LMFDB - Embedded newform 4056.2.a.bh.1.6

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			1.6
Root			$1.88227$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.12925 q^{5} +1.48962 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.12925 q^{5} +1.48962 q^{7} +1.00000 q^{9} +4.83677 q^{11} +3.12925 q^{15} +3.32731 q^{17} +3.82127 q^{19} +1.48962 q^{21} -4.77770 q^{23} +4.79219 q^{25} +1.00000 q^{27} +3.70085 q^{29} +6.48614 q^{31} +4.83677 q^{33} +4.66140 q^{35} -4.68535 q^{37} -9.72224 q^{41} -5.75872 q^{43} +3.12925 q^{45} -7.73028 q^{47} -4.78103 q^{49} +3.32731 q^{51} +8.29590 q^{53} +15.1355 q^{55} +3.82127 q^{57} -14.4763 q^{59} -7.18968 q^{61} +1.48962 q^{63} -3.81807 q^{67} -4.77770 q^{69} +5.09590 q^{71} +15.7451 q^{73} +4.79219 q^{75} +7.20496 q^{77} +11.1000 q^{79} +1.00000 q^{81} +4.06158 q^{83} +10.4120 q^{85} +3.70085 q^{87} -16.9172 q^{89} +6.48614 q^{93} +11.9577 q^{95} -16.9132 q^{97} +4.83677 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - q^5 + 5 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} - q^{5} + 5 q^{7} + 6 q^{9} + 6 q^{11} - q^{15} + 9 q^{17} - 7 q^{19} + 5 q^{21} + 12 q^{23} + 9 q^{25} + 6 q^{27} + 7 q^{29} + 11 q^{31} + 6 q^{33} + 6 q^{35} - 6 q^{37} - 13 q^{41} + 15 q^{43} - q^{45} + 9 q^{47} + 13 q^{49} + 9 q^{51} + 22 q^{53} + 3 q^{55} - 7 q^{57} + 7 q^{59} + 25 q^{61} + 5 q^{63} - 5 q^{67} + 12 q^{69} - 8 q^{71} - 15 q^{73} + 9 q^{75} + 45 q^{77} + 14 q^{79} + 6 q^{81} + 13 q^{83} + 35 q^{85} + 7 q^{87} - 33 q^{89} + 11 q^{93} + 47 q^{95} - 50 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - q^5 + 5 * q^7 + 6 * q^9 + 6 * q^11 - q^15 + 9 * q^17 - 7 * q^19 + 5 * q^21 + 12 * q^23 + 9 * q^25 + 6 * q^27 + 7 * q^29 + 11 * q^31 + 6 * q^33 + 6 * q^35 - 6 * q^37 - 13 * q^41 + 15 * q^43 - q^45 + 9 * q^47 + 13 * q^49 + 9 * q^51 + 22 * q^53 + 3 * q^55 - 7 * q^57 + 7 * q^59 + 25 * q^61 + 5 * q^63 - 5 * q^67 + 12 * q^69 - 8 * q^71 - 15 * q^73 + 9 * q^75 + 45 * q^77 + 14 * q^79 + 6 * q^81 + 13 * q^83 + 35 * q^85 + 7 * q^87 - 33 * q^89 + 11 * q^93 + 47 * q^95 - 50 * q^97 + 6 * 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
$5$	3.12925	1.39944	0.699721	−	0.714416i	$-0.253308\pi$
$5$	3.12925	1.39944	0.699721	+	0.714416i	$0.253308\pi$
$6$	0	0
$7$	1.48962	0.563024	0.281512	−	0.959558i	$-0.409164\pi$
$7$	1.48962	0.563024	0.281512	+	0.959558i	$0.409164\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.83677	1.45834	0.729171	−	0.684332i	$-0.239906\pi$
$11$	4.83677	1.45834	0.729171	+	0.684332i	$0.239906\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.12925	0.807968
$16$	0	0
$17$	3.32731	0.806991	0.403496	−	0.914982i	$-0.367795\pi$
$17$	3.32731	0.806991	0.403496	+	0.914982i	$0.367795\pi$
$18$	0	0
$19$	3.82127	0.876659	0.438330	−	0.898814i	$-0.355570\pi$
$19$	3.82127	0.876659	0.438330	+	0.898814i	$0.355570\pi$
$20$	0	0
$21$	1.48962	0.325062
$22$	0	0
$23$	−4.77770	−0.996220	−0.498110	−	0.867114i	$-0.665972\pi$
$23$	−4.77770	−0.996220	−0.498110	+	0.867114i	$0.665972\pi$
$24$	0	0
$25$	4.79219	0.958438
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.70085	0.687231	0.343615	−	0.939110i	$-0.388348\pi$
$29$	3.70085	0.687231	0.343615	+	0.939110i	$0.388348\pi$
$30$	0	0
$31$	6.48614	1.16495	0.582473	−	0.812850i	$-0.302085\pi$
$31$	6.48614	1.16495	0.582473	+	0.812850i	$0.302085\pi$
$32$	0	0
$33$	4.83677	0.841974
$34$	0	0
$35$	4.66140	0.787920
$36$	0	0
$37$	−4.68535	−0.770267	−0.385133	−	0.922861i	$-0.625845\pi$
$37$	−4.68535	−0.770267	−0.385133	+	0.922861i	$0.625845\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.72224	−1.51836	−0.759179	−	0.650882i	$-0.774399\pi$
$41$	−9.72224	−1.51836	−0.759179	+	0.650882i	$0.774399\pi$
$42$	0	0
$43$	−5.75872	−0.878197	−0.439098	−	0.898439i	$-0.644702\pi$
$43$	−5.75872	−0.878197	−0.439098	+	0.898439i	$0.644702\pi$
$44$	0	0
$45$	3.12925	0.466481
$46$	0	0
$47$	−7.73028	−1.12758	−0.563788	−	0.825919i	$-0.690657\pi$
$47$	−7.73028	−1.12758	−0.563788	+	0.825919i	$0.690657\pi$
$48$	0	0
$49$	−4.78103	−0.683004
$50$	0	0
$51$	3.32731	0.465917
$52$	0	0
$53$	8.29590	1.13953	0.569765	−	0.821808i	$-0.307034\pi$
$53$	8.29590	1.13953	0.569765	+	0.821808i	$0.307034\pi$
$54$	0	0
$55$	15.1355	2.04086
$56$	0	0
$57$	3.82127	0.506139
$58$	0	0
$59$	−14.4763	−1.88466	−0.942330	−	0.334687i	$-0.891370\pi$
$59$	−14.4763	−1.88466	−0.942330	+	0.334687i	$0.891370\pi$
$60$	0	0
$61$	−7.18968	−0.920544	−0.460272	−	0.887778i	$-0.652248\pi$
$61$	−7.18968	−0.920544	−0.460272	+	0.887778i	$0.652248\pi$
$62$	0	0
$63$	1.48962	0.187675
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.81807	−0.466452	−0.233226	−	0.972423i	$-0.574928\pi$
$67$	−3.81807	−0.466452	−0.233226	+	0.972423i	$0.574928\pi$
