LMFDB - Embedded newform 228.6.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [228,6,Mod(1,228)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(228, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("228.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 228 = 2^{2} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	228.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-36,0,-84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$36.5675109174$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3227x^{2} + 17265x + 1197450 $ x^4 - x^3 - 3227x^2 + 17265x + 1197450
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-55.6309$ of defining polynomial
Character	$\chi$	$=$	228.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -24.7456 q^{5} -94.5162 q^{7} +81.0000 q^{9} +658.815 q^{11} +137.434 q^{13} +222.710 q^{15} +434.528 q^{17} +361.000 q^{19} +850.646 q^{21} -2093.92 q^{23} -2512.66 q^{25} -729.000 q^{27} +5088.18 q^{29} -8041.00 q^{31} -5929.34 q^{33} +2338.86 q^{35} +5697.17 q^{37} -1236.90 q^{39} -19145.7 q^{41} +22560.1 q^{43} -2004.39 q^{45} -5051.26 q^{47} -7873.69 q^{49} -3910.76 q^{51} -14112.5 q^{53} -16302.8 q^{55} -3249.00 q^{57} -43554.9 q^{59} -35333.8 q^{61} -7655.81 q^{63} -3400.88 q^{65} -19068.1 q^{67} +18845.3 q^{69} +25159.9 q^{71} -60015.6 q^{73} +22613.9 q^{75} -62268.7 q^{77} +90721.4 q^{79} +6561.00 q^{81} -114736. q^{83} -10752.6 q^{85} -45793.6 q^{87} +24441.7 q^{89} -12989.7 q^{91} +72369.0 q^{93} -8933.15 q^{95} -135658. q^{97} +53364.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} - 84 q^{5} + 54 q^{7} + 324 q^{9} + 354 q^{11} + 46 q^{13} + 756 q^{15} + 2046 q^{17} + 1444 q^{19} - 486 q^{21} + 846 q^{23} + 4018 q^{25} - 2916 q^{27} - 7152 q^{29} + 238 q^{31} - 3186 q^{33}+ \cdots + 28674 q^{99}+O(q^{100}) $ 4 * q - 36 * q^3 - 84 * q^5 + 54 * q^7 + 324 * q^9 + 354 * q^11 + 46 * q^13 + 756 * q^15 + 2046 * q^17 + 1444 * q^19 - 486 * q^21 + 846 * q^23 + 4018 * q^25 - 2916 * q^27 - 7152 * q^29 + 238 * q^31 - 3186 * q^33 - 19878 * q^35 - 10326 * q^37 - 414 * q^39 - 28200 * q^41 - 18590 * q^43 - 6804 * q^45 - 18684 * q^47 - 17946 * q^49 - 18414 * q^51 - 42720 * q^53 - 13818 * q^55 - 12996 * q^57 - 68256 * q^59 - 67654 * q^61 + 4374 * q^63 - 109776 * q^65 - 81836 * q^67 - 7614 * q^69 - 62112 * q^71 - 81966 * q^73 - 36162 * q^75 - 81378 * q^77 + 8766 * q^79 + 26244 * q^81 - 55080 * q^83 - 108030 * q^85 + 64368 * q^87 - 197520 * q^89 + 25820 * q^91 - 2142 * q^93 - 30324 * q^95 - 184228 * q^97 + 28674 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−24.7456	−0.442662	−0.221331	−	0.975199i	$-0.571040\pi$
$5$	−24.7456	−0.442662	−0.221331	+	0.975199i	$0.571040\pi$
$6$	0	0
$7$	−94.5162	−0.729056	−0.364528	−	0.931192i	$-0.618770\pi$
$7$	−94.5162	−0.729056	−0.364528	+	0.931192i	$0.618770\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	658.815	1.64166	0.820828	−	0.571176i	$-0.193512\pi$
$11$	658.815	1.64166	0.820828	+	0.571176i	$0.193512\pi$
$12$	0	0
$13$	137.434	0.225546	0.112773	−	0.993621i	$-0.464027\pi$
$13$	137.434	0.225546	0.112773	+	0.993621i	$0.464027\pi$
$14$	0	0
$15$	222.710	0.255571
$16$	0	0
$17$	434.528	0.364667	0.182333	−	0.983237i	$-0.441635\pi$
$17$	434.528	0.364667	0.182333	+	0.983237i	$0.441635\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	850.646	0.420921
$22$	0	0
$23$	−2093.92	−0.825355	−0.412677	−	0.910877i	$-0.635406\pi$
$23$	−2093.92	−0.825355	−0.412677	+	0.910877i	$0.635406\pi$
$24$	0	0
$25$	−2512.66	−0.804050
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	5088.18	1.12349	0.561743	−	0.827312i	$-0.310131\pi$
$29$	5088.18	1.12349	0.561743	+	0.827312i	$0.310131\pi$
$30$	0	0
$31$	−8041.00	−1.50282	−0.751408	−	0.659838i	$-0.770625\pi$
$31$	−8041.00	−1.50282	−0.751408	+	0.659838i	$0.770625\pi$
$32$	0	0
$33$	−5929.34	−0.947810
$34$	0	0
$35$	2338.86	0.322726
$36$	0	0
$37$	5697.17	0.684155	0.342078	−	0.939672i	$-0.388869\pi$
$37$	5697.17	0.684155	0.342078	+	0.939672i	$0.388869\pi$
$38$	0	0
$39$	−1236.90	−0.130219
$40$	0	0
$41$	−19145.7	−1.77874	−0.889368	−	0.457191i	$-0.848855\pi$
$41$	−19145.7	−1.77874	−0.889368	+	0.457191i	$0.848855\pi$
$42$	0	0
$43$	22560.1	1.86067	0.930336	−	0.366707i	$-0.119515\pi$
$43$	22560.1	1.86067	0.930336	+	0.366707i	$0.119515\pi$
$44$	0	0
$45$	−2004.39	−0.147554
$46$	0	0
$47$	−5051.26	−0.333546	−0.166773	−	0.985995i	$-0.553335\pi$
$47$	−5051.26	−0.333546	−0.166773	+	0.985995i	$0.553335\pi$
$48$	0	0
$49$	−7873.69	−0.468477
$50$	0	0
$51$	−3910.76	−0.210540
$52$	0	0
$53$	−14112.5	−0.690104	−0.345052	−	0.938584i	$-0.612139\pi$
$53$	−14112.5	−0.690104	−0.345052	+	0.938584i	$0.612139\pi$
$54$	0	0
$55$	−16302.8	−0.726699
$56$	0	0
$57$	−3249.00	−0.132453
$58$	0	0
$59$	−43554.9	−1.62895	−0.814474	−	0.580201i	$-0.802974\pi$
$59$	−43554.9	−1.62895	−0.814474	+	0.580201i	$0.802974\pi$
$60$	0	0
