LMFDB - Embedded newform 77.4.a.d.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
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			$0.555307$ 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)

$f(q)$	$=$	$q-3.24550 q^{2} -5.49244 q^{3} +2.53327 q^{4} -16.0955 q^{5} +17.8257 q^{6} -7.00000 q^{7} +17.7423 q^{8} +3.16692 q^{9} +O(q^{10})$	\(q-3.24550 q^{2} -5.49244 q^{3} +2.53327 q^{4} -16.0955 q^{5} +17.8257 q^{6} -7.00000 q^{7} +17.7423 q^{8} +3.16692 q^{9} +52.2379 q^{10} -11.0000 q^{11} -13.9139 q^{12} +35.3712 q^{13} +22.7185 q^{14} +88.4036 q^{15} -77.8487 q^{16} +40.4757 q^{17} -10.2782 q^{18} +118.159 q^{19} -40.7743 q^{20} +38.4471 q^{21} +35.7005 q^{22} -174.510 q^{23} -97.4484 q^{24} +134.065 q^{25} -114.797 q^{26} +130.902 q^{27} -17.7329 q^{28} -262.725 q^{29} -286.914 q^{30} -36.1894 q^{31} +110.720 q^{32} +60.4169 q^{33} -131.364 q^{34} +112.668 q^{35} +8.02266 q^{36} +19.0464 q^{37} -383.487 q^{38} -194.274 q^{39} -285.571 q^{40} +156.996 q^{41} -124.780 q^{42} +287.182 q^{43} -27.8660 q^{44} -50.9731 q^{45} +566.371 q^{46} +397.244 q^{47} +427.580 q^{48} +49.0000 q^{49} -435.108 q^{50} -222.311 q^{51} +89.6049 q^{52} +272.483 q^{53} -424.842 q^{54} +177.050 q^{55} -124.196 q^{56} -648.984 q^{57} +852.674 q^{58} -507.466 q^{59} +223.950 q^{60} +35.5608 q^{61} +117.453 q^{62} -22.1684 q^{63} +263.448 q^{64} -569.317 q^{65} -196.083 q^{66} +979.229 q^{67} +102.536 q^{68} +958.484 q^{69} -365.666 q^{70} +750.404 q^{71} +56.1882 q^{72} +395.594 q^{73} -61.8152 q^{74} -736.344 q^{75} +299.330 q^{76} +77.0000 q^{77} +630.517 q^{78} -736.516 q^{79} +1253.01 q^{80} -804.477 q^{81} -509.531 q^{82} +582.975 q^{83} +97.3970 q^{84} -651.477 q^{85} -932.050 q^{86} +1443.00 q^{87} -195.165 q^{88} -806.201 q^{89} +165.433 q^{90} -247.598 q^{91} -442.081 q^{92} +198.768 q^{93} -1289.26 q^{94} -1901.84 q^{95} -608.123 q^{96} -957.232 q^{97} -159.030 q^{98} -34.8361 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 14 q^{3} + 26 q^{4} + 10 q^{5} + 14 q^{6} - 28 q^{7} - 18 q^{8} + 76 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 + 14 * q^3 + 26 * q^4 + 10 * q^5 + 14 * q^6 - 28 * q^7 - 18 * q^8 + 76 * q^9	\( 4 q - 2 q^{2} + 14 q^{3} + 26 q^{4} + 10 q^{5} + 14 q^{6} - 28 q^{7} - 18 q^{8} + 76 q^{9} - 2 q^{10} - 44 q^{11} + 70 q^{12} + 58 q^{13} + 14 q^{14} + 284 q^{15} + 2 q^{16} + 4 q^{17} - 62 q^{18} + 258 q^{19} + 182 q^{20} - 98 q^{21} + 22 q^{22} + 8 q^{23} - 498 q^{24} + 80 q^{25} - 482 q^{26} + 428 q^{27} - 182 q^{28} - 396 q^{29} - 628 q^{30} - 56 q^{31} + 134 q^{32} - 154 q^{33} + 472 q^{34} - 70 q^{35} - 418 q^{36} + 84 q^{37} - 942 q^{38} - 412 q^{39} - 1026 q^{40} + 52 q^{41} - 98 q^{42} + 408 q^{43} - 286 q^{44} + 826 q^{45} + 368 q^{46} + 8 q^{47} + 982 q^{48} + 196 q^{49} - 1642 q^{50} - 388 q^{51} + 2030 q^{52} + 624 q^{53} + 92 q^{54} - 110 q^{55} + 126 q^{56} + 48 q^{57} + 864 q^{58} - 238 q^{59} + 1420 q^{60} - 162 q^{61} + 688 q^{62} - 532 q^{63} - 902 q^{64} - 32 q^{65} - 154 q^{66} + 1340 q^{67} - 1384 q^{68} + 2416 q^{69} + 14 q^{70} + 1788 q^{71} - 2622 q^{72} + 1456 q^{73} + 996 q^{74} - 806 q^{75} + 3042 q^{76} + 308 q^{77} - 2632 q^{78} - 1324 q^{79} + 2342 q^{80} + 1444 q^{81} + 1984 q^{82} + 450 q^{83} - 490 q^{84} - 1736 q^{85} - 4380 q^{86} + 588 q^{87} + 198 q^{88} - 3072 q^{89} - 218 q^{90} - 406 q^{91} + 544 q^{92} - 1264 q^{93} - 1696 q^{94} + 24 q^{95} + 862 q^{96} - 652 q^{97} - 98 q^{98} - 836 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 + 14 * q^3 + 26 * q^4 + 10 * q^5 + 14 * q^6 - 28 * q^7 - 18 * q^8 + 76 * q^9 - 2 * q^10 - 44 * q^11 + 70 * q^12 + 58 * q^13 + 14 * q^14 + 284 * q^15 + 2 * q^16 + 4 * q^17 - 62 * q^18 + 258 * q^19 + 182 * q^20 - 98 * q^21 + 22 * q^22 + 8 * q^23 - 498 * q^24 + 80 * q^25 - 482 * q^26 + 428 * q^27 - 182 * q^28 - 396 * q^29 - 628 * q^30 - 56 * q^31 + 134 * q^32 - 154 * q^33 + 472 * q^34 - 70 * q^35 - 418 * q^36 + 84 * q^37 - 942 * q^38 - 412 * q^39 - 1026 * q^40 + 52 * q^41 - 98 * q^42 + 408 * q^43 - 286 * q^44 + 826 * q^45 + 368 * q^46 + 8 * q^47 + 982 * q^48 + 196 * q^49 - 1642 * q^50 - 388 * q^51 + 2030 * q^52 + 624 * q^53 + 92 * q^54 - 110 * q^55 + 126 * q^56 + 48 * q^57 + 864 * q^58 - 238 * q^59 + 1420 * q^60 - 162 * q^61 + 688 * q^62 - 532 * q^63 - 902 * q^64 - 32 * q^65 - 154 * q^66 + 1340 * q^67 - 1384 * q^68 + 2416 * q^69 + 14 * q^70 + 1788 * q^71 - 2622 * q^72 + 1456 * q^73 + 996 * q^74 - 806 * q^75 + 3042 * q^76 + 308 * q^77 - 2632 * q^78 - 1324 * q^79 + 2342 * q^80 + 1444 * q^81 + 1984 * q^82 + 450 * q^83 - 490 * q^84 - 1736 * q^85 - 4380 * q^86 + 588 * q^87 + 198 * q^88 - 3072 * q^89 - 218 * q^90 - 406 * q^91 + 544 * q^92 - 1264 * q^93 - 1696 * q^94 + 24 * q^95 + 862 * q^96 - 652 * q^97 - 98 * q^98 - 836 * 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 $

$2$	−3.24550	−1.14746	−0.573729	−	0.819045i	$-0.694504\pi$
$2$	−3.24550	−1.14746	−0.573729	+	0.819045i	$0.694504\pi$
$3$	−5.49244	−1.05702	−0.528510	−	0.848927i	$-0.677249\pi$
$3$	−5.49244	−1.05702	−0.528510	+	0.848927i	$0.677249\pi$
$4$	2.53327	0.316659
$5$	−16.0955	−1.43962	−0.719812	−	0.694169i	$-0.755772\pi$
$5$	−16.0955	−1.43962	−0.719812	+	0.694169i	$0.755772\pi$
$6$	17.8257	1.21289
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	17.7423	0.784105
$9$	3.16692	0.117293
$10$	52.2379	1.65191
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−13.9139	−0.334715
$13$	35.3712	0.754631	0.377316	−	0.926085i	$-0.376847\pi$
$13$	35.3712	0.754631	0.377316	+	0.926085i	$0.376847\pi$
$14$	22.7185	0.433698
$15$	88.4036	1.52171
$16$	−77.8487	−1.21639
$17$	40.4757	0.577459	0.288730	−	0.957411i	$-0.406767\pi$
$17$	40.4757	0.577459	0.288730	+	0.957411i	$0.406767\pi$
$18$	−10.2782	−0.134589
$19$	118.159	1.42672	0.713359	−	0.700799i	$-0.247173\pi$
$19$	118.159	1.42672	0.713359	+	0.700799i	$0.247173\pi$
$20$	−40.7743	−0.455870
$21$	38.4471	0.399516
$22$	35.7005	0.345972
$23$	−174.510	−1.58208	−0.791039	−	0.611766i	$-0.790459\pi$
$23$	−174.510	−1.58208	−0.791039	+	0.611766i	$0.790459\pi$
$24$	−97.4484	−0.828815
$25$	134.065	1.07252
$26$	−114.797	−0.865907
$27$	130.902	0.933040
$28$	−17.7329	−0.119686
$29$	−262.725	−1.68230	−0.841152	−	0.540799i	$-0.818122\pi$
$29$	−262.725	−1.68230	−0.841152	+	0.540799i	$0.818122\pi$
