LMFDB - Embedded newform 968.4.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,4,Mod(1,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("968.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	968.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:	$57.1138488856$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 193 x^{8} + 670 x^{7} + 10959 x^{6} - 33408 x^{5} - 177207 x^{4} + 365822 x^{3} + \cdots - 781744 $ x^10 - 4x^9 - 193x^8 + 670x^7 + 10959x^6 - 33408x^5 - 177207x^4 + 365822x^3 + 990744x^2 - 836120*x - 781744
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.06402$ of defining polynomial
Character	$\chi$	$=$	968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.44598 q^{3} -9.31187 q^{5} +22.5481 q^{7} -24.9091 q^{9} +O(q^{10})$	$q-1.44598 q^{3} -9.31187 q^{5} +22.5481 q^{7} -24.9091 q^{9} +9.73910 q^{13} +13.4648 q^{15} +132.634 q^{17} -29.8000 q^{19} -32.6042 q^{21} -175.208 q^{23} -38.2891 q^{25} +75.0598 q^{27} -74.0142 q^{29} -29.2295 q^{31} -209.965 q^{35} -85.4067 q^{37} -14.0826 q^{39} +282.741 q^{41} +212.780 q^{43} +231.951 q^{45} -377.373 q^{47} +165.416 q^{49} -191.787 q^{51} +717.033 q^{53} +43.0903 q^{57} +57.5530 q^{59} +146.308 q^{61} -561.653 q^{63} -90.6892 q^{65} +792.911 q^{67} +253.348 q^{69} -8.64368 q^{71} -973.110 q^{73} +55.3654 q^{75} +732.331 q^{79} +564.011 q^{81} +587.100 q^{83} -1235.07 q^{85} +107.023 q^{87} -826.421 q^{89} +219.598 q^{91} +42.2654 q^{93} +277.494 q^{95} +301.181 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) $ 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9	$ 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9} - 45 q^{13} + 120 q^{15} + 17 q^{17} + 147 q^{19} - 131 q^{21} + 164 q^{23} + 439 q^{25} + 420 q^{27} - 177 q^{29} + 275 q^{31} + 220 q^{35} + 745 q^{37} + 524 q^{39} - 967 q^{41} + 380 q^{43} - 44 q^{45} + 769 q^{47} + 503 q^{49} + 956 q^{51} + 701 q^{53} - 1293 q^{57} + 1291 q^{59} + 1359 q^{61} - 929 q^{63} - 173 q^{65} + 2260 q^{67} + 1988 q^{69} + 465 q^{71} - 111 q^{73} + 4584 q^{75} - 1827 q^{79} + 6874 q^{81} + 4947 q^{83} - 2609 q^{85} - 1303 q^{87} + 446 q^{89} + 2176 q^{91} + 4204 q^{93} - 108 q^{95} + 3511 q^{97}+O(q^{100}) $ 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9 - 45 * q^13 + 120 * q^15 + 17 * q^17 + 147 * q^19 - 131 * q^21 + 164 * q^23 + 439 * q^25 + 420 * q^27 - 177 * q^29 + 275 * q^31 + 220 * q^35 + 745 * q^37 + 524 * q^39 - 967 * q^41 + 380 * q^43 - 44 * q^45 + 769 * q^47 + 503 * q^49 + 956 * q^51 + 701 * q^53 - 1293 * q^57 + 1291 * q^59 + 1359 * q^61 - 929 * q^63 - 173 * q^65 + 2260 * q^67 + 1988 * q^69 + 465 * q^71 - 111 * q^73 + 4584 * q^75 - 1827 * q^79 + 6874 * q^81 + 4947 * q^83 - 2609 * q^85 - 1303 * q^87 + 446 * q^89 + 2176 * q^91 + 4204 * q^93 - 108 * q^95 + 3511 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.44598	−0.278280	−0.139140	−	0.990273i	$-0.544434\pi$
$3$	−1.44598	−0.278280	−0.139140	+	0.990273i	$0.544434\pi$
$4$	0	0
$5$	−9.31187	−0.832879	−0.416439	−	0.909163i	$-0.636722\pi$
$5$	−9.31187	−0.832879	−0.416439	+	0.909163i	$0.636722\pi$
$6$	0	0
$7$	22.5481	1.21748	0.608741	−	0.793369i	$-0.291675\pi$
$7$	22.5481	1.21748	0.608741	+	0.793369i	$0.291675\pi$
$8$	0	0
$9$	−24.9091	−0.922560
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	9.73910	0.207780	0.103890	−	0.994589i	$-0.466871\pi$
$13$	9.73910	0.207780	0.103890	+	0.994589i	$0.466871\pi$
$14$	0	0
$15$	13.4648	0.231773
$16$	0	0
$17$	132.634	1.89227	0.946135	−	0.323773i	$-0.104951\pi$
$17$	132.634	1.89227	0.946135	+	0.323773i	$0.104951\pi$
$18$	0	0
$19$	−29.8000	−0.359821	−0.179910	−	0.983683i	$-0.557581\pi$
$19$	−29.8000	−0.359821	−0.179910	+	0.983683i	$0.557581\pi$
$20$	0	0
$21$	−32.6042	−0.338801
$22$	0	0
$23$	−175.208	−1.58841	−0.794203	−	0.607653i	$-0.792111\pi$
$23$	−175.208	−1.58841	−0.794203	+	0.607653i	$0.792111\pi$
$24$	0	0
$25$	−38.2891	−0.306313
$26$	0	0
$27$	75.0598	0.535010
$28$	0	0
$29$	−74.0142	−0.473934	−0.236967	−	0.971518i	$-0.576153\pi$
$29$	−74.0142	−0.473934	−0.236967	+	0.971518i	$0.576153\pi$
$30$	0	0
$31$	−29.2295	−0.169348	−0.0846738	−	0.996409i	$-0.526985\pi$
$31$	−29.2295	−0.169348	−0.0846738	+	0.996409i	$0.526985\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−209.965	−1.01401
$36$	0	0
$37$	−85.4067	−0.379481	−0.189740	−	0.981834i	$-0.560765\pi$
$37$	−85.4067	−0.379481	−0.189740	+	0.981834i	$0.560765\pi$
$38$	0	0
$39$	−14.0826	−0.0578210
$40$	0	0
$41$	282.741	1.07699	0.538497	−	0.842628i	$-0.318992\pi$
$41$	282.741	1.07699	0.538497	+	0.842628i	$0.318992\pi$
$42$	0	0
$43$	212.780	0.754620	0.377310	−	0.926087i	$-0.376849\pi$
$43$	212.780	0.754620	0.377310	+	0.926087i	$0.376849\pi$
$44$	0	0
$45$	231.951	0.768381
$46$	0	0
$47$	−377.373	−1.17118	−0.585591	−	0.810607i	$-0.699137\pi$
$47$	−377.373	−1.17118	−0.585591	+	0.810607i	$0.699137\pi$