$68$	0	0
$69$	−4.77770	−0.575168
$70$	0	0
$71$	5.09590	0.604773	0.302386	−	0.953185i	$-0.402217\pi$
$71$	5.09590	0.604773	0.302386	+	0.953185i	$0.402217\pi$
$72$	0	0
$73$	15.7451	1.84282	0.921412	−	0.388588i	$-0.127037\pi$
$73$	15.7451	1.84282	0.921412	+	0.388588i	$0.127037\pi$
$74$	0	0
$75$	4.79219	0.553355
$76$	0	0
$77$	7.20496	0.821082
$78$	0	0
$79$	11.1000	1.24885	0.624424	−	0.781085i	$-0.285334\pi$
$79$	11.1000	1.24885	0.624424	+	0.781085i	$0.285334\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.06158	0.445816	0.222908	−	0.974839i	$-0.428445\pi$
$83$	4.06158	0.445816	0.222908	+	0.974839i	$0.428445\pi$
$84$	0	0
$85$	10.4120	1.12934
$86$	0	0
$87$	3.70085	0.396773
$88$	0	0
$89$	−16.9172	−1.79322	−0.896608	−	0.442824i	$-0.853977\pi$
$89$	−16.9172	−1.79322	−0.896608	+	0.442824i	$0.853977\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.48614	0.672582
$94$	0	0
$95$	11.9577	1.22683
$96$	0	0
$97$	−16.9132	−1.71728	−0.858638	−	0.512582i	$-0.828689\pi$
$97$	−16.9132	−1.71728	−0.858638	+	0.512582i	$0.828689\pi$
$98$	0	0
$99$	4.83677	0.486114
$100$	0	0
$101$	−16.1598	−1.60796	−0.803978	−	0.594659i	$-0.797287\pi$
$101$	−16.1598	−1.60796	−0.803978	+	0.594659i	$0.797287\pi$
$102$	0	0
$103$	−10.0686	−0.992091	−0.496046	−	0.868296i	$-0.665215\pi$
$103$	−10.0686	−0.992091	−0.496046	+	0.868296i	$0.665215\pi$
$104$	0	0
$105$	4.66140	0.454906
$106$	0	0
$107$	12.1710	1.17661	0.588307	−	0.808638i	$-0.299795\pi$
$107$	12.1710	1.17661	0.588307	+	0.808638i	$0.299795\pi$
$108$	0	0
$109$	−6.74502	−0.646056	−0.323028	−	0.946389i	$-0.604701\pi$
$109$	−6.74502	−0.646056	−0.323028	+	0.946389i	$0.604701\pi$
$110$	0	0
$111$	−4.68535	−0.444714
$112$	0	0
$113$	10.9767	1.03260	0.516300	−	0.856408i	$-0.327309\pi$
$113$	10.9767	1.03260	0.516300	+	0.856408i	$0.327309\pi$
$114$	0	0
$115$	−14.9506	−1.39415
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.95643	0.454356
$120$	0	0
$121$	12.3944	1.12676
$122$	0	0
$123$	−9.72224	−0.876624
$124$	0	0
$125$	−0.650283	−0.0581631
$126$	0	0
$127$	4.01527	0.356298	0.178149	−	0.984004i	$-0.442989\pi$
$127$	4.01527	0.356298	0.178149	+	0.984004i	$0.442989\pi$
$128$	0	0
$129$	−5.75872	−0.507027
$130$	0	0
$131$	−11.7892	−1.03002	−0.515012	−	0.857183i	$-0.672213\pi$
$131$	−11.7892	−1.03002	−0.515012	+	0.857183i	$0.672213\pi$
$132$	0	0
$133$	5.69225	0.493580
$134$	0	0
$135$	3.12925	0.269323
$136$	0	0
$137$	−13.3096	−1.13711	−0.568557	−	0.822644i	$-0.692498\pi$
$137$	−13.3096	−1.13711	−0.568557	+	0.822644i	$0.692498\pi$
$138$	0	0
$139$	5.14114	0.436066	0.218033	−	0.975941i	$-0.430036\pi$
$139$	5.14114	0.436066	0.218033	+	0.975941i	$0.430036\pi$
$140$	0	0
$141$	−7.73028	−0.651007
$142$	0	0
$143$	0	0
$144$	0	0
$145$	11.5809	0.961740
$146$	0	0
$147$	−4.78103	−0.394332
$148$	0	0
$149$	4.28026	0.350653	0.175326	−	0.984510i	$-0.443902\pi$
$149$	4.28026	0.350653	0.175326	+	0.984510i	$0.443902\pi$
$150$	0	0
$151$	16.5496	1.34679	0.673393	−	0.739284i	$-0.264836\pi$
$151$	16.5496	1.34679	0.673393	+	0.739284i	$0.264836\pi$
$152$	0	0
$153$	3.32731	0.268997
$154$	0	0
$155$	20.2968	1.63027
$156$	0	0
$157$	−21.5076	−1.71650	−0.858248	−	0.513236i	$-0.828447\pi$
$157$	−21.5076	−1.71650	−0.858248	+	0.513236i	$0.828447\pi$
$158$	0	0
$159$	8.29590	0.657907
$160$	0	0
$161$	−7.11697	−0.560896
$162$	0	0
$163$	−6.62970	−0.519278	−0.259639	−	0.965706i	$-0.583604\pi$
$163$	−6.62970	−0.519278	−0.259639	+	0.965706i	$0.583604\pi$
$164$	0	0
$165$	15.1355	1.17829
$166$	0	0
$167$	−4.06853	−0.314832	−0.157416	−	0.987532i	$-0.550316\pi$
$167$	−4.06853	−0.314832	−0.157416	+	0.987532i	$0.550316\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	3.82127	0.292220
$172$	0	0
$173$	0.886940	0.0674328	0.0337164	−	0.999431i	$-0.489266\pi$
$173$	0.886940	0.0674328	0.0337164	+	0.999431i	$0.489266\pi$
$174$	0	0
$175$	7.13855	0.539624
$176$	0	0
$177$	−14.4763	−1.08811
$178$	0	0
$179$	6.90942	0.516434	0.258217	−	0.966087i	$-0.416865\pi$
$179$	6.90942	0.516434	0.258217	+	0.966087i	$0.416865\pi$
$180$	0	0
$181$	4.57338	0.339937	0.169968	−	0.985450i	$-0.445633\pi$
$181$	4.57338	0.339937	0.169968	+	0.985450i	$0.445633\pi$
$182$	0	0
$183$	−7.18968	−0.531477
$184$	0	0
$185$	−14.6616	−1.07794
$186$	0	0
$187$	16.0934	1.17687
$188$	0	0
$189$	1.48962	0.108354
$190$	0	0
$191$	9.81213	0.709981	0.354991	−	0.934870i	$-0.384484\pi$
$191$	9.81213	0.709981	0.354991	+	0.934870i	$0.384484\pi$
$192$	0	0
$193$	−25.3006	−1.82118	−0.910588	−	0.413315i	$-0.864371\pi$
$193$	−25.3006	−1.82118	−0.910588	+	0.413315i	$0.864371\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.5048	1.17592	0.587959	−	0.808891i	$-0.299932\pi$
$197$	16.5048	1.17592	0.587959	+	0.808891i	$0.299932\pi$
$198$	0	0
