LMFDB - Embedded newform 1900.4.c.d.1749.2

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			1749.2
Root			$-3.40583 - 3.40583i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.4.c.d.1749.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.81167i q^{3} -31.9765i q^{7} -19.3988 q^{9} +14.6385 q^{11} -7.91818i q^{13} +29.7894i q^{17} +19.0000 q^{19} -217.813 q^{21} +182.077i q^{23} -51.7766i q^{27} +206.266 q^{29} -311.559 q^{31} -99.7130i q^{33} +204.494i q^{37} -53.9360 q^{39} -464.589 q^{41} +221.899i q^{43} +238.121i q^{47} -679.498 q^{49} +202.916 q^{51} +225.336i q^{53} -129.422i q^{57} -696.447 q^{59} -15.8359 q^{61} +620.307i q^{63} +647.975i q^{67} +1240.25 q^{69} -283.587 q^{71} -631.342i q^{73} -468.090i q^{77} -468.447 q^{79} -876.454 q^{81} +189.246i q^{83} -1405.02i q^{87} +247.467 q^{89} -253.196 q^{91} +2122.23i q^{93} -997.674i q^{97} -283.971 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 6.81167i	− 1.31091i	−0.755236	−	0.655453i	$-0.772478\pi$
$3$	− 6.81167i	− 1.31091i	0.755236	−	0.655453i	$-0.227522\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 31.9765i	− 1.72657i	−0.504717	−	0.863285i	$-0.668403\pi$
$7$	− 31.9765i	− 1.72657i	0.504717	−	0.863285i	$-0.331597\pi$
$8$	0	0
$9$	−19.3988	−0.718475
$10$	0	0
$11$	14.6385	0.401244	0.200622	−	0.979669i	$-0.435704\pi$
$11$	14.6385	0.401244	0.200622	+	0.979669i	$0.435704\pi$
$12$	0	0
$13$	− 7.91818i	− 0.168931i	−0.996426	−	0.0844657i	$-0.973082\pi$
$13$	− 7.91818i	− 0.168931i	0.996426	−	0.0844657i	$-0.0269183\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	29.7894i	0.424999i	0.977161	+	0.212500i	$0.0681605\pi$
$17$	29.7894i	0.424999i	−0.977161	+	0.212500i	$0.931840\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−217.813	−2.26337
$22$	0	0
$23$	182.077i	1.65068i	0.564637	+	0.825339i	$0.309016\pi$
$23$	182.077i	1.65068i	−0.564637	+	0.825339i	$0.690984\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 51.7766i	− 0.369052i
$28$	0	0
$29$	206.266	1.32078	0.660391	−	0.750922i	$-0.270391\pi$
$29$	206.266	1.32078	0.660391	+	0.750922i	$0.270391\pi$
$30$	0	0
$31$	−311.559	−1.80508	−0.902542	−	0.430602i	$-0.858301\pi$
$31$	−311.559	−1.80508	−0.902542	+	0.430602i	$0.858301\pi$
$32$	0	0
$33$	− 99.7130i	− 0.525994i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	204.494i	0.908610i	0.890846	+	0.454305i	$0.150112\pi$
$37$	204.494i	0.908610i	−0.890846	+	0.454305i	$0.849888\pi$
$38$	0	0
$39$	−53.9360	−0.221453
$40$	0	0
$41$	−464.589	−1.76967	−0.884837	−	0.465900i	$-0.845730\pi$
$41$	−464.589	−1.76967	−0.884837	+	0.465900i	$0.845730\pi$
$42$	0	0
$43$	221.899i	0.786959i	0.919333	+	0.393480i	$0.128729\pi$
$43$	221.899i	0.786959i	−0.919333	+	0.393480i	$0.871271\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	238.121i	0.739011i	0.929229	+	0.369505i	$0.120473\pi$
$47$	238.121i	0.739011i	−0.929229	+	0.369505i	$0.879527\pi$
$48$	0	0
$49$	−679.498	−1.98104
$50$	0	0
$51$	202.916	0.557135
$52$	0	0
$53$	225.336i	0.584006i	0.956417	+	0.292003i	$0.0943217\pi$
$53$	225.336i	0.584006i	−0.956417	+	0.292003i	$0.905678\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 129.422i	− 0.300743i
$58$	0	0
$59$	−696.447	−1.53677	−0.768387	−	0.639985i	$-0.778941\pi$
$59$	−696.447	−1.53677	−0.768387	+	0.639985i	$0.778941\pi$
$60$	0	0
$61$	−15.8359	−0.0332391	−0.0166195	−	0.999862i	$-0.505290\pi$
$61$	−15.8359	−0.0332391	−0.0166195	+	0.999862i	$0.505290\pi$
$62$	0	0
$63$	620.307i	1.24050i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	647.975i	1.18153i	0.806842	+	0.590767i	$0.201175\pi$
$67$	647.975i	1.18153i	−0.806842	+	0.590767i	$0.798825\pi$
$68$	0	0
$69$	1240.25	2.16389
$70$	0	0
$71$	−283.587	−0.474022	−0.237011	−	0.971507i	$-0.576168\pi$
$71$	−283.587	−0.474022	−0.237011	+	0.971507i	$0.576168\pi$
$72$	0	0
$73$	− 631.342i	− 1.01223i	−0.862465	−	0.506116i	$-0.831081\pi$
$73$	− 631.342i	− 1.01223i	0.862465	−	0.506116i	$-0.168919\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 468.090i	− 0.692777i
$78$	0	0
$79$	−468.447	−0.667145	−0.333572	−	0.942724i	$-0.608254\pi$
$79$	−468.447	−0.667145	−0.333572	+	0.942724i	$0.608254\pi$
$80$	0	0
$81$	−876.454	−1.20227
$82$	0	0
$83$	189.246i	0.250270i	0.992140	+	0.125135i	$0.0399364\pi$
$83$	189.246i	0.250270i	−0.992140	+	0.125135i	$0.960064\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1405.02i	− 1.73142i
$88$	0	0
$89$	247.467	0.294735	0.147368	−	0.989082i	$-0.452920\pi$
$89$	247.467	0.294735	0.147368	+	0.989082i	$0.452920\pi$
$90$	0	0
$91$	−253.196	−0.291672
$92$	0	0
