LMFDB - Embedded newform 8041.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.77655 q^{2} -0.377305 q^{3} +5.70922 q^{4} -1.07900 q^{5} +1.04760 q^{6} -3.82470 q^{7} -10.2988 q^{8} -2.85764 q^{9} +O(q^{10})$	\(q-2.77655 q^{2} -0.377305 q^{3} +5.70922 q^{4} -1.07900 q^{5} +1.04760 q^{6} -3.82470 q^{7} -10.2988 q^{8} -2.85764 q^{9} +2.99589 q^{10} +1.00000 q^{11} -2.15412 q^{12} +6.34070 q^{13} +10.6195 q^{14} +0.407111 q^{15} +17.1768 q^{16} +1.00000 q^{17} +7.93438 q^{18} +2.71507 q^{19} -6.16025 q^{20} +1.44308 q^{21} -2.77655 q^{22} +3.31642 q^{23} +3.88580 q^{24} -3.83576 q^{25} -17.6053 q^{26} +2.21011 q^{27} -21.8360 q^{28} +8.14728 q^{29} -1.13036 q^{30} -0.300360 q^{31} -27.0945 q^{32} -0.377305 q^{33} -2.77655 q^{34} +4.12685 q^{35} -16.3149 q^{36} -5.07135 q^{37} -7.53852 q^{38} -2.39238 q^{39} +11.1124 q^{40} +1.83975 q^{41} -4.00677 q^{42} +1.00000 q^{43} +5.70922 q^{44} +3.08339 q^{45} -9.20821 q^{46} +8.66756 q^{47} -6.48087 q^{48} +7.62830 q^{49} +10.6502 q^{50} -0.377305 q^{51} +36.2005 q^{52} +6.74749 q^{53} -6.13649 q^{54} -1.07900 q^{55} +39.3899 q^{56} -1.02441 q^{57} -22.6213 q^{58} +11.7431 q^{59} +2.32429 q^{60} -9.86038 q^{61} +0.833964 q^{62} +10.9296 q^{63} +40.8755 q^{64} -6.84162 q^{65} +1.04760 q^{66} -2.48384 q^{67} +5.70922 q^{68} -1.25130 q^{69} -11.4584 q^{70} +11.9966 q^{71} +29.4304 q^{72} -0.972938 q^{73} +14.0809 q^{74} +1.44725 q^{75} +15.5009 q^{76} -3.82470 q^{77} +6.64255 q^{78} -8.47241 q^{79} -18.5337 q^{80} +7.73904 q^{81} -5.10816 q^{82} -8.08161 q^{83} +8.23884 q^{84} -1.07900 q^{85} -2.77655 q^{86} -3.07401 q^{87} -10.2988 q^{88} +1.74033 q^{89} -8.56119 q^{90} -24.2513 q^{91} +18.9342 q^{92} +0.113327 q^{93} -24.0659 q^{94} -2.92956 q^{95} +10.2229 q^{96} -5.49821 q^{97} -21.1803 q^{98} -2.85764 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9}+O(q^{10}) $ 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9} + q^{10} + 82 q^{11} + 3 q^{12} + 26 q^{13} + 17 q^{14} + 66 q^{15} + 122 q^{16} + 82 q^{17} + 18 q^{18} + 12 q^{19} + 9 q^{20} + 22 q^{21} + 8 q^{22} + 50 q^{23} + 15 q^{24} + 117 q^{25} + 36 q^{26} + 30 q^{27} + 11 q^{28} + 33 q^{29} - 26 q^{30} + 40 q^{31} + 58 q^{32} + 6 q^{33} + 8 q^{34} + 16 q^{35} + 160 q^{36} + 31 q^{37} + 18 q^{38} + 41 q^{39} - 29 q^{40} + 42 q^{41} - 51 q^{42} + 82 q^{43} + 98 q^{44} - 2 q^{45} - 19 q^{46} + 84 q^{47} - 46 q^{48} + 136 q^{49} + 59 q^{50} + 6 q^{51} + 45 q^{52} + 83 q^{53} + 24 q^{54} + 11 q^{55} + 21 q^{56} + 23 q^{57} + 14 q^{58} + 96 q^{59} + 184 q^{60} - 6 q^{61} - 23 q^{62} + 8 q^{63} + 148 q^{64} + 5 q^{65} + 10 q^{66} + 78 q^{67} + 98 q^{68} + 61 q^{69} - 3 q^{70} + 155 q^{71} + 50 q^{72} - 23 q^{73} + 10 q^{74} - 19 q^{75} + 44 q^{76} + 8 q^{77} - 27 q^{78} + 31 q^{79} + 19 q^{80} + 150 q^{81} - 12 q^{82} + 54 q^{83} + 8 q^{84} + 11 q^{85} + 8 q^{86} + 20 q^{87} + 30 q^{88} + 25 q^{89} - 81 q^{90} - 14 q^{91} + 60 q^{92} + 36 q^{93} + 19 q^{94} + 111 q^{95} - 6 q^{96} + 2 q^{97} - 5 q^{98} + 108 q^{99}+O(q^{100}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9 + q^10 + 82 * q^11 + 3 * q^12 + 26 * q^13 + 17 * q^14 + 66 * q^15 + 122 * q^16 + 82 * q^17 + 18 * q^18 + 12 * q^19 + 9 * q^20 + 22 * q^21 + 8 * q^22 + 50 * q^23 + 15 * q^24 + 117 * q^25 + 36 * q^26 + 30 * q^27 + 11 * q^28 + 33 * q^29 - 26 * q^30 + 40 * q^31 + 58 * q^32 + 6 * q^33 + 8 * q^34 + 16 * q^35 + 160 * q^36 + 31 * q^37 + 18 * q^38 + 41 * q^39 - 29 * q^40 + 42 * q^41 - 51 * q^42 + 82 * q^43 + 98 * q^44 - 2 * q^45 - 19 * q^46 + 84 * q^47 - 46 * q^48 + 136 * q^49 + 59 * q^50 + 6 * q^51 + 45 * q^52 + 83 * q^53 + 24 * q^54 + 11 * q^55 + 21 * q^56 + 23 * q^57 + 14 * q^58 + 96 * q^59 + 184 * q^60 - 6 * q^61 - 23 * q^62 + 8 * q^63 + 148 * q^64 + 5 * q^65 + 10 * q^66 + 78 * q^67 + 98 * q^68 + 61 * q^69 - 3 * q^70 + 155 * q^71 + 50 * q^72 - 23 * q^73 + 10 * q^74 - 19 * q^75 + 44 * q^76 + 8 * q^77 - 27 * q^78 + 31 * q^79 + 19 * q^80 + 150 * q^81 - 12 * q^82 + 54 * q^83 + 8 * q^84 + 11 * q^85 + 8 * q^86 + 20 * q^87 + 30 * q^88 + 25 * q^89 - 81 * q^90 - 14 * q^91 + 60 * q^92 + 36 * q^93 + 19 * q^94 + 111 * q^95 - 6 * q^96 + 2 * q^97 - 5 * 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 $

$2$	−2.77655	−1.96332	−0.981658	−	0.190650i	$-0.938941\pi$
$2$	−2.77655	−1.96332	−0.981658	+	0.190650i	$0.938941\pi$
$3$	−0.377305	−0.217837	−0.108918	−	0.994051i	$-0.534739\pi$
$3$	−0.377305	−0.217837	−0.108918	+	0.994051i	$0.534739\pi$
$4$	5.70922	2.85461
$5$	−1.07900	−0.482543	−0.241272	−	0.970458i	$-0.577564\pi$
$5$	−1.07900	−0.482543	−0.241272	+	0.970458i	$0.577564\pi$
$6$	1.04760	0.427683
$7$	−3.82470	−1.44560	−0.722800	−	0.691058i	$-0.757145\pi$
$7$	−3.82470	−1.44560	−0.722800	+	0.691058i	$0.757145\pi$
$8$	−10.2988	−3.64119
$9$	−2.85764	−0.952547
$10$	2.99589	0.947385
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.15412	−0.621839
$13$	6.34070	1.75859	0.879297	−	0.476273i	$-0.158013\pi$
$13$	6.34070	1.75859	0.879297	+	0.476273i	$0.158013\pi$
$14$	10.6195	2.83817
$15$	0.407111	0.105116
$16$	17.1768	4.29419
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	7.93438	1.87015
$19$	2.71507	0.622880	0.311440	−	0.950266i	$-0.399189\pi$
$19$	2.71507	0.622880	0.311440	+	0.950266i	$0.399189\pi$
$20$	−6.16025	−1.37747
$21$	1.44308	0.314905
$22$	−2.77655	−0.591962
$23$	3.31642	0.691522	0.345761	−	0.938323i	$-0.387621\pi$
$23$	3.31642	0.691522	0.345761	+	0.938323i	$0.387621\pi$
$24$	3.88580	0.793185
$25$	−3.83576	−0.767152
$26$	−17.6053	−3.45268
$27$	2.21011	0.425337
$28$	−21.8360	−4.12662
$29$	8.14728	1.51291	0.756456	−	0.654045i	$-0.226929\pi$
$29$	8.14728	1.51291	0.756456	+	0.654045i	$0.226929\pi$
$30$	−1.13036	−0.206375
$31$	−0.300360	−0.0539462	−0.0269731	−	0.999636i	$-0.508587\pi$
$31$	−0.300360	−0.0539462	−0.0269731	+	0.999636i	$0.508587\pi$
$32$	−27.0945	−4.78967
$33$	−0.377305	−0.0656803
$34$	−2.77655	−0.476174
$35$	4.12685	0.697564
$36$	−16.3149	−2.71915
$37$	−5.07135	−0.833726	−0.416863	−	0.908969i	$-0.636870\pi$
$37$	−5.07135	−0.833726	−0.416863	+	0.908969i	$0.636870\pi$
$38$	−7.53852	−1.22291
