LMFDB - Embedded newform 8008.2.a.v.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 19 x^{9} + 33 x^{8} + 120 x^{7} - 178 x^{6} - 296 x^{5} + 380 x^{4} + 280 x^{3} + \cdots + 64 $ x^11 - 2x^10 - 19x^9 + 33x^8 + 120x^7 - 178x^6 - 296x^5 + 380x^4 + 280x^3 - 295x^2 - 87x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.29224$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.29224 q^{3} +0.279184 q^{5} -1.00000 q^{7} +2.25438 q^{9} +O(q^{10})$	$q+2.29224 q^{3} +0.279184 q^{5} -1.00000 q^{7} +2.25438 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.639958 q^{15} +2.22354 q^{17} -8.55315 q^{19} -2.29224 q^{21} -3.73720 q^{23} -4.92206 q^{25} -1.70914 q^{27} +8.08689 q^{29} -8.68510 q^{31} +2.29224 q^{33} -0.279184 q^{35} +7.36294 q^{37} -2.29224 q^{39} -0.628104 q^{41} -2.32914 q^{43} +0.629387 q^{45} -3.73720 q^{47} +1.00000 q^{49} +5.09689 q^{51} +6.34813 q^{53} +0.279184 q^{55} -19.6059 q^{57} +5.45723 q^{59} +0.113940 q^{61} -2.25438 q^{63} -0.279184 q^{65} -2.41610 q^{67} -8.56656 q^{69} -13.8194 q^{71} -7.30849 q^{73} -11.2826 q^{75} -1.00000 q^{77} -12.8219 q^{79} -10.6809 q^{81} +7.64233 q^{83} +0.620776 q^{85} +18.5371 q^{87} +8.78585 q^{89} +1.00000 q^{91} -19.9084 q^{93} -2.38790 q^{95} -9.03350 q^{97} +2.25438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * 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
$3$	2.29224	1.32343	0.661714	−	0.749757i	$-0.269829\pi$
$3$	2.29224	1.32343	0.661714	+	0.749757i	$0.269829\pi$
$4$	0	0
$5$	0.279184	0.124855	0.0624274	−	0.998050i	$-0.480116\pi$
$5$	0.279184	0.124855	0.0624274	+	0.998050i	$0.480116\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.25438	0.751460
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.639958	0.165236
$16$	0	0
$17$	2.22354	0.539287	0.269644	−	0.962960i	$-0.413094\pi$
$17$	2.22354	0.539287	0.269644	+	0.962960i	$0.413094\pi$
$18$	0	0
$19$	−8.55315	−1.96223	−0.981114	−	0.193431i	$-0.938038\pi$
$19$	−8.55315	−1.96223	−0.981114	+	0.193431i	$0.938038\pi$
$20$	0	0
$21$	−2.29224	−0.500209
$22$	0	0
$23$	−3.73720	−0.779259	−0.389630	−	0.920972i	$-0.627397\pi$
$23$	−3.73720	−0.779259	−0.389630	+	0.920972i	$0.627397\pi$
$24$	0	0
$25$	−4.92206	−0.984411
$26$	0	0
$27$	−1.70914	−0.328924
$28$	0	0
$29$	8.08689	1.50170	0.750849	−	0.660474i	$-0.229644\pi$
$29$	8.08689	1.50170	0.750849	+	0.660474i	$0.229644\pi$
$30$	0	0
$31$	−8.68510	−1.55989	−0.779945	−	0.625848i	$-0.784753\pi$
$31$	−8.68510	−1.55989	−0.779945	+	0.625848i	$0.784753\pi$
$32$	0	0
$33$	2.29224	0.399028
$34$	0	0
$35$	−0.279184	−0.0471907
$36$	0	0
$37$	7.36294	1.21046	0.605230	−	0.796051i	$-0.293081\pi$
$37$	7.36294	1.21046	0.605230	+	0.796051i	$0.293081\pi$
$38$	0	0
$39$	−2.29224	−0.367053
$40$	0	0
$41$	−0.628104	−0.0980934	−0.0490467	−	0.998796i	$-0.515618\pi$
$41$	−0.628104	−0.0980934	−0.0490467	+	0.998796i	$0.515618\pi$
$42$	0	0
$43$	−2.32914	−0.355191	−0.177596	−	0.984104i	$-0.556832\pi$
$43$	−2.32914	−0.355191	−0.177596	+	0.984104i	$0.556832\pi$
$44$	0	0
$45$	0.629387	0.0938235
$46$	0	0
$47$	−3.73720	−0.545126	−0.272563	−	0.962138i	$-0.587871\pi$
$47$	−3.73720	−0.545126	−0.272563	+	0.962138i	$0.587871\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.09689	0.713707
$52$	0	0
$53$	6.34813	0.871983	0.435991	−	0.899951i	$-0.356398\pi$
$53$	6.34813	0.871983	0.435991	+	0.899951i	$0.356398\pi$
$54$	0	0
$55$	0.279184	0.0376452
$56$	0	0
$57$	−19.6059	−2.59687
$58$	0	0
$59$	5.45723	0.710471	0.355235	−	0.934777i	$-0.384401\pi$
$59$	5.45723	0.710471	0.355235	+	0.934777i	$0.384401\pi$
$60$	0	0
$61$	0.113940	0.0145886	0.00729429	−	0.999973i	$-0.497678\pi$
$61$	0.113940	0.0145886	0.00729429	+	0.999973i	$0.497678\pi$
$62$	0	0
$63$	−2.25438	−0.284025
$64$	0	0
$65$	−0.279184	−0.0346285
$66$	0	0
$67$	−2.41610	−0.295174	−0.147587	−	0.989049i	$-0.547151\pi$
$67$	−2.41610	−0.295174	−0.147587	+	0.989049i	$0.547151\pi$
$68$	0	0
$69$	−8.56656	−1.03129
$70$	0	0
$71$	−13.8194	−1.64006	−0.820031	−	0.572319i	$-0.806044\pi$
$71$	−13.8194	−1.64006	−0.820031	+	0.572319i	$0.806044\pi$
$72$	0	0
$73$	−7.30849	−0.855395	−0.427697	−	0.903922i	$-0.640675\pi$
$73$	−7.30849	−0.855395	−0.427697	+	0.903922i	$0.640675\pi$
$74$	0	0
$75$	−11.2826	−1.30280
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−12.8219	−1.44257	−0.721287	−	0.692636i	$-0.756449\pi$
$79$	−12.8219	−1.44257	−0.721287	+	0.692636i	$0.756449\pi$
$80$	0	0
$81$	−10.6809	−1.18677
$82$	0	0
$83$	7.64233	0.838855	0.419427	−	0.907789i	$-0.362231\pi$
