LMFDB - Embedded newform 1881.2.a.n.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1881,2,Mod(1,1881)] 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("1881.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1881, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,-2,0,6,-8] 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:	$15.0198606202$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 7x^{4} + 22x^{3} - 12x^{2} - 9x - 1 $ x^7 - x^6 - 9x^5 + 7x^4 + 22x^3 - 12x^2 - 9*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.07449$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.659361 q^{2} -1.56524 q^{4} -1.36783 q^{5} -4.12178 q^{7} +2.35078 q^{8} +0.901896 q^{10} +1.00000 q^{11} +5.19538 q^{13} +2.71774 q^{14} +1.58047 q^{16} +3.22290 q^{17} +1.00000 q^{19} +2.14099 q^{20} -0.659361 q^{22} -8.38994 q^{23} -3.12903 q^{25} -3.42563 q^{26} +6.45158 q^{28} +8.06737 q^{29} +2.96753 q^{31} -5.74367 q^{32} -2.12506 q^{34} +5.63791 q^{35} +4.26722 q^{37} -0.659361 q^{38} -3.21548 q^{40} -5.15535 q^{41} +10.2105 q^{43} -1.56524 q^{44} +5.53200 q^{46} -0.149795 q^{47} +9.98904 q^{49} +2.06316 q^{50} -8.13204 q^{52} -6.14245 q^{53} -1.36783 q^{55} -9.68940 q^{56} -5.31931 q^{58} -11.8834 q^{59} -9.58057 q^{61} -1.95667 q^{62} +0.626199 q^{64} -7.10642 q^{65} -6.34315 q^{67} -5.04463 q^{68} -3.71741 q^{70} -14.3895 q^{71} +3.01996 q^{73} -2.81364 q^{74} -1.56524 q^{76} -4.12178 q^{77} +3.15936 q^{79} -2.16183 q^{80} +3.39923 q^{82} -13.1976 q^{83} -4.40840 q^{85} -6.73239 q^{86} +2.35078 q^{88} +10.4570 q^{89} -21.4142 q^{91} +13.1323 q^{92} +0.0987692 q^{94} -1.36783 q^{95} +14.2761 q^{97} -6.58638 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 2 q^{7} - 6 q^{8} + 6 q^{10} + 7 q^{11} - 7 q^{13} - 6 q^{14} - 5 q^{17} + 7 q^{19} - 20 q^{20} - 2 q^{22} - 22 q^{23} + 3 q^{25} - 10 q^{26} + 2 q^{28} - 18 q^{29} - 14 q^{32}+ \cdots + 62 q^{98}+O(q^{100}) $ 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 2 * q^7 - 6 * q^8 + 6 * q^10 + 7 * q^11 - 7 * q^13 - 6 * q^14 - 5 * q^17 + 7 * q^19 - 20 * q^20 - 2 * q^22 - 22 * q^23 + 3 * q^25 - 10 * q^26 + 2 * q^28 - 18 * q^29 - 14 * q^32 + 6 * q^34 - 8 * q^35 + 6 * q^37 - 2 * q^38 + 28 * q^40 - 6 * q^41 + 8 * q^43 + 6 * q^44 + 8 * q^46 - 14 * q^47 + 5 * q^49 - 24 * q^50 - 12 * q^52 + q^53 - 8 * q^55 - 20 * q^56 - 2 * q^58 - 23 * q^59 - 28 * q^61 - 2 * q^62 - 26 * q^64 - 10 * q^65 + 14 * q^67 - 12 * q^68 - 18 * q^70 - 31 * q^71 - 12 * q^73 - 20 * q^74 + 6 * q^76 - 2 * q^77 - 17 * q^79 + 6 * q^80 + 12 * q^82 - 5 * q^83 - 12 * q^85 - 38 * q^86 - 6 * q^88 + 3 * q^89 - 12 * q^91 - 60 * q^94 - 8 * q^95 + 12 * q^97 + 62 * 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 $

$2$	−0.659361	−0.466238	−0.233119	−	0.972448i	$-0.574893\pi$
$2$	−0.659361	−0.466238	−0.233119	+	0.972448i	$0.574893\pi$
$3$	0	0
$3$	0	0
$4$	−1.56524	−0.782622
$5$	−1.36783	−0.611714	−0.305857	−	0.952077i	$-0.598943\pi$
$5$	−1.36783	−0.611714	−0.305857	+	0.952077i	$0.598943\pi$
$6$	0	0
$7$	−4.12178	−1.55789	−0.778943	−	0.627095i	$-0.784244\pi$
$7$	−4.12178	−1.55789	−0.778943	+	0.627095i	$0.784244\pi$
$8$	2.35078	0.831127
$9$	0	0
$10$	0.901896	0.285205
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.19538	1.44094	0.720470	−	0.693486i	$-0.243926\pi$
$13$	5.19538	1.44094	0.720470	+	0.693486i	$0.243926\pi$
$14$	2.71774	0.726346
$15$	0	0
$16$	1.58047	0.395119
$17$	3.22290	0.781669	0.390835	−	0.920461i	$-0.372186\pi$
$17$	3.22290	0.781669	0.390835	+	0.920461i	$0.372186\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.14099	0.478741
$21$	0	0
$22$	−0.659361	−0.140576
$23$	−8.38994	−1.74942	−0.874712	−	0.484643i	$-0.838949\pi$
$23$	−8.38994	−1.74942	−0.874712	+	0.484643i	$0.838949\pi$
$24$	0	0
$25$	−3.12903	−0.625806
$26$	−3.42563	−0.671822
$27$	0	0
$28$	6.45158	1.21923
$29$	8.06737	1.49807	0.749037	−	0.662529i	$-0.230517\pi$
$29$	8.06737	1.49807	0.749037	+	0.662529i	$0.230517\pi$
$30$	0	0
$31$	2.96753	0.532983	0.266492	−	0.963837i	$-0.414136\pi$
$31$	2.96753	0.532983	0.266492	+	0.963837i	$0.414136\pi$
$32$	−5.74367	−1.01535
$33$	0	0
$34$	−2.12506	−0.364444
$35$	5.63791	0.952980
$36$	0	0
$37$	4.26722	0.701527	0.350763	−	0.936464i	$-0.385922\pi$
$37$	4.26722	0.701527	0.350763	+	0.936464i	$0.385922\pi$
$38$	−0.659361	−0.106962
$39$	0	0
$40$	−3.21548	−0.508412
$41$	−5.15535	−0.805130	−0.402565	−	0.915391i	$-0.631881\pi$
$41$	−5.15535	−0.805130	−0.402565	+	0.915391i	$0.631881\pi$
$42$	0	0
$43$	10.2105	1.55708	0.778542	−	0.627593i	$-0.215960\pi$
$43$	10.2105	1.55708	0.778542	+	0.627593i	$0.215960\pi$
$44$	−1.56524	−0.235969
$45$	0	0
$46$	5.53200	0.815648
$47$	−0.149795	−0.0218499	−0.0109250	−	0.999940i	$-0.503478\pi$
$47$	−0.149795	−0.0218499	−0.0109250	+	0.999940i	$0.503478\pi$
$48$	0	0
$49$	9.98904	1.42701
$50$	2.06316	0.291775
$51$	0	0
$52$	−8.13204	−1.12771
$53$	−6.14245	−0.843731	−0.421865	−	0.906659i	$-0.638624\pi$
$53$	−6.14245	−0.843731	−0.421865	+	0.906659i	$0.638624\pi$
$54$	0	0
$55$	−1.36783	−0.184439
$56$	−9.68940	−1.29480
$57$	0	0
$58$	−5.31931	−0.698459
$59$	−11.8834	−1.54709	−0.773545	−	0.633742i	$-0.781518\pi$