$39$	−2.39238	−0.383087
$40$	11.1124	1.75703
$41$	1.83975	0.287321	0.143661	−	0.989627i	$-0.454113\pi$
$41$	1.83975	0.287321	0.143661	+	0.989627i	$0.454113\pi$
$42$	−4.00677	−0.618258
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	5.70922	0.860697
$45$	3.08339	0.459645
$46$	−9.20821	−1.35768
$47$	8.66756	1.26429	0.632147	−	0.774848i	$-0.282174\pi$
$47$	8.66756	1.26429	0.632147	+	0.774848i	$0.282174\pi$
$48$	−6.48087	−0.935433
$49$	7.62830	1.08976
$50$	10.6502	1.50616
$51$	−0.377305	−0.0528332
$52$	36.2005	5.02010
$53$	6.74749	0.926840	0.463420	−	0.886139i	$-0.346622\pi$
$53$	6.74749	0.926840	0.463420	+	0.886139i	$0.346622\pi$
$54$	−6.13649	−0.835071
$55$	−1.07900	−0.145492
$56$	39.3899	5.26370
$57$	−1.02441	−0.135686
$58$	−22.6213	−2.97032
$59$	11.7431	1.52882	0.764411	−	0.644730i	$-0.223030\pi$
$59$	11.7431	1.52882	0.764411	+	0.644730i	$0.223030\pi$
$60$	2.32429	0.300064
$61$	−9.86038	−1.26249	−0.631246	−	0.775582i	$-0.717456\pi$
$61$	−9.86038	−1.26249	−0.631246	+	0.775582i	$0.717456\pi$
$62$	0.833964	0.105913
$63$	10.9296	1.37700
$64$	40.8755	5.10944
$65$	−6.84162	−0.848598
$66$	1.04760	0.128951
$67$	−2.48384	−0.303449	−0.151725	−	0.988423i	$-0.548483\pi$
$67$	−2.48384	−0.303449	−0.151725	+	0.988423i	$0.548483\pi$
$68$	5.70922	0.692345
$69$	−1.25130	−0.150639
$70$	−11.4584	−1.36954
$71$	11.9966	1.42374	0.711870	−	0.702312i	$-0.247849\pi$
$71$	11.9966	1.42374	0.711870	+	0.702312i	$0.247849\pi$
$72$	29.4304	3.46840
$73$	−0.972938	−0.113874	−0.0569369	−	0.998378i	$-0.518133\pi$
$73$	−0.972938	−0.113874	−0.0569369	+	0.998378i	$0.518133\pi$
$74$	14.0809	1.63687
$75$	1.44725	0.167114
$76$	15.5009	1.77808
$77$	−3.82470	−0.435865
$78$	6.64255	0.752120
$79$	−8.47241	−0.953220	−0.476610	−	0.879115i	$-0.658135\pi$
$79$	−8.47241	−0.953220	−0.476610	+	0.879115i	$0.658135\pi$
$80$	−18.5337	−2.07213
$81$	7.73904	0.859893
$82$	−5.10816	−0.564102
$83$	−8.08161	−0.887072	−0.443536	−	0.896257i	$-0.646276\pi$
$83$	−8.08161	−0.887072	−0.443536	+	0.896257i	$0.646276\pi$
$84$	8.23884	0.898931
$85$	−1.07900	−0.117034
$86$	−2.77655	−0.299403
$87$	−3.07401	−0.329568
$88$	−10.2988	−1.09786
$89$	1.74033	0.184475	0.0922374	−	0.995737i	$-0.470598\pi$
$89$	1.74033	0.184475	0.0922374	+	0.995737i	$0.470598\pi$
$90$	−8.56119	−0.902429
$91$	−24.2513	−2.54222
$92$	18.9342	1.97403
$93$	0.113327	0.0117515
$94$	−24.0659	−2.48221
$95$	−2.92956	−0.300566
$96$	10.2229	1.04337
$97$	−5.49821	−0.558259	−0.279129	−	0.960253i	$-0.590046\pi$
$97$	−5.49821	−0.558259	−0.279129	+	0.960253i	$0.590046\pi$
$98$	−21.1803	−2.13954
$99$	−2.85764	−0.287204
$100$	−21.8992	−2.18992
$101$	−15.7110	−1.56330	−0.781652	−	0.623715i	$-0.785623\pi$
$101$	−15.7110	−1.56330	−0.781652	+	0.623715i	$0.785623\pi$
$102$	1.04760	0.103728
$103$	−12.2569	−1.20771	−0.603855	−	0.797094i	$-0.706369\pi$
$103$	−12.2569	−1.20771	−0.603855	+	0.797094i	$0.706369\pi$
$104$	−65.3018	−6.40337
$105$	−1.55708	−0.151955
$106$	−18.7347	−1.81968
$107$	−8.84137	−0.854727	−0.427364	−	0.904080i	$-0.640558\pi$
$107$	−8.84137	−0.854727	−0.427364	+	0.904080i	$0.640558\pi$
$108$	12.6180	1.21417
$109$	7.82520	0.749519	0.374759	−	0.927122i	$-0.377725\pi$
$109$	7.82520	0.749519	0.374759	+	0.927122i	$0.377725\pi$
$110$	2.99589	0.285647
$111$	1.91345	0.181616
$112$	−65.6959	−6.20768
$113$	0.302064	0.0284158	0.0142079	−	0.999899i	$-0.495477\pi$
$113$	0.302064	0.0284158	0.0142079	+	0.999899i	$0.495477\pi$
$114$	2.84432	0.266395
$115$	−3.57842	−0.333689
$116$	46.5146	4.31877
$117$	−18.1195	−1.67514
$118$	−32.6053	−3.00156
$119$	−3.82470	−0.350609
$120$	−4.19277	−0.382746
$121$	1.00000	0.0909091
$122$	27.3778	2.47867
$123$	−0.694147	−0.0625891
$124$	−1.71482	−0.153995
$125$	9.53378	0.852727
$126$	−30.3466	−2.70349
$127$	−6.34337	−0.562883	−0.281442	−	0.959578i	$-0.590813\pi$
$127$	−6.34337	−0.562883	−0.281442	+	0.959578i	$0.590813\pi$
$128$	−59.3040	−5.24178
$129$	−0.377305	−0.0332198
$130$	18.9961	1.66607
$131$	−3.12080	−0.272665	−0.136333	−	0.990663i	$-0.543532\pi$
$131$	−3.12080	−0.272665	−0.136333	+	0.990663i	$0.543532\pi$
$132$	−2.15412	−0.187492
$133$	−10.3843	−0.900434
$134$	6.89650	0.595767
$135$	−2.38471	−0.205243
$136$	−10.2988	−0.883118
$137$	14.3215	1.22357	0.611786	−	0.791023i	$-0.290451\pi$
$137$	14.3215	1.22357	0.611786	+	0.791023i	$0.290451\pi$
$138$	3.47430	0.295752
$139$	15.5651	1.32022	0.660108	−	0.751171i	$-0.270511\pi$
$139$	15.5651	1.32022	0.660108	+	0.751171i	$0.270511\pi$
$140$	23.5611	1.99127
$141$	−3.27031	−0.275410
$142$	−33.3093	−2.79525
$143$	6.34070	0.530236
$144$	−49.0850	−4.09042
$145$	−8.79091	−0.730045
$146$	2.70141	0.223570
$147$	−2.87819	−0.237389
$148$	−28.9535	−2.37996
$149$	19.0463	1.56033	0.780167	−	0.625571i	$-0.215134\pi$
$149$	19.0463	1.56033	0.780167	+	0.625571i	$0.215134\pi$
$150$	−4.01836	−0.328098
$151$	−4.78383	−0.389302	−0.194651	−	0.980873i	$-0.562357\pi$
$151$	−4.78383	−0.389302	−0.194651	+	0.980873i	$0.562357\pi$
$152$	−27.9620	−2.26802
$153$	−2.85764	−0.231027
$154$	10.6195	0.855740
$155$	0.324088	0.0260314
$156$	−13.6586	−1.09356
$157$	−1.98360	−0.158308	−0.0791542	−	0.996862i	$-0.525222\pi$
$157$	−1.98360	−0.158308	−0.0791542	+	0.996862i	$0.525222\pi$
$158$	23.5240	1.87147
$159$	−2.54586	−0.201900
$160$	29.2349	2.31122
$161$	−12.6843	−0.999664
$162$	−21.4878	−1.68824
$163$	−0.246816	−0.0193321	−0.00966607	−	0.999953i	$-0.503077\pi$
$163$	−0.246816	−0.0193321	−0.00966607	+	0.999953i	$0.503077\pi$
$164$	10.5036	0.820190
$165$	0.407111	0.0316936
$166$	22.4390	1.74160
$167$	19.6127	1.51768	0.758839	−	0.651278i	$-0.225767\pi$
$167$	19.6127	1.51768	0.758839	+	0.651278i	$0.225767\pi$
$168$	−14.8620	−1.14663
$169$	27.2045	2.09266
$170$	2.99589	0.229775
$171$	−7.75869	−0.593322
$172$	5.70922	0.435324
$173$	−2.29051	−0.174144	−0.0870720	−	0.996202i	$-0.527751\pi$
$173$	−2.29051	−0.174144	−0.0870720	+	0.996202i	$0.527751\pi$
$174$	8.53512	0.647046
$175$	14.6706	1.10899
$176$	17.1768	1.29475
$177$	−4.43072	−0.333034
$178$	−4.83211	−0.362182
$179$	−15.3327	−1.14602	−0.573008	−	0.819549i	$-0.694224\pi$
$179$	−15.3327	−1.14602	−0.573008	+	0.819549i	$0.694224\pi$
