LMFDB - Embedded newform 6008.2.a.b.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.75295 q^{3} +1.89627 q^{5} -3.72975 q^{7} +0.0728254 q^{9} +O(q^{10})$	$q-1.75295 q^{3} +1.89627 q^{5} -3.72975 q^{7} +0.0728254 q^{9} +5.05766 q^{11} +3.05514 q^{13} -3.32407 q^{15} +0.674403 q^{17} -1.01821 q^{19} +6.53805 q^{21} -4.16593 q^{23} -1.40415 q^{25} +5.13118 q^{27} -6.08203 q^{29} +1.11741 q^{31} -8.86581 q^{33} -7.07262 q^{35} -8.11392 q^{37} -5.35549 q^{39} +9.59261 q^{41} -10.0470 q^{43} +0.138097 q^{45} -2.58604 q^{47} +6.91102 q^{49} -1.18219 q^{51} +8.96148 q^{53} +9.59070 q^{55} +1.78487 q^{57} -4.15484 q^{59} +7.74161 q^{61} -0.271620 q^{63} +5.79337 q^{65} -4.80625 q^{67} +7.30266 q^{69} -8.83390 q^{71} -0.804911 q^{73} +2.46140 q^{75} -18.8638 q^{77} +14.9038 q^{79} -9.21317 q^{81} +0.249592 q^{83} +1.27885 q^{85} +10.6615 q^{87} +12.3114 q^{89} -11.3949 q^{91} -1.95876 q^{93} -1.93081 q^{95} +6.87353 q^{97} +0.368326 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * 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
$3$	−1.75295	−1.01206	−0.506032	−	0.862514i	$-0.668888\pi$
$3$	−1.75295	−1.01206	−0.506032	+	0.862514i	$0.668888\pi$
$4$	0	0
$5$	1.89627	0.848039	0.424020	−	0.905653i	$-0.360619\pi$
$5$	1.89627	0.848039	0.424020	+	0.905653i	$0.360619\pi$
$6$	0	0
$7$	−3.72975	−1.40971	−0.704856	−	0.709350i	$-0.748989\pi$
$7$	−3.72975	−1.40971	−0.704856	+	0.709350i	$0.748989\pi$
$8$	0	0
$9$	0.0728254	0.0242751
$10$	0	0
$11$	5.05766	1.52494	0.762471	−	0.647023i	$-0.223986\pi$
$11$	5.05766	1.52494	0.762471	+	0.647023i	$0.223986\pi$
$12$	0	0
$13$	3.05514	0.847342	0.423671	−	0.905816i	$-0.360741\pi$
$13$	3.05514	0.847342	0.423671	+	0.905816i	$0.360741\pi$
$14$	0	0
$15$	−3.32407	−0.858271
$16$	0	0
$17$	0.674403	0.163567	0.0817834	−	0.996650i	$-0.473938\pi$
$17$	0.674403	0.163567	0.0817834	+	0.996650i	$0.473938\pi$
$18$	0	0
$19$	−1.01821	−0.233594	−0.116797	−	0.993156i	$-0.537263\pi$
$19$	−1.01821	−0.233594	−0.116797	+	0.993156i	$0.537263\pi$
$20$	0	0
$21$	6.53805	1.42672
$22$	0	0
$23$	−4.16593	−0.868657	−0.434329	−	0.900754i	$-0.643014\pi$
$23$	−4.16593	−0.868657	−0.434329	+	0.900754i	$0.643014\pi$
$24$	0	0
$25$	−1.40415	−0.280829
$26$	0	0
$27$	5.13118	0.987497
$28$	0	0
$29$	−6.08203	−1.12940	−0.564702	−	0.825295i	$-0.691009\pi$
$29$	−6.08203	−1.12940	−0.564702	+	0.825295i	$0.691009\pi$
$30$	0	0
$31$	1.11741	0.200693	0.100346	−	0.994953i	$-0.468005\pi$
$31$	1.11741	0.200693	0.100346	+	0.994953i	$0.468005\pi$
$32$	0	0
$33$	−8.86581	−1.54334
$34$	0	0
$35$	−7.07262	−1.19549
$36$	0	0
$37$	−8.11392	−1.33392	−0.666960	−	0.745093i	$-0.732405\pi$
$37$	−8.11392	−1.33392	−0.666960	+	0.745093i	$0.732405\pi$
$38$	0	0
$39$	−5.35549	−0.857565
$40$	0	0
$41$	9.59261	1.49811	0.749057	−	0.662505i	$-0.230507\pi$
$41$	9.59261	1.49811	0.749057	+	0.662505i	$0.230507\pi$
$42$	0	0
$43$	−10.0470	−1.53216	−0.766080	−	0.642745i	$-0.777796\pi$
$43$	−10.0470	−1.53216	−0.766080	+	0.642745i	$0.777796\pi$
$44$	0	0
$45$	0.138097	0.0205863
$46$	0	0
$47$	−2.58604	−0.377212	−0.188606	−	0.982053i	$-0.560397\pi$
$47$	−2.58604	−0.377212	−0.188606	+	0.982053i	$0.560397\pi$
$48$	0	0
$49$	6.91102	0.987289
$50$	0	0
$51$	−1.18219	−0.165540
$52$	0	0
$53$	8.96148	1.23095	0.615477	−	0.788155i	$-0.288963\pi$
$53$	8.96148	1.23095	0.615477	+	0.788155i	$0.288963\pi$
$54$	0	0
$55$	9.59070	1.29321
$56$	0	0
$57$	1.78487	0.236412
$58$	0	0
$59$	−4.15484	−0.540914	−0.270457	−	0.962732i	$-0.587175\pi$
$59$	−4.15484	−0.540914	−0.270457	+	0.962732i	$0.587175\pi$
$60$	0	0
$61$	7.74161	0.991212	0.495606	−	0.868548i	$-0.334946\pi$
$61$	7.74161	0.991212	0.495606	+	0.868548i	$0.334946\pi$
$62$	0	0
$63$	−0.271620	−0.0342210
$64$	0	0
$65$	5.79337	0.718580
$66$	0	0
$67$	−4.80625	−0.587176	−0.293588	−	0.955932i	$-0.594849\pi$
$67$	−4.80625	−0.587176	−0.293588	+	0.955932i	$0.594849\pi$
$68$	0	0
$69$	7.30266	0.879138
$70$	0	0
$71$	−8.83390	−1.04839	−0.524196	−	0.851598i	$-0.675634\pi$
$71$	−8.83390	−1.04839	−0.524196	+	0.851598i	$0.675634\pi$
$72$	0	0
$73$	−0.804911	−0.0942077	−0.0471038	−	0.998890i	$-0.514999\pi$
$73$	−0.804911	−0.0942077	−0.0471038	+	0.998890i	$0.514999\pi$
$74$	0	0
$75$	2.46140	0.284218
$76$	0	0
$77$	−18.8638	−2.14973
$78$	0	0
$79$	14.9038	1.67681	0.838403	−	0.545051i	$-0.183490\pi$
$79$	14.9038	1.67681	0.838403	+	0.545051i	$0.183490\pi$
$80$	0	0
$81$	−9.21317	−1.02369
$82$	0	0
$83$	0.249592	0.0273962	0.0136981	−	0.999906i	$-0.495640\pi$
