LMFDB - Embedded newform 2277.2.a.l.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2277,2,Mod(1,2277)] 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("2277.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2277, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2277 = 3^{2} \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2277.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,4,0,6,3,0,-3,6,0,-6,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.1819365402$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.170701.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 4x + 1 $ x^5 - x^4 - 6x^3 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 253)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.281423$ of defining polynomial
Character	$\chi$	$=$	2277.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.28142 q^{2} -0.357954 q^{4} -3.80110 q^{5} -1.85610 q^{7} -3.02154 q^{8} -4.87082 q^{10} +1.00000 q^{11} -5.27194 q^{13} -2.37845 q^{14} -3.15596 q^{16} +3.86848 q^{17} +0.195412 q^{19} +1.36062 q^{20} +1.28142 q^{22} +1.00000 q^{23} +9.44840 q^{25} -6.75559 q^{26} +0.664399 q^{28} +4.42266 q^{29} -2.08520 q^{31} +1.99896 q^{32} +4.95715 q^{34} +7.05522 q^{35} -4.77980 q^{37} +0.250406 q^{38} +11.4852 q^{40} +5.28296 q^{41} +3.21324 q^{43} -0.357954 q^{44} +1.28142 q^{46} +2.30911 q^{47} -3.55490 q^{49} +12.1074 q^{50} +1.88712 q^{52} +0.819313 q^{53} -3.80110 q^{55} +5.60827 q^{56} +5.66730 q^{58} +13.0628 q^{59} +1.76409 q^{61} -2.67202 q^{62} +8.87343 q^{64} +20.0392 q^{65} -0.481852 q^{67} -1.38474 q^{68} +9.04073 q^{70} -12.8100 q^{71} +1.48889 q^{73} -6.12494 q^{74} -0.0699487 q^{76} -1.85610 q^{77} -11.3518 q^{79} +11.9961 q^{80} +6.76970 q^{82} +15.8886 q^{83} -14.7045 q^{85} +4.11752 q^{86} -3.02154 q^{88} -1.51554 q^{89} +9.78524 q^{91} -0.357954 q^{92} +2.95895 q^{94} -0.742782 q^{95} -3.96654 q^{97} -4.55533 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{10} + 5 q^{11} - 15 q^{13} - 4 q^{14} + 8 q^{16} + 9 q^{17} - 5 q^{19} + 17 q^{20} + 4 q^{22} + 5 q^{23} + 12 q^{25} - 5 q^{26} + 8 q^{29} - 4 q^{31}+ \cdots + 33 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 6 * q^10 + 5 * q^11 - 15 * q^13 - 4 * q^14 + 8 * q^16 + 9 * q^17 - 5 * q^19 + 17 * q^20 + 4 * q^22 + 5 * q^23 + 12 * q^25 - 5 * q^26 + 8 * q^29 - 4 * q^31 + 31 * q^32 + 36 * q^34 + 20 * q^35 + 8 * q^37 + 11 * q^38 + 22 * q^40 + 6 * q^41 - 8 * q^43 + 6 * q^44 + 4 * q^46 + 34 * q^47 + 14 * q^49 - 22 * q^50 - 9 * q^52 - 2 * q^53 + 3 * q^55 - 27 * q^56 + 12 * q^58 + 13 * q^59 + 18 * q^61 - 30 * q^62 + 36 * q^64 - 4 * q^65 - 4 * q^67 + 43 * q^68 + 11 * q^70 - 10 * q^71 - 31 * q^73 - 19 * q^74 + 27 * q^76 - 3 * q^77 - 3 * q^79 - 2 * q^80 - 10 * q^82 + 43 * q^83 - 3 * q^85 + 15 * q^86 + 6 * q^88 - 15 * q^89 + 3 * q^91 + 6 * q^92 + 7 * q^94 - 3 * q^95 + 17 * q^97 + 33 * 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 $

$2$	1.28142	0.906103	0.453052	−	0.891484i	$-0.350335\pi$
$2$	1.28142	0.906103	0.453052	+	0.891484i	$0.350335\pi$
$3$	0	0
$3$	0	0
$4$	−0.357954	−0.178977
$5$	−3.80110	−1.69991	−0.849953	−	0.526859i	$-0.823370\pi$
$5$	−3.80110	−1.69991	−0.849953	+	0.526859i	$0.823370\pi$
$6$	0	0
$7$	−1.85610	−0.701539	−0.350770	−	0.936462i	$-0.614080\pi$
$7$	−1.85610	−0.701539	−0.350770	+	0.936462i	$0.614080\pi$
$8$	−3.02154	−1.06827
$9$	0	0
$10$	−4.87082	−1.54029
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.27194	−1.46217	−0.731087	−	0.682284i	$-0.760987\pi$
$13$	−5.27194	−1.46217	−0.731087	+	0.682284i	$0.760987\pi$
$14$	−2.37845	−0.635667
$15$	0	0
$16$	−3.15596	−0.788990
$17$	3.86848	0.938243	0.469122	−	0.883134i	$-0.344571\pi$
$17$	3.86848	0.938243	0.469122	+	0.883134i	$0.344571\pi$
$18$	0	0
$19$	0.195412	0.0448306	0.0224153	−	0.999749i	$-0.492864\pi$
$19$	0.195412	0.0448306	0.0224153	+	0.999749i	$0.492864\pi$
$20$	1.36062	0.304244
$21$	0	0
$22$	1.28142	0.273200
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	9.44840	1.88968
$26$	−6.75559	−1.32488
$27$	0	0
$28$	0.664399	0.125560
$29$	4.42266	0.821267	0.410633	−	0.911801i	$-0.365308\pi$
$29$	4.42266	0.821267	0.410633	+	0.911801i	$0.365308\pi$
$30$	0	0
$31$	−2.08520	−0.374512	−0.187256	−	0.982311i	$-0.559959\pi$
$31$	−2.08520	−0.374512	−0.187256	+	0.982311i	$0.559959\pi$
$32$	1.99896	0.353369
$33$	0	0
$34$	4.95715	0.850145
$35$	7.05522	1.19255
$36$	0	0
$37$	−4.77980	−0.785794	−0.392897	−	0.919583i	$-0.628527\pi$
$37$	−4.77980	−0.785794	−0.392897	+	0.919583i	$0.628527\pi$
$38$	0.250406	0.0406212
$39$	0	0
$40$	11.4852	1.81597
$41$	5.28296	0.825059	0.412530	−	0.910944i	$-0.364645\pi$
$41$	5.28296	0.825059	0.412530	+	0.910944i	$0.364645\pi$
$42$	0	0
$43$	3.21324	0.490014	0.245007	−	0.969521i	$-0.421210\pi$
$43$	3.21324	0.490014	0.245007	+	0.969521i	$0.421210\pi$
$44$	−0.357954	−0.0539637
$45$	0	0
$46$	1.28142	0.188936
$47$	2.30911	0.336819	0.168409	−	0.985717i	$-0.446137\pi$
$47$	2.30911	0.336819	0.168409	+	0.985717i	$0.446137\pi$
$48$	0	0
$49$	−3.55490	−0.507843
$50$	12.1074	1.71224
$51$	0	0
$52$	1.88712	0.261696
$53$	0.819313	0.112541	0.0562707	−	0.998416i	$-0.482079\pi$
$53$	0.819313	0.112541	0.0562707	+	0.998416i	$0.482079\pi$
