LMFDB - Embedded newform 354.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,4,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, 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("354.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	354.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:	$20.8866761420$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.47277.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 48x - 25 $ x^3 - 48*x - 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.65142$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.13667 q^{5} -6.00000 q^{6} +4.92477 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.13667 q^{5} -6.00000 q^{6} +4.92477 q^{7} +8.00000 q^{8} +9.00000 q^{9} -8.27334 q^{10} -59.9380 q^{11} -12.0000 q^{12} +45.6647 q^{13} +9.84953 q^{14} +12.4100 q^{15} +16.0000 q^{16} -95.3299 q^{17} +18.0000 q^{18} +27.3456 q^{19} -16.5467 q^{20} -14.7743 q^{21} -119.876 q^{22} -43.2915 q^{23} -24.0000 q^{24} -107.888 q^{25} +91.3294 q^{26} -27.0000 q^{27} +19.6991 q^{28} -133.487 q^{29} +24.8200 q^{30} +67.9689 q^{31} +32.0000 q^{32} +179.814 q^{33} -190.660 q^{34} -20.3721 q^{35} +36.0000 q^{36} -309.748 q^{37} +54.6912 q^{38} -136.994 q^{39} -33.0934 q^{40} -124.566 q^{41} -29.5486 q^{42} -22.2477 q^{43} -239.752 q^{44} -37.2300 q^{45} -86.5830 q^{46} -267.186 q^{47} -48.0000 q^{48} -318.747 q^{49} -215.776 q^{50} +285.990 q^{51} +182.659 q^{52} -94.7700 q^{53} -54.0000 q^{54} +247.944 q^{55} +39.3981 q^{56} -82.0368 q^{57} -266.974 q^{58} -59.0000 q^{59} +49.6401 q^{60} -39.2858 q^{61} +135.938 q^{62} +44.3229 q^{63} +64.0000 q^{64} -188.900 q^{65} +359.628 q^{66} +127.319 q^{67} -381.320 q^{68} +129.874 q^{69} -40.7443 q^{70} +536.119 q^{71} +72.0000 q^{72} +472.832 q^{73} -619.496 q^{74} +323.664 q^{75} +109.382 q^{76} -295.181 q^{77} -273.988 q^{78} -504.718 q^{79} -66.1868 q^{80} +81.0000 q^{81} -249.132 q^{82} +77.1909 q^{83} -59.0972 q^{84} +394.349 q^{85} -44.4953 q^{86} +400.461 q^{87} -479.504 q^{88} +460.203 q^{89} -74.4601 q^{90} +224.888 q^{91} -173.166 q^{92} -203.907 q^{93} -534.372 q^{94} -113.120 q^{95} -96.0000 q^{96} +678.352 q^{97} -637.493 q^{98} -539.442 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} - 18 q^{5} - 18 q^{6} + 6 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 - 18 * q^5 - 18 * q^6 + 6 * q^7 + 24 * q^8 + 27 * q^9	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} - 18 q^{5} - 18 q^{6} + 6 q^{7} + 24 q^{8} + 27 q^{9} - 36 q^{10} - 9 q^{11} - 36 q^{12} - 45 q^{13} + 12 q^{14} + 54 q^{15} + 48 q^{16} - 96 q^{17} + 54 q^{18} - 171 q^{19} - 72 q^{20} - 18 q^{21} - 18 q^{22} - 177 q^{23} - 72 q^{24} - 69 q^{25} - 90 q^{26} - 81 q^{27} + 24 q^{28} - 351 q^{29} + 108 q^{30} - 267 q^{31} + 96 q^{32} + 27 q^{33} - 192 q^{34} - 489 q^{35} + 108 q^{36} - 222 q^{37} - 342 q^{38} + 135 q^{39} - 144 q^{40} - 780 q^{41} - 36 q^{42} + 21 q^{43} - 36 q^{44} - 162 q^{45} - 354 q^{46} - 717 q^{47} - 144 q^{48} + 99 q^{49} - 138 q^{50} + 288 q^{51} - 180 q^{52} - 234 q^{53} - 162 q^{54} + 114 q^{55} + 48 q^{56} + 513 q^{57} - 702 q^{58} - 177 q^{59} + 216 q^{60} + 615 q^{61} - 534 q^{62} + 54 q^{63} + 192 q^{64} + 606 q^{65} + 54 q^{66} - 144 q^{67} - 384 q^{68} + 531 q^{69} - 978 q^{70} + 981 q^{71} + 216 q^{72} - 63 q^{73} - 444 q^{74} + 207 q^{75} - 684 q^{76} - 792 q^{77} + 270 q^{78} + 555 q^{79} - 288 q^{80} + 243 q^{81} - 1560 q^{82} + 66 q^{83} - 72 q^{84} - 807 q^{85} + 42 q^{86} + 1053 q^{87} - 72 q^{88} - 1389 q^{89} - 324 q^{90} - 222 q^{91} - 708 q^{92} + 801 q^{93} - 1434 q^{94} + 1587 q^{95} - 288 q^{96} + 960 q^{97} + 198 q^{98} - 81 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 - 18 * q^5 - 18 * q^6 + 6 * q^7 + 24 * q^8 + 27 * q^9 - 36 * q^10 - 9 * q^11 - 36 * q^12 - 45 * q^13 + 12 * q^14 + 54 * q^15 + 48 * q^16 - 96 * q^17 + 54 * q^18 - 171 * q^19 - 72 * q^20 - 18 * q^21 - 18 * q^22 - 177 * q^23 - 72 * q^24 - 69 * q^25 - 90 * q^26 - 81 * q^27 + 24 * q^28 - 351 * q^29 + 108 * q^30 - 267 * q^31 + 96 * q^32 + 27 * q^33 - 192 * q^34 - 489 * q^35 + 108 * q^36 - 222 * q^37 - 342 * q^38 + 135 * q^39 - 144 * q^40 - 780 * q^41 - 36 * q^42 + 21 * q^43 - 36 * q^44 - 162 * q^45 - 354 * q^46 - 717 * q^47 - 144 * q^48 + 99 * q^49 - 138 * q^50 + 288 * q^51 - 180 * q^52 - 234 * q^53 - 162 * q^54 + 114 * q^55 + 48 * q^56 + 513 * q^57 - 702 * q^58 - 177 * q^59 + 216 * q^60 + 615 * q^61 - 534 * q^62 + 54 * q^63 + 192 * q^64 + 606 * q^65 + 54 * q^66 - 144 * q^67 - 384 * q^68 + 531 * q^69 - 978 * q^70 + 981 * q^71 + 216 * q^72 - 63 * q^73 - 444 * q^74 + 207 * q^75 - 684 * q^76 - 792 * q^77 + 270 * q^78 + 555 * q^79 - 288 * q^80 + 243 * q^81 - 1560 * q^82 + 66 * q^83 - 72 * q^84 - 807 * q^85 + 42 * q^86 + 1053 * q^87 - 72 * q^88 - 1389 * q^89 - 324 * q^90 - 222 * q^91 - 708 * q^92 + 801 * q^93 - 1434 * q^94 + 1587 * q^95 - 288 * q^96 + 960 * q^97 + 198 * q^98 - 81 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	−4.13667	−0.369995	−0.184998	−	0.982739i	$-0.559228\pi$
$5$	−4.13667	−0.369995	−0.184998	+	0.982739i	$0.559228\pi$
$6$	−6.00000	−0.408248
$7$	4.92477	0.265912	0.132956	−	0.991122i	$-0.457553\pi$
$7$	4.92477	0.265912	0.132956	+	0.991122i	$0.457553\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−8.27334	−0.261626
$11$	−59.9380	−1.64291	−0.821454	−	0.570274i	$-0.806837\pi$
$11$	−59.9380	−1.64291	−0.821454	+	0.570274i	$0.806837\pi$
$12$	−12.0000	−0.288675
$13$	45.6647	0.974239	0.487119	−	0.873335i	$-0.338048\pi$
$13$	45.6647	0.974239	0.487119	+	0.873335i	$0.338048\pi$
$14$	9.84953	0.188028
$15$	12.4100	0.213617
$16$	16.0000	0.250000
$17$	−95.3299	−1.36005	−0.680026	−	0.733188i	$-0.738032\pi$
$17$	−95.3299	−1.36005	−0.680026	+	0.733188i	$0.738032\pi$
$18$	18.0000	0.235702
$19$	27.3456	0.330185	0.165092	−	0.986278i	$-0.447208\pi$
$19$	27.3456	0.330185	0.165092	+	0.986278i	$0.447208\pi$
$20$	−16.5467	−0.184998
$21$	−14.7743	−0.153525
$22$	−119.876	−1.16171
$23$	−43.2915	−0.392474	−0.196237	−	0.980557i	$-0.562872\pi$
$23$	−43.2915	−0.392474	−0.196237	+	0.980557i	$0.562872\pi$
$24$	−24.0000	−0.204124
$25$	−107.888	−0.863104
$26$	91.3294	0.688891
$27$	−27.0000	−0.192450
$28$	19.6991	0.132956
$29$	−133.487	−0.854757	−0.427378	−	0.904073i	$-0.640563\pi$