$83$	0.249592	0.0273962	0.0136981	+	0.999906i	$0.495640\pi$
$84$	0	0
$85$	1.27885	0.138711
$86$	0	0
$87$	10.6615	1.14303
$88$	0	0
$89$	12.3114	1.30500	0.652502	−	0.757787i	$-0.273720\pi$
$89$	12.3114	1.30500	0.652502	+	0.757787i	$0.273720\pi$
$90$	0	0
$91$	−11.3949	−1.19451
$92$	0	0
$93$	−1.95876	−0.203114
$94$	0	0
$95$	−1.93081	−0.198096
$96$	0	0
$97$	6.87353	0.697902	0.348951	−	0.937141i	$-0.386538\pi$
$97$	6.87353	0.697902	0.348951	+	0.937141i	$0.386538\pi$
$98$	0	0
$99$	0.368326	0.0370181
$100$	0	0
$101$	10.0233	0.997357	0.498679	−	0.866787i	$-0.333819\pi$
$101$	10.0233	0.997357	0.498679	+	0.866787i	$0.333819\pi$
$102$	0	0
$103$	−4.61332	−0.454564	−0.227282	−	0.973829i	$-0.572984\pi$
$103$	−4.61332	−0.454564	−0.227282	+	0.973829i	$0.572984\pi$
$104$	0	0
$105$	12.3979	1.20991
$106$	0	0
$107$	−3.80702	−0.368038	−0.184019	−	0.982923i	$-0.558911\pi$
$107$	−3.80702	−0.368038	−0.184019	+	0.982923i	$0.558911\pi$
$108$	0	0
$109$	19.4008	1.85826	0.929132	−	0.369748i	$-0.120556\pi$
$109$	19.4008	1.85826	0.929132	+	0.369748i	$0.120556\pi$
$110$	0	0
$111$	14.2233	1.35001
$112$	0	0
$113$	8.43393	0.793398	0.396699	−	0.917949i	$-0.370156\pi$
$113$	8.43393	0.793398	0.396699	+	0.917949i	$0.370156\pi$
$114$	0	0
$115$	−7.89975	−0.736656
$116$	0	0
$117$	0.222491	0.0205693
$118$	0	0
$119$	−2.51535	−0.230582
$120$	0	0
$121$	14.5799	1.32545
$122$	0	0
$123$	−16.8153	−1.51619
$124$	0	0
$125$	−12.1440	−1.08619
$126$	0	0
$127$	7.73505	0.686375	0.343188	−	0.939267i	$-0.388493\pi$
$127$	7.73505	0.686375	0.343188	+	0.939267i	$0.388493\pi$
$128$	0	0
$129$	17.6120	1.55065
$130$	0	0
$131$	−19.7474	−1.72534	−0.862671	−	0.505766i	$-0.831210\pi$
$131$	−19.7474	−1.72534	−0.862671	+	0.505766i	$0.831210\pi$
$132$	0	0
$133$	3.79767	0.329300
$134$	0	0
$135$	9.73013	0.837436
$136$	0	0
$137$	−20.4120	−1.74392	−0.871958	−	0.489581i	$-0.837150\pi$
$137$	−20.4120	−1.74392	−0.871958	+	0.489581i	$0.837150\pi$
$138$	0	0
$139$	−17.7041	−1.50165	−0.750823	−	0.660504i	$-0.770343\pi$
$139$	−17.7041	−1.50165	−0.750823	+	0.660504i	$0.770343\pi$
$140$	0	0
$141$	4.53319	0.381763
$142$	0	0
$143$	15.4518	1.29215
$144$	0	0
$145$	−11.5332	−0.957780
$146$	0	0
$147$	−12.1147	−0.999201
$148$	0	0
$149$	−17.3952	−1.42507	−0.712535	−	0.701637i	$-0.752453\pi$
$149$	−17.3952	−1.42507	−0.712535	+	0.701637i	$0.752453\pi$
$150$	0	0
$151$	−21.2179	−1.72669	−0.863343	−	0.504618i	$-0.831633\pi$
$151$	−21.2179	−1.72669	−0.863343	+	0.504618i	$0.831633\pi$
$152$	0	0
$153$	0.0491136	0.00397060
$154$	0	0
$155$	2.11892	0.170196
$156$	0	0
$157$	−20.3389	−1.62322	−0.811611	−	0.584198i	$-0.801409\pi$
$157$	−20.3389	−1.62322	−0.811611	+	0.584198i	$0.801409\pi$
$158$	0	0
$159$	−15.7090	−1.24581
$160$	0	0
$161$	15.5379	1.22456
$162$	0	0
$163$	−4.27200	−0.334609	−0.167305	−	0.985905i	$-0.553506\pi$
$163$	−4.27200	−0.334609	−0.167305	+	0.985905i	$0.553506\pi$
$164$	0	0
$165$	−16.8120	−1.30881
$166$	0	0
$167$	−4.05645	−0.313897	−0.156949	−	0.987607i	$-0.550166\pi$
$167$	−4.05645	−0.313897	−0.156949	+	0.987607i	$0.550166\pi$
$168$	0	0
$169$	−3.66614	−0.282011
$170$	0	0
$171$	−0.0741516	−0.00567051
$172$	0	0
$173$	3.26919	0.248552	0.124276	−	0.992248i	$-0.460339\pi$
$173$	3.26919	0.248552	0.124276	+	0.992248i	$0.460339\pi$
$174$	0	0
$175$	5.23712	0.395889
$176$	0	0
$177$	7.28322	0.547440
$178$	0	0
$179$	−14.9313	−1.11602	−0.558010	−	0.829834i	$-0.688435\pi$
$179$	−14.9313	−1.11602	−0.558010	+	0.829834i	$0.688435\pi$
$180$	0	0
$181$	25.0446	1.86155	0.930776	−	0.365589i	$-0.119132\pi$
$181$	25.0446	1.86155	0.930776	+	0.365589i	$0.119132\pi$
$182$	0	0
$183$	−13.5706	−1.00317
$184$	0	0
$185$	−15.3862	−1.13122
$186$	0	0
$187$	3.41090	0.249430
$188$	0	0
$189$	−19.1380	−1.39209
$190$	0	0
$191$	−23.2734	−1.68401	−0.842003	−	0.539474i	$-0.818623\pi$
$191$	−23.2734	−1.68401	−0.842003	+	0.539474i	$0.818623\pi$
$192$	0	0
$193$	−16.9863	−1.22270	−0.611351	−	0.791360i	$-0.709373\pi$
$193$	−16.9863	−1.22270	−0.611351	+	0.791360i	$0.709373\pi$
$194$	0	0
$195$	−10.1555	−0.727249
$196$	0	0
$197$	1.57953	0.112537	0.0562684	−	0.998416i	$-0.482080\pi$
$197$	1.57953	0.112537	0.0562684	+	0.998416i	$0.482080\pi$
$198$	0	0
$199$	2.57289	0.182387	0.0911937	−	0.995833i	$-0.470932\pi$
$199$	2.57289	0.182387	0.0911937	+	0.995833i	$0.470932\pi$
$200$	0	0
$201$	8.42510	0.594261
$202$	0	0
$203$	22.6844	1.59214
$204$	0	0
$205$	18.1902	1.27046
$206$	0	0
$207$	−0.303386	−0.0210868
$208$	0	0
