LMFDB - Embedded newform 3800.2.a.z.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:	$1$
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			1.2
Root			$2.43031$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43031 q^{3} -3.60737 q^{7} +2.90640 q^{9} +O(q^{10})$	$q-2.43031 q^{3} -3.60737 q^{7} +2.90640 q^{9} -2.37997 q^{11} +3.96613 q^{13} -4.28257 q^{17} +1.00000 q^{19} +8.76701 q^{21} +1.07377 q^{23} +0.227486 q^{27} +9.47021 q^{29} +6.38211 q^{31} +5.78405 q^{33} -2.04540 q^{37} -9.63892 q^{39} -4.38211 q^{41} -7.86284 q^{43} -3.83485 q^{47} +6.01310 q^{49} +10.4080 q^{51} +11.7789 q^{53} -2.43031 q^{57} +4.59593 q^{59} +6.62819 q^{61} -10.4844 q^{63} -7.02837 q^{67} -2.60959 q^{69} +4.99450 q^{71} +2.93216 q^{73} +8.58541 q^{77} +0.860616 q^{79} -9.27205 q^{81} -11.9179 q^{83} -23.0155 q^{87} +13.6460 q^{89} -14.3073 q^{91} -15.5105 q^{93} -18.5757 q^{97} -6.91712 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9	$ 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} - 34 q^{51} + 14 q^{53} - 2 q^{57} - 2 q^{59} + 10 q^{61} - 44 q^{63} - 42 q^{67} + 18 q^{69} - 8 q^{71} + 2 q^{73} - 24 q^{77} - 20 q^{79} + 6 q^{81} - 16 q^{83} - 30 q^{87} + 32 q^{89} + 10 q^{91} - 12 q^{93} - 40 q^{97} - 22 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9 - 2 * q^11 - 14 * q^13 - 10 * q^17 + 6 * q^19 + 18 * q^21 - 2 * q^23 - 2 * q^27 - 2 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^37 - 18 * q^39 + 4 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 34 * q^51 + 14 * q^53 - 2 * q^57 - 2 * q^59 + 10 * q^61 - 44 * q^63 - 42 * q^67 + 18 * q^69 - 8 * q^71 + 2 * q^73 - 24 * q^77 - 20 * q^79 + 6 * q^81 - 16 * q^83 - 30 * q^87 + 32 * q^89 + 10 * q^91 - 12 * q^93 - 40 * q^97 - 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.43031	−1.40314	−0.701569	−	0.712601i	$-0.747517\pi$
$3$	−2.43031	−1.40314	−0.701569	+	0.712601i	$0.747517\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.60737	−1.36346	−0.681728	−	0.731605i	$-0.738771\pi$
$7$	−3.60737	−1.36346	−0.681728	+	0.731605i	$0.738771\pi$
$8$	0	0
$9$	2.90640	0.968799
$10$	0	0
$11$	−2.37997	−0.717587	−0.358793	−	0.933417i	$-0.616812\pi$
$11$	−2.37997	−0.717587	−0.358793	+	0.933417i	$0.616812\pi$
$12$	0	0
$13$	3.96613	1.10001	0.550003	−	0.835162i	$-0.314626\pi$
$13$	3.96613	1.10001	0.550003	+	0.835162i	$0.314626\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.28257	−1.03868	−0.519338	−	0.854569i	$-0.673822\pi$
$17$	−4.28257	−1.03868	−0.519338	+	0.854569i	$0.673822\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	8.76701	1.91312
$22$	0	0
$23$	1.07377	0.223897	0.111948	−	0.993714i	$-0.464291\pi$
$23$	1.07377	0.223897	0.111948	+	0.993714i	$0.464291\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.227486	0.0437796
$28$	0	0
$29$	9.47021	1.75857	0.879287	−	0.476293i	$-0.158020\pi$
$29$	9.47021	1.75857	0.879287	+	0.476293i	$0.158020\pi$
$30$	0	0
$31$	6.38211	1.14626	0.573130	−	0.819464i	$-0.305729\pi$
$31$	6.38211	1.14626	0.573130	+	0.819464i	$0.305729\pi$
$32$	0	0
$33$	5.78405	1.00687
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.04540	−0.336262	−0.168131	−	0.985765i	$-0.553773\pi$
$37$	−2.04540	−0.336262	−0.168131	+	0.985765i	$0.553773\pi$
$38$	0	0
$39$	−9.63892	−1.54346
$40$	0	0
$41$	−4.38211	−0.684370	−0.342185	−	0.939633i	$-0.611167\pi$
$41$	−4.38211	−0.684370	−0.342185	+	0.939633i	$0.611167\pi$
$42$	0	0
$43$	−7.86284	−1.19907	−0.599536	−	0.800348i	$-0.704648\pi$
$43$	−7.86284	−1.19907	−0.599536	+	0.800348i	$0.704648\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.83485	−0.559371	−0.279685	−	0.960092i	$-0.590230\pi$
$47$	−3.83485	−0.559371	−0.279685	+	0.960092i	$0.590230\pi$
$48$	0	0
$49$	6.01310	0.859014
$50$	0	0
$51$	10.4080	1.45741
$52$	0	0
$53$	11.7789	1.61796	0.808980	−	0.587836i	$-0.200020\pi$
$53$	11.7789	1.61796	0.808980	+	0.587836i	$0.200020\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.43031	−0.321902
$58$	0	0
$59$	4.59593	0.598340	0.299170	−	0.954200i	$-0.403290\pi$
$59$	4.59593	0.598340	0.299170	+	0.954200i	$0.403290\pi$
$60$	0	0
$61$	6.62819	0.848653	0.424327	−	0.905509i	$-0.360511\pi$
$61$	6.62819	0.848653	0.424327	+	0.905509i	$0.360511\pi$
$62$	0	0
$63$	−10.4844	−1.32092
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.02837	−0.858652	−0.429326	−	0.903150i	$-0.641249\pi$
$67$	−7.02837	−0.858652	−0.429326	+	0.903150i	$0.641249\pi$
$68$	0	0
$69$	−2.60959	−0.314158
$70$	0	0
$71$	4.99450	0.592738	0.296369	−	0.955074i	$-0.404224\pi$
$71$	4.99450	0.592738	0.296369	+	0.955074i	$0.404224\pi$
$72$	0	0
$73$	2.93216	0.343183	0.171592	−	0.985168i	$-0.445109\pi$
