LMFDB - Embedded newform 6008.2.a.b.1.2

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			1.2
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)

$f(q)$	$=$	$q-3.25541 q^{3} +3.98380 q^{5} -3.42108 q^{7} +7.59768 q^{9} +O(q^{10})$	$q-3.25541 q^{3} +3.98380 q^{5} -3.42108 q^{7} +7.59768 q^{9} +3.25732 q^{11} -0.807608 q^{13} -12.9689 q^{15} -7.09260 q^{17} -1.26064 q^{19} +11.1370 q^{21} +4.19001 q^{23} +10.8706 q^{25} -14.9673 q^{27} +9.60617 q^{29} -3.56038 q^{31} -10.6039 q^{33} -13.6289 q^{35} -8.53329 q^{37} +2.62909 q^{39} +1.78776 q^{41} -3.07321 q^{43} +30.2676 q^{45} +1.74557 q^{47} +4.70377 q^{49} +23.0893 q^{51} -8.29290 q^{53} +12.9765 q^{55} +4.10389 q^{57} -8.47606 q^{59} -5.19121 q^{61} -25.9923 q^{63} -3.21735 q^{65} -4.78894 q^{67} -13.6402 q^{69} -3.44947 q^{71} +7.82156 q^{73} -35.3884 q^{75} -11.1435 q^{77} -10.4003 q^{79} +25.9317 q^{81} -6.41097 q^{83} -28.2555 q^{85} -31.2720 q^{87} -0.0939472 q^{89} +2.76289 q^{91} +11.5905 q^{93} -5.02213 q^{95} +7.80321 q^{97} +24.7481 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.25541	−1.87951	−0.939755	−	0.341847i	$-0.888947\pi$
$3$	−3.25541	−1.87951	−0.939755	+	0.341847i	$0.888947\pi$
$4$	0	0
$5$	3.98380	1.78161	0.890804	−	0.454387i	$-0.150142\pi$
$5$	3.98380	1.78161	0.890804	+	0.454387i	$0.150142\pi$
$6$	0	0
$7$	−3.42108	−1.29305	−0.646523	−	0.762895i	$-0.723777\pi$
$7$	−3.42108	−1.29305	−0.646523	+	0.762895i	$0.723777\pi$
$8$	0	0
$9$	7.59768	2.53256
$10$	0	0
$11$	3.25732	0.982120	0.491060	−	0.871126i	$-0.336610\pi$
$11$	3.25732	0.982120	0.491060	+	0.871126i	$0.336610\pi$
$12$	0	0
$13$	−0.807608	−0.223990	−0.111995	−	0.993709i	$-0.535724\pi$
$13$	−0.807608	−0.223990	−0.111995	+	0.993709i	$0.535724\pi$
$14$	0	0
$15$	−12.9689	−3.34855
$16$	0	0
$17$	−7.09260	−1.72021	−0.860104	−	0.510118i	$-0.829602\pi$
$17$	−7.09260	−1.72021	−0.860104	+	0.510118i	$0.829602\pi$
$18$	0	0
$19$	−1.26064	−0.289210	−0.144605	−	0.989489i	$-0.546191\pi$
$19$	−1.26064	−0.289210	−0.144605	+	0.989489i	$0.546191\pi$
$20$	0	0
$21$	11.1370	2.43029
$22$	0	0
$23$	4.19001	0.873678	0.436839	−	0.899540i	$-0.356098\pi$
$23$	4.19001	0.873678	0.436839	+	0.899540i	$0.356098\pi$
$24$	0	0
$25$	10.8706	2.17413
$26$	0	0
$27$	−14.9673	−2.88047
$28$	0	0
$29$	9.60617	1.78382	0.891910	−	0.452212i	$-0.149365\pi$
$29$	9.60617	1.78382	0.891910	+	0.452212i	$0.149365\pi$
$30$	0	0
$31$	−3.56038	−0.639463	−0.319731	−	0.947508i	$-0.603593\pi$
$31$	−3.56038	−0.639463	−0.319731	+	0.947508i	$0.603593\pi$
$32$	0	0
$33$	−10.6039	−1.84590
$34$	0	0
$35$	−13.6289	−2.30370
$36$	0	0
$37$	−8.53329	−1.40286	−0.701432	−	0.712737i	$-0.747456\pi$
$37$	−8.53329	−1.40286	−0.701432	+	0.712737i	$0.747456\pi$
$38$	0	0
$39$	2.62909	0.420992
$40$	0	0
$41$	1.78776	0.279201	0.139601	−	0.990208i	$-0.455418\pi$
$41$	1.78776	0.279201	0.139601	+	0.990208i	$0.455418\pi$
$42$	0	0
$43$	−3.07321	−0.468660	−0.234330	−	0.972157i	$-0.575290\pi$
$43$	−3.07321	−0.468660	−0.234330	+	0.972157i	$0.575290\pi$
$44$	0	0
$45$	30.2676	4.51203
$46$	0	0
$47$	1.74557	0.254617	0.127308	−	0.991863i	$-0.459366\pi$
$47$	1.74557	0.254617	0.127308	+	0.991863i	$0.459366\pi$
$48$	0	0
$49$	4.70377	0.671967
$50$	0	0
$51$	23.0893	3.23315
$52$	0	0
$53$	−8.29290	−1.13912	−0.569559	−	0.821950i	$-0.692886\pi$
$53$	−8.29290	−1.13912	−0.569559	+	0.821950i	$0.692886\pi$
$54$	0	0
$55$	12.9765	1.74975
$56$	0	0
$57$	4.10389	0.543574
$58$	0	0
$59$	−8.47606	−1.10349	−0.551745	−	0.834013i	$-0.686038\pi$
$59$	−8.47606	−1.10349	−0.551745	+	0.834013i	$0.686038\pi$
$60$	0	0
$61$	−5.19121	−0.664666	−0.332333	−	0.943162i	$-0.607836\pi$
$61$	−5.19121	−0.664666	−0.332333	+	0.943162i	$0.607836\pi$
$62$	0	0
$63$	−25.9923	−3.27472
$64$	0	0
$65$	−3.21735	−0.399063
$66$	0	0
$67$	−4.78894	−0.585062	−0.292531	−	0.956256i	$-0.594497\pi$
$67$	−4.78894	−0.585062	−0.292531	+	0.956256i	$0.594497\pi$
$68$	0	0
$69$	−13.6402	−1.64209
$70$	0	0
$71$	−3.44947	−0.409376	−0.204688	−	0.978827i	$-0.565618\pi$
$71$	−3.44947	−0.409376	−0.204688	+	0.978827i	$0.565618\pi$
$72$	0	0
$73$	7.82156	0.915445	0.457722	−	0.889095i	$-0.348665\pi$
$73$	7.82156	0.915445	0.457722	+	0.889095i	$0.348665\pi$
$74$	0	0
$75$	−35.3884	−4.08630
$76$	0	0
$77$	−11.1435	−1.26993
$78$	0	0
$79$	−10.4003	−1.17012	−0.585062	−	0.810988i	$-0.698930\pi$
$79$	−10.4003	−1.17012	−0.585062	+	0.810988i	$0.698930\pi$
$80$	0	0
$81$	25.9317	2.88130
$82$	0	0
