LMFDB - Embedded newform 1849.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, 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("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 $ x^6 - 32x^4 - 16x^3 + 251x^2 + 276x + 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.15251$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.15251 q^{2} +6.49933 q^{3} +18.5484 q^{4} -17.2665 q^{5} -33.4879 q^{6} +23.3206 q^{7} -54.3507 q^{8} +15.2413 q^{9} +O(q^{10})$	\(q-5.15251 q^{2} +6.49933 q^{3} +18.5484 q^{4} -17.2665 q^{5} -33.4879 q^{6} +23.3206 q^{7} -54.3507 q^{8} +15.2413 q^{9} +88.9661 q^{10} -60.5580 q^{11} +120.552 q^{12} +10.9419 q^{13} -120.160 q^{14} -112.221 q^{15} +131.656 q^{16} -3.57473 q^{17} -78.5308 q^{18} -33.2403 q^{19} -320.267 q^{20} +151.568 q^{21} +312.026 q^{22} +63.7158 q^{23} -353.243 q^{24} +173.134 q^{25} -56.3783 q^{26} -76.4239 q^{27} +432.560 q^{28} +89.3510 q^{29} +578.220 q^{30} +222.839 q^{31} -243.552 q^{32} -393.586 q^{33} +18.4188 q^{34} -402.666 q^{35} +282.701 q^{36} +59.6535 q^{37} +171.271 q^{38} +71.1150 q^{39} +938.449 q^{40} -143.837 q^{41} -780.957 q^{42} -1123.25 q^{44} -263.164 q^{45} -328.296 q^{46} +379.013 q^{47} +855.674 q^{48} +200.850 q^{49} -892.073 q^{50} -23.2334 q^{51} +202.955 q^{52} -150.129 q^{53} +393.775 q^{54} +1045.63 q^{55} -1267.49 q^{56} -216.040 q^{57} -460.382 q^{58} +207.310 q^{59} -2081.52 q^{60} +486.557 q^{61} -1148.18 q^{62} +355.435 q^{63} +201.659 q^{64} -188.929 q^{65} +2027.96 q^{66} +1019.41 q^{67} -66.3055 q^{68} +414.110 q^{69} +2074.74 q^{70} -13.8437 q^{71} -828.374 q^{72} -411.158 q^{73} -307.365 q^{74} +1125.25 q^{75} -616.555 q^{76} -1412.25 q^{77} -366.421 q^{78} -1315.13 q^{79} -2273.24 q^{80} -908.218 q^{81} +741.120 q^{82} +813.425 q^{83} +2811.35 q^{84} +61.7233 q^{85} +580.721 q^{87} +3291.37 q^{88} +350.573 q^{89} +1355.96 q^{90} +255.171 q^{91} +1181.83 q^{92} +1448.30 q^{93} -1952.87 q^{94} +573.946 q^{95} -1582.92 q^{96} -1187.03 q^{97} -1034.88 q^{98} -922.981 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 7 q^{3} + 22 q^{4} - 43 q^{5} - 3 q^{6} - 8 q^{7} - 54 q^{8} + 81 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 - 7 * q^3 + 22 * q^4 - 43 * q^5 - 3 * q^6 - 8 * q^7 - 54 * q^8 + 81 * q^9	\( 6 q - 6 q^{2} - 7 q^{3} + 22 q^{4} - 43 q^{5} - 3 q^{6} - 8 q^{7} - 54 q^{8} + 81 q^{9} + 57 q^{10} - 28 q^{11} + 157 q^{12} + 56 q^{13} - 184 q^{14} - 124 q^{15} - 54 q^{16} + 19 q^{17} + 81 q^{18} + 75 q^{19} - 135 q^{20} - 18 q^{21} + 504 q^{22} + 131 q^{23} - 567 q^{24} + 105 q^{25} - 44 q^{26} - 238 q^{27} + 404 q^{28} - 515 q^{29} + 396 q^{30} + 237 q^{31} - 558 q^{32} - 540 q^{33} + 107 q^{34} + 198 q^{35} + 73 q^{36} - 269 q^{37} + 527 q^{38} - 290 q^{39} + 613 q^{40} + 471 q^{41} - 362 q^{42} - 428 q^{44} - 334 q^{45} + 67 q^{46} + 415 q^{47} + 989 q^{48} + 350 q^{49} - 1335 q^{50} + 1241 q^{51} - 8 q^{52} + 450 q^{53} + 402 q^{54} + 1732 q^{55} - 780 q^{56} - 1000 q^{57} - 1055 q^{58} + 356 q^{59} - 2732 q^{60} + 1328 q^{61} - 1603 q^{62} + 2290 q^{63} + 466 q^{64} + 62 q^{65} + 156 q^{66} - 632 q^{67} + 571 q^{68} + 1130 q^{69} + 1902 q^{70} + 144 q^{71} - 567 q^{72} - 864 q^{73} + 1207 q^{74} + 2494 q^{75} - 1005 q^{76} - 2660 q^{77} + 2222 q^{78} - 1613 q^{79} - 2399 q^{80} - 102 q^{81} - 1673 q^{82} - 682 q^{83} + 3758 q^{84} - 84 q^{85} + 449 q^{87} + 608 q^{88} - 3378 q^{89} + 930 q^{90} + 3900 q^{91} + 3491 q^{92} - 1879 q^{93} - 3197 q^{94} - 79 q^{95} - 591 q^{96} - 55 q^{97} - 2398 q^{98} - 1612 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 7 * q^3 + 22 * q^4 - 43 * q^5 - 3 * q^6 - 8 * q^7 - 54 * q^8 + 81 * q^9 + 57 * q^10 - 28 * q^11 + 157 * q^12 + 56 * q^13 - 184 * q^14 - 124 * q^15 - 54 * q^16 + 19 * q^17 + 81 * q^18 + 75 * q^19 - 135 * q^20 - 18 * q^21 + 504 * q^22 + 131 * q^23 - 567 * q^24 + 105 * q^25 - 44 * q^26 - 238 * q^27 + 404 * q^28 - 515 * q^29 + 396 * q^30 + 237 * q^31 - 558 * q^32 - 540 * q^33 + 107 * q^34 + 198 * q^35 + 73 * q^36 - 269 * q^37 + 527 * q^38 - 290 * q^39 + 613 * q^40 + 471 * q^41 - 362 * q^42 - 428 * q^44 - 334 * q^45 + 67 * q^46 + 415 * q^47 + 989 * q^48 + 350 * q^49 - 1335 * q^50 + 1241 * q^51 - 8 * q^52 + 450 * q^53 + 402 * q^54 + 1732 * q^55 - 780 * q^56 - 1000 * q^57 - 1055 * q^58 + 356 * q^59 - 2732 * q^60 + 1328 * q^61 - 1603 * q^62 + 2290 * q^63 + 466 * q^64 + 62 * q^65 + 156 * q^66 - 632 * q^67 + 571 * q^68 + 1130 * q^69 + 1902 * q^70 + 144 * q^71 - 567 * q^72 - 864 * q^73 + 1207 * q^74 + 2494 * q^75 - 1005 * q^76 - 2660 * q^77 + 2222 * q^78 - 1613 * q^79 - 2399 * q^80 - 102 * q^81 - 1673 * q^82 - 682 * q^83 + 3758 * q^84 - 84 * q^85 + 449 * q^87 + 608 * q^88 - 3378 * q^89 + 930 * q^90 + 3900 * q^91 + 3491 * q^92 - 1879 * q^93 - 3197 * q^94 - 79 * q^95 - 591 * q^96 - 55 * q^97 - 2398 * q^98 - 1612 * 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 $

$2$	−5.15251	−1.82169	−0.910844	−	0.412750i	$-0.864568\pi$
$2$	−5.15251	−1.82169	−0.910844	+	0.412750i	$0.864568\pi$
$3$	6.49933	1.25080	0.625398	−	0.780306i	$-0.284937\pi$
$3$	6.49933	1.25080	0.625398	+	0.780306i	$0.284937\pi$
$4$	18.5484	2.31855
$5$	−17.2665	−1.54437	−0.772183	−	0.635400i	$-0.780835\pi$
$5$	−17.2665	−1.54437	−0.772183	+	0.635400i	$0.780835\pi$
$6$	−33.4879	−2.27856
$7$	23.3206	1.25919	0.629597	−	0.776922i	$-0.283220\pi$
$7$	23.3206	1.25919	0.629597	+	0.776922i	$0.283220\pi$
$8$	−54.3507	−2.40199
$9$	15.2413	0.564491
$10$	88.9661	2.81336
$11$	−60.5580	−1.65990	−0.829951	−	0.557836i	$-0.811632\pi$
$11$	−60.5580	−1.65990	−0.829951	+	0.557836i	$0.811632\pi$
$12$	120.552	2.90003
$13$	10.9419	0.233441	0.116721	−	0.993165i	$-0.462762\pi$
$13$	10.9419	0.233441	0.116721	+	0.993165i	$0.462762\pi$
$14$	−120.160	−2.29386
$15$	−112.221	−1.93169
$16$	131.656	2.05712
$17$	−3.57473	−0.0510000	−0.0255000	−	0.999675i	$-0.508118\pi$
$17$	−3.57473	−0.0510000	−0.0255000	+	0.999675i	$0.508118\pi$
$18$	−78.5308	−1.02833
$19$	−33.2403	−0.401361	−0.200680	−	0.979657i	$-0.564315\pi$
$19$	−33.2403	−0.401361	−0.200680	+	0.979657i	$0.564315\pi$
$20$	−320.267	−3.58069
$21$	151.568	1.57499
$22$	312.026	3.02383
$23$	63.7158	0.577637	0.288819	−	0.957384i	$-0.406738\pi$
