LMFDB - Embedded newform 6008.2.a.e.1.10

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:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-2.26403 q^{3} -1.66165 q^{5} -0.979480 q^{7} +2.12585 q^{9} +O(q^{10})$	$q-2.26403 q^{3} -1.66165 q^{5} -0.979480 q^{7} +2.12585 q^{9} +5.43078 q^{11} +0.178916 q^{13} +3.76204 q^{15} +5.72760 q^{17} -3.83378 q^{19} +2.21758 q^{21} +2.87593 q^{23} -2.23890 q^{25} +1.97911 q^{27} +8.38444 q^{29} +3.31090 q^{31} -12.2955 q^{33} +1.62756 q^{35} -0.842185 q^{37} -0.405071 q^{39} +1.22734 q^{41} -12.7081 q^{43} -3.53243 q^{45} +1.06777 q^{47} -6.04062 q^{49} -12.9675 q^{51} +8.96357 q^{53} -9.02407 q^{55} +8.67982 q^{57} +6.92183 q^{59} -4.91239 q^{61} -2.08223 q^{63} -0.297296 q^{65} +11.8860 q^{67} -6.51120 q^{69} -11.2754 q^{71} +3.04580 q^{73} +5.06896 q^{75} -5.31934 q^{77} -8.00900 q^{79} -10.8583 q^{81} +1.01756 q^{83} -9.51729 q^{85} -18.9827 q^{87} -2.43391 q^{89} -0.175244 q^{91} -7.49600 q^{93} +6.37042 q^{95} +8.02333 q^{97} +11.5450 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	−2.26403	−1.30714	−0.653570	−	0.756866i	$-0.726730\pi$
$3$	−2.26403	−1.30714	−0.653570	+	0.756866i	$0.726730\pi$
$4$	0	0
$5$	−1.66165	−0.743114	−0.371557	−	0.928410i	$-0.621176\pi$
$5$	−1.66165	−0.743114	−0.371557	+	0.928410i	$0.621176\pi$
$6$	0	0
$7$	−0.979480	−0.370209	−0.185104	−	0.982719i	$-0.559262\pi$
$7$	−0.979480	−0.370209	−0.185104	+	0.982719i	$0.559262\pi$
$8$	0	0
$9$	2.12585	0.708616
$10$	0	0
$11$	5.43078	1.63744	0.818720	−	0.574193i	$-0.194684\pi$
$11$	5.43078	1.63744	0.818720	+	0.574193i	$0.194684\pi$
$12$	0	0
$13$	0.178916	0.0496222	0.0248111	−	0.999692i	$-0.492102\pi$
$13$	0.178916	0.0496222	0.0248111	+	0.999692i	$0.492102\pi$
$14$	0	0
$15$	3.76204	0.971355
$16$	0	0
$17$	5.72760	1.38915	0.694573	−	0.719422i	$-0.255593\pi$
$17$	5.72760	1.38915	0.694573	+	0.719422i	$0.255593\pi$
$18$	0	0
$19$	−3.83378	−0.879530	−0.439765	−	0.898113i	$-0.644938\pi$
$19$	−3.83378	−0.879530	−0.439765	+	0.898113i	$0.644938\pi$
$20$	0	0
$21$	2.21758	0.483915
$22$	0	0
$23$	2.87593	0.599673	0.299837	−	0.953991i	$-0.403068\pi$
$23$	2.87593	0.599673	0.299837	+	0.953991i	$0.403068\pi$
$24$	0	0
$25$	−2.23890	−0.447781
$26$	0	0
$27$	1.97911	0.380879
$28$	0	0
$29$	8.38444	1.55695	0.778476	−	0.627674i	$-0.215993\pi$
$29$	8.38444	1.55695	0.778476	+	0.627674i	$0.215993\pi$
$30$	0	0
$31$	3.31090	0.594656	0.297328	−	0.954775i	$-0.403905\pi$
$31$	3.31090	0.594656	0.297328	+	0.954775i	$0.403905\pi$
$32$	0	0
$33$	−12.2955	−2.14036
$34$	0	0
$35$	1.62756	0.275107
$36$	0	0
$37$	−0.842185	−0.138454	−0.0692272	−	0.997601i	$-0.522053\pi$
$37$	−0.842185	−0.138454	−0.0692272	+	0.997601i	$0.522053\pi$
$38$	0	0
$39$	−0.405071	−0.0648632
$40$	0	0
$41$	1.22734	0.191678	0.0958391	−	0.995397i	$-0.469447\pi$
$41$	1.22734	0.191678	0.0958391	+	0.995397i	$0.469447\pi$
$42$	0	0
$43$	−12.7081	−1.93796	−0.968980	−	0.247139i	$-0.920510\pi$
$43$	−12.7081	−1.93796	−0.968980	+	0.247139i	$0.920510\pi$
$44$	0	0
$45$	−3.53243	−0.526583
$46$	0	0
$47$	1.06777	0.155750	0.0778748	−	0.996963i	$-0.475187\pi$
$47$	1.06777	0.155750	0.0778748	+	0.996963i	$0.475187\pi$
$48$	0	0
$49$	−6.04062	−0.862946
$50$	0	0
$51$	−12.9675	−1.81581
$52$	0	0
$53$	8.96357	1.23124	0.615621	−	0.788043i	$-0.288905\pi$
$53$	8.96357	1.23124	0.615621	+	0.788043i	$0.288905\pi$
$54$	0	0
$55$	−9.02407	−1.21681
$56$	0	0
$57$	8.67982	1.14967
$58$	0	0
$59$	6.92183	0.901145	0.450572	−	0.892740i	$-0.351220\pi$
$59$	6.92183	0.901145	0.450572	+	0.892740i	$0.351220\pi$
$60$	0	0
$61$	−4.91239	−0.628967	−0.314483	−	0.949263i	$-0.601831\pi$
$61$	−4.91239	−0.628967	−0.314483	+	0.949263i	$0.601831\pi$
$62$	0	0
$63$	−2.08223	−0.262336
$64$	0	0
$65$	−0.297296	−0.0368750
$66$	0	0
$67$	11.8860	1.45211	0.726055	−	0.687637i	$-0.241352\pi$
$67$	11.8860	1.45211	0.726055	+	0.687637i	$0.241352\pi$
$68$	0	0
$69$	−6.51120	−0.783857
$70$	0	0
$71$	−11.2754	−1.33814	−0.669071	−	0.743198i	$-0.733308\pi$
$71$	−11.2754	−1.33814	−0.669071	+	0.743198i	$0.733308\pi$
$72$	0	0
$73$	3.04580	0.356484	0.178242	−	0.983987i	$-0.442959\pi$
$73$	3.04580	0.356484	0.178242	+	0.983987i	$0.442959\pi$
$74$	0	0
$75$	5.06896	0.585313
$76$	0	0
$77$	−5.31934	−0.606195
$78$	0	0
$79$	−8.00900	−0.901083	−0.450542	−	0.892755i	$-0.648769\pi$
$79$	−8.00900	−0.901083	−0.450542	+	0.892755i	$0.648769\pi$
$80$	0	0
$81$	−10.8583	−1.20648
$82$	0	0
$83$	1.01756	0.111692	0.0558458	−	0.998439i	$-0.482214\pi$
