LMFDB - Embedded newform 338.4.a.n.1.2

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.6681389953.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 107x^{4} + 85x^{3} + 3703x^{2} - 1659x - 41951 $ x^6 - x^5 - 107x^4 + 85x^3 + 3703x^2 - 1659x - 41951
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.06450$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -3.92743 q^{3} +4.00000 q^{4} +0.152827 q^{5} +7.85487 q^{6} -33.9913 q^{7} -8.00000 q^{8} -11.5753 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.92743 q^{3} +4.00000 q^{4} +0.152827 q^{5} +7.85487 q^{6} -33.9913 q^{7} -8.00000 q^{8} -11.5753 q^{9} -0.305653 q^{10} +10.5343 q^{11} -15.7097 q^{12} +67.9825 q^{14} -0.600216 q^{15} +16.0000 q^{16} -41.3181 q^{17} +23.1505 q^{18} -131.416 q^{19} +0.611306 q^{20} +133.498 q^{21} -21.0686 q^{22} +161.225 q^{23} +31.4195 q^{24} -124.977 q^{25} +151.502 q^{27} -135.965 q^{28} -35.6032 q^{29} +1.20043 q^{30} -12.7094 q^{31} -32.0000 q^{32} -41.3728 q^{33} +82.6363 q^{34} -5.19477 q^{35} -46.3011 q^{36} -183.992 q^{37} +262.833 q^{38} -1.22261 q^{40} -443.350 q^{41} -266.997 q^{42} +466.950 q^{43} +42.1372 q^{44} -1.76901 q^{45} -322.450 q^{46} +282.293 q^{47} -62.8389 q^{48} +812.407 q^{49} +249.953 q^{50} +162.274 q^{51} +114.112 q^{53} -303.004 q^{54} +1.60992 q^{55} +271.930 q^{56} +516.129 q^{57} +71.2065 q^{58} +703.260 q^{59} -2.40086 q^{60} +600.068 q^{61} +25.4189 q^{62} +393.458 q^{63} +64.0000 q^{64} +82.7455 q^{66} +542.943 q^{67} -165.273 q^{68} -633.201 q^{69} +10.3895 q^{70} -907.027 q^{71} +92.6021 q^{72} +498.548 q^{73} +367.983 q^{74} +490.837 q^{75} -525.666 q^{76} -358.074 q^{77} -356.469 q^{79} +2.44522 q^{80} -282.481 q^{81} +886.701 q^{82} +934.114 q^{83} +533.994 q^{84} -6.31451 q^{85} -933.900 q^{86} +139.829 q^{87} -84.2744 q^{88} +581.759 q^{89} +3.53802 q^{90} +644.900 q^{92} +49.9155 q^{93} -564.585 q^{94} -20.0839 q^{95} +125.678 q^{96} -334.700 q^{97} -1624.81 q^{98} -121.937 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10}) $ 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * 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
$3$	−3.92743	−0.755835	−0.377917	−	0.925839i	$-0.623360\pi$
$3$	−3.92743	−0.755835	−0.377917	+	0.925839i	$0.623360\pi$
$4$	4.00000	0.500000
$5$	0.152827	0.0136692	0.00683461	−	0.999977i	$-0.497824\pi$
$5$	0.152827	0.0136692	0.00683461	+	0.999977i	$0.497824\pi$
$6$	7.85487	0.534456
$7$	−33.9913	−1.83536	−0.917678	−	0.397325i	$-0.869939\pi$
$7$	−33.9913	−1.83536	−0.917678	+	0.397325i	$0.869939\pi$
$8$	−8.00000	−0.353553
$9$	−11.5753	−0.428714
$10$	−0.305653	−0.00966560
$11$	10.5343	0.288746	0.144373	−	0.989523i	$-0.453883\pi$
$11$	10.5343	0.288746	0.144373	+	0.989523i	$0.453883\pi$
$12$	−15.7097	−0.377917
$13$	0	0
$13$	0	0
$14$	67.9825	1.29779
$15$	−0.600216	−0.0103317
$16$	16.0000	0.250000
$17$	−41.3181	−0.589478	−0.294739	−	0.955578i	$-0.595233\pi$
$17$	−41.3181	−0.589478	−0.294739	+	0.955578i	$0.595233\pi$
$18$	23.1505	0.303146
$19$	−131.416	−1.58679	−0.793395	−	0.608708i	$-0.791688\pi$
$19$	−131.416	−1.58679	−0.793395	+	0.608708i	$0.791688\pi$
$20$	0.611306	0.00683461
$21$	133.498	1.38723
$22$	−21.0686	−0.204175
$23$	161.225	1.46164	0.730821	−	0.682570i	$-0.239138\pi$
$23$	161.225	1.46164	0.730821	+	0.682570i	$0.239138\pi$
$24$	31.4195	0.267228
$25$	−124.977	−0.999813
$26$	0	0
$27$	151.502	1.07987
$28$	−135.965	−0.917678
$29$	−35.6032	−0.227978	−0.113989	−	0.993482i	$-0.536363\pi$
$29$	−35.6032	−0.227978	−0.113989	+	0.993482i	$0.536363\pi$
$30$	1.20043	0.00730560
$31$	−12.7094	−0.0736349	−0.0368175	−	0.999322i	$-0.511722\pi$
$31$	−12.7094	−0.0736349	−0.0368175	+	0.999322i	$0.511722\pi$
$32$	−32.0000	−0.176777
$33$	−41.3728	−0.218245
$34$	82.6363	0.416824
$35$	−5.19477	−0.0250879
$36$	−46.3011	−0.214357
$37$	−183.992	−0.817514	−0.408757	−	0.912643i	$-0.634038\pi$
$37$	−183.992	−0.817514	−0.408757	+	0.912643i	$0.634038\pi$
$38$	262.833	1.12203
$39$	0	0
$40$	−1.22261	−0.00483280
$41$	−443.350	−1.68877	−0.844386	−	0.535735i	$-0.820035\pi$
$41$	−443.350	−1.68877	−0.844386	+	0.535735i	$0.820035\pi$
$42$	−266.997	−0.980917
$43$	466.950	1.65603	0.828014	−	0.560708i	$-0.189471\pi$
$43$	466.950	1.65603	0.828014	+	0.560708i	$0.189471\pi$
$44$	42.1372	0.144373
$45$	−1.76901	−0.00586018
$46$	−322.450	−1.03354
$47$	282.293	0.876098	0.438049	−	0.898951i	$-0.355670\pi$
$47$	282.293	0.876098	0.438049	+	0.898951i	$0.355670\pi$
$48$	−62.8389	−0.188959
$49$	812.407	2.36853
$50$	249.953	0.706975
$51$	162.274	0.445548
$52$	0	0
$53$	114.112	0.295745	0.147872	−	0.989006i	$-0.452758\pi$
$53$	114.112	0.295745	0.147872	+	0.989006i	$0.452758\pi$
$54$	−303.004	−0.763585
$55$	1.60992	0.00394694
$56$	271.930	0.648896
$57$	516.129	1.19935
$58$	71.2065	0.161205
$59$	703.260	1.55181	0.775904	−	0.630851i	$-0.217294\pi$
$59$	703.260	1.55181	0.775904	+	0.630851i	$0.217294\pi$
$60$	−2.40086	−0.00516584
$61$	600.068	1.25952	0.629761	−	0.776789i	$-0.283153\pi$
$61$	600.068	1.25952	0.629761	+	0.776789i	$0.283153\pi$
$62$	25.4189	0.0520678
$63$	393.458	0.786842
$64$	64.0000	0.125000
$65$	0	0
$66$	82.7455	0.154322
$67$	542.943	0.990015	0.495007	−	0.868889i	$-0.335165\pi$
$67$	542.943	0.990015	0.495007	+	0.868889i	$0.335165\pi$
$68$	−165.273	−0.294739
$69$	−633.201	−1.10476
$70$	10.3895	0.0177398
$71$	−907.027	−1.51612	−0.758058	−	0.652187i	$-0.773852\pi$
$71$	−907.027	−1.51612	−0.758058	+	0.652187i	$0.773852\pi$
$72$	92.6021	0.151573
$73$	498.548	0.799323	0.399662	−	0.916663i	$-0.369128\pi$