$199$	26.0642	1.84764	0.923820	−	0.382826i	$-0.125049\pi$
$199$	26.0642	1.84764	0.923820	+	0.382826i	$0.125049\pi$
$200$	0	0
$201$	−3.81807	−0.269306
$202$	0	0
$203$	5.51287	0.386928
$204$	0	0
$205$	−30.4233	−2.12485
$206$	0	0
$207$	−4.77770	−0.332073
$208$	0	0
$209$	18.4826	1.27847
$210$	0	0
$211$	11.6270	0.800437	0.400218	−	0.916420i	$-0.368934\pi$
$211$	11.6270	0.800437	0.400218	+	0.916420i	$0.368934\pi$
$212$	0	0
$213$	5.09590	0.349166
$214$	0	0
$215$	−18.0205	−1.22899
$216$	0	0
$217$	9.66190	0.655893
$218$	0	0
$219$	15.7451	1.06395
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−19.7959	−1.32563	−0.662816	−	0.748782i	$-0.730639\pi$
$223$	−19.7959	−1.32563	−0.662816	+	0.748782i	$0.730639\pi$
$224$	0	0
$225$	4.79219	0.319479
$226$	0	0
$227$	4.01162	0.266261	0.133130	−	0.991099i	$-0.457497\pi$
$227$	4.01162	0.266261	0.133130	+	0.991099i	$0.457497\pi$
$228$	0	0
$229$	3.49817	0.231166	0.115583	−	0.993298i	$-0.463126\pi$
$229$	3.49817	0.231166	0.115583	+	0.993298i	$0.463126\pi$
$230$	0	0
$231$	7.20496	0.474052
$232$	0	0
$233$	8.79848	0.576408	0.288204	−	0.957569i	$-0.406942\pi$
$233$	8.79848	0.576408	0.288204	+	0.957569i	$0.406942\pi$
$234$	0	0
$235$	−24.1900	−1.57798
$236$	0	0
$237$	11.1000	0.721023
$238$	0	0
$239$	16.8723	1.09138	0.545691	−	0.837987i	$-0.316267\pi$
$239$	16.8723	1.09138	0.545691	+	0.837987i	$0.316267\pi$
$240$	0	0
$241$	−9.22284	−0.594095	−0.297048	−	0.954863i	$-0.596002\pi$
$241$	−9.22284	−0.594095	−0.297048	+	0.954863i	$0.596002\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.9610	−0.955824
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4.06158	0.257392
$250$	0	0
$251$	−3.02019	−0.190633	−0.0953164	−	0.995447i	$-0.530386\pi$
$251$	−3.02019	−0.190633	−0.0953164	+	0.995447i	$0.530386\pi$
$252$	0	0
$253$	−23.1087	−1.45283
$254$	0	0
$255$	10.4120	0.652023
$256$	0	0
$257$	24.7035	1.54096	0.770480	−	0.637464i	$-0.220017\pi$
$257$	24.7035	1.54096	0.770480	+	0.637464i	$0.220017\pi$
$258$	0	0
$259$	−6.97940	−0.433679
$260$	0	0
$261$	3.70085	0.229077
$262$	0	0
$263$	−1.12833	−0.0695759	−0.0347880	−	0.999395i	$-0.511076\pi$
$263$	−1.12833	−0.0695759	−0.0347880	+	0.999395i	$0.511076\pi$
$264$	0	0
$265$	25.9599	1.59471
$266$	0	0
$267$	−16.9172	−1.03531
$268$	0	0
$269$	11.6337	0.709318	0.354659	−	0.934996i	$-0.384597\pi$
$269$	11.6337	0.709318	0.354659	+	0.934996i	$0.384597\pi$
$270$	0	0
$271$	−13.7332	−0.834234	−0.417117	−	0.908853i	$-0.636959\pi$
$271$	−13.7332	−0.834234	−0.417117	+	0.908853i	$0.636959\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	23.1787	1.39773
$276$	0	0
$277$	−29.0414	−1.74493	−0.872463	−	0.488680i	$-0.837479\pi$
$277$	−29.0414	−1.74493	−0.872463	+	0.488680i	$0.837479\pi$
$278$	0	0
$279$	6.48614	0.388315
$280$	0	0
$281$	14.8066	0.883286	0.441643	−	0.897191i	$-0.354396\pi$
$281$	14.8066	0.883286	0.441643	+	0.897191i	$0.354396\pi$
$282$	0	0
$283$	−4.17771	−0.248339	−0.124170	−	0.992261i	$-0.539627\pi$
$283$	−4.17771	−0.248339	−0.124170	+	0.992261i	$0.539627\pi$
$284$	0	0
$285$	11.9577	0.708313
$286$	0	0
$287$	−14.4825	−0.854872
$288$	0	0
$289$	−5.92901	−0.348765
$290$	0	0
$291$	−16.9132	−0.991470
$292$	0	0
$293$	−11.2069	−0.654712	−0.327356	−	0.944901i	$-0.606158\pi$
$293$	−11.2069	−0.654712	−0.327356	+	0.944901i	$0.606158\pi$
$294$	0	0
$295$	−45.3001	−2.63747
$296$	0	0
$297$	4.83677	0.280658
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−8.57832	−0.494446
$302$	0	0
$303$	−16.1598	−0.928354
$304$	0	0
$305$	−22.4983	−1.28825
$306$	0	0
$307$	1.32884	0.0758408	0.0379204	−	0.999281i	$-0.487927\pi$
$307$	1.32884	0.0758408	0.0379204	+	0.999281i	$0.487927\pi$
$308$	0	0
$309$	−10.0686	−0.572784
$310$	0	0
$311$	3.83416	0.217415	0.108708	−	0.994074i	$-0.465329\pi$
$311$	3.83416	0.217415	0.108708	+	0.994074i	$0.465329\pi$
$312$	0	0
$313$	−3.96014	−0.223841	−0.111920	−	0.993717i	$-0.535700\pi$
$313$	−3.96014	−0.223841	−0.111920	+	0.993717i	$0.535700\pi$
$314$	0	0
$315$	4.66140	0.262640
$316$	0	0
$317$	−15.0459	−0.845063	−0.422532	−	0.906348i	$-0.638858\pi$
$317$	−15.0459	−0.845063	−0.422532	+	0.906348i	$0.638858\pi$
$318$	0	0
$319$	17.9002	1.00222
$320$	0	0
$321$	12.1710	0.679318
$322$	0	0
$323$	12.7145	0.707456
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.74502	−0.373000
$328$	0	0
$329$	−11.5152	−0.634853
$330$	0	0
$331$	14.6635	0.805980	0.402990	−	0.915204i	$-0.367971\pi$
$331$	14.6635	0.805980	0.402990	+	0.915204i	$0.367971\pi$
$332$	0	0
$333$	−4.68535	−0.256756
$334$	0	0
$335$	−11.9477	−0.652773
$336$	0	0
$337$	17.1320	0.933238	0.466619	−	0.884458i	$-0.345472\pi$
$337$	17.1320	0.933238	0.466619	+	0.884458i	$0.345472\pi$
$338$	0	0
$339$	10.9767	0.596172
$340$	0	0
$341$	31.3720	1.69889
$342$	0	0
