LMFDB - Embedded newform 1900.4.c.d.1749.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1900,4,Mod(1749,1900)] 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("1900.1749"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-208] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$112.103629011$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 92x^{5} + 1602x^{4} + 478x^{3} + 72x^{2} + 6612x + 303601 $ x^8 - 2x^7 + 2x^6 + 92x^5 + 1602x^4 + 478x^3 + 72x^2 + 6612*x + 303601
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.3
Root			$-3.15226 - 3.15226i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.4.c.d.1749.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.30451i q^{3} +2.36393i q^{7} -12.7469 q^{9} -71.0217 q^{11} +23.8865i q^{13} -102.501i q^{17} +19.0000 q^{19} +14.9035 q^{21} -95.3081i q^{23} -89.8589i q^{27} -168.054 q^{29} -240.783 q^{31} +447.757i q^{33} -354.392i q^{37} +150.593 q^{39} +466.330 q^{41} -269.628i q^{43} +227.729i q^{47} +337.412 q^{49} -646.218 q^{51} +195.612i q^{53} -119.786i q^{57} -324.959 q^{59} -365.892 q^{61} -30.1328i q^{63} +269.909i q^{67} -600.871 q^{69} +864.400 q^{71} +904.206i q^{73} -167.891i q^{77} -96.9592 q^{79} -910.683 q^{81} +1392.15i q^{83} +1059.50i q^{87} +159.411 q^{89} -56.4661 q^{91} +1518.02i q^{93} +1692.58i q^{97} +905.306 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 208 q^{9} - 120 q^{11} + 152 q^{19} - 408 q^{21} + 328 q^{29} - 1512 q^{31} + 1752 q^{39} - 352 q^{41} + 664 q^{49} - 2488 q^{51} + 288 q^{59} - 2072 q^{61} + 3320 q^{69} + 2888 q^{71} + 2112 q^{79}+ \cdots + 8296 q^{99}+O(q^{100}) $ 8 * q - 208 * q^9 - 120 * q^11 + 152 * q^19 - 408 * q^21 + 328 * q^29 - 1512 * q^31 + 1752 * q^39 - 352 * q^41 + 664 * q^49 - 2488 * q^51 + 288 * q^59 - 2072 * q^61 + 3320 * q^69 + 2888 * q^71 + 2112 * q^79 + 96 * q^81 + 3304 * q^89 - 600 * q^91 + 8296 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 6.30451i	− 1.21330i	−0.794967	−	0.606652i	$-0.792512\pi$
$3$	− 6.30451i	− 1.21330i	0.794967	−	0.606652i	$-0.207488\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.36393i	0.127640i	0.997961	+	0.0638202i	$0.0203284\pi$
$7$	2.36393i	0.127640i	−0.997961	+	0.0638202i	$0.979672\pi$
$8$	0	0
$9$	−12.7469	−0.472107
$10$	0	0
$11$	−71.0217	−1.94671	−0.973357	−	0.229297i	$-0.926357\pi$
$11$	−71.0217	−1.94671	−0.973357	+	0.229297i	$0.926357\pi$
$12$	0	0
$13$	23.8865i	0.509609i	0.966993	+	0.254805i	$0.0820112\pi$
$13$	23.8865i	0.509609i	−0.966993	+	0.254805i	$0.917989\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 102.501i	− 1.46236i	−0.682185	−	0.731180i	$-0.738970\pi$
$17$	− 102.501i	− 1.46236i	0.682185	−	0.731180i	$-0.261030\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	14.9035	0.154867
$22$	0	0
$23$	− 95.3081i	− 0.864048i	−0.901862	−	0.432024i	$-0.857800\pi$
$23$	− 95.3081i	− 0.864048i	0.901862	−	0.432024i	$-0.142200\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 89.8589i	− 0.640495i
$28$	0	0
$29$	−168.054	−1.07610	−0.538049	−	0.842913i	$-0.680839\pi$
$29$	−168.054	−1.07610	−0.538049	+	0.842913i	$0.680839\pi$
$30$	0	0
$31$	−240.783	−1.39503	−0.697514	−	0.716571i	$-0.745710\pi$
$31$	−240.783	−1.39503	−0.697514	+	0.716571i	$0.745710\pi$
$32$	0	0
$33$	447.757i	2.36196i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 354.392i	− 1.57464i	−0.616545	−	0.787320i	$-0.711468\pi$
$37$	− 354.392i	− 1.57464i	0.616545	−	0.787320i	$-0.288532\pi$
$38$	0	0
$39$	150.593	0.618311
$40$	0	0
$41$	466.330	1.77631	0.888153	−	0.459549i	$-0.151989\pi$
$41$	466.330	1.77631	0.888153	+	0.459549i	$0.151989\pi$
$42$	0	0
$43$	− 269.628i	− 0.956231i	−0.878297	−	0.478115i	$-0.841320\pi$
$43$	− 269.628i	− 0.956231i	0.878297	−	0.478115i	$-0.158680\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	227.729i	0.706759i	0.935480	+	0.353379i	$0.114967\pi$
$47$	227.729i	0.706759i	−0.935480	+	0.353379i	$0.885033\pi$
$48$	0	0
$49$	337.412	0.983708
$50$	0	0
$51$	−646.218	−1.77429
$52$	0	0
$53$	195.612i	0.506968i	0.967340	+	0.253484i	$0.0815766\pi$
$53$	195.612i	0.506968i	−0.967340	+	0.253484i	$0.918423\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 119.786i	− 0.278351i
$58$	0	0
$59$	−324.959	−0.717052	−0.358526	−	0.933520i	$-0.616721\pi$
$59$	−324.959	−0.717052	−0.358526	+	0.933520i	$0.616721\pi$
$60$	0	0
$61$	−365.892	−0.767996	−0.383998	−	0.923334i	$-0.625453\pi$
$61$	−365.892	−0.767996	−0.383998	+	0.923334i	$0.625453\pi$
$62$	0	0
$63$	− 30.1328i	− 0.0602600i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	269.909i	0.492159i	0.969250	+	0.246079i	$0.0791424\pi$
$67$	269.909i	0.492159i	−0.969250	+	0.246079i	$0.920858\pi$
$68$	0	0
$69$	−600.871	−1.04835
$70$	0	0