$73$	498.548	0.799323	0.399662	+	0.916663i	$0.369128\pi$
$74$	367.983	0.578070
$75$	490.837	0.755694
$76$	−525.666	−0.793395
$77$	−358.074	−0.529953
$78$	0	0
$79$	−356.469	−0.507670	−0.253835	−	0.967248i	$-0.581692\pi$
$79$	−356.469	−0.507670	−0.253835	+	0.967248i	$0.581692\pi$
$80$	2.44522	0.00341731
$81$	−282.481	−0.387491
$82$	886.701	1.19414
$83$	934.114	1.23533	0.617665	−	0.786442i	$-0.288079\pi$
$83$	934.114	1.23533	0.617665	+	0.786442i	$0.288079\pi$
$84$	533.994	0.693613
$85$	−6.31451	−0.00805770
$86$	−933.900	−1.17099
$87$	139.829	0.172314
$88$	−84.2744	−0.102087
$89$	581.759	0.692880	0.346440	−	0.938072i	$-0.387390\pi$
$89$	581.759	0.692880	0.346440	+	0.938072i	$0.387390\pi$
$90$	3.53802	0.00414377
$91$	0	0
$92$	644.900	0.730821
$93$	49.9155	0.0556559
$94$	−564.585	−0.619495
$95$	−20.0839	−0.0216902
$96$	125.678	0.133614
$97$	−334.700	−0.350347	−0.175173	−	0.984538i	$-0.556049\pi$
$97$	−334.700	−0.350347	−0.175173	+	0.984538i	$0.556049\pi$
$98$	−1624.81	−1.67481
$99$	−121.937	−0.123789
$100$	−499.907	−0.499907
$101$	−1359.43	−1.33929	−0.669643	−	0.742683i	$-0.733553\pi$
$101$	−1359.43	−1.33929	−0.669643	+	0.742683i	$0.733553\pi$
$102$	−324.549	−0.315050
$103$	105.794	0.101206	0.0506028	−	0.998719i	$-0.483886\pi$
$103$	105.794	0.101206	0.0506028	+	0.998719i	$0.483886\pi$
$104$	0	0
$105$	20.4021	0.0189623
$106$	−228.224	−0.209123
$107$	78.6185	0.0710312	0.0355156	−	0.999369i	$-0.488693\pi$
$107$	78.6185	0.0710312	0.0355156	+	0.999369i	$0.488693\pi$
$108$	606.007	0.539936
$109$	−650.118	−0.571285	−0.285642	−	0.958336i	$-0.592207\pi$
$109$	−650.118	−0.571285	−0.285642	+	0.958336i	$0.592207\pi$
$110$	−3.21984	−0.00279091
$111$	722.615	0.617906
$112$	−543.860	−0.458839
$113$	−1277.56	−1.06356	−0.531780	−	0.846882i	$-0.678477\pi$
$113$	−1277.56	−1.06356	−0.531780	+	0.846882i	$0.678477\pi$
$114$	−1032.26	−0.848069
$115$	24.6395	0.0199795
$116$	−142.413	−0.113989
$117$	0	0
$118$	−1406.52	−1.09729
$119$	1404.46	1.08190
$120$	4.80173	0.00365280
$121$	−1220.03	−0.916625
$122$	−1200.14	−0.890617
$123$	1741.23	1.27643
$124$	−50.8378	−0.0368175
$125$	−38.2031	−0.0273359
$126$	−786.916	−0.556381
$127$	−305.991	−0.213798	−0.106899	−	0.994270i	$-0.534092\pi$
$127$	−305.991	−0.213798	−0.106899	+	0.994270i	$0.534092\pi$
$128$	−128.000	−0.0883883
$129$	−1833.91	−1.25168
$130$	0	0
$131$	95.0876	0.0634186	0.0317093	−	0.999497i	$-0.489905\pi$
$131$	95.0876	0.0634186	0.0317093	+	0.999497i	$0.489905\pi$
$132$	−165.491	−0.109122
$133$	4467.01	2.91232
$134$	−1085.89	−0.700046
$135$	23.1535	0.0147610
$136$	330.545	0.208412
$137$	2155.03	1.34392	0.671960	−	0.740588i	$-0.265453\pi$
$137$	2155.03	1.34392	0.671960	+	0.740588i	$0.265453\pi$
$138$	1266.40	0.781183
$139$	184.637	0.112667	0.0563336	−	0.998412i	$-0.482059\pi$
$139$	184.637	0.112667	0.0563336	+	0.998412i	$0.482059\pi$
$140$	−20.7791	−0.0125439
$141$	−1108.69	−0.662186
$142$	1814.05	1.07206
$143$	0	0
$144$	−185.204	−0.107178
$145$	−5.44112	−0.00311628
$146$	−997.095	−0.565207
$147$	−3190.67	−1.79022
$148$	−735.966	−0.408757
$149$	−1449.34	−0.796877	−0.398439	−	0.917195i	$-0.630448\pi$
$149$	−1449.34	−0.796877	−0.398439	+	0.917195i	$0.630448\pi$
$150$	−981.675	−0.534356
$151$	−188.202	−0.101428	−0.0507141	−	0.998713i	$-0.516150\pi$
$151$	−188.202	−0.101428	−0.0507141	+	0.998713i	$0.516150\pi$
$152$	1051.33	0.561015
$153$	478.268	0.252717
$154$	716.148	0.374733
$155$	−1.94234	−0.00100653
$156$	0	0
$157$	−435.114	−0.221184	−0.110592	−	0.993866i	$-0.535275\pi$
$157$	−435.114	−0.221184	−0.110592	+	0.993866i	$0.535275\pi$
$158$	712.939	0.358977
$159$	−448.167	−0.223534
$160$	−4.89045	−0.00241640
$161$	−5480.24	−2.68263
$162$	564.962	0.273998
$163$	711.146	0.341725	0.170863	−	0.985295i	$-0.445345\pi$
$163$	711.146	0.341725	0.170863	+	0.985295i	$0.445345\pi$
$164$	−1773.40	−0.844386
$165$	−6.32286	−0.00298323
$166$	−1868.23	−0.873510
$167$	−2570.16	−1.19093	−0.595463	−	0.803383i	$-0.703031\pi$
$167$	−2570.16	−1.19093	−0.595463	+	0.803383i	$0.703031\pi$
$168$	−1067.99	−0.490459
$169$	0	0
$170$	12.6290	0.00569766
$171$	1521.18	0.680278
$172$	1867.80	0.828014
$173$	1094.69	0.481087	0.240543	−	0.970638i	$-0.422674\pi$
$173$	1094.69	0.481087	0.240543	+	0.970638i	$0.422674\pi$
$174$	−279.659	−0.121844
$175$	4248.12	1.83501
$176$	168.549	0.0721866
$177$	−2762.01	−1.17291
$178$	−1163.52	−0.489940
$179$	−2488.54	−1.03912	−0.519558	−	0.854435i	$-0.673903\pi$
$179$	−2488.54	−1.03912	−0.519558	+	0.854435i	$0.673903\pi$
$180$	−7.07603	−0.00293009
$181$	3327.76	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$181$	3327.76	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$182$	0	0
$183$	−2356.73	−0.951991
$184$	−1289.80	−0.516768
$185$	−28.1188	−0.0111748
$186$	−99.8310	−0.0393546
$187$	−435.258	−0.170210
$188$	1129.17	0.438049
$189$	−5149.74	−1.98195
$190$	40.1678	0.0153373
$191$	348.076	0.131863	0.0659317	−	0.997824i	$-0.478998\pi$
$191$	348.076	0.131863	0.0659317	+	0.997824i	$0.478998\pi$
$192$	−251.356	−0.0944794
$193$	3860.41	1.43978	0.719892	−	0.694086i	$-0.244191\pi$
$193$	3860.41	1.43978	0.719892	+	0.694086i	$0.244191\pi$
$194$	669.400	0.247732
$195$	0	0
$196$	3249.63	1.18427
$197$	297.824	0.107711	0.0538555	−	0.998549i	$-0.482849\pi$
$197$	297.824	0.107711	0.0538555	+	0.998549i	$0.482849\pi$
$198$	243.875	0.0875324
$199$	59.7631	0.0212889	0.0106445	−	0.999943i	$-0.496612\pi$
$199$	59.7631	0.0212889	0.0106445	+	0.999943i	$0.496612\pi$
$200$	999.813	0.353487
$201$	−2132.37	−0.748288
$202$	2718.85	0.947019
$203$	1210.20	0.418420
