LMFDB - Embedded newform 6008.2.a.e.1.30

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:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.30
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)

$f(q)$	$=$	$q+0.765267 q^{3} +3.23588 q^{5} -3.39376 q^{7} -2.41437 q^{9} +O(q^{10})$	$q+0.765267 q^{3} +3.23588 q^{5} -3.39376 q^{7} -2.41437 q^{9} -5.84859 q^{11} -4.58513 q^{13} +2.47631 q^{15} +2.17872 q^{17} +0.556540 q^{19} -2.59714 q^{21} +6.51662 q^{23} +5.47091 q^{25} -4.14344 q^{27} +8.72546 q^{29} +3.80343 q^{31} -4.47574 q^{33} -10.9818 q^{35} +8.20409 q^{37} -3.50885 q^{39} +4.82551 q^{41} +11.5830 q^{43} -7.81259 q^{45} -5.50981 q^{47} +4.51764 q^{49} +1.66730 q^{51} -13.2072 q^{53} -18.9253 q^{55} +0.425901 q^{57} -13.9115 q^{59} -3.42295 q^{61} +8.19379 q^{63} -14.8369 q^{65} +6.71360 q^{67} +4.98695 q^{69} +4.59076 q^{71} +6.05739 q^{73} +4.18671 q^{75} +19.8487 q^{77} -7.64089 q^{79} +4.07226 q^{81} +13.5018 q^{83} +7.05007 q^{85} +6.67731 q^{87} +10.9998 q^{89} +15.5608 q^{91} +2.91064 q^{93} +1.80089 q^{95} -1.70386 q^{97} +14.1206 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	0.765267	0.441827	0.220914	−	0.975293i	$-0.429096\pi$
$3$	0.765267	0.441827	0.220914	+	0.975293i	$0.429096\pi$
$4$	0	0
$5$	3.23588	1.44713	0.723564	−	0.690257i	$-0.242502\pi$
$5$	3.23588	1.44713	0.723564	+	0.690257i	$0.242502\pi$
$6$	0	0
$7$	−3.39376	−1.28272	−0.641361	−	0.767239i	$-0.721630\pi$
$7$	−3.39376	−1.28272	−0.641361	+	0.767239i	$0.721630\pi$
$8$	0	0
$9$	−2.41437	−0.804789
$10$	0	0
$11$	−5.84859	−1.76342	−0.881709	−	0.471794i	$-0.843606\pi$
$11$	−5.84859	−1.76342	−0.881709	+	0.471794i	$0.843606\pi$
$12$	0	0
$13$	−4.58513	−1.27169	−0.635843	−	0.771819i	$-0.719347\pi$
$13$	−4.58513	−1.27169	−0.635843	+	0.771819i	$0.719347\pi$
$14$	0	0
$15$	2.47631	0.639381
$16$	0	0
$17$	2.17872	0.528417	0.264209	−	0.964466i	$-0.414889\pi$
$17$	2.17872	0.528417	0.264209	+	0.964466i	$0.414889\pi$
$18$	0	0
$19$	0.556540	0.127679	0.0638395	−	0.997960i	$-0.479665\pi$
$19$	0.556540	0.127679	0.0638395	+	0.997960i	$0.479665\pi$
$20$	0	0
$21$	−2.59714	−0.566742
$22$	0	0
$23$	6.51662	1.35881	0.679404	−	0.733764i	$-0.262238\pi$
$23$	6.51662	1.35881	0.679404	+	0.733764i	$0.262238\pi$
$24$	0	0
$25$	5.47091	1.09418
$26$	0	0
$27$	−4.14344	−0.797405
$28$	0	0
$29$	8.72546	1.62028	0.810138	−	0.586239i	$-0.199392\pi$
$29$	8.72546	1.62028	0.810138	+	0.586239i	$0.199392\pi$
$30$	0	0
$31$	3.80343	0.683116	0.341558	−	0.939861i	$-0.389045\pi$
$31$	3.80343	0.683116	0.341558	+	0.939861i	$0.389045\pi$
$32$	0	0
$33$	−4.47574	−0.779126
$34$	0	0
$35$	−10.9818	−1.85626
$36$	0	0
$37$	8.20409	1.34874	0.674372	−	0.738392i	$-0.264415\pi$
$37$	8.20409	1.34874	0.674372	+	0.738392i	$0.264415\pi$
$38$	0	0
$39$	−3.50885	−0.561865
$40$	0	0
$41$	4.82551	0.753619	0.376809	−	0.926291i	$-0.377021\pi$
$41$	4.82551	0.753619	0.376809	+	0.926291i	$0.377021\pi$
$42$	0	0
$43$	11.5830	1.76640	0.883199	−	0.468998i	$-0.155385\pi$
$43$	11.5830	1.76640	0.883199	+	0.468998i	$0.155385\pi$
$44$	0	0
$45$	−7.81259	−1.16463
$46$	0	0
$47$	−5.50981	−0.803689	−0.401844	−	0.915708i	$-0.631631\pi$
$47$	−5.50981	−0.803689	−0.401844	+	0.915708i	$0.631631\pi$
$48$	0	0
$49$	4.51764	0.645377
$50$	0	0
$51$	1.66730	0.233469
$52$	0	0
$53$	−13.2072	−1.81415	−0.907077	−	0.420965i	$-0.861691\pi$
$53$	−13.2072	−1.81415	−0.907077	+	0.420965i	$0.861691\pi$
$54$	0	0
$55$	−18.9253	−2.55189
$56$	0	0
$57$	0.425901	0.0564120
$58$	0	0
$59$	−13.9115	−1.81112	−0.905558	−	0.424222i	$-0.860548\pi$
$59$	−13.9115	−1.81112	−0.905558	+	0.424222i	$0.860548\pi$
$60$	0	0
$61$	−3.42295	−0.438264	−0.219132	−	0.975695i	$-0.570322\pi$
$61$	−3.42295	−0.438264	−0.219132	+	0.975695i	$0.570322\pi$
$62$	0	0
$63$	8.19379	1.03232
$64$	0	0
$65$	−14.8369	−1.84029
$66$	0	0
$67$	6.71360	0.820197	0.410098	−	0.912041i	$-0.365494\pi$
$67$	6.71360	0.820197	0.410098	+	0.912041i	$0.365494\pi$
$68$	0	0
$69$	4.98695	0.600359
$70$	0	0
$71$	4.59076	0.544823	0.272411	−	0.962181i	$-0.412179\pi$
$71$	4.59076	0.544823	0.272411	+	0.962181i	$0.412179\pi$
$72$	0	0
$73$	6.05739	0.708963	0.354482	−	0.935063i	$-0.384657\pi$
$73$	6.05739	0.708963	0.354482	+	0.935063i	$0.384657\pi$
$74$	0	0
$75$	4.18671	0.483439
$76$	0	0
$77$	19.8487	2.26197
$78$	0	0
$79$	−7.64089	−0.859668	−0.429834	−	0.902908i	$-0.641428\pi$
$79$	−7.64089	−0.859668	−0.429834	+	0.902908i	$0.641428\pi$
$80$	0	0
$81$	4.07226	0.452473
$82$	0	0
