LMFDB - Embedded newform 2128.2.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2128,2,Mod(1,2128)] 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("2128.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2128 = 2^{4} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2128.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,3,0,-6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.9921655501$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 133)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	2128.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.30278 q^{3} -3.00000 q^{5} -1.00000 q^{7} +7.90833 q^{9} +4.30278 q^{11} +1.60555 q^{13} -9.90833 q^{15} -1.69722 q^{17} -1.00000 q^{19} -3.30278 q^{21} +3.00000 q^{23} +4.00000 q^{25} +16.2111 q^{27} -0.908327 q^{29} +2.30278 q^{31} +14.2111 q^{33} +3.00000 q^{35} -3.60555 q^{37} +5.30278 q^{39} +4.30278 q^{41} +10.0000 q^{43} -23.7250 q^{45} +8.21110 q^{47} +1.00000 q^{49} -5.60555 q^{51} +3.90833 q^{53} -12.9083 q^{55} -3.30278 q^{57} -8.21110 q^{59} +10.2111 q^{61} -7.90833 q^{63} -4.81665 q^{65} -8.90833 q^{67} +9.90833 q^{69} +2.21110 q^{71} -16.5139 q^{73} +13.2111 q^{75} -4.30278 q^{77} +3.21110 q^{79} +29.8167 q^{81} +2.09167 q^{83} +5.09167 q^{85} -3.00000 q^{87} -3.39445 q^{89} -1.60555 q^{91} +7.60555 q^{93} +3.00000 q^{95} +2.39445 q^{97} +34.0278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 6 q^{5} - 2 q^{7} + 5 q^{9} + 5 q^{11} - 4 q^{13} - 9 q^{15} - 7 q^{17} - 2 q^{19} - 3 q^{21} + 6 q^{23} + 8 q^{25} + 18 q^{27} + 9 q^{29} + q^{31} + 14 q^{33} + 6 q^{35} + 7 q^{39} + 5 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - 6 * q^5 - 2 * q^7 + 5 * q^9 + 5 * q^11 - 4 * q^13 - 9 * q^15 - 7 * q^17 - 2 * q^19 - 3 * q^21 + 6 * q^23 + 8 * q^25 + 18 * q^27 + 9 * q^29 + q^31 + 14 * q^33 + 6 * q^35 + 7 * q^39 + 5 * q^41 + 20 * q^43 - 15 * q^45 + 2 * q^47 + 2 * q^49 - 4 * q^51 - 3 * q^53 - 15 * q^55 - 3 * q^57 - 2 * q^59 + 6 * q^61 - 5 * q^63 + 12 * q^65 - 7 * q^67 + 9 * q^69 - 10 * q^71 - 15 * q^73 + 12 * q^75 - 5 * q^77 - 8 * q^79 + 38 * q^81 + 15 * q^83 + 21 * q^85 - 6 * q^87 - 14 * q^89 + 4 * q^91 + 8 * q^93 + 6 * q^95 + 12 * q^97 + 32 * 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
$3$	3.30278	1.90686	0.953429	−	0.301617i	$-0.0975264\pi$
$3$	3.30278	1.90686	0.953429	+	0.301617i	$0.0975264\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	4.30278	1.29734	0.648668	−	0.761072i	$-0.275326\pi$
$11$	4.30278	1.29734	0.648668	+	0.761072i	$0.275326\pi$
$12$	0	0
$13$	1.60555	0.445300	0.222650	−	0.974898i	$-0.428529\pi$
$13$	1.60555	0.445300	0.222650	+	0.974898i	$0.428529\pi$
$14$	0	0
$15$	−9.90833	−2.55832
$16$	0	0
$17$	−1.69722	−0.411637	−0.205819	−	0.978590i	$-0.565986\pi$
$17$	−1.69722	−0.411637	−0.205819	+	0.978590i	$0.565986\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.30278	−0.720725
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	16.2111	3.11983
$28$	0	0
$29$	−0.908327	−0.168672	−0.0843360	−	0.996437i	$-0.526877\pi$
$29$	−0.908327	−0.168672	−0.0843360	+	0.996437i	$0.526877\pi$
$30$	0	0
$31$	2.30278	0.413591	0.206795	−	0.978384i	$-0.433697\pi$
$31$	2.30278	0.413591	0.206795	+	0.978384i	$0.433697\pi$
$32$	0	0
$33$	14.2111	2.47384
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−3.60555	−0.592749	−0.296374	−	0.955072i	$-0.595778\pi$
$37$	−3.60555	−0.592749	−0.296374	+	0.955072i	$0.595778\pi$
$38$	0	0
$39$	5.30278	0.849124
$40$	0	0
$41$	4.30278	0.671981	0.335990	−	0.941865i	$-0.390929\pi$
$41$	4.30278	0.671981	0.335990	+	0.941865i	$0.390929\pi$
$42$	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
$45$	−23.7250	−3.53671
$46$	0	0
$47$	8.21110	1.19771	0.598856	−	0.800857i	$-0.295622\pi$
$47$	8.21110	1.19771	0.598856	+	0.800857i	$0.295622\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.60555	−0.784934
$52$	0	0
$53$	3.90833	0.536850	0.268425	−	0.963301i	$-0.413497\pi$
$53$	3.90833	0.536850	0.268425	+	0.963301i	$0.413497\pi$
$54$	0	0
$55$	−12.9083	−1.74056
$56$	0	0
$57$	−3.30278	−0.437463
$58$	0	0
$59$	−8.21110	−1.06899	−0.534497	−	0.845170i	$-0.679499\pi$
$59$	−8.21110	−1.06899	−0.534497	+	0.845170i	$0.679499\pi$
$60$	0	0
$61$	10.2111	1.30740	0.653699	−	0.756755i	$-0.273216\pi$
$61$	10.2111	1.30740	0.653699	+	0.756755i	$0.273216\pi$
$62$	0	0
$63$	−7.90833	−0.996356
$64$	0	0
$65$	−4.81665	−0.597432
$66$	0	0
$67$	−8.90833	−1.08833	−0.544163	−	0.838980i	$-0.683153\pi$
$67$	−8.90833	−1.08833	−0.544163	+	0.838980i	$0.683153\pi$
$68$	0	0
$69$	9.90833	1.19282
$70$	0	0
$71$	2.21110	0.262410	0.131205	−	0.991355i	$-0.458115\pi$
