LMFDB - Embedded newform 177.4.a.a.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, 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("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.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:	$10.4433380710$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 34x^{5} + 25x^{4} + 315x^{3} - 146x^{2} - 736x + 512 $ x^7 - x^6 - 34x^5 + 25x^4 + 315x^3 - 146x^2 - 736*x + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.68175$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.68175 q^{2} +3.00000 q^{3} -5.17172 q^{4} -9.78761 q^{5} +5.04525 q^{6} +6.87552 q^{7} -22.1515 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.68175 q^{2} +3.00000 q^{3} -5.17172 q^{4} -9.78761 q^{5} +5.04525 q^{6} +6.87552 q^{7} -22.1515 q^{8} +9.00000 q^{9} -16.4603 q^{10} -37.8468 q^{11} -15.5151 q^{12} -11.3334 q^{13} +11.5629 q^{14} -29.3628 q^{15} +4.12039 q^{16} -60.3439 q^{17} +15.1358 q^{18} -92.5767 q^{19} +50.6188 q^{20} +20.6266 q^{21} -63.6489 q^{22} -183.718 q^{23} -66.4546 q^{24} -29.2026 q^{25} -19.0599 q^{26} +27.0000 q^{27} -35.5582 q^{28} +265.559 q^{29} -49.3810 q^{30} +177.376 q^{31} +184.142 q^{32} -113.540 q^{33} -101.483 q^{34} -67.2949 q^{35} -46.5454 q^{36} -70.0886 q^{37} -155.691 q^{38} -34.0002 q^{39} +216.811 q^{40} +208.760 q^{41} +34.6887 q^{42} +393.127 q^{43} +195.733 q^{44} -88.0885 q^{45} -308.968 q^{46} -134.917 q^{47} +12.3612 q^{48} -295.727 q^{49} -49.1115 q^{50} -181.032 q^{51} +58.6131 q^{52} -650.432 q^{53} +45.4073 q^{54} +370.430 q^{55} -152.303 q^{56} -277.730 q^{57} +446.604 q^{58} -59.0000 q^{59} +151.856 q^{60} +22.5322 q^{61} +298.302 q^{62} +61.8797 q^{63} +276.717 q^{64} +110.927 q^{65} -190.947 q^{66} -209.476 q^{67} +312.082 q^{68} -551.155 q^{69} -113.173 q^{70} +209.160 q^{71} -199.364 q^{72} +865.282 q^{73} -117.872 q^{74} -87.6079 q^{75} +478.781 q^{76} -260.217 q^{77} -57.1798 q^{78} +843.604 q^{79} -40.3287 q^{80} +81.0000 q^{81} +351.082 q^{82} -845.384 q^{83} -106.675 q^{84} +590.623 q^{85} +661.141 q^{86} +796.677 q^{87} +838.365 q^{88} -974.766 q^{89} -148.143 q^{90} -77.9229 q^{91} +950.139 q^{92} +532.127 q^{93} -226.897 q^{94} +906.105 q^{95} +552.425 q^{96} +338.796 q^{97} -497.339 q^{98} -340.621 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9}+O(q^{10}) $ 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9} - 79 q^{10} - 131 q^{11} + 66 q^{12} - 123 q^{13} - 117 q^{14} - 84 q^{15} + 202 q^{16} - 235 q^{17} - 72 q^{18} - 80 q^{19} + 61 q^{20} - 177 q^{21} + 688 q^{22} - 274 q^{23} - 351 q^{24} + 193 q^{25} - 180 q^{26} + 189 q^{27} - 118 q^{28} - 406 q^{29} - 237 q^{30} - 346 q^{31} - 854 q^{32} - 393 q^{33} + 178 q^{34} - 424 q^{35} + 198 q^{36} - 157 q^{37} - 129 q^{38} - 369 q^{39} - 590 q^{40} - 825 q^{41} - 351 q^{42} - 815 q^{43} - 1690 q^{44} - 252 q^{45} + 1457 q^{46} - 1196 q^{47} + 606 q^{48} + 914 q^{49} + 713 q^{50} - 705 q^{51} + 1030 q^{52} - 900 q^{53} - 216 q^{54} - 1044 q^{55} + 2172 q^{56} - 240 q^{57} + 1242 q^{58} - 413 q^{59} + 183 q^{60} + 420 q^{61} + 646 q^{62} - 531 q^{63} + 3541 q^{64} + 190 q^{65} + 2064 q^{66} + 1316 q^{67} - 611 q^{68} - 822 q^{69} + 4658 q^{70} - 173 q^{71} - 1053 q^{72} - 418 q^{73} + 660 q^{74} + 579 q^{75} + 1540 q^{76} - 753 q^{77} - 540 q^{78} + 2635 q^{79} + 6155 q^{80} + 567 q^{81} - 125 q^{82} + 457 q^{83} - 354 q^{84} + 1270 q^{85} + 3482 q^{86} - 1218 q^{87} + 7685 q^{88} + 592 q^{89} - 711 q^{90} + 3179 q^{91} - 3500 q^{92} - 1038 q^{93} + 2064 q^{94} - 2250 q^{95} - 2562 q^{96} - 1906 q^{97} + 2994 q^{98} - 1179 q^{99}+O(q^{100}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9 - 79 * q^10 - 131 * q^11 + 66 * q^12 - 123 * q^13 - 117 * q^14 - 84 * q^15 + 202 * q^16 - 235 * q^17 - 72 * q^18 - 80 * q^19 + 61 * q^20 - 177 * q^21 + 688 * q^22 - 274 * q^23 - 351 * q^24 + 193 * q^25 - 180 * q^26 + 189 * q^27 - 118 * q^28 - 406 * q^29 - 237 * q^30 - 346 * q^31 - 854 * q^32 - 393 * q^33 + 178 * q^34 - 424 * q^35 + 198 * q^36 - 157 * q^37 - 129 * q^38 - 369 * q^39 - 590 * q^40 - 825 * q^41 - 351 * q^42 - 815 * q^43 - 1690 * q^44 - 252 * q^45 + 1457 * q^46 - 1196 * q^47 + 606 * q^48 + 914 * q^49 + 713 * q^50 - 705 * q^51 + 1030 * q^52 - 900 * q^53 - 216 * q^54 - 1044 * q^55 + 2172 * q^56 - 240 * q^57 + 1242 * q^58 - 413 * q^59 + 183 * q^60 + 420 * q^61 + 646 * q^62 - 531 * q^63 + 3541 * q^64 + 190 * q^65 + 2064 * q^66 + 1316 * q^67 - 611 * q^68 - 822 * q^69 + 4658 * q^70 - 173 * q^71 - 1053 * q^72 - 418 * q^73 + 660 * q^74 + 579 * q^75 + 1540 * q^76 - 753 * q^77 - 540 * q^78 + 2635 * q^79 + 6155 * q^80 + 567 * q^81 - 125 * q^82 + 457 * q^83 - 354 * q^84 + 1270 * q^85 + 3482 * q^86 - 1218 * q^87 + 7685 * q^88 + 592 * q^89 - 711 * q^90 + 3179 * q^91 - 3500 * q^92 - 1038 * q^93 + 2064 * q^94 - 2250 * q^95 - 2562 * q^96 - 1906 * q^97 + 2994 * q^98 - 1179 * 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 $

$2$	1.68175	0.594588	0.297294	−	0.954786i	$-0.403916\pi$
$2$	1.68175	0.594588	0.297294	+	0.954786i	$0.403916\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−5.17172	−0.646465
$5$	−9.78761	−0.875431	−0.437715	−	0.899114i	$-0.644212\pi$
$5$	−9.78761	−0.875431	−0.437715	+	0.899114i	$0.644212\pi$
$6$	5.04525	0.343286
$7$	6.87552	0.371243	0.185622	−	0.982621i	$-0.440570\pi$
$7$	6.87552	0.371243	0.185622	+	0.982621i	$0.440570\pi$
$8$	−22.1515	−0.978969
$9$	9.00000	0.333333
$10$	−16.4603	−0.520521
$11$	−37.8468	−1.03739	−0.518693	−	0.854961i	$-0.673581\pi$
$11$	−37.8468	−1.03739	−0.518693	+	0.854961i	$0.673581\pi$
$12$	−15.5151	−0.373237
$13$	−11.3334	−0.241794	−0.120897	−	0.992665i	$-0.538577\pi$
$13$	−11.3334	−0.241794	−0.120897	+	0.992665i	$0.538577\pi$
$14$	11.5629	0.220737
$15$	−29.3628	−0.505430
$16$	4.12039	0.0643810
$17$	−60.3439	−0.860915	−0.430457	−	0.902611i	$-0.641648\pi$
$17$	−60.3439	−0.860915	−0.430457	+	0.902611i	$0.641648\pi$
$18$	15.1358	0.198196
$19$	−92.5767	−1.11782	−0.558909	−	0.829229i	$-0.688780\pi$
$19$	−92.5767	−1.11782	−0.558909	+	0.829229i	$0.688780\pi$
$20$	50.6188	0.565935
$21$	20.6266	0.214337
$22$	−63.6489	−0.616818
$23$	−183.718	−1.66556	−0.832781	−	0.553603i	$-0.813252\pi$
$23$	−183.718	−1.66556	−0.832781	+	0.553603i	$0.813252\pi$
$24$	−66.4546	−0.565208