$59$	−11.8834	−1.54709	−0.773545	+	0.633742i	$0.781518\pi$
$60$	0	0
$61$	−9.58057	−1.22667	−0.613333	−	0.789824i	$-0.710172\pi$
$61$	−9.58057	−1.22667	−0.613333	+	0.789824i	$0.710172\pi$
$62$	−1.95667	−0.248497
$63$	0	0
$64$	0.626199	0.0782748
$65$	−7.10642	−0.881443
$66$	0	0
$67$	−6.34315	−0.774940	−0.387470	−	0.921882i	$-0.626651\pi$
$67$	−6.34315	−0.774940	−0.387470	+	0.921882i	$0.626651\pi$
$68$	−5.04463	−0.611751
$69$	0	0
$70$	−3.71741	−0.444316
$71$	−14.3895	−1.70773	−0.853863	−	0.520498i	$-0.825746\pi$
$71$	−14.3895	−1.70773	−0.853863	+	0.520498i	$0.825746\pi$
$72$	0	0
$73$	3.01996	0.353460	0.176730	−	0.984259i	$-0.443448\pi$
$73$	3.01996	0.353460	0.176730	+	0.984259i	$0.443448\pi$
$74$	−2.81364	−0.327079
$75$	0	0
$76$	−1.56524	−0.179546
$77$	−4.12178	−0.469720
$78$	0	0
$79$	3.15936	0.355455	0.177728	−	0.984080i	$-0.443125\pi$
$79$	3.15936	0.355455	0.177728	+	0.984080i	$0.443125\pi$
$80$	−2.16183	−0.241700
$81$	0	0
$82$	3.39923	0.375382
$83$	−13.1976	−1.44863	−0.724315	−	0.689470i	$-0.757844\pi$
$83$	−13.1976	−1.44863	−0.724315	+	0.689470i	$0.757844\pi$
$84$	0	0
$85$	−4.40840	−0.478158
$86$	−6.73239	−0.725972
$87$	0	0
$88$	2.35078	0.250594
$89$	10.4570	1.10844	0.554221	−	0.832369i	$-0.313016\pi$
$89$	10.4570	1.10844	0.554221	+	0.832369i	$0.313016\pi$
$90$	0	0
$91$	−21.4142	−2.24482
$92$	13.1323	1.36914
$93$	0	0
$94$	0.0987692	0.0101873
$95$	−1.36783	−0.140337
$96$	0	0
$97$	14.2761	1.44952	0.724758	−	0.689004i	$-0.241952\pi$
$97$	14.2761	1.44952	0.724758	+	0.689004i	$0.241952\pi$
$98$	−6.58638	−0.665325
$99$	0	0
$100$	4.89769	0.489769
$101$	−5.19923	−0.517343	−0.258671	−	0.965965i	$-0.583285\pi$
$101$	−5.19923	−0.517343	−0.258671	+	0.965965i	$0.583285\pi$
$102$	0	0
$103$	−5.99586	−0.590789	−0.295395	−	0.955375i	$-0.595451\pi$
$103$	−5.99586	−0.590789	−0.295395	+	0.955375i	$0.595451\pi$
$104$	12.2132	1.19760
$105$	0	0
$106$	4.05009	0.393380
$107$	0.0698571	0.00675334	0.00337667	−	0.999994i	$-0.498925\pi$
$107$	0.0698571	0.00675334	0.00337667	+	0.999994i	$0.498925\pi$
$108$	0	0
$109$	−13.9994	−1.34090	−0.670450	−	0.741955i	$-0.733899\pi$
$109$	−13.9994	−1.34090	−0.670450	+	0.741955i	$0.733899\pi$
$110$	0.901896	0.0859924
$111$	0	0
$112$	−6.51436	−0.615549
$113$	−14.9533	−1.40669	−0.703346	−	0.710847i	$-0.748312\pi$
$113$	−14.9533	−1.40669	−0.703346	+	0.710847i	$0.748312\pi$
$114$	0	0
$115$	11.4760	1.07015
$116$	−12.6274	−1.17242
$117$	0	0
$118$	7.83546	0.721312
$119$	−13.2841	−1.21775
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.31705	0.571919
$123$	0	0
$124$	−4.64490	−0.417124
$125$	11.1192	0.994528
$126$	0	0
$127$	15.9404	1.41448	0.707240	−	0.706974i	$-0.249940\pi$
$127$	15.9404	1.41448	0.707240	+	0.706974i	$0.249940\pi$
$128$	11.0744	0.978851
$129$	0	0
$130$	4.68570	0.410963
$131$	1.37498	0.120132	0.0600662	−	0.998194i	$-0.480869\pi$
$131$	1.37498	0.120132	0.0600662	+	0.998194i	$0.480869\pi$
$132$	0	0
$133$	−4.12178	−0.357403
$134$	4.18243	0.361307
$135$	0	0
$136$	7.57634	0.649666
$137$	21.4695	1.83426	0.917131	−	0.398587i	$-0.130499\pi$
$137$	21.4695	1.83426	0.917131	+	0.398587i	$0.130499\pi$
$138$	0	0
$139$	−11.1461	−0.945401	−0.472700	−	0.881223i	$-0.656721\pi$
$139$	−11.1461	−0.945401	−0.472700	+	0.881223i	$0.656721\pi$
$140$	−8.82470	−0.745823
$141$	0	0
$142$	9.48790	0.796207
$143$	5.19538	0.434460
$144$	0	0
$145$	−11.0348	−0.916393
$146$	−1.99124	−0.164796
$147$	0	0
$148$	−6.67924	−0.549030
$149$	−4.58899	−0.375945	−0.187972	−	0.982174i	$-0.560192\pi$
$149$	−4.58899	−0.375945	−0.187972	+	0.982174i	$0.560192\pi$
$150$	0	0
$151$	−18.9094	−1.53882	−0.769411	−	0.638754i	$-0.779450\pi$
$151$	−18.9094	−1.53882	−0.769411	+	0.638754i	$0.779450\pi$
$152$	2.35078	0.190674
$153$	0	0
$154$	2.71774	0.219002
$155$	−4.05908	−0.326033
$156$	0	0
$157$	−16.5112	−1.31774	−0.658868	−	0.752259i	$-0.728964\pi$
$157$	−16.5112	−1.31774	−0.658868	+	0.752259i	$0.728964\pi$
$158$	−2.08316	−0.165727
$159$	0	0
$160$	7.85638	0.621102
$161$	34.5815	2.72540
$162$	0	0
$163$	3.35007	0.262398	0.131199	−	0.991356i	$-0.458117\pi$
$163$	3.35007	0.262398	0.131199	+	0.991356i	$0.458117\pi$
$164$	8.06937	0.630112
$165$	0	0
$166$	8.70201	0.675407
$167$	−3.73604	−0.289103	−0.144552	−	0.989497i	$-0.546174\pi$
$167$	−3.73604	−0.289103	−0.144552	+	0.989497i	$0.546174\pi$
$168$	0	0
$169$	13.9920	1.07631
$170$	2.90672	0.222936
$171$	0	0
$172$	−15.9819	−1.21861
$173$	−17.6066	−1.33861	−0.669303	−	0.742989i	$-0.733407\pi$
$173$	−17.6066	−1.33861	−0.669303	+	0.742989i	$0.733407\pi$
$174$	0	0
$175$	12.8972	0.974934
$176$	1.58047	0.119133
$177$	0	0
$178$	−6.89495	−0.516798
$179$	−8.22822	−0.615006	−0.307503	−	0.951547i	$-0.599493\pi$
$179$	−8.22822	−0.615006	−0.307503	+	0.951547i	$0.599493\pi$
$180$	0	0
$181$	5.61440	0.417315	0.208657	−	0.977989i	$-0.433091\pi$
$181$	5.61440	0.417315	0.208657	+	0.977989i	$0.433091\pi$
$182$	14.1197	1.04662
$183$	0	0
$184$	−19.7229	−1.45399
$185$	−5.83685	−0.429134
$186$	0	0
$187$	3.22290	0.235682
$188$	0.234466	0.0171002
$189$	0	0
$190$	0.901896	0.0654304
$191$	−22.2453	−1.60962	−0.804808	−	0.593536i	$-0.797732\pi$
