LMFDB - Embedded newform 1352.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,2,Mod(1,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1352.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:	$10.7957743533$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	1352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80194 q^{3} -2.80194 q^{5} -1.80194 q^{7} +4.85086 q^{9} +O(q^{10})$	$q-2.80194 q^{3} -2.80194 q^{5} -1.80194 q^{7} +4.85086 q^{9} +6.40581 q^{11} +7.85086 q^{15} -5.63102 q^{17} +1.66487 q^{19} +5.04892 q^{21} +3.09783 q^{23} +2.85086 q^{25} -5.18598 q^{27} +6.07606 q^{29} +3.58211 q^{31} -17.9487 q^{33} +5.04892 q^{35} +5.14675 q^{37} -7.09783 q^{41} -2.84117 q^{43} -13.5918 q^{45} -1.51573 q^{47} -3.75302 q^{49} +15.7778 q^{51} -10.7627 q^{53} -17.9487 q^{55} -4.66487 q^{57} +9.27413 q^{59} -12.8291 q^{61} -8.74094 q^{63} +2.17092 q^{67} -8.67994 q^{69} -10.9051 q^{71} -1.95108 q^{73} -7.98792 q^{75} -11.5429 q^{77} -11.5646 q^{79} -0.0217703 q^{81} +0.0392287 q^{83} +15.7778 q^{85} -17.0248 q^{87} +7.44265 q^{89} -10.0368 q^{93} -4.66487 q^{95} +0.472189 q^{97} +31.0737 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 4 q^{5} - q^{7} + q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 - 4 * q^5 - q^7 + q^9	$ 3 q - 4 q^{3} - 4 q^{5} - q^{7} + q^{9} + 6 q^{11} + 10 q^{15} - 2 q^{17} + 6 q^{19} + 6 q^{21} - 9 q^{23} - 5 q^{25} - q^{27} + 3 q^{29} + 5 q^{31} - 22 q^{33} + 6 q^{35} - 12 q^{37} - 3 q^{41} - 17 q^{43} - 13 q^{45} + 8 q^{47} - 16 q^{49} + 5 q^{51} - 15 q^{53} - 22 q^{55} - 15 q^{57} + 17 q^{59} - 28 q^{61} - 12 q^{63} + 17 q^{67} - 2 q^{69} - 7 q^{71} - 15 q^{73} - 5 q^{75} - 16 q^{77} - 13 q^{79} + 3 q^{81} + 13 q^{83} + 5 q^{85} - 4 q^{87} - 19 q^{89} - 2 q^{93} - 15 q^{95} - 5 q^{97} + 37 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 4 * q^5 - q^7 + q^9 + 6 * q^11 + 10 * q^15 - 2 * q^17 + 6 * q^19 + 6 * q^21 - 9 * q^23 - 5 * q^25 - q^27 + 3 * q^29 + 5 * q^31 - 22 * q^33 + 6 * q^35 - 12 * q^37 - 3 * q^41 - 17 * q^43 - 13 * q^45 + 8 * q^47 - 16 * q^49 + 5 * q^51 - 15 * q^53 - 22 * q^55 - 15 * q^57 + 17 * q^59 - 28 * q^61 - 12 * q^63 + 17 * q^67 - 2 * q^69 - 7 * q^71 - 15 * q^73 - 5 * q^75 - 16 * q^77 - 13 * q^79 + 3 * q^81 + 13 * q^83 + 5 * q^85 - 4 * q^87 - 19 * q^89 - 2 * q^93 - 15 * q^95 - 5 * q^97 + 37 * 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
$3$	−2.80194	−1.61770	−0.808850	−	0.588015i	$-0.799909\pi$
$3$	−2.80194	−1.61770	−0.808850	+	0.588015i	$0.799909\pi$
$4$	0	0
$5$	−2.80194	−1.25306	−0.626532	−	0.779395i	$-0.715526\pi$
$5$	−2.80194	−1.25306	−0.626532	+	0.779395i	$0.715526\pi$
$6$	0	0
$7$	−1.80194	−0.681068	−0.340534	−	0.940232i	$-0.610608\pi$
$7$	−1.80194	−0.681068	−0.340534	+	0.940232i	$0.610608\pi$
$8$	0	0
$9$	4.85086	1.61695
$10$	0	0
$11$	6.40581	1.93143	0.965713	−	0.259613i	$-0.0835951\pi$
$11$	6.40581	1.93143	0.965713	+	0.259613i	$0.0835951\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	7.85086	2.02708
$16$	0	0
$17$	−5.63102	−1.36572	−0.682862	−	0.730548i	$-0.739265\pi$
$17$	−5.63102	−1.36572	−0.682862	+	0.730548i	$0.739265\pi$
$18$	0	0
$19$	1.66487	0.381948	0.190974	−	0.981595i	$-0.438835\pi$
$19$	1.66487	0.381948	0.190974	+	0.981595i	$0.438835\pi$
$20$	0	0
$21$	5.04892	1.10176
$22$	0	0
$23$	3.09783	0.645943	0.322972	−	0.946409i	$-0.395318\pi$
$23$	3.09783	0.645943	0.322972	+	0.946409i	$0.395318\pi$