$204$	649.097	0.222774
$205$	−67.7557	−0.0230842
$206$	−211.588	−0.0715632
$207$	−1866.22	−0.626625
$208$	0	0
$209$	−1384.38	−0.458180
$210$	−40.8042	−0.0134084
$211$	−3456.10	−1.12762	−0.563809	−	0.825905i	$-0.690665\pi$
$211$	−3456.10	−1.12762	−0.563809	+	0.825905i	$0.690665\pi$
$212$	456.448	0.147872
$213$	3562.29	1.14593
$214$	−157.237	−0.0502266
$215$	71.3623	0.0226366
$216$	−1212.01	−0.381792
$217$	432.010	0.135146
$218$	1300.24	0.403959
$219$	−1958.01	−0.604156
$220$	6.43968	0.00197347
$221$	0	0
$222$	−1445.23	−0.436925
$223$	2181.50	0.655087	0.327543	−	0.944836i	$-0.393779\pi$
$223$	2181.50	0.655087	0.327543	+	0.944836i	$0.393779\pi$
$224$	1087.72	0.324448
$225$	1446.64	0.428633
$226$	2555.11	0.752051
$227$	−3155.79	−0.922720	−0.461360	−	0.887213i	$-0.652638\pi$
$227$	−3155.79	−0.922720	−0.461360	+	0.887213i	$0.652638\pi$
$228$	2064.52	0.599675
$229$	1366.34	0.394279	0.197140	−	0.980375i	$-0.436835\pi$
$229$	1366.34	0.394279	0.197140	+	0.980375i	$0.436835\pi$
$230$	−49.2789	−0.0141276
$231$	1406.31	0.400557
$232$	284.826	0.0806023
$233$	−5030.38	−1.41438	−0.707192	−	0.707022i	$-0.750038\pi$
$233$	−5030.38	−1.41438	−0.707192	+	0.707022i	$0.750038\pi$
$234$	0	0
$235$	43.1418	0.0119756
$236$	2813.04	0.775904
$237$	1400.01	0.383715
$238$	−2808.91	−0.765020
$239$	4694.83	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$239$	4694.83	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$240$	−9.60346	−0.00258292
$241$	2885.24	0.771181	0.385591	−	0.922670i	$-0.373998\pi$
$241$	2885.24	0.771181	0.385591	+	0.922670i	$0.373998\pi$
$242$	2440.06	0.648152
$243$	−2981.12	−0.786992
$244$	2400.27	0.629761
$245$	124.157	0.0323760
$246$	−3482.46	−0.902575
$247$	0	0
$248$	101.676	0.0260339
$249$	−3668.67	−0.933705
$250$	76.4061	0.0193294
$251$	3398.02	0.854507	0.427254	−	0.904132i	$-0.359481\pi$
$251$	3398.02	0.854507	0.427254	+	0.904132i	$0.359481\pi$
$252$	1573.83	0.393421
$253$	1698.39	0.422044
$254$	611.982	0.151178
$255$	24.7998	0.00609029
$256$	256.000	0.0625000
$257$	−1784.27	−0.433073	−0.216536	−	0.976275i	$-0.569476\pi$
$257$	−1784.27	−0.433073	−0.216536	+	0.976275i	$0.569476\pi$
$258$	3667.83	0.885074
$259$	6254.11	1.50043
$260$	0	0
$261$	412.117	0.0977372
$262$	−190.175	−0.0448437
$263$	−6813.66	−1.59752	−0.798761	−	0.601648i	$-0.794511\pi$
$263$	−6813.66	−1.59752	−0.798761	+	0.601648i	$0.794511\pi$
$264$	330.982	0.0771611
$265$	17.4393	0.00404260
$266$	−8934.02	−2.05932
$267$	−2284.82	−0.523703
$268$	2171.77	0.495007
$269$	4777.13	1.08278	0.541388	−	0.840773i	$-0.317899\pi$
$269$	4777.13	1.08278	0.541388	+	0.840773i	$0.317899\pi$
$270$	−46.3070	−0.0104376
$271$	−903.040	−0.202420	−0.101210	−	0.994865i	$-0.532271\pi$
$271$	−903.040	−0.202420	−0.101210	+	0.994865i	$0.532271\pi$
$272$	−661.090	−0.147369
$273$	0	0
$274$	−4310.07	−0.950294
$275$	−1316.54	−0.288692
$276$	−2532.80	−0.552380
$277$	8576.76	1.86039	0.930195	−	0.367067i	$-0.119638\pi$
$277$	8576.76	1.86039	0.930195	+	0.367067i	$0.119638\pi$
$278$	−369.275	−0.0796678
$279$	147.115	0.0315683
$280$	41.5582	0.00886991
$281$	−5614.95	−1.19203	−0.596013	−	0.802974i	$-0.703250\pi$
$281$	−5614.95	−1.19203	−0.596013	+	0.802974i	$0.703250\pi$
$282$	2217.37	0.468236
$283$	5818.89	1.22225	0.611125	−	0.791534i	$-0.290717\pi$
$283$	5818.89	1.22225	0.611125	+	0.791534i	$0.290717\pi$
$284$	−3628.11	−0.758058
$285$	78.8783	0.0163942
$286$	0	0
$287$	15070.0	3.09950
$288$	370.408	0.0757866
$289$	−3205.81	−0.652516
$290$	10.8822	0.00220354
$291$	1314.51	0.264804
$292$	1994.19	0.399662
$293$	−6537.80	−1.30356	−0.651779	−	0.758409i	$-0.725977\pi$
$293$	−6537.80	−1.30356	−0.651779	+	0.758409i	$0.725977\pi$
$294$	6381.35	1.26588
$295$	107.477	0.0212120
$296$	1471.93	0.289035
$297$	1595.97	0.311809
$298$	2898.68	0.563477
$299$	0	0
$300$	1963.35	0.377847
$301$	−15872.2	−3.03940
$302$	376.404	0.0717206
$303$	5339.06	1.01228
$304$	−2102.66	−0.396697
$305$	91.7064	0.0172167
$306$	−956.537	−0.178698
$307$	2674.74	0.497248	0.248624	−	0.968600i	$-0.420022\pi$
$307$	2674.74	0.497248	0.248624	+	0.968600i	$0.420022\pi$
$308$	−1432.30	−0.264976
$309$	−415.498	−0.0764948
$310$	3.88468	0.000711726 0
$311$	8322.18	1.51739	0.758694	−	0.651447i	$-0.225838\pi$
$311$	8322.18	1.51739	0.758694	+	0.651447i	$0.225838\pi$
$312$	0	0
$313$	−6029.12	−1.08877	−0.544387	−	0.838834i	$-0.683238\pi$
$313$	−6029.12	−1.08877	−0.544387	+	0.838834i	$0.683238\pi$
$314$	870.228	0.156401
$315$	60.1308	0.0107555
$316$	−1425.88	−0.253835
$317$	6393.40	1.13277	0.566387	−	0.824140i	$-0.308341\pi$
$317$	6393.40	1.13277	0.566387	+	0.824140i	$0.308341\pi$
$318$	896.334	0.158063
$319$	−375.055	−0.0658278
$320$	9.78090	0.00170865
$321$	−308.769	−0.0536879
$322$	10960.5	1.89691
$323$	5429.88	0.935377
$324$	−1129.92	−0.193746
$325$	0	0
$326$	−1422.29	−0.241636
$327$	2553.30	0.431797
$328$	3546.80	0.597071
$329$	−9595.49	−1.60795
$330$	12.6457	0.00210947
$331$	5445.60	0.904281	0.452141	−	0.891947i	$-0.350660\pi$
$331$	5445.60	0.904281	0.452141	+	0.891947i	$0.350660\pi$
$332$	3736.46	0.617665
$333$	2129.75	0.350479
$334$	5140.31	0.842112
$335$	82.9760	0.0135327
$336$	2135.98	0.346807
$337$	−1819.65	−0.294132	−0.147066	−	0.989127i	$-0.546983\pi$
$337$	−1819.65	−0.294132	−0.147066	+	0.989127i	$0.546983\pi$
$338$	0	0
$339$	5017.52	0.803876
$340$	−25.2580	−0.00402885
$341$	−133.885	−0.0212618
$342$	−3042.36	−0.481029
$343$	−15955.7	−2.51174
$344$	−3735.60	−0.585494
$345$	−96.7699	−0.0151012
$346$	−2189.39	−0.340180
$347$	−4445.06	−0.687675	−0.343838	−	0.939029i	$-0.611727\pi$
