LMFDB - Embedded newform 8008.2.a.j.1.1

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:	$5$
Coefficient field:	5.5.668973.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} - x^{2} + 7x - 2 $ x^5 - x^4 - 9x^3 - x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.16366$ 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)

$f(q)$	$=$	$q-3.08441 q^{3} -0.239305 q^{5} +1.00000 q^{7} +6.51357 q^{9} +O(q^{10})$	$q-3.08441 q^{3} -0.239305 q^{5} +1.00000 q^{7} +6.51357 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.738115 q^{15} +3.76069 q^{17} -6.06183 q^{19} -3.08441 q^{21} -0.585598 q^{23} -4.94273 q^{25} -10.8373 q^{27} -8.13147 q^{29} +10.9443 q^{31} +3.08441 q^{33} -0.239305 q^{35} +3.95055 q^{37} +3.08441 q^{39} +12.3960 q^{41} -9.90931 q^{43} -1.55873 q^{45} -4.96531 q^{47} +1.00000 q^{49} -11.5995 q^{51} +9.75288 q^{53} +0.239305 q^{55} +18.6971 q^{57} -7.12091 q^{59} +3.65371 q^{61} +6.51357 q^{63} +0.239305 q^{65} +9.25407 q^{67} +1.80622 q^{69} -1.55091 q^{71} +1.12994 q^{73} +15.2454 q^{75} -1.00000 q^{77} +8.56567 q^{79} +13.8859 q^{81} +2.51203 q^{83} -0.899954 q^{85} +25.0808 q^{87} -5.89301 q^{89} -1.00000 q^{91} -33.7566 q^{93} +1.45063 q^{95} +0.759157 q^{97} -6.51357 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * q^97 - 6 * 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
$3$	−3.08441	−1.78078	−0.890392	−	0.455195i	$-0.849570\pi$
$3$	−3.08441	−1.78078	−0.890392	+	0.455195i	$0.849570\pi$
$4$	0	0
$5$	−0.239305	−0.107021	−0.0535103	−	0.998567i	$-0.517041\pi$
$5$	−0.239305	−0.107021	−0.0535103	+	0.998567i	$0.517041\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.51357	2.17119
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.738115	0.190580
$16$	0	0
$17$	3.76069	0.912102	0.456051	−	0.889954i	$-0.349263\pi$
$17$	3.76069	0.912102	0.456051	+	0.889954i	$0.349263\pi$
$18$	0	0
$19$	−6.06183	−1.39068	−0.695339	−	0.718682i	$-0.744746\pi$
$19$	−6.06183	−1.39068	−0.695339	+	0.718682i	$0.744746\pi$
$20$	0	0
$21$	−3.08441	−0.673073
$22$	0	0
$23$	−0.585598	−0.122106	−0.0610528	−	0.998135i	$-0.519446\pi$
$23$	−0.585598	−0.122106	−0.0610528	+	0.998135i	$0.519446\pi$
$24$	0	0
$25$	−4.94273	−0.988547
$26$	0	0
$27$	−10.8373	−2.08564
$28$	0	0
$29$	−8.13147	−1.50998	−0.754988	−	0.655738i	$-0.772358\pi$
$29$	−8.13147	−1.50998	−0.754988	+	0.655738i	$0.772358\pi$
$30$	0	0
$31$	10.9443	1.96565	0.982825	−	0.184542i	$-0.0590803\pi$
$31$	10.9443	1.96565	0.982825	+	0.184542i	$0.0590803\pi$
$32$	0	0
$33$	3.08441	0.536926
$34$	0	0
$35$	−0.239305	−0.0404500
$36$	0	0
$37$	3.95055	0.649467	0.324733	−	0.945806i	$-0.394725\pi$
$37$	3.95055	0.649467	0.324733	+	0.945806i	$0.394725\pi$
$38$	0	0
$39$	3.08441	0.493900
$40$	0	0
$41$	12.3960	1.93593	0.967966	−	0.251081i	$-0.0807861\pi$
$41$	12.3960	1.93593	0.967966	+	0.251081i	$0.0807861\pi$
$42$	0	0
$43$	−9.90931	−1.51116	−0.755578	−	0.655059i	$-0.772644\pi$
$43$	−9.90931	−1.51116	−0.755578	+	0.655059i	$0.772644\pi$
$44$	0	0
$45$	−1.55873	−0.232362
$46$	0	0
$47$	−4.96531	−0.724265	−0.362133	−	0.932127i	$-0.617951\pi$
$47$	−4.96531	−0.724265	−0.362133	+	0.932127i	$0.617951\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−11.5995	−1.62426
$52$	0	0
$53$	9.75288	1.33966	0.669830	−	0.742514i	$-0.266367\pi$
$53$	9.75288	1.33966	0.669830	+	0.742514i	$0.266367\pi$
$54$	0	0
$55$	0.239305	0.0322679
$56$	0	0
$57$	18.6971	2.47650
$58$	0	0
$59$	−7.12091	−0.927063	−0.463531	−	0.886080i	$-0.653418\pi$
$59$	−7.12091	−0.927063	−0.463531	+	0.886080i	$0.653418\pi$
$60$	0	0
$61$	3.65371	0.467809	0.233905	−	0.972260i	$-0.424850\pi$
$61$	3.65371	0.467809	0.233905	+	0.972260i	$0.424850\pi$
$62$	0	0
$63$	6.51357	0.820633
$64$	0	0
$65$	0.239305	0.0296821
$66$	0	0
$67$	9.25407	1.13056	0.565282	−	0.824898i	$-0.308767\pi$
$67$	9.25407	1.13056	0.565282	+	0.824898i	$0.308767\pi$
$68$	0	0
$69$	1.80622	0.217444
$70$	0	0
$71$	−1.55091	−0.184059	−0.0920297	−	0.995756i	$-0.529335\pi$
$71$	−1.55091	−0.184059	−0.0920297	+	0.995756i	$0.529335\pi$
$72$	0	0
$73$	1.12994	0.132249	0.0661245	−	0.997811i	$-0.478937\pi$
$73$	1.12994	0.132249	0.0661245	+	0.997811i	$0.478937\pi$
$74$	0	0
$75$	15.2454	1.76039
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	8.56567	0.963713	0.481857	−	0.876250i	$-0.339963\pi$
$79$	8.56567	0.963713	0.481857	+	0.876250i	$0.339963\pi$
$80$	0	0
$81$	13.8859	1.54288
$82$	0	0
$83$	2.51203	0.275731	0.137866	−	0.990451i	$-0.455976\pi$
