LMFDB - Embedded newform 3249.2.a.w.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] 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("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,3,0,-3,-3,0,6,3,0,3,6,0,-6,9,0,0,-3,0,3,-6,0,-6,12,0, -3,-9,0,24,9,0,-15,3,0,6,0,0,-9,6,0,-21,12,0,9,3,0,-12,9,0,-6,18,0,6,-3, 0,-12,15,0,-9,3,0,-3,15,0,-6,3,0,-3,9,0,-6,3,0,0,0,0,9,-3,0,21,15,0,-6, 18,0,-6,0,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.87939 q^{2} +1.53209 q^{4} -0.879385 q^{5} -2.87939 q^{7} +0.879385 q^{8} +1.65270 q^{10} +1.83750 q^{11} -2.75877 q^{13} +5.41147 q^{14} -4.71688 q^{16} +7.10607 q^{17} -1.34730 q^{20} -3.45336 q^{22} -6.59627 q^{23} -4.22668 q^{25} +5.18479 q^{26} -4.41147 q^{28} +3.12836 q^{29} +7.65270 q^{31} +7.10607 q^{32} -13.3550 q^{34} +2.53209 q^{35} +2.83750 q^{37} -0.773318 q^{40} -3.98545 q^{41} -11.4534 q^{43} +2.81521 q^{44} +12.3969 q^{46} -2.20708 q^{47} +1.29086 q^{49} +7.94356 q^{50} -4.22668 q^{52} -2.70233 q^{53} -1.61587 q^{55} -2.53209 q^{56} -5.87939 q^{58} +8.41147 q^{59} +0.615867 q^{61} -14.3824 q^{62} -3.92127 q^{64} +2.42602 q^{65} -3.67499 q^{67} +10.8871 q^{68} -4.75877 q^{70} +7.45336 q^{71} -10.0077 q^{73} -5.33275 q^{74} -5.29086 q^{77} +1.61081 q^{79} +4.14796 q^{80} +7.49020 q^{82} -0.985452 q^{83} -6.24897 q^{85} +21.5253 q^{86} +1.61587 q^{88} -17.0574 q^{89} +7.94356 q^{91} -10.1061 q^{92} +4.14796 q^{94} -5.90167 q^{97} -2.42602 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 3 q^{7} - 3 q^{8} + 6 q^{10} + 3 q^{11} + 3 q^{13} + 6 q^{14} - 6 q^{16} + 9 q^{17} - 3 q^{20} + 3 q^{22} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} - 9 q^{29} + 24 q^{31} + 9 q^{32} - 15 q^{34}+ \cdots - 15 q^{98}+O(q^{100}) $ 3 * q + 3 * q^5 - 3 * q^7 - 3 * q^8 + 6 * q^10 + 3 * q^11 + 3 * q^13 + 6 * q^14 - 6 * q^16 + 9 * q^17 - 3 * q^20 + 3 * q^22 - 6 * q^23 - 6 * q^25 + 12 * q^26 - 3 * q^28 - 9 * q^29 + 24 * q^31 + 9 * q^32 - 15 * q^34 + 3 * q^35 + 6 * q^37 - 9 * q^40 + 6 * q^41 - 21 * q^43 + 12 * q^44 + 9 * q^46 + 3 * q^47 - 12 * q^49 + 9 * q^50 - 6 * q^52 + 18 * q^53 + 6 * q^55 - 3 * q^56 - 12 * q^58 + 15 * q^59 - 9 * q^61 + 3 * q^62 - 3 * q^64 + 15 * q^65 - 6 * q^67 + 3 * q^68 - 3 * q^70 + 9 * q^71 - 6 * q^73 + 3 * q^74 + 9 * q^79 - 3 * q^80 + 21 * q^82 + 15 * q^83 - 6 * q^85 + 18 * q^86 - 6 * q^88 + 9 * q^91 - 18 * q^92 - 3 * q^94 - 6 * q^97 - 15 * q^98

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 $

$2$	−1.87939	−1.32893	−0.664463	−	0.747321i	$-0.731340\pi$
$2$	−1.87939	−1.32893	−0.664463	+	0.747321i	$0.731340\pi$
$3$	0	0
$3$	0	0
$4$	1.53209	0.766044
$5$	−0.879385	−0.393273	−0.196637	−	0.980476i	$-0.563002\pi$
$5$	−0.879385	−0.393273	−0.196637	+	0.980476i	$0.563002\pi$
$6$	0	0
$7$	−2.87939	−1.08831	−0.544153	−	0.838986i	$-0.683149\pi$
$7$	−2.87939	−1.08831	−0.544153	+	0.838986i	$0.683149\pi$
$8$	0.879385	0.310910
$9$	0	0
$10$	1.65270	0.522631
$11$	1.83750	0.554026	0.277013	−	0.960866i	$-0.410655\pi$
$11$	1.83750	0.554026	0.277013	+	0.960866i	$0.410655\pi$
$12$	0	0
$13$	−2.75877	−0.765145	−0.382573	−	0.923925i	$-0.624962\pi$
$13$	−2.75877	−0.765145	−0.382573	+	0.923925i	$0.624962\pi$
$14$	5.41147	1.44628
$15$	0	0
$16$	−4.71688	−1.17922
$17$	7.10607	1.72347	0.861737	−	0.507355i	$-0.169377\pi$
$17$	7.10607	1.72347	0.861737	+	0.507355i	$0.169377\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−1.34730	−0.301265
$21$	0	0
$22$	−3.45336	−0.736260
$23$	−6.59627	−1.37542	−0.687708	−	0.725987i	$-0.741383\pi$
$23$	−6.59627	−1.37542	−0.687708	+	0.725987i	$0.741383\pi$
$24$	0	0
$25$	−4.22668	−0.845336
$26$	5.18479	1.01682
$27$	0	0
$28$	−4.41147	−0.833690
$29$	3.12836	0.580921	0.290461	−	0.956887i	$-0.406192\pi$
$29$	3.12836	0.580921	0.290461	+	0.956887i	$0.406192\pi$
$30$	0	0
$31$	7.65270	1.37447	0.687233	−	0.726437i	$-0.258825\pi$
$31$	7.65270	1.37447	0.687233	+	0.726437i	$0.258825\pi$
$32$	7.10607	1.25619
$33$	0	0
$34$	−13.3550	−2.29037
$35$	2.53209	0.428001
$36$	0	0
$37$	2.83750	0.466481	0.233241	−	0.972419i	$-0.425067\pi$
$37$	2.83750	0.466481	0.233241	+	0.972419i	$0.425067\pi$
$38$	0	0
$39$	0	0
$40$	−0.773318	−0.122272
$41$	−3.98545	−0.622423	−0.311212	−	0.950341i	$-0.600735\pi$
$41$	−3.98545	−0.622423	−0.311212	+	0.950341i	$0.600735\pi$
$42$	0	0
$43$	−11.4534	−1.74662	−0.873311	−	0.487164i	$-0.838032\pi$
$43$	−11.4534	−1.74662	−0.873311	+	0.487164i	$0.838032\pi$
$44$	2.81521	0.424408
$45$	0	0
$46$	12.3969	1.82783
$47$	−2.20708	−0.321936	−0.160968	−	0.986960i	$-0.551462\pi$
$47$	−2.20708	−0.321936	−0.160968	+	0.986960i	$0.551462\pi$
$48$	0	0
$49$	1.29086	0.184408
$50$	7.94356	1.12339
$51$	0	0
$52$	−4.22668	−0.586135
$53$	−2.70233	−0.371194	−0.185597	−	0.982626i	$-0.559422\pi$
$53$	−2.70233	−0.371194	−0.185597	+	0.982626i	$0.559422\pi$
$54$	0	0
$55$	−1.61587	−0.217883
$56$	−2.53209	−0.338365
$57$	0	0
$58$	−5.87939	−0.772001
$59$	8.41147	1.09508	0.547540	−	0.836779i	$-0.315564\pi$
$59$	8.41147	1.09508	0.547540	+	0.836779i	$0.315564\pi$
$60$	0	0
$61$	0.615867	0.0788537	0.0394268	−	0.999222i	$-0.487447\pi$
