LMFDB - Embedded newform 4056.2.a.bg.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.90211 q^{5} +2.88227 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.90211 q^{5} +2.88227 q^{7} +1.00000 q^{9} +3.21123 q^{11} +2.90211 q^{15} +5.12833 q^{17} +1.45479 q^{19} -2.88227 q^{21} -4.09235 q^{23} +3.42223 q^{25} -1.00000 q^{27} +3.45707 q^{29} -7.55490 q^{31} -3.21123 q^{33} -8.36465 q^{35} +11.4323 q^{37} -7.52765 q^{41} +6.51996 q^{43} -2.90211 q^{45} +9.09733 q^{47} +1.30747 q^{49} -5.12833 q^{51} +5.65178 q^{53} -9.31933 q^{55} -1.45479 q^{57} -7.21876 q^{59} -1.55262 q^{61} +2.88227 q^{63} +5.43273 q^{67} +4.09235 q^{69} -15.2420 q^{71} -15.4852 q^{73} -3.42223 q^{75} +9.25562 q^{77} +5.94669 q^{79} +1.00000 q^{81} -8.20762 q^{83} -14.8830 q^{85} -3.45707 q^{87} -12.7993 q^{89} +7.55490 q^{93} -4.22195 q^{95} +19.0855 q^{97} +3.21123 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.90211	−1.29786	−0.648931	−	0.760847i	$-0.724784\pi$
$5$	−2.90211	−1.29786	−0.648931	+	0.760847i	$0.724784\pi$
$6$	0	0
$7$	2.88227	1.08939	0.544697	−	0.838633i	$-0.316644\pi$
$7$	2.88227	1.08939	0.544697	+	0.838633i	$0.316644\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.21123	0.968222	0.484111	−	0.875007i	$-0.339143\pi$
$11$	3.21123	0.968222	0.484111	+	0.875007i	$0.339143\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.90211	0.749321
$16$	0	0
$17$	5.12833	1.24380	0.621902	−	0.783095i	$-0.286360\pi$
$17$	5.12833	1.24380	0.621902	+	0.783095i	$0.286360\pi$
$18$	0	0
$19$	1.45479	0.333751	0.166876	−	0.985978i	$-0.446632\pi$
$19$	1.45479	0.333751	0.166876	+	0.985978i	$0.446632\pi$
$20$	0	0
$21$	−2.88227	−0.628962
$22$	0	0
$23$	−4.09235	−0.853315	−0.426657	−	0.904413i	$-0.640309\pi$
$23$	−4.09235	−0.853315	−0.426657	+	0.904413i	$0.640309\pi$
$24$	0	0
$25$	3.42223	0.684446
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.45707	0.641961	0.320981	−	0.947086i	$-0.395988\pi$
$29$	3.45707	0.641961	0.320981	+	0.947086i	$0.395988\pi$
$30$	0	0
$31$	−7.55490	−1.35690	−0.678450	−	0.734646i	$-0.737348\pi$
$31$	−7.55490	−1.35690	−0.678450	+	0.734646i	$0.737348\pi$
$32$	0	0
$33$	−3.21123	−0.559003
$34$	0	0
$35$	−8.36465	−1.41388
$36$	0	0
$37$	11.4323	1.87946	0.939731	−	0.341914i	$-0.111075\pi$
$37$	11.4323	1.87946	0.939731	+	0.341914i	$0.111075\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.52765	−1.17562	−0.587811	−	0.808998i	$-0.700010\pi$
$41$	−7.52765	−1.17562	−0.587811	+	0.808998i	$0.700010\pi$
$42$	0	0
$43$	6.51996	0.994285	0.497143	−	0.867669i	$-0.334383\pi$
$43$	6.51996	0.994285	0.497143	+	0.867669i	$0.334383\pi$
$44$	0	0
$45$	−2.90211	−0.432621
$46$	0	0
$47$	9.09733	1.32698	0.663491	−	0.748185i	$-0.269074\pi$
$47$	9.09733	1.32698	0.663491	+	0.748185i	$0.269074\pi$
$48$	0	0
$49$	1.30747	0.186781
$50$	0	0
$51$	−5.12833	−0.718110
$52$	0	0
$53$	5.65178	0.776332	0.388166	−	0.921590i	$-0.373109\pi$
$53$	5.65178	0.776332	0.388166	+	0.921590i	$0.373109\pi$
$54$	0	0
$55$	−9.31933	−1.25662
$56$	0	0
$57$	−1.45479	−0.192691
$58$	0	0
$59$	−7.21876	−0.939803	−0.469901	−	0.882719i	$-0.655711\pi$
$59$	−7.21876	−0.939803	−0.469901	+	0.882719i	$0.655711\pi$
$60$	0	0
$61$	−1.55262	−0.198793	−0.0993965	−	0.995048i	$-0.531691\pi$
$61$	−1.55262	−0.198793	−0.0993965	+	0.995048i	$0.531691\pi$
$62$	0	0
$63$	2.88227	0.363132
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.43273	0.663714	0.331857	−	0.943330i	$-0.392325\pi$
$67$	5.43273	0.663714	0.331857	+	0.943330i	$0.392325\pi$
$68$	0	0
$69$	4.09235	0.492662
$70$	0	0
$71$	−15.2420	−1.80889	−0.904447	−	0.426585i	$-0.859716\pi$
$71$	−15.2420	−1.80889	−0.904447	+	0.426585i	$0.859716\pi$
$72$	0	0
$73$	−15.4852	−1.81241	−0.906204	−	0.422840i	$-0.861033\pi$
$73$	−15.4852	−1.81241	−0.906204	+	0.422840i	$0.861033\pi$
$74$	0	0
$75$	−3.42223	−0.395165
$76$	0	0
$77$	9.25562	1.05478
$78$	0	0
$79$	5.94669	0.669055	0.334527	−	0.942386i	$-0.391423\pi$
$79$	5.94669	0.669055	0.334527	+	0.942386i	$0.391423\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.20762	−0.900904	−0.450452	−	0.892801i	$-0.648737\pi$
$83$	−8.20762	−0.900904	−0.450452	+	0.892801i	$0.648737\pi$
$84$	0	0
$85$	−14.8830	−1.61429
$86$	0	0
$87$	−3.45707	−0.370636
$88$	0	0
$89$	−12.7993	−1.35672	−0.678362	−	0.734728i	$-0.737310\pi$
$89$	−12.7993	−1.35672	−0.678362	+	0.734728i	$0.737310\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.55490	0.783407
$94$	0	0
$95$	−4.22195	−0.433163
