LMFDB - Embedded newform 1045.2.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1045,2,Mod(1,1045)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1045.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1045, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-2,-1,4,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7281497.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 $ x^6 - 2x^5 - 5x^4 + 7x^3 + 6x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.748369$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.412130 q^{2} +1.67805 q^{3} -1.83015 q^{4} -1.00000 q^{5} -0.691575 q^{6} +0.0703171 q^{7} +1.57852 q^{8} -0.184142 q^{9} +0.412130 q^{10} -1.00000 q^{11} -3.07109 q^{12} +1.09018 q^{13} -0.0289798 q^{14} -1.67805 q^{15} +3.00974 q^{16} -2.37951 q^{17} +0.0758904 q^{18} -1.00000 q^{19} +1.83015 q^{20} +0.117996 q^{21} +0.412130 q^{22} -2.83321 q^{23} +2.64884 q^{24} +1.00000 q^{25} -0.449296 q^{26} -5.34316 q^{27} -0.128691 q^{28} -9.85701 q^{29} +0.691575 q^{30} +1.51103 q^{31} -4.39744 q^{32} -1.67805 q^{33} +0.980667 q^{34} -0.0703171 q^{35} +0.337007 q^{36} -1.38463 q^{37} +0.412130 q^{38} +1.82938 q^{39} -1.57852 q^{40} +2.59281 q^{41} -0.0486296 q^{42} -5.63972 q^{43} +1.83015 q^{44} +0.184142 q^{45} +1.16765 q^{46} -4.95295 q^{47} +5.05051 q^{48} -6.99506 q^{49} -0.412130 q^{50} -3.99294 q^{51} -1.99519 q^{52} -3.23082 q^{53} +2.20207 q^{54} +1.00000 q^{55} +0.110997 q^{56} -1.67805 q^{57} +4.06237 q^{58} +5.63037 q^{59} +3.07109 q^{60} +1.13632 q^{61} -0.622740 q^{62} -0.0129483 q^{63} -4.20717 q^{64} -1.09018 q^{65} +0.691575 q^{66} -10.5728 q^{67} +4.35486 q^{68} -4.75428 q^{69} +0.0289798 q^{70} -3.37214 q^{71} -0.290671 q^{72} -4.54954 q^{73} +0.570648 q^{74} +1.67805 q^{75} +1.83015 q^{76} -0.0703171 q^{77} -0.753942 q^{78} +2.22913 q^{79} -3.00974 q^{80} -8.41367 q^{81} -1.06857 q^{82} +2.48243 q^{83} -0.215950 q^{84} +2.37951 q^{85} +2.32430 q^{86} -16.5406 q^{87} -1.57852 q^{88} +6.45611 q^{89} -0.0758904 q^{90} +0.0766584 q^{91} +5.18520 q^{92} +2.53559 q^{93} +2.04126 q^{94} +1.00000 q^{95} -7.37913 q^{96} +9.43225 q^{97} +2.88287 q^{98} +0.184142 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21}+ \cdots - q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - q^98 - q^99

Coefficient data

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

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

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

$2$	−0.412130	−0.291420	−0.145710	−	0.989327i	$-0.546547\pi$
$2$	−0.412130	−0.291420	−0.145710	+	0.989327i	$0.546547\pi$
$3$	1.67805	0.968824	0.484412	−	0.874840i	$-0.339034\pi$
$3$	1.67805	0.968824	0.484412	+	0.874840i	$0.339034\pi$
$4$	−1.83015	−0.915075
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.691575	−0.282334
$7$	0.0703171	0.0265774	0.0132887	−	0.999912i	$-0.495770\pi$
$7$	0.0703171	0.0265774	0.0132887	+	0.999912i	$0.495770\pi$
$8$	1.57852	0.558090
$9$	−0.184142	−0.0613807
$10$	0.412130	0.130327
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−3.07109	−0.886546
$13$	1.09018	0.302362	0.151181	−	0.988506i	$-0.451692\pi$
$13$	1.09018	0.302362	0.151181	+	0.988506i	$0.451692\pi$
$14$	−0.0289798	−0.00774517
$15$	−1.67805	−0.433271
$16$	3.00974	0.752436
$17$	−2.37951	−0.577116	−0.288558	−	0.957462i	$-0.593176\pi$
$17$	−2.37951	−0.577116	−0.288558	+	0.957462i	$0.593176\pi$
$18$	0.0758904	0.0178875
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.83015	0.409234
$21$	0.117996	0.0257488
$22$	0.412130	0.0878663
$23$	−2.83321	−0.590766	−0.295383	−	0.955379i	$-0.595447\pi$
$23$	−2.83321	−0.590766	−0.295383	+	0.955379i	$0.595447\pi$
$24$	2.64884	0.540691
$25$	1.00000	0.200000
$26$	−0.449296	−0.0881142
$27$	−5.34316	−1.02829
$28$	−0.128691	−0.0243203
$29$	−9.85701	−1.83040	−0.915201	−	0.402999i	$-0.867968\pi$
$29$	−9.85701	−1.83040	−0.915201	+	0.402999i	$0.867968\pi$
$30$	0.691575	0.126264
$31$	1.51103	0.271389	0.135694	−	0.990751i	$-0.456673\pi$
$31$	1.51103	0.271389	0.135694	+	0.990751i	$0.456673\pi$
$32$	−4.39744	−0.777365
$33$	−1.67805	−0.292111
$34$	0.980667	0.168183
$35$	−0.0703171	−0.0118858
$36$	0.337007	0.0561679
$37$	−1.38463	−0.227632	−0.113816	−	0.993502i	$-0.536307\pi$
$37$	−1.38463	−0.227632	−0.113816	+	0.993502i	$0.536307\pi$
$38$	0.412130	0.0668563
$39$	1.82938	0.292935
$40$	−1.57852	−0.249586
$41$	2.59281	0.404929	0.202464	−	0.979290i	$-0.435105\pi$
$41$	2.59281	0.404929	0.202464	+	0.979290i	$0.435105\pi$
$42$	−0.0486296	−0.00750370
$43$	−5.63972	−0.860049	−0.430025	−	0.902817i	$-0.641495\pi$
$43$	−5.63972	−0.860049	−0.430025	+	0.902817i	$0.641495\pi$
$44$	1.83015	0.275905
$45$	0.184142	0.0274503
$46$	1.16765	0.172161
$47$	−4.95295	−0.722462	−0.361231	−	0.932476i	$-0.617643\pi$
$47$	−4.95295	−0.722462	−0.361231	+	0.932476i	$0.617643\pi$
$48$	5.05051	0.728978
$49$	−6.99506	−0.999294
$50$	−0.412130	−0.0582839
$51$	−3.99294	−0.559124
$52$	−1.99519	−0.276684
$53$	−3.23082	−0.443787	−0.221893	−	0.975071i	$-0.571224\pi$
$53$	−3.23082	−0.443787	−0.221893	+	0.975071i	$0.571224\pi$
$54$	2.20207	0.299664
$55$	1.00000	0.134840
$56$	0.110997	0.0148326
$57$	−1.67805	−0.222263
$58$	4.06237	0.533415
$59$	5.63037	0.733011	0.366506	−	0.930416i	$-0.380554\pi$
$59$	5.63037	0.733011	0.366506	+	0.930416i	$0.380554\pi$
$60$	3.07109	0.396475
$61$	1.13632	0.145491	0.0727457	−	0.997351i	$-0.476824\pi$
$61$	1.13632	0.145491	0.0727457	+	0.997351i	$0.476824\pi$
$62$	−0.622740	−0.0790880
$63$	−0.0129483	−0.00163134
$64$	−4.20717	−0.525897
$65$	−1.09018	−0.135220
$66$	0.691575	0.0851270
$67$	−10.5728	−1.29167	−0.645834	−	0.763478i	$-0.723490\pi$
$67$	−10.5728	−1.29167	−0.645834	+	0.763478i	$0.723490\pi$
