LMFDB - Embedded newform 2013.4.a.f.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31501 q^{2} +3.00000 q^{3} -2.64074 q^{4} -15.6387 q^{5} -6.94502 q^{6} -20.9070 q^{7} +24.6334 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.31501 q^{2} +3.00000 q^{3} -2.64074 q^{4} -15.6387 q^{5} -6.94502 q^{6} -20.9070 q^{7} +24.6334 q^{8} +9.00000 q^{9} +36.2036 q^{10} +11.0000 q^{11} -7.92222 q^{12} -40.0670 q^{13} +48.3998 q^{14} -46.9160 q^{15} -35.9006 q^{16} -24.8056 q^{17} -20.8351 q^{18} +47.0545 q^{19} +41.2976 q^{20} -62.7210 q^{21} -25.4651 q^{22} -146.135 q^{23} +73.9002 q^{24} +119.568 q^{25} +92.7554 q^{26} +27.0000 q^{27} +55.2099 q^{28} -205.373 q^{29} +108.611 q^{30} -93.5433 q^{31} -113.957 q^{32} +33.0000 q^{33} +57.4251 q^{34} +326.957 q^{35} -23.7667 q^{36} -237.268 q^{37} -108.931 q^{38} -120.201 q^{39} -385.233 q^{40} -105.148 q^{41} +145.200 q^{42} -36.0396 q^{43} -29.0481 q^{44} -140.748 q^{45} +338.305 q^{46} +404.641 q^{47} -107.702 q^{48} +94.1023 q^{49} -276.800 q^{50} -74.4167 q^{51} +105.807 q^{52} -271.833 q^{53} -62.5052 q^{54} -172.025 q^{55} -515.010 q^{56} +141.163 q^{57} +475.441 q^{58} -342.748 q^{59} +123.893 q^{60} -61.0000 q^{61} +216.554 q^{62} -188.163 q^{63} +551.016 q^{64} +626.594 q^{65} -76.3953 q^{66} -48.8587 q^{67} +65.5050 q^{68} -438.406 q^{69} -756.909 q^{70} -927.396 q^{71} +221.701 q^{72} -835.991 q^{73} +549.276 q^{74} +358.703 q^{75} -124.259 q^{76} -229.977 q^{77} +278.266 q^{78} -301.400 q^{79} +561.437 q^{80} +81.0000 q^{81} +243.418 q^{82} -975.049 q^{83} +165.630 q^{84} +387.926 q^{85} +83.4319 q^{86} -616.120 q^{87} +270.967 q^{88} -599.452 q^{89} +325.833 q^{90} +837.680 q^{91} +385.906 q^{92} -280.630 q^{93} -936.747 q^{94} -735.869 q^{95} -341.871 q^{96} -222.217 q^{97} -217.848 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9}+O(q^{10}) $ 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9} + 99 q^{10} + 418 q^{11} + 510 q^{12} + 209 q^{13} + 128 q^{14} + 105 q^{15} + 798 q^{16} + 512 q^{17} + 126 q^{18} + 487 q^{19} + 328 q^{20} + 315 q^{21} + 154 q^{22} + 417 q^{23} + 441 q^{24} + 925 q^{25} + 177 q^{26} + 1026 q^{27} + 902 q^{28} + 626 q^{29} + 297 q^{30} + 300 q^{31} + 1625 q^{32} + 1254 q^{33} - 180 q^{34} + 1086 q^{35} + 1530 q^{36} + 554 q^{37} + 845 q^{38} + 627 q^{39} + 329 q^{40} + 1378 q^{41} + 384 q^{42} + 1979 q^{43} + 1870 q^{44} + 315 q^{45} + 937 q^{46} + 1345 q^{47} + 2394 q^{48} + 2635 q^{49} + 800 q^{50} + 1536 q^{51} + 2006 q^{52} + 1497 q^{53} + 378 q^{54} + 385 q^{55} + 415 q^{56} + 1461 q^{57} + 1241 q^{58} + 2827 q^{59} + 984 q^{60} - 2318 q^{61} + 509 q^{62} + 945 q^{63} + 1003 q^{64} + 2810 q^{65} + 462 q^{66} + 369 q^{67} + 3936 q^{68} + 1251 q^{69} + 922 q^{70} + 965 q^{71} + 1323 q^{72} + 3081 q^{73} + 722 q^{74} + 2775 q^{75} + 2210 q^{76} + 1155 q^{77} + 531 q^{78} + 3795 q^{79} + 3793 q^{80} + 3078 q^{81} - 1678 q^{82} + 3869 q^{83} + 2706 q^{84} + 3553 q^{85} + 3305 q^{86} + 1878 q^{87} + 1617 q^{88} + 2849 q^{89} + 891 q^{90} + 1252 q^{91} + 4519 q^{92} + 900 q^{93} + 340 q^{94} + 1504 q^{95} + 4875 q^{96} + 2562 q^{97} + 6164 q^{98} + 3762 q^{99}+O(q^{100}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9 + 99 * q^10 + 418 * q^11 + 510 * q^12 + 209 * q^13 + 128 * q^14 + 105 * q^15 + 798 * q^16 + 512 * q^17 + 126 * q^18 + 487 * q^19 + 328 * q^20 + 315 * q^21 + 154 * q^22 + 417 * q^23 + 441 * q^24 + 925 * q^25 + 177 * q^26 + 1026 * q^27 + 902 * q^28 + 626 * q^29 + 297 * q^30 + 300 * q^31 + 1625 * q^32 + 1254 * q^33 - 180 * q^34 + 1086 * q^35 + 1530 * q^36 + 554 * q^37 + 845 * q^38 + 627 * q^39 + 329 * q^40 + 1378 * q^41 + 384 * q^42 + 1979 * q^43 + 1870 * q^44 + 315 * q^45 + 937 * q^46 + 1345 * q^47 + 2394 * q^48 + 2635 * q^49 + 800 * q^50 + 1536 * q^51 + 2006 * q^52 + 1497 * q^53 + 378 * q^54 + 385 * q^55 + 415 * q^56 + 1461 * q^57 + 1241 * q^58 + 2827 * q^59 + 984 * q^60 - 2318 * q^61 + 509 * q^62 + 945 * q^63 + 1003 * q^64 + 2810 * q^65 + 462 * q^66 + 369 * q^67 + 3936 * q^68 + 1251 * q^69 + 922 * q^70 + 965 * q^71 + 1323 * q^72 + 3081 * q^73 + 722 * q^74 + 2775 * q^75 + 2210 * q^76 + 1155 * q^77 + 531 * q^78 + 3795 * q^79 + 3793 * q^80 + 3078 * q^81 - 1678 * q^82 + 3869 * q^83 + 2706 * q^84 + 3553 * q^85 + 3305 * q^86 + 1878 * q^87 + 1617 * q^88 + 2849 * q^89 + 891 * q^90 + 1252 * q^91 + 4519 * q^92 + 900 * q^93 + 340 * q^94 + 1504 * q^95 + 4875 * q^96 + 2562 * q^97 + 6164 * q^98 + 3762 * 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 $

$2$	−2.31501	−0.818479	−0.409239	−	0.912427i	$-0.634206\pi$
$2$	−2.31501	−0.818479	−0.409239	+	0.912427i	$0.634206\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−2.64074	−0.330092
$5$	−15.6387	−1.39876	−0.699382	−	0.714748i	$-0.746541\pi$
$5$	−15.6387	−1.39876	−0.699382	+	0.714748i	$0.746541\pi$
$6$	−6.94502	−0.472549
$7$	−20.9070	−1.12887	−0.564436	−	0.825477i	$-0.690906\pi$
$7$	−20.9070	−1.12887	−0.564436	+	0.825477i	$0.690906\pi$
$8$	24.6334	1.08865
$9$	9.00000	0.333333
$10$	36.2036	1.14486
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−7.92222	−0.190579
$13$	−40.0670	−0.854814	−0.427407	−	0.904059i	$-0.640573\pi$
$13$	−40.0670	−0.854814	−0.427407	+	0.904059i	$0.640573\pi$
$14$	48.3998	0.923957
$15$	−46.9160	−0.807577
$16$	−35.9006	−0.560947
$17$	−24.8056	−0.353896	−0.176948	−	0.984220i	$-0.556622\pi$
$17$	−24.8056	−0.353896	−0.176948	+	0.984220i	$0.556622\pi$
$18$	−20.8351	−0.272826
$19$	47.0545	0.568160	0.284080	−	0.958801i	$-0.408312\pi$
$19$	47.0545	0.568160	0.284080	+	0.958801i	$0.408312\pi$
$20$	41.2976	0.461721
$21$	−62.7210	−0.651754
$22$	−25.4651	−0.246781
$23$	−146.135	−1.32484	−0.662421	−	0.749132i	$-0.730471\pi$
$23$	−146.135	−1.32484	−0.662421	+	0.749132i	$0.730471\pi$
$24$	73.9002	0.628534
$25$	119.568	0.956541
$26$	92.7554	0.699647
$27$	27.0000	0.192450
$28$	55.2099	0.372632
$29$	−205.373	−1.31506	−0.657532	−	0.753426i	$-0.728400\pi$
$29$	−205.373	−1.31506	−0.657532	+	0.753426i	$0.728400\pi$
$30$	108.611	0.660985
$31$	−93.5433	−0.541964	−0.270982	−	0.962584i	$-0.587348\pi$
$31$	−93.5433	−0.541964	−0.270982	+	0.962584i	$0.587348\pi$
$32$	−113.957	−0.629530
$33$	33.0000	0.174078
