LMFDB - Embedded newform 1024.3.d.k.511.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,3,Mod(511,1024)]

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([1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1024.511");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1024.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$27.9019790705$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.1
Root			$1.16947 - 0.795191i$ of defining polynomial
Character	$\chi$	$=$	1024.511
Dual form			1024.3.d.k.511.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.59498 q^{3} -0.0829198i q^{5} -4.61555i q^{7} +12.1138 q^{9} +O(q^{10})$	$q-4.59498 q^{3} -0.0829198i q^{5} -4.61555i q^{7} +12.1138 q^{9} -7.58925 q^{11} -15.6344i q^{13} +0.381015i q^{15} +12.8793 q^{17} -3.72446 q^{19} +21.2083i q^{21} +16.3810i q^{23} +24.9931 q^{25} -14.3080 q^{27} -36.8427i q^{29} -20.2345i q^{31} +34.8724 q^{33} -0.382720 q^{35} +58.3828i q^{37} +71.8398i q^{39} +3.29640 q^{41} -1.11292 q^{43} -1.00448i q^{45} -79.7517i q^{47} +27.6967 q^{49} -59.1801 q^{51} -1.50200i q^{53} +0.629299i q^{55} +17.1138 q^{57} -45.9850 q^{59} +21.6229i q^{61} -55.9119i q^{63} -1.29640 q^{65} -84.9421 q^{67} -75.2704i q^{69} -56.3535i q^{71} -9.70663 q^{73} -114.843 q^{75} +35.0285i q^{77} +84.4278i q^{79} -43.2796 q^{81} -37.8412 q^{83} -1.06795i q^{85} +169.292i q^{87} -115.555 q^{89} -72.1613 q^{91} +92.9772i q^{93} +0.308832i q^{95} -146.245 q^{97} -91.9348 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^9	$ 12 q + 12 q^{9} + 8 q^{17} + 20 q^{25} - 8 q^{33} + 92 q^{49} + 72 q^{57} + 24 q^{65} + 96 q^{73} - 172 q^{81} - 160 q^{89} - 8 q^{97}+O(q^{100}) $ 12 * q + 12 * q^9 + 8 * q^17 + 20 * q^25 - 8 * q^33 + 92 * q^49 + 72 * q^57 + 24 * q^65 + 96 * q^73 - 172 * q^81 - 160 * q^89 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$-1$	$-1$

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
$3$	−4.59498	−1.53166	−0.765830	−	0.643043i	$-0.777671\pi$
$3$	−4.59498	−1.53166	−0.765830	+	0.643043i	$0.777671\pi$
$4$	0	0
$5$	− 0.0829198i	− 0.0165840i	−0.999966	−	0.00829198i	$-0.997361\pi$
$5$	− 0.0829198i	− 0.0165840i	0.999966	−	0.00829198i	$-0.00263945\pi$
$6$	0	0
$7$	− 4.61555i	− 0.659364i	−0.944092	−	0.329682i	$-0.893058\pi$
$7$	− 4.61555i	− 0.659364i	0.944092	−	0.329682i	$-0.106942\pi$
$8$	0	0
$9$	12.1138	1.34598
$10$	0	0
$11$	−7.58925	−0.689931	−0.344966	−	0.938615i	$-0.612109\pi$
$11$	−7.58925	−0.689931	−0.344966	+	0.938615i	$0.612109\pi$
$12$	0	0
$13$	− 15.6344i	− 1.20265i	−0.799006	−	0.601323i	$-0.794640\pi$
$13$	− 15.6344i	− 1.20265i	0.799006	−	0.601323i	$-0.205360\pi$
$14$	0	0
$15$	0.381015i	0.0254010i
$16$	0	0
$17$	12.8793	0.757606	0.378803	−	0.925477i	$-0.376336\pi$
$17$	12.8793	0.757606	0.378803	+	0.925477i	$0.376336\pi$
$18$	0	0
$19$	−3.72446	−0.196024	−0.0980122	−	0.995185i	$-0.531248\pi$
$19$	−3.72446	−0.196024	−0.0980122	+	0.995185i	$0.531248\pi$
$20$	0	0
$21$	21.2083i	1.00992i
$22$	0	0
$23$	16.3810i	0.712218i	0.934444	+	0.356109i	$0.115897\pi$
$23$	16.3810i	0.712218i	−0.934444	+	0.356109i	$0.884103\pi$
$24$	0	0
$25$	24.9931	0.999725
$26$	0	0
$27$	−14.3080	−0.529925
$28$	0	0
$29$	− 36.8427i	− 1.27044i	−0.772331	−	0.635220i	$-0.780910\pi$
$29$	− 36.8427i	− 1.27044i	0.772331	−	0.635220i	$-0.219090\pi$
$30$	0	0
$31$	− 20.2345i	− 0.652727i	−0.945244	−	0.326363i	$-0.894177\pi$
$31$	− 20.2345i	− 0.652727i	0.945244	−	0.326363i	$-0.105823\pi$
$32$	0	0
$33$	34.8724	1.05674
$34$	0	0
$35$	−0.382720	−0.0109349
$36$	0	0
$37$	58.3828i	1.57791i	0.614449	+	0.788956i	$0.289378\pi$
$37$	58.3828i	1.57791i	−0.614449	+	0.788956i	$0.710622\pi$
$38$	0	0
$39$	71.8398i	1.84205i
$40$	0	0
$41$	3.29640	0.0804001	0.0402000	−	0.999192i	$-0.487200\pi$
$41$	3.29640	0.0804001	0.0402000	+	0.999192i	$0.487200\pi$
$42$	0	0
$43$	−1.11292	−0.0258818	−0.0129409	−	0.999916i	$-0.504119\pi$
$43$	−1.11292	−0.0258818	−0.0129409	+	0.999916i	$0.504119\pi$
$44$	0	0
$45$	− 1.00448i	− 0.0223217i
$46$	0	0
$47$	− 79.7517i	− 1.69685i	−0.529320	−	0.848423i	$-0.677553\pi$
$47$	− 79.7517i	− 1.69685i	0.529320	−	0.848423i	$-0.322447\pi$
$48$	0	0
$49$	27.6967	0.565239
$50$	0	0
$51$	−59.1801	−1.16039
$52$	0	0
$53$	− 1.50200i	− 0.0283395i	−0.999900	−	0.0141698i	$-0.995489\pi$
$53$	− 1.50200i	− 0.0283395i	0.999900	−	0.0141698i	$-0.00451053\pi$
$54$	0	0
$55$	0.629299i	0.0114418i
$56$	0	0
$57$	17.1138	0.300243
$58$	0	0
$59$	−45.9850	−0.779407	−0.389704	−	0.920940i	$-0.627423\pi$
$59$	−45.9850	−0.779407	−0.389704	+	0.920940i	$0.627423\pi$
$60$	0	0
$61$	21.6229i	0.354474i	0.984168	+	0.177237i	$0.0567160\pi$
$61$	21.6229i	0.354474i	−0.984168	+	0.177237i	$0.943284\pi$
$62$	0	0
$63$	− 55.9119i	− 0.887491i
$64$	0	0
$65$	−1.29640	−0.0199446
$66$	0	0
$67$	−84.9421	−1.26779	−0.633896	−	0.773418i	$-0.718545\pi$
$67$	−84.9421	−1.26779	−0.633896	+	0.773418i	$0.718545\pi$
$68$	0	0
$69$	− 75.2704i	− 1.09088i
$70$	0	0
$71$	− 56.3535i	− 0.793711i	−0.917881	−	0.396856i	$-0.870101\pi$
$71$	− 56.3535i	− 0.793711i	0.917881	−	0.396856i	$-0.129899\pi$
