LMFDB - Embedded newform 4056.2.a.bg.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27700337.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.16419$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.21013 q^{5} +4.16419 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.21013 q^{5} +4.16419 q^{7} +1.00000 q^{9} -6.39369 q^{11} -2.21013 q^{15} -6.12626 q^{17} +2.16122 q^{19} -4.16419 q^{21} -0.597157 q^{23} -0.115313 q^{25} -1.00000 q^{27} -3.01207 q^{29} +8.39611 q^{31} +6.39369 q^{33} +9.20340 q^{35} +6.61995 q^{37} -6.03982 q^{41} +11.6993 q^{43} +2.21013 q^{45} +8.31329 q^{47} +10.3404 q^{49} +6.12626 q^{51} +7.73983 q^{53} -14.1309 q^{55} -2.16122 q^{57} -0.123888 q^{59} +7.22283 q^{61} +4.16419 q^{63} +15.6938 q^{67} +0.597157 q^{69} +5.78151 q^{71} +0.175176 q^{73} +0.115313 q^{75} -26.6245 q^{77} -2.74761 q^{79} +1.00000 q^{81} +12.0871 q^{83} -13.5398 q^{85} +3.01207 q^{87} +2.16256 q^{89} -8.39611 q^{93} +4.77657 q^{95} -1.65890 q^{97} -6.39369 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9	$ 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9} + 8 q^{11} - q^{15} - 5 q^{17} + 19 q^{19} - 7 q^{21} - 6 q^{23} + 17 q^{25} - 6 q^{27} + 3 q^{29} + 9 q^{31} - 8 q^{33} + 4 q^{35} + 6 q^{37} - 15 q^{41} + 11 q^{43} + q^{45} + 5 q^{47} + 5 q^{49} + 5 q^{51} + 12 q^{53} + 7 q^{55} - 19 q^{57} - 5 q^{59} + 17 q^{61} + 7 q^{63} + 13 q^{67} + 6 q^{69} + 26 q^{71} - 49 q^{73} - 17 q^{75} - 25 q^{77} + 14 q^{79} + 6 q^{81} - 3 q^{83} - 19 q^{85} - 3 q^{87} + 13 q^{89} - 9 q^{93} + 43 q^{95} - 14 q^{97} + 8 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9 + 8 * q^11 - q^15 - 5 * q^17 + 19 * q^19 - 7 * q^21 - 6 * q^23 + 17 * q^25 - 6 * q^27 + 3 * q^29 + 9 * q^31 - 8 * q^33 + 4 * q^35 + 6 * q^37 - 15 * q^41 + 11 * q^43 + q^45 + 5 * q^47 + 5 * q^49 + 5 * q^51 + 12 * q^53 + 7 * q^55 - 19 * q^57 - 5 * q^59 + 17 * q^61 + 7 * q^63 + 13 * q^67 + 6 * q^69 + 26 * q^71 - 49 * q^73 - 17 * q^75 - 25 * q^77 + 14 * q^79 + 6 * q^81 - 3 * q^83 - 19 * q^85 - 3 * q^87 + 13 * q^89 - 9 * q^93 + 43 * q^95 - 14 * q^97 + 8 * 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
$5$	2.21013	0.988401	0.494201	−	0.869348i	$-0.335461\pi$
$5$	2.21013	0.988401	0.494201	+	0.869348i	$0.335461\pi$
$6$	0	0
$7$	4.16419	1.57391	0.786957	−	0.617008i	$-0.211655\pi$
$7$	4.16419	1.57391	0.786957	+	0.617008i	$0.211655\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.39369	−1.92777	−0.963885	−	0.266320i	$-0.914192\pi$
$11$	−6.39369	−1.92777	−0.963885	+	0.266320i	$0.914192\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.21013	−0.570654
$16$	0	0
$17$	−6.12626	−1.48584	−0.742918	−	0.669382i	$-0.766559\pi$
$17$	−6.12626	−1.48584	−0.742918	+	0.669382i	$0.766559\pi$
$18$	0	0
$19$	2.16122	0.495817	0.247908	−	0.968783i	$-0.420257\pi$
$19$	2.16122	0.495817	0.247908	+	0.968783i	$0.420257\pi$
$20$	0	0
$21$	−4.16419	−0.908700
$22$	0	0
$23$	−0.597157	−0.124516	−0.0622579	−	0.998060i	$-0.519830\pi$
$23$	−0.597157	−0.124516	−0.0622579	+	0.998060i	$0.519830\pi$
$24$	0	0
$25$	−0.115313	−0.0230626
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.01207	−0.559327	−0.279664	−	0.960098i	$-0.590223\pi$
$29$	−3.01207	−0.559327	−0.279664	+	0.960098i	$0.590223\pi$
$30$	0	0
$31$	8.39611	1.50799	0.753993	−	0.656882i	$-0.228125\pi$
$31$	8.39611	1.50799	0.753993	+	0.656882i	$0.228125\pi$
$32$	0	0
$33$	6.39369	1.11300
$34$	0	0
$35$	9.20340	1.55566
$36$	0	0
$37$	6.61995	1.08831	0.544157	−	0.838984i	$-0.316850\pi$
$37$	6.61995	1.08831	0.544157	+	0.838984i	$0.316850\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.03982	−0.943261	−0.471630	−	0.881796i	$-0.656334\pi$
$41$	−6.03982	−0.943261	−0.471630	+	0.881796i	$0.656334\pi$
$42$	0	0
$43$	11.6993	1.78412	0.892062	−	0.451913i	$-0.149258\pi$
$43$	11.6993	1.78412	0.892062	+	0.451913i	$0.149258\pi$
$44$	0	0
$45$	2.21013	0.329467
$46$	0	0
$47$	8.31329	1.21262	0.606309	−	0.795229i	$-0.292649\pi$
$47$	8.31329	1.21262	0.606309	+	0.795229i	$0.292649\pi$
$48$	0	0
$49$	10.3404	1.47721
$50$	0	0
$51$	6.12626	0.857848
$52$	0	0
$53$	7.73983	1.06315	0.531574	−	0.847012i	$-0.321601\pi$
$53$	7.73983	1.06315	0.531574	+	0.847012i	$0.321601\pi$
$54$	0	0
$55$	−14.1309	−1.90541
$56$	0	0
$57$	−2.16122	−0.286260
$58$	0	0
$59$	−0.123888	−0.0161289	−0.00806443	−	0.999967i	$-0.502567\pi$
$59$	−0.123888	−0.0161289	−0.00806443	+	0.999967i	$0.502567\pi$
$60$	0	0
$61$	7.22283	0.924788	0.462394	−	0.886674i	$-0.346991\pi$
$61$	7.22283	0.924788	0.462394	+	0.886674i	$0.346991\pi$
$62$	0	0
$63$	4.16419	0.524638
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.6938	1.91731	0.958653	−	0.284577i	$-0.0918530\pi$
$67$	15.6938	1.91731	0.958653	+	0.284577i	$0.0918530\pi$