$61$	0.615867	0.0788537	0.0394268	+	0.999222i	$0.487447\pi$
$62$	−14.3824	−1.82656
$63$	0	0
$64$	−3.92127	−0.490159
$65$	2.42602	0.300911
$66$	0	0
$67$	−3.67499	−0.448972	−0.224486	−	0.974477i	$-0.572070\pi$
$67$	−3.67499	−0.448972	−0.224486	+	0.974477i	$0.572070\pi$
$68$	10.8871	1.32026
$69$	0	0
$70$	−4.75877	−0.568782
$71$	7.45336	0.884551	0.442276	−	0.896879i	$-0.354171\pi$
$71$	7.45336	0.884551	0.442276	+	0.896879i	$0.354171\pi$
$72$	0	0
$73$	−10.0077	−1.17132	−0.585659	−	0.810558i	$-0.699164\pi$
$73$	−10.0077	−1.17132	−0.585659	+	0.810558i	$0.699164\pi$
$74$	−5.33275	−0.619919
$75$	0	0
$76$	0	0
$77$	−5.29086	−0.602949
$78$	0	0
$79$	1.61081	0.181231	0.0906154	−	0.995886i	$-0.471117\pi$
$79$	1.61081	0.181231	0.0906154	+	0.995886i	$0.471117\pi$
$80$	4.14796	0.463756
$81$	0	0
$82$	7.49020	0.827154
$83$	−0.985452	−0.108167	−0.0540837	−	0.998536i	$-0.517224\pi$
$83$	−0.985452	−0.108167	−0.0540837	+	0.998536i	$0.517224\pi$
$84$	0	0
$85$	−6.24897	−0.677796
$86$	21.5253	2.32113
$87$	0	0
$88$	1.61587	0.172252
$89$	−17.0574	−1.80808	−0.904039	−	0.427450i	$-0.859412\pi$
$89$	−17.0574	−1.80808	−0.904039	+	0.427450i	$0.859412\pi$
$90$	0	0
$91$	7.94356	0.832712
$92$	−10.1061	−1.05363
$93$	0	0
$94$	4.14796	0.427829
$95$	0	0
$96$	0	0
$97$	−5.90167	−0.599224	−0.299612	−	0.954061i	$-0.596857\pi$
$97$	−5.90167	−0.599224	−0.299612	+	0.954061i	$0.596857\pi$
$98$	−2.42602	−0.245065
$99$	0	0
$100$	−6.47565	−0.647565
$101$	4.30541	0.428404	0.214202	−	0.976789i	$-0.431285\pi$
$101$	4.30541	0.428404	0.214202	+	0.976789i	$0.431285\pi$
$102$	0	0
$103$	17.8084	1.75471	0.877357	−	0.479838i	$-0.159305\pi$
$103$	17.8084	1.75471	0.877357	+	0.479838i	$0.159305\pi$
$104$	−2.42602	−0.237891
$105$	0	0
$106$	5.07873	0.493289
$107$	−5.88713	−0.569130	−0.284565	−	0.958657i	$-0.591849\pi$
$107$	−5.88713	−0.569130	−0.284565	+	0.958657i	$0.591849\pi$
$108$	0	0
$109$	15.1138	1.44764	0.723820	−	0.689989i	$-0.242385\pi$
$109$	15.1138	1.44764	0.723820	+	0.689989i	$0.242385\pi$
$110$	3.03684	0.289551
$111$	0	0
$112$	13.5817	1.28335
$113$	6.08378	0.572314	0.286157	−	0.958183i	$-0.407622\pi$
$113$	6.08378	0.572314	0.286157	+	0.958183i	$0.407622\pi$
$114$	0	0
$115$	5.80066	0.540914
$116$	4.79292	0.445011
$117$	0	0
$118$	−15.8084	−1.45528
$119$	−20.4611	−1.87567
$120$	0	0
$121$	−7.62361	−0.693055
$122$	−1.15745	−0.104791
$123$	0	0
$124$	11.7246	1.05290
$125$	8.11381	0.725721
$126$	0	0
$127$	11.9213	1.05784	0.528921	−	0.848671i	$-0.322597\pi$
$127$	11.9213	1.05784	0.528921	+	0.848671i	$0.322597\pi$
$128$	−6.84255	−0.604802
$129$	0	0
$130$	−4.55943	−0.399888
$131$	2.39187	0.208979	0.104489	−	0.994526i	$-0.466679\pi$
$131$	2.39187	0.208979	0.104489	+	0.994526i	$0.466679\pi$
$132$	0	0
$133$	0	0
$134$	6.90673	0.596650
$135$	0	0
$136$	6.24897	0.535845
$137$	21.8229	1.86446	0.932230	−	0.361866i	$-0.117860\pi$
$137$	21.8229	1.86446	0.932230	+	0.361866i	$0.117860\pi$
$138$	0	0
$139$	8.26857	0.701331	0.350666	−	0.936501i	$-0.385955\pi$
$139$	8.26857	0.701331	0.350666	+	0.936501i	$0.385955\pi$
$140$	3.87939	0.327868
$141$	0	0
$142$	−14.0077	−1.17550
$143$	−5.06923	−0.423910
$144$	0	0
$145$	−2.75103	−0.228461
$146$	18.8084	1.55659
$147$	0	0
$148$	4.34730	0.357346
$149$	−7.68273	−0.629394	−0.314697	−	0.949192i	$-0.601903\pi$
$149$	−7.68273	−0.629394	−0.314697	+	0.949192i	$0.601903\pi$
$150$	0	0
$151$	−5.94356	−0.483680	−0.241840	−	0.970316i	$-0.577751\pi$
$151$	−5.94356	−0.483680	−0.241840	+	0.970316i	$0.577751\pi$
$152$	0	0
$153$	0	0
$154$	9.94356	0.801275
$155$	−6.72967	−0.540540
$156$	0	0
$157$	−10.3969	−0.829765	−0.414883	−	0.909875i	$-0.636177\pi$
$157$	−10.3969	−0.829765	−0.414883	+	0.909875i	$0.636177\pi$
$158$	−3.02734	−0.240842
$159$	0	0
$160$	−6.24897	−0.494024
$161$	18.9932	1.49687
$162$	0	0
$163$	10.0719	0.788894	0.394447	−	0.918919i	$-0.370936\pi$
$163$	10.0719	0.788894	0.394447	+	0.918919i	$0.370936\pi$
$164$	−6.10607	−0.476804
$165$	0	0
$166$	1.85204	0.143746
$167$	−8.12836	−0.628991	−0.314496	−	0.949259i	$-0.601835\pi$
$167$	−8.12836	−0.628991	−0.314496	+	0.949259i	$0.601835\pi$
$168$	0	0
$169$	−5.38919	−0.414553
$170$	11.7442	0.900741
$171$	0	0
$172$	−17.5476	−1.33799
$173$	5.31315	0.403951	0.201976	−	0.979391i	$-0.435264\pi$
$173$	5.31315	0.403951	0.201976	+	0.979391i	$0.435264\pi$
$174$	0	0
$175$	12.1702	0.919984
$176$	−8.66725	−0.653319
$177$	0	0
$178$	32.0574	2.40280
$179$	17.0273	1.27268	0.636342	−	0.771407i	$-0.280447\pi$
$179$	17.0273	1.27268	0.636342	+	0.771407i	$0.280447\pi$
$180$	0	0
$181$	−17.0692	−1.26875	−0.634373	−	0.773027i	$-0.718742\pi$
$181$	−17.0692	−1.26875	−0.634373	+	0.773027i	$0.718742\pi$
$182$	−14.9290	−1.10661
$183$	0	0
$184$	−5.80066	−0.427630
$185$	−2.49525	−0.183455
$186$	0	0
$187$	13.0574	0.954850
$188$	−3.38144	−0.246617
$189$	0	0
$190$	0	0
$191$	−3.62361	−0.262195	−0.131098	−	0.991369i	$-0.541850\pi$
$191$	−3.62361	−0.262195	−0.131098	+	0.991369i	$0.541850\pi$
$192$	0	0
$193$	−5.12330	−0.368783	−0.184392	−	0.982853i	$-0.559032\pi$
$193$	−5.12330	−0.368783	−0.184392	+	0.982853i	$0.559032\pi$
