LMFDB - Embedded newform 338.4.a.j.1.1

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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -8.74094 q^{3} +4.00000 q^{4} +14.1685 q^{5} +17.4819 q^{6} -28.6504 q^{7} -8.00000 q^{8} +49.4040 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -8.74094 q^{3} +4.00000 q^{4} +14.1685 q^{5} +17.4819 q^{6} -28.6504 q^{7} -8.00000 q^{8} +49.4040 q^{9} -28.3370 q^{10} -9.49349 q^{11} -34.9638 q^{12} +57.3008 q^{14} -123.846 q^{15} +16.0000 q^{16} +30.6104 q^{17} -98.8080 q^{18} +153.860 q^{19} +56.6741 q^{20} +250.431 q^{21} +18.9870 q^{22} -36.0030 q^{23} +69.9275 q^{24} +75.7470 q^{25} -195.832 q^{27} -114.602 q^{28} -49.2567 q^{29} +247.692 q^{30} +166.984 q^{31} -32.0000 q^{32} +82.9820 q^{33} -61.2209 q^{34} -405.934 q^{35} +197.616 q^{36} -23.8808 q^{37} -307.720 q^{38} -113.348 q^{40} +125.474 q^{41} -500.863 q^{42} -434.774 q^{43} -37.9739 q^{44} +699.982 q^{45} +72.0061 q^{46} -186.017 q^{47} -139.855 q^{48} +477.845 q^{49} -151.494 q^{50} -267.564 q^{51} -400.631 q^{53} +391.664 q^{54} -134.509 q^{55} +229.203 q^{56} -1344.88 q^{57} +98.5134 q^{58} -408.208 q^{59} -495.385 q^{60} -603.762 q^{61} -333.968 q^{62} -1415.44 q^{63} +64.0000 q^{64} -165.964 q^{66} -287.548 q^{67} +122.442 q^{68} +314.700 q^{69} +811.868 q^{70} -961.803 q^{71} -395.232 q^{72} +963.198 q^{73} +47.7616 q^{74} -662.100 q^{75} +615.440 q^{76} +271.992 q^{77} -1043.18 q^{79} +226.696 q^{80} +377.848 q^{81} -250.948 q^{82} +313.304 q^{83} +1001.73 q^{84} +433.705 q^{85} +869.549 q^{86} +430.550 q^{87} +75.9479 q^{88} +675.754 q^{89} -1399.96 q^{90} -144.012 q^{92} -1459.60 q^{93} +372.034 q^{94} +2179.97 q^{95} +279.710 q^{96} -272.435 q^{97} -955.691 q^{98} -469.016 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9	\( 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9} - 24 q^{10} + 82 q^{11} - 48 q^{12} + 54 q^{14} - 90 q^{15} + 48 q^{16} - 90 q^{17} - 18 q^{18} + 130 q^{19} + 48 q^{20} + 234 q^{21} - 164 q^{22} + 19 q^{23} + 96 q^{24} - 61 q^{25} - 69 q^{27} - 108 q^{28} - 101 q^{29} + 180 q^{30} + 519 q^{31} - 96 q^{32} - 146 q^{33} + 180 q^{34} - 458 q^{35} + 36 q^{36} + 84 q^{37} - 260 q^{38} - 96 q^{40} + 187 q^{41} - 468 q^{42} - 1205 q^{43} + 328 q^{44} + 645 q^{45} - 38 q^{46} - 536 q^{47} - 192 q^{48} - 184 q^{49} + 122 q^{50} - 207 q^{51} - 1095 q^{53} + 138 q^{54} - 526 q^{55} + 216 q^{56} - 1409 q^{57} + 202 q^{58} - 1413 q^{59} - 360 q^{60} - 1108 q^{61} - 1038 q^{62} - 1404 q^{63} + 192 q^{64} + 292 q^{66} - 1605 q^{67} - 360 q^{68} - 314 q^{69} + 916 q^{70} - 909 q^{71} - 72 q^{72} + 287 q^{73} - 168 q^{74} - 505 q^{75} + 520 q^{76} + 480 q^{77} - 1961 q^{79} + 192 q^{80} + 915 q^{81} - 374 q^{82} + 191 q^{83} + 936 q^{84} + 67 q^{85} + 2410 q^{86} + 1636 q^{87} - 656 q^{88} + 1091 q^{89} - 1290 q^{90} + 76 q^{92} - 1614 q^{93} + 1072 q^{94} + 1829 q^{95} + 384 q^{96} - 947 q^{97} + 368 q^{98} - 2057 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9 - 24 * q^10 + 82 * q^11 - 48 * q^12 + 54 * q^14 - 90 * q^15 + 48 * q^16 - 90 * q^17 - 18 * q^18 + 130 * q^19 + 48 * q^20 + 234 * q^21 - 164 * q^22 + 19 * q^23 + 96 * q^24 - 61 * q^25 - 69 * q^27 - 108 * q^28 - 101 * q^29 + 180 * q^30 + 519 * q^31 - 96 * q^32 - 146 * q^33 + 180 * q^34 - 458 * q^35 + 36 * q^36 + 84 * q^37 - 260 * q^38 - 96 * q^40 + 187 * q^41 - 468 * q^42 - 1205 * q^43 + 328 * q^44 + 645 * q^45 - 38 * q^46 - 536 * q^47 - 192 * q^48 - 184 * q^49 + 122 * q^50 - 207 * q^51 - 1095 * q^53 + 138 * q^54 - 526 * q^55 + 216 * q^56 - 1409 * q^57 + 202 * q^58 - 1413 * q^59 - 360 * q^60 - 1108 * q^61 - 1038 * q^62 - 1404 * q^63 + 192 * q^64 + 292 * q^66 - 1605 * q^67 - 360 * q^68 - 314 * q^69 + 916 * q^70 - 909 * q^71 - 72 * q^72 + 287 * q^73 - 168 * q^74 - 505 * q^75 + 520 * q^76 + 480 * q^77 - 1961 * q^79 + 192 * q^80 + 915 * q^81 - 374 * q^82 + 191 * q^83 + 936 * q^84 + 67 * q^85 + 2410 * q^86 + 1636 * q^87 - 656 * q^88 + 1091 * q^89 - 1290 * q^90 + 76 * q^92 - 1614 * q^93 + 1072 * q^94 + 1829 * q^95 + 384 * q^96 - 947 * q^97 + 368 * q^98 - 2057 * 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
$3$	−8.74094	−1.68219	−0.841097	−	0.540884i	$-0.818090\pi$
$3$	−8.74094	−1.68219	−0.841097	+	0.540884i	$0.818090\pi$
$4$	4.00000	0.500000
$5$	14.1685	1.26727	0.633636	−	0.773632i	$-0.281562\pi$
$5$	14.1685	1.26727	0.633636	+	0.773632i	$0.281562\pi$
$6$	17.4819	1.18949
$7$	−28.6504	−1.54698	−0.773488	−	0.633811i	$-0.781490\pi$
$7$	−28.6504	−1.54698	−0.773488	+	0.633811i	$0.781490\pi$
$8$	−8.00000	−0.353553
$9$	49.4040	1.82978
$10$	−28.3370	−0.896096
$11$	−9.49349	−0.260218	−0.130109	−	0.991500i	$-0.541533\pi$
$11$	−9.49349	−0.260218	−0.130109	+	0.991500i	$0.541533\pi$
$12$	−34.9638	−0.841097
$13$	0	0
$13$	0	0
$14$	57.3008	1.09388
$15$	−123.846	−2.13180
$16$	16.0000	0.250000
$17$	30.6104	0.436713	0.218356	−	0.975869i	$-0.429930\pi$
$17$	30.6104	0.436713	0.218356	+	0.975869i	$0.429930\pi$
$18$	−98.8080	−1.29385
$19$	153.860	1.85778	0.928892	−	0.370351i	$-0.120763\pi$
$19$	153.860	1.85778	0.928892	+	0.370351i	$0.120763\pi$
$20$	56.6741	0.633636
$21$	250.431	2.60231
$22$	18.9870	0.184002
$23$	−36.0030	−0.326398	−0.163199	−	0.986593i	$-0.552181\pi$
$23$	−36.0030	−0.326398	−0.163199	+	0.986593i	$0.552181\pi$
$24$	69.9275	0.594746
$25$	75.7470	0.605976
$26$	0	0
$27$	−195.832	−1.39585
$28$	−114.602	−0.773488
$29$	−49.2567	−0.315405	−0.157702	−	0.987487i	$-0.550409\pi$
$29$	−49.2567	−0.315405	−0.157702	+	0.987487i	$0.550409\pi$
$30$	247.692	1.50741
$31$	166.984	0.967459	0.483730	−	0.875217i	$-0.339282\pi$
$31$	166.984	0.967459	0.483730	+	0.875217i	$0.339282\pi$
$32$	−32.0000	−0.176777
$33$	82.9820	0.437737
$34$	−61.2209	−0.308803
$35$	−405.934	−1.96044
$36$	197.616	0.914889
$37$	−23.8808	−0.106107	−0.0530537	−	0.998592i	$-0.516895\pi$
$37$	−23.8808	−0.106107	−0.0530537	+	0.998592i	$0.516895\pi$
$38$	−307.720	−1.31365
$39$	0	0
$40$	−113.348	−0.448048
$41$	125.474	0.477946	0.238973	−	0.971026i	$-0.423189\pi$
