LMFDB - Embedded newform 338.4.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	338.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:	$19.9426455819$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1859472.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 59x^{2} + 60x + 792 $ x^4 - 2x^3 - 59x^2 + 60*x + 792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$6.87513$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +7.87513 q^{3} +4.00000 q^{4} -3.30629 q^{5} -15.7503 q^{6} -31.3904 q^{7} -8.00000 q^{8} +35.0177 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +7.87513 q^{3} +4.00000 q^{4} -3.30629 q^{5} -15.7503 q^{6} -31.3904 q^{7} -8.00000 q^{8} +35.0177 q^{9} +6.61258 q^{10} +15.6107 q^{11} +31.5005 q^{12} +62.7808 q^{14} -26.0375 q^{15} +16.0000 q^{16} -107.728 q^{17} -70.0354 q^{18} +60.7786 q^{19} -13.2252 q^{20} -247.203 q^{21} -31.2213 q^{22} -124.308 q^{23} -63.0011 q^{24} -114.068 q^{25} +63.1404 q^{27} -125.562 q^{28} +58.2002 q^{29} +52.0749 q^{30} -200.934 q^{31} -32.0000 q^{32} +122.936 q^{33} +215.455 q^{34} +103.786 q^{35} +140.071 q^{36} -105.014 q^{37} -121.557 q^{38} +26.4503 q^{40} +221.080 q^{41} +494.407 q^{42} +112.068 q^{43} +62.4426 q^{44} -115.779 q^{45} +248.616 q^{46} -512.102 q^{47} +126.002 q^{48} +642.357 q^{49} +228.137 q^{50} -848.370 q^{51} -221.755 q^{53} -126.281 q^{54} -51.6133 q^{55} +251.123 q^{56} +478.639 q^{57} -116.400 q^{58} +556.924 q^{59} -104.150 q^{60} -458.209 q^{61} +401.868 q^{62} -1099.22 q^{63} +64.0000 q^{64} -245.872 q^{66} -529.740 q^{67} -430.911 q^{68} -978.942 q^{69} -207.571 q^{70} +67.5508 q^{71} -280.142 q^{72} +104.504 q^{73} +210.028 q^{74} -898.304 q^{75} +243.114 q^{76} -490.025 q^{77} +611.085 q^{79} -52.9006 q^{80} -448.239 q^{81} -442.161 q^{82} -491.565 q^{83} -988.814 q^{84} +356.179 q^{85} -224.136 q^{86} +458.334 q^{87} -124.885 q^{88} +377.155 q^{89} +231.557 q^{90} -497.232 q^{92} -1582.38 q^{93} +1024.20 q^{94} -200.951 q^{95} -252.004 q^{96} -573.312 q^{97} -1284.71 q^{98} +546.649 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 6 q^{3} + 16 q^{4} + 4 q^{5} - 12 q^{6} - 20 q^{7} - 32 q^{8} + 22 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 + 6 * q^3 + 16 * q^4 + 4 * q^5 - 12 * q^6 - 20 * q^7 - 32 * q^8 + 22 * q^9	\( 4 q - 8 q^{2} + 6 q^{3} + 16 q^{4} + 4 q^{5} - 12 q^{6} - 20 q^{7} - 32 q^{8} + 22 q^{9} - 8 q^{10} - 36 q^{11} + 24 q^{12} + 40 q^{14} - 164 q^{15} + 64 q^{16} + 112 q^{17} - 44 q^{18} - 144 q^{19} + 16 q^{20} - 344 q^{21} + 72 q^{22} - 230 q^{23} - 48 q^{24} + 90 q^{25} + 234 q^{27} - 80 q^{28} - 32 q^{29} + 328 q^{30} - 532 q^{31} - 128 q^{32} + 600 q^{33} - 224 q^{34} + 128 q^{35} + 88 q^{36} - 340 q^{37} + 288 q^{38} - 32 q^{40} - 352 q^{41} + 688 q^{42} - 114 q^{43} - 144 q^{44} - 596 q^{45} + 460 q^{46} - 1284 q^{47} + 96 q^{48} - 110 q^{49} - 180 q^{50} - 650 q^{51} + 18 q^{53} - 468 q^{54} - 1248 q^{55} + 160 q^{56} - 144 q^{57} + 64 q^{58} + 988 q^{59} - 656 q^{60} - 900 q^{61} + 1064 q^{62} - 1160 q^{63} + 256 q^{64} - 1200 q^{66} - 1672 q^{67} + 448 q^{68} - 2402 q^{69} - 256 q^{70} - 528 q^{71} - 176 q^{72} - 1624 q^{73} + 680 q^{74} - 862 q^{75} - 576 q^{76} - 1170 q^{77} + 392 q^{79} + 64 q^{80} - 1868 q^{81} + 704 q^{82} - 152 q^{83} - 1376 q^{84} - 940 q^{85} + 228 q^{86} - 1598 q^{87} + 288 q^{88} + 816 q^{89} + 1192 q^{90} - 920 q^{92} + 176 q^{93} + 2568 q^{94} - 816 q^{95} - 192 q^{96} + 192 q^{97} + 220 q^{98} + 1656 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 + 6 * q^3 + 16 * q^4 + 4 * q^5 - 12 * q^6 - 20 * q^7 - 32 * q^8 + 22 * q^9 - 8 * q^10 - 36 * q^11 + 24 * q^12 + 40 * q^14 - 164 * q^15 + 64 * q^16 + 112 * q^17 - 44 * q^18 - 144 * q^19 + 16 * q^20 - 344 * q^21 + 72 * q^22 - 230 * q^23 - 48 * q^24 + 90 * q^25 + 234 * q^27 - 80 * q^28 - 32 * q^29 + 328 * q^30 - 532 * q^31 - 128 * q^32 + 600 * q^33 - 224 * q^34 + 128 * q^35 + 88 * q^36 - 340 * q^37 + 288 * q^38 - 32 * q^40 - 352 * q^41 + 688 * q^42 - 114 * q^43 - 144 * q^44 - 596 * q^45 + 460 * q^46 - 1284 * q^47 + 96 * q^48 - 110 * q^49 - 180 * q^50 - 650 * q^51 + 18 * q^53 - 468 * q^54 - 1248 * q^55 + 160 * q^56 - 144 * q^57 + 64 * q^58 + 988 * q^59 - 656 * q^60 - 900 * q^61 + 1064 * q^62 - 1160 * q^63 + 256 * q^64 - 1200 * q^66 - 1672 * q^67 + 448 * q^68 - 2402 * q^69 - 256 * q^70 - 528 * q^71 - 176 * q^72 - 1624 * q^73 + 680 * q^74 - 862 * q^75 - 576 * q^76 - 1170 * q^77 + 392 * q^79 + 64 * q^80 - 1868 * q^81 + 704 * q^82 - 152 * q^83 - 1376 * q^84 - 940 * q^85 + 228 * q^86 - 1598 * q^87 + 288 * q^88 + 816 * q^89 + 1192 * q^90 - 920 * q^92 + 176 * q^93 + 2568 * q^94 - 816 * q^95 - 192 * q^96 + 192 * q^97 + 220 * q^98 + 1656 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	7.87513	1.51557	0.757785	−	0.652504i	$-0.226282\pi$
$3$	7.87513	1.51557	0.757785	+	0.652504i	$0.226282\pi$
$4$	4.00000	0.500000
$5$	−3.30629	−0.295723	−0.147862	−	0.989008i	$-0.547239\pi$
$5$	−3.30629	−0.295723	−0.147862	+	0.989008i	$0.547239\pi$
$6$	−15.7503	−1.07167
$7$	−31.3904	−1.69492	−0.847461	−	0.530858i	$-0.821870\pi$
$7$	−31.3904	−1.69492	−0.847461	+	0.530858i	$0.821870\pi$
$8$	−8.00000	−0.353553
$9$	35.0177	1.29695
$10$	6.61258	0.209108
$11$	15.6107	0.427890	0.213945	−	0.976846i	$-0.431369\pi$
$11$	15.6107	0.427890	0.213945	+	0.976846i	$0.431369\pi$
$12$	31.5005	0.757785
$13$	0	0
$13$	0	0
$14$	62.7808	1.19849
$15$	−26.0375	−0.448189
$16$	16.0000	0.250000
$17$	−107.728	−1.53693	−0.768465	−	0.639892i	$-0.778979\pi$
$17$	−107.728	−1.53693	−0.768465	+	0.639892i	$0.778979\pi$
$18$	−70.0354	−0.917083
$19$	60.7786	0.733872	0.366936	−	0.930246i	$-0.380407\pi$
$19$	60.7786	0.733872	0.366936	+	0.930246i	$0.380407\pi$
$20$	−13.2252	−0.147862
$21$	−247.203	−2.56877
$22$	−31.2213	−0.302564
$23$	−124.308	−1.12696	−0.563479	−	0.826131i	$-0.690537\pi$
$23$	−124.308	−1.12696	−0.563479	+	0.826131i	$0.690537\pi$
$24$	−63.0011	−0.535835
$25$	−114.068	−0.912548
$26$	0	0
$27$	63.1404	0.450051
$28$	−125.562	−0.847461
$29$	58.2002	0.372672	0.186336	−	0.982486i	$-0.440339\pi$
$29$	58.2002	0.372672	0.186336	+	0.982486i	$0.440339\pi$
$30$	52.0749	0.316918
$31$	−200.934	−1.16416	−0.582078	−	0.813133i	$-0.697760\pi$
