LMFDB - Embedded newform 1682.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,4,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, 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("1682.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1682.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:	$99.2412126297$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.19816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 42x - 54 $ x^3 - x^2 - 42*x - 54
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.39712$ of defining polynomial
Character	$\chi$	$=$	1682.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -2.39712 q^{3} +4.00000 q^{4} -3.28077 q^{5} +4.79424 q^{6} +33.9461 q^{7} -8.00000 q^{8} -21.2538 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -2.39712 q^{3} +4.00000 q^{4} -3.28077 q^{5} +4.79424 q^{6} +33.9461 q^{7} -8.00000 q^{8} -21.2538 q^{9} +6.56153 q^{10} -14.5759 q^{11} -9.58848 q^{12} -86.5287 q^{13} -67.8922 q^{14} +7.86439 q^{15} +16.0000 q^{16} -102.550 q^{17} +42.5076 q^{18} +105.688 q^{19} -13.1231 q^{20} -81.3729 q^{21} +29.1519 q^{22} -135.456 q^{23} +19.1770 q^{24} -114.237 q^{25} +173.057 q^{26} +115.670 q^{27} +135.784 q^{28} -15.7288 q^{30} -223.883 q^{31} -32.0000 q^{32} +34.9403 q^{33} +205.100 q^{34} -111.369 q^{35} -85.0153 q^{36} +239.723 q^{37} -211.377 q^{38} +207.420 q^{39} +26.2461 q^{40} -219.331 q^{41} +162.746 q^{42} -18.9838 q^{43} -58.3037 q^{44} +69.7288 q^{45} +270.911 q^{46} -147.922 q^{47} -38.3539 q^{48} +809.338 q^{49} +228.473 q^{50} +245.824 q^{51} -346.115 q^{52} +613.202 q^{53} -231.340 q^{54} +47.8202 q^{55} -271.569 q^{56} -253.348 q^{57} +184.206 q^{59} +31.4576 q^{60} +13.6914 q^{61} +447.766 q^{62} -721.484 q^{63} +64.0000 q^{64} +283.880 q^{65} -69.8805 q^{66} +328.733 q^{67} -410.199 q^{68} +324.703 q^{69} +222.738 q^{70} +5.15186 q^{71} +170.031 q^{72} +428.481 q^{73} -479.445 q^{74} +273.839 q^{75} +422.753 q^{76} -494.796 q^{77} -414.840 q^{78} +392.819 q^{79} -52.4922 q^{80} +296.578 q^{81} +438.661 q^{82} -454.172 q^{83} -325.492 q^{84} +336.442 q^{85} +37.9675 q^{86} +116.607 q^{88} +811.929 q^{89} -139.458 q^{90} -2937.31 q^{91} -541.822 q^{92} +536.675 q^{93} +295.844 q^{94} -346.738 q^{95} +76.7078 q^{96} +11.3513 q^{97} -1618.68 q^{98} +309.794 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9	\( 3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9} - 40 q^{10} - 10 q^{11} - 8 q^{12} - 4 q^{13} - 48 q^{14} + 130 q^{15} + 48 q^{16} + 66 q^{17} - 10 q^{18} + 164 q^{19} + 80 q^{20} + 88 q^{21} + 20 q^{22} - 204 q^{23} + 16 q^{24} + 79 q^{25} + 8 q^{26} + 142 q^{27} + 96 q^{28} - 260 q^{30} + 86 q^{31} - 96 q^{32} - 130 q^{33} - 132 q^{34} + 24 q^{35} + 20 q^{36} + 42 q^{37} - 328 q^{38} + 394 q^{39} - 160 q^{40} - 562 q^{41} - 176 q^{42} - 18 q^{43} - 40 q^{44} + 422 q^{45} + 408 q^{46} - 654 q^{47} - 32 q^{48} + 539 q^{49} - 158 q^{50} + 556 q^{51} - 16 q^{52} + 712 q^{53} - 284 q^{54} - 142 q^{55} - 192 q^{56} + 828 q^{57} + 184 q^{59} + 520 q^{60} - 322 q^{61} - 172 q^{62} - 784 q^{63} + 192 q^{64} + 1494 q^{65} + 260 q^{66} - 228 q^{67} + 264 q^{68} - 684 q^{69} - 48 q^{70} - 52 q^{71} - 40 q^{72} + 494 q^{73} - 84 q^{74} + 3048 q^{75} + 656 q^{76} - 872 q^{77} - 788 q^{78} + 2110 q^{79} + 320 q^{80} - 1513 q^{81} + 1124 q^{82} - 288 q^{83} + 352 q^{84} + 2704 q^{85} + 36 q^{86} + 80 q^{88} - 914 q^{89} - 844 q^{90} - 2984 q^{91} - 816 q^{92} - 62 q^{93} + 1308 q^{94} + 1900 q^{95} + 64 q^{96} - 218 q^{97} - 1078 q^{98} + 304 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9 - 40 * q^10 - 10 * q^11 - 8 * q^12 - 4 * q^13 - 48 * q^14 + 130 * q^15 + 48 * q^16 + 66 * q^17 - 10 * q^18 + 164 * q^19 + 80 * q^20 + 88 * q^21 + 20 * q^22 - 204 * q^23 + 16 * q^24 + 79 * q^25 + 8 * q^26 + 142 * q^27 + 96 * q^28 - 260 * q^30 + 86 * q^31 - 96 * q^32 - 130 * q^33 - 132 * q^34 + 24 * q^35 + 20 * q^36 + 42 * q^37 - 328 * q^38 + 394 * q^39 - 160 * q^40 - 562 * q^41 - 176 * q^42 - 18 * q^43 - 40 * q^44 + 422 * q^45 + 408 * q^46 - 654 * q^47 - 32 * q^48 + 539 * q^49 - 158 * q^50 + 556 * q^51 - 16 * q^52 + 712 * q^53 - 284 * q^54 - 142 * q^55 - 192 * q^56 + 828 * q^57 + 184 * q^59 + 520 * q^60 - 322 * q^61 - 172 * q^62 - 784 * q^63 + 192 * q^64 + 1494 * q^65 + 260 * q^66 - 228 * q^67 + 264 * q^68 - 684 * q^69 - 48 * q^70 - 52 * q^71 - 40 * q^72 + 494 * q^73 - 84 * q^74 + 3048 * q^75 + 656 * q^76 - 872 * q^77 - 788 * q^78 + 2110 * q^79 + 320 * q^80 - 1513 * q^81 + 1124 * q^82 - 288 * q^83 + 352 * q^84 + 2704 * q^85 + 36 * q^86 + 80 * q^88 - 914 * q^89 - 844 * q^90 - 2984 * q^91 - 816 * q^92 - 62 * q^93 + 1308 * q^94 + 1900 * q^95 + 64 * q^96 - 218 * q^97 - 1078 * q^98 + 304 * 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
$3$	−2.39712	−0.461326	−0.230663	−	0.973034i	$-0.574089\pi$
$3$	−2.39712	−0.461326	−0.230663	+	0.973034i	$0.574089\pi$
$4$	4.00000	0.500000
$5$	−3.28077	−0.293441	−0.146720	−	0.989178i	$-0.546872\pi$
$5$	−3.28077	−0.293441	−0.146720	+	0.989178i	$0.546872\pi$
$6$	4.79424	0.326207
$7$	33.9461	1.83292	0.916459	−	0.400130i	$-0.131035\pi$
$7$	33.9461	1.83292	0.916459	+	0.400130i	$0.131035\pi$
$8$	−8.00000	−0.353553
$9$	−21.2538	−0.787178
$10$	6.56153	0.207494
$11$	−14.5759	−0.399528	−0.199764	−	0.979844i	$-0.564018\pi$
$11$	−14.5759	−0.399528	−0.199764	+	0.979844i	$0.564018\pi$
$12$	−9.58848	−0.230663
$13$	−86.5287	−1.84606	−0.923029	−	0.384730i	$-0.874294\pi$
$13$	−86.5287	−1.84606	−0.923029	+	0.384730i	$0.874294\pi$
$14$	−67.8922	−1.29607
$15$	7.86439	0.135372
$16$	16.0000	0.250000
$17$	−102.550	−1.46306	−0.731529	−	0.681810i	$-0.761193\pi$
$17$	−102.550	−1.46306	−0.731529	+	0.681810i	$0.761193\pi$
$18$	42.5076	0.556619
$19$	105.688	1.27613	0.638067	−	0.769981i	$-0.279734\pi$
$19$	105.688	1.27613	0.638067	+	0.769981i	$0.279734\pi$
$20$	−13.1231	−0.146720
$21$	−81.3729	−0.845572
$22$	29.1519	0.282509
$23$	−135.456	−1.22802	−0.614010	−	0.789299i	$-0.710444\pi$
$23$	−135.456	−1.22802	−0.614010	+	0.789299i	$0.710444\pi$
$24$	19.1770	0.163103
$25$	−114.237	−0.913893
$26$	173.057	1.30536
$27$	115.670	0.824472
$28$	135.784	0.916459
$29$	0	0
$29$	0	0