$71$	2.21110	0.262410	0.131205	+	0.991355i	$0.458115\pi$
$72$	0	0
$73$	−16.5139	−1.93280	−0.966402	−	0.257037i	$-0.917254\pi$
$73$	−16.5139	−1.93280	−0.966402	+	0.257037i	$0.917254\pi$
$74$	0	0
$75$	13.2111	1.52549
$76$	0	0
$77$	−4.30278	−0.490347
$78$	0	0
$79$	3.21110	0.361277	0.180639	−	0.983550i	$-0.442184\pi$
$79$	3.21110	0.361277	0.180639	+	0.983550i	$0.442184\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	2.09167	0.229591	0.114795	−	0.993389i	$-0.463379\pi$
$83$	2.09167	0.229591	0.114795	+	0.993389i	$0.463379\pi$
$84$	0	0
$85$	5.09167	0.552269
$86$	0	0
$87$	−3.00000	−0.321634
$88$	0	0
$89$	−3.39445	−0.359811	−0.179905	−	0.983684i	$-0.557579\pi$
$89$	−3.39445	−0.359811	−0.179905	+	0.983684i	$0.557579\pi$
$90$	0	0
$91$	−1.60555	−0.168308
$92$	0	0
$93$	7.60555	0.788659
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	2.39445	0.243119	0.121560	−	0.992584i	$-0.461210\pi$
$97$	2.39445	0.243119	0.121560	+	0.992584i	$0.461210\pi$
$98$	0	0
$99$	34.0278	3.41992
$100$	0	0
$101$	−11.6056	−1.15480	−0.577398	−	0.816463i	$-0.695932\pi$
$101$	−11.6056	−1.15480	−0.577398	+	0.816463i	$0.695932\pi$
$102$	0	0
$103$	1.78890	0.176265	0.0881327	−	0.996109i	$-0.471910\pi$
$103$	1.78890	0.176265	0.0881327	+	0.996109i	$0.471910\pi$
$104$	0	0
$105$	9.90833	0.966954
$106$	0	0
$107$	20.2111	1.95388	0.976941	−	0.213512i	$-0.0684901\pi$
$107$	20.2111	1.95388	0.976941	+	0.213512i	$0.0684901\pi$
$108$	0	0
$109$	4.21110	0.403350	0.201675	−	0.979452i	$-0.435361\pi$
$109$	4.21110	0.403350	0.201675	+	0.979452i	$0.435361\pi$
$110$	0	0
$111$	−11.9083	−1.13029
$112$	0	0
$113$	13.3028	1.25142	0.625710	−	0.780056i	$-0.284809\pi$
$113$	13.3028	1.25142	0.625710	+	0.780056i	$0.284809\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	0	0
$117$	12.6972	1.17386
$118$	0	0
$119$	1.69722	0.155584
$120$	0	0
$121$	7.51388	0.683080
$122$	0	0
$123$	14.2111	1.28137
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	−9.81665	−0.871087	−0.435544	−	0.900168i	$-0.643444\pi$
$127$	−9.81665	−0.871087	−0.435544	+	0.900168i	$0.643444\pi$
$128$	0	0
$129$	33.0278	2.90793
$130$	0	0
$131$	2.48612	0.217213	0.108607	−	0.994085i	$-0.465361\pi$
$131$	2.48612	0.217213	0.108607	+	0.994085i	$0.465361\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−48.6333	−4.18569
$136$	0	0
$137$	−21.6333	−1.84826	−0.924129	−	0.382080i	$-0.875208\pi$
$137$	−21.6333	−1.84826	−0.924129	+	0.382080i	$0.875208\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	27.1194	2.28387
$142$	0	0
$143$	6.90833	0.577703
$144$	0	0
$145$	2.72498	0.226297
$146$	0	0
$147$	3.30278	0.272408
$148$	0	0
$149$	8.21110	0.672680	0.336340	−	0.941741i	$-0.390811\pi$
$149$	8.21110	0.672680	0.336340	+	0.941741i	$0.390811\pi$
$150$	0	0
$151$	4.90833	0.399434	0.199717	−	0.979854i	$-0.435998\pi$
$151$	4.90833	0.399434	0.199717	+	0.979854i	$0.435998\pi$
$152$	0	0
$153$	−13.4222	−1.08512
$154$	0	0
$155$	−6.90833	−0.554890
$156$	0	0
$157$	−13.5139	−1.07852	−0.539262	−	0.842138i	$-0.681297\pi$
$157$	−13.5139	−1.07852	−0.539262	+	0.842138i	$0.681297\pi$
$158$	0	0
$159$	12.9083	1.02370
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−9.30278	−0.728650	−0.364325	−	0.931272i	$-0.618700\pi$
$163$	−9.30278	−0.728650	−0.364325	+	0.931272i	$0.618700\pi$
$164$	0	0
$165$	−42.6333	−3.31900
$166$	0	0
$167$	−11.6056	−0.898065	−0.449032	−	0.893516i	$-0.648231\pi$
$167$	−11.6056	−0.898065	−0.449032	+	0.893516i	$0.648231\pi$
$168$	0	0
$169$	−10.4222	−0.801708
$170$	0	0
$171$	−7.90833	−0.604765
$172$	0	0
$173$	−13.8167	−1.05046	−0.525230	−	0.850960i	$-0.676021\pi$
$173$	−13.8167	−1.05046	−0.525230	+	0.850960i	$0.676021\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−27.1194	−2.03842
$178$	0	0
$179$	−20.7250	−1.54906	−0.774529	−	0.632539i	$-0.782013\pi$
$179$	−20.7250	−1.54906	−0.774529	+	0.632539i	$0.782013\pi$
$180$	0	0
$181$	−14.3028	−1.06312	−0.531558	−	0.847022i	$-0.678393\pi$
$181$	−14.3028	−1.06312	−0.531558	+	0.847022i	$0.678393\pi$
$182$	0	0
$183$	33.7250	2.49302
$184$	0	0
$185$	10.8167	0.795256
$186$	0	0
$187$	−7.30278	−0.534032
$188$	0	0
$189$	−16.2111	−1.17918
$190$	0	0
$191$	14.3305	1.03692	0.518460	−	0.855102i	$-0.326505\pi$
$191$	14.3305	1.03692	0.518460	+	0.855102i	$0.326505\pi$
$192$	0	0
$193$	2.11943	0.152560	0.0762799	−	0.997086i	$-0.475696\pi$
$193$	2.11943	0.152560	0.0762799	+	0.997086i	$0.475696\pi$
$194$	0	0
$195$	−15.9083	−1.13922
$196$	0	0
$197$	3.90833	0.278457	0.139228	−	0.990260i	$-0.455538\pi$
$197$	3.90833	0.278457	0.139228	+	0.990260i	$0.455538\pi$
$198$	0	0
$199$	−15.4222	−1.09325	−0.546626	−	0.837377i	$-0.684088\pi$
