LMFDB - Embedded newform 2214.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2214,2,Mod(1,2214)] 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("2214.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2214, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2214 = 2 \cdot 3^{3} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2214.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,3,-1,0,1,-3,0,1,1] 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:	$17.6788790075$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2292.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 1 $ x^3 - x^2 - 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.18296$ of defining polynomial
Character	$\chi$	$=$	2214.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +3.18296 q^{5} -3.18296 q^{7} -1.00000 q^{8} -3.18296 q^{10} -3.18296 q^{11} -2.56561 q^{13} +3.18296 q^{14} +1.00000 q^{16} -2.00000 q^{17} +4.74857 q^{19} +3.18296 q^{20} +3.18296 q^{22} +6.93152 q^{23} +5.13122 q^{25} +2.56561 q^{26} -3.18296 q^{28} -10.1830 q^{29} +2.43439 q^{31} -1.00000 q^{32} +2.00000 q^{34} -10.1312 q^{35} -4.74857 q^{37} -4.74857 q^{38} -3.18296 q^{40} -1.00000 q^{41} +4.00000 q^{43} -3.18296 q^{44} -6.93152 q^{46} -5.61735 q^{47} +3.13122 q^{49} -5.13122 q^{50} -2.56561 q^{52} -12.2974 q^{53} -10.1312 q^{55} +3.18296 q^{56} +10.1830 q^{58} -13.4971 q^{59} -9.87978 q^{61} -2.43439 q^{62} +1.00000 q^{64} -8.16622 q^{65} +5.49713 q^{67} -2.00000 q^{68} +10.1312 q^{70} +8.24570 q^{71} +11.6283 q^{73} +4.74857 q^{74} +4.74857 q^{76} +10.1312 q^{77} -7.31417 q^{79} +3.18296 q^{80} +1.00000 q^{82} -8.87978 q^{83} -6.36591 q^{85} -4.00000 q^{86} +3.18296 q^{88} -16.3659 q^{89} +8.16622 q^{91} +6.93152 q^{92} +5.61735 q^{94} +15.1145 q^{95} +8.98326 q^{97} -3.13122 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - q^{5} + q^{7} - 3 q^{8} + q^{10} + q^{11} - 6 q^{13} - q^{14} + 3 q^{16} - 6 q^{17} + 2 q^{19} - q^{20} - q^{22} - 2 q^{23} + 12 q^{25} + 6 q^{26} + q^{28} - 20 q^{29}+ \cdots - 6 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - q^5 + q^7 - 3 * q^8 + q^10 + q^11 - 6 * q^13 - q^14 + 3 * q^16 - 6 * q^17 + 2 * q^19 - q^20 - q^22 - 2 * q^23 + 12 * q^25 + 6 * q^26 + q^28 - 20 * q^29 + 9 * q^31 - 3 * q^32 + 6 * q^34 - 27 * q^35 - 2 * q^37 - 2 * q^38 + q^40 - 3 * q^41 + 12 * q^43 + q^44 + 2 * q^46 - 8 * q^47 + 6 * q^49 - 12 * q^50 - 6 * q^52 + 7 * q^53 - 27 * q^55 - q^56 + 20 * q^58 - 16 * q^59 - 14 * q^61 - 9 * q^62 + 3 * q^64 + 16 * q^65 - 8 * q^67 - 6 * q^68 + 27 * q^70 - 12 * q^71 + 7 * q^73 + 2 * q^74 + 2 * q^76 + 27 * q^77 - 8 * q^79 - q^80 + 3 * q^82 - 11 * q^83 + 2 * q^85 - 12 * q^86 - q^88 - 28 * q^89 - 16 * q^91 - 2 * q^92 + 8 * q^94 + 12 * q^95 - 3 * q^97 - 6 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.18296	1.42346	0.711731	−	0.702452i	$-0.247912\pi$
$5$	3.18296	1.42346	0.711731	+	0.702452i	$0.247912\pi$
$6$	0	0
$7$	−3.18296	−1.20304	−0.601522	−	0.798856i	$-0.705439\pi$
$7$	−3.18296	−1.20304	−0.601522	+	0.798856i	$0.705439\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.18296	−1.00654
$11$	−3.18296	−0.959698	−0.479849	−	0.877351i	$-0.659309\pi$
$11$	−3.18296	−0.959698	−0.479849	+	0.877351i	$0.659309\pi$
$12$	0	0
$13$	−2.56561	−0.711572	−0.355786	−	0.934568i	$-0.615787\pi$
$13$	−2.56561	−0.711572	−0.355786	+	0.934568i	$0.615787\pi$
$14$	3.18296	0.850681
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	4.74857	1.08940	0.544698	−	0.838632i	$-0.316644\pi$
$19$	4.74857	1.08940	0.544698	+	0.838632i	$0.316644\pi$
$20$	3.18296	0.711731
$21$	0	0
$22$	3.18296	0.678609
$23$	6.93152	1.44532	0.722661	−	0.691202i	$-0.242919\pi$
$23$	6.93152	1.44532	0.722661	+	0.691202i	$0.242919\pi$
$24$	0	0
$25$	5.13122	1.02624
$26$	2.56561	0.503157
$27$	0	0
$28$	−3.18296	−0.601522
$29$	−10.1830	−1.89093	−0.945464	−	0.325727i	$-0.894391\pi$
$29$	−10.1830	−1.89093	−0.945464	+	0.325727i	$0.894391\pi$
$30$	0	0
$31$	2.43439	0.437230	0.218615	−	0.975811i	$-0.429846\pi$
$31$	2.43439	0.437230	0.218615	+	0.975811i	$0.429846\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	−10.1312	−1.71249
$36$	0	0
$37$	−4.74857	−0.780659	−0.390330	−	0.920675i	$-0.627639\pi$
$37$	−4.74857	−0.780659	−0.390330	+	0.920675i	$0.627639\pi$
$38$	−4.74857	−0.770319
$39$	0	0
$40$	−3.18296	−0.503270
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−3.18296	−0.479849
$45$	0	0
$46$	−6.93152	−1.02200
$47$	−5.61735	−0.819375	−0.409687	−	0.912226i	$-0.634362\pi$
$47$	−5.61735	−0.819375	−0.409687	+	0.912226i	$0.634362\pi$
$48$	0	0
$49$	3.13122	0.447317
$50$	−5.13122	−0.725664
$51$	0	0
$52$	−2.56561	−0.355786
$53$	−12.2974	−1.68918	−0.844591	−	0.535411i	$-0.820157\pi$
$53$	−12.2974	−1.68918	−0.844591	+	0.535411i	$0.820157\pi$
$54$	0	0
$55$	−10.1312	−1.36609
$56$	3.18296	0.425341
$57$	0	0
$58$	10.1830	1.33709
$59$	−13.4971	−1.75718	−0.878588	−	0.477580i	$-0.841514\pi$
$59$	−13.4971	−1.75718	−0.878588	+	0.477580i	$0.841514\pi$
$60$	0	0
$61$	−9.87978	−1.26498	−0.632488	−	0.774570i	$-0.717966\pi$