$93$	2122.23i	2.36630i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 997.674i	− 1.04431i	−0.852850	−	0.522157i	$-0.825128\pi$
$97$	− 997.674i	− 1.04431i	0.852850	−	0.522157i	$-0.174872\pi$
$98$	0	0
$99$	−283.971	−0.288284
$100$	0	0
$101$	1963.22	1.93414	0.967070	−	0.254512i	$-0.0819148\pi$
$101$	1963.22	1.93414	0.967070	+	0.254512i	$0.0819148\pi$
$102$	0	0
$103$	847.380i	0.810629i	0.914177	+	0.405315i	$0.132838\pi$
$103$	847.380i	0.810629i	−0.914177	+	0.405315i	$0.867162\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1153.67i	1.04234i	0.853454	+	0.521168i	$0.174503\pi$
$107$	1153.67i	1.04234i	−0.853454	+	0.521168i	$0.825497\pi$
$108$	0	0
$109$	−1462.58	−1.28522	−0.642612	−	0.766192i	$-0.722149\pi$
$109$	−1462.58	−1.28522	−0.642612	+	0.766192i	$0.722149\pi$
$110$	0	0
$111$	1392.94	1.19110
$112$	0	0
$113$	− 107.523i	− 0.0895124i	−0.998998	−	0.0447562i	$-0.985749\pi$
$113$	− 107.523i	− 0.0895124i	0.998998	−	0.0447562i	$-0.0142511\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	153.603i	0.121373i
$118$	0	0
$119$	952.562	0.733791
$120$	0	0
$121$	−1116.71	−0.839003
$122$	0	0
$123$	3164.63i	2.31988i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	426.879i	0.298263i	0.988817	+	0.149131i	$0.0476478\pi$
$127$	426.879i	0.298263i	−0.988817	+	0.149131i	$0.952352\pi$
$128$	0	0
$129$	1511.50	1.03163
$130$	0	0
$131$	−947.862	−0.632176	−0.316088	−	0.948730i	$-0.602370\pi$
$131$	−947.862	−0.632176	−0.316088	+	0.948730i	$0.602370\pi$
$132$	0	0
$133$	− 607.554i	− 0.396102i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	366.102i	0.228308i	0.993463	+	0.114154i	$0.0364157\pi$
$137$	366.102i	0.228308i	−0.993463	+	0.114154i	$0.963584\pi$
$138$	0	0
$139$	−873.537	−0.533039	−0.266519	−	0.963830i	$-0.585874\pi$
$139$	−873.537	−0.533039	−0.266519	+	0.963830i	$0.585874\pi$
$140$	0	0
$141$	1622.00	0.968774
$142$	0	0
$143$	− 115.911i	− 0.0677828i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4628.52i	2.59696i
$148$	0	0
$149$	−1286.49	−0.707339	−0.353669	−	0.935371i	$-0.615066\pi$
$149$	−1286.49	−0.707339	−0.353669	+	0.935371i	$0.615066\pi$
$150$	0	0
$151$	−3568.13	−1.92298	−0.961491	−	0.274835i	$-0.911377\pi$
$151$	−3568.13	−1.92298	−0.961491	+	0.274835i	$0.911377\pi$
$152$	0	0
$153$	− 577.880i	− 0.305352i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1502.64i	− 0.763843i	−0.924195	−	0.381922i	$-0.875262\pi$
$157$	− 1502.64i	− 0.763843i	0.924195	−	0.381922i	$-0.124738\pi$
$158$	0	0
$159$	1534.92	0.765577
$160$	0	0
$161$	5822.18	2.85001
$162$	0	0
$163$	426.394i	0.204894i	0.994738	+	0.102447i	$0.0326672\pi$
$163$	426.394i	0.204894i	−0.994738	+	0.102447i	$0.967333\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 242.170i	− 0.112213i	−0.998425	−	0.0561067i	$-0.982131\pi$
$167$	− 242.170i	− 0.112213i	0.998425	−	0.0561067i	$-0.0178687\pi$
$168$	0	0
$169$	2134.30	0.971462
$170$	0	0
$171$	−368.578	−0.164830
$172$	0	0
$173$	− 3764.36i	− 1.65433i	−0.561960	−	0.827164i	$-0.689953\pi$
$173$	− 3764.36i	− 1.65433i	0.561960	−	0.827164i	$-0.310047\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4743.97i	2.01457i
$178$	0	0
$179$	1392.73	0.581551	0.290775	−	0.956791i	$-0.406087\pi$
$179$	1392.73	0.581551	0.290775	+	0.956791i	$0.406087\pi$
$180$	0	0
$181$	1581.95	0.649644	0.324822	−	0.945775i	$-0.394696\pi$
$181$	1581.95	0.649644	0.324822	+	0.945775i	$0.394696\pi$
$182$	0	0
$183$	107.869i	0.0435733i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	436.074i	0.170529i
$188$	0	0
$189$	−1655.64	−0.637195
$190$	0	0
$191$	−4295.69	−1.62736	−0.813678	−	0.581316i	$-0.802538\pi$
$191$	−4295.69	−1.62736	−0.813678	+	0.581316i	$0.802538\pi$
$192$	0	0
$193$	− 4238.41i	− 1.58076i	−0.612615	−	0.790382i	$-0.709882\pi$
$193$	− 4238.41i	− 1.58076i	0.612615	−	0.790382i	$-0.290118\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1646.93i	0.595627i	0.954624	+	0.297814i	$0.0962574\pi$
$197$	1646.93i	0.595627i	−0.954624	+	0.297814i	$0.903743\pi$
$198$	0	0
$199$	−1414.24	−0.503782	−0.251891	−	0.967756i	$-0.581052\pi$
$199$	−1414.24	−0.503782	−0.251891	+	0.967756i	$0.581052\pi$
$200$	0	0
$201$	4413.79	1.54888
$202$	0	0
$203$	− 6595.68i	− 2.28042i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3532.08i	− 1.18597i
$208$	0	0
$209$	278.132	0.0920518
$210$	0	0
$211$	−3885.88	−1.26784	−0.633922	−	0.773397i	$-0.718556\pi$
$211$	−3885.88	−1.26784	−0.633922	+	0.773397i	$0.718556\pi$
$212$	0	0
$213$	1931.70i	0.621399i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9962.57i	3.11660i
$218$	0	0
$219$	−4300.49	−1.32694
$220$	0	0
$221$	235.878	0.0717957
$222$	0	0
