LMFDB - Embedded newform 215.4.a.d.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [215,4,Mod(1,215)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(215, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("215.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 215 = 5 \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	215.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:	$12.6854106512$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 101 x^{13} + 85 x^{12} + 4104 x^{11} - 2826 x^{10} - 85598 x^{9} + 46269 x^{8} + \cdots + 110160 $ x^15 - x^14 - 101x^13 + 85x^12 + 4104x^11 - 2826x^10 - 85598x^9 + 46269x^8 + 964385x^7 - 383175x^6 - 5573805x^5 + 1453572x^4 + 13183622x^3 - 1740884x^2 - 2016128*x + 110160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$4.04752$ of defining polynomial
Character	$\chi$	$=$	215.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.04752 q^{2} -4.47643 q^{3} +8.38240 q^{4} +5.00000 q^{5} -18.1184 q^{6} +35.2850 q^{7} +1.54775 q^{8} -6.96159 q^{9} +O(q^{10})$	\(q+4.04752 q^{2} -4.47643 q^{3} +8.38240 q^{4} +5.00000 q^{5} -18.1184 q^{6} +35.2850 q^{7} +1.54775 q^{8} -6.96159 q^{9} +20.2376 q^{10} -21.9191 q^{11} -37.5232 q^{12} +70.3707 q^{13} +142.817 q^{14} -22.3821 q^{15} -60.7946 q^{16} +107.764 q^{17} -28.1772 q^{18} +54.4585 q^{19} +41.9120 q^{20} -157.951 q^{21} -88.7178 q^{22} +67.2934 q^{23} -6.92841 q^{24} +25.0000 q^{25} +284.827 q^{26} +152.027 q^{27} +295.773 q^{28} -163.908 q^{29} -90.5921 q^{30} -4.41635 q^{31} -258.449 q^{32} +98.1191 q^{33} +436.177 q^{34} +176.425 q^{35} -58.3548 q^{36} +147.963 q^{37} +220.422 q^{38} -315.009 q^{39} +7.73877 q^{40} -397.564 q^{41} -639.308 q^{42} +43.0000 q^{43} -183.734 q^{44} -34.8080 q^{45} +272.371 q^{46} -495.142 q^{47} +272.143 q^{48} +902.031 q^{49} +101.188 q^{50} -482.398 q^{51} +589.875 q^{52} -177.278 q^{53} +615.330 q^{54} -109.595 q^{55} +54.6125 q^{56} -243.780 q^{57} -663.422 q^{58} -126.013 q^{59} -187.616 q^{60} -297.662 q^{61} -17.8752 q^{62} -245.640 q^{63} -559.721 q^{64} +351.854 q^{65} +397.139 q^{66} -880.592 q^{67} +903.322 q^{68} -301.234 q^{69} +714.083 q^{70} -229.051 q^{71} -10.7748 q^{72} -394.100 q^{73} +598.882 q^{74} -111.911 q^{75} +456.493 q^{76} -773.414 q^{77} -1275.01 q^{78} +32.3392 q^{79} -303.973 q^{80} -492.573 q^{81} -1609.15 q^{82} -424.143 q^{83} -1324.01 q^{84} +538.821 q^{85} +174.043 q^{86} +733.725 q^{87} -33.9253 q^{88} -196.658 q^{89} -140.886 q^{90} +2483.03 q^{91} +564.080 q^{92} +19.7694 q^{93} -2004.10 q^{94} +272.293 q^{95} +1156.93 q^{96} +1515.69 q^{97} +3650.98 q^{98} +152.592 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{2} + 7 q^{3} + 83 q^{4} + 75 q^{5} + 40 q^{6} + 36 q^{7} + 33 q^{8} + 242 q^{9}+O(q^{10}) $ 15 * q + q^2 + 7 * q^3 + 83 * q^4 + 75 * q^5 + 40 * q^6 + 36 * q^7 + 33 * q^8 + 242 * q^9	\( 15 q + q^{2} + 7 q^{3} + 83 q^{4} + 75 q^{5} + 40 q^{6} + 36 q^{7} + 33 q^{8} + 242 q^{9} + 5 q^{10} + 87 q^{11} - 37 q^{12} - 35 q^{13} + 237 q^{14} + 35 q^{15} + 139 q^{16} + 144 q^{17} - 325 q^{18} + 293 q^{19} + 415 q^{20} + 208 q^{21} + 25 q^{22} + 22 q^{23} + 480 q^{24} + 375 q^{25} + 127 q^{26} + 202 q^{27} + 187 q^{28} + 495 q^{29} + 200 q^{30} + 408 q^{31} + 395 q^{32} + 308 q^{33} + 812 q^{34} + 180 q^{35} + 1283 q^{36} + 1013 q^{37} + 251 q^{38} + 654 q^{39} + 165 q^{40} + 820 q^{41} + 431 q^{42} + 645 q^{43} + 1244 q^{44} + 1210 q^{45} + 1416 q^{46} - 622 q^{47} + 70 q^{48} + 1525 q^{49} + 25 q^{50} + 1760 q^{51} - 593 q^{52} + 276 q^{53} - 349 q^{54} + 435 q^{55} + 822 q^{56} - 566 q^{57} + 1403 q^{58} - 1405 q^{59} - 185 q^{60} + 1969 q^{61} - 3297 q^{62} - 2587 q^{63} - 1709 q^{64} - 175 q^{65} - 3111 q^{66} - 439 q^{67} - 3412 q^{68} + 10 q^{69} + 1185 q^{70} + 494 q^{71} - 6476 q^{72} + 958 q^{73} - 6273 q^{74} + 175 q^{75} + 1089 q^{76} - 918 q^{77} - 5438 q^{78} - 1846 q^{79} + 695 q^{80} + 3655 q^{81} - 2693 q^{82} - 1524 q^{83} - 6283 q^{84} + 720 q^{85} + 43 q^{86} - 2702 q^{87} - 2010 q^{88} + 1380 q^{89} - 1625 q^{90} + 3625 q^{91} - 3658 q^{92} + 1610 q^{93} - 274 q^{94} + 1465 q^{95} - 367 q^{96} + 3240 q^{97} - 6137 q^{98} - 3769 q^{99}+O(q^{100}) \) 15 * q + q^2 + 7 * q^3 + 83 * q^4 + 75 * q^5 + 40 * q^6 + 36 * q^7 + 33 * q^8 + 242 * q^9 + 5 * q^10 + 87 * q^11 - 37 * q^12 - 35 * q^13 + 237 * q^14 + 35 * q^15 + 139 * q^16 + 144 * q^17 - 325 * q^18 + 293 * q^19 + 415 * q^20 + 208 * q^21 + 25 * q^22 + 22 * q^23 + 480 * q^24 + 375 * q^25 + 127 * q^26 + 202 * q^27 + 187 * q^28 + 495 * q^29 + 200 * q^30 + 408 * q^31 + 395 * q^32 + 308 * q^33 + 812 * q^34 + 180 * q^35 + 1283 * q^36 + 1013 * q^37 + 251 * q^38 + 654 * q^39 + 165 * q^40 + 820 * q^41 + 431 * q^42 + 645 * q^43 + 1244 * q^44 + 1210 * q^45 + 1416 * q^46 - 622 * q^47 + 70 * q^48 + 1525 * q^49 + 25 * q^50 + 1760 * q^51 - 593 * q^52 + 276 * q^53 - 349 * q^54 + 435 * q^55 + 822 * q^56 - 566 * q^57 + 1403 * q^58 - 1405 * q^59 - 185 * q^60 + 1969 * q^61 - 3297 * q^62 - 2587 * q^63 - 1709 * q^64 - 175 * q^65 - 3111 * q^66 - 439 * q^67 - 3412 * q^68 + 10 * q^69 + 1185 * q^70 + 494 * q^71 - 6476 * q^72 + 958 * q^73 - 6273 * q^74 + 175 * q^75 + 1089 * q^76 - 918 * q^77 - 5438 * q^78 - 1846 * q^79 + 695 * q^80 + 3655 * q^81 - 2693 * q^82 - 1524 * q^83 - 6283 * q^84 + 720 * q^85 + 43 * q^86 - 2702 * q^87 - 2010 * q^88 + 1380 * q^89 - 1625 * q^90 + 3625 * q^91 - 3658 * q^92 + 1610 * q^93 - 274 * q^94 + 1465 * q^95 - 367 * q^96 + 3240 * q^97 - 6137 * q^98 - 3769 * 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 $

$2$	4.04752	1.43101	0.715507	−	0.698606i	$-0.246196\pi$
$2$	4.04752	1.43101	0.715507	+	0.698606i	$0.246196\pi$
$3$	−4.47643	−0.861489	−0.430744	−	0.902474i	$-0.641749\pi$
$3$	−4.47643	−0.861489	−0.430744	+	0.902474i	$0.641749\pi$
$4$	8.38240	1.04780
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−18.1184	−1.23280
$7$	35.2850	1.90521	0.952605	−	0.304209i	$-0.0983922\pi$
$7$	35.2850	1.90521	0.952605	+	0.304209i	$0.0983922\pi$
$8$	1.54775	0.0684017
$9$	−6.96159	−0.257837
$10$	20.2376	0.639969
$11$	−21.9191	−0.600804	−0.300402	−	0.953813i	$-0.597121\pi$
$11$	−21.9191	−0.600804	−0.300402	+	0.953813i	$0.597121\pi$
$12$	−37.5232	−0.902668
$13$	70.3707	1.50133	0.750666	−	0.660681i	$-0.229733\pi$
$13$	70.3707	1.50133	0.750666	+	0.660681i	$0.229733\pi$
$14$	142.817	2.72638
$15$	−22.3821	−0.385270
