LMFDB - Embedded newform 6864.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 858)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.70156 q^{15} +4.00000 q^{17} -6.00000 q^{19} -1.70156 q^{21} +0.298438 q^{23} +8.70156 q^{25} -1.00000 q^{27} -0.298438 q^{29} -5.40312 q^{31} -1.00000 q^{33} -6.29844 q^{35} +7.40312 q^{37} +1.00000 q^{39} -0.298438 q^{41} -9.10469 q^{43} -3.70156 q^{45} -4.00000 q^{47} -4.10469 q^{49} -4.00000 q^{51} +7.40312 q^{53} -3.70156 q^{55} +6.00000 q^{57} +13.7016 q^{59} +4.29844 q^{61} +1.70156 q^{63} +3.70156 q^{65} +5.10469 q^{67} -0.298438 q^{69} +11.4031 q^{71} -1.70156 q^{73} -8.70156 q^{75} +1.70156 q^{77} +3.40312 q^{79} +1.00000 q^{81} -15.4031 q^{83} -14.8062 q^{85} +0.298438 q^{87} +1.40312 q^{89} -1.70156 q^{91} +5.40312 q^{93} +22.2094 q^{95} +8.80625 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + q^{15} + 8 q^{17} - 12 q^{19} + 3 q^{21} + 7 q^{23} + 11 q^{25} - 2 q^{27} - 7 q^{29} + 2 q^{31} - 2 q^{33} - 19 q^{35} + 2 q^{37} + 2 q^{39} - 7 q^{41} + q^{43} - q^{45} - 8 q^{47} + 11 q^{49} - 8 q^{51} + 2 q^{53} - q^{55} + 12 q^{57} + 21 q^{59} + 15 q^{61} - 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 10 q^{71} + 3 q^{73} - 11 q^{75} - 3 q^{77} - 6 q^{79} + 2 q^{81} - 18 q^{83} - 4 q^{85} + 7 q^{87} - 10 q^{89} + 3 q^{91} - 2 q^{93} + 6 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + q^15 + 8 * q^17 - 12 * q^19 + 3 * q^21 + 7 * q^23 + 11 * q^25 - 2 * q^27 - 7 * q^29 + 2 * q^31 - 2 * q^33 - 19 * q^35 + 2 * q^37 + 2 * q^39 - 7 * q^41 + q^43 - q^45 - 8 * q^47 + 11 * q^49 - 8 * q^51 + 2 * q^53 - q^55 + 12 * q^57 + 21 * q^59 + 15 * q^61 - 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 10 * q^71 + 3 * q^73 - 11 * q^75 - 3 * q^77 - 6 * q^79 + 2 * q^81 - 18 * q^83 - 4 * q^85 + 7 * q^87 - 10 * q^89 + 3 * q^91 - 2 * q^93 + 6 * q^95 - 8 * q^97 + 2 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.70156	−1.65539	−0.827694	−	0.561179i	$-0.810348\pi$
$5$	−3.70156	−1.65539	−0.827694	+	0.561179i	$0.810348\pi$
$6$	0	0
$7$	1.70156	0.643130	0.321565	−	0.946888i	$-0.395791\pi$
$7$	1.70156	0.643130	0.321565	+	0.946888i	$0.395791\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	3.70156	0.955739
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−1.70156	−0.371311
$22$	0	0
$23$	0.298438	0.0622286	0.0311143	−	0.999516i	$-0.490094\pi$
$23$	0.298438	0.0622286	0.0311143	+	0.999516i	$0.490094\pi$
$24$	0	0
$25$	8.70156	1.74031
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.298438	−0.0554185	−0.0277093	−	0.999616i	$-0.508821\pi$
$29$	−0.298438	−0.0554185	−0.0277093	+	0.999616i	$0.508821\pi$
$30$	0	0
$31$	−5.40312	−0.970430	−0.485215	−	0.874395i	$-0.661259\pi$
$31$	−5.40312	−0.970430	−0.485215	+	0.874395i	$0.661259\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−6.29844	−1.06463
$36$	0	0
$37$	7.40312	1.21707	0.608533	−	0.793529i	$-0.291758\pi$
$37$	7.40312	1.21707	0.608533	+	0.793529i	$0.291758\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−0.298438	−0.0466082	−0.0233041	−	0.999728i	$-0.507419\pi$
$41$	−0.298438	−0.0466082	−0.0233041	+	0.999728i	$0.507419\pi$
$42$	0	0
$43$	−9.10469	−1.38845	−0.694226	−	0.719757i	$-0.744253\pi$
$43$	−9.10469	−1.38845	−0.694226	+	0.719757i	$0.744253\pi$
$44$	0	0
$45$	−3.70156	−0.551796
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−4.10469	−0.586384
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	7.40312	1.01690	0.508449	−	0.861092i	$-0.330219\pi$
$53$	7.40312	1.01690	0.508449	+	0.861092i	$0.330219\pi$
$54$	0	0
$55$	−3.70156	−0.499119
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	13.7016	1.78379	0.891896	−	0.452241i	$-0.149375\pi$
$59$	13.7016	1.78379	0.891896	+	0.452241i	$0.149375\pi$
$60$	0	0
$61$	4.29844	0.550359	0.275179	−	0.961393i	$-0.411263\pi$
$61$	4.29844	0.550359	0.275179	+	0.961393i	$0.411263\pi$
$62$	0	0
$63$	1.70156	0.214377
$64$	0	0
$65$	3.70156	0.459122
$66$	0	0
$67$	5.10469	0.623637	0.311818	−	0.950142i	$-0.399062\pi$
$67$	5.10469	0.623637	0.311818	+	0.950142i	$0.399062\pi$
$68$	0	0
$69$	−0.298438	−0.0359277
$70$	0	0
$71$	11.4031	1.35330	0.676651	−	0.736304i	$-0.263431\pi$
$71$	11.4031	1.35330	0.676651	+	0.736304i	$0.263431\pi$
$72$	0	0
$73$	−1.70156	−0.199153	−0.0995764	−	0.995030i	$-0.531749\pi$
$73$	−1.70156	−0.199153	−0.0995764	+	0.995030i	$0.531749\pi$
$74$	0	0
$75$	−8.70156	−1.00477
$76$	0	0
$77$	1.70156	0.193911
