LMFDB - Embedded newform 3800.2.a.z.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.253565184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 $ x^6 - 2x^5 - 11x^4 + 20x^3 + 22x^2 - 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.84742$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.84742 q^{3} +0.145034 q^{7} +5.10782 q^{9} +O(q^{10})$	$q+2.84742 q^{3} +0.145034 q^{7} +5.10782 q^{9} -5.71774 q^{11} -5.24345 q^{13} -7.15378 q^{17} +1.00000 q^{19} +0.412974 q^{21} +0.622762 q^{23} +6.00186 q^{27} -5.46811 q^{29} -3.77513 q^{31} -16.2808 q^{33} +5.03553 q^{37} -14.9303 q^{39} +5.77513 q^{41} +3.32308 q^{43} -5.85683 q^{47} -6.97897 q^{49} -20.3699 q^{51} +6.97219 q^{53} +2.84742 q^{57} +9.09089 q^{59} -8.16707 q^{61} +0.740810 q^{63} -13.6583 q^{67} +1.77327 q^{69} +2.41484 q^{71} -7.44385 q^{73} -0.829270 q^{77} -9.69485 q^{79} +1.76638 q^{81} +2.17116 q^{83} -15.5700 q^{87} +3.90782 q^{89} -0.760481 q^{91} -10.7494 q^{93} -4.98328 q^{97} -29.2052 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9	$ 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} - 34 q^{51} + 14 q^{53} - 2 q^{57} - 2 q^{59} + 10 q^{61} - 44 q^{63} - 42 q^{67} + 18 q^{69} - 8 q^{71} + 2 q^{73} - 24 q^{77} - 20 q^{79} + 6 q^{81} - 16 q^{83} - 30 q^{87} + 32 q^{89} + 10 q^{91} - 12 q^{93} - 40 q^{97} - 22 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9 - 2 * q^11 - 14 * q^13 - 10 * q^17 + 6 * q^19 + 18 * q^21 - 2 * q^23 - 2 * q^27 - 2 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^37 - 18 * q^39 + 4 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 34 * q^51 + 14 * q^53 - 2 * q^57 - 2 * q^59 + 10 * q^61 - 44 * q^63 - 42 * q^67 + 18 * q^69 - 8 * q^71 + 2 * q^73 - 24 * q^77 - 20 * q^79 + 6 * q^81 - 16 * q^83 - 30 * q^87 + 32 * q^89 + 10 * q^91 - 12 * q^93 - 40 * q^97 - 22 * 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
$3$	2.84742	1.64396	0.821980	−	0.569516i	$-0.192869\pi$
$3$	2.84742	1.64396	0.821980	+	0.569516i	$0.192869\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.145034	0.0548179	0.0274089	−	0.999624i	$-0.491274\pi$
$7$	0.145034	0.0548179	0.0274089	+	0.999624i	$0.491274\pi$
$8$	0	0
$9$	5.10782	1.70261
$10$	0	0
$11$	−5.71774	−1.72396	−0.861982	−	0.506938i	$-0.830777\pi$
$11$	−5.71774	−1.72396	−0.861982	+	0.506938i	$0.830777\pi$
$12$	0	0
$13$	−5.24345	−1.45427	−0.727136	−	0.686493i	$-0.759149\pi$
$13$	−5.24345	−1.45427	−0.727136	+	0.686493i	$0.759149\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.15378	−1.73505	−0.867524	−	0.497396i	$-0.834290\pi$
$17$	−7.15378	−1.73505	−0.867524	+	0.497396i	$0.834290\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.412974	0.0901184
$22$	0	0
$23$	0.622762	0.129855	0.0649274	−	0.997890i	$-0.479318\pi$
$23$	0.622762	0.129855	0.0649274	+	0.997890i	$0.479318\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	6.00186	1.15506
$28$	0	0
$29$	−5.46811	−1.01540	−0.507702	−	0.861533i	$-0.669505\pi$
$29$	−5.46811	−1.01540	−0.507702	+	0.861533i	$0.669505\pi$
$30$	0	0
$31$	−3.77513	−0.678033	−0.339017	−	0.940780i	$-0.610094\pi$
$31$	−3.77513	−0.678033	−0.339017	+	0.940780i	$0.610094\pi$
$32$	0	0
$33$	−16.2808	−2.83413
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.03553	0.827836	0.413918	−	0.910314i	$-0.364160\pi$
$37$	5.03553	0.827836	0.413918	+	0.910314i	$0.364160\pi$
$38$	0	0
$39$	−14.9303	−2.39077
$40$	0	0
$41$	5.77513	0.901924	0.450962	−	0.892543i	$-0.351081\pi$
$41$	5.77513	0.901924	0.450962	+	0.892543i	$0.351081\pi$
$42$	0	0
$43$	3.32308	0.506765	0.253382	−	0.967366i	$-0.418457\pi$
$43$	3.32308	0.506765	0.253382	+	0.967366i	$0.418457\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.85683	−0.854306	−0.427153	−	0.904179i	$-0.640483\pi$
$47$	−5.85683	−0.854306	−0.427153	+	0.904179i	$0.640483\pi$
$48$	0	0
$49$	−6.97897	−0.996995
$50$	0	0
$51$	−20.3699	−2.85235
$52$	0	0
$53$	6.97219	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$53$	6.97219	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.84742	0.377150
$58$	0	0
$59$	9.09089	1.18353	0.591767	−	0.806109i	$-0.298431\pi$
$59$	9.09089	1.18353	0.591767	+	0.806109i	$0.298431\pi$
$60$	0	0
$61$	−8.16707	−1.04569	−0.522843	−	0.852429i	$-0.675129\pi$
$61$	−8.16707	−1.04569	−0.522843	+	0.852429i	$0.675129\pi$
$62$	0	0
$63$	0.740810	0.0933333
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.6583	−1.66863	−0.834313	−	0.551291i	$-0.814135\pi$
$67$	−13.6583	−1.66863	−0.834313	+	0.551291i	$0.814135\pi$
$68$	0	0
$69$	1.77327	0.213476
$70$	0	0
$71$	2.41484	0.286588	0.143294	−	0.989680i	$-0.454230\pi$
$71$	2.41484	0.286588	0.143294	+	0.989680i	$0.454230\pi$
