LMFDB - Embedded newform 1386.2.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1386,2,Mod(1,1386)] 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("1386.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1386, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,0,0,2,2,0,0,-2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1386.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +3.46410 q^{10} -1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +5.46410 q^{19} +3.46410 q^{20} -1.00000 q^{22} -6.92820 q^{23} +7.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} +5.46410 q^{31} +1.00000 q^{32} -3.46410 q^{34} +3.46410 q^{35} -4.92820 q^{37} +5.46410 q^{38} +3.46410 q^{40} -3.46410 q^{41} -10.9282 q^{43} -1.00000 q^{44} -6.92820 q^{46} -9.46410 q^{47} +1.00000 q^{49} +7.00000 q^{50} +2.00000 q^{52} +0.928203 q^{53} -3.46410 q^{55} +1.00000 q^{56} +6.00000 q^{58} -6.92820 q^{59} +2.00000 q^{61} +5.46410 q^{62} +1.00000 q^{64} +6.92820 q^{65} +14.9282 q^{67} -3.46410 q^{68} +3.46410 q^{70} +12.0000 q^{71} -0.535898 q^{73} -4.92820 q^{74} +5.46410 q^{76} -1.00000 q^{77} -10.9282 q^{79} +3.46410 q^{80} -3.46410 q^{82} -4.39230 q^{83} -12.0000 q^{85} -10.9282 q^{86} -1.00000 q^{88} +0.928203 q^{89} +2.00000 q^{91} -6.92820 q^{92} -9.46410 q^{94} +18.9282 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} - 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{19} - 2 q^{22} + 14 q^{25} + 4 q^{26} + 2 q^{28} + 12 q^{29} + 4 q^{31} + 2 q^{32} + 4 q^{37} + 4 q^{38} - 8 q^{43}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 - 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^19 - 2 * q^22 + 14 * q^25 + 4 * q^26 + 2 * q^28 + 12 * q^29 + 4 * q^31 + 2 * q^32 + 4 * q^37 + 4 * q^38 - 8 * q^43 - 2 * q^44 - 12 * q^47 + 2 * q^49 + 14 * q^50 + 4 * q^52 - 12 * q^53 + 2 * q^56 + 12 * q^58 + 4 * q^61 + 4 * q^62 + 2 * q^64 + 16 * q^67 + 24 * q^71 - 8 * q^73 + 4 * q^74 + 4 * q^76 - 2 * q^77 - 8 * q^79 + 12 * q^83 - 24 * q^85 - 8 * q^86 - 2 * q^88 - 12 * q^89 + 4 * q^91 - 12 * q^94 + 24 * q^95 + 4 * q^97 + 2 * 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
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	3.46410	1.09545
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	3.46410	0.774597
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	2.00000	0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.46410	−0.594089
$35$	3.46410	0.585540
$36$	0	0
$37$	−4.92820	−0.810192	−0.405096	−	0.914274i	$-0.632762\pi$
$37$	−4.92820	−0.810192	−0.405096	+	0.914274i	$0.632762\pi$
$38$	5.46410	0.886394
$39$	0	0
$40$	3.46410	0.547723
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−10.9282	−1.66654	−0.833268	−	0.552870i	$-0.813533\pi$
$43$	−10.9282	−1.66654	−0.833268	+	0.552870i	$0.813533\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−6.92820	−1.02151
$47$	−9.46410	−1.38048	−0.690241	−	0.723580i	$-0.742495\pi$
$47$	−9.46410	−1.38048	−0.690241	+	0.723580i	$0.742495\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.00000	0.989949
$51$	0	0
$52$	2.00000	0.277350
$53$	0.928203	0.127499	0.0637493	−	0.997966i	$-0.479694\pi$
$53$	0.928203	0.127499	0.0637493	+	0.997966i	$0.479694\pi$
$54$	0	0
$55$	−3.46410	−0.467099
$56$	1.00000	0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	5.46410	0.693942
$63$	0	0
$64$	1.00000	0.125000
$65$	6.92820	0.859338
$66$	0	0
$67$	14.9282	1.82377	0.911885	−	0.410445i	$-0.134627\pi$
$67$	14.9282	1.82377	0.911885	+	0.410445i	$0.134627\pi$
$68$	−3.46410	−0.420084
$69$	0	0
$70$	3.46410	0.414039
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−0.535898	−0.0627222	−0.0313611	−	0.999508i	$-0.509984\pi$
$73$	−0.535898	−0.0627222	−0.0313611	+	0.999508i	$0.509984\pi$
$74$	−4.92820	−0.572892
$75$	0	0
$76$	5.46410	0.626775
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−10.9282	−1.22952	−0.614759	−	0.788715i	$-0.710747\pi$
$79$	−10.9282	−1.22952	−0.614759	+	0.788715i	$0.710747\pi$
$80$	3.46410	0.387298
$81$	0	0
$82$	−3.46410	−0.382546
$83$	−4.39230	−0.482118	−0.241059	−	0.970510i	$-0.577495\pi$
$83$	−4.39230	−0.482118	−0.241059	+	0.970510i	$0.577495\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	−10.9282	−1.17842
$87$	0	0
$88$	−1.00000	−0.106600
$89$	0.928203	0.0983893	0.0491947	−	0.998789i	$-0.484335\pi$
$89$	0.928203	0.0983893	0.0491947	+	0.998789i	$0.484335\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−6.92820	−0.722315
$93$	0	0
$94$	−9.46410	−0.976148
$95$	18.9282	1.94199
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	7.00000	0.700000