$83$	2.51203	0.275731	0.137866	+	0.990451i	$0.455976\pi$
$84$	0	0
$85$	−0.899954	−0.0976137
$86$	0	0
$87$	25.0808	2.68894
$88$	0	0
$89$	−5.89301	−0.624658	−0.312329	−	0.949974i	$-0.601109\pi$
$89$	−5.89301	−0.624658	−0.312329	+	0.949974i	$0.601109\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−33.7566	−3.50040
$94$	0	0
$95$	1.45063	0.148831
$96$	0	0
$97$	0.759157	0.0770807	0.0385404	−	0.999257i	$-0.487729\pi$
$97$	0.759157	0.0770807	0.0385404	+	0.999257i	$0.487729\pi$
$98$	0	0
$99$	−6.51357	−0.654638
$100$	0	0
$101$	−17.9462	−1.78571	−0.892856	−	0.450343i	$-0.851302\pi$
$101$	−17.9462	−1.78571	−0.892856	+	0.450343i	$0.851302\pi$
$102$	0	0
$103$	−17.3229	−1.70687	−0.853437	−	0.521197i	$-0.825486\pi$
$103$	−17.3229	−1.70687	−0.853437	+	0.521197i	$0.825486\pi$
$104$	0	0
$105$	0.738115	0.0720326
$106$	0	0
$107$	−8.22720	−0.795353	−0.397677	−	0.917526i	$-0.630183\pi$
$107$	−8.22720	−0.795353	−0.397677	+	0.917526i	$0.630183\pi$
$108$	0	0
$109$	−16.6882	−1.59844	−0.799220	−	0.601038i	$-0.794754\pi$
$109$	−16.6882	−1.59844	−0.799220	+	0.601038i	$0.794754\pi$
$110$	0	0
$111$	−12.1851	−1.15656
$112$	0	0
$113$	19.0018	1.78754	0.893770	−	0.448526i	$-0.148051\pi$
$113$	19.0018	1.78754	0.893770	+	0.448526i	$0.148051\pi$
$114$	0	0
$115$	0.140137	0.0130678
$116$	0	0
$117$	−6.51357	−0.602180
$118$	0	0
$119$	3.76069	0.344742
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−38.2344	−3.44748
$124$	0	0
$125$	2.37935	0.212815
$126$	0	0
$127$	−17.0640	−1.51419	−0.757093	−	0.653307i	$-0.773381\pi$
$127$	−17.0640	−1.51419	−0.757093	+	0.653307i	$0.773381\pi$
$128$	0	0
$129$	30.5644	2.69104
$130$	0	0
$131$	0.919785	0.0803620	0.0401810	−	0.999192i	$-0.487207\pi$
$131$	0.919785	0.0803620	0.0401810	+	0.999192i	$0.487207\pi$
$132$	0	0
$133$	−6.06183	−0.525627
$134$	0	0
$135$	2.59342	0.223206
$136$	0	0
$137$	−17.0777	−1.45905	−0.729523	−	0.683956i	$-0.760258\pi$
$137$	−17.0777	−1.45905	−0.729523	+	0.683956i	$0.760258\pi$
$138$	0	0
$139$	10.6921	0.906893	0.453447	−	0.891283i	$-0.350194\pi$
$139$	10.6921	0.906893	0.453447	+	0.891283i	$0.350194\pi$
$140$	0	0
$141$	15.3151	1.28976
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	1.94590	0.161598
$146$	0	0
$147$	−3.08441	−0.254398
$148$	0	0
$149$	11.7568	0.963154	0.481577	−	0.876404i	$-0.340064\pi$
$149$	11.7568	0.963154	0.481577	+	0.876404i	$0.340064\pi$
$150$	0	0
$151$	−15.8260	−1.28790	−0.643949	−	0.765068i	$-0.722705\pi$
$151$	−15.8260	−1.28790	−0.643949	+	0.765068i	$0.722705\pi$
$152$	0	0
$153$	24.4956	1.98035
$154$	0	0
$155$	−2.61902	−0.210365
$156$	0	0
$157$	23.0558	1.84005	0.920027	−	0.391854i	$-0.128166\pi$
$157$	23.0558	1.84005	0.920027	+	0.391854i	$0.128166\pi$
$158$	0	0
$159$	−30.0818	−2.38565
$160$	0	0
$161$	−0.585598	−0.0461516
$162$	0	0
$163$	14.8870	1.16604	0.583020	−	0.812458i	$-0.301871\pi$
$163$	14.8870	1.16604	0.583020	+	0.812458i	$0.301871\pi$
$164$	0	0
$165$	−0.738115	−0.0574621
$166$	0	0
$167$	4.83272	0.373967	0.186984	−	0.982363i	$-0.440129\pi$
$167$	4.83272	0.373967	0.186984	+	0.982363i	$0.440129\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−39.4841	−3.01943
$172$	0	0
$173$	−0.201964	−0.0153550	−0.00767752	−	0.999971i	$-0.502444\pi$
$173$	−0.201964	−0.0153550	−0.00767752	+	0.999971i	$0.502444\pi$
$174$	0	0
$175$	−4.94273	−0.373635
$176$	0	0
$177$	21.9638	1.65090
$178$	0	0
$179$	−17.6835	−1.32173	−0.660864	−	0.750506i	$-0.729810\pi$
$179$	−17.6835	−1.32173	−0.660864	+	0.750506i	$0.729810\pi$
$180$	0	0
$181$	18.3135	1.36123	0.680616	−	0.732640i	$-0.261712\pi$
$181$	18.3135	1.36123	0.680616	+	0.732640i	$0.261712\pi$
$182$	0	0
$183$	−11.2695	−0.833067
$184$	0	0
$185$	−0.945388	−0.0695063
$186$	0	0
$187$	−3.76069	−0.275009
$188$	0	0
$189$	−10.8373	−0.788296
$190$	0	0
$191$	9.08478	0.657351	0.328676	−	0.944443i	$-0.393398\pi$
$191$	9.08478	0.657351	0.328676	+	0.944443i	$0.393398\pi$
$192$	0	0
$193$	−24.3788	−1.75482	−0.877412	−	0.479738i	$-0.840732\pi$
$193$	−24.3788	−1.75482	−0.877412	+	0.479738i	$0.840732\pi$
$194$	0	0
$195$	−0.738115	−0.0528575
$196$	0	0
$197$	5.39448	0.384341	0.192170	−	0.981362i	$-0.438447\pi$
$197$	5.39448	0.384341	0.192170	+	0.981362i	$0.438447\pi$
$198$	0	0
$199$	−0.417056	−0.0295643	−0.0147822	−	0.999891i	$-0.504705\pi$
$199$	−0.417056	−0.0295643	−0.0147822	+	0.999891i	$0.504705\pi$
$200$	0	0
$201$	−28.5433	−2.01329
$202$	0	0
$203$	−8.13147	−0.570718
$204$	0	0
$205$	−2.96643	−0.207184
$206$	0	0
$207$	−3.81433	−0.265115
