LMFDB - Embedded newform 968.6.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,6,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, 6, names="a")

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

chi := DirichletCharacter("968.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$155.251537579$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 2635 x^{14} + 10644 x^{13} + 2721739 x^{12} - 11107836 x^{11} + \cdots + 53\!\cdots\!20 $ x^16 - 4x^15 - 2635x^14 + 10644x^13 + 2721739x^12 - 11107836x^11 - 1388443249x^10 + 5594180788x^9 + 362606893936x^8 - 1331049359080x^7 - 46300877961632x^6 + 122447405593024x^5 + 2801116058142464x^4 - 3524304397180928x^3 - 71736438677110016x^2 + 11226318210301440*x + 533879212527569920
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26}\cdot 11^{8} $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-23.8164$ 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)

$f(q)$	$=$	$q-24.4344 q^{3} -57.4032 q^{5} -156.812 q^{7} +354.040 q^{9} +1107.14 q^{13} +1402.61 q^{15} -1734.64 q^{17} -726.877 q^{19} +3831.61 q^{21} +4796.66 q^{23} +170.132 q^{25} -2713.20 q^{27} -4674.99 q^{29} -2000.06 q^{31} +9001.52 q^{35} -10313.3 q^{37} -27052.3 q^{39} -2626.79 q^{41} -9573.28 q^{43} -20323.0 q^{45} +3281.20 q^{47} +7783.01 q^{49} +42384.9 q^{51} -25960.9 q^{53} +17760.8 q^{57} -4423.25 q^{59} -15468.1 q^{61} -55517.7 q^{63} -63553.5 q^{65} -18768.3 q^{67} -117204. q^{69} -30765.2 q^{71} -62882.9 q^{73} -4157.08 q^{75} +65539.3 q^{79} -19736.4 q^{81} +8408.81 q^{83} +99573.9 q^{85} +114230. q^{87} +774.583 q^{89} -173613. q^{91} +48870.2 q^{93} +41725.1 q^{95} -134159. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{3} + 81 q^{5} + 47 q^{7} + 1416 q^{9} - 859 q^{13} - 738 q^{15} - 1226 q^{17} - 616 q^{19} - 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} - 1613 q^{29} + 18511 q^{31} + 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) $ 16 * q + 12 * q^3 + 81 * q^5 + 47 * q^7 + 1416 * q^9 - 859 * q^13 - 738 * q^15 - 1226 * q^17 - 616 * q^19 - 1141 * q^21 + 2258 * q^23 + 10307 * q^25 + 564 * q^27 - 1613 * q^29 + 18511 * q^31 + 23544 * q^35 + 8553 * q^37 - 12020 * q^39 - 25514 * q^41 + 18931 * q^43 + 23278 * q^45 + 38781 * q^47 + 94523 * q^49 + 68584 * q^51 + 779 * q^53 - 19293 * q^57 + 50316 * q^59 - 53153 * q^61 + 54911 * q^63 - 51517 * q^65 + 10491 * q^67 - 95356 * q^69 - 91677 * q^71 - 97660 * q^73 - 3771 * q^75 + 96003 * q^79 + 120400 * q^81 - 4552 * q^83 + 129679 * q^85 - 77219 * q^87 + 6697 * q^89 + 217380 * q^91 + 132586 * q^93 + 54168 * q^95 + 171314 * 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
$3$	−24.4344	−1.56747	−0.783734	−	0.621097i	$-0.786687\pi$
$3$	−24.4344	−1.56747	−0.783734	+	0.621097i	$0.786687\pi$
$4$	0	0
$5$	−57.4032	−1.02686	−0.513430	−	0.858131i	$-0.671625\pi$
$5$	−57.4032	−1.02686	−0.513430	+	0.858131i	$0.671625\pi$
$6$	0	0
$7$	−156.812	−1.20958	−0.604790	−	0.796385i	$-0.706743\pi$
$7$	−156.812	−1.20958	−0.604790	+	0.796385i	$0.706743\pi$
$8$	0	0
$9$	354.040	1.45695
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1107.14	1.81696	0.908478	−	0.417933i	$-0.137245\pi$
$13$	1107.14	1.81696	0.908478	+	0.417933i	$0.137245\pi$
$14$	0	0
$15$	1402.61	1.60957
$16$	0	0
$17$	−1734.64	−1.45575	−0.727875	−	0.685710i	$-0.759492\pi$
$17$	−1734.64	−1.45575	−0.727875	+	0.685710i	$0.759492\pi$
$18$	0	0
$19$	−726.877	−0.461931	−0.230966	−	0.972962i	$-0.574188\pi$
$19$	−726.877	−0.461931	−0.230966	+	0.972962i	$0.574188\pi$
$20$	0	0
$21$	3831.61	1.89598
$22$	0	0
$23$	4796.66	1.89069	0.945343	−	0.326078i	$-0.105727\pi$
$23$	4796.66	1.89069	0.945343	+	0.326078i	$0.105727\pi$
$24$	0	0
$25$	170.132	0.0544424
$26$	0	0
$27$	−2713.20	−0.716262
$28$	0	0
$29$	−4674.99	−1.03225	−0.516125	−	0.856513i	$-0.672626\pi$
$29$	−4674.99	−1.03225	−0.516125	+	0.856513i	$0.672626\pi$
$30$	0	0
$31$	−2000.06	−0.373799	−0.186899	−	0.982379i	$-0.559844\pi$
$31$	−2000.06	−0.373799	−0.186899	+	0.982379i	$0.559844\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9001.52	1.24207
$36$	0	0
$37$	−10313.3	−1.23850	−0.619248	−	0.785195i	$-0.712563\pi$
$37$	−10313.3	−1.23850	−0.619248	+	0.785195i	$0.712563\pi$
$38$	0	0
$39$	−27052.3	−2.84802
$40$	0	0
$41$	−2626.79	−0.244042	−0.122021	−	0.992527i	$-0.538938\pi$
$41$	−2626.79	−0.244042	−0.122021	+	0.992527i	$0.538938\pi$
$42$	0	0
$43$	−9573.28	−0.789569	−0.394784	−	0.918774i	$-0.629181\pi$
$43$	−9573.28	−0.789569	−0.394784	+	0.918774i	$0.629181\pi$
$44$	0	0
$45$	−20323.0	−1.49609
$46$	0	0
$47$	3281.20	0.216665	0.108332	−	0.994115i	$-0.465449\pi$
$47$	3281.20	0.216665	0.108332	+	0.994115i	$0.465449\pi$
$48$	0	0
$49$	7783.01	0.463082
$50$	0	0
$51$	42384.9	2.28184
$52$	0	0
$53$	−25960.9	−1.26949	−0.634747	−	0.772720i	$-0.718896\pi$