$191$	−22.2453	−1.60962	−0.804808	+	0.593536i	$0.797732\pi$
$192$	0	0
$193$	−15.5672	−1.12055	−0.560275	−	0.828306i	$-0.689305\pi$
$193$	−15.5672	−1.12055	−0.560275	+	0.828306i	$0.689305\pi$
$194$	−9.41308	−0.675820
$195$	0	0
$196$	−15.6353	−1.11681
$197$	6.72177	0.478906	0.239453	−	0.970908i	$-0.423032\pi$
$197$	6.72177	0.478906	0.239453	+	0.970908i	$0.423032\pi$
$198$	0	0
$199$	0.465250	0.0329807	0.0164904	−	0.999864i	$-0.494751\pi$
$199$	0.465250	0.0329807	0.0164904	+	0.999864i	$0.494751\pi$
$200$	−7.35566	−0.520124
$201$	0	0
$202$	3.42817	0.241205
$203$	−33.2519	−2.33383
$204$	0	0
$205$	7.05166	0.492509
$206$	3.95343	0.275449
$207$	0	0
$208$	8.21117	0.569342
$209$	1.00000	0.0691714
$210$	0	0
$211$	9.51178	0.654818	0.327409	−	0.944883i	$-0.393825\pi$
$211$	9.51178	0.654818	0.327409	+	0.944883i	$0.393825\pi$
$212$	9.61443	0.660322
$213$	0	0
$214$	−0.0460610	−0.00314867
$215$	−13.9662	−0.952490
$216$	0	0
$217$	−12.2315	−0.830327
$218$	9.23066	0.625179
$219$	0	0
$220$	2.14099	0.144346
$221$	16.7442	1.12634
$222$	0	0
$223$	0.879682	0.0589078	0.0294539	−	0.999566i	$-0.490623\pi$
$223$	0.879682	0.0589078	0.0294539	+	0.999566i	$0.490623\pi$
$224$	23.6741	1.58179
$225$	0	0
$226$	9.85965	0.655854
$227$	−19.8435	−1.31706	−0.658529	−	0.752556i	$-0.728821\pi$
$227$	−19.8435	−1.31706	−0.658529	+	0.752556i	$0.728821\pi$
$228$	0	0
$229$	27.5646	1.82152	0.910760	−	0.412936i	$-0.135497\pi$
$229$	27.5646	1.82152	0.910760	+	0.412936i	$0.135497\pi$
$230$	−7.56686	−0.498944
$231$	0	0
$232$	18.9646	1.24509
$233$	9.60936	0.629530	0.314765	−	0.949170i	$-0.398074\pi$
$233$	9.60936	0.629530	0.314765	+	0.949170i	$0.398074\pi$
$234$	0	0
$235$	0.204895	0.0133659
$236$	18.6004	1.21079
$237$	0	0
$238$	8.75901	0.567762
$239$	23.2890	1.50644	0.753219	−	0.657770i	$-0.228500\pi$
$239$	23.2890	1.50644	0.753219	+	0.657770i	$0.228500\pi$
$240$	0	0
$241$	−0.0990583	−0.00638091	−0.00319045	−	0.999995i	$-0.501016\pi$
$241$	−0.0990583	−0.00638091	−0.00319045	+	0.999995i	$0.501016\pi$
$242$	−0.659361	−0.0423853
$243$	0	0
$244$	14.9959	0.960016
$245$	−13.6634	−0.872920
$246$	0	0
$247$	5.19538	0.330574
$248$	6.97600	0.442977
$249$	0	0
$250$	−7.33154	−0.463687
$251$	−13.8105	−0.871708	−0.435854	−	0.900017i	$-0.643554\pi$
$251$	−13.8105	−0.871708	−0.435854	+	0.900017i	$0.643554\pi$
$252$	0	0
$253$	−8.38994	−0.527471
$254$	−10.5105	−0.659485
$255$	0	0
$256$	−8.55445	−0.534653
$257$	−2.16556	−0.135084	−0.0675419	−	0.997716i	$-0.521516\pi$
$257$	−2.16556	−0.135084	−0.0675419	+	0.997716i	$0.521516\pi$
$258$	0	0
$259$	−17.5885	−1.09290
$260$	11.1233	0.689837
$261$	0	0
$262$	−0.906607	−0.0560104
$263$	7.69195	0.474306	0.237153	−	0.971472i	$-0.423786\pi$
$263$	7.69195	0.474306	0.237153	+	0.971472i	$0.423786\pi$
$264$	0	0
$265$	8.40186	0.516122
$266$	2.71774	0.166635
$267$	0	0
$268$	9.92858	0.606485
$269$	0.550251	0.0335494	0.0167747	−	0.999859i	$-0.494660\pi$
$269$	0.550251	0.0335494	0.0167747	+	0.999859i	$0.494660\pi$
$270$	0	0
$271$	6.38378	0.387787	0.193894	−	0.981023i	$-0.437888\pi$
$271$	6.38378	0.387787	0.193894	+	0.981023i	$0.437888\pi$
$272$	5.09372	0.308852
$273$	0	0
$274$	−14.1561	−0.855203
$275$	−3.12903	−0.188688
$276$	0	0
$277$	−29.2996	−1.76044	−0.880221	−	0.474563i	$-0.842606\pi$
$277$	−29.2996	−1.76044	−0.880221	+	0.474563i	$0.842606\pi$
$278$	7.34931	0.440782
$279$	0	0
$280$	13.2535	0.792047
$281$	−14.4530	−0.862196	−0.431098	−	0.902305i	$-0.641874\pi$
$281$	−14.4530	−0.862196	−0.431098	+	0.902305i	$0.641874\pi$
$282$	0	0
$283$	0.779649	0.0463453	0.0231727	−	0.999731i	$-0.492623\pi$
$283$	0.779649	0.0463453	0.0231727	+	0.999731i	$0.492623\pi$
$284$	22.5231	1.33650
$285$	0	0
$286$	−3.42563	−0.202562
$287$	21.2492	1.25430
$288$	0	0
$289$	−6.61289	−0.388994
$290$	7.27593	0.427257
$291$	0	0
$292$	−4.72697	−0.276625
$293$	−10.7836	−0.629983	−0.314992	−	0.949094i	$-0.602002\pi$
$293$	−10.7836	−0.629983	−0.314992	+	0.949094i	$0.602002\pi$
$294$	0	0
$295$	16.2545	0.946376
$296$	10.0313	0.583058
$297$	0	0
$298$	3.02580	0.175280
$299$	−43.5890	−2.52081
$300$	0	0
$301$	−42.0853	−2.42576
$302$	12.4681	0.717458
$303$	0	0
$304$	1.58047	0.0906464
$305$	13.1046	0.750369
$306$	0	0
$307$	9.09308	0.518969	0.259485	−	0.965747i	$-0.416447\pi$
$307$	9.09308	0.518969	0.259485	+	0.965747i	$0.416447\pi$
$308$	6.45158	0.367613
$309$	0	0
$310$	2.67640	0.152009
$311$	−8.54476	−0.484529	−0.242264	−	0.970210i	$-0.577890\pi$
$311$	−8.54476	−0.484529	−0.242264	+	0.970210i	$0.577890\pi$
$312$	0	0
$313$	−28.0173	−1.58363	−0.791815	−	0.610761i	$-0.790864\pi$
$313$	−28.0173	−1.58363	−0.791815	+	0.610761i	$0.790864\pi$
$314$	10.8868	0.614379
$315$	0	0
$316$	−4.94516	−0.278187
$317$	−13.8377	−0.777201	−0.388600	−	0.921406i	$-0.627041\pi$
$317$	−13.8377	−0.777201	−0.388600	+	0.921406i	$0.627041\pi$
$318$	0	0
$319$	8.06737	0.451686
$320$	−0.856536	−0.0478818
$321$	0	0
$322$	−22.8017	−1.27069
$323$	3.22290	0.179327
$324$	0	0
$325$	−16.2565	−0.901749
$326$	−2.20890	−0.122340
$327$	0	0
$328$	−12.1191	−0.669165
$329$	0.617423	0.0340396
$330$	0	0
$331$	4.27770	0.235124	0.117562	−	0.993066i	$-0.462492\pi$