$23$	63.7158	0.577637	0.288819	+	0.957384i	$0.406738\pi$
$24$	−353.243	−3.00440
$25$	173.134	1.38507
$26$	−56.3783	−0.425257
$27$	−76.4239	−0.544733
$28$	432.560	2.91950
$29$	89.3510	0.572140	0.286070	−	0.958209i	$-0.407651\pi$
$29$	89.3510	0.572140	0.286070	+	0.958209i	$0.407651\pi$
$30$	578.220	3.51893
$31$	222.839	1.29106	0.645532	−	0.763733i	$-0.276636\pi$
$31$	222.839	1.29106	0.645532	+	0.763733i	$0.276636\pi$
$32$	−243.552	−1.34545
$33$	−393.586	−2.07620
$34$	18.4188	0.0929061
$35$	−402.666	−1.94466
$36$	282.701	1.30880
$37$	59.6535	0.265053	0.132527	−	0.991179i	$-0.457691\pi$
$37$	59.6535	0.265053	0.132527	+	0.991179i	$0.457691\pi$
$38$	171.271	0.731154
$39$	71.1150	0.291987
$40$	938.449	3.70955
$41$	−143.837	−0.547890	−0.273945	−	0.961745i	$-0.588329\pi$
$41$	−143.837	−0.547890	−0.273945	+	0.961745i	$0.588329\pi$
$42$	−780.957	−2.86915
$43$	0	0
$43$	0	0
$44$	−1123.25	−3.84857
$45$	−263.164	−0.871782
$46$	−328.296	−1.05228
$47$	379.013	1.17627	0.588135	−	0.808763i	$-0.299862\pi$
$47$	379.013	1.17627	0.588135	+	0.808763i	$0.299862\pi$
$48$	855.674	2.57304
$49$	200.850	0.585569
$50$	−892.073	−2.52316
$51$	−23.2334	−0.0637906
$52$	202.955	0.541245
$53$	−150.129	−0.389090	−0.194545	−	0.980894i	$-0.562323\pi$
$53$	−150.129	−0.389090	−0.194545	+	0.980894i	$0.562323\pi$
$54$	393.775	0.992333
$55$	1045.63	2.56350
$56$	−1267.49	−3.02457
$57$	−216.040	−0.502021
$58$	−460.382	−1.04226
$59$	207.310	0.457448	0.228724	−	0.973491i	$-0.426545\pi$
$59$	207.310	0.457448	0.228724	+	0.973491i	$0.426545\pi$
$60$	−2081.52	−4.47871
$61$	486.557	1.02127	0.510633	−	0.859799i	$-0.329411\pi$
$61$	486.557	1.02127	0.510633	+	0.859799i	$0.329411\pi$
$62$	−1148.18	−2.35192
$63$	355.435	0.710804
$64$	201.659	0.393866
$65$	−188.929	−0.360519
$66$	2027.96	3.78219
$67$	1019.41	1.85882	0.929408	−	0.369054i	$-0.120318\pi$
$67$	1019.41	1.85882	0.929408	+	0.369054i	$0.120318\pi$
$68$	−66.3055	−0.118246
$69$	414.110	0.722507
$70$	2074.74	3.54256
$71$	−13.8437	−0.0231400	−0.0115700	−	0.999933i	$-0.503683\pi$
$71$	−13.8437	−0.0231400	−0.0115700	+	0.999933i	$0.503683\pi$
$72$	−828.374	−1.35590
$73$	−411.158	−0.659211	−0.329605	−	0.944119i	$-0.606916\pi$
$73$	−411.158	−0.659211	−0.329605	+	0.944119i	$0.606916\pi$
$74$	−307.365	−0.482844
$75$	1125.25	1.73244
$76$	−616.555	−0.930575
$77$	−1412.25	−2.09014
$78$	−366.421	−0.531910
$79$	−1315.13	−1.87296	−0.936481	−	0.350718i	$-0.885938\pi$
$79$	−1315.13	−1.87296	−0.936481	+	0.350718i	$0.885938\pi$
$80$	−2273.24	−3.17695
$81$	−908.218	−1.24584
$82$	741.120	0.998085
$83$	813.425	1.07572	0.537861	−	0.843033i	$-0.319232\pi$
$83$	813.425	1.07572	0.537861	+	0.843033i	$0.319232\pi$
$84$	2811.35	3.65170
$85$	61.7233	0.0787627
$86$	0	0
$87$	580.721	0.715630
$88$	3291.37	3.98706
$89$	350.573	0.417535	0.208768	−	0.977965i	$-0.433055\pi$
$89$	350.573	0.417535	0.208768	+	0.977965i	$0.433055\pi$
$90$	1355.96	1.58811
$91$	255.171	0.293948
$92$	1181.83	1.33928
$93$	1448.30	1.61486
$94$	−1952.87	−2.14280
$95$	573.946	0.619848
$96$	−1582.92	−1.68288
$97$	−1187.03	−1.24252	−0.621262	−	0.783603i	$-0.713380\pi$
$97$	−1187.03	−1.24252	−0.621262	+	0.783603i	$0.713380\pi$
$98$	−1034.88	−1.06672
$99$	−922.981	−0.937001
$100$	3211.35	3.21135
$101$	469.014	0.462066	0.231033	−	0.972946i	$-0.425790\pi$
$101$	469.014	0.462066	0.231033	+	0.972946i	$0.425790\pi$
$102$	119.710	0.116207
$103$	−1299.82	−1.24345	−0.621725	−	0.783235i	$-0.713568\pi$
$103$	−1299.82	−1.24345	−0.621725	+	0.783235i	$0.713568\pi$
$104$	−594.700	−0.560722
$105$	−2617.06	−2.43237
$106$	773.541	0.708801
$107$	−1480.92	−1.33800	−0.669000	−	0.743263i	$-0.733277\pi$
$107$	−1480.92	−1.33800	−0.669000	+	0.743263i	$0.733277\pi$
$108$	−1417.54	−1.26299
$109$	350.586	0.308073	0.154037	−	0.988065i	$-0.450773\pi$
$109$	350.586	0.308073	0.154037	+	0.988065i	$0.450773\pi$
$110$	−5387.61	−4.66990
$111$	387.707	0.331528
$112$	3070.29	2.59031
$113$	−785.106	−0.653598	−0.326799	−	0.945094i	$-0.605970\pi$
$113$	−785.106	−0.653598	−0.326799	+	0.945094i	$0.605970\pi$
$114$	1113.15	0.914525
$115$	−1100.15	−0.892084
$116$	1657.32	1.32653
$117$	166.768	0.131776
$118$	−1068.17	−0.833329
$119$	−83.3649	−0.0642189
$120$	6099.29	4.63989
$121$	2336.27	1.75528
$122$	−2506.99	−1.86043
$123$	−934.841	−0.685299
$124$	4133.30	2.99340
$125$	−831.100	−0.594687
$126$	−1831.39	−1.29486
$127$	−731.562	−0.511147	−0.255573	−	0.966790i	$-0.582264\pi$
$127$	−731.562	−0.511147	−0.255573	+	0.966790i	$0.582264\pi$
$128$	909.364	0.627947
$129$	0	0
$130$	973.458	0.656753
$131$	−2462.56	−1.64240	−0.821202	−	0.570638i	$-0.806696\pi$
$131$	−2462.56	−1.64240	−0.821202	+	0.570638i	$0.806696\pi$
$132$	−7300.40	−4.81377
$133$	−775.184	−0.505391
$134$	−5252.52	−3.38618
$135$	1319.58	0.841267
$136$	194.289	0.122501
$137$	−2384.64	−1.48710	−0.743552	−	0.668678i	$-0.766861\pi$
$137$	−2384.64	−1.48710	−0.743552	+	0.668678i	$0.766861\pi$
$138$	−2133.71	−1.31618
$139$	−3086.87	−1.88363	−0.941816	−	0.336129i	$-0.890882\pi$
$139$	−3086.87	−1.88363	−0.941816	+	0.336129i	$0.890882\pi$
$140$	−7468.81	−4.50878
$141$	2463.33	1.47127
$142$	71.3297	0.0421539
$143$	−662.619	−0.387490
$144$	2006.60	1.16123
$145$	−1542.78	−0.883594
$146$	2118.50	1.20088
$147$	1305.39	0.732427
$148$	1106.48	0.614539
$149$	1445.58	0.794808	0.397404	−	0.917644i	$-0.369911\pi$
$149$	1445.58	0.794808	0.397404	+	0.917644i	$0.369911\pi$
$150$	−5797.88	−3.15596
$151$	825.878	0.445093	0.222546	−	0.974922i	$-0.428563\pi$
$151$	825.878	0.445093	0.222546	+	0.974922i	$0.428563\pi$
$152$	1806.64	0.964063
$153$	−54.4834	−0.0287891
$154$	7276.63	3.80758
$155$	−3847.65	−1.99388
$156$	1319.07	0.676987
$157$	−1187.82	−0.603809	−0.301904	−	0.953338i	$-0.597622\pi$
$157$	−1187.82	−0.603809	−0.301904	+	0.953338i	$0.597622\pi$
$158$	6776.24	3.41195
$159$	−975.737	−0.486673
$160$	4205.30	2.07786
$161$	1485.89	0.727357
$162$	4679.61	2.26953
$163$	−1514.60	−0.727809	−0.363904	−	0.931436i	$-0.618557\pi$
$163$	−1514.60	−0.727809	−0.363904	+	0.931436i	$0.618557\pi$
$164$	−2667.94	−1.27031
$165$	6795.88	3.20641
$166$	−4191.18	−1.95963
$167$	−3354.48	−1.55435	−0.777177	−	0.629282i	$-0.783349\pi$
$167$	−3354.48	−1.55435	−0.777177	+	0.629282i	$0.783349\pi$
$168$	−8237.84	−3.78312
$169$	−2077.27	−0.945505
