LMFDB - Embedded newform 6009.2.a.c.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6009,2,Mod(1,6009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6009 = 3 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6009.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:	$47.9821065746$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.89797 q^{2} +1.00000 q^{3} +1.60229 q^{4} +4.27518 q^{5} -1.89797 q^{6} -0.912076 q^{7} +0.754847 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.89797 q^{2} +1.00000 q^{3} +1.60229 q^{4} +4.27518 q^{5} -1.89797 q^{6} -0.912076 q^{7} +0.754847 q^{8} +1.00000 q^{9} -8.11416 q^{10} +3.48975 q^{11} +1.60229 q^{12} -0.0151214 q^{13} +1.73109 q^{14} +4.27518 q^{15} -4.63725 q^{16} -3.66149 q^{17} -1.89797 q^{18} +3.18871 q^{19} +6.85007 q^{20} -0.912076 q^{21} -6.62343 q^{22} +0.460551 q^{23} +0.754847 q^{24} +13.2772 q^{25} +0.0287000 q^{26} +1.00000 q^{27} -1.46141 q^{28} -5.53924 q^{29} -8.11416 q^{30} +3.55308 q^{31} +7.29166 q^{32} +3.48975 q^{33} +6.94939 q^{34} -3.89929 q^{35} +1.60229 q^{36} -2.90616 q^{37} -6.05208 q^{38} -0.0151214 q^{39} +3.22711 q^{40} +2.56776 q^{41} +1.73109 q^{42} +5.25870 q^{43} +5.59157 q^{44} +4.27518 q^{45} -0.874111 q^{46} -12.3653 q^{47} -4.63725 q^{48} -6.16812 q^{49} -25.1997 q^{50} -3.66149 q^{51} -0.0242289 q^{52} +14.1769 q^{53} -1.89797 q^{54} +14.9193 q^{55} -0.688478 q^{56} +3.18871 q^{57} +10.5133 q^{58} -5.03300 q^{59} +6.85007 q^{60} +6.37469 q^{61} -6.74363 q^{62} -0.912076 q^{63} -4.56485 q^{64} -0.0646469 q^{65} -6.62343 q^{66} -6.19408 q^{67} -5.86675 q^{68} +0.460551 q^{69} +7.40073 q^{70} -7.62614 q^{71} +0.754847 q^{72} +9.15719 q^{73} +5.51581 q^{74} +13.2772 q^{75} +5.10923 q^{76} -3.18291 q^{77} +0.0287000 q^{78} +14.1369 q^{79} -19.8251 q^{80} +1.00000 q^{81} -4.87354 q^{82} +7.73090 q^{83} -1.46141 q^{84} -15.6535 q^{85} -9.98086 q^{86} -5.53924 q^{87} +2.63423 q^{88} +7.04418 q^{89} -8.11416 q^{90} +0.0137919 q^{91} +0.737934 q^{92} +3.55308 q^{93} +23.4690 q^{94} +13.6323 q^{95} +7.29166 q^{96} +8.87541 q^{97} +11.7069 q^{98} +3.48975 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9}+O(q^{10}) $ 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9} + 13 q^{10} + 40 q^{11} + 107 q^{12} + 6 q^{13} + 37 q^{14} + 34 q^{15} + 133 q^{16} + 77 q^{17} + 17 q^{18} + 34 q^{19} + 55 q^{20} + 22 q^{21} + 8 q^{22} + 83 q^{23} + 51 q^{24} + 110 q^{25} + 22 q^{26} + 92 q^{27} + 32 q^{28} + 97 q^{29} + 13 q^{30} + 44 q^{31} + 104 q^{32} + 40 q^{33} + 20 q^{34} + 80 q^{35} + 107 q^{36} + 12 q^{37} + 54 q^{38} + 6 q^{39} + 23 q^{40} + 67 q^{41} + 37 q^{42} + 30 q^{43} + 87 q^{44} + 34 q^{45} + 33 q^{46} + 69 q^{47} + 133 q^{48} + 112 q^{49} + 58 q^{50} + 77 q^{51} - 3 q^{52} + 113 q^{53} + 17 q^{54} + 42 q^{55} + 92 q^{56} + 34 q^{57} - 30 q^{58} + 72 q^{59} + 55 q^{60} + 19 q^{61} + 60 q^{62} + 22 q^{63} + 147 q^{64} + 74 q^{65} + 8 q^{66} + 26 q^{67} + 171 q^{68} + 83 q^{69} - 35 q^{70} + 134 q^{71} + 51 q^{72} - 17 q^{73} + 95 q^{74} + 110 q^{75} + 27 q^{76} + 108 q^{77} + 22 q^{78} + 159 q^{79} + 79 q^{80} + 92 q^{81} - 64 q^{82} + 73 q^{83} + 32 q^{84} - 4 q^{85} + 22 q^{86} + 97 q^{87} - 16 q^{88} + 50 q^{89} + 13 q^{90} + 17 q^{91} + 154 q^{92} + 44 q^{93} + 8 q^{94} + 155 q^{95} + 104 q^{96} - 20 q^{97} + 63 q^{98} + 40 q^{99}+O(q^{100}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9 + 13 * q^10 + 40 * q^11 + 107 * q^12 + 6 * q^13 + 37 * q^14 + 34 * q^15 + 133 * q^16 + 77 * q^17 + 17 * q^18 + 34 * q^19 + 55 * q^20 + 22 * q^21 + 8 * q^22 + 83 * q^23 + 51 * q^24 + 110 * q^25 + 22 * q^26 + 92 * q^27 + 32 * q^28 + 97 * q^29 + 13 * q^30 + 44 * q^31 + 104 * q^32 + 40 * q^33 + 20 * q^34 + 80 * q^35 + 107 * q^36 + 12 * q^37 + 54 * q^38 + 6 * q^39 + 23 * q^40 + 67 * q^41 + 37 * q^42 + 30 * q^43 + 87 * q^44 + 34 * q^45 + 33 * q^46 + 69 * q^47 + 133 * q^48 + 112 * q^49 + 58 * q^50 + 77 * q^51 - 3 * q^52 + 113 * q^53 + 17 * q^54 + 42 * q^55 + 92 * q^56 + 34 * q^57 - 30 * q^58 + 72 * q^59 + 55 * q^60 + 19 * q^61 + 60 * q^62 + 22 * q^63 + 147 * q^64 + 74 * q^65 + 8 * q^66 + 26 * q^67 + 171 * q^68 + 83 * q^69 - 35 * q^70 + 134 * q^71 + 51 * q^72 - 17 * q^73 + 95 * q^74 + 110 * q^75 + 27 * q^76 + 108 * q^77 + 22 * q^78 + 159 * q^79 + 79 * q^80 + 92 * q^81 - 64 * q^82 + 73 * q^83 + 32 * q^84 - 4 * q^85 + 22 * q^86 + 97 * q^87 - 16 * q^88 + 50 * q^89 + 13 * q^90 + 17 * q^91 + 154 * q^92 + 44 * q^93 + 8 * q^94 + 155 * q^95 + 104 * q^96 - 20 * q^97 + 63 * q^98 + 40 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.89797	−1.34207	−0.671033	−	0.741427i	$-0.734149\pi$
$2$	−1.89797	−1.34207	−0.671033	+	0.741427i	$0.734149\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.60229	0.801143
$5$	4.27518	1.91192	0.955960	−	0.293498i	$-0.0948195\pi$
$5$	4.27518	1.91192	0.955960	+	0.293498i	$0.0948195\pi$
$6$	−1.89797	−0.774843
$7$	−0.912076	−0.344732	−0.172366	−	0.985033i	$-0.555141\pi$
$7$	−0.912076	−0.344732	−0.172366	+	0.985033i	$0.555141\pi$
$8$	0.754847	0.266879
$9$	1.00000	0.333333
$10$	−8.11416	−2.56592
$11$	3.48975	1.05220	0.526099	−	0.850423i	$-0.323654\pi$
$11$	3.48975	1.05220	0.526099	+	0.850423i	$0.323654\pi$
$12$	1.60229	0.462540
$13$	−0.0151214	−0.00419393	−0.00209697	−	0.999998i	$-0.500667\pi$
$13$	−0.0151214	−0.00419393	−0.00209697	+	0.999998i	$0.500667\pi$
$14$	1.73109	0.462654
$15$	4.27518	1.10385
$16$	−4.63725	−1.15931
$17$	−3.66149	−0.888041	−0.444020	−	0.896017i	$-0.646448\pi$
$17$	−3.66149	−0.888041	−0.444020	+	0.896017i	$0.646448\pi$
$18$	−1.89797	−0.447356
$19$	3.18871	0.731541	0.365771	−	0.930705i	$-0.380805\pi$
$19$	3.18871	0.731541	0.365771	+	0.930705i	$0.380805\pi$
$20$	6.85007	1.53172
$21$	−0.912076	−0.199031
$22$	−6.62343	−1.41212
$23$	0.460551	0.0960315	0.0480157	−	0.998847i	$-0.484710\pi$
$23$	0.460551	0.0960315	0.0480157	+	0.998847i	$0.484710\pi$
$24$	0.754847	0.154083
$25$	13.2772	2.65544
$26$	0.0287000	0.00562854
$27$	1.00000	0.192450
$28$	−1.46141	−0.276180
$29$	−5.53924	−1.02861	−0.514305	−	0.857607i	$-0.671950\pi$
$29$	−5.53924	−1.02861	−0.514305	+	0.857607i	$0.671950\pi$
$30$	−8.11416	−1.48144
$31$	3.55308	0.638152	0.319076	−	0.947729i	$-0.396628\pi$
$31$	3.55308	0.638152	0.319076	+	0.947729i	$0.396628\pi$
