LMFDB - Embedded newform 3800.2.a.z.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3800,2,Mod(1,3800)] 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("3800.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-2,0,0,0,-6,0,8,0,-2,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.3431527681$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.253565184.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 $ x^6 - 2x^5 - 11x^4 + 20x^3 + 22x^2 - 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.59277$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59277 q^{3} -1.66290 q^{7} -0.463073 q^{9} +5.56511 q^{11} -6.31979 q^{13} -4.12142 q^{17} +1.00000 q^{19} -2.64862 q^{21} +1.82315 q^{23} -5.51589 q^{27} -4.08942 q^{29} +6.61202 q^{31} +8.86396 q^{33} -9.66787 q^{37} -10.0660 q^{39} -4.61202 q^{41} +3.75231 q^{43} +3.85299 q^{47} -4.23477 q^{49} -6.56449 q^{51} -5.24594 q^{53} +1.59277 q^{57} -11.5291 q^{59} +8.02587 q^{61} +0.770043 q^{63} -0.155284 q^{67} +2.90387 q^{69} -12.1645 q^{71} -0.795627 q^{73} -9.25422 q^{77} -7.18555 q^{79} -7.39634 q^{81} -5.88500 q^{83} -6.51351 q^{87} +18.5476 q^{89} +10.5092 q^{91} +10.5315 q^{93} -2.77634 q^{97} -2.57705 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47}+ \cdots - 22 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9 - 2 * q^11 - 14 * q^13 - 10 * q^17 + 6 * q^19 + 18 * q^21 - 2 * q^23 - 2 * q^27 - 2 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^37 - 18 * q^39 + 4 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 34 * q^51 + 14 * q^53 - 2 * q^57 - 2 * q^59 + 10 * q^61 - 44 * q^63 - 42 * q^67 + 18 * q^69 - 8 * q^71 + 2 * q^73 - 24 * q^77 - 20 * q^79 + 6 * q^81 - 16 * q^83 - 30 * q^87 + 32 * q^89 + 10 * q^91 - 12 * q^93 - 40 * q^97 - 22 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.59277	0.919588	0.459794	−	0.888026i	$-0.347923\pi$
$3$	1.59277	0.919588	0.459794	+	0.888026i	$0.347923\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.66290	−0.628516	−0.314258	−	0.949338i	$-0.601756\pi$
$7$	−1.66290	−0.628516	−0.314258	+	0.949338i	$0.601756\pi$
$8$	0	0
$9$	−0.463073	−0.154358
$10$	0	0
$11$	5.56511	1.67794	0.838972	−	0.544174i	$-0.183157\pi$
$11$	5.56511	1.67794	0.838972	+	0.544174i	$0.183157\pi$
$12$	0	0
$13$	−6.31979	−1.75280	−0.876398	−	0.481588i	$-0.840060\pi$
$13$	−6.31979	−1.75280	−0.876398	+	0.481588i	$0.840060\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.12142	−0.999592	−0.499796	−	0.866143i	$-0.666592\pi$
$17$	−4.12142	−0.999592	−0.499796	+	0.866143i	$0.666592\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.64862	−0.577976
$22$	0	0
$23$	1.82315	0.380154	0.190077	−	0.981769i	$-0.439126\pi$
$23$	1.82315	0.380154	0.190077	+	0.981769i	$0.439126\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.51589	−1.06153
$28$	0	0
$29$	−4.08942	−0.759385	−0.379693	−	0.925113i	$-0.623970\pi$
$29$	−4.08942	−0.759385	−0.379693	+	0.925113i	$0.623970\pi$
$30$	0	0
$31$	6.61202	1.18755	0.593777	−	0.804630i	$-0.297636\pi$
$31$	6.61202	1.18755	0.593777	+	0.804630i	$0.297636\pi$
$32$	0	0
$33$	8.86396	1.54302
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.66787	−1.58939	−0.794694	−	0.607010i	$-0.792369\pi$
$37$	−9.66787	−1.58939	−0.794694	+	0.607010i	$0.792369\pi$
$38$	0	0
$39$	−10.0660	−1.61185
$40$	0	0
$41$	−4.61202	−0.720277	−0.360138	−	0.932899i	$-0.617271\pi$
$41$	−4.61202	−0.720277	−0.360138	+	0.932899i	$0.617271\pi$
$42$	0	0
$43$	3.75231	0.572222	0.286111	−	0.958196i	$-0.407637\pi$
$43$	3.75231	0.572222	0.286111	+	0.958196i	$0.407637\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.85299	0.562017	0.281008	−	0.959705i	$-0.409331\pi$
$47$	3.85299	0.562017	0.281008	+	0.959705i	$0.409331\pi$
$48$	0	0
$49$	−4.23477	−0.604967
$50$	0	0
$51$	−6.56449	−0.919213
$52$	0	0
$53$	−5.24594	−0.720585	−0.360293	−	0.932839i	$-0.617323\pi$
$53$	−5.24594	−0.720585	−0.360293	+	0.932839i	$0.617323\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.59277	0.210968
$58$	0	0
$59$	−11.5291	−1.50097	−0.750483	−	0.660890i	$-0.770179\pi$
$59$	−11.5291	−1.50097	−0.750483	+	0.660890i	$0.770179\pi$
$60$	0	0
$61$	8.02587	1.02761	0.513804	−	0.857908i	$-0.328236\pi$
$61$	8.02587	1.02761	0.513804	+	0.857908i	$0.328236\pi$
$62$	0	0
$63$	0.770043	0.0970164
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.155284	−0.0189710	−0.00948548	−	0.999955i	$-0.503019\pi$
$67$	−0.155284	−0.0189710	−0.00948548	+	0.999955i	$0.503019\pi$
$68$	0	0
$69$	2.90387	0.349585
$70$	0	0
$71$	−12.1645	−1.44366	−0.721831	−	0.692069i	$-0.756699\pi$
$71$	−12.1645	−1.44366	−0.721831	+	0.692069i	$0.756699\pi$