$53$	−25960.9	−1.26949	−0.634747	+	0.772720i	$0.718896\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	17760.8	0.724062
$58$	0	0
$59$	−4423.25	−0.165429	−0.0827145	−	0.996573i	$-0.526359\pi$
$59$	−4423.25	−0.165429	−0.0827145	+	0.996573i	$0.526359\pi$
$60$	0	0
$61$	−15468.1	−0.532245	−0.266122	−	0.963939i	$-0.585743\pi$
$61$	−15468.1	−0.532245	−0.266122	+	0.963939i	$0.585743\pi$
$62$	0	0
$63$	−55517.7	−1.76230
$64$	0	0
$65$	−63553.5	−1.86576
$66$	0	0
$67$	−18768.3	−0.510786	−0.255393	−	0.966837i	$-0.582205\pi$
$67$	−18768.3	−0.510786	−0.255393	+	0.966837i	$0.582205\pi$
$68$	0	0
$69$	−117204.	−2.96359
$70$	0	0
$71$	−30765.2	−0.724291	−0.362146	−	0.932122i	$-0.617956\pi$
$71$	−30765.2	−0.724291	−0.362146	+	0.932122i	$0.617956\pi$
$72$	0	0
$73$	−62882.9	−1.38110	−0.690550	−	0.723284i	$-0.742632\pi$
$73$	−62882.9	−1.38110	−0.690550	+	0.723284i	$0.742632\pi$
$74$	0	0
$75$	−4157.08	−0.0853367
$76$	0	0
$77$	0	0
$78$	0	0
$79$	65539.3	1.18150	0.590750	−	0.806854i	$-0.298832\pi$
$79$	65539.3	1.18150	0.590750	+	0.806854i	$0.298832\pi$
$80$	0	0
$81$	−19736.4	−0.334237
$82$	0	0
$83$	8408.81	0.133980	0.0669899	−	0.997754i	$-0.478660\pi$
$83$	8408.81	0.133980	0.0669899	+	0.997754i	$0.478660\pi$
$84$	0	0
$85$	99573.9	1.49485
$86$	0	0
$87$	114230.	1.61802
$88$	0	0
$89$	774.583	0.0103656	0.00518278	−	0.999987i	$-0.498350\pi$
$89$	774.583	0.0103656	0.00518278	+	0.999987i	$0.498350\pi$
$90$	0	0
$91$	−173613.	−2.19775
$92$	0	0
$93$	48870.2	0.585918
$94$	0	0
$95$	41725.1	0.474339
$96$	0	0
$97$	−134159.	−1.44774	−0.723871	−	0.689935i	$-0.757639\pi$
$97$	−134159.	−1.44774	−0.723871	+	0.689935i	$0.757639\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	25403.6	0.247794	0.123897	−	0.992295i	$-0.460461\pi$
$101$	25403.6	0.247794	0.123897	+	0.992295i	$0.460461\pi$
$102$	0	0
$103$	−35684.3	−0.331424	−0.165712	−	0.986174i	$-0.552992\pi$
$103$	−35684.3	−0.331424	−0.165712	+	0.986174i	$0.552992\pi$
$104$	0	0
$105$	−219947.	−1.94690
$106$	0	0
$107$	89865.2	0.758808	0.379404	−	0.925231i	$-0.376129\pi$
$107$	89865.2	0.758808	0.379404	+	0.925231i	$0.376129\pi$
$108$	0	0
$109$	−12869.0	−0.103747	−0.0518737	−	0.998654i	$-0.516519\pi$
$109$	−12869.0	−0.103747	−0.0518737	+	0.998654i	$0.516519\pi$
$110$	0	0
$111$	252000.	1.94130
$112$	0	0
$113$	−90276.2	−0.665085	−0.332542	−	0.943088i	$-0.607906\pi$
$113$	−90276.2	−0.665085	−0.332542	+	0.943088i	$0.607906\pi$
$114$	0	0
$115$	−275344.	−1.94147
$116$	0	0
$117$	391972.	2.64722
$118$	0	0
$119$	272012.	1.76085
$120$	0	0
$121$	0	0
$122$	0	0
$123$	64184.0	0.382529
$124$	0	0
$125$	169619.	0.970956
$126$	0	0
$127$	30566.5	0.168165	0.0840826	−	0.996459i	$-0.473204\pi$
$127$	30566.5	0.168165	0.0840826	+	0.996459i	$0.473204\pi$
$128$	0	0
$129$	233917.	1.23762
$130$	0	0
$131$	−323898.	−1.64904	−0.824518	−	0.565836i	$-0.808553\pi$
$131$	−323898.	−1.64904	−0.824518	+	0.565836i	$0.808553\pi$
$132$	0	0
$133$	113983.	0.558742
$134$	0	0
$135$	155746.	0.735501
$136$	0	0
$137$	−184607.	−0.840326	−0.420163	−	0.907449i	$-0.638027\pi$
$137$	−184607.	−0.840326	−0.420163	+	0.907449i	$0.638027\pi$
$138$	0	0
$139$	274059.	1.20312	0.601558	−	0.798829i	$-0.294547\pi$
$139$	274059.	1.20312	0.601558	+	0.798829i	$0.294547\pi$
$140$	0	0
$141$	−80174.2	−0.339615
$142$	0	0
$143$	0	0
$144$	0	0
$145$	268359.	1.05998
$146$	0	0
$147$	−190173.	−0.725866
$148$	0	0
$149$	−215947.	−0.796859	−0.398430	−	0.917199i	$-0.630445\pi$
$149$	−215947.	−0.796859	−0.398430	+	0.917199i	$0.630445\pi$
$150$	0	0
$151$	−401718.	−1.43377	−0.716883	−	0.697193i	$-0.754432\pi$
$151$	−401718.	−1.43377	−0.716883	+	0.697193i	$0.754432\pi$
$152$	0	0
$153$	−614132.	−2.12096
$154$	0	0
$155$	114810.	0.383839
$156$	0	0
$157$	326553.	1.05732	0.528658	−	0.848835i	$-0.322696\pi$
$157$	326553.	1.05732	0.528658	+	0.848835i	$0.322696\pi$
$158$	0	0
$159$	634340.	1.98989
$160$	0	0
$161$	−752174.	−2.28693
$162$	0	0
$163$	−76329.8	−0.225022	−0.112511	−	0.993650i	$-0.535889\pi$
$163$	−76329.8	−0.225022	−0.112511	+	0.993650i	$0.535889\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−617643.	−1.71375	−0.856873	−	0.515527i	$-0.827596\pi$
$167$	−617643.	−1.71375	−0.856873	+	0.515527i	$0.827596\pi$
$168$	0	0
$169$	854467.	2.30133
$170$	0	0
$171$	−257344.	−0.673013
$172$	0	0
$173$	−302852.	−0.769335	−0.384668	−	0.923055i	$-0.625684\pi$
$173$	−302852.	−0.769335	−0.384668	+	0.923055i	$0.625684\pi$
$174$	0	0
$175$	−26678.8	−0.0658524
$176$	0	0
$177$	108080.	0.259305
$178$	0	0
$179$	−846750.	−1.97525	−0.987626	−	0.156826i	$-0.949874\pi$
$179$	−846750.	−1.97525	−0.987626	+	0.156826i	$0.949874\pi$
$180$	0	0
$181$	512903.	1.16369	0.581847	−	0.813298i	$-0.302330\pi$
$181$	512903.	1.16369	0.581847	+	0.813298i	$0.302330\pi$
$182$	0	0
$183$	377953.	0.834276
$184$	0	0
$185$	592019.	1.27176
$186$	0	0
$187$	0	0
$188$	0	0
$189$	425462.	0.866376
$190$	0	0
$191$	332097.	0.658690	0.329345	−	0.944210i	$-0.393172\pi$