$68$	0	0
$69$	0.597157	0.0718892
$70$	0	0
$71$	5.78151	0.686139	0.343070	−	0.939310i	$-0.388533\pi$
$71$	5.78151	0.686139	0.343070	+	0.939310i	$0.388533\pi$
$72$	0	0
$73$	0.175176	0.0205028	0.0102514	−	0.999947i	$-0.496737\pi$
$73$	0.175176	0.0205028	0.0102514	+	0.999947i	$0.496737\pi$
$74$	0	0
$75$	0.115313	0.0133152
$76$	0	0
$77$	−26.6245	−3.03414
$78$	0	0
$79$	−2.74761	−0.309131	−0.154565	−	0.987983i	$-0.549398\pi$
$79$	−2.74761	−0.309131	−0.154565	+	0.987983i	$0.549398\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0871	1.32673	0.663364	−	0.748296i	$-0.269128\pi$
$83$	12.0871	1.32673	0.663364	+	0.748296i	$0.269128\pi$
$84$	0	0
$85$	−13.5398	−1.46860
$86$	0	0
$87$	3.01207	0.322928
$88$	0	0
$89$	2.16256	0.229231	0.114616	−	0.993410i	$-0.463436\pi$
$89$	2.16256	0.229231	0.114616	+	0.993410i	$0.463436\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.39611	−0.870636
$94$	0	0
$95$	4.77657	0.490066
$96$	0	0
$97$	−1.65890	−0.168436	−0.0842180	−	0.996447i	$-0.526839\pi$
$97$	−1.65890	−0.168436	−0.0842180	+	0.996447i	$0.526839\pi$
$98$	0	0
$99$	−6.39369	−0.642590
$100$	0	0
$101$	4.98447	0.495973	0.247987	−	0.968763i	$-0.420231\pi$
$101$	4.98447	0.495973	0.247987	+	0.968763i	$0.420231\pi$
$102$	0	0
$103$	−12.7046	−1.25182	−0.625910	−	0.779896i	$-0.715272\pi$
$103$	−12.7046	−1.25182	−0.625910	+	0.779896i	$0.715272\pi$
$104$	0	0
$105$	−9.20340	−0.898160
$106$	0	0
$107$	13.3085	1.28658	0.643289	−	0.765623i	$-0.277569\pi$
$107$	13.3085	1.28658	0.643289	+	0.765623i	$0.277569\pi$
$108$	0	0
$109$	−7.53981	−0.722182	−0.361091	−	0.932530i	$-0.617596\pi$
$109$	−7.53981	−0.722182	−0.361091	+	0.932530i	$0.617596\pi$
$110$	0	0
$111$	−6.61995	−0.628338
$112$	0	0
$113$	1.08176	0.101763	0.0508817	−	0.998705i	$-0.483797\pi$
$113$	1.08176	0.101763	0.0508817	+	0.998705i	$0.483797\pi$
$114$	0	0
$115$	−1.31980	−0.123072
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−25.5109	−2.33858
$120$	0	0
$121$	29.8792	2.71629
$122$	0	0
$123$	6.03982	0.544592
$124$	0	0
$125$	−11.3055	−1.01120
$126$	0	0
$127$	7.14678	0.634174	0.317087	−	0.948396i	$-0.397295\pi$
$127$	7.14678	0.634174	0.317087	+	0.948396i	$0.397295\pi$
$128$	0	0
$129$	−11.6993	−1.03006
$130$	0	0
$131$	1.44604	0.126341	0.0631706	−	0.998003i	$-0.479879\pi$
$131$	1.44604	0.126341	0.0631706	+	0.998003i	$0.479879\pi$
$132$	0	0
$133$	8.99970	0.780373
$134$	0	0
$135$	−2.21013	−0.190218
$136$	0	0
$137$	−16.0847	−1.37421	−0.687105	−	0.726559i	$-0.741119\pi$
$137$	−16.0847	−1.37421	−0.687105	+	0.726559i	$0.741119\pi$
$138$	0	0
$139$	−4.29714	−0.364479	−0.182239	−	0.983254i	$-0.558335\pi$
$139$	−4.29714	−0.364479	−0.182239	+	0.983254i	$0.558335\pi$
$140$	0	0
$141$	−8.31329	−0.700106
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.65708	−0.552840
$146$	0	0
$147$	−10.3404	−0.852865
$148$	0	0
$149$	18.8594	1.54503	0.772513	−	0.634999i	$-0.219000\pi$
$149$	18.8594	1.54503	0.772513	+	0.634999i	$0.219000\pi$
$150$	0	0
$151$	−18.7795	−1.52825	−0.764126	−	0.645067i	$-0.776829\pi$
$151$	−18.7795	−1.52825	−0.764126	+	0.645067i	$0.776829\pi$
$152$	0	0
$153$	−6.12626	−0.495279
$154$	0	0
$155$	18.5565	1.49050
$156$	0	0
$157$	10.7571	0.858511	0.429256	−	0.903183i	$-0.358776\pi$
$157$	10.7571	0.858511	0.429256	+	0.903183i	$0.358776\pi$
$158$	0	0
$159$	−7.73983	−0.613809
$160$	0	0
$161$	−2.48667	−0.195977
$162$	0	0
$163$	−9.43050	−0.738654	−0.369327	−	0.929300i	$-0.620412\pi$
$163$	−9.43050	−0.738654	−0.369327	+	0.929300i	$0.620412\pi$
$164$	0	0
$165$	14.1309	1.10009
$166$	0	0
$167$	5.96738	0.461770	0.230885	−	0.972981i	$-0.425838\pi$
$167$	5.96738	0.461770	0.230885	+	0.972981i	$0.425838\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	2.16122	0.165272
$172$	0	0
$173$	6.42891	0.488781	0.244390	−	0.969677i	$-0.421412\pi$
$173$	6.42891	0.488781	0.244390	+	0.969677i	$0.421412\pi$
$174$	0	0
$175$	−0.480185	−0.0362985
$176$	0	0
$177$	0.123888	0.00931200
$178$	0	0
$179$	9.76570	0.729923	0.364961	−	0.931023i	$-0.381082\pi$
$179$	9.76570	0.729923	0.364961	+	0.931023i	$0.381082\pi$
$180$	0	0
$181$	11.3219	0.841548	0.420774	−	0.907166i	$-0.361759\pi$
$181$	11.3219	0.841548	0.420774	+	0.907166i	$0.361759\pi$
$182$	0	0
$183$	−7.22283	−0.533927
$184$	0	0
$185$	14.6310	1.07569
$186$	0	0
$187$	39.1694	2.86435
$188$	0	0
$189$	−4.16419	−0.302900
$190$	0	0
$191$	4.13842	0.299446	0.149723	−	0.988728i	$-0.452162\pi$
$191$	4.13842	0.299446	0.149723	+	0.988728i	$0.452162\pi$
$192$	0	0
$193$	−22.2221	−1.59959	−0.799793	−	0.600276i	$-0.795057\pi$
$193$	−22.2221	−1.59959	−0.799793	+	0.600276i	$0.795057\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.43229	−0.529529	−0.264764	−	0.964313i	$-0.585294\pi$
$197$	−7.43229	−0.529529	−0.264764	+	0.964313i	$0.585294\pi$
