LMFDB - Embedded newform 6008.2.a.b.1.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.28
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.320895 q^{3} +1.20543 q^{5} +2.62503 q^{7} -2.89703 q^{9} +O(q^{10})$	$q+0.320895 q^{3} +1.20543 q^{5} +2.62503 q^{7} -2.89703 q^{9} +1.62663 q^{11} -3.31735 q^{13} +0.386816 q^{15} -2.66730 q^{17} +4.21646 q^{19} +0.842357 q^{21} -1.23332 q^{23} -3.54694 q^{25} -1.89232 q^{27} -9.10559 q^{29} +5.03784 q^{31} +0.521975 q^{33} +3.16428 q^{35} -8.64255 q^{37} -1.06452 q^{39} -6.06053 q^{41} -5.22644 q^{43} -3.49216 q^{45} -2.33691 q^{47} -0.109239 q^{49} -0.855921 q^{51} +1.51550 q^{53} +1.96078 q^{55} +1.35304 q^{57} +9.59427 q^{59} +7.73345 q^{61} -7.60477 q^{63} -3.99883 q^{65} -11.1614 q^{67} -0.395764 q^{69} -7.47848 q^{71} -3.57210 q^{73} -1.13819 q^{75} +4.26993 q^{77} +2.62845 q^{79} +8.08384 q^{81} -13.9519 q^{83} -3.21523 q^{85} -2.92193 q^{87} +11.3260 q^{89} -8.70812 q^{91} +1.61662 q^{93} +5.08265 q^{95} -16.2165 q^{97} -4.71238 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.320895	0.185269	0.0926343	−	0.995700i	$-0.470471\pi$
$3$	0.320895	0.185269	0.0926343	+	0.995700i	$0.470471\pi$
$4$	0	0
$5$	1.20543	0.539084	0.269542	−	0.962989i	$-0.413128\pi$
$5$	1.20543	0.539084	0.269542	+	0.962989i	$0.413128\pi$
$6$	0	0
$7$	2.62503	0.992166	0.496083	−	0.868275i	$-0.334771\pi$
$7$	2.62503	0.992166	0.496083	+	0.868275i	$0.334771\pi$
$8$	0	0
$9$	−2.89703	−0.965676
$10$	0	0
$11$	1.62663	0.490446	0.245223	−	0.969467i	$-0.421139\pi$
$11$	1.62663	0.490446	0.245223	+	0.969467i	$0.421139\pi$
$12$	0	0
$13$	−3.31735	−0.920067	−0.460033	−	0.887902i	$-0.652163\pi$
$13$	−3.31735	−0.920067	−0.460033	+	0.887902i	$0.652163\pi$
$14$	0	0
$15$	0.386816	0.0998753
$16$	0	0
$17$	−2.66730	−0.646914	−0.323457	−	0.946243i	$-0.604845\pi$
$17$	−2.66730	−0.646914	−0.323457	+	0.946243i	$0.604845\pi$
$18$	0	0
$19$	4.21646	0.967323	0.483661	−	0.875255i	$-0.339307\pi$
$19$	4.21646	0.967323	0.483661	+	0.875255i	$0.339307\pi$
$20$	0	0
$21$	0.842357	0.183817
$22$	0	0
$23$	−1.23332	−0.257164	−0.128582	−	0.991699i	$-0.541043\pi$
$23$	−1.23332	−0.257164	−0.128582	+	0.991699i	$0.541043\pi$
$24$	0	0
$25$	−3.54694	−0.709388
$26$	0	0
$27$	−1.89232	−0.364178
$28$	0	0
$29$	−9.10559	−1.69086	−0.845432	−	0.534082i	$-0.820657\pi$
$29$	−9.10559	−1.69086	−0.845432	+	0.534082i	$0.820657\pi$
$30$	0	0
$31$	5.03784	0.904823	0.452412	−	0.891809i	$-0.350564\pi$
$31$	5.03784	0.904823	0.452412	+	0.891809i	$0.350564\pi$
$32$	0	0
$33$	0.521975	0.0908642
$34$	0	0
$35$	3.16428	0.534861
$36$	0	0
$37$	−8.64255	−1.42083	−0.710414	−	0.703784i	$-0.751492\pi$
$37$	−8.64255	−1.42083	−0.710414	+	0.703784i	$0.751492\pi$
$38$	0	0
$39$	−1.06452	−0.170459
$40$	0	0
$41$	−6.06053	−0.946496	−0.473248	−	0.880929i	$-0.656919\pi$
$41$	−6.06053	−0.946496	−0.473248	+	0.880929i	$0.656919\pi$
$42$	0	0
$43$	−5.22644	−0.797025	−0.398513	−	0.917163i	$-0.630473\pi$
$43$	−5.22644	−0.797025	−0.398513	+	0.917163i	$0.630473\pi$
$44$	0	0
$45$	−3.49216	−0.520580
$46$	0	0
$47$	−2.33691	−0.340874	−0.170437	−	0.985369i	$-0.554518\pi$
$47$	−2.33691	−0.340874	−0.170437	+	0.985369i	$0.554518\pi$
$48$	0	0
$49$	−0.109239	−0.0156056
$50$	0	0
$51$	−0.855921	−0.119853
$52$	0	0
$53$	1.51550	0.208170	0.104085	−	0.994568i	$-0.466809\pi$
$53$	1.51550	0.208170	0.104085	+	0.994568i	$0.466809\pi$
$54$	0	0
$55$	1.96078	0.264392
$56$	0	0
$57$	1.35304	0.179215
$58$	0	0
$59$	9.59427	1.24907	0.624534	−	0.780998i	$-0.285289\pi$
$59$	9.59427	1.24907	0.624534	+	0.780998i	$0.285289\pi$
$60$	0	0
$61$	7.73345	0.990167	0.495083	−	0.868845i	$-0.335138\pi$
$61$	7.73345	0.990167	0.495083	+	0.868845i	$0.335138\pi$
$62$	0	0
$63$	−7.60477	−0.958111
$64$	0	0
$65$	−3.99883	−0.495993
$66$	0	0
$67$	−11.1614	−1.36358	−0.681792	−	0.731546i	$-0.738799\pi$
$67$	−11.1614	−1.36358	−0.681792	+	0.731546i	$0.738799\pi$
$68$	0	0
$69$	−0.395764	−0.0476444
$70$	0	0
$71$	−7.47848	−0.887532	−0.443766	−	0.896143i	$-0.646358\pi$
$71$	−7.47848	−0.887532	−0.443766	+	0.896143i	$0.646358\pi$
$72$	0	0
$73$	−3.57210	−0.418083	−0.209042	−	0.977907i	$-0.567034\pi$
$73$	−3.57210	−0.418083	−0.209042	+	0.977907i	$0.567034\pi$
$74$	0	0
$75$	−1.13819	−0.131427
$76$	0	0
$77$	4.26993	0.486604
$78$	0	0
$79$	2.62845	0.295723	0.147862	−	0.989008i	$-0.452761\pi$
$79$	2.62845	0.295723	0.147862	+	0.989008i	$0.452761\pi$
$80$	0	0
$81$	8.08384	0.898205
$82$	0	0
$83$	−13.9519	−1.53142	−0.765709	−	0.643187i	$-0.777612\pi$
