LMFDB - Embedded newform 8967.2.a.bi.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8967,2,Mod(1,8967)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8967, 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("8967.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8967 = 3 \cdot 7^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8967.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:	$71.6018554925$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 23 x^{17} + 104 x^{16} + 198 x^{15} - 1098 x^{14} - 729 x^{13} + 6066 x^{12} + \cdots + 350 $ x^19 - 4x^18 - 23x^17 + 104x^16 + 198x^15 - 1098x^14 - 729x^13 + 6066x^12 + 529x^11 - 18926x^10 + 4020x^9 + 33838x^8 - 12274x^7 - 33918x^6 + 14107x^5 + 18090x^4 - 7188x^3 - 4490x^2 + 1343x + 350
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$1.66531$ of defining polynomial
Character	$\chi$	$=$	8967.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.66531 q^{2} -1.00000 q^{3} +0.773271 q^{4} -2.48363 q^{5} -1.66531 q^{6} -2.04289 q^{8} +1.00000 q^{9} -4.13602 q^{10} -1.08933 q^{11} -0.773271 q^{12} -6.85952 q^{13} +2.48363 q^{15} -4.94859 q^{16} -5.32797 q^{17} +1.66531 q^{18} -6.15321 q^{19} -1.92052 q^{20} -1.81408 q^{22} -2.44675 q^{23} +2.04289 q^{24} +1.16840 q^{25} -11.4232 q^{26} -1.00000 q^{27} -4.13329 q^{29} +4.13602 q^{30} +3.16639 q^{31} -4.15518 q^{32} +1.08933 q^{33} -8.87274 q^{34} +0.773271 q^{36} +0.0678240 q^{37} -10.2470 q^{38} +6.85952 q^{39} +5.07378 q^{40} -7.06260 q^{41} -0.385534 q^{43} -0.842347 q^{44} -2.48363 q^{45} -4.07460 q^{46} +7.37734 q^{47} +4.94859 q^{48} +1.94576 q^{50} +5.32797 q^{51} -5.30426 q^{52} +0.597533 q^{53} -1.66531 q^{54} +2.70549 q^{55} +6.15321 q^{57} -6.88323 q^{58} +0.468004 q^{59} +1.92052 q^{60} -1.00000 q^{61} +5.27303 q^{62} +2.97750 q^{64} +17.0365 q^{65} +1.81408 q^{66} +6.82635 q^{67} -4.11996 q^{68} +2.44675 q^{69} +7.17809 q^{71} -2.04289 q^{72} -16.3585 q^{73} +0.112948 q^{74} -1.16840 q^{75} -4.75810 q^{76} +11.4232 q^{78} -4.89698 q^{79} +12.2905 q^{80} +1.00000 q^{81} -11.7614 q^{82} +12.9769 q^{83} +13.2327 q^{85} -0.642035 q^{86} +4.13329 q^{87} +2.22538 q^{88} -10.1560 q^{89} -4.13602 q^{90} -1.89200 q^{92} -3.16639 q^{93} +12.2856 q^{94} +15.2823 q^{95} +4.15518 q^{96} +1.03060 q^{97} -1.08933 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{2} - 19 q^{3} + 24 q^{4} + 2 q^{5} - 4 q^{6} + 12 q^{8} + 19 q^{9} - 10 q^{10} + 4 q^{11} - 24 q^{12} - 2 q^{15} + 34 q^{16} - 5 q^{17} + 4 q^{18} - 18 q^{19} + 24 q^{20} + 6 q^{22} + 18 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) $ 19 * q + 4 * q^2 - 19 * q^3 + 24 * q^4 + 2 * q^5 - 4 * q^6 + 12 * q^8 + 19 * q^9 - 10 * q^10 + 4 * q^11 - 24 * q^12 - 2 * q^15 + 34 * q^16 - 5 * q^17 + 4 * q^18 - 18 * q^19 + 24 * q^20 + 6 * q^22 + 18 * q^23 - 12 * q^24 + 49 * q^25 + 8 * q^26 - 19 * q^27 + 20 * q^29 + 10 * q^30 - 29 * q^31 + 38 * q^32 - 4 * q^33 + 16 * q^34 + 24 * q^36 + 22 * q^37 - 30 * q^38 - 22 * q^40 + 20 * q^41 + 24 * q^43 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 34 * q^48 + 26 * q^50 + 5 * q^51 - 4 * q^52 + 16 * q^53 - 4 * q^54 - 10 * q^55 + 18 * q^57 + 16 * q^58 - 7 * q^59 - 24 * q^60 - 19 * q^61 + 44 * q^62 + 60 * q^64 + 42 * q^65 - 6 * q^66 - 14 * q^68 - 18 * q^69 + 8 * q^71 + 12 * q^72 + 2 * q^73 - 46 * q^74 - 49 * q^75 - 46 * q^76 - 8 * q^78 + 34 * q^79 - 4 * q^80 + 19 * q^81 - 34 * q^82 + 6 * q^83 + 28 * q^85 + 60 * q^86 - 20 * q^87 - 18 * q^88 - 22 * q^89 - 10 * q^90 + 46 * q^92 + 29 * q^93 - 14 * q^95 - 38 * q^96 + 12 * q^97 + 4 * 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 $

$2$	1.66531	1.17755	0.588777	−	0.808295i	$-0.299609\pi$
$2$	1.66531	1.17755	0.588777	+	0.808295i	$0.299609\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.773271	0.386635
$5$	−2.48363	−1.11071	−0.555356	−	0.831613i	$-0.687418\pi$
$5$	−2.48363	−1.11071	−0.555356	+	0.831613i	$0.687418\pi$
$6$	−1.66531	−0.679862
$7$	0	0
$7$	0	0
$8$	−2.04289	−0.722271
$9$	1.00000	0.333333
$10$	−4.13602	−1.30792
$11$	−1.08933	−0.328446	−0.164223	−	0.986423i	$-0.552512\pi$
$11$	−1.08933	−0.328446	−0.164223	+	0.986423i	$0.552512\pi$
$12$	−0.773271	−0.223224
$13$	−6.85952	−1.90249	−0.951244	−	0.308440i	$-0.900193\pi$
$13$	−6.85952	−1.90249	−0.951244	+	0.308440i	$0.900193\pi$
$14$	0	0
$15$	2.48363	0.641270
$16$	−4.94859	−1.23715
$17$	−5.32797	−1.29222	−0.646111	−	0.763244i	$-0.723606\pi$
$17$	−5.32797	−1.29222	−0.646111	+	0.763244i	$0.723606\pi$
$18$	1.66531	0.392518
$19$	−6.15321	−1.41164	−0.705822	−	0.708389i	$-0.749422\pi$
$19$	−6.15321	−1.41164	−0.705822	+	0.708389i	$0.749422\pi$
$20$	−1.92052	−0.429440
$21$	0	0
$22$	−1.81408	−0.386763
$23$	−2.44675	−0.510182	−0.255091	−	0.966917i	$-0.582105\pi$
$23$	−2.44675	−0.510182	−0.255091	+	0.966917i	$0.582105\pi$
$24$	2.04289	0.417003
$25$	1.16840	0.233680
$26$	−11.4232	−2.24028
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.13329	−0.767533	−0.383767	−	0.923430i	$-0.625373\pi$
$29$	−4.13329	−0.767533	−0.383767	+	0.923430i	$0.625373\pi$
$30$	4.13602	0.755130
$31$	3.16639	0.568700	0.284350	−	0.958721i	$-0.408222\pi$
$31$	3.16639	0.568700	0.284350	+	0.958721i	$0.408222\pi$
$32$	−4.15518	−0.734540
$33$	1.08933	0.189628
$34$	−8.87274	−1.52166
$35$	0	0
$36$	0.773271	0.128878
$37$	0.0678240	0.0111502	0.00557510	−	0.999984i	$-0.498225\pi$
$37$	0.0678240	0.0111502	0.00557510	+	0.999984i	$0.498225\pi$
$38$	−10.2470	−1.66229
$39$	6.85952	1.09840
$40$	5.07378	0.802234
$41$	−7.06260	−1.10299	−0.551496	−	0.834177i	$-0.685943\pi$
$41$	−7.06260	−1.10299	−0.551496	+	0.834177i	$0.685943\pi$
$42$	0	0
$43$	−0.385534	−0.0587933	−0.0293967	−	0.999568i	$-0.509359\pi$