$48$	0	0
$49$	165.416	0.482261
$50$	0	0
$51$	−191.787	−0.526580
$52$	0	0
$53$	717.033	1.85834	0.929171	−	0.369650i	$-0.120523\pi$
$53$	717.033	1.85834	0.929171	+	0.369650i	$0.120523\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	43.0903	0.100131
$58$	0	0
$59$	57.5530	0.126996	0.0634979	−	0.997982i	$-0.479774\pi$
$59$	57.5530	0.126996	0.0634979	+	0.997982i	$0.479774\pi$
$60$	0	0
$61$	146.308	0.307096	0.153548	−	0.988141i	$-0.450930\pi$
$61$	146.308	0.307096	0.153548	+	0.988141i	$0.450930\pi$
$62$	0	0
$63$	−561.653	−1.12320
$64$	0	0
$65$	−90.6892	−0.173056
$66$	0	0
$67$	792.911	1.44581	0.722907	−	0.690945i	$-0.242806\pi$
$67$	792.911	1.44581	0.722907	+	0.690945i	$0.242806\pi$
$68$	0	0
$69$	253.348	0.442021
$70$	0	0
$71$	−8.64368	−0.0144481	−0.00722406	−	0.999974i	$-0.502300\pi$
$71$	−8.64368	−0.0144481	−0.00722406	+	0.999974i	$0.502300\pi$
$72$	0	0
$73$	−973.110	−1.56019	−0.780095	−	0.625661i	$-0.784829\pi$
$73$	−973.110	−1.56019	−0.780095	+	0.625661i	$0.784829\pi$
$74$	0	0
$75$	55.3654	0.0852407
$76$	0	0
$77$	0	0
$78$	0	0
$79$	732.331	1.04296	0.521479	−	0.853264i	$-0.325380\pi$
$79$	732.331	1.04296	0.521479	+	0.853264i	$0.325380\pi$
$80$	0	0
$81$	564.011	0.773678
$82$	0	0
$83$	587.100	0.776417	0.388209	−	0.921571i	$-0.373094\pi$
$83$	587.100	0.776417	0.388209	+	0.921571i	$0.373094\pi$
$84$	0	0
$85$	−1235.07	−1.57603
$86$	0	0
$87$	107.023	0.131886
$88$	0	0
$89$	−826.421	−0.984275	−0.492137	−	0.870517i	$-0.663784\pi$
$89$	−826.421	−0.984275	−0.492137	+	0.870517i	$0.663784\pi$
$90$	0	0
$91$	219.598	0.252968
$92$	0	0
$93$	42.2654	0.0471260
$94$	0	0
$95$	277.494	0.299687
$96$	0	0
$97$	301.181	0.315261	0.157630	−	0.987498i	$-0.449615\pi$
$97$	301.181	0.315261	0.157630	+	0.987498i	$0.449615\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−107.349	−0.105758	−0.0528792	−	0.998601i	$-0.516840\pi$
$101$	−107.349	−0.105758	−0.0528792	+	0.998601i	$0.516840\pi$
$102$	0	0
$103$	1430.19	1.36816	0.684081	−	0.729406i	$-0.260204\pi$
$103$	1430.19	1.36816	0.684081	+	0.729406i	$0.260204\pi$
$104$	0	0
$105$	303.606	0.282180
$106$	0	0
$107$	−173.227	−0.156509	−0.0782544	−	0.996933i	$-0.524935\pi$
$107$	−173.227	−0.156509	−0.0782544	+	0.996933i	$0.524935\pi$
$108$	0	0
$109$	1291.01	1.13446	0.567231	−	0.823559i	$-0.308015\pi$
$109$	1291.01	1.13446	0.567231	+	0.823559i	$0.308015\pi$
$110$	0	0
$111$	123.497	0.105602
$112$	0	0
$113$	−1081.41	−0.900268	−0.450134	−	0.892961i	$-0.648624\pi$
$113$	−1081.41	−0.900268	−0.450134	+	0.892961i	$0.648624\pi$
$114$	0	0
$115$	1631.51	1.32295
$116$	0	0
$117$	−242.592	−0.191690
$118$	0	0
$119$	2990.65	2.30380
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−408.839	−0.299706
$124$	0	0
$125$	1520.53	1.08800
$126$	0	0
$127$	1624.07	1.13474	0.567372	−	0.823461i	$-0.307960\pi$
$127$	1624.07	1.13474	0.567372	+	0.823461i	$0.307960\pi$
$128$	0	0
$129$	−307.677	−0.209995
$130$	0	0
$131$	2677.34	1.78565	0.892827	−	0.450400i	$-0.148719\pi$
$131$	2677.34	1.78565	0.892827	+	0.450400i	$0.148719\pi$
$132$	0	0
$133$	−671.933	−0.438075
$134$	0	0
$135$	−698.947	−0.445598
$136$	0	0
$137$	2502.94	1.56088	0.780441	−	0.625230i	$-0.214995\pi$
$137$	2502.94	1.56088	0.780441	+	0.625230i	$0.214995\pi$
$138$	0	0
$139$	−376.872	−0.229970	−0.114985	−	0.993367i	$-0.536682\pi$
$139$	−376.872	−0.229970	−0.114985	+	0.993367i	$0.536682\pi$
$140$	0	0
$141$	545.675	0.325916
$142$	0	0
$143$	0	0
$144$	0	0
$145$	689.210	0.394730
$146$	0	0
$147$	−239.188	−0.134204
$148$	0	0
$149$	1776.45	0.976727	0.488363	−	0.872640i	$-0.337594\pi$
$149$	1776.45	0.976727	0.488363	+	0.872640i	$0.337594\pi$
$150$	0	0
$151$	2338.99	1.26056	0.630279	−	0.776369i	$-0.282941\pi$
$151$	2338.99	1.26056	0.630279	+	0.776369i	$0.282941\pi$
$152$	0	0
$153$	−3303.81	−1.74573
$154$	0	0
$155$	272.181	0.141046
$156$	0	0
$157$	2749.43	1.39763	0.698816	−	0.715302i	$-0.253711\pi$
$157$	2749.43	1.39763	0.698816	+	0.715302i	$0.253711\pi$
$158$	0	0
$159$	−1036.82	−0.517139
$160$	0	0
$161$	−3950.60	−1.93385
$162$	0	0
$163$	1646.61	0.791242	0.395621	−	0.918414i	$-0.370529\pi$
$163$	1646.61	0.791242	0.395621	+	0.918414i	$0.370529\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1946.55	−0.901969	−0.450984	−	0.892532i	$-0.648927\pi$
$167$	−1946.55	−0.901969	−0.450984	+	0.892532i	$0.648927\pi$
$168$	0	0
$169$	−2102.15	−0.956827
$170$	0	0
$171$	742.292	0.331956
$172$	0	0
$173$	−4118.51	−1.80997	−0.904984	−	0.425445i	$-0.860118\pi$
$173$	−4118.51	−1.80997	−0.904984	+	0.425445i	$0.860118\pi$
$174$	0	0
$175$	−863.346	−0.372930
$176$	0	0
$177$	−83.2207	−0.0353404
$178$	0	0
$179$	−2017.03	−0.842234	−0.421117	−	0.907006i	$-0.638362\pi$
$179$	−2017.03	−0.842234	−0.421117	+	0.907006i	$0.638362\pi$
$180$	0	0
$181$	2586.69	1.06225	0.531126	−	0.847293i	$-0.321769\pi$
$181$	2586.69	1.06225	0.531126	+	0.847293i	$0.321769\pi$
$182$	0	0
$183$	−211.559	−0.0854586