$83$	−13.9519	−1.53142	−0.765709	+	0.643187i	$0.777612\pi$
$84$	0	0
$85$	−3.21523	−0.348741
$86$	0	0
$87$	−2.92193	−0.313264
$88$	0	0
$89$	11.3260	1.20055	0.600276	−	0.799793i	$-0.295057\pi$
$89$	11.3260	1.20055	0.600276	+	0.799793i	$0.295057\pi$
$90$	0	0
$91$	−8.70812	−0.912859
$92$	0	0
$93$	1.61662	0.167635
$94$	0	0
$95$	5.08265	0.521468
$96$	0	0
$97$	−16.2165	−1.64654	−0.823269	−	0.567651i	$-0.807852\pi$
$97$	−16.2165	−1.64654	−0.823269	+	0.567651i	$0.807852\pi$
$98$	0	0
$99$	−4.71238	−0.473612
$100$	0	0
$101$	3.00960	0.299467	0.149733	−	0.988726i	$-0.452158\pi$
$101$	3.00960	0.299467	0.149733	+	0.988726i	$0.452158\pi$
$102$	0	0
$103$	3.21638	0.316919	0.158460	−	0.987365i	$-0.449347\pi$
$103$	3.21638	0.316919	0.158460	+	0.987365i	$0.449347\pi$
$104$	0	0
$105$	1.01540	0.0990930
$106$	0	0
$107$	13.5902	1.31381	0.656907	−	0.753972i	$-0.271864\pi$
$107$	13.5902	1.31381	0.656907	+	0.753972i	$0.271864\pi$
$108$	0	0
$109$	1.40152	0.134241	0.0671205	−	0.997745i	$-0.478619\pi$
$109$	1.40152	0.134241	0.0671205	+	0.997745i	$0.478619\pi$
$110$	0	0
$111$	−2.77335	−0.263235
$112$	0	0
$113$	5.17404	0.486733	0.243366	−	0.969934i	$-0.421748\pi$
$113$	5.17404	0.486733	0.243366	+	0.969934i	$0.421748\pi$
$114$	0	0
$115$	−1.48667	−0.138633
$116$	0	0
$117$	9.61044	0.888486
$118$	0	0
$119$	−7.00172	−0.641847
$120$	0	0
$121$	−8.35409	−0.759463
$122$	0	0
$123$	−1.94479	−0.175356
$124$	0	0
$125$	−10.3027	−0.921504
$126$	0	0
$127$	−14.5742	−1.29325	−0.646624	−	0.762809i	$-0.723820\pi$
$127$	−14.5742	−1.29325	−0.646624	+	0.762809i	$0.723820\pi$
$128$	0	0
$129$	−1.67714	−0.147664
$130$	0	0
$131$	5.71162	0.499027	0.249513	−	0.968371i	$-0.419729\pi$
$131$	5.71162	0.499027	0.249513	+	0.968371i	$0.419729\pi$
$132$	0	0
$133$	11.0683	0.959745
$134$	0	0
$135$	−2.28106	−0.196323
$136$	0	0
$137$	0.504909	0.0431373	0.0215686	−	0.999767i	$-0.493134\pi$
$137$	0.504909	0.0431373	0.0215686	+	0.999767i	$0.493134\pi$
$138$	0	0
$139$	0.0652126	0.00553126	0.00276563	−	0.999996i	$-0.499120\pi$
$139$	0.0652126	0.00553126	0.00276563	+	0.999996i	$0.499120\pi$
$140$	0	0
$141$	−0.749903	−0.0631532
$142$	0	0
$143$	−5.39608	−0.451243
$144$	0	0
$145$	−10.9761	−0.911518
$146$	0	0
$147$	−0.0350544	−0.00289124
$148$	0	0
$149$	−15.1474	−1.24092	−0.620461	−	0.784238i	$-0.713054\pi$
$149$	−15.1474	−1.24092	−0.620461	+	0.784238i	$0.713054\pi$
$150$	0	0
$151$	−7.53357	−0.613074	−0.306537	−	0.951859i	$-0.599170\pi$
$151$	−7.53357	−0.613074	−0.306537	+	0.951859i	$0.599170\pi$
$152$	0	0
$153$	7.72723	0.624709
$154$	0	0
$155$	6.07276	0.487776
$156$	0	0
$157$	2.54733	0.203299	0.101650	−	0.994820i	$-0.467588\pi$
$157$	2.54733	0.203299	0.101650	+	0.994820i	$0.467588\pi$
$158$	0	0
$159$	0.486316	0.0385674
$160$	0	0
$161$	−3.23749	−0.255150
$162$	0	0
$163$	−22.3582	−1.75123	−0.875614	−	0.483011i	$-0.839543\pi$
$163$	−22.3582	−1.75123	−0.875614	+	0.483011i	$0.839543\pi$
$164$	0	0
$165$	0.629204	0.0489835
$166$	0	0
$167$	4.90327	0.379427	0.189713	−	0.981840i	$-0.439244\pi$
$167$	4.90327	0.379427	0.189713	+	0.981840i	$0.439244\pi$
$168$	0	0
$169$	−1.99521	−0.153478
$170$	0	0
$171$	−12.2152	−0.934120
$172$	0	0
$173$	4.74521	0.360771	0.180386	−	0.983596i	$-0.442265\pi$
$173$	4.74521	0.360771	0.180386	+	0.983596i	$0.442265\pi$
$174$	0	0
$175$	−9.31081	−0.703831
$176$	0	0
$177$	3.07875	0.231413
$178$	0	0
$179$	12.1050	0.904770	0.452385	−	0.891823i	$-0.350573\pi$
$179$	12.1050	0.904770	0.452385	+	0.891823i	$0.350573\pi$
$180$	0	0
$181$	−3.83231	−0.284853	−0.142427	−	0.989805i	$-0.545491\pi$
$181$	−3.83231	−0.284853	−0.142427	+	0.989805i	$0.545491\pi$
$182$	0	0
$183$	2.48162	0.183447
$184$	0	0
$185$	−10.4180	−0.765945
$186$	0	0
$187$	−4.33869	−0.317277
$188$	0	0
$189$	−4.96740	−0.361325
$190$	0	0
$191$	19.9858	1.44612	0.723060	−	0.690786i	$-0.242735\pi$
$191$	19.9858	1.44612	0.723060	+	0.690786i	$0.242735\pi$
$192$	0	0
$193$	18.4999	1.33165	0.665825	−	0.746108i	$-0.268080\pi$
$193$	18.4999	1.33165	0.665825	+	0.746108i	$0.268080\pi$
$194$	0	0
$195$	−1.28320	−0.0918920
$196$	0	0
$197$	−19.7677	−1.40839	−0.704196	−	0.710006i	$-0.748692\pi$
$197$	−19.7677	−1.40839	−0.704196	+	0.710006i	$0.748692\pi$
$198$	0	0
$199$	13.2396	0.938527	0.469264	−	0.883058i	$-0.344519\pi$
$199$	13.2396	0.938527	0.469264	+	0.883058i	$0.344519\pi$
$200$	0	0
$201$	−3.58164	−0.252629
$202$	0	0
$203$	−23.9024	−1.67762
$204$	0	0
$205$	−7.30554	−0.510241
$206$	0	0
$207$	3.57295	0.248337
$208$	0	0
