LMFDB - Embedded newform 1386.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(1,1386)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.16228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.16228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.16228 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -0.837722 q^{17} -1.16228 q^{19} -3.16228 q^{20} -1.00000 q^{22} -6.32456 q^{23} +5.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} -4.00000 q^{29} +2.83772 q^{31} -1.00000 q^{32} +0.837722 q^{34} +3.16228 q^{35} -4.32456 q^{37} +1.16228 q^{38} +3.16228 q^{40} +0.837722 q^{41} +6.32456 q^{43} +1.00000 q^{44} +6.32456 q^{46} +7.48683 q^{47} +1.00000 q^{49} -5.00000 q^{50} +2.00000 q^{52} +8.32456 q^{53} -3.16228 q^{55} +1.00000 q^{56} +4.00000 q^{58} +14.3246 q^{59} +10.0000 q^{61} -2.83772 q^{62} +1.00000 q^{64} -6.32456 q^{65} -8.32456 q^{67} -0.837722 q^{68} -3.16228 q^{70} +8.00000 q^{71} +13.4868 q^{73} +4.32456 q^{74} -1.16228 q^{76} -1.00000 q^{77} +8.32456 q^{79} -3.16228 q^{80} -0.837722 q^{82} -1.16228 q^{83} +2.64911 q^{85} -6.32456 q^{86} -1.00000 q^{88} -3.67544 q^{89} -2.00000 q^{91} -6.32456 q^{92} -7.48683 q^{94} +3.67544 q^{95} +14.6491 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	$ 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} - 8 q^{17} + 4 q^{19} - 2 q^{22} + 10 q^{25} - 4 q^{26} - 2 q^{28} - 8 q^{29} + 12 q^{31} - 2 q^{32} + 8 q^{34} + 4 q^{37} - 4 q^{38} + 8 q^{41} + 2 q^{44} - 4 q^{47} + 2 q^{49} - 10 q^{50} + 4 q^{52} + 4 q^{53} + 2 q^{56} + 8 q^{58} + 16 q^{59} + 20 q^{61} - 12 q^{62} + 2 q^{64} - 4 q^{67} - 8 q^{68} + 16 q^{71} + 8 q^{73} - 4 q^{74} + 4 q^{76} - 2 q^{77} + 4 q^{79} - 8 q^{82} + 4 q^{83} - 20 q^{85} - 2 q^{88} - 20 q^{89} - 4 q^{91} + 4 q^{94} + 20 q^{95} + 4 q^{97} - 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 - 8 * q^17 + 4 * q^19 - 2 * q^22 + 10 * q^25 - 4 * q^26 - 2 * q^28 - 8 * q^29 + 12 * q^31 - 2 * q^32 + 8 * q^34 + 4 * q^37 - 4 * q^38 + 8 * q^41 + 2 * q^44 - 4 * q^47 + 2 * q^49 - 10 * q^50 + 4 * q^52 + 4 * q^53 + 2 * q^56 + 8 * q^58 + 16 * q^59 + 20 * q^61 - 12 * q^62 + 2 * q^64 - 4 * q^67 - 8 * q^68 + 16 * q^71 + 8 * q^73 - 4 * q^74 + 4 * q^76 - 2 * q^77 + 4 * q^79 - 8 * q^82 + 4 * q^83 - 20 * q^85 - 2 * q^88 - 20 * q^89 - 4 * q^91 + 4 * q^94 + 20 * q^95 + 4 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

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
$5$	−3.16228	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$5$	−3.16228	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.16228	1.00000
$11$	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
$17$	−0.837722	−0.203178	−0.101589	−	0.994826i	$-0.532393\pi$
$17$	−0.837722	−0.203178	−0.101589	+	0.994826i	$0.532393\pi$
$18$	0	0
$19$	−1.16228	−0.266645	−0.133322	−	0.991073i	$-0.542565\pi$
$19$	−1.16228	−0.266645	−0.133322	+	0.991073i	$0.542565\pi$
$20$	−3.16228	−0.707107
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−6.32456	−1.31876	−0.659380	−	0.751809i	$-0.729181\pi$
$23$	−6.32456	−1.31876	−0.659380	+	0.751809i	$0.729181\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	2.83772	0.509670	0.254835	−	0.966985i	$-0.417979\pi$
$31$	2.83772	0.509670	0.254835	+	0.966985i	$0.417979\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.837722	0.143668
$35$	3.16228	0.534522
$36$	0	0
$37$	−4.32456	−0.710953	−0.355476	−	0.934685i	$-0.615681\pi$
$37$	−4.32456	−0.710953	−0.355476	+	0.934685i	$0.615681\pi$
$38$	1.16228	0.188546
$39$	0	0
$40$	3.16228	0.500000
$41$	0.837722	0.130830	0.0654151	−	0.997858i	$-0.479163\pi$
$41$	0.837722	0.130830	0.0654151	+	0.997858i	$0.479163\pi$
$42$	0	0
$43$	6.32456	0.964486	0.482243	−	0.876038i	$-0.339822\pi$
$43$	6.32456	0.964486	0.482243	+	0.876038i	$0.339822\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	6.32456	0.932505
$47$	7.48683	1.09207	0.546033	−	0.837763i	$-0.316137\pi$
$47$	7.48683	1.09207	0.546033	+	0.837763i	$0.316137\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.00000	−0.707107
$51$	0	0
$52$	2.00000	0.277350
$53$	8.32456	1.14347	0.571733	−	0.820440i	$-0.306271\pi$
$53$	8.32456	1.14347	0.571733	+	0.820440i	$0.306271\pi$
$54$	0	0
$55$	−3.16228	−0.426401
$56$	1.00000	0.133631
$57$	0	0
$58$	4.00000	0.525226
$59$	14.3246	1.86490	0.932449	−	0.361301i	$-0.117667\pi$
$59$	14.3246	1.86490	0.932449	+	0.361301i	$0.117667\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−2.83772	−0.360391
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.32456	−0.784465
$66$	0	0
$67$	−8.32456	−1.01701	−0.508503	−	0.861060i	$-0.669801\pi$
$67$	−8.32456	−1.01701	−0.508503	+	0.861060i	$0.669801\pi$
$68$	−0.837722	−0.101589
$69$	0	0
$70$	−3.16228	−0.377964
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	13.4868	1.57851	0.789257	−	0.614063i	$-0.210466\pi$
$73$	13.4868	1.57851	0.789257	+	0.614063i	$0.210466\pi$