$61$	−35333.8	−1.21581	−0.607906	−	0.794009i	$-0.707990\pi$
$61$	−35333.8	−1.21581	−0.607906	+	0.794009i	$0.707990\pi$
$62$	0	0
$63$	−7655.81	−0.243019
$64$	0	0
$65$	−3400.88	−0.0998407
$66$	0	0
$67$	−19068.1	−0.518943	−0.259471	−	0.965751i	$-0.583548\pi$
$67$	−19068.1	−0.518943	−0.259471	+	0.965751i	$0.583548\pi$
$68$	0	0
$69$	18845.3	0.476519
$70$	0	0
$71$	25159.9	0.592329	0.296164	−	0.955137i	$-0.404292\pi$
$71$	25159.9	0.592329	0.296164	+	0.955137i	$0.404292\pi$
$72$	0	0
$73$	−60015.6	−1.31813	−0.659063	−	0.752088i	$-0.729047\pi$
$73$	−60015.6	−1.31813	−0.659063	+	0.752088i	$0.729047\pi$
$74$	0	0
$75$	22613.9	0.464219
$76$	0	0
$77$	−62268.7	−1.19686
$78$	0	0
$79$	90721.4	1.63547	0.817734	−	0.575597i	$-0.195230\pi$
$79$	90721.4	1.63547	0.817734	+	0.575597i	$0.195230\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−114736.	−1.82812	−0.914060	−	0.405579i	$-0.867070\pi$
$83$	−114736.	−1.82812	−0.914060	+	0.405579i	$0.867070\pi$
$84$	0	0
$85$	−10752.6	−0.161424
$86$	0	0
$87$	−45793.6	−0.648645
$88$	0	0
$89$	24441.7	0.327082	0.163541	−	0.986537i	$-0.447708\pi$
$89$	24441.7	0.327082	0.163541	+	0.986537i	$0.447708\pi$
$90$	0	0
$91$	−12989.7	−0.164436
$92$	0	0
$93$	72369.0	0.867651
$94$	0	0
$95$	−8933.15	−0.101554
$96$	0	0
$97$	−135658.	−1.46391	−0.731957	−	0.681351i	$-0.761393\pi$
$97$	−135658.	−1.46391	−0.731957	+	0.681351i	$0.761393\pi$
$98$	0	0
$99$	53364.1	0.547219
$100$	0	0
$101$	172258.	1.68026	0.840131	−	0.542383i	$-0.182478\pi$
$101$	172258.	1.68026	0.840131	+	0.542383i	$0.182478\pi$
$102$	0	0
$103$	39546.8	0.367298	0.183649	−	0.982992i	$-0.441209\pi$
$103$	39546.8	0.367298	0.183649	+	0.982992i	$0.441209\pi$
$104$	0	0
$105$	−21049.7	−0.186326
$106$	0	0
$107$	−136247.	−1.15045	−0.575226	−	0.817995i	$-0.695086\pi$
$107$	−136247.	−1.15045	−0.575226	+	0.817995i	$0.695086\pi$
$108$	0	0
$109$	−104857.	−0.845337	−0.422668	−	0.906284i	$-0.638907\pi$
$109$	−104857.	−0.845337	−0.422668	+	0.906284i	$0.638907\pi$
$110$	0	0
$111$	−51274.5	−0.394997
$112$	0	0
$113$	−137587.	−1.01363	−0.506817	−	0.862054i	$-0.669178\pi$
$113$	−137587.	−1.01363	−0.506817	+	0.862054i	$0.669178\pi$
$114$	0	0
$115$	51815.2	0.365353
$116$	0	0
$117$	11132.1	0.0751820
$118$	0	0
$119$	−41070.0	−0.265862
$120$	0	0
$121$	272987.	1.69503
$122$	0	0
$123$	172311.	1.02695
$124$	0	0
$125$	139507.	0.798585
$126$	0	0
$127$	194206.	1.06845	0.534225	−	0.845343i	$-0.320604\pi$
$127$	194206.	1.06845	0.534225	+	0.845343i	$0.320604\pi$
$128$	0	0
$129$	−203041.	−1.07426
$130$	0	0
$131$	105884.	0.539081	0.269540	−	0.962989i	$-0.413128\pi$
$131$	105884.	0.539081	0.269540	+	0.962989i	$0.413128\pi$
$132$	0	0
$133$	−34120.3	−0.167257
$134$	0	0
$135$	18039.5	0.0851903
$136$	0	0
$137$	−274466.	−1.24936	−0.624679	−	0.780882i	$-0.714770\pi$
$137$	−274466.	−1.24936	−0.624679	+	0.780882i	$0.714770\pi$
$138$	0	0
$139$	22232.9	0.0976019	0.0488010	−	0.998809i	$-0.484460\pi$
$139$	22232.9	0.0976019	0.0488010	+	0.998809i	$0.484460\pi$
$140$	0	0
$141$	45461.4	0.192573
$142$	0	0
$143$	90543.6	0.370269
$144$	0	0
$145$	−125910.	−0.497324
$146$	0	0
$147$	70863.2	0.270475
$148$	0	0
$149$	−427802.	−1.57862	−0.789309	−	0.613996i	$-0.789561\pi$
$149$	−427802.	−1.57862	−0.789309	+	0.613996i	$0.789561\pi$
$150$	0	0
$151$	242128.	0.864177	0.432089	−	0.901831i	$-0.357777\pi$
$151$	242128.	0.864177	0.432089	+	0.901831i	$0.357777\pi$
$152$	0	0
$153$	35196.8	0.121556
$154$	0	0
$155$	198979.	0.665239
$156$	0	0
$157$	447146.	1.44777	0.723885	−	0.689920i	$-0.242354\pi$
$157$	447146.	1.44777	0.723885	+	0.689920i	$0.242354\pi$
$158$	0	0
$159$	127013.	0.398432
$160$	0	0
$161$	197909.	0.601730
$162$	0	0
$163$	19737.8	0.0581874	0.0290937	−	0.999577i	$-0.490738\pi$
$163$	19737.8	0.0581874	0.0290937	+	0.999577i	$0.490738\pi$
$164$	0	0
$165$	146725.	0.419560
$166$	0	0
$167$	−102269.	−0.283762	−0.141881	−	0.989884i	$-0.545315\pi$
$167$	−102269.	−0.283762	−0.141881	+	0.989884i	$0.545315\pi$
$168$	0	0
$169$	−352405.	−0.949129
$170$	0	0
$171$	29241.0	0.0764719
$172$	0	0
$173$	−513999.	−1.30571	−0.652855	−	0.757483i	$-0.726429\pi$
$173$	−513999.	−1.30571	−0.652855	+	0.757483i	$0.726429\pi$
$174$	0	0
$175$	237487.	0.586198
$176$	0	0
$177$	391994.	0.940473
$178$	0	0
$179$	338975.	0.790743	0.395371	−	0.918521i	$-0.370616\pi$
$179$	338975.	0.790743	0.395371	+	0.918521i	$0.370616\pi$
$180$	0	0
$181$	230824.	0.523702	0.261851	−	0.965108i	$-0.415667\pi$
$181$	230824.	0.523702	0.261851	+	0.965108i	$0.415667\pi$
$182$	0	0
$183$	318005.	0.701949
$184$	0	0
$185$	−140980.	−0.302849
$186$	0	0
$187$	286274.	0.598657
$188$	0	0
$189$	68902.3	0.140307
$190$	0	0
$191$	120609.	0.239219	0.119610	−	0.992821i	$-0.461836\pi$
$191$	120609.	0.239219	0.119610	+	0.992821i	$0.461836\pi$
$192$	0	0
$193$	−203725.	−0.393687	−0.196844	−	0.980435i	$-0.563069\pi$
$193$	−203725.	−0.393687	−0.196844	+	0.980435i	$0.563069\pi$
