LMFDB - Embedded newform 8670.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$69.2302985525$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -0.347296 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.18479 q^{11} -1.00000 q^{12} +1.53209 q^{13} +0.347296 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -3.69459 q^{19} +1.00000 q^{20} +0.347296 q^{21} -1.18479 q^{22} -2.22668 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.53209 q^{26} -1.00000 q^{27} -0.347296 q^{28} +4.45336 q^{29} +1.00000 q^{30} -3.75877 q^{31} -1.00000 q^{32} -1.18479 q^{33} -0.347296 q^{35} +1.00000 q^{36} +10.5175 q^{37} +3.69459 q^{38} -1.53209 q^{39} -1.00000 q^{40} -9.82295 q^{41} -0.347296 q^{42} -3.06418 q^{43} +1.18479 q^{44} +1.00000 q^{45} +2.22668 q^{46} +1.32770 q^{47} -1.00000 q^{48} -6.87939 q^{49} -1.00000 q^{50} +1.53209 q^{52} +7.18479 q^{53} +1.00000 q^{54} +1.18479 q^{55} +0.347296 q^{56} +3.69459 q^{57} -4.45336 q^{58} +3.55438 q^{59} -1.00000 q^{60} +3.06418 q^{61} +3.75877 q^{62} -0.347296 q^{63} +1.00000 q^{64} +1.53209 q^{65} +1.18479 q^{66} -2.24123 q^{67} +2.22668 q^{69} +0.347296 q^{70} +2.36959 q^{71} -1.00000 q^{72} +4.00000 q^{73} -10.5175 q^{74} -1.00000 q^{75} -3.69459 q^{76} -0.411474 q^{77} +1.53209 q^{78} +13.1925 q^{79} +1.00000 q^{80} +1.00000 q^{81} +9.82295 q^{82} -0.822948 q^{83} +0.347296 q^{84} +3.06418 q^{86} -4.45336 q^{87} -1.18479 q^{88} +3.41147 q^{89} -1.00000 q^{90} -0.532089 q^{91} -2.22668 q^{92} +3.75877 q^{93} -1.32770 q^{94} -3.69459 q^{95} +1.00000 q^{96} +4.32501 q^{97} +6.87939 q^{98} +1.18479 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{12} - 3 q^{15} + 3 q^{16} - 3 q^{18} - 9 q^{19} + 3 q^{20} + 3 q^{24} + 3 q^{25} - 3 q^{27} + 3 q^{30} - 3 q^{32}+ \cdots + 15 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 - 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^12 - 3 * q^15 + 3 * q^16 - 3 * q^18 - 9 * q^19 + 3 * q^20 + 3 * q^24 + 3 * q^25 - 3 * q^27 + 3 * q^30 - 3 * q^32 + 3 * q^36 + 9 * q^37 + 9 * q^38 - 3 * q^40 - 9 * q^41 + 3 * q^45 - 3 * q^48 - 15 * q^49 - 3 * q^50 + 18 * q^53 + 3 * q^54 + 9 * q^57 - 3 * q^60 + 3 * q^64 - 18 * q^67 - 3 * q^72 + 12 * q^73 - 9 * q^74 - 3 * q^75 - 9 * q^76 + 9 * q^77 + 12 * q^79 + 3 * q^80 + 3 * q^81 + 9 * q^82 + 18 * q^83 - 3 * q^90 + 3 * q^91 - 9 * q^95 + 3 * q^96 + 18 * q^97 + 15 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−0.347296	−0.131266	−0.0656328	−	0.997844i	$-0.520907\pi$
$7$	−0.347296	−0.131266	−0.0656328	+	0.997844i	$0.520907\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.18479	0.357228	0.178614	−	0.983919i	$-0.442839\pi$
$11$	1.18479	0.357228	0.178614	+	0.983919i	$0.442839\pi$
$12$	−1.00000	−0.288675
$13$	1.53209	0.424925	0.212463	−	0.977169i	$-0.431852\pi$
$13$	1.53209	0.424925	0.212463	+	0.977169i	$0.431852\pi$
$14$	0.347296	0.0928189
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	−3.69459	−0.847598	−0.423799	−	0.905756i	$-0.639304\pi$
$19$	−3.69459	−0.847598	−0.423799	+	0.905756i	$0.639304\pi$
$20$	1.00000	0.223607
$21$	0.347296	0.0757863
$22$	−1.18479	−0.252599
$23$	−2.22668	−0.464295	−0.232148	−	0.972681i	$-0.574575\pi$
$23$	−2.22668	−0.464295	−0.232148	+	0.972681i	$0.574575\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−1.53209	−0.300467
$27$	−1.00000	−0.192450
$28$	−0.347296	−0.0656328
$29$	4.45336	0.826969	0.413484	−	0.910511i	$-0.364312\pi$
$29$	4.45336	0.826969	0.413484	+	0.910511i	$0.364312\pi$
$30$	1.00000	0.182574
$31$	−3.75877	−0.675095	−0.337548	−	0.941308i	$-0.609597\pi$
$31$	−3.75877	−0.675095	−0.337548	+	0.941308i	$0.609597\pi$
$32$	−1.00000	−0.176777
$33$	−1.18479	−0.206246
$34$	0	0
$35$	−0.347296	−0.0587038
$36$	1.00000	0.166667
$37$	10.5175	1.72907	0.864537	−	0.502570i	$-0.167612\pi$
$37$	10.5175	1.72907	0.864537	+	0.502570i	$0.167612\pi$
$38$	3.69459	0.599342
$39$	−1.53209	−0.245331
$40$	−1.00000	−0.158114
$41$	−9.82295	−1.53409	−0.767043	−	0.641595i	$-0.778273\pi$
$41$	−9.82295	−1.53409	−0.767043	+	0.641595i	$0.778273\pi$
$42$	−0.347296	−0.0535890
$43$	−3.06418	−0.467283	−0.233641	−	0.972323i	$-0.575064\pi$
$43$	−3.06418	−0.467283	−0.233641	+	0.972323i	$0.575064\pi$
$44$	1.18479	0.178614
$45$	1.00000	0.149071
$46$	2.22668	0.328306
$47$	1.32770	0.193664	0.0968322	−	0.995301i	$-0.469129\pi$
$47$	1.32770	0.193664	0.0968322	+	0.995301i	$0.469129\pi$
$48$	−1.00000	−0.144338
$49$	−6.87939	−0.982769
$50$	−1.00000	−0.141421
$51$	0	0
$52$	1.53209	0.212463
$53$	7.18479	0.986907	0.493454	−	0.869772i	$-0.335734\pi$
$53$	7.18479	0.986907	0.493454	+	0.869772i	$0.335734\pi$
$54$	1.00000	0.136083
$55$	1.18479	0.159757
$56$	0.347296	0.0464094
$57$	3.69459	0.489361
$58$	−4.45336	−0.584755
$59$	3.55438	0.462741	0.231370	−	0.972866i	$-0.425679\pi$
$59$	3.55438	0.462741	0.231370	+	0.972866i	$0.425679\pi$
$60$	−1.00000	−0.129099
$61$	3.06418	0.392328	0.196164	−	0.980571i	$-0.437152\pi$
$61$	3.06418	0.392328	0.196164	+	0.980571i	$0.437152\pi$
$62$	3.75877	0.477364
$63$	−0.347296	−0.0437552
$64$	1.00000	0.125000
$65$	1.53209	0.190032
$66$	1.18479	0.145838
$67$	−2.24123	−0.273810	−0.136905	−	0.990584i	$-0.543715\pi$
$67$	−2.24123	−0.273810	−0.136905	+	0.990584i	$0.543715\pi$
$68$	0	0
$69$	2.22668	0.268061
$70$	0.347296	0.0415099
$71$	2.36959	0.281218	0.140609	−	0.990065i	$-0.455094\pi$
$71$	2.36959	0.281218	0.140609	+	0.990065i	$0.455094\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−10.5175	−1.22264
$75$	−1.00000	−0.115470
$76$	−3.69459	−0.423799
$77$	−0.411474	−0.0468918
$78$	1.53209	0.173475
