LMFDB - Embedded newform 968.4.a.r.1.10

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			1.10
Root			$10.8099$ 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)

$f(q)$	$=$	$q+10.1919 q^{3} +14.0462 q^{5} -11.0557 q^{7} +76.8751 q^{9} +O(q^{10})$	$q+10.1919 q^{3} +14.0462 q^{5} -11.0557 q^{7} +76.8751 q^{9} +44.4119 q^{13} +143.158 q^{15} +48.0812 q^{17} +17.2285 q^{19} -112.679 q^{21} +33.1493 q^{23} +72.2965 q^{25} +508.323 q^{27} -266.033 q^{29} -140.335 q^{31} -155.291 q^{35} -15.3852 q^{37} +452.642 q^{39} -422.714 q^{41} +225.105 q^{43} +1079.81 q^{45} +273.342 q^{47} -220.772 q^{49} +490.040 q^{51} -461.725 q^{53} +175.592 q^{57} -26.2624 q^{59} +283.401 q^{61} -849.908 q^{63} +623.820 q^{65} +262.674 q^{67} +337.855 q^{69} +319.830 q^{71} +158.123 q^{73} +736.839 q^{75} -529.657 q^{79} +3105.15 q^{81} +1406.10 q^{83} +675.360 q^{85} -2711.38 q^{87} -71.9709 q^{89} -491.005 q^{91} -1430.28 q^{93} +241.996 q^{95} -726.164 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	10.1919	1.96143	0.980717	−	0.195432i	$-0.0626107\pi$
$3$	10.1919	1.96143	0.980717	+	0.195432i	$0.0626107\pi$
$4$	0	0
$5$	14.0462	1.25633	0.628166	−	0.778079i	$-0.283806\pi$
$5$	14.0462	1.25633	0.628166	+	0.778079i	$0.283806\pi$
$6$	0	0
$7$	−11.0557	−0.596952	−0.298476	−	0.954417i	$-0.596478\pi$
$7$	−11.0557	−0.596952	−0.298476	+	0.954417i	$0.596478\pi$
$8$	0	0
$9$	76.8751	2.84723
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	44.4119	0.947512	0.473756	−	0.880656i	$-0.342898\pi$
$13$	44.4119	0.947512	0.473756	+	0.880656i	$0.342898\pi$
$14$	0	0
$15$	143.158	2.46421
$16$	0	0
$17$	48.0812	0.685966	0.342983	−	0.939342i	$-0.388563\pi$
$17$	48.0812	0.685966	0.342983	+	0.939342i	$0.388563\pi$
$18$	0	0
$19$	17.2285	0.208026	0.104013	−	0.994576i	$-0.466832\pi$
$19$	17.2285	0.208026	0.104013	+	0.994576i	$0.466832\pi$
$20$	0	0
$21$	−112.679	−1.17088
$22$	0	0
$23$	33.1493	0.300526	0.150263	−	0.988646i	$-0.451988\pi$
$23$	33.1493	0.300526	0.150263	+	0.988646i	$0.451988\pi$
$24$	0	0
$25$	72.2965	0.578372
$26$	0	0
$27$	508.323	3.62321
$28$	0	0
$29$	−266.033	−1.70348	−0.851741	−	0.523962i	$-0.824453\pi$
$29$	−266.033	−1.70348	−0.851741	+	0.523962i	$0.824453\pi$
$30$	0	0
$31$	−140.335	−0.813059	−0.406529	−	0.913638i	$-0.633261\pi$
$31$	−140.335	−0.813059	−0.406529	+	0.913638i	$0.633261\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−155.291	−0.749970
$36$	0	0
$37$	−15.3852	−0.0683598	−0.0341799	−	0.999416i	$-0.510882\pi$
$37$	−15.3852	−0.0683598	−0.0341799	+	0.999416i	$0.510882\pi$
$38$	0	0
$39$	452.642	1.85848
$40$	0	0
$41$	−422.714	−1.61017	−0.805083	−	0.593162i	$-0.797879\pi$
$41$	−422.714	−1.61017	−0.805083	+	0.593162i	$0.797879\pi$
$42$	0	0
$43$	225.105	0.798331	0.399166	−	0.916879i	$-0.369300\pi$
$43$	225.105	0.798331	0.399166	+	0.916879i	$0.369300\pi$
$44$	0	0
$45$	1079.81	3.57706
$46$	0	0
$47$	273.342	0.848319	0.424159	−	0.905588i	$-0.360570\pi$
$47$	273.342	0.848319	0.424159	+	0.905588i	$0.360570\pi$
$48$	0	0
$49$	−220.772	−0.643649
$50$	0	0
$51$	490.040	1.34548
$52$	0	0
$53$	−461.725	−1.19666	−0.598328	−	0.801251i	$-0.704168\pi$
$53$	−461.725	−1.19666	−0.598328	+	0.801251i	$0.704168\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	175.592	0.408030
$58$	0	0
$59$	−26.2624	−0.0579504	−0.0289752	−	0.999580i	$-0.509224\pi$
$59$	−26.2624	−0.0579504	−0.0289752	+	0.999580i	$0.509224\pi$
$60$	0	0
$61$	283.401	0.594849	0.297425	−	0.954745i	$-0.403872\pi$
$61$	283.401	0.594849	0.297425	+	0.954745i	$0.403872\pi$
$62$	0	0
$63$	−849.908	−1.69966
$64$	0	0
$65$	623.820	1.19039
$66$	0	0
$67$	262.674	0.478967	0.239484	−	0.970900i	$-0.423022\pi$
$67$	262.674	0.478967	0.239484	+	0.970900i	$0.423022\pi$
$68$	0	0
$69$	337.855	0.589463
$70$	0	0
$71$	319.830	0.534603	0.267301	−	0.963613i	$-0.413868\pi$
$71$	319.830	0.534603	0.267301	+	0.963613i	$0.413868\pi$
$72$	0	0
$73$	158.123	0.253520	0.126760	−	0.991933i	$-0.459542\pi$
$73$	158.123	0.253520	0.126760	+	0.991933i	$0.459542\pi$
$74$	0	0
$75$	736.839	1.13444
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−529.657	−0.754318	−0.377159	−	0.926149i	$-0.623099\pi$
$79$	−529.657	−0.754318	−0.377159	+	0.926149i	$0.623099\pi$
$80$	0	0
$81$	3105.15	4.25947
$82$	0	0
$83$	1406.10	1.85952	0.929759	−	0.368168i	$-0.120015\pi$
$83$	1406.10	1.85952	0.929759	+	0.368168i	$0.120015\pi$
$84$	0	0
$85$	675.360	0.861801
$86$	0	0
$87$	−2711.38	−3.34127
$88$	0	0
$89$	−71.9709	−0.0857180	−0.0428590	−	0.999081i	$-0.513647\pi$
