LMFDB - Embedded newform 8008.2.a.v.1.11

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			1.11
Root			$-2.90994$ 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)

$f(q)$	$=$	$q+2.90994 q^{3} -1.16917 q^{5} -1.00000 q^{7} +5.46776 q^{9} +O(q^{10})$	$q+2.90994 q^{3} -1.16917 q^{5} -1.00000 q^{7} +5.46776 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.40221 q^{15} -5.91520 q^{17} -2.04963 q^{19} -2.90994 q^{21} +5.20762 q^{23} -3.63305 q^{25} +7.18104 q^{27} -4.12176 q^{29} +0.961296 q^{31} +2.90994 q^{33} +1.16917 q^{35} -7.61804 q^{37} -2.90994 q^{39} -7.64480 q^{41} -1.91716 q^{43} -6.39273 q^{45} +5.20762 q^{47} +1.00000 q^{49} -17.2129 q^{51} -3.94714 q^{53} -1.16917 q^{55} -5.96430 q^{57} +1.97452 q^{59} -7.98894 q^{61} -5.46776 q^{63} +1.16917 q^{65} -4.62609 q^{67} +15.1539 q^{69} +11.1079 q^{71} +3.77526 q^{73} -10.5720 q^{75} -1.00000 q^{77} +6.29845 q^{79} +4.49313 q^{81} -17.2308 q^{83} +6.91586 q^{85} -11.9941 q^{87} +1.13850 q^{89} +1.00000 q^{91} +2.79732 q^{93} +2.39636 q^{95} -13.5854 q^{97} +5.46776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.90994	1.68006	0.840028	−	0.542543i	$-0.182539\pi$
$3$	2.90994	1.68006	0.840028	+	0.542543i	$0.182539\pi$
$4$	0	0
$5$	−1.16917	−0.522868	−0.261434	−	0.965221i	$-0.584195\pi$
$5$	−1.16917	−0.522868	−0.261434	+	0.965221i	$0.584195\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.46776	1.82259
$10$	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
$15$	−3.40221	−0.878447
$16$	0	0
$17$	−5.91520	−1.43465	−0.717323	−	0.696741i	$-0.754633\pi$
$17$	−5.91520	−1.43465	−0.717323	+	0.696741i	$0.754633\pi$
$18$	0	0
$19$	−2.04963	−0.470217	−0.235109	−	0.971969i	$-0.575545\pi$
$19$	−2.04963	−0.470217	−0.235109	+	0.971969i	$0.575545\pi$
$20$	0	0
$21$	−2.90994	−0.635001
$22$	0	0
$23$	5.20762	1.08586	0.542932	−	0.839777i	$-0.317314\pi$
$23$	5.20762	1.08586	0.542932	+	0.839777i	$0.317314\pi$
$24$	0	0
$25$	−3.63305	−0.726609
$26$	0	0
$27$	7.18104	1.38199
$28$	0	0
$29$	−4.12176	−0.765392	−0.382696	−	0.923874i	$-0.625004\pi$
$29$	−4.12176	−0.765392	−0.382696	+	0.923874i	$0.625004\pi$
$30$	0	0
$31$	0.961296	0.172654	0.0863269	−	0.996267i	$-0.472487\pi$
$31$	0.961296	0.172654	0.0863269	+	0.996267i	$0.472487\pi$
$32$	0	0
$33$	2.90994	0.506556
$34$	0	0
$35$	1.16917	0.197625
$36$	0	0
$37$	−7.61804	−1.25240	−0.626199	−	0.779663i	$-0.715390\pi$
$37$	−7.61804	−1.25240	−0.626199	+	0.779663i	$0.715390\pi$
$38$	0	0
$39$	−2.90994	−0.465964
$40$	0	0
$41$	−7.64480	−1.19392	−0.596958	−	0.802272i	$-0.703624\pi$
$41$	−7.64480	−1.19392	−0.596958	+	0.802272i	$0.703624\pi$
$42$	0	0
$43$	−1.91716	−0.292363	−0.146182	−	0.989258i	$-0.546698\pi$
$43$	−1.91716	−0.292363	−0.146182	+	0.989258i	$0.546698\pi$
$44$	0	0
$45$	−6.39273	−0.952972
$46$	0	0
$47$	5.20762	0.759609	0.379804	−	0.925067i	$-0.375991\pi$
$47$	5.20762	0.759609	0.379804	+	0.925067i	$0.375991\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−17.2129	−2.41028
$52$	0	0
$53$	−3.94714	−0.542182	−0.271091	−	0.962554i	$-0.587384\pi$
$53$	−3.94714	−0.542182	−0.271091	+	0.962554i	$0.587384\pi$
$54$	0	0
$55$	−1.16917	−0.157651
$56$	0	0
$57$	−5.96430	−0.789991
$58$	0	0
$59$	1.97452	0.257061	0.128531	−	0.991706i	$-0.458974\pi$
$59$	1.97452	0.257061	0.128531	+	0.991706i	$0.458974\pi$
$60$	0	0
$61$	−7.98894	−1.02288	−0.511439	−	0.859319i	$-0.670887\pi$
$61$	−7.98894	−1.02288	−0.511439	+	0.859319i	$0.670887\pi$
$62$	0	0
$63$	−5.46776	−0.688873
$64$	0	0
$65$	1.16917	0.145017
$66$	0	0
$67$	−4.62609	−0.565167	−0.282583	−	0.959243i	$-0.591191\pi$
$67$	−4.62609	−0.565167	−0.282583	+	0.959243i	$0.591191\pi$
$68$	0	0
$69$	15.1539	1.82431
$70$	0	0
$71$	11.1079	1.31826	0.659130	−	0.752029i	$-0.270925\pi$
$71$	11.1079	1.31826	0.659130	+	0.752029i	$0.270925\pi$
$72$	0	0
$73$	3.77526	0.441861	0.220930	−	0.975290i	$-0.429091\pi$
$73$	3.77526	0.441861	0.220930	+	0.975290i	$0.429091\pi$
$74$	0	0
$75$	−10.5720	−1.22074
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	6.29845	0.708631	0.354315	−	0.935126i	$-0.384714\pi$
$79$	6.29845	0.708631	0.354315	+	0.935126i	$0.384714\pi$
$80$	0	0
$81$	4.49313	0.499236
$82$	0	0
$83$	−17.2308	−1.89133	−0.945666	−	0.325141i	$-0.894588\pi$
$83$	−17.2308	−1.89133	−0.945666	+	0.325141i	$0.894588\pi$
$84$	0	0
$85$	6.91586	0.750130
$86$	0	0
$87$	−11.9941	−1.28590
$88$	0	0
$89$	1.13850	0.120681	0.0603403	−	0.998178i	$-0.480781\pi$
