LMFDB - Embedded newform 867.4.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,4,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("867.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	867.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:	$51.1546559750$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 36x^{6} + 116x^{5} + 431x^{4} - 860x^{3} - 1756x^{2} + 480x + 544 $ x^8 - 4x^7 - 36x^6 + 116x^5 + 431x^4 - 860x^3 - 1756x^2 + 480*x + 544
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.635039$ of defining polynomial
Character	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.364961 q^{2} -3.00000 q^{3} -7.86680 q^{4} +10.0004 q^{5} +1.09488 q^{6} -26.8907 q^{7} +5.79077 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.364961 q^{2} -3.00000 q^{3} -7.86680 q^{4} +10.0004 q^{5} +1.09488 q^{6} -26.8907 q^{7} +5.79077 q^{8} +9.00000 q^{9} -3.64977 q^{10} +43.1919 q^{11} +23.6004 q^{12} +17.7892 q^{13} +9.81405 q^{14} -30.0013 q^{15} +60.8210 q^{16} -3.28465 q^{18} -106.342 q^{19} -78.6715 q^{20} +80.6720 q^{21} -15.7634 q^{22} -132.011 q^{23} -17.3723 q^{24} -24.9913 q^{25} -6.49238 q^{26} -27.0000 q^{27} +211.544 q^{28} -122.268 q^{29} +10.9493 q^{30} -59.2658 q^{31} -68.5235 q^{32} -129.576 q^{33} -268.918 q^{35} -70.8012 q^{36} +356.174 q^{37} +38.8107 q^{38} -53.3677 q^{39} +57.9102 q^{40} -325.926 q^{41} -29.4421 q^{42} -17.0616 q^{43} -339.782 q^{44} +90.0039 q^{45} +48.1791 q^{46} -112.081 q^{47} -182.463 q^{48} +380.108 q^{49} +9.12085 q^{50} -139.944 q^{52} +361.383 q^{53} +9.85395 q^{54} +431.938 q^{55} -155.718 q^{56} +319.026 q^{57} +44.6229 q^{58} +596.494 q^{59} +236.014 q^{60} +779.529 q^{61} +21.6297 q^{62} -242.016 q^{63} -461.560 q^{64} +177.900 q^{65} +47.2901 q^{66} -872.117 q^{67} +396.034 q^{69} +98.1448 q^{70} +490.150 q^{71} +52.1169 q^{72} +542.221 q^{73} -129.990 q^{74} +74.9738 q^{75} +836.572 q^{76} -1161.46 q^{77} +19.4771 q^{78} +265.798 q^{79} +608.237 q^{80} +81.0000 q^{81} +118.950 q^{82} -663.019 q^{83} -634.631 q^{84} +6.22684 q^{86} +366.803 q^{87} +250.114 q^{88} -1023.92 q^{89} -32.8479 q^{90} -478.364 q^{91} +1038.51 q^{92} +177.797 q^{93} +40.9051 q^{94} -1063.47 q^{95} +205.570 q^{96} +1471.49 q^{97} -138.725 q^{98} +388.727 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9} - 72 q^{10} + 12 q^{11} - 72 q^{12} - 60 q^{13} + 56 q^{14} - 72 q^{15} - 100 q^{16} - 36 q^{18} - 140 q^{19} + 224 q^{20} - 84 q^{21} + 484 q^{22} - 96 q^{23} + 144 q^{24} + 220 q^{25} + 500 q^{26} - 216 q^{27} + 348 q^{28} + 380 q^{29} + 216 q^{30} + 124 q^{31} - 448 q^{32} - 36 q^{33} - 592 q^{35} + 216 q^{36} + 1260 q^{37} + 84 q^{38} + 180 q^{39} - 836 q^{40} + 284 q^{41} - 168 q^{42} - 76 q^{43} - 1460 q^{44} + 216 q^{45} + 1056 q^{46} - 184 q^{47} + 300 q^{48} - 536 q^{49} - 360 q^{50} - 1580 q^{52} + 1064 q^{53} + 108 q^{54} - 2148 q^{55} + 244 q^{56} + 420 q^{57} - 532 q^{58} + 1224 q^{59} - 672 q^{60} + 324 q^{61} + 3068 q^{62} + 252 q^{63} - 876 q^{64} + 652 q^{65} - 1452 q^{66} - 624 q^{67} + 288 q^{69} - 1932 q^{70} + 2636 q^{71} - 432 q^{72} + 1640 q^{73} - 120 q^{74} - 660 q^{75} - 1172 q^{76} + 504 q^{77} - 1500 q^{78} + 3788 q^{79} + 1740 q^{80} + 648 q^{81} + 1916 q^{82} + 2032 q^{83} - 1044 q^{84} + 1612 q^{86} - 1140 q^{87} + 4576 q^{88} + 1304 q^{89} - 648 q^{90} - 2872 q^{91} - 1744 q^{92} - 372 q^{93} - 208 q^{94} - 748 q^{95} + 1344 q^{96} + 2376 q^{97} + 604 q^{98} + 108 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9 - 72 * q^10 + 12 * q^11 - 72 * q^12 - 60 * q^13 + 56 * q^14 - 72 * q^15 - 100 * q^16 - 36 * q^18 - 140 * q^19 + 224 * q^20 - 84 * q^21 + 484 * q^22 - 96 * q^23 + 144 * q^24 + 220 * q^25 + 500 * q^26 - 216 * q^27 + 348 * q^28 + 380 * q^29 + 216 * q^30 + 124 * q^31 - 448 * q^32 - 36 * q^33 - 592 * q^35 + 216 * q^36 + 1260 * q^37 + 84 * q^38 + 180 * q^39 - 836 * q^40 + 284 * q^41 - 168 * q^42 - 76 * q^43 - 1460 * q^44 + 216 * q^45 + 1056 * q^46 - 184 * q^47 + 300 * q^48 - 536 * q^49 - 360 * q^50 - 1580 * q^52 + 1064 * q^53 + 108 * q^54 - 2148 * q^55 + 244 * q^56 + 420 * q^57 - 532 * q^58 + 1224 * q^59 - 672 * q^60 + 324 * q^61 + 3068 * q^62 + 252 * q^63 - 876 * q^64 + 652 * q^65 - 1452 * q^66 - 624 * q^67 + 288 * q^69 - 1932 * q^70 + 2636 * q^71 - 432 * q^72 + 1640 * q^73 - 120 * q^74 - 660 * q^75 - 1172 * q^76 + 504 * q^77 - 1500 * q^78 + 3788 * q^79 + 1740 * q^80 + 648 * q^81 + 1916 * q^82 + 2032 * q^83 - 1044 * q^84 + 1612 * q^86 - 1140 * q^87 + 4576 * q^88 + 1304 * q^89 - 648 * q^90 - 2872 * q^91 - 1744 * q^92 - 372 * q^93 - 208 * q^94 - 748 * q^95 + 1344 * q^96 + 2376 * q^97 + 604 * q^98 + 108 * 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 $

$2$	−0.364961	−0.129033	−0.0645166	−	0.997917i	$-0.520551\pi$
$2$	−0.364961	−0.129033	−0.0645166	+	0.997917i	$0.520551\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.86680	−0.983350
$5$	10.0004	0.894466	0.447233	−	0.894417i	$-0.352409\pi$
$5$	10.0004	0.894466	0.447233	+	0.894417i	$0.352409\pi$
$6$	1.09488	0.0744974
$7$	−26.8907	−1.45196	−0.725980	−	0.687716i	$-0.758613\pi$
$7$	−26.8907	−1.45196	−0.725980	+	0.687716i	$0.758613\pi$
$8$	5.79077	0.255918
$9$	9.00000	0.333333
$10$	−3.64977	−0.115416
$11$	43.1919	1.18389	0.591947	−	0.805977i	$-0.298359\pi$
$11$	43.1919	1.18389	0.591947	+	0.805977i	$0.298359\pi$
$12$	23.6004	0.567738
$13$	17.7892	0.379527	0.189763	−	0.981830i	$-0.439228\pi$
$13$	17.7892	0.379527	0.189763	+	0.981830i	$0.439228\pi$
$14$	9.81405	0.187351
$15$	−30.0013	−0.516420
$16$	60.8210	0.950328
$17$	0	0
$17$	0	0
$18$	−3.28465	−0.0430111
$19$	−106.342	−1.28403	−0.642014	−	0.766693i	$-0.721901\pi$
$19$	−106.342	−1.28403	−0.642014	+	0.766693i	$0.721901\pi$
$20$	−78.6715	−0.879574
$21$	80.6720	0.838289
$22$	−15.7634	−0.152762
$23$	−132.011	−1.19680	−0.598398	−	0.801199i	$-0.704196\pi$
$23$	−132.011	−1.19680	−0.598398	+	0.801199i	$0.704196\pi$
$24$	−17.3723	−0.147754
$25$	−24.9913	−0.199930
$26$	−6.49238	−0.0489716
$27$	−27.0000	−0.192450
$28$	211.544	1.42778
$29$	−122.268	−0.782914	−0.391457	−	0.920196i	$-0.628029\pi$
$29$	−122.268	−0.782914	−0.391457	+	0.920196i	$0.628029\pi$
