LMFDB - Embedded newform 8954.2.a.bt.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8954,2,Mod(1,8954)] 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("8954.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8954, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8954 = 2 \cdot 11^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8954.a (trivial)

Newform invariants

comment:select newform

sage:traces = [22,22,5,22,8] 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:	$71.4980499699$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 814)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8954.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.123112 q^{3} +1.00000 q^{4} -3.89599 q^{5} +0.123112 q^{6} +1.02869 q^{7} +1.00000 q^{8} -2.98484 q^{9} -3.89599 q^{10} +0.123112 q^{12} -0.0128920 q^{13} +1.02869 q^{14} -0.479643 q^{15} +1.00000 q^{16} -6.03448 q^{17} -2.98484 q^{18} +4.80831 q^{19} -3.89599 q^{20} +0.126644 q^{21} +1.10089 q^{23} +0.123112 q^{24} +10.1787 q^{25} -0.0128920 q^{26} -0.736806 q^{27} +1.02869 q^{28} +2.66171 q^{29} -0.479643 q^{30} -9.54125 q^{31} +1.00000 q^{32} -6.03448 q^{34} -4.00776 q^{35} -2.98484 q^{36} -1.00000 q^{37} +4.80831 q^{38} -0.00158717 q^{39} -3.89599 q^{40} +1.78744 q^{41} +0.126644 q^{42} +0.161529 q^{43} +11.6289 q^{45} +1.10089 q^{46} -5.43218 q^{47} +0.123112 q^{48} -5.94180 q^{49} +10.1787 q^{50} -0.742917 q^{51} -0.0128920 q^{52} -8.50735 q^{53} -0.736806 q^{54} +1.02869 q^{56} +0.591961 q^{57} +2.66171 q^{58} +8.09590 q^{59} -0.479643 q^{60} +5.07782 q^{61} -9.54125 q^{62} -3.07048 q^{63} +1.00000 q^{64} +0.0502272 q^{65} +5.89099 q^{67} -6.03448 q^{68} +0.135533 q^{69} -4.00776 q^{70} -11.7721 q^{71} -2.98484 q^{72} +0.543581 q^{73} -1.00000 q^{74} +1.25312 q^{75} +4.80831 q^{76} -0.00158717 q^{78} +14.2093 q^{79} -3.89599 q^{80} +8.86382 q^{81} +1.78744 q^{82} +1.59761 q^{83} +0.126644 q^{84} +23.5103 q^{85} +0.161529 q^{86} +0.327688 q^{87} -18.5606 q^{89} +11.6289 q^{90} -0.0132619 q^{91} +1.10089 q^{92} -1.17464 q^{93} -5.43218 q^{94} -18.7331 q^{95} +0.123112 q^{96} +12.2796 q^{97} -5.94180 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 22 q^{2} + 5 q^{3} + 22 q^{4} + 8 q^{5} + 5 q^{6} + 7 q^{7} + 22 q^{8} + 35 q^{9} + 8 q^{10} + 5 q^{12} + 9 q^{13} + 7 q^{14} + 16 q^{15} + 22 q^{16} + 19 q^{17} + 35 q^{18} + 8 q^{20} + 15 q^{21}+ \cdots + 57 q^{98}+O(q^{100}) $ 22 * q + 22 * q^2 + 5 * q^3 + 22 * q^4 + 8 * q^5 + 5 * q^6 + 7 * q^7 + 22 * q^8 + 35 * q^9 + 8 * q^10 + 5 * q^12 + 9 * q^13 + 7 * q^14 + 16 * q^15 + 22 * q^16 + 19 * q^17 + 35 * q^18 + 8 * q^20 + 15 * q^21 + 18 * q^23 + 5 * q^24 + 26 * q^25 + 9 * q^26 + 17 * q^27 + 7 * q^28 + 6 * q^29 + 16 * q^30 + 32 * q^31 + 22 * q^32 + 19 * q^34 + 14 * q^35 + 35 * q^36 - 22 * q^37 - 2 * q^39 + 8 * q^40 + 10 * q^41 + 15 * q^42 - 2 * q^43 + 83 * q^45 + 18 * q^46 - 16 * q^47 + 5 * q^48 + 57 * q^49 + 26 * q^50 - 30 * q^51 + 9 * q^52 + 6 * q^53 + 17 * q^54 + 7 * q^56 + 66 * q^57 + 6 * q^58 + 51 * q^59 + 16 * q^60 + 32 * q^62 - 3 * q^63 + 22 * q^64 + 34 * q^65 + 22 * q^67 + 19 * q^68 + 27 * q^69 + 14 * q^70 + 24 * q^71 + 35 * q^72 - 8 * q^73 - 22 * q^74 + 16 * q^75 - 2 * q^78 + 39 * q^79 + 8 * q^80 + 86 * q^81 + 10 * q^82 - 18 * q^83 + 15 * q^84 + 34 * q^85 - 2 * q^86 - 10 * q^87 + 45 * q^89 + 83 * q^90 + 8 * q^91 + 18 * q^92 + 33 * q^93 - 16 * q^94 - 39 * q^95 + 5 * q^96 + 47 * q^97 + 57 * q^98

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.123112	0.0710788	0.0355394	−	0.999368i	$-0.488685\pi$
$3$	0.123112	0.0710788	0.0355394	+	0.999368i	$0.488685\pi$
$4$	1.00000	0.500000
$5$	−3.89599	−1.74234	−0.871169	−	0.490983i	$-0.836638\pi$
$5$	−3.89599	−1.74234	−0.871169	+	0.490983i	$0.836638\pi$
$6$	0.123112	0.0502603
$7$	1.02869	0.388808	0.194404	−	0.980922i	$-0.437723\pi$
$7$	1.02869	0.388808	0.194404	+	0.980922i	$0.437723\pi$
$8$	1.00000	0.353553
$9$	−2.98484	−0.994948
$10$	−3.89599	−1.23202
$11$	0	0
$11$	0	0
$12$	0.123112	0.0355394
$13$	−0.0128920	−0.00357561	−0.00178780	−	0.999998i	$-0.500569\pi$
$13$	−0.0128920	−0.00357561	−0.00178780	+	0.999998i	$0.500569\pi$
$14$	1.02869	0.274929
$15$	−0.479643	−0.123843
$16$	1.00000	0.250000
$17$	−6.03448	−1.46358	−0.731788	−	0.681532i	$-0.761314\pi$
$17$	−6.03448	−1.46358	−0.731788	+	0.681532i	$0.761314\pi$
$18$	−2.98484	−0.703534
$19$	4.80831	1.10310	0.551551	−	0.834141i	$-0.314036\pi$
$19$	4.80831	1.10310	0.551551	+	0.834141i	$0.314036\pi$
$20$	−3.89599	−0.871169
$21$	0.126644	0.0276360
$22$	0	0
$23$	1.10089	0.229552	0.114776	−	0.993391i	$-0.463385\pi$
$23$	1.10089	0.229552	0.114776	+	0.993391i	$0.463385\pi$
$24$	0.123112	0.0251301
$25$	10.1787	2.03574
$26$	−0.0128920	−0.00252834
$27$	−0.736806	−0.141798
$28$	1.02869	0.194404
$29$	2.66171	0.494267	0.247133	−	0.968981i	$-0.420511\pi$
$29$	2.66171	0.494267	0.247133	+	0.968981i	$0.420511\pi$
$30$	−0.479643	−0.0875704
$31$	−9.54125	−1.71366	−0.856830	−	0.515599i	$-0.827569\pi$
$31$	−9.54125	−1.71366	−0.856830	+	0.515599i	$0.827569\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.03448	−1.03491
$35$	−4.00776	−0.677435
$36$	−2.98484	−0.497474
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	4.80831	0.780011
$39$	−0.00158717	−0.000254150 0
$40$	−3.89599	−0.616010
$41$	1.78744	0.279152	0.139576	−	0.990211i	$-0.455426\pi$
$41$	1.78744	0.279152	0.139576	+	0.990211i	$0.455426\pi$
$42$	0.126644	0.0195416
$43$	0.161529	0.0246329	0.0123165	−	0.999924i	$-0.496079\pi$
$43$	0.161529	0.0246329	0.0123165	+	0.999924i	$0.496079\pi$
$44$	0	0
$45$	11.6289	1.73354
$46$	1.10089	0.162318
$47$	−5.43218	−0.792365	−0.396182	−	0.918172i	$-0.629665\pi$
$47$	−5.43218	−0.792365	−0.396182	+	0.918172i	$0.629665\pi$
$48$	0.123112	0.0177697
$49$	−5.94180	−0.848828
$50$	10.1787	1.43949
$51$	−0.742917	−0.104029
