LMFDB - Embedded newform 1024.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,4,Mod(1,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1024.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1024.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:	$60.4179558459$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 512)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685 q^{3} -1.41421 q^{5} +8.00000 q^{7} +5.00000 q^{9} +O(q^{10})$	$q+5.65685 q^{3} -1.41421 q^{5} +8.00000 q^{7} +5.00000 q^{9} +5.65685 q^{11} -4.24264 q^{13} -8.00000 q^{15} -24.0000 q^{17} -130.108 q^{19} +45.2548 q^{21} -184.000 q^{23} -123.000 q^{25} -124.451 q^{27} +97.5807 q^{29} +176.000 q^{31} +32.0000 q^{33} -11.3137 q^{35} -304.056 q^{37} -24.0000 q^{39} +200.000 q^{41} -141.421 q^{43} -7.07107 q^{45} +208.000 q^{47} -279.000 q^{49} -135.765 q^{51} +371.938 q^{53} -8.00000 q^{55} -736.000 q^{57} -152.735 q^{59} +479.418 q^{61} +40.0000 q^{63} +6.00000 q^{65} -684.479 q^{67} -1040.86 q^{69} -936.000 q^{71} +406.000 q^{73} -695.793 q^{75} +45.2548 q^{77} +768.000 q^{79} -839.000 q^{81} +514.774 q^{83} +33.9411 q^{85} +552.000 q^{87} -1306.00 q^{89} -33.9411 q^{91} +995.606 q^{93} +184.000 q^{95} -472.000 q^{97} +28.2843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{7} + 10 q^{9}+O(q^{10}) $ 2 * q + 16 * q^7 + 10 * q^9	$ 2 q + 16 q^{7} + 10 q^{9} - 16 q^{15} - 48 q^{17} - 368 q^{23} - 246 q^{25} + 352 q^{31} + 64 q^{33} - 48 q^{39} + 400 q^{41} + 416 q^{47} - 558 q^{49} - 16 q^{55} - 1472 q^{57} + 80 q^{63} + 12 q^{65} - 1872 q^{71} + 812 q^{73} + 1536 q^{79} - 1678 q^{81} + 1104 q^{87} - 2612 q^{89} + 368 q^{95} - 944 q^{97}+O(q^{100}) $ 2 * q + 16 * q^7 + 10 * q^9 - 16 * q^15 - 48 * q^17 - 368 * q^23 - 246 * q^25 + 352 * q^31 + 64 * q^33 - 48 * q^39 + 400 * q^41 + 416 * q^47 - 558 * q^49 - 16 * q^55 - 1472 * q^57 + 80 * q^63 + 12 * q^65 - 1872 * q^71 + 812 * q^73 + 1536 * q^79 - 1678 * q^81 + 1104 * q^87 - 2612 * q^89 + 368 * q^95 - 944 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	5.65685	1.08866	0.544331	−	0.838870i	$-0.316784\pi$
$3$	5.65685	1.08866	0.544331	+	0.838870i	$0.316784\pi$
$4$	0	0
$5$	−1.41421	−0.126491	−0.0632456	−	0.997998i	$-0.520145\pi$
$5$	−1.41421	−0.126491	−0.0632456	+	0.997998i	$0.520145\pi$
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	5.00000	0.185185
$10$	0	0
$11$	5.65685	0.155055	0.0775275	−	0.996990i	$-0.475297\pi$
$11$	5.65685	0.155055	0.0775275	+	0.996990i	$0.475297\pi$
$12$	0	0
$13$	−4.24264	−0.0905151	−0.0452576	−	0.998975i	$-0.514411\pi$
$13$	−4.24264	−0.0905151	−0.0452576	+	0.998975i	$0.514411\pi$
$14$	0	0
$15$	−8.00000	−0.137706
$16$	0	0
$17$	−24.0000	−0.342403	−0.171202	−	0.985236i	$-0.554765\pi$
$17$	−24.0000	−0.342403	−0.171202	+	0.985236i	$0.554765\pi$
$18$	0	0
$19$	−130.108	−1.57099	−0.785493	−	0.618870i	$-0.787591\pi$
$19$	−130.108	−1.57099	−0.785493	+	0.618870i	$0.787591\pi$
$20$	0	0
$21$	45.2548	0.470258
$22$	0	0
$23$	−184.000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−184.000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−123.000	−0.984000
$26$	0	0
$27$	−124.451	−0.887058
$28$	0	0
$29$	97.5807	0.624838	0.312419	−	0.949944i	$-0.398861\pi$
$29$	97.5807	0.624838	0.312419	+	0.949944i	$0.398861\pi$
$30$	0	0
$31$	176.000	1.01969	0.509847	−	0.860265i	$-0.329702\pi$
$31$	176.000	1.01969	0.509847	+	0.860265i	$0.329702\pi$
$32$	0	0
$33$	32.0000	0.168803
$34$	0	0
$35$	−11.3137	−0.0546390
$36$	0	0
$37$	−304.056	−1.35099	−0.675493	−	0.737366i	$-0.736069\pi$
$37$	−304.056	−1.35099	−0.675493	+	0.737366i	$0.736069\pi$
$38$	0	0
$39$	−24.0000	−0.0985404
$40$	0	0
$41$	200.000	0.761823	0.380912	−	0.924611i	$-0.375610\pi$
$41$	200.000	0.761823	0.380912	+	0.924611i	$0.375610\pi$
$42$	0	0
$43$	−141.421	−0.501548	−0.250774	−	0.968046i	$-0.580685\pi$
$43$	−141.421	−0.501548	−0.250774	+	0.968046i	$0.580685\pi$
$44$	0	0
$45$	−7.07107	−0.0234243
$46$	0	0
$47$	208.000	0.645530	0.322765	−	0.946479i	$-0.395388\pi$
$47$	208.000	0.645530	0.322765	+	0.946479i	$0.395388\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	−135.765	−0.372761
$52$	0	0
$53$	371.938	0.963955	0.481978	−	0.876183i	$-0.339919\pi$
$53$	371.938	0.963955	0.481978	+	0.876183i	$0.339919\pi$
$54$	0	0
$55$	−8.00000	−0.0196131
$56$	0	0
$57$	−736.000	−1.71027
$58$	0	0
$59$	−152.735	−0.337024	−0.168512	−	0.985700i	$-0.553896\pi$
$59$	−152.735	−0.337024	−0.168512	+	0.985700i	$0.553896\pi$
$60$	0	0
$61$	479.418	1.00628	0.503141	−	0.864204i	$-0.332178\pi$
$61$	479.418	1.00628	0.503141	+	0.864204i	$0.332178\pi$
$62$	0	0
$63$	40.0000	0.0799925
$64$	0	0
$65$	6.00000	0.0114494
$66$	0	0
$67$	−684.479	−1.24810	−0.624048	−	0.781386i	$-0.714513\pi$
$67$	−684.479	−1.24810	−0.624048	+	0.781386i	$0.714513\pi$
$68$	0	0
$69$	−1040.86	−1.81601
