LMFDB - Embedded newform 1024.4.a.n.1.7

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 36x^{8} + 405x^{6} - 1380x^{4} + 420x^{2} - 32 $ x^10 - 36x^8 + 405x^6 - 1380x^4 + 420x^2 - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 16)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-3.82089$ 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)

$f(q)$	$=$	$q+2.80518 q^{3} +0.844070 q^{5} -29.0828 q^{7} -19.1310 q^{9} +O(q^{10})$	$q+2.80518 q^{3} +0.844070 q^{5} -29.0828 q^{7} -19.1310 q^{9} +17.1532 q^{11} +68.6824 q^{13} +2.36777 q^{15} -86.7193 q^{17} +77.5614 q^{19} -81.5826 q^{21} +70.2145 q^{23} -124.288 q^{25} -129.406 q^{27} +89.6641 q^{29} -8.86868 q^{31} +48.1178 q^{33} -24.5480 q^{35} +30.7089 q^{37} +192.667 q^{39} -153.274 q^{41} -171.050 q^{43} -16.1479 q^{45} +99.9792 q^{47} +502.812 q^{49} -243.263 q^{51} +550.315 q^{53} +14.4785 q^{55} +217.574 q^{57} +459.364 q^{59} -0.479693 q^{61} +556.383 q^{63} +57.9728 q^{65} +799.438 q^{67} +196.964 q^{69} +419.500 q^{71} +374.833 q^{73} -348.649 q^{75} -498.864 q^{77} +705.750 q^{79} +153.530 q^{81} +1339.39 q^{83} -73.1972 q^{85} +251.524 q^{87} +4.72918 q^{89} -1997.48 q^{91} -24.8782 q^{93} +65.4673 q^{95} +379.542 q^{97} -328.157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 28 q^{7} + 54 q^{9}+O(q^{10}) $ 10 * q + 28 * q^7 + 54 * q^9	$ 10 q + 28 q^{7} + 54 q^{9} + 124 q^{15} + 4 q^{17} + 276 q^{23} + 50 q^{25} + 368 q^{31} - 4 q^{33} + 732 q^{39} + 944 q^{47} - 94 q^{49} + 1380 q^{55} + 108 q^{57} + 2628 q^{63} - 492 q^{65} + 3468 q^{71} - 296 q^{73} + 4416 q^{79} - 482 q^{81} + 6036 q^{87} + 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q + 28 * q^7 + 54 * q^9 + 124 * q^15 + 4 * q^17 + 276 * q^23 + 50 * q^25 + 368 * q^31 - 4 * q^33 + 732 * q^39 + 944 * q^47 - 94 * q^49 + 1380 * q^55 + 108 * q^57 + 2628 * q^63 - 492 * q^65 + 3468 * q^71 - 296 * q^73 + 4416 * q^79 - 482 * q^81 + 6036 * q^87 + 88 * q^89 + 6900 * q^95 - 4 * 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
$3$	2.80518	0.539857	0.269929	−	0.962880i	$-0.413000\pi$
$3$	2.80518	0.539857	0.269929	+	0.962880i	$0.413000\pi$
$4$	0	0
$5$	0.844070	0.0754959	0.0377480	−	0.999287i	$-0.487982\pi$
$5$	0.844070	0.0754959	0.0377480	+	0.999287i	$0.487982\pi$
$6$	0	0
$7$	−29.0828	−1.57033	−0.785163	−	0.619289i	$-0.787421\pi$
$7$	−29.0828	−1.57033	−0.785163	+	0.619289i	$0.787421\pi$
$8$	0	0
$9$	−19.1310	−0.708554
$10$	0	0
$11$	17.1532	0.470171	0.235086	−	0.971975i	$-0.424463\pi$
$11$	17.1532	0.470171	0.235086	+	0.971975i	$0.424463\pi$
$12$	0	0
$13$	68.6824	1.46531	0.732657	−	0.680598i	$-0.238280\pi$
$13$	68.6824	1.46531	0.732657	+	0.680598i	$0.238280\pi$
$14$	0	0
$15$	2.36777	0.0407570
$16$	0	0
$17$	−86.7193	−1.23721	−0.618604	−	0.785703i	$-0.712301\pi$
$17$	−86.7193	−1.23721	−0.618604	+	0.785703i	$0.712301\pi$
$18$	0	0
$19$	77.5614	0.936517	0.468258	−	0.883592i	$-0.344882\pi$
$19$	77.5614	0.936517	0.468258	+	0.883592i	$0.344882\pi$
$20$	0	0
$21$	−81.5826	−0.847752
$22$	0	0
$23$	70.2145	0.636554	0.318277	−	0.947998i	$-0.396896\pi$
$23$	70.2145	0.636554	0.318277	+	0.947998i	$0.396896\pi$
$24$	0	0
$25$	−124.288	−0.994300
$26$	0	0
$27$	−129.406	−0.922375
$28$	0	0
$29$	89.6641	0.574145	0.287072	−	0.957909i	$-0.407318\pi$
$29$	89.6641	0.574145	0.287072	+	0.957909i	$0.407318\pi$
$30$	0	0
$31$	−8.86868	−0.0513826	−0.0256913	−	0.999670i	$-0.508179\pi$
$31$	−8.86868	−0.0513826	−0.0256913	+	0.999670i	$0.508179\pi$
$32$	0	0
$33$	48.1178	0.253825
$34$	0	0
$35$	−24.5480	−0.118553
$36$	0	0
$37$	30.7089	0.136446	0.0682231	−	0.997670i	$-0.478267\pi$
$37$	30.7089	0.136446	0.0682231	+	0.997670i	$0.478267\pi$
$38$	0	0
$39$	192.667	0.791060
$40$	0	0
$41$	−153.274	−0.583840	−0.291920	−	0.956443i	$-0.594294\pi$
$41$	−153.274	−0.583840	−0.291920	+	0.956443i	$0.594294\pi$
$42$	0	0
$43$	−171.050	−0.606625	−0.303312	−	0.952891i	$-0.598093\pi$
$43$	−171.050	−0.606625	−0.303312	+	0.952891i	$0.598093\pi$
$44$	0	0
$45$	−16.1479	−0.0534930
$46$	0	0
$47$	99.9792	0.310286	0.155143	−	0.987892i	$-0.450416\pi$
$47$	99.9792	0.310286	0.155143	+	0.987892i	$0.450416\pi$
$48$	0	0
$49$	502.812	1.46592
$50$	0	0
$51$	−243.263	−0.667915
$52$	0	0
$53$	550.315	1.42626	0.713128	−	0.701034i	$-0.247278\pi$
$53$	550.315	1.42626	0.713128	+	0.701034i	$0.247278\pi$
$54$	0	0
$55$	14.4785	0.0354960
$56$	0	0
$57$	217.574	0.505585
$58$	0	0
$59$	459.364	1.01363	0.506814	−	0.862055i	$-0.330823\pi$
$59$	459.364	1.01363	0.506814	+	0.862055i	$0.330823\pi$
$60$	0	0
$61$	−0.479693	−0.00100686	−0.000503430	−	1.00000i	$-0.500160\pi$
$61$	−0.479693	−0.00100686	−0.000503430		1.00000i	$0.500160\pi$
$62$	0	0
$63$	556.383	1.11266
$64$	0	0
$65$	57.9728	0.110625
$66$	0	0
$67$	799.438	1.45771	0.728857	−	0.684666i	$-0.240052\pi$
$67$	799.438	1.45771	0.728857	+	0.684666i	$0.240052\pi$
