LMFDB - Embedded newform 862.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [862,2,Mod(1,862)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(862, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("862.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 862 = 2 \cdot 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	862.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:	$6.88310465423$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.11017801.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 13x^{3} + 3x^{2} - 10x - 1 $ x^6 - 2x^5 - 6x^4 + 13x^3 + 3x^2 - 10*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.32324$ of defining polynomial
Character	$\chi$	$=$	862.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.32324 q^{3} +1.00000 q^{4} +2.36571 q^{5} +1.32324 q^{6} +2.18258 q^{7} -1.00000 q^{8} -1.24903 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.32324 q^{3} +1.00000 q^{4} +2.36571 q^{5} +1.32324 q^{6} +2.18258 q^{7} -1.00000 q^{8} -1.24903 q^{9} -2.36571 q^{10} -5.72343 q^{11} -1.32324 q^{12} +0.593025 q^{13} -2.18258 q^{14} -3.13040 q^{15} +1.00000 q^{16} -7.42893 q^{17} +1.24903 q^{18} -3.56128 q^{19} +2.36571 q^{20} -2.88808 q^{21} +5.72343 q^{22} +4.42537 q^{23} +1.32324 q^{24} +0.596577 q^{25} -0.593025 q^{26} +5.62250 q^{27} +2.18258 q^{28} -3.86409 q^{29} +3.13040 q^{30} +8.47915 q^{31} -1.00000 q^{32} +7.57347 q^{33} +7.42893 q^{34} +5.16334 q^{35} -1.24903 q^{36} -7.14962 q^{37} +3.56128 q^{38} -0.784714 q^{39} -2.36571 q^{40} -12.6822 q^{41} +2.88808 q^{42} +1.56872 q^{43} -5.72343 q^{44} -2.95485 q^{45} -4.42537 q^{46} -4.71439 q^{47} -1.32324 q^{48} -2.23635 q^{49} -0.596577 q^{50} +9.83026 q^{51} +0.593025 q^{52} +2.42851 q^{53} -5.62250 q^{54} -13.5400 q^{55} -2.18258 q^{56} +4.71244 q^{57} +3.86409 q^{58} +12.6925 q^{59} -3.13040 q^{60} -3.35111 q^{61} -8.47915 q^{62} -2.72611 q^{63} +1.00000 q^{64} +1.40292 q^{65} -7.57347 q^{66} +8.04239 q^{67} -7.42893 q^{68} -5.85584 q^{69} -5.16334 q^{70} -7.37588 q^{71} +1.24903 q^{72} -15.2546 q^{73} +7.14962 q^{74} -0.789415 q^{75} -3.56128 q^{76} -12.4918 q^{77} +0.784714 q^{78} -8.80507 q^{79} +2.36571 q^{80} -3.69282 q^{81} +12.6822 q^{82} -11.5468 q^{83} -2.88808 q^{84} -17.5747 q^{85} -1.56872 q^{86} +5.11312 q^{87} +5.72343 q^{88} +12.3809 q^{89} +2.95485 q^{90} +1.29432 q^{91} +4.42537 q^{92} -11.2200 q^{93} +4.71439 q^{94} -8.42496 q^{95} +1.32324 q^{96} +0.965753 q^{97} +2.23635 q^{98} +7.14875 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} - 6 q^{8} - 2 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 - 6 * q^8 - 2 * q^9	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} - 6 q^{8} - 2 q^{9} + 2 q^{10} - 6 q^{11} - 2 q^{12} - 9 q^{13} - 3 q^{14} - 3 q^{15} + 6 q^{16} - 17 q^{17} + 2 q^{18} + q^{19} - 2 q^{20} - 8 q^{21} + 6 q^{22} - 20 q^{23} + 2 q^{24} + 10 q^{25} + 9 q^{26} + 7 q^{27} + 3 q^{28} + q^{29} + 3 q^{30} + 8 q^{31} - 6 q^{32} - 5 q^{33} + 17 q^{34} - 14 q^{35} - 2 q^{36} - 3 q^{37} - q^{38} - 13 q^{39} + 2 q^{40} - 19 q^{41} + 8 q^{42} - 21 q^{43} - 6 q^{44} - 16 q^{45} + 20 q^{46} - 10 q^{47} - 2 q^{48} - 7 q^{49} - 10 q^{50} - 2 q^{51} - 9 q^{52} - q^{53} - 7 q^{54} + 3 q^{55} - 3 q^{56} - 11 q^{57} - q^{58} + 11 q^{59} - 3 q^{60} - 2 q^{61} - 8 q^{62} + 6 q^{64} - 30 q^{65} + 5 q^{66} + 4 q^{67} - 17 q^{68} + 16 q^{69} + 14 q^{70} - 4 q^{71} + 2 q^{72} - 39 q^{73} + 3 q^{74} - 21 q^{75} + q^{76} - 33 q^{77} + 13 q^{78} - 16 q^{79} - 2 q^{80} - 22 q^{81} + 19 q^{82} - 20 q^{83} - 8 q^{84} - 16 q^{85} + 21 q^{86} + q^{87} + 6 q^{88} - 17 q^{89} + 16 q^{90} + 20 q^{91} - 20 q^{92} - 5 q^{93} + 10 q^{94} - 16 q^{95} + 2 q^{96} - 29 q^{97} + 7 q^{98} - 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 - 6 * q^8 - 2 * q^9 + 2 * q^10 - 6 * q^11 - 2 * q^12 - 9 * q^13 - 3 * q^14 - 3 * q^15 + 6 * q^16 - 17 * q^17 + 2 * q^18 + q^19 - 2 * q^20 - 8 * q^21 + 6 * q^22 - 20 * q^23 + 2 * q^24 + 10 * q^25 + 9 * q^26 + 7 * q^27 + 3 * q^28 + q^29 + 3 * q^30 + 8 * q^31 - 6 * q^32 - 5 * q^33 + 17 * q^34 - 14 * q^35 - 2 * q^36 - 3 * q^37 - q^38 - 13 * q^39 + 2 * q^40 - 19 * q^41 + 8 * q^42 - 21 * q^43 - 6 * q^44 - 16 * q^45 + 20 * q^46 - 10 * q^47 - 2 * q^48 - 7 * q^49 - 10 * q^50 - 2 * q^51 - 9 * q^52 - q^53 - 7 * q^54 + 3 * q^55 - 3 * q^56 - 11 * q^57 - q^58 + 11 * q^59 - 3 * q^60 - 2 * q^61 - 8 * q^62 + 6 * q^64 - 30 * q^65 + 5 * q^66 + 4 * q^67 - 17 * q^68 + 16 * q^69 + 14 * q^70 - 4 * q^71 + 2 * q^72 - 39 * q^73 + 3 * q^74 - 21 * q^75 + q^76 - 33 * q^77 + 13 * q^78 - 16 * q^79 - 2 * q^80 - 22 * q^81 + 19 * q^82 - 20 * q^83 - 8 * q^84 - 16 * q^85 + 21 * q^86 + q^87 + 6 * q^88 - 17 * q^89 + 16 * q^90 + 20 * q^91 - 20 * q^92 - 5 * q^93 + 10 * q^94 - 16 * q^95 + 2 * q^96 - 29 * q^97 + 7 * q^98 - 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.32324	−0.763974	−0.381987	−	0.924168i	$-0.624760\pi$
$3$	−1.32324	−0.763974	−0.381987	+	0.924168i	$0.624760\pi$
$4$	1.00000	0.500000
$5$	2.36571	1.05798	0.528989	−	0.848629i	$-0.322571\pi$
$5$	2.36571	1.05798	0.528989	+	0.848629i	$0.322571\pi$
$6$	1.32324	0.540211
$7$	2.18258	0.824937	0.412469	−	0.910972i	$-0.364667\pi$
$7$	2.18258	0.824937	0.412469	+	0.910972i	$0.364667\pi$
$8$	−1.00000	−0.353553
$9$	−1.24903	−0.416344
$10$	−2.36571	−0.748103
$11$	−5.72343	−1.72568	−0.862839	−	0.505479i	$-0.831316\pi$
$11$	−5.72343	−1.72568	−0.862839	+	0.505479i	$0.831316\pi$
$12$	−1.32324	−0.381987
$13$	0.593025	0.164475	0.0822377	−	0.996613i	$-0.473793\pi$
$13$	0.593025	0.164475	0.0822377	+	0.996613i	$0.473793\pi$
$14$	−2.18258	−0.583319
$15$	−3.13040	−0.808267
$16$	1.00000	0.250000
$17$	−7.42893	−1.80178	−0.900890	−	0.434048i	$-0.857085\pi$
$17$	−7.42893	−1.80178	−0.900890	+	0.434048i	$0.857085\pi$
$18$	1.24903	0.294400
$19$	−3.56128	−0.817015	−0.408507	−	0.912755i	$-0.633951\pi$
$19$	−3.56128	−0.817015	−0.408507	+	0.912755i	$0.633951\pi$
$20$	2.36571	0.528989
$21$	−2.88808	−0.630230
$22$	5.72343	1.22024
$23$	4.42537	0.922754	0.461377	−	0.887204i	$-0.347356\pi$
$23$	4.42537	0.922754	0.461377	+	0.887204i	$0.347356\pi$
$24$	1.32324	0.270105
$25$	0.596577	0.119315
$26$	−0.593025	−0.116302
$27$	5.62250	1.08205
$28$	2.18258	0.412469