$78$	0	0
$79$	3.40312	0.382881	0.191441	−	0.981504i	$-0.438684\pi$
$79$	3.40312	0.382881	0.191441	+	0.981504i	$0.438684\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.4031	−1.69071	−0.845356	−	0.534203i	$-0.820612\pi$
$83$	−15.4031	−1.69071	−0.845356	+	0.534203i	$0.820612\pi$
$84$	0	0
$85$	−14.8062	−1.60596
$86$	0	0
$87$	0.298438	0.0319959
$88$	0	0
$89$	1.40312	0.148731	0.0743654	−	0.997231i	$-0.476307\pi$
$89$	1.40312	0.148731	0.0743654	+	0.997231i	$0.476307\pi$
$90$	0	0
$91$	−1.70156	−0.178372
$92$	0	0
$93$	5.40312	0.560278
$94$	0	0
$95$	22.2094	2.27863
$96$	0	0
$97$	8.80625	0.894139	0.447070	−	0.894499i	$-0.352468\pi$
$97$	8.80625	0.894139	0.447070	+	0.894499i	$0.352468\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	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
$103$	10.2984	1.01474	0.507368	−	0.861730i	$-0.330619\pi$
$103$	10.2984	1.01474	0.507368	+	0.861730i	$0.330619\pi$
$104$	0	0
$105$	6.29844	0.614665
$106$	0	0
$107$	−11.7016	−1.13123	−0.565616	−	0.824669i	$-0.691362\pi$
$107$	−11.7016	−1.13123	−0.565616	+	0.824669i	$0.691362\pi$
$108$	0	0
$109$	−1.40312	−0.134395	−0.0671975	−	0.997740i	$-0.521406\pi$
$109$	−1.40312	−0.134395	−0.0671975	+	0.997740i	$0.521406\pi$
$110$	0	0
$111$	−7.40312	−0.702673
$112$	0	0
$113$	19.1047	1.79722	0.898609	−	0.438751i	$-0.144579\pi$
$113$	19.1047	1.79722	0.898609	+	0.438751i	$0.144579\pi$
$114$	0	0
$115$	−1.10469	−0.103013
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	6.80625	0.623928
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.298438	0.0269092
$124$	0	0
$125$	−13.7016	−1.22550
$126$	0	0
$127$	3.40312	0.301978	0.150989	−	0.988535i	$-0.451754\pi$
$127$	3.40312	0.301978	0.150989	+	0.988535i	$0.451754\pi$
$128$	0	0
$129$	9.10469	0.801623
$130$	0	0
$131$	−19.1047	−1.66918	−0.834592	−	0.550868i	$-0.814297\pi$
$131$	−19.1047	−1.66918	−0.834592	+	0.550868i	$0.814297\pi$
$132$	0	0
$133$	−10.2094	−0.885265
$134$	0	0
$135$	3.70156	0.318580
$136$	0	0
$137$	−12.8062	−1.09411	−0.547056	−	0.837096i	$-0.684251\pi$
$137$	−12.8062	−1.09411	−0.547056	+	0.837096i	$0.684251\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	1.10469	0.0917392
$146$	0	0
$147$	4.10469	0.338549
$148$	0	0
$149$	16.2094	1.32792	0.663962	−	0.747767i	$-0.268874\pi$
$149$	16.2094	1.32792	0.663962	+	0.747767i	$0.268874\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	20.0000	1.60644
$156$	0	0
$157$	−12.8062	−1.02205	−0.511025	−	0.859566i	$-0.670734\pi$
$157$	−12.8062	−1.02205	−0.511025	+	0.859566i	$0.670734\pi$
$158$	0	0
$159$	−7.40312	−0.587106
$160$	0	0
$161$	0.507811	0.0400211
$162$	0	0
$163$	−17.1047	−1.33974	−0.669871	−	0.742477i	$-0.733651\pi$
$163$	−17.1047	−1.33974	−0.669871	+	0.742477i	$0.733651\pi$
$164$	0	0
$165$	3.70156	0.288166
$166$	0	0
$167$	1.70156	0.131671	0.0658354	−	0.997830i	$-0.479029\pi$
$167$	1.70156	0.131671	0.0658354	+	0.997830i	$0.479029\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	−22.5078	−1.71124	−0.855619	−	0.517607i	$-0.826823\pi$
$173$	−22.5078	−1.71124	−0.855619	+	0.517607i	$0.826823\pi$
$174$	0	0
$175$	14.8062	1.11925
$176$	0	0
$177$	−13.7016	−1.02987
$178$	0	0
$179$	10.8062	0.807697	0.403848	−	0.914826i	$-0.367672\pi$
$179$	10.8062	0.807697	0.403848	+	0.914826i	$0.367672\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	−4.29844	−0.317750
$184$	0	0
$185$	−27.4031	−2.01472
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	−1.70156	−0.123770
$190$	0	0
$191$	10.5078	0.760318	0.380159	−	0.924921i	$-0.375869\pi$
$191$	10.5078	0.760318	0.380159	+	0.924921i	$0.375869\pi$
$192$	0	0
$193$	−8.59688	−0.618817	−0.309408	−	0.950929i	$-0.600131\pi$
$193$	−8.59688	−0.618817	−0.309408	+	0.950929i	$0.600131\pi$
$194$	0	0
$195$	−3.70156	−0.265074
$196$	0	0
$197$	−6.59688	−0.470008	−0.235004	−	0.971994i	$-0.575510\pi$
$197$	−6.59688	−0.470008	−0.235004	+	0.971994i	$0.575510\pi$
$198$	0	0
$199$	2.89531	0.205243	0.102622	−	0.994720i	$-0.467277\pi$
$199$	2.89531	0.205243	0.102622	+	0.994720i	$0.467277\pi$
$200$	0	0
$201$	−5.10469	−0.360057
$202$	0	0
$203$	−0.507811	−0.0356413
$204$	0	0
$205$	1.10469	0.0771546
$206$	0	0
$207$	0.298438	0.0207429
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	6.80625	0.468561	0.234281	−	0.972169i	$-0.424727\pi$
$211$	6.80625	0.468561	0.234281	+	0.972169i	$0.424727\pi$
$212$	0	0
$213$	−11.4031	−0.781329
$214$	0	0
$215$	33.7016	2.29843
$216$	0	0
$217$	−9.19375	−0.624113