$209$	6.85861	0.474420
$210$	0	0
$211$	14.0915	0.970099	0.485050	−	0.874487i	$-0.338802\pi$
$211$	14.0915	0.970099	0.485050	+	0.874487i	$0.338802\pi$
$212$	0	0
$213$	−2.39980	−0.164432
$214$	0	0
$215$	−6.30010	−0.429663
$216$	0	0
$217$	13.2245	0.897736
$218$	0	0
$219$	−1.14627	−0.0774577
$220$	0	0
$221$	8.84834	0.595204
$222$	0	0
$223$	−4.51822	−0.302563	−0.151281	−	0.988491i	$-0.548340\pi$
$223$	−4.51822	−0.302563	−0.151281	+	0.988491i	$0.548340\pi$
$224$	0	0
$225$	10.2756	0.685039
$226$	0	0
$227$	−8.24400	−0.547174	−0.273587	−	0.961847i	$-0.588210\pi$
$227$	−8.24400	−0.547174	−0.273587	+	0.961847i	$0.588210\pi$
$228$	0	0
$229$	24.4909	1.61840	0.809201	−	0.587532i	$-0.199900\pi$
$229$	24.4909	1.61840	0.809201	+	0.587532i	$0.199900\pi$
$230$	0	0
$231$	1.37020	0.0901525
$232$	0	0
$233$	14.5272	0.951706	0.475853	−	0.879525i	$-0.342139\pi$
$233$	14.5272	0.951706	0.475853	+	0.879525i	$0.342139\pi$
$234$	0	0
$235$	−2.81698	−0.183760
$236$	0	0
$237$	0.843454	0.0547882
$238$	0	0
$239$	−17.7484	−1.14805	−0.574026	−	0.818837i	$-0.694619\pi$
$239$	−17.7484	−1.14805	−0.574026	+	0.818837i	$0.694619\pi$
$240$	0	0
$241$	21.6534	1.39482	0.697409	−	0.716674i	$-0.254336\pi$
$241$	21.6534	1.39482	0.697409	+	0.716674i	$0.254336\pi$
$242$	0	0
$243$	8.27103	0.530587
$244$	0	0
$245$	−0.131680	−0.00841275
$246$	0	0
$247$	−13.9875	−0.890002
$248$	0	0
$249$	−4.47708	−0.283724
$250$	0	0
$251$	0.705326	0.0445198	0.0222599	−	0.999752i	$-0.492914\pi$
$251$	0.705326	0.0445198	0.0222599	+	0.999752i	$0.492914\pi$
$252$	0	0
$253$	−2.00614	−0.126125
$254$	0	0
$255$	−1.03175	−0.0646108
$256$	0	0
$257$	−0.809803	−0.0505141	−0.0252571	−	0.999681i	$-0.508040\pi$
$257$	−0.809803	−0.0505141	−0.0252571	+	0.999681i	$0.508040\pi$
$258$	0	0
$259$	−22.6869	−1.40970
$260$	0	0
$261$	26.3791	1.63283
$262$	0	0
$263$	−14.1848	−0.874672	−0.437336	−	0.899298i	$-0.644078\pi$
$263$	−14.1848	−0.874672	−0.437336	+	0.899298i	$0.644078\pi$
$264$	0	0
$265$	1.82683	0.112221
$266$	0	0
$267$	3.63445	0.222425
$268$	0	0
$269$	0.154171	0.00939995	0.00469997	−	0.999989i	$-0.498504\pi$
$269$	0.154171	0.00939995	0.00469997	+	0.999989i	$0.498504\pi$
$270$	0	0
$271$	16.4281	0.997934	0.498967	−	0.866621i	$-0.333713\pi$
$271$	16.4281	0.997934	0.498967	+	0.866621i	$0.333713\pi$
$272$	0	0
$273$	−2.79439	−0.169124
$274$	0	0
$275$	−5.76955	−0.347917
$276$	0	0
$277$	−29.8540	−1.79375	−0.896877	−	0.442281i	$-0.854170\pi$
$277$	−29.8540	−1.79375	−0.896877	+	0.442281i	$0.854170\pi$
$278$	0	0
$279$	−14.5948	−0.873766
$280$	0	0
$281$	14.8895	0.888233	0.444117	−	0.895969i	$-0.353518\pi$
$281$	14.8895	0.888233	0.444117	+	0.895969i	$0.353518\pi$
$282$	0	0
$283$	25.5236	1.51722	0.758610	−	0.651545i	$-0.225879\pi$
$283$	25.5236	1.51722	0.758610	+	0.651545i	$0.225879\pi$
$284$	0	0
$285$	1.63099	0.0966117
$286$	0	0
$287$	−15.9091	−0.939082
$288$	0	0
$289$	−9.88554	−0.581502
$290$	0	0
$291$	−5.20380	−0.305052
$292$	0	0
$293$	6.46654	0.377780	0.188890	−	0.981998i	$-0.439511\pi$
$293$	6.46654	0.377780	0.188890	+	0.981998i	$0.439511\pi$
$294$	0	0
$295$	11.5652	0.673352
$296$	0	0
$297$	−3.07810	−0.178610
$298$	0	0
$299$	4.09134	0.236608
$300$	0	0
$301$	−13.7195	−0.790782
$302$	0	0
$303$	0.965765	0.0554818
$304$	0	0
$305$	9.32212	0.533783
$306$	0	0
$307$	−18.9859	−1.08358	−0.541791	−	0.840513i	$-0.682254\pi$
$307$	−18.9859	−1.08358	−0.541791	+	0.840513i	$0.682254\pi$
$308$	0	0
$309$	1.03212	0.0587152
$310$	0	0
$311$	6.11973	0.347018	0.173509	−	0.984832i	$-0.444489\pi$
$311$	6.11973	0.347018	0.173509	+	0.984832i	$0.444489\pi$
$312$	0	0
$313$	5.73279	0.324037	0.162018	−	0.986788i	$-0.448200\pi$
$313$	5.73279	0.324037	0.162018	+	0.986788i	$0.448200\pi$
$314$	0	0
$315$	−9.16701	−0.516502
$316$	0	0
$317$	−16.8637	−0.947158	−0.473579	−	0.880751i	$-0.657038\pi$
$317$	−16.8637	−0.947158	−0.473579	+	0.880751i	$0.657038\pi$
$318$	0	0
$319$	−14.8114	−0.829278
$320$	0	0
$321$	4.36102	0.243408
$322$	0	0
$323$	−11.2466	−0.625775
$324$	0	0
$325$	11.7664	0.652684
$326$	0	0
$327$	0.449739	0.0248706
$328$	0	0
$329$	−6.13446	−0.338204
$330$	0	0
$331$	13.6510	0.750325	0.375162	−	0.926959i	$-0.377587\pi$
$331$	13.6510	0.750325	0.375162	+	0.926959i	$0.377587\pi$
$332$	0	0
$333$	25.0377	1.37206
$334$	0	0
$335$	−13.4543	−0.735087
$336$	0	0
$337$	−18.3359	−0.998821	−0.499410	−	0.866366i	$-0.666450\pi$
$337$	−18.3359	−0.998821	−0.499410	+	0.866366i	$0.666450\pi$
$338$	0	0
$339$	1.66032	0.0901763
$340$	0	0