$208$	0	0
$209$	6.06183	0.419305
$210$	0	0
$211$	13.9629	0.961248	0.480624	−	0.876927i	$-0.340410\pi$
$211$	13.9629	0.961248	0.480624	+	0.876927i	$0.340410\pi$
$212$	0	0
$213$	4.78364	0.327770
$214$	0	0
$215$	2.37135	0.161725
$216$	0	0
$217$	10.9443	0.742946
$218$	0	0
$219$	−3.48518	−0.235507
$220$	0	0
$221$	−3.76069	−0.252972
$222$	0	0
$223$	−14.5562	−0.974753	−0.487377	−	0.873192i	$-0.662046\pi$
$223$	−14.5562	−0.974753	−0.487377	+	0.873192i	$0.662046\pi$
$224$	0	0
$225$	−32.1948	−2.14632
$226$	0	0
$227$	−26.0029	−1.72587	−0.862936	−	0.505314i	$-0.831377\pi$
$227$	−26.0029	−1.72587	−0.862936	+	0.505314i	$0.831377\pi$
$228$	0	0
$229$	−0.718664	−0.0474906	−0.0237453	−	0.999718i	$-0.507559\pi$
$229$	−0.718664	−0.0474906	−0.0237453	+	0.999718i	$0.507559\pi$
$230$	0	0
$231$	3.08441	0.202939
$232$	0	0
$233$	10.0922	0.661161	0.330581	−	0.943778i	$-0.392755\pi$
$233$	10.0922	0.661161	0.330581	+	0.943778i	$0.392755\pi$
$234$	0	0
$235$	1.18822	0.0775113
$236$	0	0
$237$	−26.4200	−1.71617
$238$	0	0
$239$	−19.6127	−1.26864	−0.634321	−	0.773070i	$-0.718720\pi$
$239$	−19.6127	−1.26864	−0.634321	+	0.773070i	$0.718720\pi$
$240$	0	0
$241$	5.51320	0.355137	0.177568	−	0.984108i	$-0.443177\pi$
$241$	5.51320	0.355137	0.177568	+	0.984108i	$0.443177\pi$
$242$	0	0
$243$	−10.3179	−0.661893
$244$	0	0
$245$	−0.239305	−0.0152886
$246$	0	0
$247$	6.06183	0.385705
$248$	0	0
$249$	−7.74813	−0.491018
$250$	0	0
$251$	13.2459	0.836072	0.418036	−	0.908430i	$-0.362719\pi$
$251$	13.2459	0.836072	0.418036	+	0.908430i	$0.362719\pi$
$252$	0	0
$253$	0.585598	0.0368162
$254$	0	0
$255$	2.77582	0.173829
$256$	0	0
$257$	30.2753	1.88852	0.944262	−	0.329196i	$-0.106778\pi$
$257$	30.2753	1.88852	0.944262	+	0.329196i	$0.106778\pi$
$258$	0	0
$259$	3.95055	0.245475
$260$	0	0
$261$	−52.9649	−3.27845
$262$	0	0
$263$	0.123655	0.00762488	0.00381244	−	0.999993i	$-0.498786\pi$
$263$	0.123655	0.00762488	0.00381244	+	0.999993i	$0.498786\pi$
$264$	0	0
$265$	−2.33391	−0.143371
$266$	0	0
$267$	18.1765	1.11238
$268$	0	0
$269$	11.0766	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$269$	11.0766	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$270$	0	0
$271$	22.5014	1.36686	0.683432	−	0.730014i	$-0.260487\pi$
$271$	22.5014	1.36686	0.683432	+	0.730014i	$0.260487\pi$
$272$	0	0
$273$	3.08441	0.186677
$274$	0	0
$275$	4.94273	0.298058
$276$	0	0
$277$	−2.57386	−0.154648	−0.0773241	−	0.997006i	$-0.524638\pi$
$277$	−2.57386	−0.154648	−0.0773241	+	0.997006i	$0.524638\pi$
$278$	0	0
$279$	71.2863	4.26780
$280$	0	0
$281$	−5.59178	−0.333578	−0.166789	−	0.985993i	$-0.553340\pi$
$281$	−5.59178	−0.333578	−0.166789	+	0.985993i	$0.553340\pi$
$282$	0	0
$283$	21.0842	1.25333	0.626664	−	0.779290i	$-0.284420\pi$
$283$	21.0842	1.25333	0.626664	+	0.779290i	$0.284420\pi$
$284$	0	0
$285$	−4.47432	−0.265036
$286$	0	0
$287$	12.3960	0.731714
$288$	0	0
$289$	−2.85717	−0.168069
$290$	0	0
$291$	−2.34155	−0.137264
$292$	0	0
$293$	14.8727	0.868870	0.434435	−	0.900703i	$-0.356948\pi$
$293$	14.8727	0.868870	0.434435	+	0.900703i	$0.356948\pi$
$294$	0	0
$295$	1.70407	0.0992148
$296$	0	0
$297$	10.8373	0.628843
$298$	0	0
$299$	0.585598	0.0338660
$300$	0	0
$301$	−9.90931	−0.571163
$302$	0	0
$303$	55.3533	3.17997
$304$	0	0
$305$	−0.874351	−0.0500652
$306$	0	0
$307$	0.156067	0.00890724	0.00445362	−	0.999990i	$-0.498582\pi$
$307$	0.156067	0.00890724	0.00445362	+	0.999990i	$0.498582\pi$
$308$	0	0
$309$	53.4308	3.03957
$310$	0	0
$311$	−14.5146	−0.823049	−0.411525	−	0.911399i	$-0.635004\pi$
$311$	−14.5146	−0.823049	−0.411525	+	0.911399i	$0.635004\pi$
$312$	0	0
$313$	3.33890	0.188726	0.0943628	−	0.995538i	$-0.469919\pi$
$313$	3.33890	0.188726	0.0943628	+	0.995538i	$0.469919\pi$
$314$	0	0
$315$	−1.55873	−0.0878245
$316$	0	0
$317$	9.19653	0.516528	0.258264	−	0.966074i	$-0.416850\pi$
$317$	9.19653	0.516528	0.258264	+	0.966074i	$0.416850\pi$
$318$	0	0
$319$	8.13147	0.455275
$320$	0	0
$321$	25.3760	1.41635
$322$	0	0
$323$	−22.7967	−1.26844
$324$	0	0
$325$	4.94273	0.274173
$326$	0	0
$327$	51.4732	2.84648
$328$	0	0
$329$	−4.96531	−0.273747
$330$	0	0
$331$	−3.63695	−0.199905	−0.0999525	−	0.994992i	$-0.531869\pi$
$331$	−3.63695	−0.199905	−0.0999525	+	0.994992i	$0.531869\pi$
$332$	0	0
$333$	25.7322	1.41012
$334$	0	0
$335$	−2.21455	−0.120994
$336$	0	0
$337$	−23.2482	−1.26641	−0.633205	−	0.773984i	$-0.718261\pi$
$337$	−23.2482	−1.26641	−0.633205	+	0.773984i	$0.718261\pi$
$338$	0	0
$339$	−58.6093	−3.18322
$340$	0	0