$194$	0	0
$195$	30607.9	0.0576430
$196$	0	0
$197$	−377358.	−0.692768	−0.346384	−	0.938093i	$-0.612591\pi$
$197$	−377358.	−0.692768	−0.346384	+	0.938093i	$0.612591\pi$
$198$	0	0
$199$	669459.	1.19837	0.599186	−	0.800610i	$-0.295491\pi$
$199$	669459.	1.19837	0.599186	+	0.800610i	$0.295491\pi$
$200$	0	0
$201$	171612.	0.299612
$202$	0	0
$203$	−480915.	−0.819084
$204$	0	0
$205$	473771.	0.787379
$206$	0	0
$207$	−169608.	−0.275118
$208$	0	0
$209$	237832.	0.376622
$210$	0	0
$211$	−470731.	−0.727891	−0.363946	−	0.931420i	$-0.618571\pi$
$211$	−470731.	−0.727891	−0.363946	+	0.931420i	$0.618571\pi$
$212$	0	0
$213$	−226439.	−0.341981
$214$	0	0
$215$	−558262.	−0.823649
$216$	0	0
$217$	760005.	1.09564
$218$	0	0
$219$	540140.	0.761020
$220$	0	0
$221$	59718.9	0.0822491
$222$	0	0
$223$	1.11952e6	1.50754	0.753769	−	0.657139i	$-0.228234\pi$
$223$	1.11952e6	1.50754	0.753769	+	0.657139i	$0.228234\pi$
$224$	0	0
$225$	−203525.	−0.268017
$226$	0	0
$227$	−473873.	−0.610376	−0.305188	−	0.952292i	$-0.598719\pi$
$227$	−473873.	−0.610376	−0.305188	+	0.952292i	$0.598719\pi$
$228$	0	0
$229$	738423.	0.930500	0.465250	−	0.885179i	$-0.345964\pi$
$229$	738423.	0.930500	0.465250	+	0.885179i	$0.345964\pi$
$230$	0	0
$231$	560419.	0.691007
$232$	0	0
$233$	694.225	0.000837742 0	0.000418871	−	1.00000i	$-0.499867\pi$
$233$	694.225	0.000837742 0	0.000418871		1.00000i	$0.499867\pi$
$234$	0	0
$235$	124996.	0.147648
$236$	0	0
$237$	−816492.	−0.944238
$238$	0	0
$239$	−833340.	−0.943686	−0.471843	−	0.881682i	$-0.656411\pi$
$239$	−833340.	−0.943686	−0.471843	+	0.881682i	$0.656411\pi$
$240$	0	0
$241$	96678.1	0.107222	0.0536112	−	0.998562i	$-0.482927\pi$
$241$	96678.1	0.107222	0.0536112	+	0.998562i	$0.482927\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	194839.	0.207377
$246$	0	0
$247$	49613.6	0.0517438
$248$	0	0
$249$	1.03262e6	1.05547
$250$	0	0
$251$	537911.	0.538922	0.269461	−	0.963011i	$-0.413154\pi$
$251$	537911.	0.538922	0.269461	+	0.963011i	$0.413154\pi$
$252$	0	0
$253$	−1.37951e6	−1.35495
$254$	0	0
$255$	96773.8	0.0931982
$256$	0	0
$257$	−708849.	−0.669454	−0.334727	−	0.942315i	$-0.608644\pi$
$257$	−708849.	−0.669454	−0.334727	+	0.942315i	$0.608644\pi$
$258$	0	0
$259$	−538474.	−0.498788
$260$	0	0
$261$	412143.	0.374495
$262$	0	0
$263$	−951984.	−0.848672	−0.424336	−	0.905505i	$-0.639493\pi$
$263$	−951984.	−0.848672	−0.424336	+	0.905505i	$0.639493\pi$
$264$	0	0
$265$	349222.	0.305483
$266$	0	0
$267$	−219976.	−0.188841
$268$	0	0
$269$	−658815.	−0.555115	−0.277558	−	0.960709i	$-0.589525\pi$
$269$	−658815.	−0.555115	−0.277558	+	0.960709i	$0.589525\pi$
$270$	0	0
$271$	−1.98273e6	−1.63999	−0.819995	−	0.572370i	$-0.806024\pi$
$271$	−1.98273e6	−1.63999	−0.819995	+	0.572370i	$0.806024\pi$
$272$	0	0
$273$	116908.	0.0949371
$274$	0	0
$275$	−1.65538e6	−1.31997
$276$	0	0
$277$	−1.65627e6	−1.29697	−0.648486	−	0.761227i	$-0.724597\pi$
$277$	−1.65627e6	−1.29697	−0.648486	+	0.761227i	$0.724597\pi$
$278$	0	0
$279$	−651321.	−0.500939
$280$	0	0
$281$	−1.29006e6	−0.974641	−0.487321	−	0.873223i	$-0.662026\pi$
$281$	−1.29006e6	−0.974641	−0.487321	+	0.873223i	$0.662026\pi$
$282$	0	0
$283$	528290.	0.392109	0.196054	−	0.980593i	$-0.437187\pi$
$283$	528290.	0.392109	0.196054	+	0.980593i	$0.437187\pi$
$284$	0	0
$285$	80398.3	0.0586320
$286$	0	0
$287$	1.80958e6	1.29680
$288$	0	0
$289$	−1.23104e6	−0.867018
$290$	0	0
$291$	1.22092e6	0.845191
$292$	0	0
$293$	−2.58910e6	−1.76189	−0.880947	−	0.473214i	$-0.843094\pi$
$293$	−2.58910e6	−1.76189	−0.880947	+	0.473214i	$0.843094\pi$
$294$	0	0
$295$	1.07779e6	0.721073
$296$	0	0
$297$	−480276.	−0.315937
$298$	0	0
$299$	−287776.	−0.186156
$300$	0	0
$301$	−2.13230e6	−1.35654
$302$	0	0
$303$	−1.55033e6	−0.970100
$304$	0	0
$305$	874356.	0.538194
$306$	0	0
$307$	−2.88263e6	−1.74559	−0.872796	−	0.488086i	$-0.837695\pi$
$307$	−2.88263e6	−1.74559	−0.872796	+	0.488086i	$0.837695\pi$
$308$	0	0
$309$	−355921.	−0.212059
$310$	0	0
$311$	2.30024e6	1.34857	0.674284	−	0.738472i	$-0.264452\pi$
$311$	2.30024e6	1.34857	0.674284	+	0.738472i	$0.264452\pi$
$312$	0	0
$313$	−681327.	−0.393093	−0.196546	−	0.980495i	$-0.562973\pi$
$313$	−681327.	−0.393093	−0.196546	+	0.980495i	$0.562973\pi$
$314$	0	0
$315$	189447.	0.107575
$316$	0	0
$317$	−508450.	−0.284184	−0.142092	−	0.989853i	$-0.545383\pi$
$317$	−508450.	−0.284184	−0.142092	+	0.989853i	$0.545383\pi$
$318$	0	0
$319$	3.35217e6	1.84438
$320$	0	0
$321$	1.22623e6	0.664214
$322$	0	0
$323$	156865.	0.0836602
$324$	0	0
$325$	−345324.	−0.181350
$326$	0	0
$327$	943710.	0.488055
$328$	0	0
$329$	477426.	0.243174
$330$	0	0
$331$	2.62164e6	1.31523	0.657617	−	0.753352i	$-0.271564\pi$
$331$	2.62164e6	1.31523	0.657617	+	0.753352i	$0.271564\pi$
$332$	0	0
$333$	461470.	0.228052
$334$	0	0
$335$	471850.	0.229716
$336$	0	0
$337$	−721890.	−0.346255	−0.173128	−	0.984899i	$-0.555387\pi$