$101$	19.8564	1.97579	0.987893	−	0.155136i	$-0.0495815\pi$
$101$	19.8564	1.97579	0.987893	+	0.155136i	$0.0495815\pi$
$102$	0	0
$103$	−13.4641	−1.32666	−0.663329	−	0.748328i	$-0.730857\pi$
$103$	−13.4641	−1.32666	−0.663329	+	0.748328i	$0.730857\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	0.928203	0.0901551
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	15.8564	1.51877	0.759384	−	0.650643i	$-0.225500\pi$
$109$	15.8564	1.51877	0.759384	+	0.650643i	$0.225500\pi$
$110$	−3.46410	−0.330289
$111$	0	0
$112$	1.00000	0.0944911
$113$	−7.85641	−0.739069	−0.369534	−	0.929217i	$-0.620483\pi$
$113$	−7.85641	−0.739069	−0.369534	+	0.929217i	$0.620483\pi$
$114$	0	0
$115$	−24.0000	−2.23801
$116$	6.00000	0.557086
$117$	0	0
$118$	−6.92820	−0.637793
$119$	−3.46410	−0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	0	0
$124$	5.46410	0.490691
$125$	6.92820	0.619677
$126$	0	0
$127$	−10.9282	−0.969721	−0.484861	−	0.874591i	$-0.661130\pi$
$127$	−10.9282	−0.969721	−0.484861	+	0.874591i	$0.661130\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	6.92820	0.607644
$131$	4.39230	0.383757	0.191879	−	0.981419i	$-0.438542\pi$
$131$	4.39230	0.383757	0.191879	+	0.981419i	$0.438542\pi$
$132$	0	0
$133$	5.46410	0.473798
$134$	14.9282	1.28960
$135$	0	0
$136$	−3.46410	−0.297044
$137$	−7.85641	−0.671218	−0.335609	−	0.942001i	$-0.608942\pi$
$137$	−7.85641	−0.671218	−0.335609	+	0.942001i	$0.608942\pi$
$138$	0	0
$139$	−13.4641	−1.14201	−0.571005	−	0.820947i	$-0.693446\pi$
$139$	−13.4641	−1.14201	−0.571005	+	0.820947i	$0.693446\pi$
$140$	3.46410	0.292770
$141$	0	0
$142$	12.0000	1.00702
$143$	−2.00000	−0.167248
$144$	0	0
$145$	20.7846	1.72607
$146$	−0.535898	−0.0443513
$147$	0	0
$148$	−4.92820	−0.405096
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	5.46410	0.443197
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	18.9282	1.52035
$156$	0	0
$157$	−2.39230	−0.190927	−0.0954634	−	0.995433i	$-0.530433\pi$
$157$	−2.39230	−0.190927	−0.0954634	+	0.995433i	$0.530433\pi$
$158$	−10.9282	−0.869401
$159$	0	0
$160$	3.46410	0.273861
$161$	−6.92820	−0.546019
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	−4.39230	−0.340909
$167$	−18.9282	−1.46471	−0.732354	−	0.680924i	$-0.761578\pi$
$167$	−18.9282	−1.46471	−0.732354	+	0.680924i	$0.761578\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	0	0
$172$	−10.9282	−0.833268
$173$	0.928203	0.0705700	0.0352850	−	0.999377i	$-0.488766\pi$
$173$	0.928203	0.0705700	0.0352850	+	0.999377i	$0.488766\pi$
$174$	0	0
$175$	7.00000	0.529150
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	0.928203	0.0695718
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	6.39230	0.475136	0.237568	−	0.971371i	$-0.423650\pi$
$181$	6.39230	0.475136	0.237568	+	0.971371i	$0.423650\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	−6.92820	−0.510754
$185$	−17.0718	−1.25514
$186$	0	0
$187$	3.46410	0.253320
$188$	−9.46410	−0.690241
$189$	0	0
$190$	18.9282	1.37320
$191$	−20.7846	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$191$	−20.7846	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$192$	0	0
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	0.392305	0.0278098	0.0139049	−	0.999903i	$-0.495574\pi$
$199$	0.392305	0.0278098	0.0139049	+	0.999903i	$0.495574\pi$
$200$	7.00000	0.494975
$201$	0	0
$202$	19.8564	1.39709
$203$	6.00000	0.421117
$204$	0	0
$205$	−12.0000	−0.838116
$206$	−13.4641	−0.938088
$207$	0	0
$208$	2.00000	0.138675
$209$	−5.46410	−0.377960
$210$	0	0
$211$	16.7846	1.15550	0.577750	−	0.816214i	$-0.303931\pi$
$211$	16.7846	1.15550	0.577750	+	0.816214i	$0.303931\pi$
$212$	0.928203	0.0637493
$213$	0	0
$214$	6.92820	0.473602
$215$	−37.8564	−2.58179
$216$	0	0
$217$	5.46410	0.370927
$218$	15.8564	1.07393
$219$	0	0
$220$	−3.46410	−0.233550
$221$	−6.92820	−0.466041
$222$	0	0
$223$	−8.39230	−0.561990	−0.280995	−	0.959709i	$-0.590664\pi$
$223$	−8.39230	−0.561990	−0.280995	+	0.959709i	$0.590664\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−7.85641	−0.522600
$227$	4.39230	0.291528	0.145764	−	0.989319i	$-0.453436\pi$
$227$	4.39230	0.291528	0.145764	+	0.989319i	$0.453436\pi$
$228$	0	0
$229$	−21.3205	−1.40890	−0.704449	−	0.709754i	$-0.748806\pi$
$229$	−21.3205	−1.40890	−0.704449	+	0.709754i	$0.748806\pi$
$230$	−24.0000	−1.58251
$231$	0	0
$232$	6.00000	0.393919
$233$	7.85641	0.514690	0.257345	−	0.966320i	$-0.417152\pi$
$233$	7.85641	0.514690	0.257345	+	0.966320i	$0.417152\pi$
$234$	0	0
$235$	−32.7846	−2.13863
$236$	−6.92820	−0.450988
$237$	0	0
$238$	−3.46410	−0.224544