$83$	13.5018	1.48202	0.741008	−	0.671497i	$-0.234348\pi$
$83$	13.5018	1.48202	0.741008	+	0.671497i	$0.234348\pi$
$84$	0	0
$85$	7.05007	0.764688
$86$	0	0
$87$	6.67731	0.715882
$88$	0	0
$89$	10.9998	1.16598	0.582991	−	0.812479i	$-0.301882\pi$
$89$	10.9998	1.16598	0.582991	+	0.812479i	$0.301882\pi$
$90$	0	0
$91$	15.5608	1.63122
$92$	0	0
$93$	2.91064	0.301819
$94$	0	0
$95$	1.80089	0.184768
$96$	0	0
$97$	−1.70386	−0.173001	−0.0865005	−	0.996252i	$-0.527568\pi$
$97$	−1.70386	−0.173001	−0.0865005	+	0.996252i	$0.527568\pi$
$98$	0	0
$99$	14.1206	1.41918
$100$	0	0
$101$	16.0381	1.59585	0.797927	−	0.602754i	$-0.205930\pi$
$101$	16.0381	1.59585	0.797927	+	0.602754i	$0.205930\pi$
$102$	0	0
$103$	−14.6823	−1.44669	−0.723345	−	0.690486i	$-0.757397\pi$
$103$	−14.6823	−1.44669	−0.723345	+	0.690486i	$0.757397\pi$
$104$	0	0
$105$	−8.40402	−0.820148
$106$	0	0
$107$	13.4319	1.29851	0.649254	−	0.760572i	$-0.275081\pi$
$107$	13.4319	1.29851	0.649254	+	0.760572i	$0.275081\pi$
$108$	0	0
$109$	12.4824	1.19560	0.597799	−	0.801646i	$-0.296042\pi$
$109$	12.4824	1.19560	0.597799	+	0.801646i	$0.296042\pi$
$110$	0	0
$111$	6.27832	0.595912
$112$	0	0
$113$	6.66217	0.626725	0.313362	−	0.949634i	$-0.398545\pi$
$113$	6.66217	0.626725	0.313362	+	0.949634i	$0.398545\pi$
$114$	0	0
$115$	21.0870	1.96637
$116$	0	0
$117$	11.0702	1.02344
$118$	0	0
$119$	−7.39406	−0.677813
$120$	0	0
$121$	23.2060	2.10964
$122$	0	0
$123$	3.69281	0.332969
$124$	0	0
$125$	1.52381	0.136294
$126$	0	0
$127$	−15.6062	−1.38482	−0.692412	−	0.721503i	$-0.743452\pi$
$127$	−15.6062	−1.38482	−0.692412	+	0.721503i	$0.743452\pi$
$128$	0	0
$129$	8.86413	0.780443
$130$	0	0
$131$	−7.53692	−0.658504	−0.329252	−	0.944242i	$-0.606797\pi$
$131$	−7.53692	−0.658504	−0.329252	+	0.944242i	$0.606797\pi$
$132$	0	0
$133$	−1.88876	−0.163777
$134$	0	0
$135$	−13.4077	−1.15395
$136$	0	0
$137$	5.17007	0.441708	0.220854	−	0.975307i	$-0.429115\pi$
$137$	5.17007	0.441708	0.220854	+	0.975307i	$0.429115\pi$
$138$	0	0
$139$	−4.77033	−0.404614	−0.202307	−	0.979322i	$-0.564844\pi$
$139$	−4.77033	−0.404614	−0.202307	+	0.979322i	$0.564844\pi$
$140$	0	0
$141$	−4.21648	−0.355092
$142$	0	0
$143$	26.8165	2.24251
$144$	0	0
$145$	28.2345	2.34475
$146$	0	0
$147$	3.45720	0.285145
$148$	0	0
$149$	12.9819	1.06352	0.531759	−	0.846896i	$-0.321531\pi$
$149$	12.9819	1.06352	0.531759	+	0.846896i	$0.321531\pi$
$150$	0	0
$151$	−19.2409	−1.56580	−0.782902	−	0.622145i	$-0.786261\pi$
$151$	−19.2409	−1.56580	−0.782902	+	0.622145i	$0.786261\pi$
$152$	0	0
$153$	−5.26023	−0.425264
$154$	0	0
$155$	12.3074	0.988557
$156$	0	0
$157$	5.20359	0.415292	0.207646	−	0.978204i	$-0.433420\pi$
$157$	5.20359	0.415292	0.207646	+	0.978204i	$0.433420\pi$
$158$	0	0
$159$	−10.1071	−0.801543
$160$	0	0
$161$	−22.1159	−1.74297
$162$	0	0
$163$	−11.6864	−0.915348	−0.457674	−	0.889120i	$-0.651317\pi$
$163$	−11.6864	−0.915348	−0.457674	+	0.889120i	$0.651317\pi$
$164$	0	0
$165$	−14.4829	−1.12750
$166$	0	0
$167$	10.8175	0.837082	0.418541	−	0.908198i	$-0.362542\pi$
$167$	10.8175	0.837082	0.418541	+	0.908198i	$0.362542\pi$
$168$	0	0
$169$	8.02338	0.617183
$170$	0	0
$171$	−1.34369	−0.102755
$172$	0	0
$173$	−15.3019	−1.16338	−0.581690	−	0.813410i	$-0.697608\pi$
$173$	−15.3019	−1.16338	−0.581690	+	0.813410i	$0.697608\pi$
$174$	0	0
$175$	−18.5670	−1.40353
$176$	0	0
$177$	−10.6460	−0.800201
$178$	0	0
$179$	15.4080	1.15165	0.575823	−	0.817574i	$-0.304682\pi$
$179$	15.4080	1.15165	0.575823	+	0.817574i	$0.304682\pi$
$180$	0	0
$181$	13.7078	1.01889	0.509445	−	0.860503i	$-0.329851\pi$
$181$	13.7078	1.01889	0.509445	+	0.860503i	$0.329851\pi$
$182$	0	0
$183$	−2.61947	−0.193637
$184$	0	0
$185$	26.5474	1.95181
$186$	0	0
$187$	−12.7424	−0.931820
$188$	0	0
$189$	14.0619	1.02285
$190$	0	0
$191$	14.0063	1.01346	0.506729	−	0.862105i	$-0.330854\pi$
$191$	14.0063	1.01346	0.506729	+	0.862105i	$0.330854\pi$
$192$	0	0
$193$	7.81914	0.562834	0.281417	−	0.959586i	$-0.409196\pi$
$193$	7.81914	0.562834	0.281417	+	0.959586i	$0.409196\pi$
$194$	0	0
$195$	−11.3542	−0.813091
$196$	0	0
$197$	1.11344	0.0793291	0.0396645	−	0.999213i	$-0.487371\pi$
$197$	1.11344	0.0793291	0.0396645	+	0.999213i	$0.487371\pi$
$198$	0	0
$199$	16.2889	1.15469	0.577346	−	0.816500i	$-0.304089\pi$
$199$	16.2889	1.15469	0.577346	+	0.816500i	$0.304089\pi$
$200$	0	0
$201$	5.13770	0.362385
$202$	0	0
$203$	−29.6121	−2.07837
$204$	0	0
$205$	15.6148	1.09058
$206$	0	0
$207$	−15.7335	−1.09355
$208$	0	0