$191$	332097.	0.658690	0.329345	+	0.944210i	$0.393172\pi$
$192$	0	0
$193$	−109663.	−0.211918	−0.105959	−	0.994371i	$-0.533791\pi$
$193$	−109663.	−0.211918	−0.105959	+	0.994371i	$0.533791\pi$
$194$	0	0
$195$	1.55289e6	2.92452
$196$	0	0
$197$	199830.	0.366855	0.183428	−	0.983033i	$-0.441281\pi$
$197$	199830.	0.366855	0.183428	+	0.983033i	$0.441281\pi$
$198$	0	0
$199$	−814887.	−1.45870	−0.729348	−	0.684143i	$-0.760176\pi$
$199$	−814887.	−1.45870	−0.729348	+	0.684143i	$0.760176\pi$
$200$	0	0
$201$	458593.	0.800640
$202$	0	0
$203$	733094.	1.24859
$204$	0	0
$205$	150786.	0.250597
$206$	0	0
$207$	1.69821e6	2.75464
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−366055.	−0.566030	−0.283015	−	0.959115i	$-0.591335\pi$
$211$	−366055.	−0.566030	−0.283015	+	0.959115i	$0.591335\pi$
$212$	0	0
$213$	751728.	1.13530
$214$	0	0
$215$	549538.	0.810777
$216$	0	0
$217$	313633.	0.452139
$218$	0	0
$219$	1.53651e6	2.16483
$220$	0	0
$221$	−1.92049e6	−2.64503
$222$	0	0
$223$	1.08456e6	1.46047	0.730233	−	0.683199i	$-0.239412\pi$
$223$	1.08456e6	1.46047	0.730233	+	0.683199i	$0.239412\pi$
$224$	0	0
$225$	60233.7	0.0793201
$226$	0	0
$227$	490127.	0.631312	0.315656	−	0.948874i	$-0.397775\pi$
$227$	490127.	0.631312	0.315656	+	0.948874i	$0.397775\pi$
$228$	0	0
$229$	−212187.	−0.267380	−0.133690	−	0.991023i	$-0.542683\pi$
$229$	−212187.	−0.267380	−0.133690	+	0.991023i	$0.542683\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−542640.	−0.654820	−0.327410	−	0.944882i	$-0.606176\pi$
$233$	−542640.	−0.654820	−0.327410	+	0.944882i	$0.606176\pi$
$234$	0	0
$235$	−188352.	−0.222485
$236$	0	0
$237$	−1.60141e6	−1.85196
$238$	0	0
$239$	337506.	0.382197	0.191098	−	0.981571i	$-0.438795\pi$
$239$	337506.	0.382197	0.191098	+	0.981571i	$0.438795\pi$
$240$	0	0
$241$	−540867.	−0.599857	−0.299929	−	0.953962i	$-0.596963\pi$
$241$	−540867.	−0.599857	−0.299929	+	0.953962i	$0.596963\pi$
$242$	0	0
$243$	1.14155e6	1.24017
$244$	0	0
$245$	−446770.	−0.475520
$246$	0	0
$247$	−804756.	−0.839308
$248$	0	0
$249$	−205464.	−0.210009
$250$	0	0
$251$	363862.	0.364546	0.182273	−	0.983248i	$-0.441655\pi$
$251$	363862.	0.364546	0.182273	+	0.983248i	$0.441655\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.43303e6	−2.34313
$256$	0	0
$257$	−1.95166e6	−1.84320	−0.921598	−	0.388145i	$-0.873116\pi$
$257$	−1.95166e6	−1.84320	−0.921598	+	0.388145i	$0.873116\pi$
$258$	0	0
$259$	1.61725e6	1.49806
$260$	0	0
$261$	−1.65513e6	−1.50394
$262$	0	0
$263$	878155.	0.782856	0.391428	−	0.920209i	$-0.371981\pi$
$263$	878155.	0.782856	0.391428	+	0.920209i	$0.371981\pi$
$264$	0	0
$265$	1.49024e6	1.30359
$266$	0	0
$267$	−18926.5	−0.0162477
$268$	0	0
$269$	1.88478e6	1.58810	0.794052	−	0.607849i	$-0.207968\pi$
$269$	1.88478e6	1.58810	0.794052	+	0.607849i	$0.207968\pi$
$270$	0	0
$271$	1.59141e6	1.31631	0.658155	−	0.752883i	$-0.271337\pi$
$271$	1.59141e6	1.31631	0.658155	+	0.752883i	$0.271337\pi$
$272$	0	0
$273$	4.24213e6	3.44491
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.79542e6	−1.40594	−0.702969	−	0.711220i	$-0.748143\pi$
$277$	−1.79542e6	−1.40594	−0.702969	+	0.711220i	$0.748143\pi$
$278$	0	0
$279$	−708100.	−0.544608
$280$	0	0
$281$	−1.46354e6	−1.10570	−0.552851	−	0.833280i	$-0.686460\pi$
$281$	−1.46354e6	−1.10570	−0.552851	+	0.833280i	$0.686460\pi$
$282$	0	0
$283$	171872.	0.127567	0.0637836	−	0.997964i	$-0.479683\pi$
$283$	171872.	0.127567	0.0637836	+	0.997964i	$0.479683\pi$
$284$	0	0
$285$	−1.01953e6	−0.743511
$286$	0	0
$287$	411912.	0.295189
$288$	0	0
$289$	1.58912e6	1.11921
$290$	0	0
$291$	3.27810e6	2.26929
$292$	0	0
$293$	−110511.	−0.0752031	−0.0376015	−	0.999293i	$-0.511972\pi$
$293$	−110511.	−0.0752031	−0.0376015	+	0.999293i	$0.511972\pi$
$294$	0	0
$295$	253909.	0.169873
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.31058e6	3.43529
$300$	0	0
$301$	1.50121e6	0.955046
$302$	0	0
$303$	−620721.	−0.388410
$304$	0	0
$305$	887917.	0.546541
$306$	0	0
$307$	−79497.8	−0.0481404	−0.0240702	−	0.999710i	$-0.507663\pi$
$307$	−79497.8	−0.0481404	−0.0240702	+	0.999710i	$0.507663\pi$
$308$	0	0
$309$	871925.	0.519497
$310$	0	0
$311$	366690.	0.214980	0.107490	−	0.994206i	$-0.465719\pi$
$311$	366690.	0.214980	0.107490	+	0.994206i	$0.465719\pi$
$312$	0	0
$313$	2.52426e6	1.45638	0.728188	−	0.685377i	$-0.240363\pi$
$313$	2.52426e6	1.45638	0.728188	+	0.685377i	$0.240363\pi$
$314$	0	0
$315$	3.18690e6	1.80964
$316$	0	0
$317$	2.41151e6	1.34785	0.673924	−	0.738801i	$-0.264608\pi$
$317$	2.41151e6	1.34785	0.673924	+	0.738801i	$0.264608\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−2.19580e6	−1.18941
$322$	0	0
$323$	1.26087e6	0.672456
$324$	0	0
$325$	188361.	0.0989194
$326$	0	0
$327$	314445.	0.162621
$328$	0	0
$329$	−514532.	−0.262073
$330$	0	0
$331$	−779307.	−0.390966	−0.195483	−	0.980707i	$-0.562627\pi$
$331$	−779307.	−0.390966	−0.195483	+	0.980707i	$0.562627\pi$
