LMFDB - Embedded newform 3800.2.a.o.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.41421 q^{3} +4.82843 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} +4.82843 q^{7} -1.00000 q^{9} +4.82843 q^{11} +0.585786 q^{13} +2.82843 q^{17} -1.00000 q^{19} +6.82843 q^{21} -7.65685 q^{23} -5.65685 q^{27} +3.65685 q^{29} +6.82843 q^{31} +6.82843 q^{33} -0.585786 q^{37} +0.828427 q^{39} +0.828427 q^{41} +8.82843 q^{43} -0.828427 q^{47} +16.3137 q^{49} +4.00000 q^{51} +11.8995 q^{53} -1.41421 q^{57} -6.82843 q^{59} -5.65685 q^{61} -4.82843 q^{63} -9.89949 q^{67} -10.8284 q^{69} -8.48528 q^{71} +1.17157 q^{73} +23.3137 q^{77} +8.48528 q^{79} -5.00000 q^{81} -15.6569 q^{83} +5.17157 q^{87} -13.3137 q^{89} +2.82843 q^{91} +9.65685 q^{93} -1.07107 q^{97} -4.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^7 - 2 * q^9	$ 2 q + 4 q^{7} - 2 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{19} + 8 q^{21} - 4 q^{23} - 4 q^{29} + 8 q^{31} + 8 q^{33} - 4 q^{37} - 4 q^{39} - 4 q^{41} + 12 q^{43} + 4 q^{47} + 10 q^{49} + 8 q^{51} + 4 q^{53} - 8 q^{59} - 4 q^{63} - 16 q^{69} + 8 q^{73} + 24 q^{77} - 10 q^{81} - 20 q^{83} + 16 q^{87} - 4 q^{89} + 8 q^{93} + 12 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^7 - 2 * q^9 + 4 * q^11 + 4 * q^13 - 2 * q^19 + 8 * q^21 - 4 * q^23 - 4 * q^29 + 8 * q^31 + 8 * q^33 - 4 * q^37 - 4 * q^39 - 4 * q^41 + 12 * q^43 + 4 * q^47 + 10 * q^49 + 8 * q^51 + 4 * q^53 - 8 * q^59 - 4 * q^63 - 16 * q^69 + 8 * q^73 + 24 * q^77 - 10 * q^81 - 20 * q^83 + 16 * q^87 - 4 * q^89 + 8 * q^93 + 12 * q^97 - 4 * 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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0	0
$13$	0.585786	0.162468	0.0812340	−	0.996695i	$-0.474114\pi$
$13$	0.585786	0.162468	0.0812340	+	0.996695i	$0.474114\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	6.82843	1.49008
$22$	0	0
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	6.82843	1.22642	0.613211	−	0.789919i	$-0.289878\pi$
$31$	6.82843	1.22642	0.613211	+	0.789919i	$0.289878\pi$
$32$	0	0
$33$	6.82843	1.18868
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.585786	−0.0963027	−0.0481513	−	0.998840i	$-0.515333\pi$
$37$	−0.585786	−0.0963027	−0.0481513	+	0.998840i	$0.515333\pi$
$38$	0	0
$39$	0.828427	0.132655
$40$	0	0
$41$	0.828427	0.129379	0.0646893	−	0.997905i	$-0.479394\pi$
$41$	0.828427	0.129379	0.0646893	+	0.997905i	$0.479394\pi$
$42$	0	0
$43$	8.82843	1.34632	0.673161	−	0.739496i	$-0.264936\pi$
$43$	8.82843	1.34632	0.673161	+	0.739496i	$0.264936\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.828427	−0.120839	−0.0604193	−	0.998173i	$-0.519244\pi$
$47$	−0.828427	−0.120839	−0.0604193	+	0.998173i	$0.519244\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	11.8995	1.63452	0.817261	−	0.576268i	$-0.195492\pi$
$53$	11.8995	1.63452	0.817261	+	0.576268i	$0.195492\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.41421	−0.187317
$58$	0	0
$59$	−6.82843	−0.888985	−0.444493	−	0.895782i	$-0.646616\pi$
$59$	−6.82843	−0.888985	−0.444493	+	0.895782i	$0.646616\pi$
$60$	0	0
$61$	−5.65685	−0.724286	−0.362143	−	0.932123i	$-0.617955\pi$
$61$	−5.65685	−0.724286	−0.362143	+	0.932123i	$0.617955\pi$
$62$	0	0
$63$	−4.82843	−0.608325
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.89949	−1.20942	−0.604708	−	0.796447i	$-0.706710\pi$
$67$	−9.89949	−1.20942	−0.604708	+	0.796447i	$0.706710\pi$
$68$	0	0
$69$	−10.8284	−1.30359
$70$	0	0
$71$	−8.48528	−1.00702	−0.503509	−	0.863990i	$-0.667958\pi$
$71$	−8.48528	−1.00702	−0.503509	+	0.863990i	$0.667958\pi$
$72$	0	0
$73$	1.17157	0.137122	0.0685611	−	0.997647i	$-0.478159\pi$
$73$	1.17157	0.137122	0.0685611	+	0.997647i	$0.478159\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	23.3137	2.65684
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−15.6569	−1.71856	−0.859282	−	0.511503i	$-0.829089\pi$
$83$	−15.6569	−1.71856	−0.859282	+	0.511503i	$0.829089\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.17157	0.554451
$88$	0	0
$89$	−13.3137	−1.41125	−0.705625	−	0.708585i	$-0.749334\pi$
$89$	−13.3137	−1.41125	−0.705625	+	0.708585i	$0.749334\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	9.65685	1.00137
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.07107	−0.108750	−0.0543752	−	0.998521i	$-0.517317\pi$
