LMFDB - Embedded newform 1900.4.c.d.1749.1

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			1749.1
Root			$4.98345 - 4.98345i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.4.c.d.1749.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.96691i q^{3} +1.49735i q^{7} -72.3392 q^{9} -48.4512 q^{11} +61.3319i q^{13} -75.6585i q^{17} +19.0000 q^{19} +14.9240 q^{21} +136.212i q^{23} +451.892i q^{27} -2.11680 q^{29} +78.1380 q^{31} +482.908i q^{33} +295.250i q^{37} +611.289 q^{39} -337.387 q^{41} -517.650i q^{43} -399.581i q^{47} +340.758 q^{49} -754.081 q^{51} +457.884i q^{53} -189.371i q^{57} +305.196 q^{59} -267.687 q^{61} -108.317i q^{63} -493.582i q^{67} +1357.61 q^{69} -127.630 q^{71} -321.806i q^{73} -72.5485i q^{77} +533.196 q^{79} +2550.81 q^{81} +841.170i q^{83} +21.0980i q^{87} +1156.42 q^{89} -91.8354 q^{91} -778.794i q^{93} -374.645i q^{97} +3504.92 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 9.96691i	− 1.91813i	−0.283182	−	0.959066i	$-0.591390\pi$
$3$	− 9.96691i	− 1.91813i	0.283182	−	0.959066i	$-0.408610\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.49735i	0.0808494i	0.999183	+	0.0404247i	$0.0128711\pi$
$7$	1.49735i	0.0808494i	−0.999183	+	0.0404247i	$0.987129\pi$
$8$	0	0
$9$	−72.3392	−2.67923
$10$	0	0
$11$	−48.4512	−1.32805	−0.664026	−	0.747709i	$-0.731154\pi$
$11$	−48.4512	−1.32805	−0.664026	+	0.747709i	$0.731154\pi$
$12$	0	0
$13$	61.3319i	1.30849i	0.756281	+	0.654246i	$0.227014\pi$
$13$	61.3319i	1.30849i	−0.756281	+	0.654246i	$0.772986\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 75.6585i	− 1.07940i	−0.841856	−	0.539702i	$-0.818537\pi$
$17$	− 75.6585i	− 1.07940i	0.841856	−	0.539702i	$-0.181463\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	14.9240	0.155080
$22$	0	0
$23$	136.212i	1.23487i	0.786620	+	0.617437i	$0.211829\pi$
$23$	136.212i	1.23487i	−0.786620	+	0.617437i	$0.788171\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	451.892i	3.22099i
$28$	0	0
$29$	−2.11680	−0.0135545	−0.00677724	−	0.999977i	$-0.502157\pi$
$29$	−2.11680	−0.0135545	−0.00677724	+	0.999977i	$0.502157\pi$
$30$	0	0
$31$	78.1380	0.452710	0.226355	−	0.974045i	$-0.427319\pi$
$31$	78.1380	0.452710	0.226355	+	0.974045i	$0.427319\pi$
$32$	0	0
$33$	482.908i	2.54738i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	295.250i	1.31186i	0.754823	+	0.655929i	$0.227723\pi$
$37$	295.250i	1.31186i	−0.754823	+	0.655929i	$0.772277\pi$
$38$	0	0
$39$	611.289	2.50986
$40$	0	0
$41$	−337.387	−1.28515	−0.642574	−	0.766224i	$-0.722133\pi$
$41$	−337.387	−1.28515	−0.642574	+	0.766224i	$0.722133\pi$
$42$	0	0
$43$	− 517.650i	− 1.83583i	−0.396771	−	0.917917i	$-0.629869\pi$
$43$	− 517.650i	− 1.83583i	0.396771	−	0.917917i	$-0.370131\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 399.581i	− 1.24010i	−0.784561	−	0.620052i	$-0.787111\pi$
$47$	− 399.581i	− 1.24010i	0.784561	−	0.620052i	$-0.212889\pi$
$48$	0	0
$49$	340.758	0.993463
$50$	0	0
$51$	−754.081	−2.07044
$52$	0	0
$53$	457.884i	1.18670i	0.804944	+	0.593351i	$0.202195\pi$
$53$	457.884i	1.18670i	−0.804944	+	0.593351i	$0.797805\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 189.371i	− 0.440050i
$58$	0	0
$59$	305.196	0.673443	0.336722	−	0.941604i	$-0.390682\pi$
$59$	305.196	0.673443	0.336722	+	0.941604i	$0.390682\pi$
$60$	0	0
$61$	−267.687	−0.561866	−0.280933	−	0.959727i	$-0.590644\pi$
$61$	−267.687	−0.561866	−0.280933	+	0.959727i	$0.590644\pi$
$62$	0	0
$63$	− 108.317i	− 0.216614i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 493.582i	− 0.900010i	−0.893026	−	0.450005i	$-0.851422\pi$
$67$	− 493.582i	− 0.900010i	0.893026	−	0.450005i	$-0.148578\pi$
$68$	0	0
$69$	1357.61	2.36865
$70$	0	0
$71$	−127.630	−0.213336	−0.106668	−	0.994295i	$-0.534018\pi$
$71$	−127.630	−0.213336	−0.106668	+	0.994295i	$0.534018\pi$
$72$	0	0
$73$	− 321.806i	− 0.515953i	−0.966151	−	0.257977i	$-0.916944\pi$
$73$	− 321.806i	− 0.515953i	0.966151	−	0.257977i	$-0.0830557\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 72.5485i	− 0.107372i
$78$	0	0
$79$	533.196	0.759358	0.379679	−	0.925118i	$-0.376034\pi$
$79$	533.196	0.759358	0.379679	+	0.925118i	$0.376034\pi$
$80$	0	0
$81$	2550.81	3.49905
$82$	0	0
$83$	841.170i	1.11241i	0.831044	+	0.556207i	$0.187744\pi$
$83$	841.170i	1.11241i	−0.831044	+	0.556207i	$0.812256\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	21.0980i	0.0259993i
$88$	0	0
$89$	1156.42	1.37730	0.688652	−	0.725092i	$-0.258203\pi$
$89$	1156.42	1.37730	0.688652	+	0.725092i	$0.258203\pi$
$90$	0	0
$91$	−91.8354	−0.105791
$92$	0	0