$89$	−71.9709	−0.0857180	−0.0428590	+	0.999081i	$0.513647\pi$
$90$	0	0
$91$	−491.005	−0.565619
$92$	0	0
$93$	−1430.28	−1.59476
$94$	0	0
$95$	241.996	0.261350
$96$	0	0
$97$	−726.164	−0.760112	−0.380056	−	0.924964i	$-0.624095\pi$
$97$	−726.164	−0.760112	−0.380056	+	0.924964i	$0.624095\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1250.44	1.23192	0.615958	−	0.787779i	$-0.288769\pi$
$101$	1250.44	1.23192	0.615958	+	0.787779i	$0.288769\pi$
$102$	0	0
$103$	−712.002	−0.681122	−0.340561	−	0.940222i	$-0.610617\pi$
$103$	−712.002	−0.681122	−0.340561	+	0.940222i	$0.610617\pi$
$104$	0	0
$105$	−1582.71	−1.47102
$106$	0	0
$107$	−705.469	−0.637386	−0.318693	−	0.947858i	$-0.603244\pi$
$107$	−705.469	−0.637386	−0.318693	+	0.947858i	$0.603244\pi$
$108$	0	0
$109$	−416.995	−0.366430	−0.183215	−	0.983073i	$-0.558650\pi$
$109$	−416.995	−0.366430	−0.183215	+	0.983073i	$0.558650\pi$
$110$	0	0
$111$	−156.805	−0.134083
$112$	0	0
$113$	−55.3964	−0.0461173	−0.0230586	−	0.999734i	$-0.507340\pi$
$113$	−55.3964	−0.0461173	−0.0230586	+	0.999734i	$0.507340\pi$
$114$	0	0
$115$	465.622	0.377561
$116$	0	0
$117$	3414.17	2.69778
$118$	0	0
$119$	−531.572	−0.409488
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−4308.26	−3.15824
$124$	0	0
$125$	−740.286	−0.529705
$126$	0	0
$127$	−1545.89	−1.08012	−0.540062	−	0.841625i	$-0.681599\pi$
$127$	−1545.89	−1.08012	−0.540062	+	0.841625i	$0.681599\pi$
$128$	0	0
$129$	2294.25	1.56588
$130$	0	0
$131$	742.370	0.495124	0.247562	−	0.968872i	$-0.420371\pi$
$131$	742.370	0.495124	0.247562	+	0.968872i	$0.420371\pi$
$132$	0	0
$133$	−190.473	−0.124182
$134$	0	0
$135$	7140.02	4.55196
$136$	0	0
$137$	−601.260	−0.374957	−0.187479	−	0.982269i	$-0.560032\pi$
$137$	−601.260	−0.374957	−0.187479	+	0.982269i	$0.560032\pi$
$138$	0	0
$139$	1087.69	0.663717	0.331858	−	0.943329i	$-0.392324\pi$
$139$	1087.69	0.663717	0.331858	+	0.943329i	$0.392324\pi$
$140$	0	0
$141$	2785.87	1.66392
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3736.75	−2.14014
$146$	0	0
$147$	−2250.08	−1.26248
$148$	0	0
$149$	−269.129	−0.147972	−0.0739862	−	0.997259i	$-0.523572\pi$
$149$	−269.129	−0.147972	−0.0739862	+	0.997259i	$0.523572\pi$
$150$	0	0
$151$	2740.25	1.47681	0.738407	−	0.674356i	$-0.235579\pi$
$151$	2740.25	1.47681	0.738407	+	0.674356i	$0.235579\pi$
$152$	0	0
$153$	3696.25	1.95310
$154$	0	0
$155$	−1971.17	−1.02147
$156$	0	0
$157$	−1036.70	−0.526992	−0.263496	−	0.964660i	$-0.584876\pi$
$157$	−1036.70	−0.526992	−0.263496	+	0.964660i	$0.584876\pi$
$158$	0	0
$159$	−4705.86	−2.34716
$160$	0	0
$161$	−366.489	−0.179400
$162$	0	0
$163$	−3093.62	−1.48657	−0.743286	−	0.668974i	$-0.766734\pi$
$163$	−3093.62	−1.48657	−0.743286	+	0.668974i	$0.766734\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	349.719	0.162048	0.0810242	−	0.996712i	$-0.474181\pi$
$167$	349.719	0.162048	0.0810242	+	0.996712i	$0.474181\pi$
$168$	0	0
$169$	−224.581	−0.102222
$170$	0	0
$171$	1324.45	0.592297
$172$	0	0
$173$	−3938.96	−1.73106	−0.865531	−	0.500855i	$-0.833019\pi$
$173$	−3938.96	−1.73106	−0.865531	+	0.500855i	$0.833019\pi$
$174$	0	0
$175$	−799.288	−0.345260
$176$	0	0
$177$	−267.664	−0.113666
$178$	0	0
$179$	1300.19	0.542907	0.271454	−	0.962452i	$-0.412496\pi$
$179$	1300.19	0.542907	0.271454	+	0.962452i	$0.412496\pi$
$180$	0	0
$181$	−1178.10	−0.483797	−0.241898	−	0.970302i	$-0.577770\pi$
$181$	−1178.10	−0.483797	−0.241898	+	0.970302i	$0.577770\pi$
$182$	0	0
$183$	2888.40	1.16676
$184$	0	0
$185$	−216.104	−0.0858827
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−5619.86	−2.16288
$190$	0	0
$191$	4695.27	1.77873	0.889367	−	0.457194i	$-0.151146\pi$
$191$	4695.27	1.77873	0.889367	+	0.457194i	$0.151146\pi$
$192$	0	0
$193$	948.709	0.353832	0.176916	−	0.984226i	$-0.443388\pi$
$193$	948.709	0.353832	0.176916	+	0.984226i	$0.443388\pi$
$194$	0	0
$195$	6357.92	2.33487
$196$	0	0
$197$	−3776.20	−1.36570	−0.682851	−	0.730557i	$-0.739260\pi$
$197$	−3776.20	−1.36570	−0.682851	+	0.730557i	$0.739260\pi$
$198$	0	0
$199$	−2303.22	−0.820457	−0.410229	−	0.911983i	$-0.634551\pi$
$199$	−2303.22	−0.820457	−0.410229	+	0.911983i	$0.634551\pi$
$200$	0	0
$201$	2677.16	0.939463
$202$	0	0
$203$	2941.18	1.01690
$204$	0	0
$205$	−5937.54	−2.02290
$206$	0	0
$207$	2548.36	0.855666
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4911.72	1.60255	0.801273	−	0.598299i	$-0.204157\pi$
$211$	4911.72	1.60255	0.801273	+	0.598299i	$0.204157\pi$
$212$	0	0
$213$	3259.68	1.04859
$214$	0	0
$215$	3161.88	1.00297
