LMFDB - Embedded newform 3800.2.a.bb.1.5

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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 $ x^6 - 2x^5 - 12x^4 + 16x^3 + 33x^2 - 4*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.22174$ 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)

$f(q)$	$=$	$q+1.22174 q^{3} +3.08602 q^{7} -1.50735 q^{9} -3.83920 q^{11} +5.85633 q^{13} +6.54564 q^{17} +1.00000 q^{19} +3.77031 q^{21} +4.37163 q^{23} -5.50681 q^{27} -4.41841 q^{29} +0.451431 q^{31} -4.69051 q^{33} +2.49067 q^{37} +7.15491 q^{39} -1.03924 q^{41} +3.19209 q^{43} -3.70641 q^{47} +2.52349 q^{49} +7.99707 q^{51} +5.19667 q^{53} +1.22174 q^{57} -7.18818 q^{59} +5.02265 q^{61} -4.65171 q^{63} -13.1631 q^{67} +5.34099 q^{69} +12.2993 q^{71} -2.49886 q^{73} -11.8478 q^{77} +11.0313 q^{79} -2.20584 q^{81} +0.245834 q^{83} -5.39814 q^{87} -13.4321 q^{89} +18.0727 q^{91} +0.551531 q^{93} +11.0992 q^{97} +5.78702 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9 + 3 * q^11 + 3 * q^13 + 2 * q^17 + 6 * q^19 + 11 * q^21 - 4 * q^23 - 20 * q^27 + 7 * q^29 + 5 * q^31 + 16 * q^33 + 8 * q^39 + 11 * q^41 + 7 * q^43 - 20 * q^47 - 2 * q^49 + 13 * q^51 + 7 * q^53 - 2 * q^57 - 4 * q^59 + 13 * q^61 + q^63 - 25 * q^67 + 7 * q^69 + 29 * q^71 + 19 * q^73 - 24 * q^77 + 28 * q^79 + 38 * q^81 + 15 * q^83 - 57 * q^87 - 12 * q^89 + 27 * q^93 + 13 * q^97 + 10 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.22174	0.705372	0.352686	−	0.935742i	$-0.385268\pi$
$3$	1.22174	0.705372	0.352686	+	0.935742i	$0.385268\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.08602	1.16640	0.583202	−	0.812327i	$-0.301800\pi$
$7$	3.08602	1.16640	0.583202	+	0.812327i	$0.301800\pi$
$8$	0	0
$9$	−1.50735	−0.502450
$10$	0	0
$11$	−3.83920	−1.15756	−0.578781	−	0.815483i	$-0.696472\pi$
$11$	−3.83920	−1.15756	−0.578781	+	0.815483i	$0.696472\pi$
$12$	0	0
$13$	5.85633	1.62425	0.812126	−	0.583482i	$-0.198310\pi$
$13$	5.85633	1.62425	0.812126	+	0.583482i	$0.198310\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.54564	1.58755	0.793775	−	0.608211i	$-0.208113\pi$
$17$	6.54564	1.58755	0.793775	+	0.608211i	$0.208113\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.77031	0.822749
$22$	0	0
$23$	4.37163	0.911547	0.455774	−	0.890096i	$-0.349363\pi$
$23$	4.37163	0.911547	0.455774	+	0.890096i	$0.349363\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.50681	−1.05979
$28$	0	0
$29$	−4.41841	−0.820477	−0.410239	−	0.911978i	$-0.634555\pi$
$29$	−4.41841	−0.820477	−0.410239	+	0.911978i	$0.634555\pi$
$30$	0	0
$31$	0.451431	0.0810793	0.0405397	−	0.999178i	$-0.487092\pi$
$31$	0.451431	0.0810793	0.0405397	+	0.999178i	$0.487092\pi$
$32$	0	0
$33$	−4.69051	−0.816512
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.49067	0.409463	0.204732	−	0.978818i	$-0.434368\pi$
$37$	2.49067	0.409463	0.204732	+	0.978818i	$0.434368\pi$
$38$	0	0
$39$	7.15491	1.14570
$40$	0	0
$41$	−1.03924	−0.162302	−0.0811508	−	0.996702i	$-0.525860\pi$
$41$	−1.03924	−0.162302	−0.0811508	+	0.996702i	$0.525860\pi$
$42$	0	0
$43$	3.19209	0.486789	0.243394	−	0.969927i	$-0.421739\pi$
$43$	3.19209	0.486789	0.243394	+	0.969927i	$0.421739\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.70641	−0.540635	−0.270317	−	0.962771i	$-0.587129\pi$
$47$	−3.70641	−0.540635	−0.270317	+	0.962771i	$0.587129\pi$
$48$	0	0
$49$	2.52349	0.360499
$50$	0	0
$51$	7.99707	1.11981
$52$	0	0
$53$	5.19667	0.713817	0.356908	−	0.934139i	$-0.383831\pi$
$53$	5.19667	0.713817	0.356908	+	0.934139i	$0.383831\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.22174	0.161823
$58$	0	0
$59$	−7.18818	−0.935821	−0.467910	−	0.883776i	$-0.654993\pi$
$59$	−7.18818	−0.935821	−0.467910	+	0.883776i	$0.654993\pi$
$60$	0	0
$61$	5.02265	0.643085	0.321542	−	0.946895i	$-0.395799\pi$
$61$	5.02265	0.643085	0.321542	+	0.946895i	$0.395799\pi$
$62$	0	0
$63$	−4.65171	−0.586060
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.1631	−1.60813	−0.804064	−	0.594542i	$-0.797333\pi$
$67$	−13.1631	−1.60813	−0.804064	+	0.594542i	$0.797333\pi$
$68$	0	0
$69$	5.34099	0.642980
$70$	0	0
$71$	12.2993	1.45965	0.729827	−	0.683632i	$-0.239601\pi$
$71$	12.2993	1.45965	0.729827	+	0.683632i	$0.239601\pi$
$72$	0	0
$73$	−2.49886	−0.292470	−0.146235	−	0.989250i	$-0.546715\pi$
$73$	−2.49886	−0.292470	−0.146235	+	0.989250i	$0.546715\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−11.8478	−1.35019
$78$	0	0
$79$	11.0313	1.24112	0.620558	−	0.784160i	$-0.286906\pi$
$79$	11.0313	1.24112	0.620558	+	0.784160i	$0.286906\pi$
