LMFDB - Embedded newform 6008.2.a.d.1.6

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:	$0$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.54750 q^{3} -2.18138 q^{5} -2.46793 q^{7} +3.48974 q^{9} +O(q^{10})$	$q-2.54750 q^{3} -2.18138 q^{5} -2.46793 q^{7} +3.48974 q^{9} +5.27702 q^{11} +2.99790 q^{13} +5.55706 q^{15} +4.41465 q^{17} +7.60548 q^{19} +6.28704 q^{21} +4.58382 q^{23} -0.241577 q^{25} -1.24761 q^{27} +1.08137 q^{29} -9.31795 q^{31} -13.4432 q^{33} +5.38349 q^{35} +2.04301 q^{37} -7.63713 q^{39} +2.65339 q^{41} +5.35134 q^{43} -7.61245 q^{45} +10.9824 q^{47} -0.909334 q^{49} -11.2463 q^{51} +2.86374 q^{53} -11.5112 q^{55} -19.3749 q^{57} -0.454535 q^{59} -5.54848 q^{61} -8.61243 q^{63} -6.53955 q^{65} -14.0342 q^{67} -11.6773 q^{69} +9.55610 q^{71} -2.14864 q^{73} +0.615416 q^{75} -13.0233 q^{77} +4.17062 q^{79} -7.29093 q^{81} +1.52485 q^{83} -9.63003 q^{85} -2.75478 q^{87} +4.51844 q^{89} -7.39859 q^{91} +23.7374 q^{93} -16.5905 q^{95} +4.92843 q^{97} +18.4154 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.54750	−1.47080	−0.735399	−	0.677634i	$-0.763005\pi$
$3$	−2.54750	−1.47080	−0.735399	+	0.677634i	$0.763005\pi$
$4$	0	0
$5$	−2.18138	−0.975543	−0.487772	−	0.872971i	$-0.662190\pi$
$5$	−2.18138	−0.975543	−0.487772	+	0.872971i	$0.662190\pi$
$6$	0	0
$7$	−2.46793	−0.932789	−0.466394	−	0.884577i	$-0.654447\pi$
$7$	−2.46793	−0.932789	−0.466394	+	0.884577i	$0.654447\pi$
$8$	0	0
$9$	3.48974	1.16325
$10$	0	0
$11$	5.27702	1.59108	0.795541	−	0.605900i	$-0.207187\pi$
$11$	5.27702	1.59108	0.795541	+	0.605900i	$0.207187\pi$
$12$	0	0
$13$	2.99790	0.831467	0.415733	−	0.909487i	$-0.363525\pi$
$13$	2.99790	0.831467	0.415733	+	0.909487i	$0.363525\pi$
$14$	0	0
$15$	5.55706	1.43483
$16$	0	0
$17$	4.41465	1.07071	0.535355	−	0.844627i	$-0.320178\pi$
$17$	4.41465	1.07071	0.535355	+	0.844627i	$0.320178\pi$
$18$	0	0
$19$	7.60548	1.74482	0.872409	−	0.488777i	$-0.162557\pi$
$19$	7.60548	1.74482	0.872409	+	0.488777i	$0.162557\pi$
$20$	0	0
$21$	6.28704	1.37194
$22$	0	0
$23$	4.58382	0.955792	0.477896	−	0.878416i	$-0.341400\pi$
$23$	4.58382	0.955792	0.477896	+	0.878416i	$0.341400\pi$
$24$	0	0
$25$	−0.241577	−0.0483154
$26$	0	0
$27$	−1.24761	−0.240103
$28$	0	0
$29$	1.08137	0.200805	0.100403	−	0.994947i	$-0.467987\pi$
$29$	1.08137	0.200805	0.100403	+	0.994947i	$0.467987\pi$
$30$	0	0
$31$	−9.31795	−1.67355	−0.836776	−	0.547545i	$-0.815563\pi$
$31$	−9.31795	−1.67355	−0.836776	+	0.547545i	$0.815563\pi$
$32$	0	0
$33$	−13.4432	−2.34016
$34$	0	0
$35$	5.38349	0.909976
$36$	0	0
$37$	2.04301	0.335869	0.167935	−	0.985798i	$-0.446290\pi$
$37$	2.04301	0.335869	0.167935	+	0.985798i	$0.446290\pi$
$38$	0	0
$39$	−7.63713	−1.22292
$40$	0	0
$41$	2.65339	0.414391	0.207195	−	0.978300i	$-0.433566\pi$
$41$	2.65339	0.414391	0.207195	+	0.978300i	$0.433566\pi$
$42$	0	0
$43$	5.35134	0.816072	0.408036	−	0.912966i	$-0.366214\pi$
$43$	5.35134	0.816072	0.408036	+	0.912966i	$0.366214\pi$
$44$	0	0
$45$	−7.61245	−1.13480
$46$	0	0
$47$	10.9824	1.60195	0.800977	−	0.598695i	$-0.204314\pi$
$47$	10.9824	1.60195	0.800977	+	0.598695i	$0.204314\pi$
$48$	0	0
$49$	−0.909334	−0.129905
$50$	0	0
$51$	−11.2463	−1.57480
$52$	0	0
$53$	2.86374	0.393366	0.196683	−	0.980467i	$-0.436983\pi$
$53$	2.86374	0.393366	0.196683	+	0.980467i	$0.436983\pi$
$54$	0	0
$55$	−11.5112	−1.55217
$56$	0	0
$57$	−19.3749	−2.56627
$58$	0	0
$59$	−0.454535	−0.0591754	−0.0295877	−	0.999562i	$-0.509419\pi$
$59$	−0.454535	−0.0591754	−0.0295877	+	0.999562i	$0.509419\pi$
$60$	0	0
$61$	−5.54848	−0.710410	−0.355205	−	0.934788i	$-0.615589\pi$
$61$	−5.54848	−0.710410	−0.355205	+	0.934788i	$0.615589\pi$
$62$	0	0
$63$	−8.61243	−1.08506
$64$	0	0
$65$	−6.53955	−0.811132
$66$	0	0
$67$	−14.0342	−1.71455	−0.857273	−	0.514862i	$-0.827843\pi$
$67$	−14.0342	−1.71455	−0.857273	+	0.514862i	$0.827843\pi$
$68$	0	0
$69$	−11.6773	−1.40578
$70$	0	0
$71$	9.55610	1.13410	0.567050	−	0.823683i	$-0.308085\pi$
$71$	9.55610	1.13410	0.567050	+	0.823683i	$0.308085\pi$
$72$	0	0
$73$	−2.14864	−0.251480	−0.125740	−	0.992063i	$-0.540130\pi$
$73$	−2.14864	−0.251480	−0.125740	+	0.992063i	$0.540130\pi$
$74$	0	0
$75$	0.615416	0.0710622
$76$	0	0
$77$	−13.0233	−1.48414
$78$	0	0
$79$	4.17062	0.469232	0.234616	−	0.972088i	$-0.424617\pi$
$79$	4.17062	0.469232	0.234616	+	0.972088i	$0.424617\pi$
$80$	0	0
$81$	−7.29093	−0.810103
$82$	0	0
$83$	1.52485	0.167374	0.0836868	−	0.996492i	$-0.473330\pi$