$337$	−721890.	−0.346255	−0.173128	+	0.984899i	$0.555387\pi$
$338$	0	0
$339$	1.23828e6	0.585221
$340$	0	0
$341$	−5.29753e6	−2.46711
$342$	0	0
$343$	2.33272e6	1.07060
$344$	0	0
$345$	−466337.	−0.210937
$346$	0	0
$347$	647898.	0.288857	0.144428	−	0.989515i	$-0.453866\pi$
$347$	647898.	0.288857	0.144428	+	0.989515i	$0.453866\pi$
$348$	0	0
$349$	−1.00852e6	−0.443220	−0.221610	−	0.975135i	$-0.571131\pi$
$349$	−1.00852e6	−0.443220	−0.221610	+	0.975135i	$0.571131\pi$
$350$	0	0
$351$	−100189.	−0.0434064
$352$	0	0
$353$	53568.7	0.0228810	0.0114405	−	0.999935i	$-0.496358\pi$
$353$	53568.7	0.0228810	0.0114405	+	0.999935i	$0.496358\pi$
$354$	0	0
$355$	−622595.	−0.262201
$356$	0	0
$357$	369630.	0.153496
$358$	0	0
$359$	−3.68653e6	−1.50967	−0.754835	−	0.655914i	$-0.772283\pi$
$359$	−3.68653e6	−1.50967	−0.754835	+	0.655914i	$0.772283\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−2.45688e6	−0.978628
$364$	0	0
$365$	1.48512e6	0.583484
$366$	0	0
$367$	−1.03597e6	−0.401496	−0.200748	−	0.979643i	$-0.564337\pi$
$367$	−1.03597e6	−0.401496	−0.200748	+	0.979643i	$0.564337\pi$
$368$	0	0
$369$	−1.55080e6	−0.592912
$370$	0	0
$371$	1.33386e6	0.503125
$372$	0	0
$373$	3.28124e6	1.22114	0.610571	−	0.791962i	$-0.290940\pi$
$373$	3.28124e6	1.22114	0.610571	+	0.791962i	$0.290940\pi$
$374$	0	0
$375$	−1.25556e6	−0.461063
$376$	0	0
$377$	699288.	0.253398
$378$	0	0
$379$	4.81519e6	1.72193	0.860964	−	0.508665i	$-0.169861\pi$
$379$	4.81519e6	1.72193	0.860964	+	0.508665i	$0.169861\pi$
$380$	0	0
$381$	−1.74786e6	−0.616870
$382$	0	0
$383$	−1.68869e6	−0.588239	−0.294120	−	0.955769i	$-0.595026\pi$
$383$	−1.68869e6	−0.588239	−0.294120	+	0.955769i	$0.595026\pi$
$384$	0	0
$385$	1.54087e6	0.529804
$386$	0	0
$387$	1.82737e6	0.620224
$388$	0	0
$389$	3.02387e6	1.01318	0.506592	−	0.862186i	$-0.330905\pi$
$389$	3.02387e6	1.01318	0.506592	+	0.862186i	$0.330905\pi$
$390$	0	0
$391$	−909868.	−0.300979
$392$	0	0
$393$	−952960.	−0.311239
$394$	0	0
$395$	−2.24495e6	−0.723959
$396$	0	0
$397$	−1.89205e6	−0.602499	−0.301250	−	0.953545i	$-0.597404\pi$
$397$	−1.89205e6	−0.602499	−0.301250	+	0.953545i	$0.597404\pi$
$398$	0	0
$399$	307083.	0.0965659
$400$	0	0
$401$	1.61806e6	0.502496	0.251248	−	0.967923i	$-0.419159\pi$
$401$	1.61806e6	0.502496	0.251248	+	0.967923i	$0.419159\pi$
$402$	0	0
$403$	−1.10511e6	−0.338954
$404$	0	0
$405$	−162356.	−0.0491847
$406$	0	0
$407$	3.75338e6	1.12315
$408$	0	0
$409$	−4.29421e6	−1.26933	−0.634666	−	0.772787i	$-0.718862\pi$
$409$	−4.29421e6	−1.26933	−0.634666	+	0.772787i	$0.718862\pi$
$410$	0	0
$411$	2.47019e6	0.721317
$412$	0	0
$413$	4.11664e6	1.18759
$414$	0	0
$415$	2.83921e6	0.809239
$416$	0	0
$417$	−200096.	−0.0563505
$418$	0	0
$419$	324278.	0.0902366	0.0451183	−	0.998982i	$-0.485634\pi$
$419$	324278.	0.0902366	0.0451183	+	0.998982i	$0.485634\pi$
$420$	0	0
$421$	−3.20973e6	−0.882599	−0.441299	−	0.897360i	$-0.645482\pi$
$421$	−3.20973e6	−0.882599	−0.441299	+	0.897360i	$0.645482\pi$
$422$	0	0
$423$	−409152.	−0.111182
$424$	0	0
$425$	−1.09182e6	−0.293210
$426$	0	0
$427$	3.33962e6	0.886395
$428$	0	0
$429$	−814892.	−0.213775
$430$	0	0
$431$	4.83335e6	1.25330	0.626650	−	0.779301i	$-0.284426\pi$
$431$	4.83335e6	1.25330	0.626650	+	0.779301i	$0.284426\pi$
$432$	0	0
$433$	3.81696e6	0.978358	0.489179	−	0.872183i	$-0.337296\pi$
$433$	3.81696e6	0.978358	0.489179	+	0.872183i	$0.337296\pi$
$434$	0	0
$435$	1.13319e6	0.287130
$436$	0	0
$437$	−755905.	−0.189349
$438$	0	0
$439$	1.69417e6	0.419561	0.209781	−	0.977748i	$-0.432725\pi$
$439$	1.69417e6	0.419561	0.209781	+	0.977748i	$0.432725\pi$
$440$	0	0
$441$	−637769.	−0.156159
$442$	0	0
$443$	6.75042e6	1.63426	0.817131	−	0.576452i	$-0.195563\pi$
$443$	6.75042e6	1.63426	0.817131	+	0.576452i	$0.195563\pi$
$444$	0	0
$445$	−604825.	−0.144787
$446$	0	0
$447$	3.85022e6	0.911415
$448$	0	0
$449$	−3.29004e6	−0.770168	−0.385084	−	0.922881i	$-0.625828\pi$
$449$	−3.29004e6	−0.770168	−0.385084	+	0.922881i	$0.625828\pi$
$450$	0	0
$451$	−1.26135e7	−2.92007
$452$	0	0
$453$	−2.17915e6	−0.498933
$454$	0	0
$455$	321438.	0.0727895
$456$	0	0
$457$	4.33168e6	0.970210	0.485105	−	0.874456i	$-0.338781\pi$
$457$	4.33168e6	0.970210	0.485105	+	0.874456i	$0.338781\pi$
$458$	0	0
$459$	−316771.	−0.0701801
$460$	0	0
$461$	3.48683e6	0.764150	0.382075	−	0.924131i	$-0.375210\pi$
$461$	3.48683e6	0.764150	0.382075	+	0.924131i	$0.375210\pi$
$462$	0	0
$463$	−2.62867e6	−0.569879	−0.284940	−	0.958545i	$-0.591974\pi$
$463$	−2.62867e6	−0.569879	−0.284940	+	0.958545i	$0.591974\pi$
$464$	0	0
$465$	−1.79081e6	−0.384076
$466$	0	0
$467$	−3.21047e6	−0.681204	−0.340602	−	0.940208i	$-0.610631\pi$
$467$	−3.21047e6	−0.681204	−0.340602	+	0.940208i	$0.610631\pi$
$468$	0	0
$469$	1.80224e6	0.378338
$470$	0	0
$471$	−4.02431e6	−0.835871
$472$	0	0
$473$	1.48629e7	3.05458
$474$	0	0
$475$	−907069.	−0.184462
$476$	0	0
$477$	−1.14311e6	−0.230035
$478$	0	0