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	9.60770	0.618886	0.309443	−	0.950918i	$-0.399857\pi$
$241$	9.60770	0.618886	0.309443	+	0.950918i	$0.399857\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	2.00000	0.128037
$245$	3.46410	0.221313
$246$	0	0
$247$	10.9282	0.695345
$248$	5.46410	0.346971
$249$	0	0
$250$	6.92820	0.438178
$251$	−30.9282	−1.95217	−0.976085	−	0.217387i	$-0.930247\pi$
$251$	−30.9282	−1.95217	−0.976085	+	0.217387i	$0.930247\pi$
$252$	0	0
$253$	6.92820	0.435572
$254$	−10.9282	−0.685696
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.9282	−0.806439	−0.403220	−	0.915103i	$-0.632109\pi$
$257$	−12.9282	−0.806439	−0.403220	+	0.915103i	$0.632109\pi$
$258$	0	0
$259$	−4.92820	−0.306224
$260$	6.92820	0.429669
$261$	0	0
$262$	4.39230	0.271357
$263$	5.07180	0.312740	0.156370	−	0.987699i	$-0.450021\pi$
$263$	5.07180	0.312740	0.156370	+	0.987699i	$0.450021\pi$
$264$	0	0
$265$	3.21539	0.197520
$266$	5.46410	0.335026
$267$	0	0
$268$	14.9282	0.911885
$269$	−24.2487	−1.47847	−0.739235	−	0.673448i	$-0.764813\pi$
$269$	−24.2487	−1.47847	−0.739235	+	0.673448i	$0.764813\pi$
$270$	0	0
$271$	−24.7846	−1.50556	−0.752779	−	0.658273i	$-0.771287\pi$
$271$	−24.7846	−1.50556	−0.752779	+	0.658273i	$0.771287\pi$
$272$	−3.46410	−0.210042
$273$	0	0
$274$	−7.85641	−0.474623
$275$	−7.00000	−0.422116
$276$	0	0
$277$	−11.8564	−0.712382	−0.356191	−	0.934413i	$-0.615925\pi$
$277$	−11.8564	−0.712382	−0.356191	+	0.934413i	$0.615925\pi$
$278$	−13.4641	−0.807523
$279$	0	0
$280$	3.46410	0.207020
$281$	7.85641	0.468674	0.234337	−	0.972155i	$-0.424708\pi$
$281$	7.85641	0.468674	0.234337	+	0.972155i	$0.424708\pi$
$282$	0	0
$283$	−13.4641	−0.800358	−0.400179	−	0.916437i	$-0.631052\pi$
$283$	−13.4641	−0.800358	−0.400179	+	0.916437i	$0.631052\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−3.46410	−0.204479
$288$	0	0
$289$	−5.00000	−0.294118
$290$	20.7846	1.22051
$291$	0	0
$292$	−0.535898	−0.0313611
$293$	24.9282	1.45632	0.728161	−	0.685407i	$-0.240375\pi$
$293$	24.9282	1.45632	0.728161	+	0.685407i	$0.240375\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	−4.92820	−0.286446
$297$	0	0
$298$	6.00000	0.347571
$299$	−13.8564	−0.801337
$300$	0	0
$301$	−10.9282	−0.629891
$302$	−16.0000	−0.920697
$303$	0	0
$304$	5.46410	0.313388
$305$	6.92820	0.396708
$306$	0	0
$307$	10.5359	0.601315	0.300658	−	0.953732i	$-0.402794\pi$
$307$	10.5359	0.601315	0.300658	+	0.953732i	$0.402794\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	18.9282	1.07505
$311$	−28.3923	−1.60998	−0.804990	−	0.593288i	$-0.797829\pi$
$311$	−28.3923	−1.60998	−0.804990	+	0.593288i	$0.797829\pi$
$312$	0	0
$313$	−16.9282	−0.956839	−0.478419	−	0.878132i	$-0.658790\pi$
$313$	−16.9282	−0.956839	−0.478419	+	0.878132i	$0.658790\pi$
$314$	−2.39230	−0.135006
$315$	0	0
$316$	−10.9282	−0.614759
$317$	−12.9282	−0.726120	−0.363060	−	0.931766i	$-0.618268\pi$
$317$	−12.9282	−0.726120	−0.363060	+	0.931766i	$0.618268\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	3.46410	0.193649
$321$	0	0
$322$	−6.92820	−0.386094
$323$	−18.9282	−1.05319
$324$	0	0
$325$	14.0000	0.776580
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−3.46410	−0.191273
$329$	−9.46410	−0.521773
$330$	0	0
$331$	−9.07180	−0.498631	−0.249316	−	0.968422i	$-0.580206\pi$
$331$	−9.07180	−0.498631	−0.249316	+	0.968422i	$0.580206\pi$
$332$	−4.39230	−0.241059
$333$	0	0
$334$	−18.9282	−1.03571
$335$	51.7128	2.82537
$336$	0	0
$337$	20.9282	1.14003	0.570016	−	0.821634i	$-0.306937\pi$
$337$	20.9282	1.14003	0.570016	+	0.821634i	$0.306937\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	−12.0000	−0.650791
$341$	−5.46410	−0.295898
$342$	0	0
$343$	1.00000	0.0539949
$344$	−10.9282	−0.589209
$345$	0	0
$346$	0.928203	0.0499005
$347$	20.7846	1.11578	0.557888	−	0.829916i	$-0.311612\pi$
$347$	20.7846	1.11578	0.557888	+	0.829916i	$0.311612\pi$
$348$	0	0
$349$	10.7846	0.577287	0.288643	−	0.957437i	$-0.406796\pi$
$349$	10.7846	0.577287	0.288643	+	0.957437i	$0.406796\pi$
$350$	7.00000	0.374166
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−12.9282	−0.688099	−0.344049	−	0.938952i	$-0.611799\pi$
$353$	−12.9282	−0.688099	−0.344049	+	0.938952i	$0.611799\pi$
$354$	0	0
$355$	41.5692	2.20627
$356$	0.928203	0.0491947
$357$	0	0
$358$	20.7846	1.09850
$359$	5.07180	0.267679	0.133840	−	0.991003i	$-0.457269\pi$
$359$	5.07180	0.267679	0.133840	+	0.991003i	$0.457269\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	6.39230	0.335972
$363$	0	0
$364$	2.00000	0.104828
$365$	−1.85641	−0.0971688
$366$	0	0