$341$	8.19469	0.443767
$342$	0	0
$343$	−18.6619	−1.00765
$344$	0	0
$345$	−0.477066	−0.0256844
$346$	0	0
$347$	30.0533	1.61335	0.806673	−	0.590998i	$-0.201266\pi$
$347$	30.0533	1.61335	0.806673	+	0.590998i	$0.201266\pi$
$348$	0	0
$349$	14.3946	0.770527	0.385263	−	0.922807i	$-0.374111\pi$
$349$	14.3946	0.770527	0.385263	+	0.922807i	$0.374111\pi$
$350$	0	0
$351$	6.27750	0.335068
$352$	0	0
$353$	−29.3940	−1.56448	−0.782241	−	0.622975i	$-0.785924\pi$
$353$	−29.3940	−1.56448	−0.782241	+	0.622975i	$0.785924\pi$
$354$	0	0
$355$	−9.01477	−0.478454
$356$	0	0
$357$	−2.24681	−0.118914
$358$	0	0
$359$	−20.1770	−1.06490	−0.532450	−	0.846462i	$-0.678728\pi$
$359$	−20.1770	−1.06490	−0.532450	+	0.846462i	$0.678728\pi$
$360$	0	0
$361$	−1.22144	−0.0642862
$362$	0	0
$363$	−2.68078	−0.140705
$364$	0	0
$365$	−4.30592	−0.225382
$366$	0	0
$367$	0.0988134	0.00515801	0.00257901	−	0.999997i	$-0.499179\pi$
$367$	0.0988134	0.00515801	0.00257901	+	0.999997i	$0.499179\pi$
$368$	0	0
$369$	17.5575	0.914008
$370$	0	0
$371$	3.97823	0.206539
$372$	0	0
$373$	−23.1809	−1.20026	−0.600130	−	0.799902i	$-0.704885\pi$
$373$	−23.1809	−1.20026	−0.600130	+	0.799902i	$0.704885\pi$
$374$	0	0
$375$	−3.30609	−0.170726
$376$	0	0
$377$	30.2064	1.55571
$378$	0	0
$379$	−25.1812	−1.29347	−0.646735	−	0.762715i	$-0.723866\pi$
$379$	−25.1812	−1.29347	−0.646735	+	0.762715i	$0.723866\pi$
$380$	0	0
$381$	−4.67677	−0.239598
$382$	0	0
$383$	−4.72699	−0.241538	−0.120769	−	0.992681i	$-0.538536\pi$
$383$	−4.72699	−0.241538	−0.120769	+	0.992681i	$0.538536\pi$
$384$	0	0
$385$	5.14710	0.262321
$386$	0	0
$387$	15.1411	0.769668
$388$	0	0
$389$	2.41392	0.122390	0.0611952	−	0.998126i	$-0.480509\pi$
$389$	2.41392	0.122390	0.0611952	+	0.998126i	$0.480509\pi$
$390$	0	0
$391$	3.28962	0.166363
$392$	0	0
$393$	1.83283	0.0924540
$394$	0	0
$395$	3.16840	0.159420
$396$	0	0
$397$	17.1842	0.862451	0.431226	−	0.902244i	$-0.358081\pi$
$397$	17.1842	0.862451	0.431226	+	0.902244i	$0.358081\pi$
$398$	0	0
$399$	3.55177	0.177811
$400$	0	0
$401$	−1.65099	−0.0824465	−0.0412233	−	0.999150i	$-0.513125\pi$
$401$	−1.65099	−0.0824465	−0.0412233	+	0.999150i	$0.513125\pi$
$402$	0	0
$403$	−16.7123	−0.832498
$404$	0	0
$405$	9.74450	0.484208
$406$	0	0
$407$	−14.0582	−0.696839
$408$	0	0
$409$	5.54443	0.274155	0.137077	−	0.990560i	$-0.456229\pi$
$409$	5.54443	0.274155	0.137077	+	0.990560i	$0.456229\pi$
$410$	0	0
$411$	0.162022	0.00799198
$412$	0	0
$413$	25.1852	1.23928
$414$	0	0
$415$	−16.8180	−0.825563
$416$	0	0
$417$	0.0209264	0.00102477
$418$	0	0
$419$	−16.5645	−0.809228	−0.404614	−	0.914487i	$-0.632594\pi$
$419$	−16.5645	−0.809228	−0.404614	+	0.914487i	$0.632594\pi$
$420$	0	0
$421$	−35.0445	−1.70797	−0.853983	−	0.520302i	$-0.825820\pi$
$421$	−35.0445	−1.70797	−0.853983	+	0.520302i	$0.825820\pi$
$422$	0	0
$423$	6.77010	0.329174
$424$	0	0
$425$	9.46074	0.458913
$426$	0	0
$427$	20.3005	0.982410
$428$	0	0
$429$	−1.73157	−0.0836012
$430$	0	0
$431$	−13.1111	−0.631542	−0.315771	−	0.948836i	$-0.602263\pi$
$431$	−13.1111	−0.631542	−0.315771	+	0.948836i	$0.602263\pi$
$432$	0	0
$433$	−6.73845	−0.323829	−0.161915	−	0.986805i	$-0.551767\pi$
$433$	−6.73845	−0.323829	−0.161915	+	0.986805i	$0.551767\pi$
$434$	0	0
$435$	−3.52218	−0.168876
$436$	0	0
$437$	−5.20023	−0.248761
$438$	0	0
$439$	28.3510	1.35312	0.676560	−	0.736387i	$-0.263470\pi$
$439$	28.3510	1.35312	0.676560	+	0.736387i	$0.263470\pi$
$440$	0	0
$441$	0.316470	0.0150700
$442$	0	0
$443$	30.8531	1.46588	0.732939	−	0.680295i	$-0.238148\pi$
$443$	30.8531	1.46588	0.732939	+	0.680295i	$0.238148\pi$
$444$	0	0
$445$	13.6527	0.647198
$446$	0	0
$447$	−4.86071	−0.229904
$448$	0	0
$449$	−34.1984	−1.61392	−0.806962	−	0.590603i	$-0.798890\pi$
$449$	−34.1984	−1.61392	−0.806962	+	0.590603i	$0.798890\pi$
$450$	0	0
$451$	−9.85822	−0.464205
$452$	0	0
$453$	−2.41748	−0.113583
$454$	0	0
$455$	−10.4970	−0.492108
$456$	0	0
$457$	13.8695	0.648787	0.324394	−	0.945922i	$-0.394840\pi$
$457$	13.8695	0.648787	0.324394	+	0.945922i	$0.394840\pi$
$458$	0	0
$459$	5.04739	0.235592
$460$	0	0
$461$	−29.4101	−1.36976	−0.684882	−	0.728654i	$-0.740146\pi$
$461$	−29.4101	−1.36976	−0.684882	+	0.728654i	$0.740146\pi$
$462$	0	0
$463$	10.1122	0.469953	0.234976	−	0.972001i	$-0.424499\pi$
$463$	10.1122	0.469953	0.234976	+	0.972001i	$0.424499\pi$
$464$	0	0
$465$	1.94872	0.0903695
$466$	0	0
$467$	−14.1149	−0.653161	−0.326581	−	0.945169i	$-0.605897\pi$
$467$	−14.1149	−0.653161	−0.326581	+	0.945169i	$0.605897\pi$