$31$	−200.934	−1.16416	−0.582078	+	0.813133i	$0.697760\pi$
$32$	−32.0000	−0.176777
$33$	122.936	0.648497
$34$	215.455	1.08677
$35$	103.786	0.501228
$36$	140.071	0.648476
$37$	−105.014	−0.466599	−0.233300	−	0.972405i	$-0.574952\pi$
$37$	−105.014	−0.466599	−0.233300	+	0.972405i	$0.574952\pi$
$38$	−121.557	−0.518926
$39$	0	0
$40$	26.4503	0.104554
$41$	221.080	0.842120	0.421060	−	0.907033i	$-0.361658\pi$
$41$	221.080	0.842120	0.421060	+	0.907033i	$0.361658\pi$
$42$	494.407	1.81640
$43$	112.068	0.397446	0.198723	−	0.980056i	$-0.436321\pi$
$43$	112.068	0.397446	0.198723	+	0.980056i	$0.436321\pi$
$44$	62.4426	0.213945
$45$	−115.779	−0.383539
$46$	248.616	0.796879
$47$	−512.102	−1.58931	−0.794657	−	0.607058i	$-0.792349\pi$
$47$	−512.102	−1.58931	−0.794657	+	0.607058i	$0.792349\pi$
$48$	126.002	0.378892
$49$	642.357	1.87276
$50$	228.137	0.645269
$51$	−848.370	−2.32932
$52$	0	0
$53$	−221.755	−0.574724	−0.287362	−	0.957822i	$-0.592778\pi$
$53$	−221.755	−0.574724	−0.287362	+	0.957822i	$0.592778\pi$
$54$	−126.281	−0.318234
$55$	−51.6133	−0.126537
$56$	251.123	0.599245
$57$	478.639	1.11223
$58$	−116.400	−0.263519
$59$	556.924	1.22890	0.614452	−	0.788954i	$-0.289377\pi$
$59$	556.924	1.22890	0.614452	+	0.788954i	$0.289377\pi$
$60$	−104.150	−0.224095
$61$	−458.209	−0.961765	−0.480882	−	0.876785i	$-0.659684\pi$
$61$	−458.209	−0.961765	−0.480882	+	0.876785i	$0.659684\pi$
$62$	401.868	0.823182
$63$	−1099.22	−2.19823
$64$	64.0000	0.125000
$65$	0	0
$66$	−245.872	−0.458557
$67$	−529.740	−0.965941	−0.482971	−	0.875636i	$-0.660442\pi$
$67$	−529.740	−0.965941	−0.482971	+	0.875636i	$0.660442\pi$
$68$	−430.911	−0.768465
$69$	−978.942	−1.70798
$70$	−207.571	−0.354422
$71$	67.5508	0.112913	0.0564564	−	0.998405i	$-0.482020\pi$
$71$	67.5508	0.112913	0.0564564	+	0.998405i	$0.482020\pi$
$72$	−280.142	−0.458542
$73$	104.504	0.167552	0.0837759	−	0.996485i	$-0.473302\pi$
$73$	104.504	0.167552	0.0837759	+	0.996485i	$0.473302\pi$
$74$	210.028	0.329935
$75$	−898.304	−1.38303
$76$	243.114	0.366936
$77$	−490.025	−0.725240
$78$	0	0
$79$	611.085	0.870284	0.435142	−	0.900362i	$-0.356698\pi$
$79$	611.085	0.870284	0.435142	+	0.900362i	$0.356698\pi$
$80$	−52.9006	−0.0739308
$81$	−448.239	−0.614868
$82$	−442.161	−0.595469
$83$	−491.565	−0.650075	−0.325038	−	0.945701i	$-0.605377\pi$
$83$	−491.565	−0.650075	−0.325038	+	0.945701i	$0.605377\pi$
$84$	−988.814	−1.28439
$85$	356.179	0.454506
$86$	−224.136	−0.281037
$87$	458.334	0.564811
$88$	−124.885	−0.151282
$89$	377.155	0.449195	0.224598	−	0.974452i	$-0.427893\pi$
$89$	377.155	0.449195	0.224598	+	0.974452i	$0.427893\pi$
$90$	231.557	0.271203
$91$	0	0
$92$	−497.232	−0.563479
$93$	−1582.38	−1.76436
$94$	1024.20	1.12382
$95$	−200.951	−0.217023
$96$	−252.004	−0.267917
$97$	−573.312	−0.600113	−0.300057	−	0.953921i	$-0.597006\pi$
$97$	−573.312	−0.600113	−0.300057	+	0.953921i	$0.597006\pi$
$98$	−1284.71	−1.32424
$99$	546.649	0.554953
$100$	−456.274	−0.456274
$101$	955.502	0.941347	0.470673	−	0.882307i	$-0.344011\pi$
$101$	955.502	0.941347	0.470673	+	0.882307i	$0.344011\pi$
$102$	1696.74	1.64708
$103$	1478.96	1.41481	0.707407	−	0.706806i	$-0.249865\pi$
$103$	1478.96	1.41481	0.707407	+	0.706806i	$0.249865\pi$
$104$	0	0
$105$	817.326	0.759646
$106$	443.510	0.406391
$107$	213.116	0.192548	0.0962741	−	0.995355i	$-0.469307\pi$
$107$	213.116	0.192548	0.0962741	+	0.995355i	$0.469307\pi$
$108$	252.562	0.225026
$109$	−84.4197	−0.0741829	−0.0370915	−	0.999312i	$-0.511809\pi$
$109$	−84.4197	−0.0741829	−0.0370915	+	0.999312i	$0.511809\pi$
$110$	103.227	0.0894753
$111$	−826.998	−0.707164
$112$	−502.246	−0.423730
$113$	−786.418	−0.654690	−0.327345	−	0.944905i	$-0.606154\pi$
$113$	−786.418	−0.654690	−0.327345	+	0.944905i	$0.606154\pi$
$114$	−957.278	−0.786468
$115$	410.998	0.333268
$116$	232.801	0.186336
$117$	0	0
$118$	−1113.85	−0.868966
$119$	3381.61	2.60498
$120$	208.300	0.158459
$121$	−1087.31	−0.816910
$122$	916.418	0.680070
$123$	1741.04	1.27629
$124$	−803.736	−0.582078
$125$	790.429	0.565585
$126$	2198.44	1.55438
$127$	−732.455	−0.511770	−0.255885	−	0.966707i	$-0.582367\pi$
$127$	−732.455	−0.511770	−0.255885	+	0.966707i	$0.582367\pi$
$128$	−128.000	−0.0883883
$129$	882.549	0.602357
$130$	0	0
$131$	1862.41	1.24213	0.621066	−	0.783758i	$-0.286700\pi$
$131$	1862.41	1.24213	0.621066	+	0.783758i	$0.286700\pi$
$132$	491.744	0.324249
$133$	−1907.86	−1.24385
$134$	1059.48	0.683024
$135$	−208.760	−0.133091
$136$	861.822	0.543387
$137$	887.456	0.553434	0.276717	−	0.960951i	$-0.410754\pi$
$137$	887.456	0.553434	0.276717	+	0.960951i	$0.410754\pi$
$138$	1957.88	1.20773
$139$	−1452.12	−0.886095	−0.443047	−	0.896498i	$-0.646103\pi$
$139$	−1452.12	−0.886095	−0.443047	+	0.896498i	$0.646103\pi$
$140$	415.143	0.250614
$141$	−4032.87	−2.40872
$142$	−135.102	−0.0798414
$143$	0	0
$144$	560.283	0.324238
$145$	−192.427	−0.110208
$146$	−209.008	−0.118477
$147$	5058.64	2.83830
$148$	−420.055	−0.233300
$149$	1542.70	0.848208	0.424104	−	0.905613i	$-0.360589\pi$
$149$	1542.70	0.848208	0.424104	+	0.905613i	$0.360589\pi$
$150$	1796.61	0.977950
$151$	−1623.81	−0.875125	−0.437562	−	0.899188i	$-0.644158\pi$
$151$	−1623.81	−0.875125	−0.437562	+	0.899188i	$0.644158\pi$
$152$	−486.229	−0.259463
$153$	−3772.38	−1.99332
$154$	980.050	0.512822
$155$	664.346	0.344268
$156$	0	0
$157$	−2795.18	−1.42089	−0.710445	−	0.703753i	$-0.751506\pi$
$157$	−2795.18	−1.42089	−0.710445	+	0.703753i	$0.751506\pi$
$158$	−1222.17	−0.615383
$159$	−1746.35	−0.871034
$160$	105.801	0.0522770
$161$	3902.08	1.91010
$162$	896.477	0.434777
$163$	−289.696	−0.139207	−0.0696036	−	0.997575i	$-0.522173\pi$
$163$	−289.696	−0.139207	−0.0696036	+	0.997575i	$0.522173\pi$
$164$	884.321	0.421060
$165$	−406.462	−0.191776
$166$	983.130	0.459673
$167$	2328.12	1.07877	0.539387	−	0.842058i	$-0.318656\pi$
$167$	2328.12	1.07877	0.539387	+	0.842058i	$0.318656\pi$
$168$	1977.63	0.908198
$169$	0	0
$170$	−712.357	−0.321384
$171$	2128.33	0.951796
$172$	448.271	0.198723
$173$	1063.88	0.467546	0.233773	−	0.972291i	$-0.424893\pi$
$173$	1063.88	0.467546	0.233773	+	0.972291i	$0.424893\pi$
$174$	−916.668	−0.399382
$175$	3580.65	1.54670
$176$	249.771	0.106973