$96$	0	0
$97$	19.0855	1.93784	0.968918	−	0.247383i	$-0.0795707\pi$
$97$	19.0855	1.93784	0.968918	+	0.247383i	$0.0795707\pi$
$98$	0	0
$99$	3.21123	0.322741
$100$	0	0
$101$	3.22891	0.321289	0.160644	−	0.987012i	$-0.448643\pi$
$101$	3.22891	0.321289	0.160644	+	0.987012i	$0.448643\pi$
$102$	0	0
$103$	4.73825	0.466874	0.233437	−	0.972372i	$-0.425003\pi$
$103$	4.73825	0.466874	0.233437	+	0.972372i	$0.425003\pi$
$104$	0	0
$105$	8.36465	0.816307
$106$	0	0
$107$	11.8166	1.14236	0.571179	−	0.820825i	$-0.306486\pi$
$107$	11.8166	1.14236	0.571179	+	0.820825i	$0.306486\pi$
$108$	0	0
$109$	7.73028	0.740426	0.370213	−	0.928947i	$-0.379285\pi$
$109$	7.73028	0.740426	0.370213	+	0.928947i	$0.379285\pi$
$110$	0	0
$111$	−11.4323	−1.08511
$112$	0	0
$113$	−15.2560	−1.43517	−0.717583	−	0.696473i	$-0.754752\pi$
$113$	−15.2560	−1.43517	−0.717583	+	0.696473i	$0.754752\pi$
$114$	0	0
$115$	11.8765	1.10749
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.7812	1.35499
$120$	0	0
$121$	−0.688007	−0.0625461
$122$	0	0
$123$	7.52765	0.678745
$124$	0	0
$125$	4.57885	0.409545
$126$	0	0
$127$	20.9008	1.85464	0.927321	−	0.374267i	$-0.122106\pi$
$127$	20.9008	1.85464	0.927321	+	0.374267i	$0.122106\pi$
$128$	0	0
$129$	−6.51996	−0.574051
$130$	0	0
$131$	10.6280	0.928577	0.464289	−	0.885684i	$-0.346310\pi$
$131$	10.6280	0.928577	0.464289	+	0.885684i	$0.346310\pi$
$132$	0	0
$133$	4.19309	0.363587
$134$	0	0
$135$	2.90211	0.249774
$136$	0	0
$137$	−14.1395	−1.20802	−0.604008	−	0.796978i	$-0.706431\pi$
$137$	−14.1395	−1.20802	−0.604008	+	0.796978i	$0.706431\pi$
$138$	0	0
$139$	−0.519687	−0.0440792	−0.0220396	−	0.999757i	$-0.507016\pi$
$139$	−0.519687	−0.0440792	−0.0220396	+	0.999757i	$0.507016\pi$
$140$	0	0
$141$	−9.09733	−0.766133
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.0328	−0.833177
$146$	0	0
$147$	−1.30747	−0.107838
$148$	0	0
$149$	13.3709	1.09539	0.547695	−	0.836678i	$-0.315505\pi$
$149$	13.3709	1.09539	0.547695	+	0.836678i	$0.315505\pi$
$150$	0	0
$151$	21.4627	1.74661	0.873305	−	0.487173i	$-0.161972\pi$
$151$	21.4627	1.74661	0.873305	+	0.487173i	$0.161972\pi$
$152$	0	0
$153$	5.12833	0.414601
$154$	0	0
$155$	21.9251	1.76107
$156$	0	0
$157$	14.4785	1.15551	0.577753	−	0.816212i	$-0.303930\pi$
$157$	14.4785	1.15551	0.577753	+	0.816212i	$0.303930\pi$
$158$	0	0
$159$	−5.65178	−0.448215
$160$	0	0
$161$	−11.7953	−0.929597
$162$	0	0
$163$	1.80785	0.141602	0.0708011	−	0.997490i	$-0.477444\pi$
$163$	1.80785	0.141602	0.0708011	+	0.997490i	$0.477444\pi$
$164$	0	0
$165$	9.31933	0.725509
$166$	0	0
$167$	2.39600	0.185408	0.0927041	−	0.995694i	$-0.470449\pi$
$167$	2.39600	0.185408	0.0927041	+	0.995694i	$0.470449\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.45479	0.111250
$172$	0	0
$173$	−14.7926	−1.12466	−0.562332	−	0.826912i	$-0.690096\pi$
$173$	−14.7926	−1.12466	−0.562332	+	0.826912i	$0.690096\pi$
$174$	0	0
$175$	9.86379	0.745632
$176$	0	0
$177$	7.21876	0.542595
$178$	0	0
$179$	−9.51339	−0.711064	−0.355532	−	0.934664i	$-0.615700\pi$
$179$	−9.51339	−0.711064	−0.355532	+	0.934664i	$0.615700\pi$
$180$	0	0
$181$	22.7383	1.69013	0.845063	−	0.534667i	$-0.179563\pi$
$181$	22.7383	1.69013	0.845063	+	0.534667i	$0.179563\pi$
$182$	0	0
$183$	1.55262	0.114773
$184$	0	0
$185$	−33.1778	−2.43928
$186$	0	0
$187$	16.4682	1.20428
$188$	0	0
$189$	−2.88227	−0.209654
$190$	0	0
$191$	2.11482	0.153023	0.0765115	−	0.997069i	$-0.475622\pi$
$191$	2.11482	0.153023	0.0765115	+	0.997069i	$0.475622\pi$
$192$	0	0
$193$	1.36260	0.0980818	0.0490409	−	0.998797i	$-0.484384\pi$
$193$	1.36260	0.0980818	0.0490409	+	0.998797i	$0.484384\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.59989	0.185234	0.0926171	−	0.995702i	$-0.470477\pi$
$197$	2.59989	0.185234	0.0926171	+	0.995702i	$0.470477\pi$
$198$	0	0
$199$	25.9200	1.83742	0.918709	−	0.394936i	$-0.129233\pi$
$199$	25.9200	1.83742	0.918709	+	0.394936i	$0.129233\pi$
$200$	0	0
$201$	−5.43273	−0.383196
$202$	0	0
$203$	9.96419	0.699349
$204$	0	0
$205$	21.8461	1.52579
$206$	0	0
$207$	−4.09235	−0.284438
$208$	0	0
$209$	4.67166	0.323145
$210$	0	0
$211$	21.2034	1.45970	0.729850	−	0.683608i	$-0.239590\pi$
$211$	21.2034	1.45970	0.729850	+	0.683608i	$0.239590\pi$
$212$	0	0
$213$	15.2420	1.04437
$214$	0	0
$215$	−18.9216	−1.29045
$216$	0	0
$217$	−21.7753	−1.47820
$218$	0	0
$219$	15.4852	1.04639
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.03414	0.471041	0.235521	−	0.971869i	$-0.424320\pi$
