LMFDB - Embedded newform 8009.2.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8009,2,Mod(1,8009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8009, 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("8009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8009 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8009.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:	$63.9521869788$
Analytic rank:	$0$
Dimension:	$361$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80354 q^{2} -2.63483 q^{3} +5.85984 q^{4} +1.64291 q^{5} +7.38685 q^{6} +0.300217 q^{7} -10.8212 q^{8} +3.94232 q^{9} +O(q^{10})$	\(q-2.80354 q^{2} -2.63483 q^{3} +5.85984 q^{4} +1.64291 q^{5} +7.38685 q^{6} +0.300217 q^{7} -10.8212 q^{8} +3.94232 q^{9} -4.60596 q^{10} -2.34583 q^{11} -15.4397 q^{12} +5.70130 q^{13} -0.841672 q^{14} -4.32878 q^{15} +18.6180 q^{16} -8.03641 q^{17} -11.0525 q^{18} -4.22401 q^{19} +9.62717 q^{20} -0.791022 q^{21} +6.57664 q^{22} +2.98455 q^{23} +28.5120 q^{24} -2.30085 q^{25} -15.9838 q^{26} -2.48286 q^{27} +1.75922 q^{28} +6.07465 q^{29} +12.1359 q^{30} -0.222781 q^{31} -30.5539 q^{32} +6.18087 q^{33} +22.5304 q^{34} +0.493230 q^{35} +23.1014 q^{36} -6.44397 q^{37} +11.8422 q^{38} -15.0220 q^{39} -17.7782 q^{40} -7.24084 q^{41} +2.21766 q^{42} +4.36362 q^{43} -13.7462 q^{44} +6.47688 q^{45} -8.36729 q^{46} -3.39513 q^{47} -49.0552 q^{48} -6.90987 q^{49} +6.45053 q^{50} +21.1746 q^{51} +33.4087 q^{52} -10.9260 q^{53} +6.96080 q^{54} -3.85399 q^{55} -3.24871 q^{56} +11.1295 q^{57} -17.0305 q^{58} +6.17525 q^{59} -25.3660 q^{60} +5.75341 q^{61} +0.624577 q^{62} +1.18355 q^{63} +48.4231 q^{64} +9.36672 q^{65} -17.3283 q^{66} -0.502281 q^{67} -47.0920 q^{68} -7.86377 q^{69} -1.38279 q^{70} -3.79005 q^{71} -42.6607 q^{72} -12.6963 q^{73} +18.0659 q^{74} +6.06235 q^{75} -24.7520 q^{76} -0.704260 q^{77} +42.1147 q^{78} +7.45220 q^{79} +30.5877 q^{80} -5.28506 q^{81} +20.3000 q^{82} -12.0447 q^{83} -4.63526 q^{84} -13.2031 q^{85} -12.2336 q^{86} -16.0057 q^{87} +25.3847 q^{88} -10.4734 q^{89} -18.1582 q^{90} +1.71163 q^{91} +17.4889 q^{92} +0.586991 q^{93} +9.51839 q^{94} -6.93966 q^{95} +80.5043 q^{96} -1.86055 q^{97} +19.3721 q^{98} -9.24803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) $ 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * 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 $

$2$	−2.80354	−1.98240	−0.991201	−	0.132365i	$-0.957743\pi$
$2$	−2.80354	−1.98240	−0.991201	+	0.132365i	$0.957743\pi$
$3$	−2.63483	−1.52122	−0.760610	−	0.649210i	$-0.775100\pi$
$3$	−2.63483	−1.52122	−0.760610	+	0.649210i	$0.775100\pi$
$4$	5.85984	2.92992
$5$	1.64291	0.734731	0.367365	−	0.930077i	$-0.380260\pi$
$5$	1.64291	0.734731	0.367365	+	0.930077i	$0.380260\pi$
$6$	7.38685	3.01567
$7$	0.300217	0.113472	0.0567358	−	0.998389i	$-0.481931\pi$
$7$	0.300217	0.113472	0.0567358	+	0.998389i	$0.481931\pi$
$8$	−10.8212	−3.82587
$9$	3.94232	1.31411
$10$	−4.60596	−1.45653
$11$	−2.34583	−0.707295	−0.353648	−	0.935379i	$-0.615059\pi$
$11$	−2.34583	−0.707295	−0.353648	+	0.935379i	$0.615059\pi$
$12$	−15.4397	−4.45705
$13$	5.70130	1.58126	0.790629	−	0.612296i	$-0.209754\pi$
$13$	5.70130	1.58126	0.790629	+	0.612296i	$0.209754\pi$
$14$	−0.841672	−0.224946
$15$	−4.32878	−1.11769
$16$	18.6180	4.65450
$17$	−8.03641	−1.94911	−0.974557	−	0.224139i	$-0.928043\pi$
$17$	−8.03641	−1.94911	−0.974557	+	0.224139i	$0.928043\pi$
$18$	−11.0525	−2.60509
$19$	−4.22401	−0.969055	−0.484527	−	0.874776i	$-0.661008\pi$
$19$	−4.22401	−0.969055	−0.484527	+	0.874776i	$0.661008\pi$
$20$	9.62717	2.15270
$21$	−0.791022	−0.172615
$22$	6.57664	1.40214
$23$	2.98455	0.622321	0.311160	−	0.950357i	$-0.399282\pi$
$23$	2.98455	0.622321	0.311160	+	0.950357i	$0.399282\pi$
$24$	28.5120	5.81999
$25$	−2.30085	−0.460170
$26$	−15.9838	−3.13469
$27$	−2.48286	−0.477827
$28$	1.75922	0.332462
$29$	6.07465	1.12803	0.564017	−	0.825763i	$-0.309255\pi$
$29$	6.07465	1.12803	0.564017	+	0.825763i	$0.309255\pi$
$30$	12.1359	2.21570
$31$	−0.222781	−0.0400127	−0.0200064	−	0.999800i	$-0.506369\pi$
$31$	−0.222781	−0.0400127	−0.0200064	+	0.999800i	$0.506369\pi$
$32$	−30.5539	−5.40122
$33$	6.18087	1.07595
$34$	22.5304	3.86393
$35$	0.493230	0.0833710
$36$	23.1014	3.85023
$37$	−6.44397	−1.05938	−0.529691	−	0.848190i	$-0.677692\pi$
$37$	−6.44397	−1.05938	−0.529691	+	0.848190i	$0.677692\pi$
$38$	11.8422	1.92106
$39$	−15.0220	−2.40544
$40$	−17.7782	−2.81099
$41$	−7.24084	−1.13083	−0.565415	−	0.824807i	$-0.691284\pi$
$41$	−7.24084	−1.13083	−0.565415	+	0.824807i	$0.691284\pi$
$42$	2.21766	0.342192
$43$	4.36362	0.665446	0.332723	−	0.943025i	$-0.392033\pi$
$43$	4.36362	0.665446	0.332723	+	0.943025i	$0.392033\pi$
$44$	−13.7462	−2.07232
$45$	6.47688	0.965516
$46$	−8.36729	−1.23369
$47$	−3.39513	−0.495231	−0.247615	−	0.968858i	$-0.579647\pi$
$47$	−3.39513	−0.495231	−0.247615	+	0.968858i	$0.579647\pi$
$48$	−49.0552	−7.08052
$49$	−6.90987	−0.987124
$50$	6.45053	0.912243
$51$	21.1746	2.96503
$52$	33.4087	4.63295
$53$	−10.9260	−1.50080	−0.750399	−	0.660985i	$-0.770139\pi$
$53$	−10.9260	−1.50080	−0.750399	+	0.660985i	$0.770139\pi$
$54$	6.96080	0.947245
$55$	−3.85399	−0.519672
$56$	−3.24871	−0.434128
$57$	11.1295	1.47414
$58$	−17.0305	−2.23622
$59$	6.17525	0.803950	0.401975	−	0.915651i	$-0.368324\pi$
$59$	6.17525	0.803950	0.401975	+	0.915651i	$0.368324\pi$
$60$	−25.3660	−3.27473
$61$	5.75341	0.736648	0.368324	−	0.929697i	$-0.379932\pi$
$61$	5.75341	0.736648	0.368324	+	0.929697i	$0.379932\pi$
$62$	0.624577	0.0793213
$63$	1.18355	0.149114
$64$	48.4231	6.05288
