LMFDB - Embedded newform 177.4.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.3
Root			$1.46227$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46227 q^{2} +3.00000 q^{3} -1.93724 q^{4} -11.2702 q^{5} -7.38680 q^{6} +18.6588 q^{7} +24.4681 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.46227 q^{2} +3.00000 q^{3} -1.93724 q^{4} -11.2702 q^{5} -7.38680 q^{6} +18.6588 q^{7} +24.4681 q^{8} +9.00000 q^{9} +27.7502 q^{10} -16.5085 q^{11} -5.81171 q^{12} +24.7044 q^{13} -45.9430 q^{14} -33.8105 q^{15} -44.7492 q^{16} -125.357 q^{17} -22.1604 q^{18} -104.913 q^{19} +21.8330 q^{20} +55.9764 q^{21} +40.6484 q^{22} +116.317 q^{23} +73.4044 q^{24} +2.01688 q^{25} -60.8289 q^{26} +27.0000 q^{27} -36.1465 q^{28} -295.871 q^{29} +83.2506 q^{30} -62.8500 q^{31} -85.5606 q^{32} -49.5255 q^{33} +308.663 q^{34} -210.288 q^{35} -17.4351 q^{36} +318.173 q^{37} +258.325 q^{38} +74.1133 q^{39} -275.760 q^{40} -488.331 q^{41} -137.829 q^{42} -127.236 q^{43} +31.9809 q^{44} -101.432 q^{45} -286.404 q^{46} -221.541 q^{47} -134.248 q^{48} +5.15084 q^{49} -4.96610 q^{50} -376.072 q^{51} -47.8584 q^{52} -40.8953 q^{53} -66.4812 q^{54} +186.054 q^{55} +456.546 q^{56} -314.740 q^{57} +728.514 q^{58} -59.0000 q^{59} +65.4990 q^{60} -237.039 q^{61} +154.754 q^{62} +167.929 q^{63} +568.667 q^{64} -278.423 q^{65} +121.945 q^{66} +532.451 q^{67} +242.847 q^{68} +348.952 q^{69} +517.785 q^{70} +634.893 q^{71} +220.213 q^{72} +375.980 q^{73} -783.427 q^{74} +6.05065 q^{75} +203.242 q^{76} -308.029 q^{77} -182.487 q^{78} -323.957 q^{79} +504.332 q^{80} +81.0000 q^{81} +1202.40 q^{82} +1174.38 q^{83} -108.440 q^{84} +1412.80 q^{85} +313.290 q^{86} -887.613 q^{87} -403.933 q^{88} -1444.33 q^{89} +249.752 q^{90} +460.955 q^{91} -225.334 q^{92} -188.550 q^{93} +545.494 q^{94} +1182.39 q^{95} -256.682 q^{96} -1046.28 q^{97} -12.6827 q^{98} -148.577 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9}+O(q^{10}) $ 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9} - 79 q^{10} - 131 q^{11} + 66 q^{12} - 123 q^{13} - 117 q^{14} - 84 q^{15} + 202 q^{16} - 235 q^{17} - 72 q^{18} - 80 q^{19} + 61 q^{20} - 177 q^{21} + 688 q^{22} - 274 q^{23} - 351 q^{24} + 193 q^{25} - 180 q^{26} + 189 q^{27} - 118 q^{28} - 406 q^{29} - 237 q^{30} - 346 q^{31} - 854 q^{32} - 393 q^{33} + 178 q^{34} - 424 q^{35} + 198 q^{36} - 157 q^{37} - 129 q^{38} - 369 q^{39} - 590 q^{40} - 825 q^{41} - 351 q^{42} - 815 q^{43} - 1690 q^{44} - 252 q^{45} + 1457 q^{46} - 1196 q^{47} + 606 q^{48} + 914 q^{49} + 713 q^{50} - 705 q^{51} + 1030 q^{52} - 900 q^{53} - 216 q^{54} - 1044 q^{55} + 2172 q^{56} - 240 q^{57} + 1242 q^{58} - 413 q^{59} + 183 q^{60} + 420 q^{61} + 646 q^{62} - 531 q^{63} + 3541 q^{64} + 190 q^{65} + 2064 q^{66} + 1316 q^{67} - 611 q^{68} - 822 q^{69} + 4658 q^{70} - 173 q^{71} - 1053 q^{72} - 418 q^{73} + 660 q^{74} + 579 q^{75} + 1540 q^{76} - 753 q^{77} - 540 q^{78} + 2635 q^{79} + 6155 q^{80} + 567 q^{81} - 125 q^{82} + 457 q^{83} - 354 q^{84} + 1270 q^{85} + 3482 q^{86} - 1218 q^{87} + 7685 q^{88} + 592 q^{89} - 711 q^{90} + 3179 q^{91} - 3500 q^{92} - 1038 q^{93} + 2064 q^{94} - 2250 q^{95} - 2562 q^{96} - 1906 q^{97} + 2994 q^{98} - 1179 q^{99}+O(q^{100}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9 - 79 * q^10 - 131 * q^11 + 66 * q^12 - 123 * q^13 - 117 * q^14 - 84 * q^15 + 202 * q^16 - 235 * q^17 - 72 * q^18 - 80 * q^19 + 61 * q^20 - 177 * q^21 + 688 * q^22 - 274 * q^23 - 351 * q^24 + 193 * q^25 - 180 * q^26 + 189 * q^27 - 118 * q^28 - 406 * q^29 - 237 * q^30 - 346 * q^31 - 854 * q^32 - 393 * q^33 + 178 * q^34 - 424 * q^35 + 198 * q^36 - 157 * q^37 - 129 * q^38 - 369 * q^39 - 590 * q^40 - 825 * q^41 - 351 * q^42 - 815 * q^43 - 1690 * q^44 - 252 * q^45 + 1457 * q^46 - 1196 * q^47 + 606 * q^48 + 914 * q^49 + 713 * q^50 - 705 * q^51 + 1030 * q^52 - 900 * q^53 - 216 * q^54 - 1044 * q^55 + 2172 * q^56 - 240 * q^57 + 1242 * q^58 - 413 * q^59 + 183 * q^60 + 420 * q^61 + 646 * q^62 - 531 * q^63 + 3541 * q^64 + 190 * q^65 + 2064 * q^66 + 1316 * q^67 - 611 * q^68 - 822 * q^69 + 4658 * q^70 - 173 * q^71 - 1053 * q^72 - 418 * q^73 + 660 * q^74 + 579 * q^75 + 1540 * q^76 - 753 * q^77 - 540 * q^78 + 2635 * q^79 + 6155 * q^80 + 567 * q^81 - 125 * q^82 + 457 * q^83 - 354 * q^84 + 1270 * q^85 + 3482 * q^86 - 1218 * q^87 + 7685 * q^88 + 592 * q^89 - 711 * q^90 + 3179 * q^91 - 3500 * q^92 - 1038 * q^93 + 2064 * q^94 - 2250 * q^95 - 2562 * q^96 - 1906 * q^97 + 2994 * q^98 - 1179 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.46227	−0.870543	−0.435272	−	0.900299i	$-0.643348\pi$
$2$	−2.46227	−0.870543	−0.435272	+	0.900299i	$0.643348\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−1.93724	−0.242155
$5$	−11.2702	−1.00804	−0.504018	−	0.863693i	$-0.668145\pi$
$5$	−11.2702	−1.00804	−0.504018	+	0.863693i	$0.668145\pi$
$6$	−7.38680	−0.502608
$7$	18.6588	1.00748	0.503740	−	0.863855i	$-0.331957\pi$
$7$	18.6588	1.00748	0.503740	+	0.863855i	$0.331957\pi$
$8$	24.4681	1.08135
$9$	9.00000	0.333333
$10$	27.7502	0.877538
$11$	−16.5085	−0.452500	−0.226250	−	0.974069i	$-0.572647\pi$
$11$	−16.5085	−0.452500	−0.226250	+	0.974069i	$0.572647\pi$
$12$	−5.81171	−0.139808
$13$	24.7044	0.527060	0.263530	−	0.964651i	$-0.415113\pi$
$13$	24.7044	0.527060	0.263530	+	0.964651i	$0.415113\pi$
$14$	−45.9430	−0.877055
$15$	−33.8105	−0.581989
$16$	−44.7492	−0.699206
$17$	−125.357	−1.78845	−0.894223	−	0.447621i	$-0.852271\pi$
$17$	−125.357	−1.78845	−0.894223	+	0.447621i	$0.852271\pi$
$18$	−22.1604	−0.290181
$19$	−104.913	−1.26678	−0.633388	−	0.773834i	$-0.718336\pi$
$19$	−104.913	−1.26678	−0.633388	+	0.773834i	$0.718336\pi$
$20$	21.8330	0.244100
$21$	55.9764	0.581669
$22$	40.6484	0.393921
$23$	116.317	1.05451	0.527257	−	0.849706i	$-0.323221\pi$
$23$	116.317	1.05451	0.527257	+	0.849706i	$0.323221\pi$
$24$	73.4044	0.624317
$25$	2.01688	0.0161351
$26$	−60.8289	−0.458828
$27$	27.0000	0.192450
$28$	−36.1465	−0.243966
$29$	−295.871	−1.89455	−0.947274	−	0.320425i	$-0.896174\pi$
$29$	−295.871	−1.89455	−0.947274	+	0.320425i	$0.896174\pi$
$30$	83.2506	0.506647
$31$	−62.8500	−0.364135	−0.182068	−	0.983286i	$-0.558279\pi$
$31$	−62.8500	−0.364135	−0.182068	+	0.983286i	$0.558279\pi$