$216$	0	0
$217$	1551.50	0.485357
$218$	0	0
$219$	1611.58	0.497262
$220$	0	0
$221$	2135.38	0.649960
$222$	0	0
$223$	−137.994	−0.0414383	−0.0207191	−	0.999785i	$-0.506596\pi$
$223$	−137.994	−0.0414383	−0.0207191	+	0.999785i	$0.506596\pi$
$224$	0	0
$225$	5557.80	1.64675
$226$	0	0
$227$	−2476.43	−0.724081	−0.362040	−	0.932162i	$-0.617920\pi$
$227$	−2476.43	−0.724081	−0.362040	+	0.932162i	$0.617920\pi$
$228$	0	0
$229$	4455.46	1.28570	0.642850	−	0.765992i	$-0.277752\pi$
$229$	4455.46	1.28570	0.642850	+	0.765992i	$0.277752\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3070.23	0.863250	0.431625	−	0.902053i	$-0.357940\pi$
$233$	3070.23	0.863250	0.431625	+	0.902053i	$0.357940\pi$
$234$	0	0
$235$	3839.42	1.06577
$236$	0	0
$237$	−5398.22	−1.47955
$238$	0	0
$239$	309.379	0.0837325	0.0418662	−	0.999123i	$-0.486670\pi$
$239$	309.379	0.0837325	0.0418662	+	0.999123i	$0.486670\pi$
$240$	0	0
$241$	−2932.68	−0.783861	−0.391931	−	0.919995i	$-0.628193\pi$
$241$	−2932.68	−0.783861	−0.391931	+	0.919995i	$0.628193\pi$
$242$	0	0
$243$	17922.7	4.73146
$244$	0	0
$245$	−3101.01	−0.808637
$246$	0	0
$247$	765.152	0.197107
$248$	0	0
$249$	14330.9	3.64732
$250$	0	0
$251$	−3474.75	−0.873803	−0.436901	−	0.899509i	$-0.643924\pi$
$251$	−3474.75	−0.873803	−0.436901	+	0.899509i	$0.643924\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6883.21	1.69037
$256$	0	0
$257$	5491.12	1.33279	0.666394	−	0.745600i	$-0.267837\pi$
$257$	5491.12	1.33279	0.666394	+	0.745600i	$0.267837\pi$
$258$	0	0
$259$	170.094	0.0408075
$260$	0	0
$261$	−20451.3	−4.85020
$262$	0	0
$263$	−5151.47	−1.20781	−0.603904	−	0.797057i	$-0.706389\pi$
$263$	−5151.47	−1.20781	−0.603904	+	0.797057i	$0.706389\pi$
$264$	0	0
$265$	−6485.49	−1.50340
$266$	0	0
$267$	−733.521	−0.168130
$268$	0	0
$269$	2134.25	0.483746	0.241873	−	0.970308i	$-0.422238\pi$
$269$	2134.25	0.483746	0.241873	+	0.970308i	$0.422238\pi$
$270$	0	0
$271$	191.078	0.0428307	0.0214154	−	0.999771i	$-0.493183\pi$
$271$	191.078	0.0428307	0.0214154	+	0.999771i	$0.493183\pi$
$272$	0	0
$273$	−5004.28	−1.10942
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3924.12	−0.851182	−0.425591	−	0.904916i	$-0.639934\pi$
$277$	−3924.12	−0.851182	−0.425591	+	0.904916i	$0.639934\pi$
$278$	0	0
$279$	−10788.2	−2.31496
$280$	0	0
$281$	6114.95	1.29817	0.649087	−	0.760714i	$-0.275151\pi$
$281$	6114.95	1.29817	0.649087	+	0.760714i	$0.275151\pi$
$282$	0	0
$283$	−3996.43	−0.839446	−0.419723	−	0.907652i	$-0.637873\pi$
$283$	−3996.43	−0.839446	−0.419723	+	0.907652i	$0.637873\pi$
$284$	0	0
$285$	2466.40	0.512621
$286$	0	0
$287$	4673.40	0.961192
$288$	0	0
$289$	−2601.19	−0.529451
$290$	0	0
$291$	−7401.00	−1.49091
$292$	0	0
$293$	−4498.38	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$293$	−4498.38	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$294$	0	0
$295$	−368.887	−0.0728049
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1472.22	0.284752
$300$	0	0
$301$	−2488.70	−0.476565
$302$	0	0
$303$	12744.4	2.41632
$304$	0	0
$305$	3980.72	0.747329
$306$	0	0
$307$	3002.65	0.558210	0.279105	−	0.960261i	$-0.409962\pi$
$307$	3002.65	0.558210	0.279105	+	0.960261i	$0.409962\pi$
$308$	0	0
$309$	−7256.66	−1.33598
$310$	0	0
$311$	−6125.36	−1.11684	−0.558420	−	0.829558i	$-0.688592\pi$
$311$	−6125.36	−1.11684	−0.558420	+	0.829558i	$0.688592\pi$
$312$	0	0
$313$	8418.73	1.52030	0.760152	−	0.649746i	$-0.225125\pi$
$313$	8418.73	1.52030	0.760152	+	0.649746i	$0.225125\pi$
$314$	0	0
$315$	−11938.0	−2.13533
$316$	0	0
$317$	−7734.97	−1.37047	−0.685235	−	0.728322i	$-0.740301\pi$
$317$	−7734.97	−1.37047	−0.685235	+	0.728322i	$0.740301\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−7190.08	−1.25019
$322$	0	0
$323$	828.369	0.142699
$324$	0	0
$325$	3210.82	0.548014
$326$	0	0
$327$	−4249.98	−0.718728
$328$	0	0
$329$	−3021.98	−0.506405
$330$	0	0
$331$	1037.33	0.172257	0.0861285	−	0.996284i	$-0.472550\pi$
$331$	1037.33	0.172257	0.0861285	+	0.996284i	$0.472550\pi$
$332$	0	0
$333$	−1182.74	−0.194636
$334$	0	0
$335$	3689.59	0.601742
$336$	0	0
$337$	−3566.13	−0.576438	−0.288219	−	0.957564i	$-0.593063\pi$
$337$	−3566.13	−0.576438	−0.288219	+	0.957564i	$0.593063\pi$
$338$	0	0
$339$	−564.595	−0.0904560
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6232.89	0.981179
$344$	0	0
$345$	4745.58	0.740561
$346$	0	0
$347$	7158.51	1.10746	0.553730	−	0.832696i	$-0.313204\pi$
$347$	7158.51	1.10746	0.553730	+	0.832696i	$0.313204\pi$
$348$	0	0
$349$	508.815	0.0780408	0.0390204	−	0.999238i	$-0.487576\pi$
