LMFDB - Embedded newform 3800.2.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3800,2,Mod(1,3800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [6,0,-2,0,0,0,-6,0,8,0,-2,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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			1.3
Root			$0.664406$ 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)

$f(q)$	$=$	$q-0.664406 q^{3} +0.345799 q^{7} -2.55856 q^{9} +3.51609 q^{11} +3.22452 q^{13} -4.88550 q^{17} +1.00000 q^{19} -0.229751 q^{21} -7.62861 q^{23} +3.69315 q^{27} -1.73968 q^{29} -4.76164 q^{31} -2.33611 q^{33} +1.86749 q^{37} -2.14239 q^{39} +6.76164 q^{41} -0.606115 q^{43} -3.34735 q^{47} -6.88042 q^{49} +3.24596 q^{51} +0.107390 q^{53} -0.664406 q^{57} +12.3233 q^{59} -8.41231 q^{61} -0.884749 q^{63} -2.23888 q^{67} +5.06850 q^{69} -0.536605 q^{71} -5.57710 q^{73} +1.21586 q^{77} -2.67119 q^{79} +5.22194 q^{81} -1.79847 q^{83} +1.15586 q^{87} +8.94482 q^{89} +1.11504 q^{91} +3.16367 q^{93} -10.1560 q^{97} -8.99614 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$3$	−0.664406	−0.383595	−0.191798	−	0.981434i	$-0.561432\pi$
$3$	−0.664406	−0.383595	−0.191798	+	0.981434i	$0.561432\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.345799	0.130700	0.0653499	−	0.997862i	$-0.479184\pi$
$7$	0.345799	0.130700	0.0653499	+	0.997862i	$0.479184\pi$
$8$	0	0
$9$	−2.55856	−0.852855
$10$	0	0
$11$	3.51609	1.06014	0.530071	−	0.847954i	$-0.322165\pi$
$11$	3.51609	1.06014	0.530071	+	0.847954i	$0.322165\pi$
$12$	0	0
$13$	3.22452	0.894320	0.447160	−	0.894454i	$-0.352435\pi$
$13$	3.22452	0.894320	0.447160	+	0.894454i	$0.352435\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.88550	−1.18491	−0.592454	−	0.805604i	$-0.701841\pi$
$17$	−4.88550	−1.18491	−0.592454	+	0.805604i	$0.701841\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.229751	−0.0501358
$22$	0	0
$23$	−7.62861	−1.59067	−0.795337	−	0.606167i	$-0.792706\pi$
$23$	−7.62861	−1.59067	−0.795337	+	0.606167i	$0.792706\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.69315	0.710746
$28$	0	0
$29$	−1.73968	−0.323051	−0.161526	−	0.986869i	$-0.551641\pi$
$29$	−1.73968	−0.323051	−0.161526	+	0.986869i	$0.551641\pi$
$30$	0	0
$31$	−4.76164	−0.855216	−0.427608	−	0.903964i	$-0.640644\pi$
$31$	−4.76164	−0.855216	−0.427608	+	0.903964i	$0.640644\pi$
$32$	0	0
$33$	−2.33611	−0.406665
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.86749	0.307013	0.153506	−	0.988148i	$-0.450943\pi$
$37$	1.86749	0.307013	0.153506	+	0.988148i	$0.450943\pi$
$38$	0	0
$39$	−2.14239	−0.343057
$40$	0	0
$41$	6.76164	1.05599	0.527996	−	0.849247i	$-0.322944\pi$
$41$	6.76164	1.05599	0.527996	+	0.849247i	$0.322944\pi$
$42$	0	0
$43$	−0.606115	−0.0924317	−0.0462158	−	0.998931i	$-0.514716\pi$
$43$	−0.606115	−0.0924317	−0.0462158	+	0.998931i	$0.514716\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.34735	−0.488261	−0.244130	−	0.969742i	$-0.578502\pi$
$47$	−3.34735	−0.488261	−0.244130	+	0.969742i	$0.578502\pi$
$48$	0	0
$49$	−6.88042	−0.982918
$50$	0	0
$51$	3.24596	0.454525
$52$	0	0
$53$	0.107390	0.0147512	0.00737559	−	0.999973i	$-0.497652\pi$
$53$	0.107390	0.0147512	0.00737559	+	0.999973i	$0.497652\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.664406	−0.0880028
$58$	0	0
$59$	12.3233	1.60436	0.802179	−	0.597084i	$-0.203674\pi$
$59$	12.3233	1.60436	0.802179	+	0.597084i	$0.203674\pi$
$60$	0	0
$61$	−8.41231	−1.07709	−0.538543	−	0.842598i	$-0.681025\pi$
$61$	−8.41231	−1.07709	−0.538543	+	0.842598i	$0.681025\pi$
$62$	0	0
$63$	−0.884749	−0.111468
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.23888	−0.273522	−0.136761	−	0.990604i	$-0.543669\pi$
$67$	−2.23888	−0.273522	−0.136761	+	0.990604i	$0.543669\pi$
$68$	0	0
$69$	5.06850	0.610175
$70$	0	0
$71$	−0.536605	−0.0636833	−0.0318417	−	0.999493i	$-0.510137\pi$
$71$	−0.536605	−0.0636833	−0.0318417	+	0.999493i	$0.510137\pi$
$72$	0	0
$73$	−5.57710	−0.652750	−0.326375	−	0.945240i	$-0.605827\pi$
$73$	−5.57710	−0.652750	−0.326375	+	0.945240i	$0.605827\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.21586	0.138560
$78$	0	0
$79$	−2.67119	−0.300532	−0.150266	−	0.988646i	$-0.548013\pi$
$79$	−2.67119	−0.300532	−0.150266	+	0.988646i	$0.548013\pi$
$80$	0	0
$81$	5.22194	0.580216
$82$	0	0
$83$	−1.79847	−0.197408	−0.0987038	−	0.995117i	$-0.531470\pi$
$83$	−1.79847	−0.197408	−0.0987038	+	0.995117i	$0.531470\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.15586	0.123921
$88$	0	0