$30$	−286.914	−1.74610
$31$	−36.1894	−0.209671	−0.104836	−	0.994490i	$-0.533432\pi$
$31$	−36.1894	−0.209671	−0.104836	+	0.994490i	$0.533432\pi$
$32$	110.720	0.611647
$33$	60.4169	0.318704
$34$	−131.364	−0.662610
$35$	112.668	0.544127
$36$	8.02266	0.0371420
$37$	19.0464	0.0846274	0.0423137	−	0.999104i	$-0.486527\pi$
$37$	19.0464	0.0846274	0.0423137	+	0.999104i	$0.486527\pi$
$38$	−383.487	−1.63710
$39$	−194.274	−0.797661
$40$	−285.571	−1.12882
$41$	156.996	0.598017	0.299008	−	0.954250i	$-0.403344\pi$
$41$	156.996	0.598017	0.299008	+	0.954250i	$0.403344\pi$
$42$	−124.780	−0.458428
$43$	287.182	1.01849	0.509243	−	0.860623i	$-0.329926\pi$
$43$	287.182	1.01849	0.509243	+	0.860623i	$0.329926\pi$
$44$	−27.8660	−0.0954763
$45$	−50.9731	−0.168858
$46$	566.371	1.81537
$47$	397.244	1.23285	0.616425	−	0.787413i	$-0.288580\pi$
$47$	397.244	1.23285	0.616425	+	0.787413i	$0.288580\pi$
$48$	427.580	1.28575
$49$	49.0000	0.142857
$50$	−435.108	−1.23067
$51$	−222.311	−0.610386
$52$	89.6049	0.238961
$53$	272.483	0.706196	0.353098	−	0.935586i	$-0.385128\pi$
$53$	272.483	0.706196	0.353098	+	0.935586i	$0.385128\pi$
$54$	−424.842	−1.07062
$55$	177.050	0.434063
$56$	−124.196	−0.296364
$57$	−648.984	−1.50807
$58$	852.674	1.93037
$59$	−507.466	−1.11977	−0.559885	−	0.828570i	$-0.689155\pi$
$59$	−507.466	−1.11977	−0.559885	+	0.828570i	$0.689155\pi$
$60$	223.950	0.481865
$61$	35.5608	0.0746409	0.0373205	−	0.999303i	$-0.488118\pi$
$61$	35.5608	0.0746409	0.0373205	+	0.999303i	$0.488118\pi$
$62$	117.453	0.240589
$63$	−22.1684	−0.0443326
$64$	263.448	0.514547
$65$	−569.317	−1.08639
$66$	−196.083	−0.365699
$67$	979.229	1.78555	0.892775	−	0.450502i	$-0.148755\pi$
$67$	979.229	1.78555	0.892775	+	0.450502i	$0.148755\pi$
$68$	102.536	0.182858
$69$	958.484	1.67229
$70$	−365.666	−0.624363
$71$	750.404	1.25432	0.627159	−	0.778891i	$-0.284218\pi$
$71$	750.404	1.25432	0.627159	+	0.778891i	$0.284218\pi$
$72$	56.1882	0.0919701
$73$	395.594	0.634257	0.317129	−	0.948383i	$-0.397281\pi$
$73$	395.594	0.634257	0.317129	+	0.948383i	$0.397281\pi$
$74$	−61.8152	−0.0971063
$75$	−736.344	−1.13368
$76$	299.330	0.451783
$77$	77.0000	0.113961
$78$	630.517	0.915282
$79$	−736.516	−1.04892	−0.524459	−	0.851436i	$-0.675732\pi$
$79$	−736.516	−1.04892	−0.524459	+	0.851436i	$0.675732\pi$
$80$	1253.01	1.75114
$81$	−804.477	−1.10354
$82$	−509.531	−0.686199
$83$	582.975	0.770962	0.385481	−	0.922716i	$-0.374036\pi$
$83$	582.975	0.770962	0.385481	+	0.922716i	$0.374036\pi$
$84$	97.3970	0.126511
$85$	−651.477	−0.831325
$86$	−932.050	−1.16867
$87$	1443.00	1.77823
$88$	−195.165	−0.236416
$89$	−806.201	−0.960192	−0.480096	−	0.877216i	$-0.659398\pi$
$89$	−806.201	−0.960192	−0.480096	+	0.877216i	$0.659398\pi$
$90$	165.433	0.193758
$91$	−247.598	−0.285224
$92$	−442.081	−0.500979
$93$	198.768	0.221627
$94$	−1289.26	−1.41464
$95$	−1901.84	−2.05394
$96$	−608.123	−0.646524
$97$	−957.232	−1.00198	−0.500991	−	0.865453i	$-0.667031\pi$
$97$	−957.232	−1.00198	−0.500991	+	0.865453i	$0.667031\pi$
$98$	−159.030	−0.163923
$99$	−34.8361	−0.0353652
$100$	339.623	0.339623
$101$	−996.143	−0.981386	−0.490693	−	0.871333i	$-0.663256\pi$
$101$	−996.143	−0.981386	−0.490693	+	0.871333i	$0.663256\pi$
$102$	721.509	0.700393
$103$	1338.55	1.28050	0.640248	−	0.768169i	$-0.278832\pi$
$103$	1338.55	1.28050	0.640248	+	0.768169i	$0.278832\pi$
$104$	627.565	0.591710
$105$	−618.825	−0.575154
$106$	−884.343	−0.810330
$107$	1449.25	1.30939	0.654693	−	0.755895i	$-0.272798\pi$
$107$	1449.25	1.30939	0.654693	+	0.755895i	$0.272798\pi$
$108$	331.610	0.295456
$109$	−654.535	−0.575166	−0.287583	−	0.957756i	$-0.592852\pi$
$109$	−654.535	−0.575166	−0.287583	+	0.957756i	$0.592852\pi$
$110$	−574.617	−0.498069
$111$	−104.611	−0.0894529
$112$	544.941	0.459751
$113$	−1160.63	−0.966223	−0.483111	−	0.875559i	$-0.660493\pi$
$113$	−1160.63	−0.966223	−0.483111	+	0.875559i	$0.660493\pi$
$114$	2106.28	1.73045
$115$	2808.82	2.27760
$116$	−665.554	−0.532717
$117$	112.018	0.0885131
$118$	1646.98	1.28489
$119$	−283.330	−0.218259
$120$	1568.48	1.19318
$121$	121.000	0.0909091
$122$	−115.413	−0.0856473
$123$	−862.293	−0.632116
$124$	−91.6776	−0.0663943
$125$	−145.906	−0.104402
$126$	71.9476	0.0508698
$127$	−1055.41	−0.737420	−0.368710	−	0.929545i	$-0.620200\pi$
$127$	−1055.41	−0.737420	−0.368710	+	0.929545i	$0.620200\pi$
$128$	−1740.78	−1.20207
$129$	−1577.33	−1.07656
$130$	1847.72	1.24658
$131$	2657.40	1.77235	0.886175	−	0.463351i	$-0.153353\pi$
$131$	2657.40	1.77235	0.886175	+	0.463351i	$0.153353\pi$
$132$	153.052	0.100920
$133$	−827.116	−0.539249
$134$	−3178.09	−2.04884
$135$	−2106.93	−1.34323
$136$	718.131	0.452788
$137$	−147.314	−0.0918676	−0.0459338	−	0.998944i	$-0.514626\pi$
$137$	−147.314	−0.0918676	−0.0459338	+	0.998944i	$0.514626\pi$
$138$	−3110.76	−1.91888
$139$	902.634	0.550794	0.275397	−	0.961331i	$-0.411191\pi$
$139$	902.634	0.550794	0.275397	+	0.961331i	$0.411191\pi$
$140$	285.420	0.172303
$141$	−2181.84	−1.30315
$142$	−2435.44	−1.43928
$143$	−389.083	−0.227530
$144$	−246.540	−0.142674
$145$	4228.69	2.42189
$146$	−1283.90	−0.727783
$147$	−269.130	−0.151003
$148$	48.2498	0.0267980
$149$	1212.63	0.666727	0.333363	−	0.942798i	$-0.391816\pi$
$149$	1212.63	0.666727	0.333363	+	0.942798i	$0.391816\pi$
$150$	2389.81	1.30085
$151$	2565.33	1.38254	0.691270	−	0.722597i	$-0.257052\pi$
$151$	2565.33	1.38254	0.691270	+	0.722597i	$0.257052\pi$
$152$	2096.42	1.11870
$153$	128.183	0.0677320
$154$	−249.904	−0.130765
$155$	582.486	0.301848
$156$	−492.150	−0.252587
$157$	702.237	0.356972	0.178486	−	0.983942i	$-0.442880\pi$
$157$	702.237	0.356972	0.178486	+	0.983942i	$0.442880\pi$
$158$	2390.36	1.20359
$159$	−1496.60	−0.746464
$160$	−1782.09	−0.880542
$161$	1221.57	0.597969
$162$	2610.93	1.26626
$163$	−1146.27	−0.550814	−0.275407	−	0.961328i	$-0.588813\pi$
$163$	−1146.27	−0.550814	−0.275407	+	0.961328i	$0.588813\pi$
$164$	397.714	0.189368
$165$	−972.439	−0.458814
$166$	−1892.05	−0.884646
$167$	−3255.04	−1.50828	−0.754140	−	0.656714i	$-0.771946\pi$
$167$	−3255.04	−1.50828	−0.754140	+	0.656714i	$0.771946\pi$
$168$	682.138	0.313263
$169$	−945.879	−0.430532
$170$	2114.37	0.953910
$171$	374.201	0.167344
$172$	727.511	0.322513
$173$	4024.24	1.76854	0.884269	−	0.466977i	$-0.154657\pi$