$93$	− 778.794i	− 0.868357i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 374.645i	− 0.392159i	−0.980588	−	0.196080i	$-0.937179\pi$
$97$	− 374.645i	− 0.392159i	0.980588	−	0.196080i	$-0.0628211\pi$
$98$	0	0
$99$	3504.92	3.55816
$100$	0	0
$101$	1476.58	1.45470	0.727351	−	0.686266i	$-0.240751\pi$
$101$	1476.58	1.45470	0.727351	+	0.686266i	$0.240751\pi$
$102$	0	0
$103$	− 2023.24i	− 1.93550i	−0.251919	−	0.967748i	$-0.581062\pi$
$103$	− 2023.24i	− 1.93550i	0.251919	−	0.967748i	$-0.418938\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	541.583i	0.489316i	0.969609	+	0.244658i	$0.0786757\pi$
$107$	541.583i	0.489316i	−0.969609	+	0.244658i	$0.921324\pi$
$108$	0	0
$109$	−1716.60	−1.50844	−0.754222	−	0.656619i	$-0.771986\pi$
$109$	−1716.60	−1.50844	−0.754222	+	0.656619i	$0.771986\pi$
$110$	0	0
$111$	2942.73	2.51632
$112$	0	0
$113$	915.575i	0.762213i	0.924531	+	0.381106i	$0.124457\pi$
$113$	915.575i	0.762213i	−0.924531	+	0.381106i	$0.875543\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4436.70i	− 3.50575i
$118$	0	0
$119$	113.287	0.0872692
$120$	0	0
$121$	1016.52	0.763724
$122$	0	0
$123$	3362.71i	2.46508i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 84.3820i	− 0.0589582i	−0.999565	−	0.0294791i	$-0.990615\pi$
$127$	− 84.3820i	− 0.0589582i	0.999565	−	0.0294791i	$-0.00938485\pi$
$128$	0	0
$129$	−5159.37	−3.52137
$130$	0	0
$131$	−1932.94	−1.28917	−0.644586	−	0.764532i	$-0.722970\pi$
$131$	−1932.94	−1.28917	−0.644586	+	0.764532i	$0.722970\pi$
$132$	0	0
$133$	28.4497i	0.0185481i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1511.12i	0.942364i	0.882036	+	0.471182i	$0.156173\pi$
$137$	1511.12i	0.942364i	−0.882036	+	0.471182i	$0.843827\pi$
$138$	0	0
$139$	2103.82	1.28377	0.641883	−	0.766803i	$-0.278154\pi$
$139$	2103.82	1.28377	0.641883	+	0.766803i	$0.278154\pi$
$140$	0	0
$141$	−3982.59	−2.37868
$142$	0	0
$143$	− 2971.60i	− 1.73775i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3396.30i	− 1.90559i
$148$	0	0
$149$	526.829	0.289661	0.144831	−	0.989456i	$-0.453736\pi$
$149$	526.829	0.289661	0.144831	+	0.989456i	$0.453736\pi$
$150$	0	0
$151$	693.542	0.373772	0.186886	−	0.982382i	$-0.440160\pi$
$151$	693.542	0.373772	0.186886	+	0.982382i	$0.440160\pi$
$152$	0	0
$153$	5473.08i	2.89197i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1022.47i	0.519759i	0.965641	+	0.259879i	$0.0836828\pi$
$157$	1022.47i	0.519759i	−0.965641	+	0.259879i	$0.916317\pi$
$158$	0	0
$159$	4563.68	2.27625
$160$	0	0
$161$	−203.957	−0.0998389
$162$	0	0
$163$	1555.34i	0.747384i	0.927553	+	0.373692i	$0.121908\pi$
$163$	1555.34i	0.747384i	−0.927553	+	0.373692i	$0.878092\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2339.95i	1.08425i	0.840296	+	0.542127i	$0.182381\pi$
$167$	2339.95i	1.08425i	−0.840296	+	0.542127i	$0.817619\pi$
$168$	0	0
$169$	−1564.60	−0.712153
$170$	0	0
$171$	−1374.45	−0.614658
$172$	0	0
$173$	− 925.676i	− 0.406808i	−0.979095	−	0.203404i	$-0.934799\pi$
$173$	− 925.676i	− 0.406808i	0.979095	−	0.203404i	$-0.0652005\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3041.86i	− 1.29175i
$178$	0	0
$179$	2947.07	1.23058	0.615292	−	0.788300i	$-0.289038\pi$
$179$	2947.07	1.23058	0.615292	+	0.788300i	$0.289038\pi$
$180$	0	0
$181$	−2771.66	−1.13821	−0.569105	−	0.822265i	$-0.692710\pi$
$181$	−2771.66	−1.13821	−0.569105	+	0.822265i	$0.692710\pi$
$182$	0	0
$183$	2668.01i	1.07773i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3665.74i	1.43351i
$188$	0	0
$189$	−676.642	−0.260415
$190$	0	0
$191$	3979.61	1.50761	0.753807	−	0.657096i	$-0.228215\pi$
$191$	3979.61	1.50761	0.753807	+	0.657096i	$0.228215\pi$
$192$	0	0
$193$	− 4349.70i	− 1.62227i	−0.584858	−	0.811135i	$-0.698850\pi$
$193$	− 4349.70i	− 1.62227i	0.584858	−	0.811135i	$-0.301150\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	138.507i	0.0500924i	0.999686	+	0.0250462i	$0.00797328\pi$
$197$	138.507i	0.0500924i	−0.999686	+	0.0250462i	$0.992027\pi$
$198$	0	0
$199$	950.363	0.338540	0.169270	−	0.985570i	$-0.445859\pi$
$199$	950.363	0.338540	0.169270	+	0.985570i	$0.445859\pi$
$200$	0	0
$201$	−4919.49	−1.72634
$202$	0	0
$203$	− 3.16960i	− 0.00109587i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 9853.46i	− 3.30851i
$208$	0	0
$209$	−920.572	−0.304676
$210$	0	0
$211$	224.573	0.0732714	0.0366357	−	0.999329i	$-0.488336\pi$
$211$	224.573	0.0732714	0.0366357	+	0.999329i	$0.488336\pi$
$212$	0	0
$213$	1272.07i	0.409206i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	117.000i	0.0366013i
$218$	0	0
$219$	−3207.41	−0.989666
$220$	0	0
$221$	4640.28	1.41239