$25$	−29.2026	−0.233621
$26$	−19.0599	−0.143768
$27$	27.0000	0.192450
$28$	−35.5582	−0.239995
$29$	265.559	1.70045	0.850225	−	0.526419i	$-0.176466\pi$
$29$	265.559	1.70045	0.850225	+	0.526419i	$0.176466\pi$
$30$	−49.3810	−0.300523
$31$	177.376	1.02767	0.513833	−	0.857890i	$-0.328225\pi$
$31$	177.376	1.02767	0.513833	+	0.857890i	$0.328225\pi$
$32$	184.142	1.01725
$33$	−113.540	−0.598935
$34$	−101.483	−0.511890
$35$	−67.2949	−0.324998
$36$	−46.5454	−0.215488
$37$	−70.0886	−0.311419	−0.155709	−	0.987803i	$-0.549766\pi$
$37$	−70.0886	−0.311419	−0.155709	+	0.987803i	$0.549766\pi$
$38$	−155.691	−0.664642
$39$	−34.0002	−0.139600
$40$	216.811	0.857019
$41$	208.760	0.795190	0.397595	−	0.917561i	$-0.369845\pi$
$41$	208.760	0.795190	0.397595	+	0.917561i	$0.369845\pi$
$42$	34.6887	0.127442
$43$	393.127	1.39421	0.697107	−	0.716967i	$-0.254470\pi$
$43$	393.127	1.39421	0.697107	+	0.716967i	$0.254470\pi$
$44$	195.733	0.670633
$45$	−88.0885	−0.291810
$46$	−308.968	−0.990323
$47$	−134.917	−0.418717	−0.209358	−	0.977839i	$-0.567138\pi$
$47$	−134.917	−0.418717	−0.209358	+	0.977839i	$0.567138\pi$
$48$	12.3612	0.0371704
$49$	−295.727	−0.862179
$50$	−49.1115	−0.138908
$51$	−181.032	−0.497049
$52$	58.6131	0.156311
$53$	−650.432	−1.68573	−0.842865	−	0.538126i	$-0.819133\pi$
$53$	−650.432	−1.68573	−0.842865	+	0.538126i	$0.819133\pi$
$54$	45.4073	0.114429
$55$	370.430	0.908160
$56$	−152.303	−0.363435
$57$	−277.730	−0.645373
$58$	446.604	1.01107
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	151.856	0.326743
$61$	22.5322	0.0472943	0.0236472	−	0.999720i	$-0.492472\pi$
$61$	22.5322	0.0472943	0.0236472	+	0.999720i	$0.492472\pi$
$62$	298.302	0.611038
$63$	61.8797	0.123748
$64$	276.717	0.540463
$65$	110.927	0.211674
$66$	−190.947	−0.356120
$67$	−209.476	−0.381964	−0.190982	−	0.981594i	$-0.561167\pi$
$67$	−209.476	−0.381964	−0.190982	+	0.981594i	$0.561167\pi$
$68$	312.082	0.556551
$69$	−551.155	−0.961612
$70$	−113.173	−0.193240
$71$	209.160	0.349616	0.174808	−	0.984603i	$-0.444070\pi$
$71$	209.160	0.349616	0.174808	+	0.984603i	$0.444070\pi$
$72$	−199.364	−0.326323
$73$	865.282	1.38731	0.693655	−	0.720308i	$-0.255999\pi$
$73$	865.282	1.38731	0.693655	+	0.720308i	$0.255999\pi$
$74$	−117.872	−0.185166
$75$	−87.6079	−0.134881
$76$	478.781	0.722630
$77$	−260.217	−0.385122
$78$	−57.1798	−0.0830043
$79$	843.604	1.20143	0.600715	−	0.799464i	$-0.294883\pi$
$79$	843.604	1.20143	0.600715	+	0.799464i	$0.294883\pi$
$80$	−40.3287	−0.0563611
$81$	81.0000	0.111111
$82$	351.082	0.472811
$83$	−845.384	−1.11799	−0.558994	−	0.829172i	$-0.688812\pi$
$83$	−845.384	−1.11799	−0.558994	+	0.829172i	$0.688812\pi$
$84$	−106.675	−0.138561
$85$	590.623	0.753671
$86$	661.141	0.828984
$87$	796.677	0.981755
$88$	838.365	1.01557
$89$	−974.766	−1.16096	−0.580478	−	0.814276i	$-0.697134\pi$
$89$	−974.766	−1.16096	−0.580478	+	0.814276i	$0.697134\pi$
$90$	−148.143	−0.173507
$91$	−77.9229	−0.0897642
$92$	950.139	1.07673
$93$	532.127	0.593323
$94$	−226.897	−0.248964
$95$	906.105	0.978573
$96$	552.425	0.587309
$97$	338.796	0.354634	0.177317	−	0.984154i	$-0.443258\pi$
$97$	338.796	0.354634	0.177317	+	0.984154i	$0.443258\pi$
$98$	−497.339	−0.512641
$99$	−340.621	−0.345795
$100$	151.028	0.151028
$101$	−683.879	−0.673747	−0.336874	−	0.941550i	$-0.609369\pi$
$101$	−683.879	−0.673747	−0.336874	+	0.941550i	$0.609369\pi$
$102$	−304.450	−0.295540
$103$	−1235.54	−1.18196	−0.590979	−	0.806687i	$-0.701258\pi$
$103$	−1235.54	−1.18196	−0.590979	+	0.806687i	$0.701258\pi$
$104$	251.052	0.236708
$105$	−201.885	−0.187637
$106$	−1093.86	−1.00231
$107$	431.478	0.389837	0.194918	−	0.980819i	$-0.437556\pi$
$107$	431.478	0.389837	0.194918	+	0.980819i	$0.437556\pi$
$108$	−139.636	−0.124412
$109$	427.874	0.375989	0.187995	−	0.982170i	$-0.439801\pi$
$109$	427.874	0.375989	0.187995	+	0.982170i	$0.439801\pi$
$110$	622.971	0.539981
$111$	−210.266	−0.179798
$112$	28.3298	0.0239010
$113$	−1934.70	−1.61063	−0.805317	−	0.592845i	$-0.798005\pi$
$113$	−1934.70	−1.61063	−0.805317	+	0.592845i	$0.798005\pi$
$114$	−467.073	−0.383731
$115$	1798.16	1.45808
$116$	−1373.40	−1.09928
$117$	−102.001	−0.0805979
$118$	−99.2233	−0.0774088
$119$	−414.896	−0.319609
$120$	650.432	0.494800
$121$	101.382	0.0761698
$122$	37.8935	0.0281207
$123$	626.280	0.459103
$124$	−917.337	−0.664349
$125$	1509.28	1.07995
$126$	104.066	0.0735789
$127$	−1334.46	−0.932395	−0.466198	−	0.884681i	$-0.654376\pi$
$127$	−1334.46	−0.932395	−0.466198	+	0.884681i	$0.654376\pi$
$128$	−1007.76	−0.695896
$129$	1179.38	0.804950
$130$	186.551	0.125859
$131$	−739.638	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$131$	−739.638	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$132$	587.199	0.387190
$133$	−636.513	−0.414982
$134$	−352.287	−0.227111
$135$	−264.266	−0.168477
$136$	1336.71	0.842809
$137$	−1262.30	−0.787196	−0.393598	−	0.919283i	$-0.628770\pi$
$137$	−1262.30	−0.787196	−0.393598	+	0.919283i	$0.628770\pi$
$138$	−926.905	−0.571763
$139$	2630.69	1.60527	0.802634	−	0.596471i	$-0.203431\pi$
$139$	2630.69	1.60527	0.802634	+	0.596471i	$0.203431\pi$
$140$	348.030	0.210099
$141$	−404.752	−0.241746
$142$	351.755	0.207877
$143$	428.933	0.250833
$144$	37.0835	0.0214603
$145$	−2599.19	−1.48863
$146$	1455.19	0.824878
$147$	−887.182	−0.497779
$148$	362.478	0.201321
$149$	−434.666	−0.238988	−0.119494	−	0.992835i	$-0.538127\pi$
$149$	−434.666	−0.238988	−0.119494	+	0.992835i	$0.538127\pi$
$150$	−147.335	−0.0801988
$151$	−1585.40	−0.854425	−0.427213	−	0.904151i	$-0.640504\pi$
$151$	−1585.40	−0.854425	−0.427213	+	0.904151i	$0.640504\pi$
$152$	2050.72	1.09431
$153$	−543.095	−0.286972
$154$	−437.619	−0.228989
$155$	−1736.09	−0.899650
$156$	175.839	0.0902462
$157$	3744.07	1.90324	0.951622	−	0.307270i	$-0.0994155\pi$
$157$	3744.07	1.90324	0.951622	+	0.307270i	$0.0994155\pi$
$158$	1418.73	0.714356
$159$	−1951.29	−0.973256
$160$	−1802.31	−0.890531
$161$	−1263.16	−0.618328
$162$	136.222	0.0660654
$163$	−1331.85	−0.639993	−0.319996	−	0.947419i	$-0.603682\pi$
$163$	−1331.85	−0.639993	−0.319996	+	0.947419i	$0.603682\pi$
$164$	−1079.65	−0.514062
$165$	1111.29	0.524326
$166$	−1421.73	−0.664743
$167$	16.5584	0.00767260	0.00383630	−	0.999993i	$-0.498779\pi$
$167$	16.5584	0.00767260	0.00383630	+	0.999993i	$0.498779\pi$