$83$	7.64233	0.838855	0.419427	+	0.907789i	$0.362231\pi$
$84$	0	0
$85$	0.620776	0.0673326
$86$	0	0
$87$	18.5371	1.98739
$88$	0	0
$89$	8.78585	0.931299	0.465649	−	0.884969i	$-0.345821\pi$
$89$	8.78585	0.931299	0.465649	+	0.884969i	$0.345821\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−19.9084	−2.06440
$94$	0	0
$95$	−2.38790	−0.244994
$96$	0	0
$97$	−9.03350	−0.917213	−0.458606	−	0.888640i	$-0.651651\pi$
$97$	−9.03350	−0.917213	−0.458606	+	0.888640i	$0.651651\pi$
$98$	0	0
$99$	2.25438	0.226574
$100$	0	0
$101$	1.56739	0.155962	0.0779808	−	0.996955i	$-0.475153\pi$
$101$	1.56739	0.155962	0.0779808	+	0.996955i	$0.475153\pi$
$102$	0	0
$103$	−5.72438	−0.564040	−0.282020	−	0.959409i	$-0.591004\pi$
$103$	−5.72438	−0.564040	−0.282020	+	0.959409i	$0.591004\pi$
$104$	0	0
$105$	−0.639958	−0.0624535
$106$	0	0
$107$	−0.400752	−0.0387421	−0.0193711	−	0.999812i	$-0.506166\pi$
$107$	−0.400752	−0.0387421	−0.0193711	+	0.999812i	$0.506166\pi$
$108$	0	0
$109$	−20.3077	−1.94512	−0.972561	−	0.232649i	$-0.925261\pi$
$109$	−20.3077	−1.94512	−0.972561	+	0.232649i	$0.925261\pi$
$110$	0	0
$111$	16.8776	1.60195
$112$	0	0
$113$	−12.1185	−1.14001	−0.570007	−	0.821640i	$-0.693059\pi$
$113$	−12.1185	−1.14001	−0.570007	+	0.821640i	$0.693059\pi$
$114$	0	0
$115$	−1.04337	−0.0972943
$116$	0	0
$117$	−2.25438	−0.208418
$118$	0	0
$119$	−2.22354	−0.203831
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.43977	−0.129819
$124$	0	0
$125$	−2.77008	−0.247763
$126$	0	0
$127$	−11.2734	−1.00035	−0.500174	−	0.865925i	$-0.666731\pi$
$127$	−11.2734	−1.00035	−0.500174	+	0.865925i	$0.666731\pi$
$128$	0	0
$129$	−5.33896	−0.470070
$130$	0	0
$131$	0.756782	0.0661204	0.0330602	−	0.999453i	$-0.489475\pi$
$131$	0.756782	0.0661204	0.0330602	+	0.999453i	$0.489475\pi$
$132$	0	0
$133$	8.55315	0.741652
$134$	0	0
$135$	−0.477165	−0.0410678
$136$	0	0
$137$	−7.77046	−0.663875	−0.331938	−	0.943301i	$-0.607702\pi$
$137$	−7.77046	−0.663875	−0.331938	+	0.943301i	$0.607702\pi$
$138$	0	0
$139$	−13.7690	−1.16787	−0.583936	−	0.811800i	$-0.698488\pi$
$139$	−13.7690	−1.16787	−0.583936	+	0.811800i	$0.698488\pi$
$140$	0	0
$141$	−8.56656	−0.721435
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	2.25773	0.187494
$146$	0	0
$147$	2.29224	0.189061
$148$	0	0
$149$	6.78265	0.555656	0.277828	−	0.960631i	$-0.410385\pi$
$149$	6.78265	0.555656	0.277828	+	0.960631i	$0.410385\pi$
$150$	0	0
$151$	−15.5829	−1.26812	−0.634060	−	0.773284i	$-0.718613\pi$
$151$	−15.5829	−1.26812	−0.634060	+	0.773284i	$0.718613\pi$
$152$	0	0
$153$	5.01270	0.405253
$154$	0	0
$155$	−2.42474	−0.194760
$156$	0	0
$157$	1.46924	0.117258	0.0586290	−	0.998280i	$-0.481327\pi$
$157$	1.46924	0.117258	0.0586290	+	0.998280i	$0.481327\pi$
$158$	0	0
$159$	14.5515	1.15401
$160$	0	0
$161$	3.73720	0.294532
$162$	0	0
$163$	11.7669	0.921657	0.460828	−	0.887489i	$-0.347552\pi$
$163$	11.7669	0.921657	0.460828	+	0.887489i	$0.347552\pi$
$164$	0	0
$165$	0.639958	0.0498206
$166$	0	0
$167$	10.7199	0.829530	0.414765	−	0.909929i	$-0.363864\pi$
$167$	10.7199	0.829530	0.414765	+	0.909929i	$0.363864\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−19.2821	−1.47454
$172$	0	0
$173$	4.17187	0.317181	0.158591	−	0.987344i	$-0.449305\pi$
$173$	4.17187	0.317181	0.158591	+	0.987344i	$0.449305\pi$
$174$	0	0
$175$	4.92206	0.372072
$176$	0	0
$177$	12.5093	0.940256
$178$	0	0
$179$	15.5429	1.16173	0.580867	−	0.813999i	$-0.302714\pi$
$179$	15.5429	1.16173	0.580867	+	0.813999i	$0.302714\pi$
$180$	0	0
$181$	22.9274	1.70418	0.852089	−	0.523398i	$-0.175336\pi$
$181$	22.9274	1.70418	0.852089	+	0.523398i	$0.175336\pi$
$182$	0	0
$183$	0.261179	0.0193069
$184$	0	0
$185$	2.05561	0.151132
$186$	0	0
$187$	2.22354	0.162601
$188$	0	0
$189$	1.70914	0.124322
$190$	0	0
$191$	6.07976	0.439916	0.219958	−	0.975509i	$-0.429408\pi$
$191$	6.07976	0.439916	0.219958	+	0.975509i	$0.429408\pi$
$192$	0	0
$193$	0.753290	0.0542230	0.0271115	−	0.999632i	$-0.491369\pi$
$193$	0.753290	0.0542230	0.0271115	+	0.999632i	$0.491369\pi$
$194$	0	0
$195$	−0.639958	−0.0458283
$196$	0	0
$197$	4.08285	0.290891	0.145446	−	0.989366i	$-0.453538\pi$
$197$	4.08285	0.290891	0.145446	+	0.989366i	$0.453538\pi$
$198$	0	0
$199$	25.1545	1.78316	0.891579	−	0.452865i	$-0.149598\pi$
$199$	25.1545	1.78316	0.891579	+	0.452865i	$0.149598\pi$
$200$	0	0
$201$	−5.53830	−0.390641
$202$	0	0
$203$	−8.08689	−0.567589
$204$	0	0
$205$	−0.175357	−0.0122474
$206$	0	0
$207$	−8.42506	−0.585582
$208$	0	0