$54$	0	0
$55$	−3.80110	−0.512541
$56$	5.60827	0.749437
$57$	0	0
$58$	5.66730	0.744152
$59$	13.0628	1.70063	0.850313	−	0.526278i	$-0.176413\pi$
$59$	13.0628	1.70063	0.850313	+	0.526278i	$0.176413\pi$
$60$	0	0
$61$	1.76409	0.225869	0.112934	−	0.993602i	$-0.463975\pi$
$61$	1.76409	0.225869	0.112934	+	0.993602i	$0.463975\pi$
$62$	−2.67202	−0.339347
$63$	0	0
$64$	8.87343	1.10918
$65$	20.0392	2.48556
$66$	0	0
$67$	−0.481852	−0.0588676	−0.0294338	−	0.999567i	$-0.509370\pi$
$67$	−0.481852	−0.0588676	−0.0294338	+	0.999567i	$0.509370\pi$
$68$	−1.38474	−0.167924
$69$	0	0
$70$	9.04073	1.08057
$71$	−12.8100	−1.52027	−0.760134	−	0.649767i	$-0.774867\pi$
$71$	−12.8100	−1.52027	−0.760134	+	0.649767i	$0.774867\pi$
$72$	0	0
$73$	1.48889	0.174262	0.0871309	−	0.996197i	$-0.472230\pi$
$73$	1.48889	0.174262	0.0871309	+	0.996197i	$0.472230\pi$
$74$	−6.12494	−0.712010
$75$	0	0
$76$	−0.0699487	−0.00802366
$77$	−1.85610	−0.211522
$78$	0	0
$79$	−11.3518	−1.27718	−0.638589	−	0.769548i	$-0.720482\pi$
$79$	−11.3518	−1.27718	−0.638589	+	0.769548i	$0.720482\pi$
$80$	11.9961	1.34121
$81$	0	0
$82$	6.76970	0.747589
$83$	15.8886	1.74401	0.872003	−	0.489501i	$-0.162821\pi$
$83$	15.8886	1.74401	0.872003	+	0.489501i	$0.162821\pi$
$84$	0	0
$85$	−14.7045	−1.59492
$86$	4.11752	0.444003
$87$	0	0
$88$	−3.02154	−0.322097
$89$	−1.51554	−0.160647	−0.0803233	−	0.996769i	$-0.525595\pi$
$89$	−1.51554	−0.160647	−0.0803233	+	0.996769i	$0.525595\pi$
$90$	0	0
$91$	9.78524	1.02577
$92$	−0.357954	−0.0373193
$93$	0	0
$94$	2.95895	0.305192
$95$	−0.742782	−0.0762079
$96$	0	0
$97$	−3.96654	−0.402742	−0.201371	−	0.979515i	$-0.564540\pi$
$97$	−3.96654	−0.402742	−0.201371	+	0.979515i	$0.564540\pi$
$98$	−4.55533	−0.460158
$99$	0	0
$100$	−3.38210	−0.338210
$101$	−6.46599	−0.643390	−0.321695	−	0.946843i	$-0.604253\pi$
$101$	−6.46599	−0.643390	−0.321695	+	0.946843i	$0.604253\pi$
$102$	0	0
$103$	16.7876	1.65413	0.827065	−	0.562106i	$-0.190009\pi$
$103$	16.7876	1.65413	0.827065	+	0.562106i	$0.190009\pi$
$104$	15.9294	1.56200
$105$	0	0
$106$	1.04989	0.101974
$107$	−5.73210	−0.554143	−0.277072	−	0.960849i	$-0.589364\pi$
$107$	−5.73210	−0.554143	−0.277072	+	0.960849i	$0.589364\pi$
$108$	0	0
$109$	4.12494	0.395098	0.197549	−	0.980293i	$-0.436702\pi$
$109$	4.12494	0.395098	0.197549	+	0.980293i	$0.436702\pi$
$110$	−4.87082	−0.464415
$111$	0	0
$112$	5.85777	0.553507
$113$	−4.53730	−0.426834	−0.213417	−	0.976961i	$-0.568459\pi$
$113$	−4.53730	−0.426834	−0.213417	+	0.976961i	$0.568459\pi$
$114$	0	0
$115$	−3.80110	−0.354455
$116$	−1.58311	−0.146988
$117$	0	0
$118$	16.7389	1.54094
$119$	−7.18027	−0.658214
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.26055	0.204660
$123$	0	0
$124$	0.746405	0.0670292
$125$	−16.9088	−1.51237
$126$	0	0
$127$	−13.7670	−1.22163	−0.610813	−	0.791775i	$-0.709157\pi$
$127$	−13.7670	−1.22163	−0.610813	+	0.791775i	$0.709157\pi$
$128$	7.37271	0.651661
$129$	0	0
$130$	25.6787	2.25217
$131$	−6.96530	−0.608561	−0.304280	−	0.952583i	$-0.598416\pi$
$131$	−6.96530	−0.608561	−0.304280	+	0.952583i	$0.598416\pi$
$132$	0	0
$133$	−0.362704	−0.0314504
$134$	−0.617456	−0.0533401
$135$	0	0
$136$	−11.6887	−1.00230
$137$	14.9495	1.27723	0.638613	−	0.769528i	$-0.279508\pi$
$137$	14.9495	1.27723	0.638613	+	0.769528i	$0.279508\pi$
$138$	0	0
$139$	8.06585	0.684137	0.342068	−	0.939675i	$-0.388873\pi$
$139$	8.06585	0.684137	0.342068	+	0.939675i	$0.388873\pi$
$140$	−2.52545	−0.213439
$141$	0	0
$142$	−16.4150	−1.37752
$143$	−5.27194	−0.440862
$144$	0	0
$145$	−16.8110	−1.39608
$146$	1.90790	0.157899
$147$	0	0
$148$	1.71095	0.140639
$149$	19.1168	1.56611	0.783056	−	0.621951i	$-0.213660\pi$
$149$	19.1168	1.56611	0.783056	+	0.621951i	$0.213660\pi$
$150$	0	0
$151$	−0.136973	−0.0111467	−0.00557333	−	0.999984i	$-0.501774\pi$
$151$	−0.136973	−0.0111467	−0.00557333	+	0.999984i	$0.501774\pi$
$152$	−0.590445	−0.0478914
$153$	0	0
$154$	−2.37845	−0.191661
$155$	7.92605	0.636635
$156$	0	0
$157$	−0.0660065	−0.00526789	−0.00263395	−	0.999997i	$-0.500838\pi$
$157$	−0.0660065	−0.00526789	−0.00263395	+	0.999997i	$0.500838\pi$
$158$	−14.5465	−1.15725
$159$	0	0
$160$	−7.59824	−0.600693
$161$	−1.85610	−0.146281
$162$	0	0
$163$	16.7047	1.30841	0.654207	−	0.756316i	$-0.273003\pi$
$163$	16.7047	1.30841	0.654207	+	0.756316i	$0.273003\pi$
$164$	−1.89106	−0.147667
$165$	0	0
$166$	20.3601	1.58025
$167$	−12.4899	−0.966495	−0.483248	−	0.875484i	$-0.660543\pi$
$167$	−12.4899	−0.966495	−0.483248	+	0.875484i	$0.660543\pi$
$168$	0	0
$169$	14.7934	1.13795
$170$	−18.8427	−1.44517
$171$	0	0
$172$	−1.15019	−0.0877014
$173$	20.3094	1.54410	0.772048	−	0.635564i	$-0.219232\pi$
$173$	20.3094	1.54410	0.772048	+	0.635564i	$0.219232\pi$
$174$	0	0
$175$	−17.5371	−1.32568
$176$	−3.15596	−0.237889
$177$	0	0
$178$	−1.94204	−0.145562
$179$	−3.24632	−0.242641	−0.121321	−	0.992613i	$-0.538713\pi$
$179$	−3.24632	−0.242641	−0.121321	+	0.992613i	$0.538713\pi$
$180$	0	0
$181$	10.7943	0.802333	0.401167	−	0.916005i	$-0.368605\pi$
$181$	10.7943	0.802333	0.401167	+	0.916005i	$0.368605\pi$
$182$	12.5390	0.929455
$183$	0	0
$184$	−3.02154	−0.222751
$185$	18.1685	1.33578
$186$	0	0
$187$	3.86848	0.282891
