LMFDB - Embedded newform 616.4.a.i.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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 47x^{4} + 10x^{3} + 612x^{2} + 240x - 1440 $ x^6 - x^5 - 47x^4 + 10x^3 + 612x^2 + 240x - 1440
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.68968$ 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)

$f(q)$	$=$	$q+3.73933 q^{3} +19.7791 q^{5} +7.00000 q^{7} -13.0174 q^{9} -11.0000 q^{11} -19.4047 q^{13} +73.9605 q^{15} +70.8148 q^{17} +18.1735 q^{19} +26.1753 q^{21} +150.203 q^{23} +266.213 q^{25} -149.638 q^{27} +36.1152 q^{29} +312.071 q^{31} -41.1326 q^{33} +138.454 q^{35} -341.208 q^{37} -72.5606 q^{39} +181.399 q^{41} +334.357 q^{43} -257.473 q^{45} -255.999 q^{47} +49.0000 q^{49} +264.800 q^{51} -674.234 q^{53} -217.570 q^{55} +67.9566 q^{57} +396.486 q^{59} +495.662 q^{61} -91.1220 q^{63} -383.808 q^{65} -820.499 q^{67} +561.660 q^{69} -259.971 q^{71} +362.323 q^{73} +995.456 q^{75} -77.0000 q^{77} -39.4291 q^{79} -208.076 q^{81} +649.668 q^{83} +1400.65 q^{85} +135.047 q^{87} -386.836 q^{89} -135.833 q^{91} +1166.94 q^{93} +359.455 q^{95} +336.297 q^{97} +143.192 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} + 42 q^{7} + 60 q^{9} - 66 q^{11} - 6 q^{13} + 126 q^{15} - 14 q^{17} + 80 q^{19} + 254 q^{23} + 220 q^{25} + 90 q^{27} + 132 q^{29} - 52 q^{31} + 14 q^{35} - 518 q^{37} + 332 q^{39} + 486 q^{41}+ \cdots - 660 q^{99}+O(q^{100}) $ 6 * q + 2 * q^5 + 42 * q^7 + 60 * q^9 - 66 * q^11 - 6 * q^13 + 126 * q^15 - 14 * q^17 + 80 * q^19 + 254 * q^23 + 220 * q^25 + 90 * q^27 + 132 * q^29 - 52 * q^31 + 14 * q^35 - 518 * q^37 + 332 * q^39 + 486 * q^41 + 428 * q^43 - 244 * q^45 + 790 * q^47 + 294 * q^49 + 640 * q^51 - 40 * q^53 - 22 * q^55 + 2276 * q^57 + 436 * q^59 + 1034 * q^61 + 420 * q^63 + 800 * q^65 + 562 * q^67 + 530 * q^69 + 2474 * q^71 - 902 * q^73 + 1986 * q^75 - 462 * q^77 + 1636 * q^79 + 570 * q^81 + 3016 * q^83 + 236 * q^85 + 4340 * q^87 + 1750 * q^89 - 42 * q^91 + 150 * q^93 + 3628 * q^95 - 1250 * q^97 - 660 * 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
$3$	3.73933	0.719634	0.359817	−	0.933023i	$-0.382839\pi$
$3$	3.73933	0.719634	0.359817	+	0.933023i	$0.382839\pi$
$4$	0	0
$5$	19.7791	1.76910	0.884548	−	0.466449i	$-0.154467\pi$
$5$	19.7791	1.76910	0.884548	+	0.466449i	$0.154467\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−13.0174	−0.482127
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−19.4047	−0.413992	−0.206996	−	0.978342i	$-0.566369\pi$
$13$	−19.4047	−0.413992	−0.206996	+	0.978342i	$0.566369\pi$
$14$	0	0
$15$	73.9605	1.27310
$16$	0	0
$17$	70.8148	1.01030	0.505151	−	0.863031i	$-0.331437\pi$
$17$	70.8148	1.01030	0.505151	+	0.863031i	$0.331437\pi$
$18$	0	0
$19$	18.1735	0.219436	0.109718	−	0.993963i	$-0.465005\pi$
$19$	18.1735	0.219436	0.109718	+	0.993963i	$0.465005\pi$
$20$	0	0
$21$	26.1753	0.271996
$22$	0	0
$23$	150.203	1.36172	0.680860	−	0.732413i	$-0.261606\pi$
$23$	150.203	1.36172	0.680860	+	0.732413i	$0.261606\pi$
$24$	0	0
$25$	266.213	2.12970
$26$	0	0
$27$	−149.638	−1.06659
$28$	0	0
$29$	36.1152	0.231256	0.115628	−	0.993293i	$-0.463112\pi$
$29$	36.1152	0.231256	0.115628	+	0.993293i	$0.463112\pi$
$30$	0	0
$31$	312.071	1.80805	0.904026	−	0.427477i	$-0.140598\pi$
$31$	312.071	1.80805	0.904026	+	0.427477i	$0.140598\pi$
$32$	0	0
$33$	−41.1326	−0.216978
$34$	0	0
$35$	138.454	0.668656
$36$	0	0
$37$	−341.208	−1.51606	−0.758031	−	0.652219i	$-0.773838\pi$
$37$	−341.208	−1.51606	−0.758031	+	0.652219i	$0.773838\pi$
$38$	0	0
$39$	−72.5606	−0.297923
$40$	0	0
$41$	181.399	0.690968	0.345484	−	0.938425i	$-0.387715\pi$
$41$	181.399	0.690968	0.345484	+	0.938425i	$0.387715\pi$
$42$	0	0
$43$	334.357	1.18579	0.592894	−	0.805280i	$-0.297985\pi$
$43$	334.357	1.18579	0.592894	+	0.805280i	$0.297985\pi$
$44$	0	0
$45$	−257.473	−0.852930
$46$	0	0
$47$	−255.999	−0.794497	−0.397248	−	0.917711i	$-0.630035\pi$
$47$	−255.999	−0.794497	−0.397248	+	0.917711i	$0.630035\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	264.800	0.727047
$52$	0	0
$53$	−674.234	−1.74742	−0.873710	−	0.486448i	$-0.838292\pi$
$53$	−674.234	−1.74742	−0.873710	+	0.486448i	$0.838292\pi$
$54$	0	0
$55$	−217.570	−0.533403
$56$	0	0
$57$	67.9566	0.157913
$58$	0	0
$59$	396.486	0.874883	0.437442	−	0.899247i	$-0.355885\pi$
$59$	396.486	0.874883	0.437442	+	0.899247i	$0.355885\pi$
$60$	0	0
$61$	495.662	1.04038	0.520189	−	0.854051i	$-0.325862\pi$
$61$	495.662	1.04038	0.520189	+	0.854051i	$0.325862\pi$
$62$	0	0
$63$	−91.1220	−0.182227
$64$	0	0
$65$	−383.808	−0.732392
$66$	0	0
$67$	−820.499	−1.49612	−0.748059	−	0.663632i	$-0.769014\pi$
$67$	−820.499	−1.49612	−0.748059	+	0.663632i	$0.769014\pi$