$16$	−60.7946	−0.949916
$17$	107.764	1.53745	0.768725	−	0.639580i	$-0.220892\pi$
$17$	107.764	1.53745	0.768725	+	0.639580i	$0.220892\pi$
$18$	−28.1772	−0.368968
$19$	54.4585	0.657560	0.328780	−	0.944407i	$-0.393363\pi$
$19$	54.4585	0.657560	0.328780	+	0.944407i	$0.393363\pi$
$20$	41.9120	0.468590
$21$	−157.951	−1.64132
$22$	−88.7178	−0.859759
$23$	67.2934	0.610071	0.305036	−	0.952341i	$-0.401332\pi$
$23$	67.2934	0.610071	0.305036	+	0.952341i	$0.401332\pi$
$24$	−6.92841	−0.0589273
$25$	25.0000	0.200000
$26$	284.827	2.14843
$27$	152.027	1.08361
$28$	295.773	1.99628
$29$	−163.908	−1.04955	−0.524777	−	0.851240i	$-0.675851\pi$
$29$	−163.908	−1.04955	−0.524777	+	0.851240i	$0.675851\pi$
$30$	−90.5921	−0.551326
$31$	−4.41635	−0.0255871	−0.0127935	−	0.999918i	$-0.504072\pi$
$31$	−4.41635	−0.0255871	−0.0127935	+	0.999918i	$0.504072\pi$
$32$	−258.449	−1.42774
$33$	98.1191	0.517586
$34$	436.177	2.20011
$35$	176.425	0.852036
$36$	−58.3548	−0.270161
$37$	147.963	0.657431	0.328715	−	0.944429i	$-0.393384\pi$
$37$	147.963	0.657431	0.328715	+	0.944429i	$0.393384\pi$
$38$	220.422	0.940977
$39$	−315.009	−1.29338
$40$	7.73877	0.0305902
$41$	−397.564	−1.51437	−0.757183	−	0.653202i	$-0.773425\pi$
$41$	−397.564	−1.51437	−0.757183	+	0.653202i	$0.773425\pi$
$42$	−639.308	−2.34875
$43$	43.0000	0.152499
$43$	43.0000	0.152499
$44$	−183.734	−0.629522
$45$	−34.8080	−0.115308
$46$	272.371	0.873020
$47$	−495.142	−1.53668	−0.768340	−	0.640042i	$-0.778917\pi$
$47$	−495.142	−1.53668	−0.768340	+	0.640042i	$0.778917\pi$
$48$	272.143	0.818342
$49$	902.031	2.62983
$50$	101.188	0.286203
$51$	−482.398	−1.32450
$52$	589.875	1.57310
$53$	−177.278	−0.459453	−0.229726	−	0.973255i	$-0.573783\pi$
$53$	−177.278	−0.459453	−0.229726	+	0.973255i	$0.573783\pi$
$54$	615.330	1.55066
$55$	−109.595	−0.268688
$56$	54.6125	0.130320
$57$	−243.780	−0.566481
$58$	−663.422	−1.50192
$59$	−126.013	−0.278060	−0.139030	−	0.990288i	$-0.544398\pi$
$59$	−126.013	−0.278060	−0.139030	+	0.990288i	$0.544398\pi$
$60$	−187.616	−0.403685
$61$	−297.662	−0.624781	−0.312391	−	0.949954i	$-0.601130\pi$
$61$	−297.662	−0.624781	−0.312391	+	0.949954i	$0.601130\pi$
$62$	−17.8752	−0.0366154
$63$	−245.640	−0.491233
$64$	−559.721	−1.09321
$65$	351.854	0.671417
$66$	397.139	0.740673
$67$	−880.592	−1.60569	−0.802847	−	0.596185i	$-0.796682\pi$
$67$	−880.592	−1.60569	−0.802847	+	0.596185i	$0.796682\pi$
$68$	903.322	1.61094
$69$	−301.234	−0.525570
$70$	714.083	1.21928
$71$	−229.051	−0.382864	−0.191432	−	0.981506i	$-0.561313\pi$
$71$	−229.051	−0.382864	−0.191432	+	0.981506i	$0.561313\pi$
$72$	−10.7748	−0.0176365
$73$	−394.100	−0.631862	−0.315931	−	0.948782i	$-0.602317\pi$
$73$	−394.100	−0.631862	−0.315931	+	0.948782i	$0.602317\pi$
$74$	598.882	0.940792
$75$	−111.911	−0.172298
$76$	456.493	0.688991
$77$	−773.414	−1.14466
$78$	−1275.01	−1.85085
$79$	32.3392	0.0460563	0.0230281	−	0.999735i	$-0.492669\pi$
$79$	32.3392	0.0460563	0.0230281	+	0.999735i	$0.492669\pi$
$80$	−303.973	−0.424815
$81$	−492.573	−0.675683
$82$	−1609.15	−2.16708
$83$	−424.143	−0.560912	−0.280456	−	0.959867i	$-0.590486\pi$
$83$	−424.143	−0.560912	−0.280456	+	0.959867i	$0.590486\pi$
$84$	−1324.01	−1.71977
$85$	538.821	0.687568
$86$	174.043	0.218228
$87$	733.725	0.904179
$88$	−33.9253	−0.0410960
$89$	−196.658	−0.234222	−0.117111	−	0.993119i	$-0.537363\pi$
$89$	−196.658	−0.234222	−0.117111	+	0.993119i	$0.537363\pi$
$90$	−140.886	−0.165007
$91$	2483.03	2.86036
$92$	564.080	0.639233
$93$	19.7694	0.0220430
$94$	−2004.10	−2.19901
$95$	272.293	0.294070
$96$	1156.93	1.22999
$97$	1515.69	1.58655	0.793275	−	0.608863i	$-0.208374\pi$
$97$	1515.69	1.58655	0.793275	+	0.608863i	$0.208374\pi$
$98$	3650.98	3.76332
$99$	152.592	0.154909
$100$	209.560	0.209560
$101$	1105.03	1.08866	0.544331	−	0.838870i	$-0.316784\pi$
$101$	1105.03	1.08866	0.544331	+	0.838870i	$0.316784\pi$
$102$	−1952.52	−1.89537
$103$	510.496	0.488356	0.244178	−	0.969730i	$-0.421482\pi$
$103$	510.496	0.488356	0.244178	+	0.969730i	$0.421482\pi$
$104$	108.917	0.102694
$105$	−789.754	−0.734020
$106$	−717.535	−0.657483
$107$	1365.04	1.23330	0.616651	−	0.787237i	$-0.288489\pi$
$107$	1365.04	1.23330	0.616651	+	0.787237i	$0.288489\pi$
$108$	1274.35	1.13541
$109$	−822.014	−0.722337	−0.361168	−	0.932501i	$-0.617622\pi$
$109$	−822.014	−0.722337	−0.361168	+	0.932501i	$0.617622\pi$
$110$	−443.589	−0.384496
$111$	−662.345	−0.566369
$112$	−2145.14	−1.80979
$113$	1152.54	0.959484	0.479742	−	0.877410i	$-0.340730\pi$
$113$	1152.54	0.959484	0.479742	+	0.877410i	$0.340730\pi$
$114$	−986.702	−0.810641
$115$	336.467	0.272832
$116$	−1373.95	−1.09972
$117$	−489.892	−0.387099
$118$	−510.041	−0.397908
$119$	3802.46	2.92916
$120$	−34.6421	−0.0263531
$121$	−850.555	−0.639034
$122$	−1204.79	−0.894070
$123$	1779.67	1.30461
$124$	−37.0196	−0.0268101
$125$	125.000	0.0894427
$126$	−994.231	−0.702962
$127$	−888.312	−0.620669	−0.310334	−	0.950627i	$-0.600441\pi$
$127$	−888.312	−0.620669	−0.310334	+	0.950627i	$0.600441\pi$
$128$	−197.886	−0.136647
$129$	−192.486	−0.131376
$130$	1424.13	0.960806
$131$	−1787.49	−1.19216	−0.596082	−	0.802923i	$-0.703277\pi$
$131$	−1787.49	−1.19216	−0.596082	+	0.802923i	$0.703277\pi$
$132$	822.473	0.542326
$133$	1921.57	1.25279
$134$	−3564.21	−2.29777
$135$	760.133	0.484606
$136$	166.792	0.105164
$137$	738.607	0.460609	0.230304	−	0.973119i	$-0.426028\pi$
$137$	738.607	0.460609	0.230304	+	0.973119i	$0.426028\pi$
$138$	−1219.25	−0.752097
$139$	−1479.35	−0.902711	−0.451355	−	0.892344i	$-0.649059\pi$
$139$	−1479.35	−0.902711	−0.451355	+	0.892344i	$0.649059\pi$
$140$	1478.86	0.892763
$141$	2216.47	1.32383
$142$	−927.088	−0.547884
$143$	−1542.46	−0.902007
$144$	423.227	0.244923
$145$	−819.542	−0.469374
$146$	−1595.13	−0.904203
$147$	−4037.87	−2.26557
$148$	1240.28	0.688856
$149$	−3540.65	−1.94672	−0.973359	−	0.229286i	$-0.926361\pi$
$149$	−3540.65	−1.94672	−0.973359	+	0.229286i	$0.926361\pi$
$150$	−452.960	−0.246560
$151$	1729.42	0.932041	0.466021	−	0.884774i	$-0.345687\pi$
$151$	1729.42	0.932041	0.466021	+	0.884774i	$0.345687\pi$
$152$	84.2884	0.0449782
$153$	−750.210	−0.396411
$154$	−3130.41	−1.63802
$155$	−22.0817	−0.0114429
$156$	−2640.53	−1.35520
$157$	3023.38	1.53689	0.768445	−	0.639916i	$-0.221031\pi$
$157$	3023.38	1.53689	0.768445	+	0.639916i	$0.221031\pi$
$158$	130.894	0.0659072
$159$	793.572	0.395813
$160$	−1292.25	−0.638506