$61$	−9.87978	−1.26498	−0.632488	+	0.774570i	$0.717966\pi$
$62$	−2.43439	−0.309168
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.16622	−1.01289
$66$	0	0
$67$	5.49713	0.671581	0.335791	−	0.941937i	$-0.390997\pi$
$67$	5.49713	0.671581	0.335791	+	0.941937i	$0.390997\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	10.1312	1.21091
$71$	8.24570	0.978584	0.489292	−	0.872120i	$-0.337255\pi$
$71$	8.24570	0.978584	0.489292	+	0.872120i	$0.337255\pi$
$72$	0	0
$73$	11.6283	1.36100	0.680498	−	0.732750i	$-0.261764\pi$
$73$	11.6283	1.36100	0.680498	+	0.732750i	$0.261764\pi$
$74$	4.74857	0.552009
$75$	0	0
$76$	4.74857	0.544698
$77$	10.1312	1.15456
$78$	0	0
$79$	−7.31417	−0.822909	−0.411454	−	0.911430i	$-0.634979\pi$
$79$	−7.31417	−0.822909	−0.411454	+	0.911430i	$0.634979\pi$
$80$	3.18296	0.355865
$81$	0	0
$82$	1.00000	0.110432
$83$	−8.87978	−0.974683	−0.487341	−	0.873212i	$-0.662033\pi$
$83$	−8.87978	−0.974683	−0.487341	+	0.873212i	$0.662033\pi$
$84$	0	0
$85$	−6.36591	−0.690480
$86$	−4.00000	−0.431331
$87$	0	0
$88$	3.18296	0.339304
$89$	−16.3659	−1.73478	−0.867392	−	0.497626i	$-0.834205\pi$
$89$	−16.3659	−1.73478	−0.867392	+	0.497626i	$0.834205\pi$
$90$	0	0
$91$	8.16622	0.856053
$92$	6.93152	0.722661
$93$	0	0
$94$	5.61735	0.579385
$95$	15.1145	1.55071
$96$	0	0
$97$	8.98326	0.912112	0.456056	−	0.889951i	$-0.349262\pi$
$97$	8.98326	0.912112	0.456056	+	0.889951i	$0.349262\pi$
$98$	−3.13122	−0.316301
$99$	0	0
$100$	5.13122	0.513122
$101$	−15.7486	−1.56704	−0.783520	−	0.621366i	$-0.786578\pi$
$101$	−15.7486	−1.56704	−0.783520	+	0.621366i	$0.786578\pi$
$102$	0	0
$103$	6.86878	0.676801	0.338401	−	0.941002i	$-0.390114\pi$
$103$	6.86878	0.676801	0.338401	+	0.941002i	$0.390114\pi$
$104$	2.56561	0.251579
$105$	0	0
$106$	12.2974	1.19443
$107$	7.06274	0.682781	0.341390	−	0.939922i	$-0.389102\pi$
$107$	7.06274	0.682781	0.341390	+	0.939922i	$0.389102\pi$
$108$	0	0
$109$	−12.5656	−1.20357	−0.601783	−	0.798659i	$-0.705543\pi$
$109$	−12.5656	−1.20357	−0.601783	+	0.798659i	$0.705543\pi$
$110$	10.1312	0.965974
$111$	0	0
$112$	−3.18296	−0.300761
$113$	9.13122	0.858993	0.429496	−	0.903069i	$-0.358691\pi$
$113$	9.13122	0.858993	0.429496	+	0.903069i	$0.358691\pi$
$114$	0	0
$115$	22.0627	2.05736
$116$	−10.1830	−0.945464
$117$	0	0
$118$	13.4971	1.24251
$119$	6.36591	0.583562
$120$	0	0
$121$	−0.868784	−0.0789804
$122$	9.87978	0.894473
$123$	0	0
$124$	2.43439	0.218615
$125$	0.417655	0.0373562
$126$	0	0
$127$	−5.38265	−0.477633	−0.238817	−	0.971065i	$-0.576759\pi$
$127$	−5.38265	−0.477633	−0.238817	+	0.971065i	$0.576759\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	8.16622	0.716225
$131$	−4.69682	−0.410363	−0.205182	−	0.978724i	$-0.565779\pi$
$131$	−4.69682	−0.410363	−0.205182	+	0.978724i	$0.565779\pi$
$132$	0	0
$133$	−15.1145	−1.31059
$134$	−5.49713	−0.474880
$135$	0	0
$136$	2.00000	0.171499
$137$	12.1662	1.03943	0.519715	−	0.854340i	$-0.326038\pi$
$137$	12.1662	1.03943	0.519715	+	0.854340i	$0.326038\pi$
$138$	0	0
$139$	−11.6968	−0.992112	−0.496056	−	0.868291i	$-0.665219\pi$
$139$	−11.6968	−0.992112	−0.496056	+	0.868291i	$0.665219\pi$
$140$	−10.1312	−0.856244
$141$	0	0
$142$	−8.24570	−0.691963
$143$	8.16622	0.682894
$144$	0	0
$145$	−32.4119	−2.69166
$146$	−11.6283	−0.962369
$147$	0	0
$148$	−4.74857	−0.390330
$149$	−15.2457	−1.24898	−0.624488	−	0.781034i	$-0.714692\pi$
$149$	−15.2457	−1.24898	−0.624488	+	0.781034i	$0.714692\pi$
$150$	0	0
$151$	14.8630	1.20954	0.604769	−	0.796401i	$-0.293266\pi$
$151$	14.8630	1.20954	0.604769	+	0.796401i	$0.293266\pi$
$152$	−4.74857	−0.385159
$153$	0	0
$154$	−10.1312	−0.816397
$155$	7.74857	0.622380
$156$	0	0
$157$	0.365914	0.0292031	0.0146016	−	0.999893i	$-0.495352\pi$
$157$	0.365914	0.0292031	0.0146016	+	0.999893i	$0.495352\pi$
$158$	7.31417	0.581884
$159$	0	0
$160$	−3.18296	−0.251635
$161$	−22.0627	−1.73879
$162$	0	0
$163$	−13.2974	−1.04154	−0.520768	−	0.853698i	$-0.674354\pi$
$163$	−13.2974	−1.04154	−0.520768	+	0.853698i	$0.674354\pi$
$164$	−1.00000	−0.0780869
$165$	0	0
$166$	8.87978	0.689205
$167$	−3.13122	−0.242301	−0.121150	−	0.992634i	$-0.538658\pi$
$167$	−3.13122	−0.242301	−0.121150	+	0.992634i	$0.538658\pi$
$168$	0	0
$169$	−6.41766	−0.493666
$170$	6.36591	0.488243
$171$	0	0
$172$	4.00000	0.304997
$173$	4.39365	0.334043	0.167021	−	0.985953i	$-0.446585\pi$
$173$	4.39365	0.334043	0.167021	+	0.985953i	$0.446585\pi$
$174$	0	0
$175$	−16.3324	−1.23462
$176$	−3.18296	−0.239924
$177$	0	0
$178$	16.3659	1.22668
$179$	17.8113	1.33128	0.665640	−	0.746273i	$-0.268159\pi$
$179$	17.8113	1.33128	0.665640	+	0.746273i	$0.268159\pi$
$180$	0	0
$181$	12.7946	0.951013	0.475506	−	0.879712i	$-0.342265\pi$
$181$	12.7946	0.951013	0.475506	+	0.879712i	$0.342265\pi$
$182$	−8.16622	−0.605321
$183$	0	0
$184$	−6.93152	−0.510999
$185$	−15.1145	−1.11124
$186$	0	0
$187$	6.36591	0.465522
$188$	−5.61735	−0.409687
$189$	0	0
$190$	−15.1145	−1.09652
$191$	4.86878	0.352293	0.176146	−	0.984364i	$-0.443637\pi$
$191$	4.86878	0.352293	0.176146	+	0.984364i	$0.443637\pi$
$192$	0	0
$193$	−25.5321	−1.83784	−0.918922	−	0.394440i	$-0.870939\pi$