$30$	10.9493	0.0666354
$31$	−59.2658	−0.343369	−0.171685	−	0.985152i	$-0.554921\pi$
$31$	−59.2658	−0.343369	−0.171685	+	0.985152i	$0.554921\pi$
$32$	−68.5235	−0.378542
$33$	−129.576	−0.683522
$34$	0	0
$35$	−268.918	−1.29873
$36$	−70.8012	−0.327783
$37$	356.174	1.58256	0.791278	−	0.611456i	$-0.209416\pi$
$37$	356.174	1.58256	0.791278	+	0.611456i	$0.209416\pi$
$38$	38.8107	0.165682
$39$	−53.3677	−0.219120
$40$	57.9102	0.228910
$41$	−325.926	−1.24149	−0.620745	−	0.784012i	$-0.713170\pi$
$41$	−325.926	−1.24149	−0.620745	+	0.784012i	$0.713170\pi$
$42$	−29.4421	−0.108167
$43$	−17.0616	−0.0605088	−0.0302544	−	0.999542i	$-0.509632\pi$
$43$	−17.0616	−0.0605088	−0.0302544	+	0.999542i	$0.509632\pi$
$44$	−339.782	−1.16418
$45$	90.0039	0.298155
$46$	48.1791	0.154426
$47$	−112.081	−0.347843	−0.173922	−	0.984759i	$-0.555644\pi$
$47$	−112.081	−0.347843	−0.173922	+	0.984759i	$0.555644\pi$
$48$	−182.463	−0.548672
$49$	380.108	1.10819
$50$	9.12085	0.0257977
$51$	0	0
$52$	−139.944	−0.373208
$53$	361.383	0.936598	0.468299	−	0.883570i	$-0.344867\pi$
$53$	361.383	0.936598	0.468299	+	0.883570i	$0.344867\pi$
$54$	9.85395	0.0248325
$55$	431.938	1.05895
$56$	−155.718	−0.371583
$57$	319.026	0.741334
$58$	44.6229	0.101022
$59$	596.494	1.31622	0.658109	−	0.752922i	$-0.271356\pi$
$59$	596.494	1.31622	0.658109	+	0.752922i	$0.271356\pi$
$60$	236.014	0.507822
$61$	779.529	1.63620	0.818102	−	0.575073i	$-0.195026\pi$
$61$	779.529	1.63620	0.818102	+	0.575073i	$0.195026\pi$
$62$	21.6297	0.0443061
$63$	−242.016	−0.483986
$64$	−461.560	−0.901484
$65$	177.900	0.339474
$66$	47.2901	0.0881971
$67$	−872.117	−1.59024	−0.795119	−	0.606453i	$-0.792592\pi$
$67$	−872.117	−1.59024	−0.795119	+	0.606453i	$0.792592\pi$
$68$	0	0
$69$	396.034	0.690970
$70$	98.1448	0.167579
$71$	490.150	0.819298	0.409649	−	0.912243i	$-0.365651\pi$
$71$	490.150	0.819298	0.409649	+	0.912243i	$0.365651\pi$
$72$	52.1169	0.0853061
$73$	542.221	0.869345	0.434672	−	0.900589i	$-0.356864\pi$
$73$	542.221	0.869345	0.434672	+	0.900589i	$0.356864\pi$
$74$	−129.990	−0.204203
$75$	74.9738	0.115430
$76$	836.572	1.26265
$77$	−1161.46	−1.71897
$78$	19.4771	0.0282737
$79$	265.798	0.378540	0.189270	−	0.981925i	$-0.439388\pi$
$79$	265.798	0.378540	0.189270	+	0.981925i	$0.439388\pi$
$80$	608.237	0.850037
$81$	81.0000	0.111111
$82$	118.950	0.160194
$83$	−663.019	−0.876817	−0.438409	−	0.898776i	$-0.644458\pi$
$83$	−663.019	−0.876817	−0.438409	+	0.898776i	$0.644458\pi$
$84$	−634.631	−0.824332
$85$	0	0
$86$	6.22684	0.00780764
$87$	366.803	0.452016
$88$	250.114	0.302980
$89$	−1023.92	−1.21949	−0.609747	−	0.792596i	$-0.708729\pi$
$89$	−1023.92	−1.21949	−0.609747	+	0.792596i	$0.708729\pi$
$90$	−32.8479	−0.0384720
$91$	−478.364	−0.551057
$92$	1038.51	1.17687
$93$	177.797	0.198244
$94$	40.9051	0.0448834
$95$	−1063.47	−1.14852
$96$	205.570	0.218551
$97$	1471.49	1.54028	0.770141	−	0.637873i	$-0.220186\pi$
$97$	1471.49	1.54028	0.770141	+	0.637873i	$0.220186\pi$
$98$	−138.725	−0.142993
$99$	388.727	0.394632
$100$	196.601	0.196601
$101$	−312.917	−0.308281	−0.154140	−	0.988049i	$-0.549261\pi$
$101$	−312.917	−0.308281	−0.154140	+	0.988049i	$0.549261\pi$
$102$	0	0
$103$	35.0733	0.0335522	0.0167761	−	0.999859i	$-0.494660\pi$
$103$	35.0733	0.0335522	0.0167761	+	0.999859i	$0.494660\pi$
$104$	103.013	0.0971278
$105$	806.755	0.749821
$106$	−131.891	−0.120852
$107$	777.759	0.702699	0.351349	−	0.936244i	$-0.385723\pi$
$107$	777.759	0.702699	0.351349	+	0.936244i	$0.385723\pi$
$108$	212.404	0.189246
$109$	−154.568	−0.135825	−0.0679124	−	0.997691i	$-0.521634\pi$
$109$	−154.568	−0.135825	−0.0679124	+	0.997691i	$0.521634\pi$
$110$	−157.641	−0.136640
$111$	−1068.52	−0.913690
$112$	−1635.52	−1.37984
$113$	1894.32	1.57701	0.788505	−	0.615028i	$-0.210855\pi$
$113$	1894.32	1.57701	0.788505	+	0.615028i	$0.210855\pi$
$114$	−116.432	−0.0956568
$115$	−1320.17	−1.07049
$116$	961.855	0.769879
$117$	160.103	0.126509
$118$	−217.697	−0.169836
$119$	0	0
$120$	−173.731	−0.132161
$121$	534.539	0.401607
$122$	−284.498	−0.211125
$123$	977.778	0.716775
$124$	466.232	0.337652
$125$	−1499.98	−1.07330
$126$	88.3264	0.0624504
$127$	534.333	0.373342	0.186671	−	0.982423i	$-0.440230\pi$
$127$	534.333	0.373342	0.186671	+	0.982423i	$0.440230\pi$
$128$	716.639	0.494864
$129$	51.1849	0.0349348
$130$	−64.9266	−0.0438034
$131$	2873.67	1.91659	0.958295	−	0.285781i	$-0.0922529\pi$
$131$	2873.67	1.91659	0.958295	+	0.285781i	$0.0922529\pi$
$132$	1019.35	0.672142
$133$	2859.61	1.86436
$134$	318.289	0.205194
$135$	−270.012	−0.172140
$136$	0	0
$137$	1116.39	0.696204	0.348102	−	0.937457i	$-0.386826\pi$
$137$	1116.39	0.696204	0.348102	+	0.937457i	$0.386826\pi$
$138$	−144.537	−0.0891582
$139$	1523.92	0.929907	0.464953	−	0.885335i	$-0.346071\pi$
$139$	1523.92	0.929907	0.464953	+	0.885335i	$0.346071\pi$
$140$	2115.53	1.27711
$141$	336.242	0.200827
$142$	−178.886	−0.105717
$143$	768.351	0.449320
$144$	547.389	0.316776
$145$	−1222.73	−0.700290
$146$	−197.890	−0.112174
$147$	−1140.32	−0.639811
$148$	−2801.95	−1.55621
$149$	−398.661	−0.219192	−0.109596	−	0.993976i	$-0.534956\pi$
$149$	−398.661	−0.219192	−0.109596	+	0.993976i	$0.534956\pi$
$150$	−27.3625	−0.0148943
$151$	1003.92	0.541046	0.270523	−	0.962714i	$-0.412803\pi$
$151$	1003.92	0.541046	0.270523	+	0.962714i	$0.412803\pi$
$152$	−615.802	−0.328606
$153$	0	0
$154$	423.887	0.221804
$155$	−592.684	−0.307132
$156$	419.833	0.215472
$157$	2862.33	1.45503	0.727513	−	0.686094i	$-0.240676\pi$
$157$	2862.33	1.45503	0.727513	+	0.686094i	$0.240676\pi$
$158$	−97.0060	−0.0488442
$159$	−1084.15	−0.540745
$160$	−685.264	−0.338593
$161$	3549.88	1.73770
$162$	−29.5619	−0.0143370
$163$	189.481	0.0910510	0.0455255	−	0.998963i	$-0.485504\pi$
$163$	189.481	0.0910510	0.0455255	+	0.998963i	$0.485504\pi$
$164$	2564.00	1.22082
$165$	−1295.81	−0.611387
$166$	241.976	0.113139
$167$	−2565.49	−1.18876	−0.594382	−	0.804183i	$-0.702603\pi$
$167$	−2565.49	−1.18876	−0.594382	+	0.804183i	$0.702603\pi$
$168$	467.153	0.214533
$169$	−1880.54	−0.855960
$170$	0	0
$171$	−957.078	−0.428009
$172$	134.221	0.0595013