$161$	2374.45	1.16231
$162$	−1993.70	−0.966912
$163$	−2337.01	−1.12300	−0.561498	−	0.827478i	$-0.689775\pi$
$163$	−2337.01	−1.12300	−0.561498	+	0.827478i	$0.689775\pi$
$164$	−3332.54	−1.58675
$165$	490.595	0.231472
$166$	−1716.73	−0.802673
$167$	−765.636	−0.354771	−0.177385	−	0.984141i	$-0.556764\pi$
$167$	−765.636	−0.354771	−0.177385	+	0.984141i	$0.556764\pi$
$168$	−244.469	−0.112269
$169$	2755.04	1.25400
$170$	2180.89	0.983919
$171$	−379.118	−0.169543
$172$	360.443	0.159788
$173$	−3740.17	−1.64370	−0.821849	−	0.569705i	$-0.807057\pi$
$173$	−3740.17	−1.64370	−0.821849	+	0.569705i	$0.807057\pi$
$174$	2969.76	1.29389
$175$	882.125	0.381042
$176$	1332.56	0.570713
$177$	564.090	0.239546
$178$	−795.978	−0.335175
$179$	1984.89	0.828815	0.414408	−	0.910091i	$-0.363989\pi$
$179$	1984.89	0.828815	0.414408	+	0.910091i	$0.363989\pi$
$180$	−291.774	−0.120820
$181$	3272.15	1.34374	0.671870	−	0.740669i	$-0.265492\pi$
$181$	3272.15	1.34374	0.671870	+	0.740669i	$0.265492\pi$
$182$	10050.1	4.09321
$183$	1332.46	0.538242
$184$	104.154	0.0417299
$185$	739.814	0.294012
$186$	80.0172	0.0315438
$187$	−2362.09	−0.923706
$188$	−4150.48	−1.61013
$189$	5364.26	2.06451
$190$	1102.11	0.420818
$191$	2851.55	1.08027	0.540134	−	0.841579i	$-0.318374\pi$
$191$	2851.55	1.08027	0.540134	+	0.841579i	$0.318374\pi$
$192$	2505.55	0.941784
$193$	1437.13	0.535996	0.267998	−	0.963419i	$-0.413638\pi$
$193$	1437.13	0.535996	0.267998	+	0.963419i	$0.413638\pi$
$194$	6134.80	2.27038
$195$	−1575.05	−0.578418
$196$	7561.18	2.75553
$197$	2659.10	0.961691	0.480846	−	0.876805i	$-0.340330\pi$
$197$	2659.10	0.961691	0.480846	+	0.876805i	$0.340330\pi$
$198$	617.617	0.221677
$199$	−3007.35	−1.07128	−0.535642	−	0.844445i	$-0.679930\pi$
$199$	−3007.35	−1.07128	−0.535642	+	0.844445i	$0.679930\pi$
$200$	38.6939	0.0136803
$201$	3941.91	1.38329
$202$	4472.64	1.55789
$203$	−5783.51	−1.99962
$204$	−4043.65	−1.38781
$205$	−1987.82	−0.677245
$206$	2066.24	0.698844
$207$	−468.469	−0.157299
$208$	−4278.16	−1.42614
$209$	−1193.68	−0.395065
$210$	−3196.54	−1.05039
$211$	−4234.51	−1.38159	−0.690795	−	0.723050i	$-0.742739\pi$
$211$	−4234.51	−1.38159	−0.690795	+	0.723050i	$0.742739\pi$
$212$	−1486.01	−0.481414
$213$	1025.33	0.329833
$214$	5525.02	1.76487
$215$	215.000	0.0681994
$216$	235.300	0.0741210
$217$	−155.831	−0.0487487
$218$	−3327.12	−1.03367
$219$	1764.16	0.544342
$220$	−918.671	−0.281531
$221$	7583.44	2.30822
$222$	−2680.85	−0.810482
$223$	−240.530	−0.0722291	−0.0361145	−	0.999348i	$-0.511498\pi$
$223$	−240.530	−0.0722291	−0.0361145	+	0.999348i	$0.511498\pi$
$224$	−9119.38	−2.72015
$225$	−174.040	−0.0515674
$226$	4664.92	1.37303
$227$	−5343.90	−1.56250	−0.781249	−	0.624219i	$-0.785417\pi$
$227$	−5343.90	−1.56250	−0.781249	+	0.624219i	$0.785417\pi$
$228$	−2043.46	−0.593558
$229$	2036.67	0.587715	0.293858	−	0.955849i	$-0.405061\pi$
$229$	2036.67	0.587715	0.293858	+	0.955849i	$0.405061\pi$
$230$	1361.86	0.390427
$231$	3462.13	0.986110
$232$	−253.690	−0.0717913
$233$	2974.63	0.836372	0.418186	−	0.908361i	$-0.362666\pi$
$233$	2974.63	0.836372	0.418186	+	0.908361i	$0.362666\pi$
$234$	−1982.85	−0.553944
$235$	−2475.71	−0.687224
$236$	−1056.29	−0.291351
$237$	−144.764	−0.0396770
$238$	15390.5	4.19167
$239$	2022.60	0.547411	0.273705	−	0.961814i	$-0.411751\pi$
$239$	2022.60	0.547411	0.273705	+	0.961814i	$0.411751\pi$
$240$	1360.71	0.365974
$241$	−5525.71	−1.47694	−0.738470	−	0.674286i	$-0.764451\pi$
$241$	−5525.71	−1.47694	−0.738470	+	0.674286i	$0.764451\pi$
$242$	−3442.64	−0.914467
$243$	−1899.75	−0.501519
$244$	−2495.12	−0.654645
$245$	4510.15	1.17609
$246$	7203.23	1.86691
$247$	3832.29	0.987216
$248$	−6.83542	−0.00175020
$249$	1898.64	0.483220
$250$	505.940	0.127994
$251$	−6544.91	−1.64586	−0.822930	−	0.568142i	$-0.807662\pi$
$251$	−6544.91	−1.64586	−0.822930	+	0.568142i	$0.807662\pi$
$252$	−2059.05	−0.514714
$253$	−1475.01	−0.366533
$254$	−3595.46	−0.888186
$255$	−2411.99	−0.592332
$256$	3676.82	0.897661
$257$	3003.85	0.729086	0.364543	−	0.931187i	$-0.381225\pi$
$257$	3003.85	0.729086	0.364543	+	0.931187i	$0.381225\pi$
$258$	−779.092	−0.188001
$259$	5220.87	1.25254
$260$	2949.38	0.703510
$261$	1141.06	0.270613
$262$	−7234.89	−1.70600
$263$	123.301	0.0289090	0.0144545	−	0.999896i	$-0.495399\pi$
$263$	123.301	0.0289090	0.0144545	+	0.999896i	$0.495399\pi$
$264$	151.864	0.0354038
$265$	−886.389	−0.205473
$266$	7777.58	1.79276
$267$	880.327	0.201780
$268$	−7381.47	−1.68245
$269$	7004.23	1.58757	0.793783	−	0.608201i	$-0.208109\pi$
$269$	7004.23	1.58757	0.793783	+	0.608201i	$0.208109\pi$
$270$	3076.65	0.693478
$271$	−2280.47	−0.511175	−0.255588	−	0.966786i	$-0.582269\pi$
$271$	−2280.47	−0.511175	−0.255588	+	0.966786i	$0.582269\pi$
$272$	−6551.48	−1.46045
$273$	−11115.1	−2.46416
$274$	2989.52	0.659137
$275$	−547.976	−0.120161
$276$	−2525.06	−0.550692
$277$	−5160.72	−1.11941	−0.559707	−	0.828690i	$-0.689086\pi$
$277$	−5160.72	−1.11941	−0.559707	+	0.828690i	$0.689086\pi$
$278$	−5987.69	−1.29179
$279$	30.7448	0.00659729
$280$	273.063	0.0582807
$281$	5153.09	1.09398	0.546989	−	0.837140i	$-0.315774\pi$
$281$	5153.09	1.09398	0.546989	+	0.837140i	$0.315774\pi$
$282$	8971.20	1.89442
$283$	2144.33	0.450414	0.225207	−	0.974311i	$-0.427694\pi$
$283$	2144.33	0.450414	0.225207	+	0.974311i	$0.427694\pi$
$284$	−1920.00	−0.401165
$285$	−1218.90	−0.253338
$286$	−6243.13	−1.29078
$287$	−14028.0	−2.88519
$288$	1799.22	0.368125
$289$	6700.11	1.36375
$290$	−3317.11	−0.671681
$291$	−6784.90	−1.36680
$292$	−3303.50	−0.662065
$293$	2639.76	0.526336	0.263168	−	0.964750i	$-0.415233\pi$
$293$	2639.76	0.526336	0.263168	+	0.964750i	$0.415233\pi$
$294$	−16343.4	−3.24206
$295$	−630.067	−0.124352
$296$	229.010	0.0449694
$297$	−3332.28	−0.651039
$298$	−14330.8	−2.78578
$299$	4735.49	0.915920
$300$	−938.080	−0.180534
$301$	1517.25	0.290542
$302$	6999.86	1.33376
$303$	−4946.60	−0.937870
$304$	−3310.78	−0.624627
$305$	−1488.31	−0.279411
$306$	−3036.49	−0.567270
$307$	5758.95	1.07062	0.535310	−	0.844656i	$-0.320195\pi$
$307$	5758.95	1.07062	0.535310	+	0.844656i	$0.320195\pi$
$308$	−6483.06	−1.19937
$309$	−2285.20	−0.420714
$310$	−89.3762	−0.0163749
$311$	967.329	0.176374	0.0881868	−	0.996104i	$-0.471893\pi$
$311$	967.329	0.176374	0.0881868	+	0.996104i	$0.471893\pi$
$312$	−487.557	−0.0884696
$313$	−1380.96	−0.249381	−0.124691	−	0.992196i	$-0.539794\pi$
