LMFDB - Embedded newform 968.4.a.r.1.9

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			1.9
Root			$7.99327$ 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)

$f(q)$	$=$	$q+9.61131 q^{3} -16.7145 q^{5} -19.9185 q^{7} +65.3772 q^{9} -39.6266 q^{13} -160.648 q^{15} +2.06033 q^{17} +101.454 q^{19} -191.443 q^{21} +110.549 q^{23} +154.375 q^{25} +368.855 q^{27} +33.3622 q^{29} +173.705 q^{31} +332.928 q^{35} +251.438 q^{37} -380.863 q^{39} +192.707 q^{41} -304.185 q^{43} -1092.75 q^{45} -34.1111 q^{47} +53.7459 q^{49} +19.8025 q^{51} +216.628 q^{53} +975.109 q^{57} -16.0600 q^{59} +531.144 q^{61} -1302.21 q^{63} +662.339 q^{65} +636.628 q^{67} +1062.52 q^{69} +280.744 q^{71} +656.752 q^{73} +1483.74 q^{75} -806.016 q^{79} +1780.00 q^{81} -638.367 q^{83} -34.4374 q^{85} +320.654 q^{87} +1144.68 q^{89} +789.301 q^{91} +1669.53 q^{93} -1695.76 q^{95} +840.210 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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
$3$	9.61131	1.84970	0.924848	−	0.380336i	$-0.124192\pi$
$3$	9.61131	1.84970	0.924848	+	0.380336i	$0.124192\pi$
$4$	0	0
$5$	−16.7145	−1.49499	−0.747496	−	0.664267i	$-0.768744\pi$
$5$	−16.7145	−1.49499	−0.747496	+	0.664267i	$0.768744\pi$
$6$	0	0
$7$	−19.9185	−1.07550	−0.537748	−	0.843105i	$-0.680725\pi$
$7$	−19.9185	−1.07550	−0.537748	+	0.843105i	$0.680725\pi$
$8$	0	0
$9$	65.3772	2.42138
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−39.6266	−0.845418	−0.422709	−	0.906266i	$-0.638921\pi$
$13$	−39.6266	−0.845418	−0.422709	+	0.906266i	$0.638921\pi$
$14$	0	0
$15$	−160.648	−2.76528
$16$	0	0
$17$	2.06033	0.0293943	0.0146972	−	0.999892i	$-0.495322\pi$
$17$	2.06033	0.0293943	0.0146972	+	0.999892i	$0.495322\pi$
$18$	0	0
$19$	101.454	1.22501	0.612506	−	0.790466i	$-0.290161\pi$
$19$	101.454	1.22501	0.612506	+	0.790466i	$0.290161\pi$
$20$	0	0
$21$	−191.443	−1.98934
$22$	0	0
$23$	110.549	1.00222	0.501109	−	0.865384i	$-0.332926\pi$
$23$	110.549	1.00222	0.501109	+	0.865384i	$0.332926\pi$
$24$	0	0
$25$	154.375	1.23500
$26$	0	0
$27$	368.855	2.62912
$28$	0	0
$29$	33.3622	0.213628	0.106814	−	0.994279i	$-0.465935\pi$
$29$	33.3622	0.213628	0.106814	+	0.994279i	$0.465935\pi$
$30$	0	0
$31$	173.705	1.00640	0.503198	−	0.864171i	$-0.332157\pi$
$31$	173.705	1.00640	0.503198	+	0.864171i	$0.332157\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	332.928	1.60786
$36$	0	0
$37$	251.438	1.11719	0.558597	−	0.829439i	$-0.311340\pi$
$37$	251.438	1.11719	0.558597	+	0.829439i	$0.311340\pi$
$38$	0	0
$39$	−380.863	−1.56377
$40$	0	0
$41$	192.707	0.734043	0.367021	−	0.930213i	$-0.380378\pi$
$41$	192.707	0.734043	0.367021	+	0.930213i	$0.380378\pi$
$42$	0	0
$43$	−304.185	−1.07879	−0.539393	−	0.842054i	$-0.681346\pi$
$43$	−304.185	−1.07879	−0.539393	+	0.842054i	$0.681346\pi$
$44$	0	0
$45$	−1092.75	−3.61994
$46$	0	0
$47$	−34.1111	−0.105864	−0.0529321	−	0.998598i	$-0.516857\pi$
$47$	−34.1111	−0.105864	−0.0529321	+	0.998598i	$0.516857\pi$
$48$	0	0
$49$	53.7459	0.156694
$50$	0	0
$51$	19.8025	0.0543706
$52$	0	0
$53$	216.628	0.561436	0.280718	−	0.959790i	$-0.409427\pi$
$53$	216.628	0.561436	0.280718	+	0.959790i	$0.409427\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	975.109	2.26590
$58$	0	0
$59$	−16.0600	−0.0354378	−0.0177189	−	0.999843i	$-0.505640\pi$
$59$	−16.0600	−0.0354378	−0.0177189	+	0.999843i	$0.505640\pi$
$60$	0	0
$61$	531.144	1.11485	0.557427	−	0.830226i	$-0.311789\pi$
$61$	531.144	1.11485	0.557427	+	0.830226i	$0.311789\pi$
$62$	0	0
$63$	−1302.21	−2.60419
$64$	0	0
$65$	662.339	1.26389
$66$	0	0
$67$	636.628	1.16084	0.580422	−	0.814316i	$-0.302888\pi$
$67$	636.628	1.16084	0.580422	+	0.814316i	$0.302888\pi$
$68$	0	0
$69$	1062.52	1.85380
$70$	0	0
$71$	280.744	0.469271	0.234635	−	0.972083i	$-0.424610\pi$
$71$	280.744	0.469271	0.234635	+	0.972083i	$0.424610\pi$
$72$	0	0
$73$	656.752	1.05297	0.526486	−	0.850184i	$-0.323509\pi$
$73$	656.752	1.05297	0.526486	+	0.850184i	$0.323509\pi$
$74$	0	0
$75$	1483.74	2.28437
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−806.016	−1.14790	−0.573948	−	0.818891i	$-0.694589\pi$
$79$	−806.016	−1.14790	−0.573948	+	0.818891i	$0.694589\pi$
$80$	0	0
$81$	1780.00	2.44170
$82$	0	0
$83$	−638.367	−0.844215	−0.422108	−	0.906546i	$-0.638710\pi$
$83$	−638.367	−0.844215	−0.422108	+	0.906546i	$0.638710\pi$
$84$	0	0
$85$	−34.4374	−0.0439443
$86$	0	0
$87$	320.654	0.395146
$88$	0	0
$89$	1144.68	1.36333	0.681663	−	0.731666i	$-0.261257\pi$
