LMFDB - Embedded newform 2277.2.a.m.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)

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")

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);

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 = [6,-3,0,5,-3,0,-1,-3,0,-6,-6] 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:	$1$
Dimension:	$6$
Coefficient field:	6.6.8639957.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 4x^{4} + 10x^{3} + 6x^{2} - 7x - 2 $ x^6 - 3x^5 - 4x^4 + 10x^3 + 6x^2 - 7*x - 2
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			$-0.255514$ 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)

$f(q)$	$=$	\(q-1.25551 q^{2} -0.423685 q^{4} -4.04297 q^{5} +0.798084 q^{7} +3.04297 q^{8} +5.07601 q^{10} -1.00000 q^{11} -3.19289 q^{13} -1.00201 q^{14} -2.97312 q^{16} -0.503752 q^{17} +3.98359 q^{19} +1.71294 q^{20} +1.25551 q^{22} +1.00000 q^{23} +11.3456 q^{25} +4.00871 q^{26} -0.338136 q^{28} +6.16897 q^{29} -3.28168 q^{31} -2.35314 q^{32} +0.632468 q^{34} -3.22663 q^{35} +5.17919 q^{37} -5.00145 q^{38} -12.3026 q^{40} +11.8209 q^{41} -7.08974 q^{43} +0.423685 q^{44} -1.25551 q^{46} -1.22773 q^{47} -6.36306 q^{49} -14.2446 q^{50} +1.35278 q^{52} +3.00201 q^{53} +4.04297 q^{55} +2.42855 q^{56} -7.74523 q^{58} -9.03304 q^{59} +2.63648 q^{61} +4.12020 q^{62} +8.90065 q^{64} +12.9087 q^{65} +8.99026 q^{67} +0.213432 q^{68} +4.05108 q^{70} -9.80465 q^{71} +1.31433 q^{73} -6.50255 q^{74} -1.68779 q^{76} -0.798084 q^{77} +15.6898 q^{79} +12.0202 q^{80} -14.8413 q^{82} +2.92233 q^{83} +2.03665 q^{85} +8.90127 q^{86} -3.04297 q^{88} -6.20642 q^{89} -2.54819 q^{91} -0.423685 q^{92} +1.54144 q^{94} -16.1055 q^{95} -15.3359 q^{97} +7.98891 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 5 q^{4} - 3 q^{5} - q^{7} - 3 q^{8} - 6 q^{10} - 6 q^{11} - 3 q^{13} + 8 q^{14} - q^{16} - 5 q^{17} + q^{19} + 7 q^{20} + 3 q^{22} + 6 q^{23} + 3 q^{25} - 15 q^{26} - 6 q^{29} - 8 q^{31}+ \cdots - 34 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 5 * q^4 - 3 * q^5 - q^7 - 3 * q^8 - 6 * q^10 - 6 * q^11 - 3 * q^13 + 8 * q^14 - q^16 - 5 * q^17 + q^19 + 7 * q^20 + 3 * q^22 + 6 * q^23 + 3 * q^25 - 15 * q^26 - 6 * q^29 - 8 * q^31 + 6 * q^32 - 6 * q^34 - 12 * q^35 + 2 * q^37 - 9 * q^38 - 30 * q^40 + 2 * q^41 + 10 * q^43 - 5 * q^44 - 3 * q^46 - 14 * q^47 - 5 * q^49 - 9 * q^50 - 11 * q^52 + 4 * q^53 + 3 * q^55 + 13 * q^56 - 6 * q^58 - 39 * q^59 - 22 * q^61 - 33 * q^64 + 16 * q^65 + 8 * q^67 - 17 * q^68 + 19 * q^70 - 18 * q^71 - 9 * q^73 - 15 * q^74 - 25 * q^76 + q^77 - 15 * q^79 + 16 * q^80 - 8 * q^82 - 43 * q^83 - 21 * q^85 + 55 * q^86 + 3 * q^88 - 17 * q^89 - 9 * q^91 + 5 * q^92 + 63 * q^94 + 21 * q^95 + 19 * q^97 - 34 * 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 $

$2$	−1.25551	−0.887782	−0.443891	−	0.896081i	$-0.646402\pi$
$2$	−1.25551	−0.887782	−0.443891	+	0.896081i	$0.646402\pi$
$3$	0	0
$3$	0	0
$4$	−0.423685	−0.211842
$5$	−4.04297	−1.80807	−0.904036	−	0.427457i	$-0.859409\pi$
$5$	−4.04297	−1.80807	−0.904036	+	0.427457i	$0.859409\pi$
$6$	0	0
$7$	0.798084	0.301647	0.150824	−	0.988561i	$-0.451807\pi$
$7$	0.798084	0.301647	0.150824	+	0.988561i	$0.451807\pi$
$8$	3.04297	1.07585
$9$	0	0
$10$	5.07601	1.60517
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.19289	−0.885547	−0.442774	−	0.896633i	$-0.646005\pi$
$13$	−3.19289	−0.885547	−0.442774	+	0.896633i	$0.646005\pi$
$14$	−1.00201	−0.267797
$15$	0	0
$16$	−2.97312	−0.743281
$17$	−0.503752	−0.122178	−0.0610889	−	0.998132i	$-0.519457\pi$
$17$	−0.503752	−0.122178	−0.0610889	+	0.998132i	$0.519457\pi$
$18$	0	0
$19$	3.98359	0.913898	0.456949	−	0.889493i	$-0.348942\pi$
$19$	3.98359	0.913898	0.456949	+	0.889493i	$0.348942\pi$
$20$	1.71294	0.383026
$21$	0	0
$22$	1.25551	0.267676
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	11.3456	2.26912
$26$	4.00871	0.786173
$27$	0	0
$28$	−0.338136	−0.0639017
$29$	6.16897	1.14555	0.572774	−	0.819713i	$-0.305867\pi$
$29$	6.16897	1.14555	0.572774	+	0.819713i	$0.305867\pi$
$30$	0	0
$31$	−3.28168	−0.589407	−0.294704	−	0.955589i	$-0.595221\pi$
$31$	−3.28168	−0.589407	−0.294704	+	0.955589i	$0.595221\pi$
$32$	−2.35314	−0.415981
$33$	0	0
$34$	0.632468	0.108467
$35$	−3.22663	−0.545400
$36$	0	0
$37$	5.17919	0.851454	0.425727	−	0.904852i	$-0.360018\pi$
$37$	5.17919	0.851454	0.425727	+	0.904852i	$0.360018\pi$
$38$	−5.00145	−0.811343
$39$	0	0
$40$	−12.3026	−1.94522
$41$	11.8209	1.84611	0.923056	−	0.384666i	$-0.125683\pi$
$41$	11.8209	1.84611	0.923056	+	0.384666i	$0.125683\pi$
$42$	0	0
$43$	−7.08974	−1.08118	−0.540588	−	0.841288i	$-0.681798\pi$
$43$	−7.08974	−1.08118	−0.540588	+	0.841288i	$0.681798\pi$
$44$	0.423685	0.0638728
$45$	0	0
$46$	−1.25551	−0.185115
$47$	−1.22773	−0.179083	−0.0895416	−	0.995983i	$-0.528540\pi$
$47$	−1.22773	−0.179083	−0.0895416	+	0.995983i	$0.528540\pi$
$48$	0	0
$49$	−6.36306	−0.909009
$50$	−14.2446	−2.01449
$51$	0	0
$52$	1.35278	0.187596
$53$	3.00201	0.412357	0.206179	−	0.978514i	$-0.433897\pi$
$53$	3.00201	0.412357	0.206179	+	0.978514i	$0.433897\pi$
$54$	0	0
$55$	4.04297	0.545154
$56$	2.42855	0.324528
$57$	0	0