$313$	−1380.96	−0.249381	−0.124691	+	0.992196i	$0.539794\pi$
$314$	12237.2	2.19931
$315$	−1228.20	−0.219686
$316$	271.080	0.0482578
$317$	−367.287	−0.0650753	−0.0325376	−	0.999471i	$-0.510359\pi$
$317$	−367.287	−0.0650753	−0.0325376	+	0.999471i	$0.510359\pi$
$318$	3211.99	0.566414
$319$	3592.72	0.630576
$320$	−2798.60	−0.488896
$321$	−6110.50	−1.06248
$322$	9610.62	1.66329
$323$	5868.67	1.01097
$324$	−4128.94	−0.707981
$325$	1759.27	0.300267
$326$	−9459.08	−1.60702
$327$	3679.69	0.622285
$328$	−615.331	−0.103585
$329$	−17471.1	−2.92770
$330$	1985.69	0.331239
$331$	−9138.75	−1.51756	−0.758778	−	0.651349i	$-0.774203\pi$
$331$	−9138.75	−1.51756	−0.758778	+	0.651349i	$0.774203\pi$
$332$	−3555.33	−0.587724
$333$	−1030.06	−0.169510
$334$	−3098.92	−0.507681
$335$	−4402.96	−0.718088
$336$	9602.55	1.55911
$337$	2278.19	0.368252	0.184126	−	0.982903i	$-0.441055\pi$
$337$	2278.19	0.368252	0.184126	+	0.982903i	$0.441055\pi$
$338$	11151.1	1.79449
$339$	−5159.25	−0.826584
$340$	4516.61	0.720434
$341$	96.8021	0.0153728
$342$	−1534.49	−0.242619
$343$	19725.4	3.10516
$344$	66.5535	0.0104312
$345$	−1506.17	−0.235042
$346$	−15138.4	−2.35215
$347$	−12314.0	−1.90505	−0.952526	−	0.304459i	$-0.901524\pi$
$347$	−12314.0	−1.90505	−0.952526	+	0.304459i	$0.901524\pi$
$348$	6150.37	0.947398
$349$	5093.12	0.781170	0.390585	−	0.920567i	$-0.372273\pi$
$349$	5093.12	0.781170	0.390585	+	0.920567i	$0.372273\pi$
$350$	3570.42	0.545276
$351$	10698.2	1.62686
$352$	5664.96	0.857794
$353$	−9361.75	−1.41155	−0.705773	−	0.708438i	$-0.749400\pi$
$353$	−9361.75	−1.41155	−0.705773	+	0.708438i	$0.749400\pi$
$354$	2283.16	0.342793
$355$	−1145.26	−0.171222
$356$	−1648.47	−0.245418
$357$	−17021.4	−2.52344
$358$	8033.89	1.18605
$359$	−6955.89	−1.02261	−0.511306	−	0.859399i	$-0.670838\pi$
$359$	−6955.89	−1.02261	−0.511306	+	0.859399i	$0.670838\pi$
$360$	−53.8742	−0.00788728
$361$	−3893.27	−0.567615
$362$	13244.1	1.92291
$363$	3807.45	0.550521
$364$	20813.7	2.99708
$365$	−1970.50	−0.282577
$366$	5393.16	0.770232
$367$	−620.019	−0.0881873	−0.0440936	−	0.999027i	$-0.514040\pi$
$367$	−620.019	−0.0881873	−0.0440936	+	0.999027i	$0.514040\pi$
$368$	−4091.08	−0.579516
$369$	2767.68	0.390460
$370$	2994.41	0.420735
$371$	−6255.25	−0.875354
$372$	165.715	0.0230966
$373$	−7527.83	−1.04498	−0.522488	−	0.852647i	$-0.674996\pi$
$373$	−7527.83	−1.04498	−0.522488	+	0.852647i	$0.674996\pi$
$374$	−9560.59	−1.32184
$375$	−559.553	−0.0770539
$376$	−766.359	−0.105112
$377$	−11534.4	−1.57573
$378$	21711.9	2.95434
$379$	−478.221	−0.0648141	−0.0324070	−	0.999475i	$-0.510317\pi$
$379$	−478.221	−0.0648141	−0.0324070	+	0.999475i	$0.510317\pi$
$380$	2282.46	0.308126
$381$	3976.47	0.534699
$382$	11541.7	1.54588
$383$	6247.78	0.833542	0.416771	−	0.909012i	$-0.363162\pi$
$383$	6247.78	0.833542	0.416771	+	0.909012i	$0.363162\pi$
$384$	885.824	0.117720
$385$	−3867.07	−0.511907
$386$	5816.83	0.767017
$387$	−299.349	−0.0393197
$388$	12705.1	1.66239
$389$	5939.31	0.774125	0.387063	−	0.922053i	$-0.373490\pi$
$389$	5939.31	0.774125	0.387063	+	0.922053i	$0.373490\pi$
$390$	−6375.03	−0.827724
$391$	7251.81	0.937954
$392$	1396.12	0.179885
$393$	8001.56	1.02704
$394$	10762.8	1.37619
$395$	161.696	0.0205970
$396$	1279.08	0.162314
$397$	14686.0	1.85659	0.928296	−	0.371842i	$-0.121274\pi$
$397$	14686.0	1.85659	0.928296	+	0.371842i	$0.121274\pi$
$398$	−12172.3	−1.53302
$399$	−8601.76	−1.07926
$400$	−1519.87	−0.189983
$401$	6580.66	0.819507	0.409753	−	0.912196i	$-0.365615\pi$
$401$	6580.66	0.819507	0.409753	+	0.912196i	$0.365615\pi$
$402$	15954.9	1.97950
$403$	−310.781	−0.0384147
$404$	9262.82	1.14070
$405$	−2462.87	−0.302175
$406$	−23408.9	−2.86148
$407$	−3243.21	−0.394987
$408$	−746.634	−0.0905978
$409$	−12135.7	−1.46717	−0.733586	−	0.679596i	$-0.762155\pi$
$409$	−12135.7	−1.46717	−0.733586	+	0.679596i	$0.762155\pi$
$410$	−8045.73	−0.969147
$411$	−3306.32	−0.396809
$412$	4279.18	0.511700
$413$	−4446.38	−0.529763
$414$	−1896.14	−0.225097
$415$	−2120.71	−0.250848
$416$	−18187.3	−2.14352
$417$	6622.20	0.777675
$418$	−4831.44	−0.565343
$419$	−1561.29	−0.182038	−0.0910190	−	0.995849i	$-0.529012\pi$
$419$	−1561.29	−0.182038	−0.0910190	+	0.995849i	$0.529012\pi$
$420$	−6620.03	−0.769105
$421$	516.899	0.0598387	0.0299194	−	0.999552i	$-0.490475\pi$
$421$	516.899	0.0598387	0.0299194	+	0.999552i	$0.490475\pi$
$422$	−17139.3	−1.97708
$423$	3446.98	0.396213
$424$	−274.383	−0.0314274
$425$	2694.10	0.307490
$426$	4150.04	0.471996
$427$	−10503.0	−1.19034
$428$	11442.3	1.29225
$429$	6904.71	0.777069
$430$	870.216	0.0975943
$431$	−13618.7	−1.52201	−0.761007	−	0.648743i	$-0.775295\pi$
$431$	−13618.7	−1.52201	−0.761007	+	0.648743i	$0.775295\pi$
$432$	−9242.40	−1.02934
$433$	9864.27	1.09480	0.547398	−	0.836873i	$-0.315618\pi$
$433$	9864.27	1.09480	0.547398	+	0.836873i	$0.315618\pi$
$434$	−630.727	−0.0697601
$435$	3668.62	0.404361
$436$	−6890.45	−0.756864
$437$	3664.70	0.401159
$438$	7140.47	0.778961
$439$	3507.60	0.381341	0.190670	−	0.981654i	$-0.438934\pi$
$439$	3507.60	0.381341	0.190670	+	0.981654i	$0.438934\pi$
$440$	−169.627	−0.0183787
$441$	−6279.57	−0.678066
$442$	30694.1	3.30310
$443$	−4414.65	−0.473469	−0.236734	−	0.971574i	$-0.576077\pi$
$443$	−4414.65	−0.473469	−0.236734	+	0.971574i	$0.576077\pi$
$444$	−5552.04	−0.593442
$445$	−983.292	−0.104747
$446$	−973.549	−0.103361
$447$	15849.4	1.67708
$448$	−19749.7	−2.08279
$449$	18901.6	1.98669	0.993344	−	0.115184i	$-0.0367457\pi$
$449$	18901.6	1.98669	0.993344	+	0.115184i	$0.0367457\pi$
$450$	−704.429	−0.0737936
$451$	8714.23	0.909838
$452$	9661.03	1.00535
$453$	−7741.62	−0.802943
$454$	−21629.5	−2.23596
$455$	12415.2	1.27919
$456$	−377.311	−0.0387483
$457$	13249.2	1.35617	0.678086	−	0.734983i	$-0.262810\pi$
$457$	13249.2	1.35617	0.678086	+	0.734983i	$0.262810\pi$
$458$	8243.45	0.841028
$459$	16383.0	1.66600
$460$	2820.40	0.285873
$461$	−3962.15	−0.400294	−0.200147	−	0.979766i	$-0.564142\pi$
$461$	−3962.15	−0.400294	−0.200147	+	0.979766i	$0.564142\pi$
$462$	14013.0	1.41114
$463$	16243.9	1.63049	0.815245	−	0.579117i	$-0.196602\pi$
$463$	16243.9	1.63049	0.815245	+	0.579117i	$0.196602\pi$
$464$	9964.75	0.996987
$465$	98.8472	0.00985792
$466$	12039.9	1.19686
$467$	−14413.7	−1.42824	−0.714119	−	0.700024i	$-0.753173\pi$
$467$	−14413.7	−1.42824	−0.714119	+	0.700024i	$0.753173\pi$
$468$	−4106.47	−0.405602