$29$	−3.86409	−0.717544	−0.358772	−	0.933425i	$-0.616804\pi$
$29$	−3.86409	−0.717544	−0.358772	+	0.933425i	$0.616804\pi$
$30$	3.13040	0.571531
$31$	8.47915	1.52290	0.761450	−	0.648224i	$-0.224488\pi$
$31$	8.47915	1.52290	0.761450	+	0.648224i	$0.224488\pi$
$32$	−1.00000	−0.176777
$33$	7.57347	1.31837
$34$	7.42893	1.27405
$35$	5.16334	0.872764
$36$	−1.24903	−0.208172
$37$	−7.14962	−1.17539	−0.587695	−	0.809083i	$-0.699964\pi$
$37$	−7.14962	−1.17539	−0.587695	+	0.809083i	$0.699964\pi$
$38$	3.56128	0.577717
$39$	−0.784714	−0.125655
$40$	−2.36571	−0.374051
$41$	−12.6822	−1.98062	−0.990310	−	0.138872i	$-0.955652\pi$
$41$	−12.6822	−1.98062	−0.990310	+	0.138872i	$0.955652\pi$
$42$	2.88808	0.445640
$43$	1.56872	0.239228	0.119614	−	0.992820i	$-0.461834\pi$
$43$	1.56872	0.239228	0.119614	+	0.992820i	$0.461834\pi$
$44$	−5.72343	−0.862839
$45$	−2.95485	−0.440483
$46$	−4.42537	−0.652486
$47$	−4.71439	−0.687665	−0.343832	−	0.939031i	$-0.611725\pi$
$47$	−4.71439	−0.687665	−0.343832	+	0.939031i	$0.611725\pi$
$48$	−1.32324	−0.190993
$49$	−2.23635	−0.319479
$50$	−0.596577	−0.0843687
$51$	9.83026	1.37651
$52$	0.593025	0.0822377
$53$	2.42851	0.333582	0.166791	−	0.985992i	$-0.446660\pi$
$53$	2.42851	0.333582	0.166791	+	0.985992i	$0.446660\pi$
$54$	−5.62250	−0.765125
$55$	−13.5400	−1.82573
$56$	−2.18258	−0.291659
$57$	4.71244	0.624178
$58$	3.86409	0.507380
$59$	12.6925	1.65242	0.826210	−	0.563363i	$-0.190493\pi$
$59$	12.6925	1.65242	0.826210	+	0.563363i	$0.190493\pi$
$60$	−3.13040	−0.404133
$61$	−3.35111	−0.429066	−0.214533	−	0.976717i	$-0.568823\pi$
$61$	−3.35111	−0.429066	−0.214533	+	0.976717i	$0.568823\pi$
$62$	−8.47915	−1.07685
$63$	−2.72611	−0.343458
$64$	1.00000	0.125000
$65$	1.40292	0.174011
$66$	−7.57347	−0.932230
$67$	8.04239	0.982534	0.491267	−	0.871009i	$-0.336534\pi$
$67$	8.04239	0.982534	0.491267	+	0.871009i	$0.336534\pi$
$68$	−7.42893	−0.900890
$69$	−5.85584	−0.704960
$70$	−5.16334	−0.617138
$71$	−7.37588	−0.875356	−0.437678	−	0.899132i	$-0.644199\pi$
$71$	−7.37588	−0.875356	−0.437678	+	0.899132i	$0.644199\pi$
$72$	1.24903	0.147200
$73$	−15.2546	−1.78541	−0.892706	−	0.450639i	$-0.851196\pi$
$73$	−15.2546	−1.78541	−0.892706	+	0.450639i	$0.851196\pi$
$74$	7.14962	0.831126
$75$	−0.789415	−0.0911538
$76$	−3.56128	−0.408507
$77$	−12.4918	−1.42358
$78$	0.784714	0.0888514
$79$	−8.80507	−0.990648	−0.495324	−	0.868708i	$-0.664951\pi$
$79$	−8.80507	−0.990648	−0.495324	+	0.868708i	$0.664951\pi$
$80$	2.36571	0.264494
$81$	−3.69282	−0.410313
$82$	12.6822	1.40051
$83$	−11.5468	−1.26742	−0.633710	−	0.773570i	$-0.718469\pi$
$83$	−11.5468	−1.26742	−0.633710	+	0.773570i	$0.718469\pi$
$84$	−2.88808	−0.315115
$85$	−17.5747	−1.90624
$86$	−1.56872	−0.169160
$87$	5.11312	0.548184
$88$	5.72343	0.610119
$89$	12.3809	1.31237	0.656185	−	0.754600i	$-0.272169\pi$
$89$	12.3809	1.31237	0.656185	+	0.754600i	$0.272169\pi$
$90$	2.95485	0.311468
$91$	1.29432	0.135682
$92$	4.42537	0.461377
$93$	−11.2200	−1.16346
$94$	4.71439	0.486252
$95$	−8.42496	−0.864383
$96$	1.32324	0.135053
$97$	0.965753	0.0980574	0.0490287	−	0.998797i	$-0.484387\pi$
$97$	0.965753	0.0980574	0.0490287	+	0.998797i	$0.484387\pi$
$98$	2.23635	0.225906
$99$	7.14875	0.718476
$100$	0.596577	0.0596577
$101$	3.52582	0.350832	0.175416	−	0.984494i	$-0.443873\pi$
$101$	3.52582	0.350832	0.175416	+	0.984494i	$0.443873\pi$
$102$	−9.83026	−0.973341
$103$	−1.91457	−0.188648	−0.0943239	−	0.995542i	$-0.530069\pi$
$103$	−1.91457	−0.188648	−0.0943239	+	0.995542i	$0.530069\pi$
$104$	−0.593025	−0.0581508
$105$	−6.83235	−0.666769
$106$	−2.42851	−0.235878
$107$	−17.7035	−1.71146	−0.855732	−	0.517420i	$-0.826893\pi$
$107$	−17.7035	−1.71146	−0.855732	+	0.517420i	$0.826893\pi$
$108$	5.62250	0.541025
$109$	19.7191	1.88874	0.944372	−	0.328879i	$-0.106671\pi$
$109$	19.7191	1.88874	0.944372	+	0.328879i	$0.106671\pi$
$110$	13.5400	1.29098
$111$	9.46067	0.897967
$112$	2.18258	0.206234
$113$	−3.94098	−0.370737	−0.185368	−	0.982669i	$-0.559348\pi$
$113$	−3.94098	−0.370737	−0.185368	+	0.982669i	$0.559348\pi$
$114$	−4.71244	−0.441360
$115$	10.4691	0.976253
$116$	−3.86409	−0.358772
$117$	−0.740707	−0.0684784
$118$	−12.6925	−1.16844
$119$	−16.2142	−1.48635
$120$	3.13040	0.285765
$121$	21.7576	1.97797
$122$	3.35111	0.303396
$123$	16.7816	1.51314
$124$	8.47915	0.761450
$125$	−10.4172	−0.931744
$126$	2.72611	0.242861
$127$	−22.2288	−1.97249	−0.986245	−	0.165292i	$-0.947143\pi$
$127$	−22.2288	−1.97249	−0.986245	+	0.165292i	$0.947143\pi$
$128$	−1.00000	−0.0883883
$129$	−2.07580	−0.182764
$130$	−1.40292	−0.123045
$131$	14.6039	1.27594	0.637972	−	0.770059i	$-0.279773\pi$
$131$	14.6039	1.27594	0.637972	+	0.770059i	$0.279773\pi$
$132$	7.57347	0.659186
$133$	−7.77278	−0.673986
$134$	−8.04239	−0.694756
$135$	13.3012	1.14478
$136$	7.42893	0.637025
$137$	12.2027	1.04255	0.521275	−	0.853389i	$-0.325457\pi$
$137$	12.2027	1.04255	0.521275	+	0.853389i	$0.325457\pi$
$138$	5.85584	0.498482
$139$	8.96165	0.760118	0.380059	−	0.924962i	$-0.375904\pi$
$139$	8.96165	0.760118	0.380059	+	0.924962i	$0.375904\pi$
$140$	5.16334	0.436382
$141$	6.23828	0.525358
$142$	7.37588	0.618970
$143$	−3.39413	−0.283832
$144$	−1.24903	−0.104086
$145$	−9.14131	−0.759145
$146$	15.2546	1.26248
$147$	2.95923	0.244073
$148$	−7.14962	−0.587695
$149$	−8.36208	−0.685048	−0.342524	−	0.939509i	$-0.611282\pi$
$149$	−8.36208	−0.685048	−0.342524	+	0.939509i	$0.611282\pi$
$150$	0.789415	0.0644555
$151$	6.26793	0.510077	0.255039	−	0.966931i	$-0.417912\pi$
$151$	6.26793	0.510077	0.255039	+	0.966931i	$0.417912\pi$
$152$	3.56128	0.288858
$153$	9.27897	0.750161
$154$	12.4918	1.00662
$155$	20.0592	1.61119
$156$	−0.784714	−0.0628274
$157$	22.9217	1.82935	0.914676	−	0.404187i	$-0.132445\pi$
$157$	22.9217	1.82935	0.914676	+	0.404187i	$0.132445\pi$
$158$	8.80507	0.700494
$159$	−3.21351	−0.254848
$160$	−2.36571	−0.187026
$161$	9.65873	0.761214
$162$	3.69282	0.290135
$163$	−19.9480	−1.56245	−0.781224	−	0.624251i	$-0.785404\pi$
$163$	−19.9480	−1.56245	−0.781224	+	0.624251i	$0.785404\pi$
$164$	−12.6822	−0.990310
$165$	17.9166	1.39481
$166$	11.5468	0.896202
$167$	−4.57145	−0.353750	−0.176875	−	0.984233i	$-0.556599\pi$
$167$	−4.57145	−0.353750	−0.176875	+	0.984233i	$0.556599\pi$