$218$	0	0
$219$	1.70156	0.114981
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−21.4031	−1.43326	−0.716630	−	0.697454i	$-0.754316\pi$
$223$	−21.4031	−1.43326	−0.716630	+	0.697454i	$0.754316\pi$
$224$	0	0
$225$	8.70156	0.580104
$226$	0	0
$227$	−22.8062	−1.51370	−0.756852	−	0.653586i	$-0.773264\pi$
$227$	−22.8062	−1.51370	−0.756852	+	0.653586i	$0.773264\pi$
$228$	0	0
$229$	−5.70156	−0.376770	−0.188385	−	0.982095i	$-0.560325\pi$
$229$	−5.70156	−0.376770	−0.188385	+	0.982095i	$0.560325\pi$
$230$	0	0
$231$	−1.70156	−0.111955
$232$	0	0
$233$	19.4031	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$233$	19.4031	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$234$	0	0
$235$	14.8062	0.965853
$236$	0	0
$237$	−3.40312	−0.221057
$238$	0	0
$239$	−1.70156	−0.110065	−0.0550325	−	0.998485i	$-0.517526\pi$
$239$	−1.70156	−0.110065	−0.0550325	+	0.998485i	$0.517526\pi$
$240$	0	0
$241$	−27.4031	−1.76519	−0.882595	−	0.470134i	$-0.844206\pi$
$241$	−27.4031	−1.76519	−0.882595	+	0.470134i	$0.844206\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.1938	0.970693
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	15.4031	0.976133
$250$	0	0
$251$	7.40312	0.467281	0.233641	−	0.972323i	$-0.424936\pi$
$251$	7.40312	0.467281	0.233641	+	0.972323i	$0.424936\pi$
$252$	0	0
$253$	0.298438	0.0187626
$254$	0	0
$255$	14.8062	0.927203
$256$	0	0
$257$	3.70156	0.230897	0.115449	−	0.993313i	$-0.463169\pi$
$257$	3.70156	0.230897	0.115449	+	0.993313i	$0.463169\pi$
$258$	0	0
$259$	12.5969	0.782732
$260$	0	0
$261$	−0.298438	−0.0184728
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−27.4031	−1.68336
$266$	0	0
$267$	−1.40312	−0.0858698
$268$	0	0
$269$	30.8062	1.87829	0.939145	−	0.343521i	$-0.111620\pi$
$269$	30.8062	1.87829	0.939145	+	0.343521i	$0.111620\pi$
$270$	0	0
$271$	−26.8062	−1.62836	−0.814182	−	0.580610i	$-0.802814\pi$
$271$	−26.8062	−1.62836	−0.814182	+	0.580610i	$0.802814\pi$
$272$	0	0
$273$	1.70156	0.102983
$274$	0	0
$275$	8.70156	0.524724
$276$	0	0
$277$	15.1047	0.907553	0.453776	−	0.891116i	$-0.350077\pi$
$277$	15.1047	0.907553	0.453776	+	0.891116i	$0.350077\pi$
$278$	0	0
$279$	−5.40312	−0.323477
$280$	0	0
$281$	−19.7016	−1.17530	−0.587648	−	0.809116i	$-0.699946\pi$
$281$	−19.7016	−1.17530	−0.587648	+	0.809116i	$0.699946\pi$
$282$	0	0
$283$	5.70156	0.338923	0.169461	−	0.985537i	$-0.445797\pi$
$283$	5.70156	0.338923	0.169461	+	0.985537i	$0.445797\pi$
$284$	0	0
$285$	−22.2094	−1.31557
$286$	0	0
$287$	−0.507811	−0.0299751
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−8.80625	−0.516231
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	−50.7172	−2.95287
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−0.298438	−0.0172591
$300$	0	0
$301$	−15.4922	−0.892955
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−15.9109	−0.911057
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	0	0
$309$	−10.2984	−0.585858
$310$	0	0
$311$	25.4031	1.44048	0.720240	−	0.693725i	$-0.244032\pi$
$311$	25.4031	1.44048	0.720240	+	0.693725i	$0.244032\pi$
$312$	0	0
$313$	−29.9109	−1.69067	−0.845333	−	0.534240i	$-0.820598\pi$
$313$	−29.9109	−1.69067	−0.845333	+	0.534240i	$0.820598\pi$
$314$	0	0
$315$	−6.29844	−0.354877
$316$	0	0
$317$	−13.9109	−0.781316	−0.390658	−	0.920536i	$-0.627752\pi$
$317$	−13.9109	−0.781316	−0.390658	+	0.920536i	$0.627752\pi$
$318$	0	0
$319$	−0.298438	−0.0167093
$320$	0	0
$321$	11.7016	0.653118
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−8.70156	−0.482676
$326$	0	0
$327$	1.40312	0.0775929
$328$	0	0
$329$	−6.80625	−0.375241
$330$	0	0
$331$	−24.5078	−1.34707	−0.673536	−	0.739155i	$-0.735225\pi$
$331$	−24.5078	−1.34707	−0.673536	+	0.739155i	$0.735225\pi$
$332$	0	0
$333$	7.40312	0.405689
$334$	0	0
$335$	−18.8953	−1.03236
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−19.1047	−1.03762
$340$	0	0
$341$	−5.40312	−0.292596
$342$	0	0
$343$	−18.8953	−1.02025
$344$	0	0
$345$	1.10469	0.0594743
$346$	0	0
$347$	−13.4031	−0.719517	−0.359759	−	0.933045i	$-0.617141\pi$
$347$	−13.4031	−0.719517	−0.359759	+	0.933045i	$0.617141\pi$
$348$	0	0
$349$	8.20937	0.439438	0.219719	−	0.975563i	$-0.429486\pi$
$349$	8.20937	0.439438	0.219719	+	0.975563i	$0.429486\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	−42.2094	−2.24024
$356$	0	0
$357$	−6.80625	−0.360225
$358$	0	0
$359$	−14.2984	−0.754643	−0.377321	−	0.926082i	$-0.623155\pi$