$32$	7.29166	1.28900
$33$	3.48975	0.607487
$34$	6.94939	1.19181
$35$	−3.89929	−0.659100
$36$	1.60229	0.267048
$37$	−2.90616	−0.477770	−0.238885	−	0.971048i	$-0.576782\pi$
$37$	−2.90616	−0.477770	−0.238885	+	0.971048i	$0.576782\pi$
$38$	−6.05208	−0.981777
$39$	−0.0151214	−0.00242137
$40$	3.22711	0.510251
$41$	2.56776	0.401017	0.200509	−	0.979692i	$-0.435741\pi$
$41$	2.56776	0.401017	0.200509	+	0.979692i	$0.435741\pi$
$42$	1.73109	0.267113
$43$	5.25870	0.801945	0.400972	−	0.916090i	$-0.368672\pi$
$43$	5.25870	0.801945	0.400972	+	0.916090i	$0.368672\pi$
$44$	5.59157	0.842961
$45$	4.27518	0.637306
$46$	−0.874111	−0.128881
$47$	−12.3653	−1.80367	−0.901835	−	0.432080i	$-0.857780\pi$
$47$	−12.3653	−1.80367	−0.901835	+	0.432080i	$0.857780\pi$
$48$	−4.63725	−0.669329
$49$	−6.16812	−0.881160
$50$	−25.1997	−3.56377
$51$	−3.66149	−0.512711
$52$	−0.0242289	−0.00335994
$53$	14.1769	1.94734	0.973672	−	0.227953i	$-0.0732034\pi$
$53$	14.1769	1.94734	0.973672	+	0.227953i	$0.0732034\pi$
$54$	−1.89797	−0.258281
$55$	14.9193	2.01172
$56$	−0.688478	−0.0920017
$57$	3.18871	0.422355
$58$	10.5133	1.38046
$59$	−5.03300	−0.655241	−0.327621	−	0.944809i	$-0.606247\pi$
$59$	−5.03300	−0.655241	−0.327621	+	0.944809i	$0.606247\pi$
$60$	6.85007	0.884340
$61$	6.37469	0.816196	0.408098	−	0.912938i	$-0.366192\pi$
$61$	6.37469	0.816196	0.408098	+	0.912938i	$0.366192\pi$
$62$	−6.74363	−0.856442
$63$	−0.912076	−0.114911
$64$	−4.56485	−0.570606
$65$	−0.0646469	−0.00801846
$66$	−6.62343	−0.815288
$67$	−6.19408	−0.756728	−0.378364	−	0.925657i	$-0.623513\pi$
$67$	−6.19408	−0.756728	−0.378364	+	0.925657i	$0.623513\pi$
$68$	−5.86675	−0.711448
$69$	0.460551	0.0554438
$70$	7.40073	0.884556
$71$	−7.62614	−0.905057	−0.452528	−	0.891750i	$-0.649478\pi$
$71$	−7.62614	−0.905057	−0.452528	+	0.891750i	$0.649478\pi$
$72$	0.754847	0.0889596
$73$	9.15719	1.07177	0.535884	−	0.844291i	$-0.319978\pi$
$73$	9.15719	1.07177	0.535884	+	0.844291i	$0.319978\pi$
$74$	5.51581	0.641200
$75$	13.2772	1.53312
$76$	5.10923	0.586069
$77$	−3.18291	−0.362727
$78$	0.0287000	0.00324964
$79$	14.1369	1.59053	0.795264	−	0.606264i	$-0.207332\pi$
$79$	14.1369	1.59053	0.795264	+	0.606264i	$0.207332\pi$
$80$	−19.8251	−2.21651
$81$	1.00000	0.111111
$82$	−4.87354	−0.538192
$83$	7.73090	0.848576	0.424288	−	0.905527i	$-0.360524\pi$
$83$	7.73090	0.848576	0.424288	+	0.905527i	$0.360524\pi$
$84$	−1.46141	−0.159453
$85$	−15.6535	−1.69786
$86$	−9.98086	−1.07626
$87$	−5.53924	−0.593868
$88$	2.63423	0.280809
$89$	7.04418	0.746681	0.373341	−	0.927694i	$-0.378212\pi$
$89$	7.04418	0.746681	0.373341	+	0.927694i	$0.378212\pi$
$90$	−8.11416	−0.855308
$91$	0.0137919	0.00144578
$92$	0.737934	0.0769350
$93$	3.55308	0.368437
$94$	23.4690	2.42065
$95$	13.6323	1.39865
$96$	7.29166	0.744202
$97$	8.87541	0.901161	0.450581	−	0.892736i	$-0.351217\pi$
$97$	8.87541	0.901161	0.450581	+	0.892736i	$0.351217\pi$
$98$	11.7069	1.18258
$99$	3.48975	0.350733
$100$	21.2738	2.12738
$101$	1.70732	0.169885	0.0849423	−	0.996386i	$-0.472929\pi$
$101$	1.70732	0.169885	0.0849423	+	0.996386i	$0.472929\pi$
$102$	6.94939	0.688092
$103$	−5.11330	−0.503829	−0.251914	−	0.967750i	$-0.581060\pi$
$103$	−5.11330	−0.503829	−0.251914	+	0.967750i	$0.581060\pi$
$104$	−0.0114144	−0.00111927
$105$	−3.89929	−0.380532
$106$	−26.9073	−2.61347
$107$	18.0983	1.74963	0.874813	−	0.484461i	$-0.160984\pi$
$107$	18.0983	1.74963	0.874813	+	0.484461i	$0.160984\pi$
$108$	1.60229	0.154180
$109$	−8.92675	−0.855027	−0.427514	−	0.904009i	$-0.640610\pi$
$109$	−8.92675	−0.855027	−0.427514	+	0.904009i	$0.640610\pi$
$110$	−28.3164	−2.69986
$111$	−2.90616	−0.275841
$112$	4.22952	0.399652
$113$	−10.0630	−0.946643	−0.473322	−	0.880890i	$-0.656945\pi$
$113$	−10.0630	−0.946643	−0.473322	+	0.880890i	$0.656945\pi$
$114$	−6.05208	−0.566829
$115$	1.96894	0.183604
$116$	−8.87544	−0.824064
$117$	−0.0151214	−0.00139798
$118$	9.55248	0.879377
$119$	3.33955	0.306136
$120$	3.22711	0.294593
$121$	1.17832	0.107120
$122$	−12.0990	−1.09539
$123$	2.56776	0.231528
$124$	5.69305	0.511251
$125$	35.3864	3.16506
$126$	1.73109	0.154218
$127$	−2.38347	−0.211499	−0.105749	−	0.994393i	$-0.533724\pi$
$127$	−2.38347	−0.211499	−0.105749	+	0.994393i	$0.533724\pi$
$128$	−5.91938	−0.523204
$129$	5.25870	0.463003
$130$	0.122698	0.0107613
$131$	21.8407	1.90823	0.954117	−	0.299434i	$-0.0967978\pi$
$131$	21.8407	1.90823	0.954117	+	0.299434i	$0.0967978\pi$
$132$	5.59157	0.486684
$133$	−2.90835	−0.252186
$134$	11.7562	1.01558
$135$	4.27518	0.367949
$136$	−2.76386	−0.236999
$137$	5.95524	0.508790	0.254395	−	0.967100i	$-0.418124\pi$
$137$	5.95524	0.508790	0.254395	+	0.967100i	$0.418124\pi$
$138$	−0.874111	−0.0744093
$139$	0.921192	0.0781345	0.0390672	−	0.999237i	$-0.487561\pi$
$139$	0.921192	0.0781345	0.0390672	+	0.999237i	$0.487561\pi$
$140$	−6.24778	−0.528034
$141$	−12.3653	−1.04135
$142$	14.4742	1.21465
$143$	−0.0527700	−0.00441285
$144$	−4.63725	−0.386438
$145$	−23.6812	−1.96662
$146$	−17.3801	−1.43838
$147$	−6.16812	−0.508738
$148$	−4.65651	−0.382763
$149$	6.80098	0.557158	0.278579	−	0.960413i	$-0.410137\pi$
$149$	6.80098	0.557158	0.278579	+	0.960413i	$0.410137\pi$
$150$	−25.1997	−2.05754
$151$	−9.70971	−0.790165	−0.395082	−	0.918646i	$-0.629284\pi$
$151$	−9.70971	−0.790165	−0.395082	+	0.918646i	$0.629284\pi$
$152$	2.40699	0.195233
$153$	−3.66149	−0.296014
$154$	6.04107	0.486803
$155$	15.1901	1.22009
$156$	−0.0242289	−0.00193986
$157$	16.8618	1.34572	0.672861	−	0.739769i	$-0.265065\pi$
$157$	16.8618	1.34572	0.672861	+	0.739769i	$0.265065\pi$
$158$	−26.8314	−2.13459
$159$	14.1769	1.12430
$160$	31.1732	2.46446
$161$	−0.420057	−0.0331051
$162$	−1.89797	−0.149119
$163$	−7.08143	−0.554660	−0.277330	−	0.960775i	$-0.589450\pi$
$163$	−7.08143	−0.554660	−0.277330	+	0.960775i	$0.589450\pi$
$164$	4.11429	0.321272
$165$	14.9193	1.16147
$166$	−14.6730	−1.13885
$167$	2.76466	0.213936	0.106968	−	0.994262i	$-0.465886\pi$
$167$	2.76466	0.213936	0.106968	+	0.994262i	$0.465886\pi$
$168$	−0.688478	−0.0531172
$169$	−12.9998	−0.999982
$170$	29.7099	2.27864
$171$	3.18871	0.243847
$172$	8.42595	0.642473
$173$	7.33110	0.557373	0.278687	−	0.960382i	$-0.410101\pi$
$173$	7.33110	0.557373	0.278687	+	0.960382i	$0.410101\pi$
