LMFDB - Embedded newform 4400.2.a.bu.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] 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("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,0,0,5,0,5,0,2,0,-2] 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:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1100)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79129 q^{3} +4.79129 q^{7} +0.208712 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.79129 q^{17} +2.58258 q^{19} -8.58258 q^{21} -0.791288 q^{23} +5.00000 q^{27} +2.20871 q^{29} -0.582576 q^{31} -1.79129 q^{33} +6.58258 q^{37} +1.79129 q^{39} -10.5826 q^{41} +10.0000 q^{43} +10.5826 q^{47} +15.9564 q^{49} +6.79129 q^{51} -2.37386 q^{53} -4.62614 q^{57} +1.41742 q^{59} +8.79129 q^{61} +1.00000 q^{63} +4.00000 q^{67} +1.41742 q^{69} -16.7477 q^{71} -3.20871 q^{73} +4.79129 q^{77} -16.5390 q^{79} -9.58258 q^{81} -12.9564 q^{83} -3.95644 q^{87} +3.79129 q^{89} -4.79129 q^{91} +1.04356 q^{93} -10.7913 q^{97} +0.208712 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 5 q^{7} + 5 q^{9} + 2 q^{11} - 2 q^{13} - 3 q^{17} - 4 q^{19} - 8 q^{21} + 3 q^{23} + 10 q^{27} + 9 q^{29} + 8 q^{31} + q^{33} + 4 q^{37} - q^{39} - 12 q^{41} + 20 q^{43} + 12 q^{47} + 9 q^{49}+ \cdots + 5 q^{99}+O(q^{100}) $ 2 * q + q^3 + 5 * q^7 + 5 * q^9 + 2 * q^11 - 2 * q^13 - 3 * q^17 - 4 * q^19 - 8 * q^21 + 3 * q^23 + 10 * q^27 + 9 * q^29 + 8 * q^31 + q^33 + 4 * q^37 - q^39 - 12 * q^41 + 20 * q^43 + 12 * q^47 + 9 * q^49 + 9 * q^51 + 9 * q^53 - 23 * q^57 + 12 * q^59 + 13 * q^61 + 2 * q^63 + 8 * q^67 + 12 * q^69 - 6 * q^71 - 11 * q^73 + 5 * q^77 - q^79 - 10 * q^81 - 3 * q^83 + 15 * q^87 + 3 * q^89 - 5 * q^91 + 25 * q^93 - 17 * q^97 + 5 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.79129	−1.03420	−0.517100	−	0.855925i	$-0.672989\pi$
$3$	−1.79129	−1.03420	−0.517100	+	0.855925i	$0.672989\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.79129	1.81094	0.905468	−	0.424414i	$-0.139520\pi$
$7$	4.79129	1.81094	0.905468	+	0.424414i	$0.139520\pi$
$8$	0	0
$9$	0.208712	0.0695707
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.79129	−0.919522	−0.459761	−	0.888043i	$-0.652065\pi$
$17$	−3.79129	−0.919522	−0.459761	+	0.888043i	$0.652065\pi$
$18$	0	0
$19$	2.58258	0.592483	0.296242	−	0.955113i	$-0.404267\pi$
$19$	2.58258	0.592483	0.296242	+	0.955113i	$0.404267\pi$
$20$	0	0
$21$	−8.58258	−1.87287
$22$	0	0
$23$	−0.791288	−0.164995	−0.0824975	−	0.996591i	$-0.526290\pi$
$23$	−0.791288	−0.164995	−0.0824975	+	0.996591i	$0.526290\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	2.20871	0.410148	0.205074	−	0.978747i	$-0.434257\pi$
$29$	2.20871	0.410148	0.205074	+	0.978747i	$0.434257\pi$
$30$	0	0
$31$	−0.582576	−0.104634	−0.0523168	−	0.998631i	$-0.516661\pi$
$31$	−0.582576	−0.104634	−0.0523168	+	0.998631i	$0.516661\pi$
$32$	0	0
$33$	−1.79129	−0.311823
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.58258	1.08217	0.541084	−	0.840968i	$-0.318014\pi$
$37$	6.58258	1.08217	0.541084	+	0.840968i	$0.318014\pi$
$38$	0	0
$39$	1.79129	0.286836
$40$	0	0
$41$	−10.5826	−1.65272	−0.826360	−	0.563142i	$-0.809593\pi$
$41$	−10.5826	−1.65272	−0.826360	+	0.563142i	$0.809593\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.5826	1.54363	0.771814	−	0.635849i	$-0.219350\pi$
$47$	10.5826	1.54363	0.771814	+	0.635849i	$0.219350\pi$
$48$	0	0
$49$	15.9564	2.27949
$50$	0	0
$51$	6.79129	0.950971
$52$	0	0
$53$	−2.37386	−0.326075	−0.163038	−	0.986620i	$-0.552129\pi$
$53$	−2.37386	−0.326075	−0.163038	+	0.986620i	$0.552129\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.62614	−0.612747
$58$	0	0
$59$	1.41742	0.184533	0.0922665	−	0.995734i	$-0.470589\pi$
$59$	1.41742	0.184533	0.0922665	+	0.995734i	$0.470589\pi$
$60$	0	0
$61$	8.79129	1.12561	0.562805	−	0.826590i	$-0.309722\pi$
$61$	8.79129	1.12561	0.562805	+	0.826590i	$0.309722\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	1.41742	0.170638
$70$	0	0
$71$	−16.7477	−1.98759	−0.993795	−	0.111229i	$-0.964521\pi$
$71$	−16.7477	−1.98759	−0.993795	+	0.111229i	$0.964521\pi$
$72$	0	0
$73$	−3.20871	−0.375551	−0.187776	−	0.982212i	$-0.560128\pi$
$73$	−3.20871	−0.375551	−0.187776	+	0.982212i	$0.560128\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.79129	0.546018
$78$	0	0
$79$	−16.5390	−1.86078	−0.930392	−	0.366565i	$-0.880534\pi$
$79$	−16.5390	−1.86078	−0.930392	+	0.366565i	$0.880534\pi$