$188$	−0.826557	−0.0602829
$189$	0	0
$190$	−0.951818	−0.0690522
$191$	−14.7625	−1.06817	−0.534087	−	0.845429i	$-0.679345\pi$
$191$	−14.7625	−1.06817	−0.534087	+	0.845429i	$0.679345\pi$
$192$	0	0
$193$	10.4829	0.754579	0.377289	−	0.926095i	$-0.376856\pi$
$193$	10.4829	0.754579	0.377289	+	0.926095i	$0.376856\pi$
$194$	−5.08282	−0.364925
$195$	0	0
$196$	1.27249	0.0908923
$197$	−6.05359	−0.431301	−0.215650	−	0.976471i	$-0.569187\pi$
$197$	−6.05359	−0.431301	−0.215650	+	0.976471i	$0.569187\pi$
$198$	0	0
$199$	7.18451	0.509297	0.254648	−	0.967034i	$-0.418040\pi$
$199$	7.18451	0.509297	0.254648	+	0.967034i	$0.418040\pi$
$200$	−28.5487	−2.01870
$201$	0	0
$202$	−8.28567	−0.582978
$203$	−8.20889	−0.576151
$204$	0	0
$205$	−20.0811	−1.40252
$206$	21.5120	1.49881
$207$	0	0
$208$	16.6380	1.15364
$209$	0.195412	0.0135169
$210$	0	0
$211$	−23.1873	−1.59628	−0.798140	−	0.602472i	$-0.794183\pi$
$211$	−23.1873	−1.59628	−0.798140	+	0.602472i	$0.794183\pi$
$212$	−0.293277	−0.0201423
$213$	0	0
$214$	−7.34525	−0.502111
$215$	−12.2139	−0.832978
$216$	0	0
$217$	3.87033	0.262735
$218$	5.28580	0.357999
$219$	0	0
$220$	1.36062	0.0917331
$221$	−20.3944	−1.37187
$222$	0	0
$223$	11.3673	0.761209	0.380605	−	0.924738i	$-0.375716\pi$
$223$	11.3673	0.761209	0.380605	+	0.924738i	$0.375716\pi$
$224$	−3.71026	−0.247902
$225$	0	0
$226$	−5.81421	−0.386755
$227$	25.1503	1.66929	0.834643	−	0.550791i	$-0.185674\pi$
$227$	25.1503	1.66929	0.834643	+	0.550791i	$0.185674\pi$
$228$	0	0
$229$	−27.7272	−1.83227	−0.916133	−	0.400874i	$-0.868707\pi$
$229$	−27.7272	−1.83227	−0.916133	+	0.400874i	$0.868707\pi$
$230$	−4.87082	−0.321173
$231$	0	0
$232$	−13.3632	−0.877339
$233$	−25.1145	−1.64531	−0.822654	−	0.568543i	$-0.807507\pi$
$233$	−25.1145	−1.64531	−0.822654	+	0.568543i	$0.807507\pi$
$234$	0	0
$235$	−8.77718	−0.572560
$236$	−4.67587	−0.304373
$237$	0	0
$238$	−9.20096	−0.596410
$239$	−2.60259	−0.168348	−0.0841739	−	0.996451i	$-0.526825\pi$
$239$	−2.60259	−0.168348	−0.0841739	+	0.996451i	$0.526825\pi$
$240$	0	0
$241$	22.7867	1.46782	0.733911	−	0.679246i	$-0.237693\pi$
$241$	22.7867	1.46782	0.733911	+	0.679246i	$0.237693\pi$
$242$	1.28142	0.0823730
$243$	0	0
$244$	−0.631464	−0.0404253
$245$	13.5125	0.863285
$246$	0	0
$247$	−1.03020	−0.0655502
$248$	6.30050	0.400082
$249$	0	0
$250$	−21.6674	−1.37036
$251$	−7.91155	−0.499373	−0.249686	−	0.968327i	$-0.580328\pi$
$251$	−7.91155	−0.499373	−0.249686	+	0.968327i	$0.580328\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	−17.6414	−1.10692
$255$	0	0
$256$	−8.29930	−0.518706
$257$	−7.26003	−0.452868	−0.226434	−	0.974027i	$-0.572707\pi$
$257$	−7.26003	−0.452868	−0.226434	+	0.974027i	$0.572707\pi$
$258$	0	0
$259$	8.87177	0.551265
$260$	−7.17312	−0.444858
$261$	0	0
$262$	−8.92549	−0.551419
$263$	−3.69770	−0.228010	−0.114005	−	0.993480i	$-0.536368\pi$
$263$	−3.69770	−0.228010	−0.114005	+	0.993480i	$0.536368\pi$
$264$	0	0
$265$	−3.11430	−0.191310
$266$	−0.464778	−0.0284973
$267$	0	0
$268$	0.172481	0.0105360
$269$	31.9537	1.94825	0.974126	−	0.226005i	$-0.0725666\pi$
$269$	31.9537	1.94825	0.974126	+	0.226005i	$0.0725666\pi$
$270$	0	0
$271$	0.579921	0.0352277	0.0176138	−	0.999845i	$-0.494393\pi$
$271$	0.579921	0.0352277	0.0176138	+	0.999845i	$0.494393\pi$
$272$	−12.2088	−0.740264
$273$	0	0
$274$	19.1567	1.15730
$275$	9.44840	0.569760
$276$	0	0
$277$	6.91340	0.415386	0.207693	−	0.978194i	$-0.433404\pi$
$277$	6.91340	0.415386	0.207693	+	0.978194i	$0.433404\pi$
$278$	10.3358	0.619898
$279$	0	0
$280$	−21.3176	−1.27397
$281$	16.0724	0.958799	0.479399	−	0.877597i	$-0.340855\pi$
$281$	16.0724	0.958799	0.479399	+	0.877597i	$0.340855\pi$
$282$	0	0
$283$	3.57436	0.212473	0.106237	−	0.994341i	$-0.466120\pi$
$283$	3.57436	0.212473	0.106237	+	0.994341i	$0.466120\pi$
$284$	4.58540	0.272093
$285$	0	0
$286$	−6.75559	−0.399466
$287$	−9.80569	−0.578811
$288$	0	0
$289$	−2.03490	−0.119700
$290$	−21.5420	−1.26499
$291$	0	0
$292$	−0.532956	−0.0311889
$293$	14.7794	0.863422	0.431711	−	0.902012i	$-0.357910\pi$
$293$	14.7794	0.863422	0.431711	+	0.902012i	$0.357910\pi$
$294$	0	0
$295$	−49.6529	−2.89090
$296$	14.4423	0.839444
$297$	0	0
$298$	24.4968	1.41906
$299$	−5.27194	−0.304884
$300$	0	0
$301$	−5.96408	−0.343764
$302$	−0.175520	−0.0101000
$303$	0	0
$304$	−0.616713	−0.0353709
$305$	−6.70549	−0.383955
$306$	0	0
$307$	8.71106	0.497167	0.248583	−	0.968611i	$-0.420035\pi$
$307$	8.71106	0.497167	0.248583	+	0.968611i	$0.420035\pi$
$308$	0.664399	0.0378576
$309$	0	0
$310$	10.1566	0.576857
$311$	13.6516	0.774113	0.387057	−	0.922056i	$-0.373492\pi$
$311$	13.6516	0.774113	0.387057	+	0.922056i	$0.373492\pi$
$312$	0	0
$313$	0.389578	0.0220202	0.0110101	−	0.999939i	$-0.496495\pi$
$313$	0.389578	0.0220202	0.0110101	+	0.999939i	$0.496495\pi$
$314$	−0.0845823	−0.00477326
$315$	0	0
$316$	4.06343	0.228586
$317$	−2.69883	−0.151582	−0.0757908	−	0.997124i	$-0.524148\pi$
$317$	−2.69883	−0.151582	−0.0757908	+	0.997124i	$0.524148\pi$
$318$	0	0
$319$	4.42266	0.247621
$320$	−33.7288	−1.88550
$321$	0	0
$322$	−2.37845	−0.132546
$323$	0.755947	0.0420620
$324$	0	0
$325$	−49.8114	−2.76304
$326$	21.4058	1.18556
$327$	0	0
$328$	−15.9627	−0.881390
