LMFDB - Embedded newform 43.4.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [43,4,Mod(1,43)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(43, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("43.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	43.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:	$2.53708213025$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 $ x^6 - 32x^4 - 16x^3 + 251x^2 + 276x + 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.299707$ of defining polynomial
Character	$\chi$	$=$	43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29971 q^{2} +1.43046 q^{3} -6.31076 q^{4} +20.4116 q^{5} +1.85918 q^{6} +29.9522 q^{7} -18.5998 q^{8} -24.9538 q^{9} +O(q^{10})$	\(q+1.29971 q^{2} +1.43046 q^{3} -6.31076 q^{4} +20.4116 q^{5} +1.85918 q^{6} +29.9522 q^{7} -18.5998 q^{8} -24.9538 q^{9} +26.5291 q^{10} -22.8719 q^{11} -9.02731 q^{12} -44.4397 q^{13} +38.9292 q^{14} +29.1981 q^{15} +26.3118 q^{16} -13.0970 q^{17} -32.4326 q^{18} +5.41527 q^{19} -128.813 q^{20} +42.8456 q^{21} -29.7268 q^{22} -175.226 q^{23} -26.6063 q^{24} +291.634 q^{25} -57.7586 q^{26} -74.3180 q^{27} -189.021 q^{28} +165.972 q^{29} +37.9489 q^{30} -155.581 q^{31} +182.996 q^{32} -32.7174 q^{33} -17.0222 q^{34} +611.374 q^{35} +157.477 q^{36} -95.3991 q^{37} +7.03827 q^{38} -63.5693 q^{39} -379.652 q^{40} +189.928 q^{41} +55.6867 q^{42} -43.0000 q^{43} +144.339 q^{44} -509.347 q^{45} -227.743 q^{46} -37.2349 q^{47} +37.6381 q^{48} +554.137 q^{49} +379.039 q^{50} -18.7348 q^{51} +280.448 q^{52} +559.862 q^{53} -96.5916 q^{54} -466.852 q^{55} -557.106 q^{56} +7.74635 q^{57} +215.715 q^{58} -82.3042 q^{59} -184.262 q^{60} -640.304 q^{61} -202.210 q^{62} -747.422 q^{63} +27.3469 q^{64} -907.085 q^{65} -42.5231 q^{66} -509.592 q^{67} +82.6519 q^{68} -250.655 q^{69} +794.607 q^{70} +792.932 q^{71} +464.135 q^{72} +612.727 q^{73} -123.991 q^{74} +417.171 q^{75} -34.1745 q^{76} -685.065 q^{77} -82.6215 q^{78} +237.047 q^{79} +537.066 q^{80} +567.443 q^{81} +246.850 q^{82} +418.683 q^{83} -270.388 q^{84} -267.330 q^{85} -55.8874 q^{86} +237.417 q^{87} +425.413 q^{88} +113.788 q^{89} -662.002 q^{90} -1331.07 q^{91} +1105.81 q^{92} -222.553 q^{93} -48.3944 q^{94} +110.534 q^{95} +261.769 q^{96} +1649.00 q^{97} +720.216 q^{98} +570.740 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 7 q^{3} + 22 q^{4} + 43 q^{5} - 3 q^{6} + 8 q^{7} + 54 q^{8} + 81 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 7 * q^3 + 22 * q^4 + 43 * q^5 - 3 * q^6 + 8 * q^7 + 54 * q^8 + 81 * q^9	\( 6 q + 6 q^{2} + 7 q^{3} + 22 q^{4} + 43 q^{5} - 3 q^{6} + 8 q^{7} + 54 q^{8} + 81 q^{9} + 57 q^{10} - 28 q^{11} - 157 q^{12} + 56 q^{13} - 184 q^{14} - 124 q^{15} - 54 q^{16} + 19 q^{17} - 81 q^{18} - 75 q^{19} + 135 q^{20} - 18 q^{21} - 504 q^{22} + 131 q^{23} - 567 q^{24} + 105 q^{25} + 44 q^{26} + 238 q^{27} - 404 q^{28} + 515 q^{29} - 396 q^{30} + 237 q^{31} + 558 q^{32} + 540 q^{33} - 107 q^{34} + 198 q^{35} + 73 q^{36} + 269 q^{37} + 527 q^{38} + 290 q^{39} + 613 q^{40} + 471 q^{41} + 362 q^{42} - 258 q^{43} - 428 q^{44} + 334 q^{45} - 67 q^{46} + 415 q^{47} - 989 q^{48} + 350 q^{49} + 1335 q^{50} - 1241 q^{51} - 8 q^{52} + 450 q^{53} + 402 q^{54} - 1732 q^{55} - 780 q^{56} - 1000 q^{57} - 1055 q^{58} + 356 q^{59} - 2732 q^{60} - 1328 q^{61} + 1603 q^{62} - 2290 q^{63} + 466 q^{64} - 62 q^{65} + 156 q^{66} - 632 q^{67} + 571 q^{68} - 1130 q^{69} - 1902 q^{70} - 144 q^{71} + 567 q^{72} + 864 q^{73} + 1207 q^{74} - 2494 q^{75} + 1005 q^{76} + 2660 q^{77} + 2222 q^{78} - 1613 q^{79} + 2399 q^{80} - 102 q^{81} + 1673 q^{82} - 682 q^{83} + 3758 q^{84} + 84 q^{85} - 258 q^{86} + 449 q^{87} - 608 q^{88} + 3378 q^{89} + 930 q^{90} - 3900 q^{91} + 3491 q^{92} + 1879 q^{93} + 3197 q^{94} - 79 q^{95} - 591 q^{96} - 55 q^{97} + 2398 q^{98} - 1612 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 7 * q^3 + 22 * q^4 + 43 * q^5 - 3 * q^6 + 8 * q^7 + 54 * q^8 + 81 * q^9 + 57 * q^10 - 28 * q^11 - 157 * q^12 + 56 * q^13 - 184 * q^14 - 124 * q^15 - 54 * q^16 + 19 * q^17 - 81 * q^18 - 75 * q^19 + 135 * q^20 - 18 * q^21 - 504 * q^22 + 131 * q^23 - 567 * q^24 + 105 * q^25 + 44 * q^26 + 238 * q^27 - 404 * q^28 + 515 * q^29 - 396 * q^30 + 237 * q^31 + 558 * q^32 + 540 * q^33 - 107 * q^34 + 198 * q^35 + 73 * q^36 + 269 * q^37 + 527 * q^38 + 290 * q^39 + 613 * q^40 + 471 * q^41 + 362 * q^42 - 258 * q^43 - 428 * q^44 + 334 * q^45 - 67 * q^46 + 415 * q^47 - 989 * q^48 + 350 * q^49 + 1335 * q^50 - 1241 * q^51 - 8 * q^52 + 450 * q^53 + 402 * q^54 - 1732 * q^55 - 780 * q^56 - 1000 * q^57 - 1055 * q^58 + 356 * q^59 - 2732 * q^60 - 1328 * q^61 + 1603 * q^62 - 2290 * q^63 + 466 * q^64 - 62 * q^65 + 156 * q^66 - 632 * q^67 + 571 * q^68 - 1130 * q^69 - 1902 * q^70 - 144 * q^71 + 567 * q^72 + 864 * q^73 + 1207 * q^74 - 2494 * q^75 + 1005 * q^76 + 2660 * q^77 + 2222 * q^78 - 1613 * q^79 + 2399 * q^80 - 102 * q^81 + 1673 * q^82 - 682 * q^83 + 3758 * q^84 + 84 * q^85 - 258 * q^86 + 449 * q^87 - 608 * q^88 + 3378 * q^89 + 930 * q^90 - 3900 * q^91 + 3491 * q^92 + 1879 * q^93 + 3197 * q^94 - 79 * q^95 - 591 * q^96 - 55 * q^97 + 2398 * q^98 - 1612 * 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 $

$2$	1.29971	0.459516	0.229758	−	0.973248i	$-0.426207\pi$
$2$	1.29971	0.459516	0.229758	+	0.973248i	$0.426207\pi$
$3$	1.43046	0.275293	0.137646	−	0.990481i	$-0.456046\pi$
$3$	1.43046	0.275293	0.137646	+	0.990481i	$0.456046\pi$
$4$	−6.31076	−0.788845
$5$	20.4116	1.82567	0.912835	−	0.408329i	$-0.133888\pi$
$5$	20.4116	1.82567	0.912835	+	0.408329i	$0.133888\pi$
$6$	1.85918	0.126501
$7$	29.9522	1.61727	0.808635	−	0.588311i	$-0.200207\pi$
$7$	29.9522	1.61727	0.808635	+	0.588311i	$0.200207\pi$
$8$	−18.5998	−0.822003
$9$	−24.9538	−0.924214
$10$	26.5291	0.838924
$11$	−22.8719	−0.626922	−0.313461	−	0.949601i	$-0.601488\pi$
$11$	−22.8719	−0.626922	−0.313461	+	0.949601i	$0.601488\pi$
$12$	−9.02731	−0.217163
$13$	−44.4397	−0.948104	−0.474052	−	0.880497i	$-0.657209\pi$
$13$	−44.4397	−0.948104	−0.474052	+	0.880497i	$0.657209\pi$
$14$	38.9292	0.743161
$15$	29.1981	0.502594
$16$	26.3118	0.411122
$17$	−13.0970	−0.186852	−0.0934260	−	0.995626i	$-0.529782\pi$
$17$	−13.0970	−0.186852	−0.0934260	+	0.995626i	$0.529782\pi$
$18$	−32.4326	−0.424691
$19$	5.41527	0.0653868	0.0326934	−	0.999465i	$-0.489592\pi$