$367$	19.3205	1.00852	0.504261	−	0.863551i	$-0.331765\pi$
$367$	19.3205	1.00852	0.504261	+	0.863551i	$0.331765\pi$
$368$	−6.92820	−0.361158
$369$	0	0
$370$	−17.0718	−0.887520
$371$	0.928203	0.0481899
$372$	0	0
$373$	29.7128	1.53847	0.769236	−	0.638965i	$-0.220637\pi$
$373$	29.7128	1.53847	0.769236	+	0.638965i	$0.220637\pi$
$374$	3.46410	0.179124
$375$	0	0
$376$	−9.46410	−0.488074
$377$	12.0000	0.618031
$378$	0	0
$379$	−17.8564	−0.917222	−0.458611	−	0.888637i	$-0.651653\pi$
$379$	−17.8564	−0.917222	−0.458611	+	0.888637i	$0.651653\pi$
$380$	18.9282	0.970996
$381$	0	0
$382$	−20.7846	−1.06343
$383$	−14.5359	−0.742750	−0.371375	−	0.928483i	$-0.621114\pi$
$383$	−14.5359	−0.742750	−0.371375	+	0.928483i	$0.621114\pi$
$384$	0	0
$385$	−3.46410	−0.176547
$386$	26.0000	1.32337
$387$	0	0
$388$	2.00000	0.101535
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	1.00000	0.0505076
$393$	0	0
$394$	−18.0000	−0.906827
$395$	−37.8564	−1.90476
$396$	0	0
$397$	−2.39230	−0.120066	−0.0600332	−	0.998196i	$-0.519121\pi$
$397$	−2.39230	−0.120066	−0.0600332	+	0.998196i	$0.519121\pi$
$398$	0.392305	0.0196645
$399$	0	0
$400$	7.00000	0.350000
$401$	−4.14359	−0.206921	−0.103461	−	0.994634i	$-0.532992\pi$
$401$	−4.14359	−0.206921	−0.103461	+	0.994634i	$0.532992\pi$
$402$	0	0
$403$	10.9282	0.544373
$404$	19.8564	0.987893
$405$	0	0
$406$	6.00000	0.297775
$407$	4.92820	0.244282
$408$	0	0
$409$	−9.32051	−0.460869	−0.230435	−	0.973088i	$-0.574015\pi$
$409$	−9.32051	−0.460869	−0.230435	+	0.973088i	$0.574015\pi$
$410$	−12.0000	−0.592638
$411$	0	0
$412$	−13.4641	−0.663329
$413$	−6.92820	−0.340915
$414$	0	0
$415$	−15.2154	−0.746894
$416$	2.00000	0.0980581
$417$	0	0
$418$	−5.46410	−0.267258
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−37.7128	−1.83801	−0.919005	−	0.394246i	$-0.871006\pi$
$421$	−37.7128	−1.83801	−0.919005	+	0.394246i	$0.871006\pi$
$422$	16.7846	0.817062
$423$	0	0
$424$	0.928203	0.0450775
$425$	−24.2487	−1.17624
$426$	0	0
$427$	2.00000	0.0967868
$428$	6.92820	0.334887
$429$	0	0
$430$	−37.8564	−1.82560
$431$	18.9282	0.911739	0.455870	−	0.890047i	$-0.349328\pi$
$431$	18.9282	0.911739	0.455870	+	0.890047i	$0.349328\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	5.46410	0.262285
$435$	0	0
$436$	15.8564	0.759384
$437$	−37.8564	−1.81092
$438$	0	0
$439$	−24.7846	−1.18290	−0.591452	−	0.806340i	$-0.701445\pi$
$439$	−24.7846	−1.18290	−0.591452	+	0.806340i	$0.701445\pi$
$440$	−3.46410	−0.165145
$441$	0	0
$442$	−6.92820	−0.329541
$443$	20.7846	0.987507	0.493753	−	0.869602i	$-0.335625\pi$
$443$	20.7846	0.987507	0.493753	+	0.869602i	$0.335625\pi$
$444$	0	0
$445$	3.21539	0.152424
$446$	−8.39230	−0.397387
$447$	0	0
$448$	1.00000	0.0472456
$449$	19.8564	0.937082	0.468541	−	0.883442i	$-0.344780\pi$
$449$	19.8564	0.937082	0.468541	+	0.883442i	$0.344780\pi$
$450$	0	0
$451$	3.46410	0.163118
$452$	−7.85641	−0.369534
$453$	0	0
$454$	4.39230	0.206141
$455$	6.92820	0.324799
$456$	0	0
$457$	15.8564	0.741731	0.370866	−	0.928687i	$-0.379061\pi$
$457$	15.8564	0.741731	0.370866	+	0.928687i	$0.379061\pi$
$458$	−21.3205	−0.996242
$459$	0	0
$460$	−24.0000	−1.11901
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	7.85641	0.363941
$467$	15.7128	0.727102	0.363551	−	0.931574i	$-0.381564\pi$
$467$	15.7128	0.727102	0.363551	+	0.931574i	$0.381564\pi$
$468$	0	0
$469$	14.9282	0.689320
$470$	−32.7846	−1.51224
$471$	0	0
$472$	−6.92820	−0.318896
$473$	10.9282	0.502479
$474$	0	0
$475$	38.2487	1.75497
$476$	−3.46410	−0.158777
$477$	0	0
$478$	24.0000	1.09773
$479$	13.8564	0.633115	0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564	0.633115	0.316558	+	0.948573i	$0.397473\pi$
$480$	0	0
$481$	−9.85641	−0.449413
$482$	9.60770	0.437619
$483$	0	0
$484$	1.00000	0.0454545
$485$	6.92820	0.314594
$486$	0	0
$487$	40.7846	1.84813	0.924064	−	0.382239i	$-0.124847\pi$
$487$	40.7846	1.84813	0.924064	+	0.382239i	$0.124847\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	3.46410	0.156492
$491$	1.85641	0.0837785	0.0418892	−	0.999122i	$-0.486662\pi$
$491$	1.85641	0.0837785	0.0418892	+	0.999122i	$0.486662\pi$
$492$	0	0
$493$	−20.7846	−0.936092
$494$	10.9282	0.491683
$495$	0	0
$496$	5.46410	0.245345
$497$	12.0000	0.538274
$498$	0	0
$499$	−9.07180	−0.406109	−0.203055	−	0.979167i	$-0.565087\pi$
$499$	−9.07180	−0.406109	−0.203055	+	0.979167i	$0.565087\pi$
$500$	6.92820	0.309839
$501$	0	0
$502$	−30.9282	−1.38039
$503$	−18.9282	−0.843967	−0.421983	−	0.906604i	$-0.638666\pi$
$503$	−18.9282	−0.843967	−0.421983	+	0.906604i	$0.638666\pi$
$504$	0	0
$505$	68.7846	3.06087
$506$	6.92820	0.307996
