LMFDB - Embedded newform 8008.2.a.y.1.14

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + \cdots - 128 $ x^14 - 3x^13 - 27x^12 + 78x^11 + 273x^10 - 750x^9 - 1306x^8 + 3378x^7 + 2996x^6 - 7275x^5 - 2804x^4 + 6417x^3 + 538x^2 - 1032*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-3.22058$ 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)

$f(q)$	$=$	$q+3.22058 q^{3} -3.58838 q^{5} +1.00000 q^{7} +7.37212 q^{9} +O(q^{10})$	$q+3.22058 q^{3} -3.58838 q^{5} +1.00000 q^{7} +7.37212 q^{9} -1.00000 q^{11} -1.00000 q^{13} -11.5566 q^{15} -4.82146 q^{17} -7.88691 q^{19} +3.22058 q^{21} +4.80149 q^{23} +7.87644 q^{25} +14.0808 q^{27} +9.62720 q^{29} -5.44581 q^{31} -3.22058 q^{33} -3.58838 q^{35} +0.187963 q^{37} -3.22058 q^{39} +1.50581 q^{41} -9.53445 q^{43} -26.4539 q^{45} -3.00380 q^{47} +1.00000 q^{49} -15.5279 q^{51} -2.34802 q^{53} +3.58838 q^{55} -25.4004 q^{57} -4.59689 q^{59} +5.05035 q^{61} +7.37212 q^{63} +3.58838 q^{65} -12.9657 q^{67} +15.4636 q^{69} -13.4722 q^{71} +12.2013 q^{73} +25.3667 q^{75} -1.00000 q^{77} +11.0938 q^{79} +23.2318 q^{81} -13.8010 q^{83} +17.3012 q^{85} +31.0052 q^{87} -9.44120 q^{89} -1.00000 q^{91} -17.5387 q^{93} +28.3012 q^{95} -0.672558 q^{97} -7.37212 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * 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
$3$	3.22058	1.85940	0.929701	−	0.368316i	$-0.120065\pi$
$3$	3.22058	1.85940	0.929701	+	0.368316i	$0.120065\pi$
$4$	0	0
$5$	−3.58838	−1.60477	−0.802385	−	0.596806i	$-0.796436\pi$
$5$	−3.58838	−1.60477	−0.802385	+	0.596806i	$0.796436\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	7.37212	2.45737
$10$	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
$15$	−11.5566	−2.98391
$16$	0	0
$17$	−4.82146	−1.16938	−0.584688	−	0.811258i	$-0.698783\pi$
$17$	−4.82146	−1.16938	−0.584688	+	0.811258i	$0.698783\pi$
$18$	0	0
$19$	−7.88691	−1.80938	−0.904690	−	0.426070i	$-0.859898\pi$
$19$	−7.88691	−1.80938	−0.904690	+	0.426070i	$0.859898\pi$
$20$	0	0
$21$	3.22058	0.702788
$22$	0	0
$23$	4.80149	1.00118	0.500590	−	0.865685i	$-0.333116\pi$
$23$	4.80149	1.00118	0.500590	+	0.865685i	$0.333116\pi$
$24$	0	0
$25$	7.87644	1.57529
$26$	0	0
$27$	14.0808	2.70984
$28$	0	0
$29$	9.62720	1.78773	0.893863	−	0.448340i	$-0.147985\pi$
$29$	9.62720	1.78773	0.893863	+	0.448340i	$0.147985\pi$
$30$	0	0
$31$	−5.44581	−0.978096	−0.489048	−	0.872257i	$-0.662656\pi$
$31$	−5.44581	−0.978096	−0.489048	+	0.872257i	$0.662656\pi$
$32$	0	0
$33$	−3.22058	−0.560631
$34$	0	0
$35$	−3.58838	−0.606546
$36$	0	0
$37$	0.187963	0.0309009	0.0154504	−	0.999881i	$-0.495082\pi$
$37$	0.187963	0.0309009	0.0154504	+	0.999881i	$0.495082\pi$
$38$	0	0
$39$	−3.22058	−0.515705
$40$	0	0
$41$	1.50581	0.235169	0.117584	−	0.993063i	$-0.462485\pi$
$41$	1.50581	0.235169	0.117584	+	0.993063i	$0.462485\pi$
$42$	0	0
$43$	−9.53445	−1.45399	−0.726995	−	0.686643i	$-0.759084\pi$
$43$	−9.53445	−1.45399	−0.726995	+	0.686643i	$0.759084\pi$
$44$	0	0
$45$	−26.4539	−3.94352
$46$	0	0
$47$	−3.00380	−0.438150	−0.219075	−	0.975708i	$-0.570304\pi$
$47$	−3.00380	−0.438150	−0.219075	+	0.975708i	$0.570304\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−15.5279	−2.17434
$52$	0	0
$53$	−2.34802	−0.322526	−0.161263	−	0.986911i	$-0.551557\pi$
$53$	−2.34802	−0.322526	−0.161263	+	0.986911i	$0.551557\pi$
$54$	0	0
$55$	3.58838	0.483857
$56$	0	0
$57$	−25.4004	−3.36436
$58$	0	0
$59$	−4.59689	−0.598464	−0.299232	−	0.954180i	$-0.596731\pi$
$59$	−4.59689	−0.598464	−0.299232	+	0.954180i	$0.596731\pi$
$60$	0	0
$61$	5.05035	0.646632	0.323316	−	0.946291i	$-0.395202\pi$
$61$	5.05035	0.646632	0.323316	+	0.946291i	$0.395202\pi$
$62$	0	0
$63$	7.37212	0.928800
$64$	0	0
$65$	3.58838	0.445083
$66$	0	0
$67$	−12.9657	−1.58401	−0.792006	−	0.610513i	$-0.790963\pi$
$67$	−12.9657	−1.58401	−0.792006	+	0.610513i	$0.790963\pi$
$68$	0	0
$69$	15.4636	1.86160
$70$	0	0
$71$	−13.4722	−1.59886	−0.799430	−	0.600759i	$-0.794865\pi$
$71$	−13.4722	−1.59886	−0.799430	+	0.600759i	$0.794865\pi$
$72$	0	0
$73$	12.2013	1.42805	0.714027	−	0.700118i	$-0.246869\pi$
$73$	12.2013	1.42805	0.714027	+	0.700118i	$0.246869\pi$
$74$	0	0
$75$	25.3667	2.92909
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	11.0938	1.24815	0.624073	−	0.781366i	$-0.285477\pi$
$79$	11.0938	1.24815	0.624073	+	0.781366i	$0.285477\pi$
$80$	0	0
$81$	23.2318	2.58131
$82$	0	0
$83$	−13.8010	−1.51485	−0.757427	−	0.652920i	$-0.773544\pi$