$70$	0	0
$71$	−936.000	−1.56455	−0.782273	−	0.622936i	$-0.785940\pi$
$71$	−936.000	−1.56455	−0.782273	+	0.622936i	$0.785940\pi$
$72$	0	0
$73$	406.000	0.650941	0.325471	−	0.945552i	$-0.394477\pi$
$73$	406.000	0.650941	0.325471	+	0.945552i	$0.394477\pi$
$74$	0	0
$75$	−695.793	−1.07124
$76$	0	0
$77$	45.2548	0.0669775
$78$	0	0
$79$	768.000	1.09376	0.546878	−	0.837212i	$-0.315816\pi$
$79$	768.000	1.09376	0.546878	+	0.837212i	$0.315816\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	0	0
$83$	514.774	0.680768	0.340384	−	0.940286i	$-0.389443\pi$
$83$	514.774	0.680768	0.340384	+	0.940286i	$0.389443\pi$
$84$	0	0
$85$	33.9411	0.0433110
$86$	0	0
$87$	552.000	0.680237
$88$	0	0
$89$	−1306.00	−1.55546	−0.777729	−	0.628600i	$-0.783628\pi$
$89$	−1306.00	−1.55546	−0.777729	+	0.628600i	$0.783628\pi$
$90$	0	0
$91$	−33.9411	−0.0390989
$92$	0	0
$93$	995.606	1.11010
$94$	0	0
$95$	184.000	0.198716
$96$	0	0
$97$	−472.000	−0.494065	−0.247033	−	0.969007i	$-0.579456\pi$
$97$	−472.000	−0.494065	−0.247033	+	0.969007i	$0.579456\pi$
$98$	0	0
$99$	28.2843	0.0287139
$100$	0	0
$101$	−1322.29	−1.30270	−0.651350	−	0.758777i	$-0.725797\pi$
$101$	−1322.29	−1.30270	−0.651350	+	0.758777i	$0.725797\pi$
$102$	0	0
$103$	952.000	0.910712	0.455356	−	0.890309i	$-0.349512\pi$
$103$	952.000	0.910712	0.455356	+	0.890309i	$0.349512\pi$
$104$	0	0
$105$	−64.0000	−0.0594834
$106$	0	0
$107$	820.244	0.741084	0.370542	−	0.928816i	$-0.379172\pi$
$107$	820.244	0.741084	0.370542	+	0.928816i	$0.379172\pi$
$108$	0	0
$109$	422.850	0.371575	0.185787	−	0.982590i	$-0.440516\pi$
$109$	422.850	0.371575	0.185787	+	0.982590i	$0.440516\pi$
$110$	0	0
$111$	−1720.00	−1.47077
$112$	0	0
$113$	−1298.00	−1.08058	−0.540290	−	0.841479i	$-0.681686\pi$
$113$	−1298.00	−1.08058	−0.540290	+	0.841479i	$0.681686\pi$
$114$	0	0
$115$	260.215	0.211002
$116$	0	0
$117$	−21.2132	−0.0167621
$118$	0	0
$119$	−192.000	−0.147904
$120$	0	0
$121$	−1299.00	−0.975958
$122$	0	0
$123$	1131.37	0.829368
$124$	0	0
$125$	350.725	0.250958
$126$	0	0
$127$	−1744.00	−1.21854	−0.609272	−	0.792962i	$-0.708538\pi$
$127$	−1744.00	−1.21854	−0.609272	+	0.792962i	$0.708538\pi$
$128$	0	0
$129$	−800.000	−0.546016
$130$	0	0
$131$	1465.13	0.977165	0.488582	−	0.872518i	$-0.337514\pi$
$131$	1465.13	0.977165	0.488582	+	0.872518i	$0.337514\pi$
$132$	0	0
$133$	−1040.86	−0.678602
$134$	0	0
$135$	176.000	0.112205
$136$	0	0
$137$	−1400.00	−0.873066	−0.436533	−	0.899688i	$-0.643794\pi$
$137$	−1400.00	−0.873066	−0.436533	+	0.899688i	$0.643794\pi$
$138$	0	0
$139$	1781.91	1.08733	0.543667	−	0.839301i	$-0.317035\pi$
$139$	1781.91	1.08733	0.543667	+	0.839301i	$0.317035\pi$
$140$	0	0
$141$	1176.63	0.702764
$142$	0	0
$143$	−24.0000	−0.0140348
$144$	0	0
$145$	−138.000	−0.0790364
$146$	0	0
$147$	−1578.26	−0.885530
$148$	0	0
$149$	−2238.70	−1.23088	−0.615441	−	0.788183i	$-0.711022\pi$
$149$	−2238.70	−1.23088	−0.615441	+	0.788183i	$0.711022\pi$
$150$	0	0
$151$	−1304.00	−0.702768	−0.351384	−	0.936231i	$-0.614289\pi$
$151$	−1304.00	−0.702768	−0.351384	+	0.936231i	$0.614289\pi$
$152$	0	0
$153$	−120.000	−0.0634080
$154$	0	0
$155$	−248.902	−0.128982
$156$	0	0
$157$	2312.24	1.17539	0.587697	−	0.809081i	$-0.300035\pi$
$157$	2312.24	1.17539	0.587697	+	0.809081i	$0.300035\pi$
$158$	0	0
$159$	2104.00	1.04942
$160$	0	0
$161$	−1472.00	−0.720558
$162$	0	0
$163$	2155.26	1.03566	0.517832	−	0.855483i	$-0.326739\pi$
$163$	2155.26	1.03566	0.517832	+	0.855483i	$0.326739\pi$
$164$	0	0
$165$	−45.2548	−0.0213520
$166$	0	0
$167$	3848.00	1.78304	0.891519	−	0.452984i	$-0.149641\pi$
$167$	3848.00	1.78304	0.891519	+	0.452984i	$0.149641\pi$
$168$	0	0
$169$	−2179.00	−0.991807
$170$	0	0
$171$	−650.538	−0.290923
$172$	0	0
$173$	3005.20	1.32070	0.660351	−	0.750957i	$-0.270408\pi$
$173$	3005.20	1.32070	0.660351	+	0.750957i	$0.270408\pi$
$174$	0	0
$175$	−984.000	−0.425048
$176$	0	0
$177$	−864.000	−0.366905
$178$	0	0
$179$	1125.71	0.470055	0.235027	−	0.971989i	$-0.424482\pi$
$179$	1125.71	0.470055	0.235027	+	0.971989i	$0.424482\pi$
$180$	0	0
$181$	−2125.56	−0.872883	−0.436442	−	0.899733i	$-0.643761\pi$
$181$	−2125.56	−0.872883	−0.436442	+	0.899733i	$0.643761\pi$
$182$	0	0
$183$	2712.00	1.09550
$184$	0	0
$185$	430.000	0.170888
$186$	0	0
$187$	−135.765	−0.0530914
$188$	0	0
$189$	−995.606	−0.383173
$190$	0	0
$191$	−3696.00	−1.40017	−0.700087	−	0.714058i	$-0.746855\pi$
$191$	−3696.00	−1.40017	−0.700087	+	0.714058i	$0.746855\pi$
$192$	0	0
$193$	−2680.00	−0.999537	−0.499768	−	0.866159i	$-0.666582\pi$
$193$	−2680.00	−0.999537	−0.499768	+	0.866159i	$0.666582\pi$
$194$	0	0
$195$	33.9411	0.0124645
$196$	0	0
$197$	1062.07	0.384110	0.192055	−	0.981384i	$-0.438485\pi$
$197$	1062.07	0.384110	0.192055	+	0.981384i	$0.438485\pi$
$198$	0	0
$199$	−2680.00	−0.954674	−0.477337	−	0.878720i	$-0.658398\pi$
