LMFDB - Embedded newform 6008.2.a.e.1.29

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			1.29
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)

$f(q)$	$=$	$q+0.416230 q^{3} +3.95581 q^{5} +2.52758 q^{7} -2.82675 q^{9} +O(q^{10})$	$q+0.416230 q^{3} +3.95581 q^{5} +2.52758 q^{7} -2.82675 q^{9} +3.42008 q^{11} -3.31267 q^{13} +1.64653 q^{15} -7.23883 q^{17} +4.52109 q^{19} +1.05206 q^{21} +7.20387 q^{23} +10.6484 q^{25} -2.42527 q^{27} +7.29186 q^{29} -4.45184 q^{31} +1.42354 q^{33} +9.99863 q^{35} -6.53567 q^{37} -1.37884 q^{39} +4.03924 q^{41} -9.48256 q^{43} -11.1821 q^{45} -0.0683152 q^{47} -0.611316 q^{49} -3.01302 q^{51} +9.07518 q^{53} +13.5292 q^{55} +1.88181 q^{57} +8.40101 q^{59} +3.48581 q^{61} -7.14486 q^{63} -13.1043 q^{65} +7.25056 q^{67} +2.99847 q^{69} +3.26703 q^{71} +11.2983 q^{73} +4.43219 q^{75} +8.64454 q^{77} +10.5574 q^{79} +7.47078 q^{81} +1.52092 q^{83} -28.6354 q^{85} +3.03510 q^{87} -8.80313 q^{89} -8.37306 q^{91} -1.85299 q^{93} +17.8845 q^{95} -2.44088 q^{97} -9.66772 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.416230	0.240311	0.120155	−	0.992755i	$-0.461661\pi$
$3$	0.416230	0.240311	0.120155	+	0.992755i	$0.461661\pi$
$4$	0	0
$5$	3.95581	1.76909	0.884545	−	0.466455i	$-0.154469\pi$
$5$	3.95581	1.76909	0.884545	+	0.466455i	$0.154469\pi$
$6$	0	0
$7$	2.52758	0.955337	0.477669	−	0.878540i	$-0.341482\pi$
$7$	2.52758	0.955337	0.477669	+	0.878540i	$0.341482\pi$
$8$	0	0
$9$	−2.82675	−0.942251
$10$	0	0
$11$	3.42008	1.03119	0.515596	−	0.856832i	$-0.327570\pi$
$11$	3.42008	1.03119	0.515596	+	0.856832i	$0.327570\pi$
$12$	0	0
$13$	−3.31267	−0.918771	−0.459385	−	0.888237i	$-0.651930\pi$
$13$	−3.31267	−0.918771	−0.459385	+	0.888237i	$0.651930\pi$
$14$	0	0
$15$	1.64653	0.425131
$16$	0	0
$17$	−7.23883	−1.75567	−0.877837	−	0.478959i	$-0.841014\pi$
$17$	−7.23883	−1.75567	−0.877837	+	0.478959i	$0.841014\pi$
$18$	0	0
$19$	4.52109	1.03721	0.518604	−	0.855014i	$-0.326452\pi$
$19$	4.52109	1.03721	0.518604	+	0.855014i	$0.326452\pi$
$20$	0	0
$21$	1.05206	0.229578
$22$	0	0
$23$	7.20387	1.50211	0.751055	−	0.660240i	$-0.229545\pi$
$23$	7.20387	1.50211	0.751055	+	0.660240i	$0.229545\pi$
$24$	0	0
$25$	10.6484	2.12968
$26$	0	0
$27$	−2.42527	−0.466744
$28$	0	0
$29$	7.29186	1.35406	0.677032	−	0.735953i	$-0.263266\pi$
$29$	7.29186	1.35406	0.677032	+	0.735953i	$0.263266\pi$
$30$	0	0
$31$	−4.45184	−0.799573	−0.399787	−	0.916608i	$-0.630916\pi$
$31$	−4.45184	−0.799573	−0.399787	+	0.916608i	$0.630916\pi$
$32$	0	0
$33$	1.42354	0.247807
$34$	0	0
$35$	9.99863	1.69008
$36$	0	0
$37$	−6.53567	−1.07446	−0.537229	−	0.843436i	$-0.680529\pi$
$37$	−6.53567	−1.07446	−0.537229	+	0.843436i	$0.680529\pi$
$38$	0	0
$39$	−1.37884	−0.220790
$40$	0	0
$41$	4.03924	0.630823	0.315412	−	0.948955i	$-0.397857\pi$
$41$	4.03924	0.630823	0.315412	+	0.948955i	$0.397857\pi$
$42$	0	0
$43$	−9.48256	−1.44608	−0.723038	−	0.690808i	$-0.757255\pi$
$43$	−9.48256	−1.44608	−0.723038	+	0.690808i	$0.757255\pi$
$44$	0	0
$45$	−11.1821	−1.66693
$46$	0	0
$47$	−0.0683152	−0.00996480	−0.00498240	−	0.999988i	$-0.501586\pi$
$47$	−0.0683152	−0.00996480	−0.00498240	+	0.999988i	$0.501586\pi$
$48$	0	0
$49$	−0.611316	−0.0873308
$50$	0	0
$51$	−3.01302	−0.421907
$52$	0	0
$53$	9.07518	1.24657	0.623286	−	0.781994i	$-0.285797\pi$
$53$	9.07518	1.24657	0.623286	+	0.781994i	$0.285797\pi$
$54$	0	0
$55$	13.5292	1.82427
$56$	0	0
$57$	1.88181	0.249252
$58$	0	0
$59$	8.40101	1.09372	0.546859	−	0.837225i	$-0.315823\pi$
$59$	8.40101	1.09372	0.546859	+	0.837225i	$0.315823\pi$
$60$	0	0
$61$	3.48581	0.446312	0.223156	−	0.974783i	$-0.428364\pi$
$61$	3.48581	0.446312	0.223156	+	0.974783i	$0.428364\pi$
$62$	0	0
$63$	−7.14486	−0.900167
$64$	0	0
$65$	−13.1043	−1.62539
$66$	0	0
$67$	7.25056	0.885797	0.442899	−	0.896572i	$-0.353950\pi$
$67$	7.25056	0.885797	0.442899	+	0.896572i	$0.353950\pi$
$68$	0	0
$69$	2.99847	0.360973
$70$	0	0
$71$	3.26703	0.387725	0.193862	−	0.981029i	$-0.437898\pi$
$71$	3.26703	0.387725	0.193862	+	0.981029i	$0.437898\pi$
$72$	0	0
$73$	11.2983	1.32236	0.661182	−	0.750225i	$-0.270055\pi$
$73$	11.2983	1.32236	0.661182	+	0.750225i	$0.270055\pi$
$74$	0	0
$75$	4.43219	0.511785
$76$	0	0
$77$	8.64454	0.985137
$78$	0	0
$79$	10.5574	1.18780	0.593899	−	0.804540i	$-0.297588\pi$
$79$	10.5574	1.18780	0.593899	+	0.804540i	$0.297588\pi$
$80$	0	0
$81$	7.47078	0.830087
$82$	0	0
$83$	1.52092	0.166942	0.0834712	−	0.996510i	$-0.473399\pi$