$72$	0	0
$73$	−0.795627	−0.0931211	−0.0465606	−	0.998915i	$-0.514826\pi$
$73$	−0.795627	−0.0931211	−0.0465606	+	0.998915i	$0.514826\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.25422	−1.05462
$78$	0	0
$79$	−7.18555	−0.808437	−0.404219	−	0.914662i	$-0.632456\pi$
$79$	−7.18555	−0.808437	−0.404219	+	0.914662i	$0.632456\pi$
$80$	0	0
$81$	−7.39634	−0.821816
$82$	0	0
$83$	−5.88500	−0.645963	−0.322981	−	0.946405i	$-0.604685\pi$
$83$	−5.88500	−0.645963	−0.322981	+	0.946405i	$0.604685\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.51351	−0.698322
$88$	0	0
$89$	18.5476	1.96604	0.983021	−	0.183492i	$-0.0587401\pi$
$89$	18.5476	1.96604	0.983021	+	0.183492i	$0.0587401\pi$
$90$	0	0
$91$	10.5092	1.10166
$92$	0	0
$93$	10.5315	1.09206
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.77634	−0.281894	−0.140947	−	0.990017i	$-0.545015\pi$
$97$	−2.77634	−0.281894	−0.140947	+	0.990017i	$0.545015\pi$
$98$	0	0
$99$	−2.57705	−0.259004
$100$	0	0
$101$	−13.6991	−1.36311	−0.681557	−	0.731765i	$-0.738697\pi$
$101$	−13.6991	−1.36311	−0.681557	+	0.731765i	$0.738697\pi$
$102$	0	0
$103$	−7.42279	−0.731389	−0.365695	−	0.930735i	$-0.619169\pi$
$103$	−7.42279	−0.731389	−0.365695	+	0.930735i	$0.619169\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.8010	1.33419	0.667097	−	0.744971i	$-0.267537\pi$
$107$	13.8010	1.33419	0.667097	+	0.744971i	$0.267537\pi$
$108$	0	0
$109$	−12.7543	−1.22164	−0.610818	−	0.791771i	$-0.709159\pi$
$109$	−12.7543	−1.22164	−0.610818	+	0.791771i	$0.709159\pi$
$110$	0	0
$111$	−15.3987	−1.46158
$112$	0	0
$113$	−7.81913	−0.735562	−0.367781	−	0.929912i	$-0.619882\pi$
$113$	−7.81913	−0.735562	−0.367781	+	0.929912i	$0.619882\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.92653	0.270558
$118$	0	0
$119$	6.85351	0.628260
$120$	0	0
$121$	19.9705	1.81550
$122$	0	0
$123$	−7.34591	−0.662358
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.07017	−0.449905	−0.224952	−	0.974370i	$-0.572223\pi$
$127$	−5.07017	−0.449905	−0.224952	+	0.974370i	$0.572223\pi$
$128$	0	0
$129$	5.97658	0.526209
$130$	0	0
$131$	10.8720	0.949895	0.474947	−	0.880014i	$-0.342467\pi$
$131$	10.8720	0.949895	0.474947	+	0.880014i	$0.342467\pi$
$132$	0	0
$133$	−1.66290	−0.144192
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.53861	−0.644067	−0.322033	−	0.946728i	$-0.604366\pi$
$137$	−7.53861	−0.644067	−0.322033	+	0.946728i	$0.604366\pi$
$138$	0	0
$139$	−0.885954	−0.0751457	−0.0375728	−	0.999294i	$-0.511963\pi$
$139$	−0.885954	−0.0751457	−0.0375728	+	0.999294i	$0.511963\pi$
$140$	0	0
$141$	6.13694	0.516824
$142$	0	0
$143$	−35.1704	−2.94109
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.74503	−0.556321
$148$	0	0
$149$	12.2594	1.00433	0.502166	−	0.864771i	$-0.332537\pi$
$149$	12.2594	1.00433	0.502166	+	0.864771i	$0.332537\pi$
$150$	0	0
$151$	−2.95140	−0.240181	−0.120091	−	0.992763i	$-0.538319\pi$
$151$	−2.95140	−0.240181	−0.120091	+	0.992763i	$0.538319\pi$
$152$	0	0
$153$	1.90852	0.154295
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.7951	1.26058	0.630292	−	0.776358i	$-0.282935\pi$
$157$	15.7951	1.26058	0.630292	+	0.776358i	$0.282935\pi$
$158$	0	0
$159$	−8.35559	−0.662642
$160$	0	0
$161$	−3.03172	−0.238933
$162$	0	0
$163$	20.4459	1.60144	0.800721	−	0.599037i	$-0.204450\pi$
$163$	20.4459	1.60144	0.800721	+	0.599037i	$0.204450\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.6083	−1.13043	−0.565213	−	0.824945i	$-0.691206\pi$
$167$	−14.6083	−1.13043	−0.565213	+	0.824945i	$0.691206\pi$
$168$	0	0
$169$	26.9398	2.07229
$170$	0	0
$171$	−0.463073	−0.0354121
$172$	0	0
$173$	−19.4244	−1.47681	−0.738406	−	0.674356i	$-0.764421\pi$
$173$	−19.4244	−1.47681	−0.738406	+	0.674356i	$0.764421\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.3633	−1.38027
$178$	0	0
$179$	−7.21080	−0.538960	−0.269480	−	0.963006i	$-0.586852\pi$
$179$	−7.21080	−0.538960	−0.269480	+	0.963006i	$0.586852\pi$
$180$	0	0
$181$	−10.1237	−0.752488	−0.376244	−	0.926521i	$-0.622785\pi$
$181$	−10.1237	−0.752488	−0.376244	+	0.926521i	$0.622785\pi$
$182$	0	0
$183$	12.7834	0.944976
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−22.9362	−1.67726
$188$	0	0
$189$	9.17236	0.667191
$190$	0	0
$191$	−24.4017	−1.76564	−0.882821	−	0.469709i	$-0.844359\pi$
$191$	−24.4017	−1.76564	−0.882821	+	0.469709i	$0.844359\pi$
$192$	0	0
$193$	−19.1845	−1.38093	−0.690465	−	0.723366i	$-0.742594\pi$
$193$	−19.1845	−1.38093	−0.690465	+	0.723366i	$0.742594\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.9453	−0.851066	−0.425533	−	0.904943i	$-0.639913\pi$
$197$	−11.9453	−0.851066	−0.425533	+	0.904943i	$0.639913\pi$