$331$	4.27770	0.235124	0.117562	+	0.993066i	$0.462492\pi$
$332$	20.6575	1.13373
$333$	0	0
$334$	2.46340	0.134791
$335$	8.67638	0.474041
$336$	0	0
$337$	26.0324	1.41807	0.709037	−	0.705171i	$-0.249130\pi$
$337$	26.0324	1.41807	0.709037	+	0.705171i	$0.249130\pi$
$338$	−9.22578	−0.501816
$339$	0	0
$340$	6.90022	0.374217
$341$	2.96753	0.160701
$342$	0	0
$343$	−12.3202	−0.665226
$344$	24.0026	1.29413
$345$	0	0
$346$	11.6091	0.624110
$347$	2.04257	0.109651	0.0548256	−	0.998496i	$-0.482540\pi$
$347$	2.04257	0.109651	0.0548256	+	0.998496i	$0.482540\pi$
$348$	0	0
$349$	−10.6796	−0.571665	−0.285833	−	0.958280i	$-0.592270\pi$
$349$	−10.6796	−0.571665	−0.285833	+	0.958280i	$0.592270\pi$
$350$	−8.50388	−0.454551
$351$	0	0
$352$	−5.74367	−0.306138
$353$	29.2609	1.55740	0.778701	−	0.627396i	$-0.215879\pi$
$353$	29.2609	1.55740	0.778701	+	0.627396i	$0.215879\pi$
$354$	0	0
$355$	19.6825	1.04464
$356$	−16.3678	−0.867491
$357$	0	0
$358$	5.42537	0.286740
$359$	20.5749	1.08590	0.542952	−	0.839764i	$-0.317307\pi$
$359$	20.5749	1.08590	0.542952	+	0.839764i	$0.317307\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−3.70191	−0.194568
$363$	0	0
$364$	33.5185	1.75684
$365$	−4.13081	−0.216216
$366$	0	0
$367$	−21.4180	−1.11801	−0.559005	−	0.829164i	$-0.688817\pi$
$367$	−21.4180	−1.11801	−0.559005	+	0.829164i	$0.688817\pi$
$368$	−13.2601	−0.691230
$369$	0	0
$370$	3.84859	0.200079
$371$	25.3178	1.31444
$372$	0	0
$373$	−30.6079	−1.58482	−0.792410	−	0.609989i	$-0.791174\pi$
$373$	−30.6079	−1.58482	−0.792410	+	0.609989i	$0.791174\pi$
$374$	−2.12506	−0.109884
$375$	0	0
$376$	−0.352136	−0.0181600
$377$	41.9131	2.15863
$378$	0	0
$379$	5.38689	0.276706	0.138353	−	0.990383i	$-0.455819\pi$
$379$	5.38689	0.276706	0.138353	+	0.990383i	$0.455819\pi$
$380$	2.14099	0.109831
$381$	0	0
$382$	14.6677	0.750464
$383$	31.5772	1.61352	0.806761	−	0.590878i	$-0.201219\pi$
$383$	31.5772	1.61352	0.806761	+	0.590878i	$0.201219\pi$
$384$	0	0
$385$	5.63791	0.287334
$386$	10.2644	0.522444
$387$	0	0
$388$	−22.3455	−1.13442
$389$	−9.30869	−0.471969	−0.235985	−	0.971757i	$-0.575832\pi$
$389$	−9.30869	−0.471969	−0.235985	+	0.971757i	$0.575832\pi$
$390$	0	0
$391$	−27.0400	−1.36747
$392$	23.4820	1.18602
$393$	0	0
$394$	−4.43207	−0.223284
$395$	−4.32148	−0.217437
$396$	0	0
$397$	10.2531	0.514588	0.257294	−	0.966333i	$-0.417169\pi$
$397$	10.2531	0.514588	0.257294	+	0.966333i	$0.417169\pi$
$398$	−0.306768	−0.0153769
$399$	0	0
$400$	−4.94535	−0.247268
$401$	−17.5959	−0.878696	−0.439348	−	0.898317i	$-0.644790\pi$
$401$	−17.5959	−0.878696	−0.439348	+	0.898317i	$0.644790\pi$
$402$	0	0
$403$	15.4174	0.767997
$404$	8.13806	0.404884
$405$	0	0
$406$	21.9250	1.08812
$407$	4.26722	0.211518
$408$	0	0
$409$	−23.6042	−1.16715	−0.583576	−	0.812058i	$-0.698347\pi$
$409$	−23.6042	−1.16715	−0.583576	+	0.812058i	$0.698347\pi$
$410$	−4.64959	−0.229627
$411$	0	0
$412$	9.38497	0.462365
$413$	48.9808	2.41019
$414$	0	0
$415$	18.0522	0.886147
$416$	−29.8405	−1.46305
$417$	0	0
$418$	−0.659361	−0.0322504
$419$	−36.8265	−1.79909	−0.899546	−	0.436825i	$-0.856103\pi$
$419$	−36.8265	−1.79909	−0.899546	+	0.436825i	$0.856103\pi$
$420$	0	0
$421$	8.77615	0.427723	0.213862	−	0.976864i	$-0.431396\pi$
$421$	8.77615	0.427723	0.213862	+	0.976864i	$0.431396\pi$
$422$	−6.27169	−0.305301
$423$	0	0
$424$	−14.4396	−0.701247
$425$	−10.0846	−0.489173
$426$	0	0
$427$	39.4890	1.91100
$428$	−0.109343	−0.00528531
$429$	0	0
$430$	9.20879	0.444087
$431$	−13.7236	−0.661044	−0.330522	−	0.943798i	$-0.607225\pi$
$431$	−13.7236	−0.661044	−0.330522	+	0.943798i	$0.607225\pi$
$432$	0	0
$433$	−14.1199	−0.678557	−0.339279	−	0.940686i	$-0.610183\pi$
$433$	−14.1199	−0.678557	−0.339279	+	0.940686i	$0.610183\pi$
$434$	8.06496	0.387130
$435$	0	0
$436$	21.9125	1.04942
$437$	−8.38994	−0.401345
$438$	0	0
$439$	20.1196	0.960257	0.480128	−	0.877198i	$-0.340590\pi$
$439$	20.1196	0.960257	0.480128	+	0.877198i	$0.340590\pi$
$440$	−3.21548	−0.153292
$441$	0	0
$442$	−11.0405	−0.525142
$443$	1.19847	0.0569411	0.0284706	−	0.999595i	$-0.490936\pi$
$443$	1.19847	0.0569411	0.0284706	+	0.999595i	$0.490936\pi$
$444$	0	0
$445$	−14.3035	−0.678050
$446$	−0.580027	−0.0274651
$447$	0	0
$448$	−2.58105	−0.121943
$449$	13.6222	0.642872	0.321436	−	0.946931i	$-0.395834\pi$
$449$	13.6222	0.642872	0.321436	+	0.946931i	$0.395834\pi$
$450$	0	0
$451$	−5.15535	−0.242756
$452$	23.4056	1.10091
$453$	0	0
$454$	13.0840	0.614063
$455$	29.2911	1.37319
$456$	0	0
$457$	33.2894	1.55721	0.778606	−	0.627513i	$-0.215927\pi$
$457$	33.2894	1.55721	0.778606	+	0.627513i	$0.215927\pi$
$458$	−18.1750	−0.849262
$459$	0	0
$460$	−17.9628	−0.837520
$461$	−26.2316	−1.22173	−0.610864	−	0.791735i	$-0.709178\pi$
$461$	−26.2316	−1.22173	−0.610864	+	0.791735i	$0.709178\pi$
$462$	0	0
$463$	−24.7451	−1.15000	−0.575001	−	0.818152i	$-0.694998\pi$
$463$	−24.7451	−1.15000	−0.575001	+	0.818152i	$0.694998\pi$
$464$	12.7503	0.591917
$465$	0	0
$466$	−6.33603	−0.293511
$467$	8.19382	0.379165	0.189582	−	0.981865i	$-0.439287\pi$
$467$	8.19382	0.379165	0.189582	+	0.981865i	$0.439287\pi$
$468$	0	0