$68$	0	0
$69$	561.660	0.979940
$70$	0	0
$71$	−259.971	−0.434547	−0.217274	−	0.976111i	$-0.569716\pi$
$71$	−259.971	−0.434547	−0.217274	+	0.976111i	$0.569716\pi$
$72$	0	0
$73$	362.323	0.580914	0.290457	−	0.956888i	$-0.406193\pi$
$73$	362.323	0.580914	0.290457	+	0.956888i	$0.406193\pi$
$74$	0	0
$75$	995.456	1.53261
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−39.4291	−0.0561535	−0.0280767	−	0.999606i	$-0.508938\pi$
$79$	−39.4291	−0.0561535	−0.0280767	+	0.999606i	$0.508938\pi$
$80$	0	0
$81$	−208.076	−0.285426
$82$	0	0
$83$	649.668	0.859160	0.429580	−	0.903029i	$-0.358662\pi$
$83$	649.668	0.859160	0.429580	+	0.903029i	$0.358662\pi$
$84$	0	0
$85$	1400.65	1.78732
$86$	0	0
$87$	135.047	0.166420
$88$	0	0
$89$	−386.836	−0.460725	−0.230363	−	0.973105i	$-0.573991\pi$
$89$	−386.836	−0.460725	−0.230363	+	0.973105i	$0.573991\pi$
$90$	0	0
$91$	−135.833	−0.156474
$92$	0	0
$93$	1166.94	1.30114
$94$	0	0
$95$	359.455	0.388203
$96$	0	0
$97$	336.297	0.352019	0.176009	−	0.984389i	$-0.443681\pi$
$97$	336.297	0.352019	0.176009	+	0.984389i	$0.443681\pi$
$98$	0	0
$99$	143.192	0.145367
$100$	0	0
$101$	590.589	0.581840	0.290920	−	0.956747i	$-0.406039\pi$
$101$	590.589	0.581840	0.290920	+	0.956747i	$0.406039\pi$
$102$	0	0
$103$	698.936	0.668623	0.334312	−	0.942463i	$-0.391496\pi$
$103$	698.936	0.668623	0.334312	+	0.942463i	$0.391496\pi$
$104$	0	0
$105$	517.724	0.481187
$106$	0	0
$107$	489.378	0.442149	0.221075	−	0.975257i	$-0.429044\pi$
$107$	489.378	0.442149	0.221075	+	0.975257i	$0.429044\pi$
$108$	0	0
$109$	−2146.51	−1.88623	−0.943113	−	0.332474i	$-0.892117\pi$
$109$	−2146.51	−1.88623	−0.943113	+	0.332474i	$0.892117\pi$
$110$	0	0
$111$	−1275.89	−1.09101
$112$	0	0
$113$	−1657.61	−1.37996	−0.689978	−	0.723830i	$-0.742380\pi$
$113$	−1657.61	−1.37996	−0.689978	+	0.723830i	$0.742380\pi$
$114$	0	0
$115$	2970.89	2.40901
$116$	0	0
$117$	252.600	0.199597
$118$	0	0
$119$	495.704	0.381858
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	678.308	0.497244
$124$	0	0
$125$	2793.06	1.99855
$126$	0	0
$127$	110.623	0.0772929	0.0386464	−	0.999253i	$-0.487695\pi$
$127$	110.623	0.0772929	0.0386464	+	0.999253i	$0.487695\pi$
$128$	0	0
$129$	1250.27	0.853333
$130$	0	0
$131$	−2789.56	−1.86050	−0.930249	−	0.366930i	$-0.880409\pi$
$131$	−2789.56	−1.86050	−0.930249	+	0.366930i	$0.880409\pi$
$132$	0	0
$133$	127.214	0.0829390
$134$	0	0
$135$	−2959.71	−1.88690
$136$	0	0
$137$	253.276	0.157948	0.0789739	−	0.996877i	$-0.474836\pi$
$137$	253.276	0.157948	0.0789739	+	0.996877i	$0.474836\pi$
$138$	0	0
$139$	1417.34	0.864873	0.432437	−	0.901664i	$-0.357654\pi$
$139$	1417.34	0.864873	0.432437	+	0.901664i	$0.357654\pi$
$140$	0	0
$141$	−957.265	−0.571747
$142$	0	0
$143$	213.452	0.124823
$144$	0	0
$145$	714.326	0.409114
$146$	0	0
$147$	183.227	0.102805
$148$	0	0
$149$	−2943.98	−1.61866	−0.809329	−	0.587355i	$-0.800169\pi$
$149$	−2943.98	−1.61866	−0.809329	+	0.587355i	$0.800169\pi$
$150$	0	0
$151$	1090.43	0.587668	0.293834	−	0.955856i	$-0.405069\pi$
$151$	1090.43	0.587668	0.293834	+	0.955856i	$0.405069\pi$
$152$	0	0
$153$	−921.827	−0.487094
$154$	0	0
$155$	6172.48	3.19862
$156$	0	0
$157$	1219.91	0.620126	0.310063	−	0.950716i	$-0.399650\pi$
$157$	1219.91	0.620126	0.310063	+	0.950716i	$0.399650\pi$
$158$	0	0
$159$	−2521.18	−1.25750
$160$	0	0
$161$	1051.42	0.514682
$162$	0	0
$163$	−57.0393	−0.0274090	−0.0137045	−	0.999906i	$-0.504362\pi$
$163$	−57.0393	−0.0274090	−0.0137045	+	0.999906i	$0.504362\pi$
$164$	0	0
$165$	−813.566	−0.383855
$166$	0	0
$167$	2091.75	0.969249	0.484625	−	0.874722i	$-0.338956\pi$
$167$	2091.75	0.969249	0.484625	+	0.874722i	$0.338956\pi$
$168$	0	0
$169$	−1820.46	−0.828610
$170$	0	0
$171$	−236.572	−0.105796
$172$	0	0
$173$	−333.342	−0.146495	−0.0732473	−	0.997314i	$-0.523336\pi$
$173$	−333.342	−0.146495	−0.0732473	+	0.997314i	$0.523336\pi$
$174$	0	0
$175$	1863.49	0.804952
$176$	0	0
$177$	1482.59	0.629596
$178$	0	0
$179$	648.025	0.270590	0.135295	−	0.990805i	$-0.456802\pi$
$179$	648.025	0.270590	0.135295	+	0.990805i	$0.456802\pi$
$180$	0	0
$181$	−3579.98	−1.47015	−0.735077	−	0.677984i	$-0.762854\pi$
$181$	−3579.98	−1.47015	−0.735077	+	0.677984i	$0.762854\pi$
$182$	0	0
$183$	1853.44	0.748691
$184$	0	0
$185$	−6748.79	−2.68206
$186$	0	0
$187$	−778.963	−0.304617
$188$	0	0
$189$	−1047.47	−0.403133
$190$	0	0
$191$	−1058.90	−0.401148	−0.200574	−	0.979679i	$-0.564281\pi$
$191$	−1058.90	−0.401148	−0.200574	+	0.979679i	$0.564281\pi$
$192$	0	0
$193$	−1106.19	−0.412566	−0.206283	−	0.978492i	$-0.566137\pi$
$193$	−1106.19	−0.412566	−0.206283	+	0.978492i	$0.566137\pi$
$194$	0	0
$195$	−1435.18	−0.527054
$196$	0	0