$173$	4024.24	1.76854	0.884269	+	0.466977i	$0.154657\pi$
$174$	−4683.26	−2.04044
$175$	−938.455	−0.405374
$176$	856.336	0.366754
$177$	2787.23	1.18362
$178$	2616.52	1.10178
$179$	706.090	0.294836	0.147418	−	0.989074i	$-0.452904\pi$
$179$	706.090	0.294836	0.147418	+	0.989074i	$0.452904\pi$
$180$	−129.129	−0.0534705
$181$	−1268.90	−0.521087	−0.260544	−	0.965462i	$-0.583902\pi$
$181$	−1268.90	−0.521087	−0.260544	+	0.965462i	$0.583902\pi$
$182$	803.581	0.327282
$183$	−195.316	−0.0788970
$184$	−3096.20	−1.24051
$185$	−306.562	−0.121832
$186$	−645.102	−0.254307
$187$	−445.233	−0.174110
$188$	1006.33	0.390393
$189$	−916.313	−0.352656
$190$	6172.41	2.35681
$191$	4864.58	1.84287	0.921436	−	0.388529i	$-0.127017\pi$
$191$	4864.58	1.84287	0.921436	+	0.388529i	$0.127017\pi$
$192$	−1446.97	−0.543887
$193$	−2675.49	−0.997855	−0.498928	−	0.866644i	$-0.666273\pi$
$193$	−2675.49	−0.997855	−0.498928	+	0.866644i	$0.666273\pi$
$194$	3106.70	1.14973
$195$	3126.94	1.14833
$196$	124.130	0.0452370
$197$	1627.73	0.588684	0.294342	−	0.955700i	$-0.404899\pi$
$197$	1627.73	0.588684	0.294342	+	0.955700i	$0.404899\pi$
$198$	113.060	0.0405801
$199$	−2254.07	−0.802947	−0.401474	−	0.915871i	$-0.631502\pi$
$199$	−2254.07	−0.802947	−0.401474	+	0.915871i	$0.631502\pi$
$200$	2378.62	0.840968
$201$	−5378.36	−1.88736
$202$	3232.98	1.12610
$203$	1839.08	0.635851
$204$	−563.173	−0.193284
$205$	−2526.93	−0.860920
$206$	−4344.26	−1.46931
$207$	−552.657	−0.185567
$208$	−2753.60	−0.917923
$209$	−1299.75	−0.430172
$210$	2008.40	0.659965
$211$	3112.07	1.01537	0.507687	−	0.861542i	$-0.330501\pi$
$211$	3112.07	1.01537	0.507687	+	0.861542i	$0.330501\pi$
$212$	690.273	0.223623
$213$	−4121.55	−1.32584
$214$	−4703.54	−1.50247
$215$	−4622.34	−1.46624
$216$	2322.49	0.731601
$217$	253.326	0.0792482
$218$	2124.30	0.659979
$219$	−2172.78	−0.670423
$220$	448.517	0.137450
$221$	1431.67	0.435769
$222$	339.516	0.102643
$223$	−3558.38	−1.06855	−0.534275	−	0.845311i	$-0.679415\pi$
$223$	−3558.38	−1.06855	−0.534275	+	0.845311i	$0.679415\pi$
$224$	−775.039	−0.231181
$225$	424.572	0.125799
$226$	3766.84	1.10870
$227$	−2330.51	−0.681416	−0.340708	−	0.940169i	$-0.610667\pi$
$227$	−2330.51	−0.681416	−0.340708	+	0.940169i	$0.610667\pi$
$228$	−1644.05	−0.477544
$229$	676.106	0.195102	0.0975509	−	0.995231i	$-0.468899\pi$
$229$	676.106	0.195102	0.0975509	+	0.995231i	$0.468899\pi$
$230$	−9116.02	−2.61345
$231$	−422.918	−0.120459
$232$	−4661.34	−1.31910
$233$	1620.20	0.455548	0.227774	−	0.973714i	$-0.426855\pi$
$233$	1620.20	0.455548	0.227774	+	0.973714i	$0.426855\pi$
$234$	−363.553	−0.101565
$235$	−6393.84	−1.77484
$236$	−1285.55	−0.354586
$237$	4045.27	1.10873
$238$	919.548	0.250443
$239$	−2001.39	−0.541669	−0.270835	−	0.962626i	$-0.587300\pi$
$239$	−2001.39	−0.541669	−0.270835	+	0.962626i	$0.587300\pi$
$240$	−6882.10	−1.85099
$241$	−1586.39	−0.424018	−0.212009	−	0.977268i	$-0.568001\pi$
$241$	−1586.39	−0.424018	−0.212009	+	0.977268i	$0.568001\pi$
$242$	−392.706	−0.104314
$243$	884.196	0.233420
$244$	90.0853	0.0236357
$245$	−788.679	−0.205661
$246$	2798.57	0.725327
$247$	4179.44	1.07665
$248$	−642.081	−0.164404
$249$	−3201.96	−0.814923
$250$	473.537	0.119796
$251$	2612.23	0.656902	0.328451	−	0.944521i	$-0.393473\pi$
$251$	2612.23	0.656902	0.328451	+	0.944521i	$0.393473\pi$
$252$	−56.1586	−0.0140383
$253$	1919.61	0.477014
$254$	3425.33	0.846158
$255$	3578.20	0.878727
$256$	3542.12	0.864775
$257$	4198.23	1.01898	0.509491	−	0.860476i	$-0.329834\pi$
$257$	4198.23	1.01898	0.509491	+	0.860476i	$0.329834\pi$
$258$	5119.23	1.23531
$259$	−133.325	−0.0319861
$260$	−1442.24	−0.344014
$261$	−832.028	−0.197323
$262$	−8624.58	−2.03370
$263$	−5170.31	−1.21222	−0.606112	−	0.795380i	$-0.707272\pi$
$263$	−5170.31	−1.21222	−0.606112	+	0.795380i	$0.707272\pi$
$264$	1071.93	0.249897
$265$	−4385.74	−1.01666
$266$	2684.41	0.618765
$267$	4428.01	1.01494
$268$	2480.66	0.565411
$269$	1648.20	0.373578	0.186789	−	0.982400i	$-0.440192\pi$
$269$	1648.20	0.373578	0.186789	+	0.982400i	$0.440192\pi$
$270$	6838.04	1.54130
$271$	2562.79	0.574459	0.287230	−	0.957862i	$-0.407266\pi$
$271$	2562.79	0.574459	0.287230	+	0.957862i	$0.407266\pi$
$272$	−3150.98	−0.702413
$273$	1359.92	0.301487
$274$	478.107	0.105414
$275$	−1474.72	−0.323377
$276$	2428.10	0.529545
$277$	2762.94	0.599310	0.299655	−	0.954048i	$-0.403128\pi$
$277$	2762.94	0.599310	0.299655	+	0.954048i	$0.403128\pi$
$278$	−2929.50	−0.632013
$279$	−114.609	−0.0245930
$280$	1998.99	0.426653
$281$	6453.68	1.37009	0.685044	−	0.728502i	$-0.259783\pi$
$281$	6453.68	1.37009	0.685044	+	0.728502i	$0.259783\pi$
$282$	7081.16	1.49531
$283$	4540.78	0.953787	0.476893	−	0.878961i	$-0.341763\pi$
$283$	4540.78	0.953787	0.476893	+	0.878961i	$0.341763\pi$
$284$	1900.98	0.397191
$285$	10445.7	2.17106
$286$	1262.77	0.261081
$287$	−1098.97	−0.226029
$288$	350.641	0.0717420
$289$	−3274.72	−0.666541
$290$	−13724.2	−2.77901
$291$	5257.54	1.05912
$292$	1002.15	0.200843
$293$	1916.34	0.382094	0.191047	−	0.981581i	$-0.438812\pi$
$293$	1916.34	0.382094	0.191047	+	0.981581i	$0.438812\pi$
$294$	873.460	0.173270
$295$	8167.92	1.61205
$296$	337.927	0.0663567
$297$	−1439.92	−0.281322
$298$	−3935.58	−0.765040
$299$	−6172.61	−1.19388
$300$	−1865.36	−0.358989
$301$	−2010.28	−0.384951
$302$	−8325.78	−1.58641
$303$	5471.26	1.03735
$304$	−9198.56	−1.73544
$305$	−572.369	−0.107455
$306$	−416.019	−0.0777196
$307$	6876.30	1.27834	0.639171	−	0.769064i	$-0.279278\pi$
$307$	6876.30	1.27834	0.639171	+	0.769064i	$0.279278\pi$
$308$	195.062	0.0360867
$309$	−7351.89	−1.35351
$310$	−1890.46	−0.346357
$311$	8469.32	1.54422	0.772108	−	0.635492i	$-0.219203\pi$
$311$	8469.32	1.54422	0.772108	+	0.635492i	$0.219203\pi$
$312$	−3446.86	−0.625450
$313$	−2882.28	−0.520498	−0.260249	−	0.965542i	$-0.583805\pi$
$313$	−2882.28	−0.520498	−0.260249	+	0.965542i	$0.583805\pi$
$314$	−2279.11	−0.409610
$315$	356.812	0.0638224
$316$	−1865.80	−0.332150
$317$	9795.31	1.73552	0.867759	−	0.496985i	$-0.165560\pi$
$317$	9795.31	1.73552	0.867759	+	0.496985i	$0.165560\pi$
$318$	4857.20	0.856535
$319$	2889.98	0.507234
$320$	−4240.33	−0.740755
$321$	−7959.92	−1.38405
$322$	−3964.60	−0.686144
$323$	4782.59	0.823871
$324$	−2037.96	−0.349445
$325$	4742.04	0.809357
$326$	3720.22	0.632036
$327$	3595.00	0.607963