$65$	9.36672	1.16180
$66$	−17.3283	−2.13297
$67$	−0.502281	−0.0613634	−0.0306817	−	0.999529i	$-0.509768\pi$
$67$	−0.502281	−0.0613634	−0.0306817	+	0.999529i	$0.509768\pi$
$68$	−47.0920	−5.71075
$69$	−7.86377	−0.946686
$70$	−1.38279	−0.165275
$71$	−3.79005	−0.449797	−0.224898	−	0.974382i	$-0.572205\pi$
$71$	−3.79005	−0.449797	−0.224898	+	0.974382i	$0.572205\pi$
$72$	−42.6607	−5.02761
$73$	−12.6963	−1.48599	−0.742997	−	0.669295i	$-0.766596\pi$
$73$	−12.6963	−1.48599	−0.742997	+	0.669295i	$0.766596\pi$
$74$	18.0659	2.10012
$75$	6.06235	0.700020
$76$	−24.7520	−2.83925
$77$	−0.704260	−0.0802579
$78$	42.1147	4.76855
$79$	7.45220	0.838438	0.419219	−	0.907885i	$-0.362304\pi$
$79$	7.45220	0.838438	0.419219	+	0.907885i	$0.362304\pi$
$80$	30.5877	3.41981
$81$	−5.28506	−0.587228
$82$	20.3000	2.24176
$83$	−12.0447	−1.32208	−0.661040	−	0.750351i	$-0.729885\pi$
$83$	−12.0447	−1.32208	−0.661040	+	0.750351i	$0.729885\pi$
$84$	−4.63526	−0.505748
$85$	−13.2031	−1.43207
$86$	−12.2336	−1.31918
$87$	−16.0057	−1.71599
$88$	25.3847	2.70602
$89$	−10.4734	−1.11017	−0.555087	−	0.831792i	$-0.687315\pi$
$89$	−10.4734	−1.11017	−0.555087	+	0.831792i	$0.687315\pi$
$90$	−18.1582	−1.91404
$91$	1.71163	0.179428
$92$	17.4889	1.82335
$93$	0.586991	0.0608681
$94$	9.51839	0.981747
$95$	−6.93966	−0.711995
$96$	80.5043	8.21644
$97$	−1.86055	−0.188910	−0.0944550	−	0.995529i	$-0.530111\pi$
$97$	−1.86055	−0.188910	−0.0944550	+	0.995529i	$0.530111\pi$
$98$	19.3721	1.95688
$99$	−9.24803	−0.929462
$100$	−13.4826	−1.34826
$101$	10.6209	1.05682	0.528410	−	0.848989i	$-0.322788\pi$
$101$	10.6209	1.05682	0.528410	+	0.848989i	$0.322788\pi$
$102$	−59.3637	−5.87788
$103$	2.07256	0.204215	0.102108	−	0.994773i	$-0.467441\pi$
$103$	2.07256	0.204215	0.102108	+	0.994773i	$0.467441\pi$
$104$	−61.6950	−6.04969
$105$	−1.29958	−0.126826
$106$	30.6314	2.97519
$107$	2.37243	0.229351	0.114676	−	0.993403i	$-0.463417\pi$
$107$	2.37243	0.229351	0.114676	+	0.993403i	$0.463417\pi$
$108$	−14.5492	−1.39999
$109$	12.1282	1.16167	0.580833	−	0.814022i	$-0.302727\pi$
$109$	12.1282	1.16167	0.580833	+	0.814022i	$0.302727\pi$
$110$	10.8048	1.03020
$111$	16.9788	1.61155
$112$	5.58945	0.528153
$113$	6.76506	0.636403	0.318202	−	0.948023i	$-0.396921\pi$
$113$	6.76506	0.636403	0.318202	+	0.948023i	$0.396921\pi$
$114$	−31.2021	−2.92235
$115$	4.90334	0.457238
$116$	35.5964	3.30504
$117$	22.4764	2.07794
$118$	−17.3126	−1.59375
$119$	−2.41267	−0.221169
$120$	46.8426	4.27613
$121$	−5.49707	−0.499733
$122$	−16.1299	−1.46033
$123$	19.0784	1.72024
$124$	−1.30546	−0.117234
$125$	−11.9946	−1.07283
$126$	−3.31814	−0.295604
$127$	1.08622	0.0963869	0.0481934	−	0.998838i	$-0.484654\pi$
$127$	1.08622	0.0963869	0.0481934	+	0.998838i	$0.484654\pi$
$128$	−74.6482	−6.59803
$129$	−11.4974	−1.01229
$130$	−26.2600	−2.30315
$131$	21.9731	1.91980	0.959898	−	0.280350i	$-0.0904505\pi$
$131$	21.9731	1.91980	0.959898	+	0.280350i	$0.0904505\pi$
$132$	36.2189	3.15245
$133$	−1.26812	−0.109960
$134$	1.40817	0.121647
$135$	−4.07911	−0.351074
$136$	86.9636	7.45706
$137$	−22.4575	−1.91868	−0.959339	−	0.282256i	$-0.908917\pi$
$137$	−22.4575	−1.91868	−0.959339	+	0.282256i	$0.908917\pi$
$138$	22.0464	1.87671
$139$	−2.61524	−0.221821	−0.110911	−	0.993830i	$-0.535377\pi$
$139$	−2.61524	−0.221821	−0.110911	+	0.993830i	$0.535377\pi$
$140$	2.89025	0.244270
$141$	8.94559	0.753355
$142$	10.6256	0.891678
$143$	−13.3743	−1.11842
$144$	73.3982	6.11652
$145$	9.98009	0.828801
$146$	35.5947	2.94584
$147$	18.2063	1.50163
$148$	−37.7606	−3.10390
$149$	14.1547	1.15959	0.579797	−	0.814761i	$-0.303132\pi$
$149$	14.1547	1.15959	0.579797	+	0.814761i	$0.303132\pi$
$150$	−16.9960	−1.38772
$151$	−0.853729	−0.0694755	−0.0347377	−	0.999396i	$-0.511060\pi$
$151$	−0.853729	−0.0694755	−0.0347377	+	0.999396i	$0.511060\pi$
$152$	45.7089	3.70748
$153$	−31.6821	−2.56135
$154$	1.97442	0.159103
$155$	−0.366009	−0.0293986
$156$	−88.0262	−7.04774
$157$	11.7899	0.940936	0.470468	−	0.882417i	$-0.344085\pi$
$157$	11.7899	0.940936	0.470468	+	0.882417i	$0.344085\pi$
$158$	−20.8925	−1.66212
$159$	28.7881	2.28304
$160$	−50.1973	−3.96844
$161$	0.896013	0.0706157
$162$	14.8169	1.16412
$163$	25.3163	1.98293	0.991464	−	0.130384i	$-0.0416209\pi$
$163$	25.3163	1.98293	0.991464	+	0.130384i	$0.0416209\pi$
$164$	−42.4301	−3.31324
$165$	10.1546	0.790535
$166$	33.7678	2.62089
$167$	15.7269	1.21698	0.608492	−	0.793560i	$-0.291775\pi$
$167$	15.7269	1.21698	0.608492	+	0.793560i	$0.291775\pi$
$168$	8.55980	0.660403
$169$	19.5049	1.50037
$170$	37.0154	2.83895
$171$	−16.6524	−1.27344
$172$	25.5701	1.94970
$173$	12.5482	0.954019	0.477009	−	0.878898i	$-0.341721\pi$
$173$	12.5482	0.954019	0.477009	+	0.878898i	$0.341721\pi$
$174$	44.8725	3.40177
$175$	−0.690756	−0.0522162
$176$	−43.6747	−3.29211
$177$	−16.2707	−1.22298
$178$	29.3625	2.20081
$179$	−0.514927	−0.0384874	−0.0192437	−	0.999815i	$-0.506126\pi$
$179$	−0.514927	−0.0384874	−0.0192437	+	0.999815i	$0.506126\pi$
$180$	37.9534	2.82888
$181$	16.1745	1.20224	0.601122	−	0.799157i	$-0.294721\pi$
$181$	16.1745	1.20224	0.601122	+	0.799157i	$0.294721\pi$
$182$	−4.79862	−0.355698
$183$	−15.1592	−1.12060
$184$	−32.2964	−2.38092
$185$	−10.5869	−0.778361
$186$	−1.64565	−0.120665
$187$	18.8521	1.37860
$188$	−19.8949	−1.45099
$189$	−0.745398	−0.0542197
$190$	19.4556	1.41146
$191$	−9.65848	−0.698863	−0.349431	−	0.936962i	$-0.613625\pi$
$191$	−9.65848	−0.698863	−0.349431	+	0.936962i	$0.613625\pi$
$192$	−127.587	−9.20776
$193$	−5.58895	−0.402301	−0.201151	−	0.979560i	$-0.564468\pi$
$193$	−5.58895	−0.402301	−0.201151	+	0.979560i	$0.564468\pi$