$19$	5.41527	0.0653868	0.0326934	+	0.999465i	$0.489592\pi$
$20$	−128.813	−1.44017
$21$	42.8456	0.445223
$22$	−29.7268	−0.288081
$23$	−175.226	−1.58857	−0.794287	−	0.607542i	$-0.792155\pi$
$23$	−175.226	−1.58857	−0.794287	+	0.607542i	$0.792155\pi$
$24$	−26.6063	−0.226291
$25$	291.634	2.33307
$26$	−57.7586	−0.435669
$27$	−74.3180	−0.529722
$28$	−189.021	−1.27578
$29$	165.972	1.06277	0.531383	−	0.847132i	$-0.321672\pi$
$29$	165.972	1.06277	0.531383	+	0.847132i	$0.321672\pi$
$30$	37.9489	0.230950
$31$	−155.581	−0.901395	−0.450697	−	0.892677i	$-0.648825\pi$
$31$	−155.581	−0.901395	−0.450697	+	0.892677i	$0.648825\pi$
$32$	182.996	1.01092
$33$	−32.7174	−0.172587
$34$	−17.0222	−0.0858615
$35$	611.374	2.95260
$36$	157.477	0.729062
$37$	−95.3991	−0.423879	−0.211939	−	0.977283i	$-0.567978\pi$
$37$	−95.3991	−0.423879	−0.211939	+	0.977283i	$0.567978\pi$
$38$	7.03827	0.0300463
$39$	−63.5693	−0.261006
$40$	−379.652	−1.50071
$41$	189.928	0.723457	0.361728	−	0.932284i	$-0.382187\pi$
$41$	189.928	0.723457	0.361728	+	0.932284i	$0.382187\pi$
$42$	55.6867	0.204587
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	144.339	0.494544
$45$	−509.347	−1.68731
$46$	−227.743	−0.729975
$47$	−37.2349	−0.115559	−0.0577794	−	0.998329i	$-0.518402\pi$
$47$	−37.2349	−0.115559	−0.0577794	+	0.998329i	$0.518402\pi$
$48$	37.6381	0.113179
$49$	554.137	1.61556
$50$	379.039	1.07208
$51$	−18.7348	−0.0514390
$52$	280.448	0.747907
$53$	559.862	1.45100	0.725500	−	0.688223i	$-0.241609\pi$
$53$	559.862	1.45100	0.725500	+	0.688223i	$0.241609\pi$
$54$	−96.5916	−0.243416
$55$	−466.852	−1.14455
$56$	−557.106	−1.32940
$57$	7.74635	0.0180005
$58$	215.715	0.488358
$59$	−82.3042	−0.181612	−0.0908059	−	0.995869i	$-0.528944\pi$
$59$	−82.3042	−0.181612	−0.0908059	+	0.995869i	$0.528944\pi$
$60$	−184.262	−0.396469
$61$	−640.304	−1.34398	−0.671988	−	0.740562i	$-0.734559\pi$
$61$	−640.304	−1.34398	−0.671988	+	0.740562i	$0.734559\pi$
$62$	−202.210	−0.414205
$63$	−747.422	−1.49470
$64$	27.3469	0.0534120
$65$	−907.085	−1.73092
$66$	−42.5231	−0.0793065
$67$	−509.592	−0.929203	−0.464602	−	0.885520i	$-0.653802\pi$
$67$	−509.592	−0.929203	−0.464602	+	0.885520i	$0.653802\pi$
$68$	82.6519	0.147397
$69$	−250.655	−0.437323
$70$	794.607	1.35677
$71$	792.932	1.32540	0.662702	−	0.748883i	$-0.269410\pi$
$71$	792.932	1.32540	0.662702	+	0.748883i	$0.269410\pi$
$72$	464.135	0.759706
$73$	612.727	0.982387	0.491194	−	0.871050i	$-0.336561\pi$
$73$	612.727	0.982387	0.491194	+	0.871050i	$0.336561\pi$
$74$	−123.991	−0.194779
$75$	417.171	0.642277
$76$	−34.1745	−0.0515800
$77$	−685.065	−1.01390
$78$	−82.6215	−0.119936
$79$	237.047	0.337594	0.168797	−	0.985651i	$-0.446012\pi$
$79$	237.047	0.337594	0.168797	+	0.985651i	$0.446012\pi$
$80$	537.066	0.750573
$81$	567.443	0.778385
$82$	246.850	0.332440
$83$	418.683	0.553691	0.276846	−	0.960914i	$-0.410711\pi$
$83$	418.683	0.553691	0.276846	+	0.960914i	$0.410711\pi$
$84$	−270.388	−0.351212
$85$	−267.330	−0.341130
$86$	−55.8874	−0.0700755
$87$	237.417	0.292572
$88$	425.413	0.515331
$89$	113.788	0.135523	0.0677615	−	0.997702i	$-0.478414\pi$
$89$	113.788	0.135523	0.0677615	+	0.997702i	$0.478414\pi$
$90$	−662.002	−0.775346
$91$	−1331.07	−1.53334
$92$	1105.81	1.25314
$93$	−222.553	−0.248147
$94$	−48.3944	−0.0531011
$95$	110.534	0.119375
$96$	261.769	0.278299
$97$	1649.00	1.72609	0.863045	−	0.505127i	$-0.168554\pi$
$97$	1649.00	1.72609	0.863045	+	0.505127i	$0.168554\pi$
$98$	720.216	0.742376
$99$	570.740	0.579410
$100$	−1840.43	−1.84043
$101$	−165.065	−0.162619	−0.0813097	−	0.996689i	$-0.525910\pi$
$101$	−165.065	−0.162619	−0.0813097	+	0.996689i	$0.525910\pi$
$102$	−24.3497	−0.0236370
$103$	−1665.32	−1.59309	−0.796547	−	0.604576i	$-0.793342\pi$
$103$	−1665.32	−1.59309	−0.796547	+	0.604576i	$0.793342\pi$
$104$	826.569	0.779344
$105$	874.547	0.812829
$106$	727.657	0.666757
$107$	−631.683	−0.570721	−0.285360	−	0.958420i	$-0.592113\pi$
$107$	−631.683	−0.570721	−0.285360	+	0.958420i	$0.592113\pi$
$108$	469.003	0.417869
$109$	839.368	0.737586	0.368793	−	0.929512i	$-0.379771\pi$
$109$	839.368	0.737586	0.368793	+	0.929512i	$0.379771\pi$
$110$	−606.771	−0.525940
$111$	−136.465	−0.116691
$112$	788.097	0.664895
$113$	−1080.84	−0.899796	−0.449898	−	0.893080i	$-0.648540\pi$
$113$	−1080.84	−0.899796	−0.449898	+	0.893080i	$0.648540\pi$
$114$	10.0680	0.00827152
$115$	−3576.65	−2.90021
$116$	−1047.41	−0.838358
$117$	1108.94	0.876251
$118$	−106.971	−0.0834535
$119$	−392.284	−0.302190
$120$	−543.078	−0.413133
$121$	−807.876	−0.606969
$122$	−832.207	−0.617578
$123$	271.685	0.199162
$124$	981.837	0.711061
$125$	3401.26	2.43375
$126$	−971.429	−0.686840
$127$	594.837	0.415616	0.207808	−	0.978170i	$-0.433367\pi$
$127$	594.837	0.415616	0.207808	+	0.978170i	$0.433367\pi$
$128$	−1428.43	−0.986376
$129$	−61.5099	−0.0419818
$130$	−1178.95	−0.795387
$131$	1987.90	1.32583	0.662916	−	0.748694i	$-0.269319\pi$
$131$	1987.90	1.32583	0.662916	+	0.748694i	$0.269319\pi$
$132$	206.472	0.136144
$133$	162.200	0.105748
$134$	−662.321	−0.426984
$135$	−1516.95	−0.967098
$136$	243.601	0.153593
$137$	1717.01	1.07076	0.535379	−	0.844612i	$-0.320169\pi$
$137$	1717.01	1.07076	0.535379	+	0.844612i	$0.320169\pi$
$138$	−325.778	−0.200957
$139$	−1735.09	−1.05876	−0.529381	−	0.848384i	$-0.677576\pi$
$139$	−1735.09	−1.05876	−0.529381	+	0.848384i	$0.677576\pi$
$140$	−3858.23	−2.32914
$141$	−53.2631	−0.0318125
$142$	1030.58	0.609044
$143$	1016.42	0.594387
$144$	−656.579	−0.379964
$145$	3387.76	1.94026
$146$	796.366	0.451423
$147$	792.673	0.444752
$148$	602.041	0.334375
$149$	456.600	0.251048	0.125524	−	0.992091i	$-0.459939\pi$
$149$	456.600	0.251048	0.125524	+	0.992091i	$0.459939\pi$
$150$	542.201	0.295137
$151$	−483.618	−0.260638	−0.130319	−	0.991472i	$-0.541600\pi$
$151$	−483.618	−0.260638	−0.130319	+	0.991472i	$0.541600\pi$
$152$	−100.723	−0.0537481
$153$	326.819	0.172691
$154$	−890.384	−0.465904
$155$	−3175.67	−1.64565
$156$	401.171	0.205893
$157$	2300.64	1.16950	0.584749	−	0.811214i	$-0.301193\pi$
$157$	2300.64	1.16950	0.584749	+	0.811214i	$0.301193\pi$
$158$	308.092	0.155130
$159$	800.862	0.399450
$160$	3735.24	1.84561
$161$	−5248.42	−2.56915
$162$	737.510	0.357680
$163$	314.362	0.151060	0.0755299	−	0.997144i	$-0.475935\pi$
$163$	314.362	0.151060	0.0755299	+	0.997144i	$0.475935\pi$
$164$	−1198.59	−0.570695
$165$	−667.815	−0.315087
$166$	544.165	0.254430
