LMFDB - Embedded newform 6008.2.a.b.1.15

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			1.15
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)

$f(q)$	$=$	$q-1.61777 q^{3} -2.18000 q^{5} -2.82839 q^{7} -0.382806 q^{9} +O(q^{10})$	$q-1.61777 q^{3} -2.18000 q^{5} -2.82839 q^{7} -0.382806 q^{9} -0.331382 q^{11} +1.18515 q^{13} +3.52675 q^{15} -0.850397 q^{17} -2.54729 q^{19} +4.57569 q^{21} -2.89409 q^{23} -0.247601 q^{25} +5.47262 q^{27} +3.27328 q^{29} +9.17864 q^{31} +0.536102 q^{33} +6.16588 q^{35} +4.19824 q^{37} -1.91730 q^{39} +10.8726 q^{41} +5.52475 q^{43} +0.834518 q^{45} -3.07403 q^{47} +0.999768 q^{49} +1.37575 q^{51} -7.07359 q^{53} +0.722413 q^{55} +4.12095 q^{57} -5.25476 q^{59} +5.14026 q^{61} +1.08272 q^{63} -2.58362 q^{65} -12.7145 q^{67} +4.68198 q^{69} +6.59360 q^{71} +8.22764 q^{73} +0.400563 q^{75} +0.937277 q^{77} -7.17156 q^{79} -7.70504 q^{81} +5.27522 q^{83} +1.85387 q^{85} -5.29544 q^{87} +15.0525 q^{89} -3.35205 q^{91} -14.8490 q^{93} +5.55310 q^{95} -11.5270 q^{97} +0.126855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.61777	−0.934022	−0.467011	−	0.884251i	$-0.654669\pi$
$3$	−1.61777	−0.934022	−0.467011	+	0.884251i	$0.654669\pi$
$4$	0	0
$5$	−2.18000	−0.974926	−0.487463	−	0.873144i	$-0.662078\pi$
$5$	−2.18000	−0.974926	−0.487463	+	0.873144i	$0.662078\pi$
$6$	0	0
$7$	−2.82839	−1.06903	−0.534515	−	0.845159i	$-0.679506\pi$
$7$	−2.82839	−1.06903	−0.534515	+	0.845159i	$0.679506\pi$
$8$	0	0
$9$	−0.382806	−0.127602
$10$	0	0
$11$	−0.331382	−0.0999155	−0.0499578	−	0.998751i	$-0.515909\pi$
$11$	−0.331382	−0.0999155	−0.0499578	+	0.998751i	$0.515909\pi$
$12$	0	0
$13$	1.18515	0.328700	0.164350	−	0.986402i	$-0.447447\pi$
$13$	1.18515	0.328700	0.164350	+	0.986402i	$0.447447\pi$
$14$	0	0
$15$	3.52675	0.910602
$16$	0	0
$17$	−0.850397	−0.206252	−0.103126	−	0.994668i	$-0.532884\pi$
$17$	−0.850397	−0.206252	−0.103126	+	0.994668i	$0.532884\pi$
$18$	0	0
$19$	−2.54729	−0.584389	−0.292195	−	0.956359i	$-0.594386\pi$
$19$	−2.54729	−0.584389	−0.292195	+	0.956359i	$0.594386\pi$
$20$	0	0
$21$	4.57569	0.998498
$22$	0	0
$23$	−2.89409	−0.603459	−0.301730	−	0.953394i	$-0.597564\pi$
$23$	−2.89409	−0.603459	−0.301730	+	0.953394i	$0.597564\pi$
$24$	0	0
$25$	−0.247601	−0.0495202
$26$	0	0
$27$	5.47262	1.05321
$28$	0	0
$29$	3.27328	0.607834	0.303917	−	0.952699i	$-0.401706\pi$
$29$	3.27328	0.607834	0.303917	+	0.952699i	$0.401706\pi$
$30$	0	0
$31$	9.17864	1.64853	0.824266	−	0.566202i	$-0.191588\pi$
$31$	9.17864	1.64853	0.824266	+	0.566202i	$0.191588\pi$
$32$	0	0
$33$	0.536102	0.0933234
$34$	0	0
$35$	6.16588	1.04222
$36$	0	0
$37$	4.19824	0.690186	0.345093	−	0.938569i	$-0.387847\pi$
$37$	4.19824	0.690186	0.345093	+	0.938569i	$0.387847\pi$
$38$	0	0
$39$	−1.91730	−0.307014
$40$	0	0
$41$	10.8726	1.69801	0.849007	−	0.528382i	$-0.177201\pi$
$41$	10.8726	1.69801	0.849007	+	0.528382i	$0.177201\pi$
$42$	0	0
$43$	5.52475	0.842516	0.421258	−	0.906941i	$-0.361589\pi$
$43$	5.52475	0.842516	0.421258	+	0.906941i	$0.361589\pi$
$44$	0	0
$45$	0.834518	0.124403
$46$	0	0
$47$	−3.07403	−0.448393	−0.224196	−	0.974544i	$-0.571976\pi$
$47$	−3.07403	−0.448393	−0.224196	+	0.974544i	$0.571976\pi$
$48$	0	0
$49$	0.999768	0.142824
$50$	0	0
$51$	1.37575	0.192644
$52$	0	0
$53$	−7.07359	−0.971632	−0.485816	−	0.874061i	$-0.661478\pi$
$53$	−7.07359	−0.971632	−0.485816	+	0.874061i	$0.661478\pi$
$54$	0	0
$55$	0.722413	0.0974102
$56$	0	0
$57$	4.12095	0.545833
$58$	0	0
$59$	−5.25476	−0.684111	−0.342056	−	0.939680i	$-0.611123\pi$
$59$	−5.25476	−0.684111	−0.342056	+	0.939680i	$0.611123\pi$
$60$	0	0
$61$	5.14026	0.658143	0.329072	−	0.944305i	$-0.393264\pi$
$61$	5.14026	0.658143	0.329072	+	0.944305i	$0.393264\pi$
$62$	0	0
$63$	1.08272	0.136410
$64$	0	0
$65$	−2.58362	−0.320459
$66$	0	0
$67$	−12.7145	−1.55332	−0.776660	−	0.629920i	$-0.783088\pi$
$67$	−12.7145	−1.55332	−0.776660	+	0.629920i	$0.783088\pi$
$68$	0	0
$69$	4.68198	0.563644
$70$	0	0
$71$	6.59360	0.782516	0.391258	−	0.920281i	$-0.372040\pi$
$71$	6.59360	0.782516	0.391258	+	0.920281i	$0.372040\pi$
$72$	0	0
$73$	8.22764	0.962973	0.481486	−	0.876454i	$-0.340097\pi$
$73$	8.22764	0.962973	0.481486	+	0.876454i	$0.340097\pi$
$74$	0	0
$75$	0.400563	0.0462530
$76$	0	0
$77$	0.937277	0.106813
$78$	0	0
$79$	−7.17156	−0.806864	−0.403432	−	0.915010i	$-0.632183\pi$
$79$	−7.17156	−0.806864	−0.403432	+	0.915010i	$0.632183\pi$
$80$	0	0
$81$	−7.70504	−0.856116
$82$	0	0
$83$	5.27522	0.579030	0.289515	−	0.957173i	$-0.406506\pi$