$194$	5.21612	0.374496
$195$	−24.6797	−1.76735
$196$	−40.4907	−2.89219
$197$	5.33960	0.380431	0.190215	−	0.981742i	$-0.439081\pi$
$197$	5.33960	0.380431	0.190215	+	0.981742i	$0.439081\pi$
$198$	25.9272	1.84257
$199$	12.5625	0.890530	0.445265	−	0.895399i	$-0.353110\pi$
$199$	12.5625	0.890530	0.445265	+	0.895399i	$0.353110\pi$
$200$	24.8980	1.76055
$201$	1.32343	0.0933472
$202$	−29.7761	−2.09504
$203$	1.82371	0.128000
$204$	124.079	8.68730
$205$	−11.8960	−0.830855
$206$	−5.81051	−0.404837
$207$	11.7660	0.817797
$208$	106.147	7.35996
$209$	9.90883	0.685408
$210$	3.64341	0.251419
$211$	11.2671	0.775660	0.387830	−	0.921731i	$-0.373225\pi$
$211$	11.2671	0.775660	0.387830	+	0.921731i	$0.373225\pi$
$212$	−64.0244	−4.39721
$213$	9.98615	0.684239
$214$	−6.65120	−0.454667
$215$	7.16903	0.488924
$216$	26.8675	1.82810
$217$	−0.0668829	−0.00454030
$218$	−34.0018	−2.30289
$219$	33.4527	2.26052
$220$	−22.5837	−1.52260
$221$	−45.8180	−3.08205
$222$	−47.6007	−3.19475
$223$	18.6044	1.24584	0.622922	−	0.782284i	$-0.285945\pi$
$223$	18.6044	1.24584	0.622922	+	0.782284i	$0.285945\pi$
$224$	−9.17281	−0.612884
$225$	−9.07070	−0.604713
$226$	−18.9661	−1.26161
$227$	−24.0463	−1.59601	−0.798003	−	0.602653i	$-0.794110\pi$
$227$	−24.0463	−1.59601	−0.798003	+	0.602653i	$0.794110\pi$
$228$	65.2173	4.31912
$229$	0.820996	0.0542529	0.0271265	−	0.999632i	$-0.491364\pi$
$229$	0.820996	0.0542529	0.0271265	+	0.999632i	$0.491364\pi$
$230$	−13.7467	−0.906430
$231$	1.85560	0.122090
$232$	−65.7350	−4.31571
$233$	11.0272	0.722413	0.361207	−	0.932486i	$-0.382365\pi$
$233$	11.0272	0.722413	0.361207	+	0.932486i	$0.382365\pi$
$234$	−63.0134	−4.11932
$235$	−5.57789	−0.363862
$236$	36.1860	2.35551
$237$	−19.6353	−1.27545
$238$	6.76401	0.438446
$239$	−24.1466	−1.56191	−0.780956	−	0.624586i	$-0.785268\pi$
$239$	−24.1466	−1.56191	−0.780956	+	0.624586i	$0.785268\pi$
$240$	−80.5933	−5.20227
$241$	−24.9558	−1.60754	−0.803772	−	0.594938i	$-0.797177\pi$
$241$	−24.9558	−1.60754	−0.803772	+	0.594938i	$0.797177\pi$
$242$	15.4112	0.990672
$243$	21.3738	1.37113
$244$	33.7140	2.15832
$245$	−11.3523	−0.725271
$246$	−53.4870	−3.41021
$247$	−24.0824	−1.53232
$248$	2.41076	0.153084
$249$	31.7358	2.01117
$250$	33.6274	2.12679
$251$	6.67484	0.421312	0.210656	−	0.977560i	$-0.432440\pi$
$251$	6.67484	0.421312	0.210656	+	0.977560i	$0.432440\pi$
$252$	6.93543	0.436891
$253$	−7.00125	−0.440165
$254$	−3.04527	−0.191078
$255$	34.7879	2.17850
$256$	112.433	7.02707
$257$	−25.8776	−1.61420	−0.807102	−	0.590413i	$-0.798965\pi$
$257$	−25.8776	−1.61420	−0.807102	+	0.590413i	$0.798965\pi$
$258$	32.2334	2.00676
$259$	−1.93459	−0.120210
$260$	54.8874	3.40397
$261$	23.9482	1.48236
$262$	−61.6024	−3.80581
$263$	−6.02969	−0.371807	−0.185903	−	0.982568i	$-0.559521\pi$
$263$	−6.02969	−0.371807	−0.185903	+	0.982568i	$0.559521\pi$
$264$	−66.8844	−4.11645
$265$	−17.9504	−1.10268
$266$	3.55523	0.217985
$267$	27.5955	1.68882
$268$	−2.94329	−0.179790
$269$	13.2646	0.808758	0.404379	−	0.914592i	$-0.367488\pi$
$269$	13.2646	0.808758	0.404379	+	0.914592i	$0.367488\pi$
$270$	11.4360	0.695970
$271$	−0.416023	−0.0252716	−0.0126358	−	0.999920i	$-0.504022\pi$
$271$	−0.416023	−0.0252716	−0.0126358	+	0.999920i	$0.504022\pi$
$272$	−149.622	−9.07215
$273$	−4.50985	−0.272949
$274$	62.9606	3.80359
$275$	5.39742	0.325476
$276$	−46.0804	−2.77371
$277$	−8.90993	−0.535346	−0.267673	−	0.963510i	$-0.586255\pi$
$277$	−8.90993	−0.535346	−0.267673	+	0.963510i	$0.586255\pi$
$278$	7.33192	0.439739
$279$	−0.878276	−0.0525810
$280$	−5.33734	−0.318967
$281$	−11.7536	−0.701164	−0.350582	−	0.936532i	$-0.614016\pi$
$281$	−11.7536	−0.701164	−0.350582	+	0.936532i	$0.614016\pi$
$282$	−25.0793	−1.49345
$283$	−4.38276	−0.260528	−0.130264	−	0.991479i	$-0.541582\pi$
$283$	−4.38276	−0.260528	−0.130264	+	0.991479i	$0.541582\pi$
$284$	−22.2091	−1.31787
$285$	18.2848	1.08310
$286$	37.4954	2.21715
$287$	−2.17383	−0.128317
$288$	−120.453	−7.09778
$289$	47.5838	2.79905
$290$	−27.9796	−1.64302
$291$	4.90223	0.287374
$292$	−74.3985	−4.35384
$293$	−4.23463	−0.247390	−0.123695	−	0.992320i	$-0.539474\pi$
$293$	−4.23463	−0.247390	−0.123695	+	0.992320i	$0.539474\pi$
$294$	−51.0422	−2.97684
$295$	10.1454	0.590687
$296$	69.7316	4.05306
$297$	5.82438	0.337965
$298$	−39.6831	−2.29878
$299$	17.0158	0.984049
$300$	35.5244	2.05100
$301$	1.31004	0.0755092
$302$	2.39346	0.137728
$303$	−27.9843	−1.60765
$304$	−78.6427	−4.51047
$305$	9.45232	0.541238
$306$	88.8221	5.07762
$307$	−25.9262	−1.47968	−0.739842	−	0.672780i	$-0.765100\pi$
$307$	−25.9262	−1.47968	−0.739842	+	0.672780i	$0.765100\pi$
$308$	−4.12685	−0.235149
$309$	−5.46084	−0.310656
$310$	1.02612	0.0582798
$311$	27.6651	1.56874	0.784372	−	0.620291i	$-0.212986\pi$
$311$	27.6651	1.56874	0.784372	+	0.620291i	$0.212986\pi$
$312$	162.556	9.20290
$313$	−15.0451	−0.850400	−0.425200	−	0.905099i	$-0.639796\pi$
$313$	−15.0451	−0.850400	−0.425200	+	0.905099i	$0.639796\pi$
$314$	−33.0534	−1.86531
$315$	1.94447	0.109559
$316$	43.6687	2.45655
$317$	−3.61171	−0.202854	−0.101427	−	0.994843i	$-0.532341\pi$
$317$	−3.61171	−0.202854	−0.101427	+	0.994843i	$0.532341\pi$
$318$	−80.7085	−4.52591
$319$	−14.2501	−0.797853
$320$	79.5547	4.44724
$321$	−6.25095	−0.348894
$322$	−2.51201	−0.139989
$323$	33.9459	1.88880
$324$	−30.9696	−1.72053
$325$	−13.1179	−0.727648
$326$	−70.9753	−3.93096
$327$	−31.9556	−1.76715
$328$	78.3546	4.32641
$329$	−1.01928	−0.0561946
$330$	−28.4688	−1.56716
$331$	−15.2868	−0.840240	−0.420120	−	0.907469i	$-0.638012\pi$
$331$	−15.2868	−0.840240	−0.420120	+	0.907469i	$0.638012\pi$