$469$	26.1451	1.20727
$470$	−0.135100	−0.00623169
$471$	0	0
$472$	−27.9353	−1.28583
$473$	10.2105	0.469478
$474$	0	0
$475$	−3.12903	−0.143570
$476$	20.7928	0.953038
$477$	0	0
$478$	−15.3558	−0.702359
$479$	−27.8192	−1.27109	−0.635547	−	0.772062i	$-0.719225\pi$
$479$	−27.8192	−1.27109	−0.635547	+	0.772062i	$0.719225\pi$
$480$	0	0
$481$	22.1698	1.01086
$482$	0.0653152	0.00297502
$483$	0	0
$484$	−1.56524	−0.0711474
$485$	−19.5273	−0.886689
$486$	0	0
$487$	−6.32344	−0.286542	−0.143271	−	0.989683i	$-0.545762\pi$
$487$	−6.32344	−0.286542	−0.143271	+	0.989683i	$0.545762\pi$
$488$	−22.5218	−1.01952
$489$	0	0
$490$	9.00908	0.406989
$491$	35.9788	1.62370	0.811850	−	0.583866i	$-0.198461\pi$
$491$	35.9788	1.62370	0.811850	+	0.583866i	$0.198461\pi$
$492$	0	0
$493$	26.0004	1.17100
$494$	−3.42563	−0.154126
$495$	0	0
$496$	4.69010	0.210592
$497$	59.3105	2.66044
$498$	0	0
$499$	20.5152	0.918384	0.459192	−	0.888337i	$-0.348139\pi$
$499$	20.5152	0.918384	0.459192	+	0.888337i	$0.348139\pi$
$500$	−17.4042	−0.778340
$501$	0	0
$502$	9.10607	0.406424
$503$	−32.9161	−1.46766	−0.733829	−	0.679335i	$-0.762268\pi$
$503$	−32.9161	−1.46766	−0.733829	+	0.679335i	$0.762268\pi$
$504$	0	0
$505$	7.11169	0.316466
$506$	5.53200	0.245927
$507$	0	0
$508$	−24.9506	−1.10700
$509$	−8.08795	−0.358492	−0.179246	−	0.983804i	$-0.557366\pi$
$509$	−8.08795	−0.358492	−0.179246	+	0.983804i	$0.557366\pi$
$510$	0	0
$511$	−12.4476	−0.550650
$512$	−16.5084	−0.729576
$513$	0	0
$514$	1.42788	0.0629813
$515$	8.20134	0.361394
$516$	0	0
$517$	−0.149795	−0.00658800
$518$	11.5972	0.509551
$519$	0	0
$520$	−16.7056	−0.732591
$521$	−31.2247	−1.36798	−0.683990	−	0.729491i	$-0.739757\pi$
$521$	−31.2247	−1.36798	−0.683990	+	0.729491i	$0.739757\pi$
$522$	0	0
$523$	−15.6373	−0.683771	−0.341886	−	0.939742i	$-0.611066\pi$
$523$	−15.6373	−0.683771	−0.341886	+	0.939742i	$0.611066\pi$
$524$	−2.15218	−0.0940183
$525$	0	0
$526$	−5.07177	−0.221140
$527$	9.56405	0.416617
$528$	0	0
$529$	47.3911	2.06048
$530$	−5.53985	−0.240636
$531$	0	0
$532$	6.45158	0.279712
$533$	−26.7840	−1.16014
$534$	0	0
$535$	−0.0955530	−0.00413111
$536$	−14.9114	−0.644073
$537$	0	0
$538$	−0.362814	−0.0156420
$539$	9.98904	0.430258
$540$	0	0
$541$	−31.7133	−1.36346	−0.681730	−	0.731604i	$-0.738772\pi$
$541$	−31.7133	−1.36346	−0.681730	+	0.731604i	$0.738772\pi$
$542$	−4.20922	−0.180801
$543$	0	0
$544$	−18.5113	−0.793665
$545$	19.1489	0.820247
$546$	0	0
$547$	17.2492	0.737524	0.368762	−	0.929524i	$-0.379782\pi$
$547$	17.2492	0.737524	0.368762	+	0.929524i	$0.379782\pi$
$548$	−33.6050	−1.43553
$549$	0	0
$550$	2.06316	0.0879734
$551$	8.06737	0.343682
$552$	0	0
$553$	−13.0222	−0.553759
$554$	19.3190	0.820786
$555$	0	0
$556$	17.4464	0.739891
$557$	20.5077	0.868938	0.434469	−	0.900687i	$-0.356936\pi$
$557$	20.5077	0.868938	0.434469	+	0.900687i	$0.356936\pi$
$558$	0	0
$559$	53.0474	2.24366
$560$	8.91057	0.376540
$561$	0	0
$562$	9.52977	0.401989
$563$	29.9098	1.26055	0.630275	−	0.776372i	$-0.282942\pi$
$563$	29.9098	1.26055	0.630275	+	0.776372i	$0.282942\pi$
$564$	0	0
$565$	20.4537	0.860494
$566$	−0.514070	−0.0216080
$567$	0	0
$568$	−33.8267	−1.41934
$569$	−25.5933	−1.07293	−0.536464	−	0.843923i	$-0.680240\pi$
$569$	−25.5933	−1.07293	−0.536464	+	0.843923i	$0.680240\pi$
$570$	0	0
$571$	−16.8243	−0.704076	−0.352038	−	0.935986i	$-0.614511\pi$
$571$	−16.8243	−0.704076	−0.352038	+	0.935986i	$0.614511\pi$
$572$	−8.13204	−0.340018
$573$	0	0
$574$	−14.0109	−0.584803
$575$	26.2524	1.09480
$576$	0	0
$577$	16.9276	0.704706	0.352353	−	0.935867i	$-0.385382\pi$
$577$	16.9276	0.704706	0.352353	+	0.935867i	$0.385382\pi$
$578$	4.36028	0.181364
$579$	0	0
$580$	17.2722	0.717189
$581$	54.3977	2.25680
$582$	0	0
$583$	−6.14245	−0.254394
$584$	7.09927	0.293770
$585$	0	0
$586$	7.11027	0.293722
$587$	−17.9979	−0.742852	−0.371426	−	0.928463i	$-0.621131\pi$
$587$	−17.9979	−0.742852	−0.371426	+	0.928463i	$0.621131\pi$
$588$	0	0
$589$	2.96753	0.122275
$590$	−10.7176	−0.441237
$591$	0	0
$592$	6.74423	0.277186
$593$	−24.6139	−1.01077	−0.505387	−	0.862893i	$-0.668650\pi$
$593$	−24.6139	−1.01077	−0.505387	+	0.862893i	$0.668650\pi$
$594$	0	0
$595$	18.1704	0.744915
$596$	7.18289	0.294223
$597$	0	0
$598$	28.7408	1.17530
$599$	−37.6230	−1.53723	−0.768616	−	0.639710i	$-0.779054\pi$
$599$	−37.6230	−1.53723	−0.768616	+	0.639710i	$0.779054\pi$
$600$	0	0
$601$	3.54574	0.144634	0.0723169	−	0.997382i	$-0.476961\pi$
$601$	3.54574	0.144634	0.0723169	+	0.997382i	$0.476961\pi$
$602$	27.7494	1.13098
$603$	0	0
$604$	29.5977	1.20432
$605$	−1.36783	−0.0556104
$606$	0	0
$607$	−28.2160	−1.14525	−0.572626	−	0.819817i	$-0.694075\pi$
$607$	−28.2160	−1.14525	−0.572626	+	0.819817i	$0.694075\pi$
$608$	−5.74367	−0.232936
$609$	0	0
$610$	−8.64068	−0.349851
$611$	−0.778245	−0.0314844
$612$	0	0
$613$	40.5857	1.63924	0.819620	−	0.572907i	$-0.194184\pi$
$613$	40.5857	1.63924	0.819620	+	0.572907i	$0.194184\pi$
$614$	−5.99562	−0.241963
$615$	0	0
$616$	−9.68940	−0.390397
$617$	−17.6747	−0.711557	−0.355778	−	0.934570i	$-0.615784\pi$
$617$	−17.6747	−0.711557	−0.355778	+	0.934570i	$0.615784\pi$
$618$	0	0