$468$	0	0
$469$	−29.2990	−1.35290
$470$	0	0
$471$	0.817425	0.0376650
$472$	0	0
$473$	−8.50147	−0.390898
$474$	0	0
$475$	−14.9555	−0.686208
$476$	0	0
$477$	−4.39045	−0.201025
$478$	0	0
$479$	7.59718	0.347124	0.173562	−	0.984823i	$-0.444472\pi$
$479$	7.59718	0.347124	0.173562	+	0.984823i	$0.444472\pi$
$480$	0	0
$481$	28.6704	1.30726
$482$	0	0
$483$	−1.03889	−0.0472712
$484$	0	0
$485$	−19.5479	−0.887623
$486$	0	0
$487$	−25.6079	−1.16040	−0.580202	−	0.814473i	$-0.697026\pi$
$487$	−25.6079	−1.16040	−0.580202	+	0.814473i	$0.697026\pi$
$488$	0	0
$489$	−7.17462	−0.324448
$490$	0	0
$491$	−0.943905	−0.0425978	−0.0212989	−	0.999773i	$-0.506780\pi$
$491$	−0.943905	−0.0425978	−0.0212989	+	0.999773i	$0.506780\pi$
$492$	0	0
$493$	24.2873	1.09384
$494$	0	0
$495$	−5.68044	−0.255317
$496$	0	0
$497$	−19.6312	−0.880580
$498$	0	0
$499$	−32.2486	−1.44364	−0.721822	−	0.692078i	$-0.756695\pi$
$499$	−32.2486	−1.44364	−0.721822	+	0.692078i	$0.756695\pi$
$500$	0	0
$501$	1.57343	0.0702959
$502$	0	0
$503$	25.0618	1.11745	0.558726	−	0.829352i	$-0.311290\pi$
$503$	25.0618	1.11745	0.558726	+	0.829352i	$0.311290\pi$
$504$	0	0
$505$	3.62786	0.161438
$506$	0	0
$507$	−0.640252	−0.0284346
$508$	0	0
$509$	13.8515	0.613958	0.306979	−	0.951716i	$-0.400682\pi$
$509$	13.8515	0.613958	0.306979	+	0.951716i	$0.400682\pi$
$510$	0	0
$511$	−9.37687	−0.414808
$512$	0	0
$513$	−7.97891	−0.352278
$514$	0	0
$515$	3.87712	0.170846
$516$	0	0
$517$	−3.80129	−0.167180
$518$	0	0
$519$	1.52271	0.0668396
$520$	0	0
$521$	−5.29758	−0.232091	−0.116046	−	0.993244i	$-0.537022\pi$
$521$	−5.29758	−0.232091	−0.116046	+	0.993244i	$0.537022\pi$
$522$	0	0
$523$	−2.42411	−0.105999	−0.0529994	−	0.998595i	$-0.516878\pi$
$523$	−2.42411	−0.105999	−0.0529994	+	0.998595i	$0.516878\pi$
$524$	0	0
$525$	−2.98779	−0.130398
$526$	0	0
$527$	−13.4374	−0.585343
$528$	0	0
$529$	−21.4789	−0.933867
$530$	0	0
$531$	−27.7949	−1.20619
$532$	0	0
$533$	20.1049	0.870839
$534$	0	0
$535$	16.3820	0.708256
$536$	0	0
$537$	3.88443	0.167626
$538$	0	0
$539$	−0.177692	−0.00765373
$540$	0	0
$541$	−10.2679	−0.441453	−0.220727	−	0.975336i	$-0.570843\pi$
$541$	−10.2679	−0.441453	−0.220727	+	0.975336i	$0.570843\pi$
$542$	0	0
$543$	−1.22977	−0.0527744
$544$	0	0
$545$	1.68943	0.0723671
$546$	0	0
$547$	−40.1700	−1.71755	−0.858773	−	0.512356i	$-0.828773\pi$
$547$	−40.1700	−1.71755	−0.858773	+	0.512356i	$0.828773\pi$
$548$	0	0
$549$	−22.4040	−0.956180
$550$	0	0
$551$	−38.3934	−1.63561
$552$	0	0
$553$	6.89974	0.293407
$554$	0	0
$555$	−3.34307	−0.141906
$556$	0	0
$557$	1.69529	0.0718317	0.0359159	−	0.999355i	$-0.488565\pi$
$557$	1.69529	0.0718317	0.0359159	+	0.999355i	$0.488565\pi$
$558$	0	0
$559$	17.3379	0.733316
$560$	0	0
$561$	−1.39226	−0.0587814
$562$	0	0
$563$	−0.796711	−0.0335774	−0.0167887	−	0.999859i	$-0.505344\pi$
$563$	−0.796711	−0.0335774	−0.0167887	+	0.999859i	$0.505344\pi$
$564$	0	0
$565$	6.23694	0.262390
$566$	0	0
$567$	21.2203	0.891169
$568$	0	0
$569$	−34.3408	−1.43964	−0.719820	−	0.694161i	$-0.755776\pi$
$569$	−34.3408	−1.43964	−0.719820	+	0.694161i	$0.755776\pi$
$570$	0	0
$571$	−13.1193	−0.549027	−0.274513	−	0.961583i	$-0.588517\pi$
$571$	−13.1193	−0.549027	−0.274513	+	0.961583i	$0.588517\pi$
$572$	0	0
$573$	6.41333	0.267921
$574$	0	0
$575$	4.37450	0.182429
$576$	0	0
$577$	−33.1973	−1.38202	−0.691011	−	0.722844i	$-0.742834\pi$
$577$	−33.1973	−1.38202	−0.691011	+	0.722844i	$0.742834\pi$
$578$	0	0
$579$	5.93651	0.246713
$580$	0	0
$581$	−36.6240	−1.51942
$582$	0	0
$583$	2.46515	0.102096
$584$	0	0
$585$	11.5847	0.478969
$586$	0	0
$587$	45.0798	1.86064	0.930321	−	0.366745i	$-0.119528\pi$
$587$	45.0798	1.86064	0.930321	+	0.366745i	$0.119528\pi$
$588$	0	0
$589$	21.2419	0.875257
$590$	0	0
$591$	−6.34336	−0.260931
$592$	0	0
$593$	−35.5467	−1.45973	−0.729865	−	0.683592i	$-0.760417\pi$
$593$	−35.5467	−1.45973	−0.729865	+	0.683592i	$0.760417\pi$
$594$	0	0
$595$	−8.44007	−0.346009
$596$	0	0
$597$	4.24850	0.173880
$598$	0	0
$599$	2.70384	0.110476	0.0552379	−	0.998473i	$-0.482408\pi$
$599$	2.70384	0.110476	0.0552379	+	0.998473i	$0.482408\pi$
$600$	0	0
$601$	−11.1644	−0.455406	−0.227703	−	0.973731i	$-0.573122\pi$
$601$	−11.1644	−0.455406	−0.227703	+	0.973731i	$0.573122\pi$
$602$	0	0
$603$	32.3349	1.31678
$604$	0	0
$605$	−10.0703	−0.409414
$606$	0	0
$607$	−13.2066	−0.536038	−0.268019	−	0.963414i	$-0.586369\pi$
$607$	−13.2066	−0.536038	−0.268019	+	0.963414i	$0.586369\pi$
$608$	0	0
$609$	−7.67015	−0.310810