$174$	10.5133	0.797011
$175$	−12.1098	−0.915414
$176$	−16.1828	−1.21983
$177$	−5.03300	−0.378304
$178$	−13.3696	−1.00210
$179$	7.14917	0.534354	0.267177	−	0.963647i	$-0.413909\pi$
$179$	7.14917	0.534354	0.267177	+	0.963647i	$0.413909\pi$
$180$	6.85007	0.510574
$181$	−0.817707	−0.0607797	−0.0303898	−	0.999538i	$-0.509675\pi$
$181$	−0.817707	−0.0607797	−0.0303898	+	0.999538i	$0.509675\pi$
$182$	−0.0261766	−0.00194034
$183$	6.37469	0.471231
$184$	0.347646	0.0256288
$185$	−12.4244	−0.913458
$186$	−6.74363	−0.494467
$187$	−12.7777	−0.934395
$188$	−19.8128	−1.44500
$189$	−0.912076	−0.0663437
$190$	−25.8737	−1.87708
$191$	26.2775	1.90138	0.950688	−	0.310148i	$-0.100379\pi$
$191$	26.2775	1.90138	0.950688	+	0.310148i	$0.100379\pi$
$192$	−4.56485	−0.329440
$193$	−7.46275	−0.537180	−0.268590	−	0.963255i	$-0.586558\pi$
$193$	−7.46275	−0.537180	−0.268590	+	0.963255i	$0.586558\pi$
$194$	−16.8453	−1.20942
$195$	−0.0646469	−0.00462946
$196$	−9.88309	−0.705935
$197$	−16.7300	−1.19197	−0.595983	−	0.802997i	$-0.703237\pi$
$197$	−16.7300	−1.19197	−0.595983	+	0.802997i	$0.703237\pi$
$198$	−6.62343	−0.470707
$199$	14.1171	1.00073	0.500367	−	0.865814i	$-0.333199\pi$
$199$	14.1171	1.00073	0.500367	+	0.865814i	$0.333199\pi$
$200$	10.0222	0.708680
$201$	−6.19408	−0.436897
$202$	−3.24044	−0.227996
$203$	5.05220	0.354595
$204$	−5.86675	−0.410755
$205$	10.9777	0.766713
$206$	9.70489	0.676172
$207$	0.460551	0.0320105
$208$	0.0701219	0.00486208
$209$	11.1278	0.769726
$210$	7.40073	0.510699
$211$	0.378577	0.0260623	0.0130312	−	0.999915i	$-0.495852\pi$
$211$	0.378577	0.0260623	0.0130312	+	0.999915i	$0.495852\pi$
$212$	22.7154	1.56010
$213$	−7.62614	−0.522535
$214$	−34.3500	−2.34812
$215$	22.4819	1.53325
$216$	0.754847	0.0513609
$217$	−3.24068	−0.219991
$218$	16.9427	1.14750
$219$	9.15719	0.618786
$220$	23.9050	1.61167
$221$	0.0553669	0.00372438
$222$	5.51581	0.370197
$223$	0.967468	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$223$	0.967468	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$224$	−6.65055	−0.444359
$225$	13.2772	0.885145
$226$	19.0992	1.27046
$227$	−14.2181	−0.943689	−0.471845	−	0.881682i	$-0.656412\pi$
$227$	−14.2181	−0.943689	−0.471845	+	0.881682i	$0.656412\pi$
$228$	5.10923	0.338367
$229$	6.53996	0.432173	0.216086	−	0.976374i	$-0.430671\pi$
$229$	6.53996	0.432173	0.216086	+	0.976374i	$0.430671\pi$
$230$	−3.73698	−0.246409
$231$	−3.18291	−0.209420
$232$	−4.18128	−0.274514
$233$	−17.9287	−1.17455	−0.587274	−	0.809388i	$-0.699799\pi$
$233$	−17.9287	−1.17455	−0.587274	+	0.809388i	$0.699799\pi$
$234$	0.0287000	0.00187618
$235$	−52.8641	−3.44847
$236$	−8.06431	−0.524942
$237$	14.1369	0.918291
$238$	−6.33837	−0.410855
$239$	15.6637	1.01320	0.506600	−	0.862181i	$-0.330902\pi$
$239$	15.6637	1.01320	0.506600	+	0.862181i	$0.330902\pi$
$240$	−19.8251	−1.27970
$241$	11.9899	0.772339	0.386169	−	0.922428i	$-0.373798\pi$
$241$	11.9899	0.772339	0.386169	+	0.922428i	$0.373798\pi$
$242$	−2.23642	−0.143763
$243$	1.00000	0.0641500
$244$	10.2141	0.653890
$245$	−26.3698	−1.68471
$246$	−4.87354	−0.310725
$247$	−0.0482180	−0.00306803
$248$	2.68203	0.170309
$249$	7.73090	0.489926
$250$	−67.1624	−4.24772
$251$	−18.7734	−1.18497	−0.592483	−	0.805583i	$-0.701852\pi$
$251$	−18.7734	−1.18497	−0.592483	+	0.805583i	$0.701852\pi$
$252$	−1.46141	−0.0920600
$253$	1.60720	0.101044
$254$	4.52375	0.283846
$255$	−15.6535	−0.980261
$256$	20.3645	1.27278
$257$	−20.7350	−1.29342	−0.646708	−	0.762738i	$-0.723855\pi$
$257$	−20.7350	−1.29342	−0.646708	+	0.762738i	$0.723855\pi$
$258$	−9.98086	−0.621381
$259$	2.65064	0.164703
$260$	−0.103583	−0.00642394
$261$	−5.53924	−0.342870
$262$	−41.4530	−2.56098
$263$	−10.7434	−0.662467	−0.331234	−	0.943549i	$-0.607465\pi$
$263$	−10.7434	−0.662467	−0.331234	+	0.943549i	$0.607465\pi$
$264$	2.63423	0.162125
$265$	60.6087	3.72316
$266$	5.51996	0.338450
$267$	7.04418	0.431097
$268$	−9.92470	−0.606247
$269$	32.5866	1.98684	0.993419	−	0.114539i	$-0.0365392\pi$
$269$	32.5866	1.98684	0.993419	+	0.114539i	$0.0365392\pi$
$270$	−8.11416	−0.493812
$271$	5.80427	0.352584	0.176292	−	0.984338i	$-0.443590\pi$
$271$	5.80427	0.352584	0.176292	+	0.984338i	$0.443590\pi$
$272$	16.9792	1.02952
$273$	0.0137919	0.000834724 0
$274$	−11.3029	−0.682831
$275$	46.3340	2.79404
$276$	0.737934	0.0444184
$277$	−14.1539	−0.850424	−0.425212	−	0.905094i	$-0.639800\pi$
$277$	−14.1539	−0.850424	−0.425212	+	0.905094i	$0.639800\pi$
$278$	−1.74839	−0.104862
$279$	3.55308	0.212717
$280$	−2.94337	−0.175900
$281$	−9.23782	−0.551082	−0.275541	−	0.961289i	$-0.588857\pi$
$281$	−9.23782	−0.551082	−0.275541	+	0.961289i	$0.588857\pi$
$282$	23.4690	1.39756
$283$	−10.7076	−0.636502	−0.318251	−	0.948006i	$-0.603096\pi$
$283$	−10.7076	−0.636502	−0.318251	+	0.948006i	$0.603096\pi$
$284$	−12.2193	−0.725080
$285$	13.6323	0.807510
$286$	0.100156	0.00592234
$287$	−2.34200	−0.138244
$288$	7.29166	0.429665
$289$	−3.59352	−0.211384
$290$	44.9463	2.63934
$291$	8.87541	0.520286
$292$	14.6725	0.858640
$293$	−8.85139	−0.517104	−0.258552	−	0.965997i	$-0.583245\pi$
$293$	−8.85139	−0.517104	−0.258552	+	0.965997i	$0.583245\pi$
$294$	11.7069	0.682760
$295$	−21.5170	−1.25277
$296$	−2.19371	−0.127507
$297$	3.48975	0.202496
$298$	−12.9081	−0.747744
$299$	−0.00696419	−0.000402750 0
$300$	21.2738	1.22825
$301$	−4.79634	−0.276456
$302$	18.4287	1.06045
$303$	1.70732	0.0980829
$304$	−14.7869	−0.848085
$305$	27.2530	1.56050
$306$	6.94939	0.397270
$307$	4.18644	0.238933	0.119466	−	0.992838i	$-0.461882\pi$
$307$	4.18644	0.238933	0.119466	+	0.992838i	$0.461882\pi$
$308$	−5.09994	−0.290596
$309$	−5.11330	−0.290886
$310$	−28.8303	−1.63745
$311$	−5.36139	−0.304016	−0.152008	−	0.988379i	$-0.548574\pi$
$311$	−5.36139	−0.304016	−0.152008	+	0.988379i	$0.548574\pi$
$312$	−0.0114144	−0.000646212 0
$313$	−9.16913	−0.518270	−0.259135	−	0.965841i	$-0.583437\pi$
$313$	−9.16913	−0.518270	−0.259135	+	0.965841i	$0.583437\pi$
$314$	−32.0032	−1.80605
$315$	−3.89929	−0.219700
$316$	22.6514	1.27424
$317$	−10.4728	−0.588214	−0.294107	−	0.955773i	$-0.595022\pi$
$317$	−10.4728	−0.588214	−0.294107	+	0.955773i	$0.595022\pi$
$318$	−26.9073	−1.50889
$319$	−19.3305	−1.08230
$320$	−19.5156	−1.09095
$321$	18.0983	1.01015