$223$	4163.11i	1.25015i	0.780566	+	0.625073i	$0.214931\pi$
$223$	4163.11i	1.25015i	−0.780566	+	0.625073i	$0.785069\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	711.305i	0.207978i	0.994578	+	0.103989i	$0.0331606\pi$
$227$	711.305i	0.207978i	−0.994578	+	0.103989i	$0.966839\pi$
$228$	0	0
$229$	5100.38	1.47180	0.735900	−	0.677090i	$-0.236759\pi$
$229$	5100.38	1.47180	0.735900	+	0.677090i	$0.236759\pi$
$230$	0	0
$231$	−3188.47	−0.908165
$232$	0	0
$233$	1717.10i	0.482794i	0.970426	+	0.241397i	$0.0776056\pi$
$233$	1717.10i	0.482794i	−0.970426	+	0.241397i	$0.922394\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3190.91i	0.874564i
$238$	0	0
$239$	5152.34	1.39446	0.697232	−	0.716846i	$-0.254415\pi$
$239$	5152.34	1.39446	0.697232	+	0.716846i	$0.254415\pi$
$240$	0	0
$241$	−1319.27	−0.352622	−0.176311	−	0.984335i	$-0.556416\pi$
$241$	−1319.27	−0.352622	−0.176311	+	0.984335i	$0.556416\pi$
$242$	0	0
$243$	4572.14i	1.20701i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 150.445i	− 0.0387555i
$248$	0	0
$249$	1289.08	0.328081
$250$	0	0
$251$	4115.65	1.03497	0.517486	−	0.855692i	$-0.326868\pi$
$251$	4115.65	1.03497	0.517486	+	0.855692i	$0.326868\pi$
$252$	0	0
$253$	2665.34i	0.662326i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	908.155i	0.220425i	0.993908	+	0.110212i	$0.0351531\pi$
$257$	908.155i	0.220425i	−0.993908	+	0.110212i	$0.964847\pi$
$258$	0	0
$259$	6539.00	1.56878
$260$	0	0
$261$	−4001.32	−0.948949
$262$	0	0
$263$	− 3489.71i	− 0.818192i	−0.912491	−	0.409096i	$-0.865844\pi$
$263$	− 3489.71i	− 0.818192i	0.912491	−	0.409096i	$-0.134156\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1685.66i	− 0.386370i
$268$	0	0
$269$	5222.40	1.18370	0.591851	−	0.806048i	$-0.298398\pi$
$269$	5222.40	1.18370	0.591851	+	0.806048i	$0.298398\pi$
$270$	0	0
$271$	−5379.46	−1.20583	−0.602913	−	0.797807i	$-0.705993\pi$
$271$	−5379.46	−1.20583	−0.602913	+	0.797807i	$0.705993\pi$
$272$	0	0
$273$	1724.69i	0.382354i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	341.942i	0.0741707i	0.999312	+	0.0370853i	$0.0118073\pi$
$277$	341.942i	0.0741707i	−0.999312	+	0.0370853i	$0.988193\pi$
$278$	0	0
$279$	6043.88	1.29691
$280$	0	0
$281$	−1283.14	−0.272406	−0.136203	−	0.990681i	$-0.543490\pi$
$281$	−1283.14	−0.272406	−0.136203	+	0.990681i	$0.543490\pi$
$282$	0	0
$283$	6307.15i	1.32481i	0.749146	+	0.662405i	$0.230464\pi$
$283$	6307.15i	1.32481i	−0.749146	+	0.662405i	$0.769536\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14855.9i	3.05547i
$288$	0	0
$289$	4025.59	0.819375
$290$	0	0
$291$	−6795.82	−1.36900
$292$	0	0
$293$	− 2296.22i	− 0.457838i	−0.973446	−	0.228919i	$-0.926481\pi$
$293$	− 2296.22i	− 0.457838i	0.973446	−	0.228919i	$-0.0735190\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 757.935i	− 0.148080i
$298$	0	0
$299$	1441.72	0.278851
$300$	0	0
$301$	7095.55	1.35874
$302$	0	0
$303$	− 13372.8i	− 2.53548i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1669.11i	− 0.310297i	−0.987891	−	0.155148i	$-0.950414\pi$
$307$	− 1669.11i	− 0.310297i	0.987891	−	0.155148i	$-0.0495856\pi$
$308$	0	0
$309$	5772.07	1.06266
$310$	0	0
$311$	−1012.63	−0.184633	−0.0923165	−	0.995730i	$-0.529427\pi$
$311$	−1012.63	−0.184633	−0.0923165	+	0.995730i	$0.529427\pi$
$312$	0	0
$313$	8009.89i	1.44647i	0.690601	+	0.723236i	$0.257346\pi$
$313$	8009.89i	1.44647i	−0.690601	+	0.723236i	$0.742654\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4150.59i	− 0.735395i	−0.929945	−	0.367697i	$-0.880146\pi$
$317$	− 4150.59i	− 0.735395i	0.929945	−	0.367697i	$-0.119854\pi$
$318$	0	0
$319$	3019.44	0.529956
$320$	0	0
$321$	7858.45	1.36640
$322$	0	0
$323$	565.999i	0.0975016i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9962.58i	1.68481i
$328$	0	0
$329$	7614.28	1.27595
$330$	0	0
$331$	8764.01	1.45533	0.727664	−	0.685934i	$-0.240606\pi$
$331$	8764.01	1.45533	0.727664	+	0.685934i	$0.240606\pi$
$332$	0	0
$333$	− 3966.94i	− 0.652814i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4814.14i	− 0.778169i	−0.921202	−	0.389084i	$-0.872791\pi$
$337$	− 4814.14i	− 0.778169i	0.921202	−	0.389084i	$-0.127209\pi$
$338$	0	0
$339$	−732.411	−0.117342
$340$	0	0
$341$	−4560.77	−0.724280
$342$	0	0
$343$	10760.0i	1.69384i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5210.58i	0.806106i	0.915177	+	0.403053i	$0.132051\pi$
$347$	5210.58i	0.806106i	−0.915177	+	0.403053i	$0.867949\pi$
$348$	0	0
$349$	−9425.26	−1.44562	−0.722812	−	0.691045i	$-0.757151\pi$
$349$	−9425.26	−1.44562	−0.722812	+	0.691045i	$0.757151\pi$
$350$	0	0
$351$	−409.977	−0.0623445
$352$	0	0
$353$	− 11426.8i	− 1.72291i	−0.507836	−	0.861454i	$-0.669554\pi$