$168$	2.88808	0.222820
$169$	−12.6483	−0.972948
$170$	17.5747	1.34792
$171$	4.44816	0.340159
$172$	1.56872	0.119614
$173$	−8.83938	−0.672046	−0.336023	−	0.941854i	$-0.609082\pi$
$173$	−8.83938	−0.672046	−0.336023	+	0.941854i	$0.609082\pi$
$174$	−5.11312	−0.387625
$175$	1.30208	0.0984277
$176$	−5.72343	−0.431420
$177$	−16.7952	−1.26240
$178$	−12.3809	−0.927985
$179$	−5.09347	−0.380704	−0.190352	−	0.981716i	$-0.560963\pi$
$179$	−5.09347	−0.380704	−0.190352	+	0.981716i	$0.560963\pi$
$180$	−2.95485	−0.220241
$181$	−12.1346	−0.901957	−0.450978	−	0.892535i	$-0.648925\pi$
$181$	−12.1346	−0.901957	−0.450978	+	0.892535i	$0.648925\pi$
$182$	−1.29432	−0.0959416
$183$	4.43433	0.327795
$184$	−4.42537	−0.326243
$185$	−16.9139	−1.24354
$186$	11.2200	0.822687
$187$	42.5189	3.10929
$188$	−4.71439	−0.343832
$189$	12.2715	0.892623
$190$	8.42496	0.611211
$191$	1.18950	0.0860693	0.0430347	−	0.999074i	$-0.486297\pi$
$191$	1.18950	0.0860693	0.0430347	+	0.999074i	$0.486297\pi$
$192$	−1.32324	−0.0954967
$193$	3.45647	0.248802	0.124401	−	0.992232i	$-0.460299\pi$
$193$	3.45647	0.248802	0.124401	+	0.992232i	$0.460299\pi$
$194$	−0.965753	−0.0693370
$195$	−1.85641	−0.132940
$196$	−2.23635	−0.159739
$197$	−20.6176	−1.46895	−0.734473	−	0.678638i	$-0.762571\pi$
$197$	−20.6176	−1.46895	−0.734473	+	0.678638i	$0.762571\pi$
$198$	−7.14875	−0.508039
$199$	5.29993	0.375702	0.187851	−	0.982198i	$-0.439848\pi$
$199$	5.29993	0.375702	0.187851	+	0.982198i	$0.439848\pi$
$200$	−0.596577	−0.0421844
$201$	−10.6420	−0.750630
$202$	−3.52582	−0.248076
$203$	−8.43368	−0.591928
$204$	9.83026	0.688256
$205$	−30.0023	−2.09545
$206$	1.91457	0.133394
$207$	−5.52744	−0.384183
$208$	0.593025	0.0411189
$209$	20.3827	1.40990
$210$	6.83235	0.471477
$211$	−3.18579	−0.219319	−0.109660	−	0.993969i	$-0.534976\pi$
$211$	−3.18579	−0.219319	−0.109660	+	0.993969i	$0.534976\pi$
$212$	2.42851	0.166791
$213$	9.76007	0.668749
$214$	17.7035	1.21019
$215$	3.71114	0.253098
$216$	−5.62250	−0.382562
$217$	18.5064	1.25630
$218$	−19.7191	−1.33554
$219$	20.1855	1.36401
$220$	−13.5400	−0.912864
$221$	−4.40554	−0.296348
$222$	−9.46067	−0.634958
$223$	19.7205	1.32058	0.660290	−	0.751010i	$-0.270433\pi$
$223$	19.7205	1.32058	0.660290	+	0.751010i	$0.270433\pi$
$224$	−2.18258	−0.145830
$225$	−0.745144	−0.0496763
$226$	3.94098	0.262150
$227$	17.9668	1.19250	0.596248	−	0.802800i	$-0.296657\pi$
$227$	17.9668	1.19250	0.596248	+	0.802800i	$0.296657\pi$
$228$	4.71244	0.312089
$229$	17.6093	1.16366	0.581829	−	0.813311i	$-0.302337\pi$
$229$	17.6093	1.16366	0.581829	+	0.813311i	$0.302337\pi$
$230$	−10.4691	−0.690315
$231$	16.5297	1.08757
$232$	3.86409	0.253690
$233$	11.8245	0.774650	0.387325	−	0.921943i	$-0.373399\pi$
$233$	11.8245	0.774650	0.387325	+	0.921943i	$0.373399\pi$
$234$	0.740707	0.0484215
$235$	−11.1529	−0.727533
$236$	12.6925	0.826210
$237$	11.6512	0.756829
$238$	16.2142	1.05101
$239$	12.3656	0.799865	0.399933	−	0.916545i	$-0.369034\pi$
$239$	12.3656	0.799865	0.399933	+	0.916545i	$0.369034\pi$
$240$	−3.13040	−0.202067
$241$	−13.7153	−0.883481	−0.441741	−	0.897143i	$-0.645639\pi$
$241$	−13.7153	−0.883481	−0.441741	+	0.897143i	$0.645639\pi$
$242$	−21.7576	−1.39863
$243$	−11.9810	−0.768581
$244$	−3.35111	−0.214533
$245$	−5.29056	−0.338001
$246$	−16.7816	−1.06995
$247$	−2.11193	−0.134379
$248$	−8.47915	−0.538426
$249$	15.2791	0.968276
$250$	10.4172	0.658843
$251$	−2.70050	−0.170454	−0.0852269	−	0.996362i	$-0.527162\pi$
$251$	−2.70050	−0.170454	−0.0852269	+	0.996362i	$0.527162\pi$
$252$	−2.72611	−0.171729
$253$	−25.3283	−1.59238
$254$	22.2288	1.39476
$255$	23.2555	1.45632
$256$	1.00000	0.0625000
$257$	10.3529	0.645798	0.322899	−	0.946433i	$-0.395343\pi$
$257$	10.3529	0.645798	0.322899	+	0.946433i	$0.395343\pi$
$258$	2.07580	0.129233
$259$	−15.6046	−0.969623
$260$	1.40292	0.0870056
$261$	4.82638	0.298745
$262$	−14.6039	−0.902229
$263$	3.77652	0.232870	0.116435	−	0.993198i	$-0.462853\pi$
$263$	3.77652	0.232870	0.116435	+	0.993198i	$0.462853\pi$
$264$	−7.57347	−0.466115
$265$	5.74515	0.352922
$266$	7.77278	0.476580
$267$	−16.3829	−1.00262
$268$	8.04239	0.491267
$269$	−4.24877	−0.259052	−0.129526	−	0.991576i	$-0.541346\pi$
$269$	−4.24877	−0.259052	−0.129526	+	0.991576i	$0.541346\pi$
$270$	−13.3012	−0.809484
$271$	24.6332	1.49636	0.748179	−	0.663497i	$-0.230928\pi$
$271$	24.6332	1.49636	0.748179	+	0.663497i	$0.230928\pi$
$272$	−7.42893	−0.450445
$273$	−1.71270	−0.103657
$274$	−12.2027	−0.737194
$275$	−3.41447	−0.205900
$276$	−5.85584	−0.352480
$277$	23.1348	1.39004	0.695018	−	0.718992i	$-0.255396\pi$
$277$	23.1348	1.39004	0.695018	+	0.718992i	$0.255396\pi$
$278$	−8.96165	−0.537484
$279$	−10.5907	−0.634051
$280$	−5.16334	−0.308569
$281$	−18.9945	−1.13312	−0.566558	−	0.824022i	$-0.691725\pi$
$281$	−18.9945	−1.13312	−0.566558	+	0.824022i	$0.691725\pi$
$282$	−6.23828	−0.371484
$283$	−1.54158	−0.0916376	−0.0458188	−	0.998950i	$-0.514590\pi$
$283$	−1.54158	−0.0916376	−0.0458188	+	0.998950i	$0.514590\pi$
$284$	−7.37588	−0.437678
$285$	11.1483	0.660366
$286$	3.39413	0.200699
$287$	−27.6798	−1.63389
$288$	1.24903	0.0736000
$289$	38.1889	2.24641
$290$	9.14131	0.536796
$291$	−1.27792	−0.0749132
$292$	−15.2546	−0.892706
$293$	−12.0162	−0.701991	−0.350996	−	0.936377i	$-0.614157\pi$
$293$	−12.0162	−0.701991	−0.350996	+	0.936377i	$0.614157\pi$
$294$	−2.95923	−0.172586
$295$	30.0267	1.74822
$296$	7.14962	0.415563
$297$	−32.1799	−1.86727
$298$	8.36208	0.484402
$299$	2.62436	0.151770
$300$	−0.789415	−0.0455769
$301$	3.42386	0.197348
$302$	−6.26793	−0.360679
$303$	−4.66550	−0.268026
$304$	−3.56128	−0.204254
$305$	−7.92776	−0.453942
$306$	−9.27897	−0.530444
$307$	21.2277	1.21153	0.605763	−	0.795645i	$-0.292868\pi$
$307$	21.2277	1.21153	0.605763	+	0.795645i	$0.292868\pi$
$308$	−12.4918	−0.711788
$309$	2.53343	0.144122
$310$	−20.0592	−1.13929
$311$	2.44107	0.138420	0.0692101	−	0.997602i	$-0.477952\pi$
$311$	2.44107	0.138420	0.0692101	+	0.997602i	$0.477952\pi$
$312$	0.784714	0.0444257
$313$	−25.8091	−1.45882	−0.729409	−	0.684078i	$-0.760205\pi$
$313$	−25.8091	−1.45882	−0.729409	+	0.684078i	$0.760205\pi$
$314$	−22.9217	−1.29355
$315$	−6.44919	−0.363371
$316$	−8.80507	−0.495324
$317$	−9.80038	−0.550444	−0.275222	−	0.961381i	$-0.588751\pi$