$83$	5.27522	0.579030	0.289515	+	0.957173i	$0.406506\pi$
$84$	0	0
$85$	1.85387	0.201080
$86$	0	0
$87$	−5.29544	−0.567730
$88$	0	0
$89$	15.0525	1.59556	0.797781	−	0.602947i	$-0.206007\pi$
$89$	15.0525	1.59556	0.797781	+	0.602947i	$0.206007\pi$
$90$	0	0
$91$	−3.35205	−0.351391
$92$	0	0
$93$	−14.8490	−1.53977
$94$	0	0
$95$	5.55310	0.569736
$96$	0	0
$97$	−11.5270	−1.17039	−0.585194	−	0.810894i	$-0.698982\pi$
$97$	−11.5270	−1.17039	−0.585194	+	0.810894i	$0.698982\pi$
$98$	0	0
$99$	0.126855	0.0127494
$100$	0	0
$101$	5.19217	0.516641	0.258320	−	0.966059i	$-0.416831\pi$
$101$	5.19217	0.516641	0.258320	+	0.966059i	$0.416831\pi$
$102$	0	0
$103$	−5.44827	−0.536834	−0.268417	−	0.963303i	$-0.586500\pi$
$103$	−5.44827	−0.536834	−0.268417	+	0.963303i	$0.586500\pi$
$104$	0	0
$105$	−9.97500	−0.973461
$106$	0	0
$107$	−11.5635	−1.11788	−0.558942	−	0.829207i	$-0.688793\pi$
$107$	−11.5635	−1.11788	−0.558942	+	0.829207i	$0.688793\pi$
$108$	0	0
$109$	2.31632	0.221864	0.110932	−	0.993828i	$-0.464616\pi$
$109$	2.31632	0.221864	0.110932	+	0.993828i	$0.464616\pi$
$110$	0	0
$111$	−6.79180	−0.644649
$112$	0	0
$113$	9.68156	0.910765	0.455382	−	0.890296i	$-0.349503\pi$
$113$	9.68156	0.910765	0.455382	+	0.890296i	$0.349503\pi$
$114$	0	0
$115$	6.30911	0.588328
$116$	0	0
$117$	−0.453682	−0.0419429
$118$	0	0
$119$	2.40525	0.220489
$120$	0	0
$121$	−10.8902	−0.990017
$122$	0	0
$123$	−17.5894	−1.58598
$124$	0	0
$125$	11.4398	1.02320
$126$	0	0
$127$	−7.05321	−0.625871	−0.312935	−	0.949774i	$-0.601312\pi$
$127$	−7.05321	−0.625871	−0.312935	+	0.949774i	$0.601312\pi$
$128$	0	0
$129$	−8.93779	−0.786929
$130$	0	0
$131$	7.97466	0.696749	0.348374	−	0.937355i	$-0.386734\pi$
$131$	7.97466	0.696749	0.348374	+	0.937355i	$0.386734\pi$
$132$	0	0
$133$	7.20473	0.624729
$134$	0	0
$135$	−11.9303	−1.02680
$136$	0	0
$137$	11.9659	1.02232	0.511160	−	0.859486i	$-0.329216\pi$
$137$	11.9659	1.02232	0.511160	+	0.859486i	$0.329216\pi$
$138$	0	0
$139$	5.48764	0.465456	0.232728	−	0.972542i	$-0.425235\pi$
$139$	5.48764	0.465456	0.232728	+	0.972542i	$0.425235\pi$
$140$	0	0
$141$	4.97308	0.418809
$142$	0	0
$143$	−0.392737	−0.0328423
$144$	0	0
$145$	−7.13576	−0.592593
$146$	0	0
$147$	−1.61740	−0.133401
$148$	0	0
$149$	6.73467	0.551726	0.275863	−	0.961197i	$-0.411036\pi$
$149$	6.73467	0.551726	0.275863	+	0.961197i	$0.411036\pi$
$150$	0	0
$151$	−3.35880	−0.273335	−0.136668	−	0.990617i	$-0.543639\pi$
$151$	−3.35880	−0.273335	−0.136668	+	0.990617i	$0.543639\pi$
$152$	0	0
$153$	0.325538	0.0263182
$154$	0	0
$155$	−20.0094	−1.60720
$156$	0	0
$157$	−2.51309	−0.200567	−0.100283	−	0.994959i	$-0.531975\pi$
$157$	−2.51309	−0.200567	−0.100283	+	0.994959i	$0.531975\pi$
$158$	0	0
$159$	11.4435	0.907526
$160$	0	0
$161$	8.18560	0.645116
$162$	0	0
$163$	17.4645	1.36793	0.683963	−	0.729517i	$-0.260255\pi$
$163$	17.4645	1.36793	0.683963	+	0.729517i	$0.260255\pi$
$164$	0	0
$165$	−1.16870	−0.0909833
$166$	0	0
$167$	7.00372	0.541964	0.270982	−	0.962584i	$-0.412652\pi$
$167$	7.00372	0.541964	0.270982	+	0.962584i	$0.412652\pi$
$168$	0	0
$169$	−11.5954	−0.891956
$170$	0	0
$171$	0.975120	0.0745693
$172$	0	0
$173$	−20.4743	−1.55663	−0.778317	−	0.627871i	$-0.783926\pi$
$173$	−20.4743	−1.55663	−0.778317	+	0.627871i	$0.783926\pi$
$174$	0	0
$175$	0.700312	0.0529386
$176$	0	0
$177$	8.50101	0.638975
$178$	0	0
$179$	−5.71742	−0.427340	−0.213670	−	0.976906i	$-0.568542\pi$
$179$	−5.71742	−0.427340	−0.213670	+	0.976906i	$0.568542\pi$
$180$	0	0
$181$	0.109937	0.00817157	0.00408578	−	0.999992i	$-0.498699\pi$
$181$	0.109937	0.00817157	0.00408578	+	0.999992i	$0.498699\pi$
$182$	0	0
$183$	−8.31578	−0.614720
$184$	0	0
$185$	−9.15215	−0.672880
$186$	0	0
$187$	0.281807	0.0206077
$188$	0	0
$189$	−15.4787	−1.12591
$190$	0	0
$191$	4.31899	0.312511	0.156256	−	0.987717i	$-0.450058\pi$
$191$	4.31899	0.312511	0.156256	+	0.987717i	$0.450058\pi$
$192$	0	0
$193$	22.6709	1.63189	0.815944	−	0.578131i	$-0.196218\pi$
$193$	22.6709	1.63189	0.815944	+	0.578131i	$0.196218\pi$
$194$	0	0
$195$	4.17971	0.299315
$196$	0	0
$197$	−2.32993	−0.166000	−0.0830002	−	0.996550i	$-0.526450\pi$
$197$	−2.32993	−0.166000	−0.0830002	+	0.996550i	$0.526450\pi$
$198$	0	0
$199$	−18.2582	−1.29429	−0.647145	−	0.762367i	$-0.724037\pi$
$199$	−18.2582	−1.29429	−0.647145	+	0.762367i	$0.724037\pi$
$200$	0	0
$201$	20.5691	1.45084
$202$	0	0
$203$	−9.25811	−0.649792
$204$	0	0
$205$	−23.7022	−1.65544
$206$	0	0
$207$	1.10788	0.0770027
$208$	0	0
$209$	0.844128	0.0583896