$167$	−282.441	−0.130874	−0.0654369	−	0.997857i	$-0.520844\pi$
$167$	−282.441	−0.130874	−0.0654369	+	0.997857i	$0.520844\pi$
$168$	−796.919	−0.365974
$169$	−222.114	−0.101099
$170$	−347.451	−0.156755
$171$	−135.131	−0.0604314
$172$	271.363	0.120298
$173$	1774.10	0.779667	0.389834	−	0.920885i	$-0.372532\pi$
$173$	1774.10	0.779667	0.389834	+	0.920885i	$0.372532\pi$
$174$	308.572	0.134441
$175$	8735.09	3.77320
$176$	−601.801	−0.257741
$177$	−117.733	−0.0499964
$178$	147.892	0.0622750
$179$	890.113	0.371677	0.185839	−	0.982580i	$-0.440500\pi$
$179$	890.113	0.371677	0.185839	+	0.982580i	$0.440500\pi$
$180$	3214.37	1.33103
$181$	−2883.82	−1.18427	−0.592135	−	0.805838i	$-0.701715\pi$
$181$	−2883.82	−1.18427	−0.592135	+	0.805838i	$0.701715\pi$
$182$	−1730.00	−0.704594
$183$	−915.931	−0.369987
$184$	3259.17	1.30581
$185$	−1947.25	−0.773863
$186$	−289.254	−0.114028
$187$	299.553	0.117142
$188$	234.980	0.0911580
$189$	−2225.99	−0.856704
$190$	143.662	0.0548546
$191$	1120.48	0.424478	0.212239	−	0.977218i	$-0.431924\pi$
$191$	1120.48	0.424478	0.212239	+	0.977218i	$0.431924\pi$
$192$	39.1188	0.0147039
$193$	−2777.72	−1.03598	−0.517992	−	0.855385i	$-0.673320\pi$
$193$	−2777.72	−1.03598	−0.517992	+	0.855385i	$0.673320\pi$
$194$	2143.22	0.793166
$195$	−1297.55	−0.476511
$196$	−3497.03	−1.27443
$197$	−1266.30	−0.457971	−0.228985	−	0.973430i	$-0.573541\pi$
$197$	−1266.30	−0.457971	−0.228985	+	0.973430i	$0.573541\pi$
$198$	741.795	0.266248
$199$	−5113.40	−1.82150	−0.910752	−	0.412955i	$-0.864497\pi$
$199$	−5113.40	−1.82150	−0.910752	+	0.412955i	$0.864497\pi$
$200$	−5424.33	−1.91779
$201$	−728.953	−0.255803
$202$	−214.536	−0.0747262
$203$	4971.23	1.71878
$204$	118.231	0.0405774
$205$	3876.73	1.32079
$206$	−2164.43	−0.732052
$207$	4372.56	1.46818
$208$	−1169.29	−0.389786
$209$	−123.858	−0.0409924
$210$	1136.66	0.373508
$211$	−1318.35	−0.430138	−0.215069	−	0.976599i	$-0.568998\pi$
$211$	−1318.35	−0.430138	−0.215069	+	0.976599i	$0.568998\pi$
$212$	−3533.15	−1.14461
$213$	1134.26	0.364874
$214$	−821.003	−0.262255
$215$	−877.699	−0.278412
$216$	1382.30	0.435433
$217$	−4660.01	−1.45780
$218$	1090.93	0.338932
$219$	876.483	0.270444
$220$	2946.19	0.902874
$221$	582.026	0.177155
$222$	−177.365	−0.0536213
$223$	−812.504	−0.243988	−0.121994	−	0.992531i	$-0.538929\pi$
$223$	−812.504	−0.243988	−0.121994	+	0.992531i	$0.538929\pi$
$224$	5481.14	1.63493
$225$	−7277.36	−2.15626
$226$	−1404.78	−0.413470
$227$	1890.51	0.552763	0.276382	−	0.961048i	$-0.410865\pi$
$227$	1890.51	0.552763	0.276382	+	0.961048i	$0.410865\pi$
$228$	−48.8853	−0.0141996
$229$	1368.43	0.394884	0.197442	−	0.980315i	$-0.436737\pi$
$229$	1368.43	0.394884	0.197442	+	0.980315i	$0.436737\pi$
$230$	−4648.60	−1.33269
$231$	−979.960	−0.279120
$232$	−3087.05	−0.873597
$233$	−3535.33	−0.994023	−0.497012	−	0.867744i	$-0.665569\pi$
$233$	−3535.33	−0.994023	−0.497012	+	0.867744i	$0.665569\pi$
$234$	1441.29	0.402651
$235$	−760.023	−0.210972
$236$	519.402	0.143264
$237$	339.088	0.0929371
$238$	−509.854	−0.138861
$239$	1515.24	0.410095	0.205047	−	0.978752i	$-0.434265\pi$
$239$	1515.24	0.410095	0.205047	+	0.978752i	$0.434265\pi$
$240$	768.253	0.206627
$241$	−6717.46	−1.79548	−0.897738	−	0.440530i	$-0.854791\pi$
$241$	−6717.46	−1.79548	−0.897738	+	0.440530i	$0.854791\pi$
$242$	−1050.00	−0.278912
$243$	2818.29	0.744006
$244$	4040.80	1.06019
$245$	11310.8	2.94948
$246$	353.110	0.0915183
$247$	−240.653	−0.0619934
$248$	2893.78	0.740949
$249$	598.910	0.152427
$250$	4420.65	1.11835
$251$	−2291.38	−0.576217	−0.288109	−	0.957598i	$-0.593026\pi$
$251$	−2291.38	−0.576217	−0.288109	+	0.957598i	$0.593026\pi$
$252$	4716.80	1.17909
$253$	4007.76	0.995912
$254$	773.114	0.190982
$255$	−382.406	−0.0939107
$256$	−2075.31	−0.506668
$257$	−2236.18	−0.542760	−0.271380	−	0.962472i	$-0.587480\pi$
$257$	−2236.18	−0.542760	−0.271380	+	0.962472i	$0.587480\pi$
$258$	−79.9449	−0.0192913
$259$	−2857.42	−0.685526
$260$	5724.40	1.36543
$261$	−4141.63	−0.982224
$262$	2583.69	0.609241
$263$	−6393.87	−1.49910	−0.749550	−	0.661948i	$-0.769730\pi$
$263$	−6393.87	−1.49910	−0.749550	+	0.661948i	$0.769730\pi$
$264$	608.538	0.141867
$265$	11427.7	2.64905
$266$	210.812	0.0485929
$267$	162.770	0.0373085
$268$	3215.92	0.732997
$269$	−2688.95	−0.609473	−0.304736	−	0.952437i	$-0.598568\pi$
$269$	−2688.95	−0.609473	−0.304736	+	0.952437i	$0.598568\pi$
$270$	−1971.59	−0.444397
$271$	−1057.95	−0.237143	−0.118572	−	0.992945i	$-0.537832\pi$
$271$	−1057.95	−0.237143	−0.118572	+	0.992945i	$0.537832\pi$
$272$	−344.605	−0.0768189
$273$	−1904.04	−0.422117
$274$	2231.61	0.492030
$275$	−6670.22	−1.46265
$276$	1581.82	0.344980
$277$	1904.17	0.413035	0.206518	−	0.978443i	$-0.433787\pi$
$277$	1904.17	0.413035	0.206518	+	0.978443i	$0.433787\pi$
$278$	−2255.10	−0.486518
$279$	3882.34	0.833081
$280$	−11371.4	−2.42705
$281$	6520.37	1.38424	0.692122	−	0.721781i	$-0.256676\pi$
$281$	6520.37	1.38424	0.692122	+	0.721781i	$0.256676\pi$
$282$	−69.2264	−0.0146183
$283$	130.376	0.0273854	0.0136927	−	0.999906i	$-0.495641\pi$
$283$	130.376	0.0273854	0.0136927	+	0.999906i	$0.495641\pi$
$284$	−5004.00	−1.04554
$285$	158.115	0.0328630
$286$	1321.05	0.273130
$287$	5688.76	1.17002
$288$	−4566.44	−0.934306
$289$	−4741.47	−0.965086
$290$	4403.09	0.891581
$291$	2358.84	0.475180
$292$	−3866.77	−0.774951
$293$	−321.150	−0.0640333	−0.0320166	−	0.999487i	$-0.510193\pi$
$293$	−321.150	−0.0640333	−0.0320166	+	0.999487i	$0.510193\pi$
$294$	1030.24	0.204371
$295$	−1679.96	−0.331563
$296$	1774.40	0.348430
$297$	1699.79	0.332094
$298$	593.446	0.115361
$299$	7787.00	1.50613
$300$	−2632.67	−0.506657
$301$	−1287.95	−0.246631
$302$	−628.562	−0.119767
$303$	−236.119	−0.0447679
$304$	142.486	0.0268819
$305$	−13069.6	−2.45366
$306$	424.769	0.0793544
$307$	3944.87	0.733374	0.366687	−	0.930344i	$-0.380492\pi$
$307$	3944.87	0.733374	0.366687	+	0.930344i	$0.380492\pi$
$308$	4323.28	0.799811
$309$	−2382.18	−0.438567
$310$	−4127.44	−0.756202
$311$	7046.04	1.28471	0.642354	−	0.766408i	$-0.277958\pi$
$311$	7046.04	1.28471	0.642354	+	0.766408i	$0.277958\pi$
$312$	1182.38	0.214548
$313$	−230.283	−0.0415858	−0.0207929	−	0.999784i	$-0.506619\pi$
$313$	−230.283	−0.0415858	−0.0207929	+	0.999784i	$0.506619\pi$
$314$	2990.16	0.537403
$315$	−15256.1	−2.72883
$316$	−1495.95	−0.266309
$317$	6574.02	1.16477	0.582387	−	0.812911i	$-0.302119\pi$