$223$	7.03414	0.471041	0.235521	+	0.971869i	$0.424320\pi$
$224$	0	0
$225$	3.42223	0.228149
$226$	0	0
$227$	−0.265626	−0.0176302	−0.00881512	−	0.999961i	$-0.502806\pi$
$227$	−0.265626	−0.0176302	−0.00881512	+	0.999961i	$0.502806\pi$
$228$	0	0
$229$	−17.5065	−1.15686	−0.578430	−	0.815732i	$-0.696334\pi$
$229$	−17.5065	−1.15686	−0.578430	+	0.815732i	$0.696334\pi$
$230$	0	0
$231$	−9.25562	−0.608975
$232$	0	0
$233$	9.12549	0.597831	0.298915	−	0.954280i	$-0.403375\pi$
$233$	9.12549	0.597831	0.298915	+	0.954280i	$0.403375\pi$
$234$	0	0
$235$	−26.4014	−1.72224
$236$	0	0
$237$	−5.94669	−0.386279
$238$	0	0
$239$	−1.10416	−0.0714221	−0.0357111	−	0.999362i	$-0.511370\pi$
$239$	−1.10416	−0.0714221	−0.0357111	+	0.999362i	$0.511370\pi$
$240$	0	0
$241$	−9.71751	−0.625960	−0.312980	−	0.949760i	$-0.601327\pi$
$241$	−9.71751	−0.625960	−0.312980	+	0.949760i	$0.601327\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.79442	−0.242417
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.20762	0.520137
$250$	0	0
$251$	13.6458	0.861318	0.430659	−	0.902515i	$-0.358281\pi$
$251$	13.6458	0.861318	0.430659	+	0.902515i	$0.358281\pi$
$252$	0	0
$253$	−13.1415	−0.826198
$254$	0	0
$255$	14.8830	0.932008
$256$	0	0
$257$	20.5623	1.28264	0.641320	−	0.767274i	$-0.278387\pi$
$257$	20.5623	1.28264	0.641320	+	0.767274i	$0.278387\pi$
$258$	0	0
$259$	32.9510	2.04748
$260$	0	0
$261$	3.45707	0.213987
$262$	0	0
$263$	8.81494	0.543553	0.271776	−	0.962360i	$-0.412389\pi$
$263$	8.81494	0.543553	0.271776	+	0.962360i	$0.412389\pi$
$264$	0	0
$265$	−16.4021	−1.00757
$266$	0	0
$267$	12.7993	0.783305
$268$	0	0
$269$	16.3100	0.994441	0.497221	−	0.867624i	$-0.334354\pi$
$269$	16.3100	0.994441	0.497221	+	0.867624i	$0.334354\pi$
$270$	0	0
$271$	7.59360	0.461278	0.230639	−	0.973039i	$-0.425918\pi$
$271$	7.59360	0.461278	0.230639	+	0.973039i	$0.425918\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.9896	0.662696
$276$	0	0
$277$	−5.02572	−0.301966	−0.150983	−	0.988536i	$-0.548244\pi$
$277$	−5.02572	−0.301966	−0.150983	+	0.988536i	$0.548244\pi$
$278$	0	0
$279$	−7.55490	−0.452300
$280$	0	0
$281$	25.7781	1.53779	0.768897	−	0.639373i	$-0.220806\pi$
$281$	25.7781	1.53779	0.768897	+	0.639373i	$0.220806\pi$
$282$	0	0
$283$	17.3692	1.03249	0.516245	−	0.856441i	$-0.327329\pi$
$283$	17.3692	1.03249	0.516245	+	0.856441i	$0.327329\pi$
$284$	0	0
$285$	4.22195	0.250087
$286$	0	0
$287$	−21.6967	−1.28072
$288$	0	0
$289$	9.29979	0.547046
$290$	0	0
$291$	−19.0855	−1.11881
$292$	0	0
$293$	−15.2944	−0.893511	−0.446756	−	0.894656i	$-0.647421\pi$
$293$	−15.2944	−0.893511	−0.446756	+	0.894656i	$0.647421\pi$
$294$	0	0
$295$	20.9496	1.21973
$296$	0	0
$297$	−3.21123	−0.186334
$298$	0	0
$299$	0	0
$300$	0	0
$301$	18.7923	1.08317
$302$	0	0
$303$	−3.22891	−0.185496
$304$	0	0
$305$	4.50588	0.258006
$306$	0	0
$307$	−21.6255	−1.23423	−0.617116	−	0.786872i	$-0.711699\pi$
$307$	−21.6255	−1.23423	−0.617116	+	0.786872i	$0.711699\pi$
$308$	0	0
$309$	−4.73825	−0.269550
$310$	0	0
$311$	−13.2236	−0.749842	−0.374921	−	0.927057i	$-0.622330\pi$
$311$	−13.2236	−0.749842	−0.374921	+	0.927057i	$0.622330\pi$
$312$	0	0
$313$	1.89385	0.107047	0.0535233	−	0.998567i	$-0.482955\pi$
$313$	1.89385	0.107047	0.0535233	+	0.998567i	$0.482955\pi$
$314$	0	0
$315$	−8.36465	−0.471295
$316$	0	0
$317$	6.76995	0.380238	0.190119	−	0.981761i	$-0.439113\pi$
$317$	6.76995	0.380238	0.190119	+	0.981761i	$0.439113\pi$
$318$	0	0
$319$	11.1014	0.621561
$320$	0	0
$321$	−11.8166	−0.659541
$322$	0	0
$323$	7.46063	0.415121
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7.73028	−0.427485
$328$	0	0
$329$	26.2209	1.44561
$330$	0	0
$331$	12.1917	0.670117	0.335059	−	0.942197i	$-0.391244\pi$
$331$	12.1917	0.670117	0.335059	+	0.942197i	$0.391244\pi$
$332$	0	0
$333$	11.4323	0.626487
$334$	0	0
$335$	−15.7664	−0.861409
$336$	0	0
$337$	−25.8451	−1.40787	−0.703936	−	0.710263i	$-0.748576\pi$
$337$	−25.8451	−1.40787	−0.703936	+	0.710263i	$0.748576\pi$
$338$	0	0
$339$	15.2560	0.828594
$340$	0	0
$341$	−24.2605	−1.31378
$342$	0	0
$343$	−16.4074	−0.885916
$344$	0	0
$345$	−11.8765	−0.639407
$346$	0	0
$347$	9.45029	0.507318	0.253659	−	0.967294i	$-0.418366\pi$
$347$	9.45029	0.507318	0.253659	+	0.967294i	$0.418366\pi$
$348$	0	0
$349$	−11.0197	−0.589873	−0.294937	−	0.955517i	$-0.595299\pi$
$349$	−11.0197	−0.589873	−0.294937	+	0.955517i	$0.595299\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.35211	0.0719653	0.0359827	−	0.999352i	$-0.488544\pi$