$83$	−13.8010	−1.51485	−0.757427	+	0.652920i	$0.773544\pi$
$84$	0	0
$85$	17.3012	1.87658
$86$	0	0
$87$	31.0052	3.32410
$88$	0	0
$89$	−9.44120	−1.00077	−0.500383	−	0.865804i	$-0.666807\pi$
$89$	−9.44120	−1.00077	−0.500383	+	0.865804i	$0.666807\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−17.5387	−1.81867
$94$	0	0
$95$	28.3012	2.90364
$96$	0	0
$97$	−0.672558	−0.0682879	−0.0341439	−	0.999417i	$-0.510870\pi$
$97$	−0.672558	−0.0682879	−0.0341439	+	0.999417i	$0.510870\pi$
$98$	0	0
$99$	−7.37212	−0.740926
$100$	0	0
$101$	−16.9285	−1.68445	−0.842223	−	0.539130i	$-0.818753\pi$
$101$	−16.9285	−1.68445	−0.842223	+	0.539130i	$0.818753\pi$
$102$	0	0
$103$	−9.87422	−0.972935	−0.486468	−	0.873699i	$-0.661715\pi$
$103$	−9.87422	−0.972935	−0.486468	+	0.873699i	$0.661715\pi$
$104$	0	0
$105$	−11.5566	−1.12781
$106$	0	0
$107$	−8.81921	−0.852585	−0.426292	−	0.904585i	$-0.640181\pi$
$107$	−8.81921	−0.852585	−0.426292	+	0.904585i	$0.640181\pi$
$108$	0	0
$109$	19.4616	1.86409	0.932043	−	0.362347i	$-0.118024\pi$
$109$	19.4616	1.86409	0.932043	+	0.362347i	$0.118024\pi$
$110$	0	0
$111$	0.605348	0.0574571
$112$	0	0
$113$	−18.4208	−1.73289	−0.866444	−	0.499275i	$-0.833600\pi$
$113$	−18.4208	−1.73289	−0.866444	+	0.499275i	$0.833600\pi$
$114$	0	0
$115$	−17.2296	−1.60666
$116$	0	0
$117$	−7.37212	−0.681553
$118$	0	0
$119$	−4.82146	−0.441982
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.84959	0.437273
$124$	0	0
$125$	−10.3218	−0.923207
$126$	0	0
$127$	−8.56887	−0.760364	−0.380182	−	0.924912i	$-0.624139\pi$
$127$	−8.56887	−0.760364	−0.380182	+	0.924912i	$0.624139\pi$
$128$	0	0
$129$	−30.7064	−2.70355
$130$	0	0
$131$	10.7365	0.938052	0.469026	−	0.883184i	$-0.344605\pi$
$131$	10.7365	0.938052	0.469026	+	0.883184i	$0.344605\pi$
$132$	0	0
$133$	−7.88691	−0.683882
$134$	0	0
$135$	−50.5271	−4.34868
$136$	0	0
$137$	−4.31232	−0.368427	−0.184213	−	0.982886i	$-0.558974\pi$
$137$	−4.31232	−0.368427	−0.184213	+	0.982886i	$0.558974\pi$
$138$	0	0
$139$	−5.63319	−0.477801	−0.238900	−	0.971044i	$-0.576787\pi$
$139$	−5.63319	−0.477801	−0.238900	+	0.971044i	$0.576787\pi$
$140$	0	0
$141$	−9.67398	−0.814696
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−34.5460	−2.86889
$146$	0	0
$147$	3.22058	0.265629
$148$	0	0
$149$	17.7571	1.45471	0.727357	−	0.686259i	$-0.240748\pi$
$149$	17.7571	1.45471	0.727357	+	0.686259i	$0.240748\pi$
$150$	0	0
$151$	−7.42961	−0.604613	−0.302307	−	0.953211i	$-0.597757\pi$
$151$	−7.42961	−0.604613	−0.302307	+	0.953211i	$0.597757\pi$
$152$	0	0
$153$	−35.5444	−2.87359
$154$	0	0
$155$	19.5416	1.56962
$156$	0	0
$157$	−22.8574	−1.82422	−0.912111	−	0.409944i	$-0.865548\pi$
$157$	−22.8574	−1.82422	−0.912111	+	0.409944i	$0.865548\pi$
$158$	0	0
$159$	−7.56200	−0.599705
$160$	0	0
$161$	4.80149	0.378410
$162$	0	0
$163$	−11.0878	−0.868463	−0.434232	−	0.900801i	$-0.642980\pi$
$163$	−11.0878	−0.868463	−0.434232	+	0.900801i	$0.642980\pi$
$164$	0	0
$165$	11.5566	0.899684
$166$	0	0
$167$	−17.3294	−1.34099	−0.670495	−	0.741914i	$-0.733918\pi$
$167$	−17.3294	−1.34099	−0.670495	+	0.741914i	$0.733918\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−58.1432	−4.44632
$172$	0	0
$173$	1.42083	0.108024	0.0540118	−	0.998540i	$-0.482799\pi$
$173$	1.42083	0.108024	0.0540118	+	0.998540i	$0.482799\pi$
$174$	0	0
$175$	7.87644	0.595403
$176$	0	0
$177$	−14.8046	−1.11279
$178$	0	0
$179$	−8.73218	−0.652674	−0.326337	−	0.945254i	$-0.605814\pi$
$179$	−8.73218	−0.652674	−0.326337	+	0.945254i	$0.605814\pi$
$180$	0	0
$181$	−2.81515	−0.209249	−0.104624	−	0.994512i	$-0.533364\pi$
$181$	−2.81515	−0.209249	−0.104624	+	0.994512i	$0.533364\pi$
$182$	0	0
$183$	16.2651	1.20235
$184$	0	0
$185$	−0.674480	−0.0495888
$186$	0	0
$187$	4.82146	0.352580
$188$	0	0
$189$	14.0808	1.02422
$190$	0	0
$191$	26.2757	1.90125	0.950623	−	0.310349i	$-0.100446\pi$
$191$	26.2757	1.90125	0.950623	+	0.310349i	$0.100446\pi$
$192$	0	0
$193$	21.2260	1.52788	0.763941	−	0.645286i	$-0.223262\pi$
$193$	21.2260	1.52788	0.763941	+	0.645286i	$0.223262\pi$
$194$	0	0
$195$	11.5566	0.827589
$196$	0	0
$197$	−3.19961	−0.227963	−0.113982	−	0.993483i	$-0.536360\pi$
$197$	−3.19961	−0.227963	−0.113982	+	0.993483i	$0.536360\pi$
$198$	0	0
$199$	6.44127	0.456609	0.228305	−	0.973590i	$-0.426682\pi$
$199$	6.44127	0.456609	0.228305	+	0.973590i	$0.426682\pi$
$200$	0	0
$201$	−41.7570	−2.94532
$202$	0	0
$203$	9.62720	0.675697
$204$	0	0
$205$	−5.40342	−0.377392
$206$	0	0
$207$	35.3972	2.46027
$208$	0	0
$209$	7.88691	0.545549