$177$	4385.85	1.86249
$178$	−754.310	−0.317629
$179$	2471.00	1.03180	0.515898	−	0.856650i	$-0.327459\pi$
$179$	2471.00	1.03180	0.515898	+	0.856650i	$0.327459\pi$
$180$	−463.114	−0.191769
$181$	2196.50	0.902014	0.451007	−	0.892520i	$-0.351065\pi$
$181$	2196.50	0.902014	0.451007	+	0.892520i	$0.351065\pi$
$182$	0	0
$183$	−3608.46	−1.45762
$184$	994.464	0.398440
$185$	347.206	0.137984
$186$	3164.76	1.24759
$187$	−1681.70	−0.657637
$188$	−2048.41	−0.794657
$189$	−1982.00	−0.762802
$190$	401.903	0.153458
$191$	2051.20	0.777068	0.388534	−	0.921434i	$-0.372982\pi$
$191$	2051.20	0.777068	0.388534	+	0.921434i	$0.372982\pi$
$192$	504.008	0.189446
$193$	3753.14	1.39978	0.699889	−	0.714252i	$-0.253233\pi$
$193$	3753.14	1.39978	0.699889	+	0.714252i	$0.253233\pi$
$194$	1146.62	0.424344
$195$	0	0
$196$	2569.43	0.936380
$197$	746.515	0.269985	0.134992	−	0.990847i	$-0.456899\pi$
$197$	746.515	0.269985	0.134992	+	0.990847i	$0.456899\pi$
$198$	−1093.30	−0.392411
$199$	−2657.29	−0.946585	−0.473293	−	0.880905i	$-0.656935\pi$
$199$	−2657.29	−0.946585	−0.473293	+	0.880905i	$0.656935\pi$
$200$	912.548	0.322634
$201$	−4171.77	−1.46395
$202$	−1911.00	−0.665633
$203$	−1826.93	−0.631651
$204$	−3393.48	−1.16466
$205$	−730.955	−0.249035
$206$	−2957.91	−1.00043
$207$	−4352.98	−1.46161
$208$	0	0
$209$	948.794	0.314016
$210$	−1634.65	−0.537151
$211$	−1507.34	−0.491800	−0.245900	−	0.969295i	$-0.579083\pi$
$211$	−1507.34	−0.491800	−0.245900	+	0.969295i	$0.579083\pi$
$212$	−887.020	−0.287362
$213$	531.972	0.171127
$214$	−426.231	−0.136152
$215$	−370.528	−0.117534
$216$	−505.124	−0.159117
$217$	6307.40	1.97315
$218$	168.839	0.0524553
$219$	822.984	0.253937
$220$	−206.453	−0.0632686
$221$	0	0
$222$	1654.00	0.500040
$223$	−1979.06	−0.594294	−0.297147	−	0.954832i	$-0.596035\pi$
$223$	−1979.06	−0.594294	−0.297147	+	0.954832i	$0.596035\pi$
$224$	1004.49	0.299623
$225$	−3994.42	−1.18353
$226$	1572.84	0.462936
$227$	−2110.79	−0.617172	−0.308586	−	0.951197i	$-0.599856\pi$
$227$	−2110.79	−0.617172	−0.308586	+	0.951197i	$0.599856\pi$
$228$	1914.56	0.556117
$229$	−5326.76	−1.53713	−0.768563	−	0.639774i	$-0.779028\pi$
$229$	−5326.76	−1.53713	−0.768563	+	0.639774i	$0.779028\pi$
$230$	−821.996	−0.235656
$231$	−3859.01	−1.09915
$232$	−465.601	−0.131760
$233$	−148.949	−0.0418797	−0.0209398	−	0.999781i	$-0.506666\pi$
$233$	−148.949	−0.0418797	−0.0209398	+	0.999781i	$0.506666\pi$
$234$	0	0
$235$	1693.16	0.469997
$236$	2227.70	0.614452
$237$	4812.37	1.31898
$238$	−6763.23	−1.84200
$239$	−3081.18	−0.833912	−0.416956	−	0.908927i	$-0.636903\pi$
$239$	−3081.18	−0.833912	−0.416956	+	0.908927i	$0.636903\pi$
$240$	−416.599	−0.112047
$241$	−2570.29	−0.687001	−0.343500	−	0.939153i	$-0.611613\pi$
$241$	−2570.29	−0.687001	−0.343500	+	0.939153i	$0.611613\pi$
$242$	2174.61	0.577643
$243$	−5234.73	−1.38193
$244$	−1832.84	−0.480882
$245$	−2123.82	−0.553819
$246$	−3482.07	−0.902475
$247$	0	0
$248$	1607.47	0.411591
$249$	−3871.14	−0.985234
$250$	−1580.86	−0.399929
$251$	−2043.76	−0.513949	−0.256974	−	0.966418i	$-0.582726\pi$
$251$	−2043.76	−0.513949	−0.256974	+	0.966418i	$0.582726\pi$
$252$	−4396.88	−1.09912
$253$	−1940.53	−0.482214
$254$	1464.91	0.361876
$255$	2804.95	0.688836
$256$	256.000	0.0625000
$257$	7718.93	1.87352	0.936758	−	0.349977i	$-0.113811\pi$
$257$	7718.93	1.87352	0.936758	+	0.349977i	$0.113811\pi$
$258$	−1765.10	−0.425931
$259$	3296.43	0.790849
$260$	0	0
$261$	2038.04	0.483338
$262$	−3724.82	−0.878320
$263$	1484.52	0.348059	0.174030	−	0.984740i	$-0.444321\pi$
$263$	1484.52	0.348059	0.174030	+	0.984740i	$0.444321\pi$
$264$	−983.488	−0.229278
$265$	733.185	0.169959
$266$	3815.73	0.879538
$267$	2970.15	0.680787
$268$	−2118.96	−0.482971
$269$	−2266.54	−0.513730	−0.256865	−	0.966447i	$-0.582690\pi$
$269$	−2266.54	−0.513730	−0.256865	+	0.966447i	$0.582690\pi$
$270$	417.521	0.0941093
$271$	5267.43	1.18071	0.590357	−	0.807142i	$-0.298987\pi$
$271$	5267.43	1.18071	0.590357	+	0.807142i	$0.298987\pi$
$272$	−1723.64	−0.384232
$273$	0	0
$274$	−1774.91	−0.391337
$275$	−1780.68	−0.390470
$276$	−3915.77	−0.853991
$277$	−4815.44	−1.04452	−0.522260	−	0.852787i	$-0.674911\pi$
$277$	−4815.44	−1.04452	−0.522260	+	0.852787i	$0.674911\pi$
$278$	2904.24	0.626564
$279$	−7036.25	−1.50985
$280$	−830.285	−0.177211
$281$	−1983.72	−0.421134	−0.210567	−	0.977579i	$-0.567531\pi$
$281$	−1983.72	−0.421134	−0.210567	+	0.977579i	$0.567531\pi$
$282$	8065.74	1.70322
$283$	−827.101	−0.173732	−0.0868658	−	0.996220i	$-0.527685\pi$
$283$	−827.101	−0.173732	−0.0868658	+	0.996220i	$0.527685\pi$
$284$	270.203	0.0564564
$285$	−1582.52	−0.328913
$286$	0	0
$287$	−6939.80	−1.42733
$288$	−1120.57	−0.229271
$289$	6692.26	1.36215
$290$	384.853	0.0779288
$291$	−4514.91	−0.909514
$292$	418.017	0.0837759
$293$	1186.60	0.236593	0.118296	−	0.992978i	$-0.462257\pi$
$293$	1186.60	0.236593	0.118296	+	0.992978i	$0.462257\pi$
$294$	−10117.3	−2.00698
$295$	−1841.35	−0.363416
$296$	840.111	0.164968
$297$	985.664	0.192573
$298$	−3085.40	−0.599774
$299$	0	0
$300$	−3593.22	−0.691515
$301$	−3517.85	−0.673640
$302$	3247.62	0.618807
$303$	7524.71	1.42668
$304$	972.457	0.183468
$305$	1514.97	0.284416
$306$	7544.75	1.40949
$307$	6191.17	1.15097	0.575486	−	0.817811i	$-0.304813\pi$
$307$	6191.17	1.15097	0.575486	+	0.817811i	$0.304813\pi$
$308$	−1960.10	−0.362620
$309$	11647.0	2.14425
$310$	−1328.69	−0.243434
$311$	−10609.3	−1.93440	−0.967198	−	0.254024i	$-0.918246\pi$
$311$	−10609.3	−1.93440	−0.967198	+	0.254024i	$0.918246\pi$
$312$	0	0
$313$	9833.23	1.77574	0.887871	−	0.460093i	$-0.152184\pi$
$313$	9833.23	1.77574	0.887871	+	0.460093i	$0.152184\pi$
$314$	5590.36	1.00472
$315$	3634.34	0.650069
$316$	2444.34	0.435142
$317$	4674.74	0.828263	0.414131	−	0.910217i	$-0.364085\pi$
$317$	4674.74	0.828263	0.414131	+	0.910217i	$0.364085\pi$
$318$	3492.70	0.615914
$319$	908.543	0.159463
$320$	−211.602	−0.0369654
$321$	1678.31	0.291820
$322$	−7804.15	−1.35065
$323$	−6547.54	−1.12791
$324$	−1792.95	−0.307434
$325$	0	0
$326$	579.393	0.0984344
$327$	−664.816	−0.112429
$328$	−1768.64	−0.297735
$329$	16075.1	2.69376
$330$	812.924	0.135606
$331$	−10413.4	−1.72922	−0.864608	−	0.502447i	$-0.832433\pi$