$89$	1144.68	1.36333	0.681663	+	0.731666i	$0.261257\pi$
$90$	0	0
$91$	789.301	0.909244
$92$	0	0
$93$	1669.53	1.86153
$94$	0	0
$95$	−1695.76	−1.83138
$96$	0	0
$97$	840.210	0.879489	0.439744	−	0.898123i	$-0.355069\pi$
$97$	840.210	0.879489	0.439744	+	0.898123i	$0.355069\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1332.91	−1.31316	−0.656579	−	0.754257i	$-0.727997\pi$
$101$	−1332.91	−1.31316	−0.656579	+	0.754257i	$0.727997\pi$
$102$	0	0
$103$	95.0184	0.0908975	0.0454487	−	0.998967i	$-0.485528\pi$
$103$	95.0184	0.0908975	0.0454487	+	0.998967i	$0.485528\pi$
$104$	0	0
$105$	3199.87	2.97405
$106$	0	0
$107$	1442.16	1.30298	0.651490	−	0.758658i	$-0.274144\pi$
$107$	1442.16	1.30298	0.651490	+	0.758658i	$0.274144\pi$
$108$	0	0
$109$	708.872	0.622914	0.311457	−	0.950260i	$-0.399183\pi$
$109$	708.872	0.622914	0.311457	+	0.950260i	$0.399183\pi$
$110$	0	0
$111$	2416.65	2.06647
$112$	0	0
$113$	−1671.13	−1.39121	−0.695603	−	0.718426i	$-0.744863\pi$
$113$	−1671.13	−1.39121	−0.695603	+	0.718426i	$0.744863\pi$
$114$	0	0
$115$	−1847.77	−1.49831
$116$	0	0
$117$	−2590.67	−2.04708
$118$	0	0
$119$	−41.0387	−0.0316135
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1852.16	1.35776
$124$	0	0
$125$	−490.987	−0.351322
$126$	0	0
$127$	−1451.44	−1.01413	−0.507066	−	0.861907i	$-0.669270\pi$
$127$	−1451.44	−1.01413	−0.507066	+	0.861907i	$0.669270\pi$
$128$	0	0
$129$	−2923.62	−1.99543
$130$	0	0
$131$	2344.03	1.56335	0.781674	−	0.623687i	$-0.214366\pi$
$131$	2344.03	1.56335	0.781674	+	0.623687i	$0.214366\pi$
$132$	0	0
$133$	−2020.82	−1.31750
$134$	0	0
$135$	−6165.23	−3.93051
$136$	0	0
$137$	57.6575	0.0359563	0.0179782	−	0.999838i	$-0.494277\pi$
$137$	57.6575	0.0359563	0.0179782	+	0.999838i	$0.494277\pi$
$138$	0	0
$139$	−1356.63	−0.827824	−0.413912	−	0.910317i	$-0.635838\pi$
$139$	−1356.63	−0.827824	−0.413912	+	0.910317i	$0.635838\pi$
$140$	0	0
$141$	−327.852	−0.195817
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−557.632	−0.319371
$146$	0	0
$147$	516.569	0.289836
$148$	0	0
$149$	2653.93	1.45918	0.729591	−	0.683884i	$-0.239710\pi$
$149$	2653.93	1.45918	0.729591	+	0.683884i	$0.239710\pi$
$150$	0	0
$151$	−2433.76	−1.31163	−0.655816	−	0.754921i	$-0.727675\pi$
$151$	−2433.76	−1.31163	−0.655816	+	0.754921i	$0.727675\pi$
$152$	0	0
$153$	134.699	0.0711748
$154$	0	0
$155$	−2903.39	−1.50455
$156$	0	0
$157$	−1382.84	−0.702948	−0.351474	−	0.936198i	$-0.614319\pi$
$157$	−1382.84	−0.702948	−0.351474	+	0.936198i	$0.614319\pi$
$158$	0	0
$159$	2082.08	1.03849
$160$	0	0
$161$	−2201.96	−1.07788
$162$	0	0
$163$	1902.14	0.914031	0.457016	−	0.889459i	$-0.348918\pi$
$163$	1902.14	0.914031	0.457016	+	0.889459i	$0.348918\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2150.04	−0.996256	−0.498128	−	0.867103i	$-0.665979\pi$
$167$	−2150.04	−0.996256	−0.498128	+	0.867103i	$0.665979\pi$
$168$	0	0
$169$	−626.735	−0.285269
$170$	0	0
$171$	6632.81	2.96622
$172$	0	0
$173$	−528.511	−0.232265	−0.116133	−	0.993234i	$-0.537050\pi$
$173$	−528.511	−0.232265	−0.116133	+	0.993234i	$0.537050\pi$
$174$	0	0
$175$	−3074.91	−1.32824
$176$	0	0
$177$	−154.357	−0.0655492
$178$	0	0
$179$	−3929.46	−1.64079	−0.820395	−	0.571797i	$-0.806246\pi$
$179$	−3929.46	−1.64079	−0.820395	+	0.571797i	$0.806246\pi$
$180$	0	0
$181$	−540.832	−0.222098	−0.111049	−	0.993815i	$-0.535421\pi$
$181$	−540.832	−0.222098	−0.111049	+	0.993815i	$0.535421\pi$
$182$	0	0
$183$	5104.99	2.06214
$184$	0	0
$185$	−4202.67	−1.67020
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−7347.04	−2.82761
$190$	0	0
$191$	−119.463	−0.0452569	−0.0226285	−	0.999744i	$-0.507203\pi$
$191$	−119.463	−0.0452569	−0.0226285	+	0.999744i	$0.507203\pi$
$192$	0	0
$193$	653.656	0.243788	0.121894	−	0.992543i	$-0.461103\pi$
$193$	653.656	0.243788	0.121894	+	0.992543i	$0.461103\pi$
$194$	0	0
$195$	6365.94	2.33782
$196$	0	0
$197$	749.399	0.271028	0.135514	−	0.990775i	$-0.456731\pi$
$197$	749.399	0.271028	0.135514	+	0.990775i	$0.456731\pi$
$198$	0	0
$199$	1836.21	0.654098	0.327049	−	0.945007i	$-0.393946\pi$
$199$	1836.21	0.654098	0.327049	+	0.945007i	$0.393946\pi$
$200$	0	0
$201$	6118.83	2.14721
$202$	0	0
$203$	−664.524	−0.229756
$204$	0	0
$205$	−3221.00	−1.09739
$206$	0	0
$207$	7227.36	2.42675
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4003.19	1.30612	0.653059	−	0.757307i	$-0.273485\pi$
$211$	4003.19	1.30612	0.653059	+	0.757307i	$0.273485\pi$
$212$	0	0
$213$	2698.32	0.868009
$214$	0	0
$215$	5084.31	1.61278
$216$	0	0
$217$	−3459.93	−1.08238