$83$	1.52092	0.166942	0.0834712	+	0.996510i	$0.473399\pi$
$84$	0	0
$85$	−28.6354	−3.10595
$86$	0	0
$87$	3.03510	0.325396
$88$	0	0
$89$	−8.80313	−0.933130	−0.466565	−	0.884487i	$-0.654509\pi$
$89$	−8.80313	−0.933130	−0.466565	+	0.884487i	$0.654509\pi$
$90$	0	0
$91$	−8.37306	−0.877736
$92$	0	0
$93$	−1.85299	−0.192146
$94$	0	0
$95$	17.8845	1.83492
$96$	0	0
$97$	−2.44088	−0.247833	−0.123917	−	0.992293i	$-0.539546\pi$
$97$	−2.44088	−0.247833	−0.123917	+	0.992293i	$0.539546\pi$
$98$	0	0
$99$	−9.66772	−0.971642
$100$	0	0
$101$	9.56249	0.951503	0.475752	−	0.879580i	$-0.342176\pi$
$101$	9.56249	0.951503	0.475752	+	0.879580i	$0.342176\pi$
$102$	0	0
$103$	−18.1103	−1.78446	−0.892232	−	0.451578i	$-0.850861\pi$
$103$	−18.1103	−1.78446	−0.892232	+	0.451578i	$0.850861\pi$
$104$	0	0
$105$	4.16174	0.406144
$106$	0	0
$107$	8.61886	0.833217	0.416608	−	0.909086i	$-0.363219\pi$
$107$	8.61886	0.833217	0.416608	+	0.909086i	$0.363219\pi$
$108$	0	0
$109$	7.48252	0.716695	0.358348	−	0.933588i	$-0.383340\pi$
$109$	7.48252	0.716695	0.358348	+	0.933588i	$0.383340\pi$
$110$	0	0
$111$	−2.72035	−0.258204
$112$	0	0
$113$	20.0448	1.88565	0.942826	−	0.333284i	$-0.108157\pi$
$113$	20.0448	1.88565	0.942826	+	0.333284i	$0.108157\pi$
$114$	0	0
$115$	28.4971	2.65737
$116$	0	0
$117$	9.36411	0.865712
$118$	0	0
$119$	−18.2968	−1.67726
$120$	0	0
$121$	0.696945	0.0633587
$122$	0	0
$123$	1.68125	0.151594
$124$	0	0
$125$	22.3440	1.99851
$126$	0	0
$127$	20.1367	1.78685	0.893423	−	0.449217i	$-0.148297\pi$
$127$	20.1367	1.78685	0.893423	+	0.449217i	$0.148297\pi$
$128$	0	0
$129$	−3.94693	−0.347508
$130$	0	0
$131$	−3.66923	−0.320582	−0.160291	−	0.987070i	$-0.551243\pi$
$131$	−3.66923	−0.320582	−0.160291	+	0.987070i	$0.551243\pi$
$132$	0	0
$133$	11.4274	0.990884
$134$	0	0
$135$	−9.59390	−0.825712
$136$	0	0
$137$	11.6612	0.996285	0.498143	−	0.867095i	$-0.334016\pi$
$137$	11.6612	0.996285	0.498143	+	0.867095i	$0.334016\pi$
$138$	0	0
$139$	4.78719	0.406045	0.203022	−	0.979174i	$-0.434924\pi$
$139$	4.78719	0.406045	0.203022	+	0.979174i	$0.434924\pi$
$140$	0	0
$141$	−0.0284349	−0.00239465
$142$	0	0
$143$	−11.3296	−0.947430
$144$	0	0
$145$	28.8452	2.39546
$146$	0	0
$147$	−0.254448	−0.0209865
$148$	0	0
$149$	−2.28138	−0.186898	−0.0934492	−	0.995624i	$-0.529789\pi$
$149$	−2.28138	−0.186898	−0.0934492	+	0.995624i	$0.529789\pi$
$150$	0	0
$151$	−12.6729	−1.03130	−0.515651	−	0.856799i	$-0.672450\pi$
$151$	−12.6729	−1.03130	−0.515651	+	0.856799i	$0.672450\pi$
$152$	0	0
$153$	20.4624	1.65429
$154$	0	0
$155$	−17.6106	−1.41452
$156$	0	0
$157$	2.94428	0.234979	0.117489	−	0.993074i	$-0.462515\pi$
$157$	2.94428	0.234979	0.117489	+	0.993074i	$0.462515\pi$
$158$	0	0
$159$	3.77737	0.299565
$160$	0	0
$161$	18.2084	1.43502
$162$	0	0
$163$	−11.7301	−0.918773	−0.459387	−	0.888236i	$-0.651931\pi$
$163$	−11.7301	−0.918773	−0.459387	+	0.888236i	$0.651931\pi$
$164$	0	0
$165$	5.63125	0.438392
$166$	0	0
$167$	3.66363	0.283500	0.141750	−	0.989902i	$-0.454727\pi$
$167$	3.66363	0.283500	0.141750	+	0.989902i	$0.454727\pi$
$168$	0	0
$169$	−2.02619	−0.155861
$170$	0	0
$171$	−12.7800	−0.977310
$172$	0	0
$173$	−19.7203	−1.49930	−0.749652	−	0.661832i	$-0.769779\pi$
$173$	−19.7203	−1.49930	−0.749652	+	0.661832i	$0.769779\pi$
$174$	0	0
$175$	26.9147	2.03456
$176$	0	0
$177$	3.49676	0.262832
$178$	0	0
$179$	−21.3864	−1.59850	−0.799248	−	0.601002i	$-0.794768\pi$
$179$	−21.3864	−1.59850	−0.799248	+	0.601002i	$0.794768\pi$
$180$	0	0
$181$	−20.5412	−1.52681	−0.763406	−	0.645919i	$-0.776475\pi$
$181$	−20.5412	−1.52681	−0.763406	+	0.645919i	$0.776475\pi$
$182$	0	0
$183$	1.45090	0.107254
$184$	0	0
$185$	−25.8538	−1.90081
$186$	0	0
$187$	−24.7574	−1.81044
$188$	0	0
$189$	−6.13008	−0.445898
$190$	0	0
$191$	−6.33691	−0.458523	−0.229261	−	0.973365i	$-0.573631\pi$
$191$	−6.33691	−0.458523	−0.229261	+	0.973365i	$0.573631\pi$
$192$	0	0
$193$	−2.00200	−0.144107	−0.0720536	−	0.997401i	$-0.522955\pi$
$193$	−2.00200	−0.144107	−0.0720536	+	0.997401i	$0.522955\pi$
$194$	0	0
$195$	−5.45441	−0.390598
$196$	0	0
$197$	−8.57793	−0.611152	−0.305576	−	0.952168i	$-0.598849\pi$
$197$	−8.57793	−0.611152	−0.305576	+	0.952168i	$0.598849\pi$
$198$	0	0
$199$	−11.4115	−0.808942	−0.404471	−	0.914551i	$-0.632544\pi$
$199$	−11.4115	−0.808942	−0.404471	+	0.914551i	$0.632544\pi$
$200$	0	0
$201$	3.01790	0.212867
$202$	0	0
$203$	18.4308	1.29359
$204$	0	0
$205$	15.9784	1.11598
$206$	0	0
$207$	−20.3635	−1.41536
$208$	0	0