$41$	125.474	0.477946	0.238973	+	0.971026i	$0.423189\pi$
$42$	−500.863	−1.84011
$43$	−434.774	−1.54192	−0.770959	−	0.636885i	$-0.780223\pi$
$43$	−434.774	−1.54192	−0.770959	+	0.636885i	$0.780223\pi$
$44$	−37.9739	−0.130109
$45$	699.982	2.31883
$46$	72.0061	0.230798
$47$	−186.017	−0.577306	−0.288653	−	0.957434i	$-0.593207\pi$
$47$	−186.017	−0.577306	−0.288653	+	0.957434i	$0.593207\pi$
$48$	−139.855	−0.420549
$49$	477.845	1.39314
$50$	−151.494	−0.428490
$51$	−267.564	−0.734636
$52$	0	0
$53$	−400.631	−1.03832	−0.519159	−	0.854678i	$-0.673755\pi$
$53$	−400.631	−1.03832	−0.519159	+	0.854678i	$0.673755\pi$
$54$	391.664	0.987014
$55$	−134.509	−0.329766
$56$	229.203	0.546939
$57$	−1344.88	−3.12515
$58$	98.5134	0.223025
$59$	−408.208	−0.900748	−0.450374	−	0.892840i	$-0.648709\pi$
$59$	−408.208	−0.900748	−0.450374	+	0.892840i	$0.648709\pi$
$60$	−495.385	−1.06590
$61$	−603.762	−1.26728	−0.633638	−	0.773630i	$-0.718439\pi$
$61$	−603.762	−1.26728	−0.633638	+	0.773630i	$0.718439\pi$
$62$	−333.968	−0.684097
$63$	−1415.44	−2.83062
$64$	64.0000	0.125000
$65$	0	0
$66$	−165.964	−0.309527
$67$	−287.548	−0.524322	−0.262161	−	0.965024i	$-0.584435\pi$
$67$	−287.548	−0.524322	−0.262161	+	0.965024i	$0.584435\pi$
$68$	122.442	0.218356
$69$	314.700	0.549065
$70$	811.868	1.38624
$71$	−961.803	−1.60768	−0.803838	−	0.594848i	$-0.797212\pi$
$71$	−961.803	−1.60768	−0.803838	+	0.594848i	$0.797212\pi$
$72$	−395.232	−0.646924
$73$	963.198	1.54430	0.772149	−	0.635441i	$-0.219182\pi$
$73$	963.198	1.54430	0.772149	+	0.635441i	$0.219182\pi$
$74$	47.7616	0.0750293
$75$	−662.100	−1.01937
$76$	615.440	0.928892
$77$	271.992	0.402550
$78$	0	0
$79$	−1043.18	−1.48566	−0.742828	−	0.669482i	$-0.766516\pi$
$79$	−1043.18	−1.48566	−0.742828	+	0.669482i	$0.766516\pi$
$80$	226.696	0.316818
$81$	377.848	0.518310
$82$	−250.948	−0.337959
$83$	313.304	0.414332	0.207166	−	0.978306i	$-0.433576\pi$
$83$	313.304	0.414332	0.207166	+	0.978306i	$0.433576\pi$
$84$	1001.73	1.30116
$85$	433.705	0.553434
$86$	869.549	1.09030
$87$	430.550	0.530572
$88$	75.9479	0.0920008
$89$	675.754	0.804829	0.402414	−	0.915458i	$-0.368171\pi$
$89$	675.754	0.804829	0.402414	+	0.915458i	$0.368171\pi$
$90$	−1399.96	−1.63966
$91$	0	0
$92$	−144.012	−0.163199
$93$	−1459.60	−1.62745
$94$	372.034	0.408217
$95$	2179.97	2.35432
$96$	279.710	0.297373
$97$	−272.435	−0.285171	−0.142585	−	0.989783i	$-0.545542\pi$
$97$	−272.435	−0.285171	−0.142585	+	0.989783i	$0.545542\pi$
$98$	−955.691	−0.985095
$99$	−469.016	−0.476140
$100$	302.988	0.302988
$101$	−661.561	−0.651760	−0.325880	−	0.945411i	$-0.605661\pi$
$101$	−661.561	−0.651760	−0.325880	+	0.945411i	$0.605661\pi$
$102$	535.128	0.519466
$103$	−315.353	−0.301676	−0.150838	−	0.988558i	$-0.548197\pi$
$103$	−315.353	−0.301676	−0.150838	+	0.988558i	$0.548197\pi$
$104$	0	0
$105$	3548.24	3.29784
$106$	801.262	0.734202
$107$	2121.32	1.91660	0.958299	−	0.285769i	$-0.0922490\pi$
$107$	2121.32	1.91660	0.958299	+	0.285769i	$0.0922490\pi$
$108$	−783.328	−0.697924
$109$	884.432	0.777185	0.388593	−	0.921410i	$-0.372961\pi$
$109$	884.432	0.777185	0.388593	+	0.921410i	$0.372961\pi$
$110$	269.017	0.233180
$111$	208.740	0.178493
$112$	−458.406	−0.386744
$113$	−132.205	−0.110060	−0.0550299	−	0.998485i	$-0.517525\pi$
$113$	−132.205	−0.110060	−0.0550299	+	0.998485i	$0.517525\pi$
$114$	2689.76	2.20982
$115$	−510.110	−0.413635
$116$	−197.027	−0.157702
$117$	0	0
$118$	816.415	0.636925
$119$	−877.001	−0.675585
$120$	990.769	0.753704
$121$	−1240.87	−0.932287
$122$	1207.52	0.896099
$123$	−1096.76	−0.803998
$124$	667.937	0.483730
$125$	−697.842	−0.499335
$126$	2830.89	2.00155
$127$	−1872.81	−1.30854	−0.654271	−	0.756260i	$-0.727024\pi$
$127$	−1872.81	−1.30854	−0.654271	+	0.756260i	$0.727024\pi$
$128$	−128.000	−0.0883883
$129$	3800.34	2.59381
$130$	0	0
$131$	−1850.21	−1.23400	−0.617000	−	0.786963i	$-0.711652\pi$
$131$	−1850.21	−1.23400	−0.617000	+	0.786963i	$0.711652\pi$
$132$	331.928	0.218868
$133$	−4408.15	−2.87395
$134$	575.096	0.370751
$135$	−2774.65	−1.76892
$136$	−244.883	−0.154401
$137$	967.375	0.603273	0.301637	−	0.953423i	$-0.402467\pi$
$137$	967.375	0.603273	0.301637	+	0.953423i	$0.402467\pi$
$138$	−629.401	−0.388247
$139$	−103.597	−0.0632157	−0.0316078	−	0.999500i	$-0.510063\pi$
$139$	−103.597	−0.0632157	−0.0316078	+	0.999500i	$0.510063\pi$
$140$	−1623.74	−0.980219
$141$	1625.96	0.971140
$142$	1923.61	1.13680
$143$	0	0
$144$	790.464	0.457445
$145$	−697.894	−0.399703
$146$	−1926.40	−1.09198
$147$	−4176.82	−2.34352
$148$	−95.5231	−0.0530537
$149$	−1917.44	−1.05424	−0.527122	−	0.849789i	$-0.676729\pi$
$149$	−1917.44	−1.05424	−0.527122	+	0.849789i	$0.676729\pi$
$150$	1324.20	0.720803
$151$	−2035.28	−1.09688	−0.548439	−	0.836191i	$-0.684778\pi$
$151$	−2035.28	−1.09688	−0.548439	+	0.836191i	$0.684778\pi$
$152$	−1230.88	−0.656826
$153$	1512.28	0.799088
$154$	−543.984	−0.284646
$155$	2365.92	1.22603
$156$	0	0
$157$	−615.393	−0.312826	−0.156413	−	0.987692i	$-0.549993\pi$
$157$	−615.393	−0.312826	−0.156413	+	0.987692i	$0.549993\pi$
$158$	2086.36	1.05052
$159$	3501.89	1.74665
$160$	−453.393	−0.224024
$161$	1031.50	0.504930
$162$	−755.696	−0.366501
$163$	−1160.09	−0.557456	−0.278728	−	0.960370i	$-0.589913\pi$
$163$	−1160.09	−0.557456	−0.278728	+	0.960370i	$0.589913\pi$
$164$	501.897	0.238973
$165$	1175.73	0.554731
$166$	−626.608	−0.292977
$167$	2029.26	0.940295	0.470147	−	0.882588i	$-0.344201\pi$
$167$	2029.26	0.940295	0.470147	+	0.882588i	$0.344201\pi$
$168$	−2003.45	−0.920057
$169$	0	0
$170$	−867.409	−0.391337
$171$	7601.30	3.39933
$172$	−1739.10	−0.770959
$173$	−3073.60	−1.35076	−0.675380	−	0.737469i	$-0.736021\pi$
$173$	−3073.60	−1.35076	−0.675380	+	0.737469i	$0.736021\pi$
$174$	−861.099	−0.375171
$175$	−2170.18	−0.937431
$176$	−151.896	−0.0650544
$177$	3568.12	1.51523
$178$	−1351.51	−0.569100
$179$	2313.89	0.966190	0.483095	−	0.875568i	$-0.339513\pi$
$179$	2313.89	0.966190	0.483095	+	0.875568i	$0.339513\pi$
$180$	2799.93	1.15941
$181$	−1670.01	−0.685805	−0.342903	−	0.939371i	$-0.611410\pi$