$80$	0	0
$81$	−2.20584	−0.245093
$82$	0	0
$83$	0.245834	0.0269838	0.0134919	−	0.999909i	$-0.495705\pi$
$83$	0.245834	0.0269838	0.0134919	+	0.999909i	$0.495705\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.39814	−0.578742
$88$	0	0
$89$	−13.4321	−1.42380	−0.711898	−	0.702283i	$-0.752164\pi$
$89$	−13.4321	−1.42380	−0.711898	+	0.702283i	$0.752164\pi$
$90$	0	0
$91$	18.0727	1.89454
$92$	0	0
$93$	0.551531	0.0571911
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.0992	1.12696	0.563478	−	0.826131i	$-0.309463\pi$
$97$	11.0992	1.12696	0.563478	+	0.826131i	$0.309463\pi$
$98$	0	0
$99$	5.78702	0.581618
$100$	0	0
$101$	16.0460	1.59664	0.798318	−	0.602236i	$-0.205724\pi$
$101$	16.0460	1.59664	0.798318	+	0.602236i	$0.205724\pi$
$102$	0	0
$103$	15.4569	1.52301	0.761507	−	0.648157i	$-0.224460\pi$
$103$	15.4569	1.52301	0.761507	+	0.648157i	$0.224460\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.23615	0.892892	0.446446	−	0.894811i	$-0.352690\pi$
$107$	9.23615	0.892892	0.446446	+	0.894811i	$0.352690\pi$
$108$	0	0
$109$	−9.87147	−0.945515	−0.472758	−	0.881192i	$-0.656741\pi$
$109$	−9.87147	−0.945515	−0.472758	+	0.881192i	$0.656741\pi$
$110$	0	0
$111$	3.04295	0.288824
$112$	0	0
$113$	18.4191	1.73272	0.866360	−	0.499421i	$-0.166454\pi$
$113$	18.4191	1.73272	0.866360	+	0.499421i	$0.166454\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.82754	−0.816106
$118$	0	0
$119$	20.1999	1.85173
$120$	0	0
$121$	3.73946	0.339951
$122$	0	0
$123$	−1.26968	−0.114483
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.1753	1.25786	0.628928	−	0.777464i	$-0.283494\pi$
$127$	14.1753	1.25786	0.628928	+	0.777464i	$0.283494\pi$
$128$	0	0
$129$	3.89990	0.343367
$130$	0	0
$131$	1.40424	0.122689	0.0613446	−	0.998117i	$-0.480461\pi$
$131$	1.40424	0.122689	0.0613446	+	0.998117i	$0.480461\pi$
$132$	0	0
$133$	3.08602	0.267592
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.4212	−1.14665	−0.573327	−	0.819326i	$-0.694348\pi$
$137$	−13.4212	−1.14665	−0.573327	+	0.819326i	$0.694348\pi$
$138$	0	0
$139$	3.42929	0.290868	0.145434	−	0.989368i	$-0.453542\pi$
$139$	3.42929	0.290868	0.145434	+	0.989368i	$0.453542\pi$
$140$	0	0
$141$	−4.52827	−0.381349
$142$	0	0
$143$	−22.4836	−1.88017
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.08305	0.254286
$148$	0	0
$149$	−20.3518	−1.66729	−0.833643	−	0.552304i	$-0.813749\pi$
$149$	−20.3518	−1.66729	−0.833643	+	0.552304i	$0.813749\pi$
$150$	0	0
$151$	−2.34058	−0.190474	−0.0952369	−	0.995455i	$-0.530361\pi$
$151$	−2.34058	−0.190474	−0.0952369	+	0.995455i	$0.530361\pi$
$152$	0	0
$153$	−9.86658	−0.797665
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.5097	1.39742	0.698711	−	0.715404i	$-0.253757\pi$
$157$	17.5097	1.39742	0.698711	+	0.715404i	$0.253757\pi$
$158$	0	0
$159$	6.34897	0.503506
$160$	0	0
$161$	13.4909	1.06323
$162$	0	0
$163$	−6.65575	−0.521319	−0.260659	−	0.965431i	$-0.583940\pi$
$163$	−6.65575	−0.521319	−0.260659	+	0.965431i	$0.583940\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.32539	−0.644238	−0.322119	−	0.946699i	$-0.604395\pi$
$167$	−8.32539	−0.644238	−0.322119	+	0.946699i	$0.604395\pi$
$168$	0	0
$169$	21.2965	1.63820
$170$	0	0
$171$	−1.50735	−0.115270
$172$	0	0
$173$	20.0259	1.52254	0.761272	−	0.648433i	$-0.224575\pi$
$173$	20.0259	1.52254	0.761272	+	0.648433i	$0.224575\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.78208	−0.660102
$178$	0	0
$179$	−12.1689	−0.909544	−0.454772	−	0.890608i	$-0.650279\pi$
$179$	−12.1689	−0.909544	−0.454772	+	0.890608i	$0.650279\pi$
$180$	0	0
$181$	3.76508	0.279856	0.139928	−	0.990162i	$-0.455313\pi$
$181$	3.76508	0.279856	0.139928	+	0.990162i	$0.455313\pi$
$182$	0	0
$183$	6.13638	0.453614
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−25.1300	−1.83769
$188$	0	0
$189$	−16.9941	−1.23614
$190$	0	0
$191$	−13.0616	−0.945106	−0.472553	−	0.881302i	$-0.656667\pi$
$191$	−13.0616	−0.945106	−0.472553	+	0.881302i	$0.656667\pi$
$192$	0	0
$193$	9.44847	0.680116	0.340058	−	0.940405i	$-0.389553\pi$
$193$	9.44847	0.680116	0.340058	+	0.940405i	$0.389553\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.3270	−1.02075	−0.510377	−	0.859951i	$-0.670494\pi$
$197$	−14.3270	−1.02075	−0.510377	+	0.859951i	$0.670494\pi$
$198$	0	0
$199$	−16.1608	−1.14561	−0.572805	−	0.819692i	$-0.694145\pi$
$199$	−16.1608	−1.14561	−0.572805	+	0.819692i	$0.694145\pi$
$200$	0	0
$201$	−16.0819	−1.13433
$202$	0	0
$203$	−13.6353	−0.957008
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.58958	−0.458007
$208$	0	0