$209$	15.4625	1.06956
$210$	0	0
$211$	−26.1333	−1.79909	−0.899544	−	0.436831i	$-0.856101\pi$
$211$	−26.1333	−1.79909	−0.899544	+	0.436831i	$0.856101\pi$
$212$	0	0
$213$	1.35984	0.0931744
$214$	0	0
$215$	−37.5112	−2.55824
$216$	0	0
$217$	−11.2524	−0.763862
$218$	0	0
$219$	4.70269	0.317778
$220$	0	0
$221$	23.9799	1.61306
$222$	0	0
$223$	17.3328	1.16069	0.580346	−	0.814370i	$-0.302917\pi$
$223$	17.3328	1.16069	0.580346	+	0.814370i	$0.302917\pi$
$224$	0	0
$225$	−30.1004	−2.00669
$226$	0	0
$227$	−0.144305	−0.00957789	−0.00478894	−	0.999989i	$-0.501524\pi$
$227$	−0.144305	−0.00957789	−0.00478894	+	0.999989i	$0.501524\pi$
$228$	0	0
$229$	−14.5673	−0.962637	−0.481319	−	0.876546i	$-0.659842\pi$
$229$	−14.5673	−0.962637	−0.481319	+	0.876546i	$0.659842\pi$
$230$	0	0
$231$	3.59812	0.236739
$232$	0	0
$233$	18.4715	1.21011	0.605055	−	0.796184i	$-0.293151\pi$
$233$	18.4715	1.21011	0.605055	+	0.796184i	$0.293151\pi$
$234$	0	0
$235$	−0.270242	−0.0176286
$236$	0	0
$237$	4.39430	0.285440
$238$	0	0
$239$	−24.6859	−1.59680	−0.798400	−	0.602127i	$-0.794320\pi$
$239$	−24.6859	−1.59680	−0.798400	+	0.602127i	$0.794320\pi$
$240$	0	0
$241$	3.94882	0.254366	0.127183	−	0.991879i	$-0.459406\pi$
$241$	3.94882	0.254366	0.127183	+	0.991879i	$0.459406\pi$
$242$	0	0
$243$	10.3854	0.666223
$244$	0	0
$245$	−2.41825	−0.154496
$246$	0	0
$247$	−14.9769	−0.952957
$248$	0	0
$249$	0.633052	0.0401180
$250$	0	0
$251$	−10.4129	−0.657259	−0.328629	−	0.944459i	$-0.606587\pi$
$251$	−10.4129	−0.657259	−0.328629	+	0.944459i	$0.606587\pi$
$252$	0	0
$253$	24.6378	1.54897
$254$	0	0
$255$	−11.9189	−0.746392
$256$	0	0
$257$	−1.99666	−0.124548	−0.0622740	−	0.998059i	$-0.519835\pi$
$257$	−1.99666	−0.124548	−0.0622740	+	0.998059i	$0.519835\pi$
$258$	0	0
$259$	−16.5195	−1.02647
$260$	0	0
$261$	−20.6123	−1.27587
$262$	0	0
$263$	−4.62185	−0.284996	−0.142498	−	0.989795i	$-0.545513\pi$
$263$	−4.62185	−0.284996	−0.142498	+	0.989795i	$0.545513\pi$
$264$	0	0
$265$	35.8997	2.20530
$266$	0	0
$267$	−3.66413	−0.224241
$268$	0	0
$269$	14.2497	0.868822	0.434411	−	0.900715i	$-0.356957\pi$
$269$	14.2497	0.868822	0.434411	+	0.900715i	$0.356957\pi$
$270$	0	0
$271$	−21.6060	−1.31247	−0.656235	−	0.754557i	$-0.727852\pi$
$271$	−21.6060	−1.31247	−0.656235	+	0.754557i	$0.727852\pi$
$272$	0	0
$273$	−3.48512	−0.210929
$274$	0	0
$275$	36.4184	2.19611
$276$	0	0
$277$	−14.3677	−0.863273	−0.431637	−	0.902048i	$-0.642064\pi$
$277$	−14.3677	−0.863273	−0.431637	+	0.902048i	$0.642064\pi$
$278$	0	0
$279$	12.5842	0.753398
$280$	0	0
$281$	3.43523	0.204929	0.102464	−	0.994737i	$-0.467327\pi$
$281$	3.43523	0.204929	0.102464	+	0.994737i	$0.467327\pi$
$282$	0	0
$283$	3.31349	0.196966	0.0984832	−	0.995139i	$-0.468601\pi$
$283$	3.31349	0.196966	0.0984832	+	0.995139i	$0.468601\pi$
$284$	0	0
$285$	7.44409	0.440950
$286$	0	0
$287$	10.2095	0.602649
$288$	0	0
$289$	35.4007	2.08239
$290$	0	0
$291$	−1.01597	−0.0595570
$292$	0	0
$293$	7.90794	0.461987	0.230993	−	0.972955i	$-0.425802\pi$
$293$	7.90794	0.461987	0.230993	+	0.972955i	$0.425802\pi$
$294$	0	0
$295$	33.2328	1.93489
$296$	0	0
$297$	−8.29462	−0.481303
$298$	0	0
$299$	−23.8641	−1.38009
$300$	0	0
$301$	−23.9680	−1.38149
$302$	0	0
$303$	3.98020	0.228656
$304$	0	0
$305$	13.7892	0.789566
$306$	0	0
$307$	10.4809	0.598175	0.299087	−	0.954226i	$-0.403318\pi$
$307$	10.4809	0.598175	0.299087	+	0.954226i	$0.403318\pi$
$308$	0	0
$309$	−7.53807	−0.428826
$310$	0	0
$311$	−4.32281	−0.245124	−0.122562	−	0.992461i	$-0.539111\pi$
$311$	−4.32281	−0.245124	−0.122562	+	0.992461i	$0.539111\pi$
$312$	0	0
$313$	16.4570	0.930205	0.465103	−	0.885257i	$-0.346017\pi$
$313$	16.4570	0.930205	0.465103	+	0.885257i	$0.346017\pi$
$314$	0	0
$315$	−28.2637	−1.59248
$316$	0	0
$317$	−24.1651	−1.35725	−0.678623	−	0.734487i	$-0.737423\pi$
$317$	−24.1651	−1.35725	−0.678623	+	0.734487i	$0.737423\pi$
$318$	0	0
$319$	24.9388	1.39630
$320$	0	0
$321$	3.58743	0.200231
$322$	0	0
$323$	−32.7274	−1.82100
$324$	0	0
$325$	−35.2747	−1.95669
$326$	0	0
$327$	3.11445	0.172230
$328$	0	0
$329$	−0.172672	−0.00951974
$330$	0	0
$331$	13.1035	0.720234	0.360117	−	0.932907i	$-0.382737\pi$
$331$	13.1035	0.720234	0.360117	+	0.932907i	$0.382737\pi$
$332$	0	0
$333$	18.4747	1.01241
$334$	0	0
$335$	28.6818	1.56705
$336$	0	0
$337$	7.16212	0.390146	0.195073	−	0.980789i	$-0.437506\pi$
$337$	7.16212	0.390146	0.195073	+	0.980789i	$0.437506\pi$
$338$	0	0
$339$	8.34324	0.453143
$340$	0	0
$341$	−15.2256	−0.824514
