339
$483$	25968.8	2.44642
$484$	−4718.84	−0.443167
$485$	−21814.7	−2.04238
$486$	776.740	0.0724972
$487$	−4077.04	−0.379360	−0.189680	−	0.981846i	$-0.560745\pi$
$487$	−4077.04	−0.379360	−0.189680	+	0.981846i	$0.560745\pi$
$488$	−5978.17	−0.554547
$489$	−18661.2	−1.72574
$490$	−2162.52	−0.199373
$491$	−16633.6	−1.52884	−0.764422	−	0.644717i	$-0.776975\pi$
$491$	−16633.6	−1.52884	−0.764422	+	0.644717i	$0.776975\pi$
$492$	−4435.26	−0.406417
$493$	2081.73	0.190175
$494$	0	0
$495$	−11334.6	−1.02919
$496$	2954.92	0.267499
$497$	−2807.46	−0.253384
$498$	20754.5	1.86753
$499$	13454.7	1.20704	0.603522	−	0.797347i	$-0.293764\pi$
$499$	13454.7	1.20704	0.603522	+	0.797347i	$0.293764\pi$
$500$	−3480.28	−0.311285
$501$	7373.54	0.657536
$502$	−4200.62	−0.373472
$503$	−13103.9	−1.16158	−0.580791	−	0.814053i	$-0.697256\pi$
$503$	−13103.9	−1.16158	−0.580791	+	0.814053i	$0.697256\pi$
$504$	−7127.15	−0.629898
$505$	−15311.1	−1.34918
$506$	−4258.91	−0.374173
$507$	0	0
$508$	−4510.34	−0.393925
$509$	9909.43	0.862923	0.431462	−	0.902131i	$-0.357998\pi$
$509$	9909.43	0.862923	0.431462	+	0.902131i	$0.357998\pi$
$510$	−20854.8	−1.81072
$511$	−260.724	−0.0225709
$512$	−512.000	−0.0441942
$513$	26338.4	2.26680
$514$	4241.49	0.363977
$515$	−7127.08	−0.609819
$516$	−7219.62	−0.615942
$517$	4700.22	0.399836
$518$	−5191.59	−0.440358
$519$	−27863.4	−2.35659
$520$	0	0
$521$	20443.7	1.71911	0.859553	−	0.511046i	$-0.170742\pi$
$521$	20443.7	1.71911	0.859553	+	0.511046i	$0.170742\pi$
$522$	3294.77	0.276261
$523$	−14935.5	−1.24873	−0.624365	−	0.781133i	$-0.714642\pi$
$523$	−14935.5	−1.24873	−0.624365	+	0.781133i	$0.714642\pi$
$524$	7955.70	0.663257
$525$	26282.0	2.18484
$526$	7713.08	0.639365
$527$	−12411.9	−1.02594
$528$	1762.25	0.145251
$529$	17806.0	1.46347
$530$	13683.5	1.12146
$531$	−5889.04	−0.481285
$532$	7526.64	0.613385
$533$	0	0
$534$	23966.4	1.94218
$535$	11853.9	0.957923
$536$	−413.947	−0.0333578
$537$	−27882.0	−2.24059
$538$	−12486.3	−1.00060
$539$	−767.567	−0.0613385
$540$	−16250.5	−1.29502
$541$	−16177.8	−1.28565	−0.642827	−	0.766011i	$-0.722239\pi$
$541$	−16177.8	−1.28565	−0.642827	+	0.766011i	$0.722239\pi$
$542$	4451.64	0.352794
$543$	−2753.25	−0.217593
$544$	2150.62	0.169499
$545$	−15983.7	−1.25627
$546$	0	0
$547$	−11055.5	−0.864163	−0.432081	−	0.901835i	$-0.642221\pi$
$547$	−11055.5	−0.864163	−0.432081	+	0.901835i	$0.642221\pi$
$548$	−6573.45	−0.512416
$549$	39743.2	3.08962
$550$	−4310.28	−0.334166
$551$	−3479.44	−0.269019
$552$	−12402.3	−0.956295
$553$	−5449.49	−0.419052
$554$	9254.88	0.709751
$555$	−24043.2	−1.83888
$556$	−9221.99	−0.703416
$557$	5605.62	0.426423	0.213211	−	0.977006i	$-0.431608\pi$
$557$	5605.62	0.426423	0.213211	+	0.977006i	$0.431608\pi$
$558$	−19644.5	−1.49035