$469$	−31071.7	−3.05918
$470$	−10020.5	−0.983427
$471$	−13533.9	−1.32401
$472$	−195.038	−0.0190198
$473$	−942.519	−0.0916218
$474$	−585.936	−0.0567783
$475$	1361.46	0.131512
$476$	31873.7	3.06918
$477$	1234.14	0.118464
$478$	8186.51	0.783352
$479$	9358.60	0.892704	0.446352	−	0.894857i	$-0.352723\pi$
$479$	9358.60	0.892704	0.446352	+	0.894857i	$0.352723\pi$
$480$	5784.65	0.550066
$481$	10412.3	0.987023
$482$	−22365.4	−2.11352
$483$	−10629.0	−1.00132
$484$	−7129.69	−0.669580
$485$	7578.47	0.709527
$486$	−7689.27	−0.717680
$487$	1232.17	0.114651	0.0573256	−	0.998356i	$-0.481743\pi$
$487$	1232.17	0.114651	0.0573256	+	0.998356i	$0.481743\pi$
$488$	−460.707	−0.0427361
$489$	10461.4	0.967449
$490$	18254.9	1.68301
$491$	−13427.0	−1.23412	−0.617058	−	0.786917i	$-0.711676\pi$
$491$	−13427.0	−1.23412	−0.617058	+	0.786917i	$0.711676\pi$
$492$	14917.9	1.36697
$493$	−17663.5	−1.61363
$494$	15511.2	1.41272
$495$	762.958	0.0692776
$496$	268.490	0.0243056
$497$	−8082.06	−0.729437
$498$	7684.80	0.691494
$499$	5135.13	0.460682	0.230341	−	0.973110i	$-0.426016\pi$
$499$	5135.13	0.460682	0.230341	+	0.973110i	$0.426016\pi$
$500$	1047.80	0.0937180
$501$	3427.31	0.305631
$502$	−26490.6	−2.35525
$503$	−12121.7	−1.07451	−0.537256	−	0.843419i	$-0.680539\pi$
$503$	−12121.7	−1.07451	−0.537256	+	0.843419i	$0.680539\pi$
$504$	−380.190	−0.0336012
$505$	5525.16	0.486865
$506$	−5970.12	−0.524514
$507$	−12332.7	−1.08031
$508$	−7446.18	−0.650337
$509$	−202.127	−0.0176014	−0.00880070	−	0.999961i	$-0.502801\pi$
$509$	−202.127	−0.0176014	−0.00880070	+	0.999961i	$0.502801\pi$
$510$	−9762.58	−0.847636
$511$	−13905.8	−1.20383
$512$	16465.1	1.42121
$513$	8279.14	0.712540
$514$	12158.1	1.04333
$515$	2552.48	0.218400
$516$	−1613.50	−0.137656
$517$	10853.1	0.923243
$518$	21131.6	1.79241
$519$	16742.6	1.41603
$520$	544.583	0.0459261
$521$	4956.22	0.416768	0.208384	−	0.978047i	$-0.433180\pi$
$521$	4956.22	0.416768	0.208384	+	0.978047i	$0.433180\pi$
$522$	4618.48	0.387252
$523$	6931.85	0.579557	0.289779	−	0.957094i	$-0.406418\pi$
$523$	6931.85	0.579557	0.289779	+	0.957094i	$0.406418\pi$
$524$	−14983.4	−1.24915
$525$	−3948.77	−0.328264
$526$	499.064	0.0413692
$527$	−475.924	−0.0393388
$528$	−5965.11	−0.491663
$529$	−7638.60	−0.627813
$530$	−3587.68	−0.294035
$531$	877.254	0.0716941
$532$	16107.3	1.31267
$533$	−27976.9	−2.27357
$534$	3563.14	0.288749
$535$	6825.19	0.551549
$536$	−1362.94	−0.109832
$537$	−8885.23	−0.714015
$538$	28349.7	2.27183
$539$	−19771.7	−1.58001
$540$	6371.74	0.507770
$541$	7161.41	0.569118	0.284559	−	0.958658i	$-0.408153\pi$
$541$	7161.41	0.569118	0.284559	+	0.958658i	$0.408153\pi$
$542$	−9230.23	−0.731498
$543$	−14647.5	−1.15762
$544$	−27851.6	−2.19508
$545$	−4110.07	−0.323039
$546$	−44988.6	−3.52625
$547$	7432.45	0.580966	0.290483	−	0.956880i	$-0.406184\pi$
$547$	7432.45	0.580966	0.290483	+	0.956880i	$0.406184\pi$
$548$	6191.29	0.482626
$549$	2072.20	0.161092
$550$	−2217.94	−0.171952
$551$	−8926.21	−0.690144
$552$	−466.236	−0.0359499
$553$	1141.09	0.0877469
$554$	−20888.1	−1.60190
$555$	−3311.72	−0.253288
$556$	−12400.5	−0.945860
$557$	−11940.5	−0.908320	−0.454160	−	0.890920i	$-0.650061\pi$
$557$	−11940.5	−0.908320	−0.454160	+	0.890920i	$0.650061\pi$
$558$	124.440	0.00944081
$559$	3025.94	0.228951
$560$	−10725.7	−0.809362
$561$	10573.7	0.795762
$562$	20857.2	1.56550
$563$	6917.09	0.517799	0.258899	−	0.965904i	$-0.416640\pi$
$563$	6917.09	0.517799	0.258899	+	0.965904i	$0.416640\pi$
$564$	18579.3	1.38711
$565$	5762.69	0.429094
$566$	8679.22	0.644549
$567$	−17380.4	−1.28732
$568$	−354.515	−0.0261886
$569$	15803.8	1.16437	0.582186	−	0.813055i	$-0.302197\pi$
$569$	15803.8	1.16437	0.582186	+	0.813055i	$0.302197\pi$
$570$	−4933.51	−0.362530
$571$	−14949.1	−1.09562	−0.547812	−	0.836601i	$-0.684539\pi$
$571$	−14949.1	−1.09562	−0.547812	+	0.836601i	$0.684539\pi$
$572$	−12929.5	−0.945122
$573$	−12764.8	−0.930639
$574$	−56778.7	−4.12874
$575$	1682.33	0.122014
$576$	3896.55	0.281869
$577$	−1029.70	−0.0742928	−0.0371464	−	0.999310i	$-0.511827\pi$
$577$	−1029.70	−0.0742928	−0.0371464	+	0.999310i	$0.511827\pi$
$578$	27118.8	1.95155
$579$	−6433.23	−0.461755
$580$	−6869.73	−0.491810
$581$	−14965.9	−1.06866
$582$	−27462.0	−1.95590
$583$	3885.76	0.276041
$584$	−609.970	−0.0432205
$585$	−2449.46	−0.173116
$586$	10684.5	0.753193
$587$	25190.1	1.77122	0.885609	−	0.464432i	$-0.153741\pi$
$587$	25190.1	1.77122	0.885609	+	0.464432i	$0.153741\pi$
$588$	−33847.1	−2.37386
$589$	−240.508	−0.0168250
$590$	−2550.21	−0.177950
$591$	−11903.3	−0.828486
$592$	−8995.34	−0.624504
$593$	25617.2	1.77398	0.886992	−	0.461785i	$-0.152791\pi$
$593$	25617.2	1.77398	0.886992	+	0.461785i	$0.152791\pi$
$594$	−13487.5	−0.931645
$595$	19012.3	1.30996
$596$	−29679.1	−2.03977
$597$	13462.2	0.922899
$598$	19167.0	1.31069
$599$	5615.73	0.383059	0.191530	−	0.981487i	$-0.438655\pi$
$599$	5615.73	0.383059	0.191530	+	0.981487i	$0.438655\pi$
$600$	−173.210	−0.0117855
$601$	16414.7	1.11409	0.557045	−	0.830482i	$-0.311935\pi$
$601$	16414.7	1.11409	0.557045	+	0.830482i	$0.311935\pi$
$602$	6141.11	0.415769
$603$	6130.33	0.414007
$604$	14496.7	0.976592
$605$	−4252.77	−0.285785
$606$	−20021.4	−1.34211
$607$	9880.24	0.660670	0.330335	−	0.943864i	$-0.392838\pi$
$607$	9880.24	0.660670	0.330335	+	0.943864i	$0.392838\pi$
$608$	−14074.8	−0.938827
$609$	25889.5	1.72265
$610$	−6023.95	−0.399840
$611$	−34843.5	−2.30707
$612$	−6288.56	−0.415359
$613$	11909.7	0.784714	0.392357	−	0.919813i	$-0.371660\pi$
$613$	11909.7	0.784714	0.392357	+	0.919813i	$0.371660\pi$
$614$	23309.4	1.53207
$615$	8898.33	0.583439
$616$	−1197.05	−0.0782966
$617$	2191.99	0.143025	0.0715124	−	0.997440i	$-0.477217\pi$
$617$	2191.99	0.143025	0.0715124	+	0.997440i	$0.477217\pi$
$618$	−9249.39	−0.602047
$619$	−3676.20	−0.238706	−0.119353	−	0.992852i	$-0.538082\pi$
$619$	−3676.20	−0.238706	−0.119353	+	0.992852i	$0.538082\pi$
$620$	−185.098	−0.0119898
$621$	10230.4	0.661081
$622$	3915.28	0.252393
$623$	−6939.09	−0.446242
$624$	19150.9	1.22860
$625$	625.000	0.0400000
$626$	−5589.45	−0.356868
$627$	5343.42	0.340344
$628$	25343.1	1.61035
$629$	15945.1	1.01077
$630$	−4971.16	−0.314374
$631$	−209.068	−0.0131900	−0.00659499	−	0.999978i	$-0.502099\pi$
$631$	−209.068	−0.0131900	−0.00659499	+	0.999978i	$0.502099\pi$
$632$	50.0532	0.00315033
$633$	18955.5	1.19023
$634$	−1486.60	−0.0931236