$79$	13.1925	1.48428	0.742138	−	0.670247i	$-0.233812\pi$
$79$	13.1925	1.48428	0.742138	+	0.670247i	$0.233812\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	9.82295	1.08476
$83$	−0.822948	−0.0903303	−0.0451652	−	0.998980i	$-0.514381\pi$
$83$	−0.822948	−0.0903303	−0.0451652	+	0.998980i	$0.514381\pi$
$84$	0.347296	0.0378931
$85$	0	0
$86$	3.06418	0.330419
$87$	−4.45336	−0.477451
$88$	−1.18479	−0.126299
$89$	3.41147	0.361616	0.180808	−	0.983518i	$-0.442129\pi$
$89$	3.41147	0.361616	0.180808	+	0.983518i	$0.442129\pi$
$90$	−1.00000	−0.105409
$91$	−0.532089	−0.0557781
$92$	−2.22668	−0.232148
$93$	3.75877	0.389766
$94$	−1.32770	−0.136941
$95$	−3.69459	−0.379057
$96$	1.00000	0.102062
$97$	4.32501	0.439138	0.219569	−	0.975597i	$-0.429535\pi$
$97$	4.32501	0.439138	0.219569	+	0.975597i	$0.429535\pi$
$98$	6.87939	0.694923
$99$	1.18479	0.119076
$100$	1.00000	0.100000
$101$	5.27631	0.525013	0.262506	−	0.964930i	$-0.415451\pi$
$101$	5.27631	0.525013	0.262506	+	0.964930i	$0.415451\pi$
$102$	0	0
$103$	−3.69459	−0.364039	−0.182020	−	0.983295i	$-0.558263\pi$
$103$	−3.69459	−0.364039	−0.182020	+	0.983295i	$0.558263\pi$
$104$	−1.53209	−0.150234
$105$	0.347296	0.0338927
$106$	−7.18479	−0.697849
$107$	−4.73917	−0.458153	−0.229076	−	0.973408i	$-0.573571\pi$
$107$	−4.73917	−0.458153	−0.229076	+	0.973408i	$0.573571\pi$
$108$	−1.00000	−0.0962250
$109$	−7.27631	−0.696944	−0.348472	−	0.937319i	$-0.613299\pi$
$109$	−7.27631	−0.696944	−0.348472	+	0.937319i	$0.613299\pi$
$110$	−1.18479	−0.112966
$111$	−10.5175	−0.998281
$112$	−0.347296	−0.0328164
$113$	7.54664	0.709928	0.354964	−	0.934880i	$-0.384493\pi$
$113$	7.54664	0.709928	0.354964	+	0.934880i	$0.384493\pi$
$114$	−3.69459	−0.346030
$115$	−2.22668	−0.207639
$116$	4.45336	0.413484
$117$	1.53209	0.141642
$118$	−3.55438	−0.327207
$119$	0	0
$120$	1.00000	0.0912871
$121$	−9.59627	−0.872388
$122$	−3.06418	−0.277418
$123$	9.82295	0.885705
$124$	−3.75877	−0.337548
$125$	1.00000	0.0894427
$126$	0.347296	0.0309396
$127$	−11.3405	−1.00631	−0.503153	−	0.864197i	$-0.667827\pi$
$127$	−11.3405	−1.00631	−0.503153	+	0.864197i	$0.667827\pi$
$128$	−1.00000	−0.0883883
$129$	3.06418	0.269786
$130$	−1.53209	−0.134373
$131$	2.22668	0.194546	0.0972730	−	0.995258i	$-0.468988\pi$
$131$	2.22668	0.194546	0.0972730	+	0.995258i	$0.468988\pi$
$132$	−1.18479	−0.103123
$133$	1.28312	0.111260
$134$	2.24123	0.193613
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	5.17705	0.442305	0.221153	−	0.975239i	$-0.429018\pi$
$137$	5.17705	0.442305	0.221153	+	0.975239i	$0.429018\pi$
$138$	−2.22668	−0.189548
$139$	3.69459	0.313371	0.156686	−	0.987649i	$-0.449919\pi$
$139$	3.69459	0.313371	0.156686	+	0.987649i	$0.449919\pi$
$140$	−0.347296	−0.0293519
$141$	−1.32770	−0.111812
$142$	−2.36959	−0.198851
$143$	1.81521	0.151795
$144$	1.00000	0.0833333
$145$	4.45336	0.369832
$146$	−4.00000	−0.331042
$147$	6.87939	0.567402
$148$	10.5175	0.864537
$149$	2.36959	0.194124	0.0970620	−	0.995278i	$-0.469055\pi$
$149$	2.36959	0.194124	0.0970620	+	0.995278i	$0.469055\pi$
$150$	1.00000	0.0816497
$151$	2.49794	0.203280	0.101640	−	0.994821i	$-0.467591\pi$
$151$	2.49794	0.203280	0.101640	+	0.994821i	$0.467591\pi$
$152$	3.69459	0.299671
$153$	0	0
$154$	0.411474	0.0331575
$155$	−3.75877	−0.301912
$156$	−1.53209	−0.122665
$157$	22.8999	1.82761	0.913806	−	0.406150i	$-0.133129\pi$
$157$	22.8999	1.82761	0.913806	+	0.406150i	$0.133129\pi$
$158$	−13.1925	−1.04954
$159$	−7.18479	−0.569791
$160$	−1.00000	−0.0790569
$161$	0.773318	0.0609460
$162$	−1.00000	−0.0785674
$163$	18.4688	1.44659	0.723296	−	0.690538i	$-0.242626\pi$
$163$	18.4688	1.44659	0.723296	+	0.690538i	$0.242626\pi$
$164$	−9.82295	−0.767043
$165$	−1.18479	−0.0922360
$166$	0.822948	0.0638732
$167$	1.18479	0.0916820	0.0458410	−	0.998949i	$-0.485403\pi$
$167$	1.18479	0.0916820	0.0458410	+	0.998949i	$0.485403\pi$
$168$	−0.347296	−0.0267945
$169$	−10.6527	−0.819439
$170$	0	0
$171$	−3.69459	−0.282533
$172$	−3.06418	−0.233641
$173$	22.5963	1.71796	0.858981	−	0.512007i	$-0.171098\pi$
$173$	22.5963	1.71796	0.858981	+	0.512007i	$0.171098\pi$
$174$	4.45336	0.337609
$175$	−0.347296	−0.0262531
$176$	1.18479	0.0893071
$177$	−3.55438	−0.267163
$178$	−3.41147	−0.255701
$179$	−11.7050	−0.874874	−0.437437	−	0.899249i	$-0.644114\pi$
$179$	−11.7050	−0.874874	−0.437437	+	0.899249i	$0.644114\pi$
$180$	1.00000	0.0745356
$181$	4.61081	0.342719	0.171360	−	0.985209i	$-0.445184\pi$
$181$	4.61081	0.342719	0.171360	+	0.985209i	$0.445184\pi$
$182$	0.532089	0.0394411
$183$	−3.06418	−0.226511
$184$	2.22668	0.164153
$185$	10.5175	0.773265
$186$	−3.75877	−0.275606
$187$	0	0
$188$	1.32770	0.0968322
$189$	0.347296	0.0252621
$190$	3.69459	0.268034
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	−9.47296	−0.681879	−0.340939	−	0.940085i	$-0.610745\pi$
$193$	−9.47296	−0.681879	−0.340939	+	0.940085i	$0.610745\pi$
$194$	−4.32501	−0.310517
$195$	−1.53209	−0.109715
$196$	−6.87939	−0.491385
$197$	−0.0760373	−0.00541744	−0.00270872	−	0.999996i	$-0.500862\pi$
$197$	−0.0760373	−0.00541744	−0.00270872	+	0.999996i	$0.500862\pi$
$198$	−1.18479	−0.0841995
$199$	0.807467	0.0572398	0.0286199	−	0.999590i	$-0.490889\pi$
$199$	0.807467	0.0572398	0.0286199	+	0.999590i	$0.490889\pi$
$200$	−1.00000	−0.0707107
$201$	2.24123	0.158084
$202$	−5.27631	−0.371240
$203$	−1.54664	−0.108553
$204$	0	0
$205$	−9.82295	−0.686064
$206$	3.69459	0.257414
$207$	−2.22668	−0.154765
$208$	1.53209	0.106231
$209$	−4.37733	−0.302786
$210$	−0.347296	−0.0239657
$211$	−18.7324	−1.28959	−0.644795	−	0.764356i	$-0.723057\pi$