$328$	2785.47	0.468908
$329$	−2780.71	−0.465974
$330$	3156.05	0.526470
$331$	1813.70	0.301178	0.150589	−	0.988596i	$-0.451883\pi$
$331$	1813.70	0.301178	0.150589	+	0.988596i	$0.451883\pi$
$332$	1476.84	0.244132
$333$	60.3184	0.00992621
$334$	10564.2	1.73069
$335$	−15761.2	−2.57052
$336$	−2993.06	−0.485966
$337$	−11964.2	−1.93393	−0.966964	−	0.254913i	$-0.917953\pi$
$337$	−11964.2	−1.93393	−0.966964	+	0.254913i	$0.917953\pi$
$338$	3069.85	0.494017
$339$	6374.71	1.02132
$340$	−1650.37	−0.263247
$341$	398.083	0.0632182
$342$	−1214.47	−0.192020
$343$	−343.000	−0.0539949
$344$	5095.26	0.798599
$345$	−15427.3	−2.40747
$346$	−13060.7	−2.02932
$347$	−2283.89	−0.353330	−0.176665	−	0.984271i	$-0.556531\pi$
$347$	−2283.89	−0.353330	−0.176665	+	0.984271i	$0.556531\pi$
$348$	3655.52	0.563093
$349$	−2472.48	−0.379223	−0.189612	−	0.981859i	$-0.560723\pi$
$349$	−2472.48	−0.379223	−0.189612	+	0.981859i	$0.560723\pi$
$350$	3045.76	0.465150
$351$	4630.15	0.704101
$352$	−1217.92	−0.184418
$353$	10190.0	1.53642	0.768211	−	0.640197i	$-0.221147\pi$
$353$	10190.0	1.53642	0.768211	+	0.640197i	$0.221147\pi$
$354$	−9045.95	−1.35815
$355$	−12078.1	−1.80575
$356$	−2042.33	−0.304054
$357$	1556.17	0.230704
$358$	−2291.62	−0.338312
$359$	−43.4294	−0.00638473	−0.00319236	−	0.999995i	$-0.501016\pi$
$359$	−43.4294	−0.00638473	−0.00319236	+	0.999995i	$0.501016\pi$
$360$	−904.378	−0.132402
$361$	7102.66	1.03552
$362$	4118.23	0.597926
$363$	−664.585	−0.0960928
$364$	−627.234	−0.0903187
$365$	−6367.28	−0.913093
$366$	633.897	0.0905310
$367$	8775.78	1.24821	0.624104	−	0.781342i	$-0.285464\pi$
$367$	8775.78	1.24821	0.624104	+	0.781342i	$0.285464\pi$
$368$	13585.4	1.92442
$369$	497.194	0.0701433
$370$	994.946	0.139797
$371$	−1907.38	−0.266917
$372$	503.534	0.0701801
$373$	−2751.89	−0.382004	−0.191002	−	0.981590i	$-0.561174\pi$
$373$	−2751.89	−0.382004	−0.191002	+	0.981590i	$0.561174\pi$
$374$	1445.00	0.199784
$375$	801.379	0.110355
$376$	7048.01	0.966684
$377$	−9292.90	−1.26952
$378$	2973.89	0.404658
$379$	2605.59	0.353141	0.176570	−	0.984288i	$-0.443500\pi$
$379$	2605.59	0.353141	0.176570	+	0.984288i	$0.443500\pi$
$380$	−4817.87	−0.650399
$381$	5796.77	0.779468
$382$	−15788.0	−2.11462
$383$	−14360.5	−1.91590	−0.957949	−	0.286940i	$-0.907362\pi$
$383$	−14360.5	−1.91590	−0.957949	+	0.286940i	$0.907362\pi$
$384$	9561.14	1.27061
$385$	−1239.35	−0.164060
$386$	8683.31	1.14500
$387$	909.482	0.119461
$388$	−2424.93	−0.317287
$389$	−10607.9	−1.38262	−0.691311	−	0.722557i	$-0.742967\pi$
$389$	−10607.9	−1.38262	−0.691311	+	0.722557i	$0.742967\pi$
$390$	−10148.5	−1.31766
$391$	−7063.40	−0.913585
$392$	869.371	0.112015
$393$	−14595.6	−1.87341
$394$	−5282.79	−0.675490
$395$	11854.6	1.51005
$396$	−88.2493	−0.0111987
$397$	7362.35	0.930745	0.465373	−	0.885115i	$-0.345920\pi$
$397$	7362.35	0.930745	0.465373	+	0.885115i	$0.345920\pi$
$398$	7315.57	0.921348
$399$	4542.89	0.569997
$400$	−10436.8	−1.30460
$401$	−264.514	−0.0329406	−0.0164703	−	0.999864i	$-0.505243\pi$
$401$	−264.514	−0.0329406	−0.0164703	+	0.999864i	$0.505243\pi$
$402$	17455.5	2.16567
$403$	−1280.06	−0.158224
$404$	−2523.50	−0.310765
$405$	12948.5	1.58868
$406$	−5968.72	−0.729612
$407$	−209.511	−0.0255161
$408$	−3944.29	−0.478607
$409$	−1244.03	−0.150400	−0.0751999	−	0.997168i	$-0.523959\pi$
$409$	−1244.03	−0.150400	−0.0751999	+	0.997168i	$0.523959\pi$
$410$	8201.16	0.987869
$411$	809.112	0.0971060
$412$	3390.91	0.405481
$413$	3552.26	0.423233
$414$	1793.65	0.212930
$415$	−9383.27	−1.10990
$416$	3916.30	0.461568
$417$	−4957.66	−0.582201
$418$	4218.35	0.493604
$419$	3974.76	0.463436	0.231718	−	0.972783i	$-0.425565\pi$
$419$	3974.76	0.463436	0.231718	+	0.972783i	$0.425565\pi$
$420$	−1567.65	−0.182128
$421$	−14910.2	−1.72608	−0.863041	−	0.505134i	$-0.831443\pi$
$421$	−14910.2	−1.72608	−0.863041	+	0.505134i	$0.831443\pi$
$422$	−10100.2	−1.16510
$423$	1258.04	0.144605
$424$	4834.46	0.553731
$425$	5426.38	0.619336
$426$	13376.5	1.52135
$427$	−248.926	−0.0282116
$428$	3671.35	0.414629
$429$	2137.02	0.240504
$430$	15001.8	1.68244
$431$	−10622.4	−1.18715	−0.593574	−	0.804779i	$-0.702284\pi$
$431$	−10622.4	−1.18715	−0.593574	+	0.804779i	$0.702284\pi$
$432$	−10190.5	−1.13494
$433$	17440.5	1.93565	0.967823	−	0.251630i	$-0.0809667\pi$
$433$	17440.5	1.93565	0.967823	+	0.251630i	$0.0809667\pi$
$434$	−822.168	−0.0909340
$435$	−23225.8	−2.55999
$436$	−1658.12	−0.182132
$437$	−20620.0	−2.25718
$438$	7051.75	0.769282
$439$	13627.1	1.48151	0.740756	−	0.671774i	$-0.234467\pi$
$439$	13627.1	1.48151	0.740756	+	0.671774i	$0.234467\pi$
$440$	3141.28	0.340351
$441$	155.179	0.0167562
$442$	−4646.50	−0.500026
$443$	2135.29	0.229008	0.114504	−	0.993423i	$-0.463472\pi$
$443$	2135.29	0.229008	0.114504	+	0.993423i	$0.463472\pi$
$444$	−265.009	−0.0283261
$445$	12976.2	1.38232
$446$	11548.7	1.22612
$447$	−6660.28	−0.704744
$448$	−1844.14	−0.194481
$449$	17780.8	1.86889	0.934443	−	0.356113i	$-0.115898\pi$
$449$	17780.8	1.86889	0.934443	+	0.356113i	$0.115898\pi$
$450$	−1377.95	−0.144349
$451$	−1726.96	−0.180309
$452$	−2940.20	−0.305963
$453$	−14089.9	−1.46137
$454$	7563.67	0.781896
$455$	3985.22	0.410615
$456$	−11514.4	−1.18249
$457$	10357.0	1.06013	0.530064	−	0.847957i	$-0.322168\pi$
$457$	10357.0	1.06013	0.530064	+	0.847957i	$0.322168\pi$
$458$	−2194.30	−0.223871
$459$	5298.35	0.538792
$460$	7115.51	0.721222
$461$	−19679.5	−1.98821	−0.994106	−	0.108410i	$-0.965424\pi$
$461$	−19679.5	−1.98821	−0.994106	+	0.108410i	$0.965424\pi$
$462$	1372.58	0.138221
$463$	−7171.43	−0.719838	−0.359919	−	0.932984i	$-0.617196\pi$
$463$	−7171.43	−0.719838	−0.359919	+	0.932984i	$0.617196\pi$
$464$	20452.8	2.04633
$465$	−3199.27	−0.319059
$466$	−5258.36	−0.522722
$467$	−12192.8	−1.20817	−0.604085	−	0.796920i	$-0.706461\pi$
$467$	−12192.8	−1.20817	−0.604085	+	0.796920i	$0.706461\pi$
$468$	283.771	0.0280285
$469$	−6854.61	−0.674875
$470$	20751.2	2.03656
$471$	−3856.99	−0.377327
$472$	−9003.60	−0.878017
$473$	−3159.00	−0.307085
$474$	−13128.9	−1.27222
$475$	15841.1	1.53018
$476$	−717.752	−0.0691137
$477$	862.930	0.0828319
$478$	6495.50	0.621542
$479$	8475.11	0.808429	0.404215	−	0.914664i	$-0.367545\pi$
$479$	8475.11	0.808429	0.404215	+	0.914664i	$0.367545\pi$
$480$	9788.04	0.930751
$481$	673.695	0.0638624