$317$	6574.02	1.16477	0.582387	+	0.812911i	$0.302119\pi$
$318$	1040.89	0.183553
$319$	−3796.10	−0.666271
$320$	558.195	0.0975126
$321$	−903.600	−0.157115
$322$	−6821.41	−1.18057
$323$	−70.9237	−0.0122177
$324$	−3581.00	−0.614025
$325$	−12960.1	−2.21199
$326$	408.579	0.0694144
$327$	1200.68	0.203052
$328$	−3532.62	−0.594683
$329$	−1115.27	−0.186890
$330$	−867.964	−0.144787
$331$	8326.42	1.38266	0.691331	−	0.722538i	$-0.257025\pi$
$331$	8326.42	1.38266	0.691331	+	0.722538i	$0.257025\pi$
$332$	−2642.21	−0.436777
$333$	2380.57	0.391755
$334$	−367.091	−0.0601386
$335$	−10401.6	−1.69642
$336$	1127.34	0.183041
$337$	9021.52	1.45826	0.729130	−	0.684375i	$-0.239925\pi$
$337$	9021.52	1.45826	0.729130	+	0.684375i	$0.239925\pi$
$338$	−288.684	−0.0464566
$339$	−1546.10	−0.247707
$340$	1687.06	0.269099
$341$	3558.44	0.565104
$342$	−175.631	−0.0277692
$343$	6324.03	0.995526
$344$	799.791	0.125354
$345$	−5116.27	−0.798408
$346$	2305.81	0.358270
$347$	133.107	0.0205925	0.0102962	−	0.999947i	$-0.496723\pi$
$347$	133.107	0.0205925	0.0102962	+	0.999947i	$0.496723\pi$
$348$	−1498.28	−0.230794
$349$	−1322.67	−0.202868	−0.101434	−	0.994842i	$-0.532343\pi$
$349$	−1322.67	−0.202868	−0.101434	+	0.994842i	$0.532343\pi$
$350$	11353.1	1.73385
$351$	3302.67	0.502232
$352$	−4185.47	−0.633768
$353$	12515.7	1.88709	0.943543	−	0.331251i	$-0.107471\pi$
$353$	12515.7	1.88709	0.943543	+	0.331251i	$0.107471\pi$
$354$	−153.019	−0.0229741
$355$	16185.0	2.41975
$356$	−718.091	−0.106907
$357$	−561.148	−0.0831907
$358$	1156.89	0.170792
$359$	−12654.2	−1.86035	−0.930174	−	0.367118i	$-0.880344\pi$
$359$	−12654.2	−1.86035	−0.930174	+	0.367118i	$0.880344\pi$
$360$	9473.75	1.38697
$361$	−6829.67	−0.995725
$362$	−3748.13	−0.544191
$363$	−1155.64	−0.167094
$364$	8400.05	1.20957
$365$	12506.7	1.79351
$366$	−1190.44	−0.170015
$367$	−13307.8	−1.89281	−0.946407	−	0.322976i	$-0.895317\pi$
$367$	−13307.8	−1.89281	−0.946407	+	0.322976i	$0.895317\pi$
$368$	−4610.52	−0.653098
$369$	−4739.41	−0.668629
$370$	−2530.86	−0.355602
$371$	16769.1	2.34666
$372$	1404.48	0.195750
$373$	−6552.61	−0.909602	−0.454801	−	0.890593i	$-0.650290\pi$
$373$	−6552.61	−0.909602	−0.454801	+	0.890593i	$0.650290\pi$
$374$	389.331	0.0538284
$375$	4865.38	0.669993
$376$	692.561	0.0949896
$377$	−7375.74	−1.00761
$378$	−2893.14	−0.393669
$379$	11236.0	1.52284	0.761418	−	0.648262i	$-0.224504\pi$
$379$	11236.0	1.52284	0.761418	+	0.648262i	$0.224504\pi$
$380$	−697.556	−0.0941681
$381$	850.893	0.114416
$382$	1456.30	0.195054
$383$	6353.36	0.847628	0.423814	−	0.905749i	$-0.360691\pi$
$383$	6353.36	0.847628	0.423814	+	0.905749i	$0.360691\pi$
$384$	−2043.31	−0.271542
$385$	−13983.3	−1.85105
$386$	−3610.23	−0.476051
$387$	1073.01	0.140941
$388$	−10406.5	−1.36162
$389$	3788.91	0.493844	0.246922	−	0.969035i	$-0.420581\pi$
$389$	3788.91	0.493844	0.246922	+	0.969035i	$0.420581\pi$
$390$	−1686.44	−0.218964
$391$	2294.94	0.296828
$392$	−10306.8	−1.32799
$393$	2843.62	0.364992
$394$	−1645.82	−0.210445
$395$	4838.52	0.616335
$396$	−3601.81	−0.457065
$397$	−2601.66	−0.328901	−0.164451	−	0.986385i	$-0.552585\pi$
$397$	−2601.66	−0.328901	−0.164451	+	0.986385i	$0.552585\pi$
$398$	−6645.92	−0.837010
$399$	232.021	0.0291117
$400$	7673.41	0.959176
$401$	−1698.36	−0.211501	−0.105750	−	0.994393i	$-0.533724\pi$
$401$	−1698.36	−0.211501	−0.105750	+	0.994393i	$0.533724\pi$
$402$	−947.426	−0.117546
$403$	6913.99	0.854616
$404$	1041.68	0.128281
$405$	11582.4	1.42107
$406$	6461.15	0.789807
$407$	2181.96	0.265739
$408$	348.463	0.0422830
$409$	−5511.51	−0.666324	−0.333162	−	0.942870i	$-0.608116\pi$
$409$	−5511.51	−0.666324	−0.333162	+	0.942870i	$0.608116\pi$
$410$	5038.61	0.606925
$411$	2456.12	0.294772
$412$	10509.4	1.25670
$413$	−2465.20	−0.293715
$414$	5683.05	0.674653
$415$	8545.99	1.01086
$416$	−8132.29	−0.958457
$417$	−2481.98	−0.291470
$418$	−160.979	−0.0188367
$419$	−6137.26	−0.715573	−0.357786	−	0.933803i	$-0.616468\pi$
$419$	−6137.26	−0.715573	−0.357786	+	0.933803i	$0.616468\pi$
$420$	−5519.06	−0.641197
$421$	9168.60	1.06140	0.530701	−	0.847559i	$-0.321929\pi$
$421$	9168.60	1.06140	0.530701	+	0.847559i	$0.321929\pi$
$422$	−1713.47	−0.197655
$423$	929.150	0.106801
$424$	−10413.3	−1.19273
$425$	−3819.52	−0.435939
$426$	1474.21	0.167666
$427$	−19178.5	−2.17357
$428$	3986.40	0.450210
$429$	1453.95	0.163630
$430$	−1140.75	−0.127935
$431$	12535.8	1.40099	0.700494	−	0.713658i	$-0.252963\pi$
$431$	12535.8	1.40099	0.700494	+	0.713658i	$0.252963\pi$
$432$	−1955.44	−0.217780
$433$	7163.93	0.795095	0.397548	−	0.917582i	$-0.369861\pi$
$433$	7163.93	0.795095	0.397548	+	0.917582i	$0.369861\pi$
$434$	−6056.65	−0.669881
$435$	4846.06	0.534140
$436$	−5297.05	−0.581841
$437$	−948.898	−0.103872
$438$	1139.17	0.124273
$439$	−2988.31	−0.324884	−0.162442	−	0.986718i	$-0.551937\pi$
$439$	−2988.31	−0.324884	−0.162442	+	0.986718i	$0.551937\pi$
$440$	8683.36	0.940825
$441$	−13827.8	−1.49312
$442$	756.463	0.0814056
$443$	−11257.1	−1.20732	−0.603660	−	0.797242i	$-0.706292\pi$
$443$	−11257.1	−1.20732	−0.603660	+	0.797242i	$0.706292\pi$
$444$	861.198	0.0920510
$445$	2322.60	0.247420
$446$	−1056.02	−0.112116
$447$	653.150	0.0691117
$448$	819.102	0.0863816
$449$	−4960.39	−0.521371	−0.260685	−	0.965424i	$-0.583948\pi$
$449$	−4960.39	−0.521371	−0.260685	+	0.965424i	$0.583948\pi$
$450$	−9458.44	−0.990834
$451$	−4344.01	−0.453551
$452$	6820.92	0.709799
$453$	−691.798	−0.0717517
$454$	2457.10	0.254004
$455$	−27169.2	−2.79937
$456$	−144.081	−0.0147965
$457$	−4911.00	−0.502685	−0.251343	−	0.967898i	$-0.580872\pi$
$457$	−4911.00	−0.502685	−0.251343	+	0.967898i	$0.580872\pi$
$458$	1778.56	0.181455
$459$	973.341	0.0989797
$460$	22571.4	2.28782
$461$	−3089.54	−0.312135	−0.156068	−	0.987746i	$-0.549882\pi$
$461$	−3089.54	−0.312135	−0.156068	+	0.987746i	$0.549882\pi$
$462$	−1273.66	−0.128260
$463$	2821.14	0.283174	0.141587	−	0.989926i	$-0.454779\pi$
$463$	2821.14	0.283174	0.141587	+	0.989926i	$0.454779\pi$
$464$	4367.02	0.436926
$465$	−4542.67	−0.453035
$466$	−4594.90	−0.456770
$467$	483.354	0.0478950	0.0239475	−	0.999713i	$-0.492377\pi$
$467$	483.354	0.0478950	0.0239475	+	0.999713i	$0.492377\pi$
$468$	−6998.24	−0.691226
$469$	−15263.4	−1.50277
$470$	−987.808	−0.0969451
$471$	3290.99	0.321955
$472$	1530.84	0.149285
$473$	983.492	0.0956047
$474$	440.715	0.0427061
$475$	1579.28	0.152552
$476$	2475.61	0.238381