$559$	0	0
$560$	−4643.84	−0.350425
$561$	−7402.24	−0.557082
$562$	10832.4	0.813052
$563$	−7419.32	−0.555395	−0.277697	−	0.960669i	$-0.589571\pi$
$563$	−7419.32	−0.555395	−0.277697	+	0.960669i	$0.589571\pi$
$564$	13687.4	1.02188
$565$	−30535.4	−2.27369
$566$	−12758.7	−0.947505
$567$	11116.1	0.823341
$568$	1340.80	0.0990468
$569$	−9744.44	−0.717941	−0.358970	−	0.933349i	$-0.616872\pi$
$569$	−9744.44	−0.717941	−0.358970	+	0.933349i	$0.616872\pi$
$570$	34857.2	2.56141
$571$	6351.23	0.465483	0.232741	−	0.972539i	$-0.425230\pi$
$571$	6351.23	0.465483	0.232741	+	0.972539i	$0.425230\pi$
$572$	0	0
$573$	−9504.35	−0.692932
$574$	4148.44	0.301659
$575$	30334.6	2.20007
$576$	3403.81	0.246225
$577$	86.5008	0.00624104	0.00312052	−	0.999995i	$-0.499007\pi$
$577$	86.5008	0.00624104	0.00312052	+	0.999995i	$0.499007\pi$
$578$	792.443	0.0570264
$579$	6830.25	0.490251
$580$	2146.77	0.153689
$581$	−19412.3	−1.38616
$582$	−22548.0	−1.60592
$583$	4856.83	0.345025
$584$	124.517	0.00882289
$585$	0	0
$586$	1050.05	0.0740225
$587$	−1367.02	−0.0961209	−0.0480605	−	0.998844i	$-0.515304\pi$
$587$	−1367.02	−0.0961209	−0.0480605	+	0.998844i	$0.515304\pi$
$588$	−2235.21	−0.156766
$589$	20745.6	1.45128
$590$	−3837.12	−0.267749
$591$	−10259.9	−0.714102
$592$	2479.42	0.172134
$593$	1643.31	0.113799	0.0568995	−	0.998380i	$-0.481879\pi$
$593$	1643.31	0.113799	0.0568995	+	0.998380i	$0.481879\pi$
$594$	−5767.97	−0.398422
$595$	19506.1	1.34399
$596$	−3243.66	−0.222929
$597$	−13631.6	−0.934515
$598$	0	0
$599$	3295.56	0.224796	0.112398	−	0.993663i	$-0.464147\pi$
$599$	3295.56	0.224796	0.112398	+	0.993663i	$0.464147\pi$
$600$	−12551.8	−0.854045
$601$	−28175.8	−1.91234	−0.956169	−	0.292815i	$-0.905408\pi$
$601$	−28175.8	−1.91234	−0.956169	+	0.292815i	$0.905408\pi$
$602$	6752.73	0.457177
$603$	2751.94	0.185850
$604$	1355.53	0.0913172
$605$	20440.5	1.37360
$606$	−15825.8	−1.06086
$607$	10696.0	0.715215	0.357608	−	0.933872i	$-0.383593\pi$
$607$	10696.0	0.715215	0.357608	+	0.933872i	$0.383593\pi$
$608$	−3594.60	−0.239770
$609$	−4646.17	−0.309150
$610$	25895.5	1.71882
$611$	0	0
$612$	−14297.5	−0.944349
$613$	7253.09	0.477895	0.238947	−	0.971033i	$-0.423198\pi$
$613$	7253.09	0.477895	0.238947	+	0.971033i	$0.423198\pi$
$614$	−7194.41	−0.472871
$615$	19212.1	1.25969
$616$	−1648.29	−0.107811
$617$	−18354.9	−1.19763	−0.598817	−	0.800886i	$-0.704362\pi$
$617$	−18354.9	−1.19763	−0.598817	+	0.800886i	$0.704362\pi$
$618$	−7366.65	−0.479499
$619$	11835.1	0.768489	0.384244	−	0.923231i	$-0.374462\pi$
$619$	11835.1	0.768489	0.384244	+	0.923231i	$0.374462\pi$
$620$	−12799.8	−0.829114
$621$	40593.4	2.62312
$622$	−4337.13	−0.279587
$623$	−22416.5	−1.44157
$624$	0	0
$625$	−6826.48	−0.436895
$626$	−18828.7	−1.20215
$627$	12372.3	0.788039