$32$	−85.5606	−0.472660
$33$	−49.5255	−0.261251
$34$	308.663	1.55692
$35$	−210.288	−1.01558
$36$	−17.4351	−0.0807182
$37$	318.173	1.41371	0.706856	−	0.707358i	$-0.250113\pi$
$37$	318.173	1.41371	0.706856	+	0.707358i	$0.250113\pi$
$38$	258.325	1.10278
$39$	74.1133	0.304298
$40$	−275.760	−1.09004
$41$	−488.331	−1.86011	−0.930055	−	0.367421i	$-0.880241\pi$
$41$	−488.331	−1.86011	−0.930055	+	0.367421i	$0.880241\pi$
$42$	−137.829	−0.506368
$43$	−127.236	−0.451241	−0.225620	−	0.974215i	$-0.572441\pi$
$43$	−127.236	−0.451241	−0.225620	+	0.974215i	$0.572441\pi$
$44$	31.9809	0.109575
$45$	−101.432	−0.336012
$46$	−286.404	−0.918000
$47$	−221.541	−0.687555	−0.343778	−	0.939051i	$-0.611707\pi$
$47$	−221.541	−0.687555	−0.343778	+	0.939051i	$0.611707\pi$
$48$	−134.248	−0.403687
$49$	5.15084	0.0150170
$50$	−4.96610	−0.0140463
$51$	−376.072	−1.03256
$52$	−47.8584	−0.127630
$53$	−40.8953	−0.105989	−0.0529944	−	0.998595i	$-0.516877\pi$
$53$	−40.8953	−0.105989	−0.0529944	+	0.998595i	$0.516877\pi$
$54$	−66.4812	−0.167536
$55$	186.054	0.456136
$56$	456.546	1.08944
$57$	−314.740	−0.731374
$58$	728.514	1.64929
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	65.4990	0.140931
$61$	−237.039	−0.497536	−0.248768	−	0.968563i	$-0.580026\pi$
$61$	−237.039	−0.497536	−0.248768	+	0.968563i	$0.580026\pi$
$62$	154.754	0.316996
$63$	167.929	0.335827
$64$	568.667	1.11068
$65$	−278.423	−0.531295
$66$	121.945	0.227430
$67$	532.451	0.970884	0.485442	−	0.874269i	$-0.338659\pi$
$67$	532.451	0.970884	0.485442	+	0.874269i	$0.338659\pi$
$68$	242.847	0.433081
$69$	348.952	0.608824
$70$	517.785	0.884103
$71$	634.893	1.06124	0.530619	−	0.847610i	$-0.321960\pi$
$71$	634.893	1.06124	0.530619	+	0.847610i	$0.321960\pi$
$72$	220.213	0.360450
$73$	375.980	0.602809	0.301405	−	0.953496i	$-0.402544\pi$
$73$	375.980	0.602809	0.301405	+	0.953496i	$0.402544\pi$
$74$	−783.427	−1.23070
$75$	6.05065	0.00931558
$76$	203.242	0.306756
$77$	−308.029	−0.455885
$78$	−182.487	−0.264905
$79$	−323.957	−0.461367	−0.230683	−	0.973029i	$-0.574096\pi$
$79$	−323.957	−0.461367	−0.230683	+	0.973029i	$0.574096\pi$
$80$	504.332	0.704825
$81$	81.0000	0.111111
$82$	1202.40	1.61931
$83$	1174.38	1.55308	0.776538	−	0.630070i	$-0.216974\pi$
$83$	1174.38	1.55308	0.776538	+	0.630070i	$0.216974\pi$
$84$	−108.440	−0.140854
$85$	1412.80	1.80282
$86$	313.290	0.392824
$87$	−887.613	−1.09382
$88$	−403.933	−0.489311
$89$	−1444.33	−1.72021	−0.860107	−	0.510113i	$-0.829604\pi$
$89$	−1444.33	−1.72021	−0.860107	+	0.510113i	$0.829604\pi$
$90$	249.752	0.292513
$91$	460.955	0.531003
$92$	−225.334	−0.255355
$93$	−188.550	−0.210234
$94$	545.494	0.598547
$95$	1182.39	1.27696
$96$	−256.682	−0.272890
$97$	−1046.28	−1.09519	−0.547595	−	0.836744i	$-0.684456\pi$
$97$	−1046.28	−1.09519	−0.547595	+	0.836744i	$0.684456\pi$
$98$	−12.6827	−0.0130730
$99$	−148.577	−0.150833
$100$	−3.90718	−0.00390718
$101$	−1182.97	−1.16545	−0.582724	−	0.812670i	$-0.698013\pi$
$101$	−1182.97	−1.16545	−0.582724	+	0.812670i	$0.698013\pi$
$102$	925.989	0.898888
$103$	1084.27	1.03725	0.518625	−	0.855002i	$-0.326444\pi$
$103$	1084.27	1.03725	0.518625	+	0.855002i	$0.326444\pi$
$104$	604.472	0.569936
$105$	−630.864	−0.586343
$106$	100.695	0.0922678
$107$	−555.182	−0.501603	−0.250801	−	0.968039i	$-0.580694\pi$
$107$	−555.182	−0.501603	−0.250801	+	0.968039i	$0.580694\pi$
$108$	−52.3054	−0.0466027
$109$	−1089.62	−0.957491	−0.478746	−	0.877954i	$-0.658908\pi$
$109$	−1089.62	−0.957491	−0.478746	+	0.877954i	$0.658908\pi$
$110$	−458.114	−0.397086
$111$	954.519	0.816207
$112$	−834.967	−0.704437
$113$	514.974	0.428714	0.214357	−	0.976755i	$-0.431234\pi$
$113$	514.974	0.428714	0.214357	+	0.976755i	$0.431234\pi$
$114$	774.974	0.636692
$115$	−1310.92	−1.06299
$116$	573.173	0.458774
$117$	222.340	0.175687
$118$	145.274	0.113335
$119$	−2339.02	−1.80183
$120$	−827.281	−0.629334
$121$	−1058.47	−0.795243
$122$	583.653	0.433127
$123$	−1464.99	−1.07393
$124$	121.755	0.0881771
$125$	1386.04	0.991771
$126$	−413.487	−0.292352
$127$	2240.79	1.56565	0.782827	−	0.622240i	$-0.213777\pi$
$127$	2240.79	1.56565	0.782827	+	0.622240i	$0.213777\pi$
$128$	−715.725	−0.494233
$129$	−381.709	−0.260524
$130$	685.553	0.462515
$131$	943.152	0.629035	0.314517	−	0.949252i	$-0.398157\pi$
$131$	943.152	0.629035	0.314517	+	0.949252i	$0.398157\pi$
$132$	95.9427	0.0632632
$133$	−1957.56	−1.27625
$134$	−1311.04	−0.845196
$135$	−304.295	−0.193996
$136$	−3067.26	−1.93394
$137$	1837.48	1.14588	0.572942	−	0.819596i	$-0.305802\pi$
$137$	1837.48	1.14588	0.572942	+	0.819596i	$0.305802\pi$
$138$	−859.213	−0.530007
$139$	−404.410	−0.246774	−0.123387	−	0.992359i	$-0.539376\pi$
$139$	−404.410	−0.246774	−0.123387	+	0.992359i	$0.539376\pi$
$140$	407.378	0.245926
$141$	−664.623	−0.396960
$142$	−1563.28	−0.923853
$143$	−407.834	−0.238495
$144$	−402.743	−0.233069
$145$	3334.52	1.90977
$146$	−925.762	−0.524772
$147$	15.4525	0.00867008
$148$	−616.377	−0.342337
$149$	2398.95	1.31899	0.659496	−	0.751708i	$-0.270770\pi$
$149$	2398.95	1.31899	0.659496	+	0.751708i	$0.270770\pi$
$150$	−14.8983	−0.00810961
$151$	−1819.90	−0.980805	−0.490403	−	0.871496i	$-0.663150\pi$
$151$	−1819.90	−0.980805	−0.490403	+	0.871496i	$0.663150\pi$
$152$	−2567.03	−1.36983
$153$	−1128.22	−0.596149
$154$	758.450	0.396868
$155$	708.331	0.367061
$156$	−143.575	−0.0736872
$157$	−3018.40	−1.53436	−0.767181	−	0.641430i	$-0.778341\pi$
$157$	−3018.40	−1.53436	−0.767181	+	0.641430i	$0.778341\pi$
$158$	797.668	0.401640
$159$	−122.686	−0.0611926
$160$	964.283	0.476458
$161$	2170.34	1.06240
$162$	−199.444	−0.0967270
$163$	1249.65	0.600493	0.300246	−	0.953862i	$-0.402931\pi$
$163$	1249.65	0.600493	0.300246	+	0.953862i	$0.402931\pi$
$164$	946.013	0.450434
$165$	558.162	0.263350
$166$	−2891.65	−1.35202
$167$	−2805.96	−1.30019	−0.650095	−	0.759853i	$-0.725271\pi$
$167$	−2805.96	−1.30019	−0.650095	+	0.759853i	$0.725271\pi$
$168$	1369.64	0.628987
$169$	−1586.69	−0.722208
$170$	−3478.69	−1.56943
$171$	−944.219	−0.422259
$172$	246.487	0.109270
$173$	1191.77	0.523751	0.261875	−	0.965102i	$-0.415659\pi$
$173$	1191.77	0.523751	0.261875	+	0.965102i	$0.415659\pi$
$174$	2185.54	0.952215
$175$	37.6326	0.0162558