$30$	−15.7288	−0.0957223
$31$	−223.883	−1.29712	−0.648558	−	0.761165i	$-0.724628\pi$
$31$	−223.883	−1.29712	−0.648558	+	0.761165i	$0.724628\pi$
$32$	−32.0000	−0.176777
$33$	34.9403	0.184313
$34$	205.100	1.03454
$35$	−111.369	−0.537852
$36$	−85.0153	−0.393589
$37$	239.723	1.06514	0.532569	−	0.846386i	$-0.321227\pi$
$37$	239.723	1.06514	0.532569	+	0.846386i	$0.321227\pi$
$38$	−211.377	−0.902363
$39$	207.420	0.851634
$40$	26.2461	0.103747
$41$	−219.331	−0.835456	−0.417728	−	0.908572i	$-0.637173\pi$
$41$	−219.331	−0.835456	−0.417728	+	0.908572i	$0.637173\pi$
$42$	162.746	0.597910
$43$	−18.9838	−0.0673255	−0.0336627	−	0.999433i	$-0.510717\pi$
$43$	−18.9838	−0.0673255	−0.0336627	+	0.999433i	$0.510717\pi$
$44$	−58.3037	−0.199764
$45$	69.7288	0.230990
$46$	270.911	0.868341
$47$	−147.922	−0.459077	−0.229539	−	0.973300i	$-0.573722\pi$
$47$	−147.922	−0.459077	−0.229539	+	0.973300i	$0.573722\pi$
$48$	−38.3539	−0.115331
$49$	809.338	2.35959
$50$	228.473	0.646220
$51$	245.824	0.674947
$52$	−346.115	−0.923029
$53$	613.202	1.58924	0.794621	−	0.607106i	$-0.207670\pi$
$53$	613.202	1.58924	0.794621	+	0.607106i	$0.207670\pi$
$54$	−231.340	−0.582990
$55$	47.8202	0.117238
$56$	−271.569	−0.648034
$57$	−253.348	−0.588714
$58$	0	0
$59$	184.206	0.406468	0.203234	−	0.979130i	$-0.434855\pi$
$59$	184.206	0.406468	0.203234	+	0.979130i	$0.434855\pi$
$60$	31.4576	0.0676859
$61$	13.6914	0.0287377	0.0143689	−	0.999897i	$-0.495426\pi$
$61$	13.6914	0.0287377	0.0143689	+	0.999897i	$0.495426\pi$
$62$	447.766	0.917200
$63$	−721.484	−1.44283
$64$	64.0000	0.125000
$65$	283.880	0.541708
$66$	−69.8805	−0.130329
$67$	328.733	0.599421	0.299710	−	0.954030i	$-0.403110\pi$
$67$	328.733	0.599421	0.299710	+	0.954030i	$0.403110\pi$
$68$	−410.199	−0.731529
$69$	324.703	0.566517
$70$	222.738	0.380319
$71$	5.15186	0.00861145	0.00430573	−	0.999991i	$-0.498629\pi$
$71$	5.15186	0.00861145	0.00430573	+	0.999991i	$0.498629\pi$
$72$	170.031	0.278310
$73$	428.481	0.686984	0.343492	−	0.939156i	$-0.388390\pi$
$73$	428.481	0.686984	0.343492	+	0.939156i	$0.388390\pi$
$74$	−479.445	−0.753167
$75$	273.839	0.421602
$76$	422.753	0.638067
$77$	−494.796	−0.732302
$78$	−414.840	−0.602197
$79$	392.819	0.559437	0.279719	−	0.960082i	$-0.409759\pi$
$79$	392.819	0.559437	0.279719	+	0.960082i	$0.409759\pi$
$80$	−52.4922	−0.0733601
$81$	296.578	0.406828
$82$	438.661	0.590757
$83$	−454.172	−0.600624	−0.300312	−	0.953841i	$-0.597091\pi$
$83$	−454.172	−0.600624	−0.300312	+	0.953841i	$0.597091\pi$
$84$	−325.492	−0.422786
$85$	336.442	0.429321
$86$	37.9675	0.0476063
$87$	0	0
$88$	116.607	0.141254
$89$	811.929	0.967015	0.483507	−	0.875340i	$-0.339363\pi$
$89$	811.929	0.967015	0.483507	+	0.875340i	$0.339363\pi$
$90$	−139.458	−0.163335
$91$	−2937.31	−3.38367
$92$	−541.822	−0.614010
$93$	536.675	0.598394
$94$	295.844	0.324617
$95$	−346.738	−0.374470
$96$	76.7078	0.0815517
$97$	11.3513	0.0118820	0.00594098	−	0.999982i	$-0.498109\pi$
$97$	11.3513	0.0118820	0.00594098	+	0.999982i	$0.498109\pi$
$98$	−1618.68	−1.66848
$99$	309.794	0.314500
$100$	−456.946	−0.456946
$101$	−1548.46	−1.52552	−0.762762	−	0.646679i	$-0.776157\pi$
$101$	−1548.46	−1.52552	−0.762762	+	0.646679i	$0.776157\pi$
$102$	−491.649	−0.477259
$103$	861.888	0.824508	0.412254	−	0.911069i	$-0.364742\pi$
$103$	861.888	0.824508	0.412254	+	0.911069i	$0.364742\pi$
$104$	692.230	0.652680
$105$	266.965	0.248125
$106$	−1226.40	−1.12376
$107$	−1374.40	−1.24176	−0.620880	−	0.783906i	$-0.713225\pi$
$107$	−1374.40	−1.24176	−0.620880	+	0.783906i	$0.713225\pi$
$108$	462.681	0.412236
$109$	−467.521	−0.410829	−0.205415	−	0.978675i	$-0.565854\pi$
$109$	−467.521	−0.410829	−0.205415	+	0.978675i	$0.565854\pi$
$110$	−95.6404	−0.0828996
$111$	−574.644	−0.491376
$112$	543.138	0.458229
$113$	−655.223	−0.545471	−0.272736	−	0.962089i	$-0.587928\pi$
$113$	−655.223	−0.545471	−0.272736	+	0.962089i	$0.587928\pi$
$114$	506.695	0.416284
$115$	444.398	0.360351
$116$	0	0
$117$	1839.07	1.45318
$118$	−368.412	−0.287416
$119$	−3481.17	−2.68166
$120$	−62.9151	−0.0478611
$121$	−1118.54	−0.840377
$122$	−27.3828	−0.0203207
$123$	525.762	0.385418
$124$	−895.533	−0.648558
$125$	784.879	0.561614
$126$	1442.97	1.02024
$127$	−830.607	−0.580350	−0.290175	−	0.956974i	$-0.593714\pi$
$127$	−830.607	−0.580350	−0.290175	+	0.956974i	$0.593714\pi$
$128$	−128.000	−0.0883883
$129$	45.5063	0.0310590
$130$	−567.761	−0.383046
$131$	−783.195	−0.522351	−0.261176	−	0.965291i	$-0.584110\pi$
$131$	−783.195	−0.522351	−0.261176	+	0.965291i	$0.584110\pi$
$132$	139.761	0.0921563
$133$	3587.71	2.33905
$134$	−657.467	−0.423854
$135$	−379.487	−0.241933
$136$	820.399	0.517269
$137$	1362.28	0.849545	0.424773	−	0.905300i	$-0.360354\pi$
$137$	1362.28	0.849545	0.424773	+	0.905300i	$0.360354\pi$
$138$	−649.407	−0.400588
$139$	−874.798	−0.533809	−0.266904	−	0.963723i	$-0.586001\pi$
$139$	−874.798	−0.533809	−0.266904	+	0.963723i	$0.586001\pi$
$140$	−445.477	−0.268926
$141$	354.587	0.211784
$142$	−10.3037	−0.00608922
$143$	1261.24	0.737552
$144$	−340.061	−0.196795
$145$	0	0
$146$	−856.961	−0.485771
$147$	−1940.08	−1.08854
$148$	958.890	0.532569
$149$	1938.20	1.06566	0.532830	−	0.846222i	$-0.321129\pi$
$149$	1938.20	1.06566	0.532830	+	0.846222i	$0.321129\pi$
$150$	−547.678	−0.298118
$151$	−1460.33	−0.787018	−0.393509	−	0.919321i	$-0.628739\pi$
$151$	−1460.33	−0.787018	−0.393509	+	0.919321i	$0.628739\pi$
$152$	−845.506	−0.451182
$153$	2179.58	1.15169
$154$	989.592	0.517816
$155$	734.508	0.380627
$156$	829.679	0.425817
$157$	−421.035	−0.214027	−0.107014	−	0.994258i	$-0.534129\pi$
$157$	−421.035	−0.214027	−0.107014	+	0.994258i	$0.534129\pi$
$158$	−785.637	−0.395582
$159$	−1469.92	−0.733158
$160$	104.984	0.0518735
$161$	−4598.19	−2.25086
$162$	−593.155	−0.287671
$163$	796.182	0.382588	0.191294	−	0.981533i	$-0.438732\pi$
$163$	796.182	0.382588	0.191294	+	0.981533i	$0.438732\pi$
$164$	−877.323	−0.417728
$165$	−114.631	−0.0540848
$166$	908.344	0.424706
$167$	3625.72	1.68004	0.840020	−	0.542556i	$-0.182543\pi$
$167$	3625.72	1.68004	0.840020	+	0.542556i	$0.182543\pi$
$168$	650.983	0.298955
$169$	5290.22	2.40793
$170$	−672.884	−0.303576
$171$	−2246.28	−1.00455