$343$	−17.5493	−0.947572
$344$	0	0
$345$	−14.9506	−0.804914
$346$	0	0
$347$	−19.3813	−1.04044	−0.520222	−	0.854031i	$-0.674151\pi$
$347$	−19.3813	−1.04044	−0.520222	+	0.854031i	$0.674151\pi$
$348$	0	0
$349$	30.3434	1.62425	0.812123	−	0.583486i	$-0.198312\pi$
$349$	30.3434	1.62425	0.812123	+	0.583486i	$0.198312\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.10175	−0.111865	−0.0559324	−	0.998435i	$-0.517813\pi$
$353$	−2.10175	−0.111865	−0.0559324	+	0.998435i	$0.517813\pi$
$354$	0	0
$355$	15.9463	0.846344
$356$	0	0
$357$	4.95643	0.262322
$358$	0	0
$359$	32.0220	1.69005	0.845027	−	0.534723i	$-0.179584\pi$
$359$	32.0220	1.69005	0.845027	+	0.534723i	$0.179584\pi$
$360$	0	0
$361$	−4.39790	−0.231468
$362$	0	0
$363$	12.3944	0.650535
$364$	0	0
$365$	49.2703	2.57892
$366$	0	0
$367$	31.3627	1.63712	0.818560	−	0.574421i	$-0.194773\pi$
$367$	31.3627	1.63712	0.818560	+	0.574421i	$0.194773\pi$
$368$	0	0
$369$	−9.72224	−0.506119
$370$	0	0
$371$	12.3577	0.641582
$372$	0	0
$373$	11.2351	0.581732	0.290866	−	0.956764i	$-0.406057\pi$
$373$	11.2351	0.581732	0.290866	+	0.956764i	$0.406057\pi$
$374$	0	0
$375$	−0.650283	−0.0335805
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−18.3499	−0.942571	−0.471286	−	0.881981i	$-0.656210\pi$
$379$	−18.3499	−0.942571	−0.471286	+	0.881981i	$0.656210\pi$
$380$	0	0
$381$	4.01527	0.205709
$382$	0	0
$383$	28.2718	1.44462	0.722310	−	0.691569i	$-0.243080\pi$
$383$	28.2718	1.44462	0.722310	+	0.691569i	$0.243080\pi$
$384$	0	0
$385$	22.5461	1.14906
$386$	0	0
$387$	−5.75872	−0.292732
$388$	0	0
$389$	−8.65286	−0.438717	−0.219359	−	0.975644i	$-0.570396\pi$
$389$	−8.65286	−0.438717	−0.219359	+	0.975644i	$0.570396\pi$
$390$	0	0
$391$	−15.8969	−0.803941
$392$	0	0
$393$	−11.7892	−0.594685
$394$	0	0
$395$	34.7347	1.74769
$396$	0	0
$397$	23.3710	1.17296	0.586478	−	0.809965i	$-0.300514\pi$
$397$	23.3710	1.17296	0.586478	+	0.809965i	$0.300514\pi$
$398$	0	0
$399$	5.69225	0.284969
$400$	0	0
$401$	31.7237	1.58421	0.792103	−	0.610387i	$-0.208986\pi$
$401$	31.7237	1.58421	0.792103	+	0.610387i	$0.208986\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.12925	0.155494
$406$	0	0
$407$	−22.6620	−1.12331
$408$	0	0
$409$	−10.1860	−0.503663	−0.251832	−	0.967771i	$-0.581033\pi$
$409$	−10.1860	−0.503663	−0.251832	+	0.967771i	$0.581033\pi$
$410$	0	0
$411$	−13.3096	−0.656513
$412$	0	0
$413$	−21.5643	−1.06111
$414$	0	0
$415$	12.7097	0.623894
$416$	0	0
$417$	5.14114	0.251763
$418$	0	0
$419$	15.2901	0.746972	0.373486	−	0.927636i	$-0.378162\pi$
$419$	15.2901	0.746972	0.373486	+	0.927636i	$0.378162\pi$
$420$	0	0
$421$	3.07704	0.149966	0.0749828	−	0.997185i	$-0.476110\pi$
$421$	3.07704	0.149966	0.0749828	+	0.997185i	$0.476110\pi$
$422$	0	0
$423$	−7.73028	−0.375859
$424$	0	0
$425$	15.9451	0.773451
$426$	0	0
$427$	−10.7099	−0.518289
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.41397	−0.0681086	−0.0340543	−	0.999420i	$-0.510842\pi$
$431$	−1.41397	−0.0681086	−0.0340543	+	0.999420i	$0.510842\pi$
$432$	0	0
$433$	28.8920	1.38846	0.694231	−	0.719752i	$-0.255744\pi$
$433$	28.8920	1.38846	0.694231	+	0.719752i	$0.255744\pi$
$434$	0	0
$435$	11.5809	0.555261
$436$	0	0
$437$	−18.2569	−0.873346
$438$	0	0
$439$	−25.9827	−1.24009	−0.620044	−	0.784567i	$-0.712885\pi$
$439$	−25.9827	−1.24009	−0.620044	+	0.784567i	$0.712885\pi$
$440$	0	0
$441$	−4.78103	−0.227668
$442$	0	0
$443$	−12.1177	−0.575728	−0.287864	−	0.957671i	$-0.592945\pi$
$443$	−12.1177	−0.575728	−0.287864	+	0.957671i	$0.592945\pi$
$444$	0	0
$445$	−52.9380	−2.50950
$446$	0	0
$447$	4.28026	0.202450
$448$	0	0
$449$	3.12274	0.147371	0.0736856	−	0.997282i	$-0.476524\pi$
$449$	3.12274	0.147371	0.0736856	+	0.997282i	$0.476524\pi$
$450$	0	0
$451$	−47.0242	−2.21428
$452$	0	0
$453$	16.5496	0.777568
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.9608	−1.63540	−0.817698	−	0.575647i	$-0.804750\pi$
$457$	−34.9608	−1.63540	−0.817698	+	0.575647i	$0.804750\pi$
$458$	0	0
$459$	3.32731	0.155306
$460$	0	0
$461$	−22.1820	−1.03312	−0.516560	−	0.856251i	$-0.672788\pi$
$461$	−22.1820	−1.03312	−0.516560	+	0.856251i	$0.672788\pi$
$462$	0	0
$463$	10.8864	0.505935	0.252968	−	0.967475i	$-0.418593\pi$
$463$	10.8864	0.505935	0.252968	+	0.967475i	$0.418593\pi$
$464$	0	0
$465$	20.2968	0.941239
$466$	0	0
$467$	−11.2373	−0.519999	−0.259999	−	0.965609i	$-0.583722\pi$
$467$	−11.2373	−0.519999	−0.259999	+	0.965609i	$0.583722\pi$
$468$	0	0
$469$	−5.68749	−0.262624
$470$	0	0
$471$	−21.5076	−0.991019
$472$	0	0
$473$	−27.8536	−1.28071
$474$	0	0
$475$	18.3123	0.840224
$476$	0	0
$477$	8.29590	0.379843
$478$	0	0
$479$	24.0400	1.09841	0.549207	−	0.835686i	$-0.314930\pi$
$479$	24.0400	1.09841	0.549207	+	0.835686i	$0.314930\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−7.11697	−0.323833