$168$	−456.910	−0.209830
$169$	−2068.55	−0.941536
$170$	993.280	0.448124
$171$	−833.191	−0.372606
$172$	−2033.14	−0.901310
$173$	−2397.07	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$173$	−2397.07	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$174$	1339.81	0.583740
$175$	−200.783	−0.0867302
$176$	−155.944	−0.0667880
$177$	−177.000	−0.0751646
$178$	−1639.31	−0.690291
$179$	1223.00	0.510680	0.255340	−	0.966851i	$-0.417813\pi$
$179$	1223.00	0.510680	0.255340	+	0.966851i	$0.417813\pi$
$180$	455.569	0.188645
$181$	−4266.97	−1.75227	−0.876137	−	0.482063i	$-0.839888\pi$
$181$	−4266.97	−1.75227	−0.876137	+	0.482063i	$0.839888\pi$
$182$	−131.047	−0.0533728
$183$	67.5966	0.0273054
$184$	4069.64	1.63053
$185$	686.000	0.272626
$186$	894.905	0.352783
$187$	2283.83	0.893101
$188$	697.753	0.270686
$189$	185.639	0.0714458
$190$	1523.84	0.581848
$191$	−3080.45	−1.16698	−0.583491	−	0.812120i	$-0.698313\pi$
$191$	−3080.45	−1.16698	−0.583491	+	0.812120i	$0.698313\pi$
$192$	830.152	0.312037
$193$	3469.71	1.29407	0.647033	−	0.762462i	$-0.276009\pi$
$193$	3469.71	1.29407	0.647033	+	0.762462i	$0.276009\pi$
$194$	569.770	0.210862
$195$	332.781	0.122210
$196$	1529.42	0.557368
$197$	−1487.74	−0.538055	−0.269027	−	0.963133i	$-0.586702\pi$
$197$	−1487.74	−0.538055	−0.269027	+	0.963133i	$0.586702\pi$
$198$	−572.840	−0.205606
$199$	−3027.73	−1.07854	−0.539271	−	0.842132i	$-0.681300\pi$
$199$	−3027.73	−1.07854	−0.539271	+	0.842132i	$0.681300\pi$
$200$	646.883	0.228708
$201$	−628.429	−0.220527
$202$	−1150.11	−0.400602
$203$	1825.86	0.631280
$204$	936.245	0.321325
$205$	−2043.26	−0.696134
$206$	−2077.87	−0.702779
$207$	−1653.46	−0.555187
$208$	−46.6980	−0.0155669
$209$	3503.74	1.15961
$210$	−339.520	−0.111567
$211$	−1655.90	−0.540269	−0.270135	−	0.962823i	$-0.587068\pi$
$211$	−1655.90	−0.540269	−0.270135	+	0.962823i	$0.587068\pi$
$212$	3363.85	1.08976
$213$	627.480	0.201851
$214$	725.638	0.231792
$215$	−3847.77	−1.22054
$216$	−598.091	−0.188403
$217$	1219.55	0.381514
$218$	719.576	0.223559
$219$	2595.85	0.800964
$220$	−1915.76	−0.587093
$221$	683.901	0.208164
$222$	−353.615	−0.106906
$223$	−1225.20	−0.367916	−0.183958	−	0.982934i	$-0.558891\pi$
$223$	−1225.20	−0.367916	−0.183958	+	0.982934i	$0.558891\pi$
$224$	1266.07	0.377647
$225$	−262.824	−0.0778737
$226$	−3253.69	−0.957664
$227$	−657.677	−0.192298	−0.0961488	−	0.995367i	$-0.530652\pi$
$227$	−657.677	−0.192298	−0.0961488	+	0.995367i	$0.530652\pi$
$228$	1436.34	0.417211
$229$	1407.97	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$229$	1407.97	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$230$	3024.06	0.866959
$231$	−780.650	−0.222351
$232$	−5882.54	−1.66469
$233$	6393.48	1.79764	0.898821	−	0.438315i	$-0.144425\pi$
$233$	6393.48	1.79764	0.898821	+	0.438315i	$0.144425\pi$
$234$	−171.539	−0.0479226
$235$	1320.52	0.366558
$236$	305.131	0.0841625
$237$	2530.81	0.693645
$238$	−697.751	−0.190036
$239$	1941.60	0.525489	0.262744	−	0.964865i	$-0.415372\pi$
$239$	1941.60	0.525489	0.262744	+	0.964865i	$0.415372\pi$
$240$	−120.986	−0.0325401
$241$	−3701.61	−0.989385	−0.494692	−	0.869068i	$-0.664719\pi$
$241$	−3701.61	−0.989385	−0.494692	+	0.869068i	$0.664719\pi$
$242$	170.499	0.0452897
$243$	243.000	0.0641500
$244$	−116.530	−0.0305741
$245$	2894.46	0.754778
$246$	1053.25	0.272978
$247$	1049.21	0.270282
$248$	−3929.15	−1.00605
$249$	−2536.15	−0.645471
$250$	2538.22	0.642126
$251$	−239.067	−0.0601186	−0.0300593	−	0.999548i	$-0.509570\pi$
$251$	−239.067	−0.0601186	−0.0300593	+	0.999548i	$0.509570\pi$
$252$	−320.024	−0.0799985
$253$	6953.15	1.72783
$254$	−2244.23	−0.554392
$255$	1771.87	0.435132
$256$	−3908.55	−0.954235
$257$	6633.12	1.60997	0.804985	−	0.593295i	$-0.202173\pi$
$257$	6633.12	1.60997	0.804985	+	0.593295i	$0.202173\pi$
$258$	1983.42	0.478614
$259$	−481.895	−0.115612
$260$	−573.682	−0.136839
$261$	2390.03	0.566817
$262$	−1243.89	−0.293311
$263$	−2010.45	−0.471368	−0.235684	−	0.971830i	$-0.575733\pi$
$263$	−2010.45	−0.471368	−0.235684	+	0.971830i	$0.575733\pi$
$264$	2515.10	0.586339
$265$	6366.17	1.47574
$266$	−1070.46	−0.246744
$267$	−2924.30	−0.670278
$268$	1083.35	0.246926
$269$	5923.22	1.34255	0.671273	−	0.741210i	$-0.265748\pi$
$269$	5923.22	1.34255	0.671273	+	0.741210i	$0.265748\pi$
$270$	−444.429	−0.100174
$271$	−1908.94	−0.427896	−0.213948	−	0.976845i	$-0.568632\pi$
$271$	−1908.94	−0.427896	−0.213948	+	0.976845i	$0.568632\pi$
$272$	−248.640	−0.0554266
$273$	−233.769	−0.0518254
$274$	−2122.88	−0.468057
$275$	1105.23	0.242355
$276$	2850.42	0.621648
$277$	7685.54	1.66707	0.833536	−	0.552464i	$-0.186312\pi$
$277$	7685.54	1.66707	0.833536	+	0.552464i	$0.186312\pi$
$278$	4424.17	0.954474
$279$	1596.38	0.342555
$280$	1490.69	0.318162
$281$	1223.20	0.259680	0.129840	−	0.991535i	$-0.458554\pi$
$281$	1223.20	0.259680	0.129840	+	0.991535i	$0.458554\pi$
$282$	−680.691	−0.143740
$283$	−1543.59	−0.324229	−0.162115	−	0.986772i	$-0.551831\pi$
$283$	−1543.59	−0.324229	−0.162115	+	0.986772i	$0.551831\pi$
$284$	−1081.72	−0.226014
$285$	2718.32	0.564979
$286$	721.358	0.149143
$287$	1435.33	0.295209
$288$	1657.28	0.339083
$289$	−1271.61	−0.258826
$290$	−4371.18	−0.885120
$291$	1016.39	0.204748
$292$	−4474.99	−0.896846
$293$	−3987.52	−0.795063	−0.397532	−	0.917589i	$-0.630133\pi$
$293$	−3987.52	−0.795063	−0.397532	+	0.917589i	$0.630133\pi$
$294$	−1492.02	−0.295974
$295$	577.469	0.113971
$296$	1552.57	0.304869
$297$	−1021.86	−0.199645
$298$	−731.000	−0.142100
$299$	2082.15	0.402722
$300$	453.083	0.0871959
$301$	2702.95	0.517593
$302$	−2666.25	−0.508031
$303$	−2051.64	−0.388988
$304$	−381.452	−0.0719663
$305$	−220.537	−0.0414029
$306$	−913.350	−0.170630
$307$	−895.829	−0.166540	−0.0832698	−	0.996527i	$-0.526536\pi$
$307$	−895.829	−0.166540	−0.0832698	+	0.996527i	$0.526536\pi$
$308$	1345.77	0.248968
$309$	−3706.63	−0.682404
$310$	−2919.66	−0.534921
$311$	4166.91	0.759754	0.379877	−	0.925037i	$-0.375966\pi$
$311$	4166.91	0.759754	0.379877	+	0.925037i	$0.375966\pi$
$312$	753.156	0.136664
$313$	−3619.78	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$313$	−3619.78	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$314$	6296.59	1.13165
$315$	−605.654	−0.108333
$316$	−4362.88	−0.776681