$72$	0	0
$73$	−7.44385	−0.871237	−0.435619	−	0.900131i	$-0.643470\pi$
$73$	−7.44385	−0.871237	−0.435619	+	0.900131i	$0.643470\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.829270	−0.0945041
$78$	0	0
$79$	−9.69485	−1.09076	−0.545378	−	0.838190i	$-0.683614\pi$
$79$	−9.69485	−1.09076	−0.545378	+	0.838190i	$0.683614\pi$
$80$	0	0
$81$	1.76638	0.196264
$82$	0	0
$83$	2.17116	0.238316	0.119158	−	0.992875i	$-0.461981\pi$
$83$	2.17116	0.238316	0.119158	+	0.992875i	$0.461981\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−15.5700	−1.66928
$88$	0	0
$89$	3.90782	0.414228	0.207114	−	0.978317i	$-0.433593\pi$
$89$	3.90782	0.414228	0.207114	+	0.978317i	$0.433593\pi$
$90$	0	0
$91$	−0.760481	−0.0797201
$92$	0	0
$93$	−10.7494	−1.11466
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.98328	−0.505976	−0.252988	−	0.967469i	$-0.581413\pi$
$97$	−4.98328	−0.505976	−0.252988	+	0.967469i	$0.581413\pi$
$98$	0	0
$99$	−29.2052	−2.93524
$100$	0	0
$101$	−5.35148	−0.532493	−0.266246	−	0.963905i	$-0.585783\pi$
$101$	−5.35148	−0.532493	−0.266246	+	0.963905i	$0.585783\pi$
$102$	0	0
$103$	3.59495	0.354221	0.177111	−	0.984191i	$-0.443325\pi$
$103$	3.59495	0.354221	0.177111	+	0.984191i	$0.443325\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.3518	1.48412	0.742059	−	0.670335i	$-0.233850\pi$
$107$	15.3518	1.48412	0.742059	+	0.670335i	$0.233850\pi$
$108$	0	0
$109$	2.20105	0.210823	0.105411	−	0.994429i	$-0.466384\pi$
$109$	2.20105	0.210823	0.105411	+	0.994429i	$0.466384\pi$
$110$	0	0
$111$	14.3383	1.36093
$112$	0	0
$113$	12.3613	1.16286	0.581428	−	0.813598i	$-0.302494\pi$
$113$	12.3613	1.16286	0.581428	+	0.813598i	$0.302494\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−26.7826	−2.47605
$118$	0	0
$119$	−1.03754	−0.0951116
$120$	0	0
$121$	21.6926	1.97205
$122$	0	0
$123$	16.4442	1.48273
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.0907	−1.60529	−0.802644	−	0.596459i	$-0.796574\pi$
$127$	−18.0907	−1.60529	−0.802644	+	0.596459i	$0.796574\pi$
$128$	0	0
$129$	9.46222	0.833102
$130$	0	0
$131$	12.1244	1.05931	0.529655	−	0.848213i	$-0.322321\pi$
$131$	12.1244	1.05931	0.529655	+	0.848213i	$0.322321\pi$
$132$	0	0
$133$	0.145034	0.0125761
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.0385	1.71201	0.856004	−	0.516969i	$-0.172940\pi$
$137$	20.0385	1.71201	0.856004	+	0.516969i	$0.172940\pi$
$138$	0	0
$139$	11.3996	0.966905	0.483452	−	0.875371i	$-0.339383\pi$
$139$	11.3996	0.966905	0.483452	+	0.875371i	$0.339383\pi$
$140$	0	0
$141$	−16.6769	−1.40445
$142$	0	0
$143$	29.9807	2.50711
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−19.8721	−1.63902
$148$	0	0
$149$	−15.5818	−1.27651	−0.638257	−	0.769823i	$-0.720344\pi$
$149$	−15.5818	−1.27651	−0.638257	+	0.769823i	$0.720344\pi$
$150$	0	0
$151$	22.3717	1.82058	0.910292	−	0.413966i	$-0.135857\pi$
$151$	22.3717	1.82058	0.910292	+	0.413966i	$0.135857\pi$
$152$	0	0
$153$	−36.5403	−2.95410
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−23.1047	−1.84395	−0.921976	−	0.387246i	$-0.873426\pi$
$157$	−23.1047	−1.84395	−0.921976	+	0.387246i	$0.873426\pi$
$158$	0	0
$159$	19.8528	1.57443
$160$	0	0
$161$	0.0903219	0.00711836
$162$	0	0
$163$	−11.3524	−0.889186	−0.444593	−	0.895733i	$-0.646652\pi$
$163$	−11.3524	−0.889186	−0.444593	+	0.895733i	$0.646652\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.09989	−0.472024	−0.236012	−	0.971750i	$-0.575840\pi$
$167$	−6.09989	−0.472024	−0.236012	+	0.971750i	$0.575840\pi$
$168$	0	0
$169$	14.4938	1.11491
$170$	0	0
$171$	5.10782	0.390605
$172$	0	0
$173$	10.8951	0.828338	0.414169	−	0.910200i	$-0.364072\pi$
$173$	10.8951	0.828338	0.414169	+	0.910200i	$0.364072\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	25.8856	1.94568
$178$	0	0
$179$	4.46123	0.333448	0.166724	−	0.986004i	$-0.446681\pi$
$179$	4.46123	0.333448	0.166724	+	0.986004i	$0.446681\pi$
$180$	0	0
$181$	3.58784	0.266682	0.133341	−	0.991070i	$-0.457429\pi$
$181$	3.58784	0.266682	0.133341	+	0.991070i	$0.457429\pi$
$182$	0	0
$183$	−23.2551	−1.71907
$184$	0	0
$185$	0	0
$186$	0	0
$187$	40.9035	2.99116
$188$	0	0
$189$	0.870477	0.0633179
$190$	0	0
$191$	8.00777	0.579422	0.289711	−	0.957114i	$-0.406441\pi$
$191$	8.00777	0.579422	0.289711	+	0.957114i	$0.406441\pi$
$192$	0	0
$193$	1.45641	0.104835	0.0524173	−	0.998625i	$-0.483307\pi$
$193$	1.45641	0.104835	0.0524173	+	0.998625i	$0.483307\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.3510	−1.87743	−0.938716	−	0.344692i	$-0.887983\pi$
$197$	−26.3510	−1.87743	−0.938716	+	0.344692i	$0.887983\pi$
$198$	0	0
$199$	12.1739	0.862985	0.431492	−	0.902117i	$-0.357987\pi$
$199$	12.1739	0.862985	0.431492	+	0.902117i	$0.357987\pi$