$173$	815.067	0.358199	0.179099	−	0.983831i	$-0.442682\pi$
$173$	815.067	0.358199	0.179099	+	0.983831i	$0.442682\pi$
$174$	−133.869	−0.0583251
$175$	672.032	0.290291
$176$	2626.97	1.12509
$177$	−1789.48	−0.759919
$178$	373.690	0.157355
$179$	2642.68	1.10348	0.551740	−	0.834016i	$-0.313964\pi$
$179$	2642.68	1.10348	0.551740	+	0.834016i	$0.313964\pi$
$180$	−708.043	−0.293191
$181$	2501.60	1.02731	0.513654	−	0.857998i	$-0.328292\pi$
$181$	2501.60	1.02731	0.513654	+	0.857998i	$0.328292\pi$
$182$	174.584	0.0711047
$183$	−2338.59	−0.944663
$184$	−764.448	−0.306282
$185$	3561.89	1.41554
$186$	−64.8891	−0.0255801
$187$	0	0
$188$	881.716	0.342052
$189$	726.048	0.279430
$190$	388.124	0.148197
$191$	2193.29	0.830894	0.415447	−	0.909617i	$-0.363625\pi$
$191$	2193.29	0.830894	0.415447	+	0.909617i	$0.363625\pi$
$192$	1384.68	0.520472
$193$	1458.18	0.543847	0.271923	−	0.962319i	$-0.412340\pi$
$193$	1458.18	0.543847	0.271923	+	0.962319i	$0.412340\pi$
$194$	−537.038	−0.198748
$195$	−533.700	−0.195995
$196$	−2990.23	−1.08973
$197$	−23.7041	−0.00857284	−0.00428642	−	0.999991i	$-0.501364\pi$
$197$	−23.7041	−0.00857284	−0.00428642	+	0.999991i	$0.501364\pi$
$198$	−141.870	−0.0509206
$199$	−1898.90	−0.676431	−0.338215	−	0.941069i	$-0.609823\pi$
$199$	−1898.90	−0.676431	−0.338215	+	0.941069i	$0.609823\pi$
$200$	−144.719	−0.0511658
$201$	2616.35	0.918125
$202$	114.202	0.0397785
$203$	3287.86	1.13676
$204$	0	0
$205$	−3259.40	−1.11047
$206$	−12.8004	−0.00432935
$207$	−1188.10	−0.398932
$208$	1081.96	0.360675
$209$	−4593.11	−1.52015
$210$	−294.434	−0.0967519
$211$	565.982	0.184663	0.0923313	−	0.995728i	$-0.470568\pi$
$211$	565.982	0.184663	0.0923313	+	0.995728i	$0.470568\pi$
$212$	−2842.93	−0.921004
$213$	−1470.45	−0.473022
$214$	−283.852	−0.0906715
$215$	−170.624	−0.0541230
$216$	−156.351	−0.0492515
$217$	1593.70	0.498558
$218$	56.4112	0.0175259
$219$	−1626.66	−0.501916
$220$	−3397.97	−1.04132
$221$	0	0
$222$	389.969	0.117896
$223$	−855.576	−0.256922	−0.128461	−	0.991715i	$-0.541004\pi$
$223$	−855.576	−0.256922	−0.128461	+	0.991715i	$0.541004\pi$
$224$	1842.64	0.549628
$225$	−224.921	−0.0666434
$226$	−691.352	−0.203487
$227$	1964.25	0.574324	0.287162	−	0.957882i	$-0.407288\pi$
$227$	1964.25	0.574324	0.287162	+	0.957882i	$0.407288\pi$
$228$	−2509.72	−0.728991
$229$	−2054.94	−0.592988	−0.296494	−	0.955035i	$-0.595817\pi$
$229$	−2054.94	−0.592988	−0.296494	+	0.955035i	$0.595817\pi$
$230$	481.812	0.138129
$231$	3484.38	0.992446
$232$	−708.023	−0.200362
$233$	5417.56	1.52325	0.761623	−	0.648021i	$-0.224403\pi$
$233$	5417.56	1.52325	0.761623	+	0.648021i	$0.224403\pi$
$234$	−58.4314	−0.0163239
$235$	−1120.86	−0.311134
$236$	−4692.50	−1.29430
$237$	−797.394	−0.218550
$238$	0	0
$239$	−5766.53	−1.56069	−0.780347	−	0.625347i	$-0.784957\pi$
$239$	−5766.53	−1.56069	−0.780347	+	0.625347i	$0.784957\pi$
$240$	−1824.71	−0.490769
$241$	−840.126	−0.224553	−0.112277	−	0.993677i	$-0.535814\pi$
$241$	−840.126	−0.224553	−0.112277	+	0.993677i	$0.535814\pi$
$242$	−195.086	−0.0518207
$243$	−243.000	−0.0641500
$244$	−6132.40	−1.60896
$245$	3801.24	0.991234
$246$	−356.851	−0.0924878
$247$	−1891.74	−0.487323
$248$	−343.194	−0.0878745
$249$	1989.06	0.506231
$250$	547.434	0.138491
$251$	−7407.85	−1.86287	−0.931433	−	0.363914i	$-0.881440\pi$
$251$	−7407.85	−1.86287	−0.931433	+	0.363914i	$0.881440\pi$
$252$	1903.89	0.475928
$253$	−5701.83	−1.41688
$254$	−195.011	−0.0481735
$255$	0	0
$256$	3430.93	0.837630
$257$	3136.07	0.761178	0.380589	−	0.924744i	$-0.375721\pi$
$257$	3136.07	0.761178	0.380589	+	0.924744i	$0.375721\pi$
$258$	−18.6805	−0.00450775
$259$	−9577.75	−2.29781
$260$	−1399.51	−0.333822
$261$	−1100.41	−0.260971
$262$	−1048.78	−0.247304
$263$	5797.03	1.35916	0.679582	−	0.733600i	$-0.262161\pi$
$263$	5797.03	1.35916	0.679582	+	0.733600i	$0.262161\pi$
$264$	−750.343	−0.174926
$265$	3613.98	0.837756
$266$	−1043.65	−0.240564
$267$	3071.75	0.704076
$268$	6860.77	1.56376
$269$	−6301.32	−1.42825	−0.714123	−	0.700020i	$-0.753174\pi$
$269$	−6301.32	−1.42825	−0.714123	+	0.700020i	$0.753174\pi$
$270$	98.5438	0.0222118
$271$	1080.05	0.242097	0.121048	−	0.992647i	$-0.461374\pi$
$271$	1080.05	0.242097	0.121048	+	0.992647i	$0.461374\pi$
$272$	0	0
$273$	1435.09	0.318153
$274$	−407.440	−0.0898335
$275$	−1079.42	−0.236696
$276$	−3115.53	−0.679466
$277$	626.484	0.135891	0.0679455	−	0.997689i	$-0.478356\pi$
$277$	626.484	0.135891	0.0679455	+	0.997689i	$0.478356\pi$
$278$	−556.171	−0.119989
$279$	−533.392	−0.114456
$280$	−1557.24	−0.332368
$281$	5149.02	1.09311	0.546557	−	0.837422i	$-0.315938\pi$
$281$	5149.02	1.09311	0.546557	+	0.837422i	$0.315938\pi$
$282$	−122.715	−0.0259134
$283$	−1362.15	−0.286117	−0.143059	−	0.989714i	$-0.545694\pi$
$283$	−1362.15	−0.286117	−0.143059	+	0.989714i	$0.545694\pi$
$284$	−3855.92	−0.805657
$285$	3190.40	0.663098
$286$	−280.418	−0.0579772
$287$	8764.37	1.80259
$288$	−616.711	−0.126181
$289$	0	0
$290$	446.249	0.0903608
$291$	−4414.48	−0.889283
$292$	−4265.55	−0.854871
$293$	−4189.65	−0.835366	−0.417683	−	0.908593i	$-0.637158\pi$
$293$	−4189.65	−0.835366	−0.417683	+	0.908593i	$0.637158\pi$
$294$	416.174	0.0825569
$295$	5965.20	1.17731
$296$	2062.52	0.405005
$297$	−1166.18	−0.227841
$298$	145.496	0.0282830
$299$	−2348.38	−0.454216
$300$	−589.804	−0.113508
$301$	458.799	0.0878563
$302$	−366.392	−0.0698129
$303$	938.750	0.177986
$304$	−6467.83	−1.22025
$305$	7795.63	1.46353
$306$	0	0
$307$	6407.13	1.19112	0.595560	−	0.803311i	$-0.296930\pi$
$307$	6407.13	1.19112	0.595560	+	0.803311i	$0.296930\pi$
$308$	9136.96	1.69035
$309$	−105.220	−0.0193714
$310$	216.307	0.0396303
$311$	4657.82	0.849263	0.424631	−	0.905366i	$-0.360404\pi$
$311$	4657.82	0.849263	0.424631	+	0.905366i	$0.360404\pi$
$312$	−309.040	−0.0560767
$313$	−6234.98	−1.12595	−0.562974	−	0.826475i	$-0.690343\pi$
$313$	−6234.98	−1.12595	−0.562974	+	0.826475i	$0.690343\pi$
$314$	−1044.64	−0.187747
$315$	−2420.26	−0.432909
$316$	−2090.98	−0.372237
$317$	−8331.70	−1.47620	−0.738100	−	0.674692i	$-0.764277\pi$
$317$	−8331.70	−1.47620	−0.738100	+	0.674692i	$0.764277\pi$
$318$	395.672	0.0697741
$319$	−5280.97	−0.926888