$74$	4.32456	0.502719
$75$	0	0
$76$	−1.16228	−0.133322
$77$	−1.00000	−0.113961
$78$	0	0
$79$	8.32456	0.936586	0.468293	−	0.883573i	$-0.344869\pi$
$79$	8.32456	0.936586	0.468293	+	0.883573i	$0.344869\pi$
$80$	−3.16228	−0.353553
$81$	0	0
$82$	−0.837722	−0.0925110
$83$	−1.16228	−0.127577	−0.0637883	−	0.997963i	$-0.520318\pi$
$83$	−1.16228	−0.127577	−0.0637883	+	0.997963i	$0.520318\pi$
$84$	0	0
$85$	2.64911	0.287336
$86$	−6.32456	−0.681994
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−3.67544	−0.389596	−0.194798	−	0.980843i	$-0.562405\pi$
$89$	−3.67544	−0.389596	−0.194798	+	0.980843i	$0.562405\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−6.32456	−0.659380
$93$	0	0
$94$	−7.48683	−0.772208
$95$	3.67544	0.377093
$96$	0	0
$97$	14.6491	1.48739	0.743696	−	0.668518i	$-0.233071\pi$
$97$	14.6491	1.48739	0.743696	+	0.668518i	$0.233071\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	5.00000	0.500000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−17.8114	−1.75501	−0.877504	−	0.479569i	$-0.840793\pi$
$103$	−17.8114	−1.75501	−0.877504	+	0.479569i	$0.840793\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−8.32456	−0.808552
$107$	1.67544	0.161971	0.0809857	−	0.996715i	$-0.474193\pi$
$107$	1.67544	0.161971	0.0809857	+	0.996715i	$0.474193\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	3.16228	0.301511
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	20.0000	1.86501
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−14.3246	−1.31868
$119$	0.837722	0.0767939
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.0000	−0.905357
$123$	0	0
$124$	2.83772	0.254835
$125$	0	0
$126$	0	0
$127$	−16.9737	−1.50617	−0.753085	−	0.657924i	$-0.771435\pi$
$127$	−16.9737	−1.50617	−0.753085	+	0.657924i	$0.771435\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.32456	0.554700
$131$	1.16228	0.101549	0.0507743	−	0.998710i	$-0.483831\pi$
$131$	1.16228	0.101549	0.0507743	+	0.998710i	$0.483831\pi$
$132$	0	0
$133$	1.16228	0.100782
$134$	8.32456	0.719132
$135$	0	0
$136$	0.837722	0.0718341
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	11.4868	0.974300	0.487150	−	0.873318i	$-0.338036\pi$
$139$	11.4868	0.974300	0.487150	+	0.873318i	$0.338036\pi$
$140$	3.16228	0.267261
$141$	0	0
$142$	−8.00000	−0.671345
$143$	2.00000	0.167248
$144$	0	0
$145$	12.6491	1.05045
$146$	−13.4868	−1.11618
$147$	0	0
$148$	−4.32456	−0.355476
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−4.64911	−0.378339	−0.189170	−	0.981944i	$-0.560580\pi$
$151$	−4.64911	−0.378339	−0.189170	+	0.981944i	$0.560580\pi$
$152$	1.16228	0.0942732
$153$	0	0
$154$	1.00000	0.0805823
$155$	−8.97367	−0.720782
$156$	0	0
$157$	13.4868	1.07637	0.538183	−	0.842828i	$-0.319111\pi$
$157$	13.4868	1.07637	0.538183	+	0.842828i	$0.319111\pi$
$158$	−8.32456	−0.662266
$159$	0	0
$160$	3.16228	0.250000
$161$	6.32456	0.498445
$162$	0	0
$163$	8.64911	0.677451	0.338725	−	0.940885i	$-0.390004\pi$
$163$	8.64911	0.677451	0.338725	+	0.940885i	$0.390004\pi$
$164$	0.837722	0.0654151
$165$	0	0
$166$	1.16228	0.0902102
$167$	−6.32456	−0.489409	−0.244704	−	0.969598i	$-0.578691\pi$
$167$	−6.32456	−0.489409	−0.244704	+	0.969598i	$0.578691\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−2.64911	−0.203178
$171$	0	0
$172$	6.32456	0.482243
$173$	−0.324555	−0.0246755	−0.0123377	−	0.999924i	$-0.503927\pi$
$173$	−0.324555	−0.0246755	−0.0123377	+	0.999924i	$0.503927\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	1.00000	0.0753778
$177$	0	0
$178$	3.67544	0.275486
$179$	16.3246	1.22015	0.610077	−	0.792342i	$-0.291138\pi$
$179$	16.3246	1.22015	0.610077	+	0.792342i	$0.291138\pi$
$180$	0	0
$181$	−1.48683	−0.110515	−0.0552577	−	0.998472i	$-0.517598\pi$
$181$	−1.48683	−0.110515	−0.0552577	+	0.998472i	$0.517598\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	6.32456	0.466252
$185$	13.6754	1.00544
$186$	0	0
$187$	−0.837722	−0.0612603
$188$	7.48683	0.546033
$189$	0	0
$190$	−3.67544	−0.266645
$191$	18.3246	1.32592	0.662959	−	0.748656i	$-0.269300\pi$
$191$	18.3246	1.32592	0.662959	+	0.748656i	$0.269300\pi$
$192$	0	0
$193$	−14.6491	−1.05447	−0.527233	−	0.849721i	$-0.676771\pi$
$193$	−14.6491	−1.05447	−0.527233	+	0.849721i	$0.676771\pi$
$194$	−14.6491	−1.05174
$195$	0	0
$196$	1.00000	0.0714286
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	0.513167	0.0363774	0.0181887	−	0.999835i	$-0.494210\pi$
$199$	0.513167	0.0363774	0.0181887	+	0.999835i	$0.494210\pi$
$200$	−5.00000	−0.353553
$201$	0	0
$202$	−6.00000	−0.422159
$203$	4.00000	0.280745
$204$	0	0
$205$	−2.64911	−0.185022
$206$	17.8114	1.24098
$207$	0	0
$208$	2.00000	0.138675
$209$	−1.16228	−0.0803964
$210$	0	0
$211$	−5.67544	−0.390714	−0.195357	−	0.980732i	$-0.562587\pi$
$211$	−5.67544	−0.390714	−0.195357	+	0.980732i	$0.562587\pi$
$212$	8.32456	0.571733
$213$	0	0
$214$	−1.67544	−0.114531
$215$	−20.0000	−1.36399
$216$	0	0