$619$	5.87606	0.236179	0.118089	−	0.993003i	$-0.462323\pi$
$619$	5.87606	0.236179	0.118089	+	0.993003i	$0.462323\pi$
$620$	6.35345	0.255161
$621$	0	0
$622$	5.63408	0.225906
$623$	−43.1015	−1.72683
$624$	0	0
$625$	0.435972	0.0174389
$626$	18.4735	0.738349
$627$	0	0
$628$	25.8440	1.03129
$629$	13.7528	0.548362
$630$	0	0
$631$	−37.5236	−1.49379	−0.746895	−	0.664942i	$-0.768456\pi$
$631$	−37.5236	−1.49379	−0.746895	+	0.664942i	$0.768456\pi$
$632$	7.42696	0.295428
$633$	0	0
$634$	9.12401	0.362361
$635$	−21.8038	−0.865257
$636$	0	0
$637$	51.8969	2.05623
$638$	−5.31931	−0.210593
$639$	0	0
$640$	−15.1480	−0.598777
$641$	29.3913	1.16089	0.580444	−	0.814300i	$-0.302879\pi$
$641$	29.3913	1.16089	0.580444	+	0.814300i	$0.302879\pi$
$642$	0	0
$643$	29.5287	1.16450	0.582249	−	0.813010i	$-0.302173\pi$
$643$	29.5287	1.16450	0.582249	+	0.813010i	$0.302173\pi$
$644$	−54.1284	−2.13296
$645$	0	0
$646$	−2.12506	−0.0836092
$647$	25.1754	0.989746	0.494873	−	0.868965i	$-0.335215\pi$
$647$	25.1754	0.989746	0.494873	+	0.868965i	$0.335215\pi$
$648$	0	0
$649$	−11.8834	−0.466465
$650$	10.7189	0.420430
$651$	0	0
$652$	−5.24368	−0.205358
$653$	34.5485	1.35199	0.675993	−	0.736908i	$-0.263715\pi$
$653$	34.5485	1.35199	0.675993	+	0.736908i	$0.263715\pi$
$654$	0	0
$655$	−1.88074	−0.0734867
$656$	−8.14789	−0.318122
$657$	0	0
$658$	−0.407105	−0.0158706
$659$	11.1326	0.433666	0.216833	−	0.976209i	$-0.430427\pi$
$659$	11.1326	0.433666	0.216833	+	0.976209i	$0.430427\pi$
$660$	0	0
$661$	3.98975	0.155183	0.0775916	−	0.996985i	$-0.475277\pi$
$661$	3.98975	0.155183	0.0775916	+	0.996985i	$0.475277\pi$
$662$	−2.82055	−0.109624
$663$	0	0
$664$	−31.0248	−1.20399
$665$	5.63791	0.218629
$666$	0	0
$667$	−67.6848	−2.62076
$668$	5.84781	0.226259
$669$	0	0
$670$	−5.72087	−0.221016
$671$	−9.58057	−0.369854
$672$	0	0
$673$	−17.1495	−0.661064	−0.330532	−	0.943795i	$-0.607228\pi$
$673$	−17.1495	−0.661064	−0.330532	+	0.943795i	$0.607228\pi$
$674$	−17.1647	−0.661160
$675$	0	0
$676$	−21.9009	−0.842342
$677$	−29.9605	−1.15148	−0.575739	−	0.817634i	$-0.695285\pi$
$677$	−29.9605	−1.15148	−0.575739	+	0.817634i	$0.695285\pi$
$678$	0	0
$679$	−58.8428	−2.25818
$680$	−10.3632	−0.397410
$681$	0	0
$682$	−1.95667	−0.0749248
$683$	23.1120	0.884358	0.442179	−	0.896927i	$-0.354206\pi$
$683$	23.1120	0.884358	0.442179	+	0.896927i	$0.354206\pi$
$684$	0	0
$685$	−29.3667	−1.12204
$686$	8.12343	0.310154
$687$	0	0
$688$	16.1374	0.615233
$689$	−31.9124	−1.21577
$690$	0	0
$691$	−44.4094	−1.68941	−0.844707	−	0.535230i	$-0.820225\pi$
$691$	−44.4094	−1.68941	−0.844707	+	0.535230i	$0.820225\pi$
$692$	27.5586	1.04762
$693$	0	0
$694$	−1.34679	−0.0511236
$695$	15.2460	0.578315
$696$	0	0
$697$	−16.6152	−0.629345
$698$	7.04170	0.266532
$699$	0	0
$700$	−20.1872	−0.763004
$701$	−46.5664	−1.75879	−0.879394	−	0.476094i	$-0.842052\pi$
$701$	−46.5664	−1.75879	−0.879394	+	0.476094i	$0.842052\pi$
$702$	0	0
$703$	4.26722	0.160941
$704$	0.626199	0.0236008
$705$	0	0
$706$	−19.2935	−0.726120
$707$	21.4301	0.805961
$708$	0	0
$709$	22.9753	0.862855	0.431427	−	0.902148i	$-0.358010\pi$
$709$	22.9753	0.862855	0.431427	+	0.902148i	$0.358010\pi$
$710$	−12.9779	−0.487051
$711$	0	0
$712$	24.5822	0.921256
$713$	−24.8974	−0.932414
$714$	0	0
$715$	−7.10642	−0.265765
$716$	12.8792	0.481317
$717$	0	0
$718$	−13.5663	−0.506290
$719$	7.31949	0.272971	0.136485	−	0.990642i	$-0.456419\pi$
$719$	7.31949	0.272971	0.136485	+	0.990642i	$0.456419\pi$
$720$	0	0
$721$	24.7136	0.920382
$722$	−0.659361	−0.0245389
$723$	0	0
$724$	−8.78790	−0.326600
$725$	−25.2430	−0.937503
$726$	0	0
$727$	0.412821	0.0153107	0.00765535	−	0.999971i	$-0.497563\pi$
$727$	0.412821	0.0153107	0.00765535	+	0.999971i	$0.497563\pi$
$728$	−50.3401	−1.86573
$729$	0	0
$730$	2.72369	0.100808
$731$	32.9074	1.21712
$732$	0	0
$733$	39.4330	1.45649	0.728245	−	0.685316i	$-0.240336\pi$
$733$	39.4330	1.45649	0.728245	+	0.685316i	$0.240336\pi$
$734$	14.1222	0.521259
$735$	0	0
$736$	48.1890	1.77627
$737$	−6.34315	−0.233653
$738$	0	0
$739$	−23.5277	−0.865479	−0.432740	−	0.901519i	$-0.642453\pi$
$739$	−23.5277	−0.865479	−0.432740	+	0.901519i	$0.642453\pi$
$740$	9.13609	0.335849
$741$	0	0
$742$	−16.6936	−0.612840
$743$	0.119519	0.00438473	0.00219237	−	0.999998i	$-0.499302\pi$
$743$	0.119519	0.00438473	0.00219237	+	0.999998i	$0.499302\pi$
$744$	0	0
$745$	6.27698	0.229971
$746$	20.1817	0.738904
$747$	0	0
$748$	−5.04463	−0.184450
$749$	−0.287935	−0.0105209
$750$	0	0
$751$	−8.19273	−0.298957	−0.149478	−	0.988765i	$-0.547759\pi$
$751$	−8.19273	−0.298957	−0.149478	+	0.988765i	$0.547759\pi$
$752$	−0.236748	−0.00863330
$753$	0	0
$754$	−27.6358	−1.00644
$755$	25.8649	0.941319
$756$	0	0
$757$	−2.40721	−0.0874917	−0.0437459	−	0.999043i	$-0.513929\pi$
$757$	−2.40721	−0.0874917	−0.0437459	+	0.999043i	$0.513929\pi$
$758$	−3.55190	−0.129011
$759$	0	0
$760$	−3.21548	−0.116638
$761$	−39.4490	−1.43002	−0.715012	−	0.699112i	$-0.753579\pi$
$761$	−39.4490	−1.43002	−0.715012	+	0.699112i	$0.753579\pi$
$762$	0	0
$763$	57.7024	2.08897
$764$	34.8193	1.25972
$765$	0	0
$766$	−20.8208	−0.752286
$767$	−61.7389	−2.22926
$768$	0	0