$29$	−133.487	−0.854757	−0.427378	+	0.904073i	$0.640563\pi$
$30$	24.8200	0.151050
$31$	67.9689	0.393793	0.196896	−	0.980424i	$-0.436914\pi$
$31$	67.9689	0.393793	0.196896	+	0.980424i	$0.436914\pi$
$32$	32.0000	0.176777
$33$	179.814	0.948534
$34$	−190.660	−0.961703
$35$	−20.3721	−0.0983863
$36$	36.0000	0.166667
$37$	−309.748	−1.37628	−0.688138	−	0.725580i	$-0.741572\pi$
$37$	−309.748	−1.37628	−0.688138	+	0.725580i	$0.741572\pi$
$38$	54.6912	0.233476
$39$	−136.994	−0.562477
$40$	−33.0934	−0.130813
$41$	−124.566	−0.474486	−0.237243	−	0.971450i	$-0.576244\pi$
$41$	−124.566	−0.474486	−0.237243	+	0.971450i	$0.576244\pi$
$42$	−29.5486	−0.108558
$43$	−22.2477	−0.0789008	−0.0394504	−	0.999222i	$-0.512561\pi$
$43$	−22.2477	−0.0789008	−0.0394504	+	0.999222i	$0.512561\pi$
$44$	−239.752	−0.821454
$45$	−37.2300	−0.123332
$46$	−86.5830	−0.277521
$47$	−267.186	−0.829215	−0.414608	−	0.910000i	$-0.636081\pi$
$47$	−267.186	−0.829215	−0.414608	+	0.910000i	$0.636081\pi$
$48$	−48.0000	−0.144338
$49$	−318.747	−0.929291
$50$	−215.776	−0.610306
$51$	285.990	0.785227
$52$	182.659	0.487119
$53$	−94.7700	−0.245616	−0.122808	−	0.992430i	$-0.539190\pi$
$53$	−94.7700	−0.245616	−0.122808	+	0.992430i	$0.539190\pi$
$54$	−54.0000	−0.136083
$55$	247.944	0.607868
$56$	39.3981	0.0940142
$57$	−82.0368	−0.190632
$58$	−266.974	−0.604404
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	49.6401	0.106808
$61$	−39.2858	−0.0824594	−0.0412297	−	0.999150i	$-0.513128\pi$
$61$	−39.2858	−0.0824594	−0.0412297	+	0.999150i	$0.513128\pi$
$62$	135.938	0.278453
$63$	44.3229	0.0886374
$64$	64.0000	0.125000
$65$	−188.900	−0.360464
$66$	359.628	0.670715
$67$	127.319	0.232157	0.116079	−	0.993240i	$-0.462968\pi$
$67$	127.319	0.232157	0.116079	+	0.993240i	$0.462968\pi$
$68$	−381.320	−0.680026
$69$	129.874	0.226595
$70$	−40.7443	−0.0695696
$71$	536.119	0.896135	0.448068	−	0.894000i	$-0.352112\pi$
$71$	536.119	0.896135	0.448068	+	0.894000i	$0.352112\pi$
$72$	72.0000	0.117851
$73$	472.832	0.758093	0.379047	−	0.925378i	$-0.376252\pi$
$73$	472.832	0.758093	0.379047	+	0.925378i	$0.376252\pi$
$74$	−619.496	−0.973174
$75$	323.664	0.498313
$76$	109.382	0.165092
$77$	−295.181	−0.436870
$78$	−273.988	−0.397731
$79$	−504.718	−0.718800	−0.359400	−	0.933184i	$-0.617019\pi$
$79$	−504.718	−0.718800	−0.359400	+	0.933184i	$0.617019\pi$
$80$	−66.1868	−0.0924988
$81$	81.0000	0.111111
$82$	−249.132	−0.335512
$83$	77.1909	0.102082	0.0510410	−	0.998697i	$-0.483746\pi$
$83$	77.1909	0.102082	0.0510410	+	0.998697i	$0.483746\pi$
$84$	−59.0972	−0.0767623
$85$	394.349	0.503213
$86$	−44.4953	−0.0557913
$87$	400.461	0.493494
$88$	−479.504	−0.580856
$89$	460.203	0.548106	0.274053	−	0.961715i	$-0.411636\pi$
$89$	460.203	0.548106	0.274053	+	0.961715i	$0.411636\pi$
$90$	−74.4601	−0.0872087
$91$	224.888	0.259062
$92$	−173.166	−0.196237
$93$	−203.907	−0.227356
$94$	−534.372	−0.586344
$95$	−113.120	−0.122167
$96$	−96.0000	−0.102062
$97$	678.352	0.710064	0.355032	−	0.934854i	$-0.384470\pi$
$97$	678.352	0.710064	0.355032	+	0.934854i	$0.384470\pi$
$98$	−637.493	−0.657108
$99$	−539.442	−0.547636
$100$	−431.552	−0.431552
$101$	−498.676	−0.491288	−0.245644	−	0.969360i	$-0.578999\pi$
$101$	−498.676	−0.491288	−0.245644	+	0.969360i	$0.578999\pi$
$102$	571.979	0.555239
$103$	−1316.39	−1.25930	−0.629650	−	0.776879i	$-0.716802\pi$
$103$	−1316.39	−1.25930	−0.629650	+	0.776879i	$0.716802\pi$
$104$	365.317	0.344445
$105$	61.1164	0.0568034
$106$	−189.540	−0.173677
$107$	1135.98	1.02635	0.513173	−	0.858285i	$-0.328470\pi$
$107$	1135.98	1.02635	0.513173	+	0.858285i	$0.328470\pi$
$108$	−108.000	−0.0962250
$109$	−172.553	−0.151629	−0.0758147	−	0.997122i	$-0.524156\pi$
$109$	−172.553	−0.151629	−0.0758147	+	0.997122i	$0.524156\pi$
$110$	495.888	0.429828
$111$	929.243	0.794593
$112$	78.7962	0.0664781
$113$	1920.77	1.59904	0.799518	−	0.600642i	$-0.205088\pi$
$113$	1920.77	1.59904	0.799518	+	0.600642i	$0.205088\pi$
$114$	−164.074	−0.134797
$115$	179.083	0.145213
$116$	−533.949	−0.427378
$117$	410.982	0.324746
$118$	−118.000	−0.0920575
$119$	−469.477	−0.361655
$120$	99.2801	0.0755250
$121$	2261.57	1.69915
$122$	−78.5715	−0.0583076
$123$	373.698	0.273945
$124$	271.876	0.196896
$125$	963.381	0.689339
$126$	88.6458	0.0626761
$127$	2155.10	1.50578	0.752892	−	0.658144i	$-0.228658\pi$
$127$	2155.10	1.50578	0.752892	+	0.658144i	$0.228658\pi$
$128$	128.000	0.0883883
$129$	66.7430	0.0455534
$130$	−377.800	−0.254886
$131$	−200.153	−0.133492	−0.0667459	−	0.997770i	$-0.521262\pi$
$131$	−200.153	−0.133492	−0.0667459	+	0.997770i	$0.521262\pi$
$132$	719.256	0.474267
$133$	134.671	0.0878002
$134$	254.639	0.164160
$135$	111.690	0.0712056
$136$	−762.639	−0.480851
$137$	−1274.56	−0.794840	−0.397420	−	0.917637i	$-0.630094\pi$
$137$	−1274.56	−0.794840	−0.397420	+	0.917637i	$0.630094\pi$
$138$	259.749	0.160227
$139$	2099.27	1.28099	0.640496	−	0.767962i	$-0.278729\pi$
$139$	2099.27	1.28099	0.640496	+	0.767962i	$0.278729\pi$
$140$	−81.4886	−0.0491931
$141$	801.559	0.478748
$142$	1072.24	0.633663
$143$	−2737.05	−1.60059
$144$	144.000	0.0833333
$145$	552.193	0.316256
$146$	945.664	0.536053
$147$	956.240	0.536526
$148$	−1238.99	−0.688138
$149$	−676.501	−0.371954	−0.185977	−	0.982554i	$-0.559545\pi$
$149$	−676.501	−0.371954	−0.185977	+	0.982554i	$0.559545\pi$
$150$	647.328	0.352361
$151$	164.141	0.0884608	0.0442304	−	0.999021i	$-0.485916\pi$
$151$	164.141	0.0884608	0.0442304	+	0.999021i	$0.485916\pi$
$152$	218.765	0.116738
$153$	−857.969	−0.453351
$154$	−590.361	−0.308914
$155$	−281.165	−0.145701
$156$	−547.976	−0.281239
$157$	−2798.57	−1.42261	−0.711306	−	0.702882i	$-0.751896\pi$
$157$	−2798.57	−1.42261	−0.711306	+	0.702882i	$0.751896\pi$
$158$	−1009.44	−0.508268
$159$	284.310	0.141807
$160$	−132.374	−0.0654065
$161$	−213.200	−0.104364
$162$	162.000	0.0785674
$163$	−880.588	−0.423147	−0.211573	−	0.977362i	$-0.567859\pi$
$163$	−880.588	−0.423147	−0.211573	+	0.977362i	$0.567859\pi$
$164$	−498.264	−0.237243
$165$	−743.832	−0.350953
$166$	154.382	0.0721829
$167$	−2085.49	−0.966347	−0.483174	−	0.875525i	$-0.660516\pi$
$167$	−2085.49	−0.966347	−0.483174	+	0.875525i	$0.660516\pi$
$168$	−118.194	−0.0542791
$169$	−111.737	−0.0508588
$170$	788.697	0.355825
$171$	246.110	0.110062
$172$	−88.9906	−0.0394504