$83$	1.01756	0.111692	0.0558458	+	0.998439i	$0.482214\pi$
$84$	0	0
$85$	−9.51729	−1.03229
$86$	0	0
$87$	−18.9827	−2.03516
$88$	0	0
$89$	−2.43391	−0.257994	−0.128997	−	0.991645i	$-0.541176\pi$
$89$	−2.43391	−0.257994	−0.128997	+	0.991645i	$0.541176\pi$
$90$	0	0
$91$	−0.175244	−0.0183706
$92$	0	0
$93$	−7.49600	−0.777299
$94$	0	0
$95$	6.37042	0.653592
$96$	0	0
$97$	8.02333	0.814645	0.407323	−	0.913284i	$-0.366462\pi$
$97$	8.02333	0.814645	0.407323	+	0.913284i	$0.366462\pi$
$98$	0	0
$99$	11.5450	1.16032
$100$	0	0
$101$	9.21962	0.917386	0.458693	−	0.888595i	$-0.348318\pi$
$101$	9.21962	0.917386	0.458693	+	0.888595i	$0.348318\pi$
$102$	0	0
$103$	−15.4123	−1.51862	−0.759312	−	0.650727i	$-0.774464\pi$
$103$	−15.4123	−1.51862	−0.759312	+	0.650727i	$0.774464\pi$
$104$	0	0
$105$	−3.68484	−0.359604
$106$	0	0
$107$	−16.7577	−1.62003	−0.810014	−	0.586411i	$-0.800540\pi$
$107$	−16.7577	−1.62003	−0.810014	+	0.586411i	$0.800540\pi$
$108$	0	0
$109$	−14.2665	−1.36648	−0.683242	−	0.730192i	$-0.739431\pi$
$109$	−14.2665	−1.36648	−0.683242	+	0.730192i	$0.739431\pi$
$110$	0	0
$111$	1.90673	0.180979
$112$	0	0
$113$	−3.25926	−0.306606	−0.153303	−	0.988179i	$-0.548991\pi$
$113$	−3.25926	−0.306606	−0.153303	+	0.988179i	$0.548991\pi$
$114$	0	0
$115$	−4.77880	−0.445626
$116$	0	0
$117$	0.380347	0.0351631
$118$	0	0
$119$	−5.61007	−0.514274
$120$	0	0
$121$	18.4933	1.68121
$122$	0	0
$123$	−2.77874	−0.250550
$124$	0	0
$125$	12.0286	1.07587
$126$	0	0
$127$	3.20059	0.284007	0.142003	−	0.989866i	$-0.454646\pi$
$127$	3.20059	0.284007	0.142003	+	0.989866i	$0.454646\pi$
$128$	0	0
$129$	28.7715	2.53319
$130$	0	0
$131$	18.8360	1.64571	0.822855	−	0.568252i	$-0.192380\pi$
$131$	18.8360	1.64571	0.822855	+	0.568252i	$0.192380\pi$
$132$	0	0
$133$	3.75511	0.325610
$134$	0	0
$135$	−3.28859	−0.283037
$136$	0	0
$137$	20.8844	1.78428	0.892139	−	0.451761i	$-0.149204\pi$
$137$	20.8844	1.78428	0.892139	+	0.451761i	$0.149204\pi$
$138$	0	0
$139$	−17.2344	−1.46180	−0.730902	−	0.682482i	$-0.760900\pi$
$139$	−17.2344	−1.46180	−0.730902	+	0.682482i	$0.760900\pi$
$140$	0	0
$141$	−2.41746	−0.203587
$142$	0	0
$143$	0.971650	0.0812535
$144$	0	0
$145$	−13.9320	−1.15699
$146$	0	0
$147$	13.6762	1.12799
$148$	0	0
$149$	−2.22406	−0.182202	−0.0911011	−	0.995842i	$-0.529039\pi$
$149$	−2.22406	−0.182202	−0.0911011	+	0.995842i	$0.529039\pi$
$150$	0	0
$151$	9.05213	0.736652	0.368326	−	0.929697i	$-0.379931\pi$
$151$	9.05213	0.736652	0.368326	+	0.929697i	$0.379931\pi$
$152$	0	0
$153$	12.1760	0.984372
$154$	0	0
$155$	−5.50158	−0.441897
$156$	0	0
$157$	8.73106	0.696815	0.348407	−	0.937343i	$-0.386723\pi$
$157$	8.73106	0.696815	0.348407	+	0.937343i	$0.386723\pi$
$158$	0	0
$159$	−20.2938	−1.60941
$160$	0	0
$161$	−2.81692	−0.222004
$162$	0	0
$163$	3.61608	0.283234	0.141617	−	0.989922i	$-0.454770\pi$
$163$	3.61608	0.283234	0.141617	+	0.989922i	$0.454770\pi$
$164$	0	0
$165$	20.4308	1.59054
$166$	0	0
$167$	−3.49432	−0.270399	−0.135199	−	0.990818i	$-0.543168\pi$
$167$	−3.49432	−0.270399	−0.135199	+	0.990818i	$0.543168\pi$
$168$	0	0
$169$	−12.9680	−0.997538
$170$	0	0
$171$	−8.15004	−0.623250
$172$	0	0
$173$	0.0360332	0.00273955	0.00136978	−	0.999999i	$-0.499564\pi$
$173$	0.0360332	0.00273955	0.00136978	+	0.999999i	$0.499564\pi$
$174$	0	0
$175$	2.19296	0.165772
$176$	0	0
$177$	−15.6712	−1.17792
$178$	0	0
$179$	2.51588	0.188046	0.0940229	−	0.995570i	$-0.470027\pi$
$179$	2.51588	0.188046	0.0940229	+	0.995570i	$0.470027\pi$
$180$	0	0
$181$	16.8809	1.25475	0.627375	−	0.778717i	$-0.284129\pi$
$181$	16.8809	1.25475	0.627375	+	0.778717i	$0.284129\pi$
$182$	0	0
$183$	11.1218	0.822148
$184$	0	0
$185$	1.39942	0.102887
$186$	0	0
$187$	31.1053	2.27464
$188$	0	0
$189$	−1.93850	−0.141005
$190$	0	0
$191$	−2.66698	−0.192976	−0.0964879	−	0.995334i	$-0.530761\pi$
$191$	−2.66698	−0.192976	−0.0964879	+	0.995334i	$0.530761\pi$
$192$	0	0
$193$	−10.5894	−0.762239	−0.381120	−	0.924526i	$-0.624461\pi$
$193$	−10.5894	−0.762239	−0.381120	+	0.924526i	$0.624461\pi$
$194$	0	0
$195$	0.673088	0.0482008
$196$	0	0
$197$	18.2533	1.30050	0.650248	−	0.759722i	$-0.274665\pi$
$197$	18.2533	1.30050	0.650248	+	0.759722i	$0.274665\pi$
$198$	0	0
$199$	−1.52577	−0.108159	−0.0540796	−	0.998537i	$-0.517222\pi$
$199$	−1.52577	−0.108159	−0.0540796	+	0.998537i	$0.517222\pi$
$200$	0	0
$201$	−26.9104	−1.89811
$202$	0	0
$203$	−8.21240	−0.576397
$204$	0	0
$205$	−2.03941	−0.142439
$206$	0	0
$207$	6.11379	0.424938