$180$	17.6038	1.31211
$181$	4.90643	0.364692	0.182346	−	0.983234i	$-0.441631\pi$
$181$	4.90643	0.364692	0.182346	+	0.983234i	$0.441631\pi$
$182$	67.3348	4.99119
$183$	3.72037	0.275017
$184$	−34.1553	−2.51796
$185$	5.47199	0.402309
$186$	−0.314658	−0.0230719
$187$	1.00000	0.0731272
$188$	49.4850	3.60907
$189$	−8.45302	−0.614867
$190$	8.13406	0.590107
$191$	−16.4020	−1.18681	−0.593404	−	0.804904i	$-0.702216\pi$
$191$	−16.4020	−1.18681	−0.593404	+	0.804904i	$0.702216\pi$
$192$	−15.4225	−1.11302
$193$	−10.4332	−0.751000	−0.375500	−	0.926822i	$-0.622529\pi$
$193$	−10.4332	−0.751000	−0.375500	+	0.926822i	$0.622529\pi$
$194$	15.2661	1.09604
$195$	2.58137	0.184856
$196$	43.5517	3.11083
$197$	−10.0979	−0.719446	−0.359723	−	0.933059i	$-0.617129\pi$
$197$	−10.0979	−0.719446	−0.359723	+	0.933059i	$0.617129\pi$
$198$	7.93438	0.563872
$199$	−19.0556	−1.35082	−0.675409	−	0.737444i	$-0.736033\pi$
$199$	−19.0556	−1.35082	−0.675409	+	0.737444i	$0.736033\pi$
$200$	39.5038	2.79334
$201$	0.937163	0.0661024
$202$	43.6224	3.06926
$203$	−31.1609	−2.18706
$204$	−2.15412	−0.150818
$205$	−1.98509	−0.138645
$206$	34.0319	2.37112
$207$	−9.47715	−0.658707
$208$	108.913	7.55174
$209$	2.71507	0.187805
$210$	4.32330	0.298336
$211$	9.27076	0.638225	0.319113	−	0.947717i	$-0.396615\pi$
$211$	9.27076	0.638225	0.319113	+	0.947717i	$0.396615\pi$
$212$	38.5229	2.64577
$213$	−4.52639	−0.310143
$214$	24.5485	1.67810
$215$	−1.07900	−0.0735872
$216$	−22.7616	−1.54873
$217$	1.14878	0.0779846
$218$	−21.7271	−1.47154
$219$	0.367094	0.0248059
$220$	−6.16025	−0.415324
$221$	6.34070	0.426522
$222$	−5.31277	−0.356570
$223$	−14.0343	−0.939809	−0.469904	−	0.882717i	$-0.655712\pi$
$223$	−14.0343	−0.939809	−0.469904	+	0.882717i	$0.655712\pi$
$224$	103.628	6.92394
$225$	10.9612	0.730748
$226$	−0.838696	−0.0557892
$227$	−18.6858	−1.24022	−0.620110	−	0.784515i	$-0.712912\pi$
$227$	−18.6858	−1.24022	−0.620110	+	0.784515i	$0.712912\pi$
$228$	−5.84857	−0.387331
$229$	−6.49404	−0.429139	−0.214569	−	0.976709i	$-0.568835\pi$
$229$	−6.49404	−0.429139	−0.214569	+	0.976709i	$0.568835\pi$
$230$	9.93565	0.655138
$231$	1.44308	0.0949474
$232$	−83.9075	−5.50879
$233$	−24.7451	−1.62111	−0.810553	−	0.585665i	$-0.800833\pi$
$233$	−24.7451	−1.62111	−0.810553	+	0.585665i	$0.800833\pi$
$234$	50.3095	3.28884
$235$	−9.35230	−0.610077
$236$	67.0439	4.36419
$237$	3.19668	0.207647
$238$	10.6195	0.688357
$239$	−6.01216	−0.388894	−0.194447	−	0.980913i	$-0.562291\pi$
$239$	−6.01216	−0.388894	−0.194447	+	0.980913i	$0.562291\pi$
$240$	6.99286	0.451387
$241$	−13.1084	−0.844388	−0.422194	−	0.906506i	$-0.638740\pi$
$241$	−13.1084	−0.844388	−0.422194	+	0.906506i	$0.638740\pi$
$242$	−2.77655	−0.178483
$243$	−9.55032	−0.612653
$244$	−56.2951	−3.60393
$245$	−8.23093	−0.525855
$246$	1.92733	0.122882
$247$	17.2155	1.09539
$248$	3.09336	0.196428
$249$	3.04923	0.193237
$250$	−26.4710	−1.67417
$251$	26.0697	1.64550	0.822751	−	0.568402i	$-0.192438\pi$
$251$	26.0697	1.64550	0.822751	+	0.568402i	$0.192438\pi$
$252$	62.3996	3.93080
$253$	3.31642	0.208502
$254$	17.6127	1.10512
$255$	0.407111	0.0254943
$256$	82.9093	5.18183
$257$	14.4621	0.902121	0.451061	−	0.892493i	$-0.351046\pi$
$257$	14.4621	0.902121	0.451061	+	0.892493i	$0.351046\pi$
$258$	1.04760	0.0652210
$259$	19.3964	1.20523
$260$	−39.0603	−2.42242
$261$	−23.2820	−1.44112
$262$	8.66504	0.535328
$263$	−11.1975	−0.690467	−0.345233	−	0.938517i	$-0.612200\pi$
$263$	−11.1975	−0.690467	−0.345233	+	0.938517i	$0.612200\pi$
$264$	3.88580	0.239154
$265$	−7.28054	−0.447240
$266$	28.8326	1.76784
$267$	−0.656635	−0.0401854
$268$	−14.1808	−0.866229
$269$	18.3467	1.11862	0.559308	−	0.828960i	$-0.311067\pi$
$269$	18.3467	1.11862	0.559308	+	0.828960i	$0.311067\pi$
$270$	6.62127	0.402958
$271$	22.1695	1.34670	0.673349	−	0.739324i	$-0.264855\pi$
$271$	22.1695	1.34670	0.673349	+	0.739324i	$0.264855\pi$
$272$	17.1768	1.04149
$273$	9.15011	0.553790
$274$	−39.7644	−2.40226
$275$	−3.83576	−0.231305
$276$	−7.14396	−0.430016
$277$	−11.2740	−0.677392	−0.338696	−	0.940896i	$-0.609986\pi$
$277$	−11.2740	−0.677392	−0.338696	+	0.940896i	$0.609986\pi$
$278$	−43.2173	−2.59200
$279$	0.858321	0.0513863
$280$	−42.5017	−2.53996
$281$	7.26596	0.433451	0.216725	−	0.976233i	$-0.430462\pi$
$281$	7.26596	0.433451	0.216725	+	0.976233i	$0.430462\pi$
$282$	9.08018	0.540717
$283$	−13.2482	−0.787527	−0.393763	−	0.919212i	$-0.628827\pi$
$283$	−13.2482	−0.787527	−0.393763	+	0.919212i	$0.628827\pi$
$284$	68.4915	4.06422
$285$	1.10534	0.0654744
$286$	−17.6053	−1.04102
$287$	−7.03650	−0.415351
$288$	77.4262	4.56238
$289$	1.00000	0.0588235
$290$	24.4084	1.43331
$291$	2.07450	0.121609
$292$	−5.55472	−0.325065
$293$	−22.5600	−1.31797	−0.658984	−	0.752157i	$-0.729013\pi$
$293$	−22.5600	−1.31797	−0.658984	+	0.752157i	$0.729013\pi$
$294$	7.99144	0.466070
$295$	−12.6708	−0.737722
$296$	52.2290	3.03575
$297$	2.21011	0.128244
$298$	−52.8830	−3.06343
$299$	21.0285	1.21611
$300$	8.26267	0.477045
$301$	−3.82470	−0.220452
$302$	13.2825	0.764324
$303$	5.92783	0.340545
$304$	46.6361	2.67476
$305$	10.6393	0.609207
$306$	7.93438	0.453578
$307$	29.4190	1.67903	0.839515	−	0.543337i	$-0.182839\pi$
$307$	29.4190	1.67903	0.839515	+	0.543337i	$0.182839\pi$
$308$	−21.8360	−1.24422
$309$	4.62459	0.263084
$310$	−0.899846	−0.0511078
$311$	16.0107	0.907881	0.453940	−	0.891032i	$-0.350018\pi$
$311$	16.0107	0.907881	0.453940	+	0.891032i	$0.350018\pi$
$312$	24.6387	1.39489
$313$	24.5730	1.38895	0.694473	−	0.719519i	$-0.255637\pi$
$313$	24.5730	1.38895	0.694473	+	0.719519i	$0.255637\pi$
$314$	5.50756	0.310810
$315$	−11.7930	−0.664463
$316$	−48.3708	−2.72107
$317$	22.8723	1.28464	0.642319	−	0.766437i	$-0.277972\pi$
$317$	22.8723	1.28464	0.642319	+	0.766437i	$0.277972\pi$
$318$	7.06870	0.396393
$319$	8.14728	0.456160
$320$	−44.1047	−2.46553
$321$	3.33589	0.186191
$322$	35.2186	1.96266
$323$	2.71507	0.151071
$324$	44.1839	2.45466
$325$	−24.3214	−1.34911
$326$	0.685297	0.0379551
$327$	−2.95248	−0.163273
$328$	−18.9473	−1.04619
$329$	−33.1508	−1.82766
$330$	−1.13036	−0.0622245