$209$	−8.55315	−0.591634
$210$	0	0
$211$	−4.72132	−0.325029	−0.162515	−	0.986706i	$-0.551960\pi$
$211$	−4.72132	−0.325029	−0.162515	+	0.986706i	$0.551960\pi$
$212$	0	0
$213$	−31.6775	−2.17050
$214$	0	0
$215$	−0.650260	−0.0443473
$216$	0	0
$217$	8.68510	0.589583
$218$	0	0
$219$	−16.7528	−1.13205
$220$	0	0
$221$	−2.22354	−0.149571
$222$	0	0
$223$	14.7656	0.988775	0.494387	−	0.869242i	$-0.335392\pi$
$223$	14.7656	0.988775	0.494387	+	0.869242i	$0.335392\pi$
$224$	0	0
$225$	−11.0962	−0.739746
$226$	0	0
$227$	−12.4810	−0.828396	−0.414198	−	0.910187i	$-0.635938\pi$
$227$	−12.4810	−0.828396	−0.414198	+	0.910187i	$0.635938\pi$
$228$	0	0
$229$	−19.4633	−1.28617	−0.643086	−	0.765794i	$-0.722346\pi$
$229$	−19.4633	−1.28617	−0.643086	+	0.765794i	$0.722346\pi$
$230$	0	0
$231$	−2.29224	−0.150819
$232$	0	0
$233$	1.30972	0.0858029	0.0429014	−	0.999079i	$-0.486340\pi$
$233$	1.30972	0.0858029	0.0429014	+	0.999079i	$0.486340\pi$
$234$	0	0
$235$	−1.04337	−0.0680617
$236$	0	0
$237$	−29.3909	−1.90914
$238$	0	0
$239$	−17.2472	−1.11563	−0.557816	−	0.829965i	$-0.688360\pi$
$239$	−17.2472	−1.11563	−0.557816	+	0.829965i	$0.688360\pi$
$240$	0	0
$241$	−2.50501	−0.161362	−0.0806810	−	0.996740i	$-0.525710\pi$
$241$	−2.50501	−0.161362	−0.0806810	+	0.996740i	$0.525710\pi$
$242$	0	0
$243$	−19.3558	−1.24168
$244$	0	0
$245$	0.279184	0.0178364
$246$	0	0
$247$	8.55315	0.544224
$248$	0	0
$249$	17.5181	1.11016
$250$	0	0
$251$	−1.65297	−0.104334	−0.0521672	−	0.998638i	$-0.516613\pi$
$251$	−1.65297	−0.104334	−0.0521672	+	0.998638i	$0.516613\pi$
$252$	0	0
$253$	−3.73720	−0.234956
$254$	0	0
$255$	1.42297	0.0891098
$256$	0	0
$257$	18.1451	1.13186	0.565930	−	0.824453i	$-0.308517\pi$
$257$	18.1451	1.13186	0.565930	+	0.824453i	$0.308517\pi$
$258$	0	0
$259$	−7.36294	−0.457511
$260$	0	0
$261$	18.2309	1.12847
$262$	0	0
$263$	10.8985	0.672032	0.336016	−	0.941856i	$-0.390920\pi$
$263$	10.8985	0.672032	0.336016	+	0.941856i	$0.390920\pi$
$264$	0	0
$265$	1.77230	0.108871
$266$	0	0
$267$	20.1393	1.23251
$268$	0	0
$269$	21.4420	1.30734	0.653672	−	0.756778i	$-0.273228\pi$
$269$	21.4420	1.30734	0.653672	+	0.756778i	$0.273228\pi$
$270$	0	0
$271$	−18.8573	−1.14550	−0.572750	−	0.819730i	$-0.694123\pi$
$271$	−18.8573	−1.14550	−0.572750	+	0.819730i	$0.694123\pi$
$272$	0	0
$273$	2.29224	0.138733
$274$	0	0
$275$	−4.92206	−0.296811
$276$	0	0
$277$	0.187651	0.0112749	0.00563743	−	0.999984i	$-0.498206\pi$
$277$	0.187651	0.0112749	0.00563743	+	0.999984i	$0.498206\pi$
$278$	0	0
$279$	−19.5795	−1.17220
$280$	0	0
$281$	−31.4355	−1.87529	−0.937643	−	0.347600i	$-0.886997\pi$
$281$	−31.4355	−1.87529	−0.937643	+	0.347600i	$0.886997\pi$
$282$	0	0
$283$	−1.92359	−0.114345	−0.0571727	−	0.998364i	$-0.518209\pi$
$283$	−1.92359	−0.114345	−0.0571727	+	0.998364i	$0.518209\pi$
$284$	0	0
$285$	−5.47366	−0.324231
$286$	0	0
$287$	0.628104	0.0370758
$288$	0	0
$289$	−12.0559	−0.709170
$290$	0	0
$291$	−20.7070	−1.21386
$292$	0	0
$293$	15.2224	0.889304	0.444652	−	0.895703i	$-0.353327\pi$
$293$	15.2224	0.889304	0.444652	+	0.895703i	$0.353327\pi$
$294$	0	0
$295$	1.52357	0.0887057
$296$	0	0
$297$	−1.70914	−0.0991744
$298$	0	0
$299$	3.73720	0.216128
$300$	0	0
$301$	2.32914	0.134250
$302$	0	0
$303$	3.59285	0.206404
$304$	0	0
$305$	0.0318104	0.00182146
$306$	0	0
$307$	−29.0297	−1.65681	−0.828405	−	0.560130i	$-0.810751\pi$
$307$	−29.0297	−1.65681	−0.828405	+	0.560130i	$0.810751\pi$
$308$	0	0
$309$	−13.1217	−0.746466
$310$	0	0
$311$	18.8919	1.07126	0.535631	−	0.844452i	$-0.320074\pi$
$311$	18.8919	1.07126	0.535631	+	0.844452i	$0.320074\pi$
$312$	0	0
$313$	−11.8187	−0.668032	−0.334016	−	0.942567i	$-0.608404\pi$
$313$	−11.8187	−0.668032	−0.334016	+	0.942567i	$0.608404\pi$
$314$	0	0
$315$	−0.629387	−0.0354619
$316$	0	0
$317$	29.8309	1.67547	0.837735	−	0.546076i	$-0.183879\pi$
$317$	29.8309	1.67547	0.837735	+	0.546076i	$0.183879\pi$
$318$	0	0
$319$	8.08689	0.452779
$320$	0	0
$321$	−0.918621	−0.0512724
$322$	0	0
$323$	−19.0183	−1.05820
$324$	0	0
$325$	4.92206	0.273027
$326$	0	0
$327$	−46.5501	−2.57423
$328$	0	0
$329$	3.73720	0.206038
$330$	0	0
$331$	0.536178	0.0294710	0.0147355	−	0.999891i	$-0.495309\pi$
$331$	0.536178	0.0294710	0.0147355	+	0.999891i	$0.495309\pi$
$332$	0	0
$333$	16.5989	0.909612
$334$	0	0
$335$	−0.674537	−0.0368539
$336$	0	0
$337$	−19.4470	−1.05935	−0.529674	−	0.848201i	$-0.677686\pi$
$337$	−19.4470	−1.05935	−0.529674	+	0.848201i	$0.677686\pi$
$338$	0	0
$339$	−27.7786	−1.50873
$340$	0	0
$341$	−8.68510	−0.470325
