LMFDB - Embedded newform 8670.2.a.bw.1.4

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:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84776$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.69552 q^{11} +1.00000 q^{12} +2.72965 q^{13} -2.61313 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -0.663643 q^{19} -1.00000 q^{20} +2.61313 q^{21} +1.69552 q^{22} -0.352746 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.72965 q^{26} +1.00000 q^{27} +2.61313 q^{28} +7.18759 q^{29} +1.00000 q^{30} +9.50756 q^{31} -1.00000 q^{32} -1.69552 q^{33} -2.61313 q^{35} +1.00000 q^{36} +0.498858 q^{37} +0.663643 q^{38} +2.72965 q^{39} +1.00000 q^{40} -7.67459 q^{41} -2.61313 q^{42} +9.41838 q^{43} -1.69552 q^{44} -1.00000 q^{45} +0.352746 q^{46} +3.33636 q^{47} +1.00000 q^{48} -0.171573 q^{49} -1.00000 q^{50} +2.72965 q^{52} +5.06147 q^{53} -1.00000 q^{54} +1.69552 q^{55} -2.61313 q^{56} -0.663643 q^{57} -7.18759 q^{58} -10.7161 q^{59} -1.00000 q^{60} -7.35237 q^{61} -9.50756 q^{62} +2.61313 q^{63} +1.00000 q^{64} -2.72965 q^{65} +1.69552 q^{66} +3.14423 q^{67} -0.352746 q^{69} +2.61313 q^{70} +1.71644 q^{71} -1.00000 q^{72} -15.0656 q^{73} -0.498858 q^{74} +1.00000 q^{75} -0.663643 q^{76} -4.43060 q^{77} -2.72965 q^{78} +2.62445 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.67459 q^{82} -11.3910 q^{83} +2.61313 q^{84} -9.41838 q^{86} +7.18759 q^{87} +1.69552 q^{88} +5.25903 q^{89} +1.00000 q^{90} +7.13291 q^{91} -0.352746 q^{92} +9.50756 q^{93} -3.33636 q^{94} +0.663643 q^{95} -1.00000 q^{96} +11.9268 q^{97} +0.171573 q^{98} -1.69552 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{10} + 8 q^{11} + 4 q^{12} - 4 q^{15} + 4 q^{16} - 4 q^{18} - 4 q^{20} - 8 q^{22} - 8 q^{23} - 4 q^{24} + 4 q^{25} + 4 q^{27}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 - 4 * q^6 - 4 * q^8 + 4 * q^9 + 4 * q^10 + 8 * q^11 + 4 * q^12 - 4 * q^15 + 4 * q^16 - 4 * q^18 - 4 * q^20 - 8 * q^22 - 8 * q^23 - 4 * q^24 + 4 * q^25 + 4 * q^27 + 4 * q^30 + 8 * q^31 - 4 * q^32 + 8 * q^33 + 4 * q^36 + 8 * q^37 + 4 * q^40 - 8 * q^41 - 8 * q^43 + 8 * q^44 - 4 * q^45 + 8 * q^46 + 16 * q^47 + 4 * q^48 - 12 * q^49 - 4 * q^50 + 8 * q^53 - 4 * q^54 - 8 * q^55 + 8 * q^59 - 4 * q^60 + 8 * q^61 - 8 * q^62 + 4 * q^64 - 8 * q^66 + 40 * q^67 - 8 * q^69 - 4 * q^72 - 8 * q^73 - 8 * q^74 + 4 * q^75 - 16 * q^77 + 24 * q^79 - 4 * q^80 + 4 * q^81 + 8 * q^82 - 16 * q^83 + 8 * q^86 - 8 * q^88 + 8 * q^89 + 4 * q^90 + 32 * q^91 - 8 * q^92 + 8 * q^93 - 16 * q^94 - 4 * q^96 - 8 * q^97 + 12 * q^98 + 8 * q^99

Coefficient data

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

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

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

$2$	−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
$7$	2.61313	0.987669	0.493834	−	0.869556i	$-0.335595\pi$
$7$	2.61313	0.987669	0.493834	+	0.869556i	$0.335595\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.69552	−0.511218	−0.255609	−	0.966780i	$-0.582276\pi$
$11$	−1.69552	−0.511218	−0.255609	+	0.966780i	$0.582276\pi$
$12$	1.00000	0.288675
$13$	2.72965	0.757068	0.378534	−	0.925587i	$-0.376428\pi$
$13$	2.72965	0.757068	0.378534	+	0.925587i	$0.376428\pi$
$14$	−2.61313	−0.698387
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	−0.663643	−0.152250	−0.0761250	−	0.997098i	$-0.524255\pi$
$19$	−0.663643	−0.152250	−0.0761250	+	0.997098i	$0.524255\pi$
$20$	−1.00000	−0.223607
$21$	2.61313	0.570231
$22$	1.69552	0.361486
$23$	−0.352746	−0.0735526	−0.0367763	−	0.999324i	$-0.511709\pi$
$23$	−0.352746	−0.0735526	−0.0367763	+	0.999324i	$0.511709\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.72965	−0.535328
$27$	1.00000	0.192450
$28$	2.61313	0.493834
$29$	7.18759	1.33470	0.667351	−	0.744744i	$-0.267428\pi$
$29$	7.18759	1.33470	0.667351	+	0.744744i	$0.267428\pi$
$30$	1.00000	0.182574
$31$	9.50756	1.70761	0.853804	−	0.520595i	$-0.174290\pi$
$31$	9.50756	1.70761	0.853804	+	0.520595i	$0.174290\pi$
$32$	−1.00000	−0.176777
$33$	−1.69552	−0.295152
$34$	0	0
$35$	−2.61313	−0.441699
$36$	1.00000	0.166667
$37$	0.498858	0.0820118	0.0410059	−	0.999159i	$-0.486944\pi$
$37$	0.498858	0.0820118	0.0410059	+	0.999159i	$0.486944\pi$
$38$	0.663643	0.107657
$39$	2.72965	0.437093
$40$	1.00000	0.158114
$41$	−7.67459	−1.19857	−0.599285	−	0.800536i	$-0.704548\pi$
$41$	−7.67459	−1.19857	−0.599285	+	0.800536i	$0.704548\pi$
$42$	−2.61313	−0.403214
$43$	9.41838	1.43629	0.718144	−	0.695894i	$-0.244992\pi$
$43$	9.41838	1.43629	0.718144	+	0.695894i	$0.244992\pi$
$44$	−1.69552	−0.255609
$45$	−1.00000	−0.149071
$46$	0.352746	0.0520096
$47$	3.33636	0.486658	0.243329	−	0.969944i	$-0.421761\pi$
$47$	3.33636	0.486658	0.243329	+	0.969944i	$0.421761\pi$
$48$	1.00000	0.144338
$49$	−0.171573	−0.0245104
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.72965	0.378534
$53$	5.06147	0.695246	0.347623	−	0.937634i	$-0.386989\pi$
$53$	5.06147	0.695246	0.347623	+	0.937634i	$0.386989\pi$
$54$	−1.00000	−0.136083
$55$	1.69552	0.228624
$56$	−2.61313	−0.349194
$57$	−0.663643	−0.0879016
$58$	−7.18759	−0.943777
$59$	−10.7161	−1.39511	−0.697557	−	0.716530i	$-0.745729\pi$
$59$	−10.7161	−1.39511	−0.697557	+	0.716530i	$0.745729\pi$
$60$	−1.00000	−0.129099
$61$	−7.35237	−0.941375	−0.470687	−	0.882300i	$-0.655994\pi$
$61$	−7.35237	−0.941375	−0.470687	+	0.882300i	$0.655994\pi$
$62$	−9.50756	−1.20746
$63$	2.61313	0.329223
$64$	1.00000	0.125000
$65$	−2.72965	−0.338571
$66$	1.69552	0.208704
$67$	3.14423	0.384129	0.192065	−	0.981382i	$-0.438482\pi$
$67$	3.14423	0.384129	0.192065	+	0.981382i	$0.438482\pi$
$68$	0	0
$69$	−0.352746	−0.0424656
$70$	2.61313	0.312328