$341$	−10.9443	−0.592666
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.432239	−0.0232709
$346$	0	0
$347$	26.7052	1.43361	0.716806	−	0.697272i	$-0.245603\pi$
$347$	26.7052	1.43361	0.716806	+	0.697272i	$0.245603\pi$
$348$	0	0
$349$	−18.0921	−0.968448	−0.484224	−	0.874944i	$-0.660898\pi$
$349$	−18.0921	−0.968448	−0.484224	+	0.874944i	$0.660898\pi$
$350$	0	0
$351$	10.8373	0.578451
$352$	0	0
$353$	8.64743	0.460256	0.230128	−	0.973160i	$-0.426086\pi$
$353$	8.64743	0.460256	0.230128	+	0.973160i	$0.426086\pi$
$354$	0	0
$355$	0.371141	0.0196981
$356$	0	0
$357$	−11.5995	−0.613911
$358$	0	0
$359$	1.58100	0.0834421	0.0417211	−	0.999129i	$-0.486716\pi$
$359$	1.58100	0.0834421	0.0417211	+	0.999129i	$0.486716\pi$
$360$	0	0
$361$	17.7458	0.933987
$362$	0	0
$363$	−3.08441	−0.161889
$364$	0	0
$365$	−0.270400	−0.0141534
$366$	0	0
$367$	10.3396	0.539722	0.269861	−	0.962899i	$-0.413022\pi$
$367$	10.3396	0.539722	0.269861	+	0.962899i	$0.413022\pi$
$368$	0	0
$369$	80.7423	4.20328
$370$	0	0
$371$	9.75288	0.506344
$372$	0	0
$373$	−2.41790	−0.125194	−0.0625970	−	0.998039i	$-0.519938\pi$
$373$	−2.41790	−0.125194	−0.0625970	+	0.998039i	$0.519938\pi$
$374$	0	0
$375$	−7.33888	−0.378978
$376$	0	0
$377$	8.13147	0.418792
$378$	0	0
$379$	11.1875	0.574663	0.287332	−	0.957831i	$-0.407232\pi$
$379$	11.1875	0.574663	0.287332	+	0.957831i	$0.407232\pi$
$380$	0	0
$381$	52.6324	2.69644
$382$	0	0
$383$	11.8116	0.603546	0.301773	−	0.953380i	$-0.402422\pi$
$383$	11.8116	0.603546	0.301773	+	0.953380i	$0.402422\pi$
$384$	0	0
$385$	0.239305	0.0121961
$386$	0	0
$387$	−64.5450	−3.28101
$388$	0	0
$389$	−13.4022	−0.679519	−0.339759	−	0.940512i	$-0.610346\pi$
$389$	−13.4022	−0.679519	−0.339759	+	0.940512i	$0.610346\pi$
$390$	0	0
$391$	−2.20226	−0.111373
$392$	0	0
$393$	−2.83699	−0.143107
$394$	0	0
$395$	−2.04981	−0.103137
$396$	0	0
$397$	35.3488	1.77410	0.887052	−	0.461670i	$-0.152750\pi$
$397$	35.3488	1.77410	0.887052	+	0.461670i	$0.152750\pi$
$398$	0	0
$399$	18.6971	0.936028
$400$	0	0
$401$	−15.8356	−0.790790	−0.395395	−	0.918511i	$-0.629392\pi$
$401$	−15.8356	−0.790790	−0.395395	+	0.918511i	$0.629392\pi$
$402$	0	0
$403$	−10.9443	−0.545173
$404$	0	0
$405$	−3.32296	−0.165119
$406$	0	0
$407$	−3.95055	−0.195822
$408$	0	0
$409$	9.58053	0.473727	0.236863	−	0.971543i	$-0.423881\pi$
$409$	9.58053	0.473727	0.236863	+	0.971543i	$0.423881\pi$
$410$	0	0
$411$	52.6746	2.59825
$412$	0	0
$413$	−7.12091	−0.350397
$414$	0	0
$415$	−0.601142	−0.0295089
$416$	0	0
$417$	−32.9788	−1.61498
$418$	0	0
$419$	25.2866	1.23533	0.617665	−	0.786442i	$-0.288079\pi$
$419$	25.2866	1.23533	0.617665	+	0.786442i	$0.288079\pi$
$420$	0	0
$421$	3.85091	0.187682	0.0938409	−	0.995587i	$-0.470085\pi$
$421$	3.85091	0.187682	0.0938409	+	0.995587i	$0.470085\pi$
$422$	0	0
$423$	−32.3419	−1.57252
$424$	0	0
$425$	−18.5881	−0.901656
$426$	0	0
$427$	3.65371	0.176815
$428$	0	0
$429$	−3.08441	−0.148917
$430$	0	0
$431$	4.80461	0.231430	0.115715	−	0.993282i	$-0.463084\pi$
$431$	4.80461	0.231430	0.115715	+	0.993282i	$0.463084\pi$
$432$	0	0
$433$	40.7174	1.95675	0.978376	−	0.206835i	$-0.0663164\pi$
$433$	40.7174	1.95675	0.978376	+	0.206835i	$0.0663164\pi$
$434$	0	0
$435$	−6.00196	−0.287772
$436$	0	0
$437$	3.54979	0.169810
$438$	0	0
$439$	−11.3669	−0.542513	−0.271257	−	0.962507i	$-0.587439\pi$
$439$	−11.3669	−0.542513	−0.271257	+	0.962507i	$0.587439\pi$
$440$	0	0
$441$	6.51357	0.310170
$442$	0	0
$443$	−17.6240	−0.837342	−0.418671	−	0.908138i	$-0.637504\pi$
$443$	−17.6240	−0.837342	−0.418671	+	0.908138i	$0.637504\pi$
$444$	0	0
$445$	1.41023	0.0668512
$446$	0	0
$447$	−36.2627	−1.71517
$448$	0	0
$449$	5.97663	0.282055	0.141027	−	0.990006i	$-0.454959\pi$
$449$	5.97663	0.282055	0.141027	+	0.990006i	$0.454959\pi$
$450$	0	0
$451$	−12.3960	−0.583705
$452$	0	0
$453$	48.8137	2.29347
$454$	0	0
$455$	0.239305	0.0112188
$456$	0	0
$457$	1.75372	0.0820355	0.0410177	−	0.999158i	$-0.486940\pi$
$457$	1.75372	0.0820355	0.0410177	+	0.999158i	$0.486940\pi$
$458$	0	0
$459$	−40.7557	−1.90231
$460$	0	0
$461$	35.6034	1.65822	0.829109	−	0.559088i	$-0.188848\pi$
$461$	35.6034	1.65822	0.829109	+	0.559088i	$0.188848\pi$
$462$	0	0
$463$	13.1131	0.609417	0.304708	−	0.952446i	$-0.401441\pi$
$463$	13.1131	0.609417	0.304708	+	0.952446i	$0.401441\pi$
$464$	0	0
$465$	8.07813	0.374614
$466$	0	0
$467$	32.7788	1.51682	0.758411	−	0.651776i	$-0.225976\pi$
$467$	32.7788	1.51682	0.758411	+	0.651776i	$0.225976\pi$
$468$	0	0
$469$	9.25407	0.427313
