LMFDB - Embedded newform 1600.4.a.ck.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1600.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:	$94.4030560092$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.35890$ of defining polynomial
Character	$\chi$	$=$	1600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.71780 q^{3} +8.71780 q^{7} +49.0000 q^{9} +O(q^{10})$	$q-8.71780 q^{3} +8.71780 q^{7} +49.0000 q^{9} +20.0000 q^{11} -52.3068 q^{13} +69.7424 q^{17} +84.0000 q^{19} -76.0000 q^{21} +61.0246 q^{23} -191.792 q^{27} +6.00000 q^{29} +224.000 q^{31} -174.356 q^{33} -122.049 q^{37} +456.000 q^{39} +266.000 q^{41} +305.123 q^{43} -374.865 q^{47} -267.000 q^{49} -608.000 q^{51} -366.148 q^{53} -732.295 q^{57} +28.0000 q^{59} -182.000 q^{61} +427.172 q^{63} +427.172 q^{67} -532.000 q^{69} -408.000 q^{71} -1081.01 q^{73} +174.356 q^{77} +48.0000 q^{79} +349.000 q^{81} +200.509 q^{83} -52.3068 q^{87} +1526.00 q^{89} -456.000 q^{91} -1952.79 q^{93} +557.939 q^{97} +980.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 98 q^{9}+O(q^{10}) $ 2 * q + 98 * q^9	$ 2 q + 98 q^{9} + 40 q^{11} + 168 q^{19} - 152 q^{21} + 12 q^{29} + 448 q^{31} + 912 q^{39} + 532 q^{41} - 534 q^{49} - 1216 q^{51} + 56 q^{59} - 364 q^{61} - 1064 q^{69} - 816 q^{71} + 96 q^{79} + 698 q^{81} + 3052 q^{89} - 912 q^{91} + 1960 q^{99}+O(q^{100}) $ 2 * q + 98 * q^9 + 40 * q^11 + 168 * q^19 - 152 * q^21 + 12 * q^29 + 448 * q^31 + 912 * q^39 + 532 * q^41 - 534 * q^49 - 1216 * q^51 + 56 * q^59 - 364 * q^61 - 1064 * q^69 - 816 * q^71 + 96 * q^79 + 698 * q^81 + 3052 * q^89 - 912 * q^91 + 1960 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−8.71780	−1.67774	−0.838870	−	0.544331i	$-0.816784\pi$
$3$	−8.71780	−1.67774	−0.838870	+	0.544331i	$0.816784\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	8.71780	0.470717	0.235358	−	0.971909i	$-0.424374\pi$
$7$	8.71780	0.470717	0.235358	+	0.971909i	$0.424374\pi$
$8$	0	0
$9$	49.0000	1.81481
$10$	0	0
$11$	20.0000	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	0	0
$13$	−52.3068	−1.11595	−0.557973	−	0.829859i	$-0.688421\pi$
$13$	−52.3068	−1.11595	−0.557973	+	0.829859i	$0.688421\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	69.7424	0.995001	0.497500	−	0.867464i	$-0.334251\pi$
$17$	69.7424	0.995001	0.497500	+	0.867464i	$0.334251\pi$
$18$	0	0
$19$	84.0000	1.01426	0.507130	−	0.861870i	$-0.330707\pi$
$19$	84.0000	1.01426	0.507130	+	0.861870i	$0.330707\pi$
$20$	0	0
$21$	−76.0000	−0.789741
$22$	0	0
$23$	61.0246	0.553239	0.276620	−	0.960979i	$-0.410786\pi$
$23$	61.0246	0.553239	0.276620	+	0.960979i	$0.410786\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−191.792	−1.36705
$28$	0	0
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	224.000	1.29779	0.648897	−	0.760877i	$-0.275231\pi$
$31$	224.000	1.29779	0.648897	+	0.760877i	$0.275231\pi$
$32$	0	0
$33$	−174.356	−0.919742
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−122.049	−0.542291	−0.271145	−	0.962538i	$-0.587402\pi$
$37$	−122.049	−0.542291	−0.271145	+	0.962538i	$0.587402\pi$
$38$	0	0
$39$	456.000	1.87227
$40$	0	0
$41$	266.000	1.01322	0.506612	−	0.862174i	$-0.330898\pi$
$41$	266.000	1.01322	0.506612	+	0.862174i	$0.330898\pi$
$42$	0	0
$43$	305.123	1.08211	0.541056	−	0.840987i	$-0.318025\pi$
$43$	305.123	1.08211	0.541056	+	0.840987i	$0.318025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−374.865	−1.16340	−0.581699	−	0.813404i	$-0.697612\pi$
$47$	−374.865	−1.16340	−0.581699	+	0.813404i	$0.697612\pi$
$48$	0	0
$49$	−267.000	−0.778426
$50$	0	0
$51$	−608.000	−1.66935
$52$	0	0
$53$	−366.148	−0.948948	−0.474474	−	0.880270i	$-0.657362\pi$
$53$	−366.148	−0.948948	−0.474474	+	0.880270i	$0.657362\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−732.295	−1.70166
$58$	0	0
$59$	28.0000	0.0617846	0.0308923	−	0.999523i	$-0.490165\pi$
$59$	28.0000	0.0617846	0.0308923	+	0.999523i	$0.490165\pi$
$60$	0	0
$61$	−182.000	−0.382012	−0.191006	−	0.981589i	$-0.561175\pi$
$61$	−182.000	−0.382012	−0.191006	+	0.981589i	$0.561175\pi$
$62$	0	0
$63$	427.172	0.854264
$64$	0	0
$65$	0	0
$66$	0	0
$67$	427.172	0.778916	0.389458	−	0.921044i	$-0.372662\pi$
$67$	427.172	0.778916	0.389458	+	0.921044i	$0.372662\pi$
$68$	0	0
$69$	−532.000	−0.928192
$70$	0	0
$71$	−408.000	−0.681982	−0.340991	−	0.940067i	$-0.610762\pi$
$71$	−408.000	−0.681982	−0.340991	+	0.940067i	$0.610762\pi$
$72$	0	0
$73$	−1081.01	−1.73318	−0.866591	−	0.499019i	$-0.833694\pi$
$73$	−1081.01	−1.73318	−0.866591	+	0.499019i	$0.833694\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	174.356	0.258048
$78$	0	0
$79$	48.0000	0.0683598	0.0341799	−	0.999416i	$-0.489118\pi$
$79$	48.0000	0.0683598	0.0341799	+	0.999416i	$0.489118\pi$