$72$	0	0
$73$	−9.70663	−0.132968	−0.0664838	−	0.997788i	$-0.521178\pi$
$73$	−9.70663	−0.132968	−0.0664838	+	0.997788i	$0.521178\pi$
$74$	0	0
$75$	−114.843	−1.53124
$76$	0	0
$77$	35.0285i	0.454916i
$78$	0	0
$79$	84.4278i	1.06871i	0.845261	+	0.534353i	$0.179445\pi$
$79$	84.4278i	1.06871i	−0.845261	+	0.534353i	$0.820555\pi$
$80$	0	0
$81$	−43.2796	−0.534316
$82$	0	0
$83$	−37.8412	−0.455917	−0.227959	−	0.973671i	$-0.573205\pi$
$83$	−37.8412	−0.455917	−0.227959	+	0.973671i	$0.573205\pi$
$84$	0	0
$85$	− 1.06795i	− 0.0125641i
$86$	0	0
$87$	169.292i	1.94588i
$88$	0	0
$89$	−115.555	−1.29838	−0.649188	−	0.760628i	$-0.724891\pi$
$89$	−115.555	−1.29838	−0.649188	+	0.760628i	$0.724891\pi$
$90$	0	0
$91$	−72.1613	−0.792982
$92$	0	0
$93$	92.9772i	0.999755i
$94$	0	0
$95$	0.308832i	0.00325086i
$96$	0	0
$97$	−146.245	−1.50768	−0.753841	−	0.657056i	$-0.771801\pi$
$97$	−146.245	−1.50768	−0.753841	+	0.657056i	$0.771801\pi$
$98$	0	0
$99$	−91.9348	−0.928635
$100$	0	0
$101$	− 76.1631i	− 0.754090i	−0.926195	−	0.377045i	$-0.876940\pi$
$101$	− 76.1631i	− 0.754090i	0.926195	−	0.377045i	$-0.123060\pi$
$102$	0	0
$103$	158.184i	1.53577i	0.640588	+	0.767885i	$0.278691\pi$
$103$	158.184i	1.53577i	−0.640588	+	0.767885i	$0.721309\pi$
$104$	0	0
$105$	1.75859	0.0167485
$106$	0	0
$107$	−81.4600	−0.761309	−0.380654	−	0.924717i	$-0.624301\pi$
$107$	−81.4600	−0.761309	−0.380654	+	0.924717i	$0.624301\pi$
$108$	0	0
$109$	− 80.4398i	− 0.737979i	−0.929433	−	0.368990i	$-0.879704\pi$
$109$	− 80.4398i	− 0.737979i	0.929433	−	0.368990i	$-0.120296\pi$
$110$	0	0
$111$	− 268.268i	− 2.41683i
$112$	0	0
$113$	−135.731	−1.20116	−0.600580	−	0.799565i	$-0.705064\pi$
$113$	−135.731	−1.20116	−0.600580	+	0.799565i	$0.705064\pi$
$114$	0	0
$115$	1.35831	0.0118114
$116$	0	0
$117$	− 189.393i	− 1.61874i
$118$	0	0
$119$	− 59.4450i	− 0.499538i
$120$	0	0
$121$	−63.4034	−0.523995
$122$	0	0
$123$	−15.1469	−0.123146
$124$	0	0
$125$	− 4.14542i	− 0.0331634i
$126$	0	0
$127$	− 166.552i	− 1.31144i	−0.755006	−	0.655718i	$-0.772366\pi$
$127$	− 166.552i	− 1.31144i	0.755006	−	0.655718i	$-0.227634\pi$
$128$	0	0
$129$	5.11383	0.0396421
$130$	0	0
$131$	31.4729	0.240251	0.120126	−	0.992759i	$-0.461670\pi$
$131$	31.4729	0.240251	0.120126	+	0.992759i	$0.461670\pi$
$132$	0	0
$133$	17.1904i	0.129251i
$134$	0	0
$135$	1.18641i	0.00878826i
$136$	0	0
$137$	−174.890	−1.27657	−0.638285	−	0.769800i	$-0.720356\pi$
$137$	−174.890	−1.27657	−0.638285	+	0.769800i	$0.720356\pi$
$138$	0	0
$139$	141.265	1.01629	0.508146	−	0.861271i	$-0.330331\pi$
$139$	141.265	1.01629	0.508146	+	0.861271i	$0.330331\pi$
$140$	0	0
$141$	366.457i	2.59899i
$142$	0	0
$143$	118.653i	0.829744i
$144$	0	0
$145$	−3.05499	−0.0210689
$146$	0	0
$147$	−127.266	−0.865754
$148$	0	0
$149$	105.905i	0.710770i	0.934720	+	0.355385i	$0.115650\pi$
$149$	105.905i	0.710770i	−0.934720	+	0.355385i	$0.884350\pi$
$150$	0	0
$151$	70.0357i	0.463813i	0.972738	+	0.231906i	$0.0744963\pi$
$151$	70.0357i	0.463813i	−0.972738	+	0.231906i	$0.925504\pi$
$152$	0	0
$153$	156.018	1.01972
$154$	0	0
$155$	−1.67784	−0.0108248
$156$	0	0
$157$	− 41.7628i	− 0.266005i	−0.991116	−	0.133002i	$-0.957538\pi$
$157$	− 41.7628i	− 0.266005i	0.991116	−	0.133002i	$-0.0424618\pi$
$158$	0	0
$159$	6.90164i	0.0434065i
$160$	0	0
$161$	75.6074	0.469611
$162$	0	0
$163$	67.5980	0.414711	0.207356	−	0.978266i	$-0.433514\pi$
$163$	67.5980	0.414711	0.207356	+	0.978266i	$0.433514\pi$
$164$	0	0
$165$	− 2.89161i	− 0.0175249i
$166$	0	0
$167$	156.268i	0.935734i	0.883799	+	0.467867i	$0.154977\pi$
$167$	156.268i	0.935734i	−0.883799	+	0.467867i	$0.845023\pi$
$168$	0	0
$169$	−75.4347	−0.446359
$170$	0	0
$171$	−45.1175	−0.263845
$172$	0	0
$173$	− 268.847i	− 1.55403i	−0.629484	−	0.777014i	$-0.716734\pi$
$173$	− 268.847i	− 1.55403i	0.629484	−	0.777014i	$-0.283266\pi$
$174$	0	0
$175$	− 115.357i	− 0.659183i
$176$	0	0
$177$	211.300	1.19379
$178$	0	0
$179$	−76.7563	−0.428806	−0.214403	−	0.976745i	$-0.568781\pi$
$179$	−76.7563	−0.428806	−0.214403	+	0.976745i	$0.568781\pi$
$180$	0	0
$181$	− 27.9086i	− 0.154191i	−0.997024	−	0.0770955i	$-0.975435\pi$
$181$	− 27.9086i	− 0.154191i	0.997024	−	0.0770955i	$-0.0245646\pi$
$182$	0	0
$183$	− 99.3569i	− 0.542934i
$184$	0	0
$185$	4.84109	0.0261680
$186$	0	0
$187$	−97.7441	−0.522696
$188$	0	0
$189$	66.0391i	0.349413i
$190$	0	0
$191$	166.552i	0.872002i	0.899946	+	0.436001i	$0.143606\pi$
$191$	166.552i	0.872002i	−0.899946	+	0.436001i	$0.856394\pi$
$192$	0	0
$193$	2.18257	0.0113087	0.00565434	−	0.999984i	$-0.498200\pi$
$193$	2.18257	0.0113087	0.00565434	+	0.999984i	$0.498200\pi$
$194$	0	0
$195$	5.95694	0.0305484
$196$	0	0
$197$	95.3619i	0.484071i	0.970267	+	0.242035i	$0.0778150\pi$
$197$	95.3619i	0.484071i	−0.970267	+	0.242035i	$0.922185\pi$
$198$	0	0
$199$	− 222.906i	− 1.12013i	−0.828449	−	0.560065i	$-0.810776\pi$
$199$	− 222.906i	− 1.12013i	0.828449	−	0.560065i	$-0.189224\pi$
$200$	0	0
$201$	390.307	1.94183
$202$	0	0
$203$	−170.049	−0.837682
$204$	0	0
$205$	− 0.273337i	− 0.00133335i