$71$	864.400	1.44486	0.722432	−	0.691442i	$-0.243024\pi$
$71$	864.400	1.44486	0.722432	+	0.691442i	$0.243024\pi$
$72$	0	0
$73$	904.206i	1.44972i	0.688898	+	0.724858i	$0.258095\pi$
$73$	904.206i	1.44972i	−0.688898	+	0.724858i	$0.741905\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 167.891i	− 0.248479i
$78$	0	0
$79$	−96.9592	−0.138086	−0.0690428	−	0.997614i	$-0.521994\pi$
$79$	−96.9592	−0.138086	−0.0690428	+	0.997614i	$0.521994\pi$
$80$	0	0
$81$	−910.683	−1.24922
$82$	0	0
$83$	1392.15i	1.84106i	0.390673	+	0.920530i	$0.372242\pi$
$83$	1392.15i	1.84106i	−0.390673	+	0.920530i	$0.627758\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1059.50i	1.30563i
$88$	0	0
$89$	159.411	0.189860	0.0949301	−	0.995484i	$-0.469737\pi$
$89$	159.411	0.189860	0.0949301	+	0.995484i	$0.469737\pi$
$90$	0	0
$91$	−56.4661	−0.0650468
$92$	0	0
$93$	1518.02i	1.69259i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1692.58i	1.77171i	0.463961	+	0.885855i	$0.346428\pi$
$97$	1692.58i	1.77171i	−0.463961	+	0.885855i	$0.653572\pi$
$98$	0	0
$99$	905.306	0.919057
$100$	0	0
$101$	−1433.84	−1.41260	−0.706301	−	0.707912i	$-0.749637\pi$
$101$	−1433.84	−1.41260	−0.706301	+	0.707912i	$0.749637\pi$
$102$	0	0
$103$	1813.28i	1.73464i	0.497751	+	0.867320i	$0.334159\pi$
$103$	1813.28i	1.73464i	−0.497751	+	0.867320i	$0.665841\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	21.8923i	0.0197795i	0.999951	+	0.00988977i	$0.00314806\pi$
$107$	21.8923i	0.0197795i	−0.999951	+	0.00988977i	$0.996852\pi$
$108$	0	0
$109$	725.107	0.637180	0.318590	−	0.947893i	$-0.396791\pi$
$109$	725.107	0.637180	0.318590	+	0.947893i	$0.396791\pi$
$110$	0	0
$111$	−2234.27	−1.91052
$112$	0	0
$113$	− 933.597i	− 0.777216i	−0.921403	−	0.388608i	$-0.872956\pi$
$113$	− 933.597i	− 0.777216i	0.921403	−	0.388608i	$-0.127044\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 304.479i	− 0.240590i
$118$	0	0
$119$	242.305	0.186656
$120$	0	0
$121$	3713.08	2.78969
$122$	0	0
$123$	− 2939.98i	− 2.15520i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1956.65i	− 1.36712i	−0.729894	−	0.683560i	$-0.760431\pi$
$127$	− 1956.65i	− 1.36712i	0.729894	−	0.683560i	$-0.239569\pi$
$128$	0	0
$129$	−1699.87	−1.16020
$130$	0	0
$131$	−1685.60	−1.12421	−0.562105	−	0.827066i	$-0.690008\pi$
$131$	−1685.60	−1.12421	−0.562105	+	0.827066i	$0.690008\pi$
$132$	0	0
$133$	44.9148i	0.0292827i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1793.02i	1.11816i	0.829113	+	0.559081i	$0.188846\pi$
$137$	1793.02i	1.11816i	−0.829113	+	0.559081i	$0.811154\pi$
$138$	0	0
$139$	−683.284	−0.416946	−0.208473	−	0.978028i	$-0.566849\pi$
$139$	−683.284	−0.416946	−0.208473	+	0.978028i	$0.566849\pi$
$140$	0	0
$141$	1435.72	0.857513
$142$	0	0
$143$	− 1696.46i	− 0.992063i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2127.22i	− 1.19354i
$148$	0	0
$149$	−2944.72	−1.61907	−0.809534	−	0.587073i	$-0.800280\pi$
$149$	−2944.72	−1.61907	−0.809534	+	0.587073i	$0.800280\pi$
$150$	0	0
$151$	1162.60	0.626561	0.313281	−	0.949661i	$-0.398572\pi$
$151$	1162.60	0.626561	0.313281	+	0.949661i	$0.398572\pi$
$152$	0	0
$153$	1306.57i	0.690390i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 7.74394i	− 0.00393652i	−0.999998	−	0.00196826i	$-0.999373\pi$
$157$	− 7.74394i	− 0.00393652i	0.999998	−	0.00196826i	$-0.000626517\pi$
$158$	0	0
$159$	1233.24	0.615107
$160$	0	0
$161$	225.302	0.110288
$162$	0	0
$163$	478.327i	0.229850i	0.993374	+	0.114925i	$0.0366627\pi$
$163$	478.327i	0.229850i	−0.993374	+	0.114925i	$0.963337\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1528.78i	0.708387i	0.935172	+	0.354194i	$0.115245\pi$
$167$	1528.78i	0.708387i	−0.935172	+	0.354194i	$0.884755\pi$
$168$	0	0
$169$	1626.44	0.740298
$170$	0	0
$171$	−242.191	−0.108309
$172$	0	0
$173$	1222.36i	0.537194i	0.963253	+	0.268597i	$0.0865601\pi$
$173$	1222.36i	0.537194i	−0.963253	+	0.268597i	$0.913440\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2048.71i	0.870002i
$178$	0	0
$179$	−2835.03	−1.18380	−0.591899	−	0.806012i	$-0.701622\pi$
$179$	−2835.03	−1.18380	−0.591899	+	0.806012i	$0.701622\pi$
$180$	0	0
$181$	3208.77	1.31771	0.658856	−	0.752269i	$-0.271041\pi$
$181$	3208.77	1.31771	0.658856	+	0.752269i	$0.271041\pi$
$182$	0	0
$183$	2306.77i	0.931812i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7279.78i	2.84679i
$188$	0	0
$189$	212.421	0.0817531
$190$	0	0
$191$	−3376.35	−1.27908	−0.639540	−	0.768758i	$-0.720875\pi$
$191$	−3376.35	−1.27908	−0.639540	+	0.768758i	$0.720875\pi$
$192$	0	0
$193$	− 2893.76i	− 1.07926i	−0.841902	−	0.539631i	$-0.818564\pi$
$193$	− 2893.76i	− 1.07926i	0.841902	−	0.539631i	$-0.181436\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1295.88i	0.468668i	0.972156	+	0.234334i	$0.0752910\pi$