$470$	0	0
$471$	−71.1135	−3.27674
$472$	0	0
$473$	9.90931	0.455631
$474$	0	0
$475$	29.9620	1.37475
$476$	0	0
$477$	63.5260	2.90866
$478$	0	0
$479$	4.80860	0.219711	0.109855	−	0.993948i	$-0.464961\pi$
$479$	4.80860	0.219711	0.109855	+	0.993948i	$0.464961\pi$
$480$	0	0
$481$	−3.95055	−0.180130
$482$	0	0
$483$	1.80622	0.0821860
$484$	0	0
$485$	−0.181670	−0.00824922
$486$	0	0
$487$	−18.1641	−0.823093	−0.411546	−	0.911389i	$-0.635011\pi$
$487$	−18.1641	−0.823093	−0.411546	+	0.911389i	$0.635011\pi$
$488$	0	0
$489$	−45.9176	−2.07647
$490$	0	0
$491$	23.4791	1.05960	0.529799	−	0.848124i	$-0.322268\pi$
$491$	23.4791	1.05960	0.529799	+	0.848124i	$0.322268\pi$
$492$	0	0
$493$	−30.5800	−1.37725
$494$	0	0
$495$	1.55873	0.0700597
$496$	0	0
$497$	−1.55091	−0.0695679
$498$	0	0
$499$	−12.8108	−0.573492	−0.286746	−	0.958007i	$-0.592574\pi$
$499$	−12.8108	−0.573492	−0.286746	+	0.958007i	$0.592574\pi$
$500$	0	0
$501$	−14.9061	−0.665955
$502$	0	0
$503$	38.8430	1.73192	0.865961	−	0.500112i	$-0.166708\pi$
$503$	38.8430	1.73192	0.865961	+	0.500112i	$0.166708\pi$
$504$	0	0
$505$	4.29461	0.191108
$506$	0	0
$507$	−3.08441	−0.136983
$508$	0	0
$509$	18.5986	0.824369	0.412185	−	0.911100i	$-0.364766\pi$
$509$	18.5986	0.824369	0.412185	+	0.911100i	$0.364766\pi$
$510$	0	0
$511$	1.12994	0.0499854
$512$	0	0
$513$	65.6937	2.90045
$514$	0	0
$515$	4.14545	0.182670
$516$	0	0
$517$	4.96531	0.218374
$518$	0	0
$519$	0.622939	0.0273440
$520$	0	0
$521$	−12.9859	−0.568923	−0.284461	−	0.958688i	$-0.591815\pi$
$521$	−12.9859	−0.568923	−0.284461	+	0.958688i	$0.591815\pi$
$522$	0	0
$523$	−37.1971	−1.62652	−0.813258	−	0.581904i	$-0.802308\pi$
$523$	−37.1971	−1.62652	−0.813258	+	0.581904i	$0.802308\pi$
$524$	0	0
$525$	15.2454	0.665364
$526$	0	0
$527$	41.1581	1.79287
$528$	0	0
$529$	−22.6571	−0.985090
$530$	0	0
$531$	−46.3825	−2.01283
$532$	0	0
$533$	−12.3960	−0.536931
$534$	0	0
$535$	1.96881	0.0851191
$536$	0	0
$537$	54.5431	2.35371
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	2.99343	0.128697	0.0643487	−	0.997927i	$-0.479503\pi$
$541$	2.99343	0.128697	0.0643487	+	0.997927i	$0.479503\pi$
$542$	0	0
$543$	−56.4863	−2.42406
$544$	0	0
$545$	3.99357	0.171066
$546$	0	0
$547$	−0.986469	−0.0421784	−0.0210892	−	0.999778i	$-0.506713\pi$
$547$	−0.986469	−0.0421784	−0.0210892	+	0.999778i	$0.506713\pi$
$548$	0	0
$549$	23.7987	1.01570
$550$	0	0
$551$	49.2916	2.09989
$552$	0	0
$553$	8.56567	0.364249
$554$	0	0
$555$	2.91596	0.123776
$556$	0	0
$557$	−27.8975	−1.18205	−0.591027	−	0.806652i	$-0.701277\pi$
$557$	−27.8975	−1.18205	−0.591027	+	0.806652i	$0.701277\pi$
$558$	0	0
$559$	9.90931	0.419119
$560$	0	0
$561$	11.5995	0.489732
$562$	0	0
$563$	26.3621	1.11103	0.555515	−	0.831506i	$-0.312521\pi$
$563$	26.3621	1.11103	0.555515	+	0.831506i	$0.312521\pi$
$564$	0	0
$565$	−4.54723	−0.191303
$566$	0	0
$567$	13.8859	0.583152
$568$	0	0
$569$	−17.0711	−0.715657	−0.357828	−	0.933787i	$-0.616483\pi$
$569$	−17.0711	−0.715657	−0.357828	+	0.933787i	$0.616483\pi$
$570$	0	0
$571$	−5.90345	−0.247052	−0.123526	−	0.992341i	$-0.539420\pi$
$571$	−5.90345	−0.247052	−0.123526	+	0.992341i	$0.539420\pi$
$572$	0	0
$573$	−28.0212	−1.17060
$574$	0	0
$575$	2.89446	0.120707
$576$	0	0
$577$	8.10924	0.337592	0.168796	−	0.985651i	$-0.446012\pi$
$577$	8.10924	0.337592	0.168796	+	0.985651i	$0.446012\pi$
$578$	0	0
$579$	75.1941	3.12496
$580$	0	0
$581$	2.51203	0.104217
$582$	0	0
$583$	−9.75288	−0.403923
$584$	0	0
$585$	1.55873	0.0644456
$586$	0	0
$587$	9.75194	0.402506	0.201253	−	0.979539i	$-0.435499\pi$
$587$	9.75194	0.402506	0.201253	+	0.979539i	$0.435499\pi$
$588$	0	0
$589$	−66.3423	−2.73359
$590$	0	0
$591$	−16.6388	−0.684427
$592$	0	0
$593$	−1.53386	−0.0629883	−0.0314941	−	0.999504i	$-0.510027\pi$
$593$	−1.53386	−0.0629883	−0.0314941	+	0.999504i	$0.510027\pi$
$594$	0	0
$595$	−0.899954	−0.0368945
$596$	0	0
$597$	1.28637	0.0526477
$598$	0	0
$599$	−6.82185	−0.278733	−0.139367	−	0.990241i	$-0.544507\pi$
$599$	−6.82185	−0.278733	−0.139367	+	0.990241i	$0.544507\pi$
$600$	0	0
$601$	13.7594	0.561257	0.280628	−	0.959817i	$-0.409457\pi$
$601$	13.7594	0.561257	0.280628	+	0.959817i	$0.409457\pi$
$602$	0	0
$603$	60.2770	2.45467
$604$	0	0
$605$	−0.239305	−0.00972914
$606$	0	0
$607$	−14.6714	−0.595495	−0.297748	−	0.954645i	$-0.596235\pi$
$607$	−14.6714	−0.595495	−0.297748	+	0.954645i	$0.596235\pi$
$608$	0	0
$609$	25.0808	1.01632
$610$	0	0
$611$	4.96531	0.200875
$612$	0	0