$484$	0	0
$485$	−52.9256	−2.40323
$486$	0	0
$487$	−6.19612	−0.280773	−0.140387	−	0.990097i	$-0.544835\pi$
$487$	−6.19612	−0.280773	−0.140387	+	0.990097i	$0.544835\pi$
$488$	0	0
$489$	−6.62970	−0.299805
$490$	0	0
$491$	15.4948	0.699272	0.349636	−	0.936886i	$-0.386305\pi$
$491$	15.4948	0.699272	0.349636	+	0.936886i	$0.386305\pi$
$492$	0	0
$493$	12.3139	0.554589
$494$	0	0
$495$	15.1355	0.680288
$496$	0	0
$497$	7.59097	0.340502
$498$	0	0
$499$	36.6496	1.64066	0.820330	−	0.571890i	$-0.193790\pi$
$499$	36.6496	1.64066	0.820330	+	0.571890i	$0.193790\pi$
$500$	0	0
$501$	−4.06853	−0.181768
$502$	0	0
$503$	36.1335	1.61111	0.805556	−	0.592520i	$-0.201867\pi$
$503$	36.1335	1.61111	0.805556	+	0.592520i	$0.201867\pi$
$504$	0	0
$505$	−50.5679	−2.25024
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.8488	−0.746809	−0.373404	−	0.927669i	$-0.621810\pi$
$509$	−16.8488	−0.746809	−0.373404	+	0.927669i	$0.621810\pi$
$510$	0	0
$511$	23.4542	1.03755
$512$	0	0
$513$	3.82127	0.168713
$514$	0	0
$515$	−31.5072	−1.38837
$516$	0	0
$517$	−37.3896	−1.64439
$518$	0	0
$519$	0.886940	0.0389323
$520$	0	0
$521$	−11.9718	−0.524496	−0.262248	−	0.965001i	$-0.584464\pi$
$521$	−11.9718	−0.524496	−0.262248	+	0.965001i	$0.584464\pi$
$522$	0	0
$523$	−23.1292	−1.01137	−0.505684	−	0.862719i	$-0.668760\pi$
$523$	−23.1292	−1.01137	−0.505684	+	0.862719i	$0.668760\pi$
$524$	0	0
$525$	7.13855	0.311552
$526$	0	0
$527$	21.5814	0.940101
$528$	0	0
$529$	−0.173551	−0.00754568
$530$	0	0
$531$	−14.4763	−0.628220
$532$	0	0
$533$	0	0
$534$	0	0
$535$	38.0860	1.64660
$536$	0	0
$537$	6.90942	0.298163
$538$	0	0
$539$	−23.1247	−0.996053
$540$	0	0
$541$	−27.1983	−1.16935	−0.584673	−	0.811269i	$-0.698777\pi$
$541$	−27.1983	−1.16935	−0.584673	+	0.811269i	$0.698777\pi$
$542$	0	0
$543$	4.57338	0.196262
$544$	0	0
$545$	−21.1068	−0.904118
$546$	0	0
$547$	38.0521	1.62699	0.813494	−	0.581573i	$-0.197563\pi$
$547$	38.0521	1.62699	0.813494	+	0.581573i	$0.197563\pi$
$548$	0	0
$549$	−7.18968	−0.306848
$550$	0	0
$551$	14.1419	0.602467
$552$	0	0
$553$	16.5348	0.703132
$554$	0	0
$555$	−14.6616	−0.622351
$556$	0	0
$557$	3.34390	0.141686	0.0708429	−	0.997487i	$-0.477431\pi$
$557$	3.34390	0.141686	0.0708429	+	0.997487i	$0.477431\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	16.0934	0.679466
$562$	0	0
$563$	3.58489	0.151085	0.0755425	−	0.997143i	$-0.475931\pi$
$563$	3.58489	0.151085	0.0755425	+	0.997143i	$0.475931\pi$
$564$	0	0
$565$	34.3488	1.44506
$566$	0	0
$567$	1.48962	0.0625582
$568$	0	0
$569$	33.3537	1.39826	0.699130	−	0.714994i	$-0.253571\pi$
$569$	33.3537	1.39826	0.699130	+	0.714994i	$0.253571\pi$
$570$	0	0
$571$	−3.35708	−0.140489	−0.0702446	−	0.997530i	$-0.522378\pi$
$571$	−3.35708	−0.140489	−0.0702446	+	0.997530i	$0.522378\pi$
$572$	0	0
$573$	9.81213	0.409908
$574$	0	0
$575$	−22.8957	−0.954816
$576$	0	0
$577$	−13.4308	−0.559132	−0.279566	−	0.960127i	$-0.590191\pi$
$577$	−13.4308	−0.559132	−0.279566	+	0.960127i	$0.590191\pi$
$578$	0	0
$579$	−25.3006	−1.05146
$580$	0	0
$581$	6.05022	0.251005
$582$	0	0
$583$	40.1254	1.66182
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.5648	1.09645	0.548224	−	0.836331i	$-0.315304\pi$
$587$	26.5648	1.09645	0.548224	+	0.836331i	$0.315304\pi$
$588$	0	0
$589$	24.7853	1.02126
$590$	0	0
$591$	16.5048	0.678916
$592$	0	0
$593$	−15.4999	−0.636504	−0.318252	−	0.948006i	$-0.603096\pi$
$593$	−15.4999	−0.636504	−0.318252	+	0.948006i	$0.603096\pi$
$594$	0	0
$595$	15.5099	0.635844
$596$	0	0
$597$	26.0642	1.06674
$598$	0	0
$599$	−29.2092	−1.19346	−0.596728	−	0.802444i	$-0.703533\pi$
$599$	−29.2092	−1.19346	−0.596728	+	0.802444i	$0.703533\pi$
$600$	0	0
$601$	−11.0346	−0.450112	−0.225056	−	0.974346i	$-0.572256\pi$
$601$	−11.0346	−0.450112	−0.225056	+	0.974346i	$0.572256\pi$
$602$	0	0
$603$	−3.81807	−0.155484
$604$	0	0
$605$	38.7850	1.57684
$606$	0	0
$607$	−23.3186	−0.946474	−0.473237	−	0.880935i	$-0.656915\pi$
$607$	−23.3186	−0.946474	−0.473237	+	0.880935i	$0.656915\pi$
$608$	0	0
$609$	5.51287	0.223393
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−16.5992	−0.670434	−0.335217	−	0.942141i	$-0.608810\pi$
$613$	−16.5992	−0.670434	−0.335217	+	0.942141i	$0.608810\pi$
$614$	0	0
$615$	−30.4233	−1.22679
$616$	0	0
$617$	−19.8524	−0.799226	−0.399613	−	0.916684i	$-0.630856\pi$
$617$	−19.8524	−0.799226	−0.399613	+	0.916684i	$0.630856\pi$
$618$	0	0
$619$	10.8532	0.436228	0.218114	−	0.975923i	$-0.430010\pi$
$619$	10.8532	0.436228	0.218114	+	0.975923i	$0.430010\pi$
$620$	0	0
$621$	−4.77770	−0.191723
$622$	0	0
$623$	−25.2002	−1.00962
$624$	0	0
$625$	−25.9959	−1.03983
$626$	0	0
$627$	18.4826	0.738124
$628$	0	0
$629$	−15.5896	−0.621598
$630$	0	0
