LMFDB - Embedded newform 354.4.a.h.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 44x^{2} + 19x + 3 $ x^4 - x^3 - 44x^2 + 19x + 3
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			1.4
Root			$6.93810$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +17.3705 q^{5} -6.00000 q^{6} +29.1373 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +17.3705 q^{5} -6.00000 q^{6} +29.1373 q^{7} +8.00000 q^{8} +9.00000 q^{9} +34.7410 q^{10} +34.2581 q^{11} -12.0000 q^{12} -37.9155 q^{13} +58.2746 q^{14} -52.1115 q^{15} +16.0000 q^{16} -84.9002 q^{17} +18.0000 q^{18} -8.58000 q^{19} +69.4820 q^{20} -87.4118 q^{21} +68.5162 q^{22} -103.119 q^{23} -24.0000 q^{24} +176.734 q^{25} -75.8310 q^{26} -27.0000 q^{27} +116.549 q^{28} +169.530 q^{29} -104.223 q^{30} +117.912 q^{31} +32.0000 q^{32} -102.774 q^{33} -169.800 q^{34} +506.129 q^{35} +36.0000 q^{36} -173.663 q^{37} -17.1600 q^{38} +113.746 q^{39} +138.964 q^{40} +346.519 q^{41} -174.824 q^{42} +26.7692 q^{43} +137.032 q^{44} +156.334 q^{45} -206.238 q^{46} -472.136 q^{47} -48.0000 q^{48} +505.981 q^{49} +353.468 q^{50} +254.701 q^{51} -151.662 q^{52} -514.004 q^{53} -54.0000 q^{54} +595.081 q^{55} +233.098 q^{56} +25.7400 q^{57} +339.059 q^{58} +59.0000 q^{59} -208.446 q^{60} +133.802 q^{61} +235.823 q^{62} +262.236 q^{63} +64.0000 q^{64} -658.611 q^{65} -205.549 q^{66} -266.749 q^{67} -339.601 q^{68} +309.357 q^{69} +1012.26 q^{70} +222.827 q^{71} +72.0000 q^{72} +950.821 q^{73} -347.325 q^{74} -530.203 q^{75} -34.3200 q^{76} +998.189 q^{77} +227.493 q^{78} -1216.98 q^{79} +277.928 q^{80} +81.0000 q^{81} +693.037 q^{82} +183.540 q^{83} -349.647 q^{84} -1474.76 q^{85} +53.5385 q^{86} -508.589 q^{87} +274.065 q^{88} +654.925 q^{89} +312.669 q^{90} -1104.75 q^{91} -412.476 q^{92} -353.735 q^{93} -944.272 q^{94} -149.039 q^{95} -96.0000 q^{96} -422.708 q^{97} +1011.96 q^{98} +308.323 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} + 22 q^{5} - 24 q^{6} + 13 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q + 8 * q^2 - 12 * q^3 + 16 * q^4 + 22 * q^5 - 24 * q^6 + 13 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} + 22 q^{5} - 24 q^{6} + 13 q^{7} + 32 q^{8} + 36 q^{9} + 44 q^{10} + 24 q^{11} - 48 q^{12} + 20 q^{13} + 26 q^{14} - 66 q^{15} + 64 q^{16} + 91 q^{17} + 72 q^{18} + 141 q^{19} + 88 q^{20} - 39 q^{21} + 48 q^{22} + 13 q^{23} - 96 q^{24} + 278 q^{25} + 40 q^{26} - 108 q^{27} + 52 q^{28} + 295 q^{29} - 132 q^{30} + 311 q^{31} + 128 q^{32} - 72 q^{33} + 182 q^{34} + 551 q^{35} + 144 q^{36} + 609 q^{37} + 282 q^{38} - 60 q^{39} + 176 q^{40} + 677 q^{41} - 78 q^{42} + 170 q^{43} + 96 q^{44} + 198 q^{45} + 26 q^{46} + 17 q^{47} - 192 q^{48} + 651 q^{49} + 556 q^{50} - 273 q^{51} + 80 q^{52} + 166 q^{53} - 216 q^{54} + 108 q^{55} + 104 q^{56} - 423 q^{57} + 590 q^{58} + 236 q^{59} - 264 q^{60} + 651 q^{61} + 622 q^{62} + 117 q^{63} + 256 q^{64} + 700 q^{65} - 144 q^{66} - 894 q^{67} + 364 q^{68} - 39 q^{69} + 1102 q^{70} + 298 q^{71} + 288 q^{72} + 887 q^{73} + 1218 q^{74} - 834 q^{75} + 564 q^{76} - 79 q^{77} - 120 q^{78} - 784 q^{79} + 352 q^{80} + 324 q^{81} + 1354 q^{82} + 971 q^{83} - 156 q^{84} - 799 q^{85} + 340 q^{86} - 885 q^{87} + 192 q^{88} + 1321 q^{89} + 396 q^{90} - 2673 q^{91} + 52 q^{92} - 933 q^{93} + 34 q^{94} - 3133 q^{95} - 384 q^{96} - 1922 q^{97} + 1302 q^{98} + 216 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 - 12 * q^3 + 16 * q^4 + 22 * q^5 - 24 * q^6 + 13 * q^7 + 32 * q^8 + 36 * q^9 + 44 * q^10 + 24 * q^11 - 48 * q^12 + 20 * q^13 + 26 * q^14 - 66 * q^15 + 64 * q^16 + 91 * q^17 + 72 * q^18 + 141 * q^19 + 88 * q^20 - 39 * q^21 + 48 * q^22 + 13 * q^23 - 96 * q^24 + 278 * q^25 + 40 * q^26 - 108 * q^27 + 52 * q^28 + 295 * q^29 - 132 * q^30 + 311 * q^31 + 128 * q^32 - 72 * q^33 + 182 * q^34 + 551 * q^35 + 144 * q^36 + 609 * q^37 + 282 * q^38 - 60 * q^39 + 176 * q^40 + 677 * q^41 - 78 * q^42 + 170 * q^43 + 96 * q^44 + 198 * q^45 + 26 * q^46 + 17 * q^47 - 192 * q^48 + 651 * q^49 + 556 * q^50 - 273 * q^51 + 80 * q^52 + 166 * q^53 - 216 * q^54 + 108 * q^55 + 104 * q^56 - 423 * q^57 + 590 * q^58 + 236 * q^59 - 264 * q^60 + 651 * q^61 + 622 * q^62 + 117 * q^63 + 256 * q^64 + 700 * q^65 - 144 * q^66 - 894 * q^67 + 364 * q^68 - 39 * q^69 + 1102 * q^70 + 298 * q^71 + 288 * q^72 + 887 * q^73 + 1218 * q^74 - 834 * q^75 + 564 * q^76 - 79 * q^77 - 120 * q^78 - 784 * q^79 + 352 * q^80 + 324 * q^81 + 1354 * q^82 + 971 * q^83 - 156 * q^84 - 799 * q^85 + 340 * q^86 - 885 * q^87 + 192 * q^88 + 1321 * q^89 + 396 * q^90 - 2673 * q^91 + 52 * q^92 - 933 * q^93 + 34 * q^94 - 3133 * q^95 - 384 * q^96 - 1922 * q^97 + 1302 * q^98 + 216 * 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
$5$	17.3705	1.55366	0.776832	−	0.629708i	$-0.216825\pi$
$5$	17.3705	1.55366	0.776832	+	0.629708i	$0.216825\pi$
$6$	−6.00000	−0.408248
$7$	29.1373	1.57327	0.786633	−	0.617421i	$-0.211823\pi$
$7$	29.1373	1.57327	0.786633	+	0.617421i	$0.211823\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	34.7410	1.09861
$11$	34.2581	0.939019	0.469510	−	0.882927i	$-0.344431\pi$
$11$	34.2581	0.939019	0.469510	+	0.882927i	$0.344431\pi$
$12$	−12.0000	−0.288675
$13$	−37.9155	−0.808913	−0.404456	−	0.914557i	$-0.632539\pi$
$13$	−37.9155	−0.808913	−0.404456	+	0.914557i	$0.632539\pi$
$14$	58.2746	1.11247
$15$	−52.1115	−0.897009
$16$	16.0000	0.250000
$17$	−84.9002	−1.21125	−0.605627	−	0.795748i	$-0.707078\pi$
$17$	−84.9002	−1.21125	−0.605627	+	0.795748i	$0.707078\pi$
$18$	18.0000	0.235702
$19$	−8.58000	−0.103599	−0.0517997	−	0.998657i	$-0.516496\pi$
$19$	−8.58000	−0.103599	−0.0517997	+	0.998657i	$0.516496\pi$
$20$	69.4820	0.776832
$21$	−87.4118	−0.908325
$22$	68.5162	0.663987
$23$	−103.119	−0.934862	−0.467431	−	0.884030i	$-0.654820\pi$
$23$	−103.119	−0.934862	−0.467431	+	0.884030i	$0.654820\pi$
$24$	−24.0000	−0.204124
$25$	176.734	1.41387
$26$	−75.8310	−0.571988
$27$	−27.0000	−0.192450
$28$	116.549	0.786633
$29$	169.530	1.08555	0.542774	−	0.839879i	$-0.317374\pi$
$29$	169.530	1.08555	0.542774	+	0.839879i	$0.317374\pi$
$30$	−104.223	−0.634281
$31$	117.912	0.683147	0.341574	−	0.939855i	$-0.389040\pi$
$31$	117.912	0.683147	0.341574	+	0.939855i	$0.389040\pi$