$43$	−0.385534	−0.0587933	−0.0293967	+	0.999568i	$0.509359\pi$
$44$	−0.842347	−0.126989
$45$	−2.48363	−0.370237
$46$	−4.07460	−0.600767
$47$	7.37734	1.07610	0.538048	−	0.842914i	$-0.319162\pi$
$47$	7.37734	1.07610	0.538048	+	0.842914i	$0.319162\pi$
$48$	4.94859	0.714268
$49$	0	0
$50$	1.94576	0.275172
$51$	5.32797	0.746065
$52$	−5.30426	−0.735569
$53$	0.597533	0.0820775	0.0410387	−	0.999158i	$-0.486933\pi$
$53$	0.597533	0.0820775	0.0410387	+	0.999158i	$0.486933\pi$
$54$	−1.66531	−0.226621
$55$	2.70549	0.364808
$56$	0	0
$57$	6.15321	0.815013
$58$	−6.88323	−0.903812
$59$	0.468004	0.0609289	0.0304644	−	0.999536i	$-0.490301\pi$
$59$	0.468004	0.0609289	0.0304644	+	0.999536i	$0.490301\pi$
$60$	1.92052	0.247937
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	5.27303	0.669675
$63$	0	0
$64$	2.97750	0.372188
$65$	17.0365	2.11311
$66$	1.81408	0.223298
$67$	6.82635	0.833972	0.416986	−	0.908913i	$-0.363086\pi$
$67$	6.82635	0.833972	0.416986	+	0.908913i	$0.363086\pi$
$68$	−4.11996	−0.499619
$69$	2.44675	0.294554
$70$	0	0
$71$	7.17809	0.851882	0.425941	−	0.904751i	$-0.359943\pi$
$71$	7.17809	0.851882	0.425941	+	0.904751i	$0.359943\pi$
$72$	−2.04289	−0.240757
$73$	−16.3585	−1.91462	−0.957310	−	0.289063i	$-0.906656\pi$
$73$	−16.3585	−1.91462	−0.957310	+	0.289063i	$0.906656\pi$
$74$	0.112948	0.0131300
$75$	−1.16840	−0.134915
$76$	−4.75810	−0.545791
$77$	0	0
$78$	11.4232	1.29343
$79$	−4.89698	−0.550953	−0.275477	−	0.961308i	$-0.588836\pi$
$79$	−4.89698	−0.550953	−0.275477	+	0.961308i	$0.588836\pi$
$80$	12.2905	1.37412
$81$	1.00000	0.111111
$82$	−11.7614	−1.29883
$83$	12.9769	1.42440	0.712199	−	0.701977i	$-0.247699\pi$
$83$	12.9769	1.42440	0.712199	+	0.701977i	$0.247699\pi$
$84$	0	0
$85$	13.2327	1.43529
$86$	−0.642035	−0.0692324
$87$	4.13329	0.443135
$88$	2.22538	0.237227
$89$	−10.1560	−1.07654	−0.538268	−	0.842773i	$-0.680921\pi$
$89$	−10.1560	−1.07654	−0.538268	+	0.842773i	$0.680921\pi$
$90$	−4.13602	−0.435975
$91$	0	0
$92$	−1.89200	−0.197254
$93$	−3.16639	−0.328339
$94$	12.2856	1.26716
$95$	15.2823	1.56793
$96$	4.15518	0.424087
$97$	1.03060	0.104642	0.0523210	−	0.998630i	$-0.483338\pi$
$97$	1.03060	0.104642	0.0523210	+	0.998630i	$0.483338\pi$
$98$	0	0
$99$	−1.08933	−0.109482
$100$	0.903491	0.0903491
$101$	−8.06469	−0.802467	−0.401233	−	0.915976i	$-0.631418\pi$
$101$	−8.06469	−0.802467	−0.401233	+	0.915976i	$0.631418\pi$
$102$	8.87274	0.878532
$103$	5.04626	0.497222	0.248611	−	0.968603i	$-0.420026\pi$
$103$	5.04626	0.497222	0.248611	+	0.968603i	$0.420026\pi$
$104$	14.0132	1.37411
$105$	0	0
$106$	0.995080	0.0966507
$107$	−20.3918	−1.97135	−0.985675	−	0.168658i	$-0.946057\pi$
$107$	−20.3918	−1.97135	−0.985675	+	0.168658i	$0.946057\pi$
$108$	−0.773271	−0.0744080
$109$	−9.36765	−0.897258	−0.448629	−	0.893718i	$-0.648087\pi$
$109$	−9.36765	−0.897258	−0.448629	+	0.893718i	$0.648087\pi$
$110$	4.50549	0.429582
$111$	−0.0678240	−0.00643757
$112$	0	0
$113$	−1.38232	−0.130038	−0.0650189	−	0.997884i	$-0.520711\pi$
$113$	−1.38232	−0.130038	−0.0650189	+	0.997884i	$0.520711\pi$
$114$	10.2470	0.959722
$115$	6.07681	0.566665
$116$	−3.19615	−0.296755
$117$	−6.85952	−0.634162
$118$	0.779373	0.0717471
$119$	0	0
$120$	−5.07378	−0.463170
$121$	−9.81336	−0.892123
$122$	−1.66531	−0.150770
$123$	7.06260	0.636813
$124$	2.44847	0.219879
$125$	9.51626	0.851160
$126$	0	0
$127$	12.2641	1.08826	0.544132	−	0.839000i	$-0.316859\pi$
$127$	12.2641	1.08826	0.544132	+	0.839000i	$0.316859\pi$
$128$	13.2688	1.17281
$129$	0.385534	0.0339443
$130$	28.3711	2.48831
$131$	−1.92063	−0.167806	−0.0839032	−	0.996474i	$-0.526739\pi$
$131$	−1.92063	−0.167806	−0.0839032	+	0.996474i	$0.526739\pi$
$132$	0.842347	0.0733169
$133$	0	0
$134$	11.3680	0.982047
$135$	2.48363	0.213757
$136$	10.8844	0.933334
$137$	−4.07959	−0.348543	−0.174271	−	0.984698i	$-0.555757\pi$
$137$	−4.07959	−0.348543	−0.174271	+	0.984698i	$0.555757\pi$
$138$	4.07460	0.346853
$139$	−12.7187	−1.07878	−0.539391	−	0.842055i	$-0.681346\pi$
$139$	−12.7187	−1.07878	−0.539391	+	0.842055i	$0.681346\pi$
$140$	0	0
$141$	−7.37734	−0.621284
$142$	11.9538	1.00314
$143$	7.47228	0.624864
$144$	−4.94859	−0.412383
$145$	10.2656	0.852508
$146$	−27.2421	−2.25457
$147$	0	0
$148$	0.0524463	0.00431106
$149$	−15.6338	−1.28077	−0.640386	−	0.768053i	$-0.721226\pi$
$149$	−15.6338	−1.28077	−0.640386	+	0.768053i	$0.721226\pi$
$150$	−1.94576	−0.158870
$151$	−2.65913	−0.216397	−0.108199	−	0.994129i	$-0.534508\pi$
$151$	−2.65913	−0.216397	−0.108199	+	0.994129i	$0.534508\pi$
$152$	12.5703	1.01959
$153$	−5.32797	−0.430741
$154$	0	0
$155$	−7.86412	−0.631662
$156$	5.30426	0.424681
$157$	−20.7962	−1.65971	−0.829857	−	0.557976i	$-0.811578\pi$
$157$	−20.7962	−1.65971	−0.829857	+	0.557976i	$0.811578\pi$
$158$	−8.15501	−0.648778
$159$	−0.597533	−0.0473874
$160$	10.3199	0.815862
$161$	0	0
$162$	1.66531	0.130839
$163$	−8.62747	−0.675756	−0.337878	−	0.941190i	$-0.609709\pi$
$163$	−8.62747	−0.675756	−0.337878	+	0.941190i	$0.609709\pi$
$164$	−5.46130	−0.426456
$165$	−2.70549	−0.210622
$166$	21.6106	1.67731
$167$	5.40537	0.418280	0.209140	−	0.977886i	$-0.432934\pi$
$167$	5.40537	0.418280	0.209140	+	0.977886i	$0.432934\pi$
$168$	0	0
$169$	34.0530	2.61946
$170$	22.0366	1.69013
$171$	−6.15321	−0.470548
$172$	−0.298122	−0.0227316
$173$	−6.69610	−0.509095	−0.254548	−	0.967060i	$-0.581927\pi$
$173$	−6.69610	−0.509095	−0.254548	+	0.967060i	$0.581927\pi$
$174$	6.88323	0.521816
$175$	0	0
$176$	5.39066	0.406336
$177$	−0.468004	−0.0351773
$178$	−16.9130	−1.26768
$179$	20.7593	1.55162	0.775810	−	0.630966i	$-0.217341\pi$
$179$	20.7593	1.55162	0.775810	+	0.630966i	$0.217341\pi$