$71$	1.71644	0.203704	0.101852	−	0.994800i	$-0.467523\pi$
$71$	1.71644	0.203704	0.101852	+	0.994800i	$0.467523\pi$
$72$	−1.00000	−0.117851
$73$	−15.0656	−1.76330	−0.881649	−	0.471905i	$-0.843566\pi$
$73$	−15.0656	−1.76330	−0.881649	+	0.471905i	$0.843566\pi$
$74$	−0.498858	−0.0579911
$75$	1.00000	0.115470
$76$	−0.663643	−0.0761250
$77$	−4.43060	−0.504914
$78$	−2.72965	−0.309072
$79$	2.62445	0.295274	0.147637	−	0.989042i	$-0.452833\pi$
$79$	2.62445	0.295274	0.147637	+	0.989042i	$0.452833\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	7.67459	0.847517
$83$	−11.3910	−1.25033	−0.625164	−	0.780493i	$-0.714968\pi$
$83$	−11.3910	−1.25033	−0.625164	+	0.780493i	$0.714968\pi$
$84$	2.61313	0.285115
$85$	0	0
$86$	−9.41838	−1.01561
$87$	7.18759	0.770590
$88$	1.69552	0.180743
$89$	5.25903	0.557456	0.278728	−	0.960370i	$-0.410087\pi$
$89$	5.25903	0.557456	0.278728	+	0.960370i	$0.410087\pi$
$90$	1.00000	0.105409
$91$	7.13291	0.747732
$92$	−0.352746	−0.0367763
$93$	9.50756	0.985888
$94$	−3.33636	−0.344119
$95$	0.663643	0.0680883
$96$	−1.00000	−0.102062
$97$	11.9268	1.21099	0.605493	−	0.795850i	$-0.292976\pi$
$97$	11.9268	1.21099	0.605493	+	0.795850i	$0.292976\pi$
$98$	0.171573	0.0173315
$99$	−1.69552	−0.170406
$100$	1.00000	0.100000
$101$	10.1921	1.01415	0.507077	−	0.861901i	$-0.330726\pi$
$101$	10.1921	1.01415	0.507077	+	0.861901i	$0.330726\pi$
$102$	0	0
$103$	13.4238	1.32269	0.661344	−	0.750083i	$-0.269986\pi$
$103$	13.4238	1.32269	0.661344	+	0.750083i	$0.269986\pi$
$104$	−2.72965	−0.267664
$105$	−2.61313	−0.255015
$106$	−5.06147	−0.491613
$107$	−17.1193	−1.65499	−0.827494	−	0.561474i	$-0.810234\pi$
$107$	−17.1193	−1.65499	−0.827494	+	0.561474i	$0.810234\pi$
$108$	1.00000	0.0962250
$109$	−1.82164	−0.174481	−0.0872407	−	0.996187i	$-0.527805\pi$
$109$	−1.82164	−0.174481	−0.0872407	+	0.996187i	$0.527805\pi$
$110$	−1.69552	−0.161661
$111$	0.498858	0.0473495
$112$	2.61313	0.246917
$113$	15.3292	1.44205	0.721025	−	0.692909i	$-0.243671\pi$
$113$	15.3292	1.44205	0.721025	+	0.692909i	$0.243671\pi$
$114$	0.663643	0.0621558
$115$	0.352746	0.0328937
$116$	7.18759	0.667351
$117$	2.72965	0.252356
$118$	10.7161	0.986494
$119$	0	0
$120$	1.00000	0.0912871
$121$	−8.12522	−0.738656
$122$	7.35237	0.665653
$123$	−7.67459	−0.691995
$124$	9.50756	0.853804
$125$	−1.00000	−0.0894427
$126$	−2.61313	−0.232796
$127$	3.26492	0.289714	0.144857	−	0.989453i	$-0.453728\pi$
$127$	3.26492	0.289714	0.144857	+	0.989453i	$0.453728\pi$
$128$	−1.00000	−0.0883883
$129$	9.41838	0.829242
$130$	2.72965	0.239406
$131$	4.91588	0.429503	0.214751	−	0.976669i	$-0.431106\pi$
$131$	4.91588	0.429503	0.214751	+	0.976669i	$0.431106\pi$
$132$	−1.69552	−0.147576
$133$	−1.73418	−0.150373
$134$	−3.14423	−0.271620
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	8.99321	0.768342	0.384171	−	0.923262i	$-0.374487\pi$
$137$	8.99321	0.768342	0.384171	+	0.923262i	$0.374487\pi$
$138$	0.352746	0.0300277
$139$	13.1967	1.11933	0.559663	−	0.828720i	$-0.310931\pi$
$139$	13.1967	1.11933	0.559663	+	0.828720i	$0.310931\pi$
$140$	−2.61313	−0.220849
$141$	3.33636	0.280972
$142$	−1.71644	−0.144041
$143$	−4.62816	−0.387027
$144$	1.00000	0.0833333
$145$	−7.18759	−0.596897
$146$	15.0656	1.24684
$147$	−0.171573	−0.0141511
$148$	0.498858	0.0410059
$149$	5.57484	0.456708	0.228354	−	0.973578i	$-0.426666\pi$
$149$	5.57484	0.456708	0.228354	+	0.973578i	$0.426666\pi$
$150$	−1.00000	−0.0816497
$151$	−1.67043	−0.135938	−0.0679689	−	0.997687i	$-0.521652\pi$
$151$	−1.67043	−0.135938	−0.0679689	+	0.997687i	$0.521652\pi$
$152$	0.663643	0.0538285
$153$	0	0
$154$	4.43060	0.357028
$155$	−9.50756	−0.763665
$156$	2.72965	0.218547
$157$	−4.56036	−0.363956	−0.181978	−	0.983303i	$-0.558250\pi$
$157$	−4.56036	−0.363956	−0.181978	+	0.983303i	$0.558250\pi$
$158$	−2.62445	−0.208790
$159$	5.06147	0.401400
$160$	1.00000	0.0790569
$161$	−0.921770	−0.0726457
$162$	−1.00000	−0.0785674
$163$	10.2581	0.803479	0.401739	−	0.915754i	$-0.368406\pi$
$163$	10.2581	0.803479	0.401739	+	0.915754i	$0.368406\pi$
$164$	−7.67459	−0.599285
$165$	1.69552	0.131996
$166$	11.3910	0.884116
$167$	10.1416	0.784781	0.392391	−	0.919799i	$-0.371648\pi$
$167$	10.1416	0.784781	0.392391	+	0.919799i	$0.371648\pi$
$168$	−2.61313	−0.201607
$169$	−5.54903	−0.426849
$170$	0	0
$171$	−0.663643	−0.0507500
$172$	9.41838	0.718144
$173$	−12.2362	−0.930303	−0.465152	−	0.885231i	$-0.654000\pi$
$173$	−12.2362	−0.930303	−0.465152	+	0.885231i	$0.654000\pi$
$174$	−7.18759	−0.544890
$175$	2.61313	0.197534
$176$	−1.69552	−0.127804
$177$	−10.7161	−0.805469
$178$	−5.25903	−0.394181
$179$	−16.2764	−1.21655	−0.608277	−	0.793724i	$-0.708139\pi$
$179$	−16.2764	−1.21655	−0.608277	+	0.793724i	$0.708139\pi$
$180$	−1.00000	−0.0745356
$181$	13.4457	0.999412	0.499706	−	0.866195i	$-0.333441\pi$
$181$	13.4457	0.999412	0.499706	+	0.866195i	$0.333441\pi$
$182$	−7.13291	−0.528726
$183$	−7.35237	−0.543503
$184$	0.352746	0.0260048
$185$	−0.498858	−0.0366768
$186$	−9.50756	−0.697128
$187$	0	0
$188$	3.33636	0.243329
$189$	2.61313	0.190077
$190$	−0.663643	−0.0481457
$191$	15.8049	1.14360	0.571800	−	0.820393i	$-0.306245\pi$
$191$	15.8049	1.14360	0.571800	+	0.820393i	$0.306245\pi$
$192$	1.00000	0.0721688
$193$	−7.34502	−0.528706	−0.264353	−	0.964426i	$-0.585158\pi$
$193$	−7.34502	−0.528706	−0.264353	+	0.964426i	$0.585158\pi$
$194$	−11.9268	−0.856297
$195$	−2.72965	−0.195474
$196$	−0.171573	−0.0122552
$197$	−14.5922	−1.03965	−0.519826	−	0.854272i	$-0.674003\pi$
$197$	−14.5922	−1.03965	−0.519826	+	0.854272i	$0.674003\pi$
$198$	1.69552	0.120495
$199$	24.2587	1.71965	0.859825	−	0.510588i	$-0.170572\pi$
$199$	24.2587	1.71965	0.859825	+	0.510588i	$0.170572\pi$