$210$	0	0
$211$	−4.29768	−0.295864	−0.147932	−	0.988998i	$-0.547262\pi$
$211$	−4.29768	−0.295864	−0.147932	+	0.988998i	$0.547262\pi$
$212$	0	0
$213$	−43.3884	−2.97292
$214$	0	0
$215$	34.2132	2.33332
$216$	0	0
$217$	−5.44581	−0.369686
$218$	0	0
$219$	39.2952	2.65532
$220$	0	0
$221$	4.82146	0.324326
$222$	0	0
$223$	17.0231	1.13995	0.569974	−	0.821662i	$-0.306953\pi$
$223$	17.0231	1.13995	0.569974	+	0.821662i	$0.306953\pi$
$224$	0	0
$225$	58.0661	3.87107
$226$	0	0
$227$	8.75499	0.581089	0.290545	−	0.956861i	$-0.406164\pi$
$227$	8.75499	0.581089	0.290545	+	0.956861i	$0.406164\pi$
$228$	0	0
$229$	−2.48184	−0.164005	−0.0820023	−	0.996632i	$-0.526131\pi$
$229$	−2.48184	−0.164005	−0.0820023	+	0.996632i	$0.526131\pi$
$230$	0	0
$231$	−3.22058	−0.211898
$232$	0	0
$233$	−28.5143	−1.86804	−0.934018	−	0.357225i	$-0.883723\pi$
$233$	−28.5143	−1.86804	−0.934018	+	0.357225i	$0.883723\pi$
$234$	0	0
$235$	10.7788	0.703130
$236$	0	0
$237$	35.7283	2.32080
$238$	0	0
$239$	24.4973	1.58460	0.792300	−	0.610131i	$-0.208883\pi$
$239$	24.4973	1.58460	0.792300	+	0.610131i	$0.208883\pi$
$240$	0	0
$241$	−16.3070	−1.05042	−0.525212	−	0.850971i	$-0.676014\pi$
$241$	−16.3070	−1.05042	−0.525212	+	0.850971i	$0.676014\pi$
$242$	0	0
$243$	32.5776	2.08985
$244$	0	0
$245$	−3.58838	−0.229253
$246$	0	0
$247$	7.88691	0.501832
$248$	0	0
$249$	−44.4471	−2.81672
$250$	0	0
$251$	−15.5380	−0.980750	−0.490375	−	0.871511i	$-0.663140\pi$
$251$	−15.5380	−0.980750	−0.490375	+	0.871511i	$0.663140\pi$
$252$	0	0
$253$	−4.80149	−0.301867
$254$	0	0
$255$	55.7199	3.48931
$256$	0	0
$257$	−2.16513	−0.135057	−0.0675285	−	0.997717i	$-0.521511\pi$
$257$	−2.16513	−0.135057	−0.0675285	+	0.997717i	$0.521511\pi$
$258$	0	0
$259$	0.187963	0.0116794
$260$	0	0
$261$	70.9729	4.39311
$262$	0	0
$263$	−24.7513	−1.52623	−0.763117	−	0.646261i	$-0.776332\pi$
$263$	−24.7513	−1.52623	−0.763117	+	0.646261i	$0.776332\pi$
$264$	0	0
$265$	8.42560	0.517580
$266$	0	0
$267$	−30.4061	−1.86083
$268$	0	0
$269$	−8.63021	−0.526193	−0.263097	−	0.964769i	$-0.584744\pi$
$269$	−8.63021	−0.526193	−0.263097	+	0.964769i	$0.584744\pi$
$270$	0	0
$271$	15.5050	0.941860	0.470930	−	0.882171i	$-0.343918\pi$
$271$	15.5050	0.941860	0.470930	+	0.882171i	$0.343918\pi$
$272$	0	0
$273$	−3.22058	−0.194918
$274$	0	0
$275$	−7.87644	−0.474967
$276$	0	0
$277$	−4.26881	−0.256488	−0.128244	−	0.991743i	$-0.540934\pi$
$277$	−4.26881	−0.256488	−0.128244	+	0.991743i	$0.540934\pi$
$278$	0	0
$279$	−40.1472	−2.40355
$280$	0	0
$281$	30.5752	1.82397	0.911983	−	0.410228i	$-0.134551\pi$
$281$	30.5752	1.82397	0.911983	+	0.410228i	$0.134551\pi$
$282$	0	0
$283$	9.50838	0.565215	0.282607	−	0.959236i	$-0.408801\pi$
$283$	9.50838	0.565215	0.282607	+	0.959236i	$0.408801\pi$
$284$	0	0
$285$	91.1462	5.39903
$286$	0	0
$287$	1.50581	0.0888853
$288$	0	0
$289$	6.24647	0.367439
$290$	0	0
$291$	−2.16602	−0.126975
$292$	0	0
$293$	9.94798	0.581167	0.290584	−	0.956850i	$-0.406151\pi$
$293$	9.94798	0.581167	0.290584	+	0.956850i	$0.406151\pi$
$294$	0	0
$295$	16.4954	0.960398
$296$	0	0
$297$	−14.0808	−0.817048
$298$	0	0
$299$	−4.80149	−0.277677
$300$	0	0
$301$	−9.53445	−0.549556
$302$	0	0
$303$	−54.5194	−3.13206
$304$	0	0
$305$	−18.1226	−1.03770
$306$	0	0
$307$	1.51711	0.0865863	0.0432931	−	0.999062i	$-0.486215\pi$
$307$	1.51711	0.0865863	0.0432931	+	0.999062i	$0.486215\pi$
$308$	0	0
$309$	−31.8007	−1.80908
$310$	0	0
$311$	−24.5405	−1.39156	−0.695781	−	0.718254i	$-0.744942\pi$
$311$	−24.5405	−1.39156	−0.695781	+	0.718254i	$0.744942\pi$
$312$	0	0
$313$	−16.8630	−0.953152	−0.476576	−	0.879133i	$-0.658122\pi$
$313$	−16.8630	−0.953152	−0.476576	+	0.879133i	$0.658122\pi$
$314$	0	0
$315$	−26.4539	−1.49051
$316$	0	0
$317$	−4.33688	−0.243583	−0.121792	−	0.992556i	$-0.538864\pi$
$317$	−4.33688	−0.243583	−0.121792	+	0.992556i	$0.538864\pi$
$318$	0	0
$319$	−9.62720	−0.539020
$320$	0	0
$321$	−28.4029	−1.58530
$322$	0	0
$323$	38.0264	2.11585
$324$	0	0
$325$	−7.87644	−0.436906
$326$	0	0
$327$	62.6777	3.46609
$328$	0	0
$329$	−3.00380	−0.165605
$330$	0	0
$331$	5.17809	0.284613	0.142307	−	0.989823i	$-0.454548\pi$
$331$	5.17809	0.284613	0.142307	+	0.989823i	$0.454548\pi$
$332$	0	0
$333$	1.38568	0.0759350
$334$	0	0
$335$	46.5258	2.54198
$336$	0	0
$337$	14.9945	0.816801	0.408401	−	0.912803i	$-0.366087\pi$
$337$	14.9945	0.816801	0.408401	+	0.912803i	$0.366087\pi$
$338$	0	0
$339$	−59.3258	−3.22213
$340$	0	0
$341$	5.44581	0.294907