$24$	0	0
$25$	2.85086	0.570171
$26$	0	0
$27$	−5.18598	−0.998042
$28$	0	0
$29$	6.07606	1.12830	0.564148	−	0.825673i	$-0.309205\pi$
$29$	6.07606	1.12830	0.564148	+	0.825673i	$0.309205\pi$
$30$	0	0
$31$	3.58211	0.643365	0.321683	−	0.946848i	$-0.395752\pi$
$31$	3.58211	0.643365	0.321683	+	0.946848i	$0.395752\pi$
$32$	0	0
$33$	−17.9487	−3.12447
$34$	0	0
$35$	5.04892	0.853423
$36$	0	0
$37$	5.14675	0.846121	0.423060	−	0.906101i	$-0.360956\pi$
$37$	5.14675	0.846121	0.423060	+	0.906101i	$0.360956\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.09783	−1.10850	−0.554248	−	0.832352i	$-0.686994\pi$
$41$	−7.09783	−1.10850	−0.554248	+	0.832352i	$0.686994\pi$
$42$	0	0
$43$	−2.84117	−0.433274	−0.216637	−	0.976252i	$-0.569509\pi$
$43$	−2.84117	−0.433274	−0.216637	+	0.976252i	$0.569509\pi$
$44$	0	0
$45$	−13.5918	−2.02615
$46$	0	0
$47$	−1.51573	−0.221092	−0.110546	−	0.993871i	$-0.535260\pi$
$47$	−1.51573	−0.221092	−0.110546	+	0.993871i	$0.535260\pi$
$48$	0	0
$49$	−3.75302	−0.536146
$50$	0	0
$51$	15.7778	2.20933
$52$	0	0
$53$	−10.7627	−1.47837	−0.739186	−	0.673501i	$-0.764790\pi$
$53$	−10.7627	−1.47837	−0.739186	+	0.673501i	$0.764790\pi$
$54$	0	0
$55$	−17.9487	−2.42020
$56$	0	0
$57$	−4.66487	−0.617878
$58$	0	0
$59$	9.27413	1.20739	0.603694	−	0.797216i	$-0.293695\pi$
$59$	9.27413	1.20739	0.603694	+	0.797216i	$0.293695\pi$
$60$	0	0
$61$	−12.8291	−1.64260	−0.821298	−	0.570499i	$-0.806750\pi$
$61$	−12.8291	−1.64260	−0.821298	+	0.570499i	$0.806750\pi$
$62$	0	0
$63$	−8.74094	−1.10125
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.17092	0.265220	0.132610	−	0.991168i	$-0.457664\pi$
$67$	2.17092	0.265220	0.132610	+	0.991168i	$0.457664\pi$
$68$	0	0
$69$	−8.67994	−1.04494
$70$	0	0
$71$	−10.9051	−1.29420	−0.647102	−	0.762404i	$-0.724019\pi$
$71$	−10.9051	−1.29420	−0.647102	+	0.762404i	$0.724019\pi$
$72$	0	0
$73$	−1.95108	−0.228357	−0.114178	−	0.993460i	$-0.536424\pi$
$73$	−1.95108	−0.228357	−0.114178	+	0.993460i	$0.536424\pi$
$74$	0	0
$75$	−7.98792	−0.922365
$76$	0	0
$77$	−11.5429	−1.31543
$78$	0	0
$79$	−11.5646	−1.30112	−0.650562	−	0.759453i	$-0.725467\pi$
$79$	−11.5646	−1.30112	−0.650562	+	0.759453i	$0.725467\pi$
$80$	0	0
$81$	−0.0217703	−0.00241892
$82$	0	0
$83$	0.0392287	0.00430590	0.00215295	−	0.999998i	$-0.499315\pi$
$83$	0.0392287	0.00430590	0.00215295	+	0.999998i	$0.499315\pi$
$84$	0	0
$85$	15.7778	1.71134
$86$	0	0
$87$	−17.0248	−1.82525
$88$	0	0
$89$	7.44265	0.788919	0.394460	−	0.918913i	$-0.370932\pi$
$89$	7.44265	0.788919	0.394460	+	0.918913i	$0.370932\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.0368	−1.04077
$94$	0	0
$95$	−4.66487	−0.478606
$96$	0	0
$97$	0.472189	0.0479435	0.0239718	−	0.999713i	$-0.492369\pi$
$97$	0.472189	0.0479435	0.0239718	+	0.999713i	$0.492369\pi$
$98$	0	0
$99$	31.0737	3.12302
$100$	0	0
$101$	6.63102	0.659811	0.329906	−	0.944014i	$-0.392983\pi$
$101$	6.63102	0.659811	0.329906	+	0.944014i	$0.392983\pi$
$102$	0	0
$103$	1.21313	0.119533	0.0597665	−	0.998212i	$-0.480964\pi$
$103$	1.21313	0.119533	0.0597665	+	0.998212i	$0.480964\pi$
$104$	0	0
$105$	−14.1468	−1.38058
$106$	0	0
$107$	−4.46681	−0.431823	−0.215912	−	0.976413i	$-0.569272\pi$
$107$	−4.46681	−0.431823	−0.215912	+	0.976413i	$0.569272\pi$
$108$	0	0
$109$	−1.16421	−0.111511	−0.0557556	−	0.998444i	$-0.517757\pi$
$109$	−1.16421	−0.111511	−0.0557556	+	0.998444i	$0.517757\pi$
