LMFDB - Embedded newform 2013.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.61297 q^{2} +3.00000 q^{3} +23.5055 q^{4} +13.7147 q^{5} -16.8389 q^{6} +34.2924 q^{7} -87.0318 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.61297 q^{2} +3.00000 q^{3} +23.5055 q^{4} +13.7147 q^{5} -16.8389 q^{6} +34.2924 q^{7} -87.0318 q^{8} +9.00000 q^{9} -76.9805 q^{10} +11.0000 q^{11} +70.5164 q^{12} -25.5063 q^{13} -192.482 q^{14} +41.1442 q^{15} +300.463 q^{16} -130.348 q^{17} -50.5168 q^{18} -117.514 q^{19} +322.371 q^{20} +102.877 q^{21} -61.7427 q^{22} -141.022 q^{23} -261.095 q^{24} +63.0942 q^{25} +143.166 q^{26} +27.0000 q^{27} +806.058 q^{28} +72.2543 q^{29} -230.941 q^{30} -99.2898 q^{31} -990.238 q^{32} +33.0000 q^{33} +731.641 q^{34} +470.311 q^{35} +211.549 q^{36} -166.592 q^{37} +659.604 q^{38} -76.5190 q^{39} -1193.62 q^{40} -206.647 q^{41} -577.446 q^{42} +172.235 q^{43} +258.560 q^{44} +123.433 q^{45} +791.554 q^{46} -554.557 q^{47} +901.390 q^{48} +832.966 q^{49} -354.146 q^{50} -391.045 q^{51} -599.538 q^{52} +198.211 q^{53} -151.550 q^{54} +150.862 q^{55} -2984.53 q^{56} -352.543 q^{57} -405.561 q^{58} +373.596 q^{59} +967.114 q^{60} +61.0000 q^{61} +557.311 q^{62} +308.631 q^{63} +3154.47 q^{64} -349.813 q^{65} -185.228 q^{66} +273.813 q^{67} -3063.90 q^{68} -423.067 q^{69} -2639.84 q^{70} -48.9014 q^{71} -783.286 q^{72} -1025.09 q^{73} +935.077 q^{74} +189.282 q^{75} -2762.23 q^{76} +377.216 q^{77} +429.499 q^{78} +446.621 q^{79} +4120.78 q^{80} +81.0000 q^{81} +1159.91 q^{82} -1050.85 q^{83} +2418.17 q^{84} -1787.69 q^{85} -966.749 q^{86} +216.763 q^{87} -957.349 q^{88} -482.691 q^{89} -692.824 q^{90} -874.673 q^{91} -3314.79 q^{92} -297.869 q^{93} +3112.72 q^{94} -1611.68 q^{95} -2970.71 q^{96} -1217.43 q^{97} -4675.42 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 14 q^{2} + 108 q^{3} + 146 q^{4} - 65 q^{5} - 42 q^{6} - 105 q^{7} - 189 q^{8} + 324 q^{9}+O(q^{10}) $ 36 * q - 14 * q^2 + 108 * q^3 + 146 * q^4 - 65 * q^5 - 42 * q^6 - 105 * q^7 - 189 * q^8 + 324 * q^9	\( 36 q - 14 q^{2} + 108 q^{3} + 146 q^{4} - 65 q^{5} - 42 q^{6} - 105 q^{7} - 189 q^{8} + 324 q^{9} - 161 q^{10} + 396 q^{11} + 438 q^{12} - 233 q^{13} - 264 q^{14} - 195 q^{15} + 574 q^{16} - 556 q^{17} - 126 q^{18} - 615 q^{19} - 136 q^{20} - 315 q^{21} - 154 q^{22} - 457 q^{23} - 567 q^{24} + 863 q^{25} + 115 q^{26} + 972 q^{27} - 424 q^{28} - 754 q^{29} - 483 q^{30} - 508 q^{31} - 1511 q^{32} + 1188 q^{33} - 860 q^{34} - 826 q^{35} + 1314 q^{36} - 412 q^{37} - 599 q^{38} - 699 q^{39} - 2791 q^{40} - 2066 q^{41} - 792 q^{42} - 2063 q^{43} + 1606 q^{44} - 585 q^{45} - 787 q^{46} - 1815 q^{47} + 1722 q^{48} + 2825 q^{49} + 808 q^{50} - 1668 q^{51} - 2882 q^{52} - 759 q^{53} - 378 q^{54} - 715 q^{55} - 1749 q^{56} - 1845 q^{57} - 335 q^{58} - 2337 q^{59} - 408 q^{60} + 2196 q^{61} - 1689 q^{62} - 945 q^{63} + 4723 q^{64} - 3550 q^{65} - 462 q^{66} - 1331 q^{67} - 6166 q^{68} - 1371 q^{69} - 1750 q^{70} - 361 q^{71} - 1701 q^{72} - 4627 q^{73} - 3394 q^{74} + 2589 q^{75} - 7214 q^{76} - 1155 q^{77} + 345 q^{78} - 2583 q^{79} - 2643 q^{80} + 2916 q^{81} + 1090 q^{82} - 6123 q^{83} - 1272 q^{84} + 295 q^{85} + 613 q^{86} - 2262 q^{87} - 2079 q^{88} - 2485 q^{89} - 1449 q^{90} - 3156 q^{91} - 6291 q^{92} - 1524 q^{93} - 1744 q^{94} - 5572 q^{95} - 4533 q^{96} - 2558 q^{97} - 2314 q^{98} + 3564 q^{99}+O(q^{100}) \) 36 * q - 14 * q^2 + 108 * q^3 + 146 * q^4 - 65 * q^5 - 42 * q^6 - 105 * q^7 - 189 * q^8 + 324 * q^9 - 161 * q^10 + 396 * q^11 + 438 * q^12 - 233 * q^13 - 264 * q^14 - 195 * q^15 + 574 * q^16 - 556 * q^17 - 126 * q^18 - 615 * q^19 - 136 * q^20 - 315 * q^21 - 154 * q^22 - 457 * q^23 - 567 * q^24 + 863 * q^25 + 115 * q^26 + 972 * q^27 - 424 * q^28 - 754 * q^29 - 483 * q^30 - 508 * q^31 - 1511 * q^32 + 1188 * q^33 - 860 * q^34 - 826 * q^35 + 1314 * q^36 - 412 * q^37 - 599 * q^38 - 699 * q^39 - 2791 * q^40 - 2066 * q^41 - 792 * q^42 - 2063 * q^43 + 1606 * q^44 - 585 * q^45 - 787 * q^46 - 1815 * q^47 + 1722 * q^48 + 2825 * q^49 + 808 * q^50 - 1668 * q^51 - 2882 * q^52 - 759 * q^53 - 378 * q^54 - 715 * q^55 - 1749 * q^56 - 1845 * q^57 - 335 * q^58 - 2337 * q^59 - 408 * q^60 + 2196 * q^61 - 1689 * q^62 - 945 * q^63 + 4723 * q^64 - 3550 * q^65 - 462 * q^66 - 1331 * q^67 - 6166 * q^68 - 1371 * q^69 - 1750 * q^70 - 361 * q^71 - 1701 * q^72 - 4627 * q^73 - 3394 * q^74 + 2589 * q^75 - 7214 * q^76 - 1155 * q^77 + 345 * q^78 - 2583 * q^79 - 2643 * q^80 + 2916 * q^81 + 1090 * q^82 - 6123 * q^83 - 1272 * q^84 + 295 * q^85 + 613 * q^86 - 2262 * q^87 - 2079 * q^88 - 2485 * q^89 - 1449 * q^90 - 3156 * q^91 - 6291 * q^92 - 1524 * q^93 - 1744 * q^94 - 5572 * q^95 - 4533 * q^96 - 2558 * q^97 - 2314 * q^98 + 3564 * 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 $

$2$	−5.61297	−1.98449	−0.992243	−	0.124315i	$-0.960327\pi$
$2$	−5.61297	−1.98449	−0.992243	+	0.124315i	$0.960327\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	23.5055	2.93818
$5$	13.7147	1.22668	0.613342	−	0.789817i	$-0.289825\pi$
$5$	13.7147	1.22668	0.613342	+	0.789817i	$0.289825\pi$
$6$	−16.8389	−1.14574
$7$	34.2924	1.85161	0.925807	−	0.377997i	$-0.123387\pi$
$7$	34.2924	1.85161	0.925807	+	0.377997i	$0.123387\pi$
$8$	−87.0318	−3.84630
$9$	9.00000	0.333333
$10$	−76.9805	−2.43434
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	70.5164	1.69636
$13$	−25.5063	−0.544168	−0.272084	−	0.962273i	$-0.587713\pi$
$13$	−25.5063	−0.544168	−0.272084	+	0.962273i	$0.587713\pi$
$14$	−192.482	−3.67450
$15$	41.1442	0.708226
$16$	300.463	4.69474
$17$	−130.348	−1.85965	−0.929826	−	0.368000i	$-0.880043\pi$
$17$	−130.348	−1.85965	−0.929826	+	0.368000i	$0.880043\pi$
$18$	−50.5168	−0.661495
$19$	−117.514	−1.41893	−0.709463	−	0.704742i	$-0.751063\pi$
$19$	−117.514	−1.41893	−0.709463	+	0.704742i	$0.751063\pi$
$20$	322.371	3.60422
$21$	102.877	1.06903
$22$	−61.7427	−0.598345
$23$	−141.022	−1.27849	−0.639243	−	0.769005i	$-0.720752\pi$
$23$	−141.022	−1.27849	−0.639243	+	0.769005i	$0.720752\pi$
$24$	−261.095	−2.22066
$25$	63.0942	0.504753
$26$	143.166	1.07989
$27$	27.0000	0.192450
$28$	806.058	5.44038
$29$	72.2543	0.462665	0.231333	−	0.972875i	$-0.425691\pi$