$317$	865.480	0.153344	0.0766722	−	0.997056i	$-0.475570\pi$
$317$	865.480	0.153344	0.0766722	+	0.997056i	$0.475570\pi$
$318$	−3281.59	−0.578687
$319$	−10050.6	−1.76402
$320$	−2708.40	−0.473138
$321$	1294.43	0.225072
$322$	−2124.32	−0.367651
$323$	5586.44	0.962347
$324$	−418.909	−0.0718294
$325$	330.965	0.0564881
$326$	−2239.84	−0.380532
$327$	1283.62	0.217078
$328$	−4624.35	−0.778467
$329$	−927.626	−0.155446
$330$	1868.91	0.311758
$331$	8140.89	1.35185	0.675927	−	0.736968i	$-0.263743\pi$
$331$	8140.89	1.35185	0.675927	+	0.736968i	$0.263743\pi$
$332$	4372.09	0.722739
$333$	−630.797	−0.103806
$334$	27.8470	0.00456204
$335$	2050.27	0.334383
$336$	84.9894	0.0137993
$337$	7396.34	1.19556	0.597781	−	0.801659i	$-0.296049\pi$
$337$	7396.34	1.19556	0.597781	+	0.801659i	$0.296049\pi$
$338$	−3478.79	−0.559826
$339$	−5804.11	−0.929900
$340$	−3054.53	−0.487222
$341$	−6713.11	−1.06609
$342$	−1401.22	−0.221547
$343$	−4391.58	−0.691321
$344$	−8708.36	−1.36489
$345$	5394.49	0.841825
$346$	−4031.27	−0.626365
$347$	−1341.98	−0.207612	−0.103806	−	0.994598i	$-0.533102\pi$
$347$	−1341.98	−0.207612	−0.103806	+	0.994598i	$0.533102\pi$
$348$	−4120.19	−0.634670
$349$	2193.32	0.336407	0.168203	−	0.985752i	$-0.446203\pi$
$349$	2193.32	0.336407	0.168203	+	0.985752i	$0.446203\pi$
$350$	−337.667	−0.0515688
$351$	−306.002	−0.0465332
$352$	−6969.18	−1.05528
$353$	−636.290	−0.0959386	−0.0479693	−	0.998849i	$-0.515275\pi$
$353$	−636.290	−0.0959386	−0.0479693	+	0.998849i	$0.515275\pi$
$354$	−297.670	−0.0446920
$355$	−2047.18	−0.306064
$356$	5041.22	0.750516
$357$	−1244.69	−0.184526
$358$	2056.79	0.303644
$359$	−7124.07	−1.04734	−0.523668	−	0.851922i	$-0.675437\pi$
$359$	−7124.07	−1.04734	−0.523668	+	0.851922i	$0.675437\pi$
$360$	1951.30	0.285673
$361$	1711.45	0.249519
$362$	−7175.98	−1.04188
$363$	304.146	0.0439766
$364$	402.995	0.0580294
$365$	−8469.04	−1.21449
$366$	113.681	0.0162355
$367$	−8714.69	−1.23952	−0.619759	−	0.784792i	$-0.712770\pi$
$367$	−8714.69	−1.23952	−0.619759	+	0.784792i	$0.712770\pi$
$368$	−756.990	−0.107231
$369$	1878.84	0.265063
$370$	1153.68	0.162100
$371$	−4472.05	−0.625815
$372$	−2752.01	−0.383562
$373$	9123.29	1.26645	0.633225	−	0.773967i	$-0.281731\pi$
$373$	9123.29	1.26645	0.633225	+	0.773967i	$0.281731\pi$
$374$	3840.82	0.531027
$375$	4527.83	0.623509
$376$	2988.62	0.409911
$377$	−3009.68	−0.411158
$378$	312.198	0.0424808
$379$	8539.81	1.15742	0.578708	−	0.815535i	$-0.303557\pi$
$379$	8539.81	1.15742	0.578708	+	0.815535i	$0.303557\pi$
$380$	−4686.12	−0.632613
$381$	−4003.38	−0.538319
$382$	−5180.54	−0.693873
$383$	−8691.08	−1.15951	−0.579757	−	0.814790i	$-0.696852\pi$
$383$	−8691.08	−1.15951	−0.579757	+	0.814790i	$0.696852\pi$
$384$	−3023.29	−0.401776
$385$	2546.90	0.337148
$386$	5835.18	0.769437
$387$	3538.14	0.464738
$388$	−1752.16	−0.229259
$389$	−10079.9	−1.31381	−0.656904	−	0.753974i	$-0.728134\pi$
$389$	−10079.9	−1.31381	−0.656904	+	0.753974i	$0.728134\pi$
$390$	559.654	0.0726645
$391$	11086.3	1.43391
$392$	6550.81	0.844046
$393$	−2218.91	−0.284808
$394$	−2502.00	−0.319921
$395$	−8256.87	−1.05177
$396$	1761.60	0.223544
$397$	−1829.51	−0.231286	−0.115643	−	0.993291i	$-0.536893\pi$
$397$	−1829.51	−0.231286	−0.115643	+	0.993291i	$0.536893\pi$
$398$	−5091.88	−0.641288
$399$	−1909.54	−0.239590
$400$	−120.326	−0.0150408
$401$	−10919.6	−1.35985	−0.679926	−	0.733281i	$-0.737988\pi$
$401$	−10919.6	−1.35985	−0.679926	+	0.733281i	$0.737988\pi$
$402$	−1056.86	−0.131123
$403$	−2010.27	−0.248483
$404$	3536.83	0.435554
$405$	−792.797	−0.0972701
$406$	3070.63	0.375352
$407$	2652.63	0.323061
$408$	4010.13	0.486596
$409$	6441.50	0.778757	0.389378	−	0.921078i	$-0.372690\pi$
$409$	6441.50	0.778757	0.389378	+	0.921078i	$0.372690\pi$
$410$	−3436.25	−0.413913
$411$	−3786.91	−0.454488
$412$	6389.88	0.764094
$413$	−405.656	−0.0483317
$414$	−2780.71	−0.330108
$415$	8274.29	0.978721
$416$	−2086.95	−0.245964
$417$	7892.08	0.926802
$418$	5892.41	0.689491
$419$	6784.29	0.791012	0.395506	−	0.918463i	$-0.370569\pi$
$419$	6784.29	0.791012	0.395506	+	0.918463i	$0.370569\pi$
$420$	1044.09	0.121301
$421$	827.842	0.0958350	0.0479175	−	0.998851i	$-0.484742\pi$
$421$	827.842	0.0958350	0.0479175	+	0.998851i	$0.484742\pi$
$422$	−2784.81	−0.321238
$423$	−1214.25	−0.139572
$424$	14408.1	1.65028
$425$	1762.20	0.201128
$426$	1055.26	0.120018
$427$	154.921	0.0175577
$428$	−2231.48	−0.252016
$429$	1286.80	0.144819
$430$	−6470.99	−0.725718
$431$	1719.44	0.192163	0.0960815	−	0.995373i	$-0.469369\pi$
$431$	1719.44	0.192163	0.0960815	+	0.995373i	$0.469369\pi$
$432$	111.250	0.0123901
$433$	11944.5	1.32567	0.662836	−	0.748765i	$-0.269353\pi$
$433$	11944.5	1.32567	0.662836	+	0.748765i	$0.269353\pi$
$434$	2050.98	0.226844
$435$	−7797.56	−0.859459
$436$	−2212.84	−0.243064
$437$	17008.0	1.86180
$438$	4365.56	0.476244
$439$	316.306	0.0343882	0.0171941	−	0.999852i	$-0.494527\pi$
$439$	316.306	0.0343882	0.0171941	+	0.999852i	$0.494527\pi$
$440$	−8205.59	−0.889060
$441$	−2661.55	−0.287393
$442$	1150.15	0.123772
$443$	−9043.41	−0.969899	−0.484950	−	0.874542i	$-0.661162\pi$
$443$	−9043.41	−0.969899	−0.484950	+	0.874542i	$0.661162\pi$
$444$	1087.44	0.116233
$445$	9540.63	1.01634
$446$	−2060.47	−0.218758
$447$	−1304.00	−0.137980
$448$	1902.57	0.200643
$449$	9058.55	0.952114	0.476057	−	0.879414i	$-0.342066\pi$
$449$	9058.55	0.952114	0.476057	+	0.879414i	$0.342066\pi$
$450$	−442.004	−0.0463028
$451$	−7900.90	−0.824920
$452$	10005.7	1.04122
$453$	−4756.21	−0.493303
$454$	−1106.05	−0.114338
$455$	762.680	0.0785824
$456$	6152.15	0.631800
$457$	14344.7	1.46831	0.734155	−	0.678982i	$-0.237579\pi$
$457$	14344.7	1.46831	0.734155	+	0.678982i	$0.237579\pi$
$458$	2367.85	0.241578
$459$	−1629.29	−0.165683
$460$	−9299.59	−0.942599
$461$	−10742.1	−1.08527	−0.542634	−	0.839969i	$-0.682573\pi$
$461$	−10742.1	−1.08527	−0.542634	+	0.839969i	$0.682573\pi$
$462$	−1312.86	−0.132207
$463$	−17574.2	−1.76402	−0.882009	−	0.471232i	$-0.843809\pi$
$463$	−17574.2	−1.76402	−0.882009	+	0.471232i	$0.843809\pi$
$464$	1094.21	0.109477
$465$	−5208.26	−0.519413
$466$	10752.2	1.06886
$467$	8958.29	0.887667	0.443833	−	0.896109i	$-0.353618\pi$
$467$	8958.29	0.887667	0.443833	+	0.896109i	$0.353618\pi$
$468$	527.518	0.0521037
$469$	−1440.26	−0.141802
$470$	2220.78	0.217951