$83$	1.52485	0.167374	0.0836868	+	0.996492i	$0.473330\pi$
$84$	0	0
$85$	−9.63003	−1.04452
$86$	0	0
$87$	−2.75478	−0.295344
$88$	0	0
$89$	4.51844	0.478953	0.239477	−	0.970902i	$-0.423024\pi$
$89$	4.51844	0.478953	0.239477	+	0.970902i	$0.423024\pi$
$90$	0	0
$91$	−7.39859	−0.775583
$92$	0	0
$93$	23.7374	2.46146
$94$	0	0
$95$	−16.5905	−1.70215
$96$	0	0
$97$	4.92843	0.500406	0.250203	−	0.968193i	$-0.419503\pi$
$97$	4.92843	0.500406	0.250203	+	0.968193i	$0.419503\pi$
$98$	0	0
$99$	18.4154	1.85082
$100$	0	0
$101$	−13.0813	−1.30163	−0.650817	−	0.759235i	$-0.725574\pi$
$101$	−13.0813	−1.30163	−0.650817	+	0.759235i	$0.725574\pi$
$102$	0	0
$103$	14.0009	1.37955	0.689777	−	0.724022i	$-0.257708\pi$
$103$	14.0009	1.37955	0.689777	+	0.724022i	$0.257708\pi$
$104$	0	0
$105$	−13.7144	−1.33839
$106$	0	0
$107$	2.13667	0.206560	0.103280	−	0.994652i	$-0.467066\pi$
$107$	2.13667	0.206560	0.103280	+	0.994652i	$0.467066\pi$
$108$	0	0
$109$	−12.2394	−1.17232	−0.586162	−	0.810194i	$-0.699362\pi$
$109$	−12.2394	−1.17232	−0.586162	+	0.810194i	$0.699362\pi$
$110$	0	0
$111$	−5.20457	−0.493996
$112$	0	0
$113$	−0.416531	−0.0391839	−0.0195920	−	0.999808i	$-0.506237\pi$
$113$	−0.416531	−0.0391839	−0.0195920	+	0.999808i	$0.506237\pi$
$114$	0	0
$115$	−9.99905	−0.932416
$116$	0	0
$117$	10.4619	0.967201
$118$	0	0
$119$	−10.8950	−0.998746
$120$	0	0
$121$	16.8470	1.53154
$122$	0	0
$123$	−6.75951	−0.609485
$124$	0	0
$125$	11.4339	1.02268
$126$	0	0
$127$	−13.1762	−1.16920	−0.584598	−	0.811323i	$-0.698748\pi$
$127$	−13.1762	−1.16920	−0.584598	+	0.811323i	$0.698748\pi$
$128$	0	0
$129$	−13.6325	−1.20028
$130$	0	0
$131$	9.99976	0.873683	0.436841	−	0.899539i	$-0.356097\pi$
$131$	9.99976	0.873683	0.436841	+	0.899539i	$0.356097\pi$
$132$	0	0
$133$	−18.7698	−1.62755
$134$	0	0
$135$	2.72152	0.234231
$136$	0	0
$137$	0.309147	0.0264122	0.0132061	−	0.999913i	$-0.495796\pi$
$137$	0.309147	0.0264122	0.0132061	+	0.999913i	$0.495796\pi$
$138$	0	0
$139$	2.78407	0.236141	0.118071	−	0.993005i	$-0.462329\pi$
$139$	2.78407	0.236141	0.118071	+	0.993005i	$0.462329\pi$
$140$	0	0
$141$	−27.9777	−2.35615
$142$	0	0
$143$	15.8200	1.32293
$144$	0	0
$145$	−2.35888	−0.195894
$146$	0	0
$147$	2.31652	0.191064
$148$	0	0
$149$	−14.3759	−1.17772	−0.588860	−	0.808235i	$-0.700423\pi$
$149$	−14.3759	−1.17772	−0.588860	+	0.808235i	$0.700423\pi$
$150$	0	0
$151$	10.8791	0.885331	0.442665	−	0.896687i	$-0.354033\pi$
$151$	10.8791	0.885331	0.442665	+	0.896687i	$0.354033\pi$
$152$	0	0
$153$	15.4060	1.24550
$154$	0	0
$155$	20.3260	1.63262
$156$	0	0
$157$	14.7632	1.17823	0.589115	−	0.808049i	$-0.299476\pi$
$157$	14.7632	1.17823	0.589115	+	0.808049i	$0.299476\pi$
$158$	0	0
$159$	−7.29538	−0.578561
$160$	0	0
$161$	−11.3125	−0.891552
$162$	0	0
$163$	16.4767	1.29055	0.645276	−	0.763950i	$-0.276743\pi$
$163$	16.4767	1.29055	0.645276	+	0.763950i	$0.276743\pi$
$164$	0	0
$165$	29.3247	2.28293
$166$	0	0
$167$	15.3836	1.19042	0.595208	−	0.803571i	$-0.297069\pi$
$167$	15.3836	1.19042	0.595208	+	0.803571i	$0.297069\pi$
$168$	0	0
$169$	−4.01262	−0.308663
$170$	0	0
$171$	26.5412	2.02965
$172$	0	0
$173$	−17.4422	−1.32610	−0.663052	−	0.748573i	$-0.730739\pi$
$173$	−17.4422	−1.32610	−0.663052	+	0.748573i	$0.730739\pi$
$174$	0	0
$175$	0.596194	0.0450681
$176$	0	0
$177$	1.15793	0.0870351
$178$	0	0
$179$	0.296579	0.0221674	0.0110837	−	0.999939i	$-0.496472\pi$
$179$	0.296579	0.0221674	0.0110837	+	0.999939i	$0.496472\pi$
$180$	0	0
$181$	18.9775	1.41058	0.705292	−	0.708917i	$-0.250816\pi$
$181$	18.9775	1.41058	0.705292	+	0.708917i	$0.250816\pi$
$182$	0	0
$183$	14.1347	1.04487
$184$	0	0
$185$	−4.45659	−0.327655
$186$	0	0
$187$	23.2962	1.70359
$188$	0	0
$189$	3.07902	0.223966
$190$	0	0
$191$	−2.80983	−0.203312	−0.101656	−	0.994820i	$-0.532414\pi$
$191$	−2.80983	−0.203312	−0.101656	+	0.994820i	$0.532414\pi$
$192$	0	0
$193$	25.4626	1.83284	0.916421	−	0.400217i	$-0.131065\pi$
$193$	25.4626	1.83284	0.916421	+	0.400217i	$0.131065\pi$
$194$	0	0
$195$	16.6595	1.19301
$196$	0	0
$197$	−9.87860	−0.703821	−0.351911	−	0.936034i	$-0.614468\pi$
$197$	−9.87860	−0.703821	−0.351911	+	0.936034i	$0.614468\pi$
$198$	0	0
$199$	12.5186	0.887422	0.443711	−	0.896170i	$-0.353662\pi$
$199$	12.5186	0.887422	0.443711	+	0.896170i	$0.353662\pi$
$200$	0	0
$201$	35.7520	2.52175
$202$	0	0
$203$	−2.66874	−0.187309
$204$	0	0
$205$	−5.78806	−0.404256
$206$	0	0
$207$	15.9963	1.11182
$208$	0	0
$209$	40.1343	2.77615
$210$	0	0