$197$	−2183.49	−0.789681	−0.394841	−	0.918750i	$-0.629200\pi$
$197$	−2183.49	−0.789681	−0.394841	+	0.918750i	$0.629200\pi$
$198$	0	0
$199$	−275.244	−0.0980479	−0.0490239	−	0.998798i	$-0.515611\pi$
$199$	−275.244	−0.0980479	−0.0490239	+	0.998798i	$0.515611\pi$
$200$	0	0
$201$	−3068.11	−1.07666
$202$	0	0
$203$	252.807	0.0874066
$204$	0	0
$205$	3587.90	1.22239
$206$	0	0
$207$	−1955.26	−0.656523
$208$	0	0
$209$	−199.908	−0.0661624
$210$	0	0
$211$	−184.206	−0.0601006	−0.0300503	−	0.999548i	$-0.509567\pi$
$211$	−184.206	−0.0601006	−0.0300503	+	0.999548i	$0.509567\pi$
$212$	0	0
$213$	−972.116	−0.312715
$214$	0	0
$215$	6613.27	2.09777
$216$	0	0
$217$	2184.50	0.683380
$218$	0	0
$219$	1354.85	0.418046
$220$	0	0
$221$	−1374.14	−0.418257
$222$	0	0
$223$	−762.855	−0.229079	−0.114539	−	0.993419i	$-0.536539\pi$
$223$	−762.855	−0.229079	−0.114539	+	0.993419i	$0.536539\pi$
$224$	0	0
$225$	−3465.41	−1.02679
$226$	0	0
$227$	5280.62	1.54399	0.771997	−	0.635626i	$-0.219258\pi$
$227$	5280.62	1.54399	0.771997	+	0.635626i	$0.219258\pi$
$228$	0	0
$229$	5266.64	1.51978	0.759890	−	0.650052i	$-0.225253\pi$
$229$	5266.64	1.51978	0.759890	+	0.650052i	$0.225253\pi$
$230$	0	0
$231$	−287.928	−0.0820099
$232$	0	0
$233$	−6792.89	−1.90994	−0.954972	−	0.296695i	$-0.904116\pi$
$233$	−6792.89	−1.90994	−0.954972	+	0.296695i	$0.904116\pi$
$234$	0	0
$235$	−5063.44	−1.40554
$236$	0	0
$237$	−147.438	−0.0404099
$238$	0	0
$239$	6317.30	1.70976	0.854879	−	0.518828i	$-0.173631\pi$
$239$	6317.30	1.70976	0.854879	+	0.518828i	$0.173631\pi$
$240$	0	0
$241$	1271.42	0.339831	0.169915	−	0.985459i	$-0.445651\pi$
$241$	1271.42	0.339831	0.169915	+	0.985459i	$0.445651\pi$
$242$	0	0
$243$	3262.17	0.861187
$244$	0	0
$245$	969.176	0.252728
$246$	0	0
$247$	−352.651	−0.0908448
$248$	0	0
$249$	2429.32	0.618281
$250$	0	0
$251$	−760.172	−0.191162	−0.0955810	−	0.995422i	$-0.530471\pi$
$251$	−760.172	−0.191162	−0.0955810	+	0.995422i	$0.530471\pi$
$252$	0	0
$253$	−1652.24	−0.410574
$254$	0	0
$255$	5237.50	1.28622
$256$	0	0
$257$	−1133.67	−0.275160	−0.137580	−	0.990491i	$-0.543932\pi$
$257$	−1133.67	−0.275160	−0.137580	+	0.990491i	$0.543932\pi$
$258$	0	0
$259$	−2388.46	−0.573018
$260$	0	0
$261$	−470.127	−0.111495
$262$	0	0
$263$	−7535.49	−1.76676	−0.883381	−	0.468656i	$-0.844739\pi$
$263$	−7535.49	−1.76676	−0.883381	+	0.468656i	$0.844739\pi$
$264$	0	0
$265$	−13335.7	−3.09135
$266$	0	0
$267$	−1446.51	−0.331554
$268$	0	0
$269$	−6111.16	−1.38514	−0.692572	−	0.721349i	$-0.743523\pi$
$269$	−6111.16	−1.38514	−0.692572	+	0.721349i	$0.743523\pi$
$270$	0	0
$271$	3162.44	0.708872	0.354436	−	0.935080i	$-0.384673\pi$
$271$	3162.44	0.708872	0.354436	+	0.935080i	$0.384673\pi$
$272$	0	0
$273$	−507.924	−0.112604
$274$	0	0
$275$	−2928.34	−0.642129
$276$	0	0
$277$	2078.80	0.450914	0.225457	−	0.974253i	$-0.427612\pi$
$277$	2078.80	0.450914	0.225457	+	0.974253i	$0.427612\pi$
$278$	0	0
$279$	−4062.36	−0.871711
$280$	0	0
$281$	−8558.16	−1.81686	−0.908429	−	0.418040i	$-0.862717\pi$
$281$	−8558.16	−1.81686	−0.908429	+	0.418040i	$0.862717\pi$
$282$	0	0
$283$	383.249	0.0805011	0.0402505	−	0.999190i	$-0.487184\pi$
$283$	383.249	0.0805011	0.0402505	+	0.999190i	$0.487184\pi$
$284$	0	0
$285$	1344.12	0.279364
$286$	0	0
$287$	1269.79	0.261161
$288$	0	0
$289$	101.739	0.0207082
$290$	0	0
$291$	1257.52	0.253324
$292$	0	0
$293$	−3728.36	−0.743389	−0.371695	−	0.928355i	$-0.621223\pi$
$293$	−3728.36	−0.743389	−0.371695	+	0.928355i	$0.621223\pi$
$294$	0	0
$295$	7842.14	1.54775
$296$	0	0
$297$	1646.02	0.321589
$298$	0	0
$299$	−2914.65	−0.563742
$300$	0	0
$301$	2340.50	0.448186
$302$	0	0
$303$	2208.41	0.418712
$304$	0	0
$305$	9803.75	1.84053
$306$	0	0
$307$	−7148.19	−1.32889	−0.664444	−	0.747338i	$-0.731332\pi$
$307$	−7148.19	−1.32889	−0.664444	+	0.747338i	$0.731332\pi$
$308$	0	0
$309$	2613.55	0.481164
$310$	0	0
$311$	−5641.46	−1.02861	−0.514305	−	0.857607i	$-0.671950\pi$
$311$	−5641.46	−1.02861	−0.514305	+	0.857607i	$0.671950\pi$
$312$	0	0
$313$	9533.22	1.72156	0.860781	−	0.508975i	$-0.169975\pi$
$313$	9533.22	1.72156	0.860781	+	0.508975i	$0.169975\pi$
$314$	0	0
$315$	−1802.31	−0.322377
$316$	0	0
$317$	−3133.29	−0.555153	−0.277576	−	0.960704i	$-0.589531\pi$
$317$	−3133.29	−0.555153	−0.277576	+	0.960704i	$0.589531\pi$
$318$	0	0
$319$	−397.267	−0.0697263
$320$	0	0
$321$	1829.95	0.318186
$322$	0	0
$323$	1286.95	0.221696
$324$	0	0
$325$	−5165.78	−0.881680
$326$	0	0
$327$	−8026.51	−1.35739
$328$	0	0
$329$	−1792.00	−0.300292
$330$	0	0
$331$	−11538.6	−1.91608	−0.958038	−	0.286640i	$-0.907462\pi$
$331$	−11538.6	−1.91608	−0.958038	+	0.286640i	$0.907462\pi$
$332$	0	0
$333$	4441.66	0.730935
$334$	0	0
$335$	−16228.7	−2.64678
$336$	0	0