$173$	−3679.81	−1.61717	−0.808585	−	0.588379i	$-0.799766\pi$
$173$	−3679.81	−1.61717	−0.808585	+	0.588379i	$0.799766\pi$
$174$	800.923	0.348953
$175$	−531.323	−0.229510
$176$	−959.008	−0.410727
$177$	177.000	0.0751646
$178$	920.407	0.387570
$179$	1631.90	0.681419	0.340709	−	0.940169i	$-0.389333\pi$
$179$	1631.90	0.681419	0.340709	+	0.940169i	$0.389333\pi$
$180$	−148.920	−0.0616659
$181$	1833.42	0.752914	0.376457	−	0.926434i	$-0.377142\pi$
$181$	1833.42	0.752914	0.376457	+	0.926434i	$0.377142\pi$
$182$	449.776	0.183185
$183$	117.857	0.0476080
$184$	−346.332	−0.138760
$185$	1281.32	0.509216
$186$	−407.813	−0.160765
$187$	5713.89	2.23444
$188$	−1068.74	−0.414608
$189$	−132.969	−0.0511749
$190$	−226.240	−0.0863850
$191$	3597.34	1.36280	0.681399	−	0.731912i	$-0.261372\pi$
$191$	3597.34	1.36280	0.681399	+	0.731912i	$0.261372\pi$
$192$	−192.000	−0.0721688
$193$	2401.83	0.895789	0.447894	−	0.894086i	$-0.352174\pi$
$193$	2401.83	0.895789	0.447894	+	0.894086i	$0.352174\pi$
$194$	1356.70	0.502091
$195$	566.699	0.208114
$196$	−1274.99	−0.464645
$197$	−1901.54	−0.687713	−0.343856	−	0.939022i	$-0.611733\pi$
$197$	−1901.54	−0.687713	−0.343856	+	0.939022i	$0.611733\pi$
$198$	−1078.88	−0.387237
$199$	3029.94	1.07933	0.539666	−	0.841879i	$-0.318551\pi$
$199$	3029.94	1.07933	0.539666	+	0.841879i	$0.318551\pi$
$200$	−863.104	−0.305153
$201$	−381.958	−0.134036
$202$	−997.351	−0.347393
$203$	−657.393	−0.227290
$204$	1143.96	0.392613
$205$	515.288	0.175558
$206$	−2632.78	−0.890460
$207$	−389.623	−0.130825
$208$	730.635	0.243560
$209$	−1639.04	−0.542463
$210$	122.233	0.0401660
$211$	2321.66	0.757487	0.378743	−	0.925502i	$-0.376356\pi$
$211$	2321.66	0.757487	0.378743	+	0.925502i	$0.376356\pi$
$212$	−379.080	−0.122808
$213$	−1608.36	−0.517384
$214$	2271.95	0.725736
$215$	92.0313	0.0291929
$216$	−216.000	−0.0680414
$217$	334.731	0.104714
$218$	−345.107	−0.107218
$219$	−1418.50	−0.437685
$220$	991.776	0.303934
$221$	−4353.21	−1.32502
$222$	1858.49	0.561862
$223$	4710.44	1.41450	0.707252	−	0.706961i	$-0.249934\pi$
$223$	4710.44	1.41450	0.707252	+	0.706961i	$0.249934\pi$
$224$	157.592	0.0470071
$225$	−970.991	−0.287701
$226$	3841.54	1.13069
$227$	−5490.64	−1.60540	−0.802701	−	0.596381i	$-0.796605\pi$
$227$	−5490.64	−1.60540	−0.802701	+	0.596381i	$0.796605\pi$
$228$	−328.147	−0.0953161
$229$	−2856.79	−0.824375	−0.412188	−	0.911099i	$-0.635235\pi$
$229$	−2856.79	−0.824375	−0.412188	+	0.911099i	$0.635235\pi$
$230$	358.165	0.102681
$231$	885.542	0.252227
$232$	−1067.90	−0.302202
$233$	−4261.97	−1.19833	−0.599164	−	0.800626i	$-0.704500\pi$
$233$	−4261.97	−1.19833	−0.599164	+	0.800626i	$0.704500\pi$
$234$	821.964	0.229630
$235$	1105.26	0.306806
$236$	−236.000	−0.0650945
$237$	1514.15	0.414999
$238$	−938.955	−0.255729
$239$	5909.62	1.59942	0.799711	−	0.600386i	$-0.204986\pi$
$239$	5909.62	1.59942	0.799711	+	0.600386i	$0.204986\pi$
$240$	198.560	0.0534042
$241$	−4089.72	−1.09312	−0.546560	−	0.837420i	$-0.684063\pi$
$241$	−4089.72	−1.09312	−0.546560	+	0.837420i	$0.684063\pi$
$242$	4523.13	1.20148
$243$	−243.000	−0.0641500
$244$	−157.143	−0.0412297
$245$	1318.55	0.343833
$246$	747.395	0.193708
$247$	1248.73	0.321679
$248$	543.751	0.139227
$249$	−231.573	−0.0589371
$250$	1926.76	0.487437
$251$	−1874.32	−0.471338	−0.235669	−	0.971833i	$-0.575728\pi$
$251$	−1874.32	−0.471338	−0.235669	+	0.971833i	$0.575728\pi$
$252$	177.292	0.0443187
$253$	2594.81	0.644799
$254$	4310.21	1.06475
$255$	−1183.05	−0.290530
$256$	256.000	0.0625000
$257$	1768.40	0.429221	0.214610	−	0.976700i	$-0.431152\pi$
$257$	1768.40	0.429221	0.214610	+	0.976700i	$0.431152\pi$
$258$	133.486	0.0322111
$259$	−1525.44	−0.365969
$260$	−755.599	−0.180232
$261$	−1201.38	−0.284919
$262$	−400.305	−0.0943929
$263$	3054.51	0.716156	0.358078	−	0.933692i	$-0.383432\pi$
$263$	3054.51	0.716156	0.358078	+	0.933692i	$0.383432\pi$
$264$	1438.51	0.335357
$265$	392.032	0.0908768
$266$	269.341	0.0620841
$267$	−1380.61	−0.316449
$268$	509.278	0.116079
$269$	−6316.20	−1.43162	−0.715809	−	0.698296i	$-0.753942\pi$
$269$	−6316.20	−1.43162	−0.715809	+	0.698296i	$0.753942\pi$
$270$	223.380	0.0503500
$271$	4338.15	0.972414	0.486207	−	0.873844i	$-0.338380\pi$
$271$	4338.15	0.972414	0.486207	+	0.873844i	$0.338380\pi$
$272$	−1525.28	−0.340013
$273$	−674.664	−0.149570
$274$	−2549.12	−0.562037
$275$	6466.59	1.41800
$276$	519.498	0.113297
$277$	−6481.84	−1.40598	−0.702989	−	0.711201i	$-0.748152\pi$
$277$	−6481.84	−1.40598	−0.702989	+	0.711201i	$0.748152\pi$
$278$	4198.54	0.905798
$279$	611.720	0.131264
$280$	−162.977	−0.0347848
$281$	−5074.87	−1.07737	−0.538686	−	0.842507i	$-0.681079\pi$
$281$	−5074.87	−1.07737	−0.538686	+	0.842507i	$0.681079\pi$
$282$	1603.12	0.338526
$283$	−5600.47	−1.17637	−0.588186	−	0.808725i	$-0.700158\pi$
$283$	−5600.47	−1.17637	−0.588186	+	0.808725i	$0.700158\pi$
$284$	2144.48	0.448068
$285$	339.359	0.0705330
$286$	−5474.10	−1.13178
$287$	−613.458	−0.126172
$288$	288.000	0.0589256
$289$	4174.79	0.849743
$290$	1104.39	0.223627
$291$	−2035.05	−0.409955
$292$	1891.33	0.379047
$293$	−401.418	−0.0800378	−0.0400189	−	0.999199i	$-0.512742\pi$
$293$	−401.418	−0.0800378	−0.0400189	+	0.999199i	$0.512742\pi$
$294$	1912.48	0.379381
$295$	244.064	0.0481693
$296$	−2477.98	−0.486587
$297$	1618.33	0.316178
$298$	−1353.00	−0.263011
$299$	−1976.89	−0.382363
$300$	1294.66	0.249157
$301$	−109.564	−0.0209807
$302$	328.281	0.0625512
$303$	1496.03	0.283645
$304$	437.530	0.0825462
$305$	162.512	0.0305096
$306$	−1715.94	−0.320568
$307$	−4786.84	−0.889901	−0.444950	−	0.895555i	$-0.646779\pi$
$307$	−4786.84	−0.889901	−0.444950	+	0.895555i	$0.646779\pi$
$308$	−1180.72	−0.218435
$309$	3949.17	0.727057
$310$	−562.330	−0.103026
$311$	−8620.56	−1.57179	−0.785896	−	0.618359i	$-0.787798\pi$
$311$	−8620.56	−1.57179	−0.785896	+	0.618359i	$0.787798\pi$
$312$	−1095.95	−0.198866
$313$	−2176.98	−0.393132	−0.196566	−	0.980491i	$-0.562979\pi$
$313$	−2176.98	−0.393132	−0.196566	+	0.980491i	$0.562979\pi$
$314$	−5597.14	−1.00594
$315$	−183.349	−0.0327954
$316$	−2018.87	−0.359400
$317$	−874.305	−0.154908	−0.0774540	−	0.996996i	$-0.524679\pi$
$317$	−874.305	−0.154908	−0.0774540	+	0.996996i	$0.524679\pi$
$318$	568.620	0.100272
$319$	8000.96	1.40429
$320$	−264.747	−0.0462494
