LMFDB - Embedded newform 968.4.a.r.1.7

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			1.7
Root			$3.51662$ 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)

$f(q)$	$=$	$q+2.89859 q^{3} -6.53312 q^{5} +24.8165 q^{7} -18.5982 q^{9} +O(q^{10})$	$q+2.89859 q^{3} -6.53312 q^{5} +24.8165 q^{7} -18.5982 q^{9} +72.3177 q^{13} -18.9368 q^{15} +5.55671 q^{17} -39.2124 q^{19} +71.9327 q^{21} +172.047 q^{23} -82.3183 q^{25} -132.170 q^{27} -13.5982 q^{29} -105.198 q^{31} -162.129 q^{35} +435.420 q^{37} +209.619 q^{39} -65.7572 q^{41} -230.717 q^{43} +121.504 q^{45} +245.343 q^{47} +272.857 q^{49} +16.1066 q^{51} +300.122 q^{53} -113.661 q^{57} -87.9961 q^{59} +333.935 q^{61} -461.542 q^{63} -472.460 q^{65} -345.430 q^{67} +498.692 q^{69} +656.985 q^{71} +1063.24 q^{73} -238.607 q^{75} -535.693 q^{79} +119.044 q^{81} +31.5363 q^{83} -36.3026 q^{85} -39.4157 q^{87} +1203.72 q^{89} +1794.67 q^{91} -304.924 q^{93} +256.179 q^{95} -522.820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.89859	0.557833	0.278917	−	0.960315i	$-0.410025\pi$
$3$	2.89859	0.557833	0.278917	+	0.960315i	$0.410025\pi$
$4$	0	0
$5$	−6.53312	−0.584340	−0.292170	−	0.956366i	$-0.594377\pi$
$5$	−6.53312	−0.584340	−0.292170	+	0.956366i	$0.594377\pi$
$6$	0	0
$7$	24.8165	1.33996	0.669982	−	0.742377i	$-0.266302\pi$
$7$	24.8165	1.33996	0.669982	+	0.742377i	$0.266302\pi$
$8$	0	0
$9$	−18.5982	−0.688822
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	72.3177	1.54287	0.771435	−	0.636308i	$-0.219539\pi$
$13$	72.3177	1.54287	0.771435	+	0.636308i	$0.219539\pi$
$14$	0	0
$15$	−18.9368	−0.325965
$16$	0	0
$17$	5.55671	0.0792764	0.0396382	−	0.999214i	$-0.487379\pi$
$17$	5.55671	0.0792764	0.0396382	+	0.999214i	$0.487379\pi$
$18$	0	0
$19$	−39.2124	−0.473470	−0.236735	−	0.971574i	$-0.576077\pi$
$19$	−39.2124	−0.473470	−0.236735	+	0.971574i	$0.576077\pi$
$20$	0	0
$21$	71.9327	0.747477
$22$	0	0
$23$	172.047	1.55975	0.779874	−	0.625936i	$-0.215283\pi$
$23$	172.047	1.55975	0.779874	+	0.625936i	$0.215283\pi$
$24$	0	0
$25$	−82.3183	−0.658546
$26$	0	0
$27$	−132.170	−0.942081
$28$	0	0
$29$	−13.5982	−0.0870735	−0.0435367	−	0.999052i	$-0.513863\pi$
$29$	−13.5982	−0.0870735	−0.0435367	+	0.999052i	$0.513863\pi$
$30$	0	0
$31$	−105.198	−0.609485	−0.304743	−	0.952435i	$-0.598570\pi$
$31$	−105.198	−0.609485	−0.304743	+	0.952435i	$0.598570\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−162.129	−0.782995
$36$	0	0
$37$	435.420	1.93466	0.967332	−	0.253512i	$-0.0815859\pi$
$37$	435.420	1.93466	0.967332	+	0.253512i	$0.0815859\pi$
$38$	0	0
$39$	209.619	0.860665
$40$	0	0
$41$	−65.7572	−0.250477	−0.125238	−	0.992127i	$-0.539970\pi$
$41$	−65.7572	−0.250477	−0.125238	+	0.992127i	$0.539970\pi$
$42$	0	0
$43$	−230.717	−0.818234	−0.409117	−	0.912482i	$-0.634163\pi$
$43$	−230.717	−0.818234	−0.409117	+	0.912482i	$0.634163\pi$
$44$	0	0
$45$	121.504	0.402506
$46$	0	0
$47$	245.343	0.761425	0.380712	−	0.924693i	$-0.375679\pi$
$47$	245.343	0.761425	0.380712	+	0.924693i	$0.375679\pi$
$48$	0	0
$49$	272.857	0.795503
$50$	0	0
$51$	16.1066	0.0442230
$52$	0	0
$53$	300.122	0.777828	0.388914	−	0.921274i	$-0.372850\pi$
$53$	300.122	0.777828	0.388914	+	0.921274i	$0.372850\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−113.661	−0.264118
$58$	0	0
$59$	−87.9961	−0.194171	−0.0970857	−	0.995276i	$-0.530952\pi$
$59$	−87.9961	−0.194171	−0.0970857	+	0.995276i	$0.530952\pi$
$60$	0	0
$61$	333.935	0.700919	0.350459	−	0.936578i	$-0.386025\pi$
$61$	333.935	0.700919	0.350459	+	0.936578i	$0.386025\pi$
$62$	0	0
$63$	−461.542	−0.922996
$64$	0	0
$65$	−472.460	−0.901561
$66$	0	0
$67$	−345.430	−0.629866	−0.314933	−	0.949114i	$-0.601982\pi$
$67$	−345.430	−0.629866	−0.314933	+	0.949114i	$0.601982\pi$
$68$	0	0
$69$	498.692	0.870080
$70$	0	0
$71$	656.985	1.09817	0.549083	−	0.835768i	$-0.314977\pi$
$71$	656.985	1.09817	0.549083	+	0.835768i	$0.314977\pi$
$72$	0	0
$73$	1063.24	1.70470	0.852352	−	0.522969i	$-0.175176\pi$
$73$	1063.24	1.70470	0.852352	+	0.522969i	$0.175176\pi$
$74$	0	0
$75$	−238.607	−0.367359
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−535.693	−0.762914	−0.381457	−	0.924387i	$-0.624578\pi$
$79$	−535.693	−0.762914	−0.381457	+	0.924387i	$0.624578\pi$
$80$	0	0
$81$	119.044	0.163297
$82$	0	0
$83$	31.5363	0.0417055	0.0208528	−	0.999783i	$-0.493362\pi$
$83$	31.5363	0.0417055	0.0208528	+	0.999783i	$0.493362\pi$
$84$	0	0
$85$	−36.3026	−0.0463244
$86$	0	0
$87$	−39.4157	−0.0485725
$88$	0	0