$199$	−15.4222	−1.09325	−0.546626	+	0.837377i	$0.684088\pi$
$200$	0	0
$201$	−29.4222	−2.07528
$202$	0	0
$203$	0.908327	0.0637521
$204$	0	0
$205$	−12.9083	−0.901557
$206$	0	0
$207$	23.7250	1.64900
$208$	0	0
$209$	−4.30278	−0.297629
$210$	0	0
$211$	11.3028	0.778115	0.389058	−	0.921213i	$-0.372801\pi$
$211$	11.3028	0.778115	0.389058	+	0.921213i	$0.372801\pi$
$212$	0	0
$213$	7.30278	0.500378
$214$	0	0
$215$	−30.0000	−2.04598
$216$	0	0
$217$	−2.30278	−0.156323
$218$	0	0
$219$	−54.5416	−3.68558
$220$	0	0
$221$	−2.72498	−0.183302
$222$	0	0
$223$	8.81665	0.590407	0.295203	−	0.955434i	$-0.404613\pi$
$223$	8.81665	0.590407	0.295203	+	0.955434i	$0.404613\pi$
$224$	0	0
$225$	31.6333	2.10889
$226$	0	0
$227$	−0.119429	−0.00792681	−0.00396341	−	0.999992i	$-0.501262\pi$
$227$	−0.119429	−0.00792681	−0.00396341	+	0.999992i	$0.501262\pi$
$228$	0	0
$229$	−15.2111	−1.00518	−0.502589	−	0.864525i	$-0.667619\pi$
$229$	−15.2111	−1.00518	−0.502589	+	0.864525i	$0.667619\pi$
$230$	0	0
$231$	−14.2111	−0.935022
$232$	0	0
$233$	−12.9083	−0.845653	−0.422826	−	0.906211i	$-0.638962\pi$
$233$	−12.9083	−0.845653	−0.422826	+	0.906211i	$0.638962\pi$
$234$	0	0
$235$	−24.6333	−1.60690
$236$	0	0
$237$	10.6056	0.688905
$238$	0	0
$239$	−1.42221	−0.0919948	−0.0459974	−	0.998942i	$-0.514647\pi$
$239$	−1.42221	−0.0919948	−0.0459974	+	0.998942i	$0.514647\pi$
$240$	0	0
$241$	−13.3944	−0.862812	−0.431406	−	0.902158i	$-0.641982\pi$
$241$	−13.3944	−0.862812	−0.431406	+	0.902158i	$0.641982\pi$
$242$	0	0
$243$	49.8444	3.19752
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−1.60555	−0.102159
$248$	0	0
$249$	6.90833	0.437797
$250$	0	0
$251$	5.09167	0.321384	0.160692	−	0.987005i	$-0.448627\pi$
$251$	5.09167	0.321384	0.160692	+	0.987005i	$0.448627\pi$
$252$	0	0
$253$	12.9083	0.811540
$254$	0	0
$255$	16.8167	1.05310
$256$	0	0
$257$	9.90833	0.618064	0.309032	−	0.951052i	$-0.399995\pi$
$257$	9.90833	0.618064	0.309032	+	0.951052i	$0.399995\pi$
$258$	0	0
$259$	3.60555	0.224038
$260$	0	0
$261$	−7.18335	−0.444638
$262$	0	0
$263$	13.3028	0.820284	0.410142	−	0.912022i	$-0.365479\pi$
$263$	13.3028	0.820284	0.410142	+	0.912022i	$0.365479\pi$
$264$	0	0
$265$	−11.7250	−0.720260
$266$	0	0
$267$	−11.2111	−0.686108
$268$	0	0
$269$	−11.7250	−0.714885	−0.357442	−	0.933935i	$-0.616351\pi$
$269$	−11.7250	−0.714885	−0.357442	+	0.933935i	$0.616351\pi$
$270$	0	0
$271$	−7.09167	−0.430788	−0.215394	−	0.976527i	$-0.569104\pi$
$271$	−7.09167	−0.430788	−0.215394	+	0.976527i	$0.569104\pi$
$272$	0	0
$273$	−5.30278	−0.320939
$274$	0	0
$275$	17.2111	1.03787
$276$	0	0
$277$	−6.60555	−0.396889	−0.198445	−	0.980112i	$-0.563589\pi$
$277$	−6.60555	−0.396889	−0.198445	+	0.980112i	$0.563589\pi$
$278$	0	0
$279$	18.2111	1.09027
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	0	0
$283$	12.3305	0.732974	0.366487	−	0.930423i	$-0.380560\pi$
$283$	12.3305	0.732974	0.366487	+	0.930423i	$0.380560\pi$
$284$	0	0
$285$	9.90833	0.586919
$286$	0	0
$287$	−4.30278	−0.253985
$288$	0	0
$289$	−14.1194	−0.830555
$290$	0	0
$291$	7.90833	0.463594
$292$	0	0
$293$	30.6333	1.78962	0.894808	−	0.446450i	$-0.147312\pi$
$293$	30.6333	1.78962	0.894808	+	0.446450i	$0.147312\pi$
$294$	0	0
$295$	24.6333	1.43421
$296$	0	0
$297$	69.7527	4.04746
$298$	0	0
$299$	4.81665	0.278554
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	−38.3305	−2.20203
$304$	0	0
$305$	−30.6333	−1.75406
$306$	0	0
$307$	−18.3028	−1.04459	−0.522297	−	0.852763i	$-0.674925\pi$
$307$	−18.3028	−1.04459	−0.522297	+	0.852763i	$0.674925\pi$
$308$	0	0
$309$	5.90833	0.336113
$310$	0	0
$311$	−15.5139	−0.879711	−0.439856	−	0.898068i	$-0.644970\pi$
$311$	−15.5139	−0.879711	−0.439856	+	0.898068i	$0.644970\pi$
$312$	0	0
$313$	9.02776	0.510279	0.255139	−	0.966904i	$-0.417879\pi$
$313$	9.02776	0.510279	0.255139	+	0.966904i	$0.417879\pi$
$314$	0	0
$315$	23.7250	1.33675
$316$	0	0
$317$	23.2111	1.30367	0.651833	−	0.758363i	$-0.274000\pi$
$317$	23.2111	1.30367	0.651833	+	0.758363i	$0.274000\pi$
$318$	0	0
$319$	−3.90833	−0.218824
$320$	0	0
$321$	66.7527	3.72577
$322$	0	0
$323$	1.69722	0.0944361
$324$	0	0
$325$	6.42221	0.356240
$326$	0	0
$327$	13.9083	0.769132
$328$	0	0
$329$	−8.21110	−0.452693
$330$	0	0
$331$	−15.3028	−0.841117	−0.420558	−	0.907266i	$-0.638166\pi$
$331$	−15.3028	−0.841117	−0.420558	+	0.907266i	$0.638166\pi$
$332$	0	0
$333$	−28.5139	−1.56255
$334$	0	0
$335$	26.7250	1.46014
$336$	0	0
$337$	−24.7250	−1.34686	−0.673428	−	0.739253i	$-0.735179\pi$
$337$	−24.7250	−1.34686	−0.673428	+	0.739253i	$0.735179\pi$
$338$	0	0
$339$	43.9361	2.38628
$340$	0	0
$341$	9.90833	0.536566
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−29.7250	−1.60034