$331$	−10413.4	−1.72922	−0.864608	+	0.502447i	$0.832433\pi$
$332$	−1966.26	−0.325038
$333$	−3677.34	−0.605157
$334$	−4656.24	−0.762809
$335$	1751.47	0.285651
$336$	−3955.26	−0.642193
$337$	4873.47	0.787759	0.393879	−	0.919162i	$-0.371133\pi$
$337$	4873.47	0.787759	0.393879	+	0.919162i	$0.371133\pi$
$338$	0	0
$339$	−6193.15	−0.992229
$340$	1424.71	0.227253
$341$	−3136.71	−0.498131
$342$	−4256.65	−0.673021
$343$	−9396.92	−1.47926
$344$	−896.542	−0.140518
$345$	3236.66	0.505090
$346$	−2127.77	−0.330605
$347$	12121.7	1.87529	0.937645	−	0.347593i	$-0.113001\pi$
$347$	12121.7	1.87529	0.937645	+	0.347593i	$0.113001\pi$
$348$	1833.34	0.282406
$349$	−8872.17	−1.36079	−0.680396	−	0.732845i	$-0.738192\pi$
$349$	−8872.17	−1.36079	−0.680396	+	0.732845i	$0.738192\pi$
$350$	−7161.31	−1.09368
$351$	0	0
$352$	−499.541	−0.0756410
$353$	4541.69	0.684787	0.342393	−	0.939557i	$-0.388762\pi$
$353$	4541.69	0.684787	0.342393	+	0.939557i	$0.388762\pi$
$354$	−8771.70	−1.31698
$355$	−223.343	−0.0333910
$356$	1508.62	0.224598
$357$	26630.7	3.94802
$358$	−4942.00	−0.729589
$359$	−1398.34	−0.205576	−0.102788	−	0.994703i	$-0.532776\pi$
$359$	−1398.34	−0.205576	−0.102788	+	0.994703i	$0.532776\pi$
$360$	926.229	0.135601
$361$	−3164.97	−0.461433
$362$	−4393.00	−0.637820
$363$	−8562.69	−1.23808
$364$	0	0
$365$	−345.521	−0.0495490
$366$	7216.91	1.03069
$367$	4038.51	0.574411	0.287205	−	0.957869i	$-0.407274\pi$
$367$	4038.51	0.574411	0.287205	+	0.957869i	$0.407274\pi$
$368$	−1988.93	−0.281739
$369$	7741.72	1.09219
$370$	−694.412	−0.0975696
$371$	6960.97	0.974112
$372$	−6329.53	−0.882179
$373$	630.268	0.0874908	0.0437454	−	0.999043i	$-0.486071\pi$
$373$	630.268	0.0874908	0.0437454	+	0.999043i	$0.486071\pi$
$374$	3363.40	0.465020
$375$	6224.73	0.857184
$376$	4096.82	0.561908
$377$	0	0
$378$	3964.01	0.539382
$379$	−4918.92	−0.666670	−0.333335	−	0.942808i	$-0.608174\pi$
$379$	−4918.92	−0.666670	−0.333335	+	0.942808i	$0.608174\pi$
$380$	−803.806	−0.108511
$381$	−5768.18	−0.775624
$382$	−4102.41	−0.549470
$383$	−10025.1	−1.33748	−0.668742	−	0.743494i	$-0.733167\pi$
$383$	−10025.1	−1.33748	−0.668742	+	0.743494i	$0.733167\pi$
$384$	−1008.02	−0.133959
$385$	1620.16	0.214471
$386$	−7506.29	−0.989792
$387$	3924.36	0.515468
$388$	−2293.25	−0.300057
$389$	13329.5	1.73735	0.868676	−	0.495380i	$-0.164971\pi$
$389$	13329.5	1.73735	0.868676	+	0.495380i	$0.164971\pi$
$390$	0	0
$391$	13391.4	1.73205
$392$	−5138.85	−0.662121
$393$	14666.7	1.88254
$394$	−1493.03	−0.190908
$395$	−2020.42	−0.257363
$396$	2186.60	0.277476
$397$	−463.262	−0.0585653	−0.0292827	−	0.999571i	$-0.509322\pi$
$397$	−463.262	−0.0585653	−0.0292827	+	0.999571i	$0.509322\pi$
$398$	5314.59	0.669337
$399$	−15024.7	−1.88515
$400$	−1825.10	−0.228137
$401$	14052.6	1.75001	0.875007	−	0.484110i	$-0.160857\pi$
$401$	14052.6	1.75001	0.875007	+	0.484110i	$0.160857\pi$
$402$	8343.55	1.03517
$403$	0	0
$404$	3822.01	0.470673
$405$	1482.01	0.181831
$406$	3653.85	0.446644
$407$	−1639.34	−0.199653
$408$	6786.96	0.823540
$409$	−4439.54	−0.536726	−0.268363	−	0.963318i	$-0.586483\pi$
$409$	−4439.54	−0.536726	−0.268363	+	0.963318i	$0.586483\pi$
$410$	1461.91	0.176094
$411$	6988.83	0.838768
$412$	5915.83	0.707407
$413$	−17482.1	−2.08290
$414$	8705.96	1.03351
$415$	1625.25	0.192242
$416$	0	0
$417$	−11435.6	−1.34294
$418$	−1897.59	−0.222043
$419$	−13679.0	−1.59490	−0.797451	−	0.603383i	$-0.793819\pi$
$419$	−13679.0	−1.59490	−0.797451	+	0.603383i	$0.793819\pi$
$420$	3269.30	0.379823
$421$	−11133.2	−1.28884	−0.644419	−	0.764673i	$-0.722901\pi$
$421$	−11133.2	−1.28884	−0.644419	+	0.764673i	$0.722901\pi$
$422$	3014.68	0.347755
$423$	−17932.6	−2.06126
$424$	1774.04	0.203196
$425$	12288.3	1.40252
$426$	−1063.94	−0.121005
$427$	14383.4	1.63012
$428$	852.463	0.0962741
$429$	0	0
$430$	741.057	0.0831091
$431$	1454.09	0.162508	0.0812539	−	0.996693i	$-0.474108\pi$
$431$	1454.09	0.162508	0.0812539	+	0.996693i	$0.474108\pi$
$432$	1010.25	0.112513
$433$	−7688.57	−0.853323	−0.426662	−	0.904411i	$-0.640310\pi$
$433$	−7688.57	−0.853323	−0.426662	+	0.904411i	$0.640310\pi$
$434$	−12614.8	−1.39523
$435$	−1515.38	−0.167028
$436$	−337.679	−0.0370915
$437$	−7555.26	−0.827042
$438$	−1645.97	−0.179560
$439$	−8641.61	−0.939503	−0.469751	−	0.882799i	$-0.655656\pi$
$439$	−8641.61	−0.939503	−0.469751	+	0.882799i	$0.655656\pi$
$440$	412.907	0.0447376
$441$	22493.9	2.42888
$442$	0	0
$443$	−13563.6	−1.45469	−0.727346	−	0.686271i	$-0.759246\pi$
$443$	−13563.6	−1.45469	−0.727346	+	0.686271i	$0.759246\pi$
$444$	−3307.99	−0.353582
$445$	−1246.98	−0.132837
$446$	3958.11	0.420229
$447$	12149.0	1.28552
$448$	−2008.99	−0.211865
$449$	−18797.3	−1.97572	−0.987862	−	0.155336i	$-0.950354\pi$
$449$	−18797.3	−1.97572	−0.987862	+	0.155336i	$0.950354\pi$
$450$	7988.83	0.836882
$451$	3451.21	0.360335
$452$	−3145.67	−0.327345
$453$	−12787.7	−1.32631
$454$	4221.58	0.436406
$455$	0	0
$456$	−3829.11	−0.393234
$457$	−7225.60	−0.739605	−0.369802	−	0.929110i	$-0.620575\pi$
$457$	−7225.60	−0.739605	−0.369802	+	0.929110i	$0.620575\pi$
$458$	10653.5	1.08691
$459$	−6801.97	−0.691697
$460$	1643.99	0.166634
$461$	2295.36	0.231899	0.115950	−	0.993255i	$-0.463009\pi$
$461$	2295.36	0.231899	0.115950	+	0.993255i	$0.463009\pi$
$462$	7718.02	0.777218
$463$	9457.75	0.949328	0.474664	−	0.880167i	$-0.342570\pi$
$463$	9457.75	0.949328	0.474664	+	0.880167i	$0.342570\pi$
$464$	931.203	0.0931681
$465$	5231.81	0.521762
$466$	297.898	0.0296134
$467$	−13310.9	−1.31896	−0.659481	−	0.751721i	$-0.729224\pi$
$467$	−13310.9	−1.31896	−0.659481	+	0.751721i	$0.729224\pi$
$468$	0	0
$469$	16628.8	1.63720
$470$	−3386.31	−0.332338
$471$	−22012.4	−2.15346
$472$	−4455.39	−0.434483
$473$	1749.45	0.170063
$474$	−9624.74	−0.932656
$475$	−6932.92	−0.669693
$476$	13526.5	1.30249
$477$	−7765.35	−0.745389
$478$	6162.36	0.589665
$479$	−7969.25	−0.760176	−0.380088	−	0.924950i	$-0.624106\pi$
$479$	−7969.25	−0.760176	−0.380088	+	0.924950i	$0.624106\pi$
$480$	833.198	0.0792294
$481$	0	0
$482$	5140.59	0.485783
$483$	30729.4	2.89490
$484$	−4349.23	−0.408455
$485$	1895.53	0.177468
$486$	10469.5	0.977169
$487$	−1415.47	−0.131706	−0.0658532	−	0.997829i	$-0.520977\pi$
$487$	−1415.47	−0.131706	−0.0658532	+	0.997829i	$0.520977\pi$