$479$	8.61671e6	1.71594	0.857972	−	0.513697i	$-0.171725\pi$
$479$	8.61671e6	1.71594	0.857972	+	0.513697i	$0.171725\pi$
$480$	0	0
$481$	782983.	0.154309
$482$	0	0
$483$	−1.78118e6	−0.347409
$484$	0	0
$485$	3.35693e6	0.648019
$486$	0	0
$487$	−1.01040e7	−1.93051	−0.965256	−	0.261307i	$-0.915846\pi$
$487$	−1.01040e7	−1.93051	−0.965256	+	0.261307i	$0.915846\pi$
$488$	0	0
$489$	−177640.	−0.0335945
$490$	0	0
$491$	1.39338e6	0.260835	0.130418	−	0.991459i	$-0.458368\pi$
$491$	1.39338e6	0.260835	0.130418	+	0.991459i	$0.458368\pi$
$492$	0	0
$493$	2.21096e6	0.409698
$494$	0	0
$495$	−1.32052e6	−0.242233
$496$	0	0
$497$	−2.37802e6	−0.431841
$498$	0	0
$499$	4.06131e6	0.730154	0.365077	−	0.930977i	$-0.381043\pi$
$499$	4.06131e6	0.730154	0.365077	+	0.930977i	$0.381043\pi$
$500$	0	0
$501$	920424.	0.163830
$502$	0	0
$503$	−1.55388e6	−0.273841	−0.136920	−	0.990582i	$-0.543720\pi$
$503$	−1.55388e6	−0.273841	−0.136920	+	0.990582i	$0.543720\pi$
$504$	0	0
$505$	−4.26263e6	−0.743788
$506$	0	0
$507$	3.17164e6	0.547980
$508$	0	0
$509$	6.04592e6	1.03435	0.517176	−	0.855879i	$-0.326983\pi$
$509$	6.04592e6	1.03435	0.517176	+	0.855879i	$0.326983\pi$
$510$	0	0
$511$	5.67245e6	0.960988
$512$	0	0
$513$	−263169.	−0.0441511
$514$	0	0
$515$	−978608.	−0.162589
$516$	0	0
$517$	−3.32785e6	−0.547567
$518$	0	0
$519$	4.62599e6	0.753853
$520$	0	0
$521$	1.81343e6	0.292690	0.146345	−	0.989234i	$-0.453249\pi$
$521$	1.81343e6	0.292690	0.146345	+	0.989234i	$0.453249\pi$
$522$	0	0
$523$	1.02245e7	1.63451	0.817257	−	0.576273i	$-0.195494\pi$
$523$	1.02245e7	1.63451	0.817257	+	0.576273i	$0.195494\pi$
$524$	0	0
$525$	−2.13738e6	−0.338442
$526$	0	0
$527$	−3.49404e6	−0.548027
$528$	0	0
$529$	−2.05184e6	−0.318790
$530$	0	0
$531$	−3.52795e6	−0.542982
$532$	0	0
$533$	−2.63127e6	−0.401187
$534$	0	0
$535$	3.37152e6	0.509261
$536$	0	0
$537$	−3.05078e6	−0.456536
$538$	0	0
$539$	−5.18731e6	−0.769078
$540$	0	0
$541$	4.43625e6	0.651663	0.325832	−	0.945428i	$-0.394356\pi$
$541$	4.43625e6	0.651663	0.325832	+	0.945428i	$0.394356\pi$
$542$	0	0
$543$	−2.07742e6	−0.302360
$544$	0	0
$545$	2.59474e6	0.374198
$546$	0	0
$547$	−5.73672e6	−0.819776	−0.409888	−	0.912136i	$-0.634432\pi$
$547$	−5.73672e6	−0.819776	−0.409888	+	0.912136i	$0.634432\pi$
$548$	0	0
$549$	−2.86204e6	−0.405271
$550$	0	0
$551$	1.83683e6	0.257745
$552$	0	0
$553$	−8.57464e6	−1.19235
$554$	0	0
$555$	1.26882e6	0.174850
$556$	0	0
$557$	5.14777e6	0.703042	0.351521	−	0.936180i	$-0.385665\pi$
$557$	5.14777e6	0.703042	0.351521	+	0.936180i	$0.385665\pi$
$558$	0	0
$559$	3.10052e6	0.419668
$560$	0	0
$561$	−2.57647e6	−0.345635
$562$	0	0
$563$	−1.19950e7	−1.59489	−0.797444	−	0.603393i	$-0.793815\pi$
$563$	−1.19950e7	−1.59489	−0.797444	+	0.603393i	$0.793815\pi$
$564$	0	0
$565$	3.40466e6	0.448697
$566$	0	0
$567$	−620121.	−0.0810063
$568$	0	0
$569$	1.15979e6	0.150176	0.0750879	−	0.997177i	$-0.476076\pi$
$569$	1.15979e6	0.150176	0.0750879	+	0.997177i	$0.476076\pi$
$570$	0	0
$571$	−1.09686e7	−1.40787	−0.703935	−	0.710264i	$-0.748575\pi$
$571$	−1.09686e7	−1.40787	−0.703935	+	0.710264i	$0.748575\pi$
$572$	0	0
$573$	−1.08548e6	−0.138113
$574$	0	0
$575$	5.26130e6	0.663627
$576$	0	0
$577$	5.93291e6	0.741871	0.370935	−	0.928659i	$-0.379037\pi$
$577$	5.93291e6	0.741871	0.370935	+	0.928659i	$0.379037\pi$
$578$	0	0
$579$	1.83353e6	0.227295
$580$	0	0
$581$	1.08444e7	1.33280
$582$	0	0
$583$	−9.29754e6	−1.13291
$584$	0	0
$585$	−275471.	−0.0332802
$586$	0	0
$587$	−8.15129e6	−0.976408	−0.488204	−	0.872730i	$-0.662348\pi$
$587$	−8.15129e6	−0.976408	−0.488204	+	0.872730i	$0.662348\pi$
$588$	0	0
$589$	−2.90280e6	−0.344770
$590$	0	0
$591$	3.39622e6	0.399970
$592$	0	0
$593$	9.40914e6	1.09879	0.549393	−	0.835564i	$-0.314859\pi$
$593$	9.40914e6	1.09879	0.549393	+	0.835564i	$0.314859\pi$
$594$	0	0
$595$	1.01630e6	0.117687
$596$	0	0
$597$	−6.02513e6	−0.691880
$598$	0	0
$599$	−7.54781e6	−0.859516	−0.429758	−	0.902944i	$-0.641401\pi$
$599$	−7.54781e6	−0.859516	−0.429758	+	0.902944i	$0.641401\pi$
$600$	0	0
$601$	7.57114e6	0.855018	0.427509	−	0.904011i	$-0.359391\pi$
$601$	7.57114e6	0.855018	0.427509	+	0.904011i	$0.359391\pi$
$602$	0	0
$603$	−1.54451e6	−0.172981
$604$	0	0
$605$	−6.75521e6	−0.750327
$606$	0	0
$607$	−9.98970e6	−1.10048	−0.550238	−	0.835008i	$-0.685463\pi$
$607$	−9.98970e6	−1.10048	−0.550238	+	0.835008i	$0.685463\pi$
$608$	0	0
$609$	4.32824e6	0.472899
$610$	0	0
$611$	−694214.	−0.0752299
$612$	0	0
$613$	−374884.	−0.0402945	−0.0201472	−	0.999797i	$-0.506413\pi$
$613$	−374884.	−0.0402945	−0.0201472	+	0.999797i	$0.506413\pi$
$614$	0	0
$615$	−4.26394e6	−0.454594
$616$	0	0
$617$	1.11186e7	1.17581	0.587904	−	0.808931i	$-0.299953\pi$
$617$	1.11186e7	1.17581	0.587904	+	0.808931i	$0.299953\pi$
$618$	0	0
$619$	−1.16852e7	−1.22578	−0.612888	−	0.790170i	$-0.709992\pi$
$619$	−1.16852e7	−1.22578	−0.612888	+	0.790170i	$0.709992\pi$
$620$	0	0