$346$	0	0
$347$	−25.5416	−1.37115	−0.685573	−	0.728004i	$-0.740448\pi$
$347$	−25.5416	−1.37115	−0.685573	+	0.728004i	$0.740448\pi$
$348$	0	0
$349$	−10.5139	−0.562795	−0.281397	−	0.959591i	$-0.590798\pi$
$349$	−10.5139	−0.562795	−0.281397	+	0.959591i	$0.590798\pi$
$350$	0	0
$351$	26.0278	1.38926
$352$	0	0
$353$	−0.908327	−0.0483454	−0.0241727	−	0.999708i	$-0.507695\pi$
$353$	−0.908327	−0.0483454	−0.0241727	+	0.999708i	$0.507695\pi$
$354$	0	0
$355$	−6.63331	−0.352059
$356$	0	0
$357$	5.60555	0.296677
$358$	0	0
$359$	−28.5416	−1.50637	−0.753185	−	0.657809i	$-0.771483\pi$
$359$	−28.5416	−1.50637	−0.753185	+	0.657809i	$0.771483\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	24.8167	1.30254
$364$	0	0
$365$	49.5416	2.59313
$366$	0	0
$367$	−36.0278	−1.88063	−0.940317	−	0.340300i	$-0.889471\pi$
$367$	−36.0278	−1.88063	−0.940317	+	0.340300i	$0.889471\pi$
$368$	0	0
$369$	34.0278	1.77141
$370$	0	0
$371$	−3.90833	−0.202910
$372$	0	0
$373$	−1.11943	−0.0579619	−0.0289809	−	0.999580i	$-0.509226\pi$
$373$	−1.11943	−0.0579619	−0.0289809	+	0.999580i	$0.509226\pi$
$374$	0	0
$375$	9.90833	0.511664
$376$	0	0
$377$	−1.45837	−0.0751096
$378$	0	0
$379$	14.8167	0.761080	0.380540	−	0.924764i	$-0.375738\pi$
$379$	14.8167	0.761080	0.380540	+	0.924764i	$0.375738\pi$
$380$	0	0
$381$	−32.4222	−1.66104
$382$	0	0
$383$	36.6333	1.87187	0.935937	−	0.352167i	$-0.114555\pi$
$383$	36.6333	1.87187	0.935937	+	0.352167i	$0.114555\pi$
$384$	0	0
$385$	12.9083	0.657869
$386$	0	0
$387$	79.0833	4.02003
$388$	0	0
$389$	37.1472	1.88344	0.941719	−	0.336402i	$-0.109210\pi$
$389$	37.1472	1.88344	0.941719	+	0.336402i	$0.109210\pi$
$390$	0	0
$391$	−5.09167	−0.257497
$392$	0	0
$393$	8.21110	0.414195
$394$	0	0
$395$	−9.63331	−0.484704
$396$	0	0
$397$	0.972244	0.0487955	0.0243978	−	0.999702i	$-0.492233\pi$
$397$	0.972244	0.0487955	0.0243978	+	0.999702i	$0.492233\pi$
$398$	0	0
$399$	3.30278	0.165346
$400$	0	0
$401$	−21.5139	−1.07435	−0.537176	−	0.843470i	$-0.680509\pi$
$401$	−21.5139	−1.07435	−0.537176	+	0.843470i	$0.680509\pi$
$402$	0	0
$403$	3.69722	0.184172
$404$	0	0
$405$	−89.4500	−4.44480
$406$	0	0
$407$	−15.5139	−0.768994
$408$	0	0
$409$	10.0917	0.499001	0.249501	−	0.968375i	$-0.419734\pi$
$409$	10.0917	0.499001	0.249501	+	0.968375i	$0.419734\pi$
$410$	0	0
$411$	−71.4500	−3.52437
$412$	0	0
$413$	8.21110	0.404042
$414$	0	0
$415$	−6.27502	−0.308029
$416$	0	0
$417$	−16.5139	−0.808688
$418$	0	0
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−22.3944	−1.09144	−0.545719	−	0.837968i	$-0.683744\pi$
$421$	−22.3944	−1.09144	−0.545719	+	0.837968i	$0.683744\pi$
$422$	0	0
$423$	64.9361	3.15730
$424$	0	0
$425$	−6.78890	−0.329310
$426$	0	0
$427$	−10.2111	−0.494150
$428$	0	0
$429$	22.8167	1.10160
$430$	0	0
$431$	−11.2111	−0.540020	−0.270010	−	0.962858i	$-0.587027\pi$
$431$	−11.2111	−0.540020	−0.270010	+	0.962858i	$0.587027\pi$
$432$	0	0
$433$	−29.4222	−1.41394	−0.706970	−	0.707243i	$-0.749939\pi$
$433$	−29.4222	−1.41394	−0.706970	+	0.707243i	$0.749939\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	−3.00000	−0.143509
$438$	0	0
$439$	−37.2111	−1.77599	−0.887995	−	0.459854i	$-0.847902\pi$
$439$	−37.2111	−1.77599	−0.887995	+	0.459854i	$0.847902\pi$
$440$	0	0
$441$	7.90833	0.376587
$442$	0	0
$443$	−9.11943	−0.433277	−0.216639	−	0.976252i	$-0.569509\pi$
$443$	−9.11943	−0.433277	−0.216639	+	0.976252i	$0.569509\pi$
$444$	0	0
$445$	10.1833	0.482737
$446$	0	0
$447$	27.1194	1.28270
$448$	0	0
$449$	−18.5139	−0.873724	−0.436862	−	0.899529i	$-0.643910\pi$
$449$	−18.5139	−0.873724	−0.436862	+	0.899529i	$0.643910\pi$
$450$	0	0
$451$	18.5139	0.871784
$452$	0	0
$453$	16.2111	0.761664
$454$	0	0
$455$	4.81665	0.225808
$456$	0	0
$457$	18.6972	0.874619	0.437310	−	0.899311i	$-0.355931\pi$
$457$	18.6972	0.874619	0.437310	+	0.899311i	$0.355931\pi$
$458$	0	0
$459$	−27.5139	−1.28424
$460$	0	0
$461$	−9.90833	−0.461477	−0.230738	−	0.973016i	$-0.574114\pi$
$461$	−9.90833	−0.461477	−0.230738	+	0.973016i	$0.574114\pi$
$462$	0	0
$463$	−28.4500	−1.32218	−0.661091	−	0.750306i	$-0.729906\pi$
$463$	−28.4500	−1.32218	−0.661091	+	0.750306i	$0.729906\pi$
$464$	0	0
$465$	−22.8167	−1.05810
$466$	0	0
$467$	37.5416	1.73722	0.868610	−	0.495497i	$-0.165014\pi$
$467$	37.5416	1.73722	0.868610	+	0.495497i	$0.165014\pi$
$468$	0	0
$469$	8.90833	0.411348
$470$	0	0
$471$	−44.6333	−2.05659
$472$	0	0
$473$	43.0278	1.97842
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	30.9083	1.41520
$478$	0	0
$479$	4.30278	0.196599	0.0982994	−	0.995157i	$-0.468660\pi$
$479$	4.30278	0.196599	0.0982994	+	0.995157i	$0.468660\pi$
