LMFDB - Embedded newform 968.4.a.r.1.2

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			1.2
Root			$-9.47148$ 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)

$f(q)$	$=$	$q-7.85344 q^{3} +1.64930 q^{5} -8.79751 q^{7} +34.6766 q^{9} +O(q^{10})$	$q-7.85344 q^{3} +1.64930 q^{5} -8.79751 q^{7} +34.6766 q^{9} -73.5571 q^{13} -12.9527 q^{15} -3.58093 q^{17} +28.2574 q^{19} +69.0907 q^{21} +118.763 q^{23} -122.280 q^{25} -60.2875 q^{27} -200.671 q^{29} -196.482 q^{31} -14.5098 q^{35} -6.67144 q^{37} +577.677 q^{39} -270.511 q^{41} +300.982 q^{43} +57.1921 q^{45} -501.546 q^{47} -265.604 q^{49} +28.1226 q^{51} -183.329 q^{53} -221.918 q^{57} +601.459 q^{59} +58.2961 q^{61} -305.067 q^{63} -121.318 q^{65} -535.606 q^{67} -932.700 q^{69} +289.540 q^{71} -836.495 q^{73} +960.317 q^{75} -1263.77 q^{79} -462.803 q^{81} +1183.37 q^{83} -5.90603 q^{85} +1575.96 q^{87} +1081.40 q^{89} +647.120 q^{91} +1543.06 q^{93} +46.6050 q^{95} +1616.96 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−7.85344	−1.51140	−0.755698	−	0.654920i	$-0.772702\pi$
$3$	−7.85344	−1.51140	−0.755698	+	0.654920i	$0.772702\pi$
$4$	0	0
$5$	1.64930	0.147518	0.0737590	−	0.997276i	$-0.476500\pi$
$5$	1.64930	0.147518	0.0737590	+	0.997276i	$0.476500\pi$
$6$	0	0
$7$	−8.79751	−0.475021	−0.237510	−	0.971385i	$-0.576331\pi$
$7$	−8.79751	−0.475021	−0.237510	+	0.971385i	$0.576331\pi$
$8$	0	0
$9$	34.6766	1.28432
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−73.5571	−1.56931	−0.784657	−	0.619930i	$-0.787161\pi$
$13$	−73.5571	−1.56931	−0.784657	+	0.619930i	$0.787161\pi$
$14$	0	0
$15$	−12.9527	−0.222958
$16$	0	0
$17$	−3.58093	−0.0510884	−0.0255442	−	0.999674i	$-0.508132\pi$
$17$	−3.58093	−0.0510884	−0.0255442	+	0.999674i	$0.508132\pi$
$18$	0	0
$19$	28.2574	0.341195	0.170597	−	0.985341i	$-0.445430\pi$
$19$	28.2574	0.341195	0.170597	+	0.985341i	$0.445430\pi$
$20$	0	0
$21$	69.0907	0.717945
$22$	0	0
$23$	118.763	1.07669	0.538344	−	0.842725i	$-0.319050\pi$
$23$	118.763	1.07669	0.538344	+	0.842725i	$0.319050\pi$
$24$	0	0
$25$	−122.280	−0.978238
$26$	0	0
$27$	−60.2875	−0.429716
$28$	0	0
$29$	−200.671	−1.28496	−0.642478	−	0.766304i	$-0.722094\pi$
$29$	−200.671	−1.28496	−0.642478	+	0.766304i	$0.722094\pi$
$30$	0	0
$31$	−196.482	−1.13836	−0.569182	−	0.822212i	$-0.692740\pi$
$31$	−196.482	−1.13836	−0.569182	+	0.822212i	$0.692740\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−14.5098	−0.0700742
$36$	0	0
$37$	−6.67144	−0.0296426	−0.0148213	−	0.999890i	$-0.504718\pi$
$37$	−6.67144	−0.0296426	−0.0148213	+	0.999890i	$0.504718\pi$
$38$	0	0
$39$	577.677	2.37185
$40$	0	0
$41$	−270.511	−1.03041	−0.515204	−	0.857067i	$-0.672284\pi$
$41$	−270.511	−1.03041	−0.515204	+	0.857067i	$0.672284\pi$
$42$	0	0
$43$	300.982	1.06743	0.533713	−	0.845666i	$-0.320796\pi$
$43$	300.982	1.06743	0.533713	+	0.845666i	$0.320796\pi$
$44$	0	0
$45$	57.1921	0.189460
$46$	0	0
$47$	−501.546	−1.55655	−0.778276	−	0.627922i	$-0.783906\pi$
$47$	−501.546	−1.55655	−0.778276	+	0.627922i	$0.783906\pi$
$48$	0	0
$49$	−265.604	−0.774355
$50$	0	0
$51$	28.1226	0.0772147
$52$	0	0
$53$	−183.329	−0.475136	−0.237568	−	0.971371i	$-0.576350\pi$
$53$	−183.329	−0.475136	−0.237568	+	0.971371i	$0.576350\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−221.918	−0.515680
$58$	0	0
$59$	601.459	1.32717	0.663587	−	0.748099i	$-0.269033\pi$
$59$	601.459	1.32717	0.663587	+	0.748099i	$0.269033\pi$
$60$	0	0
$61$	58.2961	0.122361	0.0611807	−	0.998127i	$-0.480513\pi$
$61$	58.2961	0.122361	0.0611807	+	0.998127i	$0.480513\pi$
$62$	0	0
$63$	−305.067	−0.610078
$64$	0	0
$65$	−121.318	−0.231502
$66$	0	0
$67$	−535.606	−0.976637	−0.488318	−	0.872666i	$-0.662389\pi$
$67$	−535.606	−0.976637	−0.488318	+	0.872666i	$0.662389\pi$
$68$	0	0
$69$	−932.700	−1.62730
$70$	0	0
$71$	289.540	0.483974	0.241987	−	0.970280i	$-0.422201\pi$
$71$	289.540	0.483974	0.241987	+	0.970280i	$0.422201\pi$
$72$	0	0
$73$	−836.495	−1.34115	−0.670577	−	0.741840i	$-0.733954\pi$
$73$	−836.495	−1.34115	−0.670577	+	0.741840i	$0.733954\pi$
$74$	0	0
$75$	960.317	1.47851
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1263.77	−1.79981	−0.899907	−	0.436083i	$-0.856365\pi$
$79$	−1263.77	−1.79981	−0.899907	+	0.436083i	$0.856365\pi$
$80$	0	0
$81$	−462.803	−0.634846
$82$	0	0
$83$	1183.37	1.56496	0.782480	−	0.622676i	$-0.213954\pi$
$83$	1183.37	1.56496	0.782480	+	0.622676i	$0.213954\pi$
$84$	0	0
$85$	−5.90603	−0.00753646
$86$	0	0
$87$	1575.96	1.94208
$88$	0	0