$211$	−2.31941	−0.159675	−0.0798375	−	0.996808i	$-0.525440\pi$
$211$	−2.31941	−0.159675	−0.0798375	+	0.996808i	$0.525440\pi$
$212$	0	0
$213$	−24.3441	−1.66803
$214$	0	0
$215$	−11.6733	−0.796114
$216$	0	0
$217$	22.9960	1.56107
$218$	0	0
$219$	5.47366	0.369876
$220$	0	0
$221$	13.2347	0.890259
$222$	0	0
$223$	−16.5040	−1.10519	−0.552594	−	0.833450i	$-0.686362\pi$
$223$	−16.5040	−1.10519	−0.552594	+	0.833450i	$0.686362\pi$
$224$	0	0
$225$	−0.843041	−0.0562027
$226$	0	0
$227$	−1.29398	−0.0858848	−0.0429424	−	0.999078i	$-0.513673\pi$
$227$	−1.29398	−0.0858848	−0.0429424	+	0.999078i	$0.513673\pi$
$228$	0	0
$229$	1.25304	0.0828032	0.0414016	−	0.999143i	$-0.486818\pi$
$229$	1.25304	0.0828032	0.0414016	+	0.999143i	$0.486818\pi$
$230$	0	0
$231$	33.1768	2.18288
$232$	0	0
$233$	−20.1643	−1.32101	−0.660503	−	0.750823i	$-0.729657\pi$
$233$	−20.1643	−1.32101	−0.660503	+	0.750823i	$0.729657\pi$
$234$	0	0
$235$	−23.9569	−1.56278
$236$	0	0
$237$	−10.6246	−0.690145
$238$	0	0
$239$	20.3834	1.31850	0.659248	−	0.751926i	$-0.270875\pi$
$239$	20.3834	1.31850	0.659248	+	0.751926i	$0.270875\pi$
$240$	0	0
$241$	−18.5855	−1.19720	−0.598600	−	0.801048i	$-0.704276\pi$
$241$	−18.5855	−1.19720	−0.598600	+	0.801048i	$0.704276\pi$
$242$	0	0
$243$	22.3165	1.43160
$244$	0	0
$245$	1.98360	0.126728
$246$	0	0
$247$	22.8004	1.45076
$248$	0	0
$249$	−3.88454	−0.246173
$250$	0	0
$251$	−14.6999	−0.927852	−0.463926	−	0.885874i	$-0.653560\pi$
$251$	−14.6999	−0.927852	−0.463926	+	0.885874i	$0.653560\pi$
$252$	0	0
$253$	24.1889	1.52074
$254$	0	0
$255$	24.5325	1.53628
$256$	0	0
$257$	18.3862	1.14690	0.573449	−	0.819241i	$-0.305605\pi$
$257$	18.3862	1.14690	0.573449	+	0.819241i	$0.305605\pi$
$258$	0	0
$259$	−5.04201	−0.313295
$260$	0	0
$261$	3.77370	0.233586
$262$	0	0
$263$	−14.2284	−0.877358	−0.438679	−	0.898644i	$-0.644554\pi$
$263$	−14.2284	−0.877358	−0.438679	+	0.898644i	$0.644554\pi$
$264$	0	0
$265$	−6.24692	−0.383745
$266$	0	0
$267$	−11.5107	−0.704443
$268$	0	0
$269$	−6.04264	−0.368427	−0.184213	−	0.982886i	$-0.558974\pi$
$269$	−6.04264	−0.368427	−0.184213	+	0.982886i	$0.558974\pi$
$270$	0	0
$271$	−2.68686	−0.163215	−0.0816075	−	0.996665i	$-0.526005\pi$
$271$	−2.68686	−0.163215	−0.0816075	+	0.996665i	$0.526005\pi$
$272$	0	0
$273$	18.8479	1.14073
$274$	0	0
$275$	−1.27481	−0.0768737
$276$	0	0
$277$	−28.6772	−1.72305	−0.861523	−	0.507718i	$-0.830489\pi$
$277$	−28.6772	−1.72305	−0.861523	+	0.507718i	$0.830489\pi$
$278$	0	0
$279$	−32.5172	−1.94676
$280$	0	0
$281$	−19.1951	−1.14508	−0.572542	−	0.819876i	$-0.694043\pi$
$281$	−19.1951	−1.14508	−0.572542	+	0.819876i	$0.694043\pi$
$282$	0	0
$283$	17.0681	1.01459	0.507297	−	0.861771i	$-0.330645\pi$
$283$	17.0681	1.01459	0.507297	+	0.861771i	$0.330645\pi$
$284$	0	0
$285$	42.2641	2.50351
$286$	0	0
$287$	−6.54839	−0.386539
$288$	0	0
$289$	2.48914	0.146420
$290$	0	0
$291$	−12.5552	−0.735997
$292$	0	0
$293$	−15.0109	−0.876949	−0.438474	−	0.898744i	$-0.644481\pi$
$293$	−15.0109	−0.876949	−0.438474	+	0.898744i	$0.644481\pi$
$294$	0	0
$295$	0.991514	0.0577282
$296$	0	0
$297$	−6.58368	−0.382024
$298$	0	0
$299$	13.7418	0.794709
$300$	0	0
$301$	−13.2067	−0.761223
$302$	0	0
$303$	33.3245	1.91444
$304$	0	0
$305$	12.1033	0.693036
$306$	0	0
$307$	−18.2771	−1.04313	−0.521565	−	0.853211i	$-0.674652\pi$
$307$	−18.2771	−1.04313	−0.521565	+	0.853211i	$0.674652\pi$
$308$	0	0
$309$	−35.6674	−2.02905
$310$	0	0
$311$	−15.1784	−0.860690	−0.430345	−	0.902665i	$-0.641608\pi$
$311$	−15.1784	−0.860690	−0.430345	+	0.902665i	$0.641608\pi$
$312$	0	0
$313$	−3.56957	−0.201764	−0.100882	−	0.994898i	$-0.532166\pi$
$313$	−3.56957	−0.201764	−0.100882	+	0.994898i	$0.532166\pi$
$314$	0	0
$315$	18.7870	1.05853
$316$	0	0
$317$	−5.03124	−0.282582	−0.141291	−	0.989968i	$-0.545125\pi$
$317$	−5.03124	−0.282582	−0.141291	+	0.989968i	$0.545125\pi$
$318$	0	0
$319$	5.70641	0.319498
$320$	0	0
$321$	−5.44316	−0.303808
$322$	0	0
$323$	33.5756	1.86819
$324$	0	0
$325$	−0.724222	−0.0401726
$326$	0	0
$327$	31.1799	1.72425
$328$	0	0
$329$	−27.1039	−1.49429
$330$	0	0
$331$	5.62268	0.309050	0.154525	−	0.987989i	$-0.450615\pi$
$331$	5.62268	0.309050	0.154525	+	0.987989i	$0.450615\pi$
$332$	0	0
$333$	7.12959	0.390699
$334$	0	0
$335$	30.6139	1.67261
$336$	0	0
$337$	30.8419	1.68007	0.840033	−	0.542535i	$-0.182535\pi$
$337$	30.8419	1.68007	0.840033	+	0.542535i	$0.182535\pi$
$338$	0	0
$339$	1.06111	0.0576316
$340$	0	0
$341$	−49.1710	−2.66276