$317$	−9.80038	−0.550444	−0.275222	+	0.961381i	$0.588751\pi$
$318$	3.21351	0.180205
$319$	22.1158	1.23825
$320$	2.36571	0.132247
$321$	23.4260	1.30751
$322$	−9.65873	−0.538260
$323$	26.4565	1.47208
$324$	−3.69282	−0.205157
$325$	0.353785	0.0196245
$326$	19.9480	1.10482
$327$	−26.0931	−1.44295
$328$	12.6822	0.700255
$329$	−10.2895	−0.567280
$330$	−17.9166	−0.986278
$331$	8.07682	0.443942	0.221971	−	0.975053i	$-0.428751\pi$
$331$	8.07682	0.443942	0.221971	+	0.975053i	$0.428751\pi$
$332$	−11.5468	−0.633710
$333$	8.93011	0.489367
$334$	4.57145	0.250139
$335$	19.0259	1.03950
$336$	−2.88808	−0.157558
$337$	18.9338	1.03139	0.515696	−	0.856772i	$-0.327533\pi$
$337$	18.9338	1.03139	0.515696	+	0.856772i	$0.327533\pi$
$338$	12.6483	0.687978
$339$	5.21487	0.283233
$340$	−17.5747	−0.953121
$341$	−48.5298	−2.62804
$342$	−4.44816	−0.240529
$343$	−20.1591	−1.08849
$344$	−1.56872	−0.0845798
$345$	−13.8532	−0.745831
$346$	8.83938	0.475208
$347$	5.40719	0.290273	0.145137	−	0.989412i	$-0.453638\pi$
$347$	5.40719	0.290273	0.145137	+	0.989412i	$0.453638\pi$
$348$	5.11312	0.274092
$349$	15.9428	0.853396	0.426698	−	0.904394i	$-0.359677\pi$
$349$	15.9428	0.853396	0.426698	+	0.904394i	$0.359677\pi$
$350$	−1.30208	−0.0695989
$351$	3.33428	0.177971
$352$	5.72343	0.305060
$353$	−29.7808	−1.58507	−0.792536	−	0.609825i	$-0.791240\pi$
$353$	−29.7808	−1.58507	−0.792536	+	0.609825i	$0.791240\pi$
$354$	16.7952	0.892655
$355$	−17.4492	−0.926107
$356$	12.3809	0.656185
$357$	21.4553	1.13554
$358$	5.09347	0.269198
$359$	−6.81817	−0.359849	−0.179925	−	0.983680i	$-0.557585\pi$
$359$	−6.81817	−0.359849	−0.179925	+	0.983680i	$0.557585\pi$
$360$	2.95485	0.155734
$361$	−6.31726	−0.332487
$362$	12.1346	0.637780
$363$	−28.7906	−1.51111
$364$	1.29432	0.0678409
$365$	−36.0879	−1.88893
$366$	−4.43433	−0.231786
$367$	−7.69745	−0.401803	−0.200902	−	0.979611i	$-0.564387\pi$
$367$	−7.69745	−0.401803	−0.200902	+	0.979611i	$0.564387\pi$
$368$	4.42537	0.230689
$369$	15.8404	0.824620
$370$	16.9139	0.879312
$371$	5.30042	0.275184
$372$	−11.2200	−0.581728
$373$	20.9846	1.08654	0.543271	−	0.839557i	$-0.317185\pi$
$373$	20.9846	1.08654	0.543271	+	0.839557i	$0.317185\pi$
$374$	−42.5189	−2.19860
$375$	13.7845	0.711828
$376$	4.71439	0.243126
$377$	−2.29150	−0.118018
$378$	−12.2715	−0.631180
$379$	12.4153	0.637730	0.318865	−	0.947800i	$-0.396698\pi$
$379$	12.4153	0.637730	0.318865	+	0.947800i	$0.396698\pi$
$380$	−8.42496	−0.432191
$381$	29.4141	1.50693
$382$	−1.18950	−0.0608602
$383$	−23.3716	−1.19423	−0.597116	−	0.802155i	$-0.703687\pi$
$383$	−23.3716	−1.19423	−0.597116	+	0.802155i	$0.703687\pi$
$384$	1.32324	0.0675264
$385$	−29.5520	−1.50611
$386$	−3.45647	−0.175929
$387$	−1.95938	−0.0996011
$388$	0.965753	0.0490287
$389$	5.61282	0.284581	0.142291	−	0.989825i	$-0.454553\pi$
$389$	5.61282	0.284581	0.142291	+	0.989825i	$0.454553\pi$
$390$	1.85641	0.0940028
$391$	−32.8758	−1.66260
$392$	2.23635	0.112953
$393$	−19.3244	−0.974788
$394$	20.6176	1.03870
$395$	−20.8302	−1.04808
$396$	7.14875	0.359238
$397$	36.8075	1.84731	0.923657	−	0.383219i	$-0.125185\pi$
$397$	36.8075	1.84731	0.923657	+	0.383219i	$0.125185\pi$
$398$	−5.29993	−0.265661
$399$	10.2853	0.514907
$400$	0.596577	0.0298289
$401$	14.0637	0.702306	0.351153	−	0.936318i	$-0.385790\pi$
$401$	14.0637	0.702306	0.351153	+	0.936318i	$0.385790\pi$
$402$	10.6420	0.530775
$403$	5.02834	0.250480
$404$	3.52582	0.175416
$405$	−8.73613	−0.434102
$406$	8.43368	0.418556
$407$	40.9203	2.02834
$408$	−9.83026	−0.486670
$409$	−11.0305	−0.545422	−0.272711	−	0.962096i	$-0.587920\pi$
$409$	−11.0305	−0.545422	−0.272711	+	0.962096i	$0.587920\pi$
$410$	30.0023	1.48171
$411$	−16.1472	−0.796480
$412$	−1.91457	−0.0943239
$413$	27.7023	1.36314
$414$	5.52744	0.271659
$415$	−27.3162	−1.34090
$416$	−0.593025	−0.0290754
$417$	−11.8584	−0.580710
$418$	−20.3827	−0.996953
$419$	21.0666	1.02917	0.514586	−	0.857439i	$-0.327945\pi$
$419$	21.0666	1.02917	0.514586	+	0.857439i	$0.327945\pi$
$420$	−6.83235	−0.333385
$421$	−29.5821	−1.44174	−0.720871	−	0.693069i	$-0.756258\pi$
$421$	−29.5821	−1.44174	−0.720871	+	0.693069i	$0.756258\pi$
$422$	3.18579	0.155082
$423$	5.88843	0.286305
$424$	−2.42851	−0.117939
$425$	−4.43193	−0.214980
$426$	−9.76007	−0.472877
$427$	−7.31407	−0.353953
$428$	−17.7035	−0.855732
$429$	4.49126	0.216840
$430$	−3.71114	−0.178967
$431$	−1.00000	−0.0481683
$431$	−1.00000	−0.0481683
$432$	5.62250	0.270512
$433$	−9.98168	−0.479689	−0.239844	−	0.970811i	$-0.577096\pi$
$433$	−9.98168	−0.479689	−0.239844	+	0.970811i	$0.577096\pi$
$434$	−18.5064	−0.888336
$435$	12.0962	0.579966
$436$	19.7191	0.944372
$437$	−15.7600	−0.753904
$438$	−20.1855	−0.964499
$439$	−27.2149	−1.29890	−0.649448	−	0.760406i	$-0.725000\pi$
$439$	−27.2149	−1.29890	−0.649448	+	0.760406i	$0.725000\pi$
$440$	13.5400	0.645492
$441$	2.79328	0.133013
$442$	4.40554	0.209550
$443$	24.2809	1.15362	0.576809	−	0.816879i	$-0.304298\pi$
$443$	24.2809	1.15362	0.576809	+	0.816879i	$0.304298\pi$
$444$	9.46067	0.448983
$445$	29.2895	1.38846
$446$	−19.7205	−0.933792
$447$	11.0650	0.523359
$448$	2.18258	0.103117
$449$	26.3995	1.24587	0.622935	−	0.782273i	$-0.285940\pi$
$449$	26.3995	1.24587	0.622935	+	0.782273i	$0.285940\pi$
$450$	0.745144	0.0351264
$451$	72.5854	3.41791
$452$	−3.94098	−0.185368
$453$	−8.29398	−0.389685
$454$	−17.9668	−0.843222
$455$	3.06199	0.143548
$456$	−4.71244	−0.220680
$457$	10.6999	0.500519	0.250259	−	0.968179i	$-0.419484\pi$
$457$	10.6999	0.500519	0.250259	+	0.968179i	$0.419484\pi$
$458$	−17.6093	−0.822830
$459$	−41.7691	−1.94961
$460$	10.4691	0.488126
$461$	−18.6750	−0.869782	−0.434891	−	0.900483i	$-0.643213\pi$
$461$	−18.6750	−0.869782	−0.434891	+	0.900483i	$0.643213\pi$
$462$	−16.5297	−0.769031
$463$	−14.4416	−0.671159	−0.335580	−	0.942012i	$-0.608932\pi$
$463$	−14.4416	−0.671159	−0.335580	+	0.942012i	$0.608932\pi$
$464$	−3.86409	−0.179386
$465$	−26.5431	−1.23091
$466$	−11.8245	−0.547760
$467$	−30.2858	−1.40146	−0.700730	−	0.713427i	$-0.747142\pi$
$467$	−30.2858	−1.40146	−0.700730	+	0.713427i	$0.747142\pi$
$468$	−0.740707	−0.0342392
$469$	17.5531	0.810529
$470$	11.1529	0.514444
$471$	−30.3310	−1.39758
$472$	−12.6925	−0.584218