$480$	0	0
$481$	−5.78890	−0.263951
$482$	0	0
$483$	−9.90833	−0.450844
$484$	0	0
$485$	−7.18335	−0.326179
$486$	0	0
$487$	−33.4222	−1.51450	−0.757252	−	0.653122i	$-0.773459\pi$
$487$	−33.4222	−1.51450	−0.757252	+	0.653122i	$0.773459\pi$
$488$	0	0
$489$	−30.7250	−1.38943
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	1.54163	0.0694317
$494$	0	0
$495$	−102.083	−4.58830
$496$	0	0
$497$	−2.21110	−0.0991815
$498$	0	0
$499$	26.9361	1.20582	0.602912	−	0.797807i	$-0.294007\pi$
$499$	26.9361	1.20582	0.602912	+	0.797807i	$0.294007\pi$
$500$	0	0
$501$	−38.3305	−1.71248
$502$	0	0
$503$	22.4222	0.999757	0.499878	−	0.866096i	$-0.333378\pi$
$503$	22.4222	0.999757	0.499878	+	0.866096i	$0.333378\pi$
$504$	0	0
$505$	34.8167	1.54932
$506$	0	0
$507$	−34.4222	−1.52874
$508$	0	0
$509$	−9.63331	−0.426989	−0.213494	−	0.976944i	$-0.568485\pi$
$509$	−9.63331	−0.426989	−0.213494	+	0.976944i	$0.568485\pi$
$510$	0	0
$511$	16.5139	0.730531
$512$	0	0
$513$	−16.2111	−0.715738
$514$	0	0
$515$	−5.36669	−0.236485
$516$	0	0
$517$	35.3305	1.55384
$518$	0	0
$519$	−45.6333	−2.00308
$520$	0	0
$521$	−6.63331	−0.290610	−0.145305	−	0.989387i	$-0.546416\pi$
$521$	−6.63331	−0.290610	−0.145305	+	0.989387i	$0.546416\pi$
$522$	0	0
$523$	41.6611	1.82171	0.910856	−	0.412725i	$-0.135423\pi$
$523$	41.6611	1.82171	0.910856	+	0.412725i	$0.135423\pi$
$524$	0	0
$525$	−13.2111	−0.576580
$526$	0	0
$527$	−3.90833	−0.170249
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−64.9361	−2.81799
$532$	0	0
$533$	6.90833	0.299233
$534$	0	0
$535$	−60.6333	−2.62141
$536$	0	0
$537$	−68.4500	−2.95383
$538$	0	0
$539$	4.30278	0.185334
$540$	0	0
$541$	31.2111	1.34187	0.670935	−	0.741516i	$-0.265893\pi$
$541$	31.2111	1.34187	0.670935	+	0.741516i	$0.265893\pi$
$542$	0	0
$543$	−47.2389	−2.02721
$544$	0	0
$545$	−12.6333	−0.541151
$546$	0	0
$547$	28.5139	1.21917	0.609583	−	0.792722i	$-0.291337\pi$
$547$	28.5139	1.21917	0.609583	+	0.792722i	$0.291337\pi$
$548$	0	0
$549$	80.7527	3.44644
$550$	0	0
$551$	0.908327	0.0386960
$552$	0	0
$553$	−3.21110	−0.136550
$554$	0	0
$555$	35.7250	1.51644
$556$	0	0
$557$	24.1194	1.02197	0.510987	−	0.859589i	$-0.329280\pi$
$557$	24.1194	1.02197	0.510987	+	0.859589i	$0.329280\pi$
$558$	0	0
$559$	16.0555	0.679076
$560$	0	0
$561$	−24.1194	−1.01832
$562$	0	0
$563$	9.78890	0.412553	0.206276	−	0.978494i	$-0.433865\pi$
$563$	9.78890	0.412553	0.206276	+	0.978494i	$0.433865\pi$
$564$	0	0
$565$	−39.9083	−1.67896
$566$	0	0
$567$	−29.8167	−1.25218
$568$	0	0
$569$	28.8167	1.20806	0.604028	−	0.796963i	$-0.293561\pi$
$569$	28.8167	1.20806	0.604028	+	0.796963i	$0.293561\pi$
$570$	0	0
$571$	9.60555	0.401980	0.200990	−	0.979593i	$-0.435584\pi$
$571$	9.60555	0.401980	0.200990	+	0.979593i	$0.435584\pi$
$572$	0	0
$573$	47.3305	1.97726
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	1.48612	0.0618681	0.0309340	−	0.999521i	$-0.490152\pi$
$577$	1.48612	0.0618681	0.0309340	+	0.999521i	$0.490152\pi$
$578$	0	0
$579$	7.00000	0.290910
$580$	0	0
$581$	−2.09167	−0.0867772
$582$	0	0
$583$	16.8167	0.696475
$584$	0	0
$585$	−38.0917	−1.57490
$586$	0	0
$587$	28.4222	1.17311	0.586555	−	0.809909i	$-0.300484\pi$
$587$	28.4222	1.17311	0.586555	+	0.809909i	$0.300484\pi$
$588$	0	0
$589$	−2.30278	−0.0948842
$590$	0	0
$591$	12.9083	0.530978
$592$	0	0
$593$	16.8167	0.690577	0.345289	−	0.938497i	$-0.387781\pi$
$593$	16.8167	0.690577	0.345289	+	0.938497i	$0.387781\pi$
$594$	0	0
$595$	−5.09167	−0.208738
$596$	0	0
$597$	−50.9361	−2.08468
$598$	0	0
$599$	−9.90833	−0.404843	−0.202422	−	0.979298i	$-0.564881\pi$
$599$	−9.90833	−0.404843	−0.202422	+	0.979298i	$0.564881\pi$
$600$	0	0
$601$	−9.48612	−0.386947	−0.193473	−	0.981106i	$-0.561975\pi$
$601$	−9.48612	−0.386947	−0.193473	+	0.981106i	$0.561975\pi$
$602$	0	0
$603$	−70.4500	−2.86894
$604$	0	0
$605$	−22.5416	−0.916448
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	13.1833	0.533341
$612$	0	0
$613$	31.7250	1.28136	0.640680	−	0.767808i	$-0.278653\pi$
$613$	31.7250	1.28136	0.640680	+	0.767808i	$0.278653\pi$
$614$	0	0
$615$	−42.6333	−1.71914
$616$	0	0
$617$	−36.1194	−1.45411	−0.727057	−	0.686577i	$-0.759112\pi$
$617$	−36.1194	−1.45411	−0.727057	+	0.686577i	$0.759112\pi$
$618$	0	0
$619$	1.66947	0.0671016	0.0335508	−	0.999437i	$-0.489318\pi$
$619$	1.66947	0.0671016	0.0335508	+	0.999437i	$0.489318\pi$
$620$	0	0
$621$	48.6333	1.95159
$622$	0	0
$623$	3.39445	0.135996
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−14.2111	−0.567537
$628$	0	0
$629$	6.11943	0.243998
$630$	0	0
$631$	37.2389	1.48246	0.741228	−	0.671254i	$-0.234244\pi$
$631$	37.2389	1.48246	0.741228	+	0.671254i	$0.234244\pi$