$80$	0	0
$81$	−9.58258	−1.06473
$82$	0	0
$83$	−12.9564	−1.42215	−0.711077	−	0.703114i	$-0.751792\pi$
$83$	−12.9564	−1.42215	−0.711077	+	0.703114i	$0.751792\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.95644	−0.424175
$88$	0	0
$89$	3.79129	0.401876	0.200938	−	0.979604i	$-0.435601\pi$
$89$	3.79129	0.401876	0.200938	+	0.979604i	$0.435601\pi$
$90$	0	0
$91$	−4.79129	−0.502263
$92$	0	0
$93$	1.04356	0.108212
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.7913	−1.09569	−0.547845	−	0.836580i	$-0.684552\pi$
$97$	−10.7913	−1.09569	−0.547845	+	0.836580i	$0.684552\pi$
$98$	0	0
$99$	0.208712	0.0209764
$100$	0	0
$101$	−3.62614	−0.360814	−0.180407	−	0.983592i	$-0.557741\pi$
$101$	−3.62614	−0.360814	−0.180407	+	0.983592i	$0.557741\pi$
$102$	0	0
$103$	16.9564	1.67077	0.835384	−	0.549667i	$-0.185245\pi$
$103$	16.9564	1.67077	0.835384	+	0.549667i	$0.185245\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.74773	0.749001	0.374501	−	0.927227i	$-0.377814\pi$
$107$	7.74773	0.749001	0.374501	+	0.927227i	$0.377814\pi$
$108$	0	0
$109$	−0.208712	−0.0199910	−0.00999550	−	0.999950i	$-0.503182\pi$
$109$	−0.208712	−0.0199910	−0.00999550	+	0.999950i	$0.503182\pi$
$110$	0	0
$111$	−11.7913	−1.11918
$112$	0	0
$113$	10.5826	0.995525	0.497762	−	0.867313i	$-0.334155\pi$
$113$	10.5826	0.995525	0.497762	+	0.867313i	$0.334155\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.208712	−0.0192954
$118$	0	0
$119$	−18.1652	−1.66520
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	18.9564	1.70924
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.62614	−0.233032	−0.116516	−	0.993189i	$-0.537173\pi$
$127$	−2.62614	−0.233032	−0.116516	+	0.993189i	$0.537173\pi$
$128$	0	0
$129$	−17.9129	−1.57714
$130$	0	0
$131$	14.3739	1.25585	0.627925	−	0.778274i	$-0.283904\pi$
$131$	14.3739	1.25585	0.627925	+	0.778274i	$0.283904\pi$
$132$	0	0
$133$	12.3739	1.07295
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.5390	1.75477	0.877383	−	0.479790i	$-0.159287\pi$
$137$	20.5390	1.75477	0.877383	+	0.479790i	$0.159287\pi$
$138$	0	0
$139$	17.7477	1.50534	0.752671	−	0.658396i	$-0.228765\pi$
$139$	17.7477	1.50534	0.752671	+	0.658396i	$0.228765\pi$
$140$	0	0
$141$	−18.9564	−1.59642
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−28.5826	−2.35745
$148$	0	0
$149$	−15.1652	−1.24238	−0.621189	−	0.783661i	$-0.713350\pi$
$149$	−15.1652	−1.24238	−0.621189	+	0.783661i	$0.713350\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	−0.791288	−0.0639718
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.16515	−0.0929892	−0.0464946	−	0.998919i	$-0.514805\pi$
$157$	−1.16515	−0.0929892	−0.0464946	+	0.998919i	$0.514805\pi$
$158$	0	0
$159$	4.25227	0.337227
$160$	0	0
$161$	−3.79129	−0.298795
$162$	0	0
$163$	7.62614	0.597325	0.298663	−	0.954359i	$-0.403459\pi$
$163$	7.62614	0.597325	0.298663	+	0.954359i	$0.403459\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.41742	0.573978	0.286989	−	0.957934i	$-0.407346\pi$
$167$	7.41742	0.573978	0.286989	+	0.957934i	$0.407346\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0.539015	0.0412195
$172$	0	0
$173$	13.7477	1.04522	0.522610	−	0.852572i	$-0.324958\pi$
$173$	13.7477	1.04522	0.522610	+	0.852572i	$0.324958\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.53901	−0.190844
$178$	0	0
$179$	19.1216	1.42921	0.714607	−	0.699526i	$-0.246605\pi$
$179$	19.1216	1.42921	0.714607	+	0.699526i	$0.246605\pi$
$180$	0	0
$181$	10.3739	0.771083	0.385542	−	0.922690i	$-0.374015\pi$
$181$	10.3739	0.771083	0.385542	+	0.922690i	$0.374015\pi$
$182$	0	0
$183$	−15.7477	−1.16411
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.79129	−0.277246
$188$	0	0
$189$	23.9564	1.74257
$190$	0	0
$191$	−21.7913	−1.57676	−0.788381	−	0.615187i	$-0.789080\pi$
$191$	−21.7913	−1.57676	−0.788381	+	0.615187i	$0.789080\pi$
$192$	0	0
$193$	11.1652	0.803685	0.401843	−	0.915709i	$-0.368370\pi$
$193$	11.1652	0.803685	0.401843	+	0.915709i	$0.368370\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.53901	−0.394638	−0.197319	−	0.980339i	$-0.563224\pi$
$197$	−5.53901	−0.394638	−0.197319	+	0.980339i	$0.563224\pi$
$198$	0	0
$199$	15.3739	1.08982	0.544912	−	0.838493i	$-0.316563\pi$
$199$	15.3739	1.08982	0.544912	+	0.838493i	$0.316563\pi$
$200$	0	0
$201$	−7.16515	−0.505391
$202$	0	0
$203$	10.5826	0.742751
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.165151	−0.0114788
$208$	0	0
$209$	2.58258	0.178640