$170$	−318.030	−0.143481
$171$	−506.625	−0.226565
$172$	0	0
$173$	1654.64	0.727166	0.363583	−	0.931562i	$-0.381553\pi$
$173$	1654.64	0.727166	0.363583	+	0.931562i	$0.381553\pi$
$174$	−2992.17	−1.30366
$175$	4037.58	1.74407
$176$	−7972.81	−3.41462
$177$	1347.38	0.572175
$178$	−1806.33	−0.760619
$179$	1334.05	0.557049	0.278524	−	0.960429i	$-0.410155\pi$
$179$	1334.05	0.557049	0.278524	+	0.960429i	$0.410155\pi$
$180$	−4881.27	−2.02127
$181$	2771.58	1.13817	0.569087	−	0.822277i	$-0.307297\pi$
$181$	2771.58	1.13817	0.569087	+	0.822277i	$0.307297\pi$
$182$	−1314.77	−0.535481
$183$	3162.29	1.27740
$184$	−3463.00	−1.38748
$185$	−1030.01	−0.409339
$186$	−7462.39	−2.94177
$187$	216.479	0.0846550
$188$	7030.08	2.72724
$189$	−1782.25	−0.685924
$190$	−2957.26	−1.12917
$191$	−81.5135	−0.0308802	−0.0154401	−	0.999881i	$-0.504915\pi$
$191$	−81.5135	−0.0308802	−0.0154401	+	0.999881i	$0.504915\pi$
$192$	1310.65	0.492646
$193$	3305.29	1.23275	0.616373	−	0.787454i	$-0.288601\pi$
$193$	3305.29	1.23275	0.616373	+	0.787454i	$0.288601\pi$
$194$	6116.20	2.26349
$195$	−1227.91	−0.450936
$196$	3725.45	1.35767
$197$	2241.99	0.810840	0.405420	−	0.914131i	$-0.367125\pi$
$197$	2241.99	0.810840	0.405420	+	0.914131i	$0.367125\pi$
$198$	4755.67	1.70692
$199$	4074.29	1.45135	0.725675	−	0.688037i	$-0.241528\pi$
$199$	4074.29	1.45135	0.725675	+	0.688037i	$0.241528\pi$
$200$	−9409.94	−3.32691
$201$	6625.48	2.32500
$202$	−2416.60	−0.841740
$203$	2083.72	0.720435
$204$	−430.941	−0.147902
$205$	2483.56	0.846143
$206$	6697.36	2.26518
$207$	971.109	0.326071
$208$	1440.56	0.480217
$209$	2012.97	0.666220
$210$	13484.4	4.43102
$211$	−4267.62	−1.39239	−0.696196	−	0.717851i	$-0.745126\pi$
$211$	−4267.62	−1.39239	−0.696196	+	0.717851i	$0.745126\pi$
$212$	−2784.65	−0.902125
$213$	−89.9746	−0.0289435
$214$	7630.46	2.43742
$215$	0	0
$216$	4153.69	1.30844
$217$	5196.73	1.62570
$218$	−1806.40	−0.561214
$219$	−2672.25	−0.824538
$220$	19394.7	5.94360
$221$	−39.1143	−0.0119055
$222$	−1997.67	−0.603940
$223$	889.370	0.267070	0.133535	−	0.991044i	$-0.457367\pi$
$223$	889.370	0.267070	0.133535	+	0.991044i	$0.457367\pi$
$224$	−5679.78	−1.69418
$225$	2638.78	0.781859
$226$	4045.27	1.19065
$227$	−232.389	−0.0679479	−0.0339739	−	0.999423i	$-0.510816\pi$
$227$	−232.389	−0.0679479	−0.0339739	+	0.999423i	$0.510816\pi$
$228$	−4007.19	−1.16396
$229$	851.806	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$229$	851.806	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$230$	5668.54	1.62510
$231$	−9178.67	−2.61434
$232$	−4856.29	−1.37427
$233$	2083.62	0.585848	0.292924	−	0.956136i	$-0.405372\pi$
$233$	2083.62	0.585848	0.292924	+	0.956136i	$0.405372\pi$
$234$	−859.276	−0.240054
$235$	−6544.24	−1.81659
$236$	3845.27	1.06062
$237$	−8547.48	−2.34269
$238$	429.539	0.116987
$239$	986.983	0.267124	0.133562	−	0.991040i	$-0.457358\pi$
$239$	986.983	0.267124	0.133562	+	0.991040i	$0.457358\pi$
$240$	−14774.5	−3.97372
$241$	−14.8772	−0.00397644	−0.00198822	−	0.999998i	$-0.500633\pi$
$241$	−14.8772	−0.00397644	−0.00198822	+	0.999998i	$0.500633\pi$
$242$	−12037.7	−3.19757
$243$	−3839.36	−1.01356
$244$	9024.85	2.36786
$245$	−3467.99	−0.904333
$246$	4816.78	1.24840
$247$	−363.712	−0.0936942
$248$	−12111.4	−3.10112
$249$	5286.72	1.34551
$250$	4282.26	1.08333
$251$	−3027.27	−0.761274	−0.380637	−	0.924725i	$-0.624295\pi$
$251$	−3027.27	−0.761274	−0.380637	+	0.924725i	$0.624295\pi$
$252$	6592.76	1.64803
$253$	−3858.50	−0.958822
$254$	3769.38	0.931150
$255$	401.160	0.0985161
$256$	−6298.79	−1.53779
$257$	−5759.92	−1.39803	−0.699016	−	0.715106i	$-0.746378\pi$
$257$	−5759.92	−1.39803	−0.699016	+	0.715106i	$0.746378\pi$
$258$	0	0
$259$	1391.15	0.333753
$260$	−3504.32	−0.835881
$261$	1361.82	0.322968
$262$	12688.4	2.99195
$263$	−2545.09	−0.596719	−0.298359	−	0.954454i	$-0.596439\pi$
$263$	−2545.09	−0.596719	−0.298359	+	0.954454i	$0.596439\pi$
$264$	21391.7	4.98700
$265$	2592.21	0.600898
$266$	3994.15	0.920665
$267$	2278.49	0.522252
$268$	18908.4	4.30976
$269$	6757.89	1.53173	0.765866	−	0.643001i	$-0.222311\pi$
$269$	6757.89	1.53173	0.765866	+	0.643001i	$0.222311\pi$
$270$	−6799.13	−1.53253
$271$	2485.54	0.557143	0.278571	−	0.960415i	$-0.410139\pi$
$271$	2485.54	0.557143	0.278571	+	0.960415i	$0.410139\pi$
$272$	−470.634	−0.104913
$273$	1658.44	0.367669
$274$	12286.9	2.70904
$275$	−10484.6	−2.29908
$276$	7681.07	1.67517
$277$	2090.19	0.453385	0.226692	−	0.973966i	$-0.427209\pi$
$277$	2090.19	0.453385	0.226692	+	0.973966i	$0.427209\pi$
$278$	15905.1	3.43139
$279$	3396.34	0.728795
$280$	21885.2	4.67104
$281$	−6049.22	−1.28422	−0.642111	−	0.766612i	$-0.721941\pi$
$281$	−6049.22	−1.28422	−0.642111	+	0.766612i	$0.721941\pi$
$282$	−12692.3	−2.68020
$283$	−650.183	−0.136570	−0.0682851	−	0.997666i	$-0.521753\pi$
$283$	−650.183	−0.136570	−0.0682851	+	0.997666i	$0.521753\pi$
$284$	−256.778	−0.0536513
$285$	3730.26	0.775304
$286$	3414.16	0.705885
$287$	−3354.35	−0.689900
$288$	−3712.04	−0.759494
$289$	−4900.22	−0.997399
$290$	7949.21	1.60963
$291$	−7714.92	−1.55415
$292$	−7626.31	−1.52841
$293$	−2172.20	−0.433110	−0.216555	−	0.976270i	$-0.569482\pi$
$293$	−2172.20	−0.433110	−0.216555	+	0.976270i	$0.569482\pi$
$294$	−6726.04	−1.33425
$295$	−3579.53	−0.706468
$296$	−3242.21	−0.636654
$297$	4628.08	0.904203
$298$	−7448.36	−1.44789
$299$	697.171	0.134844
$300$	20871.6	4.01674
$301$	0	0
$302$	−4255.35	−0.810820
$303$	3048.28	0.577950
$304$	−4376.28	−0.825648
$305$	−8401.16	−1.57721
$306$	280.727	0.0524447
$307$	1257.47	0.233771	0.116885	−	0.993145i	$-0.462709\pi$
$307$	1257.47	0.233771	0.116885	+	0.993145i	$0.462709\pi$
$308$	−26194.9	−4.84609
$309$	−8447.98	−1.55530
$310$	19825.1	3.63222
$311$	−1316.02	−0.239950	−0.119975	−	0.992777i	$-0.538281\pi$
$311$	−1316.02	−0.239950	−0.119975	+	0.992777i	$0.538281\pi$
$312$	−3865.15	−0.701350
$313$	−3137.09	−0.566514	−0.283257	−	0.959044i	$-0.591415\pi$
$313$	−3137.09	−0.566514	−0.283257	+	0.959044i	$0.591415\pi$
$314$	6120.24	1.09995
$315$	−6137.14	−1.09774
$316$	−24393.6	−4.34256
$317$	571.518	0.101261	0.0506303	−	0.998717i	$-0.483877\pi$
$317$	571.518	0.101261	0.0506303	+	0.998717i	$0.483877\pi$
$318$	5027.50	0.886566
$319$	−5410.92	−0.949696
$320$	−3481.96	−0.608273