$332$	0	0
$333$	−3.65133e6	−1.80443
$334$	0	0
$335$	1.07736e6	0.524506
$336$	0	0
$337$	287133.	0.137724	0.0688619	−	0.997626i	$-0.478063\pi$
$337$	287133.	0.137724	0.0688619	+	0.997626i	$0.478063\pi$
$338$	0	0
$339$	2.20584e6	1.04250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.41507e6	0.649445
$344$	0	0
$345$	6.72786e6	3.04319
$346$	0	0
$347$	−1.49615e6	−0.667038	−0.333519	−	0.942743i	$-0.608236\pi$
$347$	−1.49615e6	−0.667038	−0.333519	+	0.942743i	$0.608236\pi$
$348$	0	0
$349$	−395477.	−0.173803	−0.0869016	−	0.996217i	$-0.527697\pi$
$349$	−395477.	−0.173803	−0.0869016	+	0.996217i	$0.527697\pi$
$350$	0	0
$351$	−3.00389e6	−1.30142
$352$	0	0
$353$	−414432.	−0.177017	−0.0885087	−	0.996075i	$-0.528210\pi$
$353$	−414432.	−0.177017	−0.0885087	+	0.996075i	$0.528210\pi$
$354$	0	0
$355$	1.76602e6	0.743746
$356$	0	0
$357$	−6.64646e6	−2.76007
$358$	0	0
$359$	1.82171e6	0.746005	0.373003	−	0.927830i	$-0.378328\pi$
$359$	1.82171e6	0.746005	0.373003	+	0.927830i	$0.378328\pi$
$360$	0	0
$361$	−1.94775e6	−0.786620
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.60968e6	1.41820
$366$	0	0
$367$	−988223.	−0.382992	−0.191496	−	0.981493i	$-0.561334\pi$
$367$	−988223.	−0.382992	−0.191496	+	0.981493i	$0.561334\pi$
$368$	0	0
$369$	−929988.	−0.355559
$370$	0	0
$371$	4.07099e6	1.53555
$372$	0	0
$373$	1.33761e6	0.497802	0.248901	−	0.968529i	$-0.419931\pi$
$373$	1.33761e6	0.497802	0.248901	+	0.968529i	$0.419931\pi$
$374$	0	0
$375$	−4.14454e6	−1.52194
$376$	0	0
$377$	−5.17587e6	−1.87555
$378$	0	0
$379$	1.22299e6	0.437345	0.218673	−	0.975798i	$-0.429827\pi$
$379$	1.22299e6	0.437345	0.218673	+	0.975798i	$0.429827\pi$
$380$	0	0
$381$	−746873.	−0.263593
$382$	0	0
$383$	−2.28304e6	−0.795273	−0.397637	−	0.917543i	$-0.630169\pi$
$383$	−2.28304e6	−0.795273	−0.397637	+	0.917543i	$0.630169\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.38933e6	−1.15037
$388$	0	0
$389$	−1.32923e6	−0.445377	−0.222688	−	0.974890i	$-0.571483\pi$
$389$	−1.32923e6	−0.445377	−0.222688	+	0.974890i	$0.571483\pi$
$390$	0	0
$391$	−8.32048e6	−2.75237
$392$	0	0
$393$	7.91425e6	2.58481
$394$	0	0
$395$	−3.76217e6	−1.21324
$396$	0	0
$397$	2.18246e6	0.694975	0.347488	−	0.937685i	$-0.387035\pi$
$397$	2.18246e6	0.694975	0.347488	+	0.937685i	$0.387035\pi$
$398$	0	0
$399$	−2.78511e6	−0.875810
$400$	0	0
$401$	−2.24138e6	−0.696072	−0.348036	−	0.937481i	$-0.613151\pi$
$401$	−2.24138e6	−0.696072	−0.348036	+	0.937481i	$0.613151\pi$
$402$	0	0
$403$	−2.21434e6	−0.679176
$404$	0	0
$405$	1.13293e6	0.343215
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−4.39799e6	−1.30001	−0.650003	−	0.759931i	$-0.725232\pi$
$409$	−4.39799e6	−1.30001	−0.650003	+	0.759931i	$0.725232\pi$
$410$	0	0
$411$	4.51077e6	1.31718
$412$	0	0
$413$	693619.	0.200100
$414$	0	0
$415$	−482693.	−0.137579
$416$	0	0
$417$	−6.69647e6	−1.88585
$418$	0	0
$419$	5.12999e6	1.42752	0.713759	−	0.700391i	$-0.246991\pi$
$419$	5.12999e6	1.42752	0.713759	+	0.700391i	$0.246991\pi$
$420$	0	0
$421$	−1.24163e6	−0.341419	−0.170709	−	0.985321i	$-0.554606\pi$
$421$	−1.24163e6	−0.341419	−0.170709	+	0.985321i	$0.554606\pi$
$422$	0	0
$423$	1.16168e6	0.315671
$424$	0	0
$425$	−295118.	−0.0792545
$426$	0	0
$427$	2.42558e6	0.643792
$428$	0	0
$429$	0	0
$430$	0	0
$431$	270238.	0.0700734	0.0350367	−	0.999386i	$-0.488845\pi$
$431$	270238.	0.0700734	0.0350367	+	0.999386i	$0.488845\pi$
$432$	0	0
$433$	−2.59493e6	−0.665128	−0.332564	−	0.943081i	$-0.607914\pi$
$433$	−2.59493e6	−0.665128	−0.332564	+	0.943081i	$0.607914\pi$
$434$	0	0
$435$	−6.55720e6	−1.66148
$436$	0	0
$437$	−3.48658e6	−0.873367
$438$	0	0
$439$	−6.26302e6	−1.55104	−0.775519	−	0.631325i	$-0.782512\pi$
$439$	−6.26302e6	−1.55104	−0.775519	+	0.631325i	$0.782512\pi$
$440$	0	0
$441$	2.75550e6	0.674689
$442$	0	0
$443$	98175.1	0.0237680	0.0118840	−	0.999929i	$-0.496217\pi$
$443$	98175.1	0.0237680	0.0118840	+	0.999929i	$0.496217\pi$
$444$	0	0
$445$	−44463.6	−0.0106440
$446$	0	0
$447$	5.27654e6	1.24905
$448$	0	0
$449$	−8.28515e6	−1.93948	−0.969738	−	0.244148i	$-0.921492\pi$
$449$	−8.28515e6	−1.93948	−0.969738	+	0.244148i	$0.921492\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	9.81573e6	2.24738
$454$	0	0
$455$	9.96595e6	2.25678
$456$	0	0
$457$	1.55619e6	0.348556	0.174278	−	0.984696i	$-0.444241\pi$
$457$	1.55619e6	0.348556	0.174278	+	0.984696i	$0.444241\pi$
$458$	0	0
$459$	4.70642e6	1.04270
$460$	0	0
$461$	−601195.	−0.131754	−0.0658769	−	0.997828i	$-0.520984\pi$
$461$	−601195.	−0.131754	−0.0658769	+	0.997828i	$0.520984\pi$
$462$	0	0
$463$	−1.17902e6	−0.255605	−0.127802	−	0.991800i	$-0.540792\pi$
$463$	−1.17902e6	−0.255605	−0.127802	+	0.991800i	$0.540792\pi$
$464$	0	0
$465$	−2.80531e6	−0.601656
$466$	0	0
$467$	−3.20338e6	−0.679697	−0.339849	−	0.940480i	$-0.610376\pi$
$467$	−3.20338e6	−0.679697	−0.339849	+	0.940480i	$0.610376\pi$