$176$	738.743	0.316391
$177$	−177.000	−0.0751646
$178$	3556.34	1.49752
$179$	3035.70	1.26759	0.633796	−	0.773500i	$-0.281496\pi$
$179$	3035.70	1.26759	0.633796	+	0.773500i	$0.281496\pi$
$180$	196.497	0.0813668
$181$	2731.80	1.12184	0.560920	−	0.827870i	$-0.310448\pi$
$181$	2731.80	1.12184	0.560920	+	0.827870i	$0.310448\pi$
$182$	−1135.00	−0.462261
$183$	−711.117	−0.287253
$184$	2846.07	1.14030
$185$	−3585.87	−1.42507
$186$	464.261	0.183017
$187$	2069.46	0.809273
$188$	429.178	0.166495
$189$	503.788	0.193890
$190$	−2911.36	−1.11164
$191$	1111.96	0.421248	0.210624	−	0.977567i	$-0.432450\pi$
$191$	1111.96	0.421248	0.210624	+	0.977567i	$0.432450\pi$
$192$	1706.00	0.641250
$193$	−2285.89	−0.852549	−0.426274	−	0.904594i	$-0.640174\pi$
$193$	−2285.89	−0.852549	−0.426274	+	0.904594i	$0.640174\pi$
$194$	2576.21	0.953410
$195$	−835.270	−0.306743
$196$	−9.97839	−0.00363644
$197$	625.776	0.226318	0.113159	−	0.993577i	$-0.463903\pi$
$197$	625.776	0.226318	0.113159	+	0.993577i	$0.463903\pi$
$198$	365.835	0.131307
$199$	−4797.28	−1.70890	−0.854448	−	0.519537i	$-0.826104\pi$
$199$	−4797.28	−1.70890	−0.854448	+	0.519537i	$0.826104\pi$
$200$	49.3493	0.0174476
$201$	1597.35	0.560540
$202$	2912.80	1.01457
$203$	−5520.60	−1.90872
$204$	728.540	0.250039
$205$	5503.58	1.87506
$206$	−2669.77	−0.902970
$207$	1046.86	0.351505
$208$	−1105.50	−0.368524
$209$	1731.96	0.573217
$210$	1553.36	0.510437
$211$	−3653.40	−1.19199	−0.595997	−	0.802987i	$-0.703243\pi$
$211$	−3653.40	−1.19199	−0.595997	+	0.802987i	$0.703243\pi$
$212$	79.2239	0.0256657
$213$	1904.68	0.612706
$214$	1367.01	0.436667
$215$	1433.97	0.454866
$216$	660.640	0.208106
$217$	−1172.71	−0.366859
$218$	2682.93	0.833538
$219$	1127.94	0.348032
$220$	−360.430	−0.110456
$221$	−3096.88	−0.942619
$222$	−2350.28	−0.710543
$223$	−3631.49	−1.09051	−0.545253	−	0.838272i	$-0.683566\pi$
$223$	−3631.49	−1.09051	−0.545253	+	0.838272i	$0.683566\pi$
$224$	−1596.46	−0.476196
$225$	18.1519	0.00537835
$226$	−1268.00	−0.373214
$227$	−3959.13	−1.15761	−0.578803	−	0.815467i	$-0.696480\pi$
$227$	−3959.13	−1.15761	−0.578803	+	0.815467i	$0.696480\pi$
$228$	609.726	0.177106
$229$	2725.06	0.786362	0.393181	−	0.919461i	$-0.371375\pi$
$229$	2725.06	0.786362	0.393181	+	0.919461i	$0.371375\pi$
$230$	3227.83	0.925376
$231$	−924.087	−0.263206
$232$	−7239.41	−2.04867
$233$	1249.69	0.351374	0.175687	−	0.984446i	$-0.443785\pi$
$233$	1249.69	0.351374	0.175687	+	0.984446i	$0.443785\pi$
$234$	−547.461	−0.152943
$235$	2496.81	0.693080
$236$	114.297	0.0315259
$237$	−971.870	−0.266370
$238$	5759.28	1.56857
$239$	4000.87	1.08282	0.541412	−	0.840757i	$-0.317890\pi$
$239$	4000.87	1.08282	0.541412	+	0.840757i	$0.317890\pi$
$240$	1512.99	0.406931
$241$	1847.78	0.493884	0.246942	−	0.969030i	$-0.420574\pi$
$241$	1847.78	0.493884	0.246942	+	0.969030i	$0.420574\pi$
$242$	2606.23	0.692294
$243$	243.000	0.0641500
$244$	459.201	0.120481
$245$	−58.0508	−0.0151377
$246$	3607.20	0.934906
$247$	−2591.82	−0.667667
$248$	−1537.82	−0.393758
$249$	3523.15	0.896669
$250$	−3412.81	−0.863379
$251$	2059.75	0.517970	0.258985	−	0.965881i	$-0.416612\pi$
$251$	2059.75	0.517970	0.258985	+	0.965881i	$0.416612\pi$
$252$	−325.319	−0.0813220
$253$	−1920.22	−0.477168
$254$	−5517.42	−1.36297
$255$	4238.39	1.04086
$256$	−2787.03	−0.680426
$257$	464.966	0.112855	0.0564275	−	0.998407i	$-0.482029\pi$
$257$	464.966	0.112855	0.0564275	+	0.998407i	$0.482029\pi$
$258$	939.869	0.226797
$259$	5936.73	1.42429
$260$	539.372	0.128656
$261$	−2662.84	−0.631516
$262$	−2322.29	−0.547602
$263$	−2946.12	−0.690742	−0.345371	−	0.938466i	$-0.612247\pi$
$263$	−2946.12	−0.690742	−0.345371	+	0.938466i	$0.612247\pi$
$264$	−1211.80	−0.282504
$265$	460.897	0.106840
$266$	4820.03	1.11103
$267$	−4333.00	−0.993166
$268$	−1031.48	−0.235104
$269$	4266.67	0.967076	0.483538	−	0.875323i	$-0.339351\pi$
$269$	4266.67	0.967076	0.483538	+	0.875323i	$0.339351\pi$
$270$	749.255	0.168882
$271$	−1985.24	−0.444999	−0.222499	−	0.974933i	$-0.571422\pi$
$271$	−1985.24	−0.444999	−0.222499	+	0.974933i	$0.571422\pi$
$272$	5609.64	1.25049
$273$	1382.87	0.306574
$274$	−4524.36	−0.997542
$275$	−33.2957	−0.00730112
$276$	−676.002	−0.147430
$277$	−763.770	−0.165670	−0.0828349	−	0.996563i	$-0.526397\pi$
$277$	−763.770	−0.165670	−0.0828349	+	0.996563i	$0.526397\pi$
$278$	995.766	0.214828
$279$	−565.650	−0.121378
$280$	−5145.36	−1.09819
$281$	1976.32	0.419564	0.209782	−	0.977748i	$-0.432725\pi$
$281$	1976.32	0.419564	0.209782	+	0.977748i	$0.432725\pi$
$282$	1636.48	0.345571
$283$	4445.80	0.933835	0.466918	−	0.884301i	$-0.345364\pi$
$283$	4445.80	0.933835	0.466918	+	0.884301i	$0.345364\pi$
$284$	−1229.94	−0.256984
$285$	3547.17	0.737251
$286$	1004.20	0.207620
$287$	−9111.67	−1.87402
$288$	−770.045	−0.157553
$289$	10801.4	2.19854
$290$	−8210.48	−1.66254
$291$	−3138.83	−0.632308
$292$	−728.362	−0.145973
$293$	152.195	0.0303459	0.0151729	−	0.999885i	$-0.495170\pi$
$293$	152.195	0.0303459	0.0151729	+	0.999885i	$0.495170\pi$
$294$	−38.0482	−0.00754768
$295$	664.940	0.131235
$296$	7785.10	1.52872
$297$	−445.730	−0.0870837
$298$	−5906.86	−1.14824
$299$	2873.55	0.555792
$300$	−11.7215	−0.00225581
$301$	−2374.08	−0.454616
$302$	4481.09	0.853833
$303$	−3548.92	−0.672872
$304$	4694.79	0.885738
$305$	2671.47	0.501534
$306$	2777.97	0.518973
$307$	1890.01	0.351363	0.175682	−	0.984447i	$-0.443787\pi$
$307$	1890.01	0.351363	0.175682	+	0.984447i	$0.443787\pi$
$308$	596.725	0.110395
$309$	3252.82	0.598856
$310$	−1744.10	−0.319543
$311$	−8583.58	−1.56505	−0.782524	−	0.622620i	$-0.786068\pi$
$311$	−8583.58	−1.56505	−0.782524	+	0.622620i	$0.786068\pi$
$312$	1813.42	0.329053
$313$	8124.32	1.46714	0.733568	−	0.679616i	$-0.237853\pi$
$313$	8124.32	1.46714	0.733568	+	0.679616i	$0.237853\pi$
$314$	7432.12	1.33573
$315$	−1892.59	−0.338525
$316$	627.581	0.111722
$317$	3908.89	0.692572	0.346286	−	0.938129i	$-0.387443\pi$
$317$	3908.89	0.692572	0.346286	+	0.938129i	$0.387443\pi$
$318$	302.086	0.0532708
$319$	4884.39	0.857284
$320$	−6408.97	−1.11960
$321$	−1665.55	−0.289600
$322$	−5343.96	−0.924867
$323$	13151.6	2.26556
$324$	−156.916	−0.0269061
$325$	49.8259	0.00850414
$326$	−3076.98	−0.522755
$327$	−3268.86	−0.552808