$80$	0	0
$81$	349.000	0.478738
$82$	0	0
$83$	200.509	0.265166	0.132583	−	0.991172i	$-0.457673\pi$
$83$	200.509	0.265166	0.132583	+	0.991172i	$0.457673\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−52.3068	−0.0644583
$88$	0	0
$89$	1526.00	1.81748	0.908740	−	0.417363i	$-0.137046\pi$
$89$	1526.00	1.81748	0.908740	+	0.417363i	$0.137046\pi$
$90$	0	0
$91$	−456.000	−0.525294
$92$	0	0
$93$	−1952.79	−2.17736
$94$	0	0
$95$	0	0
$96$	0	0
$97$	557.939	0.584022	0.292011	−	0.956415i	$-0.405676\pi$
$97$	557.939	0.584022	0.292011	+	0.956415i	$0.405676\pi$
$98$	0	0
$99$	980.000	0.994886
$100$	0	0
$101$	−1246.00	−1.22754	−0.613770	−	0.789485i	$-0.710348\pi$
$101$	−1246.00	−1.22754	−0.613770	+	0.789485i	$0.710348\pi$
$102$	0	0
$103$	−845.626	−0.808952	−0.404476	−	0.914549i	$-0.632546\pi$
$103$	−845.626	−0.808952	−0.404476	+	0.914549i	$0.632546\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1281.52	−1.15784	−0.578920	−	0.815384i	$-0.696526\pi$
$107$	−1281.52	−1.15784	−0.578920	+	0.815384i	$0.696526\pi$
$108$	0	0
$109$	902.000	0.792623	0.396312	−	0.918116i	$-0.370290\pi$
$109$	902.000	0.792623	0.396312	+	0.918116i	$0.370290\pi$
$110$	0	0
$111$	1064.00	0.909824
$112$	0	0
$113$	−1464.59	−1.21927	−0.609633	−	0.792684i	$-0.708683\pi$
$113$	−1464.59	−1.21927	−0.609633	+	0.792684i	$0.708683\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2563.03	−2.02523
$118$	0	0
$119$	608.000	0.468364
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	−2318.93	−1.69993
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2624.06	1.83344	0.916722	−	0.399525i	$-0.130825\pi$
$127$	2624.06	1.83344	0.916722	+	0.399525i	$0.130825\pi$
$128$	0	0
$129$	−2660.00	−1.81550
$130$	0	0
$131$	2940.00	1.96083	0.980416	−	0.196938i	$-0.0630997\pi$
$131$	2940.00	1.96083	0.980416	+	0.196938i	$0.0630997\pi$
$132$	0	0
$133$	732.295	0.477429
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−732.295	−0.456673	−0.228336	−	0.973582i	$-0.573329\pi$
$137$	−732.295	−0.456673	−0.228336	+	0.973582i	$0.573329\pi$
$138$	0	0
$139$	364.000	0.222116	0.111058	−	0.993814i	$-0.464576\pi$
$139$	364.000	0.222116	0.111058	+	0.993814i	$0.464576\pi$
$140$	0	0
$141$	3268.00	1.95188
$142$	0	0
$143$	−1046.14	−0.611764
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2327.65	1.30600
$148$	0	0
$149$	254.000	0.139654	0.0698272	−	0.997559i	$-0.477755\pi$
$149$	254.000	0.139654	0.0698272	+	0.997559i	$0.477755\pi$
$150$	0	0
$151$	2360.00	1.27188	0.635941	−	0.771738i	$-0.280612\pi$
$151$	2360.00	1.27188	0.635941	+	0.771738i	$0.280612\pi$
$152$	0	0
$153$	3417.38	1.80574
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2214.32	1.12562	0.562809	−	0.826587i	$-0.309721\pi$
$157$	2214.32	1.12562	0.562809	+	0.826587i	$0.309721\pi$
$158$	0	0
$159$	3192.00	1.59209
$160$	0	0
$161$	532.000	0.260419
$162$	0	0
$163$	1525.61	0.733100	0.366550	−	0.930398i	$-0.380539\pi$
$163$	1525.61	0.733100	0.366550	+	0.930398i	$0.380539\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3059.95	−1.41788	−0.708940	−	0.705269i	$-0.750826\pi$
$167$	−3059.95	−1.41788	−0.708940	+	0.705269i	$0.750826\pi$
$168$	0	0
$169$	539.000	0.245335
$170$	0	0
$171$	4116.00	1.84069
$172$	0	0
$173$	1900.48	0.835207	0.417604	−	0.908629i	$-0.362870\pi$
$173$	1900.48	0.835207	0.417604	+	0.908629i	$0.362870\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−244.098	−0.103659
$178$	0	0
$179$	1972.00	0.823431	0.411716	−	0.911312i	$-0.364930\pi$
$179$	1972.00	0.823431	0.411716	+	0.911312i	$0.364930\pi$
$180$	0	0
$181$	1330.00	0.546177	0.273089	−	0.961989i	$-0.411955\pi$
$181$	1330.00	0.546177	0.273089	+	0.961989i	$0.411955\pi$
$182$	0	0
$183$	1586.64	0.640917
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1394.85	0.545462
$188$	0	0
$189$	−1672.00	−0.643493
$190$	0	0
$191$	−1728.00	−0.654627	−0.327313	−	0.944916i	$-0.606143\pi$
$191$	−1728.00	−0.654627	−0.327313	+	0.944916i	$0.606143\pi$
$192$	0	0
$193$	−976.393	−0.364157	−0.182079	−	0.983284i	$-0.558283\pi$
$193$	−976.393	−0.364157	−0.182079	+	0.983284i	$0.558283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3783.52	1.36835	0.684175	−	0.729318i	$-0.260162\pi$
$197$	3783.52	1.36835	0.684175	+	0.729318i	$0.260162\pi$
$198$	0	0
$199$	−1512.00	−0.538607	−0.269304	−	0.963055i	$-0.586794\pi$
$199$	−1512.00	−0.538607	−0.269304	+	0.963055i	$0.586794\pi$
$200$	0	0
$201$	−3724.00	−1.30682
$202$	0	0
$203$	52.3068	0.0180848
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2990.20	1.00403
$208$	0	0
$209$	1680.00	0.556019
$210$	0	0
$211$	1644.00	0.536387	0.268193	−	0.963365i	$-0.413573\pi$
$211$	1644.00	0.536387	0.268193	+	0.963365i	$0.413573\pi$
$212$	0	0
$213$	3556.86	1.14419
