LMFDB - Embedded newform 4019.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4019,2,Mod(1,4019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4019, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4019.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:	$32.0918765724$
Analytic rank:	$0$
Dimension:	$186$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79148 q^{2} -3.39321 q^{3} +5.79236 q^{4} +3.31167 q^{5} +9.47208 q^{6} -0.0588258 q^{7} -10.5863 q^{8} +8.51387 q^{9} +O(q^{10})$	\(q-2.79148 q^{2} -3.39321 q^{3} +5.79236 q^{4} +3.31167 q^{5} +9.47208 q^{6} -0.0588258 q^{7} -10.5863 q^{8} +8.51387 q^{9} -9.24445 q^{10} +3.18832 q^{11} -19.6547 q^{12} +1.79597 q^{13} +0.164211 q^{14} -11.2372 q^{15} +17.9667 q^{16} +6.19854 q^{17} -23.7663 q^{18} +4.49804 q^{19} +19.1824 q^{20} +0.199608 q^{21} -8.90012 q^{22} -5.64628 q^{23} +35.9215 q^{24} +5.96714 q^{25} -5.01342 q^{26} -18.7097 q^{27} -0.340740 q^{28} +0.468620 q^{29} +31.3684 q^{30} +3.70270 q^{31} -28.9811 q^{32} -10.8186 q^{33} -17.3031 q^{34} -0.194812 q^{35} +49.3154 q^{36} -2.54153 q^{37} -12.5562 q^{38} -6.09411 q^{39} -35.0583 q^{40} -1.69797 q^{41} -0.557203 q^{42} +10.8120 q^{43} +18.4679 q^{44} +28.1951 q^{45} +15.7615 q^{46} +8.29508 q^{47} -60.9648 q^{48} -6.99654 q^{49} -16.6571 q^{50} -21.0329 q^{51} +10.4029 q^{52} +13.4286 q^{53} +52.2278 q^{54} +10.5586 q^{55} +0.622747 q^{56} -15.2628 q^{57} -1.30814 q^{58} +7.23695 q^{59} -65.0898 q^{60} +8.88227 q^{61} -10.3360 q^{62} -0.500835 q^{63} +44.9667 q^{64} +5.94766 q^{65} +30.2000 q^{66} -4.59088 q^{67} +35.9042 q^{68} +19.1590 q^{69} +0.543813 q^{70} +1.08524 q^{71} -90.1303 q^{72} +10.0813 q^{73} +7.09463 q^{74} -20.2478 q^{75} +26.0542 q^{76} -0.187555 q^{77} +17.0116 q^{78} -0.0659532 q^{79} +59.4997 q^{80} +37.9444 q^{81} +4.73984 q^{82} -15.5012 q^{83} +1.15620 q^{84} +20.5275 q^{85} -30.1815 q^{86} -1.59013 q^{87} -33.7525 q^{88} -0.766469 q^{89} -78.7061 q^{90} -0.105649 q^{91} -32.7053 q^{92} -12.5640 q^{93} -23.1555 q^{94} +14.8960 q^{95} +98.3389 q^{96} +8.17877 q^{97} +19.5307 q^{98} +27.1449 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) $ 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * 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 $

$2$	−2.79148	−1.97387	−0.986937	−	0.161106i	$-0.948494\pi$
$2$	−2.79148	−1.97387	−0.986937	+	0.161106i	$0.948494\pi$
$3$	−3.39321	−1.95907	−0.979535	−	0.201273i	$-0.935492\pi$
$3$	−3.39321	−1.95907	−0.979535	+	0.201273i	$0.935492\pi$
$4$	5.79236	2.89618
$5$	3.31167	1.48102	0.740511	−	0.672044i	$-0.234583\pi$
$5$	3.31167	1.48102	0.740511	+	0.672044i	$0.234583\pi$
$6$	9.47208	3.86696
$7$	−0.0588258	−0.0222341	−0.0111170	−	0.999938i	$-0.503539\pi$
$7$	−0.0588258	−0.0222341	−0.0111170	+	0.999938i	$0.503539\pi$
$8$	−10.5863	−3.74282
$9$	8.51387	2.83796
$10$	−9.24445	−2.92335
$11$	3.18832	0.961314	0.480657	−	0.876909i	$-0.340398\pi$
$11$	3.18832	0.961314	0.480657	+	0.876909i	$0.340398\pi$
$12$	−19.6547	−5.67382
$13$	1.79597	0.498113	0.249056	−	0.968489i	$-0.419880\pi$
$13$	1.79597	0.498113	0.249056	+	0.968489i	$0.419880\pi$
$14$	0.164211	0.0438873
$15$	−11.2372	−2.90143
$16$	17.9667	4.49167
$17$	6.19854	1.50337	0.751683	−	0.659524i	$-0.229242\pi$
$17$	6.19854	1.50337	0.751683	+	0.659524i	$0.229242\pi$
$18$	−23.7663	−5.60177
$19$	4.49804	1.03192	0.515960	−	0.856612i	$-0.327435\pi$
$19$	4.49804	1.03192	0.515960	+	0.856612i	$0.327435\pi$
$20$	19.1824	4.28931
$21$	0.199608	0.0435581
$22$	−8.90012	−1.89751
$23$	−5.64628	−1.17733	−0.588666	−	0.808377i	$-0.700347\pi$
$23$	−5.64628	−1.17733	−0.588666	+	0.808377i	$0.700347\pi$
$24$	35.9215	7.33245
$25$	5.96714	1.19343
$26$	−5.01342	−0.983212
$27$	−18.7097	−3.60069
$28$	−0.340740	−0.0643939
$29$	0.468620	0.0870206	0.0435103	−	0.999053i	$-0.486146\pi$
$29$	0.468620	0.0870206	0.0435103	+	0.999053i	$0.486146\pi$
$30$	31.3684	5.72705
$31$	3.70270	0.665025	0.332513	−	0.943099i	$-0.392104\pi$
$31$	3.70270	0.665025	0.332513	+	0.943099i	$0.392104\pi$
$32$	−28.9811	−5.12318
$33$	−10.8186	−1.88328
$34$	−17.3031	−2.96746
$35$	−0.194812	−0.0329292
$36$	49.3154	8.21923
$37$	−2.54153	−0.417825	−0.208912	−	0.977934i	$-0.566992\pi$
$37$	−2.54153	−0.417825	−0.208912	+	0.977934i	$0.566992\pi$
$38$	−12.5562	−2.03688
$39$	−6.09411	−0.975838
$40$	−35.0583	−5.54320
$41$	−1.69797	−0.265178	−0.132589	−	0.991171i	$-0.542329\pi$
$41$	−1.69797	−0.265178	−0.132589	+	0.991171i	$0.542329\pi$
$42$	−0.557203	−0.0859782
$43$	10.8120	1.64882	0.824409	−	0.565995i	$-0.191508\pi$
$43$	10.8120	1.64882	0.824409	+	0.565995i	$0.191508\pi$
$44$	18.4679	2.78414
$45$	28.1951	4.20308
$46$	15.7615	2.32390
$47$	8.29508	1.20996	0.604981	−	0.796240i	$-0.293181\pi$
$47$	8.29508	1.20996	0.604981	+	0.796240i	$0.293181\pi$
$48$	−60.9648	−8.79951
$49$	−6.99654	−0.999506
$50$	−16.6571	−2.35568
$51$	−21.0329	−2.94520
$52$	10.4029	1.44262
$53$	13.4286	1.84456	0.922278	−	0.386527i	$-0.126325\pi$
$53$	13.4286	1.84456	0.922278	+	0.386527i	$0.126325\pi$
$54$	52.2278	7.10730
$55$	10.5586	1.42373
$56$	0.622747	0.0832181
$57$	−15.2628	−2.02161
$58$	−1.30814	−0.171768
$59$	7.23695	0.942171	0.471085	−	0.882088i	$-0.343862\pi$
$59$	7.23695	0.942171	0.471085	+	0.882088i	$0.343862\pi$
$60$	−65.0898	−8.40305
$61$	8.88227	1.13726	0.568629	−	0.822594i	$-0.307474\pi$
$61$	8.88227	1.13726	0.568629	+	0.822594i	$0.307474\pi$
$62$	−10.3360	−1.31268
$63$	−0.500835	−0.0630993
$64$	44.9667	5.62084
$65$	5.94766	0.737716
$66$	30.2000	3.71736
$67$	−4.59088	−0.560866	−0.280433	−	0.959874i	$-0.590478\pi$
$67$	−4.59088	−0.560866	−0.280433	+	0.959874i	$0.590478\pi$