$322$	0.797255	0.0444293
$323$	−11.6754	−0.649638
$324$	1.60229	0.0890159
$325$	−0.200770	−0.0111367
$326$	13.4403	0.744391
$327$	−8.92675	−0.493650
$328$	1.93827	0.107023
$329$	11.2781	0.621783
$330$	−28.3164	−1.55876
$331$	12.5877	0.691883	0.345941	−	0.938256i	$-0.387560\pi$
$331$	12.5877	0.691883	0.345941	+	0.938256i	$0.387560\pi$
$332$	12.3871	0.679831
$333$	−2.90616	−0.159257
$334$	−5.24724	−0.287116
$335$	−26.4808	−1.44680
$336$	4.22952	0.230739
$337$	9.41689	0.512971	0.256485	−	0.966548i	$-0.417435\pi$
$337$	9.41689	0.512971	0.256485	+	0.966548i	$0.417435\pi$
$338$	24.6732	1.34204
$339$	−10.0630	−0.546545
$340$	−25.0814	−1.36023
$341$	12.3993	0.671462
$342$	−6.05208	−0.327259
$343$	12.0103	0.648496
$344$	3.96952	0.214022
$345$	1.96894	0.106004
$346$	−13.9142	−0.748032
$347$	−7.90340	−0.424277	−0.212138	−	0.977240i	$-0.568043\pi$
$347$	−7.90340	−0.424277	−0.212138	+	0.977240i	$0.568043\pi$
$348$	−8.87544	−0.475774
$349$	−6.93063	−0.370988	−0.185494	−	0.982645i	$-0.559389\pi$
$349$	−6.93063	−0.370988	−0.185494	+	0.982645i	$0.559389\pi$
$350$	22.9840	1.22855
$351$	−0.0151214	−0.000807123 0
$352$	25.4461	1.35628
$353$	−24.4312	−1.30034	−0.650171	−	0.759788i	$-0.725303\pi$
$353$	−24.4312	−1.30034	−0.650171	+	0.759788i	$0.725303\pi$
$354$	9.55248	0.507709
$355$	−32.6031	−1.73040
$356$	11.2868	0.598199
$357$	3.33955	0.176748
$358$	−13.5689	−0.717139
$359$	−4.08927	−0.215823	−0.107912	−	0.994160i	$-0.534416\pi$
$359$	−4.08927	−0.215823	−0.107912	+	0.994160i	$0.534416\pi$
$360$	3.22711	0.170084
$361$	−8.83210	−0.464848
$362$	1.55198	0.0815704
$363$	1.17832	0.0618460
$364$	0.0220986	0.00115828
$365$	39.1487	2.04913
$366$	−12.0990	−0.632423
$367$	−36.2439	−1.89192	−0.945959	−	0.324287i	$-0.894876\pi$
$367$	−36.2439	−1.89192	−0.945959	+	0.324287i	$0.894876\pi$
$368$	−2.13569	−0.111330
$369$	2.56776	0.133672
$370$	23.5811	1.22592
$371$	−12.9304	−0.671312
$372$	5.69305	0.295171
$373$	10.0558	0.520670	0.260335	−	0.965518i	$-0.416167\pi$
$373$	10.0558	0.520670	0.260335	+	0.965518i	$0.416167\pi$
$374$	24.2516	1.25402
$375$	35.3864	1.82735
$376$	−9.33395	−0.481362
$377$	0.0837612	0.00431392
$378$	1.73109	0.0890377
$379$	−29.0846	−1.49398	−0.746988	−	0.664837i	$-0.768501\pi$
$379$	−29.0846	−1.49398	−0.746988	+	0.664837i	$0.768501\pi$
$380$	21.8429	1.12052
$381$	−2.38347	−0.122109
$382$	−49.8740	−2.55177
$383$	25.5865	1.30741	0.653704	−	0.756751i	$-0.273214\pi$
$383$	25.5865	1.30741	0.653704	+	0.756751i	$0.273214\pi$
$384$	−5.91938	−0.302072
$385$	−13.6075	−0.693504
$386$	14.1641	0.720932
$387$	5.25870	0.267315
$388$	14.2210	0.721960
$389$	−31.4782	−1.59601	−0.798003	−	0.602653i	$-0.794110\pi$
$389$	−31.4782	−1.59601	−0.798003	+	0.602653i	$0.794110\pi$
$390$	0.122698	0.00621305
$391$	−1.68630	−0.0852799
$392$	−4.65599	−0.235163
$393$	21.8407	1.10172
$394$	31.7531	1.59970
$395$	60.4379	3.04096
$396$	5.59157	0.280987
$397$	−37.0304	−1.85850	−0.929250	−	0.369450i	$-0.879546\pi$
$397$	−37.0304	−1.85850	−0.929250	+	0.369450i	$0.879546\pi$
$398$	−26.7938	−1.34305
$399$	−2.90835	−0.145600
$400$	−61.5696	−3.07848
$401$	−20.6813	−1.03278	−0.516388	−	0.856355i	$-0.672724\pi$
$401$	−20.6813	−1.03278	−0.516388	+	0.856355i	$0.672724\pi$
$402$	11.7562	0.586345
$403$	−0.0537277	−0.00267637
$404$	2.73561	0.136102
$405$	4.27518	0.212435
$406$	−9.58892	−0.475890
$407$	−10.1418	−0.502709
$408$	−2.76386	−0.136832
$409$	26.9344	1.33182	0.665909	−	0.746033i	$-0.268044\pi$
$409$	26.9344	1.33182	0.665909	+	0.746033i	$0.268044\pi$
$410$	−20.8353	−1.02898
$411$	5.95524	0.293750
$412$	−8.19298	−0.403639
$413$	4.59048	0.225883
$414$	−0.874111	−0.0429602
$415$	33.0510	1.62241
$416$	−0.110260	−0.00540596
$417$	0.921192	0.0451110
$418$	−21.1202	−1.03302
$419$	0.505740	0.0247070	0.0123535	−	0.999924i	$-0.496068\pi$
$419$	0.505740	0.0247070	0.0123535	+	0.999924i	$0.496068\pi$
$420$	−6.24778	−0.304860
$421$	−20.1894	−0.983973	−0.491986	−	0.870603i	$-0.663729\pi$
$421$	−20.1894	−0.983973	−0.491986	+	0.870603i	$0.663729\pi$
$422$	−0.718528	−0.0349774
$423$	−12.3653	−0.601224
$424$	10.7014	0.519705
$425$	−48.6142	−2.35813
$426$	14.4742	0.701276
$427$	−5.81420	−0.281369
$428$	28.9986	1.40170
$429$	−0.0527700	−0.00254776
$430$	−42.6700	−2.05773
$431$	−12.8532	−0.619116	−0.309558	−	0.950881i	$-0.600181\pi$
$431$	−12.8532	−0.619116	−0.309558	+	0.950881i	$0.600181\pi$
$432$	−4.63725	−0.223110
$433$	8.51162	0.409043	0.204521	−	0.978862i	$-0.434436\pi$
$433$	8.51162	0.409043	0.204521	+	0.978862i	$0.434436\pi$
$434$	6.15071	0.295243
$435$	−23.6812	−1.13543
$436$	−14.3032	−0.684999
$437$	1.46856	0.0702510
$438$	−17.3801	−0.830452
$439$	−20.3232	−0.969974	−0.484987	−	0.874521i	$-0.661176\pi$
$439$	−20.3232	−0.969974	−0.484987	+	0.874521i	$0.661176\pi$
$440$	11.2618	0.536885
$441$	−6.16812	−0.293720
$442$	−0.105085	−0.00499837
$443$	−5.67895	−0.269815	−0.134907	−	0.990858i	$-0.543074\pi$
$443$	−5.67895	−0.269815	−0.134907	+	0.990858i	$0.543074\pi$
$444$	−4.65651	−0.220988
$445$	30.1151	1.42759
$446$	−1.83622	−0.0869477
$447$	6.80098	0.321676
$448$	4.16349	0.196706
$449$	23.4140	1.10498	0.552488	−	0.833521i	$-0.313679\pi$
$449$	23.4140	1.10498	0.552488	+	0.833521i	$0.313679\pi$
$450$	−25.1997	−1.18792
$451$	8.96084	0.421950
$452$	−16.1237	−0.758397
$453$	−9.70971	−0.456202
$454$	26.9855	1.26649
$455$	0.0589629	0.00276422
$456$	2.40699	0.112718
$457$	−28.2116	−1.31968	−0.659841	−	0.751405i	$-0.729377\pi$
$457$	−28.2116	−1.31968	−0.659841	+	0.751405i	$0.729377\pi$
$458$	−12.4126	−0.580005
$459$	−3.66149	−0.170904
$460$	3.15480	0.147093
$461$	17.6570	0.822367	0.411184	−	0.911553i	$-0.365115\pi$
$461$	17.6570	0.822367	0.411184	+	0.911553i	$0.365115\pi$
$462$	6.04107	0.281056
$463$	36.5171	1.69709	0.848546	−	0.529122i	$-0.177479\pi$
$463$	36.5171	1.69709	0.848546	+	0.529122i	$0.177479\pi$
$464$	25.6868	1.19248
$465$	15.1901	0.704422
$466$	34.0281	1.57632
$467$	23.1486	1.07119	0.535594	−	0.844476i	$-0.320088\pi$
$467$	23.1486	1.07119	0.535594	+	0.844476i	$0.320088\pi$
$468$	−0.0242289	−0.00111998
$469$	5.64947	0.260868
$470$	100.334	4.62808
$471$	16.8618	0.776952
$472$	−3.79915	−0.174870
$473$	18.3515	0.843805
$474$	−26.8314	−1.23241
$475$	42.3371	1.94256
$476$	5.35092	0.245259