$68$	4.35486	0.528104
$69$	−4.75428	−0.572348
$70$	0.0289798	0.00346374
$71$	−3.37214	−0.400199	−0.200099	−	0.979776i	$-0.564127\pi$
$71$	−3.37214	−0.400199	−0.200099	+	0.979776i	$0.564127\pi$
$72$	−0.290671	−0.0342560
$73$	−4.54954	−0.532483	−0.266242	−	0.963906i	$-0.585782\pi$
$73$	−4.54954	−0.532483	−0.266242	+	0.963906i	$0.585782\pi$
$74$	0.570648	0.0663365
$75$	1.67805	0.193765
$76$	1.83015	0.209933
$77$	−0.0703171	−0.00801338
$78$	−0.753942	−0.0853671
$79$	2.22913	0.250797	0.125398	−	0.992106i	$-0.459979\pi$
$79$	2.22913	0.250797	0.125398	+	0.992106i	$0.459979\pi$
$80$	−3.00974	−0.336500
$81$	−8.41367	−0.934852
$82$	−1.06857	−0.118004
$83$	2.48243	0.272482	0.136241	−	0.990676i	$-0.456498\pi$
$83$	2.48243	0.272482	0.136241	+	0.990676i	$0.456498\pi$
$84$	−0.215950	−0.0235621
$85$	2.37951	0.258094
$86$	2.32430	0.250635
$87$	−16.5406	−1.77334
$88$	−1.57852	−0.168271
$89$	6.45611	0.684346	0.342173	−	0.939637i	$-0.388837\pi$
$89$	6.45611	0.684346	0.342173	+	0.939637i	$0.388837\pi$
$90$	−0.0758904	−0.00799955
$91$	0.0766584	0.00803599
$92$	5.18520	0.540595
$93$	2.53559	0.262928
$94$	2.04126	0.210540
$95$	1.00000	0.102598
$96$	−7.37913	−0.753130
$97$	9.43225	0.957700	0.478850	−	0.877897i	$-0.341054\pi$
$97$	9.43225	0.957700	0.478850	+	0.877897i	$0.341054\pi$
$98$	2.88287	0.291214
$99$	0.184142	0.0185070
$100$	−1.83015	−0.183015
$101$	2.07500	0.206470	0.103235	−	0.994657i	$-0.467081\pi$
$101$	2.07500	0.206470	0.103235	+	0.994657i	$0.467081\pi$
$102$	1.64561	0.162940
$103$	13.9955	1.37902	0.689510	−	0.724276i	$-0.257826\pi$
$103$	13.9955	1.37902	0.689510	+	0.724276i	$0.257826\pi$
$104$	1.72087	0.168745
$105$	−0.117996	−0.0115152
$106$	1.33152	0.129328
$107$	2.15806	0.208628	0.104314	−	0.994544i	$-0.466735\pi$
$107$	2.15806	0.208628	0.104314	+	0.994544i	$0.466735\pi$
$108$	9.77877	0.940963
$109$	−2.10949	−0.202053	−0.101026	−	0.994884i	$-0.532213\pi$
$109$	−2.10949	−0.202053	−0.101026	+	0.994884i	$0.532213\pi$
$110$	−0.412130	−0.0392950
$111$	−2.32348	−0.220535
$112$	0.211637	0.0199978
$113$	1.73202	0.162935	0.0814673	−	0.996676i	$-0.474039\pi$
$113$	1.73202	0.162935	0.0814673	+	0.996676i	$0.474039\pi$
$114$	0.691575	0.0647719
$115$	2.83321	0.264199
$116$	18.0398	1.67495
$117$	−0.200748	−0.0185592
$118$	−2.32044	−0.213614
$119$	−0.167320	−0.0153382
$120$	−2.64884	−0.241804
$121$	1.00000	0.0909091
$122$	−0.468313	−0.0423991
$123$	4.35087	0.392305
$124$	−2.76541	−0.248341
$125$	−1.00000	−0.0894427
$126$	0.00533639	0.000475404 0
$127$	2.45825	0.218134	0.109067	−	0.994034i	$-0.465214\pi$
$127$	2.45825	0.218134	0.109067	+	0.994034i	$0.465214\pi$
$128$	10.5288	0.930622
$129$	−9.46374	−0.833236
$130$	0.449296	0.0394059
$131$	−18.9151	−1.65262	−0.826312	−	0.563212i	$-0.809565\pi$
$131$	−18.9151	−1.65262	−0.826312	+	0.563212i	$0.809565\pi$
$132$	3.07109	0.267304
$133$	−0.0703171	−0.00609727
$134$	4.35735	0.376418
$135$	5.34316	0.459866
$136$	−3.75610	−0.322083
$137$	−15.5938	−1.33227	−0.666134	−	0.745832i	$-0.732052\pi$
$137$	−15.5938	−1.33227	−0.666134	+	0.745832i	$0.732052\pi$
$138$	1.95938	0.166793
$139$	9.69714	0.822501	0.411250	−	0.911522i	$-0.365092\pi$
$139$	9.69714	0.822501	0.411250	+	0.911522i	$0.365092\pi$
$140$	0.128691	0.0108764
$141$	−8.31131	−0.699938
$142$	1.38976	0.116626
$143$	−1.09018	−0.0911656
$144$	−0.554220	−0.0461850
$145$	9.85701	0.818580
$146$	1.87500	0.155176
$147$	−11.7381	−0.968139
$148$	2.53408	0.208300
$149$	8.64937	0.708584	0.354292	−	0.935135i	$-0.384722\pi$
$149$	8.64937	0.708584	0.354292	+	0.935135i	$0.384722\pi$
$150$	−0.691575	−0.0564669
$151$	8.67013	0.705565	0.352782	−	0.935705i	$-0.385236\pi$
$151$	8.67013	0.705565	0.352782	+	0.935705i	$0.385236\pi$
$152$	−1.57852	−0.128035
$153$	0.438168	0.0354238
$154$	0.0289798	0.00233526
$155$	−1.51103	−0.121369
$156$	−3.34804	−0.268058
$157$	1.46525	0.116939	0.0584697	−	0.998289i	$-0.481378\pi$
$157$	1.46525	0.116939	0.0584697	+	0.998289i	$0.481378\pi$
$158$	−0.918691	−0.0730872
$159$	−5.42148	−0.429951
$160$	4.39744	0.347648
$161$	−0.199223	−0.0157010
$162$	3.46752	0.272434
$163$	8.65423	0.677852	0.338926	−	0.940813i	$-0.389936\pi$
$163$	8.65423	0.677852	0.338926	+	0.940813i	$0.389936\pi$
$164$	−4.74523	−0.370540
$165$	1.67805	0.130636
$166$	−1.02308	−0.0794065
$167$	13.6449	1.05587	0.527936	−	0.849284i	$-0.322966\pi$
$167$	13.6449	1.05587	0.527936	+	0.849284i	$0.322966\pi$
$168$	0.186258	0.0143701
$169$	−11.8115	−0.908577
$170$	−0.980667	−0.0752137
$171$	0.184142	0.0140817
$172$	10.3215	0.787009
$173$	2.05153	0.155975	0.0779876	−	0.996954i	$-0.475151\pi$
$173$	2.05153	0.155975	0.0779876	+	0.996954i	$0.475151\pi$
$174$	6.81686	0.516785
$175$	0.0703171	0.00531547
$176$	−3.00974	−0.226868
$177$	9.44805	0.710159
$178$	−2.66075	−0.199432
$179$	−22.4395	−1.67720	−0.838602	−	0.544744i	$-0.816627\pi$
$179$	−22.4395	−1.67720	−0.838602	+	0.544744i	$0.816627\pi$
$180$	−0.337007	−0.0251190
$181$	−13.8894	−1.03239	−0.516195	−	0.856471i	$-0.672652\pi$
$181$	−13.8894	−1.03239	−0.516195	+	0.856471i	$0.672652\pi$
$182$	−0.0315932	−0.00234184
$183$	1.90681	0.140956
$184$	−4.47228	−0.329701
$185$	1.38463	0.101800
$186$	−1.04499	−0.0766224
$187$	2.37951	0.174007
$188$	9.06464	0.661107
$189$	−0.375715	−0.0273293
$190$	−0.412130	−0.0298990
$191$	−14.8007	−1.07094	−0.535472	−	0.844553i	$-0.679866\pi$
$191$	−14.8007	−1.07094	−0.535472	+	0.844553i	$0.679866\pi$
$192$	−7.05985	−0.509501
$193$	9.31714	0.670662	0.335331	−	0.942100i	$-0.391152\pi$
$193$	9.31714	0.670662	0.335331	+	0.942100i	$0.391152\pi$
$194$	−3.88731	−0.279093
$195$	−1.82938	−0.131005
$196$	12.8020	0.914428
$197$	−12.3492	−0.879844	−0.439922	−	0.898036i	$-0.644994\pi$
$197$	−12.3492	−0.879844	−0.439922	+	0.898036i	$0.644994\pi$