$110$	0	0
$111$	−14.4209	−1.36877
$112$	0	0
$113$	−5.14914	−0.484391	−0.242195	−	0.970227i	$-0.577868\pi$
$113$	−5.14914	−0.484391	−0.242195	+	0.970227i	$0.577868\pi$
$114$	0	0
$115$	−8.67994	−0.809409
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.1468	0.930151
$120$	0	0
$121$	30.0344	2.73040
$122$	0	0
$123$	19.8877	1.79321
$124$	0	0
$125$	6.02177	0.538604
$126$	0	0
$127$	−10.3623	−0.919503	−0.459752	−	0.888048i	$-0.652062\pi$
$127$	−10.3623	−0.919503	−0.459752	+	0.888048i	$0.652062\pi$
$128$	0	0
$129$	7.96077	0.700907
$130$	0	0
$131$	12.8334	1.12126	0.560630	−	0.828067i	$-0.310559\pi$
$131$	12.8334	1.12126	0.560630	+	0.828067i	$0.310559\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	14.5308	1.25061
$136$	0	0
$137$	−0.792249	−0.0676864	−0.0338432	−	0.999427i	$-0.510775\pi$
$137$	−0.792249	−0.0676864	−0.0338432	+	0.999427i	$0.510775\pi$
$138$	0	0
$139$	−5.89440	−0.499956	−0.249978	−	0.968252i	$-0.580423\pi$
$139$	−5.89440	−0.499956	−0.249978	+	0.968252i	$0.580423\pi$
$140$	0	0
$141$	4.24698	0.357660
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−17.0248	−1.41383
$146$	0	0
$147$	10.5157	0.867323
$148$	0	0
$149$	−4.74094	−0.388393	−0.194196	−	0.980963i	$-0.562210\pi$
$149$	−4.74094	−0.388393	−0.194196	+	0.980963i	$0.562210\pi$
$150$	0	0
$151$	5.37196	0.437164	0.218582	−	0.975819i	$-0.429857\pi$
$151$	5.37196	0.437164	0.218582	+	0.975819i	$0.429857\pi$
$152$	0	0
$153$	−27.3153	−2.20831
$154$	0	0
$155$	−10.0368	−0.806178
$156$	0	0
$157$	−11.2295	−0.896213	−0.448107	−	0.893980i	$-0.647902\pi$
$157$	−11.2295	−0.896213	−0.448107	+	0.893980i	$0.647902\pi$
$158$	0	0
$159$	30.1564	2.39156
$160$	0	0
$161$	−5.58211	−0.439932
$162$	0	0
$163$	1.03923	0.0813987	0.0406993	−	0.999171i	$-0.487041\pi$
$163$	1.03923	0.0813987	0.0406993	+	0.999171i	$0.487041\pi$
$164$	0	0
$165$	50.2911	3.91516
$166$	0	0
$167$	−2.81163	−0.217570	−0.108785	−	0.994065i	$-0.534696\pi$
$167$	−2.81163	−0.217570	−0.108785	+	0.994065i	$0.534696\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	8.07606	0.617592
$172$	0	0
$173$	−21.1836	−1.61056	−0.805279	−	0.592896i	$-0.797985\pi$
$173$	−21.1836	−1.61056	−0.805279	+	0.592896i	$0.797985\pi$
$174$	0	0
$175$	−5.13706	−0.388325
$176$	0	0
$177$	−25.9855	−1.95319
$178$	0	0
$179$	18.3569	1.37206	0.686029	−	0.727574i	$-0.259352\pi$
$179$	18.3569	1.37206	0.686029	+	0.727574i	$0.259352\pi$
$180$	0	0
$181$	−20.4969	−1.52353	−0.761763	−	0.647856i	$-0.775666\pi$
$181$	−20.4969	−1.52353	−0.761763	+	0.647856i	$0.775666\pi$
$182$	0	0
$183$	35.9463	2.65723
$184$	0	0
$185$	−14.4209	−1.06024
$186$	0	0
$187$	−36.0713	−2.63779
$188$	0	0
$189$	9.34481	0.679735
$190$	0	0
$191$	−3.43296	−0.248400	−0.124200	−	0.992257i	$-0.539636\pi$
$191$	−3.43296	−0.248400	−0.124200	+	0.992257i	$0.539636\pi$
$192$	0	0
$193$	1.68233	0.121097	0.0605485	−	0.998165i	$-0.480715\pi$
$193$	1.68233	0.121097	0.0605485	+	0.998165i	$0.480715\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	26.9517	1.92023	0.960114	−	0.279610i	$-0.0902052\pi$
$197$	26.9517	1.92023	0.960114	+	0.279610i	$0.0902052\pi$
$198$	0	0
$199$	−20.0804	−1.42346	−0.711730	−	0.702453i	$-0.752088\pi$
$199$	−20.0804	−1.42346	−0.711730	+	0.702453i	$0.752088\pi$
$200$	0	0
$201$	−6.08277	−0.429045
$202$	0	0
$203$	−10.9487	−0.768447
$204$	0	0
$205$	19.8877	1.38902
$206$	0	0
$207$	15.0271	1.04446
$208$	0	0