$34$	57.4251	0.289656
$35$	326.957	1.57902
$36$	−23.7667	−0.110031
$37$	−237.268	−1.05423	−0.527116	−	0.849794i	$-0.676727\pi$
$37$	−237.268	−1.05423	−0.527116	+	0.849794i	$0.676727\pi$
$38$	−108.931	−0.465027
$39$	−120.201	−0.493527
$40$	−385.233	−1.52277
$41$	−105.148	−0.400519	−0.200260	−	0.979743i	$-0.564179\pi$
$41$	−105.148	−0.400519	−0.200260	+	0.979743i	$0.564179\pi$
$42$	145.200	0.533447
$43$	−36.0396	−0.127814	−0.0639068	−	0.997956i	$-0.520356\pi$
$43$	−36.0396	−0.127814	−0.0639068	+	0.997956i	$0.520356\pi$
$44$	−29.0481	−0.0995266
$45$	−140.748	−0.466255
$46$	338.305	1.08435
$47$	404.641	1.25581	0.627904	−	0.778291i	$-0.283913\pi$
$47$	404.641	1.25581	0.627904	+	0.778291i	$0.283913\pi$
$48$	−107.702	−0.323863
$49$	94.1023	0.274351
$50$	−276.800	−0.782909
$51$	−74.4167	−0.204322
$52$	105.807	0.282168
$53$	−271.833	−0.704511	−0.352255	−	0.935904i	$-0.614585\pi$
$53$	−271.833	−0.704511	−0.352255	+	0.935904i	$0.614585\pi$
$54$	−62.5052	−0.157516
$55$	−172.025	−0.421743
$56$	−515.010	−1.22895
$57$	141.163	0.328027
$58$	475.441	1.07635
$59$	−342.748	−0.756304	−0.378152	−	0.925744i	$-0.623440\pi$
$59$	−342.748	−0.756304	−0.378152	+	0.925744i	$0.623440\pi$
$60$	123.893	0.266575
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	216.554	0.443586
$63$	−188.163	−0.376290
$64$	551.016	1.07620
$65$	626.594	1.19568
$66$	−76.3953	−0.142479
$67$	−48.8587	−0.0890901	−0.0445451	−	0.999007i	$-0.514184\pi$
$67$	−48.8587	−0.0890901	−0.0445451	+	0.999007i	$0.514184\pi$
$68$	65.5050	0.116818
$69$	−438.406	−0.764897
$70$	−756.909	−1.29240
$71$	−927.396	−1.55016	−0.775082	−	0.631861i	$-0.782291\pi$
$71$	−927.396	−1.55016	−0.775082	+	0.631861i	$0.782291\pi$
$72$	221.701	0.362884
$73$	−835.991	−1.34035	−0.670174	−	0.742204i	$-0.733780\pi$
$73$	−835.991	−1.34035	−0.670174	+	0.742204i	$0.733780\pi$
$74$	549.276	0.862866
$75$	358.703	0.552259
$76$	−124.259	−0.187545
$77$	−229.977	−0.340368
$78$	278.266	0.403942
$79$	−301.400	−0.429243	−0.214621	−	0.976697i	$-0.568852\pi$
$79$	−301.400	−0.429243	−0.214621	+	0.976697i	$0.568852\pi$
$80$	561.437	0.784632
$81$	81.0000	0.111111
$82$	243.418	0.327817
$83$	−975.049	−1.28946	−0.644732	−	0.764408i	$-0.723031\pi$
$83$	−975.049	−1.28946	−0.644732	+	0.764408i	$0.723031\pi$
$84$	165.630	0.215139
$85$	387.926	0.495017
$86$	83.4319	0.104613
$87$	−616.120	−0.759253
$88$	270.967	0.328241
$89$	−599.452	−0.713953	−0.356976	−	0.934113i	$-0.616192\pi$
$89$	−599.452	−0.713953	−0.356976	+	0.934113i	$0.616192\pi$
$90$	325.833	0.381620
$91$	837.680	0.964975
$92$	385.906	0.437320
$93$	−280.630	−0.312903
$94$	−936.747	−1.02785
$95$	−735.869	−0.794721
$96$	−341.871	−0.363459
$97$	−222.217	−0.232605	−0.116302	−	0.993214i	$-0.537104\pi$
$97$	−222.217	−0.232605	−0.116302	+	0.993214i	$0.537104\pi$
$98$	−217.848	−0.224550
$99$	99.0000	0.100504
$100$	−315.747	−0.315747
$101$	−1046.10	−1.03060	−0.515301	−	0.857009i	$-0.672320\pi$
$101$	−1046.10	−1.03060	−0.515301	+	0.857009i	$0.672320\pi$
$102$	172.275	0.167233
$103$	−1829.67	−1.75032	−0.875158	−	0.483837i	$-0.839243\pi$
$103$	−1829.67	−1.75032	−0.875158	+	0.483837i	$0.839243\pi$
$104$	−986.986	−0.930596
$105$	980.872	0.911650
$106$	629.294	0.576627
$107$	−867.125	−0.783440	−0.391720	−	0.920084i	$-0.628120\pi$
$107$	−867.125	−0.783440	−0.391720	+	0.920084i	$0.628120\pi$
$108$	−71.3000	−0.0635263
$109$	−170.370	−0.149711	−0.0748553	−	0.997194i	$-0.523849\pi$
$109$	−170.370	−0.149711	−0.0748553	+	0.997194i	$0.523849\pi$
$110$	398.240	0.345188
$111$	−711.803	−0.608661
$112$	750.573	0.633237
$113$	1178.82	0.981366	0.490683	−	0.871338i	$-0.336747\pi$
$113$	1178.82	0.981366	0.490683	+	0.871338i	$0.336747\pi$
$114$	−326.794	−0.268483
$115$	2285.36	1.85314
$116$	542.337	0.434093
$117$	−360.603	−0.284938
$118$	793.463	0.619019
$119$	518.610	0.399503
$120$	−1155.70	−0.879171
$121$	121.000	0.0909091
$122$	141.215	0.104795
$123$	−315.443	−0.231240
$124$	247.024	0.178898
$125$	84.9549	0.0607888
$126$	435.599	0.307986
$127$	−2628.07	−1.83625	−0.918125	−	0.396291i	$-0.870297\pi$
$127$	−2628.07	−1.83625	−0.918125	+	0.396291i	$0.870297\pi$
$128$	−363.950	−0.251320
$129$	−108.119	−0.0737932
$130$	−1450.57	−0.978642
$131$	1121.47	0.747964	0.373982	−	0.927436i	$-0.377992\pi$
$131$	1121.47	0.747964	0.373982	+	0.927436i	$0.377992\pi$
$132$	−87.1444	−0.0574617
$133$	−983.767	−0.641379
$134$	113.108	0.0729184
$135$	−422.244	−0.269192
$136$	−611.045	−0.385270
$137$	205.060	0.127879	0.0639397	−	0.997954i	$-0.479633\pi$
$137$	205.060	0.127879	0.0639397	+	0.997954i	$0.479633\pi$
$138$	1014.91	0.626052
$139$	744.828	0.454500	0.227250	−	0.973836i	$-0.427027\pi$
$139$	744.828	0.454500	0.227250	+	0.973836i	$0.427027\pi$
$140$	−863.409	−0.521224
$141$	1213.92	0.725041
$142$	2146.93	1.26878
$143$	−440.737	−0.257736
$144$	−323.105	−0.186982
$145$	3211.76	1.83946
$146$	1935.33	1.09705
$147$	282.307	0.158397
$148$	626.562	0.347994
$149$	−783.355	−0.430705	−0.215352	−	0.976536i	$-0.569090\pi$
$149$	−783.355	−0.430705	−0.215352	+	0.976536i	$0.569090\pi$
$150$	−830.400	−0.452012
$151$	2844.14	1.53280	0.766400	−	0.642363i	$-0.222046\pi$
$151$	2844.14	1.53280	0.766400	+	0.642363i	$0.222046\pi$
$152$	1159.11	0.618528
$153$	−223.250	−0.117965
$154$	532.398	0.278584
$155$	1462.89	0.758080
$156$	317.420	0.162910
$157$	1403.98	0.713695	0.356848	−	0.934163i	$-0.383852\pi$
$157$	1403.98	0.713695	0.356848	+	0.934163i	$0.383852\pi$
$158$	697.744	0.351326
$159$	−815.498	−0.406749
$160$	1782.14	0.880563
$161$	3055.25	1.49558
$162$	−187.516	−0.0909421
$163$	−163.737	−0.0786802	−0.0393401	−	0.999226i	$-0.512526\pi$
$163$	−163.737	−0.0786802	−0.0393401	+	0.999226i	$0.512526\pi$
$164$	277.667	0.132208
$165$	−516.076	−0.243494
$166$	2257.25	1.05540
$167$	904.024	0.418895	0.209448	−	0.977820i	$-0.432833\pi$
$167$	904.024	0.418895	0.209448	+	0.977820i	$0.432833\pi$
$168$	−1545.03	−0.709534
$169$	−591.636	−0.269293
$170$	−898.051	−0.405161
$171$	423.490	0.189387
$172$	95.1711	0.0421903
$173$	3516.77	1.54552	0.772761	−	0.634697i	$-0.218875\pi$
$173$	3516.77	1.54552	0.772761	+	0.634697i	$0.218875\pi$
$174$	1426.32	0.621432
$175$	−2499.80	−1.07981
$176$	−394.906	−0.169132
$177$	−1028.24	−0.436652