$89$	8.94482	0.948149	0.474075	−	0.880485i	$-0.342783\pi$
$89$	8.94482	0.948149	0.474075	+	0.880485i	$0.342783\pi$
$90$	0	0
$91$	1.11504	0.116887
$92$	0	0
$93$	3.16367	0.328057
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.1560	−1.03119	−0.515594	−	0.856833i	$-0.672429\pi$
$97$	−10.1560	−1.03119	−0.515594	+	0.856833i	$0.672429\pi$
$98$	0	0
$99$	−8.99614	−0.904146
$100$	0	0
$101$	16.5089	1.64270	0.821350	−	0.570425i	$-0.193221\pi$
$101$	16.5089	1.64270	0.821350	+	0.570425i	$0.193221\pi$
$102$	0	0
$103$	−11.5212	−1.13522	−0.567610	−	0.823298i	$-0.692132\pi$
$103$	−11.5212	−1.13522	−0.567610	+	0.823298i	$0.692132\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.0340	−0.970027	−0.485014	−	0.874507i	$-0.661185\pi$
$107$	−10.0340	−0.970027	−0.485014	+	0.874507i	$0.661185\pi$
$108$	0	0
$109$	12.2737	1.17560	0.587802	−	0.809005i	$-0.299993\pi$
$109$	12.2737	1.17560	0.587802	+	0.809005i	$0.299993\pi$
$110$	0	0
$111$	−1.24077	−0.117769
$112$	0	0
$113$	0.700723	0.0659185	0.0329593	−	0.999457i	$-0.489507\pi$
$113$	0.700723	0.0659185	0.0329593	+	0.999457i	$0.489507\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.25014	−0.762725
$118$	0	0
$119$	−1.68940	−0.154867
$120$	0	0
$121$	1.36289	0.123899
$122$	0	0
$123$	−4.49248	−0.405073
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.8369	−1.05036	−0.525178	−	0.850992i	$-0.676001\pi$
$127$	−11.8369	−1.05036	−0.525178	+	0.850992i	$0.676001\pi$
$128$	0	0
$129$	0.402707	0.0354563
$130$	0	0
$131$	−22.0556	−1.92701	−0.963503	−	0.267699i	$-0.913737\pi$
$131$	−22.0556	−1.92701	−0.963503	+	0.267699i	$0.913737\pi$
$132$	0	0
$133$	0.345799	0.0299846
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.4858	−1.66478	−0.832390	−	0.554190i	$-0.813028\pi$
$137$	−19.4858	−1.66478	−0.832390	+	0.554190i	$0.813028\pi$
$138$	0	0
$139$	−14.5867	−1.23723	−0.618613	−	0.785696i	$-0.712305\pi$
$139$	−14.5867	−1.23723	−0.618613	+	0.785696i	$0.712305\pi$
$140$	0	0
$141$	2.22400	0.187294
$142$	0	0
$143$	11.3377	0.948106
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.57140	0.377043
$148$	0	0
$149$	15.3918	1.26094	0.630471	−	0.776213i	$-0.282862\pi$
$149$	15.3918	1.26094	0.630471	+	0.776213i	$0.282862\pi$
$150$	0	0
$151$	−3.55281	−0.289124	−0.144562	−	0.989496i	$-0.546177\pi$
$151$	−3.55281	−0.289124	−0.144562	+	0.989496i	$0.546177\pi$
$152$	0	0
$153$	12.4999	1.01055
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.9812	−0.876393	−0.438196	−	0.898879i	$-0.644383\pi$
$157$	−10.9812	−0.876393	−0.438196	+	0.898879i	$0.644383\pi$
$158$	0	0
$159$	−0.0713507	−0.00565848
$160$	0	0
$161$	−2.63797	−0.207901
$162$	0	0
$163$	−7.88687	−0.617747	−0.308874	−	0.951103i	$-0.599952\pi$
$163$	−7.88687	−0.617747	−0.308874	+	0.951103i	$0.599952\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.1924	−1.09824	−0.549121	−	0.835743i	$-0.685037\pi$
$167$	−14.1924	−1.09824	−0.549121	+	0.835743i	$0.685037\pi$
$168$	0	0
$169$	−2.60248	−0.200191
$170$	0	0
$171$	−2.55856	−0.195658
$172$	0	0
$173$	0.654185	0.0497368	0.0248684	−	0.999691i	$-0.492083\pi$
$173$	0.654185	0.0497368	0.0248684	+	0.999691i	$0.492083\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.18768	−0.615424
$178$	0	0
$179$	0.893129	0.0667556	0.0333778	−	0.999443i	$-0.489374\pi$
$179$	0.893129	0.0667556	0.0333778	+	0.999443i	$0.489374\pi$
$180$	0	0
$181$	−20.8542	−1.55008	−0.775039	−	0.631914i	$-0.782270\pi$
$181$	−20.8542	−1.55008	−0.775039	+	0.631914i	$0.782270\pi$
$182$	0	0
$183$	5.58920	0.413165
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−17.1779	−1.25617
$188$	0	0
$189$	1.27709	0.0928944
$190$	0	0
$191$	−13.8797	−1.00430	−0.502151	−	0.864780i	$-0.667458\pi$
$191$	−13.8797	−1.00430	−0.502151	+	0.864780i	$0.667458\pi$
$192$	0	0
$193$	−9.98613	−0.718817	−0.359409	−	0.933180i	$-0.617022\pi$
$193$	−9.98613	−0.718817	−0.359409	+	0.933180i	$0.617022\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.3247	0.878099	0.439050	−	0.898463i	$-0.355315\pi$
$197$	12.3247	0.878099	0.439050	+	0.898463i	$0.355315\pi$
$198$	0	0
$199$	−16.0086	−1.13482	−0.567410	−	0.823435i	$-0.692055\pi$
$199$	−16.0086	−1.13482	−0.567410	+	0.823435i	$0.692055\pi$
$200$	0	0
$201$	1.48752	0.104922
$202$	0	0
$203$	−0.601581	−0.0422227
$204$	0	0
$205$	0	0
$206$	0	0
$207$	19.5183	1.35661
$208$	0	0
$209$	3.51609	0.243213
$210$	0	0
$211$	19.0558	1.31185	0.655926	−	0.754825i	$-0.272278\pi$
$211$	19.0558	1.31185	0.655926	+	0.754825i	$0.272278\pi$
$212$	0	0