$68$	0	0
$69$	196.964	0.343648
$70$	0	0
$71$	419.500	0.701205	0.350602	−	0.936524i	$-0.385977\pi$
$71$	419.500	0.701205	0.350602	+	0.936524i	$0.385977\pi$
$72$	0	0
$73$	374.833	0.600971	0.300485	−	0.953786i	$-0.402851\pi$
$73$	374.833	0.600971	0.300485	+	0.953786i	$0.402851\pi$
$74$	0	0
$75$	−348.649	−0.536780
$76$	0	0
$77$	−498.864	−0.738322
$78$	0	0
$79$	705.750	1.00510	0.502551	−	0.864547i	$-0.332395\pi$
$79$	705.750	1.00510	0.502551	+	0.864547i	$0.332395\pi$
$80$	0	0
$81$	153.530	0.210604
$82$	0	0
$83$	1339.39	1.77129	0.885646	−	0.464361i	$-0.153716\pi$
$83$	1339.39	1.77129	0.885646	+	0.464361i	$0.153716\pi$
$84$	0	0
$85$	−73.1972	−0.0934041
$86$	0	0
$87$	251.524	0.309956
$88$	0	0
$89$	4.72918	0.00563249	0.00281625	−	0.999996i	$-0.499104\pi$
$89$	4.72918	0.00563249	0.00281625	+	0.999996i	$0.499104\pi$
$90$	0	0
$91$	−1997.48	−2.30102
$92$	0	0
$93$	−24.8782	−0.0277393
$94$	0	0
$95$	65.4673	0.0707032
$96$	0	0
$97$	379.542	0.397285	0.198643	−	0.980072i	$-0.436347\pi$
$97$	379.542	0.397285	0.198643	+	0.980072i	$0.436347\pi$
$98$	0	0
$99$	−328.157	−0.333142
$100$	0	0
$101$	552.964	0.544772	0.272386	−	0.962188i	$-0.412187\pi$
$101$	552.964	0.544772	0.272386	+	0.962188i	$0.412187\pi$
$102$	0	0
$103$	307.935	0.294580	0.147290	−	0.989093i	$-0.452945\pi$
$103$	307.935	0.294580	0.147290	+	0.989093i	$0.452945\pi$
$104$	0	0
$105$	−68.8615	−0.0640018
$106$	0	0
$107$	−850.801	−0.768692	−0.384346	−	0.923189i	$-0.625573\pi$
$107$	−850.801	−0.768692	−0.384346	+	0.923189i	$0.625573\pi$
$108$	0	0
$109$	1341.93	1.17921	0.589605	−	0.807692i	$-0.299283\pi$
$109$	1341.93	1.17921	0.589605	+	0.807692i	$0.299283\pi$
$110$	0	0
$111$	86.1439	0.0736614
$112$	0	0
$113$	−1824.02	−1.51849	−0.759244	−	0.650807i	$-0.774431\pi$
$113$	−1824.02	−1.51849	−0.759244	+	0.650807i	$0.774431\pi$
$114$	0	0
$115$	59.2660	0.0480573
$116$	0	0
$117$	−1313.96	−1.03825
$118$	0	0
$119$	2522.04	1.94282
$120$	0	0
$121$	−1036.77	−0.778939
$122$	0	0
$123$	−429.962	−0.315190
$124$	0	0
$125$	−210.416	−0.150562
$126$	0	0
$127$	988.748	0.690844	0.345422	−	0.938447i	$-0.387736\pi$
$127$	988.748	0.690844	0.345422	+	0.938447i	$0.387736\pi$
$128$	0	0
$129$	−479.826	−0.327491
$130$	0	0
$131$	1122.28	0.748505	0.374252	−	0.927327i	$-0.377899\pi$
$131$	1122.28	0.748505	0.374252	+	0.927327i	$0.377899\pi$
$132$	0	0
$133$	−2255.71	−1.47064
$134$	0	0
$135$	−109.227	−0.0696356
$136$	0	0
$137$	−1595.30	−0.994856	−0.497428	−	0.867505i	$-0.665722\pi$
$137$	−1595.30	−0.994856	−0.497428	+	0.867505i	$0.665722\pi$
$138$	0	0
$139$	−392.721	−0.239641	−0.119821	−	0.992796i	$-0.538232\pi$
$139$	−392.721	−0.239641	−0.119821	+	0.992796i	$0.538232\pi$
$140$	0	0
$141$	280.460	0.167510
$142$	0	0
$143$	1178.12	0.688948
$144$	0	0
$145$	75.6828	0.0433456
$146$	0	0
$147$	1410.48	0.791390
$148$	0	0
$149$	−839.014	−0.461307	−0.230653	−	0.973036i	$-0.574086\pi$
$149$	−839.014	−0.461307	−0.230653	+	0.973036i	$0.574086\pi$
$150$	0	0
$151$	−160.655	−0.0865821	−0.0432911	−	0.999063i	$-0.513784\pi$
$151$	−160.655	−0.0865821	−0.0432911	+	0.999063i	$0.513784\pi$
$152$	0	0
$153$	1659.02	0.876629
$154$	0	0
$155$	−7.48579	−0.00387918
$156$	0	0
$157$	−998.098	−0.507369	−0.253684	−	0.967287i	$-0.581642\pi$
$157$	−998.098	−0.507369	−0.253684	+	0.967287i	$0.581642\pi$
$158$	0	0
$159$	1543.73	0.769975
$160$	0	0
$161$	−2042.04	−0.999598
$162$	0	0
$163$	2648.31	1.27259	0.636294	−	0.771447i	$-0.280467\pi$
$163$	2648.31	1.27259	0.636294	+	0.771447i	$0.280467\pi$
$164$	0	0
$165$	40.6148	0.0191628
$166$	0	0
$167$	3852.19	1.78498	0.892490	−	0.451066i	$-0.148956\pi$
$167$	3852.19	1.78498	0.892490	+	0.451066i	$0.148956\pi$
$168$	0	0
$169$	2520.28	1.14715
$170$	0	0
$171$	−1483.83	−0.663573
$172$	0	0
$173$	−3713.17	−1.63183	−0.815917	−	0.578170i	$-0.803767\pi$
$173$	−3713.17	−1.63183	−0.815917	+	0.578170i	$0.803767\pi$
$174$	0	0
$175$	3614.64	1.56138
$176$	0	0
$177$	1288.60	0.547215
$178$	0	0
$179$	1749.01	0.730318	0.365159	−	0.930945i	$-0.381015\pi$
$179$	1749.01	0.730318	0.365159	+	0.930945i	$0.381015\pi$
$180$	0	0
$181$	2227.25	0.914640	0.457320	−	0.889302i	$-0.348809\pi$
$181$	2227.25	0.914640	0.457320	+	0.889302i	$0.348809\pi$
$182$	0	0
$183$	−1.34563	−0.000543560 0
$184$	0	0
$185$	25.9204	0.0103011
$186$	0	0
$187$	−1487.51	−0.581699
$188$	0	0
$189$	3763.48	1.44843
$190$	0	0
$191$	3585.92	1.35847	0.679236	−	0.733920i	$-0.262311\pi$
$191$	3585.92	1.35847	0.679236	+	0.733920i	$0.262311\pi$
$192$	0	0
$193$	523.601	0.195283	0.0976415	−	0.995222i	$-0.468870\pi$
$193$	523.601	0.195283	0.0976415	+	0.995222i	$0.468870\pi$
$194$	0	0
$195$	162.624	0.0597218
$196$	0	0
$197$	1591.89	0.575724	0.287862	−	0.957672i	$-0.407056\pi$
$197$	1591.89	0.575724	0.287862	+	0.957672i	$0.407056\pi$
$198$	0	0
$199$	2312.48	0.823757	0.411878	−	0.911239i	$-0.364873\pi$