$178$	1387.74	0.584355
$179$	−1242.06	−0.518637	−0.259318	−	0.965792i	$-0.583498\pi$
$179$	−1242.06	−0.518637	−0.259318	+	0.965792i	$0.583498\pi$
$180$	371.679	0.153907
$181$	−1612.67	−0.662259	−0.331129	−	0.943585i	$-0.607430\pi$
$181$	−1612.67	−0.662259	−0.331129	+	0.943585i	$0.607430\pi$
$182$	−1939.24	−0.789812
$183$	−183.000	−0.0739221
$184$	−3599.81	−1.44229
$185$	3710.55	1.47462
$186$	649.661	0.256104
$187$	−272.861	−0.106704
$188$	−1068.55	−0.414533
$189$	−564.489	−0.217251
$190$	1703.54	0.650463
$191$	2829.61	1.07195	0.535977	−	0.844233i	$-0.319943\pi$
$191$	2829.61	1.07195	0.535977	+	0.844233i	$0.319943\pi$
$192$	1653.05	0.621346
$193$	3050.28	1.13764	0.568819	−	0.822463i	$-0.307401\pi$
$193$	3050.28	1.13764	0.568819	+	0.822463i	$0.307401\pi$
$194$	514.433	0.190382
$195$	1879.78	0.690328
$196$	−248.500	−0.0905612
$197$	−2289.61	−0.828061	−0.414030	−	0.910263i	$-0.635879\pi$
$197$	−2289.61	−0.828061	−0.414030	+	0.910263i	$0.635879\pi$
$198$	−229.186	−0.0822602
$199$	−1207.60	−0.430174	−0.215087	−	0.976595i	$-0.569003\pi$
$199$	−1207.60	−0.430174	−0.215087	+	0.976595i	$0.569003\pi$
$200$	2945.36	1.04134
$201$	−146.576	−0.0514362
$202$	2421.73	0.843526
$203$	4293.74	1.48454
$204$	196.515	0.0674452
$205$	1644.37	0.560232
$206$	4235.70	1.43260
$207$	−1315.22	−0.441614
$208$	1438.43	0.479505
$209$	517.599	0.171307
$210$	−2270.73	−0.746167
$211$	−3548.30	−1.15770	−0.578850	−	0.815434i	$-0.696498\pi$
$211$	−3548.30	−1.15770	−0.578850	+	0.815434i	$0.696498\pi$
$212$	717.839	0.232554
$213$	−2782.19	−0.894988
$214$	2007.40	0.641229
$215$	563.610	0.178781
$216$	665.102	0.209511
$217$	1955.71	0.611808
$218$	394.407	0.122535
$219$	−2507.97	−0.773850
$220$	454.274	0.139214
$221$	993.885	0.302515
$222$	1647.83	0.498176
$223$	1272.61	0.382155	0.191077	−	0.981575i	$-0.438802\pi$
$223$	1272.61	0.382155	0.191077	+	0.981575i	$0.438802\pi$
$224$	2382.50	0.710658
$225$	1076.11	0.318847
$226$	−2728.98	−0.803227
$227$	6150.72	1.79840	0.899202	−	0.437533i	$-0.144148\pi$
$227$	6150.72	1.79840	0.899202	+	0.437533i	$0.144148\pi$
$228$	−372.776	−0.108279
$229$	−5764.67	−1.66349	−0.831746	−	0.555156i	$-0.812659\pi$
$229$	−5764.67	−1.66349	−0.831746	+	0.555156i	$0.812659\pi$
$230$	−5290.63	−1.51676
$231$	−689.931	−0.196511
$232$	−5059.04	−1.43165
$233$	4108.69	1.15523	0.577616	−	0.816309i	$-0.303983\pi$
$233$	4108.69	1.15523	0.577616	+	0.816309i	$0.303983\pi$
$234$	834.799	0.233216
$235$	−6328.05	−1.75658
$236$	905.107	0.249650
$237$	−904.201	−0.247823
$238$	−1200.59	−0.326985
$239$	4110.78	1.11257	0.556286	−	0.830991i	$-0.312226\pi$
$239$	4110.78	1.11257	0.556286	+	0.830991i	$0.312226\pi$
$240$	1684.31	0.453007
$241$	−5286.08	−1.41289	−0.706445	−	0.707768i	$-0.749702\pi$
$241$	−5286.08	−1.41289	−0.706445	+	0.707768i	$0.749702\pi$
$242$	−280.116	−0.0744072
$243$	243.000	0.0641500
$244$	161.085	0.0422640
$245$	−1471.63	−0.383752
$246$	730.253	0.189265
$247$	−1885.33	−0.485671
$248$	−2304.29	−0.590010
$249$	−2925.15	−0.744473
$250$	−196.671	−0.0497543
$251$	−1483.26	−0.372999	−0.186499	−	0.982455i	$-0.559714\pi$
$251$	−1483.26	−0.372999	−0.186499	+	0.982455i	$0.559714\pi$
$252$	496.889	0.124211
$253$	−1607.49	−0.399455
$254$	6084.01	1.50293
$255$	1163.78	0.285798
$256$	−3565.58	−0.870503
$257$	2821.06	0.684720	0.342360	−	0.939569i	$-0.388774\pi$
$257$	2821.06	0.684720	0.342360	+	0.939569i	$0.388774\pi$
$258$	250.296	0.0603982
$259$	4960.55	1.19009
$260$	−1654.67	−0.394686
$261$	−1848.36	−0.438355
$262$	−2596.21	−0.612193
$263$	−2023.23	−0.474364	−0.237182	−	0.971465i	$-0.576224\pi$
$263$	−2023.23	−0.474364	−0.237182	+	0.971465i	$0.576224\pi$
$264$	812.902	0.189510
$265$	4251.10	0.985444
$266$	2277.43	0.524955
$267$	−1798.36	−0.412201
$268$	129.023	0.0294080
$269$	8415.29	1.90740	0.953698	−	0.300767i	$-0.0972428\pi$
$269$	8415.29	1.90740	0.953698	+	0.300767i	$0.0972428\pi$
$270$	977.498	0.220328
$271$	−4863.20	−1.09010	−0.545052	−	0.838402i	$-0.683490\pi$
$271$	−4863.20	−1.09010	−0.545052	+	0.838402i	$0.683490\pi$
$272$	890.534	0.198517
$273$	2513.04	0.557129
$274$	−474.716	−0.104667
$275$	1315.24	0.288408
$276$	1157.72	0.252487
$277$	2558.45	0.554954	0.277477	−	0.960732i	$-0.410502\pi$
$277$	2558.45	0.554954	0.277477	+	0.960732i	$0.410502\pi$
$278$	−1724.28	−0.371999
$279$	−841.890	−0.180655
$280$	8054.07	1.71901
$281$	−809.967	−0.171952	−0.0859761	−	0.996297i	$-0.527401\pi$
$281$	−809.967	−0.171952	−0.0859761	+	0.996297i	$0.527401\pi$
$282$	−2810.24	−0.593431
$283$	−1433.36	−0.301075	−0.150537	−	0.988604i	$-0.548100\pi$
$283$	−1433.36	−0.301075	−0.150537	+	0.988604i	$0.548100\pi$
$284$	2449.01	0.511697
$285$	−2207.61	−0.458833
$286$	1020.31	0.210952
$287$	2198.32	0.452135
$288$	−1025.61	−0.209843
$289$	−4297.68	−0.874758
$290$	−7435.26	−1.50556
$291$	−666.650	−0.134295
$292$	2207.63	0.442438
$293$	−3703.03	−0.738339	−0.369170	−	0.929362i	$-0.620358\pi$
$293$	−3703.03	−0.738339	−0.369170	+	0.929362i	$0.620358\pi$
$294$	−653.543	−0.129644
$295$	5360.11	1.05789
$296$	−5844.71	−1.14769
$297$	297.000	0.0580259
$298$	1813.47	0.352523
$299$	5855.21	1.13249
$300$	−947.241	−0.182297
$301$	753.479	0.144285
$302$	−6584.21	−1.25457
$303$	−3138.30	−0.595018
$304$	−1689.28	−0.318707
$305$	953.958	0.179093
$306$	516.826	0.0965522
$307$	−1188.94	−0.221031	−0.110516	−	0.993874i	$-0.535250\pi$
$307$	−1188.94	−0.221031	−0.110516	+	0.993874i	$0.535250\pi$
$308$	607.309	0.112353
$309$	−5489.01	−1.01055
$310$	−3386.61	−0.620472
$311$	−6705.20	−1.22256	−0.611282	−	0.791413i	$-0.709346\pi$
$311$	−6705.20	−1.22256	−0.611282	+	0.791413i	$0.709346\pi$
$312$	−2960.96	−0.537280
$313$	8884.75	1.60446	0.802230	−	0.597015i	$-0.203647\pi$
$313$	8884.75	1.60446	0.802230	+	0.597015i	$0.203647\pi$
$314$	−3250.23	−0.584144
$315$	2942.62	0.526342
$316$	795.920	0.141690
$317$	1563.46	0.277012	0.138506	−	0.990362i	$-0.455770\pi$
$317$	1563.46	0.277012	0.138506	+	0.990362i	$0.455770\pi$
$318$	1887.88	0.332916
$319$	−2259.11	−0.396507
$320$	−8617.15	−1.50535
$321$	−2601.37	−0.452319
$322$	−7072.93	−1.22410
$323$	−1167.21	−0.201069
$324$	−213.900	−0.0366769
$325$	−4790.72	−0.817665
$326$	379.052	0.0643981
$327$	−511.109	−0.0864355