$337$	−2167.24	−0.350318	−0.175159	−	0.984540i	$-0.556044\pi$
$337$	−2167.24	−0.350318	−0.175159	+	0.984540i	$0.556044\pi$
$338$	0	0
$339$	−6198.35	−0.993063
$340$	0	0
$341$	−3432.78	−0.545148
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	11109.1	1.73361
$346$	0	0
$347$	−620.740	−0.0960318	−0.0480159	−	0.998847i	$-0.515290\pi$
$347$	−620.740	−0.0960318	−0.0480159	+	0.998847i	$0.515290\pi$
$348$	0	0
$349$	−156.193	−0.0239564	−0.0119782	−	0.999928i	$-0.503813\pi$
$349$	−156.193	−0.0239564	−0.0119782	+	0.999928i	$0.503813\pi$
$350$	0	0
$351$	2903.69	0.441560
$352$	0	0
$353$	2174.30	0.327836	0.163918	−	0.986474i	$-0.447587\pi$
$353$	2174.30	0.327836	0.163918	+	0.986474i	$0.447587\pi$
$354$	0	0
$355$	−5141.99	−0.768756
$356$	0	0
$357$	1853.60	0.274798
$358$	0	0
$359$	5993.02	0.881058	0.440529	−	0.897738i	$-0.354791\pi$
$359$	5993.02	0.881058	0.440529	+	0.897738i	$0.354791\pi$
$360$	0	0
$361$	−6528.72	−0.951848
$362$	0	0
$363$	452.459	0.0654213
$364$	0	0
$365$	7166.43	1.02769
$366$	0	0
$367$	−10822.6	−1.53934	−0.769669	−	0.638443i	$-0.779579\pi$
$367$	−10822.6	−1.53934	−0.769669	+	0.638443i	$0.779579\pi$
$368$	0	0
$369$	−2361.34	−0.333134
$370$	0	0
$371$	−4719.64	−0.660462
$372$	0	0
$373$	11115.4	1.54298	0.771491	−	0.636240i	$-0.219511\pi$
$373$	11115.4	1.54298	0.771491	+	0.636240i	$0.219511\pi$
$374$	0	0
$375$	10444.2	1.43823
$376$	0	0
$377$	−700.805	−0.0957382
$378$	0	0
$379$	−11919.0	−1.61540	−0.807702	−	0.589591i	$-0.799289\pi$
$379$	−11919.0	−1.61540	−0.807702	+	0.589591i	$0.799289\pi$
$380$	0	0
$381$	413.655	0.0556226
$382$	0	0
$383$	3655.90	0.487749	0.243875	−	0.969807i	$-0.421581\pi$
$383$	3655.90	0.487749	0.243875	+	0.969807i	$0.421581\pi$
$384$	0	0
$385$	−1522.99	−0.201607
$386$	0	0
$387$	−4352.47	−0.571701
$388$	0	0
$389$	4389.79	0.572162	0.286081	−	0.958205i	$-0.407647\pi$
$389$	4389.79	0.572162	0.286081	+	0.958205i	$0.407647\pi$
$390$	0	0
$391$	10636.6	1.37575
$392$	0	0
$393$	−10431.1	−1.33888
$394$	0	0
$395$	−779.873	−0.0993409
$396$	0	0
$397$	−7234.62	−0.914598	−0.457299	−	0.889313i	$-0.651183\pi$
$397$	−7234.62	−0.914598	−0.457299	+	0.889313i	$0.651183\pi$
$398$	0	0
$399$	475.696	0.0596857
$400$	0	0
$401$	14345.9	1.78653	0.893266	−	0.449528i	$-0.148408\pi$
$401$	14345.9	1.78653	0.893266	+	0.449528i	$0.148408\pi$
$402$	0	0
$403$	−6055.65	−0.748520
$404$	0	0
$405$	−4115.55	−0.504946
$406$	0	0
$407$	3753.29	0.457110
$408$	0	0
$409$	−15651.9	−1.89227	−0.946135	−	0.323773i	$-0.895048\pi$
$409$	−15651.9	−1.89227	−0.946135	+	0.323773i	$0.895048\pi$
$410$	0	0
$411$	947.082	0.113665
$412$	0	0
$413$	2775.40	0.330675
$414$	0	0
$415$	12849.8	1.51994
$416$	0	0
$417$	5299.91	0.622392
$418$	0	0
$419$	−7162.22	−0.835077	−0.417538	−	0.908659i	$-0.637107\pi$
$419$	−7162.22	−0.835077	−0.417538	+	0.908659i	$0.637107\pi$
$420$	0	0
$421$	−5125.05	−0.593301	−0.296651	−	0.954986i	$-0.595870\pi$
$421$	−5125.05	−0.593301	−0.296651	+	0.954986i	$0.595870\pi$
$422$	0	0
$423$	3332.45	0.383048
$424$	0	0
$425$	18851.8	2.15164
$426$	0	0
$427$	3469.64	0.393226
$428$	0	0
$429$	798.166	0.0898271
$430$	0	0
$431$	15125.2	1.69038	0.845192	−	0.534463i	$-0.179486\pi$
$431$	15125.2	1.69038	0.845192	+	0.534463i	$0.179486\pi$
$432$	0	0
$433$	−9009.60	−0.999940	−0.499970	−	0.866043i	$-0.666656\pi$
$433$	−9009.60	−0.999940	−0.499970	+	0.866043i	$0.666656\pi$
$434$	0	0
$435$	2671.10	0.294413
$436$	0	0
$437$	2729.72	0.298810
$438$	0	0
$439$	8544.58	0.928954	0.464477	−	0.885585i	$-0.346242\pi$
$439$	8544.58	0.928954	0.464477	+	0.885585i	$0.346242\pi$
$440$	0	0
$441$	−637.854	−0.0688753
$442$	0	0
$443$	6815.43	0.730950	0.365475	−	0.930821i	$-0.380907\pi$
$443$	6815.43	0.730950	0.365475	+	0.930821i	$0.380907\pi$
$444$	0	0
$445$	−7651.27	−0.815068
$446$	0	0
$447$	−11008.5	−1.16484
$448$	0	0
$449$	10587.3	1.11279	0.556397	−	0.830917i	$-0.312184\pi$
$449$	10587.3	1.11279	0.556397	+	0.830917i	$0.312184\pi$
$450$	0	0
$451$	−1995.38	−0.208335
$452$	0	0
$453$	4077.47	0.422906
$454$	0	0
$455$	−2686.65	−0.276818
$456$	0	0
$457$	−7189.63	−0.735923	−0.367962	−	0.929841i	$-0.619944\pi$
$457$	−7189.63	−0.735923	−0.367962	+	0.929841i	$0.619944\pi$
$458$	0	0
$459$	−10596.6	−1.07758
$460$	0	0
$461$	12715.2	1.28461	0.642304	−	0.766450i	$-0.277979\pi$
$461$	12715.2	1.28461	0.642304	+	0.766450i	$0.277979\pi$
$462$	0	0
$463$	−2900.83	−0.291173	−0.145586	−	0.989346i	$-0.546507\pi$
$463$	−2900.83	−0.291173	−0.145586	+	0.989346i	$0.546507\pi$
$464$	0	0
$465$	23080.9	2.30183
$466$	0	0
$467$	5640.30	0.558891	0.279445	−	0.960162i	$-0.409849\pi$
$467$	5640.30	0.558891	0.279445	+	0.960162i	$0.409849\pi$
$468$	0	0
$469$	−5743.49	−0.565480
$470$	0	0
$471$	4561.66	0.446263