$342$	0	0
$343$	−19.2382	−1.03877
$344$	0	0
$345$	11.8614	0.638594
$346$	0	0
$347$	−2.74633	−0.147431	−0.0737154	−	0.997279i	$-0.523486\pi$
$347$	−2.74633	−0.147431	−0.0737154	+	0.997279i	$0.523486\pi$
$348$	0	0
$349$	−21.7346	−1.16343	−0.581715	−	0.813393i	$-0.697618\pi$
$349$	−21.7346	−1.16343	−0.581715	+	0.813393i	$0.697618\pi$
$350$	0	0
$351$	8.03414	0.428830
$352$	0	0
$353$	32.7124	1.74110	0.870551	−	0.492078i	$-0.163762\pi$
$353$	32.7124	1.74110	0.870551	+	0.492078i	$0.163762\pi$
$354$	0	0
$355$	12.9237	0.685920
$356$	0	0
$357$	−7.61567	−0.403064
$358$	0	0
$359$	16.6087	0.876573	0.438287	−	0.898835i	$-0.355585\pi$
$359$	16.6087	0.876573	0.438287	+	0.898835i	$0.355585\pi$
$360$	0	0
$361$	1.44023	0.0758015
$362$	0	0
$363$	0.290090	0.0152258
$364$	0	0
$365$	44.6938	2.33938
$366$	0	0
$367$	−3.70718	−0.193513	−0.0967566	−	0.995308i	$-0.530847\pi$
$367$	−3.70718	−0.193513	−0.0967566	+	0.995308i	$0.530847\pi$
$368$	0	0
$369$	−11.4179	−0.594394
$370$	0	0
$371$	22.9383	1.19090
$372$	0	0
$373$	−5.39864	−0.279531	−0.139765	−	0.990185i	$-0.544635\pi$
$373$	−5.39864	−0.279531	−0.139765	+	0.990185i	$0.544635\pi$
$374$	0	0
$375$	9.30024	0.480262
$376$	0	0
$377$	−24.1556	−1.24407
$378$	0	0
$379$	−5.09265	−0.261592	−0.130796	−	0.991409i	$-0.541753\pi$
$379$	−5.09265	−0.261592	−0.130796	+	0.991409i	$0.541753\pi$
$380$	0	0
$381$	8.38152	0.429398
$382$	0	0
$383$	32.8186	1.67695	0.838476	−	0.544938i	$-0.183447\pi$
$383$	32.8186	1.67695	0.838476	+	0.544938i	$0.183447\pi$
$384$	0	0
$385$	34.1961	1.74280
$386$	0	0
$387$	26.8048	1.36257
$388$	0	0
$389$	−35.8011	−1.81519	−0.907594	−	0.419849i	$-0.862083\pi$
$389$	−35.8011	−1.81519	−0.907594	+	0.419849i	$0.862083\pi$
$390$	0	0
$391$	−52.1476	−2.63722
$392$	0	0
$393$	−1.52725	−0.0770394
$394$	0	0
$395$	41.7629	2.10132
$396$	0	0
$397$	33.2770	1.67012	0.835061	−	0.550157i	$-0.185432\pi$
$397$	33.2770	1.67012	0.835061	+	0.550157i	$0.185432\pi$
$398$	0	0
$399$	4.75644	0.238120
$400$	0	0
$401$	−21.2385	−1.06060	−0.530299	−	0.847811i	$-0.677920\pi$
$401$	−21.2385	−1.06060	−0.530299	+	0.847811i	$0.677920\pi$
$402$	0	0
$403$	14.7475	0.734624
$404$	0	0
$405$	29.5530	1.46850
$406$	0	0
$407$	−22.3525	−1.10797
$408$	0	0
$409$	14.5580	0.719849	0.359924	−	0.932981i	$-0.382802\pi$
$409$	14.5580	0.719849	0.359924	+	0.932981i	$0.382802\pi$
$410$	0	0
$411$	4.85376	0.239418
$412$	0	0
$413$	21.2343	1.04487
$414$	0	0
$415$	6.01645	0.295336
$416$	0	0
$417$	1.99258	0.0975769
$418$	0	0
$419$	26.7092	1.30483	0.652416	−	0.757861i	$-0.273756\pi$
$419$	26.7092	1.30483	0.652416	+	0.757861i	$0.273756\pi$
$420$	0	0
$421$	−27.3395	−1.33245	−0.666223	−	0.745752i	$-0.732090\pi$
$421$	−27.3395	−1.33245	−0.666223	+	0.745752i	$0.732090\pi$
$422$	0	0
$423$	0.193110	0.00938934
$424$	0	0
$425$	−77.0820	−3.73902
$426$	0	0
$427$	8.81067	0.426378
$428$	0	0
$429$	−4.71573	−0.227678
$430$	0	0
$431$	−25.8344	−1.24440	−0.622199	−	0.782859i	$-0.713761\pi$
$431$	−25.8344	−1.24440	−0.622199	+	0.782859i	$0.713761\pi$
$432$	0	0
$433$	4.10488	0.197268	0.0986340	−	0.995124i	$-0.468553\pi$
$433$	4.10488	0.197268	0.0986340	+	0.995124i	$0.468553\pi$
$434$	0	0
$435$	12.0062	0.575656
$436$	0	0
$437$	32.5693	1.55800
$438$	0	0
$439$	−28.5009	−1.36027	−0.680137	−	0.733085i	$-0.738080\pi$
$439$	−28.5009	−1.36027	−0.680137	+	0.733085i	$0.738080\pi$
$440$	0	0
$441$	1.72804	0.0822876
$442$	0	0
$443$	−2.09069	−0.0993316	−0.0496658	−	0.998766i	$-0.515816\pi$
$443$	−2.09069	−0.0993316	−0.0496658	+	0.998766i	$0.515816\pi$
$444$	0	0
$445$	−34.8235	−1.65079
$446$	0	0
$447$	−0.949582	−0.0449137
$448$	0	0
$449$	−20.2633	−0.956285	−0.478143	−	0.878282i	$-0.658690\pi$
$449$	−20.2633	−0.956285	−0.478143	+	0.878282i	$0.658690\pi$
$450$	0	0
$451$	13.8145	0.650500
$452$	0	0
$453$	−5.27483	−0.247833
$454$	0	0
$455$	−33.1222	−1.55279
$456$	0	0
$457$	−15.3056	−0.715968	−0.357984	−	0.933728i	$-0.616536\pi$
$457$	−15.3056	−0.715968	−0.357984	+	0.933728i	$0.616536\pi$
$458$	0	0
$459$	17.5561	0.819450
$460$	0	0
$461$	36.0225	1.67773	0.838867	−	0.544336i	$-0.183218\pi$
$461$	36.0225	1.67773	0.838867	+	0.544336i	$0.183218\pi$
$462$	0	0
$463$	−22.9678	−1.06740	−0.533702	−	0.845673i	$-0.679200\pi$
$463$	−22.9678	−1.06740	−0.533702	+	0.845673i	$0.679200\pi$
$464$	0	0
$465$	−7.33007	−0.339924
$466$	0	0
$467$	1.03068	0.0476941	0.0238471	−	0.999716i	$-0.492409\pi$
$467$	1.03068	0.0476941	0.0238471	+	0.999716i	$0.492409\pi$
$468$	0	0
$469$	18.3264	0.846235