$331$	25.0669	1.37780	0.688901	−	0.724856i	$-0.258094\pi$
$331$	25.0669	1.37780	0.688901	+	0.724856i	$0.258094\pi$
$332$	−46.1397	−2.53225
$333$	14.4921	0.794163
$334$	−54.4557	−2.97968
$335$	2.68006	0.146427
$336$	24.7874	1.35226
$337$	0.0132308	0.000720728 0	0.000360364	−	1.00000i	$-0.499885\pi$
$337$	0.0132308	0.000720728 0	0.000360364		1.00000i	$0.499885\pi$
$338$	−75.5347	−4.10854
$339$	−0.113970	−0.00619001
$340$	−6.16025	−0.334086
$341$	−0.300360	−0.0162654
$342$	21.5424	1.16488
$343$	−2.40306	−0.129753
$344$	−10.2988	−0.555276
$345$	1.35015	0.0726898
$346$	6.35970	0.341900
$347$	5.44151	0.292116	0.146058	−	0.989276i	$-0.453341\pi$
$347$	5.44151	0.292116	0.146058	+	0.989276i	$0.453341\pi$
$348$	−17.5502	−0.940788
$349$	22.4302	1.20066	0.600329	−	0.799753i	$-0.295036\pi$
$349$	22.4302	1.20066	0.600329	+	0.799753i	$0.295036\pi$
$350$	−40.7337	−2.17731
$351$	14.0137	0.747995
$352$	−27.0945	−1.44414
$353$	−3.94113	−0.209765	−0.104883	−	0.994485i	$-0.533447\pi$
$353$	−3.94113	−0.209765	−0.104883	+	0.994485i	$0.533447\pi$
$354$	12.3021	0.653850
$355$	−12.9444	−0.687016
$356$	9.93594	0.526604
$357$	1.44308	0.0763756
$358$	42.5719	2.24999
$359$	3.72413	0.196552	0.0982761	−	0.995159i	$-0.468667\pi$
$359$	3.72413	0.196552	0.0982761	+	0.995159i	$0.468667\pi$
$360$	−31.7554	−1.67365
$361$	−11.6284	−0.612021
$362$	−13.6229	−0.716006
$363$	−0.377305	−0.0198034
$364$	−138.456	−7.25706
$365$	1.04980	0.0549490
$366$	−10.3298	−0.539946
$367$	11.2058	0.584938	0.292469	−	0.956275i	$-0.405523\pi$
$367$	11.2058	0.584938	0.292469	+	0.956275i	$0.405523\pi$
$368$	56.9654	2.96953
$369$	−5.25735	−0.273687
$370$	−15.1932	−0.789859
$371$	−25.8071	−1.33984
$372$	0.647010	0.0335459
$373$	−11.5166	−0.596309	−0.298154	−	0.954518i	$-0.596371\pi$
$373$	−11.5166	−0.596309	−0.298154	+	0.954518i	$0.596371\pi$
$374$	−2.77655	−0.143572
$375$	−3.59714	−0.185755
$376$	−89.2658	−4.60353
$377$	51.6595	2.66060
$378$	23.4702	1.20718
$379$	24.0052	1.23307	0.616533	−	0.787329i	$-0.288537\pi$
$379$	24.0052	1.23307	0.616533	+	0.787329i	$0.288537\pi$
$380$	−16.7255	−0.858000
$381$	2.39338	0.122617
$382$	45.5410	2.33008
$383$	−2.74070	−0.140043	−0.0700215	−	0.997545i	$-0.522307\pi$
$383$	−2.74070	−0.140043	−0.0700215	+	0.997545i	$0.522307\pi$
$384$	22.3757	1.14185
$385$	4.12685	0.210324
$386$	28.9684	1.47445
$387$	−2.85764	−0.145262
$388$	−31.3905	−1.59361
$389$	−22.2080	−1.12599	−0.562996	−	0.826460i	$-0.690351\pi$
$389$	−22.2080	−1.12599	−0.562996	+	0.826460i	$0.690351\pi$
$390$	−7.16731	−0.362931
$391$	3.31642	0.167719
$392$	−78.5626	−3.96801
$393$	1.17749	0.0593966
$394$	28.0373	1.41250
$395$	9.14172	0.459970
$396$	−16.3149	−0.819855
$397$	−15.3192	−0.768850	−0.384425	−	0.923156i	$-0.625600\pi$
$397$	−15.3192	−0.768850	−0.384425	+	0.923156i	$0.625600\pi$
$398$	52.9089	2.65208
$399$	3.91805	0.196148
$400$	−65.8859	−3.29430
$401$	12.8519	0.641792	0.320896	−	0.947114i	$-0.396016\pi$
$401$	12.8519	0.641792	0.320896	+	0.947114i	$0.396016\pi$
$402$	−2.60208	−0.129780
$403$	−1.90449	−0.0948695
$404$	−89.6976	−4.46262
$405$	−8.35042	−0.414936
$406$	86.5197	4.29390
$407$	−5.07135	−0.251378
$408$	3.88580	0.192376
$409$	−20.5660	−1.01693	−0.508463	−	0.861084i	$-0.669786\pi$
$409$	−20.5660	−1.01693	−0.508463	+	0.861084i	$0.669786\pi$
$410$	5.51170	0.272204
$411$	−5.40358	−0.266539
$412$	−69.9774	−3.44754
$413$	−44.9138	−2.21006
$414$	26.3138	1.29325
$415$	8.72006	0.428051
$416$	−171.798	−8.42308
$417$	−5.87279	−0.287592
$418$	−7.53852	−0.368721
$419$	2.77441	0.135539	0.0677693	−	0.997701i	$-0.478412\pi$
$419$	2.77441	0.135539	0.0677693	+	0.997701i	$0.478412\pi$
$420$	−8.88970	−0.433773
$421$	28.6953	1.39852	0.699262	−	0.714865i	$-0.253512\pi$
$421$	28.6953	1.39852	0.699262	+	0.714865i	$0.253512\pi$
$422$	−25.7407	−1.25304
$423$	−24.7688	−1.20430
$424$	−69.4913	−3.37480
$425$	−3.83576	−0.186062
$426$	12.5677	0.608909
$427$	37.7130	1.82506
$428$	−50.4773	−2.43991
$429$	−2.39238	−0.115505
$430$	2.99589	0.144475
$431$	−37.2012	−1.79192	−0.895960	−	0.444134i	$-0.853511\pi$
$431$	−37.2012	−1.79192	−0.895960	+	0.444134i	$0.853511\pi$
$432$	37.9626	1.82648
$433$	0.674910	0.0324341	0.0162171	−	0.999868i	$-0.494838\pi$
$433$	0.674910	0.0324341	0.0162171	+	0.999868i	$0.494838\pi$
$434$	−3.18966	−0.153108
$435$	3.31685	0.159031
$436$	44.6758	2.13958
$437$	9.00432	0.430735
$438$	−1.01925	−0.0487019
$439$	−30.1215	−1.43762	−0.718810	−	0.695207i	$-0.755313\pi$
$439$	−30.1215	−1.43762	−0.718810	+	0.695207i	$0.755313\pi$
$440$	11.1124	0.529765
$441$	−21.7990	−1.03805
$442$	−17.6053	−0.837397
$443$	4.46539	0.212157	0.106079	−	0.994358i	$-0.466170\pi$
$443$	4.46539	0.212157	0.106079	+	0.994358i	$0.466170\pi$
$444$	10.9243	0.518443
$445$	−1.87782	−0.0890170
$446$	38.9670	1.84514
$447$	−7.18626	−0.339898
$448$	−156.337	−7.38621
$449$	−31.0984	−1.46762	−0.733812	−	0.679352i	$-0.762261\pi$
$449$	−31.0984	−1.46762	−0.733812	+	0.679352i	$0.762261\pi$
$450$	−30.4344	−1.43469
$451$	1.83975	0.0866306
$452$	1.72455	0.0811161
$453$	1.80496	0.0848044
$454$	51.8821	2.43495
$455$	26.1671	1.22673
$456$	10.5502	0.494059
$457$	22.7678	1.06503	0.532517	−	0.846419i	$-0.321246\pi$
$457$	22.7678	1.06503	0.532517	+	0.846419i	$0.321246\pi$
$458$	18.0310	0.842535
$459$	2.21011	0.103159
$460$	−20.4300	−0.952553
$461$	33.5055	1.56051	0.780253	−	0.625464i	$-0.215090\pi$
$461$	33.5055	1.56051	0.780253	+	0.625464i	$0.215090\pi$
$462$	−4.00677	−0.186412
$463$	17.7999	0.827229	0.413615	−	0.910452i	$-0.364266\pi$
$463$	17.7999	0.827229	0.413615	+	0.910452i	$0.364266\pi$
$464$	139.944	6.49673
$465$	−0.122280	−0.00567059
$466$	68.7060	3.18274
$467$	1.32975	0.0615337	0.0307669	−	0.999527i	$-0.490205\pi$
$467$	1.32975	0.0615337	0.0307669	+	0.999527i	$0.490205\pi$
$468$	−103.448	−4.78188
$469$	9.49993	0.438666
$470$	25.9671	1.19777
$471$	0.748421	0.0344854
$472$	−120.940	−5.56672
$473$	1.00000	0.0459800
$474$	−8.87573	−0.407676
$475$	−10.4144	−0.477843
$476$	−21.8360	−1.00085
$477$	−19.2819	−0.882858
$478$	16.6931	0.763523
$479$	5.93727	0.271281	0.135640	−	0.990758i	$-0.456691\pi$