$610$	0	0
$611$	7.75236	0.313627
$612$	0	0
$613$	4.14238	0.167309	0.0836546	−	0.996495i	$-0.473341\pi$
$613$	4.14238	0.167309	0.0836546	+	0.996495i	$0.473341\pi$
$614$	0	0
$615$	−2.34431	−0.0945316
$616$	0	0
$617$	−13.2622	−0.533915	−0.266958	−	0.963708i	$-0.586018\pi$
$617$	−13.2622	−0.533915	−0.266958	+	0.963708i	$0.586018\pi$
$618$	0	0
$619$	−22.7925	−0.916107	−0.458053	−	0.888925i	$-0.651453\pi$
$619$	−22.7925	−0.916107	−0.458053	+	0.888925i	$0.651453\pi$
$620$	0	0
$621$	2.33383	0.0936535
$622$	0	0
$623$	29.7310	1.19115
$624$	0	0
$625$	5.31551	0.212620
$626$	0	0
$627$	2.20089	0.0878951
$628$	0	0
$629$	23.0522	0.919153
$630$	0	0
$631$	20.5766	0.819143	0.409572	−	0.912278i	$-0.365678\pi$
$631$	20.5766	0.819143	0.409572	+	0.912278i	$0.365678\pi$
$632$	0	0
$633$	4.52189	0.179729
$634$	0	0
$635$	−17.5681	−0.697169
$636$	0	0
$637$	0.362385	0.0143582
$638$	0	0
$639$	21.6654	0.857068
$640$	0	0
$641$	44.1596	1.74420	0.872099	−	0.489329i	$-0.162758\pi$
$641$	44.1596	1.74420	0.872099	+	0.489329i	$0.162758\pi$
$642$	0	0
$643$	24.0553	0.948650	0.474325	−	0.880350i	$-0.342692\pi$
$643$	24.0553	0.948650	0.474325	+	0.880350i	$0.342692\pi$
$644$	0	0
$645$	−2.02167	−0.0796031
$646$	0	0
$647$	14.9928	0.589426	0.294713	−	0.955586i	$-0.404776\pi$
$647$	14.9928	0.589426	0.294713	+	0.955586i	$0.404776\pi$
$648$	0	0
$649$	15.6063	0.612600
$650$	0	0
$651$	4.24366	0.166322
$652$	0	0
$653$	13.9919	0.547545	0.273773	−	0.961794i	$-0.411728\pi$
$653$	13.9919	0.547545	0.273773	+	0.961794i	$0.411728\pi$
$654$	0	0
$655$	6.88496	0.269018
$656$	0	0
$657$	10.3485	0.403733
$658$	0	0
$659$	37.9086	1.47671	0.738354	−	0.674413i	$-0.235603\pi$
$659$	37.9086	1.47671	0.738354	+	0.674413i	$0.235603\pi$
$660$	0	0
$661$	24.0458	0.935273	0.467637	−	0.883921i	$-0.345106\pi$
$661$	24.0458	0.935273	0.467637	+	0.883921i	$0.345106\pi$
$662$	0	0
$663$	2.83939	0.110273
$664$	0	0
$665$	13.3421	0.517383
$666$	0	0
$667$	11.2301	0.434830
$668$	0	0
$669$	−1.44987	−0.0560553
$670$	0	0
$671$	12.5794	0.485623
$672$	0	0
$673$	43.8276	1.68943	0.844714	−	0.535217i	$-0.179770\pi$
$673$	43.8276	1.68943	0.844714	+	0.535217i	$0.179770\pi$
$674$	0	0
$675$	6.71196	0.258344
$676$	0	0
$677$	4.73446	0.181960	0.0909801	−	0.995853i	$-0.471000\pi$
$677$	4.73446	0.181960	0.0909801	+	0.995853i	$0.471000\pi$
$678$	0	0
$679$	−42.5688	−1.63364
$680$	0	0
$681$	−2.64546	−0.101374
$682$	0	0
$683$	−27.1872	−1.04029	−0.520144	−	0.854079i	$-0.674122\pi$
$683$	−27.1872	−1.04029	−0.520144	+	0.854079i	$0.674122\pi$
$684$	0	0
$685$	0.608631	0.0232546
$686$	0	0
$687$	7.85899	0.299839
$688$	0	0
$689$	−5.02744	−0.191530
$690$	0	0
$691$	16.2101	0.616663	0.308331	−	0.951279i	$-0.400229\pi$
$691$	16.2101	0.616663	0.308331	+	0.951279i	$0.400229\pi$
$692$	0	0
$693$	−12.3701	−0.469902
$694$	0	0
$695$	0.0786092	0.00298182
$696$	0	0
$697$	16.1652	0.612302
$698$	0	0
$699$	4.66169	0.176321
$700$	0	0
$701$	34.7625	1.31296	0.656481	−	0.754343i	$-0.272044\pi$
$701$	34.7625	1.31296	0.656481	+	0.754343i	$0.272044\pi$
$702$	0	0
$703$	−36.4410	−1.37440
$704$	0	0
$705$	−0.903955	−0.0340449
$706$	0	0
$707$	7.90028	0.297121
$708$	0	0
$709$	0.560357	0.0210447	0.0105223	−	0.999945i	$-0.496651\pi$
$709$	0.560357	0.0210447	0.0105223	+	0.999945i	$0.496651\pi$
$710$	0	0
$711$	−7.61468	−0.285573
$712$	0	0
$713$	−6.21325	−0.232688
$714$	0	0
$715$	−6.50459	−0.243258
$716$	0	0
$717$	−5.69538	−0.212698
$718$	0	0
$719$	−10.7798	−0.402020	−0.201010	−	0.979589i	$-0.564422\pi$
$719$	−10.7798	−0.402020	−0.201010	+	0.979589i	$0.564422\pi$
$720$	0	0
$721$	8.44308	0.314437
$722$	0	0
$723$	6.94845	0.258416
$724$	0	0
$725$	32.2970	1.19948
$726$	0	0
$727$	−25.5181	−0.946413	−0.473207	−	0.880951i	$-0.656904\pi$
$727$	−25.5181	−0.946413	−0.473207	+	0.880951i	$0.656904\pi$
$728$	0	0
$729$	−21.5974	−0.799904
$730$	0	0
$731$	13.9405	0.515607
$732$	0	0
$733$	49.6083	1.83233	0.916163	−	0.400806i	$-0.131270\pi$
$733$	49.6083	1.83233	0.916163	+	0.400806i	$0.131270\pi$
$734$	0	0
$735$	−0.0422555	−0.00155862
$736$	0	0
$737$	−18.1554	−0.668765
$738$	0	0
$739$	0.0908565	0.00334221	0.00167111	−	0.999999i	$-0.499468\pi$
$739$	0.0908565	0.00334221	0.00167111	+	0.999999i	$0.499468\pi$
$740$	0	0
$741$	−4.48850	−0.164889
$742$	0	0
$743$	−31.7804	−1.16591	−0.582955	−	0.812504i	$-0.698104\pi$
$743$	−31.7804	−1.16591	−0.582955	+	0.812504i	$0.698104\pi$
$744$	0	0
$745$	−18.2591	−0.668961
$746$	0	0
$747$	40.4190	1.47885
$748$	0	0