$89$	1203.72	1.43364	0.716818	−	0.697261i	$-0.245598\pi$
$89$	1203.72	1.43364	0.716818	+	0.697261i	$0.245598\pi$
$90$	0	0
$91$	1794.67	2.06739
$92$	0	0
$93$	−304.924	−0.339991
$94$	0	0
$95$	256.179	0.276668
$96$	0	0
$97$	−522.820	−0.547261	−0.273630	−	0.961835i	$-0.588224\pi$
$97$	−522.820	−0.547261	−0.273630	+	0.961835i	$0.588224\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1597.19	−1.57353	−0.786766	−	0.617251i	$-0.788246\pi$
$101$	−1597.19	−1.57353	−0.786766	+	0.617251i	$0.788246\pi$
$102$	0	0
$103$	1241.40	1.18756	0.593782	−	0.804626i	$-0.297634\pi$
$103$	1241.40	1.18756	0.593782	+	0.804626i	$0.297634\pi$
$104$	0	0
$105$	−469.945	−0.436781
$106$	0	0
$107$	1529.38	1.38179	0.690893	−	0.722957i	$-0.257217\pi$
$107$	1529.38	1.38179	0.690893	+	0.722957i	$0.257217\pi$
$108$	0	0
$109$	1511.94	1.32860	0.664300	−	0.747466i	$-0.268730\pi$
$109$	1511.94	1.32860	0.664300	+	0.747466i	$0.268730\pi$
$110$	0	0
$111$	1262.10	1.07922
$112$	0	0
$113$	−460.951	−0.383740	−0.191870	−	0.981420i	$-0.561455\pi$
$113$	−460.951	−0.383740	−0.191870	+	0.981420i	$0.561455\pi$
$114$	0	0
$115$	−1124.00	−0.911424
$116$	0	0
$117$	−1344.98	−1.06276
$118$	0	0
$119$	137.898	0.106228
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−190.603	−0.139724
$124$	0	0
$125$	1354.44	0.969155
$126$	0	0
$127$	2120.77	1.48179	0.740896	−	0.671620i	$-0.234401\pi$
$127$	2120.77	1.48179	0.740896	+	0.671620i	$0.234401\pi$
$128$	0	0
$129$	−668.755	−0.456439
$130$	0	0
$131$	−1534.70	−1.02357	−0.511785	−	0.859113i	$-0.671016\pi$
$131$	−1534.70	−1.02357	−0.511785	+	0.859113i	$0.671016\pi$
$132$	0	0
$133$	−973.113	−0.634433
$134$	0	0
$135$	863.485	0.550496
$136$	0	0
$137$	−3053.34	−1.90412	−0.952061	−	0.305907i	$-0.901040\pi$
$137$	−3053.34	−1.90412	−0.952061	+	0.305907i	$0.901040\pi$
$138$	0	0
$139$	2949.55	1.79984	0.899920	−	0.436055i	$-0.143625\pi$
$139$	2949.55	1.79984	0.899920	+	0.436055i	$0.143625\pi$
$140$	0	0
$141$	711.149	0.424748
$142$	0	0
$143$	0	0
$144$	0	0
$145$	88.8390	0.0508805
$146$	0	0
$147$	790.901	0.443758
$148$	0	0
$149$	−811.445	−0.446149	−0.223074	−	0.974801i	$-0.571609\pi$
$149$	−811.445	−0.446149	−0.223074	+	0.974801i	$0.571609\pi$
$150$	0	0
$151$	1054.39	0.568247	0.284123	−	0.958788i	$-0.408297\pi$
$151$	1054.39	0.568247	0.284123	+	0.958788i	$0.408297\pi$
$152$	0	0
$153$	−103.345	−0.0546073
$154$	0	0
$155$	687.269	0.356147
$156$	0	0
$157$	2778.76	1.41254	0.706271	−	0.707942i	$-0.250376\pi$
$157$	2778.76	1.41254	0.706271	+	0.707942i	$0.250376\pi$
$158$	0	0
$159$	869.929	0.433898
$160$	0	0
$161$	4269.59	2.09001
$162$	0	0
$163$	1230.64	0.591357	0.295678	−	0.955288i	$-0.404454\pi$
$163$	1230.64	0.591357	0.295678	+	0.955288i	$0.404454\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1643.34	−0.761469	−0.380734	−	0.924684i	$-0.624329\pi$
$167$	−1643.34	−0.761469	−0.380734	+	0.924684i	$0.624329\pi$
$168$	0	0
$169$	3032.85	1.38045
$170$	0	0
$171$	729.279	0.326137
$172$	0	0
$173$	−1309.02	−0.575278	−0.287639	−	0.957739i	$-0.592870\pi$
$173$	−1309.02	−0.575278	−0.287639	+	0.957739i	$0.592870\pi$
$174$	0	0
$175$	−2042.85	−0.882428
$176$	0	0
$177$	−255.064	−0.108315
$178$	0	0
$179$	1741.28	0.727091	0.363546	−	0.931576i	$-0.381566\pi$
$179$	1741.28	0.727091	0.363546	+	0.931576i	$0.381566\pi$
$180$	0	0
$181$	1921.53	0.789095	0.394548	−	0.918876i	$-0.370901\pi$
$181$	1921.53	0.789095	0.394548	+	0.918876i	$0.370901\pi$
$182$	0	0
$183$	967.941	0.390996
$184$	0	0
$185$	−2844.65	−1.13050
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3280.00	−1.26235
$190$	0	0
$191$	−4304.52	−1.63070	−0.815351	−	0.578967i	$-0.803456\pi$
$191$	−4304.52	−1.63070	−0.815351	+	0.578967i	$0.803456\pi$
$192$	0	0
$193$	−1727.47	−0.644282	−0.322141	−	0.946692i	$-0.604402\pi$
$193$	−1727.47	−0.644282	−0.322141	+	0.946692i	$0.604402\pi$
$194$	0	0
$195$	−1369.47	−0.502921
$196$	0	0
$197$	−3375.18	−1.22067	−0.610334	−	0.792145i	$-0.708965\pi$
$197$	−3375.18	−1.22067	−0.610334	+	0.792145i	$0.708965\pi$
$198$	0	0
$199$	662.783	0.236098	0.118049	−	0.993008i	$-0.462336\pi$
$199$	662.783	0.236098	0.118049	+	0.993008i	$0.462336\pi$
$200$	0	0
$201$	−1001.26	−0.351360
$202$	0	0
$203$	−337.460	−0.116675
$204$	0	0
$205$	429.600	0.146364
$206$	0	0
$207$	−3199.76	−1.07439
$208$	0	0
$209$	0	0
$210$	0	0
$211$	824.764	0.269095	0.134548	−	0.990907i	$-0.457042\pi$
$211$	824.764	0.269095	0.134548	+	0.990907i	$0.457042\pi$
$212$	0	0
$213$	1904.33	0.612593
$214$	0	0
$215$	1507.31	0.478127
$216$	0	0
$217$	−2610.63	−0.816688