$198$	0	0
$199$	9.20530	0.652546	0.326273	−	0.945276i	$-0.394207\pi$
$199$	9.20530	0.652546	0.326273	+	0.945276i	$0.394207\pi$
$200$	0	0
$201$	−0.247332	−0.0174455
$202$	0	0
$203$	6.80028	0.477286
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.844253	−0.0586796
$208$	0	0
$209$	5.56511	0.384947
$210$	0	0
$211$	18.2971	1.25962	0.629811	−	0.776749i	$-0.283133\pi$
$211$	18.2971	1.25962	0.629811	+	0.776749i	$0.283133\pi$
$212$	0	0
$213$	−19.3753	−1.32757
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.9951	−0.746397
$218$	0	0
$219$	−1.26725	−0.0856331
$220$	0	0
$221$	26.0466	1.75208
$222$	0	0
$223$	1.57405	0.105406	0.0527032	−	0.998610i	$-0.483216\pi$
$223$	1.57405	0.105406	0.0527032	+	0.998610i	$0.483216\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.6034	0.969260	0.484630	−	0.874719i	$-0.338954\pi$
$227$	14.6034	0.969260	0.484630	+	0.874719i	$0.338954\pi$
$228$	0	0
$229$	9.33328	0.616761	0.308380	−	0.951263i	$-0.400213\pi$
$229$	9.33328	0.616761	0.308380	+	0.951263i	$0.400213\pi$
$230$	0	0
$231$	−14.7399	−0.969812
$232$	0	0
$233$	−1.02066	−0.0668658	−0.0334329	−	0.999441i	$-0.510644\pi$
$233$	−1.02066	−0.0668658	−0.0334329	+	0.999441i	$0.510644\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.4449	−0.743429
$238$	0	0
$239$	3.66778	0.237249	0.118624	−	0.992939i	$-0.462152\pi$
$239$	3.66778	0.237249	0.118624	+	0.992939i	$0.462152\pi$
$240$	0	0
$241$	−23.1700	−1.49251	−0.746254	−	0.665661i	$-0.768150\pi$
$241$	−23.1700	−1.49251	−0.746254	+	0.665661i	$0.768150\pi$
$242$	0	0
$243$	4.76697	0.305801
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.31979	−0.402119
$248$	0	0
$249$	−9.37347	−0.594020
$250$	0	0
$251$	−29.6256	−1.86995	−0.934977	−	0.354709i	$-0.884580\pi$
$251$	−29.6256	−1.86995	−0.934977	+	0.354709i	$0.884580\pi$
$252$	0	0
$253$	10.1461	0.637877
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.9768	0.871848	0.435924	−	0.899984i	$-0.356422\pi$
$257$	13.9768	0.871848	0.435924	+	0.899984i	$0.356422\pi$
$258$	0	0
$259$	16.0767	0.998956
$260$	0	0
$261$	1.89370	0.117217
$262$	0	0
$263$	11.4699	0.707264	0.353632	−	0.935385i	$-0.384947\pi$
$263$	11.4699	0.707264	0.353632	+	0.935385i	$0.384947\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	29.5421	1.80795
$268$	0	0
$269$	16.0864	0.980808	0.490404	−	0.871495i	$-0.336849\pi$
$269$	16.0864	0.980808	0.490404	+	0.871495i	$0.336849\pi$
$270$	0	0
$271$	15.3369	0.931649	0.465824	−	0.884877i	$-0.345758\pi$
$271$	15.3369	0.931649	0.465824	+	0.884877i	$0.345758\pi$
$272$	0	0
$273$	16.7387	1.01307
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.0546196	0.00328177	0.00164089	−	0.999999i	$-0.499478\pi$
$277$	0.0546196	0.00328177	0.00164089	+	0.999999i	$0.499478\pi$
$278$	0	0
$279$	−3.06185	−0.183308
$280$	0	0
$281$	20.3315	1.21288	0.606438	−	0.795130i	$-0.292598\pi$
$281$	20.3315	1.21288	0.606438	+	0.795130i	$0.292598\pi$
$282$	0	0
$283$	22.3909	1.33100	0.665501	−	0.746397i	$-0.268218\pi$
$283$	22.3909	1.33100	0.665501	+	0.746397i	$0.268218\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.66932	0.452706
$288$	0	0
$289$	−0.0138666	−0.000815685 0
$290$	0	0
$291$	−4.42208	−0.259227
$292$	0	0
$293$	−13.3983	−0.782736	−0.391368	−	0.920234i	$-0.627998\pi$
$293$	−13.3983	−0.782736	−0.391368	+	0.920234i	$0.627998\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−30.6966	−1.78119
$298$	0	0
$299$	−11.5219	−0.666332
$300$	0	0
$301$	−6.23971	−0.359651
$302$	0	0
$303$	−21.8196	−1.25350
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.60404	−0.433985	−0.216993	−	0.976173i	$-0.569625\pi$
$307$	−7.60404	−0.433985	−0.216993	+	0.976173i	$0.569625\pi$
$308$	0	0
$309$	−11.8228	−0.672577
$310$	0	0
$311$	31.7283	1.79915	0.899573	−	0.436770i	$-0.143877\pi$
$311$	31.7283	1.79915	0.899573	+	0.436770i	$0.143877\pi$
$312$	0	0
$313$	−26.2396	−1.48315	−0.741575	−	0.670870i	$-0.765921\pi$
$313$	−26.2396	−1.48315	−0.741575	+	0.670870i	$0.765921\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.269210	−0.0151203	−0.00756016	−	0.999971i	$-0.502406\pi$
$317$	−0.269210	−0.0151203	−0.00756016	+	0.999971i	$0.502406\pi$
$318$	0	0
$319$	−22.7581	−1.27421
$320$	0	0
$321$	21.9819	1.22691
$322$	0	0
$323$	−4.12142	−0.229322
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−20.3146	−1.12340
$328$	0	0
$329$	−6.40713	−0.353237
$330$	0	0
$331$	−9.69472	−0.532870	−0.266435	−	0.963853i	$-0.585846\pi$
$331$	−9.69472	−0.532870	−0.266435	+	0.963853i	$0.585846\pi$
$332$	0	0
$333$	4.47693	0.245334
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.4219	1.00350	0.501751	−	0.865012i	$-0.332689\pi$