$172$	−75.9350	−0.0336627
$173$	587.037	0.257986	0.128993	−	0.991645i	$-0.458825\pi$
$173$	587.037	0.257986	0.128993	+	0.991645i	$0.458825\pi$
$174$	0	0
$175$	−3877.89	−1.67509
$176$	−233.215	−0.0998820
$177$	−441.564	−0.187514
$178$	−1623.86	−0.683783
$179$	−908.839	−0.379496	−0.189748	−	0.981833i	$-0.560767\pi$
$179$	−908.839	−0.379496	−0.189748	+	0.981833i	$0.560767\pi$
$180$	278.915	0.115495
$181$	2139.11	0.878447	0.439223	−	0.898378i	$-0.355254\pi$
$181$	2139.11	0.878447	0.439223	+	0.898378i	$0.355254\pi$
$182$	5874.63	2.39262
$183$	−32.8199	−0.0132575
$184$	1083.64	0.434170
$185$	−786.473	−0.312555
$186$	−1073.35	−0.423128
$187$	1494.76	0.584533
$188$	−591.688	−0.229539
$189$	3926.55	1.51119
$190$	693.477	0.264790
$191$	1715.31	0.649819	0.324910	−	0.945745i	$-0.394666\pi$
$191$	1715.31	0.649819	0.324910	+	0.945745i	$0.394666\pi$
$192$	−153.416	−0.0576657
$193$	2509.88	0.936090	0.468045	−	0.883705i	$-0.344958\pi$
$193$	2509.88	0.936090	0.468045	+	0.883705i	$0.344958\pi$
$194$	−22.7026	−0.00840181
$195$	−680.496	−0.249904
$196$	3237.35	1.17979
$197$	−688.689	−0.249071	−0.124536	−	0.992215i	$-0.539744\pi$
$197$	−688.689	−0.249071	−0.124536	+	0.992215i	$0.539744\pi$
$198$	−619.588	−0.222385
$199$	1106.28	0.394079	0.197040	−	0.980396i	$-0.436867\pi$
$199$	1106.28	0.394079	0.197040	+	0.980396i	$0.436867\pi$
$200$	913.893	0.323110
$201$	−788.014	−0.276528
$202$	3096.93	1.07871
$203$	0	0
$204$	983.297	0.337473
$205$	719.572	0.245157
$206$	−1723.78	−0.583015
$207$	2878.95	0.966670
$208$	−1384.46	−0.461514
$209$	−1540.51	−0.509851
$210$	−533.931	−0.175451
$211$	2094.08	0.683234	0.341617	−	0.939839i	$-0.389025\pi$
$211$	2094.08	0.683234	0.341617	+	0.939839i	$0.389025\pi$
$212$	2452.81	0.794621
$213$	−12.3496	−0.00397269
$214$	2748.80	0.878057
$215$	62.2812	0.0197560
$216$	−925.362	−0.291495
$217$	−7599.96	−2.37751
$218$	935.042	0.290500
$219$	−1027.12	−0.316924
$220$	191.281	0.0586189
$221$	8873.51	2.70089
$222$	1149.29	0.347455
$223$	6033.11	1.81169	0.905845	−	0.423610i	$-0.139237\pi$
$223$	6033.11	1.81169	0.905845	+	0.423610i	$0.139237\pi$
$224$	−1086.28	−0.324017
$225$	2427.96	0.719396
$226$	1310.45	0.385706
$227$	189.492	0.0554053	0.0277027	−	0.999616i	$-0.491181\pi$
$227$	189.492	0.0554053	0.0277027	+	0.999616i	$0.491181\pi$
$228$	−1013.39	−0.294357
$229$	2347.80	0.677498	0.338749	−	0.940877i	$-0.389996\pi$
$229$	2347.80	0.677498	0.338749	+	0.940877i	$0.389996\pi$
$230$	−888.796	−0.254806
$231$	1186.09	0.337830
$232$	0	0
$233$	551.777	0.155142	0.0775710	−	0.996987i	$-0.475284\pi$
$233$	551.777	0.155142	0.0775710	+	0.996987i	$0.475284\pi$
$234$	−3678.13	−1.02755
$235$	485.297	0.134712
$236$	736.825	0.203234
$237$	−941.633	−0.258083
$238$	6962.33	1.89622
$239$	−4533.45	−1.22696	−0.613482	−	0.789708i	$-0.710232\pi$
$239$	−4533.45	−1.22696	−0.613482	+	0.789708i	$0.710232\pi$
$240$	125.830	0.0338429
$241$	1148.28	0.306917	0.153458	−	0.988155i	$-0.450959\pi$
$241$	1148.28	0.306917	0.153458	+	0.988155i	$0.450959\pi$
$242$	2237.08	0.594237
$243$	−3834.03	−1.01215
$244$	54.7655	0.0143689
$245$	−2655.25	−0.692398
$246$	−1051.52	−0.272531
$247$	−9145.07	−2.35582
$248$	1791.07	0.458600
$249$	1088.70	0.277084
$250$	−1569.76	−0.397121
$251$	7236.42	1.81976	0.909878	−	0.414876i	$-0.136175\pi$
$251$	7236.42	1.81976	0.909878	+	0.414876i	$0.136175\pi$
$252$	−2885.94	−0.721416
$253$	1974.39	0.490628
$254$	1661.21	0.410369
$255$	−806.492	−0.198057
$256$	256.000	0.0625000
$257$	5051.89	1.22618	0.613089	−	0.790013i	$-0.289927\pi$
$257$	5051.89	1.22618	0.613089	+	0.790013i	$0.289927\pi$
$258$	−91.0127	−0.0219620
$259$	8137.64	1.95231
$260$	1135.52	0.270854
$261$	0	0
$262$	1566.39	0.369358
$263$	−6368.81	−1.49322	−0.746611	−	0.665260i	$-0.768320\pi$
$263$	−6368.81	−1.49322	−0.746611	+	0.665260i	$0.768320\pi$
$264$	−279.522	−0.0651644
$265$	−2011.77	−0.466348
$266$	−7175.41	−1.65396
$267$	−1946.29	−0.446109
$268$	1314.93	0.299710
$269$	−3299.89	−0.747948	−0.373974	−	0.927439i	$-0.622005\pi$
$269$	−3299.89	−0.747948	−0.373974	+	0.927439i	$0.622005\pi$
$270$	758.973	0.171073
$271$	2235.21	0.501031	0.250515	−	0.968113i	$-0.419400\pi$
$271$	2235.21	0.501031	0.250515	+	0.968113i	$0.419400\pi$
$272$	−1640.80	−0.365765
$273$	7041.09	1.56098
$274$	−2724.57	−0.600719
$275$	1665.10	0.365126
$276$	1298.81	0.283259
$277$	−2354.61	−0.510739	−0.255369	−	0.966844i	$-0.582197\pi$
$277$	−2354.61	−0.510739	−0.255369	+	0.966844i	$0.582197\pi$
$278$	1749.60	0.377460
$279$	4758.37	1.02106
$280$	890.953	0.190159
$281$	−6858.62	−1.45605	−0.728027	−	0.685549i	$-0.759562\pi$
$281$	−6858.62	−1.45605	−0.728027	+	0.685549i	$0.759562\pi$
$282$	−709.174	−0.149754
$283$	−3031.29	−0.636719	−0.318360	−	0.947970i	$-0.603132\pi$
$283$	−3031.29	−0.636719	−0.318360	+	0.947970i	$0.603132\pi$
$284$	20.6074	0.00430573
$285$	831.174	0.172753
$286$	−2522.47	−0.521528
$287$	−7445.42	−1.53132
$288$	680.122	0.139155
$289$	5603.47	1.14054
$290$	0	0
$291$	−27.2104	−0.00548145
$292$	1713.92	0.343492
$293$	−1921.35	−0.383094	−0.191547	−	0.981483i	$-0.561350\pi$
$293$	−1921.35	−0.383094	−0.191547	+	0.981483i	$0.561350\pi$
$294$	3880.16	0.769713
$295$	−604.337	−0.119274
$296$	−1917.78	−0.376583
$297$	−1686.00	−0.329400
$298$	−3876.39	−0.753535
$299$	11720.8	2.26699
$300$	1095.36	0.210801
$301$	−644.424	−0.123402
$302$	2920.65	0.556506
$303$	3711.86	0.703764
$304$	1691.01	0.319034
$305$	−44.9182	−0.00843282
$306$	−4359.15	−0.814366
$307$	8734.92	1.62387	0.811935	−	0.583747i	$-0.198414\pi$
$307$	8734.92	1.62387	0.811935	+	0.583747i	$0.198414\pi$
$308$	−1979.18	−0.366151
$309$	−2066.05	−0.380367
$310$	−1469.02	−0.269144
$311$	198.321	0.0361599	0.0180800	−	0.999837i	$-0.494245\pi$
$311$	198.321	0.0361599	0.0180800	+	0.999837i	$0.494245\pi$
$312$	−1659.36	−0.301098
$313$	−3759.23	−0.678864	−0.339432	−	0.940631i	$-0.610235\pi$
$313$	−3759.23	−0.678864	−0.339432	+	0.940631i	$0.610235\pi$
$314$	842.070	0.151340
$315$	2367.02	0.423386
$316$	1571.27	0.279719
$317$	5543.84	0.982249	0.491124	−	0.871090i	$-0.336586\pi$
$317$	5543.84	0.982249	0.491124	+	0.871090i	$0.336586\pi$
$318$	2939.84	0.518421
$319$	0	0
$320$	−209.969	−0.0366801
$321$	3294.60	0.572856