$208$	0	0
$209$	−20.8204	−1.44018
$210$	0	0
$211$	−14.0381	−0.966423	−0.483212	−	0.875504i	$-0.660530\pi$
$211$	−14.0381	−0.966423	−0.483212	+	0.875504i	$0.660530\pi$
$212$	0	0
$213$	25.5279	1.74914
$214$	0	0
$215$	21.1164	1.44013
$216$	0	0
$217$	−3.24296	−0.220147
$218$	0	0
$219$	−6.89579	−0.465974
$220$	0	0
$221$	1.02476	0.0689325
$222$	0	0
$223$	0.250411	0.0167688	0.00838438	−	0.999965i	$-0.497331\pi$
$223$	0.250411	0.0167688	0.00838438	+	0.999965i	$0.497331\pi$
$224$	0	0
$225$	−4.75957	−0.317305
$226$	0	0
$227$	−3.34582	−0.222070	−0.111035	−	0.993817i	$-0.535417\pi$
$227$	−3.34582	−0.222070	−0.111035	+	0.993817i	$0.535417\pi$
$228$	0	0
$229$	18.2687	1.20723	0.603615	−	0.797276i	$-0.293726\pi$
$229$	18.2687	1.20723	0.603615	+	0.797276i	$0.293726\pi$
$230$	0	0
$231$	12.0432	0.792381
$232$	0	0
$233$	−0.764762	−0.0501012	−0.0250506	−	0.999686i	$-0.507975\pi$
$233$	−0.764762	−0.0501012	−0.0250506	+	0.999686i	$0.507975\pi$
$234$	0	0
$235$	−1.77426	−0.115740
$236$	0	0
$237$	18.1327	1.17784
$238$	0	0
$239$	−17.8285	−1.15323	−0.576613	−	0.817017i	$-0.695626\pi$
$239$	−17.8285	−1.15323	−0.576613	+	0.817017i	$0.695626\pi$
$240$	0	0
$241$	20.9025	1.34645	0.673224	−	0.739439i	$-0.264909\pi$
$241$	20.9025	1.34645	0.673224	+	0.739439i	$0.264909\pi$
$242$	0	0
$243$	18.6463	1.19616
$244$	0	0
$245$	10.0374	0.641267
$246$	0	0
$247$	−0.685923	−0.0436443
$248$	0	0
$249$	−2.30379	−0.145997
$250$	0	0
$251$	3.75897	0.237264	0.118632	−	0.992938i	$-0.462149\pi$
$251$	3.75897	0.237264	0.118632	+	0.992938i	$0.462149\pi$
$252$	0	0
$253$	15.6185	0.981929
$254$	0	0
$255$	21.5475	1.34935
$256$	0	0
$257$	22.3760	1.39578	0.697890	−	0.716205i	$-0.254123\pi$
$257$	22.3760	1.39578	0.697890	+	0.716205i	$0.254123\pi$
$258$	0	0
$259$	0.824903	0.0512570
$260$	0	0
$261$	17.8241	1.10328
$262$	0	0
$263$	13.8294	0.852754	0.426377	−	0.904545i	$-0.359790\pi$
$263$	13.8294	0.852754	0.426377	+	0.904545i	$0.359790\pi$
$264$	0	0
$265$	−14.8944	−0.914953
$266$	0	0
$267$	5.51046	0.337235
$268$	0	0
$269$	17.2641	1.05261	0.526305	−	0.850296i	$-0.323577\pi$
$269$	17.2641	1.05261	0.526305	+	0.850296i	$0.323577\pi$
$270$	0	0
$271$	3.32946	0.202250	0.101125	−	0.994874i	$-0.467756\pi$
$271$	3.32946	0.202250	0.101125	+	0.994874i	$0.467756\pi$
$272$	0	0
$273$	0.396759	0.0240129
$274$	0	0
$275$	−12.1590	−0.733215
$276$	0	0
$277$	13.1006	0.787138	0.393569	−	0.919295i	$-0.371240\pi$
$277$	13.1006	0.787138	0.393569	+	0.919295i	$0.371240\pi$
$278$	0	0
$279$	7.03848	0.421383
$280$	0	0
$281$	−16.6320	−0.992180	−0.496090	−	0.868271i	$-0.665231\pi$
$281$	−16.6320	−0.992180	−0.496090	+	0.868271i	$0.665231\pi$
$282$	0	0
$283$	3.62675	0.215588	0.107794	−	0.994173i	$-0.465621\pi$
$283$	3.62675	0.215588	0.107794	+	0.994173i	$0.465621\pi$
$284$	0	0
$285$	−14.4229	−0.854336
$286$	0	0
$287$	−1.20215	−0.0709609
$288$	0	0
$289$	15.8054	0.929727
$290$	0	0
$291$	−18.1651	−1.06486
$292$	0	0
$293$	6.69323	0.391023	0.195511	−	0.980701i	$-0.437363\pi$
$293$	6.69323	0.391023	0.195511	+	0.980701i	$0.437363\pi$
$294$	0	0
$295$	−11.5017	−0.669654
$296$	0	0
$297$	10.7481	0.623667
$298$	0	0
$299$	0.514549	0.0297571
$300$	0	0
$301$	12.4473	0.717450
$302$	0	0
$303$	−20.8735	−1.19915
$304$	0	0
$305$	8.16269	0.467394
$306$	0	0
$307$	−14.4802	−0.826426	−0.413213	−	0.910634i	$-0.635594\pi$
$307$	−14.4802	−0.826426	−0.413213	+	0.910634i	$0.635594\pi$
$308$	0	0
$309$	34.8941	1.98505
$310$	0	0
$311$	7.71988	0.437754	0.218877	−	0.975752i	$-0.429761\pi$
$311$	7.71988	0.437754	0.218877	+	0.975752i	$0.429761\pi$
$312$	0	0
$313$	12.8620	0.727001	0.363501	−	0.931594i	$-0.381581\pi$
$313$	12.8620	0.727001	0.363501	+	0.931594i	$0.381581\pi$
$314$	0	0
$315$	3.45994	0.194946
$316$	0	0
$317$	29.2697	1.64395	0.821975	−	0.569524i	$-0.192872\pi$
$317$	29.2697	1.64395	0.821975	+	0.569524i	$0.192872\pi$
$318$	0	0
$319$	45.5340	2.54942
$320$	0	0
$321$	37.9400	2.11760
$322$	0	0
$323$	−21.9584	−1.22180
$324$	0	0
$325$	−0.400575	−0.0222199
$326$	0	0
$327$	32.2999	1.78619
$328$	0	0
$329$	−1.04585	−0.0576598
$330$	0	0
$331$	15.4862	0.851199	0.425600	−	0.904912i	$-0.360063\pi$
$331$	15.4862	0.851199	0.425600	+	0.904912i	$0.360063\pi$
$332$	0	0
$333$	−1.79036	−0.0981110
$334$	0	0
$335$	−19.7505	−1.07908
$336$	0	0
$337$	−5.24216	−0.285559	−0.142779	−	0.989755i	$-0.545604\pi$
$337$	−5.24216	−0.285559	−0.142779	+	0.989755i	$0.545604\pi$
$338$	0	0
$339$	7.37908	0.400777
$340$	0	0
$341$	17.9808	0.973714