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	30.0000	2.05557
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.79129	−0.189485
$218$	0	0
$219$	5.74773	0.388395
$220$	0	0
$221$	3.79129	0.255030
$222$	0	0
$223$	11.4174	0.764567	0.382284	−	0.924045i	$-0.375138\pi$
$223$	11.4174	0.764567	0.382284	+	0.924045i	$0.375138\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.5390	1.36322	0.681611	−	0.731715i	$-0.261280\pi$
$227$	20.5390	1.36322	0.681611	+	0.731715i	$0.261280\pi$
$228$	0	0
$229$	−21.3739	−1.41242	−0.706212	−	0.708000i	$-0.749598\pi$
$229$	−21.3739	−1.41242	−0.706212	+	0.708000i	$0.749598\pi$
$230$	0	0
$231$	−8.58258	−0.564692
$232$	0	0
$233$	11.2087	0.734307	0.367154	−	0.930160i	$-0.380332\pi$
$233$	11.2087	0.734307	0.367154	+	0.930160i	$0.380332\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	29.6261	1.92442
$238$	0	0
$239$	6.79129	0.439292	0.219646	−	0.975580i	$-0.429510\pi$
$239$	6.79129	0.439292	0.219646	+	0.975580i	$0.429510\pi$
$240$	0	0
$241$	18.1216	1.16731	0.583657	−	0.812000i	$-0.301621\pi$
$241$	18.1216	1.16731	0.583657	+	0.812000i	$0.301621\pi$
$242$	0	0
$243$	2.16515	0.138895
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.58258	−0.164325
$248$	0	0
$249$	23.2087	1.47079
$250$	0	0
$251$	−20.5390	−1.29641	−0.648206	−	0.761465i	$-0.724480\pi$
$251$	−20.5390	−1.29641	−0.648206	+	0.761465i	$0.724480\pi$
$252$	0	0
$253$	−0.791288	−0.0497478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	31.5390	1.95974
$260$	0	0
$261$	0.460985	0.0285343
$262$	0	0
$263$	−5.83485	−0.359792	−0.179896	−	0.983686i	$-0.557576\pi$
$263$	−5.83485	−0.359792	−0.179896	+	0.983686i	$0.557576\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.79129	−0.415620
$268$	0	0
$269$	29.7042	1.81109	0.905547	−	0.424245i	$-0.139460\pi$
$269$	29.7042	1.81109	0.905547	+	0.424245i	$0.139460\pi$
$270$	0	0
$271$	11.7477	0.713624	0.356812	−	0.934176i	$-0.383864\pi$
$271$	11.7477	0.713624	0.356812	+	0.934176i	$0.383864\pi$
$272$	0	0
$273$	8.58258	0.519441
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.582576	0.0350036	0.0175018	−	0.999847i	$-0.494429\pi$
$277$	0.582576	0.0350036	0.0175018	+	0.999847i	$0.494429\pi$
$278$	0	0
$279$	−0.121591	−0.00727944
$280$	0	0
$281$	13.7477	0.820121	0.410060	−	0.912058i	$-0.365508\pi$
$281$	13.7477	0.820121	0.410060	+	0.912058i	$0.365508\pi$
$282$	0	0
$283$	−19.7042	−1.17129	−0.585646	−	0.810567i	$-0.699159\pi$
$283$	−19.7042	−1.17129	−0.585646	+	0.810567i	$0.699159\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−50.7042	−2.99297
$288$	0	0
$289$	−2.62614	−0.154479
$290$	0	0
$291$	19.3303	1.13316
$292$	0	0
$293$	−21.1652	−1.23648	−0.618241	−	0.785989i	$-0.712154\pi$
$293$	−21.1652	−1.23648	−0.618241	+	0.785989i	$0.712154\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	0.791288	0.0457614
$300$	0	0
$301$	47.9129	2.76165
$302$	0	0
$303$	6.49545	0.373154
$304$	0	0
$305$	0	0
$306$	0	0
$307$	31.7913	1.81442	0.907212	−	0.420673i	$-0.138206\pi$
$307$	31.7913	1.81442	0.907212	+	0.420673i	$0.138206\pi$
$308$	0	0
$309$	−30.3739	−1.72791
$310$	0	0
$311$	24.1652	1.37028	0.685140	−	0.728411i	$-0.259741\pi$
$311$	24.1652	1.37028	0.685140	+	0.728411i	$0.259741\pi$
$312$	0	0
$313$	−20.7477	−1.17273	−0.586365	−	0.810047i	$-0.699442\pi$
$313$	−20.7477	−1.17273	−0.586365	+	0.810047i	$0.699442\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.37386	0.301826	0.150913	−	0.988547i	$-0.451779\pi$
$317$	5.37386	0.301826	0.150913	+	0.988547i	$0.451779\pi$
$318$	0	0
$319$	2.20871	0.123664
$320$	0	0
$321$	−13.8784	−0.774617
$322$	0	0
$323$	−9.79129	−0.544802
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.373864	0.0206747
$328$	0	0
$329$	50.7042	2.79541
$330$	0	0
$331$	19.3303	1.06249	0.531245	−	0.847218i	$-0.321724\pi$
$331$	19.3303	1.06249	0.531245	+	0.847218i	$0.321724\pi$
$332$	0	0
$333$	1.37386	0.0752873
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−18.9564	−1.02957
$340$	0	0
$341$	−0.582576	−0.0315482
$342$	0	0
$343$	42.9129	2.31708
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.3739	−1.41582	−0.707912	−	0.706301i	$-0.750362\pi$
$347$	−26.3739	−1.41582	−0.707912	+	0.706301i	$0.750362\pi$
$348$	0	0
$349$	−5.41742	−0.289988	−0.144994	−	0.989433i	$-0.546316\pi$
$349$	−5.41742	−0.289988	−0.144994	+	0.989433i	$0.546316\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−0.165151	−0.00879012	−0.00439506	−	0.999990i	$-0.501399\pi$