$209$	−3.83920	−0.265563
$210$	0	0
$211$	4.61746	0.317879	0.158940	−	0.987288i	$-0.449193\pi$
$211$	4.61746	0.317879	0.158940	+	0.987288i	$0.449193\pi$
$212$	0	0
$213$	15.0265	1.02960
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.39312	0.0945713
$218$	0	0
$219$	−3.05296	−0.206300
$220$	0	0
$221$	38.3334	2.57858
$222$	0	0
$223$	−20.2785	−1.35795	−0.678975	−	0.734161i	$-0.737576\pi$
$223$	−20.2785	−1.35795	−0.678975	+	0.734161i	$0.737576\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.5708	−0.767982	−0.383991	−	0.923337i	$-0.625451\pi$
$227$	−11.5708	−0.767982	−0.383991	+	0.923337i	$0.625451\pi$
$228$	0	0
$229$	14.0792	0.930378	0.465189	−	0.885211i	$-0.345986\pi$
$229$	14.0792	0.930378	0.465189	+	0.885211i	$0.345986\pi$
$230$	0	0
$231$	−14.4750	−0.952383
$232$	0	0
$233$	3.95281	0.258957	0.129479	−	0.991582i	$-0.458670\pi$
$233$	3.95281	0.258957	0.129479	+	0.991582i	$0.458670\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.4774	0.875449
$238$	0	0
$239$	18.0661	1.16860	0.584300	−	0.811538i	$-0.301369\pi$
$239$	18.0661	1.16860	0.584300	+	0.811538i	$0.301369\pi$
$240$	0	0
$241$	23.0999	1.48800	0.743998	−	0.668182i	$-0.232927\pi$
$241$	23.0999	1.48800	0.743998	+	0.668182i	$0.232927\pi$
$242$	0	0
$243$	13.8255	0.886904
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.85633	0.372629
$248$	0	0
$249$	0.300345	0.0190336
$250$	0	0
$251$	−13.0987	−0.826784	−0.413392	−	0.910553i	$-0.635656\pi$
$251$	−13.0987	−0.826784	−0.413392	+	0.910553i	$0.635656\pi$
$252$	0	0
$253$	−16.7836	−1.05517
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.5191	−0.843296	−0.421648	−	0.906760i	$-0.638548\pi$
$257$	−13.5191	−0.843296	−0.421648	+	0.906760i	$0.638548\pi$
$258$	0	0
$259$	7.68624	0.477600
$260$	0	0
$261$	6.66009	0.412249
$262$	0	0
$263$	−0.357938	−0.0220714	−0.0110357	−	0.999939i	$-0.503513\pi$
$263$	−0.357938	−0.0220714	−0.0110357	+	0.999939i	$0.503513\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−16.4105	−1.00431
$268$	0	0
$269$	15.9556	0.972829	0.486415	−	0.873728i	$-0.338304\pi$
$269$	15.9556	0.972829	0.486415	+	0.873728i	$0.338304\pi$
$270$	0	0
$271$	0.795153	0.0483021	0.0241511	−	0.999708i	$-0.492312\pi$
$271$	0.795153	0.0483021	0.0241511	+	0.999708i	$0.492312\pi$
$272$	0	0
$273$	22.0802	1.33635
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−12.9640	−0.778934	−0.389467	−	0.921040i	$-0.627341\pi$
$277$	−12.9640	−0.778934	−0.389467	+	0.921040i	$0.627341\pi$
$278$	0	0
$279$	−0.680465	−0.0407383
$280$	0	0
$281$	−26.2769	−1.56755	−0.783775	−	0.621045i	$-0.786709\pi$
$281$	−26.2769	−1.56755	−0.783775	+	0.621045i	$0.786709\pi$
$282$	0	0
$283$	27.4599	1.63232	0.816160	−	0.577826i	$-0.196099\pi$
$283$	27.4599	1.63232	0.816160	+	0.577826i	$0.196099\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.20710	−0.189309
$288$	0	0
$289$	25.8454	1.52032
$290$	0	0
$291$	13.5604	0.794923
$292$	0	0
$293$	−19.7572	−1.15423	−0.577114	−	0.816663i	$-0.695821\pi$
$293$	−19.7572	−1.15423	−0.577114	+	0.816663i	$0.695821\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	21.1418	1.22677
$298$	0	0
$299$	25.6017	1.48058
$300$	0	0
$301$	9.85083	0.567792
$302$	0	0
$303$	19.6040	1.12622
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−28.9367	−1.65151	−0.825753	−	0.564032i	$-0.809250\pi$
$307$	−28.9367	−1.65151	−0.825753	+	0.564032i	$0.809250\pi$
$308$	0	0
$309$	18.8843	1.07429
$310$	0	0
$311$	−28.0895	−1.59281	−0.796404	−	0.604764i	$-0.793267\pi$
$311$	−28.0895	−1.59281	−0.796404	+	0.604764i	$0.793267\pi$
$312$	0	0
$313$	29.7936	1.68403	0.842016	−	0.539453i	$-0.181369\pi$
$313$	29.7936	1.68403	0.842016	+	0.539453i	$0.181369\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.13494	−0.344572	−0.172286	−	0.985047i	$-0.555115\pi$
$317$	−6.13494	−0.344572	−0.172286	+	0.985047i	$0.555115\pi$
$318$	0	0
$319$	16.9631	0.949754
$320$	0	0
$321$	11.2842	0.629821
$322$	0	0
$323$	6.54564	0.364209
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.0604	−0.666940
$328$	0	0
$329$	−11.4380	−0.630599
$330$	0	0
$331$	−17.1019	−0.940003	−0.470002	−	0.882666i	$-0.655747\pi$
$331$	−17.1019	−0.940003	−0.470002	+	0.882666i	$0.655747\pi$
$332$	0	0
$333$	−3.75431	−0.205735
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.0521	0.928885	0.464442	−	0.885603i	$-0.346255\pi$
$337$	17.0521	0.928885	0.464442	+	0.885603i	$0.346255\pi$
$338$	0	0
$339$	22.5033	1.22221
$340$	0	0
$341$	−1.73313	−0.0938544
$342$	0	0
$343$	−13.8146	−0.745917
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.1351	−1.40300	−0.701502	−	0.712667i	$-0.747487\pi$