$197$	1295.88i	0.468668i	−0.972156	+	0.234334i	$0.924709\pi$
$198$	0	0
$199$	2203.78	0.785033	0.392517	−	0.919745i	$-0.371605\pi$
$199$	2203.78	0.785033	0.392517	+	0.919745i	$0.371605\pi$
$200$	0	0
$201$	1701.65	0.597138
$202$	0	0
$203$	− 397.269i	− 0.137354i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1214.88i	0.407923i
$208$	0	0
$209$	−1349.41	−0.446607
$210$	0	0
$211$	219.761	0.0717014	0.0358507	−	0.999357i	$-0.488586\pi$
$211$	219.761	0.0717014	0.0358507	+	0.999357i	$0.488586\pi$
$212$	0	0
$213$	− 5449.62i	− 1.75306i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 569.195i	− 0.178062i
$218$	0	0
$219$	5700.58	1.75895
$220$	0	0
$221$	2448.39	0.745232
$222$	0	0
$223$	957.362i	0.287487i	0.989615	+	0.143744i	$0.0459141\pi$
$223$	957.362i	0.287487i	−0.989615	+	0.143744i	$0.954086\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3875.01i	1.13301i	0.824058	+	0.566506i	$0.191705\pi$
$227$	3875.01i	1.13301i	−0.824058	+	0.566506i	$0.808295\pi$
$228$	0	0
$229$	4515.86	1.30313	0.651564	−	0.758594i	$-0.274113\pi$
$229$	4515.86	1.30313	0.651564	+	0.758594i	$0.274113\pi$
$230$	0	0
$231$	−1058.47	−0.301481
$232$	0	0
$233$	− 3593.62i	− 1.01041i	−0.862999	−	0.505206i	$-0.831417\pi$
$233$	− 3593.62i	− 1.01041i	0.862999	−	0.505206i	$-0.168583\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	611.280i	0.167540i
$238$	0	0
$239$	3637.90	0.984585	0.492293	−	0.870430i	$-0.336159\pi$
$239$	3637.90	0.984585	0.492293	+	0.870430i	$0.336159\pi$
$240$	0	0
$241$	−302.100	−0.0807467	−0.0403734	−	0.999185i	$-0.512855\pi$
$241$	−302.100	−0.0807467	−0.0403734	+	0.999185i	$0.512855\pi$
$242$	0	0
$243$	3315.22i	0.875191i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	453.844i	0.116912i
$248$	0	0
$249$	8776.80	2.23376
$250$	0	0
$251$	−5361.70	−1.34832	−0.674158	−	0.738587i	$-0.735494\pi$
$251$	−5361.70	−1.34832	−0.674158	+	0.738587i	$0.735494\pi$
$252$	0	0
$253$	6768.94i	1.68205i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3845.53i	0.933376i	0.884422	+	0.466688i	$0.154553\pi$
$257$	3845.53i	0.933376i	−0.884422	+	0.466688i	$0.845447\pi$
$258$	0	0
$259$	837.759	0.200988
$260$	0	0
$261$	2142.17	0.508034
$262$	0	0
$263$	− 34.6476i	− 0.00812343i	−0.999992	−	0.00406172i	$-0.998707\pi$
$263$	− 34.6476i	− 0.00812343i	0.999992	−	0.00406172i	$-0.00129289\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1005.01i	− 0.230358i
$268$	0	0
$269$	6566.14	1.48827	0.744135	−	0.668030i	$-0.232862\pi$
$269$	6566.14	1.48827	0.744135	+	0.668030i	$0.232862\pi$
$270$	0	0
$271$	−6197.27	−1.38914	−0.694571	−	0.719424i	$-0.744406\pi$
$271$	−6197.27	−1.38914	−0.694571	+	0.719424i	$0.744406\pi$
$272$	0	0
$273$	355.991i	0.0789215i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3224.93i	0.699521i	0.936839	+	0.349761i	$0.113737\pi$
$277$	3224.93i	0.699521i	−0.936839	+	0.349761i	$0.886263\pi$
$278$	0	0
$279$	3069.23	0.658602
$280$	0	0
$281$	−4859.78	−1.03171	−0.515854	−	0.856677i	$-0.672525\pi$
$281$	−4859.78	−1.03171	−0.515854	+	0.856677i	$0.672525\pi$
$282$	0	0
$283$	6136.80i	1.28903i	0.764592	+	0.644514i	$0.222940\pi$
$283$	6136.80i	1.28903i	−0.764592	+	0.644514i	$0.777060\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1102.37i	0.226728i
$288$	0	0
$289$	−5593.43	−1.13850
$290$	0	0
$291$	10670.9	2.14962
$292$	0	0
$293$	− 8952.76i	− 1.78507i	−0.450976	−	0.892536i	$-0.648924\pi$
$293$	− 8952.76i	− 1.78507i	0.450976	−	0.892536i	$-0.351076\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6381.93i	1.24686i
$298$	0	0
$299$	2276.58	0.440327
$300$	0	0
$301$	637.384	0.122054
$302$	0	0
$303$	9039.69i	1.71392i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6931.29i	1.28857i	0.764787	+	0.644283i	$0.222844\pi$
$307$	6931.29i	1.28857i	−0.764787	+	0.644283i	$0.777156\pi$
$308$	0	0
$309$	11431.9	2.10465
$310$	0	0
$311$	−5360.65	−0.977410	−0.488705	−	0.872449i	$-0.662531\pi$
$311$	−5360.65	−0.977410	−0.488705	+	0.872449i	$0.662531\pi$
$312$	0	0
$313$	− 2057.51i	− 0.371557i	−0.982592	−	0.185778i	$-0.940519\pi$
$313$	− 2057.51i	− 0.371557i	0.982592	−	0.185778i	$-0.0594807\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1903.50i	0.337259i	0.985680	+	0.168629i	$0.0539341\pi$
$317$	1903.50i	0.337259i	−0.985680	+	0.168629i	$0.946066\pi$
$318$	0	0
$319$	11935.5	2.09486
$320$	0	0
$321$	138.021	0.0239986
$322$	0	0
$323$	− 1947.52i	− 0.335488i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4571.45i	− 0.773094i
$328$	0	0
$329$	−538.336	−0.0902110
$330$	0	0
$331$	871.868	0.144780	0.0723901	−	0.997376i	$-0.476937\pi$
$331$	871.868	0.144780	0.0723901	+	0.997376i	$0.476937\pi$
$332$	0	0
$333$	4517.39i	0.743398i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7801.47i	− 1.26105i	−0.776170	−	0.630524i	$-0.782840\pi$