$472$	0	0
$473$	−3677.92	−0.357529
$474$	0	0
$475$	4838.01	0.467333
$476$	0	0
$477$	8776.80	0.842478
$478$	0	0
$479$	7611.96	0.726095	0.363047	−	0.931771i	$-0.381736\pi$
$479$	7611.96	0.726095	0.363047	+	0.931771i	$0.381736\pi$
$480$	0	0
$481$	6621.05	0.627638
$482$	0	0
$483$	3931.62	0.370383
$484$	0	0
$485$	6651.65	0.622755
$486$	0	0
$487$	−1984.97	−0.184698	−0.0923488	−	0.995727i	$-0.529437\pi$
$487$	−1984.97	−0.184698	−0.0923488	+	0.995727i	$0.529437\pi$
$488$	0	0
$489$	−213.288	−0.0197244
$490$	0	0
$491$	7934.81	0.729313	0.364657	−	0.931142i	$-0.381186\pi$
$491$	7934.81	0.729313	0.364657	+	0.931142i	$0.381186\pi$
$492$	0	0
$493$	2557.49	0.233638
$494$	0	0
$495$	2832.20	0.257168
$496$	0	0
$497$	−1819.80	−0.164243
$498$	0	0
$499$	−13405.5	−1.20263	−0.601313	−	0.799013i	$-0.705356\pi$
$499$	−13405.5	−1.20263	−0.601313	+	0.799013i	$0.705356\pi$
$500$	0	0
$501$	7821.74	0.697505
$502$	0	0
$503$	9355.38	0.829296	0.414648	−	0.909982i	$-0.363905\pi$
$503$	9355.38	0.829296	0.414648	+	0.909982i	$0.363905\pi$
$504$	0	0
$505$	11681.3	1.02933
$506$	0	0
$507$	−6807.28	−0.596296
$508$	0	0
$509$	22230.5	1.93585	0.967927	−	0.251233i	$-0.0808361\pi$
$509$	22230.5	1.93585	0.967927	+	0.251233i	$0.0808361\pi$
$510$	0	0
$511$	2536.26	0.219565
$512$	0	0
$513$	−2719.45	−0.234048
$514$	0	0
$515$	13824.3	1.18286
$516$	0	0
$517$	2815.99	0.239550
$518$	0	0
$519$	−1246.48	−0.105422
$520$	0	0
$521$	−15130.6	−1.27233	−0.636165	−	0.771553i	$-0.719480\pi$
$521$	−15130.6	−1.27233	−0.636165	+	0.771553i	$0.719480\pi$
$522$	0	0
$523$	20491.1	1.71322	0.856610	−	0.515964i	$-0.172566\pi$
$523$	20491.1	1.71322	0.856610	+	0.515964i	$0.172566\pi$
$524$	0	0
$525$	6968.20	0.579270
$526$	0	0
$527$	22099.3	1.82668
$528$	0	0
$529$	10394.1	0.854283
$530$	0	0
$531$	−5161.23	−0.421805
$532$	0	0
$533$	−3519.99	−0.286055
$534$	0	0
$535$	9679.46	0.782205
$536$	0	0
$537$	2423.18	0.194726
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	16251.4	1.29150	0.645752	−	0.763547i	$-0.276544\pi$
$541$	16251.4	1.29150	0.645752	+	0.763547i	$0.276544\pi$
$542$	0	0
$543$	−13386.7	−1.05797
$544$	0	0
$545$	−42456.1	−3.33691
$546$	0	0
$547$	−17652.3	−1.37981	−0.689906	−	0.723899i	$-0.742348\pi$
$547$	−17652.3	−1.37981	−0.689906	+	0.723899i	$0.742348\pi$
$548$	0	0
$549$	−6452.25	−0.501594
$550$	0	0
$551$	656.339	0.0507459
$552$	0	0
$553$	−276.004	−0.0212240
$554$	0	0
$555$	−25235.9	−1.93010
$556$	0	0
$557$	−14086.4	−1.07156	−0.535782	−	0.844356i	$-0.679983\pi$
$557$	−14086.4	−1.07156	−0.535782	+	0.844356i	$0.679983\pi$
$558$	0	0
$559$	−6488.09	−0.490907
$560$	0	0
$561$	−2912.80	−0.219213
$562$	0	0
$563$	−9787.92	−0.732702	−0.366351	−	0.930477i	$-0.619393\pi$
$563$	−9787.92	−0.732702	−0.366351	+	0.930477i	$0.619393\pi$
$564$	0	0
$565$	−32786.1	−2.44127
$566$	0	0
$567$	−1456.53	−0.107881
$568$	0	0
$569$	24992.9	1.84140	0.920700	−	0.390270i	$-0.127618\pi$
$569$	24992.9	1.84140	0.920700	+	0.390270i	$0.127618\pi$
$570$	0	0
$571$	−14480.8	−1.06130	−0.530649	−	0.847592i	$-0.678052\pi$
$571$	−14480.8	−1.06130	−0.530649	+	0.847592i	$0.678052\pi$
$572$	0	0
$573$	−3959.57	−0.288679
$574$	0	0
$575$	39986.1	2.90006
$576$	0	0
$577$	2348.23	0.169425	0.0847125	−	0.996405i	$-0.473003\pi$
$577$	2348.23	0.169425	0.0847125	+	0.996405i	$0.473003\pi$
$578$	0	0
$579$	−4136.40	−0.296896
$580$	0	0
$581$	4547.67	0.324732
$582$	0	0
$583$	7416.58	0.526867
$584$	0	0
$585$	4996.19	0.353106
$586$	0	0
$587$	−9755.88	−0.685977	−0.342988	−	0.939340i	$-0.611439\pi$
$587$	−9755.88	−0.685977	−0.342988	+	0.939340i	$0.611439\pi$
$588$	0	0
$589$	5671.42	0.396752
$590$	0	0
$591$	−8164.78	−0.568281
$592$	0	0
$593$	12901.1	0.893396	0.446698	−	0.894685i	$-0.352600\pi$
$593$	12901.1	0.893396	0.446698	+	0.894685i	$0.352600\pi$
$594$	0	0
$595$	9804.57	0.675543
$596$	0	0
$597$	−1029.23	−0.0705586
$598$	0	0
$599$	−13063.4	−0.891082	−0.445541	−	0.895262i	$-0.646989\pi$
$599$	−13063.4	−0.891082	−0.445541	+	0.895262i	$0.646989\pi$
$600$	0	0
$601$	−18806.4	−1.27642	−0.638212	−	0.769861i	$-0.720326\pi$
$601$	−18806.4	−1.27642	−0.638212	+	0.769861i	$0.720326\pi$
$602$	0	0
$603$	10680.8	0.721319
$604$	0	0
$605$	2393.27	0.160827
$606$	0	0
$607$	−10192.9	−0.681573	−0.340787	−	0.940141i	$-0.610693\pi$
$607$	−10192.9	−0.681573	−0.340787	+	0.940141i	$0.610693\pi$
$608$	0	0
$609$	945.326	0.0629007
$610$	0	0
$611$	4967.59	0.328915
$612$	0	0
$613$	−8546.34	−0.563105	−0.281553	−	0.959546i	$-0.590849\pi$
$613$	−8546.34	−0.563105	−0.281553	+	0.959546i	$0.590849\pi$
$614$	0	0
$615$	13416.3	0.879672
$616$	0	0
$617$	−83.9771	−0.00547940	−0.00273970	−	0.999996i	$-0.500872\pi$
$617$	−83.9771	−0.00547940	−0.00273970	+	0.999996i	$0.500872\pi$