$217$	−2.83772	−0.192637
$218$	8.00000	0.541828
$219$	0	0
$220$	−3.16228	−0.213201
$221$	−1.67544	−0.112703
$222$	0	0
$223$	7.48683	0.501355	0.250678	−	0.968071i	$-0.419347\pi$
$223$	7.48683	0.501355	0.250678	+	0.968071i	$0.419347\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	2.00000	0.133038
$227$	11.4868	0.762408	0.381204	−	0.924491i	$-0.375510\pi$
$227$	11.4868	0.762408	0.381204	+	0.924491i	$0.375510\pi$
$228$	0	0
$229$	−6.51317	−0.430402	−0.215201	−	0.976570i	$-0.569041\pi$
$229$	−6.51317	−0.430402	−0.215201	+	0.976570i	$0.569041\pi$
$230$	−20.0000	−1.31876
$231$	0	0
$232$	4.00000	0.262613
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−23.6754	−1.54442
$236$	14.3246	0.932449
$237$	0	0
$238$	−0.837722	−0.0543015
$239$	12.6491	0.818203	0.409101	−	0.912489i	$-0.365842\pi$
$239$	12.6491	0.818203	0.409101	+	0.912489i	$0.365842\pi$
$240$	0	0
$241$	7.16228	0.461363	0.230681	−	0.973029i	$-0.425904\pi$
$241$	7.16228	0.461363	0.230681	+	0.973029i	$0.425904\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	10.0000	0.640184
$245$	−3.16228	−0.202031
$246$	0	0
$247$	−2.32456	−0.147908
$248$	−2.83772	−0.180196
$249$	0	0
$250$	0	0
$251$	26.3246	1.66159	0.830796	−	0.556578i	$-0.187886\pi$
$251$	26.3246	1.66159	0.830796	+	0.556578i	$0.187886\pi$
$252$	0	0
$253$	−6.32456	−0.397621
$254$	16.9737	1.06502
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.3246	−1.26781	−0.633905	−	0.773411i	$-0.718549\pi$
$257$	−20.3246	−1.26781	−0.633905	+	0.773411i	$0.718549\pi$
$258$	0	0
$259$	4.32456	0.268715
$260$	−6.32456	−0.392232
$261$	0	0
$262$	−1.16228	−0.0718058
$263$	12.9737	0.799991	0.399995	−	0.916517i	$-0.369012\pi$
$263$	12.9737	0.799991	0.399995	+	0.916517i	$0.369012\pi$
$264$	0	0
$265$	−26.3246	−1.61710
$266$	−1.16228	−0.0712638
$267$	0	0
$268$	−8.32456	−0.508503
$269$	−4.83772	−0.294961	−0.147481	−	0.989065i	$-0.547116\pi$
$269$	−4.83772	−0.294961	−0.147481	+	0.989065i	$0.547116\pi$
$270$	0	0
$271$	6.32456	0.384189	0.192095	−	0.981376i	$-0.438472\pi$
$271$	6.32456	0.384189	0.192095	+	0.981376i	$0.438472\pi$
$272$	−0.837722	−0.0507944
$273$	0	0
$274$	−18.0000	−1.08742
$275$	5.00000	0.301511
$276$	0	0
$277$	−24.0000	−1.44202	−0.721010	−	0.692925i	$-0.756322\pi$
$277$	−24.0000	−1.44202	−0.721010	+	0.692925i	$0.756322\pi$
$278$	−11.4868	−0.688934
$279$	0	0
$280$	−3.16228	−0.188982
$281$	2.64911	0.158033	0.0790163	−	0.996873i	$-0.474822\pi$
$281$	2.64911	0.158033	0.0790163	+	0.996873i	$0.474822\pi$
$282$	0	0
$283$	3.48683	0.207271	0.103635	−	0.994615i	$-0.466953\pi$
$283$	3.48683	0.207271	0.103635	+	0.994615i	$0.466953\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−0.837722	−0.0494492
$288$	0	0
$289$	−16.2982	−0.958719
$290$	−12.6491	−0.742781
$291$	0	0
$292$	13.4868	0.789257
$293$	12.9737	0.757930	0.378965	−	0.925411i	$-0.376280\pi$
$293$	12.9737	0.757930	0.378965	+	0.925411i	$0.376280\pi$
$294$	0	0
$295$	−45.2982	−2.63736
$296$	4.32456	0.251360
$297$	0	0
$298$	2.00000	0.115857
$299$	−12.6491	−0.731517
$300$	0	0
$301$	−6.32456	−0.364541
$302$	4.64911	0.267526
$303$	0	0
$304$	−1.16228	−0.0666612
$305$	−31.6228	−1.81071
$306$	0	0
$307$	−14.8377	−0.846834	−0.423417	−	0.905935i	$-0.639169\pi$
$307$	−14.8377	−0.846834	−0.423417	+	0.905935i	$0.639169\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	8.97367	0.509670
$311$	−1.81139	−0.102714	−0.0513572	−	0.998680i	$-0.516355\pi$
$311$	−1.81139	−0.102714	−0.0513572	+	0.998680i	$0.516355\pi$
$312$	0	0
$313$	7.67544	0.433842	0.216921	−	0.976189i	$-0.430399\pi$
$313$	7.67544	0.433842	0.216921	+	0.976189i	$0.430399\pi$
$314$	−13.4868	−0.761106
$315$	0	0
$316$	8.32456	0.468293
$317$	24.9737	1.40266	0.701330	−	0.712836i	$-0.252590\pi$
$317$	24.9737	1.40266	0.701330	+	0.712836i	$0.252590\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	−3.16228	−0.176777
$321$	0	0
$322$	−6.32456	−0.352454
$323$	0.973666	0.0541762
$324$	0	0
$325$	10.0000	0.554700
$326$	−8.64911	−0.479030
$327$	0	0
$328$	−0.837722	−0.0462555
$329$	−7.48683	−0.412762
$330$	0	0
$331$	28.9737	1.59254	0.796268	−	0.604944i	$-0.206804\pi$
$331$	28.9737	1.59254	0.796268	+	0.604944i	$0.206804\pi$
$332$	−1.16228	−0.0637883
$333$	0	0
$334$	6.32456	0.346064
$335$	26.3246	1.43826
$336$	0	0
$337$	−32.3246	−1.76083	−0.880415	−	0.474203i	$-0.842736\pi$
$337$	−32.3246	−1.76083	−0.880415	+	0.474203i	$0.842736\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	2.64911	0.143668
$341$	2.83772	0.153671
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.32456	−0.340997
$345$	0	0
$346$	0.324555	0.0174482
$347$	1.67544	0.0899426	0.0449713	−	0.998988i	$-0.485680\pi$
$347$	1.67544	0.0899426	0.0449713	+	0.998988i	$0.485680\pi$
$348$	0	0
$349$	−28.9737	−1.55092	−0.775462	−	0.631394i	$-0.782483\pi$
$349$	−28.9737	−1.55092	−0.775462	+	0.631394i	$0.782483\pi$
$350$	5.00000	0.267261