$58$	−7.74523	−1.01700
$59$	−9.03304	−1.17600	−0.588001	−	0.808861i	$-0.700085\pi$
$59$	−9.03304	−1.17600	−0.588001	+	0.808861i	$0.700085\pi$
$60$	0	0
$61$	2.63648	0.337567	0.168783	−	0.985653i	$-0.446016\pi$
$61$	2.63648	0.337567	0.168783	+	0.985653i	$0.446016\pi$
$62$	4.12020	0.523266
$63$	0	0
$64$	8.90065	1.11258
$65$	12.9087	1.60113
$66$	0	0
$67$	8.99026	1.09834	0.549168	−	0.835712i	$-0.314945\pi$
$67$	8.99026	1.09834	0.549168	+	0.835712i	$0.314945\pi$
$68$	0.213432	0.0258824
$69$	0	0
$70$	4.05108	0.484197
$71$	−9.80465	−1.16360	−0.581799	−	0.813333i	$-0.697651\pi$
$71$	−9.80465	−1.16360	−0.581799	+	0.813333i	$0.697651\pi$
$72$	0	0
$73$	1.31433	0.153830	0.0769151	−	0.997038i	$-0.475493\pi$
$73$	1.31433	0.153830	0.0769151	+	0.997038i	$0.475493\pi$
$74$	−6.50255	−0.755906
$75$	0	0
$76$	−1.68779	−0.193602
$77$	−0.798084	−0.0909501
$78$	0	0
$79$	15.6898	1.76524	0.882618	−	0.470091i	$-0.155779\pi$
$79$	15.6898	1.76524	0.882618	+	0.470091i	$0.155779\pi$
$80$	12.0202	1.34390
$81$	0	0
$82$	−14.8413	−1.63895
$83$	2.92233	0.320767	0.160384	−	0.987055i	$-0.448727\pi$
$83$	2.92233	0.320767	0.160384	+	0.987055i	$0.448727\pi$
$84$	0	0
$85$	2.03665	0.220906
$86$	8.90127	0.959849
$87$	0	0
$88$	−3.04297	−0.324382
$89$	−6.20642	−0.657879	−0.328940	−	0.944351i	$-0.606691\pi$
$89$	−6.20642	−0.657879	−0.328940	+	0.944351i	$0.606691\pi$
$90$	0	0
$91$	−2.54819	−0.267123
$92$	−0.423685	−0.0441722
$93$	0	0
$94$	1.54144	0.158987
$95$	−16.1055	−1.65239
$96$	0	0
$97$	−15.3359	−1.55712	−0.778561	−	0.627569i	$-0.784050\pi$
$97$	−15.3359	−1.55712	−0.778561	+	0.627569i	$0.784050\pi$
$98$	7.98891	0.807002
$99$	0	0
$100$	−4.80696	−0.480696
$101$	5.69653	0.566826	0.283413	−	0.958998i	$-0.408533\pi$
$101$	5.69653	0.566826	0.283413	+	0.958998i	$0.408533\pi$
$102$	0	0
$103$	0.992886	0.0978319	0.0489160	−	0.998803i	$-0.484423\pi$
$103$	0.992886	0.0978319	0.0489160	+	0.998803i	$0.484423\pi$
$104$	−9.71585	−0.952718
$105$	0	0
$106$	−3.76906	−0.366084
$107$	5.98921	0.578999	0.289500	−	0.957178i	$-0.406511\pi$
$107$	5.98921	0.578999	0.289500	+	0.957178i	$0.406511\pi$
$108$	0	0
$109$	−15.3585	−1.47108	−0.735541	−	0.677480i	$-0.763072\pi$
$109$	−15.3585	−1.47108	−0.735541	+	0.677480i	$0.763072\pi$
$110$	−5.07601	−0.483978
$111$	0	0
$112$	−2.37280	−0.224209
$113$	−15.7470	−1.48135	−0.740675	−	0.671863i	$-0.765494\pi$
$113$	−15.7470	−1.48135	−0.740675	+	0.671863i	$0.765494\pi$
$114$	0	0
$115$	−4.04297	−0.377009
$116$	−2.61370	−0.242676
$117$	0	0
$118$	11.3411	1.04403
$119$	−0.402037	−0.0368546
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.31014	−0.299686
$123$	0	0
$124$	1.39040	0.124861
$125$	−25.6551	−2.29466
$126$	0	0
$127$	9.06863	0.804710	0.402355	−	0.915484i	$-0.368192\pi$
$127$	9.06863	0.804710	0.402355	+	0.915484i	$0.368192\pi$
$128$	−6.46860	−0.571749
$129$	0	0
$130$	−16.2071	−1.42146
$131$	−10.9241	−0.954442	−0.477221	−	0.878783i	$-0.658356\pi$
$131$	−10.9241	−0.954442	−0.477221	+	0.878783i	$0.658356\pi$
$132$	0	0
$133$	3.17924	0.275675
$134$	−11.2874	−0.975083
$135$	0	0
$136$	−1.53290	−0.131445
$137$	−22.4269	−1.91606	−0.958029	−	0.286672i	$-0.907451\pi$
$137$	−22.4269	−1.91606	−0.958029	+	0.286672i	$0.907451\pi$
$138$	0	0
$139$	−0.760327	−0.0644901	−0.0322451	−	0.999480i	$-0.510266\pi$
$139$	−0.760327	−0.0644901	−0.0322451	+	0.999480i	$0.510266\pi$
$140$	1.36707	0.115539
$141$	0	0
$142$	12.3099	1.03302
$143$	3.19289	0.267002
$144$	0	0
$145$	−24.9410	−2.07123
$146$	−1.65015	−0.136568
$147$	0	0
$148$	−2.19434	−0.180374
$149$	−5.59496	−0.458357	−0.229179	−	0.973384i	$-0.573604\pi$
$149$	−5.59496	−0.458357	−0.229179	+	0.973384i	$0.573604\pi$
$150$	0	0
$151$	7.13755	0.580845	0.290423	−	0.956898i	$-0.406204\pi$
$151$	7.13755	0.580845	0.290423	+	0.956898i	$0.406204\pi$
$152$	12.1219	0.983219
$153$	0	0
$154$	1.00201	0.0807439
$155$	13.2677	1.06569
$156$	0	0
$157$	−18.8473	−1.50418	−0.752090	−	0.659061i	$-0.770954\pi$
$157$	−18.8473	−1.50418	−0.752090	+	0.659061i	$0.770954\pi$
$158$	−19.6987	−1.56715
$159$	0	0
$160$	9.51369	0.752123
$161$	0.798084	0.0628978
$162$	0	0
$163$	−19.5160	−1.52861	−0.764305	−	0.644854i	$-0.776918\pi$
$163$	−19.5160	−1.52861	−0.764305	+	0.644854i	$0.776918\pi$
$164$	−5.00833	−0.391084
$165$	0	0
$166$	−3.66903	−0.284772
$167$	−20.5303	−1.58868	−0.794342	−	0.607471i	$-0.792184\pi$
$167$	−20.5303	−1.58868	−0.794342	+	0.607471i	$0.792184\pi$
$168$	0	0
$169$	−2.80548	−0.215806
$170$	−2.55705	−0.196117
$171$	0	0
$172$	3.00381	0.229039
$173$	−4.67283	−0.355268	−0.177634	−	0.984097i	$-0.556844\pi$
$173$	−4.67283	−0.355268	−0.177634	+	0.984097i	$0.556844\pi$
$174$	0	0
$175$	9.05475	0.684475
$176$	2.97312	0.224108
$177$	0	0
$178$	7.79225	0.584054
$179$	4.68491	0.350166	0.175083	−	0.984554i	$-0.443981\pi$
$179$	4.68491	0.350166	0.175083	+	0.984554i	$0.443981\pi$
$180$	0	0
$181$	10.2400	0.761133	0.380566	−	0.924754i	$-0.375729\pi$
$181$	10.2400	0.761133	0.380566	+	0.924754i	$0.375729\pi$
$182$	3.19929	0.237147
$183$	0	0
$184$	3.04297	0.224331
$185$	−20.9393	−1.53949
$186$	0	0
$187$	0.503752	0.0368380
$188$	0.520171	0.0379374
$189$	0	0
$190$	20.2207	1.46697