$83$	−6.41097	−0.703695	−0.351848	−	0.936057i	$-0.614447\pi$
$83$	−6.41097	−0.703695	−0.351848	+	0.936057i	$0.614447\pi$
$84$	0	0
$85$	−28.2555	−3.06474
$86$	0	0
$87$	−31.2720	−3.35271
$88$	0	0
$89$	−0.0939472	−0.00995838	−0.00497919	−	0.999988i	$-0.501585\pi$
$89$	−0.0939472	−0.00995838	−0.00497919	+	0.999988i	$0.501585\pi$
$90$	0	0
$91$	2.76289	0.289629
$92$	0	0
$93$	11.5905	1.20188
$94$	0	0
$95$	−5.02213	−0.515260
$96$	0	0
$97$	7.80321	0.792295	0.396148	−	0.918187i	$-0.370347\pi$
$97$	7.80321	0.792295	0.396148	+	0.918187i	$0.370347\pi$
$98$	0	0
$99$	24.7481	2.48728
$100$	0	0
$101$	9.94811	0.989874	0.494937	−	0.868929i	$-0.335191\pi$
$101$	9.94811	0.989874	0.494937	+	0.868929i	$0.335191\pi$
$102$	0	0
$103$	14.3593	1.41486	0.707431	−	0.706782i	$-0.249854\pi$
$103$	14.3593	1.41486	0.707431	+	0.706782i	$0.249854\pi$
$104$	0	0
$105$	44.3676	4.32983
$106$	0	0
$107$	16.7197	1.61635	0.808177	−	0.588939i	$-0.200454\pi$
$107$	16.7197	1.61635	0.808177	+	0.588939i	$0.200454\pi$
$108$	0	0
$109$	−10.7314	−1.02788	−0.513939	−	0.857827i	$-0.671814\pi$
$109$	−10.7314	−1.02788	−0.513939	+	0.857827i	$0.671814\pi$
$110$	0	0
$111$	27.7793	2.63670
$112$	0	0
$113$	−16.2319	−1.52697	−0.763486	−	0.645824i	$-0.776514\pi$
$113$	−16.2319	−1.52697	−0.763486	+	0.645824i	$0.776514\pi$
$114$	0	0
$115$	16.6922	1.55655
$116$	0	0
$117$	−6.13595	−0.567269
$118$	0	0
$119$	24.2643	2.22431
$120$	0	0
$121$	−0.389852	−0.0354411
$122$	0	0
$123$	−5.81989	−0.524762
$124$	0	0
$125$	23.3875	2.09184
$126$	0	0
$127$	8.83449	0.783934	0.391967	−	0.919979i	$-0.371795\pi$
$127$	8.83449	0.783934	0.391967	+	0.919979i	$0.371795\pi$
$128$	0	0
$129$	10.0045	0.880851
$130$	0	0
$131$	14.0418	1.22684	0.613418	−	0.789759i	$-0.289794\pi$
$131$	14.0418	1.22684	0.613418	+	0.789759i	$0.289794\pi$
$132$	0	0
$133$	4.31274	0.373962
$134$	0	0
$135$	−59.6268	−5.13186
$136$	0	0
$137$	13.2854	1.13505	0.567526	−	0.823356i	$-0.307901\pi$
$137$	13.2854	1.13505	0.567526	+	0.823356i	$0.307901\pi$
$138$	0	0
$139$	−7.12458	−0.604299	−0.302149	−	0.953261i	$-0.597704\pi$
$139$	−7.12458	−0.604299	−0.302149	+	0.953261i	$0.597704\pi$
$140$	0	0
$141$	−5.68253	−0.478555
$142$	0	0
$143$	−2.63064	−0.219985
$144$	0	0
$145$	38.2690	3.17807
$146$	0	0
$147$	−15.3127	−1.26297
$148$	0	0
$149$	6.10732	0.500332	0.250166	−	0.968203i	$-0.419515\pi$
$149$	6.10732	0.500332	0.250166	+	0.968203i	$0.419515\pi$
$150$	0	0
$151$	4.91160	0.399700	0.199850	−	0.979827i	$-0.435955\pi$
$151$	4.91160	0.399700	0.199850	+	0.979827i	$0.435955\pi$
$152$	0	0
$153$	−53.8873	−4.35653
$154$	0	0
$155$	−14.1838	−1.13927
$156$	0	0
$157$	−9.18561	−0.733092	−0.366546	−	0.930400i	$-0.619460\pi$
$157$	−9.18561	−0.733092	−0.366546	+	0.930400i	$0.619460\pi$
$158$	0	0
$159$	26.9968	2.14098
$160$	0	0
$161$	−14.3343	−1.12970
$162$	0	0
$163$	11.3050	0.885476	0.442738	−	0.896651i	$-0.354007\pi$
$163$	11.3050	0.885476	0.442738	+	0.896651i	$0.354007\pi$
$164$	0	0
$165$	−42.2439	−3.28868
$166$	0	0
$167$	−24.2822	−1.87901	−0.939507	−	0.342529i	$-0.888716\pi$
$167$	−24.2822	−1.87901	−0.939507	+	0.342529i	$0.888716\pi$
$168$	0	0
$169$	−12.3478	−0.949828
$170$	0	0
$171$	−9.57794	−0.732443
$172$	0	0
$173$	−9.43435	−0.717280	−0.358640	−	0.933476i	$-0.616759\pi$
$173$	−9.43435	−0.717280	−0.358640	+	0.933476i	$0.616759\pi$
$174$	0	0
$175$	−37.1893	−2.81125
$176$	0	0
$177$	27.5930	2.07402
$178$	0	0
$179$	−19.6876	−1.47152	−0.735760	−	0.677242i	$-0.763175\pi$
$179$	−19.6876	−1.47152	−0.735760	+	0.677242i	$0.763175\pi$
$180$	0	0
$181$	11.8912	0.883865	0.441933	−	0.897048i	$-0.354293\pi$
$181$	11.8912	0.883865	0.441933	+	0.897048i	$0.354293\pi$
$182$	0	0
$183$	16.8995	1.24925
$184$	0	0
$185$	−33.9949	−2.49935
$186$	0	0
$187$	−23.1029	−1.68945
$188$	0	0
$189$	51.2044	3.72457
$190$	0	0
$191$	21.5931	1.56242	0.781211	−	0.624267i	$-0.214602\pi$
$191$	21.5931	1.56242	0.781211	+	0.624267i	$0.214602\pi$
$192$	0	0
$193$	7.38434	0.531536	0.265768	−	0.964037i	$-0.414374\pi$
$193$	7.38434	0.531536	0.265768	+	0.964037i	$0.414374\pi$
$194$	0	0
$195$	10.4738	0.750043
$196$	0	0
$197$	3.94819	0.281297	0.140648	−	0.990060i	$-0.455081\pi$
$197$	3.94819	0.281297	0.140648	+	0.990060i	$0.455081\pi$
$198$	0	0
$199$	5.88359	0.417077	0.208538	−	0.978014i	$-0.433129\pi$
$199$	5.88359	0.417077	0.208538	+	0.978014i	$0.433129\pi$
$200$	0	0
$201$	15.5900	1.09963
$202$	0	0
$203$	−32.8634	−2.30656
$204$	0	0
$205$	7.12207	0.497427
$206$	0	0
$207$	31.8344	2.21264