$89$	1.13850	0.120681	0.0603403	+	0.998178i	$0.480781\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	2.79732	0.290068
$94$	0	0
$95$	2.39636	0.245861
$96$	0	0
$97$	−13.5854	−1.37939	−0.689696	−	0.724099i	$-0.742256\pi$
$97$	−13.5854	−1.37939	−0.689696	+	0.724099i	$0.742256\pi$
$98$	0	0
$99$	5.46776	0.549531
$100$	0	0
$101$	18.3121	1.82212	0.911062	−	0.412268i	$-0.135263\pi$
$101$	18.3121	1.82212	0.911062	+	0.412268i	$0.135263\pi$
$102$	0	0
$103$	−5.45742	−0.537735	−0.268868	−	0.963177i	$-0.586649\pi$
$103$	−5.45742	−0.537735	−0.268868	+	0.963177i	$0.586649\pi$
$104$	0	0
$105$	3.40221	0.332022
$106$	0	0
$107$	−4.38564	−0.423976	−0.211988	−	0.977272i	$-0.567994\pi$
$107$	−4.38564	−0.423976	−0.211988	+	0.977272i	$0.567994\pi$
$108$	0	0
$109$	−16.1408	−1.54601	−0.773005	−	0.634400i	$-0.781247\pi$
$109$	−16.1408	−1.54601	−0.773005	+	0.634400i	$0.781247\pi$
$110$	0	0
$111$	−22.1680	−2.10410
$112$	0	0
$113$	−3.48934	−0.328249	−0.164125	−	0.986440i	$-0.552480\pi$
$113$	−3.48934	−0.328249	−0.164125	+	0.986440i	$0.552480\pi$
$114$	0	0
$115$	−6.08858	−0.567763
$116$	0	0
$117$	−5.46776	−0.505495
$118$	0	0
$119$	5.91520	0.542245
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−22.2459	−2.00585
$124$	0	0
$125$	10.0935	0.902788
$126$	0	0
$127$	18.4815	1.63996	0.819982	−	0.572389i	$-0.193983\pi$
$127$	18.4815	1.63996	0.819982	+	0.572389i	$0.193983\pi$
$128$	0	0
$129$	−5.57881	−0.491187
$130$	0	0
$131$	−13.7542	−1.20171	−0.600856	−	0.799357i	$-0.705173\pi$
$131$	−13.7542	−1.20171	−0.600856	+	0.799357i	$0.705173\pi$
$132$	0	0
$133$	2.04963	0.177725
$134$	0	0
$135$	−8.39584	−0.722599
$136$	0	0
$137$	−9.28537	−0.793302	−0.396651	−	0.917969i	$-0.629828\pi$
$137$	−9.28537	−0.793302	−0.396651	+	0.917969i	$0.629828\pi$
$138$	0	0
$139$	9.48179	0.804235	0.402118	−	0.915588i	$-0.368274\pi$
$139$	9.48179	0.804235	0.402118	+	0.915588i	$0.368274\pi$
$140$	0	0
$141$	15.1539	1.27619
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	4.81903	0.400199
$146$	0	0
$147$	2.90994	0.240008
$148$	0	0
$149$	−18.5929	−1.52319	−0.761594	−	0.648054i	$-0.775583\pi$
$149$	−18.5929	−1.52319	−0.761594	+	0.648054i	$0.775583\pi$
$150$	0	0
$151$	−0.264310	−0.0215092	−0.0107546	−	0.999942i	$-0.503423\pi$
$151$	−0.264310	−0.0215092	−0.0107546	+	0.999942i	$0.503423\pi$
$152$	0	0
$153$	−32.3429	−2.61477
$154$	0	0
$155$	−1.12392	−0.0902752
$156$	0	0
$157$	2.64728	0.211276	0.105638	−	0.994405i	$-0.466312\pi$
$157$	2.64728	0.211276	0.105638	+	0.994405i	$0.466312\pi$
$158$	0	0
$159$	−11.4860	−0.910896
$160$	0	0
$161$	−5.20762	−0.410418
$162$	0	0
$163$	−15.2628	−1.19547	−0.597737	−	0.801692i	$-0.703933\pi$
$163$	−15.2628	−1.19547	−0.597737	+	0.801692i	$0.703933\pi$
$164$	0	0
$165$	−3.40221	−0.264862
$166$	0	0
$167$	11.3724	0.880023	0.440012	−	0.897992i	$-0.354974\pi$
$167$	11.3724	0.880023	0.440012	+	0.897992i	$0.354974\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−11.2069	−0.857012
$172$	0	0
$173$	−6.81083	−0.517818	−0.258909	−	0.965902i	$-0.583363\pi$
$173$	−6.81083	−0.517818	−0.258909	+	0.965902i	$0.583363\pi$
$174$	0	0
$175$	3.63305	0.274632
$176$	0	0
$177$	5.74575	0.431877
$178$	0	0
$179$	5.92885	0.443143	0.221571	−	0.975144i	$-0.428881\pi$
$179$	5.92885	0.443143	0.221571	+	0.975144i	$0.428881\pi$
$180$	0	0
$181$	9.93972	0.738813	0.369407	−	0.929268i	$-0.379561\pi$
$181$	9.93972	0.738813	0.369407	+	0.929268i	$0.379561\pi$
$182$	0	0
$183$	−23.2473	−1.71849
$184$	0	0
$185$	8.90677	0.654839
$186$	0	0
$187$	−5.91520	−0.432562
$188$	0	0
$189$	−7.18104	−0.522344
$190$	0	0
$191$	4.74438	0.343291	0.171646	−	0.985159i	$-0.445092\pi$
$191$	4.74438	0.343291	0.171646	+	0.985159i	$0.445092\pi$
$192$	0	0
$193$	−18.7613	−1.35047	−0.675236	−	0.737602i	$-0.735958\pi$
$193$	−18.7613	−1.35047	−0.675236	+	0.737602i	$0.735958\pi$
$194$	0	0
$195$	3.40221	0.243637
$196$	0	0
$197$	−23.8790	−1.70131	−0.850655	−	0.525725i	$-0.823794\pi$
$197$	−23.8790	−1.70131	−0.850655	+	0.525725i	$0.823794\pi$
$198$	0	0
$199$	−2.28067	−0.161672	−0.0808362	−	0.996727i	$-0.525759\pi$
$199$	−2.28067	−0.161672	−0.0808362	+	0.996727i	$0.525759\pi$
$200$	0	0
$201$	−13.4617	−0.949512
$202$	0	0
$203$	4.12176	0.289291
$204$	0	0
$205$	8.93805	0.624261
$206$	0	0
$207$	28.4740	1.97908
$208$	0	0
$209$	−2.04963	−0.141776
$210$	0	0
$211$	3.20483	0.220630	0.110315	−	0.993897i	$-0.464814\pi$
$211$	3.20483	0.220630	0.110315	+	0.993897i	$0.464814\pi$