$337$	18.4219	1.00350	0.501751	+	0.865012i	$0.332689\pi$
$338$	0	0
$339$	−12.4541	−0.676414
$340$	0	0
$341$	36.7967	1.99265
$342$	0	0
$343$	18.6823	1.00875
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.85913	0.475583	0.237791	−	0.971316i	$-0.423577\pi$
$347$	8.85913	0.475583	0.237791	+	0.971316i	$0.423577\pi$
$348$	0	0
$349$	−6.36382	−0.340647	−0.170324	−	0.985388i	$-0.554481\pi$
$349$	−6.36382	−0.340647	−0.170324	+	0.985388i	$0.554481\pi$
$350$	0	0
$351$	34.8593	1.86065
$352$	0	0
$353$	26.4213	1.40627	0.703133	−	0.711058i	$-0.251784\pi$
$353$	26.4213	1.40627	0.703133	+	0.711058i	$0.251784\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.9161	0.577740
$358$	0	0
$359$	0.904490	0.0477372	0.0238686	−	0.999715i	$-0.492402\pi$
$359$	0.904490	0.0477372	0.0238686	+	0.999715i	$0.492402\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	31.8085	1.66951
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−20.6657	−1.07874	−0.539370	−	0.842069i	$-0.681338\pi$
$367$	−20.6657	−1.07874	−0.539370	+	0.842069i	$0.681338\pi$
$368$	0	0
$369$	2.13570	0.111180
$370$	0	0
$371$	8.72347	0.452900
$372$	0	0
$373$	−23.7919	−1.23190	−0.615949	−	0.787786i	$-0.711227\pi$
$373$	−23.7919	−1.23190	−0.615949	+	0.787786i	$0.711227\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	25.8443	1.33105
$378$	0	0
$379$	27.9667	1.43655	0.718276	−	0.695758i	$-0.244931\pi$
$379$	27.9667	1.43655	0.718276	+	0.695758i	$0.244931\pi$
$380$	0	0
$381$	−8.07563	−0.413727
$382$	0	0
$383$	−12.1083	−0.618707	−0.309353	−	0.950947i	$-0.600113\pi$
$383$	−12.1083	−0.618707	−0.309353	+	0.950947i	$0.600113\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.73760	−0.0883269
$388$	0	0
$389$	0.871634	0.0441936	0.0220968	−	0.999756i	$-0.492966\pi$
$389$	0.871634	0.0441936	0.0220968	+	0.999756i	$0.492966\pi$
$390$	0	0
$391$	−7.51398	−0.379998
$392$	0	0
$393$	17.3167	0.873512
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.7833	1.04308	0.521542	−	0.853226i	$-0.325357\pi$
$397$	20.7833	1.04308	0.521542	+	0.853226i	$0.325357\pi$
$398$	0	0
$399$	−2.64862	−0.132597
$400$	0	0
$401$	−16.0237	−0.800184	−0.400092	−	0.916475i	$-0.631022\pi$
$401$	−16.0237	−0.800184	−0.400092	+	0.916475i	$0.631022\pi$
$402$	0	0
$403$	−41.7866	−2.08154
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−53.8028	−2.66691
$408$	0	0
$409$	20.1554	0.996621	0.498311	−	0.866999i	$-0.333954\pi$
$409$	20.1554	0.996621	0.498311	+	0.866999i	$0.333954\pi$
$410$	0	0
$411$	−12.0073	−0.592276
$412$	0	0
$413$	19.1718	0.943382
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.41112	−0.0691031
$418$	0	0
$419$	5.15507	0.251842	0.125921	−	0.992040i	$-0.459811\pi$
$419$	5.15507	0.251842	0.125921	+	0.992040i	$0.459811\pi$
$420$	0	0
$421$	26.5417	1.29356	0.646781	−	0.762676i	$-0.276115\pi$
$421$	26.5417	1.29356	0.646781	+	0.762676i	$0.276115\pi$
$422$	0	0
$423$	−1.78422	−0.0867516
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−13.3462	−0.645868
$428$	0	0
$429$	−56.0184	−2.70460
$430$	0	0
$431$	−21.2788	−1.02496	−0.512481	−	0.858699i	$-0.671274\pi$
$431$	−21.2788	−1.02496	−0.512481	+	0.858699i	$0.671274\pi$
$432$	0	0
$433$	6.44963	0.309949	0.154975	−	0.987918i	$-0.450470\pi$
$433$	6.44963	0.309949	0.154975	+	0.987918i	$0.450470\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.82315	0.0872132
$438$	0	0
$439$	22.2324	1.06110	0.530548	−	0.847655i	$-0.321986\pi$
$439$	22.2324	1.06110	0.530548	+	0.847655i	$0.321986\pi$
$440$	0	0
$441$	1.96101	0.0933813
$442$	0	0
$443$	−7.56611	−0.359477	−0.179738	−	0.983714i	$-0.557525\pi$
$443$	−7.56611	−0.359477	−0.179738	+	0.983714i	$0.557525\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	19.5265	0.923571
$448$	0	0
$449$	22.5411	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$449$	22.5411	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$450$	0	0
$451$	−25.6664	−1.20858
$452$	0	0
$453$	−4.70091	−0.220868
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.9942	−1.40307	−0.701536	−	0.712634i	$-0.747502\pi$
$457$	−29.9942	−1.40307	−0.701536	+	0.712634i	$0.747502\pi$
$458$	0	0
$459$	22.7333	1.06110
$460$	0	0
$461$	−35.7493	−1.66501	−0.832506	−	0.554016i	$-0.813095\pi$
$461$	−35.7493	−1.66501	−0.832506	+	0.554016i	$0.813095\pi$
$462$	0	0
$463$	9.81829	0.456295	0.228147	−	0.973627i	$-0.426733\pi$
$463$	9.81829	0.456295	0.228147	+	0.973627i	$0.426733\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.96046	−0.0907194	−0.0453597	−	0.998971i	$-0.514443\pi$
$467$	−1.96046	−0.0907194	−0.0453597	+	0.998971i	$0.514443\pi$
$468$	0	0
$469$	0.258222	0.0119236
$470$	0	0
$471$	25.1580	1.15922
$472$	0	0