$208$	0	0
$209$	−4.10631	−0.284039
$210$	0	0
$211$	11.8761	0.817585	0.408793	−	0.912627i	$-0.365950\pi$
$211$	11.8761	0.817585	0.408793	+	0.912627i	$0.365950\pi$
$212$	0	0
$213$	11.2294	0.769427
$214$	0	0
$215$	−12.2430	−0.834969
$216$	0	0
$217$	12.1803	0.826854
$218$	0	0
$219$	−25.4624	−1.72059
$220$	0	0
$221$	5.72804	0.385310
$222$	0	0
$223$	−10.5414	−0.705904	−0.352952	−	0.935641i	$-0.614822\pi$
$223$	−10.5414	−0.705904	−0.352952	+	0.935641i	$0.614822\pi$
$224$	0	0
$225$	82.5917	5.50612
$226$	0	0
$227$	−18.0890	−1.20061	−0.600304	−	0.799772i	$-0.704954\pi$
$227$	−18.0890	−1.20061	−0.600304	+	0.799772i	$0.704954\pi$
$228$	0	0
$229$	−7.44054	−0.491685	−0.245842	−	0.969310i	$-0.579065\pi$
$229$	−7.44054	−0.491685	−0.245842	+	0.969310i	$0.579065\pi$
$230$	0	0
$231$	36.2768	2.38684
$232$	0	0
$233$	−11.0306	−0.722641	−0.361321	−	0.932442i	$-0.617674\pi$
$233$	−11.0306	−0.722641	−0.361321	+	0.932442i	$0.617674\pi$
$234$	0	0
$235$	6.95398	0.453628
$236$	0	0
$237$	33.8572	2.19926
$238$	0	0
$239$	−22.1481	−1.43264	−0.716321	−	0.697771i	$-0.754175\pi$
$239$	−22.1481	−1.43264	−0.716321	+	0.697771i	$0.754175\pi$
$240$	0	0
$241$	−25.1368	−1.61921	−0.809603	−	0.586978i	$-0.800317\pi$
$241$	−25.1368	−1.61921	−0.809603	+	0.586978i	$0.800317\pi$
$242$	0	0
$243$	−39.5164	−2.53498
$244$	0	0
$245$	18.7389	1.19718
$246$	0	0
$247$	1.01810	0.0647803
$248$	0	0
$249$	20.8703	1.32260
$250$	0	0
$251$	−28.8358	−1.82010	−0.910050	−	0.414499i	$-0.863957\pi$
$251$	−28.8358	−1.82010	−0.910050	+	0.414499i	$0.863957\pi$
$252$	0	0
$253$	13.6482	0.858056
$254$	0	0
$255$	91.9832	5.76021
$256$	0	0
$257$	−31.6103	−1.97180	−0.985898	−	0.167348i	$-0.946480\pi$
$257$	−31.6103	−1.97180	−0.985898	+	0.167348i	$0.946480\pi$
$258$	0	0
$259$	29.1930	1.81397
$260$	0	0
$261$	72.9846	4.51764
$262$	0	0
$263$	−12.1291	−0.747910	−0.373955	−	0.927447i	$-0.621999\pi$
$263$	−12.1291	−0.747910	−0.373955	+	0.927447i	$0.621999\pi$
$264$	0	0
$265$	−33.0373	−2.02946
$266$	0	0
$267$	0.305836	0.0187169
$268$	0	0
$269$	1.17950	0.0719153	0.0359577	−	0.999353i	$-0.488552\pi$
$269$	1.17950	0.0719153	0.0359577	+	0.999353i	$0.488552\pi$
$270$	0	0
$271$	−2.48302	−0.150833	−0.0754163	−	0.997152i	$-0.524029\pi$
$271$	−2.48302	−0.150833	−0.0754163	+	0.997152i	$0.524029\pi$
$272$	0	0
$273$	−8.99433	−0.544362
$274$	0	0
$275$	35.4092	2.13526
$276$	0	0
$277$	−15.2937	−0.918909	−0.459454	−	0.888201i	$-0.651955\pi$
$277$	−15.2937	−0.918909	−0.459454	+	0.888201i	$0.651955\pi$
$278$	0	0
$279$	−27.0506	−1.61948
$280$	0	0
$281$	14.3825	0.857985	0.428993	−	0.903308i	$-0.358869\pi$
$281$	14.3825	0.857985	0.428993	+	0.903308i	$0.358869\pi$
$282$	0	0
$283$	26.7379	1.58941	0.794703	−	0.606999i	$-0.207627\pi$
$283$	26.7379	1.58941	0.794703	+	0.606999i	$0.207627\pi$
$284$	0	0
$285$	16.3491	0.968436
$286$	0	0
$287$	−6.11606	−0.361020
$288$	0	0
$289$	33.3050	1.95912
$290$	0	0
$291$	−25.4026	−1.48913
$292$	0	0
$293$	2.21933	0.129655	0.0648273	−	0.997896i	$-0.479350\pi$
$293$	2.21933	0.129655	0.0648273	+	0.997896i	$0.479350\pi$
$294$	0	0
$295$	−33.7669	−1.96599
$296$	0	0
$297$	−48.7534	−2.82896
$298$	0	0
$299$	−3.38389	−0.195695
$300$	0	0
$301$	10.5137	0.605999
$302$	0	0
$303$	−32.3852	−1.86048
$304$	0	0
$305$	−20.6807	−1.18417
$306$	0	0
$307$	−5.05256	−0.288365	−0.144182	−	0.989551i	$-0.546055\pi$
$307$	−5.05256	−0.288365	−0.144182	+	0.989551i	$0.546055\pi$
$308$	0	0
$309$	−46.7453	−2.65925
$310$	0	0
$311$	−27.1784	−1.54114	−0.770572	−	0.637353i	$-0.780029\pi$
$311$	−27.1784	−1.54114	−0.770572	+	0.637353i	$0.780029\pi$
$312$	0	0
$313$	−2.21976	−0.125468	−0.0627342	−	0.998030i	$-0.519982\pi$
$313$	−2.21976	−0.125468	−0.0627342	+	0.998030i	$0.519982\pi$
$314$	0	0
$315$	−103.548	−5.83426
$316$	0	0
$317$	22.1714	1.24527	0.622635	−	0.782513i	$-0.286062\pi$
$317$	22.1714	1.24527	0.622635	+	0.782513i	$0.286062\pi$
$318$	0	0
$319$	31.2904	1.75193
$320$	0	0
$321$	−54.4295	−3.03796
$322$	0	0
$323$	8.94121	0.497502
$324$	0	0
$325$	−8.77922	−0.486984
$326$	0	0
$327$	34.9350	1.93191
$328$	0	0
$329$	−5.97171	−0.329231
$330$	0	0
$331$	34.0036	1.86901	0.934504	−	0.355952i	$-0.115843\pi$
$331$	34.0036	1.86901	0.934504	+	0.355952i	$0.115843\pi$
$332$	0	0
$333$	−64.8332	−3.55284
$334$	0	0
$335$	−19.0782	−1.04235
$336$	0	0
$337$	−28.7726	−1.56734	−0.783672	−	0.621174i	$-0.786656\pi$
$337$	−28.7726	−1.56734	−0.783672	+	0.621174i	$0.786656\pi$
$338$	0	0
$339$	52.8416	2.86996