$477$	14.1769	0.649115
$478$	−29.7292	−1.35978
$479$	−2.22544	−0.101683	−0.0508415	−	0.998707i	$-0.516190\pi$
$479$	−2.22544	−0.101683	−0.0508415	+	0.998707i	$0.516190\pi$
$480$	31.1732	1.42285
$481$	0.0439454	0.00200374
$482$	−22.7565	−1.03653
$483$	−0.420057	−0.0191133
$484$	1.88801	0.0858188
$485$	37.9440	1.72295
$486$	−1.89797	−0.0860936
$487$	−23.4061	−1.06063	−0.530315	−	0.847801i	$-0.677926\pi$
$487$	−23.4061	−1.06063	−0.530315	+	0.847801i	$0.677926\pi$
$488$	4.81192	0.217825
$489$	−7.08143	−0.320233
$490$	50.0491	2.26099
$491$	−13.4532	−0.607133	−0.303567	−	0.952810i	$-0.598178\pi$
$491$	−13.4532	−0.607133	−0.303567	+	0.952810i	$0.598178\pi$
$492$	4.11429	0.185487
$493$	20.2818	0.913448
$494$	0.0915162	0.00411751
$495$	14.9193	0.670572
$496$	−16.4765	−0.739818
$497$	6.95562	0.312002
$498$	−14.6730	−0.657513
$499$	−26.6774	−1.19425	−0.597123	−	0.802150i	$-0.703689\pi$
$499$	−26.6774	−1.19425	−0.597123	+	0.802150i	$0.703689\pi$
$500$	56.6992	2.53567
$501$	2.76466	0.123516
$502$	35.6313	1.59030
$503$	42.9362	1.91443	0.957216	−	0.289376i	$-0.0934477\pi$
$503$	42.9362	1.91443	0.957216	+	0.289376i	$0.0934477\pi$
$504$	−0.688478	−0.0306672
$505$	7.29910	0.324806
$506$	−3.05042	−0.135608
$507$	−12.9998	−0.577340
$508$	−3.81900	−0.169441
$509$	30.6581	1.35890	0.679448	−	0.733724i	$-0.262219\pi$
$509$	30.6581	1.35890	0.679448	+	0.733724i	$0.262219\pi$
$510$	29.7099	1.31558
$511$	−8.35205	−0.369473
$512$	−26.8124	−1.18495
$513$	3.18871	0.140785
$514$	39.3545	1.73585
$515$	−21.8603	−0.963280
$516$	8.42595	0.370932
$517$	−43.1519	−1.89782
$518$	−5.03084	−0.221042
$519$	7.33110	0.321800
$520$	−0.0487986	−0.00213996
$521$	20.2985	0.889294	0.444647	−	0.895706i	$-0.353329\pi$
$521$	20.2985	0.889294	0.444647	+	0.895706i	$0.353329\pi$
$522$	10.5133	0.460155
$523$	34.1409	1.49288	0.746439	−	0.665454i	$-0.231762\pi$
$523$	34.1409	1.49288	0.746439	+	0.665454i	$0.231762\pi$
$524$	34.9951	1.52877
$525$	−12.1098	−0.528515
$526$	20.3907	0.889076
$527$	−13.0095	−0.566705
$528$	−16.1828	−0.704267
$529$	−22.7879	−0.990778
$530$	−115.033	−4.99674
$531$	−5.03300	−0.218414
$532$	−4.66001	−0.202037
$533$	−0.0388283	−0.00168184
$534$	−13.3696	−0.578561
$535$	77.3734	3.34514
$536$	−4.67559	−0.201955
$537$	7.14917	0.308509
$538$	−61.8483	−2.66647
$539$	−21.5252	−0.927154
$540$	6.85007	0.294780
$541$	−4.88697	−0.210107	−0.105054	−	0.994467i	$-0.533501\pi$
$541$	−4.88697	−0.210107	−0.105054	+	0.994467i	$0.533501\pi$
$542$	−11.0163	−0.473191
$543$	−0.817707	−0.0350912
$544$	−26.6983	−1.14468
$545$	−38.1635	−1.63474
$546$	−0.0261766	−0.00112026
$547$	−27.1705	−1.16173	−0.580863	−	0.814002i	$-0.697285\pi$
$547$	−27.1705	−1.16173	−0.580863	+	0.814002i	$0.697285\pi$
$548$	9.54200	0.407614
$549$	6.37469	0.272065
$550$	−87.9404	−3.74979
$551$	−17.6630	−0.752471
$552$	0.347646	0.0147968
$553$	−12.8939	−0.548306
$554$	26.8636	1.14133
$555$	−12.4244	−0.527385
$556$	1.47601	0.0625969
$557$	10.7616	0.455982	0.227991	−	0.973663i	$-0.426784\pi$
$557$	10.7616	0.455982	0.227991	+	0.973663i	$0.426784\pi$
$558$	−6.74363	−0.285481
$559$	−0.0795192	−0.00336330
$560$	18.0820	0.764103
$561$	−12.7777	−0.539473
$562$	17.5331	0.739589
$563$	30.8846	1.30163	0.650815	−	0.759237i	$-0.274428\pi$
$563$	30.8846	1.30163	0.650815	+	0.759237i	$0.274428\pi$
$564$	−19.8128	−0.834271
$565$	−43.0210	−1.80991
$566$	20.3227	0.854228
$567$	−0.912076	−0.0383036
$568$	−5.75657	−0.241540
$569$	30.5172	1.27935	0.639675	−	0.768646i	$-0.279069\pi$
$569$	30.5172	1.27935	0.639675	+	0.768646i	$0.279069\pi$
$570$	−25.8737	−1.08373
$571$	−6.83628	−0.286089	−0.143045	−	0.989716i	$-0.545689\pi$
$571$	−6.83628	−0.286089	−0.143045	+	0.989716i	$0.545689\pi$
$572$	−0.0845526	−0.00353532
$573$	26.2775	1.09776
$574$	4.44504	0.185532
$575$	6.11481	0.255005
$576$	−4.56485	−0.190202
$577$	−22.0087	−0.916235	−0.458117	−	0.888892i	$-0.651476\pi$
$577$	−22.0087	−0.916235	−0.458117	+	0.888892i	$0.651476\pi$
$578$	6.82039	0.283691
$579$	−7.46275	−0.310141
$580$	−37.9441	−1.57554
$581$	−7.05116	−0.292532
$582$	−16.8453	−0.698258
$583$	49.4737	2.04899
$584$	6.91228	0.286032
$585$	−0.0646469	−0.00267282
$586$	16.7997	0.693988
$587$	9.75031	0.402438	0.201219	−	0.979546i	$-0.435510\pi$
$587$	9.75031	0.402438	0.201219	+	0.979546i	$0.435510\pi$
$588$	−9.88309	−0.407572
$589$	11.3298	0.466834
$590$	40.8386	1.68130
$591$	−16.7300	−0.688182
$592$	13.4766	0.553885
$593$	−5.63754	−0.231506	−0.115753	−	0.993278i	$-0.536928\pi$
$593$	−5.63754	−0.231506	−0.115753	+	0.993278i	$0.536928\pi$
$594$	−6.62343	−0.271763
$595$	14.2772	0.585308
$596$	10.8971	0.446364
$597$	14.1171	0.577774
$598$	0.0132178	0.000540517 0
$599$	−25.8892	−1.05780	−0.528902	−	0.848683i	$-0.677396\pi$
$599$	−25.8892	−1.05780	−0.528902	+	0.848683i	$0.677396\pi$
$600$	10.0222	0.409156
$601$	−25.7738	−1.05134	−0.525668	−	0.850690i	$-0.676184\pi$
$601$	−25.7738	−1.05134	−0.525668	+	0.850690i	$0.676184\pi$
$602$	9.10330	0.371023
$603$	−6.19408	−0.252243
$604$	−15.5577	−0.633035
$605$	5.03755	0.204806
$606$	−3.24044	−0.131634
$607$	22.3126	0.905642	0.452821	−	0.891602i	$-0.350418\pi$
$607$	22.3126	0.905642	0.452821	+	0.891602i	$0.350418\pi$
$608$	23.2510	0.942954
$609$	5.05220	0.204726
$610$	−51.7253	−2.09430
$611$	0.186982	0.00756447
$612$	−5.86675	−0.237149
$613$	46.5270	1.87921	0.939605	−	0.342261i	$-0.111193\pi$
$613$	46.5270	1.87921	0.939605	+	0.342261i	$0.111193\pi$
$614$	−7.94574	−0.320664
$615$	10.9777	0.442662
$616$	−2.40261	−0.0968040
$617$	−17.2892	−0.696039	−0.348019	−	0.937487i	$-0.613146\pi$
$617$	−17.2892	−0.696039	−0.348019	+	0.937487i	$0.613146\pi$
$618$	9.70489	0.390388
$619$	32.8298	1.31954	0.659771	−	0.751466i	$-0.270653\pi$
$619$	32.8298	1.31954	0.659771	+	0.751466i	$0.270653\pi$
$620$	24.3388	0.977471
$621$	0.460551	0.0184813
$622$	10.1757	0.408010
$623$	−6.42482	−0.257405
$624$	0.0701219	0.00280712
$625$	84.8975	3.39590
$626$	17.4027	0.695552
$627$	11.1278	0.444402
$628$	27.0175	1.07812
$629$	10.6409	0.424280
$630$	7.40073	0.294852
$631$	17.6785	0.703768	0.351884	−	0.936044i	$-0.385541\pi$
$631$	17.6785	0.703768	0.351884	+	0.936044i	$0.385541\pi$
$632$	10.6712	0.424478
$633$	0.378577	0.0150471
$634$	19.8771	0.789422