$321$	−9624.99	−1.67356
$322$	−7656.07	−1.32502
$323$	118.825	0.0204694
$324$	−16846.0	−2.88854
$325$	1894.41	0.323332
$326$	7804.01	1.32584
$327$	2278.57	0.385337
$328$	7817.62	1.31602
$329$	8838.80	1.48115
$330$	−35015.8	−5.84109
$331$	2257.30	0.374841	0.187420	−	0.982280i	$-0.439987\pi$
$331$	2257.30	0.374841	0.187420	+	0.982280i	$0.439987\pi$
$332$	15087.7	2.49412
$333$	909.194	0.149620
$334$	17284.0	2.83155
$335$	−17601.7	−2.87069
$336$	19954.8	3.23995
$337$	−9222.66	−1.49077	−0.745386	−	0.666633i	$-0.767735\pi$
$337$	−9222.66	−1.49077	−0.745386	+	0.666633i	$0.767735\pi$
$338$	10703.2	1.72242
$339$	−5102.66	−0.817517
$340$	1144.87	0.182615
$341$	−13494.7	−2.14304
$342$	2610.39	0.412730
$343$	−3315.02	−0.521849
$344$	0	0
$345$	−7150.24	−1.11582
$346$	−8525.54	−1.32467
$347$	−8840.48	−1.36767	−0.683835	−	0.729636i	$-0.739689\pi$
$347$	−8840.48	−1.36767	−0.683835	+	0.729636i	$0.739689\pi$
$348$	10771.4	1.65922
$349$	−1431.64	−0.219581	−0.109791	−	0.993955i	$-0.535018\pi$
$349$	−1431.64	−0.219581	−0.109791	+	0.993955i	$0.535018\pi$
$350$	−20803.7	−3.17715
$351$	−836.222	−0.127163
$352$	14749.0	2.23331
$353$	−8308.06	−1.25267	−0.626336	−	0.779553i	$-0.715446\pi$
$353$	−8308.06	−1.25267	−0.626336	+	0.779553i	$0.715446\pi$
$354$	−6942.37	−1.04232
$355$	239.032	0.0357367
$356$	6502.56	0.968076
$357$	−541.816	−0.0803247
$358$	−6873.72	−1.01477
$359$	11642.2	1.71157	0.855786	−	0.517330i	$-0.173074\pi$
$359$	11642.2	1.71157	0.855786	+	0.517330i	$0.173074\pi$
$360$	14303.2	2.09401
$361$	−5754.08	−0.838909
$362$	−14280.6	−2.07340
$363$	15184.2	2.19549
$364$	4733.02	0.681532
$365$	7099.27	1.01806
$366$	−16293.8	−2.32702
$367$	4373.93	0.622119	0.311059	−	0.950391i	$-0.399316\pi$
$367$	4373.93	0.622119	0.311059	+	0.950391i	$0.399316\pi$
$368$	8388.55	1.18827
$369$	−2192.25	−0.309279
$370$	5307.14	0.745689
$371$	−3501.09	−0.489940
$372$	26863.7	3.74413
$373$	−7970.63	−1.10644	−0.553222	−	0.833034i	$-0.686602\pi$
$373$	−7970.63	−1.10644	−0.553222	+	0.833034i	$0.686602\pi$
$374$	−1115.41	−0.154215
$375$	−5401.59	−0.743832
$376$	−20599.6	−2.82538
$377$	977.669	0.133561
$378$	9183.07	1.24954
$379$	−2020.56	−0.273850	−0.136925	−	0.990581i	$-0.543722\pi$
$379$	−2020.56	−0.273850	−0.136925	+	0.990581i	$0.543722\pi$
$380$	10645.8	1.43715
$381$	−4754.66	−0.639340
$382$	420.000	0.0562540
$383$	−11625.0	−1.55094	−0.775468	−	0.631386i	$-0.782486\pi$
$383$	−11625.0	−1.55094	−0.775468	+	0.631386i	$0.782486\pi$
$384$	5910.26	0.785434
$385$	24384.7	3.22794
$386$	−17030.5	−2.24568
$387$	0	0
$388$	−22017.6	−2.88086
$389$	−6212.26	−0.809702	−0.404851	−	0.914383i	$-0.632677\pi$
$389$	−6212.26	−0.809702	−0.404851	+	0.914383i	$0.632677\pi$
$390$	6326.82	0.821464
$391$	−227.767	−0.0294595
$392$	−10916.4	−1.40653
$393$	−16005.0	−2.05431
$394$	−11551.9	−1.47710
$395$	22707.8	2.89254
$396$	−17119.8	−2.17248
$397$	−1647.75	−0.208307	−0.104154	−	0.994561i	$-0.533213\pi$
$397$	−1647.75	−0.208307	−0.104154	+	0.994561i	$0.533213\pi$
$398$	−20992.8	−2.64391
$399$	−5038.18	−0.632141
$400$	22794.0	2.84925
$401$	6893.37	0.858450	0.429225	−	0.903198i	$-0.358787\pi$
$401$	6893.37	0.858450	0.429225	+	0.903198i	$0.358787\pi$
$402$	−34137.8	−4.23543
$403$	2438.28	0.301388
$404$	8699.46	1.07132
$405$	15681.8	1.92404
$406$	−10736.4	−1.31241
$407$	−3612.49	−0.439962
$408$	1262.75	0.153224
$409$	2962.23	0.358125	0.179062	−	0.983838i	$-0.442694\pi$
$409$	2962.23	0.358125	0.179062	+	0.983838i	$0.442694\pi$
$410$	−12796.6	−1.54141
$411$	−15498.5	−1.86006
$412$	−24109.6	−2.88300
$413$	4834.59	0.576016
$414$	−5003.65	−0.594000
$415$	−14045.0	−1.66131
$416$	−2664.92	−0.314083
$417$	−20062.6	−2.35604
$418$	−10371.8	−1.21365
$419$	5520.25	0.643633	0.321816	−	0.946802i	$-0.395707\pi$
$419$	5520.25	0.643633	0.321816	+	0.946802i	$0.395707\pi$
$420$	−48542.2	−5.63957
$421$	−11017.9	−1.27548	−0.637742	−	0.770250i	$-0.720132\pi$
$421$	−11017.9	−1.27548	−0.637742	+	0.770250i	$0.720132\pi$
$422$	21989.0	2.53651
$423$	5776.63	0.663994
$424$	8159.61	0.934589
$425$	−618.906	−0.0706385
$426$	463.595	0.0527260
$427$	11346.8	1.28597
$428$	−27468.7	−3.10222
$429$	−4306.58	−0.484671
$430$	0	0
$431$	8825.55	0.986338	0.493169	−	0.869934i	$-0.335838\pi$
$431$	8825.55	0.986338	0.493169	+	0.869934i	$0.335838\pi$
$432$	−10061.6	−1.12058
$433$	−4570.88	−0.507303	−0.253652	−	0.967296i	$-0.581632\pi$
$433$	−4570.88	−0.507303	−0.253652	+	0.967296i	$0.581632\pi$
$434$	−26776.2	−2.96152
$435$	−10027.0	−1.10520
$436$	6502.80	0.714283
$437$	−2117.93	−0.231841
$438$	13768.8	1.50205
$439$	−3059.73	−0.332649	−0.166325	−	0.986071i	$-0.553190\pi$
$439$	−3059.73	−0.332649	−0.166325	+	0.986071i	$0.553190\pi$
$440$	−56830.6	−6.15749
$441$	3061.21	0.330549
$442$	201.537	0.0216881
$443$	−2764.55	−0.296496	−0.148248	−	0.988950i	$-0.547363\pi$
$443$	−2764.55	−0.296496	−0.148248	+	0.988950i	$0.547363\pi$
$444$	7191.35	0.768663
$445$	−6053.18	−0.644828
$446$	−4582.49	−0.486518
$447$	9395.29	0.994143
$448$	4702.82	0.495954
$449$	−7567.33	−0.795377	−0.397688	−	0.917521i	$-0.630188\pi$
$449$	−7567.33	−0.795377	−0.397688	+	0.917521i	$0.630188\pi$
$450$	−13596.3	−1.42430
$451$	8710.46	0.909444
$452$	−14562.4	−1.51540
$453$	5367.65	0.556720
$454$	1197.38	0.123780
$455$	−4405.93	−0.453963
$456$	11741.9	1.20585
$457$	8079.05	0.826963	0.413482	−	0.910512i	$-0.364313\pi$
$457$	8079.05	0.826963	0.413482	+	0.910512i	$0.364313\pi$
$458$	−4388.94	−0.447777
$459$	273.195	0.0277813
$460$	−20406.0	−2.06834
$461$	14466.4	1.46154	0.730770	−	0.682624i	$-0.239161\pi$
$461$	14466.4	1.46154	0.730770	+	0.682624i	$0.239161\pi$
$462$	47293.2	4.76251
$463$	7966.79	0.799672	0.399836	−	0.916587i	$-0.369067\pi$
$463$	7966.79	0.799672	0.399836	+	0.916587i	$0.369067\pi$
$464$	11763.6	1.17696
$465$	−25007.2	−2.49393
$466$	−10735.9	−1.06723
$467$	−6737.29	−0.667590	−0.333795	−	0.942646i	$-0.608329\pi$
$467$	−6737.29	−0.667590	−0.333795	+	0.942646i	$0.608329\pi$
$468$	3093.29	0.305528
$469$	23773.2	2.34061
$470$	33719.3	3.30927
$471$	−7720.00	−0.755242
$472$	−11267.4	−1.09878
$473$	0	0
$474$	44041.0	4.26766
$475$	−5755.02	−0.555912
$476$	−1546.28	−0.148895
$477$	−2288.15	−0.219638
$478$	−5085.44	−0.486616
$479$	−18115.7	−1.72803	−0.864014	−	0.503468i	$-0.832057\pi$