$32$	32.0000	0.176777
$33$	−102.774	−0.542143
$34$	−169.800	−0.856486
$35$	506.129	2.44433
$36$	36.0000	0.166667
$37$	−173.663	−0.771621	−0.385811	−	0.922578i	$-0.626078\pi$
$37$	−173.663	−0.771621	−0.385811	+	0.922578i	$0.626078\pi$
$38$	−17.1600	−0.0732558
$39$	113.746	0.467026
$40$	138.964	0.549303
$41$	346.519	1.31993	0.659965	−	0.751296i	$-0.270571\pi$
$41$	346.519	1.31993	0.659965	+	0.751296i	$0.270571\pi$
$42$	−174.824	−0.642283
$43$	26.7692	0.0949365	0.0474683	−	0.998873i	$-0.484885\pi$
$43$	26.7692	0.0949365	0.0474683	+	0.998873i	$0.484885\pi$
$44$	137.032	0.469510
$45$	156.334	0.517888
$46$	−206.238	−0.661047
$47$	−472.136	−1.46528	−0.732639	−	0.680617i	$-0.761712\pi$
$47$	−472.136	−1.46528	−0.732639	+	0.680617i	$0.761712\pi$
$48$	−48.0000	−0.144338
$49$	505.981	1.47516
$50$	353.468	0.999760
$51$	254.701	0.699318
$52$	−151.662	−0.404456
$53$	−514.004	−1.33215	−0.666074	−	0.745885i	$-0.732027\pi$
$53$	−514.004	−1.33215	−0.666074	+	0.745885i	$0.732027\pi$
$54$	−54.0000	−0.136083
$55$	595.081	1.45892
$56$	233.098	0.556233
$57$	25.7400	0.0598131
$58$	339.059	0.767598
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	−208.446	−0.448504
$61$	133.802	0.280846	0.140423	−	0.990092i	$-0.455154\pi$
$61$	133.802	0.280846	0.140423	+	0.990092i	$0.455154\pi$
$62$	235.823	0.483058
$63$	262.236	0.524422
$64$	64.0000	0.125000
$65$	−658.611	−1.25678
$66$	−205.549	−0.383353
$67$	−266.749	−0.486397	−0.243198	−	0.969977i	$-0.578197\pi$
$67$	−266.749	−0.486397	−0.243198	+	0.969977i	$0.578197\pi$
$68$	−339.601	−0.605627
$69$	309.357	0.539743
$70$	1012.26	1.72840
$71$	222.827	0.372461	0.186230	−	0.982506i	$-0.440373\pi$
$71$	222.827	0.372461	0.186230	+	0.982506i	$0.440373\pi$
$72$	72.0000	0.117851
$73$	950.821	1.52445	0.762227	−	0.647310i	$-0.224106\pi$
$73$	950.821	1.52445	0.762227	+	0.647310i	$0.224106\pi$
$74$	−347.325	−0.545618
$75$	−530.203	−0.816300
$76$	−34.3200	−0.0517997
$77$	998.189	1.47733
$78$	227.493	0.330237
$79$	−1216.98	−1.73317	−0.866586	−	0.499028i	$-0.833690\pi$
$79$	−1216.98	−1.73317	−0.866586	+	0.499028i	$0.833690\pi$
$80$	277.928	0.388416
$81$	81.0000	0.111111
$82$	693.037	0.933332
$83$	183.540	0.242724	0.121362	−	0.992608i	$-0.461274\pi$
$83$	183.540	0.242724	0.121362	+	0.992608i	$0.461274\pi$
$84$	−349.647	−0.454163
$85$	−1474.76	−1.88188
$86$	53.5385	0.0671303
$87$	−508.589	−0.626741
$88$	274.065	0.331993
$89$	654.925	0.780022	0.390011	−	0.920810i	$-0.372471\pi$
$89$	654.925	0.780022	0.390011	+	0.920810i	$0.372471\pi$
$90$	312.669	0.366202
$91$	−1104.75	−1.27263
$92$	−412.476	−0.467431
$93$	−353.735	−0.394415
$94$	−944.272	−1.03611
$95$	−149.039	−0.160959
$96$	−96.0000	−0.102062
$97$	−422.708	−0.442469	−0.221235	−	0.975221i	$-0.571009\pi$
$97$	−422.708	−0.442469	−0.221235	+	0.975221i	$0.571009\pi$
$98$	1011.96	1.04310
$99$	308.323	0.313006
$100$	706.937	0.706937
$101$	−149.751	−0.147532	−0.0737662	−	0.997276i	$-0.523502\pi$
$101$	−149.751	−0.147532	−0.0737662	+	0.997276i	$0.523502\pi$
$102$	509.401	0.494493
$103$	−596.202	−0.570345	−0.285173	−	0.958476i	$-0.592051\pi$
$103$	−596.202	−0.570345	−0.285173	+	0.958476i	$0.592051\pi$
$104$	−303.324	−0.285994
$105$	−1518.39	−1.41123
$106$	−1028.01	−0.941971
$107$	−2113.18	−1.90924	−0.954622	−	0.297820i	$-0.903740\pi$
$107$	−2113.18	−1.90924	−0.954622	+	0.297820i	$0.903740\pi$
$108$	−108.000	−0.0962250
$109$	989.673	0.869665	0.434833	−	0.900511i	$-0.356808\pi$
$109$	989.673	0.869665	0.434833	+	0.900511i	$0.356808\pi$
$110$	1190.16	1.03161
$111$	520.988	0.445496
$112$	466.197	0.393316
$113$	1390.37	1.15748	0.578738	−	0.815514i	$-0.303545\pi$
$113$	1390.37	1.15748	0.578738	+	0.815514i	$0.303545\pi$
$114$	51.4800	0.0422942
$115$	−1791.23	−1.45246
$116$	678.119	0.542774
$117$	−341.239	−0.269638
$118$	118.000	0.0920575
$119$	−2473.76	−1.90563
$120$	−416.892	−0.317140
$121$	−157.381	−0.118243
$122$	267.604	0.198588
$123$	−1039.56	−0.762062
$124$	471.647	0.341574
$125$	898.649	0.643021
$126$	524.471	0.370822
$127$	−703.565	−0.491585	−0.245792	−	0.969322i	$-0.579048\pi$
$127$	−703.565	−0.491585	−0.245792	+	0.969322i	$0.579048\pi$
$128$	128.000	0.0883883
$129$	−80.3077	−0.0548116
$130$	−1317.22	−0.888677
$131$	2588.74	1.72656	0.863281	−	0.504723i	$-0.168405\pi$
$131$	2588.74	1.72656	0.863281	+	0.504723i	$0.168405\pi$
$132$	−411.097	−0.271072
$133$	−249.998	−0.162989
$134$	−533.498	−0.343934
$135$	−469.003	−0.299003
$136$	−679.202	−0.428243
$137$	−1459.48	−0.910161	−0.455080	−	0.890450i	$-0.650390\pi$
$137$	−1459.48	−0.910161	−0.455080	+	0.890450i	$0.650390\pi$
$138$	618.714	0.381656
$139$	−2841.33	−1.73380	−0.866902	−	0.498479i	$-0.833892\pi$
$139$	−2841.33	−1.73380	−0.866902	+	0.498479i	$0.833892\pi$
$140$	2024.52	1.22216
$141$	1416.41	0.845979
$142$	445.655	0.263370
$143$	−1298.91	−0.759584
$144$	144.000	0.0833333
$145$	2944.82	1.68658
$146$	1901.64	1.07795
$147$	−1517.94	−0.851686
$148$	−694.651	−0.385811
$149$	1815.14	0.998003	0.499001	−	0.866601i	$-0.333700\pi$
$149$	1815.14	0.998003	0.499001	+	0.866601i	$0.333700\pi$
$150$	−1060.41	−0.577212
$151$	−242.349	−0.130610	−0.0653049	−	0.997865i	$-0.520802\pi$
$151$	−242.349	−0.130610	−0.0653049	+	0.997865i	$0.520802\pi$
$152$	−68.6400	−0.0366279
$153$	−764.102	−0.403752
$154$	1996.38	1.04463
$155$	2048.19	1.06138
$156$	454.986	0.233513
$157$	387.440	0.196950	0.0984748	−	0.995140i	$-0.468604\pi$
$157$	387.440	0.196950	0.0984748	+	0.995140i	$0.468604\pi$
$158$	−2433.95	−1.22554
$159$	1542.01	0.769116
$160$	555.856	0.274652
$161$	−3004.61	−1.47079
$162$	162.000	0.0785674
$163$	−2658.41	−1.27744	−0.638721	−	0.769438i	$-0.720536\pi$
$163$	−2658.41	−1.27744	−0.638721	+	0.769438i	$0.720536\pi$
$164$	1386.07	0.659965
$165$	−1785.24	−0.842309
$166$	367.079	0.171632
$167$	2489.17	1.15340	0.576700	−	0.816956i	$-0.304340\pi$
$167$	2489.17	1.15340	0.576700	+	0.816956i	$0.304340\pi$
$168$	−699.295	−0.321141
$169$	−759.416	−0.345661
$170$	−2949.52	−1.33069
$171$	−77.2200	−0.0345331
$172$	107.077	0.0474683
$173$	3249.10	1.42789	0.713945	−	0.700202i	$-0.246907\pi$
$173$	3249.10	1.42789	0.713945	+	0.700202i	$0.246907\pi$
$174$	−1017.18	−0.443173
$175$	5149.55	2.22440
$176$	548.130	0.234755