$193$	−25.5321	−1.83784	−0.918922	+	0.394440i	$0.870939\pi$
$194$	−8.98326	−0.644961
$195$	0	0
$196$	3.13122	0.223658
$197$	10.6801	0.760925	0.380462	−	0.924796i	$-0.375765\pi$
$197$	10.6801	0.760925	0.380462	+	0.924796i	$0.375765\pi$
$198$	0	0
$199$	21.1255	1.49755	0.748773	−	0.662827i	$-0.230643\pi$
$199$	21.1255	1.49755	0.748773	+	0.662827i	$0.230643\pi$
$200$	−5.13122	−0.362832
$201$	0	0
$202$	15.7486	1.10807
$203$	32.4119	2.27487
$204$	0	0
$205$	−3.18296	−0.222307
$206$	−6.86878	−0.478571
$207$	0	0
$208$	−2.56561	−0.177893
$209$	−15.1145	−1.04549
$210$	0	0
$211$	−5.25143	−0.361524	−0.180762	−	0.983527i	$-0.557856\pi$
$211$	−5.25143	−0.361524	−0.180762	+	0.983527i	$0.557856\pi$
$212$	−12.2974	−0.844591
$213$	0	0
$214$	−7.06274	−0.482799
$215$	12.7318	0.868303
$216$	0	0
$217$	−7.74857	−0.526007
$218$	12.5656	0.851050
$219$	0	0
$220$	−10.1312	−0.683046
$221$	5.13122	0.345163
$222$	0	0
$223$	−17.0977	−1.14495	−0.572475	−	0.819922i	$-0.694017\pi$
$223$	−17.0977	−1.14495	−0.572475	+	0.819922i	$0.694017\pi$
$224$	3.18296	0.212670
$225$	0	0
$226$	−9.13122	−0.607399
$227$	3.68009	0.244256	0.122128	−	0.992514i	$-0.461028\pi$
$227$	3.68009	0.244256	0.122128	+	0.992514i	$0.461028\pi$
$228$	0	0
$229$	−16.3659	−1.08149	−0.540745	−	0.841187i	$-0.681858\pi$
$229$	−16.3659	−1.08149	−0.540745	+	0.841187i	$0.681858\pi$
$230$	−22.0627	−1.45477
$231$	0	0
$232$	10.1830	0.668544
$233$	−11.8630	−0.777174	−0.388587	−	0.921412i	$-0.627037\pi$
$233$	−11.8630	−0.777174	−0.388587	+	0.921412i	$0.627037\pi$
$234$	0	0
$235$	−17.8798	−1.16635
$236$	−13.4971	−0.878588
$237$	0	0
$238$	−6.36591	−0.412641
$239$	−12.7318	−0.823554	−0.411777	−	0.911285i	$-0.635092\pi$
$239$	−12.7318	−0.823554	−0.411777	+	0.911285i	$0.635092\pi$
$240$	0	0
$241$	28.5489	1.83899	0.919497	−	0.393096i	$-0.128596\pi$
$241$	28.5489	1.83899	0.919497	+	0.393096i	$0.128596\pi$
$242$	0.868784	0.0558475
$243$	0	0
$244$	−9.87978	−0.632488
$245$	9.96653	0.636738
$246$	0	0
$247$	−12.1830	−0.775183
$248$	−2.43439	−0.154584
$249$	0	0
$250$	−0.417655	−0.0264148
$251$	−17.6801	−1.11596	−0.557979	−	0.829855i	$-0.688423\pi$
$251$	−17.6801	−1.11596	−0.557979	+	0.829855i	$0.688423\pi$
$252$	0	0
$253$	−22.0627	−1.38707
$254$	5.38265	0.337738
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.0627	1.50099	0.750496	−	0.660875i	$-0.229815\pi$
$257$	24.0627	1.50099	0.750496	+	0.660875i	$0.229815\pi$
$258$	0	0
$259$	15.1145	0.939168
$260$	−8.16622	−0.506447
$261$	0	0
$262$	4.69682	0.290171
$263$	0.731829	0.0451265	0.0225632	−	0.999745i	$-0.492817\pi$
$263$	0.731829	0.0451265	0.0225632	+	0.999745i	$0.492817\pi$
$264$	0	0
$265$	−39.1422	−2.40449
$266$	15.1145	0.926728
$267$	0	0
$268$	5.49713	0.335791
$269$	0.582345	0.0355062	0.0177531	−	0.999842i	$-0.494349\pi$
$269$	0.582345	0.0355062	0.0177531	+	0.999842i	$0.494349\pi$
$270$	0	0
$271$	9.34918	0.567922	0.283961	−	0.958836i	$-0.408351\pi$
$271$	9.34918	0.567922	0.283961	+	0.958836i	$0.408351\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−12.1662	−0.734988
$275$	−16.3324	−0.984883
$276$	0	0
$277$	5.98326	0.359499	0.179750	−	0.983712i	$-0.442471\pi$
$277$	5.98326	0.359499	0.179750	+	0.983712i	$0.442471\pi$
$278$	11.6968	0.701529
$279$	0	0
$280$	10.1312	0.605456
$281$	14.5321	0.866914	0.433457	−	0.901174i	$-0.357294\pi$
$281$	14.5321	0.866914	0.433457	+	0.901174i	$0.357294\pi$
$282$	0	0
$283$	−4.19969	−0.249646	−0.124823	−	0.992179i	$-0.539836\pi$
$283$	−4.19969	−0.249646	−0.124823	+	0.992179i	$0.539836\pi$
$284$	8.24570	0.489292
$285$	0	0
$286$	−8.16622	−0.482879
$287$	3.18296	0.187884
$288$	0	0
$289$	−13.0000	−0.764706
$290$	32.4119	1.90329
$291$	0	0
$292$	11.6283	0.680498
$293$	8.10348	0.473410	0.236705	−	0.971582i	$-0.423932\pi$
$293$	8.10348	0.473410	0.236705	+	0.971582i	$0.423932\pi$
$294$	0	0
$295$	−42.9608	−2.50127
$296$	4.74857	0.276005
$297$	0	0
$298$	15.2457	0.883159
$299$	−17.7836	−1.02845
$300$	0	0
$301$	−12.7318	−0.733850
$302$	−14.8630	−0.855272
$303$	0	0
$304$	4.74857	0.272349
$305$	−31.4469	−1.80065
$306$	0	0
$307$	6.82804	0.389697	0.194848	−	0.980833i	$-0.437579\pi$
$307$	6.82804	0.389697	0.194848	+	0.980833i	$0.437579\pi$
$308$	10.1312	0.577280
$309$	0	0
$310$	−7.74857	−0.440089
$311$	−23.7428	−1.34633	−0.673166	−	0.739491i	$-0.735066\pi$
$311$	−23.7428	−1.34633	−0.673166	+	0.739491i	$0.735066\pi$
$312$	0	0
$313$	−1.93152	−0.109176	−0.0545880	−	0.998509i	$-0.517385\pi$
$313$	−1.93152	−0.109176	−0.0545880	+	0.998509i	$0.517385\pi$
$314$	−0.365914	−0.0206497
$315$	0	0
$316$	−7.31417	−0.411454
$317$	−19.6911	−1.10596	−0.552981	−	0.833194i	$-0.686510\pi$
$317$	−19.6911	−1.10596	−0.552981	+	0.833194i	$0.686510\pi$
$318$	0	0
$319$	32.4119	1.81472
$320$	3.18296	0.177933
$321$	0	0
$322$	22.0627	1.22951
$323$	−9.49713	−0.528434
$324$	0	0
$325$	−13.1647	−0.730246
$326$	13.2974	0.736477
$327$	0	0
$328$	1.00000	0.0552158
$329$	17.8798	0.985744
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−8.87978	−0.487341
$333$	0	0
$334$	3.13122	0.171333
$335$	17.4971	0.955970