$209$	10.6649	0.737705
$210$	0	0
$211$	−8.26444	−0.568947	−0.284474	−	0.958684i	$-0.591819\pi$
$211$	−8.26444	−0.568947	−0.284474	+	0.958684i	$0.591819\pi$
$212$	0	0
$213$	30.5555	2.09363
$214$	0	0
$215$	7.96077	0.542920
$216$	0	0
$217$	−6.45473	−0.438176
$218$	0	0
$219$	5.46681	0.369413
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−23.3080	−1.56082	−0.780409	−	0.625270i	$-0.784989\pi$
$223$	−23.3080	−1.56082	−0.780409	+	0.625270i	$0.784989\pi$
$224$	0	0
$225$	13.8291	0.921939
$226$	0	0
$227$	14.2252	0.944160	0.472080	−	0.881556i	$-0.343503\pi$
$227$	14.2252	0.944160	0.472080	+	0.881556i	$0.343503\pi$
$228$	0	0
$229$	−26.8713	−1.77571	−0.887853	−	0.460128i	$-0.847804\pi$
$229$	−26.8713	−1.77571	−0.887853	+	0.460128i	$0.847804\pi$
$230$	0	0
$231$	32.3424	2.12798
$232$	0	0
$233$	−2.05861	−0.134864	−0.0674319	−	0.997724i	$-0.521481\pi$
$233$	−2.05861	−0.134864	−0.0674319	+	0.997724i	$0.521481\pi$
$234$	0	0
$235$	4.24698	0.277042
$236$	0	0
$237$	32.4034	2.10483
$238$	0	0
$239$	−9.03923	−0.584699	−0.292350	−	0.956312i	$-0.594437\pi$
$239$	−9.03923	−0.584699	−0.292350	+	0.956312i	$0.594437\pi$
$240$	0	0
$241$	−8.28382	−0.533607	−0.266804	−	0.963751i	$-0.585968\pi$
$241$	−8.28382	−0.533607	−0.266804	+	0.963751i	$0.585968\pi$
$242$	0	0
$243$	15.6189	1.00196
$244$	0	0
$245$	10.5157	0.671825
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.109916	−0.00696566
$250$	0	0
$251$	5.28621	0.333663	0.166831	−	0.985985i	$-0.446646\pi$
$251$	5.28621	0.333663	0.166831	+	0.985985i	$0.446646\pi$
$252$	0	0
$253$	19.8442	1.24759
$254$	0	0
$255$	−44.2083	−2.76843
$256$	0	0
$257$	19.4601	1.21389	0.606944	−	0.794745i	$-0.292395\pi$
$257$	19.4601	1.21389	0.606944	+	0.794745i	$0.292395\pi$
$258$	0	0
$259$	−9.27413	−0.576266
$260$	0	0
$261$	29.4741	1.82440
$262$	0	0
$263$	−19.3967	−1.19605	−0.598026	−	0.801476i	$-0.704048\pi$
$263$	−19.3967	−1.19605	−0.598026	+	0.801476i	$0.704048\pi$
$264$	0	0
$265$	30.1564	1.85250
$266$	0	0
$267$	−20.8538	−1.27623
$268$	0	0
$269$	−3.61356	−0.220323	−0.110161	−	0.993914i	$-0.535137\pi$
$269$	−3.61356	−0.220323	−0.110161	+	0.993914i	$0.535137\pi$
$270$	0	0
$271$	−3.14808	−0.191232	−0.0956161	−	0.995418i	$-0.530482\pi$
$271$	−3.14808	−0.191232	−0.0956161	+	0.995418i	$0.530482\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.2620	1.10124
$276$	0	0
$277$	9.70709	0.583242	0.291621	−	0.956534i	$-0.405805\pi$
$277$	9.70709	0.583242	0.291621	+	0.956534i	$0.405805\pi$
$278$	0	0
$279$	17.3763	1.04029
$280$	0	0
$281$	−12.3569	−0.737151	−0.368575	−	0.929598i	$-0.620154\pi$
$281$	−12.3569	−0.737151	−0.368575	+	0.929598i	$0.620154\pi$
$282$	0	0
$283$	−14.5133	−0.862728	−0.431364	−	0.902178i	$-0.641968\pi$
$283$	−14.5133	−0.862728	−0.431364	+	0.902178i	$0.641968\pi$
$284$	0	0
$285$	13.0707	0.774241
$286$	0	0
$287$	12.7899	0.754961
$288$	0	0
$289$	14.7084	0.865201
$290$	0	0
$291$	−1.32304	−0.0775582
$292$	0	0
$293$	−14.0814	−0.822647	−0.411323	−	0.911490i	$-0.634933\pi$
$293$	−14.0814	−0.822647	−0.411323	+	0.911490i	$0.634933\pi$
$294$	0	0
$295$	−25.9855	−1.51294
$296$	0	0
$297$	−33.2204	−1.92764
$298$	0	0
$299$	0	0
$300$	0	0
$301$	5.11960	0.295089
$302$	0	0
$303$	−18.5797	−1.06738
$304$	0	0
$305$	35.9463	2.05828
$306$	0	0
$307$	19.9952	1.14119	0.570594	−	0.821233i	$-0.306713\pi$
$307$	19.9952	1.14119	0.570594	+	0.821233i	$0.306713\pi$
$308$	0	0
$309$	−3.39911	−0.193369