$177$	−177.000	−0.0751646
$178$	1309.85	0.551559
$179$	2940.15	1.22769	0.613847	−	0.789425i	$-0.289621\pi$
$179$	2940.15	1.22769	0.613847	+	0.789425i	$0.289621\pi$
$180$	625.338	0.258944
$181$	−1789.70	−0.734958	−0.367479	−	0.930032i	$-0.619779\pi$
$181$	−1789.70	−0.734958	−0.367479	+	0.930032i	$0.619779\pi$
$182$	−2209.51	−0.899888
$183$	−401.406	−0.162146
$184$	−824.953	−0.330523
$185$	−3016.61	−1.19884
$186$	−707.470	−0.278894
$187$	−2908.52	−1.13739
$188$	−1888.54	−0.732639
$189$	−786.707	−0.302775
$190$	−298.078	−0.113815
$191$	−534.945	−0.202656	−0.101328	−	0.994853i	$-0.532309\pi$
$191$	−534.945	−0.202656	−0.101328	+	0.994853i	$0.532309\pi$
$192$	−192.000	−0.0721688
$193$	2365.19	0.882124	0.441062	−	0.897477i	$-0.354602\pi$
$193$	2365.19	0.882124	0.441062	+	0.897477i	$0.354602\pi$
$194$	−845.416	−0.312873
$195$	1975.83	0.725602
$196$	2023.92	0.737582
$197$	−2812.15	−1.01704	−0.508522	−	0.861049i	$-0.669808\pi$
$197$	−2812.15	−1.01704	−0.508522	+	0.861049i	$0.669808\pi$
$198$	616.646	0.221329
$199$	−2808.88	−1.00059	−0.500293	−	0.865856i	$-0.666774\pi$
$199$	−2808.88	−1.00059	−0.500293	+	0.865856i	$0.666774\pi$
$200$	1413.87	0.499880
$201$	800.247	0.280821
$202$	−299.502	−0.104321
$203$	4939.64	1.70785
$204$	1018.80	0.349659
$205$	6019.20	2.05073
$206$	−1192.40	−0.403295
$207$	−928.072	−0.311621
$208$	−606.648	−0.202228
$209$	−293.935	−0.0972818
$210$	−3036.77	−0.997892
$211$	−630.688	−0.205774	−0.102887	−	0.994693i	$-0.532808\pi$
$211$	−630.688	−0.205774	−0.102887	+	0.994693i	$0.532808\pi$
$212$	−2056.02	−0.666074
$213$	−668.482	−0.215040
$214$	−4226.37	−1.35004
$215$	464.995	0.147500
$216$	−216.000	−0.0680414
$217$	3435.63	1.07477
$218$	1979.35	0.614946
$219$	−2852.46	−0.880144
$220$	2380.32	0.729461
$221$	3219.03	0.979799
$222$	1041.98	0.315013
$223$	−6344.93	−1.90533	−0.952664	−	0.304026i	$-0.901669\pi$
$223$	−6344.93	−1.90533	−0.952664	+	0.304026i	$0.901669\pi$
$224$	932.393	0.278117
$225$	1590.61	0.471291
$226$	2780.73	0.818459
$227$	−3146.10	−0.919885	−0.459942	−	0.887949i	$-0.652130\pi$
$227$	−3146.10	−0.919885	−0.459942	+	0.887949i	$0.652130\pi$
$228$	102.960	0.0299065
$229$	−5682.84	−1.63988	−0.819940	−	0.572450i	$-0.805993\pi$
$229$	−5682.84	−1.63988	−0.819940	+	0.572450i	$0.805993\pi$
$230$	−3582.46	−1.02705
$231$	−2994.57	−0.852935
$232$	1356.24	0.383799
$233$	4997.43	1.40512	0.702560	−	0.711625i	$-0.252040\pi$
$233$	4997.43	1.40512	0.702560	+	0.711625i	$0.252040\pi$
$234$	−682.479	−0.190663
$235$	−8201.23	−2.27655
$236$	236.000	0.0650945
$237$	3650.93	1.00065
$238$	−4947.52	−1.34748
$239$	−5519.90	−1.49394	−0.746972	−	0.664856i	$-0.768493\pi$
$239$	−5519.90	−1.49394	−0.746972	+	0.664856i	$0.768493\pi$
$240$	−833.784	−0.224252
$241$	−4940.83	−1.32061	−0.660304	−	0.750998i	$-0.729573\pi$
$241$	−4940.83	−1.32061	−0.660304	+	0.750998i	$0.729573\pi$
$242$	−314.762	−0.0836102
$243$	−243.000	−0.0641500
$244$	535.208	0.140423
$245$	8789.15	2.29191
$246$	−2079.11	−0.538859
$247$	325.315	0.0838028
$248$	943.294	0.241529
$249$	−550.619	−0.140137
$250$	1797.30	0.454684
$251$	2797.43	0.703474	0.351737	−	0.936099i	$-0.385591\pi$
$251$	2797.43	0.703474	0.351737	+	0.936099i	$0.385591\pi$
$252$	1048.94	0.262211
$253$	−3532.67	−0.877853
$254$	−1407.13	−0.347603
$255$	4424.28	1.08651
$256$	256.000	0.0625000
$257$	−3940.27	−0.956370	−0.478185	−	0.878259i	$-0.658705\pi$
$257$	−3940.27	−0.956370	−0.478185	+	0.878259i	$0.658705\pi$
$258$	−160.615	−0.0387577
$259$	−5060.06	−1.21396
$260$	−2634.44	−0.628389
$261$	1525.77	0.361849
$262$	5177.49	1.22086
$263$	6130.23	1.43728	0.718642	−	0.695380i	$-0.244764\pi$
$263$	6130.23	1.43728	0.718642	+	0.695380i	$0.244764\pi$
$264$	−822.195	−0.191677
$265$	−8928.50	−2.06971
$266$	−499.996	−0.115251
$267$	−1964.78	−0.450346
$268$	−1067.00	−0.243198
$269$	−2577.21	−0.584146	−0.292073	−	0.956396i	$-0.594345\pi$
$269$	−2577.21	−0.584146	−0.292073	+	0.956396i	$0.594345\pi$
$270$	−938.007	−0.211427
$271$	632.343	0.141742	0.0708710	−	0.997485i	$-0.477422\pi$
$271$	632.343	0.141742	0.0708710	+	0.997485i	$0.477422\pi$
$272$	−1358.40	−0.302814
$273$	3314.26	0.734756
$274$	−2918.97	−0.643581
$275$	6054.58	1.32765
$276$	1237.43	0.269871
$277$	4444.11	0.963973	0.481986	−	0.876179i	$-0.339916\pi$
$277$	4444.11	0.963973	0.481986	+	0.876179i	$0.339916\pi$
$278$	−5682.67	−1.22598
$279$	1061.21	0.227716
$280$	4049.03	0.864200
$281$	−702.072	−0.149047	−0.0745233	−	0.997219i	$-0.523744\pi$
$281$	−702.072	−0.149047	−0.0745233	+	0.997219i	$0.523744\pi$
$282$	2832.81	0.598197
$283$	8489.01	1.78311	0.891553	−	0.452916i	$-0.149616\pi$
$283$	8489.01	1.78311	0.891553	+	0.452916i	$0.149616\pi$
$284$	891.309	0.186230
$285$	447.117	0.0929295
$286$	−2597.83	−0.537107
$287$	10096.6	2.07660
$288$	288.000	0.0589256
$289$	2295.05	0.467138
$290$	5889.63	1.19259
$291$	1268.12	0.255460
$292$	3803.28	0.762227
$293$	5446.22	1.08591	0.542955	−	0.839762i	$-0.317306\pi$
$293$	5446.22	1.08591	0.542955	+	0.839762i	$0.317306\pi$
$294$	−3035.89	−0.602233
$295$	1024.86	0.202270
$296$	−1389.30	−0.272809
$297$	−924.969	−0.180714
$298$	3630.29	0.705695
$299$	3909.81	0.756221
$300$	−2120.81	−0.408150
$301$	779.983	0.149360
$302$	−484.698	−0.0923551
$303$	449.253	0.0851779
$304$	−137.280	−0.0258998
$305$	2324.21	0.436340
$306$	−1528.20	−0.285495
$307$	−3753.63	−0.697822	−0.348911	−	0.937156i	$-0.613448\pi$
$307$	−3753.63	−0.697822	−0.348911	+	0.937156i	$0.613448\pi$
$308$	3992.75	0.738663
$309$	1788.61	0.329289
$310$	4096.37	0.750510
$311$	9088.46	1.65710	0.828552	−	0.559912i	$-0.189165\pi$
$311$	9088.46	1.65710	0.828552	+	0.559912i	$0.189165\pi$
$312$	909.972	0.165119
$313$	8278.14	1.49491	0.747457	−	0.664310i	$-0.231275\pi$
$313$	8278.14	1.49491	0.747457	+	0.664310i	$0.231275\pi$
$314$	774.881	0.139264
$315$	4555.16	0.814776
$316$	−4867.91	−0.866586
$317$	−8949.11	−1.58559	−0.792795	−	0.609488i	$-0.791375\pi$
$317$	−8949.11	−1.58559	−0.792795	+	0.609488i	$0.791375\pi$
$318$	3084.02	0.543847
$319$	5807.77	1.01935
$320$	1111.71	0.194208
$321$	6339.55	1.10230
$322$	−6009.22	−1.04000
$323$	728.444	0.125485
$324$	324.000	0.0555556
$325$	−6700.96	−1.14370
$326$	−5316.83	−0.903288
$327$	−2969.02	−0.502101