$471$	11232.2	1.09884
$472$	1306.94	0.127451
$473$	−14878.6	−1.44634
$474$	4256.19	0.412434
$475$	2703.49	0.261146
$476$	2145.72	0.206616
$477$	−5853.88	−0.561910
$478$	3265.29	0.312450
$479$	6909.71	0.659108	0.329554	−	0.944137i	$-0.393102\pi$
$479$	6909.71	0.659108	0.329554	+	0.944137i	$0.393102\pi$
$480$	−5406.92	−0.514148
$481$	794.342	0.0752991
$482$	−6225.19	−0.588277
$483$	−3789.47	−0.356992
$484$	−524.319	−0.0492411
$485$	−3316.01	−0.310458
$486$	408.665	0.0381429
$487$	−15008.4	−1.39650	−0.698248	−	0.715856i	$-0.746037\pi$
$487$	−15008.4	−1.39650	−0.698248	+	0.715856i	$0.746037\pi$
$488$	−499.123	−0.0462997
$489$	−3995.56	−0.369500
$490$	4867.76	0.448782
$491$	10939.9	1.00552	0.502761	−	0.864425i	$-0.332318\pi$
$491$	10939.9	1.00552	0.502761	+	0.864425i	$0.332318\pi$
$492$	−3238.94	−0.296794
$493$	−16024.9	−1.46394
$494$	1764.51	0.160706
$495$	3333.87	0.302720
$496$	730.857	0.0661622
$497$	1438.08	0.129792
$498$	−4265.18	−0.383789
$499$	−1960.94	−0.175919	−0.0879596	−	0.996124i	$-0.528035\pi$
$499$	−1960.94	−0.175919	−0.0879596	+	0.996124i	$0.528035\pi$
$500$	−7805.55	−0.698149
$501$	49.6751	0.00442978
$502$	−402.051	−0.0357458
$503$	−7812.26	−0.692508	−0.346254	−	0.938141i	$-0.612547\pi$
$503$	−7812.26	−0.692508	−0.346254	+	0.938141i	$0.612547\pi$
$504$	−1370.73	−0.121145
$505$	6693.54	0.589819
$506$	11693.5	1.02735
$507$	−6205.66	−0.543596
$508$	6901.45	0.602761
$509$	11915.7	1.03763	0.518816	−	0.854886i	$-0.326373\pi$
$509$	11915.7	1.03763	0.518816	+	0.854886i	$0.326373\pi$
$510$	2979.84	0.258725
$511$	5949.26	0.515029
$512$	1488.92	0.128519
$513$	−2499.57	−0.215124
$514$	11155.2	0.957270
$515$	12093.0	1.03472
$516$	−6099.42	−0.520372
$517$	5106.19	0.434371
$518$	−810.428	−0.0687416
$519$	−7191.20	−0.608205
$520$	−2457.20	−0.207222
$521$	12795.0	1.07593	0.537963	−	0.842969i	$-0.319194\pi$
$521$	12795.0	1.07593	0.537963	+	0.842969i	$0.319194\pi$
$522$	4019.43	0.337023
$523$	−18908.5	−1.58090	−0.790452	−	0.612524i	$-0.790154\pi$
$523$	−18908.5	−1.58090	−0.790452	+	0.612524i	$0.790154\pi$
$524$	3825.20	0.318902
$525$	−602.350	−0.0500737
$526$	−3381.08	−0.280270
$527$	−10703.5	−0.884732
$528$	−467.831	−0.0385601
$529$	21585.4	1.77409
$530$	10706.3	0.877457
$531$	−531.000	−0.0433963
$532$	3291.86	0.268271
$533$	−2365.96	−0.192272
$534$	−4917.94	−0.398539
$535$	−4223.14	−0.341275
$536$	4640.22	0.373931
$537$	3669.01	0.294841
$538$	9961.37	0.798263
$539$	11192.3	0.894412
$540$	1366.71	0.108914
$541$	−13497.9	−1.07268	−0.536340	−	0.844002i	$-0.680194\pi$
$541$	−13497.9	−1.07268	−0.536340	+	0.844002i	$0.680194\pi$
$542$	−3210.36	−0.254422
$543$	−12800.9	−1.01168
$544$	−11111.8	−0.875765
$545$	−4187.86	−0.329153
$546$	−393.141	−0.0308148
$547$	7859.46	0.614344	0.307172	−	0.951654i	$-0.400617\pi$
$547$	7859.46	0.614344	0.307172	+	0.951654i	$0.400617\pi$
$548$	6528.27	0.508894
$549$	202.790	0.0157648
$550$	1858.72	0.144102
$551$	−24584.6	−1.90080
$552$	12208.9	0.941388
$553$	5800.22	0.446022
$554$	12925.2	0.991222
$555$	2058.00	0.157400
$556$	−13605.2	−1.03775
$557$	20367.3	1.54936	0.774678	−	0.632356i	$-0.217912\pi$
$557$	20367.3	1.54936	0.774678	+	0.632356i	$0.217912\pi$
$558$	2684.72	0.203679
$559$	−4455.46	−0.337112
$560$	−277.281	−0.0209237
$561$	6851.48	0.515632
$562$	2057.12	0.154403
$563$	3392.89	0.253985	0.126992	−	0.991904i	$-0.459468\pi$
$563$	3392.89	0.253985	0.126992	+	0.991904i	$0.459468\pi$
$564$	2093.26	0.156280
$565$	18936.1	1.41000
$566$	−2595.93	−0.192783
$567$	556.917	0.0412492
$568$	−4633.21	−0.342263
$569$	8441.27	0.621927	0.310964	−	0.950422i	$-0.399348\pi$
$569$	8441.27	0.621927	0.310964	+	0.950422i	$0.399348\pi$
$570$	4571.53	0.335930
$571$	1840.67	0.134903	0.0674514	−	0.997723i	$-0.478513\pi$
$571$	1840.67	0.134903	0.0674514	+	0.997723i	$0.478513\pi$
$572$	−2218.32	−0.162155
$573$	−9241.34	−0.673757
$574$	2413.87	0.175528
$575$	5365.06	0.389110
$576$	2490.46	0.180154
$577$	−14007.9	−1.01067	−0.505335	−	0.862923i	$-0.668631\pi$
$577$	−14007.9	−1.01067	−0.505335	+	0.862923i	$0.668631\pi$
$578$	−2138.53	−0.153895
$579$	10409.1	0.747130
$580$	13442.3	0.962344
$581$	−5812.45	−0.415045
$582$	1709.31	0.121741
$583$	24616.8	1.74875
$584$	−19167.3	−1.35813
$585$	998.342	0.0705579
$586$	−6706.01	−0.472735
$587$	−26870.9	−1.88940	−0.944702	−	0.327931i	$-0.893649\pi$
$587$	−26870.9	−1.88940	−0.944702	+	0.327931i	$0.893649\pi$
$588$	4588.25	0.321797
$589$	−16420.9	−1.14874
$590$	971.159	0.0677661
$591$	−4463.21	−0.310646
$592$	−288.792	−0.0200495
$593$	−7529.46	−0.521413	−0.260707	−	0.965418i	$-0.583956\pi$
$593$	−7529.46	−0.521413	−0.260707	+	0.965418i	$0.583956\pi$
$594$	−1718.52	−0.118707
$595$	4060.84	0.279795
$596$	2247.97	0.154497
$597$	−9083.18	−0.622696
$598$	3501.66	0.239454
$599$	3363.55	0.229434	0.114717	−	0.993398i	$-0.463404\pi$
$599$	3363.55	0.229434	0.114717	+	0.993398i	$0.463404\pi$
$600$	1940.65	0.132045
$601$	13338.7	0.905316	0.452658	−	0.891684i	$-0.350476\pi$
$601$	13338.7	0.905316	0.452658	+	0.891684i	$0.350476\pi$
$602$	4545.68	0.307755
$603$	−1885.29	−0.127321
$604$	8199.25	0.552356
$605$	−992.288	−0.0666814
$606$	−3450.34	−0.231288
$607$	−7189.25	−0.480729	−0.240365	−	0.970683i	$-0.577267\pi$
$607$	−7189.25	−0.480729	−0.240365	+	0.970683i	$0.577267\pi$
$608$	−17047.2	−1.13710
$609$	5477.57	0.364470
$610$	−370.887	−0.0246177
$611$	1529.07	0.101243
$612$	2808.73	0.185517
$613$	−22156.6	−1.45986	−0.729932	−	0.683520i	$-0.760448\pi$
$613$	−22156.6	−1.45986	−0.729932	+	0.683520i	$0.760448\pi$
$614$	−1506.56	−0.0990225
$615$	−6129.78	−0.401913
$616$	5764.20	0.377023
$617$	−15462.4	−1.00890	−0.504452	−	0.863440i	$-0.668305\pi$
$617$	−15462.4	−1.00890	−0.504452	+	0.863440i	$0.668305\pi$
$618$	−6233.62	−0.405749
$619$	−18599.5	−1.20772	−0.603858	−	0.797092i	$-0.706371\pi$
$619$	−18599.5	−1.20772	−0.603858	+	0.797092i	$0.706371\pi$
$620$	8978.54	0.581592
$621$	−4960.39	−0.320537
$622$	7007.69	0.451741
$623$	−6702.02	−0.430997
$624$	−140.094	−0.00898757
$625$	−11121.9	−0.711800
$626$	−6087.57	−0.388671
$627$	10511.2	0.669501
$628$	−19363.3	−1.23038
$629$	4229.42	0.268105
$630$	−1018.56	−0.0644133
$631$	−14886.6	−0.939187	−0.469594	−	0.882883i	$-0.655600\pi$
$631$	−14886.6	−0.939187	−0.469594	+	0.882883i	$0.655600\pi$
$632$	−18687.1	−1.17616