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−55.4891	−2.98743
$346$	0	0
$347$	11.8587	0.636607	0.318303	−	0.947989i	$-0.396887\pi$
$347$	11.8587	0.636607	0.318303	+	0.947989i	$0.396887\pi$
$348$	0	0
$349$	2.50061	0.133854	0.0669272	−	0.997758i	$-0.478680\pi$
$349$	2.50061	0.133854	0.0669272	+	0.997758i	$0.478680\pi$
$350$	0	0
$351$	−14.0808	−0.751575
$352$	0	0
$353$	−2.57532	−0.137071	−0.0685353	−	0.997649i	$-0.521833\pi$
$353$	−2.57532	−0.137071	−0.0685353	+	0.997649i	$0.521833\pi$
$354$	0	0
$355$	48.3434	2.56580
$356$	0	0
$357$	−15.5279	−0.821823
$358$	0	0
$359$	−27.7361	−1.46386	−0.731928	−	0.681382i	$-0.761379\pi$
$359$	−27.7361	−1.46386	−0.731928	+	0.681382i	$0.761379\pi$
$360$	0	0
$361$	43.2033	2.27386
$362$	0	0
$363$	3.22058	0.169036
$364$	0	0
$365$	−43.7828	−2.29170
$366$	0	0
$367$	−2.26063	−0.118004	−0.0590019	−	0.998258i	$-0.518792\pi$
$367$	−2.26063	−0.118004	−0.0590019	+	0.998258i	$0.518792\pi$
$368$	0	0
$369$	11.1010	0.577897
$370$	0	0
$371$	−2.34802	−0.121903
$372$	0	0
$373$	15.5037	0.802750	0.401375	−	0.915914i	$-0.368532\pi$
$373$	15.5037	0.802750	0.401375	+	0.915914i	$0.368532\pi$
$374$	0	0
$375$	−33.2420	−1.71661
$376$	0	0
$377$	−9.62720	−0.495826
$378$	0	0
$379$	18.4062	0.945462	0.472731	−	0.881207i	$-0.343268\pi$
$379$	18.4062	0.945462	0.472731	+	0.881207i	$0.343268\pi$
$380$	0	0
$381$	−27.5967	−1.41382
$382$	0	0
$383$	21.6395	1.10573	0.552864	−	0.833272i	$-0.313535\pi$
$383$	21.6395	1.10573	0.552864	+	0.833272i	$0.313535\pi$
$384$	0	0
$385$	3.58838	0.182881
$386$	0	0
$387$	−70.2891	−3.57300
$388$	0	0
$389$	−18.7136	−0.948816	−0.474408	−	0.880305i	$-0.657338\pi$
$389$	−18.7136	−0.948816	−0.474408	+	0.880305i	$0.657338\pi$
$390$	0	0
$391$	−23.1502	−1.17076
$392$	0	0
$393$	34.5777	1.74422
$394$	0	0
$395$	−39.8086	−2.00299
$396$	0	0
$397$	−8.96224	−0.449802	−0.224901	−	0.974382i	$-0.572206\pi$
$397$	−8.96224	−0.449802	−0.224901	+	0.974382i	$0.572206\pi$
$398$	0	0
$399$	−25.4004	−1.27161
$400$	0	0
$401$	31.1663	1.55637	0.778186	−	0.628034i	$-0.216140\pi$
$401$	31.1663	1.55637	0.778186	+	0.628034i	$0.216140\pi$
$402$	0	0
$403$	5.44581	0.271275
$404$	0	0
$405$	−83.3645	−4.14241
$406$	0	0
$407$	−0.187963	−0.00931696
$408$	0	0
$409$	22.9295	1.13379	0.566896	−	0.823789i	$-0.308144\pi$
$409$	22.9295	1.13379	0.566896	+	0.823789i	$0.308144\pi$
$410$	0	0
$411$	−13.8882	−0.685053
$412$	0	0
$413$	−4.59689	−0.226198
$414$	0	0
$415$	49.5231	2.43099
$416$	0	0
$417$	−18.1421	−0.888423
$418$	0	0
$419$	8.40201	0.410465	0.205233	−	0.978713i	$-0.434205\pi$
$419$	8.40201	0.410465	0.205233	+	0.978713i	$0.434205\pi$
$420$	0	0
$421$	26.9322	1.31260	0.656298	−	0.754502i	$-0.272122\pi$
$421$	26.9322	1.31260	0.656298	+	0.754502i	$0.272122\pi$
$422$	0	0
$423$	−22.1444	−1.07670
$424$	0	0
$425$	−37.9760	−1.84210
$426$	0	0
$427$	5.05035	0.244404
$428$	0	0
$429$	3.22058	0.155491
$430$	0	0
$431$	−37.7147	−1.81665	−0.908326	−	0.418263i	$-0.862639\pi$
$431$	−37.7147	−1.81665	−0.908326	+	0.418263i	$0.862639\pi$
$432$	0	0
$433$	2.03235	0.0976686	0.0488343	−	0.998807i	$-0.484449\pi$
$433$	2.03235	0.0976686	0.0488343	+	0.998807i	$0.484449\pi$
$434$	0	0
$435$	−111.258	−5.33442
$436$	0	0
$437$	−37.8689	−1.81152
$438$	0	0
$439$	34.1673	1.63072	0.815358	−	0.578957i	$-0.196540\pi$
$439$	34.1673	1.63072	0.815358	+	0.578957i	$0.196540\pi$
$440$	0	0
$441$	7.37212	0.351053
$442$	0	0
$443$	−13.1136	−0.623045	−0.311522	−	0.950239i	$-0.600839\pi$
$443$	−13.1136	−0.623045	−0.311522	+	0.950239i	$0.600839\pi$
$444$	0	0
$445$	33.8786	1.60600
$446$	0	0
$447$	57.1880	2.70490
$448$	0	0
$449$	−19.2230	−0.907189	−0.453595	−	0.891208i	$-0.649859\pi$
$449$	−19.2230	−0.907189	−0.453595	+	0.891208i	$0.649859\pi$
$450$	0	0
$451$	−1.50581	−0.0709060
$452$	0	0
$453$	−23.9276	−1.12422
$454$	0	0
$455$	3.58838	0.168226
$456$	0	0
$457$	−1.52855	−0.0715024	−0.0357512	−	0.999361i	$-0.511382\pi$
$457$	−1.52855	−0.0715024	−0.0357512	+	0.999361i	$0.511382\pi$
$458$	0	0
$459$	−67.8898	−3.16882
$460$	0	0
$461$	5.36243	0.249753	0.124877	−	0.992172i	$-0.460147\pi$
$461$	5.36243	0.249753	0.124877	+	0.992172i	$0.460147\pi$
$462$	0	0
$463$	12.2844	0.570906	0.285453	−	0.958393i	$-0.407856\pi$
$463$	12.2844	0.570906	0.285453	+	0.958393i	$0.407856\pi$
$464$	0	0
$465$	62.9353	2.91855
$466$	0	0
$467$	−17.8548	−0.826221	−0.413111	−	0.910681i	$-0.635558\pi$
$467$	−17.8548	−0.826221	−0.413111	+	0.910681i	$0.635558\pi$
$468$	0	0
$469$	−12.9657	−0.598700
$470$	0	0
$471$	−73.6141	−3.39196