$198$	0	0
$199$	1.90179	0.134814	0.0674070	−	0.997726i	$-0.478527\pi$
$199$	1.90179	0.134814	0.0674070	+	0.997726i	$0.478527\pi$
$200$	0	0
$201$	−15.6938	−1.10696
$202$	0	0
$203$	−12.5428	−0.880333
$204$	0	0
$205$	−13.3488	−0.932320
$206$	0	0
$207$	−0.597157	−0.0415053
$208$	0	0
$209$	−13.8181	−0.955820
$210$	0	0
$211$	−4.42755	−0.304805	−0.152403	−	0.988318i	$-0.548701\pi$
$211$	−4.42755	−0.304805	−0.152403	+	0.988318i	$0.548701\pi$
$212$	0	0
$213$	−5.78151	−0.396143
$214$	0	0
$215$	25.8570	1.76343
$216$	0	0
$217$	34.9630	2.37344
$218$	0	0
$219$	−0.175176	−0.0118373
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−22.4142	−1.50097	−0.750484	−	0.660889i	$-0.770179\pi$
$223$	−22.4142	−1.50097	−0.750484	+	0.660889i	$0.770179\pi$
$224$	0	0
$225$	−0.115313	−0.00768753
$226$	0	0
$227$	20.0669	1.33189	0.665944	−	0.746002i	$-0.268029\pi$
$227$	20.0669	1.33189	0.665944	+	0.746002i	$0.268029\pi$
$228$	0	0
$229$	−9.30219	−0.614706	−0.307353	−	0.951596i	$-0.599443\pi$
$229$	−9.30219	−0.614706	−0.307353	+	0.951596i	$0.599443\pi$
$230$	0	0
$231$	26.6245	1.75176
$232$	0	0
$233$	4.72277	0.309399	0.154699	−	0.987962i	$-0.450559\pi$
$233$	4.72277	0.309399	0.154699	+	0.987962i	$0.450559\pi$
$234$	0	0
$235$	18.3735	1.19855
$236$	0	0
$237$	2.74761	0.178477
$238$	0	0
$239$	−2.97518	−0.192448	−0.0962242	−	0.995360i	$-0.530677\pi$
$239$	−2.97518	−0.192448	−0.0962242	+	0.995360i	$0.530677\pi$
$240$	0	0
$241$	−21.0710	−1.35730	−0.678652	−	0.734460i	$-0.737436\pi$
$241$	−21.0710	−1.35730	−0.678652	+	0.734460i	$0.737436\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	22.8537	1.46007
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−12.0871	−0.765987
$250$	0	0
$251$	−2.37444	−0.149874	−0.0749368	−	0.997188i	$-0.523875\pi$
$251$	−2.37444	−0.149874	−0.0749368	+	0.997188i	$0.523875\pi$
$252$	0	0
$253$	3.81803	0.240038
$254$	0	0
$255$	13.5398	0.847898
$256$	0	0
$257$	2.67332	0.166757	0.0833785	−	0.996518i	$-0.473429\pi$
$257$	2.67332	0.166757	0.0833785	+	0.996518i	$0.473429\pi$
$258$	0	0
$259$	27.5667	1.71291
$260$	0	0
$261$	−3.01207	−0.186442
$262$	0	0
$263$	−3.28318	−0.202450	−0.101225	−	0.994864i	$-0.532276\pi$
$263$	−3.28318	−0.202450	−0.101225	+	0.994864i	$0.532276\pi$
$264$	0	0
$265$	17.1061	1.05082
$266$	0	0
$267$	−2.16256	−0.132347
$268$	0	0
$269$	−3.99992	−0.243880	−0.121940	−	0.992537i	$-0.538911\pi$
$269$	−3.99992	−0.243880	−0.121940	+	0.992537i	$0.538911\pi$
$270$	0	0
$271$	−17.9337	−1.08940	−0.544698	−	0.838632i	$-0.683356\pi$
$271$	−17.9337	−1.08940	−0.544698	+	0.838632i	$0.683356\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.737275	0.0444594
$276$	0	0
$277$	−19.1361	−1.14978	−0.574889	−	0.818232i	$-0.694955\pi$
$277$	−19.1361	−1.14978	−0.574889	+	0.818232i	$0.694955\pi$
$278$	0	0
$279$	8.39611	0.502662
$280$	0	0
$281$	11.0487	0.659110	0.329555	−	0.944136i	$-0.393101\pi$
$281$	11.0487	0.659110	0.329555	+	0.944136i	$0.393101\pi$
$282$	0	0
$283$	28.2843	1.68133	0.840663	−	0.541559i	$-0.182166\pi$
$283$	28.2843	1.68133	0.840663	+	0.541559i	$0.182166\pi$
$284$	0	0
$285$	−4.77657	−0.282940
$286$	0	0
$287$	−25.1509	−1.48461
$288$	0	0
$289$	20.5310	1.20771
$290$	0	0
$291$	1.65890	0.0972466
$292$	0	0
$293$	−23.9911	−1.40157	−0.700787	−	0.713371i	$-0.747167\pi$
$293$	−23.9911	−1.40157	−0.700787	+	0.713371i	$0.747167\pi$
$294$	0	0
$295$	−0.273809	−0.0159418
$296$	0	0
$297$	6.39369	0.370999
$298$	0	0
$299$	0	0
$300$	0	0
$301$	48.7180	2.80806
$302$	0	0
$303$	−4.98447	−0.286350
$304$	0	0
$305$	15.9634	0.914062
$306$	0	0
$307$	16.0060	0.913511	0.456755	−	0.889592i	$-0.349011\pi$
$307$	16.0060	0.913511	0.456755	+	0.889592i	$0.349011\pi$
$308$	0	0
$309$	12.7046	0.722738
$310$	0	0
$311$	−0.740869	−0.0420108	−0.0210054	−	0.999779i	$-0.506687\pi$
$311$	−0.740869	−0.0420108	−0.0210054	+	0.999779i	$0.506687\pi$
$312$	0	0
$313$	16.1617	0.913511	0.456756	−	0.889592i	$-0.349011\pi$
$313$	16.1617	0.913511	0.456756	+	0.889592i	$0.349011\pi$
$314$	0	0
$315$	9.20340	0.518553
$316$	0	0
$317$	19.8836	1.11678	0.558388	−	0.829580i	$-0.311420\pi$
$317$	19.8836	1.11678	0.558388	+	0.829580i	$0.311420\pi$
$318$	0	0
$319$	19.2582	1.07825
$320$	0	0
$321$	−13.3085	−0.742806
$322$	0	0
$323$	−13.2402	−0.736703
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.53981	0.416952
$328$	0	0
$329$	34.6181	1.90856
$330$	0	0
$331$	−24.8901	−1.36808	−0.684041	−	0.729444i	$-0.739779\pi$
$331$	−24.8901	−1.36808	−0.684041	+	0.729444i	$0.739779\pi$
$332$	0	0
$333$	6.61995	0.362771
$334$	0	0
$335$	34.6855	1.89507
$336$	0	0
$337$	−8.49975	−0.463011	−0.231505	−	0.972834i	$-0.574365\pi$
$337$	−8.49975	−0.463011	−0.231505	+	0.972834i	$0.574365\pi$
$338$	0	0
$339$	−1.08176	−0.0587531