$473$	−8.97846	−0.412830
$474$	−11.6512	−0.535159
$475$	−2.12458	−0.0974824
$476$	−16.2142	−0.743177
$477$	−3.03329	−0.138885
$478$	−12.3656	−0.565590
$479$	0.0429468	0.00196229	0.000981144	−	1.00000i	$-0.499688\pi$
$479$	0.0429468	0.00196229	0.000981144		1.00000i	$0.499688\pi$
$480$	3.13040	0.142883
$481$	−4.23990	−0.193323
$482$	13.7153	0.624716
$483$	−12.7808	−0.581548
$484$	21.7576	0.988983
$485$	2.28469	0.103742
$486$	11.9810	0.543469
$487$	−41.6315	−1.88650	−0.943251	−	0.332082i	$-0.892249\pi$
$487$	−41.6315	−1.88650	−0.943251	+	0.332082i	$0.892249\pi$
$488$	3.35111	0.151698
$489$	26.3960	1.19367
$490$	5.29056	0.239003
$491$	−1.62707	−0.0734288	−0.0367144	−	0.999326i	$-0.511689\pi$
$491$	−1.62707	−0.0734288	−0.0367144	+	0.999326i	$0.511689\pi$
$492$	16.7816	0.756571
$493$	28.7060	1.29286
$494$	2.11193	0.0950202
$495$	16.9119	0.760131
$496$	8.47915	0.380725
$497$	−16.0984	−0.722114
$498$	−15.2791	−0.684674
$499$	4.86937	0.217983	0.108992	−	0.994043i	$-0.465238\pi$
$499$	4.86937	0.217983	0.108992	+	0.994043i	$0.465238\pi$
$500$	−10.4172	−0.465872
$501$	6.04913	0.270255
$502$	2.70050	0.120529
$503$	20.7085	0.923345	0.461672	−	0.887050i	$-0.347250\pi$
$503$	20.7085	0.923345	0.461672	+	0.887050i	$0.347250\pi$
$504$	2.72611	0.121431
$505$	8.34105	0.371172
$506$	25.3283	1.12598
$507$	16.7368	0.743306
$508$	−22.2288	−0.986245
$509$	23.7710	1.05363	0.526815	−	0.849980i	$-0.323386\pi$
$509$	23.7710	1.05363	0.526815	+	0.849980i	$0.323386\pi$
$510$	−23.2555	−1.02977
$511$	−33.2943	−1.47285
$512$	−1.00000	−0.0441942
$513$	−20.0233	−0.884050
$514$	−10.3529	−0.456648
$515$	−4.52930	−0.199585
$516$	−2.07580	−0.0913819
$517$	26.9825	1.18669
$518$	15.6046	0.685627
$519$	11.6966	0.513425
$520$	−1.40292	−0.0615223
$521$	−3.43809	−0.150625	−0.0753126	−	0.997160i	$-0.523995\pi$
$521$	−3.43809	−0.150625	−0.0753126	+	0.997160i	$0.523995\pi$
$522$	−4.82638	−0.211245
$523$	0.672956	0.0294263	0.0147131	−	0.999892i	$-0.495316\pi$
$523$	0.672956	0.0294263	0.0147131	+	0.999892i	$0.495316\pi$
$524$	14.6039	0.637972
$525$	−1.72296	−0.0751962
$526$	−3.77652	−0.164664
$527$	−62.9910	−2.74393
$528$	7.57347	0.329593
$529$	−3.41606	−0.148525
$530$	−5.74515	−0.249553
$531$	−15.8533	−0.687975
$532$	−7.77278	−0.336993
$533$	−7.52083	−0.325763
$534$	16.3829	0.708956
$535$	−41.8814	−1.81069
$536$	−8.04239	−0.347378
$537$	6.73990	0.290848
$538$	4.24877	0.183178
$539$	12.7996	0.551318
$540$	13.3012	0.572392
$541$	15.9853	0.687261	0.343631	−	0.939105i	$-0.388343\pi$
$541$	15.9853	0.687261	0.343631	+	0.939105i	$0.388343\pi$
$542$	−24.6332	−1.05808
$543$	16.0570	0.689071
$544$	7.42893	0.318513
$545$	46.6496	1.99825
$546$	1.71270	0.0732968
$547$	−28.7279	−1.22831	−0.614157	−	0.789184i	$-0.710504\pi$
$547$	−28.7279	−1.22831	−0.614157	+	0.789184i	$0.710504\pi$
$548$	12.2027	0.521275
$549$	4.18565	0.178639
$550$	3.41447	0.145593
$551$	13.7611	0.586244
$552$	5.85584	0.249241
$553$	−19.2178	−0.817222
$554$	−23.1348	−0.982904
$555$	22.3812	0.950028
$556$	8.96165	0.380059
$557$	3.90284	0.165369	0.0826844	−	0.996576i	$-0.473651\pi$
$557$	3.90284	0.165369	0.0826844	+	0.996576i	$0.473651\pi$
$558$	10.5907	0.448342
$559$	0.930290	0.0393471
$560$	5.16334	0.218191
$561$	−56.2628	−2.37542
$562$	18.9945	0.801234
$563$	−4.35913	−0.183715	−0.0918576	−	0.995772i	$-0.529280\pi$
$563$	−4.35913	−0.183715	−0.0918576	+	0.995772i	$0.529280\pi$
$564$	6.23828	0.262679
$565$	−9.32322	−0.392231
$566$	1.54158	0.0647976
$567$	−8.05986	−0.338482
$568$	7.37588	0.309485
$569$	−20.2849	−0.850388	−0.425194	−	0.905102i	$-0.639794\pi$
$569$	−20.2849	−0.850388	−0.425194	+	0.905102i	$0.639794\pi$
$570$	−11.1483	−0.466949
$571$	22.4985	0.941535	0.470767	−	0.882257i	$-0.343977\pi$
$571$	22.4985	0.941535	0.470767	+	0.882257i	$0.343977\pi$
$572$	−3.39413	−0.141916
$573$	−1.57400	−0.0657547
$574$	27.6798	1.15533
$575$	2.64008	0.110099
$576$	−1.24903	−0.0520430
$577$	−16.7727	−0.698257	−0.349128	−	0.937075i	$-0.613522\pi$
$577$	−16.7727	−0.698257	−0.349128	+	0.937075i	$0.613522\pi$
$578$	−38.1889	−1.58845
$579$	−4.57374	−0.190078
$580$	−9.14131	−0.379572
$581$	−25.2017	−1.04554
$582$	1.27792	0.0529717
$583$	−13.8994	−0.575655
$584$	15.2546	0.631239
$585$	−1.75230	−0.0724486
$586$	12.0162	0.496383
$587$	14.1911	0.585728	0.292864	−	0.956154i	$-0.405392\pi$
$587$	14.1911	0.585728	0.292864	+	0.956154i	$0.405392\pi$
$588$	2.95923	0.122037
$589$	−30.1967	−1.24423
$590$	−30.0267	−1.23618
$591$	27.2821	1.12224
$592$	−7.14962	−0.293847
$593$	13.6825	0.561874	0.280937	−	0.959726i	$-0.409355\pi$
$593$	13.6825	0.561874	0.280937	+	0.959726i	$0.409355\pi$
$594$	32.1799	1.32036
$595$	−38.3581	−1.57253
$596$	−8.36208	−0.342524
$597$	−7.01308	−0.287026
$598$	−2.62436	−0.107318
$599$	17.5662	0.717734	0.358867	−	0.933389i	$-0.383163\pi$
$599$	17.5662	0.717734	0.358867	+	0.933389i	$0.383163\pi$
$600$	0.789415	0.0322277
$601$	−33.7469	−1.37657	−0.688283	−	0.725442i	$-0.741635\pi$
$601$	−33.7469	−1.37657	−0.688283	+	0.725442i	$0.741635\pi$
$602$	−3.42386	−0.139546
$603$	−10.0452	−0.409072
$604$	6.26793	0.255039
$605$	51.4722	2.09264
$606$	4.66550	0.189523
$607$	30.9610	1.25667	0.628334	−	0.777944i	$-0.283737\pi$
$607$	30.9610	1.25667	0.628334	+	0.777944i	$0.283737\pi$
$608$	3.56128	0.144429
$609$	11.1598	0.452218
$610$	7.92776	0.320986
$611$	−2.79575	−0.113104
$612$	9.27897	0.375080
$613$	25.8630	1.04460	0.522298	−	0.852763i	$-0.325075\pi$
$613$	25.8630	1.04460	0.522298	+	0.852763i	$0.325075\pi$
$614$	−21.2277	−0.856679
$615$	39.7003	1.60087
$616$	12.4918	0.503310
$617$	0.765560	0.0308203	0.0154101	−	0.999881i	$-0.495095\pi$
$617$	0.765560	0.0308203	0.0154101	+	0.999881i	$0.495095\pi$
$618$	−2.53343	−0.101910
$619$	36.1339	1.45234	0.726172	−	0.687513i	$-0.241298\pi$
$619$	36.1339	1.45234	0.726172	+	0.687513i	$0.241298\pi$
$620$	20.0592	0.805597
$621$	24.8816	0.998466
$622$	−2.44107	−0.0978778
$623$	27.0222	1.08262
$624$	−0.784714	−0.0314137
$625$	−27.6270	−1.10508
$626$	25.8091	1.03154
$627$	−26.9713	−1.07713
$628$	22.9217	0.914676
$629$	53.1140	2.11779
$630$	6.44919	0.256942
$631$	−15.0437	−0.598880	−0.299440	−	0.954115i	$-0.596800\pi$
$631$	−15.0437	−0.598880	−0.299440	+	0.954115i	$0.596800\pi$