$353$	−0.165151	−0.00879012	−0.00439506	+	0.999990i	$0.501399\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	32.5390	1.72215
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−12.3303	−0.648963
$362$	0	0
$363$	−1.79129	−0.0940182
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−22.3739	−1.16791	−0.583953	−	0.811787i	$-0.698495\pi$
$367$	−22.3739	−1.16791	−0.583953	+	0.811787i	$0.698495\pi$
$368$	0	0
$369$	−2.20871	−0.114981
$370$	0	0
$371$	−11.3739	−0.590502
$372$	0	0
$373$	17.3303	0.897329	0.448665	−	0.893700i	$-0.351900\pi$
$373$	17.3303	0.897329	0.448665	+	0.893700i	$0.351900\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.20871	−0.113754
$378$	0	0
$379$	31.0000	1.59236	0.796182	−	0.605058i	$-0.206850\pi$
$379$	31.0000	1.59236	0.796182	+	0.605058i	$0.206850\pi$
$380$	0	0
$381$	4.70417	0.241002
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.08712	0.106094
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	−25.7477	−1.29880
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.7913	0.591788	0.295894	−	0.955221i	$-0.404383\pi$
$397$	11.7913	0.591788	0.295894	+	0.955221i	$0.404383\pi$
$398$	0	0
$399$	−22.1652	−1.10965
$400$	0	0
$401$	−33.1652	−1.65619	−0.828094	−	0.560589i	$-0.810575\pi$
$401$	−33.1652	−1.65619	−0.828094	+	0.560589i	$0.810575\pi$
$402$	0	0
$403$	0.582576	0.0290202
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.58258	0.326286
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	−36.7913	−1.81478
$412$	0	0
$413$	6.79129	0.334177
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−31.7913	−1.55683
$418$	0	0
$419$	−27.0000	−1.31904	−0.659518	−	0.751689i	$-0.729240\pi$
$419$	−27.0000	−1.31904	−0.659518	+	0.751689i	$0.729240\pi$
$420$	0	0
$421$	11.6261	0.566623	0.283312	−	0.959028i	$-0.408567\pi$
$421$	11.6261	0.566623	0.283312	+	0.959028i	$0.408567\pi$
$422$	0	0
$423$	2.20871	0.107391
$424$	0	0
$425$	0	0
$426$	0	0
$427$	42.1216	2.03841
$428$	0	0
$429$	1.79129	0.0864842
$430$	0	0
$431$	34.9129	1.68169	0.840847	−	0.541273i	$-0.182057\pi$
$431$	34.9129	1.68169	0.840847	+	0.541273i	$0.182057\pi$
$432$	0	0
$433$	23.3303	1.12118	0.560591	−	0.828093i	$-0.310574\pi$
$433$	23.3303	1.12118	0.560591	+	0.828093i	$0.310574\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.04356	−0.0977568
$438$	0	0
$439$	10.9564	0.522922	0.261461	−	0.965214i	$-0.415796\pi$
$439$	10.9564	0.522922	0.261461	+	0.965214i	$0.415796\pi$
$440$	0	0
$441$	3.33030	0.158586
$442$	0	0
$443$	−31.5826	−1.50053	−0.750267	−	0.661135i	$-0.770075\pi$
$443$	−31.5826	−1.50053	−0.750267	+	0.661135i	$0.770075\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	27.1652	1.28487
$448$	0	0
$449$	30.7913	1.45313	0.726565	−	0.687097i	$-0.241115\pi$
$449$	30.7913	1.45313	0.726565	+	0.687097i	$0.241115\pi$
$450$	0	0
$451$	−10.5826	−0.498314
$452$	0	0
$453$	8.95644	0.420810
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.373864	−0.0174886	−0.00874430	−	0.999962i	$-0.502783\pi$
$457$	−0.373864	−0.0174886	−0.00874430	+	0.999962i	$0.502783\pi$
$458$	0	0
$459$	−18.9564	−0.884811
$460$	0	0
$461$	30.4955	1.42031	0.710157	−	0.704043i	$-0.248624\pi$
$461$	30.4955	1.42031	0.710157	+	0.704043i	$0.248624\pi$
$462$	0	0
$463$	29.7477	1.38249	0.691247	−	0.722619i	$-0.257062\pi$
$463$	29.7477	1.38249	0.691247	+	0.722619i	$0.257062\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.41742	−0.0655906	−0.0327953	−	0.999462i	$-0.510441\pi$
$467$	−1.41742	−0.0655906	−0.0327953	+	0.999462i	$0.510441\pi$
$468$	0	0
$469$	19.1652	0.884964
$470$	0	0
$471$	2.08712	0.0961695
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.495454	−0.0226853
$478$	0	0
$479$	18.1652	0.829987	0.414993	−	0.909824i	$-0.363784\pi$
$479$	18.1652	0.829987	0.414993	+	0.909824i	$0.363784\pi$
$480$	0	0
$481$	−6.58258	−0.300140
$482$	0	0
$483$	6.79129	0.309014
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.25227	0.102060	0.0510301	−	0.998697i	$-0.483750\pi$
$487$	2.25227	0.102060	0.0510301	+	0.998697i	$0.483750\pi$
$488$	0	0
$489$	−13.6606	−0.617754
$490$	0	0
$491$	16.5826	0.748361	0.374181	−	0.927356i	$-0.377924\pi$
$491$	16.5826	0.748361	0.374181	+	0.927356i	$0.377924\pi$
$492$	0	0
$493$	−8.37386	−0.377140
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−80.2432	−3.59940
$498$	0	0
$499$	7.95644	0.356179	0.178090	−	0.984014i	$-0.443008\pi$
$499$	7.95644	0.356179	0.178090	+	0.984014i	$0.443008\pi$