$470$	0	0
$471$	1.22550	0.0564679
$472$	0	0
$473$	−32.4311	−1.49118
$474$	0	0
$475$	48.1423	2.20892
$476$	0	0
$477$	−25.6533	−1.17458
$478$	0	0
$479$	37.6486	1.72021	0.860103	−	0.510120i	$-0.170399\pi$
$479$	37.6486	1.72021	0.860103	+	0.510120i	$0.170399\pi$
$480$	0	0
$481$	21.6506	0.987180
$482$	0	0
$483$	7.57888	0.344851
$484$	0	0
$485$	−9.65563	−0.438439
$486$	0	0
$487$	26.2919	1.19140	0.595700	−	0.803207i	$-0.296875\pi$
$487$	26.2919	1.19140	0.595700	+	0.803207i	$0.296875\pi$
$488$	0	0
$489$	−4.88243	−0.220791
$490$	0	0
$491$	−2.92374	−0.131947	−0.0659733	−	0.997821i	$-0.521015\pi$
$491$	−2.92374	−0.131947	−0.0659733	+	0.997821i	$0.521015\pi$
$492$	0	0
$493$	−52.7846	−2.37730
$494$	0	0
$495$	−38.2436	−1.71892
$496$	0	0
$497$	8.25769	0.370408
$498$	0	0
$499$	5.40783	0.242088	0.121044	−	0.992647i	$-0.461376\pi$
$499$	5.40783	0.242088	0.121044	+	0.992647i	$0.461376\pi$
$500$	0	0
$501$	1.52491	0.0681282
$502$	0	0
$503$	13.6924	0.610513	0.305257	−	0.952270i	$-0.401258\pi$
$503$	13.6924	0.610513	0.305257	+	0.952270i	$0.401258\pi$
$504$	0	0
$505$	37.8273	1.68329
$506$	0	0
$507$	−0.843362	−0.0374550
$508$	0	0
$509$	38.1325	1.69020	0.845098	−	0.534612i	$-0.179542\pi$
$509$	38.1325	1.69020	0.845098	+	0.534612i	$0.179542\pi$
$510$	0	0
$511$	28.5574	1.26330
$512$	0	0
$513$	−10.9649	−0.484111
$514$	0	0
$515$	−71.6409	−3.15688
$516$	0	0
$517$	−0.233643	−0.0102756
$518$	0	0
$519$	−8.20818	−0.360299
$520$	0	0
$521$	−21.2455	−0.930781	−0.465390	−	0.885105i	$-0.654086\pi$
$521$	−21.2455	−0.930781	−0.465390	+	0.885105i	$0.654086\pi$
$522$	0	0
$523$	2.96494	0.129648	0.0648238	−	0.997897i	$-0.479351\pi$
$523$	2.96494	0.129648	0.0648238	+	0.997897i	$0.479351\pi$
$524$	0	0
$525$	11.2027	0.488927
$526$	0	0
$527$	32.2261	1.40379
$528$	0	0
$529$	28.8957	1.25633
$530$	0	0
$531$	−23.7476	−1.03056
$532$	0	0
$533$	−13.3807	−0.579582
$534$	0	0
$535$	34.0945	1.47404
$536$	0	0
$537$	−8.90167	−0.384136
$538$	0	0
$539$	−2.09075	−0.0900549
$540$	0	0
$541$	−1.64629	−0.0707795	−0.0353898	−	0.999374i	$-0.511267\pi$
$541$	−1.64629	−0.0707795	−0.0353898	+	0.999374i	$0.511267\pi$
$542$	0	0
$543$	−8.54986	−0.366910
$544$	0	0
$545$	29.5994	1.26790
$546$	0	0
$547$	−22.8332	−0.976279	−0.488139	−	0.872766i	$-0.662324\pi$
$547$	−22.8332	−0.976279	−0.488139	+	0.872766i	$0.662324\pi$
$548$	0	0
$549$	−9.85351	−0.420537
$550$	0	0
$551$	32.9671	1.40445
$552$	0	0
$553$	26.6847	1.13475
$554$	0	0
$555$	−10.7612	−0.456786
$556$	0	0
$557$	2.83084	0.119947	0.0599733	−	0.998200i	$-0.480898\pi$
$557$	2.83084	0.119947	0.0599733	+	0.998200i	$0.480898\pi$
$558$	0	0
$559$	31.4126	1.32861
$560$	0	0
$561$	−10.3048	−0.435068
$562$	0	0
$563$	−16.2716	−0.685766	−0.342883	−	0.939378i	$-0.611403\pi$
$563$	−16.2716	−0.685766	−0.342883	+	0.939378i	$0.611403\pi$
$564$	0	0
$565$	79.2932	3.33589
$566$	0	0
$567$	18.8830	0.793013
$568$	0	0
$569$	−27.0818	−1.13533	−0.567664	−	0.823260i	$-0.692153\pi$
$569$	−27.0818	−1.13533	−0.567664	+	0.823260i	$0.692153\pi$
$570$	0	0
$571$	−30.6590	−1.28304	−0.641520	−	0.767107i	$-0.721696\pi$
$571$	−30.6590	−1.28304	−0.641520	+	0.767107i	$0.721696\pi$
$572$	0	0
$573$	−2.63761	−0.110188
$574$	0	0
$575$	76.7097	3.19901
$576$	0	0
$577$	2.19730	0.0914749	0.0457374	−	0.998953i	$-0.485436\pi$
$577$	2.19730	0.0914749	0.0457374	+	0.998953i	$0.485436\pi$
$578$	0	0
$579$	−0.833293	−0.0346305
$580$	0	0
$581$	3.84425	0.159486
$582$	0	0
$583$	31.0378	1.28546
$584$	0	0
$585$	37.0426	1.53152
$586$	0	0
$587$	4.26846	0.176178	0.0880891	−	0.996113i	$-0.471924\pi$
$587$	4.26846	0.176178	0.0880891	+	0.996113i	$0.471924\pi$
$588$	0	0
$589$	−20.1271	−0.829324
$590$	0	0
$591$	−3.57040	−0.146867
$592$	0	0
$593$	−12.2142	−0.501578	−0.250789	−	0.968042i	$-0.580690\pi$
$593$	−12.2142	−0.501578	−0.250789	+	0.968042i	$0.580690\pi$
$594$	0	0
$595$	−72.3784	−2.96723
$596$	0	0
$597$	−4.74983	−0.194398
$598$	0	0
$599$	−33.3761	−1.36371	−0.681856	−	0.731487i	$-0.738827\pi$
$599$	−33.3761	−1.36371	−0.681856	+	0.731487i	$0.738827\pi$
$600$	0	0
$601$	40.8036	1.66441	0.832206	−	0.554466i	$-0.187078\pi$
$601$	40.8036	1.66441	0.832206	+	0.554466i	$0.187078\pi$
$602$	0	0
$603$	−20.4955	−0.834643
$604$	0	0
$605$	2.75698	0.112087
$606$	0	0
$607$	−29.2049	−1.18539	−0.592695	−	0.805427i	$-0.701936\pi$
$607$	−29.2049	−1.18539	−0.592695	+	0.805427i	$0.701936\pi$
$608$	0	0
$609$	7.67146	0.310863
$610$	0	0
$611$	0.226306	0.00915536
$612$	0	0