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.39165	−0.128762
$346$	0	0
$347$	12.8081	0.687575	0.343787	−	0.939048i	$-0.388290\pi$
$347$	12.8081	0.687575	0.343787	+	0.939048i	$0.388290\pi$
$348$	0	0
$349$	−7.93120	−0.424547	−0.212274	−	0.977210i	$-0.568087\pi$
$349$	−7.93120	−0.424547	−0.212274	+	0.977210i	$0.568087\pi$
$350$	0	0
$351$	1.70914	0.0912272
$352$	0	0
$353$	−3.31648	−0.176518	−0.0882591	−	0.996098i	$-0.528130\pi$
$353$	−3.31648	−0.176518	−0.0882591	+	0.996098i	$0.528130\pi$
$354$	0	0
$355$	−3.85816	−0.204770
$356$	0	0
$357$	−5.09689	−0.269756
$358$	0	0
$359$	36.7085	1.93740	0.968700	−	0.248235i	$-0.0798506\pi$
$359$	36.7085	1.93740	0.968700	+	0.248235i	$0.0798506\pi$
$360$	0	0
$361$	54.1564	2.85034
$362$	0	0
$363$	2.29224	0.120312
$364$	0	0
$365$	−2.04041	−0.106800
$366$	0	0
$367$	17.2739	0.901688	0.450844	−	0.892603i	$-0.351123\pi$
$367$	17.2739	0.901688	0.450844	+	0.892603i	$0.351123\pi$
$368$	0	0
$369$	−1.41599	−0.0737133
$370$	0	0
$371$	−6.34813	−0.329578
$372$	0	0
$373$	−32.9384	−1.70549	−0.852744	−	0.522329i	$-0.825063\pi$
$373$	−32.9384	−1.70549	−0.852744	+	0.522329i	$0.825063\pi$
$374$	0	0
$375$	−6.34970	−0.327897
$376$	0	0
$377$	−8.08689	−0.416496
$378$	0	0
$379$	−14.5118	−0.745423	−0.372711	−	0.927947i	$-0.621572\pi$
$379$	−14.5118	−0.745423	−0.372711	+	0.927947i	$0.621572\pi$
$380$	0	0
$381$	−25.8413	−1.32389
$382$	0	0
$383$	23.6646	1.20920	0.604602	−	0.796527i	$-0.293332\pi$
$383$	23.6646	1.20920	0.604602	+	0.796527i	$0.293332\pi$
$384$	0	0
$385$	−0.279184	−0.0142285
$386$	0	0
$387$	−5.25078	−0.266912
$388$	0	0
$389$	5.96694	0.302536	0.151268	−	0.988493i	$-0.451664\pi$
$389$	5.96694	0.302536	0.151268	+	0.988493i	$0.451664\pi$
$390$	0	0
$391$	−8.30980	−0.420244
$392$	0	0
$393$	1.73473	0.0875055
$394$	0	0
$395$	−3.57966	−0.180112
$396$	0	0
$397$	13.3533	0.670182	0.335091	−	0.942186i	$-0.391233\pi$
$397$	13.3533	0.670182	0.335091	+	0.942186i	$0.391233\pi$
$398$	0	0
$399$	19.6059	0.981523
$400$	0	0
$401$	−20.7917	−1.03829	−0.519145	−	0.854686i	$-0.673750\pi$
$401$	−20.7917	−1.03829	−0.519145	+	0.854686i	$0.673750\pi$
$402$	0	0
$403$	8.68510	0.432636
$404$	0	0
$405$	−2.98194	−0.148174
$406$	0	0
$407$	7.36294	0.364967
$408$	0	0
$409$	6.18004	0.305583	0.152792	−	0.988258i	$-0.451174\pi$
$409$	6.18004	0.305583	0.152792	+	0.988258i	$0.451174\pi$
$410$	0	0
$411$	−17.8118	−0.878591
$412$	0	0
$413$	−5.45723	−0.268533
$414$	0	0
$415$	2.13362	0.104735
$416$	0	0
$417$	−31.5619	−1.54559
$418$	0	0
$419$	−16.8424	−0.822807	−0.411403	−	0.911453i	$-0.634961\pi$
$419$	−16.8424	−0.822807	−0.411403	+	0.911453i	$0.634961\pi$
$420$	0	0
$421$	32.5332	1.58557	0.792785	−	0.609502i	$-0.208631\pi$
$421$	32.5332	1.58557	0.792785	+	0.609502i	$0.208631\pi$
$422$	0	0
$423$	−8.42506	−0.409641
$424$	0	0
$425$	−10.9444	−0.530880
$426$	0	0
$427$	−0.113940	−0.00551397
$428$	0	0
$429$	−2.29224	−0.110671
$430$	0	0
$431$	−5.41035	−0.260607	−0.130304	−	0.991474i	$-0.541595\pi$
$431$	−5.41035	−0.260607	−0.130304	+	0.991474i	$0.541595\pi$
$432$	0	0
$433$	−31.0387	−1.49163	−0.745813	−	0.666155i	$-0.767939\pi$
$433$	−31.0387	−1.49163	−0.745813	+	0.666155i	$0.767939\pi$
$434$	0	0
$435$	5.17527	0.248135
$436$	0	0
$437$	31.9648	1.52908
$438$	0	0
$439$	−10.1812	−0.485924	−0.242962	−	0.970036i	$-0.578119\pi$
$439$	−10.1812	−0.485924	−0.242962	+	0.970036i	$0.578119\pi$
$440$	0	0
$441$	2.25438	0.107351
$442$	0	0
$443$	−27.3171	−1.29788	−0.648938	−	0.760841i	$-0.724786\pi$
$443$	−27.3171	−1.29788	−0.648938	+	0.760841i	$0.724786\pi$
$444$	0	0
$445$	2.45287	0.116277
$446$	0	0
$447$	15.5475	0.735371
$448$	0	0
$449$	−30.9671	−1.46143	−0.730713	−	0.682684i	$-0.760812\pi$
$449$	−30.9671	−1.46143	−0.730713	+	0.682684i	$0.760812\pi$
$450$	0	0
$451$	−0.628104	−0.0295763
$452$	0	0
$453$	−35.7198	−1.67826
$454$	0	0
$455$	0.279184	0.0130883
$456$	0	0
$457$	−2.91169	−0.136203	−0.0681015	−	0.997678i	$-0.521694\pi$
$457$	−2.91169	−0.136203	−0.0681015	+	0.997678i	$0.521694\pi$
$458$	0	0
$459$	−3.80034	−0.177385
$460$	0	0
$461$	−31.8908	−1.48530	−0.742652	−	0.669677i	$-0.766433\pi$
$461$	−31.8908	−1.48530	−0.742652	+	0.669677i	$0.766433\pi$
$462$	0	0
$463$	10.4282	0.484641	0.242320	−	0.970196i	$-0.422091\pi$
$463$	10.4282	0.484641	0.242320	+	0.970196i	$0.422091\pi$
$464$	0	0
$465$	−5.55810	−0.257751
$466$	0	0
$467$	14.1950	0.656866	0.328433	−	0.944527i	$-0.393480\pi$
$467$	14.1950	0.656866	0.328433	+	0.944527i	$0.393480\pi$
$468$	0	0