$52$	−0.0128920	−0.00178780
$53$	−8.50735	−1.16857	−0.584287	−	0.811547i	$-0.698626\pi$
$53$	−8.50735	−1.16857	−0.584287	+	0.811547i	$0.698626\pi$
$54$	−0.736806	−0.100267
$55$	0	0
$56$	1.02869	0.137464
$57$	0.591961	0.0784071
$58$	2.66171	0.349499
$59$	8.09590	1.05400	0.526998	−	0.849866i	$-0.323317\pi$
$59$	8.09590	1.05400	0.526998	+	0.849866i	$0.323317\pi$
$60$	−0.479643	−0.0619216
$61$	5.07782	0.650149	0.325074	−	0.945688i	$-0.394611\pi$
$61$	5.07782	0.650149	0.325074	+	0.945688i	$0.394611\pi$
$62$	−9.54125	−1.21174
$63$	−3.07048	−0.386844
$64$	1.00000	0.125000
$65$	0.0502272	0.00622992
$66$	0	0
$67$	5.89099	0.719699	0.359849	−	0.933010i	$-0.382828\pi$
$67$	5.89099	0.719699	0.359849	+	0.933010i	$0.382828\pi$
$68$	−6.03448	−0.731788
$69$	0.135533	0.0163163
$70$	−4.00776	−0.479019
$71$	−11.7721	−1.39710	−0.698548	−	0.715563i	$-0.746170\pi$
$71$	−11.7721	−1.39710	−0.698548	+	0.715563i	$0.746170\pi$
$72$	−2.98484	−0.351767
$73$	0.543581	0.0636213	0.0318107	−	0.999494i	$-0.489873\pi$
$73$	0.543581	0.0636213	0.0318107	+	0.999494i	$0.489873\pi$
$74$	−1.00000	−0.116248
$75$	1.25312	0.144698
$76$	4.80831	0.551551
$77$	0	0
$78$	−0.00158717	−0.000179711 0
$79$	14.2093	1.59868	0.799338	−	0.600882i	$-0.205184\pi$
$79$	14.2093	1.59868	0.799338	+	0.600882i	$0.205184\pi$
$80$	−3.89599	−0.435585
$81$	8.86382	0.984869
$82$	1.78744	0.197390
$83$	1.59761	0.175360	0.0876802	−	0.996149i	$-0.472055\pi$
$83$	1.59761	0.175360	0.0876802	+	0.996149i	$0.472055\pi$
$84$	0.126644	0.0138180
$85$	23.5103	2.55005
$86$	0.161529	0.0174181
$87$	0.327688	0.0351319
$88$	0	0
$89$	−18.5606	−1.96742	−0.983710	−	0.179761i	$-0.942467\pi$
$89$	−18.5606	−1.96742	−0.983710	+	0.179761i	$0.942467\pi$
$90$	11.6289	1.22579
$91$	−0.0132619	−0.00139023
$92$	1.10089	0.114776
$93$	−1.17464	−0.121805
$94$	−5.43218	−0.560286
$95$	−18.7331	−1.92198
$96$	0.123112	0.0125651
$97$	12.2796	1.24680	0.623401	−	0.781903i	$-0.285751\pi$
$97$	12.2796	1.24680	0.623401	+	0.781903i	$0.285751\pi$
$98$	−5.94180	−0.600212
$99$	0	0
$100$	10.1787	1.01787
$101$	11.3530	1.12967	0.564833	−	0.825205i	$-0.308941\pi$
$101$	11.3530	1.12967	0.564833	+	0.825205i	$0.308941\pi$
$102$	−0.742917	−0.0735598
$103$	14.0153	1.38097	0.690486	−	0.723346i	$-0.257397\pi$
$103$	14.0153	1.38097	0.690486	+	0.723346i	$0.257397\pi$
$104$	−0.0128920	−0.00126417
$105$	−0.493404	−0.0481513
$106$	−8.50735	−0.826307
$107$	−11.7759	−1.13842	−0.569210	−	0.822192i	$-0.692751\pi$
$107$	−11.7759	−1.13842	−0.569210	+	0.822192i	$0.692751\pi$
$108$	−0.736806	−0.0708992
$109$	3.66858	0.351386	0.175693	−	0.984445i	$-0.443783\pi$
$109$	3.66858	0.351386	0.175693	+	0.984445i	$0.443783\pi$
$110$	0	0
$111$	−0.123112	−0.0116853
$112$	1.02869	0.0972020
$113$	−13.4128	−1.26177	−0.630887	−	0.775875i	$-0.717309\pi$
$113$	−13.4128	−1.26177	−0.630887	+	0.775875i	$0.717309\pi$
$114$	0.591961	0.0554422
$115$	−4.28907	−0.399958
$116$	2.66171	0.247133
$117$	0.0384807	0.00355754
$118$	8.09590	0.745288
$119$	−6.20761	−0.569051
$120$	−0.479643	−0.0437852
$121$	0	0
$122$	5.07782	0.459725
$123$	0.220056	0.0198418
$124$	−9.54125	−0.856830
$125$	−20.1762	−1.80462
$126$	−3.07048	−0.273540
$127$	15.9977	1.41956	0.709782	−	0.704422i	$-0.248793\pi$
$127$	15.9977	1.41956	0.709782	+	0.704422i	$0.248793\pi$
$128$	1.00000	0.0883883
$129$	0.0198861	0.00175088
$130$	0.0502272	0.00440522
$131$	13.2172	1.15479	0.577395	−	0.816465i	$-0.304069\pi$
$131$	13.2172	1.15479	0.577395	+	0.816465i	$0.304069\pi$
$132$	0	0
$133$	4.94626	0.428895
$134$	5.89099	0.508904
$135$	2.87059	0.247061
$136$	−6.03448	−0.517453
$137$	4.05441	0.346391	0.173196	−	0.984887i	$-0.444591\pi$
$137$	4.05441	0.346391	0.173196	+	0.984887i	$0.444591\pi$
$138$	0.135533	0.0115374
$139$	13.0764	1.10913	0.554565	−	0.832140i	$-0.312885\pi$
$139$	13.0764	1.10913	0.554565	+	0.832140i	$0.312885\pi$
$140$	−4.00776	−0.338718
$141$	−0.668766	−0.0563203
$142$	−11.7721	−0.987896
$143$	0	0
$144$	−2.98484	−0.248737
$145$	−10.3700	−0.861180
$146$	0.543581	0.0449871
$147$	−0.731507	−0.0603337
$148$	−1.00000	−0.0821995
$149$	6.83732	0.560135	0.280068	−	0.959980i	$-0.409643\pi$
$149$	6.83732	0.560135	0.280068	+	0.959980i	$0.409643\pi$
$150$	1.25312	0.102317
$151$	−14.8152	−1.20565	−0.602823	−	0.797875i	$-0.705958\pi$
$151$	−14.8152	−1.20565	−0.602823	+	0.797875i	$0.705958\pi$
$152$	4.80831	0.390005
$153$	18.0120	1.45618
$154$	0	0
$155$	37.1726	2.98578
$156$	−0.00158717	−0.000127075 0
$157$	14.8856	1.18800	0.594001	−	0.804464i	$-0.297547\pi$
$157$	14.8856	1.18800	0.594001	+	0.804464i	$0.297547\pi$
$158$	14.2093	1.13043
$159$	−1.04736	−0.0830608
$160$	−3.89599	−0.308005
$161$	1.13248	0.0892518
$162$	8.86382	0.696408
$163$	4.46611	0.349813	0.174907	−	0.984585i	$-0.444038\pi$
$163$	4.46611	0.349813	0.174907	+	0.984585i	$0.444038\pi$
$164$	1.78744	0.139576
$165$	0	0
$166$	1.59761	0.123998
$167$	18.2113	1.40924	0.704618	−	0.709587i	$-0.251118\pi$
$167$	18.2113	1.40924	0.704618	+	0.709587i	$0.251118\pi$
$168$	0.126644	0.00977080
$169$	−12.9998	−0.999987
$170$	23.5103	1.80316
$171$	−14.3520	−1.09753
$172$	0.161529	0.0123165
$173$	24.8247	1.88739	0.943694	−	0.330819i	$-0.107325\pi$
$173$	24.8247	1.88739	0.943694	+	0.330819i	$0.107325\pi$
$174$	0.327688	0.0248420
$175$	10.4707	0.791514
$176$	0	0
$177$	0.996703	0.0749168
$178$	−18.5606	−1.39118
$179$	8.68595	0.649219	0.324609	−	0.945848i	$-0.394767\pi$
$179$	8.68595	0.649219	0.324609	+	0.945848i	$0.394767\pi$
$180$	11.6289	0.866768
$181$	−4.56819	−0.339551	−0.169776	−	0.985483i	$-0.554304\pi$
$181$	−4.56819	−0.339551	−0.169776	+	0.985483i	$0.554304\pi$
$182$	−0.0132619	−0.000983038 0
$183$	0.625141	0.0462118
$184$	1.10089	0.0811590
$185$	3.89599	0.286439
$186$	−1.17464	−0.0861290
$187$	0	0
$188$	−5.43218	−0.396182
$189$	−0.757945	−0.0551324
$190$	−18.7331	−1.35904