$222$	0	0
$223$	1323.38i	0.397399i	0.980060	+	0.198699i	$0.0636718\pi$
$223$	1323.38i	0.397399i	−0.980060	+	0.198699i	$0.936328\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3205.58i	0.937277i	0.883390	+	0.468638i	$0.155255\pi$
$227$	3205.58i	0.937277i	−0.883390	+	0.468638i	$0.844745\pi$
$228$	0	0
$229$	4464.24	1.28823	0.644117	−	0.764927i	$-0.277225\pi$
$229$	4464.24	1.28823	0.644117	+	0.764927i	$0.277225\pi$
$230$	0	0
$231$	−723.084	−0.205954
$232$	0	0
$233$	2540.52i	0.714313i	0.934045	+	0.357156i	$0.116254\pi$
$233$	2540.52i	0.714313i	−0.934045	+	0.357156i	$0.883746\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 5314.32i	− 1.45655i
$238$	0	0
$239$	−5229.76	−1.41542	−0.707709	−	0.706504i	$-0.750271\pi$
$239$	−5229.76	−1.41542	−0.707709	+	0.706504i	$0.750271\pi$
$240$	0	0
$241$	−2405.13	−0.642856	−0.321428	−	0.946934i	$-0.604163\pi$
$241$	−2405.13	−0.642856	−0.321428	+	0.946934i	$0.604163\pi$
$242$	0	0
$243$	− 13222.6i	− 3.49065i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1165.31i	0.300189i
$248$	0	0
$249$	8383.86	2.13376
$250$	0	0
$251$	1319.69	0.331865	0.165933	−	0.986137i	$-0.446937\pi$
$251$	1319.69	0.331865	0.165933	+	0.986137i	$0.446937\pi$
$252$	0	0
$253$	− 6599.62i	− 1.63998i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4340.90i	1.05361i	0.849986	+	0.526806i	$0.176610\pi$
$257$	4340.90i	1.05361i	−0.849986	+	0.526806i	$0.823390\pi$
$258$	0	0
$259$	−442.093	−0.106063
$260$	0	0
$261$	153.128	0.0363156
$262$	0	0
$263$	3283.43i	0.769828i	0.922952	+	0.384914i	$0.125769\pi$
$263$	3283.43i	0.769828i	−0.922952	+	0.384914i	$0.874231\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 11525.9i	− 2.64185i
$268$	0	0
$269$	5440.61	1.23316	0.616580	−	0.787293i	$-0.288518\pi$
$269$	5440.61	1.23316	0.616580	+	0.787293i	$0.288518\pi$
$270$	0	0
$271$	116.693	0.0261572	0.0130786	−	0.999914i	$-0.495837\pi$
$271$	116.693	0.0261572	0.0130786	+	0.999914i	$0.495837\pi$
$272$	0	0
$273$	915.315i	0.202921i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7382.82i	− 1.60141i	−0.599059	−	0.800705i	$-0.704458\pi$
$277$	− 7382.82i	− 1.60141i	0.599059	−	0.800705i	$-0.295542\pi$
$278$	0	0
$279$	−5652.44	−1.21291
$280$	0	0
$281$	−4969.40	−1.05498	−0.527490	−	0.849561i	$-0.676867\pi$
$281$	−4969.40	−1.05498	−0.527490	+	0.849561i	$0.676867\pi$
$282$	0	0
$283$	− 4452.88i	− 0.935323i	−0.883908	−	0.467661i	$-0.845097\pi$
$283$	− 4452.88i	− 0.935323i	0.883908	−	0.467661i	$-0.154903\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 505.188i	− 0.103903i
$288$	0	0
$289$	−811.206	−0.165114
$290$	0	0
$291$	−3734.06	−0.752214
$292$	0	0
$293$	− 2780.68i	− 0.554434i	−0.960807	−	0.277217i	$-0.910588\pi$
$293$	− 2780.68i	− 0.554434i	0.960807	−	0.277217i	$-0.0894121\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 21894.7i	− 4.27764i
$298$	0	0
$299$	−8354.12	−1.61582
$300$	0	0
$301$	775.104	0.148426
$302$	0	0
$303$	− 14716.9i	− 2.79031i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6687.50i	1.24324i	0.783317	+	0.621622i	$0.213526\pi$
$307$	6687.50i	1.24324i	−0.783317	+	0.621622i	$0.786474\pi$
$308$	0	0
$309$	−20165.5	−3.71254
$310$	0	0
$311$	6748.60	1.23048	0.615238	−	0.788341i	$-0.289060\pi$
$311$	6748.60	1.23048	0.615238	+	0.788341i	$0.289060\pi$
$312$	0	0
$313$	− 8514.76i	− 1.53765i	−0.639462	−	0.768823i	$-0.720843\pi$
$313$	− 8514.76i	− 1.53765i	0.639462	−	0.768823i	$-0.279157\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2191.06i	− 0.388209i	−0.980981	−	0.194105i	$-0.937820\pi$
$317$	− 2191.06i	− 0.388209i	0.980981	−	0.194105i	$-0.0621802\pi$
$318$	0	0
$319$	102.562	0.0180011
$320$	0	0
$321$	5397.91	0.938573
$322$	0	0
$323$	− 1437.51i	− 0.247632i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17109.2i	2.89340i
$328$	0	0
$329$	598.314	0.100262
$330$	0	0
$331$	1933.10	0.321005	0.160502	−	0.987035i	$-0.448689\pi$
$331$	1933.10	0.321005	0.160502	+	0.987035i	$0.448689\pi$
$332$	0	0
$333$	− 21358.1i	− 3.51477i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10819.2i	1.74884i	0.485171	+	0.874419i	$0.338757\pi$
$337$	10819.2i	1.74884i	−0.485171	+	0.874419i	$0.661243\pi$
$338$	0	0
$339$	9125.45	1.46203
$340$	0	0
$341$	−3785.88	−0.601222
$342$	0	0
$343$	1023.83i	0.161170i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1223.35i	− 0.189260i	−0.995513	−	0.0946298i	$-0.969833\pi$
$347$	− 1223.35i	− 0.189260i	0.995513	−	0.0946298i	$-0.0301667\pi$
$348$	0	0
$349$	−8569.30	−1.31434	−0.657169	−	0.753743i	$-0.728246\pi$
$349$	−8569.30	−1.31434	−0.657169	+	0.753743i	$0.728246\pi$