$200$	−1.00000	−0.0707107
$201$	3.14423	0.221777
$202$	−10.1921	−0.717115
$203$	18.7821	1.31824
$204$	0	0
$205$	7.67459	0.536017
$206$	−13.4238	−0.935281
$207$	−0.352746	−0.0245175
$208$	2.72965	0.189267
$209$	1.12522	0.0778330
$210$	2.61313	0.180323
$211$	−7.62408	−0.524863	−0.262432	−	0.964951i	$-0.584524\pi$
$211$	−7.62408	−0.524863	−0.262432	+	0.964951i	$0.584524\pi$
$212$	5.06147	0.347623
$213$	1.71644	0.117609
$214$	17.1193	1.17025
$215$	−9.41838	−0.642328
$216$	−1.00000	−0.0680414
$217$	24.8444	1.68655
$218$	1.82164	0.123377
$219$	−15.0656	−1.01804
$220$	1.69552	0.114312
$221$	0	0
$222$	−0.498858	−0.0334812
$223$	3.95365	0.264756	0.132378	−	0.991199i	$-0.457739\pi$
$223$	3.95365	0.264756	0.132378	+	0.991199i	$0.457739\pi$
$224$	−2.61313	−0.174597
$225$	1.00000	0.0666667
$226$	−15.3292	−1.01968
$227$	−14.1480	−0.939037	−0.469519	−	0.882923i	$-0.655573\pi$
$227$	−14.1480	−0.939037	−0.469519	+	0.882923i	$0.655573\pi$
$228$	−0.663643	−0.0439508
$229$	10.9385	0.722839	0.361419	−	0.932403i	$-0.382292\pi$
$229$	10.9385	0.722839	0.361419	+	0.932403i	$0.382292\pi$
$230$	−0.352746	−0.0232594
$231$	−4.43060	−0.291512
$232$	−7.18759	−0.471888
$233$	10.4325	0.683457	0.341728	−	0.939799i	$-0.388988\pi$
$233$	10.4325	0.683457	0.341728	+	0.939799i	$0.388988\pi$
$234$	−2.72965	−0.178443
$235$	−3.33636	−0.217640
$236$	−10.7161	−0.697557
$237$	2.62445	0.170476
$238$	0	0
$239$	−4.35916	−0.281971	−0.140985	−	0.990012i	$-0.545027\pi$
$239$	−4.35916	−0.281971	−0.140985	+	0.990012i	$0.545027\pi$
$240$	−1.00000	−0.0645497
$241$	−15.9632	−1.02828	−0.514142	−	0.857705i	$-0.671889\pi$
$241$	−15.9632	−1.02828	−0.514142	+	0.857705i	$0.671889\pi$
$242$	8.12522	0.522309
$243$	1.00000	0.0641500
$244$	−7.35237	−0.470687
$245$	0.171573	0.0109614
$246$	7.67459	0.489314
$247$	−1.81151	−0.115264
$248$	−9.50756	−0.603730
$249$	−11.3910	−0.721878
$250$	1.00000	0.0632456
$251$	1.62633	0.102653	0.0513265	−	0.998682i	$-0.483655\pi$
$251$	1.62633	0.102653	0.0513265	+	0.998682i	$0.483655\pi$
$252$	2.61313	0.164611
$253$	0.598087	0.0376014
$254$	−3.26492	−0.204859
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.6987	0.854502	0.427251	−	0.904133i	$-0.359482\pi$
$257$	13.6987	0.854502	0.427251	+	0.904133i	$0.359482\pi$
$258$	−9.41838	−0.586362
$259$	1.30358	0.0810005
$260$	−2.72965	−0.169285
$261$	7.18759	0.444901
$262$	−4.91588	−0.303704
$263$	29.4102	1.81351	0.906756	−	0.421655i	$-0.138551\pi$
$263$	29.4102	1.81351	0.906756	+	0.421655i	$0.138551\pi$
$264$	1.69552	0.104352
$265$	−5.06147	−0.310923
$266$	1.73418	0.106330
$267$	5.25903	0.321847
$268$	3.14423	0.192065
$269$	−18.0502	−1.10054	−0.550269	−	0.834987i	$-0.685475\pi$
$269$	−18.0502	−1.10054	−0.550269	+	0.834987i	$0.685475\pi$
$270$	1.00000	0.0608581
$271$	−3.65685	−0.222138	−0.111069	−	0.993813i	$-0.535427\pi$
$271$	−3.65685	−0.222138	−0.111069	+	0.993813i	$0.535427\pi$
$272$	0	0
$273$	7.13291	0.431703
$274$	−8.99321	−0.543300
$275$	−1.69552	−0.102244
$276$	−0.352746	−0.0212328
$277$	27.1412	1.63076	0.815379	−	0.578927i	$-0.196528\pi$
$277$	27.1412	1.63076	0.815379	+	0.578927i	$0.196528\pi$
$278$	−13.1967	−0.791483
$279$	9.50756	0.569203
$280$	2.61313	0.156164
$281$	3.45929	0.206364	0.103182	−	0.994662i	$-0.467098\pi$
$281$	3.45929	0.206364	0.103182	+	0.994662i	$0.467098\pi$
$282$	−3.33636	−0.198677
$283$	−26.1891	−1.55678	−0.778391	−	0.627780i	$-0.783964\pi$
$283$	−26.1891	−1.55678	−0.778391	+	0.627780i	$0.783964\pi$
$284$	1.71644	0.101852
$285$	0.663643	0.0393108
$286$	4.62816	0.273669
$287$	−20.0547	−1.18379
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	7.18759	0.422070
$291$	11.9268	0.699163
$292$	−15.0656	−0.881649
$293$	22.3616	1.30638	0.653189	−	0.757195i	$-0.273431\pi$
$293$	22.3616	1.30638	0.653189	+	0.757195i	$0.273431\pi$
$294$	0.171573	0.0100063
$295$	10.7161	0.623914
$296$	−0.498858	−0.0289956
$297$	−1.69552	−0.0983839
$298$	−5.57484	−0.322942
$299$	−0.962872	−0.0556843
$300$	1.00000	0.0577350
$301$	24.6114	1.41858
$302$	1.67043	0.0961225
$303$	10.1921	0.585522
$304$	−0.663643	−0.0380625
$305$	7.35237	0.420996
$306$	0	0
$307$	7.62951	0.435439	0.217720	−	0.976011i	$-0.430138\pi$
$307$	7.62951	0.435439	0.217720	+	0.976011i	$0.430138\pi$
$308$	−4.43060	−0.252457
$309$	13.4238	0.763654
$310$	9.50756	0.539993
$311$	−10.7225	−0.608016	−0.304008	−	0.952669i	$-0.598325\pi$
$311$	−10.7225	−0.608016	−0.304008	+	0.952669i	$0.598325\pi$
$312$	−2.72965	−0.154536
$313$	20.0042	1.13070	0.565351	−	0.824850i	$-0.308741\pi$
$313$	20.0042	1.13070	0.565351	+	0.824850i	$0.308741\pi$
$314$	4.56036	0.257356
$315$	−2.61313	−0.147233
$316$	2.62445	0.147637
$317$	12.3264	0.692319	0.346159	−	0.938176i	$-0.387486\pi$
$317$	12.3264	0.692319	0.346159	+	0.938176i	$0.387486\pi$
$318$	−5.06147	−0.283833
$319$	−12.1867	−0.682323
$320$	−1.00000	−0.0559017
$321$	−17.1193	−0.955508
$322$	0.921770	0.0513682
$323$	0	0
$324$	1.00000	0.0555556
$325$	2.72965	0.151414
$326$	−10.2581	−0.568145
$327$	−1.82164	−0.100737
$328$	7.67459	0.423759
$329$	8.71832	0.480657
$330$	−1.69552	−0.0933352
$331$	13.5603	0.745343	0.372671	−	0.927963i	$-0.378442\pi$
$331$	13.5603	0.745343	0.372671	+	0.927963i	$0.378442\pi$
$332$	−11.3910	−0.625164
$333$	0.498858	0.0273373
$334$	−10.1416	−0.554924
$335$	−3.14423	−0.171788
$336$	2.61313	0.142558
$337$	−15.9623	−0.869523	−0.434761	−	0.900546i	$-0.643167\pi$
$337$	−15.9623	−0.869523	−0.434761	+	0.900546i	$0.643167\pi$
$338$	5.54903	0.301828
$339$	15.3292	0.832568
$340$	0	0
$341$	−16.1202	−0.872960
$342$	0.663643	0.0358857
$343$	−18.7402	−1.01188
$344$	−9.41838	−0.507805
$345$	0.352746	0.0189912