$199$	−2680.00	−0.954674	−0.477337	+	0.878720i	$0.658398\pi$
$200$	0	0
$201$	−3872.00	−1.35876
$202$	0	0
$203$	780.646	0.269904
$204$	0	0
$205$	−282.843	−0.0963639
$206$	0	0
$207$	−920.000	−0.308910
$208$	0	0
$209$	−736.000	−0.243589
$210$	0	0
$211$	3501.59	1.14246	0.571231	−	0.820789i	$-0.306466\pi$
$211$	3501.59	1.14246	0.571231	+	0.820789i	$0.306466\pi$
$212$	0	0
$213$	−5294.82	−1.70326
$214$	0	0
$215$	200.000	0.0634413
$216$	0	0
$217$	1408.00	0.440467
$218$	0	0
$219$	2296.68	0.708655
$220$	0	0
$221$	101.823	0.0309927
$222$	0	0
$223$	2544.00	0.763941	0.381970	−	0.924175i	$-0.375246\pi$
$223$	2544.00	0.763941	0.381970	+	0.924175i	$0.375246\pi$
$224$	0	0
$225$	−615.000	−0.182222
$226$	0	0
$227$	−3750.49	−1.09660	−0.548302	−	0.836280i	$-0.684726\pi$
$227$	−3750.49	−1.09660	−0.548302	+	0.836280i	$0.684726\pi$
$228$	0	0
$229$	3460.58	0.998610	0.499305	−	0.866426i	$-0.333589\pi$
$229$	3460.58	0.998610	0.499305	+	0.866426i	$0.333589\pi$
$230$	0	0
$231$	256.000	0.0729159
$232$	0	0
$233$	−3914.00	−1.10049	−0.550246	−	0.835003i	$-0.685466\pi$
$233$	−3914.00	−1.10049	−0.550246	+	0.835003i	$0.685466\pi$
$234$	0	0
$235$	−294.156	−0.0816538
$236$	0	0
$237$	4344.46	1.19073
$238$	0	0
$239$	2272.00	0.614910	0.307455	−	0.951563i	$-0.400523\pi$
$239$	2272.00	0.614910	0.307455	+	0.951563i	$0.400523\pi$
$240$	0	0
$241$	2280.00	0.609410	0.304705	−	0.952447i	$-0.401442\pi$
$241$	2280.00	0.609410	0.304705	+	0.952447i	$0.401442\pi$
$242$	0	0
$243$	−1385.93	−0.365874
$244$	0	0
$245$	394.566	0.102889
$246$	0	0
$247$	552.000	0.142198
$248$	0	0
$249$	2912.00	0.741127
$250$	0	0
$251$	−3162.18	−0.795200	−0.397600	−	0.917559i	$-0.630157\pi$
$251$	−3162.18	−0.795200	−0.397600	+	0.917559i	$0.630157\pi$
$252$	0	0
$253$	−1040.86	−0.258650
$254$	0	0
$255$	192.000	0.0471510
$256$	0	0
$257$	−702.000	−0.170387	−0.0851937	−	0.996364i	$-0.527151\pi$
$257$	−702.000	−0.170387	−0.0851937	+	0.996364i	$0.527151\pi$
$258$	0	0
$259$	−2432.45	−0.583571
$260$	0	0
$261$	487.904	0.115711
$262$	0	0
$263$	−7528.00	−1.76501	−0.882503	−	0.470308i	$-0.844143\pi$
$263$	−7528.00	−1.76501	−0.882503	+	0.470308i	$0.844143\pi$
$264$	0	0
$265$	−526.000	−0.121932
$266$	0	0
$267$	−7387.85	−1.69337
$268$	0	0
$269$	−1319.46	−0.299067	−0.149533	−	0.988757i	$-0.547777\pi$
$269$	−1319.46	−0.299067	−0.149533	+	0.988757i	$0.547777\pi$
$270$	0	0
$271$	8272.00	1.85420	0.927100	−	0.374814i	$-0.122293\pi$
$271$	8272.00	1.85420	0.927100	+	0.374814i	$0.122293\pi$
$272$	0	0
$273$	−192.000	−0.0425655
$274$	0	0
$275$	−695.793	−0.152574
$276$	0	0
$277$	−6190.01	−1.34268	−0.671339	−	0.741150i	$-0.734281\pi$
$277$	−6190.01	−1.34268	−0.671339	+	0.741150i	$0.734281\pi$
$278$	0	0
$279$	880.000	0.188832
$280$	0	0
$281$	1818.00	0.385953	0.192976	−	0.981203i	$-0.438186\pi$
$281$	1818.00	0.385953	0.192976	+	0.981203i	$0.438186\pi$
$282$	0	0
$283$	−1046.52	−0.219820	−0.109910	−	0.993942i	$-0.535056\pi$
$283$	−1046.52	−0.219820	−0.109910	+	0.993942i	$0.535056\pi$
$284$	0	0
$285$	1040.86	0.216334
$286$	0	0
$287$	1600.00	0.329077
$288$	0	0
$289$	−4337.00	−0.882760
$290$	0	0
$291$	−2670.04	−0.537870
$292$	0	0
$293$	1876.66	0.374183	0.187092	−	0.982342i	$-0.440094\pi$
$293$	1876.66	0.374183	0.187092	+	0.982342i	$0.440094\pi$
$294$	0	0
$295$	216.000	0.0426305
$296$	0	0
$297$	−704.000	−0.137543
$298$	0	0
$299$	780.646	0.150990
$300$	0	0
$301$	−1131.37	−0.216648
$302$	0	0
$303$	−7480.00	−1.41820
$304$	0	0
$305$	−678.000	−0.127286
$306$	0	0
$307$	−3682.61	−0.684618	−0.342309	−	0.939587i	$-0.611209\pi$
$307$	−3682.61	−0.684618	−0.342309	+	0.939587i	$0.611209\pi$
$308$	0	0
$309$	5385.33	0.991458
$310$	0	0
$311$	−7688.00	−1.40176	−0.700879	−	0.713281i	$-0.747209\pi$
$311$	−7688.00	−1.40176	−0.700879	+	0.713281i	$0.747209\pi$
$312$	0	0
$313$	680.000	0.122798	0.0613992	−	0.998113i	$-0.480444\pi$
$313$	680.000	0.122798	0.0613992	+	0.998113i	$0.480444\pi$
$314$	0	0
$315$	−56.5685	−0.0101183
$316$	0	0
$317$	−8772.37	−1.55428	−0.777138	−	0.629331i	$-0.783329\pi$
$317$	−8772.37	−1.55428	−0.777138	+	0.629331i	$0.783329\pi$
$318$	0	0
$319$	552.000	0.0968842
$320$	0	0
$321$	4640.00	0.806790
$322$	0	0
$323$	3122.58	0.537911
$324$	0	0
$325$	521.845	0.0890669
$326$	0	0
$327$	2392.00	0.404520
$328$	0	0
$329$	1664.00	0.278843
$330$	0	0
$331$	8626.70	1.43253	0.716264	−	0.697830i	$-0.245851\pi$
$331$	8626.70	1.43253	0.716264	+	0.697830i	$0.245851\pi$
$332$	0	0
$333$	−1520.28	−0.250183
$334$	0	0
$335$	968.000	0.157873
$336$	0	0
$337$	5298.00	0.856381	0.428191	−	0.903688i	$-0.359151\pi$
$337$	5298.00	0.856381	0.428191	+	0.903688i	$0.359151\pi$
$338$	0	0
$339$	−7342.60	−1.17639
$340$	0	0
$341$	995.606	0.158109
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	1472.00	0.229710
$346$	0	0
$347$	−1216.22	−0.188157	−0.0940783	−	0.995565i	$-0.529990\pi$