$469$	2.41610	0.111565
$470$	0	0
$471$	3.36785	0.155182
$472$	0	0
$473$	−2.32914	−0.107094
$474$	0	0
$475$	42.0991	1.93164
$476$	0	0
$477$	14.3111	0.655260
$478$	0	0
$479$	30.8316	1.40873	0.704365	−	0.709838i	$-0.251232\pi$
$479$	30.8316	1.40873	0.704365	+	0.709838i	$0.251232\pi$
$480$	0	0
$481$	−7.36294	−0.335721
$482$	0	0
$483$	8.56656	0.389792
$484$	0	0
$485$	−2.52201	−0.114518
$486$	0	0
$487$	−22.2542	−1.00843	−0.504217	−	0.863577i	$-0.668219\pi$
$487$	−22.2542	−1.00843	−0.504217	+	0.863577i	$0.668219\pi$
$488$	0	0
$489$	26.9727	1.21975
$490$	0	0
$491$	3.96661	0.179010	0.0895052	−	0.995986i	$-0.471471\pi$
$491$	3.96661	0.179010	0.0895052	+	0.995986i	$0.471471\pi$
$492$	0	0
$493$	17.9815	0.809847
$494$	0	0
$495$	0.629387	0.0282888
$496$	0	0
$497$	13.8194	0.619885
$498$	0	0
$499$	−28.3583	−1.26949	−0.634745	−	0.772722i	$-0.718895\pi$
$499$	−28.3583	−1.26949	−0.634745	+	0.772722i	$0.718895\pi$
$500$	0	0
$501$	24.5726	1.09782
$502$	0	0
$503$	−4.75558	−0.212041	−0.106020	−	0.994364i	$-0.533811\pi$
$503$	−4.75558	−0.212041	−0.106020	+	0.994364i	$0.533811\pi$
$504$	0	0
$505$	0.437591	0.0194726
$506$	0	0
$507$	2.29224	0.101802
$508$	0	0
$509$	−0.591914	−0.0262361	−0.0131181	−	0.999914i	$-0.504176\pi$
$509$	−0.591914	−0.0262361	−0.0131181	+	0.999914i	$0.504176\pi$
$510$	0	0
$511$	7.30849	0.323309
$512$	0	0
$513$	14.6185	0.645424
$514$	0	0
$515$	−1.59815	−0.0704231
$516$	0	0
$517$	−3.73720	−0.164362
$518$	0	0
$519$	9.56294	0.419767
$520$	0	0
$521$	13.9905	0.612935	0.306468	−	0.951881i	$-0.400853\pi$
$521$	13.9905	0.612935	0.306468	+	0.951881i	$0.400853\pi$
$522$	0	0
$523$	12.1696	0.532138	0.266069	−	0.963954i	$-0.414275\pi$
$523$	12.1696	0.532138	0.266069	+	0.963954i	$0.414275\pi$
$524$	0	0
$525$	11.2826	0.492411
$526$	0	0
$527$	−19.3116	−0.841229
$528$	0	0
$529$	−9.03336	−0.392755
$530$	0	0
$531$	12.3027	0.533890
$532$	0	0
$533$	0.628104	0.0272062
$534$	0	0
$535$	−0.111883	−0.00483714
$536$	0	0
$537$	35.6282	1.53747
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−0.354874	−0.0152572	−0.00762860	−	0.999971i	$-0.502428\pi$
$541$	−0.354874	−0.0152572	−0.00762860	+	0.999971i	$0.502428\pi$
$542$	0	0
$543$	52.5551	2.25535
$544$	0	0
$545$	−5.66958	−0.242858
$546$	0	0
$547$	27.8547	1.19098	0.595490	−	0.803363i	$-0.296958\pi$
$547$	27.8547	1.19098	0.595490	+	0.803363i	$0.296958\pi$
$548$	0	0
$549$	0.256865	0.0109627
$550$	0	0
$551$	−69.1684	−2.94667
$552$	0	0
$553$	12.8219	0.545242
$554$	0	0
$555$	4.71197	0.200012
$556$	0	0
$557$	−17.3136	−0.733602	−0.366801	−	0.930299i	$-0.619547\pi$
$557$	−17.3136	−0.733602	−0.366801	+	0.930299i	$0.619547\pi$
$558$	0	0
$559$	2.32914	0.0985123
$560$	0	0
$561$	5.09689	0.215191
$562$	0	0
$563$	−5.13413	−0.216378	−0.108189	−	0.994130i	$-0.534505\pi$
$563$	−5.13413	−0.216378	−0.108189	+	0.994130i	$0.534505\pi$
$564$	0	0
$565$	−3.38329	−0.142336
$566$	0	0
$567$	10.6809	0.448556
$568$	0	0
$569$	−26.8307	−1.12480	−0.562401	−	0.826864i	$-0.690122\pi$
$569$	−26.8307	−1.12480	−0.562401	+	0.826864i	$0.690122\pi$
$570$	0	0
$571$	−6.71293	−0.280927	−0.140464	−	0.990086i	$-0.544859\pi$
$571$	−6.71293	−0.280927	−0.140464	+	0.990086i	$0.544859\pi$
$572$	0	0
$573$	13.9363	0.582196
$574$	0	0
$575$	18.3947	0.767112
$576$	0	0
$577$	22.7930	0.948887	0.474443	−	0.880286i	$-0.342649\pi$
$577$	22.7930	0.948887	0.474443	+	0.880286i	$0.342649\pi$
$578$	0	0
$579$	1.72672	0.0717602
$580$	0	0
$581$	−7.64233	−0.317057
$582$	0	0
$583$	6.34813	0.262913
$584$	0	0
$585$	−0.629387	−0.0260219
$586$	0	0
$587$	12.8844	0.531795	0.265897	−	0.964001i	$-0.414332\pi$
$587$	12.8844	0.531795	0.265897	+	0.964001i	$0.414332\pi$
$588$	0	0
$589$	74.2850	3.06086
$590$	0	0
$591$	9.35889	0.384973
$592$	0	0
$593$	−6.84093	−0.280923	−0.140462	−	0.990086i	$-0.544859\pi$
$593$	−6.84093	−0.280923	−0.140462	+	0.990086i	$0.544859\pi$
$594$	0	0
$595$	−0.620776	−0.0254493
$596$	0	0
$597$	57.6603	2.35988
$598$	0	0
$599$	10.7168	0.437878	0.218939	−	0.975739i	$-0.429740\pi$
$599$	10.7168	0.437878	0.218939	+	0.975739i	$0.429740\pi$
$600$	0	0
$601$	15.4963	0.632108	0.316054	−	0.948741i	$-0.397642\pi$
$601$	15.4963	0.632108	0.316054	+	0.948741i	$0.397642\pi$
$602$	0	0
$603$	−5.44681	−0.221811
$604$	0	0
$605$	0.279184	0.0113504
$606$	0	0
$607$	−39.5904	−1.60693	−0.803463	−	0.595355i	$-0.797011\pi$
$607$	−39.5904	−1.60693	−0.803463	+	0.595355i	$0.797011\pi$
$608$	0	0
$609$	−18.5371	−0.751162
$610$	0	0
$611$	3.73720	0.151191
$612$	0	0