$191$	6.39892	0.463009	0.231505	−	0.972834i	$-0.425635\pi$
$191$	6.39892	0.463009	0.231505	+	0.972834i	$0.425635\pi$
$192$	0	0
$193$	2.17363	0.156462	0.0782308	−	0.996935i	$-0.475073\pi$
$193$	2.17363	0.156462	0.0782308	+	0.996935i	$0.475073\pi$
$194$	19.2544	1.38238
$195$	0	0
$196$	2.69593	0.192566
$197$	9.15789	0.652473	0.326236	−	0.945288i	$-0.394219\pi$
$197$	9.15789	0.652473	0.326236	+	0.945288i	$0.394219\pi$
$198$	0	0
$199$	7.01847	0.497526	0.248763	−	0.968564i	$-0.419976\pi$
$199$	7.01847	0.497526	0.248763	+	0.968564i	$0.419976\pi$
$200$	34.5243	2.44124
$201$	0	0
$202$	−7.15208	−0.503218
$203$	4.92336	0.345552
$204$	0	0
$205$	−47.7915	−3.33790
$206$	−1.24658	−0.0868535
$207$	0	0
$208$	9.49284	0.658210
$209$	−3.98359	−0.275551
$210$	0	0
$211$	−14.4490	−0.994707	−0.497354	−	0.867548i	$-0.665695\pi$
$211$	−14.4490	−0.994707	−0.497354	+	0.867548i	$0.665695\pi$
$212$	−1.27190	−0.0873547
$213$	0	0
$214$	−7.51954	−0.514025
$215$	28.6636	1.95484
$216$	0	0
$217$	−2.61906	−0.177793
$218$	19.2829	1.30600
$219$	0	0
$220$	−1.71294	−0.115487
$221$	1.60842	0.108194
$222$	0	0
$223$	24.2905	1.62661	0.813305	−	0.581838i	$-0.197666\pi$
$223$	24.2905	1.62661	0.813305	+	0.581838i	$0.197666\pi$
$224$	−1.87801	−0.125480
$225$	0	0
$226$	19.7705	1.31512
$227$	−21.5364	−1.42942	−0.714709	−	0.699422i	$-0.753441\pi$
$227$	−21.5364	−1.42942	−0.714709	+	0.699422i	$0.753441\pi$
$228$	0	0
$229$	6.71062	0.443450	0.221725	−	0.975109i	$-0.428831\pi$
$229$	6.71062	0.443450	0.221725	+	0.975109i	$0.428831\pi$
$230$	5.07601	0.334702
$231$	0	0
$232$	18.7720	1.23244
$233$	15.0861	0.988326	0.494163	−	0.869369i	$-0.335475\pi$
$233$	15.0861	0.988326	0.494163	+	0.869369i	$0.335475\pi$
$234$	0	0
$235$	4.96369	0.323795
$236$	3.82716	0.249127
$237$	0	0
$238$	0.504763	0.0327189
$239$	30.6490	1.98252	0.991260	−	0.131919i	$-0.0421140\pi$
$239$	30.6490	1.98252	0.991260	+	0.131919i	$0.0421140\pi$
$240$	0	0
$241$	−13.3710	−0.861301	−0.430651	−	0.902519i	$-0.641716\pi$
$241$	−13.3710	−0.861301	−0.430651	+	0.902519i	$0.641716\pi$
$242$	−1.25551	−0.0807075
$243$	0	0
$244$	−1.11704	−0.0715109
$245$	25.7257	1.64355
$246$	0	0
$247$	−12.7191	−0.809300
$248$	−9.98606	−0.634115
$249$	0	0
$250$	32.2103	2.03716
$251$	−25.4838	−1.60852	−0.804262	−	0.594276i	$-0.797439\pi$
$251$	−25.4838	−1.60852	−0.804262	+	0.594276i	$0.797439\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	−11.3858	−0.714408
$255$	0	0
$256$	−9.67988	−0.604992
$257$	14.1016	0.879634	0.439817	−	0.898087i	$-0.355043\pi$
$257$	14.1016	0.879634	0.439817	+	0.898087i	$0.355043\pi$
$258$	0	0
$259$	4.13343	0.256839
$260$	−5.46923	−0.339187
$261$	0	0
$262$	13.7153	0.847337
$263$	10.5011	0.647527	0.323764	−	0.946138i	$-0.395052\pi$
$263$	10.5011	0.647527	0.323764	+	0.946138i	$0.395052\pi$
$264$	0	0
$265$	−12.1370	−0.745571
$266$	−3.99158	−0.244739
$267$	0	0
$268$	−3.80903	−0.232674
$269$	−20.5718	−1.25429	−0.627143	−	0.778904i	$-0.715776\pi$
$269$	−20.5718	−1.25429	−0.627143	+	0.778904i	$0.715776\pi$
$270$	0	0
$271$	−28.7365	−1.74562	−0.872810	−	0.488060i	$-0.837705\pi$
$271$	−28.7365	−1.74562	−0.872810	+	0.488060i	$0.837705\pi$
$272$	1.49772	0.0908124
$273$	0	0
$274$	28.1573	1.70104
$275$	−11.3456	−0.684166
$276$	0	0
$277$	−13.7543	−0.826416	−0.413208	−	0.910637i	$-0.635592\pi$
$277$	−13.7543	−0.826416	−0.413208	+	0.910637i	$0.635592\pi$
$278$	0.954601	0.0572532
$279$	0	0
$280$	−9.81854	−0.586770
$281$	−7.54904	−0.450338	−0.225169	−	0.974320i	$-0.572293\pi$
$281$	−7.54904	−0.450338	−0.225169	+	0.974320i	$0.572293\pi$
$282$	0	0
$283$	−27.0020	−1.60510	−0.802551	−	0.596584i	$-0.796524\pi$
$283$	−27.0020	−1.60510	−0.802551	+	0.596584i	$0.796524\pi$
$284$	4.15408	0.246499
$285$	0	0
$286$	−4.00871	−0.237040
$287$	9.43406	0.556875
$288$	0	0
$289$	−16.7462	−0.985073
$290$	31.3137	1.83881
$291$	0	0
$292$	−0.556859	−0.0325877
$293$	10.4872	0.612666	0.306333	−	0.951924i	$-0.400898\pi$
$293$	10.4872	0.612666	0.306333	+	0.951924i	$0.400898\pi$
$294$	0	0
$295$	36.5203	2.12629
$296$	15.7601	0.916038
$297$	0	0
$298$	7.02455	0.406921
$299$	−3.19289	−0.184649
$300$	0	0
$301$	−5.65821	−0.326134
$302$	−8.96129	−0.515664
$303$	0	0
$304$	−11.8437	−0.679283
$305$	−10.6592	−0.610345
$306$	0	0
$307$	19.2068	1.09619	0.548096	−	0.836415i	$-0.315353\pi$
$307$	19.2068	1.09619	0.548096	+	0.836415i	$0.315353\pi$
$308$	0.338136	0.0192671
$309$	0	0
$310$	−16.6578	−0.946101
$311$	16.6171	0.942270	0.471135	−	0.882061i	$-0.343845\pi$
$311$	16.6171	0.942270	0.471135	+	0.882061i	$0.343845\pi$
$312$	0	0
$313$	20.8071	1.17609	0.588044	−	0.808829i	$-0.299898\pi$
$313$	20.8071	1.17609	0.588044	+	0.808829i	$0.299898\pi$
$314$	23.6631	1.33538
$315$	0	0
$316$	−6.64751	−0.373952
$317$	0.842442	0.0473163	0.0236581	−	0.999720i	$-0.492469\pi$
$317$	0.842442	0.0473163	0.0236581	+	0.999720i	$0.492469\pi$
$318$	0	0
$319$	−6.16897	−0.345396
$320$	−35.9851	−2.01163
$321$	0	0
$322$	−1.00201	−0.0558396
$323$	−2.00674	−0.111658
$324$	0	0
$325$	−36.2252	−2.00941
$326$	24.5026	1.35707
$327$	0	0
$328$	35.9706	1.98614
$329$	−0.979834	−0.0540200
$330$	0	0
$331$	−10.2249	−0.562014	−0.281007	−	0.959706i	$-0.590668\pi$