$209$	−5.14976	−0.356216
$210$	0	0
$211$	11.8435	0.815341	0.407670	−	0.913129i	$-0.366341\pi$
$211$	11.8435	0.815341	0.407670	+	0.913129i	$0.366341\pi$
$212$	0	0
$213$	15.4854	1.06104
$214$	0	0
$215$	−19.0520	−1.29933
$216$	0	0
$217$	−4.16766	−0.282919
$218$	0	0
$219$	1.41097	0.0953443
$220$	0	0
$221$	2.06039	0.138597
$222$	0	0
$223$	18.8211	1.26035	0.630176	−	0.776453i	$-0.282983\pi$
$223$	18.8211	1.26035	0.630176	+	0.776453i	$0.282983\pi$
$224$	0	0
$225$	−0.102258	−0.00681717
$226$	0	0
$227$	14.0465	0.932298	0.466149	−	0.884706i	$-0.345641\pi$
$227$	14.0465	0.932298	0.466149	+	0.884706i	$0.345641\pi$
$228$	0	0
$229$	−11.6081	−0.767082	−0.383541	−	0.923524i	$-0.625296\pi$
$229$	−11.6081	−0.767082	−0.383541	+	0.923524i	$0.625296\pi$
$230$	0	0
$231$	33.0672	2.17566
$232$	0	0
$233$	−28.1822	−1.84628	−0.923140	−	0.384465i	$-0.874386\pi$
$233$	−28.1822	−1.84628	−0.923140	+	0.384465i	$0.874386\pi$
$234$	0	0
$235$	−4.90383	−0.319891
$236$	0	0
$237$	−26.1255	−1.69704
$238$	0	0
$239$	21.2865	1.37691	0.688453	−	0.725280i	$-0.258290\pi$
$239$	21.2865	1.37691	0.688453	+	0.725280i	$0.258290\pi$
$240$	0	0
$241$	10.7858	0.694775	0.347387	−	0.937722i	$-0.387069\pi$
$241$	10.7858	0.694775	0.347387	+	0.937722i	$0.387069\pi$
$242$	0	0
$243$	0.756658	0.0485396
$244$	0	0
$245$	13.1052	0.837260
$246$	0	0
$247$	−3.11077	−0.197934
$248$	0	0
$249$	−0.437521	−0.0277268
$250$	0	0
$251$	−12.9565	−0.817809	−0.408904	−	0.912577i	$-0.634089\pi$
$251$	−12.9565	−0.817809	−0.408904	+	0.912577i	$0.634089\pi$
$252$	0	0
$253$	−21.0699	−1.32465
$254$	0	0
$255$	−2.24176	−0.140385
$256$	0	0
$257$	−6.37178	−0.397461	−0.198730	−	0.980054i	$-0.563682\pi$
$257$	−6.37178	−0.397461	−0.198730	+	0.980054i	$0.563682\pi$
$258$	0	0
$259$	30.2629	1.88044
$260$	0	0
$261$	−0.442926	−0.0274164
$262$	0	0
$263$	−18.6530	−1.15019	−0.575096	−	0.818086i	$-0.695035\pi$
$263$	−18.6530	−1.15019	−0.575096	+	0.818086i	$0.695035\pi$
$264$	0	0
$265$	16.9934	1.04390
$266$	0	0
$267$	−21.5812	−1.32075
$268$	0	0
$269$	−1.13990	−0.0695012	−0.0347506	−	0.999396i	$-0.511064\pi$
$269$	−1.13990	−0.0695012	−0.0347506	+	0.999396i	$0.511064\pi$
$270$	0	0
$271$	18.3880	1.11699	0.558496	−	0.829507i	$-0.311379\pi$
$271$	18.3880	1.11699	0.558496	+	0.829507i	$0.311379\pi$
$272$	0	0
$273$	19.9746	1.20892
$274$	0	0
$275$	−7.10169	−0.428248
$276$	0	0
$277$	−20.4534	−1.22893	−0.614464	−	0.788945i	$-0.710628\pi$
$277$	−20.4534	−1.22893	−0.614464	+	0.788945i	$0.710628\pi$
$278$	0	0
$279$	0.0813759	0.00487185
$280$	0	0
$281$	−8.10109	−0.483270	−0.241635	−	0.970367i	$-0.577684\pi$
$281$	−8.10109	−0.483270	−0.241635	+	0.970367i	$0.577684\pi$
$282$	0	0
$283$	−21.0613	−1.25197	−0.625983	−	0.779837i	$-0.715302\pi$
$283$	−21.0613	−1.25197	−0.625983	+	0.779837i	$0.715302\pi$
$284$	0	0
$285$	3.38460	0.200486
$286$	0	0
$287$	−35.7780	−2.11191
$288$	0	0
$289$	−16.5452	−0.973246
$290$	0	0
$291$	−12.0489	−0.706322
$292$	0	0
$293$	−4.42170	−0.258319	−0.129159	−	0.991624i	$-0.541228\pi$
$293$	−4.42170	−0.258319	−0.129159	+	0.991624i	$0.541228\pi$
$294$	0	0
$295$	−7.87872	−0.458717
$296$	0	0
$297$	25.9518	1.50587
$298$	0	0
$299$	−12.7275	−0.736050
$300$	0	0
$301$	37.4730	2.15991
$302$	0	0
$303$	−17.5703	−1.00939
$304$	0	0
$305$	14.6802	0.840586
$306$	0	0
$307$	−20.7882	−1.18645	−0.593223	−	0.805038i	$-0.702145\pi$
$307$	−20.7882	−1.18645	−0.593223	+	0.805038i	$0.702145\pi$
$308$	0	0
$309$	8.08691	0.460048
$310$	0	0
$311$	14.8137	0.840006	0.420003	−	0.907523i	$-0.362029\pi$
$311$	14.8137	0.840006	0.420003	+	0.907523i	$0.362029\pi$
$312$	0	0
$313$	−7.25930	−0.410320	−0.205160	−	0.978728i	$-0.565772\pi$
$313$	−7.25930	−0.410320	−0.205160	+	0.978728i	$0.565772\pi$
$314$	0	0
$315$	−0.515067	−0.0290207
$316$	0	0
$317$	9.56337	0.537133	0.268566	−	0.963261i	$-0.413450\pi$
$317$	9.56337	0.537133	0.268566	+	0.963261i	$0.413450\pi$
$318$	0	0
$319$	−30.7608	−1.72228
$320$	0	0
$321$	6.67351	0.372479
$322$	0	0
$323$	−0.686684	−0.0382081
$324$	0	0
$325$	−4.28986	−0.237959
$326$	0	0
$327$	−34.0087	−1.88068
$328$	0	0
$329$	9.64527	0.531761
$330$	0	0
$331$	2.48873	0.136793	0.0683966	−	0.997658i	$-0.478212\pi$
$331$	2.48873	0.136793	0.0683966	+	0.997658i	$0.478212\pi$
$332$	0	0
$333$	−0.590899	−0.0323811
$334$	0	0
$335$	−9.11396	−0.497949
$336$	0	0
$337$	−15.6733	−0.853778	−0.426889	−	0.904304i	$-0.640391\pi$
$337$	−15.6733	−0.853778	−0.426889	+	0.904304i	$0.640391\pi$
$338$	0	0
$339$	−14.7842	−0.802970
$340$	0	0