$320$	−4615.80	−0.806347
$321$	−2333.28	−0.405703
$322$	−1295.57	−0.224221
$323$	0	0
$324$	−637.211	−0.109261
$325$	−444.576	−0.0758788
$326$	−69.1533	−0.0117486
$327$	463.703	0.0784184
$328$	−1887.36	−0.317720
$329$	3013.92	0.505054
$330$	472.922	0.0788893
$331$	−9345.96	−1.55197	−0.775983	−	0.630754i	$-0.782746\pi$
$331$	−9345.96	−1.55197	−0.775983	+	0.630754i	$0.782746\pi$
$332$	5215.84	0.862218
$333$	3205.56	0.527519
$334$	936.305	0.153390
$335$	−8721.55	−1.42241
$336$	4906.55	0.796650
$337$	10323.7	1.66875	0.834374	−	0.551199i	$-0.185829\pi$
$337$	10323.7	1.66875	0.834374	+	0.551199i	$0.185829\pi$
$338$	686.325	0.110447
$339$	−5682.95	−0.910488
$340$	0	0
$341$	−2559.80	−0.406513
$342$	349.297	0.0552275
$343$	−997.847	−0.157081
$344$	−98.8000	−0.0154853
$345$	3960.52	0.618050
$346$	−297.468	−0.0462196
$347$	11755.2	1.81859	0.909296	−	0.416149i	$-0.136621\pi$
$347$	11755.2	1.81859	0.909296	+	0.416149i	$0.136621\pi$
$348$	−2885.56	−0.444490
$349$	9777.52	1.49965	0.749826	−	0.661635i	$-0.230137\pi$
$349$	9777.52	1.49965	0.749826	+	0.661635i	$0.230137\pi$
$350$	−245.266	−0.0374571
$351$	−480.309	−0.0730399
$352$	−2959.66	−0.448154
$353$	6261.26	0.944061	0.472030	−	0.881582i	$-0.343521\pi$
$353$	6261.26	0.944061	0.472030	+	0.881582i	$0.343521\pi$
$354$	653.092	0.0980549
$355$	4901.72	0.732834
$356$	8054.96	1.19919
$357$	0	0
$358$	−964.475	−0.142386
$359$	−1578.44	−0.232052	−0.116026	−	0.993246i	$-0.537016\pi$
$359$	−1578.44	−0.232052	−0.116026	+	0.993246i	$0.537016\pi$
$360$	521.192	0.0763034
$361$	4449.63	0.648729
$362$	−912.988	−0.132557
$363$	−1603.62	−0.231868
$364$	3763.20	0.541882
$365$	5422.45	0.777599
$366$	853.494	0.121893
$367$	1315.16	0.187060	0.0935299	−	0.995616i	$-0.470185\pi$
$367$	1315.16	0.187060	0.0935299	+	0.995616i	$0.470185\pi$
$368$	−8029.07	−1.13735
$369$	−2933.33	−0.413830
$370$	−1299.95	−0.182652
$371$	−9717.82	−1.35990
$372$	−1398.70	−0.194944
$373$	3573.43	0.496046	0.248023	−	0.968754i	$-0.420219\pi$
$373$	3573.43	0.496046	0.248023	+	0.968754i	$0.420219\pi$
$374$	0	0
$375$	4499.93	0.619668
$376$	−649.033	−0.0890195
$377$	−2175.05	−0.297137
$378$	−264.979	−0.0360557
$379$	6602.93	0.894907	0.447453	−	0.894307i	$-0.352331\pi$
$379$	6602.93	0.894907	0.447453	+	0.894307i	$0.352331\pi$
$380$	8366.08	1.12940
$381$	−1603.00	−0.215549
$382$	−800.466	−0.107213
$383$	6904.25	0.921125	0.460562	−	0.887627i	$-0.347648\pi$
$383$	6904.25	0.921125	0.460562	+	0.887627i	$0.347648\pi$
$384$	−2149.92	−0.285710
$385$	−11615.1	−1.53756
$386$	−532.181	−0.0701743
$387$	−153.555	−0.0201696
$388$	−11575.9	−1.51464
$389$	2802.38	0.365260	0.182630	−	0.983182i	$-0.441539\pi$
$389$	2802.38	0.365260	0.182630	+	0.983182i	$0.441539\pi$
$390$	194.780	0.0252899
$391$	0	0
$392$	2201.11	0.283605
$393$	−8621.00	−1.10654
$394$	8.65109	0.00110618
$395$	2658.10	0.338591
$396$	−3058.04	−0.388061
$397$	−11942.4	−1.50975	−0.754874	−	0.655870i	$-0.772302\pi$
$397$	−11942.4	−1.50975	−0.754874	+	0.655870i	$0.772302\pi$
$398$	693.026	0.0872821
$399$	−8578.82	−1.07639
$400$	−1519.99	−0.189999
$401$	−2162.29	−0.269276	−0.134638	−	0.990895i	$-0.542987\pi$
$401$	−2162.29	−0.269276	−0.134638	+	0.990895i	$0.542987\pi$
$402$	−954.866	−0.118469
$403$	−1054.29	−0.130318
$404$	2461.65	0.303148
$405$	810.035	0.0993851
$406$	−1199.94	−0.146680
$407$	15383.8	1.87358
$408$	0	0
$409$	−8042.37	−0.972297	−0.486149	−	0.873876i	$-0.661599\pi$
$409$	−8042.37	−0.972297	−0.486149	+	0.873876i	$0.661599\pi$
$410$	1189.56	0.143288
$411$	−3349.18	−0.401954
$412$	−275.915	−0.0329936
$413$	−16040.1	−1.91110
$414$	433.612	0.0514755
$415$	−6630.48	−0.784283
$416$	−1218.98	−0.143667
$417$	−4571.75	−0.536882
$418$	1676.31	0.196151
$419$	5173.08	0.603154	0.301577	−	0.953442i	$-0.402487\pi$
$419$	5173.08	0.603154	0.301577	+	0.953442i	$0.402487\pi$
$420$	−6346.58	−0.737337
$421$	12268.3	1.42024	0.710121	−	0.704080i	$-0.248640\pi$
$421$	12268.3	1.42024	0.710121	+	0.704080i	$0.248640\pi$
$422$	−206.561	−0.0238276
$423$	−1008.73	−0.115948
$424$	2092.68	0.239693
$425$	0	0
$426$	536.658	0.0610356
$427$	−20962.0	−2.37570
$428$	−6118.47	−0.690999
$429$	−2305.05	−0.259415
$430$	62.2711	0.00698367
$431$	−3573.99	−0.399427	−0.199713	−	0.979854i	$-0.564001\pi$
$431$	−3573.99	−0.399427	−0.199713	+	0.979854i	$0.564001\pi$
$432$	−1642.17	−0.182891
$433$	−1236.26	−0.137207	−0.0686035	−	0.997644i	$-0.521854\pi$
$433$	−1236.26	−0.137207	−0.0686035	+	0.997644i	$0.521854\pi$
$434$	−581.637	−0.0643306
$435$	3668.19	0.404313
$436$	1215.95	0.133563
$437$	14038.4	1.53672
$438$	593.669	0.0647639
$439$	12176.2	1.32377	0.661887	−	0.749604i	$-0.269756\pi$
$439$	12176.2	1.32377	0.661887	+	0.749604i	$0.269756\pi$
$440$	2501.25	0.271006
$441$	3420.97	0.369395
$442$	0	0
$443$	781.598	0.0838258	0.0419129	−	0.999121i	$-0.486655\pi$
$443$	781.598	0.0838258	0.0419129	+	0.999121i	$0.486655\pi$
$444$	8405.85	0.898477
$445$	−10239.6	−1.09080
$446$	312.252	0.0331515
$447$	1195.98	0.126550
$448$	12411.6	1.30892
$449$	−6255.69	−0.657515	−0.328758	−	0.944414i	$-0.606630\pi$
$449$	−6255.69	−0.657515	−0.328758	+	0.944414i	$0.606630\pi$
$450$	82.0876	0.00859922
$451$	−14077.4	−1.46979
$452$	−14902.2	−1.55075
$453$	−3011.76	−0.312373
$454$	−716.873	−0.0741069
$455$	−4783.85	−0.492902
$456$	1847.41	0.189721
$457$	14465.7	1.48070	0.740349	−	0.672223i	$-0.234660\pi$
$457$	14465.7	1.48070	0.740349	+	0.672223i	$0.234660\pi$
$458$	749.973	0.0765152
$459$	0	0
$460$	10385.5	1.05267
$461$	−998.651	−0.100893	−0.0504467	−	0.998727i	$-0.516064\pi$
$461$	−998.651	−0.100893	−0.0504467	+	0.998727i	$0.516064\pi$
$462$	−1271.66	−0.128059
$463$	−13320.8	−1.33708	−0.668542	−	0.743674i	$-0.733081\pi$
$463$	−13320.8	−1.33708	−0.668542	+	0.743674i	$0.733081\pi$
$464$	−7436.44	−0.744026
$465$	1778.05	0.177323
$466$	−1977.20	−0.196549
$467$	−15366.7	−1.52267	−0.761333	−	0.648361i	$-0.775455\pi$
$467$	−15366.7	−1.52267	−0.761333	+	0.648361i	$0.775455\pi$
$468$	−1259.50	−0.124403
$469$	23451.8	2.30896
$470$	409.069	0.0401467
$471$	−8587.00	−0.840060
$472$	3454.16	0.336844
$473$	−736.925	−0.0716360
$474$	291.018	0.0282002
$475$	2657.62	0.256716