$331$	−10.2249	−0.562014	−0.281007	+	0.959706i	$0.590668\pi$
$332$	−1.23815	−0.0679521
$333$	0	0
$334$	25.7761	1.41041
$335$	−36.3473	−1.98587
$336$	0	0
$337$	29.3644	1.59958	0.799791	−	0.600279i	$-0.204944\pi$
$337$	29.3644	1.59958	0.799791	+	0.600279i	$0.204944\pi$
$338$	3.52232	0.191589
$339$	0	0
$340$	−0.862899	−0.0467973
$341$	3.28168	0.177713
$342$	0	0
$343$	−10.6648	−0.575848
$344$	−21.5739	−1.16318
$345$	0	0
$346$	5.86680	0.315401
$347$	−24.9778	−1.34088	−0.670438	−	0.741965i	$-0.733894\pi$
$347$	−24.9778	−1.34088	−0.670438	+	0.741965i	$0.733894\pi$
$348$	0	0
$349$	−30.7570	−1.64638	−0.823191	−	0.567764i	$-0.807808\pi$
$349$	−30.7570	−1.64638	−0.823191	+	0.567764i	$0.807808\pi$
$350$	−11.3684	−0.607665
$351$	0	0
$352$	2.35314	0.125423
$353$	18.1605	0.966588	0.483294	−	0.875458i	$-0.339440\pi$
$353$	18.1605	0.966588	0.483294	+	0.875458i	$0.339440\pi$
$354$	0	0
$355$	39.6399	2.10387
$356$	2.62956	0.139367
$357$	0	0
$358$	−5.88197	−0.310872
$359$	0.950635	0.0501726	0.0250863	−	0.999685i	$-0.492014\pi$
$359$	0.950635	0.0501726	0.0250863	+	0.999685i	$0.492014\pi$
$360$	0	0
$361$	−3.13102	−0.164790
$362$	−12.8565	−0.675720
$363$	0	0
$364$	1.07963	0.0565879
$365$	−5.31378	−0.278136
$366$	0	0
$367$	−28.6929	−1.49776	−0.748878	−	0.662708i	$-0.769407\pi$
$367$	−28.6929	−1.49776	−0.748878	+	0.662708i	$0.769407\pi$
$368$	−2.97312	−0.154985
$369$	0	0
$370$	26.2896	1.36673
$371$	2.39585	0.124387
$372$	0	0
$373$	8.58485	0.444507	0.222253	−	0.974989i	$-0.428659\pi$
$373$	8.58485	0.444507	0.222253	+	0.974989i	$0.428659\pi$
$374$	−0.632468	−0.0327041
$375$	0	0
$376$	−3.73595	−0.192667
$377$	−19.6968	−1.01444
$378$	0	0
$379$	10.0573	0.516610	0.258305	−	0.966063i	$-0.416836\pi$
$379$	10.0573	0.516610	0.258305	+	0.966063i	$0.416836\pi$
$380$	6.82366	0.350047
$381$	0	0
$382$	−8.03393	−0.411052
$383$	19.3152	0.986963	0.493481	−	0.869756i	$-0.335724\pi$
$383$	19.3152	0.986963	0.493481	+	0.869756i	$0.335724\pi$
$384$	0	0
$385$	3.22663	0.164444
$386$	−2.72903	−0.138904
$387$	0	0
$388$	6.49757	0.329864
$389$	−15.0719	−0.764178	−0.382089	−	0.924126i	$-0.624795\pi$
$389$	−15.0719	−0.764178	−0.382089	+	0.924126i	$0.624795\pi$
$390$	0	0
$391$	−0.503752	−0.0254758
$392$	−19.3626	−0.977959
$393$	0	0
$394$	−11.4979	−0.579254
$395$	−63.4332	−3.19167
$396$	0	0
$397$	12.9854	0.651720	0.325860	−	0.945418i	$-0.394346\pi$
$397$	12.9854	0.651720	0.325860	+	0.945418i	$0.394346\pi$
$398$	−8.81179	−0.441695
$399$	0	0
$400$	−33.7319	−1.68659
$401$	12.3840	0.618429	0.309214	−	0.950992i	$-0.399934\pi$
$401$	12.3840	0.618429	0.309214	+	0.950992i	$0.399934\pi$
$402$	0	0
$403$	10.4780	0.521948
$404$	−2.41353	−0.120078
$405$	0	0
$406$	−6.18134	−0.306775
$407$	−5.17919	−0.256723
$408$	0	0
$409$	−0.984934	−0.0487019	−0.0243509	−	0.999703i	$-0.507752\pi$
$409$	−0.984934	−0.0487019	−0.0243509	+	0.999703i	$0.507752\pi$
$410$	60.0029	2.96333
$411$	0	0
$412$	−0.420670	−0.0207249
$413$	−7.20912	−0.354738
$414$	0	0
$415$	−11.8149	−0.579970
$416$	7.51332	0.368371
$417$	0	0
$418$	5.00145	0.244629
$419$	12.0822	0.590255	0.295127	−	0.955458i	$-0.404638\pi$
$419$	12.0822	0.590255	0.295127	+	0.955458i	$0.404638\pi$
$420$	0	0
$421$	−37.5161	−1.82842	−0.914210	−	0.405240i	$-0.867188\pi$
$421$	−37.5161	−1.82842	−0.914210	+	0.405240i	$0.867188\pi$
$422$	18.1409	0.883083
$423$	0	0
$424$	9.13501	0.443635
$425$	−5.71537	−0.277236
$426$	0	0
$427$	2.10413	0.101826
$428$	−2.53754	−0.122657
$429$	0	0
$430$	−35.9876	−1.73547
$431$	8.43869	0.406478	0.203239	−	0.979129i	$-0.434853\pi$
$431$	8.43869	0.406478	0.203239	+	0.979129i	$0.434853\pi$
$432$	0	0
$433$	8.74254	0.420140	0.210070	−	0.977686i	$-0.432631\pi$
$433$	8.74254	0.420140	0.210070	+	0.977686i	$0.432631\pi$
$434$	3.28826	0.157842
$435$	0	0
$436$	6.50718	0.311637
$437$	3.98359	0.190561
$438$	0	0
$439$	2.75687	0.131578	0.0657891	−	0.997834i	$-0.479044\pi$
$439$	2.75687	0.131578	0.0657891	+	0.997834i	$0.479044\pi$
$440$	12.3026	0.586505
$441$	0	0
$442$	−2.01940	−0.0960529
$443$	−0.793474	−0.0376991	−0.0188496	−	0.999822i	$-0.506000\pi$
$443$	−0.793474	−0.0376991	−0.0188496	+	0.999822i	$0.506000\pi$
$444$	0	0
$445$	25.0924	1.18949
$446$	−30.4970	−1.44408
$447$	0	0
$448$	7.10347	0.335607
$449$	38.9043	1.83601	0.918003	−	0.396574i	$-0.129801\pi$
$449$	38.9043	1.83601	0.918003	+	0.396574i	$0.129801\pi$
$450$	0	0
$451$	−11.8209	−0.556624
$452$	6.67175	0.313813
$453$	0	0
$454$	27.0392	1.26901
$455$	10.3023	0.482977
$456$	0	0
$457$	4.02954	0.188494	0.0942470	−	0.995549i	$-0.469956\pi$
$457$	4.02954	0.188494	0.0942470	+	0.995549i	$0.469956\pi$
$458$	−8.42528	−0.393688
$459$	0	0
$460$	1.71294	0.0798664
$461$	8.05034	0.374942	0.187471	−	0.982270i	$-0.439971\pi$
$461$	8.05034	0.374942	0.187471	+	0.982270i	$0.439971\pi$
$462$	0	0
$463$	−3.83757	−0.178347	−0.0891735	−	0.996016i	$-0.528423\pi$
$463$	−3.83757	−0.178347	−0.0891735	+	0.996016i	$0.528423\pi$
$464$	−18.3411	−0.851464
$465$	0	0
$466$	−18.9409	−0.877418
$467$	−32.1096	−1.48585	−0.742927	−	0.669372i	$-0.766563\pi$
$467$	−32.1096	−1.48585	−0.742927	+	0.669372i	$0.766563\pi$
$468$	0	0
$469$	7.17498	0.331310
$470$	−6.23198	−0.287460
$471$	0	0