$353$	− 11426.8i	− 1.72291i	0.507836	−	0.861454i	$-0.330446\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6488.53i	− 0.961932i
$358$	0	0
$359$	11364.9	1.67080	0.835399	−	0.549645i	$-0.185237\pi$
$359$	11364.9	1.67080	0.835399	+	0.549645i	$0.185237\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	7606.68i	1.09985i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10840.4i	− 1.54187i	−0.636915	−	0.770934i	$-0.719790\pi$
$367$	− 10840.4i	− 1.54187i	0.636915	−	0.770934i	$-0.280210\pi$
$368$	0	0
$369$	9012.49	1.27147
$370$	0	0
$371$	7205.47	1.00833
$372$	0	0
$373$	− 9658.31i	− 1.34072i	−0.742037	−	0.670359i	$-0.766140\pi$
$373$	− 9658.31i	− 1.34072i	0.742037	−	0.670359i	$-0.233860\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1633.25i	− 0.223121i
$378$	0	0
$379$	−2901.55	−0.393253	−0.196626	−	0.980478i	$-0.562999\pi$
$379$	−2901.55	−0.393253	−0.196626	+	0.980478i	$0.562999\pi$
$380$	0	0
$381$	2907.76	0.390995
$382$	0	0
$383$	1832.81i	0.244522i	0.992498	+	0.122261i	$0.0390145\pi$
$383$	1832.81i	0.244522i	−0.992498	+	0.122261i	$0.960985\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4304.58i	− 0.565411i
$388$	0	0
$389$	−7372.36	−0.960909	−0.480455	−	0.877020i	$-0.659528\pi$
$389$	−7372.36	−0.960909	−0.480455	+	0.877020i	$0.659528\pi$
$390$	0	0
$391$	−5423.96	−0.701538
$392$	0	0
$393$	6456.52i	0.828724i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4041.80i	0.510963i	0.966814	+	0.255481i	$0.0822339\pi$
$397$	4041.80i	0.510963i	−0.966814	+	0.255481i	$0.917766\pi$
$398$	0	0
$399$	−4138.46	−0.519253
$400$	0	0
$401$	−5112.15	−0.636630	−0.318315	−	0.947985i	$-0.603117\pi$
$401$	−5112.15	−0.636630	−0.318315	+	0.947985i	$0.603117\pi$
$402$	0	0
$403$	2466.98i	0.304935i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2993.49i	0.364575i
$408$	0	0
$409$	3391.26	0.409992	0.204996	−	0.978763i	$-0.434282\pi$
$409$	3391.26	0.409992	0.204996	+	0.978763i	$0.434282\pi$
$410$	0	0
$411$	2493.76	0.299290
$412$	0	0
$413$	22270.0i	2.65335i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5950.24i	0.698764i
$418$	0	0
$419$	15057.1	1.75558	0.877791	−	0.479044i	$-0.159017\pi$
$419$	15057.1	1.75558	0.877791	+	0.479044i	$0.159017\pi$
$420$	0	0
$421$	3581.77	0.414643	0.207322	−	0.978273i	$-0.433525\pi$
$421$	3581.77	0.414643	0.207322	+	0.978273i	$0.433525\pi$
$422$	0	0
$423$	− 4619.27i	− 0.530961i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	506.378i	0.0573896i
$428$	0	0
$429$	−789.545	−0.0888568
$430$	0	0
$431$	−959.700	−0.107255	−0.0536277	−	0.998561i	$-0.517078\pi$
$431$	−959.700	−0.107255	−0.0536277	+	0.998561i	$0.517078\pi$
$432$	0	0
$433$	2273.23i	0.252297i	0.992011	+	0.126148i	$0.0402615\pi$
$433$	2273.23i	0.252297i	−0.992011	+	0.126148i	$0.959738\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3459.46i	0.378692i
$438$	0	0
$439$	−2185.56	−0.237611	−0.118805	−	0.992918i	$-0.537906\pi$
$439$	−2185.56	−0.237611	−0.118805	+	0.992918i	$0.537906\pi$
$440$	0	0
$441$	13181.5	1.42333
$442$	0	0
$443$	− 14688.9i	− 1.57537i	−0.616078	−	0.787686i	$-0.711279\pi$
$443$	− 14688.9i	− 1.57537i	0.616078	−	0.787686i	$-0.288721\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8763.15i	0.927255i
$448$	0	0
$449$	−3375.36	−0.354773	−0.177387	−	0.984141i	$-0.556764\pi$
$449$	−3375.36	−0.354773	−0.177387	+	0.984141i	$0.556764\pi$
$450$	0	0
$451$	−6800.91	−0.710072
$452$	0	0
$453$	24304.9i	2.52085i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2448.93i	− 0.250670i	−0.992114	−	0.125335i	$-0.959999\pi$
$457$	− 2448.93i	− 0.250670i	0.992114	−	0.125335i	$-0.0400006\pi$
$458$	0	0
$459$	1542.39	0.156847
$460$	0	0
$461$	5710.76	0.576956	0.288478	−	0.957487i	$-0.406851\pi$
$461$	5710.76	0.576956	0.288478	+	0.957487i	$0.406851\pi$
$462$	0	0
$463$	− 11404.1i	− 1.14470i	−0.820010	−	0.572349i	$-0.806032\pi$
$463$	− 11404.1i	− 1.14470i	0.820010	−	0.572349i	$-0.193968\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.4330i	0.00370920i	0.999998	+	0.00185460i	$0.000590337\pi$
$467$	37.4330i	0.00370920i	−0.999998	+	0.00185460i	$0.999410\pi$
$468$	0	0
$469$	20720.0	2.04000
$470$	0	0
$471$	−10235.5	−1.00133
$472$	0	0
$473$	3248.28i	0.315763i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4371.26i	− 0.419594i
$478$	0	0
$479$	−6620.24	−0.631496	−0.315748	−	0.948843i	$-0.602255\pi$
$479$	−6620.24	−0.631496	−0.315748	+	0.948843i	$0.602255\pi$
$480$	0	0
$481$	1619.22	0.153493
$482$	0	0
$483$	− 39658.8i	− 3.73610i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13877.3i	1.29125i	0.763654	+	0.645626i	$0.223404\pi$