$613$	−26.8257	−1.08348	−0.541741	−	0.840546i	$-0.682235\pi$
$613$	−26.8257	−1.08348	−0.541741	+	0.840546i	$0.682235\pi$
$614$	0	0
$615$	6.65072	0.268183
$616$	0	0
$617$	13.5959	0.547349	0.273675	−	0.961822i	$-0.411761\pi$
$617$	13.5959	0.547349	0.273675	+	0.961822i	$0.411761\pi$
$618$	0	0
$619$	−31.7095	−1.27451	−0.637257	−	0.770651i	$-0.719931\pi$
$619$	−31.7095	−1.27451	−0.637257	+	0.770651i	$0.719931\pi$
$620$	0	0
$621$	−17.4713	−0.701101
$622$	0	0
$623$	−22.2507	−0.891454
$624$	0	0
$625$	35.1464	1.40586
$626$	0	0
$627$	6.43595	0.257027
$628$	0	0
$629$	47.3106	1.88640
$630$	0	0
$631$	−34.6540	−1.37956	−0.689778	−	0.724021i	$-0.742292\pi$
$631$	−34.6540	−1.37956	−0.689778	+	0.724021i	$0.742292\pi$
$632$	0	0
$633$	−10.8775	−0.432340
$634$	0	0
$635$	79.6570	3.16109
$636$	0	0
$637$	2.02509	0.0802370
$638$	0	0
$639$	−9.23508	−0.365334
$640$	0	0
$641$	−16.3012	−0.643857	−0.321929	−	0.946764i	$-0.604331\pi$
$641$	−16.3012	−0.643857	−0.321929	+	0.946764i	$0.604331\pi$
$642$	0	0
$643$	10.2511	0.404264	0.202132	−	0.979358i	$-0.435213\pi$
$643$	10.2511	0.404264	0.202132	+	0.979358i	$0.435213\pi$
$644$	0	0
$645$	−15.6133	−0.614773
$646$	0	0
$647$	−28.0119	−1.10126	−0.550630	−	0.834749i	$-0.685612\pi$
$647$	−28.0119	−1.10126	−0.550630	+	0.834749i	$0.685612\pi$
$648$	0	0
$649$	28.7321	1.12783
$650$	0	0
$651$	−4.68359	−0.183564
$652$	0	0
$653$	−42.7086	−1.67132	−0.835659	−	0.549249i	$-0.814914\pi$
$653$	−42.7086	−1.67132	−0.835659	+	0.549249i	$0.814914\pi$
$654$	0	0
$655$	−14.5148	−0.567139
$656$	0	0
$657$	−31.9375	−1.24600
$658$	0	0
$659$	−23.5297	−0.916586	−0.458293	−	0.888801i	$-0.651539\pi$
$659$	−23.5297	−0.916586	−0.458293	+	0.888801i	$0.651539\pi$
$660$	0	0
$661$	−30.6562	−1.19239	−0.596194	−	0.802840i	$-0.703321\pi$
$661$	−30.6562	−1.19239	−0.596194	+	0.802840i	$0.703321\pi$
$662$	0	0
$663$	9.98116	0.387636
$664$	0	0
$665$	45.2047	1.75296
$666$	0	0
$667$	52.5296	2.03395
$668$	0	0
$669$	7.21445	0.278927
$670$	0	0
$671$	11.9217	0.460233
$672$	0	0
$673$	−43.7867	−1.68785	−0.843926	−	0.536459i	$-0.819762\pi$
$673$	−43.7867	−1.68785	−0.843926	+	0.536459i	$0.819762\pi$
$674$	0	0
$675$	−25.8253	−0.994015
$676$	0	0
$677$	33.6766	1.29430	0.647148	−	0.762364i	$-0.275961\pi$
$677$	33.6766	1.29430	0.647148	+	0.762364i	$0.275961\pi$
$678$	0	0
$679$	−6.16952	−0.236764
$680$	0	0
$681$	−0.0600643	−0.00230167
$682$	0	0
$683$	19.0664	0.729557	0.364778	−	0.931094i	$-0.381145\pi$
$683$	19.0664	0.729557	0.364778	+	0.931094i	$0.381145\pi$
$684$	0	0
$685$	46.1295	1.76252
$686$	0	0
$687$	−6.06337	−0.231332
$688$	0	0
$689$	−30.0631	−1.14531
$690$	0	0
$691$	18.3146	0.696722	0.348361	−	0.937361i	$-0.386738\pi$
$691$	18.3146	0.696722	0.348361	+	0.937361i	$0.386738\pi$
$692$	0	0
$693$	−24.4360	−0.928246
$694$	0	0
$695$	18.9372	0.718329
$696$	0	0
$697$	−29.2394	−1.10752
$698$	0	0
$699$	7.68841	0.290803
$700$	0	0
$701$	−18.0765	−0.682740	−0.341370	−	0.939929i	$-0.610891\pi$
$701$	−18.0765	−0.682740	−0.341370	+	0.939929i	$0.610891\pi$
$702$	0	0
$703$	−29.5483	−1.11444
$704$	0	0
$705$	−0.112483	−0.00423635
$706$	0	0
$707$	24.1700	0.909006
$708$	0	0
$709$	24.6654	0.926327	0.463164	−	0.886273i	$-0.346714\pi$
$709$	24.6654	0.926327	0.463164	+	0.886273i	$0.346714\pi$
$710$	0	0
$711$	−29.8431	−1.11920
$712$	0	0
$713$	−32.0704	−1.20105
$714$	0	0
$715$	−44.8177	−1.67609
$716$	0	0
$717$	−10.2750	−0.383728
$718$	0	0
$719$	11.6829	0.435699	0.217849	−	0.975982i	$-0.430096\pi$
$719$	11.6829	0.435699	0.217849	+	0.975982i	$0.430096\pi$
$720$	0	0
$721$	−45.7754	−1.70476
$722$	0	0
$723$	1.64362	0.0611269
$724$	0	0
$725$	77.6467	2.88372
$726$	0	0
$727$	19.9582	0.740210	0.370105	−	0.928990i	$-0.379322\pi$
$727$	19.9582	0.740210	0.370105	+	0.928990i	$0.379322\pi$
$728$	0	0
$729$	−18.0896	−0.669987
$730$	0	0
$731$	68.6426	2.53884
$732$	0	0
$733$	−48.0802	−1.77588	−0.887941	−	0.459957i	$-0.847865\pi$
$733$	−48.0802	−1.77588	−0.887941	+	0.459957i	$0.847865\pi$
$734$	0	0
$735$	−1.00655	−0.0371271
$736$	0	0
$737$	24.7975	0.913428
$738$	0	0
$739$	−43.9960	−1.61842	−0.809209	−	0.587521i	$-0.800104\pi$
$739$	−43.9960	−1.61842	−0.809209	+	0.587521i	$0.800104\pi$
$740$	0	0
$741$	−6.23384	−0.229006
$742$	0	0
$743$	16.0686	0.589500	0.294750	−	0.955574i	$-0.404764\pi$
$743$	16.0686	0.589500	0.294750	+	0.955574i	$0.404764\pi$
$744$	0	0
$745$	−9.02471	−0.330640
$746$	0	0
$747$	−4.29926	−0.157302
$748$	0	0
$749$	21.7849	0.796003
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−4.33418	−0.157946