$618$	0	0
$619$	1254.82	0.0814786	0.0407393	−	0.999170i	$-0.487029\pi$
$619$	1254.82	0.0814786	0.0407393	+	0.999170i	$0.487029\pi$
$620$	0	0
$621$	−22476.2	−1.45240
$622$	0	0
$623$	−2707.85	−0.174138
$624$	0	0
$625$	21967.6	1.40593
$626$	0	0
$627$	−747.522	−0.0476127
$628$	0	0
$629$	−24162.6	−1.53168
$630$	0	0
$631$	−10941.6	−0.690300	−0.345150	−	0.938547i	$-0.612172\pi$
$631$	−10941.6	−0.690300	−0.345150	+	0.938547i	$0.612172\pi$
$632$	0	0
$633$	−688.805	−0.0432504
$634$	0	0
$635$	2188.02	0.136739
$636$	0	0
$637$	−950.831	−0.0591418
$638$	0	0
$639$	3384.15	0.209507
$640$	0	0
$641$	10785.4	0.664585	0.332293	−	0.943176i	$-0.392178\pi$
$641$	10785.4	0.664585	0.332293	+	0.943176i	$0.392178\pi$
$642$	0	0
$643$	9787.54	0.600285	0.300142	−	0.953894i	$-0.402966\pi$
$643$	9787.54	0.600285	0.300142	+	0.953894i	$0.402966\pi$
$644$	0	0
$645$	24729.2	1.50963
$646$	0	0
$647$	11154.8	0.677808	0.338904	−	0.940821i	$-0.389944\pi$
$647$	11154.8	0.677808	0.338904	+	0.940821i	$0.389944\pi$
$648$	0	0
$649$	−4361.35	−0.263787
$650$	0	0
$651$	8168.55	0.491783
$652$	0	0
$653$	11300.3	0.677202	0.338601	−	0.940930i	$-0.390046\pi$
$653$	11300.3	0.677202	0.338601	+	0.940930i	$0.390046\pi$
$654$	0	0
$655$	−55175.0	−3.29140
$656$	0	0
$657$	−4716.52	−0.280075
$658$	0	0
$659$	−9671.17	−0.571677	−0.285839	−	0.958278i	$-0.592272\pi$
$659$	−9671.17	−0.571677	−0.285839	+	0.958278i	$0.592272\pi$
$660$	0	0
$661$	1455.20	0.0856290	0.0428145	−	0.999083i	$-0.486368\pi$
$661$	1455.20	0.0856290	0.0428145	+	0.999083i	$0.486368\pi$
$662$	0	0
$663$	−5138.36	−0.300992
$664$	0	0
$665$	2516.19	0.146727
$666$	0	0
$667$	5424.63	0.314906
$668$	0	0
$669$	−2852.56	−0.164853
$670$	0	0
$671$	−5452.28	−0.313686
$672$	0	0
$673$	13180.6	0.754943	0.377472	−	0.926021i	$-0.376794\pi$
$673$	13180.6	0.754943	0.377472	+	0.926021i	$0.376794\pi$
$674$	0	0
$675$	−39835.6	−2.27152
$676$	0	0
$677$	−22201.2	−1.26036	−0.630180	−	0.776449i	$-0.717019\pi$
$677$	−22201.2	−1.26036	−0.630180	+	0.776449i	$0.717019\pi$
$678$	0	0
$679$	2354.08	0.133051
$680$	0	0
$681$	19745.9	1.11111
$682$	0	0
$683$	−17792.8	−0.996813	−0.498406	−	0.866944i	$-0.666081\pi$
$683$	−17792.8	−0.996813	−0.498406	+	0.866944i	$0.666081\pi$
$684$	0	0
$685$	5009.57	0.279425
$686$	0	0
$687$	19693.7	1.09368
$688$	0	0
$689$	13083.3	0.723418
$690$	0	0
$691$	−4729.48	−0.260373	−0.130187	−	0.991490i	$-0.541558\pi$
$691$	−4729.48	−0.260373	−0.130187	+	0.991490i	$0.541558\pi$
$692$	0	0
$693$	1002.34	0.0549435
$694$	0	0
$695$	28033.8	1.53004
$696$	0	0
$697$	12845.7	0.698086
$698$	0	0
$699$	−25400.8	−1.37446
$700$	0	0
$701$	21835.4	1.17648	0.588241	−	0.808686i	$-0.299821\pi$
$701$	21835.4	1.17648	0.588241	+	0.808686i	$0.299821\pi$
$702$	0	0
$703$	−6200.94	−0.332678
$704$	0	0
$705$	−18933.8	−1.01147
$706$	0	0
$707$	4134.13	0.219915
$708$	0	0
$709$	−26123.5	−1.38376	−0.691881	−	0.722011i	$-0.743218\pi$
$709$	−26123.5	−1.38376	−0.691881	+	0.722011i	$0.743218\pi$
$710$	0	0
$711$	513.266	0.0270731
$712$	0	0
$713$	46874.1	2.46206
$714$	0	0
$715$	4221.89	0.220825
$716$	0	0
$717$	23622.4	1.23040
$718$	0	0
$719$	−6644.44	−0.344640	−0.172320	−	0.985041i	$-0.555126\pi$
$719$	−6644.44	−0.344640	−0.172320	+	0.985041i	$0.555126\pi$
$720$	0	0
$721$	4892.55	0.252716
$722$	0	0
$723$	4754.25	0.244554
$724$	0	0
$725$	9614.33	0.492507
$726$	0	0
$727$	36976.7	1.88637	0.943185	−	0.332268i	$-0.107814\pi$
$727$	36976.7	1.88637	0.943185	+	0.332268i	$0.107814\pi$
$728$	0	0
$729$	17816.4	0.905165
$730$	0	0
$731$	23677.4	1.19800
$732$	0	0
$733$	2768.13	0.139486	0.0697430	−	0.997565i	$-0.477782\pi$
$733$	2768.13	0.139486	0.0697430	+	0.997565i	$0.477782\pi$
$734$	0	0
$735$	3624.07	0.181872
$736$	0	0
$737$	9025.49	0.451097
$738$	0	0
$739$	−16615.1	−0.827059	−0.413529	−	0.910491i	$-0.635704\pi$
$739$	−16615.1	−0.827059	−0.413529	+	0.910491i	$0.635704\pi$
$740$	0	0
$741$	−1318.68	−0.0653750
$742$	0	0
$743$	−9526.59	−0.470386	−0.235193	−	0.971949i	$-0.575572\pi$
$743$	−9526.59	−0.470386	−0.235193	+	0.971949i	$0.575572\pi$
$744$	0	0
$745$	−58229.2	−2.86356
$746$	0	0
$747$	−8457.01	−0.414225
$748$	0	0
$749$	3425.65	0.167117
$750$	0	0
$751$	−108.237	−0.00525914	−0.00262957	−	0.999997i	$-0.500837\pi$
$751$	−108.237	−0.00525914	−0.00262957	+	0.999997i	$0.500837\pi$
$752$	0	0
$753$	−2842.53	−0.137567
$754$	0	0
$755$	21567.7	1.03964
$756$	0	0
$757$	6511.07	0.312614	0.156307	−	0.987709i	$-0.450041\pi$
$757$	6511.07	0.312614	0.156307	+	0.987709i	$0.450041\pi$
$758$	0	0
$759$	−6178.26	−0.295463
$760$	0	0
$761$	7693.24	0.366465	0.183232	−	0.983070i	$-0.441344\pi$
$761$	7693.24	0.366465	0.183232	+	0.983070i	$0.441344\pi$
$762$	0	0
$763$	−15025.6	−0.712926
