LMFDB - Embedded newform 77.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,4,Mod(1,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, 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("77.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	77.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:	$4.54314707044$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.509800.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 5x + 15 $ x^4 - x^3 - 10x^2 + 5x + 15
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			1.2
Root			$-1.11082$ of defining polynomial
Character	$\chi$	$=$	77.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.65527 q^{2} +5.17115 q^{3} +5.36103 q^{4} -10.0822 q^{5} -18.9020 q^{6} -7.00000 q^{7} +9.64616 q^{8} -0.259212 q^{9} +O(q^{10})$	\(q-3.65527 q^{2} +5.17115 q^{3} +5.36103 q^{4} -10.0822 q^{5} -18.9020 q^{6} -7.00000 q^{7} +9.64616 q^{8} -0.259212 q^{9} +36.8533 q^{10} +11.0000 q^{11} +27.7227 q^{12} -84.5724 q^{13} +25.5869 q^{14} -52.1367 q^{15} -78.1476 q^{16} -38.2525 q^{17} +0.947489 q^{18} -127.283 q^{19} -54.0511 q^{20} -36.1980 q^{21} -40.2080 q^{22} +140.378 q^{23} +49.8817 q^{24} -23.3486 q^{25} +309.135 q^{26} -140.961 q^{27} -37.5272 q^{28} -116.806 q^{29} +190.574 q^{30} +338.709 q^{31} +208.482 q^{32} +56.8826 q^{33} +139.823 q^{34} +70.5756 q^{35} -1.38964 q^{36} -75.3416 q^{37} +465.256 q^{38} -437.337 q^{39} -97.2548 q^{40} -22.4446 q^{41} +132.314 q^{42} +181.844 q^{43} +58.9713 q^{44} +2.61343 q^{45} -513.121 q^{46} +300.530 q^{47} -404.113 q^{48} +49.0000 q^{49} +85.3455 q^{50} -197.810 q^{51} -453.395 q^{52} -31.8596 q^{53} +515.253 q^{54} -110.905 q^{55} -67.5231 q^{56} -658.201 q^{57} +426.958 q^{58} -68.3030 q^{59} -279.507 q^{60} -145.315 q^{61} -1238.07 q^{62} +1.81448 q^{63} -136.877 q^{64} +852.679 q^{65} -207.922 q^{66} -668.020 q^{67} -205.073 q^{68} +725.916 q^{69} -257.973 q^{70} +727.608 q^{71} -2.50040 q^{72} -416.982 q^{73} +275.394 q^{74} -120.739 q^{75} -682.370 q^{76} -77.0000 q^{77} +1598.59 q^{78} +458.805 q^{79} +787.902 q^{80} -721.934 q^{81} +82.0412 q^{82} +355.737 q^{83} -194.059 q^{84} +385.671 q^{85} -664.690 q^{86} -604.022 q^{87} +106.108 q^{88} -1245.97 q^{89} -9.55281 q^{90} +592.007 q^{91} +752.571 q^{92} +1751.51 q^{93} -1098.52 q^{94} +1283.30 q^{95} +1078.09 q^{96} -935.338 q^{97} -179.108 q^{98} -2.85133 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 12 q^{3} + 22 q^{4} - 18 q^{5} + 2 q^{6} - 28 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 12 * q^3 + 22 * q^4 - 18 * q^5 + 2 * q^6 - 28 * q^7 - 60 * q^8 + 66 * q^9	\( 4 q - 4 q^{2} - 12 q^{3} + 22 q^{4} - 18 q^{5} + 2 q^{6} - 28 q^{7} - 60 q^{8} + 66 q^{9} - 92 q^{10} + 44 q^{11} - 186 q^{12} - 134 q^{13} + 28 q^{14} - 62 q^{15} - 6 q^{16} - 74 q^{17} - 256 q^{18} - 164 q^{19} + 116 q^{20} + 84 q^{21} - 44 q^{22} + 194 q^{23} + 570 q^{24} + 38 q^{25} + 734 q^{26} - 510 q^{27} - 154 q^{28} - 108 q^{29} + 1252 q^{30} - 412 q^{31} - 4 q^{32} - 132 q^{33} - 346 q^{34} + 126 q^{35} + 1518 q^{36} + 286 q^{37} + 224 q^{38} - 256 q^{39} - 540 q^{40} - 18 q^{41} - 14 q^{42} - 496 q^{43} + 242 q^{44} + 580 q^{45} - 284 q^{46} + 62 q^{47} - 862 q^{48} + 196 q^{49} + 212 q^{50} - 508 q^{51} - 822 q^{52} - 828 q^{53} + 2420 q^{54} - 198 q^{55} + 420 q^{56} + 700 q^{57} + 1388 q^{58} - 1224 q^{59} - 1776 q^{60} - 350 q^{61} - 878 q^{62} - 462 q^{63} - 718 q^{64} - 396 q^{65} + 22 q^{66} - 1498 q^{67} + 1058 q^{68} - 386 q^{69} + 644 q^{70} + 2326 q^{71} - 3000 q^{72} - 1630 q^{73} - 1156 q^{74} - 1362 q^{75} - 3152 q^{76} - 308 q^{77} - 2464 q^{78} - 1020 q^{79} + 3072 q^{80} + 1128 q^{81} + 2118 q^{82} - 1920 q^{83} + 1302 q^{84} + 2008 q^{85} + 1056 q^{86} + 1640 q^{87} - 660 q^{88} + 1550 q^{89} - 5780 q^{90} + 938 q^{91} + 2592 q^{92} + 6046 q^{93} - 1042 q^{94} + 2332 q^{95} + 4082 q^{96} - 2202 q^{97} - 196 q^{98} + 726 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 12 * q^3 + 22 * q^4 - 18 * q^5 + 2 * q^6 - 28 * q^7 - 60 * q^8 + 66 * q^9 - 92 * q^10 + 44 * q^11 - 186 * q^12 - 134 * q^13 + 28 * q^14 - 62 * q^15 - 6 * q^16 - 74 * q^17 - 256 * q^18 - 164 * q^19 + 116 * q^20 + 84 * q^21 - 44 * q^22 + 194 * q^23 + 570 * q^24 + 38 * q^25 + 734 * q^26 - 510 * q^27 - 154 * q^28 - 108 * q^29 + 1252 * q^30 - 412 * q^31 - 4 * q^32 - 132 * q^33 - 346 * q^34 + 126 * q^35 + 1518 * q^36 + 286 * q^37 + 224 * q^38 - 256 * q^39 - 540 * q^40 - 18 * q^41 - 14 * q^42 - 496 * q^43 + 242 * q^44 + 580 * q^45 - 284 * q^46 + 62 * q^47 - 862 * q^48 + 196 * q^49 + 212 * q^50 - 508 * q^51 - 822 * q^52 - 828 * q^53 + 2420 * q^54 - 198 * q^55 + 420 * q^56 + 700 * q^57 + 1388 * q^58 - 1224 * q^59 - 1776 * q^60 - 350 * q^61 - 878 * q^62 - 462 * q^63 - 718 * q^64 - 396 * q^65 + 22 * q^66 - 1498 * q^67 + 1058 * q^68 - 386 * q^69 + 644 * q^70 + 2326 * q^71 - 3000 * q^72 - 1630 * q^73 - 1156 * q^74 - 1362 * q^75 - 3152 * q^76 - 308 * q^77 - 2464 * q^78 - 1020 * q^79 + 3072 * q^80 + 1128 * q^81 + 2118 * q^82 - 1920 * q^83 + 1302 * q^84 + 2008 * q^85 + 1056 * q^86 + 1640 * q^87 - 660 * q^88 + 1550 * q^89 - 5780 * q^90 + 938 * q^91 + 2592 * q^92 + 6046 * q^93 - 1042 * q^94 + 2332 * q^95 + 4082 * q^96 - 2202 * q^97 - 196 * q^98 + 726 * 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 $

$2$	−3.65527	−1.29233	−0.646167	−	0.763196i	$-0.723629\pi$
$2$	−3.65527	−1.29233	−0.646167	+	0.763196i	$0.723629\pi$
$3$	5.17115	0.995188	0.497594	−	0.867410i	$-0.334217\pi$
$3$	5.17115	0.995188	0.497594	+	0.867410i	$0.334217\pi$
$4$	5.36103	0.670129
$5$	−10.0822	−0.901782	−0.450891	−	0.892579i	$-0.648894\pi$
$5$	−10.0822	−0.901782	−0.450891	+	0.892579i	$0.648894\pi$
$6$	−18.9020	−1.28612
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	9.64616	0.426304
$9$	−0.259212	−0.00960043
$10$	36.8533	1.16540
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	27.7227	0.666904
$13$	−84.5724	−1.80432	−0.902161	−	0.431400i	$-0.858020\pi$
$13$	−84.5724	−1.80432	−0.902161	+	0.431400i	$0.858020\pi$
$14$	25.5869	0.488457
$15$	−52.1367	−0.897443
$16$	−78.1476	−1.22106
$17$	−38.2525	−0.545741	−0.272871	−	0.962051i	$-0.587973\pi$
$17$	−38.2525	−0.545741	−0.272871	+	0.962051i	$0.587973\pi$
$18$	0.947489	0.0124070
$19$	−127.283	−1.53688	−0.768442	−	0.639919i	$-0.778968\pi$
$19$	−127.283	−1.53688	−0.768442	+	0.639919i	$0.778968\pi$
$20$	−54.0511	−0.604310