$479$	5.93727	0.271281	0.135640	+	0.990758i	$0.456691\pi$
$480$	−11.0305	−0.503469
$481$	−32.1560	−1.46619
$482$	36.3962	1.65780
$483$	4.78585	0.217764
$484$	5.70922	0.259510
$485$	5.93257	0.269384
$486$	26.5169	1.20283
$487$	15.4558	0.700371	0.350186	−	0.936680i	$-0.386119\pi$
$487$	15.4558	0.700371	0.350186	+	0.936680i	$0.386119\pi$
$488$	101.550	4.59697
$489$	0.0931248	0.00421125
$490$	22.8536	1.03242
$491$	35.9178	1.62095	0.810473	−	0.585776i	$-0.199210\pi$
$491$	35.9178	1.62095	0.810473	+	0.585776i	$0.199210\pi$
$492$	−3.96304	−0.178668
$493$	8.14728	0.366935
$494$	−47.7995	−2.15060
$495$	3.08339	0.138588
$496$	−5.15921	−0.231655
$497$	−45.8835	−2.05816
$498$	−8.46633	−0.379385
$499$	36.5852	1.63778	0.818889	−	0.573952i	$-0.194590\pi$
$499$	36.5852	1.63778	0.818889	+	0.573952i	$0.194590\pi$
$500$	54.4305	2.43420
$501$	−7.39997	−0.330606
$502$	−72.3837	−3.23064
$503$	−16.4617	−0.733992	−0.366996	−	0.930222i	$-0.619614\pi$
$503$	−16.4617	−0.733992	−0.366996	+	0.930222i	$0.619614\pi$
$504$	−112.562	−5.01392
$505$	16.9522	0.754362
$506$	−9.20821	−0.409355
$507$	−10.2644	−0.455857
$508$	−36.2157	−1.60681
$509$	28.4605	1.26149	0.630746	−	0.775990i	$-0.282749\pi$
$509$	28.4605	1.26149	0.630746	+	0.775990i	$0.282749\pi$
$510$	−1.13036	−0.0500534
$511$	3.72119	0.164616
$512$	−111.594	−4.93180
$513$	6.00061	0.264934
$514$	−40.1547	−1.77115
$515$	13.2252	0.582772
$516$	−2.15412	−0.0948296
$517$	8.66756	0.381199
$518$	−53.8550	−2.36625
$519$	0.864218	0.0379350
$520$	70.4607	3.08990
$521$	−38.7264	−1.69664	−0.848318	−	0.529487i	$-0.822384\pi$
$521$	−38.7264	−1.69664	−0.848318	+	0.529487i	$0.822384\pi$
$522$	64.6436	2.82937
$523$	7.89941	0.345417	0.172709	−	0.984973i	$-0.444748\pi$
$523$	7.89941	0.345417	0.172709	+	0.984973i	$0.444748\pi$
$524$	−17.8173	−0.778353
$525$	−5.53529	−0.241580
$526$	31.0904	1.35561
$527$	−0.300360	−0.0130839
$528$	−6.48087	−0.282044
$529$	−12.0013	−0.521797
$530$	20.2148	0.878074
$531$	−33.5576	−1.45627
$532$	−59.2864	−2.57039
$533$	11.6653	0.505281
$534$	1.82318	0.0788966
$535$	9.53983	0.412443
$536$	25.5806	1.10492
$537$	5.78508	0.249645
$538$	−50.9404	−2.19620
$539$	7.62830	0.328574
$540$	−13.6149	−0.585890
$541$	21.7778	0.936300	0.468150	−	0.883649i	$-0.344921\pi$
$541$	21.7778	0.936300	0.468150	+	0.883649i	$0.344921\pi$
$542$	−61.5546	−2.64400
$543$	−1.85122	−0.0794434
$544$	−27.0945	−1.16167
$545$	−8.44339	−0.361675
$546$	−25.4057	−1.08726
$547$	−33.8642	−1.44793	−0.723965	−	0.689836i	$-0.757682\pi$
$547$	−33.8642	−1.44793	−0.723965	+	0.689836i	$0.757682\pi$
$548$	81.7648	3.49282
$549$	28.1774	1.20258
$550$	10.6502	0.454125
$551$	22.1204	0.942362
$552$	12.8869	0.548505
$553$	32.4044	1.37797
$554$	31.3029	1.32993
$555$	−2.06461	−0.0876377
$556$	88.8647	3.76870
$557$	−37.5570	−1.59134	−0.795671	−	0.605729i	$-0.792882\pi$
$557$	−37.5570	−1.59134	−0.795671	+	0.605729i	$0.792882\pi$
$558$	−2.38317	−0.100888
$559$	6.34070	0.268183
$560$	70.8859	2.99547
$561$	−0.377305	−0.0159298
$562$	−20.1743	−0.851001
$563$	28.4408	1.19864	0.599319	−	0.800510i	$-0.295438\pi$
$563$	28.4408	1.19864	0.599319	+	0.800510i	$0.295438\pi$
$564$	−18.6709	−0.786188
$565$	−0.325927	−0.0137119
$566$	36.7844	1.54616
$567$	−29.5995	−1.24306
$568$	−123.551	−5.18410
$569$	−1.61319	−0.0676286	−0.0338143	−	0.999428i	$-0.510765\pi$
$569$	−1.61319	−0.0676286	−0.0338143	+	0.999428i	$0.510765\pi$
$570$	−3.06902	−0.128547
$571$	−2.37582	−0.0994249	−0.0497125	−	0.998764i	$-0.515830\pi$
$571$	−2.37582	−0.0994249	−0.0497125	+	0.998764i	$0.515830\pi$
$572$	36.2005	1.51362
$573$	6.18856	0.258531
$574$	19.5372	0.815466
$575$	−12.7210	−0.530502
$576$	−116.808	−4.86698
$577$	−30.1521	−1.25525	−0.627625	−	0.778516i	$-0.715973\pi$
$577$	−30.1521	−1.25525	−0.627625	+	0.778516i	$0.715973\pi$
$578$	−2.77655	−0.115489
$579$	3.93650	0.163596
$580$	−50.1892	−2.08400
$581$	30.9097	1.28235
$582$	−5.75995	−0.238758
$583$	6.74749	0.279453
$584$	10.0201	0.414636
$585$	19.5509	0.808330
$586$	62.6388	2.58759
$587$	39.7574	1.64096	0.820482	−	0.571673i	$-0.193705\pi$
$587$	39.7574	1.64096	0.820482	+	0.571673i	$0.193705\pi$
$588$	−16.4322	−0.677654
$589$	−0.815498	−0.0336020
$590$	35.1811	1.44838
$591$	3.80999	0.156722
$592$	−87.1095	−3.58018
$593$	−34.9683	−1.43598	−0.717988	−	0.696055i	$-0.754937\pi$
$593$	−34.9683	−1.43598	−0.717988	+	0.696055i	$0.754937\pi$
$594$	−6.13649	−0.251783
$595$	4.12685	0.169184
$596$	108.740	4.45415
$597$	7.18977	0.294258
$598$	−58.3865	−2.38760
$599$	45.1951	1.84662	0.923310	−	0.384055i	$-0.125473\pi$
$599$	45.1951	1.84662	0.923310	+	0.384055i	$0.125473\pi$
$600$	−14.9050	−0.608493
$601$	20.0870	0.819365	0.409683	−	0.912228i	$-0.365639\pi$
$601$	20.0870	0.819365	0.409683	+	0.912228i	$0.365639\pi$
$602$	10.6195	0.432817
$603$	7.09792	0.289050
$604$	−27.3119	−1.11131
$605$	−1.07900	−0.0438676
$606$	−16.4589	−0.668598
$607$	−18.5524	−0.753020	−0.376510	−	0.926412i	$-0.622876\pi$
$607$	−18.5524	−0.753020	−0.376510	+	0.926412i	$0.622876\pi$
$608$	−73.5633	−2.98339
$609$	11.7571	0.476423
$610$	−29.5407	−1.19607
$611$	54.9585	2.22338
$612$	−16.3149	−0.659491
$613$	−39.8737	−1.61049	−0.805243	−	0.592946i	$-0.797965\pi$
$613$	−39.8737	−1.61049	−0.805243	+	0.592946i	$0.797965\pi$
$614$	−81.6832	−3.29647
$615$	0.748984	0.0302020
$616$	39.3899	1.58706
$617$	−29.0402	−1.16911	−0.584557	−	0.811353i	$-0.698732\pi$
$617$	−29.0402	−1.16911	−0.584557	+	0.811353i	$0.698732\pi$
$618$	−12.8404	−0.516517
$619$	19.3293	0.776909	0.388455	−	0.921468i	$-0.373009\pi$
$619$	19.3293	0.776909	0.388455	+	0.921468i	$0.373009\pi$
$620$	1.85029	0.0743095
$621$	7.32968	0.294130
$622$	−44.4543	−1.78246
$623$	−6.65624	−0.266677
$624$	−41.0933	−1.64505
$625$	8.89185	0.355674
$626$	−68.2280	−2.72694
$627$	−1.02441	−0.0409109
$628$	−11.3248	−0.451909
$629$	−5.07135	−0.202208
$630$	32.7440	1.30455
$631$	25.2507	1.00521	0.502607	−	0.864515i	$-0.332374\pi$
$631$	25.2507	1.00521	0.502607	+	0.864515i	$0.332374\pi$
$632$	87.2559	3.47085
$633$	−3.49790	−0.139029
$634$	−63.5062	−2.52215
$635$	6.84450	0.271616
$636$	−14.5349	−0.576345