$328$	2772.15	0.466666
$329$	−13756.8	−2.30527
$330$	−3570.48	−0.595602
$331$	3316.62	0.550749	0.275374	−	0.961337i	$-0.411198\pi$
$331$	3316.62	0.550749	0.275374	+	0.961337i	$0.411198\pi$
$332$	734.158	0.121362
$333$	−1562.96	−0.257207
$334$	4978.34	0.815577
$335$	−4633.56	−0.755697
$336$	−1398.59	−0.227081
$337$	−7369.02	−1.19115	−0.595573	−	0.803301i	$-0.703075\pi$
$337$	−7369.02	−1.19115	−0.595573	+	0.803301i	$0.703075\pi$
$338$	−1518.83	−0.244419
$339$	−4171.10	−0.668269
$340$	−5899.04	−0.940942
$341$	4039.43	0.641489
$342$	−154.440	−0.0244186
$343$	4748.83	0.747559
$344$	214.154	0.0335651
$345$	5373.69	0.838579
$346$	6498.21	1.00967
$347$	−4795.39	−0.741873	−0.370936	−	0.928658i	$-0.620963\pi$
$347$	−4795.39	−0.741873	−0.370936	+	0.928658i	$0.620963\pi$
$348$	−2034.36	−0.313371
$349$	4348.08	0.666897	0.333449	−	0.942768i	$-0.391788\pi$
$349$	4348.08	0.666897	0.333449	+	0.942768i	$0.391788\pi$
$350$	10299.1	1.57289
$351$	1023.72	0.155675
$352$	1096.26	0.165997
$353$	−5661.33	−0.853605	−0.426802	−	0.904345i	$-0.640360\pi$
$353$	−5661.33	−0.853605	−0.426802	+	0.904345i	$0.640360\pi$
$354$	−354.000	−0.0531494
$355$	3870.62	0.578679
$356$	2619.70	0.390011
$357$	7421.29	1.10021
$358$	5880.30	0.868110
$359$	4745.70	0.697683	0.348842	−	0.937182i	$-0.386575\pi$
$359$	4745.70	0.697683	0.348842	+	0.937182i	$0.386575\pi$
$360$	1250.68	0.183101
$361$	−6785.38	−0.989267
$362$	−3579.40	−0.519693
$363$	472.143	0.0682674
$364$	−4419.02	−0.636317
$365$	16516.2	2.36849
$366$	−802.811	−0.114655
$367$	−6047.76	−0.860192	−0.430096	−	0.902783i	$-0.641520\pi$
$367$	−6047.76	−0.860192	−0.430096	+	0.902783i	$0.641520\pi$
$368$	−1649.91	−0.233715
$369$	3118.67	0.439977
$370$	−6033.22	−0.847708
$371$	−14976.7	−2.09582
$372$	−1414.94	−0.197208
$373$	−1108.71	−0.153906	−0.0769529	−	0.997035i	$-0.524519\pi$
$373$	−1108.71	−0.153906	−0.0769529	+	0.997035i	$0.524519\pi$
$374$	−5817.05	−0.804257
$375$	−2695.95	−0.371248
$376$	−3777.09	−0.518054
$377$	−6427.80	−0.878113
$378$	−1573.41	−0.214094
$379$	−7872.26	−1.06694	−0.533471	−	0.845818i	$-0.679113\pi$
$379$	−7872.26	−1.06694	−0.533471	+	0.845818i	$0.679113\pi$
$380$	−596.156	−0.0804793
$381$	2110.69	0.283817
$382$	−1069.89	−0.143299
$383$	9968.18	1.32990	0.664948	−	0.746890i	$-0.268454\pi$
$383$	9968.18	1.32990	0.664948	+	0.746890i	$0.268454\pi$
$384$	−384.000	−0.0510310
$385$	17339.0	2.29527
$386$	4730.37	0.623756
$387$	240.923	0.0316455
$388$	−1690.83	−0.221235
$389$	14035.8	1.82942	0.914708	−	0.404115i	$-0.132421\pi$
$389$	14035.8	1.82942	0.914708	+	0.404115i	$0.132421\pi$
$390$	3951.66	0.513078
$391$	8754.83	1.13236
$392$	4047.85	0.521549
$393$	−7766.23	−0.996831
$394$	−5624.31	−0.719158
$395$	−21139.5	−2.69277
$396$	1233.29	0.156503
$397$	13638.0	1.72411	0.862056	−	0.506813i	$-0.169176\pi$
$397$	13638.0	1.72411	0.862056	+	0.506813i	$0.169176\pi$
$398$	−5617.77	−0.707520
$399$	749.994	0.0941019
$400$	2827.75	0.353468
$401$	−4100.61	−0.510660	−0.255330	−	0.966854i	$-0.582184\pi$
$401$	−4100.61	−0.510660	−0.255330	+	0.966854i	$0.582184\pi$
$402$	1600.49	0.198571
$403$	−4470.68	−0.552606
$404$	−599.004	−0.0737662
$405$	1407.01	0.172629
$406$	9879.27	1.20764
$407$	−5949.36	−0.724567
$408$	2037.61	0.247246
$409$	336.584	0.0406920	0.0203460	−	0.999793i	$-0.493523\pi$
$409$	336.584	0.0406920	0.0203460	+	0.999793i	$0.493523\pi$
$410$	12038.4	1.45008
$411$	4378.45	0.525482
$412$	−2384.81	−0.285173
$413$	1719.10	0.204822
$414$	−1856.14	−0.220349
$415$	3188.17	0.377112
$416$	−1213.30	−0.142997
$417$	8524.00	1.00101
$418$	−587.869	−0.0687886
$419$	13335.7	1.55488	0.777439	−	0.628959i	$-0.216519\pi$
$419$	13335.7	1.55488	0.777439	+	0.628959i	$0.216519\pi$
$420$	−6073.55	−0.705616
$421$	2903.76	0.336154	0.168077	−	0.985774i	$-0.446244\pi$
$421$	2903.76	0.336154	0.168077	+	0.985774i	$0.446244\pi$
$422$	−1261.38	−0.145504
$423$	−4249.22	−0.488426
$424$	−4112.03	−0.470986
$425$	−15004.8	−1.71256
$426$	−1336.96	−0.152057
$427$	3898.62	0.441845
$428$	−8452.73	−0.954622
$429$	3896.74	0.438546
$430$	929.990	0.104298
$431$	−9141.46	−1.02164	−0.510822	−	0.859686i	$-0.670659\pi$
$431$	−9141.46	−1.02164	−0.510822	+	0.859686i	$0.670659\pi$
$432$	−432.000	−0.0481125
$433$	1104.62	0.122597	0.0612985	−	0.998119i	$-0.480476\pi$
$433$	1104.62	0.122597	0.0612985	+	0.998119i	$0.480476\pi$
$434$	6871.25	0.759979
$435$	−8834.45	−0.973746
$436$	3958.69	0.434833
$437$	884.762	0.0968510
$438$	−5704.92	−0.622356
$439$	5075.02	0.551748	0.275874	−	0.961194i	$-0.411033\pi$
$439$	5075.02	0.551748	0.275874	+	0.961194i	$0.411033\pi$
$440$	4760.65	0.515807
$441$	4553.83	0.491721
$442$	6438.07	0.692823
$443$	7830.48	0.839814	0.419907	−	0.907567i	$-0.362063\pi$
$443$	7830.48	0.839814	0.419907	+	0.907567i	$0.362063\pi$
$444$	2083.95	0.222748
$445$	11376.4	1.21189
$446$	−12689.9	−1.34727
$447$	−5445.43	−0.576197
$448$	1864.79	0.196658
$449$	−14983.6	−1.57488	−0.787438	−	0.616393i	$-0.788593\pi$
$449$	−14983.6	−1.57488	−0.787438	+	0.616393i	$0.788593\pi$
$450$	3181.22	0.333253
$451$	11871.1	1.23944
$452$	5561.47	0.578738
$453$	727.047	0.0754076
$454$	−6292.19	−0.650457
$455$	−19190.1	−1.97725
$456$	205.920	0.0211471
$457$	6129.96	0.627456	0.313728	−	0.949513i	$-0.398422\pi$
$457$	6129.96	0.627456	0.313728	+	0.949513i	$0.398422\pi$
$458$	−11365.7	−1.15957
$459$	2292.31	0.233106
$460$	−7164.92	−0.726231
$461$	45.9092	0.00463818	0.00231909	−	0.999997i	$-0.499262\pi$
$461$	45.9092	0.00463818	0.00231909	+	0.999997i	$0.499262\pi$
$462$	−5989.13	−0.603116
$463$	12471.8	1.25186	0.625931	−	0.779878i	$-0.284719\pi$
$463$	12471.8	1.25186	0.625931	+	0.779878i	$0.284719\pi$
$464$	2712.48	0.271387
$465$	−6144.56	−0.612789
$466$	9994.87	0.993569
$467$	−16130.4	−1.59835	−0.799173	−	0.601100i	$-0.794729\pi$
$467$	−16130.4	−1.59835	−0.799173	+	0.601100i	$0.794729\pi$
$468$	−1364.96	−0.134819
$469$	−7772.34	−0.765231
$470$	−16402.5	−1.60976
$471$	−1162.32	−0.113709
$472$	472.000	0.0460287
$473$	917.064	0.0891472
$474$	7301.86	0.707564
$475$	−1516.38	−0.146476
$476$	−9895.05	−0.952813
$477$	−4626.04	−0.444049
$478$	−11039.8	−1.05638
$479$	−20631.2	−1.96798	−0.983992	−	0.178214i	$-0.942968\pi$
$479$	−20631.2	−1.96798	−0.983992	+	0.178214i	$0.942968\pi$