$472$	−27.4873	−1.26520
$473$	7.08974	0.325987
$474$	0	0
$475$	45.1962	2.07375
$476$	0.170337	0.00780737
$477$	0	0
$478$	−38.4803	−1.76005
$479$	41.4512	1.89395	0.946976	−	0.321305i	$-0.104121\pi$
$479$	41.4512	1.89395	0.946976	+	0.321305i	$0.104121\pi$
$480$	0	0
$481$	−16.5366	−0.754002
$482$	16.7875	0.764648
$483$	0	0
$484$	−0.423685	−0.0192584
$485$	62.0024	2.81539
$486$	0	0
$487$	21.7836	0.987109	0.493554	−	0.869715i	$-0.335697\pi$
$487$	21.7836	0.987109	0.493554	+	0.869715i	$0.335697\pi$
$488$	8.02273	0.363172
$489$	0	0
$490$	−32.2989	−1.45912
$491$	−16.5105	−0.745110	−0.372555	−	0.928010i	$-0.621518\pi$
$491$	−16.5105	−0.745110	−0.372555	+	0.928010i	$0.621518\pi$
$492$	0	0
$493$	−3.10763	−0.139961
$494$	15.9691	0.718482
$495$	0	0
$496$	9.75684	0.438095
$497$	−7.82494	−0.350996
$498$	0	0
$499$	−5.88608	−0.263497	−0.131748	−	0.991283i	$-0.542059\pi$
$499$	−5.88608	−0.263497	−0.131748	+	0.991283i	$0.542059\pi$
$500$	10.8697	0.486106
$501$	0	0
$502$	31.9953	1.42802
$503$	−25.4624	−1.13531	−0.567656	−	0.823266i	$-0.692150\pi$
$503$	−25.4624	−1.13531	−0.567656	+	0.823266i	$0.692150\pi$
$504$	0	0
$505$	−23.0309	−1.02486
$506$	1.25551	0.0558144
$507$	0	0
$508$	−3.84224	−0.170472
$509$	37.9786	1.68337	0.841687	−	0.539966i	$-0.181563\pi$
$509$	37.9786	1.68337	0.841687	+	0.539966i	$0.181563\pi$
$510$	0	0
$511$	1.04894	0.0464025
$512$	25.0904	1.10885
$513$	0	0
$514$	−17.7048	−0.780924
$515$	−4.01421	−0.176887
$516$	0	0
$517$	1.22773	0.0539956
$518$	−5.18958	−0.228017
$519$	0	0
$520$	39.2809	1.72258
$521$	−30.5870	−1.34004	−0.670020	−	0.742343i	$-0.733714\pi$
$521$	−30.5870	−1.34004	−0.670020	+	0.742343i	$0.733714\pi$
$522$	0	0
$523$	−2.66147	−0.116378	−0.0581890	−	0.998306i	$-0.518533\pi$
$523$	−2.66147	−0.116378	−0.0581890	+	0.998306i	$0.518533\pi$
$524$	4.62837	0.202191
$525$	0	0
$526$	−13.1843	−0.574863
$527$	1.65315	0.0720125
$528$	0	0
$529$	1.00000	0.0434783
$530$	15.2382	0.661905
$531$	0	0
$532$	−1.34699	−0.0583996
$533$	−37.7427	−1.63482
$534$	0	0
$535$	−24.2142	−1.04687
$536$	27.3571	1.18165
$537$	0	0
$538$	25.8282	1.11353
$539$	6.36306	0.274076
$540$	0	0
$541$	41.1086	1.76740	0.883698	−	0.468058i	$-0.155046\pi$
$541$	41.1086	1.76740	0.883698	+	0.468058i	$0.155046\pi$
$542$	36.0791	1.54973
$543$	0	0
$544$	1.18540	0.0508236
$545$	62.0941	2.65982
$546$	0	0
$547$	−7.10820	−0.303925	−0.151962	−	0.988386i	$-0.548559\pi$
$547$	−7.10820	−0.303925	−0.151962	+	0.988386i	$0.548559\pi$
$548$	9.50192	0.405902
$549$	0	0
$550$	14.2446	0.607390
$551$	24.5746	1.04691
$552$	0	0
$553$	12.5217	0.532479
$554$	17.2687	0.733678
$555$	0	0
$556$	0.322139	0.0136617
$557$	17.5181	0.742265	0.371133	−	0.928580i	$-0.378970\pi$
$557$	17.5181	0.742265	0.371133	+	0.928580i	$0.378970\pi$
$558$	0	0
$559$	22.6367	0.957432
$560$	9.59317	0.405385
$561$	0	0
$562$	9.47793	0.399802
$563$	−32.5836	−1.37324	−0.686618	−	0.727019i	$-0.740905\pi$
$563$	−32.5836	−1.37324	−0.686618	+	0.727019i	$0.740905\pi$
$564$	0	0
$565$	63.6645	2.67839
$566$	33.9014	1.42498
$567$	0	0
$568$	−29.8353	−1.25186
$569$	−42.5643	−1.78439	−0.892194	−	0.451653i	$-0.850835\pi$
$569$	−42.5643	−1.78439	−0.892194	+	0.451653i	$0.850835\pi$
$570$	0	0
$571$	−3.70786	−0.155169	−0.0775845	−	0.996986i	$-0.524721\pi$
$571$	−3.70786	−0.155169	−0.0775845	+	0.996986i	$0.524721\pi$
$572$	−1.35278	−0.0565624
$573$	0	0
$574$	−11.8446	−0.494384
$575$	11.3456	0.473144
$576$	0	0
$577$	−6.75679	−0.281289	−0.140644	−	0.990060i	$-0.544917\pi$
$577$	−6.75679	−0.281289	−0.140644	+	0.990060i	$0.544917\pi$
$578$	21.0251	0.874530
$579$	0	0
$580$	10.5671	0.438775
$581$	2.33227	0.0967587
$582$	0	0
$583$	−3.00201	−0.124330
$584$	3.99945	0.165498
$585$	0	0
$586$	−13.1668	−0.543915
$587$	−3.71910	−0.153504	−0.0767519	−	0.997050i	$-0.524455\pi$
$587$	−3.71910	−0.153504	−0.0767519	+	0.997050i	$0.524455\pi$
$588$	0	0
$589$	−13.0729	−0.538658
$590$	−45.8517	−1.88769
$591$	0	0
$592$	−15.3984	−0.632869
$593$	−22.6333	−0.929439	−0.464719	−	0.885458i	$-0.653845\pi$
$593$	−22.6333	−0.929439	−0.464719	+	0.885458i	$0.653845\pi$
$594$	0	0
$595$	1.62542	0.0666358
$596$	2.37050	0.0970994
$597$	0	0
$598$	4.00871	0.163928
$599$	30.4468	1.24402	0.622010	−	0.783009i	$-0.286316\pi$
$599$	30.4468	1.24402	0.622010	+	0.783009i	$0.286316\pi$
$600$	0	0
$601$	11.3551	0.463186	0.231593	−	0.972813i	$-0.425606\pi$
$601$	11.3551	0.463186	0.231593	+	0.972813i	$0.425606\pi$
$602$	7.10396	0.289536
$603$	0	0
$604$	−3.02407	−0.123048
$605$	−4.04297	−0.164370
$606$	0	0
$607$	9.14323	0.371112	0.185556	−	0.982634i	$-0.440591\pi$
$607$	9.14323	0.371112	0.185556	+	0.982634i	$0.440591\pi$
$608$	−9.37396	−0.380164
$609$	0	0
$610$	13.3828	0.541853
$611$	3.92001	0.158587
$612$	0	0
$613$	−3.40904	−0.137690	−0.0688449	−	0.997627i	$-0.521931\pi$
$613$	−3.40904	−0.137690	−0.0688449	+	0.997627i	$0.521931\pi$
$614$	−24.1145	−0.973180
$615$	0	0
$616$	−2.42855	−0.0978489
$617$	−1.68640	−0.0678919	−0.0339460	−	0.999424i	$-0.510807\pi$
$617$	−1.68640	−0.0678919	−0.0339460	+	0.999424i	$0.510807\pi$
$618$	0	0
$619$	−7.71276	−0.310002	−0.155001	−	0.987914i	$-0.549538\pi$
$619$	−7.71276	−0.310002	−0.155001	+	0.987914i	$0.549538\pi$