$350$	0	0
$351$	−27715.4	−4.21464
$352$	0	0
$353$	1606.77i	0.242265i	0.992636	+	0.121133i	$0.0386526\pi$
$353$	1606.77i	0.242265i	−0.992636	+	0.121133i	$0.961347\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1129.12i	− 0.167394i
$358$	0	0
$359$	6351.38	0.933740	0.466870	−	0.884326i	$-0.345382\pi$
$359$	6351.38	0.933740	0.466870	+	0.884326i	$0.345382\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	− 10131.5i	− 1.46492i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4283.51i	− 0.609258i	−0.952471	−	0.304629i	$-0.901468\pi$
$367$	− 4283.51i	− 0.609258i	0.952471	−	0.304629i	$-0.0985324\pi$
$368$	0	0
$369$	24406.3	3.44321
$370$	0	0
$371$	−685.613	−0.0959441
$372$	0	0
$373$	− 5445.18i	− 0.755873i	−0.925832	−	0.377936i	$-0.876634\pi$
$373$	− 5445.18i	− 0.755873i	0.925832	−	0.377936i	$-0.123366\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 129.827i	− 0.0177359i
$378$	0	0
$379$	5928.19	0.803458	0.401729	−	0.915759i	$-0.368409\pi$
$379$	5928.19	0.803458	0.401729	+	0.915759i	$0.368409\pi$
$380$	0	0
$381$	−841.028	−0.113090
$382$	0	0
$383$	423.702i	0.0565279i	0.999600	+	0.0282639i	$0.00899789\pi$
$383$	423.702i	0.0565279i	−0.999600	+	0.0282639i	$0.991002\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	37446.4i	4.91863i
$388$	0	0
$389$	10967.9	1.42955	0.714775	−	0.699355i	$-0.246529\pi$
$389$	10967.9	1.42955	0.714775	+	0.699355i	$0.246529\pi$
$390$	0	0
$391$	10305.6	1.33293
$392$	0	0
$393$	19265.4i	2.47280i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 8696.10i	− 1.09936i	−0.835376	−	0.549679i	$-0.814750\pi$
$397$	− 8696.10i	− 1.09936i	0.835376	−	0.549679i	$-0.185250\pi$
$398$	0	0
$399$	283.555	0.0355778
$400$	0	0
$401$	−8952.64	−1.11490	−0.557449	−	0.830211i	$-0.688220\pi$
$401$	−8952.64	−1.11490	−0.557449	+	0.830211i	$0.688220\pi$
$402$	0	0
$403$	4792.35i	0.592367i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 14305.2i	− 1.74222i
$408$	0	0
$409$	6303.71	0.762098	0.381049	−	0.924555i	$-0.375563\pi$
$409$	6303.71	0.762098	0.381049	+	0.924555i	$0.375563\pi$
$410$	0	0
$411$	15061.2	1.80758
$412$	0	0
$413$	456.986i	0.0544475i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20968.5i	− 2.46243i
$418$	0	0
$419$	6471.76	0.754573	0.377287	−	0.926097i	$-0.376857\pi$
$419$	6471.76	0.754573	0.377287	+	0.926097i	$0.376857\pi$
$420$	0	0
$421$	−12364.7	−1.43140	−0.715700	−	0.698408i	$-0.753892\pi$
$421$	−12364.7	−1.43140	−0.715700	+	0.698408i	$0.753892\pi$
$422$	0	0
$423$	28905.4i	3.32253i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 400.822i	− 0.0454265i
$428$	0	0
$429$	−29617.7	−3.33323
$430$	0	0
$431$	4698.02	0.525048	0.262524	−	0.964925i	$-0.415445\pi$
$431$	4698.02	0.525048	0.262524	+	0.964925i	$0.415445\pi$
$432$	0	0
$433$	16146.8i	1.79207i	0.443982	+	0.896036i	$0.353565\pi$
$433$	16146.8i	1.79207i	−0.443982	+	0.896036i	$0.646435\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2588.02i	0.283300i
$438$	0	0
$439$	6858.74	0.745671	0.372836	−	0.927897i	$-0.378385\pi$
$439$	6858.74	0.745671	0.372836	+	0.927897i	$0.378385\pi$
$440$	0	0
$441$	−24650.2	−2.66172
$442$	0	0
$443$	6414.66i	0.687968i	0.938975	+	0.343984i	$0.111777\pi$
$443$	6414.66i	0.687968i	−0.938975	+	0.343984i	$0.888223\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5250.86i	− 0.555608i
$448$	0	0
$449$	−12839.6	−1.34953	−0.674765	−	0.738033i	$-0.735755\pi$
$449$	−12839.6	−1.34953	−0.674765	+	0.738033i	$0.735755\pi$
$450$	0	0
$451$	16346.8	1.70674
$452$	0	0
$453$	− 6912.47i	− 0.716945i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7033.72i	− 0.719964i	−0.932959	−	0.359982i	$-0.882783\pi$
$457$	− 7033.72i	− 0.719964i	0.932959	−	0.359982i	$-0.117217\pi$
$458$	0	0
$459$	34189.5	3.47675
$460$	0	0
$461$	7472.59	0.754953	0.377477	−	0.926019i	$-0.376792\pi$
$461$	7472.59	0.754953	0.377477	+	0.926019i	$0.376792\pi$
$462$	0	0
$463$	− 15219.6i	− 1.52768i	−0.645406	−	0.763840i	$-0.723312\pi$
$463$	− 15219.6i	− 1.52768i	0.645406	−	0.763840i	$-0.276688\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18895.9i	1.87237i	0.351503	+	0.936187i	$0.385671\pi$
$467$	18895.9i	1.87237i	−0.351503	+	0.936187i	$0.614329\pi$
$468$	0	0
$469$	739.066	0.0727653
$470$	0	0
$471$	10190.9	0.996966
$472$	0	0
$473$	25080.8i	2.43809i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 33123.0i	− 3.17945i
$478$	0	0
$479$	−14765.4	−1.40845	−0.704224	−	0.709977i	$-0.748705\pi$
$479$	−14765.4	−1.40845	−0.704224	+	0.709977i	$0.748705\pi$
$480$	0	0
$481$	−18108.2	−1.71656
$482$	0	0