$321$	−3407.93	−0.592561
$322$	−426.401	−0.0737962
$323$	−2606.85	−0.449069
$324$	324.000	0.0555556
$325$	−4926.67	−0.840869
$326$	−1761.18	−0.299210
$327$	517.660	0.0875433
$328$	−996.527	−0.167756
$329$	−1315.83	−0.220499
$330$	−1487.66	−0.248161
$331$	−9778.83	−1.62385	−0.811923	−	0.583764i	$-0.801579\pi$
$331$	−9778.83	−1.62385	−0.811923	+	0.583764i	$0.801579\pi$
$332$	308.764	0.0510410
$333$	−2787.73	−0.458759
$334$	−4170.98	−0.683311
$335$	−526.679	−0.0858971
$336$	−236.389	−0.0383811
$337$	7785.05	1.25839	0.629197	−	0.777246i	$-0.283384\pi$
$337$	7785.05	1.25839	0.629197	+	0.777246i	$0.283384\pi$
$338$	−223.474	−0.0359626
$339$	−5762.32	−0.923204
$340$	1577.39	0.251606
$341$	−4073.92	−0.646965
$342$	492.221	0.0778253
$343$	−3258.95	−0.513022
$344$	−177.981	−0.0278957
$345$	−537.248	−0.0838390
$346$	−7359.61	−1.14351
$347$	3305.04	0.511308	0.255654	−	0.966768i	$-0.417709\pi$
$347$	3305.04	0.511308	0.255654	+	0.966768i	$0.417709\pi$
$348$	1601.85	0.246747
$349$	−3171.47	−0.486432	−0.243216	−	0.969972i	$-0.578202\pi$
$349$	−3171.47	−0.486432	−0.243216	+	0.969972i	$0.578202\pi$
$350$	−1062.65	−0.162288
$351$	−1232.95	−0.187492
$352$	−1918.02	−0.290428
$353$	4078.90	0.615008	0.307504	−	0.951547i	$-0.400506\pi$
$353$	4078.90	0.615008	0.307504	+	0.951547i	$0.400506\pi$
$354$	354.000	0.0531494
$355$	−2217.75	−0.331566
$356$	1840.81	0.274053
$357$	1408.43	0.208802
$358$	3263.80	0.481836
$359$	−6229.74	−0.915859	−0.457929	−	0.888989i	$-0.651409\pi$
$359$	−6229.74	−0.915859	−0.457929	+	0.888989i	$0.651409\pi$
$360$	−297.840	−0.0436044
$361$	−6111.22	−0.890978
$362$	3666.85	0.532390
$363$	−6784.70	−0.981004
$364$	899.551	0.129531
$365$	−1955.95	−0.280491
$366$	235.715	0.0336639
$367$	4659.13	0.662683	0.331341	−	0.943511i	$-0.392499\pi$
$367$	4659.13	0.662683	0.331341	+	0.943511i	$0.392499\pi$
$368$	−692.664	−0.0981185
$369$	−1121.09	−0.158162
$370$	2562.65	0.360070
$371$	−466.720	−0.0653124
$372$	−815.627	−0.113678
$373$	6027.13	0.836656	0.418328	−	0.908296i	$-0.362616\pi$
$373$	6027.13	0.836656	0.418328	+	0.908296i	$0.362616\pi$
$374$	11427.8	1.57999
$375$	−2890.14	−0.397990
$376$	−2137.49	−0.293172
$377$	−6095.65	−0.832737
$378$	−265.937	−0.0361861
$379$	−11772.5	−1.59555	−0.797774	−	0.602957i	$-0.793989\pi$
$379$	−11772.5	−1.59555	−0.797774	+	0.602957i	$0.793989\pi$
$380$	−452.479	−0.0610834
$381$	−6465.31	−0.869364
$382$	7194.68	0.963644
$383$	−1339.37	−0.178691	−0.0893455	−	0.996001i	$-0.528478\pi$
$383$	−1339.37	−0.178691	−0.0893455	+	0.996001i	$0.528478\pi$
$384$	−384.000	−0.0510310
$385$	1221.07	0.161640
$386$	4803.65	0.633418
$387$	−200.229	−0.0263003
$388$	2713.41	0.355032
$389$	8450.15	1.10139	0.550693	−	0.834708i	$-0.314363\pi$
$389$	8450.15	1.10139	0.550693	+	0.834708i	$0.314363\pi$
$390$	1133.40	0.147159
$391$	4126.97	0.533785
$392$	−2549.97	−0.328554
$393$	600.458	0.0770715
$394$	−3803.09	−0.486286
$395$	2087.85	0.265953
$396$	−2157.77	−0.273818
$397$	−5737.93	−0.725387	−0.362693	−	0.931909i	$-0.618143\pi$
$397$	−5737.93	−0.725387	−0.362693	+	0.931909i	$0.618143\pi$
$398$	6059.89	0.763203
$399$	−404.012	−0.0506915
$400$	−1726.21	−0.215776
$401$	−9851.60	−1.22685	−0.613423	−	0.789754i	$-0.710208\pi$
$401$	−9851.60	−1.22685	−0.613423	+	0.789754i	$0.710208\pi$
$402$	−763.916	−0.0947778
$403$	3103.78	0.383648
$404$	−1994.70	−0.245644
$405$	−335.070	−0.0411106
$406$	−1314.79	−0.160719
$407$	18565.7	2.26110
$408$	2287.92	0.277620
$409$	11773.4	1.42336	0.711681	−	0.702502i	$-0.247934\pi$
$409$	11773.4	1.42336	0.711681	+	0.702502i	$0.247934\pi$
$410$	1030.58	0.124138
$411$	3823.68	0.458901
$412$	−5265.57	−0.629650
$413$	−290.561	−0.0346188
$414$	−779.247	−0.0925070
$415$	−319.314	−0.0377698
$416$	1461.27	0.172223
$417$	−6297.81	−0.739581
$418$	−3278.08	−0.383580
$419$	−4579.51	−0.533947	−0.266974	−	0.963704i	$-0.586024\pi$
$419$	−4579.51	−0.533947	−0.266974	+	0.963704i	$0.586024\pi$
$420$	244.466	0.0284017
$421$	6092.27	0.705271	0.352636	−	0.935761i	$-0.385286\pi$
$421$	6092.27	0.705271	0.352636	+	0.935761i	$0.385286\pi$
$422$	4643.32	0.535624
$423$	−2404.68	−0.276405
$424$	−758.160	−0.0868384
$425$	10284.9	1.17387
$426$	−3216.71	−0.365846
$427$	−193.473	−0.0219270
$428$	4543.91	0.513173
$429$	8211.15	0.924098
$430$	184.063	0.0206425
$431$	−904.721	−0.101111	−0.0505555	−	0.998721i	$-0.516099\pi$
$431$	−904.721	−0.101111	−0.0505555	+	0.998721i	$0.516099\pi$
$432$	−432.000	−0.0481125
$433$	4180.29	0.463953	0.231977	−	0.972721i	$-0.425481\pi$
$433$	4180.29	0.463953	0.231977	+	0.972721i	$0.425481\pi$
$434$	669.462	0.0740442
$435$	−1656.58	−0.182590
$436$	−690.213	−0.0758147
$437$	−1183.83	−0.129589
$438$	−2836.99	−0.309490
$439$	−14449.0	−1.57087	−0.785436	−	0.618943i	$-0.787561\pi$
$439$	−14449.0	−1.57087	−0.785436	+	0.618943i	$0.787561\pi$
$440$	1983.55	0.214914
$441$	−2868.72	−0.309764
$442$	−8706.42	−0.936928
$443$	16324.6	1.75080	0.875400	−	0.483399i	$-0.160598\pi$
$443$	16324.6	1.75080	0.875400	+	0.483399i	$0.160598\pi$
$444$	3716.97	0.397297
$445$	−1903.71	−0.202797
$446$	9420.89	1.00021
$447$	2029.50	0.214748
$448$	315.185	0.0332390
$449$	−3720.30	−0.391029	−0.195514	−	0.980701i	$-0.562638\pi$
$449$	−3720.30	−0.391029	−0.195514	+	0.980701i	$0.562638\pi$
$450$	−1941.98	−0.203435
$451$	7466.23	0.779537
$452$	7683.09	0.799518
$453$	−492.422	−0.0510728
$454$	−10981.3	−1.13519
$455$	−930.287	−0.0958517
$456$	−656.294	−0.0673987
$457$	−5620.34	−0.575292	−0.287646	−	0.957737i	$-0.592873\pi$
$457$	−5620.34	−0.575292	−0.287646	+	0.957737i	$0.592873\pi$
$458$	−5713.58	−0.582921
$459$	2573.91	0.261742
$460$	716.331	0.0726067
$461$	2493.55	0.251922	0.125961	−	0.992035i	$-0.459799\pi$
$461$	2493.55	0.251922	0.125961	+	0.992035i	$0.459799\pi$
$462$	1771.08	0.178351
$463$	4861.59	0.487986	0.243993	−	0.969777i	$-0.421543\pi$
$463$	4861.59	0.487986	0.243993	+	0.969777i	$0.421543\pi$
$464$	−2135.79	−0.213689
$465$	843.495	0.0841207
$466$	−8523.93	−0.847347
$467$	15501.8	1.53605	0.768027	−	0.640417i	$-0.221238\pi$
$467$	15501.8	1.53605	0.768027	+	0.640417i	$0.221238\pi$
$468$	1643.93	0.162373
$469$	627.018	0.0617335
$470$	2210.52	0.216944
$471$	8395.71	0.821346
$472$	−472.000	−0.0460287
$473$	1333.48	0.129627
$474$	3028.31	0.293449
$475$	−2950.26	−0.284984
$476$	−1877.91	−0.180827