$328$	−2590.14	−0.436027
$329$	−8459.83	−1.41765
$330$	1194.72	0.199294
$331$	200.666	0.0333221	0.0166610	−	0.999861i	$-0.494696\pi$
$331$	200.666	0.0333221	0.0166610	+	0.999861i	$0.494696\pi$
$332$	2574.85	0.425643
$333$	−2135.41	−0.351410
$334$	−2092.82	−0.342857
$335$	764.084	0.124616
$336$	2251.72	0.365599
$337$	5944.93	0.960952	0.480476	−	0.877008i	$-0.340464\pi$
$337$	5944.93	0.960952	0.480476	+	0.877008i	$0.340464\pi$
$338$	1369.64	0.220410
$339$	3536.47	0.566592
$340$	−1024.41	−0.163401
$341$	−1028.98	−0.163408
$342$	−980.383	−0.155009
$343$	5203.70	0.819165
$344$	−887.777	−0.139145
$345$	6856.09	1.06991
$346$	−8141.36	−1.26498
$347$	−1877.55	−0.290467	−0.145233	−	0.989397i	$-0.546393\pi$
$347$	−1877.55	−0.290467	−0.145233	+	0.989397i	$0.546393\pi$
$348$	1627.01	0.250624
$349$	−11647.7	−1.78650	−0.893250	−	0.449561i	$-0.851581\pi$
$349$	−11647.7	−1.78650	−0.893250	+	0.449561i	$0.851581\pi$
$350$	5787.06	0.883803
$351$	−1081.81	−0.164509
$352$	−1253.53	−0.189810
$353$	2882.39	0.434601	0.217301	−	0.976105i	$-0.430275\pi$
$353$	2882.39	0.434601	0.217301	+	0.976105i	$0.430275\pi$
$354$	2380.39	0.357391
$355$	14503.2	2.16831
$356$	1583.00	0.235671
$357$	1555.83	0.230653
$358$	2875.38	0.424493
$359$	−8906.57	−1.30939	−0.654695	−	0.755893i	$-0.727203\pi$
$359$	−8906.57	−1.30939	−0.654695	+	0.755893i	$0.727203\pi$
$360$	−3467.10	−0.507589
$361$	−4644.88	−0.677195
$362$	3733.34	0.542045
$363$	363.000	0.0524864
$364$	−2212.10	−0.318531
$365$	13073.8	1.87483
$366$	423.646	0.0605037
$367$	10351.2	1.47229	0.736145	−	0.676823i	$-0.236644\pi$
$367$	10351.2	1.47229	0.736145	+	0.676823i	$0.236644\pi$
$368$	5246.35	0.743165
$369$	−946.328	−0.133506
$370$	−8589.95	−1.20695
$371$	5683.20	0.795302
$372$	741.071	0.103287
$373$	4791.51	0.665134	0.332567	−	0.943080i	$-0.392085\pi$
$373$	4791.51	0.665134	0.332567	+	0.943080i	$0.392085\pi$
$374$	631.676	0.0873347
$375$	254.865	0.0350964
$376$	9967.69	1.36714
$377$	8228.69	1.12414
$378$	1306.80	0.177816
$379$	2310.65	0.313166	0.156583	−	0.987665i	$-0.449952\pi$
$379$	2310.65	0.313166	0.156583	+	0.987665i	$0.449952\pi$
$380$	1943.24	0.262332
$381$	−7884.22	−1.06016
$382$	−6550.56	−0.877372
$383$	−7375.44	−0.983988	−0.491994	−	0.870599i	$-0.663732\pi$
$383$	−7375.44	−0.983988	−0.491994	+	0.870599i	$0.663732\pi$
$384$	−1091.85	−0.145100
$385$	3596.53	0.476094
$386$	−7061.43	−0.931132
$387$	−324.356	−0.0426045
$388$	586.816	0.0767812
$389$	3471.27	0.452443	0.226221	−	0.974076i	$-0.427363\pi$
$389$	3471.27	0.452443	0.226221	+	0.974076i	$0.427363\pi$
$390$	−4351.71	−0.565019
$391$	3624.97	0.468856
$392$	2318.06	0.298673
$393$	3364.41	0.431837
$394$	5300.47	0.677750
$395$	4713.50	0.600409
$396$	−261.433	−0.0331755
$397$	690.482	0.0872904	0.0436452	−	0.999047i	$-0.486103\pi$
$397$	690.482	0.0872904	0.0436452	+	0.999047i	$0.486103\pi$
$398$	2795.61	0.352088
$399$	−2951.30	−0.370300
$400$	−4292.55	−0.536568
$401$	8772.99	1.09252	0.546262	−	0.837614i	$-0.316050\pi$
$401$	8772.99	1.09252	0.546262	+	0.837614i	$0.316050\pi$
$402$	339.325	0.0420995
$403$	3748.00	0.463278
$404$	2762.48	0.340194
$405$	−1266.73	−0.155418
$406$	−9940.04	−1.21506
$407$	−2609.94	−0.317863
$408$	−1833.14	−0.222436
$409$	−13903.8	−1.68093	−0.840464	−	0.541868i	$-0.817717\pi$
$409$	−13903.8	−1.68093	−0.840464	+	0.541868i	$0.817717\pi$
$410$	−3806.72	−0.458538
$411$	615.181	0.0738312
$412$	4831.68	0.577766
$413$	7165.82	0.853770
$414$	3044.74	0.361451
$415$	15248.5	1.80366
$416$	4565.92	0.538131
$417$	2234.48	0.262406
$418$	−1198.25	−0.140211
$419$	5136.86	0.598930	0.299465	−	0.954107i	$-0.403192\pi$
$419$	5136.86	0.598930	0.299465	+	0.954107i	$0.403192\pi$
$420$	−2590.23	−0.300929
$421$	−1157.15	−0.133958	−0.0669789	−	0.997754i	$-0.521336\pi$
$421$	−1157.15	−0.133958	−0.0669789	+	0.997754i	$0.521336\pi$
$422$	8214.33	0.947553
$423$	3641.77	0.418603
$424$	−6696.16	−0.766967
$425$	−2965.94	−0.338516
$426$	6440.79	0.732528
$427$	1275.33	0.144537
$428$	2289.85	0.258608
$429$	−1322.21	−0.148804
$430$	−1304.76	−0.146328
$431$	6109.05	0.682744	0.341372	−	0.939928i	$-0.389108\pi$
$431$	6109.05	0.682744	0.341372	+	0.939928i	$0.389108\pi$
$432$	−969.316	−0.107954
$433$	5099.03	0.565921	0.282961	−	0.959132i	$-0.408683\pi$
$433$	5099.03	0.565921	0.282961	+	0.959132i	$0.408683\pi$
$434$	−4527.48	−0.500752
$435$	9635.29	1.06202
$436$	449.902	0.0494184
$437$	−6876.32	−0.752721
$438$	5805.98	0.633380
$439$	−3856.51	−0.419273	−0.209637	−	0.977779i	$-0.567228\pi$
$439$	−3856.51	−0.419273	−0.209637	+	0.977779i	$0.567228\pi$
$440$	−4237.57	−0.459132
$441$	846.921	0.0914503
$442$	−2300.85	−0.247602
$443$	13925.4	1.49349	0.746746	−	0.665109i	$-0.231615\pi$
$443$	13925.4	1.49349	0.746746	+	0.665109i	$0.231615\pi$
$444$	1879.69	0.200914
$445$	9374.63	0.998652
$446$	−2946.11	−0.312786
$447$	−2350.07	−0.248667
$448$	−11520.1	−1.21490
$449$	2286.70	0.240348	0.120174	−	0.992753i	$-0.461655\pi$
$449$	2286.70	0.240348	0.120174	+	0.992753i	$0.461655\pi$
$450$	−2491.20	−0.260970
$451$	−1156.62	−0.120761
$452$	−3112.96	−0.323941
$453$	8532.42	0.884963
$454$	−14239.0	−1.47196
$455$	−13100.2	−1.34977
$456$	3477.33	0.357108
$457$	−11902.8	−1.21836	−0.609181	−	0.793031i	$-0.708502\pi$
$457$	−11902.8	−1.21836	−0.609181	+	0.793031i	$0.708502\pi$
$458$	13345.2	1.36153
$459$	−669.750	−0.0681073
$460$	−6035.05	−0.611708
$461$	3671.59	0.370939	0.185469	−	0.982650i	$-0.440619\pi$
$461$	3671.59	0.370939	0.185469	+	0.982650i	$0.440619\pi$
$462$	1597.19	0.160840
$463$	11849.8	1.18943	0.594716	−	0.803936i	$-0.297265\pi$
$463$	11849.8	1.18943	0.594716	+	0.803936i	$0.297265\pi$
$464$	7373.02	0.737681
$465$	4388.68	0.437677
$466$	−9511.64	−0.945533
$467$	−13066.3	−1.29472	−0.647360	−	0.762184i	$-0.724127\pi$
$467$	−13066.3	−1.29472	−0.647360	+	0.762184i	$0.724127\pi$
$468$	952.259	0.0940559
$469$	1021.49	0.100571
$470$	14649.5	1.43772
$471$	4211.95	0.412052
$472$	−8443.04	−0.823352
$473$	−396.435	−0.0385372
$474$	2093.23	0.202838
$475$	5626.19	0.543468
$476$	−1369.51	−0.131873
$477$	−2446.49	−0.234837
$478$	−9516.49	−0.910616
$479$	10100.1	0.963435	0.481718	−	0.876327i	$-0.340013\pi$
$479$	10100.1	0.963435	0.481718	+	0.876327i	$0.340013\pi$