$507$	0	0
$508$	−10.9282	−0.484861
$509$	41.3205	1.83150	0.915750	−	0.401749i	$-0.131598\pi$
$509$	41.3205	1.83150	0.915750	+	0.401749i	$0.131598\pi$
$510$	0	0
$511$	−0.535898	−0.0237067
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.9282	−0.570239
$515$	−46.6410	−2.05525
$516$	0	0
$517$	9.46410	0.416231
$518$	−4.92820	−0.216533
$519$	0	0
$520$	6.92820	0.303822
$521$	−36.9282	−1.61785	−0.808927	−	0.587909i	$-0.799951\pi$
$521$	−36.9282	−1.61785	−0.808927	+	0.587909i	$0.799951\pi$
$522$	0	0
$523$	24.3923	1.06660	0.533301	−	0.845926i	$-0.320952\pi$
$523$	24.3923	1.06660	0.533301	+	0.845926i	$0.320952\pi$
$524$	4.39230	0.191879
$525$	0	0
$526$	5.07180	0.221141
$527$	−18.9282	−0.824525
$528$	0	0
$529$	25.0000	1.08696
$530$	3.21539	0.139668
$531$	0	0
$532$	5.46410	0.236899
$533$	−6.92820	−0.300094
$534$	0	0
$535$	24.0000	1.03761
$536$	14.9282	0.644800
$537$	0	0
$538$	−24.2487	−1.04544
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−25.7128	−1.10548	−0.552740	−	0.833354i	$-0.686418\pi$
$541$	−25.7128	−1.10548	−0.552740	+	0.833354i	$0.686418\pi$
$542$	−24.7846	−1.06459
$543$	0	0
$544$	−3.46410	−0.148522
$545$	54.9282	2.35287
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	−7.85641	−0.335609
$549$	0	0
$550$	−7.00000	−0.298481
$551$	32.7846	1.39667
$552$	0	0
$553$	−10.9282	−0.464714
$554$	−11.8564	−0.503730
$555$	0	0
$556$	−13.4641	−0.571005
$557$	−21.7128	−0.920001	−0.460001	−	0.887919i	$-0.652151\pi$
$557$	−21.7128	−0.920001	−0.460001	+	0.887919i	$0.652151\pi$
$558$	0	0
$559$	−21.8564	−0.924427
$560$	3.46410	0.146385
$561$	0	0
$562$	7.85641	0.331403
$563$	18.2487	0.769091	0.384546	−	0.923106i	$-0.374358\pi$
$563$	18.2487	0.769091	0.384546	+	0.923106i	$0.374358\pi$
$564$	0	0
$565$	−27.2154	−1.14496
$566$	−13.4641	−0.565938
$567$	0	0
$568$	12.0000	0.503509
$569$	31.8564	1.33549	0.667745	−	0.744390i	$-0.267260\pi$
$569$	31.8564	1.33549	0.667745	+	0.744390i	$0.267260\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	−3.46410	−0.144589
$575$	−48.4974	−2.02248
$576$	0	0
$577$	−30.7846	−1.28158	−0.640790	−	0.767716i	$-0.721393\pi$
$577$	−30.7846	−1.28158	−0.640790	+	0.767716i	$0.721393\pi$
$578$	−5.00000	−0.207973
$579$	0	0
$580$	20.7846	0.863034
$581$	−4.39230	−0.182224
$582$	0	0
$583$	−0.928203	−0.0384422
$584$	−0.535898	−0.0221756
$585$	0	0
$586$	24.9282	1.02977
$587$	20.7846	0.857873	0.428936	−	0.903335i	$-0.358888\pi$
$587$	20.7846	0.857873	0.428936	+	0.903335i	$0.358888\pi$
$588$	0	0
$589$	29.8564	1.23021
$590$	−24.0000	−0.988064
$591$	0	0
$592$	−4.92820	−0.202548
$593$	20.5359	0.843308	0.421654	−	0.906757i	$-0.361450\pi$
$593$	20.5359	0.843308	0.421654	+	0.906757i	$0.361450\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	6.00000	0.245770
$597$	0	0
$598$	−13.8564	−0.566631
$599$	−1.85641	−0.0758507	−0.0379254	−	0.999281i	$-0.512075\pi$
$599$	−1.85641	−0.0758507	−0.0379254	+	0.999281i	$0.512075\pi$
$600$	0	0
$601$	18.3923	0.750238	0.375119	−	0.926977i	$-0.377602\pi$
$601$	18.3923	0.750238	0.375119	+	0.926977i	$0.377602\pi$
$602$	−10.9282	−0.445400
$603$	0	0
$604$	−16.0000	−0.651031
$605$	3.46410	0.140836
$606$	0	0
$607$	−19.7128	−0.800118	−0.400059	−	0.916489i	$-0.631010\pi$
$607$	−19.7128	−0.800118	−0.400059	+	0.916489i	$0.631010\pi$
$608$	5.46410	0.221599
$609$	0	0
$610$	6.92820	0.280515
$611$	−18.9282	−0.765753
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	10.5359	0.425194
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	19.8564	0.799389	0.399694	−	0.916648i	$-0.369116\pi$
$617$	19.8564	0.799389	0.399694	+	0.916648i	$0.369116\pi$
$618$	0	0
$619$	−31.7128	−1.27465	−0.637323	−	0.770597i	$-0.719958\pi$
$619$	−31.7128	−1.27465	−0.637323	+	0.770597i	$0.719958\pi$
$620$	18.9282	0.760175
$621$	0	0
$622$	−28.3923	−1.13843
$623$	0.928203	0.0371877
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−16.9282	−0.676587
$627$	0	0
$628$	−2.39230	−0.0954634
$629$	17.0718	0.680697
$630$	0	0
$631$	−0.784610	−0.0312348	−0.0156174	−	0.999878i	$-0.504971\pi$
$631$	−0.784610	−0.0312348	−0.0156174	+	0.999878i	$0.504971\pi$
$632$	−10.9282	−0.434701
$633$	0	0
$634$	−12.9282	−0.513445
$635$	−37.8564	−1.50229
$636$	0	0
$637$	2.00000	0.0792429
$638$	−6.00000	−0.237542
$639$	0	0
$640$	3.46410	0.136931
$641$	−31.8564	−1.25825	−0.629126	−	0.777303i	$-0.716587\pi$
$641$	−31.8564	−1.25825	−0.629126	+	0.777303i	$0.716587\pi$
$642$	0	0
$643$	14.9282	0.588711	0.294355	−	0.955696i	$-0.404895\pi$
$643$	14.9282	0.588711	0.294355	+	0.955696i	$0.404895\pi$
$644$	−6.92820	−0.273009
$645$	0	0
$646$	−18.9282	−0.744720