$194$	11.0915	0.796325
$195$	0	0
$196$	1.97771	0.141265
$197$	9.12836	0.650368	0.325184	−	0.945651i	$-0.394574\pi$
$197$	9.12836	0.650368	0.325184	+	0.945651i	$0.394574\pi$
$198$	0	0
$199$	22.0847	1.56554	0.782772	−	0.622309i	$-0.213805\pi$
$199$	22.0847	1.56554	0.782772	+	0.622309i	$0.213805\pi$
$200$	−3.71688	−0.262823
$201$	0	0
$202$	−8.09152	−0.569317
$203$	−9.00774	−0.632219
$204$	0	0
$205$	3.50475	0.244782
$206$	−33.4688	−2.33188
$207$	0	0
$208$	13.0128	0.902275
$209$	0	0
$210$	0	0
$211$	6.72369	0.462878	0.231439	−	0.972849i	$-0.425657\pi$
$211$	6.72369	0.462878	0.231439	+	0.972849i	$0.425657\pi$
$212$	−4.14022	−0.284351
$213$	0	0
$214$	11.0642	0.756332
$215$	10.0719	0.686899
$216$	0	0
$217$	−22.0351	−1.49584
$218$	−28.4047	−1.92381
$219$	0	0
$220$	−2.47565	−0.166908
$221$	−19.6040	−1.31871
$222$	0	0
$223$	−4.57667	−0.306476	−0.153238	−	0.988189i	$-0.548970\pi$
$223$	−4.57667	−0.306476	−0.153238	+	0.988189i	$0.548970\pi$
$224$	−20.4611	−1.36712
$225$	0	0
$226$	−11.4338	−0.760563
$227$	15.8425	1.05151	0.525753	−	0.850637i	$-0.323783\pi$
$227$	15.8425	1.05151	0.525753	+	0.850637i	$0.323783\pi$
$228$	0	0
$229$	3.86753	0.255573	0.127787	−	0.991802i	$-0.459213\pi$
$229$	3.86753	0.255573	0.127787	+	0.991802i	$0.459213\pi$
$230$	−10.9017	−0.718835
$231$	0	0
$232$	2.75103	0.180614
$233$	23.7246	1.55425	0.777126	−	0.629345i	$-0.216677\pi$
$233$	23.7246	1.55425	0.777126	+	0.629345i	$0.216677\pi$
$234$	0	0
$235$	1.94087	0.126609
$236$	12.8871	0.838880
$237$	0	0
$238$	38.4543	2.49262
$239$	−4.54664	−0.294097	−0.147049	−	0.989129i	$-0.546977\pi$
$239$	−4.54664	−0.294097	−0.147049	+	0.989129i	$0.546977\pi$
$240$	0	0
$241$	−1.09327	−0.0704239	−0.0352120	−	0.999380i	$-0.511211\pi$
$241$	−1.09327	−0.0704239	−0.0352120	+	0.999380i	$0.511211\pi$
$242$	14.3277	0.921019
$243$	0	0
$244$	0.943563	0.0604054
$245$	−1.13516	−0.0725229
$246$	0	0
$247$	0	0
$248$	6.72967	0.427335
$249$	0	0
$250$	−15.2490	−0.964430
$251$	−13.7733	−0.869364	−0.434682	−	0.900584i	$-0.643139\pi$
$251$	−13.7733	−0.869364	−0.434682	+	0.900584i	$0.643139\pi$
$252$	0	0
$253$	−12.1206	−0.762017
$254$	−22.4047	−1.40579
$255$	0	0
$256$	20.7023	1.29390
$257$	−12.8675	−0.802654	−0.401327	−	0.915935i	$-0.631451\pi$
$257$	−12.8675	−0.802654	−0.401327	+	0.915935i	$0.631451\pi$
$258$	0	0
$259$	−8.17024	−0.507674
$260$	3.71688	0.230511
$261$	0	0
$262$	−4.49525	−0.277718
$263$	13.8425	0.853568	0.426784	−	0.904354i	$-0.359646\pi$
$263$	13.8425	0.853568	0.426784	+	0.904354i	$0.359646\pi$
$264$	0	0
$265$	2.37639	0.145981
$266$	0	0
$267$	0	0
$268$	−5.63041	−0.343932
$269$	13.8871	0.846713	0.423357	−	0.905963i	$-0.360852\pi$
$269$	13.8871	0.846713	0.423357	+	0.905963i	$0.360852\pi$
$270$	0	0
$271$	5.89393	0.358031	0.179015	−	0.983846i	$-0.442709\pi$
$271$	5.89393	0.358031	0.179015	+	0.983846i	$0.442709\pi$
$272$	−33.5185	−2.03236
$273$	0	0
$274$	−41.0137	−2.47773
$275$	−7.76651	−0.468338
$276$	0	0
$277$	−4.53890	−0.272716	−0.136358	−	0.990660i	$-0.543540\pi$
$277$	−4.53890	−0.272716	−0.136358	+	0.990660i	$0.543540\pi$
$278$	−15.5398	−0.932017
$279$	0	0
$280$	2.22668	0.133070
$281$	17.0324	1.01607	0.508034	−	0.861337i	$-0.330373\pi$
$281$	17.0324	1.01607	0.508034	+	0.861337i	$0.330373\pi$
$282$	0	0
$283$	22.7638	1.35317	0.676584	−	0.736365i	$-0.263459\pi$
$283$	22.7638	1.35317	0.676584	+	0.736365i	$0.263459\pi$
$284$	11.4192	0.677606
$285$	0	0
$286$	9.52704	0.563345
$287$	11.4757	0.677386
$288$	0	0
$289$	33.4962	1.97036
$290$	5.17024	0.303607
$291$	0	0
$292$	−15.3327	−0.897281
$293$	5.55943	0.324785	0.162393	−	0.986726i	$-0.448079\pi$
$293$	5.55943	0.324785	0.162393	+	0.986726i	$0.448079\pi$
$294$	0	0
$295$	−7.39693	−0.430666
$296$	2.49525	0.145034
$297$	0	0
$298$	14.4388	0.836418
$299$	18.1976	1.05239
$300$	0	0
$301$	32.9786	1.90086
$302$	11.1702	0.642775
$303$	0	0
$304$	0	0
$305$	−0.541584	−0.0310110
$306$	0	0
$307$	−11.8307	−0.675213	−0.337607	−	0.941287i	$-0.609617\pi$
$307$	−11.8307	−0.675213	−0.337607	+	0.941287i	$0.609617\pi$
$308$	−8.10607	−0.461886
$309$	0	0
$310$	12.6477	0.718338
$311$	18.0428	1.02311	0.511557	−	0.859249i	$-0.329069\pi$
$311$	18.0428	1.02311	0.511557	+	0.859249i	$0.329069\pi$
$312$	0	0
$313$	6.89393	0.389668	0.194834	−	0.980836i	$-0.437583\pi$
$313$	6.89393	0.389668	0.194834	+	0.980836i	$0.437583\pi$
$314$	19.5398	1.10270
$315$	0	0
$316$	2.46791	0.138831
$317$	33.3259	1.87177	0.935886	−	0.352304i	$-0.114602\pi$
$317$	33.3259	1.87177	0.935886	+	0.352304i	$0.114602\pi$
$318$	0	0
$319$	5.74834	0.321845
$320$	3.44831	0.192766
$321$	0	0
$322$	−35.6955	−1.98923
$323$	0	0
$324$	0	0
$325$	11.6604	0.646805
$326$	−18.9290	−1.04838
$327$	0	0
$328$	−3.50475	−0.193517
$329$	6.35504	0.350365
$330$	0	0
$331$	4.94356	0.271723	0.135861	−	0.990728i	$-0.456620\pi$
$331$	4.94356	0.271723	0.135861	+	0.990728i	$0.456620\pi$
$332$	−1.50980	−0.0828610
$333$	0	0
$334$	15.2763	0.835883
$335$	3.23173	0.176568
$336$	0	0
$337$	9.41921	0.513097	0.256549	−	0.966531i	$-0.417415\pi$
$337$	9.41921	0.513097	0.256549	+	0.966531i	$0.417415\pi$