$473$	20.8821	0.960158
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.42925	0.111228
$478$	0	0
$479$	−9.88045	−0.451450	−0.225725	−	0.974191i	$-0.572475\pi$
$479$	−9.88045	−0.451450	−0.225725	+	0.974191i	$0.572475\pi$
$480$	0	0
$481$	61.0989	2.78587
$482$	0	0
$483$	−4.82884	−0.219720
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.03706	−0.409508	−0.204754	−	0.978813i	$-0.565639\pi$
$487$	−9.03706	−0.409508	−0.204754	+	0.978813i	$0.565639\pi$
$488$	0	0
$489$	32.5656	1.47267
$490$	0	0
$491$	−20.3715	−0.919353	−0.459676	−	0.888086i	$-0.652035\pi$
$491$	−20.3715	−0.919353	−0.459676	+	0.888086i	$0.652035\pi$
$492$	0	0
$493$	16.8542	0.759076
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.2283	0.907365
$498$	0	0
$499$	−13.9386	−0.623979	−0.311990	−	0.950086i	$-0.600995\pi$
$499$	−13.9386	−0.623979	−0.311990	+	0.950086i	$0.600995\pi$
$500$	0	0
$501$	−23.2678	−1.03953
$502$	0	0
$503$	39.7299	1.77147	0.885735	−	0.464192i	$-0.153655\pi$
$503$	39.7299	1.77147	0.885735	+	0.464192i	$0.153655\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	42.9090	1.90566
$508$	0	0
$509$	−12.5200	−0.554941	−0.277471	−	0.960734i	$-0.589496\pi$
$509$	−12.5200	−0.554941	−0.277471	+	0.960734i	$0.589496\pi$
$510$	0	0
$511$	1.32305	0.0585282
$512$	0	0
$513$	−5.51589	−0.243533
$514$	0	0
$515$	0	0
$516$	0	0
$517$	21.4423	0.943033
$518$	0	0
$519$	−30.9387	−1.35806
$520$	0	0
$521$	−0.0426487	−0.00186847	−0.000934237	−	1.00000i	$-0.500297\pi$
$521$	−0.0426487	−0.00186847	−0.000934237		1.00000i	$0.500297\pi$
$522$	0	0
$523$	−29.4459	−1.28758	−0.643789	−	0.765203i	$-0.722639\pi$
$523$	−29.4459	−1.28758	−0.643789	+	0.765203i	$0.722639\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−27.2509	−1.18707
$528$	0	0
$529$	−19.6761	−0.855483
$530$	0	0
$531$	5.33883	0.231686
$532$	0	0
$533$	29.1470	1.26250
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.4852	−0.495621
$538$	0	0
$539$	−23.5670	−1.01510
$540$	0	0
$541$	−20.0558	−0.862264	−0.431132	−	0.902289i	$-0.641886\pi$
$541$	−20.0558	−0.862264	−0.431132	+	0.902289i	$0.641886\pi$
$542$	0	0
$543$	−16.1248	−0.691979
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−14.8952	−0.636874	−0.318437	−	0.947944i	$-0.603158\pi$
$547$	−14.8952	−0.636874	−0.318437	+	0.947944i	$0.603158\pi$
$548$	0	0
$549$	−3.71657	−0.158619
$550$	0	0
$551$	−4.08942	−0.174215
$552$	0	0
$553$	11.9488	0.508116
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.26068	−0.392388	−0.196194	−	0.980565i	$-0.562858\pi$
$557$	−9.26068	−0.392388	−0.196194	+	0.980565i	$0.562858\pi$
$558$	0	0
$559$	−23.7138	−1.00299
$560$	0	0
$561$	−36.5322	−1.54239
$562$	0	0
$563$	−12.9472	−0.545661	−0.272830	−	0.962062i	$-0.587960\pi$
$563$	−12.9472	−0.545661	−0.272830	+	0.962062i	$0.587960\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.2994	0.516525
$568$	0	0
$569$	20.0593	0.840929	0.420464	−	0.907309i	$-0.361867\pi$
$569$	20.0593	0.840929	0.420464	+	0.907309i	$0.361867\pi$
$570$	0	0
$571$	−5.64571	−0.236266	−0.118133	−	0.992998i	$-0.537691\pi$
$571$	−5.64571	−0.236266	−0.118133	+	0.992998i	$0.537691\pi$
$572$	0	0
$573$	−38.8663	−1.62366
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−33.5402	−1.39630	−0.698148	−	0.715953i	$-0.745992\pi$
$577$	−33.5402	−1.39630	−0.698148	+	0.715953i	$0.745992\pi$
$578$	0	0
$579$	−30.5566	−1.26989
$580$	0	0
$581$	9.78616	0.405998
$582$	0	0
$583$	−29.1943	−1.20910
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.5517	1.13718	0.568591	−	0.822621i	$-0.307489\pi$
$587$	27.5517	1.13718	0.568591	+	0.822621i	$0.307489\pi$
$588$	0	0
$589$	6.61202	0.272444
$590$	0	0
$591$	−19.0261	−0.782630
$592$	0	0
$593$	24.4353	1.00344	0.501719	−	0.865031i	$-0.332701\pi$
$593$	24.4353	1.00344	0.501719	+	0.865031i	$0.332701\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.6620	0.600074
$598$	0	0
$599$	−3.51535	−0.143633	−0.0718166	−	0.997418i	$-0.522880\pi$
$599$	−3.51535	−0.143633	−0.0718166	+	0.997418i	$0.522880\pi$
$600$	0	0
$601$	−1.21824	−0.0496932	−0.0248466	−	0.999691i	$-0.507910\pi$
$601$	−1.21824	−0.0496932	−0.0248466	+	0.999691i	$0.507910\pi$
$602$	0	0
$603$	0.0719079	0.00292832
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−13.1872	−0.535251	−0.267625	−	0.963523i	$-0.586239\pi$
$607$	−13.1872	−0.535251	−0.267625	+	0.963523i	$0.586239\pi$
$608$	0	0
$609$	10.8313	0.438907
$610$	0	0
$611$	−24.3501	−0.985100
$612$	0	0
$613$	−9.67880	−0.390923	−0.195462	−	0.980711i	$-0.562621\pi$
$613$	−9.67880	−0.390923	−0.195462	+	0.980711i	$0.562621\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.4077	0.418999	0.209500	−	0.977809i	$-0.432816\pi$