$210$	0	0
$211$	−0.0484845	−0.00333781	−0.00166890	−	0.999999i	$-0.500531\pi$
$211$	−0.0484845	−0.00333781	−0.00166890	+	0.999999i	$0.500531\pi$
$212$	0	0
$213$	−10.6670	−0.730888
$214$	0	0
$215$	−12.0439	−0.821390
$216$	0	0
$217$	−25.9607	−1.76233
$218$	0	0
$219$	−13.3105	−0.899438
$220$	0	0
$221$	−1.00785	−0.0677950
$222$	0	0
$223$	−4.76174	−0.318869	−0.159435	−	0.987208i	$-0.550967\pi$
$223$	−4.76174	−0.318869	−0.159435	+	0.987208i	$0.550967\pi$
$224$	0	0
$225$	0.0947833	0.00631889
$226$	0	0
$227$	−15.2257	−1.01057	−0.505284	−	0.862953i	$-0.668612\pi$
$227$	−15.2257	−1.01057	−0.505284	+	0.862953i	$0.668612\pi$
$228$	0	0
$229$	9.93794	0.656718	0.328359	−	0.944553i	$-0.393505\pi$
$229$	9.93794	0.656718	0.328359	+	0.944553i	$0.393505\pi$
$230$	0	0
$231$	−1.51630	−0.0997654
$232$	0	0
$233$	−27.6931	−1.81423	−0.907117	−	0.420878i	$-0.861722\pi$
$233$	−27.6931	−1.81423	−0.907117	+	0.420878i	$0.861722\pi$
$234$	0	0
$235$	6.70138	0.437150
$236$	0	0
$237$	11.6020	0.753629
$238$	0	0
$239$	−16.8636	−1.09082	−0.545409	−	0.838170i	$-0.683626\pi$
$239$	−16.8636	−1.09082	−0.545409	+	0.838170i	$0.683626\pi$
$240$	0	0
$241$	2.81397	0.181264	0.0906319	−	0.995884i	$-0.471111\pi$
$241$	2.81397	0.181264	0.0906319	+	0.995884i	$0.471111\pi$
$242$	0	0
$243$	−3.95284	−0.253575
$244$	0	0
$245$	−2.17949	−0.139243
$246$	0	0
$247$	−3.01892	−0.192089
$248$	0	0
$249$	−8.53411	−0.540827
$250$	0	0
$251$	3.23936	0.204467	0.102233	−	0.994760i	$-0.467401\pi$
$251$	3.23936	0.204467	0.102233	+	0.994760i	$0.467401\pi$
$252$	0	0
$253$	0.959050	0.0602949
$254$	0	0
$255$	−2.99914	−0.187813
$256$	0	0
$257$	−19.4266	−1.21180	−0.605900	−	0.795541i	$-0.707187\pi$
$257$	−19.4266	−1.21180	−0.605900	+	0.795541i	$0.707187\pi$
$258$	0	0
$259$	−11.8742	−0.737829
$260$	0	0
$261$	−1.25303	−0.0775609
$262$	0	0
$263$	−8.47044	−0.522310	−0.261155	−	0.965297i	$-0.584103\pi$
$263$	−8.47044	−0.522310	−0.261155	+	0.965297i	$0.584103\pi$
$264$	0	0
$265$	15.4204	0.947269
$266$	0	0
$267$	−24.3516	−1.49029
$268$	0	0
$269$	−21.8233	−1.33059	−0.665294	−	0.746582i	$-0.731694\pi$
$269$	−21.8233	−1.33059	−0.665294	+	0.746582i	$0.731694\pi$
$270$	0	0
$271$	−19.7397	−1.19910	−0.599550	−	0.800337i	$-0.704654\pi$
$271$	−19.7397	−1.19910	−0.599550	+	0.800337i	$0.704654\pi$
$272$	0	0
$273$	5.42286	0.328207
$274$	0	0
$275$	0.0820506	0.00494784
$276$	0	0
$277$	20.7197	1.24492	0.622462	−	0.782650i	$-0.286133\pi$
$277$	20.7197	1.24492	0.622462	+	0.782650i	$0.286133\pi$
$278$	0	0
$279$	−3.51364	−0.210356
$280$	0	0
$281$	18.8087	1.12203	0.561017	−	0.827804i	$-0.310410\pi$
$281$	18.8087	1.12203	0.561017	+	0.827804i	$0.310410\pi$
$282$	0	0
$283$	3.11774	0.185330	0.0926652	−	0.995697i	$-0.470461\pi$
$283$	3.11774	0.185330	0.0926652	+	0.995697i	$0.470461\pi$
$284$	0	0
$285$	−8.98366	−0.532146
$286$	0	0
$287$	−30.7519	−1.81523
$288$	0	0
$289$	−16.2768	−0.957460
$290$	0	0
$291$	18.6480	1.09317
$292$	0	0
$293$	28.3597	1.65679	0.828396	−	0.560143i	$-0.189254\pi$
$293$	28.3597	1.65679	0.828396	+	0.560143i	$0.189254\pi$
$294$	0	0
$295$	11.4554	0.666957
$296$	0	0
$297$	−1.81353	−0.105232
$298$	0	0
$299$	−3.42992	−0.198357
$300$	0	0
$301$	−15.6261	−0.900674
$302$	0	0
$303$	−8.39977	−0.482554
$304$	0	0
$305$	−11.2058	−0.641641
$306$	0	0
$307$	−14.6160	−0.834180	−0.417090	−	0.908865i	$-0.636950\pi$
$307$	−14.6160	−0.834180	−0.417090	+	0.908865i	$0.636950\pi$
$308$	0	0
$309$	8.81406	0.501415
$310$	0	0
$311$	−23.4725	−1.33100	−0.665501	−	0.746397i	$-0.731782\pi$
$311$	−23.4725	−1.33100	−0.665501	+	0.746397i	$0.731782\pi$
$312$	0	0
$313$	1.09616	0.0619588	0.0309794	−	0.999520i	$-0.490137\pi$
$313$	1.09616	0.0619588	0.0309794	+	0.999520i	$0.490137\pi$
$314$	0	0
$315$	−2.36034	−0.132990
$316$	0	0
$317$	15.8084	0.887886	0.443943	−	0.896055i	$-0.353579\pi$
$317$	15.8084	0.887886	0.443943	+	0.896055i	$0.353579\pi$
$318$	0	0
$319$	−1.08471	−0.0607320
$320$	0	0
$321$	18.7071	1.04413
$322$	0	0
$323$	2.16621	0.120531
$324$	0	0
$325$	−0.293444	−0.0162773
$326$	0	0
$327$	−3.74729	−0.207226
$328$	0	0
$329$	8.69453	0.479345
$330$	0	0
$331$	−12.5388	−0.689193	−0.344597	−	0.938751i	$-0.611984\pi$
$331$	−12.5388	−0.689193	−0.344597	+	0.938751i	$0.611984\pi$
$332$	0	0
$333$	−1.60711	−0.0880692
$334$	0	0
$335$	27.7175	1.51437
$336$	0	0
$337$	−33.0821	−1.80210	−0.901049	−	0.433718i	$-0.857201\pi$
$337$	−33.0821	−1.80210	−0.901049	+	0.433718i	$0.857201\pi$
$338$	0	0
$339$	−15.6626	−0.850675
$340$	0	0
$341$	−3.04164	−0.164714