$218$	0	0
$219$	3081.91	0.950941
$220$	0	0
$221$	401.848	0.122313
$222$	0	0
$223$	2755.89	0.827570	0.413785	−	0.910375i	$-0.364207\pi$
$223$	2755.89	0.827570	0.413785	+	0.910375i	$0.364207\pi$
$224$	0	0
$225$	1530.97	0.453621
$226$	0	0
$227$	−3072.87	−0.898473	−0.449237	−	0.893413i	$-0.648304\pi$
$227$	−3072.87	−0.898473	−0.449237	+	0.893413i	$0.648304\pi$
$228$	0	0
$229$	3949.27	1.13963	0.569815	−	0.821773i	$-0.307015\pi$
$229$	3949.27	1.13963	0.569815	+	0.821773i	$0.307015\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6178.43	−1.73718	−0.868589	−	0.495532i	$-0.834973\pi$
$233$	−6178.43	−1.73718	−0.868589	+	0.495532i	$0.834973\pi$
$234$	0	0
$235$	−1602.86	−0.444931
$236$	0	0
$237$	−1552.75	−0.425579
$238$	0	0
$239$	2713.57	0.734419	0.367209	−	0.930138i	$-0.380313\pi$
$239$	2713.57	0.734419	0.367209	+	0.930138i	$0.380313\pi$
$240$	0	0
$241$	−1269.76	−0.339387	−0.169694	−	0.985497i	$-0.554278\pi$
$241$	−1269.76	−0.339387	−0.169694	+	0.985497i	$0.554278\pi$
$242$	0	0
$243$	3913.66	1.03317
$244$	0	0
$245$	−1782.61	−0.464844
$246$	0	0
$247$	−2835.75	−0.730504
$248$	0	0
$249$	91.4108	0.0232647
$250$	0	0
$251$	−2457.16	−0.617907	−0.308954	−	0.951077i	$-0.599979\pi$
$251$	−2457.16	−0.617907	−0.308954	+	0.951077i	$0.599979\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−105.226	−0.0258413
$256$	0	0
$257$	7516.44	1.82437	0.912184	−	0.409782i	$-0.134395\pi$
$257$	7516.44	1.82437	0.912184	+	0.409782i	$0.134395\pi$
$258$	0	0
$259$	10805.6	2.59238
$260$	0	0
$261$	252.903	0.0599781
$262$	0	0
$263$	5022.62	1.17760	0.588798	−	0.808280i	$-0.299601\pi$
$263$	5022.62	1.17760	0.588798	+	0.808280i	$0.299601\pi$
$264$	0	0
$265$	−1960.73	−0.454516
$266$	0	0
$267$	3489.07	0.799730
$268$	0	0
$269$	1582.30	0.358641	0.179321	−	0.983791i	$-0.442610\pi$
$269$	1582.30	0.358641	0.179321	+	0.983791i	$0.442610\pi$
$270$	0	0
$271$	−1701.67	−0.381436	−0.190718	−	0.981645i	$-0.561082\pi$
$271$	−1701.67	−0.381436	−0.190718	+	0.981645i	$0.561082\pi$
$272$	0	0
$273$	5202.01	1.15326
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5748.39	1.24688	0.623442	−	0.781869i	$-0.285734\pi$
$277$	5748.39	1.24688	0.623442	+	0.781869i	$0.285734\pi$
$278$	0	0
$279$	1956.48	0.419827
$280$	0	0
$281$	−1758.64	−0.373351	−0.186675	−	0.982422i	$-0.559771\pi$
$281$	−1758.64	−0.373351	−0.186675	+	0.982422i	$0.559771\pi$
$282$	0	0
$283$	2741.68	0.575888	0.287944	−	0.957647i	$-0.407028\pi$
$283$	2741.68	0.575888	0.287944	+	0.957647i	$0.407028\pi$
$284$	0	0
$285$	742.558	0.154335
$286$	0	0
$287$	−1631.86	−0.335630
$288$	0	0
$289$	−4882.12	−0.993715
$290$	0	0
$291$	−1515.44	−0.305280
$292$	0	0
$293$	−2059.02	−0.410544	−0.205272	−	0.978705i	$-0.565808\pi$
$293$	−2059.02	−0.410544	−0.205272	+	0.978705i	$0.565808\pi$
$294$	0	0
$295$	574.889	0.113462
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12442.0	2.40649
$300$	0	0
$301$	−5725.59	−1.09640
$302$	0	0
$303$	−4629.61	−0.877769
$304$	0	0
$305$	−2181.64	−0.409575
$306$	0	0
$307$	−1530.40	−0.284510	−0.142255	−	0.989830i	$-0.545435\pi$
$307$	−1530.40	−0.284510	−0.142255	+	0.989830i	$0.545435\pi$
$308$	0	0
$309$	3598.32	0.662463
$310$	0	0
$311$	1885.82	0.343842	0.171921	−	0.985111i	$-0.445003\pi$
$311$	1885.82	0.343842	0.171921	+	0.985111i	$0.445003\pi$
$312$	0	0
$313$	4821.63	0.870718	0.435359	−	0.900257i	$-0.356621\pi$
$313$	4821.63	0.870718	0.435359	+	0.900257i	$0.356621\pi$
$314$	0	0
$315$	3015.31	0.539344
$316$	0	0
$317$	373.450	0.0661673	0.0330836	−	0.999453i	$-0.489467\pi$
$317$	373.450	0.0661673	0.0330836	+	0.999453i	$0.489467\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4433.06	0.770807
$322$	0	0
$323$	−217.892	−0.0375351
$324$	0	0
$325$	−5953.07	−1.01605
$326$	0	0
$327$	4382.49	0.741138
$328$	0	0
$329$	6088.55	1.02028
$330$	0	0
$331$	6772.60	1.12464	0.562320	−	0.826920i	$-0.309909\pi$
$331$	6772.60	1.12464	0.562320	+	0.826920i	$0.309909\pi$
$332$	0	0
$333$	−8098.02	−1.33264
$334$	0	0
$335$	2256.74	0.368056
$336$	0	0
$337$	−7818.39	−1.26378	−0.631891	−	0.775057i	$-0.717721\pi$
$337$	−7818.39	−1.26378	−0.631891	+	0.775057i	$0.717721\pi$
$338$	0	0
$339$	−1336.11	−0.214063
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1740.69	−0.274019
$344$	0	0
$345$	−3258.02	−0.508423
$346$	0	0
$347$	−2716.50	−0.420258	−0.210129	−	0.977674i	$-0.567388\pi$
$347$	−2716.50	−0.420258	−0.210129	+	0.977674i	$0.567388\pi$
$348$	0	0
$349$	−9743.78	−1.49448	−0.747238	−	0.664556i	$-0.768621\pi$
$349$	−9743.78	−1.49448	−0.747238	+	0.664556i	$0.768621\pi$