$340$	0	0
$341$	−11.5973	−0.628029
$342$	0	0
$343$	7.85559	0.424162
$344$	0	0
$345$	−54.3398	−2.92556
$346$	0	0
$347$	3.50293	0.188047	0.0940235	−	0.995570i	$-0.470027\pi$
$347$	3.50293	0.188047	0.0940235	+	0.995570i	$0.470027\pi$
$348$	0	0
$349$	6.78478	0.363181	0.181590	−	0.983374i	$-0.441876\pi$
$349$	6.78478	0.363181	0.181590	+	0.983374i	$0.441876\pi$
$350$	0	0
$351$	12.0877	0.645196
$352$	0	0
$353$	−32.2304	−1.71545	−0.857726	−	0.514107i	$-0.828124\pi$
$353$	−32.2304	−1.71545	−0.857726	+	0.514107i	$0.828124\pi$
$354$	0	0
$355$	−13.7420	−0.729348
$356$	0	0
$357$	−78.9903	−4.18061
$358$	0	0
$359$	−4.36621	−0.230440	−0.115220	−	0.993340i	$-0.536757\pi$
$359$	−4.36621	−0.230440	−0.115220	+	0.993340i	$0.536757\pi$
$360$	0	0
$361$	−17.4108	−0.916357
$362$	0	0
$363$	1.26913	0.0666120
$364$	0	0
$365$	31.1595	1.63096
$366$	0	0
$367$	37.9134	1.97906	0.989532	−	0.144312i	$-0.0460969\pi$
$367$	37.9134	1.97906	0.989532	+	0.144312i	$0.0460969\pi$
$368$	0	0
$369$	13.5828	0.707094
$370$	0	0
$371$	28.3707	1.47293
$372$	0	0
$373$	−8.56182	−0.443314	−0.221657	−	0.975125i	$-0.571147\pi$
$373$	−8.56182	−0.443314	−0.221657	+	0.975125i	$0.571147\pi$
$374$	0	0
$375$	−76.1358	−3.93163
$376$	0	0
$377$	−7.75802	−0.399558
$378$	0	0
$379$	−24.8590	−1.27692	−0.638460	−	0.769655i	$-0.720428\pi$
$379$	−24.8590	−1.27692	−0.638460	+	0.769655i	$0.720428\pi$
$380$	0	0
$381$	−28.7599	−1.47341
$382$	0	0
$383$	−20.0236	−1.02316	−0.511579	−	0.859236i	$-0.670939\pi$
$383$	−20.0236	−1.02316	−0.511579	+	0.859236i	$0.670939\pi$
$384$	0	0
$385$	−44.3937	−2.26251
$386$	0	0
$387$	−23.3493	−1.18691
$388$	0	0
$389$	−21.9099	−1.11088	−0.555438	−	0.831558i	$-0.687449\pi$
$389$	−21.9099	−1.11088	−0.555438	+	0.831558i	$0.687449\pi$
$390$	0	0
$391$	−29.7181	−1.50291
$392$	0	0
$393$	−45.7117	−2.30585
$394$	0	0
$395$	−41.4327	−2.08470
$396$	0	0
$397$	−30.2337	−1.51739	−0.758693	−	0.651449i	$-0.774162\pi$
$397$	−30.2337	−1.51739	−0.758693	+	0.651449i	$0.774162\pi$
$398$	0	0
$399$	−14.0397	−0.702866
$400$	0	0
$401$	22.2039	1.10881	0.554404	−	0.832247i	$-0.312946\pi$
$401$	22.2039	1.10881	0.554404	+	0.832247i	$0.312946\pi$
$402$	0	0
$403$	2.87539	0.143233
$404$	0	0
$405$	103.307	5.13336
$406$	0	0
$407$	−27.7957	−1.37778
$408$	0	0
$409$	−5.76070	−0.284848	−0.142424	−	0.989806i	$-0.545490\pi$
$409$	−5.76070	−0.284848	−0.142424	+	0.989806i	$0.545490\pi$
$410$	0	0
$411$	−43.2495	−2.13334
$412$	0	0
$413$	28.9973	1.42686
$414$	0	0
$415$	−25.5400	−1.25371
$416$	0	0
$417$	23.1934	1.13579
$418$	0	0
$419$	−6.67763	−0.326224	−0.163112	−	0.986608i	$-0.552153\pi$
$419$	−6.67763	−0.326224	−0.163112	+	0.986608i	$0.552153\pi$
$420$	0	0
$421$	−35.5326	−1.73175	−0.865877	−	0.500257i	$-0.833239\pi$
$421$	−35.5326	−1.73175	−0.865877	+	0.500257i	$0.833239\pi$
$422$	0	0
$423$	13.2623	0.644833
$424$	0	0
$425$	−77.1012	−3.73996
$426$	0	0
$427$	17.7595	0.859443
$428$	0	0
$429$	8.56380	0.413464
$430$	0	0
$431$	10.1940	0.491028	0.245514	−	0.969393i	$-0.421043\pi$
$431$	10.1940	0.491028	0.245514	+	0.969393i	$0.421043\pi$
$432$	0	0
$433$	−2.87531	−0.138179	−0.0690893	−	0.997610i	$-0.522009\pi$
$433$	−2.87531	−0.138179	−0.0690893	+	0.997610i	$0.522009\pi$
$434$	0	0
$435$	−124.581	−5.97322
$436$	0	0
$437$	−5.28209	−0.252677
$438$	0	0
$439$	0.679234	0.0324181	0.0162090	−	0.999869i	$-0.494840\pi$
$439$	0.679234	0.0324181	0.0162090	+	0.999869i	$0.494840\pi$
$440$	0	0
$441$	35.7377	1.70180
$442$	0	0
$443$	−29.2162	−1.38810	−0.694052	−	0.719925i	$-0.744176\pi$
$443$	−29.2162	−1.38810	−0.694052	+	0.719925i	$0.744176\pi$
$444$	0	0
$445$	−0.374267	−0.0177419
$446$	0	0
$447$	−19.8818	−0.940379
$448$	0	0
$449$	20.9446	0.988438	0.494219	−	0.869337i	$-0.335454\pi$
$449$	20.9446	0.988438	0.494219	+	0.869337i	$0.335454\pi$
$450$	0	0
$451$	5.82331	0.274209
$452$	0	0
$453$	−15.9893	−0.751241
$454$	0	0
$455$	11.0068	0.516006
$456$	0	0
$457$	29.9887	1.40281	0.701407	−	0.712761i	$-0.252556\pi$
$457$	29.9887	1.40281	0.701407	+	0.712761i	$0.252556\pi$
$458$	0	0
$459$	106.157	4.95500
$460$	0	0
$461$	1.16824	0.0544104	0.0272052	−	0.999630i	$-0.491339\pi$
$461$	1.16824	0.0544104	0.0272052	+	0.999630i	$0.491339\pi$
$462$	0	0
$463$	23.1537	1.07604	0.538021	−	0.842931i	$-0.319172\pi$
$463$	23.1537	1.07604	0.538021	+	0.842931i	$0.319172\pi$
$464$	0	0
$465$	46.1741	2.14127
$466$	0	0
$467$	36.4043	1.68459	0.842295	−	0.539017i	$-0.181204\pi$
$467$	36.4043	1.68459	0.842295	+	0.539017i	$0.181204\pi$