$342$	0	0
$343$	16.9710	0.916346
$344$	0	0
$345$	−10.2067	−0.549511
$346$	0	0
$347$	3.10816	0.166855	0.0834275	−	0.996514i	$-0.473413\pi$
$347$	3.10816	0.166855	0.0834275	+	0.996514i	$0.473413\pi$
$348$	0	0
$349$	32.5026	1.73983	0.869913	−	0.493205i	$-0.164175\pi$
$349$	32.5026	1.73983	0.869913	+	0.493205i	$0.164175\pi$
$350$	0	0
$351$	6.48585	0.346189
$352$	0	0
$353$	−20.5439	−1.09344	−0.546721	−	0.837315i	$-0.684124\pi$
$353$	−20.5439	−1.09344	−0.546721	+	0.837315i	$0.684124\pi$
$354$	0	0
$355$	−14.3740	−0.762895
$356$	0	0
$357$	−3.89115	−0.205942
$358$	0	0
$359$	1.61257	0.0851081	0.0425540	−	0.999094i	$-0.486451\pi$
$359$	1.61257	0.0851081	0.0425540	+	0.999094i	$0.486451\pi$
$360$	0	0
$361$	−12.5113	−0.658489
$362$	0	0
$363$	17.6179	0.924698
$364$	0	0
$365$	−17.9363	−0.938827
$366$	0	0
$367$	−9.27677	−0.484244	−0.242122	−	0.970246i	$-0.577843\pi$
$367$	−9.27677	−0.484244	−0.242122	+	0.970246i	$0.577843\pi$
$368$	0	0
$369$	−4.16210	−0.216670
$370$	0	0
$371$	20.0068	1.03870
$372$	0	0
$373$	20.5507	1.06407	0.532037	−	0.846721i	$-0.321427\pi$
$373$	20.5507	1.06407	0.532037	+	0.846721i	$0.321427\pi$
$374$	0	0
$375$	−18.5070	−0.955696
$376$	0	0
$377$	3.87932	0.199795
$378$	0	0
$379$	6.56327	0.337132	0.168566	−	0.985690i	$-0.446086\pi$
$379$	6.56327	0.337132	0.168566	+	0.985690i	$0.446086\pi$
$380$	0	0
$381$	11.4105	0.584577
$382$	0	0
$383$	−9.58685	−0.489865	−0.244932	−	0.969540i	$-0.578766\pi$
$383$	−9.58685	−0.489865	−0.244932	+	0.969540i	$0.578766\pi$
$384$	0	0
$385$	−2.04326	−0.104134
$386$	0	0
$387$	−2.11491	−0.107507
$388$	0	0
$389$	18.5963	0.942869	0.471434	−	0.881901i	$-0.343736\pi$
$389$	18.5963	0.942869	0.471434	+	0.881901i	$0.343736\pi$
$390$	0	0
$391$	2.46113	0.124464
$392$	0	0
$393$	−12.9012	−0.650779
$394$	0	0
$395$	15.6340	0.786632
$396$	0	0
$397$	−0.780429	−0.0391686	−0.0195843	−	0.999808i	$-0.506234\pi$
$397$	−0.780429	−0.0391686	−0.0195843	+	0.999808i	$0.506234\pi$
$398$	0	0
$399$	−11.6556	−0.583511
$400$	0	0
$401$	17.0706	0.852463	0.426231	−	0.904614i	$-0.359841\pi$
$401$	17.0706	0.852463	0.426231	+	0.904614i	$0.359841\pi$
$402$	0	0
$403$	10.8780	0.541874
$404$	0	0
$405$	16.7970	0.834649
$406$	0	0
$407$	−1.39122	−0.0689603
$408$	0	0
$409$	−23.8796	−1.18077	−0.590385	−	0.807122i	$-0.701024\pi$
$409$	−23.8796	−1.18077	−0.590385	+	0.807122i	$0.701024\pi$
$410$	0	0
$411$	−19.3582	−0.954869
$412$	0	0
$413$	14.8625	0.731335
$414$	0	0
$415$	−11.5000	−0.564511
$416$	0	0
$417$	−8.87776	−0.434746
$418$	0	0
$419$	−4.69668	−0.229448	−0.114724	−	0.993397i	$-0.536598\pi$
$419$	−4.69668	−0.229448	−0.114724	+	0.993397i	$0.536598\pi$
$420$	0	0
$421$	34.5652	1.68461	0.842303	−	0.539005i	$-0.181200\pi$
$421$	34.5652	1.68461	0.842303	+	0.539005i	$0.181200\pi$
$422$	0	0
$423$	1.17676	0.0572159
$424$	0	0
$425$	0.210559	0.0102136
$426$	0	0
$427$	−14.5386	−0.703574
$428$	0	0
$429$	0.635359	0.0306754
$430$	0	0
$431$	−14.3310	−0.690301	−0.345150	−	0.938547i	$-0.612172\pi$
$431$	−14.3310	−0.690301	−0.345150	+	0.938547i	$0.612172\pi$
$432$	0	0
$433$	−11.0660	−0.531799	−0.265900	−	0.964001i	$-0.585669\pi$
$433$	−11.0660	−0.531799	−0.265900	+	0.964001i	$0.585669\pi$
$434$	0	0
$435$	11.5440	0.553495
$436$	0	0
$437$	7.37209	0.352655
$438$	0	0
$439$	−19.0090	−0.907248	−0.453624	−	0.891193i	$-0.649869\pi$
$439$	−19.0090	−0.907248	−0.453624	+	0.891193i	$0.649869\pi$
$440$	0	0
$441$	−0.382718	−0.0182247
$442$	0	0
$443$	11.2504	0.534525	0.267262	−	0.963624i	$-0.413881\pi$
$443$	11.2504	0.534525	0.267262	+	0.963624i	$0.413881\pi$
$444$	0	0
$445$	−32.8145	−1.55555
$446$	0	0
$447$	−10.8952	−0.515324
$448$	0	0
$449$	7.28506	0.343803	0.171902	−	0.985114i	$-0.445009\pi$
$449$	7.28506	0.343803	0.171902	+	0.985114i	$0.445009\pi$
$450$	0	0
$451$	−3.60298	−0.169658
$452$	0	0
$453$	5.43378	0.255301
$454$	0	0
$455$	7.30747	0.342580
$456$	0	0
$457$	−1.07677	−0.0503690	−0.0251845	−	0.999683i	$-0.508017\pi$
$457$	−1.07677	−0.0503690	−0.0251845	+	0.999683i	$0.508017\pi$
$458$	0	0
$459$	−4.65390	−0.217225
$460$	0	0
$461$	−11.3033	−0.526448	−0.263224	−	0.964735i	$-0.584786\pi$
$461$	−11.3033	−0.526448	−0.263224	+	0.964735i	$0.584786\pi$
$462$	0	0
$463$	40.2496	1.87056	0.935278	−	0.353914i	$-0.115149\pi$
$463$	40.2496	1.87056	0.935278	+	0.353914i	$0.115149\pi$
$464$	0	0
$465$	32.3708	1.50116
$466$	0	0
$467$	−3.99138	−0.184699	−0.0923494	−	0.995727i	$-0.529438\pi$
$467$	−3.99138	−0.184699	−0.0923494	+	0.995727i	$0.529438\pi$
$468$	0	0