$97$	−1.07107	−0.108750	−0.0543752	+	0.998521i	$0.517317\pi$
$98$	0	0
$99$	−4.82843	−0.485275
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	−9.89949	−0.975426	−0.487713	−	0.873004i	$-0.662169\pi$
$103$	−9.89949	−0.975426	−0.487713	+	0.873004i	$0.662169\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.2426	1.57024	0.785118	−	0.619347i	$-0.212602\pi$
$107$	16.2426	1.57024	0.785118	+	0.619347i	$0.212602\pi$
$108$	0	0
$109$	−10.4853	−1.00431	−0.502154	−	0.864778i	$-0.667459\pi$
$109$	−10.4853	−1.00431	−0.502154	+	0.864778i	$0.667459\pi$
$110$	0	0
$111$	−0.828427	−0.0786308
$112$	0	0
$113$	11.4142	1.07376	0.536879	−	0.843659i	$-0.319603\pi$
$113$	11.4142	1.07376	0.536879	+	0.843659i	$0.319603\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.585786	−0.0541560
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	1.17157	0.105637
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.7279	1.12942	0.564710	−	0.825289i	$-0.308988\pi$
$127$	12.7279	1.12942	0.564710	+	0.825289i	$0.308988\pi$
$128$	0	0
$129$	12.4853	1.09927
$130$	0	0
$131$	−17.6569	−1.54269	−0.771343	−	0.636419i	$-0.780415\pi$
$131$	−17.6569	−1.54269	−0.771343	+	0.636419i	$0.780415\pi$
$132$	0	0
$133$	−4.82843	−0.418678
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.82843	0.241649	0.120824	−	0.992674i	$-0.461446\pi$
$137$	2.82843	0.241649	0.120824	+	0.992674i	$0.461446\pi$
$138$	0	0
$139$	15.1716	1.28684	0.643418	−	0.765515i	$-0.277516\pi$
$139$	15.1716	1.28684	0.643418	+	0.765515i	$0.277516\pi$
$140$	0	0
$141$	−1.17157	−0.0986642
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	0	0
$146$	0	0
$147$	23.0711	1.90287
$148$	0	0
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	−2.34315	−0.190682	−0.0953412	−	0.995445i	$-0.530394\pi$
$151$	−2.34315	−0.190682	−0.0953412	+	0.995445i	$0.530394\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.48528	−0.357964	−0.178982	−	0.983852i	$-0.557280\pi$
$157$	−4.48528	−0.357964	−0.178982	+	0.983852i	$0.557280\pi$
$158$	0	0
$159$	16.8284	1.33458
$160$	0	0
$161$	−36.9706	−2.91369
$162$	0	0
$163$	14.4853	1.13457	0.567287	−	0.823520i	$-0.307993\pi$
$163$	14.4853	1.13457	0.567287	+	0.823520i	$0.307993\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.07107	−0.237646	−0.118823	−	0.992915i	$-0.537912\pi$
$167$	−3.07107	−0.237646	−0.118823	+	0.992915i	$0.537912\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−8.58579	−0.652765	−0.326383	−	0.945238i	$-0.605830\pi$
$173$	−8.58579	−0.652765	−0.326383	+	0.945238i	$0.605830\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.65685	−0.725854
$178$	0	0
$179$	22.8284	1.70628	0.853138	−	0.521685i	$-0.174696\pi$
$179$	22.8284	1.70628	0.853138	+	0.521685i	$0.174696\pi$
$180$	0	0
$181$	2.68629	0.199670	0.0998352	−	0.995004i	$-0.468168\pi$
$181$	2.68629	0.199670	0.0998352	+	0.995004i	$0.468168\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.6569	0.998688
$188$	0	0
$189$	−27.3137	−1.98678
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	−1.07107	−0.0770971	−0.0385486	−	0.999257i	$-0.512273\pi$
$193$	−1.07107	−0.0770971	−0.0385486	+	0.999257i	$0.512273\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.3431	0.736919	0.368459	−	0.929644i	$-0.379885\pi$
$197$	10.3431	0.736919	0.368459	+	0.929644i	$0.379885\pi$
$198$	0	0
$199$	−11.3137	−0.802008	−0.401004	−	0.916076i	$-0.631339\pi$
$199$	−11.3137	−0.802008	−0.401004	+	0.916076i	$0.631339\pi$
$200$	0	0
$201$	−14.0000	−0.987484
$202$	0	0
$203$	17.6569	1.23927
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.65685	0.532188
$208$	0	0
$209$	−4.82843	−0.333989
$210$	0	0
$211$	−22.6274	−1.55774	−0.778868	−	0.627188i	$-0.784206\pi$
$211$	−22.6274	−1.55774	−0.778868	+	0.627188i	$0.784206\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	0	0
$216$	0	0
$217$	32.9706	2.23819
$218$	0	0
$219$	1.65685	0.111960
$220$	0	0
$221$	1.65685	0.111452
$222$	0	0
$223$	9.89949	0.662919	0.331460	−	0.943469i	$-0.392459\pi$
$223$	9.89949	0.662919	0.331460	+	0.943469i	$0.392459\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.92893	0.327145	0.163572	−	0.986531i	$-0.447698\pi$
$227$	4.92893	0.327145	0.163572	+	0.986531i	$0.447698\pi$
$228$	0	0
$229$	3.31371	0.218976	0.109488	−	0.993988i	$-0.465079\pi$
$229$	3.31371	0.218976	0.109488	+	0.993988i	$0.465079\pi$