$213$	0.356524	0.0244286
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.64657	−0.111777
$218$	0	0
$219$	3.70546	0.250392
$220$	0	0
$221$	−15.7534	−1.05969
$222$	0	0
$223$	8.68798	0.581790	0.290895	−	0.956755i	$-0.406047\pi$
$223$	8.68798	0.581790	0.290895	+	0.956755i	$0.406047\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−28.8077	−1.91203	−0.956017	−	0.293311i	$-0.905243\pi$
$227$	−28.8077	−1.91203	−0.956017	+	0.293311i	$0.905243\pi$
$228$	0	0
$229$	8.27464	0.546803	0.273402	−	0.961900i	$-0.411851\pi$
$229$	8.27464	0.546803	0.273402	+	0.961900i	$0.411851\pi$
$230$	0	0
$231$	−0.807826	−0.0531510
$232$	0	0
$233$	29.0199	1.90116	0.950580	−	0.310480i	$-0.100490\pi$
$233$	29.0199	1.90116	0.950580	+	0.310480i	$0.100490\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.77475	0.115283
$238$	0	0
$239$	−15.3744	−0.994486	−0.497243	−	0.867611i	$-0.665654\pi$
$239$	−15.3744	−0.994486	−0.497243	+	0.867611i	$0.665654\pi$
$240$	0	0
$241$	24.8583	1.60126	0.800632	−	0.599157i	$-0.204497\pi$
$241$	24.8583	1.60126	0.800632	+	0.599157i	$0.204497\pi$
$242$	0	0
$243$	−14.5489	−0.933314
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.22452	0.205171
$248$	0	0
$249$	1.19491	0.0757246
$250$	0	0
$251$	−14.3807	−0.907702	−0.453851	−	0.891077i	$-0.649950\pi$
$251$	−14.3807	−0.907702	−0.453851	+	0.891077i	$0.649950\pi$
$252$	0	0
$253$	−26.8229	−1.68634
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.9784	−1.37097	−0.685486	−	0.728086i	$-0.740410\pi$
$257$	−21.9784	−1.37097	−0.685486	+	0.728086i	$0.740410\pi$
$258$	0	0
$259$	0.645775	0.0401265
$260$	0	0
$261$	4.45109	0.275516
$262$	0	0
$263$	2.24444	0.138398	0.0691990	−	0.997603i	$-0.477956\pi$
$263$	2.24444	0.138398	0.0691990	+	0.997603i	$0.477956\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.94300	−0.363705
$268$	0	0
$269$	19.3526	1.17995	0.589975	−	0.807421i	$-0.299137\pi$
$269$	19.3526	1.17995	0.589975	+	0.807421i	$0.299137\pi$
$270$	0	0
$271$	−4.18631	−0.254300	−0.127150	−	0.991883i	$-0.540583\pi$
$271$	−4.18631	−0.254300	−0.127150	+	0.991883i	$0.540583\pi$
$272$	0	0
$273$	−0.740837	−0.0448375
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.2206	0.914518	0.457259	−	0.889334i	$-0.348831\pi$
$277$	15.2206	0.914518	0.457259	+	0.889334i	$0.348831\pi$
$278$	0	0
$279$	12.1830	0.729375
$280$	0	0
$281$	7.96393	0.475088	0.237544	−	0.971377i	$-0.423658\pi$
$281$	7.96393	0.475088	0.237544	+	0.971377i	$0.423658\pi$
$282$	0	0
$283$	31.6286	1.88013	0.940064	−	0.340998i	$-0.110765\pi$
$283$	31.6286	1.88013	0.940064	+	0.340998i	$0.110765\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.33817	0.138018
$288$	0	0
$289$	6.86811	0.404006
$290$	0	0
$291$	6.74773	0.395559
$292$	0	0
$293$	−32.6323	−1.90640	−0.953199	−	0.302342i	$-0.902231\pi$
$293$	−32.6323	−1.90640	−0.953199	+	0.302342i	$0.902231\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	12.9854	0.753491
$298$	0	0
$299$	−24.5986	−1.42257
$300$	0	0
$301$	−0.209594	−0.0120808
$302$	0	0
$303$	−10.9686	−0.630132
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4.85468	−0.277071	−0.138536	−	0.990357i	$-0.544240\pi$
$307$	−4.85468	−0.277071	−0.138536	+	0.990357i	$0.544240\pi$
$308$	0	0
$309$	7.65477	0.435465
$310$	0	0
$311$	−0.509761	−0.0289059	−0.0144530	−	0.999896i	$-0.504601\pi$
$311$	−0.509761	−0.0289059	−0.0144530	+	0.999896i	$0.504601\pi$
$312$	0	0
$313$	−5.33353	−0.301469	−0.150734	−	0.988574i	$-0.548164\pi$
$313$	−5.33353	−0.301469	−0.150734	+	0.988574i	$0.548164\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.1325	0.849927	0.424963	−	0.905211i	$-0.360287\pi$
$317$	15.1325	0.849927	0.424963	+	0.905211i	$0.360287\pi$
$318$	0	0
$319$	−6.11689	−0.342480
$320$	0	0
$321$	6.66668	0.372098
$322$	0	0
$323$	−4.88550	−0.271836
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.15470	−0.450956
$328$	0	0
$329$	−1.15751	−0.0638156
$330$	0	0
$331$	4.21378	0.231610	0.115805	−	0.993272i	$-0.463055\pi$
$331$	4.21378	0.231610	0.115805	+	0.993272i	$0.463055\pi$
$332$	0	0
$333$	−4.77808	−0.261837
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.5707	0.902664	0.451332	−	0.892356i	$-0.350949\pi$
$337$	16.5707	0.902664	0.451332	+	0.892356i	$0.350949\pi$
$338$	0	0
$339$	−0.465565	−0.0252860
$340$	0	0
$341$	−16.7424	−0.906650
$342$	0	0
$343$	−4.79984	−0.259167
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.17123	−0.277606	−0.138803	−	0.990320i	$-0.544325\pi$
$347$	−5.17123	−0.277606	−0.138803	+	0.990320i	$0.544325\pi$