$218$	0	0
$219$	6312.24	1.94768
$220$	0	0
$221$	−81.6438	−0.0248505
$222$	0	0
$223$	3851.36	1.15653	0.578265	−	0.815849i	$-0.303730\pi$
$223$	3851.36	1.15653	0.578265	+	0.815849i	$0.303730\pi$
$224$	0	0
$225$	10092.6	2.99040
$226$	0	0
$227$	−642.350	−0.187816	−0.0939080	−	0.995581i	$-0.529936\pi$
$227$	−642.350	−0.187816	−0.0939080	+	0.995581i	$0.529936\pi$
$228$	0	0
$229$	3751.08	1.08244	0.541220	−	0.840881i	$-0.317963\pi$
$229$	3751.08	1.08244	0.541220	+	0.840881i	$0.317963\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6936.86	1.95042	0.975212	−	0.221274i	$-0.0710215\pi$
$233$	6936.86	1.95042	0.975212	+	0.221274i	$0.0710215\pi$
$234$	0	0
$235$	570.150	0.158266
$236$	0	0
$237$	−7746.86	−2.12326
$238$	0	0
$239$	−553.889	−0.149908	−0.0749542	−	0.997187i	$-0.523881\pi$
$239$	−553.889	−0.149908	−0.0749542	+	0.997187i	$0.523881\pi$
$240$	0	0
$241$	−2532.58	−0.676921	−0.338460	−	0.940981i	$-0.609906\pi$
$241$	−2532.58	−0.676921	−0.338460	+	0.940981i	$0.609906\pi$
$242$	0	0
$243$	7148.99	1.88728
$244$	0	0
$245$	−898.337	−0.234256
$246$	0	0
$247$	−4020.29	−1.03565
$248$	0	0
$249$	−6135.54	−1.56154
$250$	0	0
$251$	−3185.13	−0.800970	−0.400485	−	0.916303i	$-0.631158\pi$
$251$	−3185.13	−0.800970	−0.400485	+	0.916303i	$0.631158\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−330.989	−0.0812836
$256$	0	0
$257$	−5292.23	−1.28451	−0.642257	−	0.766489i	$-0.722002\pi$
$257$	−5292.23	−1.28451	−0.642257	+	0.766489i	$0.722002\pi$
$258$	0	0
$259$	−5008.27	−1.20154
$260$	0	0
$261$	2181.13	0.517273
$262$	0	0
$263$	4256.45	0.997962	0.498981	−	0.866613i	$-0.333708\pi$
$263$	4256.45	0.997962	0.498981	+	0.866613i	$0.333708\pi$
$264$	0	0
$265$	−3620.83	−0.839342
$266$	0	0
$267$	11001.9	2.52174
$268$	0	0
$269$	−8671.69	−1.96551	−0.982755	−	0.184913i	$-0.940800\pi$
$269$	−8671.69	−1.96551	−0.982755	+	0.184913i	$0.940800\pi$
$270$	0	0
$271$	2987.64	0.669691	0.334845	−	0.942273i	$-0.391316\pi$
$271$	2987.64	0.669691	0.334845	+	0.942273i	$0.391316\pi$
$272$	0	0
$273$	7586.21	1.68183
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7898.35	1.71323	0.856617	−	0.515953i	$-0.172562\pi$
$277$	7898.35	1.71323	0.856617	+	0.515953i	$0.172562\pi$
$278$	0	0
$279$	11356.3	2.43686
$280$	0	0
$281$	−5512.64	−1.17031	−0.585154	−	0.810922i	$-0.698966\pi$
$281$	−5512.64	−1.17031	−0.585154	+	0.810922i	$0.698966\pi$
$282$	0	0
$283$	−3273.50	−0.687595	−0.343797	−	0.939044i	$-0.611713\pi$
$283$	−3273.50	−0.687595	−0.343797	+	0.939044i	$0.611713\pi$
$284$	0	0
$285$	−16298.5	−3.38750
$286$	0	0
$287$	−3838.43	−0.789461
$288$	0	0
$289$	−4908.76	−0.999136
$290$	0	0
$291$	8075.52	1.62679
$292$	0	0
$293$	−1496.14	−0.298313	−0.149156	−	0.988814i	$-0.547656\pi$
$293$	−1496.14	−0.298313	−0.149156	+	0.988814i	$0.547656\pi$
$294$	0	0
$295$	268.435	0.0529792
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4380.66	−0.847292
$300$	0	0
$301$	6058.91	1.16023
$302$	0	0
$303$	−12811.0	−2.42895
$304$	0	0
$305$	−8877.82	−1.66670
$306$	0	0
$307$	4758.05	0.884548	0.442274	−	0.896880i	$-0.354172\pi$
$307$	4758.05	0.884548	0.442274	+	0.896880i	$0.354172\pi$
$308$	0	0
$309$	913.251	0.168133
$310$	0	0
$311$	266.930	0.0486696	0.0243348	−	0.999704i	$-0.492253\pi$
$311$	266.930	0.0486696	0.0243348	+	0.999704i	$0.492253\pi$
$312$	0	0
$313$	8946.60	1.61563	0.807814	−	0.589437i	$-0.200651\pi$
$313$	8946.60	1.61563	0.807814	+	0.589437i	$0.200651\pi$
$314$	0	0
$315$	21765.9	3.89323
$316$	0	0
$317$	−259.266	−0.0459363	−0.0229681	−	0.999736i	$-0.507312\pi$
$317$	−259.266	−0.0459363	−0.0229681	+	0.999736i	$0.507312\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	13861.0	2.41012
$322$	0	0
$323$	209.030	0.0360084
$324$	0	0
$325$	−6117.35	−1.04409
$326$	0	0
$327$	6813.19	1.15220
$328$	0	0
$329$	679.441	0.113857
$330$	0	0
$331$	−4557.71	−0.756841	−0.378421	−	0.925634i	$-0.623533\pi$
$331$	−4557.71	−0.756841	−0.378421	+	0.925634i	$0.623533\pi$
$332$	0	0
$333$	16438.3	2.70515
$334$	0	0
$335$	−10640.9	−1.73545
$336$	0	0
$337$	9720.57	1.57126	0.785628	−	0.618699i	$-0.212340\pi$
$337$	9720.57	1.57126	0.785628	+	0.618699i	$0.212340\pi$
$338$	0	0
$339$	−16061.7	−2.57331
$340$	0	0
$341$	0	0
$342$	0	0
$343$	5761.50	0.906973
$344$	0	0
$345$	−17759.5	−2.77141
$346$	0	0
$347$	5210.07	0.806027	0.403013	−	0.915194i	$-0.367963\pi$
$347$	5210.07	0.806027	0.403013	+	0.915194i	$0.367963\pi$
$348$	0	0
$349$	570.087	0.0874385	0.0437193	−	0.999044i	$-0.486079\pi$
$349$	570.087	0.0874385	0.0437193	+	0.999044i	$0.486079\pi$