$209$	−3.25497	−0.225151
$210$	0	0
$211$	−17.0068	−1.17080	−0.585398	−	0.810746i	$-0.699062\pi$
$211$	−17.0068	−1.17080	−0.585398	+	0.810746i	$0.699062\pi$
$212$	0	0
$213$	3.51316	0.240718
$214$	0	0
$215$	37.4813	2.55621
$216$	0	0
$217$	−12.9079	−0.876248
$218$	0	0
$219$	4.63552	0.313239
$220$	0	0
$221$	−9.98970	−0.671980
$222$	0	0
$223$	15.2058	1.01826	0.509129	−	0.860690i	$-0.329968\pi$
$223$	15.2058	1.01826	0.509129	+	0.860690i	$0.329968\pi$
$224$	0	0
$225$	−13.2088	−0.880585
$226$	0	0
$227$	−2.74299	−0.182059	−0.0910294	−	0.995848i	$-0.529016\pi$
$227$	−2.74299	−0.182059	−0.0910294	+	0.995848i	$0.529016\pi$
$228$	0	0
$229$	20.0154	1.32266	0.661328	−	0.750097i	$-0.269993\pi$
$229$	20.0154	1.32266	0.661328	+	0.750097i	$0.269993\pi$
$230$	0	0
$231$	15.1896	0.999402
$232$	0	0
$233$	7.83957	0.513588	0.256794	−	0.966466i	$-0.417334\pi$
$233$	7.83957	0.513588	0.256794	+	0.966466i	$0.417334\pi$
$234$	0	0
$235$	−17.8291	−1.16304
$236$	0	0
$237$	−5.84733	−0.379825
$238$	0	0
$239$	−7.51519	−0.486117	−0.243058	−	0.970012i	$-0.578151\pi$
$239$	−7.51519	−0.486117	−0.243058	+	0.970012i	$0.578151\pi$
$240$	0	0
$241$	0.466774	0.0300676	0.0150338	−	0.999887i	$-0.495214\pi$
$241$	0.466774	0.0300676	0.0150338	+	0.999887i	$0.495214\pi$
$242$	0	0
$243$	15.5467	0.997320
$244$	0	0
$245$	14.6185	0.933943
$246$	0	0
$247$	−2.55180	−0.162367
$248$	0	0
$249$	10.3325	0.654795
$250$	0	0
$251$	5.42805	0.342615	0.171308	−	0.985218i	$-0.445201\pi$
$251$	5.42805	0.342615	0.171308	+	0.985218i	$0.445201\pi$
$252$	0	0
$253$	−38.1130	−2.39615
$254$	0	0
$255$	5.39519	0.337860
$256$	0	0
$257$	0.702942	0.0438483	0.0219242	−	0.999760i	$-0.493021\pi$
$257$	0.702942	0.0438483	0.0219242	+	0.999760i	$0.493021\pi$
$258$	0	0
$259$	−27.8427	−1.73006
$260$	0	0
$261$	−21.0664	−1.30398
$262$	0	0
$263$	17.8308	1.09949	0.549746	−	0.835332i	$-0.314724\pi$
$263$	17.8308	1.09949	0.549746	+	0.835332i	$0.314724\pi$
$264$	0	0
$265$	−42.7370	−2.62531
$266$	0	0
$267$	8.41783	0.515163
$268$	0	0
$269$	20.1950	1.23131	0.615656	−	0.788015i	$-0.288891\pi$
$269$	20.1950	1.23131	0.615656	+	0.788015i	$0.288891\pi$
$270$	0	0
$271$	3.30293	0.200639	0.100319	−	0.994955i	$-0.468014\pi$
$271$	3.30293	0.200639	0.100319	+	0.994955i	$0.468014\pi$
$272$	0	0
$273$	11.9082	0.720717
$274$	0	0
$275$	−31.9971	−1.92950
$276$	0	0
$277$	−6.75489	−0.405862	−0.202931	−	0.979193i	$-0.565047\pi$
$277$	−6.75489	−0.405862	−0.202931	+	0.979193i	$0.565047\pi$
$278$	0	0
$279$	−9.18287	−0.549764
$280$	0	0
$281$	5.46524	0.326029	0.163015	−	0.986624i	$-0.447878\pi$
$281$	5.46524	0.326029	0.163015	+	0.986624i	$0.447878\pi$
$282$	0	0
$283$	7.41340	0.440681	0.220340	−	0.975423i	$-0.429283\pi$
$283$	7.41340	0.440681	0.220340	+	0.975423i	$0.429283\pi$
$284$	0	0
$285$	1.37817	0.0816355
$286$	0	0
$287$	−16.3767	−0.966683
$288$	0	0
$289$	−12.2532	−0.720775
$290$	0	0
$291$	−1.30391	−0.0764366
$292$	0	0
$293$	−9.17003	−0.535719	−0.267859	−	0.963458i	$-0.586316\pi$
$293$	−9.17003	−0.535719	−0.267859	+	0.963458i	$0.586316\pi$
$294$	0	0
$295$	−45.0158	−2.62092
$296$	0	0
$297$	24.2333	1.40616
$298$	0	0
$299$	−29.8795	−1.72798
$300$	0	0
$301$	−39.3101	−2.26580
$302$	0	0
$303$	12.2735	0.705092
$304$	0	0
$305$	−11.0762	−0.634224
$306$	0	0
$307$	4.74332	0.270715	0.135358	−	0.990797i	$-0.456782\pi$
$307$	4.74332	0.270715	0.135358	+	0.990797i	$0.456782\pi$
$308$	0	0
$309$	−11.2359	−0.639188
$310$	0	0
$311$	−0.965760	−0.0547632	−0.0273816	−	0.999625i	$-0.508717\pi$
$311$	−0.965760	−0.0547632	−0.0273816	+	0.999625i	$0.508717\pi$
$312$	0	0
$313$	29.2758	1.65477	0.827384	−	0.561637i	$-0.189828\pi$
$313$	29.2758	1.65477	0.827384	+	0.561637i	$0.189828\pi$
$314$	0	0
$315$	26.5141	1.49390
$316$	0	0
$317$	2.84503	0.159793	0.0798965	−	0.996803i	$-0.474541\pi$
$317$	2.84503	0.159793	0.0798965	+	0.996803i	$0.474541\pi$
$318$	0	0
$319$	−51.0316	−2.85722
$320$	0	0
$321$	10.2790	0.573716
$322$	0	0
$323$	1.21254	0.0674677
$324$	0	0
$325$	−25.0848	−1.39145
$326$	0	0
$327$	9.55238	0.528248
$328$	0	0
$329$	18.6990	1.03091
$330$	0	0
$331$	21.0952	1.15950	0.579749	−	0.814795i	$-0.303151\pi$
$331$	21.0952	1.15950	0.579749	+	0.814795i	$0.303151\pi$
$332$	0	0
$333$	−19.8077	−1.08545
$334$	0	0
$335$	21.7244	1.18693
$336$	0	0
$337$	−22.7765	−1.24072	−0.620359	−	0.784318i	$-0.713013\pi$
$337$	−22.7765	−1.24072	−0.620359	+	0.784318i	$0.713013\pi$
$338$	0	0
$339$	5.09834	0.276904
$340$	0	0
$341$	−22.2447	−1.20462