$230$	0	0
$231$	32.9706	2.16930
$232$	0	0
$233$	28.9706	1.89792	0.948962	−	0.315389i	$-0.102135\pi$
$233$	28.9706	1.89792	0.948962	+	0.315389i	$0.102135\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	−8.97056	−0.580257	−0.290129	−	0.956988i	$-0.593698\pi$
$239$	−8.97056	−0.580257	−0.290129	+	0.956988i	$0.593698\pi$
$240$	0	0
$241$	3.17157	0.204299	0.102149	−	0.994769i	$-0.467428\pi$
$241$	3.17157	0.204299	0.102149	+	0.994769i	$0.467428\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.585786	−0.0372727
$248$	0	0
$249$	−22.1421	−1.40320
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−36.9706	−2.32432
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.2426	−0.638918	−0.319459	−	0.947600i	$-0.603501\pi$
$257$	−10.2426	−0.638918	−0.319459	+	0.947600i	$0.603501\pi$
$258$	0	0
$259$	−2.82843	−0.175750
$260$	0	0
$261$	−3.65685	−0.226354
$262$	0	0
$263$	−5.31371	−0.327657	−0.163829	−	0.986489i	$-0.552384\pi$
$263$	−5.31371	−0.327657	−0.163829	+	0.986489i	$0.552384\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−18.8284	−1.15228
$268$	0	0
$269$	20.1421	1.22809	0.614044	−	0.789272i	$-0.289542\pi$
$269$	20.1421	1.22809	0.614044	+	0.789272i	$0.289542\pi$
$270$	0	0
$271$	19.1716	1.16459	0.582295	−	0.812978i	$-0.302155\pi$
$271$	19.1716	1.16459	0.582295	+	0.812978i	$0.302155\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.17157	0.551066	0.275533	−	0.961292i	$-0.411146\pi$
$277$	9.17157	0.551066	0.275533	+	0.961292i	$0.411146\pi$
$278$	0	0
$279$	−6.82843	−0.408807
$280$	0	0
$281$	−22.4853	−1.34136	−0.670680	−	0.741747i	$-0.733997\pi$
$281$	−22.4853	−1.34136	−0.670680	+	0.741747i	$0.733997\pi$
$282$	0	0
$283$	−2.97056	−0.176582	−0.0882908	−	0.996095i	$-0.528140\pi$
$283$	−2.97056	−0.176582	−0.0882908	+	0.996095i	$0.528140\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	−1.51472	−0.0887944
$292$	0	0
$293$	−9.75736	−0.570031	−0.285016	−	0.958523i	$-0.591999\pi$
$293$	−9.75736	−0.570031	−0.285016	+	0.958523i	$0.591999\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−27.3137	−1.58490
$298$	0	0
$299$	−4.48528	−0.259391
$300$	0	0
$301$	42.6274	2.45700
$302$	0	0
$303$	−11.3137	−0.649956
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.58579	0.147579	0.0737893	−	0.997274i	$-0.476491\pi$
$307$	2.58579	0.147579	0.0737893	+	0.997274i	$0.476491\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	0	0
$311$	−23.1716	−1.31394	−0.656970	−	0.753917i	$-0.728162\pi$
$311$	−23.1716	−1.31394	−0.656970	+	0.753917i	$0.728162\pi$
$312$	0	0
$313$	20.9706	1.18533	0.592663	−	0.805450i	$-0.298077\pi$
$313$	20.9706	1.18533	0.592663	+	0.805450i	$0.298077\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.7279	−0.602540	−0.301270	−	0.953539i	$-0.597411\pi$
$317$	−10.7279	−0.602540	−0.301270	+	0.953539i	$0.597411\pi$
$318$	0	0
$319$	17.6569	0.988594
$320$	0	0
$321$	22.9706	1.28209
$322$	0	0
$323$	−2.82843	−0.157378
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−14.8284	−0.820014
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−13.6569	−0.750649	−0.375324	−	0.926894i	$-0.622469\pi$
$331$	−13.6569	−0.750649	−0.375324	+	0.926894i	$0.622469\pi$
$332$	0	0
$333$	0.585786	0.0321009
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.8995	−0.866101	−0.433050	−	0.901370i	$-0.642563\pi$
$337$	−15.8995	−0.866101	−0.433050	+	0.901370i	$0.642563\pi$
$338$	0	0
$339$	16.1421	0.876720
$340$	0	0
$341$	32.9706	1.78546
$342$	0	0
$343$	44.9706	2.42818
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	28.6274	1.53239	0.766195	−	0.642608i	$-0.222148\pi$
$349$	28.6274	1.53239	0.766195	+	0.642608i	$0.222148\pi$
$350$	0	0
$351$	−3.31371	−0.176873
$352$	0	0
$353$	−14.1421	−0.752710	−0.376355	−	0.926476i	$-0.622823\pi$
$353$	−14.1421	−0.752710	−0.376355	+	0.926476i	$0.622823\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	19.3137	1.02219
$358$	0	0
$359$	−10.4853	−0.553392	−0.276696	−	0.960958i	$-0.589239\pi$
$359$	−10.4853	−0.553392	−0.276696	+	0.960958i	$0.589239\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	17.4142	0.914009
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.1421	1.05141	0.525705	−	0.850667i	$-0.323801\pi$
$367$	20.1421	1.05141	0.525705	+	0.850667i	$0.323801\pi$
$368$	0	0
$369$	−0.828427	−0.0431262
$370$	0	0
$371$	57.4558	2.98296