$340$	0	0
$341$	−53.6821	−2.90705
$342$	0	0
$343$	13.9102	0.751081
$344$	0	0
$345$	1.31980	0.0710554
$346$	0	0
$347$	14.9202	0.800959	0.400480	−	0.916306i	$-0.368843\pi$
$347$	14.9202	0.800959	0.400480	+	0.916306i	$0.368843\pi$
$348$	0	0
$349$	−32.8658	−1.75926	−0.879632	−	0.475655i	$-0.842211\pi$
$349$	−32.8658	−1.75926	−0.879632	+	0.475655i	$0.842211\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	29.4497	1.56745	0.783726	−	0.621107i	$-0.213317\pi$
$353$	29.4497	1.56745	0.783726	+	0.621107i	$0.213317\pi$
$354$	0	0
$355$	12.7779	0.678181
$356$	0	0
$357$	25.5109	1.35018
$358$	0	0
$359$	25.2568	1.33300	0.666502	−	0.745503i	$-0.267791\pi$
$359$	25.2568	1.33300	0.666502	+	0.745503i	$0.267791\pi$
$360$	0	0
$361$	−14.3291	−0.754166
$362$	0	0
$363$	−29.8792	−1.56825
$364$	0	0
$365$	0.387163	0.0202650
$366$	0	0
$367$	7.94301	0.414622	0.207311	−	0.978275i	$-0.433529\pi$
$367$	7.94301	0.414622	0.207311	+	0.978275i	$0.433529\pi$
$368$	0	0
$369$	−6.03982	−0.314420
$370$	0	0
$371$	32.2301	1.67330
$372$	0	0
$373$	−13.7306	−0.710942	−0.355471	−	0.934687i	$-0.615680\pi$
$373$	−13.7306	−0.710942	−0.355471	+	0.934687i	$0.615680\pi$
$374$	0	0
$375$	11.3055	0.583815
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3.04691	0.156509	0.0782547	−	0.996933i	$-0.475065\pi$
$379$	3.04691	0.156509	0.0782547	+	0.996933i	$0.475065\pi$
$380$	0	0
$381$	−7.14678	−0.366141
$382$	0	0
$383$	24.6925	1.26173	0.630865	−	0.775893i	$-0.282700\pi$
$383$	24.6925	1.26173	0.630865	+	0.775893i	$0.282700\pi$
$384$	0	0
$385$	−58.8437	−2.99895
$386$	0	0
$387$	11.6993	0.594708
$388$	0	0
$389$	−24.0807	−1.22094	−0.610470	−	0.792039i	$-0.709019\pi$
$389$	−24.0807	−1.22094	−0.610470	+	0.792039i	$0.709019\pi$
$390$	0	0
$391$	3.65834	0.185010
$392$	0	0
$393$	−1.44604	−0.0729431
$394$	0	0
$395$	−6.07259	−0.305545
$396$	0	0
$397$	29.4446	1.47778	0.738891	−	0.673825i	$-0.235350\pi$
$397$	29.4446	1.47778	0.738891	+	0.673825i	$0.235350\pi$
$398$	0	0
$399$	−8.99970	−0.450549
$400$	0	0
$401$	10.8721	0.542925	0.271463	−	0.962449i	$-0.412493\pi$
$401$	10.8721	0.542925	0.271463	+	0.962449i	$0.412493\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.21013	0.109822
$406$	0	0
$407$	−42.3259	−2.09802
$408$	0	0
$409$	8.40805	0.415751	0.207876	−	0.978155i	$-0.433345\pi$
$409$	8.40805	0.415751	0.207876	+	0.978155i	$0.433345\pi$
$410$	0	0
$411$	16.0847	0.793400
$412$	0	0
$413$	−0.515893	−0.0253854
$414$	0	0
$415$	26.7140	1.31134
$416$	0	0
$417$	4.29714	0.210432
$418$	0	0
$419$	0.830918	0.0405930	0.0202965	−	0.999794i	$-0.493539\pi$
$419$	0.830918	0.0405930	0.0202965	+	0.999794i	$0.493539\pi$
$420$	0	0
$421$	−10.3133	−0.502642	−0.251321	−	0.967904i	$-0.580865\pi$
$421$	−10.3133	−0.502642	−0.251321	+	0.967904i	$0.580865\pi$
$422$	0	0
$423$	8.31329	0.404206
$424$	0	0
$425$	0.706437	0.0342672
$426$	0	0
$427$	30.0772	1.45554
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.46321	0.359490	0.179745	−	0.983713i	$-0.442473\pi$
$431$	7.46321	0.359490	0.179745	+	0.983713i	$0.442473\pi$
$432$	0	0
$433$	17.6692	0.849129	0.424565	−	0.905398i	$-0.360427\pi$
$433$	17.6692	0.849129	0.424565	+	0.905398i	$0.360427\pi$
$434$	0	0
$435$	6.65708	0.319182
$436$	0	0
$437$	−1.29058	−0.0617370
$438$	0	0
$439$	−18.5850	−0.887015	−0.443508	−	0.896271i	$-0.646266\pi$
$439$	−18.5850	−0.887015	−0.443508	+	0.896271i	$0.646266\pi$
$440$	0	0
$441$	10.3404	0.492402
$442$	0	0
$443$	−32.9956	−1.56767	−0.783835	−	0.620969i	$-0.786739\pi$
$443$	−32.9956	−1.56767	−0.783835	+	0.620969i	$0.786739\pi$
$444$	0	0
$445$	4.77955	0.226572
$446$	0	0
$447$	−18.8594	−0.892021
$448$	0	0
$449$	−27.3012	−1.28842	−0.644210	−	0.764848i	$-0.722814\pi$
$449$	−27.3012	−1.28842	−0.644210	+	0.764848i	$0.722814\pi$
$450$	0	0
$451$	38.6167	1.81839
$452$	0	0
$453$	18.7795	0.882337
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−41.7034	−1.95080	−0.975401	−	0.220437i	$-0.929252\pi$
$457$	−41.7034	−1.95080	−0.975401	+	0.220437i	$0.929252\pi$
$458$	0	0
$459$	6.12626	0.285949
$460$	0	0
$461$	−20.4122	−0.950689	−0.475345	−	0.879800i	$-0.657677\pi$
$461$	−20.4122	−0.950689	−0.475345	+	0.879800i	$0.657677\pi$
$462$	0	0
$463$	14.9638	0.695425	0.347712	−	0.937601i	$-0.386959\pi$
$463$	14.9638	0.695425	0.347712	+	0.937601i	$0.386959\pi$
$464$	0	0
$465$	−18.5565	−0.860538
$466$	0	0
$467$	−10.6220	−0.491527	−0.245763	−	0.969330i	$-0.579039\pi$
$467$	−10.6220	−0.491527	−0.245763	+	0.969330i	$0.579039\pi$
$468$	0	0
$469$	65.3520	3.01768
$470$	0	0
$471$	−10.7571	−0.495662
$472$	0	0
$473$	−74.8016	−3.43938
$474$	0	0
$475$	−0.249216	−0.0114348
$476$	0	0
$477$	7.73983	0.354383
$478$	0	0
$479$	−35.4320	−1.61893	−0.809465	−	0.587169i	$-0.800243\pi$