$472$	0	0
$473$	9.53445	0.438394
$474$	0	0
$475$	−62.1208	−2.85030
$476$	0	0
$477$	−17.3099	−0.792567
$478$	0	0
$479$	24.1782	1.10473	0.552365	−	0.833602i	$-0.313725\pi$
$479$	24.1782	1.10473	0.552365	+	0.833602i	$0.313725\pi$
$480$	0	0
$481$	−0.187963	−0.00857036
$482$	0	0
$483$	15.4636	0.703617
$484$	0	0
$485$	2.41339	0.109586
$486$	0	0
$487$	20.1691	0.913950	0.456975	−	0.889480i	$-0.348933\pi$
$487$	20.1691	0.913950	0.456975	+	0.889480i	$0.348933\pi$
$488$	0	0
$489$	−35.7091	−1.61482
$490$	0	0
$491$	−33.2466	−1.50040	−0.750199	−	0.661212i	$-0.770042\pi$
$491$	−33.2466	−1.50040	−0.750199	+	0.661212i	$0.770042\pi$
$492$	0	0
$493$	−46.4172	−2.09052
$494$	0	0
$495$	26.4539	1.18902
$496$	0	0
$497$	−13.4722	−0.604312
$498$	0	0
$499$	16.2438	0.727174	0.363587	−	0.931560i	$-0.381552\pi$
$499$	16.2438	0.727174	0.363587	+	0.931560i	$0.381552\pi$
$500$	0	0
$501$	−55.8107	−2.49344
$502$	0	0
$503$	−5.01690	−0.223692	−0.111846	−	0.993726i	$-0.535676\pi$
$503$	−5.01690	−0.223692	−0.111846	+	0.993726i	$0.535676\pi$
$504$	0	0
$505$	60.7457	2.70315
$506$	0	0
$507$	3.22058	0.143031
$508$	0	0
$509$	4.55308	0.201812	0.100906	−	0.994896i	$-0.467826\pi$
$509$	4.55308	0.201812	0.100906	+	0.994896i	$0.467826\pi$
$510$	0	0
$511$	12.2013	0.539753
$512$	0	0
$513$	−111.054	−4.90314
$514$	0	0
$515$	35.4324	1.56134
$516$	0	0
$517$	3.00380	0.132107
$518$	0	0
$519$	4.57589	0.200859
$520$	0	0
$521$	−0.492144	−0.0215612	−0.0107806	−	0.999942i	$-0.503432\pi$
$521$	−0.492144	−0.0215612	−0.0107806	+	0.999942i	$0.503432\pi$
$522$	0	0
$523$	−25.6861	−1.12317	−0.561587	−	0.827418i	$-0.689809\pi$
$523$	−25.6861	−1.12317	−0.561587	+	0.827418i	$0.689809\pi$
$524$	0	0
$525$	25.3667	1.10709
$526$	0	0
$527$	26.2568	1.14376
$528$	0	0
$529$	0.0543117	0.00236138
$530$	0	0
$531$	−33.8888	−1.47065
$532$	0	0
$533$	−1.50581	−0.0652240
$534$	0	0
$535$	31.6466	1.36820
$536$	0	0
$537$	−28.1227	−1.21358
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−22.1339	−0.951612	−0.475806	−	0.879550i	$-0.657844\pi$
$541$	−22.1339	−0.951612	−0.475806	+	0.879550i	$0.657844\pi$
$542$	0	0
$543$	−9.06642	−0.389077
$544$	0	0
$545$	−69.8357	−2.99143
$546$	0	0
$547$	−3.06333	−0.130979	−0.0654893	−	0.997853i	$-0.520861\pi$
$547$	−3.06333	−0.130979	−0.0654893	+	0.997853i	$0.520861\pi$
$548$	0	0
$549$	37.2318	1.58902
$550$	0	0
$551$	−75.9288	−3.23468
$552$	0	0
$553$	11.0938	0.471755
$554$	0	0
$555$	−2.17222	−0.0922055
$556$	0	0
$557$	−29.2576	−1.23969	−0.619843	−	0.784726i	$-0.712804\pi$
$557$	−29.2576	−1.23969	−0.619843	+	0.784726i	$0.712804\pi$
$558$	0	0
$559$	9.53445	0.403264
$560$	0	0
$561$	15.5279	0.655588
$562$	0	0
$563$	13.6837	0.576701	0.288350	−	0.957525i	$-0.406893\pi$
$563$	13.6837	0.576701	0.288350	+	0.957525i	$0.406893\pi$
$564$	0	0
$565$	66.1009	2.78089
$566$	0	0
$567$	23.2318	0.975644
$568$	0	0
$569$	33.2646	1.39452	0.697261	−	0.716817i	$-0.254402\pi$
$569$	33.2646	1.39452	0.697261	+	0.716817i	$0.254402\pi$
$570$	0	0
$571$	−9.90886	−0.414673	−0.207336	−	0.978270i	$-0.566479\pi$
$571$	−9.90886	−0.414673	−0.207336	+	0.978270i	$0.566479\pi$
$572$	0	0
$573$	84.6231	3.53518
$574$	0	0
$575$	37.8187	1.57715
$576$	0	0
$577$	12.2505	0.509996	0.254998	−	0.966942i	$-0.417925\pi$
$577$	12.2505	0.509996	0.254998	+	0.966942i	$0.417925\pi$
$578$	0	0
$579$	68.3600	2.84095
$580$	0	0
$581$	−13.8010	−0.572561
$582$	0	0
$583$	2.34802	0.0972452
$584$	0	0
$585$	26.4539	1.09374
$586$	0	0
$587$	−34.6030	−1.42822	−0.714110	−	0.700034i	$-0.753168\pi$
$587$	−34.6030	−1.42822	−0.714110	+	0.700034i	$0.753168\pi$
$588$	0	0
$589$	42.9506	1.76975
$590$	0	0
$591$	−10.3046	−0.423875
$592$	0	0
$593$	21.5812	0.886234	0.443117	−	0.896464i	$-0.353873\pi$
$593$	21.5812	0.886234	0.443117	+	0.896464i	$0.353873\pi$
$594$	0	0
$595$	17.3012	0.709280
$596$	0	0
$597$	20.7446	0.849020
$598$	0	0
$599$	7.49682	0.306312	0.153156	−	0.988202i	$-0.451056\pi$
$599$	7.49682	0.306312	0.153156	+	0.988202i	$0.451056\pi$
$600$	0	0
$601$	42.9034	1.75007	0.875034	−	0.484062i	$-0.160839\pi$
$601$	42.9034	1.75007	0.875034	+	0.484062i	$0.160839\pi$
$602$	0	0
$603$	−95.5847	−3.89251
$604$	0	0
$605$	−3.58838	−0.145888
$606$	0	0
$607$	−18.2793	−0.741934	−0.370967	−	0.928646i	$-0.620974\pi$
$607$	−18.2793	−0.741934	−0.370967	+	0.928646i	$0.620974\pi$
$608$	0	0
$609$	31.0052	1.25639
$610$	0	0
$611$	3.00380	0.121521
$612$	0	0
$613$	−13.1318	−0.530387	−0.265193	−	0.964195i	$-0.585436\pi$
