LMFDB - Embedded newform 8008.2.a.n.1.2

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:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 63x^{3} + 282x^{2} + 3x - 116 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 63x^3 + 282x^2 + 3x - 116
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.00639$ 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)

$f(q)$	$=$	$q-3.00639 q^{3} +2.62191 q^{5} -1.00000 q^{7} +6.03838 q^{9} +O(q^{10})$	$q-3.00639 q^{3} +2.62191 q^{5} -1.00000 q^{7} +6.03838 q^{9} +1.00000 q^{11} -1.00000 q^{13} -7.88249 q^{15} -0.640640 q^{17} -5.26451 q^{19} +3.00639 q^{21} +3.73795 q^{23} +1.87442 q^{25} -9.13457 q^{27} -2.10721 q^{29} -10.8285 q^{31} -3.00639 q^{33} -2.62191 q^{35} -5.89549 q^{37} +3.00639 q^{39} +10.7215 q^{41} +2.17775 q^{43} +15.8321 q^{45} +3.39933 q^{47} +1.00000 q^{49} +1.92601 q^{51} +3.38742 q^{53} +2.62191 q^{55} +15.8272 q^{57} -3.21236 q^{59} +14.8242 q^{61} -6.03838 q^{63} -2.62191 q^{65} +7.51329 q^{67} -11.2377 q^{69} -2.60540 q^{71} -0.811910 q^{73} -5.63524 q^{75} -1.00000 q^{77} +9.43761 q^{79} +9.34693 q^{81} -16.0593 q^{83} -1.67970 q^{85} +6.33511 q^{87} -8.41905 q^{89} +1.00000 q^{91} +32.5546 q^{93} -13.8031 q^{95} +8.56933 q^{97} +6.03838 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * 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
$3$	−3.00639	−1.73574	−0.867870	−	0.496791i	$-0.834512\pi$
$3$	−3.00639	−1.73574	−0.867870	+	0.496791i	$0.834512\pi$
$4$	0	0
$5$	2.62191	1.17255	0.586277	−	0.810110i	$-0.300593\pi$
$5$	2.62191	1.17255	0.586277	+	0.810110i	$0.300593\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.03838	2.01279
$10$	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
$15$	−7.88249	−2.03525
$16$	0	0
$17$	−0.640640	−0.155378	−0.0776890	−	0.996978i	$-0.524754\pi$
$17$	−0.640640	−0.155378	−0.0776890	+	0.996978i	$0.524754\pi$
$18$	0	0
$19$	−5.26451	−1.20776	−0.603881	−	0.797075i	$-0.706380\pi$
$19$	−5.26451	−1.20776	−0.603881	+	0.797075i	$0.706380\pi$
$20$	0	0
$21$	3.00639	0.656048
$22$	0	0
$23$	3.73795	0.779416	0.389708	−	0.920939i	$-0.372576\pi$
$23$	3.73795	0.779416	0.389708	+	0.920939i	$0.372576\pi$
$24$	0	0
$25$	1.87442	0.374884
$26$	0	0
$27$	−9.13457	−1.75795
$28$	0	0
$29$	−2.10721	−0.391300	−0.195650	−	0.980674i	$-0.562682\pi$
$29$	−2.10721	−0.391300	−0.195650	+	0.980674i	$0.562682\pi$
$30$	0	0
$31$	−10.8285	−1.94485	−0.972427	−	0.233210i	$-0.925077\pi$
$31$	−10.8285	−1.94485	−0.972427	+	0.233210i	$0.925077\pi$
$32$	0	0
$33$	−3.00639	−0.523345
$34$	0	0
$35$	−2.62191	−0.443184
$36$	0	0
$37$	−5.89549	−0.969213	−0.484607	−	0.874732i	$-0.661037\pi$
$37$	−5.89549	−0.969213	−0.484607	+	0.874732i	$0.661037\pi$
$38$	0	0
$39$	3.00639	0.481408
$40$	0	0
$41$	10.7215	1.67442	0.837212	−	0.546879i	$-0.184184\pi$
$41$	10.7215	1.67442	0.837212	+	0.546879i	$0.184184\pi$
$42$	0	0
$43$	2.17775	0.332104	0.166052	−	0.986117i	$-0.446898\pi$
$43$	2.17775	0.332104	0.166052	+	0.986117i	$0.446898\pi$
$44$	0	0
$45$	15.8321	2.36011
$46$	0	0
$47$	3.39933	0.495843	0.247921	−	0.968780i	$-0.420253\pi$
$47$	3.39933	0.495843	0.247921	+	0.968780i	$0.420253\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.92601	0.269696
$52$	0	0
$53$	3.38742	0.465298	0.232649	−	0.972561i	$-0.425261\pi$
$53$	3.38742	0.465298	0.232649	+	0.972561i	$0.425261\pi$
$54$	0	0
$55$	2.62191	0.353539
$56$	0	0
$57$	15.8272	2.09636
$58$	0	0
$59$	−3.21236	−0.418213	−0.209107	−	0.977893i	$-0.567056\pi$
$59$	−3.21236	−0.418213	−0.209107	+	0.977893i	$0.567056\pi$
$60$	0	0
$61$	14.8242	1.89805	0.949024	−	0.315203i	$-0.102073\pi$
$61$	14.8242	1.89805	0.949024	+	0.315203i	$0.102073\pi$
$62$	0	0
$63$	−6.03838	−0.760765
$64$	0	0
$65$	−2.62191	−0.325208
$66$	0	0
$67$	7.51329	0.917895	0.458947	−	0.888463i	$-0.348227\pi$
$67$	7.51329	0.917895	0.458947	+	0.888463i	$0.348227\pi$
$68$	0	0
$69$	−11.2377	−1.35286
$70$	0	0
$71$	−2.60540	−0.309205	−0.154602	−	0.987977i	$-0.549410\pi$
$71$	−2.60540	−0.309205	−0.154602	+	0.987977i	$0.549410\pi$
$72$	0	0
$73$	−0.811910	−0.0950269	−0.0475135	−	0.998871i	$-0.515130\pi$
$73$	−0.811910	−0.0950269	−0.0475135	+	0.998871i	$0.515130\pi$
$74$	0	0
$75$	−5.63524	−0.650702
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	9.43761	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$79$	9.43761	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$80$	0	0
$81$	9.34693	1.03855
$82$	0	0
$83$	−16.0593	−1.76273	−0.881367	−	0.472432i	$-0.843376\pi$