$620$	−5.62134	−0.225758
$621$	0	0
$622$	−20.8630	−0.836530
$623$	−4.95324	−0.198448
$624$	0	0
$625$	46.9947	1.87979
$626$	−26.1237	−1.04411
$627$	0	0
$628$	7.98532	0.318649
$629$	−2.60903	−0.104029
$630$	0	0
$631$	−39.9860	−1.59182	−0.795908	−	0.605417i	$-0.793006\pi$
$631$	−39.9860	−1.59182	−0.795908	+	0.605417i	$0.793006\pi$
$632$	47.7435	1.89913
$633$	0	0
$634$	−1.05770	−0.0420065
$635$	−36.6642	−1.45497
$636$	0	0
$637$	20.3165	0.804970
$638$	7.74523	0.306637
$639$	0	0
$640$	26.1524	1.03376
$641$	8.13445	0.321291	0.160646	−	0.987012i	$-0.448642\pi$
$641$	8.13445	0.321291	0.160646	+	0.987012i	$0.448642\pi$
$642$	0	0
$643$	−29.6220	−1.16818	−0.584088	−	0.811690i	$-0.698548\pi$
$643$	−29.6220	−1.16818	−0.584088	+	0.811690i	$0.698548\pi$
$644$	−0.338136	−0.0133244
$645$	0	0
$646$	2.51949	0.0991281
$647$	−18.3325	−0.720727	−0.360363	−	0.932812i	$-0.617347\pi$
$647$	−18.3325	−0.720727	−0.360363	+	0.932812i	$0.617347\pi$
$648$	0	0
$649$	9.03304	0.354578
$650$	45.4813	1.78392
$651$	0	0
$652$	8.26863	0.323824
$653$	4.41518	0.172779	0.0863896	−	0.996261i	$-0.472467\pi$
$653$	4.41518	0.172779	0.0863896	+	0.996261i	$0.472467\pi$
$654$	0	0
$655$	44.1657	1.72570
$656$	−35.1449	−1.37218
$657$	0	0
$658$	1.23020	0.0479580
$659$	10.5431	0.410703	0.205351	−	0.978688i	$-0.434166\pi$
$659$	10.5431	0.410703	0.205351	+	0.978688i	$0.434166\pi$
$660$	0	0
$661$	−10.2501	−0.398681	−0.199341	−	0.979930i	$-0.563880\pi$
$661$	−10.2501	−0.398681	−0.199341	+	0.979930i	$0.563880\pi$
$662$	12.8376	0.498946
$663$	0	0
$664$	8.89256	0.345098
$665$	−12.8536	−0.498440
$666$	0	0
$667$	6.16897	0.238863
$668$	8.69838	0.336550
$669$	0	0
$670$	45.6346	1.76302
$671$	−2.63648	−0.101780
$672$	0	0
$673$	29.0456	1.11962	0.559812	−	0.828620i	$-0.310873\pi$
$673$	29.0456	1.11962	0.559812	+	0.828620i	$0.310873\pi$
$674$	−36.8674	−1.42008
$675$	0	0
$676$	1.18864	0.0457169
$677$	−21.8323	−0.839084	−0.419542	−	0.907736i	$-0.637809\pi$
$677$	−21.8323	−0.839084	−0.419542	+	0.907736i	$0.637809\pi$
$678$	0	0
$679$	−12.2393	−0.469702
$680$	6.19748	0.237663
$681$	0	0
$682$	−4.12020	−0.157771
$683$	−48.4518	−1.85396	−0.926979	−	0.375114i	$-0.877604\pi$
$683$	−48.4518	−1.85396	−0.926979	+	0.375114i	$0.877604\pi$
$684$	0	0
$685$	90.6712	3.46437
$686$	13.3899	0.511227
$687$	0	0
$688$	21.0787	0.803617
$689$	−9.58506	−0.365162
$690$	0	0
$691$	23.0694	0.877601	0.438800	−	0.898585i	$-0.355404\pi$
$691$	23.0694	0.877601	0.438800	+	0.898585i	$0.355404\pi$
$692$	1.97980	0.0752609
$693$	0	0
$694$	31.3599	1.19041
$695$	3.07398	0.116603
$696$	0	0
$697$	−5.95480	−0.225554
$698$	38.6158	1.46163
$699$	0	0
$700$	−3.83636	−0.145001
$701$	−35.6219	−1.34542	−0.672711	−	0.739905i	$-0.734870\pi$
$701$	−35.6219	−1.34542	−0.672711	+	0.739905i	$0.734870\pi$
$702$	0	0
$703$	20.6318	0.778142
$704$	−8.90065	−0.335456
$705$	0	0
$706$	−22.8008	−0.858120
$707$	4.54631	0.170982
$708$	0	0
$709$	21.1768	0.795312	0.397656	−	0.917534i	$-0.369824\pi$
$709$	21.1768	0.795312	0.397656	+	0.917534i	$0.369824\pi$
$710$	−49.7685	−1.86778
$711$	0	0
$712$	−18.8859	−0.707781
$713$	−3.28168	−0.122900
$714$	0	0
$715$	−12.9087	−0.482759
$716$	−1.98492	−0.0741801
$717$	0	0
$718$	−1.19354	−0.0445424
$719$	47.0596	1.75503	0.877513	−	0.479553i	$-0.159201\pi$
$719$	47.0596	1.75503	0.877513	+	0.479553i	$0.159201\pi$
$720$	0	0
$721$	0.792406	0.0295108
$722$	3.93104	0.146298
$723$	0	0
$724$	−4.33853	−0.161240
$725$	69.9907	2.59939
$726$	0	0
$727$	−34.5022	−1.27961	−0.639807	−	0.768535i	$-0.720986\pi$
$727$	−34.5022	−1.27961	−0.639807	+	0.768535i	$0.720986\pi$
$728$	−7.75407	−0.287385
$729$	0	0
$730$	6.67152	0.246924
$731$	3.57147	0.132096
$732$	0	0
$733$	−19.9498	−0.736863	−0.368431	−	0.929655i	$-0.620105\pi$
$733$	−19.9498	−0.736863	−0.368431	+	0.929655i	$0.620105\pi$
$734$	36.0243	1.32968
$735$	0	0
$736$	−2.35314	−0.0867380
$737$	−8.99026	−0.331160
$738$	0	0
$739$	9.42242	0.346609	0.173305	−	0.984868i	$-0.444555\pi$
$739$	9.42242	0.346609	0.173305	+	0.984868i	$0.444555\pi$
$740$	8.87166	0.326129
$741$	0	0
$742$	−3.00803	−0.110428
$743$	−7.25579	−0.266189	−0.133095	−	0.991103i	$-0.542491\pi$
$743$	−7.25579	−0.266189	−0.133095	+	0.991103i	$0.542491\pi$
$744$	0	0
$745$	22.6203	0.828742
$746$	−10.7784	−0.394625
$747$	0	0
$748$	−0.213432	−0.00780385
$749$	4.77990	0.174654
$750$	0	0
$751$	−18.9612	−0.691904	−0.345952	−	0.938252i	$-0.612444\pi$
$751$	−18.9612	−0.691904	−0.345952	+	0.938252i	$0.612444\pi$
$752$	3.65020	0.133109
$753$	0	0
$754$	24.7296	0.900600
$755$	−28.8569	−1.05021
$756$	0	0
$757$	−8.51568	−0.309508	−0.154754	−	0.987953i	$-0.549458\pi$
$757$	−8.51568	−0.309508	−0.154754	+	0.987953i	$0.549458\pi$
$758$	−12.6271	−0.458637
$759$	0	0
$760$	−49.0086	−1.77773
$761$	3.63058	0.131609	0.0658043	−	0.997833i	$-0.479039\pi$
$761$	3.63058	0.131609	0.0658043	+	0.997833i	$0.479039\pi$
$762$	0	0
$763$	−12.2574	−0.443748
$764$	−2.71112	−0.0980850
$765$	0	0
$766$	−24.2506	−0.876208
$767$	28.8414	1.04140
$768$	0	0
$769$	−11.1413	−0.401764	−0.200882	−	0.979615i	$-0.564381\pi$
$769$	−11.1413	−0.401764	−0.200882	+	0.979615i	$0.564381\pi$
$770$	−4.05108	−0.145991