$338$	10.1284	0.550910
$339$	0	0
$340$	−9.57398	−0.519222
$341$	14.0618	0.761490
$342$	0	0
$343$	16.4388	0.887613
$344$	−10.0719	−0.543041
$345$	0	0
$346$	−9.98545	−0.536821
$347$	11.8648	0.636938	0.318469	−	0.947933i	$-0.396831\pi$
$347$	11.8648	0.636938	0.318469	+	0.947933i	$0.396831\pi$
$348$	0	0
$349$	21.9418	1.17452	0.587259	−	0.809399i	$-0.300207\pi$
$349$	21.9418	1.17452	0.587259	+	0.809399i	$0.300207\pi$
$350$	−22.8726	−1.22259
$351$	0	0
$352$	13.0574	0.695960
$353$	−3.89393	−0.207253	−0.103627	−	0.994616i	$-0.533045\pi$
$353$	−3.89393	−0.207253	−0.103627	+	0.994616i	$0.533045\pi$
$354$	0	0
$355$	−6.55438	−0.347870
$356$	−26.1334	−1.38507
$357$	0	0
$358$	−32.0009	−1.69130
$359$	12.0051	0.633602	0.316801	−	0.948492i	$-0.397391\pi$
$359$	12.0051	0.633602	0.316801	+	0.948492i	$0.397391\pi$
$360$	0	0
$361$	0	0
$362$	32.0797	1.68607
$363$	0	0
$364$	12.1702	0.637894
$365$	8.80066	0.460648
$366$	0	0
$367$	−26.8033	−1.39912	−0.699562	−	0.714572i	$-0.746621\pi$
$367$	−26.8033	−1.39912	−0.699562	+	0.714572i	$0.746621\pi$
$368$	31.1138	1.62192
$369$	0	0
$370$	4.68954	0.243798
$371$	7.78106	0.403972
$372$	0	0
$373$	−28.5817	−1.47991	−0.739953	−	0.672659i	$-0.765152\pi$
$373$	−28.5817	−1.47991	−0.739953	+	0.672659i	$0.765152\pi$
$374$	−24.5398	−1.26892
$375$	0	0
$376$	−1.94087	−0.100093
$377$	−8.63041	−0.444489
$378$	0	0
$379$	−3.09833	−0.159150	−0.0795752	−	0.996829i	$-0.525356\pi$
$379$	−3.09833	−0.159150	−0.0795752	+	0.996829i	$0.525356\pi$
$380$	0	0
$381$	0	0
$382$	6.81016	0.348438
$383$	28.6236	1.46260	0.731299	−	0.682057i	$-0.238914\pi$
$383$	28.6236	1.46260	0.731299	+	0.682057i	$0.238914\pi$
$384$	0	0
$385$	4.65270	0.237124
$386$	9.62866	0.490086
$387$	0	0
$388$	−9.04189	−0.459032
$389$	22.4142	1.13644	0.568222	−	0.822876i	$-0.307632\pi$
$389$	22.4142	1.13644	0.568222	+	0.822876i	$0.307632\pi$
$390$	0	0
$391$	−46.8735	−2.37050
$392$	1.13516	0.0573344
$393$	0	0
$394$	−17.1557	−0.864292
$395$	−1.41653	−0.0712732
$396$	0	0
$397$	−13.7314	−0.689161	−0.344580	−	0.938757i	$-0.611979\pi$
$397$	−13.7314	−0.689161	−0.344580	+	0.938757i	$0.611979\pi$
$398$	−41.5057	−2.08049
$399$	0	0
$400$	19.9368	0.996838
$401$	29.2867	1.46251	0.731255	−	0.682104i	$-0.238935\pi$
$401$	29.2867	1.46251	0.731255	+	0.682104i	$0.238935\pi$
$402$	0	0
$403$	−21.1121	−1.05167
$404$	6.59627	0.328177
$405$	0	0
$406$	16.9290	0.840173
$407$	5.21389	0.258443
$408$	0	0
$409$	−2.33544	−0.115480	−0.0577400	−	0.998332i	$-0.518389\pi$
$409$	−2.33544	−0.115480	−0.0577400	+	0.998332i	$0.518389\pi$
$410$	−6.58677	−0.325297
$411$	0	0
$412$	27.2841	1.34419
$413$	−24.2199	−1.19178
$414$	0	0
$415$	0.866592	0.0425393
$416$	−19.6040	−0.961166
$417$	0	0
$418$	0	0
$419$	−2.27362	−0.111074	−0.0555369	−	0.998457i	$-0.517687\pi$
$419$	−2.27362	−0.111074	−0.0555369	+	0.998457i	$0.517687\pi$
$420$	0	0
$421$	19.0182	0.926889	0.463444	−	0.886126i	$-0.346613\pi$
$421$	19.0182	0.926889	0.463444	+	0.886126i	$0.346613\pi$
$422$	−12.6364	−0.615130
$423$	0	0
$424$	−2.37639	−0.115408
$425$	−30.0351	−1.45692
$426$	0	0
$427$	−1.77332	−0.0858169
$428$	−9.01960	−0.435979
$429$	0	0
$430$	−18.9290	−0.912838
$431$	12.7939	0.616258	0.308129	−	0.951345i	$-0.400297\pi$
$431$	12.7939	0.616258	0.308129	+	0.951345i	$0.400297\pi$
$432$	0	0
$433$	14.5749	0.700426	0.350213	−	0.936670i	$-0.386109\pi$
$433$	14.5749	0.700426	0.350213	+	0.936670i	$0.386109\pi$
$434$	41.4124	1.98786
$435$	0	0
$436$	23.1557	1.10896
$437$	0	0
$438$	0	0
$439$	24.6195	1.17502	0.587512	−	0.809215i	$-0.300107\pi$
$439$	24.6195	1.17502	0.587512	+	0.809215i	$0.300107\pi$
$440$	−1.42097	−0.0677421
$441$	0	0
$442$	36.8435	1.75247
$443$	−38.5354	−1.83087	−0.915436	−	0.402464i	$-0.868154\pi$
$443$	−38.5354	−1.83087	−0.915436	+	0.402464i	$0.868154\pi$
$444$	0	0
$445$	15.0000	0.711068
$446$	8.60132	0.407284
$447$	0	0
$448$	11.2909	0.533443
$449$	35.1807	1.66028	0.830139	−	0.557556i	$-0.188261\pi$
$449$	35.1807	1.66028	0.830139	+	0.557556i	$0.188261\pi$
$450$	0	0
$451$	−7.32325	−0.344839
$452$	9.32089	0.438418
$453$	0	0
$454$	−29.7743	−1.39737
$455$	−6.98545	−0.327483
$456$	0	0
$457$	−13.7983	−0.645457	−0.322729	−	0.946492i	$-0.604600\pi$
$457$	−13.7983	−0.645457	−0.322729	+	0.946492i	$0.604600\pi$
$458$	−7.26857	−0.339638
$459$	0	0
$460$	8.88713	0.414364
$461$	−24.2722	−1.13047	−0.565234	−	0.824930i	$-0.691214\pi$
$461$	−24.2722	−1.13047	−0.565234	+	0.824930i	$0.691214\pi$
$462$	0	0
$463$	−11.2909	−0.524731	−0.262365	−	0.964969i	$-0.584503\pi$
$463$	−11.2909	−0.524731	−0.262365	+	0.964969i	$0.584503\pi$
$464$	−14.7561	−0.685034
$465$	0	0
$466$	−44.5877	−2.06549
$467$	4.78787	0.221556	0.110778	−	0.993845i	$-0.464666\pi$
$467$	4.78787	0.221556	0.110778	+	0.993845i	$0.464666\pi$
$468$	0	0
$469$	10.5817	0.488618
$470$	−3.64765	−0.168254
$471$	0	0
$472$	7.39693	0.340471
$473$	−21.0455	−0.967674
$474$	0	0
$475$	0	0
$476$	−31.3482	−1.43684
$477$	0	0
$478$	8.54488	0.390834
$479$	16.9094	0.772611	0.386305	−	0.922371i	$-0.373751\pi$
$479$	16.9094	0.772611	0.386305	+	0.922371i	$0.373751\pi$