$482$	5148.63	0.486543
$483$	−6709.39	−0.632066
$484$	306.526	0.0287872
$485$	15407.1	1.44248
$486$	−2869.66	−0.267840
$487$	2735.29	0.254513	0.127257	−	0.991870i	$-0.459383\pi$
$487$	2735.29	0.254513	0.127257	+	0.991870i	$0.459383\pi$
$488$	630.929	0.0585263
$489$	6295.81	0.582222
$490$	2559.66	0.235987
$491$	−2233.82	−0.205317	−0.102659	−	0.994717i	$-0.532735\pi$
$491$	−2233.82	−0.205317	−0.102659	+	0.994717i	$0.532735\pi$
$492$	−2184.42	−0.200165
$493$	−10634.0	−0.971462
$494$	−13564.4	−1.23541
$495$	560.704	0.0509126
$496$	2817.30	0.255041
$497$	−5252.83	−0.474088
$498$	10392.0	0.935089
$499$	−18100.3	−1.62381	−0.811904	−	0.583791i	$-0.801569\pi$
$499$	−18100.3	−1.62381	−0.811904	+	0.583791i	$0.801569\pi$
$500$	−369.619	−0.0330597
$501$	17878.1	1.59428
$502$	−8477.99	−0.753768
$503$	6149.06	0.545076	0.272538	−	0.962145i	$-0.412137\pi$
$503$	6149.06	0.545076	0.272538	+	0.962145i	$0.412137\pi$
$504$	−393.318	−0.0347614
$505$	16033.4	1.41283
$506$	−6230.08	−0.547354
$507$	5195.18	0.455081
$508$	−2673.64	−0.233511
$509$	14193.9	1.23602	0.618008	−	0.786172i	$-0.287940\pi$
$509$	14193.9	1.23602	0.618008	+	0.786172i	$0.287940\pi$
$510$	−11613.0	−1.00830
$511$	−2769.16	−0.239727
$512$	2430.30	0.209775
$513$	15467.3	1.33118
$514$	−13625.4	−1.16924
$515$	−21544.6	−1.84343
$516$	−3995.81	−0.340903
$517$	−4369.68	−0.371718
$518$	432.706	0.0367027
$519$	−22102.9	−1.86938
$520$	−10101.0	−0.851840
$521$	10371.8	0.872163	0.436082	−	0.899907i	$-0.356366\pi$
$521$	10371.8	0.872163	0.436082	+	0.899907i	$0.356366\pi$
$522$	2700.35	0.226420
$523$	−11369.4	−0.950569	−0.475285	−	0.879832i	$-0.657655\pi$
$523$	−11369.4	−0.950569	−0.475285	+	0.879832i	$0.657655\pi$
$524$	6731.91	0.561231
$525$	5154.41	0.428489
$526$	16780.2	1.39098
$527$	−1464.79	−0.121076
$528$	−4703.37	−0.387667
$529$	18286.6	1.50297
$530$	14233.9	1.16657
$531$	−1607.10	−0.131341
$532$	−2095.31	−0.170758
$533$	5553.14	0.451282
$534$	−14371.1	−1.16460
$535$	−23326.4	−1.88503
$536$	17373.7	1.40006
$537$	−3878.16	−0.311648
$538$	−5349.24	−0.428665
$539$	−539.000	−0.0430730
$540$	−5337.43	−0.425345
$541$	5386.88	0.428096	0.214048	−	0.976823i	$-0.431335\pi$
$541$	5386.88	0.428096	0.214048	+	0.976823i	$0.431335\pi$
$542$	−8317.54	−0.659167
$543$	6969.38	0.550800
$544$	4481.47	0.353201
$545$	10535.1	0.828024
$546$	−4413.62	−0.345944
$547$	13890.8	1.08579	0.542894	−	0.839801i	$-0.317328\pi$
$547$	13890.8	1.08579	0.542894	+	0.839801i	$0.317328\pi$
$548$	−373.186	−0.0290907
$549$	112.618	0.00875487
$550$	4786.19	0.371061
$551$	−31043.5	−2.40017
$552$	17005.7	1.31125
$553$	5155.61	0.396454
$554$	−8967.12	−0.687683
$555$	1683.77	0.128779
$556$	2286.62	0.174414
$557$	17498.3	1.33111	0.665553	−	0.746351i	$-0.268196\pi$
$557$	17498.3	1.33111	0.665553	+	0.746351i	$0.268196\pi$
$558$	371.962	0.0282194
$559$	10158.0	0.768581
$560$	−8771.10	−0.661869
$561$	2445.42	0.184038
$562$	−20945.4	−1.57212
$563$	147.373	0.0110320	0.00551600	−	0.999985i	$-0.498244\pi$
$563$	147.373	0.0110320	0.00551600	+	0.999985i	$0.498244\pi$
$564$	−5527.19	−0.412654
$565$	18681.0	1.39100
$566$	−14737.1	−1.09443
$567$	5631.34	0.417097
$568$	13313.9	0.983517
$569$	−14155.3	−1.04292	−0.521459	−	0.853276i	$-0.674612\pi$
$569$	−14155.3	−1.04292	−0.521459	+	0.853276i	$0.674612\pi$
$570$	−33901.6	−2.49120
$571$	−248.361	−0.0182025	−0.00910123	−	0.999959i	$-0.502897\pi$
$571$	−248.361	−0.0182025	−0.00910123	+	0.999959i	$0.502897\pi$
$572$	−985.654	−0.0720494
$573$	−26718.4	−1.94795
$574$	3566.72	0.259359
$575$	−23395.6	−1.69681
$576$	834.318	0.0603529
$577$	19364.6	1.39715	0.698576	−	0.715536i	$-0.253817\pi$
$577$	19364.6	1.39715	0.698576	+	0.715536i	$0.253817\pi$
$578$	10628.1	0.764828
$579$	14695.0	1.05475
$580$	10712.4	0.766913
$581$	−4080.83	−0.291396
$582$	−17063.4	−1.21529
$583$	−2997.31	−0.212926
$584$	7018.73	0.497324
$585$	−1802.98	−0.127426
$586$	−6219.47	−0.438437
$587$	−9135.18	−0.642332	−0.321166	−	0.947023i	$-0.604075\pi$
$587$	−9135.18	−0.642332	−0.321166	+	0.947023i	$0.604075\pi$
$588$	−681.779	−0.0478165
$589$	−4276.12	−0.299141
$590$	−26509.0	−1.84976
$591$	−8940.20	−0.622252
$592$	−1482.74	−0.102940
$593$	−12287.6	−0.850916	−0.425458	−	0.904978i	$-0.639887\pi$
$593$	−12287.6	−0.850916	−0.425458	+	0.904978i	$0.639887\pi$
$594$	4673.26	0.322805
$595$	4560.34	0.314211
$596$	3071.91	0.211125
$597$	12380.3	0.848732
$598$	20033.2	1.36993
$599$	12351.9	0.842549	0.421275	−	0.906933i	$-0.361583\pi$
$599$	12351.9	0.842549	0.421275	+	0.906933i	$0.361583\pi$
$600$	−13064.4	−0.888921
$601$	21624.2	1.46767	0.733836	−	0.679327i	$-0.237728\pi$
$601$	21624.2	1.46767	0.733836	+	0.679327i	$0.237728\pi$
$602$	6524.35	0.441715
$603$	3101.14	0.209433
$604$	6498.68	0.437794
$605$	−1947.56	−0.130875
$606$	−17757.0	−1.19031
$607$	−2086.03	−0.139488	−0.0697442	−	0.997565i	$-0.522218\pi$
$607$	−2086.03	−0.139488	−0.0697442	+	0.997565i	$0.522218\pi$
$608$	13082.6	0.872648
$609$	−10101.0	−0.672108
$610$	1857.62	0.123300
$611$	14051.0	0.930347
$612$	324.723	0.0214480
$613$	−8338.46	−0.549408	−0.274704	−	0.961529i	$-0.588580\pi$
$613$	−8338.46	−0.549408	−0.274704	+	0.961529i	$0.588580\pi$
$614$	−22317.0	−1.46684
$615$	13879.0	0.910011
$616$	1366.15	0.0893570
$617$	−4771.20	−0.311315	−0.155657	−	0.987811i	$-0.549750\pi$
$617$	−4771.20	−0.311315	−0.155657	+	0.987811i	$0.549750\pi$
$618$	23860.6	1.55310
$619$	−16609.4	−1.07850	−0.539248	−	0.842147i	$-0.681292\pi$
$619$	−16609.4	−1.07850	−0.539248	+	0.842147i	$0.681292\pi$
$620$	1475.60	0.0955828
$621$	−22843.6	−1.47614
$622$	−27487.2	−1.77192
$623$	5643.40	0.362919
$624$	15124.0	0.970264
$625$	−14409.7	−0.922221
$626$	9354.43	0.597250
$627$	7138.82	0.454700
$628$	1778.96	0.113038
$629$	770.918	0.0488688
$630$	−1158.03	−0.0732335
$631$	17254.9	1.08860	0.544299	−	0.838891i	$-0.316796\pi$
$631$	17254.9	1.08860	0.544299	+	0.838891i	$0.316796\pi$
$632$	−13067.5	−0.822462
$633$	−17092.9	−1.07327
$634$	−31790.7	−1.99143
$635$	16987.3	1.06161
$636$	−3791.28	−0.236375
$637$	1733.19	0.107804
$638$	−9379.42	−0.582029
$639$	2376.47	0.147123
$640$	28018.7	1.73053
$641$	−7350.77	−0.452945	−0.226473	−	0.974018i	$-0.572719\pi$
$641$	−7350.77	−0.452945	−0.226473	+	0.974018i	$0.572719\pi$
$642$	25833.9	1.58814
$643$	10117.8	0.620542	0.310271	−	0.950648i	$-0.399580\pi$