$769$	−14.6494	−0.528271	−0.264136	−	0.964486i	$-0.585087\pi$
$769$	−14.6494	−0.528271	−0.264136	+	0.964486i	$0.585087\pi$
$770$	−3.71741	−0.133966
$771$	0	0
$772$	24.3664	0.876967
$773$	−15.8646	−0.570610	−0.285305	−	0.958437i	$-0.592095\pi$
$773$	−15.8646	−0.570610	−0.285305	+	0.958437i	$0.592095\pi$
$774$	0	0
$775$	−9.28548	−0.333544
$776$	33.5599	1.20473
$777$	0	0
$778$	6.13779	0.220050
$779$	−5.15535	−0.184709
$780$	0	0
$781$	−14.3895	−0.514899
$782$	17.8291	0.637567
$783$	0	0
$784$	15.7874	0.563836
$785$	22.5846	0.806077
$786$	0	0
$787$	−20.8282	−0.742446	−0.371223	−	0.928544i	$-0.621062\pi$
$787$	−20.8282	−0.742446	−0.371223	+	0.928544i	$0.621062\pi$
$788$	−10.5212	−0.374802
$789$	0	0
$790$	2.84941	0.101377
$791$	61.6344	2.19147
$792$	0	0
$793$	−49.7747	−1.76755
$794$	−6.76049	−0.239921
$795$	0	0
$796$	−0.728230	−0.0258114
$797$	26.4454	0.936743	0.468372	−	0.883532i	$-0.344841\pi$
$797$	26.4454	0.936743	0.468372	+	0.883532i	$0.344841\pi$
$798$	0	0
$799$	−0.482776	−0.0170794
$800$	17.9721	0.635410
$801$	0	0
$802$	11.6020	0.409682
$803$	3.01996	0.106572
$804$	0	0
$805$	−47.3017	−1.66717
$806$	−10.1657	−0.358070
$807$	0	0
$808$	−12.2223	−0.429977
$809$	−14.1256	−0.496629	−0.248315	−	0.968679i	$-0.579877\pi$
$809$	−14.1256	−0.496629	−0.248315	+	0.968679i	$0.579877\pi$
$810$	0	0
$811$	15.2386	0.535101	0.267550	−	0.963544i	$-0.413786\pi$
$811$	15.2386	0.535101	0.267550	+	0.963544i	$0.413786\pi$
$812$	52.0473	1.82650
$813$	0	0
$814$	−2.81364	−0.0986179
$815$	−4.58234	−0.160512
$816$	0	0
$817$	10.2105	0.357219
$818$	15.5637	0.544171
$819$	0	0
$820$	−11.0376	−0.385448
$821$	19.8272	0.691973	0.345986	−	0.938240i	$-0.387544\pi$
$821$	19.8272	0.691973	0.345986	+	0.938240i	$0.387544\pi$
$822$	0	0
$823$	28.2161	0.983553	0.491776	−	0.870721i	$-0.336348\pi$
$823$	28.2161	0.983553	0.491776	+	0.870721i	$0.336348\pi$
$824$	−14.0949	−0.491021
$825$	0	0
$826$	−32.2960	−1.12372
$827$	−32.0866	−1.11576	−0.557879	−	0.829922i	$-0.688385\pi$
$827$	−32.0866	−1.11576	−0.557879	+	0.829922i	$0.688385\pi$
$828$	0	0
$829$	34.2156	1.18836	0.594178	−	0.804334i	$-0.297477\pi$
$829$	34.2156	1.18836	0.594178	+	0.804334i	$0.297477\pi$
$830$	−11.9029	−0.413156
$831$	0	0
$832$	3.25334	0.112789
$833$	32.1937	1.11545
$834$	0	0
$835$	5.11028	0.176849
$836$	−1.56524	−0.0541351
$837$	0	0
$838$	24.2820	0.838806
$839$	11.9448	0.412380	0.206190	−	0.978512i	$-0.433893\pi$
$839$	11.9448	0.412380	0.206190	+	0.978512i	$0.433893\pi$
$840$	0	0
$841$	36.0825	1.24422
$842$	−5.78665	−0.199421
$843$	0	0
$844$	−14.8882	−0.512475
$845$	−19.1388	−0.658393
$846$	0	0
$847$	−4.12178	−0.141626
$848$	−9.70799	−0.333374
$849$	0	0
$850$	6.64936	0.228071
$851$	−35.8017	−1.22727
$852$	0	0
$853$	12.3830	0.423985	0.211992	−	0.977271i	$-0.432005\pi$
$853$	12.3830	0.423985	0.211992	+	0.977271i	$0.432005\pi$
$854$	−26.0375	−0.890984
$855$	0	0
$856$	0.164219	0.00561288
$857$	45.8270	1.56542	0.782710	−	0.622387i	$-0.213837\pi$
$857$	45.8270	1.56542	0.782710	+	0.622387i	$0.213837\pi$
$858$	0	0
$859$	43.4253	1.48165	0.740826	−	0.671697i	$-0.234434\pi$
$859$	43.4253	1.48165	0.740826	+	0.671697i	$0.234434\pi$
$860$	21.8606	0.745439
$861$	0	0
$862$	9.04882	0.308204
$863$	−31.9263	−1.08678	−0.543391	−	0.839480i	$-0.682860\pi$
$863$	−31.9263	−1.08678	−0.543391	+	0.839480i	$0.682860\pi$
$864$	0	0
$865$	24.0829	0.818845
$866$	9.31008	0.316369
$867$	0	0
$868$	19.1452	0.649832
$869$	3.15936	0.107174
$870$	0	0
$871$	−32.9551	−1.11664
$872$	−32.9095	−1.11446
$873$	0	0
$874$	5.53200	0.187123
$875$	−45.8307	−1.54936
$876$	0	0
$877$	14.9354	0.504334	0.252167	−	0.967684i	$-0.418857\pi$
$877$	14.9354	0.504334	0.252167	+	0.967684i	$0.418857\pi$
$878$	−13.2661	−0.447709
$879$	0	0
$880$	−2.16183	−0.0728752
$881$	5.66513	0.190863	0.0954316	−	0.995436i	$-0.469577\pi$
$881$	5.66513	0.190863	0.0954316	+	0.995436i	$0.469577\pi$
$882$	0	0
$883$	−8.34093	−0.280695	−0.140347	−	0.990102i	$-0.544822\pi$
$883$	−8.34093	−0.280695	−0.140347	+	0.990102i	$0.544822\pi$
$884$	−26.2088	−0.881497
$885$	0	0
$886$	−0.790225	−0.0265481
$887$	−35.2777	−1.18451	−0.592255	−	0.805750i	$-0.701762\pi$
$887$	−35.2777	−1.18451	−0.592255	+	0.805750i	$0.701762\pi$
$888$	0	0
$889$	−65.7026	−2.20360
$890$	9.43115	0.316133
$891$	0	0
$892$	−1.37692	−0.0461026
$893$	−0.149795	−0.00501271
$894$	0	0
$895$	11.2548	0.376208
$896$	−45.6464	−1.52494
$897$	0	0
$898$	−8.98196	−0.299732
$899$	23.9401	0.798448
$900$	0	0
$901$	−19.7965	−0.659518
$902$	3.39923	0.113182
$903$	0	0
$904$	−35.1521	−1.16914
$905$	−7.67957	−0.255277
$906$	0	0
$907$	18.8103	0.624586	0.312293	−	0.949986i	$-0.398903\pi$
$907$	18.8103	0.624586	0.312293	+	0.949986i	$0.398903\pi$
$908$	31.0599	1.03076
$909$	0	0
$910$	−19.3134	−0.640233
$911$	36.8573	1.22114	0.610569	−	0.791963i	$-0.290941\pi$
$911$	36.8573	1.22114	0.610569	+	0.791963i	$0.290941\pi$
$912$	0	0
$913$	−13.1976	−0.436778
$914$	−21.9497	−0.726032
$915$	0	0
$916$	−43.1453	−1.42556
$917$	−5.66736	−0.187153
$918$	0	0
$919$	6.64450	0.219182	0.109591	−	0.993977i	$-0.465046\pi$
$919$	6.64450	0.219182	0.109591	+	0.993977i	$0.465046\pi$
$920$	26.9777	0.889428