$347$	−1216.22	−0.188157	−0.0940783	+	0.995565i	$0.529990\pi$
$348$	0	0
$349$	10574.1	1.62183	0.810913	−	0.585167i	$-0.198971\pi$
$349$	10574.1	1.62183	0.810913	+	0.585167i	$0.198971\pi$
$350$	0	0
$351$	528.000	0.0802922
$352$	0	0
$353$	7646.00	1.15285	0.576424	−	0.817151i	$-0.304448\pi$
$353$	7646.00	1.15285	0.576424	+	0.817151i	$0.304448\pi$
$354$	0	0
$355$	1323.70	0.197901
$356$	0	0
$357$	−1086.12	−0.161018
$358$	0	0
$359$	2456.00	0.361066	0.180533	−	0.983569i	$-0.442218\pi$
$359$	2456.00	0.361066	0.180533	+	0.983569i	$0.442218\pi$
$360$	0	0
$361$	10069.0	1.46800
$362$	0	0
$363$	−7348.25	−1.06249
$364$	0	0
$365$	−574.171	−0.0823383
$366$	0	0
$367$	1744.00	0.248055	0.124027	−	0.992279i	$-0.460419\pi$
$367$	1744.00	0.248055	0.124027	+	0.992279i	$0.460419\pi$
$368$	0	0
$369$	1000.00	0.141078
$370$	0	0
$371$	2975.51	0.416390
$372$	0	0
$373$	11255.7	1.56246	0.781232	−	0.624240i	$-0.214591\pi$
$373$	11255.7	1.56246	0.781232	+	0.624240i	$0.214591\pi$
$374$	0	0
$375$	1984.00	0.273209
$376$	0	0
$377$	−414.000	−0.0565573
$378$	0	0
$379$	−3886.26	−0.526712	−0.263356	−	0.964699i	$-0.584829\pi$
$379$	−3886.26	−0.526712	−0.263356	+	0.964699i	$0.584829\pi$
$380$	0	0
$381$	−9865.55	−1.32658
$382$	0	0
$383$	7296.00	0.973390	0.486695	−	0.873572i	$-0.338202\pi$
$383$	7296.00	0.973390	0.486695	+	0.873572i	$0.338202\pi$
$384$	0	0
$385$	−64.0000	−0.00847206
$386$	0	0
$387$	−707.107	−0.0928792
$388$	0	0
$389$	−1118.64	−0.145803	−0.0729016	−	0.997339i	$-0.523226\pi$
$389$	−1118.64	−0.145803	−0.0729016	+	0.997339i	$0.523226\pi$
$390$	0	0
$391$	4416.00	0.571168
$392$	0	0
$393$	8288.00	1.06380
$394$	0	0
$395$	−1086.12	−0.138350
$396$	0	0
$397$	−14420.7	−1.82306	−0.911531	−	0.411230i	$-0.865099\pi$
$397$	−14420.7	−1.82306	−0.911531	+	0.411230i	$0.865099\pi$
$398$	0	0
$399$	−5888.00	−0.738769
$400$	0	0
$401$	−4728.00	−0.588791	−0.294395	−	0.955684i	$-0.595118\pi$
$401$	−4728.00	−0.588791	−0.294395	+	0.955684i	$0.595118\pi$
$402$	0	0
$403$	−746.705	−0.0922978
$404$	0	0
$405$	1186.53	0.145578
$406$	0	0
$407$	−1720.00	−0.209477
$408$	0	0
$409$	−6424.00	−0.776641	−0.388321	−	0.921524i	$-0.626945\pi$
$409$	−6424.00	−0.776641	−0.388321	+	0.921524i	$0.626945\pi$
$410$	0	0
$411$	−7919.60	−0.950474
$412$	0	0
$413$	−1221.88	−0.145581
$414$	0	0
$415$	−728.000	−0.0861111
$416$	0	0
$417$	10080.0	1.18374
$418$	0	0
$419$	8841.66	1.03089	0.515446	−	0.856922i	$-0.327626\pi$
$419$	8841.66	1.03089	0.515446	+	0.856922i	$0.327626\pi$
$420$	0	0
$421$	2532.86	0.293216	0.146608	−	0.989195i	$-0.453164\pi$
$421$	2532.86	0.293216	0.146608	+	0.989195i	$0.453164\pi$
$422$	0	0
$423$	1040.00	0.119543
$424$	0	0
$425$	2952.00	0.336925
$426$	0	0
$427$	3835.35	0.434673
$428$	0	0
$429$	−135.765	−0.0152792
$430$	0	0
$431$	9952.00	1.11223	0.556115	−	0.831105i	$-0.312291\pi$
$431$	9952.00	1.11223	0.556115	+	0.831105i	$0.312291\pi$
$432$	0	0
$433$	9928.00	1.10187	0.550934	−	0.834549i	$-0.314271\pi$
$433$	9928.00	1.10187	0.550934	+	0.834549i	$0.314271\pi$
$434$	0	0
$435$	−780.646	−0.0860439
$436$	0	0
$437$	23939.8	2.62059
$438$	0	0
$439$	7288.00	0.792340	0.396170	−	0.918177i	$-0.370339\pi$
$439$	7288.00	0.792340	0.396170	+	0.918177i	$0.370339\pi$
$440$	0	0
$441$	−1395.00	−0.150632
$442$	0	0
$443$	17394.8	1.86558	0.932791	−	0.360417i	$-0.117366\pi$
$443$	17394.8	1.86558	0.932791	+	0.360417i	$0.117366\pi$
$444$	0	0
$445$	1846.96	0.196752
$446$	0	0
$447$	−12664.0	−1.34001
$448$	0	0
$449$	4808.00	0.505353	0.252677	−	0.967551i	$-0.418689\pi$
$449$	4808.00	0.505353	0.252677	+	0.967551i	$0.418689\pi$
$450$	0	0
$451$	1131.37	0.118125
$452$	0	0
$453$	−7376.54	−0.765077
$454$	0	0
$455$	48.0000	0.00494566
$456$	0	0
$457$	14792.0	1.51409	0.757047	−	0.653361i	$-0.226642\pi$
$457$	14792.0	1.51409	0.757047	+	0.653361i	$0.226642\pi$
$458$	0	0
$459$	2986.82	0.303732
$460$	0	0
$461$	−9510.59	−0.960851	−0.480425	−	0.877036i	$-0.659518\pi$
$461$	−9510.59	−0.960851	−0.480425	+	0.877036i	$0.659518\pi$
$462$	0	0
$463$	−11200.0	−1.12421	−0.562104	−	0.827067i	$-0.690008\pi$
$463$	−11200.0	−1.12421	−0.562104	+	0.827067i	$0.690008\pi$
$464$	0	0
$465$	−1408.00	−0.140418
$466$	0	0
$467$	16433.2	1.62834	0.814171	−	0.580625i	$-0.197192\pi$
$467$	16433.2	1.62834	0.814171	+	0.580625i	$0.197192\pi$
$468$	0	0
$469$	−5475.83	−0.539127
$470$	0	0
$471$	13080.0	1.27961
$472$	0	0
$473$	−800.000	−0.0777675
$474$	0	0
$475$	16003.2	1.54585
$476$	0	0
$477$	1859.69	0.178510
$478$	0	0
$479$	−8240.00	−0.786003	−0.393001	−	0.919538i	$-0.628563\pi$
$479$	−8240.00	−0.786003	−0.393001	+	0.919538i	$0.628563\pi$
$480$	0	0
$481$	1290.00	0.122285
$482$	0	0
$483$	−8326.89	−0.784444
$484$	0	0
$485$	667.509	0.0624949
$486$	0	0
$487$	−11544.0	−1.07414	−0.537072	−	0.843536i	$-0.680470\pi$
$487$	−11544.0	−1.07414	−0.537072	+	0.843536i	$0.680470\pi$
$488$	0	0