$613$	5.56507	0.224771	0.112386	−	0.993665i	$-0.464151\pi$
$613$	5.56507	0.224771	0.112386	+	0.993665i	$0.464151\pi$
$614$	0	0
$615$	−0.401960	−0.0162086
$616$	0	0
$617$	−21.8623	−0.880141	−0.440071	−	0.897963i	$-0.645047\pi$
$617$	−21.8623	−0.880141	−0.440071	+	0.897963i	$0.645047\pi$
$618$	0	0
$619$	−16.6118	−0.667685	−0.333842	−	0.942629i	$-0.608345\pi$
$619$	−16.6118	−0.667685	−0.333842	+	0.942629i	$0.608345\pi$
$620$	0	0
$621$	6.38740	0.256317
$622$	0	0
$623$	−8.78585	−0.351998
$624$	0	0
$625$	23.8369	0.953477
$626$	0	0
$627$	−19.6059	−0.782985
$628$	0	0
$629$	16.3718	0.652785
$630$	0	0
$631$	3.57830	0.142450	0.0712249	−	0.997460i	$-0.477309\pi$
$631$	3.57830	0.142450	0.0712249	+	0.997460i	$0.477309\pi$
$632$	0	0
$633$	−10.8224	−0.430153
$634$	0	0
$635$	−3.14734	−0.124898
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−31.1542	−1.23244
$640$	0	0
$641$	−31.0054	−1.22464	−0.612319	−	0.790611i	$-0.709763\pi$
$641$	−31.0054	−1.22464	−0.612319	+	0.790611i	$0.709763\pi$
$642$	0	0
$643$	27.6660	1.09104	0.545520	−	0.838098i	$-0.316332\pi$
$643$	27.6660	1.09104	0.545520	+	0.838098i	$0.316332\pi$
$644$	0	0
$645$	−1.49055	−0.0586905
$646$	0	0
$647$	−3.56435	−0.140129	−0.0700645	−	0.997542i	$-0.522321\pi$
$647$	−3.56435	−0.140129	−0.0700645	+	0.997542i	$0.522321\pi$
$648$	0	0
$649$	5.45723	0.214215
$650$	0	0
$651$	19.9084	0.780271
$652$	0	0
$653$	23.2536	0.909983	0.454991	−	0.890496i	$-0.349642\pi$
$653$	23.2536	0.909983	0.454991	+	0.890496i	$0.349642\pi$
$654$	0	0
$655$	0.211281	0.00825545
$656$	0	0
$657$	−16.4761	−0.642795
$658$	0	0
$659$	−17.8541	−0.695498	−0.347749	−	0.937588i	$-0.613054\pi$
$659$	−17.8541	−0.695498	−0.347749	+	0.937588i	$0.613054\pi$
$660$	0	0
$661$	43.8439	1.70533	0.852666	−	0.522457i	$-0.174984\pi$
$661$	43.8439	1.70533	0.852666	+	0.522457i	$0.174984\pi$
$662$	0	0
$663$	−5.09689	−0.197947
$664$	0	0
$665$	2.38790	0.0925989
$666$	0	0
$667$	−30.2223	−1.17021
$668$	0	0
$669$	33.8462	1.30857
$670$	0	0
$671$	0.113940	0.00439862
$672$	0	0
$673$	−19.5097	−0.752046	−0.376023	−	0.926610i	$-0.622709\pi$
$673$	−19.5097	−0.752046	−0.376023	+	0.926610i	$0.622709\pi$
$674$	0	0
$675$	8.41249	0.323797
$676$	0	0
$677$	35.4639	1.36299	0.681494	−	0.731824i	$-0.261331\pi$
$677$	35.4639	1.36299	0.681494	+	0.731824i	$0.261331\pi$
$678$	0	0
$679$	9.03350	0.346674
$680$	0	0
$681$	−28.6096	−1.09632
$682$	0	0
$683$	18.7385	0.717007	0.358504	−	0.933528i	$-0.383287\pi$
$683$	18.7385	0.717007	0.358504	+	0.933528i	$0.383287\pi$
$684$	0	0
$685$	−2.16939	−0.0828881
$686$	0	0
$687$	−44.6146	−1.70215
$688$	0	0
$689$	−6.34813	−0.241844
$690$	0	0
$691$	−31.2763	−1.18981	−0.594904	−	0.803797i	$-0.702810\pi$
$691$	−31.2763	−1.18981	−0.594904	+	0.803797i	$0.702810\pi$
$692$	0	0
$693$	−2.25438	−0.0856368
$694$	0	0
$695$	−3.84409	−0.145815
$696$	0	0
$697$	−1.39661	−0.0529005
$698$	0	0
$699$	3.00221	0.113554
$700$	0	0
$701$	−1.79838	−0.0679239	−0.0339620	−	0.999423i	$-0.510813\pi$
$701$	−1.79838	−0.0679239	−0.0339620	+	0.999423i	$0.510813\pi$
$702$	0	0
$703$	−62.9763	−2.37520
$704$	0	0
$705$	−2.39165	−0.0900747
$706$	0	0
$707$	−1.56739	−0.0589479
$708$	0	0
$709$	−5.22509	−0.196232	−0.0981162	−	0.995175i	$-0.531282\pi$
$709$	−5.22509	−0.196232	−0.0981162	+	0.995175i	$0.531282\pi$
$710$	0	0
$711$	−28.9054	−1.08404
$712$	0	0
$713$	32.4579	1.21556
$714$	0	0
$715$	−0.279184	−0.0104409
$716$	0	0
$717$	−39.5349	−1.47646
$718$	0	0
$719$	−28.8074	−1.07433	−0.537167	−	0.843476i	$-0.680505\pi$
$719$	−28.8074	−1.07433	−0.537167	+	0.843476i	$0.680505\pi$
$720$	0	0
$721$	5.72438	0.213187
$722$	0	0
$723$	−5.74210	−0.213551
$724$	0	0
$725$	−39.8041	−1.47829
$726$	0	0
$727$	12.5656	0.466033	0.233016	−	0.972473i	$-0.425140\pi$
$727$	12.5656	0.466033	0.233016	+	0.972473i	$0.425140\pi$
$728$	0	0
$729$	−12.3255	−0.456501
$730$	0	0
$731$	−5.17894	−0.191550
$732$	0	0
$733$	26.1826	0.967075	0.483538	−	0.875324i	$-0.339352\pi$
$733$	26.1826	0.967075	0.483538	+	0.875324i	$0.339352\pi$
$734$	0	0
$735$	0.639958	0.0236052
$736$	0	0
$737$	−2.41610	−0.0889983
$738$	0	0
$739$	43.0901	1.58509	0.792547	−	0.609811i	$-0.208755\pi$
$739$	43.0901	1.58509	0.792547	+	0.609811i	$0.208755\pi$
$740$	0	0
$741$	19.6059	0.720241
$742$	0	0
$743$	16.2792	0.597226	0.298613	−	0.954374i	$-0.403476\pi$
$743$	16.2792	0.597226	0.298613	+	0.954374i	$0.403476\pi$
$744$	0	0
$745$	1.89361	0.0693764
$746$	0	0
$747$	17.2287	0.630366
$748$	0	0
$749$	0.400752	0.0146431
$750$	0	0
$751$	28.8174	1.05156	0.525780	−	0.850621i	$-0.323773\pi$