$184$	0	0
$185$	795.296	0.316061
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1692.45	0.651364
$190$	0	0
$191$	159.921	0.0605836	0.0302918	−	0.999541i	$-0.490356\pi$
$191$	159.921	0.0605836	0.0302918	+	0.999541i	$0.490356\pi$
$192$	0	0
$193$	1758.29	0.655774	0.327887	−	0.944717i	$-0.393663\pi$
$193$	1758.29	0.655774	0.327887	+	0.944717i	$0.393663\pi$
$194$	0	0
$195$	131.135	0.0481579
$196$	0	0
$197$	−643.574	−0.232755	−0.116378	−	0.993205i	$-0.537128\pi$
$197$	−643.574	−0.232755	−0.116378	+	0.993205i	$0.537128\pi$
$198$	0	0
$199$	2632.06	0.937596	0.468798	−	0.883305i	$-0.344687\pi$
$199$	2632.06	0.937596	0.468798	+	0.883305i	$0.344687\pi$
$200$	0	0
$201$	−1146.54	−0.402341
$202$	0	0
$203$	−1668.88	−0.577006
$204$	0	0
$205$	−2632.85	−0.897005
$206$	0	0
$207$	4364.27	1.46540
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4744.27	1.54791	0.773955	−	0.633240i	$-0.218276\pi$
$211$	4744.27	1.54791	0.773955	+	0.633240i	$0.218276\pi$
$212$	0	0
$213$	12.4986	0.00402062
$214$	0	0
$215$	−1981.38	−0.628507
$216$	0	0
$217$	−659.069	−0.206178
$218$	0	0
$219$	1407.10	0.434169
$220$	0	0
$221$	1291.74	0.393176
$222$	0	0
$223$	−3200.56	−0.961101	−0.480551	−	0.876967i	$-0.659563\pi$
$223$	−3200.56	−0.961101	−0.480551	+	0.876967i	$0.659563\pi$
$224$	0	0
$225$	953.748	0.282592
$226$	0	0
$227$	3807.18	1.11318	0.556589	−	0.830788i	$-0.312110\pi$
$227$	3807.18	1.11318	0.556589	+	0.830788i	$0.312110\pi$
$228$	0	0
$229$	517.598	0.149362	0.0746808	−	0.997207i	$-0.476206\pi$
$229$	517.598	0.149362	0.0746808	+	0.997207i	$0.476206\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5438.43	−1.52911	−0.764557	−	0.644556i	$-0.777042\pi$
$233$	−5438.43	−1.52911	−0.764557	+	0.644556i	$0.777042\pi$
$234$	0	0
$235$	3514.05	0.975452
$236$	0	0
$237$	−1058.94	−0.290234
$238$	0	0
$239$	−1697.58	−0.459444	−0.229722	−	0.973256i	$-0.573782\pi$
$239$	−1697.58	−0.459444	−0.229722	+	0.973256i	$0.573782\pi$
$240$	0	0
$241$	4141.14	1.10686	0.553432	−	0.832895i	$-0.313318\pi$
$241$	4141.14	1.10686	0.553432	+	0.832895i	$0.313318\pi$
$242$	0	0
$243$	−2842.17	−0.750309
$244$	0	0
$245$	−1540.33	−0.401665
$246$	0	0
$247$	−290.225	−0.0747635
$248$	0	0
$249$	−848.938	−0.216061
$250$	0	0
$251$	−4804.99	−1.20832	−0.604160	−	0.796863i	$-0.706491\pi$
$251$	−4804.99	−1.20832	−0.604160	+	0.796863i	$0.706491\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1785.90	0.438578
$256$	0	0
$257$	1006.53	0.244301	0.122151	−	0.992512i	$-0.461021\pi$
$257$	1006.53	0.244301	0.122151	+	0.992512i	$0.461021\pi$
$258$	0	0
$259$	−1925.76	−0.462011
$260$	0	0
$261$	1843.63	0.437233
$262$	0	0
$263$	−3.13522	−0.000735080 0	−0.000367540	−	1.00000i	$-0.500117\pi$
$263$	−3.13522	−0.000735080 0	−0.000367540		1.00000i	$0.500117\pi$
$264$	0	0
$265$	−6676.92	−1.54777
$266$	0	0
$267$	1194.99	0.273904
$268$	0	0
$269$	2530.13	0.573476	0.286738	−	0.958009i	$-0.407429\pi$
$269$	2530.13	0.573476	0.286738	+	0.958009i	$0.407429\pi$
$270$	0	0
$271$	1913.13	0.428836	0.214418	−	0.976742i	$-0.431215\pi$
$271$	1913.13	0.428836	0.214418	+	0.976742i	$0.431215\pi$
$272$	0	0
$273$	−317.535	−0.0703960
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1648.04	−0.357478	−0.178739	−	0.983897i	$-0.557202\pi$
$277$	−1648.04	−0.357478	−0.178739	+	0.983897i	$0.557202\pi$
$278$	0	0
$279$	728.082	0.156233
$280$	0	0
$281$	5157.59	1.09493	0.547466	−	0.836828i	$-0.315593\pi$
$281$	5157.59	1.09493	0.547466	+	0.836828i	$0.315593\pi$
$282$	0	0
$283$	809.972	0.170134	0.0850668	−	0.996375i	$-0.472890\pi$
$283$	809.972	0.170134	0.0850668	+	0.996375i	$0.472890\pi$
$284$	0	0
$285$	−401.252	−0.0833968
$286$	0	0
$287$	6375.27	1.31122
$288$	0	0
$289$	12678.9	2.58068
$290$	0	0
$291$	−435.503	−0.0877307
$292$	0	0
$293$	−6527.26	−1.30146	−0.650728	−	0.759311i	$-0.725536\pi$
$293$	−6527.26	−1.30146	−0.650728	+	0.759311i	$0.725536\pi$
$294$	0	0
$295$	−535.926	−0.105772
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1706.37	−0.330039
$300$	0	0
$301$	4797.78	0.918736
$302$	0	0
$303$	155.225	0.0294304
$304$	0	0
$305$	−1362.40	−0.255774
$306$	0	0
$307$	8396.59	1.56097	0.780486	−	0.625173i	$-0.214971\pi$
$307$	8396.59	1.56097	0.780486	+	0.625173i	$0.214971\pi$
$308$	0	0
$309$	−2068.03	−0.380732
$310$	0	0
$311$	−7582.82	−1.38258	−0.691290	−	0.722578i	$-0.742957\pi$
$311$	−7582.82	−1.38258	−0.691290	+	0.722578i	$0.742957\pi$
$312$	0	0
$313$	−4240.94	−0.765853	−0.382926	−	0.923779i	$-0.625084\pi$
$313$	−4240.94	−0.765853	−0.382926	+	0.923779i	$0.625084\pi$
$314$	0	0
$315$	5230.04	0.935490
$316$	0	0
$317$	−6069.66	−1.07541	−0.537706	−	0.843132i	$-0.680709\pi$
$317$	−6069.66	−1.07541	−0.537706	+	0.843132i	$0.680709\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	250.483	0.0435532
$322$	0	0
$323$	−3952.51	−0.680877
$324$	0	0
$325$	−372.901	−0.0636457
$326$	0	0
$327$	−1866.78	−0.315698
$328$	0	0