$350$	0	0
$351$	−9558.25	−1.45351
$352$	0	0
$353$	−12435.4	−1.87499	−0.937493	−	0.348003i	$-0.886860\pi$
$353$	−12435.4	−1.87499	−0.937493	+	0.348003i	$0.886860\pi$
$354$	0	0
$355$	−4292.16	−0.641702
$356$	0	0
$357$	399.709	0.0592573
$358$	0	0
$359$	−5554.35	−0.816567	−0.408284	−	0.912855i	$-0.633873\pi$
$359$	−5554.35	−0.816567	−0.408284	+	0.912855i	$0.633873\pi$
$360$	0	0
$361$	−5321.39	−0.775826
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6946.31	−0.996127
$366$	0	0
$367$	−8519.34	−1.21173	−0.605866	−	0.795567i	$-0.707173\pi$
$367$	−8519.34	−1.21173	−0.605866	+	0.795567i	$0.707173\pi$
$368$	0	0
$369$	1222.97	0.172534
$370$	0	0
$371$	7447.96	1.04226
$372$	0	0
$373$	291.030	0.0403994	0.0201997	−	0.999796i	$-0.493570\pi$
$373$	291.030	0.0403994	0.0201997	+	0.999796i	$0.493570\pi$
$374$	0	0
$375$	3925.95	0.540627
$376$	0	0
$377$	−983.393	−0.134343
$378$	0	0
$379$	5255.41	0.712275	0.356137	−	0.934434i	$-0.384094\pi$
$379$	5255.41	0.712275	0.356137	+	0.934434i	$0.384094\pi$
$380$	0	0
$381$	6147.23	0.826593
$382$	0	0
$383$	−2544.59	−0.339484	−0.169742	−	0.985489i	$-0.554293\pi$
$383$	−2544.59	−0.339484	−0.169742	+	0.985489i	$0.554293\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4290.93	0.563618
$388$	0	0
$389$	−1761.25	−0.229561	−0.114780	−	0.993391i	$-0.536616\pi$
$389$	−1761.25	−0.229561	−0.114780	+	0.993391i	$0.536616\pi$
$390$	0	0
$391$	956.013	0.123651
$392$	0	0
$393$	−4448.48	−0.570982
$394$	0	0
$395$	3499.75	0.445801
$396$	0	0
$397$	−7270.28	−0.919106	−0.459553	−	0.888150i	$-0.651990\pi$
$397$	−7270.28	−0.919106	−0.459553	+	0.888150i	$0.651990\pi$
$398$	0	0
$399$	−2820.65	−0.353908
$400$	0	0
$401$	−2664.93	−0.331870	−0.165935	−	0.986137i	$-0.553064\pi$
$401$	−2664.93	−0.331870	−0.165935	+	0.986137i	$0.553064\pi$
$402$	0	0
$403$	−7607.64	−0.940357
$404$	0	0
$405$	−777.728	−0.0954212
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8090.09	−0.978066	−0.489033	−	0.872265i	$-0.662650\pi$
$409$	−8090.09	−0.978066	−0.489033	+	0.872265i	$0.662650\pi$
$410$	0	0
$411$	−8850.39	−1.06218
$412$	0	0
$413$	−2183.75	−0.260183
$414$	0	0
$415$	−206.031	−0.0243702
$416$	0	0
$417$	8549.54	1.00401
$418$	0	0
$419$	−15142.4	−1.76552	−0.882761	−	0.469822i	$-0.844318\pi$
$419$	−15142.4	−1.76552	−0.882761	+	0.469822i	$0.844318\pi$
$420$	0	0
$421$	−9756.02	−1.12940	−0.564702	−	0.825295i	$-0.691009\pi$
$421$	−9756.02	−1.12940	−0.564702	+	0.825295i	$0.691009\pi$
$422$	0	0
$423$	−4562.94	−0.524486
$424$	0	0
$425$	−457.419	−0.0522072
$426$	0	0
$427$	8287.10	0.939206
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12030.8	1.34456	0.672280	−	0.740297i	$-0.265315\pi$
$431$	12030.8	1.34456	0.672280	+	0.740297i	$0.265315\pi$
$432$	0	0
$433$	792.640	0.0879719	0.0439860	−	0.999032i	$-0.485994\pi$
$433$	792.640	0.0879719	0.0439860	+	0.999032i	$0.485994\pi$
$434$	0	0
$435$	257.508	0.0283829
$436$	0	0
$437$	−6746.36	−0.738495
$438$	0	0
$439$	−6876.90	−0.747646	−0.373823	−	0.927500i	$-0.621953\pi$
$439$	−6876.90	−0.747646	−0.373823	+	0.927500i	$0.621953\pi$
$440$	0	0
$441$	−5074.66	−0.547960
$442$	0	0
$443$	−14408.9	−1.54535	−0.772673	−	0.634805i	$-0.781081\pi$
$443$	−14408.9	−1.54535	−0.772673	+	0.634805i	$0.781081\pi$
$444$	0	0
$445$	−7864.02	−0.837731
$446$	0	0
$447$	−2352.04	−0.248877
$448$	0	0
$449$	2578.46	0.271014	0.135507	−	0.990776i	$-0.456734\pi$
$449$	2578.46	0.271014	0.135507	+	0.990776i	$0.456734\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3056.25	0.316987
$454$	0	0
$455$	−11724.8	−1.20806
$456$	0	0
$457$	−9410.95	−0.963295	−0.481647	−	0.876365i	$-0.659961\pi$
$457$	−9410.95	−0.963295	−0.481647	+	0.876365i	$0.659961\pi$
$458$	0	0
$459$	−734.432	−0.0746848
$460$	0	0
$461$	−6410.42	−0.647643	−0.323821	−	0.946118i	$-0.604968\pi$
$461$	−6410.42	−0.647643	−0.323821	+	0.946118i	$0.604968\pi$
$462$	0	0
$463$	−15702.0	−1.57610	−0.788051	−	0.615610i	$-0.788910\pi$
$463$	−15702.0	−1.57610	−0.788051	+	0.615610i	$0.788910\pi$
$464$	0	0
$465$	1992.11	0.198671
$466$	0	0
$467$	−13928.3	−1.38014	−0.690069	−	0.723744i	$-0.742420\pi$
$467$	−13928.3	−1.38014	−0.690069	+	0.723744i	$0.742420\pi$
$468$	0	0
$469$	−8572.36	−0.843998
$470$	0	0
$471$	8054.47	0.787963
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3227.90	0.311802
$476$	0	0
$477$	−5581.72	−0.535785
$478$	0	0
$479$	7329.60	0.699160	0.349580	−	0.936906i	$-0.386324\pi$
$479$	7329.60	0.699160	0.349580	+	0.936906i	$0.386324\pi$
$480$	0	0
$481$	31488.5	2.98494