$635$	−4441.56	−0.277572
$636$	6652.03	0.414733
$637$	63476.6	3.94825
$638$	14541.6	0.902363
$639$	1594.56	0.0987165
$640$	−989.431	−0.0611105
$641$	−26729.7	−1.64705	−0.823526	−	0.567279i	$-0.807996\pi$
$641$	−26729.7	−1.64705	−0.823526	+	0.567279i	$0.807996\pi$
$642$	−24732.3	−1.52042
$643$	−19152.3	−1.17464	−0.587319	−	0.809355i	$-0.699817\pi$
$643$	−19152.3	−1.17464	−0.587319	+	0.809355i	$0.699817\pi$
$644$	19903.6	1.21787
$645$	−962.432	−0.0587531
$646$	23753.6	1.44670
$647$	−5208.81	−0.316506	−0.158253	−	0.987399i	$-0.550586\pi$
$647$	−5208.81	−0.316506	−0.158253	+	0.987399i	$0.550586\pi$
$648$	−762.382	−0.0462179
$649$	2762.09	0.167060
$650$	7120.67	0.429686
$651$	697.565	0.0419965
$652$	−19589.7	−1.17668
$653$	−853.809	−0.0511671	−0.0255836	−	0.999673i	$-0.508144\pi$
$653$	−853.809	−0.0511671	−0.0255836	+	0.999673i	$0.508144\pi$
$654$	14893.6	0.890498
$655$	−8937.44	−0.533152
$656$	24169.7	1.43852
$657$	2743.57	0.162917
$658$	−70714.5	−4.18957
$659$	6239.14	0.368805	0.184403	−	0.982851i	$-0.440965\pi$
$659$	6239.14	0.368805	0.184403	+	0.982851i	$0.440965\pi$
$660$	4112.36	0.242536
$661$	−23578.1	−1.38742	−0.693708	−	0.720257i	$-0.744024\pi$
$661$	−23578.1	−1.38742	−0.693708	+	0.720257i	$0.744024\pi$
$662$	−36989.2	−2.17164
$663$	−33946.7	−1.98851
$664$	−656.469	−0.0383674
$665$	9607.84	0.560265
$666$	−4169.17	−0.242571
$667$	−11030.0	−0.640302
$668$	−6417.86	−0.371728
$669$	1076.72	0.0622245
$670$	−17821.1	−1.02759
$671$	6524.46	0.375371
$672$	40822.2	2.34338
$673$	−21431.2	−1.22750	−0.613752	−	0.789499i	$-0.710341\pi$
$673$	−21431.2	−1.22750	−0.613752	+	0.789499i	$0.710341\pi$
$674$	9221.02	0.526974
$675$	3800.67	0.216722
$676$	23093.8	1.31394
$677$	−4319.50	−0.245217	−0.122608	−	0.992455i	$-0.539126\pi$
$677$	−4319.50	−0.245217	−0.122608	+	0.992455i	$0.539126\pi$
$678$	−20882.2	−1.18285
$679$	53481.3	3.02271
$680$	833.962	0.0470309
$681$	23921.6	1.34608
$682$	391.808	0.0219987
$683$	20556.5	1.15164	0.575821	−	0.817576i	$-0.304683\pi$
$683$	20556.5	1.15164	0.575821	+	0.817576i	$0.304683\pi$
$684$	−3177.92	−0.177647
$685$	3693.03	0.205991
$686$	79838.9	4.44353
$687$	−9116.99	−0.506310
$688$	−2614.17	−0.144861
$689$	−12475.2	−0.689791
$690$	−6096.25	−0.336348
$691$	9737.74	0.536094	0.268047	−	0.963406i	$-0.413622\pi$
$691$	9737.74	0.536094	0.268047	+	0.963406i	$0.413622\pi$
$692$	−31351.6	−1.72227
$693$	5384.19	0.295135
$694$	−49841.3	−2.72615
$695$	−7396.75	−0.403704
$696$	1135.63	0.0618474
$697$	−42843.1	−2.32826
$698$	20614.5	1.11786
$699$	−13315.7	−0.720525
$700$	7394.32	0.399256
$701$	−34574.5	−1.86286	−0.931428	−	0.363925i	$-0.881436\pi$
$701$	−34574.5	−1.86286	−0.931428	+	0.363925i	$0.881436\pi$
$702$	43301.2	2.32806
$703$	8057.84	0.432300
$704$	12268.6	0.656802
$705$	11082.3	0.592036
$706$	−37891.8	−2.01994
$707$	38991.1	2.07413
$708$	4728.42	0.250996
$709$	−6994.85	−0.370518	−0.185259	−	0.982690i	$-0.559312\pi$
$709$	−6994.85	−0.370518	−0.185259	+	0.982690i	$0.559312\pi$
$710$	−4635.44	−0.245021
$711$	−225.133	−0.0118750
$712$	−304.379	−0.0160212
$713$	−297.191	−0.0156099
$714$	−68894.5	−3.61108
$715$	−7712.30	−0.403390
$716$	16638.2	0.868432
$717$	−9054.03	−0.471588
$718$	−28154.1	−1.46337
$719$	−27154.2	−1.40846	−0.704229	−	0.709972i	$-0.748707\pi$
$719$	−27154.2	−1.40846	−0.704229	+	0.709972i	$0.748707\pi$
$720$	2116.14	0.109533
$721$	18012.9	0.930422
$722$	−15758.1	−0.812265
$723$	24735.5	1.27237
$724$	27428.4	1.40797
$725$	−4097.71	−0.209911
$726$	15410.7	0.787803
$727$	−21058.0	−1.07427	−0.537137	−	0.843495i	$-0.680494\pi$
$727$	−21058.0	−1.07427	−0.537137	+	0.843495i	$0.680494\pi$
$728$	3843.12	0.195653
$729$	21803.6	1.10774
$730$	−7975.64	−0.404372
$731$	4633.86	0.234459
$732$	11169.2	0.563970
$733$	−20006.7	−1.00814	−0.504069	−	0.863664i	$-0.668164\pi$
$733$	−20006.7	−1.00814	−0.504069	+	0.863664i	$0.668164\pi$
$734$	−2509.54	−0.126197
$735$	−20189.4	−1.01319
$736$	−17391.9	−0.871026
$737$	19301.8	0.964707
$738$	11202.2	0.558753
$739$	−4208.73	−0.209500	−0.104750	−	0.994499i	$-0.533404\pi$
$739$	−4208.73	−0.209500	−0.104750	+	0.994499i	$0.533404\pi$
$740$	6201.42	0.308066
$741$	−17154.9	−0.850476
$742$	−25318.2	−1.25264
$743$	−2209.58	−0.109101	−0.0545503	−	0.998511i	$-0.517373\pi$
$743$	−2209.58	−0.109101	−0.0545503	+	0.998511i	$0.517373\pi$
$744$	30.5983	0.00150778
$745$	−17703.2	−0.870599
$746$	−30469.0	−1.49537
$747$	2952.71	0.144624
$748$	−19800.0	−0.967858
$749$	48165.4	2.34970
$750$	−2264.80	−0.110265
$751$	40524.3	1.96905	0.984523	−	0.175253i	$-0.0560744\pi$
$751$	40524.3	1.96905	0.984523	+	0.175253i	$0.0560744\pi$
$752$	30102.0	1.45972
$753$	29297.8	1.41789
$754$	−46685.5	−2.25489
$755$	8647.10	0.416821
$756$	44965.3	2.16319
$757$	32196.0	1.54582	0.772909	−	0.634517i	$-0.218801\pi$
$757$	32196.0	1.54582	0.772909	+	0.634517i	$0.218801\pi$
$758$	−1935.61	−0.0927498
$759$	6602.77	0.315764
$760$	421.442	0.0201149
$761$	7769.67	0.370106	0.185053	−	0.982729i	$-0.440754\pi$
$761$	7769.67	0.370106	0.185053	+	0.982729i	$0.440754\pi$
$762$	16094.8	0.765162
$763$	−29004.8	−1.37620
$764$	23902.8	1.13190
$765$	−3751.05	−0.177280
$766$	25288.0	1.19281
$767$	−8867.65	−0.417461
$768$	−16459.0	−0.773325
$769$	−4179.79	−0.196004	−0.0980020	−	0.995186i	$-0.531245\pi$
$769$	−4179.79	−0.196004	−0.0980020	+	0.995186i	$0.531245\pi$
$770$	−15652.0	−0.732545
$771$	−13446.5	−0.628100
$772$	12046.6	0.561616
$773$	−8662.16	−0.403048	−0.201524	−	0.979484i	$-0.564589\pi$
$773$	−8662.16	−0.403048	−0.201524	+	0.979484i	$0.564589\pi$
$774$	−1211.62	−0.0562671
$775$	−110.409	−0.00511741
$776$	2345.92	0.108523
$777$	−23370.8	−1.07905
$778$	24039.4	1.10778
$779$	−21650.7	−0.995787
$780$	−13202.7	−0.606066
$781$	5020.58	0.230026
$782$	29351.8	1.34222
$783$	−24918.5	−1.13731
$784$	−54838.6	−2.49811
$785$	15116.9	0.687318
$786$	32386.5	1.46970
$787$	19036.1	0.862216	0.431108	−	0.902300i	$-0.358123\pi$
$787$	19036.1	0.862216	0.431108	+	0.902300i	$0.358123\pi$
$788$	22289.6	1.00766
$789$	−551.949	−0.0249048
$790$	654.468	0.0294746
$791$	40667.3	1.82802
$792$	236.174	0.0105961
$793$	−20946.7	−0.938005
$794$	59441.7	2.65681
$795$	3967.86	0.177013
$796$	−25208.8	−1.12249
$797$	5094.80	0.226433	0.113216	−	0.993570i	$-0.463885\pi$
$797$	5094.80	0.226433	0.113216	+	0.993570i	$0.463885\pi$
$798$	−34815.8	−1.54444
$799$	−53358.6	−2.36257
$800$	−6461.23	−0.285549