$336$	0	0
$337$	18.2864	0.996126	0.498063	−	0.867141i	$-0.334045\pi$
$337$	18.2864	0.996126	0.498063	+	0.867141i	$0.334045\pi$
$338$	6.41766	0.349074
$339$	0	0
$340$	−6.36591	−0.345240
$341$	−7.74857	−0.419608
$342$	0	0
$343$	12.3142	0.664903
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−4.39365	−0.236204
$347$	−16.0977	−0.864172	−0.432086	−	0.901832i	$-0.642222\pi$
$347$	−16.0977	−0.864172	−0.432086	+	0.901832i	$0.642222\pi$
$348$	0	0
$349$	1.87978	0.100622	0.0503112	−	0.998734i	$-0.483979\pi$
$349$	1.87978	0.100622	0.0503112	+	0.998734i	$0.483979\pi$
$350$	16.3324	0.873006
$351$	0	0
$352$	3.18296	0.169652
$353$	−6.74857	−0.359190	−0.179595	−	0.983741i	$-0.557479\pi$
$353$	−6.74857	−0.359190	−0.179595	+	0.983741i	$0.557479\pi$
$354$	0	0
$355$	26.2457	1.39298
$356$	−16.3659	−0.867392
$357$	0	0
$358$	−17.8113	−0.941357
$359$	−6.19969	−0.327207	−0.163604	−	0.986526i	$-0.552312\pi$
$359$	−6.19969	−0.327207	−0.163604	+	0.986526i	$0.552312\pi$
$360$	0	0
$361$	3.54887	0.186783
$362$	−12.7946	−0.672468
$363$	0	0
$364$	8.16622	0.428026
$365$	37.0125	1.93732
$366$	0	0
$367$	30.9833	1.61731	0.808657	−	0.588281i	$-0.200195\pi$
$367$	30.9833	1.61731	0.808657	+	0.588281i	$0.200195\pi$
$368$	6.93152	0.361331
$369$	0	0
$370$	15.1145	0.785764
$371$	39.1422	2.03216
$372$	0	0
$373$	17.7428	0.918689	0.459344	−	0.888258i	$-0.348084\pi$
$373$	17.7428	0.918689	0.459344	+	0.888258i	$0.348084\pi$
$374$	−6.36591	−0.329174
$375$	0	0
$376$	5.61735	0.289693
$377$	26.1255	1.34553
$378$	0	0
$379$	24.6283	1.26507	0.632537	−	0.774530i	$-0.282014\pi$
$379$	24.6283	1.26507	0.632537	+	0.774530i	$0.282014\pi$
$380$	15.1145	0.775356
$381$	0	0
$382$	−4.86878	−0.249109
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	32.2472	1.64347
$386$	25.5321	1.29955
$387$	0	0
$388$	8.98326	0.456056
$389$	10.4971	0.532226	0.266113	−	0.963942i	$-0.414261\pi$
$389$	10.4971	0.532226	0.266113	+	0.963942i	$0.414261\pi$
$390$	0	0
$391$	−13.8630	−0.701084
$392$	−3.13122	−0.158150
$393$	0	0
$394$	−10.6801	−0.538055
$395$	−23.2807	−1.17138
$396$	0	0
$397$	−34.4287	−1.72792	−0.863962	−	0.503557i	$-0.832025\pi$
$397$	−34.4287	−1.72792	−0.863962	+	0.503557i	$0.832025\pi$
$398$	−21.1255	−1.05892
$399$	0	0
$400$	5.13122	0.256561
$401$	−21.9665	−1.09696	−0.548478	−	0.836165i	$-0.684793\pi$
$401$	−21.9665	−1.09696	−0.548478	+	0.836165i	$0.684793\pi$
$402$	0	0
$403$	−6.24570	−0.311120
$404$	−15.7486	−0.783520
$405$	0	0
$406$	−32.4119	−1.60858
$407$	15.1145	0.749197
$408$	0	0
$409$	−1.31991	−0.0652655	−0.0326327	−	0.999467i	$-0.510389\pi$
$409$	−1.31991	−0.0652655	−0.0326327	+	0.999467i	$0.510389\pi$
$410$	3.18296	0.157195
$411$	0	0
$412$	6.86878	0.338401
$413$	42.9608	2.11396
$414$	0	0
$415$	−28.2640	−1.38742
$416$	2.56561	0.125789
$417$	0	0
$418$	15.1145	0.739273
$419$	−3.57661	−0.174729	−0.0873643	−	0.996176i	$-0.527844\pi$
$419$	−3.57661	−0.174729	−0.0873643	+	0.996176i	$0.527844\pi$
$420$	0	0
$421$	36.4287	1.77542	0.887712	−	0.460399i	$-0.152294\pi$
$421$	36.4287	1.77542	0.887712	+	0.460399i	$0.152294\pi$
$422$	5.25143	0.255636
$423$	0	0
$424$	12.2974	0.597216
$425$	−10.2624	−0.497801
$426$	0	0
$427$	31.4469	1.52182
$428$	7.06274	0.341390
$429$	0	0
$430$	−12.7318	−0.613983
$431$	−18.6911	−0.900318	−0.450159	−	0.892948i	$-0.648633\pi$
$431$	−18.6911	−0.900318	−0.450159	+	0.892948i	$0.648633\pi$
$432$	0	0
$433$	27.9943	1.34532	0.672659	−	0.739952i	$-0.265152\pi$
$433$	27.9943	1.34532	0.672659	+	0.739952i	$0.265152\pi$
$434$	7.74857	0.371943
$435$	0	0
$436$	−12.5656	−0.601783
$437$	32.9148	1.57453
$438$	0	0
$439$	−23.0920	−1.10212	−0.551061	−	0.834465i	$-0.685777\pi$
$439$	−23.0920	−1.10212	−0.551061	+	0.834465i	$0.685777\pi$
$440$	10.1312	0.482987
$441$	0	0
$442$	−5.13122	−0.244067
$443$	−19.7596	−0.938805	−0.469403	−	0.882984i	$-0.655531\pi$
$443$	−19.7596	−0.938805	−0.469403	+	0.882984i	$0.655531\pi$
$444$	0	0
$445$	−52.0920	−2.46940
$446$	17.0977	0.809602
$447$	0	0
$448$	−3.18296	−0.150381
$449$	−2.12548	−0.100307	−0.0501537	−	0.998742i	$-0.515971\pi$
$449$	−2.12548	−0.100307	−0.0501537	+	0.998742i	$0.515971\pi$
$450$	0	0
$451$	3.18296	0.149880
$452$	9.13122	0.429496
$453$	0	0
$454$	−3.68009	−0.172715
$455$	25.9927	1.21856
$456$	0	0
$457$	7.06274	0.330381	0.165190	−	0.986262i	$-0.447176\pi$
$457$	7.06274	0.330381	0.165190	+	0.986262i	$0.447176\pi$
$458$	16.3659	0.764729
$459$	0	0
$460$	22.0627	1.02868
$461$	26.8630	1.25114	0.625568	−	0.780169i	$-0.284867\pi$
$461$	26.8630	1.25114	0.625568	+	0.780169i	$0.284867\pi$
$462$	0	0
$463$	17.9943	0.836264	0.418132	−	0.908386i	$-0.362685\pi$
$463$	17.9943	0.836264	0.418132	+	0.908386i	$0.362685\pi$
$464$	−10.1830	−0.472732
$465$	0	0
$466$	11.8630	0.549545
$467$	12.6173	0.583861	0.291931	−	0.956439i	$-0.405702\pi$
$467$	12.6173	0.583861	0.291931	+	0.956439i	$0.405702\pi$
$468$	0	0
$469$	−17.4971	−0.807942
$470$	17.8798	0.824733
$471$	0	0
$472$	13.4971	0.621256
$473$	−12.7318	−0.585410
$474$	0	0
$475$	24.3659	1.11798
$476$	6.36591	0.291781
$477$	0	0
$478$	12.7318	0.582340