$342$	0	0
$343$	12.7730	0.689679
$344$	0	0
$345$	10.8194	0.582495
$346$	0	0
$347$	−2.71707	−0.145860	−0.0729299	−	0.997337i	$-0.523235\pi$
$347$	−2.71707	−0.145860	−0.0729299	+	0.997337i	$0.523235\pi$
$348$	0	0
$349$	−21.4831	−1.14996	−0.574981	−	0.818167i	$-0.694991\pi$
$349$	−21.4831	−1.14996	−0.574981	+	0.818167i	$0.694991\pi$
$350$	0	0
$351$	0.354093	0.0189001
$352$	0	0
$353$	8.28668	0.441056	0.220528	−	0.975381i	$-0.429222\pi$
$353$	8.28668	0.441056	0.220528	+	0.975381i	$0.429222\pi$
$354$	0	0
$355$	18.7358	0.994393
$356$	0	0
$357$	12.7014	0.672228
$358$	0	0
$359$	−18.5078	−0.976802	−0.488401	−	0.872619i	$-0.662420\pi$
$359$	−18.5078	−0.976802	−0.488401	+	0.872619i	$0.662420\pi$
$360$	0	0
$361$	−4.30210	−0.226426
$362$	0	0
$363$	−41.8695	−2.19758
$364$	0	0
$365$	−5.06106	−0.264908
$366$	0	0
$367$	−4.92044	−0.256845	−0.128422	−	0.991720i	$-0.540991\pi$
$367$	−4.92044	−0.256845	−0.128422	+	0.991720i	$0.540991\pi$
$368$	0	0
$369$	2.60914	0.135826
$370$	0	0
$371$	−8.77964	−0.455816
$372$	0	0
$373$	24.1125	1.24850	0.624249	−	0.781225i	$-0.285405\pi$
$373$	24.1125	1.24850	0.624249	+	0.781225i	$0.285405\pi$
$374$	0	0
$375$	−27.2331	−1.40631
$376$	0	0
$377$	1.50011	0.0772595
$378$	0	0
$379$	−0.0153170	−0.000786782 0	−0.000393391	−	1.00000i	$-0.500125\pi$
$379$	−0.0153170	−0.000786782 0	−0.000393391		1.00000i	$0.500125\pi$
$380$	0	0
$381$	−7.24625	−0.371237
$382$	0	0
$383$	−2.44497	−0.124932	−0.0624662	−	0.998047i	$-0.519897\pi$
$383$	−2.44497	−0.124932	−0.0624662	+	0.998047i	$0.519897\pi$
$384$	0	0
$385$	8.83890	0.450472
$386$	0	0
$387$	−27.0154	−1.37327
$388$	0	0
$389$	16.0535	0.813943	0.406972	−	0.913441i	$-0.366585\pi$
$389$	16.0535	0.813943	0.406972	+	0.913441i	$0.366585\pi$
$390$	0	0
$391$	16.4722	0.833034
$392$	0	0
$393$	−42.6454	−2.15117
$394$	0	0
$395$	13.3082	0.669608
$396$	0	0
$397$	−6.41688	−0.322054	−0.161027	−	0.986950i	$-0.551481\pi$
$397$	−6.41688	−0.322054	−0.161027	+	0.986950i	$0.551481\pi$
$398$	0	0
$399$	−8.50171	−0.425618
$400$	0	0
$401$	−33.7360	−1.68469	−0.842347	−	0.538936i	$-0.818826\pi$
$401$	−33.7360	−1.68469	−0.842347	+	0.538936i	$0.818826\pi$
$402$	0	0
$403$	0.592372	0.0295082
$404$	0	0
$405$	18.0428	0.896552
$406$	0	0
$407$	−4.57372	−0.226711
$408$	0	0
$409$	6.02435	0.297885	0.148942	−	0.988846i	$-0.452413\pi$
$409$	6.02435	0.297885	0.148942	+	0.988846i	$0.452413\pi$
$410$	0	0
$411$	−47.2831	−2.33230
$412$	0	0
$413$	−6.77979	−0.333612
$414$	0	0
$415$	−1.69083	−0.0829996
$416$	0	0
$417$	39.0193	1.91078
$418$	0	0
$419$	−5.29098	−0.258481	−0.129241	−	0.991613i	$-0.541254\pi$
$419$	−5.29098	−0.258481	−0.129241	+	0.991613i	$0.541254\pi$
$420$	0	0
$421$	9.21388	0.449057	0.224528	−	0.974468i	$-0.427916\pi$
$421$	9.21388	0.449057	0.224528	+	0.974468i	$0.427916\pi$
$422$	0	0
$423$	2.26991	0.110367
$424$	0	0
$425$	−12.8235	−0.622033
$426$	0	0
$427$	4.81159	0.232849
$428$	0	0
$429$	−2.19985	−0.106210
$430$	0	0
$431$	−19.6790	−0.947904	−0.473952	−	0.880551i	$-0.657173\pi$
$431$	−19.6790	−0.947904	−0.473952	+	0.880551i	$0.657173\pi$
$432$	0	0
$433$	27.9843	1.34484	0.672419	−	0.740171i	$-0.265255\pi$
$433$	27.9843	1.34484	0.672419	+	0.740171i	$0.265255\pi$
$434$	0	0
$435$	31.5426	1.51235
$436$	0	0
$437$	−11.0257	−0.527431
$438$	0	0
$439$	29.7508	1.41993	0.709965	−	0.704237i	$-0.248711\pi$
$439$	29.7508	1.41993	0.709965	+	0.704237i	$0.248711\pi$
$440$	0	0
$441$	−12.8414	−0.611497
$442$	0	0
$443$	7.34047	0.348756	0.174378	−	0.984679i	$-0.444208\pi$
$443$	7.34047	0.348756	0.174378	+	0.984679i	$0.444208\pi$
$444$	0	0
$445$	4.04432	0.191719
$446$	0	0
$447$	5.03535	0.238164
$448$	0	0
$449$	33.0741	1.56086	0.780432	−	0.625241i	$-0.214999\pi$
$449$	33.0741	1.56086	0.780432	+	0.625241i	$0.214999\pi$
$450$	0	0
$451$	6.66540	0.313862
$452$	0	0
$453$	−20.4943	−0.962908
$454$	0	0
$455$	0.291195	0.0136514
$456$	0	0
$457$	−14.8955	−0.696783	−0.348391	−	0.937349i	$-0.613272\pi$
$457$	−14.8955	−0.696783	−0.348391	+	0.937349i	$0.613272\pi$
$458$	0	0
$459$	11.3355	0.529097
$460$	0	0
$461$	3.78263	0.176175	0.0880874	−	0.996113i	$-0.471925\pi$
$461$	3.78263	0.176175	0.0880874	+	0.996113i	$0.471925\pi$
$462$	0	0
$463$	33.0517	1.53604	0.768020	−	0.640425i	$-0.221242\pi$
$463$	33.0517	1.53604	0.768020	+	0.640425i	$0.221242\pi$
$464$	0	0
$465$	12.4558	0.577622
$466$	0	0
$467$	26.3185	1.21788	0.608938	−	0.793218i	$-0.291596\pi$
$467$	26.3185	1.21788	0.608938	+	0.793218i	$0.291596\pi$
$468$	0	0