$483$	2032.82i	0.191504i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7099.78i	− 0.660619i	−0.943873	−	0.330310i	$-0.892847\pi$
$487$	− 7099.78i	− 0.660619i	0.943873	−	0.330310i	$-0.107153\pi$
$488$	0	0
$489$	15501.9	1.43358
$490$	0	0
$491$	561.486	0.0516080	0.0258040	−	0.999667i	$-0.491785\pi$
$491$	561.486	0.0516080	0.0258040	+	0.999667i	$0.491785\pi$
$492$	0	0
$493$	160.154i	0.0146308i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 191.106i	− 0.0172481i
$498$	0	0
$499$	5679.69	0.509535	0.254767	−	0.967002i	$-0.418001\pi$
$499$	5679.69	0.509535	0.254767	+	0.967002i	$0.418001\pi$
$500$	0	0
$501$	23322.0	2.07974
$502$	0	0
$503$	9243.46i	0.819375i	0.912226	+	0.409688i	$0.134362\pi$
$503$	9243.46i	0.819375i	−0.912226	+	0.409688i	$0.865638\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	15594.2i	1.36600i
$508$	0	0
$509$	11154.6	0.971353	0.485676	−	0.874139i	$-0.338573\pi$
$509$	11154.6	0.971353	0.485676	+	0.874139i	$0.338573\pi$
$510$	0	0
$511$	481.857	0.0417145
$512$	0	0
$513$	8585.95i	0.738945i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	19360.2i	1.64692i
$518$	0	0
$519$	−9226.12	−0.780312
$520$	0	0
$521$	6969.02	0.586024	0.293012	−	0.956109i	$-0.405342\pi$
$521$	6969.02	0.586024	0.293012	+	0.956109i	$0.405342\pi$
$522$	0	0
$523$	18107.3i	1.51391i	0.653466	+	0.756956i	$0.273314\pi$
$523$	18107.3i	1.51391i	−0.653466	+	0.756956i	$0.726686\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5911.80i	− 0.488657i
$528$	0	0
$529$	−6386.64	−0.524915
$530$	0	0
$531$	−22077.7	−1.80431
$532$	0	0
$533$	− 20692.6i	− 1.68161i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 29373.2i	− 2.36042i
$538$	0	0
$539$	−16510.1	−1.31937
$540$	0	0
$541$	12301.8	0.977629	0.488815	−	0.872388i	$-0.337429\pi$
$541$	12301.8	0.977629	0.488815	+	0.872388i	$0.337429\pi$
$542$	0	0
$543$	27624.9i	2.18324i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9010.92i	0.704350i	0.935934	+	0.352175i	$0.114558\pi$
$547$	9010.92i	0.704350i	−0.935934	+	0.352175i	$0.885442\pi$
$548$	0	0
$549$	19364.3	1.50537
$550$	0	0
$551$	−40.2192	−0.00310961
$552$	0	0
$553$	798.382i	0.0613936i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23216.7i	− 1.76611i	−0.469270	−	0.883055i	$-0.655483\pi$
$557$	− 23216.7i	− 1.76611i	0.469270	−	0.883055i	$-0.344517\pi$
$558$	0	0
$559$	31748.5	2.40218
$560$	0	0
$561$	36536.1	2.74965
$562$	0	0
$563$	− 8237.79i	− 0.616664i	−0.951279	−	0.308332i	$-0.900229\pi$
$563$	− 8237.79i	− 0.616664i	0.951279	−	0.308332i	$-0.0997707\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3819.46i	0.282896i
$568$	0	0
$569$	1168.46	0.0860886	0.0430443	−	0.999073i	$-0.486294\pi$
$569$	1168.46	0.0860886	0.0430443	+	0.999073i	$0.486294\pi$
$570$	0	0
$571$	18890.0	1.38445	0.692227	−	0.721680i	$-0.256630\pi$
$571$	18890.0	1.38445	0.692227	+	0.721680i	$0.256630\pi$
$572$	0	0
$573$	− 39664.4i	− 2.89180i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15149.9i	1.09307i	0.837437	+	0.546534i	$0.184053\pi$
$577$	15149.9i	1.09307i	−0.837437	+	0.546534i	$0.815947\pi$
$578$	0	0
$579$	−43353.1	−3.11173
$580$	0	0
$581$	−1259.53	−0.0899381
$582$	0	0
$583$	− 22185.0i	− 1.57600i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 25778.5i	− 1.81260i	−0.422640	−	0.906298i	$-0.638897\pi$
$587$	− 25778.5i	− 1.81260i	0.422640	−	0.906298i	$-0.361103\pi$
$588$	0	0
$589$	1484.62	0.103859
$590$	0	0
$591$	1380.48	0.0960838
$592$	0	0
$593$	2121.28i	0.146898i	0.997299	+	0.0734490i	$0.0234006\pi$
$593$	2121.28i	0.146898i	−0.997299	+	0.0734490i	$0.976599\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 9472.18i	− 0.649365i
$598$	0	0
$599$	−7269.05	−0.495836	−0.247918	−	0.968781i	$-0.579746\pi$
$599$	−7269.05	−0.495836	−0.247918	+	0.968781i	$0.579746\pi$
$600$	0	0
$601$	−907.680	−0.0616057	−0.0308029	−	0.999525i	$-0.509806\pi$
$601$	−907.680	−0.0616057	−0.0308029	+	0.999525i	$0.509806\pi$
$602$	0	0
$603$	35705.4i	2.41134i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11320.3i	0.756966i	0.925608	+	0.378483i	$0.123554\pi$
$607$	11320.3i	0.756966i	−0.925608	+	0.378483i	$0.876446\pi$
$608$	0	0
$609$	−31.5911	−0.00210203
$610$	0	0
$611$	24507.1	1.62267
$612$	0	0
$613$	14556.7i	0.959117i	0.877510	+	0.479558i	$0.159203\pi$
$613$	14556.7i	0.959117i	−0.877510	+	0.479558i	$0.840797\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3907.72i	− 0.254974i	−0.991840	−	0.127487i	$-0.959309\pi$
$617$	− 3907.72i	− 0.254974i	0.991840	−	0.127487i	$-0.0406911\pi$
$618$	0	0
$619$	25484.0	1.65475	0.827375	−	0.561650i	$-0.189833\pi$