$349$	508.815	0.0780408	0.0390204	+	0.999238i	$0.487576\pi$
$350$	0	0
$351$	22575.6	3.43304
$352$	0	0
$353$	6275.23	0.946166	0.473083	−	0.881018i	$-0.343141\pi$
$353$	6275.23	0.946166	0.473083	+	0.881018i	$0.343141\pi$
$354$	0	0
$355$	4492.40	0.671639
$356$	0	0
$357$	−5417.73	−0.803184
$358$	0	0
$359$	7672.75	1.12800	0.564000	−	0.825775i	$-0.309262\pi$
$359$	7672.75	1.12800	0.564000	+	0.825775i	$0.309262\pi$
$360$	0	0
$361$	−6562.18	−0.956725
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2221.04	0.318505
$366$	0	0
$367$	−7316.28	−1.04062	−0.520309	−	0.853978i	$-0.674183\pi$
$367$	−7316.28	−1.04062	−0.520309	+	0.853978i	$0.674183\pi$
$368$	0	0
$369$	−32496.2	−4.58451
$370$	0	0
$371$	5104.69	0.714346
$372$	0	0
$373$	−11492.0	−1.59526	−0.797632	−	0.603145i	$-0.793914\pi$
$373$	−11492.0	−1.59526	−0.797632	+	0.603145i	$0.793914\pi$
$374$	0	0
$375$	−7544.93	−1.03898
$376$	0	0
$377$	−11815.0	−1.61407
$378$	0	0
$379$	−7765.72	−1.05250	−0.526251	−	0.850329i	$-0.676403\pi$
$379$	−7765.72	−1.05250	−0.526251	+	0.850329i	$0.676403\pi$
$380$	0	0
$381$	−15755.6	−2.11859
$382$	0	0
$383$	8732.54	1.16504	0.582522	−	0.812815i	$-0.302066\pi$
$383$	8732.54	1.16504	0.582522	+	0.812815i	$0.302066\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17305.0	2.27303
$388$	0	0
$389$	−6582.20	−0.857920	−0.428960	−	0.903323i	$-0.641120\pi$
$389$	−6582.20	−0.857920	−0.428960	+	0.903323i	$0.641120\pi$
$390$	0	0
$391$	1593.86	0.206151
$392$	0	0
$393$	7566.17	0.971153
$394$	0	0
$395$	−7439.69	−0.947674
$396$	0	0
$397$	−1941.77	−0.245478	−0.122739	−	0.992439i	$-0.539168\pi$
$397$	−1941.77	−0.245478	−0.122739	+	0.992439i	$0.539168\pi$
$398$	0	0
$399$	−1941.29	−0.243574
$400$	0	0
$401$	7439.62	0.926476	0.463238	−	0.886234i	$-0.346687\pi$
$401$	7439.62	0.926476	0.463238	+	0.886234i	$0.346687\pi$
$402$	0	0
$403$	−6232.53	−0.770383
$404$	0	0
$405$	43615.7	5.35131
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−4451.60	−0.538185	−0.269092	−	0.963114i	$-0.586724\pi$
$409$	−4451.60	−0.538185	−0.269092	+	0.963114i	$0.586724\pi$
$410$	0	0
$411$	−6127.99	−0.735454
$412$	0	0
$413$	290.349	0.0345936
$414$	0	0
$415$	19750.5	2.33617
$416$	0	0
$417$	11085.6	1.30184
$418$	0	0
$419$	−10273.6	−1.19784	−0.598921	−	0.800808i	$-0.704404\pi$
$419$	−10273.6	−1.19784	−0.598921	+	0.800808i	$0.704404\pi$
$420$	0	0
$421$	7957.15	0.921159	0.460580	−	0.887618i	$-0.347642\pi$
$421$	7957.15	0.921159	0.460580	+	0.887618i	$0.347642\pi$
$422$	0	0
$423$	21013.2	2.41536
$424$	0	0
$425$	3476.10	0.396743
$426$	0	0
$427$	−3133.20	−0.355096
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6048.94	−0.676026	−0.338013	−	0.941141i	$-0.609755\pi$
$431$	−6048.94	−0.676026	−0.338013	+	0.941141i	$0.609755\pi$
$432$	0	0
$433$	7866.23	0.873041	0.436521	−	0.899694i	$-0.356211\pi$
$433$	7866.23	0.873041	0.436521	+	0.899694i	$0.356211\pi$
$434$	0	0
$435$	−38084.7	−4.19775
$436$	0	0
$437$	571.114	0.0625173
$438$	0	0
$439$	−1116.83	−0.121420	−0.0607098	−	0.998155i	$-0.519336\pi$
$439$	−1116.83	−0.121420	−0.0607098	+	0.998155i	$0.519336\pi$
$440$	0	0
$441$	−16971.8	−1.83261
$442$	0	0
$443$	−6.87849	−0.000737713 0	−0.000368857	−	1.00000i	$-0.500117\pi$
$443$	−6.87849	−0.000737713 0	−0.000368857		1.00000i	$0.500117\pi$
$444$	0	0
$445$	−1010.92	−0.107690
$446$	0	0
$447$	−2742.94	−0.290238
$448$	0	0
$449$	−12109.8	−1.27282	−0.636409	−	0.771352i	$-0.719581\pi$
$449$	−12109.8	−1.27282	−0.636409	+	0.771352i	$0.719581\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	27928.4	2.89667
$454$	0	0
$455$	−6896.76	−0.710605
$456$	0	0
$457$	680.336	0.0696385	0.0348193	−	0.999394i	$-0.488914\pi$
$457$	680.336	0.0696385	0.0348193	+	0.999394i	$0.488914\pi$
$458$	0	0
$459$	24440.8	2.48540
$460$	0	0
$461$	−3914.17	−0.395447	−0.197724	−	0.980258i	$-0.563355\pi$
$461$	−3914.17	−0.395447	−0.197724	+	0.980258i	$0.563355\pi$
$462$	0	0
$463$	−16656.9	−1.67195	−0.835973	−	0.548771i	$-0.815096\pi$
$463$	−16656.9	−1.67195	−0.835973	+	0.548771i	$0.815096\pi$
$464$	0	0
$465$	−20090.0	−2.00355
$466$	0	0
$467$	−11785.0	−1.16776	−0.583879	−	0.811841i	$-0.698466\pi$
$467$	−11785.0	−1.16776	−0.583879	+	0.811841i	$0.698466\pi$
$468$	0	0
$469$	−2904.05	−0.285920
$470$	0	0
$471$	−10566.0	−1.03366
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1245.56	0.120316
$476$	0	0
$477$	−35495.2	−3.40715
$478$	0	0
$479$	10581.5	1.00936	0.504678	−	0.863308i	$-0.331611\pi$
$479$	10581.5	1.00936	0.504678	+	0.863308i	$0.331611\pi$