$479$	−35.4320	−1.61893	−0.809465	+	0.587169i	$0.800243\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	2.48667	0.113148
$484$	0	0
$485$	−3.66640	−0.166482
$486$	0	0
$487$	9.23632	0.418537	0.209269	−	0.977858i	$-0.432892\pi$
$487$	9.23632	0.418537	0.209269	+	0.977858i	$0.432892\pi$
$488$	0	0
$489$	9.43050	0.426462
$490$	0	0
$491$	−17.5148	−0.790431	−0.395216	−	0.918588i	$-0.629330\pi$
$491$	−17.5148	−0.790431	−0.395216	+	0.918588i	$0.629330\pi$
$492$	0	0
$493$	18.4527	0.831069
$494$	0	0
$495$	−14.1309	−0.635137
$496$	0	0
$497$	24.0753	1.07992
$498$	0	0
$499$	4.91434	0.219996	0.109998	−	0.993932i	$-0.464916\pi$
$499$	4.91434	0.219996	0.109998	+	0.993932i	$0.464916\pi$
$500$	0	0
$501$	−5.96738	−0.266603
$502$	0	0
$503$	4.08753	0.182254	0.0911270	−	0.995839i	$-0.470953\pi$
$503$	4.08753	0.182254	0.0911270	+	0.995839i	$0.470953\pi$
$504$	0	0
$505$	11.0163	0.490220
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.42742	−0.417863	−0.208932	−	0.977930i	$-0.566999\pi$
$509$	−9.42742	−0.417863	−0.208932	+	0.977930i	$0.566999\pi$
$510$	0	0
$511$	0.729466	0.0322697
$512$	0	0
$513$	−2.16122	−0.0954200
$514$	0	0
$515$	−28.0788	−1.23730
$516$	0	0
$517$	−53.1526	−2.33765
$518$	0	0
$519$	−6.42891	−0.282198
$520$	0	0
$521$	−7.95205	−0.348386	−0.174193	−	0.984712i	$-0.555732\pi$
$521$	−7.95205	−0.348386	−0.174193	+	0.984712i	$0.555732\pi$
$522$	0	0
$523$	−24.8142	−1.08505	−0.542524	−	0.840040i	$-0.682531\pi$
$523$	−24.8142	−1.08505	−0.542524	+	0.840040i	$0.682531\pi$
$524$	0	0
$525$	0.480185	0.0209570
$526$	0	0
$527$	−51.4368	−2.24062
$528$	0	0
$529$	−22.6434	−0.984496
$530$	0	0
$531$	−0.123888	−0.00537628
$532$	0	0
$533$	0	0
$534$	0	0
$535$	29.4135	1.27166
$536$	0	0
$537$	−9.76570	−0.421421
$538$	0	0
$539$	−66.1135	−2.84771
$540$	0	0
$541$	−37.2232	−1.60035	−0.800176	−	0.599765i	$-0.795261\pi$
$541$	−37.2232	−1.60035	−0.800176	+	0.599765i	$0.795261\pi$
$542$	0	0
$543$	−11.3219	−0.485868
$544$	0	0
$545$	−16.6640	−0.713806
$546$	0	0
$547$	24.5075	1.04787	0.523934	−	0.851759i	$-0.324464\pi$
$547$	24.5075	1.04787	0.523934	+	0.851759i	$0.324464\pi$
$548$	0	0
$549$	7.22283	0.308263
$550$	0	0
$551$	−6.50973	−0.277324
$552$	0	0
$553$	−11.4416	−0.486545
$554$	0	0
$555$	−14.6310	−0.621050
$556$	0	0
$557$	27.2813	1.15594	0.577972	−	0.816056i	$-0.303844\pi$
$557$	27.2813	1.15594	0.577972	+	0.816056i	$0.303844\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−39.1694	−1.65373
$562$	0	0
$563$	−41.8400	−1.76335	−0.881673	−	0.471860i	$-0.843583\pi$
$563$	−41.8400	−1.76335	−0.881673	+	0.471860i	$0.843583\pi$
$564$	0	0
$565$	2.39083	0.100583
$566$	0	0
$567$	4.16419	0.174879
$568$	0	0
$569$	32.5357	1.36397	0.681984	−	0.731367i	$-0.261117\pi$
$569$	32.5357	1.36397	0.681984	+	0.731367i	$0.261117\pi$
$570$	0	0
$571$	−2.07093	−0.0866659	−0.0433330	−	0.999061i	$-0.513798\pi$
$571$	−2.07093	−0.0866659	−0.0433330	+	0.999061i	$0.513798\pi$
$572$	0	0
$573$	−4.13842	−0.172885
$574$	0	0
$575$	0.0688599	0.00287166
$576$	0	0
$577$	31.8908	1.32763	0.663815	−	0.747896i	$-0.268936\pi$
$577$	31.8908	1.32763	0.663815	+	0.747896i	$0.268936\pi$
$578$	0	0
$579$	22.2221	0.923521
$580$	0	0
$581$	50.3328	2.08816
$582$	0	0
$583$	−49.4861	−2.04950
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.0750	−0.869859	−0.434929	−	0.900465i	$-0.643227\pi$
$587$	−21.0750	−0.869859	−0.434929	+	0.900465i	$0.643227\pi$
$588$	0	0
$589$	18.1458	0.747685
$590$	0	0
$591$	7.43229	0.305724
$592$	0	0
$593$	12.0436	0.494571	0.247285	−	0.968943i	$-0.420461\pi$
$593$	12.0436	0.494571	0.247285	+	0.968943i	$0.420461\pi$
$594$	0	0
$595$	−56.3824	−2.31145
$596$	0	0
$597$	−1.90179	−0.0778349
$598$	0	0
$599$	41.2627	1.68595	0.842974	−	0.537955i	$-0.180803\pi$
$599$	41.2627	1.68595	0.842974	+	0.537955i	$0.180803\pi$
$600$	0	0
$601$	34.4284	1.40436	0.702181	−	0.711998i	$-0.252210\pi$
$601$	34.4284	1.40436	0.702181	+	0.711998i	$0.252210\pi$
$602$	0	0
$603$	15.6938	0.639102
$604$	0	0
$605$	66.0371	2.68479
$606$	0	0
$607$	−36.1312	−1.46652	−0.733259	−	0.679949i	$-0.762002\pi$
$607$	−36.1312	−1.46652	−0.733259	+	0.679949i	$0.762002\pi$
$608$	0	0
$609$	12.5428	0.508261
$610$	0	0
$611$	0	0
$612$	0	0
$613$	21.2607	0.858710	0.429355	−	0.903136i	$-0.358741\pi$
$613$	21.2607	0.858710	0.429355	+	0.903136i	$0.358741\pi$
$614$	0	0
$615$	13.3488	0.538275
$616$	0	0
$617$	−39.7471	−1.60016	−0.800080	−	0.599893i	$-0.795210\pi$
$617$	−39.7471	−1.60016	−0.800080	+	0.599893i	$0.795210\pi$
$618$	0	0
$619$	41.2347	1.65736	0.828681	−	0.559721i	$-0.189092\pi$
$619$	41.2347	1.65736	0.828681	+	0.559721i	$0.189092\pi$
$620$	0	0
$621$	0.597157	0.0239631
$622$	0	0
$623$	9.00531	0.360790
$624$	0	0
$625$	−24.4101	−0.976406
$626$	0	0
$627$	13.8181	0.551843
$628$	0	0