$477$	−13970.7	−1.34103
$478$	1969.37	0.188445
$479$	−5570.85	−0.531396	−0.265698	−	0.964056i	$-0.585602\pi$
$479$	−5570.85	−0.531396	−0.265698	+	0.964056i	$0.585602\pi$
$480$	5343.13	0.508082
$481$	4239.51	0.401881
$482$	−8730.73	−0.825050
$483$	−7507.68	−0.707269
$484$	5098.31	0.478805
$485$	33658.8	3.15127
$486$	3662.95	0.341883
$487$	453.859	0.0422306	0.0211153	−	0.999777i	$-0.493278\pi$
$487$	453.859	0.0422306	0.0211153	+	0.999777i	$0.493278\pi$
$488$	11909.5	1.10475
$489$	449.684	0.0415857
$490$	14700.8	1.35533
$491$	2787.17	0.256178	0.128089	−	0.991763i	$-0.459116\pi$
$491$	2787.17	0.256178	0.128089	+	0.991763i	$0.459116\pi$
$492$	−1714.54	−0.157108
$493$	−2173.73	−0.198580
$494$	−312.778	−0.0284870
$495$	11649.7	1.05781
$496$	−4093.62	−0.370583
$497$	23750.1	2.14354
$498$	778.408	0.0700428
$499$	−325.118	−0.0291669	−0.0145834	−	0.999894i	$-0.504642\pi$
$499$	−325.118	−0.0291669	−0.0145834	+	0.999894i	$0.504642\pi$
$500$	−21464.6	−1.91985
$501$	−404.022	−0.0360286
$502$	−2978.12	−0.264781
$503$	−11441.8	−1.01424	−0.507120	−	0.861875i	$-0.669290\pi$
$503$	−11441.8	−1.01424	−0.507120	+	0.861875i	$0.669290\pi$
$504$	13901.9	1.22865
$505$	−3369.24	−0.296889
$506$	5208.91	0.457637
$507$	−317.727	−0.0278318
$508$	−3753.88	−0.327857
$509$	−8398.29	−0.731332	−0.365666	−	0.930746i	$-0.619159\pi$
$509$	−8398.29	−0.731332	−0.365666	+	0.930746i	$0.619159\pi$
$510$	−497.016	−0.0431534
$511$	18352.5	1.58878
$512$	8730.11	0.753554
$513$	−402.452	−0.0346368
$514$	−2906.39	−0.249407
$515$	−33991.8	−2.90846
$516$	388.174	0.0331171
$517$	851.632	0.0724463
$518$	−3713.81	−0.315010
$519$	2537.79	0.214637
$520$	16871.6	1.42282
$521$	9762.74	0.820947	0.410474	−	0.911873i	$-0.365363\pi$
$521$	9762.74	0.820947	0.410474	+	0.911873i	$0.365363\pi$
$522$	−5382.90	−0.451347
$523$	−14406.0	−1.20445	−0.602227	−	0.798325i	$-0.705720\pi$
$523$	−14406.0	−1.20445	−0.602227	+	0.798325i	$0.705720\pi$
$524$	−12545.2	−1.04588
$525$	12495.2	1.03874
$526$	−8310.16	−0.688860
$527$	2037.65	0.168427
$528$	−860.854	−0.0709543
$529$	18537.3	1.52357
$530$	14852.6	1.21728
$531$	2053.80	0.167848
$532$	−1023.60	−0.0834188
$533$	−8440.33	−0.685912
$534$	211.554	0.0171439
$535$	−12893.7	−1.04195
$536$	9478.32	0.763808
$537$	1273.27	0.102320
$538$	−3494.85	−0.280062
$539$	−12674.2	−1.01283
$540$	9573.10	0.762890
$541$	22048.2	1.75217	0.876087	−	0.482153i	$-0.160145\pi$
$541$	22048.2	1.75217	0.876087	+	0.482153i	$0.160145\pi$
$542$	−1375.02	−0.108971
$543$	−4125.21	−0.326021
$544$	−2396.70	−0.188892
$545$	17132.8	1.34659
$546$	−2474.70	−0.193970
$547$	6165.00	0.481895	0.240947	−	0.970538i	$-0.422542\pi$
$547$	6165.00	0.481895	0.240947	+	0.970538i	$0.422542\pi$
$548$	−10835.6	−0.844662
$549$	15978.0	1.24212
$550$	−8669.33	−0.672112
$551$	898.784	0.0694909
$552$	4662.13	0.359481
$553$	7100.10	0.545980
$554$	2474.87	0.189796
$555$	−2785.47	−0.213039
$556$	10949.7	0.835200
$557$	−12519.9	−0.952400	−0.476200	−	0.879337i	$-0.657986\pi$
$557$	−12519.9	−0.952400	−0.476200	+	0.879337i	$0.657986\pi$
$558$	5045.91	0.382814
$559$	1910.91	0.144584
$560$	16086.3	1.21388
$561$	428.499	0.0322482
$562$	8474.57	0.636082
$563$	21124.7	1.58135	0.790674	−	0.612237i	$-0.209730\pi$
$563$	21124.7	1.58135	0.790674	+	0.612237i	$0.209730\pi$
$564$	336.131	0.0250951
$565$	−22061.7	−1.64273
$566$	169.451	0.0125840
$567$	16996.2	1.25886
$568$	−14748.4	−1.08949
$569$	−22782.7	−1.67856	−0.839279	−	0.543700i	$-0.817023\pi$
$569$	−22782.7	−1.67856	−0.839279	+	0.543700i	$0.817023\pi$
$570$	205.504	0.0151011
$571$	−12990.7	−0.952092	−0.476046	−	0.879420i	$-0.657930\pi$
$571$	−12990.7	−0.952092	−0.476046	+	0.879420i	$0.657930\pi$
$572$	−6414.39	−0.468879
$573$	1602.81	0.116856
$574$	7393.72	0.537645
$575$	−51101.9	−3.70626
$576$	−682.409	−0.0493641
$577$	−25759.9	−1.85858	−0.929289	−	0.369353i	$-0.879579\pi$
$577$	−25759.9	−1.85858	−0.929289	+	0.369353i	$0.879579\pi$
$578$	−6162.52	−0.443473
$579$	−3973.43	−0.285199
$580$	−21379.3	−1.53057
$581$	12540.5	0.895468
$582$	3065.80	0.218353
$583$	−12805.1	−0.909663
$584$	−11396.6	−0.807525
$585$	22635.2	1.59974
$586$	−417.400	−0.0294243
$587$	−4369.06	−0.307207	−0.153604	−	0.988133i	$-0.549088\pi$
$587$	−4369.06	−0.307207	−0.153604	+	0.988133i	$0.549088\pi$
$588$	−5002.37	−0.350840
$589$	−842.515	−0.0589393
$590$	−2183.46	−0.152358
$591$	−1811.40	−0.126076
$592$	−2510.12	−0.174266
$593$	3241.12	0.224447	0.112223	−	0.993683i	$-0.464203\pi$
$593$	3241.12	0.224447	0.112223	+	0.993683i	$0.464203\pi$
$594$	2209.23	0.152603
$595$	−8007.15	−0.551699
$596$	−2881.49	−0.198038
$597$	−7314.53	−0.501447
$598$	10120.8	0.692092
$599$	−4294.37	−0.292927	−0.146463	−	0.989216i	$-0.546789\pi$
$599$	−4294.37	−0.292927	−0.146463	+	0.989216i	$0.546789\pi$
$600$	−7759.31	−0.527954
$601$	1277.55	0.0867093	0.0433546	−	0.999060i	$-0.486195\pi$
$601$	1277.55	0.0867093	0.0433546	+	0.999060i	$0.486195\pi$
$602$	−1673.95	−0.113331
$603$	12716.3	0.858782
$604$	3052.00	0.205603
$605$	−16490.0	−1.10813
$606$	−306.886	−0.0205716
$607$	17305.3	1.15717	0.578584	−	0.815623i	$-0.303606\pi$
$607$	17305.3	1.15717	0.578584	+	0.815623i	$0.303606\pi$
$608$	990.973	0.0661008
$609$	7111.17	0.473168
$610$	−16986.7	−1.12749
$611$	1654.71	0.109562
$612$	−2062.48	−0.136227
$613$	−12812.9	−0.844225	−0.422112	−	0.906544i	$-0.638711\pi$
$613$	−12812.9	−0.844225	−0.422112	+	0.906544i	$0.638711\pi$
$614$	5127.18	0.336997
$615$	5545.52	0.363605
$616$	12742.1	0.833430
$617$	−3963.52	−0.258615	−0.129307	−	0.991605i	$-0.541275\pi$
$617$	−3963.52	−0.258615	−0.129307	+	0.991605i	$0.541275\pi$
$618$	−3096.13	−0.201529
$619$	26937.7	1.74914	0.874570	−	0.484899i	$-0.161144\pi$
$619$	26937.7	1.74914	0.874570	+	0.484899i	$0.161144\pi$
$620$	20040.9	1.29816
$621$	13022.5	0.841503
$622$	9157.79	0.590344
$623$	3408.22	0.219177
$624$	−1672.62	−0.107305
$625$	32971.0	2.11015
$626$	−299.300	−0.0191093
$627$	−177.174	−0.0112849
$628$	−14518.8	−0.922554
$629$	1249.44	0.0792026
$630$	−19828.4	−1.25394
$631$	−2401.37	−0.151501	−0.0757504	−	0.997127i	$-0.524135\pi$
$631$	−2401.37	−0.151501	−0.0757504	+	0.997127i	$0.524135\pi$
$632$	−4409.03	−0.277503
$633$	−1885.85	−0.118414
$634$	8544.30	0.535232
$635$	12141.6	0.758778
$636$	−5054.05	−0.315104
$637$	−24625.7	−1.53172
$638$	−4933.81	−0.306162
$639$	−19786.6	−1.22496
$640$	−29156.5	−1.80080