$350$	0	0
$351$	−14616.5	−2.22270
$352$	0	0
$353$	2751.02	0.414793	0.207396	−	0.978257i	$-0.433501\pi$
$353$	2751.02	0.414793	0.207396	+	0.978257i	$0.433501\pi$
$354$	0	0
$355$	−4692.51	−0.701556
$356$	0	0
$357$	−394.435	−0.0584754
$358$	0	0
$359$	−4439.60	−0.652683	−0.326341	−	0.945252i	$-0.605816\pi$
$359$	−4439.60	−0.652683	−0.326341	+	0.945252i	$0.605816\pi$
$360$	0	0
$361$	3434.00	0.500656
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10977.3	−1.57418
$366$	0	0
$367$	−9325.64	−1.32641	−0.663207	−	0.748436i	$-0.730805\pi$
$367$	−9325.64	−1.32641	−0.663207	+	0.748436i	$0.730805\pi$
$368$	0	0
$369$	12598.6	1.77739
$370$	0	0
$371$	−4314.90	−0.603823
$372$	0	0
$373$	8713.65	1.20959	0.604793	−	0.796382i	$-0.293256\pi$
$373$	8713.65	1.20959	0.604793	+	0.796382i	$0.293256\pi$
$374$	0	0
$375$	−4719.02	−0.649838
$376$	0	0
$377$	−1322.03	−0.180605
$378$	0	0
$379$	−2121.30	−0.287503	−0.143752	−	0.989614i	$-0.545917\pi$
$379$	−2121.30	−0.287503	−0.143752	+	0.989614i	$0.545917\pi$
$380$	0	0
$381$	−13950.3	−1.87584
$382$	0	0
$383$	7441.24	0.992766	0.496383	−	0.868104i	$-0.334661\pi$
$383$	7441.24	0.992766	0.496383	+	0.868104i	$0.334661\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−19886.8	−2.61215
$388$	0	0
$389$	−5713.37	−0.744676	−0.372338	−	0.928097i	$-0.621444\pi$
$389$	−5713.37	−0.744676	−0.372338	+	0.928097i	$0.621444\pi$
$390$	0	0
$391$	227.767	0.0294595
$392$	0	0
$393$	22529.2	2.89172
$394$	0	0
$395$	13472.2	1.71610
$396$	0	0
$397$	−7568.62	−0.956821	−0.478411	−	0.878136i	$-0.658787\pi$
$397$	−7568.62	−0.956821	−0.478411	+	0.878136i	$0.658787\pi$
$398$	0	0
$399$	−19422.7	−2.43697
$400$	0	0
$401$	−12593.9	−1.56835	−0.784175	−	0.620539i	$-0.786914\pi$
$401$	−12593.9	−1.56835	−0.784175	+	0.620539i	$0.786914\pi$
$402$	0	0
$403$	−6883.32	−0.850825
$404$	0	0
$405$	−29751.8	−3.65031
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−9853.39	−1.19124	−0.595622	−	0.803265i	$-0.703094\pi$
$409$	−9853.39	−1.19124	−0.595622	+	0.803265i	$0.703094\pi$
$410$	0	0
$411$	554.164	0.0665083
$412$	0	0
$413$	319.890	0.0381133
$414$	0	0
$415$	10670.0	1.26209
$416$	0	0
$417$	−13038.9	−1.53122
$418$	0	0
$419$	7338.49	0.855629	0.427814	−	0.903867i	$-0.359284\pi$
$419$	7338.49	0.855629	0.427814	+	0.903867i	$0.359284\pi$
$420$	0	0
$421$	15493.2	1.79357	0.896786	−	0.442464i	$-0.145896\pi$
$421$	15493.2	1.79357	0.896786	+	0.442464i	$0.145896\pi$
$422$	0	0
$423$	−2230.09	−0.256337
$424$	0	0
$425$	318.063	0.0363020
$426$	0	0
$427$	−10579.6	−1.19902
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7163.86	0.800629	0.400314	−	0.916378i	$-0.368901\pi$
$431$	7163.86	0.800629	0.400314	+	0.916378i	$0.368901\pi$
$432$	0	0
$433$	3459.53	0.383959	0.191979	−	0.981399i	$-0.438509\pi$
$433$	3459.53	0.383959	0.191979	+	0.981399i	$0.438509\pi$
$434$	0	0
$435$	−5359.57	−0.590740
$436$	0	0
$437$	11215.7	1.22773
$438$	0	0
$439$	10534.8	1.14533	0.572665	−	0.819790i	$-0.305910\pi$
$439$	10534.8	1.14533	0.572665	+	0.819790i	$0.305910\pi$
$440$	0	0
$441$	3513.76	0.379415
$442$	0	0
$443$	−6808.43	−0.730200	−0.365100	−	0.930968i	$-0.618965\pi$
$443$	−6808.43	−0.730200	−0.365100	+	0.930968i	$0.618965\pi$
$444$	0	0
$445$	−19132.8	−2.03816
$446$	0	0
$447$	25507.7	2.69904
$448$	0	0
$449$	−1591.63	−0.167291	−0.0836454	−	0.996496i	$-0.526656\pi$
$449$	−1591.63	−0.167291	−0.0836454	+	0.996496i	$0.526656\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−23391.6	−2.42612
$454$	0	0
$455$	−13192.8	−1.35931
$456$	0	0
$457$	9738.82	0.996855	0.498428	−	0.866931i	$-0.333911\pi$
$457$	9738.82	0.996855	0.498428	+	0.866931i	$0.333911\pi$
$458$	0	0
$459$	759.964	0.0772812
$460$	0	0
$461$	−16872.1	−1.70458	−0.852289	−	0.523071i	$-0.824786\pi$
$461$	−16872.1	−1.70458	−0.852289	+	0.523071i	$0.824786\pi$
$462$	0	0
$463$	19730.3	1.98044	0.990221	−	0.139508i	$-0.0445522\pi$
$463$	19730.3	1.98044	0.990221	+	0.139508i	$0.0445522\pi$
$464$	0	0
$465$	−27905.3	−2.78297
$466$	0	0
$467$	5845.13	0.579187	0.289593	−	0.957150i	$-0.406480\pi$
$467$	5845.13	0.579187	0.289593	+	0.957150i	$0.406480\pi$
$468$	0	0
$469$	−12680.7	−1.24848
$470$	0	0
$471$	−13290.9	−1.30024
$472$	0	0
$473$	0	0
$474$	0	0
$475$	15662.0	1.51289
$476$	0	0
$477$	14162.5	1.35945
$478$	0	0
$479$	18130.9	1.72948	0.864741	−	0.502218i	$-0.167483\pi$
$479$	18130.9	1.72948	0.864741	+	0.502218i	$0.167483\pi$
$480$	0	0
$481$	−9963.63	−0.944496
$482$	0	0
$483$	−21163.7	−1.99375
$484$	0	0
$485$	−14043.7	−1.31483