$21$	−36.1980	−0.376146
$22$	−40.2080	−0.389654
$23$	140.378	1.27265	0.636323	−	0.771423i	$-0.280455\pi$
$23$	140.378	1.27265	0.636323	+	0.771423i	$0.280455\pi$
$24$	49.8817	0.424253
$25$	−23.3486	−0.186789
$26$	309.135	2.33179
$27$	−140.961	−1.00474
$28$	−37.5272	−0.253285
$29$	−116.806	−0.747943	−0.373971	−	0.927440i	$-0.622004\pi$
$29$	−116.806	−0.747943	−0.373971	+	0.927440i	$0.622004\pi$
$30$	190.574	1.15980
$31$	338.709	1.96238	0.981192	−	0.193034i	$-0.0618327\pi$
$31$	338.709	1.96238	0.981192	+	0.193034i	$0.0618327\pi$
$32$	208.482	1.15171
$33$	56.8826	0.300061
$34$	139.823	0.705280
$35$	70.5756	0.340842
$36$	−1.38964	−0.00643352
$37$	−75.3416	−0.334759	−0.167379	−	0.985893i	$-0.553531\pi$
$37$	−75.3416	−0.334759	−0.167379	+	0.985893i	$0.553531\pi$
$38$	465.256	1.98617
$39$	−437.337	−1.79564
$40$	−97.2548	−0.384434
$41$	−22.4446	−0.0854941	−0.0427471	−	0.999086i	$-0.513611\pi$
$41$	−22.4446	−0.0854941	−0.0427471	+	0.999086i	$0.513611\pi$
$42$	132.314	0.486106
$43$	181.844	0.644906	0.322453	−	0.946585i	$-0.395492\pi$
$43$	181.844	0.644906	0.322453	+	0.946585i	$0.395492\pi$
$44$	58.9713	0.202051
$45$	2.61343	0.00865749
$46$	−513.121	−1.64468
$47$	300.530	0.932698	0.466349	−	0.884601i	$-0.345569\pi$
$47$	300.530	0.932698	0.466349	+	0.884601i	$0.345569\pi$
$48$	−404.113	−1.21518
$49$	49.0000	0.142857
$50$	85.3455	0.241394
$51$	−197.810	−0.543115
$52$	−453.395	−1.20913
$53$	−31.8596	−0.0825708	−0.0412854	−	0.999147i	$-0.513145\pi$
$53$	−31.8596	−0.0825708	−0.0412854	+	0.999147i	$0.513145\pi$
$54$	515.253	1.29846
$55$	−110.905	−0.271898
$56$	−67.5231	−0.161128
$57$	−658.201	−1.52949
$58$	426.958	0.966593
$59$	−68.3030	−0.150717	−0.0753584	−	0.997157i	$-0.524010\pi$
$59$	−68.3030	−0.150717	−0.0753584	+	0.997157i	$0.524010\pi$
$60$	−279.507	−0.601402
$61$	−145.315	−0.305012	−0.152506	−	0.988303i	$-0.548734\pi$
$61$	−145.315	−0.305012	−0.152506	+	0.988303i	$0.548734\pi$
$62$	−1238.07	−2.53606
$63$	1.81448	0.00362862
$64$	−136.877	−0.267337
$65$	852.679	1.62710
$66$	−207.922	−0.387779
$67$	−668.020	−1.21808	−0.609042	−	0.793138i	$-0.708446\pi$
$67$	−668.020	−1.21808	−0.609042	+	0.793138i	$0.708446\pi$
$68$	−205.073	−0.365717
$69$	725.916	1.26652
$70$	−257.973	−0.440481
$71$	727.608	1.21621	0.608107	−	0.793855i	$-0.291929\pi$
$71$	727.608	1.21621	0.608107	+	0.793855i	$0.291929\pi$
$72$	−2.50040	−0.00409270
$73$	−416.982	−0.668548	−0.334274	−	0.942476i	$-0.608491\pi$
$73$	−416.982	−0.668548	−0.334274	+	0.942476i	$0.608491\pi$
$74$	275.394	0.432621
$75$	−120.739	−0.185890
$76$	−682.370	−1.02991
$77$	−77.0000	−0.113961
$78$	1598.59	2.32057
$79$	458.805	0.653413	0.326707	−	0.945126i	$-0.394061\pi$
$79$	458.805	0.653413	0.326707	+	0.945126i	$0.394061\pi$
$80$	787.902	1.10113
$81$	−721.934	−0.990307
$82$	82.0412	0.110487
$83$	355.737	0.470449	0.235224	−	0.971941i	$-0.424418\pi$
$83$	355.737	0.470449	0.235224	+	0.971941i	$0.424418\pi$
$84$	−194.059	−0.252066
$85$	385.671	0.492140
$86$	−664.690	−0.833435
$87$	−604.022	−0.744344
$88$	106.108	0.128536
$89$	−1245.97	−1.48396	−0.741980	−	0.670422i	$-0.766113\pi$
$89$	−1245.97	−1.48396	−0.741980	+	0.670422i	$0.766113\pi$
$90$	−9.55281	−0.0111884
$91$	592.007	0.681969
$92$	752.571	0.852837
$93$	1751.51	1.95294
$94$	−1098.52	−1.20536
$95$	1283.30	1.38594
$96$	1078.09	1.14617
$97$	−935.338	−0.979063	−0.489532	−	0.871986i	$-0.662832\pi$
$97$	−935.338	−0.979063	−0.489532	+	0.871986i	$0.662832\pi$
$98$	−179.108	−0.184619
$99$	−2.85133	−0.00289464
$100$	−125.173	−0.125173
$101$	−533.395	−0.525493	−0.262747	−	0.964865i	$-0.584628\pi$
$101$	−533.395	−0.525493	−0.262747	+	0.964865i	$0.584628\pi$
$102$	723.048	0.701887
$103$	−738.096	−0.706086	−0.353043	−	0.935607i	$-0.614853\pi$
$103$	−738.096	−0.706086	−0.353043	+	0.935607i	$0.614853\pi$
$104$	−815.800	−0.769190
$105$	364.957	0.339202
$106$	116.456	0.106709
$107$	−2039.07	−1.84228	−0.921141	−	0.389229i	$-0.872741\pi$
$107$	−2039.07	−1.84228	−0.921141	+	0.389229i	$0.872741\pi$
$108$	−755.699	−0.673307
$109$	1488.69	1.30817	0.654085	−	0.756421i	$-0.273054\pi$
$109$	1488.69	1.30817	0.654085	+	0.756421i	$0.273054\pi$
$110$	405.387	0.351383
$111$	−389.603	−0.333148
$112$	547.033	0.461516
$113$	−532.743	−0.443507	−0.221753	−	0.975103i	$-0.571178\pi$
$113$	−532.743	−0.443507	−0.221753	+	0.975103i	$0.571178\pi$
$114$	2405.91	1.97661
$115$	−1415.32	−1.14765
$116$	−626.201	−0.501218
$117$	21.9222	0.0173223
$118$	249.666	0.194776
$119$	267.768	0.206271
$120$	−502.919	−0.382584
$121$	121.000	0.0909091
$122$	531.167	0.394177
$123$	−116.064	−0.0850828
$124$	1815.83	1.31505
$125$	1495.68	1.07023
$126$	−6.63242	−0.00468939
$127$	−2257.44	−1.57729	−0.788645	−	0.614849i	$-0.789217\pi$
$127$	−2257.44	−1.57729	−0.788645	+	0.614849i	$0.789217\pi$
$128$	−1167.53	−0.806220
$129$	940.343	0.641803
$130$	−3116.78	−2.10276
$131$	1174.21	0.783142	0.391571	−	0.920148i	$-0.371932\pi$
$131$	1174.21	0.783142	0.391571	+	0.920148i	$0.371932\pi$
$132$	304.950	0.201079
$133$	890.984	0.580888
$134$	2441.79	1.57417
$135$	1421.21	0.906059
$136$	−368.990	−0.232652
$137$	2690.08	1.67758	0.838792	−	0.544451i	$-0.183262\pi$
$137$	2690.08	1.67758	0.838792	+	0.544451i	$0.183262\pi$
$138$	−2653.42	−1.63677
$139$	17.7500	0.0108312	0.00541559	−	0.999985i	$-0.498276\pi$
$139$	17.7500	0.0108312	0.00541559	+	0.999985i	$0.498276\pi$
$140$	378.358	0.228408
$141$	1554.09	0.928210
$142$	−2659.61	−1.57175
$143$	−930.297	−0.544023
$144$	20.2568	0.0117227
$145$	1177.67	0.674482
$146$	1524.18	0.863988
$147$	253.386	0.142170
$148$	−403.908	−0.224332
$149$	1517.86	0.834550	0.417275	−	0.908780i	$-0.362985\pi$
$149$	1517.86	0.834550	0.417275	+	0.908780i	$0.362985\pi$
$150$	441.335	0.240232
$151$	1948.86	1.05031	0.525153	−	0.851008i	$-0.324008\pi$
$151$	1948.86	1.05031	0.525153	+	0.851008i	$0.324008\pi$
$152$	−1227.80	−0.655180
$153$	9.91550	0.00523935
$154$	281.456	0.147275
$155$	−3414.94	−1.76964
$156$	−2344.58	−1.20331
$157$	−1554.20	−0.790055	−0.395027	−	0.918669i	$-0.629265\pi$
$157$	−1554.20	−0.790055	−0.395027	+	0.918669i	$0.629265\pi$
$158$	−1677.06	−0.844428
$159$	−164.751	−0.0821735
$160$	−2101.96	−1.03859
$161$	−982.647	−0.481015
$162$	2638.87	1.27981
$163$	−3472.71	−1.66873	−0.834367	−	0.551209i	$-0.814167\pi$
$163$	−3472.71	−1.66873	−0.834367	+	0.551209i	$0.814167\pi$
$164$	−120.326	−0.0572921
$165$	−573.504	−0.270589