$199$	2312.48	0.823757	0.411878	+	0.911239i	$0.364873\pi$
$200$	0	0
$201$	2242.57	0.786957
$202$	0	0
$203$	−2607.69	−0.901595
$204$	0	0
$205$	−129.374	−0.0440775
$206$	0	0
$207$	−1343.27	−0.451033
$208$	0	0
$209$	1330.43	0.440323
$210$	0	0
$211$	−2006.19	−0.654558	−0.327279	−	0.944928i	$-0.606132\pi$
$211$	−2006.19	−0.654558	−0.327279	+	0.944928i	$0.606132\pi$
$212$	0	0
$213$	1176.77	0.378550
$214$	0	0
$215$	−144.378	−0.0457977
$216$	0	0
$217$	257.926	0.0806875
$218$	0	0
$219$	1051.47	0.324438
$220$	0	0
$221$	−5956.10	−1.81290
$222$	0	0
$223$	−4315.08	−1.29578	−0.647890	−	0.761734i	$-0.724349\pi$
$223$	−4315.08	−1.29578	−0.647890	+	0.761734i	$0.724349\pi$
$224$	0	0
$225$	2377.74	0.704516
$226$	0	0
$227$	991.651	0.289948	0.144974	−	0.989435i	$-0.453690\pi$
$227$	991.651	0.289948	0.144974	+	0.989435i	$0.453690\pi$
$228$	0	0
$229$	938.121	0.270711	0.135355	−	0.990797i	$-0.456782\pi$
$229$	938.121	0.270711	0.135355	+	0.990797i	$0.456782\pi$
$230$	0	0
$231$	−1399.40	−0.398588
$232$	0	0
$233$	−3490.15	−0.981318	−0.490659	−	0.871352i	$-0.663244\pi$
$233$	−3490.15	−0.981318	−0.490659	+	0.871352i	$0.663244\pi$
$234$	0	0
$235$	84.3895	0.0234254
$236$	0	0
$237$	1979.76	0.542612
$238$	0	0
$239$	−2950.43	−0.798525	−0.399263	−	0.916837i	$-0.630734\pi$
$239$	−2950.43	−0.798525	−0.399263	+	0.916837i	$0.630734\pi$
$240$	0	0
$241$	1128.96	0.301755	0.150877	−	0.988552i	$-0.451790\pi$
$241$	1128.96	0.301755	0.150877	+	0.988552i	$0.451790\pi$
$242$	0	0
$243$	3924.63	1.03607
$244$	0	0
$245$	424.409	0.110671
$246$	0	0
$247$	5327.11	1.37229
$248$	0	0
$249$	3757.23	0.956244
$250$	0	0
$251$	−6536.30	−1.64370	−0.821848	−	0.569706i	$-0.807057\pi$
$251$	−6536.30	−1.64370	−0.821848	+	0.569706i	$0.807057\pi$
$252$	0	0
$253$	1204.40	0.299289
$254$	0	0
$255$	−205.331	−0.0504249
$256$	0	0
$257$	610.977	0.148295	0.0741473	−	0.997247i	$-0.476377\pi$
$257$	610.977	0.148295	0.0741473	+	0.997247i	$0.476377\pi$
$258$	0	0
$259$	−893.101	−0.214265
$260$	0	0
$261$	−1715.36	−0.406813
$262$	0	0
$263$	−4973.57	−1.16610	−0.583048	−	0.812438i	$-0.698140\pi$
$263$	−4973.57	−1.16610	−0.583048	+	0.812438i	$0.698140\pi$
$264$	0	0
$265$	464.505	0.107677
$266$	0	0
$267$	13.2662	0.00304074
$268$	0	0
$269$	1327.12	0.300803	0.150401	−	0.988625i	$-0.451943\pi$
$269$	1327.12	0.300803	0.150401	+	0.988625i	$0.451943\pi$
$270$	0	0
$271$	4010.64	0.898999	0.449500	−	0.893280i	$-0.351602\pi$
$271$	4010.64	0.898999	0.449500	+	0.893280i	$0.351602\pi$
$272$	0	0
$273$	−5603.29	−1.24222
$274$	0	0
$275$	−2131.93	−0.467491
$276$	0	0
$277$	−4999.23	−1.08439	−0.542193	−	0.840254i	$-0.682406\pi$
$277$	−4999.23	−1.08439	−0.542193	+	0.840254i	$0.682406\pi$
$278$	0	0
$279$	169.666	0.0364074
$280$	0	0
$281$	−7468.35	−1.58550	−0.792748	−	0.609550i	$-0.791350\pi$
$281$	−7468.35	−1.58550	−0.792748	+	0.609550i	$0.791350\pi$
$282$	0	0
$283$	3180.88	0.668140	0.334070	−	0.942548i	$-0.391578\pi$
$283$	3180.88	0.668140	0.334070	+	0.942548i	$0.391578\pi$
$284$	0	0
$285$	183.648	0.0381696
$286$	0	0
$287$	4457.66	0.916819
$288$	0	0
$289$	2607.24	0.530682
$290$	0	0
$291$	1064.68	0.214477
$292$	0	0
$293$	5590.09	1.11460	0.557298	−	0.830312i	$-0.311838\pi$
$293$	5590.09	1.11460	0.557298	+	0.830312i	$0.311838\pi$
$294$	0	0
$295$	387.736	0.0765249
$296$	0	0
$297$	−2219.72	−0.433674
$298$	0	0
$299$	4822.51	0.932752
$300$	0	0
$301$	4974.62	0.952599
$302$	0	0
$303$	1551.16	0.294099
$304$	0	0
$305$	−0.404895	−7.60138e−5 0
$306$	0	0
$307$	5452.13	1.01358	0.506791	−	0.862069i	$-0.330832\pi$
$307$	5452.13	1.01358	0.506791	+	0.862069i	$0.330832\pi$
$308$	0	0
$309$	863.814	0.159031
$310$	0	0
$311$	−5194.39	−0.947096	−0.473548	−	0.880768i	$-0.657027\pi$
$311$	−5194.39	−0.947096	−0.473548	+	0.880768i	$0.657027\pi$
$312$	0	0
$313$	−4710.01	−0.850561	−0.425281	−	0.905062i	$-0.639825\pi$
$313$	−4710.01	−0.850561	−0.425281	+	0.905062i	$0.639825\pi$
$314$	0	0
$315$	469.626	0.0840014
$316$	0	0
$317$	8033.04	1.42328	0.711641	−	0.702544i	$-0.247952\pi$
$317$	8033.04	1.42328	0.711641	+	0.702544i	$0.247952\pi$
$318$	0	0
$319$	1538.03	0.269946
$320$	0	0
$321$	−2386.65	−0.414984
$322$	0	0
$323$	−6726.08	−1.15867
$324$	0	0
$325$	−8536.37	−1.45696
$326$	0	0
$327$	3764.36	0.636605
$328$	0	0
$329$	−2907.68	−0.487251
$330$	0	0
$331$	2567.92	0.426423	0.213211	−	0.977006i	$-0.431608\pi$
$331$	2567.92	0.426423	0.213211	+	0.977006i	$0.431608\pi$
$332$	0	0
$333$	−587.490	−0.0966795
$334$	0	0
$335$	674.782	0.110052
$336$	0	0
$337$	2683.29	0.433733	0.216867	−	0.976201i	$-0.430416\pi$
$337$	2683.29	0.433733	0.216867	+	0.976201i	$0.430416\pi$
$338$	0	0
$339$	−5116.69	−0.819766
$340$	0	0
$341$	−152.126	−0.0241586
$342$	0	0
$343$	−4647.79	−0.731653
$344$	0	0
$345$	166.252	0.0259440
$346$	0	0
$347$	7483.74	1.15778	0.578888	−	0.815407i	$-0.303487\pi$
$347$	7483.74	1.15778	0.578888	+	0.815407i	$0.303487\pi$