$469$	35.9614	1.66054
$470$	0	0
$471$	4.06562	0.187334
$472$	0	0
$473$	−1.83080	−0.0841804
$474$	0	0
$475$	0.630713	0.0289391
$476$	0	0
$477$	2.70782	0.123982
$478$	0	0
$479$	−28.6774	−1.31030	−0.655152	−	0.755497i	$-0.727396\pi$
$479$	−28.6774	−1.31030	−0.655152	+	0.755497i	$0.727396\pi$
$480$	0	0
$481$	4.97552	0.226864
$482$	0	0
$483$	−13.2425	−0.602552
$484$	0	0
$485$	25.1288	1.14104
$486$	0	0
$487$	7.42705	0.336552	0.168276	−	0.985740i	$-0.446180\pi$
$487$	7.42705	0.336552	0.168276	+	0.985740i	$0.446180\pi$
$488$	0	0
$489$	−28.2536	−1.27767
$490$	0	0
$491$	10.2390	0.462077	0.231039	−	0.972945i	$-0.425788\pi$
$491$	10.2390	0.462077	0.231039	+	0.972945i	$0.425788\pi$
$492$	0	0
$493$	−2.78359	−0.125367
$494$	0	0
$495$	−0.276545	−0.0124298
$496$	0	0
$497$	−18.6492	−0.836533
$498$	0	0
$499$	−20.9983	−0.940014	−0.470007	−	0.882663i	$-0.655749\pi$
$499$	−20.9983	−0.940014	−0.470007	+	0.882663i	$0.655749\pi$
$500$	0	0
$501$	−11.3304	−0.506207
$502$	0	0
$503$	−0.586864	−0.0261670	−0.0130835	−	0.999914i	$-0.504165\pi$
$503$	−0.586864	−0.0261670	−0.0130835	+	0.999914i	$0.504165\pi$
$504$	0	0
$505$	−11.3189	−0.503686
$506$	0	0
$507$	18.7588	0.833107
$508$	0	0
$509$	14.8536	0.658372	0.329186	−	0.944265i	$-0.393226\pi$
$509$	14.8536	0.658372	0.329186	+	0.944265i	$0.393226\pi$
$510$	0	0
$511$	−23.2710	−1.02945
$512$	0	0
$513$	−13.9404	−0.615482
$514$	0	0
$515$	11.8772	0.523373
$516$	0	0
$517$	1.01868	0.0448014
$518$	0	0
$519$	33.1228	1.45393
$520$	0	0
$521$	17.0182	0.745583	0.372791	−	0.927915i	$-0.378401\pi$
$521$	17.0182	0.745583	0.372791	+	0.927915i	$0.378401\pi$
$522$	0	0
$523$	−19.6229	−0.858052	−0.429026	−	0.903292i	$-0.641143\pi$
$523$	−19.6229	−0.858052	−0.429026	+	0.903292i	$0.641143\pi$
$524$	0	0
$525$	−1.13295	−0.0494458
$526$	0	0
$527$	−7.80549	−0.340013
$528$	0	0
$529$	−14.6243	−0.635837
$530$	0	0
$531$	2.01155	0.0872940
$532$	0	0
$533$	12.8856	0.558138
$534$	0	0
$535$	25.2084	1.08985
$536$	0	0
$537$	9.24949	0.399145
$538$	0	0
$539$	−0.331306	−0.0142703
$540$	0	0
$541$	−13.4535	−0.578409	−0.289205	−	0.957267i	$-0.593391\pi$
$541$	−13.4535	−0.578409	−0.289205	+	0.957267i	$0.593391\pi$
$542$	0	0
$543$	−0.177854	−0.00763243
$544$	0	0
$545$	−5.04958	−0.216300
$546$	0	0
$547$	−4.68961	−0.200513	−0.100257	−	0.994962i	$-0.531966\pi$
$547$	−4.68961	−0.200513	−0.100257	+	0.994962i	$0.531966\pi$
$548$	0	0
$549$	−1.96773	−0.0839805
$550$	0	0
$551$	−8.33802	−0.355211
$552$	0	0
$553$	20.2839	0.862561
$554$	0	0
$555$	14.8061	0.628485
$556$	0	0
$557$	−3.91801	−0.166011	−0.0830057	−	0.996549i	$-0.526452\pi$
$557$	−3.91801	−0.166011	−0.0830057	+	0.996549i	$0.526452\pi$
$558$	0	0
$559$	6.54763	0.276935
$560$	0	0
$561$	−0.455900	−0.0192481
$562$	0	0
$563$	−27.9203	−1.17670	−0.588350	−	0.808607i	$-0.700222\pi$
$563$	−27.9203	−1.17670	−0.588350	+	0.808607i	$0.700222\pi$
$564$	0	0
$565$	−21.1058	−0.887928
$566$	0	0
$567$	21.7928	0.915213
$568$	0	0
$569$	31.8580	1.33556	0.667779	−	0.744360i	$-0.267245\pi$
$569$	31.8580	1.33556	0.667779	+	0.744360i	$0.267245\pi$
$570$	0	0
$571$	−36.4804	−1.52666	−0.763329	−	0.646010i	$-0.776437\pi$
$571$	−36.4804	−1.52666	−0.763329	+	0.646010i	$0.776437\pi$
$572$	0	0
$573$	−6.98715	−0.291893
$574$	0	0
$575$	0.716579	0.0298834
$576$	0	0
$577$	−40.3484	−1.67972	−0.839862	−	0.542799i	$-0.817364\pi$
$577$	−40.3484	−1.67972	−0.839862	+	0.542799i	$0.817364\pi$
$578$	0	0
$579$	−36.6764	−1.52422
$580$	0	0
$581$	−14.9204	−0.619000
$582$	0	0
$583$	2.34406	0.0970812
$584$	0	0
$585$	0.989026	0.0408912
$586$	0	0
$587$	25.3099	1.04465	0.522325	−	0.852746i	$-0.325065\pi$
$587$	25.3099	1.04465	0.522325	+	0.852746i	$0.325065\pi$
$588$	0	0
$589$	−23.3807	−0.963385
$590$	0	0
$591$	3.76929	0.155048
$592$	0	0
$593$	−5.64223	−0.231698	−0.115849	−	0.993267i	$-0.536959\pi$
$593$	−5.64223	−0.231698	−0.115849	+	0.993267i	$0.536959\pi$
$594$	0	0
$595$	−5.24345	−0.214960
$596$	0	0
$597$	29.5376	1.20890
$598$	0	0
$599$	3.94680	0.161262	0.0806309	−	0.996744i	$-0.474306\pi$
$599$	3.94680	0.161262	0.0806309	+	0.996744i	$0.474306\pi$
$600$	0	0
$601$	−13.5665	−0.553391	−0.276695	−	0.960958i	$-0.589239\pi$
$601$	−13.5665	−0.553391	−0.276695	+	0.960958i	$0.589239\pi$
$602$	0	0
$603$	4.86718	0.198207
$604$	0	0
$605$	23.7406	0.965193
$606$	0	0
$607$	8.36799	0.339646	0.169823	−	0.985475i	$-0.445680\pi$
$607$	8.36799	0.339646	0.169823	+	0.985475i	$0.445680\pi$
$608$	0	0
$609$	14.9775	0.606920
$610$	0	0
$611$	−3.64317	−0.147387
$612$	0	0