$166$	−1300.32	−0.607977
$167$	−2228.90	−1.03280	−0.516400	−	0.856347i	$-0.672728\pi$
$167$	−2228.90	−1.03280	−0.516400	+	0.856347i	$0.672728\pi$
$168$	−349.172	−0.160353
$169$	4955.50	2.25558
$170$	−1409.73	−0.636009
$171$	32.9933	0.0147547
$172$	974.872	0.432170
$173$	−1008.04	−0.443005	−0.221502	−	0.975160i	$-0.571096\pi$
$173$	−1008.04	−0.443005	−0.221502	+	0.975160i	$0.571096\pi$
$174$	2207.87	0.961942
$175$	163.440	0.0705995
$176$	−859.624	−0.368162
$177$	−353.205	−0.149992
$178$	4554.36	1.91777
$179$	−746.246	−0.311603	−0.155802	−	0.987788i	$-0.549796\pi$
$179$	−746.246	−0.311603	−0.155802	+	0.987788i	$0.549796\pi$
$180$	14.0107	0.00580163
$181$	−2787.76	−1.14482	−0.572410	−	0.819968i	$-0.693991\pi$
$181$	−2787.76	−1.14482	−0.572410	+	0.819968i	$0.693991\pi$
$182$	−2163.95	−0.881333
$183$	−751.447	−0.303544
$184$	1354.11	0.542534
$185$	759.611	0.301880
$186$	−6402.27	−2.52385
$187$	−420.778	−0.164547
$188$	1611.15	0.625028
$189$	986.730	0.379757
$190$	−4690.82	−1.79109
$191$	−911.917	−0.345466	−0.172733	−	0.984969i	$-0.555260\pi$
$191$	−911.917	−0.345466	−0.172733	+	0.984969i	$0.555260\pi$
$192$	−707.810	−0.266051
$193$	4454.66	1.66142	0.830709	−	0.556707i	$-0.187935\pi$
$193$	4454.66	1.66142	0.830709	+	0.556707i	$0.187935\pi$
$194$	3418.92	1.26528
$195$	4409.33	1.61928
$196$	262.690	0.0957327
$197$	−1377.46	−0.498171	−0.249086	−	0.968481i	$-0.580130\pi$
$197$	−1377.46	−0.498171	−0.249086	+	0.968481i	$0.580130\pi$
$198$	10.4224	0.00374084
$199$	−94.3572	−0.0336121	−0.0168060	−	0.999859i	$-0.505350\pi$
$199$	−94.3572	−0.0336121	−0.0168060	+	0.999859i	$0.505350\pi$
$200$	−225.224	−0.0796288
$201$	−3454.43	−1.21222
$202$	1949.71	0.679113
$203$	817.643	0.282696
$204$	−1060.46	−0.363957
$205$	226.292	0.0770971
$206$	2697.95	0.912499
$207$	−36.3876	−0.0122179
$208$	6609.13	2.20318
$209$	−1400.12	−0.463388
$210$	−1334.02	−0.438362
$211$	1174.19	0.383101	0.191550	−	0.981483i	$-0.438648\pi$
$211$	1174.19	0.383101	0.191550	+	0.981483i	$0.438648\pi$
$212$	−170.800	−0.0553330
$213$	3762.57	1.21036
$214$	7453.35	2.38084
$215$	−1833.40	−0.581565
$216$	−1359.74	−0.428326
$217$	−2370.96	−0.741712
$218$	−5441.56	−1.69059
$219$	−2156.27	−0.665331
$220$	−594.563	−0.182206
$221$	3235.11	0.984692
$222$	1424.10	0.430539
$223$	88.5875	0.0266021	0.0133010	−	0.999912i	$-0.495766\pi$
$223$	88.5875	0.0266021	0.0133010	+	0.999912i	$0.495766\pi$
$224$	−1459.37	−0.435305
$225$	6.05223	0.00179325
$226$	1947.32	0.573159
$227$	883.312	0.258271	0.129135	−	0.991627i	$-0.458780\pi$
$227$	883.312	0.258271	0.129135	+	0.991627i	$0.458780\pi$
$228$	−3528.64	−1.02495
$229$	1240.26	0.357898	0.178949	−	0.983858i	$-0.442730\pi$
$229$	1240.26	0.357898	0.178949	+	0.983858i	$0.442730\pi$
$230$	5173.40	1.48315
$231$	−398.179	−0.113412
$232$	−1126.73	−0.318851
$233$	−5479.93	−1.54078	−0.770392	−	0.637571i	$-0.779939\pi$
$233$	−5479.93	−1.54078	−0.770392	+	0.637571i	$0.779939\pi$
$234$	−80.1315	−0.0223861
$235$	−3030.01	−0.841090
$236$	−366.174	−0.101000
$237$	2372.55	0.650269
$238$	−978.764	−0.266571
$239$	594.006	0.160766	0.0803830	−	0.996764i	$-0.474386\pi$
$239$	594.006	0.160766	0.0803830	+	0.996764i	$0.474386\pi$
$240$	4074.36	1.09583
$241$	308.785	0.0825336	0.0412668	−	0.999148i	$-0.486861\pi$
$241$	308.785	0.0825336	0.0412668	+	0.999148i	$0.486861\pi$
$242$	−442.288	−0.117485
$243$	72.7302	0.0192002
$244$	−779.039	−0.204397
$245$	−494.029	−0.128826
$246$	424.247	0.109955
$247$	10764.7	2.77303
$248$	3267.24	0.836573
$249$	1839.57	0.468185
$250$	−5467.14	−1.38309
$251$	3487.59	0.877031	0.438515	−	0.898724i	$-0.355505\pi$
$251$	3487.59	0.877031	0.438515	+	0.898724i	$0.355505\pi$
$252$	9.72748	0.00243164
$253$	1544.16	0.383717
$254$	8251.58	2.03839
$255$	1994.36	0.489772
$256$	5362.66	1.30924
$257$	451.445	0.109574	0.0547868	−	0.998498i	$-0.482552\pi$
$257$	451.445	0.109574	0.0547868	+	0.998498i	$0.482552\pi$
$258$	−3437.21	−0.829425
$259$	527.391	0.126527
$260$	4571.24	1.09037
$261$	30.2775	0.00718057
$262$	−4292.08	−1.01208
$263$	−5878.61	−1.37829	−0.689146	−	0.724622i	$-0.742014\pi$
$263$	−5878.61	−1.37829	−0.689146	+	0.724622i	$0.742014\pi$
$264$	548.699	0.127917
$265$	321.216	0.0744609
$266$	−3256.79	−0.750701
$267$	−6443.09	−1.47682
$268$	−3581.27	−0.816272
$269$	−52.8516	−0.0119792	−0.00598962	−	0.999982i	$-0.501907\pi$
$269$	−52.8516	−0.0119792	−0.00598962	+	0.999982i	$0.501907\pi$
$270$	−5194.90	−1.17093
$271$	−6822.19	−1.52922	−0.764610	−	0.644493i	$-0.777069\pi$
$271$	−6822.19	−1.52922	−0.764610	+	0.644493i	$0.777069\pi$
$272$	2989.34	0.666381
$273$	3061.36	0.678688
$274$	−9832.98	−2.16800
$275$	−256.835	−0.0563189
$276$	3891.66	0.848733
$277$	469.032	0.101738	0.0508689	−	0.998705i	$-0.483801\pi$
$277$	469.032	0.101738	0.0508689	+	0.998705i	$0.483801\pi$
$278$	−64.8810	−0.0139975
$279$	−87.7972	−0.0188397
$280$	680.784	0.145302
$281$	−2305.05	−0.489352	−0.244676	−	0.969605i	$-0.578682\pi$
$281$	−2305.05	−0.489352	−0.244676	+	0.969605i	$0.578682\pi$
$282$	−5680.61	−1.19956
$283$	−7370.80	−1.54823	−0.774114	−	0.633046i	$-0.781805\pi$
$283$	−7370.80	−1.54823	−0.774114	+	0.633046i	$0.781805\pi$
$284$	3900.73	0.815020
$285$	6636.14	1.37927
$286$	3400.49	0.703060
$287$	157.112	0.0323137
$288$	−54.0408	−0.0110569
$289$	−3449.74	−0.702167
$290$	−4304.69	−0.871656
$291$	−4836.77	−0.974352
$292$	−2235.45	−0.448013
$293$	1758.90	0.350702	0.175351	−	0.984506i	$-0.443894\pi$
$293$	1758.90	0.350702	0.175351	+	0.984506i	$0.443894\pi$
$294$	−926.197	−0.183731
$295$	688.646	0.135914
$296$	−726.757	−0.142709
$297$	−1550.58	−0.302941
$298$	−5548.20	−1.07852
$299$	−11872.1	−2.29626
$300$	−647.286	−0.124570
$301$	−1272.91	−0.243752
$302$	−7123.63	−1.35735
$303$	−2758.27	−0.522965
$304$	9946.89	1.87662
$305$	1465.10	0.275054
$306$	−36.2439	−0.00677099
$307$	3468.10	0.644739	0.322369	−	0.946614i	$-0.395521\pi$
$307$	3468.10	0.644739	0.322369	+	0.946614i	$0.395521\pi$
$308$	−412.799	−0.0763682
$309$	−3816.81	−0.702688
$310$	12482.5	2.28697
$311$	1983.98	0.361741	0.180870	−	0.983507i	$-0.442109\pi$
$311$	1983.98	0.361741	0.180870	+	0.983507i	$0.442109\pi$
$312$	−4218.62	−0.765488
$313$	−10094.2	−1.82287	−0.911436	−	0.411443i	$-0.865025\pi$
$313$	−10094.2	−1.82287	−0.911436	+	0.411443i	$0.865025\pi$
$314$	5681.03	1.02102
$315$	−18.2940	−0.00327223
$316$	2459.67	0.437871
$317$	−3051.34	−0.540633	−0.270316	−	0.962772i	$-0.587128\pi$