$468$	0	0
$469$	2.94310e6	0.617836
$470$	0	0
$471$	−7.97913e6	−1.65731
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−123665.	−0.0251486
$476$	0	0
$477$	−9.19121e6	−1.84960
$478$	0	0
$479$	−6.41218e6	−1.27693	−0.638465	−	0.769651i	$-0.720430\pi$
$479$	−6.41218e6	−1.27693	−0.638465	+	0.769651i	$0.720430\pi$
$480$	0	0
$481$	−1.14183e7	−2.25029
$482$	0	0
$483$	1.83789e7	3.58470
$484$	0	0
$485$	7.70118e6	1.48663
$486$	0	0
$487$	5.46574e6	1.04430	0.522152	−	0.852852i	$-0.325129\pi$
$487$	5.46574e6	1.04430	0.522152	+	0.852852i	$0.325129\pi$
$488$	0	0
$489$	1.86507e6	0.352715
$490$	0	0
$491$	4.49258e6	0.840992	0.420496	−	0.907294i	$-0.361856\pi$
$491$	4.49258e6	0.840992	0.420496	+	0.907294i	$0.361856\pi$
$492$	0	0
$493$	8.10942e6	1.50270
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.82435e6	0.876088
$498$	0	0
$499$	−6.91461e6	−1.24313	−0.621565	−	0.783363i	$-0.713503\pi$
$499$	−6.91461e6	−1.24313	−0.621565	+	0.783363i	$0.713503\pi$
$500$	0	0
$501$	1.50917e7	2.68624
$502$	0	0
$503$	−1.30731e6	−0.230387	−0.115193	−	0.993343i	$-0.536749\pi$
$503$	−1.30731e6	−0.230387	−0.115193	+	0.993343i	$0.536749\pi$
$504$	0	0
$505$	−1.45825e6	−0.254450
$506$	0	0
$507$	−2.08784e7	−3.60726
$508$	0	0
$509$	−5.48520e6	−0.938422	−0.469211	−	0.883086i	$-0.655462\pi$
$509$	−5.48520e6	−0.938422	−0.469211	+	0.883086i	$0.655462\pi$
$510$	0	0
$511$	9.86080e6	1.67055
$512$	0	0
$513$	1.97216e6	0.330864
$514$	0	0
$515$	2.04840e6	0.340326
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.40001e6	1.20591
$520$	0	0
$521$	−4.73226e6	−0.763791	−0.381895	−	0.924206i	$-0.624729\pi$
$521$	−4.73226e6	−0.763791	−0.381895	+	0.924206i	$0.624729\pi$
$522$	0	0
$523$	1.48341e6	0.237141	0.118571	−	0.992946i	$-0.462169\pi$
$523$	1.48341e6	0.237141	0.118571	+	0.992946i	$0.462169\pi$
$524$	0	0
$525$	651881.	0.103221
$526$	0	0
$527$	3.46938e6	0.544158
$528$	0	0
$529$	1.65716e7	2.57469
$530$	0	0
$531$	−1.56601e6	−0.241023
$532$	0	0
$533$	−2.90822e6	−0.443414
$534$	0	0
$535$	−5.15855e6	−0.779190
$536$	0	0
$537$	2.06898e7	3.09614
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.57610e6	−0.525310	−0.262655	−	0.964890i	$-0.584598\pi$
$541$	−3.57610e6	−0.525310	−0.262655	+	0.964890i	$0.584598\pi$
$542$	0	0
$543$	−1.25325e7	−1.82405
$544$	0	0
$545$	738720.	0.106534
$546$	0	0
$547$	−6.89639e6	−0.985493	−0.492746	−	0.870173i	$-0.664007\pi$
$547$	−6.89639e6	−0.985493	−0.492746	+	0.870173i	$0.664007\pi$
$548$	0	0
$549$	−5.47631e6	−0.775457
$550$	0	0
$551$	3.39814e6	0.476829
$552$	0	0
$553$	−1.02773e7	−1.42912
$554$	0	0
$555$	−1.44656e7	−1.99345
$556$	0	0
$557$	−3.72149e6	−0.508252	−0.254126	−	0.967171i	$-0.581788\pi$
$557$	−3.72149e6	−0.508252	−0.254126	+	0.967171i	$0.581788\pi$
$558$	0	0
$559$	−1.05990e7	−1.43461
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.39390e6	−0.717186	−0.358593	−	0.933494i	$-0.616743\pi$
$563$	−5.39390e6	−0.717186	−0.358593	+	0.933494i	$0.616743\pi$
$564$	0	0
$565$	5.18214e6	0.682949
$566$	0	0
$567$	3.09490e6	0.404286
$568$	0	0
$569$	2.90466e6	0.376110	0.188055	−	0.982159i	$-0.439782\pi$
$569$	2.90466e6	0.376110	0.188055	+	0.982159i	$0.439782\pi$
$570$	0	0
$571$	636912.	0.0817503	0.0408752	−	0.999164i	$-0.486985\pi$
$571$	636912.	0.0817503	0.0408752	+	0.999164i	$0.486985\pi$
$572$	0	0
$573$	−8.11458e6	−1.03248
$574$	0	0
$575$	816067.	0.102933
$576$	0	0
$577$	1.29002e7	1.61308	0.806540	−	0.591179i	$-0.201337\pi$
$577$	1.29002e7	1.61308	0.806540	+	0.591179i	$0.201337\pi$
$578$	0	0
$579$	2.67955e6	0.332174
$580$	0	0
$581$	−1.31860e6	−0.162059
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.25005e7	−2.71833
$586$	0	0
$587$	1.25428e7	1.50245	0.751225	−	0.660046i	$-0.229463\pi$
$587$	1.25428e7	1.50245	0.751225	+	0.660046i	$0.229463\pi$
$588$	0	0
$589$	1.45380e6	0.172669
$590$	0	0
$591$	−4.88273e6	−0.575034
$592$	0	0
$593$	1.61155e7	1.88195	0.940973	−	0.338483i	$-0.109914\pi$
$593$	1.61155e7	1.88195	0.940973	+	0.338483i	$0.109914\pi$
$594$	0	0
$595$	−1.56144e7	−1.80814
$596$	0	0
$597$	1.99113e7	2.28646
$598$	0	0
$599$	−4.33869e6	−0.494073	−0.247037	−	0.969006i	$-0.579457\pi$
$599$	−4.33869e6	−0.494073	−0.247037	+	0.969006i	$0.579457\pi$
$600$	0	0
$601$	−1.61722e7	−1.82634	−0.913172	−	0.407574i	$-0.866375\pi$
$601$	−1.61722e7	−1.82634	−0.913172	+	0.407574i	$0.866375\pi$
$602$	0	0
$603$	−6.64474e6	−0.744192
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.14078e6	0.125670	0.0628350	−	0.998024i	$-0.479986\pi$
$607$	1.14078e6	0.125670	0.0628350	+	0.998024i	$0.479986\pi$
$608$	0	0
$609$	−1.79127e7	−1.95712
$610$	0	0
$611$	3.63275e6	0.393670
$612$	0	0
$613$	8.17387e6	0.878570	0.439285	−	0.898348i	$-0.355232\pi$
$613$	8.17387e6	0.878570	0.439285	+	0.898348i	$0.355232\pi$
$614$	0	0
$615$	−3.68437e6	−0.392803
$616$	0	0
$617$	7.66601e6	0.810693	0.405346	−	0.914163i	$-0.367151\pi$