$310$	0	0
$311$	21.6256	1.22628	0.613139	−	0.789975i	$-0.289907\pi$
$311$	21.6256	1.22628	0.613139	+	0.789975i	$0.289907\pi$
$312$	0	0
$313$	1.42088	0.0803128	0.0401564	−	0.999193i	$-0.487214\pi$
$313$	1.42088	0.0803128	0.0401564	+	0.999193i	$0.487214\pi$
$314$	0	0
$315$	24.4916	1.37994
$316$	0	0
$317$	−9.65817	−0.542457	−0.271228	−	0.962515i	$-0.587430\pi$
$317$	−9.65817	−0.542457	−0.271228	+	0.962515i	$0.587430\pi$
$318$	0	0
$319$	38.9221	2.17922
$320$	0	0
$321$	12.5157	0.698560
$322$	0	0
$323$	−9.37495	−0.521636
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.26205	0.180392
$328$	0	0
$329$	2.73125	0.150579
$330$	0	0
$331$	−31.9355	−1.75534	−0.877668	−	0.479269i	$-0.840902\pi$
$331$	−31.9355	−1.75534	−0.877668	+	0.479269i	$0.840902\pi$
$332$	0	0
$333$	24.9661	1.36814
$334$	0	0
$335$	−6.08277	−0.332337
$336$	0	0
$337$	−6.75302	−0.367860	−0.183930	−	0.982939i	$-0.558882\pi$
$337$	−6.75302	−0.367860	−0.183930	+	0.982939i	$0.558882\pi$
$338$	0	0
$339$	14.4276	0.783599
$340$	0	0
$341$	22.9463	1.24261
$342$	0	0
$343$	19.3763	1.04622
$344$	0	0
$345$	24.3207	1.30938
$346$	0	0
$347$	36.0006	1.93261	0.966306	−	0.257394i	$-0.0828639\pi$
$347$	36.0006	1.93261	0.966306	+	0.257394i	$0.0828639\pi$
$348$	0	0
$349$	−25.6058	−1.37065	−0.685323	−	0.728239i	$-0.740339\pi$
$349$	−25.6058	−1.37065	−0.685323	+	0.728239i	$0.740339\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.2416	−0.651555	−0.325777	−	0.945447i	$-0.605626\pi$
$353$	−12.2416	−0.651555	−0.325777	+	0.945447i	$0.605626\pi$
$354$	0	0
$355$	30.5555	1.62172
$356$	0	0
$357$	−28.4306	−1.50471
$358$	0	0
$359$	−21.0737	−1.11223	−0.556113	−	0.831107i	$-0.687708\pi$
$359$	−21.0737	−1.11223	−0.556113	+	0.831107i	$0.687708\pi$
$360$	0	0
$361$	−16.2282	−0.854115
$362$	0	0
$363$	−84.1546	−4.41697
$364$	0	0
$365$	5.46681	0.286146
$366$	0	0
$367$	7.67456	0.400609	0.200304	−	0.979734i	$-0.435807\pi$
$367$	7.67456	0.400609	0.200304	+	0.979734i	$0.435807\pi$
$368$	0	0
$369$	−34.4306	−1.79238
$370$	0	0
$371$	19.3937	1.00687
$372$	0	0
$373$	−24.0683	−1.24621	−0.623105	−	0.782139i	$-0.714129\pi$
$373$	−24.0683	−1.24621	−0.623105	+	0.782139i	$0.714129\pi$
$374$	0	0
$375$	−16.8726	−0.871299
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−27.0103	−1.38742	−0.693712	−	0.720252i	$-0.744026\pi$
$379$	−27.0103	−1.38742	−0.693712	+	0.720252i	$0.744026\pi$
$380$	0	0
$381$	29.0344	1.48748
$382$	0	0
$383$	−0.0241632	−0.00123468	−0.000617340	−	1.00000i	$-0.500197\pi$
$383$	−0.0241632	−0.00123468	−0.000617340		1.00000i	$0.500197\pi$
$384$	0	0
$385$	32.3424	1.64832
$386$	0	0
$387$	−13.7821	−0.700583
$388$	0	0
$389$	23.0422	1.16829	0.584143	−	0.811651i	$-0.301431\pi$
$389$	23.0422	1.16829	0.584143	+	0.811651i	$0.301431\pi$
$390$	0	0
$391$	−17.4440	−0.882180
$392$	0	0
$393$	−35.9584	−1.81386
$394$	0	0
$395$	32.4034	1.63039
$396$	0	0
$397$	33.1250	1.66250	0.831248	−	0.555902i	$-0.187627\pi$
$397$	33.1250	1.66250	0.831248	+	0.555902i	$0.187627\pi$
$398$	0	0
$399$	8.40581	0.420817
$400$	0	0
$401$	26.3817	1.31744	0.658718	−	0.752390i	$-0.271099\pi$
$401$	26.3817	1.31744	0.658718	+	0.752390i	$0.271099\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.0609989	0.00303106
$406$	0	0
$407$	32.9691	1.63422
$408$	0	0
$409$	5.69202	0.281452	0.140726	−	0.990049i	$-0.455056\pi$
$409$	5.69202	0.281452	0.140726	+	0.990049i	$0.455056\pi$
$410$	0	0