$628$	−15354.4	−0.975647
$629$	−10414.6	−0.660189
$630$	30872.5	1.95237
$631$	−19152.1	−1.20830	−0.604148	−	0.796872i	$-0.706486\pi$
$631$	−19152.1	−1.20830	−0.604148	+	0.796872i	$0.706486\pi$
$632$	2602.59	0.163806
$633$	15894.6	0.998034
$634$	−1700.95	−0.106551
$635$	19537.4	1.22097
$636$	14143.4	0.881799
$637$	0	0
$638$	761.979	0.0472838
$639$	−8913.70	−0.551832
$640$	2217.82	0.136980
$641$	6798.89	0.418939	0.209469	−	0.977815i	$-0.432826\pi$
$641$	6798.89	0.418939	0.209469	+	0.977815i	$0.432826\pi$
$642$	12252.4	0.753213
$643$	18792.6	1.15258	0.576288	−	0.817247i	$-0.304501\pi$
$643$	18792.6	1.15258	0.576288	+	0.817247i	$0.304501\pi$
$644$	11600.2	0.709802
$645$	31273.1	1.90911
$646$	15098.9	0.919593
$647$	13142.1	0.798561	0.399280	−	0.916829i	$-0.369260\pi$
$647$	13142.1	0.798561	0.399280	+	0.916829i	$0.369260\pi$
$648$	−5308.89	−0.321841
$649$	−1361.95	−0.0823750
$650$	0	0
$651$	27702.0	1.66778
$652$	−8335.94	−0.500706
$653$	−12250.2	−0.734131	−0.367066	−	0.930195i	$-0.619638\pi$
$653$	−12250.2	−0.734131	−0.367066	+	0.930195i	$0.619638\pi$
$654$	−16521.0	−0.987802
$655$	−34461.6	−2.05576
$656$	−1981.23	−0.117918
$657$	−827.800	−0.0491561
$658$	−12802.2	−0.758484
$659$	−552.103	−0.0326356	−0.0163178	−	0.999867i	$-0.505194\pi$
$659$	−552.103	−0.0326356	−0.0163178	+	0.999867i	$0.505194\pi$
$660$	−7633.53	−0.450204
$661$	−22350.8	−1.31520	−0.657599	−	0.753368i	$-0.728428\pi$
$661$	−22350.8	−1.31520	−0.657599	+	0.753368i	$0.728428\pi$
$662$	3472.55	0.203874
$663$	0	0
$664$	9271.00	0.541844
$665$	−32603.0	−1.90119
$666$	−16483.3	−0.959034
$667$	−5362.59	−0.311305
$668$	3293.75	0.190777
$669$	−32689.8	−1.88918
$670$	1793.09	0.103392
$671$	9191.40	0.528808
$672$	−4799.94	−0.275538
$673$	1877.38	0.107530	0.0537650	−	0.998554i	$-0.482878\pi$
$673$	1877.38	0.107530	0.0537650	+	0.998554i	$0.482878\pi$
$674$	−22990.1	−1.31386
$675$	41083.0	2.34265
$676$	0	0
$677$	−11266.5	−0.639596	−0.319798	−	0.947486i	$-0.603615\pi$
$677$	−11266.5	−0.639596	−0.319798	+	0.947486i	$0.603615\pi$
$678$	−31561.8	−1.78780
$679$	21089.9	1.19198
$680$	−9315.82	−0.525361
$681$	−26602.9	−1.49695
$682$	−4543.16	−0.255083
$683$	−1255.53	−0.0703389	−0.0351694	−	0.999381i	$-0.511197\pi$
$683$	−1255.53	−0.0703389	−0.0351694	+	0.999381i	$0.511197\pi$
$684$	23897.1	1.33586
$685$	28474.1	1.58823
$686$	13581.8	0.755914
$687$	−53223.7	−2.95577
$688$	−3224.99	−0.178709
$689$	0	0
$690$	53722.6	2.96404
$691$	−14442.3	−0.795093	−0.397547	−	0.917582i	$-0.630138\pi$
$691$	−14442.3	−0.795093	−0.397547	+	0.917582i	$0.630138\pi$
$692$	−12446.6	−0.683739
$693$	10957.9	0.600661
$694$	16598.4	0.907876
$695$	39946.7	2.18024
$696$	2218.94	0.120846
$697$	8322.01	0.452251
$698$	5079.12	0.275426
$699$	33775.2	1.82760