$211$	−18.7324	−1.28959	−0.644795	+	0.764356i	$0.723057\pi$
$212$	7.18479	0.493454
$213$	−2.36959	−0.162361
$214$	4.73917	0.323963
$215$	−3.06418	−0.208975
$216$	1.00000	0.0680414
$217$	1.30541	0.0886168
$218$	7.27631	0.492814
$219$	−4.00000	−0.270295
$220$	1.18479	0.0798787
$221$	0	0
$222$	10.5175	0.705891
$223$	4.51249	0.302179	0.151089	−	0.988520i	$-0.451722\pi$
$223$	4.51249	0.302179	0.151089	+	0.988520i	$0.451722\pi$
$224$	0.347296	0.0232047
$225$	1.00000	0.0666667
$226$	−7.54664	−0.501995
$227$	−8.90673	−0.591160	−0.295580	−	0.955318i	$-0.595513\pi$
$227$	−8.90673	−0.591160	−0.295580	+	0.955318i	$0.595513\pi$
$228$	3.69459	0.244680
$229$	−24.4688	−1.61695	−0.808473	−	0.588533i	$-0.799706\pi$
$229$	−24.4688	−1.61695	−0.808473	+	0.588533i	$0.799706\pi$
$230$	2.22668	0.146823
$231$	0.411474	0.0270730
$232$	−4.45336	−0.292378
$233$	27.2918	1.78794	0.893972	−	0.448122i	$-0.147907\pi$
$233$	27.2918	1.78794	0.893972	+	0.448122i	$0.147907\pi$
$234$	−1.53209	−0.100156
$235$	1.32770	0.0866093
$236$	3.55438	0.231370
$237$	−13.1925	−0.856947
$238$	0	0
$239$	−17.2763	−1.11751	−0.558756	−	0.829332i	$-0.688721\pi$
$239$	−17.2763	−1.11751	−0.558756	+	0.829332i	$0.688721\pi$
$240$	−1.00000	−0.0645497
$241$	−6.95811	−0.448211	−0.224106	−	0.974565i	$-0.571946\pi$
$241$	−6.95811	−0.448211	−0.224106	+	0.974565i	$0.571946\pi$
$242$	9.59627	0.616871
$243$	−1.00000	−0.0641500
$244$	3.06418	0.196164
$245$	−6.87939	−0.439508
$246$	−9.82295	−0.626288
$247$	−5.66044	−0.360165
$248$	3.75877	0.238682
$249$	0.822948	0.0521522
$250$	−1.00000	−0.0632456
$251$	16.6287	1.04959	0.524796	−	0.851228i	$-0.324142\pi$
$251$	16.6287	1.04959	0.524796	+	0.851228i	$0.324142\pi$
$252$	−0.347296	−0.0218776
$253$	−2.63816	−0.165859
$254$	11.3405	0.711566
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.19253	−0.573414	−0.286707	−	0.958018i	$-0.592561\pi$
$257$	−9.19253	−0.573414	−0.286707	+	0.958018i	$0.592561\pi$
$258$	−3.06418	−0.190767
$259$	−3.65270	−0.226968
$260$	1.53209	0.0950161
$261$	4.45336	0.275656
$262$	−2.22668	−0.137565
$263$	11.1334	0.686515	0.343258	−	0.939241i	$-0.388470\pi$
$263$	11.1334	0.686515	0.343258	+	0.939241i	$0.388470\pi$
$264$	1.18479	0.0729189
$265$	7.18479	0.441358
$266$	−1.28312	−0.0786730
$267$	−3.41147	−0.208779
$268$	−2.24123	−0.136905
$269$	10.7392	0.654779	0.327389	−	0.944890i	$-0.393831\pi$
$269$	10.7392	0.654779	0.327389	+	0.944890i	$0.393831\pi$
$270$	1.00000	0.0608581
$271$	−10.3250	−0.627200	−0.313600	−	0.949555i	$-0.601535\pi$
$271$	−10.3250	−0.627200	−0.313600	+	0.949555i	$0.601535\pi$
$272$	0	0
$273$	0.532089	0.0322035
$274$	−5.17705	−0.312757
$275$	1.18479	0.0714457
$276$	2.22668	0.134030
$277$	23.8280	1.43169	0.715843	−	0.698261i	$-0.246043\pi$
$277$	23.8280	1.43169	0.715843	+	0.698261i	$0.246043\pi$
$278$	−3.69459	−0.221587
$279$	−3.75877	−0.225032
$280$	0.347296	0.0207549
$281$	−23.2003	−1.38401	−0.692006	−	0.721892i	$-0.743273\pi$
$281$	−23.2003	−1.38401	−0.692006	+	0.721892i	$0.743273\pi$
$282$	1.32770	0.0790631
$283$	−4.58172	−0.272355	−0.136177	−	0.990684i	$-0.543482\pi$
$283$	−4.58172	−0.272355	−0.136177	+	0.990684i	$0.543482\pi$
$284$	2.36959	0.140609
$285$	3.69459	0.218849
$286$	−1.81521	−0.107335
$287$	3.41147	0.201373
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	−4.45336	−0.261510
$291$	−4.32501	−0.253536
$292$	4.00000	0.234082
$293$	25.7219	1.50269	0.751346	−	0.659909i	$-0.229405\pi$
$293$	25.7219	1.50269	0.751346	+	0.659909i	$0.229405\pi$
$294$	−6.87939	−0.401214
$295$	3.55438	0.206944
$296$	−10.5175	−0.611320
$297$	−1.18479	−0.0687486
$298$	−2.36959	−0.137266
$299$	−3.41147	−0.197291
$300$	−1.00000	−0.0577350
$301$	1.06418	0.0613382
$302$	−2.49794	−0.143740
$303$	−5.27631	−0.303116
$304$	−3.69459	−0.211899
$305$	3.06418	0.175454
$306$	0	0
$307$	12.5526	0.716416	0.358208	−	0.933642i	$-0.383388\pi$
$307$	12.5526	0.716416	0.358208	+	0.933642i	$0.383388\pi$
$308$	−0.411474	−0.0234459
$309$	3.69459	0.210178
$310$	3.75877	0.213484
$311$	10.1676	0.576549	0.288275	−	0.957548i	$-0.406918\pi$
$311$	10.1676	0.576549	0.288275	+	0.957548i	$0.406918\pi$
$312$	1.53209	0.0867375
$313$	10.3250	0.583604	0.291802	−	0.956479i	$-0.405745\pi$
$313$	10.3250	0.583604	0.291802	+	0.956479i	$0.405745\pi$
$314$	−22.8999	−1.29232
$315$	−0.347296	−0.0195679
$316$	13.1925	0.742138
$317$	26.5621	1.49188	0.745939	−	0.666015i	$-0.232001\pi$
$317$	26.5621	1.49188	0.745939	+	0.666015i	$0.232001\pi$
$318$	7.18479	0.402903
$319$	5.27631	0.295417
$320$	1.00000	0.0559017
$321$	4.73917	0.264515
$322$	−0.773318	−0.0430953
$323$	0	0
$324$	1.00000	0.0555556
$325$	1.53209	0.0849850
$326$	−18.4688	−1.02289
$327$	7.27631	0.402381
$328$	9.82295	0.542382
$329$	−0.461104	−0.0254215
$330$	1.18479	0.0652207
$331$	−15.8203	−0.869560	−0.434780	−	0.900537i	$-0.643174\pi$
$331$	−15.8203	−0.869560	−0.434780	+	0.900537i	$0.643174\pi$
$332$	−0.822948	−0.0451652
$333$	10.5175	0.576358
$334$	−1.18479	−0.0648290
$335$	−2.24123	−0.122451
$336$	0.347296	0.0189466
$337$	−21.7452	−1.18453	−0.592267	−	0.805742i	$-0.701767\pi$
$337$	−21.7452	−1.18453	−0.592267	+	0.805742i	$0.701767\pi$
$338$	10.6527	0.579431
$339$	−7.54664	−0.409877
$340$	0	0
$341$	−4.45336	−0.241163
$342$	3.69459	0.199781
$343$	4.82026	0.260270
$344$	3.06418	0.165209
$345$	2.22668	0.119881
$346$	−22.5963	−1.21478
$347$	−28.8384	−1.54813	−0.774064	−	0.633107i	$-0.781779\pi$
$347$	−28.8384	−1.54813	−0.774064	+	0.633107i	$0.781779\pi$
$348$	−4.45336	−0.238725
$349$	6.55262	0.350754	0.175377	−	0.984501i	$-0.443886\pi$