$468$	0	0
$469$	16.3833	0.756512
$470$	0	0
$471$	29.9029	1.37785
$472$	0	0
$473$	−10.0104	−0.460280
$474$	0	0
$475$	−13.7040	−0.628781
$476$	0	0
$477$	−63.0069	−2.88489
$478$	0	0
$479$	3.83238	0.175106	0.0875529	−	0.996160i	$-0.472095\pi$
$479$	3.83238	0.175106	0.0875529	+	0.996160i	$0.472095\pi$
$480$	0	0
$481$	6.89155	0.314228
$482$	0	0
$483$	46.6642	2.12329
$484$	0	0
$485$	31.0864	1.41156
$486$	0	0
$487$	−19.7865	−0.896612	−0.448306	−	0.893880i	$-0.647973\pi$
$487$	−19.7865	−0.896612	−0.448306	+	0.893880i	$0.647973\pi$
$488$	0	0
$489$	−36.8024	−1.66426
$490$	0	0
$491$	6.69904	0.302324	0.151162	−	0.988509i	$-0.451699\pi$
$491$	6.69904	0.302324	0.151162	+	0.988509i	$0.451699\pi$
$492$	0	0
$493$	−68.1327	−3.06854
$494$	0	0
$495$	98.5914	4.43136
$496$	0	0
$497$	11.8009	0.529342
$498$	0	0
$499$	−37.1137	−1.66144	−0.830720	−	0.556691i	$-0.812071\pi$
$499$	−37.1137	−1.66144	−0.830720	+	0.556691i	$0.812071\pi$
$500$	0	0
$501$	79.0485	3.53163
$502$	0	0
$503$	−22.8289	−1.01789	−0.508946	−	0.860799i	$-0.669965\pi$
$503$	−22.8289	−1.01789	−0.508946	+	0.860799i	$0.669965\pi$
$504$	0	0
$505$	39.6313	1.76357
$506$	0	0
$507$	40.1970	1.78521
$508$	0	0
$509$	−38.3428	−1.69951	−0.849757	−	0.527175i	$-0.823251\pi$
$509$	−38.3428	−1.69951	−0.849757	+	0.527175i	$0.823251\pi$
$510$	0	0
$511$	−26.7582	−1.18371
$512$	0	0
$513$	18.8684	0.833061
$514$	0	0
$515$	57.2045	2.52073
$516$	0	0
$517$	5.68587	0.250064
$518$	0	0
$519$	30.7126	1.34814
$520$	0	0
$521$	−28.4935	−1.24832	−0.624160	−	0.781296i	$-0.714559\pi$
$521$	−28.4935	−1.24832	−0.624160	+	0.781296i	$0.714559\pi$
$522$	0	0
$523$	21.3657	0.934255	0.467128	−	0.884190i	$-0.345289\pi$
$523$	21.3657	0.934255	0.467128	+	0.884190i	$0.345289\pi$
$524$	0	0
$525$	121.066	5.28377
$526$	0	0
$527$	25.2523	1.10001
$528$	0	0
$529$	−5.44381	−0.236687
$530$	0	0
$531$	−64.3984	−2.79465
$532$	0	0
$533$	−1.44381	−0.0625383
$534$	0	0
$535$	66.6079	2.87971
$536$	0	0
$537$	64.0912	2.76574
$538$	0	0
$539$	15.3217	0.659952
$540$	0	0
$541$	8.11124	0.348730	0.174365	−	0.984681i	$-0.444213\pi$
$541$	8.11124	0.348730	0.174365	+	0.984681i	$0.444213\pi$
$542$	0	0
$543$	−38.7107	−1.66123
$544$	0	0
$545$	−42.7516	−1.83128
$546$	0	0
$547$	20.5632	0.879221	0.439610	−	0.898189i	$-0.355117\pi$
$547$	20.5632	0.879221	0.439610	+	0.898189i	$0.355117\pi$
$548$	0	0
$549$	−39.4411	−1.68331
$550$	0	0
$551$	−12.1099	−0.515900
$552$	0	0
$553$	35.5802	1.51302
$554$	0	0
$555$	110.667	4.69756
$556$	0	0
$557$	−28.3552	−1.20145	−0.600724	−	0.799456i	$-0.705121\pi$
$557$	−28.3552	−1.20145	−0.600724	+	0.799456i	$0.705121\pi$
$558$	0	0
$559$	2.48195	0.104975
$560$	0	0
$561$	75.2093	3.17534
$562$	0	0
$563$	8.95165	0.377267	0.188633	−	0.982048i	$-0.439594\pi$
$563$	8.95165	0.377267	0.188633	+	0.982048i	$0.439594\pi$
$564$	0	0
$565$	−64.6648	−2.72047
$566$	0	0
$567$	−88.7145	−3.72566
$568$	0	0
$569$	−1.24021	−0.0519923	−0.0259961	−	0.999662i	$-0.508276\pi$
$569$	−1.24021	−0.0519923	−0.0259961	+	0.999662i	$0.508276\pi$
$570$	0	0
$571$	11.0486	0.462368	0.231184	−	0.972910i	$-0.425740\pi$
$571$	11.0486	0.462368	0.231184	+	0.972910i	$0.425740\pi$
$572$	0	0
$573$	−70.2944	−2.93659
$574$	0	0
$575$	45.5481	1.89949
$576$	0	0
$577$	−35.1213	−1.46212	−0.731060	−	0.682313i	$-0.760974\pi$
$577$	−35.1213	−1.46212	−0.731060	+	0.682313i	$0.760974\pi$
$578$	0	0
$579$	−24.0390	−0.999028
$580$	0	0
$581$	21.9324	0.909910
$582$	0	0
$583$	−27.0127	−1.11875
$584$	0	0
$585$	−24.4444	−1.01065
$586$	0	0
$587$	−12.1454	−0.501293	−0.250646	−	0.968079i	$-0.580643\pi$
$587$	−12.1454	−0.501293	−0.250646	+	0.968079i	$0.580643\pi$
$588$	0	0
$589$	4.48835	0.184939
$590$	0	0
$591$	−12.8530	−0.528700
$592$	0	0
$593$	−26.7848	−1.09992	−0.549961	−	0.835191i	$-0.685357\pi$
$593$	−26.7848	−1.09992	−0.549961	+	0.835191i	$0.685357\pi$
$594$	0	0
$595$	96.6642	3.96285
$596$	0	0
$597$	−19.1535	−0.783901
$598$	0	0
$599$	−4.85522	−0.198379	−0.0991895	−	0.995069i	$-0.531625\pi$
$599$	−4.85522	−0.198379	−0.0991895	+	0.995069i	$0.531625\pi$
$600$	0	0
$601$	−39.2427	−1.60074	−0.800371	−	0.599505i	$-0.795364\pi$
$601$	−39.2427	−1.60074	−0.800371	+	0.599505i	$0.795364\pi$
$602$	0	0
$603$	−36.3848	−1.48171
$604$	0	0
$605$	−1.55309	−0.0631422
$606$	0	0
$607$	−21.4714	−0.871495	−0.435748	−	0.900069i	$-0.643516\pi$
$607$	−21.4714	−0.871495	−0.435748	+	0.900069i	$0.643516\pi$
$608$	0	0
$609$	106.984	4.33521
$610$	0	0