$486$	0	0
$487$	3459.69	0.321917	0.160959	−	0.986961i	$-0.448541\pi$
$487$	3459.69	0.321917	0.160959	+	0.986961i	$0.448541\pi$
$488$	0	0
$489$	18282.1	1.69068
$490$	0	0
$491$	−1931.83	−0.177561	−0.0887803	−	0.996051i	$-0.528297\pi$
$491$	−1931.83	−0.177561	−0.0887803	+	0.996051i	$0.528297\pi$
$492$	0	0
$493$	68.7371	0.00627944
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5592.00	−0.504699
$498$	0	0
$499$	−11715.0	−1.05097	−0.525487	−	0.850802i	$-0.676117\pi$
$499$	−11715.0	−1.05097	−0.525487	+	0.850802i	$0.676117\pi$
$500$	0	0
$501$	−20664.7	−1.84277
$502$	0	0
$503$	3327.71	0.294980	0.147490	−	0.989064i	$-0.452881\pi$
$503$	3327.71	0.294980	0.147490	+	0.989064i	$0.452881\pi$
$504$	0	0
$505$	22278.9	1.96316
$506$	0	0
$507$	−6023.75	−0.527661
$508$	0	0
$509$	−11954.6	−1.04102	−0.520511	−	0.853855i	$-0.674258\pi$
$509$	−11954.6	−1.04102	−0.520511	+	0.853855i	$0.674258\pi$
$510$	0	0
$511$	−13081.5	−1.13247
$512$	0	0
$513$	37422.0	3.22070
$514$	0	0
$515$	−1588.19	−0.135891
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−5079.68	−0.429621
$520$	0	0
$521$	−11503.0	−0.967283	−0.483642	−	0.875266i	$-0.660686\pi$
$521$	−11503.0	−0.967283	−0.483642	+	0.875266i	$0.660686\pi$
$522$	0	0
$523$	−14663.6	−1.22599	−0.612997	−	0.790085i	$-0.710036\pi$
$523$	−14663.6	−1.22599	−0.612997	+	0.790085i	$0.710036\pi$
$524$	0	0
$525$	−29553.9	−2.45684
$526$	0	0
$527$	357.889	0.0295823
$528$	0	0
$529$	54.0095	0.00443902
$530$	0	0
$531$	−1049.96	−0.0858084
$532$	0	0
$533$	−7636.31	−0.620573
$534$	0	0
$535$	−24105.0	−1.94794
$536$	0	0
$537$	−37767.2	−3.03496
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3033.26	−0.241053	−0.120527	−	0.992710i	$-0.538458\pi$
$541$	−3033.26	−0.241053	−0.120527	+	0.992710i	$0.538458\pi$
$542$	0	0
$543$	−5198.10	−0.410814
$544$	0	0
$545$	−11848.4	−0.931251
$546$	0	0
$547$	−5896.30	−0.460892	−0.230446	−	0.973085i	$-0.574018\pi$
$547$	−5896.30	−0.460892	−0.230446	+	0.973085i	$0.574018\pi$
$548$	0	0
$549$	34724.7	2.69948
$550$	0	0
$551$	3384.74	0.261696
$552$	0	0
$553$	16054.6	1.23456
$554$	0	0
$555$	−40393.1	−3.08936
$556$	0	0
$557$	−1312.52	−0.0998445	−0.0499222	−	0.998753i	$-0.515897\pi$
$557$	−1312.52	−0.0998445	−0.0499222	+	0.998753i	$0.515897\pi$
$558$	0	0
$559$	12053.8	0.912026
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11802.4	0.883502	0.441751	−	0.897138i	$-0.354358\pi$
$563$	11802.4	0.883502	0.441751	+	0.897138i	$0.354358\pi$
$564$	0	0
$565$	27932.0	2.07984
$566$	0	0
$567$	−35454.8	−2.62604
$568$	0	0
$569$	−15403.3	−1.13487	−0.567433	−	0.823420i	$-0.692063\pi$
$569$	−15403.3	−1.13487	−0.567433	+	0.823420i	$0.692063\pi$
$570$	0	0
$571$	6979.75	0.511547	0.255774	−	0.966737i	$-0.417670\pi$
$571$	6979.75	0.511547	0.255774	+	0.966737i	$0.417670\pi$
$572$	0	0
$573$	−1148.20	−0.0837116
$574$	0	0
$575$	17065.9	1.23774
$576$	0	0
$577$	−12955.9	−0.934770	−0.467385	−	0.884054i	$-0.654804\pi$
$577$	−12955.9	−0.934770	−0.467385	+	0.884054i	$0.654804\pi$
$578$	0	0
$579$	6282.49	0.450935
$580$	0	0
$581$	12715.3	0.907951
$582$	0	0
$583$	0	0
$584$	0	0
$585$	43301.9	3.06036
$586$	0	0
$587$	6028.00	0.423854	0.211927	−	0.977285i	$-0.432026\pi$
$587$	6028.00	0.423854	0.211927	+	0.977285i	$0.432026\pi$
$588$	0	0
$589$	17623.1	1.23285
$590$	0	0
$591$	7202.70	0.501319
$592$	0	0
$593$	−9230.18	−0.639187	−0.319594	−	0.947555i	$-0.603546\pi$
$593$	−9230.18	−0.639187	−0.319594	+	0.947555i	$0.603546\pi$
$594$	0	0
$595$	685.941	0.0472619
$596$	0	0
$597$	17648.4	1.20988
$598$	0	0
$599$	8942.54	0.609987	0.304994	−	0.952354i	$-0.401346\pi$
$599$	8942.54	0.609987	0.304994	+	0.952354i	$0.401346\pi$
$600$	0	0
$601$	−8034.20	−0.545295	−0.272647	−	0.962114i	$-0.587899\pi$
$601$	−8034.20	−0.545295	−0.272647	+	0.962114i	$0.587899\pi$
$602$	0	0
$603$	41621.0	2.81084
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2525.95	−0.168905	−0.0844525	−	0.996428i	$-0.526914\pi$
$607$	−2525.95	−0.168905	−0.0844525	+	0.996428i	$0.526914\pi$
$608$	0	0
$609$	−6386.94	−0.424978
$610$	0	0
$611$	1351.71	0.0894995
$612$	0	0
$613$	8748.86	0.576449	0.288224	−	0.957563i	$-0.406935\pi$
$613$	8748.86	0.576449	0.288224	+	0.957563i	$0.406935\pi$
$614$	0	0
$615$	−30958.0	−2.02983
$616$	0	0
$617$	620.479	0.0404855	0.0202427	−	0.999795i	$-0.493556\pi$
$617$	620.479	0.0404855	0.0202427	+	0.999795i	$0.493556\pi$
$618$	0	0
$619$	8314.19	0.539863	0.269932	−	0.962879i	$-0.412999\pi$
$619$	8314.19	0.539863	0.269932	+	0.962879i	$0.412999\pi$