$487$	13877.3i	1.29125i	−0.763654	+	0.645626i	$0.776596\pi$
$488$	0	0
$489$	2904.45	0.268597
$490$	0	0
$491$	−19304.1	−1.77430	−0.887150	−	0.461481i	$-0.847318\pi$
$491$	−19304.1	−1.77430	−0.887150	+	0.461481i	$0.847318\pi$
$492$	0	0
$493$	6144.55i	0.561332i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9068.13i	0.818433i
$498$	0	0
$499$	−9124.18	−0.818546	−0.409273	−	0.912412i	$-0.634218\pi$
$499$	−9124.18	−0.818546	−0.409273	+	0.912412i	$0.634218\pi$
$500$	0	0
$501$	−1649.58	−0.147101
$502$	0	0
$503$	1223.67i	0.108471i	0.998528	+	0.0542353i	$0.0172721\pi$
$503$	1223.67i	0.108471i	−0.998528	+	0.0542353i	$0.982728\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 14538.2i	− 1.27350i
$508$	0	0
$509$	−14494.0	−1.26215	−0.631074	−	0.775722i	$-0.717386\pi$
$509$	−14494.0	−1.26215	−0.631074	+	0.775722i	$0.717386\pi$
$510$	0	0
$511$	−20188.1	−1.74769
$512$	0	0
$513$	− 983.756i	− 0.0846664i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3485.74i	0.296524i
$518$	0	0
$519$	−25641.6	−2.16867
$520$	0	0
$521$	8128.22	0.683501	0.341750	−	0.939791i	$-0.388980\pi$
$521$	8128.22	0.683501	0.341750	+	0.939791i	$0.388980\pi$
$522$	0	0
$523$	18389.3i	1.53749i	0.639553	+	0.768747i	$0.279119\pi$
$523$	18389.3i	1.53749i	−0.639553	+	0.768747i	$0.720881\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9281.15i	− 0.767160i
$528$	0	0
$529$	−20984.9	−1.72474
$530$	0	0
$531$	13510.3	1.10413
$532$	0	0
$533$	3678.70i	0.298953i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9486.82i	− 0.762358i
$538$	0	0
$539$	−9946.87	−0.794883
$540$	0	0
$541$	−5025.77	−0.399399	−0.199699	−	0.979857i	$-0.563997\pi$
$541$	−5025.77	−0.399399	−0.199699	+	0.979857i	$0.563997\pi$
$542$	0	0
$543$	− 10775.7i	− 0.851623i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 12216.2i	− 0.954891i	−0.878661	−	0.477445i	$-0.841563\pi$
$547$	− 12216.2i	− 0.954891i	0.878661	−	0.477445i	$-0.158437\pi$
$548$	0	0
$549$	307.199	0.0238815
$550$	0	0
$551$	3919.06	0.303008
$552$	0	0
$553$	14979.3i	1.15187i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4377.09i	− 0.332968i	−0.986044	−	0.166484i	$-0.946759\pi$
$557$	− 4377.09i	− 0.332968i	0.986044	−	0.166484i	$-0.0532415\pi$
$558$	0	0
$559$	1757.03	0.132942
$560$	0	0
$561$	2970.39	0.223547
$562$	0	0
$563$	− 2452.77i	− 0.183609i	−0.995777	−	0.0918045i	$-0.970737\pi$
$563$	− 2452.77i	− 0.183609i	0.995777	−	0.0918045i	$-0.0292635\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	28025.9i	2.07580i
$568$	0	0
$569$	−15130.0	−1.11473	−0.557366	−	0.830267i	$-0.688188\pi$
$569$	−15130.0	−1.11473	−0.557366	+	0.830267i	$0.688188\pi$
$570$	0	0
$571$	−23717.3	−1.73824	−0.869122	−	0.494597i	$-0.835316\pi$
$571$	−23717.3	−1.73824	−0.869122	+	0.494597i	$0.835316\pi$
$572$	0	0
$573$	29260.8i	2.13331i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 14138.3i	− 1.02008i	−0.860150	−	0.510041i	$-0.829630\pi$
$577$	− 14138.3i	− 1.02008i	0.860150	−	0.510041i	$-0.170370\pi$
$578$	0	0
$579$	−28870.6	−2.07223
$580$	0	0
$581$	6051.42	0.432109
$582$	0	0
$583$	3298.60i	0.234329i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23783.2i	1.67230i	0.548502	+	0.836149i	$0.315198\pi$
$587$	23783.2i	1.67230i	−0.548502	+	0.836149i	$0.684802\pi$
$588$	0	0
$589$	−5919.62	−0.414115
$590$	0	0
$591$	11218.3	0.780811
$592$	0	0
$593$	− 19528.9i	− 1.35237i	−0.736731	−	0.676186i	$-0.763632\pi$
$593$	− 19528.9i	− 1.35237i	0.736731	−	0.676186i	$-0.236368\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9633.31i	0.660411i
$598$	0	0
$599$	10823.7	0.738303	0.369151	−	0.929369i	$-0.379648\pi$
$599$	10823.7	0.738303	0.369151	+	0.929369i	$0.379648\pi$
$600$	0	0
$601$	−8575.23	−0.582015	−0.291007	−	0.956721i	$-0.593990\pi$
$601$	−8575.23	−0.582015	−0.291007	+	0.956721i	$0.593990\pi$
$602$	0	0
$603$	− 12570.0i	− 0.848903i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	976.558i	0.0653003i	0.999467	+	0.0326502i	$0.0103947\pi$
$607$	976.558i	0.0653003i	−0.999467	+	0.0326502i	$0.989605\pi$
$608$	0	0
$609$	−44927.6	−2.98942
$610$	0	0
$611$	1885.48	0.124842
$612$	0	0
$613$	− 1469.78i	− 0.0968413i	−0.998827	−	0.0484207i	$-0.984581\pi$
$613$	− 1469.78i	− 0.0968413i	0.998827	−	0.0484207i	$-0.0154188\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2057.79i	− 0.134268i	−0.997744	−	0.0671341i	$-0.978614\pi$
$617$	− 2057.79i	− 0.134268i	0.997744	−	0.0671341i	$-0.0213855\pi$
$618$	0	0
$619$	18583.0	1.20665	0.603323	−	0.797497i	$-0.293843\pi$
$619$	18583.0	1.20665	0.603323	+	0.797497i	$0.293843\pi$
$620$	0	0
$621$	9427.32	0.609187
$622$	0	0