$337$	− 7801.47i	− 1.26105i	0.776170	−	0.630524i	$-0.217160\pi$
$338$	0	0
$339$	−5885.88	−0.943000
$340$	0	0
$341$	17100.8	2.71572
$342$	0	0
$343$	1608.45i	0.253201i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12554.0i	− 1.94217i	−0.238743	−	0.971083i	$-0.576735\pi$
$347$	− 12554.0i	− 1.94217i	0.238743	−	0.971083i	$-0.423265\pi$
$348$	0	0
$349$	2561.47	0.392872	0.196436	−	0.980517i	$-0.437063\pi$
$349$	2561.47	0.392872	0.196436	+	0.980517i	$0.437063\pi$
$350$	0	0
$351$	2146.42	0.326402
$352$	0	0
$353$	− 3225.71i	− 0.486366i	−0.969980	−	0.243183i	$-0.921809\pi$
$353$	− 3225.71i	− 0.486366i	0.969980	−	0.243183i	$-0.0781915\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1527.62i	− 0.226471i
$358$	0	0
$359$	3557.82	0.523049	0.261524	−	0.965197i	$-0.415775\pi$
$359$	3557.82	0.523049	0.261524	+	0.965197i	$0.415775\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	− 23409.2i	− 3.38474i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	533.130i	0.0758287i	0.999281	+	0.0379144i	$0.0120714\pi$
$367$	533.130i	0.0758287i	−0.999281	+	0.0379144i	$0.987929\pi$
$368$	0	0
$369$	−5944.26	−0.838606
$370$	0	0
$371$	−462.413	−0.0647097
$372$	0	0
$373$	12023.0i	1.66897i	0.551030	+	0.834485i	$0.314235\pi$
$373$	12023.0i	1.66897i	−0.551030	+	0.834485i	$0.685765\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4014.22i	− 0.548390i
$378$	0	0
$379$	−1087.45	−0.147383	−0.0736917	−	0.997281i	$-0.523478\pi$
$379$	−1087.45	−0.147383	−0.0736917	+	0.997281i	$0.523478\pi$
$380$	0	0
$381$	−12335.7	−1.65873
$382$	0	0
$383$	3136.55i	0.418460i	0.977866	+	0.209230i	$0.0670957\pi$
$383$	3136.55i	0.418460i	−0.977866	+	0.209230i	$0.932904\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3436.92i	0.451443i
$388$	0	0
$389$	11874.3	1.54769	0.773846	−	0.633374i	$-0.218330\pi$
$389$	11874.3	1.54769	0.773846	+	0.633374i	$0.218330\pi$
$390$	0	0
$391$	−9769.16	−1.26355
$392$	0	0
$393$	10626.9i	1.36401i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1842.00i	− 0.232865i	−0.993199	−	0.116433i	$-0.962854\pi$
$397$	− 1842.00i	− 0.232865i	0.993199	−	0.116433i	$-0.0371459\pi$
$398$	0	0
$399$	283.166	0.0355289
$400$	0	0
$401$	−6946.08	−0.865015	−0.432507	−	0.901630i	$-0.642371\pi$
$401$	−6946.08	−0.865015	−0.432507	+	0.901630i	$0.642371\pi$
$402$	0	0
$403$	− 5751.46i	− 0.710919i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25169.5i	3.06537i
$408$	0	0
$409$	−8473.65	−1.02444	−0.512219	−	0.858855i	$-0.671176\pi$
$409$	−8473.65	−1.02444	−0.512219	+	0.858855i	$0.671176\pi$
$410$	0	0
$411$	11304.1	1.35667
$412$	0	0
$413$	− 768.182i	− 0.0915249i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4307.78i	0.505882i
$418$	0	0
$419$	−5811.67	−0.677610	−0.338805	−	0.940857i	$-0.610023\pi$
$419$	−5811.67	−0.677610	−0.338805	+	0.940857i	$0.610023\pi$
$420$	0	0
$421$	−3107.46	−0.359735	−0.179867	−	0.983691i	$-0.557567\pi$
$421$	−3107.46	−0.359735	−0.179867	+	0.983691i	$0.557567\pi$
$422$	0	0
$423$	− 2902.83i	− 0.333666i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 864.946i	− 0.0980273i
$428$	0	0
$429$	−10695.4	−1.20367
$430$	0	0
$431$	1371.57	0.153286	0.0766430	−	0.997059i	$-0.475580\pi$
$431$	1371.57	0.153286	0.0766430	+	0.997059i	$0.475580\pi$
$432$	0	0
$433$	− 3102.80i	− 0.344368i	−0.985065	−	0.172184i	$-0.944918\pi$
$433$	− 3102.80i	− 0.344368i	0.985065	−	0.172184i	$-0.0550823\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1810.85i	− 0.198226i
$438$	0	0
$439$	8445.75	0.918209	0.459104	−	0.888382i	$-0.348170\pi$
$439$	8445.75	0.918209	0.459104	+	0.888382i	$0.348170\pi$
$440$	0	0
$441$	−4300.95	−0.464415
$442$	0	0
$443$	7881.39i	0.845273i	0.906299	+	0.422637i	$0.138895\pi$
$443$	7881.39i	0.845273i	−0.906299	+	0.422637i	$0.861105\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	18565.0i	1.96442i
$448$	0	0
$449$	4284.49	0.450329	0.225164	−	0.974321i	$-0.427708\pi$
$449$	4284.49	0.450329	0.225164	+	0.974321i	$0.427708\pi$
$450$	0	0
$451$	−33119.5	−3.45796
$452$	0	0
$453$	− 7329.61i	− 0.760210i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2802.70i	0.286882i	0.989659	+	0.143441i	$0.0458167\pi$
$457$	2802.70i	0.286882i	−0.989659	+	0.143441i	$0.954183\pi$
$458$	0	0
$459$	−9210.62	−0.936634
$460$	0	0
$461$	14035.5	1.41801	0.709003	−	0.705206i	$-0.249145\pi$
$461$	14035.5	1.41801	0.709003	+	0.705206i	$0.249145\pi$
$462$	0	0
$463$	11778.4i	1.18227i	0.806574	+	0.591133i	$0.201319\pi$
$463$	11778.4i	1.18227i	−0.806574	+	0.591133i	$0.798681\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9235.51i	− 0.915136i	−0.889175	−	0.457568i	$-0.848721\pi$
$467$	− 9235.51i	− 0.915136i	0.889175	−	0.457568i	$-0.151279\pi$
$468$	0	0
$469$	−638.047	−0.0628194
$470$	0	0