$347$	−26.1351	−1.40300	−0.701502	+	0.712667i	$0.747487\pi$
$348$	0	0
$349$	−0.577782	−0.0309279	−0.0154640	−	0.999880i	$-0.504923\pi$
$349$	−0.577782	−0.0309279	−0.0154640	+	0.999880i	$0.504923\pi$
$350$	0	0
$351$	−32.2497	−1.72136
$352$	0	0
$353$	30.0746	1.60071	0.800355	−	0.599527i	$-0.204645\pi$
$353$	30.0746	1.60071	0.800355	+	0.599527i	$0.204645\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.6791	1.30616
$358$	0	0
$359$	−25.6793	−1.35530	−0.677651	−	0.735383i	$-0.737002\pi$
$359$	−25.6793	−1.35530	−0.677651	+	0.735383i	$0.737002\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	4.56865	0.239792
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.57520	0.395422	0.197711	−	0.980260i	$-0.436649\pi$
$367$	7.57520	0.395422	0.197711	+	0.980260i	$0.436649\pi$
$368$	0	0
$369$	1.56650	0.0815485
$370$	0	0
$371$	16.0370	0.832599
$372$	0	0
$373$	−26.2562	−1.35949	−0.679747	−	0.733447i	$-0.737910\pi$
$373$	−26.2562	−1.35949	−0.679747	+	0.733447i	$0.737910\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−25.8756	−1.33266
$378$	0	0
$379$	−15.6277	−0.802742	−0.401371	−	0.915915i	$-0.631466\pi$
$379$	−15.6277	−0.802742	−0.401371	+	0.915915i	$0.631466\pi$
$380$	0	0
$381$	17.3185	0.887256
$382$	0	0
$383$	−10.7189	−0.547711	−0.273856	−	0.961771i	$-0.588299\pi$
$383$	−10.7189	−0.547711	−0.273856	+	0.961771i	$0.588299\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.81159	−0.244587
$388$	0	0
$389$	7.83988	0.397498	0.198749	−	0.980050i	$-0.436312\pi$
$389$	7.83988	0.397498	0.198749	+	0.980050i	$0.436312\pi$
$390$	0	0
$391$	28.6151	1.44713
$392$	0	0
$393$	1.71562	0.0865416
$394$	0	0
$395$	0	0
$396$	0	0
$397$	37.9045	1.90237	0.951187	−	0.308614i	$-0.0998651\pi$
$397$	37.9045	1.90237	0.951187	+	0.308614i	$0.0998651\pi$
$398$	0	0
$399$	3.77031	0.188752
$400$	0	0
$401$	18.3901	0.918355	0.459178	−	0.888344i	$-0.348144\pi$
$401$	18.3901	0.918355	0.459178	+	0.888344i	$0.348144\pi$
$402$	0	0
$403$	2.64372	0.131693
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.56218	−0.473979
$408$	0	0
$409$	−23.2744	−1.15085	−0.575423	−	0.817856i	$-0.695163\pi$
$409$	−23.2744	−1.15085	−0.575423	+	0.817856i	$0.695163\pi$
$410$	0	0
$411$	−16.3973	−0.808818
$412$	0	0
$413$	−22.1828	−1.09155
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.18970	0.205170
$418$	0	0
$419$	27.5055	1.34373	0.671866	−	0.740673i	$-0.265493\pi$
$419$	27.5055	1.34373	0.671866	+	0.740673i	$0.265493\pi$
$420$	0	0
$421$	−9.70130	−0.472812	−0.236406	−	0.971654i	$-0.575970\pi$
$421$	−9.70130	−0.472812	−0.236406	+	0.971654i	$0.575970\pi$
$422$	0	0
$423$	5.58686	0.271642
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.5000	0.750097
$428$	0	0
$429$	−27.4691	−1.32622
$430$	0	0
$431$	15.6303	0.752886	0.376443	−	0.926440i	$-0.377147\pi$
$431$	15.6303	0.752886	0.376443	+	0.926440i	$0.377147\pi$
$432$	0	0
$433$	−17.3839	−0.835416	−0.417708	−	0.908581i	$-0.637166\pi$
$433$	−17.3839	−0.835416	−0.417708	+	0.908581i	$0.637166\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.37163	0.209123
$438$	0	0
$439$	−25.3238	−1.20864	−0.604320	−	0.796742i	$-0.706555\pi$
$439$	−25.3238	−1.20864	−0.604320	+	0.796742i	$0.706555\pi$
$440$	0	0
$441$	−3.80379	−0.181133
$442$	0	0
$443$	−6.49504	−0.308589	−0.154294	−	0.988025i	$-0.549310\pi$
$443$	−6.49504	−0.308589	−0.154294	+	0.988025i	$0.549310\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−24.8646	−1.17606
$448$	0	0
$449$	−3.09007	−0.145829	−0.0729147	−	0.997338i	$-0.523230\pi$
$449$	−3.09007	−0.145829	−0.0729147	+	0.997338i	$0.523230\pi$
$450$	0	0
$451$	3.98984	0.187874
$452$	0	0
$453$	−2.85958	−0.134355
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.7215	1.81131	0.905657	−	0.424012i	$-0.139378\pi$
$457$	38.7215	1.81131	0.905657	+	0.424012i	$0.139378\pi$
$458$	0	0
$459$	−36.0456	−1.68246
$460$	0	0
$461$	−21.9817	−1.02379	−0.511895	−	0.859048i	$-0.671056\pi$
$461$	−21.9817	−1.02379	−0.511895	+	0.859048i	$0.671056\pi$
$462$	0	0
$463$	−42.4385	−1.97228	−0.986142	−	0.165901i	$-0.946947\pi$
$463$	−42.4385	−1.97228	−0.986142	+	0.165901i	$0.946947\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.4303	0.667754	0.333877	−	0.942617i	$-0.391643\pi$
$467$	14.4303	0.667754	0.333877	+	0.942617i	$0.391643\pi$
$468$	0	0
$469$	−40.6215	−1.87573
$470$	0	0
$471$	21.3922	0.985703
$472$	0	0
$473$	−12.2551	−0.563488
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.83320	−0.358658
$478$	0	0
$479$	−13.0612	−0.596781	−0.298390	−	0.954444i	$-0.596450\pi$