$359$	−14.2984	−0.754643	−0.377321	+	0.926082i	$0.623155\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	6.29844	0.329675
$366$	0	0
$367$	−2.80625	−0.146485	−0.0732425	−	0.997314i	$-0.523335\pi$
$367$	−2.80625	−0.146485	−0.0732425	+	0.997314i	$0.523335\pi$
$368$	0	0
$369$	−0.298438	−0.0155361
$370$	0	0
$371$	12.5969	0.653997
$372$	0	0
$373$	−33.3141	−1.72494	−0.862468	−	0.506111i	$-0.831083\pi$
$373$	−33.3141	−1.72494	−0.862468	+	0.506111i	$0.831083\pi$
$374$	0	0
$375$	13.7016	0.707546
$376$	0	0
$377$	0.298438	0.0153703
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−3.40312	−0.174347
$382$	0	0
$383$	−8.59688	−0.439280	−0.219640	−	0.975581i	$-0.570488\pi$
$383$	−8.59688	−0.439280	−0.219640	+	0.975581i	$0.570488\pi$
$384$	0	0
$385$	−6.29844	−0.320998
$386$	0	0
$387$	−9.10469	−0.462817
$388$	0	0
$389$	−7.40312	−0.375353	−0.187677	−	0.982231i	$-0.560096\pi$
$389$	−7.40312	−0.375353	−0.187677	+	0.982231i	$0.560096\pi$
$390$	0	0
$391$	1.19375	0.0603706
$392$	0	0
$393$	19.1047	0.963704
$394$	0	0
$395$	−12.5969	−0.633818
$396$	0	0
$397$	11.9109	0.597793	0.298896	−	0.954286i	$-0.403382\pi$
$397$	11.9109	0.597793	0.298896	+	0.954286i	$0.403382\pi$
$398$	0	0
$399$	10.2094	0.511108
$400$	0	0
$401$	12.2094	0.609707	0.304853	−	0.952399i	$-0.401393\pi$
$401$	12.2094	0.609707	0.304853	+	0.952399i	$0.401393\pi$
$402$	0	0
$403$	5.40312	0.269149
$404$	0	0
$405$	−3.70156	−0.183932
$406$	0	0
$407$	7.40312	0.366959
$408$	0	0
$409$	4.50781	0.222897	0.111448	−	0.993770i	$-0.464451\pi$
$409$	4.50781	0.222897	0.111448	+	0.993770i	$0.464451\pi$
$410$	0	0
$411$	12.8062	0.631686
$412$	0	0
$413$	23.3141	1.14721
$414$	0	0
$415$	57.0156	2.79879
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	10.8062	0.527920	0.263960	−	0.964534i	$-0.414971\pi$
$419$	10.8062	0.527920	0.263960	+	0.964534i	$0.414971\pi$
$420$	0	0
$421$	−28.5078	−1.38939	−0.694693	−	0.719307i	$-0.744460\pi$
$421$	−28.5078	−1.38939	−0.694693	+	0.719307i	$0.744460\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	34.8062	1.68835
$426$	0	0
$427$	7.31406	0.353952
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−29.6125	−1.42638	−0.713192	−	0.700969i	$-0.752751\pi$
$431$	−29.6125	−1.42638	−0.713192	+	0.700969i	$0.752751\pi$
$432$	0	0
$433$	27.1047	1.30257	0.651284	−	0.758834i	$-0.274231\pi$
$433$	27.1047	1.30257	0.651284	+	0.758834i	$0.274231\pi$
$434$	0	0
$435$	−1.10469	−0.0529657
$436$	0	0
$437$	−1.79063	−0.0856573
$438$	0	0
$439$	−14.8062	−0.706664	−0.353332	−	0.935498i	$-0.614951\pi$
$439$	−14.8062	−0.706664	−0.353332	+	0.935498i	$0.614951\pi$
$440$	0	0
$441$	−4.10469	−0.195461
$442$	0	0
$443$	−30.8062	−1.46365	−0.731825	−	0.681493i	$-0.761331\pi$
$443$	−30.8062	−1.46365	−0.731825	+	0.681493i	$0.761331\pi$
$444$	0	0
$445$	−5.19375	−0.246207
$446$	0	0
$447$	−16.2094	−0.766677
$448$	0	0
$449$	−31.0156	−1.46372	−0.731859	−	0.681456i	$-0.761347\pi$
$449$	−31.0156	−1.46372	−0.731859	+	0.681456i	$0.761347\pi$
$450$	0	0
$451$	−0.298438	−0.0140529
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.29844	0.295275
$456$	0	0
$457$	−15.9109	−0.744282	−0.372141	−	0.928176i	$-0.621376\pi$
$457$	−15.9109	−0.744282	−0.372141	+	0.928176i	$0.621376\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	4.20937	0.196050	0.0980250	−	0.995184i	$-0.468747\pi$
$461$	4.20937	0.196050	0.0980250	+	0.995184i	$0.468747\pi$
$462$	0	0
$463$	−20.8062	−0.966948	−0.483474	−	0.875359i	$-0.660625\pi$
$463$	−20.8062	−0.966948	−0.483474	+	0.875359i	$0.660625\pi$
$464$	0	0
$465$	−20.0000	−0.927478
$466$	0	0
$467$	36.4187	1.68526	0.842629	−	0.538494i	$-0.181007\pi$
$467$	36.4187	1.68526	0.842629	+	0.538494i	$0.181007\pi$
$468$	0	0
$469$	8.68594	0.401079
$470$	0	0
$471$	12.8062	0.590081
$472$	0	0
$473$	−9.10469	−0.418634
$474$	0	0
$475$	−52.2094	−2.39553
$476$	0	0
$477$	7.40312	0.338966
$478$	0	0
$479$	−7.49219	−0.342327	−0.171163	−	0.985243i	$-0.554753\pi$
$479$	−7.49219	−0.342327	−0.171163	+	0.985243i	$0.554753\pi$
$480$	0	0
$481$	−7.40312	−0.337553
$482$	0	0
$483$	−0.507811	−0.0231062
$484$	0	0
$485$	−32.5969	−1.48015
$486$	0	0
$487$	39.6125	1.79501	0.897507	−	0.441001i	$-0.145376\pi$
$487$	39.6125	1.79501	0.897507	+	0.441001i	$0.145376\pi$
$488$	0	0
$489$	17.1047	0.773501
$490$	0	0
$491$	−3.10469	−0.140113	−0.0700563	−	0.997543i	$-0.522318\pi$
$491$	−3.10469	−0.140113	−0.0700563	+	0.997543i	$0.522318\pi$
$492$	0	0
$493$	−1.19375	−0.0537639
$494$	0	0
$495$	−3.70156	−0.166373