$73$	2.93216	0.343183	0.171592	+	0.985168i	$0.445109\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.58541	0.978398
$78$	0	0
$79$	0.860616	0.0968268	0.0484134	−	0.998827i	$-0.484584\pi$
$79$	0.860616	0.0968268	0.0484134	+	0.998827i	$0.484584\pi$
$80$	0	0
$81$	−9.27205	−1.03023
$82$	0	0
$83$	−11.9179	−1.30816	−0.654081	−	0.756424i	$-0.726945\pi$
$83$	−11.9179	−1.30816	−0.654081	+	0.756424i	$0.726945\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−23.0155	−2.46752
$88$	0	0
$89$	13.6460	1.44647	0.723237	−	0.690600i	$-0.242654\pi$
$89$	13.6460	1.44647	0.723237	+	0.690600i	$0.242654\pi$
$90$	0	0
$91$	−14.3073	−1.49981
$92$	0	0
$93$	−15.5105	−1.60836
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.5757	−1.88608	−0.943040	−	0.332681i	$-0.892047\pi$
$97$	−18.5757	−1.88608	−0.943040	+	0.332681i	$0.892047\pi$
$98$	0	0
$99$	−6.91712	−0.695197
$100$	0	0
$101$	−9.82645	−0.977769	−0.488884	−	0.872349i	$-0.662596\pi$
$101$	−9.82645	−0.977769	−0.488884	+	0.872349i	$0.662596\pi$
$102$	0	0
$103$	8.85269	0.872282	0.436141	−	0.899878i	$-0.356345\pi$
$103$	8.85269	0.872282	0.436141	+	0.899878i	$0.356345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.58269	−0.636372	−0.318186	−	0.948028i	$-0.603074\pi$
$107$	−6.58269	−0.636372	−0.318186	+	0.948028i	$0.603074\pi$
$108$	0	0
$109$	3.40919	0.326541	0.163271	−	0.986581i	$-0.447796\pi$
$109$	3.40919	0.326541	0.163271	+	0.986581i	$0.447796\pi$
$110$	0	0
$111$	4.97096	0.471823
$112$	0	0
$113$	6.58065	0.619055	0.309528	−	0.950890i	$-0.399829\pi$
$113$	6.58065	0.619055	0.309528	+	0.950890i	$0.399829\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.5271	1.06569
$118$	0	0
$119$	15.4488	1.41619
$120$	0	0
$121$	−5.33576	−0.485069
$122$	0	0
$123$	10.6499	0.960267
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.2826	1.08991	0.544953	−	0.838466i	$-0.316547\pi$
$127$	12.2826	1.08991	0.544953	+	0.838466i	$0.316547\pi$
$128$	0	0
$129$	19.1091	1.68246
$130$	0	0
$131$	−10.0308	−0.876395	−0.438198	−	0.898879i	$-0.644383\pi$
$131$	−10.0308	−0.876395	−0.438198	+	0.898879i	$0.644383\pi$
$132$	0	0
$133$	−3.60737	−0.312798
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.1273	1.46329	0.731644	−	0.681687i	$-0.238753\pi$
$137$	17.1273	1.46329	0.731644	+	0.681687i	$0.238753\pi$
$138$	0	0
$139$	−5.50494	−0.466923	−0.233461	−	0.972366i	$-0.575005\pi$
$139$	−5.50494	−0.466923	−0.233461	+	0.972366i	$0.575005\pi$
$140$	0	0
$141$	9.31987	0.784875
$142$	0	0
$143$	−9.43926	−0.789350
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−14.6137	−1.20532
$148$	0	0
$149$	−21.4593	−1.75801	−0.879007	−	0.476808i	$-0.841794\pi$
$149$	−21.4593	−1.75801	−0.879007	+	0.476808i	$0.841794\pi$
$150$	0	0
$151$	−14.1805	−1.15399	−0.576996	−	0.816747i	$-0.695775\pi$
$151$	−14.1805	−1.15399	−0.576996	+	0.816747i	$0.695775\pi$
$152$	0	0
$153$	−12.4469	−1.00627
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.69891	−0.215397	−0.107698	−	0.994184i	$-0.534348\pi$
$157$	−2.69891	−0.215397	−0.107698	+	0.994184i	$0.534348\pi$
$158$	0	0
$159$	−28.6264	−2.27022
$160$	0	0
$161$	−3.87349	−0.305273
$162$	0	0
$163$	11.1955	0.876898	0.438449	−	0.898756i	$-0.355528\pi$
$163$	11.1955	0.876898	0.438449	+	0.898756i	$0.355528\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.71331	0.751639	0.375819	−	0.926693i	$-0.377361\pi$
$167$	9.71331	0.751639	0.375819	+	0.926693i	$0.377361\pi$
$168$	0	0
$169$	2.73019	0.210015
$170$	0	0
$171$	2.90640	0.222258
$172$	0	0
$173$	14.2783	1.08556	0.542781	−	0.839874i	$-0.317371\pi$
$173$	14.2783	1.08556	0.542781	+	0.839874i	$0.317371\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.1695	−0.839554
$178$	0	0
$179$	−17.1327	−1.28056	−0.640278	−	0.768143i	$-0.721181\pi$
$179$	−17.1327	−1.28056	−0.640278	+	0.768143i	$0.721181\pi$
$180$	0	0
$181$	−23.6883	−1.76074	−0.880370	−	0.474288i	$-0.842705\pi$
$181$	−23.6883	−1.76074	−0.880370	+	0.474288i	$0.842705\pi$
$182$	0	0
$183$	−16.1085	−1.19078
$184$	0	0
$185$	0	0
$186$	0	0
$187$	10.1924	0.745341
$188$	0	0
$189$	−0.820624	−0.0596916
$190$	0	0
$191$	25.2506	1.82707	0.913535	−	0.406761i	$-0.133342\pi$
$191$	25.2506	1.82707	0.913535	+	0.406761i	$0.133342\pi$
$192$	0	0
$193$	−16.6814	−1.20076	−0.600378	−	0.799716i	$-0.704983\pi$
$193$	−16.6814	−1.20076	−0.600378	+	0.799716i	$0.704983\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.1471	−1.22168	−0.610839	−	0.791755i	$-0.709168\pi$
$197$	−17.1471	−1.22168	−0.610839	+	0.791755i	$0.709168\pi$
$198$	0	0
$199$	22.1273	1.56856	0.784280	−	0.620407i	$-0.213033\pi$
$199$	22.1273	1.56856	0.784280	+	0.620407i	$0.213033\pi$
$200$	0	0