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1952.79	0.610893
$218$	0	0
$219$	9424.00	2.90783
$220$	0	0
$221$	−3648.00	−1.11037
$222$	0	0
$223$	−1089.72	−0.327235	−0.163617	−	0.986524i	$-0.552316\pi$
$223$	−1089.72	−0.327235	−0.163617	+	0.986524i	$0.552316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	985.111	0.288036	0.144018	−	0.989575i	$-0.453998\pi$
$227$	985.111	0.288036	0.144018	+	0.989575i	$0.453998\pi$
$228$	0	0
$229$	3934.00	1.13522	0.567611	−	0.823297i	$-0.307868\pi$
$229$	3934.00	1.13522	0.567611	+	0.823297i	$0.307868\pi$
$230$	0	0
$231$	−1520.00	−0.432938
$232$	0	0
$233$	4637.87	1.30402	0.652010	−	0.758210i	$-0.273926\pi$
$233$	4637.87	1.30402	0.652010	+	0.758210i	$0.273926\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−418.454	−0.114690
$238$	0	0
$239$	3856.00	1.04361	0.521807	−	0.853063i	$-0.325258\pi$
$239$	3856.00	1.04361	0.521807	+	0.853063i	$0.325258\pi$
$240$	0	0
$241$	994.000	0.265681	0.132841	−	0.991137i	$-0.457590\pi$
$241$	994.000	0.265681	0.132841	+	0.991137i	$0.457590\pi$
$242$	0	0
$243$	2135.86	0.563850
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4393.77	−1.13186
$248$	0	0
$249$	−1748.00	−0.444880
$250$	0	0
$251$	−6300.00	−1.58427	−0.792136	−	0.610344i	$-0.791031\pi$
$251$	−6300.00	−1.58427	−0.792136	+	0.610344i	$0.791031\pi$
$252$	0	0
$253$	1220.49	0.303287
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6695.27	1.62506	0.812528	−	0.582922i	$-0.198091\pi$
$257$	6695.27	1.62506	0.812528	+	0.582922i	$0.198091\pi$
$258$	0	0
$259$	−1064.00	−0.255265
$260$	0	0
$261$	294.000	0.0697247
$262$	0	0
$263$	−1403.57	−0.329078	−0.164539	−	0.986371i	$-0.552614\pi$
$263$	−1403.57	−0.329078	−0.164539	+	0.986371i	$0.552614\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−13303.4	−3.04926
$268$	0	0
$269$	−5082.00	−1.15188	−0.575939	−	0.817493i	$-0.695363\pi$
$269$	−5082.00	−1.15188	−0.575939	+	0.817493i	$0.695363\pi$
$270$	0	0
$271$	−2800.00	−0.627631	−0.313815	−	0.949484i	$-0.601607\pi$
$271$	−2800.00	−0.627631	−0.313815	+	0.949484i	$0.601607\pi$
$272$	0	0
$273$	3975.32	0.881308
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1586.64	−0.344159	−0.172079	−	0.985083i	$-0.555049\pi$
$277$	−1586.64	−0.344159	−0.172079	+	0.985083i	$0.555049\pi$
$278$	0	0
$279$	10976.0	2.35525
$280$	0	0
$281$	−1254.00	−0.266218	−0.133109	−	0.991101i	$-0.542496\pi$
$281$	−1254.00	−0.266218	−0.133109	+	0.991101i	$0.542496\pi$
$282$	0	0
$283$	−4646.59	−0.976010	−0.488005	−	0.872841i	$-0.662275\pi$
$283$	−4646.59	−0.976010	−0.488005	+	0.872841i	$0.662275\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2318.93	0.476942
$288$	0	0
$289$	−49.0000	−0.00997354
$290$	0	0
$291$	−4864.00	−0.979838
$292$	0	0
$293$	6538.35	1.30367	0.651833	−	0.758362i	$-0.274000\pi$
$293$	6538.35	1.30367	0.651833	+	0.758362i	$0.274000\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3835.83	−0.749419
$298$	0	0
$299$	−3192.00	−0.617385
$300$	0	0
$301$	2660.00	0.509368
$302$	0	0
$303$	10862.4	2.05950
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6355.27	1.18148	0.590741	−	0.806862i	$-0.298836\pi$
$307$	6355.27	1.18148	0.590741	+	0.806862i	$0.298836\pi$
$308$	0	0
$309$	7372.00	1.35721
$310$	0	0
$311$	5208.00	0.949577	0.474789	−	0.880100i	$-0.342524\pi$
$311$	5208.00	0.949577	0.474789	+	0.880100i	$0.342524\pi$
$312$	0	0
$313$	4568.13	0.824939	0.412469	−	0.910972i	$-0.364666\pi$
$313$	4568.13	0.824939	0.412469	+	0.910972i	$0.364666\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−854.344	−0.151371	−0.0756857	−	0.997132i	$-0.524115\pi$
$317$	−854.344	−0.151371	−0.0756857	+	0.997132i	$0.524115\pi$
$318$	0	0
$319$	120.000	0.0210618
$320$	0	0
$321$	11172.0	1.94256
$322$	0	0
$323$	5858.36	1.00919
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7863.45	−1.32982
$328$	0	0
$329$	−3268.00	−0.547631
$330$	0	0
$331$	7828.00	1.29990	0.649948	−	0.759978i	$-0.274791\pi$
$331$	7828.00	1.29990	0.649948	+	0.759978i	$0.274791\pi$
$332$	0	0
$333$	−5980.41	−0.984157
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5370.16	−0.868046	−0.434023	−	0.900902i	$-0.642906\pi$
$337$	−5370.16	−0.868046	−0.434023	+	0.900902i	$0.642906\pi$
$338$	0	0
$339$	12768.0	2.04561
$340$	0	0
$341$	4480.00	0.711453
$342$	0	0
$343$	−5317.86	−0.837135
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2746.11	−0.424838	−0.212419	−	0.977179i	$-0.568134\pi$
$347$	−2746.11	−0.424838	−0.212419	+	0.977179i	$0.568134\pi$
$348$	0	0
$349$	−8890.00	−1.36353	−0.681763	−	0.731573i	$-0.738787\pi$
$349$	−8890.00	−1.36353	−0.681763	+	0.731573i	$0.738787\pi$
$350$	0	0
$351$	10032.0	1.52555
$352$	0	0
$353$	−1534.33	−0.231344	−0.115672	−	0.993287i	$-0.536902\pi$
