LMFDB - Embedded newform 616.4.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [616,4,Mod(1,616)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(616, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("616.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 616 = 2^{3} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	616.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:	$36.3451765635$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 144x^{5} + 354x^{4} + 5172x^{3} - 6504x^{2} - 34432x + 18816 $ x^7 - 3x^6 - 144x^5 + 354x^4 + 5172x^3 - 6504x^2 - 34432x + 18816
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.19046$ of defining polynomial
Character	$\chi$	$=$	616.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.19046 q^{3} +8.53193 q^{5} +7.00000 q^{7} -16.8209 q^{9} +11.0000 q^{11} +68.7212 q^{13} +27.2208 q^{15} -30.5206 q^{17} +14.5420 q^{19} +22.3332 q^{21} +76.8536 q^{23} -52.2062 q^{25} -139.809 q^{27} +149.326 q^{29} +188.846 q^{31} +35.0951 q^{33} +59.7235 q^{35} +260.827 q^{37} +219.253 q^{39} +183.545 q^{41} -354.695 q^{43} -143.515 q^{45} +420.339 q^{47} +49.0000 q^{49} -97.3747 q^{51} -468.426 q^{53} +93.8512 q^{55} +46.3956 q^{57} +160.334 q^{59} +26.0410 q^{61} -117.747 q^{63} +586.325 q^{65} -397.860 q^{67} +245.198 q^{69} -440.130 q^{71} +409.850 q^{73} -166.562 q^{75} +77.0000 q^{77} -71.6329 q^{79} +8.10998 q^{81} +701.049 q^{83} -260.399 q^{85} +476.418 q^{87} +72.3930 q^{89} +481.049 q^{91} +602.506 q^{93} +124.071 q^{95} -168.126 q^{97} -185.030 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} + 11 q^{5} + 49 q^{7} + 108 q^{9} + 77 q^{11} + 26 q^{13} + 67 q^{15} + 44 q^{17} + 34 q^{19} + 21 q^{21} - 29 q^{23} + 182 q^{25} + 99 q^{27} + 94 q^{29} - 173 q^{31} + 33 q^{33} + 77 q^{35}+ \cdots + 1188 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 + 11 * q^5 + 49 * q^7 + 108 * q^9 + 77 * q^11 + 26 * q^13 + 67 * q^15 + 44 * q^17 + 34 * q^19 + 21 * q^21 - 29 * q^23 + 182 * q^25 + 99 * q^27 + 94 * q^29 - 173 * q^31 + 33 * q^33 + 77 * q^35 + 255 * q^37 - 60 * q^39 + 508 * q^41 + 656 * q^43 + 466 * q^45 - 18 * q^47 + 343 * q^49 + 850 * q^51 + 1806 * q^53 + 121 * q^55 + 1154 * q^57 + 665 * q^59 + 608 * q^61 + 756 * q^63 + 588 * q^65 + 669 * q^67 + 2067 * q^69 + 1169 * q^71 + 380 * q^73 + 1254 * q^75 + 539 * q^77 - 110 * q^79 + 2847 * q^81 - 496 * q^83 + 3446 * q^85 - 582 * q^87 + 1321 * q^89 + 182 * q^91 + 1493 * q^93 - 50 * q^95 + 3927 * q^97 + 1188 * q^99

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
$3$	3.19046	0.614005	0.307002	−	0.951709i	$-0.400674\pi$
$3$	3.19046	0.614005	0.307002	+	0.951709i	$0.400674\pi$
$4$	0	0
$5$	8.53193	0.763119	0.381559	−	0.924344i	$-0.375387\pi$
$5$	8.53193	0.763119	0.381559	+	0.924344i	$0.375387\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−16.8209	−0.622998
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	68.7212	1.46614	0.733071	−	0.680152i	$-0.238086\pi$
$13$	68.7212	1.46614	0.733071	+	0.680152i	$0.238086\pi$
$14$	0	0
$15$	27.2208	0.468558
$16$	0	0
$17$	−30.5206	−0.435431	−0.217715	−	0.976012i	$-0.569860\pi$
$17$	−30.5206	−0.435431	−0.217715	+	0.976012i	$0.569860\pi$
$18$	0	0
$19$	14.5420	0.175587	0.0877935	−	0.996139i	$-0.472018\pi$
$19$	14.5420	0.175587	0.0877935	+	0.996139i	$0.472018\pi$
$20$	0	0
$21$	22.3332	0.232072
$22$	0	0
$23$	76.8536	0.696742	0.348371	−	0.937357i	$-0.386735\pi$
$23$	76.8536	0.696742	0.348371	+	0.937357i	$0.386735\pi$
$24$	0	0
$25$	−52.2062	−0.417650
$26$	0	0
$27$	−139.809	−0.996529
$28$	0	0
$29$	149.326	0.956175	0.478088	−	0.878312i	$-0.341330\pi$
$29$	149.326	0.956175	0.478088	+	0.878312i	$0.341330\pi$
$30$	0	0
$31$	188.846	1.09412	0.547060	−	0.837093i	$-0.315747\pi$
$31$	188.846	1.09412	0.547060	+	0.837093i	$0.315747\pi$
$32$	0	0
$33$	35.0951	0.185129
$34$	0	0
$35$	59.7235	0.288432
$36$	0	0
$37$	260.827	1.15891	0.579456	−	0.815003i	$-0.303265\pi$
$37$	260.827	1.15891	0.579456	+	0.815003i	$0.303265\pi$
$38$	0	0
$39$	219.253	0.900218
$40$	0	0
$41$	183.545	0.699144	0.349572	−	0.936909i	$-0.386327\pi$
$41$	183.545	0.699144	0.349572	+	0.936909i	$0.386327\pi$
$42$	0	0
$43$	−354.695	−1.25792	−0.628959	−	0.777438i	$-0.716519\pi$
$43$	−354.695	−1.25792	−0.628959	+	0.777438i	$0.716519\pi$
$44$	0	0
$45$	−143.515	−0.475421
$46$	0	0
$47$	420.339	1.30453	0.652263	−	0.757993i	$-0.273820\pi$
$47$	420.339	1.30453	0.652263	+	0.757993i	$0.273820\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−97.3747	−0.267357
$52$	0	0
$53$	−468.426	−1.21402	−0.607012	−	0.794693i	$-0.707632\pi$
$53$	−468.426	−1.21402	−0.607012	+	0.794693i	$0.707632\pi$
$54$	0	0
$55$	93.8512	0.230089
$56$	0	0
$57$	46.3956	0.107811
$58$	0	0
$59$	160.334	0.353792	0.176896	−	0.984230i	$-0.443394\pi$