$180$	−1.92052	−0.143147
$181$	−15.2864	−1.13623	−0.568115	−	0.822949i	$-0.692327\pi$
$181$	−15.2864	−1.13623	−0.568115	+	0.822949i	$0.692327\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	4.99844	0.368490
$185$	−0.168449	−0.0123846
$186$	−5.27303	−0.386637
$187$	5.80392	0.424425
$188$	5.70468	0.416057
$189$	0	0
$190$	25.4498	1.84632
$191$	−14.6650	−1.06112	−0.530560	−	0.847648i	$-0.678018\pi$
$191$	−14.6650	−1.06112	−0.530560	+	0.847648i	$0.678018\pi$
$192$	−2.97750	−0.214883
$193$	24.0250	1.72936	0.864679	−	0.502325i	$-0.167522\pi$
$193$	24.0250	1.72936	0.864679	+	0.502325i	$0.167522\pi$
$194$	1.71628	0.123222
$195$	−17.0365	−1.22001
$196$	0	0
$197$	−3.17791	−0.226416	−0.113208	−	0.993571i	$-0.536113\pi$
$197$	−3.17791	−0.226416	−0.113208	+	0.993571i	$0.536113\pi$
$198$	−1.81408	−0.128921
$199$	19.0267	1.34877	0.674383	−	0.738381i	$-0.264410\pi$
$199$	19.0267	1.34877	0.674383	+	0.738381i	$0.264410\pi$
$200$	−2.38692	−0.168780
$201$	−6.82635	−0.481494
$202$	−13.4302	−0.944948
$203$	0	0
$204$	4.11996	0.288455
$205$	17.5409	1.22511
$206$	8.40360	0.585507
$207$	−2.44675	−0.170061
$208$	33.9450	2.35366
$209$	6.70288	0.463648
$210$	0	0
$211$	0.464623	0.0319860	0.0159930	−	0.999872i	$-0.494909\pi$
$211$	0.464623	0.0319860	0.0159930	+	0.999872i	$0.494909\pi$
$212$	0.462055	0.0317340
$213$	−7.17809	−0.491834
$214$	−33.9587	−2.32137
$215$	0.957522	0.0653024
$216$	2.04289	0.139001
$217$	0	0
$218$	−15.6001	−1.05657
$219$	16.3585	1.10541
$220$	2.09208	0.141048
$221$	36.5473	2.45844
$222$	−0.112948	−0.00758059
$223$	6.25421	0.418813	0.209407	−	0.977829i	$-0.432847\pi$
$223$	6.25421	0.418813	0.209407	+	0.977829i	$0.432847\pi$
$224$	0	0
$225$	1.16840	0.0778935
$226$	−2.30200	−0.153127
$227$	−22.1252	−1.46850	−0.734252	−	0.678877i	$-0.762467\pi$
$227$	−22.1252	−1.46850	−0.734252	+	0.678877i	$0.762467\pi$
$228$	4.75810	0.315113
$229$	8.80130	0.581606	0.290803	−	0.956783i	$-0.406078\pi$
$229$	8.80130	0.581606	0.290803	+	0.956783i	$0.406078\pi$
$230$	10.1198	0.667279
$231$	0	0
$232$	8.44386	0.554367
$233$	−10.9507	−0.717402	−0.358701	−	0.933453i	$-0.616780\pi$
$233$	−10.9507	−0.717402	−0.358701	+	0.933453i	$0.616780\pi$
$234$	−11.4232	−0.746761
$235$	−18.3226	−1.19523
$236$	0.361893	0.0235573
$237$	4.89698	0.318093
$238$	0	0
$239$	5.64478	0.365130	0.182565	−	0.983194i	$-0.441560\pi$
$239$	5.64478	0.365130	0.182565	+	0.983194i	$0.441560\pi$
$240$	−12.2905	−0.793346
$241$	10.4938	0.675968	0.337984	−	0.941152i	$-0.390255\pi$
$241$	10.4938	0.675968	0.337984	+	0.941152i	$0.390255\pi$
$242$	−16.3423	−1.05052
$243$	−1.00000	−0.0641500
$244$	−0.773271	−0.0495036
$245$	0	0
$246$	11.7614	0.749882
$247$	42.2081	2.68563
$248$	−6.46858	−0.410755
$249$	−12.9769	−0.822377
$250$	15.8476	1.00229
$251$	21.2920	1.34394	0.671971	−	0.740578i	$-0.265448\pi$
$251$	21.2920	1.34394	0.671971	+	0.740578i	$0.265448\pi$
$252$	0	0
$253$	2.66532	0.167567
$254$	20.4236	1.28149
$255$	−13.2327	−0.828663
$256$	16.1418	1.00886
$257$	−8.95602	−0.558661	−0.279330	−	0.960195i	$-0.590113\pi$
$257$	−8.95602	−0.558661	−0.279330	+	0.960195i	$0.590113\pi$
$258$	0.642035	0.0399713
$259$	0	0
$260$	13.1738	0.817005
$261$	−4.13329	−0.255844
$262$	−3.19846	−0.197601
$263$	−1.27133	−0.0783934	−0.0391967	−	0.999232i	$-0.512480\pi$
$263$	−1.27133	−0.0783934	−0.0391967	+	0.999232i	$0.512480\pi$
$264$	−2.22538	−0.136963
$265$	−1.48405	−0.0911644
$266$	0	0
$267$	10.1560	0.621539
$268$	5.27862	0.322443
$269$	5.05668	0.308311	0.154156	−	0.988047i	$-0.450734\pi$
$269$	5.05668	0.308311	0.154156	+	0.988047i	$0.450734\pi$
$270$	4.13602	0.251710
$271$	−19.8775	−1.20747	−0.603736	−	0.797184i	$-0.706322\pi$
$271$	−19.8775	−1.20747	−0.603736	+	0.797184i	$0.706322\pi$
$272$	26.3659	1.59867
$273$	0	0
$274$	−6.79379	−0.410428
$275$	−1.27278	−0.0767513
$276$	1.89200	0.113885
$277$	−21.8430	−1.31242	−0.656208	−	0.754580i	$-0.727841\pi$
$277$	−21.8430	−1.31242	−0.656208	+	0.754580i	$0.727841\pi$
$278$	−21.1806	−1.27033
$279$	3.16639	0.189567
$280$	0	0
$281$	13.6619	0.814999	0.407499	−	0.913205i	$-0.366401\pi$
$281$	13.6619	0.814999	0.407499	+	0.913205i	$0.366401\pi$
$282$	−12.2856	−0.731596
$283$	9.91781	0.589553	0.294776	−	0.955566i	$-0.404755\pi$
$283$	9.91781	0.589553	0.294776	+	0.955566i	$0.404755\pi$
$284$	5.55060	0.329368
$285$	−15.2823	−0.905244
$286$	12.4437	0.735811
$287$	0	0
$288$	−4.15518	−0.244847
$289$	11.3872	0.669837
$290$	17.0954	1.00387
$291$	−1.03060	−0.0604151
$292$	−12.6496	−0.740260
$293$	17.7754	1.03845	0.519225	−	0.854638i	$-0.326221\pi$
$293$	17.7754	1.03845	0.519225	+	0.854638i	$0.326221\pi$
$294$	0	0
$295$	−1.16235	−0.0676744
$296$	−0.138557	−0.00805345
$297$	1.08933	0.0632094
$298$	−26.0352	−1.50818
$299$	16.7835	0.970615
$300$	−0.903491	−0.0521631
$301$	0	0
$302$	−4.42829	−0.254820
$303$	8.06469	0.463304
$304$	30.4497	1.74641
$305$	2.48363	0.142212
$306$	−8.87274	−0.507221
$307$	−30.4965	−1.74052	−0.870262	−	0.492589i	$-0.836051\pi$
$307$	−30.4965	−1.74052	−0.870262	+	0.492589i	$0.836051\pi$
$308$	0	0
$309$	−5.04626	−0.287072
$310$	−13.0962	−0.743816
$311$	13.1943	0.748182	0.374091	−	0.927392i	$-0.377955\pi$
$311$	13.1943	0.748182	0.374091	+	0.927392i	$0.377955\pi$
$312$	−14.0132	−0.793343
$313$	−29.6447	−1.67562	−0.837809	−	0.545964i	$-0.816164\pi$
$313$	−29.6447	−1.67562	−0.837809	+	0.545964i	$0.816164\pi$
$314$	−34.6321	−1.95441
$315$	0	0
$316$	−3.78669	−0.213018
$317$	5.52243	0.310170	0.155085	−	0.987901i	$-0.450435\pi$
$317$	5.52243	0.310170	0.155085	+	0.987901i	$0.450435\pi$
$318$	−0.995080	−0.0558013
$319$	4.50252	0.252093
$320$	−7.39501	−0.413393
$321$	20.3918	1.13816
$322$	0	0
$323$	32.7841	1.82416