$349$	6.55262	0.350754	0.175377	+	0.984501i	$0.443886\pi$
$350$	0.347296	0.0185638
$351$	−1.53209	−0.0817769
$352$	−1.18479	−0.0631497
$353$	−3.72967	−0.198511	−0.0992553	−	0.995062i	$-0.531646\pi$
$353$	−3.72967	−0.198511	−0.0992553	+	0.995062i	$0.531646\pi$
$354$	3.55438	0.188913
$355$	2.36959	0.125765
$356$	3.41147	0.180808
$357$	0	0
$358$	11.7050	0.618630
$359$	−14.9067	−0.786747	−0.393373	−	0.919379i	$-0.628692\pi$
$359$	−14.9067	−0.786747	−0.393373	+	0.919379i	$0.628692\pi$
$360$	−1.00000	−0.0527046
$361$	−5.34998	−0.281578
$362$	−4.61081	−0.242339
$363$	9.59627	0.503673
$364$	−0.532089	−0.0278890
$365$	4.00000	0.209370
$366$	3.06418	0.160167
$367$	0.0469415	0.00245033	0.00122516	−	0.999999i	$-0.499610\pi$
$367$	0.0469415	0.00245033	0.00122516	+	0.999999i	$0.499610\pi$
$368$	−2.22668	−0.116074
$369$	−9.82295	−0.511362
$370$	−10.5175	−0.546781
$371$	−2.49525	−0.129547
$372$	3.75877	0.194883
$373$	13.3432	0.690884	0.345442	−	0.938440i	$-0.387729\pi$
$373$	13.3432	0.690884	0.345442	+	0.938440i	$0.387729\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−1.32770	−0.0684707
$377$	6.82295	0.351400
$378$	−0.347296	−0.0178630
$379$	25.9026	1.33053	0.665264	−	0.746608i	$-0.268319\pi$
$379$	25.9026	1.33053	0.665264	+	0.746608i	$0.268319\pi$
$380$	−3.69459	−0.189529
$381$	11.3405	0.580991
$382$	−12.0000	−0.613973
$383$	−9.38507	−0.479555	−0.239777	−	0.970828i	$-0.577074\pi$
$383$	−9.38507	−0.479555	−0.239777	+	0.970828i	$0.577074\pi$
$384$	1.00000	0.0510310
$385$	−0.411474	−0.0209707
$386$	9.47296	0.482161
$387$	−3.06418	−0.155761
$388$	4.32501	0.219569
$389$	29.3756	1.48940	0.744700	−	0.667399i	$-0.232592\pi$
$389$	29.3756	1.48940	0.744700	+	0.667399i	$0.232592\pi$
$390$	1.53209	0.0775803
$391$	0	0
$392$	6.87939	0.347461
$393$	−2.22668	−0.112321
$394$	0.0760373	0.00383071
$395$	13.1925	0.663788
$396$	1.18479	0.0595381
$397$	−4.76382	−0.239089	−0.119545	−	0.992829i	$-0.538143\pi$
$397$	−4.76382	−0.239089	−0.119545	+	0.992829i	$0.538143\pi$
$398$	−0.807467	−0.0404746
$399$	−1.28312	−0.0642363
$400$	1.00000	0.0500000
$401$	13.7219	0.685241	0.342620	−	0.939474i	$-0.388686\pi$
$401$	13.7219	0.685241	0.342620	+	0.939474i	$0.388686\pi$
$402$	−2.24123	−0.111782
$403$	−5.75877	−0.286865
$404$	5.27631	0.262506
$405$	1.00000	0.0496904
$406$	1.54664	0.0767583
$407$	12.4611	0.617674
$408$	0	0
$409$	−30.2131	−1.49394	−0.746970	−	0.664858i	$-0.768492\pi$
$409$	−30.2131	−1.49394	−0.746970	+	0.664858i	$0.768492\pi$
$410$	9.82295	0.485121
$411$	−5.17705	−0.255365
$412$	−3.69459	−0.182020
$413$	−1.23442	−0.0607420
$414$	2.22668	0.109435
$415$	−0.822948	−0.0403969
$416$	−1.53209	−0.0751168
$417$	−3.69459	−0.180925
$418$	4.37733	0.214102
$419$	32.3259	1.57923	0.789613	−	0.613605i	$-0.210281\pi$
$419$	32.3259	1.57923	0.789613	+	0.613605i	$0.210281\pi$
$420$	0.347296	0.0169463
$421$	−11.7196	−0.571177	−0.285588	−	0.958352i	$-0.592189\pi$
$421$	−11.7196	−0.571177	−0.285588	+	0.958352i	$0.592189\pi$
$422$	18.7324	0.911877
$423$	1.32770	0.0645548
$424$	−7.18479	−0.348924
$425$	0	0
$426$	2.36959	0.114807
$427$	−1.06418	−0.0514992
$428$	−4.73917	−0.229076
$429$	−1.81521	−0.0876390
$430$	3.06418	0.147768
$431$	28.5526	1.37533	0.687666	−	0.726027i	$-0.258635\pi$
$431$	28.5526	1.37533	0.687666	+	0.726027i	$0.258635\pi$
$432$	−1.00000	−0.0481125
$433$	−9.78787	−0.470375	−0.235187	−	0.971950i	$-0.575570\pi$
$433$	−9.78787	−0.470375	−0.235187	+	0.971950i	$0.575570\pi$
$434$	−1.30541	−0.0626616
$435$	−4.45336	−0.213522
$436$	−7.27631	−0.348472
$437$	8.22668	0.393536
$438$	4.00000	0.191127
$439$	18.2567	0.871345	0.435673	−	0.900105i	$-0.356511\pi$
$439$	18.2567	0.871345	0.435673	+	0.900105i	$0.356511\pi$
$440$	−1.18479	−0.0564828
$441$	−6.87939	−0.327590
$442$	0	0
$443$	−41.4748	−1.97053	−0.985264	−	0.171039i	$-0.945288\pi$
$443$	−41.4748	−1.97053	−0.985264	+	0.171039i	$0.945288\pi$
$444$	−10.5175	−0.499140
$445$	3.41147	0.161719
$446$	−4.51249	−0.213673
$447$	−2.36959	−0.112078
$448$	−0.347296	−0.0164082
$449$	6.57161	0.310134	0.155067	−	0.987904i	$-0.450441\pi$
$449$	6.57161	0.310134	0.155067	+	0.987904i	$0.450441\pi$
$450$	−1.00000	−0.0471405
$451$	−11.6382	−0.548019
$452$	7.54664	0.354964
$453$	−2.49794	−0.117363
$454$	8.90673	0.418013
$455$	−0.532089	−0.0249447
$456$	−3.69459	−0.173015
$457$	−6.08378	−0.284587	−0.142294	−	0.989824i	$-0.545448\pi$
$457$	−6.08378	−0.284587	−0.142294	+	0.989824i	$0.545448\pi$
$458$	24.4688	1.14335
$459$	0	0
$460$	−2.22668	−0.103820
$461$	19.2608	0.897066	0.448533	−	0.893766i	$-0.351947\pi$
$461$	19.2608	0.897066	0.448533	+	0.893766i	$0.351947\pi$
$462$	−0.411474	−0.0191435
$463$	25.4216	1.18144	0.590720	−	0.806876i	$-0.298844\pi$
$463$	25.4216	1.18144	0.590720	+	0.806876i	$0.298844\pi$
$464$	4.45336	0.206742
$465$	3.75877	0.174309
$466$	−27.2918	−1.26427
$467$	20.2175	0.935555	0.467777	−	0.883846i	$-0.345055\pi$
$467$	20.2175	0.935555	0.467777	+	0.883846i	$0.345055\pi$
$468$	1.53209	0.0708208
$469$	0.778371	0.0359418
$470$	−1.32770	−0.0612420
$471$	−22.8999	−1.05517
$472$	−3.55438	−0.163603
$473$	−3.63041	−0.166927
$474$	13.1925	0.605953
$475$	−3.69459	−0.169520
$476$	0	0
$477$	7.18479	0.328969
$478$	17.2763	0.790200
$479$	26.3696	1.20486	0.602429	−	0.798173i	$-0.294200\pi$
$479$	26.3696	1.20486	0.602429	+	0.798173i	$0.294200\pi$
$480$	1.00000	0.0456435
$481$	16.1138	0.734726
$482$	6.95811	0.316933
$483$	−0.773318	−0.0351872
$484$	−9.59627	−0.436194
$485$	4.32501	0.196388
$486$	1.00000	0.0453609
$487$	20.8188	0.943391	0.471696	−	0.881761i	$-0.343642\pi$