$191$	0.187966	0.0136007	0.00680037	−	0.999977i	$-0.497835\pi$
$191$	0.187966	0.0136007	0.00680037	+	0.999977i	$0.497835\pi$
$192$	0.123112	0.00888484
$193$	−19.2052	−1.38242	−0.691211	−	0.722653i	$-0.742923\pi$
$193$	−19.2052	−1.38242	−0.691211	+	0.722653i	$0.742923\pi$
$194$	12.2796	0.881622
$195$	0.00618358	0.000442815 0
$196$	−5.94180	−0.424414
$197$	−0.922455	−0.0657222	−0.0328611	−	0.999460i	$-0.510462\pi$
$197$	−0.922455	−0.0657222	−0.0328611	+	0.999460i	$0.510462\pi$
$198$	0	0
$199$	13.2663	0.940421	0.470210	−	0.882554i	$-0.344178\pi$
$199$	13.2663	0.940421	0.470210	+	0.882554i	$0.344178\pi$
$200$	10.1787	0.719744
$201$	0.725252	0.0511553
$202$	11.3530	0.798794
$203$	2.73807	0.192175
$204$	−0.742917	−0.0520146
$205$	−6.96386	−0.486377
$206$	14.0153	0.976495
$207$	−3.28600	−0.228392
$208$	−0.0128920	−0.000893902 0
$209$	0	0
$210$	−0.493404	−0.0340481
$211$	16.3303	1.12423	0.562113	−	0.827060i	$-0.309989\pi$
$211$	16.3303	1.12423	0.562113	+	0.827060i	$0.309989\pi$
$212$	−8.50735	−0.584287
$213$	−1.44929	−0.0993039
$214$	−11.7759	−0.804985
$215$	−0.629314	−0.0429189
$216$	−0.736806	−0.0501333
$217$	−9.81499	−0.666285
$218$	3.66858	0.248467
$219$	0.0669213	0.00452213
$220$	0	0
$221$	0.0777968	0.00523318
$222$	−0.123112	−0.00826274
$223$	19.6721	1.31734	0.658670	−	0.752432i	$-0.271119\pi$
$223$	19.6721	1.31734	0.658670	+	0.752432i	$0.271119\pi$
$224$	1.02869	0.0687322
$225$	−30.3819	−2.02546
$226$	−13.4128	−0.892209
$227$	7.56019	0.501788	0.250894	−	0.968015i	$-0.419275\pi$
$227$	7.56019	0.501788	0.250894	+	0.968015i	$0.419275\pi$
$228$	0.591961	0.0392035
$229$	−5.60675	−0.370505	−0.185252	−	0.982691i	$-0.559310\pi$
$229$	−5.60675	−0.370505	−0.185252	+	0.982691i	$0.559310\pi$
$230$	−4.28907	−0.282813
$231$	0	0
$232$	2.66171	0.174750
$233$	7.60058	0.497931	0.248965	−	0.968512i	$-0.419909\pi$
$233$	7.60058	0.497931	0.248965	+	0.968512i	$0.419909\pi$
$234$	0.0384807	0.00251556
$235$	21.1637	1.38057
$236$	8.09590	0.526998
$237$	1.74934	0.113632
$238$	−6.20761	−0.402380
$239$	−5.55612	−0.359395	−0.179698	−	0.983722i	$-0.557512\pi$
$239$	−5.55612	−0.359395	−0.179698	+	0.983722i	$0.557512\pi$
$240$	−0.479643	−0.0309608
$241$	−10.5708	−0.680924	−0.340462	−	0.940258i	$-0.610583\pi$
$241$	−10.5708	−0.680924	−0.340462	+	0.940258i	$0.610583\pi$
$242$	0	0
$243$	3.30166	0.211802
$244$	5.07782	0.325074
$245$	23.1492	1.47895
$246$	0.220056	0.0140302
$247$	−0.0619889	−0.00394426
$248$	−9.54125	−0.605870
$249$	0.196685	0.0124644
$250$	−20.1762	−1.27606
$251$	30.5390	1.92761	0.963803	−	0.266615i	$-0.0859051\pi$
$251$	30.5390	1.92761	0.963803	+	0.266615i	$0.0859051\pi$
$252$	−3.07048	−0.193422
$253$	0	0
$254$	15.9977	1.00378
$255$	2.89440	0.181254
$256$	1.00000	0.0625000
$257$	6.04702	0.377203	0.188601	−	0.982054i	$-0.439605\pi$
$257$	6.04702	0.377203	0.188601	+	0.982054i	$0.439605\pi$
$258$	0.0198861	0.00123806
$259$	−1.02869	−0.0639197
$260$	0.0502272	0.00311496
$261$	−7.94478	−0.491770
$262$	13.2172	0.816560
$263$	10.9524	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$263$	10.9524	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$264$	0	0
$265$	33.1445	2.03605
$266$	4.94626	0.303274
$267$	−2.28503	−0.139842
$268$	5.89099	0.359849
$269$	21.8630	1.33301	0.666505	−	0.745501i	$-0.267790\pi$
$269$	21.8630	1.33301	0.666505	+	0.745501i	$0.267790\pi$
$270$	2.87059	0.174698
$271$	25.4829	1.54798	0.773989	−	0.633199i	$-0.218259\pi$
$271$	25.4829	1.54798	0.773989	+	0.633199i	$0.218259\pi$
$272$	−6.03448	−0.365894
$273$	−0.00163270	−9.88155e−5 0
$274$	4.05441	0.244936
$275$	0	0
$276$	0.135533	0.00815814
$277$	−21.2290	−1.27553	−0.637763	−	0.770232i	$-0.720140\pi$
$277$	−21.2290	−1.27553	−0.637763	+	0.770232i	$0.720140\pi$
$278$	13.0764	0.784273
$279$	28.4791	1.70500
$280$	−4.00776	−0.239510
$281$	14.8608	0.886521	0.443260	−	0.896393i	$-0.353822\pi$
$281$	14.8608	0.886521	0.443260	+	0.896393i	$0.353822\pi$
$282$	−0.668766	−0.0398245
$283$	−21.5823	−1.28294	−0.641468	−	0.767149i	$-0.721674\pi$
$283$	−21.5823	−1.28294	−0.641468	+	0.767149i	$0.721674\pi$
$284$	−11.7721	−0.698548
$285$	−2.30627	−0.136612
$286$	0	0
$287$	1.83872	0.108536
$288$	−2.98484	−0.175884
$289$	19.4150	1.14206
$290$	−10.3700	−0.608946
$291$	1.51176	0.0886211
$292$	0.543581	0.0318107
$293$	−13.2003	−0.771170	−0.385585	−	0.922672i	$-0.626000\pi$
$293$	−13.2003	−0.771170	−0.385585	+	0.922672i	$0.626000\pi$
$294$	−0.731507	−0.0426623
$295$	−31.5415	−1.83642
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	6.83732	0.396075
$299$	−0.0141928	−0.000820789 0
$300$	1.25312	0.0723491
$301$	0.166163	0.00957747
$302$	−14.8152	−0.852521
$303$	1.39769	0.0802952
$304$	4.80831	0.275775
$305$	−19.7831	−1.13278
$306$	18.0120	1.02968
$307$	5.20755	0.297210	0.148605	−	0.988897i	$-0.452522\pi$
$307$	5.20755	0.297210	0.148605	+	0.988897i	$0.452522\pi$
$308$	0	0
$309$	1.72546	0.0981578
$310$	37.1726	2.11126
$311$	5.62942	0.319215	0.159608	−	0.987181i	$-0.448977\pi$
$311$	5.62942	0.319215	0.159608	+	0.987181i	$0.448977\pi$
$312$	−0.00158717	−8.98555e−5 0
$313$	24.9056	1.40775	0.703875	−	0.710324i	$-0.251452\pi$
$313$	24.9056	1.40775	0.703875	+	0.710324i	$0.251452\pi$
$314$	14.8856	0.840045
$315$	11.9625	0.674013
$316$	14.2093	0.799338
$317$	−26.3658	−1.48085	−0.740427	−	0.672137i	$-0.765377\pi$
$317$	−26.3658	−1.48085	−0.740427	+	0.672137i	$0.765377\pi$
$318$	−1.04736	−0.0587329
$319$	0	0
$320$	−3.89599	−0.217792
$321$	−1.44976	−0.0809175
$322$	1.13248	0.0631105
$323$	−29.0157	−1.61447
$324$	8.86382	0.492434
$325$	−0.131224	−0.00727902
$326$	4.46611	0.247355
$327$	0.451646	0.0249761
$328$	1.78744	0.0986951
$329$	−5.58803	−0.308078
$330$	0	0
$331$	−21.3680	−1.17449	−0.587247	−	0.809408i	$-0.699788\pi$
$331$	−21.3680	−1.17449	−0.587247	+	0.809408i	$0.699788\pi$