$59$	160.334	0.353792	0.176896	+	0.984230i	$0.443394\pi$
$60$	0	0
$61$	26.0410	0.0546592	0.0273296	−	0.999626i	$-0.491300\pi$
$61$	26.0410	0.0546592	0.0273296	+	0.999626i	$0.491300\pi$
$62$	0	0
$63$	−117.747	−0.235471
$64$	0	0
$65$	586.325	1.11884
$66$	0	0
$67$	−397.860	−0.725467	−0.362734	−	0.931893i	$-0.618157\pi$
$67$	−397.860	−0.725467	−0.362734	+	0.931893i	$0.618157\pi$
$68$	0	0
$69$	245.198	0.427803
$70$	0	0
$71$	−440.130	−0.735687	−0.367844	−	0.929888i	$-0.619904\pi$
$71$	−440.130	−0.735687	−0.367844	+	0.929888i	$0.619904\pi$
$72$	0	0
$73$	409.850	0.657113	0.328557	−	0.944484i	$-0.393438\pi$
$73$	409.850	0.657113	0.328557	+	0.944484i	$0.393438\pi$
$74$	0	0
$75$	−166.562	−0.256439
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	−71.6329	−0.102017	−0.0510084	−	0.998698i	$-0.516244\pi$
$79$	−71.6329	−0.102017	−0.0510084	+	0.998698i	$0.516244\pi$
$80$	0	0
$81$	8.10998	0.0111248
$82$	0	0
$83$	701.049	0.927110	0.463555	−	0.886068i	$-0.346574\pi$
$83$	701.049	0.927110	0.463555	+	0.886068i	$0.346574\pi$
$84$	0	0
$85$	−260.399	−0.332285
$86$	0	0
$87$	476.418	0.587096
$88$	0	0
$89$	72.3930	0.0862207	0.0431104	−	0.999070i	$-0.486273\pi$
$89$	72.3930	0.0862207	0.0431104	+	0.999070i	$0.486273\pi$
$90$	0	0
$91$	481.049	0.554150
$92$	0	0
$93$	602.506	0.671795
$94$	0	0
$95$	124.071	0.133994
$96$	0	0
$97$	−168.126	−0.175985	−0.0879927	−	0.996121i	$-0.528045\pi$
$97$	−168.126	−0.175985	−0.0879927	+	0.996121i	$0.528045\pi$
$98$	0	0
$99$	−185.030	−0.187841
$100$	0	0
$101$	1847.12	1.81976	0.909880	−	0.414873i	$-0.136174\pi$
$101$	1847.12	1.81976	0.909880	+	0.414873i	$0.136174\pi$
$102$	0	0
$103$	819.105	0.783581	0.391790	−	0.920055i	$-0.371856\pi$
$103$	819.105	0.783581	0.391790	+	0.920055i	$0.371856\pi$
$104$	0	0
$105$	190.546	0.177098
$106$	0	0
$107$	−1226.70	−1.10831	−0.554156	−	0.832413i	$-0.686959\pi$
$107$	−1226.70	−1.10831	−0.554156	+	0.832413i	$0.686959\pi$
$108$	0	0
$109$	1023.82	0.899676	0.449838	−	0.893110i	$-0.351482\pi$
$109$	1023.82	0.899676	0.449838	+	0.893110i	$0.351482\pi$
$110$	0	0
$111$	832.160	0.711578
$112$	0	0
$113$	1786.92	1.48760	0.743802	−	0.668400i	$-0.233021\pi$
$113$	1786.92	1.48760	0.743802	+	0.668400i	$0.233021\pi$
$114$	0	0
$115$	655.709	0.531697
$116$	0	0
$117$	−1155.96	−0.913404
$118$	0	0
$119$	−213.644	−0.164577
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	585.593	0.429278
$124$	0	0
$125$	−1511.91	−1.08184
$126$	0	0
$127$	753.691	0.526608	0.263304	−	0.964713i	$-0.415188\pi$
$127$	753.691	0.526608	0.263304	+	0.964713i	$0.415188\pi$
$128$	0	0
$129$	−1131.64	−0.772368
$130$	0	0
$131$	757.722	0.505362	0.252681	−	0.967550i	$-0.418688\pi$
$131$	757.722	0.505362	0.252681	+	0.967550i	$0.418688\pi$
$132$	0	0
$133$	101.794	0.0663657
$134$	0	0
$135$	−1192.84	−0.760470
$136$	0	0
$137$	−1087.18	−0.677989	−0.338994	−	0.940788i	$-0.610087\pi$
$137$	−1087.18	−0.677989	−0.338994	+	0.940788i	$0.610087\pi$
$138$	0	0
$139$	−2896.93	−1.76773	−0.883865	−	0.467742i	$-0.845068\pi$
$139$	−2896.93	−1.76773	−0.883865	+	0.467742i	$0.845068\pi$
$140$	0	0
$141$	1341.07	0.800985
$142$	0	0
$143$	755.934	0.442058
$144$	0	0
$145$	1274.04	0.729675
$146$	0	0
$147$	156.333	0.0877150
$148$	0	0
$149$	−2507.36	−1.37860	−0.689298	−	0.724478i	$-0.742081\pi$
$149$	−2507.36	−1.37860	−0.689298	+	0.724478i	$0.742081\pi$
$150$	0	0
$151$	−606.134	−0.326666	−0.163333	−	0.986571i	$-0.552224\pi$
$151$	−606.134	−0.326666	−0.163333	+	0.986571i	$0.552224\pi$
$152$	0	0
$153$	513.385	0.271273
$154$	0	0
$155$	1611.22	0.834944
$156$	0	0
$157$	1476.51	0.750565	0.375282	−	0.926911i	$-0.377546\pi$
$157$	1476.51	0.750565	0.375282	+	0.926911i	$0.377546\pi$
$158$	0	0
$159$	−1494.50	−0.745417
$160$	0	0
$161$	537.975	0.263344
$162$	0	0
$163$	190.979	0.0917706	0.0458853	−	0.998947i	$-0.485389\pi$
$163$	190.979	0.0917706	0.0458853	+	0.998947i	$0.485389\pi$
$164$	0	0
$165$	299.429	0.141276
$166$	0	0
$167$	−1349.51	−0.625319	−0.312660	−	0.949865i	$-0.601220\pi$
$167$	−1349.51	−0.625319	−0.312660	+	0.949865i	$0.601220\pi$
$168$	0	0
$169$	2525.61	1.14957
$170$	0	0
$171$	−244.610	−0.109390
$172$	0	0
$173$	2263.17	0.994600	0.497300	−	0.867579i	$-0.334325\pi$
$173$	2263.17	0.994600	0.497300	+	0.867579i	$0.334325\pi$
$174$	0	0
$175$	−365.444	−0.157857
$176$	0	0
$177$	511.540	0.217230
$178$	0	0
$179$	−2712.23	−1.13252	−0.566262	−	0.824225i	$-0.691611\pi$
$179$	−2712.23	−1.13252	−0.566262	+	0.824225i	$0.691611\pi$
$180$	0	0
$181$	2478.66	1.01789	0.508944	−	0.860800i	$-0.330036\pi$
$181$	2478.66	1.01789	0.508944	+	0.860800i	$0.330036\pi$
$182$	0	0
$183$	83.0829	0.0335610
$184$	0	0
$185$	2225.36	0.884388
$186$	0	0
$187$	−335.726	−0.131287
$188$	0	0
$189$	−978.664	−0.376652
$190$	0	0