$353$	−1534.33	−0.231344	−0.115672	+	0.993287i	$0.536902\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5300.42	−0.785793
$358$	0	0
$359$	9144.00	1.34429	0.672147	−	0.740417i	$-0.265372\pi$
$359$	9144.00	1.34429	0.672147	+	0.740417i	$0.265372\pi$
$360$	0	0
$361$	197.000	0.0287214
$362$	0	0
$363$	8116.27	1.17354
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10365.5	1.47431	0.737156	−	0.675722i	$-0.236168\pi$
$367$	10365.5	1.47431	0.737156	+	0.675722i	$0.236168\pi$
$368$	0	0
$369$	13034.0	1.83882
$370$	0	0
$371$	−3192.00	−0.446686
$372$	0	0
$373$	−3295.33	−0.457441	−0.228721	−	0.973492i	$-0.573454\pi$
$373$	−3295.33	−0.457441	−0.228721	+	0.973492i	$0.573454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−313.841	−0.0428743
$378$	0	0
$379$	2588.00	0.350756	0.175378	−	0.984501i	$-0.443885\pi$
$379$	2588.00	0.350756	0.175378	+	0.984501i	$0.443885\pi$
$380$	0	0
$381$	−22876.0	−3.07604
$382$	0	0
$383$	4071.21	0.543157	0.271579	−	0.962416i	$-0.412454\pi$
$383$	4071.21	0.543157	0.271579	+	0.962416i	$0.412454\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14951.0	1.96383
$388$	0	0
$389$	−5426.00	−0.707221	−0.353611	−	0.935393i	$-0.615046\pi$
$389$	−5426.00	−0.707221	−0.353611	+	0.935393i	$0.615046\pi$
$390$	0	0
$391$	4256.00	0.550474
$392$	0	0
$393$	−25630.3	−3.28977
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9816.24	1.24096	0.620482	−	0.784220i	$-0.286937\pi$
$397$	9816.24	1.24096	0.620482	+	0.784220i	$0.286937\pi$
$398$	0	0
$399$	−6384.00	−0.801002
$400$	0	0
$401$	370.000	0.0460771	0.0230386	−	0.999735i	$-0.492666\pi$
$401$	370.000	0.0460771	0.0230386	+	0.999735i	$0.492666\pi$
$402$	0	0
$403$	−11716.7	−1.44827
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2440.98	−0.297285
$408$	0	0
$409$	−4186.00	−0.506074	−0.253037	−	0.967457i	$-0.581429\pi$
$409$	−4186.00	−0.506074	−0.253037	+	0.967457i	$0.581429\pi$
$410$	0	0
$411$	6384.00	0.766179
$412$	0	0
$413$	244.098	0.0290830
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3173.28	−0.372653
$418$	0	0
$419$	−2716.00	−0.316671	−0.158336	−	0.987385i	$-0.550613\pi$
$419$	−2716.00	−0.316671	−0.158336	+	0.987385i	$0.550613\pi$
$420$	0	0
$421$	−1966.00	−0.227594	−0.113797	−	0.993504i	$-0.536301\pi$
$421$	−1966.00	−0.227594	−0.113797	+	0.993504i	$0.536301\pi$
$422$	0	0
$423$	−18368.4	−2.11135
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1586.64	−0.179819
$428$	0	0
$429$	9120.00	1.02638
$430$	0	0
$431$	11824.0	1.32144	0.660722	−	0.750631i	$-0.270250\pi$
$431$	11824.0	1.32144	0.660722	+	0.750631i	$0.270250\pi$
$432$	0	0
$433$	2580.47	0.286396	0.143198	−	0.989694i	$-0.454261\pi$
$433$	2580.47	0.286396	0.143198	+	0.989694i	$0.454261\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5126.07	0.561128
$438$	0	0
$439$	−13272.0	−1.44291	−0.721456	−	0.692461i	$-0.756527\pi$
$439$	−13272.0	−1.44291	−0.721456	+	0.692461i	$0.756527\pi$
$440$	0	0
$441$	−13083.0	−1.41270
$442$	0	0
$443$	−3112.25	−0.333787	−0.166894	−	0.985975i	$-0.553374\pi$
$443$	−3112.25	−0.333787	−0.166894	+	0.985975i	$0.553374\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2214.32	−0.234304
$448$	0	0
$449$	2094.00	0.220093	0.110047	−	0.993926i	$-0.464900\pi$
$449$	2094.00	0.220093	0.110047	+	0.993926i	$0.464900\pi$
$450$	0	0
$451$	5320.00	0.555452
$452$	0	0
$453$	−20574.0	−2.13389
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7078.85	0.724584	0.362292	−	0.932065i	$-0.381994\pi$
$457$	7078.85	0.724584	0.362292	+	0.932065i	$0.381994\pi$
$458$	0	0
$459$	−13376.0	−1.36021
$460$	0	0
$461$	9450.00	0.954730	0.477365	−	0.878705i	$-0.341592\pi$
$461$	9450.00	0.954730	0.477365	+	0.878705i	$0.341592\pi$
$462$	0	0
$463$	−3112.25	−0.312395	−0.156197	−	0.987726i	$-0.549924\pi$
$463$	−3112.25	−0.312395	−0.156197	+	0.987726i	$0.549924\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5361.45	−0.531259	−0.265630	−	0.964075i	$-0.585580\pi$
$467$	−5361.45	−0.531259	−0.265630	+	0.964075i	$0.585580\pi$
$468$	0	0
$469$	3724.00	0.366649
$470$	0	0
$471$	−19304.0	−1.88850
$472$	0	0
$473$	6102.46	0.593216
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−17941.2	−1.72216
$478$	0	0
$479$	13216.0	1.26066	0.630328	−	0.776329i	$-0.282920\pi$
$479$	13216.0	1.26066	0.630328	+	0.776329i	$0.282920\pi$
$480$	0	0
$481$	6384.00	0.605167
$482$	0	0
$483$	−4637.87	−0.436916
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2379.96	0.221450	0.110725	−	0.993851i	$-0.464683\pi$
$487$	2379.96	0.221450	0.110725	+	0.993851i	$0.464683\pi$
$488$	0	0
$489$	−13300.0	−1.22995
$490$	0	0
$491$	−12300.0	−1.13053	−0.565266	−	0.824909i	$-0.691226\pi$
$491$	−12300.0	−1.13053	−0.565266	+	0.824909i	$0.691226\pi$
$492$	0	0
$493$	418.454	0.0382277
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3556.86	−0.321020