$212$	0	0
$213$	32.3232	2.21475
$214$	0	0
$215$	2.24148	0.152867
$216$	0	0
$217$	−0.961296	−0.0652570
$218$	0	0
$219$	10.9858	0.742351
$220$	0	0
$221$	5.91520	0.397899
$222$	0	0
$223$	15.8970	1.06454	0.532271	−	0.846574i	$-0.321339\pi$
$223$	15.8970	1.06454	0.532271	+	0.846574i	$0.321339\pi$
$224$	0	0
$225$	−19.8646	−1.32431
$226$	0	0
$227$	24.4701	1.62414	0.812068	−	0.583563i	$-0.198342\pi$
$227$	24.4701	1.62414	0.812068	+	0.583563i	$0.198342\pi$
$228$	0	0
$229$	28.3405	1.87279	0.936396	−	0.350944i	$-0.114139\pi$
$229$	28.3405	1.87279	0.936396	+	0.350944i	$0.114139\pi$
$230$	0	0
$231$	−2.90994	−0.191460
$232$	0	0
$233$	0.824406	0.0540086	0.0270043	−	0.999635i	$-0.491403\pi$
$233$	0.824406	0.0540086	0.0270043	+	0.999635i	$0.491403\pi$
$234$	0	0
$235$	−6.08858	−0.397175
$236$	0	0
$237$	18.3281	1.19054
$238$	0	0
$239$	4.51005	0.291731	0.145865	−	0.989304i	$-0.453403\pi$
$239$	4.51005	0.291731	0.145865	+	0.989304i	$0.453403\pi$
$240$	0	0
$241$	10.9230	0.703614	0.351807	−	0.936073i	$-0.385567\pi$
$241$	10.9230	0.703614	0.351807	+	0.936073i	$0.385567\pi$
$242$	0	0
$243$	−8.46839	−0.543247
$244$	0	0
$245$	−1.16917	−0.0746954
$246$	0	0
$247$	2.04963	0.130415
$248$	0	0
$249$	−50.1408	−3.17754
$250$	0	0
$251$	20.3185	1.28249	0.641246	−	0.767336i	$-0.278418\pi$
$251$	20.3185	1.28249	0.641246	+	0.767336i	$0.278418\pi$
$252$	0	0
$253$	5.20762	0.327400
$254$	0	0
$255$	20.1247	1.26026
$256$	0	0
$257$	10.7720	0.671941	0.335971	−	0.941872i	$-0.390936\pi$
$257$	10.7720	0.671941	0.335971	+	0.941872i	$0.390936\pi$
$258$	0	0
$259$	7.61804	0.473362
$260$	0	0
$261$	−22.5368	−1.39499
$262$	0	0
$263$	−15.6846	−0.967152	−0.483576	−	0.875302i	$-0.660662\pi$
$263$	−15.6846	−0.967152	−0.483576	+	0.875302i	$0.660662\pi$
$264$	0	0
$265$	4.61488	0.283490
$266$	0	0
$267$	3.31296	0.202750
$268$	0	0
$269$	−21.6449	−1.31971	−0.659857	−	0.751391i	$-0.729383\pi$
$269$	−21.6449	−1.31971	−0.659857	+	0.751391i	$0.729383\pi$
$270$	0	0
$271$	−19.0786	−1.15894	−0.579470	−	0.814994i	$-0.696740\pi$
$271$	−19.0786	−1.15894	−0.579470	+	0.814994i	$0.696740\pi$
$272$	0	0
$273$	2.90994	0.176118
$274$	0	0
$275$	−3.63305	−0.219081
$276$	0	0
$277$	17.0330	1.02342	0.511708	−	0.859159i	$-0.329013\pi$
$277$	17.0330	1.02342	0.511708	+	0.859159i	$0.329013\pi$
$278$	0	0
$279$	5.25614	0.314677
$280$	0	0
$281$	−15.6379	−0.932879	−0.466439	−	0.884553i	$-0.654463\pi$
$281$	−15.6379	−0.932879	−0.466439	+	0.884553i	$0.654463\pi$
$282$	0	0
$283$	−29.4271	−1.74926	−0.874630	−	0.484790i	$-0.838896\pi$
$283$	−29.4271	−1.74926	−0.874630	+	0.484790i	$0.838896\pi$
$284$	0	0
$285$	6.97327	0.413061
$286$	0	0
$287$	7.64480	0.451258
$288$	0	0
$289$	17.9896	1.05821
$290$	0	0
$291$	−39.5328	−2.31745
$292$	0	0
$293$	−17.0330	−0.995078	−0.497539	−	0.867442i	$-0.665763\pi$
$293$	−17.0330	−0.995078	−0.497539	+	0.867442i	$0.665763\pi$
$294$	0	0
$295$	−2.30855	−0.134409
$296$	0	0
$297$	7.18104	0.416686
$298$	0	0
$299$	−5.20762	−0.301164
$300$	0	0
$301$	1.91716	0.110503
$302$	0	0
$303$	53.2872	3.06127
$304$	0	0
$305$	9.34041	0.534830
$306$	0	0
$307$	23.8369	1.36044	0.680221	−	0.733007i	$-0.261884\pi$
$307$	23.8369	1.36044	0.680221	+	0.733007i	$0.261884\pi$
$308$	0	0
$309$	−15.8808	−0.903425
$310$	0	0
$311$	23.2215	1.31677	0.658385	−	0.752681i	$-0.271240\pi$
$311$	23.2215	1.31677	0.658385	+	0.752681i	$0.271240\pi$
$312$	0	0
$313$	12.4544	0.703965	0.351982	−	0.936007i	$-0.385508\pi$
$313$	12.4544	0.703965	0.351982	+	0.936007i	$0.385508\pi$
$314$	0	0
$315$	6.39273	0.360190
$316$	0	0
$317$	−12.0519	−0.676902	−0.338451	−	0.940984i	$-0.609903\pi$
$317$	−12.0519	−0.676902	−0.338451	+	0.940984i	$0.609903\pi$
$318$	0	0
$319$	−4.12176	−0.230774
$320$	0	0
$321$	−12.7619	−0.712303
$322$	0	0
$323$	12.1240	0.674595
$324$	0	0
$325$	3.63305	0.201525
$326$	0	0
$327$	−46.9689	−2.59738
$328$	0	0
$329$	−5.20762	−0.287105
$330$	0	0
$331$	−4.32592	−0.237774	−0.118887	−	0.992908i	$-0.537933\pi$
$331$	−4.32592	−0.237774	−0.118887	+	0.992908i	$0.537933\pi$
$332$	0	0
$333$	−41.6536	−2.28260
$334$	0	0
$335$	5.40868	0.295508
$336$	0	0
$337$	−21.2166	−1.15574	−0.577871	−	0.816128i	$-0.696116\pi$
$337$	−21.2166	−1.15574	−0.577871	+	0.816128i	$0.696116\pi$
$338$	0	0
$339$	−10.1538	−0.551477
$340$	0	0
$341$	0.961296	0.0520571
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−17.7174	−0.953873
$346$	0	0
$347$	31.2424	1.67718	0.838589	−	0.544764i	$-0.183381\pi$