$632$	8.80507	0.350247
$633$	4.21557	0.167554
$634$	9.80038	0.389223
$635$	−52.5869	−2.08685
$636$	−3.21351	−0.127424
$637$	−1.32621	−0.0525464
$638$	−22.1158	−0.875574
$639$	9.21272	0.364450
$640$	−2.36571	−0.0935128
$641$	21.3475	0.843177	0.421588	−	0.906787i	$-0.361473\pi$
$641$	21.3475	0.843177	0.421588	+	0.906787i	$0.361473\pi$
$642$	−23.4260	−0.924551
$643$	−7.76561	−0.306246	−0.153123	−	0.988207i	$-0.548933\pi$
$643$	−7.76561	−0.306246	−0.153123	+	0.988207i	$0.548933\pi$
$644$	9.65873	0.380607
$645$	−4.91073	−0.193360
$646$	−26.4565	−1.04092
$647$	−35.7626	−1.40597	−0.702987	−	0.711202i	$-0.748151\pi$
$647$	−35.7626	−1.40597	−0.702987	+	0.711202i	$0.748151\pi$
$648$	3.69282	0.145068
$649$	−72.6444	−2.85154
$650$	−0.353785	−0.0138766
$651$	−24.4884	−0.959777
$652$	−19.9480	−0.781224
$653$	3.14562	0.123098	0.0615489	−	0.998104i	$-0.480396\pi$
$653$	3.14562	0.123098	0.0615489	+	0.998104i	$0.480396\pi$
$654$	26.0931	1.02032
$655$	34.5485	1.34992
$656$	−12.6822	−0.495155
$657$	19.0535	0.743346
$658$	10.2895	0.401128
$659$	44.1737	1.72076	0.860381	−	0.509652i	$-0.170226\pi$
$659$	44.1737	1.72076	0.860381	+	0.509652i	$0.170226\pi$
$660$	17.9166	0.697404
$661$	−37.4340	−1.45602	−0.728008	−	0.685569i	$-0.759554\pi$
$661$	−37.4340	−1.45602	−0.728008	+	0.685569i	$0.759554\pi$
$662$	−8.07682	−0.313915
$663$	5.82959	0.226402
$664$	11.5468	0.448101
$665$	−18.3881	−0.713061
$666$	−8.93011	−0.346035
$667$	−17.1000	−0.662116
$668$	−4.57145	−0.176875
$669$	−26.0949	−1.00889
$670$	−19.0259	−0.735036
$671$	19.1799	0.740430
$672$	2.88808	0.111410
$673$	−26.4710	−1.02038	−0.510192	−	0.860061i	$-0.670426\pi$
$673$	−26.4710	−1.02038	−0.510192	+	0.860061i	$0.670426\pi$
$674$	−18.9338	−0.729304
$675$	3.35425	0.129105
$676$	−12.6483	−0.486474
$677$	−9.66859	−0.371594	−0.185797	−	0.982588i	$-0.559487\pi$
$677$	−9.66859	−0.371594	−0.185797	+	0.982588i	$0.559487\pi$
$678$	−5.21487	−0.200276
$679$	2.10783	0.0808911
$680$	17.5747	0.673958
$681$	−23.7744	−0.911036
$682$	48.5298	1.85830
$683$	−12.9638	−0.496046	−0.248023	−	0.968754i	$-0.579781\pi$
$683$	−12.9638	−0.496046	−0.248023	+	0.968754i	$0.579781\pi$
$684$	4.44816	0.170080
$685$	28.8681	1.10299
$686$	20.1591	0.769677
$687$	−23.3014	−0.889004
$688$	1.56872	0.0598069
$689$	1.44017	0.0548660
$690$	13.8532	0.527382
$691$	−36.6557	−1.39445	−0.697224	−	0.716853i	$-0.745582\pi$
$691$	−36.6557	−1.39445	−0.697224	+	0.716853i	$0.745582\pi$
$692$	−8.83938	−0.336023
$693$	15.6027	0.592698
$694$	−5.40719	−0.205254
$695$	21.2007	0.804187
$696$	−5.11312	−0.193812
$697$	94.2148	3.56864
$698$	−15.9428	−0.603442
$699$	−15.6467	−0.591812
$700$	1.30208	0.0492139
$701$	12.1479	0.458819	0.229409	−	0.973330i	$-0.426321\pi$
$701$	12.1479	0.458819	0.229409	+	0.973330i	$0.426321\pi$
$702$	−3.33428	−0.125844
$703$	25.4618	0.960311
$704$	−5.72343	−0.215710
$705$	14.7579	0.555816
$706$	29.7808	1.12082
$707$	7.69537	0.289414
$708$	−16.7952	−0.631202
$709$	−23.8247	−0.894757	−0.447378	−	0.894345i	$-0.647642\pi$
$709$	−23.8247	−0.894757	−0.447378	+	0.894345i	$0.647642\pi$
$710$	17.4492	0.654856
$711$	10.9978	0.412451
$712$	−12.3809	−0.463993
$713$	37.5234	1.40526
$714$	−21.4553	−0.802945
$715$	−8.02953	−0.300287
$716$	−5.09347	−0.190352
$717$	−16.3627	−0.611076
$718$	6.81817	0.254452
$719$	51.7817	1.93113	0.965566	−	0.260157i	$-0.0837742\pi$
$719$	51.7817	1.93113	0.965566	+	0.260157i	$0.0837742\pi$
$720$	−2.95485	−0.110121
$721$	−4.17869	−0.155623
$722$	6.31726	0.235104
$723$	18.1487	0.674957
$724$	−12.1346	−0.450978
$725$	−2.30523	−0.0856140
$726$	28.7906	1.06852
$727$	−33.7167	−1.25048	−0.625242	−	0.780431i	$-0.715000\pi$
$727$	−33.7167	−1.25048	−0.625242	+	0.780431i	$0.715000\pi$
$728$	−1.29432	−0.0479708
$729$	26.9322	0.997489
$730$	36.0879	1.33567
$731$	−11.6539	−0.431036
$732$	4.43433	0.163898
$733$	8.77700	0.324186	0.162093	−	0.986775i	$-0.448176\pi$
$733$	8.77700	0.324186	0.162093	+	0.986775i	$0.448176\pi$
$734$	7.69745	0.284118
$735$	7.00068	0.258224
$736$	−4.42537	−0.163121
$737$	−46.0300	−1.69554
$738$	−15.8404	−0.583095
$739$	−26.4394	−0.972590	−0.486295	−	0.873795i	$-0.661652\pi$
$739$	−26.4394	−0.972590	−0.486295	+	0.873795i	$0.661652\pi$
$740$	−16.9139	−0.621768
$741$	2.79459	0.102662
$742$	−5.30042	−0.194584
$743$	−13.4713	−0.494213	−0.247106	−	0.968988i	$-0.579480\pi$
$743$	−13.4713	−0.494213	−0.247106	+	0.968988i	$0.579480\pi$
$744$	11.2200	0.411344
$745$	−19.7822	−0.724765
$746$	−20.9846	−0.768302
$747$	14.4223	0.527683
$748$	42.5189	1.55465
$749$	−38.6393	−1.41185
$750$	−13.7845	−0.503338
$751$	16.7382	0.610786	0.305393	−	0.952226i	$-0.401212\pi$
$751$	16.7382	0.610786	0.305393	+	0.952226i	$0.401212\pi$
$752$	−4.71439	−0.171916
$753$	3.57341	0.130222
$754$	2.29150	0.0834515
$755$	14.8281	0.539650
$756$	12.2715	0.446311
$757$	−32.9806	−1.19870	−0.599349	−	0.800488i	$-0.704574\pi$
$757$	−32.9806	−1.19870	−0.599349	+	0.800488i	$0.704574\pi$
$758$	−12.4153	−0.450943
$759$	33.5155	1.21653
$760$	8.42496	0.305605
$761$	−8.96721	−0.325061	−0.162531	−	0.986704i	$-0.551966\pi$
$761$	−8.96721	−0.325061	−0.162531	+	0.986704i	$0.551966\pi$
$762$	−29.4141	−1.06556
$763$	43.0384	1.55810
$764$	1.18950	0.0430347
$765$	21.9513	0.793653
$766$	23.3716	0.844450
$767$	7.52695	0.271782
$768$	−1.32324	−0.0477484
$769$	29.6941	1.07080	0.535399	−	0.844600i	$-0.320161\pi$
$769$	29.6941	1.07080	0.535399	+	0.844600i	$0.320161\pi$
$770$	29.5520	1.06498
$771$	−13.6994	−0.493372
$772$	3.45647	0.124401
$773$	−35.8550	−1.28962	−0.644808	−	0.764345i	$-0.723063\pi$
$773$	−35.8550	−1.28962	−0.644808	+	0.764345i	$0.723063\pi$
$774$	1.95938	0.0704286
$775$	5.05846	0.181705
$776$	−0.965753	−0.0346685
$777$	20.6486	0.740766
$778$	−5.61282	−0.201229
$779$	45.1648	1.61820
$780$	−1.85641	−0.0664700
$781$	42.2153	1.51058
$782$	32.8758	1.17564
$783$	−21.7258	−0.776418
$784$	−2.23635	−0.0798697
$785$	54.2261	1.93541
$786$	19.3244	0.689279
$787$	−26.6006	−0.948209	−0.474105	−	0.880469i	$-0.657228\pi$
$787$	−26.6006	−0.948209	−0.474105	+	0.880469i	$0.657228\pi$
$788$	−20.6176	−0.734473
$789$	−4.99724	−0.177907
$790$	20.8302	0.741107
$791$	−8.60150	−0.305834
$792$	−7.14875	−0.254020
$793$	−1.98729	−0.0705708
$794$	−36.8075	−1.30625