$489$	12192.0	1.12749
$490$	0	0
$491$	12665.7	1.16414	0.582072	−	0.813137i	$-0.302242\pi$
$491$	12665.7	1.16414	0.582072	+	0.813137i	$0.302242\pi$
$492$	0	0
$493$	−2341.94	−0.213946
$494$	0	0
$495$	−40.0000	−0.00363205
$496$	0	0
$497$	−7488.00	−0.675820
$498$	0	0
$499$	14385.4	1.29054	0.645269	−	0.763956i	$-0.276745\pi$
$499$	14385.4	1.29054	0.645269	+	0.763956i	$0.276745\pi$
$500$	0	0
$501$	21767.6	1.94112
$502$	0	0
$503$	8200.00	0.726879	0.363439	−	0.931618i	$-0.381602\pi$
$503$	8200.00	0.726879	0.363439	+	0.931618i	$0.381602\pi$
$504$	0	0
$505$	1870.00	0.164780
$506$	0	0
$507$	−12326.3	−1.07974
$508$	0	0
$509$	−16197.0	−1.41045	−0.705225	−	0.708984i	$-0.749154\pi$
$509$	−16197.0	−1.41045	−0.705225	+	0.708984i	$0.749154\pi$
$510$	0	0
$511$	3248.00	0.281180
$512$	0	0
$513$	16192.0	1.39356
$514$	0	0
$515$	−1346.33	−0.115197
$516$	0	0
$517$	1176.63	0.100093
$518$	0	0
$519$	17000.0	1.43780
$520$	0	0
$521$	−600.000	−0.0504539	−0.0252269	−	0.999682i	$-0.508031\pi$
$521$	−600.000	−0.0504539	−0.0252269	+	0.999682i	$0.508031\pi$
$522$	0	0
$523$	−17598.5	−1.47137	−0.735686	−	0.677323i	$-0.763140\pi$
$523$	−17598.5	−1.47137	−0.735686	+	0.677323i	$0.763140\pi$
$524$	0	0
$525$	−5566.34	−0.462734
$526$	0	0
$527$	−4224.00	−0.349147
$528$	0	0
$529$	21689.0	1.78261
$530$	0	0
$531$	−763.675	−0.0624118
$532$	0	0
$533$	−848.528	−0.0689565
$534$	0	0
$535$	−1160.00	−0.0937405
$536$	0	0
$537$	6368.00	0.511731
$538$	0	0
$539$	−1578.26	−0.126124
$540$	0	0
$541$	16499.6	1.31123	0.655614	−	0.755096i	$-0.272410\pi$
$541$	16499.6	1.31123	0.655614	+	0.755096i	$0.272410\pi$
$542$	0	0
$543$	−12024.0	−0.950275
$544$	0	0
$545$	−598.000	−0.0470009
$546$	0	0
$547$	9181.07	0.717650	0.358825	−	0.933405i	$-0.383178\pi$
$547$	9181.07	0.717650	0.358825	+	0.933405i	$0.383178\pi$
$548$	0	0
$549$	2397.09	0.186349
$550$	0	0
$551$	−12696.0	−0.981611
$552$	0	0
$553$	6144.00	0.472458
$554$	0	0
$555$	2432.45	0.186039
$556$	0	0
$557$	4159.20	0.316393	0.158197	−	0.987408i	$-0.449432\pi$
$557$	4159.20	0.316393	0.158197	+	0.987408i	$0.449432\pi$
$558$	0	0
$559$	600.000	0.0453977
$560$	0	0
$561$	−768.000	−0.0577985
$562$	0	0
$563$	−12620.4	−0.944739	−0.472370	−	0.881401i	$-0.656601\pi$
$563$	−12620.4	−0.944739	−0.472370	+	0.881401i	$0.656601\pi$
$564$	0	0
$565$	1835.65	0.136684
$566$	0	0
$567$	−6712.00	−0.497138
$568$	0	0
$569$	−4824.00	−0.355418	−0.177709	−	0.984083i	$-0.556869\pi$
$569$	−4824.00	−0.355418	−0.177709	+	0.984083i	$0.556869\pi$
$570$	0	0
$571$	9248.96	0.677858	0.338929	−	0.940812i	$-0.389935\pi$
$571$	9248.96	0.677858	0.338929	+	0.940812i	$0.389935\pi$
$572$	0	0
$573$	−20907.7	−1.52432
$574$	0	0
$575$	22632.0	1.64143
$576$	0	0
$577$	−7938.00	−0.572727	−0.286363	−	0.958121i	$-0.592446\pi$
$577$	−7938.00	−0.572727	−0.286363	+	0.958121i	$0.592446\pi$
$578$	0	0
$579$	−15160.4	−1.08816
$580$	0	0
$581$	4118.19	0.294064
$582$	0	0
$583$	2104.00	0.149466
$584$	0	0
$585$	30.0000	0.00212025
$586$	0	0
$587$	12790.1	0.899329	0.449664	−	0.893198i	$-0.351544\pi$
$587$	12790.1	0.899329	0.449664	+	0.893198i	$0.351544\pi$
$588$	0	0
$589$	−22898.9	−1.60193
$590$	0	0
$591$	6008.00	0.418166
$592$	0	0
$593$	−15346.0	−1.06271	−0.531353	−	0.847150i	$-0.678316\pi$
$593$	−15346.0	−1.06271	−0.531353	+	0.847150i	$0.678316\pi$
$594$	0	0
$595$	271.529	0.0187086
$596$	0	0
$597$	−15160.4	−1.03932
$598$	0	0
$599$	2744.00	0.187173	0.0935866	−	0.995611i	$-0.470167\pi$
$599$	2744.00	0.187173	0.0935866	+	0.995611i	$0.470167\pi$
$600$	0	0
$601$	6374.00	0.432614	0.216307	−	0.976325i	$-0.430599\pi$
$601$	6374.00	0.432614	0.216307	+	0.976325i	$0.430599\pi$
$602$	0	0
$603$	−3422.40	−0.231129
$604$	0	0
$605$	1837.06	0.123450
$606$	0	0
$607$	−9984.00	−0.667608	−0.333804	−	0.942643i	$-0.608332\pi$
$607$	−9984.00	−0.667608	−0.333804	+	0.942643i	$0.608332\pi$
$608$	0	0
$609$	4416.00	0.293835
$610$	0	0
$611$	−882.469	−0.0584303
$612$	0	0
$613$	6153.24	0.405428	0.202714	−	0.979238i	$-0.435024\pi$
$613$	6153.24	0.405428	0.202714	+	0.979238i	$0.435024\pi$
$614$	0	0
$615$	−1600.00	−0.104908
$616$	0	0
$617$	−10486.0	−0.684198	−0.342099	−	0.939664i	$-0.611138\pi$
$617$	−10486.0	−0.684198	−0.342099	+	0.939664i	$0.611138\pi$
$618$	0	0
$619$	−1578.26	−0.102481	−0.0512405	−	0.998686i	$-0.516317\pi$
$619$	−1578.26	−0.102481	−0.0512405	+	0.998686i	$0.516317\pi$
$620$	0	0
$621$	22898.9	1.47972
$622$	0	0
$623$	−10448.0	−0.671895
$624$	0	0
$625$	14879.0	0.952256
$626$	0	0
$627$	−4163.44	−0.265187
$628$	0	0
$629$	7297.34	0.462582
$630$	0	0
$631$	−26680.0	−1.68322	−0.841612	−	0.540083i	$-0.818393\pi$
$631$	−26680.0	−1.68322	−0.841612	+	0.540083i	$0.818393\pi$
$632$	0	0
$633$	19808.0	1.24376
$634$	0	0
$635$	2466.39	0.154135
$636$	0	0
$637$	1183.70	0.0736260
$638$	0	0
$639$	−4680.00	−0.289731
$640$	0	0
$641$	−2648.00	−0.163166	−0.0815832	−	0.996667i	$-0.525998\pi$