$496$	0	0
$497$	19.4031	0.870349
$498$	0	0
$499$	−11.9109	−0.533207	−0.266603	−	0.963806i	$-0.585901\pi$
$499$	−11.9109	−0.533207	−0.266603	+	0.963806i	$0.585901\pi$
$500$	0	0
$501$	−1.70156	−0.0760202
$502$	0	0
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	−22.2094	−0.988304
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	43.6125	1.93309	0.966545	−	0.256497i	$-0.0825684\pi$
$509$	43.6125	1.93309	0.966545	+	0.256497i	$0.0825684\pi$
$510$	0	0
$511$	−2.89531	−0.128081
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	−38.1203	−1.67978
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	22.5078	0.987983
$520$	0	0
$521$	−19.1047	−0.836992	−0.418496	−	0.908219i	$-0.637443\pi$
$521$	−19.1047	−0.836992	−0.418496	+	0.908219i	$0.637443\pi$
$522$	0	0
$523$	25.6125	1.11996	0.559978	−	0.828507i	$-0.310810\pi$
$523$	25.6125	1.11996	0.559978	+	0.828507i	$0.310810\pi$
$524$	0	0
$525$	−14.8062	−0.646198
$526$	0	0
$527$	−21.6125	−0.941455
$528$	0	0
$529$	−22.9109	−0.996128
$530$	0	0
$531$	13.7016	0.594597
$532$	0	0
$533$	0.298438	0.0129268
$534$	0	0
$535$	43.3141	1.87263
$536$	0	0
$537$	−10.8062	−0.466324
$538$	0	0
$539$	−4.10469	−0.176801
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	5.19375	0.222476
$546$	0	0
$547$	1.70156	0.0727535	0.0363768	−	0.999338i	$-0.488418\pi$
$547$	1.70156	0.0727535	0.0363768	+	0.999338i	$0.488418\pi$
$548$	0	0
$549$	4.29844	0.183453
$550$	0	0
$551$	1.79063	0.0762833
$552$	0	0
$553$	5.79063	0.246243
$554$	0	0
$555$	27.4031	1.16320
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	9.10469	0.385087
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−25.4031	−1.07061	−0.535307	−	0.844658i	$-0.679804\pi$
$563$	−25.4031	−1.07061	−0.535307	+	0.844658i	$0.679804\pi$
$564$	0	0
$565$	−70.7172	−2.97509
$566$	0	0
$567$	1.70156	0.0714589
$568$	0	0
$569$	−4.59688	−0.192711	−0.0963555	−	0.995347i	$-0.530719\pi$
$569$	−4.59688	−0.192711	−0.0963555	+	0.995347i	$0.530719\pi$
$570$	0	0
$571$	36.5078	1.52780	0.763902	−	0.645332i	$-0.223281\pi$
$571$	36.5078	1.52780	0.763902	+	0.645332i	$0.223281\pi$
$572$	0	0
$573$	−10.5078	−0.438970
$574$	0	0
$575$	2.59688	0.108297
$576$	0	0
$577$	8.20937	0.341761	0.170880	−	0.985292i	$-0.445339\pi$
$577$	8.20937	0.341761	0.170880	+	0.985292i	$0.445339\pi$
$578$	0	0
$579$	8.59688	0.357274
$580$	0	0
$581$	−26.2094	−1.08735
$582$	0	0
$583$	7.40312	0.306606
$584$	0	0
$585$	3.70156	0.153041
$586$	0	0
$587$	−37.7016	−1.55611	−0.778055	−	0.628196i	$-0.783794\pi$
$587$	−37.7016	−1.55611	−0.778055	+	0.628196i	$0.783794\pi$
$588$	0	0
$589$	32.4187	1.33579
$590$	0	0
$591$	6.59688	0.271359
$592$	0	0
$593$	−4.80625	−0.197369	−0.0986845	−	0.995119i	$-0.531463\pi$
$593$	−4.80625	−0.197369	−0.0986845	+	0.995119i	$0.531463\pi$
$594$	0	0
$595$	−25.1938	−1.03284
$596$	0	0
$597$	−2.89531	−0.118497
$598$	0	0
$599$	25.9109	1.05869	0.529346	−	0.848406i	$-0.322437\pi$
$599$	25.9109	1.05869	0.529346	+	0.848406i	$0.322437\pi$
$600$	0	0
$601$	25.4031	1.03622	0.518108	−	0.855315i	$-0.326637\pi$
$601$	25.4031	1.03622	0.518108	+	0.855315i	$0.326637\pi$
$602$	0	0
$603$	5.10469	0.207879
$604$	0	0
$605$	−3.70156	−0.150490
$606$	0	0
$607$	18.2094	0.739096	0.369548	−	0.929212i	$-0.379513\pi$
$607$	18.2094	0.739096	0.369548	+	0.929212i	$0.379513\pi$
$608$	0	0
$609$	0.507811	0.0205775
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	36.2094	1.46248	0.731241	−	0.682119i	$-0.238941\pi$
$613$	36.2094	1.46248	0.731241	+	0.682119i	$0.238941\pi$
$614$	0	0
$615$	−1.10469	−0.0445453
$616$	0	0
$617$	−39.6125	−1.59474	−0.797370	−	0.603491i	$-0.793776\pi$
$617$	−39.6125	−1.59474	−0.797370	+	0.603491i	$0.793776\pi$
$618$	0	0
$619$	2.29844	0.0923820	0.0461910	−	0.998933i	$-0.485292\pi$
$619$	2.29844	0.0923820	0.0461910	+	0.998933i	$0.485292\pi$
$620$	0	0
$621$	−0.298438	−0.0119759
$622$	0	0
$623$	2.38750	0.0956533
$624$	0	0
$625$	7.20937	0.288375
$626$	0	0
$627$	6.00000	0.239617
$628$	0	0
$629$	29.6125	1.18073
$630$	0	0
$631$	−48.8062	−1.94295	−0.971473	−	0.237150i	$-0.923787\pi$
$631$	−48.8062	−1.94295	−0.971473	+	0.237150i	$0.923787\pi$
$632$	0	0
$633$	−6.80625	−0.270524
$634$	0	0
$635$	−12.5969	−0.499892
$636$	0	0
$637$	4.10469	0.162634
$638$	0	0
$639$	11.4031	0.451101
$640$	0	0
$641$	−0.298438	−0.0117876	−0.00589379	−	0.999983i	$-0.501876\pi$
$641$	−0.298438	−0.0117876	−0.00589379	+	0.999983i	$0.501876\pi$
$642$	0	0