$637$	48.3688	1.91644
$638$	−22.6213	−0.895586
$639$	−34.2821	−1.35618
$640$	63.9890	2.52939
$641$	−13.0410	−0.515089	−0.257545	−	0.966266i	$-0.582913\pi$
$641$	−13.0410	−0.515089	−0.257545	+	0.966266i	$0.582913\pi$
$642$	−9.26226	−0.365552
$643$	37.4430	1.47661	0.738303	−	0.674469i	$-0.235627\pi$
$643$	37.4430	1.47661	0.738303	+	0.674469i	$0.235627\pi$
$644$	−72.4175	−2.85365
$645$	0.407111	0.0160300
$646$	−7.53852	−0.296599
$647$	−6.87204	−0.270168	−0.135084	−	0.990834i	$-0.543130\pi$
$647$	−6.87204	−0.270168	−0.135084	+	0.990834i	$0.543130\pi$
$648$	−79.7031	−3.13103
$649$	11.7431	0.460957
$650$	67.5296	2.64873
$651$	−0.433442	−0.0169879
$652$	−1.40913	−0.0551857
$653$	48.5171	1.89862	0.949310	−	0.314340i	$-0.101783\pi$
$653$	48.5171	1.89862	0.949310	+	0.314340i	$0.101783\pi$
$654$	8.19772	0.320556
$655$	3.36734	0.131573
$656$	31.6010	1.23381
$657$	2.78031	0.108470
$658$	92.0448	3.58828
$659$	35.6802	1.38990	0.694952	−	0.719056i	$-0.255426\pi$
$659$	35.6802	1.38990	0.694952	+	0.719056i	$0.255426\pi$
$660$	2.32429	0.0904728
$661$	6.17494	0.240177	0.120089	−	0.992763i	$-0.461682\pi$
$661$	6.17494	0.240177	0.120089	+	0.992763i	$0.461682\pi$
$662$	−69.5995	−2.70506
$663$	−2.39238	−0.0929122
$664$	83.2312	3.23000
$665$	11.2047	0.434499
$666$	−40.2381	−1.55919
$667$	27.0198	1.04621
$668$	111.973	4.33238
$669$	5.29522	0.204725
$670$	−7.44132	−0.287483
$671$	−9.86038	−0.380656
$672$	−39.0993	−1.50829
$673$	10.2145	0.393741	0.196870	−	0.980430i	$-0.436922\pi$
$673$	10.2145	0.393741	0.196870	+	0.980430i	$0.436922\pi$
$674$	−0.0367360	−0.00141502
$675$	−8.47747	−0.326298
$676$	155.317	5.97372
$677$	2.92324	0.112349	0.0561747	−	0.998421i	$-0.482110\pi$
$677$	2.92324	0.112349	0.0561747	+	0.998421i	$0.482110\pi$
$678$	0.316444	0.0121529
$679$	21.0290	0.807019
$680$	11.1124	0.426142
$681$	7.05024	0.270166
$682$	0.833964	0.0319341
$683$	−11.5384	−0.441503	−0.220751	−	0.975330i	$-0.570851\pi$
$683$	−11.5384	−0.441503	−0.220751	+	0.975330i	$0.570851\pi$
$684$	−44.2961	−1.69370
$685$	−15.4529	−0.590426
$686$	6.67222	0.254747
$687$	2.45023	0.0934822
$688$	17.1768	0.654858
$689$	42.7839	1.62994
$690$	−3.74877	−0.142713
$691$	−19.1189	−0.727316	−0.363658	−	0.931533i	$-0.618472\pi$
$691$	−19.1189	−0.727316	−0.363658	+	0.931533i	$0.618472\pi$
$692$	−13.0770	−0.497113
$693$	10.9296	0.415182
$694$	−15.1086	−0.573515
$695$	−16.7948	−0.637061
$696$	31.6587	1.20002
$697$	1.83975	0.0696856
$698$	−62.2784	−2.35727
$699$	9.33645	0.353137
$700$	83.7578	3.16575
$701$	−22.8964	−0.864786	−0.432393	−	0.901685i	$-0.642331\pi$
$701$	−22.8964	−0.864786	−0.432393	+	0.901685i	$0.642331\pi$
$702$	−38.9097	−1.46855
$703$	−13.7691	−0.519311
$704$	40.8755	1.54055
$705$	3.52866	0.132897
$706$	10.9427	0.411835
$707$	60.0898	2.25991
$708$	−25.2960	−0.950681
$709$	8.51075	0.319628	0.159814	−	0.987147i	$-0.448911\pi$
$709$	8.51075	0.319628	0.159814	+	0.987147i	$0.448911\pi$
$710$	35.9407	1.34883
$711$	24.2111	0.907987
$712$	−17.9234	−0.671707
$713$	−0.996120	−0.0373050
$714$	−4.00677	−0.149950
$715$	−6.84162	−0.255862
$716$	−87.5376	−3.27143
$717$	2.26842	0.0847156
$718$	−10.3402	−0.385894
$719$	−21.1661	−0.789363	−0.394682	−	0.918818i	$-0.629145\pi$
$719$	−21.1661	−0.789363	−0.394682	+	0.918818i	$0.629145\pi$
$720$	52.9627	1.97380
$721$	46.8790	1.74586
$722$	32.2868	1.20159
$723$	4.94587	0.183939
$724$	28.0119	1.04105
$725$	−31.2510	−1.16063
$726$	1.04760	0.0388802
$727$	−5.60901	−0.208027	−0.104013	−	0.994576i	$-0.533168\pi$
$727$	−5.60901	−0.208027	−0.104013	+	0.994576i	$0.533168\pi$
$728$	249.760	9.25671
$729$	−19.6137	−0.726435
$730$	−2.91482	−0.107882
$731$	1.00000	0.0369863
$732$	21.2404	0.785068
$733$	35.0686	1.29529	0.647644	−	0.761943i	$-0.275755\pi$
$733$	35.0686	1.29529	0.647644	+	0.761943i	$0.275755\pi$
$734$	−31.1135	−1.14842
$735$	3.10557	0.114551
$736$	−89.8567	−3.31216
$737$	−2.48384	−0.0914934
$738$	14.5973	0.537334
$739$	−40.0410	−1.47293	−0.736466	−	0.676475i	$-0.763507\pi$
$739$	−40.0410	−1.47293	−0.736466	+	0.676475i	$0.763507\pi$
$740$	31.2408	1.14843
$741$	−6.49547	−0.238617
$742$	71.6547	2.63053
$743$	30.1558	1.10631	0.553154	−	0.833079i	$-0.313424\pi$
$743$	30.1558	1.10631	0.553154	+	0.833079i	$0.313424\pi$
$744$	−1.16714	−0.0427893
$745$	−20.5510	−0.752929
$746$	31.9765	1.17074
$747$	23.0943	0.844978
$748$	5.70922	0.208750
$749$	33.8155	1.23559
$750$	9.98763	0.364697
$751$	−50.0634	−1.82684	−0.913419	−	0.407021i	$-0.866568\pi$
$751$	−50.0634	−1.82684	−0.913419	+	0.407021i	$0.866568\pi$
$752$	148.881	5.42912
$753$	−9.83620	−0.358451
$754$	−143.435	−5.22360
$755$	5.16175	0.187855
$756$	−48.2601	−1.75520
$757$	23.5262	0.855074	0.427537	−	0.903998i	$-0.359381\pi$
$757$	23.5262	0.855074	0.427537	+	0.903998i	$0.359381\pi$
$758$	−66.6517	−2.42090
$759$	−1.25130	−0.0454194
$760$	30.1710	1.09442
$761$	−21.0535	−0.763190	−0.381595	−	0.924330i	$-0.624625\pi$
$761$	−21.0535	−0.763190	−0.381595	+	0.924330i	$0.624625\pi$
$762$	−6.64535	−0.240735
$763$	−29.9290	−1.08350
$764$	−93.6428	−3.38788
$765$	3.08339	0.111480
$766$	7.60968	0.274949
$767$	74.4595	2.68858
$768$	−31.2821	−1.12879
$769$	11.7780	0.424725	0.212363	−	0.977191i	$-0.431884\pi$
$769$	11.7780	0.424725	0.212363	+	0.977191i	$0.431884\pi$
$770$	−11.4584	−0.412932
$771$	−5.45662	−0.196515
$772$	−59.5656	−2.14381
$773$	−8.02273	−0.288558	−0.144279	−	0.989537i	$-0.546086\pi$
$773$	−8.02273	−0.288558	−0.144279	+	0.989537i	$0.546086\pi$
$774$	7.93438	0.285195
$775$	1.15211	0.0413849
$776$	56.6252	2.03272
$777$	−7.31835	−0.262544
$778$	61.6617	2.21068
$779$	4.99506	0.178966
$780$	14.7376	0.527692
$781$	11.9966	0.429274
$782$	−9.20821	−0.329285
$783$	18.0064	0.643497
$784$	131.030	4.67963
$785$	2.14030	0.0763907
$786$	−3.26936	−0.116614
$787$	−12.6242	−0.450005	−0.225002	−	0.974358i	$-0.572239\pi$
$787$	−12.6242	−0.450005	−0.225002	+	0.974358i	$0.572239\pi$
$788$	−57.6512	−2.05374
$789$	4.22486	0.150409
$790$	−25.3824	−0.903067
$791$	−1.15530	−0.0410779
$792$	29.4304	1.04576
$793$	−62.5218	−2.22021
$794$	42.5346	1.50950
$795$	2.74698	0.0974254