$647$	−0.679492	−0.0267136	−0.0133568	−	0.999911i	$-0.504252\pi$
$647$	−0.679492	−0.0267136	−0.0133568	+	0.999911i	$0.504252\pi$
$648$	0	0
$649$	6.92820	0.271956
$650$	14.0000	0.549125
$651$	0	0
$652$	−4.00000	−0.156652
$653$	33.7128	1.31928	0.659642	−	0.751580i	$-0.270708\pi$
$653$	33.7128	1.31928	0.659642	+	0.751580i	$0.270708\pi$
$654$	0	0
$655$	15.2154	0.594514
$656$	−3.46410	−0.135250
$657$	0	0
$658$	−9.46410	−0.368949
$659$	17.0718	0.665023	0.332511	−	0.943099i	$-0.392104\pi$
$659$	17.0718	0.665023	0.332511	+	0.943099i	$0.392104\pi$
$660$	0	0
$661$	44.2487	1.72108	0.860538	−	0.509387i	$-0.170128\pi$
$661$	44.2487	1.72108	0.860538	+	0.509387i	$0.170128\pi$
$662$	−9.07180	−0.352585
$663$	0	0
$664$	−4.39230	−0.170454
$665$	18.9282	0.734004
$666$	0	0
$667$	−41.5692	−1.60957
$668$	−18.9282	−0.732354
$669$	0	0
$670$	51.7128	1.99784
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	20.9282	0.806723	0.403361	−	0.915041i	$-0.367842\pi$
$673$	20.9282	0.806723	0.403361	+	0.915041i	$0.367842\pi$
$674$	20.9282	0.806124
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−2.78461	−0.107021	−0.0535106	−	0.998567i	$-0.517041\pi$
$677$	−2.78461	−0.107021	−0.0535106	+	0.998567i	$0.517041\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	−12.0000	−0.460179
$681$	0	0
$682$	−5.46410	−0.209231
$683$	−3.21539	−0.123033	−0.0615167	−	0.998106i	$-0.519594\pi$
$683$	−3.21539	−0.123033	−0.0615167	+	0.998106i	$0.519594\pi$
$684$	0	0
$685$	−27.2154	−1.03985
$686$	1.00000	0.0381802
$687$	0	0
$688$	−10.9282	−0.416634
$689$	1.85641	0.0707235
$690$	0	0
$691$	−33.0718	−1.25811	−0.629055	−	0.777361i	$-0.716558\pi$
$691$	−33.0718	−1.25811	−0.629055	+	0.777361i	$0.716558\pi$
$692$	0.928203	0.0352850
$693$	0	0
$694$	20.7846	0.788973
$695$	−46.6410	−1.76919
$696$	0	0
$697$	12.0000	0.454532
$698$	10.7846	0.408203
$699$	0	0
$700$	7.00000	0.264575
$701$	−45.7128	−1.72655	−0.863275	−	0.504735i	$-0.831590\pi$
$701$	−45.7128	−1.72655	−0.863275	+	0.504735i	$0.831590\pi$
$702$	0	0
$703$	−26.9282	−1.01562
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−12.9282	−0.486559
$707$	19.8564	0.746777
$708$	0	0
$709$	46.7846	1.75703	0.878516	−	0.477712i	$-0.158534\pi$
$709$	46.7846	1.75703	0.878516	+	0.477712i	$0.158534\pi$
$710$	41.5692	1.56007
$711$	0	0
$712$	0.928203	0.0347859
$713$	−37.8564	−1.41773
$714$	0	0
$715$	−6.92820	−0.259100
$716$	20.7846	0.776757
$717$	0	0
$718$	5.07180	0.189278
$719$	37.1769	1.38646	0.693232	−	0.720714i	$-0.256186\pi$
$719$	37.1769	1.38646	0.693232	+	0.720714i	$0.256186\pi$
$720$	0	0
$721$	−13.4641	−0.501429
$722$	10.8564	0.404034
$723$	0	0
$724$	6.39230	0.237568
$725$	42.0000	1.55984
$726$	0	0
$727$	33.1769	1.23046	0.615232	−	0.788346i	$-0.289062\pi$
$727$	33.1769	1.23046	0.615232	+	0.788346i	$0.289062\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	−1.85641	−0.0687087
$731$	37.8564	1.40017
$732$	0	0
$733$	12.1436	0.448534	0.224267	−	0.974528i	$-0.428001\pi$
$733$	12.1436	0.448534	0.224267	+	0.974528i	$0.428001\pi$
$734$	19.3205	0.713133
$735$	0	0
$736$	−6.92820	−0.255377
$737$	−14.9282	−0.549887
$738$	0	0
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	−17.0718	−0.627572
$741$	0	0
$742$	0.928203	0.0340754
$743$	41.5692	1.52503	0.762513	−	0.646972i	$-0.223965\pi$
$743$	41.5692	1.52503	0.762513	+	0.646972i	$0.223965\pi$
$744$	0	0
$745$	20.7846	0.761489
$746$	29.7128	1.08786
$747$	0	0
$748$	3.46410	0.126660
$749$	6.92820	0.253151
$750$	0	0
$751$	−24.7846	−0.904403	−0.452202	−	0.891916i	$-0.649361\pi$
$751$	−24.7846	−0.904403	−0.452202	+	0.891916i	$0.649361\pi$
$752$	−9.46410	−0.345120
$753$	0	0
$754$	12.0000	0.437014
$755$	−55.4256	−2.01715
$756$	0	0
$757$	22.7846	0.828121	0.414060	−	0.910249i	$-0.364110\pi$
$757$	22.7846	0.828121	0.414060	+	0.910249i	$0.364110\pi$
$758$	−17.8564	−0.648574
$759$	0	0
$760$	18.9282	0.686598
$761$	25.6077	0.928278	0.464139	−	0.885762i	$-0.346364\pi$
$761$	25.6077	0.928278	0.464139	+	0.885762i	$0.346364\pi$
$762$	0	0
$763$	15.8564	0.574040
$764$	−20.7846	−0.751961
$765$	0	0
$766$	−14.5359	−0.525203
$767$	−13.8564	−0.500326
$768$	0	0
$769$	28.5359	1.02903	0.514515	−	0.857481i	$-0.327972\pi$
$769$	28.5359	1.02903	0.514515	+	0.857481i	$0.327972\pi$
$770$	−3.46410	−0.124838
$771$	0	0
$772$	26.0000	0.935760
$773$	32.5359	1.17023	0.585117	−	0.810949i	$-0.301048\pi$
$773$	32.5359	1.17023	0.585117	+	0.810949i	$0.301048\pi$
$774$	0	0
$775$	38.2487	1.37393
$776$	2.00000	0.0717958
$777$	0	0
$778$	30.0000	1.07555
$779$	−18.9282	−0.678173
$780$	0	0
$781$	−12.0000	−0.429394