$480$	5346.41	0.508394
$481$	9506.60	0.901172
$482$	12237.3	1.15642
$483$	9165.76	0.863471
$484$	−319.530	−0.0300084
$485$	3475.17	0.325360
$486$	−562.547	−0.0525054
$487$	−18498.4	−1.72124	−0.860618	−	0.509250i	$-0.829923\pi$
$487$	−18498.4	−1.72124	−0.860618	+	0.509250i	$0.829923\pi$
$488$	−1502.64	−0.139388
$489$	−491.211	−0.0454260
$490$	3406.84	0.314093
$491$	10663.9	0.980155	0.490078	−	0.871679i	$-0.336968\pi$
$491$	10663.9	0.980155	0.490078	+	0.871679i	$0.336968\pi$
$492$	833.002	0.0763306
$493$	5094.40	0.465396
$494$	4364.56	0.397511
$495$	−1548.23	−0.140581
$496$	3358.26	0.304013
$497$	19389.1	1.74994
$498$	6771.74	0.609335
$499$	−4675.53	−0.419449	−0.209725	−	0.977760i	$-0.567257\pi$
$499$	−4675.53	−0.419449	−0.209725	+	0.977760i	$0.567257\pi$
$500$	−224.344	−0.0200659
$501$	2712.07	0.241849
$502$	3433.76	0.305292
$503$	−9976.67	−0.884369	−0.442185	−	0.896924i	$-0.645796\pi$
$503$	−9976.67	−0.884369	−0.442185	+	0.896924i	$0.645796\pi$
$504$	−4635.09	−0.409650
$505$	16359.6	1.44157
$506$	3721.35	0.326945
$507$	−1774.91	−0.155476
$508$	6940.06	0.606132
$509$	13184.7	1.14814	0.574069	−	0.818807i	$-0.305364\pi$
$509$	13184.7	1.14814	0.574069	+	0.818807i	$0.305364\pi$
$510$	−2694.15	−0.233920
$511$	17478.1	1.51308
$512$	11165.9	0.963808
$513$	1270.47	0.109342
$514$	−6530.78	−0.560429
$515$	28613.6	2.44828
$516$	285.513	0.0243586
$517$	4451.05	0.378640
$518$	−11483.7	−0.974065
$519$	10550.3	0.892307
$520$	15435.1	1.30168
$521$	−23641.7	−1.98802	−0.994012	−	0.109268i	$-0.965149\pi$
$521$	−23641.7	−1.98802	−0.994012	+	0.109268i	$0.965149\pi$
$522$	4278.97	0.358784
$523$	7918.56	0.662054	0.331027	−	0.943621i	$-0.392605\pi$
$523$	7918.56	0.662054	0.331027	+	0.943621i	$0.392605\pi$
$524$	−2961.51	−0.246897
$525$	−7499.40	−0.623430
$526$	4683.80	0.388257
$527$	2320.40	0.191799
$528$	−1184.72	−0.0976483
$529$	9188.56	0.755204
$530$	−9841.32	−0.806565
$531$	−3084.73	−0.252101
$532$	2597.87	0.211714
$533$	4212.95	0.342370
$534$	4163.21	0.337378
$535$	13560.7	1.09585
$536$	−1203.56	−0.0969882
$537$	−3726.18	−0.299435
$538$	−19481.5	−1.56116
$539$	1035.13	0.0827199
$540$	1115.04	0.0888583
$541$	4544.14	0.361123	0.180562	−	0.983564i	$-0.442208\pi$
$541$	4544.14	0.361123	0.180562	+	0.983564i	$0.442208\pi$
$542$	11258.3	0.892227
$543$	−4838.01	−0.382355
$544$	2826.77	0.222788
$545$	2664.35	0.209410
$546$	−5817.71	−0.455998
$547$	−9322.89	−0.728735	−0.364368	−	0.931255i	$-0.618715\pi$
$547$	−9322.89	−0.728735	−0.364368	+	0.931255i	$0.618715\pi$
$548$	−541.511	−0.0422120
$549$	−549.000	−0.0426790
$550$	−3044.80	−0.236056
$551$	−9663.73	−0.747166
$552$	−10799.4	−0.832707
$553$	6301.37	0.484560
$554$	−5922.83	−0.454218
$555$	11131.6	0.851373
$556$	−1966.90	−0.150027
$557$	6470.29	0.492200	0.246100	−	0.969244i	$-0.420851\pi$
$557$	6470.29	0.492200	0.246100	+	0.969244i	$0.420851\pi$
$558$	1948.98	0.147862
$559$	1444.00	0.109257
$560$	−11738.0	−0.885749
$561$	−818.584	−0.0616054
$562$	1875.08	0.140739
$563$	−13862.4	−1.03771	−0.518854	−	0.854863i	$-0.673641\pi$
$563$	−13862.4	−1.03771	−0.518854	+	0.854863i	$0.673641\pi$
$564$	−3205.66	−0.239331
$565$	−18435.2	−1.37270
$566$	3318.23	0.246423
$567$	−1693.47	−0.125430
$568$	−22844.9	−1.68759
$569$	−7717.25	−0.568583	−0.284292	−	0.958738i	$-0.591758\pi$
$569$	−7717.25	−0.568583	−0.284292	+	0.958738i	$0.591758\pi$
$570$	5110.62	0.375545
$571$	−9497.91	−0.696103	−0.348052	−	0.937475i	$-0.613157\pi$
$571$	−9497.91	−0.696103	−0.348052	+	0.937475i	$0.613157\pi$
$572$	1163.87	0.0850768
$573$	8488.82	0.618893
$574$	−5089.13	−0.370063
$575$	−17473.1	−1.26726
$576$	4959.14	0.358734
$577$	−1461.99	−0.105483	−0.0527413	−	0.998608i	$-0.516796\pi$
$577$	−1461.99	−0.105483	−0.0527413	+	0.998608i	$0.516796\pi$
$578$	9949.17	0.715971
$579$	9150.85	0.656815
$580$	−8481.43	−0.607193
$581$	20385.3	1.45564
$582$	1543.30	0.109917
$583$	−2990.16	−0.212418
$584$	−20593.3	−1.45917
$585$	5639.35	0.398561
$586$	8572.55	0.604315
$587$	−17827.8	−1.25355	−0.626773	−	0.779202i	$-0.715625\pi$
$587$	−17827.8	−1.25355	−0.626773	+	0.779202i	$0.715625\pi$
$588$	−745.499	−0.0522855
$589$	−4401.63	−0.307922
$590$	−12408.7	−0.865861
$591$	−6868.83	−0.478081
$592$	8518.05	0.591367
$593$	5971.86	0.413550	0.206775	−	0.978389i	$-0.433703\pi$
$593$	5971.86	0.413550	0.206775	+	0.978389i	$0.433703\pi$
$594$	−687.557	−0.0474930
$595$	−8110.36	−0.558811
$596$	2068.64	0.142172
$597$	−3622.80	−0.248361
$598$	−13554.9	−0.926922
$599$	7782.92	0.530887	0.265444	−	0.964126i	$-0.414482\pi$
$599$	7782.92	0.530887	0.265444	+	0.964126i	$0.414482\pi$
$600$	8836.07	0.601218
$601$	−5587.03	−0.379201	−0.189600	−	0.981861i	$-0.560719\pi$
$601$	−5587.03	−0.379201	−0.189600	+	0.981861i	$0.560719\pi$
$602$	−1744.31	−0.118094
$603$	−439.728	−0.0296967
$604$	−7510.64	−0.505966
$605$	−1892.28	−0.127160
$606$	7265.18	0.487010
$607$	−23032.8	−1.54015	−0.770076	−	0.637952i	$-0.779782\pi$
$607$	−23032.8	−1.54015	−0.770076	+	0.637952i	$0.779782\pi$
$608$	−5362.19	−0.357673
$609$	12881.2	0.857099
$610$	−2208.42	−0.146584
$611$	−16212.8	−1.07348
$612$	589.545	0.0389395
$613$	11083.9	0.730301	0.365151	−	0.930948i	$-0.381017\pi$
$613$	11083.9	0.730301	0.365151	+	0.930948i	$0.381017\pi$
$614$	2752.41	0.180909
$615$	4933.10	0.323450
$616$	−5665.11	−0.370542
$617$	3856.51	0.251633	0.125816	−	0.992054i	$-0.459845\pi$
$617$	3856.51	0.251633	0.125816	+	0.992054i	$0.459845\pi$
$618$	12707.1	0.827110
$619$	−9498.20	−0.616745	−0.308372	−	0.951266i	$-0.599784\pi$
$619$	−9498.20	−0.616745	−0.308372	+	0.951266i	$0.599784\pi$
$620$	−3863.12	−0.250236
$621$	−3945.66	−0.254966
$622$	15522.6	1.00064
$623$	12532.7	0.805961
$624$	4315.28	0.276842
$625$	−16274.5	−1.04157
$626$	−20568.3	−1.31322
$627$	1552.80	0.0989039
$628$	−3707.56	−0.235585
$629$	5885.56	0.373088
$630$	−6812.18	−0.430799
$631$	3224.93	0.203459	0.101729	−	0.994812i	$-0.467562\pi$
$631$	3224.93	0.203459	0.101729	+	0.994812i	$0.467562\pi$
$632$	−7424.51	−0.467296
$633$	−10644.9	−0.668399
$634$	−3619.43	−0.226729
$635$	41099.5	2.56848
$636$	2153.52	0.134265
$637$	−3770.40	−0.234519
$638$	5229.85	0.324532
$639$	−8346.56	−0.516721
$640$	5691.69	0.351537