$641$	−22872.8	−1.40940	−0.704698	−	0.709508i	$-0.748917\pi$
$641$	−22872.8	−1.40940	−0.704698	+	0.709508i	$0.748917\pi$
$642$	−1174.42	−0.0721970
$643$	27051.9	1.65913	0.829566	−	0.558409i	$-0.188588\pi$
$643$	27051.9	1.65913	0.829566	+	0.558409i	$0.188588\pi$
$644$	33121.5	2.02666
$645$	−1255.52	−0.0766448
$646$	−92.1801	−0.00561420
$647$	12947.9	0.786760	0.393380	−	0.919376i	$-0.371306\pi$
$647$	12947.9	0.786760	0.393380	+	0.919376i	$0.371306\pi$
$648$	−10554.3	−0.639835
$649$	1882.45	0.113856
$650$	−16844.4	−1.01645
$651$	−6665.97	−0.401321
$652$	−1983.87	−0.119163
$653$	16218.4	0.971934	0.485967	−	0.873977i	$-0.338467\pi$
$653$	16218.4	0.971934	0.485967	+	0.873977i	$0.338467\pi$
$654$	1560.54	0.0933056
$655$	40576.3	2.42053
$656$	4997.34	0.297429
$657$	−15289.8	−0.907936
$658$	−1449.52	−0.0858788
$659$	−26252.1	−1.55180	−0.775901	−	0.630855i	$-0.782704\pi$
$659$	−26252.1	−1.55180	−0.775901	+	0.630855i	$0.782704\pi$
$660$	4214.42	0.248555
$661$	24649.9	1.45049	0.725243	−	0.688493i	$-0.241727\pi$
$661$	24649.9	1.45049	0.725243	+	0.688493i	$0.241727\pi$
$662$	10821.9	0.635355
$663$	832.566	0.0487695
$664$	−7787.41	−0.455136
$665$	3310.75	0.193061
$666$	3094.04	0.180018
$667$	−29082.7	−1.68828
$668$	1782.42	0.103239
$669$	−1162.26	−0.0671681
$670$	−13519.0	−0.779531
$671$	14645.0	0.842567
$672$	7840.57	0.450084
$673$	4424.70	0.253432	0.126716	−	0.991939i	$-0.459556\pi$
$673$	4424.70	0.253432	0.126716	+	0.991939i	$0.459556\pi$
$674$	11725.3	0.670094
$675$	−21673.6	−1.23588
$676$	1401.71	0.0797514
$677$	18153.6	1.03057	0.515287	−	0.857018i	$-0.327685\pi$
$677$	18153.6	1.03057	0.515287	+	0.857018i	$0.327685\pi$
$678$	−2009.48	−0.113825
$679$	49391.3	2.79155
$680$	4972.29	0.280410
$681$	2704.30	0.152172
$682$	4624.93	0.259674
$683$	19887.7	1.11417	0.557087	−	0.830454i	$-0.311919\pi$
$683$	19887.7	1.11417	0.557087	+	0.830454i	$0.311919\pi$
$684$	852.782	0.0476710
$685$	35046.9	1.95485
$686$	8219.39	0.457460
$687$	1957.49	0.108709
$688$	−1131.41	−0.0626955
$689$	−24880.1	−1.37570
$690$	−6649.65	−0.366881
$691$	8550.10	0.470711	0.235355	−	0.971909i	$-0.424375\pi$
$691$	8550.10	0.470711	0.235355	+	0.971909i	$0.424375\pi$
$692$	−11195.9	−0.615037
$693$	17095.0	0.937062
$694$	173.001	0.00946256
$695$	−35415.9	−1.93295
$696$	−4415.91	−0.240495
$697$	−2487.48	−0.135179
$698$	−1719.09	−0.0932212
$699$	−5057.17	−0.273648
$700$	−55125.1	−2.97647
$701$	−19280.9	−1.03884	−0.519421	−	0.854518i	$-0.673852\pi$
$701$	−19280.9	−1.03884	−0.519421	+	0.854518i	$0.673852\pi$
$702$	4292.50	0.230783
$703$	−516.612	−0.0277161
$704$	−625.477	−0.0334851
$705$	−1087.19	−0.0580791
$706$	16266.7	0.867146
$707$	−4944.06	−0.262999
$708$	742.986	0.0394394
$709$	19384.3	1.02679	0.513394	−	0.858153i	$-0.328388\pi$
$709$	19384.3	1.02679	0.513394	+	0.858153i	$0.328388\pi$
$710$	21035.8	1.11191
$711$	−5915.23	−0.312009
$712$	−2116.44	−0.111400
$713$	27261.9	1.43193
$714$	−729.328	−0.0382275
$715$	20746.8	1.08515
$716$	−5617.29	−0.293196
$717$	2167.49	0.112896
$718$	−16446.8	−0.854860
$719$	−11746.0	−0.609253	−0.304627	−	0.952472i	$-0.598532\pi$
$719$	−11746.0	−0.609253	−0.304627	+	0.952472i	$0.598532\pi$
$720$	−13401.8	−0.693690
$721$	−49880.0	−2.57646
$722$	−8876.58	−0.457551
$723$	−9609.08	−0.494282
$724$	18199.1	0.934206
$725$	48403.0	2.47951
$726$	−1501.99	−0.0767825
$727$	10042.8	0.512333	0.256167	−	0.966633i	$-0.417540\pi$
$727$	10042.8	0.512333	0.256167	+	0.966633i	$0.417540\pi$
$728$	24757.6	1.26041
$729$	−11289.5	−0.573566
$730$	16255.1	0.824148
$731$	563.170	0.0284947
$732$	5780.22	0.291862
$733$	−23541.9	−1.18627	−0.593136	−	0.805102i	$-0.702111\pi$
$733$	−23541.9	−1.18627	−0.593136	+	0.805102i	$0.702111\pi$
$734$	−17296.3	−0.869778
$735$	16179.7	0.811970
$736$	−32065.7	−1.60592
$737$	11655.3	0.582538
$738$	−6159.85	−0.307245
$739$	−39871.6	−1.98471	−0.992354	−	0.123422i	$-0.960613\pi$
$739$	−39871.6	−1.98471	−0.992354	+	0.123422i	$0.960613\pi$
$740$	12288.6	0.610458
$741$	−344.245	−0.0170664
$742$	21795.0	1.07833
$743$	−11387.7	−0.562281	−0.281141	−	0.959667i	$-0.590713\pi$
$743$	−11387.7	−0.562281	−0.281141	+	0.959667i	$0.590713\pi$
$744$	4139.45	0.203978
$745$	9319.94	0.458331
$746$	−8516.48	−0.417977
$747$	−10447.7	−0.511729
$748$	−1890.41	−0.0924066
$749$	−18920.3	−0.923009
$750$	6323.57	0.307872
$751$	−26156.9	−1.27094	−0.635471	−	0.772125i	$-0.719194\pi$
$751$	−26156.9	−1.27094	−0.635471	+	0.772125i	$0.719194\pi$
$752$	−979.716	−0.0475087
$753$	−3277.73	−0.158629
$754$	−9586.31	−0.463014
$755$	−9871.43	−0.475839
$756$	14047.7	0.675806
$757$	9147.11	0.439178	0.219589	−	0.975593i	$-0.429528\pi$
$757$	9147.11	0.439178	0.219589	+	0.975593i	$0.429528\pi$
$758$	14603.5	0.699767
$759$	5732.95	0.274167
$760$	−2055.92	−0.0981263
$761$	−41329.2	−1.96870	−0.984350	−	0.176225i	$-0.943611\pi$
$761$	−41329.2	−1.96870	−0.984350	+	0.176225i	$0.943611\pi$
$762$	1105.91	0.0525761
$763$	25140.9	1.19287
$764$	−7071.10	−0.334847
$765$	6670.90	0.315277
$766$	8257.51	0.389499
$767$	3657.57	0.172187
$768$	−2968.66	−0.139482
$769$	−3722.69	−0.174569	−0.0872846	−	0.996183i	$-0.527819\pi$
$769$	−3722.69	−0.174569	−0.0872846	+	0.996183i	$0.527819\pi$
$770$	−18174.2	−0.850586
$771$	−3198.78	−0.149418
$772$	17529.6	0.817231
$773$	34945.5	1.62600	0.813002	−	0.582261i	$-0.197832\pi$
$773$	34945.5	1.62600	0.813002	+	0.582261i	$0.197832\pi$
$774$	1394.60	0.0647648
$775$	−45372.8	−2.10302
$776$	−30671.1	−1.41885
$777$	−4087.43	−0.188721
$778$	4924.48	0.226929
$779$	1028.51	0.0473045
$780$	8188.54	0.375893
$781$	−18135.9	−0.830925
$782$	2982.74	0.136397
$783$	−12334.7	−0.562971
$784$	14580.3	0.664192
$785$	46959.8	2.13512
$786$	3695.88	0.167720
$787$	−26135.3	−1.18377	−0.591883	−	0.806024i	$-0.701615\pi$
$787$	−26135.3	−1.18377	−0.591883	+	0.806024i	$0.701615\pi$
$788$	7991.33	0.361268
$789$	−9146.20	−0.412691
$790$	6288.66	0.283216
$791$	−32373.6	−1.45521
$792$	−10615.7	−0.476276
$793$	28454.9	1.27423
$794$	−3381.40	−0.151135
$795$	16346.9	0.729263
$796$	32269.4	1.43688
$797$	20345.9	0.904250	0.452125	−	0.891955i	$-0.350666\pi$
$797$	20345.9	0.904250	0.452125	+	0.891955i	$0.350666\pi$
$798$	301.559	0.0133773
$799$	487.664	0.0215924
$800$	53367.8	2.35855
$801$	−2839.45	−0.125252
$802$	−2207.37	−0.0971881
$803$	−14014.2	−0.615880
$804$	4600.25	0.201789
$805$	−107129.	−4.69042
$806$	8986.16	0.392710
$807$	−3846.44	−0.167783