$635$	−10.1898	−0.404369
$636$	22.7154	0.900725
$637$	0.0932708	0.00369553
$638$	36.6887	1.45252
$639$	−7.62614	−0.301686
$640$	−25.3064	−1.00032
$641$	47.9622	1.89439	0.947196	−	0.320654i	$-0.103903\pi$
$641$	47.9622	1.89439	0.947196	+	0.320654i	$0.103903\pi$
$642$	−34.3500	−1.35568
$643$	−20.8633	−0.822769	−0.411384	−	0.911462i	$-0.634955\pi$
$643$	−20.8633	−0.822769	−0.411384	+	0.911462i	$0.634955\pi$
$644$	−0.673052	−0.0265220
$645$	22.4819	0.885225
$646$	22.1596	0.871858
$647$	36.0411	1.41692	0.708460	−	0.705751i	$-0.249390\pi$
$647$	36.0411	1.41692	0.708460	+	0.705751i	$0.249390\pi$
$648$	0.754847	0.0296532
$649$	−17.5639	−0.689443
$650$	0.381055	0.0149462
$651$	−3.24068	−0.127012
$652$	−11.3465	−0.444363
$653$	−17.4738	−0.683801	−0.341901	−	0.939736i	$-0.611071\pi$
$653$	−17.4738	−0.683801	−0.341901	+	0.939736i	$0.611071\pi$
$654$	16.9427	0.662512
$655$	93.3731	3.64839
$656$	−11.9074	−0.464905
$657$	9.15719	0.357256
$658$	−21.4055	−0.834475
$659$	−42.9487	−1.67305	−0.836523	−	0.547932i	$-0.815415\pi$
$659$	−42.9487	−1.67305	−0.836523	+	0.547932i	$0.815415\pi$
$660$	23.9050	0.930500
$661$	5.52182	0.214774	0.107387	−	0.994217i	$-0.465752\pi$
$661$	5.52182	0.214774	0.107387	+	0.994217i	$0.465752\pi$
$662$	−23.8911	−0.928553
$663$	0.0553669	0.00215027
$664$	5.83565	0.226467
$665$	−12.4337	−0.482159
$666$	5.51581	0.213733
$667$	−2.55110	−0.0987789
$668$	4.42978	0.171393
$669$	0.967468	0.0374045
$670$	50.2598	1.94171
$671$	22.2461	0.858800
$672$	−6.65055	−0.256551
$673$	34.3099	1.32255	0.661275	−	0.750144i	$-0.270016\pi$
$673$	34.3099	1.32255	0.661275	+	0.750144i	$0.270016\pi$
$674$	−17.8730	−0.688441
$675$	13.2772	0.511039
$676$	−20.8294	−0.801129
$677$	35.8319	1.37713	0.688565	−	0.725175i	$-0.258241\pi$
$677$	35.8319	1.37713	0.688565	+	0.725175i	$0.258241\pi$
$678$	19.0992	0.733500
$679$	−8.09505	−0.310659
$680$	−11.8160	−0.453124
$681$	−14.2181	−0.544839
$682$	−23.5336	−0.901147
$683$	4.18554	0.160155	0.0800775	−	0.996789i	$-0.474483\pi$
$683$	4.18554	0.160155	0.0800775	+	0.996789i	$0.474483\pi$
$684$	5.10923	0.195356
$685$	25.4597	0.972766
$686$	−22.7952	−0.870325
$687$	6.53996	0.249515
$688$	−24.3859	−0.929705
$689$	−0.214375	−0.00816703
$690$	−3.73698	−0.142265
$691$	12.3267	0.468929	0.234465	−	0.972125i	$-0.424666\pi$
$691$	12.3267	0.468929	0.234465	+	0.972125i	$0.424666\pi$
$692$	11.7465	0.446536
$693$	−3.18291	−0.120909
$694$	15.0004	0.569408
$695$	3.93826	0.149387
$696$	−4.18128	−0.158491
$697$	−9.40183	−0.356120
$698$	13.1541	0.497891
$699$	−17.9287	−0.678126
$700$	−19.4034	−0.733378
$701$	−13.4742	−0.508915	−0.254458	−	0.967084i	$-0.581897\pi$
$701$	−13.4742	−0.508915	−0.254458	+	0.967084i	$0.581897\pi$
$702$	0.0287000	0.00108321
$703$	−9.26692	−0.349509
$704$	−15.9302	−0.600391
$705$	−52.8641	−1.99098
$706$	46.3697	1.74515
$707$	−1.55720	−0.0585647
$708$	−8.06431	−0.303075
$709$	19.1207	0.718093	0.359046	−	0.933320i	$-0.383102\pi$
$709$	19.1207	0.718093	0.359046	+	0.933320i	$0.383102\pi$
$710$	61.8798	2.32231
$711$	14.1369	0.530176
$712$	5.31728	0.199273
$713$	1.63637	0.0612827
$714$	−6.33837	−0.237207
$715$	−0.225601	−0.00843701
$716$	11.4550	0.428094
$717$	15.6637	0.584971
$718$	7.76130	0.289649
$719$	−10.8516	−0.404696	−0.202348	−	0.979314i	$-0.564857\pi$
$719$	−10.8516	−0.404696	−0.202348	+	0.979314i	$0.564857\pi$
$720$	−19.8251	−0.738837
$721$	4.66372	0.173686
$722$	16.7631	0.623857
$723$	11.9899	0.445910
$724$	−1.31020	−0.0486932
$725$	−73.5454	−2.73141
$726$	−2.23642	−0.0830015
$727$	−38.2348	−1.41805	−0.709025	−	0.705184i	$-0.750865\pi$
$727$	−38.2348	−1.41805	−0.709025	+	0.705184i	$0.750865\pi$
$728$	0.0104108	0.000385849 0
$729$	1.00000	0.0370370
$730$	−74.3030	−2.75008
$731$	−19.2547	−0.712160
$732$	10.2141	0.377524
$733$	−0.581126	−0.0214644	−0.0107322	−	0.999942i	$-0.503416\pi$
$733$	−0.581126	−0.0214644	−0.0107322	+	0.999942i	$0.503416\pi$
$734$	68.7898	2.53908
$735$	−26.3698	−0.972666
$736$	3.35818	0.123784
$737$	−21.6158	−0.796227
$738$	−4.87354	−0.179397
$739$	22.0811	0.812265	0.406133	−	0.913814i	$-0.366877\pi$
$739$	22.0811	0.812265	0.406133	+	0.913814i	$0.366877\pi$
$740$	−19.9074	−0.731811
$741$	−0.0482180	−0.00177133
$742$	24.5415	0.900946
$743$	20.8385	0.764492	0.382246	−	0.924061i	$-0.375151\pi$
$743$	20.8385	0.764492	0.382246	+	0.924061i	$0.375151\pi$
$744$	2.68203	0.0983281
$745$	29.0754	1.06524
$746$	−19.0856	−0.698774
$747$	7.73090	0.282859
$748$	−20.4735	−0.748584
$749$	−16.5070	−0.603153
$750$	−67.1624	−2.45242
$751$	−3.79325	−0.138418	−0.0692088	−	0.997602i	$-0.522047\pi$
$751$	−3.79325	−0.138418	−0.0692088	+	0.997602i	$0.522047\pi$
$752$	57.3412	2.09102
$753$	−18.7734	−0.684140
$754$	−0.158976	−0.00578957
$755$	−41.5108	−1.51073
$756$	−1.46141	−0.0531509
$757$	−11.9628	−0.434797	−0.217398	−	0.976083i	$-0.569757\pi$
$757$	−11.9628	−0.434797	−0.217398	+	0.976083i	$0.569757\pi$
$758$	55.2017	2.00502
$759$	1.60720	0.0583378
$760$	10.2903	0.373269
$761$	19.2944	0.699422	0.349711	−	0.936858i	$-0.386280\pi$
$761$	19.2944	0.699422	0.349711	+	0.936858i	$0.386280\pi$
$762$	4.52375	0.163878
$763$	8.14187	0.294755
$764$	42.1042	1.52328
$765$	−15.6535	−0.565954
$766$	−48.5623	−1.75463
$767$	0.0761063	0.00274804
$768$	20.3645	0.734841
$769$	0.454654	0.0163952	0.00819761	−	0.999966i	$-0.497391\pi$
$769$	0.454654	0.0163952	0.00819761	+	0.999966i	$0.497391\pi$
$770$	25.8267	0.930728
$771$	−20.7350	−0.746754
$772$	−11.9575	−0.430358
$773$	23.1882	0.834021	0.417010	−	0.908902i	$-0.363078\pi$
$773$	23.1882	0.834021	0.417010	+	0.908902i	$0.363078\pi$
$774$	−9.98086	−0.358755
$775$	47.1749	1.69457
$776$	6.69958	0.240501
$777$	2.65064	0.0950912
$778$	59.7446	2.14195
$779$	8.18787	0.293361
$780$	−0.103583	−0.00370886
$781$	−26.6133	−0.952299
$782$	3.20055	0.114451
$783$	−5.53924	−0.197956
$784$	28.6031	1.02154
$785$	72.0874	2.57291
$786$	−41.4530	−1.47858
$787$	−24.0326	−0.856669	−0.428335	−	0.903620i	$-0.640900\pi$
$787$	−24.0326	−0.856669	−0.428335	+	0.903620i	$0.640900\pi$
$788$	−26.8063	−0.954936
$789$	−10.7434	−0.382476
$790$	−114.709	−4.08117
$791$	9.17818	0.326338
$792$	2.63423	0.0936031
$793$	−0.0963946	−0.00342307
$794$	70.2825	2.49423
$795$	60.6087	2.14957
$796$	22.6196	0.801731