$201$	17.0811	1.20481
$202$	0	0
$203$	−34.1625	−2.39774
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.12080	0.216911
$208$	0	0
$209$	−2.37997	−0.164626
$210$	0	0
$211$	−24.6690	−1.69828	−0.849141	−	0.528166i	$-0.822880\pi$
$211$	−24.6690	−1.69828	−0.849141	+	0.528166i	$0.822880\pi$
$212$	0	0
$213$	−12.1382	−0.831694
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−23.0226	−1.56288
$218$	0	0
$219$	−7.12605	−0.481534
$220$	0	0
$221$	−16.9852	−1.14255
$222$	0	0
$223$	−21.4787	−1.43832	−0.719162	−	0.694843i	$-0.755474\pi$
$223$	−21.4787	−1.43832	−0.719162	+	0.694843i	$0.755474\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.5568	1.36440	0.682201	−	0.731164i	$-0.261023\pi$
$227$	20.5568	1.36440	0.682201	+	0.731164i	$0.261023\pi$
$228$	0	0
$229$	−17.6465	−1.16611	−0.583057	−	0.812431i	$-0.698144\pi$
$229$	−17.6465	−1.16611	−0.583057	+	0.812431i	$0.698144\pi$
$230$	0	0
$231$	−20.8652	−1.37283
$232$	0	0
$233$	−15.5021	−1.01558	−0.507788	−	0.861482i	$-0.669537\pi$
$233$	−15.5021	−1.01558	−0.507788	+	0.861482i	$0.669537\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.09156	−0.135862
$238$	0	0
$239$	−28.2186	−1.82531	−0.912656	−	0.408730i	$-0.865972\pi$
$239$	−28.2186	−1.82531	−0.912656	+	0.408730i	$0.865972\pi$
$240$	0	0
$241$	23.8653	1.53730	0.768649	−	0.639671i	$-0.220929\pi$
$241$	23.8653	1.53730	0.768649	+	0.639671i	$0.220929\pi$
$242$	0	0
$243$	21.8515	1.40177
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.96613	0.252359
$248$	0	0
$249$	28.9642	1.83553
$250$	0	0
$251$	8.15793	0.514924	0.257462	−	0.966288i	$-0.417114\pi$
$251$	8.15793	0.514924	0.257462	+	0.966288i	$0.417114\pi$
$252$	0	0
$253$	−2.55554	−0.160665
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.07899	−0.379197	−0.189598	−	0.981862i	$-0.560719\pi$
$257$	−6.07899	−0.379197	−0.189598	+	0.981862i	$0.560719\pi$
$258$	0	0
$259$	7.37852	0.458479
$260$	0	0
$261$	27.5242	1.70370
$262$	0	0
$263$	−8.47535	−0.522612	−0.261306	−	0.965256i	$-0.584153\pi$
$263$	−8.47535	−0.522612	−0.261306	+	0.965256i	$0.584153\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−33.1640	−2.02960
$268$	0	0
$269$	−16.6667	−1.01619	−0.508093	−	0.861302i	$-0.669649\pi$
$269$	−16.6667	−1.01619	−0.508093	+	0.861302i	$0.669649\pi$
$270$	0	0
$271$	−11.8151	−0.717718	−0.358859	−	0.933392i	$-0.616834\pi$
$271$	−11.8151	−0.717718	−0.358859	+	0.933392i	$0.616834\pi$
$272$	0	0
$273$	34.7711	2.10444
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.10858	0.126692	0.0633462	−	0.997992i	$-0.479823\pi$
$277$	2.10858	0.126692	0.0633462	+	0.997992i	$0.479823\pi$
$278$	0	0
$279$	18.5489	1.11050
$280$	0	0
$281$	−3.76859	−0.224815	−0.112408	−	0.993662i	$-0.535856\pi$
$281$	−3.76859	−0.224815	−0.112408	+	0.993662i	$0.535856\pi$
$282$	0	0
$283$	−0.0720389	−0.00428227	−0.00214114	−	0.999998i	$-0.500682\pi$
$283$	−0.0720389	−0.00428227	−0.00214114	+	0.999998i	$0.500682\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.8079	0.933109
$288$	0	0
$289$	1.34044	0.0788494
$290$	0	0
$291$	45.1447	2.64643
$292$	0	0
$293$	9.98245	0.583181	0.291590	−	0.956543i	$-0.405816\pi$
$293$	9.98245	0.583181	0.291590	+	0.956543i	$0.405816\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.541408	−0.0314157
$298$	0	0
$299$	4.25872	0.246288
$300$	0	0
$301$	28.3642	1.63488
$302$	0	0
$303$	23.8813	1.37195
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−21.0679	−1.20241	−0.601204	−	0.799096i	$-0.705312\pi$
$307$	−21.0679	−1.20241	−0.601204	+	0.799096i	$0.705312\pi$
$308$	0	0
$309$	−21.5148	−1.22393
$310$	0	0
$311$	−29.9966	−1.70095	−0.850475	−	0.526015i	$-0.823686\pi$
$311$	−29.9966	−1.70095	−0.850475	+	0.526015i	$0.823686\pi$
$312$	0	0
$313$	−5.48121	−0.309817	−0.154908	−	0.987929i	$-0.549508\pi$
$313$	−5.48121	−0.309817	−0.154908	+	0.987929i	$0.549508\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.687909	−0.0386368	−0.0193184	−	0.999813i	$-0.506150\pi$
$317$	−0.687909	−0.0386368	−0.0193184	+	0.999813i	$0.506150\pi$
$318$	0	0
$319$	−22.5388	−1.26193
$320$	0	0
$321$	15.9980	0.892919
$322$	0	0
$323$	−4.28257	−0.238289
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.28538	−0.458183
$328$	0	0
$329$	13.8337	0.762678
$330$	0	0
$331$	23.1679	1.27342	0.636712	−	0.771102i	$-0.280294\pi$
$331$	23.1679	1.27342	0.636712	+	0.771102i	$0.280294\pi$
$332$	0	0
$333$	−5.94475	−0.325771
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−28.8512	−1.57162	−0.785812	−	0.618465i	$-0.787755\pi$
$337$	−28.8512	−1.57162	−0.785812	+	0.618465i	$0.787755\pi$
$338$	0	0
$339$	−15.9930	−0.868620
$340$	0	0
$341$	−15.1892	−0.822541
$342$	0	0
$343$	3.56013	0.192229