$342$	0	0
$343$	19.5197	1.05396
$344$	0	0
$345$	25.4726	1.37140
$346$	0	0
$347$	0.0884668	0.00474915	0.00237457	−	0.999997i	$-0.499244\pi$
$347$	0.0884668	0.00474915	0.00237457	+	0.999997i	$0.499244\pi$
$348$	0	0
$349$	−26.0791	−1.39598	−0.697992	−	0.716105i	$-0.745923\pi$
$349$	−26.0791	−1.39598	−0.697992	+	0.716105i	$0.745923\pi$
$350$	0	0
$351$	−3.74021	−0.199638
$352$	0	0
$353$	17.8694	0.951092	0.475546	−	0.879691i	$-0.342251\pi$
$353$	17.8694	0.951092	0.475546	+	0.879691i	$0.342251\pi$
$354$	0	0
$355$	−20.8455	−1.10636
$356$	0	0
$357$	27.7551	1.46895
$358$	0	0
$359$	−26.5378	−1.40061	−0.700306	−	0.713843i	$-0.746953\pi$
$359$	−26.5378	−1.40061	−0.700306	+	0.713843i	$0.746953\pi$
$360$	0	0
$361$	38.8434	2.04439
$362$	0	0
$363$	−42.9176	−2.25259
$364$	0	0
$365$	4.68701	0.245329
$366$	0	0
$367$	20.5129	1.07076	0.535382	−	0.844610i	$-0.320167\pi$
$367$	20.5129	1.07076	0.535382	+	0.844610i	$0.320167\pi$
$368$	0	0
$369$	9.25966	0.482039
$370$	0	0
$371$	−7.06751	−0.366927
$372$	0	0
$373$	−22.5131	−1.16568	−0.582841	−	0.812586i	$-0.698059\pi$
$373$	−22.5131	−1.16568	−0.582841	+	0.812586i	$0.698059\pi$
$374$	0	0
$375$	−29.1278	−1.50415
$376$	0	0
$377$	3.24183	0.166963
$378$	0	0
$379$	3.71741	0.190951	0.0954753	−	0.995432i	$-0.469563\pi$
$379$	3.71741	0.190951	0.0954753	+	0.995432i	$0.469563\pi$
$380$	0	0
$381$	33.5663	1.71965
$382$	0	0
$383$	17.0227	0.869820	0.434910	−	0.900474i	$-0.356780\pi$
$383$	17.0227	0.869820	0.434910	+	0.900474i	$0.356780\pi$
$384$	0	0
$385$	28.4088	1.44785
$386$	0	0
$387$	18.6748	0.949294
$388$	0	0
$389$	−3.32784	−0.168728	−0.0843641	−	0.996435i	$-0.526886\pi$
$389$	−3.32784	−0.168728	−0.0843641	+	0.996435i	$0.526886\pi$
$390$	0	0
$391$	20.2359	1.02338
$392$	0	0
$393$	−25.4744	−1.28501
$394$	0	0
$395$	−9.09772	−0.457756
$396$	0	0
$397$	35.0138	1.75729	0.878647	−	0.477472i	$-0.158447\pi$
$397$	35.0138	1.75729	0.878647	+	0.477472i	$0.158447\pi$
$398$	0	0
$399$	47.8160	2.39379
$400$	0	0
$401$	31.8927	1.59264	0.796322	−	0.604873i	$-0.206776\pi$
$401$	31.8927	1.59264	0.796322	+	0.604873i	$0.206776\pi$
$402$	0	0
$403$	−27.9342	−1.39150
$404$	0	0
$405$	15.9043	0.790291
$406$	0	0
$407$	10.7810	0.534396
$408$	0	0
$409$	34.2479	1.69345	0.846724	−	0.532033i	$-0.178572\pi$
$409$	34.2479	1.69345	0.846724	+	0.532033i	$0.178572\pi$
$410$	0	0
$411$	−0.787551	−0.0388470
$412$	0	0
$413$	1.12176	0.0551982
$414$	0	0
$415$	−3.32627	−0.163280
$416$	0	0
$417$	−7.09240	−0.347316
$418$	0	0
$419$	21.3150	1.04131	0.520653	−	0.853768i	$-0.325689\pi$
$419$	21.3150	1.04131	0.520653	+	0.853768i	$0.325689\pi$
$420$	0	0
$421$	−27.8706	−1.35833	−0.679164	−	0.733986i	$-0.737658\pi$
$421$	−27.8706	−1.35833	−0.679164	+	0.733986i	$0.737658\pi$
$422$	0	0
$423$	38.3259	1.86347
$424$	0	0
$425$	−1.06648	−0.0517318
$426$	0	0
$427$	13.6932	0.662662
$428$	0	0
$429$	−40.3013	−1.94577
$430$	0	0
$431$	−16.1743	−0.779088	−0.389544	−	0.921008i	$-0.627367\pi$
$431$	−16.1743	−0.779088	−0.389544	+	0.921008i	$0.627367\pi$
$432$	0	0
$433$	2.03080	0.0975942	0.0487971	−	0.998809i	$-0.484461\pi$
$433$	2.03080	0.0975942	0.0487971	+	0.998809i	$0.484461\pi$
$434$	0	0
$435$	6.00923	0.288121
$436$	0	0
$437$	34.8621	1.66768
$438$	0	0
$439$	−5.28219	−0.252105	−0.126053	−	0.992024i	$-0.540231\pi$
$439$	−5.28219	−0.252105	−0.126053	+	0.992024i	$0.540231\pi$
$440$	0	0
$441$	−3.17334	−0.151111
$442$	0	0
$443$	−1.34342	−0.0638276	−0.0319138	−	0.999491i	$-0.510160\pi$
$443$	−1.34342	−0.0638276	−0.0319138	+	0.999491i	$0.510160\pi$
$444$	0	0
$445$	−9.85643	−0.467240
$446$	0	0
$447$	36.6226	1.73219
$448$	0	0
$449$	−1.15006	−0.0542748	−0.0271374	−	0.999632i	$-0.508639\pi$
$449$	−1.15006	−0.0542748	−0.0271374	+	0.999632i	$0.508639\pi$
$450$	0	0
$451$	14.0020	0.659330
$452$	0	0
$453$	−27.7145	−1.30214
$454$	0	0
$455$	16.1391	0.756615
$456$	0	0
$457$	21.6392	1.01224	0.506121	−	0.862463i	$-0.331079\pi$
$457$	21.6392	1.01224	0.506121	+	0.862463i	$0.331079\pi$
$458$	0	0
$459$	−5.50778	−0.257081
$460$	0	0
$461$	−13.7479	−0.640306	−0.320153	−	0.947366i	$-0.603734\pi$
$461$	−13.7479	−0.640306	−0.320153	+	0.947366i	$0.603734\pi$
$462$	0	0
$463$	−38.7113	−1.79907	−0.899533	−	0.436852i	$-0.856093\pi$
$463$	−38.7113	−1.79907	−0.899533	+	0.436852i	$0.856093\pi$
$464$	0	0
$465$	−51.7804	−2.40126
$466$	0	0
$467$	3.05194	0.141227	0.0706134	−	0.997504i	$-0.477504\pi$
$467$	3.05194	0.141227	0.0706134	+	0.997504i	$0.477504\pi$
$468$	0	0