$469$	−11.6421	−0.537583
$470$	0	0
$471$	−19.7674	−0.910835
$472$	0	0
$473$	−69.0146	−3.17329
$474$	0	0
$475$	8.58348	0.393837
$476$	0	0
$477$	19.0552	0.872477
$478$	0	0
$479$	6.23823	0.285032	0.142516	−	0.989793i	$-0.454481\pi$
$479$	6.23823	0.285032	0.142516	+	0.989793i	$0.454481\pi$
$480$	0	0
$481$	−0.150680	−0.00687041
$482$	0	0
$483$	6.37759	0.290191
$484$	0	0
$485$	−13.3320	−0.605375
$486$	0	0
$487$	−23.0600	−1.04495	−0.522473	−	0.852656i	$-0.674991\pi$
$487$	−23.0600	−1.04495	−0.522473	+	0.852656i	$0.674991\pi$
$488$	0	0
$489$	−8.18694	−0.370226
$490$	0	0
$491$	29.7733	1.34365	0.671825	−	0.740710i	$-0.265511\pi$
$491$	29.7733	1.34365	0.671825	+	0.740710i	$0.265511\pi$
$492$	0	0
$493$	48.0227	2.16283
$494$	0	0
$495$	−19.1838	−0.862248
$496$	0	0
$497$	11.0440	0.495392
$498$	0	0
$499$	−17.2050	−0.770201	−0.385101	−	0.922875i	$-0.625833\pi$
$499$	−17.2050	−0.770201	−0.385101	+	0.922875i	$0.625833\pi$
$500$	0	0
$501$	7.91127	0.353449
$502$	0	0
$503$	−4.89380	−0.218204	−0.109102	−	0.994031i	$-0.534798\pi$
$503$	−4.89380	−0.218204	−0.109102	+	0.994031i	$0.534798\pi$
$504$	0	0
$505$	−15.3198	−0.681723
$506$	0	0
$507$	29.3600	1.30392
$508$	0	0
$509$	6.85542	0.303861	0.151931	−	0.988391i	$-0.451451\pi$
$509$	6.85542	0.303861	0.151931	+	0.988391i	$0.451451\pi$
$510$	0	0
$511$	−2.98330	−0.131973
$512$	0	0
$513$	−7.58747	−0.334995
$514$	0	0
$515$	25.6100	1.12851
$516$	0	0
$517$	5.79879	0.255031
$518$	0	0
$519$	−0.0815804	−0.00358098
$520$	0	0
$521$	−10.1445	−0.444439	−0.222219	−	0.974997i	$-0.571330\pi$
$521$	−10.1445	−0.444439	−0.222219	+	0.974997i	$0.571330\pi$
$522$	0	0
$523$	−10.5709	−0.462234	−0.231117	−	0.972926i	$-0.574238\pi$
$523$	−10.5709	−0.462234	−0.231117	+	0.972926i	$0.574238\pi$
$524$	0	0
$525$	−4.96494	−0.216688
$526$	0	0
$527$	18.9635	0.826064
$528$	0	0
$529$	−14.7290	−0.640392
$530$	0	0
$531$	14.7148	0.638566
$532$	0	0
$533$	0.219590	0.00951150
$534$	0	0
$535$	27.8455	1.20387
$536$	0	0
$537$	−5.69604	−0.245802
$538$	0	0
$539$	−32.8052	−1.41302
$540$	0	0
$541$	−32.1438	−1.38197	−0.690984	−	0.722870i	$-0.742823\pi$
$541$	−32.1438	−1.38197	−0.690984	+	0.722870i	$0.742823\pi$
$542$	0	0
$543$	−38.2190	−1.64014
$544$	0	0
$545$	23.7060	1.01545
$546$	0	0
$547$	−19.1079	−0.816994	−0.408497	−	0.912760i	$-0.633947\pi$
$547$	−19.1079	−0.816994	−0.408497	+	0.912760i	$0.633947\pi$
$548$	0	0
$549$	−10.4430	−0.445696
$550$	0	0
$551$	−32.1441	−1.36939
$552$	0	0
$553$	7.84466	0.333589
$554$	0	0
$555$	−3.16833	−0.134488
$556$	0	0
$557$	26.1988	1.11008	0.555040	−	0.831824i	$-0.312703\pi$
$557$	26.1988	1.11008	0.555040	+	0.831824i	$0.312703\pi$
$558$	0	0
$559$	−2.27367	−0.0961659
$560$	0	0
$561$	−70.4234	−2.97328
$562$	0	0
$563$	−12.7816	−0.538679	−0.269340	−	0.963045i	$-0.586805\pi$
$563$	−12.7816	−0.538679	−0.269340	+	0.963045i	$0.586805\pi$
$564$	0	0
$565$	5.41577	0.227843
$566$	0	0
$567$	10.6355	0.446649
$568$	0	0
$569$	18.0571	0.756993	0.378496	−	0.925603i	$-0.376441\pi$
$569$	18.0571	0.756993	0.378496	+	0.925603i	$0.376441\pi$
$570$	0	0
$571$	−30.2965	−1.26787	−0.633934	−	0.773388i	$-0.718561\pi$
$571$	−30.2965	−1.26787	−0.633934	+	0.773388i	$0.718561\pi$
$572$	0	0
$573$	6.03813	0.252246
$574$	0	0
$575$	−6.43894	−0.268522
$576$	0	0
$577$	−44.7394	−1.86253	−0.931263	−	0.364347i	$-0.881292\pi$
$577$	−44.7394	−1.86253	−0.931263	+	0.364347i	$0.881292\pi$
$578$	0	0
$579$	23.9747	0.996354
$580$	0	0
$581$	−0.996678	−0.0413492
$582$	0	0
$583$	48.6791	2.01608
$584$	0	0
$585$	−0.632006	−0.0261302
$586$	0	0
$587$	29.0503	1.19903	0.599517	−	0.800362i	$-0.295359\pi$
$587$	29.0503	1.19903	0.599517	+	0.800362i	$0.295359\pi$
$588$	0	0
$589$	−12.6933	−0.523018
$590$	0	0
$591$	−41.3262	−1.69993
$592$	0	0
$593$	−1.45010	−0.0595486	−0.0297743	−	0.999557i	$-0.509479\pi$
$593$	−1.45010	−0.0595486	−0.0297743	+	0.999557i	$0.509479\pi$
$594$	0	0
$595$	9.32199	0.382164
$596$	0	0
$597$	3.45440	0.141379
$598$	0	0
$599$	−39.8479	−1.62814	−0.814071	−	0.580765i	$-0.802754\pi$
$599$	−39.8479	−1.62814	−0.814071	+	0.580765i	$0.802754\pi$
$600$	0	0
$601$	−32.8004	−1.33796	−0.668978	−	0.743283i	$-0.733268\pi$
$601$	−32.8004	−1.33796	−0.668978	+	0.743283i	$0.733268\pi$
$602$	0	0
$603$	25.2679	1.02899
$604$	0	0
$605$	−30.7295	−1.24933
$606$	0	0
$607$	15.2475	0.618876	0.309438	−	0.950920i	$-0.399859\pi$
$607$	15.2475	0.618876	0.309438	+	0.950920i	$0.399859\pi$
$608$	0	0
$609$	18.5931	0.753432
$610$	0	0
$611$	0.191040	0.00772864