$633$	−4967.70	−0.311925
$634$	1455.52	0.0911768
$635$	13061.2	0.816248
$636$	10091.5	0.629176
$637$	3351.59	0.208469
$638$	−16902.5	−1.04887
$639$	1882.44	0.116539
$640$	9863.61	0.609208
$641$	30295.2	1.86675	0.933376	−	0.358899i	$-0.116848\pi$
$641$	30295.2	1.86675	0.933376	+	0.358899i	$0.116848\pi$
$642$	2176.91	0.133825
$643$	12393.0	0.760080	0.380040	−	0.924970i	$-0.375910\pi$
$643$	12393.0	0.760080	0.380040	+	0.924970i	$0.375910\pi$
$644$	6532.70	0.399727
$645$	−11543.3	−0.704678
$646$	9395.00	0.572200
$647$	−18759.0	−1.13986	−0.569931	−	0.821692i	$-0.693030\pi$
$647$	−18759.0	−1.13986	−0.569931	+	0.821692i	$0.693030\pi$
$648$	−1794.27	−0.108774
$649$	2232.96	0.135056
$650$	556.601	0.0335872
$651$	3658.65	0.220267
$652$	6887.97	0.413733
$653$	17228.9	1.03249	0.516247	−	0.856440i	$-0.327329\pi$
$653$	17228.9	1.03249	0.516247	+	0.856440i	$0.327329\pi$
$654$	2158.73	0.129072
$655$	7239.29	0.431851
$656$	860.171	0.0511952
$657$	7787.54	0.462437
$658$	−1560.03	−0.0924262
$659$	10789.4	0.637776	0.318888	−	0.947792i	$-0.396691\pi$
$659$	10789.4	0.637776	0.318888	+	0.947792i	$0.396691\pi$
$660$	−5747.28	−0.338958
$661$	91.3456	0.00537509	0.00268754	−	0.999996i	$-0.499145\pi$
$661$	91.3456	0.00537509	0.00268754	+	0.999996i	$0.499145\pi$
$662$	13690.9	0.803797
$663$	2051.70	0.120183
$664$	18726.6	1.09448
$665$	6229.94	0.363288
$666$	−1060.84	−0.0617220
$667$	−48788.0	−2.83220
$668$	−85.6351	−0.00496006
$669$	−3675.59	−0.212416
$670$	3448.05	0.198820
$671$	−852.773	−0.0490625
$672$	3798.21	0.218034
$673$	−25294.9	−1.44881	−0.724404	−	0.689376i	$-0.757885\pi$
$673$	−25294.9	−1.44881	−0.724404	+	0.689376i	$0.757885\pi$
$674$	12438.8	0.710868
$675$	−788.471	−0.0449604
$676$	10698.0	0.608670
$677$	1529.96	0.0868557	0.0434278	−	0.999057i	$-0.486172\pi$
$677$	1529.96	0.0868557	0.0434278	+	0.999057i	$0.486172\pi$
$678$	−9761.06	−0.552908
$679$	2329.40	0.131656
$680$	−13083.2	−0.737820
$681$	−1973.03	−0.111023
$682$	−11289.8	−0.633882
$683$	7588.88	0.425154	0.212577	−	0.977144i	$-0.431814\pi$
$683$	7588.88	0.425154	0.212577	+	0.977144i	$0.431814\pi$
$684$	4309.03	0.240877
$685$	12354.9	0.689135
$686$	−7385.54	−0.411051
$687$	4223.91	0.234574
$688$	1619.83	0.0897610
$689$	7371.60	0.407599
$690$	9072.18	0.500539
$691$	−7781.95	−0.428421	−0.214211	−	0.976787i	$-0.568718\pi$
$691$	−7781.95	−0.428421	−0.214211	+	0.976787i	$0.568718\pi$
$692$	12396.9	0.681013
$693$	−2341.95	−0.128374
$694$	−2256.87	−0.123443
$695$	−25748.2	−1.40530
$696$	−17647.6	−0.961108
$697$	−12597.4	−0.684591
$698$	3688.62	0.200024
$699$	19180.4	1.03787
$700$	1038.39	0.0560680
$701$	16501.9	0.889111	0.444555	−	0.895751i	$-0.353362\pi$
$701$	16501.9	0.889111	0.444555	+	0.895751i	$0.353362\pi$
$702$	−514.618	−0.0276681
$703$	6488.57	0.348110
$704$	−10472.9	−0.560669
$705$	3961.55	0.211632
$706$	−1070.08	−0.0570440
$707$	−4702.02	−0.250124
$708$	915.394	0.0485913
$709$	−26435.2	−1.40027	−0.700136	−	0.714009i	$-0.746877\pi$
$709$	−26435.2	−1.40027	−0.700136	+	0.714009i	$0.746877\pi$
$710$	−3442.84	−0.181982
$711$	7592.44	0.400476
$712$	21592.6	1.13654
$713$	−32587.2	−1.71164
$714$	−2093.25	−0.109717
$715$	−4198.23	−0.219587
$716$	−6325.03	−0.330136
$717$	5824.81	0.303391
$718$	−11980.9	−0.622734
$719$	−9576.02	−0.496697	−0.248348	−	0.968671i	$-0.579888\pi$
$719$	−9576.02	−0.496697	−0.248348	+	0.968671i	$0.579888\pi$
$720$	−362.959	−0.0187870
$721$	−8495.00	−0.438794
$722$	2878.23	0.148361
$723$	−11104.8	−0.571222
$724$	22067.6	1.13278
$725$	−7755.02	−0.397261
$726$	511.498	0.0261480
$727$	−11408.0	−0.581979	−0.290990	−	0.956726i	$-0.593985\pi$
$727$	−11408.0	−0.581979	−0.290990	+	0.956726i	$0.593985\pi$
$728$	1726.11	0.0878764
$729$	729.000	0.0370370
$730$	−14242.8	−0.722124
$731$	−23722.8	−1.20030
$732$	−349.591	−0.0176520
$733$	−25829.0	−1.30152	−0.650762	−	0.759282i	$-0.725550\pi$
$733$	−25829.0	−1.30152	−0.650762	+	0.759282i	$0.725550\pi$
$734$	−14655.9	−0.737003
$735$	8683.39	0.435771
$736$	−33830.2	−1.69429
$737$	7928.01	0.396244
$738$	3159.74	0.157604
$739$	37906.8	1.88691	0.943454	−	0.331503i	$-0.107556\pi$
$739$	37906.8	1.88691	0.943454	+	0.331503i	$0.107556\pi$
$740$	−3547.80	−0.176243
$741$	3147.63	0.156047
$742$	−7520.88	−0.372102
$743$	17868.7	0.882286	0.441143	−	0.897437i	$-0.354573\pi$
$743$	17868.7	0.882286	0.441143	+	0.897437i	$0.354573\pi$
$744$	−11787.4	−0.580845
$745$	4254.35	0.209218
$746$	15343.1	0.753017
$747$	−7608.46	−0.372663
$748$	−11811.3	−0.577358
$749$	2966.63	0.144724
$750$	7614.67	0.370731
$751$	−12.6803	−0.000616125 0	−0.000308063	−	1.00000i	$-0.500098\pi$
$751$	−12.6803	−0.000616125 0	−0.000308063		1.00000i	$0.500098\pi$
$752$	−555.911	−0.0269574
$753$	−717.201	−0.0347095
$754$	−5061.54	−0.244470
$755$	15517.3	0.747990
$756$	−960.072	−0.0461872
$757$	39700.2	1.90611	0.953056	−	0.302793i	$-0.0979192\pi$
$757$	39700.2	1.90611	0.953056	+	0.302793i	$0.0979192\pi$
$758$	14361.8	0.688186
$759$	20859.5	0.997563
$760$	−20071.6	−0.957992
$761$	−9323.77	−0.444134	−0.222067	−	0.975031i	$-0.571280\pi$
$761$	−9323.77	−0.444134	−0.222067	+	0.975031i	$0.571280\pi$
$762$	−6732.69	−0.320078
$763$	2941.85	0.139583
$764$	15931.2	0.754412
$765$	5315.61	0.251224
$766$	−14616.2	−0.689433
$767$	668.670	0.0314789
$768$	−11725.6	−0.550928
$769$	20630.0	0.967408	0.483704	−	0.875232i	$-0.339291\pi$
$769$	20630.0	0.967408	0.483704	+	0.875232i	$0.339291\pi$
$770$	4283.25	0.200464
$771$	19899.3	0.929517
$772$	−17944.3	−0.836568
$773$	9335.33	0.434370	0.217185	−	0.976130i	$-0.430312\pi$
$773$	9335.33	0.434370	0.217185	+	0.976130i	$0.430312\pi$
$774$	5950.27	0.276328
$775$	−5179.84	−0.240084
$776$	−7504.85	−0.347176
$777$	−1445.69	−0.0667487
$778$	−16951.9	−0.781175
$779$	−19326.3	−0.888879
$780$	−1721.05	−0.0790043
$781$	−7916.04	−0.362686
$782$	18644.3	0.852584
$783$	7170.09	0.327252
$784$	−1218.51	−0.0555080
$785$	−36645.5	−1.66616
$786$	−3731.66	−0.169343
$787$	−14965.2	−0.677831	−0.338916	−	0.940817i	$-0.610060\pi$
$787$	−14965.2	−0.677831	−0.338916	+	0.940817i	$0.610060\pi$
$788$	7694.15	0.347833
$789$	−6031.36	−0.272144
$790$	−13886.0	−0.625369
$791$	−13302.1	−0.597937
$792$	7545.29	0.338523
$793$	−255.366	−0.0114355
$794$	−3076.78	−0.137520
$795$	19098.5	0.852018
$796$	15658.5	0.697239