$329$	−4.28594	−0.236291
$330$	0	0
$331$	−5.98910	−0.329191	−0.164595	−	0.986361i	$-0.552632\pi$
$331$	−5.98910	−0.329191	−0.164595	+	0.986361i	$0.552632\pi$
$332$	−5.68741	−0.312137
$333$	0	0
$334$	−16.0048	−0.875744
$335$	1.83157	0.100069
$336$	0	0
$337$	1.52298	0.0829622	0.0414811	−	0.999139i	$-0.486792\pi$
$337$	1.52298	0.0829622	0.0414811	+	0.999139i	$0.486792\pi$
$338$	18.9566	1.03110
$339$	0	0
$340$	5.26353	0.285455
$341$	−2.08520	−0.112920
$342$	0	0
$343$	19.5909	1.05781
$344$	−9.70892	−0.523470
$345$	0	0
$346$	26.0250	1.39911
$347$	4.67011	0.250705	0.125352	−	0.992112i	$-0.459994\pi$
$347$	4.67011	0.250705	0.125352	+	0.992112i	$0.459994\pi$
$348$	0	0
$349$	9.40093	0.503220	0.251610	−	0.967829i	$-0.419040\pi$
$349$	9.40093	0.503220	0.251610	+	0.967829i	$0.419040\pi$
$350$	−22.4725	−1.20121
$351$	0	0
$352$	1.99896	0.106545
$353$	−21.2570	−1.13140	−0.565699	−	0.824612i	$-0.691393\pi$
$353$	−21.2570	−1.13140	−0.565699	+	0.824612i	$0.691393\pi$
$354$	0	0
$355$	48.6921	2.58431
$356$	0.542493	0.0287521
$357$	0	0
$358$	−4.15991	−0.219858
$359$	−28.3775	−1.49771	−0.748855	−	0.662734i	$-0.769396\pi$
$359$	−28.3775	−1.49771	−0.748855	+	0.662734i	$0.769396\pi$
$360$	0	0
$361$	−18.9618	−0.997990
$362$	13.8321	0.726997
$363$	0	0
$364$	−3.50267	−0.183590
$365$	−5.65944	−0.296229
$366$	0	0
$367$	−8.92022	−0.465631	−0.232816	−	0.972521i	$-0.574794\pi$
$367$	−8.92022	−0.465631	−0.232816	+	0.972521i	$0.574794\pi$
$368$	−3.15596	−0.164516
$369$	0	0
$370$	23.2815	1.21035
$371$	−1.52073	−0.0789522
$372$	0	0
$373$	27.2725	1.41212	0.706059	−	0.708153i	$-0.250471\pi$
$373$	27.2725	1.41212	0.706059	+	0.708153i	$0.250471\pi$
$374$	4.95715	0.256328
$375$	0	0
$376$	−6.97707	−0.359815
$377$	−23.3160	−1.20084
$378$	0	0
$379$	−16.3560	−0.840152	−0.420076	−	0.907489i	$-0.637997\pi$
$379$	−16.3560	−0.840152	−0.420076	+	0.907489i	$0.637997\pi$
$380$	0.265882	0.0136395
$381$	0	0
$382$	−18.9170	−0.967876
$383$	20.3313	1.03888	0.519439	−	0.854507i	$-0.326141\pi$
$383$	20.3313	1.03888	0.519439	+	0.854507i	$0.326141\pi$
$384$	0	0
$385$	7.05522	0.359567
$386$	13.4331	0.683726
$387$	0	0
$388$	1.41984	0.0720816
$389$	38.3557	1.94471	0.972355	−	0.233509i	$-0.0750208\pi$
$389$	38.3557	1.94471	0.972355	+	0.233509i	$0.0750208\pi$
$390$	0	0
$391$	3.86848	0.195637
$392$	10.7413	0.542516
$393$	0	0
$394$	−7.75721	−0.390803
$395$	43.1494	2.17108
$396$	0	0
$397$	21.9251	1.10039	0.550195	−	0.835036i	$-0.314553\pi$
$397$	21.9251	1.10039	0.550195	+	0.835036i	$0.314553\pi$
$398$	9.20640	0.461475
$399$	0	0
$400$	−29.8188	−1.49094
$401$	−4.54119	−0.226776	−0.113388	−	0.993551i	$-0.536170\pi$
$401$	−4.54119	−0.226776	−0.113388	+	0.993551i	$0.536170\pi$
$402$	0	0
$403$	10.9930	0.547602
$404$	2.31453	0.115152
$405$	0	0
$406$	−10.5191	−0.522052
$407$	−4.77980	−0.236926
$408$	0	0
$409$	−8.51157	−0.420870	−0.210435	−	0.977608i	$-0.567488\pi$
$409$	−8.51157	−0.420870	−0.210435	+	0.977608i	$0.567488\pi$
$410$	−25.7323	−1.27083
$411$	0	0
$412$	−6.00919	−0.296052
$413$	−24.2458	−1.19306
$414$	0	0
$415$	−60.3944	−2.96465
$416$	−10.5384	−0.516687
$417$	0	0
$418$	0.250406	0.0122477
$419$	14.6838	0.717351	0.358675	−	0.933462i	$-0.383229\pi$
$419$	14.6838	0.717351	0.358675	+	0.933462i	$0.383229\pi$
$420$	0	0
$421$	21.4822	1.04698	0.523489	−	0.852032i	$-0.324630\pi$
$421$	21.4822	1.04698	0.523489	+	0.852032i	$0.324630\pi$
$422$	−29.7128	−1.44639
$423$	0	0
$424$	−2.47559	−0.120225
$425$	36.5509	1.77298
$426$	0	0
$427$	−3.27432	−0.158456
$428$	2.05183	0.0991790
$429$	0	0
$430$	−15.6511	−0.754764
$431$	12.3258	0.593711	0.296856	−	0.954922i	$-0.404062\pi$
$431$	12.3258	0.593711	0.296856	+	0.954922i	$0.404062\pi$
$432$	0	0
$433$	4.18186	0.200968	0.100484	−	0.994939i	$-0.467961\pi$
$433$	4.18186	0.200968	0.100484	+	0.994939i	$0.467961\pi$
$434$	4.95953	0.238065
$435$	0	0
$436$	−1.47654	−0.0707135
$437$	0.195412	0.00934783
$438$	0	0
$439$	−31.3602	−1.49674	−0.748370	−	0.663282i	$-0.769163\pi$
$439$	−31.3602	−1.49674	−0.748370	+	0.663282i	$0.769163\pi$
$440$	11.4852	0.547535
$441$	0	0
$442$	−26.1338	−1.24306
$443$	37.6486	1.78874	0.894370	−	0.447328i	$-0.147624\pi$
$443$	37.6486	1.78874	0.894370	+	0.447328i	$0.147624\pi$
$444$	0	0
$445$	5.76072	0.273084
$446$	14.5663	0.689734
$447$	0	0
$448$	−16.4700	−0.778132
$449$	−40.6798	−1.91980	−0.959899	−	0.280345i	$-0.909551\pi$
$449$	−40.6798	−1.91980	−0.959899	+	0.280345i	$0.909551\pi$
$450$	0	0
$451$	5.28296	0.248765
$452$	1.62415	0.0763935
$453$	0	0
$454$	32.2282	1.51255
$455$	−37.1947	−1.74372
$456$	0	0
$457$	30.7326	1.43761	0.718804	−	0.695213i	$-0.244690\pi$
$457$	30.7326	1.43761	0.718804	+	0.695213i	$0.244690\pi$
$458$	−35.5303	−1.66022
$459$	0	0
$460$	1.36062	0.0634393
$461$	20.8666	0.971856	0.485928	−	0.873999i	$-0.338482\pi$
$461$	20.8666	0.971856	0.485928	+	0.873999i	$0.338482\pi$
$462$	0	0
$463$	23.2506	1.08055	0.540274	−	0.841489i	$-0.318321\pi$
$463$	23.2506	1.08055	0.540274	+	0.841489i	$0.318321\pi$
$464$	−13.9577	−0.647971
$465$	0	0
$466$	−32.1823	−1.49082
$467$	13.0604	0.604365	0.302183	−	0.953250i	$-0.402285\pi$
$467$	13.0604	0.604365	0.302183	+	0.953250i	$0.402285\pi$
$468$	0	0
$469$	0.894364	0.0412979