$623$	− 7913.13i	− 0.508881i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1894.55i	− 0.120671i
$628$	0	0
$629$	−6091.75	−0.386159
$630$	0	0
$631$	−20172.8	−1.27269	−0.636345	−	0.771405i	$-0.719554\pi$
$631$	−20172.8	−1.27269	−0.636345	+	0.771405i	$0.719554\pi$
$632$	0	0
$633$	26469.3i	1.66203i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5380.39i	0.334660i
$638$	0	0
$639$	5501.26	0.340573
$640$	0	0
$641$	−19883.1	−1.22517	−0.612585	−	0.790404i	$-0.709870\pi$
$641$	−19883.1	−1.22517	−0.612585	+	0.790404i	$0.709870\pi$
$642$	0	0
$643$	− 21729.3i	− 1.33269i	−0.745643	−	0.666346i	$-0.767858\pi$
$643$	− 21729.3i	− 1.33269i	0.745643	−	0.666346i	$-0.232142\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18968.7i	1.15260i	0.817237	+	0.576302i	$0.195505\pi$
$647$	18968.7i	1.15260i	−0.817237	+	0.576302i	$0.804495\pi$
$648$	0	0
$649$	−10195.0	−0.616622
$650$	0	0
$651$	67861.7	4.08558
$652$	0	0
$653$	8314.64i	0.498280i	0.968467	+	0.249140i	$0.0801480\pi$
$653$	8314.64i	0.498280i	−0.968467	+	0.249140i	$0.919852\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12247.3i	0.727264i
$658$	0	0
$659$	695.894	0.0411353	0.0205677	−	0.999788i	$-0.493453\pi$
$659$	695.894	0.0411353	0.0205677	+	0.999788i	$0.493453\pi$
$660$	0	0
$661$	−31763.8	−1.86909	−0.934546	−	0.355844i	$-0.884194\pi$
$661$	−31763.8	−1.86909	−0.934546	+	0.355844i	$0.884194\pi$
$662$	0	0
$663$	− 1606.72i	− 0.0941175i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	37556.3i	2.18019i
$668$	0	0
$669$	28357.7	1.63882
$670$	0	0
$671$	−231.815	−0.0133370
$672$	0	0
$673$	15204.8i	0.870878i	0.900218	+	0.435439i	$0.143407\pi$
$673$	15204.8i	0.870878i	−0.900218	+	0.435439i	$0.856593\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8507.78i	0.482985i	0.970403	+	0.241492i	$0.0776368\pi$
$677$	8507.78i	0.482985i	−0.970403	+	0.241492i	$0.922363\pi$
$678$	0	0
$679$	−31902.1	−1.80308
$680$	0	0
$681$	4845.17	0.272639
$682$	0	0
$683$	22495.3i	1.26026i	0.776489	+	0.630131i	$0.216999\pi$
$683$	22495.3i	1.26026i	−0.776489	+	0.630131i	$0.783001\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 34742.1i	− 1.92939i
$688$	0	0
$689$	1784.25	0.0986569
$690$	0	0
$691$	−4729.40	−0.260369	−0.130184	−	0.991490i	$-0.541557\pi$
$691$	−4729.40	−0.260369	−0.130184	+	0.991490i	$0.541557\pi$
$692$	0	0
$693$	9080.40i	0.497743i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 13839.8i	− 0.752111i
$698$	0	0
$699$	11696.3	0.632898
$700$	0	0
$701$	3185.96	0.171658	0.0858289	−	0.996310i	$-0.472646\pi$
$701$	3185.96	0.171658	0.0858289	+	0.996310i	$0.472646\pi$
$702$	0	0
$703$	3885.38i	0.208450i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 62777.1i	− 3.33943i
$708$	0	0
$709$	15222.4	0.806329	0.403165	−	0.915127i	$-0.367910\pi$
$709$	15222.4	0.806329	0.403165	+	0.915127i	$0.367910\pi$
$710$	0	0
$711$	9087.33	0.479327
$712$	0	0
$713$	− 56727.6i	− 2.97961i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 35096.0i	− 1.82801i
$718$	0	0
$719$	−12520.3	−0.649413	−0.324707	−	0.945815i	$-0.605266\pi$
$719$	−12520.3	−0.649413	−0.324707	+	0.945815i	$0.605266\pi$
$720$	0	0
$721$	27096.3	1.39961
$722$	0	0
$723$	8986.46i	0.462254i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 29136.7i	− 1.48641i	−0.669064	−	0.743204i	$-0.733305\pi$
$727$	− 29136.7i	− 1.48641i	0.669064	−	0.743204i	$-0.266695\pi$
$728$	0	0
$729$	7479.68	0.380007
$730$	0	0
$731$	−6610.23	−0.334457
$732$	0	0
$733$	− 11257.4i	− 0.567261i	−0.958934	−	0.283631i	$-0.908461\pi$
$733$	− 11257.4i	− 0.567261i	0.958934	−	0.283631i	$-0.0915389\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9485.42i	0.474084i
$738$	0	0
$739$	28012.1	1.39437	0.697186	−	0.716891i	$-0.254435\pi$
$739$	28012.1	1.39437	0.697186	+	0.716891i	$0.254435\pi$
$740$	0	0
$741$	−1024.78	−0.0508048
$742$	0	0
$743$	− 19728.6i	− 0.974119i	−0.873369	−	0.487059i	$-0.838069\pi$
$743$	− 19728.6i	− 0.974119i	0.873369	−	0.487059i	$-0.161931\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3671.15i	− 0.179813i
$748$	0	0
$749$	36890.5	1.79966
$750$	0	0
$751$	−27864.0	−1.35389	−0.676946	−	0.736033i	$-0.736697\pi$
$751$	−27864.0	−1.35389	−0.676946	+	0.736033i	$0.736697\pi$
$752$	0	0
$753$	− 28034.5i	− 1.35675i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 28073.3i	− 1.34788i	−0.738788	−	0.673938i	$-0.764601\pi$
$757$	− 28073.3i	− 1.34788i	0.738788	−	0.673938i	$-0.235399\pi$
$758$	0	0
$759$	18155.4	0.868247
$760$	0	0
$761$	−6364.06	−0.303150	−0.151575	−	0.988446i	$-0.548434\pi$
$761$	−6364.06	−0.303150	−0.151575	+	0.988446i	$0.548434\pi$
$762$	0	0