$29$	72.2543	0.462665	0.231333	+	0.972875i	$0.425691\pi$
$30$	−230.941	−1.40546
$31$	−99.2898	−0.575257	−0.287629	−	0.957742i	$-0.592867\pi$
$31$	−99.2898	−0.575257	−0.287629	+	0.957742i	$0.592867\pi$
$32$	−990.238	−5.47034
$33$	33.0000	0.174078
$34$	731.641	3.69045
$35$	470.311	2.27134
$36$	211.549	0.979394
$37$	−166.592	−0.740205	−0.370103	−	0.928991i	$-0.620677\pi$
$37$	−166.592	−0.740205	−0.370103	+	0.928991i	$0.620677\pi$
$38$	659.604	2.81584
$39$	−76.5190	−0.314176
$40$	−1193.62	−4.71819
$41$	−206.647	−0.787144	−0.393572	−	0.919294i	$-0.628761\pi$
$41$	−206.647	−0.787144	−0.393572	+	0.919294i	$0.628761\pi$
$42$	−577.446	−2.12147
$43$	172.235	0.610827	0.305413	−	0.952220i	$-0.401205\pi$
$43$	172.235	0.610827	0.305413	+	0.952220i	$0.401205\pi$
$44$	258.560	0.885896
$45$	123.433	0.408895
$46$	791.554	2.53714
$47$	−554.557	−1.72107	−0.860537	−	0.509387i	$-0.829872\pi$
$47$	−554.557	−1.72107	−0.860537	+	0.509387i	$0.829872\pi$
$48$	901.390	2.71051
$49$	832.966	2.42847
$50$	−354.146	−1.00168
$51$	−391.045	−1.07367
$52$	−599.538	−1.59887
$53$	198.211	0.513705	0.256853	−	0.966451i	$-0.417315\pi$
$53$	198.211	0.513705	0.256853	+	0.966451i	$0.417315\pi$
$54$	−151.550	−0.381914
$55$	150.862	0.369859
$56$	−2984.53	−7.12186
$57$	−352.543	−0.819218
$58$	−405.561	−0.918152
$59$	373.596	0.824374	0.412187	−	0.911099i	$-0.364765\pi$
$59$	373.596	0.824374	0.412187	+	0.911099i	$0.364765\pi$
$60$	967.114	2.08090
$61$	61.0000	0.128037
$61$	61.0000	0.128037
$62$	557.311	1.14159
$63$	308.631	0.617205
$64$	3154.47	6.16108
$65$	−349.813	−0.667522
$66$	−185.228	−0.345455
$67$	273.813	0.499277	0.249638	−	0.968339i	$-0.419688\pi$
$67$	273.813	0.499277	0.249638	+	0.968339i	$0.419688\pi$
$68$	−3063.90	−5.46400
$69$	−423.067	−0.738134
$70$	−2639.84	−4.50745
$71$	−48.9014	−0.0817398	−0.0408699	−	0.999164i	$-0.513013\pi$
$71$	−48.9014	−0.0817398	−0.0408699	+	0.999164i	$0.513013\pi$
$72$	−783.286	−1.28210
$73$	−1025.09	−1.64353	−0.821767	−	0.569824i	$-0.807011\pi$
$73$	−1025.09	−1.64353	−0.821767	+	0.569824i	$0.807011\pi$
$74$	935.077	1.46893
$75$	189.282	0.291419
$76$	−2762.23	−4.16907
$77$	377.216	0.558283
$78$	429.499	0.623477
$79$	446.621	0.636061	0.318030	−	0.948081i	$-0.396979\pi$
$79$	446.621	0.636061	0.318030	+	0.948081i	$0.396979\pi$
$80$	4120.78	5.75896
$81$	81.0000	0.111111
$82$	1159.91	1.56208
$83$	−1050.85	−1.38971	−0.694855	−	0.719150i	$-0.744531\pi$
$83$	−1050.85	−1.38971	−0.694855	+	0.719150i	$0.744531\pi$
$84$	2418.17	3.14101
$85$	−1787.69	−2.28121
$86$	−966.749	−1.21218
$87$	216.763	0.267120
$88$	−957.349	−1.15970
$89$	−482.691	−0.574889	−0.287444	−	0.957797i	$-0.592806\pi$
$89$	−482.691	−0.574889	−0.287444	+	0.957797i	$0.592806\pi$
$90$	−692.824	−0.811446
$91$	−874.673	−1.00759
$92$	−3314.79	−3.75643
$93$	−297.869	−0.332125
$94$	3112.72	3.41545
$95$	−1611.68	−1.74057
$96$	−2970.71	−3.15830
$97$	−1217.43	−1.27434	−0.637171	−	0.770722i	$-0.719896\pi$
$97$	−1217.43	−1.27434	−0.637171	+	0.770722i	$0.719896\pi$
$98$	−4675.42	−4.81927
$99$	99.0000	0.100504
$100$	1483.06	1.48306
$101$	−1023.03	−1.00788	−0.503939	−	0.863739i	$-0.668116\pi$
$101$	−1023.03	−1.00788	−0.503939	+	0.863739i	$0.668116\pi$
$102$	2194.92	2.13068
$103$	1019.81	0.975578	0.487789	−	0.872962i	$-0.337804\pi$
$103$	1019.81	0.975578	0.487789	+	0.872962i	$0.337804\pi$
$104$	2219.86	2.09303
$105$	1410.93	1.31136
$106$	−1112.55	−1.01944
$107$	−1933.85	−1.74722	−0.873611	−	0.486625i	$-0.838228\pi$
$107$	−1933.85	−1.74722	−0.873611	+	0.486625i	$0.838228\pi$
$108$	634.648	0.565454
$109$	−466.056	−0.409542	−0.204771	−	0.978810i	$-0.565645\pi$
$109$	−466.056	−0.409542	−0.204771	+	0.978810i	$0.565645\pi$
$110$	−846.785	−0.733980
$111$	−499.777	−0.427358
$112$	10303.6	8.69284
$113$	1862.53	1.55055	0.775277	−	0.631622i	$-0.217610\pi$
$113$	1862.53	1.55055	0.775277	+	0.631622i	$0.217610\pi$
$114$	1978.81	1.62573
$115$	−1934.08	−1.56830
$116$	1698.37	1.35939
$117$	−229.557	−0.181389
$118$	−2096.98	−1.63596
$119$	−4469.95	−3.44336
$120$	−3580.86	−2.72405
$121$	121.000	0.0909091
$122$	−342.391	−0.254087
$123$	−619.942	−0.454458
$124$	−2333.85	−1.69021
$125$	−849.023	−0.607511
$126$	−1732.34	−1.22483
$127$	−851.380	−0.594864	−0.297432	−	0.954743i	$-0.596130\pi$
$127$	−851.380	−0.594864	−0.297432	+	0.954743i	$0.596130\pi$
$128$	−9784.07	−6.75623
$129$	516.704	0.352661
$130$	1963.49	1.32469
$131$	668.258	0.445695	0.222847	−	0.974853i	$-0.428465\pi$
$131$	668.258	0.445695	0.222847	+	0.974853i	$0.428465\pi$
$132$	775.680	0.511472
$133$	−4029.84	−2.62730
$134$	−1536.90	−0.990807
$135$	370.298	0.236075
$136$	11344.4	7.15277
$137$	1356.67	0.846046	0.423023	−	0.906119i	$-0.360969\pi$
$137$	1356.67	0.846046	0.423023	+	0.906119i	$0.360969\pi$
$138$	2374.66	1.46482
$139$	−672.028	−0.410077	−0.205038	−	0.978754i	$-0.565732\pi$
$139$	−672.028	−0.410077	−0.205038	+	0.978754i	$0.565732\pi$
$140$	11054.9	6.67363
$141$	−1663.67	−0.993663
$142$	274.482	0.162211
$143$	−280.570	−0.164073
$144$	2704.17	1.56491
$145$	990.949	0.567544
$146$	5753.81	3.26157
$147$	2498.90	1.40208
$148$	−3915.83	−2.17486
$149$	−613.725	−0.337438	−0.168719	−	0.985664i	$-0.553963\pi$
$149$	−613.725	−0.337438	−0.168719	+	0.985664i	$0.553963\pi$
$150$	−1062.44	−0.578318
$151$	2131.60	1.14879	0.574395	−	0.818578i	$-0.305237\pi$
$151$	2131.60	1.14879	0.574395	+	0.818578i	$0.305237\pi$
$152$	10227.5	5.45761
$153$	−1173.13	−0.619884
$154$	−2117.30	−1.10790
$155$	−1361.73	−0.705659
$156$	−1798.62	−0.923106
$157$	−1002.15	−0.509430	−0.254715	−	0.967016i	$-0.581982\pi$
$157$	−1002.15	−0.509430	−0.254715	+	0.967016i	$0.581982\pi$
$158$	−2506.87	−1.26225
$159$	594.633	0.296588
$160$	−13580.9	−6.71038
$161$	−4835.99	−2.36726
$162$	−454.651	−0.220498
$163$	−3165.47	−1.52110	−0.760548	−	0.649282i	$-0.775070\pi$
$163$	−3165.47	−1.52110	−0.760548	+	0.649282i	$0.775070\pi$
$164$	−4857.34	−2.31277
$165$	452.587	0.213538
$166$	5898.40	2.75786
$167$	1926.70	0.892769	0.446385	−	0.894841i	$-0.352711\pi$
$167$	1926.70	0.892769	0.446385	+	0.894841i	$0.352711\pi$
$168$	−8953.58	−4.11181
$169$	−1546.43	−0.703881
$170$	10034.3	4.52702
$171$	−1057.63	−0.472976
$172$	4048.46	1.79472
$173$	1860.52	0.817645	0.408823	−	0.912614i	$-0.365940\pi$