$771$	0	0
$772$	−0.920935	−0.0331452
$773$	−40.1436	−1.44387	−0.721933	−	0.691963i	$-0.756746\pi$
$773$	−40.1436	−1.44387	−0.721933	+	0.691963i	$0.756746\pi$
$774$	0	0
$775$	−37.2327	−1.33744
$776$	−46.6666	−1.67523
$777$	0	0
$778$	18.9230	0.678424
$779$	47.0895	1.68716
$780$	0	0
$781$	9.80465	0.350838
$782$	0.632468	0.0226170
$783$	0	0
$784$	18.9182	0.675649
$785$	76.1991	2.71966
$786$	0	0
$787$	−32.1916	−1.14751	−0.573754	−	0.819028i	$-0.694513\pi$
$787$	−32.1916	−1.14751	−0.573754	+	0.819028i	$0.694513\pi$
$788$	−3.88006	−0.138221
$789$	0	0
$790$	79.6413	2.83351
$791$	−12.5674	−0.446846
$792$	0	0
$793$	−8.41798	−0.298931
$794$	−16.3034	−0.578585
$795$	0	0
$796$	−2.97362	−0.105397
$797$	10.4731	0.370977	0.185489	−	0.982646i	$-0.440613\pi$
$797$	10.4731	0.370977	0.185489	+	0.982646i	$0.440613\pi$
$798$	0	0
$799$	0.618473	0.0218800
$800$	−26.6978	−0.943911
$801$	0	0
$802$	−15.5483	−0.549030
$803$	−1.31433	−0.0463815
$804$	0	0
$805$	−3.22663	−0.113724
$806$	−13.1553	−0.463376
$807$	0	0
$808$	17.3344	0.609821
$809$	2.51048	0.0882637	0.0441318	−	0.999026i	$-0.485948\pi$
$809$	2.51048	0.0882637	0.0441318	+	0.999026i	$0.485948\pi$
$810$	0	0
$811$	−28.1415	−0.988183	−0.494091	−	0.869410i	$-0.664499\pi$
$811$	−28.1415	−0.988183	−0.494091	+	0.869410i	$0.664499\pi$
$812$	−2.08595	−0.0732025
$813$	0	0
$814$	6.50255	0.227914
$815$	78.9026	2.76384
$816$	0	0
$817$	−28.2426	−0.988084
$818$	1.23660	0.0432367
$819$	0	0
$820$	20.2485	0.707109
$821$	−9.55356	−0.333421	−0.166711	−	0.986006i	$-0.553315\pi$
$821$	−9.55356	−0.333421	−0.166711	+	0.986006i	$0.553315\pi$
$822$	0	0
$823$	−38.7711	−1.35147	−0.675737	−	0.737143i	$-0.736174\pi$
$823$	−38.7711	−1.35147	−0.675737	+	0.737143i	$0.736174\pi$
$824$	3.02132	0.105253
$825$	0	0
$826$	9.05116	0.314930
$827$	−28.9099	−1.00530	−0.502648	−	0.864491i	$-0.667641\pi$
$827$	−28.9099	−1.00530	−0.502648	+	0.864491i	$0.667641\pi$
$828$	0	0
$829$	−13.0709	−0.453972	−0.226986	−	0.973898i	$-0.572887\pi$
$829$	−13.0709	−0.453972	−0.226986	+	0.973898i	$0.572887\pi$
$830$	14.8338	0.514887
$831$	0	0
$832$	−28.4187	−0.985243
$833$	3.20541	0.111061
$834$	0	0
$835$	83.0034	2.87245
$836$	1.68779	0.0583733
$837$	0	0
$838$	−15.1694	−0.524018
$839$	9.03560	0.311944	0.155972	−	0.987762i	$-0.450149\pi$
$839$	9.03560	0.311944	0.155972	+	0.987762i	$0.450149\pi$
$840$	0	0
$841$	9.05619	0.312282
$842$	47.1019	1.62324
$843$	0	0
$844$	6.12180	0.210721
$845$	11.3425	0.390193
$846$	0	0
$847$	0.798084	0.0274225
$848$	−8.92533	−0.306497
$849$	0	0
$850$	7.17573	0.246126
$851$	5.17919	0.177540
$852$	0	0
$853$	−13.4429	−0.460275	−0.230137	−	0.973158i	$-0.573918\pi$
$853$	−13.4429	−0.460275	−0.230137	+	0.973158i	$0.573918\pi$
$854$	−2.64177	−0.0903995
$855$	0	0
$856$	18.2250	0.622918
$857$	−40.8692	−1.39607	−0.698033	−	0.716065i	$-0.745941\pi$
$857$	−40.8692	−1.39607	−0.698033	+	0.716065i	$0.745941\pi$
$858$	0	0
$859$	33.6168	1.14699	0.573495	−	0.819209i	$-0.305587\pi$
$859$	33.6168	1.14699	0.573495	+	0.819209i	$0.305587\pi$
$860$	−12.1443	−0.414118
$861$	0	0
$862$	−10.5949	−0.360864
$863$	−6.28825	−0.214055	−0.107027	−	0.994256i	$-0.534133\pi$
$863$	−6.28825	−0.214055	−0.107027	+	0.994256i	$0.534133\pi$
$864$	0	0
$865$	18.8921	0.642351
$866$	−10.9764	−0.372993
$867$	0	0
$868$	1.10965	0.0376641
$869$	−15.6898	−0.532239
$870$	0	0
$871$	−28.7049	−0.972627
$872$	−46.7356	−1.58267
$873$	0	0
$874$	−5.00145	−0.169177
$875$	−20.4749	−0.692179
$876$	0	0
$877$	−46.4269	−1.56773	−0.783863	−	0.620934i	$-0.786754\pi$
$877$	−46.4269	−1.56773	−0.783863	+	0.620934i	$0.786754\pi$
$878$	−3.46129	−0.116813
$879$	0	0
$880$	−12.0202	−0.405202
$881$	6.31123	0.212631	0.106315	−	0.994332i	$-0.466095\pi$
$881$	6.31123	0.212631	0.106315	+	0.994332i	$0.466095\pi$
$882$	0	0
$883$	33.7642	1.13625	0.568127	−	0.822941i	$-0.307668\pi$
$883$	33.7642	1.13625	0.568127	+	0.822941i	$0.307668\pi$
$884$	−0.681464	−0.0229201
$885$	0	0
$886$	0.996218	0.0334686
$887$	−14.5976	−0.490139	−0.245069	−	0.969506i	$-0.578811\pi$
$887$	−14.5976	−0.490139	−0.245069	+	0.969506i	$0.578811\pi$
$888$	0	0
$889$	7.23753	0.242739
$890$	−31.5038	−1.05601
$891$	0	0
$892$	−10.2915	−0.344585
$893$	−4.89078	−0.163664
$894$	0	0
$895$	−18.9409	−0.633126
$896$	−5.16249	−0.172467
$897$	0	0
$898$	−48.8448	−1.62997
$899$	−20.2446	−0.675195
$900$	0	0
$901$	−1.51227	−0.0503809
$902$	14.8413	0.494161
$903$	0	0
$904$	−47.9176	−1.59371
$905$	−41.4000	−1.37618
$906$	0	0
$907$	−48.0826	−1.59656	−0.798278	−	0.602290i	$-0.794255\pi$
$907$	−48.0826	−1.59656	−0.798278	+	0.602290i	$0.794255\pi$
$908$	9.12462	0.302811
$909$	0	0
$910$	−12.9346	−0.428779
$911$	28.6152	0.948065	0.474033	−	0.880507i	$-0.342798\pi$
$911$	28.6152	0.948065	0.474033	+	0.880507i	$0.342798\pi$
$912$	0	0
$913$	−2.92233	−0.0967150
$914$	−5.05915	−0.167342
$915$	0	0
$916$	−2.84319	−0.0939416
$917$	−8.71834	−0.287905
$918$	0	0
$919$	8.89355	0.293371	0.146686	−	0.989183i	$-0.453139\pi$
$919$	8.89355	0.293371	0.146686	+	0.989183i	$0.453139\pi$
$920$	−12.3026	−0.405606
$921$	0	0
$922$	−10.1073	−0.332866
$923$	31.3051	1.03042
$924$	0	0
$925$	58.7610	1.93205
$926$	4.81812	0.158333
$927$	0	0