$763$	46768.1i	2.21903i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5514.59i	0.259609i
$768$	0	0
$769$	−29420.5	−1.37963	−0.689813	−	0.723988i	$-0.742307\pi$
$769$	−29420.5	−1.37963	−0.689813	+	0.723988i	$0.742307\pi$
$770$	0	0
$771$	6186.05	0.288956
$772$	0	0
$773$	− 15358.7i	− 0.714636i	−0.933983	−	0.357318i	$-0.883691\pi$
$773$	− 15358.7i	− 0.714636i	0.933983	−	0.357318i	$-0.116309\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 44541.5i	− 2.05652i
$778$	0	0
$779$	−8827.20	−0.405991
$780$	0	0
$781$	−4151.30	−0.190199
$782$	0	0
$783$	− 10679.8i	− 0.487438i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 36923.7i	− 1.67241i	−0.548417	−	0.836205i	$-0.684769\pi$
$787$	− 36923.7i	− 1.67241i	0.548417	−	0.836205i	$-0.315231\pi$
$788$	0	0
$789$	−23770.7	−1.07257
$790$	0	0
$791$	−3438.21	−0.154549
$792$	0	0
$793$	125.392i	0.00561512i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 32199.5i	− 1.43107i	−0.698576	−	0.715536i	$-0.746183\pi$
$797$	− 32199.5i	− 1.43107i	0.698576	−	0.715536i	$-0.253817\pi$
$798$	0	0
$799$	−7093.48	−0.314079
$800$	0	0
$801$	−4800.57	−0.211760
$802$	0	0
$803$	− 9241.92i	− 0.406152i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 35573.3i	− 1.55172i
$808$	0	0
$809$	−21103.2	−0.917118	−0.458559	−	0.888664i	$-0.651634\pi$
$809$	−21103.2	−0.917118	−0.458559	+	0.888664i	$0.651634\pi$
$810$	0	0
$811$	40963.8	1.77366	0.886828	−	0.462100i	$-0.152904\pi$
$811$	40963.8	1.77366	0.886828	+	0.462100i	$0.152904\pi$
$812$	0	0
$813$	36643.1i	1.58073i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4216.08i	0.180541i
$818$	0	0
$819$	4911.70	0.209559
$820$	0	0
$821$	−29667.8	−1.26116	−0.630580	−	0.776124i	$-0.717183\pi$
$821$	−29667.8	−1.26116	−0.630580	+	0.776124i	$0.717183\pi$
$822$	0	0
$823$	38960.0i	1.65013i	0.565036	+	0.825066i	$0.308862\pi$
$823$	38960.0i	1.65013i	−0.565036	+	0.825066i	$0.691138\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13349.6i	− 0.561320i	−0.959807	−	0.280660i	$-0.909447\pi$
$827$	− 13349.6i	− 0.561320i	0.959807	−	0.280660i	$-0.0905533\pi$
$828$	0	0
$829$	34207.3	1.43313	0.716567	−	0.697518i	$-0.245712\pi$
$829$	34207.3	1.43313	0.716567	+	0.697518i	$0.245712\pi$
$830$	0	0
$831$	2329.19	0.0972308
$832$	0	0
$833$	− 20241.8i	− 0.841943i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16131.5i	0.666171i
$838$	0	0
$839$	12722.3	0.523508	0.261754	−	0.965135i	$-0.415699\pi$
$839$	12722.3	0.523508	0.261754	+	0.965135i	$0.415699\pi$
$840$	0	0
$841$	18156.8	0.744465
$842$	0	0
$843$	8740.35i	0.357098i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	35708.6i	1.44860i
$848$	0	0
$849$	42962.2	1.73670
$850$	0	0
$851$	−37233.6	−1.49982
$852$	0	0
$853$	− 35044.9i	− 1.40670i	−0.710844	−	0.703350i	$-0.751687\pi$
$853$	− 35044.9i	− 1.40670i	0.710844	−	0.703350i	$-0.248313\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 23765.5i	− 0.947276i	−0.880720	−	0.473638i	$-0.842941\pi$
$857$	− 23765.5i	− 0.947276i	0.880720	−	0.473638i	$-0.157059\pi$
$858$	0	0
$859$	−868.267	−0.0344876	−0.0172438	−	0.999851i	$-0.505489\pi$
$859$	−868.267	−0.0344876	−0.0172438	+	0.999851i	$0.505489\pi$
$860$	0	0
$861$	101194.	4.00543
$862$	0	0
$863$	− 42333.1i	− 1.66980i	−0.550405	−	0.834898i	$-0.685527\pi$
$863$	− 42333.1i	− 1.66980i	0.550405	−	0.834898i	$-0.314473\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 27421.0i	− 1.07412i
$868$	0	0
$869$	−6857.39	−0.267688
$870$	0	0
$871$	5130.78	0.199598
$872$	0	0
$873$	19353.7i	0.750313i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10340.9i	0.398161i	0.979983	+	0.199081i	$0.0637956\pi$
$877$	10340.9i	0.398161i	−0.979983	+	0.199081i	$0.936204\pi$
$878$	0	0
$879$	−15641.1	−0.600182
$880$	0	0
$881$	−10277.7	−0.393036	−0.196518	−	0.980500i	$-0.562963\pi$
$881$	−10277.7	−0.393036	−0.196518	+	0.980500i	$0.562963\pi$
$882$	0	0
$883$	− 26761.7i	− 1.01994i	−0.860193	−	0.509968i	$-0.829657\pi$
$883$	− 26761.7i	− 1.01994i	0.860193	−	0.509968i	$-0.170343\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12470.6i	0.472067i	0.971745	+	0.236033i	$0.0758475\pi$
$887$	12470.6i	0.472067i	−0.971745	+	0.236033i	$0.924153\pi$
$888$	0	0
$889$	13650.1	0.514972
$890$	0	0
$891$	−12830.0	−0.482404
$892$	0	0
$893$	4524.30i	0.169541i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 9820.49i	− 0.365548i
$898$	0	0
$899$	−64264.0	−2.38412
$900$	0	0
$901$	−6712.63	−0.248202
$902$	0	0
$903$	− 48332.5i	− 1.78118i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33385.7i	1.22222i	0.791545	+	0.611111i	$0.209277\pi$
$907$	33385.7i	1.22222i	−0.791545	+	0.611111i	$0.790723\pi$