$173$	1860.52	0.817645	0.408823	+	0.912614i	$0.365940\pi$
$174$	−1216.68	−0.530095
$175$	2163.65	0.934608
$176$	3305.10	1.41552
$177$	1120.79	0.475952
$178$	2709.33	1.14086
$179$	675.290	0.281975	0.140988	−	0.990011i	$-0.454972\pi$
$179$	675.290	0.281975	0.140988	+	0.990011i	$0.454972\pi$
$180$	2901.34	1.20141
$181$	1364.22	0.560229	0.280114	−	0.959967i	$-0.409628\pi$
$181$	1364.22	0.560229	0.280114	+	0.959967i	$0.409628\pi$
$182$	4909.51	1.99955
$183$	183.000	0.0739221
$184$	12273.4	4.91744
$185$	−2284.77	−0.907998
$186$	1671.93	0.659097
$187$	−1433.83	−0.560706
$188$	−13035.1	−5.05683
$189$	925.894	0.356343
$190$	9046.30	3.45415
$191$	−1414.42	−0.535832	−0.267916	−	0.963442i	$-0.586335\pi$
$191$	−1414.42	−0.535832	−0.267916	+	0.963442i	$0.586335\pi$
$192$	9463.42	3.55710
$193$	5094.73	1.90014	0.950070	−	0.312038i	$-0.101012\pi$
$193$	5094.73	1.90014	0.950070	+	0.312038i	$0.101012\pi$
$194$	6833.40	2.52891
$195$	−1049.44	−0.385394
$196$	19579.3	7.13530
$197$	1607.39	0.581327	0.290664	−	0.956825i	$-0.406124\pi$
$197$	1607.39	0.581327	0.290664	+	0.956825i	$0.406124\pi$
$198$	−555.684	−0.199448
$199$	3034.18	1.08084	0.540421	−	0.841395i	$-0.318265\pi$
$199$	3034.18	1.08084	0.540421	+	0.841395i	$0.318265\pi$
$200$	−5491.20	−1.94143
$201$	821.438	0.288257
$202$	5742.26	2.00012
$203$	2477.77	0.856677
$204$	−9191.69	−3.15464
$205$	−2834.12	−0.965577
$206$	−5724.15	−1.93602
$207$	−1269.20	−0.426162
$208$	−7663.72	−2.55473
$209$	−1292.66	−0.427823
$210$	−7919.53	−2.60238
$211$	−684.062	−0.223189	−0.111594	−	0.993754i	$-0.535596\pi$
$211$	−684.062	−0.223189	−0.111594	+	0.993754i	$0.535596\pi$
$212$	4659.04	1.50936
$213$	−146.704	−0.0471925
$214$	10854.7	3.46734
$215$	2362.16	0.749291
$216$	−2349.86	−0.740220
$217$	−3404.88	−1.06515
$218$	2615.96	0.812730
$219$	−3075.28	−0.948895
$220$	3546.09	1.08671
$221$	3324.71	1.01196
$222$	2805.23	0.848085
$223$	3584.12	1.07628	0.538140	−	0.842856i	$-0.319127\pi$
$223$	3584.12	1.07628	0.538140	+	0.842856i	$0.319127\pi$
$224$	−33957.6	−10.1290
$225$	567.847	0.168251
$226$	−10454.4	−3.07705
$227$	215.889	0.0631237	0.0315618	−	0.999502i	$-0.489952\pi$
$227$	215.889	0.0631237	0.0315618	+	0.999502i	$0.489952\pi$
$228$	−8286.68	−2.40701
$229$	537.552	0.155120	0.0775599	−	0.996988i	$-0.475287\pi$
$229$	537.552	0.155120	0.0775599	+	0.996988i	$0.475287\pi$
$230$	10856.0	3.11227
$231$	1131.65	0.322325
$232$	−6288.42	−1.77955
$233$	−1638.61	−0.460724	−0.230362	−	0.973105i	$-0.573991\pi$
$233$	−1638.61	−0.460724	−0.230362	+	0.973105i	$0.573991\pi$
$234$	1288.50	0.359965
$235$	−7605.61	−2.11121
$236$	8781.55	2.42216
$237$	1339.86	0.367230
$238$	25089.7	6.83329
$239$	−4301.68	−1.16424	−0.582118	−	0.813104i	$-0.697776\pi$
$239$	−4301.68	−1.16424	−0.582118	+	0.813104i	$0.697776\pi$
$240$	12362.3	3.32494
$241$	−5957.95	−1.59247	−0.796234	−	0.604988i	$-0.793178\pi$
$241$	−5957.95	−1.59247	−0.796234	+	0.604988i	$0.793178\pi$
$242$	−679.170	−0.180408
$243$	243.000	0.0641500
$244$	1433.83	0.376196
$245$	11423.9	2.97897
$246$	3479.72	0.901865
$247$	2997.36	0.772135
$248$	8641.37	2.21261
$249$	−3152.55	−0.802350
$250$	4765.54	1.20560
$251$	2303.08	0.579161	0.289580	−	0.957154i	$-0.406484\pi$
$251$	2303.08	0.579161	0.289580	+	0.957154i	$0.406484\pi$
$252$	7254.52	1.81346
$253$	−1551.25	−0.385478
$254$	4778.77	1.18050
$255$	−5363.08	−1.31705
$256$	29681.9	7.24657
$257$	−716.075	−0.173804	−0.0869018	−	0.996217i	$-0.527697\pi$
$257$	−716.075	−0.173804	−0.0869018	+	0.996217i	$0.527697\pi$
$258$	−2900.25	−0.699851
$259$	−5712.84	−1.37057
$260$	−8222.52	−1.96130
$261$	650.289	0.154222
$262$	−3750.92	−0.884475
$263$	−2153.70	−0.504954	−0.252477	−	0.967603i	$-0.581245\pi$
$263$	−2153.70	−0.504954	−0.252477	+	0.967603i	$0.581245\pi$
$264$	−2872.05	−0.669554
$265$	2718.41	0.630154
$266$	22619.4	5.21385
$267$	−1448.07	−0.331912
$268$	6436.09	1.46697
$269$	997.198	0.226023	0.113012	−	0.993594i	$-0.463950\pi$
$269$	997.198	0.226023	0.113012	+	0.993594i	$0.463950\pi$
$270$	−2078.47	−0.468488
$271$	1100.87	0.246765	0.123382	−	0.992359i	$-0.460626\pi$
$271$	1100.87	0.246765	0.123382	+	0.992359i	$0.460626\pi$
$272$	−39164.8	−8.73058
$273$	−2624.02	−0.581732
$274$	−7614.96	−1.67897
$275$	694.036	0.152189
$276$	−9944.38	−2.16877
$277$	2224.73	0.482567	0.241283	−	0.970455i	$-0.422432\pi$
$277$	2224.73	0.482567	0.241283	+	0.970455i	$0.422432\pi$
$278$	3772.08	0.813792
$279$	−893.608	−0.191752
$280$	−40932.0	−8.73627
$281$	−3272.21	−0.694676	−0.347338	−	0.937740i	$-0.612914\pi$
$281$	−3272.21	−0.694676	−0.347338	+	0.937740i	$0.612914\pi$
$282$	9338.15	1.97191
$283$	−7393.93	−1.55309	−0.776543	−	0.630064i	$-0.783028\pi$
$283$	−7393.93	−1.55309	−0.776543	+	0.630064i	$0.783028\pi$
$284$	−1149.45	−0.240167
$285$	−4835.03	−1.00492
$286$	1574.83	0.325600
$287$	−7086.43	−1.45749
$288$	−8912.14	−1.82345
$289$	12077.7	2.45831
$290$	−5562.17	−1.12628
$291$	−3652.29	−0.735742
$292$	−24095.3	−4.82900
$293$	2964.98	0.591181	0.295590	−	0.955315i	$-0.404484\pi$
$293$	2964.98	0.591181	0.295590	+	0.955315i	$0.404484\pi$
$294$	−14026.3	−2.78241
$295$	5123.77	1.01125
$296$	14498.8	2.84705
$297$	297.000	0.0580259
$298$	3444.82	0.669641
$299$	3596.96	0.695711
$300$	4449.17	0.856244
$301$	5906.34	1.13102
$302$	−11964.6	−2.27976
$303$	−3069.10	−0.581898
$304$	−35308.7	−6.66149
$305$	836.599	0.157061
$306$	6584.77	1.23015
$307$	409.994	0.0762201	0.0381101	−	0.999274i	$-0.487866\pi$
$307$	409.994	0.0762201	0.0381101	+	0.999274i	$0.487866\pi$
$308$	8866.64	1.64034
$309$	3059.42	0.563250
$310$	7643.38	1.40037
$311$	−8311.22	−1.51539	−0.757694	−	0.652610i	$-0.773674\pi$
$311$	−8311.22	−1.51539	−0.757694	+	0.652610i	$0.773674\pi$
$312$	6659.59	1.20841
$313$	−4194.27	−0.757425	−0.378713	−	0.925514i	$-0.623633\pi$
$313$	−4194.27	−0.757425	−0.378713	+	0.925514i	$0.623633\pi$
$314$	5625.06	1.01096
$315$	4232.80	0.757115
$316$	10498.0	1.86886
$317$	191.675	0.0339607	0.0169804	−	0.999856i	$-0.494595\pi$
$317$	191.675	0.0339607	0.0169804	+	0.999856i	$0.494595\pi$
$318$	−3337.66	−0.588574
$319$	794.797	0.139499
$320$	43262.8	7.55770
$321$	−5801.56	−1.00876
$322$	27144.3	4.69780
$323$	15317.8	2.63871
$324$	1903.94	0.326465