$764$	0	0
$765$	−18232.9	−0.861716
$766$	0	0
$767$	−7693.70	−0.362195
$768$	0	0
$769$	15859.2	0.743690	0.371845	−	0.928295i	$-0.378725\pi$
$769$	15859.2	0.743690	0.371845	+	0.928295i	$0.378725\pi$
$770$	0	0
$771$	−4239.14	−0.198014
$772$	0	0
$773$	15798.8	0.735115	0.367558	−	0.930001i	$-0.380194\pi$
$773$	15798.8	0.735115	0.367558	+	0.930001i	$0.380194\pi$
$774$	0	0
$775$	83077.3	3.85061
$776$	0	0
$777$	−8931.22	−0.412363
$778$	0	0
$779$	3296.64	0.151623
$780$	0	0
$781$	2859.68	0.131021
$782$	0	0
$783$	−5404.22	−0.246655
$784$	0	0
$785$	24128.8	1.09706
$786$	0	0
$787$	−14535.5	−0.658365	−0.329182	−	0.944266i	$-0.606773\pi$
$787$	−14535.5	−0.658365	−0.329182	+	0.944266i	$0.606773\pi$
$788$	0	0
$789$	−28177.7	−1.27142
$790$	0	0
$791$	−11603.3	−0.521574
$792$	0	0
$793$	−9618.18	−0.430708
$794$	0	0
$795$	−49866.7	−2.22464
$796$	0	0
$797$	−5259.27	−0.233743	−0.116871	−	0.993147i	$-0.537287\pi$
$797$	−5259.27	−0.233743	−0.116871	+	0.993147i	$0.537287\pi$
$798$	0	0
$799$	−18128.5	−0.802681
$800$	0	0
$801$	5035.62	0.222128
$802$	0	0
$803$	−3985.56	−0.175152
$804$	0	0
$805$	20796.2	0.910522
$806$	0	0
$807$	−22851.6	−0.996796
$808$	0	0
$809$	24785.4	1.07714	0.538570	−	0.842581i	$-0.318965\pi$
$809$	24785.4	1.07714	0.538570	+	0.842581i	$0.318965\pi$
$810$	0	0
$811$	−33695.3	−1.45894	−0.729471	−	0.684012i	$-0.760234\pi$
$811$	−33695.3	−1.45894	−0.729471	+	0.684012i	$0.760234\pi$
$812$	0	0
$813$	11825.4	0.510128
$814$	0	0
$815$	−1128.19	−0.0484891
$816$	0	0
$817$	6076.42	0.260205
$818$	0	0
$819$	1768.20	0.0754406
$820$	0	0
$821$	25640.1	1.08994	0.544972	−	0.838454i	$-0.316540\pi$
$821$	25640.1	1.08994	0.544972	+	0.838454i	$0.316540\pi$
$822$	0	0
$823$	8504.28	0.360195	0.180097	−	0.983649i	$-0.442359\pi$
$823$	8504.28	0.360195	0.180097	+	0.983649i	$0.442359\pi$
$824$	0	0
$825$	−10950.0	−0.462098
$826$	0	0
$827$	18345.0	0.771365	0.385682	−	0.922632i	$-0.373966\pi$
$827$	18345.0	0.771365	0.385682	+	0.922632i	$0.373966\pi$
$828$	0	0
$829$	2469.13	0.103446	0.0517228	−	0.998661i	$-0.483529\pi$
$829$	2469.13	0.103446	0.0517228	+	0.998661i	$0.483529\pi$
$830$	0	0
$831$	7773.33	0.324493
$832$	0	0
$833$	3469.93	0.144329
$834$	0	0
$835$	41373.0	1.71470
$836$	0	0
$837$	−46697.8	−1.92845
$838$	0	0
$839$	−16861.3	−0.693824	−0.346912	−	0.937898i	$-0.612770\pi$
$839$	−16861.3	−0.693824	−0.346912	+	0.937898i	$0.612770\pi$
$840$	0	0
$841$	−23084.7	−0.946521
$842$	0	0
$843$	−32001.7	−1.30747
$844$	0	0
$845$	−36007.0	−1.46589
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	1433.09	0.0579313
$850$	0	0
$851$	−51250.6	−2.06445
$852$	0	0
$853$	7865.78	0.315732	0.157866	−	0.987461i	$-0.449539\pi$
$853$	7865.78	0.315732	0.157866	+	0.987461i	$0.449539\pi$
$854$	0	0
$855$	−4679.18	−0.187163
$856$	0	0
$857$	6288.35	0.250649	0.125324	−	0.992116i	$-0.460003\pi$
$857$	6288.35	0.250649	0.125324	+	0.992116i	$0.460003\pi$
$858$	0	0
$859$	34023.0	1.35140	0.675698	−	0.737179i	$-0.263842\pi$
$859$	34023.0	1.35140	0.675698	+	0.737179i	$0.263842\pi$
$860$	0	0
$861$	4748.16	0.187941
$862$	0	0
$863$	−19584.8	−0.772509	−0.386255	−	0.922392i	$-0.626231\pi$
$863$	−19584.8	−0.772509	−0.386255	+	0.922392i	$0.626231\pi$
$864$	0	0
$865$	−6593.21	−0.259163
$866$	0	0
$867$	380.437	0.0149023
$868$	0	0
$869$	433.721	0.0169309
$870$	0	0
$871$	15921.6	0.619381
$872$	0	0
$873$	−4377.73	−0.169718
$874$	0	0
$875$	19551.4	0.755382
$876$	0	0
$877$	−20098.0	−0.773844	−0.386922	−	0.922113i	$-0.626462\pi$
$877$	−20098.0	−0.773844	−0.386922	+	0.922113i	$0.626462\pi$
$878$	0	0
$879$	−13941.5	−0.534968
$880$	0	0
$881$	−10876.4	−0.415930	−0.207965	−	0.978136i	$-0.566684\pi$
$881$	−10876.4	−0.415930	−0.207965	+	0.978136i	$0.566684\pi$
$882$	0	0
$883$	19724.9	0.751751	0.375875	−	0.926670i	$-0.377342\pi$
$883$	19724.9	0.751751	0.375875	+	0.926670i	$0.377342\pi$
$884$	0	0
$885$	29324.3	1.11382
$886$	0	0
$887$	−39575.2	−1.49809	−0.749045	−	0.662519i	$-0.769488\pi$
$887$	−39575.2	−1.49809	−0.749045	+	0.662519i	$0.769488\pi$
$888$	0	0
$889$	774.360	0.0292140
$890$	0	0
$891$	2288.83	0.0860592
$892$	0	0
$893$	−4652.40	−0.174341
$894$	0	0
$895$	12817.4	0.478701
$896$	0	0
$897$	−10898.8	−0.405688
$898$	0	0
$899$	11270.5	0.418123
$900$	0	0
$901$	−47745.8	−1.76542
$902$	0	0
$903$	8751.88	0.322530
$904$	0	0
$905$	−70808.8	−2.60084
$906$	0	0
$907$	14301.8	0.523578	0.261789	−	0.965125i	$-0.415688\pi$
$907$	14301.8	0.523578	0.261789	+	0.965125i	$0.415688\pi$
$908$	0	0
$909$	−7687.96	−0.280521
$910$	0	0
$911$	−16663.3	−0.606016	−0.303008	−	0.952988i	$-0.597991\pi$
$911$	−16663.3	−0.606016	−0.303008	+	0.952988i	$0.597991\pi$
$912$	0	0
$913$	−7146.35	−0.259047
$914$	0	0
$915$	36659.4	1.32451
$916$	0	0