$332$	1.59761	0.0876802
$333$	2.98484	0.163568
$334$	18.2113	0.996481
$335$	−22.9512	−1.25396
$336$	0.126644	0.00690900
$337$	−11.5744	−0.630496	−0.315248	−	0.949009i	$-0.602088\pi$
$337$	−11.5744	−0.630496	−0.315248	+	0.949009i	$0.602088\pi$
$338$	−12.9998	−0.707098
$339$	−1.65128	−0.0896853
$340$	23.5103	1.27502
$341$	0	0
$342$	−14.3520	−0.776070
$343$	−13.3131	−0.718840
$344$	0.161529	0.00870905
$345$	−0.528036	−0.0284285
$346$	24.8247	1.33459
$347$	−3.77347	−0.202570	−0.101285	−	0.994857i	$-0.532295\pi$
$347$	−3.77347	−0.202570	−0.101285	+	0.994857i	$0.532295\pi$
$348$	0.327688	0.0175659
$349$	−31.7179	−1.69782	−0.848911	−	0.528536i	$-0.822741\pi$
$349$	−31.7179	−1.69782	−0.848911	+	0.528536i	$0.822741\pi$
$350$	10.4707	0.559685
$351$	0.00949894	0.000507016 0
$352$	0	0
$353$	−31.6492	−1.68452	−0.842258	−	0.539074i	$-0.818774\pi$
$353$	−31.6492	−1.68452	−0.842258	+	0.539074i	$0.818774\pi$
$354$	0.996703	0.0529742
$355$	45.8641	2.43422
$356$	−18.5606	−0.983710
$357$	−0.764231	−0.0404474
$358$	8.68595	0.459067
$359$	8.28551	0.437292	0.218646	−	0.975804i	$-0.429836\pi$
$359$	8.28551	0.437292	0.218646	+	0.975804i	$0.429836\pi$
$360$	11.6289	0.612897
$361$	4.11983	0.216833
$362$	−4.56819	−0.240099
$363$	0	0
$364$	−0.0132619	−0.000695113 0
$365$	−2.11778	−0.110850
$366$	0.625141	0.0326767
$367$	14.0906	0.735524	0.367762	−	0.929920i	$-0.380124\pi$
$367$	14.0906	0.735524	0.367762	+	0.929920i	$0.380124\pi$
$368$	1.10089	0.0573881
$369$	−5.33524	−0.277741
$370$	3.89599	0.202543
$371$	−8.75143	−0.454351
$372$	−1.17464	−0.0609024
$373$	6.02010	0.311709	0.155855	−	0.987780i	$-0.450187\pi$
$373$	6.02010	0.311709	0.155855	+	0.987780i	$0.450187\pi$
$374$	0	0
$375$	−2.48393	−0.128270
$376$	−5.43218	−0.280143
$377$	−0.0343148	−0.00176730
$378$	−0.757945	−0.0389845
$379$	23.5390	1.20912	0.604558	−	0.796561i	$-0.293350\pi$
$379$	23.5390	1.20912	0.604558	+	0.796561i	$0.293350\pi$
$380$	−18.7331	−0.960988
$381$	1.96951	0.100901
$382$	0.187966	0.00961718
$383$	−1.12110	−0.0572854	−0.0286427	−	0.999590i	$-0.509119\pi$
$383$	−1.12110	−0.0572854	−0.0286427	+	0.999590i	$0.509119\pi$
$384$	0.123112	0.00628253
$385$	0	0
$386$	−19.2052	−0.977520
$387$	−0.482138	−0.0245085
$388$	12.2796	0.623401
$389$	−11.4668	−0.581391	−0.290696	−	0.956816i	$-0.593887\pi$
$389$	−11.4668	−0.581391	−0.290696	+	0.956816i	$0.593887\pi$
$390$	0.00618358	0.000313118 0
$391$	−6.64332	−0.335967
$392$	−5.94180	−0.300106
$393$	1.62719	0.0820810
$394$	−0.922455	−0.0464726
$395$	−55.3594	−2.78543
$396$	0	0
$397$	14.2583	0.715601	0.357801	−	0.933798i	$-0.383527\pi$
$397$	14.2583	0.715601	0.357801	+	0.933798i	$0.383527\pi$
$398$	13.2663	0.664978
$399$	0.608944	0.0304853
$400$	10.1787	0.508936
$401$	−15.0727	−0.752697	−0.376349	−	0.926478i	$-0.622820\pi$
$401$	−15.0727	−0.752697	−0.376349	+	0.926478i	$0.622820\pi$
$402$	0.725252	0.0361723
$403$	0.123006	0.00612738
$404$	11.3530	0.564833
$405$	−34.5333	−1.71598
$406$	2.73807	0.135888
$407$	0	0
$408$	−0.742917	−0.0367799
$409$	15.1981	0.751496	0.375748	−	0.926722i	$-0.377386\pi$
$409$	15.1981	0.751496	0.375748	+	0.926722i	$0.377386\pi$
$410$	−6.96386	−0.343920
$411$	0.499146	0.0246211
$412$	14.0153	0.690486
$413$	8.32817	0.409803
$414$	−3.28600	−0.161498
$415$	−6.22426	−0.305537
$416$	−0.0128920	−0.000632084 0
$417$	1.60987	0.0788356
$418$	0	0
$419$	24.8206	1.21257	0.606283	−	0.795249i	$-0.292660\pi$
$419$	24.8206	1.21257	0.606283	+	0.795249i	$0.292660\pi$
$420$	−0.493404	−0.0240756
$421$	−0.0939937	−0.00458097	−0.00229049	−	0.999997i	$-0.500729\pi$
$421$	−0.0939937	−0.00458097	−0.00229049	+	0.999997i	$0.500729\pi$
$422$	16.3303	0.794948
$423$	16.2142	0.788361
$424$	−8.50735	−0.413154
$425$	−61.4233	−2.97947
$426$	−1.44929	−0.0702185
$427$	5.22351	0.252783
$428$	−11.7759	−0.569210
$429$	0	0
$430$	−0.629314	−0.0303482
$431$	−15.0602	−0.725423	−0.362712	−	0.931901i	$-0.618149\pi$
$431$	−15.0602	−0.725423	−0.362712	+	0.931901i	$0.618149\pi$
$432$	−0.736806	−0.0354496
$433$	−22.6080	−1.08647	−0.543235	−	0.839581i	$-0.682801\pi$
$433$	−22.6080	−1.08647	−0.543235	+	0.839581i	$0.682801\pi$
$434$	−9.81499	−0.471135
$435$	−1.27667	−0.0612116
$436$	3.66858	0.175693
$437$	5.29344	0.253219
$438$	0.0669213	0.00319763
$439$	15.2327	0.727019	0.363510	−	0.931590i	$-0.381578\pi$
$439$	15.2327	0.727019	0.363510	+	0.931590i	$0.381578\pi$
$440$	0	0
$441$	17.7353	0.844540
$442$	0.0777968	0.00370042
$443$	−4.85321	−0.230583	−0.115291	−	0.993332i	$-0.536780\pi$
$443$	−4.85321	−0.230583	−0.115291	+	0.993332i	$0.536780\pi$
$444$	−0.123112	−0.00584264
$445$	72.3119	3.42791
$446$	19.6721	0.931501
$447$	0.841756	0.0398137
$448$	1.02869	0.0486010
$449$	28.9173	1.36469	0.682347	−	0.731029i	$-0.260960\pi$
$449$	28.9173	1.36469	0.682347	+	0.731029i	$0.260960\pi$
$450$	−30.3819	−1.43222
$451$	0	0
$452$	−13.4128	−0.630887
$453$	−1.82393	−0.0856959
$454$	7.56019	0.354818
$455$	0.0516682	0.00242224
$456$	0.591961	0.0277211
$457$	−41.8706	−1.95862	−0.979311	−	0.202359i	$-0.935139\pi$
$457$	−41.8706	−1.95862	−0.979311	+	0.202359i	$0.935139\pi$
$458$	−5.60675	−0.261986
$459$	4.44624	0.207533
$460$	−4.28907	−0.199979
$461$	−17.2212	−0.802073	−0.401036	−	0.916062i	$-0.631350\pi$
$461$	−17.2212	−0.802073	−0.401036	+	0.916062i	$0.631350\pi$
$462$	0	0
$463$	14.3437	0.666610	0.333305	−	0.942819i	$-0.391836\pi$
$463$	14.3437	0.666610	0.333305	+	0.942819i	$0.391836\pi$
$464$	2.66171	0.123567
$465$	4.57639	0.212225
$466$	7.60058	0.352090
$467$	0.888200	0.0411010	0.0205505	−	0.999789i	$-0.493458\pi$
$467$	0.888200	0.0411010	0.0205505	+	0.999789i	$0.493458\pi$
$468$	0.0384807	0.00177877
$469$	6.06000	0.279825
$470$	21.1637	0.976209
$471$	1.83260	0.0844417
$472$	8.09590	0.372644
$473$	0	0
$474$	1.74934	0.0803499
$475$	48.9424	2.24563
$476$	−6.20761	−0.284525
$477$	25.3931	1.16267