$68$	35.9042	4.35402
$69$	19.1590	2.30648
$70$	0.543813	0.0649980
$71$	1.08524	0.128794	0.0643970	−	0.997924i	$-0.479488\pi$
$71$	1.08524	0.128794	0.0643970	+	0.997924i	$0.479488\pi$
$72$	−90.1303	−10.6220
$73$	10.0813	1.17992	0.589961	−	0.807432i	$-0.299143\pi$
$73$	10.0813	1.17992	0.589961	+	0.807432i	$0.299143\pi$
$74$	7.09463	0.824734
$75$	−20.2478	−2.33801
$76$	26.0542	2.98863
$77$	−0.187555	−0.0213739
$78$	17.0116	1.92618
$79$	−0.0659532	−0.00742032	−0.00371016	−	0.999993i	$-0.501181\pi$
$79$	−0.0659532	−0.00742032	−0.00371016	+	0.999993i	$0.501181\pi$
$80$	59.4997	6.65227
$81$	37.9444	4.21604
$82$	4.73984	0.523427
$83$	−15.5012	−1.70147	−0.850737	−	0.525592i	$-0.823844\pi$
$83$	−15.5012	−1.70147	−0.850737	+	0.525592i	$0.823844\pi$
$84$	1.15620	0.126152
$85$	20.5275	2.22652
$86$	−30.1815	−3.25456
$87$	−1.59013	−0.170479
$88$	−33.7525	−3.59802
$89$	−0.766469	−0.0812456	−0.0406228	−	0.999175i	$-0.512934\pi$
$89$	−0.766469	−0.0812456	−0.0406228	+	0.999175i	$0.512934\pi$
$90$	−78.7061	−8.29635
$91$	−0.105649	−0.0110751
$92$	−32.7053	−3.40976
$93$	−12.5640	−1.30283
$94$	−23.1555	−2.38831
$95$	14.8960	1.52830
$96$	98.3389	10.0367
$97$	8.17877	0.830429	0.415214	−	0.909724i	$-0.363707\pi$
$97$	8.17877	0.830429	0.415214	+	0.909724i	$0.363707\pi$
$98$	19.5307	1.97290
$99$	27.1449	2.72817
$100$	34.5638	3.45638
$101$	−4.24956	−0.422847	−0.211424	−	0.977395i	$-0.567810\pi$
$101$	−4.24956	−0.422847	−0.211424	+	0.977395i	$0.567810\pi$
$102$	58.7130	5.81346
$103$	8.42500	0.830140	0.415070	−	0.909790i	$-0.363757\pi$
$103$	8.42500	0.830140	0.415070	+	0.909790i	$0.363757\pi$
$104$	−19.0127	−1.86435
$105$	0.661036	0.0645106
$106$	−37.4856	−3.64092
$107$	1.31194	0.126830	0.0634149	−	0.997987i	$-0.479801\pi$
$107$	1.31194	0.126830	0.0634149	+	0.997987i	$0.479801\pi$
$108$	−108.373	−10.4282
$109$	10.7619	1.03081	0.515403	−	0.856948i	$-0.327642\pi$
$109$	10.7619	1.03081	0.515403	+	0.856948i	$0.327642\pi$
$110$	−29.4742	−2.81026
$111$	8.62394	0.818548
$112$	−1.05691	−0.0998682
$113$	1.49573	0.140707	0.0703534	−	0.997522i	$-0.477587\pi$
$113$	1.49573	0.140707	0.0703534	+	0.997522i	$0.477587\pi$
$114$	42.6058	3.99039
$115$	−18.6986	−1.74365
$116$	2.71442	0.252027
$117$	15.2907	1.41362
$118$	−20.2018	−1.85973
$119$	−0.364634	−0.0334260
$120$	118.960	10.8595
$121$	−0.834631	−0.0758755
$122$	−24.7947	−2.24481
$123$	5.76155	0.519502
$124$	21.4474	1.92603
$125$	3.20285	0.286471
$126$	1.39807	0.124550
$127$	15.7846	1.40066	0.700328	−	0.713821i	$-0.253037\pi$
$127$	15.7846	1.40066	0.700328	+	0.713821i	$0.253037\pi$
$128$	−67.5615	−5.97165
$129$	−36.6874	−3.23015
$130$	−16.6028	−1.45616
$131$	−16.8400	−1.47131	−0.735657	−	0.677354i	$-0.763127\pi$
$131$	−16.8400	−1.47131	−0.735657	+	0.677354i	$0.763127\pi$
$132$	−62.6654	−5.45432
$133$	−0.264601	−0.0229438
$134$	12.8154	1.10708
$135$	−61.9604	−5.33270
$136$	−65.6195	−5.62683
$137$	21.1332	1.80553	0.902766	−	0.430133i	$-0.141533\pi$
$137$	21.1332	1.80553	0.902766	+	0.430133i	$0.141533\pi$
$138$	−53.4820	−4.55269
$139$	−6.77180	−0.574376	−0.287188	−	0.957874i	$-0.592720\pi$
$139$	−6.77180	−0.574376	−0.287188	+	0.957874i	$0.592720\pi$
$140$	−1.12842	−0.0953688
$141$	−28.1469	−2.37040
$142$	−3.02942	−0.254223
$143$	5.72613	0.478843
$144$	152.966	12.7472
$145$	1.55191	0.128879
$146$	−28.1416	−2.32902
$147$	23.7407	1.95810
$148$	−14.7215	−1.21010
$149$	−0.650443	−0.0532864	−0.0266432	−	0.999645i	$-0.508482\pi$
$149$	−0.650443	−0.0532864	−0.0266432	+	0.999645i	$0.508482\pi$
$150$	56.5212	4.61494
$151$	−21.1962	−1.72492	−0.862461	−	0.506123i	$-0.831078\pi$
$151$	−21.1962	−1.72492	−0.862461	+	0.506123i	$0.831078\pi$
$152$	−47.6175	−3.86229
$153$	52.7736	4.26649
$154$	0.523557	0.0421894
$155$	12.2621	0.984917
$156$	−35.2992	−2.82620
$157$	−0.868505	−0.0693143	−0.0346571	−	0.999399i	$-0.511034\pi$
$157$	−0.868505	−0.0693143	−0.0346571	+	0.999399i	$0.511034\pi$
$158$	0.184107	0.0146468
$159$	−45.5660	−3.61362
$160$	−95.9757	−7.58755
$161$	0.332147	0.0261769
$162$	−105.921	−8.32194
$163$	−7.32957	−0.574096	−0.287048	−	0.957916i	$-0.592674\pi$
$163$	−7.32957	−0.574096	−0.287048	+	0.957916i	$0.592674\pi$
$164$	−9.83522	−0.768002
$165$	−35.8277	−2.78918
$166$	43.2712	3.35849
$167$	−21.3136	−1.64930	−0.824648	−	0.565646i	$-0.808627\pi$
$167$	−21.3136	−1.64930	−0.824648	+	0.565646i	$0.808627\pi$
$168$	−2.11311	−0.163030
$169$	−9.77449	−0.751884
$170$	−57.3021	−4.39487
$171$	38.2957	2.92855
$172$	62.6271	4.77527
$173$	−7.25464	−0.551560	−0.275780	−	0.961221i	$-0.588936\pi$
$173$	−7.25464	−0.551560	−0.275780	+	0.961221i	$0.588936\pi$
$174$	4.43881	0.336505
$175$	−0.351022	−0.0265348
$176$	57.2835	4.31791
$177$	−24.5565	−1.84578
$178$	2.13958	0.160369
$179$	−10.6435	−0.795532	−0.397766	−	0.917487i	$-0.630214\pi$
$179$	−10.6435	−0.795532	−0.397766	+	0.917487i	$0.630214\pi$
$180$	163.316	12.1729
$181$	−2.48701	−0.184858	−0.0924289	−	0.995719i	$-0.529463\pi$
$181$	−2.48701	−0.184858	−0.0924289	+	0.995719i	$0.529463\pi$
$182$	0.294918	0.0218608
$183$	−30.1394	−2.22797
$184$	59.7732	4.40654
$185$	−8.41670	−0.618808
$186$	35.0723	2.57162
$187$	19.7629	1.44521
$188$	48.0481	3.50427
$189$	1.10061	0.0800579
$190$	−41.5819	−3.01667
$191$	4.39238	0.317821	0.158911	−	0.987293i	$-0.449202\pi$
$191$	4.39238	0.317821	0.158911	+	0.987293i	$0.449202\pi$
$192$	−152.582	−11.0116
$193$	6.83078	0.491690	0.245845	−	0.969309i	$-0.420935\pi$
$193$	6.83078	0.491690	0.245845	+	0.969309i	$0.420935\pi$
$194$	−22.8309	−1.63916
$195$	−20.1816	−1.44524
$196$	−40.5265	−2.89475
$197$	−14.8072	−1.05497	−0.527485	−	0.849564i	$-0.676865\pi$