$797$	−4.59341	−0.162707	−0.0813535	−	0.996685i	$-0.525924\pi$
$797$	−4.59341	−0.162707	−0.0813535	+	0.996685i	$0.525924\pi$
$798$	5.51996	0.195404
$799$	45.2755	1.60173
$800$	96.8127	3.42285
$801$	7.04418	0.248894
$802$	39.2525	1.38606
$803$	31.9563	1.12771
$804$	−9.92470	−0.350017
$805$	−1.79582	−0.0632943
$806$	0.101973	0.00359186
$807$	32.5866	1.14710
$808$	1.28877	0.0453386
$809$	−28.0995	−0.987927	−0.493964	−	0.869483i	$-0.664452\pi$
$809$	−28.0995	−0.987927	−0.493964	+	0.869483i	$0.664452\pi$
$810$	−8.11416	−0.285103
$811$	40.8610	1.43482	0.717412	−	0.696649i	$-0.245326\pi$
$811$	40.8610	1.43482	0.717412	+	0.696649i	$0.245326\pi$
$812$	8.09508	0.284081
$813$	5.80427	0.203565
$814$	19.2488	0.674669
$815$	−30.2744	−1.06047
$816$	16.9792	0.594392
$817$	16.7685	0.586656
$818$	−51.1206	−1.78739
$819$	0.0137919	0.000481928 0
$820$	17.5894	0.614247
$821$	33.6926	1.17588	0.587939	−	0.808905i	$-0.299939\pi$
$821$	33.6926	1.17588	0.587939	+	0.808905i	$0.299939\pi$
$822$	−11.3029	−0.394232
$823$	−14.0989	−0.491458	−0.245729	−	0.969339i	$-0.579027\pi$
$823$	−14.0989	−0.491458	−0.245729	+	0.969339i	$0.579027\pi$
$824$	−3.85976	−0.134461
$825$	46.3340	1.61314
$826$	−8.71259	−0.303150
$827$	23.2949	0.810042	0.405021	−	0.914307i	$-0.367264\pi$
$827$	23.2949	0.810042	0.405021	+	0.914307i	$0.367264\pi$
$828$	0.737934	0.0256450
$829$	0.887490	0.0308238	0.0154119	−	0.999881i	$-0.495094\pi$
$829$	0.887490	0.0308238	0.0154119	+	0.999881i	$0.495094\pi$
$830$	−62.7297	−2.17738
$831$	−14.1539	−0.490992
$832$	0.0690271	0.00239309
$833$	22.5845	0.782506
$834$	−1.74839	−0.0605419
$835$	11.8194	0.409028
$836$	17.8299	0.616661
$837$	3.55308	0.122812
$838$	−0.959878	−0.0331584
$839$	28.3586	0.979048	0.489524	−	0.871990i	$-0.337171\pi$
$839$	28.3586	0.979048	0.489524	+	0.871990i	$0.337171\pi$
$840$	−2.94337	−0.101556
$841$	1.68313	0.0580390
$842$	38.3189	1.32056
$843$	−9.23782	−0.318168
$844$	0.606589	0.0208797
$845$	−55.5764	−1.91189
$846$	23.4690	0.806882
$847$	−1.07472	−0.0369279
$848$	−65.7417	−2.25758
$849$	−10.7076	−0.367485
$850$	92.2682	3.16477
$851$	−1.33844	−0.0458810
$852$	−12.2193	−0.418625
$853$	−13.7307	−0.470130	−0.235065	−	0.971980i	$-0.575530\pi$
$853$	−13.7307	−0.470130	−0.235065	+	0.971980i	$0.575530\pi$
$854$	11.0352	0.377616
$855$	13.6323	0.466216
$856$	13.6614	0.466938
$857$	−52.0819	−1.77908	−0.889542	−	0.456853i	$-0.848977\pi$
$857$	−52.0819	−1.77908	−0.889542	+	0.456853i	$0.848977\pi$
$858$	0.100156	0.00341926
$859$	19.0392	0.649609	0.324805	−	0.945781i	$-0.394701\pi$
$859$	19.0392	0.649609	0.324805	+	0.945781i	$0.394701\pi$
$860$	36.0225	1.22836
$861$	−2.34200	−0.0798150
$862$	24.3949	0.830894
$863$	−22.9932	−0.782698	−0.391349	−	0.920242i	$-0.627992\pi$
$863$	−22.9932	−0.782698	−0.391349	+	0.920242i	$0.627992\pi$
$864$	7.29166	0.248067
$865$	31.3418	1.06565
$866$	−16.1548	−0.548962
$867$	−3.59352	−0.122042
$868$	−5.19249	−0.176245
$869$	49.3342	1.67355
$870$	44.9463	1.52382
$871$	0.0936635	0.00317366
$872$	−6.73833	−0.228189
$873$	8.87541	0.300387
$874$	−2.78729	−0.0942815
$875$	−32.2751	−1.09110
$876$	14.6725	0.495736
$877$	−45.6160	−1.54034	−0.770171	−	0.637837i	$-0.779829\pi$
$877$	−45.6160	−1.54034	−0.770171	+	0.637837i	$0.779829\pi$
$878$	38.5728	1.30177
$879$	−8.85139	−0.298550
$880$	−69.1845	−2.33221
$881$	−11.3832	−0.383511	−0.191756	−	0.981443i	$-0.561418\pi$
$881$	−11.3832	−0.383511	−0.191756	+	0.981443i	$0.561418\pi$
$882$	11.7069	0.394192
$883$	−10.8648	−0.365628	−0.182814	−	0.983148i	$-0.558521\pi$
$883$	−10.8648	−0.365628	−0.182814	+	0.983148i	$0.558521\pi$
$884$	0.0887137	0.00298377
$885$	−21.5170	−0.723286
$886$	10.7785	0.362110
$887$	58.5439	1.96571	0.982856	−	0.184375i	$-0.0590261\pi$
$887$	58.5439	1.96571	0.982856	+	0.184375i	$0.0590261\pi$
$888$	−2.19371	−0.0736161
$889$	2.17391	0.0729105
$890$	−57.1576	−1.91593
$891$	3.48975	0.116911
$892$	1.55016	0.0519032
$893$	−39.4295	−1.31946
$894$	−12.9081	−0.431710
$895$	30.5640	1.02164
$896$	5.39892	0.180365
$897$	−0.00696419	−0.000232528 0
$898$	−44.4391	−1.48295
$899$	−19.6813	−0.656410
$900$	21.2738	0.709128
$901$	−51.9084	−1.72932
$902$	−17.0074	−0.566285
$903$	−4.79634	−0.159612
$904$	−7.59600	−0.252639
$905$	−3.49584	−0.116206
$906$	18.4287	0.612253
$907$	−7.97324	−0.264747	−0.132374	−	0.991200i	$-0.542260\pi$
$907$	−7.97324	−0.264747	−0.132374	+	0.991200i	$0.542260\pi$
$908$	−22.7815	−0.756031
$909$	1.70732	0.0566282
$910$	−0.111910	−0.00370977
$911$	−13.3089	−0.440944	−0.220472	−	0.975393i	$-0.570760\pi$
$911$	−13.3089	−0.440944	−0.220472	+	0.975393i	$0.570760\pi$
$912$	−14.7869	−0.489642
$913$	26.9789	0.892870
$914$	53.5447	1.77110
$915$	27.2530	0.900956
$916$	10.4789	0.346232
$917$	−19.9204	−0.657830
$918$	6.94939	0.229364
$919$	−2.90038	−0.0956746	−0.0478373	−	0.998855i	$-0.515233\pi$
$919$	−2.90038	−0.0956746	−0.0478373	+	0.998855i	$0.515233\pi$
$920$	1.48625	0.0490001
$921$	4.18644	0.137948
$922$	−33.5124	−1.10367
$923$	0.115318	0.00379575
$924$	−5.09994	−0.167776
$925$	−38.5856	−1.26869
$926$	−69.3082	−2.27761
$927$	−5.11330	−0.167943
$928$	−40.3902	−1.32587
$929$	−52.3696	−1.71819	−0.859096	−	0.511815i	$-0.828973\pi$
$929$	−52.3696	−1.71819	−0.859096	+	0.511815i	$0.828973\pi$
$930$	−28.8303	−0.945381
$931$	−19.6684	−0.644605
$932$	−28.7269	−0.940982
$933$	−5.36139	−0.175524
$934$	−43.9353	−1.43761
$935$	−54.6268	−1.78649
$936$	−0.0114144	−0.000373091 0
$937$	−37.8372	−1.23609	−0.618043	−	0.786144i	$-0.712074\pi$
$937$	−37.8372	−1.23609	−0.618043	+	0.786144i	$0.712074\pi$
$938$	−10.7225	−0.350103
$939$	−9.16913	−0.299223
$940$	−84.7034	−2.76272
$941$	39.7853	1.29696	0.648482	−	0.761230i	$-0.275404\pi$
$941$	39.7853	1.29696	0.648482	+	0.761230i	$0.275404\pi$
$942$	−32.0032	−1.04272
$943$	1.18259	0.0385103
$944$	23.3393	0.759629
$945$	−3.89929	−0.126844
$946$	−34.8307	−1.13244
$947$	−28.1096	−0.913439	−0.456719	−	0.889611i	$-0.650976\pi$
$947$	−28.1096	−0.913439	−0.456719	+	0.889611i	$0.650976\pi$
$948$	22.6514	0.735683
$949$	−0.138470	−0.00449493
$950$	−80.3545	−2.60705
$951$	−10.4728	−0.339605
$952$	2.52085	0.0817013
$953$	−6.97832	−0.226050	−0.113025	−	0.993592i	$-0.536054\pi$
$953$	−6.97832	−0.226050	−0.113025	+	0.993592i	$0.536054\pi$