$620$	0	0
$621$	40776.5	2.63495
$622$	0	0
$623$	−22800.3	−1.46625
$624$	0	0
$625$	−11090.3	−0.709776
$626$	0	0
$627$	0	0
$628$	0	0
$629$	518.046	0.0328392
$630$	0	0
$631$	−8321.15	−0.524976	−0.262488	−	0.964935i	$-0.584543\pi$
$631$	−8321.15	−0.524976	−0.262488	+	0.964935i	$0.584543\pi$
$632$	0	0
$633$	38475.9	2.41592
$634$	0	0
$635$	24260.2	1.51612
$636$	0	0
$637$	−2129.77	−0.132472
$638$	0	0
$639$	18354.3	1.13628
$640$	0	0
$641$	4827.58	0.297470	0.148735	−	0.988877i	$-0.452480\pi$
$641$	4827.58	0.297470	0.148735	+	0.988877i	$0.452480\pi$
$642$	0	0
$643$	21311.4	1.30706	0.653529	−	0.756902i	$-0.273288\pi$
$643$	21311.4	1.30706	0.653529	+	0.756902i	$0.273288\pi$
$644$	0	0
$645$	48866.9	2.98315
$646$	0	0
$647$	−5430.24	−0.329961	−0.164980	−	0.986297i	$-0.552756\pi$
$647$	−5430.24	−0.329961	−0.164980	+	0.986297i	$0.552756\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−33254.5	−2.00207
$652$	0	0
$653$	1410.00	0.0844986	0.0422493	−	0.999107i	$-0.486548\pi$
$653$	1410.00	0.0844986	0.0422493	+	0.999107i	$0.486548\pi$
$654$	0	0
$655$	−39179.3	−2.33719
$656$	0	0
$657$	42936.6	2.54964
$658$	0	0
$659$	−18088.9	−1.06926	−0.534630	−	0.845086i	$-0.679549\pi$
$659$	−18088.9	−1.06926	−0.534630	+	0.845086i	$0.679549\pi$
$660$	0	0
$661$	16624.5	0.978240	0.489120	−	0.872217i	$-0.337318\pi$
$661$	16624.5	0.978240	0.489120	+	0.872217i	$0.337318\pi$
$662$	0	0
$663$	−784.704	−0.0459659
$664$	0	0
$665$	33777.0	1.96965
$666$	0	0
$667$	3688.14	0.214101
$668$	0	0
$669$	37016.6	2.13923
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−461.551	−0.0264361	−0.0132180	−	0.999913i	$-0.504208\pi$
$673$	−461.551	−0.0264361	−0.0132180	+	0.999913i	$0.504208\pi$
$674$	0	0
$675$	56942.0	3.24696
$676$	0	0
$677$	−15908.5	−0.903120	−0.451560	−	0.892241i	$-0.649132\pi$
$677$	−15908.5	−0.903120	−0.451560	+	0.892241i	$0.649132\pi$
$678$	0	0
$679$	−16735.7	−0.945888
$680$	0	0
$681$	−6173.82	−0.347403
$682$	0	0
$683$	−32846.5	−1.84017	−0.920084	−	0.391721i	$-0.871880\pi$
$683$	−32846.5	−1.84017	−0.920084	+	0.391721i	$0.871880\pi$
$684$	0	0
$685$	−963.718	−0.0537544
$686$	0	0
$687$	36052.8	2.00218
$688$	0	0
$689$	−8584.21	−0.474648
$690$	0	0
$691$	1935.06	0.106531	0.0532656	−	0.998580i	$-0.483037\pi$
$691$	1935.06	0.106531	0.0532656	+	0.998580i	$0.483037\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22675.3	1.23759
$696$	0	0
$697$	397.040	0.0215767
$698$	0	0
$699$	66672.3	3.60769
$700$	0	0
$701$	5726.03	0.308515	0.154258	−	0.988031i	$-0.450701\pi$
$701$	5726.03	0.308515	0.154258	+	0.988031i	$0.450701\pi$
$702$	0	0
$703$	25509.5	1.36858
$704$	0	0
$705$	5479.89	0.292744
$706$	0	0
$707$	26549.4	1.41230
$708$	0	0
$709$	−27165.0	−1.43893	−0.719467	−	0.694526i	$-0.755614\pi$
$709$	−27165.0	−1.43893	−0.719467	+	0.694526i	$0.755614\pi$
$710$	0	0
$711$	−52695.1	−2.77949
$712$	0	0
$713$	19202.8	1.00863
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5323.60	−0.277285
$718$	0	0
$719$	−12480.0	−0.647326	−0.323663	−	0.946173i	$-0.604914\pi$
$719$	−12480.0	−0.647326	−0.323663	+	0.946173i	$0.604914\pi$
$720$	0	0
$721$	−1892.62	−0.0977599
$722$	0	0
$723$	−24341.4	−1.25210
$724$	0	0
$725$	5150.28	0.263830
$726$	0	0
$727$	4894.88	0.249712	0.124856	−	0.992175i	$-0.460153\pi$
$727$	4894.88	0.249712	0.124856	+	0.992175i	$0.460153\pi$
$728$	0	0
$729$	20651.3	1.04919
$730$	0	0
$731$	−626.722	−0.0317102
$732$	0	0
$733$	27399.7	1.38067	0.690334	−	0.723491i	$-0.257464\pi$
$733$	27399.7	1.38067	0.690334	+	0.723491i	$0.257464\pi$
$734$	0	0
$735$	−8634.19	−0.433302
$736$	0	0
$737$	0	0
$738$	0	0
$739$	14868.0	0.740093	0.370046	−	0.929013i	$-0.379342\pi$
$739$	14868.0	0.740093	0.370046	+	0.929013i	$0.379342\pi$
$740$	0	0
$741$	−38640.2	−1.91563
$742$	0	0
$743$	−32285.5	−1.59413	−0.797065	−	0.603893i	$-0.793615\pi$
$743$	−32285.5	−1.59413	−0.797065	+	0.603893i	$0.793615\pi$
$744$	0	0
$745$	−44359.1	−2.18146
$746$	0	0
$747$	−41734.6	−2.04416
$748$	0	0
$749$	−28725.6	−1.40135
$750$	0	0
$751$	14620.6	0.710406	0.355203	−	0.934789i	$-0.384412\pi$
$751$	14620.6	0.710406	0.355203	+	0.934789i	$0.384412\pi$
$752$	0	0
$753$	−30613.2	−1.48155
$754$	0	0
$755$	40679.1	1.96088
$756$	0	0
$757$	−6341.08	−0.304452	−0.152226	−	0.988346i	$-0.548644\pi$
$757$	−6341.08	−0.304452	−0.152226	+	0.988346i	$0.548644\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7962.76	−0.379303	−0.189652	−	0.981851i	$-0.560736\pi$
$761$	−7962.76	−0.379303	−0.189652	+	0.981851i	$0.560736\pi$
$762$	0	0