$347$	−4445.06	−0.687675	−0.343838	+	0.939029i	$0.611727\pi$
$348$	559.317	0.0861568
$349$	−2354.42	−0.361116	−0.180558	−	0.983564i	$-0.557790\pi$
$349$	−2354.42	−0.361116	−0.180558	+	0.983564i	$0.557790\pi$
$350$	−8496.23	−1.29755
$351$	0	0
$352$	−337.098	−0.0510436
$353$	−4320.46	−0.651430	−0.325715	−	0.945468i	$-0.605605\pi$
$353$	−4320.46	−0.651430	−0.325715	+	0.945468i	$0.605605\pi$
$354$	5524.02	0.829374
$355$	−138.618	−0.0207241
$356$	2327.04	0.346440
$357$	−5515.91	−0.817739
$358$	4977.07	0.734766
$359$	11936.7	1.75486	0.877429	−	0.479706i	$-0.159257\pi$
$359$	11936.7	1.75486	0.877429	+	0.479706i	$0.159257\pi$
$360$	14.1521	0.00207189
$361$	10411.3	1.51790
$362$	−6655.52	−0.966316
$363$	4791.58	0.692818
$364$	0	0
$365$	76.1913	0.0109261
$366$	4713.46	0.673159
$367$	2134.96	0.303663	0.151831	−	0.988406i	$-0.451483\pi$
$367$	2134.96	0.303663	0.151831	+	0.988406i	$0.451483\pi$
$368$	2579.60	0.365410
$369$	5131.90	0.724000
$370$	56.2376	0.00790177
$371$	−3878.81	−0.542797
$372$	199.662	0.0278279
$373$	9818.91	1.36301	0.681507	−	0.731812i	$-0.261325\pi$
$373$	9818.91	1.36301	0.681507	+	0.731812i	$0.261325\pi$
$374$	870.515	0.120356
$375$	150.040	0.0206614
$376$	−2258.34	−0.309748
$377$	0	0
$378$	10299.5	1.40145
$379$	2120.13	0.287345	0.143672	−	0.989625i	$-0.454109\pi$
$379$	2120.13	0.287345	0.143672	+	0.989625i	$0.454109\pi$
$380$	−80.3357	−0.0108451
$381$	1201.76	0.161596
$382$	−696.152	−0.0932414
$383$	−1662.83	−0.221846	−0.110923	−	0.993829i	$-0.535381\pi$
$383$	−1662.83	−0.221846	−0.110923	+	0.993829i	$0.535381\pi$
$384$	502.712	0.0668070
$385$	−54.7233	−0.00724404
$386$	−7720.82	−1.01808
$387$	−5405.07	−0.709961
$388$	−1338.80	−0.175173
$389$	−3332.26	−0.434324	−0.217162	−	0.976136i	$-0.569680\pi$
$389$	−3332.26	−0.434324	−0.217162	+	0.976136i	$0.569680\pi$
$390$	0	0
$391$	−6661.52	−0.861605
$392$	−6499.25	−0.837403
$393$	−373.450	−0.0479340
$394$	−595.648	−0.0761632
$395$	−54.4780	−0.00693945
$396$	−487.749	−0.0618947
$397$	−3948.49	−0.499166	−0.249583	−	0.968353i	$-0.580294\pi$
$397$	−3948.49	−0.499166	−0.249583	+	0.968353i	$0.580294\pi$
$398$	−119.526	−0.0150535
$399$	−17543.9	−2.20124
$400$	−1999.63	−0.249953
$401$	8141.53	1.01389	0.506944	−	0.861979i	$-0.330775\pi$
$401$	8141.53	1.01389	0.506944	+	0.861979i	$0.330775\pi$
$402$	4264.74	0.529119
$403$	0	0
$404$	−5437.71	−0.669643
$405$	−43.1706	−0.00529670
$406$	−2420.40	−0.295868
$407$	−1938.22	−0.236054
$408$	−1298.19	−0.157525
$409$	1558.79	0.188452	0.0942262	−	0.995551i	$-0.469962\pi$
$409$	1558.79	0.188452	0.0942262	+	0.995551i	$0.469962\pi$
$410$	135.511	0.0163230
$411$	−8463.75	−1.01578
$412$	423.176	0.0506028
$413$	−23904.7	−2.84812
$414$	3732.45	0.443091
$415$	142.757	0.0168860
$416$	0	0
$417$	−725.151	−0.0851578
$418$	2768.76	0.323982
$419$	−4323.92	−0.504146	−0.252073	−	0.967708i	$-0.581112\pi$
$419$	−4323.92	−0.504146	−0.252073	+	0.967708i	$0.581112\pi$
$420$	81.6084	0.00948115
$421$	−8525.91	−0.987001	−0.493501	−	0.869746i	$-0.664283\pi$
$421$	−8525.91	−0.987001	−0.493501	+	0.869746i	$0.664283\pi$
$422$	6912.19	0.797347
$423$	−3267.61	−0.375595
$424$	−912.895	−0.104562
$425$	5163.80	0.589368
$426$	−7124.57	−0.810298
$427$	−20397.1	−2.31167
$428$	314.474	0.0355156
$429$	0	0
$430$	−142.725	−0.0160065
$431$	7444.09	0.831947	0.415974	−	0.909377i	$-0.363441\pi$
$431$	7444.09	0.831947	0.415974	+	0.909377i	$0.363441\pi$
$432$	2424.03	0.269968
$433$	5893.14	0.654056	0.327028	−	0.945015i	$-0.393953\pi$
$433$	5893.14	0.654056	0.327028	+	0.945015i	$0.393953\pi$
$434$	−864.020	−0.0955629
$435$	21.3696	0.00235539
$436$	−2600.47	−0.285642
$437$	−21187.6	−2.31932
$438$	3916.03	0.427203
$439$	−2862.37	−0.311193	−0.155596	−	0.987821i	$-0.549730\pi$
$439$	−2862.37	−0.311193	−0.155596	+	0.987821i	$0.549730\pi$
$440$	−12.8794	−0.00139545
$441$	−9403.82	−1.01542
$442$	0	0
$443$	7709.34	0.826822	0.413411	−	0.910545i	$-0.364337\pi$
$443$	7709.34	0.826822	0.413411	+	0.910545i	$0.364337\pi$
$444$	2890.46	0.308953
$445$	88.9082	0.00947113
$446$	−4363.01	−0.463216
$447$	5692.19	0.602308
$448$	−2175.44	−0.229420
$449$	2174.88	0.228594	0.114297	−	0.993447i	$-0.463538\pi$
$449$	2174.88	0.228594	0.114297	+	0.993447i	$0.463538\pi$
$450$	−2893.28	−0.303090
$451$	−4670.38	−0.487627
$452$	−5110.22	−0.531780
$453$	739.151	0.0766630
$454$	6311.59	0.652462
$455$	0	0
$456$	−4129.03	−0.424035
$457$	4112.61	0.420963	0.210481	−	0.977598i	$-0.432497\pi$
$457$	4112.61	0.420963	0.210481	+	0.977598i	$0.432497\pi$
$458$	−2732.67	−0.278798
$459$	−6259.77	−0.636560
$460$	98.5579	0.00998975
$461$	5639.11	0.569717	0.284859	−	0.958570i	$-0.408053\pi$
$461$	5639.11	0.569717	0.284859	+	0.958570i	$0.408053\pi$
$462$	−2812.63	−0.283236
$463$	5028.22	0.504711	0.252355	−	0.967635i	$-0.418795\pi$
$463$	5028.22	0.504711	0.252355	+	0.967635i	$0.418795\pi$
$464$	−569.652	−0.0569944
$465$	7.62841	0.000760772 0
$466$	10060.8	1.00012
$467$	−7351.52	−0.728453	−0.364227	−	0.931310i	$-0.618667\pi$
$467$	−7351.52	−0.728453	−0.364227	+	0.931310i	$0.618667\pi$
$468$	0	0
$469$	−18455.3	−1.81703
$470$	−86.2837	−0.00846802
$471$	1708.88	0.167178
$472$	−5626.08	−0.548647
$473$	4918.99	0.478172
$474$	−2800.02	−0.271327
$475$	16424.0	1.58649
$476$	5617.83	0.540951
$477$	−1320.88	−0.126790
$478$	−9389.66	−0.898479
$479$	3496.15	0.333493	0.166746	−	0.986000i	$-0.446674\pi$
$479$	3496.15	0.333493	0.166746	+	0.986000i	$0.446674\pi$
$480$	19.2069	0.00182640
$481$	0	0
$482$	−5770.48	−0.545308
$483$	21523.3	2.02763
$484$	−4880.11	−0.458313
$485$	−51.1510	−0.00478896