$317$	−3051.34	−0.540633	−0.270316	+	0.962772i	$0.587128\pi$
$318$	602.209	0.106196
$319$	−1284.87	−0.225513
$320$	1380.02	0.241080
$321$	−10544.3	−1.83342
$322$	3591.84	0.621632
$323$	4868.91	0.838741
$324$	−3870.31	−0.663633
$325$	1974.65	0.337027
$326$	12693.7	2.15656
$327$	7698.23	1.30187
$328$	−216.504	−0.0364465
$329$	−2103.71	−0.352527
$330$	2096.31	0.349692
$331$	−26.9826	−0.00448066	−0.00224033	−	0.999997i	$-0.500713\pi$
$331$	−26.9826	−0.00448066	−0.00224033	+	0.999997i	$0.500713\pi$
$332$	1907.12	0.315261
$333$	19.5294	0.00321383
$334$	8147.25	1.33472
$335$	6735.13	1.09845
$336$	2828.79	0.459295
$337$	6818.62	1.10218	0.551089	−	0.834447i	$-0.314213\pi$
$337$	6818.62	1.10218	0.551089	+	0.834447i	$0.314213\pi$
$338$	−18113.7	−2.91496
$339$	−2754.90	−0.441373
$340$	2067.59	0.329797
$341$	3725.80	0.591681
$342$	−120.600	−0.0190681
$343$	−343.000	−0.0539949
$344$	1754.10	0.274926
$345$	−7318.86	−1.14213
$346$	3684.66	0.572510
$347$	11907.0	1.84208	0.921038	−	0.389473i	$-0.127343\pi$
$347$	11907.0	1.84208	0.921038	+	0.389473i	$0.127343\pi$
$348$	−3238.18	−0.498806
$349$	−4352.72	−0.667609	−0.333805	−	0.942642i	$-0.608333\pi$
$349$	−4352.72	−0.667609	−0.333805	+	0.942642i	$0.608333\pi$
$350$	−597.419	−0.0912382
$351$	11921.5	1.81288
$352$	2293.30	0.347253
$353$	−1326.31	−0.199978	−0.0999891	−	0.994989i	$-0.531881\pi$
$353$	−1326.31	−0.199978	−0.0999891	+	0.994989i	$0.531881\pi$
$354$	1291.06	0.193839
$355$	−7335.91	−1.09676
$356$	−6679.68	−0.994444
$357$	1384.67	0.205278
$358$	2727.73	0.402696
$359$	−8292.92	−1.21917	−0.609587	−	0.792719i	$-0.708665\pi$
$359$	−8292.92	−1.21917	−0.609587	+	0.792719i	$0.708665\pi$
$360$	25.2096	0.00369073
$361$	9342.06	1.36201
$362$	10190.0	1.47949
$363$	625.709	0.0904717
$364$	3173.77	0.457007
$365$	4204.10	0.602885
$366$	2746.74	0.392280
$367$	−11027.5	−1.56848	−0.784241	−	0.620457i	$-0.786947\pi$
$367$	−11027.5	−1.56848	−0.784241	+	0.620457i	$0.786947\pi$
$368$	−10970.2	−1.55397
$369$	5.81790	0.000820780 0
$370$	−2776.59	−0.390130
$371$	223.017	0.0312088
$372$	9389.92	1.30872
$373$	8245.72	1.14463	0.572316	−	0.820034i	$-0.306045\pi$
$373$	8245.72	1.14463	0.572316	+	0.820034i	$0.306045\pi$
$374$	1538.06	0.212650
$375$	7734.41	1.06508
$376$	2898.96	0.397613
$377$	9878.58	1.34953
$378$	−3606.77	−0.490773
$379$	10163.4	1.37747	0.688734	−	0.725014i	$-0.258167\pi$
$379$	10163.4	1.37747	0.688734	+	0.725014i	$0.258167\pi$
$380$	6879.81	0.928755
$381$	−11673.6	−1.56970
$382$	3333.31	0.446458
$383$	14338.9	1.91301	0.956506	−	0.291714i	$-0.0942255\pi$
$383$	14338.9	1.91301	0.956506	+	0.291714i	$0.0942255\pi$
$384$	−6037.48	−0.802341
$385$	776.332	0.102768
$386$	−16283.0	−2.14711
$387$	−47.1361	−0.00619138
$388$	−5014.37	−0.656098
$389$	−2382.91	−0.310587	−0.155294	−	0.987868i	$-0.549632\pi$
$389$	−2382.91	−0.310587	−0.155294	+	0.987868i	$0.549632\pi$
$390$	−16117.3	−2.09265
$391$	−5369.82	−0.694535
$392$	472.662	0.0609006
$393$	6072.04	0.779374
$394$	5034.98	0.643804
$395$	−4625.78	−0.589236
$396$	−15.2860	−0.00193978
$397$	−9868.22	−1.24754	−0.623768	−	0.781609i	$-0.714399\pi$
$397$	−9868.22	−1.24754	−0.623768	+	0.781609i	$0.714399\pi$
$398$	344.901	0.0434380
$399$	4607.41	0.578093
$400$	1824.64	0.228080
$401$	−5879.12	−0.732143	−0.366072	−	0.930587i	$-0.619298\pi$
$401$	−5879.12	−0.732143	−0.366072	+	0.930587i	$0.619298\pi$
$402$	12626.9	1.56660
$403$	−28645.4	−3.54077
$404$	−2859.55	−0.352148
$405$	7278.71	0.893042
$406$	−2988.71	−0.365338
$407$	−828.758	−0.100934
$408$	−1908.10	−0.231532
$409$	5680.84	0.686796	0.343398	−	0.939190i	$-0.388422\pi$
$409$	5680.84	0.686796	0.343398	+	0.939190i	$0.388422\pi$
$410$	−827.158	−0.0996352
$411$	13910.8	1.66951
$412$	−3956.96	−0.473168
$413$	478.121	0.0569656
$414$	133.007	0.0157897
$415$	−3586.62	−0.424242
$416$	−17631.8	−2.07805
$417$	91.7878	0.0107791
$418$	5117.81	0.598853
$419$	−1098.50	−0.128079	−0.0640395	−	0.997947i	$-0.520398\pi$
$419$	−1098.50	−0.128079	−0.0640395	+	0.997947i	$0.520398\pi$
$420$	1956.55	0.227309
$421$	−5265.06	−0.609509	−0.304754	−	0.952431i	$-0.598574\pi$
$421$	−5265.06	−0.609509	−0.304754	+	0.952431i	$0.598574\pi$
$422$	−4291.97	−0.495095
$423$	−77.9008	−0.00895430
$424$	−307.323	−0.0352003
$425$	893.143	0.101938
$426$	−13753.2	−1.56419
$427$	1017.21	0.115284
$428$	−10931.5	−1.23457
$429$	−4810.70	−0.541406
$430$	6701.56	0.751577
$431$	4273.45	0.477598	0.238799	−	0.971069i	$-0.423246\pi$
$431$	4273.45	0.477598	0.238799	+	0.971069i	$0.423246\pi$
$432$	11015.8	1.22685
$433$	−8560.19	−0.950061	−0.475031	−	0.879969i	$-0.657563\pi$
$433$	−8560.19	−0.950061	−0.475031	+	0.879969i	$0.657563\pi$
$434$	8666.52	0.958539
$435$	6089.89	0.671236
$436$	7980.90	0.876642
$437$	−17867.8	−1.95591
$438$	7881.77	0.859830
$439$	12664.2	1.37684	0.688419	−	0.725314i	$-0.258305\pi$
$439$	12664.2	1.37684	0.688419	+	0.725314i	$0.258305\pi$
$440$	−1069.80	−0.115911
$441$	−12.7014	−0.00137149
$442$	−11825.2	−1.27255
$443$	12368.9	1.32656	0.663279	−	0.748372i	$-0.269164\pi$
$443$	12368.9	1.32656	0.663279	+	0.748372i	$0.269164\pi$
$444$	−2088.67	−0.223252
$445$	12562.2	1.33821
$446$	−323.812	−0.0343788
$447$	7849.09	0.830535
$448$	958.136	0.101044
$449$	−2092.69	−0.219956	−0.109978	−	0.993934i	$-0.535078\pi$
$449$	−2092.69	−0.219956	−0.109978	+	0.993934i	$0.535078\pi$
$450$	−22.1225	−0.00231748
$451$	−246.891	−0.0257775
$452$	−2856.05	−0.297207
$453$	10077.9	1.04525
$454$	−3228.75	−0.333772
$455$	−5968.75	−0.614988
$456$	−6349.12	−0.652028
$457$	7825.71	0.801031	0.400515	−	0.916290i	$-0.368831\pi$
$457$	7825.71	0.801031	0.400515	+	0.916290i	$0.368831\pi$
$458$	−4533.49	−0.462525
$459$	5392.13	0.548329
$460$	−7587.60	−0.769073
$461$	4775.60	0.482477	0.241238	−	0.970466i	$-0.422446\pi$
$461$	4775.60	0.482477	0.241238	+	0.970466i	$0.422446\pi$
$462$	1455.45	0.146567
$463$	11518.3	1.15615	0.578077	−	0.815982i	$-0.303803\pi$
$463$	11518.3	1.15615	0.578077	+	0.815982i	$0.303803\pi$
$464$	9128.12	0.913280
$465$	−17659.2	−1.76113
$466$	20030.7	1.99121
$467$	−7420.17	−0.735256	−0.367628	−	0.929973i	$-0.619830\pi$
$467$	−7420.17	−0.735256	−0.367628	+	0.929973i	$0.619830\pi$
$468$	117.525	0.0116081
$469$	4676.14	0.460392
$470$	11075.5	1.08697
$471$	−8037.00	−0.786253
$472$	−658.861	−0.0642512
$473$	2000.29	0.194447
$474$	−8672.32	−0.840365
$475$	2971.89	0.287073