$411$	2.21983	0.109496
$412$	0	0
$413$	−16.7114	−0.822314
$414$	0	0
$415$	−0.109916	−0.00539558
$416$	0	0
$417$	16.5157	0.808779
$418$	0	0
$419$	−26.2978	−1.28473	−0.642366	−	0.766398i	$-0.722047\pi$
$419$	−26.2978	−1.28473	−0.642366	+	0.766398i	$0.722047\pi$
$420$	0	0
$421$	−31.2083	−1.52100	−0.760501	−	0.649337i	$-0.775046\pi$
$421$	−31.2083	−1.52100	−0.760501	+	0.649337i	$0.775046\pi$
$422$	0	0
$423$	−7.35258	−0.357495
$424$	0	0
$425$	−16.0532	−0.778696
$426$	0	0
$427$	23.1172	1.11872
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.4142	−1.56134	−0.780668	−	0.624946i	$-0.785121\pi$
$431$	−32.4142	−1.56134	−0.780668	+	0.624946i	$0.785121\pi$
$432$	0	0
$433$	−32.9976	−1.58576	−0.792882	−	0.609375i	$-0.791420\pi$
$433$	−32.9976	−1.58576	−0.792882	+	0.609375i	$0.791420\pi$
$434$	0	0
$435$	47.7023	2.28715
$436$	0	0
$437$	5.15751	0.246717
$438$	0	0
$439$	−4.65386	−0.222117	−0.111058	−	0.993814i	$-0.535424\pi$
$439$	−4.65386	−0.222117	−0.111058	+	0.993814i	$0.535424\pi$
$440$	0	0
$441$	−18.2054	−0.866922
$442$	0	0
$443$	−29.4819	−1.40073	−0.700363	−	0.713787i	$-0.746979\pi$
$443$	−29.4819	−1.40073	−0.700363	+	0.713787i	$0.746979\pi$
$444$	0	0
$445$	−20.8538	−0.988567
$446$	0	0
$447$	13.2838	0.628303
$448$	0	0
$449$	−21.3303	−1.00664	−0.503320	−	0.864100i	$-0.667888\pi$
$449$	−21.3303	−1.00664	−0.503320	+	0.864100i	$0.667888\pi$
$450$	0	0
$451$	−45.4674	−2.14098
$452$	0	0
$453$	−15.0519	−0.707200
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.5875	−0.495262	−0.247631	−	0.968854i	$-0.579652\pi$
$457$	−10.5875	−0.495262	−0.247631	+	0.968854i	$0.579652\pi$
$458$	0	0
$459$	29.2024	1.36305
$460$	0	0
$461$	6.68425	0.311317	0.155658	−	0.987811i	$-0.450250\pi$
$461$	6.68425	0.311317	0.155658	+	0.987811i	$0.450250\pi$
$462$	0	0
$463$	2.69633	0.125309	0.0626546	−	0.998035i	$-0.480043\pi$
$463$	2.69633	0.125309	0.0626546	+	0.998035i	$0.480043\pi$
$464$	0	0
$465$	28.1226	1.30415
$466$	0	0
$467$	−2.30127	−0.106490	−0.0532451	−	0.998581i	$-0.516956\pi$
$467$	−2.30127	−0.106490	−0.0532451	+	0.998581i	$0.516956\pi$
$468$	0	0
$469$	−3.91185	−0.180633
$470$	0	0
$471$	31.4644	1.44980
$472$	0	0
$473$	−18.2000	−0.836836
$474$	0	0
$475$	4.74632	0.217776
$476$	0	0
$477$	−52.2083	−2.39046
$478$	0	0
$479$	6.49694	0.296853	0.148426	−	0.988923i	$-0.452579\pi$
$479$	6.49694	0.296853	0.148426	+	0.988923i	$0.452579\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	15.6407	0.711677
$484$	0	0
$485$	−1.32304	−0.0600763
$486$	0	0
$487$	−14.8877	−0.674626	−0.337313	−	0.941393i	$-0.609518\pi$
$487$	−14.8877	−0.674626	−0.337313	+	0.941393i	$0.609518\pi$
$488$	0	0
$489$	−2.91185	−0.131679
$490$	0	0
$491$	−41.6276	−1.87863	−0.939313	−	0.343062i	$-0.888536\pi$
$491$	−41.6276	−1.87863	−0.939313	+	0.343062i	$0.888536\pi$
$492$	0	0
$493$	−34.2145	−1.54094
$494$	0	0
$495$	−87.0665	−3.91335
$496$	0	0
$497$	19.6504	0.881441
$498$	0	0
$499$	−5.79523	−0.259430	−0.129715	−	0.991551i	$-0.541406\pi$
$499$	−5.79523	−0.259430	−0.129715	+	0.991551i	$0.541406\pi$
$500$	0	0
$501$	7.87800	0.351963
$502$	0	0
$503$	29.3414	1.30827	0.654133	−	0.756379i	$-0.273034\pi$
$503$	29.3414	1.30827	0.654133	+	0.756379i	$0.273034\pi$
$504$	0	0
$505$	−18.5797	−0.826786
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.4174	−1.39255	−0.696276	−	0.717774i	$-0.745161\pi$
$509$	−31.4174	−1.39255	−0.696276	+	0.717774i	$0.745161\pi$