$479$	−18115.7	−1.72803	−0.864014	+	0.503468i	$0.832057\pi$
$480$	27331.6	2.59899
$481$	652.722	0.0618743
$482$	76.6548	0.00724384
$483$	9657.28	0.909776
$484$	43334.1	4.06969
$485$	20496.0	1.91891
$486$	19782.4	1.84639
$487$	−14966.4	−1.39260	−0.696298	−	0.717753i	$-0.745171\pi$
$487$	−14966.4	−1.39260	−0.696298	+	0.717753i	$0.745171\pi$
$488$	−26444.7	−2.45307
$489$	−9843.90	−0.910341
$490$	17868.8	1.64741
$491$	15713.8	1.44430	0.722151	−	0.691736i	$-0.243154\pi$
$491$	15713.8	1.44430	0.722151	+	0.691736i	$0.243154\pi$
$492$	−17339.8	−1.58890
$493$	−319.406	−0.0291791
$494$	1874.03	0.170682
$495$	15936.7	1.44707
$496$	29338.0	2.65588
$497$	−322.843	−0.0291378
$498$	−27239.9	−2.45110
$499$	15391.5	1.38080	0.690398	−	0.723430i	$-0.257435\pi$
$499$	15391.5	1.38080	0.690398	+	0.723430i	$0.257435\pi$
$500$	−15415.6	−1.37881
$501$	−21801.8	−1.94418
$502$	15598.1	1.38680
$503$	12150.9	1.07710	0.538548	−	0.842595i	$-0.318973\pi$
$503$	12150.9	1.07710	0.538548	+	0.842595i	$0.318973\pi$
$504$	−19318.2	−1.70734
$505$	−8098.25	−0.713599
$506$	19881.0	1.74667
$507$	−13500.9	−1.18263
$508$	−13569.3	−1.18512
$509$	16646.6	1.44960	0.724799	−	0.688960i	$-0.241933\pi$
$509$	16646.6	1.44960	0.724799	+	0.688960i	$0.241933\pi$
$510$	−2066.98	−0.179466
$511$	−9588.44	−0.830074
$512$	25179.7	2.17343
$513$	2540.35	0.218634
$514$	29678.1	2.54678
$515$	22443.5	1.92034
$516$	0	0
$517$	−22952.3	−1.95249
$518$	−7167.94	−0.607995
$519$	10754.0	0.909536
$520$	10268.4	0.865961
$521$	−16675.9	−1.40227	−0.701135	−	0.713028i	$-0.747323\pi$
$521$	−16675.9	−1.40227	−0.701135	+	0.713028i	$0.747323\pi$
$522$	−7016.81	−0.588347
$523$	−14326.5	−1.19781	−0.598903	−	0.800821i	$-0.704397\pi$
$523$	−14326.5	−1.19781	−0.598903	+	0.800821i	$0.704397\pi$
$524$	−45676.5	−3.80799
$525$	26241.5	2.18148
$526$	13113.6	1.08704
$527$	−796.588	−0.0658443
$528$	−51817.9	−4.27099
$529$	−8107.30	−0.666335
$530$	−13356.4	−1.09465
$531$	3159.67	0.258226
$532$	−14378.4	−1.17177
$533$	−1573.84	−0.127900
$534$	−11739.9	−0.951380
$535$	25570.4	2.06636
$536$	−55405.6	−4.46485
$537$	8670.44	0.696754
$538$	−34820.1	−2.79034
$539$	−12163.1	−0.971987
$540$	24476.0	1.95052
$541$	−8159.73	−0.648455	−0.324228	−	0.945979i	$-0.605104\pi$
$541$	−8159.73	−0.648455	−0.324228	+	0.945979i	$0.605104\pi$
$542$	−12806.8	−1.01494
$543$	18013.4	1.42362
$544$	870.633	0.0686178
$545$	−6053.40	−0.475778
$546$	−8545.15	−0.669778
$547$	−6089.67	−0.476006	−0.238003	−	0.971264i	$-0.576493\pi$
$547$	−6089.67	−0.476006	−0.238003	+	0.971264i	$0.576493\pi$
$548$	−44231.2	−3.44793
$549$	7415.74	0.576496
$550$	54022.2	4.18821
$551$	−2970.06	−0.229635
$552$	−22507.2	−1.73545
$553$	−30669.7	−2.35842
$554$	−10769.8	−0.825926
$555$	−6694.37	−0.512000
$556$	−57256.5	−4.36729
$557$	4160.66	0.316504	0.158252	−	0.987399i	$-0.449414\pi$
$557$	4160.66	0.316504	0.158252	+	0.987399i	$0.449414\pi$
$558$	−17499.7	−1.32764
$559$	0	0
$560$	−53013.3	−4.00039
$561$	1406.97	0.105886
$562$	31168.7	2.33945
$563$	7731.24	0.578744	0.289372	−	0.957217i	$-0.406554\pi$
$563$	7731.24	0.578744	0.289372	+	0.957217i	$0.406554\pi$
$564$	45690.8	3.41122
$565$	13556.1	1.00939
$566$	3350.08	0.248788
$567$	−21180.2	−1.56875
$568$	752.414	0.0555820
$569$	9134.32	0.672989	0.336494	−	0.941685i	$-0.390759\pi$
$569$	9134.32	0.672989	0.336494	+	0.941685i	$0.390759\pi$
$570$	−19220.2	−1.41236
$571$	−10421.5	−0.763797	−0.381898	−	0.924204i	$-0.624730\pi$
$571$	−10421.5	−0.763797	−0.381898	+	0.924204i	$0.624730\pi$
$572$	−12290.5	−0.898414
$573$	−529.783	−0.0386248
$574$	17283.4	1.25678
$575$	11031.3	0.800067
$576$	3073.54	0.222334
$577$	−806.571	−0.0581941	−0.0290971	−	0.999577i	$-0.509263\pi$
$577$	−806.571	−0.0581941	−0.0290971	+	0.999577i	$0.509263\pi$
$578$	25248.5	1.81695
$579$	21482.2	1.54191
$580$	−28616.1	−2.04866
$581$	18969.6	1.35454
$582$	39751.2	2.83117
$583$	9091.50	0.645852
$584$	22346.7	1.58341
$585$	−2879.51	−0.203510
$586$	11192.3	0.788992
$587$	−11514.2	−0.809614	−0.404807	−	0.914402i	$-0.632661\pi$
$587$	−11514.2	−0.809614	−0.404807	+	0.914402i	$0.632661\pi$
$588$	24212.9	1.69817
$589$	−7407.23	−0.518183
$590$	18443.6	1.28696
$591$	14571.5	1.01420
$592$	7853.72	0.545246
$593$	−2702.41	−0.187141	−0.0935706	−	0.995613i	$-0.529828\pi$
$593$	−2702.41	−0.187141	−0.0935706	+	0.995613i	$0.529828\pi$
$594$	−23846.2	−1.64718
$595$	1439.42	0.0991775
$596$	26813.1	1.84280
$597$	26480.1	1.81534
$598$	−3592.18	−0.245644
$599$	−27360.1	−1.86628	−0.933142	−	0.359507i	$-0.882945\pi$
$599$	−27360.1	−1.86628	−0.933142	+	0.359507i	$0.882945\pi$
$600$	−61158.3	−4.16129
$601$	11506.8	0.780985	0.390492	−	0.920606i	$-0.372305\pi$
$601$	11506.8	0.780985	0.390492	+	0.920606i	$0.372305\pi$
$602$	0	0
$603$	15537.1	1.04929
$604$	15318.7	1.03197
$605$	−40339.4	−2.71079
$606$	−15706.3	−1.05285
$607$	−8189.12	−0.547588	−0.273794	−	0.961788i	$-0.588279\pi$
$607$	−8189.12	−0.547588	−0.273794	+	0.961788i	$0.588279\pi$
$608$	8095.75	0.540010
$609$	13542.8	0.901117
$610$	43287.1	2.87318
$611$	4147.12	0.274590
$612$	−1010.58	−0.0667488
$613$	−24210.1	−1.59517	−0.797583	−	0.603209i	$-0.793889\pi$
$613$	−24210.1	−1.59517	−0.797583	+	0.603209i	$0.793889\pi$
$614$	−6479.14	−0.425858
$615$	16141.5	1.05835
$616$	76756.8	5.02048
$617$	−8779.45	−0.572848	−0.286424	−	0.958103i	$-0.592467\pi$
$617$	−8779.45	−0.572848	−0.286424	+	0.958103i	$0.592467\pi$
$618$	43528.3	2.83328
$619$	−6089.56	−0.395412	−0.197706	−	0.980261i	$-0.563349\pi$
$619$	−6089.56	−0.395412	−0.197706	+	0.980261i	$0.563349\pi$
$620$	−71367.8	−4.62290
$621$	−4869.40	−0.314658
$622$	6780.79	0.437114
$623$	8175.57	0.525758
$624$	9362.69	0.600653
$625$	−7291.47	−0.466654
$626$	16163.9	1.03201
$627$	13082.9	0.833305
$628$	−22032.1	−1.39996
$629$	−213.245	−0.0135177
$630$	31621.7	1.99974
$631$	−13768.3	−0.868634	−0.434317	−	0.900760i	$-0.643010\pi$
$631$	−13768.3	−0.868634	−0.434317	+	0.900760i	$0.643010\pi$
$632$	71478.5	4.49883
$633$	−27736.6	−1.74160
$634$	−2944.75	−0.184465
$635$	12631.5	0.789398
$636$	−18098.3	−1.12837
$637$	2197.68	0.136696
$638$	27879.8	1.73005
$639$	−210.995	−0.0130623
$640$	−15701.6	−0.969780
$641$	−393.612	−0.0242539	−0.0121269	−	0.999926i	$-0.503860\pi$