$480$	0	0
$481$	−683.287	−0.0647717
$482$	0	0
$483$	−3735.22	−0.351881
$484$	0	0
$485$	−10199.9	−0.954953
$486$	0	0
$487$	−8497.69	−0.790692	−0.395346	−	0.918532i	$-0.629375\pi$
$487$	−8497.69	−0.790692	−0.395346	+	0.918532i	$0.629375\pi$
$488$	0	0
$489$	−31529.9	−2.91581
$490$	0	0
$491$	5557.94	0.510848	0.255424	−	0.966829i	$-0.417785\pi$
$491$	5557.94	0.510848	0.255424	+	0.966829i	$0.417785\pi$
$492$	0	0
$493$	−12791.2	−1.16853
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3535.94	−0.319132
$498$	0	0
$499$	−6070.32	−0.544579	−0.272289	−	0.962215i	$-0.587781\pi$
$499$	−6070.32	−0.544579	−0.272289	+	0.962215i	$0.587781\pi$
$500$	0	0
$501$	3564.31	0.317848
$502$	0	0
$503$	−17219.7	−1.52642	−0.763210	−	0.646150i	$-0.776378\pi$
$503$	−17219.7	−1.52642	−0.763210	+	0.646150i	$0.776378\pi$
$504$	0	0
$505$	17564.0	1.54770
$506$	0	0
$507$	−2288.91	−0.200501
$508$	0	0
$509$	−8.68119	−0.000755967 0	−0.000377983	−	1.00000i	$-0.500120\pi$
$509$	−8.68119	−0.000755967 0	−0.000377983		1.00000i	$0.500120\pi$
$510$	0	0
$511$	−1748.16	−0.151339
$512$	0	0
$513$	8757.66	0.753723
$514$	0	0
$515$	−10000.9	−0.855716
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−40145.6	−3.39537
$520$	0	0
$521$	2672.96	0.224769	0.112384	−	0.993665i	$-0.464151\pi$
$521$	2672.96	0.224769	0.112384	+	0.993665i	$0.464151\pi$
$522$	0	0
$523$	15194.8	1.27040	0.635201	−	0.772347i	$-0.280917\pi$
$523$	15194.8	1.27040	0.635201	+	0.772347i	$0.280917\pi$
$524$	0	0
$525$	−8146.27	−0.677205
$526$	0	0
$527$	−6747.46	−0.557730
$528$	0	0
$529$	−11068.1	−0.909684
$530$	0	0
$531$	−2018.92	−0.164998
$532$	0	0
$533$	−18773.5	−1.52565
$534$	0	0
$535$	−9909.18	−0.800768
$536$	0	0
$537$	13251.4	1.06488
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−15772.1	−1.25341	−0.626707	−	0.779255i	$-0.715598\pi$
$541$	−15772.1	−1.25341	−0.626707	+	0.779255i	$0.715598\pi$
$542$	0	0
$543$	−12007.1	−0.948936
$544$	0	0
$545$	−5857.20	−0.460358
$546$	0	0
$547$	−5946.77	−0.464836	−0.232418	−	0.972616i	$-0.574664\pi$
$547$	−5946.77	−0.464836	−0.232418	+	0.972616i	$0.574664\pi$
$548$	0	0
$549$	21786.5	1.69367
$550$	0	0
$551$	−4583.35	−0.354369
$552$	0	0
$553$	5855.73	0.450291
$554$	0	0
$555$	−2202.52	−0.168453
$556$	0	0
$557$	−23653.3	−1.79932	−0.899661	−	0.436590i	$-0.856186\pi$
$557$	−23653.3	−1.79932	−0.899661	+	0.436590i	$0.856186\pi$
$558$	0	0
$559$	9997.36	0.756428
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2401.08	−0.179740	−0.0898700	−	0.995954i	$-0.528645\pi$
$563$	−2401.08	−0.179740	−0.0898700	+	0.995954i	$0.528645\pi$
$564$	0	0
$565$	−778.110	−0.0579386
$566$	0	0
$567$	−34329.6	−2.54270
$568$	0	0
$569$	9141.61	0.673526	0.336763	−	0.941590i	$-0.390668\pi$
$569$	9141.61	0.673526	0.336763	+	0.941590i	$0.390668\pi$
$570$	0	0
$571$	−1170.40	−0.0857785	−0.0428893	−	0.999080i	$-0.513656\pi$
$571$	−1170.40	−0.0857785	−0.0428893	+	0.999080i	$0.513656\pi$
$572$	0	0
$573$	47853.8	3.48887
$574$	0	0
$575$	2396.58	0.173816
$576$	0	0
$577$	24314.4	1.75428	0.877142	−	0.480231i	$-0.159447\pi$
$577$	24314.4	1.75428	0.877142	+	0.480231i	$0.159447\pi$
$578$	0	0
$579$	9669.16	0.694018
$580$	0	0
$581$	−15545.5	−1.11004
$582$	0	0
$583$	0	0
$584$	0	0
$585$	47956.2	3.38931
$586$	0	0
$587$	−8932.15	−0.628057	−0.314028	−	0.949414i	$-0.601679\pi$
$587$	−8932.15	−0.628057	−0.314028	+	0.949414i	$0.601679\pi$
$588$	0	0
$589$	−2417.76	−0.169138
$590$	0	0
$591$	−38486.8	−2.67874
$592$	0	0
$593$	25900.4	1.79359	0.896796	−	0.442443i	$-0.145888\pi$
$593$	25900.4	1.79359	0.896796	+	0.442443i	$0.145888\pi$
$594$	0	0
$595$	−7466.58	−0.514453
$596$	0	0
$597$	−23474.2	−1.60927
$598$	0	0
$599$	−4614.20	−0.314743	−0.157372	−	0.987539i	$-0.550302\pi$
$599$	−4614.20	−0.314743	−0.157372	+	0.987539i	$0.550302\pi$
$600$	0	0
$601$	15971.8	1.08403	0.542016	−	0.840368i	$-0.317661\pi$
$601$	15971.8	1.08403	0.542016	+	0.840368i	$0.317661\pi$
$602$	0	0
$603$	20193.1	1.36373
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18538.9	1.23965	0.619826	−	0.784739i	$-0.287203\pi$
$607$	18538.9	1.23965	0.619826	+	0.784739i	$0.287203\pi$
$608$	0	0
$609$	29976.2	1.99458
$610$	0	0
$611$	12139.6	0.803792
$612$	0	0
$613$	14197.4	0.935447	0.467723	−	0.883875i	$-0.345074\pi$
$613$	14197.4	0.935447	0.467723	+	0.883875i	$0.345074\pi$
$614$	0	0
$615$	−60514.9	−3.96780
$616$	0	0
$617$	2917.85	0.190386	0.0951929	−	0.995459i	$-0.469653\pi$
$617$	2917.85	0.190386	0.0951929	+	0.995459i	$0.469653\pi$
$618$	0	0
$619$	6239.28	0.405134	0.202567	−	0.979268i	$-0.435072\pi$