$348$	0	0
$349$	104.239	0.0159880	0.00799400	−	0.999968i	$-0.497455\pi$
$349$	104.239	0.0159880	0.00799400	+	0.999968i	$0.497455\pi$
$350$	0	0
$351$	−8887.90	−1.35157
$352$	0	0
$353$	−5067.25	−0.764030	−0.382015	−	0.924156i	$-0.624770\pi$
$353$	−5067.25	−0.764030	−0.382015	+	0.924156i	$0.624770\pi$
$354$	0	0
$355$	354.088	0.0529381
$356$	0	0
$357$	7074.79	1.04884
$358$	0	0
$359$	−970.230	−0.142637	−0.0713186	−	0.997454i	$-0.522721\pi$
$359$	−970.230	−0.142637	−0.0713186	+	0.997454i	$0.522721\pi$
$360$	0	0
$361$	−843.224	−0.122937
$362$	0	0
$363$	−2908.32	−0.420516
$364$	0	0
$365$	316.385	0.0453709
$366$	0	0
$367$	−13451.4	−1.91323	−0.956617	−	0.291347i	$-0.905896\pi$
$367$	−13451.4	−1.91323	−0.956617	+	0.291347i	$0.905896\pi$
$368$	0	0
$369$	2932.29	0.413682
$370$	0	0
$371$	−16004.7	−2.23969
$372$	0	0
$373$	8341.34	1.15790	0.578952	−	0.815362i	$-0.303462\pi$
$373$	8341.34	1.15790	0.578952	+	0.815362i	$0.303462\pi$
$374$	0	0
$375$	−590.255	−0.0812817
$376$	0	0
$377$	6158.35	0.841302
$378$	0	0
$379$	6288.62	0.852308	0.426154	−	0.904651i	$-0.359868\pi$
$379$	6288.62	0.852308	0.426154	+	0.904651i	$0.359868\pi$
$380$	0	0
$381$	2773.62	0.372957
$382$	0	0
$383$	−6417.68	−0.856209	−0.428105	−	0.903729i	$-0.640819\pi$
$383$	−6417.68	−0.856209	−0.428105	+	0.903729i	$0.640819\pi$
$384$	0	0
$385$	−421.076	−0.0557403
$386$	0	0
$387$	3272.35	0.429827
$388$	0	0
$389$	−9271.04	−1.20838	−0.604191	−	0.796840i	$-0.706503\pi$
$389$	−9271.04	−1.20838	−0.604191	+	0.796840i	$0.706503\pi$
$390$	0	0
$391$	−6088.96	−0.787549
$392$	0	0
$393$	3148.20	0.404086
$394$	0	0
$395$	595.703	0.0758812
$396$	0	0
$397$	−12590.0	−1.59163	−0.795814	−	0.605541i	$-0.792957\pi$
$397$	−12590.0	−1.59163	−0.795814	+	0.605541i	$0.792957\pi$
$398$	0	0
$399$	−6327.66	−0.793933
$400$	0	0
$401$	6425.77	0.800218	0.400109	−	0.916468i	$-0.368972\pi$
$401$	6425.77	0.800218	0.400109	+	0.916468i	$0.368972\pi$
$402$	0	0
$403$	−609.122	−0.0752917
$404$	0	0
$405$	129.590	0.0158997
$406$	0	0
$407$	526.755	0.0641530
$408$	0	0
$409$	12796.0	1.54699	0.773496	−	0.633801i	$-0.218506\pi$
$409$	12796.0	1.54699	0.773496	+	0.633801i	$0.218506\pi$
$410$	0	0
$411$	−4475.09	−0.537080
$412$	0	0
$413$	−13359.6	−1.59173
$414$	0	0
$415$	1130.54	0.133725
$416$	0	0
$417$	−1101.65	−0.129372
$418$	0	0
$419$	−9256.33	−1.07924	−0.539620	−	0.841909i	$-0.681432\pi$
$419$	−9256.33	−1.07924	−0.539620	+	0.841909i	$0.681432\pi$
$420$	0	0
$421$	9036.83	1.04615	0.523074	−	0.852287i	$-0.324785\pi$
$421$	9036.83	1.04615	0.523074	+	0.852287i	$0.324785\pi$
$422$	0	0
$423$	−1912.70	−0.219855
$424$	0	0
$425$	10778.1	1.23016
$426$	0	0
$427$	13.9508	0.00158110
$428$	0	0
$429$	3304.85	0.371934
$430$	0	0
$431$	10639.3	1.18904	0.594519	−	0.804081i	$-0.297342\pi$
$431$	10639.3	1.18904	0.594519	+	0.804081i	$0.297342\pi$
$432$	0	0
$433$	−3806.14	−0.422428	−0.211214	−	0.977440i	$-0.567742\pi$
$433$	−3806.14	−0.422428	−0.211214	+	0.977440i	$0.567742\pi$
$434$	0	0
$435$	212.304	0.0234004
$436$	0	0
$437$	5445.94	0.596143
$438$	0	0
$439$	−14102.8	−1.53323	−0.766616	−	0.642106i	$-0.778061\pi$
$439$	−14102.8	−1.53323	−0.766616	+	0.642106i	$0.778061\pi$
$440$	0	0
$441$	−9619.28	−1.03869
$442$	0	0
$443$	10836.3	1.16219	0.581095	−	0.813836i	$-0.302625\pi$
$443$	10836.3	1.16219	0.581095	+	0.813836i	$0.302625\pi$
$444$	0	0
$445$	3.99176	0.000425230 0
$446$	0	0
$447$	−2353.58	−0.249040
$448$	0	0
$449$	13679.4	1.43779	0.718897	−	0.695117i	$-0.244647\pi$
$449$	13679.4	1.43779	0.718897	+	0.695117i	$0.244647\pi$
$450$	0	0
$451$	−2629.14	−0.274505
$452$	0	0
$453$	−450.666	−0.0467420
$454$	0	0
$455$	−1686.01	−0.173718
$456$	0	0
$457$	−7913.48	−0.810016	−0.405008	−	0.914313i	$-0.632731\pi$
$457$	−7913.48	−0.810016	−0.405008	+	0.914313i	$0.632731\pi$
$458$	0	0
$459$	11222.0	1.14117
$460$	0	0
$461$	820.548	0.0828996	0.0414498	−	0.999141i	$-0.486802\pi$
$461$	820.548	0.0828996	0.0414498	+	0.999141i	$0.486802\pi$
$462$	0	0
$463$	−14236.5	−1.42899	−0.714497	−	0.699638i	$-0.753344\pi$
$463$	−14236.5	−1.42899	−0.714497	+	0.699638i	$0.753344\pi$
$464$	0	0
$465$	−20.9990	−0.00209420
$466$	0	0
$467$	11801.0	1.16935	0.584674	−	0.811269i	$-0.301223\pi$
$467$	11801.0	1.16935	0.584674	+	0.811269i	$0.301223\pi$
$468$	0	0
$469$	−23249.9	−2.28909
$470$	0	0
$471$	−2799.85	−0.273907
$472$	0	0
$473$	−2934.05	−0.285217
$474$	0	0
$475$	−9639.92	−0.931179
$476$	0	0
$477$	−10528.1	−1.01058
$478$	0	0
$479$	−5563.77	−0.530720	−0.265360	−	0.964149i	$-0.585491\pi$
$479$	−5563.77	−0.530720	−0.265360	+	0.964149i	$0.585491\pi$
$480$	0	0
$481$	2109.16	0.199936
$482$	0	0
$483$	−5728.29	−0.539640
$484$	0	0
$485$	320.360	0.0299934
$486$	0	0
$487$	−18150.5	−1.68886	−0.844432	−	0.535662i	$-0.820062\pi$
$487$	−18150.5	−1.68886	−0.844432	+	0.535662i	$0.820062\pi$
$488$	0	0
$489$	7428.99	0.687015
$490$	0	0