$181$	−1670.01	−0.685805	−0.342903	+	0.939371i	$0.611410\pi$
$182$	0	0
$183$	5277.44	2.13180
$184$	288.024	0.115399
$185$	−338.355	−0.134467
$186$	2919.20	1.15078
$187$	−290.600	−0.113640
$188$	−744.068	−0.288653
$189$	5610.67	2.15934
$190$	−4359.94	−1.66475
$191$	3825.21	1.44912	0.724561	−	0.689211i	$-0.242043\pi$
$191$	3825.21	1.44912	0.724561	+	0.689211i	$0.242043\pi$
$192$	−559.420	−0.210274
$193$	−908.832	−0.338959	−0.169480	−	0.985534i	$-0.554209\pi$
$193$	−908.832	−0.338959	−0.169480	+	0.985534i	$0.554209\pi$
$194$	544.869	0.201646
$195$	0	0
$196$	1911.38	0.696568
$197$	−1337.67	−0.483781	−0.241890	−	0.970304i	$-0.577767\pi$
$197$	−1337.67	−0.483781	−0.241890	+	0.970304i	$0.577767\pi$
$198$	938.033	0.336682
$199$	1144.58	0.407725	0.203862	−	0.979000i	$-0.434651\pi$
$199$	1144.58	0.407725	0.203862	+	0.979000i	$0.434651\pi$
$200$	−605.976	−0.214245
$201$	2513.44	0.882011
$202$	1323.12	0.460864
$203$	1411.22	0.487924
$204$	−1070.26	−0.367318
$205$	1777.78	0.605687
$206$	630.706	0.213317
$207$	−1778.69	−0.597236
$208$	0	0
$209$	−1460.67	−0.483428
$210$	−7096.49	−2.33192
$211$	−11.2252	−0.00366245	−0.00183123	−	0.999998i	$-0.500583\pi$
$211$	−11.2252	−0.00366245	−0.00183123	+	0.999998i	$0.500583\pi$
$212$	−1602.52	−0.519159
$213$	8407.06	2.70442
$214$	−4242.64	−1.35524
$215$	−6160.11	−1.95403
$216$	1566.66	0.493507
$217$	−4784.16	−1.49664
$218$	−1768.86	−0.549553
$219$	−8419.25	−2.59781
$220$	−538.035	−0.164883
$221$	0	0
$222$	−417.481	−0.126214
$223$	−2739.74	−0.822720	−0.411360	−	0.911473i	$-0.634946\pi$
$223$	−2739.74	−0.822720	−0.411360	+	0.911473i	$0.634946\pi$
$224$	916.813	0.273469
$225$	3742.21	1.10880
$226$	264.409	0.0778241
$227$	4529.75	1.32445	0.662225	−	0.749305i	$-0.269612\pi$
$227$	4529.75	1.32445	0.662225	+	0.749305i	$0.269612\pi$
$228$	−5379.52	−1.56258
$229$	−867.892	−0.250445	−0.125222	−	0.992129i	$-0.539964\pi$
$229$	−867.892	−0.250445	−0.125222	+	0.992129i	$0.539964\pi$
$230$	1020.22	0.292484
$231$	−2377.47	−0.677168
$232$	394.053	0.111512
$233$	535.138	0.150464	0.0752319	−	0.997166i	$-0.476030\pi$
$233$	535.138	0.150464	0.0752319	+	0.997166i	$0.476030\pi$
$234$	0	0
$235$	−2635.59	−0.731603
$236$	−1632.83	−0.450374
$237$	9118.36	2.49916
$238$	1754.00	0.477710
$239$	−1146.73	−0.310359	−0.155180	−	0.987886i	$-0.549596\pi$
$239$	−1146.73	−0.310359	−0.155180	+	0.987886i	$0.549596\pi$
$240$	−1981.54	−0.532949
$241$	−3353.03	−0.896215	−0.448107	−	0.893980i	$-0.647902\pi$
$241$	−3353.03	−0.896215	−0.448107	+	0.893980i	$0.647902\pi$
$242$	2481.75	0.659226
$243$	1984.72	0.523950
$244$	−2415.05	−0.633638
$245$	6770.36	1.76548
$246$	2193.52	0.568512
$247$	0	0
$248$	−1335.87	−0.342048
$249$	−2738.57	−0.696987
$250$	1395.68	0.353083
$251$	5357.96	1.34738	0.673688	−	0.739016i	$-0.264709\pi$
$251$	5357.96	1.34738	0.673688	+	0.739016i	$0.264709\pi$
$252$	−5661.78	−1.41531
$253$	341.794	0.0849345
$254$	3745.62	0.925279
$255$	−3790.99	−0.930983
$256$	256.000	0.0625000
$257$	−6292.11	−1.52720	−0.763601	−	0.645689i	$-0.776570\pi$
$257$	−6292.11	−1.52720	−0.763601	+	0.645689i	$0.776570\pi$
$258$	−7600.67	−1.83410
$259$	684.194	0.164146
$260$	0	0
$261$	−2433.48	−0.577121
$262$	3700.43	0.872569
$263$	−4797.76	−1.12488	−0.562438	−	0.826840i	$-0.690136\pi$
$263$	−4797.76	−1.12488	−0.562438	+	0.826840i	$0.690136\pi$
$264$	−663.856	−0.154763
$265$	−5676.35	−1.31583
$266$	8816.30	2.03219
$267$	−5906.72	−1.35388
$268$	−1150.19	−0.262161
$269$	6246.43	1.41581	0.707903	−	0.706310i	$-0.249642\pi$
$269$	6246.43	1.41581	0.707903	+	0.706310i	$0.249642\pi$
$270$	5549.30	1.25081
$271$	1831.15	0.410460	0.205230	−	0.978714i	$-0.434206\pi$
$271$	1831.15	0.410460	0.205230	+	0.978714i	$0.434206\pi$
$272$	489.767	0.109178
$273$	0	0
$274$	−1934.75	−0.426578
$275$	−719.103	−0.157686
$276$	1258.80	0.274532
$277$	−1601.42	−0.347365	−0.173683	−	0.984802i	$-0.555567\pi$
$277$	−1601.42	−0.347365	−0.173683	+	0.984802i	$0.555567\pi$
$278$	207.194	0.0447002
$279$	8249.69	1.77024
$280$	3247.47	0.693120
$281$	4516.64	0.958861	0.479431	−	0.877580i	$-0.340843\pi$
$281$	4516.64	0.958861	0.479431	+	0.877580i	$0.340843\pi$
$282$	−3251.93	−0.686700
$283$	7083.02	1.48778	0.743890	−	0.668302i	$-0.232978\pi$
$283$	7083.02	1.48778	0.743890	+	0.668302i	$0.232978\pi$
$284$	−3847.21	−0.803838
$285$	−19055.0	−3.96042
$286$	0	0
$287$	−3594.89	−0.739371
$288$	−1580.93	−0.323462
$289$	−3976.00	−0.809282
$290$	1395.79	0.282633
$291$	2381.34	0.479713
$292$	3852.79	0.772149
$293$	6923.52	1.38047	0.690233	−	0.723588i	$-0.257508\pi$
$293$	6923.52	1.38047	0.690233	+	0.723588i	$0.257508\pi$
$294$	8353.64	1.65712
$295$	−5783.70	−1.14149
$296$	191.046	0.0375146
$297$	1859.13	0.363224
$298$	3834.87	0.745463
$299$	0	0
$300$	−2648.40	−0.509685
$301$	12456.5	2.38531
$302$	4070.56	0.775610
$303$	5782.66	1.09639
$304$	2461.76	0.464446
$305$	−8554.41	−1.60598
$306$	−3024.56	−0.565040
$307$	−8778.50	−1.63197	−0.815986	−	0.578071i	$-0.803805\pi$
$307$	−8778.50	−1.63197	−0.815986	+	0.578071i	$0.803805\pi$
$308$	1087.97	0.201275
$309$	2756.48	0.507478
$310$	−4731.84	−0.866936
$311$	−2197.35	−0.400644	−0.200322	−	0.979730i	$-0.564199\pi$
$311$	−2197.35	−0.400644	−0.200322	+	0.979730i	$0.564199\pi$
$312$	0	0
$313$	−5052.55	−0.912419	−0.456209	−	0.889873i	$-0.650793\pi$
$313$	−5052.55	−0.912419	−0.456209	+	0.889873i	$0.650793\pi$
$314$	1230.79	0.221201
$315$	−20054.8	−3.58717
$316$	−4172.72	−0.742828
$317$	−1889.91	−0.334852	−0.167426	−	0.985885i	$-0.553545\pi$
$317$	−1889.91	−0.334852	−0.167426	+	0.985885i	$0.553545\pi$
$318$	−7003.78	−1.23507
$319$	467.618	0.0820738
$320$	906.785	0.158409
$321$	−18542.3	−3.22409
$322$	−2063.00	−0.357039
$323$	4709.72	0.811318
$324$	1511.39	0.259155
$325$	0	0
$326$	2320.18	0.394181
$327$	−7730.76	−1.30738
$328$	−1003.79	−0.168979
$329$	5329.46	0.893078
$330$	−2351.46	−0.392254
$331$	2549.39	0.423344	0.211672	−	0.977341i	$-0.432109\pi$
$331$	2549.39	0.423344	0.211672	+	0.977341i	$0.432109\pi$
$332$	1253.22	0.207166
$333$	−1179.81	−0.194153
$334$	−4058.53	−0.664889