$341$	5.65148	0.306045
$342$	0	0
$343$	0.331855	0.0179185
$344$	0	0
$345$	13.8478	0.745543
$346$	0	0
$347$	−15.2317	−0.817680	−0.408840	−	0.912606i	$-0.634067\pi$
$347$	−15.2317	−0.817680	−0.408840	+	0.912606i	$0.634067\pi$
$348$	0	0
$349$	−31.4164	−1.68168	−0.840839	−	0.541285i	$-0.817938\pi$
$349$	−31.4164	−1.68168	−0.840839	+	0.541285i	$0.817938\pi$
$350$	0	0
$351$	15.6765	0.836748
$352$	0	0
$353$	32.7629	1.74379	0.871895	−	0.489692i	$-0.162891\pi$
$353$	32.7629	1.74379	0.871895	+	0.489692i	$0.162891\pi$
$354$	0	0
$355$	−16.7515	−0.889077
$356$	0	0
$357$	4.40928	0.233364
$358$	0	0
$359$	−9.09599	−0.480068	−0.240034	−	0.970764i	$-0.577159\pi$
$359$	−9.09599	−0.480068	−0.240034	+	0.970764i	$0.577159\pi$
$360$	0	0
$361$	−17.9632	−0.945434
$362$	0	0
$363$	−25.5578	−1.34144
$364$	0	0
$365$	−1.52633	−0.0798918
$366$	0	0
$367$	12.4768	0.651286	0.325643	−	0.945493i	$-0.394419\pi$
$367$	12.4768	0.651286	0.325643	+	0.945493i	$0.394419\pi$
$368$	0	0
$369$	0.698586	0.0363669
$370$	0	0
$371$	−33.4241	−1.73529
$372$	0	0
$373$	13.0722	0.676853	0.338427	−	0.940993i	$-0.390105\pi$
$373$	13.0722	0.676853	0.338427	+	0.940993i	$0.390105\pi$
$374$	0	0
$375$	21.2878	1.09930
$376$	0	0
$377$	−18.5814	−0.956992
$378$	0	0
$379$	6.50610	0.334196	0.167098	−	0.985940i	$-0.446560\pi$
$379$	6.50610	0.334196	0.167098	+	0.985940i	$0.446560\pi$
$380$	0	0
$381$	−13.5591	−0.694656
$382$	0	0
$383$	22.3960	1.14438	0.572190	−	0.820121i	$-0.306094\pi$
$383$	22.3960	1.14438	0.572190	+	0.820121i	$0.306094\pi$
$384$	0	0
$385$	−35.7709	−1.82305
$386$	0	0
$387$	−0.731680	−0.0371934
$388$	0	0
$389$	−16.4682	−0.834969	−0.417485	−	0.908684i	$-0.637088\pi$
$389$	−16.4682	−0.834969	−0.417485	+	0.908684i	$0.637088\pi$
$390$	0	0
$391$	−2.80952	−0.142083
$392$	0	0
$393$	34.6162	1.74616
$394$	0	0
$395$	28.2616	1.42200
$396$	0	0
$397$	−3.63052	−0.182210	−0.0911052	−	0.995841i	$-0.529040\pi$
$397$	−3.63052	−0.182210	−0.0911052	+	0.995841i	$0.529040\pi$
$398$	0	0
$399$	−6.65712	−0.333273
$400$	0	0
$401$	24.0461	1.20081	0.600403	−	0.799698i	$-0.295007\pi$
$401$	24.0461	1.20081	0.600403	+	0.799698i	$0.295007\pi$
$402$	0	0
$403$	3.41384	0.170056
$404$	0	0
$405$	−17.4707	−0.868126
$406$	0	0
$407$	−41.0374	−2.03415
$408$	0	0
$409$	−27.6023	−1.36485	−0.682423	−	0.730957i	$-0.739074\pi$
$409$	−27.6023	−1.36485	−0.682423	+	0.730957i	$0.739074\pi$
$410$	0	0
$411$	35.7812	1.76496
$412$	0	0
$413$	15.4965	0.762534
$414$	0	0
$415$	0.473294	0.0232331
$416$	0	0
$417$	31.0344	1.51976
$418$	0	0
$419$	−21.0146	−1.02663	−0.513316	−	0.858200i	$-0.671583\pi$
$419$	−21.0146	−1.02663	−0.513316	+	0.858200i	$0.671583\pi$
$420$	0	0
$421$	20.1842	0.983718	0.491859	−	0.870675i	$-0.336318\pi$
$421$	20.1842	0.983718	0.491859	+	0.870675i	$0.336318\pi$
$422$	0	0
$423$	−0.188329	−0.00915688
$424$	0	0
$425$	−0.946961	−0.0459343
$426$	0	0
$427$	−28.8743	−1.39732
$428$	0	0
$429$	−27.0863	−1.30774
$430$	0	0
$431$	−31.5905	−1.52166	−0.760830	−	0.648951i	$-0.775208\pi$
$431$	−31.5905	−1.52166	−0.760830	+	0.648951i	$0.775208\pi$
$432$	0	0
$433$	15.8618	0.762272	0.381136	−	0.924519i	$-0.375533\pi$
$433$	15.8618	0.762272	0.381136	+	0.924519i	$0.375533\pi$
$434$	0	0
$435$	20.2171	0.969335
$436$	0	0
$437$	4.24180	0.202913
$438$	0	0
$439$	30.7973	1.46988	0.734939	−	0.678134i	$-0.237211\pi$
$439$	30.7973	1.46988	0.734939	+	0.678134i	$0.237211\pi$
$440$	0	0
$441$	0.503298	0.0239666
$442$	0	0
$443$	1.70757	0.0811291	0.0405646	−	0.999177i	$-0.487084\pi$
$443$	1.70757	0.0811291	0.0405646	+	0.999177i	$0.487084\pi$
$444$	0	0
$445$	23.3457	1.10669
$446$	0	0
$447$	30.4929	1.44226
$448$	0	0
$449$	−34.9107	−1.64754	−0.823769	−	0.566926i	$-0.808133\pi$
$449$	−34.9107	−1.64754	−0.823769	+	0.566926i	$0.808133\pi$
$450$	0	0
$451$	48.5161	2.28454
$452$	0	0
$453$	37.1938	1.74752
$454$	0	0
$455$	−21.6078	−1.01299
$456$	0	0
$457$	−32.2219	−1.50728	−0.753638	−	0.657290i	$-0.771703\pi$
$457$	−32.2219	−1.50728	−0.753638	+	0.657290i	$0.771703\pi$
$458$	0	0
$459$	3.46048	0.161522
$460$	0	0
$461$	−12.4947	−0.581936	−0.290968	−	0.956733i	$-0.593977\pi$
$461$	−12.4947	−0.581936	−0.290968	+	0.956733i	$0.593977\pi$
$462$	0	0
$463$	26.6131	1.23682	0.618408	−	0.785857i	$-0.287778\pi$
$463$	26.6131	1.23682	0.618408	+	0.785857i	$0.287778\pi$
$464$	0	0
$465$	−3.71435	−0.172249
$466$	0	0
$467$	13.8570	0.641224	0.320612	−	0.947211i	$-0.396111\pi$
$467$	13.8570	0.641224	0.320612	+	0.947211i	$0.396111\pi$
$468$	0	0