$763$	−14119.7	−0.669942
$764$	0	0
$765$	−2251.42	−0.106406
$766$	0	0
$767$	636.402	0.0299598
$768$	0	0
$769$	−19302.7	−0.905166	−0.452583	−	0.891722i	$-0.649497\pi$
$769$	−19302.7	−0.905166	−0.452583	+	0.891722i	$0.649497\pi$
$770$	0	0
$771$	−50865.3	−2.37596
$772$	0	0
$773$	−38799.0	−1.80531	−0.902654	−	0.430367i	$-0.858384\pi$
$773$	−38799.0	−1.80531	−0.902654	+	0.430367i	$0.858384\pi$
$774$	0	0
$775$	26815.6	1.24290
$776$	0	0
$777$	−48136.0	−2.22248
$778$	0	0
$779$	19551.0	0.899211
$780$	0	0
$781$	0	0
$782$	0	0
$783$	12305.8	0.561652
$784$	0	0
$785$	23113.5	1.05090
$786$	0	0
$787$	21242.9	0.962172	0.481086	−	0.876674i	$-0.340243\pi$
$787$	21242.9	0.962172	0.481086	+	0.876674i	$0.340243\pi$
$788$	0	0
$789$	40910.1	1.84593
$790$	0	0
$791$	33286.3	1.49624
$792$	0	0
$793$	−21047.4	−0.942517
$794$	0	0
$795$	−34800.9	−1.55253
$796$	0	0
$797$	−12121.6	−0.538732	−0.269366	−	0.963038i	$-0.586814\pi$
$797$	−12121.6	−0.538732	−0.269366	+	0.963038i	$0.586814\pi$
$798$	0	0
$799$	−70.2801	−0.00311181
$800$	0	0
$801$	74836.1	3.30113
$802$	0	0
$803$	0	0
$804$	0	0
$805$	36804.7	1.61142
$806$	0	0
$807$	−83346.3	−3.63560
$808$	0	0
$809$	−20996.0	−0.912460	−0.456230	−	0.889862i	$-0.650801\pi$
$809$	−20996.0	−0.912460	−0.456230	+	0.889862i	$0.650801\pi$
$810$	0	0
$811$	−11682.2	−0.505817	−0.252909	−	0.967490i	$-0.581387\pi$
$811$	−11682.2	−0.505817	−0.252909	+	0.967490i	$0.581387\pi$
$812$	0	0
$813$	28715.1	1.23872
$814$	0	0
$815$	−31793.3	−1.36647
$816$	0	0
$817$	−30860.9	−1.32153
$818$	0	0
$819$	51602.3	2.20162
$820$	0	0
$821$	−1982.09	−0.0842577	−0.0421288	−	0.999112i	$-0.513414\pi$
$821$	−1982.09	−0.0842577	−0.0421288	+	0.999112i	$0.513414\pi$
$822$	0	0
$823$	−23610.2	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	−23610.2	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28046.3	1.17928	0.589641	−	0.807665i	$-0.299269\pi$
$827$	28046.3	1.17928	0.589641	+	0.807665i	$0.299269\pi$
$828$	0	0
$829$	36602.2	1.53347	0.766735	−	0.641963i	$-0.221880\pi$
$829$	36602.2	1.53347	0.766735	+	0.641963i	$0.221880\pi$
$830$	0	0
$831$	75913.4	3.16896
$832$	0	0
$833$	110.734	0.00460591
$834$	0	0
$835$	35936.8	1.48939
$836$	0	0
$837$	64071.8	2.64593
$838$	0	0
$839$	−6771.55	−0.278641	−0.139321	−	0.990247i	$-0.544492\pi$
$839$	−6771.55	−0.278641	−0.139321	+	0.990247i	$0.544492\pi$
$840$	0	0
$841$	−23276.0	−0.954363
$842$	0	0
$843$	−52983.6	−2.16471
$844$	0	0
$845$	10475.6	0.426474
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−31462.6	−1.27184
$850$	0	0
$851$	27796.2	1.11967
$852$	0	0
$853$	17822.2	0.715382	0.357691	−	0.933840i	$-0.383564\pi$
$853$	17822.2	0.715382	0.357691	+	0.933840i	$0.383564\pi$
$854$	0	0
$855$	−110864.	−4.43447
$856$	0	0
$857$	−28270.3	−1.12683	−0.563416	−	0.826173i	$-0.690513\pi$
$857$	−28270.3	−1.12683	−0.563416	+	0.826173i	$0.690513\pi$
$858$	0	0
$859$	26699.7	1.06051	0.530257	−	0.847837i	$-0.322095\pi$
$859$	26699.7	1.06051	0.530257	+	0.847837i	$0.322095\pi$
$860$	0	0
$861$	−36892.3	−1.46026
$862$	0	0
$863$	13281.5	0.523877	0.261939	−	0.965085i	$-0.415638\pi$
$863$	13281.5	0.523877	0.261939	+	0.965085i	$0.415638\pi$
$864$	0	0
$865$	8833.79	0.347235
$866$	0	0
$867$	−47179.6	−1.84810
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−25227.4	−0.981398
$872$	0	0
$873$	54930.6	2.12958
$874$	0	0
$875$	9779.71	0.377845
$876$	0	0
$877$	14407.2	0.554729	0.277364	−	0.960765i	$-0.410539\pi$
$877$	14407.2	0.554729	0.277364	+	0.960765i	$0.410539\pi$
$878$	0	0
$879$	−14379.9	−0.551788
$880$	0	0
$881$	40967.4	1.56666	0.783330	−	0.621606i	$-0.213519\pi$
$881$	40967.4	1.56666	0.783330	+	0.621606i	$0.213519\pi$
$882$	0	0
$883$	−11350.8	−0.432599	−0.216299	−	0.976327i	$-0.569399\pi$
$883$	−11350.8	−0.432599	−0.216299	+	0.976327i	$0.569399\pi$
$884$	0	0
$885$	2580.01	0.0979955
$886$	0	0
$887$	24213.4	0.916580	0.458290	−	0.888803i	$-0.348462\pi$
$887$	24213.4	0.916580	0.458290	+	0.888803i	$0.348462\pi$
$888$	0	0
$889$	28910.6	1.09070
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3460.72	−0.129685
$894$	0	0
$895$	65679.0	2.45297
$896$	0	0
$897$	−42103.9	−1.56723
$898$	0	0
$899$	5795.16	0.214994
$900$	0	0
$901$	446.325	0.0165030
$902$	0	0
$903$	58234.1	2.14608
$904$	0	0
$905$	9039.74	0.332035
$906$	0	0
$907$	−45535.0	−1.66699	−0.833497	−	0.552523i	$-0.813665\pi$
$907$	−45535.0	−1.66699	−0.833497	+	0.552523i	$0.813665\pi$
$908$	0	0
$909$	−87141.6	−3.17965
$910$	0	0
$911$	−7789.66	−0.283296	−0.141648	−	0.989917i	$-0.545240\pi$