$324$	0.773271	0.0429595
$325$	−8.01467	−0.444574
$326$	−14.3674	−0.795739
$327$	9.36765	0.518032
$328$	14.4281	0.796659
$329$	0	0
$330$	−4.50549	−0.248019
$331$	−11.2755	−0.619756	−0.309878	−	0.950776i	$-0.600288\pi$
$331$	−11.2755	−0.619756	−0.309878	+	0.950776i	$0.600288\pi$
$332$	10.0346	0.550723
$333$	0.0678240	0.00371673
$334$	9.00163	0.492547
$335$	−16.9541	−0.926302
$336$	0	0
$337$	18.5025	1.00790	0.503949	−	0.863734i	$-0.331880\pi$
$337$	18.5025	1.00790	0.503949	+	0.863734i	$0.331880\pi$
$338$	56.7089	3.08456
$339$	1.38232	0.0750774
$340$	10.2324	0.554932
$341$	−3.44924	−0.186787
$342$	−10.2470	−0.554096
$343$	0	0
$344$	0.787603	0.0424647
$345$	−6.07681	−0.327164
$346$	−11.1511	−0.599487
$347$	20.1761	1.08311	0.541556	−	0.840665i	$-0.317836\pi$
$347$	20.1761	1.08311	0.541556	+	0.840665i	$0.317836\pi$
$348$	3.19615	0.171332
$349$	4.35583	0.233162	0.116581	−	0.993181i	$-0.462807\pi$
$349$	4.35583	0.233162	0.116581	+	0.993181i	$0.462807\pi$
$350$	0	0
$351$	6.85952	0.366134
$352$	4.52637	0.241256
$353$	8.57592	0.456450	0.228225	−	0.973608i	$-0.426708\pi$
$353$	8.57592	0.456450	0.228225	+	0.973608i	$0.426708\pi$
$354$	−0.779373	−0.0414232
$355$	−17.8277	−0.946195
$356$	−7.85336	−0.416227
$357$	0	0
$358$	34.5707	1.82712
$359$	26.9540	1.42258	0.711290	−	0.702899i	$-0.248111\pi$
$359$	26.9540	1.42258	0.711290	+	0.702899i	$0.248111\pi$
$360$	5.07378	0.267411
$361$	18.8620	0.992737
$362$	−25.4567	−1.33797
$363$	9.81336	0.515068
$364$	0	0
$365$	40.6285	2.12659
$366$	1.66531	0.0870474
$367$	0.772740	0.0403367	0.0201683	−	0.999797i	$-0.493580\pi$
$367$	0.772740	0.0403367	0.0201683	+	0.999797i	$0.493580\pi$
$368$	12.1080	0.631171
$369$	−7.06260	−0.367664
$370$	−0.280521	−0.0145836
$371$	0	0
$372$	−2.44847	−0.126947
$373$	−22.5979	−1.17008	−0.585038	−	0.811006i	$-0.698920\pi$
$373$	−22.5979	−1.17008	−0.585038	+	0.811006i	$0.698920\pi$
$374$	9.66535	0.499783
$375$	−9.51626	−0.491418
$376$	−15.0711	−0.777232
$377$	28.3524	1.46022
$378$	0	0
$379$	−29.7445	−1.52787	−0.763935	−	0.645293i	$-0.776735\pi$
$379$	−29.7445	−1.52787	−0.763935	+	0.645293i	$0.776735\pi$
$380$	11.8173	0.606217
$381$	−12.2641	−0.628309
$382$	−24.4218	−1.24953
$383$	−17.6483	−0.901784	−0.450892	−	0.892579i	$-0.648894\pi$
$383$	−17.6483	−0.901784	−0.450892	+	0.892579i	$0.648894\pi$
$384$	−13.2688	−0.677123
$385$	0	0
$386$	40.0092	2.03641
$387$	−0.385534	−0.0195978
$388$	0.796936	0.0404583
$389$	−22.2965	−1.13048	−0.565239	−	0.824927i	$-0.691216\pi$
$389$	−22.2965	−1.13048	−0.565239	+	0.824927i	$0.691216\pi$
$390$	−28.3711	−1.43663
$391$	13.0362	0.659269
$392$	0	0
$393$	1.92063	0.0968831
$394$	−5.29221	−0.266618
$395$	12.1623	0.611950
$396$	−0.842347	−0.0423296
$397$	17.5707	0.881847	0.440923	−	0.897545i	$-0.354651\pi$
$397$	17.5707	0.881847	0.440923	+	0.897545i	$0.354651\pi$
$398$	31.6854	1.58825
$399$	0	0
$400$	−5.78195	−0.289097
$401$	−17.8020	−0.888989	−0.444494	−	0.895782i	$-0.646617\pi$
$401$	−17.8020	−0.888989	−0.444494	+	0.895782i	$0.646617\pi$
$402$	−11.3680	−0.566985
$403$	−21.7199	−1.08194
$404$	−6.23619	−0.310262
$405$	−2.48363	−0.123412
$406$	0	0
$407$	−0.0738827	−0.00366223
$408$	−10.8844	−0.538860
$409$	−19.8338	−0.980716	−0.490358	−	0.871521i	$-0.663134\pi$
$409$	−19.8338	−0.980716	−0.490358	+	0.871521i	$0.663134\pi$
$410$	29.2110	1.44263
$411$	4.07959	0.201231
$412$	3.90212	0.192244
$413$	0	0
$414$	−4.07460	−0.200256
$415$	−32.2298	−1.58210
$416$	28.5025	1.39745
$417$	12.7187	0.622835
$418$	11.1624	0.545971
$419$	3.19419	0.156047	0.0780233	−	0.996952i	$-0.475139\pi$
$419$	3.19419	0.156047	0.0780233	+	0.996952i	$0.475139\pi$
$420$	0	0
$421$	−18.5523	−0.904185	−0.452092	−	0.891971i	$-0.649322\pi$
$421$	−18.5523	−0.904185	−0.452092	+	0.891971i	$0.649322\pi$
$422$	0.773743	0.0376652
$423$	7.37734	0.358699
$424$	−1.22069	−0.0592821
$425$	−6.22521	−0.301967
$426$	−11.9538	−0.579162
$427$	0	0
$428$	−15.7684	−0.762193
$429$	−7.47228	−0.360765
$430$	1.59457	0.0768972
$431$	−36.4548	−1.75597	−0.877983	−	0.478692i	$-0.841111\pi$
$431$	−36.4548	−1.75597	−0.877983	+	0.478692i	$0.841111\pi$
$432$	4.94859	0.238089
$433$	9.68925	0.465636	0.232818	−	0.972520i	$-0.425205\pi$
$433$	9.68925	0.465636	0.232818	+	0.972520i	$0.425205\pi$
$434$	0	0
$435$	−10.2656	−0.492196
$436$	−7.24373	−0.346912
$437$	15.0554	0.720195
$438$	27.2421	1.30168
$439$	7.18719	0.343026	0.171513	−	0.985182i	$-0.445134\pi$
$439$	7.18719	0.343026	0.171513	+	0.985182i	$0.445134\pi$
$440$	−5.52702	−0.263490
$441$	0	0
$442$	60.8627	2.89494
$443$	−29.3032	−1.39224	−0.696119	−	0.717926i	$-0.745092\pi$
$443$	−29.3032	−1.39224	−0.696119	+	0.717926i	$0.745092\pi$
$444$	−0.0524463	−0.00248899
$445$	25.2238	1.19572
$446$	10.4152	0.493175
$447$	15.6338	0.739455
$448$	0	0
$449$	10.5176	0.496355	0.248177	−	0.968715i	$-0.420168\pi$
$449$	10.5176	0.496355	0.248177	+	0.968715i	$0.420168\pi$
$450$	1.94576	0.0917238
$451$	7.69351	0.362273
$452$	−1.06891	−0.0502772
$453$	2.65913	0.124937
$454$	−36.8455	−1.72924
$455$	0	0
$456$	−12.5703	−0.588660
$457$	−9.20995	−0.430823	−0.215412	−	0.976523i	$-0.569109\pi$
$457$	−9.20995	−0.430823	−0.215412	+	0.976523i	$0.569109\pi$
$458$	14.6569	0.684873
$459$	5.32797	0.248688
$460$	4.69902	0.219093
$461$	−16.6122	−0.773709	−0.386855	−	0.922141i	$-0.626438\pi$
$461$	−16.6122	−0.773709	−0.386855	+	0.922141i	$0.626438\pi$
$462$	0	0
$463$	11.7105	0.544235	0.272118	−	0.962264i	$-0.412276\pi$
$463$	11.7105	0.544235	0.272118	+	0.962264i	$0.412276\pi$
$464$	20.4540	0.949552
$465$	7.86412	0.364690
$466$	−18.2363	−0.844780
$467$	9.67433	0.447675	0.223837	−	0.974627i	$-0.428142\pi$
$467$	9.67433	0.447675	0.223837	+	0.974627i	$0.428142\pi$