$613$	13.3896	0.540802	0.270401	−	0.962748i	$-0.412844\pi$
$613$	13.3896	0.540802	0.270401	+	0.962748i	$0.412844\pi$
$614$	0	0
$615$	9.14968	0.368951
$616$	0	0
$617$	3.40984	0.137275	0.0686375	−	0.997642i	$-0.478135\pi$
$617$	3.40984	0.137275	0.0686375	+	0.997642i	$0.478135\pi$
$618$	0	0
$619$	0.538205	0.0216323	0.0108161	−	0.999942i	$-0.496557\pi$
$619$	0.538205	0.0216323	0.0108161	+	0.999942i	$0.496557\pi$
$620$	0	0
$621$	6.34629	0.254668
$622$	0	0
$623$	−5.89301	−0.236099
$624$	0	0
$625$	24.1443	0.965771
$626$	0	0
$627$	−18.6971	−0.746692
$628$	0	0
$629$	14.8568	0.592380
$630$	0	0
$631$	26.3648	1.04957	0.524784	−	0.851236i	$-0.324146\pi$
$631$	26.3648	1.04957	0.524784	+	0.851236i	$0.324146\pi$
$632$	0	0
$633$	−43.0674	−1.71177
$634$	0	0
$635$	4.08350	0.162049
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−10.1020	−0.399628
$640$	0	0
$641$	5.68284	0.224459	0.112229	−	0.993682i	$-0.464201\pi$
$641$	5.68284	0.224459	0.112229	+	0.993682i	$0.464201\pi$
$642$	0	0
$643$	40.4740	1.59614	0.798069	−	0.602567i	$-0.205855\pi$
$643$	40.4740	1.59614	0.798069	+	0.602567i	$0.205855\pi$
$644$	0	0
$645$	−7.31421	−0.287997
$646$	0	0
$647$	−49.4134	−1.94264	−0.971320	−	0.237776i	$-0.923582\pi$
$647$	−49.4134	−1.94264	−0.971320	+	0.237776i	$0.923582\pi$
$648$	0	0
$649$	7.12091	0.279520
$650$	0	0
$651$	−33.7566	−1.32303
$652$	0	0
$653$	44.9446	1.75882	0.879408	−	0.476068i	$-0.157938\pi$
$653$	44.9446	1.75882	0.879408	+	0.476068i	$0.157938\pi$
$654$	0	0
$655$	−0.220109	−0.00860038
$656$	0	0
$657$	7.35992	0.287138
$658$	0	0
$659$	8.82285	0.343689	0.171845	−	0.985124i	$-0.445027\pi$
$659$	8.82285	0.343689	0.171845	+	0.985124i	$0.445027\pi$
$660$	0	0
$661$	−49.7308	−1.93430	−0.967152	−	0.254198i	$-0.918188\pi$
$661$	−49.7308	−1.93430	−0.967152	+	0.254198i	$0.918188\pi$
$662$	0	0
$663$	11.5995	0.450488
$664$	0	0
$665$	1.45063	0.0562529
$666$	0	0
$667$	4.76178	0.184377
$668$	0	0
$669$	44.8972	1.73582
$670$	0	0
$671$	−3.65371	−0.141050
$672$	0	0
$673$	37.2949	1.43761	0.718807	−	0.695210i	$-0.244689\pi$
$673$	37.2949	1.43761	0.718807	+	0.695210i	$0.244689\pi$
$674$	0	0
$675$	53.5658	2.06175
$676$	0	0
$677$	10.7454	0.412979	0.206489	−	0.978449i	$-0.433796\pi$
$677$	10.7454	0.412979	0.206489	+	0.978449i	$0.433796\pi$
$678$	0	0
$679$	0.759157	0.0291338
$680$	0	0
$681$	80.2035	3.07340
$682$	0	0
$683$	24.3394	0.931322	0.465661	−	0.884963i	$-0.345817\pi$
$683$	24.3394	0.931322	0.465661	+	0.884963i	$0.345817\pi$
$684$	0	0
$685$	4.08678	0.156148
$686$	0	0
$687$	2.21665	0.0845706
$688$	0	0
$689$	−9.75288	−0.371555
$690$	0	0
$691$	21.5171	0.818550	0.409275	−	0.912411i	$-0.365782\pi$
$691$	21.5171	0.818550	0.409275	+	0.912411i	$0.365782\pi$
$692$	0	0
$693$	−6.51357	−0.247430
$694$	0	0
$695$	−2.55868	−0.0970562
$696$	0	0
$697$	46.6176	1.76577
$698$	0	0
$699$	−31.1284	−1.17739
$700$	0	0
$701$	32.4519	1.22569	0.612845	−	0.790203i	$-0.290025\pi$
$701$	32.4519	1.22569	0.612845	+	0.790203i	$0.290025\pi$
$702$	0	0
$703$	−23.9476	−0.903200
$704$	0	0
$705$	−3.66497	−0.138031
$706$	0	0
$707$	−17.9462	−0.674935
$708$	0	0
$709$	1.75027	0.0657329	0.0328665	−	0.999460i	$-0.489536\pi$
$709$	1.75027	0.0657329	0.0328665	+	0.999460i	$0.489536\pi$
$710$	0	0
$711$	55.7931	2.09241
$712$	0	0
$713$	−6.40894	−0.240017
$714$	0	0
$715$	−0.239305	−0.00894950
$716$	0	0
$717$	60.4937	2.25918
$718$	0	0
$719$	45.7365	1.70568	0.852841	−	0.522170i	$-0.174878\pi$
$719$	45.7365	1.70568	0.852841	+	0.522170i	$0.174878\pi$
$720$	0	0
$721$	−17.3229	−0.645137
$722$	0	0
$723$	−17.0050	−0.632421
$724$	0	0
$725$	40.1917	1.49268
$726$	0	0
$727$	−23.4193	−0.868575	−0.434288	−	0.900774i	$-0.643000\pi$
$727$	−23.4193	−0.868575	−0.434288	+	0.900774i	$0.643000\pi$
$728$	0	0
$729$	−9.83309	−0.364189
$730$	0	0
$731$	−37.2659	−1.37833
$732$	0	0
$733$	−45.5250	−1.68150	−0.840752	−	0.541420i	$-0.817887\pi$
$733$	−45.5250	−1.68150	−0.840752	+	0.541420i	$0.817887\pi$
$734$	0	0
$735$	0.738115	0.0272258
$736$	0	0
$737$	−9.25407	−0.340878
$738$	0	0
$739$	15.2693	0.561691	0.280846	−	0.959753i	$-0.409385\pi$
$739$	15.2693	0.561691	0.280846	+	0.959753i	$0.409385\pi$
$740$	0	0
$741$	−18.6971	−0.686857
$742$	0	0
$743$	47.2611	1.73384	0.866921	−	0.498445i	$-0.166095\pi$
$743$	47.2611	1.73384	0.866921	+	0.498445i	$0.166095\pi$
$744$	0	0
$745$	−2.81346	−0.103077
$746$	0	0
$747$	16.3623	0.598665
$748$	0	0
$749$	−8.22720	−0.300615
$750$	0	0
$751$	35.9491	1.31180	0.655901	−	0.754847i	$-0.272289\pi$