$469$	17.9261	0.827750
$470$	0	0
$471$	35.6530	1.64281
$472$	0	0
$473$	−50.8145	−2.33645
$474$	0	0
$475$	1.42972	0.0655999
$476$	0	0
$477$	0.652623	0.0298816
$478$	0	0
$479$	−35.8675	−1.63883	−0.819413	−	0.573203i	$-0.805701\pi$
$479$	−35.8675	−1.63883	−0.819413	+	0.573203i	$0.805701\pi$
$480$	0	0
$481$	−24.7891	−1.13029
$482$	0	0
$483$	−27.2371	−1.23933
$484$	0	0
$485$	13.0341	0.591848
$486$	0	0
$487$	−12.3128	−0.557948	−0.278974	−	0.960299i	$-0.589994\pi$
$487$	−12.3128	−0.557948	−0.278974	+	0.960299i	$0.589994\pi$
$488$	0	0
$489$	7.48860	0.338646
$490$	0	0
$491$	16.2634	0.733955	0.366978	−	0.930230i	$-0.380393\pi$
$491$	16.2634	0.733955	0.366978	+	0.930230i	$0.380393\pi$
$492$	0	0
$493$	−4.10174	−0.184733
$494$	0	0
$495$	0.698447	0.0313928
$496$	0	0
$497$	32.9482	1.47793
$498$	0	0
$499$	28.6172	1.28108	0.640541	−	0.767924i	$-0.278710\pi$
$499$	28.6172	1.28108	0.640541	+	0.767924i	$0.278710\pi$
$500$	0	0
$501$	7.11074	0.317684
$502$	0	0
$503$	−22.5407	−1.00504	−0.502521	−	0.864565i	$-0.667594\pi$
$503$	−22.5407	−1.00504	−0.502521	+	0.864565i	$0.667594\pi$
$504$	0	0
$505$	19.0069	0.845798
$506$	0	0
$507$	6.42656	0.285414
$508$	0	0
$509$	−5.10159	−0.226124	−0.113062	−	0.993588i	$-0.536066\pi$
$509$	−5.10159	−0.226124	−0.113062	+	0.993588i	$0.536066\pi$
$510$	0	0
$511$	3.00211	0.132806
$512$	0	0
$513$	−5.22463	−0.230673
$514$	0	0
$515$	−8.74812	−0.385488
$516$	0	0
$517$	−13.0793	−0.575227
$518$	0	0
$519$	−5.73072	−0.251551
$520$	0	0
$521$	−23.5200	−1.03043	−0.515215	−	0.857061i	$-0.672288\pi$
$521$	−23.5200	−1.03043	−0.515215	+	0.857061i	$0.672288\pi$
$522$	0	0
$523$	5.18959	0.226925	0.113462	−	0.993542i	$-0.463806\pi$
$523$	5.18959	0.226925	0.113462	+	0.993542i	$0.463806\pi$
$524$	0	0
$525$	−9.18039	−0.400665
$526$	0	0
$527$	0.753585	0.0328267
$528$	0	0
$529$	−5.64499	−0.245434
$530$	0	0
$531$	−0.302578	−0.0131308
$532$	0	0
$533$	29.3067	1.26942
$534$	0	0
$535$	−7.21915	−0.312111
$536$	0	0
$537$	26.1738	1.12948
$538$	0	0
$539$	34.9536	1.50556
$540$	0	0
$541$	−23.1137	−0.993738	−0.496869	−	0.867826i	$-0.665517\pi$
$541$	−23.1137	−0.993738	−0.496869	+	0.867826i	$0.665517\pi$
$542$	0	0
$543$	−43.9019	−1.88401
$544$	0	0
$545$	36.7893	1.57588
$546$	0	0
$547$	−0.955705	−0.0408630	−0.0204315	−	0.999791i	$-0.506504\pi$
$547$	−0.955705	−0.0408630	−0.0204315	+	0.999791i	$0.506504\pi$
$548$	0	0
$549$	0.563786	0.0240618
$550$	0	0
$551$	6.19279	0.263822
$552$	0	0
$553$	−55.5873	−2.36381
$554$	0	0
$555$	26.9712	1.14486
$556$	0	0
$557$	−44.1868	−1.87225	−0.936127	−	0.351663i	$-0.885616\pi$
$557$	−44.1868	−1.87225	−0.936127	+	0.351663i	$0.885616\pi$
$558$	0	0
$559$	−30.6951	−1.29826
$560$	0	0
$561$	−5.97913	−0.252439
$562$	0	0
$563$	23.5722	0.993450	0.496725	−	0.867908i	$-0.334536\pi$
$563$	23.5722	0.993450	0.496725	+	0.867908i	$0.334536\pi$
$564$	0	0
$565$	15.9930	0.672832
$566$	0	0
$567$	34.3628	1.44310
$568$	0	0
$569$	−9.94642	−0.416976	−0.208488	−	0.978025i	$-0.566854\pi$
$569$	−9.94642	−0.416976	−0.208488	+	0.978025i	$0.566854\pi$
$570$	0	0
$571$	41.7250	1.74614	0.873069	−	0.487596i	$-0.162126\pi$
$571$	41.7250	1.74614	0.873069	+	0.487596i	$0.162126\pi$
$572$	0	0
$573$	40.7971	1.70432
$574$	0	0
$575$	5.84958	0.243945
$576$	0	0
$577$	−3.15976	−0.131542	−0.0657712	−	0.997835i	$-0.520951\pi$
$577$	−3.15976	−0.131542	−0.0657712	+	0.997835i	$0.520951\pi$
$578$	0	0
$579$	29.7761	1.23745
$580$	0	0
$581$	−0.930914	−0.0386208
$582$	0	0
$583$	45.3241	1.87713
$584$	0	0
$585$	0.421905	0.0174436
$586$	0	0
$587$	−28.4982	−1.17625	−0.588124	−	0.808771i	$-0.700133\pi$
$587$	−28.4982	−1.17625	−0.588124	+	0.808771i	$0.700133\pi$
$588$	0	0
$589$	−1.13776	−0.0468806
$590$	0	0
$591$	−2.76883	−0.113894
$592$	0	0
$593$	−36.1175	−1.48317	−0.741584	−	0.670860i	$-0.765925\pi$
$593$	−36.1175	−1.48317	−0.741584	+	0.670860i	$0.765925\pi$
$594$	0	0
$595$	−4.76980	−0.195543
$596$	0	0
$597$	−4.51014	−0.184588
$598$	0	0
$599$	−29.4776	−1.20442	−0.602211	−	0.798337i	$-0.705713\pi$
$599$	−29.4776	−1.20442	−0.602211	+	0.798337i	$0.705713\pi$
$600$	0	0
$601$	24.0688	0.981788	0.490894	−	0.871219i	$-0.336670\pi$
$601$	24.0688	0.981788	0.490894	+	0.871219i	$0.336670\pi$
$602$	0	0
$603$	−0.350017	−0.0142538
$604$	0	0
$605$	27.6475	1.12403
$606$	0	0
$607$	4.16907	0.169217	0.0846087	−	0.996414i	$-0.473036\pi$
$607$	4.16907	0.169217	0.0846087	+	0.996414i	$0.473036\pi$
$608$	0	0
$609$	−39.7646	−1.61134
$610$	0	0
$611$	−7.90070	−0.319628
$612$	0	0