$332$	−70.5801	−3.87358
$333$	−25.4042	−1.39214
$334$	−44.0910	−2.41255
$335$	−0.825202	−0.0450856
$336$	−14.7272	−0.803437
$337$	−18.3039	−0.997079	−0.498540	−	0.866867i	$-0.666130\pi$
$337$	−18.3039	−0.997079	−0.498540	+	0.866867i	$0.666130\pi$
$338$	−54.6826	−2.97434
$339$	−17.8248	−0.968108
$340$	−77.3679	−4.19586
$341$	0.522608	0.0283008
$342$	46.6857	2.52447
$343$	−4.17599	−0.225482
$344$	−47.2196	−2.54591
$345$	−12.9194	−0.695560
$346$	−35.1793	−1.89125
$347$	−12.8040	−0.687356	−0.343678	−	0.939087i	$-0.611673\pi$
$347$	−12.8040	−0.687356	−0.343678	+	0.939087i	$0.611673\pi$
$348$	−93.7905	−5.02770
$349$	−20.6391	−1.10478	−0.552392	−	0.833585i	$-0.686285\pi$
$349$	−20.6391	−1.10478	−0.552392	+	0.833585i	$0.686285\pi$
$350$	1.93656	0.103514
$351$	−14.1555	−0.755567
$352$	71.6744	3.82026
$353$	12.4087	0.660450	0.330225	−	0.943902i	$-0.392875\pi$
$353$	12.4087	0.660450	0.330225	+	0.943902i	$0.392875\pi$
$354$	45.6157	2.42444
$355$	−6.22671	−0.330480
$356$	−61.3722	−3.25272
$357$	6.35697	0.336447
$358$	1.44362	0.0762976
$359$	20.7215	1.09364	0.546820	−	0.837250i	$-0.315838\pi$
$359$	20.7215	1.09364	0.546820	+	0.837250i	$0.315838\pi$
$360$	−70.0876	−3.69394
$361$	−1.15773	−0.0609330
$362$	−45.3460	−2.38333
$363$	14.4838	0.760204
$364$	10.0299	0.525708
$365$	−20.8589	−1.09181
$366$	42.4995	2.22149
$367$	14.7934	0.772211	0.386105	−	0.922455i	$-0.373820\pi$
$367$	14.7934	0.772211	0.386105	+	0.922455i	$0.373820\pi$
$368$	55.5663	2.89659
$369$	−28.5457	−1.48603
$370$	29.6807	1.54303
$371$	−3.28017	−0.170298
$372$	3.43967	0.178339
$373$	−5.26852	−0.272793	−0.136397	−	0.990654i	$-0.543552\pi$
$373$	−5.26852	−0.272793	−0.136397	+	0.990654i	$0.543552\pi$
$374$	−52.8525	−2.73294
$375$	31.6038	1.63201
$376$	36.7394	1.89469
$377$	34.6334	1.78371
$378$	2.08975	0.107485
$379$	−22.0905	−1.13471	−0.567357	−	0.823472i	$-0.692034\pi$
$379$	−22.0905	−1.13471	−0.567357	+	0.823472i	$0.692034\pi$
$380$	−40.6653	−2.08609
$381$	−2.86202	−0.146626
$382$	27.0779	1.38543
$383$	25.1183	1.28349	0.641743	−	0.766920i	$-0.278212\pi$
$383$	25.1183	1.28349	0.641743	+	0.766920i	$0.278212\pi$
$384$	196.685	10.0371
$385$	−1.15703	−0.0589680
$386$	15.6688	0.797523
$387$	17.2028	0.874468
$388$	−10.9025	−0.553491
$389$	31.8000	1.61233	0.806163	−	0.591694i	$-0.201541\pi$
$389$	31.8000	1.61233	0.806163	+	0.591694i	$0.201541\pi$
$390$	69.1905	3.50360
$391$	−23.9850	−1.21297
$392$	74.7731	3.77661
$393$	−57.8953	−2.92043
$394$	−14.9698	−0.754167
$395$	12.2433	0.616026
$396$	−54.1920	−2.72325
$397$	10.5380	0.528887	0.264444	−	0.964401i	$-0.414812\pi$
$397$	10.5380	0.528887	0.264444	+	0.964401i	$0.414812\pi$
$398$	−35.2194	−1.76539
$399$	3.34128	0.167273
$400$	−42.8373	−2.14186
$401$	9.92306	0.495534	0.247767	−	0.968820i	$-0.420303\pi$
$401$	9.92306	0.495534	0.247767	+	0.968820i	$0.420303\pi$
$402$	−3.71028	−0.185052
$403$	−1.27014	−0.0632704
$404$	62.2368	3.09640
$405$	−8.68286	−0.431455
$406$	−5.11286	−0.253747
$407$	15.1165	0.749297
$408$	−229.134	−11.3438
$409$	7.26647	0.359304	0.179652	−	0.983730i	$-0.442503\pi$
$409$	7.26647	0.359304	0.179652	+	0.983730i	$0.442503\pi$
$410$	33.3510	1.64709
$411$	59.1718	2.91873
$412$	12.1449	0.598335
$413$	1.85392	0.0912254
$414$	−32.9866	−1.62120
$415$	−19.7884	−0.971373
$416$	−174.197	−8.54071
$417$	6.89070	0.337439
$418$	−27.7798	−1.35875
$419$	30.1552	1.47318	0.736588	−	0.676342i	$-0.236436\pi$
$419$	30.1552	1.47318	0.736588	+	0.676342i	$0.236436\pi$
$420$	−7.61530	−0.371589
$421$	7.48983	0.365032	0.182516	−	0.983203i	$-0.441576\pi$
$421$	7.48983	0.365032	0.182516	+	0.983203i	$0.441576\pi$
$422$	−31.5878	−1.53767
$423$	−13.3847	−0.650787
$424$	118.232	5.74186
$425$	18.4906	0.896925
$426$	−27.9966	−1.35644
$427$	1.72727	0.0835886
$428$	13.9020	0.671981
$429$	35.2390	1.70136
$430$	−20.0987	−0.969244
$431$	−0.709283	−0.0341650	−0.0170825	−	0.999854i	$-0.505438\pi$
$431$	−0.709283	−0.0341650	−0.0170825	+	0.999854i	$0.505438\pi$
$432$	−46.2259	−2.22405
$433$	−21.8741	−1.05120	−0.525600	−	0.850732i	$-0.676159\pi$
$433$	−21.8741	−1.05120	−0.525600	+	0.850732i	$0.676159\pi$
$434$	0.187509	0.00900071
$435$	−26.2958	−1.26079
$436$	71.0690	3.40359
$437$	−12.6068	−0.603063
$438$	−93.7859	−4.48127
$439$	31.3257	1.49510	0.747548	−	0.664208i	$-0.231231\pi$
$439$	31.3257	1.49510	0.747548	+	0.664208i	$0.231231\pi$
$440$	41.7048	1.98820
$441$	−27.2409	−1.29719
$442$	128.453	6.10986
$443$	21.5711	1.02487	0.512436	−	0.858725i	$-0.328743\pi$
$443$	21.5711	1.02487	0.512436	+	0.858725i	$0.328743\pi$
$444$	99.4928	4.72172
$445$	−17.2068	−0.815680
$446$	−52.1582	−2.46976
$447$	−37.2951	−1.76400
$448$	14.5375	0.686830
$449$	−13.4620	−0.635309	−0.317654	−	0.948207i	$-0.602895\pi$
$449$	−13.4620	−0.635309	−0.317654	+	0.948207i	$0.602895\pi$
$450$	25.4301	1.19879
$451$	16.9858	0.799830
$452$	39.6421	1.86461
$453$	2.24943	0.105687
$454$	67.4147	3.16393
$455$	2.81205	0.131831
$456$	−120.435	−5.63989
$457$	30.1541	1.41055	0.705275	−	0.708934i	$-0.250824\pi$
$457$	30.1541	1.41055	0.705275	+	0.708934i	$0.250824\pi$
$458$	−2.30169	−0.107551
$459$	19.9533	0.931339
$460$	28.7327	1.33967
$461$	−12.2591	−0.570963	−0.285482	−	0.958384i	$-0.592154\pi$
$461$	−12.2591	−0.570963	−0.285482	+	0.958384i	$0.592154\pi$
$462$	−5.20226	−0.242031
$463$	22.1368	1.02879	0.514393	−	0.857555i	$-0.328017\pi$
$463$	22.1368	1.02879	0.514393	+	0.857555i	$0.328017\pi$
$464$	113.098	5.25043
$465$	0.964372	0.0447217
$466$	−30.9151	−1.43211
$467$	−39.9167	−1.84713	−0.923563	−	0.383447i	$-0.874737\pi$