$200$	0	0
$201$	−38.8909	−2.74316
$202$	0	0
$203$	−0.793065	−0.0556622
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.18096	0.221092
$208$	0	0
$209$	−5.71774	−0.395505
$210$	0	0
$211$	−19.5088	−1.34304	−0.671521	−	0.740985i	$-0.734359\pi$
$211$	−19.5088	−1.34304	−0.671521	+	0.740985i	$0.734359\pi$
$212$	0	0
$213$	6.87606	0.471140
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.547524	−0.0371683
$218$	0	0
$219$	−21.1958	−1.43228
$220$	0	0
$221$	37.5105	2.52323
$222$	0	0
$223$	−14.9208	−0.999168	−0.499584	−	0.866265i	$-0.666514\pi$
$223$	−14.9208	−0.999168	−0.499584	+	0.866265i	$0.666514\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.4422	−0.892192	−0.446096	−	0.894985i	$-0.647186\pi$
$227$	−13.4422	−0.892192	−0.446096	+	0.894985i	$0.647186\pi$
$228$	0	0
$229$	−7.36620	−0.486772	−0.243386	−	0.969929i	$-0.578258\pi$
$229$	−7.36620	−0.486772	−0.243386	+	0.969929i	$0.578258\pi$
$230$	0	0
$231$	−2.36128	−0.155361
$232$	0	0
$233$	−4.24405	−0.278037	−0.139018	−	0.990290i	$-0.544395\pi$
$233$	−4.24405	−0.278037	−0.139018	+	0.990290i	$0.544395\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−27.6053	−1.79316
$238$	0	0
$239$	−15.3387	−0.992176	−0.496088	−	0.868272i	$-0.665231\pi$
$239$	−15.3387	−0.992176	−0.496088	+	0.868272i	$0.665231\pi$
$240$	0	0
$241$	26.7026	1.72006	0.860032	−	0.510240i	$-0.170443\pi$
$241$	26.7026	1.72006	0.860032	+	0.510240i	$0.170443\pi$
$242$	0	0
$243$	−12.9760	−0.832408
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.24345	−0.333633
$248$	0	0
$249$	6.18221	0.391781
$250$	0	0
$251$	3.64463	0.230047	0.115023	−	0.993363i	$-0.463306\pi$
$251$	3.64463	0.230047	0.115023	+	0.993363i	$0.463306\pi$
$252$	0	0
$253$	−3.56079	−0.223865
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.71103	−0.605757	−0.302879	−	0.953029i	$-0.597948\pi$
$257$	−9.71103	−0.605757	−0.302879	+	0.953029i	$0.597948\pi$
$258$	0	0
$259$	0.730325	0.0453802
$260$	0	0
$261$	−27.9302	−1.72883
$262$	0	0
$263$	1.82052	0.112258	0.0561290	−	0.998424i	$-0.482124\pi$
$263$	1.82052	0.112258	0.0561290	+	0.998424i	$0.482124\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.1272	0.680974
$268$	0	0
$269$	21.6353	1.31913	0.659563	−	0.751649i	$-0.270741\pi$
$269$	21.6353	1.31913	0.659563	+	0.751649i	$0.270741\pi$
$270$	0	0
$271$	−7.54436	−0.458287	−0.229144	−	0.973393i	$-0.573593\pi$
$271$	−7.54436	−0.458287	−0.229144	+	0.973393i	$0.573593\pi$
$272$	0	0
$273$	−2.16541	−0.131057
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.5559	0.754410	0.377205	−	0.926130i	$-0.376885\pi$
$277$	12.5559	0.754410	0.377205	+	0.926130i	$0.376885\pi$
$278$	0	0
$279$	−19.2827	−1.15442
$280$	0	0
$281$	22.1947	1.32403	0.662013	−	0.749493i	$-0.269703\pi$
$281$	22.1947	1.32403	0.662013	+	0.749493i	$0.269703\pi$
$282$	0	0
$283$	−20.6171	−1.22556	−0.612779	−	0.790254i	$-0.709948\pi$
$283$	−20.6171	−1.22556	−0.612779	+	0.790254i	$0.709948\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.837592	0.0494415
$288$	0	0
$289$	34.1766	2.01039
$290$	0	0
$291$	−14.1895	−0.831804
$292$	0	0
$293$	−22.1421	−1.29356	−0.646778	−	0.762679i	$-0.723884\pi$
$293$	−22.1421	−1.29356	−0.646778	+	0.762679i	$0.723884\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−34.3171	−1.99128
$298$	0	0
$299$	−3.26542	−0.188844
$300$	0	0
$301$	0.481961	0.0277798
$302$	0	0
$303$	−15.2379	−0.875397
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.80060	0.102766	0.0513828	−	0.998679i	$-0.483637\pi$
$307$	1.80060	0.102766	0.0513828	+	0.998679i	$0.483637\pi$
$308$	0	0
$309$	10.2364	0.582326
$310$	0	0
$311$	−9.26554	−0.525401	−0.262700	−	0.964877i	$-0.584613\pi$
$311$	−9.26554	−0.525401	−0.262700	+	0.964877i	$0.584613\pi$
$312$	0	0
$313$	4.29779	0.242925	0.121463	−	0.992596i	$-0.461242\pi$
$313$	4.29779	0.242925	0.121463	+	0.992596i	$0.461242\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.61065	0.258960	0.129480	−	0.991582i	$-0.458669\pi$
$317$	4.61065	0.258960	0.129480	+	0.991582i	$0.458669\pi$
$318$	0	0
$319$	31.2653	1.75052
$320$	0	0
$321$	43.7132	2.43983
$322$	0	0
$323$	−7.15378	−0.398047
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.26734	0.346584
$328$	0	0
$329$	−0.849442	−0.0468312
$330$	0	0
$331$	−0.934366	−0.0513574	−0.0256787	−	0.999670i	$-0.508175\pi$
$331$	−0.934366	−0.0513574	−0.0256787	+	0.999670i	$0.508175\pi$
$332$	0	0
$333$	25.7206	1.40948
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.8588	−1.62651	−0.813256	−	0.581905i	$-0.802307\pi$
$337$	−29.8588	−1.62651	−0.813256	+	0.581905i	$0.802307\pi$
$338$	0	0
$339$	35.1980	1.91169
$340$	0	0
$341$	21.5852	1.16891
$342$	0	0