$751$	35.9491	1.31180	0.655901	+	0.754847i	$0.272289\pi$
$752$	0	0
$753$	−40.8557	−1.48886
$754$	0	0
$755$	3.78724	0.137832
$756$	0	0
$757$	42.3951	1.54088	0.770438	−	0.637515i	$-0.220038\pi$
$757$	42.3951	1.54088	0.770438	+	0.637515i	$0.220038\pi$
$758$	0	0
$759$	−1.80622	−0.0655618
$760$	0	0
$761$	−30.1286	−1.09216	−0.546080	−	0.837733i	$-0.683881\pi$
$761$	−30.1286	−1.09216	−0.546080	+	0.837733i	$0.683881\pi$
$762$	0	0
$763$	−16.6882	−0.604154
$764$	0	0
$765$	−5.86191	−0.211938
$766$	0	0
$767$	7.12091	0.257121
$768$	0	0
$769$	7.67122	0.276631	0.138316	−	0.990388i	$-0.455831\pi$
$769$	7.67122	0.276631	0.138316	+	0.990388i	$0.455831\pi$
$770$	0	0
$771$	−93.3815	−3.36305
$772$	0	0
$773$	42.6322	1.53337	0.766686	−	0.642022i	$-0.221904\pi$
$773$	42.6322	1.53337	0.766686	+	0.642022i	$0.221904\pi$
$774$	0	0
$775$	−54.0946	−1.94314
$776$	0	0
$777$	−12.1851	−0.437139
$778$	0	0
$779$	−75.1425	−2.69226
$780$	0	0
$781$	1.55091	0.0554960
$782$	0	0
$783$	88.1231	3.14926
$784$	0	0
$785$	−5.51738	−0.196924
$786$	0	0
$787$	33.9665	1.21077	0.605387	−	0.795931i	$-0.293018\pi$
$787$	33.9665	1.21077	0.605387	+	0.795931i	$0.293018\pi$
$788$	0	0
$789$	−0.381402	−0.0135783
$790$	0	0
$791$	19.0018	0.675627
$792$	0	0
$793$	−3.65371	−0.129747
$794$	0	0
$795$	7.19874	0.255313
$796$	0	0
$797$	25.6048	0.906969	0.453484	−	0.891264i	$-0.350181\pi$
$797$	25.6048	0.906969	0.453484	+	0.891264i	$0.350181\pi$
$798$	0	0
$799$	−18.6730	−0.660604
$800$	0	0
$801$	−38.3845	−1.35625
$802$	0	0
$803$	−1.12994	−0.0398746
$804$	0	0
$805$	0.140137	0.00493917
$806$	0	0
$807$	−34.1647	−1.20265
$808$	0	0
$809$	−24.0188	−0.844456	−0.422228	−	0.906490i	$-0.638752\pi$
$809$	−24.0188	−0.844456	−0.422228	+	0.906490i	$0.638752\pi$
$810$	0	0
$811$	27.5291	0.966678	0.483339	−	0.875433i	$-0.339424\pi$
$811$	27.5291	0.966678	0.483339	+	0.875433i	$0.339424\pi$
$812$	0	0
$813$	−69.4035	−2.43409
$814$	0	0
$815$	−3.56254	−0.124790
$816$	0	0
$817$	60.0685	2.10153
$818$	0	0
$819$	−6.51357	−0.227603
$820$	0	0
$821$	−46.9786	−1.63957	−0.819783	−	0.572675i	$-0.805906\pi$
$821$	−46.9786	−1.63957	−0.819783	+	0.572675i	$0.805906\pi$
$822$	0	0
$823$	1.81016	0.0630982	0.0315491	−	0.999502i	$-0.489956\pi$
$823$	1.81016	0.0630982	0.0315491	+	0.999502i	$0.489956\pi$
$824$	0	0
$825$	−15.2454	−0.530777
$826$	0	0
$827$	0.390795	0.0135893	0.00679464	−	0.999977i	$-0.497837\pi$
$827$	0.390795	0.0135893	0.00679464	+	0.999977i	$0.497837\pi$
$828$	0	0
$829$	−36.8716	−1.28060	−0.640302	−	0.768123i	$-0.721191\pi$
$829$	−36.8716	−1.28060	−0.640302	+	0.768123i	$0.721191\pi$
$830$	0	0
$831$	7.93883	0.275395
$832$	0	0
$833$	3.76069	0.130300
$834$	0	0
$835$	−1.15650	−0.0400222
$836$	0	0
$837$	−118.606	−4.09963
$838$	0	0
$839$	7.46618	0.257761	0.128881	−	0.991660i	$-0.458862\pi$
$839$	7.46618	0.257761	0.128881	+	0.991660i	$0.458862\pi$
$840$	0	0
$841$	37.1209	1.28003
$842$	0	0
$843$	17.2473	0.594030
$844$	0	0
$845$	−0.239305	−0.00823235
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−65.0323	−2.23190
$850$	0	0
$851$	−2.31344	−0.0793036
$852$	0	0
$853$	−57.5071	−1.96900	−0.984502	−	0.175373i	$-0.943887\pi$
$853$	−57.5071	−1.96900	−0.984502	+	0.175373i	$0.943887\pi$
$854$	0	0
$855$	9.44876	0.323141
$856$	0	0
$857$	57.5651	1.96639	0.983193	−	0.182571i	$-0.0584420\pi$
$857$	57.5651	1.96639	0.983193	+	0.182571i	$0.0584420\pi$
$858$	0	0
$859$	−29.6181	−1.01056	−0.505279	−	0.862956i	$-0.668610\pi$
$859$	−29.6181	−1.01056	−0.505279	+	0.862956i	$0.668610\pi$
$860$	0	0
$861$	−38.2344	−1.30302
$862$	0	0
$863$	23.9466	0.815153	0.407576	−	0.913171i	$-0.366374\pi$
$863$	23.9466	0.815153	0.407576	+	0.913171i	$0.366374\pi$
$864$	0	0
$865$	0.0483310	0.00164330
$866$	0	0
$867$	8.81269	0.299295
$868$	0	0
$869$	−8.56567	−0.290571
$870$	0	0
$871$	−9.25407	−0.313562
$872$	0	0
$873$	4.94482	0.167357
$874$	0	0
$875$	2.37935	0.0804366
$876$	0	0
$877$	−15.2847	−0.516127	−0.258064	−	0.966128i	$-0.583084\pi$
$877$	−15.2847	−0.516127	−0.258064	+	0.966128i	$0.583084\pi$
$878$	0	0
$879$	−45.8734	−1.54727
$880$	0	0
$881$	−39.5337	−1.33192	−0.665962	−	0.745986i	$-0.731979\pi$
$881$	−39.5337	−1.33192	−0.665962	+	0.745986i	$0.731979\pi$
$882$	0	0
$883$	−27.9627	−0.941018	−0.470509	−	0.882395i	$-0.655930\pi$
$883$	−27.9627	−0.941018	−0.470509	+	0.882395i	$0.655930\pi$
$884$	0	0
$885$	−5.25604	−0.176680
$886$	0	0
$887$	47.9469	1.60990	0.804950	−	0.593343i	$-0.202192\pi$
$887$	47.9469	1.60990	0.804950	+	0.593343i	$0.202192\pi$
$888$	0	0