$621$	1.52647e6	0.158840
$622$	0	0
$623$	−2.31014e6	−0.238462
$624$	0	0
$625$	4.39988e6	0.450547
$626$	0	0
$627$	−2.14049e6	−0.217443
$628$	0	0
$629$	2.47558e6	0.249488
$630$	0	0
$631$	6.37053e6	0.636946	0.318473	−	0.947932i	$-0.396830\pi$
$631$	6.37053e6	0.636946	0.318473	+	0.947932i	$0.396830\pi$
$632$	0	0
$633$	4.23658e6	0.420248
$634$	0	0
$635$	−4.80574e6	−0.472962
$636$	0	0
$637$	−1.08211e6	−0.105663
$638$	0	0
$639$	2.03795e6	0.197443
$640$	0	0
$641$	1.61501e6	0.155250	0.0776249	−	0.996983i	$-0.475266\pi$
$641$	1.61501e6	0.155250	0.0776249	+	0.996983i	$0.475266\pi$
$642$	0	0
$643$	5.41614e6	0.516610	0.258305	−	0.966063i	$-0.416836\pi$
$643$	5.41614e6	0.516610	0.258305	+	0.966063i	$0.416836\pi$
$644$	0	0
$645$	5.02436e6	0.475534
$646$	0	0
$647$	39577.9	0.00371700	0.00185850	−	0.999998i	$-0.499408\pi$
$647$	39577.9	0.00371700	0.00185850	+	0.999998i	$0.499408\pi$
$648$	0	0
$649$	−2.86946e7	−2.67417
$650$	0	0
$651$	−6.84004e6	−0.632567
$652$	0	0
$653$	−3.35304e6	−0.307720	−0.153860	−	0.988093i	$-0.549171\pi$
$653$	−3.35304e6	−0.307720	−0.153860	+	0.988093i	$0.549171\pi$
$654$	0	0
$655$	−2.62017e6	−0.238631
$656$	0	0
$657$	−4.86126e6	−0.439375
$658$	0	0
$659$	−9.89931e6	−0.887956	−0.443978	−	0.896038i	$-0.646433\pi$
$659$	−9.89931e6	−0.887956	−0.443978	+	0.896038i	$0.646433\pi$
$660$	0	0
$661$	−1.64933e7	−1.46827	−0.734133	−	0.679006i	$-0.762411\pi$
$661$	−1.64933e7	−1.46827	−0.734133	+	0.679006i	$0.762411\pi$
$662$	0	0
$663$	−537470.	−0.0474865
$664$	0	0
$665$	844327.	0.0740383
$666$	0	0
$667$	−1.06542e7	−0.927274
$668$	0	0
$669$	−1.00757e7	−0.870378
$670$	0	0
$671$	−2.32785e7	−1.99594
$672$	0	0
$673$	624782.	0.0531730	0.0265865	−	0.999647i	$-0.491536\pi$
$673$	624782.	0.0531730	0.0265865	+	0.999647i	$0.491536\pi$
$674$	0	0
$675$	1.83173e6	0.154740
$676$	0	0
$677$	−2.19049e6	−0.183683	−0.0918417	−	0.995774i	$-0.529275\pi$
$677$	−2.19049e6	−0.183683	−0.0918417	+	0.995774i	$0.529275\pi$
$678$	0	0
$679$	1.28219e7	1.06728
$680$	0	0
$681$	4.26486e6	0.352400
$682$	0	0
$683$	−6.09670e6	−0.500084	−0.250042	−	0.968235i	$-0.580444\pi$
$683$	−6.09670e6	−0.500084	−0.250042	+	0.968235i	$0.580444\pi$
$684$	0	0
$685$	6.79181e6	0.553043
$686$	0	0
$687$	−6.64581e6	−0.537225
$688$	0	0
$689$	−1.93954e6	−0.155650
$690$	0	0
$691$	1.99498e7	1.58944	0.794718	−	0.606979i	$-0.207619\pi$
$691$	1.99498e7	1.58944	0.794718	+	0.606979i	$0.207619\pi$
$692$	0	0
$693$	−5.04377e6	−0.398953
$694$	0	0
$695$	−550165.	−0.0432047
$696$	0	0
$697$	−8.31935e6	−0.648646
$698$	0	0
$699$	−6248.02	−0.000483671 0
$700$	0	0
$701$	1.25829e7	0.967135	0.483567	−	0.875307i	$-0.339341\pi$
$701$	1.25829e7	0.967135	0.483567	+	0.875307i	$0.339341\pi$
$702$	0	0
$703$	2.05668e6	0.156956
$704$	0	0
$705$	−1.12497e6	−0.0852446
$706$	0	0
$707$	−1.62812e7	−1.22501
$708$	0	0
$709$	−1.51910e7	−1.13493	−0.567467	−	0.823396i	$-0.692077\pi$
$709$	−1.51910e7	−1.13493	−0.567467	+	0.823396i	$0.692077\pi$
$710$	0	0
$711$	7.34843e6	0.545156
$712$	0	0
$713$	1.68372e7	1.24036
$714$	0	0
$715$	−2.24055e6	−0.163904
$716$	0	0
$717$	7.50006e6	0.544838
$718$	0	0
$719$	1.41889e7	1.02359	0.511797	−	0.859107i	$-0.328980\pi$
$719$	1.41889e7	1.02359	0.511797	+	0.859107i	$0.328980\pi$
$720$	0	0
$721$	−3.73781e6	−0.267781
$722$	0	0
$723$	−870103.	−0.0619049
$724$	0	0
$725$	−1.27849e7	−0.903339
$726$	0	0
$727$	−1.18392e7	−0.830781	−0.415391	−	0.909643i	$-0.636355\pi$
$727$	−1.18392e7	−0.830781	−0.415391	+	0.909643i	$0.636355\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	9.80301e6	0.678525
$732$	0	0
$733$	2.22934e7	1.53256	0.766278	−	0.642509i	$-0.222106\pi$
$733$	2.22934e7	1.53256	0.766278	+	0.642509i	$0.222106\pi$
$734$	0	0
$735$	−1.75355e6	−0.119729
$736$	0	0
$737$	−1.25623e7	−0.851925
$738$	0	0
$739$	1.92718e7	1.29811	0.649055	−	0.760742i	$-0.275165\pi$
$739$	1.92718e7	1.29811	0.649055	+	0.760742i	$0.275165\pi$
$740$	0	0
$741$	−446523.	−0.0298743
$742$	0	0
$743$	2.74570e6	0.182466	0.0912328	−	0.995830i	$-0.470919\pi$
$743$	2.74570e6	0.182466	0.0912328	+	0.995830i	$0.470919\pi$
$744$	0	0
$745$	1.05862e7	0.698794
$746$	0	0
$747$	−9.29362e6	−0.609373
$748$	0	0
$749$	1.28776e7	0.838744
$750$	0	0
$751$	7.13715e6	0.461769	0.230885	−	0.972981i	$-0.425838\pi$
$751$	7.13715e6	0.461769	0.230885	+	0.972981i	$0.425838\pi$
$752$	0	0
$753$	−4.84120e6	−0.311147
$754$	0	0
$755$	−5.99159e6	−0.382538
$756$	0	0
$757$	−7.91712e6	−0.502143	−0.251072	−	0.967969i	$-0.580783\pi$
$757$	−7.91712e6	−0.502143	−0.251072	+	0.967969i	$0.580783\pi$
$758$	0	0
$759$	1.24156e7	0.782280
$760$	0	0
$761$	2.50924e7	1.57066	0.785329	−	0.619079i	$-0.212494\pi$
$761$	2.50924e7	1.57066	0.785329	+	0.619079i	$0.212494\pi$
$762$	0	0
$763$	9.91065e6	0.616298
$764$	0	0
$765$	−870965.	−0.0538080
$766$	0	0
$767$	−5.98592e6	−0.367403
$768$	0	0
$769$	1.40171e7	0.854759	0.427379	−	0.904072i	$-0.359437\pi$