$500$	0	0
$501$	−13.2867	−0.593608
$502$	0	0
$503$	−25.5826	−1.14067	−0.570335	−	0.821412i	$-0.693187\pi$
$503$	−25.5826	−1.14067	−0.570335	+	0.821412i	$0.693187\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	21.4955	0.954647
$508$	0	0
$509$	−31.1216	−1.37944	−0.689720	−	0.724076i	$-0.742266\pi$
$509$	−31.1216	−1.37944	−0.689720	+	0.724076i	$0.742266\pi$
$510$	0	0
$511$	−15.3739	−0.680100
$512$	0	0
$513$	12.9129	0.570118
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.5826	0.465421
$518$	0	0
$519$	−24.6261	−1.08097
$520$	0	0
$521$	−7.58258	−0.332199	−0.166099	−	0.986109i	$-0.553117\pi$
$521$	−7.58258	−0.332199	−0.166099	+	0.986109i	$0.553117\pi$
$522$	0	0
$523$	6.83485	0.298867	0.149434	−	0.988772i	$-0.452255\pi$
$523$	6.83485	0.298867	0.149434	+	0.988772i	$0.452255\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.20871	0.0962130
$528$	0	0
$529$	−22.3739	−0.972777
$530$	0	0
$531$	0.295834	0.0128381
$532$	0	0
$533$	10.5826	0.458382
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−34.2523	−1.47809
$538$	0	0
$539$	15.9564	0.687292
$540$	0	0
$541$	−24.3739	−1.04791	−0.523957	−	0.851745i	$-0.675545\pi$
$541$	−24.3739	−1.04791	−0.523957	+	0.851745i	$0.675545\pi$
$542$	0	0
$543$	−18.5826	−0.797455
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.37386	0.272527	0.136263	−	0.990673i	$-0.456491\pi$
$547$	6.37386	0.272527	0.136263	+	0.990673i	$0.456491\pi$
$548$	0	0
$549$	1.83485	0.0783094
$550$	0	0
$551$	5.70417	0.243006
$552$	0	0
$553$	−79.2432	−3.36976
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.6606	1.17202	0.586009	−	0.810305i	$-0.300698\pi$
$557$	27.6606	1.17202	0.586009	+	0.810305i	$0.300698\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	6.79129	0.286728
$562$	0	0
$563$	−1.12159	−0.0472694	−0.0236347	−	0.999721i	$-0.507524\pi$
$563$	−1.12159	−0.0472694	−0.0236347	+	0.999721i	$0.507524\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−45.9129	−1.92816
$568$	0	0
$569$	−3.95644	−0.165863	−0.0829313	−	0.996555i	$-0.526428\pi$
$569$	−3.95644	−0.165863	−0.0829313	+	0.996555i	$0.526428\pi$
$570$	0	0
$571$	32.2867	1.35116	0.675579	−	0.737288i	$-0.263894\pi$
$571$	32.2867	1.35116	0.675579	+	0.737288i	$0.263894\pi$
$572$	0	0
$573$	39.0345	1.63069
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10.9564	−0.456123	−0.228061	−	0.973647i	$-0.573239\pi$
$577$	−10.9564	−0.456123	−0.228061	+	0.973647i	$0.573239\pi$
$578$	0	0
$579$	−20.0000	−0.831172
$580$	0	0
$581$	−62.0780	−2.57543
$582$	0	0
$583$	−2.37386	−0.0983154
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.95644	0.287123	0.143561	−	0.989641i	$-0.454145\pi$
$587$	6.95644	0.287123	0.143561	+	0.989641i	$0.454145\pi$
$588$	0	0
$589$	−1.50455	−0.0619937
$590$	0	0
$591$	9.92197	0.408135
$592$	0	0
$593$	−13.4174	−0.550988	−0.275494	−	0.961303i	$-0.588841\pi$
$593$	−13.4174	−0.550988	−0.275494	+	0.961303i	$0.588841\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−27.5390	−1.12710
$598$	0	0
$599$	5.70417	0.233066	0.116533	−	0.993187i	$-0.462822\pi$
$599$	5.70417	0.233066	0.116533	+	0.993187i	$0.462822\pi$
$600$	0	0
$601$	−3.53901	−0.144359	−0.0721797	−	0.997392i	$-0.522996\pi$
$601$	−3.53901	−0.144359	−0.0721797	+	0.997392i	$0.522996\pi$
$602$	0	0
$603$	0.834849	0.0339977
$604$	0	0
$605$	0	0
$606$	0	0
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	0	0
$609$	−18.9564	−0.768154
$610$	0	0
$611$	−10.5826	−0.428125
$612$	0	0
$613$	−44.4519	−1.79540	−0.897698	−	0.440612i	$-0.854761\pi$
$613$	−44.4519	−1.79540	−0.897698	+	0.440612i	$0.854761\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.25227	−0.0504146	−0.0252073	−	0.999682i	$-0.508025\pi$
$617$	−1.25227	−0.0504146	−0.0252073	+	0.999682i	$0.508025\pi$
$618$	0	0
$619$	−33.7477	−1.35644	−0.678218	−	0.734861i	$-0.737247\pi$
$619$	−33.7477	−1.35644	−0.678218	+	0.734861i	$0.737247\pi$
$620$	0	0
$621$	−3.95644	−0.158766
$622$	0	0
$623$	18.1652	0.727771
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.62614	−0.184750
$628$	0	0
$629$	−24.9564	−0.995078
$630$	0	0
$631$	23.1216	0.920456	0.460228	−	0.887801i	$-0.347768\pi$
$631$	23.1216	0.920456	0.460228	+	0.887801i	$0.347768\pi$
$632$	0	0
$633$	8.95644	0.355987
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−15.9564	−0.632217
$638$	0	0
$639$	−3.49545	−0.138278
$640$	0	0
$641$	−42.1652	−1.66542	−0.832712	−	0.553707i	$-0.813213\pi$