$478$	−5.55612	−0.254131
$479$	16.7008	0.763080	0.381540	−	0.924352i	$-0.375394\pi$
$479$	16.7008	0.763080	0.381540	+	0.924352i	$0.375394\pi$
$480$	−0.479643	−0.0218926
$481$	0.0128920	0.000587826 0
$482$	−10.5708	−0.481486
$483$	0.139422	0.00634391
$484$	0	0
$485$	−47.8410	−2.17235
$486$	3.30166	0.149766
$487$	0.597149	0.0270594	0.0135297	−	0.999908i	$-0.495693\pi$
$487$	0.597149	0.0270594	0.0135297	+	0.999908i	$0.495693\pi$
$488$	5.07782	0.229862
$489$	0.549832	0.0248643
$490$	23.1492	1.04577
$491$	2.37658	0.107254	0.0536268	−	0.998561i	$-0.482922\pi$
$491$	2.37658	0.107254	0.0536268	+	0.998561i	$0.482922\pi$
$492$	0.220056	0.00992088
$493$	−16.0620	−0.723397
$494$	−0.0619889	−0.00278901
$495$	0	0
$496$	−9.54125	−0.428415
$497$	−12.1099	−0.543203
$498$	0.196685	0.00881366
$499$	10.5737	0.473345	0.236672	−	0.971590i	$-0.423943\pi$
$499$	10.5737	0.473345	0.236672	+	0.971590i	$0.423943\pi$
$500$	−20.1762	−0.902308
$501$	2.24204	0.100167
$502$	30.5390	1.36302
$503$	9.37897	0.418187	0.209094	−	0.977896i	$-0.432949\pi$
$503$	9.37897	0.418187	0.209094	+	0.977896i	$0.432949\pi$
$504$	−3.07048	−0.136770
$505$	−44.2311	−1.96826
$506$	0	0
$507$	−1.60044	−0.0710778
$508$	15.9977	0.709782
$509$	21.7630	0.964629	0.482314	−	0.875998i	$-0.339796\pi$
$509$	21.7630	0.964629	0.482314	+	0.875998i	$0.339796\pi$
$510$	2.89440	0.128166
$511$	0.559176	0.0247365
$512$	1.00000	0.0441942
$513$	−3.54279	−0.156418
$514$	6.04702	0.266723
$515$	−54.6036	−2.40612
$516$	0.0198861	0.000875438 0
$517$	0	0
$518$	−1.02869	−0.0451980
$519$	3.05622	0.134153
$520$	0.0502272	0.00220261
$521$	31.2495	1.36907	0.684534	−	0.728981i	$-0.260006\pi$
$521$	31.2495	1.36907	0.684534	+	0.728981i	$0.260006\pi$
$522$	−7.94478	−0.347734
$523$	−28.0377	−1.22601	−0.613003	−	0.790081i	$-0.710038\pi$
$523$	−28.0377	−1.22601	−0.613003	+	0.790081i	$0.710038\pi$
$524$	13.2172	0.577395
$525$	1.28907	0.0562598
$526$	10.9524	0.477546
$527$	57.5765	2.50807
$528$	0	0
$529$	−21.7880	−0.947306
$530$	33.1445	1.43971
$531$	−24.1650	−1.04867
$532$	4.94626	0.214447
$533$	−0.0230438	−0.000998137 0
$534$	−2.28503	−0.0988831
$535$	45.8788	1.98351
$536$	5.89099	0.254452
$537$	1.06935	0.0461457
$538$	21.8630	0.942580
$539$	0	0
$540$	2.87059	0.123530
$541$	18.2076	0.782805	0.391402	−	0.920220i	$-0.371990\pi$
$541$	18.2076	0.782805	0.391402	+	0.920220i	$0.371990\pi$
$542$	25.4829	1.09459
$543$	−0.562399	−0.0241349
$544$	−6.03448	−0.258726
$545$	−14.2927	−0.612233
$546$	−0.00163270	−6.98731e−5 0
$547$	4.96008	0.212078	0.106039	−	0.994362i	$-0.466183\pi$
$547$	4.96008	0.212078	0.106039	+	0.994362i	$0.466183\pi$
$548$	4.05441	0.173196
$549$	−15.1565	−0.646864
$550$	0	0
$551$	12.7983	0.545226
$552$	0.135533	0.00576868
$553$	14.6170	0.621578
$554$	−21.2290	−0.901934
$555$	0.479643	0.0203597
$556$	13.0764	0.554565
$557$	−38.0632	−1.61279	−0.806394	−	0.591378i	$-0.798584\pi$
$557$	−38.0632	−1.61279	−0.806394	+	0.591378i	$0.798584\pi$
$558$	28.4791	1.20562
$559$	−0.00208244	−8.80776e−5 0
$560$	−4.00776	−0.169359
$561$	0	0
$562$	14.8608	0.626865
$563$	11.5246	0.485705	0.242853	−	0.970063i	$-0.421917\pi$
$563$	11.5246	0.485705	0.242853	+	0.970063i	$0.421917\pi$
$564$	−0.668766	−0.0281601
$565$	52.2563	2.19844
$566$	−21.5823	−0.907173
$567$	9.11812	0.382925
$568$	−11.7721	−0.493948
$569$	10.4605	0.438527	0.219263	−	0.975666i	$-0.429635\pi$
$569$	10.4605	0.438527	0.219263	+	0.975666i	$0.429635\pi$
$570$	−2.30627	−0.0965991
$571$	20.2874	0.849003	0.424501	−	0.905427i	$-0.360449\pi$
$571$	20.2874	0.849003	0.424501	+	0.905427i	$0.360449\pi$
$572$	0	0
$573$	0.0231409	0.000966724 0
$574$	1.83872	0.0767469
$575$	11.2057	0.467309
$576$	−2.98484	−0.124368
$577$	2.52024	0.104919	0.0524594	−	0.998623i	$-0.483294\pi$
$577$	2.52024	0.104919	0.0524594	+	0.998623i	$0.483294\pi$
$578$	19.4150	0.807556
$579$	−2.36439	−0.0982609
$580$	−10.3700	−0.430590
$581$	1.64344	0.0681815
$582$	1.51176	0.0626646
$583$	0	0
$584$	0.543581	0.0224935
$585$	−0.149920	−0.00619845
$586$	−13.2003	−0.545299
$587$	−35.6393	−1.47099	−0.735496	−	0.677529i	$-0.763051\pi$
$587$	−35.6393	−1.47099	−0.735496	+	0.677529i	$0.763051\pi$
$588$	−0.731507	−0.0301668
$589$	−45.8773	−1.89034
$590$	−31.5415	−1.29854
$591$	−0.113565	−0.00467145
$592$	−1.00000	−0.0410997
$593$	33.5776	1.37887	0.689433	−	0.724350i	$-0.257860\pi$
$593$	33.5776	1.37887	0.689433	+	0.724350i	$0.257860\pi$
$594$	0	0
$595$	24.1848	0.991479
$596$	6.83732	0.280068
$597$	1.63324	0.0668439
$598$	−0.0141928	−0.000580385 0
$599$	−8.75927	−0.357894	−0.178947	−	0.983859i	$-0.557269\pi$
$599$	−8.75927	−0.357894	−0.178947	+	0.983859i	$0.557269\pi$
$600$	1.25312	0.0511585
$601$	−39.8702	−1.62634	−0.813170	−	0.582027i	$-0.802260\pi$
$601$	−39.8702	−1.62634	−0.813170	+	0.582027i	$0.802260\pi$
$602$	0.166163	0.00677230
$603$	−17.5837	−0.716063
$604$	−14.8152	−0.602823
$605$	0	0
$606$	1.39769	0.0567773
$607$	18.6487	0.756928	0.378464	−	0.925616i	$-0.376452\pi$
$607$	18.6487	0.756928	0.378464	+	0.925616i	$0.376452\pi$
$608$	4.80831	0.195003
$609$	0.337090	0.0136596
$610$	−19.7831	−0.800996
$611$	0.0700319	0.00283319
$612$	18.0120	0.728091
$613$	15.5862	0.629521	0.314760	−	0.949171i	$-0.398076\pi$
$613$	15.5862	0.629521	0.314760	+	0.949171i	$0.398076\pi$
$614$	5.20755	0.210160
$615$	−0.857335	−0.0345711
$616$	0	0
$617$	−17.2046	−0.692630	−0.346315	−	0.938118i	$-0.612567\pi$
$617$	−17.2046	−0.692630	−0.346315	+	0.938118i	$0.612567\pi$
$618$	1.72546	0.0694080
$619$	19.6523	0.789893	0.394947	−	0.918704i	$-0.370763\pi$
$619$	19.6523	0.789893	0.394947	+	0.918704i	$0.370763\pi$
$620$	37.1726	1.49289
$621$	−0.811145	−0.0325501
$622$	5.62942	0.225719
$623$	−19.0931	−0.764949
$624$	−0.00158717	−6.35375e−5 0
$625$	27.7127	1.10851
$626$	24.9056	0.995430
$627$	0	0
$628$	14.8856	0.594001