$479$	−13.0612	−0.596781	−0.298390	+	0.954444i	$0.596450\pi$
$480$	0	0
$481$	14.5862	0.665072
$482$	0	0
$483$	16.4824	0.749975
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.65696	0.392284	0.196142	−	0.980575i	$-0.437159\pi$
$487$	8.65696	0.392284	0.196142	+	0.980575i	$0.437159\pi$
$488$	0	0
$489$	−8.13160	−0.367723
$490$	0	0
$491$	−8.37358	−0.377894	−0.188947	−	0.981987i	$-0.560507\pi$
$491$	−8.37358	−0.377894	−0.188947	+	0.981987i	$0.560507\pi$
$492$	0	0
$493$	−28.9213	−1.30255
$494$	0	0
$495$	0	0
$496$	0	0
$497$	37.9557	1.70255
$498$	0	0
$499$	35.0254	1.56795	0.783977	−	0.620790i	$-0.213188\pi$
$499$	35.0254	1.56795	0.783977	+	0.620790i	$0.213188\pi$
$500$	0	0
$501$	−10.1715	−0.454427
$502$	0	0
$503$	14.7037	0.655604	0.327802	−	0.944746i	$-0.393692\pi$
$503$	14.7037	0.655604	0.327802	+	0.944746i	$0.393692\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	26.0188	1.15554
$508$	0	0
$509$	−3.73411	−0.165512	−0.0827558	−	0.996570i	$-0.526372\pi$
$509$	−3.73411	−0.165512	−0.0827558	+	0.996570i	$0.526372\pi$
$510$	0	0
$511$	−7.71153	−0.341138
$512$	0	0
$513$	−5.50681	−0.243132
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.2296	0.625819
$518$	0	0
$519$	24.4665	1.07396
$520$	0	0
$521$	−3.38053	−0.148104	−0.0740518	−	0.997254i	$-0.523593\pi$
$521$	−3.38053	−0.148104	−0.0740518	+	0.997254i	$0.523593\pi$
$522$	0	0
$523$	−13.7951	−0.603217	−0.301608	−	0.953432i	$-0.597524\pi$
$523$	−13.7951	−0.603217	−0.301608	+	0.953432i	$0.597524\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.95490	0.128718
$528$	0	0
$529$	−3.88888	−0.169082
$530$	0	0
$531$	10.8351	0.470203
$532$	0	0
$533$	−6.08611	−0.263619
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−14.8672	−0.641567
$538$	0	0
$539$	−9.68820	−0.417300
$540$	0	0
$541$	19.4420	0.835876	0.417938	−	0.908476i	$-0.362753\pi$
$541$	19.4420	0.835876	0.417938	+	0.908476i	$0.362753\pi$
$542$	0	0
$543$	4.59995	0.197403
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.2715	−0.524689	−0.262345	−	0.964974i	$-0.584496\pi$
$547$	−12.2715	−0.524689	−0.262345	+	0.964974i	$0.584496\pi$
$548$	0	0
$549$	−7.57090	−0.323118
$550$	0	0
$551$	−4.41841	−0.188230
$552$	0	0
$553$	34.0427	1.44764
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.01766	0.254977	0.127488	−	0.991840i	$-0.459308\pi$
$557$	6.01766	0.254977	0.127488	+	0.991840i	$0.459308\pi$
$558$	0	0
$559$	18.6939	0.790668
$560$	0	0
$561$	−30.7024	−1.29625
$562$	0	0
$563$	−30.1874	−1.27225	−0.636124	−	0.771587i	$-0.719463\pi$
$563$	−30.1874	−1.27225	−0.636124	+	0.771587i	$0.719463\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−6.80725	−0.285878
$568$	0	0
$569$	−12.3451	−0.517535	−0.258768	−	0.965940i	$-0.583316\pi$
$569$	−12.3451	−0.517535	−0.258768	+	0.965940i	$0.583316\pi$
$570$	0	0
$571$	3.13368	0.131140	0.0655702	−	0.997848i	$-0.479113\pi$
$571$	3.13368	0.131140	0.0655702	+	0.997848i	$0.479113\pi$
$572$	0	0
$573$	−15.9579	−0.666651
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−27.2272	−1.13348	−0.566742	−	0.823895i	$-0.691796\pi$
$577$	−27.2272	−1.13348	−0.566742	+	0.823895i	$0.691796\pi$
$578$	0	0
$579$	11.5436	0.479735
$580$	0	0
$581$	0.758647	0.0314740
$582$	0	0
$583$	−19.9510	−0.826288
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.7445	−1.18641	−0.593206	−	0.805050i	$-0.702138\pi$
$587$	−28.7445	−1.18641	−0.593206	+	0.805050i	$0.702138\pi$
$588$	0	0
$589$	0.451431	0.0186009
$590$	0	0
$591$	−17.5038	−0.720012
$592$	0	0
$593$	27.5257	1.13034	0.565172	−	0.824973i	$-0.308810\pi$
$593$	27.5257	1.13034	0.565172	+	0.824973i	$0.308810\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.7443	−0.808081
$598$	0	0
$599$	−28.2278	−1.15335	−0.576677	−	0.816972i	$-0.695651\pi$
$599$	−28.2278	−1.15335	−0.576677	+	0.816972i	$0.695651\pi$
$600$	0	0
$601$	8.32545	0.339602	0.169801	−	0.985478i	$-0.445687\pi$
$601$	8.32545	0.339602	0.169801	+	0.985478i	$0.445687\pi$
$602$	0	0
$603$	19.8414	0.808005
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.7364	0.476364	0.238182	−	0.971220i	$-0.423448\pi$
$607$	11.7364	0.476364	0.238182	+	0.971220i	$0.423448\pi$
$608$	0	0
$609$	−16.6588	−0.675047
$610$	0	0
$611$	−21.7059	−0.878128
$612$	0	0
$613$	−31.5934	−1.27604	−0.638022	−	0.770018i	$-0.720247\pi$
$613$	−31.5934	−1.27604	−0.638022	+	0.770018i	$0.720247\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.3247	1.18057	0.590284	−	0.807196i	$-0.299016\pi$
$617$	29.3247	1.18057	0.590284	+	0.807196i	$0.299016\pi$
$618$	0	0