$479$	−8.48613	−0.387741	−0.193871	−	0.981027i	$-0.562104\pi$
$479$	−8.48613	−0.387741	−0.193871	+	0.981027i	$0.562104\pi$
$480$	0	0
$481$	12.1830	0.555495
$482$	−28.5489	−1.30037
$483$	0	0
$484$	−0.868784	−0.0394902
$485$	28.5933	1.29836
$486$	0	0
$487$	−10.0460	−0.455228	−0.227614	−	0.973751i	$-0.573092\pi$
$487$	−10.0460	−0.455228	−0.227614	+	0.973751i	$0.573092\pi$
$488$	9.87978	0.447237
$489$	0	0
$490$	−9.96653	−0.450242
$491$	7.34918	0.331664	0.165832	−	0.986154i	$-0.446969\pi$
$491$	7.34918	0.331664	0.165832	+	0.986154i	$0.446969\pi$
$492$	0	0
$493$	20.3659	0.917235
$494$	12.1830	0.548137
$495$	0	0
$496$	2.43439	0.109307
$497$	−26.2457	−1.17728
$498$	0	0
$499$	−12.0167	−0.537943	−0.268972	−	0.963148i	$-0.586684\pi$
$499$	−12.0167	−0.537943	−0.268972	+	0.963148i	$0.586684\pi$
$500$	0.417655	0.0186781
$501$	0	0
$502$	17.6801	0.789101
$503$	−25.3602	−1.13075	−0.565377	−	0.824833i	$-0.691269\pi$
$503$	−25.3602	−1.13075	−0.565377	+	0.824833i	$0.691269\pi$
$504$	0	0
$505$	−50.1270	−2.23062
$506$	22.0627	0.980808
$507$	0	0
$508$	−5.38265	−0.238817
$509$	23.4287	1.03846	0.519228	−	0.854636i	$-0.326219\pi$
$509$	23.4287	1.03846	0.519228	+	0.854636i	$0.326219\pi$
$510$	0	0
$511$	−37.0125	−1.63734
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−24.0627	−1.06136
$515$	21.8630	0.963401
$516$	0	0
$517$	17.8798	0.786352
$518$	−15.1145	−0.664092
$519$	0	0
$520$	8.16622	0.358112
$521$	41.9885	1.83955	0.919775	−	0.392445i	$-0.128371\pi$
$521$	41.9885	1.83955	0.919775	+	0.392445i	$0.128371\pi$
$522$	0	0
$523$	−20.8573	−0.912026	−0.456013	−	0.889973i	$-0.650723\pi$
$523$	−20.8573	−0.912026	−0.456013	+	0.889973i	$0.650723\pi$
$524$	−4.69682	−0.205182
$525$	0	0
$526$	−0.731829	−0.0319092
$527$	−4.86878	−0.212088
$528$	0	0
$529$	25.0460	1.08896
$530$	39.1422	1.70023
$531$	0	0
$532$	−15.1145	−0.655296
$533$	2.56561	0.111129
$534$	0	0
$535$	22.4804	0.971912
$536$	−5.49713	−0.237440
$537$	0	0
$538$	−0.582345	−0.0251067
$539$	−9.96653	−0.429289
$540$	0	0
$541$	−25.7041	−1.10511	−0.552553	−	0.833478i	$-0.686346\pi$
$541$	−25.7041	−1.10511	−0.552553	+	0.833478i	$0.686346\pi$
$542$	−9.34918	−0.401582
$543$	0	0
$544$	2.00000	0.0857493
$545$	−39.9958	−1.71323
$546$	0	0
$547$	12.1035	0.517508	0.258754	−	0.965943i	$-0.416688\pi$
$547$	12.1035	0.517508	0.258754	+	0.965943i	$0.416688\pi$
$548$	12.1662	0.519715
$549$	0	0
$550$	16.3324	0.696418
$551$	−48.3544	−2.05997
$552$	0	0
$553$	23.2807	0.989996
$554$	−5.98326	−0.254205
$555$	0	0
$556$	−11.6968	−0.496056
$557$	40.1145	1.69971	0.849853	−	0.527021i	$-0.176691\pi$
$557$	40.1145	1.69971	0.849853	+	0.527021i	$0.176691\pi$
$558$	0	0
$559$	−10.2624	−0.434055
$560$	−10.1312	−0.428122
$561$	0	0
$562$	−14.5321	−0.613001
$563$	30.1312	1.26988	0.634940	−	0.772562i	$-0.281025\pi$
$563$	30.1312	1.26988	0.634940	+	0.772562i	$0.281025\pi$
$564$	0	0
$565$	29.0643	1.22274
$566$	4.19969	0.176526
$567$	0	0
$568$	−8.24570	−0.345982
$569$	−29.1145	−1.22054	−0.610271	−	0.792193i	$-0.708940\pi$
$569$	−29.1145	−1.22054	−0.610271	+	0.792193i	$0.708940\pi$
$570$	0	0
$571$	41.3769	1.73157	0.865785	−	0.500416i	$-0.166820\pi$
$571$	41.3769	1.73157	0.865785	+	0.500416i	$0.166820\pi$
$572$	8.16622	0.341447
$573$	0	0
$574$	−3.18296	−0.132854
$575$	35.5671	1.48325
$576$	0	0
$577$	4.50287	0.187457	0.0937285	−	0.995598i	$-0.470121\pi$
$577$	4.50287	0.187457	0.0937285	+	0.995598i	$0.470121\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	−32.4119	−1.34583
$581$	28.2640	1.17259
$582$	0	0
$583$	39.1422	1.62110
$584$	−11.6283	−0.481184
$585$	0	0
$586$	−8.10348	−0.334752
$587$	−19.9148	−0.821971	−0.410986	−	0.911642i	$-0.634815\pi$
$587$	−19.9148	−0.821971	−0.410986	+	0.911642i	$0.634815\pi$
$588$	0	0
$589$	11.5599	0.476316
$590$	42.9608	1.76867
$591$	0	0
$592$	−4.74857	−0.195165
$593$	26.8688	1.10337	0.551684	−	0.834053i	$-0.313985\pi$
$593$	26.8688	1.10337	0.551684	+	0.834053i	$0.313985\pi$
$594$	0	0
$595$	20.2624	0.830679
$596$	−15.2457	−0.624488
$597$	0	0
$598$	17.7836	0.727224
$599$	−14.1035	−0.576253	−0.288126	−	0.957592i	$-0.593032\pi$
$599$	−14.1035	−0.576253	−0.288126	+	0.957592i	$0.593032\pi$
$600$	0	0
$601$	−12.3549	−0.503968	−0.251984	−	0.967731i	$-0.581083\pi$
$601$	−12.3549	−0.503968	−0.251984	+	0.967731i	$0.581083\pi$
$602$	12.7318	0.518911
$603$	0	0
$604$	14.8630	0.604769
$605$	−2.76530	−0.112426
$606$	0	0
$607$	39.3602	1.59758	0.798790	−	0.601610i	$-0.205474\pi$
$607$	39.3602	1.59758	0.798790	+	0.601610i	$0.205474\pi$
$608$	−4.74857	−0.192580
$609$	0	0
$610$	31.4469	1.27325
$611$	14.4119	0.583044
$612$	0	0
$613$	4.73183	0.191117	0.0955584	−	0.995424i	$-0.469536\pi$
$613$	4.73183	0.191117	0.0955584	+	0.995424i	$0.469536\pi$
$614$	−6.82804	−0.275557
$615$	0	0
$616$	−10.1312	−0.408198
$617$	−20.0220	−0.806055	−0.403028	−	0.915188i	$-0.632042\pi$
$617$	−20.0220	−0.806055	−0.403028	+	0.915188i	$0.632042\pi$
$618$	0	0
$619$	−36.0293	−1.44814	−0.724069	−	0.689727i	$-0.757730\pi$
$619$	−36.0293	−1.44814	−0.724069	+	0.689727i	$0.757730\pi$
$620$	7.74857	0.311190
$621$	0	0