$629$	6.03448	0.240611
$630$	11.9625	0.476599
$631$	16.5308	0.658081	0.329040	−	0.944316i	$-0.393275\pi$
$631$	16.5308	0.658081	0.329040	+	0.944316i	$0.393275\pi$
$632$	14.2093	0.565217
$633$	2.01046	0.0799086
$634$	−26.3658	−1.04712
$635$	−62.3267	−2.47336
$636$	−1.04736	−0.0415304
$637$	0.0766019	0.00303508
$638$	0	0
$639$	35.1380	1.39004
$640$	−3.89599	−0.154002
$641$	11.4960	0.454063	0.227032	−	0.973887i	$-0.427098\pi$
$641$	11.4960	0.454063	0.227032	+	0.973887i	$0.427098\pi$
$642$	−1.44976	−0.0572173
$643$	20.7610	0.818733	0.409366	−	0.912370i	$-0.365750\pi$
$643$	20.7610	0.818733	0.409366	+	0.912370i	$0.365750\pi$
$644$	1.13248	0.0446259
$645$	−0.0774761	−0.00305062
$646$	−29.0157	−1.14161
$647$	24.2672	0.954042	0.477021	−	0.878892i	$-0.341717\pi$
$647$	24.2672	0.954042	0.477021	+	0.878892i	$0.341717\pi$
$648$	8.86382	0.348204
$649$	0	0
$650$	−0.131224	−0.00514705
$651$	−1.20834	−0.0473587
$652$	4.46611	0.174907
$653$	1.19574	0.0467929	0.0233965	−	0.999726i	$-0.492552\pi$
$653$	1.19574	0.0467929	0.0233965	+	0.999726i	$0.492552\pi$
$654$	0.451646	0.0176607
$655$	−51.4939	−2.01203
$656$	1.78744	0.0697879
$657$	−1.62250	−0.0632999
$658$	−5.58803	−0.217844
$659$	−12.4217	−0.483881	−0.241940	−	0.970291i	$-0.577784\pi$
$659$	−12.4217	−0.483881	−0.241940	+	0.970291i	$0.577784\pi$
$660$	0	0
$661$	−30.0096	−1.16724	−0.583620	−	0.812027i	$-0.698364\pi$
$661$	−30.0096	−1.16724	−0.583620	+	0.812027i	$0.698364\pi$
$662$	−21.3680	−0.830493
$663$	0.00957772	0.000371968 0
$664$	1.59761	0.0619992
$665$	−19.2706	−0.747280
$666$	2.98484	0.115660
$667$	2.93026	0.113460
$668$	18.2113	0.704618
$669$	2.42187	0.0936349
$670$	−22.9512	−0.886683
$671$	0	0
$672$	0.126644	0.00488540
$673$	−22.7736	−0.877858	−0.438929	−	0.898522i	$-0.644642\pi$
$673$	−22.7736	−0.877858	−0.438929	+	0.898522i	$0.644642\pi$
$674$	−11.5744	−0.445828
$675$	−7.49974	−0.288665
$676$	−12.9998	−0.499994
$677$	−39.1057	−1.50295	−0.751476	−	0.659760i	$-0.770658\pi$
$677$	−39.1057	−1.50295	−0.751476	+	0.659760i	$0.770658\pi$
$678$	−1.65128	−0.0634171
$679$	12.6319	0.484767
$680$	23.5103	0.901578
$681$	0.930751	0.0356664
$682$	0	0
$683$	−10.5213	−0.402587	−0.201293	−	0.979531i	$-0.564514\pi$
$683$	−10.5213	−0.402587	−0.201293	+	0.979531i	$0.564514\pi$
$684$	−14.3520	−0.548764
$685$	−15.7959	−0.603531
$686$	−13.3131	−0.508296
$687$	−0.690259	−0.0263350
$688$	0.161529	0.00615823
$689$	0.109677	0.00417837
$690$	−0.528036	−0.0201020
$691$	−21.3537	−0.812332	−0.406166	−	0.913799i	$-0.633135\pi$
$691$	−21.3537	−0.812332	−0.406166	+	0.913799i	$0.633135\pi$
$692$	24.8247	0.943694
$693$	0	0
$694$	−3.77347	−0.143239
$695$	−50.9457	−1.93248
$696$	0.327688	0.0124210
$697$	−10.7863	−0.408560
$698$	−31.7179	−1.20054
$699$	0.935723	0.0353923
$700$	10.4707	0.395757
$701$	−14.1746	−0.535366	−0.267683	−	0.963507i	$-0.586258\pi$
$701$	−14.1746	−0.535366	−0.267683	+	0.963507i	$0.586258\pi$
$702$	0.00949894	0.000358514 0
$703$	−4.80831	−0.181349
$704$	0	0
$705$	2.60551	0.0981290
$706$	−31.6492	−1.19113
$707$	11.6787	0.439223
$708$	0.996703	0.0374584
$709$	−16.8864	−0.634184	−0.317092	−	0.948395i	$-0.602706\pi$
$709$	−16.8864	−0.634184	−0.317092	+	0.948395i	$0.602706\pi$
$710$	45.8641	1.72125
$711$	−42.4127	−1.59060
$712$	−18.5606	−0.695588
$713$	−10.5039	−0.393374
$714$	−0.764231	−0.0286006
$715$	0	0
$716$	8.68595	0.324609
$717$	−0.684025	−0.0255454
$718$	8.28551	0.309212
$719$	−41.9579	−1.56477	−0.782383	−	0.622797i	$-0.785996\pi$
$719$	−41.9579	−1.56477	−0.782383	+	0.622797i	$0.785996\pi$
$720$	11.6289	0.433384
$721$	14.4174	0.536933
$722$	4.11983	0.153324
$723$	−1.30139	−0.0483992
$724$	−4.56819	−0.169776
$725$	27.0928	1.00620
$726$	0	0
$727$	27.4101	1.01658	0.508291	−	0.861185i	$-0.330277\pi$
$727$	27.4101	1.01658	0.508291	+	0.861185i	$0.330277\pi$
$728$	−0.0132619	−0.000491519 0
$729$	−26.1850	−0.969814
$730$	−2.11778	−0.0783827
$731$	−0.974742	−0.0360522
$732$	0.625141	0.0231059
$733$	36.4545	1.34648	0.673239	−	0.739425i	$-0.264903\pi$
$733$	36.4545	1.34648	0.673239	+	0.739425i	$0.264903\pi$
$734$	14.0906	0.520094
$735$	2.84994	0.105122
$736$	1.10089	0.0405795
$737$	0	0
$738$	−5.33524	−0.196393
$739$	3.37053	0.123987	0.0619935	−	0.998077i	$-0.480254\pi$
$739$	3.37053	0.123987	0.0619935	+	0.998077i	$0.480254\pi$
$740$	3.89599	0.143219
$741$	−0.00763158	−0.000280353 0
$742$	−8.75143	−0.321275
$743$	51.3484	1.88379	0.941896	−	0.335906i	$-0.109042\pi$
$743$	51.3484	1.88379	0.941896	+	0.335906i	$0.109042\pi$
$744$	−1.17464	−0.0430645
$745$	−26.6381	−0.975945
$746$	6.02010	0.220412
$747$	−4.76861	−0.174474
$748$	0	0
$749$	−12.1138	−0.442627
$750$	−2.48393	−0.0907005
$751$	6.76819	0.246975	0.123487	−	0.992346i	$-0.460592\pi$
$751$	6.76819	0.246975	0.123487	+	0.992346i	$0.460592\pi$
$752$	−5.43218	−0.198091
$753$	3.75972	0.137012
$754$	−0.0343148	−0.00124967
$755$	57.7200	2.10064
$756$	−0.757945	−0.0275662
$757$	22.2851	0.809965	0.404983	−	0.914324i	$-0.367278\pi$
$757$	22.2851	0.809965	0.404983	+	0.914324i	$0.367278\pi$
$758$	23.5390	0.854974
$759$	0	0
$760$	−18.7331	−0.679521
$761$	46.2761	1.67751	0.838753	−	0.544512i	$-0.183285\pi$
$761$	46.2761	1.67751	0.838753	+	0.544512i	$0.183285\pi$
$762$	1.96951	0.0713477
$763$	3.77383	0.136622
$764$	0.187966	0.00680037
$765$	−70.1745	−2.53716
$766$	−1.12110	−0.0405069
$767$	−0.104373	−0.00376868
$768$	0.123112	0.00444242
$769$	43.1253	1.55514	0.777569	−	0.628798i	$-0.216453\pi$
$769$	43.1253	1.55514	0.777569	+	0.628798i	$0.216453\pi$
$770$	0	0
$771$	0.744461	0.0268111
$772$	−19.2052	−0.691211
$773$	16.4915	0.593158	0.296579	−	0.955008i	$-0.404154\pi$
$773$	16.4915	0.593158	0.296579	+	0.955008i	$0.404154\pi$
$774$	−0.482138	−0.0173301
$775$	−97.1177	−3.48857
$776$	12.2796	0.440811
$777$	−0.126644	−0.00454333
$778$	−11.4668	−0.411106
$779$	8.59458	0.307933