$613$	−29.0558	−1.17355	−0.586777	−	0.809748i	$-0.699604\pi$
$613$	−29.0558	−1.17355	−0.586777	+	0.809748i	$0.699604\pi$
$614$	0	0
$615$	−31.8865	−1.28579
$616$	0	0
$617$	−6.36550	−0.256265	−0.128133	−	0.991757i	$-0.540898\pi$
$617$	−6.36550	−0.256265	−0.128133	+	0.991757i	$0.540898\pi$
$618$	0	0
$619$	−16.9730	−0.682202	−0.341101	−	0.940027i	$-0.610800\pi$
$619$	−16.9730	−0.682202	−0.341101	+	0.940027i	$0.610800\pi$
$620$	0	0
$621$	−21.3762	−0.857796
$622$	0	0
$623$	−45.9183	−1.83968
$624$	0	0
$625$	−16.0076	−0.640305
$626$	0	0
$627$	9.02726	0.360514
$628$	0	0
$629$	−5.47205	−0.218185
$630$	0	0
$631$	−11.5011	−0.457851	−0.228926	−	0.973444i	$-0.573521\pi$
$631$	−11.5011	−0.457851	−0.228926	+	0.973444i	$0.573521\pi$
$632$	0	0
$633$	−20.7611	−0.825178
$634$	0	0
$635$	14.6678	0.582073
$636$	0	0
$637$	21.1141	0.836572
$638$	0	0
$639$	−0.643332	−0.0254498
$640$	0	0
$641$	−11.2156	−0.442989	−0.221495	−	0.975162i	$-0.571094\pi$
$641$	−11.2156	−0.442989	−0.221495	+	0.975162i	$0.571094\pi$
$642$	0	0
$643$	28.9503	1.14169	0.570844	−	0.821058i	$-0.306616\pi$
$643$	28.9503	1.14169	0.570844	+	0.821058i	$0.306616\pi$
$644$	0	0
$645$	33.3971	1.31501
$646$	0	0
$647$	23.5389	0.925410	0.462705	−	0.886512i	$-0.346879\pi$
$647$	23.5389	0.925410	0.462705	+	0.886512i	$0.346879\pi$
$648$	0	0
$649$	−21.0138	−0.824863
$650$	0	0
$651$	7.30569	0.286333
$652$	0	0
$653$	29.7003	1.16226	0.581130	−	0.813810i	$-0.302611\pi$
$653$	29.7003	1.16226	0.581130	+	0.813810i	$0.302611\pi$
$654$	0	0
$655$	−37.4465	−1.46316
$656$	0	0
$657$	−0.0586179	−0.00228690
$658$	0	0
$659$	−7.36190	−0.286779	−0.143389	−	0.989666i	$-0.545800\pi$
$659$	−7.36190	−0.286779	−0.143389	+	0.989666i	$0.545800\pi$
$660$	0	0
$661$	28.1623	1.09539	0.547694	−	0.836679i	$-0.315506\pi$
$661$	28.1623	1.09539	0.547694	+	0.836679i	$0.315506\pi$
$662$	0	0
$663$	−3.61176	−0.140269
$664$	0	0
$665$	7.20142	0.279259
$666$	0	0
$667$	25.3373	0.981066
$668$	0	0
$669$	−32.9923	−1.27556
$670$	0	0
$671$	39.1544	1.51154
$672$	0	0
$673$	−35.0246	−1.35010	−0.675049	−	0.737773i	$-0.735878\pi$
$673$	−35.0246	−1.35010	−0.675049	+	0.737773i	$0.735878\pi$
$674$	0	0
$675$	−7.20494	−0.277318
$676$	0	0
$677$	3.62092	0.139163	0.0695815	−	0.997576i	$-0.477834\pi$
$677$	3.62092	0.139163	0.0695815	+	0.997576i	$0.477834\pi$
$678$	0	0
$679$	−25.6366	−0.983841
$680$	0	0
$681$	−24.6228	−0.943546
$682$	0	0
$683$	35.2092	1.34724	0.673621	−	0.739077i	$-0.264738\pi$
$683$	35.2092	1.34724	0.673621	+	0.739077i	$0.264738\pi$
$684$	0	0
$685$	−38.7067	−1.47891
$686$	0	0
$687$	20.3483	0.776337
$688$	0	0
$689$	27.3785	1.04304
$690$	0	0
$691$	2.61729	0.0995665	0.0497832	−	0.998760i	$-0.484147\pi$
$691$	2.61729	0.0995665	0.0497832	+	0.998760i	$0.484147\pi$
$692$	0	0
$693$	−1.37376	−0.0521849
$694$	0	0
$695$	−33.5719	−1.27345
$696$	0	0
$697$	6.46928	0.245042
$698$	0	0
$699$	49.4020	1.86855
$700$	0	0
$701$	−27.9810	−1.05683	−0.528413	−	0.848987i	$-0.677213\pi$
$701$	−27.9810	−1.05683	−0.528413	+	0.848987i	$0.677213\pi$
$702$	0	0
$703$	8.26168	0.311595
$704$	0	0
$705$	8.59617	0.323750
$706$	0	0
$707$	−37.3844	−1.40599
$708$	0	0
$709$	−21.7958	−0.818560	−0.409280	−	0.912409i	$-0.634220\pi$
$709$	−21.7958	−0.818560	−0.409280	+	0.912409i	$0.634220\pi$
$710$	0	0
$711$	1.08537	0.0407047
$712$	0	0
$713$	−4.65506	−0.174333
$714$	0	0
$715$	29.3009	1.09579
$716$	0	0
$717$	−37.3141	−1.39352
$718$	0	0
$719$	21.7011	0.809315	0.404657	−	0.914468i	$-0.367391\pi$
$719$	21.7011	0.809315	0.404657	+	0.914468i	$0.367391\pi$
$720$	0	0
$721$	17.2065	0.640804
$722$	0	0
$723$	−18.9069	−0.703157
$724$	0	0
$725$	8.54007	0.317170
$726$	0	0
$727$	23.3049	0.864330	0.432165	−	0.901795i	$-0.357750\pi$
$727$	23.3049	0.864330	0.432165	+	0.901795i	$0.357750\pi$
$728$	0	0
$729$	26.3131	0.974561
$730$	0	0
$731$	−6.77576	−0.250611
$732$	0	0
$733$	−21.0401	−0.777133	−0.388567	−	0.921421i	$-0.627030\pi$
$733$	−21.0401	−0.777133	−0.388567	+	0.921421i	$0.627030\pi$
$734$	0	0
$735$	−22.9727	−0.847361
$736$	0	0
$737$	−24.3083	−0.895409
$738$	0	0
$739$	−21.1585	−0.778328	−0.389164	−	0.921168i	$-0.627236\pi$
$739$	−21.1585	−0.778328	−0.389164	+	0.921168i	$0.627236\pi$
$740$	0	0
$741$	5.45302	0.200322
$742$	0	0
$743$	−7.87401	−0.288869	−0.144435	−	0.989514i	$-0.546136\pi$
$743$	−7.87401	−0.288869	−0.144435	+	0.989514i	$0.546136\pi$
$744$	0	0
$745$	−32.9860	−1.20851
$746$	0	0
$747$	0.0181766	0.000665047 0
$748$	0	0
$749$	14.1992	0.518828