$749$	35.6746	1.30352
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	0.226335	0.00824812
$754$	0	0
$755$	−9.08118	−0.330498
$756$	0	0
$757$	−11.3038	−0.410842	−0.205421	−	0.978674i	$-0.565856\pi$
$757$	−11.3038	−0.410842	−0.205421	+	0.978674i	$0.565856\pi$
$758$	0	0
$759$	−0.643760	−0.0233670
$760$	0	0
$761$	−41.9436	−1.52045	−0.760227	−	0.649657i	$-0.774912\pi$
$761$	−41.9436	−1.52045	−0.760227	+	0.649657i	$0.774912\pi$
$762$	0	0
$763$	3.67902	0.133189
$764$	0	0
$765$	9.31462	0.336771
$766$	0	0
$767$	−31.8275	−1.14923
$768$	0	0
$769$	19.6203	0.707526	0.353763	−	0.935335i	$-0.384902\pi$
$769$	19.6203	0.707526	0.353763	+	0.935335i	$0.384902\pi$
$770$	0	0
$771$	−0.259861	−0.00935868
$772$	0	0
$773$	47.1732	1.69670	0.848352	−	0.529433i	$-0.177595\pi$
$773$	47.1732	1.69670	0.848352	+	0.529433i	$0.177595\pi$
$774$	0	0
$775$	−17.8689	−0.641871
$776$	0	0
$777$	−7.28011	−0.261173
$778$	0	0
$779$	−25.5540	−0.915567
$780$	0	0
$781$	−12.1647	−0.435287
$782$	0	0
$783$	17.2307	0.615776
$784$	0	0
$785$	3.07063	0.109595
$786$	0	0
$787$	−25.8127	−0.920123	−0.460062	−	0.887887i	$-0.652173\pi$
$787$	−25.8127	−0.920123	−0.460062	+	0.887887i	$0.652173\pi$
$788$	0	0
$789$	−4.55182	−0.162049
$790$	0	0
$791$	13.5820	0.482920
$792$	0	0
$793$	−25.6545	−0.911019
$794$	0	0
$795$	0.586220	0.0207911
$796$	0	0
$797$	14.9003	0.527796	0.263898	−	0.964551i	$-0.414992\pi$
$797$	14.9003	0.527796	0.263898	+	0.964551i	$0.414992\pi$
$798$	0	0
$799$	6.23324	0.220516
$800$	0	0
$801$	−32.8117	−1.15934
$802$	0	0
$803$	−5.81048	−0.205047
$804$	0	0
$805$	−3.90256	−0.137547
$806$	0	0
$807$	0.0494725	0.00174152
$808$	0	0
$809$	−35.2734	−1.24015	−0.620073	−	0.784544i	$-0.712897\pi$
$809$	−35.2734	−1.24015	−0.620073	+	0.784544i	$0.712897\pi$
$810$	0	0
$811$	4.72789	0.166019	0.0830093	−	0.996549i	$-0.473547\pi$
$811$	4.72789	0.166019	0.0830093	+	0.996549i	$0.473547\pi$
$812$	0	0
$813$	5.27168	0.184886
$814$	0	0
$815$	−26.9512	−0.944059
$816$	0	0
$817$	−22.0371	−0.770981
$818$	0	0
$819$	25.2277	0.881526
$820$	0	0
$821$	31.0676	1.08427	0.542133	−	0.840293i	$-0.317617\pi$
$821$	31.0676	1.08427	0.542133	+	0.840293i	$0.317617\pi$
$822$	0	0
$823$	−30.6842	−1.06958	−0.534792	−	0.844984i	$-0.679610\pi$
$823$	−30.6842	−1.06958	−0.534792	+	0.844984i	$0.679610\pi$
$824$	0	0
$825$	−1.85142	−0.0644580
$826$	0	0
$827$	38.1091	1.32518	0.662592	−	0.748981i	$-0.269456\pi$
$827$	38.1091	1.32518	0.662592	+	0.748981i	$0.269456\pi$
$828$	0	0
$829$	−2.36830	−0.0822546	−0.0411273	−	0.999154i	$-0.513095\pi$
$829$	−2.36830	−0.0822546	−0.0411273	+	0.999154i	$0.513095\pi$
$830$	0	0
$831$	−9.57999	−0.332326
$832$	0	0
$833$	0.291374	0.0100955
$834$	0	0
$835$	5.91055	0.204543
$836$	0	0
$837$	−9.53323	−0.329517
$838$	0	0
$839$	−27.9016	−0.963271	−0.481636	−	0.876372i	$-0.659957\pi$
$839$	−27.9016	−0.963271	−0.481636	+	0.876372i	$0.659957\pi$
$840$	0	0
$841$	53.9117	1.85902
$842$	0	0
$843$	4.77796	0.164562
$844$	0	0
$845$	−2.40508	−0.0827373
$846$	0	0
$847$	−21.9297	−0.753513
$848$	0	0
$849$	8.19039	0.281093
$850$	0	0
$851$	10.6590	0.365386
$852$	0	0
$853$	−20.7080	−0.709028	−0.354514	−	0.935051i	$-0.615354\pi$
$853$	−20.7080	−0.709028	−0.354514	+	0.935051i	$0.615354\pi$
$854$	0	0
$855$	−14.7246	−0.503569
$856$	0	0
$857$	35.7308	1.22054	0.610271	−	0.792193i	$-0.291061\pi$
$857$	35.7308	1.22054	0.610271	+	0.792193i	$0.291061\pi$
$858$	0	0
$859$	31.5548	1.07663	0.538317	−	0.842742i	$-0.319060\pi$
$859$	31.5548	1.07663	0.538317	+	0.842742i	$0.319060\pi$
$860$	0	0
$861$	−5.10513	−0.173982
$862$	0	0
$863$	21.9392	0.746819	0.373410	−	0.927667i	$-0.378189\pi$
$863$	21.9392	0.746819	0.373410	+	0.927667i	$0.378189\pi$
$864$	0	0
$865$	5.72001	0.194486
$866$	0	0
$867$	−3.17221	−0.107734
$868$	0	0
$869$	4.27550	0.145036
$870$	0	0
$871$	37.0263	1.25459
$872$	0	0
$873$	46.9797	1.59002
$874$	0	0
$875$	−27.0449	−0.914285
$876$	0	0
$877$	−28.2204	−0.952934	−0.476467	−	0.879192i	$-0.658083\pi$
$877$	−28.2204	−0.952934	−0.476467	+	0.879192i	$0.658083\pi$
$878$	0	0
$879$	2.07508	0.0699907
$880$	0	0
$881$	−31.1430	−1.04924	−0.524618	−	0.851338i	$-0.675792\pi$
$881$	−31.1430	−1.04924	−0.524618	+	0.851338i	$0.675792\pi$
$882$	0	0
$883$	−7.77388	−0.261612	−0.130806	−	0.991408i	$-0.541756\pi$
$883$	−7.77388	−0.261612	−0.130806	+	0.991408i	$0.541756\pi$
$884$	0	0
$885$	3.71121	0.124751
$886$	0	0
$887$	31.4694	1.05664	0.528319	−	0.849046i	$-0.322823\pi$
$887$	31.4694	1.05664	0.528319	+	0.849046i	$0.322823\pi$