$468$	−5.30426	−0.245190
$469$	0	0
$470$	−30.5128	−1.40745
$471$	20.7962	0.958237
$472$	−0.956080	−0.0440071
$473$	0.419974	0.0193104
$474$	8.15501	0.374572
$475$	−7.18943	−0.329873
$476$	0	0
$477$	0.597533	0.0273592
$478$	9.40032	0.429961
$479$	17.7841	0.812578	0.406289	−	0.913745i	$-0.366823\pi$
$479$	17.7841	0.812578	0.406289	+	0.913745i	$0.366823\pi$
$480$	−10.3199	−0.471038
$481$	−0.465240	−0.0212131
$482$	17.4756	0.795990
$483$	0	0
$484$	−7.58838	−0.344926
$485$	−2.55964	−0.116227
$486$	−1.66531	−0.0755402
$487$	−20.4907	−0.928522	−0.464261	−	0.885698i	$-0.653680\pi$
$487$	−20.4907	−0.928522	−0.464261	+	0.885698i	$0.653680\pi$
$488$	2.04289	0.0924773
$489$	8.62747	0.390148
$490$	0	0
$491$	31.6017	1.42616	0.713082	−	0.701081i	$-0.247299\pi$
$491$	31.6017	1.42616	0.713082	+	0.701081i	$0.247299\pi$
$492$	5.46130	0.246214
$493$	22.0220	0.991823
$494$	70.2897	3.16248
$495$	2.70549	0.121603
$496$	−15.6692	−0.703566
$497$	0	0
$498$	−21.6106	−0.968394
$499$	−3.59836	−0.161085	−0.0805423	−	0.996751i	$-0.525665\pi$
$499$	−3.59836	−0.161085	−0.0805423	+	0.996751i	$0.525665\pi$
$500$	7.35864	0.329089
$501$	−5.40537	−0.241494
$502$	35.4579	1.58256
$503$	8.75034	0.390159	0.195079	−	0.980787i	$-0.437504\pi$
$503$	8.75034	0.390159	0.195079	+	0.980787i	$0.437504\pi$
$504$	0	0
$505$	20.0297	0.891309
$506$	4.43859	0.197319
$507$	−34.0530	−1.51234
$508$	9.48347	0.420761
$509$	42.6613	1.89093	0.945463	−	0.325728i	$-0.105609\pi$
$509$	42.6613	1.89093	0.945463	+	0.325728i	$0.105609\pi$
$510$	−22.0366	−0.975796
$511$	0	0
$512$	0.343452	0.0151786
$513$	6.15321	0.271671
$514$	−14.9146	−0.657854
$515$	−12.5330	−0.552271
$516$	0.298122	0.0131241
$517$	−8.03637	−0.353439
$518$	0	0
$519$	6.69610	0.293926
$520$	−34.8036	−1.52624
$521$	−27.7320	−1.21496	−0.607480	−	0.794335i	$-0.707819\pi$
$521$	−27.7320	−1.21496	−0.607480	+	0.794335i	$0.707819\pi$
$522$	−6.88323	−0.301271
$523$	20.0783	0.877962	0.438981	−	0.898496i	$-0.355340\pi$
$523$	20.0783	0.877962	0.438981	+	0.898496i	$0.355340\pi$
$524$	−1.48517	−0.0648799
$525$	0	0
$526$	−2.11716	−0.0923126
$527$	−16.8704	−0.734886
$528$	−5.39066	−0.234598
$529$	−17.0134	−0.739714
$530$	−2.47141	−0.107351
$531$	0.468004	0.0203096
$532$	0	0
$533$	48.4460	2.09843
$534$	16.9130	0.731896
$535$	50.6456	2.18960
$536$	−13.9455	−0.602353
$537$	−20.7593	−0.895829
$538$	8.42096	0.363054
$539$	0	0
$540$	1.92052	0.0826458
$541$	1.87674	0.0806875	0.0403438	−	0.999186i	$-0.487155\pi$
$541$	1.87674	0.0806875	0.0403438	+	0.999186i	$0.487155\pi$
$542$	−33.1023	−1.42186
$543$	15.2864	0.656003
$544$	22.1387	0.949188
$545$	23.2657	0.996595
$546$	0	0
$547$	17.8715	0.764129	0.382065	−	0.924136i	$-0.375213\pi$
$547$	17.8715	0.764129	0.382065	+	0.924136i	$0.375213\pi$
$548$	−3.15462	−0.134759
$549$	−1.00000	−0.0426790
$550$	−2.11957	−0.0903789
$551$	25.4330	1.08348
$552$	−4.99844	−0.212748
$553$	0	0
$554$	−36.3754	−1.54544
$555$	0.168449	0.00715028
$556$	−9.83496	−0.417095
$557$	−41.5297	−1.75967	−0.879835	−	0.475280i	$-0.842347\pi$
$557$	−41.5297	−1.75967	−0.879835	+	0.475280i	$0.842347\pi$
$558$	5.27303	0.223225
$559$	2.64457	0.111854
$560$	0	0
$561$	−5.80392	−0.245042
$562$	22.7513	0.959706
$563$	−41.1611	−1.73473	−0.867367	−	0.497669i	$-0.834189\pi$
$563$	−41.1611	−1.73473	−0.867367	+	0.497669i	$0.834189\pi$
$564$	−5.70468	−0.240210
$565$	3.43317	0.144435
$566$	16.5163	0.694231
$567$	0	0
$568$	−14.6640	−0.615289
$569$	7.24648	0.303788	0.151894	−	0.988397i	$-0.451463\pi$
$569$	7.24648	0.303788	0.151894	+	0.988397i	$0.451463\pi$
$570$	−25.4498	−1.06597
$571$	−28.8875	−1.20890	−0.604452	−	0.796642i	$-0.706608\pi$
$571$	−28.8875	−1.20890	−0.604452	+	0.796642i	$0.706608\pi$
$572$	5.77810	0.241594
$573$	14.6650	0.612638
$574$	0	0
$575$	−2.85879	−0.119220
$576$	2.97750	0.124063
$577$	−13.8805	−0.577851	−0.288926	−	0.957352i	$-0.593298\pi$
$577$	−13.8805	−0.577851	−0.288926	+	0.957352i	$0.593298\pi$
$578$	18.9633	0.788770
$579$	−24.0250	−0.998445
$580$	7.93805	0.329610
$581$	0	0
$582$	−1.71628	−0.0711421
$583$	−0.650911	−0.0269580
$584$	33.4187	1.38287
$585$	17.0365	0.704372
$586$	29.6016	1.22283
$587$	28.7169	1.18527	0.592637	−	0.805470i	$-0.298087\pi$
$587$	28.7169	1.18527	0.592637	+	0.805470i	$0.298087\pi$
$588$	0	0
$589$	−19.4834	−0.802802
$590$	−1.93567	−0.0796903
$591$	3.17791	0.130722
$592$	−0.335633	−0.0137944
$593$	−15.1742	−0.623129	−0.311565	−	0.950225i	$-0.600853\pi$
$593$	−15.1742	−0.623129	−0.311565	+	0.950225i	$0.600853\pi$
$594$	1.81408	0.0744325
$595$	0	0
$596$	−12.0892	−0.495192
$597$	−19.0267	−0.778711
$598$	27.9498	1.14295
$599$	12.4743	0.509687	0.254844	−	0.966982i	$-0.417976\pi$
$599$	12.4743	0.509687	0.254844	+	0.966982i	$0.417976\pi$
$600$	2.38692	0.0974455
$601$	−28.2571	−1.15263	−0.576315	−	0.817228i	$-0.695510\pi$
$601$	−28.2571	−1.15263	−0.576315	+	0.817228i	$0.695510\pi$
$602$	0	0
$603$	6.82635	0.277991
$604$	−2.05623	−0.0836668
$605$	24.3727	0.990892
$606$	13.4302	0.545566
$607$	0.00814922	0.000330767 0	0.000165383	−	1.00000i	$-0.499947\pi$
$607$	0.00814922	0.000330767 0	0.000165383		1.00000i	$0.499947\pi$
$608$	25.5677	1.03691
$609$	0	0
$610$	4.13602	0.167462
$611$	−50.6050	−2.04726
$612$	−4.11996	−0.166540
$613$	−39.8780	−1.61066	−0.805328	−	0.592829i	$-0.798011\pi$
$613$	−39.8780	−1.61066	−0.805328	+	0.592829i	$0.798011\pi$
$614$	−50.7862	−2.04956
$615$	−17.5409	−0.707316
$616$	0	0
$617$	19.1754	0.771973	0.385986	−	0.922504i	$-0.373861\pi$
$617$	19.1754	0.771973	0.385986	+	0.922504i	$0.373861\pi$
$618$	−8.40360	−0.338042
$619$	23.2748	0.935495	0.467747	−	0.883862i	$-0.345066\pi$
$619$	23.2748	0.935495	0.467747	+	0.883862i	$0.345066\pi$