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	22.7121	0.827675
$754$	0	0
$755$	−40.2349	−1.46430
$756$	0	0
$757$	46.6950	1.69716	0.848579	−	0.529069i	$-0.177459\pi$
$757$	46.6950	1.69716	0.848579	+	0.529069i	$0.177459\pi$
$758$	0	0
$759$	36.9344	1.34063
$760$	0	0
$761$	−37.5950	−1.36282	−0.681408	−	0.731903i	$-0.738632\pi$
$761$	−37.5950	−1.36282	−0.681408	+	0.731903i	$0.738632\pi$
$762$	0	0
$763$	−72.3603	−2.61962
$764$	0	0
$765$	0.0931329	0.00336723
$766$	0	0
$767$	−12.6936	−0.458340
$768$	0	0
$769$	52.9079	1.90791	0.953954	−	0.299953i	$-0.0969709\pi$
$769$	52.9079	1.90791	0.953954	+	0.299953i	$0.0969709\pi$
$770$	0	0
$771$	11.1694	0.402256
$772$	0	0
$773$	−0.392875	−0.0141307	−0.00706536	−	0.999975i	$-0.502249\pi$
$773$	−0.392875	−0.0141307	−0.00706536	+	0.999975i	$0.502249\pi$
$774$	0	0
$775$	−1.56901	−0.0563605
$776$	0	0
$777$	−53.0492	−1.90313
$778$	0	0
$779$	−9.76730	−0.349950
$780$	0	0
$781$	−44.6788	−1.59873
$782$	0	0
$783$	−31.2080	−1.11528
$784$	0	0
$785$	−38.5681	−1.37656
$786$	0	0
$787$	−22.8998	−0.816289	−0.408144	−	0.912917i	$-0.633824\pi$
$787$	−22.8998	−0.816289	−0.408144	+	0.912917i	$0.633824\pi$
$788$	0	0
$789$	32.6977	1.16407
$790$	0	0
$791$	−31.4564	−1.11846
$792$	0	0
$793$	23.6517	0.839896
$794$	0	0
$795$	−29.7886	−1.05649
$796$	0	0
$797$	−24.1842	−0.856648	−0.428324	−	0.903625i	$-0.640896\pi$
$797$	−24.1842	−0.856648	−0.428324	+	0.903625i	$0.640896\pi$
$798$	0	0
$799$	−1.74403	−0.0616994
$800$	0	0
$801$	0.896581	0.0316791
$802$	0	0
$803$	−4.07096	−0.143661
$804$	0	0
$805$	29.4641	1.03847
$806$	0	0
$807$	1.99819	0.0703397
$808$	0	0
$809$	31.8383	1.11938	0.559688	−	0.828704i	$-0.310921\pi$
$809$	31.8383	1.11938	0.559688	+	0.828704i	$0.310921\pi$
$810$	0	0
$811$	−17.1206	−0.601185	−0.300592	−	0.953753i	$-0.597184\pi$
$811$	−17.1206	−0.601185	−0.300592	+	0.953753i	$0.597184\pi$
$812$	0	0
$813$	−32.2332	−1.13047
$814$	0	0
$815$	−8.10089	−0.283762
$816$	0	0
$817$	10.2300	0.357903
$818$	0	0
$819$	−0.829837	−0.0289969
$820$	0	0
$821$	36.7001	1.28084	0.640422	−	0.768023i	$-0.278760\pi$
$821$	36.7001	1.28084	0.640422	+	0.768023i	$0.278760\pi$
$822$	0	0
$823$	33.3886	1.16385	0.581927	−	0.813241i	$-0.302299\pi$
$823$	33.3886	1.16385	0.581927	+	0.813241i	$0.302299\pi$
$824$	0	0
$825$	12.4489	0.433415
$826$	0	0
$827$	1.09363	0.0380292	0.0190146	−	0.999819i	$-0.493947\pi$
$827$	1.09363	0.0380292	0.0190146	+	0.999819i	$0.493947\pi$
$828$	0	0
$829$	47.3177	1.64341	0.821706	−	0.569912i	$-0.193023\pi$
$829$	47.3177	1.64341	0.821706	+	0.569912i	$0.193023\pi$
$830$	0	0
$831$	35.8538	1.24375
$832$	0	0
$833$	4.66081	0.161488
$834$	0	0
$835$	−7.69213	−0.266197
$836$	0	0
$837$	5.73364	0.198184
$838$	0	0
$839$	36.1346	1.24751	0.623753	−	0.781622i	$-0.285607\pi$
$839$	36.1346	1.24751	0.623753	+	0.781622i	$0.285607\pi$
$840$	0	0
$841$	7.99111	0.275555
$842$	0	0
$843$	14.2008	0.489101
$844$	0	0
$845$	−6.95201	−0.239156
$846$	0	0
$847$	−54.3794	−1.86850
$848$	0	0
$849$	36.9194	1.26707
$850$	0	0
$851$	33.8021	1.15872
$852$	0	0
$853$	−13.9612	−0.478023	−0.239012	−	0.971017i	$-0.576823\pi$
$853$	−13.9612	−0.478023	−0.239012	+	0.971017i	$0.576823\pi$
$854$	0	0
$855$	−0.140612	−0.00480882
$856$	0	0
$857$	−9.14438	−0.312366	−0.156183	−	0.987728i	$-0.549919\pi$
$857$	−9.14438	−0.312366	−0.156183	+	0.987728i	$0.549919\pi$
$858$	0	0
$859$	−3.71451	−0.126737	−0.0633687	−	0.997990i	$-0.520184\pi$
$859$	−3.71451	−0.126737	−0.0633687	+	0.997990i	$0.520184\pi$
$860$	0	0
$861$	62.7170	2.13739
$862$	0	0
$863$	−49.4067	−1.68182	−0.840912	−	0.541172i	$-0.817981\pi$
$863$	−49.4067	−1.68182	−0.840912	+	0.541172i	$0.817981\pi$
$864$	0	0
$865$	6.19928	0.210782
$866$	0	0
$867$	29.0028	0.984988
$868$	0	0
$869$	75.3782	2.55703
$870$	0	0
$871$	−14.6837	−0.497539
$872$	0	0
$873$	0.500568	0.0169417
$874$	0	0
$875$	45.2941	1.53122
$876$	0	0
$877$	−36.1953	−1.22223	−0.611115	−	0.791542i	$-0.709279\pi$
$877$	−36.1953	−1.22223	−0.611115	+	0.791542i	$0.709279\pi$
$878$	0	0
$879$	7.75101	0.261435
$880$	0	0
$881$	−27.8304	−0.937631	−0.468815	−	0.883296i	$-0.655319\pi$
$881$	−27.8304	−0.937631	−0.468815	+	0.883296i	$0.655319\pi$
$882$	0	0
$883$	21.2567	0.715346	0.357673	−	0.933847i	$-0.383570\pi$
$883$	21.2567	0.715346	0.357673	+	0.933847i	$0.383570\pi$
$884$	0	0
$885$	13.8110	0.464251
$886$	0	0
$887$	−36.6624	−1.23100	−0.615502	−	0.788136i	$-0.711047\pi$
$887$	−36.6624	−1.23100	−0.615502	+	0.788136i	$0.711047\pi$
$888$	0	0