$469$	34.6353	1.59931
$470$	0	0
$471$	−37.6092	−1.73294
$472$	0	0
$473$	28.2392	1.29844
$474$	0	0
$475$	−1.83731	−0.0843015
$476$	0	0
$477$	9.99373	0.457581
$478$	0	0
$479$	−29.9217	−1.36716	−0.683578	−	0.729877i	$-0.739577\pi$
$479$	−29.9217	−1.36716	−0.683578	+	0.729877i	$0.739577\pi$
$480$	0	0
$481$	6.12474	0.279264
$482$	0	0
$483$	28.8186	1.31129
$484$	0	0
$485$	−10.7508	−0.488168
$486$	0	0
$487$	−11.2962	−0.511879	−0.255939	−	0.966693i	$-0.582385\pi$
$487$	−11.2962	−0.511879	−0.255939	+	0.966693i	$0.582385\pi$
$488$	0	0
$489$	−41.9742	−1.89814
$490$	0	0
$491$	10.5096	0.474291	0.237145	−	0.971474i	$-0.423788\pi$
$491$	10.5096	0.474291	0.237145	+	0.971474i	$0.423788\pi$
$492$	0	0
$493$	4.77387	0.215004
$494$	0	0
$495$	−40.1711	−1.80556
$496$	0	0
$497$	−23.5838	−1.05788
$498$	0	0
$499$	34.3376	1.53716	0.768582	−	0.639752i	$-0.220963\pi$
$499$	34.3376	1.53716	0.768582	+	0.639752i	$0.220963\pi$
$500$	0	0
$501$	−39.1896	−1.75086
$502$	0	0
$503$	26.5105	1.18204	0.591022	−	0.806655i	$-0.298725\pi$
$503$	26.5105	1.18204	0.591022	+	0.806655i	$0.298725\pi$
$504$	0	0
$505$	28.5352	1.26980
$506$	0	0
$507$	10.2221	0.453982
$508$	0	0
$509$	−2.40400	−0.106555	−0.0532777	−	0.998580i	$-0.516967\pi$
$509$	−2.40400	−0.106555	−0.0532777	+	0.998580i	$0.516967\pi$
$510$	0	0
$511$	5.30270	0.234578
$512$	0	0
$513$	−9.48870	−0.418937
$514$	0	0
$515$	−30.5414	−1.34582
$516$	0	0
$517$	57.9546	2.54884
$518$	0	0
$519$	44.4339	1.95043
$520$	0	0
$521$	31.6716	1.38756	0.693780	−	0.720187i	$-0.255944\pi$
$521$	31.6716	1.38756	0.693780	+	0.720187i	$0.255944\pi$
$522$	0	0
$523$	−34.4102	−1.50465	−0.752326	−	0.658791i	$-0.771068\pi$
$523$	−34.4102	−1.50465	−0.752326	+	0.658791i	$0.771068\pi$
$524$	0	0
$525$	−1.51880	−0.0662860
$526$	0	0
$527$	−41.1355	−1.79189
$528$	0	0
$529$	−1.98862	−0.0864619
$530$	0	0
$531$	−1.58621	−0.0688356
$532$	0	0
$533$	7.95460	0.344552
$534$	0	0
$535$	−4.66089	−0.201508
$536$	0	0
$537$	−0.755534	−0.0326037
$538$	0	0
$539$	−4.79857	−0.206689
$540$	0	0
$541$	−28.7768	−1.23721	−0.618605	−	0.785702i	$-0.712302\pi$
$541$	−28.7768	−1.23721	−0.618605	+	0.785702i	$0.712302\pi$
$542$	0	0
$543$	−48.3451	−2.07469
$544$	0	0
$545$	26.6988	1.14365
$546$	0	0
$547$	−43.3627	−1.85406	−0.927028	−	0.374991i	$-0.877646\pi$
$547$	−43.3627	−1.85406	−0.927028	+	0.374991i	$0.877646\pi$
$548$	0	0
$549$	−19.3628	−0.826382
$550$	0	0
$551$	8.22433	0.350368
$552$	0	0
$553$	−10.2928	−0.437694
$554$	0	0
$555$	11.3532	0.481915
$556$	0	0
$557$	29.9680	1.26979	0.634893	−	0.772600i	$-0.281044\pi$
$557$	29.9680	1.26979	0.634893	+	0.772600i	$0.281044\pi$
$558$	0	0
$559$	16.0428	0.678537
$560$	0	0
$561$	−59.3470	−2.50563
$562$	0	0
$563$	−8.51241	−0.358755	−0.179378	−	0.983780i	$-0.557408\pi$
$563$	−8.51241	−0.358755	−0.179378	+	0.983780i	$0.557408\pi$
$564$	0	0
$565$	0.908612	0.0382256
$566$	0	0
$567$	17.9935	0.755656
$568$	0	0
$569$	−20.9687	−0.879054	−0.439527	−	0.898229i	$-0.644854\pi$
$569$	−20.9687	−0.879054	−0.439527	+	0.898229i	$0.644854\pi$
$570$	0	0
$571$	22.3087	0.933591	0.466795	−	0.884365i	$-0.345409\pi$
$571$	22.3087	0.933591	0.466795	+	0.884365i	$0.345409\pi$
$572$	0	0
$573$	7.15803	0.299031
$574$	0	0
$575$	−1.10734	−0.0461794
$576$	0	0
$577$	−19.6626	−0.818563	−0.409281	−	0.912408i	$-0.634221\pi$
$577$	−19.6626	−0.818563	−0.409281	+	0.912408i	$0.634221\pi$
$578$	0	0
$579$	−64.8660	−2.69574
$580$	0	0
$581$	−3.76321	−0.156124
$582$	0	0
$583$	15.1120	0.625877
$584$	0	0
$585$	−22.8213	−0.943546
$586$	0	0
$587$	30.7953	1.27106	0.635529	−	0.772077i	$-0.280782\pi$
$587$	30.7953	1.27106	0.635529	+	0.772077i	$0.280782\pi$
$588$	0	0
$589$	−70.8675	−2.92004
$590$	0	0
$591$	25.1657	1.03518
$592$	0	0
$593$	17.4124	0.715042	0.357521	−	0.933905i	$-0.383622\pi$
$593$	17.4124	0.715042	0.357521	+	0.933905i	$0.383622\pi$
$594$	0	0
$595$	23.7662	0.974320
$596$	0	0
$597$	−31.8912	−1.30522
$598$	0	0
$599$	17.7430	0.724961	0.362481	−	0.931991i	$-0.381930\pi$
$599$	17.7430	0.724961	0.362481	+	0.931991i	$0.381930\pi$
$600$	0	0
$601$	3.66228	0.149388	0.0746938	−	0.997207i	$-0.476202\pi$
$601$	3.66228	0.149388	0.0746938	+	0.997207i	$0.476202\pi$
$602$	0	0
$603$	−48.9756	−1.99444
$604$	0	0
$605$	−36.7497	−1.49409
$606$	0	0
$607$	42.5178	1.72575	0.862873	−	0.505421i	$-0.168663\pi$
$607$	42.5178	1.72575	0.862873	+	0.505421i	$0.168663\pi$
$608$	0	0
$609$	6.79861	0.275493
$610$	0	0
$611$	32.9242	1.33197
$612$	0	0