$89$	1081.40	1.28795	0.643977	−	0.765045i	$-0.277283\pi$
$89$	1081.40	1.28795	0.643977	+	0.765045i	$0.277283\pi$
$90$	0	0
$91$	647.120	0.745457
$92$	0	0
$93$	1543.06	1.72052
$94$	0	0
$95$	46.6050	0.0503324
$96$	0	0
$97$	1616.96	1.69255	0.846273	−	0.532749i	$-0.178841\pi$
$97$	1616.96	1.69255	0.846273	+	0.532749i	$0.178841\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1587.06	1.56354	0.781772	−	0.623564i	$-0.214316\pi$
$101$	1587.06	1.56354	0.781772	+	0.623564i	$0.214316\pi$
$102$	0	0
$103$	1056.28	1.01047	0.505234	−	0.862982i	$-0.331406\pi$
$103$	1056.28	1.01047	0.505234	+	0.862982i	$0.331406\pi$
$104$	0	0
$105$	113.952	0.105910
$106$	0	0
$107$	812.035	0.733667	0.366834	−	0.930287i	$-0.380442\pi$
$107$	812.035	0.733667	0.366834	+	0.930287i	$0.380442\pi$
$108$	0	0
$109$	−1099.13	−0.965848	−0.482924	−	0.875662i	$-0.660425\pi$
$109$	−1099.13	−0.965848	−0.482924	+	0.875662i	$0.660425\pi$
$110$	0	0
$111$	52.3938	0.0448018
$112$	0	0
$113$	1614.61	1.34416	0.672079	−	0.740479i	$-0.265402\pi$
$113$	1614.61	1.34416	0.672079	+	0.740479i	$0.265402\pi$
$114$	0	0
$115$	195.876	0.158831
$116$	0	0
$117$	−2550.71	−2.01550
$118$	0	0
$119$	31.5032	0.0242680
$120$	0	0
$121$	0	0
$122$	0	0
$123$	2124.44	1.55735
$124$	0	0
$125$	−407.839	−0.291826
$126$	0	0
$127$	−296.830	−0.207397	−0.103699	−	0.994609i	$-0.533068\pi$
$127$	−296.830	−0.207397	−0.103699	+	0.994609i	$0.533068\pi$
$128$	0	0
$129$	−2363.75	−1.61330
$130$	0	0
$131$	−1050.97	−0.700946	−0.350473	−	0.936573i	$-0.613979\pi$
$131$	−1050.97	−0.700946	−0.350473	+	0.936573i	$0.613979\pi$
$132$	0	0
$133$	−248.595	−0.162075
$134$	0	0
$135$	−99.4323	−0.0633909
$136$	0	0
$137$	1212.13	0.755905	0.377952	−	0.925825i	$-0.376628\pi$
$137$	1212.13	0.755905	0.377952	+	0.925825i	$0.376628\pi$
$138$	0	0
$139$	2041.28	1.24560	0.622801	−	0.782380i	$-0.285994\pi$
$139$	2041.28	1.24560	0.622801	+	0.782380i	$0.285994\pi$
$140$	0	0
$141$	3938.86	2.35257
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−330.967	−0.189554
$146$	0	0
$147$	2085.90	1.17036
$148$	0	0
$149$	−210.068	−0.115500	−0.0577498	−	0.998331i	$-0.518393\pi$
$149$	−210.068	−0.115500	−0.0577498	+	0.998331i	$0.518393\pi$
$150$	0	0
$151$	−1971.26	−1.06238	−0.531188	−	0.847254i	$-0.678254\pi$
$151$	−1971.26	−1.06238	−0.531188	+	0.847254i	$0.678254\pi$
$152$	0	0
$153$	−124.174	−0.0656137
$154$	0	0
$155$	−324.059	−0.167929
$156$	0	0
$157$	137.239	0.0697633	0.0348816	−	0.999391i	$-0.488895\pi$
$157$	137.239	0.0697633	0.0348816	+	0.999391i	$0.488895\pi$
$158$	0	0
$159$	1439.77	0.718119
$160$	0	0
$161$	−1044.82	−0.511450
$162$	0	0
$163$	3090.22	1.48494	0.742470	−	0.669880i	$-0.233654\pi$
$163$	3090.22	1.48494	0.742470	+	0.669880i	$0.233654\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2209.77	1.02393	0.511967	−	0.859005i	$-0.328917\pi$
$167$	2209.77	1.02393	0.511967	+	0.859005i	$0.328917\pi$
$168$	0	0
$169$	3213.65	1.46275
$170$	0	0
$171$	979.871	0.438202
$172$	0	0
$173$	−573.626	−0.252092	−0.126046	−	0.992024i	$-0.540229\pi$
$173$	−573.626	−0.252092	−0.126046	+	0.992024i	$0.540229\pi$
$174$	0	0
$175$	1075.76	0.464684
$176$	0	0
$177$	−4723.52	−2.00588
$178$	0	0
$179$	−1485.13	−0.620135	−0.310067	−	0.950715i	$-0.600352\pi$
$179$	−1485.13	−0.620135	−0.310067	+	0.950715i	$0.600352\pi$
$180$	0	0
$181$	3785.56	1.55458	0.777289	−	0.629143i	$-0.216594\pi$
$181$	3785.56	1.55458	0.777289	+	0.629143i	$0.216594\pi$
$182$	0	0
$183$	−457.825	−0.184936
$184$	0	0
$185$	−11.0032	−0.00437282
$186$	0	0
$187$	0	0
$188$	0	0
$189$	530.380	0.204124
$190$	0	0
$191$	−517.972	−0.196226	−0.0981130	−	0.995175i	$-0.531281\pi$
$191$	−517.972	−0.196226	−0.0981130	+	0.995175i	$0.531281\pi$
$192$	0	0
$193$	−722.016	−0.269284	−0.134642	−	0.990894i	$-0.542988\pi$
$193$	−722.016	−0.269284	−0.134642	+	0.990894i	$0.542988\pi$
$194$	0	0
$195$	952.764	0.349891
$196$	0	0
$197$	271.348	0.0981359	0.0490680	−	0.998795i	$-0.484375\pi$
$197$	271.348	0.0981359	0.0490680	+	0.998795i	$0.484375\pi$
$198$	0	0
$199$	3940.84	1.40381	0.701907	−	0.712269i	$-0.252332\pi$
$199$	3940.84	1.40381	0.701907	+	0.712269i	$0.252332\pi$
$200$	0	0
$201$	4206.35	1.47608
$202$	0	0
$203$	1765.41	0.610381
$204$	0	0
$205$	−446.155	−0.152004
$206$	0	0
$207$	4118.30	1.38281
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1900.22	−0.619982	−0.309991	−	0.950739i	$-0.600326\pi$
$211$	−1900.22	−0.619982	−0.309991	+	0.950739i	$0.600326\pi$
$212$	0	0
$213$	−2273.89	−0.731476
$214$	0	0
$215$	496.410	0.157465