$325$	−1609.30	−0.274671
$326$	17767.7	3.01859
$327$	−1398.17	−0.236449
$328$	17984.9	3.02759
$329$	−19017.1	−3.18677
$330$	−2540.36	−0.423764
$331$	−3945.20	−0.655130	−0.327565	−	0.944829i	$-0.606228\pi$
$331$	−3945.20	−0.655130	−0.327565	+	0.944829i	$0.606228\pi$
$332$	−24700.8	−4.08322
$333$	−1499.33	−0.246735
$334$	−10814.5	−1.77169
$335$	3755.27	0.612455
$336$	30910.8	5.01881
$337$	6668.15	1.07786	0.538928	−	0.842352i	$-0.318829\pi$
$337$	6668.15	1.07786	0.538928	+	0.842352i	$0.318829\pi$
$338$	8680.05	1.39684
$339$	5587.60	0.895212
$340$	−42020.5	−6.70260
$341$	−1092.19	−0.173447
$342$	5936.44	0.938613
$343$	16802.1	2.64498
$344$	−14989.9	−2.34942
$345$	−5802.25	−0.905457
$346$	−10443.0	−1.62260
$347$	346.367	0.0535848	0.0267924	−	0.999641i	$-0.491471\pi$
$347$	346.367	0.0535848	0.0267924	+	0.999641i	$0.491471\pi$
$348$	5095.11	0.784847
$349$	−3818.51	−0.585674	−0.292837	−	0.956162i	$-0.594599\pi$
$349$	−3818.51	−0.585674	−0.292837	+	0.956162i	$0.594599\pi$
$350$	−12144.5	−1.85472
$351$	−688.671	−0.104725
$352$	−10892.6	−1.64937
$353$	−8410.29	−1.26809	−0.634044	−	0.773297i	$-0.718606\pi$
$353$	−8410.29	−1.26809	−0.634044	+	0.773297i	$0.718606\pi$
$354$	−6290.95	−0.944521
$355$	−670.670	−0.100269
$356$	−11345.9	−1.68913
$357$	−13409.8	−1.98802
$358$	−3790.38	−0.559575
$359$	11095.6	1.63120	0.815601	−	0.578615i	$-0.196407\pi$
$359$	11095.6	1.63120	0.815601	+	0.578615i	$0.196407\pi$
$360$	−10742.6	−1.57273
$361$	6950.59	1.01335
$362$	−7657.31	−1.11177
$363$	363.000	0.0524864
$364$	−20559.6	−2.96048
$365$	−14058.9	−2.01610
$366$	−1027.17	−0.146697
$367$	−3966.58	−0.564180	−0.282090	−	0.959388i	$-0.591028\pi$
$367$	−3966.58	−0.564180	−0.282090	+	0.959388i	$0.591028\pi$
$368$	−42372.0	−6.00216
$369$	−1859.83	−0.262381
$370$	12824.3	1.80191
$371$	6797.12	0.951183
$372$	−7001.56	−0.975844
$373$	7611.72	1.05662	0.528311	−	0.849051i	$-0.322826\pi$
$373$	7611.72	1.05662	0.528311	+	0.849051i	$0.322826\pi$
$374$	8048.05	1.11271
$375$	−2547.07	−0.350747
$376$	48264.1	6.61976
$377$	−1842.94	−0.251768
$378$	−5197.02	−0.707158
$379$	12663.4	1.71630	0.858148	−	0.513402i	$-0.171615\pi$
$379$	12663.4	1.71630	0.858148	+	0.513402i	$0.171615\pi$
$380$	−37883.2	−5.11413
$381$	−2554.14	−0.343445
$382$	7939.11	1.06335
$383$	1698.85	0.226650	0.113325	−	0.993558i	$-0.463850\pi$
$383$	1698.85	0.226650	0.113325	+	0.993558i	$0.463850\pi$
$384$	−29352.2	−3.90071
$385$	5173.42	0.684836
$386$	−28596.6	−3.77080
$387$	1550.11	0.203609
$388$	−28616.2	−3.74425
$389$	−5513.38	−0.718611	−0.359305	−	0.933220i	$-0.616986\pi$
$389$	−5513.38	−0.718611	−0.359305	+	0.933220i	$0.616986\pi$
$390$	5890.47	0.764809
$391$	18382.0	2.37754
$392$	−72494.5	−9.34063
$393$	2004.77	0.257322
$394$	−9022.21	−1.15364
$395$	6125.29	0.780245
$396$	2327.04	0.295299
$397$	−5064.87	−0.640298	−0.320149	−	0.947367i	$-0.603733\pi$
$397$	−5064.87	−0.640298	−0.320149	+	0.947367i	$0.603733\pi$
$398$	−17030.8	−2.14491
$399$	−12089.5	−1.51688
$400$	18957.5	2.36968
$401$	−6714.50	−0.836174	−0.418087	−	0.908407i	$-0.637299\pi$
$401$	−6714.50	−0.836174	−0.418087	+	0.908407i	$0.637299\pi$
$402$	−4610.71	−0.572043
$403$	2532.52	0.313037
$404$	−24046.9	−2.96133
$405$	1110.89	0.136298
$406$	−13907.7	−1.70006
$407$	−1832.51	−0.223180
$408$	34033.3	4.12966
$409$	5452.94	0.659244	0.329622	−	0.944113i	$-0.393079\pi$
$409$	5452.94	0.659244	0.329622	+	0.944113i	$0.393079\pi$
$410$	15907.8	1.91617
$411$	4070.02	0.488465
$412$	23971.0	2.86643
$413$	12811.5	1.52642
$414$	7123.99	0.845712
$415$	−14412.2	−1.70474
$416$	25257.3	2.97679
$417$	−2016.08	−0.236758
$418$	7255.65	0.849008
$419$	2030.44	0.236739	0.118370	−	0.992970i	$-0.462233\pi$
$419$	2030.44	0.236739	0.118370	+	0.992970i	$0.462233\pi$
$420$	33164.6	3.85302
$421$	−1641.93	−0.190078	−0.0950389	−	0.995474i	$-0.530298\pi$
$421$	−1641.93	−0.190078	−0.0950389	+	0.995474i	$0.530298\pi$
$422$	3839.62	0.442914
$423$	−4991.02	−0.573692
$424$	−17250.7	−1.97586
$425$	−8224.21	−0.938666
$426$	823.446	0.0936528
$427$	2091.83	0.237075
$428$	−45456.2	−5.13366
$429$	−841.709	−0.0947275
$430$	−13258.7	−1.48696
$431$	−8339.47	−0.932015	−0.466007	−	0.884781i	$-0.654308\pi$
$431$	−8339.47	−0.932015	−0.466007	+	0.884781i	$0.654308\pi$
$432$	8112.51	0.903503
$433$	−11709.4	−1.29957	−0.649787	−	0.760116i	$-0.725142\pi$
$433$	−11709.4	−1.29957	−0.649787	+	0.760116i	$0.725142\pi$
$434$	19111.5	2.11378
$435$	2972.85	0.327672
$436$	−10954.9	−1.20331
$437$	16572.1	1.81408
$438$	17261.4	1.88307
$439$	−6853.11	−0.745059	−0.372530	−	0.928020i	$-0.621509\pi$
$439$	−6853.11	−0.745059	−0.372530	+	0.928020i	$0.621509\pi$
$440$	−13129.8	−1.42259
$441$	7496.70	0.809491
$442$	−18661.5	−2.00823
$443$	−1133.57	−0.121575	−0.0607875	−	0.998151i	$-0.519361\pi$
$443$	−1133.57	−0.121575	−0.0607875	+	0.998151i	$0.519361\pi$
$444$	−11747.5	−1.25565
$445$	−6619.98	−0.705207
$446$	−20117.6	−2.13586
$447$	−1841.17	−0.194820
$448$	108174.	11.4079
$449$	−13318.9	−1.39990	−0.699952	−	0.714190i	$-0.746795\pi$
$449$	−13318.9	−1.39990	−0.699952	+	0.714190i	$0.746795\pi$
$450$	−3187.31	−0.333892
$451$	−2273.12	−0.237333
$452$	43779.7	4.55581
$453$	6394.81	0.663255
$454$	−1211.78	−0.125268
$455$	−11995.9	−1.23599
$456$	30682.4	3.15096
$457$	−14329.8	−1.46678	−0.733391	−	0.679807i	$-0.762064\pi$
$457$	−14329.8	−1.46678	−0.733391	+	0.679807i	$0.762064\pi$
$458$	−3017.27	−0.307833
$459$	−3519.40	−0.357890
$460$	−45461.6	−4.60795
$461$	−1574.00	−0.159021	−0.0795103	−	0.996834i	$-0.525336\pi$
$461$	−1574.00	−0.159021	−0.0795103	+	0.996834i	$0.525336\pi$
$462$	−6351.91	−0.639648
$463$	6072.23	0.609504	0.304752	−	0.952432i	$-0.401426\pi$
$463$	6072.23	0.609504	0.304752	+	0.952432i	$0.401426\pi$
$464$	21709.8	2.17209
$465$	−4085.20	−0.407412
$466$	9197.46	0.914301
$467$	7029.99	0.696594	0.348297	−	0.937384i	$-0.386760\pi$
$467$	7029.99	0.696594	0.348297	+	0.937384i	$0.386760\pi$
$468$	−5395.85	−0.532955
$469$	9389.68	0.924467
$470$	42690.1	4.18968
$471$	−3006.46	−0.294120
$472$	−32514.7	−3.17079
$473$	1894.58	0.184171
$474$	−7520.62	−0.728762
$475$	−7414.46	−0.716208
$476$	−105068.	−10.1172
$477$	1783.90	0.171235
$478$	24145.2	2.31041
$479$	15431.2	1.47196	0.735979	−	0.677004i	$-0.236722\pi$