$348$	0	0
$349$	−12.5843	−0.673624	−0.336812	−	0.941572i	$-0.609349\pi$
$349$	−12.5843	−0.673624	−0.336812	+	0.941572i	$0.609349\pi$
$350$	0	0
$351$	11.9086	0.635635
$352$	0	0
$353$	13.2242	0.703853	0.351927	−	0.936028i	$-0.385527\pi$
$353$	13.2242	0.703853	0.351927	+	0.936028i	$0.385527\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.12245	0.0594063
$358$	0	0
$359$	−9.77950	−0.516142	−0.258071	−	0.966126i	$-0.583087\pi$
$359$	−9.77950	−0.516142	−0.258071	+	0.966126i	$0.583087\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.905512	−0.0475271
$364$	0	0
$365$	0	0
$366$	0	0
$367$	19.7113	1.02892	0.514460	−	0.857515i	$-0.327993\pi$
$367$	19.7113	1.02892	0.514460	+	0.857515i	$0.327993\pi$
$368$	0	0
$369$	−17.3001	−0.900607
$370$	0	0
$371$	0.0371354	0.00192798
$372$	0	0
$373$	−33.0128	−1.70934	−0.854670	−	0.519172i	$-0.826240\pi$
$373$	−33.0128	−1.70934	−0.854670	+	0.519172i	$0.826240\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.60964	−0.288911
$378$	0	0
$379$	−29.3422	−1.50721	−0.753604	−	0.657329i	$-0.771686\pi$
$379$	−29.3422	−1.50721	−0.753604	+	0.657329i	$0.771686\pi$
$380$	0	0
$381$	7.86453	0.402912
$382$	0	0
$383$	29.9938	1.53261	0.766305	−	0.642477i	$-0.222093\pi$
$383$	29.9938	1.53261	0.766305	+	0.642477i	$0.222093\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.55078	0.0788308
$388$	0	0
$389$	7.64392	0.387562	0.193781	−	0.981045i	$-0.437925\pi$
$389$	7.64392	0.387562	0.193781	+	0.981045i	$0.437925\pi$
$390$	0	0
$391$	37.2696	1.88480
$392$	0	0
$393$	14.6539	0.739190
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.48509	0.225100	0.112550	−	0.993646i	$-0.464098\pi$
$397$	4.48509	0.225100	0.112550	+	0.993646i	$0.464098\pi$
$398$	0	0
$399$	−0.229751	−0.0115019
$400$	0	0
$401$	3.11631	0.155621	0.0778105	−	0.996968i	$-0.475207\pi$
$401$	3.11631	0.155621	0.0778105	+	0.996968i	$0.475207\pi$
$402$	0	0
$403$	−15.3540	−0.764837
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.56625	0.325477
$408$	0	0
$409$	9.88207	0.488637	0.244319	−	0.969695i	$-0.421436\pi$
$409$	9.88207	0.488637	0.244319	+	0.969695i	$0.421436\pi$
$410$	0	0
$411$	12.9465	0.638602
$412$	0	0
$413$	4.26139	0.209689
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.69148	0.474594
$418$	0	0
$419$	−4.05273	−0.197989	−0.0989944	−	0.995088i	$-0.531563\pi$
$419$	−4.05273	−0.197989	−0.0989944	+	0.995088i	$0.531563\pi$
$420$	0	0
$421$	−34.5916	−1.68589	−0.842945	−	0.538000i	$-0.819180\pi$
$421$	−34.5916	−1.68589	−0.842945	+	0.538000i	$0.819180\pi$
$422$	0	0
$423$	8.56440	0.416415
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.90897	−0.140775
$428$	0	0
$429$	−7.53284	−0.363689
$430$	0	0
$431$	−22.8014	−1.09831	−0.549153	−	0.835722i	$-0.685050\pi$
$431$	−22.8014	−1.09831	−0.549153	+	0.835722i	$0.685050\pi$
$432$	0	0
$433$	−19.8043	−0.951733	−0.475867	−	0.879517i	$-0.657866\pi$
$433$	−19.8043	−0.951733	−0.475867	+	0.879517i	$0.657866\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.62861	−0.364926
$438$	0	0
$439$	−24.5069	−1.16965	−0.584825	−	0.811160i	$-0.698837\pi$
$439$	−24.5069	−1.16965	−0.584825	+	0.811160i	$0.698837\pi$
$440$	0	0
$441$	17.6040	0.838286
$442$	0	0
$443$	−29.8226	−1.41692	−0.708458	−	0.705753i	$-0.750609\pi$
$443$	−29.8226	−1.41692	−0.708458	+	0.705753i	$0.750609\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10.2264	−0.483692
$448$	0	0
$449$	−10.8675	−0.512869	−0.256435	−	0.966562i	$-0.582548\pi$
$449$	−10.8675	−0.512869	−0.256435	+	0.966562i	$0.582548\pi$
$450$	0	0
$451$	23.7745	1.11950
$452$	0	0
$453$	2.36051	0.110906
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.604334	−0.0282696	−0.0141348	−	0.999900i	$-0.504499\pi$
$457$	−0.604334	−0.0282696	−0.0141348	+	0.999900i	$0.504499\pi$
$458$	0	0
$459$	−18.0429	−0.842169
$460$	0	0
$461$	−31.2349	−1.45475	−0.727376	−	0.686239i	$-0.759261\pi$
$461$	−31.2349	−1.45475	−0.727376	+	0.686239i	$0.759261\pi$
$462$	0	0
$463$	−33.5246	−1.55802	−0.779010	−	0.627011i	$-0.784278\pi$
$463$	−33.5246	−1.55802	−0.779010	+	0.627011i	$0.784278\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.0124	1.71273	0.856364	−	0.516372i	$-0.172718\pi$
$467$	37.0124	1.71273	0.856364	+	0.516372i	$0.172718\pi$
$468$	0	0
$469$	−0.774201	−0.0357493
$470$	0	0
$471$	7.29596	0.336180
$472$	0	0
$473$	−2.13115	−0.0979906
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.274765	−0.0125806
$478$	0	0
$479$	14.1388	0.646020	0.323010	−	0.946396i	$-0.395305\pi$