$353$	1.35211	0.0719653	0.0359827	+	0.999352i	$0.488544\pi$
$354$	0	0
$355$	44.2340	2.34770
$356$	0	0
$357$	−14.7812	−0.782306
$358$	0	0
$359$	2.88474	0.152251	0.0761253	−	0.997098i	$-0.475745\pi$
$359$	2.88474	0.152251	0.0761253	+	0.997098i	$0.475745\pi$
$360$	0	0
$361$	−16.8836	−0.888610
$362$	0	0
$363$	0.688007	0.0361110
$364$	0	0
$365$	44.9398	2.35226
$366$	0	0
$367$	17.7803	0.928125	0.464063	−	0.885802i	$-0.346391\pi$
$367$	17.7803	0.928125	0.464063	+	0.885802i	$0.346391\pi$
$368$	0	0
$369$	−7.52765	−0.391874
$370$	0	0
$371$	16.2899	0.845732
$372$	0	0
$373$	−23.4655	−1.21500	−0.607499	−	0.794320i	$-0.707827\pi$
$373$	−23.4655	−1.21500	−0.607499	+	0.794320i	$0.707827\pi$
$374$	0	0
$375$	−4.57885	−0.236451
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0.167691	0.00861373	0.00430686	−	0.999991i	$-0.498629\pi$
$379$	0.167691	0.00861373	0.00430686	+	0.999991i	$0.498629\pi$
$380$	0	0
$381$	−20.9008	−1.07078
$382$	0	0
$383$	−11.6074	−0.593110	−0.296555	−	0.955016i	$-0.595838\pi$
$383$	−11.6074	−0.593110	−0.296555	+	0.955016i	$0.595838\pi$
$384$	0	0
$385$	−26.8608	−1.36895
$386$	0	0
$387$	6.51996	0.331428
$388$	0	0
$389$	−16.3009	−0.826489	−0.413245	−	0.910620i	$-0.635605\pi$
$389$	−16.3009	−0.826489	−0.413245	+	0.910620i	$0.635605\pi$
$390$	0	0
$391$	−20.9870	−1.06136
$392$	0	0
$393$	−10.6280	−0.536114
$394$	0	0
$395$	−17.2579	−0.868341
$396$	0	0
$397$	−4.75753	−0.238774	−0.119387	−	0.992848i	$-0.538093\pi$
$397$	−4.75753	−0.238774	−0.119387	+	0.992848i	$0.538093\pi$
$398$	0	0
$399$	−4.19309	−0.209917
$400$	0	0
$401$	−13.9149	−0.694875	−0.347438	−	0.937703i	$-0.612948\pi$
$401$	−13.9149	−0.694875	−0.347438	+	0.937703i	$0.612948\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.90211	−0.144207
$406$	0	0
$407$	36.7118	1.81974
$408$	0	0
$409$	−23.0508	−1.13979	−0.569894	−	0.821718i	$-0.693016\pi$
$409$	−23.0508	−1.13979	−0.569894	+	0.821718i	$0.693016\pi$
$410$	0	0
$411$	14.1395	0.697449
$412$	0	0
$413$	−20.8064	−1.02382
$414$	0	0
$415$	23.8194	1.16925
$416$	0	0
$417$	0.519687	0.0254492
$418$	0	0
$419$	7.47640	0.365246	0.182623	−	0.983183i	$-0.441541\pi$
$419$	7.47640	0.365246	0.182623	+	0.983183i	$0.441541\pi$
$420$	0	0
$421$	18.8387	0.918143	0.459071	−	0.888399i	$-0.348182\pi$
$421$	18.8387	0.918143	0.459071	+	0.888399i	$0.348182\pi$
$422$	0	0
$423$	9.09733	0.442327
$424$	0	0
$425$	17.5503	0.851317
$426$	0	0
$427$	−4.47507	−0.216564
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.27720	0.206025	0.103013	−	0.994680i	$-0.467152\pi$
$431$	4.27720	0.206025	0.103013	+	0.994680i	$0.467152\pi$
$432$	0	0
$433$	32.6883	1.57090	0.785450	−	0.618925i	$-0.212431\pi$
$433$	32.6883	1.57090	0.785450	+	0.618925i	$0.212431\pi$
$434$	0	0
$435$	10.0328	0.481035
$436$	0	0
$437$	−5.95351	−0.284795
$438$	0	0
$439$	19.0214	0.907843	0.453921	−	0.891042i	$-0.350025\pi$
$439$	19.0214	0.907843	0.453921	+	0.891042i	$0.350025\pi$
$440$	0	0
$441$	1.30747	0.0622605
$442$	0	0
$443$	−6.80019	−0.323087	−0.161543	−	0.986866i	$-0.551647\pi$
$443$	−6.80019	−0.323087	−0.161543	+	0.986866i	$0.551647\pi$
$444$	0	0
$445$	37.1450	1.76084
$446$	0	0
$447$	−13.3709	−0.632424
$448$	0	0
$449$	38.0533	1.79585	0.897923	−	0.440152i	$-0.145075\pi$
$449$	38.0533	1.79585	0.897923	+	0.440152i	$0.145075\pi$
$450$	0	0
$451$	−24.1730	−1.13826
$452$	0	0
$453$	−21.4627	−1.00841
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.2839	−1.60374	−0.801868	−	0.597502i	$-0.796160\pi$
$457$	−34.2839	−1.60374	−0.801868	+	0.597502i	$0.796160\pi$
$458$	0	0
$459$	−5.12833	−0.239370
$460$	0	0
$461$	−11.9224	−0.555282	−0.277641	−	0.960685i	$-0.589553\pi$
$461$	−11.9224	−0.555282	−0.277641	+	0.960685i	$0.589553\pi$
$462$	0	0
$463$	−21.6233	−1.00492	−0.502459	−	0.864601i	$-0.667571\pi$
$463$	−21.6233	−1.00492	−0.502459	+	0.864601i	$0.667571\pi$
$464$	0	0
$465$	−21.9251	−1.01675
$466$	0	0
$467$	−40.8792	−1.89166	−0.945831	−	0.324658i	$-0.894751\pi$
$467$	−40.8792	−1.89166	−0.945831	+	0.324658i	$0.894751\pi$
$468$	0	0
$469$	15.6586	0.723047
$470$	0	0
$471$	−14.4785	−0.667132
$472$	0	0
$473$	20.9371	0.962689
$474$	0	0
$475$	4.97862	0.228435
$476$	0	0
$477$	5.65178	0.258777
$478$	0	0
$479$	36.2333	1.65554	0.827772	−	0.561065i	$-0.189608\pi$
$479$	36.2333	1.65554	0.827772	+	0.561065i	$0.189608\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	11.7953	0.536703
$484$	0	0
$485$	−55.3881	−2.51504
$486$	0	0