$480$	−1667.57	−0.158570
$481$	6584.51	0.624174
$482$	−9881.65	−0.933811
$483$	9013.83	0.849158
$484$	−629.524	−0.0591213
$485$	−7342.65	−0.687449
$486$	−486.000	−0.0453609
$487$	−8705.95	−0.810070	−0.405035	−	0.914301i	$-0.632741\pi$
$487$	−8705.95	−0.810070	−0.405035	+	0.914301i	$0.632741\pi$
$488$	1070.42	0.0992939
$489$	7975.24	0.737532
$490$	17578.3	1.62062
$491$	−12174.3	−1.11898	−0.559490	−	0.828837i	$-0.689003\pi$
$491$	−12174.3	−1.11898	−0.559490	+	0.828837i	$0.689003\pi$
$492$	−4158.22	−0.381031
$493$	−14393.1	−1.31487
$494$	650.630	0.0592575
$495$	5355.73	0.486307
$496$	1886.59	0.170787
$497$	6492.58	0.585980
$498$	−1101.24	−0.0990916
$499$	84.8554	0.00761253	0.00380626	−	0.999993i	$-0.498788\pi$
$499$	84.8554	0.00761253	0.00380626	+	0.999993i	$0.498788\pi$
$500$	3594.60	0.321510
$501$	−7467.51	−0.665916
$502$	5594.85	0.497431
$503$	−10415.0	−0.923220	−0.461610	−	0.887083i	$-0.652728\pi$
$503$	−10415.0	−0.923220	−0.461610	+	0.887083i	$0.652728\pi$
$504$	2097.88	0.185411
$505$	−2601.25	−0.229216
$506$	−7065.33	−0.620736
$507$	2278.25	0.199567
$508$	−2814.26	−0.245792
$509$	14229.7	1.23914	0.619568	−	0.784943i	$-0.287308\pi$
$509$	14229.7	1.23914	0.619568	+	0.784943i	$0.287308\pi$
$510$	8848.56	0.768276
$511$	27704.3	2.39837
$512$	512.000	0.0441942
$513$	231.660	0.0199377
$514$	−7880.54	−0.676256
$515$	−10356.3	−0.886125
$516$	−321.231	−0.0274058
$517$	−16174.5	−1.37592
$518$	−10120.1	−0.858403
$519$	−9747.31	−0.824392
$520$	−5268.89	−0.444338
$521$	7782.48	0.654427	0.327213	−	0.944950i	$-0.393890\pi$
$521$	7782.48	0.654427	0.327213	+	0.944950i	$0.393890\pi$
$522$	3051.54	0.255866
$523$	−8333.56	−0.696752	−0.348376	−	0.937355i	$-0.613267\pi$
$523$	−8333.56	−0.696752	−0.348376	+	0.937355i	$0.613267\pi$
$524$	10355.0	0.863281
$525$	−15448.7	−1.28426
$526$	12260.5	1.01631
$527$	−10010.7	−0.827466
$528$	−1644.39	−0.135536
$529$	−1533.46	−0.126034
$530$	−17857.0	−1.46351
$531$	531.000	0.0433963
$532$	−999.992	−0.0814946
$533$	−13138.4	−1.06771
$534$	−3929.55	−0.318443
$535$	−36707.0	−2.96632
$536$	−2133.99	−0.171967
$537$	−8820.45	−0.708809
$538$	−5154.42	−0.413054
$539$	17334.0	1.38521
$540$	−1876.01	−0.149501
$541$	−21941.3	−1.74368	−0.871840	−	0.489791i	$-0.837073\pi$
$541$	−21941.3	−1.74368	−0.871840	+	0.489791i	$0.837073\pi$
$542$	1264.69	0.100227
$543$	5369.10	0.424328
$544$	−2716.81	−0.214122
$545$	17191.1	1.35117
$546$	6628.52	0.519551
$547$	−5699.52	−0.445510	−0.222755	−	0.974874i	$-0.571505\pi$
$547$	−5699.52	−0.445510	−0.222755	+	0.974874i	$0.571505\pi$
$548$	−5837.93	−0.455080
$549$	1204.22	0.0936152
$550$	12109.2	0.938794
$551$	−1454.57	−0.112462
$552$	2474.86	0.190828
$553$	−35459.4	−2.72674
$554$	8888.21	0.681632
$555$	9049.82	0.692151
$556$	−11365.3	−0.866902
$557$	15408.0	1.17210	0.586048	−	0.810277i	$-0.300683\pi$
$557$	15408.0	1.17210	0.586048	+	0.810277i	$0.300683\pi$
$558$	2122.41	0.161019
$559$	−1014.97	−0.0767953
$560$	8098.07	0.611082
$561$	8725.57	0.656673
$562$	−1404.14	−0.105392
$563$	−3619.15	−0.270922	−0.135461	−	0.990783i	$-0.543252\pi$
$563$	−3619.15	−0.270922	−0.135461	+	0.990783i	$0.543252\pi$
$564$	5665.63	0.422989
$565$	24151.4	1.79833
$566$	16978.0	1.26085
$567$	2360.12	0.174807
$568$	1782.62	0.131685
$569$	1437.31	0.105896	0.0529481	−	0.998597i	$-0.483138\pi$
$569$	1437.31	0.105896	0.0529481	+	0.998597i	$0.483138\pi$
$570$	894.233	0.0657111
$571$	−10974.4	−0.804315	−0.402157	−	0.915571i	$-0.631740\pi$
$571$	−10974.4	−0.804315	−0.402157	+	0.915571i	$0.631740\pi$
$572$	−5195.65	−0.379792
$573$	1604.83	0.117003
$574$	20193.2	1.46838
$575$	−18224.7	−1.32178
$576$	576.000	0.0416667
$577$	20053.9	1.44689	0.723446	−	0.690381i	$-0.242557\pi$
$577$	20053.9	1.44689	0.723446	+	0.690381i	$0.242557\pi$
$578$	4590.10	0.330317
$579$	−7095.56	−0.509294
$580$	11779.3	0.843288
$581$	5347.84	0.381869
$582$	2536.25	0.180637
$583$	−17608.8	−1.25091
$584$	7606.57	0.538976
$585$	−5927.50	−0.418926
$586$	10892.4	0.767854
$587$	−15322.1	−1.07736	−0.538681	−	0.842510i	$-0.681077\pi$
$587$	−15322.1	−1.07736	−0.538681	+	0.842510i	$0.681077\pi$
$588$	−6071.77	−0.425843
$589$	−1011.68	−0.0707736
$590$	2049.72	0.143026
$591$	8436.46	0.587190
$592$	−2778.60	−0.192905
$593$	15281.1	1.05821	0.529105	−	0.848556i	$-0.322528\pi$
$593$	15281.1	1.05821	0.529105	+	0.848556i	$0.322528\pi$
$594$	−1849.94	−0.127784
$595$	−42970.5	−2.96070
$596$	7260.58	0.499001
$597$	8426.65	0.577688
$598$	7819.62	0.534729
$599$	−5682.42	−0.387608	−0.193804	−	0.981040i	$-0.562083\pi$
$599$	−5682.42	−0.387608	−0.193804	+	0.981040i	$0.562083\pi$
$600$	−4241.62	−0.288606
$601$	−1423.57	−0.0966202	−0.0483101	−	0.998832i	$-0.515384\pi$
$601$	−1423.57	−0.0966202	−0.0483101	+	0.998832i	$0.515384\pi$
$602$	1559.97	0.105614
$603$	−2400.74	−0.162132
$604$	−969.396	−0.0653049
$605$	−2733.79	−0.183709
$606$	898.506	0.0602299
$607$	8125.76	0.543352	0.271676	−	0.962389i	$-0.412422\pi$
$607$	8125.76	0.543352	0.271676	+	0.962389i	$0.412422\pi$
$608$	−274.560	−0.0183139
$609$	−14818.9	−0.986030
$610$	4648.41	0.308539
$611$	17901.3	1.18528
$612$	−3056.41	−0.201876
$613$	19879.4	1.30982	0.654910	−	0.755707i	$-0.272706\pi$
$613$	19879.4	1.30982	0.654910	+	0.755707i	$0.272706\pi$
$614$	−7507.27	−0.493434
$615$	−18057.6	−1.18399
$616$	7985.51	0.522314
$617$	18716.4	1.22122	0.610611	−	0.791931i	$-0.290924\pi$
$617$	18716.4	1.22122	0.610611	+	0.791931i	$0.290924\pi$
$618$	3577.21	0.232842
$619$	24093.4	1.56445	0.782225	−	0.622996i	$-0.214085\pi$
$619$	24093.4	1.56445	0.782225	+	0.622996i	$0.214085\pi$
$620$	8192.74	0.530691
$621$	2784.22	0.179914
$622$	18176.9	1.17175
$623$	19082.7	1.22718
$624$	1819.94	0.116756
$625$	−6481.79	−0.414835
$626$	16556.3	1.05706
$627$	881.804	0.0561657
$628$	1549.76	0.0984748
$629$	14744.0	0.934630
$630$	9110.32	0.576133
$631$	22921.0	1.44607	0.723036	−	0.690810i	$-0.242746\pi$
$631$	22921.0	1.44607	0.723036	+	0.690810i	$0.242746\pi$
$632$	−9735.81	−0.612769
$633$	1892.06	0.118804
$634$	−17898.2	−1.12118
$635$	−12221.3	−0.763758
$636$	6168.05	0.384558
$637$	−19184.5	−1.19328
$638$	11615.5	0.720789
$639$	2005.45	0.124154
$640$	2223.42	0.137326