$780$	0.00618358	0.000221408 0
$781$	0	0
$782$	−6.64332	−0.237565
$783$	−1.96116	−0.0700862
$784$	−5.94180	−0.212207
$785$	−57.9942	−2.06990
$786$	1.62719	0.0580400
$787$	14.5598	0.519002	0.259501	−	0.965743i	$-0.416442\pi$
$787$	14.5598	0.519002	0.259501	+	0.965743i	$0.416442\pi$
$788$	−0.922455	−0.0328611
$789$	1.34837	0.0480032
$790$	−55.3594	−1.96960
$791$	−13.7977	−0.490588
$792$	0	0
$793$	−0.0654635	−0.00232468
$794$	14.2583	0.506006
$795$	4.08049	0.144720
$796$	13.2663	0.470210
$797$	−8.77648	−0.310879	−0.155439	−	0.987845i	$-0.549679\pi$
$797$	−8.77648	−0.310879	−0.155439	+	0.987845i	$0.549679\pi$
$798$	0.608944	0.0215564
$799$	32.7804	1.15969
$800$	10.1787	0.359872
$801$	55.4005	1.95748
$802$	−15.0727	−0.532237
$803$	0	0
$804$	0.725252	0.0255776
$805$	−4.41212	−0.155507
$806$	0.123006	0.00433271
$807$	2.69160	0.0947486
$808$	11.3530	0.399397
$809$	11.6288	0.408847	0.204424	−	0.978882i	$-0.434468\pi$
$809$	11.6288	0.408847	0.204424	+	0.978882i	$0.434468\pi$
$810$	−34.5333	−1.21338
$811$	−12.5463	−0.440560	−0.220280	−	0.975437i	$-0.570697\pi$
$811$	−12.5463	−0.440560	−0.220280	+	0.975437i	$0.570697\pi$
$812$	2.73807	0.0960875
$813$	3.13725	0.110028
$814$	0	0
$815$	−17.3999	−0.609493
$816$	−0.742917	−0.0260073
$817$	0.776680	0.0271726
$818$	15.1981	0.531388
$819$	0.0395847	0.00138320
$820$	−6.96386	−0.243188
$821$	−31.5530	−1.10121	−0.550603	−	0.834767i	$-0.685602\pi$
$821$	−31.5530	−1.10121	−0.550603	+	0.834767i	$0.685602\pi$
$822$	0.499146	0.0174097
$823$	15.6391	0.545146	0.272573	−	0.962135i	$-0.412125\pi$
$823$	15.6391	0.545146	0.272573	+	0.962135i	$0.412125\pi$
$824$	14.0153	0.488247
$825$	0	0
$826$	8.32817	0.289774
$827$	6.79187	0.236177	0.118088	−	0.993003i	$-0.462323\pi$
$827$	6.79187	0.236177	0.118088	+	0.993003i	$0.462323\pi$
$828$	−3.28600	−0.114196
$829$	−36.8175	−1.27873	−0.639363	−	0.768905i	$-0.720802\pi$
$829$	−36.8175	−1.27873	−0.639363	+	0.768905i	$0.720802\pi$
$830$	−6.22426	−0.216047
$831$	−2.61354	−0.0906629
$832$	−0.0128920	−0.000446951 0
$833$	35.8557	1.24233
$834$	1.60987	0.0557452
$835$	−70.9512	−2.45537
$836$	0	0
$837$	7.03005	0.242994
$838$	24.8206	0.857413
$839$	−37.4300	−1.29223	−0.646113	−	0.763242i	$-0.723607\pi$
$839$	−37.4300	−1.29223	−0.646113	+	0.763242i	$0.723607\pi$
$840$	−0.493404	−0.0170240
$841$	−21.9153	−0.755700
$842$	−0.0939937	−0.00323924
$843$	1.82954	0.0630128
$844$	16.3303	0.562113
$845$	50.6472	1.74232
$846$	16.2142	0.557456
$847$	0	0
$848$	−8.50735	−0.292144
$849$	−2.65705	−0.0911896
$850$	−61.4233	−2.10680
$851$	−1.10089	−0.0377382
$852$	−1.44929	−0.0496519
$853$	−36.3121	−1.24330	−0.621651	−	0.783295i	$-0.713538\pi$
$853$	−36.3121	−1.24330	−0.621651	+	0.783295i	$0.713538\pi$
$854$	5.22351	0.178745
$855$	55.9154	1.91227
$856$	−11.7759	−0.402492
$857$	1.90793	0.0651737	0.0325869	−	0.999469i	$-0.489625\pi$
$857$	1.90793	0.0651737	0.0325869	+	0.999469i	$0.489625\pi$
$858$	0	0
$859$	3.72968	0.127255	0.0636275	−	0.997974i	$-0.479733\pi$
$859$	3.72968	0.127255	0.0636275	+	0.997974i	$0.479733\pi$
$860$	−0.629314	−0.0214594
$861$	0.226369	0.00771464
$862$	−15.0602	−0.512952
$863$	−35.0188	−1.19205	−0.596026	−	0.802965i	$-0.703255\pi$
$863$	−35.0188	−1.19205	−0.596026	+	0.802965i	$0.703255\pi$
$864$	−0.736806	−0.0250667
$865$	−96.7168	−3.28847
$866$	−22.6080	−0.768250
$867$	2.39022	0.0811760
$868$	−9.81499	−0.333142
$869$	0	0
$870$	−1.27667	−0.0432831
$871$	−0.0759469	−0.00257336
$872$	3.66858	0.124234
$873$	−36.6526	−1.24050
$874$	5.29344	0.179053
$875$	−20.7551	−0.701649
$876$	0.0669213	0.00226106
$877$	18.6775	0.630693	0.315346	−	0.948977i	$-0.397879\pi$
$877$	18.6775	0.630693	0.315346	+	0.948977i	$0.397879\pi$
$878$	15.2327	0.514080
$879$	−1.62512	−0.0548138
$880$	0	0
$881$	11.2289	0.378310	0.189155	−	0.981947i	$-0.439425\pi$
$881$	11.2289	0.378310	0.189155	+	0.981947i	$0.439425\pi$
$882$	17.7353	0.597180
$883$	−8.98744	−0.302451	−0.151226	−	0.988499i	$-0.548322\pi$
$883$	−8.98744	−0.302451	−0.151226	+	0.988499i	$0.548322\pi$
$884$	0.0777968	0.00261659
$885$	−3.88314	−0.130530
$886$	−4.85321	−0.163047
$887$	−33.5316	−1.12588	−0.562941	−	0.826497i	$-0.690330\pi$
$887$	−33.5316	−1.12588	−0.562941	+	0.826497i	$0.690330\pi$
$888$	−0.123112	−0.00413137
$889$	16.4566	0.551938
$890$	72.3119	2.42390
$891$	0	0
$892$	19.6721	0.658670
$893$	−26.1196	−0.874059
$894$	0.841756	0.0281525
$895$	−33.8404	−1.13116
$896$	1.02869	0.0343661
$897$	−0.00174730	−5.83407e−5 0
$898$	28.9173	0.964984
$899$	−25.3960	−0.847005
$900$	−30.3819	−1.01273
$901$	51.3375	1.71030
$902$	0	0
$903$	0.0204567	0.000680755 0
$904$	−13.4128	−0.446104
$905$	17.7976	0.591613
$906$	−1.82393	−0.0605961
$907$	29.3372	0.974124	0.487062	−	0.873367i	$-0.338068\pi$
$907$	29.3372	0.974124	0.487062	+	0.873367i	$0.338068\pi$
$908$	7.56019	0.250894
$909$	−33.8869	−1.12396
$910$	0.0516682	0.00171279
$911$	10.9950	0.364279	0.182140	−	0.983273i	$-0.441698\pi$
$911$	10.9950	0.364279	0.182140	+	0.983273i	$0.441698\pi$
$912$	0.591961	0.0196018
$913$	0	0
$914$	−41.8706	−1.38496
$915$	−2.43554	−0.0805165
$916$	−5.60675	−0.185252
$917$	13.5964	0.448992
$918$	4.44624	0.146748
$919$	15.6338	0.515711	0.257856	−	0.966183i	$-0.416984\pi$
$919$	15.6338	0.515711	0.257856	+	0.966183i	$0.416984\pi$
$920$	−4.28907	−0.141406
$921$	0.641112	0.0211253
$922$	−17.2212	−0.567151
$923$	0.151767	0.00499547
$924$	0	0
$925$	−10.1787	−0.334674
$926$	14.3437	0.471364
$927$	−41.8336	−1.37399
$928$	2.66171	0.0873748
$929$	31.9938	1.04968	0.524841	−	0.851200i	$-0.324125\pi$
$929$	31.9938	1.04968	0.524841	+	0.851200i	$0.324125\pi$
$930$	4.57639	0.150066
$931$	−28.5700	−0.936344
$932$	7.60058	0.248965
$933$	0.693050	0.0226894
$934$	0.888200	0.0290628
$935$	0	0
$936$	0.0384807	0.00125778
$937$	−19.6334	−0.641394	−0.320697	−	0.947182i	$-0.603917\pi$