$322$	9196.38	1.59160
$323$	−10838.3	−1.86706
$324$	1186.31	0.203414
$325$	9884.75	1.68710
$326$	−1592.36	−0.270530
$327$	1120.70	0.189526
$328$	1754.65	0.295378
$329$	−5021.37	−0.841451
$330$	229.262	0.0382437
$331$	2875.96	0.477574	0.238787	−	0.971072i	$-0.423250\pi$
$331$	2875.96	0.477574	0.238787	+	0.971072i	$0.423250\pi$
$332$	−1816.69	−0.300312
$333$	−5095.02	−0.838454
$334$	−7251.44	−1.18797
$335$	−1078.50	−0.175894
$336$	−1301.97	−0.211393
$337$	−8141.60	−1.31603	−0.658014	−	0.753006i	$-0.728603\pi$
$337$	−8141.60	−1.31603	−0.658014	+	0.753006i	$0.728603\pi$
$338$	−10580.4	−1.70266
$339$	1570.65	0.251640
$340$	1345.77	0.214660
$341$	3263.31	0.518234
$342$	4492.56	0.710321
$343$	15830.3	2.49201
$344$	151.870	0.0238031
$345$	−1065.28	−0.166239
$346$	−1174.07	−0.182424
$347$	−8941.88	−1.38336	−0.691679	−	0.722205i	$-0.743129\pi$
$347$	−8941.88	−1.38336	−0.691679	+	0.722205i	$0.743129\pi$
$348$	0	0
$349$	−3061.39	−0.469548	−0.234774	−	0.972050i	$-0.575435\pi$
$349$	−3061.39	−0.469548	−0.234774	+	0.972050i	$0.575435\pi$
$350$	7755.77	1.18447
$351$	−10008.8	−1.52202
$352$	466.430	0.0706272
$353$	2871.64	0.432979	0.216490	−	0.976285i	$-0.430539\pi$
$353$	2871.64	0.432979	0.216490	+	0.976285i	$0.430539\pi$
$354$	883.129	0.132593
$355$	−16.9020	−0.00252695
$356$	3247.72	0.483507
$357$	8344.78	1.23712
$358$	1817.68	0.268344
$359$	9773.24	1.43680	0.718401	−	0.695629i	$-0.244874\pi$
$359$	9773.24	1.43680	0.718401	+	0.695629i	$0.244874\pi$
$360$	−557.830	−0.0816673
$361$	4311.01	0.628519
$362$	−4278.22	−0.621156
$363$	2681.28	0.387688
$364$	−11749.3	−1.69184
$365$	−1405.74	−0.201589
$366$	65.6398	0.00937444
$367$	6064.61	0.862588	0.431294	−	0.902211i	$-0.358057\pi$
$367$	6064.61	0.862588	0.431294	+	0.902211i	$0.358057\pi$
$368$	−2167.29	−0.307005
$369$	4661.61	0.657653
$370$	1572.95	0.221010
$371$	20815.8	2.91295
$372$	2146.70	0.299197
$373$	2684.28	0.372618	0.186309	−	0.982491i	$-0.440347\pi$
$373$	2684.28	0.372618	0.186309	+	0.982491i	$0.440347\pi$
$374$	−2989.52	−0.413327
$375$	−1881.45	−0.259087
$376$	1183.38	0.162308
$377$	0	0
$378$	−7853.10	−1.06857
$379$	−2069.54	−0.280489	−0.140244	−	0.990117i	$-0.544789\pi$
$379$	−2069.54	−0.280489	−0.140244	+	0.990117i	$0.544789\pi$
$380$	−1386.95	−0.187235
$381$	1991.06	0.267731
$382$	−3430.62	−0.459492
$383$	8550.46	1.14075	0.570376	−	0.821384i	$-0.306797\pi$
$383$	8550.46	1.14075	0.570376	+	0.821384i	$0.306797\pi$
$384$	306.831	0.0407758
$385$	1623.31	0.214887
$386$	−5019.77	−0.661916
$387$	403.477	0.0529971
$388$	45.4052	0.00594098
$389$	10974.4	1.43040	0.715199	−	0.698921i	$-0.246336\pi$
$389$	10974.4	1.43040	0.715199	+	0.698921i	$0.246336\pi$
$390$	1360.99	0.176709
$391$	13890.9	1.79666
$392$	−6474.70	−0.834239
$393$	1877.41	0.240974
$394$	1377.38	0.176120
$395$	−1288.75	−0.164162
$396$	1239.18	0.157250
$397$	973.767	0.123103	0.0615516	−	0.998104i	$-0.480395\pi$
$397$	973.767	0.123103	0.0615516	+	0.998104i	$0.480395\pi$
$398$	−2212.55	−0.278656
$399$	−8600.16	−1.07906
$400$	−1827.79	−0.228473
$401$	2196.32	0.273514	0.136757	−	0.990605i	$-0.456332\pi$
$401$	2196.32	0.273514	0.136757	+	0.990605i	$0.456332\pi$
$402$	1576.03	0.195535
$403$	19372.3	2.39455
$404$	−6193.86	−0.762762
$405$	−973.002	−0.119380
$406$	0	0
$407$	−3494.18	−0.425553
$408$	−1966.59	−0.238630
$409$	−7847.40	−0.948726	−0.474363	−	0.880329i	$-0.657322\pi$
$409$	−7847.40	−0.948726	−0.474363	+	0.880329i	$0.657322\pi$
$410$	−1439.14	−0.173352
$411$	−3265.56	−0.391917
$412$	3447.55	0.412254
$413$	6253.08	0.745022
$414$	−5757.90	−0.683539
$415$	1490.03	0.176248
$416$	2768.92	0.326340
$417$	2097.00	0.246260
$418$	3081.01	0.360519
$419$	−6943.21	−0.809541	−0.404771	−	0.914418i	$-0.632649\pi$
$419$	−6943.21	−0.809541	−0.404771	+	0.914418i	$0.632649\pi$
$420$	1067.86	0.124063
$421$	10527.3	1.21869	0.609343	−	0.792907i	$-0.291433\pi$
$421$	10527.3	1.21869	0.609343	+	0.792907i	$0.291433\pi$
$422$	−4188.16	−0.483120
$423$	3143.91	0.361376
$424$	−4905.62	−0.561882
$425$	11714.9	1.33708
$426$	24.6993	0.00280911
$427$	464.769	0.0526739
$428$	−5497.60	−0.620880
$429$	−3023.34	−0.340252
$430$	−124.562	−0.0139696
$431$	−1735.58	−0.193967	−0.0969836	−	0.995286i	$-0.530919\pi$
$431$	−1735.58	−0.193967	−0.0969836	+	0.995286i	$0.530919\pi$
$432$	1850.72	0.206118
$433$	9380.15	1.04107	0.520533	−	0.853842i	$-0.325733\pi$
$433$	9380.15	1.04107	0.520533	+	0.853842i	$0.325733\pi$
$434$	15199.9	1.68115
$435$	0	0
$436$	−1870.08	−0.205415
$437$	−14316.1	−1.56712
$438$	2054.24	0.224099
$439$	−4099.58	−0.445700	−0.222850	−	0.974853i	$-0.571536\pi$
$439$	−4099.58	−0.445700	−0.222850	+	0.974853i	$0.571536\pi$
$440$	−382.562	−0.0414498
$441$	−17201.5	−1.85741
$442$	−17747.0	−1.90982
$443$	−5718.20	−0.613273	−0.306636	−	0.951827i	$-0.599204\pi$
$443$	−5718.20	−0.613273	−0.306636	+	0.951827i	$0.599204\pi$
$444$	−2298.57	−0.245688
$445$	−2663.75	−0.283761
$446$	−12066.2	−1.28106
$447$	−4646.09	−0.491616
$448$	2172.55	0.229115
$449$	−2474.77	−0.260115	−0.130057	−	0.991506i	$-0.541516\pi$
$449$	−2474.77	−0.260115	−0.130057	+	0.991506i	$0.541516\pi$
$450$	−4855.93	−0.508690
$451$	3196.95	0.333788
$452$	−2620.89	−0.272736
$453$	3500.58	0.363072
$454$	−378.983	−0.0391775
$455$	9636.63	0.992906
$456$	2026.78	0.208142
$457$	10932.4	1.11903	0.559513	−	0.828822i	$-0.310988\pi$
$457$	10932.4	1.11903	0.559513	+	0.828822i	$0.310988\pi$
$458$	−4695.60	−0.479063
$459$	−11862.0	−1.20625
$460$	1777.59	0.180175
$461$	−16589.3	−1.67602	−0.838008	−	0.545659i	$-0.816280\pi$
$461$	−16589.3	−1.67602	−0.838008	+	0.545659i	$0.816280\pi$
$462$	−2372.17	−0.238882
$463$	894.244	0.0897604	0.0448802	−	0.998992i	$-0.485709\pi$
$463$	894.244	0.0897604	0.0448802	+	0.998992i	$0.485709\pi$
$464$	0	0
$465$	−1760.70	−0.175593
$466$	−1103.55	−0.109702
$467$	8011.09	0.793809	0.396905	−	0.917860i	$-0.370084\pi$
$467$	8011.09	0.793809	0.396905	+	0.917860i	$0.370084\pi$
$468$	7356.26	0.726588
$469$	11159.2	1.09869
$470$	−970.595	−0.0952557
$471$	1009.27	0.0987363
$472$	−1473.65	−0.143708
$473$	276.706	0.0268984
$474$	1883.27	0.182492
$475$	−12073.5	−1.16625
$476$	−13924.7	−1.34083
$477$	−13032.9	−1.25102
$478$	9066.90	0.867595