$700$	11740.1	0.633908
$701$	18393.2	0.991017	0.495509	−	0.868603i	$-0.334982\pi$
$701$	18393.2	0.991017	0.495509	+	0.868603i	$0.334982\pi$
$702$	0	0
$703$	17407.3	0.933893
$704$	787.197	0.0421429
$705$	−59289.3	−3.16733
$706$	8415.59	0.448619
$707$	14802.4	0.787411
$708$	−3966.11	−0.210530
$709$	25374.4	1.34408	0.672041	−	0.740514i	$-0.265418\pi$
$709$	25374.4	1.34408	0.672041	+	0.740514i	$0.265418\pi$
$710$	−5807.90	−0.306995
$711$	−17302.2	−0.912633
$712$	10705.8	0.563504
$713$	31973.5	1.67941
$714$	20161.8	1.05678
$715$	0	0
$716$	−12454.9	−0.650083
$717$	54240.5	2.82517
$718$	21176.8	1.10071
$719$	4117.65	0.213578	0.106789	−	0.994282i	$-0.465943\pi$
$719$	4117.65	0.213578	0.106789	+	0.994282i	$0.465943\pi$
$720$	−14744.2	−0.763173
$721$	6890.26	0.355904
$722$	−11518.6	−0.593735
$723$	13803.4	0.710031
$724$	−1229.87	−0.0631323
$725$	−5427.28	−0.278019
$726$	21127.6	1.08006
$727$	1337.37	0.0682262	0.0341131	−	0.999418i	$-0.489139\pi$
$727$	1337.37	0.0682262	0.0341131	+	0.999418i	$0.489139\pi$
$728$	0	0
$729$	−21395.2	−1.08699
$730$	−539.370	−0.0273466
$731$	13546.4	0.685405
$732$	26766.0	1.35150
$733$	7830.04	0.394555	0.197278	−	0.980348i	$-0.436790\pi$
$733$	7830.04	0.394555	0.197278	+	0.980348i	$0.436790\pi$
$734$	7883.29	0.396427
$735$	9682.22	0.485897
$736$	−5540.07	−0.277459
$737$	636.441	0.0318095
$738$	13171.3	0.656969
$739$	−15492.4	−0.771172	−0.385586	−	0.922672i	$-0.626001\pi$
$739$	−15492.4	−0.771172	−0.385586	+	0.922672i	$0.626001\pi$
$740$	−10740.1	−0.533530
$741$	0	0
$742$	−13228.8	−0.654508
$743$	20334.4	1.00403	0.502016	−	0.864858i	$-0.332592\pi$
$743$	20334.4	1.00403	0.502016	+	0.864858i	$0.332592\pi$
$744$	−13230.0	−0.651930
$745$	14050.5	0.690967
$746$	9998.37	0.490706
$747$	−61634.2	−3.01884
$748$	−3306.57	−0.161631
$749$	−11460.0	−0.559066
$750$	15582.2	0.758642
$751$	15433.6	0.749907	0.374954	−	0.927044i	$-0.377659\pi$
$751$	15433.6	0.749907	0.374954	+	0.927044i	$0.377659\pi$
$752$	6114.13	0.296488
$753$	18807.4	0.910200
$754$	0	0
$755$	−5871.71	−0.283038
$756$	15710.5	0.755802
$757$	22768.3	1.09317	0.546583	−	0.837405i	$-0.315928\pi$
$757$	22768.3	1.09317	0.546583	+	0.837405i	$0.315928\pi$
$758$	2814.02	0.134842
$759$	19068.4	0.911909
$760$	15570.6	0.743167
$761$	−9284.91	−0.442283	−0.221142	−	0.975242i	$-0.570978\pi$
$761$	−9284.91	−0.442283	−0.221142	+	0.975242i	$0.570978\pi$
$762$	20194.1	0.960046
$763$	15452.6	0.733188
$764$	−4245.58	−0.201047
$765$	61932.1	2.92701
$766$	−15009.2	−0.707971
$767$	0	0
$768$	2292.37	0.107707
$769$	−31496.8	−1.47699	−0.738495	−	0.674259i	$-0.764463\pi$
$769$	−31496.8	−1.47699	−0.738495	+	0.674259i	$0.764463\pi$
$770$	7139.87	0.334160
$771$	−18990.4	−0.887058
$772$	3051.06	0.142241