$617$	7.66601e6	0.810693	0.405346	+	0.914163i	$0.367151\pi$
$618$	0	0
$619$	1.44957e7	1.52059	0.760297	−	0.649576i	$-0.225054\pi$
$619$	1.44957e7	1.52059	0.760297	+	0.649576i	$0.225054\pi$
$620$	0	0
$621$	−1.30143e7	−1.35423
$622$	0	0
$623$	−121464.	−0.0125380
$624$	0	0
$625$	−1.02683e7	−1.05148
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.78899e7	1.80294
$630$	0	0
$631$	2.99105e6	0.299055	0.149527	−	0.988758i	$-0.452225\pi$
$631$	2.99105e6	0.299055	0.149527	+	0.988758i	$0.452225\pi$
$632$	0	0
$633$	8.94432e6	0.887234
$634$	0	0
$635$	−1.75461e6	−0.172682
$636$	0	0
$637$	8.61689e6	0.841399
$638$	0	0
$639$	−1.08921e7	−1.05526
$640$	0	0
$641$	−4.81819e6	−0.463168	−0.231584	−	0.972815i	$-0.574391\pi$
$641$	−4.81819e6	−0.463168	−0.231584	+	0.972815i	$0.574391\pi$
$642$	0	0
$643$	1.73575e7	1.65561	0.827807	−	0.561014i	$-0.189588\pi$
$643$	1.73575e7	1.65561	0.827807	+	0.561014i	$0.189588\pi$
$644$	0	0
$645$	−1.34276e7	−1.27087
$646$	0	0
$647$	−7.63214e6	−0.716780	−0.358390	−	0.933572i	$-0.616674\pi$
$647$	−7.63214e6	−0.716780	−0.358390	+	0.933572i	$0.616674\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−7.66344e6	−0.708714
$652$	0	0
$653$	1.16510e7	1.06926	0.534628	−	0.845087i	$-0.320452\pi$
$653$	1.16510e7	1.06926	0.534628	+	0.845087i	$0.320452\pi$
$654$	0	0
$655$	1.85928e7	1.69333
$656$	0	0
$657$	−2.22631e7	−2.01220
$658$	0	0
$659$	8.86205e6	0.794915	0.397458	−	0.917621i	$-0.369893\pi$
$659$	8.86205e6	0.794915	0.397458	+	0.917621i	$0.369893\pi$
$660$	0	0
$661$	−1.73414e6	−0.154376	−0.0771882	−	0.997017i	$-0.524594\pi$
$661$	−1.73414e6	−0.154376	−0.0771882	+	0.997017i	$0.524594\pi$
$662$	0	0
$663$	4.69260e7	4.14601
$664$	0	0
$665$	−6.54300e6	−0.573750
$666$	0	0
$667$	−2.24243e7	−1.95166
$668$	0	0
$669$	−2.65006e7	−2.28923
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.12145e7	−1.80549	−0.902743	−	0.430179i	$-0.858450\pi$
$673$	−2.12145e7	−1.80549	−0.902743	+	0.430179i	$0.858450\pi$
$674$	0	0
$675$	−461603.	−0.0389950
$676$	0	0
$677$	−7.11966e6	−0.597018	−0.298509	−	0.954407i	$-0.596489\pi$
$677$	−7.11966e6	−0.597018	−0.298509	+	0.954407i	$0.596489\pi$
$678$	0	0
$679$	2.10378e7	1.75116
$680$	0	0
$681$	−1.19760e7	−0.989562
$682$	0	0
$683$	−6.49102e6	−0.532428	−0.266214	−	0.963914i	$-0.585773\pi$
$683$	−6.49102e6	−0.532428	−0.266214	+	0.963914i	$0.585773\pi$
$684$	0	0
$685$	1.05971e7	0.862898
$686$	0	0
$687$	5.18466e6	0.419110
$688$	0	0
$689$	−2.87424e7	−2.30662
$690$	0	0
$691$	3.35308e6	0.267146	0.133573	−	0.991039i	$-0.457355\pi$
$691$	3.35308e6	0.267146	0.133573	+	0.991039i	$0.457355\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.57319e7	−1.23543
$696$	0	0
$697$	4.55653e6	0.355265
$698$	0	0
$699$	1.32591e7	1.02641
$700$	0	0
$701$	−2.22894e7	−1.71318	−0.856592	−	0.515995i	$-0.827422\pi$
$701$	−2.22894e7	−1.71318	−0.856592	+	0.515995i	$0.827422\pi$
$702$	0	0
$703$	7.49653e6	0.572100
$704$	0	0
$705$	4.60226e6	0.348737
$706$	0	0
$707$	−3.98359e6	−0.299727
$708$	0	0
$709$	−2.20316e7	−1.64600	−0.823001	−	0.568039i	$-0.807702\pi$
$709$	−2.20316e7	−1.64600	−0.823001	+	0.568039i	$0.807702\pi$
$710$	0	0
$711$	2.32035e7	1.72139
$712$	0	0
$713$	−9.59359e6	−0.706736
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.24677e6	−0.599081
$718$	0	0
$719$	−1.29186e7	−0.931955	−0.465977	−	0.884797i	$-0.654297\pi$
$719$	−1.29186e7	−0.931955	−0.465977	+	0.884797i	$0.654297\pi$
$720$	0	0
$721$	5.59573e6	0.400884
$722$	0	0
$723$	1.32158e7	0.940257
$724$	0	0
$725$	−795367.	−0.0561982
$726$	0	0
$727$	4.83500e6	0.339282	0.169641	−	0.985506i	$-0.445739\pi$
$727$	4.83500e6	0.339282	0.169641	+	0.985506i	$0.445739\pi$
$728$	0	0
$729$	−2.30972e7	−1.60969
$730$	0	0
$731$	1.66062e7	1.14941
$732$	0	0
$733$	6.52019e6	0.448230	0.224115	−	0.974563i	$-0.428051\pi$
$733$	6.52019e6	0.448230	0.224115	+	0.974563i	$0.428051\pi$
$734$	0	0
$735$	1.09166e7	0.745363
$736$	0	0
$737$	0	0
$738$	0	0
$739$	2.68574e7	1.80906	0.904530	−	0.426409i	$-0.140222\pi$
$739$	2.68574e7	1.80906	0.904530	+	0.426409i	$0.140222\pi$
$740$	0	0
$741$	1.96637e7	1.31559
$742$	0	0
$743$	−6.89024e6	−0.457891	−0.228946	−	0.973439i	$-0.573528\pi$
$743$	−6.89024e6	−0.457891	−0.228946	+	0.973439i	$0.573528\pi$
$744$	0	0
$745$	1.23961e7	0.818263
$746$	0	0
$747$	2.97706e6	0.195203
$748$	0	0
$749$	−1.40919e7	−0.917838
$750$	0	0
$751$	−6.59491e6	−0.426686	−0.213343	−	0.976977i	$-0.568435\pi$
$751$	−6.59491e6	−0.426686	−0.213343	+	0.976977i	$0.568435\pi$
$752$	0	0
$753$	−8.89075e6	−0.571414
$754$	0	0
$755$	2.30599e7	1.47228
$756$	0	0
$757$	2.83324e7	1.79698	0.898492	−	0.438990i	$-0.144664\pi$
$757$	2.83324e7	1.79698	0.898492	+	0.438990i	$0.144664\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.03429e6	−0.189931	−0.0949653	−	0.995481i	$-0.530274\pi$
$761$	−3.03429e6	−0.189931	−0.0949653	+	0.995481i	$0.530274\pi$
$762$	0	0