$488$	3665.67	0.340035
$489$	−2281.40	−0.210978
$490$	4247.63	0.391609
$491$	−3200.91	−0.294206	−0.147103	−	0.989121i	$-0.546995\pi$
$491$	−3200.91	−0.294206	−0.147103	+	0.989121i	$0.546995\pi$
$492$	6964.15	0.638146
$493$	−6269.77	−0.572771
$494$	0	0
$495$	−1807.38	−0.164113
$496$	−3214.94	−0.291039
$497$	−2120.45	−0.191378
$498$	7742.28	0.696666
$499$	1480.51	0.132819	0.0664097	−	0.997792i	$-0.478846\pi$
$499$	1480.51	0.132819	0.0664097	+	0.997792i	$0.478846\pi$
$500$	3161.72	0.282793
$501$	18334.2	1.63496
$502$	4087.52	0.363417
$503$	−20259.3	−1.79586	−0.897929	−	0.440140i	$-0.854929\pi$
$503$	−20259.3	−1.79586	−0.897929	+	0.440140i	$0.854929\pi$
$504$	8793.75	0.777192
$505$	−3159.17	−0.278378
$506$	3881.06	0.340977
$507$	0	0
$508$	−2929.82	−0.255885
$509$	7069.24	0.615596	0.307798	−	0.951452i	$-0.400408\pi$
$509$	7069.24	0.615596	0.307798	+	0.951452i	$0.400408\pi$
$510$	−5609.91	−0.487080
$511$	−3280.43	−0.283987
$512$	−512.000	−0.0441942
$513$	3837.59	0.330280
$514$	−15437.9	−1.32478
$515$	−4889.86	−0.418394
$516$	3530.19	0.301179
$517$	−7994.25	−0.680052
$518$	−6592.85	−0.559215
$519$	8378.22	0.708599
$520$	0	0
$521$	−13874.4	−1.16670	−0.583348	−	0.812223i	$-0.698257\pi$
$521$	−13874.4	−1.16670	−0.583348	+	0.812223i	$0.698257\pi$
$522$	−4076.07	−0.341772
$523$	−16353.8	−1.36731	−0.683653	−	0.729807i	$-0.739610\pi$
$523$	−16353.8	−1.36731	−0.683653	+	0.729807i	$0.739610\pi$
$524$	7449.63	0.621066
$525$	28198.1	2.34413
$526$	−2969.04	−0.246115
$527$	21646.2	1.78923
$528$	1966.98	0.162124
$529$	3285.48	0.270032
$530$	−1466.37	−0.120179
$531$	19502.2	1.59383
$532$	−7631.45	−0.621927
$533$	0	0
$534$	−5940.29	−0.481389
$535$	−704.622	−0.0569410
$536$	4237.92	0.341512
$537$	19459.5	1.56376
$538$	4533.08	0.363262
$539$	10027.6	0.801336
$540$	−835.042	−0.0665453
$541$	−2609.54	−0.207381	−0.103690	−	0.994610i	$-0.533065\pi$
$541$	−2609.54	−0.207381	−0.103690	+	0.994610i	$0.533065\pi$
$542$	−10534.9	−0.834891
$543$	17297.7	1.36707
$544$	3447.29	0.271693
$545$	279.116	0.0219376
$546$	0	0
$547$	−22168.6	−1.73284	−0.866419	−	0.499318i	$-0.833584\pi$
$547$	−22168.6	−1.73284	−0.866419	+	0.499318i	$0.833584\pi$
$548$	3549.82	0.276717
$549$	−16045.4	−1.24736
$550$	3561.37	0.276104
$551$	3537.32	0.273494
$552$	7831.54	0.603863
$553$	−19182.2	−1.47506
$554$	9630.88	0.738587
$555$	2734.29	0.209125
$556$	−5808.48	−0.443047
$557$	−20888.8	−1.58903	−0.794514	−	0.607246i	$-0.792274\pi$
$557$	−20888.8	−1.58903	−0.794514	+	0.607246i	$0.792274\pi$
$558$	14072.5	1.06763
$559$	0	0
$560$	1660.57	0.125307
$561$	−13243.6	−0.996695
$562$	3967.44	0.297787
$563$	12680.7	0.949246	0.474623	−	0.880189i	$-0.342584\pi$
$563$	12680.7	0.949246	0.474623	+	0.880189i	$0.342584\pi$
$564$	−16131.5	−1.20436
$565$	2600.13	0.193607
$566$	1654.20	0.122847
$567$	14070.4	1.04215
$568$	−540.407	−0.0399207
$569$	−632.685	−0.0466143	−0.0233072	−	0.999728i	$-0.507420\pi$
$569$	−632.685	−0.0466143	−0.0233072	+	0.999728i	$0.507420\pi$
$570$	3165.04	0.232577
$571$	8050.66	0.590034	0.295017	−	0.955492i	$-0.404675\pi$
$571$	8050.66	0.590034	0.295017	+	0.955492i	$0.404675\pi$
$572$	0	0
$573$	16153.5	1.17770
$574$	13879.6	1.00927
$575$	14179.6	1.02840
$576$	2241.13	0.162119
$577$	−14173.9	−1.02265	−0.511323	−	0.859389i	$-0.670844\pi$
$577$	−14173.9	−1.02265	−0.511323	+	0.859389i	$0.670844\pi$
$578$	−13384.5	−0.963187
$579$	29556.5	2.12146
$580$	−769.706	−0.0551040
$581$	15430.4	1.10183
$582$	9029.81	0.643123
$583$	−3461.74	−0.245919
$584$	−836.033	−0.0592385
$585$	0	0
$586$	−2373.19	−0.167296
$587$	−13177.1	−0.926539	−0.463269	−	0.886217i	$-0.653324\pi$
$587$	−13177.1	−0.926539	−0.463269	+	0.886217i	$0.653324\pi$
$588$	20234.6	1.41915
$589$	−12212.5	−0.854341
$590$	3682.70	0.256974
$591$	5878.91	0.409181
$592$	−1680.22	−0.116650
$593$	20637.4	1.42913	0.714566	−	0.699568i	$-0.246624\pi$
$593$	20637.4	1.42913	0.714566	+	0.699568i	$0.246624\pi$
$594$	−1971.33	−0.136169
$595$	−11180.6	−0.770352
$596$	6170.81	0.424104
$597$	−20926.5	−1.43462
$598$	0	0
$599$	286.244	0.0195252	0.00976262	−	0.999952i	$-0.496892\pi$
$599$	286.244	0.0195252	0.00976262	+	0.999952i	$0.496892\pi$
$600$	7186.43	0.488975
$601$	20321.7	1.37927	0.689633	−	0.724159i	$-0.257772\pi$
$601$	20321.7	1.37927	0.689633	+	0.724159i	$0.257772\pi$
$602$	7035.70	0.476335
$603$	−18550.3	−1.25278
$604$	−6495.24	−0.437562
$605$	3594.95	0.241579
$606$	−15049.4	−1.00881
$607$	4908.85	0.328244	0.164122	−	0.986440i	$-0.447521\pi$
$607$	4908.85	0.328244	0.164122	+	0.986440i	$0.447521\pi$
$608$	−1944.91	−0.129731
$609$	−14387.3	−0.957311
$610$	−3029.94	−0.201113
$611$	0	0
$612$	−15089.5	−0.996662
$613$	8092.58	0.533207	0.266604	−	0.963806i	$-0.414099\pi$
$613$	8092.58	0.533207	0.266604	+	0.963806i	$0.414099\pi$
$614$	−12382.3	−0.813861
$615$	−5756.37	−0.377429
$616$	3920.20	0.256411
$617$	22406.7	1.46201	0.731006	−	0.682372i	$-0.239051\pi$
$617$	22406.7	1.46201	0.731006	+	0.682372i	$0.239051\pi$
$618$	−23294.0	−1.51621
$619$	−29226.6	−1.89777	−0.948883	−	0.315627i	$-0.897785\pi$
$619$	−29226.6	−1.89777	−0.948883	+	0.315627i	$0.897785\pi$
$620$	2657.38	0.172134
$621$	−7848.86	−0.507188
$622$	21218.6	1.36782
$623$	−11839.0	−0.761351
$624$	0	0
$625$	11645.2	0.745291
$626$	−19666.5	−1.25564
$627$	7471.88	0.475914
$628$	−11180.7	−0.710445
$629$	11312.9	0.717130
$630$	−7268.67	−0.459668
$631$	−6320.05	−0.398728	−0.199364	−	0.979926i	$-0.563888\pi$
$631$	−6320.05	−0.398728	−0.199364	+	0.979926i	$0.563888\pi$
$632$	−4888.68	−0.307692
$633$	−11870.5	−0.745357
$634$	−9349.47	−0.585670
$635$	2421.71	0.151342
$636$	−6985.40	−0.435517
$637$	0	0
$638$	−1817.09	−0.112757
$639$	2365.48	0.146442
$640$	423.205	0.0261385
$641$	−8403.04	−0.517785	−0.258892	−	0.965906i	$-0.583357\pi$
$641$	−8403.04	−0.517785	−0.258892	+	0.965906i	$0.583357\pi$
$642$	−3356.63	−0.206348
$643$	−313.602	−0.0192337	−0.00961684	−	0.999954i	$-0.503061\pi$
$643$	−313.602	−0.0192337	−0.00961684	+	0.999954i	$0.503061\pi$
$644$	15608.3	0.955052
$645$	−2917.96	−0.178131
$646$	13095.1	0.797552
$647$	−8074.87	−0.490658	−0.245329	−	0.969440i	$-0.578896\pi$
$647$	−8074.87	−0.490658	−0.245329	+	0.969440i	$0.578896\pi$