$206$	0	0
$207$	198.437i	0.958632i
$208$	0	0
$209$	28.2659	0.135243
$210$	0	0
$211$	208.056	0.986047	0.493023	−	0.870016i	$-0.335892\pi$
$211$	208.056	0.986047	0.493023	+	0.870016i	$0.335892\pi$
$212$	0	0
$213$	258.943i	1.21570i
$214$	0	0
$215$	0.0922828i	0 0.000429223i
$216$	0	0
$217$	−93.3934	−0.430385
$218$	0	0
$219$	44.6018	0.203661
$220$	0	0
$221$	− 201.360i	− 0.911132i
$222$	0	0
$223$	60.7036i	0.272213i	0.990694	+	0.136107i	$0.0434590\pi$
$223$	60.7036i	0.272213i	−0.990694	+	0.136107i	$0.956541\pi$
$224$	0	0
$225$	302.762	1.34561
$226$	0	0
$227$	−318.942	−1.40503	−0.702514	−	0.711670i	$-0.747939\pi$
$227$	−318.942	−1.40503	−0.702514	+	0.711670i	$0.747939\pi$
$228$	0	0
$229$	322.152i	1.40678i	0.710804	+	0.703390i	$0.248331\pi$
$229$	322.152i	1.40678i	−0.710804	+	0.703390i	$0.751669\pi$
$230$	0	0
$231$	− 160.955i	− 0.696776i
$232$	0	0
$233$	121.053	0.519540	0.259770	−	0.965671i	$-0.416353\pi$
$233$	121.053	0.519540	0.259770	+	0.965671i	$0.416353\pi$
$234$	0	0
$235$	−6.61300	−0.0281404
$236$	0	0
$237$	− 387.944i	− 1.63689i
$238$	0	0
$239$	221.393i	0.926332i	0.886271	+	0.463166i	$0.153287\pi$
$239$	221.393i	0.926332i	−0.886271	+	0.463166i	$0.846713\pi$
$240$	0	0
$241$	−84.2667	−0.349654	−0.174827	−	0.984599i	$-0.555937\pi$
$241$	−84.2667	−0.349654	−0.174827	+	0.984599i	$0.555937\pi$
$242$	0	0
$243$	327.641	1.34832
$244$	0	0
$245$	− 2.29661i	− 0.00937391i
$246$	0	0
$247$	58.2298i	0.235748i
$248$	0	0
$249$	173.879	0.698310
$250$	0	0
$251$	249.771	0.995105	0.497553	−	0.867434i	$-0.334232\pi$
$251$	249.771	0.995105	0.497553	+	0.867434i	$0.334232\pi$
$252$	0	0
$253$	− 124.320i	− 0.491382i
$254$	0	0
$255$	4.90720i	0.0192439i
$256$	0	0
$257$	−163.001	−0.634244	−0.317122	−	0.948385i	$-0.602717\pi$
$257$	−163.001	−0.634244	−0.317122	+	0.948385i	$0.602717\pi$
$258$	0	0
$259$	269.468	1.04042
$260$	0	0
$261$	− 446.307i	− 1.70999i
$262$	0	0
$263$	− 175.001i	− 0.665404i	−0.943032	−	0.332702i	$-0.892040\pi$
$263$	− 175.001i	− 0.665404i	0.943032	−	0.332702i	$-0.107960\pi$
$264$	0	0
$265$	−0.124545	−0.000469982 0
$266$	0	0
$267$	530.975	1.98867
$268$	0	0
$269$	− 42.0713i	− 0.156399i	−0.996938	−	0.0781994i	$-0.975083\pi$
$269$	− 42.0713i	− 0.156399i	0.996938	−	0.0781994i	$-0.0249171\pi$
$270$	0	0
$271$	− 275.891i	− 1.01805i	−0.860753	−	0.509024i	$-0.830007\pi$
$271$	− 275.891i	− 1.01805i	0.860753	−	0.509024i	$-0.169993\pi$
$272$	0	0
$273$	331.580	1.21458
$274$	0	0
$275$	−189.679	−0.689742
$276$	0	0
$277$	393.629i	1.42104i	0.703676	+	0.710521i	$0.251541\pi$
$277$	393.629i	1.42104i	−0.703676	+	0.710521i	$0.748459\pi$
$278$	0	0
$279$	− 245.118i	− 0.878558i
$280$	0	0
$281$	202.356	0.720128	0.360064	−	0.932928i	$-0.382755\pi$
$281$	202.356	0.720128	0.360064	+	0.932928i	$0.382755\pi$
$282$	0	0
$283$	413.313	1.46047	0.730235	−	0.683196i	$-0.239411\pi$
$283$	413.313	1.46047	0.730235	+	0.683196i	$0.239411\pi$
$284$	0	0
$285$	− 1.41908i	− 0.00497921i
$286$	0	0
$287$	− 15.2147i	− 0.0530129i
$288$	0	0
$289$	−123.124	−0.426034
$290$	0	0
$291$	671.994	2.30926
$292$	0	0
$293$	468.345i	1.59845i	0.601034	+	0.799223i	$0.294756\pi$
$293$	468.345i	1.59845i	−0.601034	+	0.799223i	$0.705244\pi$
$294$	0	0
$295$	3.81307i	0.0129257i
$296$	0	0
$297$	108.587	0.365612
$298$	0	0
$299$	256.107	0.856547
$300$	0	0
$301$	5.13672i	0.0170655i
$302$	0	0
$303$	349.968i	1.15501i
$304$	0	0
$305$	1.79297	0.00587859
$306$	0	0
$307$	−33.5893	−0.109412	−0.0547058	−	0.998503i	$-0.517422\pi$
$307$	−33.5893	−0.109412	−0.0547058	+	0.998503i	$0.517422\pi$
$308$	0	0
$309$	− 726.853i	− 2.35228i
$310$	0	0
$311$	− 157.757i	− 0.507258i	−0.967302	−	0.253629i	$-0.918376\pi$
$311$	− 157.757i	− 0.507258i	0.967302	−	0.253629i	$-0.0816243\pi$
$312$	0	0
$313$	58.5936	0.187200	0.0936000	−	0.995610i	$-0.470163\pi$
$313$	58.5936	0.187200	0.0936000	+	0.995610i	$0.470163\pi$
$314$	0	0
$315$	−4.63621	−0.0147181
$316$	0	0
$317$	38.1895i	0.120472i	0.998184	+	0.0602358i	$0.0191853\pi$
$317$	38.1895i	0.120472i	−0.998184	+	0.0602358i	$0.980815\pi$
$318$	0	0
$319$	279.609i	0.876516i
$320$	0	0
$321$	374.307	1.16607
$322$	0	0
$323$	−47.9685	−0.148509
$324$	0	0
$325$	− 390.753i	− 1.20232i
$326$	0	0
$327$	369.619i	1.13033i
$328$	0	0
$329$	−368.098	−1.11884
$330$	0	0
$331$	−257.662	−0.778435	−0.389217	−	0.921146i	$-0.627255\pi$
$331$	−257.662	−0.778435	−0.389217	+	0.921146i	$0.627255\pi$
$332$	0	0
$333$	707.239i	2.12384i
$334$	0	0
$335$	7.04338i	0.0210250i
$336$	0	0
$337$	−510.137	−1.51376	−0.756881	−	0.653553i	$-0.773278\pi$
$337$	−510.137	−1.51376	−0.756881	+	0.653553i	$0.773278\pi$
$338$	0	0
$339$	623.681	1.83977
$340$	0	0
$341$	153.565i	0.450337i
$342$	0	0
$343$	− 353.997i	− 1.03206i
$344$	0	0
$345$	−6.24141	−0.0180910
$346$	0	0
$347$	−611.809	−1.76314	−0.881569	−	0.472055i	$-0.843512\pi$
$347$	−611.809	−1.76314	−0.881569	+	0.472055i	$0.843512\pi$
$348$	0	0
$349$	− 210.490i	− 0.603122i	−0.953447	−	0.301561i	$-0.902492\pi$
$349$	− 210.490i	− 0.603122i	0.953447	−	0.301561i	$-0.0975077\pi$
$350$	0	0