$198$	−0.0758904	−0.00539329
$199$	−3.24025	−0.229695	−0.114848	−	0.993383i	$-0.536638\pi$
$199$	−3.24025	−0.229695	−0.114848	+	0.993383i	$0.536638\pi$
$200$	1.57852	0.111618
$201$	−17.7416	−1.25140
$202$	−0.855167	−0.0601694
$203$	−0.693117	−0.0486472
$204$	7.30768	0.511640
$205$	−2.59281	−0.181090
$206$	−5.76797	−0.401874
$207$	0.521714	0.0362616
$208$	3.28117	0.227508
$209$	1.00000	0.0691714
$210$	0.0486296	0.00335576
$211$	24.9918	1.72051	0.860253	−	0.509868i	$-0.170306\pi$
$211$	24.9918	1.72051	0.860253	+	0.509868i	$0.170306\pi$
$212$	5.91287	0.406098
$213$	−5.65862	−0.387722
$214$	−0.889402	−0.0607982
$215$	5.63972	0.384626
$216$	−8.43427	−0.573879
$217$	0.106251	0.00721280
$218$	0.869384	0.0588821
$219$	−7.63436	−0.515882
$220$	−1.83015	−0.123389
$221$	−2.59410	−0.174498
$222$	0.957577	0.0642683
$223$	17.1079	1.14563	0.572814	−	0.819685i	$-0.305852\pi$
$223$	17.1079	1.14563	0.572814	+	0.819685i	$0.305852\pi$
$224$	−0.309215	−0.0206603
$225$	−0.184142	−0.0122761
$226$	−0.713816	−0.0474823
$227$	−2.43249	−0.161450	−0.0807251	−	0.996736i	$-0.525724\pi$
$227$	−2.43249	−0.161450	−0.0807251	+	0.996736i	$0.525724\pi$
$228$	3.07109	0.203388
$229$	4.21444	0.278498	0.139249	−	0.990257i	$-0.455531\pi$
$229$	4.21444	0.278498	0.139249	+	0.990257i	$0.455531\pi$
$230$	−1.16765	−0.0769927
$231$	−0.117996	−0.00776355
$232$	−15.5595	−1.02153
$233$	5.17997	0.339351	0.169676	−	0.985500i	$-0.445728\pi$
$233$	5.17997	0.339351	0.169676	+	0.985500i	$0.445728\pi$
$234$	0.0827343	0.00540851
$235$	4.95295	0.323095
$236$	−10.3044	−0.670760
$237$	3.74060	0.242978
$238$	0.0689576	0.00446986
$239$	1.21784	0.0787756	0.0393878	−	0.999224i	$-0.487459\pi$
$239$	1.21784	0.0787756	0.0393878	+	0.999224i	$0.487459\pi$
$240$	−5.05051	−0.326009
$241$	−5.39692	−0.347646	−0.173823	−	0.984777i	$-0.555612\pi$
$241$	−5.39692	−0.347646	−0.173823	+	0.984777i	$0.555612\pi$
$242$	−0.412130	−0.0264927
$243$	1.91090	0.122584
$244$	−2.07964	−0.133135
$245$	6.99506	0.446898
$246$	−1.79312	−0.114325
$247$	−1.09018	−0.0693666
$248$	2.38519	0.151460
$249$	4.16564	0.263987
$250$	0.412130	0.0260654
$251$	1.25250	0.0790573	0.0395287	−	0.999218i	$-0.487414\pi$
$251$	1.25250	0.0790573	0.0395287	+	0.999218i	$0.487414\pi$
$252$	0.0236974	0.00149279
$253$	2.83321	0.178123
$254$	−1.01312	−0.0635687
$255$	3.99294	0.250048
$256$	4.07512	0.254695
$257$	6.98317	0.435598	0.217799	−	0.975994i	$-0.430112\pi$
$257$	6.98317	0.435598	0.217799	+	0.975994i	$0.430112\pi$
$258$	3.90029	0.242821
$259$	−0.0973633	−0.00604986
$260$	1.99519	0.123737
$261$	1.81509	0.112351
$262$	7.79549	0.481607
$263$	13.1108	0.808449	0.404224	−	0.914660i	$-0.367542\pi$
$263$	13.1108	0.808449	0.404224	+	0.914660i	$0.367542\pi$
$264$	−2.64884	−0.163025
$265$	3.23082	0.198467
$266$	0.0289798	0.00177686
$267$	10.8337	0.663011
$268$	19.3497	1.18197
$269$	−2.33183	−0.142174	−0.0710871	−	0.997470i	$-0.522647\pi$
$269$	−2.33183	−0.142174	−0.0710871	+	0.997470i	$0.522647\pi$
$270$	−2.20207	−0.134014
$271$	5.01423	0.304593	0.152296	−	0.988335i	$-0.451333\pi$
$271$	5.01423	0.304593	0.152296	+	0.988335i	$0.451333\pi$
$272$	−7.16172	−0.434243
$273$	0.128637	0.00778545
$274$	6.42666	0.388249
$275$	−1.00000	−0.0603023
$276$	8.70104	0.523741
$277$	27.8630	1.67412	0.837062	−	0.547108i	$-0.184271\pi$
$277$	27.8630	1.67412	0.837062	+	0.547108i	$0.184271\pi$
$278$	−3.99648	−0.239693
$279$	−0.278244	−0.0166580
$280$	−0.110997	−0.00663333
$281$	22.1845	1.32342	0.661708	−	0.749762i	$-0.269832\pi$
$281$	22.1845	1.32342	0.661708	+	0.749762i	$0.269832\pi$
$282$	3.42534	0.203976
$283$	−18.8783	−1.12220	−0.561099	−	0.827749i	$-0.689621\pi$
$283$	−18.8783	−1.12220	−0.561099	+	0.827749i	$0.689621\pi$
$284$	6.17151	0.366212
$285$	1.67805	0.0993992
$286$	0.449296	0.0265674
$287$	0.182319	0.0107619
$288$	0.809754	0.0477152
$289$	−11.3379	−0.666937
$290$	−4.06237	−0.238550
$291$	15.8278	0.927842
$292$	8.32633	0.487262
$293$	16.2931	0.951854	0.475927	−	0.879485i	$-0.342113\pi$
$293$	16.2931	0.951854	0.475927	+	0.879485i	$0.342113\pi$
$294$	4.83761	0.282135
$295$	−5.63037	−0.327813
$296$	−2.18567	−0.127039
$297$	5.34316	0.310041
$298$	−3.56466	−0.206495
$299$	−3.08872	−0.178625
$300$	−3.07109	−0.177309
$301$	−0.396569	−0.0228578
$302$	−3.57322	−0.205615
$303$	3.48195	0.200033
$304$	−3.00974	−0.172621
$305$	−1.13632	−0.0650657
$306$	−0.180582	−0.0103232
$307$	30.9183	1.76460	0.882301	−	0.470686i	$-0.155994\pi$
$307$	30.9183	1.76460	0.882301	+	0.470686i	$0.155994\pi$
$308$	0.128691	0.00733284
$309$	23.4852	1.33603
$310$	0.622740	0.0353692
$311$	−31.5851	−1.79103	−0.895513	−	0.445035i	$-0.853191\pi$
$311$	−31.5851	−1.79103	−0.895513	+	0.445035i	$0.853191\pi$
$312$	2.88771	0.163484
$313$	−18.4889	−1.04505	−0.522527	−	0.852623i	$-0.675011\pi$
$313$	−18.4889	−1.04505	−0.522527	+	0.852623i	$0.675011\pi$
$314$	−0.603872	−0.0340784
$315$	0.0129483	0.000729556 0
$316$	−4.07964	−0.229498
$317$	−28.7562	−1.61511	−0.807555	−	0.589793i	$-0.799209\pi$
$317$	−28.7562	−1.61511	−0.807555	+	0.589793i	$0.799209\pi$
$318$	2.23435	0.125296
$319$	9.85701	0.551887
$320$	4.20717	0.235188
$321$	3.62134	0.202124
$322$	0.0821059	0.00457558
$323$	2.37951	0.132399
$324$	15.3983	0.855459
$325$	1.09018	0.0604724
$326$	−3.56667	−0.197539
$327$	−3.53984	−0.195753
$328$	4.09280	0.225987
$329$	−0.348277	−0.0192011
$330$	−0.691575	−0.0380699
$331$	−21.0396	−1.15644	−0.578221	−	0.815880i	$-0.696253\pi$
$331$	−21.0396	−1.15644	−0.578221	+	0.815880i	$0.696253\pi$
$332$	−4.54321	−0.249341
$333$	0.254969	0.0139722
$334$	−5.62346	−0.307702
$335$	10.5728	0.577652
$336$	0.355137	0.0193743
$337$	3.72856	0.203108	0.101554	−	0.994830i	$-0.467619\pi$