$889$	−17.0640	−0.572308
$890$	0	0
$891$	−13.8859	−0.465195
$892$	0	0
$893$	30.0989	1.00722
$894$	0	0
$895$	4.23175	0.141452
$896$	0	0
$897$	−1.80622	−0.0603080
$898$	0	0
$899$	−88.9931	−2.96808
$900$	0	0
$901$	36.6776	1.22191
$902$	0	0
$903$	30.5644	1.01712
$904$	0	0
$905$	−4.38252	−0.145680
$906$	0	0
$907$	−30.7725	−1.02178	−0.510892	−	0.859645i	$-0.670685\pi$
$907$	−30.7725	−1.02178	−0.510892	+	0.859645i	$0.670685\pi$
$908$	0	0
$909$	−116.894	−3.87712
$910$	0	0
$911$	3.13852	0.103984	0.0519919	−	0.998648i	$-0.483443\pi$
$911$	3.13852	0.103984	0.0519919	+	0.998648i	$0.483443\pi$
$912$	0	0
$913$	−2.51203	−0.0831361
$914$	0	0
$915$	2.69685	0.0891553
$916$	0	0
$917$	0.919785	0.0303740
$918$	0	0
$919$	−24.8000	−0.818078	−0.409039	−	0.912517i	$-0.634136\pi$
$919$	−24.8000	−0.818078	−0.409039	+	0.912517i	$0.634136\pi$
$920$	0	0
$921$	−0.481376	−0.0158619
$922$	0	0
$923$	1.55091	0.0510489
$924$	0	0
$925$	−19.5265	−0.642028
$926$	0	0
$927$	−112.834	−3.70595
$928$	0	0
$929$	0.823768	0.0270270	0.0135135	−	0.999909i	$-0.495698\pi$
$929$	0.823768	0.0270270	0.0135135	+	0.999909i	$0.495698\pi$
$930$	0	0
$931$	−6.06183	−0.198668
$932$	0	0
$933$	44.7690	1.46567
$934$	0	0
$935$	0.899954	0.0294316
$936$	0	0
$937$	18.0829	0.590742	0.295371	−	0.955383i	$-0.404557\pi$
$937$	18.0829	0.590742	0.295371	+	0.955383i	$0.404557\pi$
$938$	0	0
$939$	−10.2985	−0.336079
$940$	0	0
$941$	44.3134	1.44458	0.722288	−	0.691592i	$-0.243090\pi$
$941$	44.3134	1.44458	0.722288	+	0.691592i	$0.243090\pi$
$942$	0	0
$943$	−7.25908	−0.236388
$944$	0	0
$945$	2.59342	0.0843639
$946$	0	0
$947$	−26.3566	−0.856475	−0.428238	−	0.903666i	$-0.640865\pi$
$947$	−26.3566	−0.856475	−0.428238	+	0.903666i	$0.640865\pi$
$948$	0	0
$949$	−1.12994	−0.0366793
$950$	0	0
$951$	−28.3658	−0.919825
$952$	0	0
$953$	53.0224	1.71756	0.858782	−	0.512342i	$-0.171222\pi$
$953$	53.0224	1.71756	0.858782	+	0.512342i	$0.171222\pi$
$954$	0	0
$955$	−2.17403	−0.0703501
$956$	0	0
$957$	−25.0808	−0.810746
$958$	0	0
$959$	−17.0777	−0.551468
$960$	0	0
$961$	88.7771	2.86378
$962$	0	0
$963$	−53.5884	−1.72686
$964$	0	0
$965$	5.83397	0.187802
$966$	0	0
$967$	42.1398	1.35512	0.677562	−	0.735466i	$-0.263037\pi$
$967$	42.1398	1.35512	0.677562	+	0.735466i	$0.263037\pi$
$968$	0	0
$969$	70.3143	2.25882
$970$	0	0
$971$	36.6528	1.17624	0.588122	−	0.808772i	$-0.299867\pi$
$971$	36.6528	1.17624	0.588122	+	0.808772i	$0.299867\pi$
$972$	0	0
$973$	10.6921	0.342773
$974$	0	0
$975$	−15.2454	−0.488244
$976$	0	0
$977$	19.7408	0.631563	0.315782	−	0.948832i	$-0.397733\pi$
$977$	19.7408	0.631563	0.315782	+	0.948832i	$0.397733\pi$
$978$	0	0
$979$	5.89301	0.188341
$980$	0	0
$981$	−108.700	−3.47052
$982$	0	0
$983$	−1.49645	−0.0477292	−0.0238646	−	0.999715i	$-0.507597\pi$
$983$	−1.49645	−0.0477292	−0.0238646	+	0.999715i	$0.507597\pi$
$984$	0	0
$985$	−1.29093	−0.0411323
$986$	0	0
$987$	15.3151	0.487483
$988$	0	0
$989$	5.80287	0.184521
$990$	0	0
$991$	49.3189	1.56667	0.783333	−	0.621602i	$-0.213518\pi$
$991$	49.3189	1.56667	0.783333	+	0.621602i	$0.213518\pi$
$992$	0	0
$993$	11.2178	0.355987
$994$	0	0
$995$	0.0998037	0.00316399
$996$	0	0
$997$	−26.4510	−0.837713	−0.418857	−	0.908052i	$-0.637569\pi$
$997$	−26.4510	−0.837713	−0.418857	+	0.908052i	$0.637569\pi$
$998$	0	0
$999$	−42.8133	−1.35455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.j.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.j.1.1	✓	5	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.08441 q^{3} -0.239305 q^{5} +1.00000 q^{7} +6.51357 q^{9} +O(q^{10})\)	\(q-3.08441 q^{3} -0.239305 q^{5} +1.00000 q^{7} +6.51357 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.738115 q^{15} +3.76069 q^{17} -6.06183 q^{19} -3.08441 q^{21} -0.585598 q^{23} -4.94273 q^{25} -10.8373 q^{27} -8.13147 q^{29} +10.9443 q^{31} +3.08441 q^{33} -0.239305 q^{35} +3.95055 q^{37} +3.08441 q^{39} +12.3960 q^{41} -9.90931 q^{43} -1.55873 q^{45} -4.96531 q^{47} +1.00000 q^{49} -11.5995 q^{51} +9.75288 q^{53} +0.239305 q^{55} +18.6971 q^{57} -7.12091 q^{59} +3.65371 q^{61} +6.51357 q^{63} +0.239305 q^{65} +9.25407 q^{67} +1.80622 q^{69} -1.55091 q^{71} +1.12994 q^{73} +15.2454 q^{75} -1.00000 q^{77} +8.56567 q^{79} +13.8859 q^{81} +2.51203 q^{83} -0.899954 q^{85} +25.0808 q^{87} -5.89301 q^{89} -1.00000 q^{91} -33.7566 q^{93} +1.45063 q^{95} +0.759157 q^{97} -6.51357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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