$643$	9.19375	0.362566	0.181283	−	0.983431i	$-0.441975\pi$
$643$	9.19375	0.362566	0.181283	+	0.983431i	$0.441975\pi$
$644$	0	0
$645$	−33.7016	−1.32700
$646$	0	0
$647$	−23.0156	−0.904837	−0.452419	−	0.891806i	$-0.649439\pi$
$647$	−23.0156	−0.904837	−0.452419	+	0.891806i	$0.649439\pi$
$648$	0	0
$649$	13.7016	0.537833
$650$	0	0
$651$	9.19375	0.360332
$652$	0	0
$653$	21.0156	0.822405	0.411202	−	0.911544i	$-0.365109\pi$
$653$	21.0156	0.822405	0.411202	+	0.911544i	$0.365109\pi$
$654$	0	0
$655$	70.7172	2.76315
$656$	0	0
$657$	−1.70156	−0.0663843
$658$	0	0
$659$	−33.4031	−1.30120	−0.650600	−	0.759420i	$-0.725483\pi$
$659$	−33.4031	−1.30120	−0.650600	+	0.759420i	$0.725483\pi$
$660$	0	0
$661$	15.4031	0.599112	0.299556	−	0.954079i	$-0.403161\pi$
$661$	15.4031	0.599112	0.299556	+	0.954079i	$0.403161\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	37.7906	1.46546
$666$	0	0
$667$	−0.0890652	−0.00344862
$668$	0	0
$669$	21.4031	0.827493
$670$	0	0
$671$	4.29844	0.165939
$672$	0	0
$673$	21.4031	0.825030	0.412515	−	0.910951i	$-0.364650\pi$
$673$	21.4031	0.825030	0.412515	+	0.910951i	$0.364650\pi$
$674$	0	0
$675$	−8.70156	−0.334923
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	14.9844	0.575048
$680$	0	0
$681$	22.8062	0.873937
$682$	0	0
$683$	−2.29844	−0.0879473	−0.0439736	−	0.999033i	$-0.514002\pi$
$683$	−2.29844	−0.0879473	−0.0439736	+	0.999033i	$0.514002\pi$
$684$	0	0
$685$	47.4031	1.81118
$686$	0	0
$687$	5.70156	0.217528
$688$	0	0
$689$	−7.40312	−0.282037
$690$	0	0
$691$	−5.19375	−0.197580	−0.0987898	−	0.995108i	$-0.531497\pi$
$691$	−5.19375	−0.197580	−0.0987898	+	0.995108i	$0.531497\pi$
$692$	0	0
$693$	1.70156	0.0646370
$694$	0	0
$695$	59.2250	2.24653
$696$	0	0
$697$	−1.19375	−0.0452166
$698$	0	0
$699$	−19.4031	−0.733894
$700$	0	0
$701$	17.3141	0.653943	0.326971	−	0.945034i	$-0.393972\pi$
$701$	17.3141	0.653943	0.326971	+	0.945034i	$0.393972\pi$
$702$	0	0
$703$	−44.4187	−1.67528
$704$	0	0
$705$	−14.8062	−0.557636
$706$	0	0
$707$	10.2094	0.383963
$708$	0	0
$709$	2.29844	0.0863196	0.0431598	−	0.999068i	$-0.486258\pi$
$709$	2.29844	0.0863196	0.0431598	+	0.999068i	$0.486258\pi$
$710$	0	0
$711$	3.40312	0.127627
$712$	0	0
$713$	−1.61250	−0.0603885
$714$	0	0
$715$	3.70156	0.138431
$716$	0	0
$717$	1.70156	0.0635460
$718$	0	0
$719$	−5.91093	−0.220441	−0.110220	−	0.993907i	$-0.535156\pi$
$719$	−5.91093	−0.220441	−0.110220	+	0.993907i	$0.535156\pi$
$720$	0	0
$721$	17.5234	0.652607
$722$	0	0
$723$	27.4031	1.01913
$724$	0	0
$725$	−2.59688	−0.0964455
$726$	0	0
$727$	−41.6125	−1.54332	−0.771661	−	0.636034i	$-0.780574\pi$
$727$	−41.6125	−1.54332	−0.771661	+	0.636034i	$0.780574\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.4187	−1.34700
$732$	0	0
$733$	40.8062	1.50721	0.753607	−	0.657326i	$-0.228312\pi$
$733$	40.8062	1.50721	0.753607	+	0.657326i	$0.228312\pi$
$734$	0	0
$735$	−15.1938	−0.560430
$736$	0	0
$737$	5.10469	0.188034
$738$	0	0
$739$	9.40312	0.345900	0.172950	−	0.984931i	$-0.444670\pi$
$739$	9.40312	0.345900	0.172950	+	0.984931i	$0.444670\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	47.3141	1.73578	0.867892	−	0.496753i	$-0.165474\pi$
$743$	47.3141	1.73578	0.867892	+	0.496753i	$0.165474\pi$
$744$	0	0
$745$	−60.0000	−2.19823
$746$	0	0
$747$	−15.4031	−0.563571
$748$	0	0
$749$	−19.9109	−0.727530
$750$	0	0
$751$	40.5078	1.47815	0.739076	−	0.673623i	$-0.235263\pi$
$751$	40.5078	1.47815	0.739076	+	0.673623i	$0.235263\pi$
$752$	0	0
$753$	−7.40312	−0.269785
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−3.61250	−0.131298	−0.0656492	−	0.997843i	$-0.520912\pi$
$757$	−3.61250	−0.131298	−0.0656492	+	0.997843i	$0.520912\pi$
$758$	0	0
$759$	−0.298438	−0.0108326
$760$	0	0
$761$	−35.7016	−1.29418	−0.647090	−	0.762413i	$-0.724014\pi$
$761$	−35.7016	−1.29418	−0.647090	+	0.762413i	$0.724014\pi$
$762$	0	0
$763$	−2.38750	−0.0864334
$764$	0	0
$765$	−14.8062	−0.535321
$766$	0	0
$767$	−13.7016	−0.494735
$768$	0	0
$769$	17.1047	0.616811	0.308405	−	0.951255i	$-0.400205\pi$
$769$	17.1047	0.616811	0.308405	+	0.951255i	$0.400205\pi$
$770$	0	0
$771$	−3.70156	−0.133309
$772$	0	0
$773$	−19.6125	−0.705412	−0.352706	−	0.935734i	$-0.614739\pi$
$773$	−19.6125	−0.705412	−0.352706	+	0.935734i	$0.614739\pi$
$774$	0	0
$775$	−47.0156	−1.68885
$776$	0	0
$777$	−12.5969	−0.451910
$778$	0	0
$779$	1.79063	0.0641559
$780$	0	0
$781$	11.4031	0.408036
$782$	0	0
$783$	0.298438	0.0106653
$784$	0	0
$785$	47.4031	1.69189