$795$	−7.60222	−0.269623
$796$	5.29993	0.187851
$797$	−8.86760	−0.314106	−0.157053	−	0.987590i	$-0.550199\pi$
$797$	−8.86760	−0.314106	−0.157053	+	0.987590i	$0.550199\pi$
$798$	−10.2853	−0.364094
$799$	35.0229	1.23902
$800$	−0.596577	−0.0210922
$801$	−15.4641	−0.546398
$802$	−14.0637	−0.496605
$803$	87.3084	3.08105
$804$	−10.6420	−0.375315
$805$	22.8497	0.805347
$806$	−5.02834	−0.177116
$807$	5.62215	0.197909
$808$	−3.52582	−0.124038
$809$	45.4475	1.59785	0.798924	−	0.601432i	$-0.205403\pi$
$809$	45.4475	1.59785	0.798924	+	0.601432i	$0.205403\pi$
$810$	8.73613	0.306956
$811$	5.95170	0.208992	0.104496	−	0.994525i	$-0.466677\pi$
$811$	5.95170	0.208992	0.104496	+	0.994525i	$0.466677\pi$
$812$	−8.43368	−0.295964
$813$	−32.5956	−1.14318
$814$	−40.9203	−1.43426
$815$	−47.1911	−1.65303
$816$	9.83026	0.344128
$817$	−5.58666	−0.195453
$818$	11.0305	0.385672
$819$	−1.61665	−0.0564904
$820$	−30.0023	−1.04773
$821$	8.02465	0.280062	0.140031	−	0.990147i	$-0.455280\pi$
$821$	8.02465	0.280062	0.140031	+	0.990147i	$0.455280\pi$
$822$	16.1472	0.563197
$823$	32.4142	1.12989	0.564944	−	0.825129i	$-0.308898\pi$
$823$	32.4142	1.12989	0.564944	+	0.825129i	$0.308898\pi$
$824$	1.91457	0.0666970
$825$	4.51816	0.157302
$826$	−27.7023	−0.963887
$827$	21.4478	0.745815	0.372907	−	0.927869i	$-0.378361\pi$
$827$	21.4478	0.745815	0.372907	+	0.927869i	$0.378361\pi$
$828$	−5.52744	−0.192092
$829$	−5.93889	−0.206266	−0.103133	−	0.994668i	$-0.532887\pi$
$829$	−5.93889	−0.206266	−0.103133	+	0.994668i	$0.532887\pi$
$830$	27.3162	0.948161
$831$	−30.6129	−1.06195
$832$	0.593025	0.0205594
$833$	16.6137	0.575630
$834$	11.8584	0.410624
$835$	−10.8147	−0.374259
$836$	20.3827	0.704952
$837$	47.6740	1.64785
$838$	−21.0666	−0.727735
$839$	7.56223	0.261077	0.130539	−	0.991443i	$-0.458329\pi$
$839$	7.56223	0.261077	0.130539	+	0.991443i	$0.458329\pi$
$840$	6.83235	0.235738
$841$	−14.0688	−0.485131
$842$	29.5821	1.01947
$843$	25.1343	0.865671
$844$	−3.18579	−0.109660
$845$	−29.9222	−1.02936
$846$	−5.88843	−0.202448
$847$	47.4877	1.63170
$848$	2.42851	0.0833954
$849$	2.03989	0.0700087
$850$	4.43193	0.152014
$851$	−31.6397	−1.08460
$852$	9.76007	0.334375
$853$	−5.84755	−0.200216	−0.100108	−	0.994977i	$-0.531919\pi$
$853$	−5.84755	−0.200216	−0.100108	+	0.994977i	$0.531919\pi$
$854$	7.31407	0.250282
$855$	10.5231	0.359881
$856$	17.7035	0.605094
$857$	0.0905027	0.00309151	0.00154576	−	0.999999i	$-0.499508\pi$
$857$	0.0905027	0.00309151	0.00154576	+	0.999999i	$0.499508\pi$
$858$	−4.49126	−0.153329
$859$	48.3590	1.64999	0.824994	−	0.565142i	$-0.191179\pi$
$859$	48.3590	1.64999	0.824994	+	0.565142i	$0.191179\pi$
$860$	3.71114	0.126549
$861$	36.6271	1.24825
$862$	1.00000	0.0340601
$863$	−50.2436	−1.71031	−0.855155	−	0.518372i	$-0.826538\pi$
$863$	−50.2436	−1.71031	−0.855155	+	0.518372i	$0.826538\pi$
$864$	−5.62250	−0.191281
$865$	−20.9114	−0.711009
$866$	9.98168	0.339191
$867$	−50.5332	−1.71620
$868$	18.5064	0.628148
$869$	50.3952	1.70954
$870$	−12.0962	−0.410098
$871$	4.76933	0.161603
$872$	−19.7191	−0.667772
$873$	−1.20626	−0.0408256
$874$	15.7600	0.533090
$875$	−22.7364	−0.768630
$876$	20.1855	0.682004
$877$	−39.2087	−1.32398	−0.661992	−	0.749511i	$-0.730289\pi$
$877$	−39.2087	−1.32398	−0.661992	+	0.749511i	$0.730289\pi$
$878$	27.2149	0.918458
$879$	15.9003	0.536303
$880$	−13.5400	−0.456432
$881$	−53.7477	−1.81081	−0.905403	−	0.424554i	$-0.860431\pi$
$881$	−53.7477	−1.81081	−0.905403	+	0.424554i	$0.860431\pi$
$882$	−2.79328	−0.0940545
$883$	2.65639	0.0893947	0.0446973	−	0.999001i	$-0.485768\pi$
$883$	2.65639	0.0893947	0.0446973	+	0.999001i	$0.485768\pi$
$884$	−4.40554	−0.148174
$885$	−39.7326	−1.33560
$886$	−24.2809	−0.815732
$887$	−24.4649	−0.821451	−0.410725	−	0.911759i	$-0.634725\pi$
$887$	−24.4649	−0.821451	−0.410725	+	0.911759i	$0.634725\pi$
$888$	−9.46067	−0.317479
$889$	−48.5162	−1.62718
$890$	−29.2895	−0.981787
$891$	21.1356	0.708068
$892$	19.7205	0.660290
$893$	16.7893	0.561832
$894$	−11.0650	−0.370071
$895$	−12.0497	−0.402776
$896$	−2.18258	−0.0729148
$897$	−3.47265	−0.115949
$898$	−26.3995	−0.880963
$899$	−32.7642	−1.09275
$900$	−0.745144	−0.0248381
$901$	−18.0412	−0.601041
$902$	−72.5854	−2.41683
$903$	−4.53059	−0.150769
$904$	3.94098	0.131075
$905$	−28.7069	−0.954249
$906$	8.29398	0.275549
$907$	5.42034	0.179979	0.0899897	−	0.995943i	$-0.471317\pi$
$907$	5.42034	0.179979	0.0899897	+	0.995943i	$0.471317\pi$
$908$	17.9668	0.596248
$909$	−4.40386	−0.146067
$910$	−3.06199	−0.101504
$911$	−13.5809	−0.449955	−0.224977	−	0.974364i	$-0.572231\pi$
$911$	−13.5809	−0.449955	−0.224977	+	0.974364i	$0.572231\pi$
$912$	4.71244	0.156044
$913$	66.0870	2.18716
$914$	−10.6999	−0.353920
$915$	10.4903	0.346800
$916$	17.6093	0.581829
$917$	31.8741	1.05257
$918$	41.7691	1.37859
$919$	43.6751	1.44071	0.720354	−	0.693607i	$-0.243979\pi$
$919$	43.6751	1.44071	0.720354	+	0.693607i	$0.243979\pi$
$920$	−10.4691	−0.345157
$921$	−28.0893	−0.925575
$922$	18.6750	0.615029
$923$	−4.37408	−0.143975
$924$	16.5297	0.543787
$925$	−4.26530	−0.140242
$926$	14.4416	0.474581
$927$	2.39136	0.0785424
$928$	3.86409	0.126845
$929$	−37.0724	−1.21631	−0.608154	−	0.793819i	$-0.708090\pi$
$929$	−37.0724	−1.21631	−0.608154	+	0.793819i	$0.708090\pi$
$930$	26.5431	0.870384
$931$	7.96428	0.261019
$932$	11.8245	0.387325
$933$	−3.23012	−0.105749
$934$	30.2858	0.990981
$935$	100.587	3.28956
$936$	0.740707	0.0242108
$937$	−49.9019	−1.63022	−0.815111	−	0.579304i	$-0.803324\pi$
$937$	−49.9019	−1.63022	−0.815111	+	0.579304i	$0.803324\pi$
$938$	−17.5531	−0.573130
$939$	34.1517	1.11450
$940$	−11.1529	−0.363767
$941$	−40.2104	−1.31082	−0.655410	−	0.755273i	$-0.727504\pi$
$941$	−40.2104	−1.31082	−0.655410	+	0.755273i	$0.727504\pi$
$942$	30.3310	0.988236
$943$	−56.1233	−1.82763
$944$	12.6925	0.413105
$945$	29.0309	0.944375
$946$	8.97846	0.291915
$947$	−16.0286	−0.520861	−0.260430	−	0.965493i	$-0.583864\pi$
$947$	−16.0286	−0.520861	−0.260430	+	0.965493i	$0.583864\pi$
$948$	11.6512	0.378415
$949$	−9.04634	−0.293656
$950$	2.12458	0.0689305
$951$	12.9683	0.420525
$952$	16.2142	0.525506
$953$	15.7956	0.511671	0.255836	−	0.966720i	$-0.417649\pi$
$953$	15.7956	0.511671	0.255836	+	0.966720i	$0.417649\pi$