$191$	−1556.65	−0.589715	−0.294858	−	0.955541i	$-0.595272\pi$
$191$	−1556.65	−0.589715	−0.294858	+	0.955541i	$0.595272\pi$
$192$	0	0
$193$	1298.56	0.484311	0.242156	−	0.970237i	$-0.422146\pi$
$193$	1298.56	0.484311	0.242156	+	0.970237i	$0.422146\pi$
$194$	0	0
$195$	1870.65	0.686973
$196$	0	0
$197$	2357.82	0.852728	0.426364	−	0.904552i	$-0.359794\pi$
$197$	2357.82	0.852728	0.426364	+	0.904552i	$0.359794\pi$
$198$	0	0
$199$	−2090.77	−0.744776	−0.372388	−	0.928077i	$-0.621461\pi$
$199$	−2090.77	−0.744776	−0.372388	+	0.928077i	$0.621461\pi$
$200$	0	0
$201$	−1269.36	−0.445441
$202$	0	0
$203$	1045.28	0.361400
$204$	0	0
$205$	1565.99	0.533530
$206$	0	0
$207$	−1292.75	−0.434069
$208$	0	0
$209$	159.962	0.0529415
$210$	0	0
$211$	−6011.00	−1.96120	−0.980602	−	0.196007i	$-0.937202\pi$
$211$	−6011.00	−1.96120	−0.980602	+	0.196007i	$0.937202\pi$
$212$	0	0
$213$	−1404.22	−0.451715
$214$	0	0
$215$	−3026.23	−0.959941
$216$	0	0
$217$	1321.92	0.413539
$218$	0	0
$219$	1307.61	0.403471
$220$	0	0
$221$	−2097.41	−0.638403
$222$	0	0
$223$	−1963.16	−0.589519	−0.294760	−	0.955571i	$-0.595240\pi$
$223$	−1963.16	−0.589519	−0.294760	+	0.955571i	$0.595240\pi$
$224$	0	0
$225$	878.159	0.260195
$226$	0	0
$227$	−4697.74	−1.37357	−0.686784	−	0.726862i	$-0.740978\pi$
$227$	−4697.74	−1.37357	−0.686784	+	0.726862i	$0.740978\pi$
$228$	0	0
$229$	−1073.32	−0.309724	−0.154862	−	0.987936i	$-0.549493\pi$
$229$	−1073.32	−0.309724	−0.154862	+	0.987936i	$0.549493\pi$
$230$	0	0
$231$	245.666	0.0699723
$232$	0	0
$233$	−1521.11	−0.427688	−0.213844	−	0.976868i	$-0.568598\pi$
$233$	−1521.11	−0.427688	−0.213844	+	0.976868i	$0.568598\pi$
$234$	0	0
$235$	3586.30	0.995508
$236$	0	0
$237$	−228.542	−0.0626389
$238$	0	0
$239$	−5589.90	−1.51289	−0.756444	−	0.654058i	$-0.773065\pi$
$239$	−5589.90	−1.51289	−0.756444	+	0.654058i	$0.773065\pi$
$240$	0	0
$241$	−330.180	−0.0882520	−0.0441260	−	0.999026i	$-0.514050\pi$
$241$	−330.180	−0.0882520	−0.0441260	+	0.999026i	$0.514050\pi$
$242$	0	0
$243$	3800.72	1.00336
$244$	0	0
$245$	418.064	0.109017
$246$	0	0
$247$	999.342	0.257436
$248$	0	0
$249$	2236.67	0.569250
$250$	0	0
$251$	−3343.47	−0.840788	−0.420394	−	0.907342i	$-0.638108\pi$
$251$	−3343.47	−0.840788	−0.420394	+	0.907342i	$0.638108\pi$
$252$	0	0
$253$	845.389	0.210076
$254$	0	0
$255$	−830.794	−0.204025
$256$	0	0
$257$	4255.28	1.03283	0.516414	−	0.856339i	$-0.327266\pi$
$257$	4255.28	1.03283	0.516414	+	0.856339i	$0.327266\pi$
$258$	0	0
$259$	1825.79	0.438028
$260$	0	0
$261$	−2511.80	−0.595696
$262$	0	0
$263$	−1339.79	−0.314125	−0.157062	−	0.987589i	$-0.550202\pi$
$263$	−1339.79	−0.314125	−0.157062	+	0.987589i	$0.550202\pi$
$264$	0	0
$265$	−3996.58	−0.926445
$266$	0	0
$267$	230.967	0.0529399
$268$	0	0
$269$	2725.81	0.617827	0.308914	−	0.951090i	$-0.400035\pi$
$269$	2725.81	0.617827	0.308914	+	0.951090i	$0.400035\pi$
$270$	0	0
$271$	−3717.31	−0.833249	−0.416625	−	0.909079i	$-0.636787\pi$
$271$	−3717.31	−0.833249	−0.416625	+	0.909079i	$0.636787\pi$
$272$	0	0
$273$	1534.77	0.340250
$274$	0	0
$275$	−574.269	−0.125926
$276$	0	0
$277$	−6857.46	−1.48745	−0.743727	−	0.668483i	$-0.766944\pi$
$277$	−6857.46	−1.48745	−0.743727	+	0.668483i	$0.766944\pi$
$278$	0	0
$279$	−3176.57	−0.681635
$280$	0	0
$281$	−2213.01	−0.469812	−0.234906	−	0.972018i	$-0.575478\pi$
$281$	−2213.01	−0.469812	−0.234906	+	0.972018i	$0.575478\pi$
$282$	0	0
$283$	−6991.98	−1.46866	−0.734329	−	0.678794i	$-0.762503\pi$
$283$	−6991.98	−1.46866	−0.734329	+	0.678794i	$0.762503\pi$
$284$	0	0
$285$	395.844	0.0822728
$286$	0	0
$287$	1284.81	0.264252
$288$	0	0
$289$	−3981.50	−0.810400
$290$	0	0
$291$	−536.399	−0.108056
$292$	0	0
$293$	−8640.41	−1.72279	−0.861397	−	0.507933i	$-0.830410\pi$
$293$	−8640.41	−1.72279	−0.861397	+	0.507933i	$0.830410\pi$
$294$	0	0
$295$	1367.96	0.269985
$296$	0	0
$297$	−1537.90	−0.300465
$298$	0	0
$299$	5281.47	1.02152
$300$	0	0
$301$	−2482.87	−0.475448
$302$	0	0
$303$	5893.18	1.11734
$304$	0	0
$305$	222.180	0.0417115
$306$	0	0
$307$	−2848.00	−0.529458	−0.264729	−	0.964323i	$-0.585283\pi$
$307$	−2848.00	−0.529458	−0.264729	+	0.964323i	$0.585283\pi$
$308$	0	0
$309$	2613.32	0.481122
$310$	0	0
$311$	1412.70	0.257578	0.128789	−	0.991672i	$-0.458891\pi$
$311$	1412.70	0.257578	0.128789	+	0.991672i	$0.458891\pi$
$312$	0	0
$313$	4713.31	0.851157	0.425578	−	0.904922i	$-0.360071\pi$
$313$	4713.31	0.851157	0.425578	+	0.904922i	$0.360071\pi$
$314$	0	0
$315$	−1004.61	−0.179692
$316$	0	0
$317$	709.295	0.125672	0.0628359	−	0.998024i	$-0.479986\pi$
$317$	709.295	0.125672	0.0628359	+	0.998024i	$0.479986\pi$
$318$	0	0
$319$	1642.58	0.288298
$320$	0	0
$321$	−3913.74	−0.680509
$322$	0	0
$323$	−443.829	−0.0764560
$324$	0	0
$325$	−3587.68	−0.612334
$326$	0	0
$327$	3266.47	0.552405
$328$	0	0
$329$	2942.37	0.493064