$197$	−14.8072	−1.05497	−0.527485	+	0.849564i	$0.676865\pi$
$198$	−75.7745	−5.38506
$199$	−15.9671	−1.13188	−0.565940	−	0.824446i	$-0.691487\pi$
$199$	−15.9671	−1.13188	−0.565940	+	0.824446i	$0.691487\pi$
$200$	−63.1699	−4.46679
$201$	15.5778	1.09878
$202$	11.8626	0.834647
$203$	−0.0275670	−0.00193482
$204$	−121.830	−8.52983
$205$	−5.62310	−0.392734
$206$	−23.5182	−1.63859
$207$	−48.0717	−3.34122
$208$	32.2677	2.23736
$209$	14.3412	0.992000
$210$	−1.84527	−0.127336
$211$	9.12887	0.628458	0.314229	−	0.949347i	$-0.398254\pi$
$211$	9.12887	0.628458	0.314229	+	0.949347i	$0.398254\pi$
$212$	77.7831	5.34217
$213$	−3.68244	−0.252317
$214$	−3.66224	−0.250346
$215$	35.8058	2.44194
$216$	198.067	13.4767
$217$	−0.217815	−0.0147862
$218$	−30.0417	−2.03468
$219$	−34.2078	−2.31155
$220$	61.1595	4.12337
$221$	11.1324	0.748846
$222$	−24.0736	−1.61571
$223$	22.8130	1.52767	0.763835	−	0.645412i	$-0.223314\pi$
$223$	22.8130	1.52767	0.763835	+	0.645412i	$0.223314\pi$
$224$	1.70484	0.113909
$225$	50.8035	3.38690
$226$	−4.17531	−0.277737
$227$	−4.62736	−0.307128	−0.153564	−	0.988139i	$-0.549075\pi$
$227$	−4.62736	−0.307128	−0.153564	+	0.988139i	$0.549075\pi$
$228$	−88.4075	−5.85493
$229$	−12.8559	−0.849539	−0.424770	−	0.905302i	$-0.639645\pi$
$229$	−12.8559	−0.849539	−0.424770	+	0.905302i	$0.639645\pi$
$230$	52.1968	3.44175
$231$	0.636415	0.0418730
$232$	−4.96095	−0.325702
$233$	−3.17577	−0.208052	−0.104026	−	0.994575i	$-0.533172\pi$
$233$	−3.17577	−0.208052	−0.104026	+	0.994575i	$0.533172\pi$
$234$	−42.6836	−2.79031
$235$	27.4705	1.79198
$236$	41.9190	2.72870
$237$	0.223793	0.0145369
$238$	1.01787	0.0659786
$239$	−4.54620	−0.294069	−0.147035	−	0.989131i	$-0.546973\pi$
$239$	−4.54620	−0.294069	−0.147035	+	0.989131i	$0.546973\pi$
$240$	−201.895	−13.0323
$241$	28.8968	1.86141	0.930704	−	0.365773i	$-0.119195\pi$
$241$	28.8968	1.86141	0.930704	+	0.365773i	$0.119195\pi$
$242$	2.32986	0.149769
$243$	−72.6241	−4.65884
$244$	51.4493	3.29370
$245$	−23.1702	−1.48029
$246$	−16.0833	−1.02543
$247$	8.07835	0.514013
$248$	−39.1979	−2.48907
$249$	52.5987	3.33331
$250$	−8.94068	−0.565458
$251$	−14.3530	−0.905951	−0.452976	−	0.891523i	$-0.649638\pi$
$251$	−14.3530	−0.905951	−0.452976	+	0.891523i	$0.649638\pi$
$252$	−2.90102	−0.182747
$253$	−18.0021	−1.13179
$254$	−44.0623	−2.76472
$255$	−69.6541	−4.36191
$256$	98.6631	6.16645
$257$	−28.4347	−1.77371	−0.886853	−	0.462051i	$-0.847114\pi$
$257$	−28.4347	−1.77371	−0.886853	+	0.462051i	$0.847114\pi$
$258$	102.412	6.37591
$259$	0.149508	0.00928995
$260$	34.4510	2.13656
$261$	3.98977	0.246961
$262$	47.0084	2.90419
$263$	−4.38600	−0.270452	−0.135226	−	0.990815i	$-0.543176\pi$
$263$	−4.38600	−0.270452	−0.135226	+	0.990815i	$0.543176\pi$
$264$	114.529	7.04878
$265$	44.4710	2.73183
$266$	0.738628	0.0452882
$267$	2.60079	0.159166
$268$	−26.5920	−1.62437
$269$	−4.91349	−0.299581	−0.149790	−	0.988718i	$-0.547860\pi$
$269$	−4.91349	−0.299581	−0.149790	+	0.988718i	$0.547860\pi$
$270$	172.961	10.5261
$271$	−0.210175	−0.0127672	−0.00638361	−	0.999980i	$-0.502032\pi$
$271$	−0.210175	−0.0127672	−0.00638361	+	0.999980i	$0.502032\pi$
$272$	111.367	6.75263
$273$	0.358491	0.0216969
$274$	−58.9929	−3.56389
$275$	19.0251	1.14726
$276$	110.976	6.67997
$277$	−8.28268	−0.497658	−0.248829	−	0.968547i	$-0.580046\pi$
$277$	−8.28268	−0.497658	−0.248829	+	0.968547i	$0.580046\pi$
$278$	18.9033	1.13375
$279$	31.5243	1.88731
$280$	2.06233	0.123248
$281$	−1.29009	−0.0769601	−0.0384801	−	0.999259i	$-0.512252\pi$
$281$	−1.29009	−0.0769601	−0.0384801	+	0.999259i	$0.512252\pi$
$282$	78.5716	4.67887
$283$	21.5190	1.27917	0.639585	−	0.768720i	$-0.279106\pi$
$283$	21.5190	1.27917	0.639585	+	0.768720i	$0.279106\pi$
$284$	6.28609	0.373011
$285$	−50.5453	−2.99404
$286$	−15.9844	−0.945175
$287$	0.0998842	0.00589598
$288$	−246.741	−14.5394
$289$	21.4219	1.26011
$290$	−4.33214	−0.254392
$291$	−27.7523	−1.62687
$292$	58.3943	3.41727
$293$	−6.55371	−0.382872	−0.191436	−	0.981505i	$-0.561314\pi$
$293$	−6.55371	−0.382872	−0.191436	+	0.981505i	$0.561314\pi$
$294$	−66.2717	−3.86505
$295$	23.9664	1.39538
$296$	26.9054	1.56384
$297$	−59.6525	−3.46139
$298$	1.81570	0.105181
$299$	−10.1406	−0.586444
$300$	−117.282	−6.77129
$301$	−0.636026	−0.0366599
$302$	59.1688	3.40478
$303$	14.4197	0.828388
$304$	80.8149	4.63505
$305$	29.4151	1.68431
$306$	−147.316	−8.42151
$307$	−5.27872	−0.301272	−0.150636	−	0.988589i	$-0.548132\pi$
$307$	−5.27872	−0.301272	−0.150636	+	0.988589i	$0.548132\pi$
$308$	−1.08639	−0.0619027
$309$	−28.5878	−1.62630
$310$	−34.2295	−1.94410
$311$	−2.75283	−0.156098	−0.0780492	−	0.996950i	$-0.524869\pi$
$311$	−2.75283	−0.156098	−0.0780492	+	0.996950i	$0.524869\pi$
$312$	64.5140	3.65238
$313$	−13.8893	−0.785069	−0.392535	−	0.919737i	$-0.628402\pi$
$313$	−13.8893	−0.785069	−0.392535	+	0.919737i	$0.628402\pi$
$314$	2.42441	0.136818
$315$	−1.65860	−0.0934516
$316$	−0.382025	−0.0214906
$317$	4.71155	0.264627	0.132314	−	0.991208i	$-0.457759\pi$
$317$	4.71155	0.264627	0.132314	+	0.991208i	$0.457759\pi$
$318$	127.196	7.13282
$319$	1.49411	0.0836541
$320$	148.915	8.32459
$321$	−4.45168	−0.248468
$322$	−0.927182	−0.0516699
$323$	27.8813	1.55136
$324$	219.787	12.2104
$325$	10.7168	0.594462
$326$	20.4604	1.13319
$327$	−36.5175	−2.01942
$328$	17.9752	0.992512
$329$	−0.487965	−0.0269024
$330$	100.012	5.50550
$331$	−4.98756	−0.274141	−0.137070	−	0.990561i	$-0.543769\pi$
$331$	−4.98756	−0.274141	−0.137070	+	0.990561i	$0.543769\pi$
$332$	−89.7883	−4.92777
$333$	−21.6383	−1.18577
$334$	59.4965	3.25550
$335$	−15.2035	−0.830655