$643$	10117.8	0.620542	0.310271	+	0.950648i	$0.399580\pi$
$644$	3094.56	0.189352
$645$	25387.9	1.54984
$646$	−15521.9	−0.945358
$647$	−24590.9	−1.49423	−0.747116	−	0.664693i	$-0.768562\pi$
$647$	−24590.9	−1.49423	−0.747116	+	0.664693i	$0.768562\pi$
$648$	−14273.2	−0.865287
$649$	5582.13	0.337624
$650$	−15390.3	−0.928703
$651$	−1391.38	−0.0837670
$652$	−2903.81	−0.174420
$653$	2339.03	0.140173	0.0700867	−	0.997541i	$-0.477672\pi$
$653$	2339.03	0.140173	0.0700867	+	0.997541i	$0.477672\pi$
$654$	−11667.6	−0.697612
$655$	−42772.1	−2.55152
$656$	−12222.0	−0.727420
$657$	1252.81	0.0743940
$658$	9024.79	0.534685
$659$	−15735.7	−0.930162	−0.465081	−	0.885268i	$-0.653975\pi$
$659$	−15735.7	−0.930162	−0.465081	+	0.885268i	$0.653975\pi$
$660$	−2463.45	−0.145288
$661$	4846.75	0.285199	0.142600	−	0.989780i	$-0.454454\pi$
$661$	4846.75	0.285199	0.142600	+	0.989780i	$0.454454\pi$
$662$	−5886.36	−0.345589
$663$	−7863.39	−0.460617
$664$	10343.3	0.604515
$665$	13312.8	0.776316
$666$	−195.763	−0.0113899
$667$	45848.1	2.66154
$668$	−8245.91	−0.477611
$669$	19544.2	1.12948
$670$	51152.9	2.94957
$671$	−391.169	−0.0225051
$672$	4256.86	0.244363
$673$	15716.8	0.900205	0.450103	−	0.892977i	$-0.351387\pi$
$673$	15716.8	0.900205	0.450103	+	0.892977i	$0.351387\pi$
$674$	38830.0	2.21910
$675$	17549.4	1.00070
$676$	−2396.17	−0.136332
$677$	−856.968	−0.0486499	−0.0243249	−	0.999704i	$-0.507744\pi$
$677$	−856.968	−0.0486499	−0.0243249	+	0.999704i	$0.507744\pi$
$678$	−20689.1	−1.17192
$679$	6700.63	0.378713
$680$	−11558.7	−0.651846
$681$	12800.2	0.720271
$682$	−1291.98	−0.0725402
$683$	24804.0	1.38960	0.694801	−	0.719202i	$-0.255492\pi$
$683$	24804.0	1.38960	0.694801	+	0.719202i	$0.255492\pi$
$684$	947.953	0.0529911
$685$	2371.09	0.132255
$686$	1113.21	0.0619569
$687$	−3713.47	−0.206227
$688$	−22356.8	−1.23887
$689$	9638.04	0.532917
$690$	50069.2	2.76247
$691$	3475.97	0.191364	0.0956818	−	0.995412i	$-0.469497\pi$
$691$	3475.97	0.191364	0.0956818	+	0.995412i	$0.469497\pi$
$692$	10194.5	0.560024
$693$	243.852	0.0133668
$694$	7412.36	0.405431
$695$	−14528.3	−0.792937
$696$	25602.1	1.39432
$697$	6354.54	0.345330
$698$	8024.44	0.435143
$699$	−8898.85	−0.481524
$700$	−2377.36	−0.128366
$701$	17461.0	0.940790	0.470395	−	0.882456i	$-0.344111\pi$
$701$	17461.0	0.940790	0.470395	+	0.882456i	$0.344111\pi$
$702$	−15027.2	−0.807926
$703$	2250.52	0.120739
$704$	−2897.93	−0.155142
$705$	35117.8	1.87605
$706$	−33071.5	−1.76298
$707$	6973.00	0.370929
$708$	7060.81	0.374804
$709$	−18405.7	−0.974950	−0.487475	−	0.873137i	$-0.662082\pi$
$709$	−18405.7	−0.974950	−0.487475	+	0.873137i	$0.662082\pi$
$710$	39199.6	2.07202
$711$	−2332.48	−0.123031
$712$	−14303.8	−0.752891
$713$	6315.39	0.331716
$714$	−5050.56	−0.264723
$715$	6262.49	0.327558
$716$	1788.72	0.0933626
$717$	10992.5	0.572555
$718$	140.950	0.00732621
$719$	−892.380	−0.0462867	−0.0231434	−	0.999732i	$-0.507367\pi$
$719$	−892.380	−0.0462867	−0.0231434	+	0.999732i	$0.507367\pi$
$720$	3968.19	0.205397
$721$	−9369.83	−0.483982
$722$	−23051.7	−1.18822
$723$	8713.15	0.448196
$724$	−3214.48	−0.165007
$725$	−35222.2	−1.80431
$726$	2156.91	0.110262
$727$	27919.2	1.42430	0.712149	−	0.702028i	$-0.247722\pi$
$727$	27919.2	1.42430	0.712149	+	0.702028i	$0.247722\pi$
$728$	−4392.96	−0.223645
$729$	16864.5	0.856805
$730$	20665.0	1.04774
$731$	11623.9	0.588134
$732$	−494.788	−0.0249835
$733$	−4769.38	−0.240329	−0.120164	−	0.992754i	$-0.538342\pi$
$733$	−4769.38	−0.240329	−0.120164	+	0.992754i	$0.538342\pi$
$734$	−28481.8	−1.43227
$735$	4331.78	0.217388
$736$	−19321.7	−0.967673
$737$	−10771.5	−0.538364
$738$	−1613.64	−0.0804865
$739$	−5170.63	−0.257381	−0.128691	−	0.991685i	$-0.541077\pi$
$739$	−5170.63	−0.257381	−0.128691	+	0.991685i	$0.541077\pi$
$740$	−776.604	−0.0385791
$741$	−22955.3	−1.13804
$742$	6190.40	0.306276
$743$	−29407.9	−1.45205	−0.726024	−	0.687669i	$-0.758634\pi$
$743$	−29407.9	−1.45205	−0.726024	+	0.687669i	$0.758634\pi$
$744$	3526.59	0.173779
$745$	−19517.8	−0.959836
$746$	8931.26	0.438333
$747$	1846.23	0.0904286
$748$	−1127.90	−0.0551337
$749$	−10144.8	−0.494902
$750$	−2600.88	−0.126627
$751$	−16956.5	−0.823905	−0.411952	−	0.911205i	$-0.635153\pi$
$751$	−16956.5	−0.823905	−0.411952	+	0.911205i	$0.635153\pi$
$752$	−30924.9	−1.49962
$753$	−14347.5	−0.694360
$754$	30160.1	1.45672
$755$	−41290.3	−1.99034
$756$	−2321.27	−0.111672
$757$	−21322.0	−1.02373	−0.511864	−	0.859067i	$-0.671045\pi$
$757$	−21322.0	−1.02373	−0.511864	+	0.859067i	$0.671045\pi$
$758$	−8456.45	−0.405214
$759$	−10543.3	−0.504214
$760$	−33742.9	−1.61050
$761$	−23548.0	−1.12170	−0.560851	−	0.827917i	$-0.689526\pi$
$761$	−23548.0	−1.12170	−0.560851	+	0.827917i	$0.689526\pi$
$762$	−18813.4	−0.894406
$763$	4581.75	0.217392
$764$	12323.3	0.583563
$765$	−2063.17	−0.0975087
$766$	46607.1	2.19841
$767$	−17949.7	−0.845014
$768$	−19454.9	−0.914085
$769$	−17230.9	−0.808015	−0.404007	−	0.914756i	$-0.632383\pi$
$769$	−17230.9	−0.808015	−0.404007	+	0.914756i	$0.632383\pi$
$770$	4022.32	0.188252
$771$	−23058.5	−1.07709
$772$	−6777.75	−0.315980
$773$	12285.8	0.571655	0.285828	−	0.958281i	$-0.407732\pi$
$773$	12285.8	0.571655	0.285828	+	0.958281i	$0.407732\pi$
$774$	−2951.72	−0.137077
$775$	−4851.73	−0.224876
$776$	−16983.5	−0.785658
$777$	732.280	0.0338100
$778$	34427.8	1.58650
$779$	18550.6	0.853202
$780$	7921.39	0.363630
$781$	−8254.45	−0.378191
$782$	22924.3	1.04830
$783$	−34391.2	−1.56966
$784$	−3814.59	−0.173769
$785$	−11302.8	−0.513906
$786$	47370.0	2.14966
$787$	663.152	0.0300366	0.0150183	−	0.999887i	$-0.495219\pi$
$787$	663.152	0.0300366	0.0150183	+	0.999887i	$0.495219\pi$
$788$	4123.48	0.186412
$789$	28397.6	1.28135
$790$	−38474.1	−1.73272
$791$	8124.43	0.365198
$792$	−618.071	−0.0277300
$793$	1257.83	0.0563264
$794$	−23894.5	−1.06799
$795$	24088.4	1.07463
$796$	−5710.17	−0.254261
$797$	19216.3	0.854050	0.427025	−	0.904240i	$-0.359562\pi$
$797$	19216.3	0.854050	0.427025	+	0.904240i	$0.359562\pi$
$798$	−14743.9	−0.654048
$799$	16078.7	0.711921
$800$	14843.7	0.656004
$801$	−2553.17	−0.112624
$802$	858.480	0.0377980
$803$	−4351.53	−0.191236
$804$	−13624.9	−0.597651
$805$	−19661.7	−0.860851
$806$	4154.44	0.181556
$807$	−9052.65	−0.394880
$808$	−17673.8	−0.769509