$921$	0	0
$922$	17.2961	0.569617
$923$	−74.7592	−2.46073
$924$	0	0
$925$	−13.3523	−0.439020
$926$	16.3159	0.536175
$927$	0	0
$928$	−46.3363	−1.52106
$929$	20.2409	0.664081	0.332041	−	0.943265i	$-0.392263\pi$
$929$	20.2409	0.664081	0.332041	+	0.943265i	$0.392263\pi$
$930$	0	0
$931$	9.98904	0.327378
$932$	−15.0410	−0.492684
$933$	0	0
$934$	−5.40269	−0.176781
$935$	−4.40840	−0.144170
$936$	0	0
$937$	−17.5676	−0.573909	−0.286955	−	0.957944i	$-0.592643\pi$
$937$	−17.5676	−0.573909	−0.286955	+	0.957944i	$0.592643\pi$
$938$	−17.2390	−0.562874
$939$	0	0
$940$	−0.320711	−0.0104604
$941$	−10.3234	−0.336534	−0.168267	−	0.985741i	$-0.553817\pi$
$941$	−10.3234	−0.336534	−0.168267	+	0.985741i	$0.553817\pi$
$942$	0	0
$943$	43.2530	1.40851
$944$	−18.7814	−0.611284
$945$	0	0
$946$	−6.73239	−0.218889
$947$	−13.5385	−0.439943	−0.219972	−	0.975506i	$-0.570597\pi$
$947$	−13.5385	−0.439943	−0.219972	+	0.975506i	$0.570597\pi$
$948$	0	0
$949$	15.6899	0.509314
$950$	2.06316	0.0669377
$951$	0	0
$952$	−31.2280	−1.01210
$953$	16.4833	0.533947	0.266973	−	0.963704i	$-0.413976\pi$
$953$	16.4833	0.533947	0.266973	+	0.963704i	$0.413976\pi$
$954$	0	0
$955$	30.4279	0.984624
$956$	−36.4529	−1.17897
$957$	0	0
$958$	18.3429	0.592633
$959$	−88.4924	−2.85757
$960$	0	0
$961$	−22.1938	−0.715929
$962$	−14.6179	−0.471301
$963$	0	0
$964$	0.155050	0.00499384
$965$	21.2933	0.685457
$966$	0	0
$967$	−42.8282	−1.37726	−0.688631	−	0.725112i	$-0.741788\pi$
$967$	−42.8282	−1.37726	−0.688631	+	0.725112i	$0.741788\pi$
$968$	2.35078	0.0755570
$969$	0	0
$970$	12.8755	0.413408
$971$	−17.9785	−0.576958	−0.288479	−	0.957486i	$-0.593150\pi$
$971$	−17.9785	−0.576958	−0.288479	+	0.957486i	$0.593150\pi$
$972$	0	0
$973$	45.9418	1.47283
$974$	4.16943	0.133597
$975$	0	0
$976$	−15.1418	−0.484679
$977$	3.12664	0.100030	0.0500150	−	0.998748i	$-0.484073\pi$
$977$	3.12664	0.100030	0.0500150	+	0.998748i	$0.484073\pi$
$978$	0	0
$979$	10.4570	0.334208
$980$	21.3865	0.683166
$981$	0	0
$982$	−23.7230	−0.757031
$983$	−8.36079	−0.266668	−0.133334	−	0.991071i	$-0.542568\pi$
$983$	−8.36079	−0.266668	−0.133334	+	0.991071i	$0.542568\pi$
$984$	0	0
$985$	−9.19426	−0.292954
$986$	−17.1436	−0.545964
$987$	0	0
$988$	−8.13204	−0.258715
$989$	−85.6653	−2.72400
$990$	0	0
$991$	−30.9086	−0.981845	−0.490923	−	0.871203i	$-0.663340\pi$
$991$	−30.9086	−0.981845	−0.490923	+	0.871203i	$0.663340\pi$
$992$	−17.0445	−0.541163
$993$	0	0
$994$	−39.1070	−1.24040
$995$	−0.636385	−0.0201748
$996$	0	0
$997$	32.5554	1.03104	0.515520	−	0.856878i	$-0.327599\pi$
$997$	32.5554	1.03104	0.515520	+	0.856878i	$0.327599\pi$
$998$	−13.5269	−0.428186
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.n.1.4	✓	7
3.2	odd	2		1881.2.a.r.1.4	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1881.2.a.n.1.4	✓	7	1.1	even	1	trivial
1881.2.a.r.1.4	yes	7	3.2	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:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q-0.659361 q^{2} -1.56524 q^{4} -1.36783 q^{5} -4.12178 q^{7} +2.35078 q^{8} +0.901896 q^{10} +1.00000 q^{11} +5.19538 q^{13} +2.71774 q^{14} +1.58047 q^{16} +3.22290 q^{17} +1.00000 q^{19} +2.14099 q^{20} -0.659361 q^{22} -8.38994 q^{23} -3.12903 q^{25} -3.42563 q^{26} +6.45158 q^{28} +8.06737 q^{29} +2.96753 q^{31} -5.74367 q^{32} -2.12506 q^{34} +5.63791 q^{35} +4.26722 q^{37} -0.659361 q^{38} -3.21548 q^{40} -5.15535 q^{41} +10.2105 q^{43} -1.56524 q^{44} +5.53200 q^{46} -0.149795 q^{47} +9.98904 q^{49} +2.06316 q^{50} -8.13204 q^{52} -6.14245 q^{53} -1.36783 q^{55} -9.68940 q^{56} -5.31931 q^{58} -11.8834 q^{59} -9.58057 q^{61} -1.95667 q^{62} +0.626199 q^{64} -7.10642 q^{65} -6.34315 q^{67} -5.04463 q^{68} -3.71741 q^{70} -14.3895 q^{71} +3.01996 q^{73} -2.81364 q^{74} -1.56524 q^{76} -4.12178 q^{77} +3.15936 q^{79} -2.16183 q^{80} +3.39923 q^{82} -13.1976 q^{83} -4.40840 q^{85} -6.73239 q^{86} +2.35078 q^{88} +10.4570 q^{89} -21.4142 q^{91} +13.1323 q^{92} +0.0987692 q^{94} -1.36783 q^{95} +14.2761 q^{97} -6.58638 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 2 q^{7} - 6 q^{8} + 6 q^{10} + 7 q^{11} - 7 q^{13} - 6 q^{14} - 5 q^{17} + 7 q^{19} - 20 q^{20} - 2 q^{22} - 22 q^{23} + 3 q^{25} - 10 q^{26} + 2 q^{28} - 18 q^{29} - 14 q^{32}+ \cdots + 62 q^{98}+O(q^{100}) \) 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 2 * q^7 - 6 * q^8 + 6 * q^10 + 7 * q^11 - 7 * q^13 - 6 * q^14 - 5 * q^17 + 7 * q^19 - 20 * q^20 - 2 * q^22 - 22 * q^23 + 3 * q^25 - 10 * q^26 + 2 * q^28 - 18 * q^29 - 14 * q^32 + 6 * q^34 - 8 * q^35 + 6 * q^37 - 2 * q^38 + 28 * q^40 - 6 * q^41 + 8 * q^43 + 6 * q^44 + 8 * q^46 - 14 * q^47 + 5 * q^49 - 24 * q^50 - 12 * q^52 + q^53 - 8 * q^55 - 20 * q^56 - 2 * q^58 - 23 * q^59 - 28 * q^61 - 2 * q^62 - 26 * q^64 - 10 * q^65 + 14 * q^67 - 12 * q^68 - 18 * q^70 - 31 * q^71 - 12 * q^73 - 20 * q^74 + 6 * q^76 - 2 * q^77 - 17 * q^79 + 6 * q^80 + 12 * q^82 - 5 * q^83 - 12 * q^85 - 38 * q^86 - 6 * q^88 + 3 * q^89 - 12 * q^91 - 60 * q^94 - 8 * q^95 + 12 * q^97 + 62 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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