$471$	−48.8218	−0.00477620
$472$	0	0
$473$	19149.5i	1.86151i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2493.44i	− 0.239343i
$478$	0	0
$479$	2304.13	0.219788	0.109894	−	0.993943i	$-0.464949\pi$
$479$	2304.13	0.219788	0.109894	+	0.993943i	$0.464949\pi$
$480$	0	0
$481$	8465.18	0.802451
$482$	0	0
$483$	− 1420.42i	− 0.133812i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2092.35i	0.194689i	0.995251	+	0.0973445i	$0.0310349\pi$
$487$	2092.35i	0.194689i	−0.995251	+	0.0973445i	$0.968965\pi$
$488$	0	0
$489$	3015.62	0.278878
$490$	0	0
$491$	3305.32	0.303803	0.151901	−	0.988396i	$-0.451460\pi$
$491$	3305.32	0.303803	0.151901	+	0.988396i	$0.451460\pi$
$492$	0	0
$493$	17225.7i	1.57364i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2043.38i	0.184423i
$498$	0	0
$499$	−10747.6	−0.964188	−0.482094	−	0.876119i	$-0.660124\pi$
$499$	−10747.6	−0.964188	−0.482094	+	0.876119i	$0.660124\pi$
$500$	0	0
$501$	9638.23	0.859489
$502$	0	0
$503$	− 22508.8i	− 1.99526i	−0.0687746	−	0.997632i	$-0.521909\pi$
$503$	− 22508.8i	− 1.99526i	0.0687746	−	0.997632i	$-0.478091\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 10253.9i	− 0.898207i
$508$	0	0
$509$	17948.5	1.56297	0.781486	−	0.623923i	$-0.214462\pi$
$509$	17948.5	1.56297	0.781486	+	0.623923i	$0.214462\pi$
$510$	0	0
$511$	−2137.48	−0.185043
$512$	0	0
$513$	− 1707.32i	− 0.146940i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16173.7i	− 1.37586i
$518$	0	0
$519$	7706.41	0.651780
$520$	0	0
$521$	−16358.0	−1.37554	−0.687771	−	0.725928i	$-0.741411\pi$
$521$	−16358.0	−1.37554	−0.687771	+	0.725928i	$0.741411\pi$
$522$	0	0
$523$	3325.32i	0.278023i	0.990291	+	0.139012i	$0.0443925\pi$
$523$	3325.32i	0.278023i	−0.990291	+	0.139012i	$0.955607\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24680.4i	2.04003i
$528$	0	0
$529$	3083.37	0.253420
$530$	0	0
$531$	4142.22	0.338525
$532$	0	0
$533$	11139.0i	0.905222i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	17873.5i	1.43631i
$538$	0	0
$539$	−23963.6	−1.91500
$540$	0	0
$541$	11036.4	0.877064	0.438532	−	0.898716i	$-0.355499\pi$
$541$	11036.4	0.877064	0.438532	+	0.898716i	$0.355499\pi$
$542$	0	0
$543$	− 20229.7i	− 1.59878i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 7211.25i	− 0.563676i	−0.959462	−	0.281838i	$-0.909056\pi$
$547$	− 7211.25i	− 0.563676i	0.959462	−	0.281838i	$-0.0909441\pi$
$548$	0	0
$549$	4663.99	0.362576
$550$	0	0
$551$	−3193.03	−0.246874
$552$	0	0
$553$	− 229.205i	− 0.0176253i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3869.01i	− 0.294318i	−0.989113	−	0.147159i	$-0.952987\pi$
$557$	− 3869.01i	− 0.294318i	0.989113	−	0.147159i	$-0.0470129\pi$
$558$	0	0
$559$	6440.48	0.487304
$560$	0	0
$561$	45895.5	3.45403
$562$	0	0
$563$	6419.61i	0.480558i	0.970704	+	0.240279i	$0.0772390\pi$
$563$	6419.61i	0.480558i	−0.970704	+	0.240279i	$0.922761\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2152.79i	− 0.159451i
$568$	0	0
$569$	−24618.1	−1.81378	−0.906892	−	0.421363i	$-0.861552\pi$
$569$	−24618.1	−1.81378	−0.906892	+	0.421363i	$0.861552\pi$
$570$	0	0
$571$	−17703.2	−1.29747	−0.648735	−	0.761014i	$-0.724702\pi$
$571$	−17703.2	−1.29747	−0.648735	+	0.761014i	$0.724702\pi$
$572$	0	0
$573$	21286.2i	1.55191i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11277.4i	0.813668i	0.913502	+	0.406834i	$0.133367\pi$
$577$	11277.4i	0.813668i	−0.913502	+	0.406834i	$0.866633\pi$
$578$	0	0
$579$	−18243.8	−1.30947
$580$	0	0
$581$	−3290.94	−0.234994
$582$	0	0
$583$	− 13892.7i	− 0.986922i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14416.7i	− 1.01369i	−0.862036	−	0.506847i	$-0.830811\pi$
$587$	− 14416.7i	− 1.01369i	0.862036	−	0.506847i	$-0.169189\pi$
$588$	0	0
$589$	−4574.87	−0.320041
$590$	0	0
$591$	8169.90	0.568637
$592$	0	0
$593$	− 5188.47i	− 0.359300i	−0.983731	−	0.179650i	$-0.942503\pi$
$593$	− 5188.47i	− 0.359300i	0.983731	−	0.179650i	$-0.0574966\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 13893.7i	− 0.952484i
$598$	0	0
$599$	−16360.7	−1.11599	−0.557997	−	0.829843i	$-0.688430\pi$
$599$	−16360.7	−1.11599	−0.557997	+	0.829843i	$0.688430\pi$
$600$	0	0
$601$	6711.38	0.455512	0.227756	−	0.973718i	$-0.426861\pi$
$601$	6711.38	0.455512	0.227756	+	0.973718i	$0.426861\pi$
$602$	0	0
$603$	− 3440.50i	− 0.232352i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 14253.3i	− 0.953089i	−0.879150	−	0.476544i	$-0.841889\pi$
$607$	− 14253.3i	− 0.953089i	0.879150	−	0.476544i	$-0.158111\pi$
$608$	0	0
$609$	−2504.59	−0.166652
$610$	0	0
$611$	−5439.64	−0.360171
$612$	0	0
$613$	− 1718.21i	− 0.113210i	−0.998397	−	0.0566051i	$-0.981972\pi$
$613$	− 1718.21i	− 0.113210i	0.998397	−	0.0566051i	$-0.0180276\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4129.02i	0.269414i	0.990885	+	0.134707i	$0.0430093\pi$