$498$	0	0
$499$	−15756.0	−1.41350	−0.706749	−	0.707464i	$-0.749839\pi$
$499$	−15756.0	−1.41350	−0.706749	+	0.707464i	$0.749839\pi$
$500$	0	0
$501$	26676.0	2.37883
$502$	0	0
$503$	19170.4	1.69934	0.849670	−	0.527316i	$-0.176801\pi$
$503$	19170.4	1.69934	0.849670	+	0.527316i	$0.176801\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4698.89	−0.411608
$508$	0	0
$509$	−2394.00	−0.208472	−0.104236	−	0.994553i	$-0.533240\pi$
$509$	−2394.00	−0.208472	−0.104236	+	0.994553i	$0.533240\pi$
$510$	0	0
$511$	−9424.00	−0.815838
$512$	0	0
$513$	−16110.5	−1.38654
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−7497.31	−0.637778
$518$	0	0
$519$	−16568.0	−1.40126
$520$	0	0
$521$	6874.00	0.578033	0.289017	−	0.957324i	$-0.406672\pi$
$521$	6874.00	0.578033	0.289017	+	0.957324i	$0.406672\pi$
$522$	0	0
$523$	−16154.1	−1.35061	−0.675305	−	0.737539i	$-0.735988\pi$
$523$	−16154.1	−1.35061	−0.675305	+	0.737539i	$0.735988\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15622.3	1.29131
$528$	0	0
$529$	−8443.00	−0.693926
$530$	0	0
$531$	1372.00	0.112128
$532$	0	0
$533$	−13913.6	−1.13070
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−17191.5	−1.38150
$538$	0	0
$539$	−5340.00	−0.426735
$540$	0	0
$541$	−7174.00	−0.570119	−0.285059	−	0.958510i	$-0.592013\pi$
$541$	−7174.00	−0.570119	−0.285059	+	0.958510i	$0.592013\pi$
$542$	0	0
$543$	−11594.7	−0.916344
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4332.75	0.338674	0.169337	−	0.985558i	$-0.445837\pi$
$547$	4332.75	0.338674	0.169337	+	0.985558i	$0.445837\pi$
$548$	0	0
$549$	−8918.00	−0.693280
$550$	0	0
$551$	504.000	0.0389676
$552$	0	0
$553$	418.454	0.0321781
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9153.69	−0.696327	−0.348164	−	0.937434i	$-0.613195\pi$
$557$	−9153.69	−0.696327	−0.348164	+	0.937434i	$0.613195\pi$
$558$	0	0
$559$	−15960.0	−1.20758
$560$	0	0
$561$	−12160.0	−0.915144
$562$	0	0
$563$	13870.0	1.03828	0.519140	−	0.854689i	$-0.326252\pi$
$563$	13870.0	1.03828	0.519140	+	0.854689i	$0.326252\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3042.51	0.225350
$568$	0	0
$569$	16070.0	1.18399	0.591994	−	0.805942i	$-0.298341\pi$
$569$	16070.0	1.18399	0.591994	+	0.805942i	$0.298341\pi$
$570$	0	0
$571$	−3932.00	−0.288177	−0.144089	−	0.989565i	$-0.546025\pi$
$571$	−3932.00	−0.288177	−0.144089	+	0.989565i	$0.546025\pi$
$572$	0	0
$573$	15064.4	1.09829
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16180.2	1.16740	0.583702	−	0.811968i	$-0.301604\pi$
$577$	16180.2	1.16740	0.583702	+	0.811968i	$0.301604\pi$
$578$	0	0
$579$	8512.00	0.610961
$580$	0	0
$581$	1748.00	0.124818
$582$	0	0
$583$	−7322.95	−0.520215
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11272.1	0.792589	0.396295	−	0.918123i	$-0.370296\pi$
$587$	11272.1	0.792589	0.396295	+	0.918123i	$0.370296\pi$
$588$	0	0
$589$	18816.0	1.31630
$590$	0	0
$591$	−32984.0	−2.29574
$592$	0	0
$593$	10670.6	0.738935	0.369467	−	0.929244i	$-0.379540\pi$
$593$	10670.6	0.738935	0.369467	+	0.929244i	$0.379540\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13181.3	0.903643
$598$	0	0
$599$	−6840.00	−0.466569	−0.233284	−	0.972409i	$-0.574947\pi$
$599$	−6840.00	−0.466569	−0.233284	+	0.972409i	$0.574947\pi$
$600$	0	0
$601$	−10150.0	−0.688897	−0.344449	−	0.938805i	$-0.611934\pi$
$601$	−10150.0	−0.688897	−0.344449	+	0.938805i	$0.611934\pi$
$602$	0	0
$603$	20931.4	1.41359
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1839.46	−0.123000	−0.0615002	−	0.998107i	$-0.519588\pi$
$607$	−1839.46	−0.123000	−0.0615002	+	0.998107i	$0.519588\pi$
$608$	0	0
$609$	−456.000	−0.0303416
$610$	0	0
$611$	19608.0	1.29829
$612$	0	0
$613$	5004.02	0.329707	0.164853	−	0.986318i	$-0.447285\pi$
$613$	5004.02	0.329707	0.164853	+	0.986318i	$0.447285\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24653.9	1.60864	0.804319	−	0.594197i	$-0.202530\pi$
$617$	24653.9	1.60864	0.804319	+	0.594197i	$0.202530\pi$
$618$	0	0
$619$	27020.0	1.75448	0.877242	−	0.480049i	$-0.159381\pi$
$619$	27020.0	1.75448	0.877242	+	0.480049i	$0.159381\pi$
$620$	0	0
$621$	−11704.0	−0.756305
$622$	0	0
$623$	13303.4	0.855518
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−14645.9	−0.932856
$628$	0	0
$629$	−8512.00	−0.539580
$630$	0	0
$631$	10648.0	0.671775	0.335888	−	0.941902i	$-0.390964\pi$
$631$	10648.0	0.671775	0.335888	+	0.941902i	$0.390964\pi$
$632$	0	0
$633$	−14332.1	−0.899918
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13965.9	0.868681
$638$	0	0
$639$	−19992.0	−1.23767
$640$	0	0
$641$	1682.00	0.103643	0.0518214	−	0.998656i	$-0.483497\pi$
$641$	1682.00	0.103643	0.0518214	+	0.998656i	$0.483497\pi$
$642$	0	0
$643$	18542.8	1.13725	0.568627	−	0.822595i	$-0.307475\pi$