$613$	−13.1318	−0.530387	−0.265193	+	0.964195i	$0.585436\pi$
$614$	0	0
$615$	−17.4021	−0.701722
$616$	0	0
$617$	−28.6665	−1.15407	−0.577035	−	0.816719i	$-0.695791\pi$
$617$	−28.6665	−1.15407	−0.577035	+	0.816719i	$0.695791\pi$
$618$	0	0
$619$	−8.00413	−0.321713	−0.160856	−	0.986978i	$-0.551426\pi$
$619$	−8.00413	−0.321713	−0.160856	+	0.986978i	$0.551426\pi$
$620$	0	0
$621$	67.6086	2.71304
$622$	0	0
$623$	−9.44120	−0.378254
$624$	0	0
$625$	−2.34385	−0.0937541
$626$	0	0
$627$	25.4004	1.01439
$628$	0	0
$629$	−0.906254	−0.0361347
$630$	0	0
$631$	−2.94503	−0.117240	−0.0586198	−	0.998280i	$-0.518670\pi$
$631$	−2.94503	−0.117240	−0.0586198	+	0.998280i	$0.518670\pi$
$632$	0	0
$633$	−13.8410	−0.550131
$634$	0	0
$635$	30.7483	1.22021
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−99.3189	−3.92900
$640$	0	0
$641$	6.80551	0.268802	0.134401	−	0.990927i	$-0.457089\pi$
$641$	6.80551	0.268802	0.134401	+	0.990927i	$0.457089\pi$
$642$	0	0
$643$	−5.83786	−0.230223	−0.115111	−	0.993353i	$-0.536722\pi$
$643$	−5.83786	−0.230223	−0.115111	+	0.993353i	$0.536722\pi$
$644$	0	0
$645$	110.186	4.33858
$646$	0	0
$647$	25.5786	1.00560	0.502800	−	0.864403i	$-0.332303\pi$
$647$	25.5786	1.00560	0.502800	+	0.864403i	$0.332303\pi$
$648$	0	0
$649$	4.59689	0.180444
$650$	0	0
$651$	−17.5387	−0.687394
$652$	0	0
$653$	−13.7271	−0.537183	−0.268592	−	0.963254i	$-0.586558\pi$
$653$	−13.7271	−0.537183	−0.268592	+	0.963254i	$0.586558\pi$
$654$	0	0
$655$	−38.5266	−1.50536
$656$	0	0
$657$	89.9494	3.50926
$658$	0	0
$659$	16.9970	0.662109	0.331054	−	0.943612i	$-0.392596\pi$
$659$	16.9970	0.662109	0.331054	+	0.943612i	$0.392596\pi$
$660$	0	0
$661$	−14.9076	−0.579839	−0.289920	−	0.957051i	$-0.593629\pi$
$661$	−14.9076	−0.579839	−0.289920	+	0.957051i	$0.593629\pi$
$662$	0	0
$663$	15.5279	0.603053
$664$	0	0
$665$	28.3012	1.09747
$666$	0	0
$667$	46.2249	1.78984
$668$	0	0
$669$	54.8241	2.11962
$670$	0	0
$671$	−5.05035	−0.194967
$672$	0	0
$673$	43.1293	1.66251	0.831256	−	0.555889i	$-0.187622\pi$
$673$	43.1293	1.66251	0.831256	+	0.555889i	$0.187622\pi$
$674$	0	0
$675$	110.906	4.26879
$676$	0	0
$677$	−22.9025	−0.880213	−0.440107	−	0.897946i	$-0.645059\pi$
$677$	−22.9025	−0.880213	−0.440107	+	0.897946i	$0.645059\pi$
$678$	0	0
$679$	−0.672558	−0.0258104
$680$	0	0
$681$	28.1961	1.08048
$682$	0	0
$683$	30.1717	1.15449	0.577245	−	0.816571i	$-0.304128\pi$
$683$	30.1717	1.15449	0.577245	+	0.816571i	$0.304128\pi$
$684$	0	0
$685$	15.4742	0.591240
$686$	0	0
$687$	−7.99296	−0.304950
$688$	0	0
$689$	2.34802	0.0894526
$690$	0	0
$691$	−43.3464	−1.64897	−0.824487	−	0.565881i	$-0.808536\pi$
$691$	−43.3464	−1.64897	−0.824487	+	0.565881i	$0.808536\pi$
$692$	0	0
$693$	−7.37212	−0.280044
$694$	0	0
$695$	20.2140	0.766760
$696$	0	0
$697$	−7.26022	−0.275000
$698$	0	0
$699$	−91.8326	−3.47343
$700$	0	0
$701$	20.2983	0.766655	0.383327	−	0.923613i	$-0.374778\pi$
$701$	20.2983	0.766655	0.383327	+	0.923613i	$0.374778\pi$
$702$	0	0
$703$	−1.48244	−0.0559114
$704$	0	0
$705$	34.7139	1.30740
$706$	0	0
$707$	−16.9285	−0.636661
$708$	0	0
$709$	−35.3039	−1.32587	−0.662933	−	0.748679i	$-0.730689\pi$
$709$	−35.3039	−1.32587	−0.662933	+	0.748679i	$0.730689\pi$
$710$	0	0
$711$	81.7846	3.06716
$712$	0	0
$713$	−26.1480	−0.979250
$714$	0	0
$715$	−3.58838	−0.134198
$716$	0	0
$717$	78.8956	2.94641
$718$	0	0
$719$	−8.25158	−0.307732	−0.153866	−	0.988092i	$-0.549172\pi$
$719$	−8.25158	−0.307732	−0.153866	+	0.988092i	$0.549172\pi$
$720$	0	0
$721$	−9.87422	−0.367735
$722$	0	0
$723$	−52.5178	−1.95316
$724$	0	0
$725$	75.8281	2.81619
$726$	0	0
$727$	22.8606	0.847852	0.423926	−	0.905697i	$-0.360652\pi$
$727$	22.8606	0.847852	0.423926	+	0.905697i	$0.360652\pi$
$728$	0	0
$729$	35.2232	1.30456
$730$	0	0
$731$	45.9700	1.70026
$732$	0	0
$733$	9.50689	0.351145	0.175572	−	0.984467i	$-0.443822\pi$
$733$	9.50689	0.351145	0.175572	+	0.984467i	$0.443822\pi$
$734$	0	0
$735$	−11.5566	−0.426273
$736$	0	0
$737$	12.9657	0.477598
$738$	0	0
$739$	−19.9610	−0.734277	−0.367139	−	0.930166i	$-0.619663\pi$
$739$	−19.9610	−0.734277	−0.367139	+	0.930166i	$0.619663\pi$
$740$	0	0
$741$	25.4004	0.933107
$742$	0	0
$743$	0.175889	0.00645274	0.00322637	−	0.999995i	$-0.498973\pi$
$743$	0.175889	0.00645274	0.00322637	+	0.999995i	$0.498973\pi$
$744$	0	0
$745$	−63.7190	−2.33448
$746$	0	0
$747$	−101.742	−3.72256
$748$	0	0
$749$	−8.81921	−0.322247
$750$	0	0
$751$	1.57364	0.0574228	0.0287114	−	0.999588i	$-0.490860\pi$
$751$	1.57364	0.0574228	0.0287114	+	0.999588i	$0.490860\pi$