$479$	14.1388	0.646020	0.323010	+	0.946396i	$0.395305\pi$
$480$	0	0
$481$	6.02174	0.274568
$482$	0	0
$483$	1.75268	0.0797498
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.2708	−1.64359	−0.821793	−	0.569785i	$-0.807026\pi$
$487$	−36.2708	−1.64359	−0.821793	+	0.569785i	$0.807026\pi$
$488$	0	0
$489$	5.24009	0.236965
$490$	0	0
$491$	−12.8630	−0.580498	−0.290249	−	0.956951i	$-0.593738\pi$
$491$	−12.8630	−0.580498	−0.290249	+	0.956951i	$0.593738\pi$
$492$	0	0
$493$	8.49923	0.382786
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.185558	−0.00832339
$498$	0	0
$499$	30.9889	1.38725	0.693626	−	0.720336i	$-0.256012\pi$
$499$	30.9889	1.38725	0.693626	+	0.720336i	$0.256012\pi$
$500$	0	0
$501$	9.42953	0.421280
$502$	0	0
$503$	3.17853	0.141724	0.0708619	−	0.997486i	$-0.477425\pi$
$503$	3.17853	0.141724	0.0708619	+	0.997486i	$0.477425\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.72911	0.0767923
$508$	0	0
$509$	−10.6322	−0.471265	−0.235632	−	0.971842i	$-0.575716\pi$
$509$	−10.6322	−0.471265	−0.235632	+	0.971842i	$0.575716\pi$
$510$	0	0
$511$	−1.92856	−0.0853143
$512$	0	0
$513$	3.69315	0.163056
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−11.7696	−0.517625
$518$	0	0
$519$	−0.434645	−0.0190788
$520$	0	0
$521$	31.4792	1.37913	0.689564	−	0.724225i	$-0.257802\pi$
$521$	31.4792	1.37913	0.689564	+	0.724225i	$0.257802\pi$
$522$	0	0
$523$	2.60906	0.114086	0.0570430	−	0.998372i	$-0.481833\pi$
$523$	2.60906	0.114086	0.0570430	+	0.998372i	$0.481833\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23.2630	1.01335
$528$	0	0
$529$	35.1957	1.53025
$530$	0	0
$531$	−31.5300	−1.36828
$532$	0	0
$533$	21.8030	0.944395
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.593401	−0.0256071
$538$	0	0
$539$	−24.1922	−1.04203
$540$	0	0
$541$	8.22279	0.353525	0.176763	−	0.984254i	$-0.443438\pi$
$541$	8.22279	0.353525	0.176763	+	0.984254i	$0.443438\pi$
$542$	0	0
$543$	13.8556	0.594602
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.9776	0.640394	0.320197	−	0.947351i	$-0.396251\pi$
$547$	14.9776	0.640394	0.320197	+	0.947351i	$0.396251\pi$
$548$	0	0
$549$	21.5234	0.918598
$550$	0	0
$551$	−1.73968	−0.0741130
$552$	0	0
$553$	−0.923694	−0.0392795
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.5965	−0.957443	−0.478722	−	0.877967i	$-0.658900\pi$
$557$	−22.5965	−0.957443	−0.478722	+	0.877967i	$0.658900\pi$
$558$	0	0
$559$	−1.95443	−0.0826635
$560$	0	0
$561$	11.4131	0.481861
$562$	0	0
$563$	21.7518	0.916728	0.458364	−	0.888765i	$-0.348436\pi$
$563$	21.7518	0.916728	0.458364	+	0.888765i	$0.348436\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.80574	0.0758341
$568$	0	0
$569$	32.5606	1.36501	0.682506	−	0.730880i	$-0.260890\pi$
$569$	32.5606	1.36501	0.682506	+	0.730880i	$0.260890\pi$
$570$	0	0
$571$	−11.4285	−0.478267	−0.239134	−	0.970987i	$-0.576863\pi$
$571$	−11.4285	−0.478267	−0.239134	+	0.970987i	$0.576863\pi$
$572$	0	0
$573$	9.22179	0.385246
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.4277	0.975307	0.487654	−	0.873037i	$-0.337853\pi$
$577$	23.4277	0.975307	0.487654	+	0.873037i	$0.337853\pi$
$578$	0	0
$579$	6.63485	0.275735
$580$	0	0
$581$	−0.621909	−0.0258011
$582$	0	0
$583$	0.377594	0.0156383
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.85818	−0.0766953	−0.0383477	−	0.999264i	$-0.512209\pi$
$587$	−1.85818	−0.0766953	−0.0383477	+	0.999264i	$0.512209\pi$
$588$	0	0
$589$	−4.76164	−0.196200
$590$	0	0
$591$	−8.18862	−0.336835
$592$	0	0
$593$	−1.37788	−0.0565829	−0.0282914	−	0.999600i	$-0.509007\pi$
$593$	−1.37788	−0.0565829	−0.0282914	+	0.999600i	$0.509007\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.6362	0.435312
$598$	0	0
$599$	33.9275	1.38624	0.693120	−	0.720822i	$-0.256235\pi$
$599$	33.9275	1.38624	0.693120	+	0.720822i	$0.256235\pi$
$600$	0	0
$601$	−15.9368	−0.650076	−0.325038	−	0.945701i	$-0.605377\pi$
$601$	−15.9368	−0.650076	−0.325038	+	0.945701i	$0.605377\pi$
$602$	0	0
$603$	5.72831	0.233275
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.3921	1.31475	0.657377	−	0.753562i	$-0.271666\pi$
$607$	32.3921	1.31475	0.657377	+	0.753562i	$0.271666\pi$
$608$	0	0
$609$	0.399694	0.0161964
$610$	0	0
$611$	−10.7936	−0.436662
$612$	0	0
$613$	−8.44582	−0.341123	−0.170562	−	0.985347i	$-0.554558\pi$
$613$	−8.44582	−0.341123	−0.170562	+	0.985347i	$0.554558\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.4621	1.50817	0.754084	−	0.656778i	$-0.228081\pi$
$617$	37.4621	1.50817	0.754084	+	0.656778i	$0.228081\pi$