$486$	5962.25	0.556488
$487$	18326.0	1.70520	0.852598	−	0.522568i	$-0.175026\pi$
$487$	18326.0	1.70520	0.852598	+	0.522568i	$0.175026\pi$
$488$	−4800.55	−0.445308
$489$	−2792.98	−0.258288
$490$	−248.315	−0.0228933
$491$	14270.1	1.31161	0.655807	−	0.754929i	$-0.272329\pi$
$491$	14270.1	1.31161	0.655807	+	0.754929i	$0.272329\pi$
$492$	6964.92	0.638217
$493$	1471.06	0.134388
$494$	0	0
$495$	−18.6353	−0.00169211
$496$	−203.351	−0.0184087
$497$	30831.0	2.78261
$498$	7337.34	0.660229
$499$	−1424.52	−0.127796	−0.0638980	−	0.997956i	$-0.520353\pi$
$499$	−1424.52	−0.127796	−0.0638980	+	0.997956i	$0.520353\pi$
$500$	−152.812	−0.0136679
$501$	10094.1	0.900144
$502$	−6796.05	−0.604228
$503$	14269.1	1.26487	0.632435	−	0.774614i	$-0.282056\pi$
$503$	14269.1	1.26487	0.632435	+	0.774614i	$0.282056\pi$
$504$	−3147.66	−0.278191
$505$	−207.756	−0.0183070
$506$	−3396.79	−0.298430
$507$	0	0
$508$	−1223.96	−0.106899
$509$	12595.8	1.09685	0.548426	−	0.836199i	$-0.315227\pi$
$509$	12595.8	1.09685	0.548426	+	0.836199i	$0.315227\pi$
$510$	−49.5996	−0.00430649
$511$	−16946.3	−1.46704
$512$	−512.000	−0.0441942
$513$	−19909.8	−1.71353
$514$	3568.54	0.306229
$515$	16.1681	0.00138340
$516$	−7335.66	−0.625842
$517$	2973.76	0.252970
$518$	−12508.2	−1.06096
$519$	−4299.34	−0.363622
$520$	0	0
$521$	−14217.5	−1.19555	−0.597773	−	0.801665i	$-0.703948\pi$
$521$	−14217.5	−1.19555	−0.597773	+	0.801665i	$0.703948\pi$
$522$	−824.234	−0.0691106
$523$	1118.21	0.0934916	0.0467458	−	0.998907i	$-0.485115\pi$
$523$	1118.21	0.0934916	0.0467458	+	0.998907i	$0.485115\pi$
$524$	380.350	0.0317093
$525$	−16684.2	−1.38697
$526$	13627.3	1.12962
$527$	525.131	0.0434062
$528$	−661.964	−0.0545612
$529$	13826.5	1.13639
$530$	−34.8787	−0.00285855
$531$	−8140.43	−0.665281
$532$	17868.0	1.45616
$533$	0	0
$534$	4569.64	0.370314
$535$	12.0150	0.000970941 0
$536$	−4343.54	−0.350023
$537$	9773.56	0.785401
$538$	−9554.27	−0.765639
$539$	8558.13	0.683905
$540$	92.6140	0.00738050
$541$	−12016.8	−0.954973	−0.477487	−	0.878639i	$-0.658452\pi$
$541$	−12016.8	−0.954973	−0.477487	+	0.878639i	$0.658452\pi$
$542$	1806.08	0.143132
$543$	−13069.6	−1.03291
$544$	1322.18	0.104206
$545$	−99.3553	−0.00780902
$546$	0	0
$547$	−4885.51	−0.381882	−0.190941	−	0.981602i	$-0.561154\pi$
$547$	−4885.51	−0.381882	−0.190941	+	0.981602i	$0.561154\pi$
$548$	8620.13	0.671960
$549$	−6945.95	−0.539974
$550$	2633.08	0.204136
$551$	4678.85	0.361753
$552$	5065.61	0.390591
$553$	12116.8	0.931755
$554$	−17153.5	−1.31549
$555$	110.435	0.00844629
$556$	738.550	0.0563336
$557$	12515.0	0.952026	0.476013	−	0.879438i	$-0.342082\pi$
$557$	12515.0	0.952026	0.476013	+	0.879438i	$0.342082\pi$
$558$	−294.230	−0.0223222
$559$	0	0
$560$	−83.1163	−0.00627197
$561$	1709.45	0.128650
$562$	11229.9	0.842890
$563$	−14533.7	−1.08796	−0.543982	−	0.839097i	$-0.683084\pi$
$563$	−14533.7	−1.08796	−0.543982	+	0.839097i	$0.683084\pi$
$564$	−4434.74	−0.331093
$565$	−195.244	−0.0145380
$566$	−11637.8	−0.864262
$567$	9601.89	0.711184
$568$	7256.21	0.536028
$569$	−6413.41	−0.472520	−0.236260	−	0.971690i	$-0.575922\pi$
$569$	−6413.41	−0.472520	−0.236260	+	0.971690i	$0.575922\pi$
$570$	−157.757	−0.0115924
$571$	21173.5	1.55181	0.775905	−	0.630850i	$-0.217294\pi$
$571$	21173.5	1.55181	0.775905	+	0.630850i	$0.217294\pi$
$572$	0	0
$573$	−1367.05	−0.0996669
$574$	−30140.1	−2.19168
$575$	−20149.4	−1.46137
$576$	−740.817	−0.0535892
$577$	1040.31	0.0750584	0.0375292	−	0.999296i	$-0.488051\pi$
$577$	1040.31	0.0750584	0.0375292	+	0.999296i	$0.488051\pi$
$578$	6411.62	0.461398
$579$	−15161.5	−1.08824
$580$	−21.7645	−0.00155814
$581$	−31751.7	−2.26727
$582$	−2629.02	−0.187245
$583$	1202.09	0.0853952
$584$	−3988.38	−0.282603
$585$	0	0
$586$	13075.6	0.921755
$587$	−11575.4	−0.813916	−0.406958	−	0.913447i	$-0.633411\pi$
$587$	−11575.4	−0.813916	−0.406958	+	0.913447i	$0.633411\pi$
$588$	−12762.7	−0.895110
$589$	1670.23	0.116843
$590$	−214.954	−0.0149992
$591$	−1169.68	−0.0814118
$592$	−2943.87	−0.204379
$593$	27162.8	1.88101	0.940507	−	0.339774i	$-0.110351\pi$
$593$	27162.8	1.88101	0.940507	+	0.339774i	$0.110351\pi$
$594$	−3191.93	−0.220482
$595$	214.638	0.0147888
$596$	−5797.37	−0.398439
$597$	−234.716	−0.0160909
$598$	0	0
$599$	4086.72	0.278762	0.139381	−	0.990239i	$-0.455489\pi$
$599$	4086.72	0.278762	0.139381	+	0.990239i	$0.455489\pi$
$600$	−3926.70	−0.267178
$601$	15180.0	1.03029	0.515144	−	0.857104i	$-0.327738\pi$
$601$	15180.0	1.03029	0.515144	+	0.857104i	$0.327738\pi$
$602$	31744.4	2.14918
$603$	−6284.70	−0.424433
$604$	−752.808	−0.0507141
$605$	−186.453	−0.0125296
$606$	−10678.1	−0.715790
$607$	16314.2	1.09089	0.545447	−	0.838146i	$-0.316360\pi$
$607$	16314.2	1.09089	0.545447	+	0.838146i	$0.316360\pi$
$608$	4205.33	0.280507
$609$	−4752.98	−0.316257
$610$	−183.413	−0.0121740
$611$	0	0
$612$	1913.07	0.126359
$613$	−22822.6	−1.50374	−0.751872	−	0.659309i	$-0.770849\pi$
$613$	−22822.6	−1.50374	−0.751872	+	0.659309i	$0.770849\pi$
$614$	−5349.47	−0.351608
$615$	266.106	0.0174479
$616$	2864.59	0.187367
$617$	−14065.9	−0.917782	−0.458891	−	0.888493i	$-0.651753\pi$
$617$	−14065.9	−0.917782	−0.458891	+	0.888493i	$0.651753\pi$
$618$	830.997	0.0540900
$619$	11387.0	0.739387	0.369694	−	0.929154i	$-0.379463\pi$
$619$	11387.0	0.739387	0.369694	+	0.929154i	$0.379463\pi$
$620$	−7.76936	−0.000503266 0
$621$	24425.9	1.57838
$622$	−16644.4	−1.07296
$623$	−19774.7	−1.27168
$624$	0	0
$625$	15616.2	0.999439
$626$	12058.2	0.769879
$627$	5437.06	0.346308
$628$	−1740.46	−0.110592
$629$	7602.19	0.481906