$752$	0	0
$753$	−50.0414	−1.82361
$754$	0	0
$755$	26.6602	0.970266
$756$	0	0
$757$	50.0158	1.81785	0.908927	−	0.416955i	$-0.136903\pi$
$757$	50.0158	1.81785	0.908927	+	0.416955i	$0.136903\pi$
$758$	0	0
$759$	−15.4636	−0.561292
$760$	0	0
$761$	−16.8031	−0.609110	−0.304555	−	0.952495i	$-0.598508\pi$
$761$	−16.8031	−0.609110	−0.304555	+	0.952495i	$0.598508\pi$
$762$	0	0
$763$	19.4616	0.704558
$764$	0	0
$765$	127.547	4.61146
$766$	0	0
$767$	4.59689	0.165984
$768$	0	0
$769$	35.5212	1.28093	0.640463	−	0.767989i	$-0.278742\pi$
$769$	35.5212	1.28093	0.640463	+	0.767989i	$0.278742\pi$
$770$	0	0
$771$	−6.97297	−0.251125
$772$	0	0
$773$	−16.8047	−0.604422	−0.302211	−	0.953241i	$-0.597725\pi$
$773$	−16.8047	−0.604422	−0.302211	+	0.953241i	$0.597725\pi$
$774$	0	0
$775$	−42.8936	−1.54078
$776$	0	0
$777$	0.605348	0.0217167
$778$	0	0
$779$	−11.8762	−0.425509
$780$	0	0
$781$	13.4722	0.482074
$782$	0	0
$783$	135.558	4.84446
$784$	0	0
$785$	82.0210	2.92746
$786$	0	0
$787$	−42.6983	−1.52203	−0.761015	−	0.648734i	$-0.775299\pi$
$787$	−42.6983	−1.52203	−0.761015	+	0.648734i	$0.775299\pi$
$788$	0	0
$789$	−79.7136	−2.83788
$790$	0	0
$791$	−18.4208	−0.654970
$792$	0	0
$793$	−5.05035	−0.179343
$794$	0	0
$795$	27.1353	0.962389
$796$	0	0
$797$	54.5422	1.93198	0.965992	−	0.258573i	$-0.0832523\pi$
$797$	54.5422	1.93198	0.965992	+	0.258573i	$0.0832523\pi$
$798$	0	0
$799$	14.4827	0.512361
$800$	0	0
$801$	−69.6017	−2.45926
$802$	0	0
$803$	−12.2013	−0.430574
$804$	0	0
$805$	−17.2296	−0.607262
$806$	0	0
$807$	−27.7943	−0.978405
$808$	0	0
$809$	−20.7298	−0.728822	−0.364411	−	0.931238i	$-0.618730\pi$
$809$	−20.7298	−0.728822	−0.364411	+	0.931238i	$0.618730\pi$
$810$	0	0
$811$	17.3286	0.608490	0.304245	−	0.952594i	$-0.401596\pi$
$811$	17.3286	0.608490	0.304245	+	0.952594i	$0.401596\pi$
$812$	0	0
$813$	49.9350	1.75130
$814$	0	0
$815$	39.7872	1.39368
$816$	0	0
$817$	75.1973	2.63082
$818$	0	0
$819$	−7.37212	−0.257603
$820$	0	0
$821$	−23.2760	−0.812337	−0.406169	−	0.913798i	$-0.633135\pi$
$821$	−23.2760	−0.812337	−0.406169	+	0.913798i	$0.633135\pi$
$822$	0	0
$823$	−21.3793	−0.745237	−0.372618	−	0.927985i	$-0.621540\pi$
$823$	−21.3793	−0.745237	−0.372618	+	0.927985i	$0.621540\pi$
$824$	0	0
$825$	−25.3667	−0.883155
$826$	0	0
$827$	3.61740	0.125789	0.0628947	−	0.998020i	$-0.479967\pi$
$827$	3.61740	0.125789	0.0628947	+	0.998020i	$0.479967\pi$
$828$	0	0
$829$	32.0455	1.11299	0.556493	−	0.830852i	$-0.312147\pi$
$829$	32.0455	1.11299	0.556493	+	0.830852i	$0.312147\pi$
$830$	0	0
$831$	−13.7480	−0.476914
$832$	0	0
$833$	−4.82146	−0.167054
$834$	0	0
$835$	62.1844	2.15198
$836$	0	0
$837$	−76.6811	−2.65049
$838$	0	0
$839$	19.9950	0.690305	0.345152	−	0.938547i	$-0.387827\pi$
$839$	19.9950	0.690305	0.345152	+	0.938547i	$0.387827\pi$
$840$	0	0
$841$	63.6830	2.19597
$842$	0	0
$843$	98.4699	3.39148
$844$	0	0
$845$	−3.58838	−0.123444
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	30.6225	1.05096
$850$	0	0
$851$	0.902501	0.0309373
$852$	0	0
$853$	−2.80987	−0.0962080	−0.0481040	−	0.998842i	$-0.515318\pi$
$853$	−2.80987	−0.0962080	−0.0481040	+	0.998842i	$0.515318\pi$
$854$	0	0
$855$	208.640	7.13533
$856$	0	0
$857$	−50.6433	−1.72994	−0.864971	−	0.501822i	$-0.832663\pi$
$857$	−50.6433	−1.72994	−0.864971	+	0.501822i	$0.832663\pi$
$858$	0	0
$859$	−29.6529	−1.01174	−0.505871	−	0.862609i	$-0.668829\pi$
$859$	−29.6529	−1.01174	−0.505871	+	0.862609i	$0.668829\pi$
$860$	0	0
$861$	4.84959	0.165274
$862$	0	0
$863$	−42.5305	−1.44776	−0.723878	−	0.689928i	$-0.757642\pi$
$863$	−42.5305	−1.44776	−0.723878	+	0.689928i	$0.757642\pi$
$864$	0	0
$865$	−5.09847	−0.173353
$866$	0	0
$867$	20.1172	0.683217
$868$	0	0
$869$	−11.0938	−0.376330
$870$	0	0
$871$	12.9657	0.439326
$872$	0	0
$873$	−4.95818	−0.167809
$874$	0	0
$875$	−10.3218	−0.348939
$876$	0	0
$877$	21.2810	0.718610	0.359305	−	0.933220i	$-0.383014\pi$
$877$	21.2810	0.718610	0.359305	+	0.933220i	$0.383014\pi$
$878$	0	0
$879$	32.0383	1.08062
$880$	0	0
$881$	−12.6285	−0.425466	−0.212733	−	0.977110i	$-0.568236\pi$
$881$	−12.6285	−0.425466	−0.212733	+	0.977110i	$0.568236\pi$
$882$	0	0
$883$	37.7754	1.27124	0.635621	−	0.772001i	$-0.280744\pi$
$883$	37.7754	1.27124	0.635621	+	0.772001i	$0.280744\pi$
$884$	0	0
$885$	53.1247	1.78577
$886$	0	0
$887$	−6.70061	−0.224984	−0.112492	−	0.993653i	$-0.535883\pi$
$887$	−6.70061	−0.224984	−0.112492	+	0.993653i	$0.535883\pi$
$888$	0	0
$889$	−8.56887	−0.287391