$643$	18542.8	1.13725	0.568627	+	0.822595i	$0.307475\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30677.9	−1.86410	−0.932051	−	0.362327i	$-0.881982\pi$
$647$	−30677.9	−1.86410	−0.932051	+	0.362327i	$0.881982\pi$
$648$	0	0
$649$	560.000	0.0338705
$650$	0	0
$651$	−17024.0	−1.02492
$652$	0	0
$653$	1342.54	0.0804559	0.0402279	−	0.999191i	$-0.487192\pi$
$653$	1342.54	0.0804559	0.0402279	+	0.999191i	$0.487192\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−52969.3	−3.14540
$658$	0	0
$659$	−21356.0	−1.26238	−0.631192	−	0.775626i	$-0.717434\pi$
$659$	−21356.0	−1.26238	−0.631192	+	0.775626i	$0.717434\pi$
$660$	0	0
$661$	13762.0	0.809803	0.404901	−	0.914360i	$-0.367306\pi$
$661$	13762.0	0.809803	0.404901	+	0.914360i	$0.367306\pi$
$662$	0	0
$663$	31802.5	1.86291
$664$	0	0
$665$	0	0
$666$	0	0
$667$	366.148	0.0212553
$668$	0	0
$669$	9500.00	0.549015
$670$	0	0
$671$	−3640.00	−0.209420
$672$	0	0
$673$	5858.36	0.335547	0.167774	−	0.985826i	$-0.446342\pi$
$673$	5858.36	0.335547	0.167774	+	0.985826i	$0.446342\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29588.2	1.67972	0.839858	−	0.542807i	$-0.182638\pi$
$677$	29588.2	1.67972	0.839858	+	0.542807i	$0.182638\pi$
$678$	0	0
$679$	4864.00	0.274909
$680$	0	0
$681$	−8588.00	−0.483249
$682$	0	0
$683$	−17270.0	−0.967521	−0.483760	−	0.875201i	$-0.660729\pi$
$683$	−17270.0	−0.967521	−0.483760	+	0.875201i	$0.660729\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−34295.8	−1.90461
$688$	0	0
$689$	19152.0	1.05897
$690$	0	0
$691$	−1652.00	−0.0909480	−0.0454740	−	0.998966i	$-0.514480\pi$
$691$	−1652.00	−0.0909480	−0.0454740	+	0.998966i	$0.514480\pi$
$692$	0	0
$693$	8543.44	0.468310
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18551.5	1.00816
$698$	0	0
$699$	−40432.0	−2.18781
$700$	0	0
$701$	−28790.0	−1.55119	−0.775594	−	0.631232i	$-0.782550\pi$
$701$	−28790.0	−1.55119	−0.775594	+	0.631232i	$0.782550\pi$
$702$	0	0
$703$	−10252.1	−0.550023
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10862.4	−0.577824
$708$	0	0
$709$	−7554.00	−0.400136	−0.200068	−	0.979782i	$-0.564116\pi$
$709$	−7554.00	−0.400136	−0.200068	+	0.979782i	$0.564116\pi$
$710$	0	0
$711$	2352.00	0.124060
$712$	0	0
$713$	13669.5	0.717990
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−33615.8	−1.75092
$718$	0	0
$719$	28336.0	1.46976	0.734878	−	0.678199i	$-0.237239\pi$
$719$	28336.0	1.46976	0.734878	+	0.678199i	$0.237239\pi$
$720$	0	0
$721$	−7372.00	−0.380787
$722$	0	0
$723$	−8665.49	−0.445744
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23511.9	1.19946	0.599730	−	0.800202i	$-0.295274\pi$
$727$	23511.9	1.19946	0.599730	+	0.800202i	$0.295274\pi$
$728$	0	0
$729$	−28043.0	−1.42473
$730$	0	0
$731$	21280.0	1.07670
$732$	0	0
$733$	3365.07	0.169566	0.0847829	−	0.996399i	$-0.472980\pi$
$733$	3365.07	0.169566	0.0847829	+	0.996399i	$0.472980\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8543.44	0.427004
$738$	0	0
$739$	−14300.0	−0.711819	−0.355909	−	0.934520i	$-0.615829\pi$
$739$	−14300.0	−0.711819	−0.355909	+	0.934520i	$0.615829\pi$
$740$	0	0
$741$	38304.0	1.89896
$742$	0	0
$743$	−31671.8	−1.56383	−0.781914	−	0.623386i	$-0.785756\pi$
$743$	−31671.8	−1.56383	−0.781914	+	0.623386i	$0.785756\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9824.96	0.481227
$748$	0	0
$749$	−11172.0	−0.545015
$750$	0	0
$751$	11824.0	0.574519	0.287260	−	0.957853i	$-0.407256\pi$
$751$	11824.0	0.574519	0.287260	+	0.957853i	$0.407256\pi$
$752$	0	0
$753$	54922.1	2.65800
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1342.54	0.0644590	0.0322295	−	0.999480i	$-0.489739\pi$
$757$	1342.54	0.0644590	0.0322295	+	0.999480i	$0.489739\pi$
$758$	0	0
$759$	−10640.0	−0.508837
$760$	0	0
$761$	6762.00	0.322106	0.161053	−	0.986946i	$-0.448511\pi$
$761$	6762.00	0.322106	0.161053	+	0.986946i	$0.448511\pi$
$762$	0	0
$763$	7863.45	0.373101
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1464.59	−0.0689482
$768$	0	0
$769$	−29442.0	−1.38063	−0.690316	−	0.723508i	$-0.742528\pi$
$769$	−29442.0	−1.38063	−0.690316	+	0.723508i	$0.742528\pi$
$770$	0	0
$771$	−58368.0	−2.72642
$772$	0	0
$773$	35830.1	1.66717	0.833584	−	0.552393i	$-0.186285\pi$
$773$	35830.1	1.66717	0.833584	+	0.552393i	$0.186285\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9275.74	0.428269
$778$	0	0
$779$	22344.0	1.02767
$780$	0	0
$781$	−8160.00	−0.373864
$782$	0	0
$783$	−1150.75	−0.0525216
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−23912.9	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$787$	−23912.9	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$788$	0	0
$789$	12236.0	0.552108
$790$	0	0
$791$	−12768.0	−0.573929
$792$	0	0
$793$	9519.84	0.426304