$622$	23.7428	0.952001
$623$	52.0920	2.08702
$624$	0	0
$625$	−24.3267	−0.973068
$626$	1.93152	0.0771992
$627$	0	0
$628$	0.365914	0.0146016
$629$	9.49713	0.378675
$630$	0	0
$631$	−7.50813	−0.298894	−0.149447	−	0.988770i	$-0.547749\pi$
$631$	−7.50813	−0.298894	−0.149447	+	0.988770i	$0.547749\pi$
$632$	7.31417	0.290942
$633$	0	0
$634$	19.6911	0.782033
$635$	−17.1327	−0.679892
$636$	0	0
$637$	−8.03347	−0.318298
$638$	−32.4119	−1.28320
$639$	0	0
$640$	−3.18296	−0.125817
$641$	−3.84105	−0.151712	−0.0758562	−	0.997119i	$-0.524169\pi$
$641$	−3.84105	−0.151712	−0.0758562	+	0.997119i	$0.524169\pi$
$642$	0	0
$643$	−29.6059	−1.16754	−0.583771	−	0.811919i	$-0.698423\pi$
$643$	−29.6059	−1.16754	−0.583771	+	0.811919i	$0.698423\pi$
$644$	−22.0627	−0.869394
$645$	0	0
$646$	9.49713	0.373660
$647$	−31.8223	−1.25106	−0.625532	−	0.780199i	$-0.715118\pi$
$647$	−31.8223	−1.25106	−0.625532	+	0.780199i	$0.715118\pi$
$648$	0	0
$649$	42.9608	1.68636
$650$	13.1647	0.516362
$651$	0	0
$652$	−13.2974	−0.520768
$653$	−5.50813	−0.215550	−0.107775	−	0.994175i	$-0.534373\pi$
$653$	−5.50813	−0.215550	−0.107775	+	0.994175i	$0.534373\pi$
$654$	0	0
$655$	−14.9498	−0.584137
$656$	−1.00000	−0.0390434
$657$	0	0
$658$	−17.8798	−0.697026
$659$	15.0240	0.585252	0.292626	−	0.956227i	$-0.405471\pi$
$659$	15.0240	0.585252	0.292626	+	0.956227i	$0.405471\pi$
$660$	0	0
$661$	−4.50813	−0.175346	−0.0876729	−	0.996149i	$-0.527943\pi$
$661$	−4.50813	−0.175346	−0.0876729	+	0.996149i	$0.527943\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	8.87978	0.344602
$665$	−48.1087	−1.86558
$666$	0	0
$667$	−70.5834	−2.73300
$668$	−3.13122	−0.121150
$669$	0	0
$670$	−17.4971	−0.675973
$671$	31.4469	1.21399
$672$	0	0
$673$	−31.3952	−1.21020	−0.605098	−	0.796151i	$-0.706866\pi$
$673$	−31.3952	−1.21020	−0.605098	+	0.796151i	$0.706866\pi$
$674$	−18.2864	−0.704367
$675$	0	0
$676$	−6.41766	−0.246833
$677$	−19.0517	−0.732218	−0.366109	−	0.930572i	$-0.619310\pi$
$677$	−19.0517	−0.732218	−0.366109	+	0.930572i	$0.619310\pi$
$678$	0	0
$679$	−28.5933	−1.09731
$680$	6.36591	0.244122
$681$	0	0
$682$	7.74857	0.296708
$683$	31.4062	1.20172	0.600862	−	0.799353i	$-0.294824\pi$
$683$	31.4062	1.20172	0.600862	+	0.799353i	$0.294824\pi$
$684$	0	0
$685$	38.7246	1.47959
$686$	−12.3142	−0.470157
$687$	0	0
$688$	4.00000	0.152499
$689$	31.5504	1.20197
$690$	0	0
$691$	36.0867	1.37280	0.686402	−	0.727222i	$-0.259189\pi$
$691$	36.0867	1.37280	0.686402	+	0.727222i	$0.259189\pi$
$692$	4.39365	0.167021
$693$	0	0
$694$	16.0977	0.611062
$695$	−37.2305	−1.41223
$696$	0	0
$697$	2.00000	0.0757554
$698$	−1.87978	−0.0711508
$699$	0	0
$700$	−16.3324	−0.617308
$701$	−35.2750	−1.33232	−0.666158	−	0.745810i	$-0.732063\pi$
$701$	−35.2750	−1.33232	−0.666158	+	0.745810i	$0.732063\pi$
$702$	0	0
$703$	−22.5489	−0.850447
$704$	−3.18296	−0.119962
$705$	0	0
$706$	6.74857	0.253986
$707$	50.1270	1.88522
$708$	0	0
$709$	−10.1662	−0.381800	−0.190900	−	0.981609i	$-0.561141\pi$
$709$	−10.1662	−0.381800	−0.190900	+	0.981609i	$0.561141\pi$
$710$	−26.2457	−0.984983
$711$	0	0
$712$	16.3659	0.613339
$713$	16.8740	0.631938
$714$	0	0
$715$	25.9927	0.972073
$716$	17.8113	0.665640
$717$	0	0
$718$	6.19969	0.231371
$719$	−33.2012	−1.23820	−0.619098	−	0.785313i	$-0.712502\pi$
$719$	−33.2012	−1.23820	−0.619098	+	0.785313i	$0.712502\pi$
$720$	0	0
$721$	−21.8630	−0.814222
$722$	−3.54887	−0.132075
$723$	0	0
$724$	12.7946	0.475506
$725$	−52.2510	−1.94055
$726$	0	0
$727$	−19.1035	−0.708509	−0.354254	−	0.935149i	$-0.615265\pi$
$727$	−19.1035	−0.708509	−0.354254	+	0.935149i	$0.615265\pi$
$728$	−8.16622	−0.302660
$729$	0	0
$730$	−37.0125	−1.36990
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−9.51387	−0.351403	−0.175701	−	0.984444i	$-0.556219\pi$
$733$	−9.51387	−0.351403	−0.175701	+	0.984444i	$0.556219\pi$
$734$	−30.9833	−1.14361
$735$	0	0
$736$	−6.93152	−0.255499
$737$	−17.4971	−0.644515
$738$	0	0
$739$	15.2974	0.562725	0.281363	−	0.959602i	$-0.409214\pi$
$739$	15.2974	0.562725	0.281363	+	0.959602i	$0.409214\pi$
$740$	−15.1145	−0.555619
$741$	0	0
$742$	−39.1422	−1.43696
$743$	50.0513	1.83620	0.918101	−	0.396346i	$-0.129722\pi$
$743$	50.0513	1.83620	0.918101	+	0.396346i	$0.129722\pi$
$744$	0	0
$745$	−48.5264	−1.77787
$746$	−17.7428	−0.649611
$747$	0	0
$748$	6.36591	0.232761
$749$	−22.4804	−0.821416
$750$	0	0
$751$	−2.39365	−0.0873455	−0.0436727	−	0.999046i	$-0.513906\pi$
$751$	−2.39365	−0.0873455	−0.0436727	+	0.999046i	$0.513906\pi$
$752$	−5.61735	−0.204844
$753$	0	0
$754$	−26.1255	−0.951434
$755$	47.3084	1.72173
$756$	0	0
$757$	47.4857	1.72590	0.862948	−	0.505293i	$-0.168616\pi$
$757$	47.4857	1.72590	0.862948	+	0.505293i	$0.168616\pi$
$758$	−24.6283	−0.894542
$759$	0	0
$760$	−15.1145	−0.548260
$761$	44.3377	1.60724	0.803620	−	0.595143i	$-0.202904\pi$
$761$	44.3377	1.60724	0.803620	+	0.595143i	$0.202904\pi$
$762$	0	0
$763$	39.9958	1.44794
$764$	4.86878	0.176146
$765$	0	0
$766$	−12.0000	−0.433578
$767$	34.6283	1.25036
$768$	0	0
$769$	−19.6524	−0.708682	−0.354341	−	0.935116i	$-0.615295\pi$