$342$	0	0
$343$	8.42455	0.454883
$344$	0	0
$345$	16.1372	0.868796
$346$	0	0
$347$	−28.8258	−1.54745	−0.773726	−	0.633520i	$-0.781609\pi$
$347$	−28.8258	−1.54745	−0.773726	+	0.633520i	$0.781609\pi$
$348$	0	0
$349$	−25.0085	−1.33868	−0.669338	−	0.742958i	$-0.733422\pi$
$349$	−25.0085	−1.33868	−0.669338	+	0.742958i	$0.733422\pi$
$350$	0	0
$351$	18.9982	1.01405
$352$	0	0
$353$	−27.9977	−1.49017	−0.745084	−	0.666970i	$-0.767591\pi$
$353$	−27.9977	−1.49017	−0.745084	+	0.666970i	$0.767591\pi$
$354$	0	0
$355$	14.8551	0.788429
$356$	0	0
$357$	−5.65843	−0.299476
$358$	0	0
$359$	3.91910	0.206842	0.103421	−	0.994638i	$-0.467021\pi$
$359$	3.91910	0.206842	0.103421	+	0.994638i	$0.467021\pi$
$360$	0	0
$361$	−18.6903	−0.983698
$362$	0	0
$363$	17.7588	0.932097
$364$	0	0
$365$	19.6010	1.02596
$366$	0	0
$367$	18.8814	0.985603	0.492801	−	0.870142i	$-0.335973\pi$
$367$	18.8814	0.985603	0.492801	+	0.870142i	$0.335973\pi$
$368$	0	0
$369$	−11.6506	−0.606504
$370$	0	0
$371$	44.8223	2.32706
$372$	0	0
$373$	−15.1045	−0.782079	−0.391040	−	0.920374i	$-0.627884\pi$
$373$	−15.1045	−0.782079	−0.391040	+	0.920374i	$0.627884\pi$
$374$	0	0
$375$	1.16612	0.0602182
$376$	0	0
$377$	−40.0073	−2.06048
$378$	0	0
$379$	17.7674	0.912648	0.456324	−	0.889814i	$-0.349166\pi$
$379$	17.7674	0.912648	0.456324	+	0.889814i	$0.349166\pi$
$380$	0	0
$381$	−11.9429	−0.611853
$382$	0	0
$383$	36.4408	1.86204	0.931019	−	0.364971i	$-0.118921\pi$
$383$	36.4408	1.86204	0.931019	+	0.364971i	$0.118921\pi$
$384$	0	0
$385$	64.2281	3.27337
$386$	0	0
$387$	−27.9657	−1.42158
$388$	0	0
$389$	12.8959	0.653846	0.326923	−	0.945051i	$-0.393988\pi$
$389$	12.8959	0.653846	0.326923	+	0.945051i	$0.393988\pi$
$390$	0	0
$391$	14.1979	0.718018
$392$	0	0
$393$	−5.76776	−0.290945
$394$	0	0
$395$	−24.7250	−1.24405
$396$	0	0
$397$	−0.598243	−0.0300250	−0.0150125	−	0.999887i	$-0.504779\pi$
$397$	−0.598243	−0.0300250	−0.0150125	+	0.999887i	$0.504779\pi$
$398$	0	0
$399$	−1.44541	−0.0723610
$400$	0	0
$401$	31.5859	1.57733	0.788663	−	0.614825i	$-0.210773\pi$
$401$	31.5859	1.57733	0.788663	+	0.614825i	$0.210773\pi$
$402$	0	0
$403$	−17.4392	−0.868709
$404$	0	0
$405$	13.1773	0.654787
$406$	0	0
$407$	−47.9824	−2.37840
$408$	0	0
$409$	−38.1064	−1.88424	−0.942120	−	0.335276i	$-0.891170\pi$
$409$	−38.1064	−1.88424	−0.942120	+	0.335276i	$0.891170\pi$
$410$	0	0
$411$	3.95648	0.195159
$412$	0	0
$413$	47.2122	2.32316
$414$	0	0
$415$	43.6902	2.14467
$416$	0	0
$417$	−3.65057	−0.178769
$418$	0	0
$419$	−30.7909	−1.50424	−0.752118	−	0.659028i	$-0.770968\pi$
$419$	−30.7909	−1.50424	−0.752118	+	0.659028i	$0.770968\pi$
$420$	0	0
$421$	24.0309	1.17120	0.585598	−	0.810602i	$-0.300860\pi$
$421$	24.0309	1.17120	0.585598	+	0.810602i	$0.300860\pi$
$422$	0	0
$423$	13.3027	0.646800
$424$	0	0
$425$	11.9196	0.578185
$426$	0	0
$427$	11.6167	0.562171
$428$	0	0
$429$	20.5218	0.990803
$430$	0	0
$431$	4.84457	0.233355	0.116677	−	0.993170i	$-0.462776\pi$
$431$	4.84457	0.233355	0.116677	+	0.993170i	$0.462776\pi$
$432$	0	0
$433$	12.0471	0.578945	0.289472	−	0.957186i	$-0.406520\pi$
$433$	12.0471	0.578945	0.289472	+	0.957186i	$0.406520\pi$
$434$	0	0
$435$	21.6070	1.03597
$436$	0	0
$437$	3.62675	0.173491
$438$	0	0
$439$	−18.8132	−0.897904	−0.448952	−	0.893556i	$-0.648203\pi$
$439$	−18.8132	−0.897904	−0.448952	+	0.893556i	$0.648203\pi$
$440$	0	0
$441$	−10.9072	−0.519392
$442$	0	0
$443$	8.62168	0.409628	0.204814	−	0.978801i	$-0.434341\pi$
$443$	8.62168	0.409628	0.204814	+	0.978801i	$0.434341\pi$
$444$	0	0
$445$	35.5942	1.68733
$446$	0	0
$447$	9.93462	0.469892
$448$	0	0
$449$	4.93292	0.232799	0.116399	−	0.993202i	$-0.462865\pi$
$449$	4.93292	0.232799	0.116399	+	0.993202i	$0.462865\pi$
$450$	0	0
$451$	−28.2225	−1.32894
$452$	0	0
$453$	−14.7244	−0.691815
$454$	0	0
$455$	50.3530	2.36058
$456$	0	0
$457$	15.3039	0.715886	0.357943	−	0.933744i	$-0.383478\pi$
$457$	15.3039	0.715886	0.357943	+	0.933744i	$0.383478\pi$
$458$	0	0
$459$	−9.02739	−0.421362
$460$	0	0
$461$	0.668936	0.0311554	0.0155777	−	0.999879i	$-0.495041\pi$
$461$	0.668936	0.0311554	0.0155777	+	0.999879i	$0.495041\pi$
$462$	0	0
$463$	−29.1923	−1.35668	−0.678340	−	0.734748i	$-0.737300\pi$
$463$	−29.1923	−1.35668	−0.678340	+	0.734748i	$0.737300\pi$
$464$	0	0
$465$	9.41848	0.436771
$466$	0	0
$467$	0.0530828	0.00245638	0.00122819	−	0.999999i	$-0.499609\pi$
$467$	0.0530828	0.00245638	0.00122819	+	0.999999i	$0.499609\pi$
$468$	0	0
$469$	−22.7844	−1.05208