$917$	−19526.9	−0.703202
$918$	0	0
$919$	−32690.6	−1.17341	−0.586705	−	0.809800i	$-0.699575\pi$
$919$	−32690.6	−1.17341	−0.586705	+	0.809800i	$0.699575\pi$
$920$	0	0
$921$	−26729.4	−0.956313
$922$	0	0
$923$	5044.66	0.179899
$924$	0	0
$925$	−90834.0	−3.22876
$926$	0	0
$927$	−9098.36	−0.322362
$928$	0	0
$929$	−22269.0	−0.786463	−0.393231	−	0.919440i	$-0.628643\pi$
$929$	−22269.0	−0.786463	−0.393231	+	0.919440i	$0.628643\pi$
$930$	0	0
$931$	890.500	0.0313480
$932$	0	0
$933$	−21095.3	−0.740222
$934$	0	0
$935$	−15407.2	−0.538897
$936$	0	0
$937$	−56186.9	−1.95896	−0.979480	−	0.201543i	$-0.935404\pi$
$937$	−56186.9	−1.95896	−0.979480	+	0.201543i	$0.935404\pi$
$938$	0	0
$939$	35647.8	1.23889
$940$	0	0
$941$	27795.0	0.962901	0.481451	−	0.876473i	$-0.340110\pi$
$941$	27795.0	0.962901	0.481451	+	0.876473i	$0.340110\pi$
$942$	0	0
$943$	27246.7	0.940905
$944$	0	0
$945$	−20718.0	−0.713181
$946$	0	0
$947$	−39834.1	−1.36688	−0.683440	−	0.730007i	$-0.739517\pi$
$947$	−39834.1	−1.36688	−0.683440	+	0.730007i	$0.739517\pi$
$948$	0	0
$949$	−7030.78	−0.240494
$950$	0	0
$951$	−11716.4	−0.399506
$952$	0	0
$953$	6560.61	0.223000	0.111500	−	0.993764i	$-0.464434\pi$
$953$	6560.61	0.223000	0.111500	+	0.993764i	$0.464434\pi$
$954$	0	0
$955$	−20944.1	−0.709669
$956$	0	0
$957$	−1485.51	−0.0501774
$958$	0	0
$959$	1772.93	0.0596986
$960$	0	0
$961$	67597.4	2.26905
$962$	0	0
$963$	−6370.45	−0.213172
$964$	0	0
$965$	−21879.4	−0.729868
$966$	0	0
$967$	4283.37	0.142444	0.0712222	−	0.997460i	$-0.477310\pi$
$967$	4283.37	0.142444	0.0712222	+	0.997460i	$0.477310\pi$
$968$	0	0
$969$	4812.33	0.159540
$970$	0	0
$971$	50140.4	1.65714	0.828569	−	0.559887i	$-0.189155\pi$
$971$	50140.4	1.65714	0.828569	+	0.559887i	$0.189155\pi$
$972$	0	0
$973$	9921.40	0.326891
$974$	0	0
$975$	−19316.5	−0.634487
$976$	0	0
$977$	16287.5	0.533352	0.266676	−	0.963786i	$-0.414075\pi$
$977$	16287.5	0.533352	0.266676	+	0.963786i	$0.414075\pi$
$978$	0	0
$979$	4255.20	0.138914
$980$	0	0
$981$	27942.1	0.909400
$982$	0	0
$983$	−50147.5	−1.62712	−0.813560	−	0.581482i	$-0.802473\pi$
$983$	−50147.5	−1.62712	−0.813560	+	0.581482i	$0.802473\pi$
$984$	0	0
$985$	−43187.5	−1.39702
$986$	0	0
$987$	−6700.86	−0.216100
$988$	0	0
$989$	50221.5	1.61471
$990$	0	0
$991$	14908.0	0.477870	0.238935	−	0.971036i	$-0.423202\pi$
$991$	14908.0	0.477870	0.238935	+	0.971036i	$0.423202\pi$
$992$	0	0
$993$	−43146.8	−1.37887
$994$	0	0
$995$	−5444.08	−0.173456
$996$	0	0
$997$	20565.1	0.653263	0.326631	−	0.945152i	$-0.394086\pi$
$997$	20565.1	0.653263	0.326631	+	0.945152i	$0.394086\pi$
$998$	0	0
$999$	51057.8	1.61701

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	616.4.a.i.1.5	✓	6
4.3	odd	2		1232.4.a.bb.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.i.1.5	✓	6	1.1	even	1	trivial
1232.4.a.bb.1.2		6	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)

\(f(q)\)	\(=\)	\(q+3.73933 q^{3} +19.7791 q^{5} +7.00000 q^{7} -13.0174 q^{9} -11.0000 q^{11} -19.4047 q^{13} +73.9605 q^{15} +70.8148 q^{17} +18.1735 q^{19} +26.1753 q^{21} +150.203 q^{23} +266.213 q^{25} -149.638 q^{27} +36.1152 q^{29} +312.071 q^{31} -41.1326 q^{33} +138.454 q^{35} -341.208 q^{37} -72.5606 q^{39} +181.399 q^{41} +334.357 q^{43} -257.473 q^{45} -255.999 q^{47} +49.0000 q^{49} +264.800 q^{51} -674.234 q^{53} -217.570 q^{55} +67.9566 q^{57} +396.486 q^{59} +495.662 q^{61} -91.1220 q^{63} -383.808 q^{65} -820.499 q^{67} +561.660 q^{69} -259.971 q^{71} +362.323 q^{73} +995.456 q^{75} -77.0000 q^{77} -39.4291 q^{79} -208.076 q^{81} +649.668 q^{83} +1400.65 q^{85} +135.047 q^{87} -386.836 q^{89} -135.833 q^{91} +1166.94 q^{93} +359.455 q^{95} +336.297 q^{97} +143.192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} + 42 q^{7} + 60 q^{9} - 66 q^{11} - 6 q^{13} + 126 q^{15} - 14 q^{17} + 80 q^{19} + 254 q^{23} + 220 q^{25} + 90 q^{27} + 132 q^{29} - 52 q^{31} + 14 q^{35} - 518 q^{37} + 332 q^{39} + 486 q^{41}+ \cdots - 660 q^{99}+O(q^{100}) \) 6 * q + 2 * q^5 + 42 * q^7 + 60 * q^9 - 66 * q^11 - 6 * q^13 + 126 * q^15 - 14 * q^17 + 80 * q^19 + 254 * q^23 + 220 * q^25 + 90 * q^27 + 132 * q^29 - 52 * q^31 + 14 * q^35 - 518 * q^37 + 332 * q^39 + 486 * q^41 + 428 * q^43 - 244 * q^45 + 790 * q^47 + 294 * q^49 + 640 * q^51 - 40 * q^53 - 22 * q^55 + 2276 * q^57 + 436 * q^59 + 1034 * q^61 + 420 * q^63 + 800 * q^65 + 562 * q^67 + 530 * q^69 + 2474 * q^71 - 902 * q^73 + 1986 * q^75 - 462 * q^77 + 1636 * q^79 + 570 * q^81 + 3016 * q^83 + 236 * q^85 + 4340 * q^87 + 1750 * q^89 - 42 * q^91 + 150 * q^93 + 3628 * q^95 - 1250 * q^97 - 660 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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