$617$	10.4077	0.418999	0.209500	+	0.977809i	$0.432816\pi$
$618$	0	0
$619$	−32.7144	−1.31490	−0.657452	−	0.753496i	$-0.728366\pi$
$619$	−32.7144	−1.31490	−0.657452	+	0.753496i	$0.728366\pi$
$620$	0	0
$621$	−10.0563	−0.403546
$622$	0	0
$623$	−30.8428	−1.23569
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.86396	0.353993
$628$	0	0
$629$	39.8454	1.58874
$630$	0	0
$631$	−37.2305	−1.48212	−0.741061	−	0.671438i	$-0.765677\pi$
$631$	−37.2305	−1.48212	−0.741061	+	0.671438i	$0.765677\pi$
$632$	0	0
$633$	29.1431	1.15833
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26.7629	1.06038
$638$	0	0
$639$	5.63306	0.222840
$640$	0	0
$641$	23.3803	0.923464	0.461732	−	0.887019i	$-0.347228\pi$
$641$	23.3803	0.923464	0.461732	+	0.887019i	$0.347228\pi$
$642$	0	0
$643$	−24.6705	−0.972911	−0.486455	−	0.873705i	$-0.661710\pi$
$643$	−24.6705	−0.972911	−0.486455	+	0.873705i	$0.661710\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.2591	1.85795	0.928973	−	0.370147i	$-0.120693\pi$
$647$	47.2591	1.85795	0.928973	+	0.370147i	$0.120693\pi$
$648$	0	0
$649$	−64.1610	−2.51854
$650$	0	0
$651$	−17.5127	−0.686378
$652$	0	0
$653$	31.5361	1.23410	0.617051	−	0.786924i	$-0.288327\pi$
$653$	31.5361	1.23410	0.617051	+	0.786924i	$0.288327\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.368434	0.0143740
$658$	0	0
$659$	−21.3511	−0.831722	−0.415861	−	0.909428i	$-0.636520\pi$
$659$	−21.3511	−0.831722	−0.415861	+	0.909428i	$0.636520\pi$
$660$	0	0
$661$	−31.2161	−1.21416	−0.607082	−	0.794639i	$-0.707660\pi$
$661$	−31.2161	−1.21416	−0.607082	+	0.794639i	$0.707660\pi$
$662$	0	0
$663$	41.4863	1.61119
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.45563	−0.288683
$668$	0	0
$669$	2.50711	0.0969304
$670$	0	0
$671$	44.6649	1.72427
$672$	0	0
$673$	−4.16240	−0.160449	−0.0802243	−	0.996777i	$-0.525564\pi$
$673$	−4.16240	−0.160449	−0.0802243	+	0.996777i	$0.525564\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.5734	1.36720	0.683599	−	0.729857i	$-0.260414\pi$
$677$	35.5734	1.36720	0.683599	+	0.729857i	$0.260414\pi$
$678$	0	0
$679$	4.61677	0.177175
$680$	0	0
$681$	23.2599	0.891320
$682$	0	0
$683$	26.8628	1.02788	0.513938	−	0.857827i	$-0.328186\pi$
$683$	26.8628	1.02788	0.513938	+	0.857827i	$0.328186\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14.8658	0.567166
$688$	0	0
$689$	33.1533	1.26304
$690$	0	0
$691$	32.0064	1.21758	0.608791	−	0.793331i	$-0.291655\pi$
$691$	32.0064	1.21758	0.608791	+	0.793331i	$0.291655\pi$
$692$	0	0
$693$	4.28538	0.162788
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.0081	0.719983
$698$	0	0
$699$	−1.62568	−0.0614890
$700$	0	0
$701$	−0.822942	−0.0310821	−0.0155410	−	0.999879i	$-0.504947\pi$
$701$	−0.822942	−0.0310821	−0.0155410	+	0.999879i	$0.504947\pi$
$702$	0	0
$703$	−9.66787	−0.364631
$704$	0	0
$705$	0	0
$706$	0	0
$707$	22.7802	0.856739
$708$	0	0
$709$	24.8057	0.931599	0.465800	−	0.884890i	$-0.345767\pi$
$709$	24.8057	0.931599	0.465800	+	0.884890i	$0.345767\pi$
$710$	0	0
$711$	3.32743	0.124789
$712$	0	0
$713$	12.0547	0.451453
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.84194	0.218171
$718$	0	0
$719$	13.8765	0.517506	0.258753	−	0.965943i	$-0.416688\pi$
$719$	13.8765	0.517506	0.258753	+	0.965943i	$0.416688\pi$
$720$	0	0
$721$	12.3433	0.459690
$722$	0	0
$723$	−36.9045	−1.37249
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.7158	1.06501	0.532506	−	0.846426i	$-0.321251\pi$
$727$	28.7158	1.06501	0.532506	+	0.846426i	$0.321251\pi$
$728$	0	0
$729$	29.7817	1.10303
$730$	0	0
$731$	−15.4649	−0.571989
$732$	0	0
$733$	47.1903	1.74301	0.871506	−	0.490384i	$-0.163144\pi$
$733$	47.1903	1.74301	0.871506	+	0.490384i	$0.163144\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.864174	−0.0318322
$738$	0	0
$739$	−7.61271	−0.280038	−0.140019	−	0.990149i	$-0.544716\pi$
$739$	−7.61271	−0.280038	−0.140019	+	0.990149i	$0.544716\pi$
$740$	0	0
$741$	−10.0660	−0.369784
$742$	0	0
$743$	−15.8127	−0.580111	−0.290055	−	0.957010i	$-0.593674\pi$
$743$	−15.8127	−0.580111	−0.290055	+	0.957010i	$0.593674\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.72519	0.0997093
$748$	0	0
$749$	−22.9497	−0.838563
$750$	0	0
$751$	35.7516	1.30459	0.652296	−	0.757964i	$-0.273806\pi$
$751$	35.7516	1.30459	0.652296	+	0.757964i	$0.273806\pi$
$752$	0	0
$753$	−47.1869	−1.71959
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.8136	0.465717	0.232859	−	0.972511i	$-0.425192\pi$
$757$	12.8136	0.465717	0.232859	+	0.972511i	$0.425192\pi$
$758$	0	0
$759$	16.1604	0.586584
$760$	0	0
$761$	18.8830	0.684508	0.342254	−	0.939608i	$-0.388810\pi$