$889$	−28.8498	−0.967592
$890$	0	0
$891$	−46.5971	−1.56106
$892$	0	0
$893$	2.63313	0.0881144
$894$	0	0
$895$	−28.3139	−0.946429
$896$	0	0
$897$	22.3106	0.744930
$898$	0	0
$899$	−6.79613	−0.226664
$900$	0	0
$901$	6.04365	0.201343
$902$	0	0
$903$	−65.6882	−2.18596
$904$	0	0
$905$	47.4915	1.57867
$906$	0	0
$907$	10.8054	0.358786	0.179393	−	0.983777i	$-0.442587\pi$
$907$	10.8054	0.358786	0.179393	+	0.983777i	$0.442587\pi$
$908$	0	0
$909$	0.729952	0.0242110
$910$	0	0
$911$	22.3876	0.741733	0.370867	−	0.928686i	$-0.379061\pi$
$911$	22.3876	0.741733	0.370867	+	0.928686i	$0.379061\pi$
$912$	0	0
$913$	1.26235	0.0417776
$914$	0	0
$915$	−25.7336	−0.850728
$916$	0	0
$917$	73.6530	2.43224
$918$	0	0
$919$	26.3710	0.869898	0.434949	−	0.900455i	$-0.356766\pi$
$919$	26.3710	0.869898	0.434949	+	0.900455i	$0.356766\pi$
$920$	0	0
$921$	36.4406	1.20076
$922$	0	0
$923$	−26.9888	−0.888346
$924$	0	0
$925$	11.3931	0.374604
$926$	0	0
$927$	−0.335967	−0.0110346
$928$	0	0
$929$	−16.7360	−0.549091	−0.274545	−	0.961574i	$-0.588527\pi$
$929$	−16.7360	−0.549091	−0.274545	+	0.961574i	$0.588527\pi$
$930$	0	0
$931$	−7.03688	−0.230624
$932$	0	0
$933$	−25.9676	−0.850141
$934$	0	0
$935$	6.46800	0.211526
$936$	0	0
$937$	15.4001	0.503100	0.251550	−	0.967844i	$-0.419060\pi$
$937$	15.4001	0.503100	0.251550	+	0.967844i	$0.419060\pi$
$938$	0	0
$939$	12.7252	0.415271
$940$	0	0
$941$	55.1465	1.79773	0.898863	−	0.438230i	$-0.144395\pi$
$941$	55.1465	1.79773	0.898863	+	0.438230i	$0.144395\pi$
$942$	0	0
$943$	−39.9622	−1.30135
$944$	0	0
$945$	−36.2909	−1.18054
$946$	0	0
$947$	13.0896	0.425355	0.212678	−	0.977122i	$-0.431782\pi$
$947$	13.0896	0.425355	0.212678	+	0.977122i	$0.431782\pi$
$948$	0	0
$949$	−2.45911	−0.0798261
$950$	0	0
$951$	−16.7641	−0.543613
$952$	0	0
$953$	−3.50919	−0.113674	−0.0568369	−	0.998383i	$-0.518101\pi$
$953$	−3.50919	−0.113674	−0.0568369	+	0.998383i	$0.518101\pi$
$954$	0	0
$955$	−44.1328	−1.42810
$956$	0	0
$957$	53.9221	1.74305
$958$	0	0
$959$	76.1316	2.45842
$960$	0	0
$961$	−29.7514	−0.959722
$962$	0	0
$963$	−0.277248	−0.00893418
$964$	0	0
$965$	−32.2107	−1.03690
$966$	0	0
$967$	−24.0201	−0.772433	−0.386217	−	0.922408i	$-0.626218\pi$
$967$	−24.0201	−0.772433	−0.386217	+	0.922408i	$0.626218\pi$
$968$	0	0
$969$	1.20372	0.0386691
$970$	0	0
$971$	−3.17730	−0.101964	−0.0509822	−	0.998700i	$-0.516235\pi$
$971$	−3.17730	−0.101964	−0.0509822	+	0.998700i	$0.516235\pi$
$972$	0	0
$973$	66.0320	2.11689
$974$	0	0
$975$	7.51990	0.240830
$976$	0	0
$977$	31.0978	0.994906	0.497453	−	0.867491i	$-0.334269\pi$
$977$	31.0978	0.994906	0.497453	+	0.867491i	$0.334269\pi$
$978$	0	0
$979$	62.2667	1.99005
$980$	0	0
$981$	1.41287	0.0451096
$982$	0	0
$983$	−32.2134	−1.02745	−0.513724	−	0.857956i	$-0.671734\pi$
$983$	−32.2134	−1.02745	−0.513724	+	0.857956i	$0.671734\pi$
$984$	0	0
$985$	2.99522	0.0954355
$986$	0	0
$987$	−16.9077	−0.538177
$988$	0	0
$989$	41.8554	1.33092
$990$	0	0
$991$	−30.6297	−0.972984	−0.486492	−	0.873685i	$-0.661724\pi$
$991$	−30.6297	−0.972984	−0.486492	+	0.873685i	$0.661724\pi$
$992$	0	0
$993$	−4.36262	−0.138444
$994$	0	0
$995$	4.87890	0.154672
$996$	0	0
$997$	−52.9544	−1.67708	−0.838542	−	0.544837i	$-0.816592\pi$
$997$	−52.9544	−1.67708	−0.838542	+	0.544837i	$0.816592\pi$
$998$	0	0
$999$	−41.6340	−1.31724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.14	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.14	✓	44	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75295 q^{3} +1.89627 q^{5} -3.72975 q^{7} +0.0728254 q^{9} +O(q^{10})\)	\(q-1.75295 q^{3} +1.89627 q^{5} -3.72975 q^{7} +0.0728254 q^{9} +5.05766 q^{11} +3.05514 q^{13} -3.32407 q^{15} +0.674403 q^{17} -1.01821 q^{19} +6.53805 q^{21} -4.16593 q^{23} -1.40415 q^{25} +5.13118 q^{27} -6.08203 q^{29} +1.11741 q^{31} -8.86581 q^{33} -7.07262 q^{35} -8.11392 q^{37} -5.35549 q^{39} +9.59261 q^{41} -10.0470 q^{43} +0.138097 q^{45} -2.58604 q^{47} +6.91102 q^{49} -1.18219 q^{51} +8.96148 q^{53} +9.59070 q^{55} +1.78487 q^{57} -4.15484 q^{59} +7.74161 q^{61} -0.271620 q^{63} +5.79337 q^{65} -4.80625 q^{67} +7.30266 q^{69} -8.83390 q^{71} -0.804911 q^{73} +2.46140 q^{75} -18.8638 q^{77} +14.9038 q^{79} -9.21317 q^{81} +0.249592 q^{83} +1.27885 q^{85} +10.6615 q^{87} +12.3114 q^{89} -11.3949 q^{91} -1.95876 q^{93} -1.93081 q^{95} +6.87353 q^{97} +0.368326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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