$346$	12.2362	0.657824
$347$	−15.7547	−0.845757	−0.422878	−	0.906186i	$-0.638980\pi$
$347$	−15.7547	−0.845757	−0.422878	+	0.906186i	$0.638980\pi$
$348$	7.18759	0.385295
$349$	7.83522	0.419409	0.209705	−	0.977765i	$-0.432750\pi$
$349$	7.83522	0.419409	0.209705	+	0.977765i	$0.432750\pi$
$350$	−2.61313	−0.139677
$351$	2.72965	0.145698
$352$	1.69552	0.0903714
$353$	−5.23638	−0.278704	−0.139352	−	0.990243i	$-0.544502\pi$
$353$	−5.23638	−0.278704	−0.139352	+	0.990243i	$0.544502\pi$
$354$	10.7161	0.569553
$355$	−1.71644	−0.0910993
$356$	5.25903	0.278728
$357$	0	0
$358$	16.2764	0.860234
$359$	−22.7594	−1.20120	−0.600598	−	0.799551i	$-0.705071\pi$
$359$	−22.7594	−1.20120	−0.600598	+	0.799551i	$0.705071\pi$
$360$	1.00000	0.0527046
$361$	−18.5596	−0.976820
$362$	−13.4457	−0.706691
$363$	−8.12522	−0.426463
$364$	7.13291	0.373866
$365$	15.0656	0.788571
$366$	7.35237	0.384315
$367$	−19.7029	−1.02848	−0.514241	−	0.857646i	$-0.671926\pi$
$367$	−19.7029	−1.02848	−0.514241	+	0.857646i	$0.671926\pi$
$368$	−0.352746	−0.0183882
$369$	−7.67459	−0.399523
$370$	0.498858	0.0259344
$371$	13.2263	0.686673
$372$	9.50756	0.492944
$373$	−21.4891	−1.11266	−0.556331	−	0.830961i	$-0.687791\pi$
$373$	−21.4891	−1.11266	−0.556331	+	0.830961i	$0.687791\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−3.33636	−0.172059
$377$	19.6196	1.01046
$378$	−2.61313	−0.134405
$379$	−18.2574	−0.937819	−0.468909	−	0.883246i	$-0.655353\pi$
$379$	−18.2574	−0.937819	−0.468909	+	0.883246i	$0.655353\pi$
$380$	0.663643	0.0340442
$381$	3.26492	0.167267
$382$	−15.8049	−0.808648
$383$	−20.6584	−1.05559	−0.527797	−	0.849370i	$-0.676982\pi$
$383$	−20.6584	−1.05559	−0.527797	+	0.849370i	$0.676982\pi$
$384$	−1.00000	−0.0510310
$385$	4.43060	0.225804
$386$	7.34502	0.373852
$387$	9.41838	0.478763
$388$	11.9268	0.605493
$389$	15.6751	0.794760	0.397380	−	0.917654i	$-0.369919\pi$
$389$	15.6751	0.794760	0.397380	+	0.917654i	$0.369919\pi$
$390$	2.72965	0.138221
$391$	0	0
$392$	0.171573	0.00866574
$393$	4.91588	0.247974
$394$	14.5922	0.735144
$395$	−2.62445	−0.132050
$396$	−1.69552	−0.0852030
$397$	26.3811	1.32403	0.662014	−	0.749492i	$-0.269702\pi$
$397$	26.3811	1.32403	0.662014	+	0.749492i	$0.269702\pi$
$398$	−24.2587	−1.21598
$399$	−1.73418	−0.0868177
$400$	1.00000	0.0500000
$401$	−29.8167	−1.48898	−0.744488	−	0.667636i	$-0.767306\pi$
$401$	−29.8167	−1.48898	−0.744488	+	0.667636i	$0.767306\pi$
$402$	−3.14423	−0.156820
$403$	25.9523	1.29277
$404$	10.1921	0.507077
$405$	−1.00000	−0.0496904
$406$	−18.7821	−0.932139
$407$	−0.845823	−0.0419259
$408$	0	0
$409$	40.3288	1.99413	0.997066	−	0.0765475i	$-0.0243897\pi$
$409$	40.3288	1.99413	0.997066	+	0.0765475i	$0.0243897\pi$
$410$	−7.67459	−0.379021
$411$	8.99321	0.443602
$412$	13.4238	0.661344
$413$	−28.0024	−1.37791
$414$	0.352746	0.0173365
$415$	11.3910	0.559164
$416$	−2.72965	−0.133832
$417$	13.1967	0.646243
$418$	−1.12522	−0.0550362
$419$	14.2626	0.696775	0.348388	−	0.937351i	$-0.386729\pi$
$419$	14.2626	0.696775	0.348388	+	0.937351i	$0.386729\pi$
$420$	−2.61313	−0.127507
$421$	19.3228	0.941735	0.470867	−	0.882204i	$-0.343941\pi$
$421$	19.3228	0.941735	0.470867	+	0.882204i	$0.343941\pi$
$422$	7.62408	0.371134
$423$	3.33636	0.162219
$424$	−5.06147	−0.245807
$425$	0	0
$426$	−1.71644	−0.0831619
$427$	−19.2127	−0.929767
$428$	−17.1193	−0.827494
$429$	−4.62816	−0.223450
$430$	9.41838	0.454194
$431$	31.7349	1.52862	0.764308	−	0.644851i	$-0.223081\pi$
$431$	31.7349	1.52862	0.764308	+	0.644851i	$0.223081\pi$
$432$	1.00000	0.0481125
$433$	31.1785	1.49834	0.749172	−	0.662376i	$-0.230452\pi$
$433$	31.1785	1.49834	0.749172	+	0.662376i	$0.230452\pi$
$434$	−24.8444	−1.19257
$435$	−7.18759	−0.344618
$436$	−1.82164	−0.0872407
$437$	0.234097	0.0111984
$438$	15.0656	0.719864
$439$	−0.812941	−0.0387995	−0.0193998	−	0.999812i	$-0.506176\pi$
$439$	−0.812941	−0.0387995	−0.0193998	+	0.999812i	$0.506176\pi$
$440$	−1.69552	−0.0808307
$441$	−0.171573	−0.00817014
$442$	0	0
$443$	−2.54903	−0.121108	−0.0605541	−	0.998165i	$-0.519287\pi$
$443$	−2.54903	−0.121108	−0.0605541	+	0.998165i	$0.519287\pi$
$444$	0.498858	0.0236748
$445$	−5.25903	−0.249302
$446$	−3.95365	−0.187211
$447$	5.57484	0.263681
$448$	2.61313	0.123459
$449$	−11.3405	−0.535192	−0.267596	−	0.963531i	$-0.586229\pi$
$449$	−11.3405	−0.535192	−0.267596	+	0.963531i	$0.586229\pi$
$450$	−1.00000	−0.0471405
$451$	13.0124	0.612731
$452$	15.3292	0.721025
$453$	−1.67043	−0.0784837
$454$	14.1480	0.664000
$455$	−7.13291	−0.334396
$456$	0.663643	0.0310779
$457$	−26.0379	−1.21800	−0.609001	−	0.793169i	$-0.708430\pi$
$457$	−26.0379	−1.21800	−0.609001	+	0.793169i	$0.708430\pi$
$458$	−10.9385	−0.511124
$459$	0	0
$460$	0.352746	0.0164469
$461$	3.04245	0.141701	0.0708506	−	0.997487i	$-0.477429\pi$
$461$	3.04245	0.141701	0.0708506	+	0.997487i	$0.477429\pi$
$462$	4.43060	0.206130
$463$	10.4989	0.487923	0.243962	−	0.969785i	$-0.421553\pi$
$463$	10.4989	0.487923	0.243962	+	0.969785i	$0.421553\pi$
$464$	7.18759	0.333675
$465$	−9.50756	−0.440902
$466$	−10.4325	−0.483277
$467$	9.44646	0.437130	0.218565	−	0.975822i	$-0.429862\pi$
$467$	9.44646	0.437130	0.218565	+	0.975822i	$0.429862\pi$
$468$	2.72965	0.126178
$469$	8.21628	0.379392
$470$	3.33636	0.153895
$471$	−4.56036	−0.210130
$472$	10.7161	0.493247
$473$	−15.9690	−0.734257
$474$	−2.62445	−0.120545
$475$	−0.663643	−0.0304500
$476$	0	0
$477$	5.06147	0.231749
$478$	4.35916	0.199383
$479$	26.1727	1.19586	0.597931	−	0.801548i	$-0.295990\pi$
$479$	26.1727	1.19586	0.597931	+	0.801548i	$0.295990\pi$
$480$	1.00000	0.0456435
$481$	1.36171	0.0620885
$482$	15.9632	0.727106
$483$	−0.921770	−0.0419420
$484$	−8.12522	−0.369328