$330$	0	0
$331$	2724.16	0.452367	0.226183	−	0.974085i	$-0.427375\pi$
$331$	2724.16	0.452367	0.226183	+	0.974085i	$0.427375\pi$
$332$	0	0
$333$	−4387.36	−0.722000
$334$	0	0
$335$	−3394.51	−0.553618
$336$	0	0
$337$	−5644.70	−0.912422	−0.456211	−	0.889872i	$-0.650794\pi$
$337$	−5644.70	−0.912422	−0.456211	+	0.889872i	$0.650794\pi$
$338$	0	0
$339$	5701.10	0.913396
$340$	0	0
$341$	2077.31	0.329890
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	2092.01	0.326465
$346$	0	0
$347$	−953.311	−0.147482	−0.0737412	−	0.997277i	$-0.523494\pi$
$347$	−953.311	−0.147482	−0.0737412	+	0.997277i	$0.523494\pi$
$348$	0	0
$349$	−11589.4	−1.77756	−0.888779	−	0.458337i	$-0.848445\pi$
$349$	−11589.4	−1.77756	−0.888779	+	0.458337i	$0.848445\pi$
$350$	0	0
$351$	−9607.86	−1.46105
$352$	0	0
$353$	3423.27	0.516154	0.258077	−	0.966124i	$-0.416911\pi$
$353$	3423.27	0.516154	0.258077	+	0.966124i	$0.416911\pi$
$354$	0	0
$355$	−3755.15	−0.561417
$356$	0	0
$357$	−681.623	−0.101051
$358$	0	0
$359$	−2856.22	−0.419904	−0.209952	−	0.977712i	$-0.567331\pi$
$359$	−2856.22	−0.419904	−0.209952	+	0.977712i	$0.567331\pi$
$360$	0	0
$361$	−6647.53	−0.969169
$362$	0	0
$363$	386.046	0.0558186
$364$	0	0
$365$	3496.81	0.501455
$366$	0	0
$367$	−4092.10	−0.582032	−0.291016	−	0.956718i	$-0.593993\pi$
$367$	−4092.10	−0.582032	−0.291016	+	0.956718i	$0.593993\pi$
$368$	0	0
$369$	−3087.40	−0.435565
$370$	0	0
$371$	−3278.98	−0.458858
$372$	0	0
$373$	−5886.94	−0.817196	−0.408598	−	0.912714i	$-0.633982\pi$
$373$	−5886.94	−0.817196	−0.408598	+	0.912714i	$0.633982\pi$
$374$	0	0
$375$	−4823.69	−0.664252
$376$	0	0
$377$	10261.8	1.40189
$378$	0	0
$379$	8729.03	1.18306	0.591531	−	0.806282i	$-0.298524\pi$
$379$	8729.03	1.18306	0.591531	+	0.806282i	$0.298524\pi$
$380$	0	0
$381$	2404.62	0.323340
$382$	0	0
$383$	2232.54	0.297852	0.148926	−	0.988848i	$-0.452418\pi$
$383$	2232.54	0.297852	0.148926	+	0.988848i	$0.452418\pi$
$384$	0	0
$385$	656.958	0.0869654
$386$	0	0
$387$	5966.31	0.783681
$388$	0	0
$389$	13901.2	1.81187	0.905937	−	0.423413i	$-0.139168\pi$
$389$	13901.2	1.81187	0.905937	+	0.423413i	$0.139168\pi$
$390$	0	0
$391$	−2345.61	−0.303383
$392$	0	0
$393$	2417.48	0.310295
$394$	0	0
$395$	−611.167	−0.0778510
$396$	0	0
$397$	−3359.38	−0.424691	−0.212345	−	0.977195i	$-0.568110\pi$
$397$	−3359.38	−0.424691	−0.212345	+	0.977195i	$0.568110\pi$
$398$	0	0
$399$	324.769	0.0407488
$400$	0	0
$401$	−82.2661	−0.0102448	−0.00512241	−	0.999987i	$-0.501631\pi$
$401$	−82.2661	−0.0102448	−0.00512241	+	0.999987i	$0.501631\pi$
$402$	0	0
$403$	12977.7	1.60414
$404$	0	0
$405$	69.1937	0.00848954
$406$	0	0
$407$	2869.10	0.349425
$408$	0	0
$409$	−9945.83	−1.20242	−0.601210	−	0.799091i	$-0.705315\pi$
$409$	−9945.83	−1.20242	−0.601210	+	0.799091i	$0.705315\pi$
$410$	0	0
$411$	−3468.62	−0.416288
$412$	0	0
$413$	1122.34	0.133721
$414$	0	0
$415$	5981.30	0.707495
$416$	0	0
$417$	−9242.55	−1.08539
$418$	0	0
$419$	−2029.06	−0.236578	−0.118289	−	0.992979i	$-0.537741\pi$
$419$	−2029.06	−0.236578	−0.118289	+	0.992979i	$0.537741\pi$
$420$	0	0
$421$	5169.88	0.598490	0.299245	−	0.954176i	$-0.403265\pi$
$421$	5169.88	0.598490	0.299245	+	0.954176i	$0.403265\pi$
$422$	0	0
$423$	−7070.50	−0.812717
$424$	0	0
$425$	1593.36	0.181858
$426$	0	0
$427$	182.287	0.0206592
$428$	0	0
$429$	2411.78	0.271426
$430$	0	0
$431$	13264.9	1.48248	0.741241	−	0.671239i	$-0.234238\pi$
$431$	13264.9	1.48248	0.741241	+	0.671239i	$0.234238\pi$
$432$	0	0
$433$	−687.605	−0.0763146	−0.0381573	−	0.999272i	$-0.512149\pi$
$433$	−687.605	−0.0763146	−0.0381573	+	0.999272i	$0.512149\pi$
$434$	0	0
$435$	4064.76	0.448024
$436$	0	0
$437$	1117.60	0.122339
$438$	0	0
$439$	1455.57	0.158247	0.0791234	−	0.996865i	$-0.474788\pi$
$439$	1455.57	0.158247	0.0791234	+	0.996865i	$0.474788\pi$
$440$	0	0
$441$	−824.227	−0.0889997
$442$	0	0
$443$	12648.9	1.35658	0.678291	−	0.734794i	$-0.262721\pi$
$443$	12648.9	1.35658	0.678291	+	0.734794i	$0.262721\pi$
$444$	0	0
$445$	617.652	0.0657966
$446$	0	0
$447$	−7999.64	−0.846465
$448$	0	0
$449$	−13112.4	−1.37820	−0.689100	−	0.724666i	$-0.741994\pi$
$449$	−13112.4	−1.37820	−0.689100	+	0.724666i	$0.741994\pi$
$450$	0	0
$451$	2018.99	0.210800
$452$	0	0
$453$	−1933.85	−0.200574
$454$	0	0
$455$	4104.27	0.422882
$456$	0	0
$457$	8153.14	0.834547	0.417273	−	0.908781i	$-0.362986\pi$
$457$	8153.14	0.834547	0.417273	+	0.908781i	$0.362986\pi$
$458$	0	0
$459$	4267.05	0.433919
$460$	0	0
$461$	−10774.4	−1.08853	−0.544267	−	0.838912i	$-0.683192\pi$
$461$	−10774.4	−1.08853	−0.544267	+	0.838912i	$0.683192\pi$
$462$	0	0
$463$	4799.27	0.481730	0.240865	−	0.970559i	$-0.422569\pi$
$463$	4799.27	0.481730	0.240865	+	0.970559i	$0.422569\pi$
$464$	0	0
$465$	5140.54	0.512660
$466$	0	0
$467$	13503.2	1.33802	0.669008	−	0.743255i	$-0.266719\pi$