$619$	−37.8615	−1.52178	−0.760890	−	0.648881i	$-0.775237\pi$
$619$	−37.8615	−1.52178	−0.760890	+	0.648881i	$0.775237\pi$
$620$	0	0
$621$	−24.0737	−0.966045
$622$	0	0
$623$	−41.4516	−1.66072
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.69051	−0.187321
$628$	0	0
$629$	16.3030	0.650044
$630$	0	0
$631$	−24.0801	−0.958615	−0.479308	−	0.877647i	$-0.659112\pi$
$631$	−24.0801	−0.958615	−0.479308	+	0.877647i	$0.659112\pi$
$632$	0	0
$633$	5.64134	0.224223
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14.7784	0.585542
$638$	0	0
$639$	−18.5393	−0.733404
$640$	0	0
$641$	−48.2114	−1.90424	−0.952118	−	0.305732i	$-0.901099\pi$
$641$	−48.2114	−1.90424	−0.952118	+	0.305732i	$0.901099\pi$
$642$	0	0
$643$	18.6087	0.733856	0.366928	−	0.930249i	$-0.380410\pi$
$643$	18.6087	0.733856	0.366928	+	0.930249i	$0.380410\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.7599	0.658900	0.329450	−	0.944173i	$-0.393137\pi$
$647$	16.7599	0.658900	0.329450	+	0.944173i	$0.393137\pi$
$648$	0	0
$649$	27.5968	1.08327
$650$	0	0
$651$	1.70203	0.0667079
$652$	0	0
$653$	−2.12145	−0.0830186	−0.0415093	−	0.999138i	$-0.513217\pi$
$653$	−2.12145	−0.0830186	−0.0415093	+	0.999138i	$0.513217\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.76666	0.146951
$658$	0	0
$659$	0.175317	0.00682937	0.00341468	−	0.999994i	$-0.498913\pi$
$659$	0.175317	0.00682937	0.00341468	+	0.999994i	$0.498913\pi$
$660$	0	0
$661$	16.9306	0.658526	0.329263	−	0.944238i	$-0.393200\pi$
$661$	16.9306	0.658526	0.329263	+	0.944238i	$0.393200\pi$
$662$	0	0
$663$	46.8334	1.81886
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−19.3156	−0.747904
$668$	0	0
$669$	−24.7751	−0.957860
$670$	0	0
$671$	−19.2830	−0.744411
$672$	0	0
$673$	−21.8162	−0.840952	−0.420476	−	0.907304i	$-0.638137\pi$
$673$	−21.8162	−0.840952	−0.420476	+	0.907304i	$0.638137\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.0305	−0.500803	−0.250402	−	0.968142i	$-0.580563\pi$
$677$	−13.0305	−0.500803	−0.250402	+	0.968142i	$0.580563\pi$
$678$	0	0
$679$	34.2524	1.31449
$680$	0	0
$681$	−14.1365	−0.541713
$682$	0	0
$683$	8.41103	0.321839	0.160920	−	0.986968i	$-0.448554\pi$
$683$	8.41103	0.321839	0.160920	+	0.986968i	$0.448554\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17.2011	0.656263
$688$	0	0
$689$	30.4334	1.15942
$690$	0	0
$691$	−8.74129	−0.332534	−0.166267	−	0.986081i	$-0.553171\pi$
$691$	−8.74129	−0.332534	−0.166267	+	0.986081i	$0.553171\pi$
$692$	0	0
$693$	17.8588	0.678402
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.80247	−0.257662
$698$	0	0
$699$	4.82931	0.182661
$700$	0	0
$701$	−45.5898	−1.72190	−0.860952	−	0.508686i	$-0.830131\pi$
$701$	−45.5898	−1.72190	−0.860952	+	0.508686i	$0.830131\pi$
$702$	0	0
$703$	2.49067	0.0939373
$704$	0	0
$705$	0	0
$706$	0	0
$707$	49.5182	1.86232
$708$	0	0
$709$	−34.2985	−1.28811	−0.644053	−	0.764981i	$-0.722748\pi$
$709$	−34.2985	−1.28811	−0.644053	+	0.764981i	$0.722748\pi$
$710$	0	0
$711$	−16.6280	−0.623600
$712$	0	0
$713$	1.97349	0.0739077
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22.0721	0.824297
$718$	0	0
$719$	−15.0753	−0.562215	−0.281107	−	0.959676i	$-0.590702\pi$
$719$	−15.0753	−0.562215	−0.281107	+	0.959676i	$0.590702\pi$
$720$	0	0
$721$	47.7003	1.77645
$722$	0	0
$723$	28.2221	1.04959
$724$	0	0
$725$	0	0
$726$	0	0
$727$	15.0189	0.557021	0.278510	−	0.960433i	$-0.410159\pi$
$727$	15.0189	0.557021	0.278510	+	0.960433i	$0.410159\pi$
$728$	0	0
$729$	23.5087	0.870691
$730$	0	0
$731$	20.8942	0.772802
$732$	0	0
$733$	−32.0731	−1.18465	−0.592324	−	0.805700i	$-0.701789\pi$
$733$	−32.0731	−1.18465	−0.592324	+	0.805700i	$0.701789\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	50.5358	1.86151
$738$	0	0
$739$	−32.0663	−1.17958	−0.589788	−	0.807558i	$-0.700789\pi$
$739$	−32.0663	−1.17958	−0.589788	+	0.807558i	$0.700789\pi$
$740$	0	0
$741$	7.15491	0.262842
$742$	0	0
$743$	−14.3624	−0.526904	−0.263452	−	0.964673i	$-0.584861\pi$
$743$	−14.3624	−0.526904	−0.263452	+	0.964673i	$0.584861\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.370558	−0.0135580
$748$	0	0
$749$	28.5029	1.04147
$750$	0	0
$751$	21.2055	0.773800	0.386900	−	0.922122i	$-0.373546\pi$
$751$	21.2055	0.773800	0.386900	+	0.922122i	$0.373546\pi$
$752$	0	0
$753$	−16.0032	−0.583190
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−41.8045	−1.51941	−0.759705	−	0.650267i	$-0.774657\pi$
$757$	−41.8045	−1.51941	−0.759705	+	0.650267i	$0.774657\pi$
$758$	0	0
$759$	−20.5051	−0.744289
$760$	0	0
$761$	−1.64223	−0.0595307	−0.0297654	−	0.999557i	$-0.509476\pi$