$83$	−16.0593	−1.76273	−0.881367	+	0.472432i	$0.843376\pi$
$84$	0	0
$85$	−1.67970	−0.182189
$86$	0	0
$87$	6.33511	0.679195
$88$	0	0
$89$	−8.41905	−0.892417	−0.446209	−	0.894929i	$-0.647226\pi$
$89$	−8.41905	−0.892417	−0.446209	+	0.894929i	$0.647226\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	32.5546	3.37576
$94$	0	0
$95$	−13.8031	−1.41617
$96$	0	0
$97$	8.56933	0.870084	0.435042	−	0.900410i	$-0.356734\pi$
$97$	8.56933	0.870084	0.435042	+	0.900410i	$0.356734\pi$
$98$	0	0
$99$	6.03838	0.606880
$100$	0	0
$101$	−16.7195	−1.66365	−0.831827	−	0.555034i	$-0.812705\pi$
$101$	−16.7195	−1.66365	−0.831827	+	0.555034i	$0.812705\pi$
$102$	0	0
$103$	4.15665	0.409567	0.204784	−	0.978807i	$-0.434351\pi$
$103$	4.15665	0.409567	0.204784	+	0.978807i	$0.434351\pi$
$104$	0	0
$105$	7.88249	0.769252
$106$	0	0
$107$	11.7552	1.13642	0.568209	−	0.822884i	$-0.307636\pi$
$107$	11.7552	1.13642	0.568209	+	0.822884i	$0.307636\pi$
$108$	0	0
$109$	18.0693	1.73073	0.865363	−	0.501145i	$-0.167088\pi$
$109$	18.0693	1.73073	0.865363	+	0.501145i	$0.167088\pi$
$110$	0	0
$111$	17.7242	1.68230
$112$	0	0
$113$	0.147481	0.0138739	0.00693694	−	0.999976i	$-0.497792\pi$
$113$	0.147481	0.0138739	0.00693694	+	0.999976i	$0.497792\pi$
$114$	0	0
$115$	9.80057	0.913908
$116$	0	0
$117$	−6.03838	−0.558249
$118$	0	0
$119$	0.640640	0.0587273
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−32.2331	−2.90636
$124$	0	0
$125$	−8.19499	−0.732982
$126$	0	0
$127$	−9.12294	−0.809530	−0.404765	−	0.914421i	$-0.632647\pi$
$127$	−9.12294	−0.809530	−0.404765	+	0.914421i	$0.632647\pi$
$128$	0	0
$129$	−6.54718	−0.576447
$130$	0	0
$131$	0.635495	0.0555235	0.0277617	−	0.999615i	$-0.491162\pi$
$131$	0.635495	0.0555235	0.0277617	+	0.999615i	$0.491162\pi$
$132$	0	0
$133$	5.26451	0.456491
$134$	0	0
$135$	−23.9500	−2.06129
$136$	0	0
$137$	16.6993	1.42672	0.713358	−	0.700799i	$-0.247173\pi$
$137$	16.6993	1.42672	0.713358	+	0.700799i	$0.247173\pi$
$138$	0	0
$139$	−2.98332	−0.253042	−0.126521	−	0.991964i	$-0.540381\pi$
$139$	−2.98332	−0.253042	−0.126521	+	0.991964i	$0.540381\pi$
$140$	0	0
$141$	−10.2197	−0.860654
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−5.52493	−0.458821
$146$	0	0
$147$	−3.00639	−0.247963
$148$	0	0
$149$	−10.1030	−0.827671	−0.413835	−	0.910352i	$-0.635811\pi$
$149$	−10.1030	−0.827671	−0.413835	+	0.910352i	$0.635811\pi$
$150$	0	0
$151$	−9.67817	−0.787598	−0.393799	−	0.919196i	$-0.628839\pi$
$151$	−9.67817	−0.787598	−0.393799	+	0.919196i	$0.628839\pi$
$152$	0	0
$153$	−3.86843	−0.312744
$154$	0	0
$155$	−28.3913	−2.28045
$156$	0	0
$157$	−8.82238	−0.704102	−0.352051	−	0.935981i	$-0.614516\pi$
$157$	−8.82238	−0.704102	−0.352051	+	0.935981i	$0.614516\pi$
$158$	0	0
$159$	−10.1839	−0.807637
$160$	0	0
$161$	−3.73795	−0.294591
$162$	0	0
$163$	3.41996	0.267872	0.133936	−	0.990990i	$-0.457238\pi$
$163$	3.41996	0.267872	0.133936	+	0.990990i	$0.457238\pi$
$164$	0	0
$165$	−7.88249	−0.613651
$166$	0	0
$167$	1.94831	0.150764	0.0753822	−	0.997155i	$-0.475982\pi$
$167$	1.94831	0.150764	0.0753822	+	0.997155i	$0.475982\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−31.7891	−2.43098
$172$	0	0
$173$	24.9934	1.90022	0.950108	−	0.311922i	$-0.100973\pi$
$173$	24.9934	1.90022	0.950108	+	0.311922i	$0.100973\pi$
$174$	0	0
$175$	−1.87442	−0.141693
$176$	0	0
$177$	9.65760	0.725910
$178$	0	0
$179$	10.4326	0.779766	0.389883	−	0.920864i	$-0.372515\pi$
$179$	10.4326	0.779766	0.389883	+	0.920864i	$0.372515\pi$
$180$	0	0
$181$	−3.34840	−0.248884	−0.124442	−	0.992227i	$-0.539714\pi$
$181$	−3.34840	−0.248884	−0.124442	+	0.992227i	$0.539714\pi$
$182$	0	0
$183$	−44.5674	−3.29452
$184$	0	0
$185$	−15.4575	−1.13646
$186$	0	0
$187$	−0.640640	−0.0468482
$188$	0	0
$189$	9.13457	0.664442
$190$	0	0
$191$	0.0717404	0.00519096	0.00259548	−	0.999997i	$-0.499174\pi$
$191$	0.0717404	0.00519096	0.00259548	+	0.999997i	$0.499174\pi$
$192$	0	0
$193$	13.7086	0.986768	0.493384	−	0.869812i	$-0.335760\pi$
$193$	13.7086	0.986768	0.493384	+	0.869812i	$0.335760\pi$
$194$	0	0
$195$	7.88249	0.564477
$196$	0	0
$197$	−10.9890	−0.782934	−0.391467	−	0.920192i	$-0.628032\pi$
$197$	−10.9890	−0.782934	−0.391467	+	0.920192i	$0.628032\pi$
$198$	0	0
$199$	26.7225	1.89431	0.947154	−	0.320779i	$-0.103945\pi$
$199$	26.7225	1.89431	0.947154	+	0.320779i	$0.103945\pi$
$200$	0	0
$201$	−22.5879	−1.59323
$202$	0	0
$203$	2.10721	0.147897
$204$	0	0
$205$	28.1109	1.96335
$206$	0	0
$207$	22.5712	1.56880
$208$	0	0
$209$	−5.26451	−0.364154
$210$	0	0