$480$	0	0
$481$	−7.82800	−0.356926
$482$	2.05468	0.0935882
$483$	0	0
$484$	−11.6800	−0.530911
$485$	5.18984	0.235659
$486$	0	0
$487$	7.76382	0.351812	0.175906	−	0.984407i	$-0.443714\pi$
$487$	7.76382	0.351812	0.175906	+	0.984407i	$0.443714\pi$
$488$	0.541584	0.0245164
$489$	0	0
$490$	2.13341	0.0963775
$491$	14.5253	0.655517	0.327758	−	0.944762i	$-0.393707\pi$
$491$	14.5253	0.655517	0.327758	+	0.944762i	$0.393707\pi$
$492$	0	0
$493$	22.2303	1.00120
$494$	0	0
$495$	0	0
$496$	−36.0969	−1.62080
$497$	−21.4611	−0.962662
$498$	0	0
$499$	28.2695	1.26552	0.632758	−	0.774349i	$-0.281923\pi$
$499$	28.2695	1.26552	0.632758	+	0.774349i	$0.281923\pi$
$500$	12.4311	0.555935
$501$	0	0
$502$	25.8854	1.15532
$503$	38.0624	1.69712	0.848560	−	0.529100i	$-0.177470\pi$
$503$	38.0624	1.69712	0.848560	+	0.529100i	$0.177470\pi$
$504$	0	0
$505$	−3.78611	−0.168480
$506$	22.7793	1.01266
$507$	0	0
$508$	18.2645	0.810354
$509$	13.7520	0.609545	0.304773	−	0.952425i	$-0.401420\pi$
$509$	13.7520	0.609545	0.304773	+	0.952425i	$0.401420\pi$
$510$	0	0
$511$	28.8161	1.27475
$512$	−25.2226	−1.11469
$513$	0	0
$514$	24.1830	1.06667
$515$	−15.6604	−0.690082
$516$	0	0
$517$	−4.05550	−0.178361
$518$	15.3550	0.674662
$519$	0	0
$520$	2.13341	0.0935561
$521$	−15.3942	−0.674434	−0.337217	−	0.941427i	$-0.609486\pi$
$521$	−15.3942	−0.674434	−0.337217	+	0.941427i	$0.609486\pi$
$522$	0	0
$523$	0.260830	0.0114053	0.00570265	−	0.999984i	$-0.498185\pi$
$523$	0.260830	0.0114053	0.00570265	+	0.999984i	$0.498185\pi$
$524$	3.66456	0.160087
$525$	0	0
$526$	−26.0155	−1.13433
$527$	54.3806	2.36886
$528$	0	0
$529$	20.5107	0.891771
$530$	−4.46616	−0.193997
$531$	0	0
$532$	0	0
$533$	10.9949	0.476244
$534$	0	0
$535$	5.17705	0.223823
$536$	−3.23173	−0.139590
$537$	0	0
$538$	−26.0993	−1.12522
$539$	2.37195	0.102167
$540$	0	0
$541$	1.16519	0.0500955	0.0250478	−	0.999686i	$-0.492026\pi$
$541$	1.16519	0.0500955	0.0250478	+	0.999686i	$0.492026\pi$
$542$	−11.0770	−0.475797
$543$	0	0
$544$	50.4962	2.16501
$545$	−13.2909	−0.569318
$546$	0	0
$547$	5.52259	0.236129	0.118065	−	0.993006i	$-0.462331\pi$
$547$	5.52259	0.236129	0.118065	+	0.993006i	$0.462331\pi$
$548$	33.4347	1.42826
$549$	0	0
$550$	14.5963	0.622387
$551$	0	0
$552$	0	0
$553$	−4.63816	−0.197234
$554$	8.53033	0.362419
$555$	0	0
$556$	12.6682	0.537251
$557$	29.6560	1.25657	0.628283	−	0.777985i	$-0.283758\pi$
$557$	29.6560	1.25657	0.628283	+	0.777985i	$0.283758\pi$
$558$	0	0
$559$	31.5972	1.33642
$560$	−11.9436	−0.504708
$561$	0	0
$562$	−32.0104	−1.35028
$563$	−12.3078	−0.518711	−0.259355	−	0.965782i	$-0.583510\pi$
$563$	−12.3078	−0.518711	−0.259355	+	0.965782i	$0.583510\pi$
$564$	0	0
$565$	−5.34998	−0.225076
$566$	−42.7820	−1.79826
$567$	0	0
$568$	6.55438	0.275016
$569$	−24.3054	−1.01894	−0.509468	−	0.860490i	$-0.670158\pi$
$569$	−24.3054	−1.01894	−0.509468	+	0.860490i	$0.670158\pi$
$570$	0	0
$571$	−7.99731	−0.334677	−0.167339	−	0.985899i	$-0.553517\pi$
$571$	−7.99731	−0.334677	−0.167339	+	0.985899i	$0.553517\pi$
$572$	−7.76651	−0.324734
$573$	0	0
$574$	−21.5672	−0.900196
$575$	27.8803	1.16269
$576$	0	0
$577$	−11.6970	−0.486951	−0.243475	−	0.969907i	$-0.578287\pi$
$577$	−11.6970	−0.486951	−0.243475	+	0.969907i	$0.578287\pi$
$578$	−62.9522	−2.61847
$579$	0	0
$580$	−4.21482	−0.175011
$581$	2.83750	0.117719
$582$	0	0
$583$	−4.96553	−0.205651
$584$	−8.80066	−0.364174
$585$	0	0
$586$	−10.4483	−0.431616
$587$	−6.49289	−0.267990	−0.133995	−	0.990982i	$-0.542781\pi$
$587$	−6.49289	−0.267990	−0.133995	+	0.990982i	$0.542781\pi$
$588$	0	0
$589$	0	0
$590$	13.9017	0.572323
$591$	0	0
$592$	−13.3841	−0.550084
$593$	−22.8821	−0.939654	−0.469827	−	0.882758i	$-0.655684\pi$
$593$	−22.8821	−0.939654	−0.469827	+	0.882758i	$0.655684\pi$
$594$	0	0
$595$	17.9932	0.737649
$596$	−11.7706	−0.482144
$597$	0	0
$598$	−34.2003	−1.39855
$599$	−27.6040	−1.12787	−0.563935	−	0.825819i	$-0.690713\pi$
$599$	−27.6040	−1.12787	−0.563935	+	0.825819i	$0.690713\pi$
$600$	0	0
$601$	−14.1310	−0.576417	−0.288209	−	0.957568i	$-0.593060\pi$
$601$	−14.1310	−0.576417	−0.288209	+	0.957568i	$0.593060\pi$
$602$	−61.9796	−2.52610
$603$	0	0
$604$	−9.10607	−0.370521
$605$	6.70409	0.272560
$606$	0	0
$607$	9.50980	0.385991	0.192995	−	0.981200i	$-0.438180\pi$
$607$	9.50980	0.385991	0.192995	+	0.981200i	$0.438180\pi$
$608$	0	0
$609$	0	0
$610$	1.01785	0.0412114
$611$	6.08883	0.246328
$612$	0	0
$613$	−42.1165	−1.70107	−0.850535	−	0.525919i	$-0.823721\pi$
$613$	−42.1165	−1.70107	−0.850535	+	0.525919i	$0.823721\pi$
$614$	22.2344	0.897308
$615$	0	0
$616$	−4.65270	−0.187463
$617$	28.9617	1.16596	0.582978	−	0.812488i	$-0.301887\pi$
$617$	28.9617	1.16596	0.582978	+	0.812488i	$0.301887\pi$
$618$	0	0
$619$	30.9195	1.24276	0.621380	−	0.783509i	$-0.286572\pi$
$619$	30.9195	1.24276	0.621380	+	0.783509i	$0.286572\pi$
$620$	−10.3105	−0.414078
$621$	0	0
$622$	−33.9094	−1.35964
$623$	49.1147	1.96774
$624$	0	0
$625$	13.9982	0.559930
$626$	−12.9564	−0.517840
$627$	0	0
$628$	−15.9290	−0.635637
$629$	20.1634	0.803969
$630$	0	0