$928$	−14.5165	−0.476526
$929$	18.1893	0.596771	0.298385	−	0.954445i	$-0.403552\pi$
$929$	18.1893	0.596771	0.298385	+	0.954445i	$0.403552\pi$
$930$	0	0
$931$	−25.3478	−0.830741
$932$	−6.39176	−0.209369
$933$	0	0
$934$	40.3140	1.31912
$935$	−2.03665	−0.0666057
$936$	0	0
$937$	33.3478	1.08943	0.544713	−	0.838623i	$-0.316639\pi$
$937$	33.3478	1.08943	0.544713	+	0.838623i	$0.316639\pi$
$938$	−9.00829	−0.294131
$939$	0	0
$940$	−2.10304	−0.0685935
$941$	−29.2596	−0.953836	−0.476918	−	0.878948i	$-0.658246\pi$
$941$	−29.2596	−0.953836	−0.476918	+	0.878948i	$0.658246\pi$
$942$	0	0
$943$	11.8209	0.384941
$944$	26.8563	0.874099
$945$	0	0
$946$	−8.90127	−0.289405
$947$	−34.5660	−1.12324	−0.561621	−	0.827394i	$-0.689822\pi$
$947$	−34.5660	−1.12324	−0.561621	+	0.827394i	$0.689822\pi$
$948$	0	0
$949$	−4.19649	−0.136224
$950$	−56.7445	−1.84103
$951$	0	0
$952$	−1.22339	−0.0396502
$953$	−28.4736	−0.922350	−0.461175	−	0.887309i	$-0.652572\pi$
$953$	−28.4736	−0.922350	−0.461175	+	0.887309i	$0.652572\pi$
$954$	0	0
$955$	−25.8706	−0.837154
$956$	−12.9855	−0.419982
$957$	0	0
$958$	−52.0425	−1.68142
$959$	−17.8985	−0.577974
$960$	0	0
$961$	−20.2306	−0.652599
$962$	20.7619	0.669390
$963$	0	0
$964$	5.66508	0.182460
$965$	−8.78794	−0.282894
$966$	0	0
$967$	−10.9313	−0.351526	−0.175763	−	0.984433i	$-0.556239\pi$
$967$	−10.9313	−0.351526	−0.175763	+	0.984433i	$0.556239\pi$
$968$	3.04297	0.0978048
$969$	0	0
$970$	−77.8449	−2.49945
$971$	7.59300	0.243671	0.121835	−	0.992550i	$-0.461122\pi$
$971$	7.59300	0.243671	0.121835	+	0.992550i	$0.461122\pi$
$972$	0	0
$973$	−0.606805	−0.0194533
$974$	−27.3496	−0.876338
$975$	0	0
$976$	−7.83858	−0.250907
$977$	−23.0302	−0.736802	−0.368401	−	0.929667i	$-0.620095\pi$
$977$	−23.0302	−0.736802	−0.368401	+	0.929667i	$0.620095\pi$
$978$	0	0
$979$	6.20642	0.198358
$980$	−10.8996	−0.348174
$981$	0	0
$982$	20.7292	0.661496
$983$	−21.6864	−0.691688	−0.345844	−	0.938292i	$-0.612407\pi$
$983$	−21.6864	−0.691688	−0.345844	+	0.938292i	$0.612407\pi$
$984$	0	0
$985$	−37.0251	−1.17972
$986$	3.90168	0.124255
$987$	0	0
$988$	5.38890	0.171444
$989$	−7.08974	−0.225441
$990$	0	0
$991$	32.0034	1.01662	0.508310	−	0.861174i	$-0.330270\pi$
$991$	32.0034	1.01662	0.508310	+	0.861174i	$0.330270\pi$
$992$	7.72227	0.245182
$993$	0	0
$994$	9.82432	0.311608
$995$	−28.3755	−0.899563
$996$	0	0
$997$	−44.7240	−1.41642	−0.708211	−	0.706001i	$-0.750498\pi$
$997$	−44.7240	−1.41642	−0.708211	+	0.706001i	$0.750498\pi$
$998$	7.39005	0.233928
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2277.2.a.m.1.3		6
3.2	odd	2		253.2.a.d.1.4	✓	6
12.11	even	2		4048.2.a.bc.1.4		6
15.14	odd	2		6325.2.a.m.1.3		6
33.32	even	2		2783.2.a.h.1.3		6
69.68	even	2		5819.2.a.e.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
253.2.a.d.1.4	✓	6	3.2	odd	2
2277.2.a.m.1.3		6	1.1	even	1	trivial
2783.2.a.h.1.3		6	33.32	even	2
4048.2.a.bc.1.4		6	12.11	even	2
5819.2.a.e.1.4		6	69.68	even	2
6325.2.a.m.1.3		6	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)

\(f(q)\)	\(=\)	\(q-1.25551 q^{2} -0.423685 q^{4} -4.04297 q^{5} +0.798084 q^{7} +3.04297 q^{8} +5.07601 q^{10} -1.00000 q^{11} -3.19289 q^{13} -1.00201 q^{14} -2.97312 q^{16} -0.503752 q^{17} +3.98359 q^{19} +1.71294 q^{20} +1.25551 q^{22} +1.00000 q^{23} +11.3456 q^{25} +4.00871 q^{26} -0.338136 q^{28} +6.16897 q^{29} -3.28168 q^{31} -2.35314 q^{32} +0.632468 q^{34} -3.22663 q^{35} +5.17919 q^{37} -5.00145 q^{38} -12.3026 q^{40} +11.8209 q^{41} -7.08974 q^{43} +0.423685 q^{44} -1.25551 q^{46} -1.22773 q^{47} -6.36306 q^{49} -14.2446 q^{50} +1.35278 q^{52} +3.00201 q^{53} +4.04297 q^{55} +2.42855 q^{56} -7.74523 q^{58} -9.03304 q^{59} +2.63648 q^{61} +4.12020 q^{62} +8.90065 q^{64} +12.9087 q^{65} +8.99026 q^{67} +0.213432 q^{68} +4.05108 q^{70} -9.80465 q^{71} +1.31433 q^{73} -6.50255 q^{74} -1.68779 q^{76} -0.798084 q^{77} +15.6898 q^{79} +12.0202 q^{80} -14.8413 q^{82} +2.92233 q^{83} +2.03665 q^{85} +8.90127 q^{86} -3.04297 q^{88} -6.20642 q^{89} -2.54819 q^{91} -0.423685 q^{92} +1.54144 q^{94} -16.1055 q^{95} -15.3359 q^{97} +7.98891 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 5 q^{4} - 3 q^{5} - q^{7} - 3 q^{8} - 6 q^{10} - 6 q^{11} - 3 q^{13} + 8 q^{14} - q^{16} - 5 q^{17} + q^{19} + 7 q^{20} + 3 q^{22} + 6 q^{23} + 3 q^{25} - 15 q^{26} - 6 q^{29} - 8 q^{31}+ \cdots - 34 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 5 * q^4 - 3 * q^5 - q^7 - 3 * q^8 - 6 * q^10 - 6 * q^11 - 3 * q^13 + 8 * q^14 - q^16 - 5 * q^17 + q^19 + 7 * q^20 + 3 * q^22 + 6 * q^23 + 3 * q^25 - 15 * q^26 - 6 * q^29 - 8 * q^31 + 6 * q^32 - 6 * q^34 - 12 * q^35 + 2 * q^37 - 9 * q^38 - 30 * q^40 + 2 * q^41 + 10 * q^43 - 5 * q^44 - 3 * q^46 - 14 * q^47 - 5 * q^49 - 9 * q^50 - 11 * q^52 + 4 * q^53 + 3 * q^55 + 13 * q^56 - 6 * q^58 - 39 * q^59 - 22 * q^61 - 33 * q^64 + 16 * q^65 + 8 * q^67 - 17 * q^68 + 19 * q^70 - 18 * q^71 - 9 * q^73 - 15 * q^74 - 25 * q^76 + q^77 - 15 * q^79 + 16 * q^80 - 8 * q^82 - 43 * q^83 - 21 * q^85 + 55 * q^86 + 3 * q^88 - 17 * q^89 - 9 * q^91 + 5 * q^92 + 63 * q^94 + 21 * q^95 + 19 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more