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−20.3246	−1.08177	−0.540883	−	0.841098i	$-0.681910\pi$
$353$	−20.3246	−1.08177	−0.540883	+	0.841098i	$0.681910\pi$
$354$	0	0
$355$	−25.2982	−1.34269
$356$	−3.67544	−0.194798
$357$	0	0
$358$	−16.3246	−0.862780
$359$	−16.3246	−0.861577	−0.430789	−	0.902453i	$-0.641765\pi$
$359$	−16.3246	−0.861577	−0.430789	+	0.902453i	$0.641765\pi$
$360$	0	0
$361$	−17.6491	−0.928901
$362$	1.48683	0.0781462
$363$	0	0
$364$	−2.00000	−0.104828
$365$	−42.6491	−2.23236
$366$	0	0
$367$	5.16228	0.269469	0.134734	−	0.990882i	$-0.456982\pi$
$367$	5.16228	0.269469	0.134734	+	0.990882i	$0.456982\pi$
$368$	−6.32456	−0.329690
$369$	0	0
$370$	−13.6754	−0.710953
$371$	−8.32456	−0.432189
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0.837722	0.0433176
$375$	0	0
$376$	−7.48683	−0.386104
$377$	−8.00000	−0.412021
$378$	0	0
$379$	37.2982	1.91588	0.957940	−	0.286967i	$-0.0926470\pi$
$379$	37.2982	1.91588	0.957940	+	0.286967i	$0.0926470\pi$
$380$	3.67544	0.188546
$381$	0	0
$382$	−18.3246	−0.937566
$383$	2.83772	0.145001	0.0725004	−	0.997368i	$-0.476902\pi$
$383$	2.83772	0.145001	0.0725004	+	0.997368i	$0.476902\pi$
$384$	0	0
$385$	3.16228	0.161165
$386$	14.6491	0.745620
$387$	0	0
$388$	14.6491	0.743696
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	5.29822	0.267943
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	24.0000	1.20910
$395$	−26.3246	−1.32453
$396$	0	0
$397$	2.51317	0.126132	0.0630661	−	0.998009i	$-0.479912\pi$
$397$	2.51317	0.126132	0.0630661	+	0.998009i	$0.479912\pi$
$398$	−0.513167	−0.0257227
$399$	0	0
$400$	5.00000	0.250000
$401$	27.2982	1.36321	0.681604	−	0.731721i	$-0.261283\pi$
$401$	27.2982	1.36321	0.681604	+	0.731721i	$0.261283\pi$
$402$	0	0
$403$	5.67544	0.282714
$404$	6.00000	0.298511
$405$	0	0
$406$	−4.00000	−0.198517
$407$	−4.32456	−0.214360
$408$	0	0
$409$	20.8377	1.03036	0.515180	−	0.857082i	$-0.327725\pi$
$409$	20.8377	1.03036	0.515180	+	0.857082i	$0.327725\pi$
$410$	2.64911	0.130830
$411$	0	0
$412$	−17.8114	−0.877504
$413$	−14.3246	−0.704865
$414$	0	0
$415$	3.67544	0.180420
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	1.16228	0.0568489
$419$	−8.64911	−0.422537	−0.211268	−	0.977428i	$-0.567759\pi$
$419$	−8.64911	−0.422537	−0.211268	+	0.977428i	$0.567759\pi$
$420$	0	0
$421$	35.9473	1.75197	0.875983	−	0.482342i	$-0.160214\pi$
$421$	35.9473	1.75197	0.875983	+	0.482342i	$0.160214\pi$
$422$	5.67544	0.276276
$423$	0	0
$424$	−8.32456	−0.404276
$425$	−4.18861	−0.203178
$426$	0	0
$427$	−10.0000	−0.483934
$428$	1.67544	0.0809857
$429$	0	0
$430$	20.0000	0.964486
$431$	−28.9737	−1.39561	−0.697806	−	0.716287i	$-0.745840\pi$
$431$	−28.9737	−1.39561	−0.697806	+	0.716287i	$0.745840\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	2.83772	0.136215
$435$	0	0
$436$	−8.00000	−0.383131
$437$	7.35089	0.351641
$438$	0	0
$439$	6.32456	0.301855	0.150927	−	0.988545i	$-0.451774\pi$
$439$	6.32456	0.301855	0.150927	+	0.988545i	$0.451774\pi$
$440$	3.16228	0.150756
$441$	0	0
$442$	1.67544	0.0796928
$443$	8.32456	0.395512	0.197756	−	0.980251i	$-0.436635\pi$
$443$	8.32456	0.395512	0.197756	+	0.980251i	$0.436635\pi$
$444$	0	0
$445$	11.6228	0.550972
$446$	−7.48683	−0.354512
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0.837722	0.0394468
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	−11.4868	−0.539104
$455$	6.32456	0.296500
$456$	0	0
$457$	42.6491	1.99504	0.997521	−	0.0703747i	$-0.0224195\pi$
$457$	42.6491	1.99504	0.997521	+	0.0703747i	$0.0224195\pi$
$458$	6.51317	0.304340
$459$	0	0
$460$	20.0000	0.932505
$461$	−27.2982	−1.27140	−0.635702	−	0.771934i	$-0.719289\pi$
$461$	−27.2982	−1.27140	−0.635702	+	0.771934i	$0.719289\pi$
$462$	0	0
$463$	−3.35089	−0.155729	−0.0778645	−	0.996964i	$-0.524810\pi$
$463$	−3.35089	−0.155729	−0.0778645	+	0.996964i	$0.524810\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	6.00000	0.277945
$467$	−33.2982	−1.54086	−0.770429	−	0.637526i	$-0.779958\pi$
$467$	−33.2982	−1.54086	−0.770429	+	0.637526i	$0.779958\pi$
$468$	0	0
$469$	8.32456	0.384392
$470$	23.6754	1.09207
$471$	0	0
$472$	−14.3246	−0.659341
$473$	6.32456	0.290803
$474$	0	0
$475$	−5.81139	−0.266645
$476$	0.837722	0.0383969
$477$	0	0
$478$	−12.6491	−0.578557
$479$	−29.2982	−1.33867	−0.669335	−	0.742961i	$-0.733421\pi$
$479$	−29.2982	−1.33867	−0.669335	+	0.742961i	$0.733421\pi$
$480$	0	0
$481$	−8.64911	−0.394365
$482$	−7.16228	−0.326233
$483$	0	0
$484$	1.00000	0.0454545
$485$	−46.3246	−2.10349
$486$	0	0
$487$	−1.67544	−0.0759216	−0.0379608	−	0.999279i	$-0.512086\pi$
$487$	−1.67544	−0.0759216	−0.0379608	+	0.999279i	$0.512086\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	3.16228	0.142857
$491$	16.6491	0.751364	0.375682	−	0.926749i	$-0.377409\pi$
$491$	16.6491	0.751364	0.375682	+	0.926749i	$0.377409\pi$
$492$	0	0
$493$	3.35089	0.150916
$494$	2.32456	0.104587
$495$	0	0
$496$	2.83772	0.127417