$620$	−6.08110	−0.244223
$621$	2.44675	0.0981846
$622$	21.9727	0.881025
$623$	0	0
$624$	−33.9450	−1.35889
$625$	−29.4768	−1.17907
$626$	−49.3677	−1.97313
$627$	−6.70288	−0.267687
$628$	−16.0811	−0.641704
$629$	−0.361364	−0.0144085
$630$	0	0
$631$	−23.2569	−0.925841	−0.462921	−	0.886400i	$-0.653199\pi$
$631$	−23.2569	−0.925841	−0.462921	+	0.886400i	$0.653199\pi$
$632$	10.0040	0.397937
$633$	−0.464623	−0.0184671
$634$	9.19658	0.365243
$635$	−30.4595	−1.20875
$636$	−0.462055	−0.0183217
$637$	0	0
$638$	7.49811	0.296853
$639$	7.17809	0.283961
$640$	−32.9549	−1.30266
$641$	−30.9736	−1.22339	−0.611693	−	0.791096i	$-0.709511\pi$
$641$	−30.9736	−1.22339	−0.611693	+	0.791096i	$0.709511\pi$
$642$	33.9587	1.34024
$643$	−27.0298	−1.06595	−0.532977	−	0.846130i	$-0.678927\pi$
$643$	−27.0298	−1.06595	−0.532977	+	0.846130i	$0.678927\pi$
$644$	0	0
$645$	−0.957522	−0.0377024
$646$	54.5958	2.14804
$647$	29.7990	1.17152	0.585760	−	0.810484i	$-0.300796\pi$
$647$	29.7990	1.17152	0.585760	+	0.810484i	$0.300796\pi$
$648$	−2.04289	−0.0802523
$649$	−0.509811	−0.0200118
$650$	−13.3469	−0.523510
$651$	0	0
$652$	−6.67137	−0.261271
$653$	15.1106	0.591324	0.295662	−	0.955293i	$-0.404460\pi$
$653$	15.1106	0.591324	0.295662	+	0.955293i	$0.404460\pi$
$654$	15.6001	0.610011
$655$	4.77013	0.186385
$656$	34.9499	1.36457
$657$	−16.3585	−0.638207
$658$	0	0
$659$	35.9618	1.40087	0.700437	−	0.713714i	$-0.252989\pi$
$659$	35.9618	1.40087	0.700437	+	0.713714i	$0.252989\pi$
$660$	−2.09208	−0.0814340
$661$	44.3819	1.72626	0.863128	−	0.504986i	$-0.168502\pi$
$661$	44.3819	1.72626	0.863128	+	0.504986i	$0.168502\pi$
$662$	−18.7772	−0.729797
$663$	−36.5473	−1.41938
$664$	−26.5104	−1.02880
$665$	0	0
$666$	0.112948	0.00437665
$667$	10.1131	0.391582
$668$	4.17981	0.161722
$669$	−6.25421	−0.241802
$670$	−28.2339	−1.09077
$671$	1.08933	0.0420531
$672$	0	0
$673$	−45.3058	−1.74641	−0.873205	−	0.487353i	$-0.837962\pi$
$673$	−45.3058	−1.74641	−0.873205	+	0.487353i	$0.837962\pi$
$674$	30.8125	1.18685
$675$	−1.16840	−0.0449718
$676$	26.3321	1.01277
$677$	−15.3495	−0.589929	−0.294965	−	0.955508i	$-0.595308\pi$
$677$	−15.3495	−0.589929	−0.294965	+	0.955508i	$0.595308\pi$
$678$	2.30200	0.0884077
$679$	0	0
$680$	−27.0329	−1.03666
$681$	22.1252	0.847841
$682$	−5.74407	−0.219952
$683$	30.5362	1.16844	0.584218	−	0.811597i	$-0.301401\pi$
$683$	30.5362	1.16844	0.584218	+	0.811597i	$0.301401\pi$
$684$	−4.75810	−0.181930
$685$	10.1322	0.387130
$686$	0	0
$687$	−8.80130	−0.335790
$688$	1.90785	0.0727361
$689$	−4.09879	−0.156151
$690$	−10.1198	−0.385254
$691$	37.4178	1.42344	0.711720	−	0.702464i	$-0.247917\pi$
$691$	37.4178	1.42344	0.711720	+	0.702464i	$0.247917\pi$
$692$	−5.17790	−0.196834
$693$	0	0
$694$	33.5996	1.27542
$695$	31.5884	1.19822
$696$	−8.44386	−0.320064
$697$	37.6293	1.42531
$698$	7.25382	0.274561
$699$	10.9507	0.414192
$700$	0	0
$701$	−45.8005	−1.72986	−0.864931	−	0.501891i	$-0.832638\pi$
$701$	−45.8005	−1.72986	−0.864931	+	0.501891i	$0.832638\pi$
$702$	11.4232	0.431143
$703$	−0.417335	−0.0157401
$704$	−3.24349	−0.122243
$705$	18.3226	0.690068
$706$	14.2816	0.537495
$707$	0	0
$708$	−0.361893	−0.0136008
$709$	4.37397	0.164268	0.0821340	−	0.996621i	$-0.473826\pi$
$709$	4.37397	0.164268	0.0821340	+	0.996621i	$0.473826\pi$
$710$	−29.6887	−1.11420
$711$	−4.89698	−0.183651
$712$	20.7476	0.777551
$713$	−7.74735	−0.290141
$714$	0	0
$715$	−18.5584	−0.694043
$716$	16.0525	0.599911
$717$	−5.64478	−0.210808
$718$	44.8869	1.67516
$719$	−33.1144	−1.23496	−0.617479	−	0.786587i	$-0.711846\pi$
$719$	−33.1144	−1.23496	−0.617479	+	0.786587i	$0.711846\pi$
$720$	12.2905	0.458038
$721$	0	0
$722$	31.4112	1.16900
$723$	−10.4938	−0.390270
$724$	−11.8205	−0.439307
$725$	−4.82935	−0.179357
$726$	16.3423	0.606520
$727$	21.2510	0.788155	0.394078	−	0.919077i	$-0.371064\pi$
$727$	21.2510	0.788155	0.394078	+	0.919077i	$0.371064\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	67.6592	2.50418
$731$	2.05411	0.0759740
$732$	0.773271	0.0285809
$733$	−5.44744	−0.201206	−0.100603	−	0.994927i	$-0.532077\pi$
$733$	−5.44744	−0.201206	−0.100603	+	0.994927i	$0.532077\pi$
$734$	1.28685	0.0474986
$735$	0	0
$736$	10.1667	0.374749
$737$	−7.43616	−0.273914
$738$	−11.7614	−0.432945
$739$	−3.21230	−0.118166	−0.0590831	−	0.998253i	$-0.518818\pi$
$739$	−3.21230	−0.118166	−0.0590831	+	0.998253i	$0.518818\pi$
$740$	−0.130257	−0.00478834
$741$	−42.2081	−1.55055
$742$	0	0
$743$	31.9772	1.17313	0.586566	−	0.809902i	$-0.300480\pi$
$743$	31.9772	1.17313	0.586566	+	0.809902i	$0.300480\pi$
$744$	6.46858	0.237150
$745$	38.8286	1.42257
$746$	−37.6326	−1.37783
$747$	12.9769	0.474800
$748$	4.48800	0.164098
$749$	0	0
$750$	−15.8476	−0.578671
$751$	−39.3392	−1.43551	−0.717754	−	0.696297i	$-0.754830\pi$
$751$	−39.3392	−1.43551	−0.717754	+	0.696297i	$0.754830\pi$
$752$	−36.5075	−1.33129
$753$	−21.2920	−0.775925
$754$	47.2156	1.71949
$755$	6.60430	0.240355
$756$	0	0
$757$	4.64206	0.168718	0.0843592	−	0.996435i	$-0.473116\pi$
$757$	4.64206	0.168718	0.0843592	+	0.996435i	$0.473116\pi$
$758$	−49.5339	−1.79915
$759$	−2.66532	−0.0967449
$760$	−31.2200	−1.13247
$761$	−29.7421	−1.07815	−0.539076	−	0.842257i	$-0.681226\pi$
$761$	−29.7421	−1.07815	−0.539076	+	0.842257i	$0.681226\pi$
$762$	−20.4236	−0.739869
$763$	0	0
$764$	−11.3400	−0.410266
$765$	13.2327	0.478429
$766$	−29.3899	−1.06190
$767$	−3.21028	−0.115916
$768$	−16.1418	−0.582466
$769$	−9.80110	−0.353437	−0.176718	−	0.984261i	$-0.556548\pi$
$769$	−9.80110	−0.353437	−0.176718	+	0.984261i	$0.556548\pi$
$770$	0	0
$771$	8.95602	0.322543
$772$	18.5778	0.668631
$773$	39.9423	1.43662	0.718312	−	0.695722i	$-0.244915\pi$