$476$	1435.51	0.138228
$477$	8.25837	0.000792715 0
$478$	−2171.26	−0.207764
$479$	10159.2	0.969076	0.484538	−	0.874770i	$-0.338988\pi$
$479$	10159.2	0.969076	0.484538	+	0.874770i	$0.338988\pi$
$480$	−10869.5	−1.03359
$481$	6371.82	0.604013
$482$	−1128.69	−0.106661
$483$	−5081.41	−0.478701
$484$	648.685	0.0609208
$485$	9430.29	0.882902
$486$	−265.849	−0.0248131
$487$	−12344.9	−1.14866	−0.574331	−	0.818623i	$-0.694738\pi$
$487$	−12344.9	−1.14866	−0.574331	+	0.818623i	$0.694738\pi$
$488$	−1401.73	−0.130028
$489$	−17957.9	−1.66070
$490$	1805.81	0.166486
$491$	15344.9	1.41040	0.705200	−	0.709009i	$-0.250857\pi$
$491$	15344.9	1.41040	0.705200	+	0.709009i	$0.250857\pi$
$492$	−622.225	−0.0570164
$493$	4468.13	0.408183
$494$	−39347.8	−3.58369
$495$	28.7477	0.00261033
$496$	−26469.3	−2.39618
$497$	−5093.26	−0.459686
$498$	−6724.13	−0.605051
$499$	−5022.76	−0.450601	−0.225300	−	0.974289i	$-0.572336\pi$
$499$	−5022.76	−0.450601	−0.225300	+	0.974289i	$0.572336\pi$
$500$	8018.41	0.717188
$501$	−11526.0	−1.02783
$502$	−12748.1	−1.13342
$503$	8735.90	0.774383	0.387191	−	0.921999i	$-0.373445\pi$
$503$	8735.90	0.774383	0.387191	+	0.921999i	$0.373445\pi$
$504$	17.5028	0.00154690
$505$	5377.82	0.473881
$506$	−5644.33	−0.495891
$507$	25625.6	2.24472
$508$	−12102.2	−1.05699
$509$	−21805.5	−1.89884	−0.949420	−	0.314008i	$-0.898328\pi$
$509$	−21805.5	−1.89884	−0.949420	+	0.314008i	$0.898328\pi$
$510$	−7289.94	−0.632949
$511$	2918.87	0.252687
$512$	−10261.7	−0.885760
$513$	17942.1	1.54417
$514$	−1650.16	−0.141606
$515$	7441.66	0.636735
$516$	5041.21	0.430091
$517$	3305.83	0.281219
$518$	−1927.76	−0.163515
$519$	−5212.72	−0.440873
$520$	8225.08	0.693642
$521$	−4732.28	−0.397936	−0.198968	−	0.980006i	$-0.563759\pi$
$521$	−4732.28	−0.397936	−0.198968	+	0.980006i	$0.563759\pi$
$522$	−110.673	−0.00927970
$523$	−7511.23	−0.627998	−0.313999	−	0.949423i	$-0.601669\pi$
$523$	−7511.23	−0.627998	−0.313999	+	0.949423i	$0.601669\pi$
$524$	6295.00	0.524806
$525$	845.174	0.0702598
$526$	21487.9	1.78121
$527$	−12956.5	−1.07095
$528$	−4445.24	−0.366391
$529$	7539.02	0.619629
$530$	−1174.13	−0.0962283
$531$	17.7049	0.00144695
$532$	4776.59	0.389270
$533$	1898.20	0.154259
$534$	23551.3	1.90855
$535$	20558.4	1.66134
$536$	−6443.82	−0.519274
$537$	−3858.95	−0.310104
$538$	193.187	0.0154812
$539$	539.000	0.0430730
$540$	7619.13	0.607176
$541$	−598.410	−0.0475557	−0.0237779	−	0.999717i	$-0.507569\pi$
$541$	−598.410	−0.0475557	−0.0237779	+	0.999717i	$0.507569\pi$
$542$	24937.0	1.97626
$543$	−14415.9	−1.13931
$544$	−7974.95	−0.628535
$545$	−15009.3	−1.17968
$546$	−11190.1	−0.877092
$547$	−14042.3	−1.09763	−0.548816	−	0.835943i	$-0.684921\pi$
$547$	−14042.3	−1.09763	−0.548816	+	0.835943i	$0.684921\pi$
$548$	14421.6	1.12420
$549$	37.6674	0.00292824
$550$	938.801	0.0727829
$551$	14867.5	1.14950
$552$	7002.31	0.539924
$553$	−3211.64	−0.246967
$554$	−1714.44	−0.131479
$555$	3928.06	0.300427
$556$	95.1581	0.00725828
$557$	4965.87	0.377757	0.188878	−	0.982000i	$-0.439515\pi$
$557$	4965.87	0.377757	0.188878	+	0.982000i	$0.439515\pi$
$558$	320.923	0.0243472
$559$	−15379.0	−1.16362
$560$	−5515.32	−0.416187
$561$	−2175.90	−0.163755
$562$	8425.60	0.632406
$563$	−15362.4	−1.14999	−0.574997	−	0.818155i	$-0.694997\pi$
$563$	−15362.4	−1.14999	−0.574997	+	0.818155i	$0.694997\pi$
$564$	8331.50	0.622020
$565$	5371.24	0.399947
$566$	26942.3	2.00083
$567$	5053.54	0.374301
$568$	7018.62	0.518477
$569$	17735.3	1.30668	0.653341	−	0.757064i	$-0.273367\pi$
$569$	17735.3	1.30668	0.653341	+	0.757064i	$0.273367\pi$
$570$	−24256.9	−1.78247
$571$	−19818.8	−1.45252	−0.726262	−	0.687418i	$-0.758744\pi$
$571$	−19818.8	−1.45252	−0.726262	+	0.687418i	$0.758744\pi$
$572$	−4987.35	−0.364566
$573$	−4715.66	−0.343804
$574$	−574.288	−0.0417602
$575$	−3277.63	−0.237716
$576$	35.4800	0.00256655
$577$	6579.03	0.474677	0.237339	−	0.971427i	$-0.423725\pi$
$577$	6579.03	0.474677	0.237339	+	0.971427i	$0.423725\pi$
$578$	12609.8	0.907434
$579$	23035.7	1.65342
$580$	6313.50	0.451989
$581$	−2490.16	−0.177813
$582$	17679.7	1.25919
$583$	−350.455	−0.0248960
$584$	−4022.27	−0.285005
$585$	−221.024	−0.0156209
$586$	−6429.25	−0.453225
$587$	13901.5	0.977470	0.488735	−	0.872432i	$-0.337459\pi$
$587$	13901.5	0.977470	0.488735	+	0.872432i	$0.337459\pi$
$588$	1358.41	0.0952720
$589$	−43112.0	−3.01596
$590$	−2517.19	−0.175646
$591$	−7123.03	−0.495774
$592$	5887.76	0.408760
$593$	23928.0	1.65700	0.828502	−	0.559986i	$-0.189193\pi$
$593$	23928.0	1.65700	0.828502	+	0.559986i	$0.189193\pi$
$594$	5667.78	0.391501
$595$	−2699.70	−0.186011
$596$	8137.30	0.559256
$597$	−487.935	−0.0334503
$598$	43395.9	2.96754
$599$	24078.9	1.64247	0.821233	−	0.570594i	$-0.193287\pi$
$599$	24078.9	1.64247	0.821233	+	0.570594i	$0.193287\pi$
$600$	−1164.67	−0.0792457
$601$	−11806.6	−0.801336	−0.400668	−	0.916223i	$-0.631222\pi$
$601$	−11806.6	−0.801336	−0.400668	+	0.916223i	$0.631222\pi$
$602$	4652.83	0.315009
$603$	173.158	0.0116941
$604$	10447.9	0.703840
$605$	−1219.95	−0.0819802
$606$	10082.2	0.675846
$607$	1957.71	0.130908	0.0654539	−	0.997856i	$-0.479150\pi$
$607$	1957.71	0.130908	0.0654539	+	0.997856i	$0.479150\pi$
$608$	−26536.2	−1.77004
$609$	4228.15	0.281336
$610$	−5355.35	−0.355462
$611$	−25416.6	−1.68289
$612$	53.1573	0.00351104
$613$	−16029.2	−1.05614	−0.528069	−	0.849201i	$-0.677084\pi$
$613$	−16029.2	−1.05614	−0.528069	+	0.849201i	$0.677084\pi$
$614$	−12676.9	−0.833218
$615$	1170.19	0.0767261
$616$	−742.754	−0.0485819
$617$	−7153.99	−0.466789	−0.233394	−	0.972382i	$-0.574983\pi$
$617$	−7153.99	−0.466789	−0.233394	+	0.972382i	$0.574983\pi$
$618$	13951.5	0.908108
$619$	−11035.9	−0.716590	−0.358295	−	0.933608i	$-0.616642\pi$
$619$	−11035.9	−0.716590	−0.358295	+	0.933608i	$0.616642\pi$
$620$	−18307.6	−1.18589
$621$	−19787.9	−1.27868
$622$	−7252.00	−0.467490
$623$	8721.78	0.560884
$624$	34176.8	2.19258
$625$	−12161.3	−0.778321
$626$	36897.1	2.35576
$627$	−7240.22	−0.461158
$628$	−8332.11	−0.529438
$629$	2882.01	0.182692
$630$	66.8696	0.00422881
$631$	−4311.46	−0.272007	−0.136004	−	0.990708i	$-0.543426\pi$
$631$	−4311.46	−0.272007	−0.136004	+	0.990708i	$0.543426\pi$
$632$	4425.71	0.278553
$633$	6071.89	0.381258
$634$	11153.5	0.698678
$635$	22760.1	1.42237
$636$	−883.233	−0.0550668
$637$	−4144.05	−0.257760
$638$	4696.54	0.291439
$639$	−188.604	−0.0116762
$640$	11771.3	0.727035