$808$	3070.17	0.133674
$809$	−21515.4	−0.935034	−0.467517	−	0.883984i	$-0.654851\pi$
$809$	−21515.4	−0.935034	−0.467517	+	0.883984i	$0.654851\pi$
$810$	15053.8	0.653006
$811$	−19685.9	−0.852363	−0.426181	−	0.904638i	$-0.640142\pi$
$811$	−19685.9	−0.852363	−0.426181	+	0.904638i	$0.640142\pi$
$812$	−31372.3	−1.35585
$813$	−1513.36	−0.0652838
$814$	2835.91	0.122111
$815$	6416.64	0.275785
$816$	−492.945	−0.0211477
$817$	−232.857	−0.00997139
$818$	−7163.35	−0.306187
$819$	33215.2	1.41713
$820$	−24465.1	−1.04190
$821$	23501.4	0.999029	0.499515	−	0.866305i	$-0.333512\pi$
$821$	23501.4	0.999029	0.499515	+	0.866305i	$0.333512\pi$
$822$	3192.23	0.135452
$823$	25153.6	1.06537	0.532684	−	0.846314i	$-0.321183\pi$
$823$	25153.6	1.06537	0.532684	+	0.846314i	$0.321183\pi$
$824$	30974.6	1.30953
$825$	−9541.51	−0.402658
$826$	−3204.03	−0.134967
$827$	31043.6	1.30531	0.652655	−	0.757655i	$-0.273655\pi$
$827$	31043.6	1.30531	0.652655	+	0.757655i	$0.273655\pi$
$828$	−27594.2	−1.15817
$829$	16479.2	0.690404	0.345202	−	0.938528i	$-0.387810\pi$
$829$	16479.2	0.690404	0.345202	+	0.938528i	$0.387810\pi$
$830$	11107.3	0.464505
$831$	2723.85	0.113706
$832$	−1215.29	−0.0506401
$833$	−7257.52	−0.301871
$834$	−3225.84	−0.133935
$835$	−5765.08	−0.238933
$836$	781.636	0.0323366
$837$	11562.5	0.477489
$838$	−7976.65	−0.328817
$839$	10409.3	0.428329	0.214164	−	0.976798i	$-0.431297\pi$
$839$	10409.3	0.428329	0.214164	+	0.976798i	$0.431297\pi$
$840$	−16266.4	−0.668148
$841$	3157.71	0.129473
$842$	11916.5	0.487731
$843$	9327.15	0.381072
$844$	8319.80	0.339312
$845$	−4533.71	−0.184573
$846$	1207.62	0.0490768
$847$	−24197.7	−0.981633
$848$	14731.0	0.596537
$849$	186.499	0.00753901
$850$	−4964.26	−0.200321
$851$	16716.4	0.673363
$852$	−7158.04	−0.287829
$853$	1701.91	0.0683147	0.0341573	−	0.999416i	$-0.489125\pi$
$853$	1701.91	0.0683147	0.0341573	+	0.999416i	$0.489125\pi$
$854$	−24926.5	−0.998790
$855$	−2758.25	−0.110328
$856$	11749.2	0.469134
$857$	−13987.3	−0.557524	−0.278762	−	0.960360i	$-0.589924\pi$
$857$	−13987.3	−0.557524	−0.278762	+	0.960360i	$0.589924\pi$
$858$	1889.71	0.0751908
$859$	−845.375	−0.0335784	−0.0167892	−	0.999859i	$-0.505344\pi$
$859$	−845.375	−0.0335784	−0.0167892	+	0.999859i	$0.505344\pi$
$860$	5538.95	0.219624
$861$	8137.56	0.322099
$862$	16292.8	0.643776
$863$	−16970.3	−0.669381	−0.334691	−	0.942328i	$-0.608632\pi$
$863$	−16970.3	−0.669381	−0.334691	+	0.942328i	$0.608632\pi$
$864$	−13599.9	−0.535507
$865$	36212.3	1.42342
$866$	9311.01	0.365359
$867$	−6782.50	−0.265681
$868$	29408.2	1.14998
$869$	−5421.72	−0.211645
$870$	6298.46	0.245446
$871$	22646.1	0.880981
$872$	−15612.1	−0.606297
$873$	−41148.8	−1.59528
$874$	−1233.29	−0.0477307
$875$	101876.	3.93602
$876$	−5531.28	−0.213339
$877$	−3402.23	−0.130998	−0.0654989	−	0.997853i	$-0.520864\pi$
$877$	−3402.23	−0.130998	−0.0654989	+	0.997853i	$0.520864\pi$
$878$	−3883.92	−0.149289
$879$	−459.393	−0.0176279
$880$	−12283.7	−0.470550
$881$	−9753.40	−0.372986	−0.186493	−	0.982456i	$-0.559712\pi$
$881$	−9753.40	−0.372986	−0.186493	+	0.982456i	$0.559712\pi$
$882$	−17972.1	−0.686114
$883$	42672.3	1.62632	0.813159	−	0.582042i	$-0.197746\pi$
$883$	42672.3	1.62632	0.813159	+	0.582042i	$0.197746\pi$
$884$	−3673.03	−0.139748
$885$	−2403.12	−0.0912769
$886$	−14631.0	−0.554783
$887$	13027.3	0.493139	0.246570	−	0.969125i	$-0.420697\pi$
$887$	13027.3	0.493139	0.246570	+	0.969125i	$0.420697\pi$
$888$	2538.22	0.0959202
$889$	17816.7	0.672164
$890$	3018.71	0.113694
$891$	−12978.5	−0.487987
$892$	5127.52	0.192469
$893$	−201.637	−0.00755601
$894$	848.903	0.0317579
$895$	18168.6	0.678560
$896$	−42784.5	−1.59524
$897$	11139.0	0.414628
$898$	−6447.06	−0.239578
$899$	−25822.1	−0.957972
$900$	45925.7	1.70095
$901$	−7332.50	−0.271122
$902$	−5645.94	−0.208414
$903$	−1842.36	−0.0678958
$904$	20103.4	0.739634
$905$	−58863.5	−2.16209
$906$	−899.135	−0.0329711
$907$	−2184.13	−0.0799591	−0.0399796	−	0.999200i	$-0.512729\pi$
$907$	−2184.13	−0.0799591	−0.0399796	+	0.999200i	$0.512729\pi$
$908$	−11930.5	−0.436045
$909$	4118.99	0.150295
$910$	−35312.1	−1.28636
$911$	15349.5	0.558236	0.279118	−	0.960257i	$-0.409958\pi$
$911$	15349.5	0.558236	0.279118	+	0.960257i	$0.409958\pi$
$912$	203.820	0.00740040
$913$	−9576.07	−0.347121
$914$	−6382.87	−0.230992
$915$	−18695.6	−0.675474
$916$	−8635.84	−0.311502
$917$	59542.2	2.14423
$918$	1265.06	0.0454827
$919$	−50672.7	−1.81887	−0.909433	−	0.415850i	$-0.863484\pi$
$919$	−50672.7	−1.81887	−0.909433	+	0.415850i	$0.863484\pi$
$920$	66525.0	2.38398
$921$	5642.99	0.201892
$922$	−4015.50	−0.143431
$923$	−35237.6	−1.25662
$924$	6184.30	0.220182
$925$	−27821.6	−0.988940
$926$	3666.66	0.130123
$927$	41556.0	1.47236
$928$	30372.2	1.07437
$929$	−40538.7	−1.43168	−0.715841	−	0.698263i	$-0.753957\pi$
$929$	−40538.7	−1.43168	−0.715841	+	0.698263i	$0.753957\pi$
$930$	−5904.15	−0.208177
$931$	3000.80	0.105636
$932$	22310.7	0.784131
$933$	10079.1	0.353671
$934$	628.219	0.0220085
$935$	6114.36	0.213862
$936$	−20626.0	−0.720281
$937$	13492.4	0.470413	0.235206	−	0.971945i	$-0.424423\pi$
$937$	13492.4	0.470413	0.235206	+	0.971945i	$0.424423\pi$
$938$	−19838.0	−0.690548
$939$	−329.411	−0.0114483
$940$	4796.33	0.166424
$941$	22597.2	0.782835	0.391417	−	0.920213i	$-0.371985\pi$
$941$	22597.2	0.782835	0.391417	+	0.920213i	$0.371985\pi$
$942$	4277.32	0.147943
$943$	−33280.3	−1.14926
$944$	−2165.57	−0.0746645
$945$	−45436.0	−1.56406
$946$	1278.25	0.0439319
$947$	21796.6	0.747933	0.373967	−	0.927442i	$-0.377997\pi$
$947$	21796.6	0.747933	0.373967	+	0.927442i	$0.377997\pi$
$948$	−2139.90	−0.0733130
$949$	−27229.4	−0.931405
$950$	2052.60	0.0701000
$951$	9403.89	0.320654
$952$	7296.40	0.248401
$953$	−40073.5	−1.36213	−0.681064	−	0.732224i	$-0.738482\pi$
$953$	−40073.5	−1.36213	−0.681064	+	0.732224i	$0.738482\pi$
$954$	−18157.8	−0.616226
$955$	22870.9	0.774957
$956$	−9562.31	−0.323501
$957$	−5430.18	−0.183420
$958$	−7240.47	−0.244185
$959$	51428.2	1.73170
$960$	798.477	0.0268445
$961$	−5585.45	−0.187488
$962$	5510.12	0.184671
$963$	15762.9	0.527468
$964$	42392.3	1.41635
$965$	−56697.8	−1.89137
$966$	−9757.78	−0.325001
$967$	−20836.9	−0.692935	−0.346468	−	0.938062i	$-0.612619\pi$
$967$	−20836.9	−0.692935	−0.346468	+	0.938062i	$0.612619\pi$
$968$	15026.3	0.498930
$969$	−101.454	−0.00336343
$970$	43746.6	1.44806
$971$	−5902.12	−0.195065	−0.0975325	−	0.995232i	$-0.531095\pi$