$479$	14558.7	1.38873	0.694366	−	0.719622i	$-0.255685\pi$
$479$	14558.7	1.38873	0.694366	+	0.719622i	$0.255685\pi$
$480$	−251.660	−0.0239306
$481$	−20742.9	−1.96631
$482$	−2296.55	−0.217023
$483$	11022.4	1.03838
$484$	−4474.17	−0.420189
$485$	−37.2409	−0.00348665
$486$	7668.05	0.715700
$487$	−17666.8	−1.64386	−0.821928	−	0.569591i	$-0.807102\pi$
$487$	−17666.8	−1.64386	−0.821928	+	0.569591i	$0.807102\pi$
$488$	−109.531	−0.0101603
$489$	−1908.54	−0.176498
$490$	5310.49	0.489599
$491$	4904.36	0.450775	0.225388	−	0.974269i	$-0.427635\pi$
$491$	4904.36	0.450775	0.225388	+	0.974269i	$0.427635\pi$
$492$	2103.05	0.192709
$493$	0	0
$494$	18290.1	1.66582
$495$	−1016.36	−0.0922870
$496$	−3582.13	−0.324279
$497$	174.886	0.0157841
$498$	−2177.41	−0.195928
$499$	2496.99	0.224009	0.112005	−	0.993708i	$-0.464273\pi$
$499$	2496.99	0.224009	0.112005	+	0.993708i	$0.464273\pi$
$500$	3139.52	0.280807
$501$	−8691.29	−0.775046
$502$	−14472.8	−1.28676
$503$	968.191	0.0858241	0.0429120	−	0.999079i	$-0.486336\pi$
$503$	968.191	0.0858241	0.0429120	+	0.999079i	$0.486336\pi$
$504$	5771.87	0.510118
$505$	5080.15	0.447651
$506$	−3948.78	−0.346926
$507$	−12681.3	−1.11084
$508$	−3322.43	−0.290175
$509$	3379.07	0.294253	0.147126	−	0.989118i	$-0.452998\pi$
$509$	3379.07	0.294253	0.147126	+	0.989118i	$0.452998\pi$
$510$	1612.98	0.140047
$511$	14545.2	1.25919
$512$	−512.000	−0.0441942
$513$	12225.0	1.05214
$514$	−10103.8	−0.867039
$515$	−2827.65	−0.241944
$516$	182.025	0.0155295
$517$	2156.10	0.183414
$518$	−16275.3	−1.38049
$519$	−1407.20	−0.119016
$520$	−2271.04	−0.191523
$521$	10941.9	0.920098	0.460049	−	0.887893i	$-0.347832\pi$
$521$	10941.9	0.920098	0.460049	+	0.887893i	$0.347832\pi$
$522$	0	0
$523$	8947.20	0.748057	0.374029	−	0.927417i	$-0.377976\pi$
$523$	8947.20	0.748057	0.374029	+	0.927417i	$0.377976\pi$
$524$	−3132.78	−0.261176
$525$	9295.76	0.772762
$526$	12737.6	1.05587
$527$	22959.2	1.89776
$528$	559.044	0.0460782
$529$	6181.21	0.508031
$530$	4023.54	0.329758
$531$	−3915.09	−0.319963
$532$	14350.8	1.16952
$533$	18978.4	1.54230
$534$	3892.58	0.315447
$535$	4509.09	0.364383
$536$	−2629.87	−0.211927
$537$	2178.60	0.175071
$538$	6599.79	0.528879
$539$	−11796.8	−0.942720
$540$	−1517.95	−0.120967
$541$	−9008.10	−0.715875	−0.357938	−	0.933746i	$-0.616520\pi$
$541$	−9008.10	−0.715875	−0.357938	+	0.933746i	$0.616520\pi$
$542$	−4470.42	−0.354282
$543$	−5127.71	−0.405250
$544$	3281.59	0.258635
$545$	1533.83	0.120554
$546$	−14082.2	−1.10378
$547$	12980.3	1.01462	0.507312	−	0.861763i	$-0.330639\pi$
$547$	12980.3	1.01462	0.507312	+	0.861763i	$0.330639\pi$
$548$	5449.13	0.424773
$549$	−290.994	−0.0226217
$550$	−3330.21	−0.258183
$551$	0	0
$552$	−2597.63	−0.200294
$553$	13334.7	1.02540
$554$	4709.21	0.361147
$555$	1885.27	0.144190
$556$	−3499.19	−0.266904
$557$	−8777.82	−0.667734	−0.333867	−	0.942620i	$-0.608354\pi$
$557$	−8777.82	−0.667734	−0.333867	+	0.942620i	$0.608354\pi$
$558$	−9516.75	−0.722000
$559$	1642.64	0.124287
$560$	−1781.91	−0.134463
$561$	−3583.12	−0.269660
$562$	13717.2	1.02959
$563$	−3734.91	−0.279588	−0.139794	−	0.990181i	$-0.544644\pi$
$563$	−3734.91	−0.279588	−0.139794	+	0.990181i	$0.544644\pi$
$564$	1418.35	0.105892
$565$	2149.63	0.160063
$566$	6062.58	0.450229
$567$	10067.7	0.745682
$568$	−41.2149	−0.00304461
$569$	14373.7	1.05901	0.529504	−	0.848307i	$-0.322378\pi$
$569$	14373.7	1.05901	0.529504	+	0.848307i	$0.322378\pi$
$570$	−1662.35	−0.122155
$571$	8147.54	0.597135	0.298567	−	0.954389i	$-0.403491\pi$
$571$	8147.54	0.597135	0.298567	+	0.954389i	$0.403491\pi$
$572$	5044.95	0.368776
$573$	−4111.81	−0.299779
$574$	14890.8	1.08281
$575$	15474.0	1.12228
$576$	−1360.24	−0.0983973
$577$	−9438.43	−0.680982	−0.340491	−	0.940248i	$-0.610593\pi$
$577$	−9438.43	−0.680982	−0.340491	+	0.940248i	$0.610593\pi$
$578$	−11206.9	−0.806483
$579$	−6016.49	−0.431843
$580$	0	0
$581$	−15417.4	−1.10089
$582$	54.4208	0.00387597
$583$	−8937.99	−0.634946
$584$	−3427.84	−0.242886
$585$	−6033.54	−0.426421
$586$	3842.71	0.270889
$587$	−3498.97	−0.246027	−0.123014	−	0.992405i	$-0.539256\pi$
$587$	−3498.97	−0.246027	−0.123014	+	0.992405i	$0.539256\pi$
$588$	−7760.32	−0.544269
$589$	−23661.8	−1.65530
$590$	1208.67	0.0843396
$591$	1650.87	0.114903
$592$	3835.56	0.266285
$593$	7383.92	0.511335	0.255667	−	0.966765i	$-0.417705\pi$
$593$	7383.92	0.511335	0.255667	+	0.966765i	$0.417705\pi$
$594$	3372.00	0.232921
$595$	11420.9	0.786909
$596$	7752.79	0.532830
$597$	−2651.87	−0.181799
$598$	−23441.6	−1.60301
$599$	−14673.8	−1.00093	−0.500463	−	0.865758i	$-0.666837\pi$
$599$	−14673.8	−1.00093	−0.500463	+	0.865758i	$0.666837\pi$
$600$	−2190.71	−0.149059
$601$	10950.2	0.743209	0.371605	−	0.928391i	$-0.378808\pi$
$601$	10950.2	0.743209	0.371605	+	0.928391i	$0.378808\pi$
$602$	1288.85	0.0872584
$603$	−6986.84	−0.471851
$604$	−5841.31	−0.393509
$605$	3669.67	0.246601
$606$	−7423.71	−0.497636
$607$	−17469.1	−1.16812	−0.584059	−	0.811712i	$-0.698536\pi$
$607$	−17469.1	−1.16812	−0.584059	+	0.811712i	$0.698536\pi$
$608$	−3382.03	−0.225591
$609$	0	0
$610$	89.8364	0.00596290
$611$	12799.5	0.847484
$612$	8718.30	0.575844
$613$	5866.70	0.386548	0.193274	−	0.981145i	$-0.438089\pi$
$613$	5866.70	0.386548	0.193274	+	0.981145i	$0.438089\pi$
$614$	−17469.8	−1.14825
$615$	−1724.90	−0.113097
$616$	3958.37	0.258908
$617$	14383.8	0.938528	0.469264	−	0.883058i	$-0.344519\pi$
$617$	14383.8	0.938528	0.469264	+	0.883058i	$0.344519\pi$
$618$	4132.10	0.268960
$619$	−28631.0	−1.85909	−0.929545	−	0.368710i	$-0.879800\pi$
$619$	−28631.0	−1.85909	−0.929545	+	0.368710i	$0.879800\pi$
$620$	2938.03	0.190313
$621$	−15668.2	−1.01247
$622$	−396.642	−0.0255689
$623$	27561.8	1.77246
$624$	3318.72	0.212909
$625$	11704.6	0.749092
$626$	7518.47	0.480029
$627$	3692.78	0.235208
$628$	−1684.14	−0.107014
$629$	−24583.5	−1.55836
$630$	−4734.04	−0.299379
$631$	−2860.48	−0.180466	−0.0902329	−	0.995921i	$-0.528761\pi$
$631$	−2860.48	−0.180466	−0.0902329	+	0.995921i	$0.528761\pi$
$632$	−3142.55	−0.197791
$633$	−5019.76	−0.315194
$634$	−11087.7	−0.694555
$635$	2725.03	0.170298
$636$	−5879.68	−0.366579
$637$	−70031.0	−4.35593
$638$	0	0
$639$	−109.497	−0.00677875