$612$	0	0
$613$	21.9793	0.887734	0.443867	−	0.896093i	$-0.353606\pi$
$613$	21.9793	0.887734	0.443867	+	0.896093i	$0.353606\pi$
$614$	0	0
$615$	4.61730	0.186188
$616$	0	0
$617$	34.1104	1.37323	0.686616	−	0.727020i	$-0.259095\pi$
$617$	34.1104	1.37323	0.686616	+	0.727020i	$0.259095\pi$
$618$	0	0
$619$	44.7559	1.79889	0.899445	−	0.437034i	$-0.143971\pi$
$619$	44.7559	1.79889	0.899445	+	0.437034i	$0.143971\pi$
$620$	0	0
$621$	5.69178	0.228403
$622$	0	0
$623$	2.38397	0.0955117
$624$	0	0
$625$	−8.79278	−0.351711
$626$	0	0
$627$	47.1381	1.88252
$628$	0	0
$629$	−4.82369	−0.192333
$630$	0	0
$631$	18.7326	0.745733	0.372866	−	0.927885i	$-0.378375\pi$
$631$	18.7326	0.745733	0.372866	+	0.927885i	$0.378375\pi$
$632$	0	0
$633$	31.7828	1.26325
$634$	0	0
$635$	−5.31828	−0.211049
$636$	0	0
$637$	−1.08076	−0.0428213
$638$	0	0
$639$	−23.9698	−0.948230
$640$	0	0
$641$	17.3220	0.684180	0.342090	−	0.939667i	$-0.388865\pi$
$641$	17.3220	0.684180	0.342090	+	0.939667i	$0.388865\pi$
$642$	0	0
$643$	19.5533	0.771107	0.385554	−	0.922685i	$-0.374010\pi$
$643$	19.5533	0.771107	0.385554	+	0.922685i	$0.374010\pi$
$644$	0	0
$645$	−47.8082	−1.88245
$646$	0	0
$647$	−32.5914	−1.28130	−0.640650	−	0.767833i	$-0.721335\pi$
$647$	−32.5914	−1.28130	−0.640650	+	0.767833i	$0.721335\pi$
$648$	0	0
$649$	37.5909	1.47557
$650$	0	0
$651$	7.34218	0.287763
$652$	0	0
$653$	−43.7813	−1.71329	−0.856647	−	0.515903i	$-0.827457\pi$
$653$	−43.7813	−1.71329	−0.856647	+	0.515903i	$0.827457\pi$
$654$	0	0
$655$	−31.2989	−1.22295
$656$	0	0
$657$	6.47491	0.252610
$658$	0	0
$659$	−0.378588	−0.0147477	−0.00737384	−	0.999973i	$-0.502347\pi$
$659$	−0.378588	−0.0147477	−0.00737384	+	0.999973i	$0.502347\pi$
$660$	0	0
$661$	24.7230	0.961613	0.480807	−	0.876827i	$-0.340344\pi$
$661$	24.7230	0.961613	0.480807	+	0.876827i	$0.340344\pi$
$662$	0	0
$663$	−2.32008	−0.0901045
$664$	0	0
$665$	−6.23970	−0.241965
$666$	0	0
$667$	24.1131	0.933662
$668$	0	0
$669$	−0.566939	−0.0219191
$670$	0	0
$671$	−26.6781	−1.02990
$672$	0	0
$673$	0.660271	0.0254516	0.0127258	−	0.999919i	$-0.495949\pi$
$673$	0.660271	0.0254516	0.0127258	+	0.999919i	$0.495949\pi$
$674$	0	0
$675$	−4.43103	−0.170551
$676$	0	0
$677$	46.6556	1.79312	0.896560	−	0.442923i	$-0.146058\pi$
$677$	46.6556	1.79312	0.896560	+	0.442923i	$0.146058\pi$
$678$	0	0
$679$	−7.85869	−0.301589
$680$	0	0
$681$	7.57504	0.290276
$682$	0	0
$683$	21.9233	0.838872	0.419436	−	0.907785i	$-0.362228\pi$
$683$	21.9233	0.838872	0.419436	+	0.907785i	$0.362228\pi$
$684$	0	0
$685$	−34.7027	−1.32592
$686$	0	0
$687$	−41.3610	−1.57802
$688$	0	0
$689$	1.60372	0.0610969
$690$	0	0
$691$	10.7679	0.409629	0.204814	−	0.978801i	$-0.434341\pi$
$691$	10.7679	0.409629	0.204814	+	0.978801i	$0.434341\pi$
$692$	0	0
$693$	−11.3081	−0.429559
$694$	0	0
$695$	28.6376	1.08629
$696$	0	0
$697$	7.02971	0.266269
$698$	0	0
$699$	1.73145	0.0654893
$700$	0	0
$701$	40.4392	1.52737	0.763683	−	0.645592i	$-0.223389\pi$
$701$	40.4392	1.52737	0.763683	+	0.645592i	$0.223389\pi$
$702$	0	0
$703$	3.22875	0.121775
$704$	0	0
$705$	4.01698	0.151288
$706$	0	0
$707$	−9.03043	−0.339624
$708$	0	0
$709$	−36.3063	−1.36351	−0.681756	−	0.731579i	$-0.738784\pi$
$709$	−36.3063	−1.36351	−0.681756	+	0.731579i	$0.738784\pi$
$710$	0	0
$711$	−17.0259	−0.638522
$712$	0	0
$713$	9.52193	0.356599
$714$	0	0
$715$	−1.61455	−0.0603806
$716$	0	0
$717$	40.3642	1.50743
$718$	0	0
$719$	0.569664	0.0212449	0.0106224	−	0.999944i	$-0.496619\pi$
$719$	0.569664	0.0212449	0.0106224	+	0.999944i	$0.496619\pi$
$720$	0	0
$721$	15.0961	0.562208
$722$	0	0
$723$	−47.3239	−1.76000
$724$	0	0
$725$	−18.7720	−0.697174
$726$	0	0
$727$	7.00786	0.259907	0.129954	−	0.991520i	$-0.458517\pi$
$727$	7.00786	0.259907	0.129954	+	0.991520i	$0.458517\pi$
$728$	0	0
$729$	−9.64083	−0.357068
$730$	0	0
$731$	−72.7866	−2.69211
$732$	0	0
$733$	−16.7943	−0.620312	−0.310156	−	0.950686i	$-0.600381\pi$
$733$	−16.7943	−0.620312	−0.310156	+	0.950686i	$0.600381\pi$
$734$	0	0
$735$	−22.7251	−0.838226
$736$	0	0
$737$	64.5503	2.37774
$738$	0	0
$739$	1.10670	0.0407105	0.0203552	−	0.999793i	$-0.493520\pi$
$739$	1.10670	0.0407105	0.0203552	+	0.999793i	$0.493520\pi$
$740$	0	0
$741$	1.55295	0.0570492
$742$	0	0
$743$	17.3023	0.634758	0.317379	−	0.948299i	$-0.397197\pi$
$743$	17.3023	0.634758	0.317379	+	0.948299i	$0.397197\pi$
$744$	0	0
$745$	3.69562	0.135397
$746$	0	0
$747$	2.16318	0.0791465
$748$	0	0
$749$	16.4138	0.599748