$497$	−8.00000	−0.358849
$498$	0	0
$499$	20.9737	0.938910	0.469455	−	0.882956i	$-0.344450\pi$
$499$	20.9737	0.938910	0.469455	+	0.882956i	$0.344450\pi$
$500$	0	0
$501$	0	0
$502$	−26.3246	−1.17492
$503$	42.9737	1.91610	0.958051	−	0.286599i	$-0.0925248\pi$
$503$	42.9737	1.91610	0.958051	+	0.286599i	$0.0925248\pi$
$504$	0	0
$505$	−18.9737	−0.844317
$506$	6.32456	0.281161
$507$	0	0
$508$	−16.9737	−0.753085
$509$	−8.18861	−0.362954	−0.181477	−	0.983395i	$-0.558088\pi$
$509$	−8.18861	−0.362954	−0.181477	+	0.983395i	$0.558088\pi$
$510$	0	0
$511$	−13.4868	−0.596622
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	20.3246	0.896478
$515$	56.3246	2.48196
$516$	0	0
$517$	7.48683	0.329271
$518$	−4.32456	−0.190010
$519$	0	0
$520$	6.32456	0.277350
$521$	−7.67544	−0.336267	−0.168134	−	0.985764i	$-0.553774\pi$
$521$	−7.67544	−0.336267	−0.168134	+	0.985764i	$0.553774\pi$
$522$	0	0
$523$	40.1359	1.75502	0.877511	−	0.479556i	$-0.159202\pi$
$523$	40.1359	1.75502	0.877511	+	0.479556i	$0.159202\pi$
$524$	1.16228	0.0507743
$525$	0	0
$526$	−12.9737	−0.565679
$527$	−2.37722	−0.103553
$528$	0	0
$529$	17.0000	0.739130
$530$	26.3246	1.14347
$531$	0	0
$532$	1.16228	0.0503911
$533$	1.67544	0.0725716
$534$	0	0
$535$	−5.29822	−0.229062
$536$	8.32456	0.359566
$537$	0	0
$538$	4.83772	0.208569
$539$	1.00000	0.0430730
$540$	0	0
$541$	−19.2982	−0.829695	−0.414848	−	0.909891i	$-0.636165\pi$
$541$	−19.2982	−0.829695	−0.414848	+	0.909891i	$0.636165\pi$
$542$	−6.32456	−0.271663
$543$	0	0
$544$	0.837722	0.0359170
$545$	25.2982	1.08366
$546$	0	0
$547$	9.29822	0.397563	0.198782	−	0.980044i	$-0.436302\pi$
$547$	9.29822	0.397563	0.198782	+	0.980044i	$0.436302\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	−5.00000	−0.213201
$551$	4.64911	0.198059
$552$	0	0
$553$	−8.32456	−0.353996
$554$	24.0000	1.01966
$555$	0	0
$556$	11.4868	0.487150
$557$	−17.2982	−0.732949	−0.366475	−	0.930428i	$-0.619435\pi$
$557$	−17.2982	−0.732949	−0.366475	+	0.930428i	$0.619435\pi$
$558$	0	0
$559$	12.6491	0.535000
$560$	3.16228	0.133631
$561$	0	0
$562$	−2.64911	−0.111746
$563$	24.1359	1.01721	0.508604	−	0.861000i	$-0.330162\pi$
$563$	24.1359	1.01721	0.508604	+	0.861000i	$0.330162\pi$
$564$	0	0
$565$	6.32456	0.266076
$566$	−3.48683	−0.146563
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−1.35089	−0.0566322	−0.0283161	−	0.999599i	$-0.509015\pi$
$569$	−1.35089	−0.0566322	−0.0283161	+	0.999599i	$0.509015\pi$
$570$	0	0
$571$	−12.6491	−0.529349	−0.264674	−	0.964338i	$-0.585264\pi$
$571$	−12.6491	−0.529349	−0.264674	+	0.964338i	$0.585264\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	0.837722	0.0349659
$575$	−31.6228	−1.31876
$576$	0	0
$577$	28.3246	1.17917	0.589583	−	0.807708i	$-0.299292\pi$
$577$	28.3246	1.17917	0.589583	+	0.807708i	$0.299292\pi$
$578$	16.2982	0.677917
$579$	0	0
$580$	12.6491	0.525226
$581$	1.16228	0.0482194
$582$	0	0
$583$	8.32456	0.344768
$584$	−13.4868	−0.558089
$585$	0	0
$586$	−12.9737	−0.535937
$587$	30.9737	1.27842	0.639210	−	0.769032i	$-0.279261\pi$
$587$	30.9737	1.27842	0.639210	+	0.769032i	$0.279261\pi$
$588$	0	0
$589$	−3.29822	−0.135901
$590$	45.2982	1.86490
$591$	0	0
$592$	−4.32456	−0.177738
$593$	−40.4605	−1.66151	−0.830757	−	0.556636i	$-0.812092\pi$
$593$	−40.4605	−1.66151	−0.830757	+	0.556636i	$0.812092\pi$
$594$	0	0
$595$	−2.64911	−0.108603
$596$	−2.00000	−0.0819232
$597$	0	0
$598$	12.6491	0.517261
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	12.8377	0.523662	0.261831	−	0.965114i	$-0.415674\pi$
$601$	12.8377	0.523662	0.261831	+	0.965114i	$0.415674\pi$
$602$	6.32456	0.257770
$603$	0	0
$604$	−4.64911	−0.189170
$605$	−3.16228	−0.128565
$606$	0	0
$607$	−20.6491	−0.838122	−0.419061	−	0.907958i	$-0.637641\pi$
$607$	−20.6491	−0.838122	−0.419061	+	0.907958i	$0.637641\pi$
$608$	1.16228	0.0471366
$609$	0	0
$610$	31.6228	1.28037
$611$	14.9737	0.605770
$612$	0	0
$613$	19.2982	0.779448	0.389724	−	0.920932i	$-0.372570\pi$
$613$	19.2982	0.779448	0.389724	+	0.920932i	$0.372570\pi$
$614$	14.8377	0.598802
$615$	0	0
$616$	1.00000	0.0402911
$617$	4.64911	0.187166	0.0935831	−	0.995611i	$-0.470168\pi$
$617$	4.64911	0.187166	0.0935831	+	0.995611i	$0.470168\pi$
$618$	0	0
$619$	48.6491	1.95537	0.977686	−	0.210070i	$-0.0673691\pi$
$619$	48.6491	1.95537	0.977686	+	0.210070i	$0.0673691\pi$
$620$	−8.97367	−0.360391
$621$	0	0
$622$	1.81139	0.0726301
$623$	3.67544	0.147254
$624$	0	0
$625$	−25.0000	−1.00000
$626$	−7.67544	−0.306772
$627$	0	0
$628$	13.4868	0.538183
$629$	3.62278	0.144450
$630$	0	0
$631$	35.6228	1.41812	0.709060	−	0.705148i	$-0.249119\pi$
$631$	35.6228	1.41812	0.709060	+	0.705148i	$0.249119\pi$
$632$	−8.32456	−0.331133
$633$	0	0
$634$	−24.9737	−0.991831
$635$	53.6754	2.13005
$636$	0	0
$637$	2.00000	0.0792429
$638$	4.00000	0.158362
$639$	0	0
$640$	3.16228	0.125000
$641$	37.9473	1.49883	0.749415	−	0.662101i	$-0.230335\pi$