$328$	−11948.5	−2.01143
$329$	−4133.69	−0.692699
$330$	−1374.34	−0.229258
$331$	−865.302	−0.143690	−0.0718448	−	0.997416i	$-0.522889\pi$
$331$	−865.302	−0.143690	−0.0718448	+	0.997416i	$0.522889\pi$
$332$	−2275.06	−0.376085
$333$	2863.56	0.471237
$334$	6909.02	1.13187
$335$	−6000.81	−0.978685
$336$	−2504.90	−0.406707
$337$	508.758	0.0822369	0.0411184	−	0.999154i	$-0.486908\pi$
$337$	508.758	0.0822369	0.0411184	+	0.999154i	$0.486908\pi$
$338$	3906.86	0.628713
$339$	1544.92	0.247518
$340$	−2736.93	−0.436561
$341$	1037.56	0.164771
$342$	2324.92	0.367595
$343$	−6303.86	−0.992351
$344$	−3113.23	−0.487949
$345$	−3932.75	−0.613716
$346$	−2934.47	−0.455948
$347$	8339.96	1.29024	0.645119	−	0.764082i	$-0.276808\pi$
$347$	8339.96	1.29024	0.645119	+	0.764082i	$0.276808\pi$
$348$	1719.52	0.264873
$349$	3274.69	0.502264	0.251132	−	0.967953i	$-0.419197\pi$
$349$	3274.69	0.502264	0.251132	+	0.967953i	$0.419197\pi$
$350$	−92.6615	−0.0141513
$351$	667.020	0.101433
$352$	1412.48	0.213879
$353$	−1578.59	−0.238017	−0.119008	−	0.992893i	$-0.537972\pi$
$353$	−1578.59	−0.238017	−0.119008	+	0.992893i	$0.537972\pi$
$354$	435.821	0.0654340
$355$	−7155.35	−1.06977
$356$	2798.02	0.416558
$357$	−7017.05	−1.04028
$358$	−7474.71	−1.10349
$359$	3661.78	0.538333	0.269166	−	0.963094i	$-0.413252\pi$
$359$	3661.78	0.538333	0.269166	+	0.963094i	$0.413252\pi$
$360$	−2481.84	−0.363346
$361$	4147.79	0.604723
$362$	−6726.42	−0.976610
$363$	−3175.41	−0.459134
$364$	−892.980	−0.128585
$365$	−4237.36	−0.607653
$366$	1750.96	0.250066
$367$	−4573.82	−0.650548	−0.325274	−	0.945620i	$-0.605457\pi$
$367$	−4573.82	−0.650548	−0.325274	+	0.945620i	$0.605457\pi$
$368$	−5205.10	−0.737323
$369$	−4394.98	−0.620036
$370$	8829.36	1.24059
$371$	−763.057	−0.106782
$372$	365.266	0.0509091
$373$	−1200.09	−0.166591	−0.0832954	−	0.996525i	$-0.526545\pi$
$373$	−1200.09	−0.166591	−0.0832954	+	0.996525i	$0.526545\pi$
$374$	−5095.57	−0.704507
$375$	4158.12	0.572599
$376$	−5420.70	−0.743487
$377$	−7309.33	−0.998540
$378$	−1240.46	−0.168789
$379$	13056.0	1.76950	0.884752	−	0.466062i	$-0.154328\pi$
$379$	13056.0	1.76950	0.884752	+	0.466062i	$0.154328\pi$
$380$	−2290.57	−0.309221
$381$	6722.37	0.903930
$382$	−2737.94	−0.366715
$383$	6014.50	0.802420	0.401210	−	0.915986i	$-0.368590\pi$
$383$	6014.50	0.802420	0.401210	+	0.915986i	$0.368590\pi$
$384$	−2147.18	−0.285345
$385$	3471.54	0.459548
$386$	5628.47	0.742181
$387$	−1145.13	−0.150414
$388$	2026.89	0.265205
$389$	−6419.19	−0.836673	−0.418336	−	0.908292i	$-0.637387\pi$
$389$	−6419.19	−0.836673	−0.418336	+	0.908292i	$0.637387\pi$
$390$	2056.66	0.267033
$391$	−14581.2	−1.88594
$392$	126.031	0.0162386
$393$	2829.46	0.363173
$394$	−1540.83	−0.197020
$395$	3651.05	0.465074
$396$	287.828	0.0365250
$397$	10002.4	1.26450	0.632252	−	0.774763i	$-0.282131\pi$
$397$	10002.4	1.26450	0.632252	+	0.774763i	$0.282131\pi$
$398$	11812.2	1.48767
$399$	−5872.67	−0.736845
$400$	−90.2539	−0.0112817
$401$	−4714.22	−0.587075	−0.293537	−	0.955948i	$-0.594832\pi$
$401$	−4714.22	−0.587075	−0.293537	+	0.955948i	$0.594832\pi$
$402$	−3933.11	−0.487974
$403$	−1552.67	−0.191921
$404$	2291.70	0.282219
$405$	−912.884	−0.112004
$406$	13593.2	1.66162
$407$	−5252.56	−0.639705
$408$	−9201.77	−1.11656
$409$	−14797.6	−1.78898	−0.894489	−	0.447090i	$-0.852460\pi$
$409$	−14797.6	−1.78898	−0.894489	+	0.447090i	$0.852460\pi$
$410$	−13551.3	−1.63232
$411$	5512.43	0.661577
$412$	−2100.50	−0.251175
$413$	−1100.87	−0.131163
$414$	−2577.64	−0.306000
$415$	−13235.5	−1.56556
$416$	−2113.73	−0.249120
$417$	−1213.23	−0.142475
$418$	−4264.55	−0.499010
$419$	12527.3	1.46062	0.730311	−	0.683115i	$-0.239375\pi$
$419$	12527.3	1.46062	0.730311	+	0.683115i	$0.239375\pi$
$420$	1222.13	0.141986
$421$	−3127.91	−0.362103	−0.181051	−	0.983474i	$-0.557950\pi$
$421$	−3127.91	−0.362103	−0.181051	+	0.983474i	$0.557950\pi$
$422$	8995.65	1.03768
$423$	−1993.87	−0.229185
$424$	−1000.63	−0.114611
$425$	−252.831	−0.0288567
$426$	−4689.83	−0.533387
$427$	−4422.86	−0.501258
$428$	1075.52	0.121465
$429$	−1223.50	−0.137695
$430$	−3530.83	−0.395981
$431$	780.300	0.0872058	0.0436029	−	0.999049i	$-0.486116\pi$
$431$	780.300	0.0872058	0.0436029	+	0.999049i	$0.486116\pi$
$432$	−1208.23	−0.134562
$433$	7622.27	0.845965	0.422983	−	0.906138i	$-0.360983\pi$
$433$	7622.27	0.845965	0.422983	+	0.906138i	$0.360983\pi$
$434$	2887.52	0.319367
$435$	10003.6	1.10261
$436$	2110.85	0.231861
$437$	−12203.2	−1.33583
$438$	−2777.29	−0.302977
$439$	10167.2	1.10536	0.552682	−	0.833392i	$-0.313604\pi$
$439$	10167.2	1.10536	0.552682	+	0.833392i	$0.313604\pi$
$440$	4552.39	0.493243
$441$	46.3575	0.00500567
$442$	7625.35	0.820590
$443$	−14545.0	−1.55994	−0.779969	−	0.625819i	$-0.784765\pi$
$443$	−14545.0	−1.55994	−0.779969	+	0.625819i	$0.784765\pi$
$444$	−1849.13	−0.197648
$445$	16277.9	1.73404
$446$	8941.71	0.949332
$447$	7196.85	0.761520
$448$	10610.6	1.11899
$449$	4431.72	0.465803	0.232902	−	0.972500i	$-0.425178\pi$
$449$	4431.72	0.465803	0.232902	+	0.972500i	$0.425178\pi$
$450$	−44.6949	−0.00468209
$451$	8061.62	0.841700
$452$	−997.628	−0.103815
$453$	−5459.71	−0.566268
$454$	9748.44	1.00775
$455$	−5195.05	−0.535269
$456$	−7701.10	−0.790870
$457$	2406.49	0.246326	0.123163	−	0.992386i	$-0.460696\pi$
$457$	2406.49	0.246326	0.123163	+	0.992386i	$0.460696\pi$
$458$	−6709.82	−0.684562
$459$	−3384.65	−0.344187
$460$	2539.55	0.257407
$461$	−14856.5	−1.50094	−0.750471	−	0.660903i	$-0.770173\pi$
$461$	−14856.5	−1.50094	−0.750471	+	0.660903i	$0.770173\pi$
$462$	2275.35	0.229132
$463$	1605.14	0.161117	0.0805586	−	0.996750i	$-0.474330\pi$
$463$	1605.14	0.161117	0.0805586	+	0.996750i	$0.474330\pi$
$464$	13240.0	1.32468
$465$	2124.99	0.211923
$466$	−3077.08	−0.305886
$467$	−17904.4	−1.77412	−0.887062	−	0.461651i	$-0.847257\pi$
$467$	−17904.4	−1.77412	−0.887062	+	0.461651i	$0.847257\pi$
$468$	−430.725	−0.0425433
$469$	9934.89	0.978146
$470$	−6147.81	−0.603356
$471$	−9055.21	−0.885865
$472$	−1443.62	−0.140780
$473$	2100.48	0.204187
$474$	2393.00	0.231887
$475$	−211.598	−0.0204395
$476$	4531.23	0.436320
$477$	−368.058	−0.0353296
$478$	−9851.22	−0.942645
$479$	−10467.2	−0.998448	−0.499224	−	0.866473i	$-0.666382\pi$