$491$	16395.0	1.50692	0.753459	−	0.657495i	$-0.228384\pi$
$491$	16395.0	1.50692	0.753459	+	0.657495i	$0.228384\pi$
$492$	0	0
$493$	−7775.61	−0.710336
$494$	0	0
$495$	−276.988	−0.0251509
$496$	0	0
$497$	−12200.3	−1.10112
$498$	0	0
$499$	−4397.61	−0.394517	−0.197259	−	0.980351i	$-0.563204\pi$
$499$	−4397.61	−0.394517	−0.197259	+	0.980351i	$0.563204\pi$
$500$	0	0
$501$	10806.1	0.963635
$502$	0	0
$503$	6221.21	0.551471	0.275736	−	0.961233i	$-0.411079\pi$
$503$	6221.21	0.551471	0.275736	+	0.961233i	$0.411079\pi$
$504$	0	0
$505$	466.740	0.0411281
$506$	0	0
$507$	7069.83	0.619295
$508$	0	0
$509$	19516.8	1.69954	0.849770	−	0.527154i	$-0.176741\pi$
$509$	19516.8	1.69954	0.849770	+	0.527154i	$0.176741\pi$
$510$	0	0
$511$	−10901.2	−0.943720
$512$	0	0
$513$	−10036.9	−0.863820
$514$	0	0
$515$	259.919	0.0222396
$516$	0	0
$517$	1714.96	0.145888
$518$	0	0
$519$	−10416.1	−0.880957
$520$	0	0
$521$	6874.63	0.578086	0.289043	−	0.957316i	$-0.406663\pi$
$521$	6874.63	0.578086	0.289043	+	0.957316i	$0.406663\pi$
$522$	0	0
$523$	3261.91	0.272722	0.136361	−	0.990659i	$-0.456459\pi$
$523$	3261.91	0.272722	0.136361	+	0.990659i	$0.456459\pi$
$524$	0	0
$525$	10139.7	0.842920
$526$	0	0
$527$	769.086	0.0635710
$528$	0	0
$529$	−7236.92	−0.594799
$530$	0	0
$531$	−8788.08	−0.718211
$532$	0	0
$533$	−10527.3	−0.855509
$534$	0	0
$535$	−718.136	−0.0580332
$536$	0	0
$537$	4906.28	0.394267
$538$	0	0
$539$	8624.83	0.689235
$540$	0	0
$541$	−18724.1	−1.48801	−0.744005	−	0.668174i	$-0.767076\pi$
$541$	−18724.1	−1.48801	−0.744005	+	0.668174i	$0.767076\pi$
$542$	0	0
$543$	6247.82	0.493775
$544$	0	0
$545$	1132.69	0.0890256
$546$	0	0
$547$	−18768.5	−1.46706	−0.733530	−	0.679657i	$-0.762129\pi$
$547$	−18768.5	−1.46706	−0.733530	+	0.679657i	$0.762129\pi$
$548$	0	0
$549$	9.17700	0.000713415 0
$550$	0	0
$551$	6954.47	0.537696
$552$	0	0
$553$	−20525.2	−1.57834
$554$	0	0
$555$	72.7115	0.00556114
$556$	0	0
$557$	12021.7	0.914497	0.457249	−	0.889339i	$-0.348835\pi$
$557$	12021.7	0.914497	0.457249	+	0.889339i	$0.348835\pi$
$558$	0	0
$559$	−11748.1	−0.888896
$560$	0	0
$561$	−4172.74	−0.314034
$562$	0	0
$563$	−24504.4	−1.83434	−0.917172	−	0.398491i	$-0.869534\pi$
$563$	−24504.4	−1.83434	−0.917172	+	0.398491i	$0.869534\pi$
$564$	0	0
$565$	−1539.60	−0.114640
$566$	0	0
$567$	−4465.09	−0.330716
$568$	0	0
$569$	8998.54	0.662985	0.331492	−	0.943458i	$-0.392448\pi$
$569$	8998.54	0.662985	0.331492	+	0.943458i	$0.392448\pi$
$570$	0	0
$571$	13928.9	1.02085	0.510427	−	0.859921i	$-0.329487\pi$
$571$	13928.9	1.02085	0.510427	+	0.859921i	$0.329487\pi$
$572$	0	0
$573$	10059.1	0.733380
$574$	0	0
$575$	−8726.79	−0.632926
$576$	0	0
$577$	−20584.4	−1.48516	−0.742580	−	0.669757i	$-0.766398\pi$
$577$	−20584.4	−1.48516	−0.742580	+	0.669757i	$0.766398\pi$
$578$	0	0
$579$	1468.79	0.105425
$580$	0	0
$581$	−38953.3	−2.78151
$582$	0	0
$583$	9439.66	0.670585
$584$	0	0
$585$	−1109.08	−0.0783840
$586$	0	0
$587$	−3658.25	−0.257227	−0.128613	−	0.991695i	$-0.541053\pi$
$587$	−3658.25	−0.257227	−0.128613	+	0.991695i	$0.541053\pi$
$588$	0	0
$589$	−687.867	−0.0481207
$590$	0	0
$591$	4465.54	0.310809
$592$	0	0
$593$	6035.89	0.417984	0.208992	−	0.977917i	$-0.432982\pi$
$593$	6035.89	0.417984	0.208992	+	0.977917i	$0.432982\pi$
$594$	0	0
$595$	2128.78	0.146675
$596$	0	0
$597$	6486.93	0.444711
$598$	0	0
$599$	5427.20	0.370199	0.185100	−	0.982720i	$-0.440739\pi$
$599$	5427.20	0.370199	0.185100	+	0.982720i	$0.440739\pi$
$600$	0	0
$601$	−17725.7	−1.20307	−0.601535	−	0.798847i	$-0.705444\pi$
$601$	−17725.7	−1.20307	−0.601535	+	0.798847i	$0.705444\pi$
$602$	0	0
$603$	−15294.0	−1.03287
$604$	0	0
$605$	−875.105	−0.0588067
$606$	0	0
$607$	13487.6	0.901884	0.450942	−	0.892553i	$-0.351088\pi$
$607$	13487.6	0.901884	0.450942	+	0.892553i	$0.351088\pi$
$608$	0	0
$609$	−7315.03	−0.486732
$610$	0	0
$611$	6866.81	0.454667
$612$	0	0
$613$	23830.0	1.57012	0.785062	−	0.619417i	$-0.212631\pi$
$613$	23830.0	1.57012	0.785062	+	0.619417i	$0.212631\pi$
$614$	0	0
$615$	−362.918	−0.0237956
$616$	0	0
$617$	−535.243	−0.0349239	−0.0174620	−	0.999848i	$-0.505559\pi$
$617$	−535.243	−0.0349239	−0.0174620	+	0.999848i	$0.505559\pi$
$618$	0	0
$619$	−27847.2	−1.80820	−0.904098	−	0.427324i	$-0.859456\pi$
$619$	−27847.2	−1.80820	−0.904098	+	0.427324i	$0.859456\pi$
$620$	0	0
$621$	−9086.16	−0.587142
$622$	0	0
$623$	−137.538	−0.00884485
$624$	0	0
$625$	15358.3	0.982934
$626$	0	0
$627$	3732.08	0.237711
$628$	0	0
$629$	−2663.05	−0.168812
$630$	0	0
$631$	−11880.2	−0.749511	−0.374755	−	0.927124i	$-0.622273\pi$
$631$	−11880.2	−0.749511	−0.374755	+	0.927124i	$0.622273\pi$
$632$	0	0
$633$	−5627.72	−0.353368
$634$	0	0
$635$	834.573	0.0521560
$636$	0	0
$637$	34534.4	2.14804
$638$	0	0
$639$	−8025.45	−0.496842
$640$	0	0
$641$	18341.0	1.13015	0.565074	−	0.825040i	$-0.308848\pi$