$641$	37.9473	1.49883	0.749415	+	0.662101i	$0.230335\pi$
$642$	0	0
$643$	−2.97367	−0.117270	−0.0586350	−	0.998279i	$-0.518675\pi$
$643$	−2.97367	−0.117270	−0.0586350	+	0.998279i	$0.518675\pi$
$644$	6.32456	0.249222
$645$	0	0
$646$	−0.973666	−0.0383084
$647$	15.4868	0.608850	0.304425	−	0.952536i	$-0.401536\pi$
$647$	15.4868	0.608850	0.304425	+	0.952536i	$0.401536\pi$
$648$	0	0
$649$	14.3246	0.562288
$650$	−10.0000	−0.392232
$651$	0	0
$652$	8.64911	0.338725
$653$	−7.29822	−0.285601	−0.142801	−	0.989751i	$-0.545611\pi$
$653$	−7.29822	−0.285601	−0.142801	+	0.989751i	$0.545611\pi$
$654$	0	0
$655$	−3.67544	−0.143612
$656$	0.837722	0.0327076
$657$	0	0
$658$	7.48683	0.291867
$659$	14.9737	0.583291	0.291646	−	0.956526i	$-0.405797\pi$
$659$	14.9737	0.583291	0.291646	+	0.956526i	$0.405797\pi$
$660$	0	0
$661$	22.1359	0.860988	0.430494	−	0.902593i	$-0.358339\pi$
$661$	22.1359	0.860988	0.430494	+	0.902593i	$0.358339\pi$
$662$	−28.9737	−1.12609
$663$	0	0
$664$	1.16228	0.0451051
$665$	−3.67544	−0.142528
$666$	0	0
$667$	25.2982	0.979551
$668$	−6.32456	−0.244704
$669$	0	0
$670$	−26.3246	−1.01701
$671$	10.0000	0.386046
$672$	0	0
$673$	−49.6228	−1.91282	−0.956409	−	0.292031i	$-0.905669\pi$
$673$	−49.6228	−1.91282	−0.956409	+	0.292031i	$0.905669\pi$
$674$	32.3246	1.24510
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−4.32456	−0.166206	−0.0831031	−	0.996541i	$-0.526483\pi$
$677$	−4.32456	−0.166206	−0.0831031	+	0.996541i	$0.526483\pi$
$678$	0	0
$679$	−14.6491	−0.562181
$680$	−2.64911	−0.101589
$681$	0	0
$682$	−2.83772	−0.108662
$683$	−12.9737	−0.496424	−0.248212	−	0.968706i	$-0.579843\pi$
$683$	−12.9737	−0.496424	−0.248212	+	0.968706i	$0.579843\pi$
$684$	0	0
$685$	−56.9210	−2.17484
$686$	1.00000	0.0381802
$687$	0	0
$688$	6.32456	0.241121
$689$	16.6491	0.634281
$690$	0	0
$691$	30.9737	1.17829	0.589147	−	0.808026i	$-0.299464\pi$
$691$	30.9737	1.17829	0.589147	+	0.808026i	$0.299464\pi$
$692$	−0.324555	−0.0123377
$693$	0	0
$694$	−1.67544	−0.0635990
$695$	−36.3246	−1.37787
$696$	0	0
$697$	−0.701779	−0.0265818
$698$	28.9737	1.09667
$699$	0	0
$700$	−5.00000	−0.188982
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	5.02633	0.189572
$704$	1.00000	0.0376889
$705$	0	0
$706$	20.3246	0.764925
$707$	−6.00000	−0.225653
$708$	0	0
$709$	27.6754	1.03937	0.519687	−	0.854357i	$-0.326049\pi$
$709$	27.6754	1.03937	0.519687	+	0.854357i	$0.326049\pi$
$710$	25.2982	0.949425
$711$	0	0
$712$	3.67544	0.137743
$713$	−17.9473	−0.672133
$714$	0	0
$715$	−6.32456	−0.236525
$716$	16.3246	0.610077
$717$	0	0
$718$	16.3246	0.609227
$719$	−29.1623	−1.08757	−0.543785	−	0.839225i	$-0.683009\pi$
$719$	−29.1623	−1.08757	−0.543785	+	0.839225i	$0.683009\pi$
$720$	0	0
$721$	17.8114	0.663331
$722$	17.6491	0.656832
$723$	0	0
$724$	−1.48683	−0.0552577
$725$	−20.0000	−0.742781
$726$	0	0
$727$	4.13594	0.153394	0.0766968	−	0.997054i	$-0.475563\pi$
$727$	4.13594	0.153394	0.0766968	+	0.997054i	$0.475563\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	42.6491	1.57851
$731$	−5.29822	−0.195962
$732$	0	0
$733$	30.6491	1.13205	0.566025	−	0.824388i	$-0.308480\pi$
$733$	30.6491	1.13205	0.566025	+	0.824388i	$0.308480\pi$
$734$	−5.16228	−0.190543
$735$	0	0
$736$	6.32456	0.233126
$737$	−8.32456	−0.306639
$738$	0	0
$739$	28.6491	1.05387	0.526937	−	0.849904i	$-0.323340\pi$
$739$	28.6491	1.05387	0.526937	+	0.849904i	$0.323340\pi$
$740$	13.6754	0.502719
$741$	0	0
$742$	8.32456	0.305604
$743$	−28.6491	−1.05103	−0.525517	−	0.850783i	$-0.676128\pi$
$743$	−28.6491	−1.05103	−0.525517	+	0.850783i	$0.676128\pi$
$744$	0	0
$745$	6.32456	0.231714
$746$	10.0000	0.366126
$747$	0	0
$748$	−0.837722	−0.0306302
$749$	−1.67544	−0.0612194
$750$	0	0
$751$	18.3246	0.668673	0.334336	−	0.942454i	$-0.391488\pi$
$751$	18.3246	0.668673	0.334336	+	0.942454i	$0.391488\pi$
$752$	7.48683	0.273017
$753$	0	0
$754$	8.00000	0.291343
$755$	14.7018	0.535053
$756$	0	0
$757$	−33.6228	−1.22204	−0.611020	−	0.791615i	$-0.709241\pi$
$757$	−33.6228	−1.22204	−0.611020	+	0.791615i	$0.709241\pi$
$758$	−37.2982	−1.35473
$759$	0	0
$760$	−3.67544	−0.133322
$761$	−9.48683	−0.343897	−0.171949	−	0.985106i	$-0.555006\pi$
$761$	−9.48683	−0.343897	−0.171949	+	0.985106i	$0.555006\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	18.3246	0.662959
$765$	0	0
$766$	−2.83772	−0.102531
$767$	28.6491	1.03446
$768$	0	0
$769$	6.51317	0.234871	0.117435	−	0.993081i	$-0.462533\pi$
$769$	6.51317	0.234871	0.117435	+	0.993081i	$0.462533\pi$
$770$	−3.16228	−0.113961
$771$	0	0
$772$	−14.6491	−0.527233
$773$	50.1359	1.80326	0.901632	−	0.432503i	$-0.142370\pi$
$773$	50.1359	1.80326	0.901632	+	0.432503i	$0.142370\pi$
$774$	0	0
$775$	14.1886	0.509670
$776$	−14.6491	−0.525872
$777$	0	0
$778$	6.00000	0.215110
$779$	−0.973666	−0.0348852
$780$	0	0
$781$	8.00000	0.286263
$782$	−5.29822	−0.189464
$783$	0	0
$784$	1.00000	0.0357143
$785$	−42.6491	−1.52221
$786$	0	0