$619$	6239.28	0.405134	0.202567	+	0.979268i	$0.435072\pi$
$620$	0	0
$621$	16850.5	1.08887
$622$	0	0
$623$	795.689	0.0511695
$624$	0	0
$625$	−19435.3	−1.24386
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−739.740	−0.0468925
$630$	0	0
$631$	23610.7	1.48959	0.744793	−	0.667296i	$-0.232548\pi$
$631$	23610.7	1.48959	0.744793	+	0.667296i	$0.232548\pi$
$632$	0	0
$633$	50059.9	3.14329
$634$	0	0
$635$	−21713.9	−1.35699
$636$	0	0
$637$	−9804.89	−0.609865
$638$	0	0
$639$	24586.9	1.52214
$640$	0	0
$641$	−18752.6	−1.15551	−0.577757	−	0.816209i	$-0.696072\pi$
$641$	−18752.6	−1.15551	−0.577757	+	0.816209i	$0.696072\pi$
$642$	0	0
$643$	14869.9	0.911993	0.455996	−	0.889982i	$-0.349283\pi$
$643$	14869.9	0.911993	0.455996	+	0.889982i	$0.349283\pi$
$644$	0	0
$645$	32225.6	1.96726
$646$	0	0
$647$	−5947.94	−0.361418	−0.180709	−	0.983537i	$-0.557839\pi$
$647$	−5947.94	−0.361418	−0.180709	+	0.983537i	$0.557839\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	15812.7	0.951996
$652$	0	0
$653$	24711.9	1.48094	0.740469	−	0.672090i	$-0.234603\pi$
$653$	24711.9	1.48094	0.740469	+	0.672090i	$0.234603\pi$
$654$	0	0
$655$	10427.5	0.622040
$656$	0	0
$657$	12155.7	0.721828
$658$	0	0
$659$	−3367.98	−0.199086	−0.0995431	−	0.995033i	$-0.531738\pi$
$659$	−3367.98	−0.199086	−0.0995431	+	0.995033i	$0.531738\pi$
$660$	0	0
$661$	9382.60	0.552104	0.276052	−	0.961143i	$-0.410974\pi$
$661$	9382.60	0.552104	0.276052	+	0.961143i	$0.410974\pi$
$662$	0	0
$663$	21763.6	1.27485
$664$	0	0
$665$	−2675.43	−0.156013
$666$	0	0
$667$	−8818.79	−0.511941
$668$	0	0
$669$	−1406.42	−0.0812785
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2331.74	−0.133554	−0.0667770	−	0.997768i	$-0.521272\pi$
$673$	−2331.74	−0.133554	−0.0667770	+	0.997768i	$0.521272\pi$
$674$	0	0
$675$	36749.9	2.09556
$676$	0	0
$677$	−23491.9	−1.33363	−0.666814	−	0.745224i	$-0.732342\pi$
$677$	−23491.9	−1.33363	−0.666814	+	0.745224i	$0.732342\pi$
$678$	0	0
$679$	8028.25	0.453750
$680$	0	0
$681$	−25239.5	−1.42024
$682$	0	0
$683$	−19040.5	−1.06671	−0.533355	−	0.845891i	$-0.679069\pi$
$683$	−19040.5	−1.06671	−0.533355	+	0.845891i	$0.679069\pi$
$684$	0	0
$685$	−8445.44	−0.471071
$686$	0	0
$687$	45409.7	2.52182
$688$	0	0
$689$	−20506.1	−1.13385
$690$	0	0
$691$	34508.0	1.89978	0.949889	−	0.312588i	$-0.101196\pi$
$691$	34508.0	1.89978	0.949889	+	0.312588i	$0.101196\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15277.9	0.833849
$696$	0	0
$697$	−20324.6	−1.10452
$698$	0	0
$699$	31291.5	1.69321
$700$	0	0
$701$	19747.5	1.06399	0.531993	−	0.846749i	$-0.321443\pi$
$701$	19747.5	1.06399	0.531993	+	0.846749i	$0.321443\pi$
$702$	0	0
$703$	−265.065	−0.0142206
$704$	0	0
$705$	39131.0	2.09044
$706$	0	0
$707$	−13824.5	−0.735395
$708$	0	0
$709$	29214.8	1.54751	0.773755	−	0.633485i	$-0.218376\pi$
$709$	29214.8	1.54751	0.773755	+	0.633485i	$0.218376\pi$
$710$	0	0
$711$	−40717.5	−2.14771
$712$	0	0
$713$	−4651.99	−0.244346
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3153.16	0.164236
$718$	0	0
$719$	2227.98	0.115563	0.0577813	−	0.998329i	$-0.481597\pi$
$719$	2227.98	0.115563	0.0577813	+	0.998329i	$0.481597\pi$
$720$	0	0
$721$	7871.68	0.406597
$722$	0	0
$723$	−29889.6	−1.53749
$724$	0	0
$725$	−19233.2	−0.985246
$726$	0	0
$727$	9245.78	0.471674	0.235837	−	0.971793i	$-0.424217\pi$
$727$	9245.78	0.471674	0.235837	+	0.971793i	$0.424217\pi$
$728$	0	0
$729$	98827.9	5.02098
$730$	0	0
$731$	10823.3	0.547628
$732$	0	0
$733$	−16112.5	−0.811909	−0.405954	−	0.913893i	$-0.633061\pi$
$733$	−16112.5	−0.811909	−0.405954	+	0.913893i	$0.633061\pi$
$734$	0	0
$735$	−31605.2	−1.58609
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−8408.04	−0.418532	−0.209266	−	0.977859i	$-0.567107\pi$
$739$	−8408.04	−0.418532	−0.209266	+	0.977859i	$0.567107\pi$
$740$	0	0
$741$	7798.37	0.386613
$742$	0	0
$743$	−24642.2	−1.21673	−0.608367	−	0.793656i	$-0.708175\pi$
$743$	−24642.2	−1.21673	−0.608367	+	0.793656i	$0.708175\pi$
$744$	0	0
$745$	−3780.25	−0.185903
$746$	0	0
$747$	108094.	5.29447
$748$	0	0
$749$	7799.45	0.380488
$750$	0	0
$751$	7041.76	0.342154	0.171077	−	0.985258i	$-0.445275\pi$
$751$	7041.76	0.342154	0.171077	+	0.985258i	$0.445275\pi$
$752$	0	0
$753$	−35414.4	−1.71391
$754$	0	0
$755$	38490.2	1.85537
$756$	0	0
$757$	32553.1	1.56296	0.781481	−	0.623929i	$-0.214465\pi$
$757$	32553.1	1.56296	0.781481	+	0.623929i	$0.214465\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16905.6	−0.805292	−0.402646	−	0.915356i	$-0.631909\pi$