$640$	419.938	0.0259367
$641$	−6003.37	−0.369920	−0.184960	−	0.982746i	$-0.559216\pi$
$641$	−6003.37	−0.369920	−0.184960	+	0.982746i	$0.559216\pi$
$642$	−6589.21	−0.405071
$643$	6690.61	0.410345	0.205173	−	0.978726i	$-0.434224\pi$
$643$	6690.61	0.410345	0.205173	+	0.978726i	$0.434224\pi$
$644$	−18392.8	−1.12543
$645$	−149.296	−0.00911397
$646$	21676.6	1.32021
$647$	−8697.55	−0.528495	−0.264247	−	0.964455i	$-0.585124\pi$
$647$	−8697.55	−0.528495	−0.264247	+	0.964455i	$0.585124\pi$
$648$	−2372.62	−0.143835
$649$	−2684.98	−0.162395
$650$	−19769.5	−1.19296
$651$	18218.0	1.09681
$652$	3184.73	0.191294
$653$	−21260.9	−1.27413	−0.637063	−	0.770812i	$-0.719851\pi$
$653$	−21260.9	−1.27413	−0.637063	+	0.770812i	$0.719851\pi$
$654$	−2241.41	−0.134015
$655$	2569.48	0.153279
$656$	−3509.29	−0.208864
$657$	−9106.85	−0.540779
$658$	10042.7	0.594996
$659$	−11717.3	−0.692625	−0.346313	−	0.938119i	$-0.612566\pi$
$659$	−11717.3	−0.692625	−0.346313	+	0.938119i	$0.612566\pi$
$660$	−458.523	−0.0270424
$661$	27523.1	1.61955	0.809775	−	0.586740i	$-0.199589\pi$
$661$	27523.1	1.61955	0.809775	+	0.586740i	$0.199589\pi$
$662$	−5751.92	−0.337696
$663$	−21270.9	−1.24599
$664$	3633.37	0.212353
$665$	−11770.4	−0.686372
$666$	10190.0	0.592877
$667$	0	0
$668$	14502.9	0.840020
$669$	−14462.1	−0.835780
$670$	2156.99	0.124376
$671$	−199.565	−0.0114815
$672$	2603.93	0.149477
$673$	−30200.5	−1.72978	−0.864891	−	0.501960i	$-0.832612\pi$
$673$	−30200.5	−1.72978	−0.864891	+	0.501960i	$0.832612\pi$
$674$	16283.2	0.930572
$675$	−13213.8	−0.753479
$676$	21160.9	1.20396
$677$	−19986.2	−1.13461	−0.567306	−	0.823507i	$-0.692014\pi$
$677$	−19986.2	−1.13461	−0.567306	+	0.823507i	$0.692014\pi$
$678$	−3141.30	−0.177936
$679$	385.332	0.0217786
$680$	−2691.54	−0.151788
$681$	−454.234	−0.0255599
$682$	−6526.61	−0.366447
$683$	−6037.11	−0.338219	−0.169110	−	0.985597i	$-0.554089\pi$
$683$	−6037.11	−0.338219	−0.169110	+	0.985597i	$0.554089\pi$
$684$	−8985.12	−0.502273
$685$	−4469.33	−0.249291
$686$	−31660.7	−1.76211
$687$	−5627.96	−0.312547
$688$	−303.740	−0.0168314
$689$	−53059.6	−2.93383
$690$	2130.55	0.117549
$691$	−28078.2	−1.54579	−0.772897	−	0.634531i	$-0.781193\pi$
$691$	−28078.2	−1.54579	−0.772897	+	0.634531i	$0.781193\pi$
$692$	2348.15	0.128993
$693$	10516.3	0.576452
$694$	17883.8	0.978182
$695$	2870.01	0.156641
$696$	0	0
$697$	22492.3	1.22232
$698$	6122.78	0.332021
$699$	−1322.68	−0.0715711
$700$	−15511.5	−0.837545
$701$	32839.5	1.76938	0.884688	−	0.466184i	$-0.154372\pi$
$701$	32839.5	1.76938	0.884688	+	0.466184i	$0.154372\pi$
$702$	20017.6	1.07623
$703$	25335.9	1.35926
$704$	−932.860	−0.0499410
$705$	−1163.32	−0.0621461
$706$	−5743.27	−0.306163
$707$	−52564.3	−2.79616
$708$	−1766.26	−0.0937571
$709$	30391.9	1.60986	0.804932	−	0.593367i	$-0.202202\pi$
$709$	30391.9	1.60986	0.804932	+	0.593367i	$0.202202\pi$
$710$	33.8041	0.00178682
$711$	−8348.89	−0.440377
$712$	−6495.43	−0.341891
$713$	30326.2	1.59288
$714$	−16689.6	−0.874777
$715$	−4137.82	−0.216428
$716$	−3635.36	−0.189748
$717$	10867.2	0.566031
$718$	−19546.5	−1.01597
$719$	−28162.9	−1.46078	−0.730390	−	0.683031i	$-0.760662\pi$
$719$	−28162.9	−1.46078	−0.730390	+	0.683031i	$0.760662\pi$
$720$	1115.66	0.0577475
$721$	29257.7	1.51125
$722$	−8622.03	−0.444430
$723$	−2752.56	−0.141589
$724$	8556.44	0.439223
$725$	0	0
$726$	−5362.56	−0.274137
$727$	36419.3	1.85794	0.928968	−	0.370161i	$-0.120698\pi$
$727$	36419.3	1.85794	0.928968	+	0.370161i	$0.120698\pi$
$728$	23498.5	1.19631
$729$	1183.03	0.0601040
$730$	2811.49	0.142545
$731$	1946.78	0.0985011
$732$	−131.280	−0.00662873
$733$	−8525.28	−0.429589	−0.214794	−	0.976659i	$-0.568908\pi$
$733$	−8525.28	−0.429589	−0.214794	+	0.976659i	$0.568908\pi$
$734$	−12129.2	−0.609942
$735$	6364.95	0.319421
$736$	4334.58	0.217085
$737$	−4791.60	−0.239485
$738$	−9323.23	−0.465031
$739$	−18309.2	−0.911389	−0.455695	−	0.890136i	$-0.650609\pi$
$739$	−18309.2	−0.911389	−0.455695	+	0.890136i	$0.650609\pi$
$740$	−3145.89	−0.156277
$741$	21921.8	1.08680
$742$	−41631.6	−2.05976
$743$	−12992.8	−0.641534	−0.320767	−	0.947158i	$-0.603941\pi$
$743$	−12992.8	−0.641534	−0.320767	+	0.947158i	$0.603941\pi$
$744$	−4293.40	−0.211564
$745$	−6358.77	−0.312708
$746$	−5368.56	−0.263481
$747$	9652.88	0.472799
$748$	5979.04	0.292266
$749$	−46655.5	−2.27604
$750$	3762.90	0.183202
$751$	3443.18	0.167302	0.0836508	−	0.996495i	$-0.473342\pi$
$751$	3443.18	0.167302	0.0836508	+	0.996495i	$0.473342\pi$
$752$	−2366.75	−0.114769
$753$	−17346.6	−0.839501
$754$	0	0
$755$	4790.99	0.230943
$756$	15706.2	0.755594
$757$	−3498.73	−0.167983	−0.0839917	−	0.996466i	$-0.526767\pi$
$757$	−3498.73	−0.167983	−0.0839917	+	0.996466i	$0.526767\pi$
$758$	4139.09	0.198336
$759$	−4732.85	−0.226339
$760$	2773.91	0.132395
$761$	−21460.2	−1.02225	−0.511124	−	0.859507i	$-0.670771\pi$
$761$	−21460.2	−1.02225	−0.511124	+	0.859507i	$0.670771\pi$
$762$	−3982.13	−0.189314
$763$	−15870.5	−0.753016
$764$	6861.24	0.324910
$765$	−7150.67	−0.337952
$766$	−17100.9	−0.806634
$767$	−15939.1	−0.750363
$768$	−613.663	−0.0288329
$769$	8766.41	0.411086	0.205543	−	0.978648i	$-0.434104\pi$
$769$	8766.41	0.411086	0.205543	+	0.978648i	$0.434104\pi$
$770$	−3246.62	−0.151948
$771$	−12110.0	−0.565668
$772$	10039.5	0.468045
$773$	17969.5	0.836116	0.418058	−	0.908420i	$-0.362711\pi$
$773$	17969.5	0.836116	0.418058	+	0.908420i	$0.362711\pi$
$774$	−806.954	−0.0374746
$775$	25575.7	1.18543
$776$	−90.8104	−0.00420091
$777$	−19506.9	−0.900652
$778$	−21948.8	−1.01144
$779$	−23180.7	−1.06615
$780$	−2721.98	−0.124952
$781$	−75.0931	−0.00344052
$782$	−27781.9	−1.27043
$783$	0	0
$784$	12949.4	0.589896
$785$	1381.32	0.0628042
$786$	−3754.82	−0.170395
$787$	−2799.53	−0.126801	−0.0634005	−	0.997988i	$-0.520195\pi$
$787$	−2799.53	−0.126801	−0.0634005	+	0.997988i	$0.520195\pi$
$788$	−2754.76	−0.124536
$789$	15266.8	0.688862
$790$	2577.49	0.116080
$791$	−22242.3	−0.999803
$792$	−2478.35	−0.111192
$793$	−1184.70	−0.0530515
$794$	−1947.53	−0.0870471
$795$	4822.46	0.215138
$796$	4425.10	0.197040
$797$	21771.2	0.967597	0.483799	−	0.875179i	$-0.339257\pi$
$797$	21771.2	0.967597	0.483799	+	0.875179i	$0.339257\pi$