$343$	−2.02743	−0.109471
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.6502	−0.571732	−0.285866	−	0.958270i	$-0.592281\pi$
$347$	−10.6502	−0.571732	−0.285866	+	0.958270i	$0.592281\pi$
$348$	0	0
$349$	29.3479	1.57096	0.785480	−	0.618888i	$-0.212416\pi$
$349$	29.3479	1.57096	0.785480	+	0.618888i	$0.212416\pi$
$350$	0	0
$351$	−31.4705	−1.67977
$352$	0	0
$353$	14.3847	0.765619	0.382810	−	0.923827i	$-0.374957\pi$
$353$	14.3847	0.765619	0.382810	+	0.923827i	$0.374957\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.95433	−0.156360
$358$	0	0
$359$	−2.23773	−0.118103	−0.0590514	−	0.998255i	$-0.518808\pi$
$359$	−2.23773	−0.118103	−0.0590514	+	0.998255i	$0.518808\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	61.7680	3.24198
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.636873	0.0332445	0.0166222	−	0.999862i	$-0.494709\pi$
$367$	0.636873	0.0332445	0.0166222	+	0.999862i	$0.494709\pi$
$368$	0	0
$369$	29.4983	1.53562
$370$	0	0
$371$	1.01121	0.0524993
$372$	0	0
$373$	−16.2358	−0.840655	−0.420328	−	0.907372i	$-0.638085\pi$
$373$	−16.2358	−0.840655	−0.420328	+	0.907372i	$0.638085\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	28.6718	1.47667
$378$	0	0
$379$	−17.9624	−0.922667	−0.461333	−	0.887227i	$-0.652629\pi$
$379$	−17.9624	−0.922667	−0.461333	+	0.887227i	$0.652629\pi$
$380$	0	0
$381$	−51.5118	−2.63903
$382$	0	0
$383$	−4.06305	−0.207612	−0.103806	−	0.994598i	$-0.533102\pi$
$383$	−4.06305	−0.207612	−0.103806	+	0.994598i	$0.533102\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.9737	0.862822
$388$	0	0
$389$	−28.7507	−1.45772	−0.728859	−	0.684663i	$-0.759949\pi$
$389$	−28.7507	−1.45772	−0.728859	+	0.684663i	$0.759949\pi$
$390$	0	0
$391$	−4.45510	−0.225304
$392$	0	0
$393$	34.5232	1.74147
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.6518	0.584786	0.292393	−	0.956298i	$-0.405548\pi$
$397$	11.6518	0.584786	0.292393	+	0.956298i	$0.405548\pi$
$398$	0	0
$399$	0.412974	0.0206746
$400$	0	0
$401$	1.98190	0.0989716	0.0494858	−	0.998775i	$-0.484242\pi$
$401$	1.98190	0.0989716	0.0494858	+	0.998775i	$0.484242\pi$
$402$	0	0
$403$	19.7947	0.986045
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.7919	−1.42716
$408$	0	0
$409$	26.3984	1.30532	0.652660	−	0.757651i	$-0.273653\pi$
$409$	26.3984	1.30532	0.652660	+	0.757651i	$0.273653\pi$
$410$	0	0
$411$	57.0582	2.81447
$412$	0	0
$413$	1.31849	0.0648788
$414$	0	0
$415$	0	0
$416$	0	0
$417$	32.4596	1.58955
$418$	0	0
$419$	−16.3134	−0.796963	−0.398481	−	0.917176i	$-0.630463\pi$
$419$	−16.3134	−0.796963	−0.398481	+	0.917176i	$0.630463\pi$
$420$	0	0
$421$	−1.90815	−0.0929976	−0.0464988	−	0.998918i	$-0.514806\pi$
$421$	−1.90815	−0.0929976	−0.0464988	+	0.998918i	$0.514806\pi$
$422$	0	0
$423$	−29.9156	−1.45455
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.18451	−0.0573223
$428$	0	0
$429$	85.3678	4.12160
$430$	0	0
$431$	13.9013	0.669600	0.334800	−	0.942289i	$-0.391331\pi$
$431$	13.9013	0.669600	0.334800	+	0.942289i	$0.391331\pi$
$432$	0	0
$433$	4.93885	0.237346	0.118673	−	0.992933i	$-0.462136\pi$
$433$	4.93885	0.237346	0.118673	+	0.992933i	$0.462136\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.622762	0.0297907
$438$	0	0
$439$	−18.0918	−0.863473	−0.431736	−	0.902000i	$-0.642099\pi$
$439$	−18.0918	−0.863473	−0.431736	+	0.902000i	$0.642099\pi$
$440$	0	0
$441$	−35.6473	−1.69749
$442$	0	0
$443$	−8.13690	−0.386596	−0.193298	−	0.981140i	$-0.561918\pi$
$443$	−8.13690	−0.386596	−0.193298	+	0.981140i	$0.561918\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−44.3681	−2.09854
$448$	0	0
$449$	19.9187	0.940020	0.470010	−	0.882661i	$-0.344250\pi$
$449$	19.9187	0.940020	0.470010	+	0.882661i	$0.344250\pi$
$450$	0	0
$451$	−33.0207	−1.55488
$452$	0	0
$453$	63.7018	2.99297
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.4434	−1.00308	−0.501540	−	0.865134i	$-0.667233\pi$
$457$	−21.4434	−1.00308	−0.501540	+	0.865134i	$0.667233\pi$
$458$	0	0
$459$	−42.9360	−2.00408
$460$	0	0
$461$	11.2325	0.523148	0.261574	−	0.965183i	$-0.415758\pi$
$461$	11.2325	0.523148	0.261574	+	0.965183i	$0.415758\pi$
$462$	0	0
$463$	−37.8262	−1.75793	−0.878967	−	0.476882i	$-0.841767\pi$
$463$	−37.8262	−1.75793	−0.878967	+	0.476882i	$0.841767\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.3388	1.40391	0.701955	−	0.712221i	$-0.252311\pi$
$467$	30.3388	1.40391	0.701955	+	0.712221i	$0.252311\pi$
$468$	0	0
$469$	−1.98092	−0.0914705
$470$	0	0
$471$	−65.7888	−3.03139
$472$	0	0
$473$	−19.0005	−0.873645
$474$	0	0
$475$	0	0
$476$	0	0
$477$	35.6127	1.63059
$478$	0	0
$479$	−35.3121	−1.61345	−0.806725	−	0.590927i	$-0.798762\pi$
$479$	−35.3121	−1.61345	−0.806725	+	0.590927i	$0.798762\pi$