$329$	−8509.03	−1.42589
$330$	0	0
$331$	10582.6	1.75732	0.878661	−	0.477446i	$-0.158437\pi$
$331$	10582.6	1.75732	0.878661	+	0.477446i	$0.158437\pi$
$332$	0	0
$333$	2127.41	0.350094
$334$	0	0
$335$	−7383.48	−1.20419
$336$	0	0
$337$	6398.34	1.03424	0.517121	−	0.855912i	$-0.327004\pi$
$337$	6398.34	1.03424	0.517121	+	0.855912i	$0.327004\pi$
$338$	0	0
$339$	1563.70	0.250527
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−4004.18	−0.630337
$344$	0	0
$345$	−2359.14	−0.368150
$346$	0	0
$347$	3813.50	0.589969	0.294984	−	0.955502i	$-0.404686\pi$
$347$	3813.50	0.589969	0.294984	+	0.955502i	$0.404686\pi$
$348$	0	0
$349$	2909.92	0.446316	0.223158	−	0.974782i	$-0.428363\pi$
$349$	2909.92	0.446316	0.223158	+	0.974782i	$0.428363\pi$
$350$	0	0
$351$	731.015	0.111164
$352$	0	0
$353$	−659.888	−0.0994966	−0.0497483	−	0.998762i	$-0.515842\pi$
$353$	−659.888	−0.0994966	−0.0497483	+	0.998762i	$0.515842\pi$
$354$	0	0
$355$	80.4888	0.0120335
$356$	0	0
$357$	−4324.44	−0.641102
$358$	0	0
$359$	8543.24	1.25597	0.627987	−	0.778223i	$-0.283879\pi$
$359$	8543.24	1.25597	0.627987	+	0.778223i	$0.283879\pi$
$360$	0	0
$361$	−5970.96	−0.870529
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9061.47	1.29945
$366$	0	0
$367$	−9669.07	−1.37526	−0.687631	−	0.726060i	$-0.741349\pi$
$367$	−9669.07	−1.37526	−0.687631	+	0.726060i	$0.741349\pi$
$368$	0	0
$369$	−7042.83	−0.993592
$370$	0	0
$371$	16167.7	2.26250
$372$	0	0
$373$	−11401.5	−1.58270	−0.791350	−	0.611363i	$-0.790621\pi$
$373$	−11401.5	−1.58270	−0.791350	+	0.611363i	$0.790621\pi$
$374$	0	0
$375$	−2198.66	−0.302769
$376$	0	0
$377$	−720.831	−0.0984740
$378$	0	0
$379$	−547.975	−0.0742680	−0.0371340	−	0.999310i	$-0.511823\pi$
$379$	−547.975	−0.0742680	−0.0371340	+	0.999310i	$0.511823\pi$
$380$	0	0
$381$	−2348.37	−0.315776
$382$	0	0
$383$	−7811.49	−1.04216	−0.521082	−	0.853507i	$-0.674471\pi$
$383$	−7811.49	−1.04216	−0.521082	+	0.853507i	$0.674471\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5300.17	−0.696182
$388$	0	0
$389$	−8783.97	−1.14490	−0.572449	−	0.819940i	$-0.694006\pi$
$389$	−8783.97	−1.14490	−0.572449	+	0.819940i	$0.694006\pi$
$390$	0	0
$391$	−23238.6	−3.00569
$392$	0	0
$393$	−3871.40	−0.496911
$394$	0	0
$395$	−6819.37	−0.868658
$396$	0	0
$397$	−8738.01	−1.10466	−0.552328	−	0.833627i	$-0.686260\pi$
$397$	−8738.01	−1.10466	−0.552328	+	0.833627i	$0.686260\pi$
$398$	0	0
$399$	971.604	0.121907
$400$	0	0
$401$	−3615.02	−0.450188	−0.225094	−	0.974337i	$-0.572269\pi$
$401$	−3615.02	−0.450188	−0.225094	+	0.974337i	$0.572269\pi$
$402$	0	0
$403$	−284.669	−0.0351870
$404$	0	0
$405$	−5252.00	−0.644380
$406$	0	0
$407$	0	0
$408$	0	0
$409$	369.919	0.0447220	0.0223610	−	0.999750i	$-0.492882\pi$
$409$	369.919	0.0447220	0.0223610	+	0.999750i	$0.492882\pi$
$410$	0	0
$411$	−3619.21	−0.434362
$412$	0	0
$413$	1297.71	0.154615
$414$	0	0
$415$	−5467.00	−0.646662
$416$	0	0
$417$	544.951	0.0639960
$418$	0	0
$419$	−8238.24	−0.960535	−0.480267	−	0.877122i	$-0.659460\pi$
$419$	−8238.24	−0.960535	−0.480267	+	0.877122i	$0.659460\pi$
$420$	0	0
$421$	−9069.94	−1.04998	−0.524990	−	0.851108i	$-0.675931\pi$
$421$	−9069.94	−1.04998	−0.524990	+	0.851108i	$0.675931\pi$
$422$	0	0
$423$	9400.03	1.08049
$424$	0	0
$425$	−5078.45	−0.579626
$426$	0	0
$427$	3298.97	0.373883
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2170.93	0.242622	0.121311	−	0.992615i	$-0.461290\pi$
$431$	2170.93	0.242622	0.121311	+	0.992615i	$0.461290\pi$
$432$	0	0
$433$	1077.19	0.119553	0.0597763	−	0.998212i	$-0.480961\pi$
$433$	1077.19	0.119553	0.0597763	+	0.998212i	$0.480961\pi$
$434$	0	0
$435$	−996.587	−0.109845
$436$	0	0
$437$	5221.19	0.571541
$438$	0	0
$439$	−5546.15	−0.602969	−0.301485	−	0.953471i	$-0.597482\pi$
$439$	−5546.15	−0.602969	−0.301485	+	0.953471i	$0.597482\pi$
$440$	0	0
$441$	−4120.36	−0.444915
$442$	0	0
$443$	8767.03	0.940258	0.470129	−	0.882598i	$-0.344207\pi$
$443$	8767.03	0.940258	0.470129	+	0.882598i	$0.344207\pi$
$444$	0	0
$445$	7695.53	0.819782
$446$	0	0
$447$	−2568.72	−0.271803
$448$	0	0
$449$	−5101.68	−0.536221	−0.268111	−	0.963388i	$-0.586399\pi$
$449$	−5101.68	−0.536221	−0.268111	+	0.963388i	$0.586399\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−3382.14	−0.350788
$454$	0	0
$455$	−2044.87	−0.210692
$456$	0	0
$457$	−9871.38	−1.01042	−0.505212	−	0.862995i	$-0.668586\pi$
$457$	−9871.38	−1.01042	−0.505212	+	0.862995i	$0.668586\pi$
$458$	0	0
$459$	9955.51	1.01238
$460$	0	0
$461$	1717.54	0.173522	0.0867610	−	0.996229i	$-0.472348\pi$
$461$	1717.54	0.173522	0.0867610	+	0.996229i	$0.472348\pi$
$462$	0	0
$463$	6422.84	0.644696	0.322348	−	0.946621i	$-0.395528\pi$
$463$	6422.84	0.644696	0.322348	+	0.946621i	$0.395528\pi$
$464$	0	0
$465$	−393.570	−0.0392503
$466$	0	0
$467$	−15260.2	−1.51211	−0.756056	−	0.654507i	$-0.772876\pi$
$467$	−15260.2	−1.51211	−0.756056	+	0.654507i	$0.772876\pi$