$336$	3.58630	0.195649
$337$	22.6082	1.23155	0.615773	−	0.787923i	$-0.288844\pi$
$337$	22.6082	1.23155	0.615773	+	0.787923i	$0.288844\pi$
$338$	27.2853	1.48412
$339$	−5.07534	−0.275654
$340$	118.903	6.44840
$341$	11.8054	0.639298
$342$	−106.902	−5.78058
$343$	0.823358	0.0444572
$344$	−114.459	−6.17123
$345$	63.4483	3.41594
$346$	20.2512	1.08871
$347$	10.6305	0.570673	0.285336	−	0.958427i	$-0.407895\pi$
$347$	10.6305	0.570673	0.285336	+	0.958427i	$0.407895\pi$
$348$	−9.21058	−0.493739
$349$	9.17194	0.490963	0.245481	−	0.969401i	$-0.421054\pi$
$349$	9.17194	0.490963	0.245481	+	0.969401i	$0.421054\pi$
$350$	0.979871	0.0523763
$351$	−33.6021	−1.79355
$352$	−92.4009	−4.92499
$353$	−21.7008	−1.15502	−0.577508	−	0.816385i	$-0.695975\pi$
$353$	−21.7008	−1.15502	−0.577508	+	0.816385i	$0.695975\pi$
$354$	68.5489	3.64333
$355$	3.59395	0.190747
$356$	−4.43966	−0.235302
$357$	1.23728	0.0654838
$358$	29.7111	1.57028
$359$	−11.0528	−0.583344	−0.291672	−	0.956518i	$-0.594212\pi$
$359$	−11.0528	−0.583344	−0.291672	+	0.956518i	$0.594212\pi$
$360$	−298.482	−15.7314
$361$	1.23235	0.0648603
$362$	6.94243	0.364886
$363$	2.83208	0.148646
$364$	−0.611960	−0.0320754
$365$	33.3858	1.74749
$366$	84.1336	4.39773
$367$	−11.2090	−0.585107	−0.292553	−	0.956249i	$-0.594505\pi$
$367$	−11.2090	−0.585107	−0.292553	+	0.956249i	$0.594505\pi$
$368$	−101.445	−5.28819
$369$	−14.4563	−0.752563
$370$	23.4950	1.22145
$371$	−0.789947	−0.0410120
$372$	−72.7755	−3.77323
$373$	−28.7319	−1.48768	−0.743842	−	0.668356i	$-0.766998\pi$
$373$	−28.7319	−1.48768	−0.743842	+	0.668356i	$0.766998\pi$
$374$	−55.1678	−2.85266
$375$	−10.8679	−0.561217
$376$	−87.8141	−4.52867
$377$	0.841628	0.0433461
$378$	−3.07234	−0.158024
$379$	−20.0953	−1.03223	−0.516113	−	0.856521i	$-0.672621\pi$
$379$	−20.0953	−1.03223	−0.516113	+	0.856521i	$0.672621\pi$
$380$	86.2830	4.42622
$381$	−53.5604	−2.74398
$382$	−12.2612	−0.627339
$383$	−15.4554	−0.789736	−0.394868	−	0.918738i	$-0.629210\pi$
$383$	−15.4554	−0.789736	−0.394868	+	0.918738i	$0.629210\pi$
$384$	229.250	11.6989
$385$	−0.621121	−0.0316553
$386$	−19.0680	−0.970535
$387$	92.0521	4.67927
$388$	47.3744	2.40507
$389$	−5.73335	−0.290692	−0.145346	−	0.989381i	$-0.546430\pi$
$389$	−5.73335	−0.290692	−0.145346	+	0.989381i	$0.546430\pi$
$390$	56.3367	2.85272
$391$	−34.9987	−1.76996
$392$	74.0674	3.74097
$393$	57.1415	2.88241
$394$	41.3340	2.08238
$395$	−0.218415	−0.0109897
$396$	157.233	7.90126
$397$	32.7094	1.64164	0.820819	−	0.571188i	$-0.193517\pi$
$397$	32.7094	1.64164	0.820819	+	0.571188i	$0.193517\pi$
$398$	44.5719	2.23419
$399$	0.897846	0.0449485
$400$	107.210	5.36049
$401$	7.10338	0.354726	0.177363	−	0.984146i	$-0.443243\pi$
$401$	7.10338	0.354726	0.177363	+	0.984146i	$0.443243\pi$
$402$	−43.4852	−2.16884
$403$	6.64995	0.331257
$404$	−24.6150	−1.22464
$405$	125.659	6.24405
$406$	0.0769527	0.00381910
$407$	−8.10320	−0.401661
$408$	222.661	11.0234
$409$	−8.73243	−0.431791	−0.215895	−	0.976417i	$-0.569267\pi$
$409$	−8.73243	−0.431791	−0.215895	+	0.976417i	$0.569267\pi$
$410$	15.6968	0.775208
$411$	−71.7094	−3.53716
$412$	48.8006	2.40423
$413$	−0.425720	−0.0209483
$414$	134.191	6.59514
$415$	−51.3347	−2.51992
$416$	−52.0492	−2.55192
$417$	22.9781	1.12524
$418$	−40.0331	−1.95808
$419$	−9.58578	−0.468296	−0.234148	−	0.972201i	$-0.575230\pi$
$419$	−9.58578	−0.468296	−0.234148	+	0.972201i	$0.575230\pi$
$420$	3.82896	0.186834
$421$	8.97404	0.437368	0.218684	−	0.975796i	$-0.429824\pi$
$421$	8.97404	0.437368	0.218684	+	0.975796i	$0.429824\pi$
$422$	−25.4831	−1.24050
$423$	70.6232	3.43382
$424$	−142.159	−6.90384
$425$	36.9876	1.79416
$426$	10.2795	0.498041
$427$	−0.522507	−0.0252859
$428$	7.59921	0.367322
$429$	−19.4299	−0.938087
$430$	−99.9512	−4.82007
$431$	−8.77461	−0.422658	−0.211329	−	0.977415i	$-0.567779\pi$
$431$	−8.77461	−0.422658	−0.211329	+	0.977415i	$0.567779\pi$
$432$	−336.152	−16.1731
$433$	−20.3910	−0.979929	−0.489965	−	0.871742i	$-0.662990\pi$
$433$	−20.3910	−0.979929	−0.489965	+	0.871742i	$0.662990\pi$
$434$	0.608025	0.0291861
$435$	−5.26597	−0.252484
$436$	62.3369	2.98540
$437$	−25.3972	−1.21491
$438$	95.4905	4.56271
$439$	19.9997	0.954534	0.477267	−	0.878758i	$-0.341628\pi$
$439$	19.9997	0.954534	0.477267	+	0.878758i	$0.341628\pi$
$440$	−111.777	−5.32876
$441$	−59.5676	−2.83655
$442$	−31.0759	−1.47813
$443$	−10.1109	−0.480384	−0.240192	−	0.970725i	$-0.577210\pi$
$443$	−10.1109	−0.480384	−0.240192	+	0.970725i	$0.577210\pi$
$444$	49.9530	2.37066
$445$	−2.53829	−0.120327
$446$	−63.6820	−3.01543
$447$	2.20709	0.104392
$448$	−2.64520	−0.124974
$449$	3.76482	0.177673	0.0888365	−	0.996046i	$-0.471685\pi$
$449$	3.76482	0.177673	0.0888365	+	0.996046i	$0.471685\pi$
$450$	−141.817	−6.68531
$451$	−5.41365	−0.254919
$452$	8.66382	0.407512
$453$	71.9232	3.37925
$454$	12.9172	0.606233
$455$	−0.349876	−0.0164024
$456$	161.576	7.56650
$457$	20.1456	0.942372	0.471186	−	0.882034i	$-0.343826\pi$
$457$	20.1456	0.942372	0.471186	+	0.882034i	$0.343826\pi$
$458$	35.8869	1.67688
$459$	−115.973	−5.41315
$460$	−108.309	−5.04994
$461$	36.8199	1.71487	0.857436	−	0.514590i	$-0.172056\pi$
$461$	36.8199	1.71487	0.857436	+	0.514590i	$0.172056\pi$
$462$	−1.77654	−0.0826521
$463$	3.30313	0.153510	0.0767548	−	0.997050i	$-0.475544\pi$
$463$	3.30313	0.153510	0.0767548	+	0.997050i	$0.475544\pi$
$464$	8.41956	0.390868
$465$	−41.6079	−1.92952
$466$	8.86510	0.410668
$467$	−29.7903	−1.37853	−0.689265	−	0.724510i	$-0.742066\pi$
$467$	−29.7903	−1.37853	−0.689265	+	0.724510i	$0.742066\pi$
$468$	88.5690	4.09410
$469$	0.270063	0.0124703
$470$	−76.6835	−3.53714