$351$	223.697i	0.637313i
$352$	0	0
$353$	−268.587	−0.760869	−0.380434	−	0.924808i	$-0.624226\pi$
$353$	−268.587	−0.760869	−0.380434	+	0.924808i	$0.624226\pi$
$354$	0	0
$355$	−4.67282	−0.0131629
$356$	0	0
$357$	273.149i	0.765122i
$358$	0	0
$359$	− 628.520i	− 1.75075i	−0.483442	−	0.875376i	$-0.660614\pi$
$359$	− 628.520i	− 1.75075i	0.483442	−	0.875376i	$-0.339386\pi$
$360$	0	0
$361$	−347.128	−0.961574
$362$	0	0
$363$	291.337	0.802581
$364$	0	0
$365$	0.804872i	0.00220513i
$366$	0	0
$367$	396.386i	1.08007i	0.841643	+	0.540035i	$0.181589\pi$
$367$	396.386i	1.08007i	−0.841643	+	0.540035i	$0.818411\pi$
$368$	0	0
$369$	39.9320	0.108217
$370$	0	0
$371$	−6.93253	−0.0186861
$372$	0	0
$373$	− 189.894i	− 0.509099i	−0.967060	−	0.254549i	$-0.918073\pi$
$373$	− 189.894i	− 0.509099i	0.967060	−	0.254549i	$-0.0819271\pi$
$374$	0	0
$375$	19.0481i	0.0507950i
$376$	0	0
$377$	−576.015	−1.52789
$378$	0	0
$379$	495.668	1.30783	0.653916	−	0.756567i	$-0.273125\pi$
$379$	495.668	1.30783	0.653916	+	0.756567i	$0.273125\pi$
$380$	0	0
$381$	765.305i	2.00867i
$382$	0	0
$383$	403.778i	1.05425i	0.849787	+	0.527126i	$0.176730\pi$
$383$	403.778i	1.05425i	−0.849787	+	0.527126i	$0.823270\pi$
$384$	0	0
$385$	2.90456	0.00754431
$386$	0	0
$387$	−13.4817	−0.0348364
$388$	0	0
$389$	− 177.215i	− 0.455566i	−0.973712	−	0.227783i	$-0.926852\pi$
$389$	− 177.215i	− 0.455566i	0.973712	−	0.227783i	$-0.0731476\pi$
$390$	0	0
$391$	210.976i	0.539580i
$392$	0	0
$393$	−144.617	−0.367983
$394$	0	0
$395$	7.00074	0.0177234
$396$	0	0
$397$	98.8143i	0.248902i	0.992226	+	0.124451i	$0.0397170\pi$
$397$	98.8143i	0.248902i	−0.992226	+	0.124451i	$0.960283\pi$
$398$	0	0
$399$	− 78.9897i	− 0.197969i
$400$	0	0
$401$	−11.3010	−0.0281821	−0.0140911	−	0.999901i	$-0.504485\pi$
$401$	−11.3010	−0.0281821	−0.0140911	+	0.999901i	$0.504485\pi$
$402$	0	0
$403$	−316.355	−0.785000
$404$	0	0
$405$	3.58874i	0.00886108i
$406$	0	0
$407$	− 443.081i	− 1.08865i
$408$	0	0
$409$	−614.595	−1.50268	−0.751339	−	0.659917i	$-0.770592\pi$
$409$	−614.595	−1.50268	−0.751339	+	0.659917i	$0.770592\pi$
$410$	0	0
$411$	803.616	1.95527
$412$	0	0
$413$	212.246i	0.513913i
$414$	0	0
$415$	3.13778i	0.00756092i
$416$	0	0
$417$	−649.108	−1.55661
$418$	0	0
$419$	111.312	0.265660	0.132830	−	0.991139i	$-0.457594\pi$
$419$	111.312	0.265660	0.132830	+	0.991139i	$0.457594\pi$
$420$	0	0
$421$	− 529.790i	− 1.25841i	−0.777240	−	0.629204i	$-0.783381\pi$
$421$	− 529.790i	− 1.25841i	0.777240	−	0.629204i	$-0.216619\pi$
$422$	0	0
$423$	− 966.099i	− 2.28392i
$424$	0	0
$425$	321.894	0.757397
$426$	0	0
$427$	99.8017	0.233728
$428$	0	0
$429$	− 545.210i	− 1.27088i
$430$	0	0
$431$	616.593i	1.43061i	0.698813	+	0.715305i	$0.253712\pi$
$431$	616.593i	1.43061i	−0.698813	+	0.715305i	$0.746288\pi$
$432$	0	0
$433$	−219.246	−0.506342	−0.253171	−	0.967422i	$-0.581474\pi$
$433$	−219.246	−0.506342	−0.253171	+	0.967422i	$0.581474\pi$
$434$	0	0
$435$	14.0376	0.0322704
$436$	0	0
$437$	− 61.0105i	− 0.139612i
$438$	0	0
$439$	575.292i	1.31046i	0.755429	+	0.655231i	$0.227429\pi$
$439$	575.292i	1.31046i	−0.755429	+	0.655231i	$0.772571\pi$
$440$	0	0
$441$	335.513	0.760801
$442$	0	0
$443$	−525.939	−1.18722	−0.593610	−	0.804753i	$-0.702298\pi$
$443$	−525.939	−1.18722	−0.593610	+	0.804753i	$0.702298\pi$
$444$	0	0
$445$	9.58183i	0.0215322i
$446$	0	0
$447$	− 486.630i	− 1.08866i
$448$	0	0
$449$	−498.135	−1.10943	−0.554716	−	0.832040i	$-0.687173\pi$
$449$	−498.135	−1.10943	−0.554716	+	0.832040i	$0.687173\pi$
$450$	0	0
$451$	−25.0172	−0.0554705
$452$	0	0
$453$	− 321.812i	− 0.710403i
$454$	0	0
$455$	5.98361i	0.0131508i
$456$	0	0
$457$	−61.1711	−0.133854	−0.0669268	−	0.997758i	$-0.521319\pi$
$457$	−61.1711	−0.133854	−0.0669268	+	0.997758i	$0.521319\pi$
$458$	0	0
$459$	−184.277	−0.401474
$460$	0	0
$461$	626.756i	1.35956i	0.733418	+	0.679778i	$0.237924\pi$
$461$	626.756i	1.35956i	−0.733418	+	0.679778i	$0.762076\pi$
$462$	0	0
$463$	− 706.883i	− 1.52675i	−0.645958	−	0.763373i	$-0.723542\pi$
$463$	− 706.883i	− 1.52675i	0.645958	−	0.763373i	$-0.276458\pi$
$464$	0	0
$465$	7.70966	0.0165799
$466$	0	0
$467$	575.383	1.23208	0.616041	−	0.787714i	$-0.288735\pi$
$467$	575.383	1.23208	0.616041	+	0.787714i	$0.288735\pi$
$468$	0	0
$469$	392.054i	0.835937i
$470$	0	0
$471$	191.899i	0.407429i
$472$	0	0
$473$	8.44620	0.0178567
$474$	0	0
$475$	−93.0860	−0.195970
$476$	0	0
$477$	− 18.1949i	− 0.0381445i
$478$	0	0
$479$	− 133.063i	− 0.277793i	−0.990307	−	0.138896i	$-0.955645\pi$
$479$	− 133.063i	− 0.277793i	0.990307	−	0.138896i	$-0.0443555\pi$
$480$	0	0
$481$	912.780	1.89767
$482$	0	0
$483$	−347.414	−0.719284
$484$	0	0
$485$	12.1266i	0.0250034i
$486$	0	0
$487$	208.075i	0.427259i	0.976915	+	0.213629i	$0.0685285\pi$
$487$	208.075i	0.427259i	−0.976915	+	0.213629i	$0.931471\pi$
$488$	0	0
$489$	−310.611	−0.635197
$490$	0	0
$491$	−139.919	−0.284967	−0.142483	−	0.989797i	$-0.545509\pi$
$491$	−139.919	−0.284967	−0.142483	+	0.989797i	$0.545509\pi$
$492$	0	0
$493$	− 474.509i	− 0.962492i
$494$	0	0
$495$	7.62322i	0.0154004i
$496$	0	0