$641$	2692.13	0.165886	0.0829429	−	0.996554i	$-0.473568\pi$
$641$	2692.13	0.165886	0.0829429	+	0.996554i	$0.473568\pi$
$642$	38542.4	2.36939
$643$	19694.3	1.20788	0.603941	−	0.797029i	$-0.293596\pi$
$643$	19694.3	1.20788	0.603941	+	0.797029i	$0.293596\pi$
$644$	−5268.00	−0.322342
$645$	−9480.76	−0.578767
$646$	−17797.2	−1.08393
$647$	−21225.5	−1.28974	−0.644870	−	0.764292i	$-0.723089\pi$
$647$	−21225.5	−1.28974	−0.644870	+	0.764292i	$0.723089\pi$
$648$	−6963.89	−0.422172
$649$	−751.333	−0.0454428
$650$	−7217.88	−0.435552
$651$	−12260.6	−0.738143
$652$	−18617.3	−1.11827
$653$	−12929.7	−0.774850	−0.387425	−	0.921901i	$-0.626635\pi$
$653$	−12929.7	−0.774850	−0.387425	+	0.921901i	$0.626635\pi$
$654$	−28139.1	−1.68246
$655$	−11838.7	−0.706224
$656$	1753.99	0.104393
$657$	108.086	0.00641835
$658$	7689.64	0.455582
$659$	−20835.3	−1.23161	−0.615803	−	0.787900i	$-0.711168\pi$
$659$	−20835.3	−1.23161	−0.615803	+	0.787900i	$0.711168\pi$
$660$	−3074.57	−0.181330
$661$	−1451.06	−0.0853850	−0.0426925	−	0.999088i	$-0.513594\pi$
$661$	−1451.06	−0.0853850	−0.0426925	+	0.999088i	$0.513594\pi$
$662$	98.6288	0.00579051
$663$	16729.2	0.979954
$664$	3431.50	0.200554
$665$	−8983.10	−0.523834
$666$	−71.3853	−0.00415334
$667$	−16397.0	−0.951867
$668$	−11949.2	−0.692109
$669$	458.099	0.0264741
$670$	−24618.7	−1.41956
$671$	−1598.47	−0.0919645
$672$	−7546.63	−0.433211
$673$	−28986.0	−1.66022	−0.830111	−	0.557598i	$-0.811723\pi$
$673$	−28986.0	−1.66022	−0.830111	+	0.557598i	$0.811723\pi$
$674$	−24923.9	−1.42438
$675$	3291.25	0.187675
$676$	26566.6	1.51153
$677$	24818.6	1.40895	0.704474	−	0.709730i	$-0.251183\pi$
$677$	24818.6	1.40895	0.704474	+	0.709730i	$0.251183\pi$
$678$	10069.9	0.570401
$679$	6547.36	0.370051
$680$	3720.24	0.209801
$681$	4567.74	0.257028
$682$	−13618.8	−0.764650
$683$	−7450.70	−0.417413	−0.208706	−	0.977978i	$-0.566925\pi$
$683$	−7450.70	−0.417413	−0.208706	+	0.977978i	$0.566925\pi$
$684$	176.878	0.00988758
$685$	−27122.0	−1.51282
$686$	1253.76	0.0697795
$687$	6413.57	0.356176
$688$	−14210.7	−0.787467
$689$	2694.44	0.148984
$690$	26752.4	1.47601
$691$	−28469.1	−1.56731	−0.783657	−	0.621193i	$-0.786648\pi$
$691$	−28469.1	−1.56731	−0.783657	+	0.621193i	$0.786648\pi$
$692$	−5404.13	−0.296870
$693$	19.9593	0.00109407
$694$	−43523.3	−2.38058
$695$	−178.959	−0.00976736
$696$	−5826.49	−0.317317
$697$	858.563	0.0466577
$698$	15910.4	0.862775
$699$	−28337.6	−1.53337
$700$	876.208	0.0473108
$701$	20045.7	1.08005	0.540027	−	0.841648i	$-0.318414\pi$
$701$	20045.7	1.08005	0.540027	+	0.841648i	$0.318414\pi$
$702$	−43576.2	−2.34285
$703$	9589.73	0.514486
$704$	−1505.64	−0.0806052
$705$	−15668.6	−0.837043
$706$	4848.02	0.258439
$707$	3733.77	0.198618
$708$	−1893.54	−0.100514
$709$	15326.9	0.811868	0.405934	−	0.913902i	$-0.366946\pi$
$709$	15326.9	0.811868	0.405934	+	0.913902i	$0.366946\pi$
$710$	26814.8	1.41738
$711$	−118.928	−0.00627304
$712$	−12018.8	−0.632618
$713$	47547.3	2.49742
$714$	−5061.34	−0.265288
$715$	9379.47	0.490591
$716$	−4000.64	−0.208814
$717$	3071.70	0.159992
$718$	30312.9	1.57558
$719$	−11023.4	−0.571770	−0.285885	−	0.958264i	$-0.592288\pi$
$719$	−11023.4	−0.571770	−0.285885	+	0.958264i	$0.592288\pi$
$720$	−204.233	−0.0105713
$721$	5166.68	0.266875
$722$	−34147.8	−1.76018
$723$	1596.77	0.0821365
$724$	−14945.3	−0.767177
$725$	2727.26	0.139707
$726$	−2287.14	−0.116920
$727$	28755.9	1.46698	0.733492	−	0.679699i	$-0.237889\pi$
$727$	28755.9	1.46698	0.733492	+	0.679699i	$0.237889\pi$
$728$	5710.60	0.290726
$729$	19868.3	1.00942
$730$	−15367.2	−0.779129
$731$	−6956.00	−0.351952
$732$	−4028.53	−0.203413
$733$	21500.9	1.08343	0.541714	−	0.840563i	$-0.317775\pi$
$733$	21500.9	1.08343	0.541714	+	0.840563i	$0.317775\pi$
$734$	40308.7	2.02700
$735$	−2554.70	−0.128206
$736$	29266.3	1.46572
$737$	−7348.21	−0.367266
$738$	−21.2660	−0.00106072
$739$	−2598.63	−0.129353	−0.0646767	−	0.997906i	$-0.520602\pi$
$739$	−2598.63	−0.129353	−0.0646767	+	0.997906i	$0.520602\pi$
$740$	4072.30	0.202298
$741$	55665.7	2.75969
$742$	−815.189	−0.0403322
$743$	29920.5	1.47736	0.738678	−	0.674058i	$-0.235450\pi$
$743$	29920.5	1.47736	0.738678	+	0.674058i	$0.235450\pi$
$744$	16895.4	0.832547
$745$	−15303.4	−0.752583
$746$	−30140.4	−1.47925
$747$	−92.2112	−0.00451651
$748$	−2255.80	−0.110268
$749$	14273.5	0.696317
$750$	−28271.4	−1.37643
$751$	17763.1	0.863096	0.431548	−	0.902090i	$-0.357968\pi$
$751$	17763.1	0.863096	0.431548	+	0.902090i	$0.357968\pi$
$752$	−23485.7	−1.13888
$753$	18034.8	0.872810
$754$	−36108.9	−1.74404
$755$	−19648.9	−0.947147
$756$	5289.89	0.254486
$757$	6472.04	0.310740	0.155370	−	0.987856i	$-0.450343\pi$
$757$	6472.04	0.310740	0.155370	+	0.987856i	$0.450343\pi$
$758$	−37150.1	−1.78015
$759$	7985.08	0.381871
$760$	12378.9	0.590830
$761$	−33720.2	−1.60625	−0.803126	−	0.595809i	$-0.796831\pi$
$761$	−33720.2	−1.60625	−0.803126	+	0.595809i	$0.796831\pi$
$762$	42670.1	2.02858
$763$	−10420.8	−0.494441
$764$	−4888.81	−0.231507
$765$	−99.9703	−0.00472475
$766$	−52412.6	−2.47225
$767$	5776.55	0.271941
$768$	27731.1	1.30294
$769$	−9361.49	−0.438991	−0.219495	−	0.975614i	$-0.570441\pi$
$769$	−9361.49	−0.438991	−0.219495	+	0.975614i	$0.570441\pi$
$770$	−2837.71	−0.132810
$771$	2334.49	0.109046
$772$	23881.6	1.11336
$773$	34886.0	1.62324	0.811618	−	0.584188i	$-0.198587\pi$
$773$	34886.0	1.62324	0.811618	+	0.584188i	$0.198587\pi$
$774$	172.295	0.00800133
$775$	−7908.38	−0.366551
$776$	−9022.42	−0.417379
$777$	2727.22	0.125918
$778$	8710.20	0.401383
$779$	2856.83	0.131395
$780$	23638.6	1.08512
$781$	8003.69	0.366702
$782$	19628.2	0.897572
$783$	16465.2	0.751490
$784$	−3829.23	−0.174437
$785$	15669.8	0.712458
$786$	−22195.0	−1.00721
$787$	−9526.64	−0.431497	−0.215748	−	0.976449i	$-0.569219\pi$
$787$	−9526.64	−0.431497	−0.215748	+	0.976449i	$0.569219\pi$
$788$	−7384.58	−0.333839
$789$	−30399.2	−1.37166
$790$	16908.5	0.761490
$791$	3729.20	0.167630
$792$	−27.5044	−0.00123400
$793$	12289.7	0.550339
$794$	36071.1	1.61223
$795$	1661.05	0.0741026
$796$	−505.852	−0.0225244
$797$	−33267.9	−1.47856	−0.739278	−	0.673400i	$-0.764833\pi$
$797$	−33267.9	−1.47856	−0.739278	+	0.673400i	$0.764833\pi$
$798$	−16841.3	−0.747089
$799$	−11496.0	−0.509012
$800$	−4867.75	−0.215126
$801$	322.970	0.0142467
$802$	21489.8	0.946174
$803$	−4586.80	−0.201575
$804$	−18519.3	−0.812345
$805$	9907.27	0.433771
$806$	104707.	4.57586