$471$	2.94702	0.135792
$472$	−76.6125	−3.52637
$473$	34.4721	1.58503
$474$	−0.624714	−0.0286941
$475$	26.8404	1.23152
$476$	−2.11209	−0.0968076
$477$	114.329	5.23477
$478$	12.6906	0.580456
$479$	−25.8255	−1.18000	−0.589998	−	0.807405i	$-0.700872\pi$
$479$	−25.8255	−1.18000	−0.589998	+	0.807405i	$0.700872\pi$
$480$	325.666	14.8645
$481$	−4.56451	−0.208124
$482$	−80.6649	−3.67419
$483$	−1.12705	−0.0512823
$484$	−4.83448	−0.219749
$485$	27.0854	1.22988
$486$	202.729	9.19596
$487$	−23.4214	−1.06132	−0.530662	−	0.847584i	$-0.678057\pi$
$487$	−23.4214	−1.06132	−0.530662	+	0.847584i	$0.678057\pi$
$488$	−94.0303	−4.25655
$489$	24.8708	1.12470
$490$	64.6792	2.92191
$491$	24.0741	1.08645	0.543226	−	0.839587i	$-0.317203\pi$
$491$	24.0741	1.08645	0.543226	+	0.839587i	$0.317203\pi$
$492$	33.3730	1.50457
$493$	2.90476	0.130824
$494$	−22.5505	−1.01460
$495$	89.8950	4.04048
$496$	66.5254	2.98708
$497$	−0.0638400	−0.00286362
$498$	−146.828	−6.57953
$499$	−5.19841	−0.232713	−0.116357	−	0.993208i	$-0.537122\pi$
$499$	−5.19841	−0.232713	−0.116357	+	0.993208i	$0.537122\pi$
$500$	18.5520	0.829672
$501$	72.3215	3.23109
$502$	40.0660	1.78823
$503$	−13.4706	−0.600624	−0.300312	−	0.953841i	$-0.597091\pi$
$503$	−13.4706	−0.600624	−0.300312	+	0.953841i	$0.597091\pi$
$504$	5.30199	0.236169
$505$	−14.0731	−0.626246
$506$	50.2526	2.23400
$507$	33.1669	1.47299
$508$	91.4300	4.05655
$509$	6.64288	0.294440	0.147220	−	0.989104i	$-0.452967\pi$
$509$	6.64288	0.294440	0.147220	+	0.989104i	$0.452967\pi$
$510$	194.438	8.60986
$511$	−0.593039	−0.0262345
$512$	−140.293	−6.20014
$513$	−84.1570	−3.71562
$514$	79.3748	3.50107
$515$	27.9008	1.22946
$516$	−212.507	−9.35509
$517$	26.4473	1.16315
$518$	−0.417347	−0.0183372
$519$	24.6165	1.08055
$520$	−62.9636	−2.76114
$521$	14.5048	0.635466	0.317733	−	0.948180i	$-0.397078\pi$
$521$	14.5048	0.635466	0.317733	+	0.948180i	$0.397078\pi$
$522$	−11.1374	−0.487469
$523$	15.2881	0.668500	0.334250	−	0.942484i	$-0.391517\pi$
$523$	15.2881	0.668500	0.334250	+	0.942484i	$0.391517\pi$
$524$	−97.5431	−4.26119
$525$	1.19109	0.0519835
$526$	12.2434	0.533839
$527$	22.9514	0.999777
$528$	−194.375	−8.45909
$529$	8.88052	0.386110
$530$	−124.140	−5.39229
$531$	61.6145	2.67384
$532$	−1.53266	−0.0664494
$533$	−3.04950	−0.132088
$534$	−7.26005	−0.314173
$535$	4.34470	0.187838
$536$	48.6004	2.09922
$537$	36.1156	1.55850
$538$	13.7159	0.591334
$539$	−22.3072	−0.960839
$540$	−358.897	−15.4445
$541$	0.147364	0.00633567	0.00316783	−	0.999995i	$-0.498992\pi$
$541$	0.147364	0.00633567	0.00316783	+	0.999995i	$0.498992\pi$
$542$	0.586699	0.0252009
$543$	8.43893	0.362149
$544$	−179.640	−7.70202
$545$	35.6399	1.52665
$546$	−1.00072	−0.0428269
$547$	34.1302	1.45930	0.729650	−	0.683821i	$-0.239683\pi$
$547$	34.1302	1.45930	0.729650	+	0.683821i	$0.239683\pi$
$548$	122.411	5.22914
$549$	75.6225	3.22749
$550$	−53.1083	−2.26454
$551$	2.10787	0.0897984
$552$	−202.823	−8.63272
$553$	0.00387975	0.000164984 0
$554$	23.1209	0.982314
$555$	28.5596	1.21229
$556$	−39.2247	−1.66350
$557$	16.6720	0.706414	0.353207	−	0.935545i	$-0.385091\pi$
$557$	16.6720	0.706414	0.353207	+	0.935545i	$0.385091\pi$
$558$	−87.9995	−3.72532
$559$	19.4181	0.821297
$560$	−3.50012	−0.147907
$561$	−67.0597	−2.83126
$562$	3.60125	0.151910
$563$	17.2366	0.726437	0.363219	−	0.931704i	$-0.381678\pi$
$563$	17.2366	0.726437	0.363219	+	0.931704i	$0.381678\pi$
$564$	−163.037	−6.86510
$565$	4.95337	0.208390
$566$	−60.0698	−2.52492
$567$	−2.23211	−0.0937398
$568$	−11.4886	−0.482053
$569$	−27.4550	−1.15098	−0.575488	−	0.817810i	$-0.695188\pi$
$569$	−27.4550	−1.15098	−0.575488	+	0.817810i	$0.695188\pi$
$570$	141.096	5.90986
$571$	−7.76414	−0.324919	−0.162460	−	0.986715i	$-0.551943\pi$
$571$	−7.76414	−0.324919	−0.162460	+	0.986715i	$0.551943\pi$
$572$	33.1678	1.38681
$573$	−14.9043	−0.622634
$574$	−0.278825	−0.0116379
$575$	−33.6922	−1.40506
$576$	382.841	15.9517
$577$	−35.3062	−1.46982	−0.734909	−	0.678166i	$-0.762775\pi$
$577$	−35.3062	−1.46982	−0.734909	+	0.678166i	$0.762775\pi$
$578$	−59.7988	−2.48730
$579$	−23.1783	−0.963256
$580$	8.98924	0.373258
$581$	0.911868	0.0378307
$582$	77.4700	3.21123
$583$	42.8146	1.77320
$584$	−106.723	−4.41624
$585$	50.6376	2.09361
$586$	18.2946	0.755741
$587$	−24.3739	−1.00602	−0.503009	−	0.864281i	$-0.667774\pi$
$587$	−24.3739	−1.00602	−0.503009	+	0.864281i	$0.667774\pi$
$588$	137.515	5.67101
$589$	16.6549	0.686253
$590$	−66.9016	−2.75430
$591$	50.2440	2.06676
$592$	−45.6629	−1.87673
$593$	−36.9055	−1.51553	−0.757763	−	0.652530i	$-0.773708\pi$
$593$	−36.9055	−1.51553	−0.757763	+	0.652530i	$0.773708\pi$
$594$	166.519	6.83235
$595$	−1.20755	−0.0495046
$596$	−3.76760	−0.154327
$597$	54.1798	2.21743
$598$	28.3072	1.15757
$599$	−7.89027	−0.322388	−0.161194	−	0.986923i	$-0.551534\pi$
$599$	−7.89027	−0.322388	−0.161194	+	0.986923i	$0.551534\pi$
$600$	214.349	8.75075
$601$	−12.4167	−0.506489	−0.253244	−	0.967402i	$-0.581498\pi$
$601$	−12.4167	−0.506489	−0.253244	+	0.967402i	$0.581498\pi$
$602$	1.77545	0.0723621
$603$	−39.0862	−1.59171
$604$	−122.776	−4.99569
$605$	−2.76402	−0.112373
$606$	−40.2522	−1.63513
$607$	40.7676	1.65471	0.827353	−	0.561683i	$-0.189846\pi$
$607$	40.7676	1.65471	0.827353	+	0.561683i	$0.189846\pi$
$608$	−130.358	−5.28672
$609$	0.0935405	0.00379045
$610$	−82.1117	−3.32461
$611$	14.8977	0.602697
$612$	305.683	12.3565
$613$	−32.8429	−1.32651	−0.663256	−	0.748393i	$-0.730826\pi$
$613$	−32.8429	−1.32651	−0.663256	+	0.748393i	$0.730826\pi$
$614$	14.7354	0.594674
$615$	19.0803	0.769394
$616$	1.98552	0.0799987
$617$	−13.2654	−0.534045	−0.267022	−	0.963690i	$-0.586040\pi$