$641$	9665.78	0.595593	0.297797	−	0.954629i	$-0.403748\pi$
$641$	9665.78	0.595593	0.297797	+	0.954629i	$0.403748\pi$
$642$	6022.20	0.370214
$643$	7188.43	0.440877	0.220439	−	0.975401i	$-0.429251\pi$
$643$	7188.43	0.440877	0.220439	+	0.975401i	$0.429251\pi$
$644$	−8068.13	−0.493678
$645$	1690.83	0.103219
$646$	2702.11	0.164571
$647$	−19240.7	−1.16914	−0.584568	−	0.811345i	$-0.698736\pi$
$647$	−19240.7	−1.16914	−0.584568	+	0.811345i	$0.698736\pi$
$648$	1995.30	0.120961
$649$	−3770.22	−0.228034
$650$	11090.5	0.669241
$651$	5867.13	0.353227
$652$	432.387	0.0259717
$653$	29635.6	1.77601	0.888003	−	0.459838i	$-0.152093\pi$
$653$	29635.6	1.77601	0.888003	+	0.459838i	$0.152093\pi$
$654$	1183.22	0.0707456
$655$	−17538.3	−1.04623
$656$	3774.86	0.224670
$657$	−7523.92	−0.446782
$658$	19584.6	1.16031
$659$	27191.9	1.60736	0.803678	−	0.595065i	$-0.202874\pi$
$659$	27191.9	1.60736	0.803678	+	0.595065i	$0.202874\pi$
$660$	1362.82	0.0803754
$661$	−2751.13	−0.161886	−0.0809430	−	0.996719i	$-0.525793\pi$
$661$	−2751.13	−0.161886	−0.0809430	+	0.996719i	$0.525793\pi$
$662$	−464.544	−0.0272734
$663$	2981.65	0.174657
$664$	−24018.8	−1.40378
$665$	15384.8	0.897138
$666$	4943.49	0.287622
$667$	30012.3	1.74225
$668$	−2387.29	−0.138274
$669$	3817.84	0.220637
$670$	−1768.86	−0.101996
$671$	−671.000	−0.0386046
$672$	7147.50	0.410299
$673$	891.204	0.0510452	0.0255226	−	0.999674i	$-0.491875\pi$
$673$	891.204	0.0510452	0.0255226	+	0.999674i	$0.491875\pi$
$674$	−13762.5	−0.786519
$675$	3228.33	0.184086
$676$	1562.36	0.0888914
$677$	−33523.5	−1.90312	−0.951560	−	0.307462i	$-0.900520\pi$
$677$	−33523.5	−1.90312	−0.951560	+	0.307462i	$0.900520\pi$
$678$	−8186.95	−0.463743
$679$	4645.88	0.262581
$680$	9555.93	0.538902
$681$	18452.2	1.03831
$682$	2382.09	0.133746
$683$	−4126.18	−0.231163	−0.115581	−	0.993298i	$-0.536873\pi$
$683$	−4126.18	−0.231163	−0.115581	+	0.993298i	$0.536873\pi$
$684$	−1118.33	−0.0625151
$685$	−3206.87	−0.178873
$686$	−12046.6	−0.670469
$687$	−17294.0	−0.960418
$688$	1293.84	0.0716966
$689$	10891.5	0.602226
$690$	−15871.9	−0.875699
$691$	−5570.72	−0.306686	−0.153343	−	0.988173i	$-0.549004\pi$
$691$	−5570.72	−0.306686	−0.153343	+	0.988173i	$0.549004\pi$
$692$	−9286.88	−0.510165
$693$	−2069.79	−0.113456
$694$	4346.53	0.237741
$695$	−11648.1	−0.635738
$696$	−15177.1	−0.826562
$697$	2608.25	0.141742
$698$	26964.6	1.46221
$699$	12326.1	0.666974
$700$	6601.32	0.356438
$701$	−8766.08	−0.472312	−0.236156	−	0.971715i	$-0.575888\pi$
$701$	−8766.08	−0.472312	−0.236156	+	0.971715i	$0.575888\pi$
$702$	2504.40	0.134647
$703$	−11164.5	−0.598972
$704$	6061.18	0.324487
$705$	−18984.1	−1.01416
$706$	−6672.76	−0.355712
$707$	21870.8	1.16342
$708$	2715.32	0.144136
$709$	−19161.3	−1.01497	−0.507487	−	0.861659i	$-0.669426\pi$
$709$	−19161.3	−1.01497	−0.507487	+	0.861659i	$0.669426\pi$
$710$	−33575.1	−1.77472
$711$	−2712.60	−0.143081
$712$	−14766.5	−0.777247
$713$	13670.0	0.718016
$714$	−3601.76	−0.188785
$715$	6892.53	0.360512
$716$	3279.96	0.171198
$717$	12332.4	0.642343
$718$	20618.8	1.07171
$719$	−19422.5	−1.00742	−0.503711	−	0.863872i	$-0.668032\pi$
$719$	−19422.5	−1.00742	−0.503711	+	0.863872i	$0.668032\pi$
$720$	5052.93	0.261544
$721$	38252.9	1.97588
$722$	10752.9	0.554269
$723$	−15858.3	−0.815733
$724$	4258.64	0.218607
$725$	−24556.0	−1.25791
$726$	−840.348	−0.0429590
$727$	27570.2	1.40649	0.703246	−	0.710946i	$-0.251733\pi$
$727$	27570.2	1.40649	0.703246	+	0.710946i	$0.251733\pi$
$728$	20634.9	1.05052
$729$	729.000	0.0370370
$730$	−30265.9	−1.53451
$731$	893.982	0.0452327
$732$	483.255	0.0244011
$733$	1108.53	0.0558586	0.0279293	−	0.999610i	$-0.491109\pi$
$733$	1108.53	0.0558586	0.0279293	+	0.999610i	$0.491109\pi$
$734$	−23963.2	−1.20504
$735$	−4414.90	−0.221559
$736$	16653.2	0.834027
$737$	−537.446	−0.0268617
$738$	2190.76	0.109272
$739$	−20158.8	−1.00346	−0.501728	−	0.865025i	$-0.667302\pi$
$739$	−20158.8	−1.00346	−0.501728	+	0.865025i	$0.667302\pi$
$740$	−9798.59	−0.486761
$741$	−5655.99	−0.280402
$742$	−13156.7	−0.650938
$743$	−728.291	−0.0359602	−0.0179801	−	0.999838i	$-0.505724\pi$
$743$	−728.291	−0.0359602	−0.0179801	+	0.999838i	$0.505724\pi$
$744$	−6912.87	−0.340643
$745$	12250.6	0.602454
$746$	−11092.4	−0.544398
$747$	−8775.44	−0.429822
$748$	720.556	0.0352221
$749$	18129.0	0.884403
$750$	−590.014	−0.0287257
$751$	−12295.0	−0.597404	−0.298702	−	0.954346i	$-0.596554\pi$
$751$	−12295.0	−0.597404	−0.298702	+	0.954346i	$0.596554\pi$
$752$	−14526.9	−0.704441
$753$	−4449.79	−0.215351
$754$	−19049.5	−0.920081
$755$	−44478.6	−2.14403
$756$	1490.67	0.0717131
$757$	−23691.2	−1.13748	−0.568740	−	0.822517i	$-0.692569\pi$
$757$	−23691.2	−1.13748	−0.568740	+	0.822517i	$0.692569\pi$
$758$	−5349.17	−0.256320
$759$	−4822.47	−0.230625
$760$	−18126.9	−0.865175
$761$	24293.3	1.15720	0.578602	−	0.815610i	$-0.303598\pi$
$761$	24293.3	1.15720	0.578602	+	0.815610i	$0.303598\pi$
$762$	18252.0	0.867718
$763$	3561.92	0.169004
$764$	−7472.26	−0.353844
$765$	3491.33	0.165006
$766$	17074.2	0.805373
$767$	13732.9	0.646499
$768$	−10696.7	−0.502585
$769$	24518.2	1.14974	0.574868	−	0.818246i	$-0.305053\pi$
$769$	24518.2	1.14974	0.574868	+	0.818246i	$0.305053\pi$
$770$	−8326.00	−0.389673
$771$	8463.19	0.395323
$772$	−8055.00	−0.375526
$773$	−16298.8	−0.758380	−0.379190	−	0.925319i	$-0.623797\pi$
$773$	−16298.8	−0.758380	−0.379190	+	0.925319i	$0.623797\pi$
$774$	750.887	0.0348709
$775$	−11184.8	−0.518411
$776$	−5473.95	−0.253226
$777$	14881.7	0.687100
$778$	−8036.01	−0.370315
$779$	−4947.66	−0.227559
$780$	−4964.02	−0.227872
$781$	−10201.4	−0.467392
$782$	−8391.84	−0.383749
$783$	−5545.08	−0.253084
$784$	−3378.33	−0.153896
$785$	−21956.4	−0.998291
$786$	−7788.64	−0.353450
$787$	7161.83	0.324386	0.162193	−	0.986759i	$-0.448143\pi$
$787$	7161.83	0.324386	0.162193	+	0.986759i	$0.448143\pi$
$788$	6046.26	0.273337
$789$	−6069.70	−0.273874
$790$	−10911.8	−0.491422
$791$	−24645.6	−1.10784
$792$	2438.71	0.109414
$793$	2444.09	0.109448
$794$	−1598.47	−0.0714453
$795$	12753.3	0.568947
$796$	3188.96	0.141997
$797$	−660.416	−0.0293515	−0.0146758	−	0.999892i	$-0.504672\pi$
$797$	−660.416	−0.0293515	−0.0146758	+	0.999892i	$0.504672\pi$