$809$	−42881.8	−1.86359	−0.931794	−	0.362989i	$-0.881756\pi$
$809$	−42881.8	−1.86359	−0.931794	+	0.362989i	$0.881756\pi$
$810$	−42024.2	−1.82294
$811$	−1205.73	−0.0522058	−0.0261029	−	0.999659i	$-0.508310\pi$
$811$	−1205.73	−0.0522058	−0.0261029	+	0.999659i	$0.508310\pi$
$812$	4658.88	0.201348
$813$	−14076.0	−0.607215
$814$	679.967	0.0292787
$815$	18449.8	0.792966
$816$	17306.6	0.742466
$817$	33933.3	1.45309
$818$	4037.51	0.172577
$819$	−784.123	−0.0334548
$820$	−6401.41	−0.272618
$821$	−28577.6	−1.21482	−0.607408	−	0.794390i	$-0.707791\pi$
$821$	−28577.6	−1.21482	−0.607408	+	0.794390i	$0.707791\pi$
$822$	−2625.97	−0.111425
$823$	42524.8	1.80112	0.900561	−	0.434730i	$-0.143156\pi$
$823$	42524.8	1.80112	0.900561	+	0.434730i	$0.143156\pi$
$824$	23748.9	1.00404
$825$	8099.79	0.341816
$826$	−11528.9	−0.485643
$827$	30768.7	1.29375	0.646876	−	0.762595i	$-0.276075\pi$
$827$	30768.7	1.29375	0.646876	+	0.762595i	$0.276075\pi$
$828$	−1400.03	−0.0587614
$829$	−17583.3	−0.736661	−0.368330	−	0.929695i	$-0.620070\pi$
$829$	−17583.3	−0.736661	−0.368330	+	0.929695i	$0.620070\pi$
$830$	30453.4	1.27356
$831$	−15175.3	−0.633484
$832$	9318.48	0.388293
$833$	1983.31	0.0824942
$834$	16090.1	0.668051
$835$	52391.5	2.17136
$836$	−3292.63	−0.136218
$837$	−4737.25	−0.195631
$838$	−12900.1	−0.531773
$839$	19552.8	0.804573	0.402287	−	0.915514i	$-0.368216\pi$
$839$	19552.8	0.804573	0.402287	+	0.915514i	$0.368216\pi$
$840$	−10979.4	−0.450981
$841$	44635.5	1.83015
$842$	48391.2	1.98061
$843$	−35446.5	−1.44821
$844$	7883.72	0.321527
$845$	15224.4	0.619805
$846$	−4082.96	−0.165928
$847$	−847.000	−0.0343604
$848$	−21212.4	−0.859007
$849$	−24940.0	−1.00817
$850$	−17611.3	−0.710662
$851$	−3323.78	−0.133887
$852$	−10441.0	−0.419840
$853$	−18524.1	−0.743557	−0.371779	−	0.928321i	$-0.621252\pi$
$853$	−18524.1	−0.743557	−0.371779	+	0.928321i	$0.621252\pi$
$854$	807.889	0.0323716
$855$	−6022.95	−0.240913
$856$	25713.0	1.02670
$857$	24439.0	0.974118	0.487059	−	0.873369i	$-0.338070\pi$
$857$	24439.0	0.974118	0.487059	+	0.873369i	$0.338070\pi$
$858$	−6935.69	−0.275968
$859$	11301.4	0.448893	0.224447	−	0.974486i	$-0.427943\pi$
$859$	11301.4	0.448893	0.224447	+	0.974486i	$0.427943\pi$
$860$	−11709.6	−0.464297
$861$	6036.05	0.238918
$862$	34474.9	1.36220
$863$	−26377.3	−1.04043	−0.520217	−	0.854034i	$-0.674149\pi$
$863$	−26377.3	−1.04043	−0.520217	+	0.854034i	$0.674149\pi$
$864$	14493.4	0.570691
$865$	−64772.1	−2.54603
$866$	−56603.0	−2.22107
$867$	17986.2	0.704548
$868$	641.743	0.0250947
$869$	8101.68	0.316261
$870$	75379.5	2.93747
$871$	34636.5	1.34743
$872$	−11612.9	−0.450991
$873$	−3031.47	−0.117526
$874$	66922.1	2.59002
$875$	1021.34	0.0394601
$876$	−5504.24	−0.212296
$877$	−28425.0	−1.09446	−0.547232	−	0.836981i	$-0.684319\pi$
$877$	−28425.0	−1.09446	−0.547232	+	0.836981i	$0.684319\pi$
$878$	−44226.6	−1.69997
$879$	−10525.4	−0.403882
$880$	−13783.2	−0.527989
$881$	16897.1	0.646171	0.323086	−	0.946370i	$-0.395280\pi$
$881$	16897.1	0.646171	0.323086	+	0.946370i	$0.395280\pi$
$882$	−503.633	−0.0192270
$883$	−25538.9	−0.973331	−0.486665	−	0.873588i	$-0.661787\pi$
$883$	−25538.9	−0.973331	−0.486665	+	0.873588i	$0.661787\pi$
$884$	3626.82	0.137990
$885$	−44861.8	−1.70397
$886$	−6930.07	−0.262777
$887$	6478.48	0.245238	0.122619	−	0.992454i	$-0.460871\pi$
$887$	6478.48	0.245238	0.122619	+	0.992454i	$0.460871\pi$
$888$	−1856.04	−0.0701404
$889$	7387.85	0.278718
$890$	−42114.3	−1.58615
$891$	8849.25	0.332728
$892$	−9014.35	−0.338366
$893$	46938.1	1.75893
$894$	21615.9	0.808664
$895$	−11364.9	−0.424453
$896$	12185.5	0.454339
$897$	33902.7	1.26196
$898$	−57707.7	−2.14447
$899$	9507.86	0.352731
$900$	1075.56	0.0398355
$901$	11028.9	0.407799
$902$	5604.85	0.206897
$903$	11041.3	0.406902
$904$	−20592.3	−0.757620
$905$	20423.6	0.750171
$906$	45728.8	1.67686
$907$	−9356.17	−0.342521	−0.171260	−	0.985226i	$-0.554784\pi$
$907$	−9356.17	−0.342521	−0.171260	+	0.985226i	$0.554784\pi$
$908$	−5903.82	−0.215777
$909$	−3154.70	−0.115110
$910$	−12934.0	−0.471164
$911$	−16574.0	−0.602768	−0.301384	−	0.953503i	$-0.597449\pi$
$911$	−16574.0	−0.602768	−0.301384	+	0.953503i	$0.597449\pi$
$912$	50522.6	1.83440
$913$	−6412.73	−0.232454
$914$	−33613.6	−1.21645
$915$	3143.70	0.113582
$916$	1712.76	0.0617808
$917$	−18601.8	−0.669885
$918$	−17195.8	−0.618241
$919$	8214.92	0.294870	0.147435	−	0.989072i	$-0.452898\pi$
$919$	8214.92	0.294870	0.147435	+	0.989072i	$0.452898\pi$
$920$	49834.8	1.78588
$921$	−37767.7	−1.35123
$922$	63869.8	2.28139
$923$	26542.7	0.946548
$924$	−1071.37	−0.0381444
$925$	2553.46	0.0907646
$926$	23274.9	0.825983
$927$	4239.07	0.150193
$928$	−29088.9	−1.02898
$929$	42653.9	1.50638	0.753192	−	0.657801i	$-0.228513\pi$
$929$	42653.9	1.50638	0.753192	+	0.657801i	$0.228513\pi$
$930$	10383.2	0.366107
$931$	5789.81	0.203817
$932$	4104.41	0.144254
$933$	−46517.2	−1.63227
$934$	39571.7	1.38632
$935$	7166.25	0.250654
$936$	1987.45	0.0694035
$937$	−18484.8	−0.644473	−0.322237	−	0.946659i	$-0.604435\pi$
$937$	−18484.8	−0.644473	−0.322237	+	0.946659i	$0.604435\pi$
$938$	22246.6	0.774390
$939$	15830.7	0.550177
$940$	−16197.3	−0.562020
$941$	−7183.03	−0.248842	−0.124421	−	0.992230i	$-0.539707\pi$
$941$	−7183.03	−0.248842	−0.124421	+	0.992230i	$0.539707\pi$
$942$	12517.9	0.432966
$943$	−27397.4	−0.946109
$944$	39505.6	1.36207
$945$	14748.5	0.507692
$946$	10252.5	0.352367
$947$	41443.3	1.42210	0.711049	−	0.703143i	$-0.248221\pi$
$947$	41443.3	1.42210	0.711049	+	0.703143i	$0.248221\pi$
$948$	10247.8	0.351089
$949$	13992.6	0.478630
$950$	−51412.1	−1.75582
$951$	−53800.2	−1.83448
$952$	−5026.92	−0.171138
$953$	7981.30	0.271290	0.135645	−	0.990757i	$-0.456689\pi$
$953$	7981.30	0.271290	0.135645	+	0.990757i	$0.456689\pi$
$954$	−2800.64	−0.0950461
$955$	−78297.8	−2.65305
$956$	−5070.06	−0.171524
$957$	−15873.0	−0.536157
$958$	−27506.0	−0.927638
$959$	1031.20	0.0347227
$960$	23289.8	0.782993
$961$	−28481.3	−0.956038
$962$	−2186.48	−0.0732795
$963$	4589.65	0.153582
$964$	−4018.76	−0.134269
$965$	43063.3	1.43654
$966$	21775.3	0.725269
$967$	−18745.7	−0.623394	−0.311697	−	0.950182i	$-0.600897\pi$
$967$	−18745.7	−0.623394	−0.311697	+	0.950182i	$0.600897\pi$
$968$	2146.81	0.0712822
$969$	−26268.1	−0.870849
$970$	−50003.8	−1.65518