$347$	31.2424	1.67718	0.838589	+	0.544764i	$0.183381\pi$
$348$	0	0
$349$	31.6059	1.69182	0.845911	−	0.533324i	$-0.179057\pi$
$349$	31.6059	1.69182	0.845911	+	0.533324i	$0.179057\pi$
$350$	0	0
$351$	−7.18104	−0.383296
$352$	0	0
$353$	−15.1385	−0.805742	−0.402871	−	0.915257i	$-0.631988\pi$
$353$	−15.1385	−0.805742	−0.402871	+	0.915257i	$0.631988\pi$
$354$	0	0
$355$	−12.9870	−0.689276
$356$	0	0
$357$	17.2129	0.911002
$358$	0	0
$359$	−24.8290	−1.31042	−0.655212	−	0.755445i	$-0.727421\pi$
$359$	−24.8290	−1.31042	−0.655212	+	0.755445i	$0.727421\pi$
$360$	0	0
$361$	−14.7990	−0.778896
$362$	0	0
$363$	2.90994	0.152732
$364$	0	0
$365$	−4.41391	−0.231035
$366$	0	0
$367$	13.5980	0.709810	0.354905	−	0.934902i	$-0.384513\pi$
$367$	13.5980	0.709810	0.354905	+	0.934902i	$0.384513\pi$
$368$	0	0
$369$	−41.7999	−2.17602
$370$	0	0
$371$	3.94714	0.204926
$372$	0	0
$373$	−15.2937	−0.791879	−0.395940	−	0.918277i	$-0.629581\pi$
$373$	−15.2937	−0.791879	−0.395940	+	0.918277i	$0.629581\pi$
$374$	0	0
$375$	29.3714	1.51673
$376$	0	0
$377$	4.12176	0.212282
$378$	0	0
$379$	−23.6586	−1.21526	−0.607631	−	0.794219i	$-0.707880\pi$
$379$	−23.6586	−1.21526	−0.607631	+	0.794219i	$0.707880\pi$
$380$	0	0
$381$	53.7800	2.75523
$382$	0	0
$383$	−12.0507	−0.615763	−0.307882	−	0.951425i	$-0.599620\pi$
$383$	−12.0507	−0.615763	−0.307882	+	0.951425i	$0.599620\pi$
$384$	0	0
$385$	1.16917	0.0595863
$386$	0	0
$387$	−10.4825	−0.532858
$388$	0	0
$389$	−3.95795	−0.200676	−0.100338	−	0.994953i	$-0.531992\pi$
$389$	−3.95795	−0.200676	−0.100338	+	0.994953i	$0.531992\pi$
$390$	0	0
$391$	−30.8041	−1.55783
$392$	0	0
$393$	−40.0240	−2.01894
$394$	0	0
$395$	−7.36394	−0.370520
$396$	0	0
$397$	−17.8178	−0.894250	−0.447125	−	0.894471i	$-0.647552\pi$
$397$	−17.8178	−0.894250	−0.447125	+	0.894471i	$0.647552\pi$
$398$	0	0
$399$	5.96430	0.298589
$400$	0	0
$401$	−24.5860	−1.22777	−0.613883	−	0.789397i	$-0.710393\pi$
$401$	−24.5860	−1.22777	−0.613883	+	0.789397i	$0.710393\pi$
$402$	0	0
$403$	−0.961296	−0.0478856
$404$	0	0
$405$	−5.25322	−0.261035
$406$	0	0
$407$	−7.61804	−0.377612
$408$	0	0
$409$	13.3972	0.662451	0.331225	−	0.943552i	$-0.392538\pi$
$409$	13.3972	0.662451	0.331225	+	0.943552i	$0.392538\pi$
$410$	0	0
$411$	−27.0199	−1.33279
$412$	0	0
$413$	−1.97452	−0.0971600
$414$	0	0
$415$	20.1458	0.988916
$416$	0	0
$417$	27.5915	1.35116
$418$	0	0
$419$	4.94153	0.241410	0.120705	−	0.992688i	$-0.461485\pi$
$419$	4.94153	0.241410	0.120705	+	0.992688i	$0.461485\pi$
$420$	0	0
$421$	−21.5301	−1.04931	−0.524656	−	0.851315i	$-0.675806\pi$
$421$	−21.5301	−1.04931	−0.524656	+	0.851315i	$0.675806\pi$
$422$	0	0
$423$	28.4740	1.38445
$424$	0	0
$425$	21.4902	1.04243
$426$	0	0
$427$	7.98894	0.386612
$428$	0	0
$429$	−2.90994	−0.140493
$430$	0	0
$431$	−20.0047	−0.963595	−0.481797	−	0.876283i	$-0.660016\pi$
$431$	−20.0047	−0.963595	−0.481797	+	0.876283i	$0.660016\pi$
$432$	0	0
$433$	−29.4079	−1.41325	−0.706627	−	0.707587i	$-0.749784\pi$
$433$	−29.4079	−1.41325	−0.706627	+	0.707587i	$0.749784\pi$
$434$	0	0
$435$	14.0231	0.672357
$436$	0	0
$437$	−10.6737	−0.510592
$438$	0	0
$439$	28.5440	1.36233	0.681165	−	0.732130i	$-0.261474\pi$
$439$	28.5440	1.36233	0.681165	+	0.732130i	$0.261474\pi$
$440$	0	0
$441$	5.46776	0.260370
$442$	0	0
$443$	7.84235	0.372601	0.186301	−	0.982493i	$-0.440350\pi$
$443$	7.84235	0.372601	0.186301	+	0.982493i	$0.440350\pi$
$444$	0	0
$445$	−1.33110	−0.0631000
$446$	0	0
$447$	−54.1042	−2.55904
$448$	0	0
$449$	21.0341	0.992661	0.496331	−	0.868134i	$-0.334680\pi$
$449$	21.0341	0.992661	0.496331	+	0.868134i	$0.334680\pi$
$450$	0	0
$451$	−7.64480	−0.359979
$452$	0	0
$453$	−0.769126	−0.0361367
$454$	0	0
$455$	−1.16917	−0.0548114
$456$	0	0
$457$	32.2475	1.50847	0.754237	−	0.656602i	$-0.228007\pi$
$457$	32.2475	1.50847	0.754237	+	0.656602i	$0.228007\pi$
$458$	0	0
$459$	−42.4773	−1.98267
$460$	0	0
$461$	−38.2475	−1.78136	−0.890681	−	0.454629i	$-0.849772\pi$
$461$	−38.2475	−1.78136	−0.890681	+	0.454629i	$0.849772\pi$
$462$	0	0
$463$	−9.55243	−0.443939	−0.221970	−	0.975054i	$-0.571249\pi$
$463$	−9.55243	−0.443939	−0.221970	+	0.975054i	$0.571249\pi$
$464$	0	0
$465$	−3.27053	−0.151667
$466$	0	0
$467$	−7.95322	−0.368031	−0.184016	−	0.982923i	$-0.558910\pi$
$467$	−7.95322	−0.368031	−0.184016	+	0.982923i	$0.558910\pi$
$468$	0	0
$469$	4.62609	0.213613
$470$	0	0
$471$	7.70343	0.354955
$472$	0	0
$473$	−1.91716	−0.0881509
$474$	0	0