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.51365	−0.295988	−0.147994	−	0.988988i	$-0.547282\pi$
$347$	−5.51365	−0.295988	−0.147994	+	0.988988i	$0.547282\pi$
$348$	0	0
$349$	−6.62869	−0.354826	−0.177413	−	0.984137i	$-0.556773\pi$
$349$	−6.62869	−0.354826	−0.177413	+	0.984137i	$0.556773\pi$
$350$	0	0
$351$	0.902238	0.0481579
$352$	0	0
$353$	10.0657	0.535745	0.267872	−	0.963454i	$-0.413679\pi$
$353$	10.0657	0.535745	0.267872	+	0.963454i	$0.413679\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−37.5454	−1.98711
$358$	0	0
$359$	−23.9747	−1.26534	−0.632668	−	0.774423i	$-0.718040\pi$
$359$	−23.9747	−1.26534	−0.632668	+	0.774423i	$0.718040\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	12.9675	0.680620
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.10030	−0.109635	−0.0548174	−	0.998496i	$-0.517458\pi$
$367$	−2.10030	−0.109635	−0.0548174	+	0.998496i	$0.517458\pi$
$368$	0	0
$369$	−12.7361	−0.663017
$370$	0	0
$371$	−42.4909	−2.20602
$372$	0	0
$373$	−19.2927	−0.998939	−0.499470	−	0.866331i	$-0.666472\pi$
$373$	−19.2927	−0.998939	−0.499470	+	0.866331i	$0.666472\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	37.5601	1.93444
$378$	0	0
$379$	4.29783	0.220765	0.110382	−	0.993889i	$-0.464792\pi$
$379$	4.29783	0.220765	0.110382	+	0.993889i	$0.464792\pi$
$380$	0	0
$381$	−29.8506	−1.52929
$382$	0	0
$383$	15.6099	0.797628	0.398814	−	0.917032i	$-0.369422\pi$
$383$	15.6099	0.797628	0.398814	+	0.917032i	$0.369422\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−22.8525	−1.16166
$388$	0	0
$389$	15.4509	0.783390	0.391695	−	0.920095i	$-0.371889\pi$
$389$	15.4509	0.783390	0.391695	+	0.920095i	$0.371889\pi$
$390$	0	0
$391$	−4.59850	−0.232556
$392$	0	0
$393$	24.3779	1.22970
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−0.852887	−0.0428051	−0.0214026	−	0.999771i	$-0.506813\pi$
$397$	−0.852887	−0.0428051	−0.0214026	+	0.999771i	$0.506813\pi$
$398$	0	0
$399$	8.76701	0.438900
$400$	0	0
$401$	38.8284	1.93900	0.969500	−	0.245091i	$-0.0788180\pi$
$401$	38.8284	1.93900	0.969500	+	0.245091i	$0.0788180\pi$
$402$	0	0
$403$	25.3123	1.26089
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.86799	0.241297
$408$	0	0
$409$	6.16871	0.305023	0.152512	−	0.988302i	$-0.451264\pi$
$409$	6.16871	0.305023	0.152512	+	0.988302i	$0.451264\pi$
$410$	0	0
$411$	−41.6247	−2.05320
$412$	0	0
$413$	−16.5792	−0.815810
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.3787	0.655157
$418$	0	0
$419$	−14.0712	−0.687422	−0.343711	−	0.939075i	$-0.611684\pi$
$419$	−14.0712	−0.687422	−0.343711	+	0.939075i	$0.611684\pi$
$420$	0	0
$421$	4.85298	0.236520	0.118260	−	0.992983i	$-0.462268\pi$
$421$	4.85298	0.236520	0.118260	+	0.992983i	$0.462268\pi$
$422$	0	0
$423$	−11.1456	−0.541918
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−23.9103	−1.15710
$428$	0	0
$429$	22.9403	1.10757
$430$	0	0
$431$	−15.6171	−0.752251	−0.376126	−	0.926569i	$-0.622744\pi$
$431$	−15.6171	−0.752251	−0.376126	+	0.926569i	$0.622744\pi$
$432$	0	0
$433$	−25.3497	−1.21823	−0.609114	−	0.793082i	$-0.708475\pi$
$433$	−25.3497	−1.21823	−0.609114	+	0.793082i	$0.708475\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.07377	0.0513654
$438$	0	0
$439$	23.4184	1.11770	0.558849	−	0.829269i	$-0.311243\pi$
$439$	23.4184	1.11770	0.558849	+	0.829269i	$0.311243\pi$
$440$	0	0
$441$	17.4764	0.832211
$442$	0	0
$443$	25.1133	1.19317	0.596584	−	0.802551i	$-0.296524\pi$
$443$	25.1133	1.19317	0.596584	+	0.802551i	$0.296524\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	52.1527	2.46674
$448$	0	0
$449$	6.37270	0.300746	0.150373	−	0.988629i	$-0.451953\pi$
$449$	6.37270	0.300746	0.150373	+	0.988629i	$0.451953\pi$
$450$	0	0
$451$	10.4293	0.491095
$452$	0	0
$453$	34.4630	1.61921
$454$	0	0
$455$	0	0
$456$	0	0
$457$	40.6893	1.90336	0.951682	−	0.307085i	$-0.0993536\pi$
$457$	40.6893	1.90336	0.951682	+	0.307085i	$0.0993536\pi$
$458$	0	0
$459$	−0.974224	−0.0454729
$460$	0	0
$461$	−11.5304	−0.537024	−0.268512	−	0.963276i	$-0.586532\pi$
$461$	−11.5304	−0.537024	−0.268512	+	0.963276i	$0.586532\pi$
$462$	0	0
$463$	32.0882	1.49127	0.745633	−	0.666357i	$-0.232147\pi$
$463$	32.0882	1.49127	0.745633	+	0.666357i	$0.232147\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.6254	−0.954429	−0.477214	−	0.878787i	$-0.658353\pi$
$467$	−20.6254	−0.954429	−0.477214	+	0.878787i	$0.658353\pi$
$468$	0	0
$469$	25.3539	1.17073
$470$	0	0
$471$	6.55919	0.302231
$472$	0	0
$473$	18.7133	0.860438
$474$	0	0
$475$	0	0
$476$	0	0
$477$	34.2342	1.56748
$478$	0	0
$479$	10.8936	0.497740	0.248870	−	0.968537i	$-0.419941\pi$
$479$	10.8936	0.497740	0.248870	+	0.968537i	$0.419941\pi$