$787$	−50.4605	−1.79872	−0.899361	−	0.437206i	$-0.855968\pi$
$787$	−50.4605	−1.79872	−0.899361	+	0.437206i	$0.855968\pi$
$788$	−24.0000	−0.854965
$789$	0	0
$790$	26.3246	0.936586
$791$	2.00000	0.0711118
$792$	0	0
$793$	20.0000	0.710221
$794$	−2.51317	−0.0891890
$795$	0	0
$796$	0.513167	0.0181887
$797$	−15.1623	−0.537075	−0.268538	−	0.963269i	$-0.586540\pi$
$797$	−15.1623	−0.537075	−0.268538	+	0.963269i	$0.586540\pi$
$798$	0	0
$799$	−6.27189	−0.221883
$800$	−5.00000	−0.176777
$801$	0	0
$802$	−27.2982	−0.963934
$803$	13.4868	0.475940
$804$	0	0
$805$	−20.0000	−0.704907
$806$	−5.67544	−0.199909
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−56.3246	−1.98027	−0.990133	−	0.140131i	$-0.955248\pi$
$809$	−56.3246	−1.98027	−0.990133	+	0.140131i	$0.955248\pi$
$810$	0	0
$811$	11.4868	0.403357	0.201679	−	0.979452i	$-0.435360\pi$
$811$	11.4868	0.403357	0.201679	+	0.979452i	$0.435360\pi$
$812$	4.00000	0.140372
$813$	0	0
$814$	4.32456	0.151576
$815$	−27.3509	−0.958060
$816$	0	0
$817$	−7.35089	−0.257175
$818$	−20.8377	−0.728574
$819$	0	0
$820$	−2.64911	−0.0925110
$821$	−48.5964	−1.69603	−0.848014	−	0.529974i	$-0.822202\pi$
$821$	−48.5964	−1.69603	−0.848014	+	0.529974i	$0.822202\pi$
$822$	0	0
$823$	−46.9737	−1.63740	−0.818700	−	0.574222i	$-0.805305\pi$
$823$	−46.9737	−1.63740	−0.818700	+	0.574222i	$0.805305\pi$
$824$	17.8114	0.620489
$825$	0	0
$826$	14.3246	0.498415
$827$	41.9473	1.45865	0.729326	−	0.684167i	$-0.239834\pi$
$827$	41.9473	1.45865	0.729326	+	0.684167i	$0.239834\pi$
$828$	0	0
$829$	44.4605	1.54418	0.772088	−	0.635515i	$-0.219212\pi$
$829$	44.4605	1.54418	0.772088	+	0.635515i	$0.219212\pi$
$830$	−3.67544	−0.127577
$831$	0	0
$832$	2.00000	0.0693375
$833$	−0.837722	−0.0290254
$834$	0	0
$835$	20.0000	0.692129
$836$	−1.16228	−0.0401982
$837$	0	0
$838$	8.64911	0.298779
$839$	−22.4605	−0.775423	−0.387711	−	0.921781i	$-0.626734\pi$
$839$	−22.4605	−0.775423	−0.387711	+	0.921781i	$0.626734\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−35.9473	−1.23883
$843$	0	0
$844$	−5.67544	−0.195357
$845$	28.4605	0.979071
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	8.32456	0.285866
$849$	0	0
$850$	4.18861	0.143668
$851$	27.3509	0.937576
$852$	0	0
$853$	18.6491	0.638533	0.319267	−	0.947665i	$-0.396563\pi$
$853$	18.6491	0.638533	0.319267	+	0.947665i	$0.396563\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	−1.67544	−0.0572655
$857$	−55.8114	−1.90648	−0.953240	−	0.302213i	$-0.902275\pi$
$857$	−55.8114	−1.90648	−0.953240	+	0.302213i	$0.902275\pi$
$858$	0	0
$859$	−17.6754	−0.603078	−0.301539	−	0.953454i	$-0.597500\pi$
$859$	−17.6754	−0.603078	−0.301539	+	0.953454i	$0.597500\pi$
$860$	−20.0000	−0.681994
$861$	0	0
$862$	28.9737	0.986847
$863$	−30.3246	−1.03226	−0.516130	−	0.856510i	$-0.672628\pi$
$863$	−30.3246	−1.03226	−0.516130	+	0.856510i	$0.672628\pi$
$864$	0	0
$865$	1.02633	0.0348964
$866$	34.0000	1.15537
$867$	0	0
$868$	−2.83772	−0.0963186
$869$	8.32456	0.282391
$870$	0	0
$871$	−16.6491	−0.564134
$872$	8.00000	0.270914
$873$	0	0
$874$	−7.35089	−0.248648
$875$	0	0
$876$	0	0
$877$	−48.6491	−1.64276	−0.821382	−	0.570379i	$-0.806796\pi$
$877$	−48.6491	−1.64276	−0.821382	+	0.570379i	$0.806796\pi$
$878$	−6.32456	−0.213443
$879$	0	0
$880$	−3.16228	−0.106600
$881$	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
$883$	37.2982	1.25519	0.627593	−	0.778542i	$-0.284040\pi$
$883$	37.2982	1.25519	0.627593	+	0.778542i	$0.284040\pi$
$884$	−1.67544	−0.0563513
$885$	0	0
$886$	−8.32456	−0.279669
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	16.9737	0.569278
$890$	−11.6228	−0.389596
$891$	0	0
$892$	7.48683	0.250678
$893$	−8.70178	−0.291194
$894$	0	0
$895$	−51.6228	−1.72556
$896$	1.00000	0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−11.3509	−0.378573
$900$	0	0
$901$	−6.97367	−0.232326
$902$	−0.837722	−0.0278931
$903$	0	0
$904$	2.00000	0.0665190
$905$	4.70178	0.156292
$906$	0	0
$907$	−9.62278	−0.319519	−0.159760	−	0.987156i	$-0.551072\pi$
$907$	−9.62278	−0.319519	−0.159760	+	0.987156i	$0.551072\pi$
$908$	11.4868	0.381204
$909$	0	0
$910$	−6.32456	−0.209657
$911$	9.67544	0.320562	0.160281	−	0.987071i	$-0.448760\pi$
$911$	9.67544	0.320562	0.160281	+	0.987071i	$0.448760\pi$
$912$	0	0
$913$	−1.16228	−0.0384658
$914$	−42.6491	−1.41071
$915$	0	0
$916$	−6.51317	−0.215201
$917$	−1.16228	−0.0383818
$918$	0	0
$919$	33.2982	1.09841	0.549203	−	0.835689i	$-0.314931\pi$
$919$	33.2982	1.09841	0.549203	+	0.835689i	$0.314931\pi$
$920$	−20.0000	−0.659380
$921$	0	0
$922$	27.2982	0.899019
$923$	16.0000	0.526646
$924$	0	0
$925$	−21.6228	−0.710953
$926$	3.35089	0.110117
$927$	0	0
$928$	4.00000	0.131306
$929$	29.6228	0.971892	0.485946	−	0.873989i	$-0.338475\pi$
$929$	29.6228	0.971892	0.485946	+	0.873989i	$0.338475\pi$
$930$	0	0
$931$	−1.16228	−0.0380921
$932$	−6.00000	−0.196537
$933$	0	0
$934$	33.2982	1.08955
$935$	2.64911	0.0866352
$936$	0	0