$337$	3.72856	0.203108	0.101554	+	0.994830i	$0.467619\pi$
$338$	4.86787	0.264777
$339$	2.90642	0.157855
$340$	−4.35486	−0.236175
$341$	−1.51103	−0.0818268
$342$	−0.0758904	−0.00410368
$343$	−0.984092	−0.0531360
$344$	−8.90240	−0.479985
$345$	4.75428	0.255962
$346$	−0.845498	−0.0454542
$347$	18.0660	0.969836	0.484918	−	0.874560i	$-0.338849\pi$
$347$	18.0660	0.969836	0.484918	+	0.874560i	$0.338849\pi$
$348$	30.2717	1.62273
$349$	21.4770	1.14964	0.574820	−	0.818280i	$-0.305072\pi$
$349$	21.4770	1.14964	0.574820	+	0.818280i	$0.305072\pi$
$350$	−0.0289798	−0.00154903
$351$	−5.82501	−0.310916
$352$	4.39744	0.234384
$353$	−19.3979	−1.03245	−0.516223	−	0.856454i	$-0.672662\pi$
$353$	−19.3979	−1.03245	−0.516223	+	0.856454i	$0.672662\pi$
$354$	−3.89382	−0.206954
$355$	3.37214	0.178974
$356$	−11.8156	−0.626228
$357$	−0.280772	−0.0148600
$358$	9.24797	0.488770
$359$	−10.6909	−0.564245	−0.282123	−	0.959378i	$-0.591038\pi$
$359$	−10.6909	−0.564245	−0.282123	+	0.959378i	$0.591038\pi$
$360$	0.290671	0.0153197
$361$	1.00000	0.0526316
$362$	5.72423	0.300859
$363$	1.67805	0.0880749
$364$	−0.140296	−0.00735353
$365$	4.54954	0.238134
$366$	−0.785853	−0.0410772
$367$	26.5288	1.38479	0.692395	−	0.721518i	$-0.256556\pi$
$367$	26.5288	1.38479	0.692395	+	0.721518i	$0.256556\pi$
$368$	−8.52725	−0.444514
$369$	−0.477445	−0.0248548
$370$	−0.570648	−0.0296666
$371$	−0.227182	−0.0117947
$372$	−4.64050	−0.240599
$373$	−13.5037	−0.699194	−0.349597	−	0.936900i	$-0.613681\pi$
$373$	−13.5037	−0.699194	−0.349597	+	0.936900i	$0.613681\pi$
$374$	−0.980667	−0.0507091
$375$	−1.67805	−0.0866542
$376$	−7.81832	−0.403199
$377$	−10.7459	−0.553444
$378$	0.154843	0.00796428
$379$	−25.0206	−1.28522	−0.642611	−	0.766193i	$-0.722149\pi$
$379$	−25.0206	−1.28522	−0.642611	+	0.766193i	$0.722149\pi$
$380$	−1.83015	−0.0938847
$381$	4.12507	0.211334
$382$	6.09983	0.312094
$383$	22.3247	1.14074	0.570369	−	0.821389i	$-0.306800\pi$
$383$	22.3247	1.14074	0.570369	+	0.821389i	$0.306800\pi$
$384$	17.6678	0.901608
$385$	0.0703171	0.00358369
$386$	−3.83987	−0.195444
$387$	1.03851	0.0527904
$388$	−17.2624	−0.876367
$389$	10.1663	0.515450	0.257725	−	0.966218i	$-0.417027\pi$
$389$	10.1663	0.515450	0.257725	+	0.966218i	$0.417027\pi$
$390$	0.753942	0.0381773
$391$	6.74166	0.340940
$392$	−11.0418	−0.557696
$393$	−31.7406	−1.60110
$394$	5.08947	0.256404
$395$	−2.22913	−0.112160
$396$	−0.337007	−0.0169353
$397$	18.2304	0.914956	0.457478	−	0.889221i	$-0.348753\pi$
$397$	18.2304	0.914956	0.457478	+	0.889221i	$0.348753\pi$
$398$	1.33540	0.0669377
$399$	−0.117996	−0.00590718
$400$	3.00974	0.150487
$401$	−4.66663	−0.233040	−0.116520	−	0.993188i	$-0.537174\pi$
$401$	−4.66663	−0.233040	−0.116520	+	0.993188i	$0.537174\pi$
$402$	7.31186	0.364682
$403$	1.64730	0.0820577
$404$	−3.79755	−0.188935
$405$	8.41367	0.418078
$406$	0.285654	0.0141768
$407$	1.38463	0.0686337
$408$	−6.30293	−0.312042
$409$	−5.67179	−0.280452	−0.140226	−	0.990120i	$-0.544783\pi$
$409$	−5.67179	−0.280452	−0.140226	+	0.990120i	$0.544783\pi$
$410$	1.06857	0.0527731
$411$	−26.1672	−1.29073
$412$	−25.6139	−1.26191
$413$	0.395911	0.0194815
$414$	−0.215014	−0.0105673
$415$	−2.48243	−0.121857
$416$	−4.79401	−0.235046
$417$	16.2723	0.796858
$418$	−0.412130	−0.0201579
$419$	23.7450	1.16002	0.580009	−	0.814610i	$-0.303049\pi$
$419$	23.7450	1.16002	0.580009	+	0.814610i	$0.303049\pi$
$420$	0.215950	0.0105373
$421$	12.5779	0.613010	0.306505	−	0.951869i	$-0.400840\pi$
$421$	12.5779	0.613010	0.306505	+	0.951869i	$0.400840\pi$
$422$	−10.2999	−0.501389
$423$	0.912046	0.0443452
$424$	−5.09990	−0.247673
$425$	−2.37951	−0.115423
$426$	2.33208	0.112990
$427$	0.0799030	0.00386678
$428$	−3.94958	−0.190910
$429$	−1.82938	−0.0883234
$430$	−2.32430	−0.112087
$431$	−39.2782	−1.89197	−0.945983	−	0.324216i	$-0.894900\pi$
$431$	−39.2782	−1.89197	−0.945983	+	0.324216i	$0.894900\pi$
$432$	−16.0815	−0.773723
$433$	11.3560	0.545737	0.272868	−	0.962051i	$-0.412028\pi$
$433$	11.3560	0.545737	0.272868	+	0.962051i	$0.412028\pi$
$434$	−0.0437893	−0.00210195
$435$	16.5406	0.793060
$436$	3.86069	0.184893
$437$	2.83321	0.135531
$438$	3.14635	0.150338
$439$	−2.62398	−0.125236	−0.0626178	−	0.998038i	$-0.519945\pi$
$439$	−2.62398	−0.125236	−0.0626178	+	0.998038i	$0.519945\pi$
$440$	1.57852	0.0752529
$441$	1.28808	0.0613373
$442$	1.06910	0.0508521
$443$	−19.9485	−0.947782	−0.473891	−	0.880583i	$-0.657151\pi$
$443$	−19.9485	−0.947782	−0.473891	+	0.880583i	$0.657151\pi$
$444$	4.25232	0.201806
$445$	−6.45611	−0.306049
$446$	−7.05066	−0.333859
$447$	14.5141	0.686493
$448$	−0.295836	−0.0139769
$449$	−2.91822	−0.137719	−0.0688597	−	0.997626i	$-0.521936\pi$
$449$	−2.91822	−0.137719	−0.0688597	+	0.997626i	$0.521936\pi$
$450$	0.0758904	0.00357751
$451$	−2.59281	−0.122091
$452$	−3.16985	−0.149097
$453$	14.5489	0.683568
$454$	1.00250	0.0470498
$455$	−0.0766584	−0.00359380
$456$	−2.64884	−0.124043
$457$	−2.23340	−0.104474	−0.0522369	−	0.998635i	$-0.516635\pi$
$457$	−2.23340	−0.104474	−0.0522369	+	0.998635i	$0.516635\pi$
$458$	−1.73690	−0.0811599
$459$	12.7141	0.593443
$460$	−5.18520	−0.241761
$461$	−27.1480	−1.26441	−0.632204	−	0.774802i	$-0.717849\pi$
$461$	−27.1480	−1.26441	−0.632204	+	0.774802i	$0.717849\pi$
$462$	0.0486296	0.00226245
$463$	−0.728298	−0.0338469	−0.0169234	−	0.999857i	$-0.505387\pi$
$463$	−0.728298	−0.0338469	−0.0169234	+	0.999857i	$0.505387\pi$
$464$	−29.6671	−1.37726
$465$	−2.53559	−0.117585
$466$	−2.13482	−0.0988937
$467$	−26.7879	−1.23960	−0.619798	−	0.784761i	$-0.712785\pi$
$467$	−26.7879	−1.23960	−0.619798	+	0.784761i	$0.712785\pi$
$468$	0.367399	0.0169830
$469$	−0.743446	−0.0343291
$470$	−2.04126	−0.0941562
$471$	2.45876	0.113294
$472$	8.88764	0.409087