$631$	−16.4302	−0.654077	−0.327039	−	0.945011i	$-0.606051\pi$
$631$	−16.4302	−0.654077	−0.327039	+	0.945011i	$0.606051\pi$
$632$	0	0
$633$	11.6270	0.462132
$634$	0	0
$635$	12.5648	0.498618
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.09590	0.201591
$640$	0	0
$641$	−2.56950	−0.101489	−0.0507446	−	0.998712i	$-0.516159\pi$
$641$	−2.56950	−0.101489	−0.0507446	+	0.998712i	$0.516159\pi$
$642$	0	0
$643$	19.4067	0.765324	0.382662	−	0.923888i	$-0.375007\pi$
$643$	19.4067	0.765324	0.382662	+	0.923888i	$0.375007\pi$
$644$	0	0
$645$	−18.0205	−0.709555
$646$	0	0
$647$	40.9837	1.61124	0.805618	−	0.592436i	$-0.201834\pi$
$647$	40.9837	1.61124	0.805618	+	0.592436i	$0.201834\pi$
$648$	0	0
$649$	−70.0188	−2.74848
$650$	0	0
$651$	9.66190	0.378680
$652$	0	0
$653$	31.0938	1.21679	0.608397	−	0.793633i	$-0.291813\pi$
$653$	31.0938	1.21679	0.608397	+	0.793633i	$0.291813\pi$
$654$	0	0
$655$	−36.8912	−1.44146
$656$	0	0
$657$	15.7451	0.614274
$658$	0	0
$659$	24.1087	0.939142	0.469571	−	0.882895i	$-0.344409\pi$
$659$	24.1087	0.939142	0.469571	+	0.882895i	$0.344409\pi$
$660$	0	0
$661$	−4.19648	−0.163224	−0.0816121	−	0.996664i	$-0.526007\pi$
$661$	−4.19648	−0.163224	−0.0816121	+	0.996664i	$0.526007\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	17.8125	0.690737
$666$	0	0
$667$	−17.6816	−0.684633
$668$	0	0
$669$	−19.7959	−0.765354
$670$	0	0
$671$	−34.7748	−1.34247
$672$	0	0
$673$	−2.36676	−0.0912318	−0.0456159	−	0.998959i	$-0.514525\pi$
$673$	−2.36676	−0.0912318	−0.0456159	+	0.998959i	$0.514525\pi$
$674$	0	0
$675$	4.79219	0.184452
$676$	0	0
$677$	−7.03853	−0.270513	−0.135256	−	0.990811i	$-0.543186\pi$
$677$	−7.03853	−0.270513	−0.135256	+	0.990811i	$0.543186\pi$
$678$	0	0
$679$	−25.1943	−0.966868
$680$	0	0
$681$	4.01162	0.153726
$682$	0	0
$683$	27.1863	1.04025	0.520127	−	0.854089i	$-0.325885\pi$
$683$	27.1863	1.04025	0.520127	+	0.854089i	$0.325885\pi$
$684$	0	0
$685$	−41.6489	−1.59132
$686$	0	0
$687$	3.49817	0.133464
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−48.5884	−1.84839	−0.924194	−	0.381923i	$-0.875262\pi$
$691$	−48.5884	−1.84839	−0.924194	+	0.381923i	$0.875262\pi$
$692$	0	0
$693$	7.20496	0.273694
$694$	0	0
$695$	16.0879	0.610249
$696$	0	0
$697$	−32.3489	−1.22530
$698$	0	0
$699$	8.79848	0.332789
$700$	0	0
$701$	11.7456	0.443625	0.221812	−	0.975089i	$-0.428803\pi$
$701$	11.7456	0.443625	0.221812	+	0.975089i	$0.428803\pi$
$702$	0	0
$703$	−17.9040	−0.675261
$704$	0	0
$705$	−24.1900	−0.911046
$706$	0	0
$707$	−24.0719	−0.905318
$708$	0	0
$709$	41.9538	1.57561	0.787804	−	0.615926i	$-0.211218\pi$
$709$	41.9538	1.57561	0.787804	+	0.615926i	$0.211218\pi$
$710$	0	0
$711$	11.1000	0.416283
$712$	0	0
$713$	−30.9889	−1.16054
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.8723	0.630109
$718$	0	0
$719$	44.9578	1.67664	0.838322	−	0.545176i	$-0.183537\pi$
$719$	44.9578	1.67664	0.838322	+	0.545176i	$0.183537\pi$
$720$	0	0
$721$	−14.9985	−0.558571
$722$	0	0
$723$	−9.22284	−0.343001
$724$	0	0
$725$	17.7352	0.658668
$726$	0	0
$727$	−1.84927	−0.0685857	−0.0342928	−	0.999412i	$-0.510918\pi$
$727$	−1.84927	−0.0685857	−0.0342928	+	0.999412i	$0.510918\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.1610	−0.708697
$732$	0	0
$733$	−11.0976	−0.409898	−0.204949	−	0.978773i	$-0.565703\pi$
$733$	−11.0976	−0.409898	−0.204949	+	0.978773i	$0.565703\pi$
$734$	0	0
$735$	−14.9610	−0.551845
$736$	0	0
$737$	−18.4672	−0.680246
$738$	0	0
$739$	−43.2648	−1.59152	−0.795761	−	0.605611i	$-0.792929\pi$
$739$	−43.2648	−1.59152	−0.795761	+	0.605611i	$0.792929\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.92296	−0.0705467	−0.0352734	−	0.999378i	$-0.511230\pi$
$743$	−1.92296	−0.0705467	−0.0352734	+	0.999378i	$0.511230\pi$
$744$	0	0
$745$	13.3940	0.490718
$746$	0	0
$747$	4.06158	0.148605
$748$	0	0
$749$	18.1302	0.662462
$750$	0	0
$751$	25.2821	0.922557	0.461279	−	0.887255i	$-0.347391\pi$
$751$	25.2821	0.922557	0.461279	+	0.887255i	$0.347391\pi$
$752$	0	0
$753$	−3.02019	−0.110062
$754$	0	0
$755$	51.7878	1.88475
$756$	0	0
$757$	−1.72509	−0.0626993	−0.0313497	−	0.999508i	$-0.509981\pi$
$757$	−1.72509	−0.0626993	−0.0313497	+	0.999508i	$0.509981\pi$
$758$	0	0
$759$	−23.1087	−0.838791
$760$	0	0
$761$	−29.6005	−1.07302	−0.536509	−	0.843895i	$-0.680257\pi$
$761$	−29.6005	−1.07302	−0.536509	+	0.843895i	$0.680257\pi$
$762$	0	0
$763$	−10.0475	−0.363745
$764$	0	0
$765$	10.4120	0.376446
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0.321666	0.0115996	0.00579978	−	0.999983i	$-0.498154\pi$
$769$	0.321666	0.0115996	0.00579978	+	0.999983i	$0.498154\pi$
$770$	0	0
$771$	24.7035	0.889674
$772$	0	0
$773$	31.2148	1.12272	0.561359	−	0.827572i	$-0.310279\pi$
$773$	31.2148	1.12272	0.561359	+	0.827572i	$0.310279\pi$