$216$	0	0
$217$	1728.56	0.540747
$218$	0	0
$219$	6569.36	2.02702
$220$	0	0
$221$	263.403	0.0801737
$222$	0	0
$223$	−1859.99	−0.558540	−0.279270	−	0.960213i	$-0.590092\pi$
$223$	−1859.99	−0.558540	−0.279270	+	0.960213i	$0.590092\pi$
$224$	0	0
$225$	−4240.24	−1.25637
$226$	0	0
$227$	−686.473	−0.200717	−0.100359	−	0.994951i	$-0.531999\pi$
$227$	−686.473	−0.200717	−0.100359	+	0.994951i	$0.531999\pi$
$228$	0	0
$229$	−5626.43	−1.62360	−0.811801	−	0.583934i	$-0.801513\pi$
$229$	−5626.43	−1.62360	−0.811801	+	0.583934i	$0.801513\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5029.02	−1.41400	−0.707001	−	0.707213i	$-0.749952\pi$
$233$	−5029.02	−1.41400	−0.707001	+	0.707213i	$0.749952\pi$
$234$	0	0
$235$	−827.200	−0.229620
$236$	0	0
$237$	9924.95	2.72023
$238$	0	0
$239$	2492.39	0.674558	0.337279	−	0.941405i	$-0.390493\pi$
$239$	2492.39	0.674558	0.337279	+	0.941405i	$0.390493\pi$
$240$	0	0
$241$	−1945.60	−0.520030	−0.260015	−	0.965605i	$-0.583728\pi$
$241$	−1945.60	−0.520030	−0.260015	+	0.965605i	$0.583728\pi$
$242$	0	0
$243$	5262.36	1.38922
$244$	0	0
$245$	−438.061	−0.114231
$246$	0	0
$247$	−2078.54	−0.535442
$248$	0	0
$249$	−9293.52	−2.36527
$250$	0	0
$251$	606.548	0.152530	0.0762649	−	0.997088i	$-0.475701\pi$
$251$	606.548	0.152530	0.0762649	+	0.997088i	$0.475701\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	46.3827	0.0113906
$256$	0	0
$257$	6753.01	1.63907	0.819536	−	0.573028i	$-0.194231\pi$
$257$	6753.01	1.63907	0.819536	+	0.573028i	$0.194231\pi$
$258$	0	0
$259$	58.6920	0.0140809
$260$	0	0
$261$	−6958.59	−1.65029
$262$	0	0
$263$	−3485.41	−0.817184	−0.408592	−	0.912717i	$-0.633980\pi$
$263$	−3485.41	−0.817184	−0.408592	+	0.912717i	$0.633980\pi$
$264$	0	0
$265$	−302.365	−0.0700912
$266$	0	0
$267$	−8492.70	−1.94661
$268$	0	0
$269$	2902.87	0.657959	0.328979	−	0.944337i	$-0.393295\pi$
$269$	2902.87	0.657959	0.328979	+	0.944337i	$0.393295\pi$
$270$	0	0
$271$	2849.29	0.638680	0.319340	−	0.947640i	$-0.396539\pi$
$271$	2849.29	0.638680	0.319340	+	0.947640i	$0.396539\pi$
$272$	0	0
$273$	−5082.12	−1.12668
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5166.46	1.12066	0.560329	−	0.828270i	$-0.310675\pi$
$277$	5166.46	1.12066	0.560329	+	0.828270i	$0.310675\pi$
$278$	0	0
$279$	−6813.34	−1.46202
$280$	0	0
$281$	1599.98	0.339667	0.169834	−	0.985473i	$-0.445677\pi$
$281$	1599.98	0.339667	0.169834	+	0.985473i	$0.445677\pi$
$282$	0	0
$283$	−1611.50	−0.338493	−0.169246	−	0.985574i	$-0.554133\pi$
$283$	−1611.50	−0.338493	−0.169246	+	0.985574i	$0.554133\pi$
$284$	0	0
$285$	−366.010	−0.0760722
$286$	0	0
$287$	2379.82	0.489465
$288$	0	0
$289$	−4900.18	−0.997390
$290$	0	0
$291$	−12698.7	−2.55811
$292$	0	0
$293$	7815.50	1.55832	0.779158	−	0.626828i	$-0.215647\pi$
$293$	7815.50	1.55832	0.779158	+	0.626828i	$0.215647\pi$
$294$	0	0
$295$	991.987	0.195782
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8735.88	−1.68966
$300$	0	0
$301$	−2647.89	−0.507050
$302$	0	0
$303$	−12463.9	−2.36313
$304$	0	0
$305$	96.1478	0.0180505
$306$	0	0
$307$	−1361.80	−0.253165	−0.126583	−	0.991956i	$-0.540401\pi$
$307$	−1361.80	−0.253165	−0.126583	+	0.991956i	$0.540401\pi$
$308$	0	0
$309$	−8295.43	−1.52722
$310$	0	0
$311$	−3730.84	−0.680245	−0.340123	−	0.940381i	$-0.610469\pi$
$311$	−3730.84	−0.680245	−0.340123	+	0.940381i	$0.610469\pi$
$312$	0	0
$313$	2693.58	0.486423	0.243211	−	0.969973i	$-0.421799\pi$
$313$	2693.58	0.486423	0.243211	+	0.969973i	$0.421799\pi$
$314$	0	0
$315$	−503.148	−0.0899975
$316$	0	0
$317$	−1994.85	−0.353444	−0.176722	−	0.984261i	$-0.556549\pi$
$317$	−1994.85	−0.353444	−0.176722	+	0.984261i	$0.556549\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6377.27	−1.10886
$322$	0	0
$323$	−101.188	−0.0174311
$324$	0	0
$325$	8994.55	1.53516
$326$	0	0
$327$	8631.94	1.45978
$328$	0	0
$329$	4412.35	0.739395
$330$	0	0
$331$	2268.34	0.376675	0.188337	−	0.982104i	$-0.439690\pi$
$331$	2268.34	0.376675	0.188337	+	0.982104i	$0.439690\pi$
$332$	0	0
$333$	−231.343	−0.0380706
$334$	0	0
$335$	−883.375	−0.144072
$336$	0	0
$337$	−3036.87	−0.490886	−0.245443	−	0.969411i	$-0.578933\pi$
$337$	−3036.87	−0.490886	−0.245443	+	0.969411i	$0.578933\pi$
$338$	0	0
$339$	−12680.3	−2.03155
$340$	0	0
$341$	0	0
$342$	0	0
$343$	5354.20	0.842856
$344$	0	0
$345$	−1538.30	−0.240057
$346$	0	0
$347$	−4838.46	−0.748537	−0.374269	−	0.927320i	$-0.622106\pi$
$347$	−4838.46	−0.748537	−0.374269	+	0.927320i	$0.622106\pi$
$348$	0	0
$349$	−1536.08	−0.235600	−0.117800	−	0.993037i	$-0.537584\pi$