$372$	0	0
$373$	27.8995	1.44458	0.722291	−	0.691590i	$-0.243089\pi$
$373$	27.8995	1.44458	0.722291	+	0.691590i	$0.243089\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.14214	0.110326
$378$	0	0
$379$	−23.3137	−1.19754	−0.598772	−	0.800919i	$-0.704345\pi$
$379$	−23.3137	−1.19754	−0.598772	+	0.800919i	$0.704345\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	0	0
$383$	−13.4142	−0.685434	−0.342717	−	0.939439i	$-0.611347\pi$
$383$	−13.4142	−0.685434	−0.342717	+	0.939439i	$0.611347\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.82843	−0.448774
$388$	0	0
$389$	−9.31371	−0.472224	−0.236112	−	0.971726i	$-0.575873\pi$
$389$	−9.31371	−0.472224	−0.236112	+	0.971726i	$0.575873\pi$
$390$	0	0
$391$	−21.6569	−1.09523
$392$	0	0
$393$	−24.9706	−1.25960
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13.1716	0.661062	0.330531	−	0.943795i	$-0.392772\pi$
$397$	13.1716	0.661062	0.330531	+	0.943795i	$0.392772\pi$
$398$	0	0
$399$	−6.82843	−0.341849
$400$	0	0
$401$	−26.9706	−1.34685	−0.673423	−	0.739258i	$-0.735177\pi$
$401$	−26.9706	−1.34685	−0.673423	+	0.739258i	$0.735177\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.82843	−0.140200
$408$	0	0
$409$	−36.8284	−1.82105	−0.910524	−	0.413456i	$-0.864322\pi$
$409$	−36.8284	−1.82105	−0.910524	+	0.413456i	$0.864322\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	0	0
$413$	−32.9706	−1.62238
$414$	0	0
$415$	0	0
$416$	0	0
$417$	21.4558	1.05070
$418$	0	0
$419$	−10.6274	−0.519183	−0.259592	−	0.965718i	$-0.583588\pi$
$419$	−10.6274	−0.519183	−0.259592	+	0.965718i	$0.583588\pi$
$420$	0	0
$421$	−1.79899	−0.0876774	−0.0438387	−	0.999039i	$-0.513959\pi$
$421$	−1.79899	−0.0876774	−0.0438387	+	0.999039i	$0.513959\pi$
$422$	0	0
$423$	0.828427	0.0402795
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−27.3137	−1.32180
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	−29.9411	−1.44221	−0.721107	−	0.692824i	$-0.756366\pi$
$431$	−29.9411	−1.44221	−0.721107	+	0.692824i	$0.756366\pi$
$432$	0	0
$433$	−37.5563	−1.80484	−0.902421	−	0.430854i	$-0.858212\pi$
$433$	−37.5563	−1.80484	−0.902421	+	0.430854i	$0.858212\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.65685	0.366277
$438$	0	0
$439$	28.4853	1.35953	0.679764	−	0.733431i	$-0.262082\pi$
$439$	28.4853	1.35953	0.679764	+	0.733431i	$0.262082\pi$
$440$	0	0
$441$	−16.3137	−0.776843
$442$	0	0
$443$	16.6274	0.789992	0.394996	−	0.918683i	$-0.370746\pi$
$443$	16.6274	0.789992	0.394996	+	0.918683i	$0.370746\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.65685	0.267560
$448$	0	0
$449$	−26.4853	−1.24992	−0.624959	−	0.780658i	$-0.714884\pi$
$449$	−26.4853	−1.24992	−0.624959	+	0.780658i	$0.714884\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	−3.31371	−0.155692
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.97056	−0.419625	−0.209813	−	0.977742i	$-0.567285\pi$
$457$	−8.97056	−0.419625	−0.209813	+	0.977742i	$0.567285\pi$
$458$	0	0
$459$	−16.0000	−0.746816
$460$	0	0
$461$	−5.31371	−0.247484	−0.123742	−	0.992314i	$-0.539490\pi$
$461$	−5.31371	−0.247484	−0.123742	+	0.992314i	$0.539490\pi$
$462$	0	0
$463$	30.9706	1.43932	0.719662	−	0.694325i	$-0.244297\pi$
$463$	30.9706	1.43932	0.719662	+	0.694325i	$0.244297\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.6569	0.909611	0.454805	−	0.890591i	$-0.349709\pi$
$467$	19.6569	0.909611	0.454805	+	0.890591i	$0.349709\pi$
$468$	0	0
$469$	−47.7990	−2.20715
$470$	0	0
$471$	−6.34315	−0.292277
$472$	0	0
$473$	42.6274	1.96001
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.8995	−0.544840
$478$	0	0
$479$	−12.8284	−0.586146	−0.293073	−	0.956090i	$-0.594678\pi$
$479$	−12.8284	−0.586146	−0.293073	+	0.956090i	$0.594678\pi$
$480$	0	0
$481$	−0.343146	−0.0156461
$482$	0	0
$483$	−52.2843	−2.37902
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.27208	−0.329529	−0.164765	−	0.986333i	$-0.552686\pi$
$487$	−7.27208	−0.329529	−0.164765	+	0.986333i	$0.552686\pi$
$488$	0	0
$489$	20.4853	0.926376
$490$	0	0
$491$	6.34315	0.286262	0.143131	−	0.989704i	$-0.454283\pi$
$491$	6.34315	0.286262	0.143131	+	0.989704i	$0.454283\pi$
$492$	0	0
$493$	10.3431	0.465832
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−40.9706	−1.83778
$498$	0	0
$499$	15.4558	0.691899	0.345949	−	0.938253i	$-0.387557\pi$