$763$	2.01801e6	0.125491
$764$	0	0
$765$	3.52532e7	2.17793
$766$	0	0
$767$	−4.89716e6	−0.300577
$768$	0	0
$769$	−1.31846e7	−0.803993	−0.401996	−	0.915641i	$-0.631684\pi$
$769$	−1.31846e7	−0.803993	−0.401996	+	0.915641i	$0.631684\pi$
$770$	0	0
$771$	4.76877e7	2.88915
$772$	0	0
$773$	2.57172e7	1.54802	0.774008	−	0.633175i	$-0.218249\pi$
$773$	2.57172e7	1.54802	0.774008	+	0.633175i	$0.218249\pi$
$774$	0	0
$775$	−340275.	−0.0203505
$776$	0	0
$777$	−3.95167e7	−2.34816
$778$	0	0
$779$	1.90935e6	0.112731
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.26842e7	0.739362
$784$	0	0
$785$	−1.87452e7	−1.08572
$786$	0	0
$787$	6.29391e6	0.362229	0.181115	−	0.983462i	$-0.442029\pi$
$787$	6.29391e6	0.362229	0.181115	+	0.983462i	$0.442029\pi$
$788$	0	0
$789$	−2.14572e7	−1.22710
$790$	0	0
$791$	1.41564e7	0.804473
$792$	0	0
$793$	−1.71253e7	−0.967065
$794$	0	0
$795$	−3.64132e7	−2.04334
$796$	0	0
$797$	−2.11720e7	−1.18064	−0.590319	−	0.807170i	$-0.700998\pi$
$797$	−2.11720e7	−1.18064	−0.590319	+	0.807170i	$0.700998\pi$
$798$	0	0
$799$	−5.69170e6	−0.315410
$800$	0	0
$801$	274233.	0.0151022
$802$	0	0
$803$	0	0
$804$	0	0
$805$	4.31772e7	2.34836
$806$	0	0
$807$	−4.60534e7	−2.48930
$808$	0	0
$809$	−1.03526e7	−0.556132	−0.278066	−	0.960562i	$-0.589693\pi$
$809$	−1.03526e7	−0.556132	−0.278066	+	0.960562i	$0.589693\pi$
$810$	0	0
$811$	−2.84973e7	−1.52143	−0.760715	−	0.649086i	$-0.775152\pi$
$811$	−2.84973e7	−1.52143	−0.760715	+	0.649086i	$0.775152\pi$
$812$	0	0
$813$	−3.88851e7	−2.06327
$814$	0	0
$815$	4.38158e6	0.231066
$816$	0	0
$817$	6.95861e6	0.364726
$818$	0	0
$819$	−6.14659e7	−3.20203
$820$	0	0
$821$	−2.11560e6	−0.109541	−0.0547703	−	0.998499i	$-0.517443\pi$
$821$	−2.11560e6	−0.109541	−0.0547703	+	0.998499i	$0.517443\pi$
$822$	0	0
$823$	−3.20795e7	−1.65093	−0.825463	−	0.564456i	$-0.809086\pi$
$823$	−3.20795e7	−1.65093	−0.825463	+	0.564456i	$0.809086\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.14157e6	0.312260	0.156130	−	0.987737i	$-0.450098\pi$
$827$	6.14157e6	0.312260	0.156130	+	0.987737i	$0.450098\pi$
$828$	0	0
$829$	3.57939e7	1.80894	0.904468	−	0.426542i	$-0.140268\pi$
$829$	3.57939e7	1.80894	0.904468	+	0.426542i	$0.140268\pi$
$830$	0	0
$831$	4.38700e7	2.20376
$832$	0	0
$833$	−1.35007e7	−0.674131
$834$	0	0
$835$	3.54547e7	1.75978
$836$	0	0
$837$	5.42655e6	0.267738
$838$	0	0
$839$	−1.82982e7	−0.897438	−0.448719	−	0.893673i	$-0.648120\pi$
$839$	−1.82982e7	−0.897438	−0.448719	+	0.893673i	$0.648120\pi$
$840$	0	0
$841$	1.34434e6	0.0655420
$842$	0	0
$843$	3.57607e7	1.73315
$844$	0	0
$845$	−4.90492e7	−2.36314
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.19959e6	−0.199958
$850$	0	0
$851$	−4.94696e7	−2.34161
$852$	0	0
$853$	2.64368e7	1.24405	0.622023	−	0.782999i	$-0.286311\pi$
$853$	2.64368e7	1.24405	0.622023	+	0.782999i	$0.286311\pi$
$854$	0	0
$855$	1.47724e7	0.691090
$856$	0	0
$857$	7.98183e6	0.371236	0.185618	−	0.982622i	$-0.440571\pi$
$857$	7.98183e6	0.371236	0.185618	+	0.982622i	$0.440571\pi$
$858$	0	0
$859$	3.32876e7	1.53922	0.769609	−	0.638516i	$-0.220451\pi$
$859$	3.32876e7	1.53922	0.769609	+	0.638516i	$0.220451\pi$
$860$	0	0
$861$	−1.00648e7	−0.462699
$862$	0	0
$863$	3.90759e7	1.78600	0.893001	−	0.450056i	$-0.148596\pi$
$863$	3.90759e7	1.78600	0.893001	+	0.450056i	$0.148596\pi$
$864$	0	0
$865$	1.73847e7	0.790000
$866$	0	0
$867$	−3.88291e7	−1.75432
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−2.07792e7	−0.928076
$872$	0	0
$873$	−4.74978e7	−2.10930
$874$	0	0
$875$	−2.65983e7	−1.17445
$876$	0	0
$877$	8.06867e6	0.354244	0.177122	−	0.984189i	$-0.443321\pi$
$877$	8.06867e6	0.354244	0.177122	+	0.984189i	$0.443321\pi$
$878$	0	0
$879$	2.70026e6	0.117878
$880$	0	0
$881$	−8.26109e6	−0.358589	−0.179295	−	0.983795i	$-0.557382\pi$
$881$	−8.26109e6	−0.358589	−0.179295	+	0.983795i	$0.557382\pi$
$882$	0	0
$883$	3.47617e7	1.50037	0.750186	−	0.661227i	$-0.229964\pi$
$883$	3.47617e7	1.50037	0.750186	+	0.661227i	$0.229964\pi$
$884$	0	0
$885$	−6.20412e6	−0.266270
$886$	0	0
$887$	−2.00670e7	−0.856394	−0.428197	−	0.903685i	$-0.640851\pi$
$887$	−2.00670e7	−0.856394	−0.428197	+	0.903685i	$0.640851\pi$
$888$	0	0
$889$	−4.79319e6	−0.203409
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.38503e6	−0.100084
$894$	0	0
$895$	4.86062e7	2.02831
$896$	0	0
$897$	−1.29761e8	−5.38471
$898$	0	0
$899$	9.35024e6	0.385854
$900$	0	0
$901$	4.50329e7	1.84807
$902$	0	0
$903$	−3.66811e7	−1.49700
$904$	0	0
$905$	−2.94423e7	−1.19495
$906$	0	0
$907$	−9.14734e6	−0.369213	−0.184606	−	0.982813i	$-0.559101\pi$
$907$	−9.14734e6	−0.369213	−0.184606	+	0.982813i	$0.559101\pi$
$908$	0	0
$909$	8.99388e6	0.361025
$910$	0	0
$911$	−1.02902e7	−0.410799	−0.205400	−	0.978678i	$-0.565849\pi$
$911$	−1.02902e7	−0.410799	−0.205400	+	0.978678i	$0.565849\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−2.16957e7	−0.856685
$916$	0	0