$467$	−39.9167	−1.84713	−0.923563	+	0.383447i	$0.874737\pi$
$468$	131.708	6.08820
$469$	−0.150794	−0.00696300
$470$	15.6378	0.721320
$471$	−31.0644	−1.43137
$472$	−66.8237	−3.07581
$473$	−10.2363	−0.470667
$474$	55.0483	2.52845
$475$	9.71882	0.445930
$476$	−14.1378	−0.648007
$477$	−43.0737	−1.97221
$478$	67.6959	3.09634
$479$	−16.6431	−0.760442	−0.380221	−	0.924896i	$-0.624152\pi$
$479$	−16.6431	−0.760442	−0.380221	+	0.924896i	$0.624152\pi$
$480$	132.261	6.03687
$481$	−36.7391	−1.67516
$482$	69.9646	3.18680
$483$	−2.36084	−0.107422
$484$	−32.2119	−1.46418
$485$	−3.05671	−0.138798
$486$	−59.9223	−2.71813
$487$	−18.7521	−0.849738	−0.424869	−	0.905255i	$-0.639680\pi$
$487$	−18.7521	−0.849738	−0.424869	+	0.905255i	$0.639680\pi$
$488$	−62.2588	−2.81832
$489$	−66.7042	−3.01647
$490$	31.8266	1.43778
$491$	12.6010	0.568674	0.284337	−	0.958724i	$-0.408226\pi$
$491$	12.6010	0.568674	0.284337	+	0.958724i	$0.408226\pi$
$492$	111.796	5.04016
$493$	−48.8183	−2.19867
$494$	67.5159	3.03768
$495$	−15.1937	−0.682905
$496$	−4.14774	−0.186239
$497$	−1.13784	−0.0510391
$498$	−88.9725	−3.98695
$499$	−24.1370	−1.08052	−0.540261	−	0.841498i	$-0.681674\pi$
$499$	−24.1370	−1.08052	−0.540261	+	0.841498i	$0.681674\pi$
$500$	−70.2866	−3.14331
$501$	−41.4377	−1.85130
$502$	−18.7132	−0.835210
$503$	39.2629	1.75064	0.875322	−	0.483540i	$-0.160649\pi$
$503$	39.2629	1.75064	0.875322	+	0.483540i	$0.160649\pi$
$504$	−12.8075	−0.570491
$505$	17.4492	0.776478
$506$	19.6283	0.872583
$507$	−51.3920	−2.28240
$508$	6.36510	0.282406
$509$	−12.7302	−0.564255	−0.282128	−	0.959377i	$-0.591040\pi$
$509$	−12.7302	−0.564255	−0.282128	+	0.959377i	$0.591040\pi$
$510$	−97.5291	−4.31866
$511$	−3.81166	−0.168618
$512$	−165.914	−7.33244
$513$	10.4876	0.463040
$514$	72.5490	3.20000
$515$	3.40503	0.150043
$516$	−67.3729	−2.96592
$517$	7.96441	0.350275
$518$	5.42371	0.238304
$519$	−33.0622	−1.45127
$520$	−101.359	−4.44489
$521$	−17.5951	−0.770855	−0.385427	−	0.922738i	$-0.625946\pi$
$521$	−17.5951	−0.770855	−0.385427	+	0.922738i	$0.625946\pi$
$522$	−67.1398	−2.93863
$523$	−4.86688	−0.212814	−0.106407	−	0.994323i	$-0.533935\pi$
$523$	−4.86688	−0.212814	−0.106407	+	0.994323i	$0.533935\pi$
$524$	128.759	5.62484
$525$	1.82002	0.0794323
$526$	16.9045	0.737070
$527$	1.79036	0.0779894
$528$	115.075	5.00802
$529$	−14.0925	−0.612717
$530$	50.3246	2.18596
$531$	24.3448	1.05648
$532$	−7.43099	−0.322174
$533$	−41.2822	−1.78813
$534$	−77.3652	−3.34792
$535$	3.89769	0.168512
$536$	5.43529	0.234769
$537$	1.35674	0.0585478
$538$	−37.1879	−1.60328
$539$	16.2094	0.698188
$540$	−23.9029	−1.02862
$541$	18.5396	0.797080	0.398540	−	0.917151i	$-0.369517\pi$
$541$	18.5396	0.797080	0.398540	+	0.917151i	$0.369517\pi$
$542$	1.16634	0.0500985
$543$	−42.6171	−1.82888
$544$	245.544	10.5276
$545$	19.9255	0.853513
$546$	12.6436	0.541094
$547$	−41.4107	−1.77059	−0.885296	−	0.465028i	$-0.846044\pi$
$547$	−41.4107	−1.77059	−0.885296	+	0.465028i	$0.846044\pi$
$548$	−131.598	−5.62157
$549$	22.6818	0.968035
$550$	−15.1319	−0.645225
$551$	−25.6594	−1.09313
$552$	85.0954	3.62190
$553$	2.23728	0.0951388
$554$	24.9794	1.06127
$555$	27.8946	1.18406
$556$	−15.3249	−0.649918
$557$	39.1803	1.66012	0.830061	−	0.557673i	$-0.188306\pi$
$557$	39.1803	1.66012	0.830061	+	0.557673i	$0.188306\pi$
$558$	2.46228	0.104237
$559$	24.8783	1.05224
$560$	9.18295	0.388051
$561$	−49.6720	−2.09715
$562$	32.9518	1.38999
$563$	−16.0304	−0.675599	−0.337800	−	0.941218i	$-0.609683\pi$
$563$	−16.0304	−0.675599	−0.337800	+	0.941218i	$0.609683\pi$
$564$	52.4197	2.20727
$565$	11.1144	0.467585
$566$	12.2872	0.516471
$567$	−1.58667	−0.0666337
$568$	41.0129	1.72087
$569$	−36.5223	−1.53109	−0.765547	−	0.643380i	$-0.777532\pi$
$569$	−36.5223	−1.53109	−0.765547	+	0.643380i	$0.777532\pi$
$570$	−51.2622	−2.14714
$571$	−4.24720	−0.177740	−0.0888700	−	0.996043i	$-0.528326\pi$
$571$	−4.24720	−0.177740	−0.0888700	+	0.996043i	$0.528326\pi$
$572$	−78.3712	−3.27687
$573$	25.4484	1.06312
$574$	6.09441	0.254376
$575$	−6.86700	−0.286374
$576$	190.899	7.95414
$577$	14.8161	0.616801	0.308400	−	0.951257i	$-0.400206\pi$
$577$	14.8161	0.616801	0.308400	+	0.951257i	$0.400206\pi$
$578$	−133.403	−5.54884
$579$	14.7259	0.611988
$580$	58.4817	2.42832
$581$	−3.61603	−0.150018
$582$	−13.7436	−0.569690
$583$	25.6305	1.06151
$584$	137.390	5.68523
$585$	36.9266	1.52673
$586$	11.8719	0.490426
$587$	12.6965	0.524043	0.262021	−	0.965062i	$-0.415611\pi$
$587$	12.6965	0.524043	0.262021	+	0.965062i	$0.415611\pi$
$588$	106.686	4.39966
$589$	0.941031	0.0387745
$590$	−28.4430	−1.17098
$591$	−14.0689	−0.578718
$592$	−119.974	−4.93090
$593$	15.5511	0.638606	0.319303	−	0.947653i	$-0.396551\pi$
$593$	15.5511	0.638606	0.319303	+	0.947653i	$0.396551\pi$
$594$	−16.3289	−0.669982
$595$	−3.96379	−0.162500
$596$	82.9440	3.39752
$597$	−33.1000	−1.35469
$598$	−47.7045	−1.95078
$599$	13.0540	0.533373	0.266687	−	0.963783i	$-0.414071\pi$
$599$	13.0540	0.533373	0.266687	+	0.963783i	$0.414071\pi$
$600$	−65.6019	−2.67819
$601$	21.4526	0.875070	0.437535	−	0.899201i	$-0.355852\pi$
$601$	21.4526	0.875070	0.437535	+	0.899201i	$0.355852\pi$
$602$	−3.67274	−0.149690
$603$	−1.98016	−0.0806382
$604$	−5.00271	−0.203557
$605$	−9.03117	−0.367169
$606$	78.4550	3.18702
$607$	−10.3069	−0.418343	−0.209171	−	0.977879i	$-0.567077\pi$
$607$	−10.3069	−0.418343	−0.209171	+	0.977879i	$0.567077\pi$
$608$	129.060	5.23408
$609$	−4.80518	−0.194716
$610$	−26.5000	−1.07295
$611$	−19.3567	−0.783087
$612$	−185.652	−7.50454
$613$	6.87154	0.277539	0.138769	−	0.990325i	$-0.455685\pi$
$613$	6.87154	0.277539	0.138769	+	0.990325i	$0.455685\pi$