$470$	−11.2473	−0.518798
$471$	0	0
$472$	−39.4696	−1.81674
$473$	3.21324	0.147745
$474$	0	0
$475$	1.84633	0.0847155
$476$	2.57021	0.117805
$477$	0	0
$478$	−3.33502	−0.152540
$479$	5.70788	0.260800	0.130400	−	0.991461i	$-0.458374\pi$
$479$	5.70788	0.260800	0.130400	+	0.991461i	$0.458374\pi$
$480$	0	0
$481$	25.1988	1.14897
$482$	29.1994	1.33000
$483$	0	0
$484$	−0.357954	−0.0162707
$485$	15.0772	0.684623
$486$	0	0
$487$	23.0832	1.04600	0.523000	−	0.852333i	$-0.324813\pi$
$487$	23.0832	1.04600	0.523000	+	0.852333i	$0.324813\pi$
$488$	−5.33027	−0.241290
$489$	0	0
$490$	17.3153	0.782225
$491$	−15.6115	−0.704536	−0.352268	−	0.935899i	$-0.614589\pi$
$491$	−15.6115	−0.704536	−0.352268	+	0.935899i	$0.614589\pi$
$492$	0	0
$493$	17.1089	0.770548
$494$	−1.32012	−0.0593952
$495$	0	0
$496$	6.58079	0.295486
$497$	23.7766	1.06653
$498$	0	0
$499$	−8.07097	−0.361306	−0.180653	−	0.983547i	$-0.557821\pi$
$499$	−8.07097	−0.361306	−0.180653	+	0.983547i	$0.557821\pi$
$500$	6.05259	0.270680
$501$	0	0
$502$	−10.1380	−0.452483
$503$	34.6008	1.54277	0.771386	−	0.636367i	$-0.219564\pi$
$503$	34.6008	1.54277	0.771386	+	0.636367i	$0.219564\pi$
$504$	0	0
$505$	24.5779	1.09370
$506$	1.28142	0.0569662
$507$	0	0
$508$	4.92797	0.218643
$509$	−15.0158	−0.665563	−0.332781	−	0.943004i	$-0.607987\pi$
$509$	−15.0158	−0.665563	−0.332781	+	0.943004i	$0.607987\pi$
$510$	0	0
$511$	−2.76353	−0.122251
$512$	−25.3803	−1.12166
$513$	0	0
$514$	−9.30317	−0.410345
$515$	−63.8114	−2.81187
$516$	0	0
$517$	2.30911	0.101555
$518$	11.3685	0.499503
$519$	0	0
$520$	−60.5492	−2.65526
$521$	25.1881	1.10351	0.551755	−	0.834006i	$-0.313958\pi$
$521$	25.1881	1.10351	0.551755	+	0.834006i	$0.313958\pi$
$522$	0	0
$523$	−32.0155	−1.39994	−0.699970	−	0.714172i	$-0.746804\pi$
$523$	−32.0155	−1.39994	−0.699970	+	0.714172i	$0.746804\pi$
$524$	2.49326	0.108919
$525$	0	0
$526$	−4.73832	−0.206601
$527$	−8.06653	−0.351383
$528$	0	0
$529$	1.00000	0.0434783
$530$	−3.99073	−0.173346
$531$	0	0
$532$	0.129832	0.00562891
$533$	−27.8514	−1.20638
$534$	0	0
$535$	21.7883	0.941991
$536$	1.45593	0.0628867
$537$	0	0
$538$	40.9462	1.76532
$539$	−3.55490	−0.153120
$540$	0	0
$541$	−14.2450	−0.612439	−0.306219	−	0.951961i	$-0.599064\pi$
$541$	−14.2450	−0.612439	−0.306219	+	0.951961i	$0.599064\pi$
$542$	0.743124	0.0319199
$543$	0	0
$544$	7.73291	0.331546
$545$	−15.6793	−0.671629
$546$	0	0
$547$	9.20117	0.393414	0.196707	−	0.980462i	$-0.436975\pi$
$547$	9.20117	0.393414	0.196707	+	0.980462i	$0.436975\pi$
$548$	−5.35126	−0.228594
$549$	0	0
$550$	12.1074	0.516261
$551$	0.864241	0.0368179
$552$	0	0
$553$	21.0701	0.895990
$554$	8.85900	0.376383
$555$	0	0
$556$	−2.88721	−0.122445
$557$	17.4388	0.738907	0.369453	−	0.929249i	$-0.379545\pi$
$557$	17.4388	0.738907	0.369453	+	0.929249i	$0.379545\pi$
$558$	0	0
$559$	−16.9400	−0.716486
$560$	−22.2660	−0.940910
$561$	0	0
$562$	20.5955	0.868771
$563$	−7.12652	−0.300347	−0.150174	−	0.988660i	$-0.547983\pi$
$563$	−7.12652	−0.300347	−0.150174	+	0.988660i	$0.547983\pi$
$564$	0	0
$565$	17.2468	0.725577
$566$	4.58026	0.192523
$567$	0	0
$568$	38.7059	1.62406
$569$	−38.9361	−1.63229	−0.816143	−	0.577849i	$-0.803892\pi$
$569$	−38.9361	−1.63229	−0.816143	+	0.577849i	$0.803892\pi$
$570$	0	0
$571$	29.3981	1.23027	0.615136	−	0.788421i	$-0.289101\pi$
$571$	29.3981	1.23027	0.615136	+	0.788421i	$0.289101\pi$
$572$	1.88712	0.0789043
$573$	0	0
$574$	−12.5652	−0.524463
$575$	9.44840	0.394025
$576$	0	0
$577$	−20.3394	−0.846740	−0.423370	−	0.905957i	$-0.639153\pi$
$577$	−20.3394	−0.846740	−0.423370	+	0.905957i	$0.639153\pi$
$578$	−2.60757	−0.108460
$579$	0	0
$580$	6.01757	0.249866
$581$	−29.4909	−1.22349
$582$	0	0
$583$	0.819313	0.0339325
$584$	−4.49875	−0.186159
$585$	0	0
$586$	18.9387	0.782350
$587$	−0.714390	−0.0294860	−0.0147430	−	0.999891i	$-0.504693\pi$
$587$	−0.714390	−0.0294860	−0.0147430	+	0.999891i	$0.504693\pi$
$588$	0	0
$589$	−0.407473	−0.0167896
$590$	−63.6264	−2.61946
$591$	0	0
$592$	15.0848	0.619983
$593$	3.17833	0.130518	0.0652592	−	0.997868i	$-0.479213\pi$
$593$	3.17833	0.130518	0.0652592	+	0.997868i	$0.479213\pi$
$594$	0	0
$595$	27.2930	1.11890
$596$	−6.84296	−0.280298
$597$	0	0
$598$	−6.75559	−0.276257
$599$	−15.1347	−0.618387	−0.309194	−	0.950999i	$-0.600059\pi$
$599$	−15.1347	−0.618387	−0.309194	+	0.950999i	$0.600059\pi$
$600$	0	0
$601$	−29.8591	−1.21798	−0.608990	−	0.793178i	$-0.708425\pi$
$601$	−29.8591	−1.21798	−0.608990	+	0.793178i	$0.708425\pi$
$602$	−7.64252	−0.311486
$603$	0	0
$604$	0.0490299	0.00199500
$605$	−3.80110	−0.154537
$606$	0	0
$607$	33.7487	1.36982	0.684909	−	0.728629i	$-0.259842\pi$
$607$	33.7487	1.36982	0.684909	+	0.728629i	$0.259842\pi$
$608$	0.390620	0.0158417
$609$	0	0
$610$	−8.59257	−0.347903
$611$	−12.1735	−0.492487
$612$	0	0
$613$	35.5376	1.43535	0.717675	−	0.696378i	$-0.245206\pi$
$613$	35.5376	1.43535	0.717675	+	0.696378i	$0.245206\pi$
$614$	11.1626	0.450484
$615$	0	0
$616$	5.60827	0.225964
$617$	8.52643	0.343261	0.171631	−	0.985161i	$-0.445096\pi$
$617$	8.52643	0.343261	0.171631	+	0.985161i	$0.445096\pi$
$618$	0	0
$619$	10.7185	0.430815	0.215407	−	0.976524i	$-0.430892\pi$
$619$	10.7185	0.430815	0.215407	+	0.976524i	$0.430892\pi$