$475$	7.44640	0.341664
$476$	0	0
$477$	−21.5820	−0.988174
$478$	0	0
$479$	−16.7258	−0.764222	−0.382111	−	0.924116i	$-0.624803\pi$
$479$	−16.7258	−0.764222	−0.382111	+	0.924116i	$0.624803\pi$
$480$	0	0
$481$	7.61804	0.347353
$482$	0	0
$483$	−15.1539	−0.689525
$484$	0	0
$485$	15.8837	0.721239
$486$	0	0
$487$	22.2426	1.00791	0.503954	−	0.863730i	$-0.331878\pi$
$487$	22.2426	1.00791	0.503954	+	0.863730i	$0.331878\pi$
$488$	0	0
$489$	−44.4138	−2.00846
$490$	0	0
$491$	10.9920	0.496063	0.248031	−	0.968752i	$-0.420216\pi$
$491$	10.9920	0.496063	0.248031	+	0.968752i	$0.420216\pi$
$492$	0	0
$493$	24.3810	1.09807
$494$	0	0
$495$	−6.39273	−0.287332
$496$	0	0
$497$	−11.1079	−0.498255
$498$	0	0
$499$	−38.0374	−1.70279	−0.851394	−	0.524526i	$-0.824243\pi$
$499$	−38.0374	−1.70279	−0.851394	+	0.524526i	$0.824243\pi$
$500$	0	0
$501$	33.0930	1.47849
$502$	0	0
$503$	26.4044	1.17732	0.588658	−	0.808383i	$-0.299657\pi$
$503$	26.4044	1.17732	0.588658	+	0.808383i	$0.299657\pi$
$504$	0	0
$505$	−21.4100	−0.952730
$506$	0	0
$507$	2.90994	0.129235
$508$	0	0
$509$	−8.62431	−0.382266	−0.191133	−	0.981564i	$-0.561216\pi$
$509$	−8.62431	−0.382266	−0.191133	+	0.981564i	$0.561216\pi$
$510$	0	0
$511$	−3.77526	−0.167008
$512$	0	0
$513$	−14.7185	−0.649836
$514$	0	0
$515$	6.38064	0.281165
$516$	0	0
$517$	5.20762	0.229031
$518$	0	0
$519$	−19.8191	−0.869963
$520$	0	0
$521$	−5.13203	−0.224838	−0.112419	−	0.993661i	$-0.535860\pi$
$521$	−5.13203	−0.224838	−0.112419	+	0.993661i	$0.535860\pi$
$522$	0	0
$523$	32.3238	1.41342	0.706711	−	0.707503i	$-0.250178\pi$
$523$	32.3238	1.41342	0.706711	+	0.707503i	$0.250178\pi$
$524$	0	0
$525$	10.5720	0.461398
$526$	0	0
$527$	−5.68625	−0.247697
$528$	0	0
$529$	4.11927	0.179099
$530$	0	0
$531$	10.7962	0.468516
$532$	0	0
$533$	7.64480	0.331133
$534$	0	0
$535$	5.12755	0.221683
$536$	0	0
$537$	17.2526	0.744504
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	25.6379	1.10226	0.551130	−	0.834419i	$-0.314197\pi$
$541$	25.6379	1.10226	0.551130	+	0.834419i	$0.314197\pi$
$542$	0	0
$543$	28.9240	1.24125
$544$	0	0
$545$	18.8713	0.808359
$546$	0	0
$547$	−11.7038	−0.500417	−0.250208	−	0.968192i	$-0.580499\pi$
$547$	−11.7038	−0.500417	−0.250208	+	0.968192i	$0.580499\pi$
$548$	0	0
$549$	−43.6816	−1.86428
$550$	0	0
$551$	8.44809	0.359901
$552$	0	0
$553$	−6.29845	−0.267837
$554$	0	0
$555$	25.9182	1.10017
$556$	0	0
$557$	24.6149	1.04297	0.521484	−	0.853261i	$-0.325378\pi$
$557$	24.6149	1.04297	0.521484	+	0.853261i	$0.325378\pi$
$558$	0	0
$559$	1.91716	0.0810870
$560$	0	0
$561$	−17.2129	−0.726728
$562$	0	0
$563$	13.7661	0.580171	0.290086	−	0.957001i	$-0.406316\pi$
$563$	13.7661	0.580171	0.290086	+	0.957001i	$0.406316\pi$
$564$	0	0
$565$	4.07962	0.171631
$566$	0	0
$567$	−4.49313	−0.188694
$568$	0	0
$569$	45.6622	1.91426	0.957128	−	0.289664i	$-0.0935435\pi$
$569$	45.6622	1.91426	0.957128	+	0.289664i	$0.0935435\pi$
$570$	0	0
$571$	−10.5471	−0.441381	−0.220690	−	0.975344i	$-0.570831\pi$
$571$	−10.5471	−0.441381	−0.220690	+	0.975344i	$0.570831\pi$
$572$	0	0
$573$	13.8059	0.576748
$574$	0	0
$575$	−18.9195	−0.788998
$576$	0	0
$577$	31.1186	1.29548	0.647742	−	0.761860i	$-0.275713\pi$
$577$	31.1186	1.29548	0.647742	+	0.761860i	$0.275713\pi$
$578$	0	0
$579$	−54.5944	−2.26887
$580$	0	0
$581$	17.2308	0.714856
$582$	0	0
$583$	−3.94714	−0.163474
$584$	0	0
$585$	6.39273	0.264307
$586$	0	0
$587$	−16.8660	−0.696136	−0.348068	−	0.937469i	$-0.613162\pi$
$587$	−16.8660	−0.696136	−0.348068	+	0.937469i	$0.613162\pi$
$588$	0	0
$589$	−1.97030	−0.0811848
$590$	0	0
$591$	−69.4865	−2.85829
$592$	0	0
$593$	−34.8391	−1.43067	−0.715334	−	0.698782i	$-0.753726\pi$
$593$	−34.8391	−1.43067	−0.715334	+	0.698782i	$0.753726\pi$
$594$	0	0
$595$	−6.91586	−0.283523
$596$	0	0
$597$	−6.63662	−0.271619
$598$	0	0
$599$	23.0816	0.943088	0.471544	−	0.881843i	$-0.343697\pi$
$599$	23.0816	0.943088	0.471544	+	0.881843i	$0.343697\pi$
$600$	0	0
$601$	16.6140	0.677698	0.338849	−	0.940841i	$-0.389962\pi$
$601$	16.6140	0.677698	0.338849	+	0.940841i	$0.389962\pi$
$602$	0	0
$603$	−25.2944	−1.03007
$604$	0	0
$605$	−1.16917	−0.0475334
$606$	0	0
$607$	44.2153	1.79464	0.897322	−	0.441376i	$-0.145510\pi$
$607$	44.2153	1.79464	0.897322	+	0.441376i	$0.145510\pi$
$608$	0	0
$609$	11.9941	0.486025
$610$	0	0
$611$	−5.20762	−0.210678
$612$	0	0
$613$	−39.6930	−1.60319	−0.801593	−	0.597870i	$-0.796014\pi$
$613$	−39.6930	−1.60319	−0.801593	+	0.597870i	$0.796014\pi$