$477$	−852.930	−0.0818720
$478$	11819.2	1.13096
$479$	10437.2	0.995595	0.497798	−	0.867293i	$-0.334142\pi$
$479$	10437.2	0.995595	0.497798	+	0.867293i	$0.334142\pi$
$480$	397.121	0.0377625
$481$	−14144.5	−1.34082
$482$	−8179.44	−0.772953
$483$	639.601	0.0602544
$484$	9046.27	0.849574
$485$	−2806.12	−0.262720
$486$	−486.000	−0.0453609
$487$	−3829.15	−0.356294	−0.178147	−	0.984004i	$-0.557010\pi$
$487$	−3829.15	−0.356294	−0.178147	+	0.984004i	$0.557010\pi$
$488$	−314.286	−0.0291538
$489$	2641.76	0.244304
$490$	2637.10	0.243127
$491$	19434.8	1.78631	0.893155	−	0.449749i	$-0.148487\pi$
$491$	19434.8	1.78631	0.893155	+	0.449749i	$0.148487\pi$
$492$	1494.79	0.136972
$493$	12725.3	1.16251
$494$	2497.46	0.227461
$495$	2231.50	0.202623
$496$	1087.50	0.0984482
$497$	2640.26	0.238293
$498$	−463.146	−0.0416748
$499$	−8027.00	−0.720116	−0.360058	−	0.932930i	$-0.617243\pi$
$499$	−8027.00	−0.720116	−0.360058	+	0.932930i	$0.617243\pi$
$500$	3853.52	0.344670
$501$	6256.47	0.557921
$502$	−3748.64	−0.333286
$503$	8718.68	0.772856	0.386428	−	0.922320i	$-0.373709\pi$
$503$	8718.68	0.772856	0.386428	+	0.922320i	$0.373709\pi$
$504$	354.583	0.0313381
$505$	2062.86	0.181774
$506$	5189.61	0.455942
$507$	335.211	0.0293634
$508$	8620.41	0.752892
$509$	3736.29	0.325360	0.162680	−	0.986679i	$-0.447986\pi$
$509$	3736.29	0.325360	0.162680	+	0.986679i	$0.447986\pi$
$510$	−2366.09	−0.205436
$511$	2328.59	0.201586
$512$	512.000	0.0441942
$513$	−738.331	−0.0635441
$514$	3536.80	0.303505
$515$	5445.48	0.465935
$516$	266.972	0.0227767
$517$	16014.6	1.36232
$518$	−3050.87	−0.258779
$519$	11039.4	0.933674
$520$	−1511.20	−0.127443
$521$	−9067.03	−0.762445	−0.381222	−	0.924483i	$-0.624497\pi$
$521$	−9067.03	−0.762445	−0.381222	+	0.924483i	$0.624497\pi$
$522$	−2402.77	−0.201468
$523$	6687.68	0.559143	0.279571	−	0.960125i	$-0.409808\pi$
$523$	6687.68	0.559143	0.279571	+	0.960125i	$0.409808\pi$
$524$	−800.610	−0.0667459
$525$	1593.97	0.132508
$526$	6109.02	0.506399
$527$	−6479.47	−0.535579
$528$	2877.03	0.237133
$529$	−10292.8	−0.845964
$530$	784.064	0.0642596
$531$	−531.000	−0.0433963
$532$	538.683	0.0439001
$533$	−5688.26	−0.462263
$534$	−2761.22	−0.223763
$535$	−4699.17	−0.379743
$536$	1018.56	0.0820800
$537$	−4895.70	−0.393417
$538$	−12632.4	−1.01231
$539$	19105.0	1.52674
$540$	446.761	0.0356028
$541$	−18307.3	−1.45488	−0.727441	−	0.686170i	$-0.759291\pi$
$541$	−18307.3	−1.45488	−0.727441	+	0.686170i	$0.759291\pi$
$542$	8676.31	0.687600
$543$	−5500.27	−0.434695
$544$	−3050.56	−0.240426
$545$	713.796	0.0561022
$546$	−1349.33	−0.105762
$547$	17162.2	1.34150	0.670751	−	0.741683i	$-0.265972\pi$
$547$	17162.2	1.34150	0.670751	+	0.741683i	$0.265972\pi$
$548$	−5098.24	−0.397420
$549$	−353.572	−0.0274865
$550$	12933.2	1.00268
$551$	−3650.29	−0.282228
$552$	1039.00	0.0801134
$553$	−2485.62	−0.191138
$554$	−12963.7	−0.994176
$555$	−3843.97	−0.293996
$556$	8397.08	0.640496
$557$	−841.454	−0.0640100	−0.0320050	−	0.999488i	$-0.510189\pi$
$557$	−841.454	−0.0640100	−0.0320050	+	0.999488i	$0.510189\pi$
$558$	1223.44	0.0928178
$559$	−1015.93	−0.0768682
$560$	−325.954	−0.0245966
$561$	−17141.7	−1.29006
$562$	−10149.7	−0.761817
$563$	−22249.6	−1.66555	−0.832777	−	0.553609i	$-0.813250\pi$
$563$	−22249.6	−1.66555	−0.832777	+	0.553609i	$0.813250\pi$
$564$	3206.23	0.239374
$565$	−7945.60	−0.591636
$566$	−11200.9	−0.831821
$567$	398.906	0.0295458
$568$	4288.95	0.316832
$569$	20190.1	1.48754	0.743772	−	0.668433i	$-0.233035\pi$
$569$	20190.1	1.48754	0.743772	+	0.668433i	$0.233035\pi$
$570$	678.719	0.0498744
$571$	−1037.06	−0.0760064	−0.0380032	−	0.999278i	$-0.512100\pi$
$571$	−1037.06	−0.0760064	−0.0380032	+	0.999278i	$0.512100\pi$
$572$	−10948.2	−0.800293
$573$	−10792.0	−0.786812
$574$	−1226.92	−0.0892169
$575$	4670.63	0.338746
$576$	576.000	0.0416667
$577$	12462.1	0.899144	0.449572	−	0.893244i	$-0.351577\pi$
$577$	12462.1	0.899144	0.449572	+	0.893244i	$0.351577\pi$
$578$	8349.58	0.600859
$579$	−7205.48	−0.517184
$580$	2208.77	0.158128
$581$	380.147	0.0271449
$582$	−4070.11	−0.289882
$583$	5680.32	0.403525
$584$	3782.66	0.268027
$585$	−1700.10	−0.120155
$586$	−802.835	−0.0565952
$587$	19975.8	1.40458	0.702292	−	0.711889i	$-0.252160\pi$
$587$	19975.8	1.40458	0.702292	+	0.711889i	$0.252160\pi$
$588$	3824.96	0.268263
$589$	1858.65	0.130024
$590$	488.127	0.0340608
$591$	5704.63	0.397051
$592$	−4955.96	−0.344069
$593$	−16339.3	−1.13149	−0.565745	−	0.824581i	$-0.691411\pi$
$593$	−16339.3	−1.13149	−0.565745	+	0.824581i	$0.691411\pi$
$594$	3236.65	0.223572
$595$	1942.07	0.133811
$596$	−2706.00	−0.185977
$597$	−9089.83	−0.623153
$598$	−3953.78	−0.270372
$599$	10102.9	0.689136	0.344568	−	0.938761i	$-0.388025\pi$
$599$	10102.9	0.689136	0.344568	+	0.938761i	$0.388025\pi$
$600$	2589.31	0.176180
$601$	5853.64	0.397296	0.198648	−	0.980071i	$-0.436345\pi$
$601$	5853.64	0.397296	0.198648	+	0.980071i	$0.436345\pi$
$602$	−219.129	−0.0148356
$603$	1145.87	0.0773858
$604$	656.563	0.0442304
$605$	−9355.36	−0.628677
$606$	2992.05	0.200567
$607$	1314.88	0.0879231	0.0439615	−	0.999033i	$-0.486002\pi$
$607$	1314.88	0.0879231	0.0439615	+	0.999033i	$0.486002\pi$
$608$	875.059	0.0583690
$609$	1972.18	0.131226
$610$	325.025	0.0215735
$611$	−12201.0	−0.807854
$612$	−3431.88	−0.226675
$613$	24122.9	1.58942	0.794710	−	0.606990i	$-0.207623\pi$
$613$	24122.9	1.58942	0.794710	+	0.606990i	$0.207623\pi$
$614$	−9573.68	−0.629255
$615$	−1545.86	−0.101358
$616$	−2361.45	−0.154457
$617$	3676.04	0.239857	0.119929	−	0.992783i	$-0.461733\pi$
$617$	3676.04	0.239857	0.119929	+	0.992783i	$0.461733\pi$
$618$	7898.35	0.514107
$619$	−2708.84	−0.175893	−0.0879463	−	0.996125i	$-0.528030\pi$
$619$	−2708.84	−0.175893	−0.0879463	+	0.996125i	$0.528030\pi$
$620$	−1124.66	−0.0728507
$621$	1168.87	0.0755316
$622$	−17241.1	−1.11142
$623$	2266.39	0.145748
$624$	−2191.90	−0.140619
$625$	9500.80	0.608051
$626$	−4353.96	−0.277986
$627$	4917.12	0.313191
$628$	−11194.3	−0.711306
$629$	29528.2	1.87181
$630$	−366.699	−0.0231899
$631$	−5689.38	−0.358939	−0.179470	−	0.983764i	$-0.557438\pi$
$631$	−5689.38	−0.358939	−0.179470	+	0.983764i	$0.557438\pi$
$632$	−4037.74	−0.254134
$633$	−6964.98	−0.437335
$634$	−1748.61	−0.109537
$635$	−8914.96	−0.557133
$636$	1137.24	0.0709033