$796$	−108.793	−3.85606
$797$	−36.9899	−1.31025	−0.655124	−	0.755521i	$-0.727384\pi$
$797$	−36.9899	−1.31025	−0.655124	+	0.755521i	$0.727384\pi$
$798$	−10.8787	−0.385100
$799$	8.66756	0.306636
$800$	103.928	3.67440
$801$	−4.97324	−0.175721
$802$	−35.6838	−1.26004
$803$	−0.972938	−0.0343342
$804$	5.35047	0.188697
$805$	13.6864	0.482381
$806$	5.28792	0.186259
$807$	−6.92229	−0.243676
$808$	161.805	5.69228
$809$	40.2673	1.41572	0.707862	−	0.706351i	$-0.249660\pi$
$809$	40.2673	1.41572	0.707862	+	0.706351i	$0.249660\pi$
$810$	23.1853	0.814650
$811$	33.3352	1.17056	0.585279	−	0.810832i	$-0.300985\pi$
$811$	33.3352	1.17056	0.585279	+	0.810832i	$0.300985\pi$
$812$	−177.904	−6.24322
$813$	−8.36464	−0.293361
$814$	14.0809	0.493534
$815$	0.266314	0.00932859
$816$	−6.48087	−0.226876
$817$	2.71507	0.0949883
$818$	57.1026	1.99655
$819$	69.3014	2.42159
$820$	−11.3333	−0.395777
$821$	−34.6699	−1.20999	−0.604994	−	0.796230i	$-0.706824\pi$
$821$	−34.6699	−1.20999	−0.604994	+	0.796230i	$0.706824\pi$
$822$	15.0033	0.523300
$823$	−17.0426	−0.594066	−0.297033	−	0.954867i	$-0.595997\pi$
$823$	−17.0426	−0.594066	−0.297033	+	0.954867i	$0.595997\pi$
$824$	126.232	4.39750
$825$	1.44725	0.0503868
$826$	124.705	4.33905
$827$	−46.4347	−1.61469	−0.807346	−	0.590078i	$-0.799097\pi$
$827$	−46.4347	−1.61469	−0.807346	+	0.590078i	$0.799097\pi$
$828$	−54.1071	−1.88035
$829$	−42.4748	−1.47521	−0.737605	−	0.675233i	$-0.764043\pi$
$829$	−42.4748	−1.47521	−0.737605	+	0.675233i	$0.764043\pi$
$830$	−24.2117	−0.840399
$831$	4.25375	0.147561
$832$	259.180	8.98544
$833$	7.62830	0.264305
$834$	16.3061	0.564633
$835$	−21.1621	−0.732346
$836$	15.5009	0.536111
$837$	−0.663830	−0.0229453
$838$	−7.70328	−0.266105
$839$	−6.27558	−0.216657	−0.108329	−	0.994115i	$-0.534550\pi$
$839$	−6.27558	−0.216657	−0.108329	+	0.994115i	$0.534550\pi$
$840$	16.0361	0.553297
$841$	37.3781	1.28890
$842$	−79.6739	−2.74575
$843$	−2.74148	−0.0944215
$844$	52.9288	1.82188
$845$	−29.3537	−1.00980
$846$	68.7717	2.36442
$847$	−3.82470	−0.131418
$848$	115.900	3.98003
$849$	4.99862	0.171552
$850$	10.6502	0.365298
$851$	−16.8188	−0.576540
$852$	−25.8421	−0.885337
$853$	8.19762	0.280681	0.140341	−	0.990103i	$-0.455180\pi$
$853$	8.19762	0.280681	0.140341	+	0.990103i	$0.455180\pi$
$854$	−104.712	−3.58317
$855$	8.37163	0.286304
$856$	91.0558	3.11222
$857$	30.0186	1.02542	0.512708	−	0.858563i	$-0.328642\pi$
$857$	30.0186	1.02542	0.512708	+	0.858563i	$0.328642\pi$
$858$	6.64255	0.226773
$859$	−12.2797	−0.418979	−0.209489	−	0.977811i	$-0.567180\pi$
$859$	−12.2797	−0.418979	−0.209489	+	0.977811i	$0.567180\pi$
$860$	−6.16025	−0.210063
$861$	2.65490	0.0904788
$862$	103.291	3.51811
$863$	47.5951	1.62016	0.810078	−	0.586322i	$-0.199425\pi$
$863$	47.5951	1.62016	0.810078	+	0.586322i	$0.199425\pi$
$864$	−59.8818	−2.03722
$865$	2.47145	0.0840320
$866$	−1.87392	−0.0636784
$867$	−0.377305	−0.0128139
$868$	6.55867	0.222616
$869$	−8.47241	−0.287407
$870$	−9.20939	−0.312228
$871$	−15.7493	−0.533644
$872$	−80.5905	−2.72914
$873$	15.7119	0.531768
$874$	−25.0009	−0.845669
$875$	−36.4638	−1.23270
$876$	2.09582	0.0708112
$877$	12.7459	0.430400	0.215200	−	0.976570i	$-0.430960\pi$
$877$	12.7459	0.430400	0.215200	+	0.976570i	$0.430960\pi$
$878$	83.6338	2.82250
$879$	8.51198	0.287102
$880$	−18.5337	−0.624772
$881$	−29.9871	−1.01029	−0.505145	−	0.863034i	$-0.668561\pi$
$881$	−29.9871	−1.01029	−0.505145	+	0.863034i	$0.668561\pi$
$882$	60.5258	2.03801
$883$	2.27882	0.0766884	0.0383442	−	0.999265i	$-0.487792\pi$
$883$	2.27882	0.0766884	0.0383442	+	0.999265i	$0.487792\pi$
$884$	36.2005	1.21755
$885$	4.78075	0.160703
$886$	−12.3984	−0.416532
$887$	37.1401	1.24704	0.623521	−	0.781806i	$-0.285701\pi$
$887$	37.1401	1.24704	0.623521	+	0.781806i	$0.285701\pi$
$888$	−19.7063	−0.661299
$889$	24.2615	0.813704
$890$	5.21385	0.174769
$891$	7.73904	0.259268
$892$	−80.1252	−2.68279
$893$	23.5330	0.787503
$894$	19.9530	0.667328
$895$	16.5439	0.553003
$896$	226.820	7.57752
$897$	−7.93413	−0.264913
$898$	86.3462	2.88141
$899$	−2.44711	−0.0816158
$900$	62.5801	2.08600
$901$	6.74749	0.224792
$902$	−5.10816	−0.170083
$903$	1.44308	0.0480225
$904$	−3.11091	−0.103467
$905$	−5.29404	−0.175980
$906$	−5.01156	−0.166498
$907$	6.18018	0.205209	0.102605	−	0.994722i	$-0.467282\pi$
$907$	6.18018	0.205209	0.102605	+	0.994722i	$0.467282\pi$
$908$	−106.681	−3.54035
$909$	44.8964	1.48912
$910$	−72.6542	−2.40846
$911$	58.8512	1.94983	0.974914	−	0.222581i	$-0.0714483\pi$
$911$	58.8512	1.94983	0.974914	+	0.222581i	$0.0714483\pi$
$912$	−17.5960	−0.582662
$913$	−8.08161	−0.267462
$914$	−63.2160	−2.09100
$915$	−4.01427	−0.132708
$916$	−37.0759	−1.22502
$917$	11.9361	0.394165
$918$	−6.13649	−0.202534
$919$	19.9257	0.657289	0.328645	−	0.944454i	$-0.393408\pi$
$919$	19.9257	0.657289	0.328645	+	0.944454i	$0.393408\pi$
$920$	36.8535	1.21503
$921$	−11.0999	−0.365754
$922$	−93.0296	−3.06377
$923$	76.0671	2.50378
$924$	8.23884	0.271038
$925$	19.4525	0.639594
$926$	−49.4221	−1.62411
$927$	35.0259	1.15040
$928$	−220.746	−7.24634
$929$	42.6711	1.39999	0.699996	−	0.714147i	$-0.253185\pi$
$929$	42.6711	1.39999	0.699996	+	0.714147i	$0.253185\pi$
$930$	0.339516	0.0111332
$931$	20.7114	0.678788
$932$	−141.275	−4.62763
$933$	−6.04089	−0.197770
$934$	−3.69213	−0.120810
$935$	−1.07900	−0.0352871
$936$	186.609	6.09951
$937$	25.9637	0.848198	0.424099	−	0.905616i	$-0.360591\pi$
$937$	25.9637	0.848198	0.424099	+	0.905616i	$0.360591\pi$
$938$	−26.3770	−0.861240
$939$	−9.27149	−0.302564
$940$	−53.3943	−1.74153
$941$	−12.4935	−0.407276	−0.203638	−	0.979046i	$-0.565277\pi$
$941$	−12.4935	−0.407276	−0.203638	+	0.979046i	$0.565277\pi$
$942$	−2.07803	−0.0677058
$943$	6.10140	0.198689
$944$	201.708	6.56505
$945$	9.12080	0.296700
$946$	−2.77655	−0.0902734
$947$	43.2910	1.40677	0.703385	−	0.710809i	$-0.251671\pi$
$947$	43.2910	1.40677	0.703385	+	0.710809i	$0.251671\pi$
$948$	18.2505	0.592750
$949$	−6.16911	−0.200258
$950$	28.9160	0.938158
$951$	−8.62984	−0.279842
$952$	39.3899	1.27663
$953$	−22.6473	−0.733617	−0.366809	−	0.930296i	$-0.619550\pi$