$761$	−1.64223	−0.0595307	−0.0297654	+	0.999557i	$0.509476\pi$
$762$	0	0
$763$	−30.4635	−1.10285
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−42.0963	−1.52001
$768$	0	0
$769$	30.3955	1.09609	0.548044	−	0.836449i	$-0.315373\pi$
$769$	30.3955	1.09609	0.548044	+	0.836449i	$0.315373\pi$
$770$	0	0
$771$	−16.5168	−0.594837
$772$	0	0
$773$	47.6753	1.71476	0.857380	−	0.514684i	$-0.172091\pi$
$773$	47.6753	1.71476	0.857380	+	0.514684i	$0.172091\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.39059	0.336886
$778$	0	0
$779$	−1.03924	−0.0372346
$780$	0	0
$781$	−47.2194	−1.68964
$782$	0	0
$783$	24.3313	0.869531
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.3173	1.22328	0.611639	−	0.791137i	$-0.290510\pi$
$787$	34.3173	1.22328	0.611639	+	0.791137i	$0.290510\pi$
$788$	0	0
$789$	−0.437307	−0.0155685
$790$	0	0
$791$	56.8415	2.02105
$792$	0	0
$793$	29.4143	1.04453
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.4643	1.39790	0.698949	−	0.715172i	$-0.253651\pi$
$797$	39.4643	1.39790	0.698949	+	0.715172i	$0.253651\pi$
$798$	0	0
$799$	−24.2608	−0.858285
$800$	0	0
$801$	20.2468	0.715387
$802$	0	0
$803$	9.59363	0.338552
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.4936	0.686207
$808$	0	0
$809$	−36.3760	−1.27891	−0.639456	−	0.768827i	$-0.720841\pi$
$809$	−36.3760	−1.27891	−0.639456	+	0.768827i	$0.720841\pi$
$810$	0	0
$811$	−45.6700	−1.60369	−0.801845	−	0.597532i	$-0.796148\pi$
$811$	−45.6700	−1.60369	−0.801845	+	0.597532i	$0.796148\pi$
$812$	0	0
$813$	0.971471	0.0340710
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.19209	0.111677
$818$	0	0
$819$	−27.2419	−0.951910
$820$	0	0
$821$	18.4543	0.644061	0.322030	−	0.946729i	$-0.395635\pi$
$821$	18.4543	0.644061	0.322030	+	0.946729i	$0.395635\pi$
$822$	0	0
$823$	−13.0227	−0.453942	−0.226971	−	0.973902i	$-0.572882\pi$
$823$	−13.0227	−0.453942	−0.226971	+	0.973902i	$0.572882\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.0455	1.32297	0.661485	−	0.749958i	$-0.269926\pi$
$827$	38.0455	1.32297	0.661485	+	0.749958i	$0.269926\pi$
$828$	0	0
$829$	35.9506	1.24862	0.624308	−	0.781178i	$-0.285381\pi$
$829$	35.9506	1.24862	0.624308	+	0.781178i	$0.285381\pi$
$830$	0	0
$831$	−15.8387	−0.549438
$832$	0	0
$833$	16.5179	0.572311
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.48594	−0.0859268
$838$	0	0
$839$	−3.66477	−0.126522	−0.0632610	−	0.997997i	$-0.520150\pi$
$839$	−3.66477	−0.126522	−0.0632610	+	0.997997i	$0.520150\pi$
$840$	0	0
$841$	−9.47769	−0.326817
$842$	0	0
$843$	−32.1036	−1.10571
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.5400	0.396521
$848$	0	0
$849$	33.5488	1.15139
$850$	0	0
$851$	10.8883	0.373245
$852$	0	0
$853$	−31.5802	−1.08129	−0.540643	−	0.841252i	$-0.681819\pi$
$853$	−31.5802	−1.08129	−0.540643	+	0.841252i	$0.681819\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.40752	−0.0822392	−0.0411196	−	0.999154i	$-0.513092\pi$
$857$	−2.40752	−0.0822392	−0.0411196	+	0.999154i	$0.513092\pi$
$858$	0	0
$859$	41.4550	1.41443	0.707214	−	0.707000i	$-0.249952\pi$
$859$	41.4550	1.41443	0.707214	+	0.707000i	$0.249952\pi$
$860$	0	0
$861$	−3.91825	−0.133534
$862$	0	0
$863$	−6.99173	−0.238001	−0.119001	−	0.992894i	$-0.537969\pi$
$863$	−6.99173	−0.238001	−0.119001	+	0.992894i	$0.537969\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	31.5764	1.07239
$868$	0	0
$869$	−42.3513	−1.43667
$870$	0	0
$871$	−77.0874	−2.61201
$872$	0	0
$873$	−16.7304	−0.566239
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−39.8619	−1.34604	−0.673020	−	0.739624i	$-0.735003\pi$
$877$	−39.8619	−1.34604	−0.673020	+	0.739624i	$0.735003\pi$
$878$	0	0
$879$	−24.1382	−0.814161
$880$	0	0
$881$	24.7409	0.833541	0.416771	−	0.909012i	$-0.363162\pi$
$881$	24.7409	0.833541	0.416771	+	0.909012i	$0.363162\pi$
$882$	0	0
$883$	−31.5648	−1.06224	−0.531120	−	0.847297i	$-0.678229\pi$
$883$	−31.5648	−1.06224	−0.531120	+	0.847297i	$0.678229\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.7866	−0.395755	−0.197877	−	0.980227i	$-0.563405\pi$
$887$	−11.7866	−0.395755	−0.197877	+	0.980227i	$0.563405\pi$
$888$	0	0
$889$	43.7452	1.46717
$890$	0	0
$891$	8.46866	0.283711
$892$	0	0
$893$	−3.70641	−0.124030
$894$	0	0
$895$	0	0
$896$	0	0
$897$	31.2786	1.04436
$898$	0	0
$899$	−1.99460	−0.0665238
$900$	0	0
$901$	34.0155	1.13322
$902$	0	0
$903$	12.0352	0.400505
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.08545	−0.135655	−0.0678276	−	0.997697i	$-0.521607\pi$
$907$	−4.08545	−0.135655	−0.0678276	+	0.997697i	$0.521607\pi$