$614$	0	0
$615$	26.0092	1.04879
$616$	0	0
$617$	−11.2609	−0.453348	−0.226674	−	0.973971i	$-0.572785\pi$
$617$	−11.2609	−0.453348	−0.226674	+	0.973971i	$0.572785\pi$
$618$	0	0
$619$	−27.4814	−1.10457	−0.552285	−	0.833656i	$-0.686244\pi$
$619$	−27.4814	−1.10457	−0.552285	+	0.833656i	$0.686244\pi$
$620$	0	0
$621$	37.3961	1.50065
$622$	0	0
$623$	−1.13850	−0.0456130
$624$	0	0
$625$	6.36425	0.254570
$626$	0	0
$627$	−5.96430	−0.238191
$628$	0	0
$629$	45.0622	1.79675
$630$	0	0
$631$	9.55126	0.380230	0.190115	−	0.981762i	$-0.439114\pi$
$631$	9.55126	0.380230	0.190115	+	0.981762i	$0.439114\pi$
$632$	0	0
$633$	9.32588	0.370670
$634$	0	0
$635$	−21.6079	−0.857485
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	60.7351	2.40264
$640$	0	0
$641$	27.4180	1.08295	0.541473	−	0.840718i	$-0.317867\pi$
$641$	27.4180	1.08295	0.541473	+	0.840718i	$0.317867\pi$
$642$	0	0
$643$	−5.15156	−0.203158	−0.101579	−	0.994827i	$-0.532389\pi$
$643$	−5.15156	−0.203158	−0.101579	+	0.994827i	$0.532389\pi$
$644$	0	0
$645$	6.52257	0.256826
$646$	0	0
$647$	−42.4286	−1.66804	−0.834021	−	0.551733i	$-0.813967\pi$
$647$	−42.4286	−1.66804	−0.834021	+	0.551733i	$0.813967\pi$
$648$	0	0
$649$	1.97452	0.0775069
$650$	0	0
$651$	−2.79732	−0.109635
$652$	0	0
$653$	22.8485	0.894132	0.447066	−	0.894501i	$-0.352469\pi$
$653$	22.8485	0.894132	0.447066	+	0.894501i	$0.352469\pi$
$654$	0	0
$655$	16.0810	0.628336
$656$	0	0
$657$	20.6422	0.805330
$658$	0	0
$659$	−20.2815	−0.790053	−0.395027	−	0.918670i	$-0.629265\pi$
$659$	−20.2815	−0.790053	−0.395027	+	0.918670i	$0.629265\pi$
$660$	0	0
$661$	−6.70253	−0.260698	−0.130349	−	0.991468i	$-0.541610\pi$
$661$	−6.70253	−0.260698	−0.130349	+	0.991468i	$0.541610\pi$
$662$	0	0
$663$	17.2129	0.668493
$664$	0	0
$665$	−2.39636	−0.0929269
$666$	0	0
$667$	−21.4646	−0.831111
$668$	0	0
$669$	46.2594	1.78849
$670$	0	0
$671$	−7.98894	−0.308409
$672$	0	0
$673$	−16.2565	−0.626642	−0.313321	−	0.949647i	$-0.601442\pi$
$673$	−16.2565	−0.626642	−0.313321	+	0.949647i	$0.601442\pi$
$674$	0	0
$675$	−26.0890	−1.00417
$676$	0	0
$677$	−15.1073	−0.580620	−0.290310	−	0.956933i	$-0.593758\pi$
$677$	−15.1073	−0.580620	−0.290310	+	0.956933i	$0.593758\pi$
$678$	0	0
$679$	13.5854	0.521361
$680$	0	0
$681$	71.2065	2.72864
$682$	0	0
$683$	27.5615	1.05461	0.527306	−	0.849676i	$-0.323202\pi$
$683$	27.5615	1.05461	0.527306	+	0.849676i	$0.323202\pi$
$684$	0	0
$685$	10.8562	0.414792
$686$	0	0
$687$	82.4692	3.14640
$688$	0	0
$689$	3.94714	0.150374
$690$	0	0
$691$	19.5169	0.742459	0.371229	−	0.928541i	$-0.378936\pi$
$691$	19.5169	0.742459	0.371229	+	0.928541i	$0.378936\pi$
$692$	0	0
$693$	−5.46776	−0.207703
$694$	0	0
$695$	−11.0858	−0.420509
$696$	0	0
$697$	45.2205	1.71285
$698$	0	0
$699$	2.39897	0.0907375
$700$	0	0
$701$	−7.28846	−0.275281	−0.137641	−	0.990482i	$-0.543952\pi$
$701$	−7.28846	−0.275281	−0.137641	+	0.990482i	$0.543952\pi$
$702$	0	0
$703$	15.6142	0.588899
$704$	0	0
$705$	−17.7174	−0.667276
$706$	0	0
$707$	−18.3121	−0.688698
$708$	0	0
$709$	39.3079	1.47624	0.738120	−	0.674670i	$-0.235714\pi$
$709$	39.3079	1.47624	0.738120	+	0.674670i	$0.235714\pi$
$710$	0	0
$711$	34.4384	1.29154
$712$	0	0
$713$	5.00606	0.187478
$714$	0	0
$715$	1.16917	0.0437244
$716$	0	0
$717$	13.1240	0.490124
$718$	0	0
$719$	−2.88228	−0.107491	−0.0537454	−	0.998555i	$-0.517116\pi$
$719$	−2.88228	−0.107491	−0.0537454	+	0.998555i	$0.517116\pi$
$720$	0	0
$721$	5.45742	0.203245
$722$	0	0
$723$	31.7854	1.18211
$724$	0	0
$725$	14.9746	0.556141
$726$	0	0
$727$	−19.4261	−0.720473	−0.360237	−	0.932861i	$-0.617304\pi$
$727$	−19.4261	−0.720473	−0.360237	+	0.932861i	$0.617304\pi$
$728$	0	0
$729$	−38.1219	−1.41192
$730$	0	0
$731$	11.3403	0.419438
$732$	0	0
$733$	39.5041	1.45912	0.729558	−	0.683919i	$-0.239726\pi$
$733$	39.5041	1.45912	0.729558	+	0.683919i	$0.239726\pi$
$734$	0	0
$735$	−3.40221	−0.125492
$736$	0	0
$737$	−4.62609	−0.170404
$738$	0	0
$739$	21.1493	0.777991	0.388996	−	0.921240i	$-0.372822\pi$
$739$	21.1493	0.777991	0.388996	+	0.921240i	$0.372822\pi$
$740$	0	0
$741$	5.96430	0.219104
$742$	0	0
$743$	−23.3505	−0.856647	−0.428324	−	0.903625i	$-0.640896\pi$
$743$	−23.3505	−0.856647	−0.428324	+	0.903625i	$0.640896\pi$
$744$	0	0
$745$	21.7382	0.796426
$746$	0	0
$747$	−94.2141	−3.44712
$748$	0	0
$749$	4.38564	0.160248
$750$	0	0
$751$	28.6706	1.04620	0.523102	−	0.852270i	$-0.324775\pi$
$751$	28.6706	1.04620	0.523102	+	0.852270i	$0.324775\pi$