$211$	−17.8725	−1.23040	−0.615198	−	0.788373i	$-0.710924\pi$
$211$	−17.8725	−1.23040	−0.615198	+	0.788373i	$0.710924\pi$
$212$	0	0
$213$	7.83286	0.536699
$214$	0	0
$215$	5.70988	0.389411
$216$	0	0
$217$	10.8285	0.735085
$218$	0	0
$219$	2.44092	0.164942
$220$	0	0
$221$	0.640640	0.0430941
$222$	0	0
$223$	28.5819	1.91399	0.956993	−	0.290110i	$-0.0936920\pi$
$223$	28.5819	1.91399	0.956993	+	0.290110i	$0.0936920\pi$
$224$	0	0
$225$	11.3185	0.754565
$226$	0	0
$227$	7.63926	0.507036	0.253518	−	0.967331i	$-0.418412\pi$
$227$	7.63926	0.507036	0.253518	+	0.967331i	$0.418412\pi$
$228$	0	0
$229$	−7.03493	−0.464881	−0.232441	−	0.972611i	$-0.574671\pi$
$229$	−7.03493	−0.464881	−0.232441	+	0.972611i	$0.574671\pi$
$230$	0	0
$231$	3.00639	0.197806
$232$	0	0
$233$	−11.8395	−0.775630	−0.387815	−	0.921737i	$-0.626770\pi$
$233$	−11.8395	−0.775630	−0.387815	+	0.921737i	$0.626770\pi$
$234$	0	0
$235$	8.91273	0.581402
$236$	0	0
$237$	−28.3732	−1.84303
$238$	0	0
$239$	3.04360	0.196874	0.0984370	−	0.995143i	$-0.468616\pi$
$239$	3.04360	0.196874	0.0984370	+	0.995143i	$0.468616\pi$
$240$	0	0
$241$	15.3595	0.989394	0.494697	−	0.869066i	$-0.335279\pi$
$241$	15.3595	0.989394	0.494697	+	0.869066i	$0.335279\pi$
$242$	0	0
$243$	−0.696807	−0.0447002
$244$	0	0
$245$	2.62191	0.167508
$246$	0	0
$247$	5.26451	0.334973
$248$	0	0
$249$	48.2804	3.05965
$250$	0	0
$251$	−23.1777	−1.46297	−0.731483	−	0.681859i	$-0.761172\pi$
$251$	−23.1777	−1.46297	−0.731483	+	0.681859i	$0.761172\pi$
$252$	0	0
$253$	3.73795	0.235003
$254$	0	0
$255$	5.04984	0.316233
$256$	0	0
$257$	−11.4778	−0.715964	−0.357982	−	0.933728i	$-0.616535\pi$
$257$	−11.4778	−0.715964	−0.357982	+	0.933728i	$0.616535\pi$
$258$	0	0
$259$	5.89549	0.366328
$260$	0	0
$261$	−12.7242	−0.787606
$262$	0	0
$263$	−18.2082	−1.12277	−0.561383	−	0.827556i	$-0.689730\pi$
$263$	−18.2082	−1.12277	−0.561383	+	0.827556i	$0.689730\pi$
$264$	0	0
$265$	8.88152	0.545587
$266$	0	0
$267$	25.3109	1.54900
$268$	0	0
$269$	−8.10349	−0.494078	−0.247039	−	0.969005i	$-0.579458\pi$
$269$	−8.10349	−0.494078	−0.247039	+	0.969005i	$0.579458\pi$
$270$	0	0
$271$	16.4840	1.00133	0.500667	−	0.865640i	$-0.333088\pi$
$271$	16.4840	1.00133	0.500667	+	0.865640i	$0.333088\pi$
$272$	0	0
$273$	−3.00639	−0.181955
$274$	0	0
$275$	1.87442	0.113032
$276$	0	0
$277$	29.0211	1.74371	0.871855	−	0.489764i	$-0.162917\pi$
$277$	29.0211	1.74371	0.871855	+	0.489764i	$0.162917\pi$
$278$	0	0
$279$	−65.3865	−3.91459
$280$	0	0
$281$	−12.4263	−0.741293	−0.370646	−	0.928774i	$-0.620864\pi$
$281$	−12.4263	−0.741293	−0.370646	+	0.928774i	$0.620864\pi$
$282$	0	0
$283$	−15.1671	−0.901593	−0.450796	−	0.892627i	$-0.648860\pi$
$283$	−15.1671	−0.901593	−0.450796	+	0.892627i	$0.648860\pi$
$284$	0	0
$285$	41.4975	2.45810
$286$	0	0
$287$	−10.7215	−0.632873
$288$	0	0
$289$	−16.5896	−0.975858
$290$	0	0
$291$	−25.7628	−1.51024
$292$	0	0
$293$	13.6503	0.797456	0.398728	−	0.917069i	$-0.369452\pi$
$293$	13.6503	0.797456	0.398728	+	0.917069i	$0.369452\pi$
$294$	0	0
$295$	−8.42252	−0.490378
$296$	0	0
$297$	−9.13457	−0.530041
$298$	0	0
$299$	−3.73795	−0.216171
$300$	0	0
$301$	−2.17775	−0.125524
$302$	0	0
$303$	50.2654	2.88767
$304$	0	0
$305$	38.8678	2.22557
$306$	0	0
$307$	−8.36347	−0.477328	−0.238664	−	0.971102i	$-0.576710\pi$
$307$	−8.36347	−0.477328	−0.238664	+	0.971102i	$0.576710\pi$
$308$	0	0
$309$	−12.4965	−0.710902
$310$	0	0
$311$	10.0374	0.569169	0.284585	−	0.958651i	$-0.408144\pi$
$311$	10.0374	0.569169	0.284585	+	0.958651i	$0.408144\pi$
$312$	0	0
$313$	10.2185	0.577583	0.288792	−	0.957392i	$-0.406747\pi$
$313$	10.2185	0.577583	0.288792	+	0.957392i	$0.406747\pi$
$314$	0	0
$315$	−15.8321	−0.892038
$316$	0	0
$317$	20.0912	1.12844	0.564218	−	0.825626i	$-0.309178\pi$
$317$	20.0912	1.12844	0.564218	+	0.825626i	$0.309178\pi$
$318$	0	0
$319$	−2.10721	−0.117981
$320$	0	0
$321$	−35.3407	−1.97253
$322$	0	0
$323$	3.37265	0.187660
$324$	0	0
$325$	−1.87442	−0.103974
$326$	0	0
$327$	−54.3234	−3.00409
$328$	0	0
$329$	−3.39933	−0.187411
$330$	0	0
$331$	−0.159166	−0.00874854	−0.00437427	−	0.999990i	$-0.501392\pi$
$331$	−0.159166	−0.00874854	−0.00437427	+	0.999990i	$0.501392\pi$
$332$	0	0
$333$	−35.5993	−1.95083
$334$	0	0
$335$	19.6992	1.07628
$336$	0	0
$337$	−17.1801	−0.935858	−0.467929	−	0.883766i	$-0.655000\pi$
$337$	−17.1801	−0.935858	−0.467929	+	0.883766i	$0.655000\pi$
$338$	0	0
$339$	−0.443387	−0.0240815
$340$	0	0
$341$	−10.8285	−0.586395
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−29.4643	−1.58631