$761$	−16905.6	−0.805292	−0.402646	+	0.915356i	$0.631909\pi$
$762$	0	0
$763$	4610.17	0.218741
$764$	0	0
$765$	51918.4	2.45374
$766$	0	0
$767$	−1166.36	−0.0549086
$768$	0	0
$769$	−3363.52	−0.157727	−0.0788633	−	0.996885i	$-0.525129\pi$
$769$	−3363.52	−0.157727	−0.0788633	+	0.996885i	$0.525129\pi$
$770$	0	0
$771$	55965.0	2.61418
$772$	0	0
$773$	34397.7	1.60052	0.800258	−	0.599656i	$-0.204696\pi$
$773$	34397.7	1.60052	0.800258	+	0.599656i	$0.204696\pi$
$774$	0	0
$775$	−10145.7	−0.470250
$776$	0	0
$777$	1733.59	0.0800413
$778$	0	0
$779$	−7282.74	−0.334957
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−135230.	−6.17208
$784$	0	0
$785$	−14561.7	−0.662078
$786$	0	0
$787$	5693.87	0.257897	0.128948	−	0.991651i	$-0.458840\pi$
$787$	5693.87	0.257897	0.128948	+	0.991651i	$0.458840\pi$
$788$	0	0
$789$	−52503.4	−2.36904
$790$	0	0
$791$	612.446	0.0275298
$792$	0	0
$793$	12586.4	0.563627
$794$	0	0
$795$	−66099.6	−2.94882
$796$	0	0
$797$	−32438.0	−1.44167	−0.720837	−	0.693105i	$-0.756242\pi$
$797$	−32438.0	−1.44167	−0.720837	+	0.693105i	$0.756242\pi$
$798$	0	0
$799$	13142.6	0.581917
$800$	0	0
$801$	−5532.77	−0.244059
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−5147.78	−0.225386
$806$	0	0
$807$	21752.1	0.948837
$808$	0	0
$809$	5196.27	0.225823	0.112912	−	0.993605i	$-0.463982\pi$
$809$	5196.27	0.225823	0.112912	+	0.993605i	$0.463982\pi$
$810$	0	0
$811$	34448.6	1.49156	0.745780	−	0.666193i	$-0.232077\pi$
$811$	34448.6	1.49156	0.745780	+	0.666193i	$0.232077\pi$
$812$	0	0
$813$	1947.45	0.0840097
$814$	0	0
$815$	−43453.7	−1.86763
$816$	0	0
$817$	3878.24	0.166074
$818$	0	0
$819$	−37746.0	−1.61044
$820$	0	0
$821$	15392.4	0.654321	0.327161	−	0.944969i	$-0.393908\pi$
$821$	15392.4	0.654321	0.327161	+	0.944969i	$0.393908\pi$
$822$	0	0
$823$	−24097.5	−1.02064	−0.510319	−	0.859985i	$-0.670473\pi$
$823$	−24097.5	−1.02064	−0.510319	+	0.859985i	$0.670473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30103.8	1.26579	0.632896	−	0.774237i	$-0.281866\pi$
$827$	30103.8	1.26579	0.632896	+	0.774237i	$0.281866\pi$
$828$	0	0
$829$	5258.84	0.220322	0.110161	−	0.993914i	$-0.464863\pi$
$829$	5258.84	0.220322	0.110161	+	0.993914i	$0.464863\pi$
$830$	0	0
$831$	−39994.3	−1.66954
$832$	0	0
$833$	−10615.0	−0.441521
$834$	0	0
$835$	4912.24	0.203587
$836$	0	0
$837$	−71335.2	−2.94589
$838$	0	0
$839$	33051.0	1.36001	0.680004	−	0.733208i	$-0.261978\pi$
$839$	33051.0	1.36001	0.680004	+	0.733208i	$0.261978\pi$
$840$	0	0
$841$	46384.3	1.90185
$842$	0	0
$843$	62323.0	2.54628
$844$	0	0
$845$	−3154.52	−0.128425
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−40731.3	−1.64652
$850$	0	0
$851$	−510.009	−0.0205439
$852$	0	0
$853$	−10611.2	−0.425934	−0.212967	−	0.977059i	$-0.568313\pi$
$853$	−10611.2	−0.425934	−0.212967	+	0.977059i	$0.568313\pi$
$854$	0	0
$855$	18603.5	0.744123
$856$	0	0
$857$	−40501.4	−1.61435	−0.807177	−	0.590309i	$-0.799006\pi$
$857$	−40501.4	−1.61435	−0.807177	+	0.590309i	$0.799006\pi$
$858$	0	0
$859$	36118.3	1.43462	0.717311	−	0.696754i	$-0.245373\pi$
$859$	36118.3	1.43462	0.717311	+	0.696754i	$0.245373\pi$
$860$	0	0
$861$	47630.9	1.88531
$862$	0	0
$863$	−11073.3	−0.436776	−0.218388	−	0.975862i	$-0.570080\pi$
$863$	−11073.3	−0.436776	−0.218388	+	0.975862i	$0.570080\pi$
$864$	0	0
$865$	−55327.5	−2.17479
$866$	0	0
$867$	−26511.1	−1.03848
$868$	0	0
$869$	0	0
$870$	0	0
$871$	11665.9	0.453827
$872$	0	0
$873$	−55824.0	−2.16421
$874$	0	0
$875$	8184.38	0.316209
$876$	0	0
$877$	23052.0	0.887583	0.443792	−	0.896130i	$-0.353633\pi$
$877$	23052.0	0.887583	0.443792	+	0.896130i	$0.353633\pi$
$878$	0	0
$879$	−45847.1	−1.75925
$880$	0	0
$881$	−12918.9	−0.494041	−0.247021	−	0.969010i	$-0.579452\pi$
$881$	−12918.9	−0.494041	−0.247021	+	0.969010i	$0.579452\pi$
$882$	0	0
$883$	46652.8	1.77802	0.889010	−	0.457888i	$-0.151394\pi$
$883$	46652.8	1.77802	0.889010	+	0.457888i	$0.151394\pi$
$884$	0	0
$885$	−3759.67	−0.142802
$886$	0	0
$887$	19436.5	0.735755	0.367878	−	0.929874i	$-0.380085\pi$
$887$	19436.5	0.735755	0.367878	+	0.929874i	$0.380085\pi$
$888$	0	0
$889$	17090.9	0.644781
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4709.28	0.176473
$894$	0	0
$895$	18262.7	0.682072
$896$	0	0
$897$	15004.8	0.558523
$898$	0	0
$899$	37333.5	1.38503
$900$	0	0
$901$	−22200.3	−0.820865
$902$	0	0
$903$	−25364.6	−0.934752
$904$	0	0
$905$	−16547.8	−0.607810
$906$	0	0
$907$	2496.95	0.0914111	0.0457056	−	0.998955i	$-0.485446\pi$
$907$	2496.95	0.0914111	0.0457056	+	0.998955i	$0.485446\pi$