$797$	35050.2	1.55777	0.778885	−	0.627167i	$-0.215786\pi$
$797$	35050.2	1.55777	0.778885	+	0.627167i	$0.215786\pi$
$798$	−3211.37	−0.142458
$799$	8141.43	0.360480
$800$	−5377.43	−0.237651
$801$	−8772.90	−0.386985
$802$	−18364.1	−0.808552
$803$	−32748.2	−1.43918
$804$	3250.06	0.142563
$805$	12363.3	0.541303
$806$	−3380.77	−0.147745
$807$	17769.7	0.775120
$808$	15149.0	0.659577
$809$	−14925.1	−0.648628	−0.324314	−	0.945950i	$-0.605133\pi$
$809$	−14925.1	−0.648628	−0.324314	+	0.945950i	$0.605133\pi$
$810$	−1333.29	−0.0578357
$811$	−28928.7	−1.25256	−0.626280	−	0.779598i	$-0.715423\pi$
$811$	−28928.7	−1.25256	−0.626280	+	0.779598i	$0.715423\pi$
$812$	−9442.81	−0.408100
$813$	−5726.82	−0.247046
$814$	4461.06	0.192089
$815$	13035.7	0.560269
$816$	−745.921	−0.0320005
$817$	−36394.4	−1.55848
$818$	10833.0	0.463040
$819$	−701.307	−0.0299214
$820$	10567.2	0.450026
$821$	−1009.83	−0.0429274	−0.0214637	−	0.999770i	$-0.506833\pi$
$821$	−1009.83	−0.0429274	−0.0214637	+	0.999770i	$0.506833\pi$
$822$	−6368.63	−0.270233
$823$	4148.74	0.175718	0.0878590	−	0.996133i	$-0.471998\pi$
$823$	4148.74	0.175718	0.0878590	+	0.996133i	$0.471998\pi$
$824$	27369.2	1.15710
$825$	3315.68	0.139924
$826$	−682.211	−0.0287375
$827$	46387.4	1.95048	0.975241	−	0.221147i	$-0.0709800\pi$
$827$	46387.4	1.95048	0.975241	+	0.221147i	$0.0709800\pi$
$828$	8551.25	0.358909
$829$	16193.2	0.678423	0.339211	−	0.940710i	$-0.389840\pi$
$829$	16193.2	0.678423	0.339211	+	0.940710i	$0.389840\pi$
$830$	13915.3	0.581936
$831$	23056.6	0.962485
$832$	−3136.15	−0.130681
$833$	17845.3	0.742262
$834$	13272.5	0.551066
$835$	−162.067	−0.00671683
$836$	−18120.3	−0.749647
$837$	4789.15	0.197774
$838$	11409.5	0.470327
$839$	−26644.1	−1.09637	−0.548186	−	0.836357i	$-0.684681\pi$
$839$	−26644.1	−1.09637	−0.548186	+	0.836357i	$0.684681\pi$
$840$	4472.06	0.183691
$841$	46132.5	1.89153
$842$	1392.22	0.0569824
$843$	3669.60	0.149926
$844$	8563.84	0.349265
$845$	20246.2	0.824249
$846$	−2042.07	−0.0829881
$847$	697.054	0.0282775
$848$	−2680.03	−0.108529
$849$	−4630.77	−0.187194
$850$	2963.58	0.119588
$851$	12876.6	0.518687
$852$	−3245.15	−0.130489
$853$	−28697.2	−1.15190	−0.575952	−	0.817484i	$-0.695368\pi$
$853$	−28697.2	−1.15190	−0.575952	+	0.817484i	$0.695368\pi$
$854$	260.538	0.0104396
$855$	8154.95	0.326191
$856$	−9557.89	−0.381638
$857$	19383.8	0.772623	0.386311	−	0.922368i	$-0.373749\pi$
$857$	19383.8	0.772623	0.386311	+	0.922368i	$0.373749\pi$
$858$	2164.07	0.0861075
$859$	−18991.9	−0.754362	−0.377181	−	0.926140i	$-0.623107\pi$
$859$	−18991.9	−0.754362	−0.377181	+	0.926140i	$0.623107\pi$
$860$	19899.6	0.789035
$861$	4306.00	0.170439
$862$	2891.66	0.114258
$863$	34623.5	1.36570	0.682849	−	0.730560i	$-0.260741\pi$
$863$	34623.5	1.36570	0.682849	+	0.730560i	$0.260741\pi$
$864$	4971.83	0.195770
$865$	23461.6	0.922216
$866$	20087.7	0.788229
$867$	−3814.84	−0.149433
$868$	−6307.17	−0.246635
$869$	−31927.7	−1.24635
$870$	−13113.6	−0.511024
$871$	2374.08	0.0923565
$872$	−9478.06	−0.368082
$873$	3049.17	0.118211
$874$	28603.3	1.10700
$875$	10377.1	0.400924
$876$	−13425.0	−0.517795
$877$	−33705.2	−1.29777	−0.648884	−	0.760887i	$-0.724764\pi$
$877$	−33705.2	−1.29777	−0.648884	+	0.760887i	$0.724764\pi$
$878$	531.947	0.0204469
$879$	−11962.6	−0.459030
$880$	1526.31	0.0584683
$881$	−24525.5	−0.937896	−0.468948	−	0.883226i	$-0.655367\pi$
$881$	−24525.5	−0.937896	−0.468948	+	0.883226i	$0.655367\pi$
$882$	−4476.05	−0.170880
$883$	3141.61	0.119732	0.0598662	−	0.998206i	$-0.480933\pi$
$883$	3141.61	0.119732	0.0598662	+	0.998206i	$0.480933\pi$
$884$	−3536.94	−0.134570
$885$	1732.41	0.0658014
$886$	−15208.8	−0.576691
$887$	−5590.87	−0.211638	−0.105819	−	0.994385i	$-0.533746\pi$
$887$	−5590.87	−0.211638	−0.105819	+	0.994385i	$0.533746\pi$
$888$	4657.71	0.176016
$889$	−9175.11	−0.346145
$890$	16045.0	0.604302
$891$	−3065.59	−0.115265
$892$	6336.37	0.237845
$893$	12490.2	0.468050
$894$	−2193.00	−0.0820413
$895$	−11970.3	−0.447065
$896$	−6928.90	−0.258346
$897$	6246.45	0.232512
$898$	15234.2	0.566116
$899$	47103.7	1.74749
$900$	1359.25	0.0503426
$901$	39249.6	1.45127
$902$	−13287.3	−0.490488
$903$	8108.85	0.298832
$904$	42856.7	1.57676
$905$	41763.5	1.53399
$906$	−7998.75	−0.293312
$907$	−27732.8	−1.01527	−0.507637	−	0.861571i	$-0.669481\pi$
$907$	−27732.8	−1.01527	−0.507637	+	0.861571i	$0.669481\pi$
$908$	3401.32	0.124314
$909$	−6154.91	−0.224582
$910$	1282.64	0.0467242
$911$	7724.27	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$911$	7724.27	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$912$	−1144.36	−0.0415498
$913$	31995.1	1.15978
$914$	24124.2	0.873040
$915$	−661.610	−0.0239040
$916$	−7281.62	−0.262655
$917$	−5085.39	−0.183135
$918$	−2740.05	−0.0985133
$919$	40052.6	1.43767	0.718833	−	0.695183i	$-0.244677\pi$
$919$	40052.6	1.43767	0.718833	+	0.695183i	$0.244677\pi$
$920$	−39832.1	−1.42742
$921$	−2687.49	−0.0961517
$922$	−18065.5	−0.645288
$923$	−2370.49	−0.0845349
$924$	4037.30	0.143742
$925$	2046.77	0.0727540
$926$	−29555.3	−1.04887
$927$	−11119.9	−0.393986
$928$	48900.5	1.72978
$929$	14184.7	0.500952	0.250476	−	0.968123i	$-0.419413\pi$
$929$	14184.7	0.500952	0.250476	+	0.968123i	$0.419413\pi$
$930$	−8758.99	−0.308837
$931$	27377.5	0.963759
$932$	−33065.3	−1.16211
$933$	12500.7	0.438644
$934$	15065.6	0.527796
$935$	−22353.2	−0.781848
$936$	2259.47	0.0789028
$937$	−30556.2	−1.06534	−0.532672	−	0.846322i	$-0.678812\pi$
$937$	−30556.2	−1.06534	−0.532672	+	0.846322i	$0.678812\pi$
$938$	−2422.15	−0.0843135
$939$	−10859.4	−0.377403
$940$	−6829.34	−0.236967
$941$	−9474.36	−0.328220	−0.164110	−	0.986442i	$-0.552475\pi$
$941$	−9474.36	−0.328220	−0.164110	+	0.986442i	$0.552475\pi$
$942$	18889.8	0.653357
$943$	−38353.0	−1.32444
$944$	−243.103	−0.00838170
$945$	−1816.96	−0.0625458
$946$	−25022.1	−0.859976
$947$	55248.5	1.89581	0.947906	−	0.318549i	$-0.103196\pi$
$947$	55248.5	1.89581	0.947906	+	0.318549i	$0.103196\pi$
$948$	−13088.6	−0.448417
$949$	−9806.58	−0.335443
$950$	4546.59	0.155274
$951$	2596.44	0.0885335
$952$	9190.58	0.312887
$953$	6103.29	0.207455	0.103728	−	0.994606i	$-0.466923\pi$
$953$	6103.29	0.207455	0.103728	+	0.994606i	$0.466923\pi$
$954$	−9844.77	−0.334105
$955$	30150.2	1.02161
$956$	−10041.4	−0.339710