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−8.51043	−0.310137
$754$	0	0
$755$	−15.0415	−0.547417
$756$	0	0
$757$	10.3633	0.376659	0.188330	−	0.982106i	$-0.439693\pi$
$757$	10.3633	0.376659	0.188330	+	0.982106i	$0.439693\pi$
$758$	0	0
$759$	−35.3609	−1.28352
$760$	0	0
$761$	−34.1482	−1.23787	−0.618935	−	0.785442i	$-0.712436\pi$
$761$	−34.1482	−1.23787	−0.618935	+	0.785442i	$0.712436\pi$
$762$	0	0
$763$	13.9738	0.505884
$764$	0	0
$765$	−20.2323	−0.731501
$766$	0	0
$767$	1.23842	0.0447168
$768$	0	0
$769$	−7.14196	−0.257546	−0.128773	−	0.991674i	$-0.541104\pi$
$769$	−7.14196	−0.257546	−0.128773	+	0.991674i	$0.541104\pi$
$770$	0	0
$771$	−50.6601	−1.82448
$772$	0	0
$773$	32.5279	1.16995	0.584973	−	0.811052i	$-0.301105\pi$
$773$	32.5279	1.16995	0.584973	+	0.811052i	$0.301105\pi$
$774$	0	0
$775$	−7.41280	−0.266276
$776$	0	0
$777$	−1.86761	−0.0670001
$778$	0	0
$779$	−4.70535	−0.168587
$780$	0	0
$781$	−61.2341	−2.19113
$782$	0	0
$783$	16.5937	0.593011
$784$	0	0
$785$	−14.5080	−0.517813
$786$	0	0
$787$	8.61225	0.306994	0.153497	−	0.988149i	$-0.450947\pi$
$787$	8.61225	0.306994	0.153497	+	0.988149i	$0.450947\pi$
$788$	0	0
$789$	−31.3101	−1.11467
$790$	0	0
$791$	3.19238	0.113508
$792$	0	0
$793$	−0.878902	−0.0312107
$794$	0	0
$795$	33.7213	1.19597
$796$	0	0
$797$	−43.0676	−1.52553	−0.762767	−	0.646674i	$-0.776160\pi$
$797$	−43.0676	−1.52553	−0.762767	+	0.646674i	$0.776160\pi$
$798$	0	0
$799$	6.11573	0.216359
$800$	0	0
$801$	−5.17413	−0.182819
$802$	0	0
$803$	16.5410	0.583721
$804$	0	0
$805$	4.68074	0.164974
$806$	0	0
$807$	−39.0865	−1.37591
$808$	0	0
$809$	−10.7820	−0.379076	−0.189538	−	0.981873i	$-0.560699\pi$
$809$	−10.7820	−0.379076	−0.189538	+	0.981873i	$0.560699\pi$
$810$	0	0
$811$	−14.8144	−0.520203	−0.260101	−	0.965581i	$-0.583756\pi$
$811$	−14.8144	−0.520203	−0.260101	+	0.965581i	$0.583756\pi$
$812$	0	0
$813$	−7.53801	−0.264369
$814$	0	0
$815$	−6.00868	−0.210475
$816$	0	0
$817$	48.7199	1.70449
$818$	0	0
$819$	−0.372543	−0.0130177
$820$	0	0
$821$	−8.06427	−0.281445	−0.140722	−	0.990049i	$-0.544943\pi$
$821$	−8.06427	−0.281445	−0.140722	+	0.990049i	$0.544943\pi$
$822$	0	0
$823$	53.9244	1.87969	0.939843	−	0.341607i	$-0.110971\pi$
$823$	53.9244	1.87969	0.939843	+	0.341607i	$0.110971\pi$
$824$	0	0
$825$	27.5284	0.958414
$826$	0	0
$827$	−8.70137	−0.302576	−0.151288	−	0.988490i	$-0.548342\pi$
$827$	−8.70137	−0.302576	−0.151288	+	0.988490i	$0.548342\pi$
$828$	0	0
$829$	53.9306	1.87309	0.936544	−	0.350550i	$-0.114005\pi$
$829$	53.9306	1.87309	0.936544	+	0.350550i	$0.114005\pi$
$830$	0	0
$831$	−29.6602	−1.02890
$832$	0	0
$833$	−34.5982	−1.19876
$834$	0	0
$835$	5.80636	0.200937
$836$	0	0
$837$	6.55264	0.226492
$838$	0	0
$839$	−14.9345	−0.515598	−0.257799	−	0.966199i	$-0.582997\pi$
$839$	−14.9345	−0.515598	−0.257799	+	0.966199i	$0.582997\pi$
$840$	0	0
$841$	41.2989	1.42410
$842$	0	0
$843$	37.6553	1.29692
$844$	0	0
$845$	21.5483	0.741285
$846$	0	0
$847$	−18.1138	−0.622399
$848$	0	0
$849$	−8.21109	−0.281804
$850$	0	0
$851$	−2.42206	−0.0830273
$852$	0	0
$853$	−18.8964	−0.647000	−0.323500	−	0.946228i	$-0.604860\pi$
$853$	−18.8964	−0.647000	−0.323500	+	0.946228i	$0.604860\pi$
$854$	0	0
$855$	13.5426	0.463146
$856$	0	0
$857$	−22.8763	−0.781441	−0.390720	−	0.920509i	$-0.627774\pi$
$857$	−22.8763	−0.781441	−0.390720	+	0.920509i	$0.627774\pi$
$858$	0	0
$859$	41.9390	1.43094	0.715471	−	0.698643i	$-0.246212\pi$
$859$	41.9390	1.43094	0.715471	+	0.698643i	$0.246212\pi$
$860$	0	0
$861$	2.72172	0.0927559
$862$	0	0
$863$	32.0462	1.09087	0.545433	−	0.838154i	$-0.316365\pi$
$863$	32.0462	1.09087	0.545433	+	0.838154i	$0.316365\pi$
$864$	0	0
$865$	−0.0598747	−0.00203580
$866$	0	0
$867$	−35.7839	−1.21528
$868$	0	0
$869$	−43.4951	−1.47547
$870$	0	0
$871$	2.12659	0.0720569
$872$	0	0
$873$	17.0564	0.577271
$874$	0	0
$875$	−11.7817	−0.398295
$876$	0	0
$877$	−25.2130	−0.851382	−0.425691	−	0.904869i	$-0.639969\pi$
$877$	−25.2130	−0.851382	−0.425691	+	0.904869i	$0.639969\pi$
$878$	0	0
$879$	−15.1537	−0.511121
$880$	0	0
$881$	11.0251	0.371444	0.185722	−	0.982602i	$-0.440538\pi$
$881$	11.0251	0.371444	0.185722	+	0.982602i	$0.440538\pi$
$882$	0	0
$883$	26.2792	0.884365	0.442183	−	0.896925i	$-0.354204\pi$
$883$	26.2792	0.884365	0.442183	+	0.896925i	$0.354204\pi$
$884$	0	0
$885$	26.0402	0.875332
$886$	0	0
$887$	35.3401	1.18661	0.593303	−	0.804979i	$-0.297824\pi$
$887$	35.3401	1.18661	0.593303	+	0.804979i	$0.297824\pi$
$888$	0	0
$889$	−3.13492	−0.105142