$641$	−2648.00	−0.163166	−0.0815832	+	0.996667i	$0.525998\pi$
$642$	0	0
$643$	23707.9	1.45404	0.727020	−	0.686617i	$-0.240905\pi$
$643$	23707.9	1.45404	0.727020	+	0.686617i	$0.240905\pi$
$644$	0	0
$645$	1131.37	0.0690662
$646$	0	0
$647$	18264.0	1.10979	0.554893	−	0.831922i	$-0.312759\pi$
$647$	18264.0	1.10979	0.554893	+	0.831922i	$0.312759\pi$
$648$	0	0
$649$	−864.000	−0.0522573
$650$	0	0
$651$	7964.85	0.479519
$652$	0	0
$653$	6238.10	0.373837	0.186919	−	0.982375i	$-0.440150\pi$
$653$	6238.10	0.373837	0.186919	+	0.982375i	$0.440150\pi$
$654$	0	0
$655$	−2072.00	−0.123603
$656$	0	0
$657$	2030.00	0.120545
$658$	0	0
$659$	−31435.1	−1.85818	−0.929089	−	0.369856i	$-0.879407\pi$
$659$	−31435.1	−1.85818	−0.929089	+	0.369856i	$0.879407\pi$
$660$	0	0
$661$	−31133.9	−1.83203	−0.916013	−	0.401149i	$-0.868611\pi$
$661$	−31133.9	−1.83203	−0.916013	+	0.401149i	$0.868611\pi$
$662$	0	0
$663$	576.000	0.0337406
$664$	0	0
$665$	1472.00	0.0858372
$666$	0	0
$667$	−17954.9	−1.04230
$668$	0	0
$669$	14391.0	0.831674
$670$	0	0
$671$	2712.00	0.156029
$672$	0	0
$673$	−14072.0	−0.805997	−0.402998	−	0.915201i	$-0.632032\pi$
$673$	−14072.0	−0.805997	−0.402998	+	0.915201i	$0.632032\pi$
$674$	0	0
$675$	15307.4	0.872865
$676$	0	0
$677$	19562.8	1.11058	0.555288	−	0.831658i	$-0.312608\pi$
$677$	19562.8	1.11058	0.555288	+	0.831658i	$0.312608\pi$
$678$	0	0
$679$	−3776.00	−0.213416
$680$	0	0
$681$	−21216.0	−1.19383
$682$	0	0
$683$	−28900.9	−1.61912	−0.809561	−	0.587035i	$-0.800295\pi$
$683$	−28900.9	−1.61912	−0.809561	+	0.587035i	$0.800295\pi$
$684$	0	0
$685$	1979.90	0.110435
$686$	0	0
$687$	19576.0	1.08715
$688$	0	0
$689$	−1578.00	−0.0872526
$690$	0	0
$691$	1374.62	0.0756770	0.0378385	−	0.999284i	$-0.487953\pi$
$691$	1374.62	0.0756770	0.0378385	+	0.999284i	$0.487953\pi$
$692$	0	0
$693$	226.274	0.0124032
$694$	0	0
$695$	−2520.00	−0.137538
$696$	0	0
$697$	−4800.00	−0.260851
$698$	0	0
$699$	−22140.9	−1.19806
$700$	0	0
$701$	−36493.8	−1.96626	−0.983132	−	0.182897i	$-0.941452\pi$
$701$	−36493.8	−1.96626	−0.983132	+	0.182897i	$0.941452\pi$
$702$	0	0
$703$	39560.0	2.12238
$704$	0	0
$705$	−1664.00	−0.0888934
$706$	0	0
$707$	−10578.3	−0.562714
$708$	0	0
$709$	5850.60	0.309907	0.154953	−	0.987922i	$-0.450477\pi$
$709$	5850.60	0.309907	0.154953	+	0.987922i	$0.450477\pi$
$710$	0	0
$711$	3840.00	0.202547
$712$	0	0
$713$	−32384.0	−1.70097
$714$	0	0
$715$	33.9411	0.00177528
$716$	0	0
$717$	12852.4	0.669429
$718$	0	0
$719$	864.000	0.0448147	0.0224073	−	0.999749i	$-0.492867\pi$
$719$	864.000	0.0448147	0.0224073	+	0.999749i	$0.492867\pi$
$720$	0	0
$721$	7616.00	0.393391
$722$	0	0
$723$	12897.6	0.663441
$724$	0	0
$725$	−12002.4	−0.614840
$726$	0	0
$727$	−5416.00	−0.276298	−0.138149	−	0.990411i	$-0.544115\pi$
$727$	−5416.00	−0.276298	−0.138149	+	0.990411i	$0.544115\pi$
$728$	0	0
$729$	14813.0	0.752578
$730$	0	0
$731$	3394.11	0.171732
$732$	0	0
$733$	21322.1	1.07442	0.537210	−	0.843449i	$-0.319478\pi$
$733$	21322.1	1.07442	0.537210	+	0.843449i	$0.319478\pi$
$734$	0	0
$735$	2232.00	0.112012
$736$	0	0
$737$	−3872.00	−0.193524
$738$	0	0
$739$	−37759.5	−1.87957	−0.939787	−	0.341760i	$-0.888977\pi$
$739$	−37759.5	−1.87957	−0.939787	+	0.341760i	$0.888977\pi$
$740$	0	0
$741$	3122.58	0.154806
$742$	0	0
$743$	−18840.0	−0.930246	−0.465123	−	0.885246i	$-0.653990\pi$
$743$	−18840.0	−0.930246	−0.465123	+	0.885246i	$0.653990\pi$
$744$	0	0
$745$	3166.00	0.155696
$746$	0	0
$747$	2573.87	0.126068
$748$	0	0
$749$	6561.95	0.320118
$750$	0	0
$751$	24640.0	1.19724	0.598619	−	0.801034i	$-0.295716\pi$
$751$	24640.0	1.19724	0.598619	+	0.801034i	$0.295716\pi$
$752$	0	0
$753$	−17888.0	−0.865704
$754$	0	0
$755$	1844.13	0.0888939
$756$	0	0
$757$	−11742.2	−0.563776	−0.281888	−	0.959447i	$-0.590961\pi$
$757$	−11742.2	−0.563776	−0.281888	+	0.959447i	$0.590961\pi$
$758$	0	0
$759$	−5888.00	−0.281582
$760$	0	0
$761$	4968.00	0.236649	0.118324	−	0.992975i	$-0.462248\pi$
$761$	4968.00	0.236649	0.118324	+	0.992975i	$0.462248\pi$
$762$	0	0
$763$	3382.80	0.160505
$764$	0	0
$765$	169.706	0.00802055
$766$	0	0
$767$	648.000	0.0305058
$768$	0	0
$769$	−34808.0	−1.63226	−0.816130	−	0.577868i	$-0.803885\pi$
$769$	−34808.0	−1.63226	−0.816130	+	0.577868i	$0.803885\pi$
$770$	0	0
$771$	−3971.11	−0.185494
$772$	0	0
$773$	−2431.03	−0.113115	−0.0565577	−	0.998399i	$-0.518012\pi$
$773$	−2431.03	−0.113115	−0.0565577	+	0.998399i	$0.518012\pi$
$774$	0	0
$775$	−21648.0	−1.00338
$776$	0	0
$777$	−13760.0	−0.635312
$778$	0	0
$779$	−26021.5	−1.19681
$780$	0	0
$781$	−5294.82	−0.242591
$782$	0	0
$783$	−12144.0	−0.554267
$784$	0	0
$785$	−3270.00	−0.148677
$786$	0	0
$787$	−18514.9	−0.838608	−0.419304	−	0.907846i	$-0.637726\pi$
$787$	−18514.9	−0.838608	−0.419304	+	0.907846i	$0.637726\pi$
$788$	0	0
$789$	−42584.8	−1.92149