$773$	18539.5	0.862639	0.431320	−	0.902199i	$-0.358048\pi$
$773$	18539.5	0.862639	0.431320	+	0.902199i	$0.358048\pi$
$774$	21440.0	0.995664
$775$	32359.2	1.49984
$776$	−10072.2	−0.465941
$777$	23244.3	1.07321
$778$	7560.16	0.348387
$779$	−13909.6	−0.639747
$780$	0	0
$781$	−2061.47	−0.0944495
$782$	23270.7	1.06414
$783$	−7262.72	−0.331480
$784$	−998.466	−0.0454840
$785$	66510.3	3.02402
$786$	−35620.0	−1.61644
$787$	−41447.4	−1.87731	−0.938654	−	0.344860i	$-0.887926\pi$
$787$	−41447.4	−1.87731	−0.938654	+	0.344860i	$0.887926\pi$
$788$	−4583.06	−0.207189
$789$	−34533.7	−1.55822
$790$	−11273.6	−0.507716
$791$	29520.8	1.32698
$792$	−5233.34	−0.234796
$793$	0	0
$794$	−18705.4	−0.836056
$795$	−61264.9	−2.73313
$796$	−6089.23	−0.271140
$797$	−6968.76	−0.309719	−0.154860	−	0.987936i	$-0.549493\pi$
$797$	−6968.76	−0.309719	−0.154860	+	0.987936i	$0.549493\pi$
$798$	−33698.9	−1.49490
$799$	−25682.0	−1.13713
$800$	−5606.89	−0.247792
$801$	−71172.5	−3.13952
$802$	−17095.1	−0.752678
$803$	−191.445	−0.00841338
$804$	1853.36	0.0812972
$805$	−50248.4	−2.20003
$806$	0	0
$807$	55904.7	2.43858
$808$	−7069.36	−0.307796
$809$	−10070.6	−0.437657	−0.218828	−	0.975763i	$-0.570224\pi$
$809$	−10070.6	−0.437657	−0.218828	+	0.975763i	$0.570224\pi$
$810$	22996.4	0.997545
$811$	−3587.33	−0.155324	−0.0776622	−	0.996980i	$-0.524746\pi$
$811$	−3587.33	−0.155324	−0.0776622	+	0.996980i	$0.524746\pi$
$812$	−2075.44	−0.0896967
$813$	−19931.3	−0.859803
$814$	−3812.09	−0.164145
$815$	36108.6	1.55194
$816$	−9628.97	−0.413090
$817$	−22641.7	−0.969563
$818$	−27126.4	−1.15948
$819$	0	0
$820$	8582.04	0.365485
$821$	−8415.43	−0.357735	−0.178868	−	0.983873i	$-0.557243\pi$
$821$	−8415.43	−0.357735	−0.178868	+	0.983873i	$0.557243\pi$
$822$	29431.3	1.24882
$823$	−24609.3	−1.04232	−0.521159	−	0.853460i	$-0.674500\pi$
$823$	−24609.3	−1.04232	−0.521159	+	0.853460i	$0.674500\pi$
$824$	−3290.68	−0.139121
$825$	19298.4	0.814404
$826$	3709.62	0.156264
$827$	−9627.34	−0.404807	−0.202403	−	0.979302i	$-0.564875\pi$
$827$	−9627.34	−0.404807	−0.202403	+	0.979302i	$0.564875\pi$
$828$	36830.7	1.54584
$829$	−35557.0	−1.48968	−0.744840	−	0.667243i	$-0.767474\pi$
$829$	−35557.0	−1.48968	−0.744840	+	0.667243i	$0.767474\pi$
$830$	−40159.0	−1.67944
$831$	−41436.8	−1.72975
$832$	0	0
$833$	4193.99	0.174446
$834$	41289.5	1.71432
$835$	−14267.5	−0.591313
$836$	5526.67	0.228641
$837$	43302.7	1.78824
$838$	−12793.1	−0.527363
$839$	−15656.8	−0.644260	−0.322130	−	0.946695i	$-0.604399\pi$
$839$	−15656.8	−0.644260	−0.322130	+	0.946695i	$0.604399\pi$
$840$	20791.8	0.854031
$841$	−23429.6	−0.960661
$842$	22468.6	0.919617
$843$	−48499.6	−1.98151
$844$	7100.12	0.289569
$845$	0	0
$846$	−40647.1	−1.65186
$847$	−19761.3	−0.801662
$848$	6317.86	0.255845
$849$	57124.4	2.30919