$470$	0	0
$471$	3.98214	0.183487
$472$	0	0
$473$	−67.7445	−3.11490
$474$	0	0
$475$	3.04478	0.139704
$476$	0	0
$477$	31.8871	1.46001
$478$	0	0
$479$	30.0569	1.37333	0.686667	−	0.726973i	$-0.259073\pi$
$479$	30.0569	1.37333	0.686667	+	0.726973i	$0.259073\pi$
$480$	0	0
$481$	−37.6168	−1.71518
$482$	0	0
$483$	−16.9245	−0.770094
$484$	0	0
$485$	−5.51349	−0.250355
$486$	0	0
$487$	−33.7655	−1.53006	−0.765031	−	0.643994i	$-0.777276\pi$
$487$	−33.7655	−1.53006	−0.765031	+	0.643994i	$0.777276\pi$
$488$	0	0
$489$	−8.94320	−0.404426
$490$	0	0
$491$	−36.1428	−1.63110	−0.815551	−	0.578685i	$-0.803566\pi$
$491$	−36.1428	−1.63110	−0.815551	+	0.578685i	$0.803566\pi$
$492$	0	0
$493$	19.0103	0.856182
$494$	0	0
$495$	45.6927	2.05373
$496$	0	0
$497$	−15.5799	−0.698856
$498$	0	0
$499$	−37.0888	−1.66032	−0.830161	−	0.557524i	$-0.811751\pi$
$499$	−37.0888	−1.66032	−0.830161	+	0.557524i	$0.811751\pi$
$500$	0	0
$501$	8.27827	0.369846
$502$	0	0
$503$	−11.5443	−0.514734	−0.257367	−	0.966314i	$-0.582855\pi$
$503$	−11.5443	−0.514734	−0.257367	+	0.966314i	$0.582855\pi$
$504$	0	0
$505$	51.8974	2.30941
$506$	0	0
$507$	6.14003	0.272688
$508$	0	0
$509$	−23.0395	−1.02121	−0.510605	−	0.859815i	$-0.670579\pi$
$509$	−23.0395	−1.02121	−0.510605	+	0.859815i	$0.670579\pi$
$510$	0	0
$511$	−20.5573	−0.909403
$512$	0	0
$513$	−2.30599	−0.101812
$514$	0	0
$515$	−47.5102	−2.09355
$516$	0	0
$517$	32.2247	1.41724
$518$	0	0
$519$	−11.7100	−0.514013
$520$	0	0
$521$	−5.09378	−0.223163	−0.111581	−	0.993755i	$-0.535592\pi$
$521$	−5.09378	−0.223163	−0.111581	+	0.993755i	$0.535592\pi$
$522$	0	0
$523$	29.5181	1.29074	0.645368	−	0.763872i	$-0.276704\pi$
$523$	29.5181	1.29074	0.645368	+	0.763872i	$0.276704\pi$
$524$	0	0
$525$	−14.2087	−0.620119
$526$	0	0
$527$	8.28661	0.360970
$528$	0	0
$529$	19.4663	0.846361
$530$	0	0
$531$	33.5873	1.45757
$532$	0	0
$533$	−22.1256	−0.958365
$534$	0	0
$535$	43.4639	1.87911
$536$	0	0
$537$	11.7912	0.508829
$538$	0	0
$539$	−26.4218	−1.13807
$540$	0	0
$541$	−2.22210	−0.0955357	−0.0477679	−	0.998858i	$-0.515211\pi$
$541$	−2.22210	−0.0955357	−0.0477679	+	0.998858i	$0.515211\pi$
$542$	0	0
$543$	10.4901	0.450173
$544$	0	0
$545$	40.3916	1.73018
$546$	0	0
$547$	−9.36339	−0.400350	−0.200175	−	0.979760i	$-0.564151\pi$
$547$	−9.36339	−0.400350	−0.200175	+	0.979760i	$0.564151\pi$
$548$	0	0
$549$	8.26425	0.352710
$550$	0	0
$551$	4.85606	0.206875
$552$	0	0
$553$	25.9314	1.10272
$554$	0	0
$555$	20.3159	0.862361
$556$	0	0
$557$	−26.4523	−1.12082	−0.560410	−	0.828215i	$-0.689356\pi$
$557$	−26.4523	−1.12082	−0.560410	+	0.828215i	$0.689356\pi$
$558$	0	0
$559$	−53.1097	−2.24630
$560$	0	0
$561$	−9.75138	−0.411703
$562$	0	0
$563$	14.8805	0.627139	0.313569	−	0.949565i	$-0.398475\pi$
$563$	14.8805	0.627139	0.313569	+	0.949565i	$0.398475\pi$
$564$	0	0
$565$	21.5580	0.906951
$566$	0	0
$567$	−13.8203	−0.580398
$568$	0	0
$569$	−27.6969	−1.16111	−0.580557	−	0.814220i	$-0.697165\pi$
$569$	−27.6969	−1.16111	−0.580557	+	0.814220i	$0.697165\pi$
$570$	0	0
$571$	−8.16086	−0.341521	−0.170761	−	0.985313i	$-0.554622\pi$
$571$	−8.16086	−0.341521	−0.170761	+	0.985313i	$0.554622\pi$
$572$	0	0
$573$	10.7185	0.447774
$574$	0	0
$575$	35.6518	1.48678
$576$	0	0
$577$	−33.2338	−1.38354	−0.691770	−	0.722118i	$-0.743169\pi$
$577$	−33.2338	−1.38354	−0.691770	+	0.722118i	$0.743169\pi$
$578$	0	0
$579$	5.98373	0.248676
$580$	0	0
$581$	−45.8219	−1.90101
$582$	0	0
$583$	77.2438	3.19911
$584$	0	0
$585$	35.8217	1.48105
$586$	0	0
$587$	11.0593	0.456464	0.228232	−	0.973607i	$-0.426705\pi$
$587$	11.0593	0.456464	0.228232	+	0.973607i	$0.426705\pi$
$588$	0	0
$589$	2.11676	0.0872195
$590$	0	0
$591$	0.852077	0.0350498
$592$	0	0
$593$	13.8183	0.567449	0.283724	−	0.958906i	$-0.408430\pi$
$593$	13.8183	0.567449	0.283724	+	0.958906i	$0.408430\pi$
$594$	0	0
$595$	−23.9263	−0.980882
$596$	0	0
$597$	12.4654	0.510174
$598$	0	0
$599$	38.4999	1.57306	0.786532	−	0.617550i	$-0.211875\pi$
$599$	38.4999	1.57306	0.786532	+	0.617550i	$0.211875\pi$
$600$	0	0
$601$	37.4916	1.52932	0.764658	−	0.644436i	$-0.222908\pi$
$601$	37.4916	1.52932	0.764658	+	0.644436i	$0.222908\pi$
$602$	0	0
$603$	−16.2091	−0.660085
$604$	0	0
$605$	75.0919	3.05292
$606$	0	0
$607$	−3.68230	−0.149460	−0.0747300	−	0.997204i	$-0.523809\pi$
$607$	−3.68230	−0.149460	−0.0747300	+	0.997204i	$0.523809\pi$
$608$	0	0
$609$	−22.6612	−0.918278
$610$	0	0
$611$	25.2632	1.02204
$612$	0	0
$613$	−14.7869	−0.597236	−0.298618	−	0.954373i	$-0.596526\pi$