$911$	−7789.66	−0.283296	−0.141648	+	0.989917i	$0.545240\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−85327.4	−3.08288
$916$	0	0
$917$	−46689.5	−1.68138
$918$	0	0
$919$	−19976.7	−0.717053	−0.358527	−	0.933520i	$-0.616721\pi$
$919$	−19976.7	−0.717053	−0.358527	+	0.933520i	$0.616721\pi$
$920$	0	0
$921$	45731.0	1.63614
$922$	0	0
$923$	−11124.9	−0.396730
$924$	0	0
$925$	38815.7	1.37973
$926$	0	0
$927$	6212.04	0.220097
$928$	0	0
$929$	47133.0	1.66457	0.832284	−	0.554349i	$-0.187033\pi$
$929$	47133.0	1.66457	0.832284	+	0.554349i	$0.187033\pi$
$930$	0	0
$931$	5452.76	0.191952
$932$	0	0
$933$	2565.55	0.0900240
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−47606.2	−1.65980	−0.829898	−	0.557916i	$-0.811601\pi$
$937$	−47606.2	−1.65980	−0.829898	+	0.557916i	$0.811601\pi$
$938$	0	0
$939$	85988.5	2.98842
$940$	0	0
$941$	27553.9	0.954550	0.477275	−	0.878754i	$-0.341625\pi$
$941$	27553.9	0.954550	0.477275	+	0.878754i	$0.341625\pi$
$942$	0	0
$943$	21303.5	0.735670
$944$	0	0
$945$	122802.	4.22725
$946$	0	0
$947$	−8056.38	−0.276449	−0.138224	−	0.990401i	$-0.544140\pi$
$947$	−8056.38	−0.276449	−0.138224	+	0.990401i	$0.544140\pi$
$948$	0	0
$949$	−26024.8	−0.890202
$950$	0	0
$951$	−2491.88	−0.0849682
$952$	0	0
$953$	−10193.3	−0.346479	−0.173239	−	0.984880i	$-0.555423\pi$
$953$	−10193.3	−0.346479	−0.173239	+	0.984880i	$0.555423\pi$
$954$	0	0
$955$	1996.77	0.0676587
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1148.45	−0.0386709
$960$	0	0
$961$	382.285	0.0128322
$962$	0	0
$963$	94284.4	3.15501
$964$	0	0
$965$	−10925.5	−0.364462
$966$	0	0
$967$	13913.2	0.462685	0.231343	−	0.972872i	$-0.425688\pi$
$967$	13913.2	0.462685	0.231343	+	0.972872i	$0.425688\pi$
$968$	0	0
$969$	2009.05	0.0666047
$970$	0	0
$971$	30246.6	0.999650	0.499825	−	0.866126i	$-0.333398\pi$
$971$	30246.6	0.999650	0.499825	+	0.866126i	$0.333398\pi$
$972$	0	0
$973$	27021.9	0.890322
$974$	0	0
$975$	−58795.7	−1.93125
$976$	0	0
$977$	−24322.6	−0.796467	−0.398234	−	0.917284i	$-0.630377\pi$
$977$	−24322.6	−0.796467	−0.398234	+	0.917284i	$0.630377\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	46344.1	1.50831
$982$	0	0
$983$	−16434.8	−0.533254	−0.266627	−	0.963800i	$-0.585909\pi$
$983$	−16434.8	−0.533254	−0.266627	+	0.963800i	$0.585909\pi$
$984$	0	0
$985$	−12525.8	−0.405184
$986$	0	0
$987$	6530.32	0.210600
$988$	0	0
$989$	−33627.3	−1.08118
$990$	0	0
$991$	−25604.7	−0.820745	−0.410373	−	0.911918i	$-0.634601\pi$
$991$	−25604.7	−0.820745	−0.410373	+	0.911918i	$0.634601\pi$
$992$	0	0
$993$	−43805.6	−1.39993
$994$	0	0
$995$	−30691.4	−0.977871
$996$	0	0
$997$	38070.2	1.20932	0.604661	−	0.796483i	$-0.293309\pi$
$997$	38070.2	1.20932	0.604661	+	0.796483i	$0.293309\pi$
$998$	0	0
$999$	92744.3	2.93724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.r.1.9		10
4.3	odd	2		1936.4.a.by.1.2		10
11.2	odd	10		88.4.i.b.81.5	yes	20
11.6	odd	10		88.4.i.b.25.5	✓	20
11.10	odd	2		968.4.a.s.1.9		10
44.35	even	10		176.4.m.f.81.1		20
44.39	even	10		176.4.m.f.113.1		20
44.43	even	2		1936.4.a.bx.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.b.25.5	✓	20	11.6	odd	10
88.4.i.b.81.5	yes	20	11.2	odd	10
176.4.m.f.81.1		20	44.35	even	10
176.4.m.f.113.1		20	44.39	even	10
968.4.a.r.1.9		10	1.1	even	1	trivial
968.4.a.s.1.9		10	11.10	odd	2
1936.4.a.bx.1.2		10	44.43	even	2
1936.4.a.by.1.2		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)

\(f(q)\)	\(=\)	\(q+9.61131 q^{3} -16.7145 q^{5} -19.9185 q^{7} +65.3772 q^{9} -39.6266 q^{13} -160.648 q^{15} +2.06033 q^{17} +101.454 q^{19} -191.443 q^{21} +110.549 q^{23} +154.375 q^{25} +368.855 q^{27} +33.3622 q^{29} +173.705 q^{31} +332.928 q^{35} +251.438 q^{37} -380.863 q^{39} +192.707 q^{41} -304.185 q^{43} -1092.75 q^{45} -34.1111 q^{47} +53.7459 q^{49} +19.8025 q^{51} +216.628 q^{53} +975.109 q^{57} -16.0600 q^{59} +531.144 q^{61} -1302.21 q^{63} +662.339 q^{65} +636.628 q^{67} +1062.52 q^{69} +280.744 q^{71} +656.752 q^{73} +1483.74 q^{75} -806.016 q^{79} +1780.00 q^{81} -638.367 q^{83} -34.4374 q^{85} +320.654 q^{87} +1144.68 q^{89} +789.301 q^{91} +1669.53 q^{93} -1695.76 q^{95} +840.210 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9} - 45 q^{13} + 120 q^{15} + 17 q^{17} + 147 q^{19} - 131 q^{21} + 164 q^{23} + 439 q^{25} + 420 q^{27} - 177 q^{29} + 275 q^{31} + 220 q^{35} + 745 q^{37}+ \cdots + 3511 q^{97}+O(q^{100}) \) 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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