$346$	0	0
$347$	22.3482	1.19971	0.599857	−	0.800107i	$-0.295224\pi$
$347$	22.3482	1.19971	0.599857	+	0.800107i	$0.295224\pi$
$348$	0	0
$349$	−22.6192	−1.21078	−0.605389	−	0.795930i	$-0.706982\pi$
$349$	−22.6192	−1.21078	−0.605389	+	0.795930i	$0.706982\pi$
$350$	0	0
$351$	9.13457	0.487567
$352$	0	0
$353$	−10.6490	−0.566788	−0.283394	−	0.959004i	$-0.591460\pi$
$353$	−10.6490	−0.566788	−0.283394	+	0.959004i	$0.591460\pi$
$354$	0	0
$355$	−6.83114	−0.362559
$356$	0	0
$357$	−1.92601	−0.101935
$358$	0	0
$359$	17.2862	0.912332	0.456166	−	0.889895i	$-0.349222\pi$
$359$	17.2862	0.912332	0.456166	+	0.889895i	$0.349222\pi$
$360$	0	0
$361$	8.71508	0.458688
$362$	0	0
$363$	−3.00639	−0.157795
$364$	0	0
$365$	−2.12876	−0.111424
$366$	0	0
$367$	−15.6465	−0.816739	−0.408370	−	0.912817i	$-0.633903\pi$
$367$	−15.6465	−0.816739	−0.408370	+	0.912817i	$0.633903\pi$
$368$	0	0
$369$	64.7408	3.37027
$370$	0	0
$371$	−3.38742	−0.175866
$372$	0	0
$373$	27.8567	1.44236	0.721182	−	0.692746i	$-0.243599\pi$
$373$	27.8567	1.44236	0.721182	+	0.692746i	$0.243599\pi$
$374$	0	0
$375$	24.6373	1.27227
$376$	0	0
$377$	2.10721	0.108527
$378$	0	0
$379$	28.8132	1.48004	0.740019	−	0.672586i	$-0.234817\pi$
$379$	28.8132	1.48004	0.740019	+	0.672586i	$0.234817\pi$
$380$	0	0
$381$	27.4271	1.40513
$382$	0	0
$383$	5.23069	0.267276	0.133638	−	0.991030i	$-0.457334\pi$
$383$	5.23069	0.267276	0.133638	+	0.991030i	$0.457334\pi$
$384$	0	0
$385$	−2.62191	−0.133625
$386$	0	0
$387$	13.1501	0.668458
$388$	0	0
$389$	−17.6693	−0.895868	−0.447934	−	0.894067i	$-0.647840\pi$
$389$	−17.6693	−0.895868	−0.447934	+	0.894067i	$0.647840\pi$
$390$	0	0
$391$	−2.39468	−0.121104
$392$	0	0
$393$	−1.91055	−0.0963743
$394$	0	0
$395$	24.7446	1.24504
$396$	0	0
$397$	−2.47696	−0.124315	−0.0621575	−	0.998066i	$-0.519798\pi$
$397$	−2.47696	−0.124315	−0.0621575	+	0.998066i	$0.519798\pi$
$398$	0	0
$399$	−15.8272	−0.792350
$400$	0	0
$401$	39.0987	1.95250	0.976248	−	0.216656i	$-0.0695151\pi$
$401$	39.0987	1.95250	0.976248	+	0.216656i	$0.0695151\pi$
$402$	0	0
$403$	10.8285	0.539405
$404$	0	0
$405$	24.5068	1.21775
$406$	0	0
$407$	−5.89549	−0.292229
$408$	0	0
$409$	−3.61951	−0.178973	−0.0894866	−	0.995988i	$-0.528523\pi$
$409$	−3.61951	−0.178973	−0.0894866	+	0.995988i	$0.528523\pi$
$410$	0	0
$411$	−50.2046	−2.47641
$412$	0	0
$413$	3.21236	0.158070
$414$	0	0
$415$	−42.1060	−2.06690
$416$	0	0
$417$	8.96903	0.439215
$418$	0	0
$419$	−21.4317	−1.04701	−0.523503	−	0.852024i	$-0.675375\pi$
$419$	−21.4317	−1.04701	−0.523503	+	0.852024i	$0.675375\pi$
$420$	0	0
$421$	1.84175	0.0897614	0.0448807	−	0.998992i	$-0.485709\pi$
$421$	1.84175	0.0897614	0.0448807	+	0.998992i	$0.485709\pi$
$422$	0	0
$423$	20.5264	0.998029
$424$	0	0
$425$	−1.20083	−0.0582488
$426$	0	0
$427$	−14.8242	−0.717395
$428$	0	0
$429$	3.00639	0.145150
$430$	0	0
$431$	35.4834	1.70918	0.854589	−	0.519306i	$-0.173809\pi$
$431$	35.4834	1.70918	0.854589	+	0.519306i	$0.173809\pi$
$432$	0	0
$433$	−2.68908	−0.129229	−0.0646144	−	0.997910i	$-0.520582\pi$
$433$	−2.68908	−0.129229	−0.0646144	+	0.997910i	$0.520582\pi$
$434$	0	0
$435$	16.6101	0.796393
$436$	0	0
$437$	−19.6785	−0.941349
$438$	0	0
$439$	−25.3807	−1.21135	−0.605677	−	0.795710i	$-0.707098\pi$
$439$	−25.3807	−1.21135	−0.605677	+	0.795710i	$0.707098\pi$
$440$	0	0
$441$	6.03838	0.287542
$442$	0	0
$443$	−3.24993	−0.154409	−0.0772044	−	0.997015i	$-0.524599\pi$
$443$	−3.24993	−0.154409	−0.0772044	+	0.997015i	$0.524599\pi$
$444$	0	0
$445$	−22.0740	−1.04641
$446$	0	0
$447$	30.3736	1.43662
$448$	0	0
$449$	30.7838	1.45278	0.726390	−	0.687283i	$-0.241197\pi$
$449$	30.7838	1.45278	0.726390	+	0.687283i	$0.241197\pi$
$450$	0	0
$451$	10.7215	0.504858
$452$	0	0
$453$	29.0964	1.36707
$454$	0	0
$455$	2.62191	0.122917
$456$	0	0
$457$	5.91634	0.276755	0.138377	−	0.990380i	$-0.455811\pi$
$457$	5.91634	0.276755	0.138377	+	0.990380i	$0.455811\pi$
$458$	0	0
$459$	5.85197	0.273146
$460$	0	0
$461$	−0.223628	−0.0104154	−0.00520771	−	0.999986i	$-0.501658\pi$
$461$	−0.223628	−0.0104154	−0.00520771	+	0.999986i	$0.501658\pi$
$462$	0	0
$463$	−2.64416	−0.122885	−0.0614423	−	0.998111i	$-0.519570\pi$
$463$	−2.64416	−0.122885	−0.0614423	+	0.998111i	$0.519570\pi$
$464$	0	0
$465$	85.3554	3.95826
$466$	0	0
$467$	18.9325	0.876093	0.438046	−	0.898952i	$-0.355671\pi$
$467$	18.9325	0.876093	0.438046	+	0.898952i	$0.355671\pi$
$468$	0	0
$469$	−7.51329	−0.346932
$470$	0	0
$471$	26.5235	1.22214
$472$	0	0
$473$	2.17775	0.100133
$474$	0	0