$801$	1369.06	0.0603910
$802$	26635.3	1.17273
$803$	8638.30	0.379625
$804$	33042.6	1.44941
$805$	11872.2	0.519803
$806$	−1257.89	−0.0549720
$807$	−31353.9	−1.36767
$808$	1710.32	0.0744664
$809$	−4085.55	−0.177553	−0.0887764	−	0.996052i	$-0.528296\pi$
$809$	−4085.55	−0.177553	−0.0887764	+	0.996052i	$0.528296\pi$
$810$	−9968.49	−0.432416
$811$	36520.8	1.58128	0.790641	−	0.612280i	$-0.209748\pi$
$811$	36520.8	1.58128	0.790641	+	0.612280i	$0.209748\pi$
$812$	−48479.7	−2.09520
$813$	10208.3	0.440372
$814$	−13126.9	−0.565232
$815$	−11685.0	−0.502220
$816$	29327.2	1.25816
$817$	2341.72	0.100277
$818$	−49119.6	−2.09954
$819$	−17285.9	−0.737505
$820$	−16662.7	−0.709617
$821$	14383.6	0.611440	0.305720	−	0.952121i	$-0.401103\pi$
$821$	14383.6	0.611440	0.305720	+	0.952121i	$0.401103\pi$
$822$	−13382.4	−0.567840
$823$	22404.5	0.948932	0.474466	−	0.880274i	$-0.342641\pi$
$823$	22404.5	0.948932	0.474466	+	0.880274i	$0.342641\pi$
$824$	790.123	0.0334044
$825$	2452.98	0.103517
$826$	−17996.8	−0.758098
$827$	929.994	0.0391040	0.0195520	−	0.999809i	$-0.493776\pi$
$827$	929.994	0.0391040	0.0195520	+	0.999809i	$0.493776\pi$
$828$	−3926.90	−0.164818
$829$	16490.4	0.690873	0.345437	−	0.938442i	$-0.387731\pi$
$829$	16490.4	0.690873	0.345437	+	0.938442i	$0.387731\pi$
$830$	−8583.63	−0.358966
$831$	23101.6	0.964363
$832$	−39388.0	−1.64126
$833$	97206.5	4.04323
$834$	26803.5	1.11286
$835$	−3828.18	−0.158658
$836$	−10005.9	−0.413949
$837$	−671.402	−0.0277265
$838$	−6319.34	−0.260499
$839$	41770.6	1.71881	0.859405	−	0.511295i	$-0.170834\pi$
$839$	41770.6	1.71881	0.859405	+	0.511295i	$0.170834\pi$
$840$	−1222.34	−0.0502082
$841$	2477.00	0.101562
$842$	2092.16	0.0856300
$843$	−23067.4	−0.942449
$844$	−35495.3	−1.44763
$845$	13775.2	0.560806
$846$	13951.7	0.566985
$847$	−30011.8	−1.21750
$848$	10777.5	0.436441
$849$	−9598.94	−0.388027
$850$	10904.4	0.440022
$851$	9956.92	0.401080
$852$	8594.73	0.345599
$853$	−12895.9	−0.517642	−0.258821	−	0.965925i	$-0.583334\pi$
$853$	−12895.9	−0.517642	−0.258821	+	0.965925i	$0.583334\pi$
$854$	−42511.0	−1.70339
$855$	−1895.59	−0.0758220
$856$	2112.74	0.0843600
$857$	−2463.33	−0.0981863	−0.0490931	−	0.998794i	$-0.515633\pi$
$857$	−2463.33	−0.0981863	−0.0490931	+	0.998794i	$0.515633\pi$
$858$	27946.9	1.11200
$859$	−4156.92	−0.165113	−0.0825566	−	0.996586i	$-0.526309\pi$
$859$	−4156.92	−0.165113	−0.0825566	+	0.996586i	$0.526309\pi$
$860$	1802.22	0.0714593
$861$	62795.5	2.48556
$862$	−55121.8	−2.17802
$863$	−18623.0	−0.734572	−0.367286	−	0.930108i	$-0.619713\pi$
$863$	−18623.0	−0.734572	−0.367286	+	0.930108i	$0.619713\pi$
$864$	−39291.2	−1.54712
$865$	−18700.8	−0.735084
$866$	39925.8	1.56667
$867$	−29992.5	−1.17486
$868$	−1306.23	−0.0510789
$869$	−708.845	−0.0276708
$870$	14848.8	0.578646
$871$	−61967.9	−2.41068
$872$	−1272.28	−0.0494091
$873$	−10551.6	−0.409071
$874$	14832.9	0.574063
$875$	4410.62	0.170407
$876$	14787.9	0.570361
$877$	44474.8	1.71244	0.856219	−	0.516613i	$-0.172807\pi$
$877$	44474.8	1.71244	0.856219	+	0.516613i	$0.172807\pi$
$878$	14197.1	0.545704
$879$	−11816.7	−0.453432
$880$	6662.80	0.255231
$881$	−22188.5	−0.848524	−0.424262	−	0.905539i	$-0.639467\pi$
$881$	−22188.5	−0.848524	−0.424262	+	0.905539i	$0.639467\pi$
$882$	−25416.7	−0.970322
$883$	−22096.8	−0.842149	−0.421074	−	0.907026i	$-0.638347\pi$
$883$	−22096.8	−0.842149	−0.421074	+	0.907026i	$0.638347\pi$
$884$	63567.4	2.41856
$885$	2820.45	0.107128
$886$	−17868.4	−0.677540
$887$	−2797.36	−0.105892	−0.0529459	−	0.998597i	$-0.516861\pi$
$887$	−2797.36	−0.105892	−0.0529459	+	0.998597i	$0.516861\pi$
$888$	−1025.15	−0.0387407
$889$	−31344.1	−1.18250
$890$	−3979.89	−0.149895
$891$	10796.7	0.405953
$892$	−2016.22	−0.0756816
$893$	−26964.7	−1.01046
$894$	64150.9	2.39992
$895$	9924.47	0.370657
$896$	−6982.42	−0.260342
$897$	−21198.1	−0.789055
$898$	76504.7	2.84298
$899$	723.876	0.0268550
$900$	−1458.87	−0.0540323
$901$	−19104.2	−0.706385
$902$	35271.0	1.30199
$903$	−6791.88	−0.250299
$904$	1783.85	0.0656303
$905$	16360.7	0.600938
$906$	−31334.4	−1.14902
$907$	25176.3	0.921683	0.460842	−	0.887482i	$-0.347548\pi$
$907$	25176.3	0.921683	0.460842	+	0.887482i	$0.347548\pi$
$908$	−44794.7	−1.63719
$909$	−7692.79	−0.280697
$910$	50250.5	1.83054
$911$	27032.6	0.983127	0.491563	−	0.870842i	$-0.336426\pi$
$911$	27032.6	0.983127	0.491563	+	0.870842i	$0.336426\pi$
$912$	14820.5	0.538109
$913$	9296.81	0.336998
$914$	53626.3	1.94070
$915$	6662.30	0.240709
$916$	17072.2	0.615808
$917$	−63071.5	−2.27132
$918$	66310.5	2.38407
$919$	48383.7	1.73670	0.868351	−	0.495950i	$-0.165180\pi$
$919$	48383.7	1.73670	0.868351	+	0.495950i	$0.165180\pi$
$920$	520.768	0.0186622
$921$	−25779.5	−0.922327
$922$	−16036.9	−0.572827
$923$	−16118.5	−0.574807
$924$	29021.0	1.03325
$925$	3699.07	0.131486
$926$	65747.3	2.33325
$927$	−3553.87	−0.125916
$928$	42362.0	1.49849
$929$	−16500.5	−0.582738	−0.291369	−	0.956611i	$-0.594111\pi$
$929$	−16500.5	−0.582738	−0.291369	+	0.956611i	$0.594111\pi$
$930$	400.086	0.0141068
$931$	49123.2	1.72927
$932$	24934.6	0.876350
$933$	−4330.18	−0.151944
$934$	−58339.8	−2.04383
$935$	−11810.4	−0.413094
$936$	−758.233	−0.0264782
$937$	31016.7	1.08140	0.540700	−	0.841215i	$-0.318159\pi$
$937$	31016.7	1.08140	0.540700	+	0.841215i	$0.318159\pi$
$938$	−125763.	−4.37773
$939$	6181.76	0.214839
$940$	−20752.4	−0.720073
$941$	−18073.8	−0.626131	−0.313065	−	0.949732i	$-0.601356\pi$
$941$	−18073.8	−0.626131	−0.313065	+	0.949732i	$0.601356\pi$
$942$	−54778.8	−1.89468
$943$	−26753.4	−0.923872
$944$	7660.93	0.264134
$945$	26821.3	0.923277
$946$	−3814.86	−0.131112
$947$	27378.2	0.939465	0.469732	−	0.882809i	$-0.344350\pi$
$947$	27378.2	0.939465	0.469732	+	0.882809i	$0.344350\pi$
$948$	−1213.47	−0.0415735
$949$	−27733.1	−0.948635
$950$	5510.54	0.188195
$951$	1644.13	0.0560616
$952$	5885.27	0.200360
$953$	−9637.52	−0.327586	−0.163793	−	0.986495i	$-0.552373\pi$
$953$	−9637.52	−0.327586	−0.163793	+	0.986495i	$0.552373\pi$
$954$	4995.19	0.169523
$955$	14257.8	0.483110
$956$	16954.2	0.573577
$957$	−16082.6	−0.543234
$958$	37879.1	1.27747
$959$	26061.7	0.877557
$960$	12527.8	0.421179
$961$	−29771.5	−0.999345
$962$	42143.8	1.41244
$963$	−9502.84	−0.317991
$964$	−46318.7	−1.54754
$965$	7185.67	0.239705
$966$	−43021.2	−1.43290
$967$	−6478.05	−0.215429	−0.107715	−	0.994182i	$-0.534353\pi$