$480$	0	0
$481$	−26.4035	−1.20390
$482$	0	0
$483$	0.257185	0.0117023
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−6.88482	−0.311981	−0.155991	−	0.987759i	$-0.549857\pi$
$487$	−6.88482	−0.311981	−0.155991	+	0.987759i	$0.549857\pi$
$488$	0	0
$489$	−32.3250	−1.46179
$490$	0	0
$491$	−10.1116	−0.456328	−0.228164	−	0.973623i	$-0.573272\pi$
$491$	−10.1116	−0.456328	−0.228164	+	0.973623i	$0.573272\pi$
$492$	0	0
$493$	39.1177	1.76177
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.350234	0.0157102
$498$	0	0
$499$	−36.4594	−1.63215	−0.816074	−	0.577947i	$-0.803854\pi$
$499$	−36.4594	−1.63215	−0.816074	+	0.577947i	$0.803854\pi$
$500$	0	0
$501$	−17.3690	−0.775989
$502$	0	0
$503$	−9.00269	−0.401410	−0.200705	−	0.979652i	$-0.564323\pi$
$503$	−9.00269	−0.401410	−0.200705	+	0.979652i	$0.564323\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	41.2700	1.83286
$508$	0	0
$509$	10.3542	0.458943	0.229471	−	0.973315i	$-0.426300\pi$
$509$	10.3542	0.458943	0.229471	+	0.973315i	$0.426300\pi$
$510$	0	0
$511$	−1.07961	−0.0477594
$512$	0	0
$513$	6.00186	0.264989
$514$	0	0
$515$	0	0
$516$	0	0
$517$	33.4878	1.47279
$518$	0	0
$519$	31.0229	1.36175
$520$	0	0
$521$	11.5209	0.504738	0.252369	−	0.967631i	$-0.418790\pi$
$521$	11.5209	0.504738	0.252369	+	0.967631i	$0.418790\pi$
$522$	0	0
$523$	−24.0131	−1.05002	−0.525009	−	0.851097i	$-0.675938\pi$
$523$	−24.0131	−1.05002	−0.525009	+	0.851097i	$0.675938\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27.0065	1.17642
$528$	0	0
$529$	−22.6122	−0.983138
$530$	0	0
$531$	46.4347	2.01509
$532$	0	0
$533$	−30.2816	−1.31164
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.7030	0.548175
$538$	0	0
$539$	39.9039	1.71878
$540$	0	0
$541$	−34.5131	−1.48383	−0.741916	−	0.670492i	$-0.766083\pi$
$541$	−34.5131	−1.48383	−0.741916	+	0.670492i	$0.766083\pi$
$542$	0	0
$543$	10.2161	0.438415
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.89383	0.166488	0.0832441	−	0.996529i	$-0.473472\pi$
$547$	3.89383	0.166488	0.0832441	+	0.996529i	$0.473472\pi$
$548$	0	0
$549$	−41.7159	−1.78039
$550$	0	0
$551$	−5.46811	−0.232949
$552$	0	0
$553$	−1.40609	−0.0597929
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.1816	0.473779	0.236889	−	0.971537i	$-0.423872\pi$
$557$	11.1816	0.473779	0.236889	+	0.971537i	$0.423872\pi$
$558$	0	0
$559$	−17.4244	−0.736974
$560$	0	0
$561$	116.470	4.91735
$562$	0	0
$563$	−2.54455	−0.107240	−0.0536200	−	0.998561i	$-0.517076\pi$
$563$	−2.54455	−0.107240	−0.0536200	+	0.998561i	$0.517076\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.256186	0.0107588
$568$	0	0
$569$	2.09511	0.0878314	0.0439157	−	0.999035i	$-0.486017\pi$
$569$	2.09511	0.0878314	0.0439157	+	0.999035i	$0.486017\pi$
$570$	0	0
$571$	36.2358	1.51642	0.758211	−	0.652009i	$-0.226074\pi$
$571$	36.2358	1.51642	0.758211	+	0.652009i	$0.226074\pi$
$572$	0	0
$573$	22.8015	0.952547
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.5756	0.648419	0.324209	−	0.945985i	$-0.394902\pi$
$577$	15.5756	0.648419	0.324209	+	0.945985i	$0.394902\pi$
$578$	0	0
$579$	4.14702	0.172344
$580$	0	0
$581$	0.314893	0.0130639
$582$	0	0
$583$	−39.8652	−1.65105
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.54894	0.352853	0.176426	−	0.984314i	$-0.443546\pi$
$587$	8.54894	0.352853	0.176426	+	0.984314i	$0.443546\pi$
$588$	0	0
$589$	−3.77513	−0.155551
$590$	0	0
$591$	−75.0325	−3.08642
$592$	0	0
$593$	25.1005	1.03075	0.515377	−	0.856964i	$-0.327652\pi$
$593$	25.1005	1.03075	0.515377	+	0.856964i	$0.327652\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	34.6642	1.41871
$598$	0	0
$599$	−24.9528	−1.01954	−0.509772	−	0.860310i	$-0.670270\pi$
$599$	−24.9528	−1.01954	−0.509772	+	0.860310i	$0.670270\pi$
$600$	0	0
$601$	11.9744	0.488445	0.244223	−	0.969719i	$-0.421467\pi$
$601$	11.9744	0.488445	0.244223	+	0.969719i	$0.421467\pi$
$602$	0	0
$603$	−69.7641	−2.84101
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.89412	0.279824	0.139912	−	0.990164i	$-0.455318\pi$
$607$	6.89412	0.279824	0.139912	+	0.990164i	$0.455318\pi$
$608$	0	0
$609$	−2.25819	−0.0915065
$610$	0	0
$611$	30.7100	1.24239
$612$	0	0
$613$	−25.8193	−1.04283	−0.521416	−	0.853303i	$-0.674596\pi$
$613$	−25.8193	−1.04283	−0.521416	+	0.853303i	$0.674596\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.7584	0.634408	0.317204	−	0.948357i	$-0.397256\pi$
$617$	15.7584	0.634408	0.317204	+	0.948357i	$0.397256\pi$
$618$	0	0
$619$	−25.9111	−1.04145	−0.520727	−	0.853723i	$-0.674339\pi$
$619$	−25.9111	−1.04145	−0.520727	+	0.853723i	$0.674339\pi$
$620$	0	0
$621$	3.73773	0.149990
$622$	0	0
$623$	0.566768	0.0227071