$641$	−393.612	−0.0242539	−0.0121269	+	0.999926i	$0.503860\pi$
$642$	49592.9	3.04871
$643$	−29944.9	−1.83656	−0.918281	−	0.395929i	$-0.870423\pi$
$643$	−29944.9	−1.83656	−0.918281	+	0.395929i	$0.870423\pi$
$644$	27560.9	1.68641
$645$	0	0
$646$	−612.249	−0.0372889
$647$	8372.05	0.508716	0.254358	−	0.967110i	$-0.418136\pi$
$647$	8372.05	0.508716	0.254358	+	0.967110i	$0.418136\pi$
$648$	49362.3	2.99249
$649$	−12554.3	−0.759320
$650$	−9760.97	−0.589010
$651$	33775.3	2.03342
$652$	−28093.4	−1.68746
$653$	−7203.34	−0.431682	−0.215841	−	0.976429i	$-0.569249\pi$
$653$	−7203.34	−0.431682	−0.215841	+	0.976429i	$0.569249\pi$
$654$	−11740.4	−0.701964
$655$	42519.9	2.53647
$656$	−18936.9	−1.12708
$657$	−6266.56	−0.372119
$658$	−45542.0	−2.69820
$659$	−28054.8	−1.65836	−0.829182	−	0.558979i	$-0.811193\pi$
$659$	−28054.8	−1.65836	−0.829182	+	0.558979i	$0.811193\pi$
$660$	126053.	7.43423
$661$	−11582.1	−0.681531	−0.340766	−	0.940148i	$-0.610686\pi$
$661$	−11582.1	−0.681531	−0.340766	+	0.940148i	$0.610686\pi$
$662$	−11630.8	−0.682843
$663$	−254.217	−0.0148914
$664$	−44210.2	−2.58387
$665$	13384.8	0.780509
$666$	−4684.64	−0.272561
$667$	5693.07	0.330489
$668$	−62220.1	−3.60385
$669$	5780.31	0.334050
$670$	90692.9	5.22951
$671$	−29464.9	−1.69520
$672$	−36914.7	−2.11907
$673$	−9582.65	−0.548862	−0.274431	−	0.961607i	$-0.588489\pi$
$673$	−9582.65	−0.548862	−0.274431	+	0.961607i	$0.588489\pi$
$674$	47519.9	2.71572
$675$	−13231.5	−0.754492
$676$	−38530.1	−2.19220
$677$	−8247.63	−0.468216	−0.234108	−	0.972211i	$-0.575217\pi$
$677$	−8247.63	−0.468216	−0.234108	+	0.972211i	$0.575217\pi$
$678$	26291.5	1.48926
$679$	−27682.3	−1.56458
$680$	−3354.70	−0.189187
$681$	−1510.37	−0.0849890
$682$	69531.5	3.90395
$683$	23171.1	1.29812	0.649061	−	0.760736i	$-0.275162\pi$
$683$	23171.1	1.29812	0.649061	+	0.760736i	$0.275162\pi$
$684$	−9397.08	−0.525301
$685$	41174.4	2.29663
$686$	17080.7	0.950647
$687$	5536.16	0.307450
$688$	0	0
$689$	−1642.69	−0.0908297
$690$	36841.7	2.03267
$691$	−24019.8	−1.32237	−0.661185	−	0.750223i	$-0.729946\pi$
$691$	−24019.8	−1.32237	−0.661185	+	0.750223i	$0.729946\pi$
$692$	30690.8	1.68597
$693$	−21524.5	−1.17987
$694$	45550.7	2.49147
$695$	53299.6	2.90902
$696$	−31562.6	−1.71893
$697$	514.177	0.0279424
$698$	7376.54	0.400009
$699$	13542.1	0.732776
$700$	74890.6	4.04371
$701$	−756.339	−0.0407511	−0.0203755	−	0.999792i	$-0.506486\pi$
$701$	−756.339	−0.0407511	−0.0203755	+	0.999792i	$0.506486\pi$
$702$	4308.64	0.231651
$703$	−1982.90	−0.106382
$704$	−12212.1	−0.653779
$705$	−42533.2	−2.27219
$706$	42807.4	2.28198
$707$	10937.7	0.581830
$708$	24991.7	1.32662
$709$	−26580.1	−1.40795	−0.703975	−	0.710224i	$-0.748594\pi$
$709$	−26580.1	−1.40795	−0.703975	+	0.710224i	$0.748594\pi$
$710$	−1231.62	−0.0651011
$711$	−20044.3	−1.05727
$712$	−19053.9	−1.00291
$713$	14198.3	0.745767
$714$	2791.71	0.146327
$715$	11441.1	0.598426
$716$	24744.5	1.29154
$717$	6414.72	0.334118
$718$	−59986.8	−3.11795
$719$	14125.2	0.732656	0.366328	−	0.930486i	$-0.380615\pi$
$719$	14125.2	0.732656	0.366328	+	0.930486i	$0.380615\pi$
$720$	−34647.1	−1.79336
$721$	−30312.7	−1.56575
$722$	29648.0	1.52823
$723$	−96.6916	−0.00497372
$724$	51408.3	2.63891
$725$	15469.6	0.792453
$726$	−78236.8	−3.99951
$727$	19206.2	0.979805	0.489903	−	0.871777i	$-0.337032\pi$
$727$	19206.2	0.979805	0.489903	+	0.871777i	$0.337032\pi$
$728$	−13868.8	−0.706058
$729$	−431.393	−0.0219170
$730$	−36579.1	−1.85459
$731$	0	0
$732$	58655.5	2.96170
$733$	29831.4	1.50320	0.751602	−	0.659617i	$-0.229282\pi$
$733$	29831.4	1.50320	0.751602	+	0.659617i	$0.229282\pi$
$734$	−22536.8	−1.13331
$735$	−22539.6	−1.13114
$736$	−15518.1	−0.777181
$737$	−61733.4	−3.08545
$738$	11295.6	0.563411
$739$	−11720.8	−0.583431	−0.291715	−	0.956505i	$-0.594226\pi$
$739$	−11720.8	−0.583431	−0.291715	+	0.956505i	$0.594226\pi$
$740$	−19105.0	−0.949073
$741$	−2363.89	−0.117192
$742$	18039.4	0.892518
$743$	−4257.57	−0.210222	−0.105111	−	0.994460i	$-0.533520\pi$
$743$	−4257.57	−0.210222	−0.105111	+	0.994460i	$0.533520\pi$
$744$	−78716.3	−3.87887
$745$	−24960.1	−1.22747
$746$	41068.8	2.01560
$747$	12397.6	0.607236
$748$	4015.33	0.196277
$749$	−34535.9	−1.68480
$750$	27831.8	1.35503
$751$	16414.4	0.797564	0.398782	−	0.917046i	$-0.369433\pi$
$751$	16414.4	0.797564	0.398782	+	0.917046i	$0.369433\pi$
$752$	49899.2	2.41973
$753$	−19675.2	−0.952199
$754$	−5037.45	−0.243307
$755$	−14260.1	−0.687386
$756$	−33057.9	−1.59035
$757$	22813.5	1.09534	0.547668	−	0.836696i	$-0.315516\pi$
$757$	22813.5	1.09534	0.547668	+	0.836696i	$0.315516\pi$
$758$	10410.9	0.498869
$759$	−25077.7	−1.19929
$760$	−31194.4	−1.48887
$761$	−1695.44	−0.0807617	−0.0403808	−	0.999184i	$-0.512857\pi$
$761$	−1695.44	−0.0807617	−0.0403808	+	0.999184i	$0.512857\pi$
$762$	24498.5	1.16468
$763$	8175.86	0.387924
$764$	−1511.95	−0.0715972
$765$	940.741	0.0444609
$766$	59897.9	2.82532
$767$	2268.36	0.106787
$768$	−40937.9	−1.92346
$769$	21604.9	1.01313	0.506563	−	0.862203i	$-0.330916\pi$
$769$	21604.9	1.01313	0.506563	+	0.862203i	$0.330916\pi$
$770$	−125642.	−5.88030
$771$	−37435.6	−1.74865
$772$	61307.8	2.85818
$773$	−34435.4	−1.60227	−0.801136	−	0.598482i	$-0.795771\pi$
$773$	−34435.4	−1.60227	−0.801136	+	0.598482i	$0.795771\pi$
$774$	0	0
$775$	38580.9	1.78821
$776$	64516.1	2.98453
$777$	9041.57	0.417457
$778$	32008.8	1.47503
$779$	4781.18	0.219902
$780$	−22775.8	−1.04552
$781$	838.345	0.0384102
$782$	1173.57	0.0536660
$783$	−6828.54	−0.311663
$784$	26443.1	1.20459
$785$	20509.5	0.932502
$786$	82465.9	3.74232
$787$	−12352.5	−0.559490	−0.279745	−	0.960074i	$-0.590250\pi$
$787$	−12352.5	−0.559490	−0.279745	+	0.960074i	$0.590250\pi$
$788$	41585.4	1.87997
$789$	−16541.4	−0.746374
$790$	−117002.	−5.26931
$791$	−18309.1	−0.823006
$792$	50164.7	2.25066
$793$	5323.85	0.238406
$794$	8490.03	0.379471
$795$	16847.6	0.751601
$796$	75571.5	3.36503
$797$	29811.8	1.32495	0.662477	−	0.749083i	$-0.269505\pi$
$797$	29811.8	1.32495	0.662477	+	0.749083i	$0.269505\pi$
$798$	25959.3	1.15156
$799$	−1354.87	−0.0599897
$800$	−42167.0	−1.86354
$801$	5343.17	0.235695
$802$	−35518.2	−1.56383
$803$	24898.9	1.09423
$804$	122892.	5.39063
$805$	−25656.2	−1.12331
$806$	−12563.3	−0.549035