$630$	−120.262	−0.00760530
$631$	22572.1	1.42406	0.712028	−	0.702151i	$-0.247777\pi$
$631$	22572.1	1.42406	0.712028	+	0.702151i	$0.247777\pi$
$632$	2851.75	0.179488
$633$	13573.6	0.852294
$634$	−12786.8	−0.800992
$635$	−46.7636	−0.00292245
$636$	−1792.67	−0.111767
$637$	0	0
$638$	750.110	0.0465473
$639$	10499.1	0.649980
$640$	−19.5618	−0.00120820
$641$	−21290.0	−1.31186	−0.655932	−	0.754820i	$-0.727724\pi$
$641$	−21290.0	−1.31186	−0.655932	+	0.754820i	$0.727724\pi$
$642$	617.538	0.0379630
$643$	−17817.6	−1.09278	−0.546390	−	0.837531i	$-0.683998\pi$
$643$	−17817.6	−1.09278	−0.546390	+	0.837531i	$0.683998\pi$
$644$	−21921.0	−1.34132
$645$	−280.271	−0.0171095
$646$	−10859.8	−0.661411
$647$	6318.10	0.383910	0.191955	−	0.981404i	$-0.438517\pi$
$647$	6318.10	0.383910	0.191955	+	0.981404i	$0.438517\pi$
$648$	2259.85	0.136999
$649$	7408.36	0.448079
$650$	0	0
$651$	−1696.69	−0.102148
$652$	2844.58	0.170863
$653$	5953.96	0.356809	0.178405	−	0.983957i	$-0.442906\pi$
$653$	5953.96	0.356809	0.178405	+	0.983957i	$0.442906\pi$
$654$	−5106.59	−0.305327
$655$	14.5319	0.000866883 0
$656$	−7093.60	−0.422193
$657$	−5770.82	−0.342681
$658$	19191.0	1.13699
$659$	−31915.6	−1.88658	−0.943290	−	0.331969i	$-0.892287\pi$
$659$	−31915.6	−1.88658	−0.943290	+	0.331969i	$0.892287\pi$
$660$	−25.2914	−0.00149162
$661$	22796.0	1.34139	0.670697	−	0.741731i	$-0.265995\pi$
$661$	22796.0	1.34139	0.670697	+	0.741731i	$0.265995\pi$
$662$	−10891.2	−0.639423
$663$	0	0
$664$	−7472.91	−0.436755
$665$	682.678	0.0398092
$666$	−4259.50	−0.247826
$667$	−5740.13	−0.333222
$668$	−10280.6	−0.595463
$669$	−8567.71	−0.495137
$670$	−165.952	−0.00956909
$671$	6321.30	0.363683
$672$	−4271.95	−0.245229
$673$	13804.6	0.790681	0.395341	−	0.918535i	$-0.370627\pi$
$673$	13804.6	0.790681	0.395341	+	0.918535i	$0.370627\pi$
$674$	3639.29	0.207983
$675$	−18934.2	−1.07967
$676$	0	0
$677$	21662.7	1.22978	0.614892	−	0.788611i	$-0.289199\pi$
$677$	21662.7	1.22978	0.614892	+	0.788611i	$0.289199\pi$
$678$	−10035.0	−0.568426
$679$	11376.9	0.643011
$680$	50.5161	0.00284883
$681$	12394.2	0.697424
$682$	267.770	0.0150344
$683$	13883.3	0.777788	0.388894	−	0.921282i	$-0.372857\pi$
$683$	13883.3	0.777788	0.388894	+	0.921282i	$0.372857\pi$
$684$	6084.72	0.340139
$685$	329.346	0.0183703
$686$	31911.5	1.77607
$687$	−5366.19	−0.298010
$688$	7471.20	0.414007
$689$	0	0
$690$	193.540	0.0106782
$691$	−864.998	−0.0476209	−0.0238105	−	0.999716i	$-0.507580\pi$
$691$	−864.998	−0.0476209	−0.0238105	+	0.999716i	$0.507580\pi$
$692$	4378.77	0.240543
$693$	4144.80	0.227198
$694$	8890.12	0.486260
$695$	28.2175	0.00154007
$696$	−1118.63	−0.0609221
$697$	18318.4	0.995494
$698$	4708.85	0.255347
$699$	19756.5	1.06904
$700$	16992.5	0.917507
$701$	−25377.5	−1.36732	−0.683661	−	0.729799i	$-0.739613\pi$
$701$	−25377.5	−1.36732	−0.683661	+	0.729799i	$0.739613\pi$
$702$	0	0
$703$	24179.5	1.29722
$704$	674.195	0.0360933
$705$	−169.437	−0.00905157
$706$	8640.92	0.460631
$707$	46208.6	2.45807
$708$	−11048.0	−0.586456
$709$	−18630.0	−0.986834	−0.493417	−	0.869793i	$-0.664252\pi$
$709$	−18630.0	−0.986834	−0.493417	+	0.869793i	$0.664252\pi$
$710$	277.236	0.0146542
$711$	4126.23	0.217645
$712$	−4654.07	−0.244970
$713$	−2049.08	−0.107628
$714$	11031.8	0.578229
$715$	0	0
$716$	−9954.14	−0.519558
$717$	−18438.6	−0.960395
$718$	−23873.4	−1.24087
$719$	37966.4	1.96927	0.984637	−	0.174615i	$-0.0558682\pi$
$719$	37966.4	1.96927	0.984637	+	0.174615i	$0.0558682\pi$
$720$	−28.3041	−0.00146505
$721$	−3596.07	−0.185748
$722$	−20822.6	−1.07332
$723$	−11331.6	−0.582886
$724$	13311.0	0.683288
$725$	4449.57	0.227935
$726$	−9583.16	−0.489896
$727$	22117.8	1.12834	0.564169	−	0.825659i	$-0.309197\pi$
$727$	22117.8	1.12834	0.564169	+	0.825659i	$0.309197\pi$
$728$	0	0
$729$	19335.2	0.982327
$730$	−152.383	−0.00772594
$731$	−19293.5	−0.976191
$732$	−9426.91	−0.475995
$733$	−8619.47	−0.434335	−0.217167	−	0.976134i	$-0.569682\pi$
$733$	−8619.47	−0.434335	−0.217167	+	0.976134i	$0.569682\pi$
$734$	−4269.93	−0.214722
$735$	−487.620	−0.0244709
$736$	−5159.20	−0.258384
$737$	5719.52	0.285863
$738$	−10263.8	−0.511945
$739$	−6659.12	−0.331474	−0.165737	−	0.986170i	$-0.553000\pi$
$739$	−6659.12	−0.331474	−0.165737	+	0.986170i	$0.553000\pi$
$740$	−112.475	−0.00558739
$741$	0	0
$742$	7757.62	0.383815
$743$	−16754.8	−0.827286	−0.413643	−	0.910439i	$-0.635744\pi$
$743$	−16754.8	−0.827286	−0.413643	+	0.910439i	$0.635744\pi$
$744$	−399.324	−0.0196773
$745$	−221.498	−0.0108927
$746$	−19637.8	−0.963796
$747$	−10812.6	−0.529602
$748$	−1741.03	−0.0851048
$749$	−2672.34	−0.130368
$750$	−300.080	−0.0146098
$751$	8114.13	0.394259	0.197130	−	0.980377i	$-0.436838\pi$
$751$	8114.13	0.394259	0.197130	+	0.980377i	$0.436838\pi$
$752$	4516.68	0.219025
$753$	−13345.5	−0.645867
$754$	0	0
$755$	−28.7623	−0.00138645
$756$	−20599.0	−0.990975
$757$	−30442.5	−1.46163	−0.730814	−	0.682576i	$-0.760859\pi$
$757$	−30442.5	−1.46163	−0.730814	+	0.682576i	$0.760859\pi$
$758$	−4240.26	−0.203184
$759$	−6670.33	−0.318995
$760$	160.671	0.00766864
$761$	−4266.18	−0.203218	−0.101609	−	0.994824i	$-0.532399\pi$
$761$	−4266.18	−0.203218	−0.101609	+	0.994824i	$0.532399\pi$
$762$	−2403.52	−0.114265
$763$	22098.3	1.04851
$764$	1392.30	0.0659317
$765$	73.0921	0.00345445
$766$	3325.67	0.156869
$767$	0	0
$768$	−1005.42	−0.0472397
$769$	7718.91	0.361965	0.180982	−	0.983486i	$-0.442072\pi$
$769$	7718.91	0.361965	0.180982	+	0.983486i	$0.442072\pi$
$770$	109.447	0.00512231
$771$	7007.60	0.327331
$772$	15441.6	0.719892