$487$	−11.1729	−0.506293	−0.253146	−	0.967428i	$-0.581465\pi$
$487$	−11.1729	−0.506293	−0.253146	+	0.967428i	$0.581465\pi$
$488$	0	0
$489$	−1.80785	−0.0817540
$490$	0	0
$491$	3.61988	0.163363	0.0816814	−	0.996658i	$-0.473971\pi$
$491$	3.61988	0.163363	0.0816814	+	0.996658i	$0.473971\pi$
$492$	0	0
$493$	17.7290	0.798473
$494$	0	0
$495$	−9.31933	−0.418873
$496$	0	0
$497$	−43.9316	−1.97060
$498$	0	0
$499$	12.5988	0.563998	0.281999	−	0.959415i	$-0.409003\pi$
$499$	12.5988	0.563998	0.281999	+	0.959415i	$0.409003\pi$
$500$	0	0
$501$	−2.39600	−0.107045
$502$	0	0
$503$	2.88632	0.128695	0.0643474	−	0.997928i	$-0.479503\pi$
$503$	2.88632	0.128695	0.0643474	+	0.997928i	$0.479503\pi$
$504$	0	0
$505$	−9.37065	−0.416989
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.6973	−0.740094	−0.370047	−	0.929013i	$-0.620658\pi$
$509$	−16.6973	−0.740094	−0.370047	+	0.929013i	$0.620658\pi$
$510$	0	0
$511$	−44.6326	−1.97443
$512$	0	0
$513$	−1.45479	−0.0642304
$514$	0	0
$515$	−13.7509	−0.605938
$516$	0	0
$517$	29.2136	1.28481
$518$	0	0
$519$	14.7926	0.649325
$520$	0	0
$521$	38.9887	1.70813	0.854063	−	0.520170i	$-0.174131\pi$
$521$	38.9887	1.70813	0.854063	+	0.520170i	$0.174131\pi$
$522$	0	0
$523$	−39.5816	−1.73078	−0.865392	−	0.501095i	$-0.832931\pi$
$523$	−39.5816	−1.73078	−0.865392	+	0.501095i	$0.832931\pi$
$524$	0	0
$525$	−9.86379	−0.430491
$526$	0	0
$527$	−38.7440	−1.68772
$528$	0	0
$529$	−6.25263	−0.271854
$530$	0	0
$531$	−7.21876	−0.313268
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−34.2932	−1.48262
$536$	0	0
$537$	9.51339	0.410533
$538$	0	0
$539$	4.19859	0.180846
$540$	0	0
$541$	−14.8215	−0.637227	−0.318613	−	0.947885i	$-0.603217\pi$
$541$	−14.8215	−0.637227	−0.318613	+	0.947885i	$0.603217\pi$
$542$	0	0
$543$	−22.7383	−0.975795
$544$	0	0
$545$	−22.4341	−0.960971
$546$	0	0
$547$	−35.6509	−1.52432	−0.762162	−	0.647387i	$-0.775862\pi$
$547$	−35.6509	−1.52432	−0.762162	+	0.647387i	$0.775862\pi$
$548$	0	0
$549$	−1.55262	−0.0662643
$550$	0	0
$551$	5.02930	0.214255
$552$	0	0
$553$	17.1400	0.728865
$554$	0	0
$555$	33.1778	1.40832
$556$	0	0
$557$	38.4541	1.62935	0.814677	−	0.579915i	$-0.196914\pi$
$557$	38.4541	1.62935	0.814677	+	0.579915i	$0.196914\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−16.4682	−0.695290
$562$	0	0
$563$	−25.3615	−1.06886	−0.534429	−	0.845213i	$-0.679473\pi$
$563$	−25.3615	−1.06886	−0.534429	+	0.845213i	$0.679473\pi$
$564$	0	0
$565$	44.2747	1.86265
$566$	0	0
$567$	2.88227	0.121044
$568$	0	0
$569$	−4.26905	−0.178968	−0.0894839	−	0.995988i	$-0.528522\pi$
$569$	−4.26905	−0.178968	−0.0894839	+	0.995988i	$0.528522\pi$
$570$	0	0
$571$	27.3951	1.14645	0.573224	−	0.819398i	$-0.305692\pi$
$571$	27.3951	1.14645	0.573224	+	0.819398i	$0.305692\pi$
$572$	0	0
$573$	−2.11482	−0.0883478
$574$	0	0
$575$	−14.0050	−0.584048
$576$	0	0
$577$	−17.7235	−0.737839	−0.368919	−	0.929461i	$-0.620272\pi$
$577$	−17.7235	−0.737839	−0.368919	+	0.929461i	$0.620272\pi$
$578$	0	0
$579$	−1.36260	−0.0566275
$580$	0	0
$581$	−23.6566	−0.981440
$582$	0	0
$583$	18.1492	0.751661
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.2344	1.70193	0.850964	−	0.525225i	$-0.176019\pi$
$587$	41.2344	1.70193	0.850964	+	0.525225i	$0.176019\pi$
$588$	0	0
$589$	−10.9908	−0.452867
$590$	0	0
$591$	−2.59989	−0.106945
$592$	0	0
$593$	−7.07012	−0.290335	−0.145168	−	0.989407i	$-0.546372\pi$
$593$	−7.07012	−0.290335	−0.145168	+	0.989407i	$0.546372\pi$
$594$	0	0
$595$	−42.8967	−1.75859
$596$	0	0
$597$	−25.9200	−1.06083
$598$	0	0
$599$	10.3461	0.422730	0.211365	−	0.977407i	$-0.432209\pi$
$599$	10.3461	0.422730	0.211365	+	0.977407i	$0.432209\pi$
$600$	0	0
$601$	23.8387	0.972399	0.486200	−	0.873848i	$-0.338383\pi$
$601$	23.8387	0.972399	0.486200	+	0.873848i	$0.338383\pi$
$602$	0	0
$603$	5.43273	0.221238
$604$	0	0
$605$	1.99667	0.0811762
$606$	0	0
$607$	9.51001	0.386000	0.193000	−	0.981199i	$-0.438178\pi$
$607$	9.51001	0.386000	0.193000	+	0.981199i	$0.438178\pi$
$608$	0	0
$609$	−9.96419	−0.403769
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−35.0358	−1.41508	−0.707542	−	0.706671i	$-0.750196\pi$
$613$	−35.0358	−1.41508	−0.707542	+	0.706671i	$0.750196\pi$
$614$	0	0
$615$	−21.8461	−0.880918
$616$	0	0
$617$	9.76386	0.393078	0.196539	−	0.980496i	$-0.437030\pi$
$617$	9.76386	0.393078	0.196539	+	0.980496i	$0.437030\pi$
$618$	0	0
$619$	−7.17233	−0.288280	−0.144140	−	0.989557i	$-0.546042\pi$
$619$	−7.17233	−0.288280	−0.144140	+	0.989557i	$0.546042\pi$