$908$	0	0
$909$	96127.8	3.50755
$910$	0	0
$911$	44289.3	1.61072	0.805361	−	0.592784i	$-0.201971\pi$
$911$	44289.3	1.61072	0.805361	+	0.592784i	$0.201971\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	40571.1	1.46584
$916$	0	0
$917$	−8207.42	−0.295565
$918$	0	0
$919$	7013.78	0.251755	0.125878	−	0.992046i	$-0.459825\pi$
$919$	7013.78	0.251755	0.125878	+	0.992046i	$0.459825\pi$
$920$	0	0
$921$	30602.8	1.09489
$922$	0	0
$923$	14204.3	0.506543
$924$	0	0
$925$	−1112.30	−0.0395374
$926$	0	0
$927$	−54735.2	−1.93931
$928$	0	0
$929$	−23019.8	−0.812977	−0.406488	−	0.913656i	$-0.633247\pi$
$929$	−23019.8	−0.812977	−0.406488	+	0.913656i	$0.633247\pi$
$930$	0	0
$931$	−3803.57	−0.133896
$932$	0	0
$933$	−62429.2	−2.19061
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2917.25	0.101710	0.0508551	−	0.998706i	$-0.483805\pi$
$937$	2917.25	0.101710	0.0508551	+	0.998706i	$0.483805\pi$
$938$	0	0
$939$	85803.0	2.98198
$940$	0	0
$941$	−27187.7	−0.941864	−0.470932	−	0.882170i	$-0.656082\pi$
$941$	−27187.7	−0.941864	−0.470932	+	0.882170i	$0.656082\pi$
$942$	0	0
$943$	−14012.7	−0.483897
$944$	0	0
$945$	−78937.9	−2.71730
$946$	0	0
$947$	−21246.8	−0.729069	−0.364535	−	0.931190i	$-0.618772\pi$
$947$	−21246.8	−0.729069	−0.364535	+	0.931190i	$0.618772\pi$
$948$	0	0
$949$	7022.56	0.240213
$950$	0	0
$951$	−78834.1	−2.68809
$952$	0	0
$953$	15242.1	0.518090	0.259045	−	0.965865i	$-0.416592\pi$
$953$	15242.1	0.518090	0.259045	+	0.965865i	$0.416592\pi$
$954$	0	0
$955$	65950.9	2.23468
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6647.35	0.223831
$960$	0	0
$961$	−10097.2	−0.338935
$962$	0	0
$963$	−54233.0	−1.81478
$964$	0	0
$965$	13325.8	0.444531
$966$	0	0
$967$	28185.6	0.937319	0.468660	−	0.883379i	$-0.344737\pi$
$967$	28185.6	0.937319	0.468660	+	0.883379i	$0.344737\pi$
$968$	0	0
$969$	8442.67	0.279894
$970$	0	0
$971$	−60121.5	−1.98702	−0.993508	−	0.113765i	$-0.963709\pi$
$971$	−60121.5	−1.98702	−0.993508	+	0.113765i	$0.963709\pi$
$972$	0	0
$973$	−12025.2	−0.396207
$974$	0	0
$975$	32724.4	1.07489
$976$	0	0
$977$	33718.3	1.10414	0.552070	−	0.833798i	$-0.313838\pi$
$977$	33718.3	1.10414	0.552070	+	0.833798i	$0.313838\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−32056.5	−1.04331
$982$	0	0
$983$	−7454.43	−0.241871	−0.120936	−	0.992660i	$-0.538589\pi$
$983$	−7454.43	−0.241871	−0.120936	+	0.992660i	$0.538589\pi$
$984$	0	0
$985$	−53041.4	−1.71578
$986$	0	0
$987$	−30799.8	−0.993281
$988$	0	0
$989$	7462.09	0.239920
$990$	0	0
$991$	−15927.5	−0.510550	−0.255275	−	0.966869i	$-0.582166\pi$
$991$	−15927.5	−0.510550	−0.255275	+	0.966869i	$0.582166\pi$
$992$	0	0
$993$	10572.4	0.337871
$994$	0	0
$995$	−32351.6	−1.03077
$996$	0	0
$997$	−45031.5	−1.43045	−0.715227	−	0.698892i	$-0.753677\pi$
$997$	−45031.5	−1.43045	−0.715227	+	0.698892i	$0.753677\pi$
$998$	0	0
$999$	−7820.66	−0.247682

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.r.1.10		10
4.3	odd	2		1936.4.a.by.1.1		10
11.7	odd	10		88.4.i.b.49.1	yes	20
11.8	odd	10		88.4.i.b.9.1	✓	20
11.10	odd	2		968.4.a.s.1.10		10
44.7	even	10		176.4.m.f.49.5		20
44.19	even	10		176.4.m.f.97.5		20
44.43	even	2		1936.4.a.bx.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.b.9.1	✓	20	11.8	odd	10
88.4.i.b.49.1	yes	20	11.7	odd	10
176.4.m.f.49.5		20	44.7	even	10
176.4.m.f.97.5		20	44.19	even	10
968.4.a.r.1.10		10	1.1	even	1	trivial
968.4.a.s.1.10		10	11.10	odd	2
1936.4.a.bx.1.1		10	44.43	even	2
1936.4.a.by.1.1		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)

\(f(q)\)	\(=\)	\(q+10.1919 q^{3} +14.0462 q^{5} -11.0557 q^{7} +76.8751 q^{9} +O(q^{10})\)	\(q+10.1919 q^{3} +14.0462 q^{5} -11.0557 q^{7} +76.8751 q^{9} +44.4119 q^{13} +143.158 q^{15} +48.0812 q^{17} +17.2285 q^{19} -112.679 q^{21} +33.1493 q^{23} +72.2965 q^{25} +508.323 q^{27} -266.033 q^{29} -140.335 q^{31} -155.291 q^{35} -15.3852 q^{37} +452.642 q^{39} -422.714 q^{41} +225.105 q^{43} +1079.81 q^{45} +273.342 q^{47} -220.772 q^{49} +490.040 q^{51} -461.725 q^{53} +175.592 q^{57} -26.2624 q^{59} +283.401 q^{61} -849.908 q^{63} +623.820 q^{65} +262.674 q^{67} +337.855 q^{69} +319.830 q^{71} +158.123 q^{73} +736.839 q^{75} -529.657 q^{79} +3105.15 q^{81} +1406.10 q^{83} +675.360 q^{85} -2711.38 q^{87} -71.9709 q^{89} -491.005 q^{91} -1430.28 q^{93} +241.996 q^{95} -726.164 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) \) 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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