$773$	39.9423	1.43662	0.718312	+	0.695722i	$0.244915\pi$
$774$	−0.642035	−0.0230775
$775$	3.69961	0.132894
$776$	−2.10541	−0.0755798
$777$	0	0
$778$	−37.1307	−1.33120
$779$	43.4577	1.55703
$780$	−13.1738	−0.471698
$781$	−7.81931	−0.279797
$782$	21.7094	0.776325
$783$	4.13329	0.147712
$784$	0	0
$785$	51.6499	1.84346
$786$	3.19846	0.114085
$787$	1.53745	0.0548042	0.0274021	−	0.999624i	$-0.491277\pi$
$787$	1.53745	0.0548042	0.0274021	+	0.999624i	$0.491277\pi$
$788$	−2.45738	−0.0875406
$789$	1.27133	0.0452605
$790$	20.2540	0.720605
$791$	0	0
$792$	2.22538	0.0790755
$793$	6.85952	0.243589
$794$	29.2607	1.03842
$795$	1.48405	0.0526338
$796$	14.7128	0.521481
$797$	−36.1021	−1.27880	−0.639401	−	0.768874i	$-0.720818\pi$
$797$	−36.1021	−1.27880	−0.639401	+	0.768874i	$0.720818\pi$
$798$	0	0
$799$	−39.3062	−1.39055
$800$	−4.85492	−0.171648
$801$	−10.1560	−0.358846
$802$	−29.6459	−1.04683
$803$	17.8198	0.628849
$804$	−5.27862	−0.186162
$805$	0	0
$806$	−36.1704	−1.27405
$807$	−5.05668	−0.178004
$808$	16.4753	0.579598
$809$	−33.6350	−1.18255	−0.591273	−	0.806472i	$-0.701374\pi$
$809$	−33.6350	−1.18255	−0.591273	+	0.806472i	$0.701374\pi$
$810$	−4.13602	−0.145325
$811$	−49.0189	−1.72129	−0.860643	−	0.509209i	$-0.829938\pi$
$811$	−49.0189	−1.72129	−0.860643	+	0.509209i	$0.829938\pi$
$812$	0	0
$813$	19.8775	0.697134
$814$	−0.123038	−0.00431248
$815$	21.4274	0.750570
$816$	−26.3659	−0.922993
$817$	2.37227	0.0829952
$818$	−33.0294	−1.15485
$819$	0	0
$820$	13.5638	0.473670
$821$	−3.58484	−0.125112	−0.0625558	−	0.998041i	$-0.519925\pi$
$821$	−3.58484	−0.125112	−0.0625558	+	0.998041i	$0.519925\pi$
$822$	6.79379	0.236961
$823$	−21.0239	−0.732848	−0.366424	−	0.930448i	$-0.619418\pi$
$823$	−21.0239	−0.732848	−0.366424	+	0.930448i	$0.619418\pi$
$824$	−10.3089	−0.359129
$825$	1.27278	0.0443124
$826$	0	0
$827$	−15.8507	−0.551183	−0.275591	−	0.961275i	$-0.588874\pi$
$827$	−15.8507	−0.551183	−0.275591	+	0.961275i	$0.588874\pi$
$828$	−1.89200	−0.0657515
$829$	−0.230495	−0.00800543	−0.00400272	−	0.999992i	$-0.501274\pi$
$829$	−0.230495	−0.00800543	−0.00400272	+	0.999992i	$0.501274\pi$
$830$	−53.6727	−1.86301
$831$	21.8430	0.757724
$832$	−20.4242	−0.708083
$833$	0	0
$834$	21.1806	0.733423
$835$	−13.4249	−0.464588
$836$	5.18314	0.179263
$837$	−3.16639	−0.109446
$838$	5.31934	0.183753
$839$	24.4708	0.844827	0.422413	−	0.906403i	$-0.361183\pi$
$839$	24.4708	0.844827	0.422413	+	0.906403i	$0.361183\pi$
$840$	0	0
$841$	−11.9159	−0.410893
$842$	−30.8954	−1.06473
$843$	−13.6619	−0.470540
$844$	0.359279	0.0123669
$845$	−84.5748	−2.90946
$846$	12.2856	0.422387
$847$	0	0
$848$	−2.95695	−0.101542
$849$	−9.91781	−0.340378
$850$	−10.3669	−0.355583
$851$	−0.165948	−0.00568863
$852$	−5.55060	−0.190161
$853$	−12.3208	−0.421856	−0.210928	−	0.977502i	$-0.567649\pi$
$853$	−12.3208	−0.421856	−0.210928	+	0.977502i	$0.567649\pi$
$854$	0	0
$855$	15.2823	0.522643
$856$	41.6582	1.42385
$857$	21.7617	0.743366	0.371683	−	0.928360i	$-0.378781\pi$
$857$	21.7617	0.743366	0.371683	+	0.928360i	$0.378781\pi$
$858$	−12.4437	−0.424821
$859$	32.2956	1.10191	0.550955	−	0.834535i	$-0.314264\pi$
$859$	32.2956	1.10191	0.550955	+	0.834535i	$0.314264\pi$
$860$	0.740423	0.0252482
$861$	0	0
$862$	−60.7087	−2.06775
$863$	−33.4015	−1.13700	−0.568500	−	0.822683i	$-0.692476\pi$
$863$	−33.4015	−1.13700	−0.568500	+	0.822683i	$0.692476\pi$
$864$	4.15518	0.141362
$865$	16.6306	0.565458
$866$	16.1356	0.548311
$867$	−11.3872	−0.386731
$868$	0	0
$869$	5.33443	0.180958
$870$	−17.0954	−0.579587
$871$	−46.8255	−1.58662
$872$	19.1371	0.648063
$873$	1.03060	0.0348807
$874$	25.0719	0.848070
$875$	0	0
$876$	12.6496	0.427389
$877$	27.2853	0.921357	0.460679	−	0.887567i	$-0.347606\pi$
$877$	27.2853	0.921357	0.460679	+	0.887567i	$0.347606\pi$
$878$	11.9689	0.403932
$879$	−17.7754	−0.599549
$880$	−13.3884	−0.451322
$881$	−0.454864	−0.0153248	−0.00766239	−	0.999971i	$-0.502439\pi$
$881$	−0.454864	−0.0153248	−0.00766239	+	0.999971i	$0.502439\pi$
$882$	0	0
$883$	−44.8031	−1.50774	−0.753872	−	0.657021i	$-0.771816\pi$
$883$	−44.8031	−1.50774	−0.753872	+	0.657021i	$0.771816\pi$
$884$	28.2609	0.950518
$885$	1.16235	0.0390718
$886$	−48.7991	−1.63944
$887$	18.8285	0.632200	0.316100	−	0.948726i	$-0.397627\pi$
$887$	18.8285	0.632200	0.316100	+	0.948726i	$0.397627\pi$
$888$	0.138557	0.00464966
$889$	0	0
$890$	42.0055	1.40803
$891$	−1.08933	−0.0364940
$892$	4.83620	0.161928
$893$	−45.3943	−1.51906
$894$	26.0352	0.870748
$895$	−51.5583	−1.72340
$896$	0	0
$897$	−16.7835	−0.560385
$898$	17.5150	0.584485
$899$	−13.0876	−0.436496
$900$	0.903491	0.0301164
$901$	−3.18364	−0.106062
$902$	12.8121	0.426596
$903$	0	0
$904$	2.82393	0.0939225
$905$	37.9657	1.26202
$906$	4.42829	0.147120
$907$	−18.1507	−0.602686	−0.301343	−	0.953516i	$-0.597435\pi$
$907$	−18.1507	−0.602686	−0.301343	+	0.953516i	$0.597435\pi$
$908$	−17.1088	−0.567775
$909$	−8.06469	−0.267489
$910$	0	0
$911$	3.80550	0.126082	0.0630409	−	0.998011i	$-0.479920\pi$
$911$	3.80550	0.126082	0.0630409	+	0.998011i	$0.479920\pi$
$912$	−30.4497	−1.00829
$913$	−14.1361	−0.467838
$914$	−15.3375	−0.507318
$915$	−2.48363	−0.0821062
$916$	6.80578	0.224869
$917$	0	0
$918$	8.87274	0.292844
$919$	54.7353	1.80555	0.902776	−	0.430111i	$-0.141526\pi$
$919$	54.7353	1.80555	0.902776	+	0.430111i	$0.141526\pi$
$920$	−12.4142	−0.409286
$921$	30.4965	1.00489
$922$	−27.6646	−0.911085
$923$	−49.2382	−1.62069
$924$	0	0
$925$	0.0792457	0.00260558
$926$	19.5017	0.640867
$927$	5.04626	0.165741
$928$	17.1746	0.563783
$929$	−17.3605	−0.569581	−0.284790	−	0.958590i	$-0.591924\pi$
$929$	−17.3605	−0.569581	−0.284790	+	0.958590i	$0.591924\pi$