$850$	23551.4	0.950360
$851$	26828.4	1.08069
$852$	−6003.14	−0.241390
$853$	−5908.12	−0.237152	−0.118576	−	0.992945i	$-0.537833\pi$
$853$	−5908.12	−0.237152	−0.118576	+	0.992945i	$0.537833\pi$
$854$	−25035.1	−1.00314
$855$	−103515.	−4.14050
$856$	5473.12	0.218537
$857$	16128.5	0.642867	0.321434	−	0.946932i	$-0.395835\pi$
$857$	16128.5	0.642867	0.321434	+	0.946932i	$0.395835\pi$
$858$	0	0
$859$	−48567.5	−1.92911	−0.964553	−	0.263889i	$-0.914995\pi$
$859$	−48567.5	−1.92911	−0.964553	+	0.263889i	$0.914995\pi$
$860$	13969.6	0.553908
$861$	−18573.8	−0.735183
$862$	−18791.8	−0.742521
$863$	33998.4	1.34104	0.670521	−	0.741890i	$-0.266071\pi$
$863$	33998.4	1.34104	0.670521	+	0.741890i	$0.266071\pi$
$864$	−7503.08	−0.295440
$865$	53914.6	2.11925
$866$	−32220.7	−1.26432
$867$	−3548.00	−0.138981
$868$	12374.4	0.483889
$869$	−4001.46	−0.156203
$870$	−9611.73	−0.374561
$871$	0	0
$872$	−7379.91	−0.286600
$873$	66960.5	2.59596
$874$	−38895.2	−1.50532
$875$	−14574.5	−0.563096
$876$	−557.501	−0.0215025
$877$	44774.8	1.72399	0.861995	−	0.506917i	$-0.169215\pi$
$877$	44774.8	1.72399	0.861995	+	0.506917i	$0.169215\pi$
$878$	−23909.6	−0.919033
$879$	−4701.38	−0.180402
$880$	−3409.89	−0.130622
$881$	−3101.61	−0.118611	−0.0593053	−	0.998240i	$-0.518889\pi$
$881$	−3101.61	−0.118611	−0.0593053	+	0.998240i	$0.518889\pi$
$882$	6637.86	0.253411
$883$	26016.4	0.991531	0.495765	−	0.868456i	$-0.334888\pi$
$883$	26016.4	0.991531	0.495765	+	0.868456i	$0.334888\pi$
$884$	0	0
$885$	17179.9	0.652538
$886$	−17248.1	−0.654018
$887$	−23730.5	−0.898299	−0.449150	−	0.893457i	$-0.648273\pi$
$887$	−23730.5	−0.898299	−0.449150	+	0.893457i	$0.648273\pi$
$888$	−11101.1	−0.419513
$889$	−18888.2	−0.712586
$890$	−46373.9	−1.74658
$891$	8162.38	0.306902
$892$	−14602.5	−0.548125
$893$	42925.4	1.60856
$894$	14522.8	0.543306
$895$	53950.4	2.01493
$896$	−2144.13	−0.0799445
$897$	0	0
$898$	21622.1	0.803495
$899$	−5720.51	−0.212224
$900$	37275.0	1.38056
$901$	−26537.8	−0.981244
$902$	3046.12	0.112444
$903$	−30233.9	−1.11420
$904$	−14098.6	−0.518710
$905$	5327.41	0.195679
$906$	−6069.09	−0.222552
$907$	−51850.6	−1.89820	−0.949102	−	0.314969i	$-0.898006\pi$
$907$	−51850.6	−1.89820	−0.949102	+	0.314969i	$0.898006\pi$
$908$	−11883.5	−0.434325
$909$	46997.6	1.71486
$910$	0	0
$911$	5955.68	0.216598	0.108299	−	0.994118i	$-0.465460\pi$
$911$	5955.68	0.216598	0.108299	+	0.994118i	$0.465460\pi$
$912$	16094.1	0.584350
$913$	−14254.1	−0.516694
$914$	17102.1	0.618914
$915$	−115942.	−4.18898
$916$	−23775.0	−0.857584
$917$	33316.5	1.19979
$918$	31516.2	1.13311
$919$	−30266.8	−1.08641	−0.543204	−	0.839601i	$-0.682789\pi$
$919$	−30266.8	−1.08641	−0.543204	+	0.839601i	$0.682789\pi$
$920$	23997.8	0.859983
$921$	32211.5	1.15245