$971$	−5902.12	−0.195065	−0.0975325	+	0.995232i	$0.531095\pi$
$972$	−17785.6	−0.586906
$973$	−51969.7	−1.71230
$974$	589.884	0.0194056
$975$	−18539.0	−0.608946
$976$	−16847.5	−0.552538
$977$	17199.6	0.563218	0.281609	−	0.959529i	$-0.409132\pi$
$977$	17199.6	0.563218	0.281609	+	0.959529i	$0.409132\pi$
$978$	584.457	0.0191093
$979$	−2602.56	−0.0849623
$980$	−71379.9	−2.32668
$981$	−20945.4	−0.681687
$982$	3622.51	0.117718
$983$	8184.53	0.265561	0.132780	−	0.991145i	$-0.457610\pi$
$983$	8184.53	0.265561	0.132780	+	0.991145i	$0.457610\pi$
$984$	−5053.28	−0.163712
$985$	−25847.3	−0.836104
$986$	−2825.22	−0.0912507
$987$	−1595.35	−0.0514494
$988$	1518.70	0.0489032
$989$	7534.73	0.242255
$990$	15141.2	0.486081
$991$	−43473.7	−1.39353	−0.696765	−	0.717300i	$-0.745378\pi$
$991$	−43473.7	−1.39353	−0.696765	+	0.717300i	$0.745378\pi$
$992$	−28470.8	−0.911238
$993$	11910.6	0.380637
$994$	30868.2	0.984989
$995$	−104373.	−3.32546
$996$	−3779.58	−0.120242
$997$	42253.7	1.34221	0.671107	−	0.741361i	$-0.265819\pi$
$997$	42253.7	1.34221	0.671107	+	0.741361i	$0.265819\pi$
$998$	−422.558	−0.0134026
$999$	7089.87	0.224538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.a.b.1.3	✓	6
3.2	odd	2		387.4.a.h.1.4		6
4.3	odd	2		688.4.a.i.1.4		6
5.4	even	2		1075.4.a.b.1.4		6
7.6	odd	2		2107.4.a.c.1.3		6
43.42	odd	2		1849.4.a.c.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.3	✓	6	1.1	even	1	trivial
387.4.a.h.1.4		6	3.2	odd	2
688.4.a.i.1.4		6	4.3	odd	2
1075.4.a.b.1.4		6	5.4	even	2
1849.4.a.c.1.4		6	43.42	odd	2
2107.4.a.c.1.3		6	7.6	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:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.a (trivial)

\(f(q)\)	\(=\)	\(q+1.29971 q^{2} +1.43046 q^{3} -6.31076 q^{4} +20.4116 q^{5} +1.85918 q^{6} +29.9522 q^{7} -18.5998 q^{8} -24.9538 q^{9} +O(q^{10})\)	\(q+1.29971 q^{2} +1.43046 q^{3} -6.31076 q^{4} +20.4116 q^{5} +1.85918 q^{6} +29.9522 q^{7} -18.5998 q^{8} -24.9538 q^{9} +26.5291 q^{10} -22.8719 q^{11} -9.02731 q^{12} -44.4397 q^{13} +38.9292 q^{14} +29.1981 q^{15} +26.3118 q^{16} -13.0970 q^{17} -32.4326 q^{18} +5.41527 q^{19} -128.813 q^{20} +42.8456 q^{21} -29.7268 q^{22} -175.226 q^{23} -26.6063 q^{24} +291.634 q^{25} -57.7586 q^{26} -74.3180 q^{27} -189.021 q^{28} +165.972 q^{29} +37.9489 q^{30} -155.581 q^{31} +182.996 q^{32} -32.7174 q^{33} -17.0222 q^{34} +611.374 q^{35} +157.477 q^{36} -95.3991 q^{37} +7.03827 q^{38} -63.5693 q^{39} -379.652 q^{40} +189.928 q^{41} +55.6867 q^{42} -43.0000 q^{43} +144.339 q^{44} -509.347 q^{45} -227.743 q^{46} -37.2349 q^{47} +37.6381 q^{48} +554.137 q^{49} +379.039 q^{50} -18.7348 q^{51} +280.448 q^{52} +559.862 q^{53} -96.5916 q^{54} -466.852 q^{55} -557.106 q^{56} +7.74635 q^{57} +215.715 q^{58} -82.3042 q^{59} -184.262 q^{60} -640.304 q^{61} -202.210 q^{62} -747.422 q^{63} +27.3469 q^{64} -907.085 q^{65} -42.5231 q^{66} -509.592 q^{67} +82.6519 q^{68} -250.655 q^{69} +794.607 q^{70} +792.932 q^{71} +464.135 q^{72} +612.727 q^{73} -123.991 q^{74} +417.171 q^{75} -34.1745 q^{76} -685.065 q^{77} -82.6215 q^{78} +237.047 q^{79} +537.066 q^{80} +567.443 q^{81} +246.850 q^{82} +418.683 q^{83} -270.388 q^{84} -267.330 q^{85} -55.8874 q^{86} +237.417 q^{87} +425.413 q^{88} +113.788 q^{89} -662.002 q^{90} -1331.07 q^{91} +1105.81 q^{92} -222.553 q^{93} -48.3944 q^{94} +110.534 q^{95} +261.769 q^{96} +1649.00 q^{97} +720.216 q^{98} +570.740 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 7 q^{3} + 22 q^{4} + 43 q^{5} - 3 q^{6} + 8 q^{7} + 54 q^{8} + 81 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 7 * q^3 + 22 * q^4 + 43 * q^5 - 3 * q^6 + 8 * q^7 + 54 * q^8 + 81 * q^9	\( 6 q + 6 q^{2} + 7 q^{3} + 22 q^{4} + 43 q^{5} - 3 q^{6} + 8 q^{7} + 54 q^{8} + 81 q^{9} + 57 q^{10} - 28 q^{11} - 157 q^{12} + 56 q^{13} - 184 q^{14} - 124 q^{15} - 54 q^{16} + 19 q^{17} - 81 q^{18} - 75 q^{19} + 135 q^{20} - 18 q^{21} - 504 q^{22} + 131 q^{23} - 567 q^{24} + 105 q^{25} + 44 q^{26} + 238 q^{27} - 404 q^{28} + 515 q^{29} - 396 q^{30} + 237 q^{31} + 558 q^{32} + 540 q^{33} - 107 q^{34} + 198 q^{35} + 73 q^{36} + 269 q^{37} + 527 q^{38} + 290 q^{39} + 613 q^{40} + 471 q^{41} + 362 q^{42} - 258 q^{43} - 428 q^{44} + 334 q^{45} - 67 q^{46} + 415 q^{47} - 989 q^{48} + 350 q^{49} + 1335 q^{50} - 1241 q^{51} - 8 q^{52} + 450 q^{53} + 402 q^{54} - 1732 q^{55} - 780 q^{56} - 1000 q^{57} - 1055 q^{58} + 356 q^{59} - 2732 q^{60} - 1328 q^{61} + 1603 q^{62} - 2290 q^{63} + 466 q^{64} - 62 q^{65} + 156 q^{66} - 632 q^{67} + 571 q^{68} - 1130 q^{69} - 1902 q^{70} - 144 q^{71} + 567 q^{72} + 864 q^{73} + 1207 q^{74} - 2494 q^{75} + 1005 q^{76} + 2660 q^{77} + 2222 q^{78} - 1613 q^{79} + 2399 q^{80} - 102 q^{81} + 1673 q^{82} - 682 q^{83} + 3758 q^{84} + 84 q^{85} - 258 q^{86} + 449 q^{87} - 608 q^{88} + 3378 q^{89} + 930 q^{90} - 3900 q^{91} + 3491 q^{92} + 1879 q^{93} + 3197 q^{94} - 79 q^{95} - 591 q^{96} - 55 q^{97} + 2398 q^{98} - 1612 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 7 * q^3 + 22 * q^4 + 43 * q^5 - 3 * q^6 + 8 * q^7 + 54 * q^8 + 81 * q^9 + 57 * q^10 - 28 * q^11 - 157 * q^12 + 56 * q^13 - 184 * q^14 - 124 * q^15 - 54 * q^16 + 19 * q^17 - 81 * q^18 - 75 * q^19 + 135 * q^20 - 18 * q^21 - 504 * q^22 + 131 * q^23 - 567 * q^24 + 105 * q^25 + 44 * q^26 + 238 * q^27 - 404 * q^28 + 515 * q^29 - 396 * q^30 + 237 * q^31 + 558 * q^32 + 540 * q^33 - 107 * q^34 + 198 * q^35 + 73 * q^36 + 269 * q^37 + 527 * q^38 + 290 * q^39 + 613 * q^40 + 471 * q^41 + 362 * q^42 - 258 * q^43 - 428 * q^44 + 334 * q^45 - 67 * q^46 + 415 * q^47 - 989 * q^48 + 350 * q^49 + 1335 * q^50 - 1241 * q^51 - 8 * q^52 + 450 * q^53 + 402 * q^54 - 1732 * q^55 - 780 * q^56 - 1000 * q^57 - 1055 * q^58 + 356 * q^59 - 2732 * q^60 - 1328 * q^61 + 1603 * q^62 - 2290 * q^63 + 466 * q^64 - 62 * q^65 + 156 * q^66 - 632 * q^67 + 571 * q^68 - 1130 * q^69 - 1902 * q^70 - 144 * q^71 + 567 * q^72 + 864 * q^73 + 1207 * q^74 - 2494 * q^75 + 1005 * q^76 + 2660 * q^77 + 2222 * q^78 - 1613 * q^79 + 2399 * q^80 - 102 * q^81 + 1673 * q^82 - 682 * q^83 + 3758 * q^84 + 84 * q^85 - 258 * q^86 + 449 * q^87 - 608 * q^88 + 3378 * q^89 + 930 * q^90 - 3900 * q^91 + 3491 * q^92 + 1879 * q^93 + 3197 * q^94 - 79 * q^95 - 591 * q^96 - 55 * q^97 + 2398 * q^98 - 1612 * q^99

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