$349$	−1536.08	−0.235600	−0.117800	+	0.993037i	$0.537584\pi$
$350$	0	0
$351$	4434.57	0.674359
$352$	0	0
$353$	−8873.96	−1.33800	−0.668999	−	0.743264i	$-0.733277\pi$
$353$	−8873.96	−1.33800	−0.668999	+	0.743264i	$0.733277\pi$
$354$	0	0
$355$	477.540	0.0713948
$356$	0	0
$357$	−247.409	−0.0366786
$358$	0	0
$359$	3512.00	0.516313	0.258156	−	0.966103i	$-0.416885\pi$
$359$	3512.00	0.516313	0.258156	+	0.966103i	$0.416885\pi$
$360$	0	0
$361$	−6060.52	−0.883586
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1379.63	−0.197844
$366$	0	0
$367$	4679.04	0.665515	0.332758	−	0.943012i	$-0.392021\pi$
$367$	4679.04	0.665515	0.332758	+	0.943012i	$0.392021\pi$
$368$	0	0
$369$	−9380.40	−1.32337
$370$	0	0
$371$	1612.84	0.225700
$372$	0	0
$373$	2049.52	0.284504	0.142252	−	0.989830i	$-0.454566\pi$
$373$	2049.52	0.284504	0.142252	+	0.989830i	$0.454566\pi$
$374$	0	0
$375$	3202.94	0.441064
$376$	0	0
$377$	14760.8	2.01650
$378$	0	0
$379$	3483.59	0.472137	0.236068	−	0.971736i	$-0.424141\pi$
$379$	3483.59	0.472137	0.236068	+	0.971736i	$0.424141\pi$
$380$	0	0
$381$	2331.14	0.313459
$382$	0	0
$383$	5012.17	0.668694	0.334347	−	0.942450i	$-0.391484\pi$
$383$	5012.17	0.668694	0.334347	+	0.942450i	$0.391484\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10437.0	1.37091
$388$	0	0
$389$	−11490.2	−1.49763	−0.748814	−	0.662780i	$-0.769376\pi$
$389$	−11490.2	−1.49763	−0.748814	+	0.662780i	$0.769376\pi$
$390$	0	0
$391$	−425.282	−0.0550063
$392$	0	0
$393$	8253.76	1.05941
$394$	0	0
$395$	−2084.34	−0.265505
$396$	0	0
$397$	10661.3	1.34780	0.673901	−	0.738821i	$-0.264617\pi$
$397$	10661.3	1.34780	0.673901	+	0.738821i	$0.264617\pi$
$398$	0	0
$399$	1952.33	0.244959
$400$	0	0
$401$	−1909.17	−0.237754	−0.118877	−	0.992909i	$-0.537929\pi$
$401$	−1909.17	−0.237754	−0.118877	+	0.992909i	$0.537929\pi$
$402$	0	0
$403$	14452.7	1.78645
$404$	0	0
$405$	−763.302	−0.0936513
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8038.89	−0.971876	−0.485938	−	0.873993i	$-0.661522\pi$
$409$	−8038.89	−0.971876	−0.485938	+	0.873993i	$0.661522\pi$
$410$	0	0
$411$	−9519.37	−1.14247
$412$	0	0
$413$	−5291.34	−0.630435
$414$	0	0
$415$	1951.73	0.230860
$416$	0	0
$417$	−16031.0	−1.88260
$418$	0	0
$419$	2645.72	0.308477	0.154239	−	0.988034i	$-0.450708\pi$
$419$	2645.72	0.308477	0.154239	+	0.988034i	$0.450708\pi$
$420$	0	0
$421$	16704.4	1.93378	0.966891	−	0.255190i	$-0.0821381\pi$
$421$	16704.4	1.93378	0.966891	+	0.255190i	$0.0821381\pi$
$422$	0	0
$423$	−17391.9	−1.99911
$424$	0	0
$425$	437.875	0.0499766
$426$	0	0
$427$	−512.860	−0.0581242
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1936.74	0.216449	0.108225	−	0.994126i	$-0.465483\pi$
$431$	1936.74	0.216449	0.108225	+	0.994126i	$0.465483\pi$
$432$	0	0
$433$	11920.6	1.32302	0.661512	−	0.749935i	$-0.269915\pi$
$433$	11920.6	1.32302	0.661512	+	0.749935i	$0.269915\pi$
$434$	0	0
$435$	2599.23	0.286491
$436$	0	0
$437$	3355.94	0.367361
$438$	0	0
$439$	−11458.8	−1.24578	−0.622891	−	0.782308i	$-0.714042\pi$
$439$	−11458.8	−1.24578	−0.622891	+	0.782308i	$0.714042\pi$
$440$	0	0
$441$	−9210.23	−0.994518
$442$	0	0
$443$	6694.97	0.718030	0.359015	−	0.933332i	$-0.383113\pi$
$443$	6694.97	0.718030	0.359015	+	0.933332i	$0.383113\pi$
$444$	0	0
$445$	1783.55	0.189997
$446$	0	0
$447$	1649.76	0.174566
$448$	0	0
$449$	−9301.59	−0.977660	−0.488830	−	0.872379i	$-0.662576\pi$
$449$	−9301.59	−0.977660	−0.488830	+	0.872379i	$0.662576\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	15481.2	1.60567
$454$	0	0
$455$	1067.30	0.109968
$456$	0	0
$457$	−15528.5	−1.58948	−0.794740	−	0.606950i	$-0.792393\pi$
$457$	−15528.5	−1.58948	−0.794740	+	0.606950i	$0.792393\pi$
$458$	0	0
$459$	215.885	0.0219535
$460$	0	0
$461$	−16197.6	−1.63644	−0.818219	−	0.574907i	$-0.805038\pi$
$461$	−16197.6	−1.63644	−0.818219	+	0.574907i	$0.805038\pi$
$462$	0	0
$463$	2418.31	0.242740	0.121370	−	0.992607i	$-0.461271\pi$
$463$	2418.31	0.242740	0.121370	+	0.992607i	$0.461271\pi$
$464$	0	0
$465$	2544.98	0.253808
$466$	0	0
$467$	16774.1	1.66212	0.831062	−	0.556179i	$-0.187733\pi$
$467$	16774.1	1.66212	0.831062	+	0.556179i	$0.187733\pi$
$468$	0	0
$469$	4712.00	0.463923
$470$	0	0
$471$	−1077.80	−0.105440
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−3455.31	−0.333770
$476$	0	0
$477$	−6357.23	−0.610226
$478$	0	0
$479$	−5378.09	−0.513009	−0.256504	−	0.966543i	$-0.582571\pi$
$479$	−5378.09	−0.513009	−0.256504	+	0.966543i	$0.582571\pi$