$473$	5.63972	0.259315
$474$	−1.54161	−0.0708086
$475$	−1.00000	−0.0458831
$476$	0.306221	0.0140356
$477$	0.594929	0.0272399
$478$	−0.501909	−0.0229568
$479$	−31.4796	−1.43834	−0.719170	−	0.694835i	$-0.755478\pi$
$479$	−31.4796	−1.43834	−0.719170	+	0.694835i	$0.755478\pi$
$480$	7.37913	0.336810
$481$	−1.50950	−0.0688273
$482$	2.22423	0.101311
$483$	−0.334307	−0.0152115
$484$	−1.83015	−0.0831886
$485$	−9.43225	−0.428296
$486$	−0.787538	−0.0357235
$487$	−18.5946	−0.842603	−0.421302	−	0.906921i	$-0.638427\pi$
$487$	−18.5946	−0.842603	−0.421302	+	0.906921i	$0.638427\pi$
$488$	1.79371	0.0811974
$489$	14.5223	0.656719
$490$	−2.88287	−0.130235
$491$	−21.5567	−0.972839	−0.486420	−	0.873725i	$-0.661697\pi$
$491$	−21.5567	−0.972839	−0.486420	+	0.873725i	$0.661697\pi$
$492$	−7.96274	−0.358988
$493$	23.4549	1.05635
$494$	0.449296	0.0202148
$495$	−0.184142	−0.00827657
$496$	4.54781	0.204203
$497$	−0.237119	−0.0106362
$498$	−1.71678	−0.0769309
$499$	5.83099	0.261031	0.130515	−	0.991446i	$-0.458337\pi$
$499$	5.83099	0.261031	0.130515	+	0.991446i	$0.458337\pi$
$500$	1.83015	0.0818468
$501$	22.8968	1.02295
$502$	−0.516194	−0.0230389
$503$	−16.7197	−0.745494	−0.372747	−	0.927933i	$-0.621584\pi$
$503$	−16.7197	−0.745494	−0.372747	+	0.927933i	$0.621584\pi$
$504$	−0.0204392	−0.000910433 0
$505$	−2.07500	−0.0923361
$506$	−1.16765	−0.0519084
$507$	−19.8203	−0.880251
$508$	−4.49897	−0.199609
$509$	−17.7373	−0.786190	−0.393095	−	0.919498i	$-0.628596\pi$
$509$	−17.7373	−0.786190	−0.393095	+	0.919498i	$0.628596\pi$
$510$	−1.64561	−0.0728688
$511$	−0.319910	−0.0141520
$512$	−22.7370	−1.00484
$513$	5.34316	0.235906
$514$	−2.87797	−0.126942
$515$	−13.9955	−0.616717
$516$	17.3201	0.762473
$517$	4.95295	0.217830
$518$	0.0401263	0.00176305
$519$	3.44258	0.151112
$520$	−1.72087	−0.0754652
$521$	−29.8773	−1.30895	−0.654474	−	0.756085i	$-0.727110\pi$
$521$	−29.8773	−1.30895	−0.654474	+	0.756085i	$0.727110\pi$
$522$	−0.748052	−0.0327414
$523$	−27.0125	−1.18117	−0.590586	−	0.806975i	$-0.701103\pi$
$523$	−27.0125	−1.18117	−0.590586	+	0.806975i	$0.701103\pi$
$524$	34.6175	1.51227
$525$	0.117996	0.00514976
$526$	−5.40336	−0.235598
$527$	−3.59551	−0.156623
$528$	−5.05051	−0.219795
$529$	−14.9729	−0.650996
$530$	−1.33152	−0.0578373
$531$	−1.03679	−0.0449927
$532$	0.128691	0.00557945
$533$	2.82663	0.122435
$534$	−4.46488	−0.193214
$535$	−2.15806	−0.0933012
$536$	−16.6893	−0.720868
$537$	−37.6546	−1.62492
$538$	0.961017	0.0414324
$539$	6.99506	0.301298
$540$	−9.77877	−0.420811
$541$	30.2095	1.29881	0.649404	−	0.760444i	$-0.275019\pi$
$541$	30.2095	1.29881	0.649404	+	0.760444i	$0.275019\pi$
$542$	−2.06651	−0.0887643
$543$	−23.3071	−1.00020
$544$	10.4638	0.448630
$545$	2.10949	0.0903607
$546$	−0.0530150	−0.00226883
$547$	4.79311	0.204939	0.102469	−	0.994736i	$-0.467326\pi$
$547$	4.79311	0.204939	0.102469	+	0.994736i	$0.467326\pi$
$548$	28.5390	1.21912
$549$	−0.209245	−0.00893036
$550$	0.412130	0.0175733
$551$	9.85701	0.419923
$552$	−7.50472	−0.319422
$553$	0.156746	0.00666552
$554$	−11.4832	−0.487873
$555$	2.32348	0.0986264
$556$	−17.7472	−0.752650
$557$	−2.86504	−0.121396	−0.0606978	−	0.998156i	$-0.519333\pi$
$557$	−2.86504	−0.121396	−0.0606978	+	0.998156i	$0.519333\pi$
$558$	0.114673	0.00485448
$559$	−6.14832	−0.260046
$560$	−0.211637	−0.00894327
$561$	3.99294	0.168582
$562$	−9.14288	−0.385669
$563$	−41.4795	−1.74815	−0.874076	−	0.485790i	$-0.838532\pi$
$563$	−41.4795	−1.74815	−0.874076	+	0.485790i	$0.838532\pi$
$564$	15.2109	0.640496
$565$	−1.73202	−0.0728665
$566$	7.78030	0.327030
$567$	−0.591625	−0.0248459
$568$	−5.32298	−0.223347
$569$	9.90541	0.415256	0.207628	−	0.978208i	$-0.433426\pi$
$569$	9.90541	0.415256	0.207628	+	0.978208i	$0.433426\pi$
$570$	−0.691575	−0.0289669
$571$	−17.6387	−0.738156	−0.369078	−	0.929398i	$-0.620326\pi$
$571$	−17.6387	−0.738156	−0.369078	+	0.929398i	$0.620326\pi$
$572$	1.99519	0.0834233
$573$	−24.8364	−1.03756
$574$	−0.0751390	−0.00313624
$575$	−2.83321	−0.118153
$576$	0.774717	0.0322799
$577$	−27.8986	−1.16143	−0.580716	−	0.814106i	$-0.697227\pi$
$577$	−27.8986	−1.16143	−0.580716	+	0.814106i	$0.697227\pi$
$578$	4.67270	0.194359
$579$	15.6346	0.649754
$580$	−18.0398	−0.749062
$581$	0.174557	0.00724184
$582$	−6.52311	−0.270392
$583$	3.23082	0.133807
$584$	−7.18153	−0.297174
$585$	0.200748	0.00829992
$586$	−6.71488	−0.277389
$587$	−16.0386	−0.661984	−0.330992	−	0.943634i	$-0.607383\pi$
$587$	−16.0386	−0.661984	−0.330992	+	0.943634i	$0.607383\pi$
$588$	21.4824	0.885920
$589$	−1.51103	−0.0622609
$590$	2.32044	0.0955311
$591$	−20.7226	−0.852414
$592$	−4.16739	−0.171279
$593$	29.1140	1.19557	0.597784	−	0.801657i	$-0.296048\pi$
$593$	29.1140	1.19557	0.597784	+	0.801657i	$0.296048\pi$
$594$	−2.20207	−0.0903521
$595$	0.167320	0.00685946
$596$	−15.8296	−0.648407
$597$	−5.43730	−0.222534
$598$	1.27295	0.0520549
$599$	−26.0938	−1.06616	−0.533082	−	0.846064i	$-0.678966\pi$
$599$	−26.0938	−1.06616	−0.533082	+	0.846064i	$0.678966\pi$
$600$	2.64884	0.108138
$601$	15.2644	0.622647	0.311324	−	0.950304i	$-0.399228\pi$
$601$	15.2644	0.622647	0.311324	+	0.950304i	$0.399228\pi$
$602$	0.163438	0.00666123
$603$	1.94689	0.0792835
$604$	−15.8676	−0.645644
$605$	−1.00000	−0.0406558
$606$	−1.43502	−0.0582935
$607$	29.2038	1.18535	0.592674	−	0.805443i	$-0.298072\pi$
$607$	29.2038	1.18535	0.592674	+	0.805443i	$0.298072\pi$
$608$	4.39744	0.178340
$609$	−1.16309	−0.0471306
$610$	0.468313	0.0189614
$611$	−5.39961	−0.218445
$612$	−0.801912	−0.0324154
$613$	−22.1940	−0.896409	−0.448204	−	0.893931i	$-0.647936\pi$
$613$	−22.1940	−0.896409	−0.448204	+	0.893931i	$0.647936\pi$
$614$	−12.7424	−0.514240
$615$	−4.35087	−0.175444
$616$	−0.110997	−0.00447219