$761$	18.8830	0.684508	0.342254	+	0.939608i	$0.388810\pi$
$762$	0	0
$763$	21.2090	0.767818
$764$	0	0
$765$	0	0
$766$	0	0
$767$	72.8618	2.63089
$768$	0	0
$769$	−16.0228	−0.577798	−0.288899	−	0.957360i	$-0.593289\pi$
$769$	−16.0228	−0.577798	−0.288899	+	0.957360i	$0.593289\pi$
$770$	0	0
$771$	22.2618	0.801741
$772$	0	0
$773$	−9.91039	−0.356452	−0.178226	−	0.983990i	$-0.557036\pi$
$773$	−9.91039	−0.356452	−0.178226	+	0.983990i	$0.557036\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	25.6065	0.918628
$778$	0	0
$779$	−4.61202	−0.165243
$780$	0	0
$781$	−67.6969	−2.42238
$782$	0	0
$783$	22.5568	0.806113
$784$	0	0
$785$	0	0
$786$	0	0
$787$	30.9671	1.10386	0.551929	−	0.833891i	$-0.313892\pi$
$787$	30.9671	1.10386	0.551929	+	0.833891i	$0.313892\pi$
$788$	0	0
$789$	18.2689	0.650392
$790$	0	0
$791$	13.0024	0.462313
$792$	0	0
$793$	−50.7219	−1.80119
$794$	0	0
$795$	0	0
$796$	0	0
$797$	54.8059	1.94132	0.970662	−	0.240448i	$-0.0772945\pi$
$797$	54.8059	1.94132	0.970662	+	0.240448i	$0.0772945\pi$
$798$	0	0
$799$	−15.8798	−0.561787
$800$	0	0
$801$	−8.58890	−0.303474
$802$	0	0
$803$	−4.42776	−0.156252
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25.6221	0.901939
$808$	0	0
$809$	25.6862	0.903080	0.451540	−	0.892251i	$-0.350875\pi$
$809$	25.6862	0.903080	0.451540	+	0.892251i	$0.350875\pi$
$810$	0	0
$811$	8.01224	0.281348	0.140674	−	0.990056i	$-0.455073\pi$
$811$	8.01224	0.281348	0.140674	+	0.990056i	$0.455073\pi$
$812$	0	0
$813$	24.4282	0.856733
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.75231	0.131277
$818$	0	0
$819$	−4.86652	−0.170050
$820$	0	0
$821$	−48.8946	−1.70643	−0.853217	−	0.521555i	$-0.825352\pi$
$821$	−48.8946	−1.70643	−0.853217	+	0.521555i	$0.825352\pi$
$822$	0	0
$823$	−39.8314	−1.38843	−0.694217	−	0.719766i	$-0.744249\pi$
$823$	−39.8314	−1.38843	−0.694217	+	0.719766i	$0.744249\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.8758	−1.00411	−0.502056	−	0.864835i	$-0.667423\pi$
$827$	−28.8758	−1.00411	−0.502056	+	0.864835i	$0.667423\pi$
$828$	0	0
$829$	−2.35519	−0.0817991	−0.0408995	−	0.999163i	$-0.513022\pi$
$829$	−2.35519	−0.0817991	−0.0408995	+	0.999163i	$0.513022\pi$
$830$	0	0
$831$	0.0869966	0.00301788
$832$	0	0
$833$	17.4533	0.604720
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−36.4712	−1.26063
$838$	0	0
$839$	26.0230	0.898415	0.449207	−	0.893428i	$-0.351706\pi$
$839$	26.0230	0.898415	0.449207	+	0.893428i	$0.351706\pi$
$840$	0	0
$841$	−12.2767	−0.423334
$842$	0	0
$843$	32.3835	1.11535
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−33.2089	−1.14107
$848$	0	0
$849$	35.6636	1.22397
$850$	0	0
$851$	−17.6260	−0.604211
$852$	0	0
$853$	−37.7907	−1.29393	−0.646964	−	0.762520i	$-0.723962\pi$
$853$	−37.7907	−1.29393	−0.646964	+	0.762520i	$0.723962\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.7796	0.675657	0.337828	−	0.941208i	$-0.390308\pi$
$857$	19.7796	0.675657	0.337828	+	0.941208i	$0.390308\pi$
$858$	0	0
$859$	22.0641	0.752816	0.376408	−	0.926454i	$-0.377159\pi$
$859$	22.0641	0.752816	0.376408	+	0.926454i	$0.377159\pi$
$860$	0	0
$861$	12.2155	0.416303
$862$	0	0
$863$	32.3505	1.10122	0.550612	−	0.834761i	$-0.314394\pi$
$863$	32.3505	1.10122	0.550612	+	0.834761i	$0.314394\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−0.0220864	−0.000750094 0
$868$	0	0
$869$	−39.9884	−1.35651
$870$	0	0
$871$	0.981363	0.0332522
$872$	0	0
$873$	1.28565	0.0435126
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.5876	0.762728	0.381364	−	0.924425i	$-0.375454\pi$
$877$	22.5876	0.762728	0.381364	+	0.924425i	$0.375454\pi$
$878$	0	0
$879$	−21.3404	−0.719794
$880$	0	0
$881$	31.2658	1.05337	0.526686	−	0.850060i	$-0.323434\pi$
$881$	31.2658	1.05337	0.526686	+	0.850060i	$0.323434\pi$
$882$	0	0
$883$	−22.4584	−0.755786	−0.377893	−	0.925849i	$-0.623351\pi$
$883$	−22.4584	−0.755786	−0.377893	+	0.925849i	$0.623351\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.4883	−0.486470	−0.243235	−	0.969967i	$-0.578209\pi$
$887$	−14.4883	−0.486470	−0.243235	+	0.969967i	$0.578209\pi$
$888$	0	0
$889$	8.43117	0.282772
$890$	0	0
$891$	−41.1615	−1.37896
$892$	0	0
$893$	3.85299	0.128935
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−18.3519	−0.612751
$898$	0	0
$899$	−27.0393	−0.901811
$900$	0	0
$901$	21.6207	0.720291
$902$	0	0
$903$	−9.93845	−0.330731
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.58465	0.318253	0.159127	−	0.987258i	$-0.449132\pi$
$907$	9.58465	0.318253	0.159127	+	0.987258i	$0.449132\pi$
$908$	0	0
$909$	6.34370	0.210407
$910$	0	0
$911$	29.8122	0.987721	0.493860	−	0.869541i	$-0.335585\pi$
$911$	29.8122	0.987721	0.493860	+	0.869541i	$0.335585\pi$