$611$	−1.40973	−0.0570317
$612$	0	0
$613$	7.89473	0.318865	0.159433	−	0.987209i	$-0.449034\pi$
$613$	7.89473	0.318865	0.159433	+	0.987209i	$0.449034\pi$
$614$	0	0
$615$	−23.1853	−0.934920
$616$	0	0
$617$	6.97253	0.280704	0.140352	−	0.990102i	$-0.455177\pi$
$617$	6.97253	0.280704	0.140352	+	0.990102i	$0.455177\pi$
$618$	0	0
$619$	19.0118	0.764149	0.382075	−	0.924131i	$-0.375210\pi$
$619$	19.0118	0.764149	0.382075	+	0.924131i	$0.375210\pi$
$620$	0	0
$621$	−62.7133	−2.51660
$622$	0	0
$623$	0.321401	0.0128766
$624$	0	0
$625$	38.8177	1.55271
$626$	0	0
$627$	13.3677	0.533855
$628$	0	0
$629$	60.5232	2.41322
$630$	0	0
$631$	−16.2520	−0.646983	−0.323492	−	0.946231i	$-0.604857\pi$
$631$	−16.2520	−0.646983	−0.323492	+	0.946231i	$0.604857\pi$
$632$	0	0
$633$	−38.6616	−1.53666
$634$	0	0
$635$	35.1948	1.39666
$636$	0	0
$637$	−3.79880	−0.150514
$638$	0	0
$639$	−26.2079	−1.03677
$640$	0	0
$641$	−36.8993	−1.45744	−0.728718	−	0.684814i	$-0.759883\pi$
$641$	−36.8993	−1.45744	−0.728718	+	0.684814i	$0.759883\pi$
$642$	0	0
$643$	9.10483	0.359059	0.179530	−	0.983753i	$-0.442542\pi$
$643$	9.10483	0.359059	0.179530	+	0.983753i	$0.442542\pi$
$644$	0	0
$645$	39.8561	1.56933
$646$	0	0
$647$	−45.4351	−1.78624	−0.893120	−	0.449819i	$-0.851488\pi$
$647$	−45.4351	−1.78624	−0.893120	+	0.449819i	$0.851488\pi$
$648$	0	0
$649$	−27.6093	−1.08376
$650$	0	0
$651$	−39.6519	−1.55408
$652$	0	0
$653$	−37.4827	−1.46681	−0.733405	−	0.679792i	$-0.762070\pi$
$653$	−37.4827	−1.46681	−0.733405	+	0.679792i	$0.762070\pi$
$654$	0	0
$655$	55.9396	2.18574
$656$	0	0
$657$	59.4258	2.31842
$658$	0	0
$659$	−5.97908	−0.232912	−0.116456	−	0.993196i	$-0.537153\pi$
$659$	−5.97908	−0.232912	−0.116456	+	0.993196i	$0.537153\pi$
$660$	0	0
$661$	49.2370	1.91510	0.957548	−	0.288272i	$-0.0930808\pi$
$661$	49.2370	1.91510	0.957548	+	0.288272i	$0.0930808\pi$
$662$	0	0
$663$	−18.6471	−0.724194
$664$	0	0
$665$	17.1811	0.666254
$666$	0	0
$667$	40.2499	1.55848
$668$	0	0
$669$	34.3166	1.32675
$670$	0	0
$671$	−16.9094	−0.652781
$672$	0	0
$673$	−51.1888	−1.97318	−0.986592	−	0.163206i	$-0.947816\pi$
$673$	−51.1888	−1.97318	−0.986592	+	0.163206i	$0.947816\pi$
$674$	0	0
$675$	−162.705	−6.26250
$676$	0	0
$677$	−5.66213	−0.217613	−0.108807	−	0.994063i	$-0.534703\pi$
$677$	−5.66213	−0.217613	−0.108807	+	0.994063i	$0.534703\pi$
$678$	0	0
$679$	−26.6954	−1.02447
$680$	0	0
$681$	58.8870	2.25656
$682$	0	0
$683$	44.1051	1.68764	0.843818	−	0.536630i	$-0.180303\pi$
$683$	44.1051	1.68764	0.843818	+	0.536630i	$0.180303\pi$
$684$	0	0
$685$	52.9265	2.02222
$686$	0	0
$687$	24.2220	0.924127
$688$	0	0
$689$	6.69741	0.255151
$690$	0	0
$691$	−7.24429	−0.275586	−0.137793	−	0.990461i	$-0.544001\pi$
$691$	−7.24429	−0.275586	−0.137793	+	0.990461i	$0.544001\pi$
$692$	0	0
$693$	−84.6652	−3.21616
$694$	0	0
$695$	−28.3829	−1.07662
$696$	0	0
$697$	−12.6799	−0.480284
$698$	0	0
$699$	35.9092	1.35821
$700$	0	0
$701$	9.55968	0.361064	0.180532	−	0.983569i	$-0.442218\pi$
$701$	9.55968	0.361064	0.180532	+	0.983569i	$0.442218\pi$
$702$	0	0
$703$	10.7574	0.405723
$704$	0	0
$705$	−22.6380	−0.852598
$706$	0	0
$707$	−34.0333	−1.27995
$708$	0	0
$709$	−21.4083	−0.804004	−0.402002	−	0.915639i	$-0.631686\pi$
$709$	−21.4083	−0.804004	−0.402002	+	0.915639i	$0.631686\pi$
$710$	0	0
$711$	−79.0181	−2.96341
$712$	0	0
$713$	−14.9180	−0.558684
$714$	0	0
$715$	−10.4799	−0.391927
$716$	0	0
$717$	72.1012	2.69267
$718$	0	0
$719$	−4.49117	−0.167492	−0.0837461	−	0.996487i	$-0.526688\pi$
$719$	−4.49117	−0.167492	−0.0837461	+	0.996487i	$0.526688\pi$
$720$	0	0
$721$	−49.1242	−1.82948
$722$	0	0
$723$	81.8306	3.04331
$724$	0	0
$725$	104.425	3.87826
$726$	0	0
$727$	8.41611	0.312136	0.156068	−	0.987746i	$-0.450118\pi$
$727$	8.41611	0.312136	0.156068	+	0.987746i	$0.450118\pi$
$728$	0	0
$729$	50.8468	1.88321
$730$	0	0
$731$	21.7970	0.806193
$732$	0	0
$733$	46.4050	1.71401	0.857003	−	0.515311i	$-0.172324\pi$
$733$	46.4050	1.71401	0.857003	+	0.515311i	$0.172324\pi$
$734$	0	0
$735$	−61.0026	−2.25012
$736$	0	0
$737$	−15.5991	−0.574601
$738$	0	0
$739$	6.26522	0.230470	0.115235	−	0.993338i	$-0.463238\pi$
$739$	6.26522	0.230470	0.115235	+	0.993338i	$0.463238\pi$
$740$	0	0
$741$	−3.31434	−0.121755
$742$	0	0
$743$	−15.7833	−0.579034	−0.289517	−	0.957173i	$-0.593495\pi$
$743$	−15.7833	−0.579034	−0.289517	+	0.957173i	$0.593495\pi$
$744$	0	0
$745$	24.3303	0.891395
$746$	0	0
$747$	−48.7085	−1.78215
$748$	0	0
$749$	−57.1994	−2.09002