$476$	0	0
$477$	3252.44	0.312199
$478$	2104.56	0.201381
$479$	−17189.6	−1.63969	−0.819847	−	0.572583i	$-0.805941\pi$
$479$	−17189.6	−1.63969	−0.819847	+	0.572583i	$0.805941\pi$
$480$	2055.79	0.195487
$481$	6336.06	0.600623
$482$	306.614	0.0289748
$483$	−10649.6	−1.00326
$484$	−4205.12	−0.394921
$485$	14715.6	1.37773
$486$	88.6856	0.00827749
$487$	−13940.1	−1.29710	−0.648550	−	0.761172i	$-0.724624\pi$
$487$	−13940.1	−1.29710	−0.648550	+	0.761172i	$0.724624\pi$
$488$	4514.07	0.418734
$489$	−568.444	−0.0525683
$490$	−1387.31	−0.127902
$491$	−17592.7	−1.61700	−0.808499	−	0.588498i	$-0.799719\pi$
$491$	−17592.7	−1.61700	−0.808499	+	0.588498i	$0.799719\pi$
$492$	−7691.99	−0.704841
$493$	0	0
$494$	690.413	0.0628809
$495$	3887.44	0.352985
$496$	−3604.61	−0.326314
$497$	−13180.5	−1.18959
$498$	−725.929	−0.0653206
$499$	−7306.44	−0.655473	−0.327737	−	0.944769i	$-0.606286\pi$
$499$	−7306.44	−0.655473	−0.327737	+	0.944769i	$0.606286\pi$
$500$	11800.0	1.05543
$501$	7696.47	0.686334
$502$	2703.58	0.240372
$503$	9467.89	0.839269	0.419635	−	0.907693i	$-0.362158\pi$
$503$	9467.89	0.839269	0.419635	+	0.907693i	$0.362158\pi$
$504$	−1401.46	−0.123861
$505$	−3129.30	−0.275747
$506$	2080.95	0.182825
$507$	5641.63	0.494188
$508$	−4203.49	−0.367126
$509$	11603.2	1.01042	0.505208	−	0.862997i	$-0.331416\pi$
$509$	11603.2	1.01042	0.505208	+	0.862997i	$0.331416\pi$
$510$	0	0
$511$	−14580.7	−1.26225
$512$	−6985.27	−0.602946
$513$	2871.24	0.247111
$514$	−1144.54	−0.0982173
$515$	350.748	0.0300113
$516$	−402.662	−0.0343531
$517$	−4840.97	−0.411810
$518$	3495.51	0.296494
$519$	−2445.20	−0.206806
$520$	1030.18	0.0868775
$521$	13931.0	1.17146	0.585729	−	0.810507i	$-0.300808\pi$
$521$	13931.0	1.17146	0.585729	+	0.810507i	$0.300808\pi$
$522$	401.606	0.0336740
$523$	−8648.66	−0.723096	−0.361548	−	0.932353i	$-0.617752\pi$
$523$	−8648.66	−0.723096	−0.361548	+	0.932353i	$0.617752\pi$
$524$	−22606.6	−1.88468
$525$	−2016.10	−0.167599
$526$	−2115.69	−0.175377
$527$	0	0
$528$	−7880.92	−0.649570
$529$	5260.04	0.432320
$530$	−1318.96	−0.108098
$531$	5368.45	0.438740
$532$	−22496.0	−1.83332
$533$	−5797.98	−0.471179
$534$	−1121.07	−0.0908492
$535$	7777.93	0.628540
$536$	−5050.23	−0.406971
$537$	−7928.03	−0.637095
$538$	2299.74	0.184291
$539$	16417.6	1.31198
$540$	2124.13	0.169274
$541$	−6546.86	−0.520280	−0.260140	−	0.965571i	$-0.583769\pi$
$541$	−6546.86	−0.520280	−0.260140	+	0.965571i	$0.583769\pi$
$542$	−394.176	−0.0312386
$543$	−7504.81	−0.593116
$544$	0	0
$545$	−1545.74	−0.121491
$546$	−523.753	−0.0410523
$547$	20384.5	1.59338	0.796689	−	0.604390i	$-0.206583\pi$
$547$	20384.5	1.59338	0.796689	+	0.604390i	$0.206583\pi$
$548$	−8782.45	−0.684613
$549$	7015.76	0.545401
$550$	393.947	0.0305417
$551$	13002.2	1.00528
$552$	2293.34	0.176832
$553$	−7147.49	−0.549624
$554$	−228.642	−0.0175344
$555$	−10685.7	−0.817265
$556$	−11988.4	−0.914424
$557$	−8560.07	−0.651171	−0.325585	−	0.945513i	$-0.605561\pi$
$557$	−8560.07	−0.651171	−0.325585	+	0.945513i	$0.605561\pi$
$558$	194.667	0.0147687
$559$	−303.514	−0.0229647
$560$	−16355.9	−1.23422
$561$	0	0
$562$	−1879.19	−0.141048
$563$	−2446.20	−0.183117	−0.0915587	−	0.995800i	$-0.529185\pi$
$563$	−2446.20	−0.183117	−0.0915587	+	0.995800i	$0.529185\pi$
$564$	−2645.15	−0.197484
$565$	18944.0	1.41058
$566$	497.130	0.0369186
$567$	−2178.14	−0.161329
$568$	2838.35	0.209673
$569$	−4431.69	−0.326513	−0.163257	−	0.986584i	$-0.552200\pi$
$569$	−4431.69	−0.326513	−0.163257	+	0.986584i	$0.552200\pi$
$570$	−1164.37	−0.0855617
$571$	−12448.7	−0.912366	−0.456183	−	0.889886i	$-0.650784\pi$
$571$	−12448.7	−0.912366	−0.456183	+	0.889886i	$0.650784\pi$
$572$	−6044.46	−0.441839
$573$	−6579.87	−0.479717
$574$	−3198.65	−0.232595
$575$	3299.14	0.239276
$576$	−4154.04	−0.300495
$577$	12488.3	0.901029	0.450515	−	0.892769i	$-0.351240\pi$
$577$	12488.3	0.901029	0.450515	+	0.892769i	$0.351240\pi$
$578$	0	0
$579$	−4374.55	−0.313990
$580$	9618.97	0.688631
$581$	17829.0	1.27310
$582$	1611.11	0.114747
$583$	15608.8	1.10883
$584$	3139.88	0.222481
$585$	1601.10	0.113158
$586$	1529.06	0.107790
$587$	7481.51	0.526056	0.263028	−	0.964788i	$-0.415279\pi$
$587$	7481.51	0.526056	0.263028	+	0.964788i	$0.415279\pi$
$588$	8970.69	0.629159
$589$	6302.44	0.440896
$590$	−2177.07	−0.151913
$591$	71.1124	0.00494953
$592$	21662.9	1.50395
$593$	15754.8	1.09102	0.545508	−	0.838105i	$-0.316337\pi$
$593$	15754.8	1.09102	0.545508	+	0.838105i	$0.316337\pi$
$594$	425.611	0.0293990
$595$	0	0
$596$	3136.18	0.215542
$597$	5696.71	0.390537
$598$	857.069	0.0586090
$599$	10310.0	0.703265	0.351632	−	0.936138i	$-0.385627\pi$
$599$	10310.0	0.703265	0.351632	+	0.936138i	$0.385627\pi$
$600$	434.156	0.0295406
$601$	22040.0	1.49589	0.747946	−	0.663759i	$-0.231040\pi$
$601$	22040.0	1.49589	0.747946	+	0.663759i	$0.231040\pi$
$602$	−167.444	−0.0113364
$603$	−7849.05	−0.530080
$604$	−7897.65	−0.532038
$605$	5345.63	0.359224
$606$	−342.607	−0.0229661
$607$	−2475.34	−0.165520	−0.0827601	−	0.996569i	$-0.526374\pi$
$607$	−2475.34	−0.165520	−0.0827601	+	0.996569i	$0.526374\pi$
$608$	7286.92	0.486059
$609$	−9863.57	−0.656309
$610$	−2845.10	−0.188844
$611$	−1993.83	−0.132016
$612$	0	0
$613$	15264.9	1.00578	0.502890	−	0.864351i	$-0.332270\pi$
$613$	15264.9	1.00578	0.502890	+	0.864351i	$0.332270\pi$
$614$	−2338.35	−0.153694
$615$	9778.21	0.641131
$616$	−6725.74	−0.439915
$617$	6826.01	0.445389	0.222694	−	0.974888i	$-0.428515\pi$
$617$	6826.01	0.445389	0.222694	+	0.974888i	$0.428515\pi$
$618$	38.4012	0.00249955
$619$	−5171.62	−0.335807	−0.167904	−	0.985803i	$-0.553700\pi$
$619$	−5171.62	−0.335807	−0.167904	+	0.985803i	$0.553700\pi$
$620$	4662.53	0.302019
$621$	3564.31	0.230323
$622$	−1699.92	−0.109583
$623$	27533.8	1.77066
$624$	−3245.88	−0.208236
$625$	−11876.5	−0.760098
$626$	2275.52	0.145285
$627$	13779.3	0.877662
$628$	−22517.4	−1.43080
$629$	0	0
$630$	883.303	0.0558597
$631$	17726.6	1.11836	0.559180	−	0.829046i	$-0.311116\pi$
$631$	17726.6	1.11836	0.559180	+	0.829046i	$0.311116\pi$
$632$	1539.18	0.0968752
$633$	−1697.95	−0.106615
$634$	3040.75	0.190479
$635$	5343.56	0.333942
$636$	8528.78	0.531742