$467$	13503.2	1.33802	0.669008	+	0.743255i	$0.266719\pi$
$468$	0	0
$469$	−2785.02	−0.274201
$470$	0	0
$471$	4710.76	0.460850
$472$	0	0
$473$	−3901.65	−0.379277
$474$	0	0
$475$	−759.181	−0.0733339
$476$	0	0
$477$	7879.37	0.756335
$478$	0	0
$479$	9189.09	0.876535	0.438267	−	0.898845i	$-0.355592\pi$
$479$	9189.09	0.876535	0.438267	+	0.898845i	$0.355592\pi$
$480$	0	0
$481$	17924.4	1.69913
$482$	0	0
$483$	1716.39	0.161694
$484$	0	0
$485$	−1434.44	−0.134298
$486$	0	0
$487$	−16306.7	−1.51731	−0.758654	−	0.651494i	$-0.774143\pi$
$487$	−16306.7	−1.51731	−0.758654	+	0.651494i	$0.774143\pi$
$488$	0	0
$489$	609.311	0.0563476
$490$	0	0
$491$	10336.0	0.950012	0.475006	−	0.879982i	$-0.342446\pi$
$491$	10336.0	0.950012	0.475006	+	0.879982i	$0.342446\pi$
$492$	0	0
$493$	−4557.50	−0.416348
$494$	0	0
$495$	−1578.67	−0.143345
$496$	0	0
$497$	−3080.91	−0.278064
$498$	0	0
$499$	11993.1	1.07592	0.537961	−	0.842970i	$-0.319195\pi$
$499$	11993.1	1.07592	0.537961	+	0.842970i	$0.319195\pi$
$500$	0	0
$501$	−4305.56	−0.383949
$502$	0	0
$503$	−14930.0	−1.32346	−0.661728	−	0.749744i	$-0.730176\pi$
$503$	−14930.0	−1.32346	−0.661728	+	0.749744i	$0.730176\pi$
$504$	0	0
$505$	15759.5	1.38869
$506$	0	0
$507$	8057.86	0.705843
$508$	0	0
$509$	−9086.79	−0.791287	−0.395643	−	0.918404i	$-0.629478\pi$
$509$	−9086.79	−0.791287	−0.395643	+	0.918404i	$0.629478\pi$
$510$	0	0
$511$	2868.95	0.248365
$512$	0	0
$513$	−2033.10	−0.174978
$514$	0	0
$515$	6988.54	0.597965
$516$	0	0
$517$	4623.73	0.393329
$518$	0	0
$519$	7220.56	0.610689
$520$	0	0
$521$	21210.9	1.78362	0.891811	−	0.452408i	$-0.149435\pi$
$521$	21210.9	1.78362	0.891811	+	0.452408i	$0.149435\pi$
$522$	0	0
$523$	10266.9	0.858392	0.429196	−	0.903211i	$-0.358797\pi$
$523$	10266.9	0.858392	0.429196	+	0.903211i	$0.358797\pi$
$524$	0	0
$525$	−1165.93	−0.0969249
$526$	0	0
$527$	−5763.69	−0.476414
$528$	0	0
$529$	−6260.53	−0.514550
$530$	0	0
$531$	−2696.97	−0.220412
$532$	0	0
$533$	12613.4	1.02504
$534$	0	0
$535$	−10466.1	−0.845774
$536$	0	0
$537$	−8653.28	−0.695376
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	20174.0	1.60323	0.801616	−	0.597839i	$-0.203974\pi$
$541$	20174.0	1.60323	0.801616	+	0.597839i	$0.203974\pi$
$542$	0	0
$543$	7908.09	0.624988
$544$	0	0
$545$	8735.20	0.686559
$546$	0	0
$547$	6521.92	0.509794	0.254897	−	0.966968i	$-0.417959\pi$
$547$	6521.92	0.509794	0.254897	+	0.966968i	$0.417959\pi$
$548$	0	0
$549$	−438.035	−0.0340526
$550$	0	0
$551$	2171.49	0.167892
$552$	0	0
$553$	−501.431	−0.0385588
$554$	0	0
$555$	7099.93	0.543018
$556$	0	0
$557$	−5459.87	−0.415336	−0.207668	−	0.978199i	$-0.566587\pi$
$557$	−5459.87	−0.415336	−0.207668	+	0.978199i	$0.566587\pi$
$558$	0	0
$559$	−24375.1	−1.84429
$560$	0	0
$561$	−1071.12	−0.0806110
$562$	0	0
$563$	−4928.99	−0.368973	−0.184487	−	0.982835i	$-0.559062\pi$
$563$	−4928.99	−0.368973	−0.184487	+	0.982835i	$0.559062\pi$
$564$	0	0
$565$	15245.9	1.13522
$566$	0	0
$567$	56.7699	0.00420478
$568$	0	0
$569$	8485.96	0.625220	0.312610	−	0.949882i	$-0.398797\pi$
$569$	8485.96	0.625220	0.312610	+	0.949882i	$0.398797\pi$
$570$	0	0
$571$	−25006.4	−1.83273	−0.916363	−	0.400348i	$-0.868889\pi$
$571$	−25006.4	−1.83273	−0.916363	+	0.400348i	$0.868889\pi$
$572$	0	0
$573$	−4966.45	−0.362088
$574$	0	0
$575$	−4012.24	−0.290994
$576$	0	0
$577$	−24688.3	−1.78126	−0.890630	−	0.454730i	$-0.849736\pi$
$577$	−24688.3	−1.78126	−0.890630	+	0.454730i	$0.849736\pi$
$578$	0	0
$579$	4142.99	0.297369
$580$	0	0
$581$	4907.34	0.350415
$582$	0	0
$583$	−5152.69	−0.366042
$584$	0	0
$585$	−9862.54	−0.697035
$586$	0	0
$587$	−6033.27	−0.424224	−0.212112	−	0.977245i	$-0.568034\pi$
$587$	−6033.27	−0.424224	−0.212112	+	0.977245i	$0.568034\pi$
$588$	0	0
$589$	2746.19	0.192113
$590$	0	0
$591$	7522.52	0.523579
$592$	0	0
$593$	−8105.88	−0.561330	−0.280665	−	0.959806i	$-0.590555\pi$
$593$	−8105.88	−0.561330	−0.280665	+	0.959806i	$0.590555\pi$
$594$	0	0
$595$	−1822.79	−0.125592
$596$	0	0
$597$	−6670.51	−0.457296
$598$	0	0
$599$	13393.5	0.913595	0.456798	−	0.889571i	$-0.348996\pi$
$599$	13393.5	0.913595	0.456798	+	0.889571i	$0.348996\pi$
$600$	0	0
$601$	3006.70	0.204069	0.102035	−	0.994781i	$-0.467465\pi$
$601$	3006.70	0.204069	0.102035	+	0.994781i	$0.467465\pi$
$602$	0	0
$603$	6692.38	0.451965
$604$	0	0
$605$	1032.36	0.0693744
$606$	0	0
$607$	−7506.67	−0.501955	−0.250977	−	0.967993i	$-0.580752\pi$
$607$	−7506.67	−0.501955	−0.250977	+	0.967993i	$0.580752\pi$
$608$	0	0
$609$	3334.93	0.221902
$610$	0	0
$611$	28886.2	1.91262
$612$	0	0
$613$	−13157.8	−0.866950	−0.433475	−	0.901166i	$-0.642713\pi$
$613$	−13157.8	−0.866950	−0.433475	+	0.901166i	$0.642713\pi$
$614$	0	0
$615$	4996.24	0.327590
$616$	0	0
$617$	23420.9	1.52818	0.764092	−	0.645107i	$-0.223187\pi$