$617$	18.6664	0.751482	0.375741	−	0.926725i	$-0.377388\pi$
$617$	18.6664	0.751482	0.375741	+	0.926725i	$0.377388\pi$
$618$	−9.67896	−0.389345
$619$	−31.8733	−1.28110	−0.640549	−	0.767917i	$-0.721293\pi$
$619$	−31.8733	−1.28110	−0.640549	+	0.767917i	$0.721293\pi$
$620$	2.76541	0.111061
$621$	15.1383	0.607479
$622$	13.0172	0.521940
$623$	0.453975	0.0181881
$624$	5.50597	0.220415
$625$	1.00000	0.0400000
$626$	7.61982	0.304549
$627$	1.67805	0.0670149
$628$	−2.68162	−0.107008
$629$	3.29475	0.131370
$630$	−0.00533639	−0.000212607 0
$631$	−34.0268	−1.35458	−0.677292	−	0.735714i	$-0.736847\pi$
$631$	−34.0268	−1.35458	−0.677292	+	0.735714i	$0.736847\pi$
$632$	3.51872	0.139967
$633$	41.9375	1.66687
$634$	11.8513	0.470675
$635$	−2.45825	−0.0975527
$636$	9.92211	0.393437
$637$	−7.62588	−0.302148
$638$	−4.06237	−0.160831
$639$	0.620952	0.0245645
$640$	−10.5288	−0.416187
$641$	12.2428	0.483561	0.241781	−	0.970331i	$-0.422269\pi$
$641$	12.2428	0.483561	0.241781	+	0.970331i	$0.422269\pi$
$642$	−1.49246	−0.0589028
$643$	25.2441	0.995531	0.497765	−	0.867312i	$-0.334154\pi$
$643$	25.2441	0.995531	0.497765	+	0.867312i	$0.334154\pi$
$644$	0.364609	0.0143676
$645$	9.46374	0.372634
$646$	−0.980667	−0.0385838
$647$	20.4406	0.803604	0.401802	−	0.915727i	$-0.368384\pi$
$647$	20.4406	0.803604	0.401802	+	0.915727i	$0.368384\pi$
$648$	−13.2811	−0.521732
$649$	−5.63037	−0.221011
$650$	−0.449296	−0.0176228
$651$	0.178295	0.00698793
$652$	−15.8385	−0.620285
$653$	−24.2656	−0.949587	−0.474793	−	0.880097i	$-0.657477\pi$
$653$	−24.2656	−0.949587	−0.474793	+	0.880097i	$0.657477\pi$
$654$	1.45887	0.0570464
$655$	18.9151	0.739076
$656$	7.80369	0.304683
$657$	0.837761	0.0326842
$658$	0.143535	0.00559559
$659$	15.1423	0.589859	0.294929	−	0.955519i	$-0.404704\pi$
$659$	15.1423	0.589859	0.294929	+	0.955519i	$0.404704\pi$
$660$	−3.07109	−0.119542
$661$	25.5680	0.994480	0.497240	−	0.867613i	$-0.334347\pi$
$661$	25.5680	0.994480	0.497240	+	0.867613i	$0.334347\pi$
$662$	8.67106	0.337010
$663$	−4.35303	−0.169058
$664$	3.91855	0.152069
$665$	0.0703171	0.00272678
$666$	−0.105080	−0.00407178
$667$	27.9270	1.08134
$668$	−24.9721	−0.966201
$669$	28.7079	1.10991
$670$	−4.35735	−0.168339
$671$	−1.13632	−0.0438673
$672$	−0.518879	−0.0200162
$673$	−37.5675	−1.44812	−0.724061	−	0.689736i	$-0.757727\pi$
$673$	−37.5675	−1.44812	−0.724061	+	0.689736i	$0.757727\pi$
$674$	−1.53665	−0.0591896
$675$	−5.34316	−0.205658
$676$	21.6168	0.831416
$677$	23.6420	0.908637	0.454319	−	0.890839i	$-0.349883\pi$
$677$	23.6420	0.908637	0.454319	+	0.890839i	$0.349883\pi$
$678$	−1.19782	−0.0460020
$679$	0.663249	0.0254531
$680$	3.75610	0.144040
$681$	−4.08185	−0.156417
$682$	0.622740	0.0238459
$683$	14.6976	0.562389	0.281195	−	0.959651i	$-0.409269\pi$
$683$	14.6976	0.562389	0.281195	+	0.959651i	$0.409269\pi$
$684$	−0.337007	−0.0128858
$685$	15.5938	0.595808
$686$	0.405573	0.0154849
$687$	7.07205	0.269816
$688$	−16.9741	−0.647132
$689$	−3.52218	−0.134184
$690$	−1.95938	−0.0745923
$691$	−7.76870	−0.295535	−0.147768	−	0.989022i	$-0.547209\pi$
$691$	−7.76870	−0.295535	−0.147768	+	0.989022i	$0.547209\pi$
$692$	−3.75461	−0.142729
$693$	0.0129483	0.000491866 0
$694$	−7.44555	−0.282629
$695$	−9.69714	−0.367834
$696$	−26.1096	−0.989682
$697$	−6.16962	−0.233691
$698$	−8.85132	−0.335028
$699$	8.69227	0.328772
$700$	−0.128691	−0.00486405
$701$	21.9159	0.827751	0.413876	−	0.910333i	$-0.364175\pi$
$701$	21.9159	0.827751	0.413876	+	0.910333i	$0.364175\pi$
$702$	2.40066	0.0906070
$703$	1.38463	0.0522224
$704$	4.20717	0.158564
$705$	8.31131	0.313022
$706$	7.99445	0.300875
$707$	0.145908	0.00548742
$708$	−17.2913	−0.649848
$709$	−2.41310	−0.0906260	−0.0453130	−	0.998973i	$-0.514429\pi$
$709$	−2.41310	−0.0906260	−0.0453130	+	0.998973i	$0.514429\pi$
$710$	−1.38976	−0.0521566
$711$	−0.410477	−0.0153941
$712$	10.1911	0.381927
$713$	−4.28107	−0.160327
$714$	0.115715	0.00433051
$715$	1.09018	0.0407705
$716$	41.0676	1.53477
$717$	2.04360	0.0763197
$718$	4.40604	0.164432
$719$	25.0038	0.932484	0.466242	−	0.884657i	$-0.345608\pi$
$719$	25.0038	0.932484	0.466242	+	0.884657i	$0.345608\pi$
$720$	0.554220	0.0206546
$721$	0.984125	0.0366507
$722$	−0.412130	−0.0153379
$723$	−9.05632	−0.336808
$724$	25.4197	0.944714
$725$	−9.85701	−0.366080
$726$	−0.691575	−0.0256668
$727$	−17.0950	−0.634017	−0.317009	−	0.948423i	$-0.602678\pi$
$727$	−17.0950	−0.634017	−0.317009	+	0.948423i	$0.602678\pi$
$728$	0.121007	0.00448481
$729$	28.4476	1.05361
$730$	−1.87500	−0.0693969
$731$	13.4198	0.496348
$732$	−3.48975	−0.128985
$733$	4.33014	0.159937	0.0799686	−	0.996797i	$-0.474518\pi$
$733$	4.33014	0.159937	0.0799686	+	0.996797i	$0.474518\pi$
$734$	−10.9333	−0.403555
$735$	11.7381	0.432965
$736$	12.4589	0.459241
$737$	10.5728	0.389453
$738$	0.196769	0.00724318
$739$	−45.3068	−1.66664	−0.833319	−	0.552793i	$-0.813562\pi$
$739$	−45.3068	−1.66664	−0.833319	+	0.552793i	$0.813562\pi$
$740$	−2.53408	−0.0931547
$741$	−1.82938	−0.0672040
$742$	0.0936283	0.00343720
$743$	−29.9590	−1.09909	−0.549544	−	0.835465i	$-0.685198\pi$
$743$	−29.9590	−1.09909	−0.549544	+	0.835465i	$0.685198\pi$
$744$	4.00247	0.146738
$745$	−8.64937	−0.316888
$746$	5.56527	0.203759
$747$	−0.457119	−0.0167251
$748$	−4.35486	−0.159229
$749$	0.151749	0.00554478
$750$	0.691575	0.0252527
$751$	19.1315	0.698117	0.349058	−	0.937101i	$-0.386502\pi$
$751$	19.1315	0.698117	0.349058	+	0.937101i	$0.386502\pi$
$752$	−14.9071	−0.543606
$753$	2.10177	0.0765926
$754$	4.42872	0.161284
$755$	−8.67013	−0.315538
$756$	0.687615	0.0250083
$757$	1.82770	0.0664290	0.0332145	−	0.999448i	$-0.489426\pi$
$757$	1.82770	0.0664290	0.0332145	+	0.999448i	$0.489426\pi$
$758$	10.3117	0.374539
$759$	4.75428	0.172569