$487$	20.8188	0.943391	0.471696	+	0.881761i	$0.343642\pi$
$488$	−3.06418	−0.138709
$489$	−18.4688	−0.835190
$490$	6.87939	0.310779
$491$	25.5262	1.15198	0.575991	−	0.817456i	$-0.304616\pi$
$491$	25.5262	1.15198	0.575991	+	0.817456i	$0.304616\pi$
$492$	9.82295	0.442853
$493$	0	0
$494$	5.66044	0.254675
$495$	1.18479	0.0532525
$496$	−3.75877	−0.168774
$497$	−0.822948	−0.0369143
$498$	−0.822948	−0.0368772
$499$	−31.3688	−1.40426	−0.702129	−	0.712049i	$-0.747767\pi$
$499$	−31.3688	−1.40426	−0.702129	+	0.712049i	$0.747767\pi$
$500$	1.00000	0.0447214
$501$	−1.18479	−0.0529326
$502$	−16.6287	−0.742173
$503$	4.74834	0.211718	0.105859	−	0.994381i	$-0.466241\pi$
$503$	4.74834	0.211718	0.105859	+	0.994381i	$0.466241\pi$
$504$	0.347296	0.0154698
$505$	5.27631	0.234793
$506$	2.63816	0.117280
$507$	10.6527	0.473103
$508$	−11.3405	−0.503153
$509$	24.6709	1.09352	0.546759	−	0.837290i	$-0.315862\pi$
$509$	24.6709	1.09352	0.546759	+	0.837290i	$0.315862\pi$
$510$	0	0
$511$	−1.38919	−0.0614539
$512$	−1.00000	−0.0441942
$513$	3.69459	0.163120
$514$	9.19253	0.405465
$515$	−3.69459	−0.162803
$516$	3.06418	0.134893
$517$	1.57304	0.0691824
$518$	3.65270	0.160491
$519$	−22.5963	−0.991866
$520$	−1.53209	−0.0671865
$521$	12.9162	0.565870	0.282935	−	0.959139i	$-0.408692\pi$
$521$	12.9162	0.565870	0.282935	+	0.959139i	$0.408692\pi$
$522$	−4.45336	−0.194918
$523$	3.36009	0.146926	0.0734632	−	0.997298i	$-0.476595\pi$
$523$	3.36009	0.146926	0.0734632	+	0.997298i	$0.476595\pi$
$524$	2.22668	0.0972730
$525$	0.347296	0.0151573
$526$	−11.1334	−0.485440
$527$	0	0
$528$	−1.18479	−0.0515615
$529$	−18.0419	−0.784430
$530$	−7.18479	−0.312087
$531$	3.55438	0.154247
$532$	1.28312	0.0556302
$533$	−15.0496	−0.651872
$534$	3.41147	0.147629
$535$	−4.73917	−0.204892
$536$	2.24123	0.0968064
$537$	11.7050	0.505109
$538$	−10.7392	−0.462999
$539$	−8.15064	−0.351073
$540$	−1.00000	−0.0430331
$541$	−25.5276	−1.09752	−0.548760	−	0.835980i	$-0.684900\pi$
$541$	−25.5276	−1.09752	−0.548760	+	0.835980i	$0.684900\pi$
$542$	10.3250	0.443497
$543$	−4.61081	−0.197869
$544$	0	0
$545$	−7.27631	−0.311683
$546$	−0.532089	−0.0227713
$547$	−30.9513	−1.32338	−0.661691	−	0.749777i	$-0.730161\pi$
$547$	−30.9513	−1.32338	−0.661691	+	0.749777i	$0.730161\pi$
$548$	5.17705	0.221153
$549$	3.06418	0.130776
$550$	−1.18479	−0.0505197
$551$	−16.4534	−0.700937
$552$	−2.22668	−0.0947739
$553$	−4.58172	−0.194834
$554$	−23.8280	−1.01235
$555$	−10.5175	−0.446445
$556$	3.69459	0.156686
$557$	−3.41147	−0.144549	−0.0722744	−	0.997385i	$-0.523026\pi$
$557$	−3.41147	−0.144549	−0.0722744	+	0.997385i	$0.523026\pi$
$558$	3.75877	0.159121
$559$	−4.69459	−0.198560
$560$	−0.347296	−0.0146759
$561$	0	0
$562$	23.2003	0.978644
$563$	23.0250	0.970387	0.485194	−	0.874407i	$-0.338749\pi$
$563$	23.0250	0.970387	0.485194	+	0.874407i	$0.338749\pi$
$564$	−1.32770	−0.0559061
$565$	7.54664	0.317489
$566$	4.58172	0.192584
$567$	−0.347296	−0.0145851
$568$	−2.36959	−0.0994256
$569$	−6.72967	−0.282123	−0.141061	−	0.990001i	$-0.545051\pi$
$569$	−6.72967	−0.282123	−0.141061	+	0.990001i	$0.545051\pi$
$570$	−3.69459	−0.154749
$571$	39.9427	1.67155	0.835776	−	0.549071i	$-0.185018\pi$
$571$	39.9427	1.67155	0.835776	+	0.549071i	$0.185018\pi$
$572$	1.81521	0.0758976
$573$	−12.0000	−0.501307
$574$	−3.41147	−0.142392
$575$	−2.22668	−0.0928590
$576$	1.00000	0.0416667
$577$	−29.2472	−1.21758	−0.608789	−	0.793332i	$-0.708344\pi$
$577$	−29.2472	−1.21758	−0.608789	+	0.793332i	$0.708344\pi$
$578$	0	0
$579$	9.47296	0.393683
$580$	4.45336	0.184916
$581$	0.285807	0.0118573
$582$	4.32501	0.179277
$583$	8.51249	0.352551
$584$	−4.00000	−0.165521
$585$	1.53209	0.0633441
$586$	−25.7219	−1.06256
$587$	1.79797	0.0742102	0.0371051	−	0.999311i	$-0.488186\pi$
$587$	1.79797	0.0742102	0.0371051	+	0.999311i	$0.488186\pi$
$588$	6.87939	0.283701
$589$	13.8871	0.572209
$590$	−3.55438	−0.146331
$591$	0.0760373	0.00312776
$592$	10.5175	0.432268
$593$	19.3601	0.795024	0.397512	−	0.917597i	$-0.369874\pi$
$593$	19.3601	0.795024	0.397512	+	0.917597i	$0.369874\pi$
$594$	1.18479	0.0486126
$595$	0	0
$596$	2.36959	0.0970620
$597$	−0.807467	−0.0330474
$598$	3.41147	0.139506
$599$	−28.2668	−1.15495	−0.577475	−	0.816408i	$-0.695962\pi$
$599$	−28.2668	−1.15495	−0.577475	+	0.816408i	$0.695962\pi$
$600$	1.00000	0.0408248
$601$	29.3233	1.19612	0.598060	−	0.801451i	$-0.295938\pi$
$601$	29.3233	1.19612	0.598060	+	0.801451i	$0.295938\pi$
$602$	−1.06418	−0.0433726
$603$	−2.24123	−0.0912699
$604$	2.49794	0.101640
$605$	−9.59627	−0.390144
$606$	5.27631	0.214336
$607$	9.64496	0.391477	0.195738	−	0.980656i	$-0.437290\pi$
$607$	9.64496	0.391477	0.195738	+	0.980656i	$0.437290\pi$
$608$	3.69459	0.149836
$609$	1.54664	0.0626729
$610$	−3.06418	−0.124065
$611$	2.03415	0.0822928
$612$	0	0
$613$	−9.76744	−0.394503	−0.197252	−	0.980353i	$-0.563202\pi$
$613$	−9.76744	−0.394503	−0.197252	+	0.980353i	$0.563202\pi$
$614$	−12.5526	−0.506583
$615$	9.82295	0.396100
$616$	0.411474	0.0165788
$617$	20.9067	0.841673	0.420837	−	0.907136i	$-0.361737\pi$
$617$	20.9067	0.841673	0.420837	+	0.907136i	$0.361737\pi$
$618$	−3.69459	−0.148618
$619$	25.5107	1.02536	0.512681	−	0.858579i	$-0.328652\pi$
$619$	25.5107	1.02536	0.512681	+	0.858579i	$0.328652\pi$
$620$	−3.75877	−0.150956
$621$	2.22668	0.0893537
$622$	−10.1676	−0.407682
$623$	−1.18479	−0.0474677
$624$	−1.53209	−0.0613326
$625$	1.00000	0.0400000
$626$	−10.3250	−0.412670
$627$	4.37733	0.174814
$628$	22.8999	0.913806
$629$	0	0
$630$	0.347296	0.0138366
$631$	−9.16344	−0.364791	−0.182395	−	0.983225i	$-0.558385\pi$