$790$	0	0
$791$	−10384.0	−0.466767
$792$	0	0
$793$	−2034.00	−0.0910838
$794$	0	0
$795$	−2975.51	−0.132743
$796$	0	0
$797$	−2077.48	−0.0923314	−0.0461657	−	0.998934i	$-0.514700\pi$
$797$	−2077.48	−0.0923314	−0.0461657	+	0.998934i	$0.514700\pi$
$798$	0	0
$799$	−4992.00	−0.221032
$800$	0	0
$801$	−6530.00	−0.288048
$802$	0	0
$803$	2296.68	0.100932
$804$	0	0
$805$	2081.72	0.0911442
$806$	0	0
$807$	−7464.00	−0.325583
$808$	0	0
$809$	−24056.0	−1.04544	−0.522722	−	0.852503i	$-0.675083\pi$
$809$	−24056.0	−1.04544	−0.522722	+	0.852503i	$0.675083\pi$
$810$	0	0
$811$	12699.6	0.549870	0.274935	−	0.961463i	$-0.411344\pi$
$811$	12699.6	0.549870	0.274935	+	0.961463i	$0.411344\pi$
$812$	0	0
$813$	46793.5	2.01860
$814$	0	0
$815$	−3048.00	−0.131002
$816$	0	0
$817$	18400.0	0.787925
$818$	0	0
$819$	−169.706	−0.00724053
$820$	0	0
$821$	15328.7	0.651613	0.325806	−	0.945437i	$-0.394364\pi$
$821$	15328.7	0.651613	0.325806	+	0.945437i	$0.394364\pi$
$822$	0	0
$823$	37528.0	1.58948	0.794741	−	0.606949i	$-0.207607\pi$
$823$	37528.0	1.58948	0.794741	+	0.606949i	$0.207607\pi$
$824$	0	0
$825$	−3936.00	−0.166102
$826$	0	0
$827$	2754.89	0.115837	0.0579183	−	0.998321i	$-0.481554\pi$
$827$	2754.89	0.115837	0.0579183	+	0.998321i	$0.481554\pi$
$828$	0	0
$829$	36027.1	1.50938	0.754688	−	0.656084i	$-0.227788\pi$
$829$	36027.1	1.50938	0.754688	+	0.656084i	$0.227788\pi$
$830$	0	0
$831$	−35016.0	−1.46172
$832$	0	0
$833$	6696.00	0.278515
$834$	0	0
$835$	−5441.89	−0.225538
$836$	0	0
$837$	−21903.3	−0.904528
$838$	0	0
$839$	6600.00	0.271582	0.135791	−	0.990738i	$-0.456642\pi$
$839$	6600.00	0.271582	0.135791	+	0.990738i	$0.456642\pi$
$840$	0	0
$841$	−14867.0	−0.609578
$842$	0	0
$843$	10284.2	0.420172
$844$	0	0
$845$	3081.57	0.125455
$846$	0	0
$847$	−10392.0	−0.421574
$848$	0	0
$849$	−5920.00	−0.239310
$850$	0	0
$851$	55946.3	2.25360
$852$	0	0
$853$	−33161.9	−1.33112	−0.665558	−	0.746346i	$-0.731806\pi$
$853$	−33161.9	−1.33112	−0.665558	+	0.746346i	$0.731806\pi$
$854$	0	0
$855$	920.000	0.0367992
$856$	0	0
$857$	−15480.0	−0.617021	−0.308510	−	0.951221i	$-0.599830\pi$
$857$	−15480.0	−0.617021	−0.308510	+	0.951221i	$0.599830\pi$
$858$	0	0
$859$	31955.6	1.26928	0.634639	−	0.772809i	$-0.281149\pi$
$859$	31955.6	1.26928	0.634639	+	0.772809i	$0.281149\pi$
$860$	0	0
$861$	9050.97	0.358253
$862$	0	0
$863$	−1392.00	−0.0549064	−0.0274532	−	0.999623i	$-0.508740\pi$
$863$	−1392.00	−0.0549064	−0.0274532	+	0.999623i	$0.508740\pi$
$864$	0	0
$865$	−4250.00	−0.167057
$866$	0	0
$867$	−24533.8	−0.961027
$868$	0	0
$869$	4344.46	0.169592
$870$	0	0
$871$	2904.00	0.112972
$872$	0	0
$873$	−2360.00	−0.0914936
$874$	0	0
$875$	2805.80	0.108404
$876$	0	0
$877$	4065.86	0.156550	0.0782751	−	0.996932i	$-0.475059\pi$
$877$	4065.86	0.156550	0.0782751	+	0.996932i	$0.475059\pi$
$878$	0	0
$879$	10616.0	0.407359
$880$	0	0
$881$	−49518.0	−1.89365	−0.946824	−	0.321752i	$-0.895728\pi$
$881$	−49518.0	−1.89365	−0.946824	+	0.321752i	$0.895728\pi$
$882$	0	0
$883$	−4655.59	−0.177433	−0.0887164	−	0.996057i	$-0.528276\pi$
$883$	−4655.59	−0.177433	−0.0887164	+	0.996057i	$0.528276\pi$
$884$	0	0
$885$	1221.88	0.0464102
$886$	0	0
$887$	33720.0	1.27645	0.638223	−	0.769852i	$-0.279670\pi$
$887$	33720.0	1.27645	0.638223	+	0.769852i	$0.279670\pi$
$888$	0	0
$889$	−13952.0	−0.526361
$890$	0	0
$891$	−4746.10	−0.178452
$892$	0	0
$893$	−27062.4	−1.01412
$894$	0	0
$895$	−1592.00	−0.0594578
$896$	0	0
$897$	4416.00	0.164377
$898$	0	0
$899$	17174.2	0.637143
$900$	0	0
$901$	−8926.52	−0.330061
$902$	0	0
$903$	−6400.00	−0.235857
$904$	0	0
$905$	3006.00	0.110412
$906$	0	0
$907$	−16998.8	−0.622313	−0.311156	−	0.950359i	$-0.600716\pi$
$907$	−16998.8	−0.622313	−0.311156	+	0.950359i	$0.600716\pi$
$908$	0	0
$909$	−6611.45	−0.241241
$910$	0	0
$911$	18128.0	0.659284	0.329642	−	0.944106i	$-0.393072\pi$
$911$	18128.0	0.659284	0.329642	+	0.944106i	$0.393072\pi$
$912$	0	0
$913$	2912.00	0.105557
$914$	0	0
$915$	−3835.35	−0.138571
$916$	0	0
$917$	11721.0	0.422096
$918$	0	0
$919$	13640.0	0.489600	0.244800	−	0.969574i	$-0.421278\pi$
$919$	13640.0	0.489600	0.244800	+	0.969574i	$0.421278\pi$
$920$	0	0
$921$	−20832.0	−0.745318
$922$	0	0
$923$	3971.11	0.141615
$924$	0	0
$925$	37398.9	1.32937
$926$	0	0
$927$	4760.00	0.168650
$928$	0	0
$929$	−49848.0	−1.76045	−0.880226	−	0.474555i	$-0.842609\pi$
$929$	−49848.0	−1.76045	−0.880226	+	0.474555i	$0.842609\pi$
$930$	0	0
$931$	36300.0	1.27786
$932$	0	0
$933$	−43489.9	−1.52604
$934$	0	0
$935$	192.000	0.00671558
$936$	0	0
$937$	−4490.00	−0.156544	−0.0782721	−	0.996932i	$-0.524940\pi$
$937$	−4490.00	−0.156544	−0.0782721	+	0.996932i	$0.524940\pi$
$938$	0	0
$939$	3846.66	0.133686
$940$	0	0
$941$	−1070.56	−0.0370874	−0.0185437	−	0.999828i	$-0.505903\pi$
$941$	−1070.56	−0.0370874	−0.0185437	+	0.999828i	$0.505903\pi$
$942$	0	0