$641$	18341.0	1.13015	0.565074	+	0.825040i	$0.308848\pi$
$642$	0	0
$643$	9936.56	0.609424	0.304712	−	0.952445i	$-0.401440\pi$
$643$	9936.56	0.609424	0.304712	+	0.952445i	$0.401440\pi$
$644$	0	0
$645$	−405.007	−0.0247242
$646$	0	0
$647$	21429.7	1.30215	0.651073	−	0.759015i	$-0.274319\pi$
$647$	21429.7	1.30215	0.651073	+	0.759015i	$0.274319\pi$
$648$	0	0
$649$	7879.56	0.476579
$650$	0	0
$651$	723.530	0.0435597
$652$	0	0
$653$	−12277.1	−0.735742	−0.367871	−	0.929877i	$-0.619913\pi$
$653$	−12277.1	−0.735742	−0.367871	+	0.929877i	$0.619913\pi$
$654$	0	0
$655$	947.284	0.0565091
$656$	0	0
$657$	−7170.92	−0.425821
$658$	0	0
$659$	5871.25	0.347058	0.173529	−	0.984829i	$-0.444483\pi$
$659$	5871.25	0.347058	0.173529	+	0.984829i	$0.444483\pi$
$660$	0	0
$661$	17308.5	1.01849	0.509246	−	0.860621i	$-0.329924\pi$
$661$	17308.5	1.01849	0.509246	+	0.860621i	$0.329924\pi$
$662$	0	0
$663$	−16707.9	−0.978706
$664$	0	0
$665$	−1903.98	−0.111027
$666$	0	0
$667$	6295.72	0.365474
$668$	0	0
$669$	−12104.6	−0.699536
$670$	0	0
$671$	−8.22827	−0.000473396 0
$672$	0	0
$673$	6528.62	0.373937	0.186969	−	0.982366i	$-0.440134\pi$
$673$	6528.62	0.373937	0.186969	+	0.982366i	$0.440134\pi$
$674$	0	0
$675$	16083.5	0.917118
$676$	0	0
$677$	20110.9	1.14169	0.570847	−	0.821057i	$-0.306615\pi$
$677$	20110.9	1.14169	0.570847	+	0.821057i	$0.306615\pi$
$678$	0	0
$679$	−11038.2	−0.623867
$680$	0	0
$681$	2781.76	0.156530
$682$	0	0
$683$	30291.8	1.69705	0.848524	−	0.529157i	$-0.177492\pi$
$683$	30291.8	1.69705	0.848524	+	0.529157i	$0.177492\pi$
$684$	0	0
$685$	−1346.54	−0.0751076
$686$	0	0
$687$	2631.60	0.146145
$688$	0	0
$689$	37797.0	2.08991
$690$	0	0
$691$	23387.1	1.28753	0.643767	−	0.765222i	$-0.277371\pi$
$691$	23387.1	1.28753	0.643767	+	0.765222i	$0.277371\pi$
$692$	0	0
$693$	9543.74	0.523141
$694$	0	0
$695$	−331.484	−0.0180920
$696$	0	0
$697$	13291.8	0.722331
$698$	0	0
$699$	−9790.49	−0.529772
$700$	0	0
$701$	−4280.14	−0.230612	−0.115306	−	0.993330i	$-0.536785\pi$
$701$	−4280.14	−0.230612	−0.115306	+	0.993330i	$0.536785\pi$
$702$	0	0
$703$	2381.82	0.127784
$704$	0	0
$705$	236.728	0.0126464
$706$	0	0
$707$	−16081.8	−0.855470
$708$	0	0
$709$	−5630.18	−0.298231	−0.149115	−	0.988820i	$-0.547643\pi$
$709$	−5630.18	−0.298231	−0.149115	+	0.988820i	$0.547643\pi$
$710$	0	0
$711$	−13501.7	−0.712170
$712$	0	0
$713$	−622.710	−0.0327078
$714$	0	0
$715$	994.419	0.0520128
$716$	0	0
$717$	−8276.49	−0.431090
$718$	0	0
$719$	5682.25	0.294732	0.147366	−	0.989082i	$-0.452921\pi$
$719$	5682.25	0.294732	0.147366	+	0.989082i	$0.452921\pi$
$720$	0	0
$721$	−8955.64	−0.462587
$722$	0	0
$723$	3166.94	0.162904
$724$	0	0
$725$	−11144.1	−0.570872
$726$	0	0
$727$	18883.0	0.963317	0.481658	−	0.876359i	$-0.340035\pi$
$727$	18883.0	0.963317	0.481658	+	0.876359i	$0.340035\pi$
$728$	0	0
$729$	6863.99	0.348727
$730$	0	0
$731$	14833.3	0.750521
$732$	0	0
$733$	−35302.3	−1.77888	−0.889441	−	0.457049i	$-0.848906\pi$
$733$	−35302.3	−1.77888	−0.889441	+	0.457049i	$0.848906\pi$
$734$	0	0
$735$	1190.54	0.0597467
$736$	0	0
$737$	13712.9	0.685375
$738$	0	0
$739$	8771.48	0.436623	0.218311	−	0.975879i	$-0.429945\pi$
$739$	8771.48	0.436623	0.218311	+	0.975879i	$0.429945\pi$
$740$	0	0
$741$	14943.5	0.740841
$742$	0	0
$743$	30.9140	0.00152641	0.000763205	−	1.00000i	$-0.499757\pi$
$743$	30.9140	0.00152641	0.000763205		1.00000i	$0.499757\pi$
$744$	0	0
$745$	−708.187	−0.0348268
$746$	0	0
$747$	−25623.8	−1.25506
$748$	0	0
$749$	24743.7	1.20710
$750$	0	0
$751$	16318.5	0.792905	0.396453	−	0.918055i	$-0.370241\pi$
$751$	16318.5	0.792905	0.396453	+	0.918055i	$0.370241\pi$
$752$	0	0
$753$	−18335.5	−0.887361
$754$	0	0
$755$	−135.604	−0.00653660
$756$	0	0
$757$	13936.5	0.669129	0.334564	−	0.942373i	$-0.391411\pi$
$757$	13936.5	0.669129	0.334564	+	0.942373i	$0.391411\pi$
$758$	0	0
$759$	3378.57	0.161573
$760$	0	0
$761$	−3823.42	−0.182127	−0.0910637	−	0.995845i	$-0.529027\pi$
$761$	−3823.42	−0.182127	−0.0910637	+	0.995845i	$0.529027\pi$
$762$	0	0
$763$	−39027.2	−1.85174
$764$	0	0
$765$	1400.33	0.0661819
$766$	0	0
$767$	31550.2	1.48528
$768$	0	0
$769$	31689.1	1.48601	0.743003	−	0.669288i	$-0.233401\pi$
$769$	31689.1	1.48601	0.743003	+	0.669288i	$0.233401\pi$
$770$	0	0
$771$	1713.90	0.0800578
$772$	0	0
$773$	1846.74	0.0859282	0.0429641	−	0.999077i	$-0.486320\pi$
$773$	1846.74	0.0859282	0.0429641	+	0.999077i	$0.486320\pi$
$774$	0	0
$775$	1102.27	0.0510898
$776$	0	0
$777$	−2505.31	−0.115672
$778$	0	0
$779$	−11888.2	−0.546776
$780$	0	0
$781$	7195.77	0.329686
$782$	0	0
$783$	−11603.0	−0.529577
$784$	0	0
$785$	−842.465	−0.0383043
$786$	0	0
$787$	−20364.0	−0.922363	−0.461181	−	0.887306i	$-0.652574\pi$
$787$	−20364.0	−0.922363	−0.461181	+	0.887306i	$0.652574\pi$
$788$	0	0
$789$	−13951.7	−0.629525
$790$	0	0
$791$	53047.6	2.38452
$792$	0	0
$793$	−32.9465	−0.00147537
$794$	0	0
$795$	1303.02	0.0581300