$335$	−4074.13	−0.664458
$336$	4006.90	0.650579
$337$	140.649	0.0227349	0.0113674	−	0.999935i	$-0.496382\pi$
$337$	140.649	0.0227349	0.0113674	+	0.999935i	$0.496382\pi$
$338$	0	0
$339$	1155.59	0.185142
$340$	1734.82	0.276717
$341$	−1585.26	−0.251750
$342$	−15202.6	−2.40369
$343$	−3863.38	−0.608171
$344$	3478.19	0.545150
$345$	4458.84	0.695814
$346$	6147.20	0.955132
$347$	140.513	0.0217382	0.0108691	−	0.999941i	$-0.496540\pi$
$347$	140.513	0.0217382	0.0108691	+	0.999941i	$0.496540\pi$
$348$	1722.20	0.265286
$349$	−12273.2	−1.88244	−0.941220	−	0.337794i	$-0.890319\pi$
$349$	−12273.2	−1.88244	−0.941220	+	0.337794i	$0.890319\pi$
$350$	4340.37	0.662864
$351$	0	0
$352$	303.792	0.0460004
$353$	−5887.97	−0.887776	−0.443888	−	0.896082i	$-0.646401\pi$
$353$	−5887.97	−0.887776	−0.443888	+	0.896082i	$0.646401\pi$
$354$	−7136.24	−1.07143
$355$	−13627.3	−2.03736
$356$	2703.02	0.402414
$357$	7665.81	1.13646
$358$	−4627.77	−0.683199
$359$	5902.88	0.867806	0.433903	−	0.900960i	$-0.357136\pi$
$359$	5902.88	0.867806	0.433903	+	0.900960i	$0.357136\pi$
$360$	−5599.85	−0.819828
$361$	16813.9	2.45136
$362$	3340.02	0.484937
$363$	10846.4	1.56829
$364$	0	0
$365$	13647.1	1.95704
$366$	−10554.9	−1.50741
$367$	−3318.93	−0.472062	−0.236031	−	0.971746i	$-0.575847\pi$
$367$	−3318.93	−0.472062	−0.236031	+	0.971746i	$0.575847\pi$
$368$	−576.049	−0.0815995
$369$	6198.93	0.874535
$370$	676.711	0.0950825
$371$	11478.2	1.60625
$372$	−5838.39	−0.813727
$373$	242.215	0.0336232	0.0168116	−	0.999859i	$-0.494648\pi$
$373$	242.215	0.0336232	0.0168116	+	0.999859i	$0.494648\pi$
$374$	581.199	0.0803559
$375$	6099.79	0.839979
$376$	1488.14	0.204108
$377$	0	0
$378$	−11221.3	−1.52689
$379$	12904.3	1.74894	0.874469	−	0.485081i	$-0.161210\pi$
$379$	12904.3	1.74894	0.874469	+	0.485081i	$0.161210\pi$
$380$	8719.87	1.17716
$381$	16370.1	2.20122
$382$	−7650.41	−1.02468
$383$	4591.02	0.612507	0.306253	−	0.951950i	$-0.400925\pi$
$383$	4591.02	0.612507	0.306253	+	0.951950i	$0.400925\pi$
$384$	1118.84	0.148686
$385$	3853.73	0.510141
$386$	1817.66	0.239680
$387$	−21479.6	−2.82137
$388$	−1089.74	−0.142585
$389$	−13973.1	−1.82125	−0.910623	−	0.413239i	$-0.864397\pi$
$389$	−13973.1	−1.82125	−0.910623	+	0.413239i	$0.864397\pi$
$390$	0	0
$391$	−1102.07	−0.142542
$392$	−3822.76	−0.492548
$393$	16172.6	2.07583
$394$	2675.33	0.342085
$395$	−14780.3	−1.88273
$396$	−1876.07	−0.238070
$397$	−2617.73	−0.330932	−0.165466	−	0.986215i	$-0.552913\pi$
$397$	−2617.73	−0.330932	−0.165466	+	0.986215i	$0.552913\pi$
$398$	−2289.16	−0.288305
$399$	38531.4	4.83454
$400$	1211.95	0.151494
$401$	−7889.19	−0.982463	−0.491231	−	0.871029i	$-0.663453\pi$
$401$	−7889.19	−0.982463	−0.491231	+	0.871029i	$0.663453\pi$
$402$	−5026.87	−0.623676
$403$	0	0
$404$	−2646.24	−0.325880
$405$	5353.55	0.656839
$406$	−2822.45	−0.345014
$407$	226.712	0.0276110
$408$	2140.51	0.259733
$409$	−7731.54	−0.934719	−0.467359	−	0.884067i	$-0.654795\pi$
$409$	−7731.54	−0.934719	−0.467359	+	0.884067i	$0.654795\pi$
$410$	−3555.57	−0.428285
$411$	−8455.76	−1.01482
$412$	−1261.41	−0.150838
$413$	11695.3	1.39344
$414$	3557.39	0.422309
$415$	4439.05	0.525071
$416$	0	0
$417$	905.535	0.106341
$418$	2921.33	0.341835
$419$	2756.41	0.321383	0.160691	−	0.987005i	$-0.448628\pi$
$419$	2756.41	0.321383	0.160691	+	0.987005i	$0.448628\pi$
$420$	14193.0	1.64892
$421$	11815.2	1.36778	0.683890	−	0.729585i	$-0.260287\pi$
$421$	11815.2	1.36778	0.683890	+	0.729585i	$0.260287\pi$
$422$	22.4505	0.00258974
$423$	−9189.98	−1.05634
$424$	3205.05	0.367101
$425$	2318.65	0.264638
$426$	−16814.1	−1.91232
$427$	17298.0	1.96044
$428$	8485.29	0.958299
$429$	0	0
$430$	12320.2	1.38171
$431$	−10601.2	−1.18479	−0.592393	−	0.805649i	$-0.701817\pi$
$431$	−10601.2	−1.18479	−0.592393	+	0.805649i	$0.701817\pi$
$432$	−3133.31	−0.348962
$433$	−9229.72	−1.02437	−0.512185	−	0.858875i	$-0.671164\pi$
$433$	−9229.72	−1.02437	−0.512185	+	0.858875i	$0.671164\pi$
$434$	9568.32	1.05828
$435$	6100.25	0.672379
$436$	3537.73	0.388593
$437$	−5539.42	−0.606377
$438$	16838.5	1.83693
$439$	7201.66	0.782953	0.391476	−	0.920188i	$-0.371964\pi$
$439$	7201.66	0.782953	0.391476	+	0.920188i	$0.371964\pi$
$440$	1076.07	0.116590
$441$	23607.5	2.54913
$442$	0	0
$443$	−3544.70	−0.380167	−0.190083	−	0.981768i	$-0.560876\pi$
$443$	−3544.70	−0.380167	−0.190083	+	0.981768i	$0.560876\pi$
$444$	834.962	0.0892467
$445$	9574.43	1.01994
$446$	5479.48	0.581751
$447$	16760.2	1.77344
$448$	−1833.63	−0.193372
$449$	−18325.4	−1.92612	−0.963059	−	0.269289i	$-0.913211\pi$
$449$	−18325.4	−1.92612	−0.963059	+	0.269289i	$0.913211\pi$
$450$	−7484.41	−0.784041
$451$	−1191.19	−0.124370
$452$	−528.818	−0.0550299
$453$	17790.2	1.84516
$454$	−9059.50	−0.936527
$455$	0	0
$456$	10759.0	1.10491
$457$	−5383.84	−0.551084	−0.275542	−	0.961289i	$-0.588857\pi$
$457$	−5383.84	−0.551084	−0.275542	+	0.961289i	$0.588857\pi$
$458$	1735.78	0.177091
$459$	−5994.51	−0.609585
$460$	−2040.44	−0.206817
$461$	−10621.1	−1.07305	−0.536525	−	0.843885i	$-0.680263\pi$
$461$	−10621.1	−1.07305	−0.536525	+	0.843885i	$0.680263\pi$
$462$	4754.93	0.478830
$463$	10407.3	1.04464	0.522318	−	0.852751i	$-0.325067\pi$
$463$	10407.3	1.04464	0.522318	+	0.852751i	$0.325067\pi$
$464$	−788.107	−0.0788512
$465$	−20680.3	−2.06243
$466$	−1070.28	−0.106394
$467$	−15784.0	−1.56402	−0.782010	−	0.623266i	$-0.785805\pi$
$467$	−15784.0	−1.56402	−0.782010	+	0.623266i	$0.785805\pi$
$468$	0	0
$469$	8238.36	0.811113
$470$	5271.17	0.517321
$471$	5379.11	0.526234
$472$	3265.66	0.318462
$473$	4127.52	0.401234
$474$	−18236.7	−1.76717
$475$	11654.4	1.12577
$476$	−3508.00	−0.337792
$477$	−19792.8	−1.89989
$478$	2293.46	0.219457
$479$	−18263.7	−1.74215	−0.871075	−	0.491149i	$-0.836577\pi$
$479$	−18263.7	−1.74215	−0.871075	+	0.491149i	$0.836577\pi$
$480$	3963.08	0.376852
$481$	0	0
$482$	6706.06	0.633720
$483$	−9016.29	−0.849390
$484$	−4963.49	−0.466143
$485$	−3860.00	−0.361389
$486$	−3969.44	−0.370489
$487$	−5369.01	−0.499575	−0.249788	−	0.968301i	$-0.580361\pi$