$485$	−11.9268	−0.541570
$486$	−1.00000	−0.0453609
$487$	4.62220	0.209452	0.104726	−	0.994501i	$-0.466603\pi$
$487$	4.62220	0.209452	0.104726	+	0.994501i	$0.466603\pi$
$488$	7.35237	0.332826
$489$	10.2581	0.463889
$490$	−0.171573	−0.00775087
$491$	30.3175	1.36821	0.684105	−	0.729384i	$-0.260193\pi$
$491$	30.3175	1.36821	0.684105	+	0.729384i	$0.260193\pi$
$492$	−7.67459	−0.345997
$493$	0	0
$494$	1.81151	0.0815037
$495$	1.69552	0.0762079
$496$	9.50756	0.426902
$497$	4.48528	0.201192
$498$	11.3910	0.510445
$499$	41.7854	1.87057	0.935286	−	0.353893i	$-0.115142\pi$
$499$	41.7854	1.87057	0.935286	+	0.353893i	$0.115142\pi$
$500$	−1.00000	−0.0447214
$501$	10.1416	0.453094
$502$	−1.62633	−0.0725866
$503$	26.3056	1.17291	0.586455	−	0.809982i	$-0.300523\pi$
$503$	26.3056	1.17291	0.586455	+	0.809982i	$0.300523\pi$
$504$	−2.61313	−0.116398
$505$	−10.1921	−0.453544
$506$	−0.598087	−0.0265882
$507$	−5.54903	−0.246441
$508$	3.26492	0.144857
$509$	40.0002	1.77298	0.886489	−	0.462751i	$-0.153137\pi$
$509$	40.0002	1.77298	0.886489	+	0.462751i	$0.153137\pi$
$510$	0	0
$511$	−39.3684	−1.74155
$512$	−1.00000	−0.0441942
$513$	−0.663643	−0.0293005
$514$	−13.6987	−0.604224
$515$	−13.4238	−0.591524
$516$	9.41838	0.414621
$517$	−5.65685	−0.248788
$518$	−1.30358	−0.0572760
$519$	−12.2362	−0.537111
$520$	2.72965	0.119703
$521$	−24.9555	−1.09332	−0.546661	−	0.837354i	$-0.684101\pi$
$521$	−24.9555	−1.09332	−0.546661	+	0.837354i	$0.684101\pi$
$522$	−7.18759	−0.314592
$523$	−13.1627	−0.575564	−0.287782	−	0.957696i	$-0.592918\pi$
$523$	−13.1627	−0.575564	−0.287782	+	0.957696i	$0.592918\pi$
$524$	4.91588	0.214751
$525$	2.61313	0.114046
$526$	−29.4102	−1.28235
$527$	0	0
$528$	−1.69552	−0.0737880
$529$	−22.8756	−0.994590
$530$	5.06147	0.219856
$531$	−10.7161	−0.465038
$532$	−1.73418	−0.0751863
$533$	−20.9489	−0.907399
$534$	−5.25903	−0.227580
$535$	17.1193	0.740133
$536$	−3.14423	−0.135810
$537$	−16.2764	−0.702378
$538$	18.0502	0.778198
$539$	0.290905	0.0125302
$540$	−1.00000	−0.0430331
$541$	17.8049	0.765491	0.382746	−	0.923854i	$-0.374979\pi$
$541$	17.8049	0.765491	0.382746	+	0.923854i	$0.374979\pi$
$542$	3.65685	0.157075
$543$	13.4457	0.577011
$544$	0	0
$545$	1.82164	0.0780304
$546$	−7.13291	−0.305260
$547$	22.3320	0.954848	0.477424	−	0.878673i	$-0.341571\pi$
$547$	22.3320	0.954848	0.477424	+	0.878673i	$0.341571\pi$
$548$	8.99321	0.384171
$549$	−7.35237	−0.313792
$550$	1.69552	0.0722971
$551$	−4.76999	−0.203208
$552$	0.352746	0.0150139
$553$	6.85802	0.291633
$554$	−27.1412	−1.15312
$555$	−0.498858	−0.0211754
$556$	13.1967	0.559663
$557$	8.08746	0.342677	0.171338	−	0.985212i	$-0.445191\pi$
$557$	8.08746	0.342677	0.171338	+	0.985212i	$0.445191\pi$
$558$	−9.50756	−0.402487
$559$	25.7088	1.08737
$560$	−2.61313	−0.110425
$561$	0	0
$562$	−3.45929	−0.145921
$563$	26.8176	1.13023	0.565113	−	0.825014i	$-0.308833\pi$
$563$	26.8176	1.13023	0.565113	+	0.825014i	$0.308833\pi$
$564$	3.33636	0.140486
$565$	−15.3292	−0.644904
$566$	26.1891	1.10081
$567$	2.61313	0.109741
$568$	−1.71644	−0.0720203
$569$	45.8132	1.92059	0.960295	−	0.278988i	$-0.0899989\pi$
$569$	45.8132	1.92059	0.960295	+	0.278988i	$0.0899989\pi$
$570$	−0.663643	−0.0277969
$571$	−31.5847	−1.32178	−0.660889	−	0.750484i	$-0.729820\pi$
$571$	−31.5847	−1.32178	−0.660889	+	0.750484i	$0.729820\pi$
$572$	−4.62816	−0.193513
$573$	15.8049	0.660258
$574$	20.0547	0.837066
$575$	−0.352746	−0.0147105
$576$	1.00000	0.0416667
$577$	16.0696	0.668988	0.334494	−	0.942398i	$-0.391435\pi$
$577$	16.0696	0.668988	0.334494	+	0.942398i	$0.391435\pi$
$578$	0	0
$579$	−7.34502	−0.305249
$580$	−7.18759	−0.298448
$581$	−29.7662	−1.23491
$582$	−11.9268	−0.494383
$583$	−8.58181	−0.355422
$584$	15.0656	0.623420
$585$	−2.72965	−0.112857
$586$	−22.3616	−0.923749
$587$	36.3926	1.50208	0.751041	−	0.660255i	$-0.229552\pi$
$587$	36.3926	1.50208	0.751041	+	0.660255i	$0.229552\pi$
$588$	−0.171573	−0.00707555
$589$	−6.30962	−0.259983
$590$	−10.7161	−0.441174
$591$	−14.5922	−0.600243
$592$	0.498858	0.0205030
$593$	−3.18440	−0.130768	−0.0653839	−	0.997860i	$-0.520827\pi$
$593$	−3.18440	−0.130768	−0.0653839	+	0.997860i	$0.520827\pi$
$594$	1.69552	0.0695680
$595$	0	0
$596$	5.57484	0.228354
$597$	24.2587	0.992841
$598$	0.962872	0.0393748
$599$	30.6501	1.25233	0.626164	−	0.779691i	$-0.284624\pi$
$599$	30.6501	1.25233	0.626164	+	0.779691i	$0.284624\pi$
$600$	−1.00000	−0.0408248
$601$	−7.47346	−0.304849	−0.152424	−	0.988315i	$-0.548708\pi$
$601$	−7.47346	−0.304849	−0.152424	+	0.988315i	$0.548708\pi$
$602$	−24.6114	−1.00309
$603$	3.14423	0.128043
$604$	−1.67043	−0.0679689
$605$	8.12522	0.330337
$606$	−10.1921	−0.414027
$607$	33.6346	1.36519	0.682593	−	0.730798i	$-0.260852\pi$
$607$	33.6346	1.36519	0.682593	+	0.730798i	$0.260852\pi$
$608$	0.663643	0.0269143
$609$	18.7821	0.761088
$610$	−7.35237	−0.297689
$611$	9.10707	0.368433
$612$	0	0
$613$	−20.7742	−0.839062	−0.419531	−	0.907741i	$-0.637805\pi$
$613$	−20.7742	−0.839062	−0.419531	+	0.907741i	$0.637805\pi$
$614$	−7.62951	−0.307902
$615$	7.67459	0.309469
$616$	4.43060	0.178514
$617$	−48.2451	−1.94227	−0.971137	−	0.238523i	$-0.923337\pi$
$617$	−48.2451	−1.94227	−0.971137	+	0.238523i	$0.923337\pi$
$618$	−13.4238	−0.539985
$619$	−27.8891	−1.12096	−0.560480	−	0.828168i	$-0.689383\pi$
$619$	−27.8891	−1.12096	−0.560480	+	0.828168i	$0.689383\pi$
$620$	−9.50756	−0.381833
$621$	−0.352746	−0.0141552
$622$	10.7225	0.429932
$623$	13.7425	0.550582
$624$	2.72965	0.109273
$625$	1.00000	0.0400000
$626$	−20.0042	−0.799527
$627$	1.12522	0.0449369
$628$	−4.56036	−0.181978
$629$	0	0
$630$	2.61313	0.104109
$631$	16.7776	0.667904	0.333952	−	0.942590i	$-0.391618\pi$