$967$	−6478.05	−0.215429	−0.107715	+	0.994182i	$0.534353\pi$
$968$	−1316.45	−0.0437111
$969$	−26270.7	−0.870935
$970$	30674.0	1.01534
$971$	−28924.6	−0.955959	−0.477979	−	0.878371i	$-0.658631\pi$
$971$	−28924.6	−0.955959	−0.477979	+	0.878371i	$0.658631\pi$
$972$	−15924.5	−0.525491
$973$	−52198.8	−1.71985
$974$	4987.25	0.164067
$975$	−7875.24	−0.258676
$976$	18096.2	0.593489
$977$	−14362.8	−0.470325	−0.235162	−	0.971956i	$-0.575562\pi$
$977$	−14362.8	−0.470325	−0.235162	+	0.971956i	$0.575562\pi$
$978$	42342.9	1.38443
$979$	4310.57	0.140721
$980$	37805.9	1.23231
$981$	5722.53	0.186245
$982$	−54345.9	−1.76604
$983$	39806.8	1.29160	0.645798	−	0.763508i	$-0.276525\pi$
$983$	39806.8	1.29160	0.645798	+	0.763508i	$0.276525\pi$
$984$	2754.49	0.0892376
$985$	13295.5	0.430081
$986$	−71493.1	−2.30913
$987$	78208.1	2.52218
$988$	32123.7	1.03440
$989$	2893.62	0.0930350
$990$	3088.09	0.0991372
$991$	35617.3	1.14170	0.570848	−	0.821056i	$-0.306615\pi$
$991$	35617.3	1.14170	0.570848	+	0.821056i	$0.306615\pi$
$992$	1141.40	0.0365318
$993$	40909.0	1.30736
$994$	−32712.3	−1.04383
$995$	−15036.8	−0.479093
$996$	15915.2	0.506317
$997$	−5365.76	−0.170447	−0.0852234	−	0.996362i	$-0.527160\pi$
$997$	−5365.76	−0.170447	−0.0852234	+	0.996362i	$0.527160\pi$
$998$	20784.5	0.659242
$999$	22494.3	0.712400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	215.4.a.d.1.12	✓	15
3.2	odd	2		1935.4.a.n.1.4		15
5.4	even	2		1075.4.a.f.1.4		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.4.a.d.1.12	✓	15	1.1	even	1	trivial
1075.4.a.f.1.4		15	5.4	even	2
1935.4.a.n.1.4		15	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 215 = 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	215.a (trivial)

\(f(q)\)	\(=\)	\(q+4.04752 q^{2} -4.47643 q^{3} +8.38240 q^{4} +5.00000 q^{5} -18.1184 q^{6} +35.2850 q^{7} +1.54775 q^{8} -6.96159 q^{9} +O(q^{10})\)	\(q+4.04752 q^{2} -4.47643 q^{3} +8.38240 q^{4} +5.00000 q^{5} -18.1184 q^{6} +35.2850 q^{7} +1.54775 q^{8} -6.96159 q^{9} +20.2376 q^{10} -21.9191 q^{11} -37.5232 q^{12} +70.3707 q^{13} +142.817 q^{14} -22.3821 q^{15} -60.7946 q^{16} +107.764 q^{17} -28.1772 q^{18} +54.4585 q^{19} +41.9120 q^{20} -157.951 q^{21} -88.7178 q^{22} +67.2934 q^{23} -6.92841 q^{24} +25.0000 q^{25} +284.827 q^{26} +152.027 q^{27} +295.773 q^{28} -163.908 q^{29} -90.5921 q^{30} -4.41635 q^{31} -258.449 q^{32} +98.1191 q^{33} +436.177 q^{34} +176.425 q^{35} -58.3548 q^{36} +147.963 q^{37} +220.422 q^{38} -315.009 q^{39} +7.73877 q^{40} -397.564 q^{41} -639.308 q^{42} +43.0000 q^{43} -183.734 q^{44} -34.8080 q^{45} +272.371 q^{46} -495.142 q^{47} +272.143 q^{48} +902.031 q^{49} +101.188 q^{50} -482.398 q^{51} +589.875 q^{52} -177.278 q^{53} +615.330 q^{54} -109.595 q^{55} +54.6125 q^{56} -243.780 q^{57} -663.422 q^{58} -126.013 q^{59} -187.616 q^{60} -297.662 q^{61} -17.8752 q^{62} -245.640 q^{63} -559.721 q^{64} +351.854 q^{65} +397.139 q^{66} -880.592 q^{67} +903.322 q^{68} -301.234 q^{69} +714.083 q^{70} -229.051 q^{71} -10.7748 q^{72} -394.100 q^{73} +598.882 q^{74} -111.911 q^{75} +456.493 q^{76} -773.414 q^{77} -1275.01 q^{78} +32.3392 q^{79} -303.973 q^{80} -492.573 q^{81} -1609.15 q^{82} -424.143 q^{83} -1324.01 q^{84} +538.821 q^{85} +174.043 q^{86} +733.725 q^{87} -33.9253 q^{88} -196.658 q^{89} -140.886 q^{90} +2483.03 q^{91} +564.080 q^{92} +19.7694 q^{93} -2004.10 q^{94} +272.293 q^{95} +1156.93 q^{96} +1515.69 q^{97} +3650.98 q^{98} +152.592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} + 7 q^{3} + 83 q^{4} + 75 q^{5} + 40 q^{6} + 36 q^{7} + 33 q^{8} + 242 q^{9}+O(q^{10}) \) 15 * q + q^2 + 7 * q^3 + 83 * q^4 + 75 * q^5 + 40 * q^6 + 36 * q^7 + 33 * q^8 + 242 * q^9	\( 15 q + q^{2} + 7 q^{3} + 83 q^{4} + 75 q^{5} + 40 q^{6} + 36 q^{7} + 33 q^{8} + 242 q^{9} + 5 q^{10} + 87 q^{11} - 37 q^{12} - 35 q^{13} + 237 q^{14} + 35 q^{15} + 139 q^{16} + 144 q^{17} - 325 q^{18} + 293 q^{19} + 415 q^{20} + 208 q^{21} + 25 q^{22} + 22 q^{23} + 480 q^{24} + 375 q^{25} + 127 q^{26} + 202 q^{27} + 187 q^{28} + 495 q^{29} + 200 q^{30} + 408 q^{31} + 395 q^{32} + 308 q^{33} + 812 q^{34} + 180 q^{35} + 1283 q^{36} + 1013 q^{37} + 251 q^{38} + 654 q^{39} + 165 q^{40} + 820 q^{41} + 431 q^{42} + 645 q^{43} + 1244 q^{44} + 1210 q^{45} + 1416 q^{46} - 622 q^{47} + 70 q^{48} + 1525 q^{49} + 25 q^{50} + 1760 q^{51} - 593 q^{52} + 276 q^{53} - 349 q^{54} + 435 q^{55} + 822 q^{56} - 566 q^{57} + 1403 q^{58} - 1405 q^{59} - 185 q^{60} + 1969 q^{61} - 3297 q^{62} - 2587 q^{63} - 1709 q^{64} - 175 q^{65} - 3111 q^{66} - 439 q^{67} - 3412 q^{68} + 10 q^{69} + 1185 q^{70} + 494 q^{71} - 6476 q^{72} + 958 q^{73} - 6273 q^{74} + 175 q^{75} + 1089 q^{76} - 918 q^{77} - 5438 q^{78} - 1846 q^{79} + 695 q^{80} + 3655 q^{81} - 2693 q^{82} - 1524 q^{83} - 6283 q^{84} + 720 q^{85} + 43 q^{86} - 2702 q^{87} - 2010 q^{88} + 1380 q^{89} - 1625 q^{90} + 3625 q^{91} - 3658 q^{92} + 1610 q^{93} - 274 q^{94} + 1465 q^{95} - 367 q^{96} + 3240 q^{97} - 6137 q^{98} - 3769 q^{99}+O(q^{100}) \) 15 * q + q^2 + 7 * q^3 + 83 * q^4 + 75 * q^5 + 40 * q^6 + 36 * q^7 + 33 * q^8 + 242 * q^9 + 5 * q^10 + 87 * q^11 - 37 * q^12 - 35 * q^13 + 237 * q^14 + 35 * q^15 + 139 * q^16 + 144 * q^17 - 325 * q^18 + 293 * q^19 + 415 * q^20 + 208 * q^21 + 25 * q^22 + 22 * q^23 + 480 * q^24 + 375 * q^25 + 127 * q^26 + 202 * q^27 + 187 * q^28 + 495 * q^29 + 200 * q^30 + 408 * q^31 + 395 * q^32 + 308 * q^33 + 812 * q^34 + 180 * q^35 + 1283 * q^36 + 1013 * q^37 + 251 * q^38 + 654 * q^39 + 165 * q^40 + 820 * q^41 + 431 * q^42 + 645 * q^43 + 1244 * q^44 + 1210 * q^45 + 1416 * q^46 - 622 * q^47 + 70 * q^48 + 1525 * q^49 + 25 * q^50 + 1760 * q^51 - 593 * q^52 + 276 * q^53 - 349 * q^54 + 435 * q^55 + 822 * q^56 - 566 * q^57 + 1403 * q^58 - 1405 * q^59 - 185 * q^60 + 1969 * q^61 - 3297 * q^62 - 2587 * q^63 - 1709 * q^64 - 175 * q^65 - 3111 * q^66 - 439 * q^67 - 3412 * q^68 + 10 * q^69 + 1185 * q^70 + 494 * q^71 - 6476 * q^72 + 958 * q^73 - 6273 * q^74 + 175 * q^75 + 1089 * q^76 - 918 * q^77 - 5438 * q^78 - 1846 * q^79 + 695 * q^80 + 3655 * q^81 - 2693 * q^82 - 1524 * q^83 - 6283 * q^84 + 720 * q^85 + 43 * q^86 - 2702 * q^87 - 2010 * q^88 + 1380 * q^89 - 1625 * q^90 + 3625 * q^91 - 3658 * q^92 + 1610 * q^93 - 274 * q^94 + 1465 * q^95 - 367 * q^96 + 3240 * q^97 - 6137 * q^98 - 3769 * q^99

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