$480$	0	0
$481$	−8.11234	−0.369891
$482$	0	0
$483$	9.41376	0.428341
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.05668	0.229140	0.114570	−	0.993415i	$-0.463451\pi$
$487$	5.05668	0.229140	0.114570	+	0.993415i	$0.463451\pi$
$488$	0	0
$489$	−27.2085	−1.23041
$490$	0	0
$491$	−25.5833	−1.15456	−0.577279	−	0.816547i	$-0.695886\pi$
$491$	−25.5833	−1.15456	−0.577279	+	0.816547i	$0.695886\pi$
$492$	0	0
$493$	−40.5569	−1.82659
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.0170	−0.808172
$498$	0	0
$499$	−34.0476	−1.52418	−0.762090	−	0.647471i	$-0.775827\pi$
$499$	−34.0476	−1.52418	−0.762090	+	0.647471i	$0.775827\pi$
$500$	0	0
$501$	−23.6063	−1.05465
$502$	0	0
$503$	5.70145	0.254215	0.127108	−	0.991889i	$-0.459431\pi$
$503$	5.70145	0.254215	0.127108	+	0.991889i	$0.459431\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.63521	−0.294680
$508$	0	0
$509$	−27.9604	−1.23932	−0.619661	−	0.784870i	$-0.712730\pi$
$509$	−27.9604	−1.23932	−0.619661	+	0.784870i	$0.712730\pi$
$510$	0	0
$511$	−10.5774	−0.467916
$512$	0	0
$513$	0.227486	0.0100437
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.12682	0.401397
$518$	0	0
$519$	−34.7008	−1.52319
$520$	0	0
$521$	−0.419591	−0.0183826	−0.00919131	−	0.999958i	$-0.502926\pi$
$521$	−0.419591	−0.0183826	−0.00919131	+	0.999958i	$0.502926\pi$
$522$	0	0
$523$	2.23302	0.0976433	0.0488217	−	0.998808i	$-0.484453\pi$
$523$	2.23302	0.0976433	0.0488217	+	0.998808i	$0.484453\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−27.3319	−1.19059
$528$	0	0
$529$	−21.8470	−0.949870
$530$	0	0
$531$	13.3576	0.579671
$532$	0	0
$533$	−17.3800	−0.752812
$534$	0	0
$535$	0	0
$536$	0	0
$537$	41.6377	1.79680
$538$	0	0
$539$	−14.3110	−0.616417
$540$	0	0
$541$	39.9938	1.71947	0.859734	−	0.510741i	$-0.170629\pi$
$541$	39.9938	1.71947	0.859734	+	0.510741i	$0.170629\pi$
$542$	0	0
$543$	57.5699	2.47056
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−15.5651	−0.665516	−0.332758	−	0.943012i	$-0.607979\pi$
$547$	−15.5651	−0.665516	−0.332758	+	0.943012i	$0.607979\pi$
$548$	0	0
$549$	19.2642	0.822174
$550$	0	0
$551$	9.47021	0.403444
$552$	0	0
$553$	−3.10456	−0.132019
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.68789	−0.113890	−0.0569448	−	0.998377i	$-0.518136\pi$
$557$	−2.68789	−0.113890	−0.0569448	+	0.998377i	$0.518136\pi$
$558$	0	0
$559$	−31.1851	−1.31899
$560$	0	0
$561$	−24.7706	−1.04582
$562$	0	0
$563$	−12.7089	−0.535618	−0.267809	−	0.963472i	$-0.586300\pi$
$563$	−12.7089	−0.535618	−0.267809	+	0.963472i	$0.586300\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	33.4477	1.40467
$568$	0	0
$569$	28.9522	1.21374	0.606870	−	0.794801i	$-0.292425\pi$
$569$	28.9522	1.21374	0.606870	+	0.794801i	$0.292425\pi$
$570$	0	0
$571$	8.89885	0.372405	0.186203	−	0.982511i	$-0.440382\pi$
$571$	8.89885	0.372405	0.186203	+	0.982511i	$0.440382\pi$
$572$	0	0
$573$	−61.3667	−2.56363
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25.3754	−1.05639	−0.528196	−	0.849123i	$-0.677131\pi$
$577$	−25.3754	−1.05639	−0.528196	+	0.849123i	$0.677131\pi$
$578$	0	0
$579$	40.5411	1.68483
$580$	0	0
$581$	42.9924	1.78362
$582$	0	0
$583$	−28.0334	−1.16103
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.7611	1.59984	0.799920	−	0.600107i	$-0.204875\pi$
$587$	38.7611	1.59984	0.799920	+	0.600107i	$0.204875\pi$
$588$	0	0
$589$	6.38211	0.262970
$590$	0	0
$591$	41.6727	1.71418
$592$	0	0
$593$	−35.5132	−1.45835	−0.729176	−	0.684327i	$-0.760096\pi$
$593$	−35.5132	−1.45835	−0.729176	+	0.684327i	$0.760096\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−53.7761	−2.20091
$598$	0	0
$599$	−22.7248	−0.928512	−0.464256	−	0.885701i	$-0.653678\pi$
$599$	−22.7248	−0.928512	−0.464256	+	0.885701i	$0.653678\pi$
$600$	0	0
$601$	−25.3951	−1.03589	−0.517944	−	0.855415i	$-0.673302\pi$
$601$	−25.3951	−1.03589	−0.517944	+	0.855415i	$0.673302\pi$
$602$	0	0
$603$	−20.4272	−0.831861
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.2509	0.740779	0.370390	−	0.928876i	$-0.379224\pi$
$607$	18.2509	0.740779	0.370390	+	0.928876i	$0.379224\pi$
$608$	0	0
$609$	83.0254	3.36436
$610$	0	0
$611$	−15.2095	−0.615312
$612$	0	0
$613$	3.43574	0.138768	0.0693842	−	0.997590i	$-0.477897\pi$
$613$	3.43574	0.138768	0.0693842	+	0.997590i	$0.477897\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.7976	−0.474952	−0.237476	−	0.971393i	$-0.576320\pi$
$617$	−11.7976	−0.474952	−0.237476	+	0.971393i	$0.576320\pi$
$618$	0	0
$619$	27.6561	1.11159	0.555797	−	0.831318i	$-0.312413\pi$
$619$	27.6561	1.11159	0.555797	+	0.831318i	$0.312413\pi$
$620$	0	0
$621$	0.244267	0.00980211
$622$	0	0
$623$	−49.2261	−1.97220