$479$	−10467.2	−0.998448	−0.499224	+	0.866473i	$0.666382\pi$
$480$	2892.85	0.275083
$481$	7860.29	0.745111
$482$	−4549.74	−0.429948
$483$	6511.02	0.613378
$484$	2050.51	0.192572
$485$	11791.7	1.10399
$486$	−598.331	−0.0558454
$487$	−15284.5	−1.42219	−0.711097	−	0.703093i	$-0.751802\pi$
$487$	−15284.5	−1.42219	−0.711097	+	0.703093i	$0.751802\pi$
$488$	−5799.90	−0.538011
$489$	3748.96	0.346695
$490$	142.937	0.0131780
$491$	−3589.22	−0.329897	−0.164949	−	0.986302i	$-0.552746\pi$
$491$	−3589.22	−0.329897	−0.164949	+	0.986302i	$0.552746\pi$
$492$	2838.04	0.260058
$493$	37089.6	3.38830
$494$	6381.76	0.581233
$495$	1674.48	0.152045
$496$	2812.49	0.254606
$497$	11846.3	1.06918
$498$	−8674.94	−0.780589
$499$	−3237.65	−0.290455	−0.145228	−	0.989398i	$-0.546391\pi$
$499$	−3237.65	−0.290455	−0.145228	+	0.989398i	$0.546391\pi$
$500$	−2685.09	−0.240162
$501$	−8417.88	−0.750665
$502$	−5071.67	−0.450915
$503$	1612.24	0.142915	0.0714576	−	0.997444i	$-0.477235\pi$
$503$	1612.24	0.142915	0.0714576	+	0.997444i	$0.477235\pi$
$504$	4108.92	0.363146
$505$	13332.3	1.17481
$506$	4728.11	0.415395
$507$	−4760.07	−0.416967
$508$	−4340.94	−0.379130
$509$	9148.28	0.796641	0.398320	−	0.917246i	$-0.369593\pi$
$509$	9148.28	0.796641	0.398320	+	0.917246i	$0.369593\pi$
$510$	−10436.1	−0.906111
$511$	7015.33	0.607319
$512$	12588.2	1.08657
$513$	−2832.66	−0.243791
$514$	−1144.87	−0.0982452
$515$	−12220.0	−1.04558
$516$	739.460	0.0630871
$517$	3657.32	0.311119
$518$	−14617.8	−1.23990
$519$	3575.32	0.302388
$520$	−6812.50	−0.574515
$521$	−1715.13	−0.144225	−0.0721124	−	0.997397i	$-0.522974\pi$
$521$	−1715.13	−0.144225	−0.0721124	+	0.997397i	$0.522974\pi$
$522$	6556.62	0.549762
$523$	−7402.28	−0.618889	−0.309445	−	0.950918i	$-0.600143\pi$
$523$	−7402.28	−0.618889	−0.309445	+	0.950918i	$0.600143\pi$
$524$	−1827.11	−0.152324
$525$	112.898	0.00938526
$526$	7254.13	0.601321
$527$	7878.71	0.651237
$528$	2216.23	0.182669
$529$	1362.70	0.112000
$530$	−1134.85	−0.0930092
$531$	−531.000	−0.0433963
$532$	3792.25	0.309051
$533$	−12063.9	−0.980389
$534$	10669.0	0.864594
$535$	6257.00	0.505633
$536$	13028.1	1.04986
$537$	9107.11	0.731845
$538$	−10505.7	−0.841882
$539$	−85.0327	−0.00679521
$540$	589.491	0.0469772
$541$	−20535.4	−1.63195	−0.815974	−	0.578089i	$-0.803799\pi$
$541$	−20535.4	−1.63195	−0.815974	+	0.578089i	$0.803799\pi$
$542$	4888.19	0.387391
$543$	8195.40	0.647694
$544$	10725.6	0.845327
$545$	12280.2	0.965185
$546$	−3404.99	−0.266886
$547$	−1648.99	−0.128895	−0.0644475	−	0.997921i	$-0.520528\pi$
$547$	−1648.99	−0.128895	−0.0644475	+	0.997921i	$0.520528\pi$
$548$	−3559.63	−0.277481
$549$	−2133.35	−0.165845
$550$	81.9830	0.00635594
$551$	31040.8	2.39997
$552$	8538.20	0.658351
$553$	−6044.64	−0.464818
$554$	1880.61	0.144223
$555$	−10757.6	−0.822765
$556$	783.439	0.0597575
$557$	−10612.2	−0.807275	−0.403638	−	0.914919i	$-0.632254\pi$
$557$	−10612.2	−0.807275	−0.403638	+	0.914919i	$0.632254\pi$
$558$	1392.78	0.105665
$559$	−3143.30	−0.237831
$560$	9410.22	0.710097
$561$	6208.38	0.467234
$562$	−4866.23	−0.365249
$563$	9821.65	0.735228	0.367614	−	0.929979i	$-0.380175\pi$
$563$	9821.65	0.735228	0.367614	+	0.929979i	$0.380175\pi$
$564$	1287.53	0.0961258
$565$	−5803.85	−0.432159
$566$	−10946.7	−0.812944
$567$	1511.36	0.111942
$568$	15534.6	1.14757
$569$	18417.7	1.35696	0.678479	−	0.734620i	$-0.262640\pi$
$569$	18417.7	1.35696	0.678479	+	0.734620i	$0.262640\pi$
$570$	−8734.09	−0.641808
$571$	−18415.0	−1.34964	−0.674818	−	0.737984i	$-0.735778\pi$
$571$	−18415.0	−1.34964	−0.674818	+	0.737984i	$0.735778\pi$
$572$	790.070	0.0577526
$573$	3335.87	0.243208
$574$	22435.4	1.63142
$575$	234.598	0.0170146
$576$	5118.00	0.370226
$577$	15969.8	1.15222	0.576110	−	0.817372i	$-0.304570\pi$
$577$	15969.8	1.15222	0.576110	+	0.817372i	$0.304570\pi$
$578$	−26596.0	−1.91393
$579$	−6857.67	−0.492219
$580$	−6459.76	−0.462460
$581$	21912.6	1.56469
$582$	7728.64	0.550451
$583$	675.121	0.0479599
$584$	9199.52	0.651847
$585$	−2505.81	−0.177098
$586$	−374.745	−0.0264174
$587$	−8593.97	−0.604278	−0.302139	−	0.953264i	$-0.597701\pi$
$587$	−8593.97	−0.604278	−0.302139	+	0.953264i	$0.597701\pi$
$588$	−29.9352	−0.00209950
$589$	6593.80	0.461278
$590$	−1637.26	−0.114246
$591$	1877.33	0.130665
$592$	−14238.0	−0.988476
$593$	−1632.32	−0.113038	−0.0565188	−	0.998402i	$-0.518000\pi$
$593$	−1632.32	−0.113038	−0.0565188	+	0.998402i	$0.518000\pi$
$594$	1097.51	0.0758102
$595$	26361.1	1.81630
$596$	−4647.34	−0.319400
$597$	−14391.8	−0.986631
$598$	−7075.45	−0.483841
$599$	−5743.37	−0.391766	−0.195883	−	0.980627i	$-0.562757\pi$
$599$	−5743.37	−0.391766	−0.195883	+	0.980627i	$0.562757\pi$
$600$	148.048	0.0100734
$601$	10777.0	0.731454	0.365727	−	0.930722i	$-0.380820\pi$
$601$	10777.0	0.731454	0.365727	+	0.930722i	$0.380820\pi$
$602$	5845.61	0.395763
$603$	4792.06	0.323628
$604$	3525.58	0.237507
$605$	11929.1	0.801633
$606$	8738.40	0.585764
$607$	2064.93	0.138077	0.0690386	−	0.997614i	$-0.478007\pi$
$607$	2064.93	0.138077	0.0690386	+	0.997614i	$0.478007\pi$
$608$	8976.44	0.598754
$609$	−16561.8	−1.10200
$610$	−6577.88	−0.436607
$611$	−5473.05	−0.362383
$612$	2185.62	0.144360
$613$	−7430.28	−0.489569	−0.244785	−	0.969577i	$-0.578717\pi$
$613$	−7430.28	−0.489569	−0.244785	+	0.969577i	$0.578717\pi$
$614$	−4653.71	−0.305877
$615$	16510.7	1.08256
$616$	−7536.90	−0.492971
$617$	−26320.0	−1.71735	−0.858675	−	0.512520i	$-0.828712\pi$
$617$	−26320.0	−1.71735	−0.858675	+	0.512520i	$0.828712\pi$
$618$	−8009.32	−0.521330
$619$	−1075.60	−0.0698416	−0.0349208	−	0.999390i	$-0.511118\pi$
$619$	−1075.60	−0.0698416	−0.0349208	+	0.999390i	$0.511118\pi$
$620$	−1372.21	−0.0888856
$621$	3140.57	0.202941
$622$	21135.1	1.36244
$623$	−26949.5	−1.73308
$624$	−3316.51	−0.212767
$625$	−15873.0	−1.01587
$626$	−20004.3	−1.27721
$627$	5195.89	0.330947
$628$	5847.37	0.371553
$629$	−39885.3	−2.52835
$630$	4660.07	0.294701
$631$	−16073.1	−1.01404	−0.507022	−	0.861933i	$-0.669254\pi$
$631$	−16073.1	−1.01404	−0.507022	+	0.861933i	$0.669254\pi$
$632$	−7926.62	−0.498898
$633$	−10960.2	−0.688197
$634$	−9624.74	−0.602913
$635$	−25254.1	−1.57823
$636$	237.672	0.0148181
$637$	127.249	0.00791487
$638$	−12026.7	−0.746302
$639$	5714.03	0.353746