$620$	0	0
$621$	4.09235	0.164221
$622$	0	0
$623$	−36.8911	−1.47801
$624$	0	0
$625$	−30.3995	−1.21598
$626$	0	0
$627$	−4.67166	−0.186568
$628$	0	0
$629$	58.6287	2.33768
$630$	0	0
$631$	40.6899	1.61984	0.809921	−	0.586540i	$-0.199510\pi$
$631$	40.6899	1.61984	0.809921	+	0.586540i	$0.199510\pi$
$632$	0	0
$633$	−21.2034	−0.842758
$634$	0	0
$635$	−60.6563	−2.40707
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−15.2420	−0.602965
$640$	0	0
$641$	10.5847	0.418072	0.209036	−	0.977908i	$-0.432967\pi$
$641$	10.5847	0.418072	0.209036	+	0.977908i	$0.432967\pi$
$642$	0	0
$643$	−4.18603	−0.165081	−0.0825405	−	0.996588i	$-0.526303\pi$
$643$	−4.18603	−0.165081	−0.0825405	+	0.996588i	$0.526303\pi$
$644$	0	0
$645$	18.9216	0.745039
$646$	0	0
$647$	19.5833	0.769899	0.384949	−	0.922938i	$-0.374219\pi$
$647$	19.5833	0.769899	0.384949	+	0.922938i	$0.374219\pi$
$648$	0	0
$649$	−23.1811	−0.909938
$650$	0	0
$651$	21.7753	0.853439
$652$	0	0
$653$	19.4027	0.759285	0.379643	−	0.925133i	$-0.376047\pi$
$653$	19.4027	0.759285	0.379643	+	0.925133i	$0.376047\pi$
$654$	0	0
$655$	−30.8438	−1.20517
$656$	0	0
$657$	−15.4852	−0.604136
$658$	0	0
$659$	−7.34984	−0.286309	−0.143155	−	0.989700i	$-0.545725\pi$
$659$	−7.34984	−0.286309	−0.143155	+	0.989700i	$0.545725\pi$
$660$	0	0
$661$	−34.3362	−1.33553	−0.667763	−	0.744374i	$-0.732748\pi$
$661$	−34.3362	−1.33553	−0.667763	+	0.744374i	$0.732748\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.1688	−0.471886
$666$	0	0
$667$	−14.1475	−0.547795
$668$	0	0
$669$	−7.03414	−0.271956
$670$	0	0
$671$	−4.98583	−0.192476
$672$	0	0
$673$	39.0529	1.50538	0.752689	−	0.658376i	$-0.228756\pi$
$673$	39.0529	1.50538	0.752689	+	0.658376i	$0.228756\pi$
$674$	0	0
$675$	−3.42223	−0.131722
$676$	0	0
$677$	−32.8563	−1.26277	−0.631385	−	0.775469i	$-0.717513\pi$
$677$	−32.8563	−1.26277	−0.631385	+	0.775469i	$0.717513\pi$
$678$	0	0
$679$	55.0094	2.11107
$680$	0	0
$681$	0.265626	0.0101788
$682$	0	0
$683$	26.6796	1.02087	0.510433	−	0.859918i	$-0.329485\pi$
$683$	26.6796	1.02087	0.510433	+	0.859918i	$0.329485\pi$
$684$	0	0
$685$	41.0343	1.56784
$686$	0	0
$687$	17.5065	0.667913
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−2.63574	−0.100268	−0.0501342	−	0.998742i	$-0.515965\pi$
$691$	−2.63574	−0.100268	−0.0501342	+	0.998742i	$0.515965\pi$
$692$	0	0
$693$	9.25562	0.351592
$694$	0	0
$695$	1.50819	0.0572088
$696$	0	0
$697$	−38.6043	−1.46224
$698$	0	0
$699$	−9.12549	−0.345158
$700$	0	0
$701$	−6.91385	−0.261132	−0.130566	−	0.991440i	$-0.541680\pi$
$701$	−6.91385	−0.261132	−0.130566	+	0.991440i	$0.541680\pi$
$702$	0	0
$703$	16.6316	0.627273
$704$	0	0
$705$	26.4014	0.994335
$706$	0	0
$707$	9.30659	0.350010
$708$	0	0
$709$	−8.13495	−0.305514	−0.152757	−	0.988264i	$-0.548815\pi$
$709$	−8.13495	−0.305514	−0.152757	+	0.988264i	$0.548815\pi$
$710$	0	0
$711$	5.94669	0.223018
$712$	0	0
$713$	30.9173	1.15786
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.10416	0.0412356
$718$	0	0
$719$	−24.9984	−0.932284	−0.466142	−	0.884710i	$-0.654356\pi$
$719$	−24.9984	−0.932284	−0.466142	+	0.884710i	$0.654356\pi$
$720$	0	0
$721$	13.6569	0.508610
$722$	0	0
$723$	9.71751	0.361398
$724$	0	0
$725$	11.8309	0.439388
$726$	0	0
$727$	−3.14024	−0.116465	−0.0582324	−	0.998303i	$-0.518546\pi$
$727$	−3.14024	−0.116465	−0.0582324	+	0.998303i	$0.518546\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	33.4365	1.23669
$732$	0	0
$733$	33.6263	1.24202	0.621008	−	0.783804i	$-0.286723\pi$
$733$	33.6263	1.24202	0.621008	+	0.783804i	$0.286723\pi$
$734$	0	0
$735$	3.79442	0.139959
$736$	0	0
$737$	17.4458	0.642623
$738$	0	0
$739$	41.8106	1.53803	0.769013	−	0.639233i	$-0.220748\pi$
$739$	41.8106	1.53803	0.769013	+	0.639233i	$0.220748\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	45.9161	1.68450	0.842250	−	0.539088i	$-0.181231\pi$
$743$	45.9161	1.68450	0.842250	+	0.539088i	$0.181231\pi$
$744$	0	0
$745$	−38.8039	−1.42167
$746$	0	0
$747$	−8.20762	−0.300301
$748$	0	0
$749$	34.0588	1.24448
$750$	0	0
$751$	−10.9653	−0.400131	−0.200066	−	0.979783i	$-0.564115\pi$
$751$	−10.9653	−0.400131	−0.200066	+	0.979783i	$0.564115\pi$
$752$	0	0
$753$	−13.6458	−0.497282
$754$	0	0
$755$	−62.2871	−2.26686
$756$	0	0
$757$	25.5124	0.927263	0.463631	−	0.886028i	$-0.346546\pi$
$757$	25.5124	0.927263	0.463631	+	0.886028i	$0.346546\pi$
$758$	0	0
$759$	13.1415	0.477006
$760$	0	0
$761$	24.6168	0.892357	0.446179	−	0.894944i	$-0.352785\pi$