$487$	−5369.01	−0.499575	−0.249788	+	0.968301i	$0.580361\pi$
$488$	4830.09	0.448049
$489$	10140.3	0.937750
$490$	−13540.7	−1.24838
$491$	−1491.33	−0.137073	−0.0685366	−	0.997649i	$-0.521833\pi$
$491$	−1491.33	−0.137073	−0.0685366	+	0.997649i	$0.521833\pi$
$492$	−4387.05	−0.401999
$493$	−1507.77	−0.137741
$494$	0	0
$495$	−6645.27	−0.603399
$496$	2671.75	0.241865
$497$	27556.0	2.48704
$498$	5477.14	0.492844
$499$	−10067.4	−0.903163	−0.451582	−	0.892230i	$-0.649140\pi$
$499$	−10067.4	−0.903163	−0.451582	+	0.892230i	$0.649140\pi$
$500$	−2791.37	−0.249668
$501$	−17737.7	−1.58176
$502$	−10715.9	−0.952738
$503$	4683.07	0.415124	0.207562	−	0.978222i	$-0.433447\pi$
$503$	4683.07	0.415124	0.207562	+	0.978222i	$0.433447\pi$
$504$	11323.6	1.00078
$505$	−9373.34	−0.825957
$506$	−683.589	−0.0600577
$507$	0	0
$508$	−7491.23	−0.654271
$509$	11545.0	1.00535	0.502673	−	0.864476i	$-0.332350\pi$
$509$	11545.0	1.00535	0.502673	+	0.864476i	$0.332350\pi$
$510$	7581.97	0.658305
$511$	−27596.0	−2.38899
$512$	−512.000	−0.0441942
$513$	−30130.7	−2.59318
$514$	12584.2	1.07989
$515$	−4468.09	−0.382306
$516$	15201.3	1.29690
$517$	1765.95	0.150225
$518$	−1368.39	−0.116069
$519$	26866.2	2.27224
$520$	0	0
$521$	19889.6	1.67252	0.836258	−	0.548336i	$-0.184739\pi$
$521$	19889.6	1.67252	0.836258	+	0.548336i	$0.184739\pi$
$522$	4866.96	0.408086
$523$	−5090.07	−0.425570	−0.212785	−	0.977099i	$-0.568253\pi$
$523$	−5090.07	−0.425570	−0.212785	+	0.977099i	$0.568253\pi$
$524$	−7400.86	−0.617000
$525$	18969.4	1.57694
$526$	9595.51	0.795407
$527$	5111.46	0.422502
$528$	1327.71	0.109434
$529$	−10870.8	−0.893464
$530$	11352.7	0.930433
$531$	−20167.1	−1.64817
$532$	−17632.6	−1.43697
$533$	0	0
$534$	11813.4	0.957337
$535$	30056.0	2.42885
$536$	2300.38	0.185376
$537$	−20225.5	−1.62532
$538$	−12492.9	−1.00113
$539$	−4536.42	−0.362518
$540$	−11098.6	−0.884459
$541$	−5958.49	−0.473523	−0.236761	−	0.971568i	$-0.576086\pi$
$541$	−5958.49	−0.473523	−0.236761	+	0.971568i	$0.576086\pi$
$542$	−3662.30	−0.290239
$543$	14597.4	1.15366
$544$	−979.534	−0.0772007
$545$	12531.1	0.984905
$546$	0	0
$547$	2475.52	0.193502	0.0967509	−	0.995309i	$-0.469155\pi$
$547$	2475.52	0.193502	0.0967509	+	0.995309i	$0.469155\pi$
$548$	3869.50	0.301637
$549$	−29828.3	−2.31883
$550$	1438.21	0.111501
$551$	−7578.63	−0.585954
$552$	−2517.60	−0.194124
$553$	29887.5	2.29827
$554$	3202.85	0.245624
$555$	2957.54	0.226199
$556$	−414.388	−0.0316078
$557$	23311.5	1.77332	0.886662	−	0.462417i	$-0.153018\pi$
$557$	23311.5	1.77332	0.886662	+	0.462417i	$0.153018\pi$
$558$	−16499.4	−1.25175
$559$	0	0
$560$	−6494.94	−0.490110
$561$	2540.11	0.191165
$562$	−9033.27	−0.678017
$563$	24626.2	1.84347	0.921733	−	0.387825i	$-0.126773\pi$
$563$	24626.2	1.84347	0.921733	+	0.387825i	$0.126773\pi$
$564$	6503.85	0.485570
$565$	−1873.14	−0.139476
$566$	−14166.0	−1.05202
$567$	−10825.5	−0.801813
$568$	7694.42	0.568399
$569$	−13198.6	−0.972431	−0.486215	−	0.873839i	$-0.661623\pi$
$569$	−13198.6	−0.972431	−0.486215	+	0.873839i	$0.661623\pi$
$570$	38109.9	2.80044
$571$	20934.2	1.53427	0.767137	−	0.641483i	$-0.221681\pi$
$571$	20934.2	1.53427	0.767137	+	0.641483i	$0.221681\pi$
$572$	0	0
$573$	−33435.9	−2.43770
$574$	7189.77	0.522814
$575$	−2727.12	−0.197789
$576$	3161.86	0.228722
$577$	−17083.2	−1.23255	−0.616275	−	0.787531i	$-0.711359\pi$
$577$	−17083.2	−1.23255	−0.616275	+	0.787531i	$0.711359\pi$
$578$	7952.00	0.572249
$579$	7944.04	0.570195
$580$	−2791.58	−0.199852
$581$	−8976.28	−0.640962
$582$	−4762.67	−0.339208
$583$	3803.38	0.270189
$584$	−7705.58	−0.545992
$585$	0	0
$586$	−13847.0	−0.976136
$587$	−4458.82	−0.313518	−0.156759	−	0.987637i	$-0.550105\pi$
$587$	−4458.82	−0.313518	−0.156759	+	0.987637i	$0.550105\pi$
$588$	−16707.3	−1.17176
$589$	25692.2	1.79733
$590$	11567.4	0.807156
$591$	11692.5	0.813813
$592$	−382.092	−0.0265269
$593$	3217.47	0.222809	0.111404	−	0.993775i	$-0.464465\pi$
$593$	3217.47	0.222809	0.111404	+	0.993775i	$0.464465\pi$
$594$	−3718.26	−0.256838
$595$	−12425.8	−0.856149
$596$	−7669.74	−0.527122
$597$	−10004.7	−0.685872
$598$	0	0
$599$	−4495.37	−0.306637	−0.153319	−	0.988177i	$-0.548996\pi$
$599$	−4495.37	−0.306637	−0.153319	+	0.988177i	$0.548996\pi$
$600$	5296.80	0.360402
$601$	−10918.5	−0.741054	−0.370527	−	0.928822i	$-0.620823\pi$
$601$	−10918.5	−0.741054	−0.370527	+	0.928822i	$0.620823\pi$
$602$	−24912.9	−1.68667
$603$	−14206.0	−0.959392
$604$	−8141.11	−0.548439
$605$	−17581.3	−1.18146
$606$	−11565.3	−0.775263
$607$	12487.0	0.834980	0.417490	−	0.908682i	$-0.362910\pi$
$607$	12487.0	0.834980	0.417490	+	0.908682i	$0.362910\pi$
$608$	−4923.52	−0.328413
$609$	−12335.4	−0.820782
$610$	17108.8	1.13560
$611$	0	0
$612$	6049.11	0.399544
$613$	−961.468	−0.0633496	−0.0316748	−	0.999498i	$-0.510084\pi$
$613$	−961.468	−0.0633496	−0.0316748	+	0.999498i	$0.510084\pi$
$614$	17557.0	1.15398
$615$	−15539.5	−1.01888
$616$	−2175.94	−0.142323
$617$	15433.5	1.00701	0.503507	−	0.863991i	$-0.332043\pi$
$617$	15433.5	1.00701	0.503507	+	0.863991i	$0.332043\pi$
$618$	−5512.96	−0.358841
$619$	−24569.6	−1.59537	−0.797686	−	0.603073i	$-0.793943\pi$
$619$	−24569.6	−1.59537	−0.797686	+	0.603073i	$0.793943\pi$
$620$	9463.67	0.613017
$621$	7050.55	0.455602
$622$	4394.70	0.283298
$623$	−19360.6	−1.24505
$624$	0	0
$625$	−19355.8	−1.23877
$626$	10105.1	0.645177
$627$	12767.6	0.813220
$628$	−2461.57	−0.156413
$629$	−731.001	−0.0463385
$630$	40109.5	2.53651
$631$	−13322.2	−0.840489	−0.420245	−	0.907411i	$-0.638056\pi$
$631$	−13322.2	−0.840489	−0.420245	+	0.907411i	$0.638056\pi$
$632$	8345.43	0.525259
$633$	98.1191	0.00616096
$634$	3779.82	0.236776
$635$	−26534.9	−1.65828
$636$	14007.6	0.873327
$637$	0	0
$638$	−935.235	−0.0580350
$639$	−47516.9	−2.94169
$640$	−1813.57	−0.112012
$641$	14894.6	0.917784	0.458892	−	0.888492i	$-0.348246\pi$
$641$	14894.6	0.917784	0.458892	+	0.888492i	$0.348246\pi$
$642$	37084.7	2.27978
$643$	−9247.56	−0.567167	−0.283583	−	0.958948i	$-0.591523\pi$
$643$	−9247.56	−0.567167	−0.283583	+	0.958948i	$0.591523\pi$
$644$	4126.01	0.252465