$773$	36026.8	1.67632	0.838159	−	0.545426i	$-0.183632\pi$
$773$	36026.8	1.67632	0.838159	+	0.545426i	$0.183632\pi$
$774$	10810.1	0.502019
$775$	1588.38	0.0736212
$776$	2677.60	0.123866
$777$	−24562.6	−1.13408
$778$	6664.52	0.307114
$779$	58263.5	2.67973
$780$	0	0
$781$	−9554.89	−0.437773
$782$	13323.0	0.609247
$783$	−5393.96	−0.246187
$784$	12998.5	0.592133
$785$	−66.4969	−0.00302341
$786$	746.900	0.0338945
$787$	−15831.3	−0.717060	−0.358530	−	0.933518i	$-0.616722\pi$
$787$	−15831.3	−0.717060	−0.358530	+	0.933518i	$0.616722\pi$
$788$	1191.30	0.0538555
$789$	26760.2	1.20746
$790$	108.956	0.00490693
$791$	43425.8	1.95201
$792$	975.498	0.0437662
$793$	0	0
$794$	7896.98	0.352964
$795$	−68.4918	−0.00305554
$796$	239.052	0.0106445
$797$	−3142.78	−0.139678	−0.0698388	−	0.997558i	$-0.522248\pi$
$797$	−3142.78	−0.139678	−0.0698388	+	0.997558i	$0.522248\pi$
$798$	35087.8	1.55651
$799$	−11663.8	−0.516441
$800$	3999.25	0.176744
$801$	−6734.01	−0.297047
$802$	−16283.1	−0.716926
$803$	5251.85	0.230802
$804$	−8529.48	−0.374144
$805$	−837.527	−0.0366695
$806$	0	0
$807$	−18761.9	−0.818401
$808$	10875.4	0.473509
$809$	19323.3	0.839767	0.419884	−	0.907578i	$-0.362071\pi$
$809$	19323.3	0.839767	0.419884	+	0.907578i	$0.362071\pi$
$810$	86.3412	0.00374534
$811$	−16218.2	−0.702218	−0.351109	−	0.936335i	$-0.614195\pi$
$811$	−16218.2	−0.702218	−0.351109	+	0.936335i	$0.614195\pi$
$812$	4840.80	0.209210
$813$	3546.63	0.152996
$814$	3876.44	0.166916
$815$	108.682	0.00467112
$816$	2596.39	0.111387
$817$	−61364.9	−2.62777
$818$	−3117.57	−0.133256
$819$	0	0
$820$	−271.023	−0.0115421
$821$	−22452.4	−0.954437	−0.477218	−	0.878785i	$-0.658355\pi$
$821$	−22452.4	−0.954437	−0.477218	+	0.878785i	$0.658355\pi$
$822$	16927.5	0.718266
$823$	40384.9	1.71049	0.855243	−	0.518227i	$-0.173408\pi$
$823$	40384.9	1.71049	0.855243	+	0.518227i	$0.173408\pi$
$824$	−846.351	−0.0357816
$825$	5170.63	0.218204
$826$	47809.4	2.01393
$827$	−10768.9	−0.452807	−0.226404	−	0.974034i	$-0.572697\pi$
$827$	−10768.9	−0.452807	−0.226404	+	0.974034i	$0.572697\pi$
$828$	−7464.89	−0.313313
$829$	26899.9	1.12699	0.563494	−	0.826120i	$-0.309457\pi$
$829$	26899.9	1.12699	0.563494	+	0.826120i	$0.309457\pi$
$830$	−285.515	−0.0119402
$831$	−33684.7	−1.40615
$832$	0	0
$833$	−33567.1	−1.39620
$834$	1450.30	0.0602157
$835$	−392.788	−0.0162790
$836$	−5537.52	−0.229090
$837$	−1925.50	−0.0795163
$838$	8647.83	0.356485
$839$	23526.4	0.968082	0.484041	−	0.875045i	$-0.339169\pi$
$839$	23526.4	0.968082	0.484041	+	0.875045i	$0.339169\pi$
$840$	−163.217	−0.00670419
$841$	−23121.4	−0.948026
$842$	17051.8	0.697915
$843$	22052.3	0.900976
$844$	−13824.4	−0.563809
$845$	0	0
$846$	6535.23	0.265586
$847$	41470.3	1.68233
$848$	1825.79	0.0739362
$849$	−22853.3	−0.923820
$850$	−10327.6	−0.416746
$851$	−29664.0	−1.19491
$852$	14249.1	0.572967
$853$	10246.9	0.411310	0.205655	−	0.978625i	$-0.434067\pi$
$853$	10246.9	0.411310	0.205655	+	0.978625i	$0.434067\pi$
$854$	40794.2	1.63460
$855$	232.477	0.00929887
$856$	−628.948	−0.0251133
$857$	44057.2	1.75609	0.878043	−	0.478581i	$-0.158849\pi$
$857$	44057.2	1.75609	0.878043	+	0.478581i	$0.158849\pi$
$858$	0	0
$859$	−38844.4	−1.54290	−0.771452	−	0.636288i	$-0.780469\pi$
$859$	−38844.4	−1.54290	−0.771452	+	0.636288i	$0.780469\pi$
$860$	285.449	0.0113183
$861$	−59186.6	−2.34271
$862$	−14888.2	−0.588276
$863$	3538.63	0.139578	0.0697892	−	0.997562i	$-0.477767\pi$
$863$	3538.63	0.139578	0.0697892	+	0.997562i	$0.477767\pi$
$864$	−4848.06	−0.190896
$865$	167.298	0.00657608
$866$	−11786.3	−0.462488
$867$	12590.6	0.493194
$868$	1728.04	0.0675732
$869$	−3755.15	−0.146588
$870$	−42.7393	−0.00166551
$871$	0	0
$872$	5200.95	0.201980
$873$	3874.24	0.150198
$874$	42375.2	1.64000
$875$	1298.57	0.0501711
$876$	−7832.05	−0.302078
$877$	−13821.9	−0.532192	−0.266096	−	0.963947i	$-0.585734\pi$
$877$	−13821.9	−0.532192	−0.266096	+	0.963947i	$0.585734\pi$
$878$	5724.75	0.220047
$879$	25676.8	0.985275
$880$	25.7587	0.000986735 0
$881$	−17424.0	−0.666323	−0.333161	−	0.942870i	$-0.608115\pi$
$881$	−17424.0	−0.666323	−0.333161	+	0.942870i	$0.608115\pi$
$882$	18807.6	0.718012
$883$	−28891.7	−1.10111	−0.550557	−	0.834797i	$-0.685585\pi$
$883$	−28891.7	−1.10111	−0.550557	+	0.834797i	$0.685585\pi$
$884$	0	0
$885$	−422.108	−0.0160328
$886$	−15418.7	−0.584651
$887$	32638.5	1.23551	0.617753	−	0.786372i	$-0.288043\pi$
$887$	32638.5	1.23551	0.617753	+	0.786372i	$0.288043\pi$
$888$	−5780.92	−0.218463
$889$	10401.0	0.392395
$890$	−177.816	−0.00669710
$891$	−2975.74	−0.111887
$892$	8726.02	0.327543
$893$	−37097.9	−1.39018
$894$	−11384.4	−0.425896
$895$	−380.314	−0.0142039
$896$	4350.88	0.162224
$897$	0	0
$898$	−4349.76	−0.161641
$899$	452.497	0.0167871
$900$	5786.55	0.214317
$901$	−4714.89	−0.174335
$902$	9340.77	0.344804
$903$	62337.1	2.29729
$904$	10220.4	0.376026
$905$	508.570	0.0186800
$906$	−1478.30	−0.0542089
$907$	−26009.1	−0.952169	−0.476085	−	0.879400i	$-0.657944\pi$
$907$	−26009.1	−0.952169	−0.476085	+	0.879400i	$0.657944\pi$
$908$	−12623.2	−0.461360
$909$	15735.7	0.574170
$910$	0	0
$911$	−4030.56	−0.146584	−0.0732922	−	0.997311i	$-0.523351\pi$
$911$	−4030.56	−0.146584	−0.0732922	+	0.997311i	$0.523351\pi$
$912$	8258.07	0.299838
$913$	9840.24	0.356697
$914$	−8225.23	−0.297666
$915$	−360.171	−0.0130130
$916$	5465.34	0.197140
$917$	−3232.15	−0.116396
$918$	12519.5	0.450116
$919$	1967.32	0.0706157	0.0353079	−	0.999376i	$-0.488759\pi$
$919$	1967.32	0.0706157	0.0353079	+	0.999376i	$0.488759\pi$
$920$	−197.116	−0.00706382
$921$	−10504.8	−0.375838
$922$	−11278.2	−0.402851