$641$	29002.5	1.78710	0.893550	−	0.448964i	$-0.148207\pi$
$641$	29002.5	1.78710	0.893550	+	0.448964i	$0.148207\pi$
$642$	12679.1	0.779446
$643$	2024.75	0.124181	0.0620904	−	0.998071i	$-0.480223\pi$
$643$	2024.75	0.124181	0.0620904	+	0.998071i	$0.480223\pi$
$644$	−12018.4	−0.735393
$645$	−1394.99	−0.0851589
$646$	1456.89	0.0887314
$647$	8460.57	0.514095	0.257047	−	0.966399i	$-0.417250\pi$
$647$	8460.57	0.514095	0.257047	+	0.966399i	$0.417250\pi$
$648$	648.000	0.0392837
$649$	2021.23	0.122250
$650$	−13401.9	−0.808718
$651$	−10306.9	−0.620520
$652$	−10633.7	−0.638721
$653$	−15136.5	−0.907103	−0.453551	−	0.891230i	$-0.649843\pi$
$653$	−15136.5	−0.907103	−0.453551	+	0.891230i	$0.649843\pi$
$654$	−5938.04	−0.355039
$655$	44967.8	2.68250
$656$	5544.30	0.329983
$657$	8557.39	0.508151
$658$	−27513.5	−1.63007
$659$	7902.37	0.467121	0.233560	−	0.972342i	$-0.424962\pi$
$659$	7902.37	0.467121	0.233560	+	0.972342i	$0.424962\pi$
$660$	−7140.97	−0.421154
$661$	6588.47	0.387688	0.193844	−	0.981032i	$-0.437904\pi$
$661$	6588.47	0.387688	0.193844	+	0.981032i	$0.437904\pi$
$662$	6633.24	0.389438
$663$	−9657.10	−0.565687
$664$	1468.32	0.0858159
$665$	−4342.59	−0.253231
$666$	−3125.93	−0.181873
$667$	−17481.8	−1.01484
$668$	9956.68	0.576700
$669$	19034.8	1.10004
$670$	−9267.13	−0.534359
$671$	4583.80	0.263719
$672$	−2797.18	−0.160571
$673$	−19281.7	−1.10439	−0.552194	−	0.833716i	$-0.686209\pi$
$673$	−19281.7	−1.10439	−0.552194	+	0.833716i	$0.686209\pi$
$674$	−14738.0	−0.842267
$675$	−4771.82	−0.272100
$676$	−3037.66	−0.172830
$677$	−8752.86	−0.496898	−0.248449	−	0.968645i	$-0.579921\pi$
$677$	−8752.86	−0.496898	−0.248449	+	0.968645i	$0.579921\pi$
$678$	−8342.20	−0.472537
$679$	−12316.6	−0.696121
$680$	−11798.1	−0.665346
$681$	9438.29	0.531096
$682$	8078.87	0.453601
$683$	3472.38	0.194534	0.0972671	−	0.995258i	$-0.468990\pi$
$683$	3472.38	0.194534	0.0972671	+	0.995258i	$0.468990\pi$
$684$	−308.880	−0.0172666
$685$	−25351.9	−1.41408
$686$	9497.66	0.528604
$687$	17048.5	0.946785
$688$	428.308	0.0237341
$689$	19488.7	1.07759
$690$	10747.4	0.592965
$691$	13653.8	0.751687	0.375843	−	0.926683i	$-0.377353\pi$
$691$	13653.8	0.751687	0.375843	+	0.926683i	$0.377353\pi$
$692$	12996.4	0.713945
$693$	8983.70	0.492442
$694$	−9590.77	−0.524583
$695$	−49355.4	−2.69375
$696$	−4068.71	−0.221586
$697$	−29419.5	−1.59877
$698$	8696.15	0.471568
$699$	−14992.3	−0.811246
$700$	20598.2	1.11220
$701$	−17065.6	−0.919483	−0.459741	−	0.888053i	$-0.652058\pi$
$701$	−17065.6	−0.919483	−0.459741	+	0.888053i	$0.652058\pi$
$702$	2047.44	0.110079
$703$	1490.03	0.0799394
$704$	2192.52	0.117377
$705$	24603.7	1.31437
$706$	−11322.7	−0.603590
$707$	−4363.34	−0.232108
$708$	−708.000	−0.0375823
$709$	19602.7	1.03836	0.519179	−	0.854666i	$-0.326238\pi$
$709$	19602.7	1.03836	0.519179	+	0.854666i	$0.326238\pi$
$710$	7741.24	0.409188
$711$	−10952.8	−0.577724
$712$	5239.40	0.275779
$713$	−12158.9	−0.638648
$714$	14842.6	0.777968
$715$	−22562.8	−1.18014
$716$	11760.6	0.613847
$717$	16559.7	0.862528
$718$	9491.40	0.493337
$719$	28943.2	1.50125	0.750624	−	0.660729i	$-0.229753\pi$
$719$	28943.2	1.50125	0.750624	+	0.660729i	$0.229753\pi$
$720$	2501.35	0.129472
$721$	−17371.7	−0.897304
$722$	−13570.8	−0.699518
$723$	14822.5	0.762454
$724$	−7158.80	−0.367479
$725$	29961.7	1.53483
$726$	944.286	0.0482724
$727$	−4459.41	−0.227497	−0.113749	−	0.993510i	$-0.536286\pi$
$727$	−4459.41	−0.227497	−0.113749	+	0.993510i	$0.536286\pi$
$728$	−8838.03	−0.449944
$729$	729.000	0.0370370
$730$	33032.5	1.67478
$731$	−2272.71	−0.114992
$732$	−1605.62	−0.0810731
$733$	−2564.77	−0.129239	−0.0646194	−	0.997910i	$-0.520583\pi$
$733$	−2564.77	−0.129239	−0.0646194	+	0.997910i	$0.520583\pi$
$734$	−12095.5	−0.608248
$735$	−26367.4	−1.32323
$736$	−3299.81	−0.165262
$737$	−9138.32	−0.456736
$738$	6237.34	0.311111
$739$	9845.29	0.490074	0.245037	−	0.969514i	$-0.421200\pi$
$739$	9845.29	0.490074	0.245037	+	0.969514i	$0.421200\pi$
$740$	−12066.4	−0.599420
$741$	−975.945	−0.0483836
$742$	−29953.4	−1.48197
$743$	−37299.7	−1.84171	−0.920856	−	0.389903i	$-0.872508\pi$
$743$	−37299.7	−1.84171	−0.920856	+	0.389903i	$0.872508\pi$
$744$	−2829.88	−0.139447
$745$	31530.0	1.55056
$746$	−2217.42	−0.108828
$747$	1651.86	0.0809080
$748$	−11634.1	−0.568696
$749$	−61572.4	−3.00375
$750$	−5391.89	−0.262512
$751$	−43.6748	−0.00212212	−0.00106106	−	0.999999i	$-0.500338\pi$
$751$	−43.6748	−0.00212212	−0.00106106	+	0.999999i	$0.500338\pi$
$752$	−7554.17	−0.366320
$753$	−8392.28	−0.406151
$754$	−12855.6	−0.620920
$755$	−4209.72	−0.202924
$756$	−3146.83	−0.151388
$757$	781.640	0.0375286	0.0187643	−	0.999824i	$-0.494027\pi$
$757$	781.640	0.0375286	0.0187643	+	0.999824i	$0.494027\pi$
$758$	−15744.5	−0.754442
$759$	10598.0	0.506829
$760$	−1192.31	−0.0569075
$761$	26687.0	1.27123	0.635613	−	0.772008i	$-0.280747\pi$
$761$	26687.0	1.27123	0.635613	+	0.772008i	$0.280747\pi$
$762$	4221.39	0.200689
$763$	28836.4	1.36821
$764$	−2139.78	−0.101328
$765$	−13272.8	−0.627295
$766$	19936.4	0.940378
$767$	−2237.01	−0.105311
$768$	−768.000	−0.0360844
$769$	25366.1	1.18950	0.594750	−	0.803910i	$-0.297251\pi$
$769$	25366.1	1.18950	0.594750	+	0.803910i	$0.297251\pi$
$770$	34678.1	1.62300
$771$	11820.8	0.552161
$772$	9460.75	0.441062
$773$	34042.0	1.58397	0.791983	−	0.610543i	$-0.209049\pi$
$773$	34042.0	1.58397	0.791983	+	0.610543i	$0.209049\pi$
$774$	481.846	0.0223768
$775$	20839.0	0.965884
$776$	−3381.67	−0.156436
$777$	15180.2	0.700883
$778$	28071.6	1.29359
$779$	−2973.13	−0.136744
$780$	7903.33	0.362801
$781$	7633.64	0.349748
$782$	17509.7	0.800696
$783$	−4577.30	−0.208914
$784$	8095.70	0.368791
$785$	6730.03	0.305994
$786$	−15532.5	−0.704866
$787$	29184.8	1.32189	0.660944	−	0.750435i	$-0.270156\pi$
$787$	29184.8	1.32189	0.660944	+	0.750435i	$0.270156\pi$
$788$	−11248.6	−0.508522
$789$	−18390.7	−0.829817
$790$	−42279.0	−1.90407
$791$	40511.5	1.82102
$792$	2466.58	0.110664
$793$	−5073.16	−0.227179
$794$	27276.0	1.21913
$795$	26785.5	1.19495
$796$	−11235.5	−0.500293
$797$	34093.0	1.51523	0.757613	−	0.652704i	$-0.226366\pi$
$797$	34093.0	1.51523	0.757613	+	0.652704i	$0.226366\pi$
$798$	1499.99	0.0665401
$799$	40084.4	1.77483
$800$	5655.49	0.249940