$632$	0	0
$633$	37.3305	1.48376
$634$	0	0
$635$	29.4500	1.16869
$636$	0	0
$637$	1.60555	0.0636143
$638$	0	0
$639$	17.4861	0.691740
$640$	0	0
$641$	−0.908327	−0.0358768	−0.0179384	−	0.999839i	$-0.505710\pi$
$641$	−0.908327	−0.0358768	−0.0179384	+	0.999839i	$0.505710\pi$
$642$	0	0
$643$	−37.6056	−1.48302	−0.741509	−	0.670943i	$-0.765890\pi$
$643$	−37.6056	−1.48302	−0.741509	+	0.670943i	$0.765890\pi$
$644$	0	0
$645$	−99.0833	−3.90140
$646$	0	0
$647$	13.4222	0.527681	0.263841	−	0.964566i	$-0.415011\pi$
$647$	13.4222	0.527681	0.263841	+	0.964566i	$0.415011\pi$
$648$	0	0
$649$	−35.3305	−1.38684
$650$	0	0
$651$	−7.60555	−0.298085
$652$	0	0
$653$	4.81665	0.188490	0.0942451	−	0.995549i	$-0.469956\pi$
$653$	4.81665	0.188490	0.0942451	+	0.995549i	$0.469956\pi$
$654$	0	0
$655$	−7.45837	−0.291422
$656$	0	0
$657$	−130.597	−5.09508
$658$	0	0
$659$	23.3305	0.908828	0.454414	−	0.890790i	$-0.349849\pi$
$659$	23.3305	0.908828	0.454414	+	0.890790i	$0.349849\pi$
$660$	0	0
$661$	−15.2111	−0.591643	−0.295822	−	0.955243i	$-0.595593\pi$
$661$	−15.2111	−0.591643	−0.295822	+	0.955243i	$0.595593\pi$
$662$	0	0
$663$	−9.00000	−0.349531
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	−2.72498	−0.105512
$668$	0	0
$669$	29.1194	1.12582
$670$	0	0
$671$	43.9361	1.69613
$672$	0	0
$673$	4.09167	0.157722	0.0788612	−	0.996886i	$-0.474872\pi$
$673$	4.09167	0.157722	0.0788612	+	0.996886i	$0.474872\pi$
$674$	0	0
$675$	64.8444	2.49586
$676$	0	0
$677$	−28.9361	−1.11210	−0.556052	−	0.831147i	$-0.687684\pi$
$677$	−28.9361	−1.11210	−0.556052	+	0.831147i	$0.687684\pi$
$678$	0	0
$679$	−2.39445	−0.0918905
$680$	0	0
$681$	−0.394449	−0.0151153
$682$	0	0
$683$	2.21110	0.0846055	0.0423027	−	0.999105i	$-0.486531\pi$
$683$	2.21110	0.0846055	0.0423027	+	0.999105i	$0.486531\pi$
$684$	0	0
$685$	64.8999	2.47970
$686$	0	0
$687$	−50.2389	−1.91673
$688$	0	0
$689$	6.27502	0.239059
$690$	0	0
$691$	20.1833	0.767811	0.383905	−	0.923372i	$-0.374579\pi$
$691$	20.1833	0.767811	0.383905	+	0.923372i	$0.374579\pi$
$692$	0	0
$693$	−34.0278	−1.29261
$694$	0	0
$695$	15.0000	0.568982
$696$	0	0
$697$	−7.30278	−0.276612
$698$	0	0
$699$	−42.6333	−1.61254
$700$	0	0
$701$	6.78890	0.256413	0.128207	−	0.991747i	$-0.459078\pi$
$701$	6.78890	0.256413	0.128207	+	0.991747i	$0.459078\pi$
$702$	0	0
$703$	3.60555	0.135986
$704$	0	0
$705$	−81.3583	−3.06413
$706$	0	0
$707$	11.6056	0.436472
$708$	0	0
$709$	−4.39445	−0.165037	−0.0825185	−	0.996590i	$-0.526296\pi$
$709$	−4.39445	−0.165037	−0.0825185	+	0.996590i	$0.526296\pi$
$710$	0	0
$711$	25.3944	0.952366
$712$	0	0
$713$	6.90833	0.258719
$714$	0	0
$715$	−20.7250	−0.775070
$716$	0	0
$717$	−4.69722	−0.175421
$718$	0	0
$719$	−15.6333	−0.583024	−0.291512	−	0.956567i	$-0.594158\pi$
$719$	−15.6333	−0.583024	−0.291512	+	0.956567i	$0.594158\pi$
$720$	0	0
$721$	−1.78890	−0.0666220
$722$	0	0
$723$	−44.2389	−1.64526
$724$	0	0
$725$	−3.63331	−0.134938
$726$	0	0
$727$	−51.6611	−1.91600	−0.958001	−	0.286764i	$-0.907421\pi$
$727$	−51.6611	−1.91600	−0.958001	+	0.286764i	$0.907421\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	−16.9722	−0.627741
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	0	0
$735$	−9.90833	−0.365474
$736$	0	0
$737$	−38.3305	−1.41192
$738$	0	0
$739$	47.4222	1.74445	0.872227	−	0.489101i	$-0.162675\pi$
$739$	47.4222	1.74445	0.872227	+	0.489101i	$0.162675\pi$
$740$	0	0
$741$	−5.30278	−0.194802
$742$	0	0
$743$	−30.6333	−1.12383	−0.561914	−	0.827196i	$-0.689935\pi$
$743$	−30.6333	−1.12383	−0.561914	+	0.827196i	$0.689935\pi$
$744$	0	0
$745$	−24.6333	−0.902495
$746$	0	0
$747$	16.5416	0.605227
$748$	0	0
$749$	−20.2111	−0.738498
$750$	0	0
$751$	49.3583	1.80111	0.900555	−	0.434743i	$-0.143161\pi$
$751$	49.3583	1.80111	0.900555	+	0.434743i	$0.143161\pi$
$752$	0	0
$753$	16.8167	0.612833
$754$	0	0
$755$	−14.7250	−0.535897
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	42.6333	1.54749
$760$	0	0
$761$	−9.78890	−0.354847	−0.177424	−	0.984135i	$-0.556776\pi$
$761$	−9.78890	−0.354847	−0.177424	+	0.984135i	$0.556776\pi$
$762$	0	0
$763$	−4.21110	−0.152452
$764$	0	0
$765$	40.2666	1.45584
$766$	0	0
$767$	−13.1833	−0.476023
$768$	0	0
$769$	−10.7889	−0.389058	−0.194529	−	0.980897i	$-0.562318\pi$
$769$	−10.7889	−0.389058	−0.194529	+	0.980897i	$0.562318\pi$
$770$	0	0
$771$	32.7250	1.17856
$772$	0	0
$773$	−43.8167	−1.57598	−0.787988	−	0.615691i	$-0.788877\pi$
$773$	−43.8167	−1.57598	−0.787988	+	0.615691i	$0.788877\pi$
$774$	0	0
$775$	9.21110	0.330873
$776$	0	0
$777$	11.9083	0.427209
$778$	0	0
$779$	−4.30278	−0.154163
$780$	0	0
$781$	9.51388	0.340433
$782$	0	0
$783$	−14.7250	−0.526228