$479$	15431.2	1.47196	0.735979	+	0.677004i	$0.236722\pi$
$480$	−40742.6	−3.87424
$481$	4249.16	0.402796
$482$	33441.8	3.16023
$483$	−14508.0	−1.36674
$484$	2844.16	0.267108
$485$	−16696.7	−1.56322
$486$	−1363.95	−0.127305
$487$	−3351.31	−0.311832	−0.155916	−	0.987770i	$-0.549833\pi$
$487$	−3351.31	−0.311832	−0.155916	+	0.987770i	$0.549833\pi$
$488$	−5308.94	−0.492468
$489$	−9496.40	−0.878205
$490$	−64122.1	−5.91172
$491$	18529.5	1.70310	0.851551	−	0.524272i	$-0.175662\pi$
$491$	18529.5	1.70310	0.851551	+	0.524272i	$0.175662\pi$
$492$	−14572.0	−1.33528
$493$	−9418.22	−0.860396
$494$	−16824.1	−1.53229
$495$	1357.76	0.123286
$496$	−29832.9	−2.70068
$497$	−1676.94	−0.151351
$498$	17695.2	1.59225
$499$	7951.04	0.713301	0.356651	−	0.934238i	$-0.383919\pi$
$499$	7951.04	0.713301	0.356651	+	0.934238i	$0.383919\pi$
$500$	−19956.7	−1.78498
$501$	5780.10	0.515441
$502$	−12927.2	−1.14934
$503$	7431.14	0.658724	0.329362	−	0.944204i	$-0.393166\pi$
$503$	7431.14	0.658724	0.329362	+	0.944204i	$0.393166\pi$
$504$	−26860.7	−2.37395
$505$	−14030.6	−1.23635
$506$	8707.10	0.764976
$507$	−4639.28	−0.406386
$508$	−20012.1	−1.74782
$509$	−6555.22	−0.570835	−0.285418	−	0.958403i	$-0.592132\pi$
$509$	−6555.22	−0.570835	−0.285418	+	0.958403i	$0.592132\pi$
$510$	30102.8	2.61368
$511$	−35152.8	−3.04319
$512$	−88331.3	−7.62447
$513$	−3172.88	−0.273073
$514$	4019.31	0.344911
$515$	13986.4	1.19673
$516$	12145.4	1.03618
$517$	−6100.13	−0.518924
$518$	32066.0	2.71988
$519$	5581.56	0.472068
$520$	30444.8	2.56749
$521$	20988.6	1.76493	0.882464	−	0.470380i	$-0.155883\pi$
$521$	20988.6	1.76493	0.882464	+	0.470380i	$0.155883\pi$
$522$	−3650.05	−0.306051
$523$	−3007.92	−0.251486	−0.125743	−	0.992063i	$-0.540131\pi$
$523$	−3007.92	−0.251486	−0.125743	+	0.992063i	$0.540131\pi$
$524$	15707.7	1.30953
$525$	6490.94	0.539596
$526$	12088.7	1.00207
$527$	12942.2	1.06978
$528$	9915.29	0.817249
$529$	7720.28	0.634526
$530$	−15258.4	−1.25053
$531$	3362.36	0.274791
$532$	−94723.3	−7.71950
$533$	5270.82	0.428339
$534$	8127.99	0.658675
$535$	−26522.3	−2.14329
$536$	−23830.4	−1.92037
$537$	2025.87	0.162798
$538$	−5597.25	−0.448540
$539$	9162.63	0.732212
$540$	8704.03	0.693633
$541$	6114.39	0.485912	0.242956	−	0.970037i	$-0.421883\pi$
$541$	6114.39	0.485912	0.242956	+	0.970037i	$0.421883\pi$
$542$	−6179.17	−0.489701
$543$	4092.65	0.323448
$544$	129076.	10.1729
$545$	−6391.84	−0.502379
$546$	14728.5	1.15444
$547$	19489.0	1.52338	0.761690	−	0.647942i	$-0.224370\pi$
$547$	19489.0	1.52338	0.761690	+	0.647942i	$0.224370\pi$
$548$	31889.2	2.48584
$549$	549.000	0.0426790
$550$	−3895.60	−0.302017
$551$	−8490.91	−0.656488
$552$	36820.3	2.83908
$553$	15315.7	1.17774
$554$	−12487.3	−0.957647
$555$	−6854.31	−0.524233
$556$	−15796.3	−1.20488
$557$	−20599.9	−1.56705	−0.783526	−	0.621359i	$-0.786581\pi$
$557$	−20599.9	−1.56705	−0.783526	+	0.621359i	$0.786581\pi$
$558$	5015.80	0.380530
$559$	−4393.08	−0.332392
$560$	141311.	10.6634
$561$	−4301.49	−0.323724
$562$	18366.8	1.37857
$563$	−1359.94	−0.101802	−0.0509010	−	0.998704i	$-0.516209\pi$
$563$	−1359.94	−0.101802	−0.0509010	+	0.998704i	$0.516209\pi$
$564$	−39105.4	−2.91956
$565$	25544.2	1.90204
$566$	41501.9	3.08208
$567$	2777.68	0.205735
$568$	4255.97	0.314396
$569$	6758.54	0.497948	0.248974	−	0.968510i	$-0.419907\pi$
$569$	6758.54	0.497948	0.248974	+	0.968510i	$0.419907\pi$
$570$	27138.9	1.99425
$571$	24316.0	1.78213	0.891063	−	0.453879i	$-0.149960\pi$
$571$	24316.0	1.78213	0.891063	+	0.453879i	$0.149960\pi$
$572$	−6594.92	−0.482076
$573$	−4243.27	−0.309363
$574$	39775.9	2.89236
$575$	−8897.68	−0.645320
$576$	28390.3	2.05369
$577$	−11528.5	−0.831780	−0.415890	−	0.909415i	$-0.636530\pi$
$577$	−11528.5	−0.831780	−0.415890	+	0.909415i	$0.636530\pi$
$578$	−67791.5	−4.87847
$579$	15284.2	1.09705
$580$	23292.7	1.66755
$581$	−36036.2	−2.57321
$582$	20500.2	1.46007
$583$	2180.32	0.154888
$584$	89215.6	6.32152
$585$	−3148.32	−0.222507
$586$	−16642.3	−1.17319
$587$	19302.8	1.35726	0.678629	−	0.734481i	$-0.262574\pi$
$587$	19302.8	1.35726	0.678629	+	0.734481i	$0.262574\pi$
$588$	58737.8	4.11957
$589$	11668.0	0.816248
$590$	−28759.6	−2.00680
$591$	4822.16	0.335629
$592$	−50054.8	−3.47507
$593$	21093.4	1.46071	0.730357	−	0.683065i	$-0.239354\pi$
$593$	21093.4	1.46071	0.730357	+	0.683065i	$0.239354\pi$
$594$	−1667.05	−0.115152
$595$	−61304.2	−4.22391
$596$	−14425.9	−0.991455
$597$	9102.55	0.624024
$598$	−20189.7	−1.38063
$599$	−1824.39	−0.124445	−0.0622226	−	0.998062i	$-0.519819\pi$
$599$	−1824.39	−0.124445	−0.0622226	+	0.998062i	$0.519819\pi$
$600$	−16473.6	−1.12089
$601$	−14398.4	−0.977241	−0.488621	−	0.872496i	$-0.662500\pi$
$601$	−14398.4	−0.977241	−0.488621	+	0.872496i	$0.662500\pi$
$602$	−33152.1	−2.24448
$603$	2464.31	0.166426
$604$	50104.3	3.37536
$605$	1659.48	0.111517
$606$	17226.8	1.15477
$607$	−19455.4	−1.30094	−0.650470	−	0.759532i	$-0.725428\pi$
$607$	−19455.4	−1.30094	−0.650470	+	0.759532i	$0.725428\pi$
$608$	116367.	7.76202
$609$	7433.31	0.494603
$610$	−4695.81	−0.311685
$611$	14144.7	0.936554
$612$	−27575.1	−1.82133
$613$	−23711.3	−1.56230	−0.781151	−	0.624342i	$-0.785367\pi$
$613$	−23711.3	−1.56230	−0.781151	+	0.624342i	$0.785367\pi$
$614$	−2301.28	−0.151258
$615$	−8502.35	−0.557476
$616$	−32829.8	−2.14732
$617$	7953.90	0.518982	0.259491	−	0.965746i	$-0.416445\pi$
$617$	7953.90	0.518982	0.259491	+	0.965746i	$0.416445\pi$
$618$	−17172.4	−1.11776
$619$	7762.59	0.504047	0.252023	−	0.967721i	$-0.418904\pi$
$619$	7762.59	0.504047	0.252023	+	0.967721i	$0.418904\pi$
$620$	−32008.2	−2.07336
$621$	−3807.60	−0.246045
$622$	46650.6	3.00727
$623$	−16552.6	−1.06447
$624$	−22991.2	−1.47497
$625$	−19530.9	−1.24998
$626$	23542.3	1.50310
$627$	−3877.97	−0.247003
$628$	−23556.1	−1.49680
$629$	21715.0	1.37652
$630$	−23758.6	−1.50248
$631$	−15349.2	−0.968372	−0.484186	−	0.874965i	$-0.660884\pi$
$631$	−15349.2	−0.968372	−0.484186	+	0.874965i	$0.660884\pi$
$632$	−38870.2	−2.44648
$633$	−2052.19	−0.128858
$634$	−1075.87	−0.0673946
$635$	−11676.5	−0.729710
$636$	13977.1	0.871429
$637$	−21245.9	−1.32150
$638$	−4461.18	−0.276833
$639$	−440.112	−0.0272466
$640$	−134186.	−8.28776
$641$	14894.6	0.917785	0.458893	−	0.888492i	$-0.348246\pi$