$480$	0	0
$481$	490.732	0.0465186
$482$	0	0
$483$	8205.44	0.773003
$484$	0	0
$485$	2666.85	0.249681
$486$	0	0
$487$	1686.67	0.156941	0.0784705	−	0.996916i	$-0.474996\pi$
$487$	1686.67	0.156941	0.0784705	+	0.996916i	$0.474996\pi$
$488$	0	0
$489$	−24268.9	−2.24433
$490$	0	0
$491$	19376.7	1.78098	0.890488	−	0.455007i	$-0.150363\pi$
$491$	19376.7	1.78098	0.890488	+	0.455007i	$0.150363\pi$
$492$	0	0
$493$	718.589	0.0656463
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2547.23	−0.229898
$498$	0	0
$499$	12289.2	1.10249	0.551244	−	0.834344i	$-0.314153\pi$
$499$	12289.2	1.10249	0.551244	+	0.834344i	$0.314153\pi$
$500$	0	0
$501$	−17354.3	−1.54757
$502$	0	0
$503$	3853.13	0.341556	0.170778	−	0.985310i	$-0.445372\pi$
$503$	3853.13	0.341556	0.170778	+	0.985310i	$0.445372\pi$
$504$	0	0
$505$	2617.54	0.230651
$506$	0	0
$507$	−25238.2	−2.21079
$508$	0	0
$509$	−14567.8	−1.26858	−0.634289	−	0.773096i	$-0.718707\pi$
$509$	−14567.8	−1.26858	−0.634289	+	0.773096i	$0.718707\pi$
$510$	0	0
$511$	7359.07	0.637076
$512$	0	0
$513$	−1703.57	−0.146617
$514$	0	0
$515$	1742.12	0.149062
$516$	0	0
$517$	0	0
$518$	0	0
$519$	4504.94	0.381011
$520$	0	0
$521$	−14374.2	−1.20872	−0.604360	−	0.796711i	$-0.706571\pi$
$521$	−14374.2	−1.20872	−0.604360	+	0.796711i	$0.706571\pi$
$522$	0	0
$523$	−13972.0	−1.16817	−0.584086	−	0.811691i	$-0.698547\pi$
$523$	−13972.0	−1.16817	−0.584086	+	0.811691i	$0.698547\pi$
$524$	0	0
$525$	−8448.40	−0.702321
$526$	0	0
$527$	703.589	0.0581572
$528$	0	0
$529$	1937.70	0.159258
$530$	0	0
$531$	20856.5	1.70451
$532$	0	0
$533$	19898.0	1.61703
$534$	0	0
$535$	1339.29	0.108229
$536$	0	0
$537$	11663.4	0.937269
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18047.2	1.43421	0.717106	−	0.696964i	$-0.245466\pi$
$541$	18047.2	1.43421	0.717106	+	0.696964i	$0.245466\pi$
$542$	0	0
$543$	−29729.7	−2.34958
$544$	0	0
$545$	−1812.79	−0.142480
$546$	0	0
$547$	6078.91	0.475165	0.237583	−	0.971367i	$-0.423645\pi$
$547$	6078.91	0.475165	0.237583	+	0.971367i	$0.423645\pi$
$548$	0	0
$549$	2021.51	0.157151
$550$	0	0
$551$	−5670.46	−0.438420
$552$	0	0
$553$	11118.0	0.854949
$554$	0	0
$555$	86.4131	0.00660907
$556$	0	0
$557$	−8418.07	−0.640368	−0.320184	−	0.947355i	$-0.603745\pi$
$557$	−8418.07	−0.640368	−0.320184	+	0.947355i	$0.603745\pi$
$558$	0	0
$559$	−22139.4	−1.67513
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3113.82	−0.233094	−0.116547	−	0.993185i	$-0.537183\pi$
$563$	−3113.82	−0.233094	−0.116547	+	0.993185i	$0.537183\pi$
$564$	0	0
$565$	2662.98	0.198288
$566$	0	0
$567$	4071.51	0.301565
$568$	0	0
$569$	13725.5	1.01125	0.505625	−	0.862754i	$-0.331262\pi$
$569$	13725.5	1.01125	0.505625	+	0.862754i	$0.331262\pi$
$570$	0	0
$571$	12613.6	0.924452	0.462226	−	0.886762i	$-0.347051\pi$
$571$	12613.6	0.924452	0.462226	+	0.886762i	$0.347051\pi$
$572$	0	0
$573$	4067.87	0.296575
$574$	0	0
$575$	−14522.3	−1.05326
$576$	0	0
$577$	389.165	0.0280783	0.0140391	−	0.999901i	$-0.495531\pi$
$577$	389.165	0.0280783	0.0140391	+	0.999901i	$0.495531\pi$
$578$	0	0
$579$	5670.31	0.406995
$580$	0	0
$581$	−10410.7	−0.743389
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4206.89	−0.297322
$586$	0	0
$587$	23471.5	1.65038	0.825188	−	0.564858i	$-0.191069\pi$
$587$	23471.5	1.65038	0.825188	+	0.564858i	$0.191069\pi$
$588$	0	0
$589$	−5552.09	−0.388404
$590$	0	0
$591$	−2131.02	−0.148322
$592$	0	0
$593$	−12843.2	−0.889385	−0.444692	−	0.895683i	$-0.646687\pi$
$593$	−12843.2	−0.889385	−0.444692	+	0.895683i	$0.646687\pi$
$594$	0	0
$595$	51.9583	0.00357997
$596$	0	0
$597$	−30949.2	−2.12172
$598$	0	0
$599$	−24232.4	−1.65293	−0.826467	−	0.562985i	$-0.809653\pi$
$599$	−24232.4	−1.65293	−0.826467	+	0.562985i	$0.809653\pi$
$600$	0	0
$601$	−1000.83	−0.0679277	−0.0339638	−	0.999423i	$-0.510813\pi$
$601$	−1000.83	−0.0679277	−0.0339638	+	0.999423i	$0.510813\pi$
$602$	0	0
$603$	−18573.0	−1.25431
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26628.0	−1.78056	−0.890278	−	0.455418i	$-0.849490\pi$
$607$	−26628.0	−1.78056	−0.890278	+	0.455418i	$0.849490\pi$
$608$	0	0
$609$	−13864.5	−0.922527
$610$	0	0
$611$	36892.3	2.44272
$612$	0	0
$613$	3456.70	0.227756	0.113878	−	0.993495i	$-0.463673\pi$
$613$	3456.70	0.227756	0.113878	+	0.993495i	$0.463673\pi$
$614$	0	0
$615$	3503.85	0.229738
$616$	0	0
$617$	18763.1	1.22427	0.612135	−	0.790753i	$-0.290311\pi$
$617$	18763.1	1.22427	0.612135	+	0.790753i	$0.290311\pi$
$618$	0	0
$619$	10525.0	0.683414	0.341707	−	0.939806i	$-0.388995\pi$