$510$	0	0
$511$	3.51573	0.155527
$512$	0	0
$513$	−8.63401	−0.381201
$514$	0	0
$515$	−3.39911	−0.149783
$516$	0	0
$517$	−9.70948	−0.427022
$518$	0	0
$519$	59.3551	2.60540
$520$	0	0
$521$	37.2150	1.63042	0.815210	−	0.579165i	$-0.196621\pi$
$521$	37.2150	1.63042	0.815210	+	0.579165i	$0.196621\pi$
$522$	0	0
$523$	22.5924	0.987896	0.493948	−	0.869491i	$-0.335553\pi$
$523$	22.5924	0.987896	0.493948	+	0.869491i	$0.335553\pi$
$524$	0	0
$525$	14.3937	0.628194
$526$	0	0
$527$	−20.1709	−0.878659
$528$	0	0
$529$	−13.4034	−0.582757
$530$	0	0
$531$	44.9874	1.95229
$532$	0	0
$533$	0	0
$534$	0	0
$535$	12.5157	0.541102
$536$	0	0
$537$	−51.4349	−2.21958
$538$	0	0
$539$	−24.0411	−1.03553
$540$	0	0
$541$	24.6340	1.05910	0.529549	−	0.848279i	$-0.322361\pi$
$541$	24.6340	1.05910	0.529549	+	0.848279i	$0.322361\pi$
$542$	0	0
$543$	57.4312	2.46461
$544$	0	0
$545$	3.26205	0.139731
$546$	0	0
$547$	21.0248	0.898954	0.449477	−	0.893292i	$-0.351610\pi$
$547$	21.0248	0.898954	0.449477	+	0.893292i	$0.351610\pi$
$548$	0	0
$549$	−62.2320	−2.65600
$550$	0	0
$551$	10.1159	0.430951
$552$	0	0
$553$	20.8388	0.886155
$554$	0	0
$555$	40.4064	1.71516
$556$	0	0
$557$	−14.8629	−0.629763	−0.314881	−	0.949131i	$-0.601965\pi$
$557$	−14.8629	−0.629763	−0.314881	+	0.949131i	$0.601965\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	101.069	4.26716
$562$	0	0
$563$	−28.9105	−1.21843	−0.609217	−	0.793004i	$-0.708516\pi$
$563$	−28.9105	−1.21843	−0.609217	+	0.793004i	$0.708516\pi$
$564$	0	0
$565$	14.4276	0.606973
$566$	0	0
$567$	0.0392287	0.00164745
$568$	0	0
$569$	30.3317	1.27157	0.635785	−	0.771866i	$-0.280677\pi$
$569$	30.3317	1.27157	0.635785	+	0.771866i	$0.280677\pi$
$570$	0	0
$571$	31.9017	1.33504	0.667522	−	0.744590i	$-0.267355\pi$
$571$	31.9017	1.33504	0.667522	+	0.744590i	$0.267355\pi$
$572$	0	0
$573$	9.61894	0.401837
$574$	0	0
$575$	8.83148	0.368298
$576$	0	0
$577$	43.8364	1.82493	0.912466	−	0.409152i	$-0.134175\pi$
$577$	43.8364	1.82493	0.912466	+	0.409152i	$0.134175\pi$
$578$	0	0
$579$	−4.71379	−0.195898
$580$	0	0
$581$	−0.0706876	−0.00293262
$582$	0	0
$583$	−68.9439	−2.85536
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.11231	0.0871843	0.0435922	−	0.999049i	$-0.486120\pi$
$587$	2.11231	0.0871843	0.0435922	+	0.999049i	$0.486120\pi$
$588$	0	0
$589$	5.96376	0.245732
$590$	0	0
$591$	−75.5169	−3.10635
$592$	0	0
$593$	1.98062	0.0813344	0.0406672	−	0.999173i	$-0.487052\pi$
$593$	1.98062	0.0813344	0.0406672	+	0.999173i	$0.487052\pi$
$594$	0	0
$595$	−28.4306	−1.16554
$596$	0	0
$597$	56.2640	2.30273
$598$	0	0
$599$	−14.6045	−0.596722	−0.298361	−	0.954453i	$-0.596440\pi$
$599$	−14.6045	−0.596722	−0.298361	+	0.954453i	$0.596440\pi$
$600$	0	0
$601$	15.7982	0.644423	0.322211	−	0.946668i	$-0.395574\pi$
$601$	15.7982	0.644423	0.322211	+	0.946668i	$0.395574\pi$
$602$	0	0
$603$	10.5308	0.428847
$604$	0	0
$605$	−84.1546	−3.42137
$606$	0	0
$607$	25.9995	1.05529	0.527644	−	0.849466i	$-0.323075\pi$
$607$	25.9995	1.05529	0.527644	+	0.849466i	$0.323075\pi$
$608$	0	0
$609$	30.6775	1.24312
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−12.4558	−0.503085	−0.251542	−	0.967846i	$-0.580938\pi$
$613$	−12.4558	−0.503085	−0.251542	+	0.967846i	$0.580938\pi$
$614$	0	0
$615$	−55.7241	−2.24701
$616$	0	0
$617$	−38.0723	−1.53273	−0.766367	−	0.642403i	$-0.777938\pi$
$617$	−38.0723	−1.53273	−0.766367	+	0.642403i	$0.777938\pi$