$499$	15.4558	0.691899	0.345949	+	0.938253i	$0.387557\pi$
$500$	0	0
$501$	−4.34315	−0.194037
$502$	0	0
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−17.8995	−0.794944
$508$	0	0
$509$	−43.9411	−1.94766	−0.973828	−	0.227286i	$-0.927015\pi$
$509$	−43.9411	−1.94766	−0.973828	+	0.227286i	$0.927015\pi$
$510$	0	0
$511$	5.65685	0.250244
$512$	0	0
$513$	5.65685	0.249756
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	−12.1421	−0.532981
$520$	0	0
$521$	−24.3431	−1.06649	−0.533246	−	0.845960i	$-0.679028\pi$
$521$	−24.3431	−1.06649	−0.533246	+	0.845960i	$0.679028\pi$
$522$	0	0
$523$	34.5858	1.51233	0.756165	−	0.654381i	$-0.227071\pi$
$523$	34.5858	1.51233	0.756165	+	0.654381i	$0.227071\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.3137	0.841318
$528$	0	0
$529$	35.6274	1.54902
$530$	0	0
$531$	6.82843	0.296328
$532$	0	0
$533$	0.485281	0.0210199
$534$	0	0
$535$	0	0
$536$	0	0
$537$	32.2843	1.39317
$538$	0	0
$539$	78.7696	3.39284
$540$	0	0
$541$	4.00000	0.171973	0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000	0.171973	0.0859867	+	0.996296i	$0.472596\pi$
$542$	0	0
$543$	3.79899	0.163030
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.24264	−0.352430	−0.176215	−	0.984352i	$-0.556385\pi$
$547$	−8.24264	−0.352430	−0.176215	+	0.984352i	$0.556385\pi$
$548$	0	0
$549$	5.65685	0.241429
$550$	0	0
$551$	−3.65685	−0.155787
$552$	0	0
$553$	40.9706	1.74225
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.8284	0.628301	0.314150	−	0.949373i	$-0.398280\pi$
$557$	14.8284	0.628301	0.314150	+	0.949373i	$0.398280\pi$
$558$	0	0
$559$	5.17157	0.218734
$560$	0	0
$561$	19.3137	0.815425
$562$	0	0
$563$	−45.4142	−1.91398	−0.956990	−	0.290119i	$-0.906305\pi$
$563$	−45.4142	−1.91398	−0.956990	+	0.290119i	$0.906305\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−24.1421	−1.01387
$568$	0	0
$569$	11.4558	0.480254	0.240127	−	0.970741i	$-0.422811\pi$
$569$	11.4558	0.480254	0.240127	+	0.970741i	$0.422811\pi$
$570$	0	0
$571$	−47.4558	−1.98597	−0.992983	−	0.118260i	$-0.962268\pi$
$571$	−47.4558	−1.98597	−0.992983	+	0.118260i	$0.962268\pi$
$572$	0	0
$573$	27.3137	1.14105
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.34315	0.264069	0.132034	−	0.991245i	$-0.457849\pi$
$577$	6.34315	0.264069	0.132034	+	0.991245i	$0.457849\pi$
$578$	0	0
$579$	−1.51472	−0.0629496
$580$	0	0
$581$	−75.5980	−3.13633
$582$	0	0
$583$	57.4558	2.37958
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	−6.82843	−0.281360
$590$	0	0
$591$	14.6274	0.601692
$592$	0	0
$593$	−14.6274	−0.600676	−0.300338	−	0.953833i	$-0.597099\pi$
$593$	−14.6274	−0.600676	−0.300338	+	0.953833i	$0.597099\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	45.4558	1.85728	0.928638	−	0.370988i	$-0.120981\pi$
$599$	45.4558	1.85728	0.928638	+	0.370988i	$0.120981\pi$
$600$	0	0
$601$	−11.8579	−0.483692	−0.241846	−	0.970315i	$-0.577753\pi$
$601$	−11.8579	−0.483692	−0.241846	+	0.970315i	$0.577753\pi$
$602$	0	0
$603$	9.89949	0.403139
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−41.2132	−1.67279	−0.836396	−	0.548125i	$-0.815342\pi$
$607$	−41.2132	−1.67279	−0.836396	+	0.548125i	$0.815342\pi$
$608$	0	0
$609$	24.9706	1.01186
$610$	0	0
$611$	−0.485281	−0.0196324
$612$	0	0
$613$	−24.4853	−0.988951	−0.494476	−	0.869192i	$-0.664640\pi$
$613$	−24.4853	−0.988951	−0.494476	+	0.869192i	$0.664640\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.82843	−0.274902	−0.137451	−	0.990509i	$-0.543891\pi$
$617$	−6.82843	−0.274902	−0.137451	+	0.990509i	$0.543891\pi$
$618$	0	0
$619$	31.1716	1.25289	0.626446	−	0.779465i	$-0.284509\pi$
$619$	31.1716	1.25289	0.626446	+	0.779465i	$0.284509\pi$
$620$	0	0
$621$	43.3137	1.73812
$622$	0	0
$623$	−64.2843	−2.57549
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.82843	−0.272701
$628$	0	0
$629$	−1.65685	−0.0660631
$630$	0	0
$631$	−21.7990	−0.867804	−0.433902	−	0.900960i	$-0.642864\pi$
$631$	−21.7990	−0.867804	−0.433902	+	0.900960i	$0.642864\pi$
$632$	0	0
$633$	−32.0000	−1.27189
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.55635	0.378636
$638$	0	0
$639$	8.48528	0.335673
$640$	0	0
$641$	−11.8579	−0.468357	−0.234179	−	0.972194i	$-0.575240\pi$
$641$	−11.8579	−0.468357	−0.234179	+	0.972194i	$0.575240\pi$
$642$	0	0
$643$	10.9706	0.432637	0.216318	−	0.976323i	$-0.430595\pi$