$798$	17200.3	0.763013
$799$	15169.4	0.671657
$800$	3655.57	0.161555
$801$	−17256.6	−0.761213
$802$	−4392.65	−0.193404
$803$	−6245.50	−0.274469
$804$	−3152.05	−0.138264
$805$	15085.6	0.660493
$806$	−38744.7	−1.69320
$807$	7910.24	0.345048
$808$	12387.7	0.539354
$809$	28676.9	1.24626	0.623130	−	0.782118i	$-0.285861\pi$
$809$	28676.9	1.24626	0.623130	+	0.782118i	$0.285861\pi$
$810$	1946.00	0.0844143
$811$	12021.0	0.520486	0.260243	−	0.965543i	$-0.416197\pi$
$811$	12021.0	0.520486	0.260243	+	0.965543i	$0.416197\pi$
$812$	0	0
$813$	−5358.07	−0.231139
$814$	6988.36	0.300911
$815$	−2612.09	−0.112267
$816$	3933.19	0.168737
$817$	−2006.36	−0.0859164
$818$	15694.8	0.670850
$819$	62429.1	2.66355
$820$	2878.29	0.122578
$821$	−14455.7	−0.614504	−0.307252	−	0.951628i	$-0.599409\pi$
$821$	−14455.7	−0.614504	−0.307252	+	0.951628i	$0.599409\pi$
$822$	6531.11	0.277127
$823$	43192.9	1.82941	0.914707	−	0.404117i	$-0.132421\pi$
$823$	43192.9	1.82941	0.914707	+	0.404117i	$0.132421\pi$
$824$	−6895.10	−0.291508
$825$	−3991.46	−0.168442
$826$	−12506.2	−0.526810
$827$	−10474.2	−0.440415	−0.220207	−	0.975453i	$-0.570673\pi$
$827$	−10474.2	−0.440415	−0.220207	+	0.975453i	$0.570673\pi$
$828$	11515.8	0.483335
$829$	13114.2	0.549425	0.274713	−	0.961526i	$-0.411417\pi$
$829$	13114.2	0.549425	0.274713	+	0.961526i	$0.411417\pi$
$830$	−2980.06	−0.124626
$831$	5644.28	0.235617
$832$	−5537.84	−0.230757
$833$	−82997.5	−3.45221
$834$	−4193.99	−0.174132
$835$	−11895.1	−0.492992
$836$	−6162.02	−0.254926
$837$	−25896.6	−1.06944
$838$	13886.4	0.572432
$839$	−4806.78	−0.197793	−0.0988965	−	0.995098i	$-0.531531\pi$
$839$	−4806.78	−0.197793	−0.0988965	+	0.995098i	$0.531531\pi$
$840$	−2135.72	−0.0877255
$841$	0	0
$842$	−21054.5	−0.861741
$843$	16440.9	0.671715
$844$	8376.32	0.341617
$845$	−17356.0	−0.706584
$846$	−6287.81	−0.255531
$847$	−37970.1	−1.54034
$848$	9811.23	0.397310
$849$	7266.37	0.293735
$850$	−23429.9	−0.945457
$851$	−32471.8	−1.30801
$852$	−49.3985	−0.00198634
$853$	−49150.9	−1.97291	−0.986456	−	0.164026i	$-0.947552\pi$
$853$	−49150.9	−1.97291	−0.986456	+	0.164026i	$0.947552\pi$
$854$	−929.538	−0.0372461
$855$	7369.52	0.294774
$856$	10995.2	0.439028
$857$	44886.9	1.78916	0.894578	−	0.446911i	$-0.147476\pi$
$857$	44886.9	1.78916	0.894578	+	0.446911i	$0.147476\pi$
$858$	6046.67	0.240594
$859$	28327.3	1.12516	0.562582	−	0.826742i	$-0.309808\pi$
$859$	28327.3	1.12516	0.562582	+	0.826742i	$0.309808\pi$
$860$	249.125	0.00987801
$861$	17847.6	0.706438
$862$	3471.16	0.137155
$863$	28693.4	1.13179	0.565895	−	0.824477i	$-0.308531\pi$
$863$	28693.4	1.13179	0.565895	+	0.824477i	$0.308531\pi$
$864$	−3701.45	−0.145747
$865$	−1925.93	−0.0757036
$866$	−18760.3	−0.736144
$867$	−13432.2	−0.526160
$868$	−30399.8	−1.18875
$869$	−5725.70	−0.223511
$870$	0	0
$871$	−28444.9	−1.10657
$872$	3740.17	0.145250
$873$	−241.258	−0.00935322
$874$	28632.1	1.10812
$875$	26643.6	1.02939
$876$	−4108.48	−0.158462
$877$	−29417.1	−1.13266	−0.566332	−	0.824177i	$-0.691638\pi$
$877$	−29417.1	−1.13266	−0.566332	+	0.824177i	$0.691638\pi$
$878$	8199.15	0.315157
$879$	4605.71	0.176731
$880$	765.123	0.0293094
$881$	6154.67	0.235365	0.117682	−	0.993051i	$-0.462454\pi$
$881$	6154.67	0.235365	0.117682	+	0.993051i	$0.462454\pi$
$882$	34403.0	1.31339
$883$	28371.7	1.08130	0.540648	−	0.841249i	$-0.318179\pi$
$883$	28371.7	1.08130	0.540648	+	0.841249i	$0.318179\pi$
$884$	35494.0	1.35045
$885$	1448.67	0.0550243
$886$	11436.4	0.433649
$887$	−36685.3	−1.38869	−0.694346	−	0.719641i	$-0.744306\pi$
$887$	−36685.3	−1.38869	−0.694346	+	0.719641i	$0.744306\pi$
$888$	4597.15	0.173728
$889$	−28195.9	−1.06373
$890$	5327.50	0.200650
$891$	−4322.90	−0.162539
$892$	24132.4	0.905845
$893$	−15633.6	−0.585845
$894$	9292.18	0.347625
$895$	2981.69	0.111360
$896$	−4345.10	−0.162009
$897$	−28096.2	−1.04582
$898$	4949.54	0.183929
$899$	0	0
$900$	9711.85	0.359698
$901$	−62883.8	−2.32515
$902$	−6393.90	−0.236024
$903$	1544.76	0.0569285
$904$	5241.79	0.192853
$905$	−7017.92	−0.257772
$906$	−7001.16	−0.256730
$907$	38189.3	1.39808	0.699039	−	0.715084i	$-0.253612\pi$
$907$	38189.3	1.39808	0.699039	+	0.715084i	$0.253612\pi$
$908$	757.967	0.0277027
$909$	32910.8	1.20086
$910$	−19273.3	−0.702091
$911$	−6696.40	−0.243537	−0.121768	−	0.992559i	$-0.538856\pi$
$911$	−6696.40	−0.243537	−0.121768	+	0.992559i	$0.538856\pi$
$912$	−4053.56	−0.147179
$913$	6619.98	0.239966
$914$	−21864.7	−0.791271
$915$	107.674	0.00389028
$916$	9391.20	0.338749
$917$	−26586.4	−0.957427
$918$	23723.9	0.852948
$919$	48279.1	1.73295	0.866474	−	0.499222i	$-0.166381\pi$
$919$	48279.1	1.73295	0.866474	+	0.499222i	$0.166381\pi$
$920$	−3555.18	−0.127403
$921$	−20938.7	−0.749134
$922$	33178.7	1.18512
$923$	−445.784	−0.0158972
$924$	4744.34	0.168915
$925$	−27385.1	−0.973423
$926$	−1788.49	−0.0634702
$927$	−18318.4	−0.649035
$928$	0	0
$929$	22375.0	0.790206	0.395103	−	0.918637i	$-0.370709\pi$
$929$	22375.0	0.790206	0.395103	+	0.918637i	$0.370709\pi$
$930$	3521.41	0.124163
$931$	85537.5	3.01115
$932$	2207.11	0.0775710
$933$	−475.399	−0.0166815
$934$	−16022.2	−0.561308
$935$	−4903.95	−0.171526
$936$	−14712.5	−0.513776
$937$	−27929.2	−0.973754	−0.486877	−	0.873470i	$-0.661864\pi$
$937$	−27929.2	−0.973754	−0.486877	+	0.873470i	$0.661864\pi$
$938$	−22318.4	−0.776890
$939$	9011.33	0.313178
$940$	1941.19	0.0673560
$941$	−11456.3	−0.396881	−0.198440	−	0.980113i	$-0.563588\pi$
$941$	−11456.3	−0.396881	−0.198440	+	0.980113i	$0.563588\pi$
$942$	−2018.54	−0.0698171
$943$	29709.6	1.02596
$944$	2947.30	0.101617
$945$	−12882.1	−0.443444
$946$	−553.412	−0.0190200
$947$	38730.6	1.32901	0.664506	−	0.747283i	$-0.268642\pi$
$947$	38730.6	1.32901	0.664506	+	0.747283i	$0.268642\pi$
$948$	−3766.53	−0.129041
$949$	−37075.9	−1.26821
$950$	24146.9	0.824663
$951$	−13289.2	−0.453137
$952$	27849.3	0.948112
$953$	56157.0	1.90882	0.954408	−	0.298504i	$-0.0964877\pi$
$953$	56157.0	1.90882	0.954408	+	0.298504i	$0.0964877\pi$
$954$	26065.8	0.884602
$955$	−5627.53	−0.190683
$956$	−18133.8	−0.613482
$957$	0	0
$958$	−29117.4	−0.981982
$959$	46244.2	1.55715
$960$	503.321	0.0169215
$961$	20332.7	0.682511
$962$	41485.8	1.39039
$963$	29211.3	0.977487