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	93.8723	3.42090
$754$	0	0
$755$	19.5668	0.712109
$756$	0	0
$757$	21.3890	0.777396	0.388698	−	0.921365i	$-0.372925\pi$
$757$	21.3890	0.777396	0.388698	+	0.921365i	$0.372925\pi$
$758$	0	0
$759$	−44.4305	−1.61273
$760$	0	0
$761$	−20.1231	−0.729462	−0.364731	−	0.931113i	$-0.618839\pi$
$761$	−20.1231	−0.729462	−0.364731	+	0.931113i	$0.618839\pi$
$762$	0	0
$763$	36.7128	1.32909
$764$	0	0
$765$	−214.676	−7.76164
$766$	0	0
$767$	6.84533	0.247171
$768$	0	0
$769$	46.8679	1.69010	0.845050	−	0.534687i	$-0.179571\pi$
$769$	46.8679	1.69010	0.845050	+	0.534687i	$0.179571\pi$
$770$	0	0
$771$	102.904	3.70601
$772$	0	0
$773$	−32.4710	−1.16790	−0.583951	−	0.811789i	$-0.698494\pi$
$773$	−32.4710	−1.16790	−0.583951	+	0.811789i	$0.698494\pi$
$774$	0	0
$775$	−38.7036	−1.39027
$776$	0	0
$777$	−95.0352	−3.40937
$778$	0	0
$779$	−2.25372	−0.0807479
$780$	0	0
$781$	−11.2360	−0.402056
$782$	0	0
$783$	−143.779	−5.13823
$784$	0	0
$785$	−36.5936	−1.30608
$786$	0	0
$787$	31.3602	1.11787	0.558936	−	0.829211i	$-0.311210\pi$
$787$	31.3602	1.11787	0.558936	+	0.829211i	$0.311210\pi$
$788$	0	0
$789$	39.4851	1.40571
$790$	0	0
$791$	55.5307	1.97444
$792$	0	0
$793$	4.19246	0.148879
$794$	0	0
$795$	107.550	3.81440
$796$	0	0
$797$	−3.28579	−0.116388	−0.0581942	−	0.998305i	$-0.518534\pi$
$797$	−3.28579	−0.116388	−0.0581942	+	0.998305i	$0.518534\pi$
$798$	0	0
$799$	−12.3806	−0.437994
$800$	0	0
$801$	−0.713781	−0.0252202
$802$	0	0
$803$	25.4774	0.899076
$804$	0	0
$805$	−57.1051	−2.01269
$806$	0	0
$807$	−3.83975	−0.135166
$808$	0	0
$809$	27.9216	0.981672	0.490836	−	0.871252i	$-0.336691\pi$
$809$	27.9216	0.981672	0.490836	+	0.871252i	$0.336691\pi$
$810$	0	0
$811$	2.37264	0.0833145	0.0416573	−	0.999132i	$-0.486736\pi$
$811$	2.37264	0.0833145	0.0416573	+	0.999132i	$0.486736\pi$
$812$	0	0
$813$	8.08323	0.283491
$814$	0	0
$815$	45.0369	1.57757
$816$	0	0
$817$	3.87421	0.135541
$818$	0	0
$819$	20.9916	0.733504
$820$	0	0
$821$	27.0229	0.943105	0.471553	−	0.881838i	$-0.343694\pi$
$821$	27.0229	0.943105	0.471553	+	0.881838i	$0.343694\pi$
$822$	0	0
$823$	13.1553	0.458566	0.229283	−	0.973360i	$-0.426362\pi$
$823$	13.1553	0.458566	0.229283	+	0.973360i	$0.426362\pi$
$824$	0	0
$825$	−115.271	−4.01324
$826$	0	0
$827$	−1.32207	−0.0459730	−0.0229865	−	0.999736i	$-0.507317\pi$
$827$	−1.32207	−0.0459730	−0.0229865	+	0.999736i	$0.507317\pi$
$828$	0	0
$829$	35.1684	1.22145	0.610725	−	0.791843i	$-0.290878\pi$
$829$	35.1684	1.22145	0.610725	+	0.791843i	$0.290878\pi$
$830$	0	0
$831$	49.7872	1.72710
$832$	0	0
$833$	−33.3619	−1.15592
$834$	0	0
$835$	−96.7354	−3.34767
$836$	0	0
$837$	53.2894	1.84195
$838$	0	0
$839$	−50.9475	−1.75890	−0.879451	−	0.475989i	$-0.842090\pi$
$839$	−50.9475	−1.75890	−0.879451	+	0.475989i	$0.842090\pi$
$840$	0	0
$841$	63.2785	2.18202
$842$	0	0
$843$	−46.8208	−1.61259
$844$	0	0
$845$	−49.1910	−1.69222
$846$	0	0
$847$	1.33371	0.0458270
$848$	0	0
$849$	−87.0429	−2.98730
$850$	0	0
$851$	−35.7546	−1.22565
$852$	0	0
$853$	−56.3181	−1.92829	−0.964147	−	0.265370i	$-0.914506\pi$
$853$	−56.3181	−1.92829	−0.964147	+	0.265370i	$0.914506\pi$
$854$	0	0
$855$	−38.1566	−1.30493
$856$	0	0
$857$	50.7152	1.73240	0.866199	−	0.499700i	$-0.166556\pi$
$857$	50.7152	1.73240	0.866199	+	0.499700i	$0.166556\pi$
$858$	0	0
$859$	−49.1843	−1.67815	−0.839073	−	0.544018i	$-0.816902\pi$
$859$	−49.1843	−1.67815	−0.839073	+	0.544018i	$0.816902\pi$
$860$	0	0
$861$	19.9103	0.678541
$862$	0	0
$863$	33.3076	1.13380	0.566902	−	0.823786i	$-0.308142\pi$
$863$	33.3076	1.13380	0.566902	+	0.823786i	$0.308142\pi$
$864$	0	0
$865$	−37.5845	−1.27791
$866$	0	0
$867$	−108.421	−3.68218
$868$	0	0
$869$	−33.8771	−1.14920
$870$	0	0
$871$	3.86759	0.131048
$872$	0	0
$873$	59.2863	2.00654
$874$	0	0
$875$	−80.0103	−2.70484
$876$	0	0
$877$	32.6366	1.10206	0.551030	−	0.834486i	$-0.314235\pi$
$877$	32.6366	1.10206	0.551030	+	0.834486i	$0.314235\pi$
$878$	0	0
$879$	−7.22482	−0.243687
$880$	0	0
$881$	−8.33467	−0.280802	−0.140401	−	0.990095i	$-0.544839\pi$
$881$	−8.33467	−0.280802	−0.140401	+	0.990095i	$0.544839\pi$
$882$	0	0
$883$	44.1851	1.48695	0.743474	−	0.668765i	$-0.233177\pi$
$883$	44.1851	1.48695	0.743474	+	0.668765i	$0.233177\pi$
$884$	0	0
$885$	109.925	3.69509
$886$	0	0
$887$	−24.0589	−0.807820	−0.403910	−	0.914799i	$-0.632349\pi$
$887$	−24.0589	−0.807820	−0.403910	+	0.914799i	$0.632349\pi$
$888$	0	0