$782$	24.0000	0.858238
$783$	0	0
$784$	1.00000	0.0357143
$785$	−8.28719	−0.295782
$786$	0	0
$787$	33.1769	1.18263	0.591315	−	0.806441i	$-0.298609\pi$
$787$	33.1769	1.18263	0.591315	+	0.806441i	$0.298609\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	−37.8564	−1.34687
$791$	−7.85641	−0.279342
$792$	0	0
$793$	4.00000	0.142044
$794$	−2.39230	−0.0848997
$795$	0	0
$796$	0.392305	0.0139049
$797$	32.5359	1.15248	0.576240	−	0.817280i	$-0.304519\pi$
$797$	32.5359	1.15248	0.576240	+	0.817280i	$0.304519\pi$
$798$	0	0
$799$	32.7846	1.15984
$800$	7.00000	0.247487
$801$	0	0
$802$	−4.14359	−0.146315
$803$	0.535898	0.0189114
$804$	0	0
$805$	−24.0000	−0.845889
$806$	10.9282	0.384930
$807$	0	0
$808$	19.8564	0.698546
$809$	−11.0718	−0.389264	−0.194632	−	0.980876i	$-0.562351\pi$
$809$	−11.0718	−0.389264	−0.194632	+	0.980876i	$0.562351\pi$
$810$	0	0
$811$	−17.1769	−0.603163	−0.301582	−	0.953440i	$-0.597515\pi$
$811$	−17.1769	−0.603163	−0.301582	+	0.953440i	$0.597515\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	4.92820	0.172733
$815$	−13.8564	−0.485369
$816$	0	0
$817$	−59.7128	−2.08909
$818$	−9.32051	−0.325884
$819$	0	0
$820$	−12.0000	−0.419058
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	40.7846	1.42166	0.710831	−	0.703363i	$-0.248319\pi$
$823$	40.7846	1.42166	0.710831	+	0.703363i	$0.248319\pi$
$824$	−13.4641	−0.469044
$825$	0	0
$826$	−6.92820	−0.241063
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	44.2487	1.53682	0.768411	−	0.639957i	$-0.221048\pi$
$829$	44.2487	1.53682	0.768411	+	0.639957i	$0.221048\pi$
$830$	−15.2154	−0.528134
$831$	0	0
$832$	2.00000	0.0693375
$833$	−3.46410	−0.120024
$834$	0	0
$835$	−65.5692	−2.26912
$836$	−5.46410	−0.188980
$837$	0	0
$838$	36.0000	1.24360
$839$	33.4641	1.15531	0.577655	−	0.816281i	$-0.303968\pi$
$839$	33.4641	1.15531	0.577655	+	0.816281i	$0.303968\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−37.7128	−1.29967
$843$	0	0
$844$	16.7846	0.577750
$845$	−31.1769	−1.07252
$846$	0	0
$847$	1.00000	0.0343604
$848$	0.928203	0.0318746
$849$	0	0
$850$	−24.2487	−0.831724
$851$	34.1436	1.17043
$852$	0	0
$853$	−49.7128	−1.70213	−0.851067	−	0.525057i	$-0.824044\pi$
$853$	−49.7128	−1.70213	−0.851067	+	0.525057i	$0.824044\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	6.92820	0.236801
$857$	5.32051	0.181745	0.0908725	−	0.995863i	$-0.471034\pi$
$857$	5.32051	0.181745	0.0908725	+	0.995863i	$0.471034\pi$
$858$	0	0
$859$	52.7846	1.80099	0.900494	−	0.434869i	$-0.143205\pi$
$859$	52.7846	1.80099	0.900494	+	0.434869i	$0.143205\pi$
$860$	−37.8564	−1.29089
$861$	0	0
$862$	18.9282	0.644697
$863$	−44.7846	−1.52449	−0.762243	−	0.647291i	$-0.775902\pi$
$863$	−44.7846	−1.52449	−0.762243	+	0.647291i	$0.775902\pi$
$864$	0	0
$865$	3.21539	0.109327
$866$	2.00000	0.0679628
$867$	0	0
$868$	5.46410	0.185464
$869$	10.9282	0.370714
$870$	0	0
$871$	29.8564	1.01165
$872$	15.8564	0.536966
$873$	0	0
$874$	−37.8564	−1.28051
$875$	6.92820	0.234216
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	−24.7846	−0.836440
$879$	0	0
$880$	−3.46410	−0.116775
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	−41.8564	−1.40858	−0.704290	−	0.709912i	$-0.748735\pi$
$883$	−41.8564	−1.40858	−0.704290	+	0.709912i	$0.748735\pi$
$884$	−6.92820	−0.233021
$885$	0	0
$886$	20.7846	0.698273
$887$	−10.1436	−0.340589	−0.170294	−	0.985393i	$-0.554472\pi$
$887$	−10.1436	−0.340589	−0.170294	+	0.985393i	$0.554472\pi$
$888$	0	0
$889$	−10.9282	−0.366520
$890$	3.21539	0.107780
$891$	0	0
$892$	−8.39230	−0.280995
$893$	−51.7128	−1.73050
$894$	0	0
$895$	72.0000	2.40669
$896$	1.00000	0.0334077
$897$	0	0
$898$	19.8564	0.662617
$899$	32.7846	1.09343
$900$	0	0
$901$	−3.21539	−0.107120
$902$	3.46410	0.115342
$903$	0	0
$904$	−7.85641	−0.261300
$905$	22.1436	0.736078
$906$	0	0
$907$	14.9282	0.495683	0.247841	−	0.968801i	$-0.420279\pi$
$907$	14.9282	0.495683	0.247841	+	0.968801i	$0.420279\pi$
$908$	4.39230	0.145764
$909$	0	0
$910$	6.92820	0.229668
$911$	48.4974	1.60679	0.803396	−	0.595446i	$-0.203024\pi$
$911$	48.4974	1.60679	0.803396	+	0.595446i	$0.203024\pi$
$912$	0	0
$913$	4.39230	0.145364
$914$	15.8564	0.524483
$915$	0	0
$916$	−21.3205	−0.704449
$917$	4.39230	0.145047
$918$	0	0
$919$	21.8564	0.720976	0.360488	−	0.932764i	$-0.382610\pi$
$919$	21.8564	0.720976	0.360488	+	0.932764i	$0.382610\pi$
$920$	−24.0000	−0.791257
$921$	0	0
$922$	6.00000	0.197599
$923$	24.0000	0.789970
$924$	0	0
$925$	−34.4974	−1.13427
$926$	8.00000	0.262896
$927$	0	0
$928$	6.00000	0.196960
$929$	−40.6410	−1.33339	−0.666694	−	0.745331i	$-0.732291\pi$