$641$	−42.1652	−1.66542	−0.832712	+	0.553707i	$0.813213\pi$
$642$	0	0
$643$	−11.4955	−0.453336	−0.226668	−	0.973972i	$-0.572783\pi$
$643$	−11.4955	−0.453336	−0.226668	+	0.973972i	$0.572783\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.3303	0.602696	0.301348	−	0.953514i	$-0.402563\pi$
$647$	15.3303	0.602696	0.301348	+	0.953514i	$0.402563\pi$
$648$	0	0
$649$	1.41742	0.0556388
$650$	0	0
$651$	5.00000	0.195965
$652$	0	0
$653$	−23.7042	−0.927616	−0.463808	−	0.885936i	$-0.653517\pi$
$653$	−23.7042	−0.927616	−0.463808	+	0.885936i	$0.653517\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.669697	−0.0261274
$658$	0	0
$659$	−32.5390	−1.26754	−0.633770	−	0.773522i	$-0.718493\pi$
$659$	−32.5390	−1.26754	−0.633770	+	0.773522i	$0.718493\pi$
$660$	0	0
$661$	21.4174	0.833041	0.416521	−	0.909126i	$-0.363249\pi$
$661$	21.4174	0.833041	0.416521	+	0.909126i	$0.363249\pi$
$662$	0	0
$663$	−6.79129	−0.263752
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.74773	−0.0676723
$668$	0	0
$669$	−20.4519	−0.790716
$670$	0	0
$671$	8.79129	0.339384
$672$	0	0
$673$	6.91288	0.266472	0.133236	−	0.991084i	$-0.457463\pi$
$673$	6.91288	0.266472	0.133236	+	0.991084i	$0.457463\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.58258	0.291422	0.145711	−	0.989327i	$-0.453453\pi$
$677$	7.58258	0.291422	0.145711	+	0.989327i	$0.453453\pi$
$678$	0	0
$679$	−51.7042	−1.98422
$680$	0	0
$681$	−36.7913	−1.40985
$682$	0	0
$683$	10.9129	0.417570	0.208785	−	0.977962i	$-0.433049\pi$
$683$	10.9129	0.417570	0.208785	+	0.977962i	$0.433049\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	38.2867	1.46073
$688$	0	0
$689$	2.37386	0.0904370
$690$	0	0
$691$	−9.12159	−0.347002	−0.173501	−	0.984834i	$-0.555508\pi$
$691$	−9.12159	−0.347002	−0.173501	+	0.984834i	$0.555508\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	40.1216	1.51971
$698$	0	0
$699$	−20.0780	−0.759421
$700$	0	0
$701$	39.1652	1.47925	0.739624	−	0.673021i	$-0.235004\pi$
$701$	39.1652	1.47925	0.739624	+	0.673021i	$0.235004\pi$
$702$	0	0
$703$	17.0000	0.641167
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.3739	−0.653411
$708$	0	0
$709$	−25.4955	−0.957502	−0.478751	−	0.877951i	$-0.658910\pi$
$709$	−25.4955	−0.957502	−0.478751	+	0.877951i	$0.658910\pi$
$710$	0	0
$711$	−3.45189	−0.129456
$712$	0	0
$713$	0.460985	0.0172640
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12.1652	−0.454316
$718$	0	0
$719$	48.4955	1.80858	0.904288	−	0.426924i	$-0.140403\pi$
$719$	48.4955	1.80858	0.904288	+	0.426924i	$0.140403\pi$
$720$	0	0
$721$	81.2432	3.02565
$722$	0	0
$723$	−32.4610	−1.20724
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−18.1216	−0.672093	−0.336046	−	0.941845i	$-0.609090\pi$
$727$	−18.1216	−0.672093	−0.336046	+	0.941845i	$0.609090\pi$
$728$	0	0
$729$	24.8693	0.921086
$730$	0	0
$731$	−37.9129	−1.40226
$732$	0	0
$733$	−51.2432	−1.89271	−0.946355	−	0.323129i	$-0.895265\pi$
$733$	−51.2432	−1.89271	−0.946355	+	0.323129i	$0.895265\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	−0.878409	−0.0323128	−0.0161564	−	0.999869i	$-0.505143\pi$
$739$	−0.878409	−0.0323128	−0.0161564	+	0.999869i	$0.505143\pi$
$740$	0	0
$741$	4.62614	0.169945
$742$	0	0
$743$	−35.2087	−1.29168	−0.645841	−	0.763472i	$-0.723493\pi$
$743$	−35.2087	−1.29168	−0.645841	+	0.763472i	$0.723493\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.70417	−0.0989403
$748$	0	0
$749$	37.1216	1.35639
$750$	0	0
$751$	−17.7913	−0.649213	−0.324607	−	0.945849i	$-0.605232\pi$
$751$	−17.7913	−0.649213	−0.324607	+	0.945849i	$0.605232\pi$
$752$	0	0
$753$	36.7913	1.34075
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.3303	1.06603	0.533014	−	0.846106i	$-0.321059\pi$
$757$	29.3303	1.06603	0.533014	+	0.846106i	$0.321059\pi$
$758$	0	0
$759$	1.41742	0.0514492
$760$	0	0
$761$	−6.33030	−0.229473	−0.114737	−	0.993396i	$-0.536602\pi$
$761$	−6.33030	−0.229473	−0.114737	+	0.993396i	$0.536602\pi$
$762$	0	0
$763$	−1.00000	−0.0362024
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.41742	−0.0511802
$768$	0	0
$769$	−14.2523	−0.513950	−0.256975	−	0.966418i	$-0.582726\pi$
$769$	−14.2523	−0.513950	−0.256975	+	0.966418i	$0.582726\pi$
$770$	0	0
$771$	−32.2432	−1.16121
$772$	0	0
$773$	30.7913	1.10749	0.553743	−	0.832688i	$-0.313199\pi$
$773$	30.7913	1.10749	0.553743	+	0.832688i	$0.313199\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−56.4955	−2.02676
$778$	0	0
$779$	−27.3303	−0.979210
$780$	0	0