$648$	3585.91	0.217389
$649$	8693.95	0.525836
$650$	0	0
$651$	49671.6	2.99045
$652$	−1158.79	−0.0696036
$653$	−18166.7	−1.08870	−0.544348	−	0.838859i	$-0.683223\pi$
$653$	−18166.7	−1.08870	−0.544348	+	0.838859i	$0.683223\pi$
$654$	1329.63	0.0794996
$655$	−6157.66	−0.367327
$656$	3537.28	0.210530
$657$	3659.50	0.217307
$658$	−32150.2	−1.90478
$659$	−19645.4	−1.16127	−0.580633	−	0.814165i	$-0.697195\pi$
$659$	−19645.4	−1.16127	−0.580633	+	0.814165i	$0.697195\pi$
$660$	−1625.85	−0.0958879
$661$	−10987.3	−0.646529	−0.323265	−	0.946309i	$-0.604780\pi$
$661$	−10987.3	−0.646529	−0.323265	+	0.946309i	$0.604780\pi$
$662$	20826.7	1.22274
$663$	0	0
$664$	3932.52	0.229836
$665$	6307.94	0.367837
$666$	7354.69	0.427910
$667$	−7234.75	−0.419986
$668$	9312.48	0.539387
$669$	−15585.3	−0.900693
$670$	−3502.95	−0.201986
$671$	−7152.95	−0.411530
$672$	7910.51	0.454099
$673$	34257.1	1.96213	0.981065	−	0.193680i	$-0.0620422\pi$
$673$	34257.1	1.96213	0.981065	+	0.193680i	$0.0620422\pi$
$674$	−9746.94	−0.557030
$675$	−7202.33	−0.410693
$676$	0	0
$677$	25142.1	1.42731	0.713656	−	0.700497i	$-0.247038\pi$
$677$	25142.1	1.42731	0.713656	+	0.700497i	$0.247038\pi$
$678$	12386.3	0.701612
$679$	17996.5	1.01715
$680$	−2849.43	−0.160692
$681$	−16622.7	−0.935367
$682$	6273.43	0.352232
$683$	−10005.0	−0.560511	−0.280255	−	0.959925i	$-0.590419\pi$
$683$	−10005.0	−0.560511	−0.280255	+	0.959925i	$0.590419\pi$
$684$	8513.30	0.475898
$685$	−2934.18	−0.163663
$686$	18793.8	1.04599
$687$	−41948.9	−2.32962
$688$	1793.08	0.0993615
$689$	0	0
$690$	−6473.33	−0.357153
$691$	−7337.18	−0.403936	−0.201968	−	0.979392i	$-0.564734\pi$
$691$	−7337.18	−0.403936	−0.201968	+	0.979392i	$0.564734\pi$
$692$	4255.53	0.233773
$693$	−17159.5	−0.940602
$694$	−24243.4	−1.32603
$695$	4801.12	0.262039
$696$	−3666.67	−0.199691
$697$	−23816.5	−1.29428
$698$	17744.3	0.962225
$699$	−1172.99	−0.0634716
$700$	14322.6	0.773349
$701$	−7785.35	−0.419470	−0.209735	−	0.977758i	$-0.567260\pi$
$701$	−7785.35	−0.419470	−0.209735	+	0.977758i	$0.567260\pi$
$702$	0	0
$703$	−6382.59	−0.342424
$704$	999.082	0.0534863
$705$	13333.8	0.712314
$706$	−9083.38	−0.484217
$707$	−29993.6	−1.59551
$708$	17543.4	0.931245
$709$	−597.025	−0.0316245	−0.0158122	−	0.999875i	$-0.505033\pi$
$709$	−597.025	−0.0316245	−0.0158122	+	0.999875i	$0.505033\pi$
$710$	446.685	0.0236110
$711$	21398.8	1.12872
$712$	−3017.24	−0.158814
$713$	24977.7	1.31195
$714$	−53261.3	−2.79167
$715$	0	0
$716$	9884.01	0.515898
$717$	−24264.7	−1.26385
$718$	2796.69	0.145364
$719$	−12013.6	−0.623132	−0.311566	−	0.950224i	$-0.600854\pi$
$719$	−12013.6	−0.623132	−0.311566	+	0.950224i	$0.600854\pi$
$720$	−1852.46	−0.0958847
$721$	−46425.0	−2.39800
$722$	6329.93	0.326282
$723$	−20241.4	−1.04120
$724$	8786.00	0.451007
$725$	−6638.80	−0.340081
$726$	17125.4	0.875458
$727$	1137.34	0.0580216	0.0290108	−	0.999579i	$-0.490764\pi$
$727$	1137.34	0.0580216	0.0290108	+	0.999579i	$0.490764\pi$
$728$	0	0
$729$	−29121.7	−1.47954
$730$	691.042	0.0350364
$731$	−12072.8	−0.610846
$732$	−14433.8	−0.728811
$733$	24930.1	1.25623	0.628113	−	0.778122i	$-0.283828\pi$
$733$	24930.1	1.25623	0.628113	+	0.778122i	$0.283828\pi$
$734$	−8077.03	−0.406170
$735$	−16725.3	−0.839351
$736$	3977.86	0.199220
$737$	−8269.60	−0.413317
$738$	−15483.4	−0.772295
$739$	−11115.7	−0.553312	−0.276656	−	0.960969i	$-0.589226\pi$
$739$	−11115.7	−0.553312	−0.276656	+	0.960969i	$0.589226\pi$
$740$	1388.82	0.0689921
$741$	0	0
$742$	−13921.9	−0.688802
$743$	−12327.2	−0.608671	−0.304336	−	0.952565i	$-0.598434\pi$
$743$	−12327.2	−0.608671	−0.304336	+	0.952565i	$0.598434\pi$
$744$	12659.1	0.623795
$745$	−5100.62	−0.250835
$746$	−1260.54	−0.0618653
$747$	−17213.5	−0.843116
$748$	−6726.80	−0.328819
$749$	−6689.78	−0.326354
$750$	−12449.5	−0.606120
$751$	−13804.1	−0.670729	−0.335364	−	0.942088i	$-0.608859\pi$
$751$	−13804.1	−0.670729	−0.335364	+	0.942088i	$0.608859\pi$
$752$	−8193.64	−0.397329
$753$	−16094.9	−0.778925
$754$	0	0
$755$	5368.78	0.258795
$756$	−7928.01	−0.381401
$757$	901.030	0.0432609	0.0216304	−	0.999766i	$-0.493114\pi$
$757$	901.030	0.0432609	0.0216304	+	0.999766i	$0.493114\pi$
$758$	9837.84	0.471407
$759$	−15281.9	−0.730829
$760$	1607.61	0.0767292
$761$	29724.6	1.41592	0.707962	−	0.706251i	$-0.249615\pi$
$761$	29724.6	1.41592	0.707962	+	0.706251i	$0.249615\pi$
$762$	11536.4	0.548449
$763$	2649.97	0.125734
$764$	8204.82	0.388534
$765$	12472.6	0.589472
$766$	20050.1	0.945745
$767$	0	0
$768$	2016.03	0.0947231
$769$	−26017.4	−1.22004	−0.610021	−	0.792385i	$-0.708839\pi$
$769$	−26017.4	−1.22004	−0.610021	+	0.792385i	$0.708839\pi$
$770$	−3240.33	−0.151654
$771$	60787.6	2.83945
$772$	15012.6	0.699889
$773$	−3917.05	−0.182259	−0.0911296	−	0.995839i	$-0.529048\pi$
$773$	−3917.05	−0.182259	−0.0911296	+	0.995839i	$0.529048\pi$
$774$	−7848.71	−0.364491
$775$	22920.2	1.06235
$776$	4586.49	0.212172
$777$	25959.8	1.19859
$778$	−26658.9	−1.22849
$779$	13436.9	0.618008
$780$	0	0
$781$	1054.51	0.0483143
$782$	−26782.8	−1.22475
$783$	3674.78	0.167722
$784$	10277.7	0.468190
$785$	9241.67	0.420190
$786$	−29333.4	−1.33116
$787$	1797.29	0.0814058	0.0407029	−	0.999171i	$-0.487040\pi$
$787$	1797.29	0.0814058	0.0407029	+	0.999171i	$0.487040\pi$
$788$	2986.06	0.134992
$789$	11690.8	0.527508
$790$	4040.84	0.181983
$791$	24686.0	1.10965
$792$	−4373.20	−0.196205
$793$	0	0
$794$	926.524	0.0414120
$795$	5773.93	0.257585
$796$	−10629.2	−0.473293
$797$	−35290.6	−1.56845	−0.784226	−	0.620475i	$-0.786940\pi$
$797$	−35290.6	−1.56845	−0.784226	+	0.620475i	$0.786940\pi$
$798$	30049.3	1.33300
$799$	55167.6	2.44266
$800$	3650.19	0.161317
$801$	13207.1	0.582584
$802$	−28105.3	−1.23745
$803$	1631.38	0.0716938
$804$	−16687.1	−0.731976
$805$	−12901.4	−0.564862
$806$	0	0
$807$	−17849.3	−0.778594
$808$	−7644.02	−0.332816
$809$	18697.8	0.812585	0.406292	−	0.913743i	$-0.366821\pi$
$809$	18697.8	0.812585	0.406292	+	0.913743i	$0.366821\pi$
$810$	−2964.01	−0.128574
$811$	22245.0	0.963165	0.481582	−	0.876401i	$-0.340062\pi$
$811$	22245.0	0.963165	0.481582	+	0.876401i	$0.340062\pi$
$812$	−7307.70	−0.315825
$813$	41481.7	1.78946
$814$	3278.67	0.141176
$815$	957.820	0.0411668
$816$	−13573.9	−0.582331