$497$	−260.102	−0.523345
$498$	0	0
$499$	405.987	0.813602	0.406801	−	0.913517i	$-0.366644\pi$
$499$	405.987	0.813602	0.406801	+	0.913517i	$0.366644\pi$
$500$	0	0
$501$	− 718.046i	− 1.43323i
$502$	0	0
$503$	− 78.7359i	− 0.156533i	−0.996932	−	0.0782663i	$-0.975062\pi$
$503$	− 78.7359i	− 0.156533i	0.996932	−	0.0782663i	$-0.0249384\pi$
$504$	0	0
$505$	−6.31543	−0.0125058
$506$	0	0
$507$	346.621	0.683670
$508$	0	0
$509$	− 342.914i	− 0.673701i	−0.941558	−	0.336850i	$-0.890638\pi$
$509$	− 342.914i	− 0.673701i	0.941558	−	0.336850i	$-0.109362\pi$
$510$	0	0
$511$	44.8014i	0.0876740i
$512$	0	0
$513$	53.2895	0.103878
$514$	0	0
$515$	13.1166	0.0254691
$516$	0	0
$517$	605.255i	1.17071i
$518$	0	0
$519$	1235.34i	2.38024i
$520$	0	0
$521$	561.306	1.07736	0.538681	−	0.842510i	$-0.318923\pi$
$521$	561.306	1.07736	0.538681	+	0.842510i	$0.318923\pi$
$522$	0	0
$523$	560.243	1.07121	0.535605	−	0.844469i	$-0.320084\pi$
$523$	560.243	1.07121	0.535605	+	0.844469i	$0.320084\pi$
$524$	0	0
$525$	530.063i	1.00964i
$526$	0	0
$527$	− 260.607i	− 0.494510i
$528$	0	0
$529$	260.662	0.492745
$530$	0	0
$531$	−557.055	−1.04907
$532$	0	0
$533$	− 51.5373i	− 0.0966929i
$534$	0	0
$535$	6.75465i	0.0126255i
$536$	0	0
$537$	352.694	0.656785
$538$	0	0
$539$	−210.197	−0.389976
$540$	0	0
$541$	− 31.9227i	− 0.0590069i	−0.999565	−	0.0295035i	$-0.990607\pi$
$541$	− 31.9227i	− 0.0590069i	0.999565	−	0.0295035i	$-0.00939260\pi$
$542$	0	0
$543$	128.239i	0.236168i
$544$	0	0
$545$	−6.67005	−0.0122386
$546$	0	0
$547$	850.839	1.55547	0.777733	−	0.628595i	$-0.216370\pi$
$547$	850.839	1.55547	0.777733	+	0.628595i	$0.216370\pi$
$548$	0	0
$549$	261.937i	0.477116i
$550$	0	0
$551$	137.219i	0.249037i
$552$	0	0
$553$	389.681	0.704666
$554$	0	0
$555$	−22.2447	−0.0400805
$556$	0	0
$557$	− 711.183i	− 1.27681i	−0.769701	−	0.638405i	$-0.779594\pi$
$557$	− 711.183i	− 1.27681i	0.769701	−	0.638405i	$-0.220406\pi$
$558$	0	0
$559$	17.3998i	0.0311266i
$560$	0	0
$561$	449.132	0.800592
$562$	0	0
$563$	−927.685	−1.64775	−0.823876	−	0.566770i	$-0.808193\pi$
$563$	−927.685	−1.64775	−0.823876	+	0.566770i	$0.808193\pi$
$564$	0	0
$565$	11.2548i	0.0199200i
$566$	0	0
$567$	199.759i	0.352309i
$568$	0	0
$569$	649.911	1.14220	0.571099	−	0.820881i	$-0.306517\pi$
$569$	649.911	1.14220	0.571099	+	0.820881i	$0.306517\pi$
$570$	0	0
$571$	−381.438	−0.668018	−0.334009	−	0.942570i	$-0.608402\pi$
$571$	−381.438	−0.668018	−0.334009	+	0.942570i	$0.608402\pi$
$572$	0	0
$573$	− 765.305i	− 1.33561i
$574$	0	0
$575$	409.413i	0.712022i
$576$	0	0
$577$	−142.675	−0.247271	−0.123635	−	0.992328i	$-0.539455\pi$
$577$	−142.675	−0.247271	−0.123635	+	0.992328i	$0.539455\pi$
$578$	0	0
$579$	−10.0289	−0.0173210
$580$	0	0
$581$	174.658i	0.300616i
$582$	0	0
$583$	11.3990i	0.0195523i
$584$	0	0
$585$	−15.7044	−0.0268451
$586$	0	0
$587$	−972.803	−1.65725	−0.828623	−	0.559807i	$-0.810875\pi$
$587$	−972.803	−1.65725	−0.828623	+	0.559807i	$0.810875\pi$
$588$	0	0
$589$	75.3628i	0.127950i
$590$	0	0
$591$	− 438.186i	− 0.741431i
$592$	0	0
$593$	58.8678	0.0992711	0.0496355	−	0.998767i	$-0.484194\pi$
$593$	58.8678	0.0992711	0.0496355	+	0.998767i	$0.484194\pi$
$594$	0	0
$595$	−4.92917	−0.00828432
$596$	0	0
$597$	1024.25i	1.71566i
$598$	0	0
$599$	− 670.449i	− 1.11928i	−0.828735	−	0.559641i	$-0.810939\pi$
$599$	− 670.449i	− 1.11928i	0.828735	−	0.559641i	$-0.189061\pi$
$600$	0	0
$601$	910.721	1.51534	0.757671	−	0.652636i	$-0.226337\pi$
$601$	910.721	1.51534	0.757671	+	0.652636i	$0.226337\pi$
$602$	0	0
$603$	−1028.97	−1.70642
$604$	0	0
$605$	5.25739i	0.00868991i
$606$	0	0
$607$	− 761.794i	− 1.25501i	−0.778611	−	0.627507i	$-0.784075\pi$
$607$	− 761.794i	− 1.25501i	0.778611	−	0.627507i	$-0.215925\pi$
$608$	0	0
$609$	781.374	1.28304
$610$	0	0
$611$	−1246.87	−2.04071
$612$	0	0
$613$	386.642i	0.630738i	0.948969	+	0.315369i	$0.102128\pi$
$613$	386.642i	0.630738i	−0.948969	+	0.315369i	$0.897872\pi$
$614$	0	0
$615$	1.25598i	0.00204224i
$616$	0	0
$617$	−1088.68	−1.76448	−0.882238	−	0.470804i	$-0.843964\pi$
$617$	−1088.68	−1.76448	−0.882238	+	0.470804i	$0.843964\pi$
$618$	0	0
$619$	−182.857	−0.295407	−0.147703	−	0.989032i	$-0.547188\pi$
$619$	−182.857	−0.295407	−0.147703	+	0.989032i	$0.547188\pi$
$620$	0	0
$621$	− 234.379i	− 0.377422i
$622$	0	0
$623$	533.351i	0.856102i
$624$	0	0
$625$	624.484	0.999175
$626$	0	0
$627$	−129.881	−0.207147
$628$	0	0
$629$	751.929i	1.19544i
$630$	0	0
$631$	455.029i	0.721123i	0.932735	+	0.360562i	$0.117415\pi$
$631$	455.029i	0.721123i	−0.932735	+	0.360562i	$0.882585\pi$
$632$	0	0
$633$	−956.012	−1.51029
$634$	0	0
$635$	−13.8105	−0.0217488
$636$	0	0
$637$	− 433.022i	− 0.679783i
$638$	0	0
$639$	− 682.657i	− 1.06832i
$640$	0	0
$641$	798.626	1.24591	0.622953	−	0.782259i	$-0.285933\pi$
$641$	798.626	1.24591	0.622953	+	0.782259i	$0.285933\pi$
$642$	0	0
$643$	−432.350	−0.672395	−0.336198	−	0.941791i	$-0.609141\pi$
$643$	−432.350	−0.672395	−0.336198	+	0.941791i	$0.609141\pi$
$644$	0	0
$645$	− 0.424038i	0 0.000657423i
$646$	0	0