$618$	0	0
$619$	−7.35835	−0.295757	−0.147878	−	0.989006i	$-0.547244\pi$
$619$	−7.35835	−0.295757	−0.147878	+	0.989006i	$0.547244\pi$
$620$	0	0
$621$	−28.1736	−1.13057
$622$	0	0
$623$	3.09311	0.123923
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.33611	−0.0932954
$628$	0	0
$629$	−9.12360	−0.363782
$630$	0	0
$631$	35.8314	1.42642	0.713212	−	0.700948i	$-0.247240\pi$
$631$	35.8314	1.42642	0.713212	+	0.700948i	$0.247240\pi$
$632$	0	0
$633$	−12.6608	−0.503220
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−22.1860	−0.879043
$638$	0	0
$639$	1.37294	0.0543126
$640$	0	0
$641$	15.5827	0.615479	0.307740	−	0.951471i	$-0.400427\pi$
$641$	15.5827	0.615479	0.307740	+	0.951471i	$0.400427\pi$
$642$	0	0
$643$	44.5775	1.75797	0.878983	−	0.476852i	$-0.158222\pi$
$643$	44.5775	1.75797	0.878983	+	0.476852i	$0.158222\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.9836	0.549752	0.274876	−	0.961480i	$-0.411363\pi$
$647$	13.9836	0.549752	0.274876	+	0.961480i	$0.411363\pi$
$648$	0	0
$649$	43.3298	1.70085
$650$	0	0
$651$	1.09399	0.0428770
$652$	0	0
$653$	18.9179	0.740314	0.370157	−	0.928969i	$-0.379304\pi$
$653$	18.9179	0.740314	0.370157	+	0.928969i	$0.379304\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.2694	0.556701
$658$	0	0
$659$	35.9816	1.40164	0.700822	−	0.713337i	$-0.252817\pi$
$659$	35.9816	1.40164	0.700822	+	0.713337i	$0.252817\pi$
$660$	0	0
$661$	7.69177	0.299175	0.149588	−	0.988748i	$-0.452205\pi$
$661$	7.69177	0.299175	0.149588	+	0.988748i	$0.452205\pi$
$662$	0	0
$663$	10.4667	0.406491
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.2714	0.513870
$668$	0	0
$669$	−5.77235	−0.223172
$670$	0	0
$671$	−29.5785	−1.14186
$672$	0	0
$673$	−19.0426	−0.734040	−0.367020	−	0.930213i	$-0.619622\pi$
$673$	−19.0426	−0.734040	−0.367020	+	0.930213i	$0.619622\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.61140	0.177230	0.0886152	−	0.996066i	$-0.471756\pi$
$677$	4.61140	0.177230	0.0886152	+	0.996066i	$0.471756\pi$
$678$	0	0
$679$	−3.51194	−0.134776
$680$	0	0
$681$	19.1400	0.733447
$682$	0	0
$683$	37.5087	1.43523	0.717614	−	0.696441i	$-0.245234\pi$
$683$	37.5087	1.43523	0.717614	+	0.696441i	$0.245234\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5.49772	−0.209751
$688$	0	0
$689$	0.346282	0.0131923
$690$	0	0
$691$	−18.3081	−0.696473	−0.348236	−	0.937407i	$-0.613219\pi$
$691$	−18.3081	−0.696473	−0.348236	+	0.937407i	$0.613219\pi$
$692$	0	0
$693$	−3.11086	−0.118172
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−33.0340	−1.25125
$698$	0	0
$699$	−19.2810	−0.729276
$700$	0	0
$701$	6.22677	0.235182	0.117591	−	0.993062i	$-0.462483\pi$
$701$	6.22677	0.235182	0.117591	+	0.993062i	$0.462483\pi$
$702$	0	0
$703$	1.86749	0.0704335
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.70877	0.214700
$708$	0	0
$709$	−30.0878	−1.12997	−0.564986	−	0.825100i	$-0.691119\pi$
$709$	−30.0878	−1.12997	−0.564986	+	0.825100i	$0.691119\pi$
$710$	0	0
$711$	6.83440	0.256310
$712$	0	0
$713$	36.3247	1.36037
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.2148	0.381480
$718$	0	0
$719$	−32.7649	−1.22193	−0.610963	−	0.791659i	$-0.709218\pi$
$719$	−32.7649	−1.22193	−0.610963	+	0.791659i	$0.709218\pi$
$720$	0	0
$721$	−3.98403	−0.148373
$722$	0	0
$723$	−16.5160	−0.614237
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−24.9335	−0.924733	−0.462367	−	0.886689i	$-0.653000\pi$
$727$	−24.9335	−0.924733	−0.462367	+	0.886689i	$0.653000\pi$
$728$	0	0
$729$	−5.99942	−0.222201
$730$	0	0
$731$	2.96117	0.109523
$732$	0	0
$733$	−16.3618	−0.604338	−0.302169	−	0.953254i	$-0.597711\pi$
$733$	−16.3618	−0.604338	−0.302169	+	0.953254i	$0.597711\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.87209	−0.289972
$738$	0	0
$739$	14.0239	0.515876	0.257938	−	0.966162i	$-0.416957\pi$
$739$	14.0239	0.515876	0.257938	+	0.966162i	$0.416957\pi$
$740$	0	0
$741$	−2.14239	−0.0787027
$742$	0	0
$743$	48.2145	1.76882	0.884408	−	0.466714i	$-0.154562\pi$
$743$	48.2145	1.76882	0.884408	+	0.466714i	$0.154562\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.60150	0.168360
$748$	0	0
$749$	−3.46976	−0.126782
$750$	0	0
$751$	−29.4639	−1.07515	−0.537577	−	0.843214i	$-0.680660\pi$
$751$	−29.4639	−1.07515	−0.537577	+	0.843214i	$0.680660\pi$
$752$	0	0
$753$	9.55464	0.348190
$754$	0	0
$755$	0	0
$756$	0	0
$757$	17.8966	0.650464	0.325232	−	0.945634i	$-0.394558\pi$
$757$	17.8966	0.650464	0.325232	+	0.945634i	$0.394558\pi$
$758$	0	0
$759$	17.8213	0.646872