$929$	−40.6410	−1.33339	−0.666694	+	0.745331i	$0.732291\pi$
$930$	0	0
$931$	5.46410	0.179079
$932$	7.85641	0.257345
$933$	0	0
$934$	15.7128	0.514139
$935$	12.0000	0.392442
$936$	0	0
$937$	−14.3923	−0.470176	−0.235088	−	0.971974i	$-0.575538\pi$
$937$	−14.3923	−0.470176	−0.235088	+	0.971974i	$0.575538\pi$
$938$	14.9282	0.487423
$939$	0	0
$940$	−32.7846	−1.06932
$941$	11.0718	0.360930	0.180465	−	0.983581i	$-0.442240\pi$
$941$	11.0718	0.360930	0.180465	+	0.983581i	$0.442240\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	−6.92820	−0.225494
$945$	0	0
$946$	10.9282	0.355307
$947$	−49.8564	−1.62012	−0.810058	−	0.586350i	$-0.800564\pi$
$947$	−49.8564	−1.62012	−0.810058	+	0.586350i	$0.800564\pi$
$948$	0	0
$949$	−1.07180	−0.0347920
$950$	38.2487	1.24095
$951$	0	0
$952$	−3.46410	−0.112272
$953$	26.7846	0.867639	0.433819	−	0.901000i	$-0.357166\pi$
$953$	26.7846	0.867639	0.433819	+	0.901000i	$0.357166\pi$
$954$	0	0
$955$	−72.0000	−2.32987
$956$	24.0000	0.776215
$957$	0	0
$958$	13.8564	0.447680
$959$	−7.85641	−0.253697
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	−9.85641	−0.317783
$963$	0	0
$964$	9.60770	0.309443
$965$	90.0666	2.89935
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	6.92820	0.222451
$971$	15.7128	0.504248	0.252124	−	0.967695i	$-0.418871\pi$
$971$	15.7128	0.504248	0.252124	+	0.967695i	$0.418871\pi$
$972$	0	0
$973$	−13.4641	−0.431639
$974$	40.7846	1.30682
$975$	0	0
$976$	2.00000	0.0640184
$977$	47.5692	1.52187	0.760937	−	0.648826i	$-0.224740\pi$
$977$	47.5692	1.52187	0.760937	+	0.648826i	$0.224740\pi$
$978$	0	0
$979$	−0.928203	−0.0296655
$980$	3.46410	0.110657
$981$	0	0
$982$	1.85641	0.0592403
$983$	−23.3205	−0.743809	−0.371904	−	0.928271i	$-0.621295\pi$
$983$	−23.3205	−0.743809	−0.371904	+	0.928271i	$0.621295\pi$
$984$	0	0
$985$	−62.3538	−1.98676
$986$	−20.7846	−0.661917
$987$	0	0
$988$	10.9282	0.347672
$989$	75.7128	2.40753
$990$	0	0
$991$	−2.14359	−0.0680935	−0.0340467	−	0.999420i	$-0.510840\pi$
$991$	−2.14359	−0.0680935	−0.0340467	+	0.999420i	$0.510840\pi$
$992$	5.46410	0.173485
$993$	0	0
$994$	12.0000	0.380617
$995$	1.35898	0.0430827
$996$	0	0
$997$	24.6410	0.780389	0.390194	−	0.920732i	$-0.372408\pi$
$997$	24.6410	0.780389	0.390194	+	0.920732i	$0.372408\pi$
$998$	−9.07180	−0.287163
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.a.p.1.2		2
3.2	odd	2		462.2.a.h.1.1	✓	2
7.6	odd	2		9702.2.a.dd.1.1		2
12.11	even	2		3696.2.a.bc.1.1		2
21.20	even	2		3234.2.a.x.1.2		2
33.32	even	2		5082.2.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.h.1.1	✓	2	3.2	odd	2
1386.2.a.p.1.2		2	1.1	even	1	trivial
3234.2.a.x.1.2		2	21.20	even	2
3696.2.a.bc.1.1		2	12.11	even	2
5082.2.a.bu.1.1		2	33.32	even	2
9702.2.a.dd.1.1		2	7.6	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +3.46410 q^{10} -1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +5.46410 q^{19} +3.46410 q^{20} -1.00000 q^{22} -6.92820 q^{23} +7.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} +5.46410 q^{31} +1.00000 q^{32} -3.46410 q^{34} +3.46410 q^{35} -4.92820 q^{37} +5.46410 q^{38} +3.46410 q^{40} -3.46410 q^{41} -10.9282 q^{43} -1.00000 q^{44} -6.92820 q^{46} -9.46410 q^{47} +1.00000 q^{49} +7.00000 q^{50} +2.00000 q^{52} +0.928203 q^{53} -3.46410 q^{55} +1.00000 q^{56} +6.00000 q^{58} -6.92820 q^{59} +2.00000 q^{61} +5.46410 q^{62} +1.00000 q^{64} +6.92820 q^{65} +14.9282 q^{67} -3.46410 q^{68} +3.46410 q^{70} +12.0000 q^{71} -0.535898 q^{73} -4.92820 q^{74} +5.46410 q^{76} -1.00000 q^{77} -10.9282 q^{79} +3.46410 q^{80} -3.46410 q^{82} -4.39230 q^{83} -12.0000 q^{85} -10.9282 q^{86} -1.00000 q^{88} +0.928203 q^{89} +2.00000 q^{91} -6.92820 q^{92} -9.46410 q^{94} +18.9282 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} - 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{19} - 2 q^{22} + 14 q^{25} + 4 q^{26} + 2 q^{28} + 12 q^{29} + 4 q^{31} + 2 q^{32} + 4 q^{37} + 4 q^{38} - 8 q^{43}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 - 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^19 - 2 * q^22 + 14 * q^25 + 4 * q^26 + 2 * q^28 + 12 * q^29 + 4 * q^31 + 2 * q^32 + 4 * q^37 + 4 * q^38 - 8 * q^43 - 2 * q^44 - 12 * q^47 + 2 * q^49 + 14 * q^50 + 4 * q^52 - 12 * q^53 + 2 * q^56 + 12 * q^58 + 4 * q^61 + 4 * q^62 + 2 * q^64 + 16 * q^67 + 24 * q^71 - 8 * q^73 + 4 * q^74 + 4 * q^76 - 2 * q^77 - 8 * q^79 + 12 * q^83 - 24 * q^85 - 8 * q^86 - 2 * q^88 - 12 * q^89 + 4 * q^91 - 12 * q^94 + 24 * q^95 + 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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