$922$	4884.93	0.174487
$923$	0	0
$924$	7379.88	0.262749
$925$	27152.0	0.965139
$926$	−22617.1	−0.802639
$927$	21876.6	0.775105
$928$	991.196	0.0350621
$929$	7850.26	0.277243	0.138622	−	0.990345i	$-0.455733\pi$
$929$	7850.26	0.277243	0.138622	+	0.990345i	$0.455733\pi$
$930$	57308.2	2.02066
$931$	−7009.92	−0.246768
$932$	15087.3	0.530260
$933$	19418.6	0.681389
$934$	−32446.7	−1.13671
$935$	14323.0	0.500976
$936$	0	0
$937$	37712.4	1.31485	0.657424	−	0.753521i	$-0.271646\pi$
$937$	37712.4	1.31485	0.657424	+	0.753521i	$0.271646\pi$
$938$	−1733.50	−0.0603422
$939$	84301.4	2.92979
$940$	−26484.5	−0.918966
$941$	−30329.3	−1.05070	−0.525349	−	0.850887i	$-0.676065\pi$
$941$	−30329.3	−1.05070	−0.525349	+	0.850887i	$0.676065\pi$
$942$	68746.0	2.37778
$943$	−21437.8	−0.740307
$944$	−1771.66	−0.0610831
$945$	−68053.0	−2.34261
$946$	4958.41	0.170414
$947$	−29041.1	−0.996526	−0.498263	−	0.867026i	$-0.666029\pi$
$947$	−29041.1	−0.996526	−0.498263	+	0.867026i	$0.666029\pi$
$948$	−11652.5	−0.399216
$949$	0	0
$950$	−39364.3	−1.34436
$951$	7615.66	0.259679
$952$	9006.27	0.306612
$953$	5133.10	0.174478	0.0872390	−	0.996187i	$-0.472196\pi$
$953$	5133.10	0.174478	0.0872390	+	0.996187i	$0.472196\pi$
$954$	−42001.5	−1.42542
$955$	18390.5	0.623144
$956$	24229.2	0.819694
$957$	−3411.60	−0.115237
$958$	−13640.4	−0.460021
$959$	−27528.0	−0.926929
$960$	−9929.83	−0.333837
$961$	4316.53	0.144894
$962$	0	0
$963$	−36385.7	−1.21756
$964$	6165.95	0.206008
$965$	−13216.2	−0.440876
$966$	−51937.5	−1.72988
$967$	990.387	0.0329356	0.0164678	−	0.999864i	$-0.494758\pi$
$967$	990.387	0.0329356	0.0164678	+	0.999864i	$0.494758\pi$
$968$	9437.69	0.313367
$969$	−67602.1	−2.24117
$970$	43629.5	1.44418
$971$	−3535.60	−0.116852	−0.0584258	−	0.998292i	$-0.518608\pi$
$971$	−3535.60	−0.116852	−0.0584258	+	0.998292i	$0.518608\pi$
$972$	−1553.48	−0.0512632
$973$	−38619.4	−1.27244
$974$	8154.09	0.268248
$975$	0	0
$976$	11956.3	0.392124
$977$	−619.679	−0.0202920	−0.0101460	−	0.999949i	$-0.503230\pi$
$977$	−619.679	−0.0202920	−0.0101460	+	0.999949i	$0.503230\pi$
$978$	37322.4	1.22029
$979$	−16460.0	−0.537349
$980$	4325.03	0.140978
$981$	49062.1	1.59677
$982$	33267.1	1.08106
$983$	25222.6	0.818387	0.409194	−	0.912448i	$-0.365810\pi$
$983$	25222.6	0.818387	0.409194	+	0.912448i	$0.365810\pi$
$984$	8870.53	0.287380
$985$	19852.4	0.642182
$986$	−4163.46	−0.134474
$987$	57319.3	1.84852
$988$	0	0
$989$	−34895.9	−1.12197
$990$	22669.1	0.727750
$991$	−14421.7	−0.462280	−0.231140	−	0.972920i	$-0.574246\pi$
$991$	−14421.7	−0.462280	−0.231140	+	0.972920i	$0.574246\pi$
$992$	−5909.83	−0.189151
$993$	−15547.6	−0.496867
$994$	5614.92	0.179169
$995$	26376.6	0.840397
$996$	−41508.9	−1.32054
$997$	34940.1	1.10989	0.554946	−	0.831886i	$-0.312739\pi$