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17139.2	0.761733	0.380867	−	0.924630i	$-0.375626\pi$
$797$	17139.2	0.761733	0.380867	+	0.924630i	$0.375626\pi$
$798$	0	0
$799$	−26144.0	−1.15758
$800$	0	0
$801$	74774.0	3.29839
$802$	0	0
$803$	−21620.1	−0.950135
$804$	0	0
$805$	0	0
$806$	0	0
$807$	44303.8	1.93255
$808$	0	0
$809$	8646.00	0.375744	0.187872	−	0.982193i	$-0.439841\pi$
$809$	8646.00	0.375744	0.187872	+	0.982193i	$0.439841\pi$
$810$	0	0
$811$	−18956.0	−0.820759	−0.410379	−	0.911915i	$-0.634604\pi$
$811$	−18956.0	−0.820759	−0.410379	+	0.911915i	$0.634604\pi$
$812$	0	0
$813$	24409.8	1.05300
$814$	0	0
$815$	0	0
$816$	0	0
$817$	25630.3	1.09754
$818$	0	0
$819$	−22344.0	−0.953312
$820$	0	0
$821$	−13726.0	−0.583484	−0.291742	−	0.956497i	$-0.594235\pi$
$821$	−13726.0	−0.583484	−0.291742	+	0.956497i	$0.594235\pi$
$822$	0	0
$823$	45951.5	1.94626	0.973128	−	0.230264i	$-0.0739590\pi$
$823$	45951.5	1.94626	0.973128	+	0.230264i	$0.0739590\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2135.86	0.0898079	0.0449040	−	0.998991i	$-0.485702\pi$
$827$	2135.86	0.0898079	0.0449040	+	0.998991i	$0.485702\pi$
$828$	0	0
$829$	−8778.00	−0.367759	−0.183880	−	0.982949i	$-0.558866\pi$
$829$	−8778.00	−0.367759	−0.183880	+	0.982949i	$0.558866\pi$
$830$	0	0
$831$	13832.0	0.577409
$832$	0	0
$833$	−18621.2	−0.774534
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−42961.3	−1.77415
$838$	0	0
$839$	−18088.0	−0.744299	−0.372150	−	0.928173i	$-0.621379\pi$
$839$	−18088.0	−0.744299	−0.372150	+	0.928173i	$0.621379\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	10932.1	0.446646
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8116.27	−0.329254
$848$	0	0
$849$	40508.0	1.63749
$850$	0	0
$851$	−7448.00	−0.300017
$852$	0	0
$853$	22927.8	0.920320	0.460160	−	0.887836i	$-0.347792\pi$
$853$	22927.8	0.920320	0.460160	+	0.887836i	$0.347792\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4498.38	0.179302	0.0896510	−	0.995973i	$-0.471425\pi$
$857$	4498.38	0.179302	0.0896510	+	0.995973i	$0.471425\pi$
$858$	0	0
$859$	−40516.0	−1.60930	−0.804650	−	0.593750i	$-0.797647\pi$
$859$	−40516.0	−1.60930	−0.804650	+	0.593750i	$0.797647\pi$
$860$	0	0
$861$	−20216.0	−0.800185
$862$	0	0
$863$	−28010.3	−1.10484	−0.552422	−	0.833564i	$-0.686296\pi$
$863$	−28010.3	−1.10484	−0.552422	+	0.833564i	$0.686296\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	427.172	0.0167330
$868$	0	0
$869$	960.000	0.0374750
$870$	0	0
$871$	−22344.0	−0.869228
$872$	0	0
$873$	27339.0	1.05989
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33807.6	1.30171	0.650856	−	0.759201i	$-0.274410\pi$
$877$	33807.6	1.30171	0.650856	+	0.759201i	$0.274410\pi$
$878$	0	0
$879$	−57000.0	−2.18722
$880$	0	0
$881$	19362.0	0.740434	0.370217	−	0.928945i	$-0.379283\pi$
$881$	19362.0	0.740434	0.370217	+	0.928945i	$0.379283\pi$
$882$	0	0
$883$	4942.99	0.188386	0.0941930	−	0.995554i	$-0.469973\pi$
$883$	4942.99	0.188386	0.0941930	+	0.995554i	$0.469973\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44016.2	1.66620	0.833099	−	0.553124i	$-0.186564\pi$
$887$	44016.2	1.66620	0.833099	+	0.553124i	$0.186564\pi$
$888$	0	0
$889$	22876.0	0.863033
$890$	0	0
$891$	6980.00	0.262445
$892$	0	0
$893$	−31488.7	−1.17999
$894$	0	0
$895$	0	0
$896$	0	0
$897$	27827.2	1.03581
$898$	0	0
$899$	1344.00	0.0498609
$900$	0	0
$901$	−25536.0	−0.944204
$902$	0	0
$903$	−23189.3	−0.854588
$904$	0	0
$905$	0	0
$906$	0	0
$907$	42656.2	1.56160	0.780802	−	0.624778i	$-0.214811\pi$
$907$	42656.2	1.56160	0.780802	+	0.624778i	$0.214811\pi$
$908$	0	0
$909$	−61054.0	−2.22776
$910$	0	0
$911$	15888.0	0.577819	0.288909	−	0.957356i	$-0.406707\pi$
$911$	15888.0	0.577819	0.288909	+	0.957356i	$0.406707\pi$
$912$	0	0
$913$	4010.19	0.145365
$914$	0	0
$915$	0	0
$916$	0	0
$917$	25630.3	0.922997
$918$	0	0
$919$	7944.00	0.285145	0.142573	−	0.989784i	$-0.454463\pi$
$919$	7944.00	0.285145	0.142573	+	0.989784i	$0.454463\pi$
$920$	0	0
$921$	−55404.0	−1.98222
$922$	0	0
$923$	21341.2	0.761054
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−41435.7	−1.46810
$928$	0	0
$929$	20622.0	0.728295	0.364147	−	0.931341i	$-0.381360\pi$
$929$	20622.0	0.728295	0.364147	+	0.931341i	$0.381360\pi$
$930$	0	0
$931$	−22428.0	−0.789525
$932$	0	0
$933$	−45402.3	−1.59315
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−41182.9	−1.43584	−0.717922	−	0.696123i	$-0.754907\pi$
$937$	−41182.9	−1.43584	−0.717922	+	0.696123i	$0.754907\pi$
$938$	0	0
$939$	−39824.0	−1.38403
$940$	0	0
$941$	−40502.0	−1.40311	−0.701556	−	0.712615i	$-0.747511\pi$
$941$	−40502.0	−1.40311	−0.701556	+	0.712615i	$0.747511\pi$
$942$	0	0
$943$	16232.5	0.560556
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3844.55	0.131923	0.0659615	−	0.997822i	$-0.478989\pi$