$908$	0	0
$909$	−24.1869	−0.802230
$910$	0	0
$911$	4.88428	0.161823	0.0809116	−	0.996721i	$-0.474217\pi$
$911$	4.88428	0.161823	0.0809116	+	0.996721i	$0.474217\pi$
$912$	0	0
$913$	−0.943806	−0.0312354
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.33351	0.143105
$918$	0	0
$919$	−53.8155	−1.77521	−0.887605	−	0.460606i	$-0.847632\pi$
$919$	−53.8155	−1.77521	−0.887605	+	0.460606i	$0.847632\pi$
$920$	0	0
$921$	−35.3532	−1.16493
$922$	0	0
$923$	72.0285	2.37085
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−23.2990	−0.765239
$928$	0	0
$929$	−43.3501	−1.42227	−0.711135	−	0.703055i	$-0.751819\pi$
$929$	−43.3501	−1.42227	−0.711135	+	0.703055i	$0.751819\pi$
$930$	0	0
$931$	2.52349	0.0827042
$932$	0	0
$933$	−34.3181	−1.12352
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.5106	1.35609	0.678046	−	0.735019i	$-0.262827\pi$
$937$	41.5106	1.35609	0.678046	+	0.735019i	$0.262827\pi$
$938$	0	0
$939$	36.4000	1.18787
$940$	0	0
$941$	10.2485	0.334091	0.167045	−	0.985949i	$-0.446577\pi$
$941$	10.2485	0.334091	0.167045	+	0.985949i	$0.446577\pi$
$942$	0	0
$943$	−4.54316	−0.147946
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.3985	0.597871	0.298936	−	0.954273i	$-0.403368\pi$
$947$	18.3985	0.597871	0.298936	+	0.954273i	$0.403368\pi$
$948$	0	0
$949$	−14.6341	−0.475044
$950$	0	0
$951$	−7.49530	−0.243052
$952$	0	0
$953$	26.4170	0.855732	0.427866	−	0.903842i	$-0.359266\pi$
$953$	26.4170	0.855732	0.427866	+	0.903842i	$0.359266\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	20.7246	0.669930
$958$	0	0
$959$	−41.4182	−1.33746
$960$	0	0
$961$	−30.7962	−0.993426
$962$	0	0
$963$	−13.9221	−0.448634
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−56.2597	−1.80919	−0.904596	−	0.426271i	$-0.859827\pi$
$967$	−56.2597	−1.80919	−0.904596	+	0.426271i	$0.859827\pi$
$968$	0	0
$969$	7.99707	0.256903
$970$	0	0
$971$	−38.5451	−1.23697	−0.618485	−	0.785796i	$-0.712253\pi$
$971$	−38.5451	−1.23697	−0.618485	+	0.785796i	$0.712253\pi$
$972$	0	0
$973$	10.5828	0.339270
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.9699	0.926831	0.463415	−	0.886141i	$-0.346624\pi$
$977$	28.9699	0.926831	0.463415	+	0.886141i	$0.346624\pi$
$978$	0	0
$979$	51.5684	1.64813
$980$	0	0
$981$	14.8798	0.475075
$982$	0	0
$983$	1.75595	0.0560061	0.0280030	−	0.999608i	$-0.491085\pi$
$983$	1.75595	0.0560061	0.0280030	+	0.999608i	$0.491085\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13.9743	−0.444807
$988$	0	0
$989$	13.9546	0.443731
$990$	0	0
$991$	1.13587	0.0360822	0.0180411	−	0.999837i	$-0.494257\pi$
$991$	1.13587	0.0360822	0.0180411	+	0.999837i	$0.494257\pi$
$992$	0	0
$993$	−20.8940	−0.663052
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−16.2745	−0.515420	−0.257710	−	0.966222i	$-0.582968\pi$
$997$	−16.2745	−0.515420	−0.257710	+	0.966222i	$0.582968\pi$
$998$	0	0
$999$	−13.7156	−0.433944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.bb.1.5	✓	6
4.3	odd	2		7600.2.a.cm.1.2		6
5.2	odd	4		3800.2.d.p.3649.4		12
5.3	odd	4		3800.2.d.p.3649.9		12
5.4	even	2		3800.2.a.bd.1.2	yes	6
20.19	odd	2		7600.2.a.ci.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.5	✓	6	1.1	even	1	trivial
3800.2.a.bd.1.2	yes	6	5.4	even	2
3800.2.d.p.3649.4		12	5.2	odd	4
3800.2.d.p.3649.9		12	5.3	odd	4
7600.2.a.ci.1.5		6	20.19	odd	2
7600.2.a.cm.1.2		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)

\(f(q)\)	\(=\)	\(q+1.22174 q^{3} +3.08602 q^{7} -1.50735 q^{9} -3.83920 q^{11} +5.85633 q^{13} +6.54564 q^{17} +1.00000 q^{19} +3.77031 q^{21} +4.37163 q^{23} -5.50681 q^{27} -4.41841 q^{29} +0.451431 q^{31} -4.69051 q^{33} +2.49067 q^{37} +7.15491 q^{39} -1.03924 q^{41} +3.19209 q^{43} -3.70641 q^{47} +2.52349 q^{49} +7.99707 q^{51} +5.19667 q^{53} +1.22174 q^{57} -7.18818 q^{59} +5.02265 q^{61} -4.65171 q^{63} -13.1631 q^{67} +5.34099 q^{69} +12.2993 q^{71} -2.49886 q^{73} -11.8478 q^{77} +11.0313 q^{79} -2.20584 q^{81} +0.245834 q^{83} -5.39814 q^{87} -13.4321 q^{89} +18.0727 q^{91} +0.551531 q^{93} +11.0992 q^{97} +5.78702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9 + 3 * q^11 + 3 * q^13 + 2 * q^17 + 6 * q^19 + 11 * q^21 - 4 * q^23 - 20 * q^27 + 7 * q^29 + 5 * q^31 + 16 * q^33 + 8 * q^39 + 11 * q^41 + 7 * q^43 - 20 * q^47 - 2 * q^49 + 13 * q^51 + 7 * q^53 - 2 * q^57 - 4 * q^59 + 13 * q^61 + q^63 - 25 * q^67 + 7 * q^69 + 29 * q^71 + 19 * q^73 - 24 * q^77 + 28 * q^79 + 38 * q^81 + 15 * q^83 - 57 * q^87 - 12 * q^89 + 27 * q^93 + 13 * q^97 + 10 * q^99

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