$890$	0	0
$891$	−58.9691	−1.97554
$892$	0	0
$893$	−4.09358	−0.136986
$894$	0	0
$895$	−4.18052	−0.139740
$896$	0	0
$897$	−1.16496	−0.0388967
$898$	0	0
$899$	27.7601	0.925851
$900$	0	0
$901$	51.3397	1.71037
$902$	0	0
$903$	−28.1811	−0.937807
$904$	0	0
$905$	−28.0503	−0.932423
$906$	0	0
$907$	−32.3037	−1.07263	−0.536313	−	0.844019i	$-0.680183\pi$
$907$	−32.3037	−1.07263	−0.536313	+	0.844019i	$0.680183\pi$
$908$	0	0
$909$	19.5995	0.650075
$910$	0	0
$911$	35.2316	1.16727	0.583637	−	0.812015i	$-0.301629\pi$
$911$	35.2316	1.16727	0.583637	+	0.812015i	$0.301629\pi$
$912$	0	0
$913$	5.52613	0.182888
$914$	0	0
$915$	−18.4806	−0.610950
$916$	0	0
$917$	−18.4495	−0.609256
$918$	0	0
$919$	39.6703	1.30860	0.654302	−	0.756233i	$-0.272963\pi$
$919$	39.6703	1.30860	0.654302	+	0.756233i	$0.272963\pi$
$920$	0	0
$921$	32.7836	1.08026
$922$	0	0
$923$	−2.01734	−0.0664016
$924$	0	0
$925$	1.88557	0.0619972
$926$	0	0
$927$	−32.7643	−1.07612
$928$	0	0
$929$	−34.4231	−1.12938	−0.564692	−	0.825302i	$-0.691005\pi$
$929$	−34.4231	−1.12938	−0.564692	+	0.825302i	$0.691005\pi$
$930$	0	0
$931$	23.1584	0.758987
$932$	0	0
$933$	−17.4781	−0.572206
$934$	0	0
$935$	−51.6862	−1.69032
$936$	0	0
$937$	−23.8436	−0.778935	−0.389468	−	0.921040i	$-0.627341\pi$
$937$	−23.8436	−0.778935	−0.389468	+	0.921040i	$0.627341\pi$
$938$	0	0
$939$	−29.1199	−0.950292
$940$	0	0
$941$	−49.5011	−1.61369	−0.806844	−	0.590764i	$-0.798826\pi$
$941$	−49.5011	−1.61369	−0.806844	+	0.590764i	$0.798826\pi$
$942$	0	0
$943$	3.52974	0.114944
$944$	0	0
$945$	3.22111	0.104783
$946$	0	0
$947$	−42.9809	−1.39669	−0.698346	−	0.715760i	$-0.746080\pi$
$947$	−42.9809	−1.39669	−0.698346	+	0.715760i	$0.746080\pi$
$948$	0	0
$949$	0.544941	0.0176895
$950$	0	0
$951$	−66.2676	−2.14887
$952$	0	0
$953$	−38.4370	−1.24510	−0.622548	−	0.782582i	$-0.713902\pi$
$953$	−38.4370	−1.24510	−0.622548	+	0.782582i	$0.713902\pi$
$954$	0	0
$955$	4.43160	0.143403
$956$	0	0
$957$	−103.091	−3.33245
$958$	0	0
$959$	−20.4559	−0.660555
$960$	0	0
$961$	−20.0379	−0.646384
$962$	0	0
$963$	−35.6243	−1.14798
$964$	0	0
$965$	17.5959	0.566431
$966$	0	0
$967$	−5.47739	−0.176141	−0.0880704	−	0.996114i	$-0.528070\pi$
$967$	−5.47739	−0.176141	−0.0880704	+	0.996114i	$0.528070\pi$
$968$	0	0
$969$	49.7145	1.59706
$970$	0	0
$971$	29.0325	0.931697	0.465848	−	0.884865i	$-0.345749\pi$
$971$	29.0325	0.931697	0.465848	+	0.884865i	$0.345749\pi$
$972$	0	0
$973$	16.8808	0.541173
$974$	0	0
$975$	0.906915	0.0290445
$976$	0	0
$977$	40.0075	1.27995	0.639977	−	0.768394i	$-0.278944\pi$
$977$	40.0075	1.27995	0.639977	+	0.768394i	$0.278944\pi$
$978$	0	0
$979$	−13.2180	−0.422450
$980$	0	0
$981$	−30.3284	−0.968313
$982$	0	0
$983$	−24.4076	−0.778481	−0.389241	−	0.921136i	$-0.627263\pi$
$983$	−24.4076	−0.778481	−0.389241	+	0.921136i	$0.627263\pi$
$984$	0	0
$985$	−30.3307	−0.966417
$986$	0	0
$987$	2.36785	0.0753695
$988$	0	0
$989$	−36.5475	−1.16214
$990$	0	0
$991$	37.6584	1.19626	0.598129	−	0.801400i	$-0.295911\pi$
$991$	37.6584	1.19626	0.598129	+	0.801400i	$0.295911\pi$
$992$	0	0
$993$	−35.0613	−1.11264
$994$	0	0
$995$	2.53531	0.0803747
$996$	0	0
$997$	54.7618	1.73432	0.867161	−	0.498028i	$-0.165942\pi$
$997$	54.7618	1.73432	0.867161	+	0.498028i	$0.165942\pi$
$998$	0	0
$999$	−1.66677	−0.0527344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.10	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.10	✓	50	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)

\(f(q)\)	\(=\)	\(q-2.26403 q^{3} -1.66165 q^{5} -0.979480 q^{7} +2.12585 q^{9} +O(q^{10})\)	\(q-2.26403 q^{3} -1.66165 q^{5} -0.979480 q^{7} +2.12585 q^{9} +5.43078 q^{11} +0.178916 q^{13} +3.76204 q^{15} +5.72760 q^{17} -3.83378 q^{19} +2.21758 q^{21} +2.87593 q^{23} -2.23890 q^{25} +1.97911 q^{27} +8.38444 q^{29} +3.31090 q^{31} -12.2955 q^{33} +1.62756 q^{35} -0.842185 q^{37} -0.405071 q^{39} +1.22734 q^{41} -12.7081 q^{43} -3.53243 q^{45} +1.06777 q^{47} -6.04062 q^{49} -12.9675 q^{51} +8.96357 q^{53} -9.02407 q^{55} +8.67982 q^{57} +6.92183 q^{59} -4.91239 q^{61} -2.08223 q^{63} -0.297296 q^{65} +11.8860 q^{67} -6.51120 q^{69} -11.2754 q^{71} +3.04580 q^{73} +5.06896 q^{75} -5.31934 q^{77} -8.00900 q^{79} -10.8583 q^{81} +1.01756 q^{83} -9.51729 q^{85} -18.9827 q^{87} -2.43391 q^{89} -0.175244 q^{91} -7.49600 q^{93} +6.37042 q^{95} +8.02333 q^{97} +11.5450 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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