$647$	− 1161.90i	− 1.79583i	−0.440167	−	0.897916i	$-0.645081\pi$
$647$	− 1161.90i	− 1.79583i	0.440167	−	0.897916i	$-0.354919\pi$
$648$	0	0
$649$	348.992	0.537738
$650$	0	0
$651$	429.141	0.659203
$652$	0	0
$653$	− 109.660i	− 0.167932i	−0.996469	−	0.0839660i	$-0.973241\pi$
$653$	− 109.660i	− 0.167932i	0.996469	−	0.0839660i	$-0.0267587\pi$
$654$	0	0
$655$	− 2.60973i	− 0.00398431i
$656$	0	0
$657$	−117.584	−0.178972
$658$	0	0
$659$	−1183.69	−1.79619	−0.898093	−	0.439805i	$-0.855048\pi$
$659$	−1183.69	−1.79619	−0.898093	+	0.439805i	$0.855048\pi$
$660$	0	0
$661$	− 171.220i	− 0.259032i	−0.991577	−	0.129516i	$-0.958658\pi$
$661$	− 171.220i	− 0.259032i	0.991577	−	0.129516i	$-0.0413424\pi$
$662$	0	0
$663$	925.246i	1.39554i
$664$	0	0
$665$	1.42543	0.00214350
$666$	0	0
$667$	603.522	0.904830
$668$	0	0
$669$	− 278.932i	− 0.416938i
$670$	0	0
$671$	− 164.102i	− 0.244563i
$672$	0	0
$673$	−954.371	−1.41808	−0.709042	−	0.705166i	$-0.750872\pi$
$673$	−954.371	−1.41808	−0.709042	+	0.705166i	$0.750872\pi$
$674$	0	0
$675$	−357.601	−0.529779
$676$	0	0
$677$	347.155i	0.512784i	0.966573	+	0.256392i	$0.0825337\pi$
$677$	347.155i	0.512784i	−0.966573	+	0.256392i	$0.917466\pi$
$678$	0	0
$679$	675.002i	0.994112i
$680$	0	0
$681$	1465.53	2.15203
$682$	0	0
$683$	−1288.73	−1.88687	−0.943435	−	0.331557i	$-0.892426\pi$
$683$	−1288.73	−1.88687	−0.943435	+	0.331557i	$0.892426\pi$
$684$	0	0
$685$	14.5018i	0.0211706i
$686$	0	0
$687$	− 1480.28i	− 2.15471i
$688$	0	0
$689$	−23.4828	−0.0340824
$690$	0	0
$691$	673.385	0.974508	0.487254	−	0.873260i	$-0.337999\pi$
$691$	673.385	0.974508	0.487254	+	0.873260i	$0.337999\pi$
$692$	0	0
$693$	424.330i	0.612308i
$694$	0	0
$695$	− 11.7136i	− 0.0168541i
$696$	0	0
$697$	42.4553	0.0609115
$698$	0	0
$699$	−556.235	−0.795758
$700$	0	0
$701$	1322.24i	1.88622i	0.332479	+	0.943111i	$0.392115\pi$
$701$	1322.24i	1.88622i	−0.332479	+	0.943111i	$0.607885\pi$
$702$	0	0
$703$	− 217.444i	− 0.309309i
$704$	0	0
$705$	30.3866	0.0431015
$706$	0	0
$707$	−351.534	−0.497220
$708$	0	0
$709$	8.33746i	0.0117595i	0.999983	+	0.00587973i	$0.00187159\pi$
$709$	8.33746i	0.0117595i	−0.999983	+	0.00587973i	$0.998128\pi$
$710$	0	0
$711$	1022.74i	1.43846i
$712$	0	0
$713$	331.462	0.464884
$714$	0	0
$715$	9.83871	0.0137604
$716$	0	0
$717$	− 1017.30i	− 1.41883i
$718$	0	0
$719$	19.5965i	0.0272552i	0.999907	+	0.0136276i	$0.00433793\pi$
$719$	19.5965i	0.0272552i	−0.999907	+	0.0136276i	$0.995662\pi$
$720$	0	0
$721$	730.107	1.01263
$722$	0	0
$723$	387.204	0.535551
$724$	0	0
$725$	− 920.815i	− 1.27009i
$726$	0	0
$727$	741.995i	1.02063i	0.859989	+	0.510313i	$0.170470\pi$
$727$	741.995i	1.02063i	−0.859989	+	0.510313i	$0.829530\pi$
$728$	0	0
$729$	−1115.99	−1.53084
$730$	0	0
$731$	−14.3336	−0.0196082
$732$	0	0
$733$	493.938i	0.673858i	0.941530	+	0.336929i	$0.109388\pi$
$733$	493.938i	0.673858i	−0.941530	+	0.336929i	$0.890612\pi$
$734$	0	0
$735$	10.5529i	0.0143576i
$736$	0	0
$737$	644.646	0.874690
$738$	0	0
$739$	−507.607	−0.686883	−0.343442	−	0.939174i	$-0.611593\pi$
$739$	−507.607	−0.686883	−0.343442	+	0.939174i	$0.611593\pi$
$740$	0	0
$741$	− 267.565i	− 0.361086i
$742$	0	0
$743$	856.214i	1.15237i	0.817318	+	0.576187i	$0.195460\pi$
$743$	856.214i	1.15237i	−0.817318	+	0.576187i	$0.804540\pi$
$744$	0	0
$745$	8.78160	0.0117874
$746$	0	0
$747$	−458.401	−0.613656
$748$	0	0
$749$	375.983i	0.501979i
$750$	0	0
$751$	− 442.218i	− 0.588839i	−0.955676	−	0.294420i	$-0.904874\pi$
$751$	− 442.218i	− 0.588839i	0.955676	−	0.294420i	$-0.0951264\pi$
$752$	0	0
$753$	−1147.69	−1.52416
$754$	0	0
$755$	5.80735	0.00769185
$756$	0	0
$757$	− 691.831i	− 0.913911i	−0.889490	−	0.456955i	$-0.848940\pi$
$757$	− 691.831i	− 0.913911i	0.889490	−	0.456955i	$-0.151060\pi$
$758$	0	0
$759$	571.246i	0.752629i
$760$	0	0
$761$	404.015	0.530899	0.265450	−	0.964125i	$-0.414480\pi$
$761$	404.015	0.530899	0.265450	+	0.964125i	$0.414480\pi$
$762$	0	0
$763$	−371.274	−0.486597
$764$	0	0
$765$	− 12.9369i	− 0.0169110i
$766$	0	0
$767$	718.949i	0.937352i
$768$	0	0
$769$	−387.336	−0.503688	−0.251844	−	0.967768i	$-0.581037\pi$
$769$	−387.336	−0.503688	−0.251844	+	0.967768i	$0.581037\pi$
$770$	0	0
$771$	748.985	0.971446
$772$	0	0
$773$	− 1358.21i	− 1.75706i	−0.477689	−	0.878529i	$-0.658526\pi$
$773$	− 1358.21i	− 1.75706i	0.477689	−	0.878529i	$-0.341474\pi$
$774$	0	0
$775$	− 505.724i	− 0.652547i
$776$	0	0
$777$	−1238.20	−1.59357
$778$	0	0
$779$	−12.2773	−0.0157604
$780$	0	0
$781$	427.681i	0.547606i
$782$	0	0
$783$	527.145i	0.673238i
$784$	0	0
$785$	−3.46296	−0.00441141
$786$	0	0
$787$	421.965	0.536169	0.268085	−	0.963395i	$-0.413609\pi$
$787$	421.965	0.536169	0.268085	+	0.963395i	$0.413609\pi$
$788$	0	0
$789$	804.127i	1.01917i
$790$	0	0
$791$	626.473i	0.792002i
$792$	0	0
$793$	338.062	0.426308
$794$	0	0
$795$	0.572282	0.000719852 0
$796$	0	0
$797$	− 1230.50i	− 1.54391i	−0.635676	−	0.771956i	$-0.719279\pi$
$797$	− 1230.50i	− 1.54391i	0.635676	−	0.771956i	$-0.280721\pi$
$798$	0	0
$799$	− 1027.15i	− 1.28554i
$800$	0	0
$801$	−1399.82	−1.74759
$802$	0	0