$619$	25484.0	1.65475	0.827375	+	0.561650i	$0.189833\pi$
$620$	0	0
$621$	−61553.0	−3.97752
$622$	0	0
$623$	1731.57i	0.111354i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9175.26i	0.584409i
$628$	0	0
$629$	22338.1	1.41603
$630$	0	0
$631$	−14851.8	−0.936992	−0.468496	−	0.883466i	$-0.655204\pi$
$631$	−14851.8	−0.936992	−0.468496	+	0.883466i	$0.655204\pi$
$632$	0	0
$633$	− 2238.30i	− 0.140544i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	20899.3i	1.29994i
$638$	0	0
$639$	9232.63	0.571576
$640$	0	0
$641$	19959.4	1.22987	0.614937	−	0.788576i	$-0.289181\pi$
$641$	19959.4	1.22987	0.614937	+	0.788576i	$0.289181\pi$
$642$	0	0
$643$	27074.1i	1.66050i	0.557393	+	0.830249i	$0.311802\pi$
$643$	27074.1i	1.66050i	−0.557393	+	0.830249i	$0.688198\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6870.00i	0.417446i	0.977975	+	0.208723i	$0.0669307\pi$
$647$	6870.00i	0.417446i	−0.977975	+	0.208723i	$0.933069\pi$
$648$	0	0
$649$	−14787.1	−0.894368
$650$	0	0
$651$	1166.13	0.0702062
$652$	0	0
$653$	− 20974.4i	− 1.25696i	−0.777827	−	0.628479i	$-0.783678\pi$
$653$	− 20974.4i	− 1.25696i	0.777827	−	0.628479i	$-0.216322\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	23279.2i	1.38236i
$658$	0	0
$659$	−19828.8	−1.17211	−0.586054	−	0.810272i	$-0.699319\pi$
$659$	−19828.8	−1.17211	−0.586054	+	0.810272i	$0.699319\pi$
$660$	0	0
$661$	17849.1	1.05030	0.525150	−	0.851010i	$-0.324009\pi$
$661$	17849.1	1.05030	0.525150	+	0.851010i	$0.324009\pi$
$662$	0	0
$663$	− 46249.2i	− 2.70916i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 288.333i	− 0.0167381i
$668$	0	0
$669$	13190.0	0.762264
$670$	0	0
$671$	12969.8	0.746188
$672$	0	0
$673$	− 16613.9i	− 0.951590i	−0.879556	−	0.475795i	$-0.842160\pi$
$673$	− 16613.9i	− 0.951590i	0.879556	−	0.475795i	$-0.157840\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28743.9i	1.63178i	0.578204	+	0.815892i	$0.303754\pi$
$677$	28743.9i	1.63178i	−0.578204	+	0.815892i	$0.696246\pi$
$678$	0	0
$679$	560.976	0.0317059
$680$	0	0
$681$	31949.7	1.79782
$682$	0	0
$683$	28596.8i	1.60209i	0.598605	+	0.801045i	$0.295722\pi$
$683$	28596.8i	1.60209i	−0.598605	+	0.801045i	$0.704278\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 44494.7i	− 2.47100i
$688$	0	0
$689$	−28082.9	−1.55279
$690$	0	0
$691$	8506.40	0.468305	0.234152	−	0.972200i	$-0.424769\pi$
$691$	8506.40	0.468305	0.234152	+	0.972200i	$0.424769\pi$
$692$	0	0
$693$	5248.10i	0.287675i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	25526.2i	1.38719i
$698$	0	0
$699$	25321.1	1.37015
$700$	0	0
$701$	16521.5	0.890172	0.445086	−	0.895488i	$-0.353173\pi$
$701$	16521.5	0.890172	0.445086	+	0.895488i	$0.353173\pi$
$702$	0	0
$703$	5609.74i	0.300961i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2210.96i	0.117612i
$708$	0	0
$709$	137.576	0.00728742	0.00364371	−	0.999993i	$-0.498840\pi$
$709$	137.576	0.00728742	0.00364371	+	0.999993i	$0.498840\pi$
$710$	0	0
$711$	−38571.0	−2.03449
$712$	0	0
$713$	10643.3i	0.559040i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	52124.5i	2.71496i
$718$	0	0
$719$	1636.71	0.0848944	0.0424472	−	0.999099i	$-0.486485\pi$
$719$	1636.71	0.0848944	0.0424472	+	0.999099i	$0.486485\pi$
$720$	0	0
$721$	3029.51	0.156484
$722$	0	0
$723$	23971.7i	1.23308i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	38694.7i	1.97401i	0.160683	+	0.987006i	$0.448630\pi$
$727$	38694.7i	1.97401i	−0.160683	+	0.987006i	$0.551370\pi$
$728$	0	0
$729$	−62916.4	−3.19648
$730$	0	0
$731$	−39164.6	−1.98161
$732$	0	0
$733$	8097.37i	0.408026i	0.978968	+	0.204013i	$0.0653985\pi$
$733$	8097.37i	0.408026i	−0.978968	+	0.204013i	$0.934601\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23914.6i	1.19526i
$738$	0	0
$739$	32818.8	1.63364	0.816821	−	0.576892i	$-0.195735\pi$
$739$	32818.8	1.63364	0.816821	+	0.576892i	$0.195735\pi$
$740$	0	0
$741$	11614.5	0.575802
$742$	0	0
$743$	27374.0i	1.35162i	0.737075	+	0.675811i	$0.236207\pi$
$743$	27374.0i	1.35162i	−0.737075	+	0.675811i	$0.763793\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 60849.6i	− 2.98042i
$748$	0	0
$749$	−810.940	−0.0395609
$750$	0	0
$751$	7836.90	0.380789	0.190395	−	0.981708i	$-0.439023\pi$
$751$	7836.90	0.380789	0.190395	+	0.981708i	$0.439023\pi$
$752$	0	0
$753$	− 13153.2i	− 0.636561i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34474.9i	1.65523i	0.561293	+	0.827617i	$0.310304\pi$
$757$	34474.9i	1.65523i	−0.561293	+	0.827617i	$0.689696\pi$
$758$	0	0
$759$	−65777.8	−3.14570
$760$	0	0
$761$	3521.23	0.167733	0.0838664	−	0.996477i	$-0.473273\pi$
$761$	3521.23	0.167733	0.0838664	+	0.996477i	$0.473273\pi$