$614$	72.6850	2.93333
$615$	31.3440	1.26391
$616$	7.62094	0.307056
$617$	−34.5757	−1.39197	−0.695983	−	0.718059i	$-0.745031\pi$
$617$	−34.5757	−1.39197	−0.695983	+	0.718059i	$0.745031\pi$
$618$	15.3097	0.615846
$619$	26.0118	1.04550	0.522751	−	0.852486i	$-0.324906\pi$
$619$	26.0118	1.04550	0.522751	+	0.852486i	$0.324906\pi$
$620$	−2.14476	−0.0861354
$621$	−7.41021	−0.297362
$622$	−77.5602	−3.10988
$623$	−3.14429	−0.125973
$624$	−279.679	−11.1961
$625$	−8.20182	−0.328073
$626$	42.1796	1.68584
$627$	−26.1081	−1.04266
$628$	69.0869	2.75687
$629$	51.7864	2.06486
$630$	−5.45140	−0.217189
$631$	7.69445	0.306311	0.153156	−	0.988202i	$-0.451056\pi$
$631$	7.69445	0.306311	0.153156	+	0.988202i	$0.451056\pi$
$632$	−80.6418	−3.20776
$633$	−29.6869	−1.17995
$634$	10.1256	0.402138
$635$	1.78457	0.0708184
$636$	168.693	6.68913
$637$	−39.3953	−1.56090
$638$	39.9507	1.58166
$639$	−14.9416	−0.591081
$640$	−122.640	−4.84778
$641$	−37.2016	−1.46937	−0.734687	−	0.678407i	$-0.762671\pi$
$641$	−37.2016	−1.46937	−0.734687	+	0.678407i	$0.762671\pi$
$642$	17.5248	0.691648
$643$	14.8628	0.586131	0.293065	−	0.956092i	$-0.405325\pi$
$643$	14.8628	0.586131	0.293065	+	0.956092i	$0.405325\pi$
$644$	5.25049	0.206898
$645$	−18.8892	−0.743760
$646$	−95.1686	−3.74436
$647$	12.2885	0.483112	0.241556	−	0.970387i	$-0.422342\pi$
$647$	12.2885	0.483112	0.241556	+	0.970387i	$0.422342\pi$
$648$	57.1907	2.24666
$649$	−14.4861	−0.568630
$650$	36.7764	1.44249
$651$	0.176225	0.00690680
$652$	148.349	5.80981
$653$	30.9886	1.21268	0.606338	−	0.795207i	$-0.292638\pi$
$653$	30.9886	1.21268	0.606338	+	0.795207i	$0.292638\pi$
$654$	89.5889	3.50320
$655$	36.0997	1.41053
$656$	−134.810	−5.26345
$657$	−50.0531	−1.95276
$658$	2.85759	0.111400
$659$	5.60833	0.218470	0.109235	−	0.994016i	$-0.465160\pi$
$659$	5.60833	0.218470	0.109235	+	0.994016i	$0.465160\pi$
$660$	59.5043	2.31620
$661$	−27.3906	−1.06537	−0.532685	−	0.846313i	$-0.678817\pi$
$661$	−27.3906	−1.06537	−0.532685	+	0.846313i	$0.678817\pi$
$662$	42.8572	1.66569
$663$	120.723	4.68848
$664$	130.338	5.05811
$665$	−2.08341	−0.0807911
$666$	71.2218	2.75979
$667$	18.1301	0.701999
$668$	92.1570	3.56566
$669$	−49.0195	−1.89520
$670$	2.31349	0.0893778
$671$	−13.4965	−0.521028
$672$	24.1688	0.932332
$673$	5.79254	0.223286	0.111643	−	0.993748i	$-0.464389\pi$
$673$	5.79254	0.223286	0.111643	+	0.993748i	$0.464389\pi$
$674$	51.3158	1.97661
$675$	5.71270	0.219882
$676$	114.295	4.39597
$677$	−10.1688	−0.390821	−0.195410	−	0.980722i	$-0.562604\pi$
$677$	−10.1688	−0.390821	−0.195410	+	0.980722i	$0.562604\pi$
$678$	49.9724	1.91918
$679$	−0.558569	−0.0214359
$680$	142.873	5.47894
$681$	63.3578	2.42788
$682$	−1.46515	−0.0561036
$683$	−50.3838	−1.92788	−0.963941	−	0.266117i	$-0.914259\pi$
$683$	−50.3838	−1.92788	−0.963941	+	0.266117i	$0.914259\pi$
$684$	−97.5804	−3.73108
$685$	−36.8957	−1.40971
$686$	11.7075	0.446996
$687$	−2.16318	−0.0825306
$688$	81.2419	3.09732
$689$	−62.2923	−2.37315
$690$	36.2202	1.37888
$691$	18.0800	0.687796	0.343898	−	0.939007i	$-0.388253\pi$
$691$	18.0800	0.687796	0.343898	+	0.939007i	$0.388253\pi$
$692$	73.5301	2.79520
$693$	−2.77642	−0.105468
$694$	35.8966	1.36262
$695$	−4.29659	−0.162979
$696$	173.200	6.56514
$697$	58.1903	2.20412
$698$	57.8624	2.19012
$699$	−29.0547	−1.09895
$700$	−4.04772	−0.152989
$701$	42.2785	1.59684	0.798419	−	0.602103i	$-0.205670\pi$
$701$	42.2785	1.59684	0.798419	+	0.602103i	$0.205670\pi$
$702$	39.6856	1.49784
$703$	27.2194	1.02660
$704$	−113.592	−4.28118
$705$	14.6968	0.553513
$706$	−34.7884	−1.30928
$707$	3.18858	0.119919
$708$	−95.3438	−3.58324
$709$	−0.368055	−0.0138226	−0.00691130	−	0.999976i	$-0.502200\pi$
$709$	−0.368055	−0.0138226	−0.00691130	+	0.999976i	$0.502200\pi$
$710$	17.4568	0.655143
$711$	29.3790	1.10180
$712$	113.334	4.24739
$713$	−0.664901	−0.0249007
$714$	−17.8220	−0.666972
$715$	−21.9728	−0.821735
$716$	−3.01739	−0.112765
$717$	63.6221	2.37601
$718$	−58.0936	−2.16804
$719$	25.3209	0.944309	0.472154	−	0.881516i	$-0.343476\pi$
$719$	25.3209	0.944309	0.472154	+	0.881516i	$0.343476\pi$
$720$	120.586	4.49399
$721$	0.622219	0.0231726
$722$	3.24573	0.120794
$723$	65.7542	2.44543
$724$	94.7801	3.52248
$725$	−13.9769	−0.519088
$726$	−40.6060	−1.50703
$727$	17.6678	0.655262	0.327631	−	0.944806i	$-0.393750\pi$
$727$	17.6678	0.655262	0.327631	+	0.944806i	$0.393750\pi$
$728$	−18.5219	−0.686467
$729$	−40.4611	−1.49856
$730$	58.4788	2.16440
$731$	−35.0678	−1.29703
$732$	−88.8307	−3.28328
$733$	51.6103	1.90627	0.953134	−	0.302548i	$-0.0978372\pi$
$733$	51.6103	1.90627	0.953134	+	0.302548i	$0.0978372\pi$
$734$	−41.4740	−1.53083
$735$	29.9113	1.10330
$736$	−91.1895	−3.36129
$737$	1.17827	0.0434021
$738$	80.0291	2.94591
$739$	−19.2707	−0.708884	−0.354442	−	0.935078i	$-0.615329\pi$
$739$	−19.2707	−0.708884	−0.354442	+	0.935078i	$0.615329\pi$
$740$	−62.0373	−2.28054
$741$	63.4529	2.33100
$742$	9.19608	0.337599
$743$	42.1169	1.54512	0.772560	−	0.634942i	$-0.218976\pi$
$743$	42.1169	1.54512	0.772560	+	0.634942i	$0.218976\pi$
$744$	−6.35195	−0.232874
$745$	23.2548	0.851990
$746$	14.7705	0.540786
$747$	−47.4842	−1.73736
$748$	110.470	4.03918
$749$	0.712245	0.0260249
$750$	−88.6025	−3.23531
$751$	31.8879	1.16361	0.581804	−	0.813329i	$-0.302347\pi$
$751$	31.8879	1.16361	0.581804	+	0.813329i	$0.302347\pi$
$752$	−63.2106	−2.30505
$753$	−17.5871	−0.640908
$754$	−97.0961	−3.53603
$755$	−1.40260	−0.0510458
$756$	−4.36791	−0.158859
$757$	−0.309712	−0.0112567	−0.00562835	−	0.999984i	$-0.501792\pi$
$757$	−0.309712	−0.0112567	−0.00562835	+	0.999984i	$0.501792\pi$