$468$	0	0
$469$	17878.6	1.76025
$470$	0	0
$471$	−3975.63	−0.388933
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1141.02	0.110218
$476$	0	0
$477$	−17860.7	−1.71443
$478$	0	0
$479$	−5076.96	−0.484284	−0.242142	−	0.970241i	$-0.577850\pi$
$479$	−5076.96	−0.484284	−0.242142	+	0.970241i	$0.577850\pi$
$480$	0	0
$481$	−831.784	−0.0788485
$482$	0	0
$483$	5712.50	0.538153
$484$	0	0
$485$	−2804.56	−0.262574
$486$	0	0
$487$	231.720	0.0215611	0.0107805	−	0.999942i	$-0.496568\pi$
$487$	231.720	0.0215611	0.0107805	+	0.999942i	$0.496568\pi$
$488$	0	0
$489$	−2380.97	−0.220187
$490$	0	0
$491$	19995.8	1.83788	0.918938	−	0.394403i	$-0.129049\pi$
$491$	19995.8	1.83788	0.918938	+	0.394403i	$0.129049\pi$
$492$	0	0
$493$	−9816.83	−0.896811
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−194.898	−0.0175903
$498$	0	0
$499$	14448.3	1.29618	0.648090	−	0.761563i	$-0.275568\pi$
$499$	14448.3	1.29618	0.648090	+	0.761563i	$0.275568\pi$
$500$	0	0
$501$	2814.68	0.251000
$502$	0	0
$503$	14629.8	1.29684	0.648418	−	0.761284i	$-0.275431\pi$
$503$	14629.8	1.29684	0.648418	+	0.761284i	$0.275431\pi$
$504$	0	0
$505$	999.617	0.0880839
$506$	0	0
$507$	3039.68	0.266266
$508$	0	0
$509$	16720.8	1.45606	0.728032	−	0.685543i	$-0.240435\pi$
$509$	16720.8	1.45606	0.728032	+	0.685543i	$0.240435\pi$
$510$	0	0
$511$	−21941.8	−1.89950
$512$	0	0
$513$	−2236.78	−0.192507
$514$	0	0
$515$	−13317.7	−1.13951
$516$	0	0
$517$	0	0
$518$	0	0
$519$	5955.30	0.503678
$520$	0	0
$521$	6618.34	0.556535	0.278268	−	0.960504i	$-0.410240\pi$
$521$	6618.34	0.556535	0.278268	+	0.960504i	$0.410240\pi$
$522$	0	0
$523$	15303.7	1.27951	0.639754	−	0.768580i	$-0.279036\pi$
$523$	15303.7	1.27951	0.639754	+	0.768580i	$0.279036\pi$
$524$	0	0
$525$	1248.38	0.103779
$526$	0	0
$527$	−3876.84	−0.320451
$528$	0	0
$529$	18530.7	1.52303
$530$	0	0
$531$	−1433.59	−0.117161
$532$	0	0
$533$	2753.64	0.223778
$534$	0	0
$535$	1613.06	0.130353
$536$	0	0
$537$	2916.59	0.234377
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10924.1	−0.868139	−0.434069	−	0.900879i	$-0.642923\pi$
$541$	−10924.1	−0.868139	−0.434069	+	0.900879i	$0.642923\pi$
$542$	0	0
$543$	−3740.32	−0.295603
$544$	0	0
$545$	−12021.7	−0.944869
$546$	0	0
$547$	−18891.6	−1.47668	−0.738341	−	0.674428i	$-0.764390\pi$
$547$	−18891.6	−1.47668	−0.738341	+	0.674428i	$0.764390\pi$
$548$	0	0
$549$	−3644.41	−0.283314
$550$	0	0
$551$	2205.62	0.170531
$552$	0	0
$553$	16512.7	1.26978
$554$	0	0
$555$	−1149.99	−0.0879535
$556$	0	0
$557$	1899.03	0.144460	0.0722301	−	0.997388i	$-0.476988\pi$
$557$	1899.03	0.144460	0.0722301	+	0.997388i	$0.476988\pi$
$558$	0	0
$559$	2072.29	0.156795
$560$	0	0
$561$	0	0
$562$	0	0
$563$	951.735	0.0712449	0.0356224	−	0.999365i	$-0.488659\pi$
$563$	951.735	0.0712449	0.0356224	+	0.999365i	$0.488659\pi$
$564$	0	0
$565$	10069.9	0.749815
$566$	0	0
$567$	12717.4	0.941939
$568$	0	0
$569$	−2310.66	−0.170243	−0.0851213	−	0.996371i	$-0.527128\pi$
$569$	−2310.66	−0.170243	−0.0851213	+	0.996371i	$0.527128\pi$
$570$	0	0
$571$	−3452.17	−0.253011	−0.126505	−	0.991966i	$-0.540376\pi$
$571$	−3452.17	−0.253011	−0.126505	+	0.991966i	$0.540376\pi$
$572$	0	0
$573$	−231.243	−0.0168592
$574$	0	0
$575$	6708.55	0.486549
$576$	0	0
$577$	−1066.18	−0.0769250	−0.0384625	−	0.999260i	$-0.512246\pi$
$577$	−1066.18	−0.0769250	−0.0384625	+	0.999260i	$0.512246\pi$
$578$	0	0
$579$	−2542.46	−0.182489
$580$	0	0
$581$	13238.0	0.945274
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2258.99	0.159654
$586$	0	0
$587$	13017.1	0.915287	0.457643	−	0.889136i	$-0.348694\pi$
$587$	13017.1	0.915287	0.457643	+	0.889136i	$0.348694\pi$
$588$	0	0
$589$	871.040	0.0609348
$590$	0	0
$591$	930.598	0.0647710
$592$	0	0
$593$	13757.5	0.952701	0.476350	−	0.879256i	$-0.341959\pi$
$593$	13757.5	0.952701	0.476350	+	0.879256i	$0.341959\pi$
$594$	0	0
$595$	−27848.6	−1.91879
$596$	0	0
$597$	−3805.91	−0.260914
$598$	0	0
$599$	22565.6	1.53924	0.769620	−	0.638502i	$-0.220445\pi$
$599$	22565.6	1.53924	0.769620	+	0.638502i	$0.220445\pi$
$600$	0	0
$601$	−7655.50	−0.519592	−0.259796	−	0.965664i	$-0.583655\pi$
$601$	−7655.50	−0.519592	−0.259796	+	0.965664i	$0.583655\pi$
$602$	0	0
$603$	−19750.7	−1.33385
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7760.09	0.518900	0.259450	−	0.965757i	$-0.416459\pi$
$607$	7760.09	0.518900	0.259450	+	0.965757i	$0.416459\pi$
$608$	0	0
$609$	2413.17	0.160569
$610$	0	0
$611$	−3675.27	−0.243348
$612$	0	0
$613$	−6010.18	−0.396002	−0.198001	−	0.980202i	$-0.563445\pi$
$613$	−6010.18	−0.396002	−0.198001	+	0.980202i	$0.563445\pi$
$614$	0	0
$615$	3807.06	0.249618
$616$	0	0
$617$	5960.28	0.388901	0.194450	−	0.980912i	$-0.437708\pi$
$617$	5960.28	0.388901	0.194450	+	0.980912i	$0.437708\pi$
$618$	0	0
$619$	−10354.9	−0.672370	−0.336185	−	0.941796i	$-0.609137\pi$
$619$	−10354.9	−0.672370	−0.336185	+	0.941796i	$0.609137\pi$
$620$	0	0