$769$	1.40171e7	0.854759	0.427379	+	0.904072i	$0.359437\pi$
$770$	0	0
$771$	6.37964e6	0.386509
$772$	0	0
$773$	1.05759e7	0.636602	0.318301	−	0.947990i	$-0.396888\pi$
$773$	1.05759e7	0.636602	0.318301	+	0.947990i	$0.396888\pi$
$774$	0	0
$775$	2.02043e7	1.20834
$776$	0	0
$777$	4.84627e6	0.287975
$778$	0	0
$779$	−6.91160e6	−0.408070
$780$	0	0
$781$	1.65757e7	0.972400
$782$	0	0
$783$	−3.70928e6	−0.216215
$784$	0	0
$785$	−1.10649e7	−0.640873
$786$	0	0
$787$	−2.39638e7	−1.37917	−0.689587	−	0.724203i	$-0.742208\pi$
$787$	−2.39638e7	−1.37917	−0.689587	+	0.724203i	$0.742208\pi$
$788$	0	0
$789$	8.56785e6	0.489981
$790$	0	0
$791$	1.30042e7	0.738996
$792$	0	0
$793$	−4.85607e6	−0.274222
$794$	0	0
$795$	−3.14300e6	−0.176371
$796$	0	0
$797$	−2.38767e7	−1.33146	−0.665730	−	0.746193i	$-0.731880\pi$
$797$	−2.38767e7	−1.33146	−0.665730	+	0.746193i	$0.731880\pi$
$798$	0	0
$799$	−2.19492e6	−0.121633
$800$	0	0
$801$	1.97978e6	0.109027
$802$	0	0
$803$	−3.95392e7	−2.16391
$804$	0	0
$805$	−4.89738e6	−0.266363
$806$	0	0
$807$	5.92934e6	0.320496
$808$	0	0
$809$	−2.72754e7	−1.46521	−0.732605	−	0.680655i	$-0.761695\pi$
$809$	−2.72754e7	−1.46521	−0.732605	+	0.680655i	$0.761695\pi$
$810$	0	0
$811$	−1.08456e7	−0.579030	−0.289515	−	0.957173i	$-0.593494\pi$
$811$	−1.08456e7	−0.579030	−0.289515	+	0.957173i	$0.593494\pi$
$812$	0	0
$813$	1.78446e7	0.946849
$814$	0	0
$815$	−488422.	−0.0257573
$816$	0	0
$817$	8.14420e6	0.426868
$818$	0	0
$819$	−1.05217e6	−0.0548119
$820$	0	0
$821$	3.00464e7	1.55573	0.777865	−	0.628432i	$-0.216303\pi$
$821$	3.00464e7	1.55573	0.777865	+	0.628432i	$0.216303\pi$
$822$	0	0
$823$	8.61101e6	0.443154	0.221577	−	0.975143i	$-0.428880\pi$
$823$	8.61101e6	0.443154	0.221577	+	0.975143i	$0.428880\pi$
$824$	0	0
$825$	1.48984e7	0.762087
$826$	0	0
$827$	1.89682e7	0.964411	0.482205	−	0.876058i	$-0.339836\pi$
$827$	1.89682e7	0.964411	0.482205	+	0.876058i	$0.339836\pi$
$828$	0	0
$829$	1.66045e7	0.839152	0.419576	−	0.907720i	$-0.362179\pi$
$829$	1.66045e7	0.839152	0.419576	+	0.907720i	$0.362179\pi$
$830$	0	0
$831$	1.49064e7	0.748807
$832$	0	0
$833$	−3.42134e6	−0.170838
$834$	0	0
$835$	2.53071e6	0.125611
$836$	0	0
$837$	5.86189e6	0.289217
$838$	0	0
$839$	−1.11312e7	−0.545931	−0.272966	−	0.962024i	$-0.588005\pi$
$839$	−1.11312e7	−0.545931	−0.272966	+	0.962024i	$0.588005\pi$
$840$	0	0
$841$	5.37843e6	0.262220
$842$	0	0
$843$	1.16106e7	0.562709
$844$	0	0
$845$	8.72046e6	0.420143
$846$	0	0
$847$	−2.58017e7	−1.23578
$848$	0	0
$849$	−4.75461e6	−0.226384
$850$	0	0
$851$	−1.19294e7	−0.564671
$852$	0	0
$853$	3.57571e7	1.68263	0.841316	−	0.540543i	$-0.181781\pi$
$853$	3.57571e7	1.68263	0.841316	+	0.540543i	$0.181781\pi$
$854$	0	0
$855$	−723585.	−0.0338512
$856$	0	0
$857$	−1.04845e6	−0.0487634	−0.0243817	−	0.999703i	$-0.507762\pi$
$857$	−1.04845e6	−0.0487634	−0.0243817	+	0.999703i	$0.507762\pi$
$858$	0	0
$859$	2.89884e7	1.34042	0.670212	−	0.742170i	$-0.266203\pi$
$859$	2.89884e7	1.34042	0.670212	+	0.742170i	$0.266203\pi$
$860$	0	0
$861$	−1.62862e7	−0.748707
$862$	0	0
$863$	1.95329e7	0.892771	0.446386	−	0.894841i	$-0.352711\pi$
$863$	1.95329e7	0.892771	0.446386	+	0.894841i	$0.352711\pi$
$864$	0	0
$865$	1.27192e7	0.577989
$866$	0	0
$867$	1.10794e7	0.500573
$868$	0	0
$869$	5.97687e7	2.68487
$870$	0	0
$871$	−2.62060e6	−0.117045
$872$	0	0
$873$	−1.09883e7	−0.487971
$874$	0	0
$875$	−1.31857e7	−0.582213
$876$	0	0
$877$	1.97678e7	0.867879	0.433940	−	0.900942i	$-0.357123\pi$
$877$	1.97678e7	0.867879	0.433940	+	0.900942i	$0.357123\pi$
$878$	0	0
$879$	2.33019e7	1.01723
$880$	0	0
$881$	1.18863e6	0.0515950	0.0257975	−	0.999667i	$-0.491787\pi$
$881$	1.18863e6	0.0515950	0.0257975	+	0.999667i	$0.491787\pi$
$882$	0	0
$883$	−2.65395e7	−1.14549	−0.572744	−	0.819734i	$-0.694121\pi$
$883$	−2.65395e7	−1.14549	−0.572744	+	0.819734i	$0.694121\pi$
$884$	0	0
$885$	−9.70011e6	−0.416312
$886$	0	0
$887$	677347.	0.0289069	0.0144535	−	0.999896i	$-0.495399\pi$
$887$	677347.	0.0289069	0.0144535	+	0.999896i	$0.495399\pi$
$888$	0	0
$889$	−1.83556e7	−0.778960
$890$	0	0
$891$	4.32249e6	0.182406
$892$	0	0
$893$	−1.82351e6	−0.0765206
$894$	0	0
$895$	−8.38813e6	−0.350032
$896$	0	0
$897$	2.58998e6	0.107477
$898$	0	0
$899$	−4.09141e7	−1.68839
$900$	0	0
$901$	−6.13229e6	−0.251658
$902$	0	0
$903$	1.91907e7	0.783196
$904$	0	0
$905$	−5.71187e6	−0.231823
$906$	0	0
$907$	3.80294e7	1.53497	0.767487	−	0.641064i	$-0.221507\pi$
$907$	3.80294e7	1.53497	0.767487	+	0.641064i	$0.221507\pi$
$908$	0	0
$909$	1.39529e7	0.560087
$910$	0	0
$911$	−2.88019e7	−1.14981	−0.574904	−	0.818221i	$-0.694961\pi$
$911$	−2.88019e7	−1.14981	−0.574904	+	0.818221i	$0.694961\pi$
$912$	0	0
$913$	−7.55899e7	−3.00114
$914$	0	0
$915$	−7.86920e6	−0.310726
$916$	0	0
$917$	−1.00078e7	−0.393020
$918$	0	0
$919$	2.34690e7	0.916656	0.458328	−	0.888783i	$-0.348449\pi$
$919$	2.34690e7	0.916656	0.458328	+	0.888783i	$0.348449\pi$