$643$	10.9706	0.432637	0.216318	+	0.976323i	$0.430595\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.2843	0.404316	0.202158	−	0.979353i	$-0.435204\pi$
$647$	10.2843	0.404316	0.202158	+	0.979353i	$0.435204\pi$
$648$	0	0
$649$	−32.9706	−1.29421
$650$	0	0
$651$	46.6274	1.82747
$652$	0	0
$653$	−2.82843	−0.110685	−0.0553425	−	0.998467i	$-0.517625\pi$
$653$	−2.82843	−0.110685	−0.0553425	+	0.998467i	$0.517625\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.17157	−0.0457074
$658$	0	0
$659$	−32.4853	−1.26545	−0.632723	−	0.774378i	$-0.718063\pi$
$659$	−32.4853	−1.26545	−0.632723	+	0.774378i	$0.718063\pi$
$660$	0	0
$661$	27.4558	1.06791	0.533954	−	0.845513i	$-0.320705\pi$
$661$	27.4558	1.06791	0.533954	+	0.845513i	$0.320705\pi$
$662$	0	0
$663$	2.34315	0.0910002
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28.0000	−1.08416
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	−27.3137	−1.05443
$672$	0	0
$673$	−21.7574	−0.838685	−0.419342	−	0.907828i	$-0.637739\pi$
$673$	−21.7574	−0.838685	−0.419342	+	0.907828i	$0.637739\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.1005	−0.465060	−0.232530	−	0.972589i	$-0.574700\pi$
$677$	−12.1005	−0.465060	−0.232530	+	0.972589i	$0.574700\pi$
$678$	0	0
$679$	−5.17157	−0.198467
$680$	0	0
$681$	6.97056	0.267113
$682$	0	0
$683$	−35.0711	−1.34196	−0.670979	−	0.741477i	$-0.734126\pi$
$683$	−35.0711	−1.34196	−0.670979	+	0.741477i	$0.734126\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.68629	0.178793
$688$	0	0
$689$	6.97056	0.265557
$690$	0	0
$691$	38.7696	1.47486	0.737432	−	0.675422i	$-0.236038\pi$
$691$	38.7696	1.47486	0.737432	+	0.675422i	$0.236038\pi$
$692$	0	0
$693$	−23.3137	−0.885615
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.34315	0.0887530
$698$	0	0
$699$	40.9706	1.54965
$700$	0	0
$701$	31.3137	1.18270	0.591351	−	0.806414i	$-0.298595\pi$
$701$	31.3137	1.18270	0.591351	+	0.806414i	$0.298595\pi$
$702$	0	0
$703$	0.585786	0.0220934
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−38.6274	−1.45273
$708$	0	0
$709$	33.3137	1.25112	0.625561	−	0.780175i	$-0.284870\pi$
$709$	33.3137	1.25112	0.625561	+	0.780175i	$0.284870\pi$
$710$	0	0
$711$	−8.48528	−0.318223
$712$	0	0
$713$	−52.2843	−1.95806
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12.6863	−0.473778
$718$	0	0
$719$	2.48528	0.0926854	0.0463427	−	0.998926i	$-0.485243\pi$
$719$	2.48528	0.0926854	0.0463427	+	0.998926i	$0.485243\pi$
$720$	0	0
$721$	−47.7990	−1.78013
$722$	0	0
$723$	4.48528	0.166809
$724$	0	0
$725$	0	0
$726$	0	0
$727$	31.4558	1.16663	0.583316	−	0.812245i	$-0.301755\pi$
$727$	31.4558	1.16663	0.583316	+	0.812245i	$0.301755\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	24.9706	0.923570
$732$	0	0
$733$	12.6863	0.468579	0.234289	−	0.972167i	$-0.424724\pi$
$733$	12.6863	0.468579	0.234289	+	0.972167i	$0.424724\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−47.7990	−1.76070
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−0.828427	−0.0304330
$742$	0	0
$743$	2.10051	0.0770601	0.0385300	−	0.999257i	$-0.487732\pi$
$743$	2.10051	0.0770601	0.0385300	+	0.999257i	$0.487732\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.6569	0.572854
$748$	0	0
$749$	78.4264	2.86564
$750$	0	0
$751$	43.5980	1.59091	0.795456	−	0.606011i	$-0.207231\pi$
$751$	43.5980	1.59091	0.795456	+	0.606011i	$0.207231\pi$
$752$	0	0
$753$	5.65685	0.206147
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−38.6274	−1.40394	−0.701969	−	0.712208i	$-0.747695\pi$
$757$	−38.6274	−1.40394	−0.701969	+	0.712208i	$0.747695\pi$
$758$	0	0
$759$	−52.2843	−1.89780
$760$	0	0
$761$	−24.2843	−0.880304	−0.440152	−	0.897923i	$-0.645075\pi$
$761$	−24.2843	−0.880304	−0.440152	+	0.897923i	$0.645075\pi$
$762$	0	0
$763$	−50.6274	−1.83284
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−36.2843	−1.30844	−0.654222	−	0.756302i	$-0.727004\pi$
$769$	−36.2843	−1.30844	−0.654222	+	0.756302i	$0.727004\pi$
$770$	0	0
$771$	−14.4853	−0.521675
$772$	0	0
$773$	−13.0711	−0.470134	−0.235067	−	0.971979i	$-0.575531\pi$
$773$	−13.0711	−0.470134	−0.235067	+	0.971979i	$0.575531\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	−0.828427	−0.0296815
$780$	0	0
$781$	−40.9706	−1.46604
$782$	0	0
$783$	−20.6863	−0.739268
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−28.5269	−1.01687	−0.508437	−	0.861099i	$-0.669777\pi$