$619$	10525.0	0.683414	0.341707	+	0.939806i	$0.388995\pi$
$620$	0	0
$621$	−7159.93	−0.462670
$622$	0	0
$623$	−9513.61	−0.611805
$624$	0	0
$625$	14612.3	0.935189
$626$	0	0
$627$	0	0
$628$	0	0
$629$	23.8899	0.00151439
$630$	0	0
$631$	17373.0	1.09605	0.548026	−	0.836461i	$-0.315379\pi$
$631$	17373.0	1.09605	0.548026	+	0.836461i	$0.315379\pi$
$632$	0	0
$633$	14923.2	0.937039
$634$	0	0
$635$	−489.563	−0.0305948
$636$	0	0
$637$	19537.1	1.21521
$638$	0	0
$639$	10040.3	0.621576
$640$	0	0
$641$	−12593.6	−0.776003	−0.388002	−	0.921659i	$-0.626835\pi$
$641$	−12593.6	−0.776003	−0.388002	+	0.921659i	$0.626835\pi$
$642$	0	0
$643$	−22882.6	−1.40342	−0.701712	−	0.712461i	$-0.747581\pi$
$643$	−22882.6	−1.40342	−0.701712	+	0.712461i	$0.747581\pi$
$644$	0	0
$645$	−3898.53	−0.237991
$646$	0	0
$647$	13195.4	0.801799	0.400899	−	0.916122i	$-0.368698\pi$
$647$	13195.4	0.801799	0.400899	+	0.916122i	$0.368698\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−13575.1	−0.817282
$652$	0	0
$653$	−974.479	−0.0583986	−0.0291993	−	0.999574i	$-0.509296\pi$
$653$	−974.479	−0.0583986	−0.0291993	+	0.999574i	$0.509296\pi$
$654$	0	0
$655$	−1733.37	−0.103402
$656$	0	0
$657$	−29006.8	−1.72247
$658$	0	0
$659$	−27639.1	−1.63379	−0.816894	−	0.576788i	$-0.804306\pi$
$659$	−27639.1	−1.63379	−0.816894	+	0.576788i	$0.804306\pi$
$660$	0	0
$661$	11266.4	0.662955	0.331477	−	0.943463i	$-0.392453\pi$
$661$	11266.4	0.662955	0.331477	+	0.943463i	$0.392453\pi$
$662$	0	0
$663$	−2068.62	−0.121174
$664$	0	0
$665$	−410.008	−0.0239089
$666$	0	0
$667$	−23832.4	−1.38350
$668$	0	0
$669$	14607.4	0.844175
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−14560.1	−0.833954	−0.416977	−	0.908917i	$-0.636910\pi$
$673$	−14560.1	−0.833954	−0.416977	+	0.908917i	$0.636910\pi$
$674$	0	0
$675$	7371.94	0.420365
$676$	0	0
$677$	21615.7	1.22712	0.613558	−	0.789649i	$-0.289738\pi$
$677$	21615.7	1.22712	0.613558	+	0.789649i	$0.289738\pi$
$678$	0	0
$679$	−14225.2	−0.803995
$680$	0	0
$681$	5391.18	0.303363
$682$	0	0
$683$	−114.888	−0.00643642	−0.00321821	−	0.999995i	$-0.501024\pi$
$683$	−114.888	−0.00643642	−0.00321821	+	0.999995i	$0.501024\pi$
$684$	0	0
$685$	1999.16	0.111510
$686$	0	0
$687$	44186.8	2.45391
$688$	0	0
$689$	13485.2	0.745638
$690$	0	0
$691$	−2430.19	−0.133790	−0.0668950	−	0.997760i	$-0.521309\pi$
$691$	−2430.19	−0.133790	−0.0668950	+	0.997760i	$0.521309\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3366.68	0.183749
$696$	0	0
$697$	968.680	0.0526419
$698$	0	0
$699$	39495.1	2.13712
$700$	0	0
$701$	15328.3	0.825878	0.412939	−	0.910759i	$-0.364502\pi$
$701$	15328.3	0.825878	0.412939	+	0.910759i	$0.364502\pi$
$702$	0	0
$703$	−188.518	−0.0101139
$704$	0	0
$705$	6496.37	0.347046
$706$	0	0
$707$	−13962.1	−0.742716
$708$	0	0
$709$	−16076.1	−0.851553	−0.425776	−	0.904828i	$-0.639999\pi$
$709$	−16076.1	−0.851553	−0.425776	+	0.904828i	$0.639999\pi$
$710$	0	0
$711$	−43823.2	−2.31153
$712$	0	0
$713$	−23334.9	−1.22566
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19573.8	−1.01952
$718$	0	0
$719$	−5312.96	−0.275577	−0.137789	−	0.990462i	$-0.543999\pi$
$719$	−5312.96	−0.275577	−0.137789	+	0.990462i	$0.543999\pi$
$720$	0	0
$721$	−9292.63	−0.479994
$722$	0	0
$723$	15279.7	0.785972
$724$	0	0
$725$	24538.0	1.25699
$726$	0	0
$727$	11345.2	0.578775	0.289388	−	0.957212i	$-0.406548\pi$
$727$	11345.2	0.578775	0.289388	+	0.957212i	$0.406548\pi$
$728$	0	0
$729$	−28832.0	−1.46482
$730$	0	0
$731$	−1077.79	−0.0545331
$732$	0	0
$733$	22813.9	1.14959	0.574796	−	0.818296i	$-0.305081\pi$
$733$	22813.9	1.14959	0.574796	+	0.818296i	$0.305081\pi$
$734$	0	0
$735$	3440.29	0.172649
$736$	0	0
$737$	0	0
$738$	0	0
$739$	14194.1	0.706548	0.353274	−	0.935520i	$-0.385068\pi$
$739$	14194.1	0.706548	0.353274	+	0.935520i	$0.385068\pi$
$740$	0	0
$741$	16323.7	0.809264
$742$	0	0
$743$	−11532.1	−0.569412	−0.284706	−	0.958615i	$-0.591896\pi$
$743$	−11532.1	−0.569412	−0.284706	+	0.958615i	$0.591896\pi$
$744$	0	0
$745$	−346.466	−0.0170383
$746$	0	0
$747$	41035.2	2.00990
$748$	0	0
$749$	−7143.89	−0.348507
$750$	0	0
$751$	−13785.0	−0.669800	−0.334900	−	0.942254i	$-0.608703\pi$
$751$	−13785.0	−0.669800	−0.334900	+	0.942254i	$0.608703\pi$
$752$	0	0
$753$	−4763.49	−0.230533
$754$	0	0
$755$	−3251.20	−0.156720
$756$	0	0
$757$	9335.04	0.448201	0.224100	−	0.974566i	$-0.428056\pi$
$757$	9335.04	0.448201	0.224100	+	0.974566i	$0.428056\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9927.54	0.472895	0.236447	−	0.971644i	$-0.424017\pi$