$957$	−30151.7	−1.01846
$958$	11620.4	0.391898
$959$	−8678.99	−0.292241
$960$	−8125.21	−0.273167
$961$	1671.17	0.0560963
$962$	1335.88	0.0447720
$963$	3883.30	0.129946
$964$	19143.7	0.639602
$965$	−33960.1	−1.13287
$966$	−6372.95	−0.212263
$967$	−22070.6	−0.733965	−0.366982	−	0.930228i	$-0.619609\pi$
$967$	−22070.6	−0.733965	−0.366982	+	0.930228i	$0.619609\pi$
$968$	−2245.77	−0.0745678
$969$	16759.3	0.555611
$970$	−5576.69	−0.184595
$971$	−46982.7	−1.55278	−0.776389	−	0.630254i	$-0.782951\pi$
$971$	−46982.7	−1.55278	−0.776389	+	0.630254i	$0.782951\pi$
$972$	−1256.73	−0.0414707
$973$	18087.4	0.595945
$974$	−25240.3	−0.830341
$975$	992.895	0.0326134
$976$	92.8414	0.00304486
$977$	−38521.7	−1.26143	−0.630715	−	0.776014i	$-0.717238\pi$
$977$	−38521.7	−1.26143	−0.630715	+	0.776014i	$0.717238\pi$
$978$	−6719.53	−0.219700
$979$	36891.8	1.20436
$980$	−14969.3	−0.487937
$981$	3850.86	0.125330
$982$	18398.2	0.597872
$983$	−17793.1	−0.577327	−0.288664	−	0.957431i	$-0.593211\pi$
$983$	−17793.1	−0.577327	−0.288664	+	0.957431i	$0.593211\pi$
$984$	−13873.1	−0.449448
$985$	14561.4	0.471030
$986$	−26949.8	−0.870443
$987$	−2782.88	−0.0897466
$988$	−5426.21	−0.174727
$989$	−72224.5	−2.32215
$990$	5606.74	0.179994
$991$	−52191.4	−1.67297	−0.836485	−	0.547990i	$-0.815393\pi$
$991$	−52191.4	−1.67297	−0.836485	+	0.547990i	$0.815393\pi$
$992$	32662.3	1.04539
$993$	24422.7	0.780494
$994$	2418.50	0.0771731
$995$	29634.2	0.944189
$996$	13116.3	0.417274
$997$	6732.09	0.213849	0.106924	−	0.994267i	$-0.465900\pi$
$997$	6732.09	0.213849	0.106924	+	0.994267i	$0.465900\pi$
$998$	−3297.81	−0.104600
$999$	−1892.39	−0.0599326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.a.1.6	✓	7
3.2	odd	2		531.4.a.d.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.6	✓	7	1.1	even	1	trivial
531.4.a.d.1.2		7	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q+1.68175 q^{2} +3.00000 q^{3} -5.17172 q^{4} -9.78761 q^{5} +5.04525 q^{6} +6.87552 q^{7} -22.1515 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.68175 q^{2} +3.00000 q^{3} -5.17172 q^{4} -9.78761 q^{5} +5.04525 q^{6} +6.87552 q^{7} -22.1515 q^{8} +9.00000 q^{9} -16.4603 q^{10} -37.8468 q^{11} -15.5151 q^{12} -11.3334 q^{13} +11.5629 q^{14} -29.3628 q^{15} +4.12039 q^{16} -60.3439 q^{17} +15.1358 q^{18} -92.5767 q^{19} +50.6188 q^{20} +20.6266 q^{21} -63.6489 q^{22} -183.718 q^{23} -66.4546 q^{24} -29.2026 q^{25} -19.0599 q^{26} +27.0000 q^{27} -35.5582 q^{28} +265.559 q^{29} -49.3810 q^{30} +177.376 q^{31} +184.142 q^{32} -113.540 q^{33} -101.483 q^{34} -67.2949 q^{35} -46.5454 q^{36} -70.0886 q^{37} -155.691 q^{38} -34.0002 q^{39} +216.811 q^{40} +208.760 q^{41} +34.6887 q^{42} +393.127 q^{43} +195.733 q^{44} -88.0885 q^{45} -308.968 q^{46} -134.917 q^{47} +12.3612 q^{48} -295.727 q^{49} -49.1115 q^{50} -181.032 q^{51} +58.6131 q^{52} -650.432 q^{53} +45.4073 q^{54} +370.430 q^{55} -152.303 q^{56} -277.730 q^{57} +446.604 q^{58} -59.0000 q^{59} +151.856 q^{60} +22.5322 q^{61} +298.302 q^{62} +61.8797 q^{63} +276.717 q^{64} +110.927 q^{65} -190.947 q^{66} -209.476 q^{67} +312.082 q^{68} -551.155 q^{69} -113.173 q^{70} +209.160 q^{71} -199.364 q^{72} +865.282 q^{73} -117.872 q^{74} -87.6079 q^{75} +478.781 q^{76} -260.217 q^{77} -57.1798 q^{78} +843.604 q^{79} -40.3287 q^{80} +81.0000 q^{81} +351.082 q^{82} -845.384 q^{83} -106.675 q^{84} +590.623 q^{85} +661.141 q^{86} +796.677 q^{87} +838.365 q^{88} -974.766 q^{89} -148.143 q^{90} -77.9229 q^{91} +950.139 q^{92} +532.127 q^{93} -226.897 q^{94} +906.105 q^{95} +552.425 q^{96} +338.796 q^{97} -497.339 q^{98} -340.621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9}+O(q^{10}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9} - 79 q^{10} - 131 q^{11} + 66 q^{12} - 123 q^{13} - 117 q^{14} - 84 q^{15} + 202 q^{16} - 235 q^{17} - 72 q^{18} - 80 q^{19} + 61 q^{20} - 177 q^{21} + 688 q^{22} - 274 q^{23} - 351 q^{24} + 193 q^{25} - 180 q^{26} + 189 q^{27} - 118 q^{28} - 406 q^{29} - 237 q^{30} - 346 q^{31} - 854 q^{32} - 393 q^{33} + 178 q^{34} - 424 q^{35} + 198 q^{36} - 157 q^{37} - 129 q^{38} - 369 q^{39} - 590 q^{40} - 825 q^{41} - 351 q^{42} - 815 q^{43} - 1690 q^{44} - 252 q^{45} + 1457 q^{46} - 1196 q^{47} + 606 q^{48} + 914 q^{49} + 713 q^{50} - 705 q^{51} + 1030 q^{52} - 900 q^{53} - 216 q^{54} - 1044 q^{55} + 2172 q^{56} - 240 q^{57} + 1242 q^{58} - 413 q^{59} + 183 q^{60} + 420 q^{61} + 646 q^{62} - 531 q^{63} + 3541 q^{64} + 190 q^{65} + 2064 q^{66} + 1316 q^{67} - 611 q^{68} - 822 q^{69} + 4658 q^{70} - 173 q^{71} - 1053 q^{72} - 418 q^{73} + 660 q^{74} + 579 q^{75} + 1540 q^{76} - 753 q^{77} - 540 q^{78} + 2635 q^{79} + 6155 q^{80} + 567 q^{81} - 125 q^{82} + 457 q^{83} - 354 q^{84} + 1270 q^{85} + 3482 q^{86} - 1218 q^{87} + 7685 q^{88} + 592 q^{89} - 711 q^{90} + 3179 q^{91} - 3500 q^{92} - 1038 q^{93} + 2064 q^{94} - 2250 q^{95} - 2562 q^{96} - 1906 q^{97} + 2994 q^{98} - 1179 q^{99}+O(q^{100}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9 - 79 * q^10 - 131 * q^11 + 66 * q^12 - 123 * q^13 - 117 * q^14 - 84 * q^15 + 202 * q^16 - 235 * q^17 - 72 * q^18 - 80 * q^19 + 61 * q^20 - 177 * q^21 + 688 * q^22 - 274 * q^23 - 351 * q^24 + 193 * q^25 - 180 * q^26 + 189 * q^27 - 118 * q^28 - 406 * q^29 - 237 * q^30 - 346 * q^31 - 854 * q^32 - 393 * q^33 + 178 * q^34 - 424 * q^35 + 198 * q^36 - 157 * q^37 - 129 * q^38 - 369 * q^39 - 590 * q^40 - 825 * q^41 - 351 * q^42 - 815 * q^43 - 1690 * q^44 - 252 * q^45 + 1457 * q^46 - 1196 * q^47 + 606 * q^48 + 914 * q^49 + 713 * q^50 - 705 * q^51 + 1030 * q^52 - 900 * q^53 - 216 * q^54 - 1044 * q^55 + 2172 * q^56 - 240 * q^57 + 1242 * q^58 - 413 * q^59 + 183 * q^60 + 420 * q^61 + 646 * q^62 - 531 * q^63 + 3541 * q^64 + 190 * q^65 + 2064 * q^66 + 1316 * q^67 - 611 * q^68 - 822 * q^69 + 4658 * q^70 - 173 * q^71 - 1053 * q^72 - 418 * q^73 + 660 * q^74 + 579 * q^75 + 1540 * q^76 - 753 * q^77 - 540 * q^78 + 2635 * q^79 + 6155 * q^80 + 567 * q^81 - 125 * q^82 + 457 * q^83 - 354 * q^84 + 1270 * q^85 + 3482 * q^86 - 1218 * q^87 + 7685 * q^88 + 592 * q^89 - 711 * q^90 + 3179 * q^91 - 3500 * q^92 - 1038 * q^93 + 2064 * q^94 - 2250 * q^95 - 2562 * q^96 - 1906 * q^97 + 2994 * q^98 - 1179 * q^99

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