$801$	5894.33	0.260007
$802$	−8201.22	−0.361091
$803$	32573.3	1.43149
$804$	3200.99	0.140411
$805$	−52191.6	−2.28511
$806$	−8941.36	−0.390752
$807$	7731.63	0.337257
$808$	−1198.01	−0.0521606
$809$	3117.30	0.135474	0.0677369	−	0.997703i	$-0.478422\pi$
$809$	3117.30	0.135474	0.0677369	+	0.997703i	$0.478422\pi$
$810$	2814.02	0.122067
$811$	−1122.19	−0.0485887	−0.0242944	−	0.999705i	$-0.507734\pi$
$811$	−1122.19	−0.0485887	−0.0242944	+	0.999705i	$0.507734\pi$
$812$	19758.5	0.853927
$813$	−1897.03	−0.0818348
$814$	−11898.7	−0.512346
$815$	−46178.0	−1.98472
$816$	4075.21	0.174830
$817$	−229.680	−0.00983536
$818$	673.168	0.0287736
$819$	−9942.79	−0.424211
$820$	24076.8	1.02536
$821$	29677.1	1.26156	0.630779	−	0.775962i	$-0.282735\pi$
$821$	29677.1	1.26156	0.630779	+	0.775962i	$0.282735\pi$
$822$	8756.90	0.371572
$823$	−27566.6	−1.16757	−0.583786	−	0.811908i	$-0.698429\pi$
$823$	−27566.6	−1.16757	−0.583786	+	0.811908i	$0.698429\pi$
$824$	−4769.62	−0.201647
$825$	−18163.7	−0.766522
$826$	3438.20	0.144831
$827$	17934.9	0.754122	0.377061	−	0.926188i	$-0.376935\pi$
$827$	17934.9	0.754122	0.377061	+	0.926188i	$0.376935\pi$
$828$	−3712.29	−0.155810
$829$	−7735.33	−0.324076	−0.162038	−	0.986785i	$-0.551807\pi$
$829$	−7735.33	−0.324076	−0.162038	+	0.986785i	$0.551807\pi$
$830$	6376.35	0.266658
$831$	−13332.3	−0.556550
$832$	−2426.59	−0.101114
$833$	−42957.9	−1.78680
$834$	17048.0	0.707823
$835$	43238.1	1.79200
$836$	−1175.74	−0.0486409
$837$	−3183.62	−0.131472
$838$	26671.5	1.09946
$839$	−46742.8	−1.92341	−0.961704	−	0.274091i	$-0.911623\pi$
$839$	−46742.8	−1.92341	−0.961704	+	0.274091i	$0.911623\pi$
$840$	−12147.1	−0.498946
$841$	4351.33	0.178414
$842$	5807.53	0.237697
$843$	2106.22	0.0860521
$844$	−2522.75	−0.102887
$845$	−13191.4	−0.537041
$846$	−8498.44	−0.345369
$847$	−4585.65	−0.186027
$848$	−8224.06	−0.333037
$849$	−25467.0	−1.02948
$850$	−30009.6	−1.21096
$851$	17907.9	0.721359
$852$	−2673.93	−0.107520
$853$	−33867.5	−1.35944	−0.679719	−	0.733472i	$-0.737898\pi$
$853$	−33867.5	−1.35944	−0.679719	+	0.733472i	$0.737898\pi$
$854$	7797.25	0.312431
$855$	−1341.35	−0.0536529
$856$	−16905.5	−0.675020
$857$	−51.2174	−0.00204149	−0.00102074	−	0.999999i	$-0.500325\pi$
$857$	−51.2174	−0.00204149	−0.00102074	+	0.999999i	$0.500325\pi$
$858$	7793.48	0.310099
$859$	11206.0	0.445103	0.222551	−	0.974921i	$-0.428561\pi$
$859$	11206.0	0.445103	0.222551	+	0.974921i	$0.428561\pi$
$860$	1859.98	0.0737498
$861$	−30289.8	−1.19893
$862$	−18282.9	−0.722411
$863$	37668.2	1.48579	0.742896	−	0.669406i	$-0.233451\pi$
$863$	37668.2	1.48579	0.742896	+	0.669406i	$0.233451\pi$
$864$	−864.000	−0.0340207
$865$	56438.6	2.21846
$866$	2209.23	0.0866891
$867$	−6885.15	−0.269702
$868$	13742.5	0.537386
$869$	−41691.3	−1.62748
$870$	−17668.9	−0.688542
$871$	10113.9	0.393452
$872$	7917.39	0.307473
$873$	−3804.37	−0.147490
$874$	1769.52	0.0684840
$875$	26184.2	1.01164
$876$	−11409.8	−0.440072
$877$	−20656.9	−0.795363	−0.397681	−	0.917524i	$-0.630185\pi$
$877$	−20656.9	−0.795363	−0.397681	+	0.917524i	$0.630185\pi$
$878$	10150.0	0.390145
$879$	−16338.7	−0.626950
$880$	9521.29	0.364730
$881$	−17503.6	−0.669367	−0.334684	−	0.942331i	$-0.608629\pi$
$881$	−17503.6	−0.669367	−0.334684	+	0.942331i	$0.608629\pi$
$882$	9107.66	0.347699
$883$	35401.3	1.34921	0.674603	−	0.738181i	$-0.264315\pi$
$883$	35401.3	1.34921	0.674603	+	0.738181i	$0.264315\pi$
$884$	12876.1	0.489900
$885$	−3074.58	−0.116781
$886$	15661.0	0.593838
$887$	−17056.3	−0.645654	−0.322827	−	0.946458i	$-0.604633\pi$
$887$	−17056.3	−0.645654	−0.322827	+	0.946458i	$0.604633\pi$
$888$	4167.91	0.157506
$889$	−20500.0	−0.773394
$890$	22752.8	0.856937
$891$	2774.91	0.104335
$892$	−25379.7	−0.952664
$893$	4050.92	0.151802
$894$	−10890.9	−0.407433
$895$	51071.9	1.90742
$896$	3729.57	0.139058
$897$	−11729.4	−0.436604
$898$	−29967.2	−1.11361
$899$	19989.5	0.741589
$900$	6362.43	0.235646
$901$	43639.1	1.61357
$902$	23742.2	0.876416
$903$	−2339.95	−0.0862332
$904$	11122.9	0.409229
$905$	−31088.0	−1.14188
$906$	1454.09	0.0533212
$907$	13277.4	0.486075	0.243037	−	0.970017i	$-0.421856\pi$
$907$	13277.4	0.486075	0.243037	+	0.970017i	$0.421856\pi$
$908$	−12584.4	−0.459942
$909$	−1347.76	−0.0491775
$910$	−38380.3	−1.39812
$911$	−9407.42	−0.342132	−0.171066	−	0.985260i	$-0.554721\pi$
$911$	−9407.42	−0.342132	−0.171066	+	0.985260i	$0.554721\pi$
$912$	411.840	0.0149533
$913$	6287.72	0.227922
$914$	12259.9	0.443679
$915$	−6972.62	−0.251921
$916$	−22731.3	−0.819940
$917$	75429.0	2.71634
$918$	4584.61	0.164831
$919$	−39183.0	−1.40645	−0.703224	−	0.710968i	$-0.748257\pi$
$919$	−39183.0	−1.40645	−0.703224	+	0.710968i	$0.748257\pi$
$920$	−14329.8	−0.513523
$921$	11260.9	0.402887
$922$	91.8183	0.00327969
$923$	−8448.60	−0.301288
$924$	−11978.3	−0.426467
$925$	−30692.1	−1.09097
$926$	24943.5	0.885200
$927$	−5365.82	−0.190115
$928$	5424.95	0.191900
$929$	−13093.0	−0.462396	−0.231198	−	0.972907i	$-0.574265\pi$
$929$	−13093.0	−0.462396	−0.231198	+	0.972907i	$0.574265\pi$
$930$	−12289.1	−0.433307
$931$	−4341.32	−0.152826
$932$	19989.7	0.702560
$933$	−27265.4	−0.956730
$934$	−32260.9	−1.13020
$935$	−50522.5	−1.76713
$936$	−2729.91	−0.0953313
$937$	37433.0	1.30510	0.652552	−	0.757744i	$-0.273699\pi$
$937$	37433.0	1.30510	0.652552	+	0.757744i	$0.273699\pi$
$938$	−15544.7	−0.541100
$939$	−24834.4	−0.863089
$940$	−32804.9	−1.13828
$941$	−22450.6	−0.777756	−0.388878	−	0.921289i	$-0.627137\pi$
$941$	−22450.6	−0.777756	−0.388878	+	0.921289i	$0.627137\pi$
$942$	−2324.64	−0.0804044
$943$	−35732.7	−1.23395
$944$	944.000	0.0325472
$945$	−13665.5	−0.470411
$946$	1834.13	0.0630366
$947$	39917.7	1.36975	0.684875	−	0.728661i	$-0.259857\pi$
$947$	39917.7	1.36975	0.684875	+	0.728661i	$0.259857\pi$
$948$	14603.7	0.500324
$949$	−36050.8	−1.23315
$950$	−3032.76	−0.103574
$951$	26847.3	0.915441
$952$	−19790.1	−0.673740
$953$	41081.5	1.39639	0.698195	−	0.715908i	$-0.253987\pi$
$953$	41081.5	1.39639	0.698195	+	0.715908i	$0.253987\pi$
$954$	−9252.07	−0.313990
$955$	−9292.26	−0.314859
$956$	−22079.6	−0.746972
$957$	−17423.3	−0.588522
$958$	−41262.4	−1.39157
$959$	−42525.4	−1.43192
$960$	−3335.14	−0.112126
$961$	−15887.8	−0.533310
$962$	13169.0	0.441358