$930$	13.0962	0.429442
$931$	0	0
$932$	−8.46783	−0.277373
$933$	−13.1943	−0.431963
$934$	16.1108	0.527162
$935$	−14.4148	−0.471413
$936$	14.0132	0.458037
$937$	46.0922	1.50577	0.752883	−	0.658155i	$-0.228663\pi$
$937$	46.0922	1.50577	0.752883	+	0.658155i	$0.228663\pi$
$938$	0	0
$939$	29.6447	0.967418
$940$	−14.1683	−0.462119
$941$	2.34375	0.0764039	0.0382020	−	0.999270i	$-0.487837\pi$
$941$	2.34375	0.0764039	0.0382020	+	0.999270i	$0.487837\pi$
$942$	34.6321	1.12838
$943$	17.2804	0.562727
$944$	−2.31596	−0.0753781
$945$	0	0
$946$	0.699388	0.0227391
$947$	−51.9602	−1.68848	−0.844241	−	0.535964i	$-0.819948\pi$
$947$	−51.9602	−1.68848	−0.844241	+	0.535964i	$0.819948\pi$
$948$	3.78669	0.122986
$949$	112.212	3.64254
$950$	−11.9727	−0.388444
$951$	−5.52243	−0.179077
$952$	0	0
$953$	17.8478	0.578148	0.289074	−	0.957307i	$-0.406653\pi$
$953$	17.8478	0.578148	0.289074	+	0.957307i	$0.406653\pi$
$954$	0.995080	0.0322169
$955$	36.4223	1.17860
$956$	4.36494	0.141172
$957$	−4.50252	−0.145546
$958$	29.6162	0.956856
$959$	0	0
$960$	7.39501	0.238673
$961$	−20.9740	−0.676580
$962$	−0.774770	−0.0249796
$963$	−20.3918	−0.657116
$964$	8.11458	0.261353
$965$	−59.6692	−1.92082
$966$	0	0
$967$	−1.28496	−0.0413215	−0.0206608	−	0.999787i	$-0.506577\pi$
$967$	−1.28496	−0.0413215	−0.0206608	+	0.999787i	$0.506577\pi$
$968$	20.0476	0.644355
$969$	−32.7841	−1.05318
$970$	−4.26260	−0.136864
$971$	27.6157	0.886229	0.443114	−	0.896465i	$-0.353874\pi$
$971$	27.6157	0.886229	0.443114	+	0.896465i	$0.353874\pi$
$972$	−0.773271	−0.0248027
$973$	0	0
$974$	−34.1234	−1.09339
$975$	8.01467	0.256675
$976$	4.94859	0.158401
$977$	−19.5229	−0.624594	−0.312297	−	0.949985i	$-0.601098\pi$
$977$	−19.5229	−0.624594	−0.312297	+	0.949985i	$0.601098\pi$
$978$	14.3674	0.459420
$979$	11.0633	0.353584
$980$	0	0
$981$	−9.36765	−0.299086
$982$	52.6267	1.67939
$983$	9.61078	0.306536	0.153268	−	0.988185i	$-0.451020\pi$
$983$	9.61078	0.306536	0.153268	+	0.988185i	$0.451020\pi$
$984$	−14.4281	−0.459951
$985$	7.89273	0.251483
$986$	36.6736	1.16793
$987$	0	0
$988$	32.6382	1.03836
$989$	0.943304	0.0299953
$990$	4.50549	0.143194
$991$	30.5066	0.969075	0.484537	−	0.874771i	$-0.338988\pi$
$991$	30.5066	0.969075	0.484537	+	0.874771i	$0.338988\pi$
$992$	−13.1569	−0.417733
$993$	11.2755	0.357816
$994$	0	0
$995$	−47.2552	−1.49809
$996$	−10.0346	−0.317960
$997$	−24.0879	−0.762872	−0.381436	−	0.924395i	$-0.624570\pi$
$997$	−24.0879	−0.762872	−0.381436	+	0.924395i	$0.624570\pi$
$998$	−5.99240	−0.189686
$999$	−0.0678240	−0.00214586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8967.2.a.bi.1.14	✓	19
7.6	odd	2		8967.2.a.bj.1.14	yes	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8967.2.a.bi.1.14	✓	19	1.1	even	1	trivial
8967.2.a.bj.1.14	yes	19	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8967 = 3 \cdot 7^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8967.a (trivial)

\(f(q)\)	\(=\)	\(q+1.66531 q^{2} -1.00000 q^{3} +0.773271 q^{4} -2.48363 q^{5} -1.66531 q^{6} -2.04289 q^{8} +1.00000 q^{9} -4.13602 q^{10} -1.08933 q^{11} -0.773271 q^{12} -6.85952 q^{13} +2.48363 q^{15} -4.94859 q^{16} -5.32797 q^{17} +1.66531 q^{18} -6.15321 q^{19} -1.92052 q^{20} -1.81408 q^{22} -2.44675 q^{23} +2.04289 q^{24} +1.16840 q^{25} -11.4232 q^{26} -1.00000 q^{27} -4.13329 q^{29} +4.13602 q^{30} +3.16639 q^{31} -4.15518 q^{32} +1.08933 q^{33} -8.87274 q^{34} +0.773271 q^{36} +0.0678240 q^{37} -10.2470 q^{38} +6.85952 q^{39} +5.07378 q^{40} -7.06260 q^{41} -0.385534 q^{43} -0.842347 q^{44} -2.48363 q^{45} -4.07460 q^{46} +7.37734 q^{47} +4.94859 q^{48} +1.94576 q^{50} +5.32797 q^{51} -5.30426 q^{52} +0.597533 q^{53} -1.66531 q^{54} +2.70549 q^{55} +6.15321 q^{57} -6.88323 q^{58} +0.468004 q^{59} +1.92052 q^{60} -1.00000 q^{61} +5.27303 q^{62} +2.97750 q^{64} +17.0365 q^{65} +1.81408 q^{66} +6.82635 q^{67} -4.11996 q^{68} +2.44675 q^{69} +7.17809 q^{71} -2.04289 q^{72} -16.3585 q^{73} +0.112948 q^{74} -1.16840 q^{75} -4.75810 q^{76} +11.4232 q^{78} -4.89698 q^{79} +12.2905 q^{80} +1.00000 q^{81} -11.7614 q^{82} +12.9769 q^{83} +13.2327 q^{85} -0.642035 q^{86} +4.13329 q^{87} +2.22538 q^{88} -10.1560 q^{89} -4.13602 q^{90} -1.89200 q^{92} -3.16639 q^{93} +12.2856 q^{94} +15.2823 q^{95} +4.15518 q^{96} +1.03060 q^{97} -1.08933 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{2} - 19 q^{3} + 24 q^{4} + 2 q^{5} - 4 q^{6} + 12 q^{8} + 19 q^{9} - 10 q^{10} + 4 q^{11} - 24 q^{12} - 2 q^{15} + 34 q^{16} - 5 q^{17} + 4 q^{18} - 18 q^{19} + 24 q^{20} + 6 q^{22} + 18 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) 19 * q + 4 * q^2 - 19 * q^3 + 24 * q^4 + 2 * q^5 - 4 * q^6 + 12 * q^8 + 19 * q^9 - 10 * q^10 + 4 * q^11 - 24 * q^12 - 2 * q^15 + 34 * q^16 - 5 * q^17 + 4 * q^18 - 18 * q^19 + 24 * q^20 + 6 * q^22 + 18 * q^23 - 12 * q^24 + 49 * q^25 + 8 * q^26 - 19 * q^27 + 20 * q^29 + 10 * q^30 - 29 * q^31 + 38 * q^32 - 4 * q^33 + 16 * q^34 + 24 * q^36 + 22 * q^37 - 30 * q^38 - 22 * q^40 + 20 * q^41 + 24 * q^43 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 34 * q^48 + 26 * q^50 + 5 * q^51 - 4 * q^52 + 16 * q^53 - 4 * q^54 - 10 * q^55 + 18 * q^57 + 16 * q^58 - 7 * q^59 - 24 * q^60 - 19 * q^61 + 44 * q^62 + 60 * q^64 + 42 * q^65 - 6 * q^66 - 14 * q^68 - 18 * q^69 + 8 * q^71 + 12 * q^72 + 2 * q^73 - 46 * q^74 - 49 * q^75 - 46 * q^76 - 8 * q^78 + 34 * q^79 - 4 * q^80 + 19 * q^81 - 34 * q^82 + 6 * q^83 + 28 * q^85 + 60 * q^86 - 20 * q^87 - 18 * q^88 - 22 * q^89 - 10 * q^90 + 46 * q^92 + 29 * q^93 - 14 * q^95 - 38 * q^96 + 12 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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