$920$	0	0
$921$	2.59436e7	1.00782
$922$	0	0
$923$	3.45782e6	0.133597
$924$	0	0
$925$	−1.43150e7	−0.550095
$926$	0	0
$927$	3.20329e6	0.122433
$928$	0	0
$929$	−3.78887e7	−1.44036	−0.720179	−	0.693789i	$-0.755940\pi$
$929$	−3.78887e7	−1.44036	−0.720179	+	0.693789i	$0.755940\pi$
$930$	0	0
$931$	−2.84240e6	−0.107476
$932$	0	0
$933$	−2.07022e7	−0.778596
$934$	0	0
$935$	−7.08401e6	−0.265003
$936$	0	0
$937$	2.20817e7	0.821642	0.410821	−	0.911716i	$-0.365242\pi$
$937$	2.20817e7	0.821642	0.410821	+	0.911716i	$0.365242\pi$
$938$	0	0
$939$	6.13194e6	0.226952
$940$	0	0
$941$	−2.31653e7	−0.852834	−0.426417	−	0.904527i	$-0.640224\pi$
$941$	−2.31653e7	−0.852834	−0.426417	+	0.904527i	$0.640224\pi$
$942$	0	0
$943$	4.00896e7	1.46809
$944$	0	0
$945$	−1.70503e6	−0.0621086
$946$	0	0
$947$	1.73951e7	0.630305	0.315153	−	0.949041i	$-0.397944\pi$
$947$	1.73951e7	0.630305	0.315153	+	0.949041i	$0.397944\pi$
$948$	0	0
$949$	−8.24817e6	−0.297298
$950$	0	0
$951$	4.57605e6	0.164074
$952$	0	0
$953$	−1.59515e7	−0.568943	−0.284472	−	0.958684i	$-0.591818\pi$
$953$	−1.59515e7	−0.568943	−0.284472	+	0.958684i	$0.591818\pi$
$954$	0	0
$955$	−2.98454e6	−0.105893
$956$	0	0
$957$	−3.01695e7	−1.06485
$958$	0	0
$959$	2.59415e7	0.910852
$960$	0	0
$961$	3.60285e7	1.25846
$962$	0	0
$963$	−1.10360e7	−0.383484
$964$	0	0
$965$	5.04129e6	0.174270
$966$	0	0
$967$	3.70532e7	1.27426	0.637132	−	0.770754i	$-0.280120\pi$
$967$	3.70532e7	1.27426	0.637132	+	0.770754i	$0.280120\pi$
$968$	0	0
$969$	−1.41178e6	−0.0483013
$970$	0	0
$971$	−5.35847e7	−1.82387	−0.911933	−	0.410338i	$-0.865411\pi$
$971$	−5.35847e7	−1.82387	−0.911933	+	0.410338i	$0.865411\pi$
$972$	0	0
$973$	−2.10137e6	−0.0711573
$974$	0	0
$975$	3.10792e6	0.104703
$976$	0	0
$977$	1.45719e7	0.488403	0.244202	−	0.969724i	$-0.421474\pi$
$977$	1.45719e7	0.488403	0.244202	+	0.969724i	$0.421474\pi$
$978$	0	0
$979$	1.61026e7	0.536957
$980$	0	0
$981$	−8.49339e6	−0.281779
$982$	0	0
$983$	−1.50226e7	−0.495864	−0.247932	−	0.968777i	$-0.579751\pi$
$983$	−1.50226e7	−0.495864	−0.247932	+	0.968777i	$0.579751\pi$
$984$	0	0
$985$	9.33793e6	0.306662
$986$	0	0
$987$	−4.29683e6	−0.140396
$988$	0	0
$989$	−4.72391e7	−1.53571
$990$	0	0
$991$	−7.45464e6	−0.241125	−0.120563	−	0.992706i	$-0.538470\pi$
$991$	−7.45464e6	−0.241125	−0.120563	+	0.992706i	$0.538470\pi$
$992$	0	0
$993$	−2.35948e7	−0.759351
$994$	0	0
$995$	−1.65661e7	−0.530474
$996$	0	0
$997$	4.85048e7	1.54542	0.772710	−	0.634759i	$-0.218901\pi$
$997$	4.85048e7	1.54542	0.772710	+	0.634759i	$0.218901\pi$
$998$	0	0
$999$	−4.15323e6	−0.131666

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	228.6.a.b.1.2	✓	4
3.2	odd	2		684.6.a.d.1.3		4
4.3	odd	2		912.6.a.p.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.6.a.b.1.2	✓	4	1.1	even	1	trivial
684.6.a.d.1.3		4	3.2	odd	2
912.6.a.p.1.2		4	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:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	228.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -24.7456 q^{5} -94.5162 q^{7} +81.0000 q^{9} +658.815 q^{11} +137.434 q^{13} +222.710 q^{15} +434.528 q^{17} +361.000 q^{19} +850.646 q^{21} -2093.92 q^{23} -2512.66 q^{25} -729.000 q^{27} +5088.18 q^{29} -8041.00 q^{31} -5929.34 q^{33} +2338.86 q^{35} +5697.17 q^{37} -1236.90 q^{39} -19145.7 q^{41} +22560.1 q^{43} -2004.39 q^{45} -5051.26 q^{47} -7873.69 q^{49} -3910.76 q^{51} -14112.5 q^{53} -16302.8 q^{55} -3249.00 q^{57} -43554.9 q^{59} -35333.8 q^{61} -7655.81 q^{63} -3400.88 q^{65} -19068.1 q^{67} +18845.3 q^{69} +25159.9 q^{71} -60015.6 q^{73} +22613.9 q^{75} -62268.7 q^{77} +90721.4 q^{79} +6561.00 q^{81} -114736. q^{83} -10752.6 q^{85} -45793.6 q^{87} +24441.7 q^{89} -12989.7 q^{91} +72369.0 q^{93} -8933.15 q^{95} -135658. q^{97} +53364.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} - 84 q^{5} + 54 q^{7} + 324 q^{9} + 354 q^{11} + 46 q^{13} + 756 q^{15} + 2046 q^{17} + 1444 q^{19} - 486 q^{21} + 846 q^{23} + 4018 q^{25} - 2916 q^{27} - 7152 q^{29} + 238 q^{31} - 3186 q^{33}+ \cdots + 28674 q^{99}+O(q^{100}) \) 4 * q - 36 * q^3 - 84 * q^5 + 54 * q^7 + 324 * q^9 + 354 * q^11 + 46 * q^13 + 756 * q^15 + 2046 * q^17 + 1444 * q^19 - 486 * q^21 + 846 * q^23 + 4018 * q^25 - 2916 * q^27 - 7152 * q^29 + 238 * q^31 - 3186 * q^33 - 19878 * q^35 - 10326 * q^37 - 414 * q^39 - 28200 * q^41 - 18590 * q^43 - 6804 * q^45 - 18684 * q^47 - 17946 * q^49 - 18414 * q^51 - 42720 * q^53 - 13818 * q^55 - 12996 * q^57 - 68256 * q^59 - 67654 * q^61 + 4374 * q^63 - 109776 * q^65 - 81836 * q^67 - 7614 * q^69 - 62112 * q^71 - 81966 * q^73 - 36162 * q^75 - 81378 * q^77 + 8766 * q^79 + 26244 * q^81 - 55080 * q^83 - 108030 * q^85 + 64368 * q^87 - 197520 * q^89 + 25820 * q^91 - 2142 * q^93 - 30324 * q^95 - 184228 * q^97 + 28674 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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