$641$	14894.6	0.917785	0.458893	+	0.888492i	$0.348246\pi$
$642$	32564.0	2.00187
$643$	−20049.9	−1.22969	−0.614845	−	0.788648i	$-0.710781\pi$
$643$	−20049.9	−1.22969	−0.614845	+	0.788648i	$0.710781\pi$
$644$	−113672.	−6.95545
$645$	7086.47	0.432604
$646$	−85978.2	−5.23648
$647$	−18550.9	−1.12722	−0.563610	−	0.826041i	$-0.690588\pi$
$647$	−18550.9	−1.12722	−0.563610	+	0.826041i	$0.690588\pi$
$648$	−7049.57	−0.427366
$649$	4109.56	0.248558
$650$	9032.96	0.545080
$651$	−10214.6	−0.614967
$652$	−74405.8	−4.46926
$653$	10765.9	0.645181	0.322591	−	0.946539i	$-0.395446\pi$
$653$	10765.9	0.645181	0.322591	+	0.946539i	$0.395446\pi$
$654$	7847.88	0.469230
$655$	9164.99	0.546726
$656$	−62090.0	−3.69544
$657$	−9225.83	−0.547845
$658$	106742.	6.32409
$659$	−24415.1	−1.44321	−0.721606	−	0.692304i	$-0.756596\pi$
$659$	−24415.1	−1.44321	−0.721606	+	0.692304i	$0.756596\pi$
$660$	10638.3	0.627415
$661$	27762.0	1.63361	0.816806	−	0.576913i	$-0.195743\pi$
$661$	27762.0	1.63361	0.816806	+	0.576913i	$0.195743\pi$
$662$	22144.3	1.30010
$663$	9974.12	0.584257
$664$	91457.5	5.34524
$665$	−55268.2	−3.22287
$666$	8415.70	0.489642
$667$	−10189.5	−0.591511
$668$	45288.0	2.62312
$669$	10752.4	0.621390
$670$	−21078.2	−1.21541
$671$	671.000	0.0386046
$672$	−101873.	−5.84796
$673$	−24256.1	−1.38931	−0.694653	−	0.719345i	$-0.744442\pi$
$673$	−24256.1	−1.38931	−0.694653	+	0.719345i	$0.744442\pi$
$674$	−37428.2	−2.13899
$675$	1703.54	0.0971398
$676$	−36349.5	−2.06813
$677$	14322.1	0.813063	0.406531	−	0.913637i	$-0.366738\pi$
$677$	14322.1	0.813063	0.406531	+	0.913637i	$0.366738\pi$
$678$	−31363.1	−1.77654
$679$	−41748.5	−2.35959
$680$	155586.	8.77419
$681$	647.668	0.0364445
$682$	6130.42	0.344202
$683$	21223.4	1.18901	0.594503	−	0.804094i	$-0.297349\pi$
$683$	21223.4	1.18901	0.594503	+	0.804094i	$0.297349\pi$
$684$	−24860.0	−1.38969
$685$	18606.4	1.03783
$686$	−94309.8	−5.24893
$687$	1612.66	0.0895585
$688$	51750.2	2.86767
$689$	−5055.64	−0.279542
$690$	32567.9	1.79687
$691$	4685.46	0.257950	0.128975	−	0.991648i	$-0.458831\pi$
$691$	4685.46	0.257950	0.128975	+	0.991648i	$0.458831\pi$
$692$	43732.4	2.40239
$693$	3394.94	0.186094
$694$	−1944.15	−0.106338
$695$	−9216.69	−0.503035
$696$	−18865.3	−1.02742
$697$	26936.1	1.46381
$698$	21433.2	1.16226
$699$	−4915.82	−0.265999
$700$	50857.6	2.74605
$701$	3566.89	0.192182	0.0960909	−	0.995373i	$-0.469366\pi$
$701$	3566.89	0.192182	0.0960909	+	0.995373i	$0.469366\pi$
$702$	3865.49	0.207826
$703$	19577.0	1.05030
$704$	34699.2	1.85764
$705$	−22816.8	−1.21891
$706$	47206.8	2.51650
$707$	−35082.2	−1.86620
$708$	26344.6	1.39844
$709$	−27486.2	−1.45594	−0.727972	−	0.685606i	$-0.759537\pi$
$709$	−27486.2	−1.45594	−0.727972	+	0.685606i	$0.759537\pi$
$710$	3764.45	0.198982
$711$	4019.59	0.212020
$712$	42009.4	2.21119
$713$	14002.1	0.735458
$714$	75269.1	3.94520
$715$	−3847.94	−0.201266
$716$	15873.0	0.828494
$717$	−12905.0	−0.672172
$718$	−62279.1	−3.23710
$719$	12393.1	0.642817	0.321408	−	0.946941i	$-0.395844\pi$
$719$	12393.1	0.642817	0.321408	+	0.946941i	$0.395844\pi$
$720$	37087.0	1.91965
$721$	34971.6	1.80639
$722$	−39013.5	−2.01099
$723$	−17873.8	−0.919412
$724$	32066.5	1.64605
$725$	4558.82	0.233532
$726$	−2037.51	−0.104158
$727$	14052.9	0.716909	0.358455	−	0.933547i	$-0.383304\pi$
$727$	14052.9	0.716909	0.358455	+	0.933547i	$0.383304\pi$
$728$	76124.3	3.87549
$729$	729.000	0.0370370
$730$	78912.1	4.00091
$731$	−22450.5	−1.13593
$732$	4301.50	0.217197
$733$	−8440.15	−0.425299	−0.212649	−	0.977129i	$-0.568209\pi$
$733$	−8440.15	−0.425299	−0.212649	+	0.977129i	$0.568209\pi$
$734$	22264.3	1.11961
$735$	34271.8	1.71991
$736$	139646.	6.99376
$737$	3011.94	0.150538
$738$	10439.2	0.520692
$739$	19494.6	0.970393	0.485196	−	0.874405i	$-0.338748\pi$
$739$	19494.6	0.970393	0.485196	+	0.874405i	$0.338748\pi$
$740$	−53704.6	−2.66786
$741$	8992.07	0.445792
$742$	−38152.1	−1.88761
$743$	−14148.8	−0.698612	−0.349306	−	0.937009i	$-0.613583\pi$
$743$	−14148.8	−0.698612	−0.349306	+	0.937009i	$0.613583\pi$
$744$	25924.1	1.27745
$745$	−8417.08	−0.413930
$746$	−42724.4	−2.09685
$747$	−9457.66	−0.463237
$748$	−33702.9	−1.64746
$749$	−66316.4	−3.23518
$750$	14296.6	0.696052
$751$	4974.36	0.241700	0.120850	−	0.992671i	$-0.461438\pi$
$751$	4974.36	0.241700	0.120850	+	0.992671i	$0.461438\pi$
$752$	−166624.	−8.08000
$753$	6909.25	0.334379
$754$	10344.4	0.499629
$755$	29234.4	1.40920
$756$	21763.6	1.04700
$757$	5354.36	0.257077	0.128539	−	0.991704i	$-0.458971\pi$
$757$	5354.36	0.257077	0.128539	+	0.991704i	$0.458971\pi$
$758$	−71079.5	−3.40597
$759$	−4653.74	−0.222556
$760$	140267.	6.69477
$761$	23455.6	1.11730	0.558649	−	0.829404i	$-0.311320\pi$
$761$	23455.6	1.11730	0.558649	+	0.829404i	$0.311320\pi$
$762$	14336.3	0.681562
$763$	−15982.2	−0.758314
$764$	−33246.6	−1.57437
$765$	−16089.2	−0.760402
$766$	−9535.59	−0.449784
$767$	−9529.07	−0.448598
$768$	89045.8	4.18381
$769$	−17123.4	−0.802973	−0.401487	−	0.915865i	$-0.631506\pi$
$769$	−17123.4	−0.802973	−0.401487	+	0.915865i	$0.631506\pi$
$770$	−29038.3	−1.35905
$771$	−2148.22	−0.100346
$772$	119754.	5.58296
$773$	22029.5	1.02503	0.512514	−	0.858679i	$-0.328714\pi$
$773$	22029.5	1.02503	0.512514	+	0.858679i	$0.328714\pi$
$774$	−8700.74	−0.404059
$775$	−6264.61	−0.290363
$776$	105955.	4.90150
$777$	−17138.5	−0.791301
$778$	30946.5	1.42607
$779$	24284.0	1.11690
$780$	−24667.5	−1.13236
$781$	−537.915	−0.0246455
$782$	−103178.	−4.71819
$783$	1950.87	0.0890399
$784$	250276.	11.4010
$785$	−13744.3	−0.624910
$786$	−11252.7	−0.510652
$787$	31617.9	1.43209	0.716047	−	0.698052i	$-0.245950\pi$
$787$	31617.9	1.43209	0.716047	+	0.698052i	$0.245950\pi$
$788$	37782.3	1.70805
$789$	−6461.11	−0.291535
$790$	−34381.1	−1.54839
$791$	63870.7	2.87103
$792$	−8616.15	−0.386567
$793$	−1555.89	−0.0696736
$794$	28429.0	1.27066
$795$	8155.24	0.363819
$796$	71319.9	3.17571
$797$	20334.8	0.903760	0.451880	−	0.892079i	$-0.350754\pi$
$797$	20334.8	0.903760	0.451880	+	0.892079i	$0.350754\pi$
$798$	67858.2	3.01022
$799$	72285.6	3.20060
$800$	−62478.2	−2.76117
$801$	−4344.22	−0.191630
$802$	37688.3	1.65938
$803$	−11276.0	−0.495544
$804$	19308.3	0.846953
$805$	−66324.3	−2.90388