$637$	−14555.5	−0.905351
$638$	16001.9	0.992981
$639$	4825.07	0.298712
$640$	−529.494	−0.0327033
$641$	22198.8	1.36787	0.683933	−	0.729545i	$-0.260268\pi$
$641$	22198.8	1.36787	0.683933	+	0.729545i	$0.260268\pi$
$642$	−6815.86	−0.419004
$643$	−19289.7	−1.18306	−0.591532	−	0.806281i	$-0.701477\pi$
$643$	−19289.7	−1.18306	−0.591532	+	0.806281i	$0.701477\pi$
$644$	−852.802	−0.0521818
$645$	−276.094	−0.0168545
$646$	−5213.71	−0.317540
$647$	−3739.45	−0.227222	−0.113611	−	0.993525i	$-0.536242\pi$
$647$	−3739.45	−0.227222	−0.113611	+	0.993525i	$0.536242\pi$
$648$	648.000	0.0392837
$649$	3536.34	0.213888
$650$	−9853.34	−0.594584
$651$	−1004.19	−0.0604569
$652$	−3522.35	−0.211573
$653$	31302.5	1.87590	0.937948	−	0.346776i	$-0.112724\pi$
$653$	31302.5	1.87590	0.937948	+	0.346776i	$0.112724\pi$
$654$	1035.32	0.0619025
$655$	827.966	0.0493913
$656$	−1993.05	−0.118621
$657$	4255.49	0.252698
$658$	−2631.66	−0.155916
$659$	−23456.0	−1.38652	−0.693259	−	0.720689i	$-0.743826\pi$
$659$	−23456.0	−1.38652	−0.693259	+	0.720689i	$0.743826\pi$
$660$	−2975.33	−0.175476
$661$	5762.52	0.339086	0.169543	−	0.985523i	$-0.445771\pi$
$661$	5762.52	0.339086	0.169543	+	0.985523i	$0.445771\pi$
$662$	−19557.7	−1.14823
$663$	13059.6	0.764998
$664$	617.527	0.0360914
$665$	−557.088	−0.0324857
$666$	−5575.46	−0.324391
$667$	5778.86	0.335470
$668$	−8341.96	−0.483174
$669$	−14131.3	−0.816665
$670$	−1053.36	−0.0607384
$671$	2354.71	0.135473
$672$	−472.777	−0.0271396
$673$	−11120.8	−0.636959	−0.318480	−	0.947930i	$-0.603172\pi$
$673$	−11120.8	−0.636959	−0.318480	+	0.947930i	$0.603172\pi$
$674$	15570.1	0.889819
$675$	2912.97	0.166104
$676$	−446.947	−0.0254294
$677$	−17906.0	−1.01652	−0.508259	−	0.861204i	$-0.669711\pi$
$677$	−17906.0	−1.01652	−0.508259	+	0.861204i	$0.669711\pi$
$678$	−11524.6	−0.652804
$679$	3340.72	0.188815
$680$	3154.79	0.177913
$681$	16471.9	0.926880
$682$	−8147.84	−0.457474
$683$	11886.7	0.665930	0.332965	−	0.942939i	$-0.391951\pi$
$683$	11886.7	0.665930	0.332965	+	0.942939i	$0.391951\pi$
$684$	984.442	0.0550308
$685$	5272.44	0.294087
$686$	−6517.89	−0.362761
$687$	8570.37	0.475953
$688$	−355.962	−0.0197252
$689$	−4327.64	−0.239289
$690$	−1074.50	−0.0592831
$691$	−23915.9	−1.31665	−0.658324	−	0.752734i	$-0.728734\pi$
$691$	−23915.9	−1.31665	−0.658324	+	0.752734i	$0.728734\pi$
$692$	−14719.2	−0.808585
$693$	−2656.63	−0.145623
$694$	6610.09	0.361550
$695$	−8683.99	−0.473961
$696$	3203.69	0.174476
$697$	11874.9	0.645326
$698$	−6342.94	−0.343960
$699$	12785.9	0.691856
$700$	−2125.29	−0.114755
$701$	−35532.3	−1.91446	−0.957229	−	0.289332i	$-0.906567\pi$
$701$	−35532.3	−1.91446	−0.957229	+	0.289332i	$0.906567\pi$
$702$	−2465.89	−0.132577
$703$	−8470.24	−0.454425
$704$	−3836.03	−0.205364
$705$	−3315.79	−0.177134
$706$	8157.80	0.434877
$707$	−2455.86	−0.130639
$708$	708.000	0.0375823
$709$	−10305.8	−0.545897	−0.272949	−	0.962029i	$-0.587999\pi$
$709$	−10305.8	−0.545897	−0.272949	+	0.962029i	$0.587999\pi$
$710$	−4435.50	−0.234452
$711$	−4542.46	−0.239600
$712$	3681.63	0.193785
$713$	−2942.47	−0.154553
$714$	2816.86	0.147645
$715$	11322.3	0.592209
$716$	6527.60	0.340709
$717$	−17728.9	−0.923426
$718$	−12459.5	−0.647610
$719$	2524.81	0.130959	0.0654795	−	0.997854i	$-0.479142\pi$
$719$	2524.81	0.130959	0.0654795	+	0.997854i	$0.479142\pi$
$720$	−595.681	−0.0308329
$721$	−6482.92	−0.334863
$722$	−12222.4	−0.630017
$723$	12269.2	0.631113
$724$	7333.70	0.376457
$725$	14401.7	0.737743
$726$	−13569.4	−0.693675
$727$	24759.5	1.26311	0.631553	−	0.775333i	$-0.282418\pi$
$727$	24759.5	1.26311	0.631553	+	0.775333i	$0.282418\pi$
$728$	1799.10	0.0915923
$729$	729.000	0.0370370
$730$	−3911.90	−0.198337
$731$	2120.87	0.107309
$732$	471.429	0.0238040
$733$	−15968.0	−0.804626	−0.402313	−	0.915502i	$-0.631794\pi$
$733$	−15968.0	−0.804626	−0.402313	+	0.915502i	$0.631794\pi$
$734$	9318.26	0.468588
$735$	−3955.65	−0.198512
$736$	−1385.33	−0.0693802
$737$	−7631.27	−0.381413
$738$	−2242.19	−0.111837
$739$	8456.17	0.420927	0.210464	−	0.977602i	$-0.432503\pi$
$739$	8456.17	0.420927	0.210464	+	0.977602i	$0.432503\pi$
$740$	5125.30	0.254608
$741$	−3746.18	−0.185721
$742$	−933.440	−0.0461828
$743$	−15616.5	−0.771081	−0.385540	−	0.922691i	$-0.625985\pi$
$743$	−15616.5	−0.771081	−0.385540	+	0.922691i	$0.625985\pi$
$744$	−1631.25	−0.0803826
$745$	2798.46	0.137621
$746$	12054.3	0.591605
$747$	694.718	0.0340273
$748$	22855.5	1.11722
$749$	5594.42	0.272918
$750$	−5780.29	−0.281422
$751$	−21411.7	−1.04038	−0.520189	−	0.854051i	$-0.674139\pi$
$751$	−21411.7	−1.04038	−0.520189	+	0.854051i	$0.674139\pi$
$752$	−4274.98	−0.207304
$753$	5622.95	0.272127
$754$	−12191.3	−0.588834
$755$	−678.996	−0.0327301
$756$	−531.875	−0.0255874
$757$	18237.2	0.875618	0.437809	−	0.899068i	$-0.355755\pi$
$757$	18237.2	0.875618	0.437809	+	0.899068i	$0.355755\pi$
$758$	−23545.0	−1.12822
$759$	−7784.42	−0.372275
$760$	−904.958	−0.0431925
$761$	−23496.5	−1.11925	−0.559624	−	0.828747i	$-0.689054\pi$
$761$	−23496.5	−1.11925	−0.559624	+	0.828747i	$0.689054\pi$
$762$	−12930.6	−0.614734
$763$	−849.785	−0.0403201
$764$	14389.4	0.681399
$765$	3549.14	0.167738
$766$	−2678.74	−0.126354
$767$	−2694.22	−0.126835
$768$	−768.000	−0.0360844
$769$	18097.5	0.848650	0.424325	−	0.905510i	$-0.360511\pi$
$769$	18097.5	0.848650	0.424325	+	0.905510i	$0.360511\pi$
$770$	2442.13	0.114297
$771$	−5305.20	−0.247811
$772$	9607.31	0.447894
$773$	−24382.4	−1.13451	−0.567253	−	0.823543i	$-0.691994\pi$
$773$	−24382.4	−1.13451	−0.567253	+	0.823543i	$0.691994\pi$
$774$	−400.458	−0.0185971
$775$	−7333.02	−0.339884
$776$	5426.81	0.251045
$777$	4576.31	0.211292
$778$	16900.3	0.778798
$779$	−3406.33	−0.156668
$780$	2266.80	0.104057
$781$	−32133.9	−1.47227
$782$	8253.95	0.377443
$783$	3604.15	0.164498
$784$	−5099.95	−0.232323
$785$	11576.8	0.526360
$786$	1200.92	0.0544978
$787$	14667.1	0.664329	0.332164	−	0.943222i	$-0.392221\pi$
$787$	14667.1	0.664329	0.332164	+	0.943222i	$0.392221\pi$
$788$	−7606.18	−0.343856
$789$	−9163.53	−0.413473
$790$	4175.71	0.188057
$791$	9459.35	0.425203
$792$	−4315.54	−0.193619
$793$	−1793.97	−0.0803352
$794$	−11475.9	−0.512926
$795$	−1176.10	−0.0524677
$796$	12119.8	0.539666
$797$	−13827.2	−0.614536	−0.307268	−	0.951623i	$-0.599415\pi$
$797$	−13827.2	−0.614536	−0.307268	+	0.951623i	$0.599415\pi$