$631$	−9.16344	−0.364791	−0.182395	+	0.983225i	$0.558385\pi$
$632$	−13.1925	−0.524771
$633$	18.7324	0.744545
$634$	−26.5621	−1.05492
$635$	−11.3405	−0.450034
$636$	−7.18479	−0.284896
$637$	−10.5398	−0.417603
$638$	−5.27631	−0.208891
$639$	2.36959	0.0937393
$640$	−1.00000	−0.0395285
$641$	−2.97359	−0.117450	−0.0587249	−	0.998274i	$-0.518703\pi$
$641$	−2.97359	−0.117450	−0.0587249	+	0.998274i	$0.518703\pi$
$642$	−4.73917	−0.187040
$643$	−26.0601	−1.02771	−0.513854	−	0.857878i	$-0.671783\pi$
$643$	−26.0601	−1.02771	−0.513854	+	0.857878i	$0.671783\pi$
$644$	0.773318	0.0304730
$645$	3.06418	0.120652
$646$	0	0
$647$	1.90848	0.0750301	0.0375151	−	0.999296i	$-0.488056\pi$
$647$	1.90848	0.0750301	0.0375151	+	0.999296i	$0.488056\pi$
$648$	−1.00000	−0.0392837
$649$	4.21120	0.165304
$650$	−1.53209	−0.0600935
$651$	−1.30541	−0.0511629
$652$	18.4688	0.723296
$653$	0.119681	0.00468348	0.00234174	−	0.999997i	$-0.499255\pi$
$653$	0.119681	0.00468348	0.00234174	+	0.999997i	$0.499255\pi$
$654$	−7.27631	−0.284526
$655$	2.22668	0.0870036
$656$	−9.82295	−0.383522
$657$	4.00000	0.156055
$658$	0.461104	0.0179757
$659$	36.6769	1.42873	0.714364	−	0.699775i	$-0.246716\pi$
$659$	36.6769	1.42873	0.714364	+	0.699775i	$0.246716\pi$
$660$	−1.18479	−0.0461180
$661$	42.2276	1.64246	0.821232	−	0.570595i	$-0.193287\pi$
$661$	42.2276	1.64246	0.821232	+	0.570595i	$0.193287\pi$
$662$	15.8203	0.614872
$663$	0	0
$664$	0.822948	0.0319366
$665$	1.28312	0.0497572
$666$	−10.5175	−0.407546
$667$	−9.91622	−0.383958
$668$	1.18479	0.0458410
$669$	−4.51249	−0.174463
$670$	2.24123	0.0865863
$671$	3.63041	0.140151
$672$	−0.347296	−0.0133972
$673$	27.8871	1.07497	0.537485	−	0.843273i	$-0.319375\pi$
$673$	27.8871	1.07497	0.537485	+	0.843273i	$0.319375\pi$
$674$	21.7452	0.837592
$675$	−1.00000	−0.0384900
$676$	−10.6527	−0.409719
$677$	−22.2344	−0.854538	−0.427269	−	0.904124i	$-0.640524\pi$
$677$	−22.2344	−0.854538	−0.427269	+	0.904124i	$0.640524\pi$
$678$	7.54664	0.289827
$679$	−1.50206	−0.0576437
$680$	0	0
$681$	8.90673	0.341306
$682$	4.45336	0.170528
$683$	−32.2823	−1.23525	−0.617624	−	0.786474i	$-0.711905\pi$
$683$	−32.2823	−1.23525	−0.617624	+	0.786474i	$0.711905\pi$
$684$	−3.69459	−0.141266
$685$	5.17705	0.197805
$686$	−4.82026	−0.184038
$687$	24.4688	0.933545
$688$	−3.06418	−0.116821
$689$	11.0077	0.419362
$690$	−2.22668	−0.0847683
$691$	31.0360	1.18067	0.590333	−	0.807160i	$-0.298996\pi$
$691$	31.0360	1.18067	0.590333	+	0.807160i	$0.298996\pi$
$692$	22.5963	0.858981
$693$	−0.411474	−0.0156306
$694$	28.8384	1.09469
$695$	3.69459	0.140144
$696$	4.45336	0.168804
$697$	0	0
$698$	−6.55262	−0.248020
$699$	−27.2918	−1.03227
$700$	−0.347296	−0.0131266
$701$	−43.8444	−1.65598	−0.827990	−	0.560742i	$-0.810516\pi$
$701$	−43.8444	−1.65598	−0.827990	+	0.560742i	$0.810516\pi$
$702$	1.53209	0.0578250
$703$	−38.8580	−1.46556
$704$	1.18479	0.0446535
$705$	−1.32770	−0.0500039
$706$	3.72967	0.140368
$707$	−1.83244	−0.0689161
$708$	−3.55438	−0.133582
$709$	−1.48845	−0.0558997	−0.0279499	−	0.999609i	$-0.508898\pi$
$709$	−1.48845	−0.0558997	−0.0279499	+	0.999609i	$0.508898\pi$
$710$	−2.36959	−0.0889289
$711$	13.1925	0.494759
$712$	−3.41147	−0.127850
$713$	8.36959	0.313443
$714$	0	0
$715$	1.81521	0.0678849
$716$	−11.7050	−0.437437
$717$	17.2763	0.645196
$718$	14.9067	0.556314
$719$	14.6209	0.545268	0.272634	−	0.962118i	$-0.412105\pi$
$719$	14.6209	0.545268	0.272634	+	0.962118i	$0.412105\pi$
$720$	1.00000	0.0372678
$721$	1.28312	0.0477858
$722$	5.34998	0.199106
$723$	6.95811	0.258775
$724$	4.61081	0.171360
$725$	4.45336	0.165394
$726$	−9.59627	−0.356151
$727$	47.2618	1.75284	0.876421	−	0.481546i	$-0.159925\pi$
$727$	47.2618	1.75284	0.876421	+	0.481546i	$0.159925\pi$
$728$	0.532089	0.0197205
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	0	0
$732$	−3.06418	−0.113255
$733$	43.8675	1.62028	0.810142	−	0.586234i	$-0.199390\pi$
$733$	43.8675	1.62028	0.810142	+	0.586234i	$0.199390\pi$
$734$	−0.0469415	−0.00173264
$735$	6.87939	0.253750
$736$	2.22668	0.0820766
$737$	−2.65539	−0.0978126
$738$	9.82295	0.361588
$739$	−2.06593	−0.0759966	−0.0379983	−	0.999278i	$-0.512098\pi$
$739$	−2.06593	−0.0759966	−0.0379983	+	0.999278i	$0.512098\pi$
$740$	10.5175	0.386632
$741$	5.66044	0.207942
$742$	2.49525	0.0916036
$743$	33.3678	1.22415	0.612073	−	0.790801i	$-0.290336\pi$
$743$	33.3678	1.22415	0.612073	+	0.790801i	$0.290336\pi$
$744$	−3.75877	−0.137803
$745$	2.36959	0.0868149
$746$	−13.3432	−0.488528
$747$	−0.822948	−0.0301101
$748$	0	0
$749$	1.64590	0.0601397
$750$	1.00000	0.0365148
$751$	−21.8580	−0.797611	−0.398805	−	0.917036i	$-0.630575\pi$
$751$	−21.8580	−0.797611	−0.398805	+	0.917036i	$0.630575\pi$
$752$	1.32770	0.0484161
$753$	−16.6287	−0.605982
$754$	−6.82295	−0.248477
$755$	2.49794	0.0909094
$756$	0.347296	0.0126310
$757$	−6.77063	−0.246083	−0.123041	−	0.992402i	$-0.539265\pi$
$757$	−6.77063	−0.246083	−0.123041	+	0.992402i	$0.539265\pi$
$758$	−25.9026	−0.940825
$759$	2.63816	0.0957590
$760$	3.69459	0.134017
$761$	29.0574	1.05333	0.526664	−	0.850073i	$-0.323442\pi$
$761$	29.0574	1.05333	0.526664	+	0.850073i	$0.323442\pi$
$762$	−11.3405	−0.410823
$763$	2.52704	0.0914849
$764$	12.0000	0.434145
$765$	0	0
$766$	9.38507	0.339096
$767$	5.44562	0.196630
$768$	−1.00000	−0.0360844
$769$	−0.874333	−0.0315292	−0.0157646	−	0.999876i	$-0.505018\pi$
$769$	−0.874333	−0.0315292	−0.0157646	+	0.999876i	$0.505018\pi$
$770$	0.411474	0.0148285
$771$	9.19253	0.331061
$772$	−9.47296	−0.340939
$773$	50.3756	1.81188	0.905942	−	0.423402i	$-0.139164\pi$