$784$	0	0
$785$	40.5416	1.44699
$786$	0	0
$787$	−11.6333	−0.414683	−0.207341	−	0.978269i	$-0.566481\pi$
$787$	−11.6333	−0.414683	−0.207341	+	0.978269i	$0.566481\pi$
$788$	0	0
$789$	43.9361	1.56417
$790$	0	0
$791$	−13.3028	−0.472992
$792$	0	0
$793$	16.3944	0.582184
$794$	0	0
$795$	−38.7250	−1.37343
$796$	0	0
$797$	36.9083	1.30736	0.653680	−	0.756771i	$-0.273224\pi$
$797$	36.9083	1.30736	0.653680	+	0.756771i	$0.273224\pi$
$798$	0	0
$799$	−13.9361	−0.493023
$800$	0	0
$801$	−26.8444	−0.948501
$802$	0	0
$803$	−71.0555	−2.50749
$804$	0	0
$805$	9.00000	0.317208
$806$	0	0
$807$	−38.7250	−1.36318
$808$	0	0
$809$	10.0278	0.352557	0.176279	−	0.984340i	$-0.443594\pi$
$809$	10.0278	0.352557	0.176279	+	0.984340i	$0.443594\pi$
$810$	0	0
$811$	−40.0555	−1.40654	−0.703270	−	0.710923i	$-0.748277\pi$
$811$	−40.0555	−1.40654	−0.703270	+	0.710923i	$0.748277\pi$
$812$	0	0
$813$	−23.4222	−0.821453
$814$	0	0
$815$	27.9083	0.977586
$816$	0	0
$817$	−10.0000	−0.349856
$818$	0	0
$819$	−12.6972	−0.443677
$820$	0	0
$821$	−8.36669	−0.292000	−0.146000	−	0.989285i	$-0.546640\pi$
$821$	−8.36669	−0.292000	−0.146000	+	0.989285i	$0.546640\pi$
$822$	0	0
$823$	34.2389	1.19349	0.596746	−	0.802430i	$-0.296460\pi$
$823$	34.2389	1.19349	0.596746	+	0.802430i	$0.296460\pi$
$824$	0	0
$825$	56.8444	1.97907
$826$	0	0
$827$	45.0000	1.56480	0.782402	−	0.622774i	$-0.213994\pi$
$827$	45.0000	1.56480	0.782402	+	0.622774i	$0.213994\pi$
$828$	0	0
$829$	−18.0555	−0.627094	−0.313547	−	0.949573i	$-0.601517\pi$
$829$	−18.0555	−0.627094	−0.313547	+	0.949573i	$0.601517\pi$
$830$	0	0
$831$	−21.8167	−0.756811
$832$	0	0
$833$	−1.69722	−0.0588053
$834$	0	0
$835$	34.8167	1.20488
$836$	0	0
$837$	37.3305	1.29033
$838$	0	0
$839$	17.2111	0.594193	0.297097	−	0.954847i	$-0.403982\pi$
$839$	17.2111	0.594193	0.297097	+	0.954847i	$0.403982\pi$
$840$	0	0
$841$	−28.1749	−0.971550
$842$	0	0
$843$	9.90833	0.341261
$844$	0	0
$845$	31.2666	1.07560
$846$	0	0
$847$	−7.51388	−0.258180
$848$	0	0
$849$	40.7250	1.39768
$850$	0	0
$851$	−10.8167	−0.370790
$852$	0	0
$853$	−36.3305	−1.24393	−0.621967	−	0.783044i	$-0.713666\pi$
$853$	−36.3305	−1.24393	−0.621967	+	0.783044i	$0.713666\pi$
$854$	0	0
$855$	23.7250	0.811377
$856$	0	0
$857$	42.9083	1.46572	0.732860	−	0.680379i	$-0.238185\pi$
$857$	42.9083	1.46572	0.732860	+	0.680379i	$0.238185\pi$
$858$	0	0
$859$	23.6972	0.808539	0.404269	−	0.914640i	$-0.367526\pi$
$859$	23.6972	0.808539	0.404269	+	0.914640i	$0.367526\pi$
$860$	0	0
$861$	−14.2111	−0.484313
$862$	0	0
$863$	−37.5416	−1.27793	−0.638966	−	0.769235i	$-0.720638\pi$
$863$	−37.5416	−1.27793	−0.638966	+	0.769235i	$0.720638\pi$
$864$	0	0
$865$	41.4500	1.40934
$866$	0	0
$867$	−46.6333	−1.58375
$868$	0	0
$869$	13.8167	0.468698
$870$	0	0
$871$	−14.3028	−0.484631
$872$	0	0
$873$	18.9361	0.640889
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−29.8167	−1.00684	−0.503418	−	0.864043i	$-0.667925\pi$
$877$	−29.8167	−1.00684	−0.503418	+	0.864043i	$0.667925\pi$
$878$	0	0
$879$	101.175	3.41255
$880$	0	0
$881$	48.7527	1.64252	0.821261	−	0.570553i	$-0.193271\pi$
$881$	48.7527	1.64252	0.821261	+	0.570553i	$0.193271\pi$
$882$	0	0
$883$	31.7889	1.06978	0.534891	−	0.844921i	$-0.320353\pi$
$883$	31.7889	1.06978	0.534891	+	0.844921i	$0.320353\pi$
$884$	0	0
$885$	81.3583	2.73483
$886$	0	0
$887$	−40.8167	−1.37049	−0.685245	−	0.728313i	$-0.740305\pi$
$887$	−40.8167	−1.37049	−0.685245	+	0.728313i	$0.740305\pi$
$888$	0	0
$889$	9.81665	0.329240
$890$	0	0
$891$	128.294	4.29802
$892$	0	0
$893$	−8.21110	−0.274774
$894$	0	0
$895$	62.1749	2.07828
$896$	0	0
$897$	15.9083	0.531164
$898$	0	0
$899$	−2.09167	−0.0697612
$900$	0	0
$901$	−6.63331	−0.220988
$902$	0	0
$903$	−33.0278	−1.09909
$904$	0	0
$905$	42.9083	1.42632
$906$	0	0
$907$	26.8167	0.890432	0.445216	−	0.895423i	$-0.353127\pi$
$907$	26.8167	0.890432	0.445216	+	0.895423i	$0.353127\pi$
$908$	0	0
$909$	−91.7805	−3.04417
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	−101.175	−3.34474
$916$	0	0
$917$	−2.48612	−0.0820990
$918$	0	0
$919$	44.0278	1.45234	0.726171	−	0.687514i	$-0.241298\pi$
$919$	44.0278	1.45234	0.726171	+	0.687514i	$0.241298\pi$
$920$	0	0
$921$	−60.4500	−1.99189
$922$	0	0
$923$	3.55004	0.116851
$924$	0	0
$925$	−14.4222	−0.474199
$926$	0	0
$927$	14.1472	0.464655
$928$	0	0
$929$	−6.51388	−0.213713	−0.106857	−	0.994274i	$-0.534079\pi$
$929$	−6.51388	−0.213713	−0.106857	+	0.994274i	$0.534079\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−51.2389	−1.67748
$934$	0	0
$935$	21.9083	0.716479
$936$	0	0
$937$	−4.11943	−0.134576	−0.0672879	−	0.997734i	$-0.521435\pi$