$617$	−13.2654	−0.534045	−0.267022	+	0.963690i	$0.586040\pi$
$618$	79.8022	3.21012
$619$	−2.69384	−0.108274	−0.0541372	−	0.998534i	$-0.517241\pi$
$619$	−2.69384	−0.108274	−0.0541372	+	0.998534i	$0.517241\pi$
$620$	71.0266	2.85250
$621$	105.640	4.23920
$622$	7.68446	0.308119
$623$	0.0450882	0.00180642
$624$	−109.491	−4.38315
$625$	−19.2289	−0.769158
$626$	38.7717	1.54963
$627$	−48.6626	−1.94340
$628$	−5.03069	−0.200746
$629$	−15.7538	−0.628144
$630$	4.62995	0.184462
$631$	−4.81315	−0.191609	−0.0958043	−	0.995400i	$-0.530542\pi$
$631$	−4.81315	−0.191609	−0.0958043	+	0.995400i	$0.530542\pi$
$632$	0.698200	0.0277729
$633$	−30.9762	−1.23119
$634$	−13.1522	−0.522341
$635$	52.2733	2.07440
$636$	−263.934	−10.4657
$637$	−12.5656	−0.497866
$638$	−4.17078	−0.165123
$639$	9.23958	0.365512
$640$	−223.741	−8.84415
$641$	10.0123	0.395461	0.197730	−	0.980256i	$-0.436643\pi$
$641$	10.0123	0.395461	0.197730	+	0.980256i	$0.436643\pi$
$642$	12.4268	0.490445
$643$	11.6628	0.459935	0.229967	−	0.973198i	$-0.426138\pi$
$643$	11.6628	0.459935	0.229967	+	0.973198i	$0.426138\pi$
$644$	1.92392	0.0758129
$645$	−121.497	−4.78392
$646$	−77.8300	−3.06218
$647$	−31.7555	−1.24844	−0.624218	−	0.781250i	$-0.714582\pi$
$647$	−31.7555	−1.24844	−0.624218	+	0.781250i	$0.714582\pi$
$648$	−401.690	−15.7799
$649$	23.0737	0.905722
$650$	−29.9158	−1.17339
$651$	0.739091	0.0289672
$652$	−42.4555	−1.66269
$653$	−8.88968	−0.347880	−0.173940	−	0.984756i	$-0.555650\pi$
$653$	−8.88968	−0.347880	−0.173940	+	0.984756i	$0.555650\pi$
$654$	101.938	3.98608
$655$	−55.7684	−2.17905
$656$	−30.5068	−1.19109
$657$	85.8306	3.34857
$658$	1.36214	0.0531019
$659$	34.6975	1.35162	0.675812	−	0.737074i	$-0.263793\pi$
$659$	34.6975	1.35162	0.675812	+	0.737074i	$0.263793\pi$
$660$	−207.527	−8.07797
$661$	−41.9538	−1.63181	−0.815907	−	0.578183i	$-0.803762\pi$
$661$	−41.9538	−1.63181	−0.815907	+	0.578183i	$0.803762\pi$
$662$	13.9227	0.541120
$663$	−37.7746	−1.46704
$664$	164.100	6.36831
$665$	−0.876270	−0.0339803
$666$	60.4027	2.34056
$667$	−2.64596	−0.102452
$668$	−123.456	−4.77666
$669$	−77.4092	−2.99281
$670$	42.4402	1.63961
$671$	28.3195	1.09326
$672$	−5.78487	−0.223156
$673$	−45.0710	−1.73736	−0.868679	−	0.495376i	$-0.835030\pi$
$673$	−45.0710	−1.73736	−0.868679	+	0.495376i	$0.835030\pi$
$674$	−63.1103	−2.43092
$675$	−111.644	−4.29716
$676$	−56.6173	−2.17759
$677$	45.4326	1.74612	0.873059	−	0.487615i	$-0.162133\pi$
$677$	45.4326	1.74612	0.873059	+	0.487615i	$0.162133\pi$
$678$	14.1677	0.544107
$679$	−0.481123	−0.0184638
$680$	−217.310	−8.33346
$681$	15.7016	0.601686
$682$	−32.9545	−1.26189
$683$	29.9300	1.14524	0.572620	−	0.819821i	$-0.305927\pi$
$683$	29.9300	1.14524	0.572620	+	0.819821i	$0.305927\pi$
$684$	221.822	8.48160
$685$	69.9861	2.67403
$686$	−2.29839	−0.0877528
$687$	43.6226	1.66431
$688$	194.256	7.40595
$689$	24.1173	0.918797
$690$	−177.115	−6.74264
$691$	−2.17052	−0.0825703	−0.0412852	−	0.999147i	$-0.513145\pi$
$691$	−2.17052	−0.0825703	−0.0412852	+	0.999147i	$0.513145\pi$
$692$	−42.0215	−1.59742
$693$	−1.59682	−0.0606583
$694$	−29.6747	−1.12644
$695$	−22.4259	−0.850664
$696$	16.8335	0.638074
$697$	−10.5249	−0.398659
$698$	−25.6033	−0.969099
$699$	10.7761	0.407588
$700$	−2.03324	−0.0768494
$701$	−20.9867	−0.792656	−0.396328	−	0.918109i	$-0.629716\pi$
$701$	−20.9867	−0.792656	−0.396328	+	0.918109i	$0.629716\pi$
$702$	93.7996	3.54024
$703$	−11.4319	−0.431162
$704$	143.368	5.40339
$705$	−93.2133	−3.51062
$706$	60.5773	2.27986
$707$	0.249984	0.00940162
$708$	−142.240	−5.34571
$709$	1.83365	0.0688642	0.0344321	−	0.999407i	$-0.489038\pi$
$709$	1.83365	0.0688642	0.0344321	+	0.999407i	$0.489038\pi$
$710$	−10.0324	−0.376510
$711$	−0.561517	−0.0210585
$712$	8.11406	0.304087
$713$	−20.9065	−0.782955
$714$	−3.45384	−0.129257
$715$	18.9630	0.709177
$716$	−61.6509	−2.30400
$717$	15.4262	0.576103
$718$	30.8537	1.15145
$719$	33.3957	1.24545	0.622725	−	0.782440i	$-0.286025\pi$
$719$	33.3957	1.24545	0.622725	+	0.782440i	$0.286025\pi$
$720$	506.573	18.8789
$721$	−0.495607	−0.0184574
$722$	−3.44007	−0.128026
$723$	−98.0530	−3.64663
$724$	−14.4056	−0.535381
$725$	2.79632	0.103853
$726$	−7.90569	−0.293408
$727$	43.1420	1.60005	0.800023	−	0.599969i	$-0.204820\pi$
$727$	43.1420	1.60005	0.800023	+	0.599969i	$0.204820\pi$
$728$	1.11844	0.0414520
$729$	132.596	4.91095
$730$	−93.1958	−3.44933
$731$	67.0187	2.47878
$732$	−174.578	−6.45260
$733$	17.7038	0.653905	0.326952	−	0.945041i	$-0.393978\pi$
$733$	17.7038	0.653905	0.326952	+	0.945041i	$0.393978\pi$
$734$	31.2898	1.15493
$735$	78.6214	2.89999
$736$	163.635	6.03168
$737$	−14.6372	−0.539168
$738$	40.3543	1.48546
$739$	38.7703	1.42619	0.713093	−	0.701069i	$-0.247294\pi$
$739$	38.7703	1.42619	0.713093	+	0.701069i	$0.247294\pi$
$740$	−48.7525	−1.79218
$741$	−27.4115	−1.00699
$742$	2.20512	0.0809525
$743$	−27.9035	−1.02368	−0.511841	−	0.859080i	$-0.671036\pi$
$743$	−27.9035	−1.02368	−0.511841	+	0.859080i	$0.671036\pi$
$744$	133.007	4.87626
$745$	−2.15405	−0.0789184
$746$	80.2046	2.93650
$747$	−131.975	−4.82871
$748$	114.474	4.18558
$749$	−0.0771758	−0.00281994
$750$	30.3376	1.10777
$751$	41.9860	1.53209	0.766045	−	0.642787i	$-0.222222\pi$
$751$	41.9860	1.53209	0.766045	+	0.642787i	$0.222222\pi$
$752$	149.035	5.43475
$753$	48.7026	1.77482
$754$	−2.34939	−0.0855597
$755$	−70.1948	−2.55465
$756$	6.37515	0.231862
$757$	−30.3527	−1.10319	−0.551594	−	0.834113i	$-0.685980\pi$
$757$	−30.3527	−1.10319	−0.551594	+	0.834113i	$0.685980\pi$
$758$	56.0956	2.03748
$759$	61.0851	2.21725
$760$	−157.693	−5.72014