$774$	0	0
$775$	31.0828	1.11653
$776$	0	0
$777$	−6.97940	−0.250384
$778$	0	0
$779$	−37.1513	−1.33108
$780$	0	0
$781$	24.6477	0.881965
$782$	0	0
$783$	3.70085	0.132258
$784$	0	0
$785$	−67.3027	−2.40214
$786$	0	0
$787$	−23.9712	−0.854480	−0.427240	−	0.904138i	$-0.640514\pi$
$787$	−23.9712	−0.854480	−0.427240	+	0.904138i	$0.640514\pi$
$788$	0	0
$789$	−1.12833	−0.0401697
$790$	0	0
$791$	16.3511	0.581378
$792$	0	0
$793$	0	0
$794$	0	0
$795$	25.9599	0.920703
$796$	0	0
$797$	−19.0863	−0.676070	−0.338035	−	0.941133i	$-0.609762\pi$
$797$	−19.0863	−0.676070	−0.338035	+	0.941133i	$0.609762\pi$
$798$	0	0
$799$	−25.7210	−0.909945
$800$	0	0
$801$	−16.9172	−0.597739
$802$	0	0
$803$	76.1554	2.68747
$804$	0	0
$805$	−22.2708	−0.784941
$806$	0	0
$807$	11.6337	0.409525
$808$	0	0
$809$	0.329314	0.0115781	0.00578904	−	0.999983i	$-0.498157\pi$
$809$	0.329314	0.0115781	0.00578904	+	0.999983i	$0.498157\pi$
$810$	0	0
$811$	−44.2741	−1.55467	−0.777337	−	0.629085i	$-0.783430\pi$
$811$	−44.2741	−1.55467	−0.777337	+	0.629085i	$0.783430\pi$
$812$	0	0
$813$	−13.7332	−0.481645
$814$	0	0
$815$	−20.7460	−0.726700
$816$	0	0
$817$	−22.0056	−0.769879
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46.0495	−1.60714	−0.803568	−	0.595212i	$-0.797068\pi$
$821$	−46.0495	−1.60714	−0.803568	+	0.595212i	$0.797068\pi$
$822$	0	0
$823$	−10.7087	−0.373281	−0.186641	−	0.982428i	$-0.559760\pi$
$823$	−10.7087	−0.373281	−0.186641	+	0.982428i	$0.559760\pi$
$824$	0	0
$825$	23.1787	0.806980
$826$	0	0
$827$	−18.6297	−0.647820	−0.323910	−	0.946088i	$-0.604997\pi$
$827$	−18.6297	−0.647820	−0.323910	+	0.946088i	$0.604997\pi$
$828$	0	0
$829$	28.8185	1.00091	0.500454	−	0.865763i	$-0.333166\pi$
$829$	28.8185	1.00091	0.500454	+	0.865763i	$0.333166\pi$
$830$	0	0
$831$	−29.0414	−1.00743
$832$	0	0
$833$	−15.9080	−0.551178
$834$	0	0
$835$	−12.7314	−0.440589
$836$	0	0
$837$	6.48614	0.224194
$838$	0	0
$839$	−48.0413	−1.65857	−0.829284	−	0.558827i	$-0.811252\pi$
$839$	−48.0413	−1.65857	−0.829284	+	0.558827i	$0.811252\pi$
$840$	0	0
$841$	−15.3037	−0.527714
$842$	0	0
$843$	14.8066	0.509965
$844$	0	0
$845$	0	0
$846$	0	0
$847$	18.4629	0.634393
$848$	0	0
$849$	−4.17771	−0.143379
$850$	0	0
$851$	22.3852	0.767355
$852$	0	0
$853$	13.2568	0.453906	0.226953	−	0.973906i	$-0.427124\pi$
$853$	13.2568	0.453906	0.226953	+	0.973906i	$0.427124\pi$
$854$	0	0
$855$	11.9577	0.408945
$856$	0	0
$857$	−39.5164	−1.34986	−0.674928	−	0.737884i	$-0.735825\pi$
$857$	−39.5164	−1.34986	−0.674928	+	0.737884i	$0.735825\pi$
$858$	0	0
$859$	−34.3382	−1.17161	−0.585803	−	0.810454i	$-0.699221\pi$
$859$	−34.3382	−1.17161	−0.585803	+	0.810454i	$0.699221\pi$
$860$	0	0
$861$	−14.4825	−0.493561
$862$	0	0
$863$	37.7547	1.28518	0.642592	−	0.766209i	$-0.277859\pi$
$863$	37.7547	1.28518	0.642592	+	0.766209i	$0.277859\pi$
$864$	0	0
$865$	2.77545	0.0943683
$866$	0	0
$867$	−5.92901	−0.201360
$868$	0	0
$869$	53.6882	1.82125
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−16.9132	−0.572426
$874$	0	0
$875$	−0.968676	−0.0327472
$876$	0	0
$877$	3.14218	0.106104	0.0530519	−	0.998592i	$-0.483105\pi$
$877$	3.14218	0.106104	0.0530519	+	0.998592i	$0.483105\pi$
$878$	0	0
$879$	−11.2069	−0.377998
$880$	0	0
$881$	−35.8867	−1.20905	−0.604527	−	0.796585i	$-0.706638\pi$
$881$	−35.8867	−1.20905	−0.604527	+	0.796585i	$0.706638\pi$
$882$	0	0
$883$	−39.2723	−1.32162	−0.660810	−	0.750554i	$-0.729787\pi$
$883$	−39.2723	−1.32162	−0.660810	+	0.750554i	$0.729787\pi$
$884$	0	0
$885$	−45.3001	−1.52274
$886$	0	0
$887$	−49.5023	−1.66212	−0.831062	−	0.556179i	$-0.812267\pi$
$887$	−49.5023	−1.66212	−0.831062	+	0.556179i	$0.812267\pi$
$888$	0	0
$889$	5.98124	0.200604
$890$	0	0
$891$	4.83677	0.162038
$892$	0	0
$893$	−29.5395	−0.988501
$894$	0	0
$895$	21.6213	0.722719
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0043	0.800587
$900$	0	0
$901$	27.6030	0.919590
$902$	0	0
$903$	−8.57832	−0.285468
$904$	0	0
$905$	14.3112	0.475722
$906$	0	0
$907$	−34.2459	−1.13712	−0.568558	−	0.822643i	$-0.692498\pi$
$907$	−34.2459	−1.13712	−0.568558	+	0.822643i	$0.692498\pi$
$908$	0	0
$909$	−16.1598	−0.535985
$910$	0	0
$911$	29.6128	0.981117	0.490558	−	0.871408i	$-0.336793\pi$
$911$	29.6128	0.981117	0.490558	+	0.871408i	$0.336793\pi$
$912$	0	0
$913$	19.6449	0.650152
$914$	0	0
$915$	−22.4983	−0.743771
$916$	0	0
$917$	−17.5614	−0.579928
$918$	0	0
$919$	−17.6710	−0.582913	−0.291457	−	0.956584i	$-0.594140\pi$
$919$	−17.6710	−0.582913	−0.291457	+	0.956584i	$0.594140\pi$
$920$	0	0
$921$	1.32884	0.0437867
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−22.4531	−0.738253
$926$	0	0
$927$	−10.0686	−0.330697
$928$	0	0
$929$	−16.3131	−0.535216	−0.267608	−	0.963528i	$-0.586233\pi$