$754$	0	0
$755$	−50.1314	−1.82447
$756$	0	0
$757$	−3.40087	−0.123607	−0.0618034	−	0.998088i	$-0.519685\pi$
$757$	−3.40087	−0.123607	−0.0618034	+	0.998088i	$0.519685\pi$
$758$	0	0
$759$	10.2550	0.372233
$760$	0	0
$761$	15.0807	0.546674	0.273337	−	0.961918i	$-0.411873\pi$
$761$	15.0807	0.546674	0.273337	+	0.961918i	$0.411873\pi$
$762$	0	0
$763$	18.9127	0.684685
$764$	0	0
$765$	80.9452	2.92658
$766$	0	0
$767$	−27.8298	−1.00488
$768$	0	0
$769$	37.8490	1.36487	0.682436	−	0.730946i	$-0.260921\pi$
$769$	37.8490	1.36487	0.682436	+	0.730946i	$0.260921\pi$
$770$	0	0
$771$	−0.831069	−0.0299302
$772$	0	0
$773$	−1.68818	−0.0607198	−0.0303599	−	0.999539i	$-0.509665\pi$
$773$	−1.68818	−0.0607198	−0.0303599	+	0.999539i	$0.509665\pi$
$774$	0	0
$775$	−47.4049	−1.70283
$776$	0	0
$777$	−6.87590	−0.246672
$778$	0	0
$779$	18.2618	0.654295
$780$	0	0
$781$	11.1735	0.399819
$782$	0	0
$783$	−17.6847	−0.632001
$784$	0	0
$785$	11.6470	0.415699
$786$	0	0
$787$	21.1509	0.753949	0.376975	−	0.926224i	$-0.376964\pi$
$787$	21.1509	0.753949	0.376975	+	0.926224i	$0.376964\pi$
$788$	0	0
$789$	−1.92376	−0.0684875
$790$	0	0
$791$	50.6648	1.80143
$792$	0	0
$793$	−11.5473	−0.410058
$794$	0	0
$795$	14.9425	0.529957
$796$	0	0
$797$	53.5742	1.89770	0.948848	−	0.315733i	$-0.102250\pi$
$797$	53.5742	1.89770	0.948848	+	0.315733i	$0.102250\pi$
$798$	0	0
$799$	0.494522	0.0174949
$800$	0	0
$801$	24.8843	0.879242
$802$	0	0
$803$	38.6410	1.36361
$804$	0	0
$805$	72.0288	2.53868
$806$	0	0
$807$	5.93117	0.208787
$808$	0	0
$809$	28.8424	1.01404	0.507022	−	0.861933i	$-0.330746\pi$
$809$	28.8424	1.01404	0.507022	+	0.861933i	$0.330746\pi$
$810$	0	0
$811$	−40.4030	−1.41874	−0.709371	−	0.704835i	$-0.751021\pi$
$811$	−40.4030	−1.41874	−0.709371	+	0.704835i	$0.751021\pi$
$812$	0	0
$813$	−8.99307	−0.315401
$814$	0	0
$815$	−46.4021	−1.62539
$816$	0	0
$817$	−42.8715	−1.49988
$818$	0	0
$819$	23.6686	0.827047
$820$	0	0
$821$	50.4957	1.76231	0.881157	−	0.472824i	$-0.156765\pi$
$821$	50.4957	1.76231	0.881157	+	0.472824i	$0.156765\pi$
$822$	0	0
$823$	47.2314	1.64638	0.823191	−	0.567764i	$-0.192192\pi$
$823$	47.2314	1.64638	0.823191	+	0.567764i	$0.192192\pi$
$824$	0	0
$825$	15.1584	0.527749
$826$	0	0
$827$	−14.3757	−0.499892	−0.249946	−	0.968260i	$-0.580413\pi$
$827$	−14.3757	−0.499892	−0.249946	+	0.968260i	$0.580413\pi$
$828$	0	0
$829$	−1.41240	−0.0490548	−0.0245274	−	0.999699i	$-0.507808\pi$
$829$	−1.41240	−0.0490548	−0.0245274	+	0.999699i	$0.507808\pi$
$830$	0	0
$831$	−5.98029	−0.207454
$832$	0	0
$833$	4.42521	0.153325
$834$	0	0
$835$	14.4926	0.501537
$836$	0	0
$837$	10.7969	0.373196
$838$	0	0
$839$	−0.522791	−0.0180488	−0.00902438	−	0.999959i	$-0.502873\pi$
$839$	−0.522791	−0.0180488	−0.00902438	+	0.999959i	$0.502873\pi$
$840$	0	0
$841$	24.1713	0.833492
$842$	0	0
$843$	1.42985	0.0492466
$844$	0	0
$845$	−8.01521	−0.275732
$846$	0	0
$847$	1.76159	0.0605289
$848$	0	0
$849$	1.37918	0.0473332
$850$	0	0
$851$	−47.0821	−1.61395
$852$	0	0
$853$	−3.31887	−0.113636	−0.0568180	−	0.998385i	$-0.518095\pi$
$853$	−3.31887	−0.113636	−0.0568180	+	0.998385i	$0.518095\pi$
$854$	0	0
$855$	−50.5552	−1.72895
$856$	0	0
$857$	4.52657	0.154625	0.0773124	−	0.997007i	$-0.475366\pi$
$857$	4.52657	0.154625	0.0773124	+	0.997007i	$0.475366\pi$
$858$	0	0
$859$	13.2237	0.451186	0.225593	−	0.974222i	$-0.427568\pi$
$859$	13.2237	0.451186	0.225593	+	0.974222i	$0.427568\pi$
$860$	0	0
$861$	4.24951	0.144823
$862$	0	0
$863$	10.4020	0.354089	0.177044	−	0.984203i	$-0.443346\pi$
$863$	10.4020	0.354089	0.177044	+	0.984203i	$0.443346\pi$
$864$	0	0
$865$	−78.0095	−2.65240
$866$	0	0
$867$	14.7348	0.500421
$868$	0	0
$869$	36.1071	1.22485
$870$	0	0
$871$	−24.0187	−0.813844
$872$	0	0
$873$	6.89975	0.233521
$874$	0	0
$875$	56.4763	1.90925
$876$	0	0
$877$	32.1045	1.08409	0.542047	−	0.840348i	$-0.317650\pi$
$877$	32.1045	1.08409	0.542047	+	0.840348i	$0.317650\pi$
$878$	0	0
$879$	3.29152	0.111020
$880$	0	0
$881$	−46.1835	−1.55596	−0.777981	−	0.628288i	$-0.783756\pi$
$881$	−46.1835	−1.55596	−0.777981	+	0.628288i	$0.783756\pi$
$882$	0	0
$883$	−15.3369	−0.516127	−0.258063	−	0.966128i	$-0.583084\pi$
$883$	−15.3369	−0.516127	−0.258063	+	0.966128i	$0.583084\pi$
$884$	0	0
$885$	13.8325	0.464974
$886$	0	0
$887$	−27.1206	−0.910620	−0.455310	−	0.890333i	$-0.650472\pi$
$887$	−27.1206	−0.910620	−0.455310	+	0.890333i	$0.650472\pi$
$888$	0	0
$889$	50.8973	1.70704
$890$	0	0