$890$	0	0
$891$	−23.2318	−0.778295
$892$	0	0
$893$	23.6907	0.792779
$894$	0	0
$895$	31.3343	1.04739
$896$	0	0
$897$	−15.4636	−0.516314
$898$	0	0
$899$	−52.4279	−1.74857
$900$	0	0
$901$	11.3209	0.377154
$902$	0	0
$903$	−30.7064	−1.02185
$904$	0	0
$905$	10.1018	0.335796
$906$	0	0
$907$	−37.0193	−1.22920	−0.614602	−	0.788837i	$-0.710683\pi$
$907$	−37.0193	−1.22920	−0.614602	+	0.788837i	$0.710683\pi$
$908$	0	0
$909$	−124.799	−4.13931
$910$	0	0
$911$	−5.08133	−0.168352	−0.0841760	−	0.996451i	$-0.526826\pi$
$911$	−5.08133	−0.168352	−0.0841760	+	0.996451i	$0.526826\pi$
$912$	0	0
$913$	13.8010	0.456745
$914$	0	0
$915$	−58.3652	−1.92949
$916$	0	0
$917$	10.7365	0.354550
$918$	0	0
$919$	−2.40857	−0.0794514	−0.0397257	−	0.999211i	$-0.512648\pi$
$919$	−2.40857	−0.0794514	−0.0397257	+	0.999211i	$0.512648\pi$
$920$	0	0
$921$	4.88599	0.160999
$922$	0	0
$923$	13.4722	0.443444
$924$	0	0
$925$	1.48048	0.0486778
$926$	0	0
$927$	−72.7939	−2.39087
$928$	0	0
$929$	49.1507	1.61258	0.806291	−	0.591519i	$-0.201471\pi$
$929$	49.1507	1.61258	0.806291	+	0.591519i	$0.201471\pi$
$930$	0	0
$931$	−7.88691	−0.258483
$932$	0	0
$933$	−79.0345	−2.58747
$934$	0	0
$935$	−17.3012	−0.565810
$936$	0	0
$937$	19.9279	0.651016	0.325508	−	0.945539i	$-0.394465\pi$
$937$	19.9279	0.651016	0.325508	+	0.945539i	$0.394465\pi$
$938$	0	0
$939$	−54.3086	−1.77229
$940$	0	0
$941$	18.6058	0.606531	0.303266	−	0.952906i	$-0.401923\pi$
$941$	18.6058	0.606531	0.303266	+	0.952906i	$0.401923\pi$
$942$	0	0
$943$	7.23015	0.235446
$944$	0	0
$945$	−50.5271	−1.64365
$946$	0	0
$947$	24.0457	0.781380	0.390690	−	0.920522i	$-0.372236\pi$
$947$	24.0457	0.781380	0.390690	+	0.920522i	$0.372236\pi$
$948$	0	0
$949$	−12.2013	−0.396071
$950$	0	0
$951$	−13.9672	−0.452919
$952$	0	0
$953$	−11.8193	−0.382865	−0.191432	−	0.981506i	$-0.561313\pi$
$953$	−11.8193	−0.382865	−0.191432	+	0.981506i	$0.561313\pi$
$954$	0	0
$955$	−94.2872	−3.05106
$956$	0	0
$957$	−31.0052	−1.00225
$958$	0	0
$959$	−4.31232	−0.139252
$960$	0	0
$961$	−1.34315	−0.0433276
$962$	0	0
$963$	−65.0163	−2.09512
$964$	0	0
$965$	−76.1669	−2.45190
$966$	0	0
$967$	−2.52709	−0.0812657	−0.0406329	−	0.999174i	$-0.512937\pi$
$967$	−2.52709	−0.0812657	−0.0406329	+	0.999174i	$0.512937\pi$
$968$	0	0
$969$	122.467	3.93421
$970$	0	0
$971$	−24.9102	−0.799407	−0.399703	−	0.916645i	$-0.630887\pi$
$971$	−24.9102	−0.799407	−0.399703	+	0.916645i	$0.630887\pi$
$972$	0	0
$973$	−5.63319	−0.180592
$974$	0	0
$975$	−25.3667	−0.812385
$976$	0	0
$977$	18.1903	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$977$	18.1903	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$978$	0	0
$979$	9.44120	0.301742
$980$	0	0
$981$	143.474	4.58076
$982$	0	0
$983$	−41.7039	−1.33015	−0.665074	−	0.746778i	$-0.731600\pi$
$983$	−41.7039	−1.33015	−0.665074	+	0.746778i	$0.731600\pi$
$984$	0	0
$985$	11.4814	0.365828
$986$	0	0
$987$	−9.67398	−0.307926
$988$	0	0
$989$	−45.7796	−1.45571
$990$	0	0
$991$	47.8239	1.51918	0.759589	−	0.650404i	$-0.225400\pi$
$991$	47.8239	1.51918	0.759589	+	0.650404i	$0.225400\pi$
$992$	0	0
$993$	16.6764	0.529210
$994$	0	0
$995$	−23.1137	−0.732753
$996$	0	0
$997$	−16.8654	−0.534134	−0.267067	−	0.963678i	$-0.586055\pi$
$997$	−16.8654	−0.534134	−0.267067	+	0.963678i	$0.586055\pi$
$998$	0	0
$999$	2.64666	0.0837365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.y.1.14	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.14	✓	14	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+3.22058 q^{3} -3.58838 q^{5} +1.00000 q^{7} +7.37212 q^{9} +O(q^{10})\)	\(q+3.22058 q^{3} -3.58838 q^{5} +1.00000 q^{7} +7.37212 q^{9} -1.00000 q^{11} -1.00000 q^{13} -11.5566 q^{15} -4.82146 q^{17} -7.88691 q^{19} +3.22058 q^{21} +4.80149 q^{23} +7.87644 q^{25} +14.0808 q^{27} +9.62720 q^{29} -5.44581 q^{31} -3.22058 q^{33} -3.58838 q^{35} +0.187963 q^{37} -3.22058 q^{39} +1.50581 q^{41} -9.53445 q^{43} -26.4539 q^{45} -3.00380 q^{47} +1.00000 q^{49} -15.5279 q^{51} -2.34802 q^{53} +3.58838 q^{55} -25.4004 q^{57} -4.59689 q^{59} +5.05035 q^{61} +7.37212 q^{63} +3.58838 q^{65} -12.9657 q^{67} +15.4636 q^{69} -13.4722 q^{71} +12.2013 q^{73} +25.3667 q^{75} -1.00000 q^{77} +11.0938 q^{79} +23.2318 q^{81} -13.8010 q^{83} +17.3012 q^{85} +31.0052 q^{87} -9.44120 q^{89} -1.00000 q^{91} -17.5387 q^{93} +28.3012 q^{95} -0.672558 q^{97} -7.37212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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