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−16.2808	−0.650194
$628$	0	0
$629$	−36.0231	−1.43633
$630$	0	0
$631$	−6.86287	−0.273207	−0.136603	−	0.990626i	$-0.543619\pi$
$631$	−6.86287	−0.273207	−0.136603	+	0.990626i	$0.543619\pi$
$632$	0	0
$633$	−55.5499	−2.20791
$634$	0	0
$635$	0	0
$636$	0	0
$637$	36.5939	1.44990
$638$	0	0
$639$	12.3346	0.487947
$640$	0	0
$641$	−35.4136	−1.39875	−0.699376	−	0.714754i	$-0.746539\pi$
$641$	−35.4136	−1.39875	−0.699376	+	0.714754i	$0.746539\pi$
$642$	0	0
$643$	−27.6618	−1.09088	−0.545438	−	0.838151i	$-0.683637\pi$
$643$	−27.6618	−1.09088	−0.545438	+	0.838151i	$0.683637\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.0871	−1.14353	−0.571765	−	0.820417i	$-0.693741\pi$
$647$	−29.0871	−1.14353	−0.571765	+	0.820417i	$0.693741\pi$
$648$	0	0
$649$	−51.9794	−2.04037
$650$	0	0
$651$	−1.55903	−0.0611033
$652$	0	0
$653$	22.7798	0.891444	0.445722	−	0.895171i	$-0.352947\pi$
$653$	22.7798	0.891444	0.445722	+	0.895171i	$0.352947\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−38.0219	−1.48337
$658$	0	0
$659$	−13.5698	−0.528605	−0.264302	−	0.964440i	$-0.585142\pi$
$659$	−13.5698	−0.528605	−0.264302	+	0.964440i	$0.585142\pi$
$660$	0	0
$661$	47.4318	1.84489	0.922443	−	0.386134i	$-0.126190\pi$
$661$	47.4318	1.84489	0.922443	+	0.386134i	$0.126190\pi$
$662$	0	0
$663$	106.808	4.14809
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.40533	−0.131855
$668$	0	0
$669$	−42.4857	−1.64259
$670$	0	0
$671$	46.6972	1.80273
$672$	0	0
$673$	4.71398	0.181711	0.0908553	−	0.995864i	$-0.471040\pi$
$673$	4.71398	0.181711	0.0908553	+	0.995864i	$0.471040\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.5524	−1.63542	−0.817711	−	0.575629i	$-0.804757\pi$
$677$	−42.5524	−1.63542	−0.817711	+	0.575629i	$0.804757\pi$
$678$	0	0
$679$	−0.722747	−0.0277365
$680$	0	0
$681$	−38.2757	−1.46673
$682$	0	0
$683$	34.6186	1.32464	0.662321	−	0.749220i	$-0.269571\pi$
$683$	34.6186	1.32464	0.662321	+	0.749220i	$0.269571\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−20.9747	−0.800235
$688$	0	0
$689$	−36.5584	−1.39276
$690$	0	0
$691$	22.4131	0.852636	0.426318	−	0.904573i	$-0.359811\pi$
$691$	22.4131	0.852636	0.426318	+	0.904573i	$0.359811\pi$
$692$	0	0
$693$	−4.23576	−0.160903
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−41.3140	−1.56488
$698$	0	0
$699$	−12.0846	−0.457082
$700$	0	0
$701$	−30.3599	−1.14668	−0.573339	−	0.819318i	$-0.694352\pi$
$701$	−30.3599	−1.14668	−0.573339	+	0.819318i	$0.694352\pi$
$702$	0	0
$703$	5.03553	0.189919
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.776149	−0.0291901
$708$	0	0
$709$	−11.2281	−0.421679	−0.210839	−	0.977521i	$-0.567620\pi$
$709$	−11.2281	−0.421679	−0.210839	+	0.977521i	$0.567620\pi$
$710$	0	0
$711$	−49.5196	−1.85713
$712$	0	0
$713$	−2.35101	−0.0880459
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−43.6757	−1.63110
$718$	0	0
$719$	−27.4485	−1.02366	−0.511829	−	0.859088i	$-0.671032\pi$
$719$	−27.4485	−1.02366	−0.511829	+	0.859088i	$0.671032\pi$
$720$	0	0
$721$	0.521392	0.0194177
$722$	0	0
$723$	76.0336	2.82772
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17.2650	0.640323	0.320162	−	0.947363i	$-0.396263\pi$
$727$	17.2650	0.640323	0.320162	+	0.947363i	$0.396263\pi$
$728$	0	0
$729$	−42.2472	−1.56471
$730$	0	0
$731$	−23.7726	−0.879261
$732$	0	0
$733$	15.6789	0.579112	0.289556	−	0.957161i	$-0.406492\pi$
$733$	15.6789	0.579112	0.289556	+	0.957161i	$0.406492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	78.0946	2.87665
$738$	0	0
$739$	47.7761	1.75747	0.878737	−	0.477307i	$-0.158387\pi$
$739$	47.7761	1.75747	0.878737	+	0.477307i	$0.158387\pi$
$740$	0	0
$741$	−14.9303	−0.548479
$742$	0	0
$743$	−37.5789	−1.37864	−0.689318	−	0.724459i	$-0.742090\pi$
$743$	−37.5789	−1.37864	−0.689318	+	0.724459i	$0.742090\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.0899	0.405758
$748$	0	0
$749$	2.22654	0.0813561
$750$	0	0
$751$	50.4411	1.84062	0.920311	−	0.391188i	$-0.127936\pi$
$751$	50.4411	1.84062	0.920311	+	0.391188i	$0.127936\pi$
$752$	0	0
$753$	10.3778	0.378188
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10.8756	−0.395280	−0.197640	−	0.980275i	$-0.563328\pi$
$757$	−10.8756	−0.395280	−0.197640	+	0.980275i	$0.563328\pi$
$758$	0	0
$759$	−10.1391	−0.368026
$760$	0	0
$761$	10.7675	0.390322	0.195161	−	0.980771i	$-0.437477\pi$
$761$	10.7675	0.390322	0.195161	+	0.980771i	$0.437477\pi$
$762$	0	0
$763$	0.319229	0.0115569
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−47.6677	−1.72118
$768$	0	0
$769$	−1.76320	−0.0635827	−0.0317914	−	0.999495i	$-0.510121\pi$
$769$	−1.76320	−0.0635827	−0.0317914	+	0.999495i	$0.510121\pi$