$637$	6761.82	0.420586
$638$	1927.35	0.119599
$639$	4411.35	0.273099
$640$	7166.70	0.442639
$641$	21430.3	1.32051	0.660253	−	0.751043i	$-0.270449\pi$
$641$	21430.3	1.32051	0.660253	+	0.751043i	$0.270449\pi$
$642$	851.555	0.0523492
$643$	−8513.99	−0.522176	−0.261088	−	0.965315i	$-0.584081\pi$
$643$	−8513.99	−0.522176	−0.261088	+	0.965315i	$0.584081\pi$
$644$	−27926.2	−1.70877
$645$	511.872	0.0312480
$646$	0	0
$647$	7856.20	0.477371	0.238685	−	0.971097i	$-0.423284\pi$
$647$	7856.20	0.477371	0.238685	+	0.971097i	$0.423284\pi$
$648$	469.052	0.0284354
$649$	25763.7	1.55826
$650$	162.253	0.00979090
$651$	−4781.09	−0.287843
$652$	−1490.61	−0.0895351
$653$	−74.8160	−0.00448358	−0.00224179	−	0.999997i	$-0.500714\pi$
$653$	−74.8160	−0.00448358	−0.00224179	+	0.999997i	$0.500714\pi$
$654$	−169.234	−0.0101186
$655$	28737.9	1.71433
$656$	−19823.2	−1.17982
$657$	4879.99	0.289782
$658$	−1099.96	−0.0651688
$659$	−9897.21	−0.585039	−0.292519	−	0.956260i	$-0.594494\pi$
$659$	−9897.21	−0.585039	−0.292519	+	0.956260i	$0.594494\pi$
$660$	10193.9	0.601208
$661$	−11872.3	−0.698608	−0.349304	−	0.937010i	$-0.613582\pi$
$661$	−11872.3	−0.698608	−0.349304	+	0.937010i	$0.613582\pi$
$662$	3410.91	0.200255
$663$	0	0
$664$	−3839.39	−0.224393
$665$	28597.3	1.66760
$666$	−1169.91	−0.0680675
$667$	16140.7	0.936989
$668$	20182.2	1.16897
$669$	2566.73	0.148334
$670$	3183.03	0.183539
$671$	33669.3	1.93709
$672$	−5527.92	−0.317328
$673$	−7894.26	−0.452157	−0.226078	−	0.974109i	$-0.572591\pi$
$673$	−7894.26	−0.452157	−0.226078	+	0.974109i	$0.572591\pi$
$674$	−3767.75	−0.215324
$675$	674.764	0.0384766
$676$	14793.9	0.841708
$677$	21385.4	1.21404	0.607022	−	0.794685i	$-0.292364\pi$
$677$	21385.4	1.21404	0.607022	+	0.794685i	$0.292364\pi$
$678$	2074.06	0.117483
$679$	−39569.4	−2.23643
$680$	0	0
$681$	−5892.74	−0.331586
$682$	934.228	0.0524537
$683$	−2907.48	−0.162887	−0.0814433	−	0.996678i	$-0.525953\pi$
$683$	−2907.48	−0.162887	−0.0814433	+	0.996678i	$0.525953\pi$
$684$	7529.15	0.420883
$685$	11164.4	0.622731
$686$	364.176	0.0202686
$687$	6164.82	0.342362
$688$	−1037.71	−0.0575032
$689$	6428.72	0.355464
$690$	−1445.44	−0.0797490
$691$	9472.52	0.521493	0.260746	−	0.965407i	$-0.416031\pi$
$691$	9472.52	0.521493	0.260746	+	0.965407i	$0.416031\pi$
$692$	−6411.97	−0.352235
$693$	−10453.1	−0.572989
$694$	−4290.19	−0.234659
$695$	15239.8	0.831770
$696$	2124.07	0.115679
$697$	0	0
$698$	−3568.41	−0.193505
$699$	−16252.7	−0.879446
$700$	−5286.74	−0.285457
$701$	1519.26	0.0818568	0.0409284	−	0.999162i	$-0.486968\pi$
$701$	1519.26	0.0818568	0.0409284	+	0.999162i	$0.486968\pi$
$702$	175.294	0.00942458
$703$	−37876.2	−2.03205
$704$	−19935.6	−1.06726
$705$	3362.57	0.179633
$706$	−2285.12	−0.121815
$707$	8414.54	0.447611
$708$	14077.5	0.747267
$709$	23240.3	1.23104	0.615522	−	0.788120i	$-0.288945\pi$
$709$	23240.3	1.23104	0.615522	+	0.788120i	$0.288945\pi$
$710$	−1788.94	−0.0945600
$711$	2392.18	0.126180
$712$	−5929.27	−0.312091
$713$	7823.77	0.410943
$714$	0	0
$715$	7683.84	0.401901
$716$	−20789.4	−1.08511
$717$	17299.6	0.901067
$718$	576.069	0.0299425
$719$	−15545.6	−0.806331	−0.403166	−	0.915127i	$-0.632090\pi$
$719$	−15545.6	−0.806331	−0.403166	+	0.915127i	$0.632090\pi$
$720$	5474.13	0.283346
$721$	−943.144	−0.0487164
$722$	−1623.94	−0.0837076
$723$	2520.38	0.129646
$724$	−19679.6	−1.01020
$725$	3055.62	0.156528
$726$	585.258	0.0299187
$727$	3665.66	0.187004	0.0935019	−	0.995619i	$-0.470194\pi$
$727$	3665.66	0.187004	0.0935019	+	0.995619i	$0.470194\pi$
$728$	−2770.10	−0.141026
$729$	729.000	0.0370370
$730$	−1978.98	−0.100336
$731$	0	0
$732$	18397.2	0.928935
$733$	−18641.2	−0.939331	−0.469665	−	0.882844i	$-0.655625\pi$
$733$	−18641.2	−0.939331	−0.469665	+	0.882844i	$0.655625\pi$
$734$	−479.984	−0.0241370
$735$	−11403.7	−0.572289
$736$	9045.88	0.453038
$737$	−37668.4	−1.88268
$738$	1070.55	0.0533979
$739$	14178.2	0.705757	0.352878	−	0.935669i	$-0.385203\pi$
$739$	14178.2	0.705757	0.352878	+	0.935669i	$0.385203\pi$
$740$	−28020.7	−1.39198
$741$	5675.23	0.281356
$742$	3546.63	0.175473
$743$	−17301.3	−0.854273	−0.427137	−	0.904187i	$-0.640478\pi$
$743$	−17301.3	−0.854273	−0.427137	+	0.904187i	$0.640478\pi$
$744$	1029.58	0.0507343
$745$	−3986.78	−0.196059
$746$	−1304.16	−0.0640064
$747$	−5967.17	−0.292272
$748$	0	0
$749$	−20914.4	−1.02029
$750$	−1642.30	−0.0799578
$751$	−28904.1	−1.40443	−0.702214	−	0.711966i	$-0.747805\pi$
$751$	−28904.1	−1.40443	−0.702214	+	0.711966i	$0.747805\pi$
$752$	−6816.86	−0.330565
$753$	22223.5	1.07553
$754$	793.808	0.0383405
$755$	10039.6	0.483947
$756$	−5711.68	−0.274777
$757$	−22263.0	−1.06891	−0.534453	−	0.845198i	$-0.679482\pi$
$757$	−22263.0	−1.06891	−0.534453	+	0.845198i	$0.679482\pi$
$758$	−2409.81	−0.115473
$759$	17105.5	0.818036
$760$	−6158.29	−0.293927
$761$	−7547.94	−0.359543	−0.179772	−	0.983708i	$-0.557536\pi$
$761$	−7547.94	−0.359543	−0.179772	+	0.983708i	$0.557536\pi$
$762$	585.033	0.0278130
$763$	4156.43	0.197212
$764$	−17254.2	−0.817060
$765$	0	0
$766$	−2519.78	−0.118856
$767$	10611.2	0.499540
$768$	−10292.8	−0.483606
$769$	7647.30	0.358607	0.179303	−	0.983794i	$-0.442616\pi$
$769$	7647.30	0.358607	0.179303	+	0.983794i	$0.442616\pi$
$770$	4239.06	0.198396
$771$	−9408.21	−0.439466
$772$	−11471.2	−0.534792
$773$	−7062.66	−0.328624	−0.164312	−	0.986408i	$-0.552540\pi$
$773$	−7062.66	−0.328624	−0.164312	+	0.986408i	$0.552540\pi$
$774$	56.0416	0.00260255
$775$	1481.13	0.0686499
$776$	8521.07	0.394186
$777$	28733.2	1.32664
$778$	−1022.76	−0.0471307
$779$	34659.6	1.59411
$780$	4198.52	0.192732
$781$	21170.5	0.969963
$782$	0	0
$783$	3301.22	0.150672
$784$	23118.5	1.05314
$785$	28624.6	1.30147
$786$	3146.33	0.142781
$787$	−33790.6	−1.53050	−0.765250	−	0.643733i	$-0.777385\pi$
$787$	−33790.6	−1.53050	−0.765250	+	0.643733i	$0.777385\pi$
$788$	186.476	0.00843010
$789$	−17391.1	−0.784713
$790$	−970.103	−0.0436895
$791$	−50939.4	−2.28976
$792$	2251.03	0.100993
$793$	13867.2	0.620983
$794$	4358.50	0.194808
$795$	−10841.9	−0.483678
$796$	14938.3	0.665168
$797$	22802.7	1.01344	0.506722	−	0.862110i	$-0.330857\pi$