$762$	0	0
$763$	− 2570.35i	− 0.121957i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18718.3i	0.881196i
$768$	0	0
$769$	−10693.7	−0.501463	−0.250731	−	0.968057i	$-0.580671\pi$
$769$	−10693.7	−0.501463	−0.250731	+	0.968057i	$0.580671\pi$
$770$	0	0
$771$	43265.4	2.02097
$772$	0	0
$773$	6642.81i	0.309088i	0.987986	+	0.154544i	$0.0493909\pi$
$773$	6642.81i	0.309088i	−0.987986	+	0.154544i	$0.950609\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4406.30i	0.203443i
$778$	0	0
$779$	−6410.36	−0.294833
$780$	0	0
$781$	6183.80	0.283321
$782$	0	0
$783$	− 956.566i	− 0.0436588i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 15297.6i	− 0.692886i	−0.938071	−	0.346443i	$-0.887389\pi$
$787$	− 15297.6i	− 0.692886i	0.938071	−	0.346443i	$-0.112611\pi$
$788$	0	0
$789$	32725.6	1.47663
$790$	0	0
$791$	−1370.94	−0.0616245
$792$	0	0
$793$	− 16417.8i	− 0.735198i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18020.7i	0.800910i	0.916316	+	0.400455i	$0.131148\pi$
$797$	18020.7i	0.800910i	−0.916316	+	0.400455i	$0.868852\pi$
$798$	0	0
$799$	−30231.7	−1.33857
$800$	0	0
$801$	−83654.4	−3.69012
$802$	0	0
$803$	15591.9i	0.685213i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 54226.1i	− 2.36536i
$808$	0	0
$809$	36204.2	1.57339	0.786695	−	0.617342i	$-0.211790\pi$
$809$	36204.2	1.57339	0.786695	+	0.617342i	$0.211790\pi$
$810$	0	0
$811$	−9798.47	−0.424255	−0.212128	−	0.977242i	$-0.568039\pi$
$811$	−9798.47	−0.424255	−0.212128	+	0.977242i	$0.568039\pi$
$812$	0	0
$813$	− 1163.07i	− 0.0501729i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9835.35i	− 0.421169i
$818$	0	0
$819$	6643.31	0.283438
$820$	0	0
$821$	−2690.03	−0.114351	−0.0571757	−	0.998364i	$-0.518210\pi$
$821$	−2690.03	−0.114351	−0.0571757	+	0.998364i	$0.518210\pi$
$822$	0	0
$823$	− 16582.9i	− 0.702361i	−0.936308	−	0.351180i	$-0.885780\pi$
$823$	− 16582.9i	− 0.702361i	0.936308	−	0.351180i	$-0.114220\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2306.52i	0.0969837i	0.998824	+	0.0484919i	$0.0154415\pi$
$827$	2306.52i	0.0969837i	−0.998824	+	0.0484919i	$0.984559\pi$
$828$	0	0
$829$	−21059.0	−0.882278	−0.441139	−	0.897439i	$-0.645425\pi$
$829$	−21059.0	−0.882278	−0.441139	+	0.897439i	$0.645425\pi$
$830$	0	0
$831$	−73583.9	−3.07172
$832$	0	0
$833$	− 25781.2i	− 1.07235i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	35309.9i	1.45817i
$838$	0	0
$839$	36496.4	1.50178	0.750891	−	0.660426i	$-0.229624\pi$
$839$	36496.4	1.50178	0.750891	+	0.660426i	$0.229624\pi$
$840$	0	0
$841$	−24384.5	−0.999816
$842$	0	0
$843$	49529.5i	2.02359i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1522.08i	0.0617466i
$848$	0	0
$849$	−44381.5	−1.79407
$850$	0	0
$851$	−40216.5	−1.61998
$852$	0	0
$853$	− 37876.2i	− 1.52035i	−0.649721	−	0.760173i	$-0.725114\pi$
$853$	− 37876.2i	− 1.52035i	0.649721	−	0.760173i	$-0.274886\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 12040.9i	− 0.479939i	−0.970780	−	0.239969i	$-0.922863\pi$
$857$	− 12040.9i	− 0.479939i	0.970780	−	0.239969i	$-0.0771374\pi$
$858$	0	0
$859$	−6165.58	−0.244898	−0.122449	−	0.992475i	$-0.539075\pi$
$859$	−6165.58	−0.244898	−0.122449	+	0.992475i	$0.539075\pi$
$860$	0	0
$861$	−5035.16	−0.199300
$862$	0	0
$863$	− 14478.7i	− 0.571103i	−0.958363	−	0.285552i	$-0.907823\pi$
$863$	− 14478.7i	− 0.571103i	0.958363	−	0.285552i	$-0.0921768\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8085.22i	0.316711i
$868$	0	0
$869$	−25834.0	−1.00847
$870$	0	0
$871$	30272.3	1.17766
$872$	0	0
$873$	27101.6i	1.05069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 17745.5i	− 0.683265i	−0.939834	−	0.341633i	$-0.889020\pi$
$877$	− 17745.5i	− 0.683265i	0.939834	−	0.341633i	$-0.110980\pi$
$878$	0	0
$879$	−27714.8	−1.06348
$880$	0	0
$881$	−37497.1	−1.43395	−0.716974	−	0.697100i	$-0.754473\pi$
$881$	−37497.1	−1.43395	−0.716974	+	0.697100i	$0.754473\pi$
$882$	0	0
$883$	19972.6i	0.761190i	0.924742	+	0.380595i	$0.124281\pi$
$883$	19972.6i	0.761190i	−0.924742	+	0.380595i	$0.875719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27566.7i	1.04352i	0.853094	+	0.521758i	$0.174724\pi$
$887$	27566.7i	1.04352i	−0.853094	+	0.521758i	$0.825276\pi$
$888$	0	0
$889$	126.350	0.00476674
$890$	0	0
$891$	−123590.	−4.64692
$892$	0	0
$893$	− 7592.04i	− 0.284499i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	83264.8i	3.09936i
$898$	0	0
$899$	−165.403	−0.00613625
$900$	0	0
$901$	34642.8	1.28093
$902$	0	0
$903$	− 7725.39i	− 0.284701i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 18665.0i	− 0.683309i	−0.939826	−	0.341654i	$-0.889013\pi$