$769$	−19.6524	−0.708682	−0.354341	+	0.935116i	$0.615295\pi$
$770$	−32.2472	−1.16211
$771$	0	0
$772$	−25.5321	−0.918922
$773$	6.36591	0.228966	0.114483	−	0.993425i	$-0.463479\pi$
$773$	6.36591	0.228966	0.114483	+	0.993425i	$0.463479\pi$
$774$	0	0
$775$	12.4914	0.448704
$776$	−8.98326	−0.322480
$777$	0	0
$778$	−10.4971	−0.376340
$779$	−4.74857	−0.170135
$780$	0	0
$781$	−26.2457	−0.939145
$782$	13.8630	0.495741
$783$	0	0
$784$	3.13122	0.111829
$785$	1.16469	0.0415695
$786$	0	0
$787$	−43.7261	−1.55867	−0.779333	−	0.626610i	$-0.784442\pi$
$787$	−43.7261	−1.55867	−0.779333	+	0.626610i	$0.784442\pi$
$788$	10.6801	0.380462
$789$	0	0
$790$	23.2807	0.828290
$791$	−29.0643	−1.03341
$792$	0	0
$793$	25.3476	0.900121
$794$	34.4287	1.22183
$795$	0	0
$796$	21.1255	0.748773
$797$	35.7204	1.26528	0.632640	−	0.774446i	$-0.281971\pi$
$797$	35.7204	1.26528	0.632640	+	0.774446i	$0.281971\pi$
$798$	0	0
$799$	11.2347	0.397455
$800$	−5.13122	−0.181416
$801$	0	0
$802$	21.9665	0.775665
$803$	−37.0125	−1.30614
$804$	0	0
$805$	−70.2248	−2.47510
$806$	6.24570	0.219995
$807$	0	0
$808$	15.7486	0.554033
$809$	36.1662	1.27154	0.635768	−	0.771880i	$-0.280683\pi$
$809$	36.1662	1.27154	0.635768	+	0.771880i	$0.280683\pi$
$810$	0	0
$811$	34.8353	1.22323	0.611617	−	0.791154i	$-0.290520\pi$
$811$	34.8353	1.22323	0.611617	+	0.791154i	$0.290520\pi$
$812$	32.4119	1.13744
$813$	0	0
$814$	−15.1145	−0.529762
$815$	−42.3252	−1.48259
$816$	0	0
$817$	18.9943	0.664525
$818$	1.31991	0.0461497
$819$	0	0
$820$	−3.18296	−0.111154
$821$	7.25669	0.253260	0.126630	−	0.991950i	$-0.459584\pi$
$821$	7.25669	0.253260	0.126630	+	0.991950i	$0.459584\pi$
$822$	0	0
$823$	47.0700	1.64076	0.820379	−	0.571821i	$-0.193763\pi$
$823$	47.0700	1.64076	0.820379	+	0.571821i	$0.193763\pi$
$824$	−6.86878	−0.239285
$825$	0	0
$826$	−42.9608	−1.49480
$827$	−46.6743	−1.62303	−0.811513	−	0.584334i	$-0.801356\pi$
$827$	−46.6743	−1.62303	−0.811513	+	0.584334i	$0.801356\pi$
$828$	0	0
$829$	19.6173	0.681339	0.340669	−	0.940183i	$-0.389346\pi$
$829$	19.6173	0.681339	0.340669	+	0.940183i	$0.389346\pi$
$830$	28.2640	0.981056
$831$	0	0
$832$	−2.56561	−0.0889465
$833$	−6.26243	−0.216980
$834$	0	0
$835$	−9.96653	−0.344906
$836$	−15.1145	−0.522745
$837$	0	0
$838$	3.57661	0.123552
$839$	1.23470	0.0426265	0.0213133	−	0.999773i	$-0.493215\pi$
$839$	1.23470	0.0426265	0.0213133	+	0.999773i	$0.493215\pi$
$840$	0	0
$841$	74.6926	2.57561
$842$	−36.4287	−1.25541
$843$	0	0
$844$	−5.25143	−0.180762
$845$	−20.4271	−0.702714
$846$	0	0
$847$	2.76530	0.0950169
$848$	−12.2974	−0.422296
$849$	0	0
$850$	10.2624	0.351999
$851$	−32.9148	−1.12830
$852$	0	0
$853$	−39.7428	−1.36077	−0.680384	−	0.732856i	$-0.738187\pi$
$853$	−39.7428	−1.36077	−0.680384	+	0.732856i	$0.738187\pi$
$854$	−31.4469	−1.07609
$855$	0	0
$856$	−7.06274	−0.241399
$857$	−28.9775	−0.989853	−0.494927	−	0.868935i	$-0.664805\pi$
$857$	−28.9775	−0.989853	−0.494927	+	0.868935i	$0.664805\pi$
$858$	0	0
$859$	−13.6226	−0.464797	−0.232399	−	0.972621i	$-0.574657\pi$
$859$	−13.6226	−0.464797	−0.232399	+	0.972621i	$0.574657\pi$
$860$	12.7318	0.434152
$861$	0	0
$862$	18.6911	0.636621
$863$	53.3267	1.81526	0.907631	−	0.419769i	$-0.137889\pi$
$863$	53.3267	1.81526	0.907631	+	0.419769i	$0.137889\pi$
$864$	0	0
$865$	13.9848	0.475497
$866$	−27.9943	−0.951284
$867$	0	0
$868$	−7.74857	−0.263003
$869$	23.2807	0.789744
$870$	0	0
$871$	−14.1035	−0.477878
$872$	12.5656	0.425525
$873$	0	0
$874$	−32.9148	−1.11336
$875$	−1.32938	−0.0449412
$876$	0	0
$877$	15.6561	0.528668	0.264334	−	0.964431i	$-0.414848\pi$
$877$	15.6561	0.528668	0.264334	+	0.964431i	$0.414848\pi$
$878$	23.0920	0.779317
$879$	0	0
$880$	−10.1312	−0.341523
$881$	−10.3659	−0.349237	−0.174618	−	0.984636i	$-0.555869\pi$
$881$	−10.3659	−0.349237	−0.174618	+	0.984636i	$0.555869\pi$
$882$	0	0
$883$	36.2510	1.21994	0.609971	−	0.792424i	$-0.291181\pi$
$883$	36.2510	1.21994	0.609971	+	0.792424i	$0.291181\pi$
$884$	5.13122	0.172581
$885$	0	0
$886$	19.7596	0.663835
$887$	−36.9608	−1.24102	−0.620511	−	0.784198i	$-0.713075\pi$
$887$	−36.9608	−1.24102	−0.620511	+	0.784198i	$0.713075\pi$
$888$	0	0
$889$	17.1327	0.574614
$890$	52.0920	1.74613
$891$	0	0
$892$	−17.0977	−0.572475
$893$	−26.6743	−0.892623
$894$	0	0
$895$	56.6926	1.89503
$896$	3.18296	0.106335
$897$	0	0
$898$	2.12548	0.0709281
$899$	−24.7893	−0.826770
$900$	0	0
$901$	24.5949	0.819374
$902$	−3.18296	−0.105981
$903$	0	0
$904$	−9.13122	−0.303700
$905$	40.7246	1.35373
$906$	0	0
$907$	−7.33091	−0.243419	−0.121709	−	0.992566i	$-0.538838\pi$
$907$	−7.33091	−0.243419	−0.121709	+	0.992566i	$0.538838\pi$
$908$	3.68009	0.122128
$909$	0	0
$910$	−25.9927	−0.861651
$911$	−49.6926	−1.64639	−0.823195	−	0.567759i	$-0.807811\pi$
$911$	−49.6926	−1.64639	−0.823195	+	0.567759i	$0.807811\pi$
$912$	0	0
$913$	28.2640	0.935401
$914$	−7.06274	−0.233615
$915$	0	0
$916$	−16.3659	−0.540745
$917$	14.9498	0.493686
$918$	0	0
$919$	2.29218	0.0756120	0.0378060	−	0.999285i	$-0.487963\pi$
$919$	2.29218	0.0756120	0.0378060	+	0.999285i	$0.487963\pi$
$920$	−22.0627	−0.727387
$921$	0	0
$922$	−26.8630	−0.884687