$798$	6832.29	0.303083
$799$	−10037.4	−0.444426
$800$	−13625.6	−0.602171
$801$	−5395.07	−0.237984
$802$	−20309.5	−0.894208
$803$	−9195.90	−0.404130
$804$	387.069	0.0169787
$805$	−47780.0	−2.09196
$806$	−8676.65	−0.379184
$807$	25245.9	1.10124
$808$	−25769.0	−1.12197
$809$	30279.3	1.31590	0.657950	−	0.753062i	$-0.271424\pi$
$809$	30279.3	1.31590	0.657950	+	0.753062i	$0.271424\pi$
$810$	2932.49	0.127207
$811$	31977.8	1.38458	0.692289	−	0.721620i	$-0.256602\pi$
$811$	31977.8	1.38458	0.692289	+	0.721620i	$0.256602\pi$
$812$	−11338.6	−0.490035
$813$	−14589.6	−0.629372
$814$	6042.04	0.260164
$815$	2560.63	0.110055
$816$	2671.60	0.114614
$817$	−1695.82	−0.0726185
$818$	32187.4	1.37580
$819$	7539.12	0.321658
$820$	−4342.35	−0.184928
$821$	30007.7	1.27561	0.637804	−	0.770199i	$-0.279843\pi$
$821$	30007.7	1.27561	0.637804	+	0.770199i	$0.279843\pi$
$822$	−1424.15	−0.0604293
$823$	−45880.9	−1.94327	−0.971633	−	0.236494i	$-0.924002\pi$
$823$	−45880.9	−1.94327	−0.971633	+	0.236494i	$0.924002\pi$
$824$	−45071.0	−1.90549
$825$	3945.73	0.166512
$826$	−16588.9	−0.698793
$827$	−26581.6	−1.11769	−0.558847	−	0.829271i	$-0.688756\pi$
$827$	−26581.6	−1.11769	−0.558847	+	0.829271i	$0.688756\pi$
$828$	3473.15	0.145773
$829$	−31471.3	−1.31851	−0.659255	−	0.751919i	$-0.729128\pi$
$829$	−31471.3	−1.31851	−0.659255	+	0.751919i	$0.729128\pi$
$830$	−35300.3	−1.47625
$831$	7675.35	0.320403
$832$	−22077.6	−0.919954
$833$	−2334.26	−0.0970917
$834$	−5172.85	−0.214774
$835$	−14137.7	−0.585936
$836$	−1366.84	−0.0565470
$837$	−2525.67	−0.104301
$838$	−11891.9	−0.490212
$839$	7651.01	0.314830	0.157415	−	0.987533i	$-0.449684\pi$
$839$	7651.01	0.314830	0.157415	+	0.987533i	$0.449684\pi$
$840$	24162.2	0.992471
$841$	17789.2	0.729394
$842$	2678.82	0.109642
$843$	−2429.90	−0.0992766
$844$	9370.13	0.382148
$845$	9252.39	0.376677
$846$	−8430.73	−0.342617
$847$	−2529.75	−0.102625
$848$	9758.95	0.395193
$849$	−4300.07	−0.173826
$850$	6866.18	0.277068
$851$	34673.2	1.39669
$852$	7347.03	0.295429
$853$	15960.5	0.640653	0.320327	−	0.947307i	$-0.396207\pi$
$853$	15960.5	0.640653	0.320327	+	0.947307i	$0.396207\pi$
$854$	−2952.39	−0.118301
$855$	−6622.82	−0.264907
$856$	−21360.2	−0.852894
$857$	−2381.94	−0.0949423	−0.0474711	−	0.998873i	$-0.515116\pi$
$857$	−2381.94	−0.0949423	−0.0474711	+	0.998873i	$0.515116\pi$
$858$	3060.93	0.121793
$859$	−39141.0	−1.55469	−0.777343	−	0.629078i	$-0.783433\pi$
$859$	−39141.0	−1.55469	−0.777343	+	0.629078i	$0.783433\pi$
$860$	−1488.35	−0.0590143
$861$	6594.96	0.261040
$862$	−14142.5	−0.558812
$863$	21196.9	0.836095	0.418047	−	0.908425i	$-0.362715\pi$
$863$	21196.9	0.836095	0.418047	+	0.908425i	$0.362715\pi$
$864$	−3076.84	−0.121153
$865$	−54997.6	−2.16182
$866$	−11804.3	−0.463195
$867$	−12893.1	−0.505042
$868$	−5164.52	−0.201953
$869$	−3315.40	−0.129422
$870$	−22305.8	−0.869237
$871$	1957.62	0.0761555
$872$	−4196.78	−0.162983
$873$	−1999.95	−0.0775350
$874$	15918.7	0.616086
$875$	−1776.15	−0.0686227
$876$	6622.90	0.255442
$877$	30035.4	1.15647	0.578235	−	0.815871i	$-0.303742\pi$
$877$	30035.4	1.15647	0.578235	+	0.815871i	$0.303742\pi$
$878$	8927.84	0.343166
$879$	−11109.1	−0.426280
$880$	6175.81	0.236575
$881$	12273.9	0.469374	0.234687	−	0.972071i	$-0.424594\pi$
$881$	12273.9	0.469374	0.234687	+	0.972071i	$0.424594\pi$
$882$	−1960.63	−0.0748501
$883$	30777.6	1.17299	0.586493	−	0.809954i	$-0.300508\pi$
$883$	30777.6	1.17299	0.586493	+	0.809954i	$0.300508\pi$
$884$	−2624.59	−0.0998581
$885$	16080.3	0.610774
$886$	−32237.5	−1.22239
$887$	−21300.5	−0.806313	−0.403156	−	0.915131i	$-0.632087\pi$
$887$	−21300.5	−0.806313	−0.403156	+	0.915131i	$0.632087\pi$
$888$	−17534.1	−0.662620
$889$	54945.1	2.07289
$890$	−21702.3	−0.817375
$891$	891.000	0.0335013
$892$	−3360.64	−0.126146
$893$	19040.2	0.713500
$894$	5440.42	0.203529
$895$	19424.2	0.725451
$896$	7609.10	0.283708
$897$	17565.6	0.653845
$898$	−5293.73	−0.196719
$899$	19211.3	0.712717
$900$	−2841.72	−0.105249
$901$	6742.96	0.249324
$902$	2677.59	0.0988404
$903$	2260.44	0.0833030
$904$	29038.4	1.06837
$905$	25220.0	0.926344
$906$	−19752.6	−0.724323
$907$	−53282.1	−1.95061	−0.975305	−	0.220862i	$-0.929113\pi$
$907$	−53282.1	−1.95061	−0.975305	+	0.220862i	$0.929113\pi$
$908$	−16242.5	−0.593640
$909$	−9414.90	−0.343534
$910$	30327.1	1.10476
$911$	−22489.7	−0.817912	−0.408956	−	0.912554i	$-0.634107\pi$
$911$	−22489.7	−0.817912	−0.408956	+	0.912554i	$0.634107\pi$
$912$	−5067.85	−0.184006
$913$	−10725.5	−0.388788
$914$	27555.2	0.997204
$915$	2861.87	0.103400
$916$	15223.0	0.549106
$917$	−23446.6	−0.844356
$918$	1550.48	0.0557444
$919$	32762.2	1.17598	0.587991	−	0.808868i	$-0.299919\pi$
$919$	32762.2	1.17598	0.587991	+	0.808868i	$0.299919\pi$
$920$	56296.2	2.01743
$921$	−3566.83	−0.127612
$922$	−8499.75	−0.303606
$923$	37158.0	1.32510
$924$	1821.93	0.0648669
$925$	−28369.5	−1.00842
$926$	−27432.4	−0.973525
$927$	−16467.0	−0.583439
$928$	23403.7	0.827872
$929$	−26529.7	−0.936932	−0.468466	−	0.883482i	$-0.655193\pi$
$929$	−26529.7	−0.936932	−0.468466	+	0.883482i	$0.655193\pi$
$930$	−10159.8	−0.358230
$931$	4427.93	0.155875
$932$	−10850.0	−0.381333
$933$	−20115.6	−0.705847
$934$	30248.5	1.05970
$935$	4267.18	0.149253
$936$	−8882.87	−0.310199
$937$	51721.4	1.80327	0.901634	−	0.432499i	$-0.142368\pi$
$937$	51721.4	1.80327	0.901634	+	0.432499i	$0.142368\pi$
$938$	−2364.75	−0.0823155
$939$	26654.3	0.926335
$940$	16710.7	0.579834
$941$	9798.40	0.339446	0.169723	−	0.985492i	$-0.445713\pi$
$941$	9798.40	0.339446	0.169723	+	0.985492i	$0.445713\pi$
$942$	−9750.70	−0.337256
$943$	15365.8	0.530625
$944$	12304.8	0.424246
$945$	8827.85	0.303883
$946$	917.751	0.0315419
$947$	−25309.9	−0.868491	−0.434246	−	0.900795i	$-0.642985\pi$
$947$	−25309.9	−0.868491	−0.434246	+	0.900795i	$0.642985\pi$
$948$	2387.76	0.0818046
$949$	33495.6	1.14575
$950$	−13024.7	−0.444817
$951$	4690.39	0.159933
$952$	12775.1	0.434920
$953$	17801.6	0.605090	0.302545	−	0.953135i	$-0.402164\pi$
$953$	17801.6	0.605090	0.302545	+	0.953135i	$0.402164\pi$
$954$	5663.65	0.192209
$955$	−44251.3	−1.49941
$956$	−10855.5	−0.367251
$957$	−6777.32	−0.228923
$958$	−23381.8	−0.788551
$959$	−4287.20	−0.144359