$817$	6811.32	0.291674
$818$	8879.08	0.379523
$819$	0	0
$820$	−2923.82	−0.124517
$821$	39291.2	1.67025	0.835123	−	0.550063i	$-0.185396\pi$
$821$	39291.2	1.67025	0.835123	+	0.550063i	$0.185396\pi$
$822$	−13977.7	−0.593098
$823$	31.5142	0.00133477	0.000667386	−	1.00000i	$-0.499788\pi$
$823$	31.5142	0.00133477	0.000667386		1.00000i	$0.499788\pi$
$824$	−11831.7	−0.500213
$825$	−14023.1	−0.591785
$826$	34964.1	1.47283
$827$	−30870.8	−1.29805	−0.649023	−	0.760769i	$-0.724822\pi$
$827$	−30870.8	−1.29805	−0.649023	+	0.760769i	$0.724822\pi$
$828$	−17411.9	−0.730805
$829$	38665.0	1.61989	0.809946	−	0.586505i	$-0.199496\pi$
$829$	38665.0	1.61989	0.809946	+	0.586505i	$0.199496\pi$
$830$	−3250.51	−0.135936
$831$	−37922.2	−1.58304
$832$	0	0
$833$	−69199.6	−2.87830
$834$	22871.3	0.949601
$835$	−7697.43	−0.319019
$836$	3795.17	0.157008
$837$	−12687.1	−0.523930
$838$	27358.1	1.12777
$839$	−9892.30	−0.407056	−0.203528	−	0.979069i	$-0.565241\pi$
$839$	−9892.30	−0.407056	−0.203528	+	0.979069i	$0.565241\pi$
$840$	−6538.61	−0.268575
$841$	−21001.7	−0.861115
$842$	22266.5	0.911346
$843$	−15622.0	−0.638258
$844$	−6029.37	−0.245900
$845$	0	0
$846$	35865.3	1.45753
$847$	34131.0	1.38460
$848$	−3548.08	−0.143681
$849$	−6513.53	−0.263302
$850$	−24576.7	−0.991733
$851$	13054.1	0.525837
$852$	2127.89	0.0855636
$853$	35366.0	1.41959	0.709794	−	0.704409i	$-0.248788\pi$
$853$	35366.0	1.41959	0.709794	+	0.704409i	$0.248788\pi$
$854$	−28766.7	−1.15267
$855$	−7036.86	−0.281468
$856$	−1704.93	−0.0680761
$857$	760.177	0.0303001	0.0151500	−	0.999885i	$-0.495177\pi$
$857$	760.177	0.0303001	0.0151500	+	0.999885i	$0.495177\pi$
$858$	0	0
$859$	27332.4	1.08565	0.542823	−	0.839847i	$-0.317355\pi$
$859$	27332.4	1.08565	0.542823	+	0.839847i	$0.317355\pi$
$860$	−1482.11	−0.0587670
$861$	−54651.8	−2.16322
$862$	−2908.17	−0.114910
$863$	35583.9	1.40358	0.701790	−	0.712384i	$-0.252385\pi$
$863$	35583.9	1.40358	0.701790	+	0.712384i	$0.252385\pi$
$864$	−2020.49	−0.0795586
$865$	−3517.50	−0.138264
$866$	15377.1	0.603391
$867$	52702.4	2.06444
$868$	25229.6	0.986576
$869$	9539.44	0.372386
$870$	3030.77	0.118107
$871$	0	0
$872$	675.358	0.0262276
$873$	−20076.1	−0.778318
$874$	15110.5	0.584807
$875$	−24811.9	−0.958622
$876$	3291.94	0.126968
$877$	−3445.04	−0.132646	−0.0663232	−	0.997798i	$-0.521127\pi$
$877$	−3445.04	−0.132646	−0.0663232	+	0.997798i	$0.521127\pi$
$878$	17283.2	0.664329
$879$	9344.60	0.358573
$880$	−825.813	−0.0316343
$881$	−20296.2	−0.776158	−0.388079	−	0.921626i	$-0.626861\pi$
$881$	−20296.2	−0.776158	−0.388079	+	0.921626i	$0.626861\pi$
$882$	−44987.7	−1.71748
$883$	2952.08	0.112509	0.0562545	−	0.998416i	$-0.482084\pi$
$883$	2952.08	0.112509	0.0562545	+	0.998416i	$0.482084\pi$
$884$	0	0
$885$	−14500.9	−0.550782
$886$	27127.3	1.02862
$887$	24754.5	0.937062	0.468531	−	0.883447i	$-0.344783\pi$
$887$	24754.5	0.937062	0.468531	+	0.883447i	$0.344783\pi$
$888$	6615.98	0.250020
$889$	22992.0	0.867411
$890$	2493.97	0.0939303
$891$	−6997.30	−0.263096
$892$	−7916.23	−0.297147
$893$	−31124.8	−1.16635
$894$	−24298.0	−0.908999
$895$	−8169.84	−0.305126
$896$	4017.97	0.149811
$897$	0	0
$898$	37594.6	1.39705
$899$	−11694.4	−0.433849
$900$	−15977.7	−0.591765
$901$	23889.1	0.883311
$902$	−6902.42	−0.254795
$903$	−27703.5	−1.02095
$904$	6291.35	0.231468
$905$	−7262.26	−0.266747
$906$	25575.4	0.937844
$907$	24207.0	0.886197	0.443099	−	0.896473i	$-0.353879\pi$
$907$	24207.0	0.886197	0.443099	+	0.896473i	$0.353879\pi$
$908$	−8443.15	−0.308586
$909$	33459.5	1.22088
$910$	0	0
$911$	−21027.3	−0.764725	−0.382363	−	0.924012i	$-0.624889\pi$
$911$	−21027.3	−0.764725	−0.382363	+	0.924012i	$0.624889\pi$
$912$	7658.23	0.278058
$913$	−7673.65	−0.278161
$914$	14451.2	0.522979
$915$	11930.6	0.431053
$916$	−21307.0	−0.768563
$917$	−58461.7	−2.10532
$918$	13603.9	0.489104
$919$	−7899.56	−0.283550	−0.141775	−	0.989899i	$-0.545281\pi$
$919$	−7899.56	−0.283550	−0.141775	+	0.989899i	$0.545281\pi$
$920$	−3287.98	−0.117828
$921$	48756.3	1.74438
$922$	−4590.72	−0.163977
$923$	0	0
$924$	−15436.0	−0.549576
$925$	11978.8	0.425794
$926$	−18915.5	−0.671276
$927$	51789.7	1.83495
$928$	−1862.41	−0.0658798
$929$	5434.65	0.191932	0.0959662	−	0.995385i	$-0.469406\pi$
$929$	5434.65	0.191932	0.0959662	+	0.995385i	$0.469406\pi$
$930$	−10463.6	−0.368942
$931$	39041.5	1.37437
$932$	−595.795	−0.0209398
$933$	−83549.5	−2.93171
$934$	26621.8	0.932647
$935$	5560.19	0.194479
$936$	0	0
$937$	8130.03	0.283454	0.141727	−	0.989906i	$-0.454734\pi$
$937$	8130.03	0.283454	0.141727	+	0.989906i	$0.454734\pi$
$938$	−33257.5	−1.15767
$939$	77438.0	2.69126
$940$	6772.63	0.234999
$941$	5409.95	0.187417	0.0937084	−	0.995600i	$-0.470128\pi$
$941$	5409.95	0.187417	0.0937084	+	0.995600i	$0.470128\pi$
$942$	44024.8	1.52272
$943$	−27482.1	−0.949034
$944$	8910.79	0.307226
$945$	6553.07	0.225578
$946$	−3498.90	−0.120253
$947$	37140.9	1.27446	0.637231	−	0.770673i	$-0.280080\pi$
$947$	37140.9	1.27446	0.637231	+	0.770673i	$0.280080\pi$
$948$	19249.5	0.659488
$949$	0	0
$950$	13865.8	0.473544
$951$	36814.2	1.25529
$952$	−27052.9	−0.920998
$953$	−3541.43	−0.120376	−0.0601879	−	0.998187i	$-0.519170\pi$
$953$	−3541.43	−0.120376	−0.0601879	+	0.998187i	$0.519170\pi$
$954$	15530.7	0.527070
$955$	−6781.87	−0.229797
$956$	−12324.7	−0.416956
$957$	7154.90	0.241677
$958$	15938.5	0.537526
$959$	−27857.6	−0.938027
$960$	−1666.40	−0.0560237
$961$	10583.5	0.355258
$962$	0	0
$963$	7462.82	0.249726
$964$	−10281.2	−0.343500
$965$	−12409.0	−0.413947
$966$	−61458.7	−2.04700
$967$	−48102.4	−1.59966	−0.799828	−	0.600229i	$-0.795076\pi$
$967$	−48102.4	−1.59966	−0.799828	+	0.600229i	$0.795076\pi$
$968$	8698.46	0.288821
$969$	−51562.7	−1.70942
$970$	−3791.07	−0.125488
$971$	35079.2	1.15937	0.579683	−	0.814842i	$-0.303176\pi$
$971$	35079.2	1.15937	0.579683	+	0.814842i	$0.303176\pi$
$972$	−20938.9	−0.690963
$973$	45582.6	1.50186
$974$	2830.94	0.0931305
$975$	0	0
$976$	−7331.35	−0.240441
$977$	42666.4	1.39715	0.698577	−	0.715535i	$-0.253817\pi$
$977$	42666.4	1.39715	0.698577	+	0.715535i	$0.253817\pi$
$978$	4562.79	0.149184
$979$	5887.64	0.192206
$980$	−8495.26	−0.276909
$981$	−2956.18	−0.0962117
$982$	6401.83	0.208035