$752$	0	0
$753$	59.1256	2.15466
$754$	0	0
$755$	0.309023	0.0112465
$756$	0	0
$757$	6.43531	0.233895	0.116948	−	0.993138i	$-0.462689\pi$
$757$	6.43531	0.233895	0.116948	+	0.993138i	$0.462689\pi$
$758$	0	0
$759$	15.1539	0.550050
$760$	0	0
$761$	−5.39718	−0.195648	−0.0978239	−	0.995204i	$-0.531188\pi$
$761$	−5.39718	−0.195648	−0.0978239	+	0.995204i	$0.531188\pi$
$762$	0	0
$763$	16.1408	0.584337
$764$	0	0
$765$	37.8143	1.36718
$766$	0	0
$767$	−1.97452	−0.0712960
$768$	0	0
$769$	1.94472	0.0701283	0.0350642	−	0.999385i	$-0.488836\pi$
$769$	1.94472	0.0701283	0.0350642	+	0.999385i	$0.488836\pi$
$770$	0	0
$771$	31.3460	1.12890
$772$	0	0
$773$	22.6995	0.816443	0.408221	−	0.912883i	$-0.366149\pi$
$773$	22.6995	0.816443	0.408221	+	0.912883i	$0.366149\pi$
$774$	0	0
$775$	−3.49243	−0.125452
$776$	0	0
$777$	22.1680	0.795274
$778$	0	0
$779$	15.6690	0.561400
$780$	0	0
$781$	11.1079	0.397470
$782$	0	0
$783$	−29.5986	−1.05777
$784$	0	0
$785$	−3.09511	−0.110469
$786$	0	0
$787$	38.4305	1.36990	0.684950	−	0.728591i	$-0.259824\pi$
$787$	38.4305	1.36990	0.684950	+	0.728591i	$0.259824\pi$
$788$	0	0
$789$	−45.6412	−1.62487
$790$	0	0
$791$	3.48934	0.124067
$792$	0	0
$793$	7.98894	0.283695
$794$	0	0
$795$	13.4290	0.476278
$796$	0	0
$797$	−9.88069	−0.349992	−0.174996	−	0.984569i	$-0.555991\pi$
$797$	−9.88069	−0.349992	−0.174996	+	0.984569i	$0.555991\pi$
$798$	0	0
$799$	−30.8041	−1.08977
$800$	0	0
$801$	6.22504	0.219951
$802$	0	0
$803$	3.77526	0.133226
$804$	0	0
$805$	6.08858	0.214594
$806$	0	0
$807$	−62.9855	−2.21719
$808$	0	0
$809$	12.3960	0.435822	0.217911	−	0.975969i	$-0.430076\pi$
$809$	12.3960	0.435822	0.217911	+	0.975969i	$0.430076\pi$
$810$	0	0
$811$	−43.1531	−1.51531	−0.757655	−	0.652655i	$-0.773655\pi$
$811$	−43.1531	−1.51531	−0.757655	+	0.652655i	$0.773655\pi$
$812$	0	0
$813$	−55.5175	−1.94708
$814$	0	0
$815$	17.8448	0.625075
$816$	0	0
$817$	3.92946	0.137474
$818$	0	0
$819$	5.46776	0.191059
$820$	0	0
$821$	3.45499	0.120580	0.0602900	−	0.998181i	$-0.480797\pi$
$821$	3.45499	0.120580	0.0602900	+	0.998181i	$0.480797\pi$
$822$	0	0
$823$	−18.5544	−0.646765	−0.323383	−	0.946268i	$-0.604820\pi$
$823$	−18.5544	−0.646765	−0.323383	+	0.946268i	$0.604820\pi$
$824$	0	0
$825$	−10.5720	−0.368068
$826$	0	0
$827$	−11.0690	−0.384906	−0.192453	−	0.981306i	$-0.561644\pi$
$827$	−11.0690	−0.384906	−0.192453	+	0.981306i	$0.561644\pi$
$828$	0	0
$829$	24.1043	0.837178	0.418589	−	0.908176i	$-0.362525\pi$
$829$	24.1043	0.837178	0.418589	+	0.908176i	$0.362525\pi$
$830$	0	0
$831$	49.5652	1.71940
$832$	0	0
$833$	−5.91520	−0.204949
$834$	0	0
$835$	−13.2963	−0.460136
$836$	0	0
$837$	6.90310	0.238606
$838$	0	0
$839$	28.4491	0.982173	0.491086	−	0.871111i	$-0.336600\pi$
$839$	28.4491	0.982173	0.491086	+	0.871111i	$0.336600\pi$
$840$	0	0
$841$	−12.0111	−0.414175
$842$	0	0
$843$	−45.5054	−1.56729
$844$	0	0
$845$	−1.16917	−0.0402206
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−85.6312	−2.93886
$850$	0	0
$851$	−39.6718	−1.35993
$852$	0	0
$853$	−9.45076	−0.323588	−0.161794	−	0.986825i	$-0.551728\pi$
$853$	−9.45076	−0.323588	−0.161794	+	0.986825i	$0.551728\pi$
$854$	0	0
$855$	13.1027	0.448104
$856$	0	0
$857$	−30.3193	−1.03569	−0.517844	−	0.855475i	$-0.673265\pi$
$857$	−30.3193	−1.03569	−0.517844	+	0.855475i	$0.673265\pi$
$858$	0	0
$859$	−19.8096	−0.675896	−0.337948	−	0.941165i	$-0.609733\pi$
$859$	−19.8096	−0.675896	−0.337948	+	0.941165i	$0.609733\pi$
$860$	0	0
$861$	22.2459	0.758139
$862$	0	0
$863$	−49.4083	−1.68188	−0.840940	−	0.541129i	$-0.817997\pi$
$863$	−49.4083	−1.68188	−0.840940	+	0.541129i	$0.817997\pi$
$864$	0	0
$865$	7.96301	0.270750
$866$	0	0
$867$	52.3485	1.77785
$868$	0	0
$869$	6.29845	0.213660
$870$	0	0
$871$	4.62609	0.156749
$872$	0	0
$873$	−74.2819	−2.51406
$874$	0	0
$875$	−10.0935	−0.341222
$876$	0	0
$877$	50.9650	1.72097	0.860483	−	0.509480i	$-0.170162\pi$
$877$	50.9650	1.72097	0.860483	+	0.509480i	$0.170162\pi$
$878$	0	0
$879$	−49.5650	−1.67179
$880$	0	0
$881$	23.3840	0.787829	0.393914	−	0.919147i	$-0.371121\pi$
$881$	23.3840	0.787829	0.393914	+	0.919147i	$0.371121\pi$
$882$	0	0
$883$	25.0697	0.843662	0.421831	−	0.906675i	$-0.361388\pi$
$883$	25.0697	0.843662	0.421831	+	0.906675i	$0.361388\pi$
$884$	0	0
$885$	−6.71775	−0.225815
$886$	0	0
$887$	−7.19577	−0.241610	−0.120805	−	0.992676i	$-0.538548\pi$
$887$	−7.19577	−0.241610	−0.120805	+	0.992676i	$0.538548\pi$
$888$	0	0