$964$	4593.11	0.153458
$965$	−8234.34	−0.274687
$966$	−22044.8	−0.734245
$967$	−7078.24	−0.235389	−0.117694	−	0.993050i	$-0.537550\pi$
$967$	−7078.24	−0.235389	−0.117694	+	0.993050i	$0.537550\pi$
$968$	8948.34	0.297118
$969$	25980.7	0.861323
$970$	74.4819	0.00246543
$971$	49719.3	1.64322	0.821610	−	0.570049i	$-0.193076\pi$
$971$	49719.3	1.64322	0.821610	+	0.570049i	$0.193076\pi$
$972$	−15336.1	−0.506076
$973$	−29696.0	−0.978427
$974$	35333.6	1.16238
$975$	−23694.9	−0.778302
$976$	219.062	0.00718444
$977$	41123.3	1.34662	0.673312	−	0.739358i	$-0.264871\pi$
$977$	41123.3	1.34662	0.673312	+	0.739358i	$0.264871\pi$
$978$	3817.09	0.124803
$979$	−11834.6	−0.386349
$980$	−10621.0	−0.346199
$981$	9936.60	0.323396
$982$	−9808.72	−0.318746
$983$	13824.2	0.448549	0.224275	−	0.974526i	$-0.427999\pi$
$983$	13824.2	0.448549	0.224275	+	0.974526i	$0.427999\pi$
$984$	−4206.10	−0.136266
$985$	2259.43	0.0730876
$986$	0	0
$987$	12036.8	0.388183
$988$	−36580.3	−1.17791
$989$	2571.46	0.0826770
$990$	2032.72	0.0652568
$991$	−13606.6	−0.436154	−0.218077	−	0.975932i	$-0.569978\pi$
$991$	−13606.6	−0.436154	−0.218077	+	0.975932i	$0.569978\pi$
$992$	7164.26	0.229300
$993$	−6894.02	−0.220317
$994$	−349.771	−0.0111610
$995$	−3629.43	−0.115639
$996$	4354.82	0.138542
$997$	41867.4	1.32995	0.664973	−	0.746868i	$-0.268443\pi$
$997$	41867.4	1.32995	0.664973	+	0.746868i	$0.268443\pi$
$998$	−4993.98	−0.158398
$999$	27728.7	0.878177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.4.a.d.1.2		3
29.28	even	2		58.4.a.d.1.2	✓	3
87.86	odd	2		522.4.a.k.1.3		3
116.115	odd	2		464.4.a.i.1.2		3
145.144	even	2		1450.4.a.h.1.2		3
232.115	odd	2		1856.4.a.s.1.2		3
232.173	even	2		1856.4.a.r.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.d.1.2	✓	3	29.28	even	2
464.4.a.i.1.2		3	116.115	odd	2
522.4.a.k.1.3		3	87.86	odd	2
1450.4.a.h.1.2		3	145.144	even	2
1682.4.a.d.1.2		3	1.1	even	1	trivial
1856.4.a.r.1.2		3	232.173	even	2
1856.4.a.s.1.2		3	232.115	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -2.39712 q^{3} +4.00000 q^{4} -3.28077 q^{5} +4.79424 q^{6} +33.9461 q^{7} -8.00000 q^{8} -21.2538 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -2.39712 q^{3} +4.00000 q^{4} -3.28077 q^{5} +4.79424 q^{6} +33.9461 q^{7} -8.00000 q^{8} -21.2538 q^{9} +6.56153 q^{10} -14.5759 q^{11} -9.58848 q^{12} -86.5287 q^{13} -67.8922 q^{14} +7.86439 q^{15} +16.0000 q^{16} -102.550 q^{17} +42.5076 q^{18} +105.688 q^{19} -13.1231 q^{20} -81.3729 q^{21} +29.1519 q^{22} -135.456 q^{23} +19.1770 q^{24} -114.237 q^{25} +173.057 q^{26} +115.670 q^{27} +135.784 q^{28} -15.7288 q^{30} -223.883 q^{31} -32.0000 q^{32} +34.9403 q^{33} +205.100 q^{34} -111.369 q^{35} -85.0153 q^{36} +239.723 q^{37} -211.377 q^{38} +207.420 q^{39} +26.2461 q^{40} -219.331 q^{41} +162.746 q^{42} -18.9838 q^{43} -58.3037 q^{44} +69.7288 q^{45} +270.911 q^{46} -147.922 q^{47} -38.3539 q^{48} +809.338 q^{49} +228.473 q^{50} +245.824 q^{51} -346.115 q^{52} +613.202 q^{53} -231.340 q^{54} +47.8202 q^{55} -271.569 q^{56} -253.348 q^{57} +184.206 q^{59} +31.4576 q^{60} +13.6914 q^{61} +447.766 q^{62} -721.484 q^{63} +64.0000 q^{64} +283.880 q^{65} -69.8805 q^{66} +328.733 q^{67} -410.199 q^{68} +324.703 q^{69} +222.738 q^{70} +5.15186 q^{71} +170.031 q^{72} +428.481 q^{73} -479.445 q^{74} +273.839 q^{75} +422.753 q^{76} -494.796 q^{77} -414.840 q^{78} +392.819 q^{79} -52.4922 q^{80} +296.578 q^{81} +438.661 q^{82} -454.172 q^{83} -325.492 q^{84} +336.442 q^{85} +37.9675 q^{86} +116.607 q^{88} +811.929 q^{89} -139.458 q^{90} -2937.31 q^{91} -541.822 q^{92} +536.675 q^{93} +295.844 q^{94} -346.738 q^{95} +76.7078 q^{96} +11.3513 q^{97} -1618.68 q^{98} +309.794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9	\( 3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9} - 40 q^{10} - 10 q^{11} - 8 q^{12} - 4 q^{13} - 48 q^{14} + 130 q^{15} + 48 q^{16} + 66 q^{17} - 10 q^{18} + 164 q^{19} + 80 q^{20} + 88 q^{21} + 20 q^{22} - 204 q^{23} + 16 q^{24} + 79 q^{25} + 8 q^{26} + 142 q^{27} + 96 q^{28} - 260 q^{30} + 86 q^{31} - 96 q^{32} - 130 q^{33} - 132 q^{34} + 24 q^{35} + 20 q^{36} + 42 q^{37} - 328 q^{38} + 394 q^{39} - 160 q^{40} - 562 q^{41} - 176 q^{42} - 18 q^{43} - 40 q^{44} + 422 q^{45} + 408 q^{46} - 654 q^{47} - 32 q^{48} + 539 q^{49} - 158 q^{50} + 556 q^{51} - 16 q^{52} + 712 q^{53} - 284 q^{54} - 142 q^{55} - 192 q^{56} + 828 q^{57} + 184 q^{59} + 520 q^{60} - 322 q^{61} - 172 q^{62} - 784 q^{63} + 192 q^{64} + 1494 q^{65} + 260 q^{66} - 228 q^{67} + 264 q^{68} - 684 q^{69} - 48 q^{70} - 52 q^{71} - 40 q^{72} + 494 q^{73} - 84 q^{74} + 3048 q^{75} + 656 q^{76} - 872 q^{77} - 788 q^{78} + 2110 q^{79} + 320 q^{80} - 1513 q^{81} + 1124 q^{82} - 288 q^{83} + 352 q^{84} + 2704 q^{85} + 36 q^{86} + 80 q^{88} - 914 q^{89} - 844 q^{90} - 2984 q^{91} - 816 q^{92} - 62 q^{93} + 1308 q^{94} + 1900 q^{95} + 64 q^{96} - 218 q^{97} - 1078 q^{98} + 304 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9 - 40 * q^10 - 10 * q^11 - 8 * q^12 - 4 * q^13 - 48 * q^14 + 130 * q^15 + 48 * q^16 + 66 * q^17 - 10 * q^18 + 164 * q^19 + 80 * q^20 + 88 * q^21 + 20 * q^22 - 204 * q^23 + 16 * q^24 + 79 * q^25 + 8 * q^26 + 142 * q^27 + 96 * q^28 - 260 * q^30 + 86 * q^31 - 96 * q^32 - 130 * q^33 - 132 * q^34 + 24 * q^35 + 20 * q^36 + 42 * q^37 - 328 * q^38 + 394 * q^39 - 160 * q^40 - 562 * q^41 - 176 * q^42 - 18 * q^43 - 40 * q^44 + 422 * q^45 + 408 * q^46 - 654 * q^47 - 32 * q^48 + 539 * q^49 - 158 * q^50 + 556 * q^51 - 16 * q^52 + 712 * q^53 - 284 * q^54 - 142 * q^55 - 192 * q^56 + 828 * q^57 + 184 * q^59 + 520 * q^60 - 322 * q^61 - 172 * q^62 - 784 * q^63 + 192 * q^64 + 1494 * q^65 + 260 * q^66 - 228 * q^67 + 264 * q^68 - 684 * q^69 - 48 * q^70 - 52 * q^71 - 40 * q^72 + 494 * q^73 - 84 * q^74 + 3048 * q^75 + 656 * q^76 - 872 * q^77 - 788 * q^78 + 2110 * q^79 + 320 * q^80 - 1513 * q^81 + 1124 * q^82 - 288 * q^83 + 352 * q^84 + 2704 * q^85 + 36 * q^86 + 80 * q^88 - 914 * q^89 - 844 * q^90 - 2984 * q^91 - 816 * q^92 - 62 * q^93 + 1308 * q^94 + 1900 * q^95 + 64 * q^96 - 218 * q^97 - 1078 * q^98 + 304 * q^99

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