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.78405	0.230993
$628$	0	0
$629$	8.75959	0.349268
$630$	0	0
$631$	7.23176	0.287892	0.143946	−	0.989586i	$-0.454021\pi$
$631$	7.23176	0.287892	0.143946	+	0.989586i	$0.454021\pi$
$632$	0	0
$633$	59.9532	2.38293
$634$	0	0
$635$	0	0
$636$	0	0
$637$	23.8487	0.944921
$638$	0	0
$639$	14.5160	0.574244
$640$	0	0
$641$	20.1802	0.797070	0.398535	−	0.917153i	$-0.369519\pi$
$641$	20.1802	0.797070	0.398535	+	0.917153i	$0.369519\pi$
$642$	0	0
$643$	−4.68579	−0.184790	−0.0923948	−	0.995722i	$-0.529452\pi$
$643$	−4.68579	−0.184790	−0.0923948	+	0.995722i	$0.529452\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.17831	0.0856383	0.0428191	−	0.999083i	$-0.486366\pi$
$647$	2.17831	0.0856383	0.0428191	+	0.999083i	$0.486366\pi$
$648$	0	0
$649$	−10.9382	−0.429361
$650$	0	0
$651$	55.9520	2.19293
$652$	0	0
$653$	−3.73150	−0.146025	−0.0730124	−	0.997331i	$-0.523261\pi$
$653$	−3.73150	−0.146025	−0.0730124	+	0.997331i	$0.523261\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.52202	0.332476
$658$	0	0
$659$	−18.1674	−0.707700	−0.353850	−	0.935302i	$-0.615128\pi$
$659$	−18.1674	−0.707700	−0.353850	+	0.935302i	$0.615128\pi$
$660$	0	0
$661$	8.87307	0.345123	0.172561	−	0.984999i	$-0.444796\pi$
$661$	8.87307	0.345123	0.172561	+	0.984999i	$0.444796\pi$
$662$	0	0
$663$	41.2794	1.60316
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.1688	0.393739
$668$	0	0
$669$	52.2000	2.01817
$670$	0	0
$671$	−15.7749	−0.608982
$672$	0	0
$673$	−35.7318	−1.37736	−0.688680	−	0.725065i	$-0.741809\pi$
$673$	−35.7318	−1.37736	−0.688680	+	0.725065i	$0.741809\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.0748	0.809971	0.404985	−	0.914323i	$-0.367277\pi$
$677$	21.0748	0.809971	0.404985	+	0.914323i	$0.367277\pi$
$678$	0	0
$679$	67.0095	2.57159
$680$	0	0
$681$	−49.9594	−1.91445
$682$	0	0
$683$	9.90904	0.379159	0.189579	−	0.981865i	$-0.439288\pi$
$683$	9.90904	0.379159	0.189579	+	0.981865i	$0.439288\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	42.8865	1.63622
$688$	0	0
$689$	46.7168	1.77977
$690$	0	0
$691$	−11.9960	−0.456348	−0.228174	−	0.973620i	$-0.573276\pi$
$691$	−11.9960	−0.456348	−0.228174	+	0.973620i	$0.573276\pi$
$692$	0	0
$693$	24.9526	0.947871
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.7667	0.710840
$698$	0	0
$699$	37.6748	1.42499
$700$	0	0
$701$	−42.9782	−1.62326	−0.811632	−	0.584169i	$-0.801420\pi$
$701$	−42.9782	−1.62326	−0.811632	+	0.584169i	$0.801420\pi$
$702$	0	0
$703$	−2.04540	−0.0771439
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35.4476	1.33314
$708$	0	0
$709$	−30.3524	−1.13991	−0.569954	−	0.821676i	$-0.693039\pi$
$709$	−30.3524	−1.13991	−0.569954	+	0.821676i	$0.693039\pi$
$710$	0	0
$711$	2.50129	0.0938057
$712$	0	0
$713$	6.85292	0.256644
$714$	0	0
$715$	0	0
$716$	0	0
$717$	68.5799	2.56116
$718$	0	0
$719$	38.0282	1.41821	0.709106	−	0.705102i	$-0.249099\pi$
$719$	38.0282	1.41821	0.709106	+	0.705102i	$0.249099\pi$
$720$	0	0
$721$	−31.9349	−1.18932
$722$	0	0
$723$	−58.0000	−2.15704
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17.1429	0.635796	0.317898	−	0.948125i	$-0.397023\pi$
$727$	17.1429	0.635796	0.317898	+	0.948125i	$0.397023\pi$
$728$	0	0
$729$	−25.2897	−0.936654
$730$	0	0
$731$	33.6732	1.24545
$732$	0	0
$733$	14.0860	0.520277	0.260139	−	0.965571i	$-0.416232\pi$
$733$	14.0860	0.520277	0.260139	+	0.965571i	$0.416232\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.7273	0.616157
$738$	0	0
$739$	−4.08166	−0.150146	−0.0750731	−	0.997178i	$-0.523919\pi$
$739$	−4.08166	−0.150146	−0.0750731	+	0.997178i	$0.523919\pi$
$740$	0	0
$741$	−9.63892	−0.354095
$742$	0	0
$743$	22.5661	0.827872	0.413936	−	0.910306i	$-0.364154\pi$
$743$	22.5661	0.827872	0.413936	+	0.910306i	$0.364154\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−34.6382	−1.26735
$748$	0	0
$749$	23.7462	0.867666
$750$	0	0
$751$	39.5626	1.44366	0.721830	−	0.692070i	$-0.243301\pi$
$751$	39.5626	1.44366	0.721830	+	0.692070i	$0.243301\pi$
$752$	0	0
$753$	−19.8263	−0.722510
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33.4582	−1.21606	−0.608029	−	0.793915i	$-0.708040\pi$
$757$	−33.4582	−1.21606	−0.608029	+	0.793915i	$0.708040\pi$
$758$	0	0
$759$	6.21074	0.225436
$760$	0	0
$761$	4.30846	0.156181	0.0780907	−	0.996946i	$-0.475118\pi$
$761$	4.30846	0.156181	0.0780907	+	0.996946i	$0.475118\pi$
$762$	0	0
$763$	−12.2982	−0.445225
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.2281	0.658178
$768$	0	0
$769$	−29.3601	−1.05875	−0.529376	−	0.848387i	$-0.677574\pi$
$769$	−29.3601	−1.05875	−0.529376	+	0.848387i	$0.677574\pi$
$770$	0	0
$771$	14.7738	0.532066