$613$	−5.01101	−0.202393	−0.101196	−	0.994866i	$-0.532267\pi$
$613$	−5.01101	−0.202393	−0.101196	+	0.994866i	$0.532267\pi$
$614$	0	0
$615$	14.7451	0.594579
$616$	0	0
$617$	0.629899	0.0253588	0.0126794	−	0.999920i	$-0.495964\pi$
$617$	0.629899	0.0253588	0.0126794	+	0.999920i	$0.495964\pi$
$618$	0	0
$619$	−7.28888	−0.292965	−0.146482	−	0.989213i	$-0.546795\pi$
$619$	−7.28888	−0.292965	−0.146482	+	0.989213i	$0.546795\pi$
$620$	0	0
$621$	−5.71883	−0.229489
$622$	0	0
$623$	−11.1512	−0.446762
$624$	0	0
$625$	−23.7338	−0.949350
$626$	0	0
$627$	−102.242	−4.08315
$628$	0	0
$629$	9.01919	0.359619
$630$	0	0
$631$	8.85240	0.352408	0.176204	−	0.984354i	$-0.443618\pi$
$631$	8.85240	0.352408	0.176204	+	0.984354i	$0.443618\pi$
$632$	0	0
$633$	5.90870	0.234850
$634$	0	0
$635$	28.7423	1.14060
$636$	0	0
$637$	−2.72609	−0.108012
$638$	0	0
$639$	33.3483	1.31924
$640$	0	0
$641$	13.9760	0.552020	0.276010	−	0.961155i	$-0.410988\pi$
$641$	13.9760	0.552020	0.276010	+	0.961155i	$0.410988\pi$
$642$	0	0
$643$	22.2105	0.875898	0.437949	−	0.899000i	$-0.355705\pi$
$643$	22.2105	0.875898	0.437949	+	0.899000i	$0.355705\pi$
$644$	0	0
$645$	29.7377	1.17092
$646$	0	0
$647$	38.6488	1.51944	0.759721	−	0.650249i	$-0.225335\pi$
$647$	38.6488	1.51944	0.759721	+	0.650249i	$0.225335\pi$
$648$	0	0
$649$	−2.39859	−0.0941530
$650$	0	0
$651$	−58.5823	−2.29602
$652$	0	0
$653$	12.6297	0.494238	0.247119	−	0.968985i	$-0.420516\pi$
$653$	12.6297	0.494238	0.247119	+	0.968985i	$0.420516\pi$
$654$	0	0
$655$	−21.8133	−0.852316
$656$	0	0
$657$	−7.49821	−0.292533
$658$	0	0
$659$	37.6872	1.46809	0.734043	−	0.679103i	$-0.237631\pi$
$659$	37.6872	1.46809	0.734043	+	0.679103i	$0.237631\pi$
$660$	0	0
$661$	1.09596	0.0426279	0.0213140	−	0.999773i	$-0.493215\pi$
$661$	1.09596	0.0426279	0.0213140	+	0.999773i	$0.493215\pi$
$662$	0	0
$663$	−33.7153	−1.30939
$664$	0	0
$665$	40.9441	1.58774
$666$	0	0
$667$	4.95680	0.191928
$668$	0	0
$669$	42.0439	1.62551
$670$	0	0
$671$	−29.2794	−1.13032
$672$	0	0
$673$	37.7771	1.45620	0.728100	−	0.685471i	$-0.240403\pi$
$673$	37.7771	1.45620	0.728100	+	0.685471i	$0.240403\pi$
$674$	0	0
$675$	0.301395	0.0116007
$676$	0	0
$677$	−32.6625	−1.25532	−0.627661	−	0.778487i	$-0.715987\pi$
$677$	−32.6625	−1.25532	−0.627661	+	0.778487i	$0.715987\pi$
$678$	0	0
$679$	−12.1630	−0.466773
$680$	0	0
$681$	3.29642	0.126319
$682$	0	0
$683$	48.3444	1.84985	0.924923	−	0.380154i	$-0.124129\pi$
$683$	48.3444	1.84985	0.924923	+	0.380154i	$0.124129\pi$
$684$	0	0
$685$	−0.674368	−0.0257663
$686$	0	0
$687$	−3.19211	−0.121787
$688$	0	0
$689$	8.58521	0.327070
$690$	0	0
$691$	1.04783	0.0398612	0.0199306	−	0.999801i	$-0.493655\pi$
$691$	1.04783	0.0398612	0.0199306	+	0.999801i	$0.493655\pi$
$692$	0	0
$693$	−45.4480	−1.72643
$694$	0	0
$695$	−6.07311	−0.230366
$696$	0	0
$697$	11.7138	0.443692
$698$	0	0
$699$	51.3685	1.94293
$700$	0	0
$701$	−10.7306	−0.405290	−0.202645	−	0.979252i	$-0.564954\pi$
$701$	−10.7306	−0.405290	−0.202645	+	0.979252i	$0.564954\pi$
$702$	0	0
$703$	15.5381	0.586031
$704$	0	0
$705$	61.0301	2.29853
$706$	0	0
$707$	32.2836	1.21415
$708$	0	0
$709$	8.59951	0.322961	0.161481	−	0.986876i	$-0.448373\pi$
$709$	8.59951	0.322961	0.161481	+	0.986876i	$0.448373\pi$
$710$	0	0
$711$	14.5544	0.545832
$712$	0	0
$713$	−42.7118	−1.59957
$714$	0	0
$715$	−34.5094	−1.29058
$716$	0	0
$717$	−51.9268	−1.93924
$718$	0	0
$719$	−31.8635	−1.18831	−0.594153	−	0.804352i	$-0.702513\pi$
$719$	−31.8635	−1.18831	−0.594153	+	0.804352i	$0.702513\pi$
$720$	0	0
$721$	−34.5533	−1.28683
$722$	0	0
$723$	47.3466	1.76084
$724$	0	0
$725$	−0.261234	−0.00970198
$726$	0	0
$727$	0.886916	0.0328939	0.0164469	−	0.999865i	$-0.494765\pi$
$727$	0.886916	0.0328939	0.0164469	+	0.999865i	$0.494765\pi$
$728$	0	0
$729$	−34.9783	−1.29549
$730$	0	0
$731$	23.6243	0.873777
$732$	0	0
$733$	18.0107	0.665239	0.332620	−	0.943061i	$-0.392067\pi$
$733$	18.0107	0.665239	0.332620	+	0.943061i	$0.392067\pi$
$734$	0	0
$735$	−5.05322	−0.186391
$736$	0	0
$737$	−74.0586	−2.72798
$738$	0	0
$739$	0.754700	0.0277621	0.0138810	−	0.999904i	$-0.495581\pi$
$739$	0.754700	0.0277621	0.0138810	+	0.999904i	$0.495581\pi$
$740$	0	0
$741$	−58.0841	−2.13377
$742$	0	0
$743$	−26.1301	−0.958622	−0.479311	−	0.877645i	$-0.659113\pi$
$743$	−26.1301	−0.958622	−0.479311	+	0.877645i	$0.659113\pi$
$744$	0	0
$745$	31.3593	1.14892
$746$	0	0
$747$	5.32132	0.194697
$748$	0	0
$749$	−5.27315	−0.192677
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	37.4481	1.36468