$613$	−14.7869	−0.597236	−0.298618	+	0.954373i	$0.596526\pi$
$614$	0	0
$615$	11.9495	0.481849
$616$	0	0
$617$	36.9347	1.48694	0.743468	−	0.668771i	$-0.233179\pi$
$617$	36.9347	1.48694	0.743468	+	0.668771i	$0.233179\pi$
$618$	0	0
$619$	14.3869	0.578258	0.289129	−	0.957290i	$-0.406634\pi$
$619$	14.3869	0.578258	0.289129	+	0.957290i	$0.406634\pi$
$620$	0	0
$621$	−27.0012	−1.08352
$622$	0	0
$623$	−37.3309	−1.49563
$624$	0	0
$625$	−22.4237	−0.896948
$626$	0	0
$627$	−2.49092	−0.0994779
$628$	0	0
$629$	17.8744	0.712699
$630$	0	0
$631$	−16.0583	−0.639269	−0.319634	−	0.947541i	$-0.603560\pi$
$631$	−16.0583	−0.639269	−0.319634	+	0.947541i	$0.603560\pi$
$632$	0	0
$633$	−13.0148	−0.517290
$634$	0	0
$635$	−50.4997	−2.00402
$636$	0	0
$637$	−20.7139	−0.820716
$638$	0	0
$639$	−11.0838	−0.438467
$640$	0	0
$641$	−11.3742	−0.449256	−0.224628	−	0.974445i	$-0.572117\pi$
$641$	−11.3742	−0.449256	−0.224628	+	0.974445i	$0.572117\pi$
$642$	0	0
$643$	−25.9574	−1.02366	−0.511829	−	0.859087i	$-0.671032\pi$
$643$	−25.9574	−1.02366	−0.511829	+	0.859087i	$0.671032\pi$
$644$	0	0
$645$	28.6832	1.12940
$646$	0	0
$647$	−41.5169	−1.63220	−0.816098	−	0.577913i	$-0.803867\pi$
$647$	−41.5169	−1.63220	−0.816098	+	0.577913i	$0.803867\pi$
$648$	0	0
$649$	81.3624	3.19375
$650$	0	0
$651$	−9.87803	−0.387150
$652$	0	0
$653$	3.62532	0.141870	0.0709349	−	0.997481i	$-0.477402\pi$
$653$	3.62532	0.141870	0.0709349	+	0.997481i	$0.477402\pi$
$654$	0	0
$655$	−24.3886	−0.952940
$656$	0	0
$657$	−14.6247	−0.570566
$658$	0	0
$659$	0.914082	0.0356076	0.0178038	−	0.999841i	$-0.494333\pi$
$659$	0.914082	0.0356076	0.0178038	+	0.999841i	$0.494333\pi$
$660$	0	0
$661$	0.630739	0.0245329	0.0122664	−	0.999925i	$-0.496095\pi$
$661$	0.630739	0.0245329	0.0122664	+	0.999925i	$0.496095\pi$
$662$	0	0
$663$	−7.64479	−0.296899
$664$	0	0
$665$	−6.11181	−0.237006
$666$	0	0
$667$	56.8605	2.20165
$668$	0	0
$669$	11.6365	0.449894
$670$	0	0
$671$	20.0194	0.772842
$672$	0	0
$673$	18.3996	0.709254	0.354627	−	0.935008i	$-0.384608\pi$
$673$	18.3996	0.709254	0.354627	+	0.935008i	$0.384608\pi$
$674$	0	0
$675$	−22.6684	−0.872506
$676$	0	0
$677$	0.413243	0.0158822	0.00794110	−	0.999968i	$-0.497472\pi$
$677$	0.413243	0.0158822	0.00794110	+	0.999968i	$0.497472\pi$
$678$	0	0
$679$	5.78251	0.221912
$680$	0	0
$681$	−2.09912	−0.0804386
$682$	0	0
$683$	37.6549	1.44083	0.720413	−	0.693545i	$-0.243952\pi$
$683$	37.6549	1.44083	0.720413	+	0.693545i	$0.243952\pi$
$684$	0	0
$685$	16.7297	0.639209
$686$	0	0
$687$	15.3171	0.584385
$688$	0	0
$689$	60.5568	2.30703
$690$	0	0
$691$	−21.3359	−0.811655	−0.405827	−	0.913950i	$-0.633017\pi$
$691$	−21.3359	−0.811655	−0.405827	+	0.913950i	$0.633017\pi$
$692$	0	0
$693$	−47.9221	−1.82041
$694$	0	0
$695$	−15.4362	−0.585528
$696$	0	0
$697$	10.5134	0.398225
$698$	0	0
$699$	5.99937	0.226917
$700$	0	0
$701$	6.62865	0.250361	0.125180	−	0.992134i	$-0.460049\pi$
$701$	6.62865	0.250361	0.125180	+	0.992134i	$0.460049\pi$
$702$	0	0
$703$	4.56590	0.172206
$704$	0	0
$705$	−13.6440	−0.513863
$706$	0	0
$707$	−54.4296	−2.04704
$708$	0	0
$709$	44.6143	1.67552	0.837762	−	0.546036i	$-0.183864\pi$
$709$	44.6143	1.67552	0.837762	+	0.546036i	$0.183864\pi$
$710$	0	0
$711$	18.4479	0.691851
$712$	0	0
$713$	24.7855	0.928224
$714$	0	0
$715$	86.7750	3.24520
$716$	0	0
$717$	−5.75113	−0.214780
$718$	0	0
$719$	−4.79209	−0.178715	−0.0893573	−	0.996000i	$-0.528481\pi$
$719$	−4.79209	−0.178715	−0.0893573	+	0.996000i	$0.528481\pi$
$720$	0	0
$721$	49.8283	1.85570
$722$	0	0
$723$	0.357207	0.0132847
$724$	0	0
$725$	47.7362	1.77288
$726$	0	0
$727$	40.2422	1.49250	0.746250	−	0.665666i	$-0.231852\pi$
$727$	40.2422	1.49250	0.746250	+	0.665666i	$0.231852\pi$
$728$	0	0
$729$	−0.319416	−0.0118302
$730$	0	0
$731$	25.2362	0.933395
$732$	0	0
$733$	−5.87345	−0.216941	−0.108470	−	0.994100i	$-0.534595\pi$
$733$	−5.87345	−0.216941	−0.108470	+	0.994100i	$0.534595\pi$
$734$	0	0
$735$	11.1871	0.412642
$736$	0	0
$737$	−39.2651	−1.44635
$738$	0	0
$739$	−7.33495	−0.269820	−0.134910	−	0.990858i	$-0.543075\pi$
$739$	−7.33495	−0.269820	−0.134910	+	0.990858i	$0.543075\pi$
$740$	0	0
$741$	−1.95281	−0.0717383
$742$	0	0
$743$	13.9483	0.511712	0.255856	−	0.966715i	$-0.417643\pi$
$743$	13.9483	0.511712	0.255856	+	0.966715i	$0.417643\pi$
$744$	0	0
$745$	42.0079	1.53905
$746$	0	0
$747$	−32.5983	−1.19271
$748$	0	0
$749$	−45.5846	−1.66563
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	4.15391	0.151377