$937$	−4.11943	−0.134576	−0.0672879	+	0.997734i	$0.521435\pi$
$938$	0	0
$939$	29.8167	0.973030
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	12.9083	0.420353
$944$	0	0
$945$	48.6333	1.58204
$946$	0	0
$947$	−26.0917	−0.847865	−0.423933	−	0.905694i	$-0.639351\pi$
$947$	−26.0917	−0.847865	−0.423933	+	0.905694i	$0.639351\pi$
$948$	0	0
$949$	−26.5139	−0.860677
$950$	0	0
$951$	76.6611	2.48591
$952$	0	0
$953$	−35.3305	−1.14447	−0.572234	−	0.820090i	$-0.693923\pi$
$953$	−35.3305	−1.14447	−0.572234	+	0.820090i	$0.693923\pi$
$954$	0	0
$955$	−42.9916	−1.39118
$956$	0	0
$957$	−12.9083	−0.417267
$958$	0	0
$959$	21.6333	0.698576
$960$	0	0
$961$	−25.6972	−0.828943
$962$	0	0
$963$	159.836	5.15064
$964$	0	0
$965$	−6.35829	−0.204681
$966$	0	0
$967$	−30.3028	−0.974472	−0.487236	−	0.873270i	$-0.661995\pi$
$967$	−30.3028	−0.974472	−0.487236	+	0.873270i	$0.661995\pi$
$968$	0	0
$969$	5.60555	0.180076
$970$	0	0
$971$	−24.6333	−0.790520	−0.395260	−	0.918569i	$-0.629346\pi$
$971$	−24.6333	−0.790520	−0.395260	+	0.918569i	$0.629346\pi$
$972$	0	0
$973$	5.00000	0.160293
$974$	0	0
$975$	21.2111	0.679299
$976$	0	0
$977$	47.4500	1.51806	0.759029	−	0.651056i	$-0.225674\pi$
$977$	47.4500	1.51806	0.759029	+	0.651056i	$0.225674\pi$
$978$	0	0
$979$	−14.6056	−0.466795
$980$	0	0
$981$	33.3028	1.06328
$982$	0	0
$983$	−23.0555	−0.735357	−0.367678	−	0.929953i	$-0.619847\pi$
$983$	−23.0555	−0.735357	−0.367678	+	0.929953i	$0.619847\pi$
$984$	0	0
$985$	−11.7250	−0.373589
$986$	0	0
$987$	−27.1194	−0.863221
$988$	0	0
$989$	30.0000	0.953945
$990$	0	0
$991$	53.0278	1.68448	0.842241	−	0.539101i	$-0.181236\pi$
$991$	53.0278	1.68448	0.842241	+	0.539101i	$0.181236\pi$
$992$	0	0
$993$	−50.5416	−1.60389
$994$	0	0
$995$	46.2666	1.46675
$996$	0	0
$997$	37.7250	1.19476	0.597381	−	0.801958i	$-0.296208\pi$
$997$	37.7250	1.19476	0.597381	+	0.801958i	$0.296208\pi$
$998$	0	0
$999$	−58.4500	−1.84927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2128.2.a.l.1.2		2
4.3	odd	2		133.2.a.b.1.2	✓	2
8.3	odd	2		8512.2.a.bh.1.2		2
8.5	even	2		8512.2.a.l.1.1		2
12.11	even	2		1197.2.a.h.1.1		2
20.19	odd	2		3325.2.a.n.1.1		2
28.3	even	6		931.2.f.g.324.1		4
28.11	odd	6		931.2.f.h.324.1		4
28.19	even	6		931.2.f.g.704.1		4
28.23	odd	6		931.2.f.h.704.1		4
28.27	even	2		931.2.a.g.1.2		2
76.75	even	2		2527.2.a.d.1.1		2
84.83	odd	2		8379.2.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.a.b.1.2	✓	2	4.3	odd	2
931.2.a.g.1.2		2	28.27	even	2
931.2.f.g.324.1		4	28.3	even	6
931.2.f.g.704.1		4	28.19	even	6
931.2.f.h.324.1		4	28.11	odd	6
931.2.f.h.704.1		4	28.23	odd	6
1197.2.a.h.1.1		2	12.11	even	2
2128.2.a.l.1.2		2	1.1	even	1	trivial
2527.2.a.d.1.1		2	76.75	even	2
3325.2.a.n.1.1		2	20.19	odd	2
8379.2.a.bf.1.1		2	84.83	odd	2
8512.2.a.l.1.1		2	8.5	even	2
8512.2.a.bh.1.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2128 = 2^{4} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2128.a (trivial)

\(f(q)\)	\(=\)	\(q+3.30278 q^{3} -3.00000 q^{5} -1.00000 q^{7} +7.90833 q^{9} +4.30278 q^{11} +1.60555 q^{13} -9.90833 q^{15} -1.69722 q^{17} -1.00000 q^{19} -3.30278 q^{21} +3.00000 q^{23} +4.00000 q^{25} +16.2111 q^{27} -0.908327 q^{29} +2.30278 q^{31} +14.2111 q^{33} +3.00000 q^{35} -3.60555 q^{37} +5.30278 q^{39} +4.30278 q^{41} +10.0000 q^{43} -23.7250 q^{45} +8.21110 q^{47} +1.00000 q^{49} -5.60555 q^{51} +3.90833 q^{53} -12.9083 q^{55} -3.30278 q^{57} -8.21110 q^{59} +10.2111 q^{61} -7.90833 q^{63} -4.81665 q^{65} -8.90833 q^{67} +9.90833 q^{69} +2.21110 q^{71} -16.5139 q^{73} +13.2111 q^{75} -4.30278 q^{77} +3.21110 q^{79} +29.8167 q^{81} +2.09167 q^{83} +5.09167 q^{85} -3.00000 q^{87} -3.39445 q^{89} -1.60555 q^{91} +7.60555 q^{93} +3.00000 q^{95} +2.39445 q^{97} +34.0278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 6 q^{5} - 2 q^{7} + 5 q^{9} + 5 q^{11} - 4 q^{13} - 9 q^{15} - 7 q^{17} - 2 q^{19} - 3 q^{21} + 6 q^{23} + 8 q^{25} + 18 q^{27} + 9 q^{29} + q^{31} + 14 q^{33} + 6 q^{35} + 7 q^{39} + 5 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - 6 * q^5 - 2 * q^7 + 5 * q^9 + 5 * q^11 - 4 * q^13 - 9 * q^15 - 7 * q^17 - 2 * q^19 - 3 * q^21 + 6 * q^23 + 8 * q^25 + 18 * q^27 + 9 * q^29 + q^31 + 14 * q^33 + 6 * q^35 + 7 * q^39 + 5 * q^41 + 20 * q^43 - 15 * q^45 + 2 * q^47 + 2 * q^49 - 4 * q^51 - 3 * q^53 - 15 * q^55 - 3 * q^57 - 2 * q^59 + 6 * q^61 - 5 * q^63 + 12 * q^65 - 7 * q^67 + 9 * q^69 - 10 * q^71 - 15 * q^73 + 12 * q^75 - 5 * q^77 - 8 * q^79 + 38 * q^81 + 15 * q^83 + 21 * q^85 - 6 * q^87 - 14 * q^89 + 4 * q^91 + 8 * q^93 + 6 * q^95 + 12 * q^97 + 32 * q^99

Introduction

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