$640$	8066.35	0.498204
$641$	−23363.5	−1.43963	−0.719815	−	0.694166i	$-0.755773\pi$
$641$	−23363.5	−1.43963	−0.719815	+	0.694166i	$0.755773\pi$
$642$	4101.02	0.252110
$643$	24799.4	1.52098	0.760492	−	0.649347i	$-0.224958\pi$
$643$	24799.4	1.52098	0.760492	+	0.649347i	$0.224958\pi$
$644$	−4204.46	−0.257266
$645$	4301.92	0.262617
$646$	−32382.9	−1.97227
$647$	2126.66	0.129224	0.0646118	−	0.997910i	$-0.479419\pi$
$647$	2126.66	0.129224	0.0646118	+	0.997910i	$0.479419\pi$
$648$	1981.92	0.120150
$649$	974.002	0.0589105
$650$	−122.685	−0.00740322
$651$	−3518.12	−0.211806
$652$	−2420.87	−0.145412
$653$	−501.178	−0.0300347	−0.0150173	−	0.999887i	$-0.504780\pi$
$653$	−501.178	−0.0300347	−0.0150173	+	0.999887i	$0.504780\pi$
$654$	8048.80	0.481243
$655$	−10629.5	−0.634089
$656$	21852.4	1.30060
$657$	3383.82	0.200936
$658$	10178.3	0.603024
$659$	−22302.4	−1.31833	−0.659164	−	0.752000i	$-0.729090\pi$
$659$	−22302.4	−1.31833	−0.659164	+	0.752000i	$0.729090\pi$
$660$	−1081.29	−0.0637715
$661$	−20096.2	−1.18253	−0.591263	−	0.806479i	$-0.701371\pi$
$661$	−20096.2	−1.18253	−0.591263	+	0.806479i	$0.701371\pi$
$662$	2130.60	0.125088
$663$	−9290.64	−0.544221
$664$	28735.0	1.67942
$665$	22062.0	1.28651
$666$	−7050.85	−0.410232
$667$	−34414.9	−1.99783
$668$	5435.81	0.314847
$669$	−10894.5	−0.629604
$670$	14775.6	0.851987
$671$	3913.16	0.225135
$672$	−4789.37	−0.274932
$673$	−21596.5	−1.23698	−0.618488	−	0.785794i	$-0.712255\pi$
$673$	−21596.5	−1.23698	−0.618488	+	0.785794i	$0.712255\pi$
$674$	−1252.70	−0.0715907
$675$	54.4558	0.00310519
$676$	3073.80	0.174886
$677$	18607.2	1.05633	0.528163	−	0.849143i	$-0.322881\pi$
$677$	18607.2	1.05633	0.528163	+	0.849143i	$0.322881\pi$
$678$	−3804.01	−0.215475
$679$	−19522.3	−1.10338
$680$	34568.5	1.94948
$681$	−11877.4	−0.668345
$682$	−2554.75	−0.143441
$683$	−3100.05	−0.173675	−0.0868376	−	0.996222i	$-0.527676\pi$
$683$	−3100.05	−0.173675	−0.0868376	+	0.996222i	$0.527676\pi$
$684$	1829.18	0.102252
$685$	−20708.7	−1.15509
$686$	15521.8	0.863884
$687$	8175.17	0.454006
$688$	5693.72	0.315510
$689$	−1010.30	−0.0558624
$690$	9683.48	0.534266
$691$	−9940.03	−0.547231	−0.273615	−	0.961839i	$-0.588220\pi$
$691$	−9940.03	−0.547231	−0.273615	+	0.961839i	$0.588220\pi$
$692$	−2308.75	−0.126829
$693$	−2772.26	−0.151962
$694$	−20535.2	−1.12321
$695$	4557.77	0.248757
$696$	−21718.2	−1.18280
$697$	61215.8	3.32671
$698$	−8063.17	−0.437243
$699$	3749.08	0.202866
$700$	−72.9033	−0.00393641
$701$	17598.8	0.948213	0.474107	−	0.880467i	$-0.342771\pi$
$701$	17598.8	0.948213	0.474107	+	0.880467i	$0.342771\pi$
$702$	−1642.38	−0.0883016
$703$	−33380.6	−1.79086
$704$	−9387.84	−0.502582
$705$	7490.42	0.400150
$706$	3886.91	0.207204
$707$	−22072.9	−1.17417
$708$	342.891	0.0182015
$709$	−22022.9	−1.16656	−0.583279	−	0.812272i	$-0.698230\pi$
$709$	−22022.9	−1.16656	−0.583279	+	0.812272i	$0.698230\pi$
$710$	17618.4	0.931277
$711$	−2915.61	−0.153789
$712$	−35340.2	−1.86015
$713$	−7310.54	−0.383986
$714$	17277.8	0.905612
$715$	4596.36	0.240411
$716$	−5880.88	−0.306953
$717$	12002.6	0.625169
$718$	−9016.29	−0.468642
$719$	30517.9	1.58293	0.791464	−	0.611216i	$-0.209319\pi$
$719$	30517.9	1.58293	0.791464	+	0.611216i	$0.209319\pi$
$720$	4538.98	0.234942
$721$	20231.2	1.04501
$722$	−10213.0	−0.526437
$723$	5543.35	0.285144
$724$	−5292.14	−0.271659
$725$	−596.737	−0.0305686
$726$	7818.70	0.399696
$727$	7485.46	0.381871	0.190936	−	0.981603i	$-0.438848\pi$
$727$	7485.46	0.381871	0.190936	+	0.981603i	$0.438848\pi$
$728$	11278.7	0.574199
$729$	729.000	0.0370370
$730$	10433.5	0.528988
$731$	15950.0	0.807020
$732$	1377.60	0.0695596
$733$	3904.12	0.196729	0.0983643	−	0.995150i	$-0.468639\pi$
$733$	3904.12	0.196729	0.0983643	+	0.995150i	$0.468639\pi$
$734$	11262.0	0.566330
$735$	−174.153	−0.00873974
$736$	−9952.17	−0.498426
$737$	−8789.97	−0.439325
$738$	10821.6	0.539768
$739$	5176.48	0.257672	0.128836	−	0.991666i	$-0.458876\pi$
$739$	5176.48	0.257672	0.128836	+	0.991666i	$0.458876\pi$
$740$	6946.67	0.345088
$741$	−7775.47	−0.385478
$742$	1878.85	0.0929580
$743$	−25011.8	−1.23498	−0.617492	−	0.786577i	$-0.711851\pi$
$743$	−25011.8	−1.23498	−0.617492	+	0.786577i	$0.711851\pi$
$744$	−4613.47	−0.227336
$745$	−27036.6	−1.32959
$746$	2954.95	0.145025
$747$	10569.5	0.517692
$748$	−4009.04	−0.195969
$749$	−10359.0	−0.505355
$750$	−10238.4	−0.498472
$751$	1794.26	0.0871817	0.0435908	−	0.999049i	$-0.486120\pi$
$751$	1794.26	0.0871817	0.0435908	+	0.999049i	$0.486120\pi$
$752$	9913.79	0.480743
$753$	6179.26	0.299050
$754$	17997.5	0.869272
$755$	20510.6	0.988686
$756$	−975.956	−0.0469513
$757$	−22317.1	−1.07150	−0.535752	−	0.844376i	$-0.679972\pi$
$757$	−22317.1	−1.07150	−0.535752	+	0.844376i	$0.679972\pi$
$758$	−32147.4	−1.54043
$759$	−5760.67	−0.275493
$760$	28930.9	1.38083
$761$	28965.0	1.37974	0.689868	−	0.723935i	$-0.257669\pi$
$761$	28965.0	1.37974	0.689868	+	0.723935i	$0.257669\pi$
$762$	−16552.3	−0.786910
$763$	−20331.0	−0.964654
$764$	−2154.12	−0.102007
$765$	12715.2	0.600939
$766$	−14809.3	−0.698541
$767$	−1457.56	−0.0686174
$768$	−8361.08	−0.392844
$769$	−31709.6	−1.48697	−0.743484	−	0.668753i	$-0.766828\pi$
$769$	−31709.6	−1.48697	−0.743484	+	0.668753i	$0.766828\pi$
$770$	−8547.87	−0.400057
$771$	1394.90	0.0651569
$772$	4428.31	0.206449
$773$	−17454.7	−0.812165	−0.406082	−	0.913836i	$-0.633105\pi$
$773$	−17454.7	−0.812165	−0.406082	+	0.913836i	$0.633105\pi$
$774$	2819.61	0.130941
$775$	−126.761	−0.00587535
$776$	−25600.5	−1.18428
$777$	17810.2	0.822312
$778$	15805.8	0.728360
$779$	51232.4	2.35634
$780$	1618.12	0.0742793
$781$	−10481.1	−0.480211
$782$	35902.8	1.64179
$783$	−7988.52	−0.364606
$784$	−230.496	−0.0105000
$785$	34018.0	1.54669
$786$	−6966.88	−0.316158
$787$	−43123.6	−1.95323	−0.976614	−	0.214999i	$-0.931025\pi$
$787$	−43123.6	−1.95323	−0.976614	+	0.214999i	$0.931025\pi$
$788$	−1212.28	−0.0548041
$789$	−8838.35	−0.398800
$790$	−8989.86	−0.404867
$791$	9608.80	0.431921
$792$	−3635.39	−0.163104
$793$	−5855.91	−0.262231
$794$	−24628.7	−1.10081
$795$	1382.69	0.0616843
$796$	9293.47	0.413817
$797$	35425.2	1.57443	0.787217	−	0.616676i	$-0.211521\pi$
$797$	35425.2	1.57443	0.787217	+	0.616676i	$0.211521\pi$