$482$	0	0
$483$	12375.8	1.16588
$484$	0	0
$485$	3415.64	0.319786
$486$	0	0
$487$	8081.93	0.752006	0.376003	−	0.926618i	$-0.377298\pi$
$487$	8081.93	0.752006	0.376003	+	0.926618i	$0.377298\pi$
$488$	0	0
$489$	3567.12	0.329879
$490$	0	0
$491$	1110.75	0.102092	0.0510462	−	0.998696i	$-0.483744\pi$
$491$	1110.75	0.102092	0.0510462	+	0.998696i	$0.483744\pi$
$492$	0	0
$493$	−75.5615	−0.00690287
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16304.0	1.47150
$498$	0	0
$499$	2498.75	0.224167	0.112083	−	0.993699i	$-0.464248\pi$
$499$	2498.75	0.224167	0.112083	+	0.993699i	$0.464248\pi$
$500$	0	0
$501$	−4763.36	−0.424773
$502$	0	0
$503$	−13776.6	−1.22121	−0.610606	−	0.791934i	$-0.709074\pi$
$503$	−13776.6	−1.22121	−0.610606	+	0.791934i	$0.709074\pi$
$504$	0	0
$505$	10434.7	0.919478
$506$	0	0
$507$	8790.97	0.770060
$508$	0	0
$509$	−7700.37	−0.670556	−0.335278	−	0.942119i	$-0.608830\pi$
$509$	−7700.37	−0.670556	−0.335278	+	0.942119i	$0.608830\pi$
$510$	0	0
$511$	26386.0	2.28424
$512$	0	0
$513$	5182.72	0.446048
$514$	0	0
$515$	−8110.24	−0.693941
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−3794.31	−0.320909
$520$	0	0
$521$	−4247.91	−0.357206	−0.178603	−	0.983921i	$-0.557158\pi$
$521$	−4247.91	−0.357206	−0.178603	+	0.983921i	$0.557158\pi$
$522$	0	0
$523$	−20697.0	−1.73043	−0.865217	−	0.501398i	$-0.832819\pi$
$523$	−20697.0	−1.73043	−0.865217	+	0.501398i	$0.832819\pi$
$524$	0	0
$525$	−5921.38	−0.492248
$526$	0	0
$527$	−584.552	−0.0483178
$528$	0	0
$529$	17433.1	1.43281
$530$	0	0
$531$	1636.57	0.133750
$532$	0	0
$533$	−4755.41	−0.386453
$534$	0	0
$535$	−9991.66	−0.807434
$536$	0	0
$537$	5047.25	0.405596
$538$	0	0
$539$	0	0
$540$	0	0
$541$	15409.5	1.22460	0.612299	−	0.790627i	$-0.290245\pi$
$541$	15409.5	1.22460	0.612299	+	0.790627i	$0.290245\pi$
$542$	0	0
$543$	5569.72	0.440184
$544$	0	0
$545$	−9877.68	−0.776355
$546$	0	0
$547$	21412.8	1.67376	0.836879	−	0.547388i	$-0.184378\pi$
$547$	21412.8	1.67376	0.836879	+	0.547388i	$0.184378\pi$
$548$	0	0
$549$	−6210.59	−0.482808
$550$	0	0
$551$	533.220	0.0412267
$552$	0	0
$553$	−13294.0	−1.02228
$554$	0	0
$555$	−8245.47	−0.630632
$556$	0	0
$557$	−9564.00	−0.727540	−0.363770	−	0.931489i	$-0.618511\pi$
$557$	−9564.00	−0.727540	−0.363770	+	0.931489i	$0.618511\pi$
$558$	0	0
$559$	−16684.9	−1.26243
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1102.58	0.0825369	0.0412684	−	0.999148i	$-0.486860\pi$
$563$	1102.58	0.0825369	0.0412684	+	0.999148i	$0.486860\pi$
$564$	0	0
$565$	3011.45	0.224235
$566$	0	0
$567$	2954.25	0.218813
$568$	0	0
$569$	7491.69	0.551965	0.275983	−	0.961163i	$-0.410997\pi$
$569$	7491.69	0.551965	0.275983	+	0.961163i	$0.410997\pi$
$570$	0	0
$571$	18622.5	1.36484	0.682422	−	0.730958i	$-0.260927\pi$
$571$	18622.5	1.36484	0.682422	+	0.730958i	$0.260927\pi$
$572$	0	0
$573$	−12477.0	−0.909660
$574$	0	0
$575$	−14162.6	−1.02717
$576$	0	0
$577$	18713.3	1.35017	0.675083	−	0.737741i	$-0.264108\pi$
$577$	18713.3	1.35017	0.675083	+	0.737741i	$0.264108\pi$
$578$	0	0
$579$	−5007.24	−0.359402
$580$	0	0
$581$	782.620	0.0558839
$582$	0	0
$583$	0	0
$584$	0	0
$585$	8786.90	0.621015
$586$	0	0
$587$	−9146.77	−0.643148	−0.321574	−	0.946885i	$-0.604212\pi$
$587$	−9146.77	−0.643148	−0.321574	+	0.946885i	$0.604212\pi$
$588$	0	0
$589$	4125.05	0.288573
$590$	0	0
$591$	−9783.25	−0.680929
$592$	0	0
$593$	−19655.7	−1.36115	−0.680576	−	0.732677i	$-0.738271\pi$
$593$	−19655.7	−1.36115	−0.680576	+	0.732677i	$0.738271\pi$
$594$	0	0
$595$	−900.904	−0.0620730
$596$	0	0
$597$	1921.13	0.131703
$598$	0	0
$599$	2612.76	0.178221	0.0891106	−	0.996022i	$-0.471598\pi$
$599$	2612.76	0.178221	0.0891106	+	0.996022i	$0.471598\pi$
$600$	0	0
$601$	−22723.2	−1.54226	−0.771130	−	0.636678i	$-0.780308\pi$
$601$	−22723.2	−1.54226	−0.771130	+	0.636678i	$0.780308\pi$
$602$	0	0
$603$	6424.38	0.433865
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24991.6	−1.67114	−0.835568	−	0.549387i	$-0.814861\pi$
$607$	−24991.6	−1.67114	−0.835568	+	0.549387i	$0.814861\pi$
$608$	0	0
$609$	−978.159	−0.0650854
$610$	0	0
$611$	17742.6	1.17478
$612$	0	0
$613$	3069.31	0.202232	0.101116	−	0.994875i	$-0.467759\pi$
$613$	3069.31	0.202232	0.101116	+	0.994875i	$0.467759\pi$
$614$	0	0
$615$	1245.23	0.0816466
$616$	0	0
$617$	4381.05	0.285858	0.142929	−	0.989733i	$-0.454348\pi$
$617$	4381.05	0.285858	0.142929	+	0.989733i	$0.454348\pi$
$618$	0	0
$619$	19150.0	1.24347	0.621733	−	0.783229i	$-0.286429\pi$
$619$	19150.0	1.24347	0.621733	+	0.783229i	$0.286429\pi$