$645$	53845.1	3.28705
$646$	−9419.44	−0.573689
$647$	2734.14	0.166136	0.0830681	−	0.996544i	$-0.473528\pi$
$647$	2734.14	0.166136	0.0830681	+	0.996544i	$0.473528\pi$
$648$	−3022.78	−0.183250
$649$	3875.31	0.234390
$650$	0	0
$651$	41818.1	2.51763
$652$	−4640.37	−0.278728
$653$	−6562.78	−0.393295	−0.196647	−	0.980474i	$-0.563005\pi$
$653$	−6562.78	−0.393295	−0.196647	+	0.980474i	$0.563005\pi$
$654$	15461.5	0.924455
$655$	−26214.8	−1.56381
$656$	2007.59	0.119486
$657$	47585.8	2.82572
$658$	−10658.9	−0.631501
$659$	−4953.84	−0.292829	−0.146414	−	0.989223i	$-0.546773\pi$
$659$	−4953.84	−0.292829	−0.146414	+	0.989223i	$0.546773\pi$
$660$	4702.93	0.277365
$661$	9347.27	0.550025	0.275013	−	0.961441i	$-0.411318\pi$
$661$	9347.27	0.550025	0.275013	+	0.961441i	$0.411318\pi$
$662$	−5098.78	−0.299350
$663$	0	0
$664$	−2506.43	−0.146489
$665$	−62456.9	−3.64207
$666$	2359.61	0.137287
$667$	1773.39	0.102947
$668$	8117.06	0.470147
$669$	23947.9	1.38398
$670$	8148.25	0.469843
$671$	5731.80	0.329767
$672$	−8013.80	−0.460029
$673$	15419.0	0.883150	0.441575	−	0.897224i	$-0.354420\pi$
$673$	15419.0	0.883150	0.441575	+	0.897224i	$0.354420\pi$
$674$	−281.298	−0.0160760
$675$	−14833.7	−0.845851
$676$	0	0
$677$	26079.0	1.48050	0.740248	−	0.672334i	$-0.234708\pi$
$677$	26079.0	1.48050	0.740248	+	0.672334i	$0.234708\pi$
$678$	−2311.18	−0.130915
$679$	7805.36	0.441152
$680$	−3469.64	−0.195668
$681$	−39594.3	−2.22798
$682$	3170.52	0.178014
$683$	−14890.0	−0.834190	−0.417095	−	0.908863i	$-0.636952\pi$
$683$	−14890.0	−0.834190	−0.417095	+	0.908863i	$0.636952\pi$
$684$	30405.2	1.69967
$685$	13706.3	0.764510
$686$	7726.75	0.430042
$687$	7586.19	0.421297
$688$	−6956.39	−0.385479
$689$	0	0
$690$	−8917.68	−0.492015
$691$	−4071.03	−0.224123	−0.112062	−	0.993701i	$-0.535745\pi$
$691$	−4071.03	−0.224123	−0.112062	+	0.993701i	$0.535745\pi$
$692$	−12294.4	−0.675380
$693$	13437.5	0.736578
$694$	−281.026	−0.0153712
$695$	−1467.82	−0.0801114
$696$	−3444.40	−0.187586
$697$	3840.82	0.208725
$698$	24546.5	1.33109
$699$	−4677.61	−0.253109
$700$	−8680.73	−0.468715
$701$	18378.1	0.990200	0.495100	−	0.868836i	$-0.335131\pi$
$701$	18378.1	0.990200	0.495100	+	0.868836i	$0.335131\pi$
$702$	0	0
$703$	−3674.29	−0.197125
$704$	−607.583	−0.0325272
$705$	23037.5	1.23070
$706$	11775.9	0.627753
$707$	18954.0	1.00826
$708$	14272.5	0.757616
$709$	14946.4	0.791710	0.395855	−	0.918313i	$-0.370448\pi$
$709$	14946.4	0.791710	0.395855	+	0.918313i	$0.370448\pi$
$710$	27254.7	1.44063
$711$	−51537.2	−2.71842
$712$	−5406.03	−0.284550
$713$	−6011.94	−0.315777
$714$	−15331.6	−0.803602
$715$	0	0
$716$	9255.55	0.483095
$717$	10023.5	0.522084
$718$	−11805.8	−0.613631
$719$	15005.8	0.778335	0.389168	−	0.921167i	$-0.372763\pi$
$719$	15005.8	0.778335	0.389168	+	0.921167i	$0.372763\pi$
$720$	11199.7	0.579706
$721$	9034.99	0.466686
$722$	−33627.8	−1.73337
$723$	29308.6	1.50761
$724$	−6680.03	−0.342903
$725$	−3731.05	−0.191128
$726$	−21692.8	−1.10895
$727$	−30390.1	−1.55035	−0.775177	−	0.631744i	$-0.782339\pi$
$727$	−30390.1	−1.55035	−0.775177	+	0.631744i	$0.782339\pi$
$728$	0	0
$729$	−27550.2	−1.39970
$730$	−27294.2	−1.38384
$731$	−13308.6	−0.673375
$732$	21109.8	1.06590
$733$	31143.3	1.56931	0.784655	−	0.619932i	$-0.212840\pi$
$733$	31143.3	1.56931	0.784655	+	0.619932i	$0.212840\pi$
$734$	6637.86	0.333798
$735$	−59179.3	−2.96988
$736$	1152.10	0.0576995
$737$	2729.83	0.136438
$738$	−12397.9	−0.618389
$739$	3884.81	0.193376	0.0966881	−	0.995315i	$-0.469175\pi$
$739$	3884.81	0.193376	0.0966881	+	0.995315i	$0.469175\pi$
$740$	−1353.42	−0.0672334
$741$	0	0
$742$	−22956.5	−1.13579
$743$	11614.8	0.573495	0.286748	−	0.958006i	$-0.407426\pi$
$743$	11614.8	0.573495	0.286748	+	0.958006i	$0.407426\pi$
$744$	11676.8	0.575392
$745$	−27167.2	−1.33601
$746$	−484.431	−0.0237752
$747$	15478.5	0.758136
$748$	−1162.40	−0.0568202
$749$	−60776.7	−2.96493
$750$	−12199.6	−0.593955
$751$	−34410.2	−1.67196	−0.835982	−	0.548757i	$-0.815101\pi$
$751$	−34410.2	−1.67196	−0.835982	+	0.548757i	$0.815101\pi$
$752$	−2976.27	−0.144326
$753$	−46833.6	−2.26655
$754$	0	0
$755$	−28836.9	−1.39004
$756$	22442.7	1.07967
$757$	−29456.8	−1.41430	−0.707150	−	0.707063i	$-0.750020\pi$
$757$	−29456.8	−1.41430	−0.707150	+	0.707063i	$0.750020\pi$
$758$	−25808.5	−1.23669
$759$	−2987.60	−0.142876
$760$	−17439.7	−0.832376
$761$	32950.1	1.56957	0.784784	−	0.619769i	$-0.212774\pi$
$761$	32950.1	1.56957	0.784784	+	0.619769i	$0.212774\pi$
$762$	−32740.2	−1.55650
$763$	−25339.3	−1.20229
$764$	15300.8	0.724561
$765$	21426.7	1.01266
$766$	−9182.04	−0.433108
$767$	0	0
$768$	−2237.68	−0.105137
$769$	10223.7	0.479424	0.239712	−	0.970844i	$-0.422947\pi$
$769$	10223.7	0.479424	0.239712	+	0.970844i	$0.422947\pi$
$770$	−7707.45	−0.360724
$771$	54998.9	2.56905
$772$	−3635.33	−0.169480
$773$	10140.1	0.471815	0.235908	−	0.971776i	$-0.424194\pi$
$773$	10140.1	0.471815	0.235908	+	0.971776i	$0.424194\pi$
$774$	42959.2	1.99501
$775$	12648.6	0.586257
$776$	2179.48	0.100823
$777$	−5980.50	−0.276125
$778$	27946.2	1.28781
$779$	19305.4	0.887920
$780$	0	0
$781$	9130.86	0.418346
$782$	2204.14	0.100793
$783$	9646.04	0.440257
$784$	7645.53	0.348284
$785$	−8719.21	−0.396435
$786$	−32345.2	−1.46783
$787$	−9088.53	−0.411653	−0.205827	−	0.978588i	$-0.565988\pi$
$787$	−9088.53	−0.411653	−0.205827	+	0.978588i	$0.565988\pi$
$788$	−5350.67	−0.241890
$789$	41936.9	1.89226
$790$	29560.6	1.33129
$791$	3787.71	0.170260
$792$	3752.13	0.168341
$793$	0	0
$794$	5235.46	0.234004
$795$	49616.6	2.21348
$796$	4578.32	0.203862
$797$	−21032.4	−0.934764	−0.467382	−	0.884055i	$-0.654803\pi$
$797$	−21032.4	−0.934764	−0.467382	+	0.884055i	$0.654803\pi$
$798$	−77062.7	−3.41853
$799$	−5694.06	−0.252117
$800$	−2423.90	−0.107122
$801$	33385.0	1.47266
$802$	15778.4	0.694706
$803$	−9144.10	−0.401854
$804$	10053.7	0.441005
$805$	14614.9	0.639883
$806$	0	0
$807$	−54599.7	−2.38166
$808$	5292.49	0.230432
$809$	−43644.9	−1.89675	−0.948376	−	0.317147i	$-0.897275\pi$
$809$	−43644.9	−1.89675	−0.948376	+	0.317147i	$0.897275\pi$