$773$	50.3756	1.81188	0.905942	+	0.423402i	$0.139164\pi$
$774$	3.06418	0.110140
$775$	−3.75877	−0.135019
$776$	−4.32501	−0.155259
$777$	3.65270	0.131040
$778$	−29.3756	−1.05317
$779$	36.2918	1.30029
$780$	−1.53209	−0.0548576
$781$	2.80747	0.100459
$782$	0	0
$783$	−4.45336	−0.159150
$784$	−6.87939	−0.245692
$785$	22.8999	0.817333
$786$	2.22668	0.0794231
$787$	44.7256	1.59429	0.797147	−	0.603785i	$-0.206342\pi$
$787$	44.7256	1.59429	0.797147	+	0.603785i	$0.206342\pi$
$788$	−0.0760373	−0.00270872
$789$	−11.1334	−0.396360
$790$	−13.1925	−0.469369
$791$	−2.62092	−0.0931892
$792$	−1.18479	−0.0420998
$793$	4.69459	0.166710
$794$	4.76382	0.169062
$795$	−7.18479	−0.254818
$796$	0.807467	0.0286199
$797$	11.7466	0.416085	0.208043	−	0.978120i	$-0.433291\pi$
$797$	11.7466	0.416085	0.208043	+	0.978120i	$0.433291\pi$
$798$	1.28312	0.0454219
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	3.41147	0.120539
$802$	−13.7219	−0.484538
$803$	4.73917	0.167242
$804$	2.24123	0.0790421
$805$	0.773318	0.0272559
$806$	5.75877	0.202844
$807$	−10.7392	−0.378037
$808$	−5.27631	−0.185620
$809$	−25.1147	−0.882987	−0.441494	−	0.897264i	$-0.645551\pi$
$809$	−25.1147	−0.882987	−0.441494	+	0.897264i	$0.645551\pi$
$810$	−1.00000	−0.0351364
$811$	−51.8376	−1.82026	−0.910132	−	0.414318i	$-0.864020\pi$
$811$	−51.8376	−1.82026	−0.910132	+	0.414318i	$0.864020\pi$
$812$	−1.54664	−0.0542763
$813$	10.3250	0.362114
$814$	−12.4611	−0.436761
$815$	18.4688	0.646935
$816$	0	0
$817$	11.3209	0.396068
$818$	30.2131	1.05638
$819$	−0.532089	−0.0185927
$820$	−9.82295	−0.343032
$821$	−3.09327	−0.107956	−0.0539780	−	0.998542i	$-0.517190\pi$
$821$	−3.09327	−0.107956	−0.0539780	+	0.998542i	$0.517190\pi$
$822$	5.17705	0.180570
$823$	32.0892	1.11856	0.559279	−	0.828980i	$-0.311078\pi$
$823$	32.0892	1.11856	0.559279	+	0.828980i	$0.311078\pi$
$824$	3.69459	0.128707
$825$	−1.18479	−0.0412492
$826$	1.23442	0.0429510
$827$	35.1242	1.22139	0.610695	−	0.791866i	$-0.290890\pi$
$827$	35.1242	1.22139	0.610695	+	0.791866i	$0.290890\pi$
$828$	−2.22668	−0.0773825
$829$	−23.5330	−0.817336	−0.408668	−	0.912683i	$-0.634007\pi$
$829$	−23.5330	−0.817336	−0.408668	+	0.912683i	$0.634007\pi$
$830$	0.822948	0.0285650
$831$	−23.8280	−0.826584
$832$	1.53209	0.0531156
$833$	0	0
$834$	3.69459	0.127933
$835$	1.18479	0.0410014
$836$	−4.37733	−0.151393
$837$	3.75877	0.129922
$838$	−32.3259	−1.11668
$839$	21.6304	0.746765	0.373382	−	0.927678i	$-0.378198\pi$
$839$	21.6304	0.746765	0.373382	+	0.927678i	$0.378198\pi$
$840$	−0.347296	−0.0119829
$841$	−9.16756	−0.316123
$842$	11.7196	0.403883
$843$	23.2003	0.799060
$844$	−18.7324	−0.644795
$845$	−10.6527	−0.366464
$846$	−1.32770	−0.0456471
$847$	3.33275	0.114515
$848$	7.18479	0.246727
$849$	4.58172	0.157244
$850$	0	0
$851$	−23.4192	−0.802800
$852$	−2.36959	−0.0811806
$853$	19.5408	0.669063	0.334531	−	0.942385i	$-0.391422\pi$
$853$	19.5408	0.669063	0.334531	+	0.942385i	$0.391422\pi$
$854$	1.06418	0.0364154
$855$	−3.69459	−0.126352
$856$	4.73917	0.161982
$857$	−10.8384	−0.370234	−0.185117	−	0.982717i	$-0.559266\pi$
$857$	−10.8384	−0.370234	−0.185117	+	0.982717i	$0.559266\pi$
$858$	1.81521	0.0619702
$859$	19.7338	0.673308	0.336654	−	0.941628i	$-0.390705\pi$
$859$	19.7338	0.673308	0.336654	+	0.941628i	$0.390705\pi$
$860$	−3.06418	−0.104488
$861$	−3.41147	−0.116263
$862$	−28.5526	−0.972506
$863$	3.70645	0.126169	0.0630846	−	0.998008i	$-0.479906\pi$
$863$	3.70645	0.126169	0.0630846	+	0.998008i	$0.479906\pi$
$864$	1.00000	0.0340207
$865$	22.5963	0.768296
$866$	9.78787	0.332605
$867$	0	0
$868$	1.30541	0.0443084
$869$	15.6304	0.530226
$870$	4.45336	0.150983
$871$	−3.43376	−0.116349
$872$	7.27631	0.246407
$873$	4.32501	0.146379
$874$	−8.22668	−0.278272
$875$	−0.347296	−0.0117408
$876$	−4.00000	−0.135147
$877$	−2.79055	−0.0942303	−0.0471152	−	0.998889i	$-0.515003\pi$
$877$	−2.79055	−0.0942303	−0.0471152	+	0.998889i	$0.515003\pi$
$878$	−18.2567	−0.616134
$879$	−25.7219	−0.867579
$880$	1.18479	0.0399393
$881$	52.1056	1.75548	0.877741	−	0.479136i	$-0.159050\pi$
$881$	52.1056	1.75548	0.877741	+	0.479136i	$0.159050\pi$
$882$	6.87939	0.231641
$883$	51.4938	1.73291	0.866453	−	0.499259i	$-0.166395\pi$
$883$	51.4938	1.73291	0.866453	+	0.499259i	$0.166395\pi$
$884$	0	0
$885$	−3.55438	−0.119479
$886$	41.4748	1.39337
$887$	5.90673	0.198328	0.0991642	−	0.995071i	$-0.468383\pi$
$887$	5.90673	0.198328	0.0991642	+	0.995071i	$0.468383\pi$
$888$	10.5175	0.352946
$889$	3.93851	0.132093
$890$	−3.41147	−0.114353
$891$	1.18479	0.0396920
$892$	4.51249	0.151089
$893$	−4.90530	−0.164149
$894$	2.36959	0.0792508
$895$	−11.7050	−0.391256
$896$	0.347296	0.0116024
$897$	3.41147	0.113906
$898$	−6.57161	−0.219298
$899$	−16.7392	−0.558283
$900$	1.00000	0.0333333
$901$	0	0
$902$	11.6382	0.387508
$903$	−1.06418	−0.0354136
$904$	−7.54664	−0.250997
$905$	4.61081	0.153269
$906$	2.49794	0.0829885
$907$	−13.4884	−0.447876	−0.223938	−	0.974603i	$-0.571891\pi$
$907$	−13.4884	−0.447876	−0.223938	+	0.974603i	$0.571891\pi$
$908$	−8.90673	−0.295580
$909$	5.27631	0.175004
$910$	0.532089	0.0176386
$911$	21.1581	0.700998	0.350499	−	0.936563i	$-0.386012\pi$
$911$	21.1581	0.700998	0.350499	+	0.936563i	$0.386012\pi$
$912$	3.69459	0.122340
$913$	−0.975023	−0.0322685
$914$	6.08378	0.201233
$915$	−3.06418	−0.101299
$916$	−24.4688	−0.808473
$917$	−0.773318	−0.0255372
$918$	0	0
$919$	−45.4100	−1.49794	−0.748970	−	0.662604i	$-0.769451\pi$
$919$	−45.4100	−1.49794	−0.748970	+	0.662604i	$0.769451\pi$
$920$	2.22668	0.0734115
$921$	−12.5526	−0.413623
$922$	−19.2608	−0.634322
$923$	3.63041	0.119497