$807$	−273.303	−0.0119216
$808$	−5145.22	−0.224020
$809$	17949.6	0.780069	0.390035	−	0.920800i	$-0.372463\pi$
$809$	17949.6	0.780069	0.390035	+	0.920800i	$0.372463\pi$
$810$	−26605.7	−1.15411
$811$	10877.9	0.470991	0.235495	−	0.971875i	$-0.424329\pi$
$811$	10877.9	0.470991	0.235495	+	0.971875i	$0.424329\pi$
$812$	4383.41	0.189443
$813$	−35278.6	−1.52186
$814$	3029.34	0.130440
$815$	35012.7	1.50484
$816$	15458.3	0.663174
$817$	−23145.7	−0.991147
$818$	−20765.0	−0.887570
$819$	−153.455	−0.00654720
$820$	1213.16	0.0516650
$821$	−4519.04	−0.192102	−0.0960509	−	0.995376i	$-0.530621\pi$
$821$	−4519.04	−0.192102	−0.0960509	+	0.995376i	$0.530621\pi$
$822$	−50847.8	−2.15757
$823$	36987.7	1.56660	0.783300	−	0.621644i	$-0.213535\pi$
$823$	36987.7	1.56660	0.783300	+	0.621644i	$0.213535\pi$
$824$	−7119.80	−0.301007
$825$	−1328.13	−0.0560480
$826$	−1747.66	−0.0736186
$827$	15325.3	0.644392	0.322196	−	0.946673i	$-0.395579\pi$
$827$	15325.3	0.644392	0.322196	+	0.946673i	$0.395579\pi$
$828$	−195.075	−0.00818760
$829$	26546.3	1.11217	0.556087	−	0.831124i	$-0.312302\pi$
$829$	26546.3	1.11217	0.556087	+	0.831124i	$0.312302\pi$
$830$	13110.1	0.548263
$831$	2425.43	0.101248
$832$	11576.0	0.482362
$833$	−1874.37	−0.0779630
$834$	−335.509	−0.0139301
$835$	22472.3	0.931361
$836$	−7506.07	−0.310530
$837$	−47744.9	−1.97169
$838$	4015.31	0.165521
$839$	11906.5	0.489938	0.244969	−	0.969531i	$-0.421222\pi$
$839$	11906.5	0.489938	0.244969	+	0.969531i	$0.421222\pi$
$840$	3520.44	0.144603
$841$	−10745.3	−0.440581
$842$	19245.2	0.787689
$843$	−11919.8	−0.486997
$844$	6294.85	0.256727
$845$	−49962.5	−2.03404
$846$	284.749	0.0115719
$847$	−847.000	−0.0343604
$848$	2489.75	0.100824
$849$	−38115.5	−1.54078
$850$	−3264.68	−0.131738
$851$	−10576.3	−0.426030
$852$	20171.2	0.811098
$853$	−6859.53	−0.275341	−0.137670	−	0.990478i	$-0.543961\pi$
$853$	−6859.53	−0.275341	−0.137670	+	0.990478i	$0.543961\pi$
$854$	−3718.17	−0.148985
$855$	−332.646	−0.0133056
$856$	−19669.2	−0.785372
$857$	5193.59	0.207013	0.103506	−	0.994629i	$-0.466994\pi$
$857$	5193.59	0.207013	0.103506	+	0.994629i	$0.466994\pi$
$858$	17584.4	0.699677
$859$	5265.73	0.209155	0.104578	−	0.994517i	$-0.466651\pi$
$859$	5265.73	0.209155	0.104578	+	0.994517i	$0.466651\pi$
$860$	−9828.89	−0.389724
$861$	812.451	0.0321583
$862$	−15620.6	−0.617216
$863$	−16016.7	−0.631767	−0.315884	−	0.948798i	$-0.602301\pi$
$863$	−16016.7	−0.631767	−0.315884	+	0.948798i	$0.602301\pi$
$864$	−29387.9	−1.15717
$865$	10163.3	0.399494
$866$	31289.8	1.22780
$867$	−17839.1	−0.698788
$868$	−12710.8	−0.497042
$869$	5046.86	0.197011
$870$	−22260.2	−0.867462
$871$	56496.0	2.19781
$872$	14360.1	0.557678
$873$	242.450	0.00939943
$874$	65311.7	2.52769
$875$	−10469.8	−0.404507
$876$	−11559.8	−0.445857
$877$	−11195.5	−0.431065	−0.215533	−	0.976497i	$-0.569149\pi$
$877$	−11195.5	−0.431065	−0.215533	+	0.976497i	$0.569149\pi$
$878$	−46291.3	−1.77933
$879$	9095.51	0.349015
$880$	8666.92	0.332002
$881$	45542.5	1.74162	0.870809	−	0.491621i	$-0.163595\pi$
$881$	45542.5	1.74162	0.870809	+	0.491621i	$0.163595\pi$
$882$	46.4270	0.00177242
$883$	10394.9	0.396167	0.198083	−	0.980185i	$-0.436528\pi$
$883$	10394.9	0.396167	0.198083	+	0.980185i	$0.436528\pi$
$884$	17343.5	0.659871
$885$	3561.09	0.135260
$886$	−45211.8	−1.71436
$887$	−4020.51	−0.152193	−0.0760967	−	0.997100i	$-0.524246\pi$
$887$	−4020.51	−0.152193	−0.0760967	+	0.997100i	$0.524246\pi$
$888$	−3758.17	−0.142022
$889$	15802.1	0.596159
$890$	−45918.1	−1.72941
$891$	−7941.28	−0.298589
$892$	474.920	0.0178268
$893$	−38252.5	−1.43345
$894$	−28690.6	−1.07333
$895$	7523.82	0.280998
$896$	8172.72	0.304723
$897$	−61392.5	−2.28521
$898$	7649.36	0.284257
$899$	−39563.3	−1.46775
$900$	32.4462	0.00120171
$901$	1218.71	0.0450623
$902$	902.453	0.0333131
$903$	−6582.40	−0.242579
$904$	−5138.93	−0.189069
$905$	28106.8	1.03238
$906$	−36837.3	−1.35082
$907$	−23907.5	−0.875231	−0.437615	−	0.899162i	$-0.644177\pi$
$907$	−23907.5	−0.875231	−0.437615	+	0.899162i	$0.644177\pi$
$908$	4735.46	0.173075
$909$	138.262	0.00504496
$910$	21817.4	0.794770
$911$	40571.4	1.47551	0.737755	−	0.675069i	$-0.235886\pi$
$911$	40571.4	1.47551	0.737755	+	0.675069i	$0.235886\pi$
$912$	51436.9	1.86759
$913$	3913.11	0.141846
$914$	−28605.1	−1.03520
$915$	7576.26	0.273731
$916$	6649.07	0.239838
$917$	−8219.50	−0.296000
$918$	−19709.7	−0.708625
$919$	20551.0	0.737667	0.368834	−	0.929495i	$-0.379757\pi$
$919$	20551.0	0.737667	0.368834	+	0.929495i	$0.379757\pi$
$920$	−13652.5	−0.489248
$921$	17934.1	0.641637
$922$	−17456.1	−0.623521
$923$	−61535.6	−2.19444
$924$	−2134.65	−0.0760008
$925$	1759.12	0.0625292
$926$	−42102.4	−1.49414
$927$	191.323	0.00677872
$928$	−24351.9	−0.861413
$929$	34068.4	1.20317	0.601587	−	0.798807i	$-0.294535\pi$
$929$	34068.4	1.20317	0.601587	+	0.798807i	$0.294535\pi$
$930$	64549.1	2.27597
$931$	−6236.89	−0.219555
$932$	−29378.1	−1.03252
$933$	10259.5	0.360000
$934$	27122.8	0.950197
$935$	4242.38	0.148386
$936$	211.465	0.00738455
$937$	15597.9	0.543824	0.271912	−	0.962322i	$-0.412344\pi$
$937$	15597.9	0.543824	0.271912	+	0.962322i	$0.412344\pi$
$938$	−17092.6	−0.594981
$939$	−52198.7	−1.81410
$940$	−16244.0	−0.563639
$941$	−22855.3	−0.791775	−0.395887	−	0.918299i	$-0.629563\pi$
$941$	−22855.3	−0.791775	−0.395887	+	0.918299i	$0.629563\pi$
$942$	29377.4	1.01610
$943$	−3150.73	−0.108804
$944$	5337.71	0.184034
$945$	−9948.44	−0.342458
$946$	−7311.59	−0.251290
$947$	36670.7	1.25833	0.629164	−	0.777272i	$-0.283397\pi$
$947$	36670.7	1.25833	0.629164	+	0.777272i	$0.283397\pi$
$948$	12719.3	0.435764
$949$	35265.1	1.20628
$950$	−10863.1	−0.370994
$951$	−15779.0	−0.538031
$952$	2582.93	0.0879341
$953$	−19922.8	−0.677192	−0.338596	−	0.940932i	$-0.609952\pi$
$953$	−19922.8	−0.677192	−0.338596	+	0.940932i	$0.609952\pi$
$954$	−30.1866	−0.00102445
$955$	9194.16	0.311535
$956$	3184.49	0.107734
$957$	−6644.24	−0.224428
$958$	−37134.8	−1.25237
$959$	−18830.6	−0.634068
$960$	7136.30	0.239920
$961$	84932.7	2.85095
$962$	−23290.8	−0.780587
$963$	528.550	0.0176867
$964$	1655.41	0.0553081
$965$	−44913.0	−1.49824
$966$	18574.0	0.618641
$967$	24523.8	0.815545	0.407772	−	0.913084i	$-0.366306\pi$
$967$	24523.8	0.815545	0.407772	+	0.913084i	$0.366306\pi$
$968$	1167.19	0.0387549
$969$	25177.9	0.834705
$970$	−34470.3	−1.14100
$971$	4493.82	0.148520	0.0742602	−	0.997239i	$-0.476340\pi$