$761$	−20.4000	−0.739498	−0.369749	−	0.929132i	$-0.620556\pi$
$761$	−20.4000	−0.739498	−0.369749	+	0.929132i	$0.620556\pi$
$762$	149.513	5.41628
$763$	−0.633079	−0.0229190
$764$	25.4422	0.920467
$765$	174.768	6.31877
$766$	43.1436	1.55884
$767$	12.9974	0.469307
$768$	−334.785	−12.0805
$769$	11.9939	0.432510	0.216255	−	0.976337i	$-0.430616\pi$
$769$	11.9939	0.432510	0.216255	+	0.976337i	$0.430616\pi$
$770$	1.73385	0.0624835
$771$	96.4848	3.47482
$772$	39.5663	1.42402
$773$	−20.4701	−0.736258	−0.368129	−	0.929775i	$-0.620001\pi$
$773$	−20.4701	−0.736258	−0.368129	+	0.929775i	$0.620001\pi$
$774$	−256.962	−9.23630
$775$	22.0946	0.793660
$776$	−86.5829	−3.10814
$777$	−0.507311	−0.0181997
$778$	16.0045	0.573790
$779$	−7.63751	−0.273642
$780$	−116.899	−4.18567
$781$	3.46008	0.123812
$782$	97.6982	3.49368
$783$	−8.76775	−0.313334
$784$	−125.705	−4.48945
$785$	−2.87620	−0.102656
$786$	−159.509	−5.68951
$787$	−11.7779	−0.419837	−0.209919	−	0.977719i	$-0.567320\pi$
$787$	−11.7779	−0.419837	−0.209919	+	0.977719i	$0.567320\pi$
$788$	−85.7687	−3.05538
$789$	14.8826	0.529835
$790$	0.609701	0.0216922
$791$	−0.0879877	−0.00312848
$792$	−287.364	−10.2110
$793$	15.9523	0.566483
$794$	−91.3076	−3.24039
$795$	−150.899	−5.35185
$796$	−92.4874	−3.27813
$797$	−32.6588	−1.15683	−0.578417	−	0.815741i	$-0.696329\pi$
$797$	−32.6588	−1.15683	−0.578417	+	0.815741i	$0.696329\pi$
$798$	−2.50632	−0.0887227
$799$	51.4174	1.81902
$800$	−172.934	−6.11415
$801$	−6.52562	−0.230571
$802$	−19.8289	−0.700184
$803$	32.1423	1.13428
$804$	90.2324	3.18225
$805$	1.09996	0.0387685
$806$	−18.5632	−0.653861
$807$	16.6725	0.586899
$808$	44.9871	1.58264
$809$	10.7998	0.379700	0.189850	−	0.981813i	$-0.439200\pi$
$809$	10.7998	0.379700	0.189850	+	0.981813i	$0.439200\pi$
$810$	−350.775	−12.3250
$811$	32.3548	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$811$	32.3548	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$812$	−0.159678	−0.00560359
$813$	0.713168	0.0250119
$814$	22.6199	0.792828
$815$	−24.2731	−0.850250
$816$	−377.893	−13.2289
$817$	48.6329	1.70145
$818$	24.3764	0.852301
$819$	−0.899486	−0.0314306
$820$	−32.5710	−1.13743
$821$	18.6941	0.652430	0.326215	−	0.945296i	$-0.394227\pi$
$821$	18.6941	0.652430	0.326215	+	0.945296i	$0.394227\pi$
$822$	200.175	6.98192
$823$	31.9265	1.11289	0.556445	−	0.830885i	$-0.312165\pi$
$823$	31.9265	1.11289	0.556445	+	0.830885i	$0.312165\pi$
$824$	−89.1895	−3.10706
$825$	−64.5563	−2.24756
$826$	1.18839	0.0413493
$827$	−21.9542	−0.763423	−0.381712	−	0.924281i	$-0.624665\pi$
$827$	−21.9542	−0.763423	−0.381712	+	0.924281i	$0.624665\pi$
$828$	−278.449	−9.67676
$829$	−6.91657	−0.240222	−0.120111	−	0.992760i	$-0.538325\pi$
$829$	−6.91657	−0.240222	−0.120111	+	0.992760i	$0.538325\pi$
$830$	143.300	4.97401
$831$	28.1049	0.974947
$832$	80.7589	2.79981
$833$	−43.3683	−1.50262
$834$	−64.1430	−2.22109
$835$	−70.5835	−2.44264
$836$	83.0692	2.87301
$837$	−69.2765	−2.39455
$838$	26.7585	0.924357
$839$	−42.0345	−1.45119	−0.725597	−	0.688120i	$-0.758436\pi$
$839$	−42.0345	−1.45119	−0.725597	+	0.688120i	$0.758436\pi$
$840$	−6.99792	−0.241451
$841$	−28.7804	−0.992427
$842$	−25.0509	−0.863309
$843$	4.37754	0.150770
$844$	52.8777	1.82013
$845$	−32.3699	−1.11356
$846$	−197.143	−6.77793
$847$	0.0490979	0.00168702
$848$	241.267	8.28515
$849$	−73.0184	−2.50598
$850$	−103.250	−3.54145
$851$	14.3502	0.491918
$852$	−21.3300	−0.730754
$853$	12.2437	0.419218	0.209609	−	0.977785i	$-0.432781\pi$
$853$	12.2437	0.419218	0.209609	+	0.977785i	$0.432781\pi$
$854$	1.45857	0.0499112
$855$	126.823	4.33724
$856$	−13.8885	−0.474701
$857$	29.2810	1.00022	0.500111	−	0.865961i	$-0.333293\pi$
$857$	29.2810	1.00022	0.500111	+	0.865961i	$0.333293\pi$
$858$	54.2383	1.85166
$859$	11.0410	0.376714	0.188357	−	0.982101i	$-0.439684\pi$
$859$	11.0410	0.376714	0.188357	+	0.982101i	$0.439684\pi$
$860$	207.400	7.07228
$861$	−0.338928	−0.0115506
$862$	24.4941	0.834273
$863$	−11.7603	−0.400324	−0.200162	−	0.979763i	$-0.564147\pi$
$863$	−11.7603	−0.400324	−0.200162	+	0.979763i	$0.564147\pi$
$864$	542.228	18.4470
$865$	−24.0250	−0.816873
$866$	56.9211	1.93426
$867$	−72.6890	−2.46865
$868$	−1.26166	−0.0428235
$869$	−0.210280	−0.00713325
$870$	14.6999	0.498372
$871$	−8.24509	−0.279374
$872$	−113.929	−3.85812
$873$	69.6330	2.35672
$874$	70.8958	2.39808
$875$	−0.188410	−0.00636942
$876$	−198.144	−6.69467
$877$	18.8586	0.636810	0.318405	−	0.947955i	$-0.396853\pi$
$877$	18.8586	0.636810	0.318405	+	0.947955i	$0.396853\pi$
$878$	−55.8288	−1.88413
$879$	22.2381	0.750073
$880$	189.704	6.39492
$881$	50.8589	1.71348	0.856740	−	0.515749i	$-0.172486\pi$
$881$	50.8589	1.71348	0.856740	+	0.515749i	$0.172486\pi$
$882$	166.282	5.59900
$883$	26.2185	0.882323	0.441161	−	0.897428i	$-0.354567\pi$
$883$	26.2185	0.882323	0.441161	+	0.897428i	$0.354567\pi$
$884$	64.4828	2.16879
$885$	−81.3229	−2.73364
$886$	28.2244	0.948218
$887$	−12.7191	−0.427066	−0.213533	−	0.976936i	$-0.568497\pi$
$887$	−12.7191	−0.427066	−0.213533	+	0.976936i	$0.568497\pi$
$888$	−91.2956	−3.06368
$889$	−0.928541	−0.0311423
$890$	7.08559	0.237509
$891$	120.979	4.05294
$892$	132.141	4.42441
$893$	37.3116	1.24858
$894$	−6.16105	−0.206056
$895$	−35.2477	−1.17820
$896$	3.97436	0.132774
$897$	34.4091	1.14888
$898$	−10.5094	−0.350704
$899$	1.73516	0.0578709
$900$	294.272	9.80906
$901$	83.2375	2.77304
$902$	15.1121	0.503178
$903$	2.15817	0.0718194
$904$	−15.8343	−0.526640
$905$	−8.23614	−0.273778
$906$	−200.772	−6.67020
$907$	−7.53515	−0.250201	−0.125100	−	0.992144i	$-0.539925\pi$
$907$	−7.53515	−0.250201	−0.125100	+	0.992144i	$0.539925\pi$