$631$	−8.57491	−0.341362	−0.170681	−	0.985326i	$-0.554597\pi$
$631$	−8.57491	−0.341362	−0.170681	+	0.985326i	$0.554597\pi$
$632$	1.41653	0.0563464
$633$	0	0
$634$	−62.6323	−2.48745
$635$	−10.4834	−0.416021
$636$	0	0
$637$	−3.56118	−0.141099
$638$	−10.8033	−0.427709
$639$	0	0
$640$	6.01724	0.237852
$641$	−26.5229	−1.04759	−0.523796	−	0.851844i	$-0.675485\pi$
$641$	−26.5229	−1.04759	−0.523796	+	0.851844i	$0.675485\pi$
$642$	0	0
$643$	−21.0137	−0.828700	−0.414350	−	0.910118i	$-0.635991\pi$
$643$	−21.0137	−0.828700	−0.414350	+	0.910118i	$0.635991\pi$
$644$	29.0993	1.14667
$645$	0	0
$646$	0	0
$647$	−38.0273	−1.49501	−0.747505	−	0.664257i	$-0.768748\pi$
$647$	−38.0273	−1.49501	−0.747505	+	0.664257i	$0.768748\pi$
$648$	0	0
$649$	15.4561	0.606703
$650$	−21.9145	−0.859556
$651$	0	0
$652$	15.4311	0.604328
$653$	−33.1917	−1.29889	−0.649446	−	0.760408i	$-0.724999\pi$
$653$	−33.1917	−1.29889	−0.649446	+	0.760408i	$0.724999\pi$
$654$	0	0
$655$	−2.10338	−0.0821858
$656$	18.7989	0.733974
$657$	0	0
$658$	−11.9436	−0.465609
$659$	39.4201	1.53559	0.767795	−	0.640695i	$-0.221354\pi$
$659$	39.4201	1.53559	0.767795	+	0.640695i	$0.221354\pi$
$660$	0	0
$661$	−1.92808	−0.0749937	−0.0374968	−	0.999297i	$-0.511938\pi$
$661$	−1.92808	−0.0749937	−0.0374968	+	0.999297i	$0.511938\pi$
$662$	−9.29086	−0.361100
$663$	0	0
$664$	−0.866592	−0.0336303
$665$	0	0
$666$	0	0
$667$	−20.6355	−0.799008
$668$	−12.4534	−0.481835
$669$	0	0
$670$	−6.07367	−0.234646
$671$	1.13165	0.0436870
$672$	0	0
$673$	13.3865	0.516012	0.258006	−	0.966143i	$-0.416935\pi$
$673$	13.3865	0.516012	0.258006	+	0.966143i	$0.416935\pi$
$674$	−17.7023	−0.681868
$675$	0	0
$676$	−8.25671	−0.317566
$677$	−11.0419	−0.424374	−0.212187	−	0.977229i	$-0.568059\pi$
$677$	−11.0419	−0.424374	−0.212187	+	0.977229i	$0.568059\pi$
$678$	0	0
$679$	16.9932	0.652139
$680$	−5.49525	−0.210733
$681$	0	0
$682$	−26.4276	−1.01196
$683$	9.16014	0.350503	0.175252	−	0.984524i	$-0.443926\pi$
$683$	9.16014	0.350503	0.175252	+	0.984524i	$0.443926\pi$
$684$	0	0
$685$	−19.1908	−0.733242
$686$	−30.8949	−1.17957
$687$	0	0
$688$	54.0242	2.05965
$689$	7.45512	0.284017
$690$	0	0
$691$	−9.79385	−0.372576	−0.186288	−	0.982495i	$-0.559646\pi$
$691$	−9.79385	−0.372576	−0.186288	+	0.982495i	$0.559646\pi$
$692$	8.14022	0.309445
$693$	0	0
$694$	−22.2986	−0.846443
$695$	−7.27126	−0.275815
$696$	0	0
$697$	−28.3209	−1.07273
$698$	−41.2371	−1.56085
$699$	0	0
$700$	18.6459	0.704749
$701$	18.2071	0.687672	0.343836	−	0.939030i	$-0.388274\pi$
$701$	18.2071	0.687672	0.343836	+	0.939030i	$0.388274\pi$
$702$	0	0
$703$	0	0
$704$	−7.20533	−0.271561
$705$	0	0
$706$	7.31820	0.275424
$707$	−12.3969	−0.466234
$708$	0	0
$709$	42.1789	1.58406	0.792031	−	0.610481i	$-0.209024\pi$
$709$	42.1789	1.58406	0.792031	+	0.610481i	$0.209024\pi$
$710$	12.3182	0.462294
$711$	0	0
$712$	−15.0000	−0.562149
$713$	−50.4793	−1.89046
$714$	0	0
$715$	4.45781	0.166713
$716$	26.0874	0.974932
$717$	0	0
$718$	−22.5621	−0.842011
$719$	4.52259	0.168664	0.0843321	−	0.996438i	$-0.473124\pi$
$719$	4.52259	0.168664	0.0843321	+	0.996438i	$0.473124\pi$
$720$	0	0
$721$	−51.2772	−1.90966
$722$	0	0
$723$	0	0
$724$	−26.1516	−0.971916
$725$	−13.2226	−0.491074
$726$	0	0
$727$	26.6932	0.989995	0.494997	−	0.868895i	$-0.335169\pi$
$727$	26.6932	0.989995	0.494997	+	0.868895i	$0.335169\pi$
$728$	6.98545	0.258898
$729$	0	0
$730$	−16.5398	−0.612167
$731$	−81.3884	−3.01026
$732$	0	0
$733$	−5.04963	−0.186512	−0.0932562	−	0.995642i	$-0.529728\pi$
$733$	−5.04963	−0.186512	−0.0932562	+	0.995642i	$0.529728\pi$
$734$	50.3738	1.85933
$735$	0	0
$736$	−46.8735	−1.72778
$737$	−6.75278	−0.248742
$738$	0	0
$739$	−8.42427	−0.309892	−0.154946	−	0.987923i	$-0.549520\pi$
$739$	−8.42427	−0.309892	−0.154946	+	0.987923i	$0.549520\pi$
$740$	−3.82295	−0.140534
$741$	0	0
$742$	−14.6236	−0.536850
$743$	−52.9846	−1.94382	−0.971909	−	0.235358i	$-0.924374\pi$
$743$	−52.9846	−1.94382	−0.971909	+	0.235358i	$0.924374\pi$
$744$	0	0
$745$	6.75608	0.247524
$746$	53.7161	1.96668
$747$	0	0
$748$	20.0051	0.731457
$749$	16.9513	0.619387
$750$	0	0
$751$	−11.9172	−0.434863	−0.217432	−	0.976076i	$-0.569768\pi$
$751$	−11.9172	−0.434863	−0.217432	+	0.976076i	$0.569768\pi$
$752$	10.4105	0.379633
$753$	0	0
$754$	16.2199	0.590693
$755$	5.22668	0.190218
$756$	0	0
$757$	−30.8256	−1.12038	−0.560188	−	0.828365i	$-0.689271\pi$
$757$	−30.8256	−1.12038	−0.560188	+	0.828365i	$0.689271\pi$
$758$	5.82295	0.211499
$759$	0	0
$760$	0	0
$761$	−43.9053	−1.59157	−0.795783	−	0.605582i	$-0.792940\pi$
$761$	−43.9053	−1.59157	−0.795783	+	0.605582i	$0.792940\pi$
$762$	0	0
$763$	−43.5185	−1.57547
$764$	−5.55169	−0.200853
$765$	0	0
$766$	−53.7948	−1.94368
$767$	−23.2053	−0.837896
$768$	0	0
$769$	−8.86247	−0.319589	−0.159794	−	0.987150i	$-0.551083\pi$
$769$	−8.86247	−0.319589	−0.159794	+	0.987150i	$0.551083\pi$
$770$	−8.74422	−0.315120
$771$	0	0
$772$	−7.84936	−0.282504
$773$	−44.6432	−1.60570	−0.802852	−	0.596178i	$-0.796685\pi$
$773$	−44.6432	−1.60570	−0.802852	+	0.596178i	$0.796685\pi$
$774$	0	0
$775$	−32.3455	−1.16189
$776$	−5.18984	−0.186305