$620$	0	0
$621$	−22739.5	−1.46941
$622$	0	0
$623$	29872.0	1.92102
$624$	0	0
$625$	1441.09	0.0922300
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2419.50	0.153373
$630$	0	0
$631$	−1197.22	−0.0755317	−0.0377658	−	0.999287i	$-0.512024\pi$
$631$	−1197.22	−0.0755317	−0.0377658	+	0.999287i	$0.512024\pi$
$632$	0	0
$633$	2390.65	0.150110
$634$	0	0
$635$	−13855.2	−0.865871
$636$	0	0
$637$	19732.4	1.22736
$638$	0	0
$639$	−12218.7	−0.756440
$640$	0	0
$641$	25681.8	1.58248	0.791239	−	0.611507i	$-0.209436\pi$
$641$	25681.8	1.58248	0.791239	+	0.611507i	$0.209436\pi$
$642$	0	0
$643$	1602.72	0.0982973	0.0491486	−	0.998791i	$-0.484349\pi$
$643$	1602.72	0.0982973	0.0491486	+	0.998791i	$0.484349\pi$
$644$	0	0
$645$	4369.06	0.266715
$646$	0	0
$647$	24909.1	1.51357	0.756784	−	0.653665i	$-0.226769\pi$
$647$	24909.1	1.51357	0.756784	+	0.653665i	$0.226769\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−7567.15	−0.455576
$652$	0	0
$653$	14496.5	0.868747	0.434373	−	0.900733i	$-0.356970\pi$
$653$	14496.5	0.868747	0.434373	+	0.900733i	$0.356970\pi$
$654$	0	0
$655$	10026.4	0.598114
$656$	0	0
$657$	−19774.4	−1.17424
$658$	0	0
$659$	−17983.5	−1.06303	−0.531515	−	0.847049i	$-0.678377\pi$
$659$	−17983.5	−1.06303	−0.531515	+	0.847049i	$0.678377\pi$
$660$	0	0
$661$	−699.376	−0.0411537	−0.0205768	−	0.999788i	$-0.506550\pi$
$661$	−699.376	−0.0411537	−0.0205768	+	0.999788i	$0.506550\pi$
$662$	0	0
$663$	1164.79	0.0682304
$664$	0	0
$665$	6357.47	0.370725
$666$	0	0
$667$	−2339.53	−0.135813
$668$	0	0
$669$	7988.19	0.461646
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−22046.4	−1.26274	−0.631371	−	0.775481i	$-0.717507\pi$
$673$	−22046.4	−1.26274	−0.631371	+	0.775481i	$0.717507\pi$
$674$	0	0
$675$	10880.0	0.620404
$676$	0	0
$677$	32013.1	1.81737	0.908687	−	0.417478i	$-0.137086\pi$
$677$	32013.1	1.81737	0.908687	+	0.417478i	$0.137086\pi$
$678$	0	0
$679$	−12974.5	−0.733310
$680$	0	0
$681$	−8906.98	−0.501199
$682$	0	0
$683$	25439.0	1.42518	0.712588	−	0.701583i	$-0.247523\pi$
$683$	25439.0	1.42518	0.712588	+	0.701583i	$0.247523\pi$
$684$	0	0
$685$	19947.9	1.11266
$686$	0	0
$687$	11447.3	0.635724
$688$	0	0
$689$	21704.1	1.20009
$690$	0	0
$691$	−20280.9	−1.11653	−0.558264	−	0.829663i	$-0.688532\pi$
$691$	−20280.9	−1.11653	−0.558264	+	0.829663i	$0.688532\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19269.8	−1.05172
$696$	0	0
$697$	−365.394	−0.0198569
$698$	0	0
$699$	−17908.7	−0.969057
$700$	0	0
$701$	−221.370	−0.0119273	−0.00596364	−	0.999982i	$-0.501898\pi$
$701$	−221.370	−0.0119273	−0.00596364	+	0.999982i	$0.501898\pi$
$702$	0	0
$703$	−17073.9	−0.916006
$704$	0	0
$705$	−4646.02	−0.248198
$706$	0	0
$707$	−39636.7	−2.10848
$708$	0	0
$709$	1994.40	0.105643	0.0528217	−	0.998604i	$-0.483178\pi$
$709$	1994.40	0.105643	0.0528217	+	0.998604i	$0.483178\pi$
$710$	0	0
$711$	9962.92	0.525512
$712$	0	0
$713$	−18098.9	−0.950643
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7865.51	0.409683
$718$	0	0
$719$	10673.7	0.553635	0.276817	−	0.960923i	$-0.410720\pi$
$719$	10673.7	0.553635	0.276817	+	0.960923i	$0.410720\pi$
$720$	0	0
$721$	30807.2	1.59129
$722$	0	0
$723$	−3680.50	−0.189321
$724$	0	0
$725$	1119.38	0.0573419
$726$	0	0
$727$	−21340.3	−1.08868	−0.544338	−	0.838866i	$-0.683219\pi$
$727$	−21340.3	−1.08868	−0.544338	+	0.838866i	$0.683219\pi$
$728$	0	0
$729$	8129.90	0.413042
$730$	0	0
$731$	−1282.03	−0.0648667
$732$	0	0
$733$	2379.68	0.119912	0.0599559	−	0.998201i	$-0.480904\pi$
$733$	2379.68	0.119912	0.0599559	+	0.998201i	$0.480904\pi$
$734$	0	0
$735$	−5167.06	−0.259306
$736$	0	0
$737$	0	0
$738$	0	0
$739$	14380.8	0.715843	0.357921	−	0.933752i	$-0.383486\pi$
$739$	14380.8	0.715843	0.357921	+	0.933752i	$0.383486\pi$
$740$	0	0
$741$	−8219.67	−0.407499
$742$	0	0
$743$	−23947.4	−1.18243	−0.591215	−	0.806514i	$-0.701352\pi$
$743$	−23947.4	−1.18243	−0.591215	+	0.806514i	$0.701352\pi$
$744$	0	0
$745$	5301.27	0.260703
$746$	0	0
$747$	−586.518	−0.0287277
$748$	0	0
$749$	37953.9	1.85154
$750$	0	0
$751$	10076.2	0.489593	0.244797	−	0.969574i	$-0.421279\pi$
$751$	10076.2	0.489593	0.244797	+	0.969574i	$0.421279\pi$
$752$	0	0
$753$	−7122.30	−0.344689
$754$	0	0
$755$	−6888.48	−0.332049
$756$	0	0
$757$	−13036.2	−0.625902	−0.312951	−	0.949769i	$-0.601318\pi$
$757$	−13036.2	−0.625902	−0.312951	+	0.949769i	$0.601318\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5895.73	−0.280841	−0.140421	−	0.990092i	$-0.544845\pi$
$761$	−5895.73	−0.280841	−0.140421	+	0.990092i	$0.544845\pi$