$613$	0.277150	0.0111940	0.00559698	−	0.999984i	$-0.498218\pi$
$613$	0.277150	0.0111940	0.00559698	+	0.999984i	$0.498218\pi$
$614$	0	0
$615$	38.3449	1.54621
$616$	0	0
$617$	27.1309	1.09225	0.546124	−	0.837704i	$-0.316103\pi$
$617$	27.1309	1.09225	0.546124	+	0.837704i	$0.316103\pi$
$618$	0	0
$619$	37.2491	1.49717	0.748583	−	0.663041i	$-0.230734\pi$
$619$	37.2491	1.49717	0.748583	+	0.663041i	$0.230734\pi$
$620$	0	0
$621$	−15.8382	−0.635567
$622$	0	0
$623$	−42.5743	−1.70570
$624$	0	0
$625$	−23.7007	−0.948028
$626$	0	0
$627$	−1.36561	−0.0545372
$628$	0	0
$629$	−3.57017	−0.142352
$630$	0	0
$631$	17.2921	0.688388	0.344194	−	0.938899i	$-0.388152\pi$
$631$	17.2921	0.688388	0.344194	+	0.938899i	$0.388152\pi$
$632$	0	0
$633$	0.0784370	0.00311759
$634$	0	0
$635$	15.3760	0.610178
$636$	0	0
$637$	1.18487	0.0469463
$638$	0	0
$639$	−2.52407	−0.0998507
$640$	0	0
$641$	−5.71059	−0.225555	−0.112777	−	0.993620i	$-0.535975\pi$
$641$	−5.71059	−0.225555	−0.112777	+	0.993620i	$0.535975\pi$
$642$	0	0
$643$	−39.8738	−1.57247	−0.786234	−	0.617929i	$-0.787972\pi$
$643$	−39.8738	−1.57247	−0.786234	+	0.617929i	$0.787972\pi$
$644$	0	0
$645$	19.4844	0.767197
$646$	0	0
$647$	−35.5414	−1.39728	−0.698639	−	0.715474i	$-0.746211\pi$
$647$	−35.5414	−1.39728	−0.698639	+	0.715474i	$0.746211\pi$
$648$	0	0
$649$	1.74133	0.0683533
$650$	0	0
$651$	41.9986	1.64606
$652$	0	0
$653$	46.0880	1.80356	0.901782	−	0.432191i	$-0.142259\pi$
$653$	46.0880	1.80356	0.901782	+	0.432191i	$0.142259\pi$
$654$	0	0
$655$	−17.3847	−0.679278
$656$	0	0
$657$	−3.14959	−0.122877
$658$	0	0
$659$	−1.34339	−0.0523311	−0.0261656	−	0.999658i	$-0.508330\pi$
$659$	−1.34339	−0.0523311	−0.0261656	+	0.999658i	$0.508330\pi$
$660$	0	0
$661$	−0.189408	−0.00736712	−0.00368356	−	0.999993i	$-0.501173\pi$
$661$	−0.189408	−0.00736712	−0.00368356	+	0.999993i	$0.501173\pi$
$662$	0	0
$663$	1.63047	0.0633221
$664$	0	0
$665$	−15.7063	−0.609065
$666$	0	0
$667$	−9.47317	−0.366803
$668$	0	0
$669$	7.70342	0.297831
$670$	0	0
$671$	−1.70339	−0.0657587
$672$	0	0
$673$	−30.8302	−1.18842	−0.594208	−	0.804311i	$-0.702534\pi$
$673$	−30.8302	−1.18842	−0.594208	+	0.804311i	$0.702534\pi$
$674$	0	0
$675$	−1.35503	−0.0521550
$676$	0	0
$677$	−26.6069	−1.02259	−0.511294	−	0.859406i	$-0.670834\pi$
$677$	−26.6069	−1.02259	−0.511294	+	0.859406i	$0.670834\pi$
$678$	0	0
$679$	32.6027	1.25118
$680$	0	0
$681$	24.6318	0.943893
$682$	0	0
$683$	−47.5141	−1.81808	−0.909038	−	0.416712i	$-0.863182\pi$
$683$	−47.5141	−1.81808	−0.909038	+	0.416712i	$0.863182\pi$
$684$	0	0
$685$	−26.0858	−0.996685
$686$	0	0
$687$	−16.0773	−0.613389
$688$	0	0
$689$	−8.38324	−0.319376
$690$	0	0
$691$	−24.2081	−0.920920	−0.460460	−	0.887680i	$-0.652316\pi$
$691$	−24.2081	−0.920920	−0.460460	+	0.887680i	$0.652316\pi$
$692$	0	0
$693$	−0.358796	−0.0136295
$694$	0	0
$695$	−11.9631	−0.453785
$696$	0	0
$697$	−9.24602	−0.350218
$698$	0	0
$699$	44.8012	1.69454
$700$	0	0
$701$	8.80260	0.332470	0.166235	−	0.986086i	$-0.446839\pi$
$701$	8.80260	0.332470	0.166235	+	0.986086i	$0.446839\pi$
$702$	0	0
$703$	−10.6941	−0.403337
$704$	0	0
$705$	−10.8413	−0.408307
$706$	0	0
$707$	−14.6855	−0.552304
$708$	0	0
$709$	23.5395	0.884045	0.442022	−	0.897004i	$-0.354261\pi$
$709$	23.5395	0.884045	0.442022	+	0.897004i	$0.354261\pi$
$710$	0	0
$711$	2.74532	0.102958
$712$	0	0
$713$	−26.5638	−0.994822
$714$	0	0
$715$	0.856166	0.0320188
$716$	0	0
$717$	27.2815	1.01885
$718$	0	0
$719$	51.9497	1.93740	0.968698	−	0.248244i	$-0.0798534\pi$
$719$	51.9497	1.93740	0.968698	+	0.248244i	$0.0798534\pi$
$720$	0	0
$721$	15.4098	0.573891
$722$	0	0
$723$	−4.55237	−0.169304
$724$	0	0
$725$	−0.810469	−0.0301001
$726$	0	0
$727$	19.7904	0.733986	0.366993	−	0.930224i	$-0.380387\pi$
$727$	19.7904	0.733986	0.366993	+	0.930224i	$0.380387\pi$
$728$	0	0
$729$	29.5099	1.09296
$730$	0	0
$731$	−4.69823	−0.173770
$732$	0	0
$733$	−17.3444	−0.640630	−0.320315	−	0.947311i	$-0.603789\pi$
$733$	−17.3444	−0.640630	−0.320315	+	0.947311i	$0.603789\pi$
$734$	0	0
$735$	3.52593	0.130056
$736$	0	0
$737$	4.21335	0.155201
$738$	0	0
$739$	25.6769	0.944539	0.472270	−	0.881454i	$-0.343435\pi$
$739$	25.6769	0.944539	0.472270	+	0.881454i	$0.343435\pi$
$740$	0	0
$741$	4.88392	0.179415
$742$	0	0
$743$	−43.4030	−1.59230	−0.796151	−	0.605098i	$-0.793134\pi$
$743$	−43.4030	−1.59230	−0.796151	+	0.605098i	$0.793134\pi$
$744$	0	0
$745$	−14.6816	−0.537891
$746$	0	0
$747$	−2.01939	−0.0738855
$748$	0	0
$749$	32.7060	1.19505