$772$	0	0
$773$	43.7346	1.57303	0.786513	−	0.617574i	$-0.211884\pi$
$773$	43.7346	1.57303	0.786513	+	0.617574i	$0.211884\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−17.9321	−0.643310
$778$	0	0
$779$	−4.38211	−0.157005
$780$	0	0
$781$	−11.8867	−0.425341
$782$	0	0
$783$	2.15434	0.0769897
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29.4309	1.04910	0.524550	−	0.851380i	$-0.324234\pi$
$787$	29.4309	1.04910	0.524550	+	0.851380i	$0.324234\pi$
$788$	0	0
$789$	20.5977	0.733298
$790$	0	0
$791$	−23.7388	−0.844055
$792$	0	0
$793$	26.2883	0.933524
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.2121	−0.751370	−0.375685	−	0.926747i	$-0.622593\pi$
$797$	−21.2121	−0.751370	−0.375685	+	0.926747i	$0.622593\pi$
$798$	0	0
$799$	16.4230	0.581005
$800$	0	0
$801$	39.6607	1.40134
$802$	0	0
$803$	−6.97844	−0.246264
$804$	0	0
$805$	0	0
$806$	0	0
$807$	40.5052	1.42585
$808$	0	0
$809$	−23.3698	−0.821638	−0.410819	−	0.911717i	$-0.634757\pi$
$809$	−23.3698	−0.821638	−0.410819	+	0.911717i	$0.634757\pi$
$810$	0	0
$811$	−15.1194	−0.530915	−0.265458	−	0.964123i	$-0.585523\pi$
$811$	−15.1194	−0.530915	−0.265458	+	0.964123i	$0.585523\pi$
$812$	0	0
$813$	28.7144	1.00706
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.86284	−0.275086
$818$	0	0
$819$	−41.5827	−1.45302
$820$	0	0
$821$	27.2818	0.952140	0.476070	−	0.879407i	$-0.342061\pi$
$821$	27.2818	0.952140	0.476070	+	0.879407i	$0.342061\pi$
$822$	0	0
$823$	18.4082	0.641670	0.320835	−	0.947135i	$-0.396036\pi$
$823$	18.4082	0.641670	0.320835	+	0.947135i	$0.396036\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.3875	0.882811	0.441405	−	0.897308i	$-0.354480\pi$
$827$	25.3875	0.882811	0.441405	+	0.897308i	$0.354480\pi$
$828$	0	0
$829$	−1.32188	−0.0459108	−0.0229554	−	0.999736i	$-0.507308\pi$
$829$	−1.32188	−0.0459108	−0.0229554	+	0.999736i	$0.507308\pi$
$830$	0	0
$831$	−5.12450	−0.177767
$832$	0	0
$833$	−25.7515	−0.892238
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.45184	0.0501828
$838$	0	0
$839$	22.9481	0.792257	0.396129	−	0.918195i	$-0.370353\pi$
$839$	22.9481	0.792257	0.396129	+	0.918195i	$0.370353\pi$
$840$	0	0
$841$	60.6849	2.09258
$842$	0	0
$843$	9.15884	0.315447
$844$	0	0
$845$	0	0
$846$	0	0
$847$	19.2481	0.661371
$848$	0	0
$849$	0.175077	0.00600862
$850$	0	0
$851$	−2.19630	−0.0752880
$852$	0	0
$853$	2.58912	0.0886498	0.0443249	−	0.999017i	$-0.485886\pi$
$853$	2.58912	0.0886498	0.0443249	+	0.999017i	$0.485886\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.62800	−0.0897709	−0.0448855	−	0.998992i	$-0.514292\pi$
$857$	−2.62800	−0.0897709	−0.0448855	+	0.998992i	$0.514292\pi$
$858$	0	0
$859$	4.38188	0.149508	0.0747539	−	0.997202i	$-0.476183\pi$
$859$	4.38188	0.149508	0.0747539	+	0.997202i	$0.476183\pi$
$860$	0	0
$861$	−38.4180	−1.30928
$862$	0	0
$863$	25.8373	0.879511	0.439755	−	0.898118i	$-0.355065\pi$
$863$	25.8373	0.879511	0.439755	+	0.898118i	$0.355065\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.25768	−0.110637
$868$	0	0
$869$	−2.04824	−0.0694816
$870$	0	0
$871$	−27.8754	−0.944523
$872$	0	0
$873$	−53.9884	−1.82723
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.72116	0.226957	0.113479	−	0.993540i	$-0.463801\pi$
$877$	6.72116	0.226957	0.113479	+	0.993540i	$0.463801\pi$
$878$	0	0
$879$	−24.2604	−0.818284
$880$	0	0
$881$	−7.25064	−0.244280	−0.122140	−	0.992513i	$-0.538976\pi$
$881$	−7.25064	−0.244280	−0.122140	+	0.992513i	$0.538976\pi$
$882$	0	0
$883$	−27.9309	−0.939949	−0.469974	−	0.882680i	$-0.655737\pi$
$883$	−27.9309	−0.939949	−0.469974	+	0.882680i	$0.655737\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.8032	−1.30288	−0.651442	−	0.758699i	$-0.725836\pi$
$887$	−38.8032	−1.30288	−0.651442	+	0.758699i	$0.725836\pi$
$888$	0	0
$889$	−44.3079	−1.48604
$890$	0	0
$891$	22.0672	0.739278
$892$	0	0
$893$	−3.83485	−0.128328
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−10.3500	−0.345576
$898$	0	0
$899$	60.4399	2.01578
$900$	0	0
$901$	−50.4441	−1.68054
$902$	0	0
$903$	−68.9336	−2.29397
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−31.8847	−1.05872	−0.529358	−	0.848399i	$-0.677567\pi$
$907$	−31.8847	−1.05872	−0.529358	+	0.848399i	$0.677567\pi$
$908$	0	0
$909$	−28.5596	−0.947261
$910$	0	0
$911$	−30.1437	−0.998707	−0.499353	−	0.866398i	$-0.666429\pi$
$911$	−30.1437	−0.998707	−0.499353	+	0.866398i	$0.666429\pi$
$912$	0	0
$913$	28.3643	0.938720
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.1848	1.19493
$918$	0	0
$919$	−55.2958	−1.82404	−0.912020	−	0.410147i	$-0.865478\pi$
$919$	−55.2958	−1.82404	−0.912020	+	0.410147i	$0.865478\pi$
$920$	0	0
$921$	51.2015	1.68715