$475$	−9.86792	−0.452771
$476$	0	0
$477$	20.4546	0.936550
$478$	0	0
$479$	26.3722	1.20498	0.602488	−	0.798128i	$-0.294176\pi$
$479$	26.3722	1.20498	0.602488	+	0.798128i	$0.294176\pi$
$480$	0	0
$481$	5.89549	0.268811
$482$	0	0
$483$	11.2377	0.511334
$484$	0	0
$485$	22.4680	1.02022
$486$	0	0
$487$	19.2556	0.872553	0.436276	−	0.899813i	$-0.356297\pi$
$487$	19.2556	0.872553	0.436276	+	0.899813i	$0.356297\pi$
$488$	0	0
$489$	−10.2817	−0.464956
$490$	0	0
$491$	23.3647	1.05443	0.527217	−	0.849731i	$-0.323236\pi$
$491$	23.3647	1.05443	0.527217	+	0.849731i	$0.323236\pi$
$492$	0	0
$493$	1.34997	0.0607994
$494$	0	0
$495$	15.8321	0.711600
$496$	0	0
$497$	2.60540	0.116868
$498$	0	0
$499$	−25.8930	−1.15913	−0.579565	−	0.814926i	$-0.696777\pi$
$499$	−25.8930	−1.15913	−0.579565	+	0.814926i	$0.696777\pi$
$500$	0	0
$501$	−5.85737	−0.261688
$502$	0	0
$503$	38.4397	1.71394	0.856972	−	0.515363i	$-0.172343\pi$
$503$	38.4397	1.71394	0.856972	+	0.515363i	$0.172343\pi$
$504$	0	0
$505$	−43.8371	−1.95073
$506$	0	0
$507$	−3.00639	−0.133518
$508$	0	0
$509$	−0.0129086	−0.000572166 0	−0.000286083	−	1.00000i	$-0.500091\pi$
$509$	−0.0129086	−0.000572166 0	−0.000286083		1.00000i	$0.500091\pi$
$510$	0	0
$511$	0.811910	0.0359168
$512$	0	0
$513$	48.0890	2.12318
$514$	0	0
$515$	10.8984	0.480240
$516$	0	0
$517$	3.39933	0.149502
$518$	0	0
$519$	−75.1400	−3.29828
$520$	0	0
$521$	−12.5988	−0.551965	−0.275983	−	0.961163i	$-0.589003\pi$
$521$	−12.5988	−0.551965	−0.275983	+	0.961163i	$0.589003\pi$
$522$	0	0
$523$	31.5723	1.38056	0.690280	−	0.723542i	$-0.257487\pi$
$523$	31.5723	1.38056	0.690280	+	0.723542i	$0.257487\pi$
$524$	0	0
$525$	5.63524	0.245942
$526$	0	0
$527$	6.93715	0.302187
$528$	0	0
$529$	−9.02775	−0.392511
$530$	0	0
$531$	−19.3975	−0.841778
$532$	0	0
$533$	−10.7215	−0.464402
$534$	0	0
$535$	30.8211	1.33251
$536$	0	0
$537$	−31.3643	−1.35347
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	35.7034	1.53501	0.767504	−	0.641045i	$-0.221499\pi$
$541$	35.7034	1.53501	0.767504	+	0.641045i	$0.221499\pi$
$542$	0	0
$543$	10.0666	0.431998
$544$	0	0
$545$	47.3762	2.02937
$546$	0	0
$547$	28.6308	1.22416	0.612082	−	0.790794i	$-0.290332\pi$
$547$	28.6308	1.22416	0.612082	+	0.790794i	$0.290332\pi$
$548$	0	0
$549$	89.5144	3.82038
$550$	0	0
$551$	11.0935	0.472597
$552$	0	0
$553$	−9.43761	−0.401328
$554$	0	0
$555$	46.4712	1.97259
$556$	0	0
$557$	23.4323	0.992860	0.496430	−	0.868077i	$-0.334644\pi$
$557$	23.4323	0.992860	0.496430	+	0.868077i	$0.334644\pi$
$558$	0	0
$559$	−2.17775	−0.0921092
$560$	0	0
$561$	1.92601	0.0813163
$562$	0	0
$563$	27.8530	1.17386	0.586932	−	0.809636i	$-0.300336\pi$
$563$	27.8530	1.17386	0.586932	+	0.809636i	$0.300336\pi$
$564$	0	0
$565$	0.386683	0.0162679
$566$	0	0
$567$	−9.34693	−0.392534
$568$	0	0
$569$	20.2369	0.848374	0.424187	−	0.905575i	$-0.360560\pi$
$569$	20.2369	0.848374	0.424187	+	0.905575i	$0.360560\pi$
$570$	0	0
$571$	−10.7462	−0.449716	−0.224858	−	0.974392i	$-0.572192\pi$
$571$	−10.7462	−0.449716	−0.224858	+	0.974392i	$0.572192\pi$
$572$	0	0
$573$	−0.215680	−0.00901015
$574$	0	0
$575$	7.00649	0.292191
$576$	0	0
$577$	40.2497	1.67561	0.837807	−	0.545966i	$-0.183837\pi$
$577$	40.2497	1.67561	0.837807	+	0.545966i	$0.183837\pi$
$578$	0	0
$579$	−41.2135	−1.71277
$580$	0	0
$581$	16.0593	0.666251
$582$	0	0
$583$	3.38742	0.140293
$584$	0	0
$585$	−15.8321	−0.654577
$586$	0	0
$587$	32.0025	1.32088	0.660442	−	0.750877i	$-0.270369\pi$
$587$	32.0025	1.32088	0.660442	+	0.750877i	$0.270369\pi$
$588$	0	0
$589$	57.0067	2.34892
$590$	0	0
$591$	33.0372	1.35897
$592$	0	0
$593$	−20.1994	−0.829491	−0.414746	−	0.909937i	$-0.636129\pi$
$593$	−20.1994	−0.829491	−0.414746	+	0.909937i	$0.636129\pi$
$594$	0	0
$595$	1.67970	0.0688610
$596$	0	0
$597$	−80.3383	−3.28803
$598$	0	0
$599$	2.86763	0.117168	0.0585840	−	0.998282i	$-0.481341\pi$
$599$	2.86763	0.117168	0.0585840	+	0.998282i	$0.481341\pi$
$600$	0	0
$601$	23.3415	0.952119	0.476059	−	0.879413i	$-0.342065\pi$
$601$	23.3415	0.952119	0.476059	+	0.879413i	$0.342065\pi$
$602$	0	0
$603$	45.3682	1.84753
$604$	0	0
$605$	2.62191	0.106596
$606$	0	0
$607$	2.78759	0.113145	0.0565724	−	0.998398i	$-0.481983\pi$
$607$	2.78759	0.113145	0.0565724	+	0.998398i	$0.481983\pi$
$608$	0	0
$609$	−6.33511	−0.256712
$610$	0	0
$611$	−3.39933	−0.137522
$612$	0	0
$613$	−24.0657	−0.972004	−0.486002	−	0.873958i	$-0.661545\pi$
$613$	−24.0657	−0.972004	−0.486002	+	0.873958i	$0.661545\pi$
$614$	0	0
$615$	−84.5125	−3.40787
$616$	0	0