$888$	0	0
$889$	−38.2576	−1.28312
$890$	0	0
$891$	13.1494	0.440521
$892$	0	0
$893$	−9.85351	−0.329735
$894$	0	0
$895$	14.5917	0.487747
$896$	0	0
$897$	1.31289	0.0438360
$898$	0	0
$899$	−45.8725	−1.52993
$900$	0	0
$901$	−4.04229	−0.134668
$902$	0	0
$903$	−4.40253	−0.146507
$904$	0	0
$905$	−4.61958	−0.153560
$906$	0	0
$907$	21.8828	0.726608	0.363304	−	0.931671i	$-0.381649\pi$
$907$	21.8828	0.726608	0.363304	+	0.931671i	$0.381649\pi$
$908$	0	0
$909$	−8.71890	−0.289188
$910$	0	0
$911$	31.3720	1.03940	0.519700	−	0.854349i	$-0.326044\pi$
$911$	31.3720	1.03940	0.519700	+	0.854349i	$0.326044\pi$
$912$	0	0
$913$	−22.6945	−0.751078
$914$	0	0
$915$	2.99142	0.0988932
$916$	0	0
$917$	14.9932	0.495118
$918$	0	0
$919$	22.7351	0.749962	0.374981	−	0.927033i	$-0.377649\pi$
$919$	22.7351	0.749962	0.374981	+	0.927033i	$0.377649\pi$
$920$	0	0
$921$	−6.09247	−0.200754
$922$	0	0
$923$	24.8087	0.816589
$924$	0	0
$925$	30.6546	1.00792
$926$	0	0
$927$	−9.31794	−0.306041
$928$	0	0
$929$	22.3903	0.734603	0.367302	−	0.930102i	$-0.380282\pi$
$929$	22.3903	0.734603	0.367302	+	0.930102i	$0.380282\pi$
$930$	0	0
$931$	−0.460604	−0.0150957
$932$	0	0
$933$	1.96379	0.0642916
$934$	0	0
$935$	−5.22998	−0.171039
$936$	0	0
$937$	9.34736	0.305365	0.152682	−	0.988275i	$-0.451209\pi$
$937$	9.34736	0.305365	0.152682	+	0.988275i	$0.451209\pi$
$938$	0	0
$939$	1.83962	0.0600338
$940$	0	0
$941$	44.7622	1.45921	0.729603	−	0.683871i	$-0.239705\pi$
$941$	44.7622	1.45921	0.729603	+	0.683871i	$0.239705\pi$
$942$	0	0
$943$	7.47455	0.243405
$944$	0	0
$945$	−5.98785	−0.194785
$946$	0	0
$947$	−26.9382	−0.875374	−0.437687	−	0.899127i	$-0.644202\pi$
$947$	−26.9382	−0.875374	−0.437687	+	0.899127i	$0.644202\pi$
$948$	0	0
$949$	11.8499	0.384664
$950$	0	0
$951$	−5.41146	−0.175479
$952$	0	0
$953$	−8.50412	−0.275475	−0.137738	−	0.990469i	$-0.543983\pi$
$953$	−8.50412	−0.275475	−0.137738	+	0.990469i	$0.543983\pi$
$954$	0	0
$955$	24.0914	0.779580
$956$	0	0
$957$	−4.75289	−0.153639
$958$	0	0
$959$	1.32540	0.0427993
$960$	0	0
$961$	−5.62013	−0.181295
$962$	0	0
$963$	−39.3712	−1.26872
$964$	0	0
$965$	22.3003	0.717871
$966$	0	0
$967$	−35.3831	−1.13784	−0.568921	−	0.822392i	$-0.692639\pi$
$967$	−35.3831	−1.13784	−0.568921	+	0.822392i	$0.692639\pi$
$968$	0	0
$969$	−3.60896	−0.115936
$970$	0	0
$971$	−14.2839	−0.458392	−0.229196	−	0.973380i	$-0.573610\pi$
$971$	−14.2839	−0.458392	−0.229196	+	0.973380i	$0.573610\pi$
$972$	0	0
$973$	0.171185	0.00548793
$974$	0	0
$975$	3.77579	0.120922
$976$	0	0
$977$	37.7758	1.20856	0.604278	−	0.796774i	$-0.293462\pi$
$977$	37.7758	1.20856	0.604278	+	0.796774i	$0.293462\pi$
$978$	0	0
$979$	18.4231	0.588806
$980$	0	0
$981$	−4.06023	−0.129633
$982$	0	0
$983$	−25.7633	−0.821723	−0.410861	−	0.911698i	$-0.634772\pi$
$983$	−25.7633	−0.821723	−0.410861	+	0.911698i	$0.634772\pi$
$984$	0	0
$985$	−23.8286	−0.759242
$986$	0	0
$987$	−1.96852	−0.0626585
$988$	0	0
$989$	6.44585	0.204966
$990$	0	0
$991$	44.0197	1.39833	0.699167	−	0.714959i	$-0.253555\pi$
$991$	44.0197	1.39833	0.699167	+	0.714959i	$0.253555\pi$
$992$	0	0
$993$	4.38052	0.139012
$994$	0	0
$995$	15.9593	0.505945
$996$	0	0
$997$	47.0752	1.49089	0.745443	−	0.666569i	$-0.232238\pi$
$997$	47.0752	1.49089	0.745443	+	0.666569i	$0.232238\pi$
$998$	0	0
$999$	16.3545	0.517434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.28	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.28	✓	44	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.320895 q^{3} +1.20543 q^{5} +2.62503 q^{7} -2.89703 q^{9} +O(q^{10})\)	\(q+0.320895 q^{3} +1.20543 q^{5} +2.62503 q^{7} -2.89703 q^{9} +1.62663 q^{11} -3.31735 q^{13} +0.386816 q^{15} -2.66730 q^{17} +4.21646 q^{19} +0.842357 q^{21} -1.23332 q^{23} -3.54694 q^{25} -1.89232 q^{27} -9.10559 q^{29} +5.03784 q^{31} +0.521975 q^{33} +3.16428 q^{35} -8.64255 q^{37} -1.06452 q^{39} -6.06053 q^{41} -5.22644 q^{43} -3.49216 q^{45} -2.33691 q^{47} -0.109239 q^{49} -0.855921 q^{51} +1.51550 q^{53} +1.96078 q^{55} +1.35304 q^{57} +9.59427 q^{59} +7.73345 q^{61} -7.60477 q^{63} -3.99883 q^{65} -11.1614 q^{67} -0.395764 q^{69} -7.47848 q^{71} -3.57210 q^{73} -1.13819 q^{75} +4.26993 q^{77} +2.62845 q^{79} +8.08384 q^{81} -13.9519 q^{83} -3.21523 q^{85} -2.92193 q^{87} +11.3260 q^{89} -8.70812 q^{91} +1.61662 q^{93} +5.08265 q^{95} -16.2165 q^{97} -4.71238 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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