$889$	−18.4815	−0.619848
$890$	0	0
$891$	4.49313	0.150525
$892$	0	0
$893$	−10.6737	−0.357181
$894$	0	0
$895$	−6.93182	−0.231705
$896$	0	0
$897$	−15.1539	−0.505973
$898$	0	0
$899$	−3.96224	−0.132148
$900$	0	0
$901$	23.3481	0.777839
$902$	0	0
$903$	5.57881	0.185651
$904$	0	0
$905$	−11.6212	−0.386302
$906$	0	0
$907$	−16.8870	−0.560724	−0.280362	−	0.959894i	$-0.590454\pi$
$907$	−16.8870	−0.560724	−0.280362	+	0.959894i	$0.590454\pi$
$908$	0	0
$909$	100.126	3.32098
$910$	0	0
$911$	18.0905	0.599366	0.299683	−	0.954039i	$-0.403119\pi$
$911$	18.0905	0.599366	0.299683	+	0.954039i	$0.403119\pi$
$912$	0	0
$913$	−17.2308	−0.570258
$914$	0	0
$915$	27.1800	0.898544
$916$	0	0
$917$	13.7542	0.454204
$918$	0	0
$919$	38.7151	1.27709	0.638547	−	0.769583i	$-0.279536\pi$
$919$	38.7151	1.27709	0.638547	+	0.769583i	$0.279536\pi$
$920$	0	0
$921$	69.3639	2.28562
$922$	0	0
$923$	−11.1079	−0.365620
$924$	0	0
$925$	27.6767	0.910004
$926$	0	0
$927$	−29.8399	−0.980070
$928$	0	0
$929$	48.5134	1.59167	0.795837	−	0.605511i	$-0.207031\pi$
$929$	48.5134	1.59167	0.795837	+	0.605511i	$0.207031\pi$
$930$	0	0
$931$	−2.04963	−0.0671739
$932$	0	0
$933$	67.5732	2.21225
$934$	0	0
$935$	6.91586	0.226173
$936$	0	0
$937$	28.2144	0.921725	0.460863	−	0.887471i	$-0.347540\pi$
$937$	28.2144	0.921725	0.460863	+	0.887471i	$0.347540\pi$
$938$	0	0
$939$	36.2416	1.18270
$940$	0	0
$941$	56.7604	1.85033	0.925167	−	0.379560i	$-0.123924\pi$
$941$	56.7604	1.85033	0.925167	+	0.379560i	$0.123924\pi$
$942$	0	0
$943$	−39.8112	−1.29643
$944$	0	0
$945$	8.39584	0.273117
$946$	0	0
$947$	20.5131	0.666586	0.333293	−	0.942823i	$-0.391840\pi$
$947$	20.5131	0.666586	0.333293	+	0.942823i	$0.391840\pi$
$948$	0	0
$949$	−3.77526	−0.122550
$950$	0	0
$951$	−35.0703	−1.13723
$952$	0	0
$953$	−41.1569	−1.33320	−0.666601	−	0.745415i	$-0.732252\pi$
$953$	−41.1569	−1.33320	−0.666601	+	0.745415i	$0.732252\pi$
$954$	0	0
$955$	−5.54697	−0.179496
$956$	0	0
$957$	−11.9941	−0.387714
$958$	0	0
$959$	9.28537	0.299840
$960$	0	0
$961$	−30.0759	−0.970191
$962$	0	0
$963$	−23.9796	−0.772732
$964$	0	0
$965$	21.9352	0.706118
$966$	0	0
$967$	−46.8348	−1.50611	−0.753053	−	0.657960i	$-0.771420\pi$
$967$	−46.8348	−1.50611	−0.753053	+	0.657960i	$0.771420\pi$
$968$	0	0
$969$	35.2800	1.13336
$970$	0	0
$971$	−27.0701	−0.868722	−0.434361	−	0.900739i	$-0.643026\pi$
$971$	−27.0701	−0.868722	−0.434361	+	0.900739i	$0.643026\pi$
$972$	0	0
$973$	−9.48179	−0.303972
$974$	0	0
$975$	10.5720	0.338573
$976$	0	0
$977$	35.8623	1.14734	0.573668	−	0.819088i	$-0.305520\pi$
$977$	35.8623	1.14734	0.573668	+	0.819088i	$0.305520\pi$
$978$	0	0
$979$	1.13850	0.0363866
$980$	0	0
$981$	−88.2542	−2.81774
$982$	0	0
$983$	21.0442	0.671207	0.335603	−	0.942003i	$-0.391060\pi$
$983$	21.0442	0.671207	0.335603	+	0.942003i	$0.391060\pi$
$984$	0	0
$985$	27.9186	0.889560
$986$	0	0
$987$	−15.1539	−0.482353
$988$	0	0
$989$	−9.98381	−0.317467
$990$	0	0
$991$	−18.4345	−0.585591	−0.292796	−	0.956175i	$-0.594586\pi$
$991$	−18.4345	−0.585591	−0.292796	+	0.956175i	$0.594586\pi$
$992$	0	0
$993$	−12.5882	−0.399474
$994$	0	0
$995$	2.66649	0.0845333
$996$	0	0
$997$	8.86721	0.280827	0.140414	−	0.990093i	$-0.455157\pi$
$997$	8.86721	0.280827	0.140414	+	0.990093i	$0.455157\pi$
$998$	0	0
$999$	−54.7054	−1.73080

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.v.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.v.1.11	✓	11	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)

\(f(q)\)	\(=\)	\(q+2.90994 q^{3} -1.16917 q^{5} -1.00000 q^{7} +5.46776 q^{9} +O(q^{10})\)	\(q+2.90994 q^{3} -1.16917 q^{5} -1.00000 q^{7} +5.46776 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.40221 q^{15} -5.91520 q^{17} -2.04963 q^{19} -2.90994 q^{21} +5.20762 q^{23} -3.63305 q^{25} +7.18104 q^{27} -4.12176 q^{29} +0.961296 q^{31} +2.90994 q^{33} +1.16917 q^{35} -7.61804 q^{37} -2.90994 q^{39} -7.64480 q^{41} -1.91716 q^{43} -6.39273 q^{45} +5.20762 q^{47} +1.00000 q^{49} -17.2129 q^{51} -3.94714 q^{53} -1.16917 q^{55} -5.96430 q^{57} +1.97452 q^{59} -7.98894 q^{61} -5.46776 q^{63} +1.16917 q^{65} -4.62609 q^{67} +15.1539 q^{69} +11.1079 q^{71} +3.77526 q^{73} -10.5720 q^{75} -1.00000 q^{77} +6.29845 q^{79} +4.49313 q^{81} -17.2308 q^{83} +6.91586 q^{85} -11.9941 q^{87} +1.13850 q^{89} +1.00000 q^{91} +2.79732 q^{93} +2.39636 q^{95} -13.5854 q^{97} +5.46776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more