$798$	14460.1	0.641455
$799$	27771.8	1.22966
$800$	−172.566	−0.00762639
$801$	−12999.0	−0.573405
$802$	11607.7	0.511074
$803$	−6206.86	−0.272771
$804$	−3094.45	−0.135737
$805$	−24460.1	−1.07094
$806$	3823.10	0.167076
$807$	12800.0	0.558342
$808$	−28945.2	−1.26026
$809$	23010.2	0.999995	0.499997	−	0.866027i	$-0.333334\pi$
$809$	23010.2	0.999995	0.499997	+	0.866027i	$0.333334\pi$
$810$	2247.77	0.0975042
$811$	21404.3	0.926767	0.463384	−	0.886158i	$-0.346635\pi$
$811$	21404.3	0.926767	0.463384	+	0.886158i	$0.346635\pi$
$812$	10694.7	0.462205
$813$	−5955.72	−0.256920
$814$	12933.2	0.556891
$815$	−14083.8	−0.605318
$816$	16828.9	0.721973
$817$	13348.8	0.571621
$818$	36435.5	1.55738
$819$	4148.60	0.177001
$820$	−10661.7	−0.454053
$821$	−1299.78	−0.0552531	−0.0276265	−	0.999618i	$-0.508795\pi$
$821$	−1299.78	−0.0552531	−0.0276265	+	0.999618i	$0.508795\pi$
$822$	−13573.1	−0.575931
$823$	−15147.7	−0.641572	−0.320786	−	0.947152i	$-0.603947\pi$
$823$	−15147.7	−0.641572	−0.320786	+	0.947152i	$0.603947\pi$
$824$	26530.2	1.12163
$825$	−99.8872	−0.00421530
$826$	2710.63	0.114183
$827$	1909.49	0.0802895	0.0401448	−	0.999194i	$-0.487218\pi$
$827$	1909.49	0.0802895	0.0401448	+	0.999194i	$0.487218\pi$
$828$	−2028.01	−0.0851185
$829$	−32675.1	−1.36894	−0.684472	−	0.729039i	$-0.739967\pi$
$829$	−32675.1	−1.36894	−0.684472	+	0.729039i	$0.739967\pi$
$830$	32589.4	1.36288
$831$	−2291.31	−0.0956495
$832$	14048.6	0.585393
$833$	−645.695	−0.0268571
$834$	2987.30	0.124031
$835$	31623.7	1.31064
$836$	−3355.22	−0.138807
$837$	−1696.95	−0.0700779
$838$	−30845.7	−1.27153
$839$	23287.4	0.958250	0.479125	−	0.877747i	$-0.340954\pi$
$839$	23287.4	0.958250	0.479125	+	0.877747i	$0.340954\pi$
$840$	−15436.1	−0.634042
$841$	63150.7	2.58931
$842$	7701.76	0.315226
$843$	5928.97	0.242235
$844$	7077.51	0.288647
$845$	17882.3	0.728011
$846$	4909.44	0.199516
$847$	−19749.8	−0.801192
$848$	1830.03	0.0741080
$849$	13337.4	0.539150
$850$	622.537	0.0251210
$851$	37009.0	1.49078
$852$	−3689.81	−0.148370
$853$	−6441.16	−0.258548	−0.129274	−	0.991609i	$-0.541265\pi$
$853$	−6441.16	−0.258548	−0.129274	+	0.991609i	$0.541265\pi$
$854$	10890.3	0.436367
$855$	10641.5	0.425652
$856$	−13584.3	−0.542408
$857$	4466.38	0.178026	0.0890131	−	0.996030i	$-0.471629\pi$
$857$	4466.38	0.178026	0.0890131	+	0.996030i	$0.471629\pi$
$858$	3012.59	0.119869
$859$	−43793.5	−1.73948	−0.869741	−	0.493509i	$-0.835714\pi$
$859$	−43793.5	−1.73948	−0.869741	+	0.493509i	$0.835714\pi$
$860$	−2777.95	−0.110148
$861$	−27335.0	−1.08197
$862$	−1921.31	−0.0759164
$863$	−698.451	−0.0275499	−0.0137749	−	0.999905i	$-0.504385\pi$
$863$	−698.451	−0.0275499	−0.0137749	+	0.999905i	$0.504385\pi$
$864$	−2310.14	−0.0909634
$865$	−13431.5	−0.527959
$866$	−18768.1	−0.736449
$867$	32404.3	1.26933
$868$	2271.81	0.0888367
$869$	5348.04	0.208769
$870$	−24631.4	−0.959867
$871$	13153.9	0.511714
$872$	−26660.9	−1.03538
$873$	−9416.50	−0.365063
$874$	30047.6	1.16290
$875$	25861.9	0.999190
$876$	−2185.09	−0.0842776
$877$	25379.0	0.977180	0.488590	−	0.872513i	$-0.337511\pi$
$877$	25379.0	0.977180	0.488590	+	0.872513i	$0.337511\pi$
$878$	−25034.4	−0.962267
$879$	456.586	0.0175202
$880$	−8325.76	−0.318933
$881$	−29738.5	−1.13725	−0.568624	−	0.822598i	$-0.692524\pi$
$881$	−29738.5	−1.13725	−0.568624	+	0.822598i	$0.692524\pi$
$882$	−114.145	−0.00435765
$883$	40030.7	1.52564	0.762820	−	0.646611i	$-0.223814\pi$
$883$	40030.7	1.52564	0.762820	+	0.646611i	$0.223814\pi$
$884$	5999.39	0.228259
$885$	1994.82	0.0757686
$886$	35813.6	1.35799
$887$	−31928.0	−1.20861	−0.604304	−	0.796754i	$-0.706549\pi$
$887$	−31928.0	−1.20861	−0.604304	+	0.796754i	$0.706549\pi$
$888$	23355.3	0.882605
$889$	41810.5	1.57737
$890$	−40080.5	−1.50955
$891$	−1337.19	−0.0502778
$892$	7035.07	0.264071
$893$	23242.6	0.870979
$894$	−17720.6	−0.662936
$895$	−34212.9	−1.27778
$896$	−13354.6	−0.497930
$897$	8620.66	0.320887
$898$	−10912.1	−0.405502
$899$	18595.5	0.689872
$900$	−35.1646	−0.00130239
$901$	5126.52	0.189555
$902$	−19849.9	−0.732736
$903$	−7122.23	−0.262473
$904$	12600.5	0.463590
$905$	−30787.9	−1.13085
$906$	13443.3	0.492961
$907$	18316.8	0.670560	0.335280	−	0.942119i	$-0.391169\pi$
$907$	18316.8	0.670560	0.335280	+	0.942119i	$0.391169\pi$
$908$	7669.78	0.280320
$909$	−10646.8	−0.388483
$910$	12791.6	0.465975
$911$	−35873.6	−1.30466	−0.652331	−	0.757934i	$-0.726209\pi$
$911$	−35873.6	−1.30466	−0.652331	+	0.757934i	$0.726209\pi$
$912$	14084.4	0.511381
$913$	−19387.3	−0.702768
$914$	−5925.43	−0.214438
$915$	8014.41	0.289561
$916$	−5279.08	−0.190421
$917$	17598.1	0.633740
$918$	8333.90	0.299629
$919$	−14992.6	−0.538151	−0.269076	−	0.963119i	$-0.586718\pi$
$919$	−14992.6	−0.538151	−0.269076	+	0.963119i	$0.586718\pi$
$920$	−32075.7	−1.14946
$921$	5670.03	0.202860
$922$	36580.6	1.30664
$923$	15684.7	0.559336
$924$	1790.18	0.0637364
$925$	641.717	0.0228103
$926$	−3952.29	−0.140260
$927$	9758.46	0.345750
$928$	25314.9	0.895477
$929$	−42810.8	−1.51192	−0.755962	−	0.654615i	$-0.772831\pi$
$929$	−42810.8	−1.51192	−0.755962	+	0.654615i	$0.772831\pi$
$930$	−5232.30	−0.184488
$931$	−540.391	−0.0190232
$932$	−2420.95	−0.0850868
$933$	−25750.7	−0.903581
$934$	44085.4	1.54445
$935$	−23323.2	−0.815776
$936$	5440.25	0.189979
$937$	−1805.57	−0.0629514	−0.0314757	−	0.999505i	$-0.510021\pi$
$937$	−1805.57	−0.0629514	−0.0314757	+	0.999505i	$0.510021\pi$
$938$	−24462.4	−0.851518
$939$	24373.0	0.847052
$940$	−4836.91	−0.167833
$941$	−48559.2	−1.68224	−0.841119	−	0.540850i	$-0.818103\pi$
$941$	−48559.2	−1.68224	−0.841119	+	0.540850i	$0.818103\pi$
$942$	22296.4	0.771183
$943$	−56801.3	−1.96151
$944$	2640.20	0.0910289
$945$	−5677.78	−0.195448
$946$	−5171.95	−0.177753
$947$	−38153.0	−1.30919	−0.654597	−	0.755978i	$-0.727161\pi$
$947$	−38153.0	−1.30919	−0.654597	+	0.755978i	$0.727161\pi$
$948$	1882.74	0.0645028
$949$	9288.37	0.317717
$950$	521.010	0.0177935
$951$	11726.7	0.399856
$952$	−57231.4	−1.94840
$953$	20745.8	0.705166	0.352583	−	0.935781i	$-0.385303\pi$
$953$	20745.8	0.705166	0.352583	+	0.935781i	$0.385303\pi$
$954$	906.257	0.0307559
$955$	−12532.0	−0.424633
$956$	−7750.64	−0.262211
$957$	14653.2	0.494953
$958$	25772.9	0.869192
$959$	34285.1	1.15446