$754$	0	0
$755$	−23.7315	−0.863679
$756$	0	0
$757$	−20.7192	−0.753053	−0.376526	−	0.926406i	$-0.622882\pi$
$757$	−20.7192	−0.753053	−0.376526	+	0.926406i	$0.622882\pi$
$758$	0	0
$759$	−61.6212	−2.23671
$760$	0	0
$761$	−26.7899	−0.971134	−0.485567	−	0.874199i	$-0.661387\pi$
$761$	−26.7899	−0.971134	−0.485567	+	0.874199i	$0.661387\pi$
$762$	0	0
$763$	30.2060	1.09353
$764$	0	0
$765$	−33.6063	−1.21504
$766$	0	0
$767$	−1.36265	−0.0492024
$768$	0	0
$769$	6.54117	0.235881	0.117940	−	0.993021i	$-0.462371\pi$
$769$	6.54117	0.235881	0.117940	+	0.993021i	$0.462371\pi$
$770$	0	0
$771$	−46.8387	−1.68686
$772$	0	0
$773$	21.0506	0.757139	0.378569	−	0.925573i	$-0.376416\pi$
$773$	21.0506	0.757139	0.378569	+	0.925573i	$0.376416\pi$
$774$	0	0
$775$	2.25100	0.0808583
$776$	0	0
$777$	12.8445	0.460794
$778$	0	0
$779$	20.1804	0.723036
$780$	0	0
$781$	50.4277	1.80445
$782$	0	0
$783$	−1.34913	−0.0482140
$784$	0	0
$785$	−32.2041	−1.14941
$786$	0	0
$787$	31.6063	1.12664	0.563322	−	0.826237i	$-0.309523\pi$
$787$	31.6063	1.12664	0.563322	+	0.826237i	$0.309523\pi$
$788$	0	0
$789$	36.2467	1.29042
$790$	0	0
$791$	1.02797	0.0365503
$792$	0	0
$793$	−16.6338	−0.590682
$794$	0	0
$795$	15.9140	0.564412
$796$	0	0
$797$	−9.75461	−0.345526	−0.172763	−	0.984963i	$-0.555269\pi$
$797$	−9.75461	−0.345526	−0.172763	+	0.984963i	$0.555269\pi$
$798$	0	0
$799$	48.4837	1.71523
$800$	0	0
$801$	15.7682	0.557141
$802$	0	0
$803$	−11.3384	−0.400125
$804$	0	0
$805$	24.6769	0.869748
$806$	0	0
$807$	15.3936	0.541881
$808$	0	0
$809$	21.7234	0.763753	0.381876	−	0.924213i	$-0.375278\pi$
$809$	21.7234	0.763753	0.381876	+	0.924213i	$0.375278\pi$
$810$	0	0
$811$	−42.1420	−1.47980	−0.739902	−	0.672714i	$-0.765128\pi$
$811$	−42.1420	−1.47980	−0.739902	+	0.672714i	$0.765128\pi$
$812$	0	0
$813$	6.84476	0.240056
$814$	0	0
$815$	−35.9419	−1.25899
$816$	0	0
$817$	40.6996	1.42390
$818$	0	0
$819$	−25.8192	−0.902194
$820$	0	0
$821$	−36.6119	−1.27777	−0.638883	−	0.769304i	$-0.720603\pi$
$821$	−36.6119	−1.27777	−0.638883	+	0.769304i	$0.720603\pi$
$822$	0	0
$823$	41.4549	1.44503	0.722513	−	0.691357i	$-0.242987\pi$
$823$	41.4549	1.44503	0.722513	+	0.691357i	$0.242987\pi$
$824$	0	0
$825$	3.24757	0.113066
$826$	0	0
$827$	−35.9808	−1.25118	−0.625588	−	0.780154i	$-0.715141\pi$
$827$	−35.9808	−1.25118	−0.625588	+	0.780154i	$0.715141\pi$
$828$	0	0
$829$	16.8665	0.585799	0.292899	−	0.956143i	$-0.405380\pi$
$829$	16.8665	0.585799	0.292899	+	0.956143i	$0.405380\pi$
$830$	0	0
$831$	73.0551	2.53425
$832$	0	0
$833$	−4.01439	−0.139090
$834$	0	0
$835$	−33.5574	−1.16130
$836$	0	0
$837$	11.6252	0.401826
$838$	0	0
$839$	−36.9569	−1.27589	−0.637947	−	0.770080i	$-0.720216\pi$
$839$	−36.9569	−1.27589	−0.637947	+	0.770080i	$0.720216\pi$
$840$	0	0
$841$	−27.8306	−0.959677
$842$	0	0
$843$	48.8995	1.68419
$844$	0	0
$845$	8.75306	0.301114
$846$	0	0
$847$	−41.5771	−1.42861
$848$	0	0
$849$	−43.4810	−1.49226
$850$	0	0
$851$	9.36480	0.321021
$852$	0	0
$853$	−0.348485	−0.0119319	−0.00596594	−	0.999982i	$-0.501899\pi$
$853$	−0.348485	−0.0119319	−0.00596594	+	0.999982i	$0.501899\pi$
$854$	0	0
$855$	−57.8964	−1.98002
$856$	0	0
$857$	−10.5538	−0.360511	−0.180255	−	0.983620i	$-0.557692\pi$
$857$	−10.5538	−0.360511	−0.180255	+	0.983620i	$0.557692\pi$
$858$	0	0
$859$	16.5184	0.563602	0.281801	−	0.959473i	$-0.409068\pi$
$859$	16.5184	0.563602	0.281801	+	0.959473i	$0.409068\pi$
$860$	0	0
$861$	16.6820	0.568521
$862$	0	0
$863$	23.3354	0.794348	0.397174	−	0.917743i	$-0.369991\pi$
$863$	23.3354	0.794348	0.397174	+	0.917743i	$0.369991\pi$
$864$	0	0
$865$	38.0480	1.29367
$866$	0	0
$867$	−6.34106	−0.215354
$868$	0	0
$869$	22.0085	0.746586
$870$	0	0
$871$	−42.0730	−1.42559
$872$	0	0
$873$	17.1989	0.582096
$874$	0	0
$875$	−28.2180	−0.953942
$876$	0	0
$877$	44.5822	1.50543	0.752717	−	0.658344i	$-0.228743\pi$
$877$	44.5822	1.50543	0.752717	+	0.658344i	$0.228743\pi$
$878$	0	0
$879$	38.2403	1.28981
$880$	0	0
$881$	17.0002	0.572753	0.286376	−	0.958117i	$-0.407549\pi$
$881$	17.0002	0.572753	0.286376	+	0.958117i	$0.407549\pi$
$882$	0	0
$883$	−49.6723	−1.67161	−0.835803	−	0.549030i	$-0.814997\pi$
$883$	−49.6723	−1.67161	−0.835803	+	0.549030i	$0.814997\pi$
$884$	0	0
$885$	−2.52588	−0.0849065
$886$	0	0
$887$	20.0844	0.674369	0.337185	−	0.941439i	$-0.390525\pi$
$887$	20.0844	0.674369	0.337185	+	0.941439i	$0.390525\pi$
$888$	0	0
$889$	32.5179	1.09061
$890$	0	0
$891$	−38.4744	−1.28894