$770$	0	0
$771$	−27.6514	−0.995841
$772$	0	0
$773$	−26.5661	−0.955517	−0.477759	−	0.878491i	$-0.658551\pi$
$773$	−26.5661	−0.955517	−0.477759	+	0.878491i	$0.658551\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.07954	0.0746032
$778$	0	0
$779$	5.77513	0.206915
$780$	0	0
$781$	−13.8074	−0.494068
$782$	0	0
$783$	−32.8189	−1.17285
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−34.4557	−1.22821	−0.614106	−	0.789223i	$-0.710483\pi$
$787$	−34.4557	−1.22821	−0.614106	+	0.789223i	$0.710483\pi$
$788$	0	0
$789$	5.18379	0.184548
$790$	0	0
$791$	1.79282	0.0637453
$792$	0	0
$793$	42.8236	1.52071
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.5121	0.407779	0.203890	−	0.978994i	$-0.434642\pi$
$797$	11.5121	0.407779	0.203890	+	0.978994i	$0.434642\pi$
$798$	0	0
$799$	41.8985	1.48226
$800$	0	0
$801$	19.9604	0.705267
$802$	0	0
$803$	42.5621	1.50198
$804$	0	0
$805$	0	0
$806$	0	0
$807$	61.6048	2.16859
$808$	0	0
$809$	35.7367	1.25643	0.628217	−	0.778038i	$-0.283785\pi$
$809$	35.7367	1.25643	0.628217	+	0.778038i	$0.283785\pi$
$810$	0	0
$811$	−47.4425	−1.66593	−0.832966	−	0.553324i	$-0.813359\pi$
$811$	−47.4425	−1.66593	−0.832966	+	0.553324i	$0.813359\pi$
$812$	0	0
$813$	−21.4820	−0.753406
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.32308	0.116260
$818$	0	0
$819$	−3.88440	−0.135732
$820$	0	0
$821$	−5.70237	−0.199014	−0.0995070	−	0.995037i	$-0.531727\pi$
$821$	−5.70237	−0.199014	−0.0995070	+	0.995037i	$0.531727\pi$
$822$	0	0
$823$	3.37125	0.117514	0.0587572	−	0.998272i	$-0.481286\pi$
$823$	3.37125	0.117514	0.0587572	+	0.998272i	$0.481286\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.22704	0.112215	0.0561075	−	0.998425i	$-0.482131\pi$
$827$	3.22704	0.112215	0.0561075	+	0.998425i	$0.482131\pi$
$828$	0	0
$829$	10.5746	0.367273	0.183636	−	0.982994i	$-0.441213\pi$
$829$	10.5746	0.367273	0.183636	+	0.982994i	$0.441213\pi$
$830$	0	0
$831$	35.7519	1.24022
$832$	0	0
$833$	49.9260	1.72983
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−22.6578	−0.783168
$838$	0	0
$839$	51.5214	1.77872	0.889359	−	0.457210i	$-0.151151\pi$
$839$	51.5214	1.77872	0.889359	+	0.457210i	$0.151151\pi$
$840$	0	0
$841$	0.900274	0.0310439
$842$	0	0
$843$	63.1977	2.17665
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.14617	0.108104
$848$	0	0
$849$	−58.7055	−2.01477
$850$	0	0
$851$	3.13593	0.107498
$852$	0	0
$853$	14.3074	0.489874	0.244937	−	0.969539i	$-0.421233\pi$
$853$	14.3074	0.489874	0.244937	+	0.969539i	$0.421233\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.55632	−0.121482	−0.0607408	−	0.998154i	$-0.519346\pi$
$857$	−3.55632	−0.121482	−0.0607408	+	0.998154i	$0.519346\pi$
$858$	0	0
$859$	13.8084	0.471135	0.235568	−	0.971858i	$-0.424305\pi$
$859$	13.8084	0.471135	0.235568	+	0.971858i	$0.424305\pi$
$860$	0	0
$861$	2.38498	0.0812799
$862$	0	0
$863$	−11.3476	−0.386277	−0.193139	−	0.981171i	$-0.561867\pi$
$863$	−11.3476	−0.386277	−0.193139	+	0.981171i	$0.561867\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	97.3154	3.30500
$868$	0	0
$869$	55.4327	1.88042
$870$	0	0
$871$	71.6166	2.42664
$872$	0	0
$873$	−25.4537	−0.861478
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.6155	1.50656	0.753279	−	0.657701i	$-0.228471\pi$
$877$	44.6155	1.50656	0.753279	+	0.657701i	$0.228471\pi$
$878$	0	0
$879$	−63.0479	−2.12655
$880$	0	0
$881$	34.3973	1.15887	0.579437	−	0.815017i	$-0.303272\pi$
$881$	34.3973	1.15887	0.579437	+	0.815017i	$0.303272\pi$
$882$	0	0
$883$	−54.2057	−1.82417	−0.912084	−	0.410004i	$-0.865527\pi$
$883$	−54.2057	−1.82417	−0.912084	+	0.410004i	$0.865527\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.69998	0.0570798	0.0285399	−	0.999593i	$-0.490914\pi$
$887$	1.69998	0.0570798	0.0285399	+	0.999593i	$0.490914\pi$
$888$	0	0
$889$	−2.62377	−0.0879984
$890$	0	0
$891$	−10.0997	−0.338353
$892$	0	0
$893$	−5.85683	−0.195991
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−9.29804	−0.310453
$898$	0	0
$899$	20.6428	0.688477
$900$	0	0
$901$	−49.8776	−1.66166
$902$	0	0
$903$	1.37235	0.0456689
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−13.1338	−0.436101	−0.218050	−	0.975938i	$-0.569970\pi$
$907$	−13.1338	−0.436101	−0.218050	+	0.975938i	$0.569970\pi$
$908$	0	0
$909$	−27.3344	−0.906626
$910$	0	0
$911$	24.7511	0.820040	0.410020	−	0.912076i	$-0.365522\pi$
$911$	24.7511	0.820040	0.410020	+	0.912076i	$0.365522\pi$
$912$	0	0
$913$	−12.4141	−0.410848
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.75845	0.0580692
$918$	0	0
$919$	−14.2789	−0.471019	−0.235510	−	0.971872i	$-0.575676\pi$
$919$	−14.2789	−0.471019	−0.235510	+	0.971872i	$0.575676\pi$
$920$	0	0
$921$	5.12707	0.168943