$631$	16.7776	0.667904	0.333952	+	0.942590i	$0.391618\pi$
$632$	−2.62445	−0.104395
$633$	−7.62408	−0.303030
$634$	−12.3264	−0.489543
$635$	−3.26492	−0.129564
$636$	5.06147	0.200700
$637$	−0.468333	−0.0185560
$638$	12.1867	0.482476
$639$	1.71644	0.0679014
$640$	1.00000	0.0395285
$641$	−19.8722	−0.784903	−0.392451	−	0.919773i	$-0.628373\pi$
$641$	−19.8722	−0.784903	−0.392451	+	0.919773i	$0.628373\pi$
$642$	17.1193	0.675646
$643$	−35.2287	−1.38928	−0.694642	−	0.719356i	$-0.744437\pi$
$643$	−35.2287	−1.38928	−0.694642	+	0.719356i	$0.744437\pi$
$644$	−0.921770	−0.0363228
$645$	−9.41838	−0.370848
$646$	0	0
$647$	−32.9106	−1.29385	−0.646925	−	0.762553i	$-0.723945\pi$
$647$	−32.9106	−1.29385	−0.646925	+	0.762553i	$0.723945\pi$
$648$	−1.00000	−0.0392837
$649$	18.1693	0.713207
$650$	−2.72965	−0.107066
$651$	24.8444	0.973730
$652$	10.2581	0.401739
$653$	−22.8730	−0.895089	−0.447544	−	0.894262i	$-0.647701\pi$
$653$	−22.8730	−0.895089	−0.447544	+	0.894262i	$0.647701\pi$
$654$	1.82164	0.0712317
$655$	−4.91588	−0.192079
$656$	−7.67459	−0.299643
$657$	−15.0656	−0.587766
$658$	−8.71832	−0.339876
$659$	14.1116	0.549710	0.274855	−	0.961486i	$-0.411370\pi$
$659$	14.1116	0.549710	0.274855	+	0.961486i	$0.411370\pi$
$660$	1.69552	0.0659980
$661$	3.01586	0.117303	0.0586517	−	0.998279i	$-0.481320\pi$
$661$	3.01586	0.117303	0.0586517	+	0.998279i	$0.481320\pi$
$662$	−13.5603	−0.527037
$663$	0	0
$664$	11.3910	0.442058
$665$	1.73418	0.0672487
$666$	−0.498858	−0.0193304
$667$	−2.53539	−0.0981708
$668$	10.1416	0.392391
$669$	3.95365	0.152857
$670$	3.14423	0.121472
$671$	12.4661	0.481248
$672$	−2.61313	−0.100804
$673$	−34.5577	−1.33210	−0.666051	−	0.745906i	$-0.732017\pi$
$673$	−34.5577	−1.33210	−0.666051	+	0.745906i	$0.732017\pi$
$674$	15.9623	0.614845
$675$	1.00000	0.0384900
$676$	−5.54903	−0.213424
$677$	−25.7812	−0.990851	−0.495425	−	0.868650i	$-0.664988\pi$
$677$	−25.7812	−0.990851	−0.495425	+	0.868650i	$0.664988\pi$
$678$	−15.3292	−0.588714
$679$	31.1663	1.19605
$680$	0	0
$681$	−14.1480	−0.542153
$682$	16.1202	0.617276
$683$	−28.2640	−1.08149	−0.540746	−	0.841186i	$-0.681858\pi$
$683$	−28.2640	−1.08149	−0.540746	+	0.841186i	$0.681858\pi$
$684$	−0.663643	−0.0253750
$685$	−8.99321	−0.343613
$686$	18.7402	0.715505
$687$	10.9385	0.417331
$688$	9.41838	0.359072
$689$	13.8160	0.526348
$690$	−0.352746	−0.0134288
$691$	−30.4989	−1.16023	−0.580116	−	0.814534i	$-0.696993\pi$
$691$	−30.4989	−1.16023	−0.580116	+	0.814534i	$0.696993\pi$
$692$	−12.2362	−0.465152
$693$	−4.43060	−0.168305
$694$	15.7547	0.598040
$695$	−13.1967	−0.500578
$696$	−7.18759	−0.272445
$697$	0	0
$698$	−7.83522	−0.296567
$699$	10.4325	0.394594
$700$	2.61313	0.0987669
$701$	14.3286	0.541185	0.270593	−	0.962694i	$-0.412780\pi$
$701$	14.3286	0.541185	0.270593	+	0.962694i	$0.412780\pi$
$702$	−2.72965	−0.103024
$703$	−0.331064	−0.0124863
$704$	−1.69552	−0.0639022
$705$	−3.33636	−0.125654
$706$	5.23638	0.197074
$707$	26.6333	1.00165
$708$	−10.7161	−0.402735
$709$	−27.6948	−1.04010	−0.520049	−	0.854136i	$-0.674087\pi$
$709$	−27.6948	−1.04010	−0.520049	+	0.854136i	$0.674087\pi$
$710$	1.71644	0.0644170
$711$	2.62445	0.0984246
$712$	−5.25903	−0.197090
$713$	−3.35375	−0.125599
$714$	0	0
$715$	4.62816	0.173084
$716$	−16.2764	−0.608277
$717$	−4.35916	−0.162796
$718$	22.7594	0.849374
$719$	32.9374	1.22836	0.614180	−	0.789166i	$-0.289487\pi$
$719$	32.9374	1.22836	0.614180	+	0.789166i	$0.289487\pi$
$720$	−1.00000	−0.0372678
$721$	35.0781	1.30638
$722$	18.5596	0.690716
$723$	−15.9632	−0.593679
$724$	13.4457	0.499706
$725$	7.18759	0.266940
$726$	8.12522	0.301555
$727$	5.23320	0.194088	0.0970442	−	0.995280i	$-0.469061\pi$
$727$	5.23320	0.194088	0.0970442	+	0.995280i	$0.469061\pi$
$728$	−7.13291	−0.264363
$729$	1.00000	0.0370370
$730$	−15.0656	−0.557604
$731$	0	0
$732$	−7.35237	−0.271752
$733$	−19.8824	−0.734373	−0.367186	−	0.930147i	$-0.619679\pi$
$733$	−19.8824	−0.734373	−0.367186	+	0.930147i	$0.619679\pi$
$734$	19.7029	0.727246
$735$	0.171573	0.00632856
$736$	0.352746	0.0130024
$737$	−5.33110	−0.196374
$738$	7.67459	0.282506
$739$	−24.5901	−0.904563	−0.452281	−	0.891875i	$-0.649390\pi$
$739$	−24.5901	−0.904563	−0.452281	+	0.891875i	$0.649390\pi$
$740$	−0.498858	−0.0183384
$741$	−1.81151	−0.0665475
$742$	−13.2263	−0.485551
$743$	−20.9700	−0.769316	−0.384658	−	0.923059i	$-0.625681\pi$
$743$	−20.9700	−0.769316	−0.384658	+	0.923059i	$0.625681\pi$
$744$	−9.50756	−0.348564
$745$	−5.57484	−0.204246
$746$	21.4891	0.786771
$747$	−11.3910	−0.416776
$748$	0	0
$749$	−44.7350	−1.63458
$750$	1.00000	0.0365148
$751$	39.6852	1.44813	0.724066	−	0.689730i	$-0.242271\pi$
$751$	39.6852	1.44813	0.724066	+	0.689730i	$0.242271\pi$
$752$	3.33636	0.121664
$753$	1.62633	0.0592667
$754$	−19.6196	−0.714503
$755$	1.67043	0.0607932
$756$	2.61313	0.0950385
$757$	17.2059	0.625357	0.312679	−	0.949859i	$-0.398774\pi$
$757$	17.2059	0.625357	0.312679	+	0.949859i	$0.398774\pi$
$758$	18.2574	0.663138
$759$	0.598087	0.0217092
$760$	−0.663643	−0.0240729
$761$	41.0176	1.48689	0.743443	−	0.668800i	$-0.233192\pi$
$761$	41.0176	1.48689	0.743443	+	0.668800i	$0.233192\pi$
$762$	−3.26492	−0.118275
$763$	−4.76017	−0.172330
$764$	15.8049	0.571800
$765$	0	0
$766$	20.6584	0.746418
$767$	−29.2511	−1.05620
$768$	1.00000	0.0360844
$769$	−44.6904	−1.61158	−0.805789	−	0.592203i	$-0.798258\pi$
$769$	−44.6904	−1.61158	−0.805789	+	0.592203i	$0.798258\pi$
$770$	−4.43060	−0.159668
$771$	13.6987	0.493347
$772$	−7.34502	−0.264353
$773$	−6.61278	−0.237845	−0.118923	−	0.992904i	$-0.537944\pi$
$773$	−6.61278	−0.237845	−0.118923	+	0.992904i	$0.537944\pi$
$774$	−9.41838	−0.338537
$775$	9.50756	0.341522