$803$	73.6660	0.0917385
$804$	0	0
$805$	− 6.26935i	− 0.00778801i
$806$	0	0
$807$	193.317i	0.239550i
$808$	0	0
$809$	107.642	0.133055	0.0665277	−	0.997785i	$-0.478808\pi$
$809$	107.642	0.133055	0.0665277	+	0.997785i	$0.478808\pi$
$810$	0	0
$811$	1173.43	1.44689	0.723445	−	0.690382i	$-0.242557\pi$
$811$	1173.43	1.44689	0.723445	+	0.690382i	$0.242557\pi$
$812$	0	0
$813$	1267.71i	1.55930i
$814$	0	0
$815$	− 5.60521i	− 0.00687756i
$816$	0	0
$817$	4.14502	0.00507346
$818$	0	0
$819$	−874.150	−1.06734
$820$	0	0
$821$	− 716.863i	− 0.873158i	−0.899666	−	0.436579i	$-0.856190\pi$
$821$	− 716.863i	− 0.873158i	0.899666	−	0.436579i	$-0.143810\pi$
$822$	0	0
$823$	− 927.304i	− 1.12674i	−0.826206	−	0.563368i	$-0.809505\pi$
$823$	− 927.304i	− 1.12674i	0.826206	−	0.563368i	$-0.190495\pi$
$824$	0	0
$825$	871.571	1.05645
$826$	0	0
$827$	27.5363	0.0332966	0.0166483	−	0.999861i	$-0.494700\pi$
$827$	27.5363	0.0332966	0.0166483	+	0.999861i	$0.494700\pi$
$828$	0	0
$829$	578.454i	0.697773i	0.937165	+	0.348886i	$0.113440\pi$
$829$	578.454i	0.697773i	−0.937165	+	0.348886i	$0.886560\pi$
$830$	0	0
$831$	− 1808.71i	− 2.17655i
$832$	0	0
$833$	356.714	0.428228
$834$	0	0
$835$	12.9577	0.0155182
$836$	0	0
$837$	289.515i	0.345896i
$838$	0	0
$839$	− 634.212i	− 0.755914i	−0.925823	−	0.377957i	$-0.876627\pi$
$839$	− 634.212i	− 0.755914i	0.925823	−	0.377957i	$-0.123373\pi$
$840$	0	0
$841$	−516.388	−0.614017
$842$	0	0
$843$	−929.822	−1.10299
$844$	0	0
$845$	6.25503i	0.00740240i
$846$	0	0
$847$	292.641i	0.345503i
$848$	0	0
$849$	−1899.16	−2.23694
$850$	0	0
$851$	−956.369	−1.12382
$852$	0	0
$853$	− 971.851i	− 1.13933i	−0.821876	−	0.569667i	$-0.807072\pi$
$853$	− 971.851i	− 1.13933i	0.821876	−	0.569667i	$-0.192928\pi$
$854$	0	0
$855$	3.74114i	0.00437560i
$856$	0	0
$857$	−995.675	−1.16181	−0.580907	−	0.813970i	$-0.697302\pi$
$857$	−995.675	−1.16181	−0.580907	+	0.813970i	$0.697302\pi$
$858$	0	0
$859$	−608.453	−0.708327	−0.354164	−	0.935183i	$-0.615234\pi$
$859$	−608.453	−0.708327	−0.354164	+	0.935183i	$0.615234\pi$
$860$	0	0
$861$	69.9112i	0.0811977i
$862$	0	0
$863$	1014.03i	1.17501i	0.809222	+	0.587503i	$0.199889\pi$
$863$	1014.03i	1.17501i	−0.809222	+	0.587503i	$0.800111\pi$
$864$	0	0
$865$	−22.2927	−0.0257719
$866$	0	0
$867$	565.751	0.652539
$868$	0	0
$869$	− 640.743i	− 0.737334i
$870$	0	0
$871$	1328.02i	1.52471i
$872$	0	0
$873$	−1771.59	−2.02931
$874$	0	0
$875$	−19.1334	−0.0218667
$876$	0	0
$877$	769.778i	0.877740i	0.898551	+	0.438870i	$0.144621\pi$
$877$	769.778i	0.877740i	−0.898551	+	0.438870i	$0.855379\pi$
$878$	0	0
$879$	− 2152.03i	− 2.44828i
$880$	0	0
$881$	645.905	0.733150	0.366575	−	0.930388i	$-0.380530\pi$
$881$	645.905	0.733150	0.366575	+	0.930388i	$0.380530\pi$
$882$	0	0
$883$	−830.076	−0.940063	−0.470032	−	0.882650i	$-0.655757\pi$
$883$	−830.076	−0.940063	−0.470032	+	0.882650i	$0.655757\pi$
$884$	0	0
$885$	− 17.5210i	− 0.0197977i
$886$	0	0
$887$	− 1221.93i	− 1.37759i	−0.724955	−	0.688797i	$-0.758139\pi$
$887$	− 1221.93i	− 1.37759i	0.724955	−	0.688797i	$-0.241861\pi$
$888$	0	0
$889$	−768.730	−0.864713
$890$	0	0
$891$	328.460	0.368642
$892$	0	0
$893$	297.032i	0.332623i
$894$	0	0
$895$	6.36462i	0.00711131i
$896$	0	0
$897$	−1176.81	−1.31194
$898$	0	0
$899$	−745.496	−0.829250
$900$	0	0
$901$	− 19.3446i	− 0.0214702i
$902$	0	0
$903$	− 23.6031i	− 0.0261386i
$904$	0	0
$905$	−2.31417	−0.00255710
$906$	0	0
$907$	438.426	0.483380	0.241690	−	0.970354i	$-0.422298\pi$
$907$	438.426	0.483380	0.241690	+	0.970354i	$0.422298\pi$
$908$	0	0
$909$	− 922.627i	− 1.01499i
$910$	0	0
$911$	1044.12i	1.14612i	0.819513	+	0.573060i	$0.194244\pi$
$911$	1044.12i	1.14612i	−0.819513	+	0.573060i	$0.805756\pi$
$912$	0	0
$913$	287.186	0.314552
$914$	0	0
$915$	−8.23866	−0.00900400
$916$	0	0
$917$	− 145.265i	− 0.158413i
$918$	0	0
$919$	188.522i	0.205138i	0.994726	+	0.102569i	$0.0327062\pi$
$919$	188.522i	0.205138i	−0.994726	+	0.102569i	$0.967294\pi$
$920$	0	0
$921$	154.342	0.167581
$922$	0	0
$923$	−881.054	−0.954554
$924$	0	0
$925$	1459.17i	1.57748i
$926$	0	0
$927$	1916.22i	2.06712i
$928$	0	0
$929$	−220.366	−0.237208	−0.118604	−	0.992942i	$-0.537842\pi$
$929$	−220.366	−0.237208	−0.118604	+	0.992942i	$0.537842\pi$
$930$	0	0
$931$	−103.155	−0.110801
$932$	0	0
$933$	724.891i	0.776947i
$934$	0	0
$935$	8.10493i	0.00866837i
$936$	0	0
$937$	558.321	0.595860	0.297930	−	0.954588i	$-0.403704\pi$
$937$	558.321	0.595860	0.297930	+	0.954588i	$0.403704\pi$
$938$	0	0
$939$	−269.236	−0.286727
$940$	0	0
$941$	1123.96i	1.19443i	0.802081	+	0.597216i	$0.203726\pi$
$941$	1123.96i	1.19443i	−0.802081	+	0.597216i	$0.796274\pi$
$942$	0	0
$943$	53.9984i	0.0572624i
$944$	0	0
$945$	5.47595	0.00579466
$946$	0	0
$947$	63.5494	0.0671060	0.0335530	−	0.999437i	$-0.489318\pi$
$947$	63.5494	0.0671060	0.0335530	+	0.999437i	$0.489318\pi$
$948$	0	0
$949$	151.757i	0.159913i
$950$	0	0
$951$	− 175.480i	− 0.184521i
$952$	0	0
$953$	−304.232	−0.319236	−0.159618	−	0.987179i	$-0.551026\pi$
$953$	−304.232	−0.319236	−0.159618	+	0.987179i	$0.551026\pi$
$954$	0	0