$797$	22802.7	1.01344	0.506722	+	0.862110i	$0.330857\pi$
$798$	3130.94	0.138890
$799$	0	0
$800$	1712.49	0.0756820
$801$	−9215.26	−0.406498
$802$	789.151	0.0347455
$803$	23419.5	1.02921
$804$	−20582.3	−0.902838
$805$	35500.3	1.55431
$806$	384.776	0.0168153
$807$	18904.0	0.824599
$808$	−1812.03	−0.0788947
$809$	−21945.4	−0.953720	−0.476860	−	0.878979i	$-0.658225\pi$
$809$	−21945.4	−0.953720	−0.476860	+	0.878979i	$0.658225\pi$
$810$	−295.631	−0.0128240
$811$	30871.9	1.33669	0.668347	−	0.743850i	$-0.267002\pi$
$811$	30871.9	1.33669	0.668347	+	0.743850i	$0.267002\pi$
$812$	−25864.9	−1.11783
$813$	−3240.14	−0.139775
$814$	−5614.50	−0.241754
$815$	1894.90	0.0814421
$816$	0	0
$817$	1814.37	0.0776950
$818$	2935.15	0.125459
$819$	−4305.28	−0.183686
$820$	25641.1	1.09198
$821$	9089.18	0.386376	0.193188	−	0.981162i	$-0.438117\pi$
$821$	9089.18	0.386376	0.193188	+	0.981162i	$0.438117\pi$
$822$	1222.32	0.0518654
$823$	27715.0	1.17386	0.586929	−	0.809639i	$-0.300337\pi$
$823$	27715.0	1.17386	0.586929	+	0.809639i	$0.300337\pi$
$824$	203.101	0.00858662
$825$	3238.26	0.136657
$826$	5854.02	0.246595
$827$	−9246.34	−0.388787	−0.194393	−	0.980924i	$-0.562274\pi$
$827$	−9246.34	−0.388787	−0.194393	+	0.980924i	$0.562274\pi$
$828$	9346.58	0.392290
$829$	−6364.44	−0.266642	−0.133321	−	0.991073i	$-0.542564\pi$
$829$	−6364.44	−0.266642	−0.133321	+	0.991073i	$0.542564\pi$
$830$	2419.87	0.101199
$831$	−1879.45	−0.0784566
$832$	−8210.80	−0.342137
$833$	0	0
$834$	1668.51	0.0692756
$835$	−25656.0	−1.06331
$836$	36133.1	1.49484
$837$	1600.18	0.0660815
$838$	−1887.98	−0.0778270
$839$	11079.2	0.455895	0.227947	−	0.973673i	$-0.426799\pi$
$839$	11079.2	0.455895	0.227947	+	0.973673i	$0.426799\pi$
$840$	4671.73	0.191893
$841$	−9439.64	−0.387045
$842$	−4477.46	−0.183258
$843$	−15447.1	−0.631109
$844$	−4452.47	−0.181588
$845$	−18806.3	−0.765627
$846$	368.146	0.0149611
$847$	−14374.1	−0.583117
$848$	21979.7	0.890076
$849$	4086.44	0.165190
$850$	0	0
$851$	−47019.0	−1.89400
$852$	11567.7	0.465146
$853$	15135.0	0.607518	0.303759	−	0.952749i	$-0.401758\pi$
$853$	15135.0	0.607518	0.303759	+	0.952749i	$0.401758\pi$
$854$	7650.33	0.306545
$855$	−9571.20	−0.382840
$856$	4503.82	0.179833
$857$	−10981.5	−0.437714	−0.218857	−	0.975757i	$-0.570233\pi$
$857$	−10981.5	−0.437714	−0.218857	+	0.975757i	$0.570233\pi$
$858$	841.255	0.0334731
$859$	−13284.3	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$859$	−13284.3	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$860$	1342.26	0.0532219
$861$	−26293.1	−1.04073
$862$	1304.37	0.0515393
$863$	15831.4	0.624457	0.312228	−	0.950007i	$-0.398925\pi$
$863$	15831.4	0.624457	0.312228	+	0.950007i	$0.398925\pi$
$864$	1850.13	0.0728505
$865$	8151.02	0.320397
$866$	451.185	0.0177043
$867$	0	0
$868$	−12537.3	−0.490257
$869$	11480.3	0.448151
$870$	−1338.75	−0.0521698
$871$	−15514.3	−0.603538
$872$	−895.065	−0.0347600
$873$	13243.4	0.513428
$874$	−5123.46	−0.198288
$875$	40335.4	1.55838
$876$	12796.6	0.493560
$877$	−3928.07	−0.151245	−0.0756223	−	0.997137i	$-0.524094\pi$
$877$	−3928.07	−0.151245	−0.0756223	+	0.997137i	$0.524094\pi$
$878$	−4443.83	−0.170811
$879$	12569.0	0.482299
$880$	26270.9	1.00635
$881$	10233.8	0.391355	0.195678	−	0.980668i	$-0.437309\pi$
$881$	10233.8	0.391355	0.195678	+	0.980668i	$0.437309\pi$
$882$	−1248.52	−0.0476643
$883$	−2485.53	−0.0947280	−0.0473640	−	0.998878i	$-0.515082\pi$
$883$	−2485.53	−0.0947280	−0.0473640	+	0.998878i	$0.515082\pi$
$884$	0	0
$885$	−17895.6	−0.679722
$886$	−285.253	−0.0108163
$887$	−32437.1	−1.22788	−0.613941	−	0.789352i	$-0.710417\pi$
$887$	−32437.1	−1.22788	−0.613941	+	0.789352i	$0.710417\pi$
$888$	−6187.56	−0.233830
$889$	−14368.6	−0.542077
$890$	3737.06	0.140749
$891$	3498.54	0.131544
$892$	6730.64	0.252644
$893$	11918.9	0.446641
$894$	−436.487	−0.0163292
$895$	26427.9	0.987026
$896$	−19270.9	−0.718522
$897$	7045.15	0.262242
$898$	2283.08	0.0848413
$899$	7246.28	0.268829
$900$	1769.41	0.0655338
$901$	0	0
$902$	5137.69	0.189652
$903$	−1376.40	−0.0507238
$904$	10969.5	0.403586
$905$	25017.1	0.918892
$906$	1099.18	0.0403065
$907$	−11877.1	−0.434811	−0.217405	−	0.976081i	$-0.569759\pi$
$907$	−11877.1	−0.434811	−0.217405	+	0.976081i	$0.569759\pi$
$908$	−15452.3	−0.564762
$909$	−2816.25	−0.102760
$910$	1745.92	0.0636008
$911$	49383.1	1.79598	0.897989	−	0.440018i	$-0.145028\pi$
$911$	49383.1	1.79598	0.897989	+	0.440018i	$0.145028\pi$
$912$	19403.5	0.704511
$913$	−28637.1	−1.03806
$914$	−5279.44	−0.191059
$915$	−23386.9	−0.844969
$916$	16165.8	0.583115
$917$	−77274.8	−2.78281
$918$	0	0
$919$	46885.8	1.68294	0.841468	−	0.540307i	$-0.181692\pi$
$919$	46885.8	1.68294	0.841468	+	0.540307i	$0.181692\pi$
$920$	−7644.81	−0.273959
$921$	−19221.4	−0.687694
$922$	364.469	0.0130186
$923$	8719.40	0.310945
$924$	−27410.9	−0.975922
$925$	−8901.24	−0.316401
$926$	4861.58	0.172528
$927$	315.660	0.0111841
$928$	8378.20	0.296366
$929$	−42305.6	−1.49408	−0.747041	−	0.664777i	$-0.768526\pi$
$929$	−42305.6	−1.49408	−0.747041	+	0.664777i	$0.768526\pi$
$930$	−648.920	−0.0228806
$931$	−40421.4	−1.42294
$932$	−42618.9	−1.49788
$933$	−13973.5	−0.490322
$934$	5608.24	0.196475
$935$	0	0
$936$	927.120	0.0323759
$937$	−2158.14	−0.0752437	−0.0376219	−	0.999292i	$-0.511978\pi$
$937$	−2158.14	−0.0752437	−0.0376219	+	0.999292i	$0.511978\pi$
$938$	−8558.99	−0.297933
$939$	18704.9	0.650066
$940$	8817.55	0.305954
$941$	19824.6	0.686783	0.343391	−	0.939192i	$-0.388424\pi$
$941$	19824.6	0.686783	0.343391	+	0.939192i	$0.388424\pi$
$942$	3133.92	0.108396
$943$	43026.0	1.48581
$944$	36279.4	1.25084
$945$	7260.79	0.249940
$946$	268.949	0.00924343
$947$	−20994.1	−0.720396	−0.360198	−	0.932876i	$-0.617291\pi$
$947$	−20994.1	−0.720396	−0.360198	+	0.932876i	$0.617291\pi$
$948$	6272.95	0.214911
$949$	9645.70	0.329939
$950$	−969.930	−0.0331249
$951$	24995.1	0.852284
$952$	0	0
$953$	48188.0	1.63794	0.818972	−	0.573833i	$-0.194544\pi$
$953$	48188.0	1.63794	0.818972	+	0.573833i	$0.194544\pi$
$954$	−1187.02	−0.0402841
$955$	21933.8	0.743207
$956$	45364.1	1.53471
$957$	15842.9	0.535139
$958$	6273.54	0.211575
$959$	−30020.6	−1.01086
$960$	13847.4	0.465545