$960$	−19226.9	−0.646402
$961$	−25840.9	−0.867405
$962$	−19354.1	−0.648651
$963$	−4996.64	−0.167201
$964$	−3579.59	−0.119596
$965$	25762.4	0.859399
$966$	−16031.9	−0.533972
$967$	−17347.2	−0.576884	−0.288442	−	0.957497i	$-0.593137\pi$
$967$	−17347.2	−0.576884	−0.288442	+	0.957497i	$0.593137\pi$
$968$	−25898.8	−0.859936
$969$	39454.9	1.30802
$970$	−29034.4	−0.961070
$971$	23425.2	0.774202	0.387101	−	0.922037i	$-0.373476\pi$
$971$	23425.2	0.774202	0.387101	+	0.922037i	$0.373476\pi$
$972$	−470.749	−0.0155342
$973$	−7545.81	−0.248620
$974$	37634.6	1.23808
$975$	149.478	0.00490987
$976$	10607.3	0.347881
$977$	−13587.7	−0.444942	−0.222471	−	0.974939i	$-0.571412\pi$
$977$	−13587.7	−0.444942	−0.222471	+	0.974939i	$0.571412\pi$
$978$	−9230.93	−0.301813
$979$	23843.8	0.778398
$980$	112.458	0.00366566
$981$	−9806.57	−0.319164
$982$	8837.63	0.287190
$983$	−42058.9	−1.36467	−0.682336	−	0.731039i	$-0.739036\pi$
$983$	−42058.9	−1.36467	−0.682336	+	0.731039i	$0.739036\pi$
$984$	−35845.6	−1.16130
$985$	−7052.61	−0.228137
$986$	−91324.5	−2.94966
$987$	−12401.1	−0.399930
$988$	5020.98	0.161679
$989$	−14799.8	−0.475839
$990$	−4123.03	−0.132362
$991$	10765.5	0.345084	0.172542	−	0.985002i	$-0.444802\pi$
$991$	10765.5	0.345084	0.172542	+	0.985002i	$0.444802\pi$
$992$	5377.48	0.172112
$993$	−2595.90	−0.0829593
$994$	−29168.8	−0.930764
$995$	54066.2	1.72263
$996$	−6825.18	−0.217133
$997$	33120.6	1.05210	0.526048	−	0.850455i	$-0.323673\pi$
$997$	33120.6	1.05210	0.526048	+	0.850455i	$0.323673\pi$
$998$	7971.96	0.252854
$999$	8590.67	0.272069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.a.1.3	✓	7
3.2	odd	2		531.4.a.d.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.3	✓	7	1.1	even	1	trivial
531.4.a.d.1.5		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46227 q^{2} +3.00000 q^{3} -1.93724 q^{4} -11.2702 q^{5} -7.38680 q^{6} +18.6588 q^{7} +24.4681 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.46227 q^{2} +3.00000 q^{3} -1.93724 q^{4} -11.2702 q^{5} -7.38680 q^{6} +18.6588 q^{7} +24.4681 q^{8} +9.00000 q^{9} +27.7502 q^{10} -16.5085 q^{11} -5.81171 q^{12} +24.7044 q^{13} -45.9430 q^{14} -33.8105 q^{15} -44.7492 q^{16} -125.357 q^{17} -22.1604 q^{18} -104.913 q^{19} +21.8330 q^{20} +55.9764 q^{21} +40.6484 q^{22} +116.317 q^{23} +73.4044 q^{24} +2.01688 q^{25} -60.8289 q^{26} +27.0000 q^{27} -36.1465 q^{28} -295.871 q^{29} +83.2506 q^{30} -62.8500 q^{31} -85.5606 q^{32} -49.5255 q^{33} +308.663 q^{34} -210.288 q^{35} -17.4351 q^{36} +318.173 q^{37} +258.325 q^{38} +74.1133 q^{39} -275.760 q^{40} -488.331 q^{41} -137.829 q^{42} -127.236 q^{43} +31.9809 q^{44} -101.432 q^{45} -286.404 q^{46} -221.541 q^{47} -134.248 q^{48} +5.15084 q^{49} -4.96610 q^{50} -376.072 q^{51} -47.8584 q^{52} -40.8953 q^{53} -66.4812 q^{54} +186.054 q^{55} +456.546 q^{56} -314.740 q^{57} +728.514 q^{58} -59.0000 q^{59} +65.4990 q^{60} -237.039 q^{61} +154.754 q^{62} +167.929 q^{63} +568.667 q^{64} -278.423 q^{65} +121.945 q^{66} +532.451 q^{67} +242.847 q^{68} +348.952 q^{69} +517.785 q^{70} +634.893 q^{71} +220.213 q^{72} +375.980 q^{73} -783.427 q^{74} +6.05065 q^{75} +203.242 q^{76} -308.029 q^{77} -182.487 q^{78} -323.957 q^{79} +504.332 q^{80} +81.0000 q^{81} +1202.40 q^{82} +1174.38 q^{83} -108.440 q^{84} +1412.80 q^{85} +313.290 q^{86} -887.613 q^{87} -403.933 q^{88} -1444.33 q^{89} +249.752 q^{90} +460.955 q^{91} -225.334 q^{92} -188.550 q^{93} +545.494 q^{94} +1182.39 q^{95} -256.682 q^{96} -1046.28 q^{97} -12.6827 q^{98} -148.577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9}+O(q^{10}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9} - 79 q^{10} - 131 q^{11} + 66 q^{12} - 123 q^{13} - 117 q^{14} - 84 q^{15} + 202 q^{16} - 235 q^{17} - 72 q^{18} - 80 q^{19} + 61 q^{20} - 177 q^{21} + 688 q^{22} - 274 q^{23} - 351 q^{24} + 193 q^{25} - 180 q^{26} + 189 q^{27} - 118 q^{28} - 406 q^{29} - 237 q^{30} - 346 q^{31} - 854 q^{32} - 393 q^{33} + 178 q^{34} - 424 q^{35} + 198 q^{36} - 157 q^{37} - 129 q^{38} - 369 q^{39} - 590 q^{40} - 825 q^{41} - 351 q^{42} - 815 q^{43} - 1690 q^{44} - 252 q^{45} + 1457 q^{46} - 1196 q^{47} + 606 q^{48} + 914 q^{49} + 713 q^{50} - 705 q^{51} + 1030 q^{52} - 900 q^{53} - 216 q^{54} - 1044 q^{55} + 2172 q^{56} - 240 q^{57} + 1242 q^{58} - 413 q^{59} + 183 q^{60} + 420 q^{61} + 646 q^{62} - 531 q^{63} + 3541 q^{64} + 190 q^{65} + 2064 q^{66} + 1316 q^{67} - 611 q^{68} - 822 q^{69} + 4658 q^{70} - 173 q^{71} - 1053 q^{72} - 418 q^{73} + 660 q^{74} + 579 q^{75} + 1540 q^{76} - 753 q^{77} - 540 q^{78} + 2635 q^{79} + 6155 q^{80} + 567 q^{81} - 125 q^{82} + 457 q^{83} - 354 q^{84} + 1270 q^{85} + 3482 q^{86} - 1218 q^{87} + 7685 q^{88} + 592 q^{89} - 711 q^{90} + 3179 q^{91} - 3500 q^{92} - 1038 q^{93} + 2064 q^{94} - 2250 q^{95} - 2562 q^{96} - 1906 q^{97} + 2994 q^{98} - 1179 q^{99}+O(q^{100}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9 - 79 * q^10 - 131 * q^11 + 66 * q^12 - 123 * q^13 - 117 * q^14 - 84 * q^15 + 202 * q^16 - 235 * q^17 - 72 * q^18 - 80 * q^19 + 61 * q^20 - 177 * q^21 + 688 * q^22 - 274 * q^23 - 351 * q^24 + 193 * q^25 - 180 * q^26 + 189 * q^27 - 118 * q^28 - 406 * q^29 - 237 * q^30 - 346 * q^31 - 854 * q^32 - 393 * q^33 + 178 * q^34 - 424 * q^35 + 198 * q^36 - 157 * q^37 - 129 * q^38 - 369 * q^39 - 590 * q^40 - 825 * q^41 - 351 * q^42 - 815 * q^43 - 1690 * q^44 - 252 * q^45 + 1457 * q^46 - 1196 * q^47 + 606 * q^48 + 914 * q^49 + 713 * q^50 - 705 * q^51 + 1030 * q^52 - 900 * q^53 - 216 * q^54 - 1044 * q^55 + 2172 * q^56 - 240 * q^57 + 1242 * q^58 - 413 * q^59 + 183 * q^60 + 420 * q^61 + 646 * q^62 - 531 * q^63 + 3541 * q^64 + 190 * q^65 + 2064 * q^66 + 1316 * q^67 - 611 * q^68 - 822 * q^69 + 4658 * q^70 - 173 * q^71 - 1053 * q^72 - 418 * q^73 + 660 * q^74 + 579 * q^75 + 1540 * q^76 - 753 * q^77 - 540 * q^78 + 2635 * q^79 + 6155 * q^80 + 567 * q^81 - 125 * q^82 + 457 * q^83 - 354 * q^84 + 1270 * q^85 + 3482 * q^86 - 1218 * q^87 + 7685 * q^88 + 592 * q^89 - 711 * q^90 + 3179 * q^91 - 3500 * q^92 - 1038 * q^93 + 2064 * q^94 - 2250 * q^95 - 2562 * q^96 - 1906 * q^97 + 2994 * q^98 - 1179 * q^99

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