$892$	0	0
$893$	83.5268	2.79512
$894$	0	0
$895$	−0.646952	−0.0216252
$896$	0	0
$897$	−35.0072	−1.16886
$898$	0	0
$899$	−10.0761	−0.336058
$900$	0	0
$901$	12.6424	0.421180
$902$	0	0
$903$	33.6441	1.11961
$904$	0	0
$905$	−41.3971	−1.37609
$906$	0	0
$907$	−1.16524	−0.0386912	−0.0193456	−	0.999813i	$-0.506158\pi$
$907$	−1.16524	−0.0386912	−0.0193456	+	0.999813i	$0.506158\pi$
$908$	0	0
$909$	−45.6502	−1.51412
$910$	0	0
$911$	43.5970	1.44443	0.722216	−	0.691668i	$-0.243124\pi$
$911$	43.5970	1.44443	0.722216	+	0.691668i	$0.243124\pi$
$912$	0	0
$913$	8.04665	0.266305
$914$	0	0
$915$	−30.8332	−1.01932
$916$	0	0
$917$	−24.6787	−0.814962
$918$	0	0
$919$	−53.5624	−1.76686	−0.883430	−	0.468564i	$-0.844772\pi$
$919$	−53.5624	−1.76686	−0.883430	+	0.468564i	$0.844772\pi$
$920$	0	0
$921$	46.5609	1.53423
$922$	0	0
$923$	28.6482	0.942966
$924$	0	0
$925$	−0.493545	−0.0162277
$926$	0	0
$927$	48.8597	1.60476
$928$	0	0
$929$	−54.1546	−1.77675	−0.888377	−	0.459115i	$-0.848167\pi$
$929$	−54.1546	−1.77675	−0.888377	+	0.459115i	$0.848167\pi$
$930$	0	0
$931$	−6.91592	−0.226660
$932$	0	0
$933$	38.6670	1.26590
$934$	0	0
$935$	−50.8179	−1.66192
$936$	0	0
$937$	18.9562	0.619271	0.309635	−	0.950855i	$-0.399793\pi$
$937$	18.9562	0.619271	0.309635	+	0.950855i	$0.399793\pi$
$938$	0	0
$939$	9.09347	0.296754
$940$	0	0
$941$	−3.41857	−0.111442	−0.0557210	−	0.998446i	$-0.517746\pi$
$941$	−3.41857	−0.111442	−0.0557210	+	0.998446i	$0.517746\pi$
$942$	0	0
$943$	12.1627	0.396071
$944$	0	0
$945$	−6.71651	−0.218488
$946$	0	0
$947$	−17.3528	−0.563889	−0.281944	−	0.959431i	$-0.590979\pi$
$947$	−17.3528	−0.563889	−0.281944	+	0.959431i	$0.590979\pi$
$948$	0	0
$949$	−6.44141	−0.209097
$950$	0	0
$951$	12.8171	0.415622
$952$	0	0
$953$	−44.0631	−1.42734	−0.713672	−	0.700480i	$-0.752969\pi$
$953$	−44.0631	−1.42734	−0.713672	+	0.700480i	$0.752969\pi$
$954$	0	0
$955$	6.12930	0.198340
$956$	0	0
$957$	−14.5371	−0.469916
$958$	0	0
$959$	−0.762953	−0.0246370
$960$	0	0
$961$	55.8242	1.80078
$962$	0	0
$963$	7.45643	0.240280
$964$	0	0
$965$	−55.5437	−1.78802
$966$	0	0
$967$	−2.25452	−0.0725005	−0.0362502	−	0.999343i	$-0.511541\pi$
$967$	−2.25452	−0.0725005	−0.0362502	+	0.999343i	$0.511541\pi$
$968$	0	0
$969$	−85.5336	−2.74774
$970$	0	0
$971$	−35.8615	−1.15085	−0.575424	−	0.817855i	$-0.695163\pi$
$971$	−35.8615	−1.15085	−0.575424	+	0.817855i	$0.695163\pi$
$972$	0	0
$973$	−6.87087	−0.220270
$974$	0	0
$975$	1.84495	0.0590858
$976$	0	0
$977$	−24.0954	−0.770881	−0.385440	−	0.922733i	$-0.625950\pi$
$977$	−24.0954	−0.770881	−0.385440	+	0.922733i	$0.625950\pi$
$978$	0	0
$979$	23.8439	0.762054
$980$	0	0
$981$	−42.7124	−1.36370
$982$	0	0
$983$	−1.44824	−0.0461918	−0.0230959	−	0.999733i	$-0.507352\pi$
$983$	−1.44824	−0.0461918	−0.0230959	+	0.999733i	$0.507352\pi$
$984$	0	0
$985$	21.5490	0.686608
$986$	0	0
$987$	69.0471	2.19779
$988$	0	0
$989$	24.5296	0.779995
$990$	0	0
$991$	−31.8953	−1.01319	−0.506593	−	0.862185i	$-0.669095\pi$
$991$	−31.8953	−1.01319	−0.506593	+	0.862185i	$0.669095\pi$
$992$	0	0
$993$	−14.3238	−0.454551
$994$	0	0
$995$	−27.3079	−0.865718
$996$	0	0
$997$	7.51845	0.238112	0.119056	−	0.992888i	$-0.462013\pi$
$997$	7.51845	0.238112	0.119056	+	0.992888i	$0.462013\pi$
$998$	0	0
$999$	−2.54889	−0.0806434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.6	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.6	✓	49	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)

\(f(q)\)	\(=\)	\(q-2.54750 q^{3} -2.18138 q^{5} -2.46793 q^{7} +3.48974 q^{9} +O(q^{10})\)	\(q-2.54750 q^{3} -2.18138 q^{5} -2.46793 q^{7} +3.48974 q^{9} +5.27702 q^{11} +2.99790 q^{13} +5.55706 q^{15} +4.41465 q^{17} +7.60548 q^{19} +6.28704 q^{21} +4.58382 q^{23} -0.241577 q^{25} -1.24761 q^{27} +1.08137 q^{29} -9.31795 q^{31} -13.4432 q^{33} +5.38349 q^{35} +2.04301 q^{37} -7.63713 q^{39} +2.65339 q^{41} +5.35134 q^{43} -7.61245 q^{45} +10.9824 q^{47} -0.909334 q^{49} -11.2463 q^{51} +2.86374 q^{53} -11.5112 q^{55} -19.3749 q^{57} -0.454535 q^{59} -5.54848 q^{61} -8.61243 q^{63} -6.53955 q^{65} -14.0342 q^{67} -11.6773 q^{69} +9.55610 q^{71} -2.14864 q^{73} +0.615416 q^{75} -13.0233 q^{77} +4.17062 q^{79} -7.29093 q^{81} +1.52485 q^{83} -9.63003 q^{85} -2.75478 q^{87} +4.51844 q^{89} -7.39859 q^{91} +23.7374 q^{93} -16.5905 q^{95} +4.92843 q^{97} +18.4154 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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