$908$	−26.8033	−0.889499
$909$	−36.1802	−1.20002
$910$	0.976671	0.0323763
$911$	−57.6440	−1.90983	−0.954916	−	0.296875i	$-0.904055\pi$
$911$	−57.6440	−1.90983	−0.954916	+	0.296875i	$0.904055\pi$
$912$	−274.222	−9.08039
$913$	−49.4226	−1.63565
$914$	−56.2360	−1.86012
$915$	−99.8117	−3.29967
$916$	−74.4657	−2.46042
$917$	0.990625	0.0327133
$918$	323.736	10.6849
$919$	−57.9051	−1.91011	−0.955057	−	0.296423i	$-0.904206\pi$
$919$	−57.9051	−1.91011	−0.955057	+	0.296423i	$0.904206\pi$
$920$	197.949	6.52618
$921$	17.9118	0.590214
$922$	−102.782	−3.38494
$923$	1.94906	0.0641540
$924$	3.68634	0.121272
$925$	−15.1657	−0.498644
$926$	−9.22062	−0.303008
$927$	71.7293	2.35590
$928$	−13.5811	−0.445822
$929$	−33.0230	−1.08345	−0.541725	−	0.840556i	$-0.682229\pi$
$929$	−33.0230	−1.08345	−0.541725	+	0.840556i	$0.682229\pi$
$930$	116.148	3.80863
$931$	−31.4707	−1.03141
$932$	−18.3952	−0.602555
$933$	9.34091	0.305808
$934$	83.1589	2.72104
$935$	65.4482	2.14038
$936$	−161.871	−5.29093
$937$	−41.3694	−1.35148	−0.675739	−	0.737141i	$-0.736175\pi$
$937$	−41.3694	−1.35148	−0.675739	+	0.737141i	$0.736175\pi$
$938$	−0.753874	−0.0246149
$939$	47.1293	1.53801
$940$	159.119	5.18990
$941$	38.5037	1.25518	0.627592	−	0.778542i	$-0.284041\pi$
$941$	38.5037	1.25518	0.627592	+	0.778542i	$0.284041\pi$
$942$	−8.22655	−0.268035
$943$	9.58719	0.312202
$944$	130.024	4.23192
$945$	3.64487	0.118568
$946$	−96.2283	−3.12865
$947$	28.1100	0.913451	0.456725	−	0.889608i	$-0.349022\pi$
$947$	28.1100	0.913451	0.456725	+	0.889608i	$0.349022\pi$
$948$	1.29629	0.0421015
$949$	18.1057	0.587734
$950$	−74.9245	−2.43087
$951$	−15.9873	−0.518423
$952$	3.86012	0.125107
$953$	−25.3834	−0.822250	−0.411125	−	0.911579i	$-0.634864\pi$
$953$	−25.3834	−0.822250	−0.411125	+	0.911579i	$0.634864\pi$
$954$	−319.147	−10.3328
$955$	14.5461	0.470700
$956$	−26.3332	−0.851678
$957$	−5.06983	−0.163884
$958$	72.0913	2.32916
$959$	−1.24318	−0.0401443
$960$	−505.299	−16.3085
$961$	−17.2900	−0.557742
$962$	12.7417	0.410810
$963$	11.1697	0.359937
$964$	167.381	5.39097
$965$	22.6213	0.728204
$966$	3.14612	0.101225
$967$	29.5088	0.948941	0.474470	−	0.880272i	$-0.342640\pi$
$967$	29.5088	0.948941	0.474470	+	0.880272i	$0.342640\pi$
$968$	8.83565	0.283988
$969$	−94.6070	−3.03921
$970$	−75.6083	−2.42764
$971$	33.8925	1.08766	0.543831	−	0.839195i	$-0.316973\pi$
$971$	33.8925	1.08766	0.543831	+	0.839195i	$0.316973\pi$
$972$	−420.665	−13.4928
$973$	0.398357	0.0127707
$974$	65.3803	2.09492
$975$	−36.3644	−1.16459
$976$	159.585	5.10820
$977$	54.2431	1.73539	0.867696	−	0.497095i	$-0.165600\pi$
$977$	54.2431	1.73539	0.867696	+	0.497095i	$0.165600\pi$
$978$	−69.4263	−2.22001
$979$	−2.44375	−0.0781025
$980$	−134.210	−4.28719
$981$	91.6257	2.92538
$982$	−67.2025	−2.14452
$983$	3.05541	0.0974525	0.0487263	−	0.998812i	$-0.484484\pi$
$983$	3.05541	0.0974525	0.0487263	+	0.998812i	$0.484484\pi$
$984$	−60.9935	−1.94440
$985$	−49.0366	−1.56243
$986$	−8.10858	−0.258230
$987$	1.65577	0.0527037
$988$	46.7927	1.48867
$989$	−61.0477	−1.94120
$990$	−250.940	−7.97539
$991$	61.3063	1.94746	0.973729	−	0.227709i	$-0.0731235\pi$
$991$	61.3063	1.94746	0.973729	+	0.227709i	$0.0731235\pi$
$992$	−107.308	−3.40704
$993$	16.9238	0.537061
$994$	0.178208	0.00565242
$995$	−52.8778	−1.67634
$996$	304.670	9.65385
$997$	−49.7062	−1.57421	−0.787106	−	0.616818i	$-0.788421\pi$
$997$	−49.7062	−1.57421	−0.787106	+	0.616818i	$0.788421\pi$
$998$	14.5113	0.459346
$999$	47.5513	1.50446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4019.2.a.b.1.2	✓	186

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4019.2.a.b.1.2	✓	186	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:	\( N \)	\(=\)	\( 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79148 q^{2} -3.39321 q^{3} +5.79236 q^{4} +3.31167 q^{5} +9.47208 q^{6} -0.0588258 q^{7} -10.5863 q^{8} +8.51387 q^{9} +O(q^{10})\)	\(q-2.79148 q^{2} -3.39321 q^{3} +5.79236 q^{4} +3.31167 q^{5} +9.47208 q^{6} -0.0588258 q^{7} -10.5863 q^{8} +8.51387 q^{9} -9.24445 q^{10} +3.18832 q^{11} -19.6547 q^{12} +1.79597 q^{13} +0.164211 q^{14} -11.2372 q^{15} +17.9667 q^{16} +6.19854 q^{17} -23.7663 q^{18} +4.49804 q^{19} +19.1824 q^{20} +0.199608 q^{21} -8.90012 q^{22} -5.64628 q^{23} +35.9215 q^{24} +5.96714 q^{25} -5.01342 q^{26} -18.7097 q^{27} -0.340740 q^{28} +0.468620 q^{29} +31.3684 q^{30} +3.70270 q^{31} -28.9811 q^{32} -10.8186 q^{33} -17.3031 q^{34} -0.194812 q^{35} +49.3154 q^{36} -2.54153 q^{37} -12.5562 q^{38} -6.09411 q^{39} -35.0583 q^{40} -1.69797 q^{41} -0.557203 q^{42} +10.8120 q^{43} +18.4679 q^{44} +28.1951 q^{45} +15.7615 q^{46} +8.29508 q^{47} -60.9648 q^{48} -6.99654 q^{49} -16.6571 q^{50} -21.0329 q^{51} +10.4029 q^{52} +13.4286 q^{53} +52.2278 q^{54} +10.5586 q^{55} +0.622747 q^{56} -15.2628 q^{57} -1.30814 q^{58} +7.23695 q^{59} -65.0898 q^{60} +8.88227 q^{61} -10.3360 q^{62} -0.500835 q^{63} +44.9667 q^{64} +5.94766 q^{65} +30.2000 q^{66} -4.59088 q^{67} +35.9042 q^{68} +19.1590 q^{69} +0.543813 q^{70} +1.08524 q^{71} -90.1303 q^{72} +10.0813 q^{73} +7.09463 q^{74} -20.2478 q^{75} +26.0542 q^{76} -0.187555 q^{77} +17.0116 q^{78} -0.0659532 q^{79} +59.4997 q^{80} +37.9444 q^{81} +4.73984 q^{82} -15.5012 q^{83} +1.15620 q^{84} +20.5275 q^{85} -30.1815 q^{86} -1.59013 q^{87} -33.7525 q^{88} -0.766469 q^{89} -78.7061 q^{90} -0.105649 q^{91} -32.7053 q^{92} -12.5640 q^{93} -23.1555 q^{94} +14.8960 q^{95} +98.3389 q^{96} +8.17877 q^{97} +19.5307 q^{98} +27.1449 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * q^99

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