$761$	24.6168	0.892357	0.446179	+	0.894944i	$0.352785\pi$
$762$	0	0
$763$	22.2807	0.806617
$764$	0	0
$765$	−14.8830	−0.538095
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−4.37892	−0.157908	−0.0789539	−	0.996878i	$-0.525158\pi$
$769$	−4.37892	−0.157908	−0.0789539	+	0.996878i	$0.525158\pi$
$770$	0	0
$771$	−20.5623	−0.740532
$772$	0	0
$773$	20.9314	0.752849	0.376425	−	0.926447i	$-0.377153\pi$
$773$	20.9314	0.752849	0.376425	+	0.926447i	$0.377153\pi$
$774$	0	0
$775$	−25.8546	−0.928726
$776$	0	0
$777$	−32.9510	−1.18211
$778$	0	0
$779$	−10.9511	−0.392365
$780$	0	0
$781$	−48.9456	−1.75141
$782$	0	0
$783$	−3.45707	−0.123545
$784$	0	0
$785$	−42.0180	−1.49969
$786$	0	0
$787$	12.4556	0.443994	0.221997	−	0.975047i	$-0.428742\pi$
$787$	12.4556	0.443994	0.221997	+	0.975047i	$0.428742\pi$
$788$	0	0
$789$	−8.81494	−0.313820
$790$	0	0
$791$	−43.9720	−1.56346
$792$	0	0
$793$	0	0
$794$	0	0
$795$	16.4021	0.581722
$796$	0	0
$797$	−47.7697	−1.69209	−0.846044	−	0.533113i	$-0.821022\pi$
$797$	−47.7697	−1.69209	−0.846044	+	0.533113i	$0.821022\pi$
$798$	0	0
$799$	46.6541	1.65050
$800$	0	0
$801$	−12.7993	−0.452242
$802$	0	0
$803$	−49.7266	−1.75481
$804$	0	0
$805$	34.2311	1.20649
$806$	0	0
$807$	−16.3100	−0.574141
$808$	0	0
$809$	6.27576	0.220644	0.110322	−	0.993896i	$-0.464812\pi$
$809$	6.27576	0.220644	0.110322	+	0.993896i	$0.464812\pi$
$810$	0	0
$811$	17.0453	0.598542	0.299271	−	0.954168i	$-0.403257\pi$
$811$	17.0453	0.598542	0.299271	+	0.954168i	$0.403257\pi$
$812$	0	0
$813$	−7.59360	−0.266319
$814$	0	0
$815$	−5.24659	−0.183780
$816$	0	0
$817$	9.48516	0.331844
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.0653567	0.00228096	0.00114048	−	0.999999i	$-0.499637\pi$
$821$	0.0653567	0.00228096	0.00114048	+	0.999999i	$0.499637\pi$
$822$	0	0
$823$	−19.6964	−0.686572	−0.343286	−	0.939231i	$-0.611540\pi$
$823$	−19.6964	−0.686572	−0.343286	+	0.939231i	$0.611540\pi$
$824$	0	0
$825$	−10.9896	−0.382608
$826$	0	0
$827$	51.4423	1.78882	0.894412	−	0.447243i	$-0.147594\pi$
$827$	51.4423	1.78882	0.894412	+	0.447243i	$0.147594\pi$
$828$	0	0
$829$	4.44809	0.154489	0.0772443	−	0.997012i	$-0.475388\pi$
$829$	4.44809	0.154489	0.0772443	+	0.997012i	$0.475388\pi$
$830$	0	0
$831$	5.02572	0.174340
$832$	0	0
$833$	6.70514	0.232319
$834$	0	0
$835$	−6.95345	−0.240634
$836$	0	0
$837$	7.55490	0.261136
$838$	0	0
$839$	−10.5097	−0.362834	−0.181417	−	0.983406i	$-0.558068\pi$
$839$	−10.5097	−0.362834	−0.181417	+	0.983406i	$0.558068\pi$
$840$	0	0
$841$	−17.0487	−0.587886
$842$	0	0
$843$	−25.7781	−0.887845
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.98302	−0.0681374
$848$	0	0
$849$	−17.3692	−0.596108
$850$	0	0
$851$	−46.7851	−1.60377
$852$	0	0
$853$	41.9887	1.43767	0.718833	−	0.695183i	$-0.244677\pi$
$853$	41.9887	1.43767	0.718833	+	0.695183i	$0.244677\pi$
$854$	0	0
$855$	−4.22195	−0.144388
$856$	0	0
$857$	−10.0693	−0.343961	−0.171981	−	0.985100i	$-0.555017\pi$
$857$	−10.0693	−0.343961	−0.171981	+	0.985100i	$0.555017\pi$
$858$	0	0
$859$	14.8702	0.507366	0.253683	−	0.967287i	$-0.418358\pi$
$859$	14.8702	0.507366	0.253683	+	0.967287i	$0.418358\pi$
$860$	0	0
$861$	21.6967	0.739422
$862$	0	0
$863$	36.2100	1.23260	0.616301	−	0.787511i	$-0.288631\pi$
$863$	36.2100	1.23260	0.616301	+	0.787511i	$0.288631\pi$
$864$	0	0
$865$	42.9298	1.45966
$866$	0	0
$867$	−9.29979	−0.315837
$868$	0	0
$869$	19.0962	0.647793
$870$	0	0
$871$	0	0
$872$	0	0
$873$	19.0855	0.645945
$874$	0	0
$875$	13.1975	0.446156
$876$	0	0
$877$	25.0918	0.847289	0.423644	−	0.905829i	$-0.360751\pi$
$877$	25.0918	0.847289	0.423644	+	0.905829i	$0.360751\pi$
$878$	0	0
$879$	15.2944	0.515869
$880$	0	0
$881$	−31.5438	−1.06274	−0.531369	−	0.847140i	$-0.678322\pi$
$881$	−31.5438	−1.06274	−0.531369	+	0.847140i	$0.678322\pi$
$882$	0	0
$883$	−42.4085	−1.42716	−0.713580	−	0.700574i	$-0.752927\pi$
$883$	−42.4085	−1.42716	−0.713580	+	0.700574i	$0.752927\pi$
$884$	0	0
$885$	−20.9496	−0.704214
$886$	0	0
$887$	−13.8142	−0.463835	−0.231917	−	0.972735i	$-0.574500\pi$
$887$	−13.8142	−0.463835	−0.231917	+	0.972735i	$0.574500\pi$
$888$	0	0
$889$	60.2416	2.02044
$890$	0	0
$891$	3.21123	0.107580
$892$	0	0
$893$	13.2347	0.442882
$894$	0	0
$895$	27.6089	0.922864
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−26.1178	−0.871077
$900$	0	0
$901$	28.9842	0.965604
$902$	0	0
$903$	−18.7923	−0.625368
$904$	0	0
$905$	−65.9891	−2.19355
$906$	0	0
$907$	−8.12843	−0.269900	−0.134950	−	0.990852i	$-0.543087\pi$