$796$	0	0
$797$	−7880.27	−0.350230	−0.175115	−	0.984548i	$-0.556030\pi$
$797$	−7880.27	−0.350230	−0.175115	+	0.984548i	$0.556030\pi$
$798$	0	0
$799$	−8670.13	−0.383889
$800$	0	0
$801$	−90.4737	−0.00399093
$802$	0	0
$803$	6429.58	0.282559
$804$	0	0
$805$	−1723.62	−0.0754656
$806$	0	0
$807$	3722.81	0.162390
$808$	0	0
$809$	5081.28	0.220826	0.110413	−	0.993886i	$-0.464783\pi$
$809$	5081.28	0.220826	0.110413	+	0.993886i	$0.464783\pi$
$810$	0	0
$811$	6354.33	0.275130	0.137565	−	0.990493i	$-0.456072\pi$
$811$	6354.33	0.275130	0.137565	+	0.990493i	$0.456072\pi$
$812$	0	0
$813$	11250.6	0.485331
$814$	0	0
$815$	2235.36	0.0960752
$816$	0	0
$817$	−13266.9	−0.568114
$818$	0	0
$819$	38213.7	1.63040
$820$	0	0
$821$	2991.83	0.127181	0.0635904	−	0.997976i	$-0.479745\pi$
$821$	2991.83	0.127181	0.0635904	+	0.997976i	$0.479745\pi$
$822$	0	0
$823$	24432.6	1.03483	0.517416	−	0.855734i	$-0.326894\pi$
$823$	24432.6	1.03483	0.517416	+	0.855734i	$0.326894\pi$
$824$	0	0
$825$	−5980.44	−0.252378
$826$	0	0
$827$	−29771.2	−1.25181	−0.625904	−	0.779900i	$-0.715270\pi$
$827$	−29771.2	−1.25181	−0.625904	+	0.779900i	$0.715270\pi$
$828$	0	0
$829$	35980.5	1.50742	0.753711	−	0.657206i	$-0.228262\pi$
$829$	35980.5	1.50742	0.753711	+	0.657206i	$0.228262\pi$
$830$	0	0
$831$	−14023.7	−0.585413
$832$	0	0
$833$	−43603.5	−1.81365
$834$	0	0
$835$	3251.52	0.134759
$836$	0	0
$837$	1147.66	0.0473941
$838$	0	0
$839$	18757.2	0.771837	0.385919	−	0.922533i	$-0.373885\pi$
$839$	18757.2	0.771837	0.385919	+	0.922533i	$0.373885\pi$
$840$	0	0
$841$	−16349.4	−0.670358
$842$	0	0
$843$	−20950.1	−0.855941
$844$	0	0
$845$	2127.29	0.0866048
$846$	0	0
$847$	30152.2	1.22319
$848$	0	0
$849$	8922.94	0.360700
$850$	0	0
$851$	2156.21	0.0868553
$852$	0	0
$853$	−8626.94	−0.346285	−0.173142	−	0.984897i	$-0.555392\pi$
$853$	−8626.94	−0.346285	−0.173142	+	0.984897i	$0.555392\pi$
$854$	0	0
$855$	−1252.45	−0.0500971
$856$	0	0
$857$	−2079.04	−0.0828687	−0.0414344	−	0.999141i	$-0.513193\pi$
$857$	−2079.04	−0.0828687	−0.0414344	+	0.999141i	$0.513193\pi$
$858$	0	0
$859$	−11740.7	−0.466343	−0.233171	−	0.972436i	$-0.574910\pi$
$859$	−11740.7	−0.466343	−0.233171	+	0.972436i	$0.574910\pi$
$860$	0	0
$861$	12504.5	0.494951
$862$	0	0
$863$	43830.8	1.72887	0.864436	−	0.502743i	$-0.167676\pi$
$863$	43830.8	1.72887	0.864436	+	0.502743i	$0.167676\pi$
$864$	0	0
$865$	−3134.18	−0.123197
$866$	0	0
$867$	7313.78	0.286493
$868$	0	0
$869$	12105.9	0.472570
$870$	0	0
$871$	54907.3	2.13601
$872$	0	0
$873$	−7261.01	−0.281498
$874$	0	0
$875$	6119.50	0.236431
$876$	0	0
$877$	3628.71	0.139718	0.0698591	−	0.997557i	$-0.477745\pi$
$877$	3628.71	0.139718	0.0698591	+	0.997557i	$0.477745\pi$
$878$	0	0
$879$	15681.2	0.601723
$880$	0	0
$881$	−26429.8	−1.01072	−0.505360	−	0.862909i	$-0.668640\pi$
$881$	−26429.8	−1.01072	−0.505360	+	0.862909i	$0.668640\pi$
$882$	0	0
$883$	29286.1	1.11614	0.558071	−	0.829793i	$-0.311542\pi$
$883$	29286.1	1.11614	0.558071	+	0.829793i	$0.311542\pi$
$884$	0	0
$885$	1087.67	0.0413125
$886$	0	0
$887$	−22350.0	−0.846042	−0.423021	−	0.906120i	$-0.639030\pi$
$887$	−22350.0	−0.846042	−0.423021	+	0.906120i	$0.639030\pi$
$888$	0	0
$889$	−28755.6	−1.08485
$890$	0	0
$891$	2633.53	0.0990197
$892$	0	0
$893$	7754.53	0.290588
$894$	0	0
$895$	1476.28	0.0551360
$896$	0	0
$897$	13528.0	0.503553
$898$	0	0
$899$	−795.202	−0.0295011
$900$	0	0
$901$	−47723.0	−1.76457
$902$	0	0
$903$	13954.7	0.514267
$904$	0	0
$905$	1879.95	0.0690516
$906$	0	0
$907$	−5779.79	−0.211593	−0.105796	−	0.994388i	$-0.533739\pi$
$907$	−5779.79	−0.211593	−0.105796	+	0.994388i	$0.533739\pi$
$908$	0	0
$909$	−10578.7	−0.386000
$910$	0	0
$911$	20024.6	0.728259	0.364130	−	0.931348i	$-0.381367\pi$
$911$	20024.6	0.728259	0.364130	+	0.931348i	$0.381367\pi$
$912$	0	0
$913$	22974.8	0.832810
$914$	0	0
$915$	−1.13580	−4.10366e−5 0
$916$	0	0
$917$	−32639.1	−1.17540
$918$	0	0
$919$	−26977.7	−0.968349	−0.484175	−	0.874971i	$-0.660880\pi$
$919$	−26977.7	−0.968349	−0.484175	+	0.874971i	$0.660880\pi$
$920$	0	0
$921$	15294.2	0.547189
$922$	0	0
$923$	28812.3	1.02748
$924$	0	0
$925$	−3816.73	−0.135668
$926$	0	0
$927$	−5891.10	−0.208726
$928$	0	0
$929$	−34371.4	−1.21387	−0.606937	−	0.794750i	$-0.707602\pi$
$929$	−34371.4	−1.21387	−0.606937	+	0.794750i	$0.707602\pi$
$930$	0	0
$931$	38998.8	1.37286
$932$	0	0
$933$	−14571.2	−0.511297
$934$	0	0
$935$	−1255.57	−0.0439159
$936$	0	0
$937$	32901.8	1.14712	0.573561	−	0.819163i	$-0.305561\pi$
$937$	32901.8	1.14712	0.573561	+	0.819163i	$0.305561\pi$
$938$	0	0
$939$	−13212.4	−0.459181
$940$	0	0
$941$	−47208.7	−1.63545	−0.817726	−	0.575607i	$-0.804766\pi$
$941$	−47208.7	−1.63545	−0.817726	+	0.575607i	$0.804766\pi$
$942$	0	0
$943$	−10762.1	−0.371646
$944$	0	0
$945$	3176.65	0.109351
$946$	0	0
$947$	−19250.2	−0.660557	−0.330279	−	0.943884i	$-0.607143\pi$