$786$	0	0
$787$	5.40312	0.192601	0.0963003	−	0.995352i	$-0.469299\pi$
$787$	5.40312	0.192601	0.0963003	+	0.995352i	$0.469299\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	32.5078	1.15584
$792$	0	0
$793$	−4.29844	−0.152642
$794$	0	0
$795$	27.4031	0.971889
$796$	0	0
$797$	20.0000	0.708436	0.354218	−	0.935163i	$-0.384747\pi$
$797$	20.0000	0.708436	0.354218	+	0.935163i	$0.384747\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	1.40312	0.0495770
$802$	0	0
$803$	−1.70156	−0.0600468
$804$	0	0
$805$	−1.87969	−0.0662505
$806$	0	0
$807$	−30.8062	−1.08443
$808$	0	0
$809$	−49.0156	−1.72330	−0.861649	−	0.507505i	$-0.830568\pi$
$809$	−49.0156	−1.72330	−0.861649	+	0.507505i	$0.830568\pi$
$810$	0	0
$811$	14.5969	0.512566	0.256283	−	0.966602i	$-0.417502\pi$
$811$	14.5969	0.512566	0.256283	+	0.966602i	$0.417502\pi$
$812$	0	0
$813$	26.8062	0.940136
$814$	0	0
$815$	63.3141	2.21779
$816$	0	0
$817$	54.6281	1.91120
$818$	0	0
$819$	−1.70156	−0.0594574
$820$	0	0
$821$	−12.8062	−0.446941	−0.223471	−	0.974711i	$-0.571739\pi$
$821$	−12.8062	−0.446941	−0.223471	+	0.974711i	$0.571739\pi$
$822$	0	0
$823$	15.3141	0.533815	0.266907	−	0.963722i	$-0.413998\pi$
$823$	15.3141	0.533815	0.266907	+	0.963722i	$0.413998\pi$
$824$	0	0
$825$	−8.70156	−0.302950
$826$	0	0
$827$	51.2250	1.78127	0.890634	−	0.454721i	$-0.150261\pi$
$827$	51.2250	1.78127	0.890634	+	0.454721i	$0.150261\pi$
$828$	0	0
$829$	−55.6125	−1.93150	−0.965751	−	0.259471i	$-0.916452\pi$
$829$	−55.6125	−1.93150	−0.965751	+	0.259471i	$0.916452\pi$
$830$	0	0
$831$	−15.1047	−0.523976
$832$	0	0
$833$	−16.4187	−0.568876
$834$	0	0
$835$	−6.29844	−0.217966
$836$	0	0
$837$	5.40312	0.186759
$838$	0	0
$839$	−15.4031	−0.531775	−0.265887	−	0.964004i	$-0.585665\pi$
$839$	−15.4031	−0.531775	−0.265887	+	0.964004i	$0.585665\pi$
$840$	0	0
$841$	−28.9109	−0.996929
$842$	0	0
$843$	19.7016	0.678558
$844$	0	0
$845$	−3.70156	−0.127338
$846$	0	0
$847$	1.70156	0.0584664
$848$	0	0
$849$	−5.70156	−0.195677
$850$	0	0
$851$	2.20937	0.0757363
$852$	0	0
$853$	−53.8219	−1.84283	−0.921413	−	0.388585i	$-0.872964\pi$
$853$	−53.8219	−1.84283	−0.921413	+	0.388585i	$0.872964\pi$
$854$	0	0
$855$	22.2094	0.759545
$856$	0	0
$857$	13.6125	0.464994	0.232497	−	0.972597i	$-0.425310\pi$
$857$	13.6125	0.464994	0.232497	+	0.972597i	$0.425310\pi$
$858$	0	0
$859$	−34.8062	−1.18757	−0.593787	−	0.804622i	$-0.702368\pi$
$859$	−34.8062	−1.18757	−0.593787	+	0.804622i	$0.702368\pi$
$860$	0	0
$861$	0.507811	0.0173061
$862$	0	0
$863$	22.8062	0.776334	0.388167	−	0.921589i	$-0.373108\pi$
$863$	22.8062	0.776334	0.388167	+	0.921589i	$0.373108\pi$
$864$	0	0
$865$	83.3141	2.83276
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	3.40312	0.115443
$870$	0	0
$871$	−5.10469	−0.172966
$872$	0	0
$873$	8.80625	0.298046
$874$	0	0
$875$	−23.3141	−0.788159
$876$	0	0
$877$	−29.4031	−0.992873	−0.496436	−	0.868073i	$-0.665358\pi$
$877$	−29.4031	−0.992873	−0.496436	+	0.868073i	$0.665358\pi$
$878$	0	0
$879$	−14.0000	−0.472208
$880$	0	0
$881$	−33.3141	−1.12238	−0.561190	−	0.827687i	$-0.689656\pi$
$881$	−33.3141	−1.12238	−0.561190	+	0.827687i	$0.689656\pi$
$882$	0	0
$883$	−24.5969	−0.827751	−0.413875	−	0.910334i	$-0.635825\pi$
$883$	−24.5969	−0.827751	−0.413875	+	0.910334i	$0.635825\pi$
$884$	0	0
$885$	50.7172	1.70484
$886$	0	0
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	5.79063	0.194211
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	24.0000	0.803129
$894$	0	0
$895$	−40.0000	−1.33705
$896$	0	0
$897$	0.298438	0.00996455
$898$	0	0
$899$	1.61250	0.0537798
$900$	0	0
$901$	29.6125	0.986535
$902$	0	0
$903$	15.4922	0.515548
$904$	0	0
$905$	81.4344	2.70697
$906$	0	0
$907$	−5.19375	−0.172456	−0.0862278	−	0.996275i	$-0.527481\pi$
$907$	−5.19375	−0.172456	−0.0862278	+	0.996275i	$0.527481\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−25.4031	−0.841643	−0.420822	−	0.907143i	$-0.638258\pi$
$911$	−25.4031	−0.841643	−0.420822	+	0.907143i	$0.638258\pi$
$912$	0	0
$913$	−15.4031	−0.509769
$914$	0	0
$915$	15.9109	0.525999
$916$	0	0
$917$	−32.5078	−1.07350
$918$	0	0
$919$	22.8062	0.752309	0.376154	−	0.926557i	$-0.377246\pi$
$919$	22.8062	0.752309	0.376154	+	0.926557i	$0.377246\pi$
$920$	0	0
$921$	10.0000	0.329511
$922$	0	0
$923$	−11.4031	−0.375338
$924$	0	0
$925$	64.4187	2.11808
$926$	0	0
$927$	10.2984	0.338245
$928$	0	0
$929$	37.4031	1.22716	0.613578	−	0.789634i	$-0.289729\pi$