$629$	−40.5555	−1.61706
$630$	0	0
$631$	−14.1526	−0.563404	−0.281702	−	0.959502i	$-0.590899\pi$
$631$	−14.1526	−0.563404	−0.281702	+	0.959502i	$0.590899\pi$
$632$	0	0
$633$	4.42755	0.175979
$634$	0	0
$635$	15.7953	0.626819
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.78151	0.228713
$640$	0	0
$641$	−1.78542	−0.0705198	−0.0352599	−	0.999378i	$-0.511226\pi$
$641$	−1.78542	−0.0705198	−0.0352599	+	0.999378i	$0.511226\pi$
$642$	0	0
$643$	−9.48841	−0.374186	−0.187093	−	0.982342i	$-0.559907\pi$
$643$	−9.48841	−0.374186	−0.187093	+	0.982342i	$0.559907\pi$
$644$	0	0
$645$	−25.8570	−1.01812
$646$	0	0
$647$	−27.3190	−1.07402	−0.537010	−	0.843576i	$-0.680446\pi$
$647$	−27.3190	−1.07402	−0.537010	+	0.843576i	$0.680446\pi$
$648$	0	0
$649$	0.792101	0.0310927
$650$	0	0
$651$	−34.9630	−1.37031
$652$	0	0
$653$	19.5623	0.765531	0.382766	−	0.923845i	$-0.374972\pi$
$653$	19.5623	0.765531	0.382766	+	0.923845i	$0.374972\pi$
$654$	0	0
$655$	3.19594	0.124876
$656$	0	0
$657$	0.175176	0.00683428
$658$	0	0
$659$	−26.0027	−1.01292	−0.506461	−	0.862263i	$-0.669046\pi$
$659$	−26.0027	−1.01292	−0.506461	+	0.862263i	$0.669046\pi$
$660$	0	0
$661$	16.9004	0.657349	0.328674	−	0.944443i	$-0.393398\pi$
$661$	16.9004	0.657349	0.328674	+	0.944443i	$0.393398\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	19.8905	0.771322
$666$	0	0
$667$	1.79868	0.0696451
$668$	0	0
$669$	22.4142	0.866584
$670$	0	0
$671$	−46.1805	−1.78278
$672$	0	0
$673$	−30.8424	−1.18889	−0.594443	−	0.804137i	$-0.702628\pi$
$673$	−30.8424	−1.18889	−0.594443	+	0.804137i	$0.702628\pi$
$674$	0	0
$675$	0.115313	0.00443840
$676$	0	0
$677$	4.73797	0.182095	0.0910474	−	0.995847i	$-0.470979\pi$
$677$	4.73797	0.182095	0.0910474	+	0.995847i	$0.470979\pi$
$678$	0	0
$679$	−6.90798	−0.265104
$680$	0	0
$681$	−20.0669	−0.768966
$682$	0	0
$683$	9.71145	0.371598	0.185799	−	0.982588i	$-0.440513\pi$
$683$	9.71145	0.371598	0.185799	+	0.982588i	$0.440513\pi$
$684$	0	0
$685$	−35.5493	−1.35827
$686$	0	0
$687$	9.30219	0.354901
$688$	0	0
$689$	0	0
$690$	0	0
$691$	12.1097	0.460675	0.230338	−	0.973111i	$-0.426017\pi$
$691$	12.1097	0.460675	0.230338	+	0.973111i	$0.426017\pi$
$692$	0	0
$693$	−26.6245	−1.01138
$694$	0	0
$695$	−9.49725	−0.360251
$696$	0	0
$697$	37.0015	1.40153
$698$	0	0
$699$	−4.72277	−0.178631
$700$	0	0
$701$	−19.2355	−0.726514	−0.363257	−	0.931689i	$-0.618335\pi$
$701$	−19.2355	−0.726514	−0.363257	+	0.931689i	$0.618335\pi$
$702$	0	0
$703$	14.3071	0.539604
$704$	0	0
$705$	−18.3735	−0.691985
$706$	0	0
$707$	20.7562	0.780619
$708$	0	0
$709$	−3.05621	−0.114778	−0.0573892	−	0.998352i	$-0.518278\pi$
$709$	−3.05621	−0.114778	−0.0573892	+	0.998352i	$0.518278\pi$
$710$	0	0
$711$	−2.74761	−0.103044
$712$	0	0
$713$	−5.01380	−0.187768
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.97518	0.111110
$718$	0	0
$719$	−26.4564	−0.986657	−0.493329	−	0.869843i	$-0.664220\pi$
$719$	−26.4564	−0.986657	−0.493329	+	0.869843i	$0.664220\pi$
$720$	0	0
$721$	−52.9042	−1.97026
$722$	0	0
$723$	21.0710	0.783640
$724$	0	0
$725$	0.347331	0.0128995
$726$	0	0
$727$	22.9357	0.850638	0.425319	−	0.905043i	$-0.360162\pi$
$727$	22.9357	0.850638	0.425319	+	0.905043i	$0.360162\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−71.6728	−2.65092
$732$	0	0
$733$	−12.9634	−0.478813	−0.239407	−	0.970919i	$-0.576953\pi$
$733$	−12.9634	−0.478813	−0.239407	+	0.970919i	$0.576953\pi$
$734$	0	0
$735$	−22.8537	−0.842973
$736$	0	0
$737$	−100.341	−3.69612
$738$	0	0
$739$	−3.43570	−0.126384	−0.0631921	−	0.998001i	$-0.520128\pi$
$739$	−3.43570	−0.126384	−0.0631921	+	0.998001i	$0.520128\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.5957	−1.52600	−0.763000	−	0.646399i	$-0.776274\pi$
$743$	−41.5957	−1.52600	−0.763000	+	0.646399i	$0.776274\pi$
$744$	0	0
$745$	41.6819	1.52711
$746$	0	0
$747$	12.0871	0.442243
$748$	0	0
$749$	55.4189	2.02496
$750$	0	0
$751$	53.0727	1.93665	0.968325	−	0.249693i	$-0.0803298\pi$
$751$	53.0727	1.93665	0.968325	+	0.249693i	$0.0803298\pi$
$752$	0	0
$753$	2.37444	0.0865295
$754$	0	0
$755$	−41.5051	−1.51053
$756$	0	0
$757$	−28.2269	−1.02592	−0.512962	−	0.858411i	$-0.671452\pi$
$757$	−28.2269	−1.02592	−0.512962	+	0.858411i	$0.671452\pi$
$758$	0	0
$759$	−3.81803	−0.138586
$760$	0	0
$761$	−6.09333	−0.220883	−0.110441	−	0.993883i	$-0.535226\pi$
$761$	−6.09333	−0.220883	−0.110441	+	0.993883i	$0.535226\pi$
$762$	0	0
$763$	−31.3972	−1.13665
$764$	0	0
$765$	−13.5398	−0.489534
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−2.12209	−0.0765244	−0.0382622	−	0.999268i	$-0.512182\pi$
$769$	−2.12209	−0.0765244	−0.0382622	+	0.999268i	$0.512182\pi$
$770$	0	0
$771$	−2.67332	−0.0962772
$772$	0	0
$773$	−17.5488	−0.631187	−0.315593	−	0.948895i	$-0.602204\pi$