$776$	−11.9268	−0.428148
$777$	1.30358	0.0467657
$778$	−15.6751	−0.561980
$779$	5.09319	0.182482
$780$	−2.72965	−0.0977370
$781$	−2.91026	−0.104137
$782$	0	0
$783$	7.18759	0.256863
$784$	−0.171573	−0.00612760
$785$	4.56036	0.162766
$786$	−4.91588	−0.175344
$787$	13.0897	0.466599	0.233299	−	0.972405i	$-0.425048\pi$
$787$	13.0897	0.466599	0.233299	+	0.972405i	$0.425048\pi$
$788$	−14.5922	−0.519826
$789$	29.4102	1.04703
$790$	2.62445	0.0933738
$791$	40.0571	1.42427
$792$	1.69552	0.0602476
$793$	−20.0694	−0.712684
$794$	−26.3811	−0.936229
$795$	−5.06147	−0.179512
$796$	24.2587	0.859825
$797$	44.2278	1.56663	0.783314	−	0.621626i	$-0.213528\pi$
$797$	44.2278	1.56663	0.783314	+	0.621626i	$0.213528\pi$
$798$	1.73418	0.0613894
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	5.25903	0.185819
$802$	29.8167	1.05287
$803$	25.5440	0.901430
$804$	3.14423	0.110889
$805$	0.921770	0.0324881
$806$	−25.9523	−0.914130
$807$	−18.0502	−0.635396
$808$	−10.1921	−0.358558
$809$	19.7519	0.694441	0.347220	−	0.937784i	$-0.387126\pi$
$809$	19.7519	0.694441	0.347220	+	0.937784i	$0.387126\pi$
$810$	1.00000	0.0351364
$811$	41.1168	1.44381	0.721903	−	0.691995i	$-0.243268\pi$
$811$	41.1168	1.44381	0.721903	+	0.691995i	$0.243268\pi$
$812$	18.7821	0.659122
$813$	−3.65685	−0.128251
$814$	0.845823	0.0296461
$815$	−10.2581	−0.359327
$816$	0	0
$817$	−6.25044	−0.218675
$818$	−40.3288	−1.41006
$819$	7.13291	0.249244
$820$	7.67459	0.268008
$821$	−32.2773	−1.12649	−0.563243	−	0.826291i	$-0.690447\pi$
$821$	−32.2773	−1.12649	−0.563243	+	0.826291i	$0.690447\pi$
$822$	−8.99321	−0.313674
$823$	−31.9842	−1.11490	−0.557450	−	0.830211i	$-0.688220\pi$
$823$	−31.9842	−1.11490	−0.557450	+	0.830211i	$0.688220\pi$
$824$	−13.4238	−0.467641
$825$	−1.69552	−0.0590304
$826$	28.0024	0.974329
$827$	−32.4518	−1.12846	−0.564229	−	0.825618i	$-0.690827\pi$
$827$	−32.4518	−1.12846	−0.564229	+	0.825618i	$0.690827\pi$
$828$	−0.352746	−0.0122588
$829$	10.6602	0.370244	0.185122	−	0.982716i	$-0.440732\pi$
$829$	10.6602	0.370244	0.185122	+	0.982716i	$0.440732\pi$
$830$	−11.3910	−0.395389
$831$	27.1412	0.941519
$832$	2.72965	0.0946335
$833$	0	0
$834$	−13.1967	−0.456963
$835$	−10.1416	−0.350965
$836$	1.12522	0.0389165
$837$	9.50756	0.328629
$838$	−14.2626	−0.492694
$839$	−18.8190	−0.649704	−0.324852	−	0.945765i	$-0.605315\pi$
$839$	−18.8190	−0.649704	−0.324852	+	0.945765i	$0.605315\pi$
$840$	2.61313	0.0901614
$841$	22.6614	0.781428
$842$	−19.3228	−0.665907
$843$	3.45929	0.119144
$844$	−7.62408	−0.262432
$845$	5.54903	0.190893
$846$	−3.33636	−0.114706
$847$	−21.2322	−0.729548
$848$	5.06147	0.173812
$849$	−26.1891	−0.898808
$850$	0	0
$851$	−0.175970	−0.00603219
$852$	1.71644	0.0588044
$853$	20.9074	0.715857	0.357929	−	0.933749i	$-0.383483\pi$
$853$	20.9074	0.715857	0.357929	+	0.933749i	$0.383483\pi$
$854$	19.2127	0.657444
$855$	0.663643	0.0226961
$856$	17.1193	0.585127
$857$	−23.2090	−0.792803	−0.396401	−	0.918077i	$-0.629741\pi$
$857$	−23.2090	−0.792803	−0.396401	+	0.918077i	$0.629741\pi$
$858$	4.62816	0.158003
$859$	43.6052	1.48779	0.743896	−	0.668295i	$-0.232976\pi$
$859$	43.6052	1.48779	0.743896	+	0.668295i	$0.232976\pi$
$860$	−9.41838	−0.321164
$861$	−20.0547	−0.683462
$862$	−31.7349	−1.08089
$863$	−34.7858	−1.18412	−0.592062	−	0.805893i	$-0.701686\pi$
$863$	−34.7858	−1.18412	−0.592062	+	0.805893i	$0.701686\pi$
$864$	−1.00000	−0.0340207
$865$	12.2362	0.416044
$866$	−31.1785	−1.05949
$867$	0	0
$868$	24.8444	0.843275
$869$	−4.44980	−0.150949
$870$	7.18759	0.243682
$871$	8.58264	0.290812
$872$	1.82164	0.0616885
$873$	11.9268	0.403662
$874$	−0.234097	−0.00791846
$875$	−2.61313	−0.0883398
$876$	−15.0656	−0.509020
$877$	3.54299	0.119638	0.0598192	−	0.998209i	$-0.480948\pi$
$877$	3.54299	0.119638	0.0598192	+	0.998209i	$0.480948\pi$
$878$	0.812941	0.0274354
$879$	22.3616	0.754238
$880$	1.69552	0.0571559
$881$	45.5245	1.53376	0.766880	−	0.641791i	$-0.221808\pi$
$881$	45.5245	1.53376	0.766880	+	0.641791i	$0.221808\pi$
$882$	0.171573	0.00577716
$883$	−28.3562	−0.954261	−0.477130	−	0.878833i	$-0.658323\pi$
$883$	−28.3562	−0.954261	−0.477130	+	0.878833i	$0.658323\pi$
$884$	0	0
$885$	10.7161	0.360217
$886$	2.54903	0.0856364
$887$	−44.7763	−1.50344	−0.751720	−	0.659482i	$-0.770776\pi$
$887$	−44.7763	−1.50344	−0.751720	+	0.659482i	$0.770776\pi$
$888$	−0.498858	−0.0167406
$889$	8.53164	0.286142
$890$	5.25903	0.176283
$891$	−1.69552	−0.0568020
$892$	3.95365	0.132378
$893$	−2.21415	−0.0740937
$894$	−5.57484	−0.186450
$895$	16.2764	0.544060
$896$	−2.61313	−0.0872984
$897$	−0.962872	−0.0321494
$898$	11.3405	0.378438
$899$	68.3364	2.27915
$900$	1.00000	0.0333333
$901$	0	0
$902$	−13.0124	−0.433266
$903$	24.6114	0.819016
$904$	−15.3292	−0.509841
$905$	−13.4457	−0.446951
$906$	1.67043	0.0554964
$907$	14.9668	0.496965	0.248482	−	0.968636i	$-0.420068\pi$
$907$	14.9668	0.496965	0.248482	+	0.968636i	$0.420068\pi$
$908$	−14.1480	−0.469519
$909$	10.1921	0.338051
$910$	7.13291	0.236454
$911$	0.483820	0.0160297	0.00801483	−	0.999968i	$-0.497449\pi$
$911$	0.483820	0.0160297	0.00801483	+	0.999968i	$0.497449\pi$
$912$	−0.663643	−0.0219754
$913$	19.3137	0.639190
$914$	26.0379	0.861258
$915$	7.35237	0.243062
$916$	10.9385	0.361419
$917$	12.8458	0.424206
$918$	0	0
$919$	54.6848	1.80388	0.901942	−	0.431856i	$-0.142141\pi$
$919$	54.6848	1.80388	0.901942	+	0.431856i	$0.142141\pi$
$920$	−0.352746	−0.0116297
$921$	7.62951	0.251401
$922$	−3.04245	−0.100198
$923$	4.68528	0.154218
$924$	−4.43060	−0.145756
$925$	0.498858	0.0164024
$926$	−10.4989	−0.345014
$927$	13.4238	0.440896
$928$	−7.18759	−0.235944
$929$	20.4766	0.671816	0.335908	−	0.941895i	$-0.390957\pi$