$917$	5.07911e7	1.99464
$918$	0	0
$919$	−2.93383e7	−1.14590	−0.572949	−	0.819591i	$-0.694201\pi$
$919$	−2.93383e7	−1.14590	−0.572949	+	0.819591i	$0.694201\pi$
$920$	0	0
$921$	1.94248e6	0.0754585
$922$	0	0
$923$	−3.40614e7	−1.31601
$924$	0	0
$925$	−1.75463e6	−0.0674267
$926$	0	0
$927$	−1.26337e7	−0.482870
$928$	0	0
$929$	5.04761e7	1.91887	0.959437	−	0.281922i	$-0.0909720\pi$
$929$	5.04761e7	1.91887	0.959437	+	0.281922i	$0.0909720\pi$
$930$	0	0
$931$	−5.65730e6	−0.213912
$932$	0	0
$933$	−8.95984e6	−0.336974
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.33405e7	1.61267	0.806334	−	0.591460i	$-0.201448\pi$
$937$	4.33405e7	1.61267	0.806334	+	0.591460i	$0.201448\pi$
$938$	0	0
$939$	−6.16788e7	−2.28282
$940$	0	0
$941$	1.71534e7	0.631503	0.315751	−	0.948842i	$-0.397743\pi$
$941$	1.71534e7	0.631503	0.315751	+	0.948842i	$0.397743\pi$
$942$	0	0
$943$	−1.25998e7	−0.461407
$944$	0	0
$945$	−2.44229e7	−0.889647
$946$	0	0
$947$	4.69793e6	0.170228	0.0851141	−	0.996371i	$-0.472875\pi$
$947$	4.69793e6	0.170228	0.0851141	+	0.996371i	$0.472875\pi$
$948$	0	0
$949$	−6.96202e7	−2.50940
$950$	0	0
$951$	−5.89238e7	−2.11271
$952$	0	0
$953$	−4.71615e7	−1.68212	−0.841058	−	0.540946i	$-0.818066\pi$
$953$	−4.71615e7	−1.68212	−0.841058	+	0.540946i	$0.818066\pi$
$954$	0	0
$955$	−1.90634e7	−0.676383
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.89487e7	1.01644
$960$	0	0
$961$	−2.46289e7	−0.860274
$962$	0	0
$963$	3.18159e7	1.10555
$964$	0	0
$965$	6.29502e6	0.217610
$966$	0	0
$967$	2.66682e6	0.0917124	0.0458562	−	0.998948i	$-0.485398\pi$
$967$	2.66682e6	0.0917124	0.0458562	+	0.998948i	$0.485398\pi$
$968$	0	0
$969$	−3.08086e7	−1.05405
$970$	0	0
$971$	−1.64763e7	−0.560803	−0.280401	−	0.959883i	$-0.590468\pi$
$971$	−1.64763e7	−0.560803	−0.280401	+	0.959883i	$0.590468\pi$
$972$	0	0
$973$	−4.29758e7	−1.45526
$974$	0	0
$975$	−4.60248e6	−0.155053
$976$	0	0
$977$	1.36143e7	0.456308	0.228154	−	0.973625i	$-0.426731\pi$
$977$	1.36143e7	0.456308	0.228154	+	0.973625i	$0.426731\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−4.55612e6	−0.151155
$982$	0	0
$983$	1.03422e7	0.341372	0.170686	−	0.985325i	$-0.445402\pi$
$983$	1.03422e7	0.341372	0.170686	+	0.985325i	$0.445402\pi$
$984$	0	0
$985$	−1.14709e7	−0.376709
$986$	0	0
$987$	1.25723e7	0.410791
$988$	0	0
$989$	−4.59198e7	−1.49283
$990$	0	0
$991$	−1.65697e7	−0.535959	−0.267979	−	0.963425i	$-0.586356\pi$
$991$	−1.65697e7	−0.535959	−0.267979	+	0.963425i	$0.586356\pi$
$992$	0	0
$993$	1.90419e7	0.612826
$994$	0	0
$995$	4.67772e7	1.49788
$996$	0	0
$997$	9.75109e6	0.310682	0.155341	−	0.987861i	$-0.450352\pi$
$997$	9.75109e6	0.310682	0.155341	+	0.987861i	$0.450352\pi$
$998$	0	0
$999$	2.79821e7	0.887088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.6.a.q.1.2		16
11.5	even	5		88.6.i.b.25.2	✓	32
11.9	even	5		88.6.i.b.81.2	yes	32
11.10	odd	2		968.6.a.p.1.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.6.i.b.25.2	✓	32	11.5	even	5
88.6.i.b.81.2	yes	32	11.9	even	5
968.6.a.p.1.2		16	11.10	odd	2
968.6.a.q.1.2		16	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q-24.4344 q^{3} -57.4032 q^{5} -156.812 q^{7} +354.040 q^{9} +1107.14 q^{13} +1402.61 q^{15} -1734.64 q^{17} -726.877 q^{19} +3831.61 q^{21} +4796.66 q^{23} +170.132 q^{25} -2713.20 q^{27} -4674.99 q^{29} -2000.06 q^{31} +9001.52 q^{35} -10313.3 q^{37} -27052.3 q^{39} -2626.79 q^{41} -9573.28 q^{43} -20323.0 q^{45} +3281.20 q^{47} +7783.01 q^{49} +42384.9 q^{51} -25960.9 q^{53} +17760.8 q^{57} -4423.25 q^{59} -15468.1 q^{61} -55517.7 q^{63} -63553.5 q^{65} -18768.3 q^{67} -117204. q^{69} -30765.2 q^{71} -62882.9 q^{73} -4157.08 q^{75} +65539.3 q^{79} -19736.4 q^{81} +8408.81 q^{83} +99573.9 q^{85} +114230. q^{87} +774.583 q^{89} -173613. q^{91} +48870.2 q^{93} +41725.1 q^{95} -134159. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{3} + 81 q^{5} + 47 q^{7} + 1416 q^{9} - 859 q^{13} - 738 q^{15} - 1226 q^{17} - 616 q^{19} - 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} - 1613 q^{29} + 18511 q^{31} + 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) \) 16 * q + 12 * q^3 + 81 * q^5 + 47 * q^7 + 1416 * q^9 - 859 * q^13 - 738 * q^15 - 1226 * q^17 - 616 * q^19 - 1141 * q^21 + 2258 * q^23 + 10307 * q^25 + 564 * q^27 - 1613 * q^29 + 18511 * q^31 + 23544 * q^35 + 8553 * q^37 - 12020 * q^39 - 25514 * q^41 + 18931 * q^43 + 23278 * q^45 + 38781 * q^47 + 94523 * q^49 + 68584 * q^51 + 779 * q^53 - 19293 * q^57 + 50316 * q^59 - 53153 * q^61 + 54911 * q^63 - 51517 * q^65 + 10491 * q^67 - 95356 * q^69 - 91677 * q^71 - 97660 * q^73 - 3771 * q^75 + 96003 * q^79 + 120400 * q^81 - 4552 * q^83 + 129679 * q^85 - 77219 * q^87 + 6697 * q^89 + 217380 * q^91 + 132586 * q^93 + 54168 * q^95 + 171314 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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