$787$	−28.5269	−1.01687	−0.508437	+	0.861099i	$0.669777\pi$
$788$	0	0
$789$	−7.51472	−0.267531
$790$	0	0
$791$	55.1127	1.95958
$792$	0	0
$793$	−3.31371	−0.117673
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38.0416	−1.34750	−0.673752	−	0.738958i	$-0.735318\pi$
$797$	−38.0416	−1.34750	−0.673752	+	0.738958i	$0.735318\pi$
$798$	0	0
$799$	−2.34315	−0.0828945
$800$	0	0
$801$	13.3137	0.470417
$802$	0	0
$803$	5.65685	0.199626
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.4853	1.00273
$808$	0	0
$809$	−35.2548	−1.23949	−0.619747	−	0.784802i	$-0.712765\pi$
$809$	−35.2548	−1.23949	−0.619747	+	0.784802i	$0.712765\pi$
$810$	0	0
$811$	−26.8284	−0.942073	−0.471037	−	0.882114i	$-0.656120\pi$
$811$	−26.8284	−0.942073	−0.471037	+	0.882114i	$0.656120\pi$
$812$	0	0
$813$	27.1127	0.950884
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.82843	−0.308868
$818$	0	0
$819$	−2.82843	−0.0988332
$820$	0	0
$821$	5.02944	0.175529	0.0877643	−	0.996141i	$-0.472028\pi$
$821$	5.02944	0.175529	0.0877643	+	0.996141i	$0.472028\pi$
$822$	0	0
$823$	26.4853	0.923219	0.461609	−	0.887083i	$-0.347272\pi$
$823$	26.4853	0.923219	0.461609	+	0.887083i	$0.347272\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33.4142	−1.16193	−0.580963	−	0.813930i	$-0.697324\pi$
$827$	−33.4142	−1.16193	−0.580963	+	0.813930i	$0.697324\pi$
$828$	0	0
$829$	30.4853	1.05880	0.529399	−	0.848373i	$-0.322418\pi$
$829$	30.4853	1.05880	0.529399	+	0.848373i	$0.322418\pi$
$830$	0	0
$831$	12.9706	0.449944
$832$	0	0
$833$	46.1421	1.59873
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−38.6274	−1.33516
$838$	0	0
$839$	8.48528	0.292944	0.146472	−	0.989215i	$-0.453208\pi$
$839$	8.48528	0.292944	0.146472	+	0.989215i	$0.453208\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	−31.7990	−1.09522
$844$	0	0
$845$	0	0
$846$	0	0
$847$	59.4558	2.04293
$848$	0	0
$849$	−4.20101	−0.144178
$850$	0	0
$851$	4.48528	0.153753
$852$	0	0
$853$	47.3137	1.61999	0.809995	−	0.586436i	$-0.199470\pi$
$853$	47.3137	1.61999	0.809995	+	0.586436i	$0.199470\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	56.6690	1.93578	0.967889	−	0.251378i	$-0.0808838\pi$
$857$	56.6690	1.93578	0.967889	+	0.251378i	$0.0808838\pi$
$858$	0	0
$859$	−13.9411	−0.475665	−0.237833	−	0.971306i	$-0.576437\pi$
$859$	−13.9411	−0.475665	−0.237833	+	0.971306i	$0.576437\pi$
$860$	0	0
$861$	5.65685	0.192785
$862$	0	0
$863$	4.04163	0.137579	0.0687894	−	0.997631i	$-0.478086\pi$
$863$	4.04163	0.137579	0.0687894	+	0.997631i	$0.478086\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−12.7279	−0.432263
$868$	0	0
$869$	40.9706	1.38983
$870$	0	0
$871$	−5.79899	−0.196491
$872$	0	0
$873$	1.07107	0.0362502
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.1838	0.816628	0.408314	−	0.912842i	$-0.366117\pi$
$877$	24.1838	0.816628	0.408314	+	0.912842i	$0.366117\pi$
$878$	0	0
$879$	−13.7990	−0.465428
$880$	0	0
$881$	20.2843	0.683394	0.341697	−	0.939810i	$-0.388998\pi$
$881$	20.2843	0.683394	0.341697	+	0.939810i	$0.388998\pi$
$882$	0	0
$883$	−42.7696	−1.43931	−0.719655	−	0.694332i	$-0.755700\pi$
$883$	−42.7696	−1.43931	−0.719655	+	0.694332i	$0.755700\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.443651	−0.0148963	−0.00744817	−	0.999972i	$-0.502371\pi$
$887$	−0.443651	−0.0148963	−0.00744817	+	0.999972i	$0.502371\pi$
$888$	0	0
$889$	61.4558	2.06116
$890$	0	0
$891$	−24.1421	−0.808792
$892$	0	0
$893$	0.828427	0.0277223
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.34315	−0.211791
$898$	0	0
$899$	24.9706	0.832815
$900$	0	0
$901$	33.6569	1.12127
$902$	0	0
$903$	60.2843	2.00613
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−3.07107	−0.101973	−0.0509866	−	0.998699i	$-0.516237\pi$
$907$	−3.07107	−0.101973	−0.0509866	+	0.998699i	$0.516237\pi$
$908$	0	0
$909$	8.00000	0.265343
$910$	0	0
$911$	22.3431	0.740261	0.370131	−	0.928980i	$-0.379313\pi$
$911$	22.3431	0.740261	0.370131	+	0.928980i	$0.379313\pi$
$912$	0	0
$913$	−75.5980	−2.50193
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−85.2548	−2.81536
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	3.65685	0.120497
$922$	0	0
$923$	−4.97056	−0.163608
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.89949	0.325142
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−16.3137	−0.534660