$922$	0	0
$923$	19.8088	0.652016
$924$	0	0
$925$	0	0
$926$	0	0
$927$	25.7294	0.845066
$928$	0	0
$929$	21.1376	0.693503	0.346751	−	0.937957i	$-0.387285\pi$
$929$	21.1376	0.693503	0.346751	+	0.937957i	$0.387285\pi$
$930$	0	0
$931$	6.01310	0.197071
$932$	0	0
$933$	72.9009	2.38667
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.59000	0.0846117	0.0423059	−	0.999105i	$-0.486530\pi$
$937$	2.59000	0.0846117	0.0423059	+	0.999105i	$0.486530\pi$
$938$	0	0
$939$	13.3210	0.434716
$940$	0	0
$941$	−35.1167	−1.14477	−0.572386	−	0.819984i	$-0.693982\pi$
$941$	−35.1167	−1.14477	−0.572386	+	0.819984i	$0.693982\pi$
$942$	0	0
$943$	−4.70538	−0.153228
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.8898	0.906295	0.453148	−	0.891435i	$-0.350301\pi$
$947$	27.8898	0.906295	0.453148	+	0.891435i	$0.350301\pi$
$948$	0	0
$949$	11.6293	0.377504
$950$	0	0
$951$	1.67183	0.0542128
$952$	0	0
$953$	47.2402	1.53026	0.765131	−	0.643875i	$-0.222674\pi$
$953$	47.2402	1.53026	0.765131	+	0.643875i	$0.222674\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	54.7762	1.77066
$958$	0	0
$959$	−61.7846	−1.99513
$960$	0	0
$961$	9.73131	0.313913
$962$	0	0
$963$	−19.1319	−0.616517
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.1848	−0.938519	−0.469259	−	0.883060i	$-0.655479\pi$
$967$	−29.1848	−0.938519	−0.469259	+	0.883060i	$0.655479\pi$
$968$	0	0
$969$	10.4080	0.334352
$970$	0	0
$971$	43.4864	1.39555	0.697773	−	0.716319i	$-0.254174\pi$
$971$	43.4864	1.39555	0.697773	+	0.716319i	$0.254174\pi$
$972$	0	0
$973$	19.8583	0.636629
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−38.2511	−1.22376	−0.611880	−	0.790951i	$-0.709587\pi$
$977$	−38.2511	−1.22376	−0.611880	+	0.790951i	$0.709587\pi$
$978$	0	0
$979$	−32.4770	−1.03797
$980$	0	0
$981$	9.90846	0.316353
$982$	0	0
$983$	53.5820	1.70900	0.854500	−	0.519451i	$-0.173864\pi$
$983$	53.5820	1.70900	0.854500	+	0.519451i	$0.173864\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−33.6202	−1.07014
$988$	0	0
$989$	−8.44289	−0.268468
$990$	0	0
$991$	2.34300	0.0744277	0.0372139	−	0.999307i	$-0.488152\pi$
$991$	2.34300	0.0744277	0.0372139	+	0.999307i	$0.488152\pi$
$992$	0	0
$993$	−56.3051	−1.78679
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−17.6187	−0.557991	−0.278995	−	0.960292i	$-0.590001\pi$
$997$	−17.6187	−0.557991	−0.278995	+	0.960292i	$0.590001\pi$
$998$	0	0
$999$	−0.465300	−0.0147214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.z.1.2		6
4.3	odd	2		7600.2.a.cn.1.5		6
5.2	odd	4		760.2.d.e.609.10	yes	12
5.3	odd	4		760.2.d.e.609.3	✓	12
5.4	even	2		3800.2.a.be.1.5		6
20.3	even	4		1520.2.d.k.609.10		12
20.7	even	4		1520.2.d.k.609.3		12
20.19	odd	2		7600.2.a.cg.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.e.609.3	✓	12	5.3	odd	4
760.2.d.e.609.10	yes	12	5.2	odd	4
1520.2.d.k.609.3		12	20.7	even	4
1520.2.d.k.609.10		12	20.3	even	4
3800.2.a.z.1.2		6	1.1	even	1	trivial
3800.2.a.be.1.5		6	5.4	even	2
7600.2.a.cg.1.2		6	20.19	odd	2
7600.2.a.cn.1.5		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.43031 q^{3} -3.60737 q^{7} +2.90640 q^{9} +O(q^{10})\)	\(q-2.43031 q^{3} -3.60737 q^{7} +2.90640 q^{9} -2.37997 q^{11} +3.96613 q^{13} -4.28257 q^{17} +1.00000 q^{19} +8.76701 q^{21} +1.07377 q^{23} +0.227486 q^{27} +9.47021 q^{29} +6.38211 q^{31} +5.78405 q^{33} -2.04540 q^{37} -9.63892 q^{39} -4.38211 q^{41} -7.86284 q^{43} -3.83485 q^{47} +6.01310 q^{49} +10.4080 q^{51} +11.7789 q^{53} -2.43031 q^{57} +4.59593 q^{59} +6.62819 q^{61} -10.4844 q^{63} -7.02837 q^{67} -2.60959 q^{69} +4.99450 q^{71} +2.93216 q^{73} +8.58541 q^{77} +0.860616 q^{79} -9.27205 q^{81} -11.9179 q^{83} -23.0155 q^{87} +13.6460 q^{89} -14.3073 q^{91} -15.5105 q^{93} -18.5757 q^{97} -6.91712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9	\( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} - 34 q^{51} + 14 q^{53} - 2 q^{57} - 2 q^{59} + 10 q^{61} - 44 q^{63} - 42 q^{67} + 18 q^{69} - 8 q^{71} + 2 q^{73} - 24 q^{77} - 20 q^{79} + 6 q^{81} - 16 q^{83} - 30 q^{87} + 32 q^{89} + 10 q^{91} - 12 q^{93} - 40 q^{97} - 22 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9 - 2 * q^11 - 14 * q^13 - 10 * q^17 + 6 * q^19 + 18 * q^21 - 2 * q^23 - 2 * q^27 - 2 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^37 - 18 * q^39 + 4 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 34 * q^51 + 14 * q^53 - 2 * q^57 - 2 * q^59 + 10 * q^61 - 44 * q^63 - 42 * q^67 + 18 * q^69 - 8 * q^71 + 2 * q^73 - 24 * q^77 - 20 * q^79 + 6 * q^81 - 16 * q^83 - 30 * q^87 + 32 * q^89 + 10 * q^91 - 12 * q^93 - 40 * q^97 - 22 * q^99

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