$773$	−17.5488	−0.631187	−0.315593	+	0.948895i	$0.602204\pi$
$774$	0	0
$775$	−0.968181	−0.0347781
$776$	0	0
$777$	−27.5667	−0.988950
$778$	0	0
$779$	−13.0533	−0.467685
$780$	0	0
$781$	−36.9652	−1.32272
$782$	0	0
$783$	3.01207	0.107643
$784$	0	0
$785$	23.7747	0.848554
$786$	0	0
$787$	−9.65059	−0.344007	−0.172003	−	0.985096i	$-0.555024\pi$
$787$	−9.65059	−0.344007	−0.172003	+	0.985096i	$0.555024\pi$
$788$	0	0
$789$	3.28318	0.116884
$790$	0	0
$791$	4.50464	0.160167
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−17.1061	−0.606689
$796$	0	0
$797$	39.1190	1.38567	0.692833	−	0.721098i	$-0.256362\pi$
$797$	39.1190	1.38567	0.692833	+	0.721098i	$0.256362\pi$
$798$	0	0
$799$	−50.9294	−1.80175
$800$	0	0
$801$	2.16256	0.0764104
$802$	0	0
$803$	−1.12002	−0.0395247
$804$	0	0
$805$	−5.49588	−0.193704
$806$	0	0
$807$	3.99992	0.140804
$808$	0	0
$809$	39.7440	1.39732	0.698662	−	0.715452i	$-0.253779\pi$
$809$	39.7440	1.39732	0.698662	+	0.715452i	$0.253779\pi$
$810$	0	0
$811$	−0.865440	−0.0303897	−0.0151949	−	0.999885i	$-0.504837\pi$
$811$	−0.865440	−0.0303897	−0.0151949	+	0.999885i	$0.504837\pi$
$812$	0	0
$813$	17.9337	0.628963
$814$	0	0
$815$	−20.8427	−0.730086
$816$	0	0
$817$	25.2847	0.884599
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.6588	0.755898	0.377949	−	0.925826i	$-0.376629\pi$
$821$	21.6588	0.755898	0.377949	+	0.925826i	$0.376629\pi$
$822$	0	0
$823$	54.5134	1.90022	0.950110	−	0.311916i	$-0.100971\pi$
$823$	54.5134	1.90022	0.950110	+	0.311916i	$0.100971\pi$
$824$	0	0
$825$	−0.737275	−0.0256686
$826$	0	0
$827$	9.42330	0.327680	0.163840	−	0.986487i	$-0.447612\pi$
$827$	9.42330	0.327680	0.163840	+	0.986487i	$0.447612\pi$
$828$	0	0
$829$	34.9856	1.21510	0.607549	−	0.794282i	$-0.292153\pi$
$829$	34.9856	1.21510	0.607549	+	0.794282i	$0.292153\pi$
$830$	0	0
$831$	19.1361	0.663824
$832$	0	0
$833$	−63.3482	−2.19489
$834$	0	0
$835$	13.1887	0.456414
$836$	0	0
$837$	−8.39611	−0.290212
$838$	0	0
$839$	28.9346	0.998932	0.499466	−	0.866333i	$-0.333530\pi$
$839$	28.9346	0.998932	0.499466	+	0.866333i	$0.333530\pi$
$840$	0	0
$841$	−19.9274	−0.687153
$842$	0	0
$843$	−11.0487	−0.380538
$844$	0	0
$845$	0	0
$846$	0	0
$847$	124.423	4.27521
$848$	0	0
$849$	−28.2843	−0.970714
$850$	0	0
$851$	−3.95315	−0.135512
$852$	0	0
$853$	26.3058	0.900694	0.450347	−	0.892854i	$-0.351300\pi$
$853$	26.3058	0.900694	0.450347	+	0.892854i	$0.351300\pi$
$854$	0	0
$855$	4.77657	0.163355
$856$	0	0
$857$	55.0780	1.88143	0.940714	−	0.339202i	$-0.110157\pi$
$857$	55.0780	1.88143	0.940714	+	0.339202i	$0.110157\pi$
$858$	0	0
$859$	28.4471	0.970601	0.485300	−	0.874347i	$-0.338710\pi$
$859$	28.4471	0.970601	0.485300	+	0.874347i	$0.338710\pi$
$860$	0	0
$861$	25.1509	0.857141
$862$	0	0
$863$	16.7549	0.570344	0.285172	−	0.958476i	$-0.407949\pi$
$863$	16.7549	0.570344	0.285172	+	0.958476i	$0.407949\pi$
$864$	0	0
$865$	14.2087	0.483112
$866$	0	0
$867$	−20.5310	−0.697271
$868$	0	0
$869$	17.5674	0.595932
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1.65890	−0.0561454
$874$	0	0
$875$	−47.0783	−1.59154
$876$	0	0
$877$	−18.6501	−0.629769	−0.314884	−	0.949130i	$-0.601966\pi$
$877$	−18.6501	−0.629769	−0.314884	+	0.949130i	$0.601966\pi$
$878$	0	0
$879$	23.9911	0.809199
$880$	0	0
$881$	4.10210	0.138203	0.0691016	−	0.997610i	$-0.477987\pi$
$881$	4.10210	0.138203	0.0691016	+	0.997610i	$0.477987\pi$
$882$	0	0
$883$	2.86219	0.0963205	0.0481602	−	0.998840i	$-0.484664\pi$
$883$	2.86219	0.0963205	0.0481602	+	0.998840i	$0.484664\pi$
$884$	0	0
$885$	0.273809	0.00920399
$886$	0	0
$887$	5.78416	0.194213	0.0971065	−	0.995274i	$-0.469041\pi$
$887$	5.78416	0.194213	0.0971065	+	0.995274i	$0.469041\pi$
$888$	0	0
$889$	29.7605	0.998136
$890$	0	0
$891$	−6.39369	−0.214197
$892$	0	0
$893$	17.9668	0.601237
$894$	0	0
$895$	21.5835	0.721457
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−25.2897	−0.843458
$900$	0	0
$901$	−47.4162	−1.57966
$902$	0	0
$903$	−48.7180	−1.62123
$904$	0	0
$905$	25.0228	0.831787
$906$	0	0
$907$	26.8975	0.893116	0.446558	−	0.894755i	$-0.352650\pi$
$907$	26.8975	0.893116	0.446558	+	0.894755i	$0.352650\pi$
$908$	0	0
$909$	4.98447	0.165324
$910$	0	0
$911$	34.3311	1.13744	0.568721	−	0.822531i	$-0.307439\pi$
$911$	34.3311	1.13744	0.568721	+	0.822531i	$0.307439\pi$
$912$	0	0
$913$	−77.2810	−2.55763
$914$	0	0
$915$	−15.9634	−0.527734
$916$	0	0
$917$	6.02158	0.198850
$918$	0	0
$919$	−15.5238	−0.512082	−0.256041	−	0.966666i	$-0.582418\pi$
$919$	−15.5238	−0.512082	−0.256041	+	0.966666i	$0.582418\pi$
$920$	0	0
$921$	−16.0060	−0.527416
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−0.763366	−0.0250993
$926$	0	0
$927$	−12.7046	−0.417273
$928$	0	0
$929$	34.5944	1.13501	0.567503	−	0.823372i	$-0.307910\pi$