$751$	28.8174	1.05156	0.525780	+	0.850621i	$0.323773\pi$
$752$	0	0
$753$	−3.78901	−0.138079
$754$	0	0
$755$	−4.35050	−0.158331
$756$	0	0
$757$	−22.0739	−0.802290	−0.401145	−	0.916015i	$-0.631388\pi$
$757$	−22.0739	−0.802290	−0.401145	+	0.916015i	$0.631388\pi$
$758$	0	0
$759$	−8.56656	−0.310947
$760$	0	0
$761$	16.4069	0.594748	0.297374	−	0.954761i	$-0.403889\pi$
$761$	16.4069	0.594748	0.297374	+	0.954761i	$0.403889\pi$
$762$	0	0
$763$	20.3077	0.735187
$764$	0	0
$765$	1.39947	0.0505978
$766$	0	0
$767$	−5.45723	−0.197049
$768$	0	0
$769$	−23.4283	−0.844846	−0.422423	−	0.906399i	$-0.638820\pi$
$769$	−23.4283	−0.844846	−0.422423	+	0.906399i	$0.638820\pi$
$770$	0	0
$771$	41.5930	1.49793
$772$	0	0
$773$	13.7376	0.494106	0.247053	−	0.969002i	$-0.420538\pi$
$773$	13.7376	0.494106	0.247053	+	0.969002i	$0.420538\pi$
$774$	0	0
$775$	42.7486	1.53557
$776$	0	0
$777$	−16.8776	−0.605482
$778$	0	0
$779$	5.37227	0.192482
$780$	0	0
$781$	−13.8194	−0.494497
$782$	0	0
$783$	−13.8216	−0.493945
$784$	0	0
$785$	0.410188	0.0146402
$786$	0	0
$787$	−39.8352	−1.41997	−0.709986	−	0.704215i	$-0.751299\pi$
$787$	−39.8352	−1.41997	−0.709986	+	0.704215i	$0.751299\pi$
$788$	0	0
$789$	24.9821	0.889386
$790$	0	0
$791$	12.1185	0.430885
$792$	0	0
$793$	−0.113940	−0.00404614
$794$	0	0
$795$	4.06253	0.144083
$796$	0	0
$797$	23.7805	0.842349	0.421174	−	0.906980i	$-0.361618\pi$
$797$	23.7805	0.842349	0.421174	+	0.906980i	$0.361618\pi$
$798$	0	0
$799$	−8.30980	−0.293979
$800$	0	0
$801$	19.8067	0.699834
$802$	0	0
$803$	−7.30849	−0.257911
$804$	0	0
$805$	1.04337	0.0367738
$806$	0	0
$807$	49.1504	1.73017
$808$	0	0
$809$	−24.7253	−0.869294	−0.434647	−	0.900601i	$-0.643127\pi$
$809$	−24.7253	−0.869294	−0.434647	+	0.900601i	$0.643127\pi$
$810$	0	0
$811$	29.7470	1.04456	0.522280	−	0.852774i	$-0.325082\pi$
$811$	29.7470	1.04456	0.522280	+	0.852774i	$0.325082\pi$
$812$	0	0
$813$	−43.2255	−1.51599
$814$	0	0
$815$	3.28514	0.115073
$816$	0	0
$817$	19.9215	0.696966
$818$	0	0
$819$	2.25438	0.0787744
$820$	0	0
$821$	18.2476	0.636847	0.318423	−	0.947949i	$-0.396847\pi$
$821$	18.2476	0.636847	0.318423	+	0.947949i	$0.396847\pi$
$822$	0	0
$823$	41.8020	1.45713	0.728563	−	0.684979i	$-0.240189\pi$
$823$	41.8020	1.45713	0.728563	+	0.684979i	$0.240189\pi$
$824$	0	0
$825$	−11.2826	−0.392808
$826$	0	0
$827$	−34.0851	−1.18526	−0.592628	−	0.805477i	$-0.701909\pi$
$827$	−34.0851	−1.18526	−0.592628	+	0.805477i	$0.701909\pi$
$828$	0	0
$829$	31.9074	1.10819	0.554095	−	0.832454i	$-0.313064\pi$
$829$	31.9074	1.10819	0.554095	+	0.832454i	$0.313064\pi$
$830$	0	0
$831$	0.430142	0.0149215
$832$	0	0
$833$	2.22354	0.0770410
$834$	0	0
$835$	2.99282	0.103571
$836$	0	0
$837$	14.8441	0.513086
$838$	0	0
$839$	−32.9192	−1.13650	−0.568248	−	0.822857i	$-0.692379\pi$
$839$	−32.9192	−1.13650	−0.568248	+	0.822857i	$0.692379\pi$
$840$	0	0
$841$	36.3979	1.25510
$842$	0	0
$843$	−72.0579	−2.48180
$844$	0	0
$845$	0.279184	0.00960422
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−4.40933	−0.151328
$850$	0	0
$851$	−27.5167	−0.943262
$852$	0	0
$853$	−37.3653	−1.27936	−0.639681	−	0.768640i	$-0.720934\pi$
$853$	−37.3653	−1.27936	−0.639681	+	0.768640i	$0.720934\pi$
$854$	0	0
$855$	−5.38324	−0.184103
$856$	0	0
$857$	10.1339	0.346169	0.173085	−	0.984907i	$-0.444627\pi$
$857$	10.1339	0.346169	0.173085	+	0.984907i	$0.444627\pi$
$858$	0	0
$859$	−18.7270	−0.638958	−0.319479	−	0.947593i	$-0.603508\pi$
$859$	−18.7270	−0.638958	−0.319479	+	0.947593i	$0.603508\pi$
$860$	0	0
$861$	1.43977	0.0490671
$862$	0	0
$863$	3.82969	0.130364	0.0651821	−	0.997873i	$-0.479237\pi$
$863$	3.82969	0.130364	0.0651821	+	0.997873i	$0.479237\pi$
$864$	0	0
$865$	1.16472	0.0396016
$866$	0	0
$867$	−27.6350	−0.938534
$868$	0	0
$869$	−12.8219	−0.434952
$870$	0	0
$871$	2.41610	0.0818665
$872$	0	0
$873$	−20.3649	−0.689249
$874$	0	0
$875$	2.77008	0.0936458
$876$	0	0
$877$	28.2904	0.955300	0.477650	−	0.878550i	$-0.341489\pi$
$877$	28.2904	0.955300	0.477650	+	0.878550i	$0.341489\pi$
$878$	0	0
$879$	34.8935	1.17693
$880$	0	0
$881$	38.8110	1.30757	0.653787	−	0.756678i	$-0.273179\pi$
$881$	38.8110	1.30757	0.653787	+	0.756678i	$0.273179\pi$
$882$	0	0
$883$	15.9988	0.538402	0.269201	−	0.963084i	$-0.413240\pi$
$883$	15.9988	0.538402	0.269201	+	0.963084i	$0.413240\pi$
$884$	0	0
$885$	3.49240	0.117396
$886$	0	0
$887$	10.4734	0.351663	0.175832	−	0.984420i	$-0.443739\pi$
$887$	10.4734	0.351663	0.175832	+	0.984420i	$0.443739\pi$
$888$	0	0