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−5.24055	−0.190976
$754$	0	0
$755$	7.32219	0.266482
$756$	0	0
$757$	32.2539	1.17229	0.586145	−	0.810206i	$-0.300645\pi$
$757$	32.2539	1.17229	0.586145	+	0.810206i	$0.300645\pi$
$758$	0	0
$759$	−1.55153	−0.0563168
$760$	0	0
$761$	−9.03410	−0.327486	−0.163743	−	0.986503i	$-0.552357\pi$
$761$	−9.03410	−0.327486	−0.163743	+	0.986503i	$0.552357\pi$
$762$	0	0
$763$	−6.55146	−0.237179
$764$	0	0
$765$	−0.709672	−0.0256582
$766$	0	0
$767$	−6.22766	−0.224868
$768$	0	0
$769$	−52.7801	−1.90330	−0.951649	−	0.307186i	$-0.900613\pi$
$769$	−52.7801	−1.90330	−0.951649	+	0.307186i	$0.900613\pi$
$770$	0	0
$771$	31.4279	1.13185
$772$	0	0
$773$	40.9954	1.47450	0.737251	−	0.675619i	$-0.236124\pi$
$773$	40.9954	1.47450	0.737251	+	0.675619i	$0.236124\pi$
$774$	0	0
$775$	−2.27264	−0.0816357
$776$	0	0
$777$	19.2098	0.689149
$778$	0	0
$779$	−27.6957	−0.992301
$780$	0	0
$781$	−2.18500	−0.0781855
$782$	0	0
$783$	17.9134	0.640174
$784$	0	0
$785$	5.47854	0.195538
$786$	0	0
$787$	19.7088	0.702544	0.351272	−	0.936273i	$-0.385749\pi$
$787$	19.7088	0.702544	0.351272	+	0.936273i	$0.385749\pi$
$788$	0	0
$789$	13.7033	0.487849
$790$	0	0
$791$	−27.3832	−0.973634
$792$	0	0
$793$	6.09196	0.216332
$794$	0	0
$795$	−24.9468	−0.884771
$796$	0	0
$797$	9.73666	0.344890	0.172445	−	0.985019i	$-0.444833\pi$
$797$	9.73666	0.344890	0.172445	+	0.985019i	$0.444833\pi$
$798$	0	0
$799$	2.61414	0.0924817
$800$	0	0
$801$	−5.76220	−0.203597
$802$	0	0
$803$	−2.72650	−0.0962160
$804$	0	0
$805$	−17.8446	−0.628940
$806$	0	0
$807$	35.3051	1.24280
$808$	0	0
$809$	27.3284	0.960814	0.480407	−	0.877046i	$-0.340489\pi$
$809$	27.3284	0.960814	0.480407	+	0.877046i	$0.340489\pi$
$810$	0	0
$811$	8.14626	0.286054	0.143027	−	0.989719i	$-0.454316\pi$
$811$	8.14626	0.286054	0.143027	+	0.989719i	$0.454316\pi$
$812$	0	0
$813$	31.9343	1.11999
$814$	0	0
$815$	−38.0726	−1.33363
$816$	0	0
$817$	−14.0732	−0.492357
$818$	0	0
$819$	1.28319	0.0448382
$820$	0	0
$821$	28.8107	1.00550	0.502750	−	0.864432i	$-0.332322\pi$
$821$	28.8107	1.00550	0.502750	+	0.864432i	$0.332322\pi$
$822$	0	0
$823$	19.1723	0.668305	0.334152	−	0.942519i	$-0.391550\pi$
$823$	19.1723	0.668305	0.334152	+	0.942519i	$0.391550\pi$
$824$	0	0
$825$	−0.132739	−0.00462139
$826$	0	0
$827$	−23.1729	−0.805800	−0.402900	−	0.915244i	$-0.631998\pi$
$827$	−23.1729	−0.805800	−0.402900	+	0.915244i	$0.631998\pi$
$828$	0	0
$829$	−21.9673	−0.762955	−0.381478	−	0.924378i	$-0.624585\pi$
$829$	−21.9673	−0.762955	−0.381478	+	0.924378i	$0.624585\pi$
$830$	0	0
$831$	−33.5197	−1.16279
$832$	0	0
$833$	−0.850200	−0.0294577
$834$	0	0
$835$	−15.2681	−0.528375
$836$	0	0
$837$	50.2312	1.73624
$838$	0	0
$839$	−40.4409	−1.39618	−0.698088	−	0.716012i	$-0.745965\pi$
$839$	−40.4409	−1.39618	−0.698088	+	0.716012i	$0.745965\pi$
$840$	0	0
$841$	−18.2856	−0.630538
$842$	0	0
$843$	−30.4282	−1.04800
$844$	0	0
$845$	25.2780	0.869591
$846$	0	0
$847$	30.8017	1.05836
$848$	0	0
$849$	−5.04380	−0.173103
$850$	0	0
$851$	−12.1501	−0.416499
$852$	0	0
$853$	13.3680	0.457710	0.228855	−	0.973460i	$-0.426502\pi$
$853$	13.3680	0.457710	0.228855	+	0.973460i	$0.426502\pi$
$854$	0	0
$855$	−2.12576	−0.0726995
$856$	0	0
$857$	40.4263	1.38094	0.690469	−	0.723362i	$-0.257404\pi$
$857$	40.4263	1.38094	0.690469	+	0.723362i	$0.257404\pi$
$858$	0	0
$859$	0.796162	0.0271647	0.0135823	−	0.999908i	$-0.495676\pi$
$859$	0.796162	0.0271647	0.0135823	+	0.999908i	$0.495676\pi$
$860$	0	0
$861$	49.7496	1.69546
$862$	0	0
$863$	24.4938	0.833778	0.416889	−	0.908958i	$-0.363120\pi$
$863$	24.4938	0.833778	0.416889	+	0.908958i	$0.363120\pi$
$864$	0	0
$865$	44.6340	1.51760
$866$	0	0
$867$	26.3322	0.894289
$868$	0	0
$869$	2.37653	0.0806182
$870$	0	0
$871$	−15.0685	−0.510577
$872$	0	0
$873$	4.41260	0.149344
$874$	0	0
$875$	−32.3561	−1.09384
$876$	0	0
$877$	−42.8172	−1.44584	−0.722918	−	0.690934i	$-0.757199\pi$
$877$	−42.8172	−1.44584	−0.722918	+	0.690934i	$0.757199\pi$
$878$	0	0
$879$	−45.8796	−1.54748
$880$	0	0
$881$	4.70084	0.158375	0.0791876	−	0.996860i	$-0.474767\pi$
$881$	4.70084	0.158375	0.0791876	+	0.996860i	$0.474767\pi$
$882$	0	0
$883$	−8.87909	−0.298805	−0.149403	−	0.988776i	$-0.547735\pi$
$883$	−8.87909	−0.298805	−0.149403	+	0.988776i	$0.547735\pi$
$884$	0	0
$885$	−18.5322	−0.622953
$886$	0	0
$887$	49.1295	1.64961	0.824803	−	0.565421i	$-0.191286\pi$
$887$	49.1295	1.64961	0.824803	+	0.565421i	$0.191286\pi$
$888$	0	0
$889$	19.9492	0.669074
$890$	0	0
$891$	2.55331	0.0855393
$892$	0	0