$761$	9927.54	0.472895	0.236447	+	0.971644i	$0.424017\pi$
$762$	0	0
$763$	9669.60	0.458798
$764$	0	0
$765$	−204.801	−0.00967920
$766$	0	0
$767$	−44241.6	−2.08275
$768$	0	0
$769$	36268.3	1.70074	0.850370	−	0.526186i	$-0.176378\pi$
$769$	36268.3	1.70074	0.850370	+	0.526186i	$0.176378\pi$
$770$	0	0
$771$	−53034.4	−2.47729
$772$	0	0
$773$	2448.19	0.113914	0.0569569	−	0.998377i	$-0.481860\pi$
$773$	2448.19	0.113914	0.0569569	+	0.998377i	$0.481860\pi$
$774$	0	0
$775$	24025.8	1.11359
$776$	0	0
$777$	−460.935	−0.0212818
$778$	0	0
$779$	−7643.95	−0.351570
$780$	0	0
$781$	0	0
$782$	0	0
$783$	12098.0	0.552166
$784$	0	0
$785$	226.348	0.0102913
$786$	0	0
$787$	27529.0	1.24689	0.623445	−	0.781867i	$-0.285733\pi$
$787$	27529.0	1.24689	0.623445	+	0.781867i	$0.285733\pi$
$788$	0	0
$789$	27372.4	1.23509
$790$	0	0
$791$	−14204.6	−0.638503
$792$	0	0
$793$	−4288.09	−0.192023
$794$	0	0
$795$	2374.61	0.105935
$796$	0	0
$797$	11243.3	0.499695	0.249847	−	0.968285i	$-0.419620\pi$
$797$	11243.3	0.499695	0.249847	+	0.968285i	$0.419620\pi$
$798$	0	0
$799$	1796.00	0.0795217
$800$	0	0
$801$	37499.2	1.65414
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1723.22	−0.0754480
$806$	0	0
$807$	−22797.5	−0.994436
$808$	0	0
$809$	−6215.60	−0.270122	−0.135061	−	0.990837i	$-0.543123\pi$
$809$	−6215.60	−0.270122	−0.135061	+	0.990837i	$0.543123\pi$
$810$	0	0
$811$	−38124.4	−1.65071	−0.825356	−	0.564612i	$-0.809026\pi$
$811$	−38124.4	−1.65071	−0.825356	+	0.564612i	$0.809026\pi$
$812$	0	0
$813$	−22376.8	−0.965299
$814$	0	0
$815$	5096.71	0.219055
$816$	0	0
$817$	8504.98	0.364200
$818$	0	0
$819$	22439.9	0.957403
$820$	0	0
$821$	−37435.9	−1.59138	−0.795690	−	0.605704i	$-0.792892\pi$
$821$	−37435.9	−1.59138	−0.795690	+	0.605704i	$0.792892\pi$
$822$	0	0
$823$	43160.2	1.82803	0.914017	−	0.405677i	$-0.132964\pi$
$823$	43160.2	1.82803	0.914017	+	0.405677i	$0.132964\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4837.15	−0.203391	−0.101695	−	0.994816i	$-0.532427\pi$
$827$	−4837.15	−0.203391	−0.101695	+	0.994816i	$0.532427\pi$
$828$	0	0
$829$	19749.0	0.827395	0.413698	−	0.910414i	$-0.364237\pi$
$829$	19749.0	0.827395	0.413698	+	0.910414i	$0.364237\pi$
$830$	0	0
$831$	−40574.5	−1.69376
$832$	0	0
$833$	951.108	0.0395605
$834$	0	0
$835$	3644.57	0.151049
$836$	0	0
$837$	11845.4	0.489173
$838$	0	0
$839$	22147.1	0.911326	0.455663	−	0.890152i	$-0.349402\pi$
$839$	22147.1	0.911326	0.455663	+	0.890152i	$0.349402\pi$
$840$	0	0
$841$	15879.9	0.651111
$842$	0	0
$843$	−12565.3	−0.513372
$844$	0	0
$845$	5300.28	0.215781
$846$	0	0
$847$	0	0
$848$	0	0
$849$	12655.8	0.511597
$850$	0	0
$851$	−792.321	−0.0319159
$852$	0	0
$853$	10980.3	0.440749	0.220374	−	0.975415i	$-0.429272\pi$
$853$	10980.3	0.440749	0.220374	+	0.975415i	$0.429272\pi$
$854$	0	0
$855$	1616.10	0.0646428
$856$	0	0
$857$	6288.67	0.250661	0.125331	−	0.992115i	$-0.460001\pi$
$857$	6288.67	0.250661	0.125331	+	0.992115i	$0.460001\pi$
$858$	0	0
$859$	6331.34	0.251482	0.125741	−	0.992063i	$-0.459869\pi$
$859$	6331.34	0.251482	0.125741	+	0.992063i	$0.459869\pi$
$860$	0	0
$861$	−18689.8	−0.739776
$862$	0	0
$863$	−2753.93	−0.108627	−0.0543134	−	0.998524i	$-0.517297\pi$
$863$	−2753.93	−0.108627	−0.0543134	+	0.998524i	$0.517297\pi$
$864$	0	0
$865$	−946.082	−0.0371881
$866$	0	0
$867$	38483.3	1.50745
$868$	0	0
$869$	0	0
$870$	0	0
$871$	39397.6	1.53265
$872$	0	0
$873$	56070.5	2.17377
$874$	0	0
$875$	3587.97	0.138623
$876$	0	0
$877$	−42373.9	−1.63154	−0.815772	−	0.578374i	$-0.803687\pi$
$877$	−42373.9	−1.63154	−0.815772	+	0.578374i	$0.803687\pi$
$878$	0	0
$879$	−61378.6	−2.35523
$880$	0	0
$881$	16567.1	0.633554	0.316777	−	0.948500i	$-0.397399\pi$
$881$	16567.1	0.633554	0.316777	+	0.948500i	$0.397399\pi$
$882$	0	0
$883$	−4948.64	−0.188601	−0.0943007	−	0.995544i	$-0.530062\pi$
$883$	−4948.64	−0.188601	−0.0943007	+	0.995544i	$0.530062\pi$
$884$	0	0
$885$	−7790.51	−0.295904
$886$	0	0
$887$	−23416.4	−0.886411	−0.443206	−	0.896420i	$-0.646159\pi$
$887$	−23416.4	−0.886411	−0.443206	+	0.896420i	$0.646159\pi$
$888$	0	0
$889$	2611.37	0.0985180
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−14172.4	−0.531087
$894$	0	0
$895$	−2449.43	−0.0914810
$896$	0	0
$897$	68606.7	2.55375
$898$	0	0
$899$	39428.4	1.46275
$900$	0	0
$901$	656.489	0.0242739
$902$	0	0
$903$	20795.1	0.766353
$904$	0	0
$905$	6243.54	0.229328
$906$	0	0
$907$	20997.1	0.768686	0.384343	−	0.923190i	$-0.374428\pi$
$907$	20997.1	0.768686	0.384343	+	0.923190i	$0.374428\pi$