$923$	0	0
$924$	5625.25	0.200278
$925$	22994.6	0.817362
$926$	−10056.4	−0.356884
$927$	−1224.59	−0.0433882
$928$	1139.30	0.0403012
$929$	36288.5	1.28158	0.640789	−	0.767717i	$-0.278607\pi$
$929$	36288.5	1.28158	0.640789	+	0.767717i	$0.278607\pi$
$930$	−15.2568	−0.000537947 0
$931$	−106764.	−3.75836
$932$	−20121.5	−0.707192
$933$	−32684.8	−1.14690
$934$	14703.0	0.515094
$935$	−66.5189	−0.00232663
$936$	0	0
$937$	3899.73	0.135964	0.0679821	−	0.997687i	$-0.478344\pi$
$937$	3899.73	0.135964	0.0679821	+	0.997687i	$0.478344\pi$
$938$	36910.6	1.28483
$939$	23679.0	0.822933
$940$	172.567	0.00598779
$941$	−32961.6	−1.14189	−0.570944	−	0.820989i	$-0.693423\pi$
$941$	−32961.6	−1.14189	−0.570944	+	0.820989i	$0.693423\pi$
$942$	−3417.76	−0.118213
$943$	−71479.2	−2.46838
$944$	11252.2	0.387952
$945$	−787.017	−0.0270917
$946$	−9837.98	−0.338119
$947$	−17974.7	−0.616788	−0.308394	−	0.951259i	$-0.599791\pi$
$947$	−17974.7	−0.616788	−0.308394	+	0.951259i	$0.599791\pi$
$948$	5600.04	0.191857
$949$	0	0
$950$	−32848.0	−1.12182
$951$	−25109.7	−0.856190
$952$	−11235.7	−0.382510
$953$	7627.67	0.259270	0.129635	−	0.991562i	$-0.458619\pi$
$953$	7627.67	0.259270	0.129635	+	0.991562i	$0.458619\pi$
$954$	2641.75	0.0896539
$955$	53.1952	0.00180247
$956$	18779.3	0.635321
$957$	1473.00	0.0497549
$958$	−6992.29	−0.235815
$959$	−73252.3	−2.46657
$960$	−38.4138	−0.00129146
$961$	−29629.5	−0.994578
$962$	0	0
$963$	−910.030	−0.0304520
$964$	11541.0	0.385591
$965$	589.973	0.0196807
$966$	−43046.6	−1.43375
$967$	−48536.0	−1.61408	−0.807038	−	0.590499i	$-0.798931\pi$
$967$	−48536.0	−1.61408	−0.807038	+	0.590499i	$0.798931\pi$
$968$	9760.23	0.324076
$969$	−21325.5	−0.706991
$970$	102.302	0.00338631
$971$	−19301.5	−0.637915	−0.318957	−	0.947769i	$-0.603333\pi$
$971$	−19301.5	−0.637915	−0.318957	+	0.947769i	$0.603333\pi$
$972$	−11924.5	−0.393496
$973$	−6276.06	−0.206784
$974$	−36652.0	−1.20576
$975$	0	0
$976$	9601.09	0.314881
$977$	−4620.95	−0.151318	−0.0756588	−	0.997134i	$-0.524106\pi$
$977$	−4620.95	−0.151318	−0.0756588	+	0.997134i	$0.524106\pi$
$978$	5585.96	0.182637
$979$	6128.42	0.200067
$980$	496.629	0.0161880
$981$	7525.29	0.244917
$982$	−28540.3	−0.927451
$983$	−17936.8	−0.581990	−0.290995	−	0.956725i	$-0.593986\pi$
$983$	−17936.8	−0.581990	−0.290995	+	0.956725i	$0.593986\pi$
$984$	−13929.8	−0.451287
$985$	45.5154	0.00147233
$986$	−2942.12	−0.0950266
$987$	37685.6	1.21535
$988$	0	0
$989$	75284.0	2.42052
$990$	37.2705	0.00119650
$991$	−45697.1	−1.46480	−0.732400	−	0.680875i	$-0.761600\pi$
$991$	−45697.1	−1.46480	−0.732400	+	0.680875i	$0.761600\pi$
$992$	406.702	0.0130169
$993$	−21387.2	−0.683487
$994$	−61662.0	−1.96760
$995$	9.13339	0.000291003 0
$996$	−14674.7	−0.466853
$997$	7979.45	0.253472	0.126736	−	0.991936i	$-0.459550\pi$
$997$	7979.45	0.253472	0.126736	+	0.991936i	$0.459550\pi$
$998$	2849.03	0.0903654
$999$	−27875.1	−0.882810

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.a.n.1.2	✓	6
13.2	odd	12		338.4.e.i.147.3		24
13.3	even	3		338.4.c.p.191.5		12
13.4	even	6		338.4.c.o.315.5		12
13.5	odd	4		338.4.b.h.337.8		12
13.6	odd	12		338.4.e.i.23.11		24
13.7	odd	12		338.4.e.i.23.3		24
13.8	odd	4		338.4.b.h.337.2		12
13.9	even	3		338.4.c.p.315.5		12
13.10	even	6		338.4.c.o.191.5		12
13.11	odd	12		338.4.e.i.147.11		24
13.12	even	2		338.4.a.o.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.4.a.n.1.2	✓	6	1.1	even	1	trivial
338.4.a.o.1.2	yes	6	13.12	even	2
338.4.b.h.337.2		12	13.8	odd	4
338.4.b.h.337.8		12	13.5	odd	4
338.4.c.o.191.5		12	13.10	even	6
338.4.c.o.315.5		12	13.4	even	6
338.4.c.p.191.5		12	13.3	even	3
338.4.c.p.315.5		12	13.9	even	3
338.4.e.i.23.3		24	13.7	odd	12
338.4.e.i.23.11		24	13.6	odd	12
338.4.e.i.147.3		24	13.2	odd	12
338.4.e.i.147.11		24	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.92743 q^{3} +4.00000 q^{4} +0.152827 q^{5} +7.85487 q^{6} -33.9913 q^{7} -8.00000 q^{8} -11.5753 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.92743 q^{3} +4.00000 q^{4} +0.152827 q^{5} +7.85487 q^{6} -33.9913 q^{7} -8.00000 q^{8} -11.5753 q^{9} -0.305653 q^{10} +10.5343 q^{11} -15.7097 q^{12} +67.9825 q^{14} -0.600216 q^{15} +16.0000 q^{16} -41.3181 q^{17} +23.1505 q^{18} -131.416 q^{19} +0.611306 q^{20} +133.498 q^{21} -21.0686 q^{22} +161.225 q^{23} +31.4195 q^{24} -124.977 q^{25} +151.502 q^{27} -135.965 q^{28} -35.6032 q^{29} +1.20043 q^{30} -12.7094 q^{31} -32.0000 q^{32} -41.3728 q^{33} +82.6363 q^{34} -5.19477 q^{35} -46.3011 q^{36} -183.992 q^{37} +262.833 q^{38} -1.22261 q^{40} -443.350 q^{41} -266.997 q^{42} +466.950 q^{43} +42.1372 q^{44} -1.76901 q^{45} -322.450 q^{46} +282.293 q^{47} -62.8389 q^{48} +812.407 q^{49} +249.953 q^{50} +162.274 q^{51} +114.112 q^{53} -303.004 q^{54} +1.60992 q^{55} +271.930 q^{56} +516.129 q^{57} +71.2065 q^{58} +703.260 q^{59} -2.40086 q^{60} +600.068 q^{61} +25.4189 q^{62} +393.458 q^{63} +64.0000 q^{64} +82.7455 q^{66} +542.943 q^{67} -165.273 q^{68} -633.201 q^{69} +10.3895 q^{70} -907.027 q^{71} +92.6021 q^{72} +498.548 q^{73} +367.983 q^{74} +490.837 q^{75} -525.666 q^{76} -358.074 q^{77} -356.469 q^{79} +2.44522 q^{80} -282.481 q^{81} +886.701 q^{82} +934.114 q^{83} +533.994 q^{84} -6.31451 q^{85} -933.900 q^{86} +139.829 q^{87} -84.2744 q^{88} +581.759 q^{89} +3.53802 q^{90} +644.900 q^{92} +49.9155 q^{93} -564.585 q^{94} -20.0839 q^{95} +125.678 q^{96} -334.700 q^{97} -1624.81 q^{98} -121.937 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * q^99

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