$621$	−13151.1	−0.849813
$622$	0	0
$623$	−18634.2	−1.19834
$624$	0	0
$625$	−9372.81	−0.599860
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−11327.9	−0.718080
$630$	0	0
$631$	−8902.87	−0.561676	−0.280838	−	0.959755i	$-0.590612\pi$
$631$	−8902.87	−0.561676	−0.280838	+	0.959755i	$0.590612\pi$
$632$	0	0
$633$	−6860.14	−0.430752
$634$	0	0
$635$	−15123.1	−0.945105
$636$	0	0
$637$	1611.00	0.100204
$638$	0	0
$639$	215.307	0.0133293
$640$	0	0
$641$	−25585.8	−1.57656	−0.788282	−	0.615315i	$-0.789029\pi$
$641$	−25585.8	−1.57656	−0.788282	+	0.615315i	$0.789029\pi$
$642$	0	0
$643$	8588.20	0.526727	0.263363	−	0.964697i	$-0.415168\pi$
$643$	8588.20	0.526727	0.263363	+	0.964697i	$0.415168\pi$
$644$	0	0
$645$	2865.04	0.174901
$646$	0	0
$647$	20014.8	1.21617	0.608087	−	0.793870i	$-0.291937\pi$
$647$	20014.8	1.21617	0.608087	+	0.793870i	$0.291937\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	953.004	0.0573751
$652$	0	0
$653$	−29328.3	−1.75759	−0.878793	−	0.477204i	$-0.841650\pi$
$653$	−29328.3	−1.75759	−0.878793	+	0.477204i	$0.841650\pi$
$654$	0	0
$655$	−24931.1	−1.48723
$656$	0	0
$657$	24239.3	1.43937
$658$	0	0
$659$	3507.31	0.207322	0.103661	−	0.994613i	$-0.466944\pi$
$659$	3507.31	0.207322	0.103661	+	0.994613i	$0.466944\pi$
$660$	0	0
$661$	22349.0	1.31509	0.657545	−	0.753415i	$-0.271595\pi$
$661$	22349.0	1.31509	0.657545	+	0.753415i	$0.271595\pi$
$662$	0	0
$663$	−1867.84	−0.109413
$664$	0	0
$665$	6256.95	0.364863
$666$	0	0
$667$	12967.9	0.752800
$668$	0	0
$669$	4627.96	0.267455
$670$	0	0
$671$	0	0
$672$	0	0
$673$	13124.3	0.751715	0.375858	−	0.926677i	$-0.377348\pi$
$673$	13124.3	0.751715	0.375858	+	0.926677i	$0.377348\pi$
$674$	0	0
$675$	−2873.97	−0.163880
$676$	0	0
$677$	−11688.1	−0.663529	−0.331765	−	0.943362i	$-0.607644\pi$
$677$	−11688.1	−0.663529	−0.331765	+	0.943362i	$0.607644\pi$
$678$	0	0
$679$	6791.05	0.383824
$680$	0	0
$681$	−5505.12	−0.309775
$682$	0	0
$683$	−23890.4	−1.33842	−0.669211	−	0.743073i	$-0.733368\pi$
$683$	−23890.4	−1.33842	−0.669211	+	0.743073i	$0.733368\pi$
$684$	0	0
$685$	−23307.1	−1.30002
$686$	0	0
$687$	−748.438	−0.0415643
$688$	0	0
$689$	6983.26	0.386126
$690$	0	0
$691$	24296.0	1.33757	0.668787	−	0.743454i	$-0.266814\pi$
$691$	24296.0	1.33757	0.668787	+	0.743454i	$0.266814\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3509.38	0.191537
$696$	0	0
$697$	37501.2	2.03796
$698$	0	0
$699$	7863.89	0.425522
$700$	0	0
$701$	−4331.47	−0.233377	−0.116689	−	0.993169i	$-0.537228\pi$
$701$	−4331.47	−0.233377	−0.116689	+	0.993169i	$0.537228\pi$
$702$	0	0
$703$	2545.12	0.136545
$704$	0	0
$705$	−5081.26	−0.271449
$706$	0	0
$707$	−2420.51	−0.128759
$708$	0	0
$709$	−23967.4	−1.26955	−0.634776	−	0.772696i	$-0.718908\pi$
$709$	−23967.4	−1.26955	−0.634776	+	0.772696i	$0.718908\pi$
$710$	0	0
$711$	−18241.7	−0.962192
$712$	0	0
$713$	5121.24	0.268993
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2454.67	0.127854
$718$	0	0
$719$	−27812.8	−1.44262	−0.721310	−	0.692612i	$-0.756460\pi$
$719$	−27812.8	−1.44262	−0.721310	+	0.692612i	$0.756460\pi$
$720$	0	0
$721$	32248.0	1.66571
$722$	0	0
$723$	−5988.02	−0.308018
$724$	0	0
$725$	2833.94	0.145172
$726$	0	0
$727$	31593.5	1.61175	0.805873	−	0.592089i	$-0.201697\pi$
$727$	31593.5	1.61175	0.805873	+	0.592089i	$0.201697\pi$
$728$	0	0
$729$	−11118.6	−0.564882
$730$	0	0
$731$	28222.0	1.42794
$732$	0	0
$733$	−20401.8	−1.02804	−0.514022	−	0.857777i	$-0.671845\pi$
$733$	−20401.8	−1.02804	−0.514022	+	0.857777i	$0.671845\pi$
$734$	0	0
$735$	2227.29	0.111775
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−22500.7	−1.12003	−0.560016	−	0.828482i	$-0.689205\pi$
$739$	−22500.7	−1.12003	−0.560016	+	0.828482i	$0.689205\pi$
$740$	0	0
$741$	419.661	0.0208052
$742$	0	0
$743$	17567.4	0.867407	0.433704	−	0.901056i	$-0.357206\pi$
$743$	17567.4	0.867407	0.433704	+	0.901056i	$0.357206\pi$
$744$	0	0
$745$	−16542.1	−0.813495
$746$	0	0
$747$	−14624.2	−0.716292
$748$	0	0
$749$	−3905.92	−0.190547
$750$	0	0
$751$	20813.7	1.01132	0.505660	−	0.862733i	$-0.331249\pi$
$751$	20813.7	1.01132	0.505660	+	0.862733i	$0.331249\pi$
$752$	0	0
$753$	6947.95	0.336251
$754$	0	0
$755$	−21780.3	−1.04989
$756$	0	0
$757$	19360.2	0.929535	0.464767	−	0.885433i	$-0.346138\pi$
$757$	19360.2	0.929535	0.464767	+	0.885433i	$0.346138\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9659.98	−0.460150	−0.230075	−	0.973173i	$-0.573897\pi$
$761$	−9659.98	−0.460150	−0.230075	+	0.973173i	$0.573897\pi$
$762$	0	0
$763$	29109.8	1.38119
$764$	0	0
$765$	30764.6	1.45398
$766$	0	0
$767$	560.514	0.0263872
$768$	0	0
$769$	12769.9	0.598824	0.299412	−	0.954124i	$-0.403209\pi$
$769$	12769.9	0.598824	0.299412	+	0.954124i	$0.403209\pi$
$770$	0	0
$771$	−1455.42	−0.0679841
$772$	0	0
$773$	9630.21	0.448091	0.224046	−	0.974579i	$-0.428074\pi$
$773$	9630.21	0.448091	0.224046	+	0.974579i	$0.428074\pi$