$908$	0	0
$909$	55033.7	2.00809
$910$	0	0
$911$	16556.3	0.602123	0.301062	−	0.953605i	$-0.402659\pi$
$911$	16556.3	0.602123	0.301062	+	0.953605i	$0.402659\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−755.091	−0.0272815
$916$	0	0
$917$	9245.95	0.332964
$918$	0	0
$919$	−48271.3	−1.73267	−0.866335	−	0.499464i	$-0.833530\pi$
$919$	−48271.3	−1.73267	−0.866335	+	0.499464i	$0.833530\pi$
$920$	0	0
$921$	10694.8	0.382633
$922$	0	0
$923$	−21297.8	−0.759506
$924$	0	0
$925$	815.782	0.0289976
$926$	0	0
$927$	36628.1	1.29776
$928$	0	0
$929$	23920.0	0.844768	0.422384	−	0.906417i	$-0.361193\pi$
$929$	23920.0	0.844768	0.422384	+	0.906417i	$0.361193\pi$
$930$	0	0
$931$	−7505.28	−0.264206
$932$	0	0
$933$	29299.9	1.02812
$934$	0	0
$935$	0	0
$936$	0	0
$937$	52713.3	1.83785	0.918927	−	0.394428i	$-0.129057\pi$
$937$	52713.3	1.83785	0.918927	+	0.394428i	$0.129057\pi$
$938$	0	0
$939$	−21153.9	−0.735177
$940$	0	0
$941$	2968.92	0.102852	0.0514261	−	0.998677i	$-0.483623\pi$
$941$	2968.92	0.102852	0.0514261	+	0.998677i	$0.483623\pi$
$942$	0	0
$943$	−32126.8	−1.10943
$944$	0	0
$945$	874.756	0.0301120
$946$	0	0
$947$	−27916.0	−0.957918	−0.478959	−	0.877837i	$-0.658986\pi$
$947$	−27916.0	−0.957918	−0.478959	+	0.877837i	$0.658986\pi$
$948$	0	0
$949$	61530.1	2.10469
$950$	0	0
$951$	15666.4	0.534194
$952$	0	0
$953$	21451.4	0.729148	0.364574	−	0.931174i	$-0.381215\pi$
$953$	21451.4	0.729148	0.364574	+	0.931174i	$0.381215\pi$
$954$	0	0
$955$	−854.293	−0.0289469
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10663.7	−0.359071
$960$	0	0
$961$	8814.34	0.295873
$962$	0	0
$963$	28158.6	0.942261
$964$	0	0
$965$	−1190.82	−0.0397243
$966$	0	0
$967$	−20189.8	−0.671417	−0.335709	−	0.941966i	$-0.608976\pi$
$967$	−20189.8	−0.671417	−0.335709	+	0.941966i	$0.608976\pi$
$968$	0	0
$969$	794.673	0.0263453
$970$	0	0
$971$	6396.14	0.211392	0.105696	−	0.994398i	$-0.466293\pi$
$971$	6396.14	0.211392	0.105696	+	0.994398i	$0.466293\pi$
$972$	0	0
$973$	−17958.2	−0.591687
$974$	0	0
$975$	−70638.2	−2.32024
$976$	0	0
$977$	31920.5	1.04527	0.522634	−	0.852557i	$-0.324949\pi$
$977$	31920.5	1.04527	0.522634	+	0.852557i	$0.324949\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−38114.0	−1.24046
$982$	0	0
$983$	−50326.5	−1.63293	−0.816463	−	0.577397i	$-0.804068\pi$
$983$	−50326.5	−1.63293	−0.816463	+	0.577397i	$0.804068\pi$
$984$	0	0
$985$	447.535	0.0144768
$986$	0	0
$987$	−34652.2	−1.11752
$988$	0	0
$989$	35745.6	1.14929
$990$	0	0
$991$	33513.9	1.07427	0.537136	−	0.843496i	$-0.319506\pi$
$991$	33513.9	1.07427	0.537136	+	0.843496i	$0.319506\pi$
$992$	0	0
$993$	−17814.3	−0.569304
$994$	0	0
$995$	6499.64	0.207088
$996$	0	0
$997$	−2545.04	−0.0808447	−0.0404223	−	0.999183i	$-0.512870\pi$
$997$	−2545.04	−0.0808447	−0.0404223	+	0.999183i	$0.512870\pi$
$998$	0	0
$999$	402.204	0.0127379

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.r.1.2		10
4.3	odd	2		1936.4.a.by.1.9		10
11.2	odd	10		88.4.i.b.81.1	yes	20
11.6	odd	10		88.4.i.b.25.1	✓	20
11.10	odd	2		968.4.a.s.1.2		10
44.35	even	10		176.4.m.f.81.5		20
44.39	even	10		176.4.m.f.113.5		20
44.43	even	2		1936.4.a.bx.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.b.25.1	✓	20	11.6	odd	10
88.4.i.b.81.1	yes	20	11.2	odd	10
176.4.m.f.81.5		20	44.35	even	10
176.4.m.f.113.5		20	44.39	even	10
968.4.a.r.1.2		10	1.1	even	1	trivial
968.4.a.s.1.2		10	11.10	odd	2
1936.4.a.bx.1.9		10	44.43	even	2
1936.4.a.by.1.9		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)

\(f(q)\)	\(=\)	\(q-7.85344 q^{3} +1.64930 q^{5} -8.79751 q^{7} +34.6766 q^{9} +O(q^{10})\)	\(q-7.85344 q^{3} +1.64930 q^{5} -8.79751 q^{7} +34.6766 q^{9} -73.5571 q^{13} -12.9527 q^{15} -3.58093 q^{17} +28.2574 q^{19} +69.0907 q^{21} +118.763 q^{23} -122.280 q^{25} -60.2875 q^{27} -200.671 q^{29} -196.482 q^{31} -14.5098 q^{35} -6.67144 q^{37} +577.677 q^{39} -270.511 q^{41} +300.982 q^{43} +57.1921 q^{45} -501.546 q^{47} -265.604 q^{49} +28.1226 q^{51} -183.329 q^{53} -221.918 q^{57} +601.459 q^{59} +58.2961 q^{61} -305.067 q^{63} -121.318 q^{65} -535.606 q^{67} -932.700 q^{69} +289.540 q^{71} -836.495 q^{73} +960.317 q^{75} -1263.77 q^{79} -462.803 q^{81} +1183.37 q^{83} -5.90603 q^{85} +1575.96 q^{87} +1081.40 q^{89} +647.120 q^{91} +1543.06 q^{93} +46.6050 q^{95} +1616.96 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) \) 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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