$929$	−16.3131	−0.535216	−0.267608	+	0.963528i	$0.586233\pi$
$930$	0	0
$931$	−18.2696	−0.598762
$932$	0	0
$933$	3.83416	0.125525
$934$	0	0
$935$	50.3604	1.64696
$936$	0	0
$937$	−40.3202	−1.31720	−0.658602	−	0.752492i	$-0.728852\pi$
$937$	−40.3202	−1.31720	−0.658602	+	0.752492i	$0.728852\pi$
$938$	0	0
$939$	−3.96014	−0.129234
$940$	0	0
$941$	10.8965	0.355216	0.177608	−	0.984101i	$-0.443164\pi$
$941$	10.8965	0.355216	0.177608	+	0.984101i	$0.443164\pi$
$942$	0	0
$943$	46.4500	1.51262
$944$	0	0
$945$	4.66140	0.151635
$946$	0	0
$947$	7.64336	0.248376	0.124188	−	0.992259i	$-0.460367\pi$
$947$	7.64336	0.248376	0.124188	+	0.992259i	$0.460367\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−15.0459	−0.487897
$952$	0	0
$953$	8.61098	0.278937	0.139468	−	0.990227i	$-0.455461\pi$
$953$	8.61098	0.278937	0.139468	+	0.990227i	$0.455461\pi$
$954$	0	0
$955$	30.7046	0.993578
$956$	0	0
$957$	17.9002	0.578630
$958$	0	0
$959$	−19.8262	−0.640222
$960$	0	0
$961$	11.0701	0.357099
$962$	0	0
$963$	12.1710	0.392205
$964$	0	0
$965$	−79.1718	−2.54863
$966$	0	0
$967$	−34.5227	−1.11017	−0.555087	−	0.831792i	$-0.687315\pi$
$967$	−34.5227	−1.11017	−0.555087	+	0.831792i	$0.687315\pi$
$968$	0	0
$969$	12.7145	0.408450
$970$	0	0
$971$	35.5678	1.14142	0.570712	−	0.821150i	$-0.306667\pi$
$971$	35.5678	1.14142	0.570712	+	0.821150i	$0.306667\pi$
$972$	0	0
$973$	7.65836	0.245516
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.2619	1.35208	0.676039	−	0.736866i	$-0.263695\pi$
$977$	42.2619	1.35208	0.676039	+	0.736866i	$0.263695\pi$
$978$	0	0
$979$	−81.8245	−2.61512
$980$	0	0
$981$	−6.74502	−0.215352
$982$	0	0
$983$	−45.7605	−1.45953	−0.729766	−	0.683697i	$-0.760371\pi$
$983$	−45.7605	−1.45953	−0.729766	+	0.683697i	$0.760371\pi$
$984$	0	0
$985$	51.6476	1.64563
$986$	0	0
$987$	−11.5152	−0.366533
$988$	0	0
$989$	27.5135	0.874877
$990$	0	0
$991$	5.73491	0.182175	0.0910877	−	0.995843i	$-0.470966\pi$
$991$	5.73491	0.182175	0.0910877	+	0.995843i	$0.470966\pi$
$992$	0	0
$993$	14.6635	0.465333
$994$	0	0
$995$	81.5613	2.58567
$996$	0	0
$997$	−53.5872	−1.69712	−0.848561	−	0.529097i	$-0.822531\pi$
$997$	−53.5872	−1.69712	−0.848561	+	0.529097i	$0.822531\pi$
$998$	0	0
$999$	−4.68535	−0.148238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.bh.1.6	✓	6
4.3	odd	2		8112.2.a.ct.1.6		6
13.5	odd	4		4056.2.c.r.337.2		12
13.8	odd	4		4056.2.c.r.337.11		12
13.12	even	2		4056.2.a.bi.1.1	yes	6
52.51	odd	2		8112.2.a.cu.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bh.1.6	✓	6	1.1	even	1	trivial
4056.2.a.bi.1.1	yes	6	13.12	even	2
4056.2.c.r.337.2		12	13.5	odd	4
4056.2.c.r.337.11		12	13.8	odd	4
8112.2.a.ct.1.6		6	4.3	odd	2
8112.2.a.cu.1.1		6	52.51	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.12925 q^{5} +1.48962 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.12925 q^{5} +1.48962 q^{7} +1.00000 q^{9} +4.83677 q^{11} +3.12925 q^{15} +3.32731 q^{17} +3.82127 q^{19} +1.48962 q^{21} -4.77770 q^{23} +4.79219 q^{25} +1.00000 q^{27} +3.70085 q^{29} +6.48614 q^{31} +4.83677 q^{33} +4.66140 q^{35} -4.68535 q^{37} -9.72224 q^{41} -5.75872 q^{43} +3.12925 q^{45} -7.73028 q^{47} -4.78103 q^{49} +3.32731 q^{51} +8.29590 q^{53} +15.1355 q^{55} +3.82127 q^{57} -14.4763 q^{59} -7.18968 q^{61} +1.48962 q^{63} -3.81807 q^{67} -4.77770 q^{69} +5.09590 q^{71} +15.7451 q^{73} +4.79219 q^{75} +7.20496 q^{77} +11.1000 q^{79} +1.00000 q^{81} +4.06158 q^{83} +10.4120 q^{85} +3.70085 q^{87} -16.9172 q^{89} +6.48614 q^{93} +11.9577 q^{95} -16.9132 q^{97} +4.83677 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - q^5 + 5 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} - q^{5} + 5 q^{7} + 6 q^{9} + 6 q^{11} - q^{15} + 9 q^{17} - 7 q^{19} + 5 q^{21} + 12 q^{23} + 9 q^{25} + 6 q^{27} + 7 q^{29} + 11 q^{31} + 6 q^{33} + 6 q^{35} - 6 q^{37} - 13 q^{41} + 15 q^{43} - q^{45} + 9 q^{47} + 13 q^{49} + 9 q^{51} + 22 q^{53} + 3 q^{55} - 7 q^{57} + 7 q^{59} + 25 q^{61} + 5 q^{63} - 5 q^{67} + 12 q^{69} - 8 q^{71} - 15 q^{73} + 9 q^{75} + 45 q^{77} + 14 q^{79} + 6 q^{81} + 13 q^{83} + 35 q^{85} + 7 q^{87} - 33 q^{89} + 11 q^{93} + 47 q^{95} - 50 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - q^5 + 5 * q^7 + 6 * q^9 + 6 * q^11 - q^15 + 9 * q^17 - 7 * q^19 + 5 * q^21 + 12 * q^23 + 9 * q^25 + 6 * q^27 + 7 * q^29 + 11 * q^31 + 6 * q^33 + 6 * q^35 - 6 * q^37 - 13 * q^41 + 15 * q^43 - q^45 + 9 * q^47 + 13 * q^49 + 9 * q^51 + 22 * q^53 + 3 * q^55 - 7 * q^57 + 7 * q^59 + 25 * q^61 + 5 * q^63 - 5 * q^67 + 12 * q^69 - 8 * q^71 - 15 * q^73 + 9 * q^75 + 45 * q^77 + 14 * q^79 + 6 * q^81 + 13 * q^83 + 35 * q^85 + 7 * q^87 - 33 * q^89 + 11 * q^93 + 47 * q^95 - 50 * q^97 + 6 * q^99

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