$620$	−2.83716	−0.113943
$621$	0	0
$622$	17.4935	0.701426
$623$	2.81299	0.112700
$624$	0	0
$625$	17.0302	0.681208
$626$	0.499214	0.0199526
$627$	0	0
$628$	0.0236273	0.000942833 0
$629$	−18.4905	−0.737265
$630$	0	0
$631$	−5.06204	−0.201517	−0.100758	−	0.994911i	$-0.532127\pi$
$631$	−5.06204	−0.201517	−0.100758	+	0.994911i	$0.532127\pi$
$632$	34.2999	1.36438
$633$	0	0
$634$	−3.45835	−0.137349
$635$	52.3299	2.07665
$636$	0	0
$637$	18.7412	0.742555
$638$	5.66730	0.224370
$639$	0	0
$640$	−28.0244	−1.10776
$641$	−6.71512	−0.265231	−0.132616	−	0.991168i	$-0.542338\pi$
$641$	−6.71512	−0.265231	−0.132616	+	0.991168i	$0.542338\pi$
$642$	0	0
$643$	10.3874	0.409638	0.204819	−	0.978800i	$-0.434339\pi$
$643$	10.3874	0.409638	0.204819	+	0.978800i	$0.434339\pi$
$644$	0.664399	0.0261810
$645$	0	0
$646$	0.968688	0.0381125
$647$	−3.54279	−0.139281	−0.0696407	−	0.997572i	$-0.522185\pi$
$647$	−3.54279	−0.139281	−0.0696407	+	0.997572i	$0.522185\pi$
$648$	0	0
$649$	13.0628	0.512758
$650$	−63.8295	−2.50360
$651$	0	0
$652$	−5.97952	−0.234176
$653$	−29.8393	−1.16770	−0.583851	−	0.811861i	$-0.698455\pi$
$653$	−29.8393	−1.16770	−0.583851	+	0.811861i	$0.698455\pi$
$654$	0	0
$655$	26.4758	1.03450
$656$	−16.6728	−0.650963
$657$	0	0
$658$	−5.49210	−0.214104
$659$	46.1786	1.79886	0.899432	−	0.437062i	$-0.143981\pi$
$659$	46.1786	1.79886	0.899432	+	0.437062i	$0.143981\pi$
$660$	0	0
$661$	−13.0358	−0.507032	−0.253516	−	0.967331i	$-0.581587\pi$
$661$	−13.0358	−0.507032	−0.253516	+	0.967331i	$0.581587\pi$
$662$	−7.67457	−0.298281
$663$	0	0
$664$	−48.0082	−1.86308
$665$	1.37868	0.0534628
$666$	0	0
$667$	4.42266	0.171246
$668$	4.47081	0.172981
$669$	0	0
$670$	2.34702	0.0906731
$671$	1.76409	0.0681020
$672$	0	0
$673$	1.49971	0.0578097	0.0289049	−	0.999582i	$-0.490798\pi$
$673$	1.49971	0.0578097	0.0289049	+	0.999582i	$0.490798\pi$
$674$	1.95159	0.0751723
$675$	0	0
$676$	−5.29536	−0.203668
$677$	−37.4924	−1.44095	−0.720474	−	0.693482i	$-0.756076\pi$
$677$	−37.4924	−1.44095	−0.720474	+	0.693482i	$0.756076\pi$
$678$	0	0
$679$	7.36230	0.282539
$680$	44.4301	1.70382
$681$	0	0
$682$	−2.67202	−0.102317
$683$	−4.06400	−0.155505	−0.0777523	−	0.996973i	$-0.524774\pi$
$683$	−4.06400	−0.155505	−0.0777523	+	0.996973i	$0.524774\pi$
$684$	0	0
$685$	−56.8248	−2.17116
$686$	25.1043	0.958486
$687$	0	0
$688$	−10.1408	−0.386616
$689$	−4.31937	−0.164555
$690$	0	0
$691$	4.81991	0.183358	0.0916790	−	0.995789i	$-0.470777\pi$
$691$	4.81991	0.183358	0.0916790	+	0.995789i	$0.470777\pi$
$692$	−7.26985	−0.276358
$693$	0	0
$694$	5.98439	0.227164
$695$	−30.6591	−1.16297
$696$	0	0
$697$	20.4370	0.774106
$698$	12.0466	0.455970
$699$	0	0
$700$	6.27750	0.237267
$701$	−5.75429	−0.217337	−0.108668	−	0.994078i	$-0.534659\pi$
$701$	−5.75429	−0.217337	−0.108668	+	0.994078i	$0.534659\pi$
$702$	0	0
$703$	−0.934031	−0.0352276
$704$	8.87343	0.334430
$705$	0	0
$706$	−27.2393	−1.02516
$707$	12.0015	0.451363
$708$	0	0
$709$	−12.4167	−0.466318	−0.233159	−	0.972439i	$-0.574906\pi$
$709$	−12.4167	−0.466318	−0.233159	+	0.972439i	$0.574906\pi$
$710$	62.3952	2.34165
$711$	0	0
$712$	4.57925	0.171615
$713$	−2.08520	−0.0780912
$714$	0	0
$715$	20.0392	0.749424
$716$	1.16203	0.0434272
$717$	0	0
$718$	−36.3636	−1.35708
$719$	12.2243	0.455888	0.227944	−	0.973674i	$-0.426800\pi$
$719$	12.2243	0.455888	0.227944	+	0.973674i	$0.426800\pi$
$720$	0	0
$721$	−31.1594	−1.16044
$722$	−24.2981	−0.904282
$723$	0	0
$724$	−3.86387	−0.143599
$725$	41.7870	1.55193
$726$	0	0
$727$	−25.2721	−0.937291	−0.468646	−	0.883386i	$-0.655258\pi$
$727$	−25.2721	−0.937291	−0.468646	+	0.883386i	$0.655258\pi$
$728$	−29.5665	−1.09581
$729$	0	0
$730$	−7.25214	−0.268414
$731$	12.4303	0.459752
$732$	0	0
$733$	2.34082	0.0864600	0.0432300	−	0.999065i	$-0.486235\pi$
$733$	2.34082	0.0864600	0.0432300	+	0.999065i	$0.486235\pi$
$734$	−11.4306	−0.421910
$735$	0	0
$736$	1.99896	0.0736825
$737$	−0.481852	−0.0177492
$738$	0	0
$739$	17.6494	0.649245	0.324623	−	0.945844i	$-0.394763\pi$
$739$	17.6494	0.649245	0.324623	+	0.945844i	$0.394763\pi$
$740$	−6.50350	−0.239073
$741$	0	0
$742$	−1.94869	−0.0715388
$743$	25.6263	0.940137	0.470069	−	0.882630i	$-0.344229\pi$
$743$	25.6263	0.940137	0.470069	+	0.882630i	$0.344229\pi$
$744$	0	0
$745$	−72.6651	−2.66224
$746$	34.9476	1.27952
$747$	0	0
$748$	−1.38474	−0.0506310
$749$	10.6393	0.388753
$750$	0	0
$751$	29.0528	1.06015	0.530077	−	0.847950i	$-0.322163\pi$
$751$	29.0528	1.06015	0.530077	+	0.847950i	$0.322163\pi$
$752$	−7.28746	−0.265746
$753$	0	0
$754$	−29.8777	−1.08808
$755$	0.520647	0.0189483
$756$	0	0
$757$	35.1534	1.27767	0.638837	−	0.769342i	$-0.279416\pi$
$757$	35.1534	1.27767	0.638837	+	0.769342i	$0.279416\pi$
$758$	−20.9590	−0.761264
$759$	0	0
$760$	2.24434	0.0814109
$761$	28.8811	1.04694	0.523469	−	0.852045i	$-0.324638\pi$
$761$	28.8811	1.04694	0.523469	+	0.852045i	$0.324638\pi$
$762$	0	0
$763$	−7.65630	−0.277177
$764$	5.28429	0.191179
$765$	0	0
$766$	26.0529	0.941331
$767$	−68.8661	−2.48661
$768$	0	0
$769$	−43.5547	−1.57062	−0.785311	−	0.619102i	$-0.787497\pi$
$769$	−43.5547	−1.57062	−0.785311	+	0.619102i	$0.787497\pi$
$770$	9.04073	0.325805
$771$	0	0
$772$	−3.75242	−0.135052