$889$	11.2734	0.378096
$890$	0	0
$891$	−10.6809	−0.357824
$892$	0	0
$893$	31.9648	1.06966
$894$	0	0
$895$	4.33934	0.145048
$896$	0	0
$897$	8.56656	0.286029
$898$	0	0
$899$	−70.2355	−2.34248
$900$	0	0
$901$	14.1153	0.470249
$902$	0	0
$903$	5.33896	0.177670
$904$	0	0
$905$	6.40095	0.212775
$906$	0	0
$907$	1.14389	0.0379823	0.0189912	−	0.999820i	$-0.493955\pi$
$907$	1.14389	0.0379823	0.0189912	+	0.999820i	$0.493955\pi$
$908$	0	0
$909$	3.53350	0.117199
$910$	0	0
$911$	−40.1015	−1.32862	−0.664312	−	0.747456i	$-0.731275\pi$
$911$	−40.1015	−1.32862	−0.664312	+	0.747456i	$0.731275\pi$
$912$	0	0
$913$	7.64233	0.252924
$914$	0	0
$915$	0.0729171	0.00241056
$916$	0	0
$917$	−0.756782	−0.0249911
$918$	0	0
$919$	35.0200	1.15520	0.577601	−	0.816319i	$-0.303989\pi$
$919$	35.0200	1.15520	0.577601	+	0.816319i	$0.303989\pi$
$920$	0	0
$921$	−66.5430	−2.19267
$922$	0	0
$923$	13.8194	0.454872
$924$	0	0
$925$	−36.2408	−1.19159
$926$	0	0
$927$	−12.9049	−0.423853
$928$	0	0
$929$	−28.5737	−0.937471	−0.468736	−	0.883338i	$-0.655290\pi$
$929$	−28.5737	−0.937471	−0.468736	+	0.883338i	$0.655290\pi$
$930$	0	0
$931$	−8.55315	−0.280318
$932$	0	0
$933$	43.3049	1.41774
$934$	0	0
$935$	0.620776	0.0203015
$936$	0	0
$937$	0.618145	0.0201939	0.0100970	−	0.999949i	$-0.496786\pi$
$937$	0.618145	0.0201939	0.0100970	+	0.999949i	$0.496786\pi$
$938$	0	0
$939$	−27.0913	−0.884091
$940$	0	0
$941$	−1.65313	−0.0538905	−0.0269453	−	0.999637i	$-0.508578\pi$
$941$	−1.65313	−0.0538905	−0.0269453	+	0.999637i	$0.508578\pi$
$942$	0	0
$943$	2.34735	0.0764402
$944$	0	0
$945$	0.477165	0.0155222
$946$	0	0
$947$	25.0407	0.813713	0.406857	−	0.913492i	$-0.366625\pi$
$947$	25.0407	0.813713	0.406857	+	0.913492i	$0.366625\pi$
$948$	0	0
$949$	7.30849	0.237244
$950$	0	0
$951$	68.3797	2.21736
$952$	0	0
$953$	−46.9384	−1.52048	−0.760242	−	0.649640i	$-0.774920\pi$
$953$	−46.9384	−1.52048	−0.760242	+	0.649640i	$0.774920\pi$
$954$	0	0
$955$	1.69737	0.0549256
$956$	0	0
$957$	18.5371	0.599220
$958$	0	0
$959$	7.77046	0.250921
$960$	0	0
$961$	44.4310	1.43326
$962$	0	0
$963$	−0.903447	−0.0291132
$964$	0	0
$965$	0.210307	0.00677001
$966$	0	0
$967$	−19.5619	−0.629067	−0.314534	−	0.949246i	$-0.601848\pi$
$967$	−19.5619	−0.629067	−0.314534	+	0.949246i	$0.601848\pi$
$968$	0	0
$969$	−43.5945	−1.40046
$970$	0	0
$971$	20.3428	0.652832	0.326416	−	0.945226i	$-0.394159\pi$
$971$	20.3428	0.652832	0.326416	+	0.945226i	$0.394159\pi$
$972$	0	0
$973$	13.7690	0.441414
$974$	0	0
$975$	11.2826	0.361331
$976$	0	0
$977$	23.5135	0.752264	0.376132	−	0.926566i	$-0.377254\pi$
$977$	23.5135	0.752264	0.376132	+	0.926566i	$0.377254\pi$
$978$	0	0
$979$	8.78585	0.280797
$980$	0	0
$981$	−45.7812	−1.46168
$982$	0	0
$983$	−48.2794	−1.53987	−0.769936	−	0.638121i	$-0.779712\pi$
$983$	−48.2794	−1.53987	−0.769936	+	0.638121i	$0.779712\pi$
$984$	0	0
$985$	1.13987	0.0363192
$986$	0	0
$987$	8.56656	0.272677
$988$	0	0
$989$	8.70447	0.276786
$990$	0	0
$991$	−15.5806	−0.494933	−0.247467	−	0.968896i	$-0.579598\pi$
$991$	−15.5806	−0.494933	−0.247467	+	0.968896i	$0.579598\pi$
$992$	0	0
$993$	1.22905	0.0390027
$994$	0	0
$995$	7.02275	0.222636
$996$	0	0
$997$	−3.57523	−0.113229	−0.0566144	−	0.998396i	$-0.518031\pi$
$997$	−3.57523	−0.113229	−0.0566144	+	0.998396i	$0.518031\pi$
$998$	0	0
$999$	−12.5843	−0.398150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.v.1.10	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.v.1.10	✓	11	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+2.29224 q^{3} +0.279184 q^{5} -1.00000 q^{7} +2.25438 q^{9} +O(q^{10})\)	\(q+2.29224 q^{3} +0.279184 q^{5} -1.00000 q^{7} +2.25438 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.639958 q^{15} +2.22354 q^{17} -8.55315 q^{19} -2.29224 q^{21} -3.73720 q^{23} -4.92206 q^{25} -1.70914 q^{27} +8.08689 q^{29} -8.68510 q^{31} +2.29224 q^{33} -0.279184 q^{35} +7.36294 q^{37} -2.29224 q^{39} -0.628104 q^{41} -2.32914 q^{43} +0.629387 q^{45} -3.73720 q^{47} +1.00000 q^{49} +5.09689 q^{51} +6.34813 q^{53} +0.279184 q^{55} -19.6059 q^{57} +5.45723 q^{59} +0.113940 q^{61} -2.25438 q^{63} -0.279184 q^{65} -2.41610 q^{67} -8.56656 q^{69} -13.8194 q^{71} -7.30849 q^{73} -11.2826 q^{75} -1.00000 q^{77} -12.8219 q^{79} -10.6809 q^{81} +7.64233 q^{83} +0.620776 q^{85} +18.5371 q^{87} +8.78585 q^{89} +1.00000 q^{91} -19.9084 q^{93} -2.38790 q^{95} -9.03350 q^{97} +2.25438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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