$937$	2.13594	0.0697782	0.0348891	−	0.999391i	$-0.488892\pi$
$937$	2.13594	0.0697782	0.0348891	+	0.999391i	$0.488892\pi$
$938$	−8.32456	−0.271806
$939$	0	0
$940$	−23.6754	−0.772208
$941$	40.3246	1.31454	0.657271	−	0.753654i	$-0.271711\pi$
$941$	40.3246	1.31454	0.657271	+	0.753654i	$0.271711\pi$
$942$	0	0
$943$	−5.29822	−0.172534
$944$	14.3246	0.466225
$945$	0	0
$946$	−6.32456	−0.205629
$947$	−53.2982	−1.73196	−0.865980	−	0.500079i	$-0.833304\pi$
$947$	−53.2982	−1.73196	−0.865980	+	0.500079i	$0.833304\pi$
$948$	0	0
$949$	26.9737	0.875602
$950$	5.81139	0.188546
$951$	0	0
$952$	−0.837722	−0.0271507
$953$	−32.9737	−1.06812	−0.534061	−	0.845446i	$-0.679335\pi$
$953$	−32.9737	−1.06812	−0.534061	+	0.845446i	$0.679335\pi$
$954$	0	0
$955$	−57.9473	−1.87513
$956$	12.6491	0.409101
$957$	0	0
$958$	29.2982	0.946583
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−22.9473	−0.740237
$962$	8.64911	0.278859
$963$	0	0
$964$	7.16228	0.230681
$965$	46.3246	1.49124
$966$	0	0
$967$	−6.70178	−0.215515	−0.107757	−	0.994177i	$-0.534367\pi$
$967$	−6.70178	−0.215515	−0.107757	+	0.994177i	$0.534367\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	46.3246	1.48739
$971$	32.6491	1.04776	0.523880	−	0.851792i	$-0.324484\pi$
$971$	32.6491	1.04776	0.523880	+	0.851792i	$0.324484\pi$
$972$	0	0
$973$	−11.4868	−0.368251
$974$	1.67544	0.0536847
$975$	0	0
$976$	10.0000	0.320092
$977$	−0.649111	−0.0207669	−0.0103834	−	0.999946i	$-0.503305\pi$
$977$	−0.649111	−0.0207669	−0.0103834	+	0.999946i	$0.503305\pi$
$978$	0	0
$979$	−3.67544	−0.117468
$980$	−3.16228	−0.101015
$981$	0	0
$982$	−16.6491	−0.531294
$983$	−3.86406	−0.123244	−0.0616221	−	0.998100i	$-0.519627\pi$
$983$	−3.86406	−0.123244	−0.0616221	+	0.998100i	$0.519627\pi$
$984$	0	0
$985$	75.8947	2.41821
$986$	−3.35089	−0.106714
$987$	0	0
$988$	−2.32456	−0.0739540
$989$	−40.0000	−1.27193
$990$	0	0
$991$	−10.7018	−0.339953	−0.169977	−	0.985448i	$-0.554369\pi$
$991$	−10.7018	−0.339953	−0.169977	+	0.985448i	$0.554369\pi$
$992$	−2.83772	−0.0900978
$993$	0	0
$994$	8.00000	0.253745
$995$	−1.62278	−0.0514455
$996$	0	0
$997$	11.0263	0.349208	0.174604	−	0.984639i	$-0.444136\pi$
$997$	11.0263	0.349208	0.174604	+	0.984639i	$0.444136\pi$
$998$	−20.9737	−0.663910
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.a.n.1.1	✓	2
3.2	odd	2		1386.2.a.o.1.2	yes	2
7.6	odd	2		9702.2.a.cn.1.2		2
21.20	even	2		9702.2.a.dc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.a.n.1.1	✓	2	1.1	even	1	trivial
1386.2.a.o.1.2	yes	2	3.2	odd	2
9702.2.a.cn.1.2		2	7.6	odd	2
9702.2.a.dc.1.1		2	21.20	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.16228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.16228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.16228 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -0.837722 q^{17} -1.16228 q^{19} -3.16228 q^{20} -1.00000 q^{22} -6.32456 q^{23} +5.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} -4.00000 q^{29} +2.83772 q^{31} -1.00000 q^{32} +0.837722 q^{34} +3.16228 q^{35} -4.32456 q^{37} +1.16228 q^{38} +3.16228 q^{40} +0.837722 q^{41} +6.32456 q^{43} +1.00000 q^{44} +6.32456 q^{46} +7.48683 q^{47} +1.00000 q^{49} -5.00000 q^{50} +2.00000 q^{52} +8.32456 q^{53} -3.16228 q^{55} +1.00000 q^{56} +4.00000 q^{58} +14.3246 q^{59} +10.0000 q^{61} -2.83772 q^{62} +1.00000 q^{64} -6.32456 q^{65} -8.32456 q^{67} -0.837722 q^{68} -3.16228 q^{70} +8.00000 q^{71} +13.4868 q^{73} +4.32456 q^{74} -1.16228 q^{76} -1.00000 q^{77} +8.32456 q^{79} -3.16228 q^{80} -0.837722 q^{82} -1.16228 q^{83} +2.64911 q^{85} -6.32456 q^{86} -1.00000 q^{88} -3.67544 q^{89} -2.00000 q^{91} -6.32456 q^{92} -7.48683 q^{94} +3.67544 q^{95} +14.6491 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 8 q^{17} + 4 q^{19} - 2 q^{22} + 10 q^{25} - 4 q^{26} - 2 q^{28} - 8 q^{29} + 12 q^{31} - 2 q^{32} + 8 q^{34} + 4 q^{37} - 4 q^{38} + 8 q^{41} + 2 q^{44} - 4 q^{47} + 2 q^{49} - 10 q^{50} + 4 q^{52} + 4 q^{53} + 2 q^{56} + 8 q^{58} + 16 q^{59} + 20 q^{61} - 12 q^{62} + 2 q^{64} - 4 q^{67} - 8 q^{68} + 16 q^{71} + 8 q^{73} - 4 q^{74} + 4 q^{76} - 2 q^{77} + 4 q^{79} - 8 q^{82} + 4 q^{83} - 20 q^{85} - 2 q^{88} - 20 q^{89} - 4 q^{91} + 4 q^{94} + 20 q^{95} + 4 q^{97} - 2 q^{98}+O(q^{100}) \) 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 - 8 * q^17 + 4 * q^19 - 2 * q^22 + 10 * q^25 - 4 * q^26 - 2 * q^28 - 8 * q^29 + 12 * q^31 - 2 * q^32 + 8 * q^34 + 4 * q^37 - 4 * q^38 + 8 * q^41 + 2 * q^44 - 4 * q^47 + 2 * q^49 - 10 * q^50 + 4 * q^52 + 4 * q^53 + 2 * q^56 + 8 * q^58 + 16 * q^59 + 20 * q^61 - 12 * q^62 + 2 * q^64 - 4 * q^67 - 8 * q^68 + 16 * q^71 + 8 * q^73 - 4 * q^74 + 4 * q^76 - 2 * q^77 + 4 * q^79 - 8 * q^82 + 4 * q^83 - 20 * q^85 - 2 * q^88 - 20 * q^89 - 4 * q^91 + 4 * q^94 + 20 * q^95 + 4 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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