$617$	23420.9	1.52818	0.764092	+	0.645107i	$0.223187\pi$
$618$	0	0
$619$	−904.750	−0.0587479	−0.0293740	−	0.999568i	$-0.509351\pi$
$619$	−904.750	−0.0587479	−0.0293740	+	0.999568i	$0.509351\pi$
$620$	0	0
$621$	−10744.8	−0.694324
$622$	0	0
$623$	506.751	0.0325884
$624$	0	0
$625$	−6373.73	−0.407919
$626$	0	0
$627$	510.351	0.0325063
$628$	0	0
$629$	−7960.60	−0.504626
$630$	0	0
$631$	−22840.2	−1.44097	−0.720486	−	0.693469i	$-0.756081\pi$
$631$	−22840.2	−1.44097	−0.720486	+	0.693469i	$0.756081\pi$
$632$	0	0
$633$	−19177.9	−1.20419
$634$	0	0
$635$	6430.43	0.401865
$636$	0	0
$637$	3367.34	0.209449
$638$	0	0
$639$	7403.40	0.458332
$640$	0	0
$641$	−23658.3	−1.45780	−0.728899	−	0.684621i	$-0.759968\pi$
$641$	−23658.3	−1.45780	−0.728899	+	0.684621i	$0.759968\pi$
$642$	0	0
$643$	1184.17	0.0726268	0.0363134	−	0.999340i	$-0.488439\pi$
$643$	1184.17	0.0726268	0.0363134	+	0.999340i	$0.488439\pi$
$644$	0	0
$645$	−9655.08	−0.589408
$646$	0	0
$647$	−8378.23	−0.509091	−0.254546	−	0.967061i	$-0.581926\pi$
$647$	−8378.23	−0.509091	−0.254546	+	0.967061i	$0.581926\pi$
$648$	0	0
$649$	1763.68	0.106672
$650$	0	0
$651$	4217.54	0.253915
$652$	0	0
$653$	29112.3	1.74464	0.872322	−	0.488932i	$-0.162613\pi$
$653$	29112.3	1.74464	0.872322	+	0.488932i	$0.162613\pi$
$654$	0	0
$655$	6464.83	0.385651
$656$	0	0
$657$	−6894.06	−0.409380
$658$	0	0
$659$	21401.7	1.26509	0.632544	−	0.774525i	$-0.282011\pi$
$659$	21401.7	1.26509	0.632544	+	0.774525i	$0.282011\pi$
$660$	0	0
$661$	−1209.18	−0.0711522	−0.0355761	−	0.999367i	$-0.511327\pi$
$661$	−1209.18	−0.0711522	−0.0355761	+	0.999367i	$0.511327\pi$
$662$	0	0
$663$	−6691.71	−0.391983
$664$	0	0
$665$	868.496	0.0506449
$666$	0	0
$667$	11476.2	0.666208
$668$	0	0
$669$	−6263.38	−0.361968
$670$	0	0
$671$	286.451	0.0164804
$672$	0	0
$673$	−12981.0	−0.743506	−0.371753	−	0.928332i	$-0.621243\pi$
$673$	−12981.0	−0.743506	−0.371753	+	0.928332i	$0.621243\pi$
$674$	0	0
$675$	7298.91	0.416200
$676$	0	0
$677$	32237.3	1.83010	0.915051	−	0.403339i	$-0.132150\pi$
$677$	32237.3	1.83010	0.915051	+	0.403339i	$0.132150\pi$
$678$	0	0
$679$	−1176.88	−0.0665162
$680$	0	0
$681$	−14988.0	−0.843377
$682$	0	0
$683$	23482.5	1.31557	0.657784	−	0.753207i	$-0.271494\pi$
$683$	23482.5	1.31557	0.657784	+	0.753207i	$0.271494\pi$
$684$	0	0
$685$	−9275.78	−0.517386
$686$	0	0
$687$	−3424.38	−0.190172
$688$	0	0
$689$	−32190.8	−1.77993
$690$	0	0
$691$	−34349.6	−1.89106	−0.945529	−	0.325538i	$-0.894455\pi$
$691$	−34349.6	−1.89106	−0.945529	+	0.325538i	$0.894455\pi$
$692$	0	0
$693$	−1295.21	−0.0709972
$694$	0	0
$695$	−24716.4	−1.34899
$696$	0	0
$697$	−5601.89	−0.304429
$698$	0	0
$699$	−4853.04	−0.262602
$700$	0	0
$701$	11366.2	0.612406	0.306203	−	0.951966i	$-0.400941\pi$
$701$	11366.2	0.612406	0.306203	+	0.951966i	$0.400941\pi$
$702$	0	0
$703$	3792.94	0.203490
$704$	0	0
$705$	11442.0	0.611247
$706$	0	0
$707$	12929.9	0.687804
$708$	0	0
$709$	25488.2	1.35011	0.675057	−	0.737766i	$-0.264119\pi$
$709$	25488.2	1.35011	0.675057	+	0.737766i	$0.264119\pi$
$710$	0	0
$711$	1204.93	0.0635563
$712$	0	0
$713$	14513.5	0.762320
$714$	0	0
$715$	6449.57	0.337343
$716$	0	0
$717$	−17834.4	−0.928921
$718$	0	0
$719$	15240.6	0.790513	0.395257	−	0.918571i	$-0.370656\pi$
$719$	15240.6	0.790513	0.395257	+	0.918571i	$0.370656\pi$
$720$	0	0
$721$	5733.73	0.296166
$722$	0	0
$723$	−1053.43	−0.0541872
$724$	0	0
$725$	−7795.73	−0.399347
$726$	0	0
$727$	25465.8	1.29914	0.649569	−	0.760302i	$-0.274949\pi$
$727$	25465.8	1.29914	0.649569	+	0.760302i	$0.274949\pi$
$728$	0	0
$729$	11907.1	0.604943
$730$	0	0
$731$	10825.5	0.547736
$732$	0	0
$733$	28441.7	1.43318	0.716588	−	0.697497i	$-0.245703\pi$
$733$	28441.7	1.43318	0.716588	+	0.697497i	$0.245703\pi$
$734$	0	0
$735$	1333.82	0.0669369
$736$	0	0
$737$	−4376.46	−0.218737
$738$	0	0
$739$	−16740.8	−0.833315	−0.416657	−	0.909064i	$-0.636799\pi$
$739$	−16740.8	−0.833315	−0.416657	+	0.909064i	$0.636799\pi$
$740$	0	0
$741$	3188.36	0.158067
$742$	0	0
$743$	10929.1	0.539638	0.269819	−	0.962911i	$-0.413036\pi$
$743$	10929.1	0.539638	0.269819	+	0.962911i	$0.413036\pi$
$744$	0	0
$745$	−21392.6	−1.05203
$746$	0	0
$747$	−11792.3	−0.577588
$748$	0	0
$749$	−8586.89	−0.418903
$750$	0	0
$751$	17745.8	0.862256	0.431128	−	0.902291i	$-0.358116\pi$
$751$	17745.8	0.862256	0.431128	+	0.902291i	$0.358116\pi$
$752$	0	0
$753$	−10667.2	−0.516248
$754$	0	0
$755$	−5171.49	−0.249285
$756$	0	0
$757$	−7640.51	−0.366842	−0.183421	−	0.983034i	$-0.558717\pi$
$757$	−7640.51	−0.366842	−0.183421	+	0.983034i	$0.558717\pi$
$758$	0	0
$759$	2697.18	0.128988
$760$	0	0
$761$	−2633.59	−0.125450	−0.0627250	−	0.998031i	$-0.519979\pi$
$761$	−2633.59	−0.125450	−0.0627250	+	0.998031i	$0.519979\pi$
$762$	0	0
$763$	7166.77	0.340045
$764$	0	0