$758$	61.9317	2.24946
$759$	18.4471	0.669587
$760$	75.0955	2.72400
$761$	−38.5220	−1.39642	−0.698211	−	0.715892i	$-0.746020\pi$
$761$	−38.5220	−1.39642	−0.698211	+	0.715892i	$0.746020\pi$
$762$	8.02378	0.290671
$763$	3.64108	0.131816
$764$	−56.5971	−2.04761
$765$	−52.0508	−1.88190
$766$	−70.4201	−2.54438
$767$	35.2070	1.27125
$768$	−296.242	−10.6897
$769$	−6.00358	−0.216495	−0.108247	−	0.994124i	$-0.534524\pi$
$769$	−6.00358	−0.216495	−0.108247	+	0.994124i	$0.534524\pi$
$770$	3.24379	0.116898
$771$	68.1832	2.45556
$772$	−32.7503	−1.17871
$773$	6.84809	0.246309	0.123154	−	0.992388i	$-0.460699\pi$
$773$	6.84809	0.246309	0.123154	+	0.992388i	$0.460699\pi$
$774$	−48.2288	−1.73355
$775$	0.512587	0.0184127
$776$	20.1334	0.722746
$777$	5.09732	0.182865
$778$	−89.1526	−3.19628
$779$	30.5854	1.09584
$780$	−144.619	−5.17819
$781$	8.89084	0.318139
$782$	67.2430	2.40460
$783$	−15.0825	−0.539005
$784$	−128.648	−4.59457
$785$	19.3697	0.691335
$786$	162.312	5.78947
$787$	−0.647354	−0.0230757	−0.0115378	−	0.999933i	$-0.503673\pi$
$787$	−0.647354	−0.0230757	−0.0115378	+	0.999933i	$0.503673\pi$
$788$	31.2892	1.11463
$789$	15.8872	0.565600
$790$	−34.3245	−1.22121
$791$	2.03099	0.0722136
$792$	100.075	3.55601
$793$	32.8019	1.16483
$794$	−29.5437	−1.04847
$795$	47.2962	1.67742
$796$	73.6140	2.60918
$797$	−3.75579	−0.133037	−0.0665184	−	0.997785i	$-0.521189\pi$
$797$	−3.75579	−0.133037	−0.0665184	+	0.997785i	$0.521189\pi$
$798$	−9.36742	−0.331603
$799$	27.2847	0.965262
$800$	70.3000	2.48548
$801$	−41.2894	−1.45889
$802$	−27.8197	−0.982347
$803$	29.7835	1.05104
$804$	7.75506	0.273500
$805$	1.47207	0.0518835
$806$	3.56090	0.125427
$807$	−34.9500	−1.23030
$808$	−114.931	−4.04326
$809$	41.6901	1.46575	0.732873	−	0.680365i	$-0.238179\pi$
$809$	41.6901	1.46575	0.732873	+	0.680365i	$0.238179\pi$
$810$	24.3428	0.855317
$811$	23.0467	0.809281	0.404640	−	0.914476i	$-0.367397\pi$
$811$	23.0467	0.809281	0.404640	+	0.914476i	$0.367397\pi$
$812$	10.6867	0.375028
$813$	1.09615	0.0384436
$814$	−42.3797	−1.48541
$815$	41.5924	1.45692
$816$	394.228	13.8007
$817$	−18.4320	−0.644854
$818$	−20.3718	−0.712284
$819$	6.74780	0.235787
$820$	−69.7088	−2.43434
$821$	−32.5831	−1.13716	−0.568579	−	0.822629i	$-0.692507\pi$
$821$	−32.5831	−1.13716	−0.568579	+	0.822629i	$0.692507\pi$
$822$	−165.891	−5.78610
$823$	34.1849	1.19161	0.595805	−	0.803129i	$-0.296833\pi$
$823$	34.1849	1.19161	0.595805	+	0.803129i	$0.296833\pi$
$824$	−22.4276	−0.781302
$825$	−14.2213	−0.495121
$826$	−5.19754	−0.180845
$827$	17.0244	0.591996	0.295998	−	0.955189i	$-0.404348\pi$
$827$	17.0244	0.591996	0.295998	+	0.955189i	$0.404348\pi$
$828$	68.9471	2.39608
$829$	19.1094	0.663697	0.331848	−	0.943333i	$-0.392328\pi$
$829$	19.1094	0.663697	0.331848	+	0.943333i	$0.392328\pi$
$830$	55.4775	1.92565
$831$	23.4761	0.814379
$832$	276.075	9.57117
$833$	55.5305	1.92402
$834$	−19.3183	−0.668940
$835$	25.8378	0.894155
$836$	58.0641	2.00819
$837$	0.553135	0.0191192
$838$	−84.5412	−2.92043
$839$	−7.17594	−0.247741	−0.123870	−	0.992298i	$-0.539531\pi$
$839$	−7.17594	−0.247741	−0.123870	+	0.992298i	$0.539531\pi$
$840$	14.0630	0.485219
$841$	7.90131	0.272459
$842$	−20.9980	−0.723640
$843$	30.9688	1.06662
$844$	66.0234	2.27262
$845$	32.0447	1.10237
$846$	37.5246	1.29012
$847$	−1.65031	−0.0567055
$848$	−203.420	−6.98546
$849$	11.5478	0.396320
$850$	−51.8391	−1.77807
$851$	−19.2323	−0.659276
$852$	58.5172	2.00477
$853$	39.1018	1.33882	0.669411	−	0.742892i	$-0.266547\pi$
$853$	39.1018	1.33882	0.669411	+	0.742892i	$0.266547\pi$
$854$	−4.84248	−0.165706
$855$	−27.3584	−0.935638
$856$	−25.6725	−0.877469
$857$	42.7697	1.46099	0.730493	−	0.682921i	$-0.239291\pi$
$857$	42.7697	1.46099	0.730493	+	0.682921i	$0.239291\pi$
$858$	−98.7940	−3.37277
$859$	31.0918	1.06084	0.530420	−	0.847735i	$-0.322034\pi$
$859$	31.0918	1.06084	0.530420	+	0.847735i	$0.322034\pi$
$860$	42.0093	1.43251
$861$	5.72766	0.195198
$862$	1.98850	0.0677287
$863$	−36.9863	−1.25903	−0.629514	−	0.776989i	$-0.716746\pi$
$863$	−36.9863	−1.25903	−0.629514	+	0.776989i	$0.716746\pi$
$864$	75.8611	2.58085
$865$	20.6155	0.700947
$866$	61.3248	2.08390
$867$	−125.375	−4.25796
$868$	−0.391923	−0.0133027
$869$	−17.4816	−0.593023
$870$	73.7214	2.49939
$871$	−2.86366	−0.0970314
$872$	−131.241	−4.44439
$873$	−7.33488	−0.248248
$874$	35.3435	1.19551
$875$	−3.60100	−0.121736
$876$	196.027	6.62315
$877$	−49.8739	−1.68412	−0.842061	−	0.539382i	$-0.818658\pi$
$877$	−49.8739	−1.68412	−0.842061	+	0.539382i	$0.818658\pi$
$878$	−87.8229	−2.96388
$879$	11.1575	0.376334
$880$	−71.7536	−2.41881
$881$	0.783766	0.0264058	0.0132029	−	0.999913i	$-0.495797\pi$
$881$	0.783766	0.0264058	0.0132029	+	0.999913i	$0.495797\pi$
$882$	76.3711	2.57155
$883$	−56.6144	−1.90523	−0.952614	−	0.304183i	$-0.901616\pi$
$883$	−56.6144	−1.90523	−0.952614	+	0.304183i	$0.901616\pi$
$884$	−268.486	−9.03016
$885$	−26.7313	−0.898564
$886$	−60.4753	−2.03171
$887$	−15.3534	−0.515515	−0.257758	−	0.966210i	$-0.582984\pi$
$887$	−15.3534	−0.515515	−0.257758	+	0.966210i	$0.582984\pi$
$888$	−183.731	−6.16560
$889$	0.326104	0.0109372
$890$	48.2399	1.61701
$891$	12.3979	0.415344
$892$	109.019	3.65022
$893$	14.3411	0.479906
$894$	104.558	3.49695
$895$	−0.845977	−0.0282779
$896$	−22.4107	−0.748689
$897$	−44.8337	−1.49695
$898$	37.7411	1.25944
$899$	−1.35332	−0.0451357
$900$	−53.1528	−1.77176
$901$	87.8056	2.92523
$902$	−47.6204	−1.58559
$903$	−3.45172	−0.114866
$904$	−73.2060	−2.43480
$905$	26.5733	0.883326
$906$	−6.30637	−0.209515
$907$	23.3492	0.775297	0.387648	−	0.921807i	$-0.373287\pi$