$893$	7.83045	0.262036
$894$	0	0
$895$	12.4640	0.416624
$896$	0	0
$897$	5.54883	0.185270
$898$	0	0
$899$	30.0443	1.00203
$900$	0	0
$901$	6.01536	0.200401
$902$	0	0
$903$	25.2795	0.841250
$904$	0	0
$905$	−0.239663	−0.00796667
$906$	0	0
$907$	59.2812	1.96840	0.984200	−	0.177060i	$-0.0566585\pi$
$907$	59.2812	1.96840	0.984200	+	0.177060i	$0.0566585\pi$
$908$	0	0
$909$	−1.98760	−0.0659245
$910$	0	0
$911$	5.71519	0.189353	0.0946763	−	0.995508i	$-0.469818\pi$
$911$	5.71519	0.189353	0.0946763	+	0.995508i	$0.469818\pi$
$912$	0	0
$913$	−1.74811	−0.0578541
$914$	0	0
$915$	18.1284	0.599307
$916$	0	0
$917$	−22.5554	−0.744845
$918$	0	0
$919$	20.3009	0.669663	0.334832	−	0.942278i	$-0.391321\pi$
$919$	20.3009	0.669663	0.334832	+	0.942278i	$0.391321\pi$
$920$	0	0
$921$	23.6454	0.779143
$922$	0	0
$923$	7.81438	0.257213
$924$	0	0
$925$	−1.03949	−0.0341781
$926$	0	0
$927$	2.08563	0.0685011
$928$	0	0
$929$	−24.1868	−0.793543	−0.396771	−	0.917918i	$-0.629869\pi$
$929$	−24.1868	−0.793543	−0.396771	+	0.917918i	$0.629869\pi$
$930$	0	0
$931$	−2.54670	−0.0834648
$932$	0	0
$933$	37.9732	1.24319
$934$	0	0
$935$	−0.614338	−0.0200910
$936$	0	0
$937$	−48.8114	−1.59460	−0.797300	−	0.603584i	$-0.793739\pi$
$937$	−48.8114	−1.59460	−0.797300	+	0.603584i	$0.793739\pi$
$938$	0	0
$939$	−1.77334	−0.0578709
$940$	0	0
$941$	22.5023	0.733555	0.366777	−	0.930309i	$-0.380461\pi$
$941$	22.5023	0.733555	0.366777	+	0.930309i	$0.380461\pi$
$942$	0	0
$943$	−31.4662	−1.02468
$944$	0	0
$945$	33.7435	1.09768
$946$	0	0
$947$	−2.60674	−0.0847075	−0.0423538	−	0.999103i	$-0.513486\pi$
$947$	−2.60674	−0.0847075	−0.0423538	+	0.999103i	$0.513486\pi$
$948$	0	0
$949$	9.75096	0.316530
$950$	0	0
$951$	−25.5744	−0.829306
$952$	0	0
$953$	−29.1285	−0.943565	−0.471783	−	0.881715i	$-0.656389\pi$
$953$	−29.1285	−0.943565	−0.471783	+	0.881715i	$0.656389\pi$
$954$	0	0
$955$	−9.41540	−0.304675
$956$	0	0
$957$	1.75481	0.0567251
$958$	0	0
$959$	−33.8443	−1.09289
$960$	0	0
$961$	53.2475	1.71766
$962$	0	0
$963$	4.42657	0.142644
$964$	0	0
$965$	−49.4226	−1.59097
$966$	0	0
$967$	−44.1094	−1.41846	−0.709231	−	0.704976i	$-0.750958\pi$
$967$	−44.1094	−1.41846	−0.709231	+	0.704976i	$0.750958\pi$
$968$	0	0
$969$	−3.50444	−0.112579
$970$	0	0
$971$	−16.3250	−0.523896	−0.261948	−	0.965082i	$-0.584365\pi$
$971$	−16.3250	−0.523896	−0.261948	+	0.965082i	$0.584365\pi$
$972$	0	0
$973$	−15.5212	−0.497586
$974$	0	0
$975$	0.474725	0.0152034
$976$	0	0
$977$	19.5231	0.624601	0.312300	−	0.949983i	$-0.398900\pi$
$977$	19.5231	0.624601	0.312300	+	0.949983i	$0.398900\pi$
$978$	0	0
$979$	−4.98814	−0.159422
$980$	0	0
$981$	−0.886703	−0.0283103
$982$	0	0
$983$	−37.9408	−1.21012	−0.605062	−	0.796179i	$-0.706852\pi$
$983$	−37.9408	−1.21012	−0.605062	+	0.796179i	$0.706852\pi$
$984$	0	0
$985$	5.07924	0.161838
$986$	0	0
$987$	−14.0658	−0.447719
$988$	0	0
$989$	−15.9891	−0.508424
$990$	0	0
$991$	30.6236	0.972792	0.486396	−	0.873739i	$-0.338311\pi$
$991$	30.6236	0.972792	0.486396	+	0.873739i	$0.338311\pi$
$992$	0	0
$993$	20.2849	0.643722
$994$	0	0
$995$	39.8029	1.26184
$996$	0	0
$997$	−41.5890	−1.31714	−0.658569	−	0.752520i	$-0.728838\pi$
$997$	−41.5890	−1.31714	−0.658569	+	0.752520i	$0.728838\pi$
$998$	0	0
$999$	22.9753	0.726907

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.15	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.15	✓	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)

\(f(q)\)	\(=\)	\(q-1.61777 q^{3} -2.18000 q^{5} -2.82839 q^{7} -0.382806 q^{9} +O(q^{10})\)	\(q-1.61777 q^{3} -2.18000 q^{5} -2.82839 q^{7} -0.382806 q^{9} -0.331382 q^{11} +1.18515 q^{13} +3.52675 q^{15} -0.850397 q^{17} -2.54729 q^{19} +4.57569 q^{21} -2.89409 q^{23} -0.247601 q^{25} +5.47262 q^{27} +3.27328 q^{29} +9.17864 q^{31} +0.536102 q^{33} +6.16588 q^{35} +4.19824 q^{37} -1.91730 q^{39} +10.8726 q^{41} +5.52475 q^{43} +0.834518 q^{45} -3.07403 q^{47} +0.999768 q^{49} +1.37575 q^{51} -7.07359 q^{53} +0.722413 q^{55} +4.12095 q^{57} -5.25476 q^{59} +5.14026 q^{61} +1.08272 q^{63} -2.58362 q^{65} -12.7145 q^{67} +4.68198 q^{69} +6.59360 q^{71} +8.22764 q^{73} +0.400563 q^{75} +0.937277 q^{77} -7.17156 q^{79} -7.70504 q^{81} +5.27522 q^{83} +1.85387 q^{85} -5.29544 q^{87} +15.0525 q^{89} -3.35205 q^{91} -14.8490 q^{93} +5.55310 q^{95} -11.5270 q^{97} +0.126855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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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