$798$	−808.024	−0.0358443
$799$	25470.8	1.12778
$800$	−3452.41	−0.152577
$801$	4141.83	0.182702
$802$	−19703.2	−0.867511
$803$	−28340.6	−1.24548
$804$	−1527.83	−0.0670180
$805$	881.940	0.0386140
$806$	6207.56	0.271280
$807$	18948.6	0.826545
$808$	−3989.40	−0.173696
$809$	−8358.77	−0.363262	−0.181631	−	0.983367i	$-0.558138\pi$
$809$	−8358.77	−0.363262	−0.181631	+	0.983367i	$0.558138\pi$
$810$	−670.141	−0.0290696
$811$	−38753.6	−1.67796	−0.838978	−	0.544166i	$-0.816846\pi$
$811$	−38753.6	−1.67796	−0.838978	+	0.544166i	$0.816846\pi$
$812$	−2629.57	−0.113645
$813$	−13014.5	−0.561423
$814$	37131.3	1.59884
$815$	3642.70	0.156562
$816$	4575.84	0.196307
$817$	−608.376	−0.0260519
$818$	23546.7	1.00647
$819$	2023.99	0.0863540
$820$	2061.15	0.0877788
$821$	−12117.1	−0.515093	−0.257546	−	0.966266i	$-0.582914\pi$
$821$	−12117.1	−0.515093	−0.257546	+	0.966266i	$0.582914\pi$
$822$	7647.37	0.324492
$823$	−2979.48	−0.126195	−0.0630974	−	0.998007i	$-0.520098\pi$
$823$	−2979.48	−0.126195	−0.0630974	+	0.998007i	$0.520098\pi$
$824$	−10531.1	−0.445230
$825$	−19399.8	−0.818683
$826$	−581.122	−0.0244792
$827$	−39638.8	−1.66672	−0.833360	−	0.552731i	$-0.813586\pi$
$827$	−39638.8	−1.66672	−0.833360	+	0.552731i	$0.813586\pi$
$828$	−1558.49	−0.0654123
$829$	−40897.1	−1.71341	−0.856703	−	0.515809i	$-0.827491\pi$
$829$	−40897.1	−1.71341	−0.856703	+	0.515809i	$0.827491\pi$
$830$	−638.627	−0.0267073
$831$	19445.5	0.811742
$832$	2922.54	0.121780
$833$	30386.1	1.26388
$834$	−12595.6	−0.522963
$835$	8626.98	0.357544
$836$	−6556.17	−0.271232
$837$	−1835.16	−0.0757854
$838$	−9159.03	−0.377558
$839$	−41236.9	−1.69685	−0.848424	−	0.529317i	$-0.822448\pi$
$839$	−41236.9	−1.69685	−0.848424	+	0.529317i	$0.822448\pi$
$840$	488.931	0.0200830
$841$	−6570.18	−0.269391
$842$	12184.5	0.498702
$843$	15224.6	0.622021
$844$	9286.64	0.378743
$845$	462.219	0.0188175
$846$	−4809.35	−0.195448
$847$	11137.7	0.451825
$848$	−1516.32	−0.0614040
$849$	16801.4	0.679179
$850$	20569.9	0.830049
$851$	13409.4	0.540152
$852$	−6433.43	−0.258692
$853$	−21964.3	−0.881646	−0.440823	−	0.897594i	$-0.645313\pi$
$853$	−21964.3	−0.881646	−0.440823	+	0.897594i	$0.645313\pi$
$854$	−386.946	−0.0155047
$855$	−1018.08	−0.0407223
$856$	9087.82	0.362868
$857$	−4175.97	−0.166451	−0.0832255	−	0.996531i	$-0.526522\pi$
$857$	−4175.97	−0.166451	−0.0832255	+	0.996531i	$0.526522\pi$
$858$	16422.3	0.653436
$859$	−12121.1	−0.481451	−0.240725	−	0.970593i	$-0.577385\pi$
$859$	−12121.1	−0.481451	−0.240725	+	0.970593i	$0.577385\pi$
$860$	368.125	0.0145965
$861$	1840.37	0.0728453
$862$	−1809.44	−0.0714963
$863$	23846.8	0.940619	0.470310	−	0.882502i	$-0.344142\pi$
$863$	23846.8	0.940619	0.470310	+	0.882502i	$0.344142\pi$
$864$	−864.000	−0.0340207
$865$	15222.2	0.598345
$866$	8360.57	0.328064
$867$	−12524.4	−0.490600
$868$	1338.92	0.0523572
$869$	30251.8	1.18092
$870$	−3313.16	−0.129111
$871$	5814.00	0.226177
$872$	−1380.43	−0.0536091
$873$	6105.16	0.236688
$874$	−2367.66	−0.0916332
$875$	4744.43	0.183304
$876$	−5673.99	−0.218843
$877$	44989.0	1.73224	0.866118	−	0.499839i	$-0.166607\pi$
$877$	44989.0	1.73224	0.866118	+	0.499839i	$0.166607\pi$
$878$	−28898.0	−1.11077
$879$	1204.25	0.0462098
$880$	3967.10	0.151967
$881$	−21788.7	−0.833235	−0.416618	−	0.909082i	$-0.636785\pi$
$881$	−21788.7	−0.833235	−0.416618	+	0.909082i	$0.636785\pi$
$882$	−5737.44	−0.219036
$883$	35282.7	1.34468	0.672342	−	0.740240i	$-0.265288\pi$
$883$	35282.7	1.34468	0.672342	+	0.740240i	$0.265288\pi$
$884$	−17412.8	−0.662508
$885$	−732.191	−0.0278105
$886$	32649.2	1.23800
$887$	25493.8	0.965047	0.482524	−	0.875883i	$-0.339720\pi$
$887$	25493.8	0.965047	0.482524	+	0.875883i	$0.339720\pi$
$888$	7433.95	0.280931
$889$	10613.4	0.400406
$890$	−3807.42	−0.143399
$891$	−4854.98	−0.182545
$892$	18841.8	0.707252
$893$	−7306.37	−0.273794
$894$	4059.00	0.151849
$895$	−6750.64	−0.252122
$896$	630.370	0.0235036
$897$	5930.68	0.220758
$898$	−7440.60	−0.276499
$899$	−9072.97	−0.336597
$900$	−3883.97	−0.143851
$901$	9034.41	0.334051
$902$	14932.5	0.551216
$903$	328.693	0.0121132
$904$	15366.2	0.565344
$905$	−7584.28	−0.278574
$906$	−984.844	−0.0361140
$907$	−4505.15	−0.164930	−0.0824648	−	0.996594i	$-0.526279\pi$
$907$	−4505.15	−0.164930	−0.0824648	+	0.996594i	$0.526279\pi$
$908$	−21962.6	−0.802701
$909$	−4488.08	−0.163763
$910$	−1860.57	−0.0677774
$911$	−13947.9	−0.507259	−0.253630	−	0.967301i	$-0.581624\pi$
$911$	−13947.9	−0.507259	−0.253630	+	0.967301i	$0.581624\pi$
$912$	−1312.59	−0.0476581
$913$	−4626.67	−0.167711
$914$	−11240.7	−0.406793
$915$	−487.537	−0.0176147
$916$	−11427.2	−0.412188
$917$	−985.705	−0.0354971
$918$	5147.81	0.185080
$919$	−16253.7	−0.583416	−0.291708	−	0.956507i	$-0.594224\pi$
$919$	−16253.7	−0.583416	−0.291708	+	0.956507i	$0.594224\pi$
$920$	1432.66	0.0513407
$921$	14360.5	0.513784
$922$	4987.09	0.178136
$923$	24481.7	0.873050
$924$	3542.17	0.126113
$925$	33418.0	1.18787
$926$	9723.19	0.345058
$927$	−11847.5	−0.419767
$928$	−4271.59	−0.151101
$929$	36648.8	1.29430	0.647152	−	0.762361i	$-0.275960\pi$
$929$	36648.8	1.29430	0.647152	+	0.762361i	$0.275960\pi$
$930$	1686.99	0.0594824
$931$	−8716.32	−0.306838
$932$	−17047.9	−0.599164
$933$	25861.7	0.907474
$934$	31003.6	1.08615
$935$	−23636.5	−0.826733
$936$	3287.86	0.114815
$937$	−25914.3	−0.903504	−0.451752	−	0.892144i	$-0.649201\pi$
$937$	−25914.3	−0.903504	−0.451752	+	0.892144i	$0.649201\pi$
$938$	1254.04	0.0436522
$939$	6530.95	0.226975
$940$	4421.05	0.153403
$941$	−10822.9	−0.374938	−0.187469	−	0.982270i	$-0.560028\pi$
$941$	−10822.9	−0.374938	−0.187469	+	0.982270i	$0.560028\pi$
$942$	16791.4	0.580779
$943$	5392.64	0.186223
$944$	−944.000	−0.0325472
$945$	550.048	0.0189345
$946$	2666.96	0.0916600
$947$	17773.2	0.609875	0.304937	−	0.952372i	$-0.401364\pi$
$947$	17773.2	0.609875	0.304937	+	0.952372i	$0.401364\pi$
$948$	6056.62	0.207500
$949$	21591.7	0.738564
$950$	−5900.52	−0.201514
$951$	2622.91	0.0894362
$952$	−3755.82	−0.127864
$953$	1448.66	0.0492411	0.0246206	−	0.999697i	$-0.492162\pi$
$953$	1448.66	0.0492411	0.0246206	+	0.999697i	$0.492162\pi$
$954$	−1705.86	−0.0578923
$955$	−14881.0	−0.504229
$956$	23638.5	0.799711
$957$	−24002.9	−0.810766
$958$	20874.5	0.703992
$959$	−6276.91	−0.211358