$773$	−31.2181	−1.12284	−0.561418	−	0.827532i	$-0.689744\pi$
$773$	−31.2181	−1.12284	−0.561418	+	0.827532i	$0.689744\pi$
$774$	0	0
$775$	−19.7018	−0.707708
$776$	11.9851	0.430239
$777$	0	0
$778$	49.1498	1.76211
$779$	1.03235	0.0369879
$780$	0	0
$781$	−12.8100	−0.458378
$782$	4.95715	0.177267
$783$	0	0
$784$	11.2191	0.400683
$785$	0.250898	0.00895492
$786$	0	0
$787$	−17.9596	−0.640190	−0.320095	−	0.947385i	$-0.603715\pi$
$787$	−17.9596	−0.640190	−0.320095	+	0.947385i	$0.603715\pi$
$788$	2.16691	0.0771930
$789$	0	0
$790$	55.2926	1.96722
$791$	8.42168	0.299441
$792$	0	0
$793$	−9.30019	−0.330259
$794$	28.0954	0.997068
$795$	0	0
$796$	−2.57173	−0.0911525
$797$	17.1452	0.607315	0.303657	−	0.952781i	$-0.401792\pi$
$797$	17.1452	0.607315	0.303657	+	0.952781i	$0.401792\pi$
$798$	0	0
$799$	8.93274	0.316018
$800$	18.8869	0.667753
$801$	0	0
$802$	−5.81919	−0.205483
$803$	1.48889	0.0525419
$804$	0	0
$805$	7.05522	0.248664
$806$	14.0867	0.496184
$807$	0	0
$808$	19.5372	0.687318
$809$	−50.8251	−1.78692	−0.893458	−	0.449146i	$-0.851728\pi$
$809$	−50.8251	−1.78692	−0.893458	+	0.449146i	$0.851728\pi$
$810$	0	0
$811$	−25.5494	−0.897162	−0.448581	−	0.893742i	$-0.648070\pi$
$811$	−25.5494	−0.897162	−0.448581	+	0.893742i	$0.648070\pi$
$812$	2.93841	0.103118
$813$	0	0
$814$	−6.12494	−0.214679
$815$	−63.4963	−2.22418
$816$	0	0
$817$	0.627906	0.0219676
$818$	−10.9069	−0.381351
$819$	0	0
$820$	7.18811	0.251020
$821$	14.0095	0.488936	0.244468	−	0.969657i	$-0.421387\pi$
$821$	14.0095	0.488936	0.244468	+	0.969657i	$0.421387\pi$
$822$	0	0
$823$	−20.8927	−0.728273	−0.364136	−	0.931346i	$-0.618636\pi$
$823$	−20.8927	−0.728273	−0.364136	+	0.931346i	$0.618636\pi$
$824$	−50.7243	−1.76707
$825$	0	0
$826$	−31.0691	−1.08103
$827$	37.9506	1.31967	0.659836	−	0.751409i	$-0.270626\pi$
$827$	37.9506	1.31967	0.659836	+	0.751409i	$0.270626\pi$
$828$	0	0
$829$	7.05288	0.244957	0.122478	−	0.992471i	$-0.460916\pi$
$829$	7.05288	0.244957	0.122478	+	0.992471i	$0.460916\pi$
$830$	−77.3908	−2.68627
$831$	0	0
$832$	−46.7802	−1.62181
$833$	−13.7520	−0.476480
$834$	0	0
$835$	47.4753	1.64295
$836$	−0.0699487	−0.00241923
$837$	0	0
$838$	18.8162	0.649994
$839$	21.9542	0.757942	0.378971	−	0.925409i	$-0.376278\pi$
$839$	21.9542	0.757942	0.378971	+	0.925409i	$0.376278\pi$
$840$	0	0
$841$	−9.44010	−0.325521
$842$	27.5278	0.948670
$843$	0	0
$844$	8.30000	0.285698
$845$	−56.2312	−1.93441
$846$	0	0
$847$	−1.85610	−0.0637763
$848$	−2.58572	−0.0887940
$849$	0	0
$850$	46.8372	1.60650
$851$	−4.77980	−0.163849
$852$	0	0
$853$	12.3263	0.422043	0.211022	−	0.977481i	$-0.432321\pi$
$853$	12.3263	0.422043	0.211022	+	0.977481i	$0.432321\pi$
$854$	−4.19580	−0.143577
$855$	0	0
$856$	17.3198	0.591977
$857$	7.42600	0.253667	0.126834	−	0.991924i	$-0.459519\pi$
$857$	7.42600	0.253667	0.126834	+	0.991924i	$0.459519\pi$
$858$	0	0
$859$	−10.8222	−0.369247	−0.184624	−	0.982809i	$-0.559107\pi$
$859$	−10.8222	−0.369247	−0.184624	+	0.982809i	$0.559107\pi$
$860$	4.37200	0.149084
$861$	0	0
$862$	15.7945	0.537964
$863$	29.5880	1.00719	0.503593	−	0.863941i	$-0.332011\pi$
$863$	29.5880	1.00719	0.503593	+	0.863941i	$0.332011\pi$
$864$	0	0
$865$	−77.1982	−2.62482
$866$	5.35874	0.182097
$867$	0	0
$868$	−1.38540	−0.0470236
$869$	−11.3518	−0.385084
$870$	0	0
$871$	2.54030	0.0860746
$872$	−12.4637	−0.422073
$873$	0	0
$874$	0.250406	0.00847010
$875$	31.3844	1.06099
$876$	0	0
$877$	−20.6128	−0.696045	−0.348023	−	0.937486i	$-0.613147\pi$
$877$	−20.6128	−0.696045	−0.348023	+	0.937486i	$0.613147\pi$
$878$	−40.1856	−1.35620
$879$	0	0
$880$	11.9961	0.404390
$881$	−0.245259	−0.00826298	−0.00413149	−	0.999991i	$-0.501315\pi$
$881$	−0.245259	−0.00826298	−0.00413149	+	0.999991i	$0.501315\pi$
$882$	0	0
$883$	−6.05542	−0.203781	−0.101891	−	0.994796i	$-0.532489\pi$
$883$	−6.05542	−0.203781	−0.101891	+	0.994796i	$0.532489\pi$
$884$	7.30026	0.245534
$885$	0	0
$886$	48.2438	1.62078
$887$	1.26377	0.0424333	0.0212167	−	0.999775i	$-0.493246\pi$
$887$	1.26377	0.0424333	0.0212167	+	0.999775i	$0.493246\pi$
$888$	0	0
$889$	25.5530	0.857019
$890$	7.38191	0.247442
$891$	0	0
$892$	−4.06897	−0.136239
$893$	0.451229	0.0150998
$894$	0	0
$895$	12.3396	0.412467
$896$	−13.6845	−0.457166
$897$	0	0
$898$	−52.1280	−1.73954
$899$	−9.22211	−0.307574
$900$	0	0
$901$	3.16949	0.105591
$902$	6.76970	0.225406
$903$	0	0
$904$	13.7096	0.455976
$905$	−41.0302	−1.36389
$906$	0	0
$907$	−13.9710	−0.463900	−0.231950	−	0.972728i	$-0.574511\pi$
$907$	−13.9710	−0.463900	−0.231950	+	0.972728i	$0.574511\pi$
$908$	−9.00267	−0.298764
$909$	0	0
$910$	−47.6622	−1.57999
$911$	32.0863	1.06307	0.531533	−	0.847037i	$-0.321616\pi$
$911$	32.0863	1.06307	0.531533	+	0.847037i	$0.321616\pi$
$912$	0	0
$913$	15.8886	0.525838
$914$	39.3814	1.30262
$915$	0	0
$916$	9.92508	0.327934
$917$	12.9283	0.426929
$918$	0	0
$919$	9.71485	0.320463	0.160232	−	0.987079i	$-0.448776\pi$
$919$	9.71485	0.320463	0.160232	+	0.987079i	$0.448776\pi$
$920$	11.4852	0.378655
$921$	0	0
$922$	26.7390	0.880601
$923$	67.5336	2.22290
$924$	0	0
$925$	−45.1614	−1.48490
$926$	29.7939	0.979088
$927$	0	0
$928$	8.84069	0.290210
$929$	37.7506	1.23856	0.619279	−	0.785171i	$-0.287425\pi$