$947$	3844.55	0.131923	0.0659615	+	0.997822i	$0.478989\pi$
$948$	0	0
$949$	56544.0	1.93414
$950$	0	0
$951$	7448.00	0.253962
$952$	0	0
$953$	−31976.9	−1.08692	−0.543459	−	0.839436i	$-0.682886\pi$
$953$	−31976.9	−1.08692	−0.543459	+	0.839436i	$0.682886\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1046.14	−0.0353362
$958$	0	0
$959$	−6384.00	−0.214964
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	−62794.3	−2.10126
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−19100.7	−0.635198	−0.317599	−	0.948225i	$-0.602877\pi$
$967$	−19100.7	−0.635198	−0.317599	+	0.948225i	$0.602877\pi$
$968$	0	0
$969$	−51072.0	−1.69316
$970$	0	0
$971$	17556.0	0.580225	0.290113	−	0.956992i	$-0.406307\pi$
$971$	17556.0	0.580225	0.290113	+	0.956992i	$0.406307\pi$
$972$	0	0
$973$	3173.28	0.104554
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4393.77	−0.143878	−0.0719392	−	0.997409i	$-0.522919\pi$
$977$	−4393.77	−0.143878	−0.0719392	+	0.997409i	$0.522919\pi$
$978$	0	0
$979$	30520.0	0.996347
$980$	0	0
$981$	44198.0	1.43846
$982$	0	0
$983$	9406.50	0.305209	0.152605	−	0.988287i	$-0.451234\pi$
$983$	9406.50	0.305209	0.152605	+	0.988287i	$0.451234\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	28489.8	0.918783
$988$	0	0
$989$	18620.0	0.598667
$990$	0	0
$991$	−28576.0	−0.915990	−0.457995	−	0.888955i	$-0.651432\pi$
$991$	−28576.0	−0.915990	−0.457995	+	0.888955i	$0.651432\pi$
$992$	0	0
$993$	−68242.9	−2.18089
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31541.0	1.00192	0.500960	−	0.865471i	$-0.332981\pi$
$997$	31541.0	1.00192	0.500960	+	0.865471i	$0.332981\pi$
$998$	0	0
$999$	23408.0	0.741338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.ck.1.1		2
4.3	odd	2		1600.4.a.cj.1.2		2
5.2	odd	4		320.4.c.b.129.2		2
5.3	odd	4		320.4.c.b.129.1		2
5.4	even	2	inner	1600.4.a.ck.1.2		2
8.3	odd	2		100.4.a.d.1.1		2
8.5	even	2		400.4.a.w.1.2		2
20.3	even	4		320.4.c.a.129.2		2
20.7	even	4		320.4.c.a.129.1		2
20.19	odd	2		1600.4.a.cj.1.1		2
24.11	even	2		900.4.a.s.1.1		2
40.3	even	4		20.4.c.a.9.1	✓	2
40.13	odd	4		80.4.c.b.49.2		2
40.19	odd	2		100.4.a.d.1.2		2
40.27	even	4		20.4.c.a.9.2	yes	2
40.29	even	2		400.4.a.w.1.1		2
40.37	odd	4		80.4.c.b.49.1		2
120.53	even	4		720.4.f.a.289.1		2
120.59	even	2		900.4.a.s.1.2		2
120.77	even	4		720.4.f.a.289.2		2
120.83	odd	4		180.4.d.a.109.1		2
120.107	odd	4		180.4.d.a.109.2		2
280.27	odd	4		980.4.e.a.589.1		2
280.83	odd	4		980.4.e.a.589.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.c.a.9.1	✓	2	40.3	even	4
20.4.c.a.9.2	yes	2	40.27	even	4
80.4.c.b.49.1		2	40.37	odd	4
80.4.c.b.49.2		2	40.13	odd	4
100.4.a.d.1.1		2	8.3	odd	2
100.4.a.d.1.2		2	40.19	odd	2
180.4.d.a.109.1		2	120.83	odd	4
180.4.d.a.109.2		2	120.107	odd	4
320.4.c.a.129.1		2	20.7	even	4
320.4.c.a.129.2		2	20.3	even	4
320.4.c.b.129.1		2	5.3	odd	4
320.4.c.b.129.2		2	5.2	odd	4
400.4.a.w.1.1		2	40.29	even	2
400.4.a.w.1.2		2	8.5	even	2
720.4.f.a.289.1		2	120.53	even	4
720.4.f.a.289.2		2	120.77	even	4
900.4.a.s.1.1		2	24.11	even	2
900.4.a.s.1.2		2	120.59	even	2
980.4.e.a.589.1		2	280.27	odd	4
980.4.e.a.589.2		2	280.83	odd	4
1600.4.a.cj.1.1		2	20.19	odd	2
1600.4.a.cj.1.2		2	4.3	odd	2
1600.4.a.ck.1.1		2	1.1	even	1	trivial
1600.4.a.ck.1.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

\(f(q)\)	\(=\)	\(q-8.71780 q^{3} +8.71780 q^{7} +49.0000 q^{9} +O(q^{10})\)	\(q-8.71780 q^{3} +8.71780 q^{7} +49.0000 q^{9} +20.0000 q^{11} -52.3068 q^{13} +69.7424 q^{17} +84.0000 q^{19} -76.0000 q^{21} +61.0246 q^{23} -191.792 q^{27} +6.00000 q^{29} +224.000 q^{31} -174.356 q^{33} -122.049 q^{37} +456.000 q^{39} +266.000 q^{41} +305.123 q^{43} -374.865 q^{47} -267.000 q^{49} -608.000 q^{51} -366.148 q^{53} -732.295 q^{57} +28.0000 q^{59} -182.000 q^{61} +427.172 q^{63} +427.172 q^{67} -532.000 q^{69} -408.000 q^{71} -1081.01 q^{73} +174.356 q^{77} +48.0000 q^{79} +349.000 q^{81} +200.509 q^{83} -52.3068 q^{87} +1526.00 q^{89} -456.000 q^{91} -1952.79 q^{93} +557.939 q^{97} +980.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 98 q^{9}+O(q^{10}) \) 2 * q + 98 * q^9	\( 2 q + 98 q^{9} + 40 q^{11} + 168 q^{19} - 152 q^{21} + 12 q^{29} + 448 q^{31} + 912 q^{39} + 532 q^{41} - 534 q^{49} - 1216 q^{51} + 56 q^{59} - 364 q^{61} - 1064 q^{69} - 816 q^{71} + 96 q^{79} + 698 q^{81} + 3052 q^{89} - 912 q^{91} + 1960 q^{99}+O(q^{100}) \) 2 * q + 98 * q^9 + 40 * q^11 + 168 * q^19 - 152 * q^21 + 12 * q^29 + 448 * q^31 + 912 * q^39 + 532 * q^41 - 534 * q^49 - 1216 * q^51 + 56 * q^59 - 364 * q^61 - 1064 * q^69 - 816 * q^71 + 96 * q^79 + 698 * q^81 + 3052 * q^89 - 912 * q^91 + 1960 * q^99

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