$617$	11.5876	0.466501	0.233250	−	0.972417i	$-0.425064\pi$
$617$	11.5876	0.466501	0.233250	+	0.972417i	$0.425064\pi$
$618$	0	0
$619$	28.4011	1.14154	0.570768	−	0.821111i	$-0.306645\pi$
$619$	28.4011	1.14154	0.570768	+	0.821111i	$0.306645\pi$
$620$	0	0
$621$	−34.1445	−1.37017
$622$	0	0
$623$	8.41905	0.337302
$624$	0	0
$625$	−30.8587	−1.23435
$626$	0	0
$627$	15.8272	0.632077
$628$	0	0
$629$	3.77689	0.150594
$630$	0	0
$631$	39.5351	1.57387	0.786934	−	0.617037i	$-0.211667\pi$
$631$	39.5351	1.57387	0.786934	+	0.617037i	$0.211667\pi$
$632$	0	0
$633$	53.7318	2.13565
$634$	0	0
$635$	−23.9195	−0.949218
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−15.7324	−0.622365
$640$	0	0
$641$	−26.5690	−1.04941	−0.524706	−	0.851284i	$-0.675825\pi$
$641$	−26.5690	−1.04941	−0.524706	+	0.851284i	$0.675825\pi$
$642$	0	0
$643$	−2.30572	−0.0909287	−0.0454644	−	0.998966i	$-0.514477\pi$
$643$	−2.30572	−0.0909287	−0.0454644	+	0.998966i	$0.514477\pi$
$644$	0	0
$645$	−17.1661	−0.675916
$646$	0	0
$647$	−0.403023	−0.0158445	−0.00792223	−	0.999969i	$-0.502522\pi$
$647$	−0.403023	−0.0158445	−0.00792223	+	0.999969i	$0.502522\pi$
$648$	0	0
$649$	−3.21236	−0.126096
$650$	0	0
$651$	−32.5546	−1.27592
$652$	0	0
$653$	−25.2173	−0.986829	−0.493415	−	0.869794i	$-0.664251\pi$
$653$	−25.2173	−0.986829	−0.493415	+	0.869794i	$0.664251\pi$
$654$	0	0
$655$	1.66621	0.0651043
$656$	0	0
$657$	−4.90263	−0.191270
$658$	0	0
$659$	−8.45316	−0.329289	−0.164644	−	0.986353i	$-0.552648\pi$
$659$	−8.45316	−0.329289	−0.164644	+	0.986353i	$0.552648\pi$
$660$	0	0
$661$	37.9491	1.47605	0.738025	−	0.674773i	$-0.235758\pi$
$661$	37.9491	1.47605	0.738025	+	0.674773i	$0.235758\pi$
$662$	0	0
$663$	−1.92601	−0.0748001
$664$	0	0
$665$	13.8031	0.535261
$666$	0	0
$667$	−7.87666	−0.304985
$668$	0	0
$669$	−85.9284	−3.32218
$670$	0	0
$671$	14.8242	0.572283
$672$	0	0
$673$	41.9745	1.61800	0.808998	−	0.587811i	$-0.200010\pi$
$673$	41.9745	1.61800	0.808998	+	0.587811i	$0.200010\pi$
$674$	0	0
$675$	−17.1220	−0.659027
$676$	0	0
$677$	6.75283	0.259532	0.129766	−	0.991545i	$-0.458577\pi$
$677$	6.75283	0.259532	0.129766	+	0.991545i	$0.458577\pi$
$678$	0	0
$679$	−8.56933	−0.328861
$680$	0	0
$681$	−22.9666	−0.880082
$682$	0	0
$683$	3.11326	0.119126	0.0595628	−	0.998225i	$-0.481029\pi$
$683$	3.11326	0.119126	0.0595628	+	0.998225i	$0.481029\pi$
$684$	0	0
$685$	43.7841	1.67290
$686$	0	0
$687$	21.1498	0.806913
$688$	0	0
$689$	−3.38742	−0.129050
$690$	0	0
$691$	−2.14985	−0.0817842	−0.0408921	−	0.999164i	$-0.513020\pi$
$691$	−2.14985	−0.0817842	−0.0408921	+	0.999164i	$0.513020\pi$
$692$	0	0
$693$	−6.03838	−0.229379
$694$	0	0
$695$	−7.82201	−0.296706
$696$	0	0
$697$	−6.86864	−0.260168
$698$	0	0
$699$	35.5941	1.34629
$700$	0	0
$701$	−34.5343	−1.30434	−0.652171	−	0.758072i	$-0.726142\pi$
$701$	−34.5343	−1.30434	−0.652171	+	0.758072i	$0.726142\pi$
$702$	0	0
$703$	31.0369	1.17058
$704$	0	0
$705$	−26.7951	−1.00916
$706$	0	0
$707$	16.7195	0.628802
$708$	0	0
$709$	21.2520	0.798135	0.399068	−	0.916922i	$-0.369334\pi$
$709$	21.2520	0.798135	0.399068	+	0.916922i	$0.369334\pi$
$710$	0	0
$711$	56.9879	2.13721
$712$	0	0
$713$	−40.4763	−1.51585
$714$	0	0
$715$	−2.62191	−0.0980539
$716$	0	0
$717$	−9.15024	−0.341722
$718$	0	0
$719$	−42.0739	−1.56909	−0.784546	−	0.620070i	$-0.787104\pi$
$719$	−42.0739	−1.56909	−0.784546	+	0.620070i	$0.787104\pi$
$720$	0	0
$721$	−4.15665	−0.154802
$722$	0	0
$723$	−46.1767	−1.71733
$724$	0	0
$725$	−3.94981	−0.146692
$726$	0	0
$727$	11.6948	0.433735	0.216868	−	0.976201i	$-0.430416\pi$
$727$	11.6948	0.433735	0.216868	+	0.976201i	$0.430416\pi$
$728$	0	0
$729$	−25.9459	−0.960959
$730$	0	0
$731$	−1.39516	−0.0516017
$732$	0	0
$733$	−9.01427	−0.332950	−0.166475	−	0.986046i	$-0.553238\pi$
$733$	−9.01427	−0.332950	−0.166475	+	0.986046i	$0.553238\pi$
$734$	0	0
$735$	−7.88249	−0.290750
$736$	0	0
$737$	7.51329	0.276756
$738$	0	0
$739$	−12.8353	−0.472153	−0.236077	−	0.971734i	$-0.575862\pi$
$739$	−12.8353	−0.472153	−0.236077	+	0.971734i	$0.575862\pi$
$740$	0	0
$741$	−15.8272	−0.581426
$742$	0	0
$743$	27.8116	1.02031	0.510154	−	0.860083i	$-0.329588\pi$
$743$	27.8116	1.02031	0.510154	+	0.860083i	$0.329588\pi$
$744$	0	0
$745$	−26.4892	−0.970489
$746$	0	0
$747$	−96.9721	−3.54802
$748$	0	0
$749$	−11.7552	−0.429526
$750$	0	0
$751$	16.1453	0.589149	0.294574	−	0.955629i	$-0.404822\pi$
$751$	16.1453	0.589149	0.294574	+	0.955629i	$0.404822\pi$
$752$	0	0
$753$	69.6814	2.53933
$754$	0	0
$755$	−25.3753	−0.923502
$756$	0	0