$810$	−10707.1	−0.464456
$811$	28688.1	1.24214	0.621069	−	0.783756i	$-0.286699\pi$
$811$	28688.1	1.24214	0.621069	+	0.783756i	$0.286699\pi$
$812$	5644.89	0.243962
$813$	−16006.0	−0.690473
$814$	−453.424	−0.0195239
$815$	−16436.8	−0.706448
$816$	−4281.02	−0.183659
$817$	−66894.3	−2.86455
$818$	15463.1	0.660946
$819$	0	0
$820$	7111.14	0.302843
$821$	16042.2	0.681947	0.340973	−	0.940073i	$-0.389243\pi$
$821$	16042.2	0.681947	0.340973	+	0.940073i	$0.389243\pi$
$822$	16911.5	0.717588
$823$	−12138.9	−0.514137	−0.257068	−	0.966393i	$-0.582757\pi$
$823$	−12138.9	−0.514137	−0.257068	+	0.966393i	$0.582757\pi$
$824$	2522.82	0.106659
$825$	6285.64	0.265258
$826$	−23390.6	−0.985307
$827$	−17136.2	−0.720538	−0.360269	−	0.932848i	$-0.617315\pi$
$827$	−17136.2	−0.720538	−0.360269	+	0.932848i	$0.617315\pi$
$828$	−7114.78	−0.298618
$829$	10947.3	0.458642	0.229321	−	0.973351i	$-0.426349\pi$
$829$	10947.3	0.458642	0.229321	+	0.973351i	$0.426349\pi$
$830$	−8878.11	−0.371281
$831$	13997.9	0.584336
$832$	0	0
$833$	14627.1	0.608400
$834$	−1811.07	−0.0751945
$835$	28751.7	1.19161
$836$	−5842.67	−0.241714
$837$	−32700.9	−1.35043
$838$	−5512.82	−0.227252
$839$	13628.7	0.560805	0.280402	−	0.959883i	$-0.409532\pi$
$839$	13628.7	0.560805	0.280402	+	0.959883i	$0.409532\pi$
$840$	−28385.9	−1.16596
$841$	−21962.8	−0.900520
$842$	−23630.3	−0.967167
$843$	−39479.6	−1.61299
$844$	−44.9010	−0.00183123
$845$	0	0
$846$	18380.0	0.746946
$847$	35551.5	1.44223
$848$	−6410.09	−0.259580
$849$	−61912.3	−2.50274
$850$	−4637.30	−0.187127
$851$	859.780	0.0346332
$852$	33628.2	1.35221
$853$	−6645.59	−0.266753	−0.133377	−	0.991065i	$-0.542582\pi$
$853$	−6645.59	−0.266753	−0.133377	+	0.991065i	$0.542582\pi$
$854$	−34596.0	−1.38624
$855$	107699.	4.30788
$856$	−16970.6	−0.677619
$857$	15552.6	0.619914	0.309957	−	0.950751i	$-0.399685\pi$
$857$	15552.6	0.619914	0.309957	+	0.950751i	$0.399685\pi$
$858$	0	0
$859$	−48006.5	−1.90682	−0.953412	−	0.301672i	$-0.902455\pi$
$859$	−48006.5	−1.90682	−0.953412	+	0.301672i	$0.902455\pi$
$860$	−24640.4	−0.977014
$861$	31422.7	1.24377
$862$	21202.4	0.837770
$863$	3572.64	0.140920	0.0704601	−	0.997515i	$-0.477553\pi$
$863$	3572.64	0.140920	0.0704601	+	0.997515i	$0.477553\pi$
$864$	6266.63	0.246753
$865$	−43548.4	−1.71178
$866$	18459.4	0.724339
$867$	34754.0	1.36137
$868$	−19136.6	−0.748318
$869$	9903.40	0.386594
$870$	−12200.5	−0.475443
$871$	0	0
$872$	−7075.45	−0.274777
$873$	−13459.4	−0.521799
$874$	11078.8	0.428773
$875$	19993.4	0.772459
$876$	−33677.0	−1.29891
$877$	2699.84	0.103953	0.0519767	−	0.998648i	$-0.483448\pi$
$877$	2699.84	0.103953	0.0519767	+	0.998648i	$0.483448\pi$
$878$	−14403.3	−0.553631
$879$	−60518.1	−2.32221
$880$	−2152.14	−0.0824416
$881$	3982.46	0.152295	0.0761477	−	0.997097i	$-0.475738\pi$
$881$	3982.46	0.152295	0.0761477	+	0.997097i	$0.475738\pi$
$882$	−47215.0	−1.80251
$883$	21896.7	0.834523	0.417261	−	0.908787i	$-0.362990\pi$
$883$	21896.7	0.834523	0.417261	+	0.908787i	$0.362990\pi$
$884$	0	0
$885$	50555.0	1.92021
$886$	7089.41	0.268819
$887$	−21219.7	−0.803254	−0.401627	−	0.915803i	$-0.631555\pi$
$887$	−21219.7	−0.803254	−0.401627	+	0.915803i	$0.631555\pi$
$888$	−1669.92	−0.0631069
$889$	53656.7	2.02428
$890$	−19148.9	−0.721204
$891$	−3587.09	−0.134873
$892$	−10959.0	−0.411360
$893$	−28620.6	−1.07251
$894$	−33520.4	−1.25401
$895$	32784.4	1.22442
$896$	3667.25	0.136735
$897$	0	0
$898$	36650.7	1.36197
$899$	−8225.08	−0.305141
$900$	14968.8	0.554401
$901$	−12263.5	−0.453447
$902$	2382.37	0.0879428
$903$	−108881.	−4.01256
$904$	1057.64	0.0389120
$905$	−23661.5	−0.869101
$906$	−35580.5	−1.30473
$907$	−42873.6	−1.56957	−0.784783	−	0.619771i	$-0.787225\pi$
$907$	−42873.6	−1.56957	−0.784783	+	0.619771i	$0.787225\pi$
$908$	18119.0	0.662225
$909$	−32683.8	−1.19258
$910$	0	0
$911$	28943.9	1.05264	0.526319	−	0.850287i	$-0.323572\pi$
$911$	28943.9	1.05264	0.526319	+	0.850287i	$0.323572\pi$
$912$	−21518.1	−0.781288
$913$	−2974.35	−0.107817
$914$	10767.7	0.389675
$915$	74773.6	2.70157
$916$	−3471.57	−0.125222
$917$	53009.4	1.90897
$918$	11989.0	0.431042
$919$	2735.02	0.0981720	0.0490860	−	0.998795i	$-0.484369\pi$
$919$	2735.02	0.0981720	0.0490860	+	0.998795i	$0.484369\pi$
$920$	4080.88	0.146242
$921$	76732.3	2.74529
$922$	21242.3	0.758760
$923$	0	0
$924$	−9509.87	−0.338584
$925$	−1808.90	−0.0642986
$926$	−20814.5	−0.738669
$927$	−15579.7	−0.552001
$928$	1576.21	0.0557562
$929$	15091.8	0.532988	0.266494	−	0.963837i	$-0.414135\pi$
$929$	15091.8	0.532988	0.266494	+	0.963837i	$0.414135\pi$
$930$	41360.7	1.45836
$931$	73521.3	2.58814
$932$	2140.55	0.0752319
$933$	19206.9	0.673961
$934$	31568.0	1.10593
$935$	−4117.37	−0.144013
$936$	0	0
$937$	−20974.5	−0.731278	−0.365639	−	0.930757i	$-0.619150\pi$
$937$	−20974.5	−0.731278	−0.365639	+	0.930757i	$0.619150\pi$
$938$	−16476.7	−0.573544
$939$	44164.0	1.53487
$940$	−10542.3	−0.365801
$941$	−886.927	−0.0307258	−0.0153629	−	0.999882i	$-0.504890\pi$
$941$	−886.927	−0.0307258	−0.0153629	+	0.999882i	$0.504890\pi$
$942$	−10758.2	−0.372104
$943$	−4517.45	−0.156000
$944$	−6531.32	−0.225187
$945$	79494.9	2.73647
$946$	−8255.05	−0.283715
$947$	43341.3	1.48723	0.743613	−	0.668610i	$-0.233110\pi$
$947$	43341.3	1.48723	0.743613	+	0.668610i	$0.233110\pi$
$948$	36473.5	1.24958
$949$	0	0
$950$	−23308.9	−0.796041
$951$	16519.6	0.563286
$952$	7016.01	0.238855
$953$	16014.9	0.544357	0.272178	−	0.962247i	$-0.412256\pi$
$953$	16014.9	0.544357	0.272178	+	0.962247i	$0.412256\pi$
$954$	39585.5	1.34343
$955$	54197.5	1.83643
$956$	−4586.92	−0.155180
$957$	−4087.42	−0.138064
$958$	36527.4	1.23189
$959$	−27715.7	−0.933249
$960$	−7926.16	−0.266475
$961$	−1907.30	−0.0640227
$962$	0	0
$963$	104802.	3.50695
$964$	−13412.1	−0.448107
$965$	−12876.8	−0.429553
$966$	18032.6	0.600610
$967$	7986.48	0.265592	0.132796	−	0.991143i	$-0.457604\pi$
$967$	7986.48	0.265592	0.132796	+	0.991143i	$0.457604\pi$
$968$	9926.99	0.329613
$969$	−41167.4	−1.36479
$970$	7720.00	0.255540
$971$	25245.4	0.834361	0.417180	−	0.908824i	$-0.363018\pi$