$932$	0	0
$933$	−32.7696	−1.07283
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.14214	−0.0699805	−0.0349903	−	0.999388i	$-0.511140\pi$
$937$	−2.14214	−0.0699805	−0.0349903	+	0.999388i	$0.511140\pi$
$938$	0	0
$939$	29.6569	0.967815
$940$	0	0
$941$	31.9411	1.04125	0.520625	−	0.853785i	$-0.325699\pi$
$941$	31.9411	1.04125	0.520625	+	0.853785i	$0.325699\pi$
$942$	0	0
$943$	−6.34315	−0.206561
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.3137	1.34252	0.671258	−	0.741224i	$-0.265754\pi$
$947$	41.3137	1.34252	0.671258	+	0.741224i	$0.265754\pi$
$948$	0	0
$949$	0.686292	0.0222780
$950$	0	0
$951$	−15.1716	−0.491972
$952$	0	0
$953$	44.3848	1.43776	0.718882	−	0.695132i	$-0.244654\pi$
$953$	44.3848	1.43776	0.718882	+	0.695132i	$0.244654\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.9706	0.807184
$958$	0	0
$959$	13.6569	0.441003
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	−16.2426	−0.523412
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−28.3431	−0.911454	−0.455727	−	0.890120i	$-0.650621\pi$
$967$	−28.3431	−0.911454	−0.455727	+	0.890120i	$0.650621\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	19.5147	0.626257	0.313129	−	0.949711i	$-0.398623\pi$
$971$	19.5147	0.626257	0.313129	+	0.949711i	$0.398623\pi$
$972$	0	0
$973$	73.2548	2.34844
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.0711	−0.546152	−0.273076	−	0.961992i	$-0.588041\pi$
$977$	−17.0711	−0.546152	−0.273076	+	0.961992i	$0.588041\pi$
$978$	0	0
$979$	−64.2843	−2.05453
$980$	0	0
$981$	10.4853	0.334769
$982$	0	0
$983$	4.52691	0.144386	0.0721930	−	0.997391i	$-0.477000\pi$
$983$	4.52691	0.144386	0.0721930	+	0.997391i	$0.477000\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.65685	−0.180060
$988$	0	0
$989$	−67.5980	−2.14949
$990$	0	0
$991$	−20.2843	−0.644351	−0.322176	−	0.946680i	$-0.604414\pi$
$991$	−20.2843	−0.644351	−0.322176	+	0.946680i	$0.604414\pi$
$992$	0	0
$993$	−19.3137	−0.612902
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.45584	0.172788	0.0863942	−	0.996261i	$-0.472466\pi$
$997$	5.45584	0.172788	0.0863942	+	0.996261i	$0.472466\pi$
$998$	0	0
$999$	3.31371	0.104841

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.o.1.2		2
4.3	odd	2		7600.2.a.z.1.1		2
5.2	odd	4		760.2.d.d.609.1	✓	4
5.3	odd	4		760.2.d.d.609.4	yes	4
5.4	even	2		3800.2.a.m.1.1		2
20.3	even	4		1520.2.d.g.609.2		4
20.7	even	4		1520.2.d.g.609.3		4
20.19	odd	2		7600.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.d.609.1	✓	4	5.2	odd	4
760.2.d.d.609.4	yes	4	5.3	odd	4
1520.2.d.g.609.2		4	20.3	even	4
1520.2.d.g.609.3		4	20.7	even	4
3800.2.a.m.1.1		2	5.4	even	2
3800.2.a.o.1.2		2	1.1	even	1	trivial
7600.2.a.z.1.1		2	4.3	odd	2
7600.2.a.bb.1.2		2	20.19	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +4.82843 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} +4.82843 q^{7} -1.00000 q^{9} +4.82843 q^{11} +0.585786 q^{13} +2.82843 q^{17} -1.00000 q^{19} +6.82843 q^{21} -7.65685 q^{23} -5.65685 q^{27} +3.65685 q^{29} +6.82843 q^{31} +6.82843 q^{33} -0.585786 q^{37} +0.828427 q^{39} +0.828427 q^{41} +8.82843 q^{43} -0.828427 q^{47} +16.3137 q^{49} +4.00000 q^{51} +11.8995 q^{53} -1.41421 q^{57} -6.82843 q^{59} -5.65685 q^{61} -4.82843 q^{63} -9.89949 q^{67} -10.8284 q^{69} -8.48528 q^{71} +1.17157 q^{73} +23.3137 q^{77} +8.48528 q^{79} -5.00000 q^{81} -15.6569 q^{83} +5.17157 q^{87} -13.3137 q^{89} +2.82843 q^{91} +9.65685 q^{93} -1.07107 q^{97} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^7 - 2 * q^9	\( 2 q + 4 q^{7} - 2 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{19} + 8 q^{21} - 4 q^{23} - 4 q^{29} + 8 q^{31} + 8 q^{33} - 4 q^{37} - 4 q^{39} - 4 q^{41} + 12 q^{43} + 4 q^{47} + 10 q^{49} + 8 q^{51} + 4 q^{53} - 8 q^{59} - 4 q^{63} - 16 q^{69} + 8 q^{73} + 24 q^{77} - 10 q^{81} - 20 q^{83} + 16 q^{87} - 4 q^{89} + 8 q^{93} + 12 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^7 - 2 * q^9 + 4 * q^11 + 4 * q^13 - 2 * q^19 + 8 * q^21 - 4 * q^23 - 4 * q^29 + 8 * q^31 + 8 * q^33 - 4 * q^37 - 4 * q^39 - 4 * q^41 + 12 * q^43 + 4 * q^47 + 10 * q^49 + 8 * q^51 + 4 * q^53 - 8 * q^59 - 4 * q^63 - 16 * q^69 + 8 * q^73 + 24 * q^77 - 10 * q^81 - 20 * q^83 + 16 * q^87 - 4 * q^89 + 8 * q^93 + 12 * q^97 - 4 * q^99

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