$971$	4493.82	0.148520	0.0742602	+	0.997239i	$0.476340\pi$
$972$	389.909	0.0128666
$973$	−124.250	−0.00409380
$974$	45123.8	1.48446
$975$	10211.2	0.335405
$976$	11356.0	0.372436
$977$	−18285.3	−0.598771	−0.299385	−	0.954132i	$-0.596782\pi$
$977$	−18285.3	−0.598771	−0.299385	+	0.954132i	$0.596782\pi$
$978$	65641.1	2.14619
$979$	−13705.7	−0.447431
$980$	−2648.51	−0.0863300
$981$	−385.885	−0.0125590
$982$	−56089.9	−1.82271
$983$	−25850.8	−0.838772	−0.419386	−	0.907808i	$-0.637755\pi$
$983$	−25850.8	−0.838772	−0.419386	+	0.907808i	$0.637755\pi$
$984$	−1119.58	−0.0362711
$985$	13887.8	0.449242
$986$	−16332.2	−0.527509
$987$	−10878.6	−0.350830
$988$	57709.7	1.85829
$989$	25526.9	0.820738
$990$	−105.081	−0.00337342
$991$	−26842.1	−0.860412	−0.430206	−	0.902731i	$-0.641559\pi$
$991$	−26842.1	−0.860412	−0.430206	+	0.902731i	$0.641559\pi$
$992$	70614.6	2.26010
$993$	−139.531	−0.00445910
$994$	18617.2	0.594068
$995$	951.331	0.0303108
$996$	9861.99	0.313744
$997$	−10468.1	−0.332526	−0.166263	−	0.986081i	$-0.553170\pi$
$997$	−10468.1	−0.332526	−0.166263	+	0.986081i	$0.553170\pi$
$998$	18359.6	0.582327
$999$	10620.3	0.336347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.a.c.1.2	✓	4
3.2	odd	2		693.4.a.m.1.3		4
4.3	odd	2		1232.4.a.w.1.1		4
5.4	even	2		1925.4.a.q.1.3		4
7.6	odd	2		539.4.a.f.1.2		4
11.10	odd	2		847.4.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.c.1.2	✓	4	1.1	even	1	trivial
539.4.a.f.1.2		4	7.6	odd	2
693.4.a.m.1.3		4	3.2	odd	2
847.4.a.e.1.3		4	11.10	odd	2
1232.4.a.w.1.1		4	4.3	odd	2
1925.4.a.q.1.3		4	5.4	even	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:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	77.a (trivial)

\(f(q)\)	\(=\)	\(q-3.65527 q^{2} +5.17115 q^{3} +5.36103 q^{4} -10.0822 q^{5} -18.9020 q^{6} -7.00000 q^{7} +9.64616 q^{8} -0.259212 q^{9} +O(q^{10})\)	\(q-3.65527 q^{2} +5.17115 q^{3} +5.36103 q^{4} -10.0822 q^{5} -18.9020 q^{6} -7.00000 q^{7} +9.64616 q^{8} -0.259212 q^{9} +36.8533 q^{10} +11.0000 q^{11} +27.7227 q^{12} -84.5724 q^{13} +25.5869 q^{14} -52.1367 q^{15} -78.1476 q^{16} -38.2525 q^{17} +0.947489 q^{18} -127.283 q^{19} -54.0511 q^{20} -36.1980 q^{21} -40.2080 q^{22} +140.378 q^{23} +49.8817 q^{24} -23.3486 q^{25} +309.135 q^{26} -140.961 q^{27} -37.5272 q^{28} -116.806 q^{29} +190.574 q^{30} +338.709 q^{31} +208.482 q^{32} +56.8826 q^{33} +139.823 q^{34} +70.5756 q^{35} -1.38964 q^{36} -75.3416 q^{37} +465.256 q^{38} -437.337 q^{39} -97.2548 q^{40} -22.4446 q^{41} +132.314 q^{42} +181.844 q^{43} +58.9713 q^{44} +2.61343 q^{45} -513.121 q^{46} +300.530 q^{47} -404.113 q^{48} +49.0000 q^{49} +85.3455 q^{50} -197.810 q^{51} -453.395 q^{52} -31.8596 q^{53} +515.253 q^{54} -110.905 q^{55} -67.5231 q^{56} -658.201 q^{57} +426.958 q^{58} -68.3030 q^{59} -279.507 q^{60} -145.315 q^{61} -1238.07 q^{62} +1.81448 q^{63} -136.877 q^{64} +852.679 q^{65} -207.922 q^{66} -668.020 q^{67} -205.073 q^{68} +725.916 q^{69} -257.973 q^{70} +727.608 q^{71} -2.50040 q^{72} -416.982 q^{73} +275.394 q^{74} -120.739 q^{75} -682.370 q^{76} -77.0000 q^{77} +1598.59 q^{78} +458.805 q^{79} +787.902 q^{80} -721.934 q^{81} +82.0412 q^{82} +355.737 q^{83} -194.059 q^{84} +385.671 q^{85} -664.690 q^{86} -604.022 q^{87} +106.108 q^{88} -1245.97 q^{89} -9.55281 q^{90} +592.007 q^{91} +752.571 q^{92} +1751.51 q^{93} -1098.52 q^{94} +1283.30 q^{95} +1078.09 q^{96} -935.338 q^{97} -179.108 q^{98} -2.85133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 12 q^{3} + 22 q^{4} - 18 q^{5} + 2 q^{6} - 28 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 12 * q^3 + 22 * q^4 - 18 * q^5 + 2 * q^6 - 28 * q^7 - 60 * q^8 + 66 * q^9	\( 4 q - 4 q^{2} - 12 q^{3} + 22 q^{4} - 18 q^{5} + 2 q^{6} - 28 q^{7} - 60 q^{8} + 66 q^{9} - 92 q^{10} + 44 q^{11} - 186 q^{12} - 134 q^{13} + 28 q^{14} - 62 q^{15} - 6 q^{16} - 74 q^{17} - 256 q^{18} - 164 q^{19} + 116 q^{20} + 84 q^{21} - 44 q^{22} + 194 q^{23} + 570 q^{24} + 38 q^{25} + 734 q^{26} - 510 q^{27} - 154 q^{28} - 108 q^{29} + 1252 q^{30} - 412 q^{31} - 4 q^{32} - 132 q^{33} - 346 q^{34} + 126 q^{35} + 1518 q^{36} + 286 q^{37} + 224 q^{38} - 256 q^{39} - 540 q^{40} - 18 q^{41} - 14 q^{42} - 496 q^{43} + 242 q^{44} + 580 q^{45} - 284 q^{46} + 62 q^{47} - 862 q^{48} + 196 q^{49} + 212 q^{50} - 508 q^{51} - 822 q^{52} - 828 q^{53} + 2420 q^{54} - 198 q^{55} + 420 q^{56} + 700 q^{57} + 1388 q^{58} - 1224 q^{59} - 1776 q^{60} - 350 q^{61} - 878 q^{62} - 462 q^{63} - 718 q^{64} - 396 q^{65} + 22 q^{66} - 1498 q^{67} + 1058 q^{68} - 386 q^{69} + 644 q^{70} + 2326 q^{71} - 3000 q^{72} - 1630 q^{73} - 1156 q^{74} - 1362 q^{75} - 3152 q^{76} - 308 q^{77} - 2464 q^{78} - 1020 q^{79} + 3072 q^{80} + 1128 q^{81} + 2118 q^{82} - 1920 q^{83} + 1302 q^{84} + 2008 q^{85} + 1056 q^{86} + 1640 q^{87} - 660 q^{88} + 1550 q^{89} - 5780 q^{90} + 938 q^{91} + 2592 q^{92} + 6046 q^{93} - 1042 q^{94} + 2332 q^{95} + 4082 q^{96} - 2202 q^{97} - 196 q^{98} + 726 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 12 * q^3 + 22 * q^4 - 18 * q^5 + 2 * q^6 - 28 * q^7 - 60 * q^8 + 66 * q^9 - 92 * q^10 + 44 * q^11 - 186 * q^12 - 134 * q^13 + 28 * q^14 - 62 * q^15 - 6 * q^16 - 74 * q^17 - 256 * q^18 - 164 * q^19 + 116 * q^20 + 84 * q^21 - 44 * q^22 + 194 * q^23 + 570 * q^24 + 38 * q^25 + 734 * q^26 - 510 * q^27 - 154 * q^28 - 108 * q^29 + 1252 * q^30 - 412 * q^31 - 4 * q^32 - 132 * q^33 - 346 * q^34 + 126 * q^35 + 1518 * q^36 + 286 * q^37 + 224 * q^38 - 256 * q^39 - 540 * q^40 - 18 * q^41 - 14 * q^42 - 496 * q^43 + 242 * q^44 + 580 * q^45 - 284 * q^46 + 62 * q^47 - 862 * q^48 + 196 * q^49 + 212 * q^50 - 508 * q^51 - 822 * q^52 - 828 * q^53 + 2420 * q^54 - 198 * q^55 + 420 * q^56 + 700 * q^57 + 1388 * q^58 - 1224 * q^59 - 1776 * q^60 - 350 * q^61 - 878 * q^62 - 462 * q^63 - 718 * q^64 - 396 * q^65 + 22 * q^66 - 1498 * q^67 + 1058 * q^68 - 386 * q^69 + 644 * q^70 + 2326 * q^71 - 3000 * q^72 - 1630 * q^73 - 1156 * q^74 - 1362 * q^75 - 3152 * q^76 - 308 * q^77 - 2464 * q^78 - 1020 * q^79 + 3072 * q^80 + 1128 * q^81 + 2118 * q^82 - 1920 * q^83 + 1302 * q^84 + 2008 * q^85 + 1056 * q^86 + 1640 * q^87 - 660 * q^88 + 1550 * q^89 - 5780 * q^90 + 938 * q^91 + 2592 * q^92 + 6046 * q^93 - 1042 * q^94 + 2332 * q^95 + 4082 * q^96 - 2202 * q^97 - 196 * q^98 + 726 * q^99

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