$757$	0.449041	0.0163207	0.00816033	−	0.999967i	$-0.497402\pi$
$757$	0.449041	0.0163207	0.00816033	+	0.999967i	$0.497402\pi$
$758$	0	0
$759$	−11.2377	−0.407904
$760$	0	0
$761$	38.6952	1.40270	0.701351	−	0.712816i	$-0.252581\pi$
$761$	38.6952	1.40270	0.701351	+	0.712816i	$0.252581\pi$
$762$	0	0
$763$	−18.0693	−0.654153
$764$	0	0
$765$	−10.1427	−0.366709
$766$	0	0
$767$	3.21236	0.115992
$768$	0	0
$769$	−19.9355	−0.718891	−0.359446	−	0.933166i	$-0.617034\pi$
$769$	−19.9355	−0.718891	−0.359446	+	0.933166i	$0.617034\pi$
$770$	0	0
$771$	34.5067	1.24273
$772$	0	0
$773$	−34.0010	−1.22293	−0.611466	−	0.791270i	$-0.709420\pi$
$773$	−34.0010	−1.22293	−0.611466	+	0.791270i	$0.709420\pi$
$774$	0	0
$775$	−20.2971	−0.729095
$776$	0	0
$777$	−17.7242	−0.635851
$778$	0	0
$779$	−56.4437	−2.02230
$780$	0	0
$781$	−2.60540	−0.0932287
$782$	0	0
$783$	19.2485	0.687885
$784$	0	0
$785$	−23.1315	−0.825599
$786$	0	0
$787$	47.6131	1.69722	0.848612	−	0.529016i	$-0.177439\pi$
$787$	47.6131	1.69722	0.848612	+	0.529016i	$0.177439\pi$
$788$	0	0
$789$	54.7409	1.94883
$790$	0	0
$791$	−0.147481	−0.00524384
$792$	0	0
$793$	−14.8242	−0.526424
$794$	0	0
$795$	−26.7013	−0.946998
$796$	0	0
$797$	20.4579	0.724657	0.362328	−	0.932050i	$-0.381982\pi$
$797$	20.4579	0.724657	0.362328	+	0.932050i	$0.381982\pi$
$798$	0	0
$799$	−2.17774	−0.0770430
$800$	0	0
$801$	−50.8374	−1.79625
$802$	0	0
$803$	−0.811910	−0.0286517
$804$	0	0
$805$	−9.80057	−0.345425
$806$	0	0
$807$	24.3622	0.857592
$808$	0	0
$809$	33.2328	1.16840	0.584201	−	0.811609i	$-0.301408\pi$
$809$	33.2328	1.16840	0.584201	+	0.811609i	$0.301408\pi$
$810$	0	0
$811$	−11.2418	−0.394752	−0.197376	−	0.980328i	$-0.563242\pi$
$811$	−11.2418	−0.394752	−0.197376	+	0.980328i	$0.563242\pi$
$812$	0	0
$813$	−49.5574	−1.73805
$814$	0	0
$815$	8.96683	0.314094
$816$	0	0
$817$	−11.4648	−0.401103
$818$	0	0
$819$	6.03838	0.210998
$820$	0	0
$821$	−29.9929	−1.04676	−0.523379	−	0.852100i	$-0.675329\pi$
$821$	−29.9929	−1.04676	−0.523379	+	0.852100i	$0.675329\pi$
$822$	0	0
$823$	18.9024	0.658897	0.329448	−	0.944174i	$-0.393137\pi$
$823$	18.9024	0.658897	0.329448	+	0.944174i	$0.393137\pi$
$824$	0	0
$825$	−5.63524	−0.196194
$826$	0	0
$827$	3.44487	0.119790	0.0598949	−	0.998205i	$-0.480923\pi$
$827$	3.44487	0.119790	0.0598949	+	0.998205i	$0.480923\pi$
$828$	0	0
$829$	−26.7370	−0.928616	−0.464308	−	0.885674i	$-0.653697\pi$
$829$	−26.7370	−0.928616	−0.464308	+	0.885674i	$0.653697\pi$
$830$	0	0
$831$	−87.2488	−3.02663
$832$	0	0
$833$	−0.640640	−0.0221968
$834$	0	0
$835$	5.10829	0.176780
$836$	0	0
$837$	98.9135	3.41895
$838$	0	0
$839$	−12.3535	−0.426491	−0.213246	−	0.976999i	$-0.568403\pi$
$839$	−12.3535	−0.426491	−0.213246	+	0.976999i	$0.568403\pi$
$840$	0	0
$841$	−24.5596	−0.846884
$842$	0	0
$843$	37.3584	1.28669
$844$	0	0
$845$	2.62191	0.0901965
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	45.5983	1.56493
$850$	0	0
$851$	−22.0370	−0.755420
$852$	0	0
$853$	−50.6873	−1.73550	−0.867750	−	0.497002i	$-0.834434\pi$
$853$	−50.6873	−1.73550	−0.867750	+	0.497002i	$0.834434\pi$
$854$	0	0
$855$	−83.3483	−2.85045
$856$	0	0
$857$	−13.4844	−0.460620	−0.230310	−	0.973117i	$-0.573974\pi$
$857$	−13.4844	−0.460620	−0.230310	+	0.973117i	$0.573974\pi$
$858$	0	0
$859$	4.09174	0.139608	0.0698041	−	0.997561i	$-0.477763\pi$
$859$	4.09174	0.139608	0.0698041	+	0.997561i	$0.477763\pi$
$860$	0	0
$861$	32.2331	1.09850
$862$	0	0
$863$	44.6753	1.52077	0.760383	−	0.649475i	$-0.225011\pi$
$863$	44.6753	1.52077	0.760383	+	0.649475i	$0.225011\pi$
$864$	0	0
$865$	65.5306	2.22811
$866$	0	0
$867$	49.8748	1.69384
$868$	0	0
$869$	9.43761	0.320149
$870$	0	0
$871$	−7.51329	−0.254578
$872$	0	0
$873$	51.7449	1.75130
$874$	0	0
$875$	8.19499	0.277041
$876$	0	0
$877$	−35.5232	−1.19953	−0.599767	−	0.800175i	$-0.704740\pi$
$877$	−35.5232	−1.19953	−0.599767	+	0.800175i	$0.704740\pi$
$878$	0	0
$879$	−41.0380	−1.38418
$880$	0	0
$881$	−15.1895	−0.511748	−0.255874	−	0.966710i	$-0.582363\pi$
$881$	−15.1895	−0.511748	−0.255874	+	0.966710i	$0.582363\pi$
$882$	0	0
$883$	−3.51926	−0.118433	−0.0592163	−	0.998245i	$-0.518860\pi$
$883$	−3.51926	−0.118433	−0.0592163	+	0.998245i	$0.518860\pi$
$884$	0	0
$885$	25.3214	0.851169
$886$	0	0
$887$	−18.4868	−0.620726	−0.310363	−	0.950618i	$-0.600451\pi$
$887$	−18.4868	−0.620726	−0.310363	+	0.950618i	$0.600451\pi$
$888$	0	0
$889$	9.12294	0.305974
$890$	0	0
$891$	9.34693	0.313134
$892$	0	0
$893$	−17.8958	−0.598860
$894$	0	0
$895$	27.3533	0.914318
$896$	0	0
$897$	11.2377	0.375217