$947$	−19250.2	−0.660557	−0.330279	+	0.943884i	$0.607143\pi$
$948$	0	0
$949$	25744.4	0.880611
$950$	0	0
$951$	22534.1	0.768369
$952$	0	0
$953$	−4572.49	−0.155422	−0.0777112	−	0.996976i	$-0.524761\pi$
$953$	−4572.49	−0.155422	−0.0777112	+	0.996976i	$0.524761\pi$
$954$	0	0
$955$	3026.77	0.102559
$956$	0	0
$957$	4314.44	0.145732
$958$	0	0
$959$	46395.7	1.56225
$960$	0	0
$961$	−29712.3	−0.997360
$962$	0	0
$963$	16276.7	0.544660
$964$	0	0
$965$	441.956	0.0147431
$966$	0	0
$967$	−33060.9	−1.09945	−0.549725	−	0.835346i	$-0.685268\pi$
$967$	−33060.9	−1.09945	−0.549725	+	0.835346i	$0.685268\pi$
$968$	0	0
$969$	−18867.9	−0.625514
$970$	0	0
$971$	−21995.9	−0.726964	−0.363482	−	0.931601i	$-0.618412\pi$
$971$	−21995.9	−0.726964	−0.363482	+	0.931601i	$0.618412\pi$
$972$	0	0
$973$	11421.4	0.376315
$974$	0	0
$975$	−23946.1	−0.786551
$976$	0	0
$977$	−1241.54	−0.0406555	−0.0203277	−	0.999793i	$-0.506471\pi$
$977$	−1241.54	−0.0406555	−0.0203277	+	0.999793i	$0.506471\pi$
$978$	0	0
$979$	81.1205	0.00264823
$980$	0	0
$981$	−25672.5	−0.835534
$982$	0	0
$983$	−10797.7	−0.350349	−0.175175	−	0.984537i	$-0.556049\pi$
$983$	−10797.7	−0.350349	−0.175175	+	0.984537i	$0.556049\pi$
$984$	0	0
$985$	1343.67	0.0434648
$986$	0	0
$987$	−8156.56	−0.263046
$988$	0	0
$989$	−12010.2	−0.386149
$990$	0	0
$991$	−10204.8	−0.327112	−0.163556	−	0.986534i	$-0.552296\pi$
$991$	−10204.8	−0.327112	−0.163556	+	0.986534i	$0.552296\pi$
$992$	0	0
$993$	7203.49	0.230207
$994$	0	0
$995$	1951.90	0.0621903
$996$	0	0
$997$	−21279.2	−0.675947	−0.337973	−	0.941156i	$-0.609741\pi$
$997$	−21279.2	−0.675947	−0.337973	+	0.941156i	$0.609741\pi$
$998$	0	0
$999$	−3973.90	−0.125855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.a.n.1.7		10
4.3	odd	2		1024.4.a.m.1.4		10
8.3	odd	2		1024.4.a.m.1.7		10
8.5	even	2	inner	1024.4.a.n.1.4		10
16.3	odd	4		1024.4.b.k.513.4		10
16.5	even	4		1024.4.b.j.513.4		10
16.11	odd	4		1024.4.b.k.513.7		10
16.13	even	4		1024.4.b.j.513.7		10
32.3	odd	8		64.4.e.a.17.2		10
32.5	even	8		128.4.e.b.97.2		10
32.11	odd	8		64.4.e.a.49.2		10
32.13	even	8		128.4.e.b.33.2		10
32.19	odd	8		128.4.e.a.33.4		10
32.21	even	8		16.4.e.a.5.4	✓	10
32.27	odd	8		128.4.e.a.97.4		10
32.29	even	8		16.4.e.a.13.4	yes	10
96.11	even	8		576.4.k.a.433.3		10
96.29	odd	8		144.4.k.a.109.2		10
96.35	even	8		576.4.k.a.145.3		10
96.53	odd	8		144.4.k.a.37.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.4	✓	10	32.21	even	8
16.4.e.a.13.4	yes	10	32.29	even	8
64.4.e.a.17.2		10	32.3	odd	8
64.4.e.a.49.2		10	32.11	odd	8
128.4.e.a.33.4		10	32.19	odd	8
128.4.e.a.97.4		10	32.27	odd	8
128.4.e.b.33.2		10	32.13	even	8
128.4.e.b.97.2		10	32.5	even	8
144.4.k.a.37.2		10	96.53	odd	8
144.4.k.a.109.2		10	96.29	odd	8
576.4.k.a.145.3		10	96.35	even	8
576.4.k.a.433.3		10	96.11	even	8
1024.4.a.m.1.4		10	4.3	odd	2
1024.4.a.m.1.7		10	8.3	odd	2
1024.4.a.n.1.4		10	8.5	even	2	inner
1024.4.a.n.1.7		10	1.1	even	1	trivial
1024.4.b.j.513.4		10	16.5	even	4
1024.4.b.j.513.7		10	16.13	even	4
1024.4.b.k.513.4		10	16.3	odd	4
1024.4.b.k.513.7		10	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)

\(f(q)\)	\(=\)	\(q+2.80518 q^{3} +0.844070 q^{5} -29.0828 q^{7} -19.1310 q^{9} +O(q^{10})\)	\(q+2.80518 q^{3} +0.844070 q^{5} -29.0828 q^{7} -19.1310 q^{9} +17.1532 q^{11} +68.6824 q^{13} +2.36777 q^{15} -86.7193 q^{17} +77.5614 q^{19} -81.5826 q^{21} +70.2145 q^{23} -124.288 q^{25} -129.406 q^{27} +89.6641 q^{29} -8.86868 q^{31} +48.1178 q^{33} -24.5480 q^{35} +30.7089 q^{37} +192.667 q^{39} -153.274 q^{41} -171.050 q^{43} -16.1479 q^{45} +99.9792 q^{47} +502.812 q^{49} -243.263 q^{51} +550.315 q^{53} +14.4785 q^{55} +217.574 q^{57} +459.364 q^{59} -0.479693 q^{61} +556.383 q^{63} +57.9728 q^{65} +799.438 q^{67} +196.964 q^{69} +419.500 q^{71} +374.833 q^{73} -348.649 q^{75} -498.864 q^{77} +705.750 q^{79} +153.530 q^{81} +1339.39 q^{83} -73.1972 q^{85} +251.524 q^{87} +4.72918 q^{89} -1997.48 q^{91} -24.8782 q^{93} +65.4673 q^{95} +379.542 q^{97} -328.157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 28 q^{7} + 54 q^{9}+O(q^{10}) \) 10 * q + 28 * q^7 + 54 * q^9	\( 10 q + 28 q^{7} + 54 q^{9} + 124 q^{15} + 4 q^{17} + 276 q^{23} + 50 q^{25} + 368 q^{31} - 4 q^{33} + 732 q^{39} + 944 q^{47} - 94 q^{49} + 1380 q^{55} + 108 q^{57} + 2628 q^{63} - 492 q^{65} + 3468 q^{71} - 296 q^{73} + 4416 q^{79} - 482 q^{81} + 6036 q^{87} + 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q + 28 * q^7 + 54 * q^9 + 124 * q^15 + 4 * q^17 + 276 * q^23 + 50 * q^25 + 368 * q^31 - 4 * q^33 + 732 * q^39 + 944 * q^47 - 94 * q^49 + 1380 * q^55 + 108 * q^57 + 2628 * q^63 - 492 * q^65 + 3468 * q^71 - 296 * q^73 + 4416 * q^79 - 482 * q^81 + 6036 * q^87 + 88 * q^89 + 6900 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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