$929$	37.4031	1.22716	0.613578	+	0.789634i	$0.289729\pi$
$930$	0	0
$931$	24.6281	0.807154
$932$	0	0
$933$	−25.4031	−0.831661
$934$	0	0
$935$	−14.8062	−0.484216
$936$	0	0
$937$	45.4031	1.48326	0.741628	−	0.670812i	$-0.234054\pi$
$937$	45.4031	1.48326	0.741628	+	0.670812i	$0.234054\pi$
$938$	0	0
$939$	29.9109	0.976106
$940$	0	0
$941$	−40.2094	−1.31079	−0.655394	−	0.755287i	$-0.727497\pi$
$941$	−40.2094	−1.31079	−0.655394	+	0.755287i	$0.727497\pi$
$942$	0	0
$943$	−0.0890652	−0.00290036
$944$	0	0
$945$	6.29844	0.204888
$946$	0	0
$947$	−1.61250	−0.0523991	−0.0261996	−	0.999657i	$-0.508341\pi$
$947$	−1.61250	−0.0523991	−0.0261996	+	0.999657i	$0.508341\pi$
$948$	0	0
$949$	1.70156	0.0552350
$950$	0	0
$951$	13.9109	0.451093
$952$	0	0
$953$	−21.6125	−0.700097	−0.350049	−	0.936731i	$-0.613835\pi$
$953$	−21.6125	−0.700097	−0.350049	+	0.936731i	$0.613835\pi$
$954$	0	0
$955$	−38.8953	−1.25862
$956$	0	0
$957$	0.298438	0.00964713
$958$	0	0
$959$	−21.7906	−0.703656
$960$	0	0
$961$	−1.80625	−0.0582661
$962$	0	0
$963$	−11.7016	−0.377078
$964$	0	0
$965$	31.8219	1.02438
$966$	0	0
$967$	−7.31406	−0.235204	−0.117602	−	0.993061i	$-0.537521\pi$
$967$	−7.31406	−0.235204	−0.117602	+	0.993061i	$0.537521\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	12.5969	0.404253	0.202126	−	0.979359i	$-0.435215\pi$
$971$	12.5969	0.404253	0.202126	+	0.979359i	$0.435215\pi$
$972$	0	0
$973$	−27.2250	−0.872793
$974$	0	0
$975$	8.70156	0.278673
$976$	0	0
$977$	−43.0156	−1.37619	−0.688096	−	0.725620i	$-0.741553\pi$
$977$	−43.0156	−1.37619	−0.688096	+	0.725620i	$0.741553\pi$
$978$	0	0
$979$	1.40312	0.0448440
$980$	0	0
$981$	−1.40312	−0.0447983
$982$	0	0
$983$	−17.0156	−0.542714	−0.271357	−	0.962479i	$-0.587472\pi$
$983$	−17.0156	−0.542714	−0.271357	+	0.962479i	$0.587472\pi$
$984$	0	0
$985$	24.4187	0.778046
$986$	0	0
$987$	6.80625	0.216645
$988$	0	0
$989$	−2.71718	−0.0864014
$990$	0	0
$991$	−39.3141	−1.24885	−0.624426	−	0.781084i	$-0.714667\pi$
$991$	−39.3141	−1.24885	−0.624426	+	0.781084i	$0.714667\pi$
$992$	0	0
$993$	24.5078	0.777732
$994$	0	0
$995$	−10.7172	−0.339758
$996$	0	0
$997$	−36.7172	−1.16284	−0.581422	−	0.813602i	$-0.697504\pi$
$997$	−36.7172	−1.16284	−0.581422	+	0.813602i	$0.697504\pi$
$998$	0	0
$999$	−7.40312	−0.234224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bd.1.1		2
4.3	odd	2		858.2.a.p.1.1	✓	2
12.11	even	2		2574.2.a.be.1.2		2
44.43	even	2		9438.2.a.ch.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.p.1.1	✓	2	4.3	odd	2
2574.2.a.be.1.2		2	12.11	even	2
6864.2.a.bd.1.1		2	1.1	even	1	trivial
9438.2.a.ch.1.1		2	44.43	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.70156 q^{15} +4.00000 q^{17} -6.00000 q^{19} -1.70156 q^{21} +0.298438 q^{23} +8.70156 q^{25} -1.00000 q^{27} -0.298438 q^{29} -5.40312 q^{31} -1.00000 q^{33} -6.29844 q^{35} +7.40312 q^{37} +1.00000 q^{39} -0.298438 q^{41} -9.10469 q^{43} -3.70156 q^{45} -4.00000 q^{47} -4.10469 q^{49} -4.00000 q^{51} +7.40312 q^{53} -3.70156 q^{55} +6.00000 q^{57} +13.7016 q^{59} +4.29844 q^{61} +1.70156 q^{63} +3.70156 q^{65} +5.10469 q^{67} -0.298438 q^{69} +11.4031 q^{71} -1.70156 q^{73} -8.70156 q^{75} +1.70156 q^{77} +3.40312 q^{79} +1.00000 q^{81} -15.4031 q^{83} -14.8062 q^{85} +0.298438 q^{87} +1.40312 q^{89} -1.70156 q^{91} +5.40312 q^{93} +22.2094 q^{95} +8.80625 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + q^{15} + 8 q^{17} - 12 q^{19} + 3 q^{21} + 7 q^{23} + 11 q^{25} - 2 q^{27} - 7 q^{29} + 2 q^{31} - 2 q^{33} - 19 q^{35} + 2 q^{37} + 2 q^{39} - 7 q^{41} + q^{43} - q^{45} - 8 q^{47} + 11 q^{49} - 8 q^{51} + 2 q^{53} - q^{55} + 12 q^{57} + 21 q^{59} + 15 q^{61} - 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 10 q^{71} + 3 q^{73} - 11 q^{75} - 3 q^{77} - 6 q^{79} + 2 q^{81} - 18 q^{83} - 4 q^{85} + 7 q^{87} - 10 q^{89} + 3 q^{91} - 2 q^{93} + 6 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + q^15 + 8 * q^17 - 12 * q^19 + 3 * q^21 + 7 * q^23 + 11 * q^25 - 2 * q^27 - 7 * q^29 + 2 * q^31 - 2 * q^33 - 19 * q^35 + 2 * q^37 + 2 * q^39 - 7 * q^41 + q^43 - q^45 - 8 * q^47 + 11 * q^49 - 8 * q^51 + 2 * q^53 - q^55 + 12 * q^57 + 21 * q^59 + 15 * q^61 - 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 10 * q^71 + 3 * q^73 - 11 * q^75 - 3 * q^77 - 6 * q^79 + 2 * q^81 - 18 * q^83 - 4 * q^85 + 7 * q^87 - 10 * q^89 + 3 * q^91 - 2 * q^93 + 6 * q^95 - 8 * q^97 + 2 * q^99

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