$807$	43921.7	1.91588
$808$	−25491.3	−1.10988
$809$	−1273.45	−0.0553424	−0.0276712	−	0.999617i	$-0.508809\pi$
$809$	−1273.45	−0.0553424	−0.0276712	+	0.999617i	$0.508809\pi$
$810$	−80800.6	−3.50499
$811$	−37825.5	−1.63777	−0.818886	−	0.573956i	$-0.805408\pi$
$811$	−37825.5	−1.63777	−0.818886	+	0.573956i	$0.805408\pi$
$812$	38649.6	1.67036
$813$	16154.3	0.696872
$814$	18613.4	0.801475
$815$	26152.0	1.12400
$816$	−3058.80	−0.131225
$817$	0	0
$818$	−15262.9	−0.652391
$819$	3889.14	0.165931
$820$	46066.1	1.96182
$821$	189.193	0.00804247	0.00402124	−	0.999992i	$-0.498720\pi$
$821$	189.193	0.00804247	0.00402124	+	0.999992i	$0.498720\pi$
$822$	79856.4	3.38846
$823$	26388.5	1.11767	0.558837	−	0.829278i	$-0.311248\pi$
$823$	26388.5	1.11767	0.558837	+	0.829278i	$0.311248\pi$
$824$	70646.4	2.98675
$825$	−68143.0	−2.87568
$826$	−24910.3	−1.04932
$827$	−37247.2	−1.56616	−0.783079	−	0.621923i	$-0.786352\pi$
$827$	−37247.2	−1.56616	−0.783079	+	0.621923i	$0.786352\pi$
$828$	18012.5	0.756012
$829$	18429.8	0.772128	0.386064	−	0.922472i	$-0.373834\pi$
$829$	18429.8	0.772128	0.386064	+	0.922472i	$0.373834\pi$
$830$	72367.2	3.02639
$831$	13584.9	0.567092
$832$	2206.54	0.0919445
$833$	−717.985	−0.0298640
$834$	103373.	4.29197
$835$	57920.2	2.40049
$836$	37337.3	1.54466
$837$	−17030.2	−0.703285
$838$	−28443.2	−1.17250
$839$	−24344.3	−1.00174	−0.500870	−	0.865523i	$-0.666986\pi$
$839$	−24344.3	−1.00174	−0.500870	+	0.865523i	$0.666986\pi$
$840$	142239.	5.84252
$841$	−16405.4	−0.672656
$842$	56769.8	2.32354
$843$	−39315.9	−1.60630
$844$	−79157.5	−3.22833
$845$	35867.4	1.46021
$846$	−29764.2	−1.20959
$847$	54483.3	2.21023
$848$	−19765.3	−0.800406
$849$	−4225.75	−0.170822
$850$	3188.92	0.128681
$851$	3800.87	0.153105
$852$	−1668.88	−0.0671068
$853$	14146.8	0.567852	0.283926	−	0.958846i	$-0.408363\pi$
$853$	14146.8	0.567852	0.283926	+	0.958846i	$0.408363\pi$
$854$	−58464.5	−2.34264
$855$	8747.66	0.349899
$856$	80489.1	3.21386
$857$	8682.05	0.346060	0.173030	−	0.984917i	$-0.444644\pi$
$857$	8682.05	0.346060	0.173030	+	0.984917i	$0.444644\pi$
$858$	22189.7	0.882919
$859$	25428.8	1.01003	0.505017	−	0.863110i	$-0.331486\pi$
$859$	25428.8	1.01003	0.505017	+	0.863110i	$0.331486\pi$
$860$	0	0
$861$	−21801.0	−0.862924
$862$	−45473.7	−1.79680
$863$	39552.6	1.56012	0.780062	−	0.625703i	$-0.215188\pi$
$863$	39552.6	1.56012	0.780062	+	0.625703i	$0.215188\pi$
$864$	18613.2	0.732909
$865$	−28569.9	−1.12301
$866$	23551.5	0.924148
$867$	−31848.1	−1.24754
$868$	96391.0	3.76927
$869$	79641.9	3.10894
$870$	51664.5	2.01332
$871$	11154.3	0.433924
$872$	−19054.6	−0.739988
$873$	−18091.9	−0.701395
$874$	10912.7	0.422342
$875$	−19381.8	−0.748826
$876$	−49565.9	−1.91173
$877$	1955.60	0.0752975	0.0376487	−	0.999291i	$-0.488013\pi$
$877$	1955.60	0.0752975	0.0376487	+	0.999291i	$0.488013\pi$
$878$	15765.3	0.605983
$879$	−14117.8	−0.541733
$880$	137663.	5.27343
$881$	3037.09	0.116143	0.0580716	−	0.998312i	$-0.481505\pi$
$881$	3037.09	0.116143	0.0580716	+	0.998312i	$0.481505\pi$
$882$	−15772.9	−0.602157
$883$	15702.3	0.598443	0.299222	−	0.954184i	$-0.403273\pi$
$883$	15702.3	0.598443	0.299222	+	0.954184i	$0.403273\pi$
$884$	−725.508	−0.0276035
$885$	−23264.5	−0.883648
$886$	14244.4	0.540123
$887$	−2964.83	−0.112231	−0.0561157	−	0.998424i	$-0.517872\pi$
$887$	−2964.83	−0.112231	−0.0561157	+	0.998424i	$0.517872\pi$
$888$	−21072.2	−0.796324
$889$	−17060.5	−0.643633
$890$	31189.1	1.17467
$891$	54999.9	2.06797
$892$	16496.4	0.619215
$893$	−12598.5	−0.472109
$894$	−48409.3	−1.81102
$895$	−23034.5	−0.860287
$896$	21206.9	0.790707
$897$	4531.14	0.168663
$898$	38990.8	1.44893
$899$	19910.9	0.738670
$900$	48945.0	1.81278
$901$	536.670	0.0198436
$902$	−44880.7	−1.65672
$903$	0	0
$904$	42671.1	1.56993
$905$	−47855.5	−1.75776
$906$	−27656.9	−1.01417
$907$	47065.4	1.72302	0.861511	−	0.507738i	$-0.169518\pi$
$907$	47065.4	1.72302	0.861511	+	0.507738i	$0.169518\pi$
$908$	−4310.43	−0.157540
$909$	7148.37	0.260832
$910$	22701.6	0.826979
$911$	40218.3	1.46267	0.731335	−	0.682018i	$-0.238898\pi$
$911$	40218.3	1.46267	0.731335	+	0.682018i	$0.238898\pi$
$912$	−28442.9	−1.03272
$913$	−49259.4	−1.78559
$914$	−41627.4	−1.50647
$915$	−54601.9	−1.97277
$916$	15799.6	0.569906
$917$	−57428.4	−2.06810
$918$	−1407.64	−0.0506090
$919$	−13015.3	−0.467175	−0.233587	−	0.972336i	$-0.575047\pi$
$919$	−13015.3	−0.467175	−0.233587	+	0.972336i	$0.575047\pi$
$920$	59794.0	2.14277
$921$	8172.72	0.292400
$922$	−74538.6	−2.66247
$923$	−151.476	−0.00540184
$924$	−170250.	−6.06147
$925$	10328.0	0.367117
$926$	−41049.0	−1.45675
$927$	−19811.0	−0.701917
$928$	−21761.6	−0.769784
$929$	17365.8	0.613298	0.306649	−	0.951823i	$-0.400792\pi$
$929$	17365.8	0.613298	0.306649	+	0.951823i	$0.400792\pi$
$930$	128850.	4.54317
$931$	−6676.32	−0.235024
$932$	38647.8	1.35832
$933$	−8553.22	−0.300128
$934$	34714.0	1.21614
$935$	−3737.84	−0.130738
$936$	−9063.98	−0.316523
$937$	41655.6	1.45232	0.726162	−	0.687523i	$-0.241302\pi$
$937$	41655.6	1.45232	0.726162	+	0.687523i	$0.241302\pi$
$938$	−122492.	−4.26386
$939$	−20389.0	−0.708593
$940$	−121385.	−4.21186
$941$	−11377.5	−0.394149	−0.197075	−	0.980388i	$-0.563144\pi$
$941$	−11377.5	−0.394149	−0.197075	+	0.980388i	$0.563144\pi$
$942$	39777.4	1.37582
$943$	−9164.66	−0.316482
$944$	27293.5	0.941027
$945$	30773.3	1.05932
$946$	0	0
$947$	39835.0	1.36691	0.683455	−	0.729992i	$-0.260476\pi$
$947$	39835.0	1.36691	0.683455	+	0.729992i	$0.260476\pi$
$948$	−158542.	−5.43165
$949$	−4498.84	−0.153887
$950$	29652.8	1.01270
$951$	3714.48	0.126656
$952$	4530.94	0.154253
$953$	−16468.3	−0.559769	−0.279885	−	0.960034i	$-0.590296\pi$
$953$	−16468.3	−0.559769	−0.279885	+	0.960034i	$0.590296\pi$
$954$	11789.7	0.400112
$955$	1407.46	0.0476903
$956$	18306.9	0.619340
$957$	−35167.3	−1.18788
$958$	93341.2	3.14793
$959$	−55611.2	−1.87255
$960$	−22630.4	−0.760826
$961$	19866.1	0.666848
$962$	−3363.16	−0.112716
$963$	−22571.1	−0.755289
$964$	−275.948	−0.00921958
$965$	−57070.9	−1.90381
$966$	−49759.3	−1.65733
$967$	55135.3	1.83354	0.916770	−	0.399416i	$-0.130787\pi$
$967$	55135.3	1.83354	0.916770	+	0.399416i	$0.130787\pi$
$968$	−126978.	−4.21615
$969$	772.284	0.0256030
$970$	−105606.	−3.49566