$953$	−22.6473	−0.733617	−0.366809	+	0.930296i	$0.619550\pi$
$954$	53.5372	1.73333
$955$	17.6978	0.572687
$956$	−34.3248	−1.11014
$957$	−3.07401	−0.0993685
$958$	−16.4851	−0.532610
$959$	−54.7755	−1.76879
$960$	16.6409	0.537083
$961$	−30.9098	−0.997090
$962$	89.2826	2.87859
$963$	25.2655	0.814168
$964$	−74.8389	−2.41040
$965$	11.2574	0.362390
$966$	−13.2881	−0.427539
$967$	20.0352	0.644289	0.322145	−	0.946690i	$-0.395596\pi$
$967$	20.0352	0.644289	0.322145	+	0.946690i	$0.395596\pi$
$968$	−10.2988	−0.331017
$969$	−1.02441	−0.0329087
$970$	−16.4721	−0.528886
$971$	22.8856	0.734433	0.367216	−	0.930136i	$-0.380311\pi$
$971$	22.8856	0.734433	0.367216	+	0.930136i	$0.380311\pi$
$972$	−54.5249	−1.74889
$973$	−59.5318	−1.90850
$974$	−42.9139	−1.37505
$975$	9.17658	0.293886
$976$	−169.369	−5.42139
$977$	−7.33985	−0.234823	−0.117411	−	0.993083i	$-0.537460\pi$
$977$	−7.33985	−0.234823	−0.117411	+	0.993083i	$0.537460\pi$
$978$	−0.258566	−0.00826802
$979$	1.74033	0.0556212
$980$	−46.9922	−1.50111
$981$	−22.3616	−0.713952
$982$	−99.7274	−3.18243
$983$	−10.7895	−0.344131	−0.172065	−	0.985086i	$-0.555044\pi$
$983$	−10.7895	−0.344131	−0.172065	+	0.985086i	$0.555044\pi$
$984$	7.14890	0.227899
$985$	10.8956	0.347164
$986$	−22.6213	−0.720409
$987$	12.5079	0.398132
$988$	98.2868	3.12692
$989$	3.31642	0.105456
$990$	−8.56119	−0.272093
$991$	−43.3699	−1.37769	−0.688846	−	0.724908i	$-0.741882\pi$
$991$	−43.3699	−1.37769	−0.688846	+	0.724908i	$0.741882\pi$
$992$	8.13808	0.258384
$993$	−9.45786	−0.300136
$994$	127.398	4.04081
$995$	20.5610	0.651828
$996$	17.4087	0.551616
$997$	19.0086	0.602009	0.301005	−	0.953623i	$-0.402678\pi$
$997$	19.0086	0.602009	0.301005	+	0.953623i	$0.402678\pi$
$998$	−101.581	−3.21548
$999$	−11.2083	−0.354614

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.j.1.1	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.j.1.1	✓	82	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.77655 q^{2} -0.377305 q^{3} +5.70922 q^{4} -1.07900 q^{5} +1.04760 q^{6} -3.82470 q^{7} -10.2988 q^{8} -2.85764 q^{9} +O(q^{10})\)	\(q-2.77655 q^{2} -0.377305 q^{3} +5.70922 q^{4} -1.07900 q^{5} +1.04760 q^{6} -3.82470 q^{7} -10.2988 q^{8} -2.85764 q^{9} +2.99589 q^{10} +1.00000 q^{11} -2.15412 q^{12} +6.34070 q^{13} +10.6195 q^{14} +0.407111 q^{15} +17.1768 q^{16} +1.00000 q^{17} +7.93438 q^{18} +2.71507 q^{19} -6.16025 q^{20} +1.44308 q^{21} -2.77655 q^{22} +3.31642 q^{23} +3.88580 q^{24} -3.83576 q^{25} -17.6053 q^{26} +2.21011 q^{27} -21.8360 q^{28} +8.14728 q^{29} -1.13036 q^{30} -0.300360 q^{31} -27.0945 q^{32} -0.377305 q^{33} -2.77655 q^{34} +4.12685 q^{35} -16.3149 q^{36} -5.07135 q^{37} -7.53852 q^{38} -2.39238 q^{39} +11.1124 q^{40} +1.83975 q^{41} -4.00677 q^{42} +1.00000 q^{43} +5.70922 q^{44} +3.08339 q^{45} -9.20821 q^{46} +8.66756 q^{47} -6.48087 q^{48} +7.62830 q^{49} +10.6502 q^{50} -0.377305 q^{51} +36.2005 q^{52} +6.74749 q^{53} -6.13649 q^{54} -1.07900 q^{55} +39.3899 q^{56} -1.02441 q^{57} -22.6213 q^{58} +11.7431 q^{59} +2.32429 q^{60} -9.86038 q^{61} +0.833964 q^{62} +10.9296 q^{63} +40.8755 q^{64} -6.84162 q^{65} +1.04760 q^{66} -2.48384 q^{67} +5.70922 q^{68} -1.25130 q^{69} -11.4584 q^{70} +11.9966 q^{71} +29.4304 q^{72} -0.972938 q^{73} +14.0809 q^{74} +1.44725 q^{75} +15.5009 q^{76} -3.82470 q^{77} +6.64255 q^{78} -8.47241 q^{79} -18.5337 q^{80} +7.73904 q^{81} -5.10816 q^{82} -8.08161 q^{83} +8.23884 q^{84} -1.07900 q^{85} -2.77655 q^{86} -3.07401 q^{87} -10.2988 q^{88} +1.74033 q^{89} -8.56119 q^{90} -24.2513 q^{91} +18.9342 q^{92} +0.113327 q^{93} -24.0659 q^{94} -2.92956 q^{95} +10.2229 q^{96} -5.49821 q^{97} -21.1803 q^{98} -2.85764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9}+O(q^{10}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9} + q^{10} + 82 q^{11} + 3 q^{12} + 26 q^{13} + 17 q^{14} + 66 q^{15} + 122 q^{16} + 82 q^{17} + 18 q^{18} + 12 q^{19} + 9 q^{20} + 22 q^{21} + 8 q^{22} + 50 q^{23} + 15 q^{24} + 117 q^{25} + 36 q^{26} + 30 q^{27} + 11 q^{28} + 33 q^{29} - 26 q^{30} + 40 q^{31} + 58 q^{32} + 6 q^{33} + 8 q^{34} + 16 q^{35} + 160 q^{36} + 31 q^{37} + 18 q^{38} + 41 q^{39} - 29 q^{40} + 42 q^{41} - 51 q^{42} + 82 q^{43} + 98 q^{44} - 2 q^{45} - 19 q^{46} + 84 q^{47} - 46 q^{48} + 136 q^{49} + 59 q^{50} + 6 q^{51} + 45 q^{52} + 83 q^{53} + 24 q^{54} + 11 q^{55} + 21 q^{56} + 23 q^{57} + 14 q^{58} + 96 q^{59} + 184 q^{60} - 6 q^{61} - 23 q^{62} + 8 q^{63} + 148 q^{64} + 5 q^{65} + 10 q^{66} + 78 q^{67} + 98 q^{68} + 61 q^{69} - 3 q^{70} + 155 q^{71} + 50 q^{72} - 23 q^{73} + 10 q^{74} - 19 q^{75} + 44 q^{76} + 8 q^{77} - 27 q^{78} + 31 q^{79} + 19 q^{80} + 150 q^{81} - 12 q^{82} + 54 q^{83} + 8 q^{84} + 11 q^{85} + 8 q^{86} + 20 q^{87} + 30 q^{88} + 25 q^{89} - 81 q^{90} - 14 q^{91} + 60 q^{92} + 36 q^{93} + 19 q^{94} + 111 q^{95} - 6 q^{96} + 2 q^{97} - 5 q^{98} + 108 q^{99}+O(q^{100}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9 + q^10 + 82 * q^11 + 3 * q^12 + 26 * q^13 + 17 * q^14 + 66 * q^15 + 122 * q^16 + 82 * q^17 + 18 * q^18 + 12 * q^19 + 9 * q^20 + 22 * q^21 + 8 * q^22 + 50 * q^23 + 15 * q^24 + 117 * q^25 + 36 * q^26 + 30 * q^27 + 11 * q^28 + 33 * q^29 - 26 * q^30 + 40 * q^31 + 58 * q^32 + 6 * q^33 + 8 * q^34 + 16 * q^35 + 160 * q^36 + 31 * q^37 + 18 * q^38 + 41 * q^39 - 29 * q^40 + 42 * q^41 - 51 * q^42 + 82 * q^43 + 98 * q^44 - 2 * q^45 - 19 * q^46 + 84 * q^47 - 46 * q^48 + 136 * q^49 + 59 * q^50 + 6 * q^51 + 45 * q^52 + 83 * q^53 + 24 * q^54 + 11 * q^55 + 21 * q^56 + 23 * q^57 + 14 * q^58 + 96 * q^59 + 184 * q^60 - 6 * q^61 - 23 * q^62 + 8 * q^63 + 148 * q^64 + 5 * q^65 + 10 * q^66 + 78 * q^67 + 98 * q^68 + 61 * q^69 - 3 * q^70 + 155 * q^71 + 50 * q^72 - 23 * q^73 + 10 * q^74 - 19 * q^75 + 44 * q^76 + 8 * q^77 - 27 * q^78 + 31 * q^79 + 19 * q^80 + 150 * q^81 - 12 * q^82 + 54 * q^83 + 8 * q^84 + 11 * q^85 + 8 * q^86 + 20 * q^87 + 30 * q^88 + 25 * q^89 - 81 * q^90 - 14 * q^91 + 60 * q^92 + 36 * q^93 + 19 * q^94 + 111 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^98 + 108 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more