$961$	−26278.6	−0.882098
$962$	−2312.42	−0.0775003
$963$	6999.83	0.234233
$964$	6609.11	0.220814
$965$	14582.5	0.486452
$966$	3886.70	0.129454
$967$	36291.2	1.20687	0.603437	−	0.797411i	$-0.293798\pi$
$967$	36291.2	1.20687	0.603437	+	0.797411i	$0.293798\pi$
$968$	3095.39	0.102779
$969$	0	0
$970$	−5370.61	−0.177773
$971$	−3922.46	−0.129637	−0.0648187	−	0.997897i	$-0.520647\pi$
$971$	−3922.46	−0.129637	−0.0648187	+	0.997897i	$0.520647\pi$
$972$	1911.63	0.0630820
$973$	−40979.2	−1.35019
$974$	5087.61	0.167369
$975$	1333.73	0.0438087
$976$	47411.7	1.55493
$977$	−47232.4	−1.54667	−0.773336	−	0.633996i	$-0.781413\pi$
$977$	−47232.4	−1.54667	−0.773336	+	0.633996i	$0.781413\pi$
$978$	207.460	0.00678307
$979$	−44224.9	−1.44375
$980$	−29903.6	−0.974731
$981$	−1391.11	−0.0452749
$982$	6420.64	0.208646
$983$	31584.5	1.02481	0.512405	−	0.858744i	$-0.328755\pi$
$983$	31584.5	1.02481	0.512405	+	0.858744i	$0.328755\pi$
$984$	5662.09	0.183436
$985$	−237.052	−0.00766811
$986$	0	0
$987$	−9041.77	−0.291593
$988$	14882.0	0.479209
$989$	2252.33	0.0724166
$990$	−1418.76	−0.0455468
$991$	20811.9	0.667116	0.333558	−	0.942730i	$-0.391751\pi$
$991$	20811.9	0.667116	0.333558	+	0.942730i	$0.391751\pi$
$992$	4061.10	0.129980
$993$	28037.9	0.896028
$994$	4810.36	0.153496
$995$	−18989.9	−0.605044
$996$	−15647.5	−0.497802
$997$	9193.85	0.292048	0.146024	−	0.989281i	$-0.453352\pi$
$997$	9193.85	0.292048	0.146024	+	0.989281i	$0.453352\pi$
$998$	2666.57	0.0845779
$999$	−9616.69	−0.304563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.4.a.p.1.5		8
17.2	even	8		51.4.e.a.4.5	✓	16
17.9	even	8		51.4.e.a.13.4	yes	16
17.16	even	2		867.4.a.q.1.5		8
51.2	odd	8		153.4.f.b.55.4		16
51.26	odd	8		153.4.f.b.64.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.e.a.4.5	✓	16	17.2	even	8
51.4.e.a.13.4	yes	16	17.9	even	8
153.4.f.b.55.4		16	51.2	odd	8
153.4.f.b.64.5		16	51.26	odd	8
867.4.a.p.1.5		8	1.1	even	1	trivial
867.4.a.q.1.5		8	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

\(f(q)\)	\(=\)	\(q-0.364961 q^{2} -3.00000 q^{3} -7.86680 q^{4} +10.0004 q^{5} +1.09488 q^{6} -26.8907 q^{7} +5.79077 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.364961 q^{2} -3.00000 q^{3} -7.86680 q^{4} +10.0004 q^{5} +1.09488 q^{6} -26.8907 q^{7} +5.79077 q^{8} +9.00000 q^{9} -3.64977 q^{10} +43.1919 q^{11} +23.6004 q^{12} +17.7892 q^{13} +9.81405 q^{14} -30.0013 q^{15} +60.8210 q^{16} -3.28465 q^{18} -106.342 q^{19} -78.6715 q^{20} +80.6720 q^{21} -15.7634 q^{22} -132.011 q^{23} -17.3723 q^{24} -24.9913 q^{25} -6.49238 q^{26} -27.0000 q^{27} +211.544 q^{28} -122.268 q^{29} +10.9493 q^{30} -59.2658 q^{31} -68.5235 q^{32} -129.576 q^{33} -268.918 q^{35} -70.8012 q^{36} +356.174 q^{37} +38.8107 q^{38} -53.3677 q^{39} +57.9102 q^{40} -325.926 q^{41} -29.4421 q^{42} -17.0616 q^{43} -339.782 q^{44} +90.0039 q^{45} +48.1791 q^{46} -112.081 q^{47} -182.463 q^{48} +380.108 q^{49} +9.12085 q^{50} -139.944 q^{52} +361.383 q^{53} +9.85395 q^{54} +431.938 q^{55} -155.718 q^{56} +319.026 q^{57} +44.6229 q^{58} +596.494 q^{59} +236.014 q^{60} +779.529 q^{61} +21.6297 q^{62} -242.016 q^{63} -461.560 q^{64} +177.900 q^{65} +47.2901 q^{66} -872.117 q^{67} +396.034 q^{69} +98.1448 q^{70} +490.150 q^{71} +52.1169 q^{72} +542.221 q^{73} -129.990 q^{74} +74.9738 q^{75} +836.572 q^{76} -1161.46 q^{77} +19.4771 q^{78} +265.798 q^{79} +608.237 q^{80} +81.0000 q^{81} +118.950 q^{82} -663.019 q^{83} -634.631 q^{84} +6.22684 q^{86} +366.803 q^{87} +250.114 q^{88} -1023.92 q^{89} -32.8479 q^{90} -478.364 q^{91} +1038.51 q^{92} +177.797 q^{93} +40.9051 q^{94} -1063.47 q^{95} +205.570 q^{96} +1471.49 q^{97} -138.725 q^{98} +388.727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9} - 72 q^{10} + 12 q^{11} - 72 q^{12} - 60 q^{13} + 56 q^{14} - 72 q^{15} - 100 q^{16} - 36 q^{18} - 140 q^{19} + 224 q^{20} - 84 q^{21} + 484 q^{22} - 96 q^{23} + 144 q^{24} + 220 q^{25} + 500 q^{26} - 216 q^{27} + 348 q^{28} + 380 q^{29} + 216 q^{30} + 124 q^{31} - 448 q^{32} - 36 q^{33} - 592 q^{35} + 216 q^{36} + 1260 q^{37} + 84 q^{38} + 180 q^{39} - 836 q^{40} + 284 q^{41} - 168 q^{42} - 76 q^{43} - 1460 q^{44} + 216 q^{45} + 1056 q^{46} - 184 q^{47} + 300 q^{48} - 536 q^{49} - 360 q^{50} - 1580 q^{52} + 1064 q^{53} + 108 q^{54} - 2148 q^{55} + 244 q^{56} + 420 q^{57} - 532 q^{58} + 1224 q^{59} - 672 q^{60} + 324 q^{61} + 3068 q^{62} + 252 q^{63} - 876 q^{64} + 652 q^{65} - 1452 q^{66} - 624 q^{67} + 288 q^{69} - 1932 q^{70} + 2636 q^{71} - 432 q^{72} + 1640 q^{73} - 120 q^{74} - 660 q^{75} - 1172 q^{76} + 504 q^{77} - 1500 q^{78} + 3788 q^{79} + 1740 q^{80} + 648 q^{81} + 1916 q^{82} + 2032 q^{83} - 1044 q^{84} + 1612 q^{86} - 1140 q^{87} + 4576 q^{88} + 1304 q^{89} - 648 q^{90} - 2872 q^{91} - 1744 q^{92} - 372 q^{93} - 208 q^{94} - 748 q^{95} + 1344 q^{96} + 2376 q^{97} + 604 q^{98} + 108 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9 - 72 * q^10 + 12 * q^11 - 72 * q^12 - 60 * q^13 + 56 * q^14 - 72 * q^15 - 100 * q^16 - 36 * q^18 - 140 * q^19 + 224 * q^20 - 84 * q^21 + 484 * q^22 - 96 * q^23 + 144 * q^24 + 220 * q^25 + 500 * q^26 - 216 * q^27 + 348 * q^28 + 380 * q^29 + 216 * q^30 + 124 * q^31 - 448 * q^32 - 36 * q^33 - 592 * q^35 + 216 * q^36 + 1260 * q^37 + 84 * q^38 + 180 * q^39 - 836 * q^40 + 284 * q^41 - 168 * q^42 - 76 * q^43 - 1460 * q^44 + 216 * q^45 + 1056 * q^46 - 184 * q^47 + 300 * q^48 - 536 * q^49 - 360 * q^50 - 1580 * q^52 + 1064 * q^53 + 108 * q^54 - 2148 * q^55 + 244 * q^56 + 420 * q^57 - 532 * q^58 + 1224 * q^59 - 672 * q^60 + 324 * q^61 + 3068 * q^62 + 252 * q^63 - 876 * q^64 + 652 * q^65 - 1452 * q^66 - 624 * q^67 + 288 * q^69 - 1932 * q^70 + 2636 * q^71 - 432 * q^72 + 1640 * q^73 - 120 * q^74 - 660 * q^75 - 1172 * q^76 + 504 * q^77 - 1500 * q^78 + 3788 * q^79 + 1740 * q^80 + 648 * q^81 + 1916 * q^82 + 2032 * q^83 - 1044 * q^84 + 1612 * q^86 - 1140 * q^87 + 4576 * q^88 + 1304 * q^89 - 648 * q^90 - 2872 * q^91 - 1744 * q^92 - 372 * q^93 - 208 * q^94 - 748 * q^95 + 1344 * q^96 + 2376 * q^97 + 604 * q^98 + 108 * q^99

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