$806$	−14215.0	−0.621217
$807$	2991.59	0.130495
$808$	89036.4	3.87660
$809$	−600.556	−0.0260994	−0.0130497	−	0.999915i	$-0.504154\pi$
$809$	−600.556	−0.0260994	−0.0130497	+	0.999915i	$0.504154\pi$
$810$	−6235.42	−0.270482
$811$	34131.7	1.47784	0.738918	−	0.673795i	$-0.235337\pi$
$811$	34131.7	1.47784	0.738918	+	0.673795i	$0.235337\pi$
$812$	58241.2	2.51707
$813$	3302.62	0.142470
$814$	10285.9	0.442898
$815$	−43413.6	−1.86590
$816$	−117495.	−5.04060
$817$	−20240.0	−0.866719
$818$	−30607.2	−1.30826
$819$	−7872.05	−0.335863
$820$	−66617.2	−2.83704
$821$	2542.74	0.108090	0.0540452	−	0.998538i	$-0.482789\pi$
$821$	2542.74	0.108090	0.0540452	+	0.998538i	$0.482789\pi$
$822$	−22844.9	−0.969352
$823$	29602.1	1.25378	0.626892	−	0.779106i	$-0.284327\pi$
$823$	29602.1	1.25378	0.626892	+	0.779106i	$0.284327\pi$
$824$	−88755.6	−3.75236
$825$	2082.11	0.0878663
$826$	−71910.5	−3.02916
$827$	−30192.4	−1.26952	−0.634760	−	0.772710i	$-0.718901\pi$
$827$	−30192.4	−1.26952	−0.634760	+	0.772710i	$0.718901\pi$
$828$	−29833.2	−1.25214
$829$	1369.08	0.0573584	0.0286792	−	0.999589i	$-0.490870\pi$
$829$	1369.08	0.0573584	0.0286792	+	0.999589i	$0.490870\pi$
$830$	80895.1	3.38302
$831$	6674.19	0.278610
$832$	−80459.1	−3.35266
$833$	−108576.	−4.51612
$834$	11316.2	0.469843
$835$	26424.2	1.09515
$836$	−30384.5	−1.25702
$837$	−2680.82	−0.110708
$838$	−11396.8	−0.469805
$839$	−20899.8	−0.859999	−0.430000	−	0.902829i	$-0.641486\pi$
$839$	−20899.8	−0.859999	−0.430000	+	0.902829i	$0.641486\pi$
$840$	−122796.	−5.04389
$841$	−19168.3	−0.785941
$842$	9216.11	0.377207
$843$	−9816.64	−0.401071
$844$	−16079.2	−0.655769
$845$	−21208.8	−0.863440
$846$	28014.4	1.13848
$847$	4149.38	0.168329
$848$	59555.1	2.41171
$849$	−22181.8	−0.896674
$850$	46162.3	1.86277
$851$	23493.2	0.946342
$852$	−3448.35	−0.138660
$853$	21666.9	0.869709	0.434854	−	0.900501i	$-0.356800\pi$
$853$	21666.9	0.869709	0.434854	+	0.900501i	$0.356800\pi$
$854$	−11741.4	−0.470472
$855$	−14505.1	−0.580192
$856$	168307.	6.72034
$857$	34775.7	1.38613	0.693067	−	0.720873i	$-0.256259\pi$
$857$	34775.7	1.38613	0.693067	+	0.720873i	$0.256259\pi$
$858$	4724.49	0.187985
$859$	18442.0	0.732519	0.366260	−	0.930513i	$-0.380638\pi$
$859$	18442.0	0.732519	0.366260	+	0.930513i	$0.380638\pi$
$860$	55523.6	2.20156
$861$	−21259.3	−0.841480
$862$	46809.2	1.84957
$863$	−792.141	−0.0312454	−0.0156227	−	0.999878i	$-0.504973\pi$
$863$	−792.141	−0.0312454	−0.0156227	+	0.999878i	$0.504973\pi$
$864$	−26736.4	−1.05277
$865$	25516.5	1.00299
$866$	65724.3	2.57899
$867$	36233.0	1.41930
$868$	−80033.3	−3.12962
$869$	4912.83	0.191780
$870$	−16686.5	−0.650259
$871$	−6983.96	−0.271690
$872$	40561.7	1.57522
$873$	−10956.9	−0.424781
$874$	−93018.9	−3.60001
$875$	−29115.0	−1.12488
$876$	−72285.8	−2.78803
$877$	−33266.7	−1.28089	−0.640443	−	0.768006i	$-0.721249\pi$
$877$	−33266.7	−1.28089	−0.640443	+	0.768006i	$0.721249\pi$
$878$	38466.3	1.47856
$879$	8894.94	0.341318
$880$	45328.5	1.73639
$881$	36315.7	1.38877	0.694385	−	0.719604i	$-0.255677\pi$
$881$	36315.7	1.38877	0.694385	+	0.719604i	$0.255677\pi$
$882$	−42078.8	−1.60642
$883$	36741.7	1.40029	0.700145	−	0.714001i	$-0.253119\pi$
$883$	36741.7	1.40029	0.700145	+	0.714001i	$0.253119\pi$
$884$	78148.8	2.97333
$885$	15371.3	0.583843
$886$	6362.72	0.241264
$887$	30706.3	1.16236	0.581182	−	0.813774i	$-0.302591\pi$
$887$	30706.3	1.16236	0.581182	+	0.813774i	$0.302591\pi$
$888$	43496.4	1.64374
$889$	−29195.8	−1.10146
$890$	37157.8	1.39947
$891$	891.000	0.0335013
$892$	84246.4	3.16231
$893$	65168.4	2.44208
$894$	10334.5	0.386618
$895$	9261.43	0.345894
$896$	−335519.	−12.5099
$897$	10790.9	0.401669
$898$	74758.5	2.77809
$899$	−7174.11	−0.266151
$900$	13347.5	0.494353
$901$	−25836.4	−0.955313
$902$	12759.0	0.470984
$903$	17719.0	0.652992
$904$	−162100.	−5.96389
$905$	18709.9	0.687224
$906$	−35893.9	−1.31622
$907$	−405.084	−0.0148297	−0.00741487	−	0.999973i	$-0.502360\pi$
$907$	−405.084	−0.0148297	−0.00741487	+	0.999973i	$0.502360\pi$
$908$	5074.58	0.185469
$909$	−9207.30	−0.335959
$910$	67332.7	2.45281
$911$	−23893.1	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$911$	−23893.1	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$912$	−105926.	−3.84601
$913$	−11559.4	−0.419013
$914$	80432.8	2.91081
$915$	2509.80	0.0906791
$916$	12635.4	0.455771
$917$	22916.2	0.825254
$918$	19754.3	0.710228
$919$	−33051.4	−1.18636	−0.593180	−	0.805070i	$-0.702128\pi$
$919$	−33051.4	−1.18636	−0.593180	+	0.805070i	$0.702128\pi$
$920$	168327.	6.03214
$921$	1229.98	0.0440057
$922$	8834.82	0.315574
$923$	1247.30	0.0444802
$924$	26599.9	0.947049
$925$	−10511.0	−0.373621
$926$	−34083.3	−1.20955
$927$	9178.26	0.325193
$928$	−71549.0	−2.53094
$929$	−16870.1	−0.595790	−0.297895	−	0.954599i	$-0.596284\pi$
$929$	−16870.1	−0.595790	−0.297895	+	0.954599i	$0.596284\pi$
$930$	22930.1	0.808504
$931$	−97885.4	−3.44583
$932$	−38516.2	−1.35369
$933$	−24933.7	−0.874910
$934$	−39459.2	−1.38238
$935$	−19664.6	−0.687809
$936$	19978.8	0.697677
$937$	−37860.3	−1.32000	−0.660000	−	0.751265i	$-0.729444\pi$
$937$	−37860.3	−1.32000	−0.660000	+	0.751265i	$0.729444\pi$
$938$	−52704.0	−1.83459
$939$	−12582.8	−0.437300
$940$	−178773.	−6.20314
$941$	15952.4	0.552639	0.276319	−	0.961066i	$-0.410885\pi$
$941$	15952.4	0.552639	0.276319	+	0.961066i	$0.410885\pi$
$942$	16875.2	0.583677
$943$	29141.9	1.00635
$944$	112252.	3.87022
$945$	12698.4	0.437120
$946$	−10634.2	−0.365485
$947$	−5738.43	−0.196910	−0.0984550	−	0.995142i	$-0.531390\pi$
$947$	−5738.43	−0.196910	−0.0984550	+	0.995142i	$0.531390\pi$
$948$	31494.1	1.07899
$949$	26146.3	0.894359
$950$	41617.2	1.42130
$951$	575.026	0.0196072
$952$	389028.	13.2442
$953$	−21938.0	−0.745689	−0.372845	−	0.927894i	$-0.621618\pi$
$953$	−21938.0	−0.745689	−0.372845	+	0.927894i	$0.621618\pi$
$954$	−10013.0	−0.339813
$955$	−19398.4	−0.657297
$956$	−101113.	−3.42074
$957$	2384.39	0.0805396
$958$	−86614.8	−2.92108
$959$	46523.5	1.56655
$960$	129788.	4.36344
$961$	−19932.5	−0.669079
$962$	−23850.4	−0.799343
$963$	−17404.7	−0.582407
$964$	−140044.	−4.67897
$965$	69873.0	2.33087
$966$	81432.8	2.71227
$967$	46466.2	1.54525	0.772623	−	0.634865i	$-0.218944\pi$