$762$	0	0
$763$	37521.0	1.78028
$764$	0	0
$765$	675.164	0.0319093
$766$	0	0
$767$	−6363.67	−0.299581
$768$	0	0
$769$	−1751.30	−0.0821241	−0.0410620	−	0.999157i	$-0.513074\pi$
$769$	−1751.30	−0.0821241	−0.0410620	+	0.999157i	$0.513074\pi$
$770$	0	0
$771$	21787.0	1.01769
$772$	0	0
$773$	21597.2	1.00491	0.502456	−	0.864603i	$-0.332430\pi$
$773$	21597.2	1.00491	0.502456	+	0.864603i	$0.332430\pi$
$774$	0	0
$775$	8659.69	0.401374
$776$	0	0
$777$	31320.9	1.44612
$778$	0	0
$779$	2578.50	0.118593
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1797.28	0.0820303
$784$	0	0
$785$	−18154.0	−0.825405
$786$	0	0
$787$	−32533.7	−1.47357	−0.736785	−	0.676127i	$-0.763657\pi$
$787$	−32533.7	−1.47357	−0.736785	+	0.676127i	$0.763657\pi$
$788$	0	0
$789$	14558.5	0.656902
$790$	0	0
$791$	−11439.2	−0.514198
$792$	0	0
$793$	24149.4	1.08143
$794$	0	0
$795$	−5683.35	−0.253544
$796$	0	0
$797$	19556.8	0.869181	0.434591	−	0.900628i	$-0.356893\pi$
$797$	19556.8	0.869181	0.434591	+	0.900628i	$0.356893\pi$
$798$	0	0
$799$	1363.30	0.0603631
$800$	0	0
$801$	−22386.9	−0.987519
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−27893.8	−1.22127
$806$	0	0
$807$	4586.43	0.200062
$808$	0	0
$809$	−7496.90	−0.325806	−0.162903	−	0.986642i	$-0.552086\pi$
$809$	−7496.90	−0.325806	−0.162903	+	0.986642i	$0.552086\pi$
$810$	0	0
$811$	−5451.43	−0.236037	−0.118018	−	0.993011i	$-0.537654\pi$
$811$	−5451.43	−0.236037	−0.118018	+	0.993011i	$0.537654\pi$
$812$	0	0
$813$	−4932.45	−0.212778
$814$	0	0
$815$	−8039.92	−0.345554
$816$	0	0
$817$	9046.98	0.387410
$818$	0	0
$819$	−33377.6	−1.42406
$820$	0	0
$821$	1322.92	0.0562365	0.0281183	−	0.999605i	$-0.491049\pi$
$821$	1322.92	0.0562365	0.0281183	+	0.999605i	$0.491049\pi$
$822$	0	0
$823$	40515.0	1.71599	0.857997	−	0.513655i	$-0.171709\pi$
$823$	40515.0	1.71599	0.857997	+	0.513655i	$0.171709\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15847.5	0.666349	0.333175	−	0.942865i	$-0.391880\pi$
$827$	15847.5	0.666349	0.333175	+	0.942865i	$0.391880\pi$
$828$	0	0
$829$	8700.90	0.364529	0.182265	−	0.983250i	$-0.441657\pi$
$829$	8700.90	0.364529	0.182265	+	0.983250i	$0.441657\pi$
$830$	0	0
$831$	16662.2	0.695554
$832$	0	0
$833$	1516.19	0.0630646
$834$	0	0
$835$	10736.1	0.444957
$836$	0	0
$837$	13904.0	0.574185
$838$	0	0
$839$	37614.6	1.54779	0.773897	−	0.633311i	$-0.218305\pi$
$839$	37614.6	1.54779	0.773897	+	0.633311i	$0.218305\pi$
$840$	0	0
$841$	−24204.1	−0.992418
$842$	0	0
$843$	−5097.57	−0.208267
$844$	0	0
$845$	−19814.0	−0.806652
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7947.01	0.321249
$850$	0	0
$851$	74912.5	3.01759
$852$	0	0
$853$	20350.4	0.816864	0.408432	−	0.912789i	$-0.366076\pi$
$853$	20350.4	0.816864	0.408432	+	0.912789i	$0.366076\pi$
$854$	0	0
$855$	−4764.47	−0.190575
$856$	0	0
$857$	−1599.70	−0.0637627	−0.0318814	−	0.999492i	$-0.510150\pi$
$857$	−1599.70	−0.0637627	−0.0318814	+	0.999492i	$0.510150\pi$
$858$	0	0
$859$	21607.1	0.858237	0.429118	−	0.903248i	$-0.358824\pi$
$859$	21607.1	0.858237	0.429118	+	0.903248i	$0.358824\pi$
$860$	0	0
$861$	−4730.10	−0.187226
$862$	0	0
$863$	−13593.7	−0.536194	−0.268097	−	0.963392i	$-0.586395\pi$
$863$	−13593.7	−0.536194	−0.268097	+	0.963392i	$0.586395\pi$
$864$	0	0
$865$	8552.00	0.336158
$866$	0	0
$867$	−14151.3	−0.554328
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24980.7	−0.971801
$872$	0	0
$873$	9723.50	0.376965
$874$	0	0
$875$	33612.3	1.29863
$876$	0	0
$877$	42783.4	1.64731	0.823657	−	0.567089i	$-0.191930\pi$
$877$	42783.4	1.64731	0.823657	+	0.567089i	$0.191930\pi$
$878$	0	0
$879$	−5968.25	−0.229015
$880$	0	0
$881$	−14607.7	−0.558621	−0.279311	−	0.960201i	$-0.590106\pi$
$881$	−14607.7	−0.558621	−0.279311	+	0.960201i	$0.590106\pi$
$882$	0	0
$883$	38184.6	1.45528	0.727641	−	0.685958i	$-0.240617\pi$
$883$	38184.6	1.45528	0.727641	+	0.685958i	$0.240617\pi$
$884$	0	0
$885$	1666.37	0.0632930
$886$	0	0
$887$	45123.4	1.70811	0.854056	−	0.520181i	$-0.174136\pi$
$887$	45123.4	1.70811	0.854056	+	0.520181i	$0.174136\pi$
$888$	0	0
$889$	52630.0	1.98555
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9620.49	−0.360512
$894$	0	0
$895$	−11376.0	−0.424869
$896$	0	0
$897$	36064.3	1.34242
$898$	0	0
$899$	1430.50	0.0530700
$900$	0	0
$901$	1667.69	0.0616634
$902$	0	0
$903$	−16596.1	−0.611611
$904$	0	0
$905$	−12553.6	−0.461100
$906$	0	0
$907$	−11808.7	−0.432306	−0.216153	−	0.976359i	$-0.569351\pi$
$907$	−11808.7	−0.432306	−0.216153	+	0.976359i	$0.569351\pi$