$781$	−16.7477	−0.599281
$782$	0	0
$783$	11.0436	0.394665
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9.74773	−0.347469	−0.173734	−	0.984793i	$-0.555583\pi$
$787$	−9.74773	−0.347469	−0.173734	+	0.984793i	$0.555583\pi$
$788$	0	0
$789$	10.4519	0.372097
$790$	0	0
$791$	50.7042	1.80283
$792$	0	0
$793$	−8.79129	−0.312188
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.2867	−0.895702	−0.447851	−	0.894108i	$-0.647811\pi$
$797$	−25.2867	−0.895702	−0.447851	+	0.894108i	$0.647811\pi$
$798$	0	0
$799$	−40.1216	−1.41940
$800$	0	0
$801$	0.791288	0.0279588
$802$	0	0
$803$	−3.20871	−0.113233
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−53.2087	−1.87304
$808$	0	0
$809$	9.49545	0.333842	0.166921	−	0.985970i	$-0.446617\pi$
$809$	9.49545	0.333842	0.166921	+	0.985970i	$0.446617\pi$
$810$	0	0
$811$	−17.6606	−0.620148	−0.310074	−	0.950712i	$-0.600354\pi$
$811$	−17.6606	−0.620148	−0.310074	+	0.950712i	$0.600354\pi$
$812$	0	0
$813$	−21.0436	−0.738030
$814$	0	0
$815$	0	0
$816$	0	0
$817$	25.8258	0.903529
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−24.6606	−0.860661	−0.430331	−	0.902671i	$-0.641603\pi$
$821$	−24.6606	−0.860661	−0.430331	+	0.902671i	$0.641603\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.5826	0.680953	0.340476	−	0.940253i	$-0.389412\pi$
$827$	19.5826	0.680953	0.340476	+	0.940253i	$0.389412\pi$
$828$	0	0
$829$	−15.0436	−0.522484	−0.261242	−	0.965273i	$-0.584132\pi$
$829$	−15.0436	−0.522484	−0.261242	+	0.965273i	$0.584132\pi$
$830$	0	0
$831$	−1.04356	−0.0362007
$832$	0	0
$833$	−60.4955	−2.09604
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.91288	−0.100684
$838$	0	0
$839$	−26.7042	−0.921930	−0.460965	−	0.887418i	$-0.652497\pi$
$839$	−26.7042	−0.921930	−0.460965	+	0.887418i	$0.652497\pi$
$840$	0	0
$841$	−24.1216	−0.831779
$842$	0	0
$843$	−24.6261	−0.848169
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.79129	0.164631
$848$	0	0
$849$	35.2958	1.21135
$850$	0	0
$851$	−5.20871	−0.178552
$852$	0	0
$853$	−38.1216	−1.30526	−0.652629	−	0.757677i	$-0.726334\pi$
$853$	−38.1216	−1.30526	−0.652629	+	0.757677i	$0.726334\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.4955	0.836749	0.418374	−	0.908275i	$-0.362600\pi$
$857$	24.4955	0.836749	0.418374	+	0.908275i	$0.362600\pi$
$858$	0	0
$859$	47.9129	1.63477	0.817383	−	0.576094i	$-0.195424\pi$
$859$	47.9129	1.63477	0.817383	+	0.576094i	$0.195424\pi$
$860$	0	0
$861$	90.8258	3.09533
$862$	0	0
$863$	−9.33030	−0.317607	−0.158804	−	0.987310i	$-0.550764\pi$
$863$	−9.33030	−0.317607	−0.158804	+	0.987310i	$0.550764\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.70417	0.159762
$868$	0	0
$869$	−16.5390	−0.561048
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	−2.25227	−0.0762279
$874$	0	0
$875$	0	0
$876$	0	0
$877$	56.6606	1.91329	0.956646	−	0.291252i	$-0.0940718\pi$
$877$	56.6606	1.91329	0.956646	+	0.291252i	$0.0940718\pi$
$878$	0	0
$879$	37.9129	1.27877
$880$	0	0
$881$	43.1216	1.45280	0.726402	−	0.687270i	$-0.241191\pi$
$881$	43.1216	1.45280	0.726402	+	0.687270i	$0.241191\pi$
$882$	0	0
$883$	−7.83485	−0.263664	−0.131832	−	0.991272i	$-0.542086\pi$
$883$	−7.83485	−0.263664	−0.131832	+	0.991272i	$0.542086\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.66970	0.0896397	0.0448198	−	0.998995i	$-0.485729\pi$
$887$	2.66970	0.0896397	0.0448198	+	0.998995i	$0.485729\pi$
$888$	0	0
$889$	−12.5826	−0.422006
$890$	0	0
$891$	−9.58258	−0.321028
$892$	0	0
$893$	27.3303	0.914574
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.41742	−0.0473264
$898$	0	0
$899$	−1.28674	−0.0429152
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	−85.8258	−2.85610
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.6606	1.35011	0.675057	−	0.737766i	$-0.264119\pi$
$907$	40.6606	1.35011	0.675057	+	0.737766i	$0.264119\pi$
$908$	0	0
$909$	−0.756819	−0.0251021
$910$	0	0
$911$	−24.3303	−0.806099	−0.403049	−	0.915178i	$-0.632050\pi$
$911$	−24.3303	−0.806099	−0.403049	+	0.915178i	$0.632050\pi$
$912$	0	0
$913$	−12.9564	−0.428796
$914$	0	0
$915$	0	0
$916$	0	0
$917$	68.8693	2.27427
$918$	0	0
$919$	−47.1652	−1.55583	−0.777917	−	0.628367i	$-0.783724\pi$
$919$	−47.1652	−1.55583	−0.777917	+	0.628367i	$0.783724\pi$
$920$	0	0
$921$	−56.9473	−1.87648
$922$	0	0
$923$	16.7477	0.551258
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.53901	0.116237
$928$	0	0
$929$	−13.7477	−0.451048	−0.225524	−	0.974238i	$-0.572409\pi$