$960$	794.241	0.0267021
$961$	−25171.2	−0.844927
$962$	−28289.1	−0.948104
$963$	10223.8	0.342115
$964$	−16358.9	−0.546560
$965$	−9935.57	−0.331438
$966$	1279.20	0.0426063
$967$	9650.60	0.320933	0.160467	−	0.987041i	$-0.448700\pi$
$967$	9650.60	0.320933	0.160467	+	0.987041i	$0.448700\pi$
$968$	18092.5	0.600740
$969$	7820.56	0.259270
$970$	−5612.24	−0.185771
$971$	−7088.02	−0.234259	−0.117129	−	0.993117i	$-0.537369\pi$
$971$	−7088.02	−0.234259	−0.117129	+	0.993117i	$0.537369\pi$
$972$	−972.000	−0.0320750
$973$	10338.4	0.340631
$974$	−7658.29	−0.251938
$975$	14780.0	0.485476
$976$	−628.572	−0.0206149
$977$	−54767.5	−1.79341	−0.896707	−	0.442624i	$-0.854048\pi$
$977$	−54767.5	−1.79341	−0.896707	+	0.442624i	$0.854048\pi$
$978$	5283.53	0.172749
$979$	−27583.7	−0.900488
$980$	5274.20	0.171917
$981$	−1552.98	−0.0505431
$982$	38869.5	1.26311
$983$	46197.9	1.49897	0.749484	−	0.662022i	$-0.230302\pi$
$983$	46197.9	1.49897	0.749484	+	0.662022i	$0.230302\pi$
$984$	2989.58	0.0968540
$985$	7866.07	0.254450
$986$	25450.6	0.822022
$987$	3947.49	0.127305
$988$	4994.91	0.160839
$989$	963.134	0.0309665
$990$	4462.99	0.143276
$991$	−34915.5	−1.11920	−0.559600	−	0.828763i	$-0.689045\pi$
$991$	−34915.5	−1.11920	−0.559600	+	0.828763i	$0.689045\pi$
$992$	2175.00	0.0696134
$993$	29336.5	0.937528
$994$	5280.52	0.168499
$995$	−12533.9	−0.399348
$996$	−926.291	−0.0294685
$997$	−35165.0	−1.11704	−0.558520	−	0.829491i	$-0.688630\pi$
$997$	−35165.0	−1.11704	−0.558520	+	0.829491i	$0.688630\pi$
$998$	−16054.0	−0.509199
$999$	8363.19	0.264864

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.4.a.g.1.2	✓	3
3.2	odd	2		1062.4.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.4.a.g.1.2	✓	3	1.1	even	1	trivial
1062.4.a.i.1.2		3	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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.13667 q^{5} -6.00000 q^{6} +4.92477 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.13667 q^{5} -6.00000 q^{6} +4.92477 q^{7} +8.00000 q^{8} +9.00000 q^{9} -8.27334 q^{10} -59.9380 q^{11} -12.0000 q^{12} +45.6647 q^{13} +9.84953 q^{14} +12.4100 q^{15} +16.0000 q^{16} -95.3299 q^{17} +18.0000 q^{18} +27.3456 q^{19} -16.5467 q^{20} -14.7743 q^{21} -119.876 q^{22} -43.2915 q^{23} -24.0000 q^{24} -107.888 q^{25} +91.3294 q^{26} -27.0000 q^{27} +19.6991 q^{28} -133.487 q^{29} +24.8200 q^{30} +67.9689 q^{31} +32.0000 q^{32} +179.814 q^{33} -190.660 q^{34} -20.3721 q^{35} +36.0000 q^{36} -309.748 q^{37} +54.6912 q^{38} -136.994 q^{39} -33.0934 q^{40} -124.566 q^{41} -29.5486 q^{42} -22.2477 q^{43} -239.752 q^{44} -37.2300 q^{45} -86.5830 q^{46} -267.186 q^{47} -48.0000 q^{48} -318.747 q^{49} -215.776 q^{50} +285.990 q^{51} +182.659 q^{52} -94.7700 q^{53} -54.0000 q^{54} +247.944 q^{55} +39.3981 q^{56} -82.0368 q^{57} -266.974 q^{58} -59.0000 q^{59} +49.6401 q^{60} -39.2858 q^{61} +135.938 q^{62} +44.3229 q^{63} +64.0000 q^{64} -188.900 q^{65} +359.628 q^{66} +127.319 q^{67} -381.320 q^{68} +129.874 q^{69} -40.7443 q^{70} +536.119 q^{71} +72.0000 q^{72} +472.832 q^{73} -619.496 q^{74} +323.664 q^{75} +109.382 q^{76} -295.181 q^{77} -273.988 q^{78} -504.718 q^{79} -66.1868 q^{80} +81.0000 q^{81} -249.132 q^{82} +77.1909 q^{83} -59.0972 q^{84} +394.349 q^{85} -44.4953 q^{86} +400.461 q^{87} -479.504 q^{88} +460.203 q^{89} -74.4601 q^{90} +224.888 q^{91} -173.166 q^{92} -203.907 q^{93} -534.372 q^{94} -113.120 q^{95} -96.0000 q^{96} +678.352 q^{97} -637.493 q^{98} -539.442 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} - 18 q^{5} - 18 q^{6} + 6 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 - 18 * q^5 - 18 * q^6 + 6 * q^7 + 24 * q^8 + 27 * q^9	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} - 18 q^{5} - 18 q^{6} + 6 q^{7} + 24 q^{8} + 27 q^{9} - 36 q^{10} - 9 q^{11} - 36 q^{12} - 45 q^{13} + 12 q^{14} + 54 q^{15} + 48 q^{16} - 96 q^{17} + 54 q^{18} - 171 q^{19} - 72 q^{20} - 18 q^{21} - 18 q^{22} - 177 q^{23} - 72 q^{24} - 69 q^{25} - 90 q^{26} - 81 q^{27} + 24 q^{28} - 351 q^{29} + 108 q^{30} - 267 q^{31} + 96 q^{32} + 27 q^{33} - 192 q^{34} - 489 q^{35} + 108 q^{36} - 222 q^{37} - 342 q^{38} + 135 q^{39} - 144 q^{40} - 780 q^{41} - 36 q^{42} + 21 q^{43} - 36 q^{44} - 162 q^{45} - 354 q^{46} - 717 q^{47} - 144 q^{48} + 99 q^{49} - 138 q^{50} + 288 q^{51} - 180 q^{52} - 234 q^{53} - 162 q^{54} + 114 q^{55} + 48 q^{56} + 513 q^{57} - 702 q^{58} - 177 q^{59} + 216 q^{60} + 615 q^{61} - 534 q^{62} + 54 q^{63} + 192 q^{64} + 606 q^{65} + 54 q^{66} - 144 q^{67} - 384 q^{68} + 531 q^{69} - 978 q^{70} + 981 q^{71} + 216 q^{72} - 63 q^{73} - 444 q^{74} + 207 q^{75} - 684 q^{76} - 792 q^{77} + 270 q^{78} + 555 q^{79} - 288 q^{80} + 243 q^{81} - 1560 q^{82} + 66 q^{83} - 72 q^{84} - 807 q^{85} + 42 q^{86} + 1053 q^{87} - 72 q^{88} - 1389 q^{89} - 324 q^{90} - 222 q^{91} - 708 q^{92} + 801 q^{93} - 1434 q^{94} + 1587 q^{95} - 288 q^{96} + 960 q^{97} + 198 q^{98} - 81 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 - 18 * q^5 - 18 * q^6 + 6 * q^7 + 24 * q^8 + 27 * q^9 - 36 * q^10 - 9 * q^11 - 36 * q^12 - 45 * q^13 + 12 * q^14 + 54 * q^15 + 48 * q^16 - 96 * q^17 + 54 * q^18 - 171 * q^19 - 72 * q^20 - 18 * q^21 - 18 * q^22 - 177 * q^23 - 72 * q^24 - 69 * q^25 - 90 * q^26 - 81 * q^27 + 24 * q^28 - 351 * q^29 + 108 * q^30 - 267 * q^31 + 96 * q^32 + 27 * q^33 - 192 * q^34 - 489 * q^35 + 108 * q^36 - 222 * q^37 - 342 * q^38 + 135 * q^39 - 144 * q^40 - 780 * q^41 - 36 * q^42 + 21 * q^43 - 36 * q^44 - 162 * q^45 - 354 * q^46 - 717 * q^47 - 144 * q^48 + 99 * q^49 - 138 * q^50 + 288 * q^51 - 180 * q^52 - 234 * q^53 - 162 * q^54 + 114 * q^55 + 48 * q^56 + 513 * q^57 - 702 * q^58 - 177 * q^59 + 216 * q^60 + 615 * q^61 - 534 * q^62 + 54 * q^63 + 192 * q^64 + 606 * q^65 + 54 * q^66 - 144 * q^67 - 384 * q^68 + 531 * q^69 - 978 * q^70 + 981 * q^71 + 216 * q^72 - 63 * q^73 - 444 * q^74 + 207 * q^75 - 684 * q^76 - 792 * q^77 + 270 * q^78 + 555 * q^79 - 288 * q^80 + 243 * q^81 - 1560 * q^82 + 66 * q^83 - 72 * q^84 - 807 * q^85 + 42 * q^86 + 1053 * q^87 - 72 * q^88 - 1389 * q^89 - 324 * q^90 - 222 * q^91 - 708 * q^92 + 801 * q^93 - 1434 * q^94 + 1587 * q^95 - 288 * q^96 + 960 * q^97 + 198 * q^98 - 81 * q^99

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