$754$	0	0
$755$	−62.2613	−2.26592
$756$	0	0
$757$	26.3916	0.959220	0.479610	−	0.877482i	$-0.340778\pi$
$757$	26.3916	0.959220	0.479610	+	0.877482i	$0.340778\pi$
$758$	0	0
$759$	−29.1667	−1.05868
$760$	0	0
$761$	−24.5870	−0.891278	−0.445639	−	0.895213i	$-0.647024\pi$
$761$	−24.5870	−0.891278	−0.445639	+	0.895213i	$0.647024\pi$
$762$	0	0
$763$	−42.3624	−1.53362
$764$	0	0
$765$	−17.0215	−0.615412
$766$	0	0
$767$	63.7858	2.30317
$768$	0	0
$769$	−31.1447	−1.12311	−0.561553	−	0.827441i	$-0.689796\pi$
$769$	−31.1447	−1.12311	−0.561553	+	0.827441i	$0.689796\pi$
$770$	0	0
$771$	0.537939	0.0193734
$772$	0	0
$773$	−26.6561	−0.958755	−0.479377	−	0.877609i	$-0.659137\pi$
$773$	−26.6561	−0.958755	−0.479377	+	0.877609i	$0.659137\pi$
$774$	0	0
$775$	20.8082	0.747453
$776$	0	0
$777$	−21.3071	−0.764389
$778$	0	0
$779$	2.68559	0.0962212
$780$	0	0
$781$	−26.8495	−0.960750
$782$	0	0
$783$	−36.1534	−1.29202
$784$	0	0
$785$	16.8382	0.600981
$786$	0	0
$787$	−7.15777	−0.255147	−0.127574	−	0.991829i	$-0.540719\pi$
$787$	−7.15777	−0.255147	−0.127574	+	0.991829i	$0.540719\pi$
$788$	0	0
$789$	13.6453	0.485786
$790$	0	0
$791$	−22.6099	−0.803914
$792$	0	0
$793$	15.6946	0.557333
$794$	0	0
$795$	−32.7052	−1.15994
$796$	0	0
$797$	−16.2400	−0.575249	−0.287624	−	0.957743i	$-0.592865\pi$
$797$	−16.2400	−0.575249	−0.287624	+	0.957743i	$0.592865\pi$
$798$	0	0
$799$	−12.0043	−0.424683
$800$	0	0
$801$	−26.5577	−0.938369
$802$	0	0
$803$	−35.4272	−1.25020
$804$	0	0
$805$	−71.5642	−2.52231
$806$	0	0
$807$	15.4546	0.544027
$808$	0	0
$809$	−36.5517	−1.28509	−0.642545	−	0.766248i	$-0.722121\pi$
$809$	−36.5517	−1.28509	−0.642545	+	0.766248i	$0.722121\pi$
$810$	0	0
$811$	0.908184	0.0318907	0.0159453	−	0.999873i	$-0.494924\pi$
$811$	0.908184	0.0318907	0.0159453	+	0.999873i	$0.494924\pi$
$812$	0	0
$813$	2.52762	0.0886476
$814$	0	0
$815$	−37.8157	−1.32463
$816$	0	0
$817$	6.44642	0.225532
$818$	0	0
$819$	−37.5696	−1.31279
$820$	0	0
$821$	−45.4633	−1.58668	−0.793340	−	0.608778i	$-0.791660\pi$
$821$	−45.4633	−1.58668	−0.793340	+	0.608778i	$0.791660\pi$
$822$	0	0
$823$	−23.2278	−0.809670	−0.404835	−	0.914390i	$-0.632671\pi$
$823$	−23.2278	−0.809670	−0.404835	+	0.914390i	$0.632671\pi$
$824$	0	0
$825$	−24.4864	−0.852505
$826$	0	0
$827$	−2.11683	−0.0736093	−0.0368046	−	0.999322i	$-0.511718\pi$
$827$	−2.11683	−0.0736093	−0.0368046	+	0.999322i	$0.511718\pi$
$828$	0	0
$829$	−5.08046	−0.176452	−0.0882259	−	0.996100i	$-0.528120\pi$
$829$	−5.08046	−0.176452	−0.0882259	+	0.996100i	$0.528120\pi$
$830$	0	0
$831$	−5.16930	−0.179321
$832$	0	0
$833$	9.84267	0.341028
$834$	0	0
$835$	35.0041	1.21137
$836$	0	0
$837$	−15.7593	−0.544720
$838$	0	0
$839$	37.1831	1.28370	0.641852	−	0.766828i	$-0.278166\pi$
$839$	37.1831	1.28370	0.641852	+	0.766828i	$0.278166\pi$
$840$	0	0
$841$	47.1336	1.62530
$842$	0	0
$843$	4.18237	0.144049
$844$	0	0
$845$	25.9627	0.893143
$846$	0	0
$847$	−78.7558	−2.70608
$848$	0	0
$849$	5.67323	0.194705
$850$	0	0
$851$	53.4629	1.83268
$852$	0	0
$853$	29.2192	1.00044	0.500222	−	0.865897i	$-0.333252\pi$
$853$	29.2192	1.00044	0.500222	+	0.865897i	$0.333252\pi$
$854$	0	0
$855$	−4.34802	−0.148699
$856$	0	0
$857$	30.3067	1.03526	0.517628	−	0.855606i	$-0.326815\pi$
$857$	30.3067	1.03526	0.517628	+	0.855606i	$0.326815\pi$
$858$	0	0
$859$	17.2880	0.589860	0.294930	−	0.955519i	$-0.404704\pi$
$859$	17.2880	0.589860	0.294930	+	0.955519i	$0.404704\pi$
$860$	0	0
$861$	−12.5325	−0.427107
$862$	0	0
$863$	−17.0801	−0.581415	−0.290707	−	0.956812i	$-0.593891\pi$
$863$	−17.0801	−0.581415	−0.290707	+	0.956812i	$0.593891\pi$
$864$	0	0
$865$	−49.5150	−1.68356
$866$	0	0
$867$	−9.37696	−0.318458
$868$	0	0
$869$	44.6885	1.51595
$870$	0	0
$871$	−30.7827	−1.04303
$872$	0	0
$873$	4.11375	0.139229
$874$	0	0
$875$	−5.17145	−0.174827
$876$	0	0
$877$	−37.7052	−1.27321	−0.636607	−	0.771188i	$-0.719663\pi$
$877$	−37.7052	−1.27321	−0.636607	+	0.771188i	$0.719663\pi$
$878$	0	0
$879$	−7.01752	−0.236695
$880$	0	0
$881$	−2.79242	−0.0940790	−0.0470395	−	0.998893i	$-0.514979\pi$
$881$	−2.79242	−0.0940790	−0.0470395	+	0.998893i	$0.514979\pi$
$882$	0	0
$883$	−4.90793	−0.165165	−0.0825825	−	0.996584i	$-0.526317\pi$
$883$	−4.90793	−0.165165	−0.0825825	+	0.996584i	$0.526317\pi$
$884$	0	0
$885$	−34.4491	−1.15799
$886$	0	0
$887$	1.70211	0.0571512	0.0285756	−	0.999592i	$-0.490903\pi$
$887$	1.70211	0.0571512	0.0285756	+	0.999592i	$0.490903\pi$
$888$	0	0