$943$	−36800.0	−1.27081
$944$	0	0
$945$	1408.00	0.0484680
$946$	0	0
$947$	−18843.0	−0.646584	−0.323292	−	0.946299i	$-0.604790\pi$
$947$	−18843.0	−0.646584	−0.323292	+	0.946299i	$0.604790\pi$
$948$	0	0
$949$	−1722.51	−0.0589200
$950$	0	0
$951$	−49624.0	−1.69208
$952$	0	0
$953$	−25272.0	−0.859014	−0.429507	−	0.903064i	$-0.641313\pi$
$953$	−25272.0	−0.859014	−0.429507	+	0.903064i	$0.641313\pi$
$954$	0	0
$955$	5226.93	0.177110
$956$	0	0
$957$	3122.58	0.105474
$958$	0	0
$959$	−11200.0	−0.377129
$960$	0	0
$961$	1185.00	0.0397771
$962$	0	0
$963$	4101.22	0.137238
$964$	0	0
$965$	3790.09	0.126433
$966$	0	0
$967$	20008.0	0.665371	0.332686	−	0.943038i	$-0.392045\pi$
$967$	20008.0	0.665371	0.332686	+	0.943038i	$0.392045\pi$
$968$	0	0
$969$	17664.0	0.585603
$970$	0	0
$971$	52852.0	1.74676	0.873379	−	0.487042i	$-0.161924\pi$
$971$	52852.0	1.74676	0.873379	+	0.487042i	$0.161924\pi$
$972$	0	0
$973$	14255.3	0.469685
$974$	0	0
$975$	2952.00	0.0969638
$976$	0	0
$977$	−53912.0	−1.76540	−0.882701	−	0.469935i	$-0.844277\pi$
$977$	−53912.0	−1.76540	−0.882701	+	0.469935i	$0.844277\pi$
$978$	0	0
$979$	−7387.85	−0.241182
$980$	0	0
$981$	2114.25	0.0688102
$982$	0	0
$983$	−27592.0	−0.895268	−0.447634	−	0.894217i	$-0.647733\pi$
$983$	−27592.0	−0.895268	−0.447634	+	0.894217i	$0.647733\pi$
$984$	0	0
$985$	−1502.00	−0.0485865
$986$	0	0
$987$	9413.01	0.303566
$988$	0	0
$989$	26021.5	0.836640
$990$	0	0
$991$	3104.00	0.0994973	0.0497486	−	0.998762i	$-0.484158\pi$
$991$	3104.00	0.0994973	0.0497486	+	0.998762i	$0.484158\pi$
$992$	0	0
$993$	48800.0	1.55954
$994$	0	0
$995$	3790.09	0.120758
$996$	0	0
$997$	−32299.2	−1.02600	−0.513002	−	0.858387i	$-0.671467\pi$
$997$	−32299.2	−1.02600	−0.513002	+	0.858387i	$0.671467\pi$
$998$	0	0
$999$	37840.0	1.19840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.a.d.1.2		2
4.3	odd	2		1024.4.a.a.1.1		2
8.3	odd	2		1024.4.a.a.1.2		2
8.5	even	2	inner	1024.4.a.d.1.1		2
16.3	odd	4		1024.4.b.d.513.1		2
16.5	even	4		1024.4.b.a.513.1		2
16.11	odd	4		1024.4.b.d.513.2		2
16.13	even	4		1024.4.b.a.513.2		2
32.3	odd	8		512.4.e.b.129.1	yes	2
32.5	even	8		512.4.e.a.385.1	yes	2
32.11	odd	8		512.4.e.b.385.1	yes	2
32.13	even	8		512.4.e.a.129.1	✓	2
32.19	odd	8		512.4.e.g.129.1	yes	2
32.21	even	8		512.4.e.h.385.1	yes	2
32.27	odd	8		512.4.e.g.385.1	yes	2
32.29	even	8		512.4.e.h.129.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.4.e.a.129.1	✓	2	32.13	even	8
512.4.e.a.385.1	yes	2	32.5	even	8
512.4.e.b.129.1	yes	2	32.3	odd	8
512.4.e.b.385.1	yes	2	32.11	odd	8
512.4.e.g.129.1	yes	2	32.19	odd	8
512.4.e.g.385.1	yes	2	32.27	odd	8
512.4.e.h.129.1	yes	2	32.29	even	8
512.4.e.h.385.1	yes	2	32.21	even	8
1024.4.a.a.1.1		2	4.3	odd	2
1024.4.a.a.1.2		2	8.3	odd	2
1024.4.a.d.1.1		2	8.5	even	2	inner
1024.4.a.d.1.2		2	1.1	even	1	trivial
1024.4.b.a.513.1		2	16.5	even	4
1024.4.b.a.513.2		2	16.13	even	4
1024.4.b.d.513.1		2	16.3	odd	4
1024.4.b.d.513.2		2	16.11	odd	4

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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q+5.65685 q^{3} -1.41421 q^{5} +8.00000 q^{7} +5.00000 q^{9} +O(q^{10})\)	\(q+5.65685 q^{3} -1.41421 q^{5} +8.00000 q^{7} +5.00000 q^{9} +5.65685 q^{11} -4.24264 q^{13} -8.00000 q^{15} -24.0000 q^{17} -130.108 q^{19} +45.2548 q^{21} -184.000 q^{23} -123.000 q^{25} -124.451 q^{27} +97.5807 q^{29} +176.000 q^{31} +32.0000 q^{33} -11.3137 q^{35} -304.056 q^{37} -24.0000 q^{39} +200.000 q^{41} -141.421 q^{43} -7.07107 q^{45} +208.000 q^{47} -279.000 q^{49} -135.765 q^{51} +371.938 q^{53} -8.00000 q^{55} -736.000 q^{57} -152.735 q^{59} +479.418 q^{61} +40.0000 q^{63} +6.00000 q^{65} -684.479 q^{67} -1040.86 q^{69} -936.000 q^{71} +406.000 q^{73} -695.793 q^{75} +45.2548 q^{77} +768.000 q^{79} -839.000 q^{81} +514.774 q^{83} +33.9411 q^{85} +552.000 q^{87} -1306.00 q^{89} -33.9411 q^{91} +995.606 q^{93} +184.000 q^{95} -472.000 q^{97} +28.2843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{7} + 10 q^{9}+O(q^{10}) \) 2 * q + 16 * q^7 + 10 * q^9	\( 2 q + 16 q^{7} + 10 q^{9} - 16 q^{15} - 48 q^{17} - 368 q^{23} - 246 q^{25} + 352 q^{31} + 64 q^{33} - 48 q^{39} + 400 q^{41} + 416 q^{47} - 558 q^{49} - 16 q^{55} - 1472 q^{57} + 80 q^{63} + 12 q^{65} - 1872 q^{71} + 812 q^{73} + 1536 q^{79} - 1678 q^{81} + 1104 q^{87} - 2612 q^{89} + 368 q^{95} - 944 q^{97}+O(q^{100}) \) 2 * q + 16 * q^7 + 10 * q^9 - 16 * q^15 - 48 * q^17 - 368 * q^23 - 246 * q^25 + 352 * q^31 + 64 * q^33 - 48 * q^39 + 400 * q^41 + 416 * q^47 - 558 * q^49 - 16 * q^55 - 1472 * q^57 + 80 * q^63 + 12 * q^65 - 1872 * q^71 + 812 * q^73 + 1536 * q^79 - 1678 * q^81 + 1104 * q^87 - 2612 * q^89 + 368 * q^95 - 944 * q^97

Introduction

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Modular forms

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