$617$	4129.02i	0.269414i	−0.990885	+	0.134707i	$0.956991\pi$
$618$	0	0
$619$	−14248.7	−0.925205	−0.462602	−	0.886566i	$-0.653084\pi$
$619$	−14248.7	−0.925205	−0.462602	+	0.886566i	$0.653084\pi$
$620$	0	0
$621$	−8564.28	−0.553418
$622$	0	0
$623$	376.838i	0.0242339i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8507.39i	0.541870i
$628$	0	0
$629$	−36325.5	−2.30269
$630$	0	0
$631$	−13858.8	−0.874339	−0.437170	−	0.899379i	$-0.644019\pi$
$631$	−13858.8	−0.874339	−0.437170	+	0.899379i	$0.644019\pi$
$632$	0	0
$633$	− 1385.49i	− 0.0869956i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8059.59i	0.501307i
$638$	0	0
$639$	−11018.4	−0.682130
$640$	0	0
$641$	−7216.21	−0.444654	−0.222327	−	0.974972i	$-0.571365\pi$
$641$	−7216.21	−0.444654	−0.222327	+	0.974972i	$0.571365\pi$
$642$	0	0
$643$	24050.7i	1.47507i	0.675310	+	0.737534i	$0.264010\pi$
$643$	24050.7i	1.47507i	−0.675310	+	0.737534i	$0.735990\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3306.55i	− 0.200918i	−0.994941	−	0.100459i	$-0.967969\pi$
$647$	− 3306.55i	− 0.200918i	0.994941	−	0.100459i	$-0.0320311\pi$
$648$	0	0
$649$	23079.1	1.39589
$650$	0	0
$651$	−3588.50	−0.216043
$652$	0	0
$653$	− 9127.27i	− 0.546980i	−0.961875	−	0.273490i	$-0.911822\pi$
$653$	− 9127.27i	− 0.546980i	0.961875	−	0.273490i	$-0.0881780\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 11525.8i	− 0.684421i
$658$	0	0
$659$	1197.51	0.0707865	0.0353933	−	0.999373i	$-0.488732\pi$
$659$	1197.51	0.0707865	0.0353933	+	0.999373i	$0.488732\pi$
$660$	0	0
$661$	5949.19	0.350071	0.175035	−	0.984562i	$-0.443996\pi$
$661$	5949.19	0.350071	0.175035	+	0.984562i	$0.443996\pi$
$662$	0	0
$663$	− 15435.9i	− 0.904193i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16016.9i	0.929801i
$668$	0	0
$669$	6035.70	0.348810
$670$	0	0
$671$	25986.3	1.49507
$672$	0	0
$673$	− 4918.69i	− 0.281726i	−0.990029	−	0.140863i	$-0.955012\pi$
$673$	− 4918.69i	− 0.281726i	0.990029	−	0.140863i	$-0.0449877\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16866.2i	0.957492i	0.877953	+	0.478746i	$0.158909\pi$
$677$	16866.2i	0.957492i	−0.877953	+	0.478746i	$0.841091\pi$
$678$	0	0
$679$	−4001.16	−0.226142
$680$	0	0
$681$	24430.1	1.37469
$682$	0	0
$683$	− 24571.6i	− 1.37658i	−0.725434	−	0.688292i	$-0.758361\pi$
$683$	− 24571.6i	− 1.37658i	0.725434	−	0.688292i	$-0.241639\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 28470.3i	− 1.58109i
$688$	0	0
$689$	−4672.48	−0.258356
$690$	0	0
$691$	−19987.2	−1.10036	−0.550181	−	0.835045i	$-0.685441\pi$
$691$	−19987.2	−1.10036	−0.550181	+	0.835045i	$0.685441\pi$
$692$	0	0
$693$	2140.08i	0.117309i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 47799.2i	− 2.59760i
$698$	0	0
$699$	−22656.0	−1.22594
$700$	0	0
$701$	−13759.4	−0.741349	−0.370674	−	0.928763i	$-0.620873\pi$
$701$	−13759.4	−0.741349	−0.370674	+	0.928763i	$0.620873\pi$
$702$	0	0
$703$	− 6733.45i	− 0.361247i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3389.51i	− 0.180305i
$708$	0	0
$709$	−12974.7	−0.687273	−0.343636	−	0.939103i	$-0.611659\pi$
$709$	−12974.7	−0.687273	−0.343636	+	0.939103i	$0.611659\pi$
$710$	0	0
$711$	1235.93	0.0651912
$712$	0	0
$713$	22948.5i	1.20537i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 22935.2i	− 1.19460i
$718$	0	0
$719$	−22316.8	−1.15755	−0.578774	−	0.815488i	$-0.696469\pi$
$719$	−22316.8	−1.15755	−0.578774	+	0.815488i	$0.696469\pi$
$720$	0	0
$721$	−4286.48	−0.221410
$722$	0	0
$723$	1904.59i	0.0979703i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 3452.80i	− 0.176145i	−0.996114	−	0.0880724i	$-0.971929\pi$
$727$	− 3452.80i	− 0.176145i	0.996114	−	0.0880724i	$-0.0280707\pi$
$728$	0	0
$729$	−3687.58	−0.187349
$730$	0	0
$731$	−27637.1	−1.39835
$732$	0	0
$733$	− 15659.5i	− 0.789080i	−0.918879	−	0.394540i	$-0.870904\pi$
$733$	− 15659.5i	− 0.789080i	0.918879	−	0.394540i	$-0.129096\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 19169.4i	− 0.958092i
$738$	0	0
$739$	13622.7	0.678105	0.339053	−	0.940767i	$-0.389893\pi$
$739$	13622.7	0.678105	0.339053	+	0.940767i	$0.389893\pi$
$740$	0	0
$741$	2861.26	0.141850
$742$	0	0
$743$	1798.53i	0.0888046i	0.999014	+	0.0444023i	$0.0141383\pi$
$743$	1798.53i	0.0888046i	−0.999014	+	0.0444023i	$0.985862\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 17745.5i	− 0.869177i
$748$	0	0
$749$	−51.7520	−0.00252467
$750$	0	0
$751$	−16836.8	−0.818085	−0.409042	−	0.912515i	$-0.634137\pi$
$751$	−16836.8	−0.818085	−0.409042	+	0.912515i	$0.634137\pi$
$752$	0	0
$753$	33802.9i	1.63592i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9512.94i	0.456742i	0.973574	+	0.228371i	$0.0733399\pi$
$757$	9512.94i	0.456742i	−0.973574	+	0.228371i	$0.926660\pi$