$960$	−25851.5	−0.869117
$961$	−21040.6	−0.706275
$962$	−22007.9	−0.737590
$963$	−7804.12	−0.261147
$964$	13959.2	0.466385
$965$	−47702.3	−1.59129
$966$	−21218.8	−0.706733
$967$	−14626.5	−0.486409	−0.243204	−	0.969975i	$-0.578199\pi$
$967$	−14626.5	−0.486409	−0.243204	+	0.969975i	$0.578199\pi$
$968$	2980.64	0.0989684
$969$	−3501.64	−0.116088
$970$	−8045.05	−0.266300
$971$	7811.41	0.258167	0.129083	−	0.991634i	$-0.458796\pi$
$971$	7811.41	0.258167	0.129083	+	0.991634i	$0.458796\pi$
$972$	−641.700	−0.0211754
$973$	−15572.1	−0.513072
$974$	42823.9	1.40880
$975$	−14372.1	−0.472079
$976$	2189.94	0.0718218
$977$	−10342.2	−0.338664	−0.169332	−	0.985559i	$-0.554161\pi$
$977$	−10342.2	−0.338664	−0.169332	+	0.985559i	$0.554161\pi$
$978$	1137.16	0.0371802
$979$	−6593.98	−0.215265
$980$	3886.20	0.126674
$981$	−1533.33	−0.0499036
$982$	−24687.1	−0.802236
$983$	28060.2	0.910460	0.455230	−	0.890374i	$-0.349557\pi$
$983$	28060.2	0.910460	0.455230	+	0.890374i	$0.349557\pi$
$984$	−7770.43	−0.251740
$985$	35806.4	1.15826
$986$	−11793.6	−0.380917
$987$	−25379.5	−0.818478
$988$	4978.67	0.160316
$989$	5266.66	0.169333
$990$	3584.16	0.115063
$991$	−39764.0	−1.27462	−0.637309	−	0.770608i	$-0.719952\pi$
$991$	−39764.0	−1.27462	−0.637309	+	0.770608i	$0.719952\pi$
$992$	10659.9	0.341182
$993$	601.998	0.0192385
$994$	−44885.8	−1.43229
$995$	18885.3	0.601712
$996$	7724.55	0.245745
$997$	−16298.8	−0.517742	−0.258871	−	0.965912i	$-0.583350\pi$
$997$	−16298.8	−0.517742	−0.258871	+	0.965912i	$0.583350\pi$
$998$	10823.9	0.343311
$999$	−6406.23	−0.202887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.f.1.12	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.f.1.12	✓	38	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31501 q^{2} +3.00000 q^{3} -2.64074 q^{4} -15.6387 q^{5} -6.94502 q^{6} -20.9070 q^{7} +24.6334 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.31501 q^{2} +3.00000 q^{3} -2.64074 q^{4} -15.6387 q^{5} -6.94502 q^{6} -20.9070 q^{7} +24.6334 q^{8} +9.00000 q^{9} +36.2036 q^{10} +11.0000 q^{11} -7.92222 q^{12} -40.0670 q^{13} +48.3998 q^{14} -46.9160 q^{15} -35.9006 q^{16} -24.8056 q^{17} -20.8351 q^{18} +47.0545 q^{19} +41.2976 q^{20} -62.7210 q^{21} -25.4651 q^{22} -146.135 q^{23} +73.9002 q^{24} +119.568 q^{25} +92.7554 q^{26} +27.0000 q^{27} +55.2099 q^{28} -205.373 q^{29} +108.611 q^{30} -93.5433 q^{31} -113.957 q^{32} +33.0000 q^{33} +57.4251 q^{34} +326.957 q^{35} -23.7667 q^{36} -237.268 q^{37} -108.931 q^{38} -120.201 q^{39} -385.233 q^{40} -105.148 q^{41} +145.200 q^{42} -36.0396 q^{43} -29.0481 q^{44} -140.748 q^{45} +338.305 q^{46} +404.641 q^{47} -107.702 q^{48} +94.1023 q^{49} -276.800 q^{50} -74.4167 q^{51} +105.807 q^{52} -271.833 q^{53} -62.5052 q^{54} -172.025 q^{55} -515.010 q^{56} +141.163 q^{57} +475.441 q^{58} -342.748 q^{59} +123.893 q^{60} -61.0000 q^{61} +216.554 q^{62} -188.163 q^{63} +551.016 q^{64} +626.594 q^{65} -76.3953 q^{66} -48.8587 q^{67} +65.5050 q^{68} -438.406 q^{69} -756.909 q^{70} -927.396 q^{71} +221.701 q^{72} -835.991 q^{73} +549.276 q^{74} +358.703 q^{75} -124.259 q^{76} -229.977 q^{77} +278.266 q^{78} -301.400 q^{79} +561.437 q^{80} +81.0000 q^{81} +243.418 q^{82} -975.049 q^{83} +165.630 q^{84} +387.926 q^{85} +83.4319 q^{86} -616.120 q^{87} +270.967 q^{88} -599.452 q^{89} +325.833 q^{90} +837.680 q^{91} +385.906 q^{92} -280.630 q^{93} -936.747 q^{94} -735.869 q^{95} -341.871 q^{96} -222.217 q^{97} -217.848 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9}+O(q^{10}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9} + 99 q^{10} + 418 q^{11} + 510 q^{12} + 209 q^{13} + 128 q^{14} + 105 q^{15} + 798 q^{16} + 512 q^{17} + 126 q^{18} + 487 q^{19} + 328 q^{20} + 315 q^{21} + 154 q^{22} + 417 q^{23} + 441 q^{24} + 925 q^{25} + 177 q^{26} + 1026 q^{27} + 902 q^{28} + 626 q^{29} + 297 q^{30} + 300 q^{31} + 1625 q^{32} + 1254 q^{33} - 180 q^{34} + 1086 q^{35} + 1530 q^{36} + 554 q^{37} + 845 q^{38} + 627 q^{39} + 329 q^{40} + 1378 q^{41} + 384 q^{42} + 1979 q^{43} + 1870 q^{44} + 315 q^{45} + 937 q^{46} + 1345 q^{47} + 2394 q^{48} + 2635 q^{49} + 800 q^{50} + 1536 q^{51} + 2006 q^{52} + 1497 q^{53} + 378 q^{54} + 385 q^{55} + 415 q^{56} + 1461 q^{57} + 1241 q^{58} + 2827 q^{59} + 984 q^{60} - 2318 q^{61} + 509 q^{62} + 945 q^{63} + 1003 q^{64} + 2810 q^{65} + 462 q^{66} + 369 q^{67} + 3936 q^{68} + 1251 q^{69} + 922 q^{70} + 965 q^{71} + 1323 q^{72} + 3081 q^{73} + 722 q^{74} + 2775 q^{75} + 2210 q^{76} + 1155 q^{77} + 531 q^{78} + 3795 q^{79} + 3793 q^{80} + 3078 q^{81} - 1678 q^{82} + 3869 q^{83} + 2706 q^{84} + 3553 q^{85} + 3305 q^{86} + 1878 q^{87} + 1617 q^{88} + 2849 q^{89} + 891 q^{90} + 1252 q^{91} + 4519 q^{92} + 900 q^{93} + 340 q^{94} + 1504 q^{95} + 4875 q^{96} + 2562 q^{97} + 6164 q^{98} + 3762 q^{99}+O(q^{100}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9 + 99 * q^10 + 418 * q^11 + 510 * q^12 + 209 * q^13 + 128 * q^14 + 105 * q^15 + 798 * q^16 + 512 * q^17 + 126 * q^18 + 487 * q^19 + 328 * q^20 + 315 * q^21 + 154 * q^22 + 417 * q^23 + 441 * q^24 + 925 * q^25 + 177 * q^26 + 1026 * q^27 + 902 * q^28 + 626 * q^29 + 297 * q^30 + 300 * q^31 + 1625 * q^32 + 1254 * q^33 - 180 * q^34 + 1086 * q^35 + 1530 * q^36 + 554 * q^37 + 845 * q^38 + 627 * q^39 + 329 * q^40 + 1378 * q^41 + 384 * q^42 + 1979 * q^43 + 1870 * q^44 + 315 * q^45 + 937 * q^46 + 1345 * q^47 + 2394 * q^48 + 2635 * q^49 + 800 * q^50 + 1536 * q^51 + 2006 * q^52 + 1497 * q^53 + 378 * q^54 + 385 * q^55 + 415 * q^56 + 1461 * q^57 + 1241 * q^58 + 2827 * q^59 + 984 * q^60 - 2318 * q^61 + 509 * q^62 + 945 * q^63 + 1003 * q^64 + 2810 * q^65 + 462 * q^66 + 369 * q^67 + 3936 * q^68 + 1251 * q^69 + 922 * q^70 + 965 * q^71 + 1323 * q^72 + 3081 * q^73 + 722 * q^74 + 2775 * q^75 + 2210 * q^76 + 1155 * q^77 + 531 * q^78 + 3795 * q^79 + 3793 * q^80 + 3078 * q^81 - 1678 * q^82 + 3869 * q^83 + 2706 * q^84 + 3553 * q^85 + 3305 * q^86 + 1878 * q^87 + 1617 * q^88 + 2849 * q^89 + 891 * q^90 + 1252 * q^91 + 4519 * q^92 + 900 * q^93 + 340 * q^94 + 1504 * q^95 + 4875 * q^96 + 2562 * q^97 + 6164 * q^98 + 3762 * q^99

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