$777$	0	0
$778$	−42.1248	−1.51025
$779$	0	0
$780$	0	0
$781$	13.6955	0.490064
$782$	88.0934	3.15021
$783$	0	0
$784$	−6.08883	−0.217458
$785$	9.14290	0.326324
$786$	0	0
$787$	−18.5054	−0.659645	−0.329822	−	0.944043i	$-0.606989\pi$
$787$	−18.5054	−0.659645	−0.329822	+	0.944043i	$0.606989\pi$
$788$	13.9855	0.498211
$789$	0	0
$790$	2.66220	0.0947168
$791$	−17.5175	−0.622852
$792$	0	0
$793$	−1.69904	−0.0603345
$794$	25.8066	0.915844
$795$	0	0
$796$	33.8357	1.19928
$797$	15.3209	0.542694	0.271347	−	0.962482i	$-0.412531\pi$
$797$	15.3209	0.542694	0.271347	+	0.962482i	$0.412531\pi$
$798$	0	0
$799$	−15.6837	−0.554848
$800$	−30.0351	−1.06190
$801$	0	0
$802$	−55.0411	−1.94357
$803$	−18.3892	−0.648940
$804$	0	0
$805$	−16.7023	−0.588680
$806$	39.6777	1.39759
$807$	0	0
$808$	3.78611	0.133195
$809$	14.0960	0.495588	0.247794	−	0.968813i	$-0.420294\pi$
$809$	14.0960	0.495588	0.247794	+	0.968813i	$0.420294\pi$
$810$	0	0
$811$	−43.7306	−1.53559	−0.767795	−	0.640696i	$-0.778646\pi$
$811$	−43.7306	−1.53559	−0.767795	+	0.640696i	$0.778646\pi$
$812$	−13.8007	−0.484308
$813$	0	0
$814$	−9.79890	−0.343451
$815$	−8.85710	−0.310251
$816$	0	0
$817$	0	0
$818$	4.38919	0.153464
$819$	0	0
$820$	5.36959	0.187514
$821$	12.8966	0.450095	0.225048	−	0.974348i	$-0.427746\pi$
$821$	12.8966	0.450095	0.225048	+	0.974348i	$0.427746\pi$
$822$	0	0
$823$	31.0428	1.08208	0.541042	−	0.840995i	$-0.318030\pi$
$823$	31.0428	1.08208	0.541042	+	0.840995i	$0.318030\pi$
$824$	15.6604	0.545557
$825$	0	0
$826$	45.5185	1.58379
$827$	−26.6049	−0.925144	−0.462572	−	0.886582i	$-0.653073\pi$
$827$	−26.6049	−0.925144	−0.462572	+	0.886582i	$0.653073\pi$
$828$	0	0
$829$	48.3664	1.67983	0.839917	−	0.542714i	$-0.182603\pi$
$829$	48.3664	1.67983	0.839917	+	0.542714i	$0.182603\pi$
$830$	−1.62866	−0.0565316
$831$	0	0
$832$	10.8179	0.375043
$833$	9.17293	0.317823
$834$	0	0
$835$	7.14796	0.247365
$836$	0	0
$837$	0	0
$838$	4.27301	0.147609
$839$	17.4929	0.603922	0.301961	−	0.953320i	$-0.402359\pi$
$839$	17.4929	0.603922	0.301961	+	0.953320i	$0.402359\pi$
$840$	0	0
$841$	−19.2134	−0.662531
$842$	−35.7425	−1.23177
$843$	0	0
$844$	10.3013	0.354585
$845$	4.73917	0.163032
$846$	0	0
$847$	21.9513	0.754256
$848$	12.7466	0.437720
$849$	0	0
$850$	56.4475	1.93613
$851$	−18.7169	−0.641606
$852$	0	0
$853$	−56.6596	−1.93999	−0.969994	−	0.243128i	$-0.921827\pi$
$853$	−56.6596	−1.93999	−0.969994	+	0.243128i	$0.921827\pi$
$854$	3.33275	0.114044
$855$	0	0
$856$	−5.17705	−0.176948
$857$	−29.1352	−0.995238	−0.497619	−	0.867396i	$-0.665792\pi$
$857$	−29.1352	−0.995238	−0.497619	+	0.867396i	$0.665792\pi$
$858$	0	0
$859$	−17.9135	−0.611202	−0.305601	−	0.952160i	$-0.598857\pi$
$859$	−17.9135	−0.611202	−0.305601	+	0.952160i	$0.598857\pi$
$860$	15.4311	0.526195
$861$	0	0
$862$	−24.0446	−0.818962
$863$	23.2746	0.792275	0.396138	−	0.918191i	$-0.370350\pi$
$863$	23.2746	0.792275	0.396138	+	0.918191i	$0.370350\pi$
$864$	0	0
$865$	−4.67230	−0.158863
$866$	−27.3919	−0.930814
$867$	0	0
$868$	−33.7597	−1.14588
$869$	2.95987	0.100407
$870$	0	0
$871$	10.1385	0.343529
$872$	13.2909	0.450085
$873$	0	0
$874$	0	0
$875$	−23.3628	−0.789806
$876$	0	0
$877$	27.6372	0.933243	0.466621	−	0.884457i	$-0.345471\pi$
$877$	27.6372	0.933243	0.466621	+	0.884457i	$0.345471\pi$
$878$	−46.2695	−1.56152
$879$	0	0
$880$	7.62185	0.256933
$881$	−15.4037	−0.518965	−0.259482	−	0.965748i	$-0.583552\pi$
$881$	−15.4037	−0.518965	−0.259482	+	0.965748i	$0.583552\pi$
$882$	0	0
$883$	22.7314	0.764974	0.382487	−	0.923961i	$-0.375068\pi$
$883$	22.7314	0.764974	0.382487	+	0.923961i	$0.375068\pi$
$884$	−30.0351	−1.01019
$885$	0	0
$886$	72.4228	2.43309
$887$	12.0564	0.404816	0.202408	−	0.979301i	$-0.435123\pi$
$887$	12.0564	0.404816	0.202408	+	0.979301i	$0.435123\pi$
$888$	0	0
$889$	−34.3259	−1.15126
$890$	−28.1908	−0.944957
$891$	0	0
$892$	−7.01186	−0.234774
$893$	0	0
$894$	0	0
$895$	−14.9736	−0.500512
$896$	19.7023	0.658209
$897$	0	0
$898$	−66.1180	−2.20639
$899$	23.9404	0.798456
$900$	0	0
$901$	−19.2030	−0.639743
$902$	13.7632	0.458265
$903$	0	0
$904$	5.34998	0.177938
$905$	15.0104	0.498964
$906$	0	0
$907$	−35.5776	−1.18133	−0.590667	−	0.806915i	$-0.701135\pi$
$907$	−35.5776	−1.18133	−0.590667	+	0.806915i	$0.701135\pi$
$908$	24.2722	0.805501
$909$	0	0
$910$	13.1284	0.435201
$911$	−7.00505	−0.232088	−0.116044	−	0.993244i	$-0.537021\pi$
$911$	−7.00505	−0.232088	−0.116044	+	0.993244i	$0.537021\pi$
$912$	0	0
$913$	−1.81076	−0.0599276
$914$	25.9323	0.857765
$915$	0	0
$916$	5.92539	0.195780
$917$	−6.88713	−0.227433
$918$	0	0
$919$	−8.36453	−0.275920	−0.137960	−	0.990438i	$-0.544055\pi$
$919$	−8.36453	−0.275920	−0.137960	+	0.990438i	$0.544055\pi$
$920$	5.10101	0.168175
$921$	0	0
$922$	45.6168	1.50231
$923$	−20.5621	−0.676810
$924$	0	0
$925$	−11.9932	−0.394334
$926$	21.2199	0.697328
$927$	0	0
$928$	22.2303	0.729745
$929$	−50.8685	−1.66894	−0.834470	−	0.551053i	$-0.814226\pi$
$929$	−50.8685	−1.66894	−0.834470	+	0.551053i	$0.814226\pi$
$930$	0	0
$931$	0	0
$932$	36.3482	1.19063
$933$	0	0
$934$	−8.99825	−0.294432
$935$	−11.4825	−0.375517