$760$	0	0
$761$	38.1996	1.38474	0.692368	−	0.721545i	$-0.256568\pi$
$761$	38.1996	1.38474	0.692368	+	0.721545i	$0.256568\pi$
$762$	0	0
$763$	4.24422	0.153651
$764$	0	0
$765$	0	0
$766$	0	0
$767$	39.7367	1.43481
$768$	0	0
$769$	39.5374	1.42576	0.712878	−	0.701289i	$-0.247392\pi$
$769$	39.5374	1.42576	0.712878	+	0.701289i	$0.247392\pi$
$770$	0	0
$771$	14.6026	0.525898
$772$	0	0
$773$	51.3427	1.84667	0.923335	−	0.383995i	$-0.125452\pi$
$773$	51.3427	1.84667	0.923335	+	0.383995i	$0.125452\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.429057	−0.0153923
$778$	0	0
$779$	6.76164	0.242261
$780$	0	0
$781$	−1.88675	−0.0675133
$782$	0	0
$783$	−6.42491	−0.229607
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41.6629	1.48512	0.742560	−	0.669779i	$-0.233611\pi$
$787$	41.6629	1.48512	0.742560	+	0.669779i	$0.233611\pi$
$788$	0	0
$789$	−1.49122	−0.0530888
$790$	0	0
$791$	0.242309	0.00861553
$792$	0	0
$793$	−27.1257	−0.963261
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.4446	−0.440812	−0.220406	−	0.975408i	$-0.570738\pi$
$797$	−12.4446	−0.440812	−0.220406	+	0.975408i	$0.570738\pi$
$798$	0	0
$799$	16.3535	0.578544
$800$	0	0
$801$	−22.8859	−0.808633
$802$	0	0
$803$	−19.6096	−0.692007
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.8580	−0.452624
$808$	0	0
$809$	0.283694	0.00997414	0.00498707	−	0.999988i	$-0.498413\pi$
$809$	0.283694	0.00997414	0.00498707	+	0.999988i	$0.498413\pi$
$810$	0	0
$811$	−46.2273	−1.62326	−0.811631	−	0.584171i	$-0.801420\pi$
$811$	−46.2273	−1.62326	−0.811631	+	0.584171i	$0.801420\pi$
$812$	0	0
$813$	2.78141	0.0975483
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.606115	−0.0212053
$818$	0	0
$819$	−2.85289	−0.0996880
$820$	0	0
$821$	24.7448	0.863600	0.431800	−	0.901969i	$-0.357879\pi$
$821$	24.7448	0.863600	0.431800	+	0.901969i	$0.357879\pi$
$822$	0	0
$823$	−29.5937	−1.03157	−0.515786	−	0.856717i	$-0.672500\pi$
$823$	−29.5937	−1.03157	−0.515786	+	0.856717i	$0.672500\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	52.8976	1.83943	0.919715	−	0.392587i	$-0.128420\pi$
$827$	52.8976	1.83943	0.919715	+	0.392587i	$0.128420\pi$
$828$	0	0
$829$	7.44924	0.258723	0.129361	−	0.991598i	$-0.458707\pi$
$829$	7.44924	0.258723	0.129361	+	0.991598i	$0.458707\pi$
$830$	0	0
$831$	−10.1127	−0.350805
$832$	0	0
$833$	33.6143	1.16467
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−17.5854	−0.607842
$838$	0	0
$839$	7.89398	0.272531	0.136265	−	0.990672i	$-0.456490\pi$
$839$	7.89398	0.272531	0.136265	+	0.990672i	$0.456490\pi$
$840$	0	0
$841$	−25.9735	−0.895638
$842$	0	0
$843$	−5.29128	−0.182242
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.471286	0.0161936
$848$	0	0
$849$	−21.0143	−0.721208
$850$	0	0
$851$	−14.2463	−0.488357
$852$	0	0
$853$	46.6687	1.59791	0.798953	−	0.601393i	$-0.205387\pi$
$853$	46.6687	1.59791	0.798953	+	0.601393i	$0.205387\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.92083	0.270570	0.135285	−	0.990807i	$-0.456805\pi$
$857$	7.92083	0.270570	0.135285	+	0.990807i	$0.456805\pi$
$858$	0	0
$859$	8.42793	0.287557	0.143779	−	0.989610i	$-0.454075\pi$
$859$	8.42793	0.287557	0.143779	+	0.989610i	$0.454075\pi$
$860$	0	0
$861$	−1.55350	−0.0529430
$862$	0	0
$863$	−28.2751	−0.962495	−0.481248	−	0.876585i	$-0.659816\pi$
$863$	−28.2751	−0.962495	−0.481248	+	0.876585i	$0.659816\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.56322	−0.154975
$868$	0	0
$869$	−9.39213	−0.318606
$870$	0	0
$871$	−7.21930	−0.244617
$872$	0	0
$873$	25.9848	0.879453
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.0169	1.08113	0.540567	−	0.841301i	$-0.318210\pi$
$877$	32.0169	1.08113	0.540567	+	0.841301i	$0.318210\pi$
$878$	0	0
$879$	21.6811	0.731286
$880$	0	0
$881$	55.6183	1.87383	0.936914	−	0.349560i	$-0.113669\pi$
$881$	55.6183	1.87383	0.936914	+	0.349560i	$0.113669\pi$
$882$	0	0
$883$	−16.1272	−0.542723	−0.271362	−	0.962477i	$-0.587474\pi$
$883$	−16.1272	−0.542723	−0.271362	+	0.962477i	$0.587474\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.9850	0.503146	0.251573	−	0.967838i	$-0.419052\pi$
$887$	14.9850	0.503146	0.251573	+	0.967838i	$0.419052\pi$
$888$	0	0
$889$	−4.09320	−0.137281
$890$	0	0
$891$	18.3608	0.615111
$892$	0	0
$893$	−3.34735	−0.112015
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.3435	0.545692
$898$	0	0
$899$	8.28375	0.276279
$900$	0	0
$901$	−0.524655	−0.0174788
$902$	0	0
$903$	0.139256	0.00463414