$889$	52.9637	1.77634
$890$	0	0
$891$	−23.8170	−0.797899
$892$	0	0
$893$	−3.06643	−0.102614
$894$	0	0
$895$	49.8583	1.66658
$896$	0	0
$897$	−22.8658	−0.763467
$898$	0	0
$899$	33.1867	1.10684
$900$	0	0
$901$	−28.7749	−0.958630
$902$	0	0
$903$	−30.0828	−1.00109
$904$	0	0
$905$	44.3566	1.47446
$906$	0	0
$907$	19.7176	0.654712	0.327356	−	0.944901i	$-0.393842\pi$
$907$	19.7176	0.654712	0.327356	+	0.944901i	$0.393842\pi$
$908$	0	0
$909$	−38.7219	−1.28432
$910$	0	0
$911$	−20.6859	−0.685354	−0.342677	−	0.939453i	$-0.611334\pi$
$911$	−20.6859	−0.685354	−0.342677	+	0.939453i	$0.611334\pi$
$912$	0	0
$913$	−78.9665	−2.61341
$914$	0	0
$915$	−8.47629	−0.280217
$916$	0	0
$917$	25.5785	0.844678
$918$	0	0
$919$	−7.46112	−0.246120	−0.123060	−	0.992399i	$-0.539271\pi$
$919$	−7.46112	−0.246120	−0.123060	+	0.992399i	$0.539271\pi$
$920$	0	0
$921$	3.62991	0.119609
$922$	0	0
$923$	−21.0492	−0.692843
$924$	0	0
$925$	44.8838	1.47577
$926$	0	0
$927$	35.4485	1.16428
$928$	0	0
$929$	31.3116	1.02730	0.513651	−	0.857999i	$-0.328293\pi$
$929$	31.3116	1.02730	0.513651	+	0.857999i	$0.328293\pi$
$930$	0	0
$931$	2.51424	0.0824010
$932$	0	0
$933$	−0.739064	−0.0241959
$934$	0	0
$935$	−41.2330	−1.34846
$936$	0	0
$937$	−19.6845	−0.643063	−0.321532	−	0.946899i	$-0.604198\pi$
$937$	−19.6845	−0.643063	−0.321532	+	0.946899i	$0.604198\pi$
$938$	0	0
$939$	22.4038	0.731122
$940$	0	0
$941$	57.1933	1.86445	0.932224	−	0.361881i	$-0.117865\pi$
$941$	57.1933	1.86445	0.932224	+	0.361881i	$0.117865\pi$
$942$	0	0
$943$	31.4460	1.02402
$944$	0	0
$945$	45.5024	1.48019
$946$	0	0
$947$	−6.14626	−0.199727	−0.0998634	−	0.995001i	$-0.531841\pi$
$947$	−6.14626	−0.199727	−0.0998634	+	0.995001i	$0.531841\pi$
$948$	0	0
$949$	−27.7739	−0.901578
$950$	0	0
$951$	2.17721	0.0706009
$952$	0	0
$953$	45.8124	1.48401	0.742005	−	0.670395i	$-0.233875\pi$
$953$	45.8124	1.48401	0.742005	+	0.670395i	$0.233875\pi$
$954$	0	0
$955$	45.3226	1.46661
$956$	0	0
$957$	−39.0528	−1.26240
$958$	0	0
$959$	−17.5460	−0.566589
$960$	0	0
$961$	−16.5339	−0.533352
$962$	0	0
$963$	−32.4295	−1.04502
$964$	0	0
$965$	25.3018	0.814494
$966$	0	0
$967$	17.1965	0.553004	0.276502	−	0.961013i	$-0.410825\pi$
$967$	17.1965	0.553004	0.276502	+	0.961013i	$0.410825\pi$
$968$	0	0
$969$	0.927920	0.0298091
$970$	0	0
$971$	37.9153	1.21676	0.608379	−	0.793646i	$-0.291820\pi$
$971$	37.9153	1.21676	0.608379	+	0.793646i	$0.291820\pi$
$972$	0	0
$973$	16.1894	0.519007
$974$	0	0
$975$	−19.1966	−0.614783
$976$	0	0
$977$	−41.0385	−1.31294	−0.656469	−	0.754353i	$-0.727951\pi$
$977$	−41.0385	−1.31294	−0.656469	+	0.754353i	$0.727951\pi$
$978$	0	0
$979$	−64.3336	−2.05611
$980$	0	0
$981$	−30.1371	−0.962204
$982$	0	0
$983$	−19.6445	−0.626563	−0.313281	−	0.949660i	$-0.601428\pi$
$983$	−19.6445	−0.626563	−0.313281	+	0.949660i	$0.601428\pi$
$984$	0	0
$985$	3.60295	0.114799
$986$	0	0
$987$	14.3097	0.455484
$988$	0	0
$989$	75.4823	2.40020
$990$	0	0
$991$	2.19516	0.0697316	0.0348658	−	0.999392i	$-0.488900\pi$
$991$	2.19516	0.0697316	0.0348658	+	0.999392i	$0.488900\pi$
$992$	0	0
$993$	16.1435	0.512298
$994$	0	0
$995$	52.7090	1.67099
$996$	0	0
$997$	8.00820	0.253622	0.126811	−	0.991927i	$-0.459526\pi$
$997$	8.00820	0.253622	0.126811	+	0.991927i	$0.459526\pi$
$998$	0	0
$999$	−33.9931	−1.07549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.30	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.30	✓	50	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)

\(f(q)\)	\(=\)	\(q+0.765267 q^{3} +3.23588 q^{5} -3.39376 q^{7} -2.41437 q^{9} +O(q^{10})\)	\(q+0.765267 q^{3} +3.23588 q^{5} -3.39376 q^{7} -2.41437 q^{9} -5.84859 q^{11} -4.58513 q^{13} +2.47631 q^{15} +2.17872 q^{17} +0.556540 q^{19} -2.59714 q^{21} +6.51662 q^{23} +5.47091 q^{25} -4.14344 q^{27} +8.72546 q^{29} +3.80343 q^{31} -4.47574 q^{33} -10.9818 q^{35} +8.20409 q^{37} -3.50885 q^{39} +4.82551 q^{41} +11.5830 q^{43} -7.81259 q^{45} -5.50981 q^{47} +4.51764 q^{49} +1.66730 q^{51} -13.2072 q^{53} -18.9253 q^{55} +0.425901 q^{57} -13.9115 q^{59} -3.42295 q^{61} +8.19379 q^{63} -14.8369 q^{65} +6.71360 q^{67} +4.98695 q^{69} +4.59076 q^{71} +6.05739 q^{73} +4.18671 q^{75} +19.8487 q^{77} -7.64089 q^{79} +4.07226 q^{81} +13.5018 q^{83} +7.05007 q^{85} +6.67731 q^{87} +10.9998 q^{89} +15.5608 q^{91} +2.91064 q^{93} +1.80089 q^{95} -1.70386 q^{97} +14.1206 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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