$923$	−21.1552	−0.696333
$924$	0	0
$925$	−24.3659	−0.801146
$926$	−17.9943	−0.591328
$927$	0	0
$928$	10.1830	0.334272
$929$	51.7554	1.69804	0.849019	−	0.528362i	$-0.177194\pi$
$929$	51.7554	1.69804	0.849019	+	0.528362i	$0.177194\pi$
$930$	0	0
$931$	14.8688	0.487305
$932$	−11.8630	−0.388587
$933$	0	0
$934$	−12.6173	−0.412852
$935$	20.2624	0.662652
$936$	0	0
$937$	−14.9593	−0.488698	−0.244349	−	0.969687i	$-0.578574\pi$
$937$	−14.9593	−0.488698	−0.244349	+	0.969687i	$0.578574\pi$
$938$	17.4971	0.571302
$939$	0	0
$940$	−17.8798	−0.583174
$941$	24.4397	0.796710	0.398355	−	0.917231i	$-0.369581\pi$
$941$	24.4397	0.796710	0.398355	+	0.917231i	$0.369581\pi$
$942$	0	0
$943$	−6.93152	−0.225721
$944$	−13.4971	−0.439294
$945$	0	0
$946$	12.7318	0.413947
$947$	11.9890	0.389590	0.194795	−	0.980844i	$-0.437596\pi$
$947$	11.9890	0.389590	0.194795	+	0.980844i	$0.437596\pi$
$948$	0	0
$949$	−29.8338	−0.968445
$950$	−24.3659	−0.790535
$951$	0	0
$952$	−6.36591	−0.206320
$953$	14.3994	0.466442	0.233221	−	0.972424i	$-0.425073\pi$
$953$	14.3994	0.466442	0.233221	+	0.972424i	$0.425073\pi$
$954$	0	0
$955$	15.4971	0.501475
$956$	−12.7318	−0.411777
$957$	0	0
$958$	8.48613	0.274175
$959$	−38.7246	−1.25048
$960$	0	0
$961$	−25.0737	−0.808830
$962$	−12.1830	−0.392794
$963$	0	0
$964$	28.5489	0.919497
$965$	−81.2677	−2.61610
$966$	0	0
$967$	−11.8353	−0.380598	−0.190299	−	0.981726i	$-0.560946\pi$
$967$	−11.8353	−0.380598	−0.190299	+	0.981726i	$0.560946\pi$
$968$	0.868784	0.0279238
$969$	0	0
$970$	−28.5933	−0.918077
$971$	15.4914	0.497142	0.248571	−	0.968614i	$-0.420039\pi$
$971$	15.4914	0.497142	0.248571	+	0.968614i	$0.420039\pi$
$972$	0	0
$973$	37.2305	1.19355
$974$	10.0460	0.321895
$975$	0	0
$976$	−9.87978	−0.316244
$977$	−49.3894	−1.58011	−0.790054	−	0.613037i	$-0.789948\pi$
$977$	−49.3894	−1.58011	−0.790054	+	0.613037i	$0.789948\pi$
$978$	0	0
$979$	52.0920	1.66487
$980$	9.96653	0.318369
$981$	0	0
$982$	−7.34918	−0.234522
$983$	−10.7026	−0.341359	−0.170679	−	0.985327i	$-0.554596\pi$
$983$	−10.7026	−0.341359	−0.170679	+	0.985327i	$0.554596\pi$
$984$	0	0
$985$	33.9943	1.08315
$986$	−20.3659	−0.648583
$987$	0	0
$988$	−12.1830	−0.387591
$989$	27.7261	0.881638
$990$	0	0
$991$	33.6409	1.06864	0.534319	−	0.845283i	$-0.320568\pi$
$991$	33.6409	1.06864	0.534319	+	0.845283i	$0.320568\pi$
$992$	−2.43439	−0.0772920
$993$	0	0
$994$	26.2457	0.832463
$995$	67.2415	2.13170
$996$	0	0
$997$	3.42291	0.108405	0.0542024	−	0.998530i	$-0.482738\pi$
$997$	3.42291	0.108405	0.0542024	+	0.998530i	$0.482738\pi$
$998$	12.0167	0.380383
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2214.2.a.m.1.3	✓	3
3.2	odd	2		2214.2.a.p.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2214.2.a.m.1.3	✓	3	1.1	even	1	trivial
2214.2.a.p.1.1	yes	3	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:	\( N \)	\(=\)	\( 2214 = 2 \cdot 3^{3} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2214.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.18296 q^{5} -3.18296 q^{7} -1.00000 q^{8} -3.18296 q^{10} -3.18296 q^{11} -2.56561 q^{13} +3.18296 q^{14} +1.00000 q^{16} -2.00000 q^{17} +4.74857 q^{19} +3.18296 q^{20} +3.18296 q^{22} +6.93152 q^{23} +5.13122 q^{25} +2.56561 q^{26} -3.18296 q^{28} -10.1830 q^{29} +2.43439 q^{31} -1.00000 q^{32} +2.00000 q^{34} -10.1312 q^{35} -4.74857 q^{37} -4.74857 q^{38} -3.18296 q^{40} -1.00000 q^{41} +4.00000 q^{43} -3.18296 q^{44} -6.93152 q^{46} -5.61735 q^{47} +3.13122 q^{49} -5.13122 q^{50} -2.56561 q^{52} -12.2974 q^{53} -10.1312 q^{55} +3.18296 q^{56} +10.1830 q^{58} -13.4971 q^{59} -9.87978 q^{61} -2.43439 q^{62} +1.00000 q^{64} -8.16622 q^{65} +5.49713 q^{67} -2.00000 q^{68} +10.1312 q^{70} +8.24570 q^{71} +11.6283 q^{73} +4.74857 q^{74} +4.74857 q^{76} +10.1312 q^{77} -7.31417 q^{79} +3.18296 q^{80} +1.00000 q^{82} -8.87978 q^{83} -6.36591 q^{85} -4.00000 q^{86} +3.18296 q^{88} -16.3659 q^{89} +8.16622 q^{91} +6.93152 q^{92} +5.61735 q^{94} +15.1145 q^{95} +8.98326 q^{97} -3.13122 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - q^{5} + q^{7} - 3 q^{8} + q^{10} + q^{11} - 6 q^{13} - q^{14} + 3 q^{16} - 6 q^{17} + 2 q^{19} - q^{20} - q^{22} - 2 q^{23} + 12 q^{25} + 6 q^{26} + q^{28} - 20 q^{29}+ \cdots - 6 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - q^5 + q^7 - 3 * q^8 + q^10 + q^11 - 6 * q^13 - q^14 + 3 * q^16 - 6 * q^17 + 2 * q^19 - q^20 - q^22 - 2 * q^23 + 12 * q^25 + 6 * q^26 + q^28 - 20 * q^29 + 9 * q^31 - 3 * q^32 + 6 * q^34 - 27 * q^35 - 2 * q^37 - 2 * q^38 + q^40 - 3 * q^41 + 12 * q^43 + q^44 + 2 * q^46 - 8 * q^47 + 6 * q^49 - 12 * q^50 - 6 * q^52 + 7 * q^53 - 27 * q^55 - q^56 + 20 * q^58 - 16 * q^59 - 14 * q^61 - 9 * q^62 + 3 * q^64 + 16 * q^65 - 8 * q^67 - 6 * q^68 + 27 * q^70 - 12 * q^71 + 7 * q^73 + 2 * q^74 + 2 * q^76 + 27 * q^77 - 8 * q^79 - q^80 + 3 * q^82 - 11 * q^83 + 2 * q^85 - 12 * q^86 - q^88 - 28 * q^89 - 16 * q^91 - 2 * q^92 + 8 * q^94 + 12 * q^95 - 3 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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