$907$	− 18665.0i	− 0.683309i	0.939826	−	0.341654i	$-0.110987\pi$
$908$	0	0
$909$	−106814.	−3.89748
$910$	0	0
$911$	41576.9	1.51208	0.756039	−	0.654526i	$-0.227132\pi$
$911$	41576.9	1.51208	0.756039	+	0.654526i	$0.227132\pi$
$912$	0	0
$913$	− 40755.7i	− 1.47735i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2894.29i	− 0.104229i
$918$	0	0
$919$	26761.7	0.960595	0.480298	−	0.877106i	$-0.340529\pi$
$919$	26761.7	0.960595	0.480298	+	0.877106i	$0.340529\pi$
$920$	0	0
$921$	66653.7	2.38471
$922$	0	0
$923$	− 7827.76i	− 0.279148i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	146360.i	5.18564i
$928$	0	0
$929$	6900.64	0.243706	0.121853	−	0.992548i	$-0.461116\pi$
$929$	6900.64	0.243706	0.121853	+	0.992548i	$0.461116\pi$
$930$	0	0
$931$	6474.40	0.227916
$932$	0	0
$933$	− 67262.7i	− 2.36022i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 9239.89i	− 0.322149i	−0.986942	−	0.161075i	$-0.948504\pi$
$937$	− 9239.89i	− 0.322149i	0.986942	−	0.161075i	$-0.0514960\pi$
$938$	0	0
$939$	−84865.9	−2.94941
$940$	0	0
$941$	22807.3	0.790113	0.395056	−	0.918657i	$-0.370725\pi$
$941$	22807.3	0.790113	0.395056	+	0.918657i	$0.370725\pi$
$942$	0	0
$943$	− 45956.1i	− 1.58700i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5222.47i	0.179205i	0.995978	+	0.0896027i	$0.0285597\pi$
$947$	5222.47i	0.179205i	−0.995978	+	0.0896027i	$0.971440\pi$
$948$	0	0
$949$	19737.0	0.675121
$950$	0	0
$951$	−21838.1	−0.744637
$952$	0	0
$953$	10078.8i	0.342585i	0.985220	+	0.171293i	$0.0547944\pi$
$953$	10078.8i	0.342585i	−0.985220	+	0.171293i	$0.945206\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1022.22i	− 0.0345284i
$958$	0	0
$959$	−2262.68	−0.0761896
$960$	0	0
$961$	−23685.5	−0.795054
$962$	0	0
$963$	− 39177.7i	− 1.31099i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 28446.6i	− 0.945998i	−0.881063	−	0.472999i	$-0.843171\pi$
$967$	− 28446.6i	− 0.945998i	0.881063	−	0.472999i	$-0.156829\pi$
$968$	0	0
$969$	−14327.5	−0.474992
$970$	0	0
$971$	54767.7	1.81007	0.905035	−	0.425336i	$-0.139844\pi$
$971$	54767.7	1.81007	0.905035	+	0.425336i	$0.139844\pi$
$972$	0	0
$973$	3150.15i	0.103792i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34932.4i	− 1.14390i	−0.820289	−	0.571949i	$-0.806188\pi$
$977$	− 34932.4i	− 1.14390i	0.820289	−	0.571949i	$-0.193812\pi$
$978$	0	0
$979$	−56029.8	−1.82913
$980$	0	0
$981$	124178.	4.04147
$982$	0	0
$983$	− 40302.3i	− 1.30767i	−0.756635	−	0.653837i	$-0.773158\pi$
$983$	− 40302.3i	− 1.30767i	0.756635	−	0.653837i	$-0.226842\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 5963.34i	− 0.192315i
$988$	0	0
$989$	70510.0	2.26703
$990$	0	0
$991$	−40726.0	−1.30545	−0.652727	−	0.757593i	$-0.726375\pi$
$991$	−40726.0	−1.30545	−0.652727	+	0.757593i	$0.726375\pi$
$992$	0	0
$993$	− 19267.0i	− 0.615730i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15904.3i	0.505210i	0.967570	+	0.252605i	$0.0812873\pi$
$997$	15904.3i	0.505210i	−0.967570	+	0.252605i	$0.918713\pi$
$998$	0	0
$999$	−133421.	−4.22548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.4.c.d.1749.1		8
5.2	odd	4		380.4.a.c.1.1	✓	4
5.3	odd	4		1900.4.a.e.1.4		4
5.4	even	2	inner	1900.4.c.d.1749.8		8
20.7	even	4		1520.4.a.s.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.4.a.c.1.1	✓	4	5.2	odd	4
1520.4.a.s.1.4		4	20.7	even	4
1900.4.a.e.1.4		4	5.3	odd	4
1900.4.c.d.1749.1		8	1.1	even	1	trivial
1900.4.c.d.1749.8		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)

\(f(q)\)	\(=\)	\(q-9.96691i q^{3} +1.49735i q^{7} -72.3392 q^{9} -48.4512 q^{11} +61.3319i q^{13} -75.6585i q^{17} +19.0000 q^{19} +14.9240 q^{21} +136.212i q^{23} +451.892i q^{27} -2.11680 q^{29} +78.1380 q^{31} +482.908i q^{33} +295.250i q^{37} +611.289 q^{39} -337.387 q^{41} -517.650i q^{43} -399.581i q^{47} +340.758 q^{49} -754.081 q^{51} +457.884i q^{53} -189.371i q^{57} +305.196 q^{59} -267.687 q^{61} -108.317i q^{63} -493.582i q^{67} +1357.61 q^{69} -127.630 q^{71} -321.806i q^{73} -72.5485i q^{77} +533.196 q^{79} +2550.81 q^{81} +841.170i q^{83} +21.0980i q^{87} +1156.42 q^{89} -91.8354 q^{91} -778.794i q^{93} -374.645i q^{97} +3504.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 208 q^{9} - 120 q^{11} + 152 q^{19} - 408 q^{21} + 328 q^{29} - 1512 q^{31} + 1752 q^{39} - 352 q^{41} + 664 q^{49} - 2488 q^{51} + 288 q^{59} - 2072 q^{61} + 3320 q^{69} + 2888 q^{71} + 2112 q^{79}+ \cdots + 8296 q^{99}+O(q^{100}) \) 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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