$997$	34940.1	1.10989	0.554946	+	0.831886i	$0.312739\pi$
$998$	−26909.4	−0.853509
$999$	36334.6	1.15073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.a.n.1.5	✓	6
13.2	odd	12		338.4.e.i.147.4		24
13.3	even	3		338.4.c.p.191.2		12
13.4	even	6		338.4.c.o.315.2		12
13.5	odd	4		338.4.b.h.337.11		12
13.6	odd	12		338.4.e.i.23.7		24
13.7	odd	12		338.4.e.i.23.4		24
13.8	odd	4		338.4.b.h.337.5		12
13.9	even	3		338.4.c.p.315.2		12
13.10	even	6		338.4.c.o.191.2		12
13.11	odd	12		338.4.e.i.147.7		24
13.12	even	2		338.4.a.o.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.4.a.n.1.5	✓	6	1.1	even	1	trivial
338.4.a.o.1.5	yes	6	13.12	even	2
338.4.b.h.337.5		12	13.8	odd	4
338.4.b.h.337.11		12	13.5	odd	4
338.4.c.o.191.2		12	13.10	even	6
338.4.c.o.315.2		12	13.4	even	6
338.4.c.p.191.2		12	13.3	even	3
338.4.c.p.315.2		12	13.9	even	3
338.4.e.i.23.4		24	13.7	odd	12
338.4.e.i.23.7		24	13.6	odd	12
338.4.e.i.147.4		24	13.2	odd	12
338.4.e.i.147.7		24	13.11	odd	12

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +8.95458 q^{3} +4.00000 q^{4} -17.3267 q^{5} -17.9092 q^{6} +16.7510 q^{7} -8.00000 q^{8} +53.1845 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +8.95458 q^{3} +4.00000 q^{4} -17.3267 q^{5} -17.9092 q^{6} +16.7510 q^{7} -8.00000 q^{8} +53.1845 q^{9} +34.6535 q^{10} +12.3000 q^{11} +35.8183 q^{12} -33.5020 q^{14} -155.154 q^{15} +16.0000 q^{16} -67.2070 q^{17} -106.369 q^{18} +112.331 q^{19} -69.3069 q^{20} +149.998 q^{21} -24.5999 q^{22} +173.127 q^{23} -71.6366 q^{24} +175.215 q^{25} +234.471 q^{27} +67.0040 q^{28} -30.9749 q^{29} +310.307 q^{30} +184.682 q^{31} -32.0000 q^{32} +110.141 q^{33} +134.414 q^{34} -290.240 q^{35} +212.738 q^{36} +154.964 q^{37} -224.662 q^{38} +138.614 q^{40} -123.827 q^{41} -299.996 q^{42} -201.562 q^{43} +49.1998 q^{44} -921.514 q^{45} -346.254 q^{46} +382.133 q^{47} +143.273 q^{48} -62.4041 q^{49} -350.431 q^{50} -601.810 q^{51} +394.866 q^{53} -468.943 q^{54} -213.118 q^{55} -134.008 q^{56} +1005.88 q^{57} +61.9498 q^{58} -110.728 q^{59} -620.614 q^{60} +747.271 q^{61} -369.364 q^{62} +890.894 q^{63} +64.0000 q^{64} -220.282 q^{66} +51.7433 q^{67} -268.828 q^{68} +1550.28 q^{69} +580.480 q^{70} -167.600 q^{71} -425.476 q^{72} -15.5647 q^{73} -309.927 q^{74} +1568.98 q^{75} +449.325 q^{76} +206.036 q^{77} -325.323 q^{79} -277.228 q^{80} +663.611 q^{81} +247.653 q^{82} -1158.87 q^{83} +599.993 q^{84} +1164.48 q^{85} +403.124 q^{86} -277.367 q^{87} -98.3996 q^{88} -1338.22 q^{89} +1843.03 q^{90} +692.509 q^{92} +1653.75 q^{93} -764.266 q^{94} -1946.33 q^{95} -286.547 q^{96} +1259.02 q^{97} +124.808 q^{98} +654.167 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * q^99

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