$907$	−8.12843	−0.269900	−0.134950	+	0.990852i	$0.543087\pi$
$908$	0	0
$909$	3.22891	0.107096
$910$	0	0
$911$	−53.9405	−1.78713	−0.893564	−	0.448935i	$-0.851803\pi$
$911$	−53.9405	−1.78713	−0.893564	+	0.448935i	$0.851803\pi$
$912$	0	0
$913$	−26.3566	−0.872275
$914$	0	0
$915$	−4.50588	−0.148960
$916$	0	0
$917$	30.6329	1.01159
$918$	0	0
$919$	10.3020	0.339833	0.169916	−	0.985458i	$-0.445650\pi$
$919$	10.3020	0.339833	0.169916	+	0.985458i	$0.445650\pi$
$920$	0	0
$921$	21.6255	0.712584
$922$	0	0
$923$	0	0
$924$	0	0
$925$	39.1241	1.28639
$926$	0	0
$927$	4.73825	0.155625
$928$	0	0
$929$	−35.7452	−1.17276	−0.586382	−	0.810035i	$-0.699448\pi$
$929$	−35.7452	−1.17276	−0.586382	+	0.810035i	$0.699448\pi$
$930$	0	0
$931$	1.90209	0.0623385
$932$	0	0
$933$	13.2236	0.432921
$934$	0	0
$935$	−47.7926	−1.56299
$936$	0	0
$937$	−6.67457	−0.218049	−0.109024	−	0.994039i	$-0.534773\pi$
$937$	−6.67457	−0.218049	−0.109024	+	0.994039i	$0.534773\pi$
$938$	0	0
$939$	−1.89385	−0.0618034
$940$	0	0
$941$	−30.1361	−0.982409	−0.491205	−	0.871044i	$-0.663443\pi$
$941$	−30.1361	−0.982409	−0.491205	+	0.871044i	$0.663443\pi$
$942$	0	0
$943$	30.8058	1.00318
$944$	0	0
$945$	8.36465	0.272102
$946$	0	0
$947$	−56.4030	−1.83285	−0.916426	−	0.400205i	$-0.868939\pi$
$947$	−56.4030	−1.83285	−0.916426	+	0.400205i	$0.868939\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.76995	−0.219531
$952$	0	0
$953$	−21.3522	−0.691667	−0.345834	−	0.938296i	$-0.612404\pi$
$953$	−21.3522	−0.691667	−0.345834	+	0.938296i	$0.612404\pi$
$954$	0	0
$955$	−6.13743	−0.198603
$956$	0	0
$957$	−11.1014	−0.358858
$958$	0	0
$959$	−40.7537	−1.31601
$960$	0	0
$961$	26.0765	0.841178
$962$	0	0
$963$	11.8166	0.380786
$964$	0	0
$965$	−3.95440	−0.127297
$966$	0	0
$967$	−17.3338	−0.557419	−0.278709	−	0.960376i	$-0.589907\pi$
$967$	−17.3338	−0.557419	−0.278709	+	0.960376i	$0.589907\pi$
$968$	0	0
$969$	−7.46063	−0.239670
$970$	0	0
$971$	28.9041	0.927577	0.463789	−	0.885946i	$-0.346490\pi$
$971$	28.9041	0.927577	0.463789	+	0.885946i	$0.346490\pi$
$972$	0	0
$973$	−1.49788	−0.0480197
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.8286	−1.11427	−0.557133	−	0.830424i	$-0.688098\pi$
$977$	−34.8286	−1.11427	−0.557133	+	0.830424i	$0.688098\pi$
$978$	0	0
$979$	−41.1015	−1.31361
$980$	0	0
$981$	7.73028	0.246809
$982$	0	0
$983$	−29.4130	−0.938130	−0.469065	−	0.883164i	$-0.655409\pi$
$983$	−29.4130	−0.938130	−0.469065	+	0.883164i	$0.655409\pi$
$984$	0	0
$985$	−7.54515	−0.240409
$986$	0	0
$987$	−26.2209	−0.834622
$988$	0	0
$989$	−26.6820	−0.848438
$990$	0	0
$991$	−2.00401	−0.0636595	−0.0318297	−	0.999493i	$-0.510133\pi$
$991$	−2.00401	−0.0636595	−0.0318297	+	0.999493i	$0.510133\pi$
$992$	0	0
$993$	−12.1917	−0.386892
$994$	0	0
$995$	−75.2225	−2.38471
$996$	0	0
$997$	−21.4173	−0.678293	−0.339146	−	0.940734i	$-0.610138\pi$
$997$	−21.4173	−0.678293	−0.339146	+	0.940734i	$0.610138\pi$
$998$	0	0
$999$	−11.4323	−0.361703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.bg.1.2	yes	6
4.3	odd	2		8112.2.a.cw.1.2		6
13.5	odd	4		4056.2.c.q.337.10		12
13.8	odd	4		4056.2.c.q.337.3		12
13.12	even	2		4056.2.a.bf.1.5	✓	6
52.51	odd	2		8112.2.a.cv.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bf.1.5	✓	6	13.12	even	2
4056.2.a.bg.1.2	yes	6	1.1	even	1	trivial
4056.2.c.q.337.3		12	13.8	odd	4
4056.2.c.q.337.10		12	13.5	odd	4
8112.2.a.cv.1.5		6	52.51	odd	2
8112.2.a.cw.1.2		6	4.3	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.90211 q^{5} +2.88227 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.90211 q^{5} +2.88227 q^{7} +1.00000 q^{9} +3.21123 q^{11} +2.90211 q^{15} +5.12833 q^{17} +1.45479 q^{19} -2.88227 q^{21} -4.09235 q^{23} +3.42223 q^{25} -1.00000 q^{27} +3.45707 q^{29} -7.55490 q^{31} -3.21123 q^{33} -8.36465 q^{35} +11.4323 q^{37} -7.52765 q^{41} +6.51996 q^{43} -2.90211 q^{45} +9.09733 q^{47} +1.30747 q^{49} -5.12833 q^{51} +5.65178 q^{53} -9.31933 q^{55} -1.45479 q^{57} -7.21876 q^{59} -1.55262 q^{61} +2.88227 q^{63} +5.43273 q^{67} +4.09235 q^{69} -15.2420 q^{71} -15.4852 q^{73} -3.42223 q^{75} +9.25562 q^{77} +5.94669 q^{79} +1.00000 q^{81} -8.20762 q^{83} -14.8830 q^{85} -3.45707 q^{87} -12.7993 q^{89} +7.55490 q^{93} -4.22195 q^{95} +19.0855 q^{97} +3.21123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9}+O(q^{10}) \) 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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