$758$	0	0
$759$	42674.9	2.04084
$760$	0	0
$761$	23379.8	1.11369	0.556844	−	0.830617i	$-0.312012\pi$
$761$	23379.8	1.11369	0.556844	+	0.830617i	$0.312012\pi$
$762$	0	0
$763$	1714.11i	0.0813300i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7762.14i	− 0.365417i
$768$	0	0
$769$	−1328.80	−0.0623115	−0.0311558	−	0.999515i	$-0.509919\pi$
$769$	−1328.80	−0.0623115	−0.0311558	+	0.999515i	$0.509919\pi$
$770$	0	0
$771$	24244.2	1.13247
$772$	0	0
$773$	17594.1i	0.818651i	0.912389	+	0.409325i	$0.134236\pi$
$773$	17594.1i	0.818651i	−0.912389	+	0.409325i	$0.865764\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 5281.66i	− 0.243859i
$778$	0	0
$779$	8860.27	0.407512
$780$	0	0
$781$	−61391.1	−2.81274
$782$	0	0
$783$	15101.2i	0.689235i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 5903.55i	− 0.267394i	−0.991022	−	0.133697i	$-0.957315\pi$
$787$	− 5903.55i	− 0.267394i	0.991022	−	0.133697i	$-0.0426848\pi$
$788$	0	0
$789$	−218.436	−0.00985619
$790$	0	0
$791$	2206.96	0.0992042
$792$	0	0
$793$	− 8739.89i	− 0.391378i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2690.65i	− 0.119583i	−0.998211	−	0.0597916i	$-0.980956\pi$
$797$	− 2690.65i	− 0.119583i	0.998211	−	0.0597916i	$-0.0190436\pi$
$798$	0	0
$799$	23342.4	1.03354
$800$	0	0
$801$	−2032.00	−0.0896344
$802$	0	0
$803$	− 64218.2i	− 2.82218i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 41396.3i	− 1.80572i
$808$	0	0
$809$	20516.0	0.891600	0.445800	−	0.895133i	$-0.352919\pi$
$809$	20516.0	0.891600	0.445800	+	0.895133i	$0.352919\pi$
$810$	0	0
$811$	4815.53	0.208503	0.104252	−	0.994551i	$-0.466755\pi$
$811$	4815.53	0.208503	0.104252	+	0.994551i	$0.466755\pi$
$812$	0	0
$813$	39070.8i	1.68545i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5122.94i	− 0.219374i
$818$	0	0
$819$	719.767	0.0307090
$820$	0	0
$821$	−20633.8	−0.877133	−0.438566	−	0.898699i	$-0.644514\pi$
$821$	−20633.8	−0.877133	−0.438566	+	0.898699i	$0.644514\pi$
$822$	0	0
$823$	− 25993.1i	− 1.10093i	−0.834860	−	0.550463i	$-0.814451\pi$
$823$	− 25993.1i	− 1.10093i	0.834860	−	0.550463i	$-0.185549\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 29216.8i	− 1.22850i	−0.789112	−	0.614249i	$-0.789459\pi$
$827$	− 29216.8i	− 1.22850i	0.789112	−	0.614249i	$-0.210541\pi$
$828$	0	0
$829$	−44420.7	−1.86103	−0.930517	−	0.366250i	$-0.880642\pi$
$829$	−44420.7	−1.86103	−0.930517	+	0.366250i	$0.880642\pi$
$830$	0	0
$831$	20331.6	0.848732
$832$	0	0
$833$	− 34585.0i	− 1.43853i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	21636.5i	0.893508i
$838$	0	0
$839$	35397.5	1.45657	0.728283	−	0.685277i	$-0.240319\pi$
$839$	35397.5	1.45657	0.728283	+	0.685277i	$0.240319\pi$
$840$	0	0
$841$	3853.17	0.157988
$842$	0	0
$843$	30638.5i	1.25178i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8777.48i	0.356078i
$848$	0	0
$849$	38689.6	1.56398
$850$	0	0
$851$	−33776.4	−1.36056
$852$	0	0
$853$	15232.6i	0.611435i	0.952122	+	0.305718i	$0.0988964\pi$
$853$	15232.6i	0.611435i	−0.952122	+	0.305718i	$0.901104\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44107.4i	1.75809i	0.476742	+	0.879043i	$0.341818\pi$
$857$	44107.4i	1.75809i	−0.476742	+	0.879043i	$0.658182\pi$
$858$	0	0
$859$	42023.7	1.66918	0.834592	−	0.550869i	$-0.185704\pi$
$859$	42023.7	1.66918	0.834592	+	0.550869i	$0.185704\pi$
$860$	0	0
$861$	6949.93	0.275091
$862$	0	0
$863$	− 14537.5i	− 0.573419i	−0.958017	−	0.286710i	$-0.907438\pi$
$863$	− 14537.5i	− 0.573419i	0.958017	−	0.286710i	$-0.0925615\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	35263.8i	1.38134i
$868$	0	0
$869$	6886.20	0.268813
$870$	0	0
$871$	−6447.18	−0.250809
$872$	0	0
$873$	− 21575.2i	− 0.836437i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 10217.5i	− 0.393411i	−0.980463	−	0.196705i	$-0.936976\pi$
$877$	− 10217.5i	− 0.393411i	0.980463	−	0.196705i	$-0.0630243\pi$
$878$	0	0
$879$	−56442.8	−2.16583
$880$	0	0
$881$	22938.9	0.877219	0.438609	−	0.898678i	$-0.355471\pi$
$881$	22938.9	0.877219	0.438609	+	0.898678i	$0.355471\pi$
$882$	0	0
$883$	− 13588.0i	− 0.517862i	−0.965896	−	0.258931i	$-0.916630\pi$
$883$	− 13588.0i	− 0.517862i	0.965896	−	0.258931i	$-0.0833702\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 14461.3i	− 0.547423i	−0.961812	−	0.273712i	$-0.911749\pi$
$887$	− 14461.3i	− 0.547423i	0.961812	−	0.273712i	$-0.0882514\pi$
$888$	0	0
$889$	4625.38	0.174500
$890$	0	0
$891$	64678.2	2.43188
$892$	0	0
$893$	4326.85i	0.162142i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 14352.7i	− 0.534251i
$898$	0	0
$899$	40464.5	1.50119
$900$	0	0
$901$	20050.4	0.741370
$902$	0	0
$903$	− 4018.39i	− 0.148088i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 13449.2i	− 0.492362i	−0.969224	−	0.246181i	$-0.920824\pi$