$954$	3.03329	0.0982064
$955$	2.81401	0.0910593
$956$	12.3656	0.399933
$957$	−29.2646	−0.945990
$958$	−0.0429468	−0.00138755
$959$	26.6334	0.860038
$960$	−3.13040	−0.101033
$961$	40.8959	1.31922
$962$	4.23990	0.136700
$963$	22.1123	0.712558
$964$	−13.7153	−0.441741
$965$	8.17699	0.263227
$966$	12.7808	0.411216
$967$	−52.7366	−1.69590	−0.847948	−	0.530080i	$-0.822162\pi$
$967$	−52.7366	−1.69590	−0.847948	+	0.530080i	$0.822162\pi$
$968$	−21.7576	−0.699316
$969$	−35.0083	−1.12463
$970$	−2.28469	−0.0733570
$971$	39.5003	1.26762	0.633812	−	0.773487i	$-0.281489\pi$
$971$	39.5003	1.26762	0.633812	+	0.773487i	$0.281489\pi$
$972$	−11.9810	−0.384291
$973$	19.5595	0.627049
$974$	41.6315	1.33396
$975$	−0.468143	−0.0149926
$976$	−3.35111	−0.107267
$977$	−22.9090	−0.732923	−0.366462	−	0.930433i	$-0.619431\pi$
$977$	−22.9090	−0.732923	−0.366462	+	0.930433i	$0.619431\pi$
$978$	−26.3960	−0.844051
$979$	−70.8610	−2.26473
$980$	−5.29056	−0.169001
$981$	−24.6298	−0.786368
$982$	1.62707	0.0519220
$983$	12.2582	0.390976	0.195488	−	0.980706i	$-0.437371\pi$
$983$	12.2582	0.390976	0.195488	+	0.980706i	$0.437371\pi$
$984$	−16.7816	−0.534976
$985$	−48.7753	−1.55411
$986$	−28.7060	−0.914187
$987$	13.6155	0.433387
$988$	−2.11193	−0.0671894
$989$	6.94218	0.220748
$990$	−16.9119	−0.537494
$991$	−39.3982	−1.25152	−0.625762	−	0.780014i	$-0.715212\pi$
$991$	−39.3982	−1.25152	−0.625762	+	0.780014i	$0.715212\pi$
$992$	−8.47915	−0.269213
$993$	−10.6876	−0.339160
$994$	16.0984	0.510612
$995$	12.5381	0.397484
$996$	15.2791	0.484138
$997$	−10.6266	−0.336549	−0.168275	−	0.985740i	$-0.553820\pi$
$997$	−10.6266	−0.336549	−0.168275	+	0.985740i	$0.553820\pi$
$998$	−4.86937	−0.154137
$999$	−40.1987	−1.27183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	862.2.a.i.1.3	✓	6
3.2	odd	2		7758.2.a.t.1.1		6
4.3	odd	2		6896.2.a.q.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
862.2.a.i.1.3	✓	6	1.1	even	1	trivial
6896.2.a.q.1.4		6	4.3	odd	2
7758.2.a.t.1.1		6	3.2	odd	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:	\( N \)	\(=\)	\( 862 = 2 \cdot 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	862.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.32324 q^{3} +1.00000 q^{4} +2.36571 q^{5} +1.32324 q^{6} +2.18258 q^{7} -1.00000 q^{8} -1.24903 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.32324 q^{3} +1.00000 q^{4} +2.36571 q^{5} +1.32324 q^{6} +2.18258 q^{7} -1.00000 q^{8} -1.24903 q^{9} -2.36571 q^{10} -5.72343 q^{11} -1.32324 q^{12} +0.593025 q^{13} -2.18258 q^{14} -3.13040 q^{15} +1.00000 q^{16} -7.42893 q^{17} +1.24903 q^{18} -3.56128 q^{19} +2.36571 q^{20} -2.88808 q^{21} +5.72343 q^{22} +4.42537 q^{23} +1.32324 q^{24} +0.596577 q^{25} -0.593025 q^{26} +5.62250 q^{27} +2.18258 q^{28} -3.86409 q^{29} +3.13040 q^{30} +8.47915 q^{31} -1.00000 q^{32} +7.57347 q^{33} +7.42893 q^{34} +5.16334 q^{35} -1.24903 q^{36} -7.14962 q^{37} +3.56128 q^{38} -0.784714 q^{39} -2.36571 q^{40} -12.6822 q^{41} +2.88808 q^{42} +1.56872 q^{43} -5.72343 q^{44} -2.95485 q^{45} -4.42537 q^{46} -4.71439 q^{47} -1.32324 q^{48} -2.23635 q^{49} -0.596577 q^{50} +9.83026 q^{51} +0.593025 q^{52} +2.42851 q^{53} -5.62250 q^{54} -13.5400 q^{55} -2.18258 q^{56} +4.71244 q^{57} +3.86409 q^{58} +12.6925 q^{59} -3.13040 q^{60} -3.35111 q^{61} -8.47915 q^{62} -2.72611 q^{63} +1.00000 q^{64} +1.40292 q^{65} -7.57347 q^{66} +8.04239 q^{67} -7.42893 q^{68} -5.85584 q^{69} -5.16334 q^{70} -7.37588 q^{71} +1.24903 q^{72} -15.2546 q^{73} +7.14962 q^{74} -0.789415 q^{75} -3.56128 q^{76} -12.4918 q^{77} +0.784714 q^{78} -8.80507 q^{79} +2.36571 q^{80} -3.69282 q^{81} +12.6822 q^{82} -11.5468 q^{83} -2.88808 q^{84} -17.5747 q^{85} -1.56872 q^{86} +5.11312 q^{87} +5.72343 q^{88} +12.3809 q^{89} +2.95485 q^{90} +1.29432 q^{91} +4.42537 q^{92} -11.2200 q^{93} +4.71439 q^{94} -8.42496 q^{95} +1.32324 q^{96} +0.965753 q^{97} +2.23635 q^{98} +7.14875 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 - 6 * q^8 - 2 * q^9	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} - 6 q^{8} - 2 q^{9} + 2 q^{10} - 6 q^{11} - 2 q^{12} - 9 q^{13} - 3 q^{14} - 3 q^{15} + 6 q^{16} - 17 q^{17} + 2 q^{18} + q^{19} - 2 q^{20} - 8 q^{21} + 6 q^{22} - 20 q^{23} + 2 q^{24} + 10 q^{25} + 9 q^{26} + 7 q^{27} + 3 q^{28} + q^{29} + 3 q^{30} + 8 q^{31} - 6 q^{32} - 5 q^{33} + 17 q^{34} - 14 q^{35} - 2 q^{36} - 3 q^{37} - q^{38} - 13 q^{39} + 2 q^{40} - 19 q^{41} + 8 q^{42} - 21 q^{43} - 6 q^{44} - 16 q^{45} + 20 q^{46} - 10 q^{47} - 2 q^{48} - 7 q^{49} - 10 q^{50} - 2 q^{51} - 9 q^{52} - q^{53} - 7 q^{54} + 3 q^{55} - 3 q^{56} - 11 q^{57} - q^{58} + 11 q^{59} - 3 q^{60} - 2 q^{61} - 8 q^{62} + 6 q^{64} - 30 q^{65} + 5 q^{66} + 4 q^{67} - 17 q^{68} + 16 q^{69} + 14 q^{70} - 4 q^{71} + 2 q^{72} - 39 q^{73} + 3 q^{74} - 21 q^{75} + q^{76} - 33 q^{77} + 13 q^{78} - 16 q^{79} - 2 q^{80} - 22 q^{81} + 19 q^{82} - 20 q^{83} - 8 q^{84} - 16 q^{85} + 21 q^{86} + q^{87} + 6 q^{88} - 17 q^{89} + 16 q^{90} + 20 q^{91} - 20 q^{92} - 5 q^{93} + 10 q^{94} - 16 q^{95} + 2 q^{96} - 29 q^{97} + 7 q^{98} - 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 - 6 * q^8 - 2 * q^9 + 2 * q^10 - 6 * q^11 - 2 * q^12 - 9 * q^13 - 3 * q^14 - 3 * q^15 + 6 * q^16 - 17 * q^17 + 2 * q^18 + q^19 - 2 * q^20 - 8 * q^21 + 6 * q^22 - 20 * q^23 + 2 * q^24 + 10 * q^25 + 9 * q^26 + 7 * q^27 + 3 * q^28 + q^29 + 3 * q^30 + 8 * q^31 - 6 * q^32 - 5 * q^33 + 17 * q^34 - 14 * q^35 - 2 * q^36 - 3 * q^37 - q^38 - 13 * q^39 + 2 * q^40 - 19 * q^41 + 8 * q^42 - 21 * q^43 - 6 * q^44 - 16 * q^45 + 20 * q^46 - 10 * q^47 - 2 * q^48 - 7 * q^49 - 10 * q^50 - 2 * q^51 - 9 * q^52 - q^53 - 7 * q^54 + 3 * q^55 - 3 * q^56 - 11 * q^57 - q^58 + 11 * q^59 - 3 * q^60 - 2 * q^61 - 8 * q^62 + 6 * q^64 - 30 * q^65 + 5 * q^66 + 4 * q^67 - 17 * q^68 + 16 * q^69 + 14 * q^70 - 4 * q^71 + 2 * q^72 - 39 * q^73 + 3 * q^74 - 21 * q^75 + q^76 - 33 * q^77 + 13 * q^78 - 16 * q^79 - 2 * q^80 - 22 * q^81 + 19 * q^82 - 20 * q^83 - 8 * q^84 - 16 * q^85 + 21 * q^86 + q^87 + 6 * q^88 - 17 * q^89 + 16 * q^90 + 20 * q^91 - 20 * q^92 - 5 * q^93 + 10 * q^94 - 16 * q^95 + 2 * q^96 - 29 * q^97 + 7 * q^98 - 4 * q^99

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