$760$	1.57852	0.0572589
$761$	14.9911	0.543426	0.271713	−	0.962378i	$-0.412410\pi$
$761$	14.9911	0.543426	0.271713	+	0.962378i	$0.412410\pi$
$762$	−1.70006	−0.0615868
$763$	−0.148333	−0.00537003
$764$	27.0876	0.979994
$765$	−0.438168	−0.0158420
$766$	−9.20066	−0.332433
$767$	6.13812	0.221635
$768$	6.83826	0.246755
$769$	22.5313	0.812501	0.406251	−	0.913762i	$-0.366836\pi$
$769$	22.5313	0.812501	0.406251	+	0.913762i	$0.366836\pi$
$770$	−0.0289798	−0.00104436
$771$	11.7181	0.422018
$772$	−17.0518	−0.613706
$773$	−32.9273	−1.18431	−0.592157	−	0.805822i	$-0.701724\pi$
$773$	−32.9273	−1.18431	−0.592157	+	0.805822i	$0.701724\pi$
$774$	−0.428000	−0.0153842
$775$	1.51103	0.0542778
$776$	14.8890	0.534483
$777$	−0.163381	−0.00586125
$778$	−4.18982	−0.150212
$779$	−2.59281	−0.0928970
$780$	3.34804	0.119879
$781$	3.37214	0.120664
$782$	−2.77844	−0.0993568
$783$	52.6675	1.88218
$784$	−21.0533	−0.751905
$785$	−1.46525	−0.0522969
$786$	13.0812	0.466593
$787$	−17.7799	−0.633786	−0.316893	−	0.948461i	$-0.602640\pi$
$787$	−17.7799	−0.633786	−0.316893	+	0.948461i	$0.602640\pi$
$788$	22.6009	0.805123
$789$	22.0007	0.783244
$790$	0.918691	0.0326856
$791$	0.121790	0.00433037
$792$	0.290671	0.0103286
$793$	1.23880	0.0439911
$794$	−7.51327	−0.266636
$795$	5.42148	0.192280
$796$	5.93014	0.210188
$797$	10.5569	0.373943	0.186972	−	0.982365i	$-0.440133\pi$
$797$	10.5569	0.373943	0.186972	+	0.982365i	$0.440133\pi$
$798$	0.0486296	0.00172147
$799$	11.7856	0.416944
$800$	−4.39744	−0.155473
$801$	−1.18884	−0.0420056
$802$	1.92326	0.0679126
$803$	4.54954	0.160550
$804$	32.4699	1.14512
$805$	0.199223	0.00702170
$806$	−0.678900	−0.0239132
$807$	−3.91293	−0.137742
$808$	3.27542	0.115229
$809$	−0.800366	−0.0281394	−0.0140697	−	0.999901i	$-0.504479\pi$
$809$	−0.800366	−0.0281394	−0.0140697	+	0.999901i	$0.504479\pi$
$810$	−3.46752	−0.121836
$811$	3.75214	0.131756	0.0658778	−	0.997828i	$-0.479015\pi$
$811$	3.75214	0.131756	0.0658778	+	0.997828i	$0.479015\pi$
$812$	1.26851	0.0445159
$813$	8.41413	0.295097
$814$	−0.570648	−0.0200012
$815$	−8.65423	−0.303145
$816$	−12.0177	−0.420705
$817$	5.63972	0.197309
$818$	2.33751	0.0817293
$819$	−0.0141160	−0.000493254 0
$820$	4.74523	0.165711
$821$	−16.4431	−0.573868	−0.286934	−	0.957950i	$-0.592636\pi$
$821$	−16.4431	−0.573868	−0.286934	+	0.957950i	$0.592636\pi$
$822$	10.7843	0.376145
$823$	42.6902	1.48809	0.744044	−	0.668131i	$-0.232905\pi$
$823$	42.6902	1.48809	0.744044	+	0.668131i	$0.232905\pi$
$824$	22.0922	0.769618
$825$	−1.67805	−0.0584223
$826$	−0.163167	−0.00567730
$827$	−11.2062	−0.389679	−0.194839	−	0.980835i	$-0.562419\pi$
$827$	−11.2062	−0.389679	−0.194839	+	0.980835i	$0.562419\pi$
$828$	−0.954814	−0.0331821
$829$	21.4252	0.744128	0.372064	−	0.928207i	$-0.378650\pi$
$829$	21.4252	0.744128	0.372064	+	0.928207i	$0.378650\pi$
$830$	1.02308	0.0355117
$831$	46.7555	1.62193
$832$	−4.58658	−0.159011
$833$	16.6448	0.576708
$834$	−6.70630	−0.232220
$835$	−13.6449	−0.472200
$836$	−1.83015	−0.0632970
$837$	−8.07366	−0.279067
$838$	−9.78601	−0.338052
$839$	−10.8785	−0.375569	−0.187784	−	0.982210i	$-0.560131\pi$
$839$	−10.8785	−0.375569	−0.187784	+	0.982210i	$0.560131\pi$
$840$	−0.186258	−0.00642653
$841$	68.1607	2.35037
$842$	−5.18373	−0.178643
$843$	37.2267	1.28216
$844$	−45.7387	−1.57439
$845$	11.8115	0.406328
$846$	−0.375881	−0.0129231
$847$	0.0703171	0.00241612
$848$	−9.72393	−0.333921
$849$	−31.6787	−1.08721
$850$	0.980667	0.0336366
$851$	3.92296	0.134477
$852$	10.3561	0.354795
$853$	10.5424	0.360966	0.180483	−	0.983578i	$-0.442234\pi$
$853$	10.5424	0.360966	0.180483	+	0.983578i	$0.442234\pi$
$854$	−0.0329304	−0.00112686
$855$	−0.184142	−0.00629752
$856$	3.40654	0.116433
$857$	−14.9488	−0.510641	−0.255320	−	0.966857i	$-0.582181\pi$
$857$	−14.9488	−0.510641	−0.255320	+	0.966857i	$0.582181\pi$
$858$	0.753942	0.0257392
$859$	12.4442	0.424590	0.212295	−	0.977206i	$-0.431906\pi$
$859$	12.4442	0.424590	0.212295	+	0.977206i	$0.431906\pi$
$860$	−10.3215	−0.351961
$861$	0.305940	0.0104264
$862$	16.1877	0.551356
$863$	−31.1531	−1.06046	−0.530232	−	0.847853i	$-0.677895\pi$
$863$	−31.1531	−1.06046	−0.530232	+	0.847853i	$0.677895\pi$
$864$	23.4962	0.799357
$865$	−2.05153	−0.0697542
$866$	−4.68016	−0.159038
$867$	−19.0256	−0.646144
$868$	−0.194456	−0.00660025
$869$	−2.22913	−0.0756181
$870$	−6.81686	−0.231113
$871$	−11.5262	−0.390551
$872$	−3.32987	−0.112764
$873$	−1.73687	−0.0587843
$874$	−1.16765	−0.0394964
$875$	−0.0703171	−0.00237715
$876$	13.9720	0.472071
$877$	41.3652	1.39680	0.698402	−	0.715706i	$-0.253895\pi$
$877$	41.3652	1.39680	0.698402	+	0.715706i	$0.253895\pi$
$878$	1.08142	0.0364961
$879$	27.3407	0.922179
$880$	3.00974	0.101458
$881$	56.9643	1.91917	0.959587	−	0.281411i	$-0.0908025\pi$
$881$	56.9643	1.91917	0.959587	+	0.281411i	$0.0908025\pi$
$882$	−0.530857	−0.0178749
$883$	−41.5298	−1.39759	−0.698795	−	0.715322i	$-0.746280\pi$
$883$	−41.5298	−1.39759	−0.698795	+	0.715322i	$0.746280\pi$
$884$	4.74759	0.159679
$885$	−9.44805	−0.317593
$886$	8.22137	0.276202
$887$	−37.4902	−1.25880	−0.629399	−	0.777082i	$-0.716699\pi$
$887$	−37.4902	−1.25880	−0.629399	+	0.777082i	$0.716699\pi$
$888$	−3.66766	−0.123079
$889$	0.172857	0.00579744
$890$	2.66075	0.0891887
$891$	8.41367	0.281868
$892$	−31.3100	−1.04834
$893$	4.95295	0.165744
$894$	−5.98169	−0.200057
$895$	22.4395	0.750069
$896$	0.740354	0.0247335
$897$	−5.18303	−0.173056
$898$	1.20268	0.0401341
$899$	−14.8942	−0.496750
$900$	0.337007	0.0112336
$901$	7.68776	0.256116
$902$	1.06857	0.0355796
$903$	−0.665463	−0.0221452
$904$	2.73402	0.0909322
$905$	13.8894	0.461699
$906$	−5.99604	−0.199205
$907$	−25.3818	−0.842789	−0.421394	−	0.906877i	$-0.638459\pi$