$907$	− 13449.2i	− 0.492362i	0.969224	−	0.246181i	$-0.0791758\pi$
$908$	0	0
$909$	18277.0	0.666899
$910$	0	0
$911$	−3924.01	−0.142709	−0.0713547	−	0.997451i	$-0.522732\pi$
$911$	−3924.01	−0.142709	−0.0713547	+	0.997451i	$0.522732\pi$
$912$	0	0
$913$	− 98872.6i	− 3.58401i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3984.65i	− 0.143495i
$918$	0	0
$919$	6652.79	0.238798	0.119399	−	0.992846i	$-0.461903\pi$
$919$	6652.79	0.238798	0.119399	+	0.992846i	$0.461903\pi$
$920$	0	0
$921$	43698.4	1.56342
$922$	0	0
$923$	20647.5i	0.736316i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 23113.7i	− 0.818936i
$928$	0	0
$929$	−24802.0	−0.875918	−0.437959	−	0.898995i	$-0.644298\pi$
$929$	−24802.0	−0.875918	−0.437959	+	0.898995i	$0.644298\pi$
$930$	0	0
$931$	6410.82	0.225678
$932$	0	0
$933$	33796.3i	1.18590i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33062.5i	1.15273i	0.817193	+	0.576364i	$0.195529\pi$
$937$	33062.5i	1.15273i	−0.817193	+	0.576364i	$0.804471\pi$
$938$	0	0
$939$	−12971.6	−0.450811
$940$	0	0
$941$	−41837.8	−1.44939	−0.724694	−	0.689071i	$-0.758019\pi$
$941$	−41837.8	−1.44939	−0.724694	+	0.689071i	$0.758019\pi$
$942$	0	0
$943$	− 44445.0i	− 1.53481i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20581.0i	0.706222i	0.935581	+	0.353111i	$0.114876\pi$
$947$	20581.0i	0.706222i	−0.935581	+	0.353111i	$0.885124\pi$
$948$	0	0
$949$	−21598.3	−0.738789
$950$	0	0
$951$	12000.6	0.409197
$952$	0	0
$953$	10663.8i	0.362470i	0.983440	+	0.181235i	$0.0580095\pi$
$953$	10663.8i	0.362470i	−0.983440	+	0.181235i	$0.941990\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 75247.4i	− 2.54170i
$958$	0	0
$959$	−4238.59	−0.142723
$960$	0	0
$961$	28185.3	0.946103
$962$	0	0
$963$	− 279.059i	− 0.00933806i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18383.1i	0.611335i	0.952138	+	0.305668i	$0.0988796\pi$
$967$	18383.1i	0.611335i	−0.952138	+	0.305668i	$0.901120\pi$
$968$	0	0
$969$	−12278.1	−0.407049
$970$	0	0
$971$	−13158.9	−0.434900	−0.217450	−	0.976071i	$-0.569774\pi$
$971$	−13158.9	−0.434900	−0.217450	+	0.976071i	$0.569774\pi$
$972$	0	0
$973$	− 1615.24i	− 0.0532191i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 256.157i	− 0.00838810i	−0.999991	−	0.00419405i	$-0.998665\pi$
$977$	− 256.157i	− 0.00838810i	0.999991	−	0.00419405i	$-0.00133501\pi$
$978$	0	0
$979$	−11321.7	−0.369603
$980$	0	0
$981$	−9242.86	−0.300817
$982$	0	0
$983$	− 22447.7i	− 0.728353i	−0.931330	−	0.364176i	$-0.881351\pi$
$983$	− 22447.7i	− 0.728353i	0.931330	−	0.364176i	$-0.118649\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3393.95i	0.109453i
$988$	0	0
$989$	−25697.8	−0.826230
$990$	0	0
$991$	10208.4	0.327226	0.163613	−	0.986525i	$-0.447685\pi$
$991$	10208.4	0.327226	0.163613	+	0.986525i	$0.447685\pi$
$992$	0	0
$993$	− 5496.71i	− 0.175662i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 43204.8i	− 1.37243i	−0.727400	−	0.686214i	$-0.759271\pi$
$997$	− 43204.8i	− 1.37243i	0.727400	−	0.686214i	$-0.240729\pi$
$998$	0	0
$999$	−31845.3	−1.00855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.4.c.d.1749.3		8
5.2	odd	4		1900.4.a.e.1.2		4
5.3	odd	4		380.4.a.c.1.3	✓	4
5.4	even	2	inner	1900.4.c.d.1749.6		8
20.3	even	4		1520.4.a.s.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.4.a.c.1.3	✓	4	5.3	odd	4
1520.4.a.s.1.2		4	20.3	even	4
1900.4.a.e.1.2		4	5.2	odd	4
1900.4.c.d.1749.3		8	1.1	even	1	trivial
1900.4.c.d.1749.6		8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.30451i q^{3} +2.36393i q^{7} -12.7469 q^{9} -71.0217 q^{11} +23.8865i q^{13} -102.501i q^{17} +19.0000 q^{19} +14.9035 q^{21} -95.3081i q^{23} -89.8589i q^{27} -168.054 q^{29} -240.783 q^{31} +447.757i q^{33} -354.392i q^{37} +150.593 q^{39} +466.330 q^{41} -269.628i q^{43} +227.729i q^{47} +337.412 q^{49} -646.218 q^{51} +195.612i q^{53} -119.786i q^{57} -324.959 q^{59} -365.892 q^{61} -30.1328i q^{63} +269.909i q^{67} -600.871 q^{69} +864.400 q^{71} +904.206i q^{73} -167.891i q^{77} -96.9592 q^{79} -910.683 q^{81} +1392.15i q^{83} +1059.50i q^{87} +159.411 q^{89} -56.4661 q^{91} +1518.02i q^{93} +1692.58i q^{97} +905.306 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 208 q^{9} - 120 q^{11} + 152 q^{19} - 408 q^{21} + 328 q^{29} - 1512 q^{31} + 1752 q^{39} - 352 q^{41} + 664 q^{49} - 2488 q^{51} + 288 q^{59} - 2072 q^{61} + 3320 q^{69} + 2888 q^{71} + 2112 q^{79}+ \cdots + 8296 q^{99}+O(q^{100}) \) 8 * q - 208 * q^9 - 120 * q^11 + 152 * q^19 - 408 * q^21 + 328 * q^29 - 1512 * q^31 + 1752 * q^39 - 352 * q^41 + 664 * q^49 - 2488 * q^51 + 288 * q^59 - 2072 * q^61 + 3320 * q^69 + 2888 * q^71 + 2112 * q^79 + 96 * q^81 + 3304 * q^89 - 600 * q^91 + 8296 * q^99

Introduction

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