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−45.0654	−1.49637	−0.748185	−	0.663490i	$-0.769075\pi$
$907$	−45.0654	−1.49637	−0.748185	+	0.663490i	$0.769075\pi$
$908$	0	0
$909$	−42.2391	−1.40098
$910$	0	0
$911$	−3.86130	−0.127930	−0.0639652	−	0.997952i	$-0.520375\pi$
$911$	−3.86130	−0.127930	−0.0639652	+	0.997952i	$0.520375\pi$
$912$	0	0
$913$	−6.32358	−0.209280
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.62680	−0.251859
$918$	0	0
$919$	−4.62296	−0.152497	−0.0762487	−	0.997089i	$-0.524294\pi$
$919$	−4.62296	−0.152497	−0.0762487	+	0.997089i	$0.524294\pi$
$920$	0	0
$921$	3.22548	0.106283
$922$	0	0
$923$	−1.73029	−0.0569533
$924$	0	0
$925$	0	0
$926$	0	0
$927$	29.4778	0.968177
$928$	0	0
$929$	−31.0613	−1.01909	−0.509543	−	0.860445i	$-0.670186\pi$
$929$	−31.0613	−1.01909	−0.509543	+	0.860445i	$0.670186\pi$
$930$	0	0
$931$	−6.88042	−0.225497
$932$	0	0
$933$	0.338689	0.0110882
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.5290	0.899334	0.449667	−	0.893196i	$-0.351543\pi$
$937$	27.5290	0.899334	0.449667	+	0.893196i	$0.351543\pi$
$938$	0	0
$939$	3.54363	0.115642
$940$	0	0
$941$	−40.2018	−1.31054	−0.655271	−	0.755394i	$-0.727445\pi$
$941$	−40.2018	−1.31054	−0.655271	+	0.755394i	$0.727445\pi$
$942$	0	0
$943$	−51.5819	−1.67974
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.8753	−0.613363	−0.306682	−	0.951812i	$-0.599219\pi$
$947$	−18.8753	−0.613363	−0.306682	+	0.951812i	$0.599219\pi$
$948$	0	0
$949$	−17.9835	−0.583768
$950$	0	0
$951$	−10.0541	−0.326028
$952$	0	0
$953$	−46.2885	−1.49943	−0.749715	−	0.661760i	$-0.769810\pi$
$953$	−46.2885	−1.49943	−0.749715	+	0.661760i	$0.769810\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.06410	0.131374
$958$	0	0
$959$	−6.73815	−0.217586
$960$	0	0
$961$	−8.32676	−0.268605
$962$	0	0
$963$	25.6727	0.827292
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.98996	0.224782	0.112391	−	0.993664i	$-0.464149\pi$
$967$	6.98996	0.224782	0.112391	+	0.993664i	$0.464149\pi$
$968$	0	0
$969$	3.24596	0.104275
$970$	0	0
$971$	2.02077	0.0648496	0.0324248	−	0.999474i	$-0.489677\pi$
$971$	2.02077	0.0648496	0.0324248	+	0.999474i	$0.489677\pi$
$972$	0	0
$973$	−5.04406	−0.161705
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.12836	0.196064	0.0980318	−	0.995183i	$-0.468745\pi$
$977$	6.12836	0.196064	0.0980318	+	0.995183i	$0.468745\pi$
$978$	0	0
$979$	31.4508	1.00517
$980$	0	0
$981$	−31.4030	−1.00262
$982$	0	0
$983$	−30.8941	−0.985370	−0.492685	−	0.870208i	$-0.663984\pi$
$983$	−30.8941	−0.985370	−0.492685	+	0.870208i	$0.663984\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.769057	0.0244793
$988$	0	0
$989$	4.62381	0.147029
$990$	0	0
$991$	−4.01803	−0.127637	−0.0638185	−	0.997962i	$-0.520328\pi$
$991$	−4.01803	−0.127637	−0.0638185	+	0.997962i	$0.520328\pi$
$992$	0	0
$993$	−2.79966	−0.0888446
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−5.96552	−0.188930	−0.0944650	−	0.995528i	$-0.530114\pi$
$997$	−5.96552	−0.188930	−0.0944650	+	0.995528i	$0.530114\pi$
$998$	0	0
$999$	6.89690	0.218208

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.z.1.3		6
4.3	odd	2		7600.2.a.cn.1.4		6
5.2	odd	4		760.2.d.e.609.8	yes	12
5.3	odd	4		760.2.d.e.609.5	✓	12
5.4	even	2		3800.2.a.be.1.4		6
20.3	even	4		1520.2.d.k.609.8		12
20.7	even	4		1520.2.d.k.609.5		12
20.19	odd	2		7600.2.a.cg.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.e.609.5	✓	12	5.3	odd	4
760.2.d.e.609.8	yes	12	5.2	odd	4
1520.2.d.k.609.5		12	20.7	even	4
1520.2.d.k.609.8		12	20.3	even	4
3800.2.a.z.1.3		6	1.1	even	1	trivial
3800.2.a.be.1.4		6	5.4	even	2
7600.2.a.cg.1.3		6	20.19	odd	2
7600.2.a.cn.1.4		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.664406 q^{3} +0.345799 q^{7} -2.55856 q^{9} +3.51609 q^{11} +3.22452 q^{13} -4.88550 q^{17} +1.00000 q^{19} -0.229751 q^{21} -7.62861 q^{23} +3.69315 q^{27} -1.73968 q^{29} -4.76164 q^{31} -2.33611 q^{33} +1.86749 q^{37} -2.14239 q^{39} +6.76164 q^{41} -0.606115 q^{43} -3.34735 q^{47} -6.88042 q^{49} +3.24596 q^{51} +0.107390 q^{53} -0.664406 q^{57} +12.3233 q^{59} -8.41231 q^{61} -0.884749 q^{63} -2.23888 q^{67} +5.06850 q^{69} -0.536605 q^{71} -5.57710 q^{73} +1.21586 q^{77} -2.67119 q^{79} +5.22194 q^{81} -1.79847 q^{83} +1.15586 q^{87} +8.94482 q^{89} +1.11504 q^{91} +3.16367 q^{93} -10.1560 q^{97} -8.99614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47}+ \cdots - 22 q^{99}+O(q^{100}) \) 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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