$955$	13.8105	0.0144612
$956$	0	0
$957$	− 1284.80i	− 1.34252i
$958$	0	0
$959$	807.213i	0.841724i
$960$	0	0
$961$	551.564	0.573948
$962$	0	0
$963$	−986.793	−1.02471
$964$	0	0
$965$	− 0.180979i	0 0.000187543i
$966$	0	0
$967$	− 834.409i	− 0.862884i	−0.902141	−	0.431442i	$-0.858005\pi$
$967$	− 834.409i	− 0.862884i	0.902141	−	0.431442i	$-0.141995\pi$
$968$	0	0
$969$	220.414	0.227465
$970$	0	0
$971$	299.105	0.308038	0.154019	−	0.988068i	$-0.450778\pi$
$971$	299.105	0.308038	0.154019	+	0.988068i	$0.450778\pi$
$972$	0	0
$973$	− 652.013i	− 0.670106i
$974$	0	0
$975$	1795.50i	1.84154i
$976$	0	0
$977$	−891.561	−0.912549	−0.456275	−	0.889839i	$-0.650817\pi$
$977$	−891.561	−0.912549	−0.456275	+	0.889839i	$0.650817\pi$
$978$	0	0
$979$	876.978	0.895790
$980$	0	0
$981$	− 974.433i	− 0.993306i
$982$	0	0
$983$	− 181.589i	− 0.184730i	−0.995725	−	0.0923648i	$-0.970557\pi$
$983$	− 181.589i	− 0.184730i	0.995725	−	0.0923648i	$-0.0294426\pi$
$984$	0	0
$985$	7.90739	0.00802781
$986$	0	0
$987$	1691.40	1.71368
$988$	0	0
$989$	− 18.2307i	− 0.0184335i
$990$	0	0
$991$	− 1140.89i	− 1.15125i	−0.817715	−	0.575624i	$-0.804759\pi$
$991$	− 1140.89i	− 1.15125i	0.817715	−	0.575624i	$-0.195241\pi$
$992$	0	0
$993$	1183.95	1.19230
$994$	0	0
$995$	−18.4833	−0.0185762
$996$	0	0
$997$	− 1050.68i	− 1.05385i	−0.849913	−	0.526923i	$-0.823346\pi$
$997$	− 1050.68i	− 1.05385i	0.849913	−	0.526923i	$-0.176654\pi$
$998$	0	0
$999$	− 835.339i	− 0.836175i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.3.d.k.511.1		12
4.3	odd	2	inner	1024.3.d.k.511.11		12
8.3	odd	2	inner	1024.3.d.k.511.2		12
8.5	even	2	inner	1024.3.d.k.511.12		12
16.3	odd	4		1024.3.c.j.1023.11		12
16.5	even	4		1024.3.c.j.1023.12		12
16.11	odd	4		1024.3.c.j.1023.2		12
16.13	even	4		1024.3.c.j.1023.1		12
32.3	odd	8		128.3.f.b.31.3		6
32.5	even	8		128.3.f.b.95.3		6
32.11	odd	8		64.3.f.a.47.3		6
32.13	even	8		64.3.f.a.15.3		6
32.19	odd	8		16.3.f.a.11.3	yes	6
32.21	even	8		16.3.f.a.3.3	✓	6
32.27	odd	8		128.3.f.a.95.1		6
32.29	even	8		128.3.f.a.31.1		6
96.5	odd	8		1152.3.m.a.991.2		6
96.11	even	8		576.3.m.a.559.2		6
96.29	odd	8		1152.3.m.b.415.2		6
96.35	even	8		1152.3.m.a.415.2		6
96.53	odd	8		144.3.m.a.19.1		6
96.59	even	8		1152.3.m.b.991.2		6
96.77	odd	8		576.3.m.a.271.2		6
96.83	even	8		144.3.m.a.91.1		6
160.19	odd	8		400.3.r.c.251.1		6
160.53	odd	8		400.3.k.c.99.1		6
160.83	even	8		400.3.k.d.299.3		6
160.117	odd	8		400.3.k.d.99.3		6
160.147	even	8		400.3.k.c.299.1		6
160.149	even	8		400.3.r.c.51.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.f.a.3.3	✓	6	32.21	even	8
16.3.f.a.11.3	yes	6	32.19	odd	8
64.3.f.a.15.3		6	32.13	even	8
64.3.f.a.47.3		6	32.11	odd	8
128.3.f.a.31.1		6	32.29	even	8
128.3.f.a.95.1		6	32.27	odd	8
128.3.f.b.31.3		6	32.3	odd	8
128.3.f.b.95.3		6	32.5	even	8
144.3.m.a.19.1		6	96.53	odd	8
144.3.m.a.91.1		6	96.83	even	8
400.3.k.c.99.1		6	160.53	odd	8
400.3.k.c.299.1		6	160.147	even	8
400.3.k.d.99.3		6	160.117	odd	8
400.3.k.d.299.3		6	160.83	even	8
400.3.r.c.51.1		6	160.149	even	8
400.3.r.c.251.1		6	160.19	odd	8
576.3.m.a.271.2		6	96.77	odd	8
576.3.m.a.559.2		6	96.11	even	8
1024.3.c.j.1023.1		12	16.13	even	4
1024.3.c.j.1023.2		12	16.11	odd	4
1024.3.c.j.1023.11		12	16.3	odd	4
1024.3.c.j.1023.12		12	16.5	even	4
1024.3.d.k.511.1		12	1.1	even	1	trivial
1024.3.d.k.511.2		12	8.3	odd	2	inner
1024.3.d.k.511.11		12	4.3	odd	2	inner
1024.3.d.k.511.12		12	8.5	even	2	inner
1152.3.m.a.415.2		6	96.35	even	8
1152.3.m.a.991.2		6	96.5	odd	8
1152.3.m.b.415.2		6	96.29	odd	8
1152.3.m.b.991.2		6	96.59	even	8

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1024.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.59498 q^{3} -0.0829198i q^{5} -4.61555i q^{7} +12.1138 q^{9} +O(q^{10})\)	\(q-4.59498 q^{3} -0.0829198i q^{5} -4.61555i q^{7} +12.1138 q^{9} -7.58925 q^{11} -15.6344i q^{13} +0.381015i q^{15} +12.8793 q^{17} -3.72446 q^{19} +21.2083i q^{21} +16.3810i q^{23} +24.9931 q^{25} -14.3080 q^{27} -36.8427i q^{29} -20.2345i q^{31} +34.8724 q^{33} -0.382720 q^{35} +58.3828i q^{37} +71.8398i q^{39} +3.29640 q^{41} -1.11292 q^{43} -1.00448i q^{45} -79.7517i q^{47} +27.6967 q^{49} -59.1801 q^{51} -1.50200i q^{53} +0.629299i q^{55} +17.1138 q^{57} -45.9850 q^{59} +21.6229i q^{61} -55.9119i q^{63} -1.29640 q^{65} -84.9421 q^{67} -75.2704i q^{69} -56.3535i q^{71} -9.70663 q^{73} -114.843 q^{75} +35.0285i q^{77} +84.4278i q^{79} -43.2796 q^{81} -37.8412 q^{83} -1.06795i q^{85} +169.292i q^{87} -115.555 q^{89} -72.1613 q^{91} +92.9772i q^{93} +0.308832i q^{95} -146.245 q^{97} -91.9348 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^9	\( 12 q + 12 q^{9} + 8 q^{17} + 20 q^{25} - 8 q^{33} + 92 q^{49} + 72 q^{57} + 24 q^{65} + 96 q^{73} - 172 q^{81} - 160 q^{89} - 8 q^{97}+O(q^{100}) \) 12 * q + 12 * q^9 + 8 * q^17 + 20 * q^25 - 8 * q^33 + 92 * q^49 + 72 * q^57 + 24 * q^65 + 96 * q^73 - 172 * q^81 - 160 * q^89 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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