$774$	0	0
$775$	1119.17	0.0518733
$776$	0	0
$777$	2784.61	0.128568
$778$	0	0
$779$	−8425.69	−0.387524
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−5555.49	−0.253559
$784$	0	0
$785$	−25602.3	−1.16406
$786$	0	0
$787$	41584.5	1.88352	0.941758	−	0.336292i	$-0.109173\pi$
$787$	41584.5	1.88352	0.941758	+	0.336292i	$0.109173\pi$
$788$	0	0
$789$	4.53348	0.000204558 0
$790$	0	0
$791$	−24383.7	−1.09606
$792$	0	0
$793$	1424.91	0.0638084
$794$	0	0
$795$	9654.72	0.430714
$796$	0	0
$797$	41357.4	1.83808	0.919042	−	0.394160i	$-0.128964\pi$
$797$	41357.4	1.83808	0.919042	+	0.394160i	$0.128964\pi$
$798$	0	0
$799$	−50052.7	−2.21619
$800$	0	0
$801$	20585.4	0.908053
$802$	0	0
$803$	0	0
$804$	0	0
$805$	36787.4	1.61067
$806$	0	0
$807$	−3658.53	−0.159587
$808$	0	0
$809$	7306.79	0.317544	0.158772	−	0.987315i	$-0.449247\pi$
$809$	7306.79	0.317544	0.158772	+	0.987315i	$0.449247\pi$
$810$	0	0
$811$	−33916.3	−1.46851	−0.734255	−	0.678874i	$-0.762468\pi$
$811$	−33916.3	−1.46851	−0.734255	+	0.678874i	$0.762468\pi$
$812$	0	0
$813$	−2766.36	−0.119336
$814$	0	0
$815$	−15333.0	−0.659009
$816$	0	0
$817$	−6340.85	−0.271528
$818$	0	0
$819$	−5469.99	−0.233379
$820$	0	0
$821$	39410.7	1.67533	0.837663	−	0.546188i	$-0.183921\pi$
$821$	39410.7	1.67533	0.837663	+	0.546188i	$0.183921\pi$
$822$	0	0
$823$	−4206.08	−0.178147	−0.0890734	−	0.996025i	$-0.528391\pi$
$823$	−4206.08	−0.178147	−0.0890734	+	0.996025i	$0.528391\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31420.2	−1.32115	−0.660573	−	0.750762i	$-0.729687\pi$
$827$	−31420.2	−1.32115	−0.660573	+	0.750762i	$0.729687\pi$
$828$	0	0
$829$	42850.6	1.79525	0.897624	−	0.440761i	$-0.145291\pi$
$829$	42850.6	1.79525	0.897624	+	0.440761i	$0.145291\pi$
$830$	0	0
$831$	2383.05	0.0994789
$832$	0	0
$833$	21939.8	0.912569
$834$	0	0
$835$	18126.0	0.751231
$836$	0	0
$837$	−2193.96	−0.0906026
$838$	0	0
$839$	−1641.17	−0.0675320	−0.0337660	−	0.999430i	$-0.510750\pi$
$839$	−1641.17	−0.0675320	−0.0337660	+	0.999430i	$0.510750\pi$
$840$	0	0
$841$	−18910.9	−0.775386
$842$	0	0
$843$	−7457.79	−0.304697
$844$	0	0
$845$	19574.9	0.796921
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1171.21	−0.0473448
$850$	0	0
$851$	14963.9	0.602769
$852$	0	0
$853$	−21113.4	−0.847489	−0.423745	−	0.905782i	$-0.639285\pi$
$853$	−21113.4	−0.847489	−0.423745	+	0.905782i	$0.639285\pi$
$854$	0	0
$855$	−6912.13	−0.276479
$856$	0	0
$857$	12058.4	0.480640	0.240320	−	0.970694i	$-0.422748\pi$
$857$	12058.4	0.480640	0.240320	+	0.970694i	$0.422748\pi$
$858$	0	0
$859$	17435.3	0.692533	0.346267	−	0.938136i	$-0.387449\pi$
$859$	17435.3	0.692533	0.346267	+	0.938136i	$0.387449\pi$
$860$	0	0
$861$	−9218.54	−0.364886
$862$	0	0
$863$	11814.0	0.465995	0.232998	−	0.972477i	$-0.425147\pi$
$863$	11814.0	0.465995	0.232998	+	0.972477i	$0.425147\pi$
$864$	0	0
$865$	38351.0	1.50748
$866$	0	0
$867$	−18333.5	−0.718152
$868$	0	0
$869$	0	0
$870$	0	0
$871$	7722.24	0.300411
$872$	0	0
$873$	−7502.16	−0.290847
$874$	0	0
$875$	34284.9	1.32462
$876$	0	0
$877$	12722.4	0.489859	0.244929	−	0.969541i	$-0.421235\pi$
$877$	12722.4	0.489859	0.244929	+	0.969541i	$0.421235\pi$
$878$	0	0
$879$	9438.31	0.362169
$880$	0	0
$881$	−13310.0	−0.508997	−0.254499	−	0.967073i	$-0.581910\pi$
$881$	−13310.0	−0.508997	−0.254499	+	0.967073i	$0.581910\pi$
$882$	0	0
$883$	29451.2	1.12244	0.561219	−	0.827667i	$-0.310333\pi$
$883$	29451.2	1.12244	0.561219	+	0.827667i	$0.310333\pi$
$884$	0	0
$885$	774.940	0.0294343
$886$	0	0
$887$	−35615.1	−1.34818	−0.674091	−	0.738648i	$-0.735465\pi$
$887$	−35615.1	−1.34818	−0.674091	+	0.738648i	$0.735465\pi$
$888$	0	0
$889$	36619.6	1.38153
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11245.7	0.421415
$894$	0	0
$895$	18782.3	0.701479
$896$	0	0
$897$	2467.38	0.0918432
$898$	0	0
$899$	2163.40	0.0802596
$900$	0	0
$901$	95103.3	3.51648
$902$	0	0
$903$	−6937.51	−0.255666
$904$	0	0
$905$	−24087.0	−0.884726
$906$	0	0
$907$	49503.8	1.81229	0.906144	−	0.422969i	$-0.139012\pi$
$907$	49503.8	1.81229	0.906144	+	0.422969i	$0.139012\pi$
$908$	0	0
$909$	2673.96	0.0975685
$910$	0	0
$911$	53045.5	1.92917	0.964586	−	0.263768i	$-0.0849653\pi$
$911$	53045.5	1.92917	0.964586	+	0.263768i	$0.0849653\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1970.01	0.0711766
$916$	0	0
$917$	60369.0	2.17400
$918$	0	0
$919$	−43818.7	−1.57285	−0.786423	−	0.617688i	$-0.788069\pi$
$919$	−43818.7	−1.57285	−0.786423	+	0.617688i	$0.788069\pi$
$920$	0	0
$921$	−12141.3	−0.434387
$922$	0	0
$923$	−84.1817	−0.00300203
$924$	0	0
$925$	3270.15	0.116240
$926$	0	0
$927$	−35624.8	−1.26221
$928$	0	0
$929$	1532.55	0.0541241	0.0270621	−	0.999634i	$-0.491385\pi$
$929$	1532.55	0.0541241	0.0270621	+	0.999634i	$0.491385\pi$
$930$	0	0
$931$	−4929.39	−0.173528
$932$	0	0
$933$	10964.6	0.384744
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8020.03	0.279619	0.139809	−	0.990178i	$-0.455351\pi$