$929$	34.5944	1.13501	0.567503	+	0.823372i	$0.307910\pi$
$930$	0	0
$931$	22.3479	0.732424
$932$	0	0
$933$	0.740869	0.0242550
$934$	0	0
$935$	86.5695	2.83113
$936$	0	0
$937$	23.1734	0.757040	0.378520	−	0.925593i	$-0.376433\pi$
$937$	23.1734	0.757040	0.378520	+	0.925593i	$0.376433\pi$
$938$	0	0
$939$	−16.1617	−0.527416
$940$	0	0
$941$	−52.0999	−1.69841	−0.849205	−	0.528064i	$-0.822918\pi$
$941$	−52.0999	−1.69841	−0.849205	+	0.528064i	$0.822918\pi$
$942$	0	0
$943$	3.60672	0.117451
$944$	0	0
$945$	−9.20340	−0.299387
$946$	0	0
$947$	−4.36829	−0.141950	−0.0709752	−	0.997478i	$-0.522611\pi$
$947$	−4.36829	−0.141950	−0.0709752	+	0.997478i	$0.522611\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−19.8836	−0.644771
$952$	0	0
$953$	−36.4597	−1.18105	−0.590523	−	0.807021i	$-0.701078\pi$
$953$	−36.4597	−1.18105	−0.590523	+	0.807021i	$0.701078\pi$
$954$	0	0
$955$	9.14646	0.295972
$956$	0	0
$957$	−19.2582	−0.622530
$958$	0	0
$959$	−66.9797	−2.16289
$960$	0	0
$961$	39.4947	1.27402
$962$	0	0
$963$	13.3085	0.428859
$964$	0	0
$965$	−49.1139	−1.58103
$966$	0	0
$967$	−0.695132	−0.0223539	−0.0111770	−	0.999938i	$-0.503558\pi$
$967$	−0.695132	−0.0223539	−0.0111770	+	0.999938i	$0.503558\pi$
$968$	0	0
$969$	13.2402	0.425335
$970$	0	0
$971$	5.54299	0.177883	0.0889415	−	0.996037i	$-0.471652\pi$
$971$	5.54299	0.177883	0.0889415	+	0.996037i	$0.471652\pi$
$972$	0	0
$973$	−17.8941	−0.573658
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.0728	−0.962113	−0.481056	−	0.876690i	$-0.659747\pi$
$977$	−30.0728	−0.962113	−0.481056	+	0.876690i	$0.659747\pi$
$978$	0	0
$979$	−13.8267	−0.441905
$980$	0	0
$981$	−7.53981	−0.240727
$982$	0	0
$983$	22.3615	0.713221	0.356610	−	0.934253i	$-0.383932\pi$
$983$	22.3615	0.713221	0.356610	+	0.934253i	$0.383932\pi$
$984$	0	0
$985$	−16.4264	−0.523387
$986$	0	0
$987$	−34.6181	−1.10191
$988$	0	0
$989$	−6.98631	−0.222152
$990$	0	0
$991$	32.1412	1.02100	0.510499	−	0.859878i	$-0.329461\pi$
$991$	32.1412	1.02100	0.510499	+	0.859878i	$0.329461\pi$
$992$	0	0
$993$	24.8901	0.789862
$994$	0	0
$995$	4.20320	0.133250
$996$	0	0
$997$	18.7280	0.593121	0.296561	−	0.955014i	$-0.404160\pi$
$997$	18.7280	0.593121	0.296561	+	0.955014i	$0.404160\pi$
$998$	0	0
$999$	−6.61995	−0.209446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.bg.1.4	yes	6
4.3	odd	2		8112.2.a.cw.1.4		6
13.5	odd	4		4056.2.c.q.337.5		12
13.8	odd	4		4056.2.c.q.337.8		12
13.12	even	2		4056.2.a.bf.1.3	✓	6
52.51	odd	2		8112.2.a.cv.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bf.1.3	✓	6	13.12	even	2
4056.2.a.bg.1.4	yes	6	1.1	even	1	trivial
4056.2.c.q.337.5		12	13.5	odd	4
4056.2.c.q.337.8		12	13.8	odd	4
8112.2.a.cv.1.3		6	52.51	odd	2
8112.2.a.cw.1.4		6	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.21013 q^{5} +4.16419 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.21013 q^{5} +4.16419 q^{7} +1.00000 q^{9} -6.39369 q^{11} -2.21013 q^{15} -6.12626 q^{17} +2.16122 q^{19} -4.16419 q^{21} -0.597157 q^{23} -0.115313 q^{25} -1.00000 q^{27} -3.01207 q^{29} +8.39611 q^{31} +6.39369 q^{33} +9.20340 q^{35} +6.61995 q^{37} -6.03982 q^{41} +11.6993 q^{43} +2.21013 q^{45} +8.31329 q^{47} +10.3404 q^{49} +6.12626 q^{51} +7.73983 q^{53} -14.1309 q^{55} -2.16122 q^{57} -0.123888 q^{59} +7.22283 q^{61} +4.16419 q^{63} +15.6938 q^{67} +0.597157 q^{69} +5.78151 q^{71} +0.175176 q^{73} +0.115313 q^{75} -26.6245 q^{77} -2.74761 q^{79} +1.00000 q^{81} +12.0871 q^{83} -13.5398 q^{85} +3.01207 q^{87} +2.16256 q^{89} -8.39611 q^{93} +4.77657 q^{95} -1.65890 q^{97} -6.39369 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9	\( 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9} + 8 q^{11} - q^{15} - 5 q^{17} + 19 q^{19} - 7 q^{21} - 6 q^{23} + 17 q^{25} - 6 q^{27} + 3 q^{29} + 9 q^{31} - 8 q^{33} + 4 q^{35} + 6 q^{37} - 15 q^{41} + 11 q^{43} + q^{45} + 5 q^{47} + 5 q^{49} + 5 q^{51} + 12 q^{53} + 7 q^{55} - 19 q^{57} - 5 q^{59} + 17 q^{61} + 7 q^{63} + 13 q^{67} + 6 q^{69} + 26 q^{71} - 49 q^{73} - 17 q^{75} - 25 q^{77} + 14 q^{79} + 6 q^{81} - 3 q^{83} - 19 q^{85} - 3 q^{87} + 13 q^{89} - 9 q^{93} + 43 q^{95} - 14 q^{97} + 8 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9 + 8 * q^11 - q^15 - 5 * q^17 + 19 * q^19 - 7 * q^21 - 6 * q^23 + 17 * q^25 - 6 * q^27 + 3 * q^29 + 9 * q^31 - 8 * q^33 + 4 * q^35 + 6 * q^37 - 15 * q^41 + 11 * q^43 + q^45 + 5 * q^47 + 5 * q^49 + 5 * q^51 + 12 * q^53 + 7 * q^55 - 19 * q^57 - 5 * q^59 + 17 * q^61 + 7 * q^63 + 13 * q^67 + 6 * q^69 + 26 * q^71 - 49 * q^73 - 17 * q^75 - 25 * q^77 + 14 * q^79 + 6 * q^81 - 3 * q^83 - 19 * q^85 - 3 * q^87 + 13 * q^89 - 9 * q^93 + 43 * q^95 - 14 * q^97 + 8 * q^99

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