$929$	20.4766	0.671816	0.335908	+	0.941895i	$0.390957\pi$
$930$	9.50756	0.311765
$931$	0.113863	0.00373171
$932$	10.4325	0.341728
$933$	−10.7225	−0.351038
$934$	−9.44646	−0.309098
$935$	0	0
$936$	−2.72965	−0.0892213
$937$	22.8503	0.746488	0.373244	−	0.927733i	$-0.378245\pi$
$937$	22.8503	0.746488	0.373244	+	0.927733i	$0.378245\pi$
$938$	−8.21628	−0.268271
$939$	20.0042	0.652811
$940$	−3.33636	−0.108820
$941$	40.3867	1.31657	0.658284	−	0.752770i	$-0.271283\pi$
$941$	40.3867	1.31657	0.658284	+	0.752770i	$0.271283\pi$
$942$	4.56036	0.148584
$943$	2.70718	0.0881580
$944$	−10.7161	−0.348778
$945$	−2.61313	−0.0850050
$946$	15.9690	0.519198
$947$	25.1193	0.816269	0.408134	−	0.912922i	$-0.366180\pi$
$947$	25.1193	0.816269	0.408134	+	0.912922i	$0.366180\pi$
$948$	2.62445	0.0852382
$949$	−41.1238	−1.33494
$950$	0.663643	0.0215314
$951$	12.3264	0.399710
$952$	0	0
$953$	55.5498	1.79944	0.899718	−	0.436473i	$-0.143772\pi$
$953$	55.5498	1.79944	0.899718	+	0.436473i	$0.143772\pi$
$954$	−5.06147	−0.163871
$955$	−15.8049	−0.511434
$956$	−4.35916	−0.140985
$957$	−12.1867	−0.393940
$958$	−26.1727	−0.845602
$959$	23.5004	0.758867
$960$	−1.00000	−0.0322749
$961$	59.3936	1.91592
$962$	−1.36171	−0.0439032
$963$	−17.1193	−0.551663
$964$	−15.9632	−0.514142
$965$	7.34502	0.236445
$966$	0.921770	0.0296575
$967$	−41.9356	−1.34856	−0.674278	−	0.738477i	$-0.735545\pi$
$967$	−41.9356	−1.34856	−0.674278	+	0.738477i	$0.735545\pi$
$968$	8.12522	0.261154
$969$	0	0
$970$	11.9268	0.382948
$971$	−8.66409	−0.278044	−0.139022	−	0.990289i	$-0.544396\pi$
$971$	−8.66409	−0.278044	−0.139022	+	0.990289i	$0.544396\pi$
$972$	1.00000	0.0320750
$973$	34.4845	1.10552
$974$	−4.62220	−0.148105
$975$	2.72965	0.0874186
$976$	−7.35237	−0.235344
$977$	−22.6138	−0.723481	−0.361740	−	0.932279i	$-0.617817\pi$
$977$	−22.6138	−0.723481	−0.361740	+	0.932279i	$0.617817\pi$
$978$	−10.2581	−0.328019
$979$	−8.91678	−0.284982
$980$	0.171573	0.00548069
$981$	−1.82164	−0.0581604
$982$	−30.3175	−0.967470
$983$	−56.0507	−1.78774	−0.893870	−	0.448326i	$-0.852020\pi$
$983$	−56.0507	−1.78774	−0.893870	+	0.448326i	$0.852020\pi$
$984$	7.67459	0.244657
$985$	14.5922	0.464946
$986$	0	0
$987$	8.71832	0.277507
$988$	−1.81151	−0.0576318
$989$	−3.32230	−0.105643
$990$	−1.69552	−0.0538871
$991$	28.8213	0.915538	0.457769	−	0.889071i	$-0.348649\pi$
$991$	28.8213	0.915538	0.457769	+	0.889071i	$0.348649\pi$
$992$	−9.50756	−0.301865
$993$	13.5603	0.430324
$994$	−4.48528	−0.142264
$995$	−24.2587	−0.769051
$996$	−11.3910	−0.360939
$997$	54.0887	1.71301	0.856503	−	0.516143i	$-0.172633\pi$
$997$	54.0887	1.71301	0.856503	+	0.516143i	$0.172633\pi$
$998$	−41.7854	−1.32269
$999$	0.498858	0.0157832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bw.1.4		4
17.2	even	8		510.2.p.d.361.3	✓	8
17.9	even	8		510.2.p.d.421.3	yes	8
17.16	even	2		8670.2.a.bt.1.1		4
51.2	odd	8		1530.2.q.i.361.1		8
51.26	odd	8		1530.2.q.i.1441.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.p.d.361.3	✓	8	17.2	even	8
510.2.p.d.421.3	yes	8	17.9	even	8
1530.2.q.i.361.1		8	51.2	odd	8
1530.2.q.i.1441.1		8	51.26	odd	8
8670.2.a.bt.1.1		4	17.16	even	2
8670.2.a.bw.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.69552 q^{11} +1.00000 q^{12} +2.72965 q^{13} -2.61313 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -0.663643 q^{19} -1.00000 q^{20} +2.61313 q^{21} +1.69552 q^{22} -0.352746 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.72965 q^{26} +1.00000 q^{27} +2.61313 q^{28} +7.18759 q^{29} +1.00000 q^{30} +9.50756 q^{31} -1.00000 q^{32} -1.69552 q^{33} -2.61313 q^{35} +1.00000 q^{36} +0.498858 q^{37} +0.663643 q^{38} +2.72965 q^{39} +1.00000 q^{40} -7.67459 q^{41} -2.61313 q^{42} +9.41838 q^{43} -1.69552 q^{44} -1.00000 q^{45} +0.352746 q^{46} +3.33636 q^{47} +1.00000 q^{48} -0.171573 q^{49} -1.00000 q^{50} +2.72965 q^{52} +5.06147 q^{53} -1.00000 q^{54} +1.69552 q^{55} -2.61313 q^{56} -0.663643 q^{57} -7.18759 q^{58} -10.7161 q^{59} -1.00000 q^{60} -7.35237 q^{61} -9.50756 q^{62} +2.61313 q^{63} +1.00000 q^{64} -2.72965 q^{65} +1.69552 q^{66} +3.14423 q^{67} -0.352746 q^{69} +2.61313 q^{70} +1.71644 q^{71} -1.00000 q^{72} -15.0656 q^{73} -0.498858 q^{74} +1.00000 q^{75} -0.663643 q^{76} -4.43060 q^{77} -2.72965 q^{78} +2.62445 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.67459 q^{82} -11.3910 q^{83} +2.61313 q^{84} -9.41838 q^{86} +7.18759 q^{87} +1.69552 q^{88} +5.25903 q^{89} +1.00000 q^{90} +7.13291 q^{91} -0.352746 q^{92} +9.50756 q^{93} -3.33636 q^{94} +0.663643 q^{95} -1.00000 q^{96} +11.9268 q^{97} +0.171573 q^{98} -1.69552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{10} + 8 q^{11} + 4 q^{12} - 4 q^{15} + 4 q^{16} - 4 q^{18} - 4 q^{20} - 8 q^{22} - 8 q^{23} - 4 q^{24} + 4 q^{25} + 4 q^{27}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 - 4 * q^6 - 4 * q^8 + 4 * q^9 + 4 * q^10 + 8 * q^11 + 4 * q^12 - 4 * q^15 + 4 * q^16 - 4 * q^18 - 4 * q^20 - 8 * q^22 - 8 * q^23 - 4 * q^24 + 4 * q^25 + 4 * q^27 + 4 * q^30 + 8 * q^31 - 4 * q^32 + 8 * q^33 + 4 * q^36 + 8 * q^37 + 4 * q^40 - 8 * q^41 - 8 * q^43 + 8 * q^44 - 4 * q^45 + 8 * q^46 + 16 * q^47 + 4 * q^48 - 12 * q^49 - 4 * q^50 + 8 * q^53 - 4 * q^54 - 8 * q^55 + 8 * q^59 - 4 * q^60 + 8 * q^61 - 8 * q^62 + 4 * q^64 - 8 * q^66 + 40 * q^67 - 8 * q^69 - 4 * q^72 - 8 * q^73 - 8 * q^74 + 4 * q^75 - 16 * q^77 + 24 * q^79 - 4 * q^80 + 4 * q^81 + 8 * q^82 - 16 * q^83 + 8 * q^86 - 8 * q^88 + 8 * q^89 + 4 * q^90 + 32 * q^91 - 8 * q^92 + 8 * q^93 - 16 * q^94 - 4 * q^96 - 8 * q^97 + 12 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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