$908$	0	0
$909$	−38084.3	−1.38963
$910$	0	0
$911$	−13763.8	−0.500565	−0.250282	−	0.968173i	$-0.580523\pi$
$911$	−13763.8	−0.500565	−0.250282	+	0.968173i	$0.580523\pi$
$912$	0	0
$913$	2770.28i	0.100420i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30309.3i	1.09150i
$918$	0	0
$919$	155.072	0.00556621	0.00278310	−	0.999996i	$-0.499114\pi$
$919$	155.072	0.00556621	0.00278310	+	0.999996i	$0.499114\pi$
$920$	0	0
$921$	−11369.4	−0.406770
$922$	0	0
$923$	2245.49i	0.0800772i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 16438.2i	− 0.582417i
$928$	0	0
$929$	9104.53	0.321539	0.160770	−	0.986992i	$-0.448602\pi$
$929$	9104.53	0.321539	0.160770	+	0.986992i	$0.448602\pi$
$930$	0	0
$931$	−12910.5	−0.454483
$932$	0	0
$933$	6897.69i	0.242037i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33827.8i	1.17941i	0.807619	+	0.589705i	$0.200756\pi$
$937$	33827.8i	1.17941i	−0.807619	+	0.589705i	$0.799244\pi$
$938$	0	0
$939$	54560.7	1.89619
$940$	0	0
$941$	−8399.15	−0.290972	−0.145486	−	0.989360i	$-0.546475\pi$
$941$	−8399.15	−0.290972	−0.145486	+	0.989360i	$0.546475\pi$
$942$	0	0
$943$	− 84590.9i	− 2.92116i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 13075.6i	− 0.448679i	−0.974511	−	0.224340i	$-0.927977\pi$
$947$	− 13075.6i	− 0.448679i	0.974511	−	0.224340i	$-0.0720225\pi$
$948$	0	0
$949$	−4999.07	−0.170998
$950$	0	0
$951$	−28272.4	−0.964034
$952$	0	0
$953$	45819.4i	1.55744i	0.627374	+	0.778718i	$0.284130\pi$
$953$	45819.4i	1.55744i	−0.627374	+	0.778718i	$0.715870\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 20567.4i	− 0.694723i
$958$	0	0
$959$	11706.7	0.394189
$960$	0	0
$961$	67277.8	2.25833
$962$	0	0
$963$	− 22379.9i	− 0.748892i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 36703.4i	− 1.22058i	−0.792177	−	0.610291i	$-0.791052\pi$
$967$	− 36703.4i	− 1.22058i	0.792177	−	0.610291i	$-0.208948\pi$
$968$	0	0
$969$	3855.40	0.127815
$970$	0	0
$971$	−24412.1	−0.806819	−0.403409	−	0.915020i	$-0.632175\pi$
$971$	−24412.1	−0.806819	−0.403409	+	0.915020i	$0.632175\pi$
$972$	0	0
$973$	27932.7i	0.920329i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32264.8i	1.05654i	0.849075	+	0.528272i	$0.177160\pi$
$977$	32264.8i	1.05654i	−0.849075	+	0.528272i	$0.822840\pi$
$978$	0	0
$979$	3622.56	0.118261
$980$	0	0
$981$	28372.3	0.923401
$982$	0	0
$983$	− 19591.4i	− 0.635676i	−0.948145	−	0.317838i	$-0.897043\pi$
$983$	− 19591.4i	− 0.635676i	0.948145	−	0.317838i	$-0.102957\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 51865.9i	− 1.67266i
$988$	0	0
$989$	−40402.6	−1.29902
$990$	0	0
$991$	20684.3	0.663026	0.331513	−	0.943451i	$-0.392441\pi$
$991$	20684.3	0.663026	0.331513	+	0.943451i	$0.392441\pi$
$992$	0	0
$993$	− 59697.6i	− 1.90780i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2597.95i	0.0825255i	0.999148	+	0.0412627i	$0.0131381\pi$
$997$	2597.95i	0.0825255i	−0.999148	+	0.0412627i	$0.986862\pi$
$998$	0	0
$999$	10588.0	0.335325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.4.c.d.1749.2		8
5.2	odd	4		1900.4.a.e.1.1		4
5.3	odd	4		380.4.a.c.1.4	✓	4
5.4	even	2	inner	1900.4.c.d.1749.7		8
20.3	even	4		1520.4.a.s.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.4.a.c.1.4	✓	4	5.3	odd	4
1520.4.a.s.1.1		4	20.3	even	4
1900.4.a.e.1.1		4	5.2	odd	4
1900.4.c.d.1749.2		8	1.1	even	1	trivial
1900.4.c.d.1749.7		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)

\(f(q)\)	\(=\)	\(q-6.81167i q^{3} -31.9765i q^{7} -19.3988 q^{9} +14.6385 q^{11} -7.91818i q^{13} +29.7894i q^{17} +19.0000 q^{19} -217.813 q^{21} +182.077i q^{23} -51.7766i q^{27} +206.266 q^{29} -311.559 q^{31} -99.7130i q^{33} +204.494i q^{37} -53.9360 q^{39} -464.589 q^{41} +221.899i q^{43} +238.121i q^{47} -679.498 q^{49} +202.916 q^{51} +225.336i q^{53} -129.422i q^{57} -696.447 q^{59} -15.8359 q^{61} +620.307i q^{63} +647.975i q^{67} +1240.25 q^{69} -283.587 q^{71} -631.342i q^{73} -468.090i q^{77} -468.447 q^{79} -876.454 q^{81} +189.246i q^{83} -1405.02i q^{87} +247.467 q^{89} -253.196 q^{91} +2122.23i q^{93} -997.674i q^{97} -283.971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 208 q^{9} - 120 q^{11} + 152 q^{19} - 408 q^{21} + 328 q^{29} - 1512 q^{31} + 1752 q^{39} - 352 q^{41} + 664 q^{49} - 2488 q^{51} + 288 q^{59} - 2072 q^{61} + 3320 q^{69} + 2888 q^{71} + 2112 q^{79}+ \cdots + 8296 q^{99}+O(q^{100}) \) 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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Modular forms

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Database

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