$898$	0	0
$899$	22.8179	0.761021
$900$	0	0
$901$	−2.17012	−0.0722971
$902$	0	0
$903$	6.54718	0.217877
$904$	0	0
$905$	−8.77920	−0.291830
$906$	0	0
$907$	−41.1041	−1.36484	−0.682419	−	0.730961i	$-0.739072\pi$
$907$	−41.1041	−1.36484	−0.682419	+	0.730961i	$0.739072\pi$
$908$	0	0
$909$	−100.959	−3.34860
$910$	0	0
$911$	−23.6347	−0.783051	−0.391525	−	0.920167i	$-0.628053\pi$
$911$	−23.6347	−0.783051	−0.391525	+	0.920167i	$0.628053\pi$
$912$	0	0
$913$	−16.0593	−0.531484
$914$	0	0
$915$	−116.852	−3.86300
$916$	0	0
$917$	−0.635495	−0.0209859
$918$	0	0
$919$	−10.3341	−0.340890	−0.170445	−	0.985367i	$-0.554520\pi$
$919$	−10.3341	−0.340890	−0.170445	+	0.985367i	$0.554520\pi$
$920$	0	0
$921$	25.1439	0.828518
$922$	0	0
$923$	2.60540	0.0857579
$924$	0	0
$925$	−11.0506	−0.363343
$926$	0	0
$927$	25.0995	0.824374
$928$	0	0
$929$	4.88488	0.160268	0.0801338	−	0.996784i	$-0.474465\pi$
$929$	4.88488	0.160268	0.0801338	+	0.996784i	$0.474465\pi$
$930$	0	0
$931$	−5.26451	−0.172537
$932$	0	0
$933$	−30.1764	−0.987930
$934$	0	0
$935$	−1.67970	−0.0549321
$936$	0	0
$937$	−10.6863	−0.349106	−0.174553	−	0.984648i	$-0.555848\pi$
$937$	−10.6863	−0.349106	−0.174553	+	0.984648i	$0.555848\pi$
$938$	0	0
$939$	−30.7208	−1.00253
$940$	0	0
$941$	−4.59053	−0.149647	−0.0748234	−	0.997197i	$-0.523839\pi$
$941$	−4.59053	−0.149647	−0.0748234	+	0.997197i	$0.523839\pi$
$942$	0	0
$943$	40.0766	1.30507
$944$	0	0
$945$	23.9500	0.779095
$946$	0	0
$947$	52.8670	1.71795	0.858973	−	0.512021i	$-0.171103\pi$
$947$	52.8670	1.71795	0.858973	+	0.512021i	$0.171103\pi$
$948$	0	0
$949$	0.811910	0.0263557
$950$	0	0
$951$	−60.4021	−1.95867
$952$	0	0
$953$	53.8607	1.74472	0.872359	−	0.488865i	$-0.162589\pi$
$953$	53.8607	1.74472	0.872359	+	0.488865i	$0.162589\pi$
$954$	0	0
$955$	0.188097	0.00608668
$956$	0	0
$957$	6.33511	0.204785
$958$	0	0
$959$	−16.6993	−0.539248
$960$	0	0
$961$	86.2560	2.78245
$962$	0	0
$963$	70.9824	2.28738
$964$	0	0
$965$	35.9428	1.15704
$966$	0	0
$967$	−13.9767	−0.449462	−0.224731	−	0.974421i	$-0.572150\pi$
$967$	−13.9767	−0.449462	−0.224731	+	0.974421i	$0.572150\pi$
$968$	0	0
$969$	−10.1395	−0.325728
$970$	0	0
$971$	−28.1830	−0.904436	−0.452218	−	0.891908i	$-0.649367\pi$
$971$	−28.1830	−0.904436	−0.452218	+	0.891908i	$0.649367\pi$
$972$	0	0
$973$	2.98332	0.0956409
$974$	0	0
$975$	5.63524	0.180472
$976$	0	0
$977$	61.0613	1.95352	0.976762	−	0.214329i	$-0.0687564\pi$
$977$	61.0613	1.95352	0.976762	+	0.214329i	$0.0687564\pi$
$978$	0	0
$979$	−8.41905	−0.269074
$980$	0	0
$981$	109.109	3.48360
$982$	0	0
$983$	−47.2785	−1.50795	−0.753975	−	0.656904i	$-0.771866\pi$
$983$	−47.2785	−1.50795	−0.753975	+	0.656904i	$0.771866\pi$
$984$	0	0
$985$	−28.8122	−0.918033
$986$	0	0
$987$	10.2197	0.325297
$988$	0	0
$989$	8.14033	0.258847
$990$	0	0
$991$	−16.9842	−0.539522	−0.269761	−	0.962927i	$-0.586945\pi$
$991$	−16.9842	−0.539522	−0.269761	+	0.962927i	$0.586945\pi$
$992$	0	0
$993$	0.478515	0.0151852
$994$	0	0
$995$	70.0641	2.22118
$996$	0	0
$997$	21.1064	0.668447	0.334224	−	0.942494i	$-0.391526\pi$
$997$	21.1064	0.668447	0.334224	+	0.942494i	$0.391526\pi$
$998$	0	0
$999$	53.8528	1.70383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.n.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.n.1.2	✓	9	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)

\(f(q)\)	\(=\)	\(q-3.00639 q^{3} +2.62191 q^{5} -1.00000 q^{7} +6.03838 q^{9} +O(q^{10})\)	\(q-3.00639 q^{3} +2.62191 q^{5} -1.00000 q^{7} +6.03838 q^{9} +1.00000 q^{11} -1.00000 q^{13} -7.88249 q^{15} -0.640640 q^{17} -5.26451 q^{19} +3.00639 q^{21} +3.73795 q^{23} +1.87442 q^{25} -9.13457 q^{27} -2.10721 q^{29} -10.8285 q^{31} -3.00639 q^{33} -2.62191 q^{35} -5.89549 q^{37} +3.00639 q^{39} +10.7215 q^{41} +2.17775 q^{43} +15.8321 q^{45} +3.39933 q^{47} +1.00000 q^{49} +1.92601 q^{51} +3.38742 q^{53} +2.62191 q^{55} +15.8272 q^{57} -3.21236 q^{59} +14.8242 q^{61} -6.03838 q^{63} -2.62191 q^{65} +7.51329 q^{67} -11.2377 q^{69} -2.60540 q^{71} -0.811910 q^{73} -5.63524 q^{75} -1.00000 q^{77} +9.43761 q^{79} +9.34693 q^{81} -16.0593 q^{83} -1.67970 q^{85} +6.33511 q^{87} -8.41905 q^{89} +1.00000 q^{91} +32.5546 q^{93} -13.8031 q^{95} +8.56933 q^{97} +6.03838 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * q^99

Introduction

L-functions

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Varieties

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Database

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