$889$	−30.2235	−1.01366
$890$	0	0
$891$	84.4680	2.82979
$892$	0	0
$893$	−2.20053	−0.0736378
$894$	0	0
$895$	−78.4314	−2.62167
$896$	0	0
$897$	11.0159	0.367811
$898$	0	0
$899$	−34.2016	−1.14069
$900$	0	0
$901$	58.8183	1.95952
$902$	0	0
$903$	−34.2263	−1.13898
$904$	0	0
$905$	47.3721	1.57470
$906$	0	0
$907$	−30.5796	−1.01538	−0.507690	−	0.861540i	$-0.669500\pi$
$907$	−30.5796	−1.01538	−0.507690	+	0.861540i	$0.669500\pi$
$908$	0	0
$909$	75.5826	2.50692
$910$	0	0
$911$	31.0213	1.02778	0.513891	−	0.857855i	$-0.328203\pi$
$911$	31.0213	1.02778	0.513891	+	0.857855i	$0.328203\pi$
$912$	0	0
$913$	−20.8826	−0.691113
$914$	0	0
$915$	67.3242	2.22567
$916$	0	0
$917$	−48.0380	−1.58635
$918$	0	0
$919$	−26.8679	−0.886289	−0.443144	−	0.896450i	$-0.646137\pi$
$919$	−26.8679	−0.886289	−0.443144	+	0.896450i	$0.646137\pi$
$920$	0	0
$921$	16.4481	0.541984
$922$	0	0
$923$	2.78582	0.0916962
$924$	0	0
$925$	−92.7624	−3.05001
$926$	0	0
$927$	109.097	3.58323
$928$	0	0
$929$	9.41665	0.308950	0.154475	−	0.987997i	$-0.450631\pi$
$929$	9.41665	0.308950	0.154475	+	0.987997i	$0.450631\pi$
$930$	0	0
$931$	−5.92975	−0.194340
$932$	0	0
$933$	88.4767	2.89660
$934$	0	0
$935$	−92.0372	−3.00994
$936$	0	0
$937$	−0.902024	−0.0294678	−0.0147339	−	0.999891i	$-0.504690\pi$
$937$	−0.902024	−0.0294678	−0.0147339	+	0.999891i	$0.504690\pi$
$938$	0	0
$939$	7.22623	0.235819
$940$	0	0
$941$	16.2562	0.529938	0.264969	−	0.964257i	$-0.414638\pi$
$941$	16.2562	0.529938	0.264969	+	0.964257i	$0.414638\pi$
$942$	0	0
$943$	7.49073	0.243932
$944$	0	0
$945$	203.988	6.63573
$946$	0	0
$947$	5.47695	0.177977	0.0889885	−	0.996033i	$-0.471637\pi$
$947$	5.47695	0.177977	0.0889885	+	0.996033i	$0.471637\pi$
$948$	0	0
$949$	−6.31676	−0.205051
$950$	0	0
$951$	−72.1769	−2.34050
$952$	0	0
$953$	50.4670	1.63479	0.817393	−	0.576081i	$-0.195419\pi$
$953$	50.4670	1.63479	0.817393	+	0.576081i	$0.195419\pi$
$954$	0	0
$955$	86.0226	2.78362
$956$	0	0
$957$	−101.863	−3.29276
$958$	0	0
$959$	−45.4505	−1.46767
$960$	0	0
$961$	−18.3237	−0.591088
$962$	0	0
$963$	127.031	4.09352
$964$	0	0
$965$	29.4177	0.946989
$966$	0	0
$967$	−1.56488	−0.0503231	−0.0251616	−	0.999683i	$-0.508010\pi$
$967$	−1.56488	−0.0503231	−0.0251616	+	0.999683i	$0.508010\pi$
$968$	0	0
$969$	−29.1073	−0.935061
$970$	0	0
$971$	17.5450	0.563046	0.281523	−	0.959554i	$-0.409160\pi$
$971$	17.5450	0.563046	0.281523	+	0.959554i	$0.409160\pi$
$972$	0	0
$973$	24.3737	0.781386
$974$	0	0
$975$	28.5799	0.915291
$976$	0	0
$977$	−5.09155	−0.162893	−0.0814466	−	0.996678i	$-0.525954\pi$
$977$	−5.09155	−0.162893	−0.0814466	+	0.996678i	$0.525954\pi$
$978$	0	0
$979$	−0.306016	−0.00978032
$980$	0	0
$981$	−81.5335	−2.60316
$982$	0	0
$983$	26.2053	0.835820	0.417910	−	0.908489i	$-0.362763\pi$
$983$	26.2053	0.835820	0.417910	+	0.908489i	$0.362763\pi$
$984$	0	0
$985$	15.7288	0.501160
$986$	0	0
$987$	19.4404	0.618794
$988$	0	0
$989$	−12.8768	−0.409458
$990$	0	0
$991$	−26.8396	−0.852588	−0.426294	−	0.904585i	$-0.640181\pi$
$991$	−26.8396	−0.852588	−0.426294	+	0.904585i	$0.640181\pi$
$992$	0	0
$993$	−110.696	−3.51282
$994$	0	0
$995$	23.4391	0.743068
$996$	0	0
$997$	25.5246	0.808373	0.404187	−	0.914676i	$-0.367555\pi$
$997$	25.5246	0.808373	0.404187	+	0.914676i	$0.367555\pi$
$998$	0	0
$999$	127.721	4.04090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.2	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-3.25541 q^{3} +3.98380 q^{5} -3.42108 q^{7} +7.59768 q^{9} +O(q^{10})\)	\(q-3.25541 q^{3} +3.98380 q^{5} -3.42108 q^{7} +7.59768 q^{9} +3.25732 q^{11} -0.807608 q^{13} -12.9689 q^{15} -7.09260 q^{17} -1.26064 q^{19} +11.1370 q^{21} +4.19001 q^{23} +10.8706 q^{25} -14.9673 q^{27} +9.60617 q^{29} -3.56038 q^{31} -10.6039 q^{33} -13.6289 q^{35} -8.53329 q^{37} +2.62909 q^{39} +1.78776 q^{41} -3.07321 q^{43} +30.2676 q^{45} +1.74557 q^{47} +4.70377 q^{49} +23.0893 q^{51} -8.29290 q^{53} +12.9765 q^{55} +4.10389 q^{57} -8.47606 q^{59} -5.19121 q^{61} -25.9923 q^{63} -3.21735 q^{65} -4.78894 q^{67} -13.6402 q^{69} -3.44947 q^{71} +7.82156 q^{73} -35.3884 q^{75} -11.1435 q^{77} -10.4003 q^{79} +25.9317 q^{81} -6.41097 q^{83} -28.2555 q^{85} -31.2720 q^{87} -0.0939472 q^{89} +2.76289 q^{91} +11.5905 q^{93} -5.02213 q^{95} +7.80321 q^{97} +24.7481 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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

Properties

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