$971$	25245.4	0.834361	0.417180	+	0.908824i	$0.363018\pi$
$972$	7938.88	0.261975
$973$	2968.09	0.0977932
$974$	10738.0	0.353253
$975$	0	0
$976$	−9660.19	−0.316819
$977$	−5231.76	−0.171319	−0.0856595	−	0.996324i	$-0.527300\pi$
$977$	−5231.76	−0.171319	−0.0856595	+	0.996324i	$0.527300\pi$
$978$	−20280.6	−0.663089
$979$	−6415.26	−0.209431
$980$	27081.5	0.882740
$981$	43694.5	1.42208
$982$	2982.67	0.0969254
$983$	22720.8	0.737212	0.368606	−	0.929586i	$-0.379835\pi$
$983$	22720.8	0.737212	0.368606	+	0.929586i	$0.379835\pi$
$984$	8774.10	0.284256
$985$	−18952.8	−0.613081
$986$	3015.54	0.0973978
$987$	−46584.5	−1.50233
$988$	0	0
$989$	15653.2	0.503279
$990$	13290.5	0.426668
$991$	19005.1	0.609200	0.304600	−	0.952480i	$-0.401477\pi$
$991$	19005.1	0.609200	0.304600	+	0.952480i	$0.401477\pi$
$992$	−5343.49	−0.171024
$993$	−22284.0	−0.712148
$994$	−55112.1	−1.75860
$995$	16217.0	0.516698
$996$	−10954.3	−0.348494
$997$	49243.2	1.56424	0.782121	−	0.623127i	$-0.214138\pi$
$997$	49243.2	1.56424	0.782121	+	0.623127i	$0.214138\pi$
$998$	20134.8	0.638633
$999$	4676.62	0.148110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.a.j.1.1	✓	3
13.2	odd	12		338.4.e.h.147.3		12
13.3	even	3		338.4.c.l.191.3		6
13.4	even	6		338.4.c.k.315.3		6
13.5	odd	4		338.4.b.f.337.4		6
13.6	odd	12		338.4.e.h.23.6		12
13.7	odd	12		338.4.e.h.23.3		12
13.8	odd	4		338.4.b.f.337.1		6
13.9	even	3		338.4.c.l.315.3		6
13.10	even	6		338.4.c.k.191.3		6
13.11	odd	12		338.4.e.h.147.6		12
13.12	even	2		338.4.a.k.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.4.a.j.1.1	✓	3	1.1	even	1	trivial
338.4.a.k.1.1	yes	3	13.12	even	2
338.4.b.f.337.1		6	13.8	odd	4
338.4.b.f.337.4		6	13.5	odd	4
338.4.c.k.191.3		6	13.10	even	6
338.4.c.k.315.3		6	13.4	even	6
338.4.c.l.191.3		6	13.3	even	3
338.4.c.l.315.3		6	13.9	even	3
338.4.e.h.23.3		12	13.7	odd	12
338.4.e.h.23.6		12	13.6	odd	12
338.4.e.h.147.3		12	13.2	odd	12
338.4.e.h.147.6		12	13.11	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -8.74094 q^{3} +4.00000 q^{4} +14.1685 q^{5} +17.4819 q^{6} -28.6504 q^{7} -8.00000 q^{8} +49.4040 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -8.74094 q^{3} +4.00000 q^{4} +14.1685 q^{5} +17.4819 q^{6} -28.6504 q^{7} -8.00000 q^{8} +49.4040 q^{9} -28.3370 q^{10} -9.49349 q^{11} -34.9638 q^{12} +57.3008 q^{14} -123.846 q^{15} +16.0000 q^{16} +30.6104 q^{17} -98.8080 q^{18} +153.860 q^{19} +56.6741 q^{20} +250.431 q^{21} +18.9870 q^{22} -36.0030 q^{23} +69.9275 q^{24} +75.7470 q^{25} -195.832 q^{27} -114.602 q^{28} -49.2567 q^{29} +247.692 q^{30} +166.984 q^{31} -32.0000 q^{32} +82.9820 q^{33} -61.2209 q^{34} -405.934 q^{35} +197.616 q^{36} -23.8808 q^{37} -307.720 q^{38} -113.348 q^{40} +125.474 q^{41} -500.863 q^{42} -434.774 q^{43} -37.9739 q^{44} +699.982 q^{45} +72.0061 q^{46} -186.017 q^{47} -139.855 q^{48} +477.845 q^{49} -151.494 q^{50} -267.564 q^{51} -400.631 q^{53} +391.664 q^{54} -134.509 q^{55} +229.203 q^{56} -1344.88 q^{57} +98.5134 q^{58} -408.208 q^{59} -495.385 q^{60} -603.762 q^{61} -333.968 q^{62} -1415.44 q^{63} +64.0000 q^{64} -165.964 q^{66} -287.548 q^{67} +122.442 q^{68} +314.700 q^{69} +811.868 q^{70} -961.803 q^{71} -395.232 q^{72} +963.198 q^{73} +47.7616 q^{74} -662.100 q^{75} +615.440 q^{76} +271.992 q^{77} -1043.18 q^{79} +226.696 q^{80} +377.848 q^{81} -250.948 q^{82} +313.304 q^{83} +1001.73 q^{84} +433.705 q^{85} +869.549 q^{86} +430.550 q^{87} +75.9479 q^{88} +675.754 q^{89} -1399.96 q^{90} -144.012 q^{92} -1459.60 q^{93} +372.034 q^{94} +2179.97 q^{95} +279.710 q^{96} -272.435 q^{97} -955.691 q^{98} -469.016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9	\( 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9} - 24 q^{10} + 82 q^{11} - 48 q^{12} + 54 q^{14} - 90 q^{15} + 48 q^{16} - 90 q^{17} - 18 q^{18} + 130 q^{19} + 48 q^{20} + 234 q^{21} - 164 q^{22} + 19 q^{23} + 96 q^{24} - 61 q^{25} - 69 q^{27} - 108 q^{28} - 101 q^{29} + 180 q^{30} + 519 q^{31} - 96 q^{32} - 146 q^{33} + 180 q^{34} - 458 q^{35} + 36 q^{36} + 84 q^{37} - 260 q^{38} - 96 q^{40} + 187 q^{41} - 468 q^{42} - 1205 q^{43} + 328 q^{44} + 645 q^{45} - 38 q^{46} - 536 q^{47} - 192 q^{48} - 184 q^{49} + 122 q^{50} - 207 q^{51} - 1095 q^{53} + 138 q^{54} - 526 q^{55} + 216 q^{56} - 1409 q^{57} + 202 q^{58} - 1413 q^{59} - 360 q^{60} - 1108 q^{61} - 1038 q^{62} - 1404 q^{63} + 192 q^{64} + 292 q^{66} - 1605 q^{67} - 360 q^{68} - 314 q^{69} + 916 q^{70} - 909 q^{71} - 72 q^{72} + 287 q^{73} - 168 q^{74} - 505 q^{75} + 520 q^{76} + 480 q^{77} - 1961 q^{79} + 192 q^{80} + 915 q^{81} - 374 q^{82} + 191 q^{83} + 936 q^{84} + 67 q^{85} + 2410 q^{86} + 1636 q^{87} - 656 q^{88} + 1091 q^{89} - 1290 q^{90} + 76 q^{92} - 1614 q^{93} + 1072 q^{94} + 1829 q^{95} + 384 q^{96} - 947 q^{97} + 368 q^{98} - 2057 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9 - 24 * q^10 + 82 * q^11 - 48 * q^12 + 54 * q^14 - 90 * q^15 + 48 * q^16 - 90 * q^17 - 18 * q^18 + 130 * q^19 + 48 * q^20 + 234 * q^21 - 164 * q^22 + 19 * q^23 + 96 * q^24 - 61 * q^25 - 69 * q^27 - 108 * q^28 - 101 * q^29 + 180 * q^30 + 519 * q^31 - 96 * q^32 - 146 * q^33 + 180 * q^34 - 458 * q^35 + 36 * q^36 + 84 * q^37 - 260 * q^38 - 96 * q^40 + 187 * q^41 - 468 * q^42 - 1205 * q^43 + 328 * q^44 + 645 * q^45 - 38 * q^46 - 536 * q^47 - 192 * q^48 - 184 * q^49 + 122 * q^50 - 207 * q^51 - 1095 * q^53 + 138 * q^54 - 526 * q^55 + 216 * q^56 - 1409 * q^57 + 202 * q^58 - 1413 * q^59 - 360 * q^60 - 1108 * q^61 - 1038 * q^62 - 1404 * q^63 + 192 * q^64 + 292 * q^66 - 1605 * q^67 - 360 * q^68 - 314 * q^69 + 916 * q^70 - 909 * q^71 - 72 * q^72 + 287 * q^73 - 168 * q^74 - 505 * q^75 + 520 * q^76 + 480 * q^77 - 1961 * q^79 + 192 * q^80 + 915 * q^81 - 374 * q^82 + 191 * q^83 + 936 * q^84 + 67 * q^85 + 2410 * q^86 + 1636 * q^87 - 656 * q^88 + 1091 * q^89 - 1290 * q^90 + 76 * q^92 - 1614 * q^93 + 1072 * q^94 + 1829 * q^95 + 384 * q^96 - 947 * q^97 + 368 * q^98 - 2057 * q^99

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