$971$	−13040.1	−0.430975	−0.215488	−	0.976507i	$-0.569134\pi$
$971$	−13040.1	−0.430975	−0.215488	+	0.976507i	$0.569134\pi$
$972$	−71214.0	−2.34999
$973$	−71987.6	−2.37186
$974$	77114.7	2.53687
$975$	12312.4	0.404423
$976$	64058.0	2.10087
$977$	−45144.9	−1.47831	−0.739157	−	0.673534i	$-0.764776\pi$
$977$	−45144.9	−1.47831	−0.739157	+	0.673534i	$0.764776\pi$
$978$	50720.8	1.65836
$979$	−21230.0	−0.693068
$980$	−64325.6	−2.09674
$981$	5343.37	0.173905
$982$	−80965.4	−2.63107
$983$	26573.6	0.862225	0.431113	−	0.902298i	$-0.358121\pi$
$983$	26573.6	0.862225	0.431113	+	0.902298i	$0.358121\pi$
$984$	50809.3	1.64608
$985$	−38711.5	−1.25223
$986$	1645.74	0.0531553
$987$	57446.3	1.85262
$988$	−6746.28	−0.217235
$989$	0	0
$990$	−82114.0	−2.63612
$991$	−20658.0	−0.662183	−0.331092	−	0.943599i	$-0.607417\pi$
$991$	−20658.0	−0.662183	−0.331092	+	0.943599i	$0.607417\pi$
$992$	−54272.8	−1.73706
$993$	14670.9	0.468850
$994$	1663.45	0.0530800
$995$	−70348.9	−2.24142
$996$	98060.1	3.11963
$997$	27284.2	0.866701	0.433350	−	0.901226i	$-0.357331\pi$
$997$	27284.2	0.866701	0.433350	+	0.901226i	$0.357331\pi$
$998$	−79304.8	−2.51538
$999$	−4558.95	−0.144383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.c.1.1		6
43.42	odd	2		43.4.a.b.1.6	✓	6
129.128	even	2		387.4.a.h.1.1		6
172.171	even	2		688.4.a.i.1.5		6
215.214	odd	2		1075.4.a.b.1.1		6
301.300	even	2		2107.4.a.c.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.6	✓	6	43.42	odd	2
387.4.a.h.1.1		6	129.128	even	2
688.4.a.i.1.5		6	172.171	even	2
1075.4.a.b.1.1		6	215.214	odd	2
1849.4.a.c.1.1		6	1.1	even	1	trivial
2107.4.a.c.1.6		6	301.300	even	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-5.15251 q^{2} +6.49933 q^{3} +18.5484 q^{4} -17.2665 q^{5} -33.4879 q^{6} +23.3206 q^{7} -54.3507 q^{8} +15.2413 q^{9} +O(q^{10})\)	\(q-5.15251 q^{2} +6.49933 q^{3} +18.5484 q^{4} -17.2665 q^{5} -33.4879 q^{6} +23.3206 q^{7} -54.3507 q^{8} +15.2413 q^{9} +88.9661 q^{10} -60.5580 q^{11} +120.552 q^{12} +10.9419 q^{13} -120.160 q^{14} -112.221 q^{15} +131.656 q^{16} -3.57473 q^{17} -78.5308 q^{18} -33.2403 q^{19} -320.267 q^{20} +151.568 q^{21} +312.026 q^{22} +63.7158 q^{23} -353.243 q^{24} +173.134 q^{25} -56.3783 q^{26} -76.4239 q^{27} +432.560 q^{28} +89.3510 q^{29} +578.220 q^{30} +222.839 q^{31} -243.552 q^{32} -393.586 q^{33} +18.4188 q^{34} -402.666 q^{35} +282.701 q^{36} +59.6535 q^{37} +171.271 q^{38} +71.1150 q^{39} +938.449 q^{40} -143.837 q^{41} -780.957 q^{42} -1123.25 q^{44} -263.164 q^{45} -328.296 q^{46} +379.013 q^{47} +855.674 q^{48} +200.850 q^{49} -892.073 q^{50} -23.2334 q^{51} +202.955 q^{52} -150.129 q^{53} +393.775 q^{54} +1045.63 q^{55} -1267.49 q^{56} -216.040 q^{57} -460.382 q^{58} +207.310 q^{59} -2081.52 q^{60} +486.557 q^{61} -1148.18 q^{62} +355.435 q^{63} +201.659 q^{64} -188.929 q^{65} +2027.96 q^{66} +1019.41 q^{67} -66.3055 q^{68} +414.110 q^{69} +2074.74 q^{70} -13.8437 q^{71} -828.374 q^{72} -411.158 q^{73} -307.365 q^{74} +1125.25 q^{75} -616.555 q^{76} -1412.25 q^{77} -366.421 q^{78} -1315.13 q^{79} -2273.24 q^{80} -908.218 q^{81} +741.120 q^{82} +813.425 q^{83} +2811.35 q^{84} +61.7233 q^{85} +580.721 q^{87} +3291.37 q^{88} +350.573 q^{89} +1355.96 q^{90} +255.171 q^{91} +1181.83 q^{92} +1448.30 q^{93} -1952.87 q^{94} +573.946 q^{95} -1582.92 q^{96} -1187.03 q^{97} -1034.88 q^{98} -922.981 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 7 q^{3} + 22 q^{4} - 43 q^{5} - 3 q^{6} - 8 q^{7} - 54 q^{8} + 81 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 - 7 * q^3 + 22 * q^4 - 43 * q^5 - 3 * q^6 - 8 * q^7 - 54 * q^8 + 81 * q^9	\( 6 q - 6 q^{2} - 7 q^{3} + 22 q^{4} - 43 q^{5} - 3 q^{6} - 8 q^{7} - 54 q^{8} + 81 q^{9} + 57 q^{10} - 28 q^{11} + 157 q^{12} + 56 q^{13} - 184 q^{14} - 124 q^{15} - 54 q^{16} + 19 q^{17} + 81 q^{18} + 75 q^{19} - 135 q^{20} - 18 q^{21} + 504 q^{22} + 131 q^{23} - 567 q^{24} + 105 q^{25} - 44 q^{26} - 238 q^{27} + 404 q^{28} - 515 q^{29} + 396 q^{30} + 237 q^{31} - 558 q^{32} - 540 q^{33} + 107 q^{34} + 198 q^{35} + 73 q^{36} - 269 q^{37} + 527 q^{38} - 290 q^{39} + 613 q^{40} + 471 q^{41} - 362 q^{42} - 428 q^{44} - 334 q^{45} + 67 q^{46} + 415 q^{47} + 989 q^{48} + 350 q^{49} - 1335 q^{50} + 1241 q^{51} - 8 q^{52} + 450 q^{53} + 402 q^{54} + 1732 q^{55} - 780 q^{56} - 1000 q^{57} - 1055 q^{58} + 356 q^{59} - 2732 q^{60} + 1328 q^{61} - 1603 q^{62} + 2290 q^{63} + 466 q^{64} + 62 q^{65} + 156 q^{66} - 632 q^{67} + 571 q^{68} + 1130 q^{69} + 1902 q^{70} + 144 q^{71} - 567 q^{72} - 864 q^{73} + 1207 q^{74} + 2494 q^{75} - 1005 q^{76} - 2660 q^{77} + 2222 q^{78} - 1613 q^{79} - 2399 q^{80} - 102 q^{81} - 1673 q^{82} - 682 q^{83} + 3758 q^{84} - 84 q^{85} + 449 q^{87} + 608 q^{88} - 3378 q^{89} + 930 q^{90} + 3900 q^{91} + 3491 q^{92} - 1879 q^{93} - 3197 q^{94} - 79 q^{95} - 591 q^{96} - 55 q^{97} - 2398 q^{98} - 1612 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 7 * q^3 + 22 * q^4 - 43 * q^5 - 3 * q^6 - 8 * q^7 - 54 * q^8 + 81 * q^9 + 57 * q^10 - 28 * q^11 + 157 * q^12 + 56 * q^13 - 184 * q^14 - 124 * q^15 - 54 * q^16 + 19 * q^17 + 81 * q^18 + 75 * q^19 - 135 * q^20 - 18 * q^21 + 504 * q^22 + 131 * q^23 - 567 * q^24 + 105 * q^25 - 44 * q^26 - 238 * q^27 + 404 * q^28 - 515 * q^29 + 396 * q^30 + 237 * q^31 - 558 * q^32 - 540 * q^33 + 107 * q^34 + 198 * q^35 + 73 * q^36 - 269 * q^37 + 527 * q^38 - 290 * q^39 + 613 * q^40 + 471 * q^41 - 362 * q^42 - 428 * q^44 - 334 * q^45 + 67 * q^46 + 415 * q^47 + 989 * q^48 + 350 * q^49 - 1335 * q^50 + 1241 * q^51 - 8 * q^52 + 450 * q^53 + 402 * q^54 + 1732 * q^55 - 780 * q^56 - 1000 * q^57 - 1055 * q^58 + 356 * q^59 - 2732 * q^60 + 1328 * q^61 - 1603 * q^62 + 2290 * q^63 + 466 * q^64 + 62 * q^65 + 156 * q^66 - 632 * q^67 + 571 * q^68 + 1130 * q^69 + 1902 * q^70 + 144 * q^71 - 567 * q^72 - 864 * q^73 + 1207 * q^74 + 2494 * q^75 - 1005 * q^76 - 2660 * q^77 + 2222 * q^78 - 1613 * q^79 - 2399 * q^80 - 102 * q^81 - 1673 * q^82 - 682 * q^83 + 3758 * q^84 - 84 * q^85 + 449 * q^87 + 608 * q^88 - 3378 * q^89 + 930 * q^90 + 3900 * q^91 + 3491 * q^92 - 1879 * q^93 - 3197 * q^94 - 79 * q^95 - 591 * q^96 - 55 * q^97 - 2398 * q^98 - 1612 * q^99

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