$924$	0.411474	0.0135365
$925$	10.5175	0.345815
$926$	−25.4216	−0.835405
$927$	−3.69459	−0.121346
$928$	−4.45336	−0.146189
$929$	17.4284	0.571807	0.285903	−	0.958258i	$-0.407706\pi$
$929$	17.4284	0.571807	0.285903	+	0.958258i	$0.407706\pi$
$930$	−3.75877	−0.123255
$931$	25.4165	0.832993
$932$	27.2918	0.893972
$933$	−10.1676	−0.332871
$934$	−20.2175	−0.661537
$935$	0	0
$936$	−1.53209	−0.0500779
$937$	−8.67911	−0.283534	−0.141767	−	0.989900i	$-0.545278\pi$
$937$	−8.67911	−0.283534	−0.141767	+	0.989900i	$0.545278\pi$
$938$	−0.778371	−0.0254147
$939$	−10.3250	−0.336944
$940$	1.32770	0.0433047
$941$	38.7202	1.26224	0.631121	−	0.775684i	$-0.282595\pi$
$941$	38.7202	1.26224	0.631121	+	0.775684i	$0.282595\pi$
$942$	22.8999	0.746120
$943$	21.8726	0.712269
$944$	3.55438	0.115685
$945$	0.347296	0.0112976
$946$	3.63041	0.118035
$947$	31.5466	1.02513	0.512564	−	0.858649i	$-0.328696\pi$
$947$	31.5466	1.02513	0.512564	+	0.858649i	$0.328696\pi$
$948$	−13.1925	−0.428474
$949$	6.12836	0.198935
$950$	3.69459	0.119868
$951$	−26.5621	−0.861336
$952$	0	0
$953$	28.0155	0.907510	0.453755	−	0.891126i	$-0.350084\pi$
$953$	28.0155	0.907510	0.453755	+	0.891126i	$0.350084\pi$
$954$	−7.18479	−0.232616
$955$	12.0000	0.388311
$956$	−17.2763	−0.558756
$957$	−5.27631	−0.170559
$958$	−26.3696	−0.851963
$959$	−1.79797	−0.0580595
$960$	−1.00000	−0.0322749
$961$	−16.8716	−0.544247
$962$	−16.1138	−0.519530
$963$	−4.73917	−0.152718
$964$	−6.95811	−0.224106
$965$	−9.47296	−0.304945
$966$	0.773318	0.0248811
$967$	−44.0215	−1.41563	−0.707817	−	0.706395i	$-0.750320\pi$
$967$	−44.0215	−1.41563	−0.707817	+	0.706395i	$0.750320\pi$
$968$	9.59627	0.308436
$969$	0	0
$970$	−4.32501	−0.138868
$971$	−37.7392	−1.21111	−0.605554	−	0.795804i	$-0.707048\pi$
$971$	−37.7392	−1.21111	−0.605554	+	0.795804i	$0.707048\pi$
$972$	−1.00000	−0.0320750
$973$	−1.28312	−0.0411349
$974$	−20.8188	−0.667078
$975$	−1.53209	−0.0490661
$976$	3.06418	0.0980819
$977$	25.1088	0.803300	0.401650	−	0.915793i	$-0.368437\pi$
$977$	25.1088	0.803300	0.401650	+	0.915793i	$0.368437\pi$
$978$	18.4688	0.590569
$979$	4.04189	0.129179
$980$	−6.87939	−0.219754
$981$	−7.27631	−0.232315
$982$	−25.5262	−0.814574
$983$	42.8877	1.36791	0.683953	−	0.729526i	$-0.260259\pi$
$983$	42.8877	1.36791	0.683953	+	0.729526i	$0.260259\pi$
$984$	−9.82295	−0.313144
$985$	−0.0760373	−0.00242275
$986$	0	0
$987$	0.461104	0.0146771
$988$	−5.66044	−0.180083
$989$	6.82295	0.216957
$990$	−1.18479	−0.0376552
$991$	−0.891245	−0.0283113	−0.0141557	−	0.999900i	$-0.504506\pi$
$991$	−0.891245	−0.0283113	−0.0141557	+	0.999900i	$0.504506\pi$
$992$	3.75877	0.119341
$993$	15.8203	0.502041
$994$	0.822948	0.0261023
$995$	0.807467	0.0255984
$996$	0.822948	0.0260761
$997$	−0.642275	−0.0203410	−0.0101705	−	0.999948i	$-0.503237\pi$
$997$	−0.642275	−0.0203410	−0.0101705	+	0.999948i	$0.503237\pi$
$998$	31.3688	0.992961
$999$	−10.5175	−0.332760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bm.1.2	✓	3
17.16	even	2		8670.2.a.bn.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8670.2.a.bm.1.2	✓	3	1.1	even	1	trivial
8670.2.a.bn.1.2	yes	3	17.16	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -0.347296 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.18479 q^{11} -1.00000 q^{12} +1.53209 q^{13} +0.347296 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -3.69459 q^{19} +1.00000 q^{20} +0.347296 q^{21} -1.18479 q^{22} -2.22668 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.53209 q^{26} -1.00000 q^{27} -0.347296 q^{28} +4.45336 q^{29} +1.00000 q^{30} -3.75877 q^{31} -1.00000 q^{32} -1.18479 q^{33} -0.347296 q^{35} +1.00000 q^{36} +10.5175 q^{37} +3.69459 q^{38} -1.53209 q^{39} -1.00000 q^{40} -9.82295 q^{41} -0.347296 q^{42} -3.06418 q^{43} +1.18479 q^{44} +1.00000 q^{45} +2.22668 q^{46} +1.32770 q^{47} -1.00000 q^{48} -6.87939 q^{49} -1.00000 q^{50} +1.53209 q^{52} +7.18479 q^{53} +1.00000 q^{54} +1.18479 q^{55} +0.347296 q^{56} +3.69459 q^{57} -4.45336 q^{58} +3.55438 q^{59} -1.00000 q^{60} +3.06418 q^{61} +3.75877 q^{62} -0.347296 q^{63} +1.00000 q^{64} +1.53209 q^{65} +1.18479 q^{66} -2.24123 q^{67} +2.22668 q^{69} +0.347296 q^{70} +2.36959 q^{71} -1.00000 q^{72} +4.00000 q^{73} -10.5175 q^{74} -1.00000 q^{75} -3.69459 q^{76} -0.411474 q^{77} +1.53209 q^{78} +13.1925 q^{79} +1.00000 q^{80} +1.00000 q^{81} +9.82295 q^{82} -0.822948 q^{83} +0.347296 q^{84} +3.06418 q^{86} -4.45336 q^{87} -1.18479 q^{88} +3.41147 q^{89} -1.00000 q^{90} -0.532089 q^{91} -2.22668 q^{92} +3.75877 q^{93} -1.32770 q^{94} -3.69459 q^{95} +1.00000 q^{96} +4.32501 q^{97} +6.87939 q^{98} +1.18479 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{12} - 3 q^{15} + 3 q^{16} - 3 q^{18} - 9 q^{19} + 3 q^{20} + 3 q^{24} + 3 q^{25} - 3 q^{27} + 3 q^{30} - 3 q^{32}+ \cdots + 15 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 - 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^12 - 3 * q^15 + 3 * q^16 - 3 * q^18 - 9 * q^19 + 3 * q^20 + 3 * q^24 + 3 * q^25 - 3 * q^27 + 3 * q^30 - 3 * q^32 + 3 * q^36 + 9 * q^37 + 9 * q^38 - 3 * q^40 - 9 * q^41 + 3 * q^45 - 3 * q^48 - 15 * q^49 - 3 * q^50 + 18 * q^53 + 3 * q^54 + 9 * q^57 - 3 * q^60 + 3 * q^64 - 18 * q^67 - 3 * q^72 + 12 * q^73 - 9 * q^74 - 3 * q^75 - 9 * q^76 + 9 * q^77 + 12 * q^79 + 3 * q^80 + 3 * q^81 + 9 * q^82 + 18 * q^83 - 3 * q^90 + 3 * q^91 - 9 * q^95 + 3 * q^96 + 18 * q^97 + 15 * q^98

Introduction

L-functions

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Database

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