$907$	23.3492	0.775297	0.387648	+	0.921807i	$0.373287\pi$
$908$	−140.907	−4.67617
$909$	41.8711	1.38878
$910$	−7.88370	−0.261342
$911$	−53.4978	−1.77246	−0.886231	−	0.463244i	$-0.846685\pi$
$911$	−53.4978	−1.77246	−0.886231	+	0.463244i	$0.846685\pi$
$912$	207.210	6.86141
$913$	28.2549	0.935101
$914$	−84.5382	−2.79628
$915$	−24.9052	−0.823342
$916$	4.81090	0.158957
$917$	6.59670	0.217842
$918$	−55.9398	−1.84629
$919$	−47.6386	−1.57145	−0.785727	−	0.618573i	$-0.787711\pi$
$919$	−47.6386	−1.57145	−0.785727	+	0.618573i	$0.787711\pi$
$920$	−53.0600	−1.74934
$921$	68.3110	2.25092
$922$	34.3689	1.13188
$923$	−21.6083	−0.711244
$924$	10.8735	0.357713
$925$	14.8266	0.487497
$926$	−62.0615	−2.03947
$927$	8.17070	0.268361
$928$	−185.604	−6.09275
$929$	26.5195	0.870076	0.435038	−	0.900412i	$-0.356735\pi$
$929$	26.5195	0.870076	0.435038	+	0.900412i	$0.356735\pi$
$930$	−2.70366	−0.0886564
$931$	29.1874	0.956577
$932$	64.6174	2.11661
$933$	−72.8928	−2.38640
$934$	111.908	3.66175
$935$	30.9722	1.01290
$936$	−243.221	−7.94994
$937$	6.79106	0.221854	0.110927	−	0.993829i	$-0.464618\pi$
$937$	6.79106	0.221854	0.110927	+	0.993829i	$0.464618\pi$
$938$	0.422756	0.0138035
$939$	39.6413	1.29365
$940$	−32.6855	−1.06608
$941$	−8.26399	−0.269398	−0.134699	−	0.990887i	$-0.543007\pi$
$941$	−8.26399	−0.269398	−0.134699	+	0.990887i	$0.543007\pi$
$942$	87.0902	2.83755
$943$	−21.6106	−0.703739
$944$	114.971	3.74198
$945$	−1.22462	−0.0398369
$946$	28.6980	0.933051
$947$	−14.9692	−0.486432	−0.243216	−	0.969972i	$-0.578202\pi$
$947$	−14.9692	−0.486432	−0.243216	+	0.969972i	$0.578202\pi$
$948$	−115.059	−3.73696
$949$	−72.3857	−2.34974
$950$	−27.2471	−0.884013
$951$	9.51624	0.308585
$952$	26.1080	0.846164
$953$	54.7200	1.77255	0.886277	−	0.463156i	$-0.153283\pi$
$953$	54.7200	1.77255	0.886277	+	0.463156i	$0.153283\pi$
$954$	120.759	3.90971
$955$	−15.8680	−0.513476
$956$	−141.495	−4.57627
$957$	37.5466	1.21371
$958$	46.6595	1.50750
$959$	−6.74215	−0.217715
$960$	−209.613	−6.76523
$961$	−30.9504	−0.998399
$962$	102.999	3.32083
$963$	9.35289	0.301393
$964$	−146.237	−4.70997
$965$	−9.18213	−0.295583
$966$	6.61871	0.212953
$967$	10.3420	0.332575	0.166287	−	0.986077i	$-0.446822\pi$
$967$	10.3420	0.332575	0.166287	+	0.986077i	$0.446822\pi$
$968$	59.4849	1.91192
$969$	−89.4416	−2.87328
$970$	8.56961	0.275154
$971$	26.3089	0.844292	0.422146	−	0.906528i	$-0.361277\pi$
$971$	26.3089	0.844292	0.422146	+	0.906528i	$0.361277\pi$
$972$	125.247	4.01730
$973$	−0.785139	−0.0251704
$974$	52.5722	1.68452
$975$	34.5633	1.10691
$976$	107.117	3.42873
$977$	−8.61281	−0.275548	−0.137774	−	0.990464i	$-0.543995\pi$
$977$	−8.61281	−0.275548	−0.137774	+	0.990464i	$0.543995\pi$
$978$	187.008	5.97985
$979$	24.5688	0.785222
$980$	−66.5225	−2.12498
$981$	47.8131	1.52656
$982$	−35.3274	−1.12734
$983$	45.0075	1.43552	0.717759	−	0.696292i	$-0.245168\pi$
$983$	45.0075	1.43552	0.717759	+	0.696292i	$0.245168\pi$
$984$	−206.451	−6.58142
$985$	8.77247	0.279514
$986$	136.864	4.35864
$987$	2.68562	0.0854843
$988$	−141.119	−4.48958
$989$	13.0234	0.414121
$990$	42.5961	1.35379
$991$	−33.6398	−1.06860	−0.534302	−	0.845294i	$-0.679425\pi$
$991$	−33.6398	−1.06860	−0.534302	+	0.845294i	$0.679425\pi$
$992$	6.80684	0.216117
$993$	40.2782	1.27819
$994$	3.18998	0.101180
$995$	20.6390	0.654300
$996$	185.966	5.89257
$997$	18.0561	0.571842	0.285921	−	0.958253i	$-0.407700\pi$
$997$	18.0561	0.571842	0.285921	+	0.958253i	$0.407700\pi$
$998$	67.6691	2.14203
$999$	15.9995	0.506202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.b.1.4	✓	361

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.b.1.4	✓	361	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:	\( N \)	\(=\)	\( 8009 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.80354 q^{2} -2.63483 q^{3} +5.85984 q^{4} +1.64291 q^{5} +7.38685 q^{6} +0.300217 q^{7} -10.8212 q^{8} +3.94232 q^{9} +O(q^{10})\)	\(q-2.80354 q^{2} -2.63483 q^{3} +5.85984 q^{4} +1.64291 q^{5} +7.38685 q^{6} +0.300217 q^{7} -10.8212 q^{8} +3.94232 q^{9} -4.60596 q^{10} -2.34583 q^{11} -15.4397 q^{12} +5.70130 q^{13} -0.841672 q^{14} -4.32878 q^{15} +18.6180 q^{16} -8.03641 q^{17} -11.0525 q^{18} -4.22401 q^{19} +9.62717 q^{20} -0.791022 q^{21} +6.57664 q^{22} +2.98455 q^{23} +28.5120 q^{24} -2.30085 q^{25} -15.9838 q^{26} -2.48286 q^{27} +1.75922 q^{28} +6.07465 q^{29} +12.1359 q^{30} -0.222781 q^{31} -30.5539 q^{32} +6.18087 q^{33} +22.5304 q^{34} +0.493230 q^{35} +23.1014 q^{36} -6.44397 q^{37} +11.8422 q^{38} -15.0220 q^{39} -17.7782 q^{40} -7.24084 q^{41} +2.21766 q^{42} +4.36362 q^{43} -13.7462 q^{44} +6.47688 q^{45} -8.36729 q^{46} -3.39513 q^{47} -49.0552 q^{48} -6.90987 q^{49} +6.45053 q^{50} +21.1746 q^{51} +33.4087 q^{52} -10.9260 q^{53} +6.96080 q^{54} -3.85399 q^{55} -3.24871 q^{56} +11.1295 q^{57} -17.0305 q^{58} +6.17525 q^{59} -25.3660 q^{60} +5.75341 q^{61} +0.624577 q^{62} +1.18355 q^{63} +48.4231 q^{64} +9.36672 q^{65} -17.3283 q^{66} -0.502281 q^{67} -47.0920 q^{68} -7.86377 q^{69} -1.38279 q^{70} -3.79005 q^{71} -42.6607 q^{72} -12.6963 q^{73} +18.0659 q^{74} +6.06235 q^{75} -24.7520 q^{76} -0.704260 q^{77} +42.1147 q^{78} +7.45220 q^{79} +30.5877 q^{80} -5.28506 q^{81} +20.3000 q^{82} -12.0447 q^{83} -4.63526 q^{84} -13.2031 q^{85} -12.2336 q^{86} -16.0057 q^{87} +25.3847 q^{88} -10.4734 q^{89} -18.1582 q^{90} +1.71163 q^{91} +17.4889 q^{92} +0.586991 q^{93} +9.51839 q^{94} -6.93966 q^{95} +80.5043 q^{96} -1.86055 q^{97} +19.3721 q^{98} -9.24803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more