$907$	−25.3818	−0.842789	−0.421394	+	0.906877i	$0.638459\pi$
$908$	4.45183	0.147739
$909$	−0.382094	−0.0126733
$910$	0.0315932	0.00104730
$911$	−13.4110	−0.444327	−0.222163	−	0.975009i	$-0.571312\pi$
$911$	−13.4110	−0.444327	−0.222163	+	0.975009i	$0.571312\pi$
$912$	−5.05051	−0.167239
$913$	−2.48243	−0.0821563
$914$	0.920449	0.0304457
$915$	−1.90681	−0.0630372
$916$	−7.71306	−0.254847
$917$	−1.33006	−0.0439224
$918$	−5.23985	−0.172941
$919$	34.5515	1.13975	0.569874	−	0.821732i	$-0.306992\pi$
$919$	34.5515	1.13975	0.569874	+	0.821732i	$0.306992\pi$
$920$	4.47228	0.147447
$921$	51.8825	1.70959
$922$	11.1885	0.368473
$923$	−3.67624	−0.121005
$924$	0.215950	0.00710423
$925$	−1.38463	−0.0455264
$926$	0.300153	0.00986365
$927$	−2.57716	−0.0846452
$928$	43.3456	1.42289
$929$	−27.2218	−0.893117	−0.446558	−	0.894755i	$-0.647350\pi$
$929$	−27.2218	−0.893117	−0.446558	+	0.894755i	$0.647350\pi$
$930$	1.04499	0.0342666
$931$	6.99506	0.229254
$932$	−9.48013	−0.310532
$933$	−53.0014	−1.73519
$934$	11.0401	0.361243
$935$	−2.37951	−0.0778183
$936$	−0.316885	−0.0103577
$937$	−1.39912	−0.0457071	−0.0228535	−	0.999739i	$-0.507275\pi$
$937$	−1.39912	−0.0457071	−0.0228535	+	0.999739i	$0.507275\pi$
$938$	0.306396	0.0100042
$939$	−31.0253	−1.01247
$940$	−9.06464	−0.295656
$941$	15.1184	0.492845	0.246423	−	0.969162i	$-0.420745\pi$
$941$	15.1184	0.492845	0.246423	+	0.969162i	$0.420745\pi$
$942$	−1.01333	−0.0330160
$943$	−7.34598	−0.239218
$944$	16.9460	0.551544
$945$	0.375715	0.0122220
$946$	−2.32430	−0.0755694
$947$	−29.9768	−0.974116	−0.487058	−	0.873370i	$-0.661930\pi$
$947$	−29.9768	−0.974116	−0.487058	+	0.873370i	$0.661930\pi$
$948$	−6.84585	−0.222343
$949$	−4.95982	−0.161003
$950$	0.412130	0.0133713
$951$	−48.2544	−1.56476
$952$	−0.264118	−0.00856012
$953$	−27.4874	−0.890405	−0.445203	−	0.895430i	$-0.646868\pi$
$953$	−27.4874	−0.890405	−0.445203	+	0.895430i	$0.646868\pi$
$954$	−0.245188	−0.00793825
$955$	14.8007	0.478941
$956$	−2.22883	−0.0720856
$957$	16.5406	0.534681
$958$	12.9737	0.419160
$959$	−1.09651	−0.0354082
$960$	7.05985	0.227856
$961$	−28.7168	−0.926348
$962$	0.622110	0.0200576
$963$	−0.397390	−0.0128057
$964$	9.87717	0.318122
$965$	−9.31714	−0.299929
$966$	0.137778	0.00443293
$967$	39.1290	1.25830	0.629152	−	0.777282i	$-0.283402\pi$
$967$	39.1290	1.25830	0.629152	+	0.777282i	$0.283402\pi$
$968$	1.57852	0.0507355
$969$	3.99294	0.128272
$970$	3.88731	0.124814
$971$	−25.6444	−0.822967	−0.411484	−	0.911417i	$-0.634989\pi$
$971$	−25.6444	−0.822967	−0.411484	+	0.911417i	$0.634989\pi$
$972$	−3.49723	−0.112174
$973$	0.681875	0.0218599
$974$	7.66340	0.245551
$975$	1.82938	0.0585871
$976$	3.42004	0.109473
$977$	47.5333	1.52072	0.760362	−	0.649500i	$-0.225022\pi$
$977$	47.5333	1.52072	0.760362	+	0.649500i	$0.225022\pi$
$978$	−5.98505	−0.191381
$979$	−6.45611	−0.206338
$980$	−12.8020	−0.408945
$981$	0.388446	0.0124021
$982$	8.88415	0.283504
$983$	6.49735	0.207233	0.103617	−	0.994617i	$-0.466959\pi$
$983$	6.49735	0.207233	0.103617	+	0.994617i	$0.466959\pi$
$984$	6.86792	0.218941
$985$	12.3492	0.393478
$986$	−9.66644	−0.307842
$987$	−0.584427	−0.0186025
$988$	1.99519	0.0634756
$989$	15.9785	0.508088
$990$	0.0758904	0.00241195
$991$	−1.15760	−0.0367724	−0.0183862	−	0.999831i	$-0.505853\pi$
$991$	−1.15760	−0.0367724	−0.0183862	+	0.999831i	$0.505853\pi$
$992$	−6.64466	−0.210968
$993$	−35.3056	−1.12039
$994$	0.0977237	0.00309961
$995$	3.24025	0.102723
$996$	−7.62374	−0.241567
$997$	45.5209	1.44166	0.720830	−	0.693112i	$-0.243761\pi$
$997$	45.5209	1.44166	0.720830	+	0.693112i	$0.243761\pi$
$998$	−2.40312	−0.0760696
$999$	7.39830	0.234072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.f.1.3	✓	6
3.2	odd	2		9405.2.a.z.1.4		6
5.4	even	2		5225.2.a.l.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.3	✓	6	1.1	even	1	trivial
5225.2.a.l.1.4		6	5.4	even	2
9405.2.a.z.1.4		6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-0.412130 q^{2} +1.67805 q^{3} -1.83015 q^{4} -1.00000 q^{5} -0.691575 q^{6} +0.0703171 q^{7} +1.57852 q^{8} -0.184142 q^{9} +0.412130 q^{10} -1.00000 q^{11} -3.07109 q^{12} +1.09018 q^{13} -0.0289798 q^{14} -1.67805 q^{15} +3.00974 q^{16} -2.37951 q^{17} +0.0758904 q^{18} -1.00000 q^{19} +1.83015 q^{20} +0.117996 q^{21} +0.412130 q^{22} -2.83321 q^{23} +2.64884 q^{24} +1.00000 q^{25} -0.449296 q^{26} -5.34316 q^{27} -0.128691 q^{28} -9.85701 q^{29} +0.691575 q^{30} +1.51103 q^{31} -4.39744 q^{32} -1.67805 q^{33} +0.980667 q^{34} -0.0703171 q^{35} +0.337007 q^{36} -1.38463 q^{37} +0.412130 q^{38} +1.82938 q^{39} -1.57852 q^{40} +2.59281 q^{41} -0.0486296 q^{42} -5.63972 q^{43} +1.83015 q^{44} +0.184142 q^{45} +1.16765 q^{46} -4.95295 q^{47} +5.05051 q^{48} -6.99506 q^{49} -0.412130 q^{50} -3.99294 q^{51} -1.99519 q^{52} -3.23082 q^{53} +2.20207 q^{54} +1.00000 q^{55} +0.110997 q^{56} -1.67805 q^{57} +4.06237 q^{58} +5.63037 q^{59} +3.07109 q^{60} +1.13632 q^{61} -0.622740 q^{62} -0.0129483 q^{63} -4.20717 q^{64} -1.09018 q^{65} +0.691575 q^{66} -10.5728 q^{67} +4.35486 q^{68} -4.75428 q^{69} +0.0289798 q^{70} -3.37214 q^{71} -0.290671 q^{72} -4.54954 q^{73} +0.570648 q^{74} +1.67805 q^{75} +1.83015 q^{76} -0.0703171 q^{77} -0.753942 q^{78} +2.22913 q^{79} -3.00974 q^{80} -8.41367 q^{81} -1.06857 q^{82} +2.48243 q^{83} -0.215950 q^{84} +2.37951 q^{85} +2.32430 q^{86} -16.5406 q^{87} -1.57852 q^{88} +6.45611 q^{89} -0.0758904 q^{90} +0.0766584 q^{91} +5.18520 q^{92} +2.53559 q^{93} +2.04126 q^{94} +1.00000 q^{95} -7.37913 q^{96} +9.43225 q^{97} +2.88287 q^{98} +0.184142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21}+ \cdots - q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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