$912$	0	0
$913$	−32.7507	−1.08389
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18.0791	−0.597025
$918$	0	0
$919$	−22.0075	−0.725961	−0.362981	−	0.931797i	$-0.618241\pi$
$919$	−22.0075	−0.725961	−0.362981	+	0.931797i	$0.618241\pi$
$920$	0	0
$921$	−12.1115	−0.399088
$922$	0	0
$923$	76.8772	2.53044
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.43729	0.112896
$928$	0	0
$929$	−22.7036	−0.744880	−0.372440	−	0.928056i	$-0.621479\pi$
$929$	−22.7036	−0.744880	−0.372440	+	0.928056i	$0.621479\pi$
$930$	0	0
$931$	−4.23477	−0.138789
$932$	0	0
$933$	50.5360	1.65447
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−10.5567	−0.344871	−0.172436	−	0.985021i	$-0.555164\pi$
$937$	−10.5567	−0.344871	−0.172436	+	0.985021i	$0.555164\pi$
$938$	0	0
$939$	−41.7937	−1.36389
$940$	0	0
$941$	−24.7961	−0.808331	−0.404166	−	0.914686i	$-0.632438\pi$
$941$	−24.7961	−0.808331	−0.404166	+	0.914686i	$0.632438\pi$
$942$	0	0
$943$	−8.40842	−0.273816
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.83000	0.189449	0.0947247	−	0.995504i	$-0.469803\pi$
$947$	5.83000	0.189449	0.0947247	+	0.995504i	$0.469803\pi$
$948$	0	0
$949$	5.02820	0.163222
$950$	0	0
$951$	−0.428790	−0.0139045
$952$	0	0
$953$	−7.15932	−0.231913	−0.115957	−	0.993254i	$-0.536993\pi$
$953$	−7.15932	−0.231913	−0.115957	+	0.993254i	$0.536993\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−36.2484	−1.17175
$958$	0	0
$959$	12.5359	0.404806
$960$	0	0
$961$	12.7188	0.410285
$962$	0	0
$963$	−6.39088	−0.205943
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10.3120	0.331612	0.165806	−	0.986158i	$-0.446977\pi$
$967$	10.3120	0.331612	0.165806	+	0.986158i	$0.446977\pi$
$968$	0	0
$969$	−6.56449	−0.210882
$970$	0	0
$971$	38.2430	1.22728	0.613639	−	0.789587i	$-0.289705\pi$
$971$	38.2430	1.22728	0.613639	+	0.789587i	$0.289705\pi$
$972$	0	0
$973$	1.47325	0.0472303
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.5114	0.592232	0.296116	−	0.955152i	$-0.404308\pi$
$977$	18.5114	0.592232	0.296116	+	0.955152i	$0.404308\pi$
$978$	0	0
$979$	103.220	3.29891
$980$	0	0
$981$	5.90615	0.188569
$982$	0	0
$983$	10.9590	0.349539	0.174769	−	0.984609i	$-0.444082\pi$
$983$	10.9590	0.349539	0.174769	+	0.984609i	$0.444082\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−10.2051	−0.324832
$988$	0	0
$989$	6.84104	0.217532
$990$	0	0
$991$	−36.8766	−1.17142	−0.585712	−	0.810519i	$-0.699185\pi$
$991$	−36.8766	−1.17142	−0.585712	+	0.810519i	$0.699185\pi$
$992$	0	0
$993$	−15.4415	−0.490021
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−36.4988	−1.15593	−0.577964	−	0.816062i	$-0.696153\pi$
$997$	−36.4988	−1.15593	−0.577964	+	0.816062i	$0.696153\pi$
$998$	0	0
$999$	53.3269	1.68719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.z.1.5		6
4.3	odd	2		7600.2.a.cn.1.2		6
5.2	odd	4		760.2.d.e.609.4	✓	12
5.3	odd	4		760.2.d.e.609.9	yes	12
5.4	even	2		3800.2.a.be.1.2		6
20.3	even	4		1520.2.d.k.609.4		12
20.7	even	4		1520.2.d.k.609.9		12
20.19	odd	2		7600.2.a.cg.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.e.609.4	✓	12	5.2	odd	4
760.2.d.e.609.9	yes	12	5.3	odd	4
1520.2.d.k.609.4		12	20.3	even	4
1520.2.d.k.609.9		12	20.7	even	4
3800.2.a.z.1.5		6	1.1	even	1	trivial
3800.2.a.be.1.2		6	5.4	even	2
7600.2.a.cg.1.5		6	20.19	odd	2
7600.2.a.cn.1.2		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.59277 q^{3} -1.66290 q^{7} -0.463073 q^{9} +5.56511 q^{11} -6.31979 q^{13} -4.12142 q^{17} +1.00000 q^{19} -2.64862 q^{21} +1.82315 q^{23} -5.51589 q^{27} -4.08942 q^{29} +6.61202 q^{31} +8.86396 q^{33} -9.66787 q^{37} -10.0660 q^{39} -4.61202 q^{41} +3.75231 q^{43} +3.85299 q^{47} -4.23477 q^{49} -6.56449 q^{51} -5.24594 q^{53} +1.59277 q^{57} -11.5291 q^{59} +8.02587 q^{61} +0.770043 q^{63} -0.155284 q^{67} +2.90387 q^{69} -12.1645 q^{71} -0.795627 q^{73} -9.25422 q^{77} -7.18555 q^{79} -7.39634 q^{81} -5.88500 q^{83} -6.51351 q^{87} +18.5476 q^{89} +10.5092 q^{91} +10.5315 q^{93} -2.77634 q^{97} -2.57705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47}+ \cdots - 22 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^7 + 8 * q^9 - 2 * q^11 - 14 * q^13 - 10 * q^17 + 6 * q^19 + 18 * q^21 - 2 * q^23 - 2 * q^27 - 2 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^37 - 18 * q^39 + 4 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 34 * q^51 + 14 * q^53 - 2 * q^57 - 2 * q^59 + 10 * q^61 - 44 * q^63 - 42 * q^67 + 18 * q^69 - 8 * q^71 + 2 * q^73 - 24 * q^77 - 20 * q^79 + 6 * q^81 - 16 * q^83 - 30 * q^87 + 32 * q^89 + 10 * q^91 - 12 * q^93 - 40 * q^97 - 22 * q^99

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