$963$	−19018.6	−0.636415
$964$	−19763.3	−0.660304
$965$	41084.5	1.37052
$966$	18027.7	0.600446
$967$	−28323.2	−0.941897	−0.470948	−	0.882161i	$-0.656088\pi$
$967$	−28323.2	−0.941897	−0.470948	+	0.882161i	$0.656088\pi$
$968$	−1259.05	−0.0418051
$969$	−2185.33	−0.0724489
$970$	−14685.3	−0.486100
$971$	32795.1	1.08388	0.541938	−	0.840418i	$-0.317691\pi$
$971$	32795.1	1.08388	0.541938	+	0.840418i	$0.317691\pi$
$972$	−972.000	−0.0320750
$973$	−82788.7	−2.72773
$974$	−17411.9	−0.572806
$975$	20102.9	0.660316
$976$	2140.83	0.0702114
$977$	−12694.6	−0.415699	−0.207849	−	0.978161i	$-0.566646\pi$
$977$	−12694.6	−0.415699	−0.207849	+	0.978161i	$0.566646\pi$
$978$	15950.5	0.521514
$979$	22436.5	0.732456
$980$	35156.6	1.14595
$981$	8907.06	0.289888
$982$	−24348.6	−0.791238
$983$	8087.36	0.262408	0.131204	−	0.991355i	$-0.458116\pi$
$983$	8087.36	0.262408	0.131204	+	0.991355i	$0.458116\pi$
$984$	−8316.45	−0.269430
$985$	−48848.5	−1.58014
$986$	−28786.2	−0.929757
$987$	41270.3	1.33095
$988$	1301.26	0.0419014
$989$	−2760.42	−0.0887525
$990$	10711.5	0.343871
$991$	−25723.9	−0.824566	−0.412283	−	0.911056i	$-0.635269\pi$
$991$	−25723.9	−0.824566	−0.412283	+	0.911056i	$0.635269\pi$
$992$	3773.17	0.120765
$993$	−9949.85	−0.317975
$994$	12985.2	0.414350
$995$	−48791.7	−1.55457
$996$	−2202.48	−0.0700684
$997$	33928.2	1.07775	0.538875	−	0.842386i	$-0.318849\pi$
$997$	33928.2	1.07775	0.538875	+	0.842386i	$0.318849\pi$
$998$	169.711	0.00538287
$999$	4688.89	0.148499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.4.a.h.1.4	✓	4
3.2	odd	2		1062.4.a.n.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.4.a.h.1.4	✓	4	1.1	even	1	trivial
1062.4.a.n.1.1		4	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +17.3705 q^{5} -6.00000 q^{6} +29.1373 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +17.3705 q^{5} -6.00000 q^{6} +29.1373 q^{7} +8.00000 q^{8} +9.00000 q^{9} +34.7410 q^{10} +34.2581 q^{11} -12.0000 q^{12} -37.9155 q^{13} +58.2746 q^{14} -52.1115 q^{15} +16.0000 q^{16} -84.9002 q^{17} +18.0000 q^{18} -8.58000 q^{19} +69.4820 q^{20} -87.4118 q^{21} +68.5162 q^{22} -103.119 q^{23} -24.0000 q^{24} +176.734 q^{25} -75.8310 q^{26} -27.0000 q^{27} +116.549 q^{28} +169.530 q^{29} -104.223 q^{30} +117.912 q^{31} +32.0000 q^{32} -102.774 q^{33} -169.800 q^{34} +506.129 q^{35} +36.0000 q^{36} -173.663 q^{37} -17.1600 q^{38} +113.746 q^{39} +138.964 q^{40} +346.519 q^{41} -174.824 q^{42} +26.7692 q^{43} +137.032 q^{44} +156.334 q^{45} -206.238 q^{46} -472.136 q^{47} -48.0000 q^{48} +505.981 q^{49} +353.468 q^{50} +254.701 q^{51} -151.662 q^{52} -514.004 q^{53} -54.0000 q^{54} +595.081 q^{55} +233.098 q^{56} +25.7400 q^{57} +339.059 q^{58} +59.0000 q^{59} -208.446 q^{60} +133.802 q^{61} +235.823 q^{62} +262.236 q^{63} +64.0000 q^{64} -658.611 q^{65} -205.549 q^{66} -266.749 q^{67} -339.601 q^{68} +309.357 q^{69} +1012.26 q^{70} +222.827 q^{71} +72.0000 q^{72} +950.821 q^{73} -347.325 q^{74} -530.203 q^{75} -34.3200 q^{76} +998.189 q^{77} +227.493 q^{78} -1216.98 q^{79} +277.928 q^{80} +81.0000 q^{81} +693.037 q^{82} +183.540 q^{83} -349.647 q^{84} -1474.76 q^{85} +53.5385 q^{86} -508.589 q^{87} +274.065 q^{88} +654.925 q^{89} +312.669 q^{90} -1104.75 q^{91} -412.476 q^{92} -353.735 q^{93} -944.272 q^{94} -149.039 q^{95} -96.0000 q^{96} -422.708 q^{97} +1011.96 q^{98} +308.323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} + 22 q^{5} - 24 q^{6} + 13 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 - 12 * q^3 + 16 * q^4 + 22 * q^5 - 24 * q^6 + 13 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} + 22 q^{5} - 24 q^{6} + 13 q^{7} + 32 q^{8} + 36 q^{9} + 44 q^{10} + 24 q^{11} - 48 q^{12} + 20 q^{13} + 26 q^{14} - 66 q^{15} + 64 q^{16} + 91 q^{17} + 72 q^{18} + 141 q^{19} + 88 q^{20} - 39 q^{21} + 48 q^{22} + 13 q^{23} - 96 q^{24} + 278 q^{25} + 40 q^{26} - 108 q^{27} + 52 q^{28} + 295 q^{29} - 132 q^{30} + 311 q^{31} + 128 q^{32} - 72 q^{33} + 182 q^{34} + 551 q^{35} + 144 q^{36} + 609 q^{37} + 282 q^{38} - 60 q^{39} + 176 q^{40} + 677 q^{41} - 78 q^{42} + 170 q^{43} + 96 q^{44} + 198 q^{45} + 26 q^{46} + 17 q^{47} - 192 q^{48} + 651 q^{49} + 556 q^{50} - 273 q^{51} + 80 q^{52} + 166 q^{53} - 216 q^{54} + 108 q^{55} + 104 q^{56} - 423 q^{57} + 590 q^{58} + 236 q^{59} - 264 q^{60} + 651 q^{61} + 622 q^{62} + 117 q^{63} + 256 q^{64} + 700 q^{65} - 144 q^{66} - 894 q^{67} + 364 q^{68} - 39 q^{69} + 1102 q^{70} + 298 q^{71} + 288 q^{72} + 887 q^{73} + 1218 q^{74} - 834 q^{75} + 564 q^{76} - 79 q^{77} - 120 q^{78} - 784 q^{79} + 352 q^{80} + 324 q^{81} + 1354 q^{82} + 971 q^{83} - 156 q^{84} - 799 q^{85} + 340 q^{86} - 885 q^{87} + 192 q^{88} + 1321 q^{89} + 396 q^{90} - 2673 q^{91} + 52 q^{92} - 933 q^{93} + 34 q^{94} - 3133 q^{95} - 384 q^{96} - 1922 q^{97} + 1302 q^{98} + 216 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 - 12 * q^3 + 16 * q^4 + 22 * q^5 - 24 * q^6 + 13 * q^7 + 32 * q^8 + 36 * q^9 + 44 * q^10 + 24 * q^11 - 48 * q^12 + 20 * q^13 + 26 * q^14 - 66 * q^15 + 64 * q^16 + 91 * q^17 + 72 * q^18 + 141 * q^19 + 88 * q^20 - 39 * q^21 + 48 * q^22 + 13 * q^23 - 96 * q^24 + 278 * q^25 + 40 * q^26 - 108 * q^27 + 52 * q^28 + 295 * q^29 - 132 * q^30 + 311 * q^31 + 128 * q^32 - 72 * q^33 + 182 * q^34 + 551 * q^35 + 144 * q^36 + 609 * q^37 + 282 * q^38 - 60 * q^39 + 176 * q^40 + 677 * q^41 - 78 * q^42 + 170 * q^43 + 96 * q^44 + 198 * q^45 + 26 * q^46 + 17 * q^47 - 192 * q^48 + 651 * q^49 + 556 * q^50 - 273 * q^51 + 80 * q^52 + 166 * q^53 - 216 * q^54 + 108 * q^55 + 104 * q^56 - 423 * q^57 + 590 * q^58 + 236 * q^59 - 264 * q^60 + 651 * q^61 + 622 * q^62 + 117 * q^63 + 256 * q^64 + 700 * q^65 - 144 * q^66 - 894 * q^67 + 364 * q^68 - 39 * q^69 + 1102 * q^70 + 298 * q^71 + 288 * q^72 + 887 * q^73 + 1218 * q^74 - 834 * q^75 + 564 * q^76 - 79 * q^77 - 120 * q^78 - 784 * q^79 + 352 * q^80 + 324 * q^81 + 1354 * q^82 + 971 * q^83 - 156 * q^84 - 799 * q^85 + 340 * q^86 - 885 * q^87 + 192 * q^88 + 1321 * q^89 + 396 * q^90 - 2673 * q^91 + 52 * q^92 - 933 * q^93 + 34 * q^94 - 3133 * q^95 - 384 * q^96 - 1922 * q^97 + 1302 * q^98 + 216 * q^99

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