$765$	4380.16	0.207013
$766$	0	0
$767$	11018.4	0.518709
$768$	0	0
$769$	10669.3	0.500317	0.250158	−	0.968205i	$-0.419517\pi$
$769$	10669.3	0.500317	0.250158	+	0.968205i	$0.419517\pi$
$770$	0	0
$771$	13576.3	0.634162
$772$	0	0
$773$	−17023.0	−0.792076	−0.396038	−	0.918234i	$-0.629615\pi$
$773$	−17023.0	−0.792076	−0.396038	+	0.918234i	$0.629615\pi$
$774$	0	0
$775$	−9858.94	−0.456959
$776$	0	0
$777$	5825.12	0.268951
$778$	0	0
$779$	2669.10	0.122761
$780$	0	0
$781$	−4841.43	−0.221818
$782$	0	0
$783$	−20877.1	−0.952856
$784$	0	0
$785$	12597.5	0.572770
$786$	0	0
$787$	20986.2	0.950543	0.475271	−	0.879839i	$-0.342350\pi$
$787$	20986.2	0.950543	0.475271	+	0.879839i	$0.342350\pi$
$788$	0	0
$789$	−4274.54	−0.192874
$790$	0	0
$791$	12508.4	0.562261
$792$	0	0
$793$	1789.57	0.0801382
$794$	0	0
$795$	−12750.9	−0.568841
$796$	0	0
$797$	−24611.2	−1.09382	−0.546910	−	0.837191i	$-0.684196\pi$
$797$	−24611.2	−1.09382	−0.546910	+	0.837191i	$0.684196\pi$
$798$	0	0
$799$	−12829.0	−0.568031
$800$	0	0
$801$	−1217.72	−0.0537153
$802$	0	0
$803$	4508.34	0.198127
$804$	0	0
$805$	4589.96	0.200963
$806$	0	0
$807$	8696.59	0.379349
$808$	0	0
$809$	−21256.6	−0.923787	−0.461893	−	0.886935i	$-0.652830\pi$
$809$	−21256.6	−0.923787	−0.461893	+	0.886935i	$0.652830\pi$
$810$	0	0
$811$	19012.5	0.823203	0.411601	−	0.911364i	$-0.364970\pi$
$811$	19012.5	0.823203	0.411601	+	0.911364i	$0.364970\pi$
$812$	0	0
$813$	−11859.9	−0.511619
$814$	0	0
$815$	1629.42	0.0700319
$816$	0	0
$817$	−5157.96	−0.220874
$818$	0	0
$819$	−8091.70	−0.345234
$820$	0	0
$821$	2422.77	0.102991	0.0514953	−	0.998673i	$-0.483601\pi$
$821$	2422.77	0.102991	0.0514953	+	0.998673i	$0.483601\pi$
$822$	0	0
$823$	−32860.8	−1.39181	−0.695903	−	0.718136i	$-0.744996\pi$
$823$	−32860.8	−1.39181	−0.695903	+	0.718136i	$0.744996\pi$
$824$	0	0
$825$	−1832.18	−0.0773193
$826$	0	0
$827$	−9714.43	−0.408469	−0.204234	−	0.978922i	$-0.565470\pi$
$827$	−9714.43	−0.408469	−0.204234	+	0.978922i	$0.565470\pi$
$828$	0	0
$829$	25958.6	1.08755	0.543775	−	0.839231i	$-0.316994\pi$
$829$	25958.6	1.08755	0.543775	+	0.839231i	$0.316994\pi$
$830$	0	0
$831$	−21878.5	−0.913304
$832$	0	0
$833$	−1495.51	−0.0622044
$834$	0	0
$835$	−11513.9	−0.477193
$836$	0	0
$837$	−26402.4	−1.09032
$838$	0	0
$839$	−15023.6	−0.618203	−0.309102	−	0.951029i	$-0.600028\pi$
$839$	−15023.6	−0.618203	−0.309102	+	0.951029i	$0.600028\pi$
$840$	0	0
$841$	−2090.83	−0.0857285
$842$	0	0
$843$	−7060.53	−0.288467
$844$	0	0
$845$	21548.3	0.877260
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	−22307.6	−0.901762
$850$	0	0
$851$	20045.5	0.807463
$852$	0	0
$853$	8558.51	0.343538	0.171769	−	0.985137i	$-0.445052\pi$
$853$	8558.51	0.343538	0.171769	+	0.985137i	$0.445052\pi$
$854$	0	0
$855$	−2086.99	−0.0834779
$856$	0	0
$857$	24270.5	0.967403	0.483702	−	0.875233i	$-0.339292\pi$
$857$	24270.5	0.967403	0.483702	+	0.875233i	$0.339292\pi$
$858$	0	0
$859$	−33934.7	−1.34789	−0.673945	−	0.738782i	$-0.735401\pi$
$859$	−33934.7	−1.34789	−0.673945	+	0.738782i	$0.735401\pi$
$860$	0	0
$861$	4099.15	0.162252
$862$	0	0
$863$	23863.8	0.941291	0.470646	−	0.882322i	$-0.344021\pi$
$863$	23863.8	0.941291	0.470646	+	0.882322i	$0.344021\pi$
$864$	0	0
$865$	19309.2	0.758998
$866$	0	0
$867$	−12702.8	−0.497590
$868$	0	0
$869$	−787.962	−0.0307593
$870$	0	0
$871$	−27341.4	−1.06364
$872$	0	0
$873$	2828.03	0.109639
$874$	0	0
$875$	−10583.4	−0.408895
$876$	0	0
$877$	−44421.4	−1.71038	−0.855191	−	0.518312i	$-0.826560\pi$
$877$	−44421.4	−1.71038	−0.855191	+	0.518312i	$0.826560\pi$
$878$	0	0
$879$	−27566.9	−1.05780
$880$	0	0
$881$	−23584.1	−0.901893	−0.450947	−	0.892551i	$-0.648914\pi$
$881$	−23584.1	−0.901893	−0.450947	+	0.892551i	$0.648914\pi$
$882$	0	0
$883$	−40074.6	−1.52731	−0.763657	−	0.645622i	$-0.776598\pi$
$883$	−40074.6	−1.52731	−0.763657	+	0.645622i	$0.776598\pi$
$884$	0	0
$885$	4364.42	0.165772
$886$	0	0
$887$	−47624.2	−1.80278	−0.901388	−	0.433012i	$-0.857451\pi$
$887$	−47624.2	−1.80278	−0.901388	+	0.433012i	$0.857451\pi$
$888$	0	0
$889$	5275.84	0.199039
$890$	0	0
$891$	89.2098	0.00335425
$892$	0	0
$893$	6112.55	0.229058
$894$	0	0
$895$	−23140.6	−0.864251
$896$	0	0
$897$	16850.3	0.627220
$898$	0	0
$899$	28199.6	1.04617
$900$	0	0
$901$	14296.6	0.528623
$902$	0	0
$903$	−7921.49	−0.291928
$904$	0	0
$905$	21147.8	0.776769
$906$	0	0
$907$	−17445.9	−0.638680	−0.319340	−	0.947640i	$-0.603461\pi$
$907$	−17445.9	−0.638680	−0.319340	+	0.947640i	$0.603461\pi$
$908$	0	0
$909$	−31070.4	−1.13371
$910$	0	0
$911$	44006.4	1.60044	0.800218	−	0.599709i	$-0.204717\pi$
$911$	44006.4	1.60044	0.800218	+	0.599709i	$0.204717\pi$
$912$	0	0
$913$	7711.54	0.279534
$914$	0	0
$915$	708.857	0.0256110
$916$	0	0
$917$	5304.05	0.191009
$918$	0	0
$919$	−5259.05	−0.188771	−0.0943853	−	0.995536i	$-0.530089\pi$