$967$	46466.2	1.54525	0.772623	+	0.634865i	$0.218944\pi$
$968$	−10530.8	−0.349663
$969$	45953.3	1.52346
$970$	93718.3	3.10218
$971$	−30502.2	−1.00810	−0.504049	−	0.863675i	$-0.668157\pi$
$971$	−30502.2	−1.00810	−0.504049	+	0.863675i	$0.668157\pi$
$972$	5711.83	0.188485
$973$	−23045.4	−0.759304
$974$	18810.8	0.618826
$975$	−4827.90	−0.158581
$976$	18328.3	0.601100
$977$	−24333.0	−0.796807	−0.398403	−	0.917210i	$-0.630436\pi$
$977$	−24333.0	−0.796807	−0.398403	+	0.917210i	$0.630436\pi$
$978$	53303.1	1.74279
$979$	−5309.60	−0.173336
$980$	268525.	8.75276
$981$	−4194.50	−0.136514
$982$	−104005.	−3.37978
$983$	−35921.7	−1.16554	−0.582769	−	0.812638i	$-0.698031\pi$
$983$	−35921.7	−1.16554	−0.582769	+	0.812638i	$0.698031\pi$
$984$	53954.7	1.74798
$985$	22044.9	0.713105
$986$	52864.2	1.70744
$987$	−57051.3	−1.83988
$988$	70454.3	2.26867
$989$	−24288.9	−0.780933
$990$	−7621.07	−0.244660
$991$	−18993.3	−0.608821	−0.304410	−	0.952541i	$-0.598459\pi$
$991$	−18993.3	−0.608821	−0.304410	+	0.952541i	$0.598459\pi$
$992$	98320.5	3.14686
$993$	−11835.6	−0.378239
$994$	9412.64	0.300353
$995$	41613.0	1.32585
$996$	−74102.3	−2.35745
$997$	−54840.5	−1.74204	−0.871022	−	0.491244i	$-0.836542\pi$
$997$	−54840.5	−1.74204	−0.871022	+	0.491244i	$0.836542\pi$
$998$	−44628.9	−1.41554
$999$	−4497.99	−0.142453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.a.1.1	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.a.1.1	✓	36	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-5.61297 q^{2} +3.00000 q^{3} +23.5055 q^{4} +13.7147 q^{5} -16.8389 q^{6} +34.2924 q^{7} -87.0318 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.61297 q^{2} +3.00000 q^{3} +23.5055 q^{4} +13.7147 q^{5} -16.8389 q^{6} +34.2924 q^{7} -87.0318 q^{8} +9.00000 q^{9} -76.9805 q^{10} +11.0000 q^{11} +70.5164 q^{12} -25.5063 q^{13} -192.482 q^{14} +41.1442 q^{15} +300.463 q^{16} -130.348 q^{17} -50.5168 q^{18} -117.514 q^{19} +322.371 q^{20} +102.877 q^{21} -61.7427 q^{22} -141.022 q^{23} -261.095 q^{24} +63.0942 q^{25} +143.166 q^{26} +27.0000 q^{27} +806.058 q^{28} +72.2543 q^{29} -230.941 q^{30} -99.2898 q^{31} -990.238 q^{32} +33.0000 q^{33} +731.641 q^{34} +470.311 q^{35} +211.549 q^{36} -166.592 q^{37} +659.604 q^{38} -76.5190 q^{39} -1193.62 q^{40} -206.647 q^{41} -577.446 q^{42} +172.235 q^{43} +258.560 q^{44} +123.433 q^{45} +791.554 q^{46} -554.557 q^{47} +901.390 q^{48} +832.966 q^{49} -354.146 q^{50} -391.045 q^{51} -599.538 q^{52} +198.211 q^{53} -151.550 q^{54} +150.862 q^{55} -2984.53 q^{56} -352.543 q^{57} -405.561 q^{58} +373.596 q^{59} +967.114 q^{60} +61.0000 q^{61} +557.311 q^{62} +308.631 q^{63} +3154.47 q^{64} -349.813 q^{65} -185.228 q^{66} +273.813 q^{67} -3063.90 q^{68} -423.067 q^{69} -2639.84 q^{70} -48.9014 q^{71} -783.286 q^{72} -1025.09 q^{73} +935.077 q^{74} +189.282 q^{75} -2762.23 q^{76} +377.216 q^{77} +429.499 q^{78} +446.621 q^{79} +4120.78 q^{80} +81.0000 q^{81} +1159.91 q^{82} -1050.85 q^{83} +2418.17 q^{84} -1787.69 q^{85} -966.749 q^{86} +216.763 q^{87} -957.349 q^{88} -482.691 q^{89} -692.824 q^{90} -874.673 q^{91} -3314.79 q^{92} -297.869 q^{93} +3112.72 q^{94} -1611.68 q^{95} -2970.71 q^{96} -1217.43 q^{97} -4675.42 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 14 q^{2} + 108 q^{3} + 146 q^{4} - 65 q^{5} - 42 q^{6} - 105 q^{7} - 189 q^{8} + 324 q^{9}+O(q^{10}) \) 36 * q - 14 * q^2 + 108 * q^3 + 146 * q^4 - 65 * q^5 - 42 * q^6 - 105 * q^7 - 189 * q^8 + 324 * q^9	\( 36 q - 14 q^{2} + 108 q^{3} + 146 q^{4} - 65 q^{5} - 42 q^{6} - 105 q^{7} - 189 q^{8} + 324 q^{9} - 161 q^{10} + 396 q^{11} + 438 q^{12} - 233 q^{13} - 264 q^{14} - 195 q^{15} + 574 q^{16} - 556 q^{17} - 126 q^{18} - 615 q^{19} - 136 q^{20} - 315 q^{21} - 154 q^{22} - 457 q^{23} - 567 q^{24} + 863 q^{25} + 115 q^{26} + 972 q^{27} - 424 q^{28} - 754 q^{29} - 483 q^{30} - 508 q^{31} - 1511 q^{32} + 1188 q^{33} - 860 q^{34} - 826 q^{35} + 1314 q^{36} - 412 q^{37} - 599 q^{38} - 699 q^{39} - 2791 q^{40} - 2066 q^{41} - 792 q^{42} - 2063 q^{43} + 1606 q^{44} - 585 q^{45} - 787 q^{46} - 1815 q^{47} + 1722 q^{48} + 2825 q^{49} + 808 q^{50} - 1668 q^{51} - 2882 q^{52} - 759 q^{53} - 378 q^{54} - 715 q^{55} - 1749 q^{56} - 1845 q^{57} - 335 q^{58} - 2337 q^{59} - 408 q^{60} + 2196 q^{61} - 1689 q^{62} - 945 q^{63} + 4723 q^{64} - 3550 q^{65} - 462 q^{66} - 1331 q^{67} - 6166 q^{68} - 1371 q^{69} - 1750 q^{70} - 361 q^{71} - 1701 q^{72} - 4627 q^{73} - 3394 q^{74} + 2589 q^{75} - 7214 q^{76} - 1155 q^{77} + 345 q^{78} - 2583 q^{79} - 2643 q^{80} + 2916 q^{81} + 1090 q^{82} - 6123 q^{83} - 1272 q^{84} + 295 q^{85} + 613 q^{86} - 2262 q^{87} - 2079 q^{88} - 2485 q^{89} - 1449 q^{90} - 3156 q^{91} - 6291 q^{92} - 1524 q^{93} - 1744 q^{94} - 5572 q^{95} - 4533 q^{96} - 2558 q^{97} - 2314 q^{98} + 3564 q^{99}+O(q^{100}) \) 36 * q - 14 * q^2 + 108 * q^3 + 146 * q^4 - 65 * q^5 - 42 * q^6 - 105 * q^7 - 189 * q^8 + 324 * q^9 - 161 * q^10 + 396 * q^11 + 438 * q^12 - 233 * q^13 - 264 * q^14 - 195 * q^15 + 574 * q^16 - 556 * q^17 - 126 * q^18 - 615 * q^19 - 136 * q^20 - 315 * q^21 - 154 * q^22 - 457 * q^23 - 567 * q^24 + 863 * q^25 + 115 * q^26 + 972 * q^27 - 424 * q^28 - 754 * q^29 - 483 * q^30 - 508 * q^31 - 1511 * q^32 + 1188 * q^33 - 860 * q^34 - 826 * q^35 + 1314 * q^36 - 412 * q^37 - 599 * q^38 - 699 * q^39 - 2791 * q^40 - 2066 * q^41 - 792 * q^42 - 2063 * q^43 + 1606 * q^44 - 585 * q^45 - 787 * q^46 - 1815 * q^47 + 1722 * q^48 + 2825 * q^49 + 808 * q^50 - 1668 * q^51 - 2882 * q^52 - 759 * q^53 - 378 * q^54 - 715 * q^55 - 1749 * q^56 - 1845 * q^57 - 335 * q^58 - 2337 * q^59 - 408 * q^60 + 2196 * q^61 - 1689 * q^62 - 945 * q^63 + 4723 * q^64 - 3550 * q^65 - 462 * q^66 - 1331 * q^67 - 6166 * q^68 - 1371 * q^69 - 1750 * q^70 - 361 * q^71 - 1701 * q^72 - 4627 * q^73 - 3394 * q^74 + 2589 * q^75 - 7214 * q^76 - 1155 * q^77 + 345 * q^78 - 2583 * q^79 - 2643 * q^80 + 2916 * q^81 + 1090 * q^82 - 6123 * q^83 - 1272 * q^84 + 295 * q^85 + 613 * q^86 - 2262 * q^87 - 2079 * q^88 - 2485 * q^89 - 1449 * q^90 - 3156 * q^91 - 6291 * q^92 - 1524 * q^93 - 1744 * q^94 - 5572 * q^95 - 4533 * q^96 - 2558 * q^97 - 2314 * q^98 + 3564 * q^99

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