$954$	−26.9073	−0.871155
$955$	112.341	3.63528
$956$	25.0977	0.811718
$957$	−19.3305	−0.624867
$958$	4.22382	0.136465
$959$	−5.43163	−0.175396
$960$	−19.5156	−0.629862
$961$	−18.3756	−0.592762
$962$	−0.0834070	−0.00268915
$963$	18.0983	0.583209
$964$	19.2113	0.618754
$965$	−31.9046	−1.02705
$966$	0.797255	0.0256513
$967$	−46.0525	−1.48095	−0.740474	−	0.672084i	$-0.765399\pi$
$967$	−46.0525	−1.48095	−0.740474	+	0.672084i	$0.765399\pi$
$968$	0.889455	0.0285882
$969$	−11.6754	−0.375069
$970$	−72.0165	−2.31231
$971$	0.0437094	0.00140270	0.000701350	−	1.00000i	$-0.499777\pi$
$971$	0.0437094	0.00140270	0.000701350		1.00000i	$0.499777\pi$
$972$	1.60229	0.0513934
$973$	−0.840197	−0.0269355
$974$	44.4240	1.42344
$975$	−0.200770	−0.00642979
$976$	−29.5611	−0.946226
$977$	−30.6044	−0.979122	−0.489561	−	0.871969i	$-0.662843\pi$
$977$	−30.6044	−0.979122	−0.489561	+	0.871969i	$0.662843\pi$
$978$	13.4403	0.429775
$979$	24.5824	0.785657
$980$	−42.2520	−1.34969
$981$	−8.92675	−0.285009
$982$	25.5337	0.814814
$983$	−41.9366	−1.33757	−0.668785	−	0.743456i	$-0.733185\pi$
$983$	−41.9366	−1.33757	−0.668785	+	0.743456i	$0.733185\pi$
$984$	1.93827	0.0617898
$985$	−71.5240	−2.27894
$986$	−38.4943	−1.22591
$987$	11.2781	0.358987
$988$	−0.0772590	−0.00245794
$989$	2.42190	0.0770119
$990$	−28.3164	−0.899953
$991$	1.94329	0.0617305	0.0308652	−	0.999524i	$-0.490174\pi$
$991$	1.94329	0.0617305	0.0308652	+	0.999524i	$0.490174\pi$
$992$	25.9079	0.822575
$993$	12.5877	0.399459
$994$	−13.2016	−0.418728
$995$	60.3531	1.91332
$996$	12.3871	0.392501
$997$	−21.1139	−0.668684	−0.334342	−	0.942452i	$-0.608514\pi$
$997$	−21.1139	−0.668684	−0.334342	+	0.942452i	$0.608514\pi$
$998$	50.6329	1.60276
$999$	−2.90616	−0.0919469

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6009.2.a.c.1.16	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6009.2.a.c.1.16	✓	92	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:	\( N \)	\(=\)	\( 6009 = 3 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.89797 q^{2} +1.00000 q^{3} +1.60229 q^{4} +4.27518 q^{5} -1.89797 q^{6} -0.912076 q^{7} +0.754847 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.89797 q^{2} +1.00000 q^{3} +1.60229 q^{4} +4.27518 q^{5} -1.89797 q^{6} -0.912076 q^{7} +0.754847 q^{8} +1.00000 q^{9} -8.11416 q^{10} +3.48975 q^{11} +1.60229 q^{12} -0.0151214 q^{13} +1.73109 q^{14} +4.27518 q^{15} -4.63725 q^{16} -3.66149 q^{17} -1.89797 q^{18} +3.18871 q^{19} +6.85007 q^{20} -0.912076 q^{21} -6.62343 q^{22} +0.460551 q^{23} +0.754847 q^{24} +13.2772 q^{25} +0.0287000 q^{26} +1.00000 q^{27} -1.46141 q^{28} -5.53924 q^{29} -8.11416 q^{30} +3.55308 q^{31} +7.29166 q^{32} +3.48975 q^{33} +6.94939 q^{34} -3.89929 q^{35} +1.60229 q^{36} -2.90616 q^{37} -6.05208 q^{38} -0.0151214 q^{39} +3.22711 q^{40} +2.56776 q^{41} +1.73109 q^{42} +5.25870 q^{43} +5.59157 q^{44} +4.27518 q^{45} -0.874111 q^{46} -12.3653 q^{47} -4.63725 q^{48} -6.16812 q^{49} -25.1997 q^{50} -3.66149 q^{51} -0.0242289 q^{52} +14.1769 q^{53} -1.89797 q^{54} +14.9193 q^{55} -0.688478 q^{56} +3.18871 q^{57} +10.5133 q^{58} -5.03300 q^{59} +6.85007 q^{60} +6.37469 q^{61} -6.74363 q^{62} -0.912076 q^{63} -4.56485 q^{64} -0.0646469 q^{65} -6.62343 q^{66} -6.19408 q^{67} -5.86675 q^{68} +0.460551 q^{69} +7.40073 q^{70} -7.62614 q^{71} +0.754847 q^{72} +9.15719 q^{73} +5.51581 q^{74} +13.2772 q^{75} +5.10923 q^{76} -3.18291 q^{77} +0.0287000 q^{78} +14.1369 q^{79} -19.8251 q^{80} +1.00000 q^{81} -4.87354 q^{82} +7.73090 q^{83} -1.46141 q^{84} -15.6535 q^{85} -9.98086 q^{86} -5.53924 q^{87} +2.63423 q^{88} +7.04418 q^{89} -8.11416 q^{90} +0.0137919 q^{91} +0.737934 q^{92} +3.55308 q^{93} +23.4690 q^{94} +13.6323 q^{95} +7.29166 q^{96} +8.87541 q^{97} +11.7069 q^{98} +3.48975 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9}+O(q^{10}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9} + 13 q^{10} + 40 q^{11} + 107 q^{12} + 6 q^{13} + 37 q^{14} + 34 q^{15} + 133 q^{16} + 77 q^{17} + 17 q^{18} + 34 q^{19} + 55 q^{20} + 22 q^{21} + 8 q^{22} + 83 q^{23} + 51 q^{24} + 110 q^{25} + 22 q^{26} + 92 q^{27} + 32 q^{28} + 97 q^{29} + 13 q^{30} + 44 q^{31} + 104 q^{32} + 40 q^{33} + 20 q^{34} + 80 q^{35} + 107 q^{36} + 12 q^{37} + 54 q^{38} + 6 q^{39} + 23 q^{40} + 67 q^{41} + 37 q^{42} + 30 q^{43} + 87 q^{44} + 34 q^{45} + 33 q^{46} + 69 q^{47} + 133 q^{48} + 112 q^{49} + 58 q^{50} + 77 q^{51} - 3 q^{52} + 113 q^{53} + 17 q^{54} + 42 q^{55} + 92 q^{56} + 34 q^{57} - 30 q^{58} + 72 q^{59} + 55 q^{60} + 19 q^{61} + 60 q^{62} + 22 q^{63} + 147 q^{64} + 74 q^{65} + 8 q^{66} + 26 q^{67} + 171 q^{68} + 83 q^{69} - 35 q^{70} + 134 q^{71} + 51 q^{72} - 17 q^{73} + 95 q^{74} + 110 q^{75} + 27 q^{76} + 108 q^{77} + 22 q^{78} + 159 q^{79} + 79 q^{80} + 92 q^{81} - 64 q^{82} + 73 q^{83} + 32 q^{84} - 4 q^{85} + 22 q^{86} + 97 q^{87} - 16 q^{88} + 50 q^{89} + 13 q^{90} + 17 q^{91} + 154 q^{92} + 44 q^{93} + 8 q^{94} + 155 q^{95} + 104 q^{96} - 20 q^{97} + 63 q^{98} + 40 q^{99}+O(q^{100}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9 + 13 * q^10 + 40 * q^11 + 107 * q^12 + 6 * q^13 + 37 * q^14 + 34 * q^15 + 133 * q^16 + 77 * q^17 + 17 * q^18 + 34 * q^19 + 55 * q^20 + 22 * q^21 + 8 * q^22 + 83 * q^23 + 51 * q^24 + 110 * q^25 + 22 * q^26 + 92 * q^27 + 32 * q^28 + 97 * q^29 + 13 * q^30 + 44 * q^31 + 104 * q^32 + 40 * q^33 + 20 * q^34 + 80 * q^35 + 107 * q^36 + 12 * q^37 + 54 * q^38 + 6 * q^39 + 23 * q^40 + 67 * q^41 + 37 * q^42 + 30 * q^43 + 87 * q^44 + 34 * q^45 + 33 * q^46 + 69 * q^47 + 133 * q^48 + 112 * q^49 + 58 * q^50 + 77 * q^51 - 3 * q^52 + 113 * q^53 + 17 * q^54 + 42 * q^55 + 92 * q^56 + 34 * q^57 - 30 * q^58 + 72 * q^59 + 55 * q^60 + 19 * q^61 + 60 * q^62 + 22 * q^63 + 147 * q^64 + 74 * q^65 + 8 * q^66 + 26 * q^67 + 171 * q^68 + 83 * q^69 - 35 * q^70 + 134 * q^71 + 51 * q^72 - 17 * q^73 + 95 * q^74 + 110 * q^75 + 27 * q^76 + 108 * q^77 + 22 * q^78 + 159 * q^79 + 79 * q^80 + 92 * q^81 - 64 * q^82 + 73 * q^83 + 32 * q^84 - 4 * q^85 + 22 * q^86 + 97 * q^87 - 16 * q^88 + 50 * q^89 + 13 * q^90 + 17 * q^91 + 154 * q^92 + 44 * q^93 + 8 * q^94 + 155 * q^95 + 104 * q^96 - 20 * q^97 + 63 * q^98 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more