$929$	−13.7477	−0.451048	−0.225524	+	0.974238i	$0.572409\pi$
$930$	0	0
$931$	41.2087	1.35056
$932$	0	0
$933$	−43.2867	−1.41714
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.8348	0.549971	0.274985	−	0.961448i	$-0.411327\pi$
$937$	16.8348	0.549971	0.274985	+	0.961448i	$0.411327\pi$
$938$	0	0
$939$	37.1652	1.21284
$940$	0	0
$941$	14.0780	0.458931	0.229465	−	0.973317i	$-0.426302\pi$
$941$	14.0780	0.458931	0.229465	+	0.973317i	$0.426302\pi$
$942$	0	0
$943$	8.37386	0.272691
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.9129	−1.32949	−0.664745	−	0.747070i	$-0.731460\pi$
$947$	−40.9129	−1.32949	−0.664745	+	0.747070i	$0.731460\pi$
$948$	0	0
$949$	3.20871	0.104159
$950$	0	0
$951$	−9.62614	−0.312149
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.95644	−0.127894
$958$	0	0
$959$	98.4083	3.17777
$960$	0	0
$961$	−30.6606	−0.989052
$962$	0	0
$963$	1.61704	0.0521085
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11.4610	−0.368560	−0.184280	−	0.982874i	$-0.558995\pi$
$967$	−11.4610	−0.368560	−0.184280	+	0.982874i	$0.558995\pi$
$968$	0	0
$969$	17.5390	0.563434
$970$	0	0
$971$	3.95644	0.126968	0.0634841	−	0.997983i	$-0.479779\pi$
$971$	3.95644	0.126968	0.0634841	+	0.997983i	$0.479779\pi$
$972$	0	0
$973$	85.0345	2.72608
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.74773	0.0559147	0.0279574	−	0.999609i	$-0.491100\pi$
$977$	1.74773	0.0559147	0.0279574	+	0.999609i	$0.491100\pi$
$978$	0	0
$979$	3.79129	0.121170
$980$	0	0
$981$	−0.0435608	−0.00139079
$982$	0	0
$983$	−24.6606	−0.786551	−0.393276	−	0.919421i	$-0.628658\pi$
$983$	−24.6606	−0.786551	−0.393276	+	0.919421i	$0.628658\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−90.8258	−2.89102
$988$	0	0
$989$	−7.91288	−0.251615
$990$	0	0
$991$	12.8693	0.408807	0.204404	−	0.978887i	$-0.434474\pi$
$991$	12.8693	0.408807	0.204404	+	0.978887i	$0.434474\pi$
$992$	0	0
$993$	−34.6261	−1.09883
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.6261	0.843258	0.421629	−	0.906768i	$-0.361458\pi$
$997$	26.6261	0.843258	0.421629	+	0.906768i	$0.361458\pi$
$998$	0	0
$999$	32.9129	1.04132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bu.1.1		2
4.3	odd	2		1100.2.a.g.1.2	✓	2
5.2	odd	4		4400.2.b.s.4049.3		4
5.3	odd	4		4400.2.b.s.4049.2		4
5.4	even	2		4400.2.a.bi.1.2		2
12.11	even	2		9900.2.a.bh.1.1		2
20.3	even	4		1100.2.b.d.749.3		4
20.7	even	4		1100.2.b.d.749.2		4
20.19	odd	2		1100.2.a.h.1.1	yes	2
60.23	odd	4		9900.2.c.x.5149.4		4
60.47	odd	4		9900.2.c.x.5149.1		4
60.59	even	2		9900.2.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.2.a.g.1.2	✓	2	4.3	odd	2
1100.2.a.h.1.1	yes	2	20.19	odd	2
1100.2.b.d.749.2		4	20.7	even	4
1100.2.b.d.749.3		4	20.3	even	4
4400.2.a.bi.1.2		2	5.4	even	2
4400.2.a.bu.1.1		2	1.1	even	1	trivial
4400.2.b.s.4049.2		4	5.3	odd	4
4400.2.b.s.4049.3		4	5.2	odd	4
9900.2.a.bh.1.1		2	12.11	even	2
9900.2.a.bz.1.2		2	60.59	even	2
9900.2.c.x.5149.1		4	60.47	odd	4
9900.2.c.x.5149.4		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79129 q^{3} +4.79129 q^{7} +0.208712 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.79129 q^{17} +2.58258 q^{19} -8.58258 q^{21} -0.791288 q^{23} +5.00000 q^{27} +2.20871 q^{29} -0.582576 q^{31} -1.79129 q^{33} +6.58258 q^{37} +1.79129 q^{39} -10.5826 q^{41} +10.0000 q^{43} +10.5826 q^{47} +15.9564 q^{49} +6.79129 q^{51} -2.37386 q^{53} -4.62614 q^{57} +1.41742 q^{59} +8.79129 q^{61} +1.00000 q^{63} +4.00000 q^{67} +1.41742 q^{69} -16.7477 q^{71} -3.20871 q^{73} +4.79129 q^{77} -16.5390 q^{79} -9.58258 q^{81} -12.9564 q^{83} -3.95644 q^{87} +3.79129 q^{89} -4.79129 q^{91} +1.04356 q^{93} -10.7913 q^{97} +0.208712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 5 q^{7} + 5 q^{9} + 2 q^{11} - 2 q^{13} - 3 q^{17} - 4 q^{19} - 8 q^{21} + 3 q^{23} + 10 q^{27} + 9 q^{29} + 8 q^{31} + q^{33} + 4 q^{37} - q^{39} - 12 q^{41} + 20 q^{43} + 12 q^{47} + 9 q^{49}+ \cdots + 5 q^{99}+O(q^{100}) \) 2 * q + q^3 + 5 * q^7 + 5 * q^9 + 2 * q^11 - 2 * q^13 - 3 * q^17 - 4 * q^19 - 8 * q^21 + 3 * q^23 + 10 * q^27 + 9 * q^29 + 8 * q^31 + q^33 + 4 * q^37 - q^39 - 12 * q^41 + 20 * q^43 + 12 * q^47 + 9 * q^49 + 9 * q^51 + 9 * q^53 - 23 * q^57 + 12 * q^59 + 13 * q^61 + 2 * q^63 + 8 * q^67 + 12 * q^69 - 6 * q^71 - 11 * q^73 + 5 * q^77 - q^79 - 10 * q^81 - 3 * q^83 + 15 * q^87 + 3 * q^89 - 5 * q^91 + 25 * q^93 - 17 * q^97 + 5 * q^99

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