$936$	0	0
$937$	23.5817	0.770381	0.385191	−	0.922837i	$-0.374136\pi$
$937$	23.5817	0.770381	0.385191	+	0.922837i	$0.374136\pi$
$938$	−19.8871	−0.649338
$939$	0	0
$940$	2.97359	0.0969879
$941$	44.6901	1.45686	0.728429	−	0.685122i	$-0.240251\pi$
$941$	44.6901	1.45686	0.728429	+	0.685122i	$0.240251\pi$
$942$	0	0
$943$	26.2891	0.856091
$944$	−39.6759	−1.29134
$945$	0	0
$946$	39.5526	1.28597
$947$	42.1786	1.37062	0.685310	−	0.728251i	$-0.259667\pi$
$947$	42.1786	1.37062	0.685310	+	0.728251i	$0.259667\pi$
$948$	0	0
$949$	27.6091	0.896228
$950$	0	0
$951$	0	0
$952$	−17.9932	−0.583163
$953$	−51.7502	−1.67635	−0.838177	−	0.545399i	$-0.816378\pi$
$953$	−51.7502	−1.67635	−0.838177	+	0.545399i	$0.816378\pi$
$954$	0	0
$955$	3.18655	0.103114
$956$	−6.96585	−0.225292
$957$	0	0
$958$	−31.7793	−1.02674
$959$	−62.8367	−2.02910
$960$	0	0
$961$	27.5639	0.889157
$962$	14.7118	0.474328
$963$	0	0
$964$	−1.67499	−0.0539479
$965$	4.50536	0.145033
$966$	0	0
$967$	17.8544	0.574159	0.287080	−	0.957907i	$-0.407316\pi$
$967$	17.8544	0.574159	0.287080	+	0.957907i	$0.407316\pi$
$968$	−6.70409	−0.215478
$969$	0	0
$970$	−9.75372	−0.313173
$971$	41.5449	1.33324	0.666619	−	0.745398i	$-0.267741\pi$
$971$	41.5449	1.33324	0.666619	+	0.745398i	$0.267741\pi$
$972$	0	0
$973$	−23.8084	−0.763262
$974$	−14.5912	−0.467533
$975$	0	0
$976$	−2.90497	−0.0929859
$977$	−12.3527	−0.395197	−0.197599	−	0.980283i	$-0.563314\pi$
$977$	−12.3527	−0.395197	−0.197599	+	0.980283i	$0.563314\pi$
$978$	0	0
$979$	−31.3429	−1.00172
$980$	−1.73917	−0.0555558
$981$	0	0
$982$	−27.2986	−0.871133
$983$	−16.2686	−0.518887	−0.259443	−	0.965758i	$-0.583539\pi$
$983$	−16.2686	−0.518887	−0.259443	+	0.965758i	$0.583539\pi$
$984$	0	0
$985$	−8.02734	−0.255772
$986$	−41.7793	−1.33052
$987$	0	0
$988$	0	0
$989$	75.5494	2.40233
$990$	0	0
$991$	11.0993	0.352580	0.176290	−	0.984338i	$-0.443590\pi$
$991$	11.0993	0.352580	0.176290	+	0.984338i	$0.443590\pi$
$992$	54.3806	1.72659
$993$	0	0
$994$	40.3337	1.27931
$995$	−19.4210	−0.615686
$996$	0	0
$997$	−15.6459	−0.495511	−0.247755	−	0.968823i	$-0.579693\pi$
$997$	−15.6459	−0.495511	−0.247755	+	0.968823i	$0.579693\pi$
$998$	−53.1293	−1.68178
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.w.1.1		3
3.2	odd	2		1083.2.a.m.1.3		3
19.6	even	9		171.2.u.a.55.1		6
19.16	even	9		171.2.u.a.28.1		6
19.18	odd	2		3249.2.a.x.1.3		3
57.35	odd	18		57.2.i.a.28.1	✓	6
57.44	odd	18		57.2.i.a.55.1	yes	6
57.56	even	2		1083.2.a.n.1.1		3
228.35	even	18		912.2.bo.b.769.1		6
228.215	even	18		912.2.bo.b.625.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.i.a.28.1	✓	6	57.35	odd	18
57.2.i.a.55.1	yes	6	57.44	odd	18
171.2.u.a.28.1		6	19.16	even	9
171.2.u.a.55.1		6	19.6	even	9
912.2.bo.b.625.1		6	228.215	even	18
912.2.bo.b.769.1		6	228.35	even	18
1083.2.a.m.1.3		3	3.2	odd	2
1083.2.a.n.1.1		3	57.56	even	2
3249.2.a.w.1.1		3	1.1	even	1	trivial
3249.2.a.x.1.3		3	19.18	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:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q-1.87939 q^{2} +1.53209 q^{4} -0.879385 q^{5} -2.87939 q^{7} +0.879385 q^{8} +1.65270 q^{10} +1.83750 q^{11} -2.75877 q^{13} +5.41147 q^{14} -4.71688 q^{16} +7.10607 q^{17} -1.34730 q^{20} -3.45336 q^{22} -6.59627 q^{23} -4.22668 q^{25} +5.18479 q^{26} -4.41147 q^{28} +3.12836 q^{29} +7.65270 q^{31} +7.10607 q^{32} -13.3550 q^{34} +2.53209 q^{35} +2.83750 q^{37} -0.773318 q^{40} -3.98545 q^{41} -11.4534 q^{43} +2.81521 q^{44} +12.3969 q^{46} -2.20708 q^{47} +1.29086 q^{49} +7.94356 q^{50} -4.22668 q^{52} -2.70233 q^{53} -1.61587 q^{55} -2.53209 q^{56} -5.87939 q^{58} +8.41147 q^{59} +0.615867 q^{61} -14.3824 q^{62} -3.92127 q^{64} +2.42602 q^{65} -3.67499 q^{67} +10.8871 q^{68} -4.75877 q^{70} +7.45336 q^{71} -10.0077 q^{73} -5.33275 q^{74} -5.29086 q^{77} +1.61081 q^{79} +4.14796 q^{80} +7.49020 q^{82} -0.985452 q^{83} -6.24897 q^{85} +21.5253 q^{86} +1.61587 q^{88} -17.0574 q^{89} +7.94356 q^{91} -10.1061 q^{92} +4.14796 q^{94} -5.90167 q^{97} -2.42602 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7} - 3 q^{8} + 6 q^{10} + 3 q^{11} + 3 q^{13} + 6 q^{14} - 6 q^{16} + 9 q^{17} - 3 q^{20} + 3 q^{22} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} - 9 q^{29} + 24 q^{31} + 9 q^{32} - 15 q^{34}+ \cdots - 15 q^{98}+O(q^{100}) \) 3 * q + 3 * q^5 - 3 * q^7 - 3 * q^8 + 6 * q^10 + 3 * q^11 + 3 * q^13 + 6 * q^14 - 6 * q^16 + 9 * q^17 - 3 * q^20 + 3 * q^22 - 6 * q^23 - 6 * q^25 + 12 * q^26 - 3 * q^28 - 9 * q^29 + 24 * q^31 + 9 * q^32 - 15 * q^34 + 3 * q^35 + 6 * q^37 - 9 * q^40 + 6 * q^41 - 21 * q^43 + 12 * q^44 + 9 * q^46 + 3 * q^47 - 12 * q^49 + 9 * q^50 - 6 * q^52 + 18 * q^53 + 6 * q^55 - 3 * q^56 - 12 * q^58 + 15 * q^59 - 9 * q^61 + 3 * q^62 - 3 * q^64 + 15 * q^65 - 6 * q^67 + 3 * q^68 - 3 * q^70 + 9 * q^71 - 6 * q^73 + 3 * q^74 + 9 * q^79 - 3 * q^80 + 21 * q^82 + 15 * q^83 - 6 * q^85 + 18 * q^86 - 6 * q^88 + 9 * q^91 - 18 * q^92 - 3 * q^94 - 6 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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