$983$	−32297.0	−1.04793	−0.523964	−	0.851740i	$-0.675547\pi$
$983$	−32297.0	−1.04793	−0.523964	+	0.851740i	$0.675547\pi$
$984$	−13928.3	−0.451237
$985$	−2468.19	−0.0798408
$986$	12539.5	0.405010
$987$	126593.	4.08259
$988$	0	0
$989$	−13930.9	−0.447905
$990$	3614.76	0.116045
$991$	−11349.2	−0.363793	−0.181897	−	0.983318i	$-0.558224\pi$
$991$	−11349.2	−0.363793	−0.181897	+	0.983318i	$0.558224\pi$
$992$	6429.89	0.205796
$993$	−82006.7	−2.62075
$994$	4240.89	0.135325
$995$	8785.77	0.279927
$996$	−15484.6	−0.492617
$997$	−30536.4	−0.970006	−0.485003	−	0.874512i	$-0.661182\pi$
$997$	−30536.4	−0.970006	−0.485003	+	0.874512i	$0.661182\pi$
$998$	−2961.03	−0.0939175
$999$	−6630.62	−0.209994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.a.l.1.4		4
13.2	odd	12		338.4.e.e.147.1		8
13.3	even	3		338.4.c.n.191.1		8
13.4	even	6		338.4.c.m.315.1		8
13.5	odd	4		338.4.b.g.337.8		8
13.6	odd	12		26.4.e.a.23.3	yes	8
13.7	odd	12		338.4.e.e.23.1		8
13.8	odd	4		338.4.b.g.337.4		8
13.9	even	3		338.4.c.n.315.1		8
13.10	even	6		338.4.c.m.191.1		8
13.11	odd	12		26.4.e.a.17.3	✓	8
13.12	even	2		338.4.a.m.1.4		4
39.11	even	12		234.4.l.b.199.2		8
39.32	even	12		234.4.l.b.127.1		8
52.11	even	12		208.4.w.d.17.4		8
52.19	even	12		208.4.w.d.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.4.e.a.17.3	✓	8	13.11	odd	12
26.4.e.a.23.3	yes	8	13.6	odd	12
208.4.w.d.17.4		8	52.11	even	12
208.4.w.d.49.4		8	52.19	even	12
234.4.l.b.127.1		8	39.32	even	12
234.4.l.b.199.2		8	39.11	even	12
338.4.a.l.1.4		4	1.1	even	1	trivial
338.4.a.m.1.4		4	13.12	even	2
338.4.b.g.337.4		8	13.8	odd	4
338.4.b.g.337.8		8	13.5	odd	4
338.4.c.m.191.1		8	13.10	even	6
338.4.c.m.315.1		8	13.4	even	6
338.4.c.n.191.1		8	13.3	even	3
338.4.c.n.315.1		8	13.9	even	3
338.4.e.e.23.1		8	13.7	odd	12
338.4.e.e.147.1		8	13.2	odd	12

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +7.87513 q^{3} +4.00000 q^{4} -3.30629 q^{5} -15.7503 q^{6} -31.3904 q^{7} -8.00000 q^{8} +35.0177 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +7.87513 q^{3} +4.00000 q^{4} -3.30629 q^{5} -15.7503 q^{6} -31.3904 q^{7} -8.00000 q^{8} +35.0177 q^{9} +6.61258 q^{10} +15.6107 q^{11} +31.5005 q^{12} +62.7808 q^{14} -26.0375 q^{15} +16.0000 q^{16} -107.728 q^{17} -70.0354 q^{18} +60.7786 q^{19} -13.2252 q^{20} -247.203 q^{21} -31.2213 q^{22} -124.308 q^{23} -63.0011 q^{24} -114.068 q^{25} +63.1404 q^{27} -125.562 q^{28} +58.2002 q^{29} +52.0749 q^{30} -200.934 q^{31} -32.0000 q^{32} +122.936 q^{33} +215.455 q^{34} +103.786 q^{35} +140.071 q^{36} -105.014 q^{37} -121.557 q^{38} +26.4503 q^{40} +221.080 q^{41} +494.407 q^{42} +112.068 q^{43} +62.4426 q^{44} -115.779 q^{45} +248.616 q^{46} -512.102 q^{47} +126.002 q^{48} +642.357 q^{49} +228.137 q^{50} -848.370 q^{51} -221.755 q^{53} -126.281 q^{54} -51.6133 q^{55} +251.123 q^{56} +478.639 q^{57} -116.400 q^{58} +556.924 q^{59} -104.150 q^{60} -458.209 q^{61} +401.868 q^{62} -1099.22 q^{63} +64.0000 q^{64} -245.872 q^{66} -529.740 q^{67} -430.911 q^{68} -978.942 q^{69} -207.571 q^{70} +67.5508 q^{71} -280.142 q^{72} +104.504 q^{73} +210.028 q^{74} -898.304 q^{75} +243.114 q^{76} -490.025 q^{77} +611.085 q^{79} -52.9006 q^{80} -448.239 q^{81} -442.161 q^{82} -491.565 q^{83} -988.814 q^{84} +356.179 q^{85} -224.136 q^{86} +458.334 q^{87} -124.885 q^{88} +377.155 q^{89} +231.557 q^{90} -497.232 q^{92} -1582.38 q^{93} +1024.20 q^{94} -200.951 q^{95} -252.004 q^{96} -573.312 q^{97} -1284.71 q^{98} +546.649 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 6 q^{3} + 16 q^{4} + 4 q^{5} - 12 q^{6} - 20 q^{7} - 32 q^{8} + 22 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 + 6 * q^3 + 16 * q^4 + 4 * q^5 - 12 * q^6 - 20 * q^7 - 32 * q^8 + 22 * q^9	\( 4 q - 8 q^{2} + 6 q^{3} + 16 q^{4} + 4 q^{5} - 12 q^{6} - 20 q^{7} - 32 q^{8} + 22 q^{9} - 8 q^{10} - 36 q^{11} + 24 q^{12} + 40 q^{14} - 164 q^{15} + 64 q^{16} + 112 q^{17} - 44 q^{18} - 144 q^{19} + 16 q^{20} - 344 q^{21} + 72 q^{22} - 230 q^{23} - 48 q^{24} + 90 q^{25} + 234 q^{27} - 80 q^{28} - 32 q^{29} + 328 q^{30} - 532 q^{31} - 128 q^{32} + 600 q^{33} - 224 q^{34} + 128 q^{35} + 88 q^{36} - 340 q^{37} + 288 q^{38} - 32 q^{40} - 352 q^{41} + 688 q^{42} - 114 q^{43} - 144 q^{44} - 596 q^{45} + 460 q^{46} - 1284 q^{47} + 96 q^{48} - 110 q^{49} - 180 q^{50} - 650 q^{51} + 18 q^{53} - 468 q^{54} - 1248 q^{55} + 160 q^{56} - 144 q^{57} + 64 q^{58} + 988 q^{59} - 656 q^{60} - 900 q^{61} + 1064 q^{62} - 1160 q^{63} + 256 q^{64} - 1200 q^{66} - 1672 q^{67} + 448 q^{68} - 2402 q^{69} - 256 q^{70} - 528 q^{71} - 176 q^{72} - 1624 q^{73} + 680 q^{74} - 862 q^{75} - 576 q^{76} - 1170 q^{77} + 392 q^{79} + 64 q^{80} - 1868 q^{81} + 704 q^{82} - 152 q^{83} - 1376 q^{84} - 940 q^{85} + 228 q^{86} - 1598 q^{87} + 288 q^{88} + 816 q^{89} + 1192 q^{90} - 920 q^{92} + 176 q^{93} + 2568 q^{94} - 816 q^{95} - 192 q^{96} + 192 q^{97} + 220 q^{98} + 1656 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 + 6 * q^3 + 16 * q^4 + 4 * q^5 - 12 * q^6 - 20 * q^7 - 32 * q^8 + 22 * q^9 - 8 * q^10 - 36 * q^11 + 24 * q^12 + 40 * q^14 - 164 * q^15 + 64 * q^16 + 112 * q^17 - 44 * q^18 - 144 * q^19 + 16 * q^20 - 344 * q^21 + 72 * q^22 - 230 * q^23 - 48 * q^24 + 90 * q^25 + 234 * q^27 - 80 * q^28 - 32 * q^29 + 328 * q^30 - 532 * q^31 - 128 * q^32 + 600 * q^33 - 224 * q^34 + 128 * q^35 + 88 * q^36 - 340 * q^37 + 288 * q^38 - 32 * q^40 - 352 * q^41 + 688 * q^42 - 114 * q^43 - 144 * q^44 - 596 * q^45 + 460 * q^46 - 1284 * q^47 + 96 * q^48 - 110 * q^49 - 180 * q^50 - 650 * q^51 + 18 * q^53 - 468 * q^54 - 1248 * q^55 + 160 * q^56 - 144 * q^57 + 64 * q^58 + 988 * q^59 - 656 * q^60 - 900 * q^61 + 1064 * q^62 - 1160 * q^63 + 256 * q^64 - 1200 * q^66 - 1672 * q^67 + 448 * q^68 - 2402 * q^69 - 256 * q^70 - 528 * q^71 - 176 * q^72 - 1624 * q^73 + 680 * q^74 - 862 * q^75 - 576 * q^76 - 1170 * q^77 + 392 * q^79 + 64 * q^80 - 1868 * q^81 + 704 * q^82 - 152 * q^83 - 1376 * q^84 - 940 * q^85 + 228 * q^86 - 1598 * q^87 + 288 * q^88 + 816 * q^89 + 1192 * q^90 - 920 * q^92 + 176 * q^93 + 2568 * q^94 - 816 * q^95 - 192 * q^96 + 192 * q^97 + 220 * q^98 + 1656 * q^99

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