$922$	0	0
$923$	−12.6621	−0.416777
$924$	0	0
$925$	0	0
$926$	0	0
$927$	18.3624	0.603100
$928$	0	0
$929$	18.6484	0.611834	0.305917	−	0.952058i	$-0.401037\pi$
$929$	18.6484	0.611834	0.305917	+	0.952058i	$0.401037\pi$
$930$	0	0
$931$	−6.97897	−0.228726
$932$	0	0
$933$	−26.3829	−0.863738
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.9120	0.585159	0.292580	−	0.956241i	$-0.405486\pi$
$937$	17.9120	0.585159	0.292580	+	0.956241i	$0.405486\pi$
$938$	0	0
$939$	12.2376	0.399360
$940$	0	0
$941$	10.7412	0.350152	0.175076	−	0.984555i	$-0.443983\pi$
$941$	10.7412	0.350152	0.175076	+	0.984555i	$0.443983\pi$
$942$	0	0
$943$	3.59653	0.117119
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−58.5220	−1.90171	−0.950855	−	0.309636i	$-0.899793\pi$
$947$	−58.5220	−1.90171	−0.950855	+	0.309636i	$0.899793\pi$
$948$	0	0
$949$	39.0315	1.26702
$950$	0	0
$951$	13.1285	0.425720
$952$	0	0
$953$	−2.93656	−0.0951244	−0.0475622	−	0.998868i	$-0.515145\pi$
$953$	−2.93656	−0.0951244	−0.0475622	+	0.998868i	$0.515145\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	89.0255	2.87779
$958$	0	0
$959$	2.90628	0.0938486
$960$	0	0
$961$	−16.7484	−0.540271
$962$	0	0
$963$	78.4144	2.52687
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−20.5192	−0.659852	−0.329926	−	0.944007i	$-0.607024\pi$
$967$	−20.5192	−0.659852	−0.329926	+	0.944007i	$0.607024\pi$
$968$	0	0
$969$	−20.3699	−0.654374
$970$	0	0
$971$	−2.37837	−0.0763255	−0.0381628	−	0.999272i	$-0.512151\pi$
$971$	−2.37837	−0.0763255	−0.0381628	+	0.999272i	$0.512151\pi$
$972$	0	0
$973$	1.65334	0.0530037
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.2822	1.48070	0.740349	−	0.672222i	$-0.234660\pi$
$977$	46.2822	1.48070	0.740349	+	0.672222i	$0.234660\pi$
$978$	0	0
$979$	−22.3439	−0.714114
$980$	0	0
$981$	11.2426	0.358948
$982$	0	0
$983$	49.2913	1.57215	0.786074	−	0.618133i	$-0.212111\pi$
$983$	49.2913	1.57215	0.786074	+	0.618133i	$0.212111\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.41872	−0.0769887
$988$	0	0
$989$	2.06949	0.0658059
$990$	0	0
$991$	6.92186	0.219880	0.109940	−	0.993938i	$-0.464934\pi$
$991$	6.92186	0.219880	0.109940	+	0.993938i	$0.464934\pi$
$992$	0	0
$993$	−2.66054	−0.0844296
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.1533	0.891624	0.445812	−	0.895127i	$-0.352915\pi$
$997$	28.1533	0.891624	0.445812	+	0.895127i	$0.352915\pi$
$998$	0	0
$999$	30.2225	0.956199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.z.1.6		6
4.3	odd	2		7600.2.a.cn.1.1		6
5.2	odd	4		760.2.d.e.609.2	✓	12
5.3	odd	4		760.2.d.e.609.11	yes	12
5.4	even	2		3800.2.a.be.1.1		6
20.3	even	4		1520.2.d.k.609.2		12
20.7	even	4		1520.2.d.k.609.11		12
20.19	odd	2		7600.2.a.cg.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.e.609.2	✓	12	5.2	odd	4
760.2.d.e.609.11	yes	12	5.3	odd	4
1520.2.d.k.609.2		12	20.3	even	4
1520.2.d.k.609.11		12	20.7	even	4
3800.2.a.z.1.6		6	1.1	even	1	trivial
3800.2.a.be.1.1		6	5.4	even	2
7600.2.a.cg.1.6		6	20.19	odd	2
7600.2.a.cn.1.1		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.84742 q^{3} +0.145034 q^{7} +5.10782 q^{9} +O(q^{10})\)	\(q+2.84742 q^{3} +0.145034 q^{7} +5.10782 q^{9} -5.71774 q^{11} -5.24345 q^{13} -7.15378 q^{17} +1.00000 q^{19} +0.412974 q^{21} +0.622762 q^{23} +6.00186 q^{27} -5.46811 q^{29} -3.77513 q^{31} -16.2808 q^{33} +5.03553 q^{37} -14.9303 q^{39} +5.77513 q^{41} +3.32308 q^{43} -5.85683 q^{47} -6.97897 q^{49} -20.3699 q^{51} +6.97219 q^{53} +2.84742 q^{57} +9.09089 q^{59} -8.16707 q^{61} +0.740810 q^{63} -13.6583 q^{67} +1.77327 q^{69} +2.41484 q^{71} -7.44385 q^{73} -0.829270 q^{77} -9.69485 q^{79} +1.76638 q^{81} +2.17116 q^{83} -15.5700 q^{87} +3.90782 q^{89} -0.760481 q^{91} -10.7494 q^{93} -4.98328 q^{97} -29.2052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9	\( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} - 34 q^{51} + 14 q^{53} - 2 q^{57} - 2 q^{59} + 10 q^{61} - 44 q^{63} - 42 q^{67} + 18 q^{69} - 8 q^{71} + 2 q^{73} - 24 q^{77} - 20 q^{79} + 6 q^{81} - 16 q^{83} - 30 q^{87} + 32 q^{89} + 10 q^{91} - 12 q^{93} - 40 q^{97} - 22 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9 - 2 * q^11 - 14 * q^13 - 10 * q^17 + 6 * q^19 + 18 * q^21 - 2 * q^23 - 2 * q^27 - 2 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^37 - 18 * q^39 + 4 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 34 * q^51 + 14 * q^53 - 2 * q^57 - 2 * q^59 + 10 * q^61 - 44 * q^63 - 42 * q^67 + 18 * q^69 - 8 * q^71 + 2 * q^73 - 24 * q^77 - 20 * q^79 + 6 * q^81 - 16 * q^83 - 30 * q^87 + 32 * q^89 + 10 * q^91 - 12 * q^93 - 40 * q^97 - 22 * q^99

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