$937$	8020.03	0.279619	0.139809	+	0.990178i	$0.455351\pi$
$938$	0	0
$939$	6132.33	0.213121
$940$	0	0
$941$	−3612.88	−0.125161	−0.0625805	−	0.998040i	$-0.519933\pi$
$941$	−3612.88	−0.125161	−0.0625805	+	0.998040i	$0.519933\pi$
$942$	0	0
$943$	−49538.4	−1.71070
$944$	0	0
$945$	−15759.9	−0.542508
$946$	0	0
$947$	−16335.4	−0.560537	−0.280268	−	0.959922i	$-0.590423\pi$
$947$	−16335.4	−0.560537	−0.280268	+	0.959922i	$0.590423\pi$
$948$	0	0
$949$	−9477.21	−0.324176
$950$	0	0
$951$	8776.63	0.299266
$952$	0	0
$953$	−9317.59	−0.316712	−0.158356	−	0.987382i	$-0.550619\pi$
$953$	−9317.59	−0.316712	−0.158356	+	0.987382i	$0.550619\pi$
$954$	0	0
$955$	−1489.16	−0.0504588
$956$	0	0
$957$	0	0
$958$	0	0
$959$	56436.5	1.90034
$960$	0	0
$961$	−28936.6	−0.971321
$962$	0	0
$963$	4314.92	0.144389
$964$	0	0
$965$	−16373.0	−0.546181
$966$	0	0
$967$	−15791.6	−0.525153	−0.262577	−	0.964911i	$-0.584572\pi$
$967$	−15791.6	−0.525153	−0.262577	+	0.964911i	$0.584572\pi$
$968$	0	0
$969$	5715.26	0.189474
$970$	0	0
$971$	31862.5	1.05305	0.526527	−	0.850159i	$-0.323494\pi$
$971$	31862.5	1.05305	0.526527	+	0.850159i	$0.323494\pi$
$972$	0	0
$973$	−8497.73	−0.279984
$974$	0	0
$975$	539.209	0.0177113
$976$	0	0
$977$	2555.12	0.0836700	0.0418350	−	0.999125i	$-0.486680\pi$
$977$	2555.12	0.0836700	0.0418350	+	0.999125i	$0.486680\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−32157.9	−1.04661
$982$	0	0
$983$	−248.443	−0.00806114	−0.00403057	−	0.999992i	$-0.501283\pi$
$983$	−248.443	−0.00806114	−0.00403057	+	0.999992i	$0.501283\pi$
$984$	0	0
$985$	5992.88	0.193857
$986$	0	0
$987$	12303.9	0.396797
$988$	0	0
$989$	−37280.7	−1.19864
$990$	0	0
$991$	−739.784	−0.0237134	−0.0118567	−	0.999930i	$-0.503774\pi$
$991$	−739.784	−0.0237134	−0.0118567	+	0.999930i	$0.503774\pi$
$992$	0	0
$993$	−15302.3	−0.489027
$994$	0	0
$995$	−24509.4	−0.780904
$996$	0	0
$997$	−45993.2	−1.46100	−0.730501	−	0.682912i	$-0.760713\pi$
$997$	−45993.2	−1.46100	−0.730501	+	0.682912i	$0.760713\pi$
$998$	0	0
$999$	−6410.61	−0.203026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.r.1.4		10
4.3	odd	2		1936.4.a.by.1.7		10
11.2	odd	10		88.4.i.b.81.2	yes	20
11.6	odd	10		88.4.i.b.25.2	✓	20
11.10	odd	2		968.4.a.s.1.4		10
44.35	even	10		176.4.m.f.81.4		20
44.39	even	10		176.4.m.f.113.4		20
44.43	even	2		1936.4.a.bx.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.b.25.2	✓	20	11.6	odd	10
88.4.i.b.81.2	yes	20	11.2	odd	10
176.4.m.f.81.4		20	44.35	even	10
176.4.m.f.113.4		20	44.39	even	10
968.4.a.r.1.4		10	1.1	even	1	trivial
968.4.a.s.1.4		10	11.10	odd	2
1936.4.a.bx.1.7		10	44.43	even	2
1936.4.a.by.1.7		10	4.3	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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q-1.44598 q^{3} -9.31187 q^{5} +22.5481 q^{7} -24.9091 q^{9} +O(q^{10})\)	\(q-1.44598 q^{3} -9.31187 q^{5} +22.5481 q^{7} -24.9091 q^{9} +9.73910 q^{13} +13.4648 q^{15} +132.634 q^{17} -29.8000 q^{19} -32.6042 q^{21} -175.208 q^{23} -38.2891 q^{25} +75.0598 q^{27} -74.0142 q^{29} -29.2295 q^{31} -209.965 q^{35} -85.4067 q^{37} -14.0826 q^{39} +282.741 q^{41} +212.780 q^{43} +231.951 q^{45} -377.373 q^{47} +165.416 q^{49} -191.787 q^{51} +717.033 q^{53} +43.0903 q^{57} +57.5530 q^{59} +146.308 q^{61} -561.653 q^{63} -90.6892 q^{65} +792.911 q^{67} +253.348 q^{69} -8.64368 q^{71} -973.110 q^{73} +55.3654 q^{75} +732.331 q^{79} +564.011 q^{81} +587.100 q^{83} -1235.07 q^{85} +107.023 q^{87} -826.421 q^{89} +219.598 q^{91} +42.2654 q^{93} +277.494 q^{95} +301.181 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) \) 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9} - 45 q^{13} + 120 q^{15} + 17 q^{17} + 147 q^{19} - 131 q^{21} + 164 q^{23} + 439 q^{25} + 420 q^{27} - 177 q^{29} + 275 q^{31} + 220 q^{35} + 745 q^{37} + 524 q^{39} - 967 q^{41} + 380 q^{43} - 44 q^{45} + 769 q^{47} + 503 q^{49} + 956 q^{51} + 701 q^{53} - 1293 q^{57} + 1291 q^{59} + 1359 q^{61} - 929 q^{63} - 173 q^{65} + 2260 q^{67} + 1988 q^{69} + 465 q^{71} - 111 q^{73} + 4584 q^{75} - 1827 q^{79} + 6874 q^{81} + 4947 q^{83} - 2609 q^{85} - 1303 q^{87} + 446 q^{89} + 2176 q^{91} + 4204 q^{93} - 108 q^{95} + 3511 q^{97}+O(q^{100}) \) 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9 - 45 * q^13 + 120 * q^15 + 17 * q^17 + 147 * q^19 - 131 * q^21 + 164 * q^23 + 439 * q^25 + 420 * q^27 - 177 * q^29 + 275 * q^31 + 220 * q^35 + 745 * q^37 + 524 * q^39 - 967 * q^41 + 380 * q^43 - 44 * q^45 + 769 * q^47 + 503 * q^49 + 956 * q^51 + 701 * q^53 - 1293 * q^57 + 1291 * q^59 + 1359 * q^61 - 929 * q^63 - 173 * q^65 + 2260 * q^67 + 1988 * q^69 + 465 * q^71 - 111 * q^73 + 4584 * q^75 - 1827 * q^79 + 6874 * q^81 + 4947 * q^83 - 2609 * q^85 - 1303 * q^87 + 446 * q^89 + 2176 * q^91 + 4204 * q^93 - 108 * q^95 + 3511 * q^97

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