$908$	0	0
$909$	29704.9	1.08388
$910$	0	0
$911$	1316.68	0.0478855	0.0239428	−	0.999713i	$-0.492378\pi$
$911$	1316.68	0.0478855	0.0239428	+	0.999713i	$0.492378\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6323.68	−0.228475
$916$	0	0
$917$	−38086.0	−1.37155
$918$	0	0
$919$	−32372.1	−1.16198	−0.580989	−	0.813911i	$-0.697334\pi$
$919$	−32372.1	−1.16198	−0.580989	+	0.813911i	$0.697334\pi$
$920$	0	0
$921$	−4436.00	−0.158709
$922$	0	0
$923$	47511.6	1.69433
$924$	0	0
$925$	−35843.0	−1.27407
$926$	0	0
$927$	−23087.8	−0.818020
$928$	0	0
$929$	−45235.1	−1.59754	−0.798770	−	0.601637i	$-0.794515\pi$
$929$	−45235.1	−1.59754	−0.798770	+	0.601637i	$0.794515\pi$
$930$	0	0
$931$	−10699.4	−0.376647
$932$	0	0
$933$	5466.21	0.191807
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6810.68	0.237455	0.118727	−	0.992927i	$-0.462119\pi$
$937$	6810.68	0.237455	0.118727	+	0.992927i	$0.462119\pi$
$938$	0	0
$939$	13975.9	0.485715
$940$	0	0
$941$	−6960.05	−0.241117	−0.120559	−	0.992706i	$-0.538469\pi$
$941$	−6960.05	−0.241117	−0.120559	+	0.992706i	$0.538469\pi$
$942$	0	0
$943$	−11313.3	−0.390681
$944$	0	0
$945$	21428.7	0.737645
$946$	0	0
$947$	17574.4	0.603052	0.301526	−	0.953458i	$-0.402504\pi$
$947$	17574.4	0.603052	0.301526	+	0.953458i	$0.402504\pi$
$948$	0	0
$949$	76891.4	2.63014
$950$	0	0
$951$	1082.48	0.0369103
$952$	0	0
$953$	−30835.4	−1.04812	−0.524058	−	0.851682i	$-0.675583\pi$
$953$	−30835.4	−1.04812	−0.524058	+	0.851682i	$0.675583\pi$
$954$	0	0
$955$	28122.0	0.952885
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−75773.3	−2.55146
$960$	0	0
$961$	−18724.5	−0.628528
$962$	0	0
$963$	−28443.8	−0.951805
$964$	0	0
$965$	11285.8	0.376480
$966$	0	0
$967$	−15521.1	−0.516160	−0.258080	−	0.966124i	$-0.583090\pi$
$967$	−15521.1	−0.516160	−0.258080	+	0.966124i	$0.583090\pi$
$968$	0	0
$969$	−631.578	−0.0209383
$970$	0	0
$971$	−12157.6	−0.401807	−0.200904	−	0.979611i	$-0.564388\pi$
$971$	−12157.6	−0.401807	−0.200904	+	0.979611i	$0.564388\pi$
$972$	0	0
$973$	73197.5	2.41172
$974$	0	0
$975$	−17255.5	−0.566788
$976$	0	0
$977$	−30691.2	−1.00501	−0.502507	−	0.864573i	$-0.667589\pi$
$977$	−30691.2	−1.00501	−0.502507	+	0.864573i	$0.667589\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−28119.3	−0.915169
$982$	0	0
$983$	14017.1	0.454808	0.227404	−	0.973800i	$-0.426976\pi$
$983$	14017.1	0.454808	0.227404	+	0.973800i	$0.426976\pi$
$984$	0	0
$985$	22050.4	0.713285
$986$	0	0
$987$	17648.2	0.569147
$988$	0	0
$989$	−39694.2	−1.27624
$990$	0	0
$991$	−45083.3	−1.44513	−0.722563	−	0.691305i	$-0.757036\pi$
$991$	−45083.3	−1.44513	−0.722563	+	0.691305i	$0.757036\pi$
$992$	0	0
$993$	19631.0	0.627362
$994$	0	0
$995$	−4330.04	−0.137961
$996$	0	0
$997$	−9691.53	−0.307857	−0.153929	−	0.988082i	$-0.549193\pi$
$997$	−9691.53	−0.307857	−0.153929	+	0.988082i	$0.549193\pi$
$998$	0	0
$999$	−57549.6	−1.82261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.r.1.7		10
4.3	odd	2		1936.4.a.by.1.4		10
11.7	odd	10		88.4.i.b.49.2	yes	20
11.8	odd	10		88.4.i.b.9.2	✓	20
11.10	odd	2		968.4.a.s.1.7		10
44.7	even	10		176.4.m.f.49.4		20
44.19	even	10		176.4.m.f.97.4		20
44.43	even	2		1936.4.a.bx.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.b.9.2	✓	20	11.8	odd	10
88.4.i.b.49.2	yes	20	11.7	odd	10
176.4.m.f.49.4		20	44.7	even	10
176.4.m.f.97.4		20	44.19	even	10
968.4.a.r.1.7		10	1.1	even	1	trivial
968.4.a.s.1.7		10	11.10	odd	2
1936.4.a.bx.1.4		10	44.43	even	2
1936.4.a.by.1.4		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)

\(f(q)\)	\(=\)	\(q+2.89859 q^{3} -6.53312 q^{5} +24.8165 q^{7} -18.5982 q^{9} +O(q^{10})\)	\(q+2.89859 q^{3} -6.53312 q^{5} +24.8165 q^{7} -18.5982 q^{9} +72.3177 q^{13} -18.9368 q^{15} +5.55671 q^{17} -39.2124 q^{19} +71.9327 q^{21} +172.047 q^{23} -82.3183 q^{25} -132.170 q^{27} -13.5982 q^{29} -105.198 q^{31} -162.129 q^{35} +435.420 q^{37} +209.619 q^{39} -65.7572 q^{41} -230.717 q^{43} +121.504 q^{45} +245.343 q^{47} +272.857 q^{49} +16.1066 q^{51} +300.122 q^{53} -113.661 q^{57} -87.9961 q^{59} +333.935 q^{61} -461.542 q^{63} -472.460 q^{65} -345.430 q^{67} +498.692 q^{69} +656.985 q^{71} +1063.24 q^{73} -238.607 q^{75} -535.693 q^{79} +119.044 q^{81} +31.5363 q^{83} -36.3026 q^{85} -39.4157 q^{87} +1203.72 q^{89} +1794.67 q^{91} -304.924 q^{93} +256.179 q^{95} -522.820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) \) 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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