$891$	25.5507	0.855980
$892$	0	0
$893$	−0.308859	−0.0103356
$894$	0	0
$895$	−84.6005	−2.82788
$896$	0	0
$897$	−9.93295	−0.331652
$898$	0	0
$899$	−32.4622	−1.08267
$900$	0	0
$901$	−65.6937	−2.18857
$902$	0	0
$903$	−9.97620	−0.331987
$904$	0	0
$905$	−81.2568	−2.70107
$906$	0	0
$907$	16.8042	0.557976	0.278988	−	0.960295i	$-0.410001\pi$
$907$	16.8042	0.557976	0.278988	+	0.960295i	$0.410001\pi$
$908$	0	0
$909$	−27.0308	−0.896554
$910$	0	0
$911$	40.9944	1.35820	0.679102	−	0.734044i	$-0.262369\pi$
$911$	40.9944	1.35820	0.679102	+	0.734044i	$0.262369\pi$
$912$	0	0
$913$	5.20166	0.172150
$914$	0	0
$915$	5.73947	0.189741
$916$	0	0
$917$	−9.27430	−0.306264
$918$	0	0
$919$	−2.07414	−0.0684197	−0.0342098	−	0.999415i	$-0.510891\pi$
$919$	−2.07414	−0.0684197	−0.0342098	+	0.999415i	$0.510891\pi$
$920$	0	0
$921$	4.36246	0.143748
$922$	0	0
$923$	−10.8226	−0.356230
$924$	0	0
$925$	−69.5944	−2.28825
$926$	0	0
$927$	51.1934	1.68141
$928$	0	0
$929$	−30.3153	−0.994612	−0.497306	−	0.867575i	$-0.665677\pi$
$929$	−30.3153	−0.994612	−0.497306	+	0.867575i	$0.665677\pi$
$930$	0	0
$931$	−2.76381	−0.0905803
$932$	0	0
$933$	−1.79929	−0.0589060
$934$	0	0
$935$	−97.9354	−3.20283
$936$	0	0
$937$	−15.0261	−0.490881	−0.245440	−	0.969412i	$-0.578933\pi$
$937$	−15.0261	−0.490881	−0.245440	+	0.969412i	$0.578933\pi$
$938$	0	0
$939$	6.84991	0.223538
$940$	0	0
$941$	−25.7935	−0.840843	−0.420422	−	0.907329i	$-0.638118\pi$
$941$	−25.7935	−0.840843	−0.420422	+	0.907329i	$0.638118\pi$
$942$	0	0
$943$	29.0981	0.947566
$944$	0	0
$945$	−24.2494	−0.788833
$946$	0	0
$947$	47.9193	1.55717	0.778584	−	0.627540i	$-0.215938\pi$
$947$	47.9193	1.55717	0.778584	+	0.627540i	$0.215938\pi$
$948$	0	0
$949$	−37.4276	−1.21495
$950$	0	0
$951$	−10.0582	−0.326161
$952$	0	0
$953$	−45.9750	−1.48928	−0.744638	−	0.667469i	$-0.767378\pi$
$953$	−45.9750	−1.48928	−0.744638	+	0.667469i	$0.767378\pi$
$954$	0	0
$955$	−25.0676	−0.811168
$956$	0	0
$957$	10.3803	0.335546
$958$	0	0
$959$	29.4747	0.951788
$960$	0	0
$961$	−11.1812	−0.360683
$962$	0	0
$963$	−24.3634	−0.785099
$964$	0	0
$965$	−7.91952	−0.254938
$966$	0	0
$967$	41.1328	1.32274	0.661371	−	0.750059i	$-0.269975\pi$
$967$	41.1328	1.32274	0.661371	+	0.750059i	$0.269975\pi$
$968$	0	0
$969$	−13.6221	−0.437606
$970$	0	0
$971$	−7.13673	−0.229028	−0.114514	−	0.993422i	$-0.536531\pi$
$971$	−7.13673	−0.229028	−0.114514	+	0.993422i	$0.536531\pi$
$972$	0	0
$973$	12.1000	0.387909
$974$	0	0
$975$	−14.6824	−0.470213
$976$	0	0
$977$	−31.7605	−1.01611	−0.508054	−	0.861325i	$-0.669635\pi$
$977$	−31.7605	−1.01611	−0.508054	+	0.861325i	$0.669635\pi$
$978$	0	0
$979$	−30.1074	−0.962237
$980$	0	0
$981$	−21.1512	−0.675306
$982$	0	0
$983$	16.0639	0.512359	0.256180	−	0.966629i	$-0.417536\pi$
$983$	16.0639	0.512359	0.256180	+	0.966629i	$0.417536\pi$
$984$	0	0
$985$	−33.9326	−1.08118
$986$	0	0
$987$	−0.0718715	−0.00228770
$988$	0	0
$989$	−68.3111	−2.17217
$990$	0	0
$991$	−19.7424	−0.627138	−0.313569	−	0.949565i	$-0.601525\pi$
$991$	−19.7424	−0.627138	−0.313569	+	0.949565i	$0.601525\pi$
$992$	0	0
$993$	5.45408	0.173080
$994$	0	0
$995$	−45.1418	−1.43109
$996$	0	0
$997$	−12.2962	−0.389426	−0.194713	−	0.980860i	$-0.562378\pi$
$997$	−12.2962	−0.389426	−0.194713	+	0.980860i	$0.562378\pi$
$998$	0	0
$999$	15.8508	0.501497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.29	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.29	✓	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)

\(f(q)\)	\(=\)	\(q+0.416230 q^{3} +3.95581 q^{5} +2.52758 q^{7} -2.82675 q^{9} +O(q^{10})\)	\(q+0.416230 q^{3} +3.95581 q^{5} +2.52758 q^{7} -2.82675 q^{9} +3.42008 q^{11} -3.31267 q^{13} +1.64653 q^{15} -7.23883 q^{17} +4.52109 q^{19} +1.05206 q^{21} +7.20387 q^{23} +10.6484 q^{25} -2.42527 q^{27} +7.29186 q^{29} -4.45184 q^{31} +1.42354 q^{33} +9.99863 q^{35} -6.53567 q^{37} -1.37884 q^{39} +4.03924 q^{41} -9.48256 q^{43} -11.1821 q^{45} -0.0683152 q^{47} -0.611316 q^{49} -3.01302 q^{51} +9.07518 q^{53} +13.5292 q^{55} +1.88181 q^{57} +8.40101 q^{59} +3.48581 q^{61} -7.14486 q^{63} -13.1043 q^{65} +7.25056 q^{67} +2.99847 q^{69} +3.26703 q^{71} +11.2983 q^{73} +4.43219 q^{75} +8.64454 q^{77} +10.5574 q^{79} +7.47078 q^{81} +1.52092 q^{83} -28.6354 q^{85} +3.03510 q^{87} -8.80313 q^{89} -8.37306 q^{91} -1.85299 q^{93} +17.8845 q^{95} -2.44088 q^{97} -9.66772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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