$919$	−5259.05	−0.188771	−0.0943853	+	0.995536i	$0.530089\pi$
$920$	0	0
$921$	−9086.42	−0.325090
$922$	0	0
$923$	−30246.3	−1.07862
$924$	0	0
$925$	−13616.8	−0.484020
$926$	0	0
$927$	−13778.1	−0.488169
$928$	0	0
$929$	−10092.4	−0.356428	−0.178214	−	0.983992i	$-0.557032\pi$
$929$	−10092.4	−0.356428	−0.178214	+	0.983992i	$0.557032\pi$
$930$	0	0
$931$	712.556	0.0250839
$932$	0	0
$933$	4507.17	0.158154
$934$	0	0
$935$	−2864.39	−0.100188
$936$	0	0
$937$	−50066.0	−1.74555	−0.872777	−	0.488119i	$-0.837683\pi$
$937$	−50066.0	−1.74555	−0.872777	+	0.488119i	$0.837683\pi$
$938$	0	0
$939$	15037.6	0.522614
$940$	0	0
$941$	−10595.4	−0.367057	−0.183529	−	0.983014i	$-0.558752\pi$
$941$	−10595.4	−0.367057	−0.183529	+	0.983014i	$0.558752\pi$
$942$	0	0
$943$	14106.1	0.487123
$944$	0	0
$945$	−8349.89	−0.287430
$946$	0	0
$947$	48636.9	1.66894	0.834471	−	0.551052i	$-0.185773\pi$
$947$	48636.9	1.66894	0.834471	+	0.551052i	$0.185773\pi$
$948$	0	0
$949$	28165.4	0.963421
$950$	0	0
$951$	2262.98	0.0771631
$952$	0	0
$953$	29709.9	1.00986	0.504931	−	0.863160i	$-0.331518\pi$
$953$	29709.9	1.00986	0.504931	+	0.863160i	$0.331518\pi$
$954$	0	0
$955$	−13281.3	−0.450023
$956$	0	0
$957$	5240.60	0.177016
$958$	0	0
$959$	−7610.29	−0.256256
$960$	0	0
$961$	5871.81	0.197100
$962$	0	0
$963$	20634.2	0.690477
$964$	0	0
$965$	11079.2	0.369587
$966$	0	0
$967$	−23662.3	−0.786897	−0.393448	−	0.919347i	$-0.628718\pi$
$967$	−23662.3	−0.786897	−0.393448	+	0.919347i	$0.628718\pi$
$968$	0	0
$969$	−1416.02	−0.0469443
$970$	0	0
$971$	−6841.96	−0.226127	−0.113063	−	0.993588i	$-0.536066\pi$
$971$	−6841.96	−0.226127	−0.113063	+	0.993588i	$0.536066\pi$
$972$	0	0
$973$	−20278.5	−0.668139
$974$	0	0
$975$	−11446.4	−0.375976
$976$	0	0
$977$	−35439.8	−1.16051	−0.580256	−	0.814434i	$-0.697047\pi$
$977$	−35439.8	−1.16051	−0.580256	+	0.814434i	$0.697047\pi$
$978$	0	0
$979$	796.323	0.0259965
$980$	0	0
$981$	−17221.7	−0.560496
$982$	0	0
$983$	20807.5	0.675133	0.337566	−	0.941302i	$-0.390396\pi$
$983$	20807.5	0.675133	0.337566	+	0.941302i	$0.390396\pi$
$984$	0	0
$985$	20116.7	0.650733
$986$	0	0
$987$	9387.52	0.302744
$988$	0	0
$989$	−27259.6	−0.876445
$990$	0	0
$991$	2744.82	0.0879840	0.0439920	−	0.999032i	$-0.485992\pi$
$991$	2744.82	0.0879840	0.0439920	+	0.999032i	$0.485992\pi$
$992$	0	0
$993$	8691.33	0.277755
$994$	0	0
$995$	−17838.3	−0.568353
$996$	0	0
$997$	8624.49	0.273962	0.136981	−	0.990574i	$-0.456260\pi$
$997$	8624.49	0.273962	0.136981	+	0.990574i	$0.456260\pi$
$998$	0	0
$999$	−36466.0	−1.15489

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	616.4.a.k.1.5	✓	7
4.3	odd	2		1232.4.a.bc.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.k.1.5	✓	7	1.1	even	1	trivial
1232.4.a.bc.1.3		7	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:	\( N \)	\(=\)	\( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	616.a (trivial)

\(f(q)\)	\(=\)	\(q+3.19046 q^{3} +8.53193 q^{5} +7.00000 q^{7} -16.8209 q^{9} +11.0000 q^{11} +68.7212 q^{13} +27.2208 q^{15} -30.5206 q^{17} +14.5420 q^{19} +22.3332 q^{21} +76.8536 q^{23} -52.2062 q^{25} -139.809 q^{27} +149.326 q^{29} +188.846 q^{31} +35.0951 q^{33} +59.7235 q^{35} +260.827 q^{37} +219.253 q^{39} +183.545 q^{41} -354.695 q^{43} -143.515 q^{45} +420.339 q^{47} +49.0000 q^{49} -97.3747 q^{51} -468.426 q^{53} +93.8512 q^{55} +46.3956 q^{57} +160.334 q^{59} +26.0410 q^{61} -117.747 q^{63} +586.325 q^{65} -397.860 q^{67} +245.198 q^{69} -440.130 q^{71} +409.850 q^{73} -166.562 q^{75} +77.0000 q^{77} -71.6329 q^{79} +8.10998 q^{81} +701.049 q^{83} -260.399 q^{85} +476.418 q^{87} +72.3930 q^{89} +481.049 q^{91} +602.506 q^{93} +124.071 q^{95} -168.126 q^{97} -185.030 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} + 11 q^{5} + 49 q^{7} + 108 q^{9} + 77 q^{11} + 26 q^{13} + 67 q^{15} + 44 q^{17} + 34 q^{19} + 21 q^{21} - 29 q^{23} + 182 q^{25} + 99 q^{27} + 94 q^{29} - 173 q^{31} + 33 q^{33} + 77 q^{35}+ \cdots + 1188 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 + 11 * q^5 + 49 * q^7 + 108 * q^9 + 77 * q^11 + 26 * q^13 + 67 * q^15 + 44 * q^17 + 34 * q^19 + 21 * q^21 - 29 * q^23 + 182 * q^25 + 99 * q^27 + 94 * q^29 - 173 * q^31 + 33 * q^33 + 77 * q^35 + 255 * q^37 - 60 * q^39 + 508 * q^41 + 656 * q^43 + 466 * q^45 - 18 * q^47 + 343 * q^49 + 850 * q^51 + 1806 * q^53 + 121 * q^55 + 1154 * q^57 + 665 * q^59 + 608 * q^61 + 756 * q^63 + 588 * q^65 + 669 * q^67 + 2067 * q^69 + 1169 * q^71 + 380 * q^73 + 1254 * q^75 + 539 * q^77 - 110 * q^79 + 2847 * q^81 - 496 * q^83 + 3446 * q^85 - 582 * q^87 + 1321 * q^89 + 182 * q^91 + 1493 * q^93 - 50 * q^95 + 3927 * q^97 + 1188 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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