$937$	−19.6334	−0.641394	−0.320697	+	0.947182i	$0.603917\pi$
$938$	6.06000	0.197866
$939$	3.06618	0.100061
$940$	21.1637	0.690284
$941$	−49.0692	−1.59961	−0.799806	−	0.600259i	$-0.795064\pi$
$941$	−49.0692	−1.59961	−0.799806	+	0.600259i	$0.795064\pi$
$942$	1.83260	0.0597093
$943$	1.96779	0.0640799
$944$	8.09590	0.263499
$945$	2.95294	0.0960593
$946$	0	0
$947$	57.7198	1.87564	0.937821	−	0.347120i	$-0.112840\pi$
$947$	57.7198	1.87564	0.937821	+	0.347120i	$0.112840\pi$
$948$	1.74934	0.0568159
$949$	−0.00700787	−0.000227485 0
$950$	48.9424	1.58790
$951$	−3.24595	−0.105257
$952$	−6.20761	−0.201190
$953$	1.32966	0.0430719	0.0215359	−	0.999768i	$-0.493144\pi$
$953$	1.32966	0.0430719	0.0215359	+	0.999768i	$0.493144\pi$
$954$	25.3931	0.822133
$955$	−0.732313	−0.0236971
$956$	−5.55612	−0.179698
$957$	0	0
$958$	16.7008	0.539579
$959$	4.17073	0.134680
$960$	−0.479643	−0.0154804
$961$	60.0355	1.93663
$962$	0.0128920	0.000415656 0
$963$	35.1492	1.13267
$964$	−10.5708	−0.340462
$965$	74.8233	2.40865
$966$	0.139422	0.00448582
$967$	45.3576	1.45860	0.729301	−	0.684193i	$-0.239846\pi$
$967$	45.3576	1.45860	0.729301	+	0.684193i	$0.239846\pi$
$968$	0	0
$969$	−3.57218	−0.114755
$970$	−47.8410	−1.53608
$971$	−16.5996	−0.532706	−0.266353	−	0.963876i	$-0.585819\pi$
$971$	−16.5996	−0.532706	−0.266353	+	0.963876i	$0.585819\pi$
$972$	3.30166	0.105901
$973$	13.4516	0.431239
$974$	0.597149	0.0191339
$975$	−0.0161553	−0.000517384 0
$976$	5.07782	0.162537
$977$	−36.5324	−1.16878	−0.584388	−	0.811475i	$-0.698665\pi$
$977$	−36.5324	−1.16878	−0.584388	+	0.811475i	$0.698665\pi$
$978$	0.549832	0.0175817
$979$	0	0
$980$	23.1492	0.739473
$981$	−10.9501	−0.349611
$982$	2.37658	0.0758398
$983$	23.7195	0.756536	0.378268	−	0.925696i	$-0.376520\pi$
$983$	23.7195	0.756536	0.378268	+	0.925696i	$0.376520\pi$
$984$	0.220056	0.00701512
$985$	3.59387	0.114510
$986$	−16.0620	−0.511519
$987$	−0.687953	−0.0218978
$988$	−0.0619889	−0.00197213
$989$	0.177826	0.00565454
$990$	0	0
$991$	3.64787	0.115878	0.0579392	−	0.998320i	$-0.481547\pi$
$991$	3.64787	0.115878	0.0579392	+	0.998320i	$0.481547\pi$
$992$	−9.54125	−0.302935
$993$	−2.63066	−0.0834816
$994$	−12.1099	−0.384102
$995$	−51.6852	−1.63853
$996$	0.196685	0.00623220
$997$	15.7670	0.499346	0.249673	−	0.968330i	$-0.419677\pi$
$997$	15.7670	0.499346	0.249673	+	0.968330i	$0.419677\pi$
$998$	10.5737	0.334705
$999$	0.736806	0.0233115

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8954.2.a.bt.1.10		22
11.3	even	5		814.2.h.c.75.7	✓	44
11.4	even	5		814.2.h.c.445.7	yes	44
11.10	odd	2		8954.2.a.bs.1.10		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
814.2.h.c.75.7	✓	44	11.3	even	5
814.2.h.c.445.7	yes	44	11.4	even	5
8954.2.a.bs.1.10		22	11.10	odd	2
8954.2.a.bt.1.10		22	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8954 = 2 \cdot 11^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8954.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.123112 q^{3} +1.00000 q^{4} -3.89599 q^{5} +0.123112 q^{6} +1.02869 q^{7} +1.00000 q^{8} -2.98484 q^{9} -3.89599 q^{10} +0.123112 q^{12} -0.0128920 q^{13} +1.02869 q^{14} -0.479643 q^{15} +1.00000 q^{16} -6.03448 q^{17} -2.98484 q^{18} +4.80831 q^{19} -3.89599 q^{20} +0.126644 q^{21} +1.10089 q^{23} +0.123112 q^{24} +10.1787 q^{25} -0.0128920 q^{26} -0.736806 q^{27} +1.02869 q^{28} +2.66171 q^{29} -0.479643 q^{30} -9.54125 q^{31} +1.00000 q^{32} -6.03448 q^{34} -4.00776 q^{35} -2.98484 q^{36} -1.00000 q^{37} +4.80831 q^{38} -0.00158717 q^{39} -3.89599 q^{40} +1.78744 q^{41} +0.126644 q^{42} +0.161529 q^{43} +11.6289 q^{45} +1.10089 q^{46} -5.43218 q^{47} +0.123112 q^{48} -5.94180 q^{49} +10.1787 q^{50} -0.742917 q^{51} -0.0128920 q^{52} -8.50735 q^{53} -0.736806 q^{54} +1.02869 q^{56} +0.591961 q^{57} +2.66171 q^{58} +8.09590 q^{59} -0.479643 q^{60} +5.07782 q^{61} -9.54125 q^{62} -3.07048 q^{63} +1.00000 q^{64} +0.0502272 q^{65} +5.89099 q^{67} -6.03448 q^{68} +0.135533 q^{69} -4.00776 q^{70} -11.7721 q^{71} -2.98484 q^{72} +0.543581 q^{73} -1.00000 q^{74} +1.25312 q^{75} +4.80831 q^{76} -0.00158717 q^{78} +14.2093 q^{79} -3.89599 q^{80} +8.86382 q^{81} +1.78744 q^{82} +1.59761 q^{83} +0.126644 q^{84} +23.5103 q^{85} +0.161529 q^{86} +0.327688 q^{87} -18.5606 q^{89} +11.6289 q^{90} -0.0132619 q^{91} +1.10089 q^{92} -1.17464 q^{93} -5.43218 q^{94} -18.7331 q^{95} +0.123112 q^{96} +12.2796 q^{97} -5.94180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 22 q^{2} + 5 q^{3} + 22 q^{4} + 8 q^{5} + 5 q^{6} + 7 q^{7} + 22 q^{8} + 35 q^{9} + 8 q^{10} + 5 q^{12} + 9 q^{13} + 7 q^{14} + 16 q^{15} + 22 q^{16} + 19 q^{17} + 35 q^{18} + 8 q^{20} + 15 q^{21}+ \cdots + 57 q^{98}+O(q^{100}) \) 22 * q + 22 * q^2 + 5 * q^3 + 22 * q^4 + 8 * q^5 + 5 * q^6 + 7 * q^7 + 22 * q^8 + 35 * q^9 + 8 * q^10 + 5 * q^12 + 9 * q^13 + 7 * q^14 + 16 * q^15 + 22 * q^16 + 19 * q^17 + 35 * q^18 + 8 * q^20 + 15 * q^21 + 18 * q^23 + 5 * q^24 + 26 * q^25 + 9 * q^26 + 17 * q^27 + 7 * q^28 + 6 * q^29 + 16 * q^30 + 32 * q^31 + 22 * q^32 + 19 * q^34 + 14 * q^35 + 35 * q^36 - 22 * q^37 - 2 * q^39 + 8 * q^40 + 10 * q^41 + 15 * q^42 - 2 * q^43 + 83 * q^45 + 18 * q^46 - 16 * q^47 + 5 * q^48 + 57 * q^49 + 26 * q^50 - 30 * q^51 + 9 * q^52 + 6 * q^53 + 17 * q^54 + 7 * q^56 + 66 * q^57 + 6 * q^58 + 51 * q^59 + 16 * q^60 + 32 * q^62 - 3 * q^63 + 22 * q^64 + 34 * q^65 + 22 * q^67 + 19 * q^68 + 27 * q^69 + 14 * q^70 + 24 * q^71 + 35 * q^72 - 8 * q^73 - 22 * q^74 + 16 * q^75 - 2 * q^78 + 39 * q^79 + 8 * q^80 + 86 * q^81 + 10 * q^82 - 18 * q^83 + 15 * q^84 + 34 * q^85 - 2 * q^86 - 10 * q^87 + 45 * q^89 + 83 * q^90 + 8 * q^91 + 18 * q^92 + 33 * q^93 - 16 * q^94 - 39 * q^95 + 5 * q^96 + 47 * q^97 + 57 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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