$971$	3096.87	0.102351	0.0511757	−	0.998690i	$-0.483703\pi$
$971$	3096.87	0.102351	0.0511757	+	0.998690i	$0.483703\pi$
$972$	2239.91	0.0739147
$973$	−6318.44	−0.208181
$974$	−8877.40	−0.292043
$975$	−26045.4	−0.855507
$976$	−2768.36	−0.0907922
$977$	19960.2	0.653618	0.326809	−	0.945090i	$-0.394027\pi$
$977$	19960.2	0.653618	0.326809	+	0.945090i	$0.394027\pi$
$978$	−20433.1	−0.668075
$979$	8868.21	0.289509
$980$	−1997.94	−0.0651243
$981$	−2072.86	−0.0674631
$982$	7249.85	0.235593
$983$	−33434.5	−1.08484	−0.542419	−	0.840108i	$-0.682491\pi$
$983$	−33434.5	−1.08484	−0.542419	+	0.840108i	$0.682491\pi$
$984$	−15299.0	−0.495645
$985$	−26199.1	−0.847485
$986$	34512.6	1.11471
$987$	15272.9	0.492544
$988$	10587.7	0.340930
$989$	−50116.1	−1.61132
$990$	−1819.76	−0.0584201
$991$	24855.2	0.796722	0.398361	−	0.917229i	$-0.369579\pi$
$991$	24855.2	0.796722	0.398361	+	0.917229i	$0.369579\pi$
$992$	−4006.88	−0.128245
$993$	−9961.63	−0.318351
$994$	17048.1	0.543996
$995$	36280.3	1.15594
$996$	−8111.43	−0.258053
$997$	−7810.38	−0.248102	−0.124051	−	0.992276i	$-0.539589\pi$
$997$	−7810.38	−0.248102	−0.124051	+	0.992276i	$0.539589\pi$
$998$	58744.5	1.86325
$999$	2493.21	0.0789607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.a.d.1.2	✓	4
3.2	odd	2		693.4.a.l.1.3		4
4.3	odd	2		1232.4.a.s.1.4		4
5.4	even	2		1925.4.a.p.1.3		4
7.6	odd	2		539.4.a.g.1.2		4
11.10	odd	2		847.4.a.d.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.2	✓	4	1.1	even	1	trivial
539.4.a.g.1.2		4	7.6	odd	2
693.4.a.l.1.3		4	3.2	odd	2
847.4.a.d.1.3		4	11.10	odd	2
1232.4.a.s.1.4		4	4.3	odd	2
1925.4.a.p.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)

\(f(q)\)	\(=\)	\(q-3.24550 q^{2} -5.49244 q^{3} +2.53327 q^{4} -16.0955 q^{5} +17.8257 q^{6} -7.00000 q^{7} +17.7423 q^{8} +3.16692 q^{9} +O(q^{10})\)	\(q-3.24550 q^{2} -5.49244 q^{3} +2.53327 q^{4} -16.0955 q^{5} +17.8257 q^{6} -7.00000 q^{7} +17.7423 q^{8} +3.16692 q^{9} +52.2379 q^{10} -11.0000 q^{11} -13.9139 q^{12} +35.3712 q^{13} +22.7185 q^{14} +88.4036 q^{15} -77.8487 q^{16} +40.4757 q^{17} -10.2782 q^{18} +118.159 q^{19} -40.7743 q^{20} +38.4471 q^{21} +35.7005 q^{22} -174.510 q^{23} -97.4484 q^{24} +134.065 q^{25} -114.797 q^{26} +130.902 q^{27} -17.7329 q^{28} -262.725 q^{29} -286.914 q^{30} -36.1894 q^{31} +110.720 q^{32} +60.4169 q^{33} -131.364 q^{34} +112.668 q^{35} +8.02266 q^{36} +19.0464 q^{37} -383.487 q^{38} -194.274 q^{39} -285.571 q^{40} +156.996 q^{41} -124.780 q^{42} +287.182 q^{43} -27.8660 q^{44} -50.9731 q^{45} +566.371 q^{46} +397.244 q^{47} +427.580 q^{48} +49.0000 q^{49} -435.108 q^{50} -222.311 q^{51} +89.6049 q^{52} +272.483 q^{53} -424.842 q^{54} +177.050 q^{55} -124.196 q^{56} -648.984 q^{57} +852.674 q^{58} -507.466 q^{59} +223.950 q^{60} +35.5608 q^{61} +117.453 q^{62} -22.1684 q^{63} +263.448 q^{64} -569.317 q^{65} -196.083 q^{66} +979.229 q^{67} +102.536 q^{68} +958.484 q^{69} -365.666 q^{70} +750.404 q^{71} +56.1882 q^{72} +395.594 q^{73} -61.8152 q^{74} -736.344 q^{75} +299.330 q^{76} +77.0000 q^{77} +630.517 q^{78} -736.516 q^{79} +1253.01 q^{80} -804.477 q^{81} -509.531 q^{82} +582.975 q^{83} +97.3970 q^{84} -651.477 q^{85} -932.050 q^{86} +1443.00 q^{87} -195.165 q^{88} -806.201 q^{89} +165.433 q^{90} -247.598 q^{91} -442.081 q^{92} +198.768 q^{93} -1289.26 q^{94} -1901.84 q^{95} -608.123 q^{96} -957.232 q^{97} -159.030 q^{98} -34.8361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 14 q^{3} + 26 q^{4} + 10 q^{5} + 14 q^{6} - 28 q^{7} - 18 q^{8} + 76 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 + 14 * q^3 + 26 * q^4 + 10 * q^5 + 14 * q^6 - 28 * q^7 - 18 * q^8 + 76 * q^9	\( 4 q - 2 q^{2} + 14 q^{3} + 26 q^{4} + 10 q^{5} + 14 q^{6} - 28 q^{7} - 18 q^{8} + 76 q^{9} - 2 q^{10} - 44 q^{11} + 70 q^{12} + 58 q^{13} + 14 q^{14} + 284 q^{15} + 2 q^{16} + 4 q^{17} - 62 q^{18} + 258 q^{19} + 182 q^{20} - 98 q^{21} + 22 q^{22} + 8 q^{23} - 498 q^{24} + 80 q^{25} - 482 q^{26} + 428 q^{27} - 182 q^{28} - 396 q^{29} - 628 q^{30} - 56 q^{31} + 134 q^{32} - 154 q^{33} + 472 q^{34} - 70 q^{35} - 418 q^{36} + 84 q^{37} - 942 q^{38} - 412 q^{39} - 1026 q^{40} + 52 q^{41} - 98 q^{42} + 408 q^{43} - 286 q^{44} + 826 q^{45} + 368 q^{46} + 8 q^{47} + 982 q^{48} + 196 q^{49} - 1642 q^{50} - 388 q^{51} + 2030 q^{52} + 624 q^{53} + 92 q^{54} - 110 q^{55} + 126 q^{56} + 48 q^{57} + 864 q^{58} - 238 q^{59} + 1420 q^{60} - 162 q^{61} + 688 q^{62} - 532 q^{63} - 902 q^{64} - 32 q^{65} - 154 q^{66} + 1340 q^{67} - 1384 q^{68} + 2416 q^{69} + 14 q^{70} + 1788 q^{71} - 2622 q^{72} + 1456 q^{73} + 996 q^{74} - 806 q^{75} + 3042 q^{76} + 308 q^{77} - 2632 q^{78} - 1324 q^{79} + 2342 q^{80} + 1444 q^{81} + 1984 q^{82} + 450 q^{83} - 490 q^{84} - 1736 q^{85} - 4380 q^{86} + 588 q^{87} + 198 q^{88} - 3072 q^{89} - 218 q^{90} - 406 q^{91} + 544 q^{92} - 1264 q^{93} - 1696 q^{94} + 24 q^{95} + 862 q^{96} - 652 q^{97} - 98 q^{98} - 836 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 + 14 * q^3 + 26 * q^4 + 10 * q^5 + 14 * q^6 - 28 * q^7 - 18 * q^8 + 76 * q^9 - 2 * q^10 - 44 * q^11 + 70 * q^12 + 58 * q^13 + 14 * q^14 + 284 * q^15 + 2 * q^16 + 4 * q^17 - 62 * q^18 + 258 * q^19 + 182 * q^20 - 98 * q^21 + 22 * q^22 + 8 * q^23 - 498 * q^24 + 80 * q^25 - 482 * q^26 + 428 * q^27 - 182 * q^28 - 396 * q^29 - 628 * q^30 - 56 * q^31 + 134 * q^32 - 154 * q^33 + 472 * q^34 - 70 * q^35 - 418 * q^36 + 84 * q^37 - 942 * q^38 - 412 * q^39 - 1026 * q^40 + 52 * q^41 - 98 * q^42 + 408 * q^43 - 286 * q^44 + 826 * q^45 + 368 * q^46 + 8 * q^47 + 982 * q^48 + 196 * q^49 - 1642 * q^50 - 388 * q^51 + 2030 * q^52 + 624 * q^53 + 92 * q^54 - 110 * q^55 + 126 * q^56 + 48 * q^57 + 864 * q^58 - 238 * q^59 + 1420 * q^60 - 162 * q^61 + 688 * q^62 - 532 * q^63 - 902 * q^64 - 32 * q^65 - 154 * q^66 + 1340 * q^67 - 1384 * q^68 + 2416 * q^69 + 14 * q^70 + 1788 * q^71 - 2622 * q^72 + 1456 * q^73 + 996 * q^74 - 806 * q^75 + 3042 * q^76 + 308 * q^77 - 2632 * q^78 - 1324 * q^79 + 2342 * q^80 + 1444 * q^81 + 1984 * q^82 + 450 * q^83 - 490 * q^84 - 1736 * q^85 - 4380 * q^86 + 588 * q^87 + 198 * q^88 - 3072 * q^89 - 218 * q^90 - 406 * q^91 + 544 * q^92 - 1264 * q^93 - 1696 * q^94 + 24 * q^95 + 862 * q^96 - 652 * q^97 - 98 * q^98 - 836 * q^99

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