$929$	37.7506	1.23856	0.619279	+	0.785171i	$0.287425\pi$
$930$	0	0
$931$	−0.694671	−0.0227669
$932$	8.98986	0.294473
$933$	0	0
$934$	16.7360	0.547617
$935$	−14.7045	−0.480888
$936$	0	0
$937$	−45.6896	−1.49262	−0.746308	−	0.665601i	$-0.768175\pi$
$937$	−45.6896	−1.49262	−0.746308	+	0.665601i	$0.768175\pi$
$938$	1.14606	0.0374202
$939$	0	0
$940$	3.14183	0.102475
$941$	−8.70849	−0.283889	−0.141944	−	0.989875i	$-0.545335\pi$
$941$	−8.70849	−0.283889	−0.141944	+	0.989875i	$0.545335\pi$
$942$	0	0
$943$	5.28296	0.172037
$944$	−41.2255	−1.34178
$945$	0	0
$946$	4.11752	0.133872
$947$	54.0484	1.75634	0.878168	−	0.478352i	$-0.158766\pi$
$947$	54.0484	1.75634	0.878168	+	0.478352i	$0.158766\pi$
$948$	0	0
$949$	−7.84936	−0.254801
$950$	2.36593	0.0767610
$951$	0	0
$952$	21.6955	0.703154
$953$	42.0502	1.36214	0.681069	−	0.732219i	$-0.261515\pi$
$953$	42.0502	1.36214	0.681069	+	0.732219i	$0.261515\pi$
$954$	0	0
$955$	56.1137	1.81580
$956$	0.931610	0.0301304
$957$	0	0
$958$	7.31421	0.236312
$959$	−27.7478	−0.896024
$960$	0	0
$961$	−26.6520	−0.859741
$962$	32.2903	1.04108
$963$	0	0
$964$	−8.15661	−0.262707
$965$	−39.8468	−1.28271
$966$	0	0
$967$	−17.3621	−0.558328	−0.279164	−	0.960243i	$-0.590057\pi$
$967$	−17.3621	−0.558328	−0.279164	+	0.960243i	$0.590057\pi$
$968$	−3.02154	−0.0971159
$969$	0	0
$970$	19.3203	0.620339
$971$	55.0793	1.76758	0.883789	−	0.467886i	$-0.154984\pi$
$971$	55.0793	1.76758	0.883789	+	0.467886i	$0.154984\pi$
$972$	0	0
$973$	−14.9710	−0.479949
$974$	29.5793	0.947783
$975$	0	0
$976$	−5.56740	−0.178208
$977$	6.16086	0.197103	0.0985517	−	0.995132i	$-0.468579\pi$
$977$	6.16086	0.197103	0.0985517	+	0.995132i	$0.468579\pi$
$978$	0	0
$979$	−1.51554	−0.0484368
$980$	−4.83688	−0.154508
$981$	0	0
$982$	−20.0049	−0.638383
$983$	−10.3104	−0.328851	−0.164425	−	0.986390i	$-0.552577\pi$
$983$	−10.3104	−0.328851	−0.164425	+	0.986390i	$0.552577\pi$
$984$	0	0
$985$	23.0103	0.733170
$986$	21.9238	0.698196
$987$	0	0
$988$	0.368765	0.0117320
$989$	3.21324	0.102175
$990$	0	0
$991$	−15.1363	−0.480820	−0.240410	−	0.970671i	$-0.577282\pi$
$991$	−15.1363	−0.480820	−0.240410	+	0.970671i	$0.577282\pi$
$992$	−4.16821	−0.132341
$993$	0	0
$994$	30.4679	0.966383
$995$	−27.3091	−0.865756
$996$	0	0
$997$	−10.2906	−0.325907	−0.162953	−	0.986634i	$-0.552102\pi$
$997$	−10.2906	−0.325907	−0.162953	+	0.986634i	$0.552102\pi$
$998$	−10.3423	−0.327381
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2277.2.a.l.1.3		5
3.2	odd	2		253.2.a.c.1.3	✓	5
12.11	even	2		4048.2.a.bb.1.5		5
15.14	odd	2		6325.2.a.l.1.3		5
33.32	even	2		2783.2.a.g.1.3		5
69.68	even	2		5819.2.a.d.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
253.2.a.c.1.3	✓	5	3.2	odd	2
2277.2.a.l.1.3		5	1.1	even	1	trivial
2783.2.a.g.1.3		5	33.32	even	2
4048.2.a.bb.1.5		5	12.11	even	2
5819.2.a.d.1.3		5	69.68	even	2
6325.2.a.l.1.3		5	15.14	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:	\( N \)	\(=\)	\( 2277 = 3^{2} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2277.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28142 q^{2} -0.357954 q^{4} -3.80110 q^{5} -1.85610 q^{7} -3.02154 q^{8} -4.87082 q^{10} +1.00000 q^{11} -5.27194 q^{13} -2.37845 q^{14} -3.15596 q^{16} +3.86848 q^{17} +0.195412 q^{19} +1.36062 q^{20} +1.28142 q^{22} +1.00000 q^{23} +9.44840 q^{25} -6.75559 q^{26} +0.664399 q^{28} +4.42266 q^{29} -2.08520 q^{31} +1.99896 q^{32} +4.95715 q^{34} +7.05522 q^{35} -4.77980 q^{37} +0.250406 q^{38} +11.4852 q^{40} +5.28296 q^{41} +3.21324 q^{43} -0.357954 q^{44} +1.28142 q^{46} +2.30911 q^{47} -3.55490 q^{49} +12.1074 q^{50} +1.88712 q^{52} +0.819313 q^{53} -3.80110 q^{55} +5.60827 q^{56} +5.66730 q^{58} +13.0628 q^{59} +1.76409 q^{61} -2.67202 q^{62} +8.87343 q^{64} +20.0392 q^{65} -0.481852 q^{67} -1.38474 q^{68} +9.04073 q^{70} -12.8100 q^{71} +1.48889 q^{73} -6.12494 q^{74} -0.0699487 q^{76} -1.85610 q^{77} -11.3518 q^{79} +11.9961 q^{80} +6.76970 q^{82} +15.8886 q^{83} -14.7045 q^{85} +4.11752 q^{86} -3.02154 q^{88} -1.51554 q^{89} +9.78524 q^{91} -0.357954 q^{92} +2.95895 q^{94} -0.742782 q^{95} -3.96654 q^{97} -4.55533 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{10} + 5 q^{11} - 15 q^{13} - 4 q^{14} + 8 q^{16} + 9 q^{17} - 5 q^{19} + 17 q^{20} + 4 q^{22} + 5 q^{23} + 12 q^{25} - 5 q^{26} + 8 q^{29} - 4 q^{31}+ \cdots + 33 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 6 * q^10 + 5 * q^11 - 15 * q^13 - 4 * q^14 + 8 * q^16 + 9 * q^17 - 5 * q^19 + 17 * q^20 + 4 * q^22 + 5 * q^23 + 12 * q^25 - 5 * q^26 + 8 * q^29 - 4 * q^31 + 31 * q^32 + 36 * q^34 + 20 * q^35 + 8 * q^37 + 11 * q^38 + 22 * q^40 + 6 * q^41 - 8 * q^43 + 6 * q^44 + 4 * q^46 + 34 * q^47 + 14 * q^49 - 22 * q^50 - 9 * q^52 - 2 * q^53 + 3 * q^55 - 27 * q^56 + 12 * q^58 + 13 * q^59 + 18 * q^61 - 30 * q^62 + 36 * q^64 - 4 * q^65 - 4 * q^67 + 43 * q^68 + 11 * q^70 - 10 * q^71 - 31 * q^73 - 19 * q^74 + 27 * q^76 - 3 * q^77 - 3 * q^79 - 2 * q^80 - 10 * q^82 + 43 * q^83 - 3 * q^85 + 15 * q^86 + 6 * q^88 - 15 * q^89 + 3 * q^91 + 6 * q^92 + 7 * q^94 - 3 * q^95 + 17 * q^97 + 33 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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