$618$	0	0
$619$	25.8678	1.03972	0.519858	−	0.854253i	$-0.325985\pi$
$619$	25.8678	1.03972	0.519858	+	0.854253i	$0.325985\pi$
$620$	0	0
$621$	−16.0653	−0.644679
$622$	0	0
$623$	−13.4112	−0.537308
$624$	0	0
$625$	−31.1269	−1.24508
$626$	0	0
$627$	−29.8823	−1.19338
$628$	0	0
$629$	−28.9815	−1.15557
$630$	0	0
$631$	−7.08144	−0.281908	−0.140954	−	0.990016i	$-0.545017\pi$
$631$	−7.08144	−0.281908	−0.140954	+	0.990016i	$0.545017\pi$
$632$	0	0
$633$	23.1564	0.920386
$634$	0	0
$635$	29.0344	1.15220
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−52.8993	−2.09266
$640$	0	0
$641$	−4.79630	−0.189442	−0.0947212	−	0.995504i	$-0.530196\pi$
$641$	−4.79630	−0.189442	−0.0947212	+	0.995504i	$0.530196\pi$
$642$	0	0
$643$	−20.3394	−0.802109	−0.401055	−	0.916054i	$-0.631356\pi$
$643$	−20.3394	−0.802109	−0.401055	+	0.916054i	$0.631356\pi$
$644$	0	0
$645$	−22.3056	−0.878282
$646$	0	0
$647$	−6.02416	−0.236834	−0.118417	−	0.992964i	$-0.537782\pi$
$647$	−6.02416	−0.236834	−0.118417	+	0.992964i	$0.537782\pi$
$648$	0	0
$649$	59.4083	2.33198
$650$	0	0
$651$	18.0858	0.708837
$652$	0	0
$653$	−19.1914	−0.751016	−0.375508	−	0.926819i	$-0.622532\pi$
$653$	−19.1914	−0.751016	−0.375508	+	0.926819i	$0.622532\pi$
$654$	0	0
$655$	−35.9584	−1.40501
$656$	0	0
$657$	−9.46442	−0.369242
$658$	0	0
$659$	37.6752	1.46762	0.733808	−	0.679357i	$-0.237741\pi$
$659$	37.6752	1.46762	0.733808	+	0.679357i	$0.237741\pi$
$660$	0	0
$661$	3.68127	0.143185	0.0715924	−	0.997434i	$-0.477192\pi$
$661$	3.68127	0.143185	0.0715924	+	0.997434i	$0.477192\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.40581	0.325963
$666$	0	0
$667$	18.8226	0.728816
$668$	0	0
$669$	65.3075	2.52493
$670$	0	0
$671$	−82.1807	−3.17255
$672$	0	0
$673$	4.62863	0.178421	0.0892103	−	0.996013i	$-0.471566\pi$
$673$	4.62863	0.178421	0.0892103	+	0.996013i	$0.471566\pi$
$674$	0	0
$675$	−14.7845	−0.569055
$676$	0	0
$677$	−1.57109	−0.0603818	−0.0301909	−	0.999544i	$-0.509612\pi$
$677$	−1.57109	−0.0603818	−0.0301909	+	0.999544i	$0.509612\pi$
$678$	0	0
$679$	−0.850855	−0.0326528
$680$	0	0
$681$	−39.8582	−1.52737
$682$	0	0
$683$	−44.1433	−1.68910	−0.844548	−	0.535480i	$-0.820131\pi$
$683$	−44.1433	−1.68910	−0.844548	+	0.535480i	$0.820131\pi$
$684$	0	0
$685$	2.21983	0.0848154
$686$	0	0
$687$	75.2917	2.87256
$688$	0	0
$689$	0	0
$690$	0	0
$691$	31.2446	1.18860	0.594300	−	0.804243i	$-0.297429\pi$
$691$	31.2446	1.18860	0.594300	+	0.804243i	$0.297429\pi$
$692$	0	0
$693$	−55.9928	−2.12699
$694$	0	0
$695$	16.5157	0.626477
$696$	0	0
$697$	39.9681	1.51390
$698$	0	0
$699$	5.76809	0.218169
$700$	0	0
$701$	6.88876	0.260185	0.130092	−	0.991502i	$-0.458473\pi$
$701$	6.88876	0.260185	0.130092	+	0.991502i	$0.458473\pi$
$702$	0	0
$703$	8.56870	0.323174
$704$	0	0
$705$	−11.8998	−0.448171
$706$	0	0
$707$	−11.9487	−0.449377
$708$	0	0
$709$	−23.4265	−0.879801	−0.439901	−	0.898046i	$-0.644986\pi$
$709$	−23.4265	−0.879801	−0.439901	+	0.898046i	$0.644986\pi$
$710$	0	0
$711$	−56.0984	−2.10386
$712$	0	0
$713$	11.0968	0.415577
$714$	0	0
$715$	0	0
$716$	0	0
$717$	25.3274	0.945867
$718$	0	0
$719$	16.1666	0.602913	0.301456	−	0.953480i	$-0.402527\pi$
$719$	16.1666	0.602913	0.301456	+	0.953480i	$0.402527\pi$
$720$	0	0
$721$	−2.18598	−0.0814102
$722$	0	0
$723$	23.2107	0.863217
$724$	0	0
$725$	17.3220	0.643322
$726$	0	0
$727$	−45.2054	−1.67657	−0.838287	−	0.545229i	$-0.816443\pi$
$727$	−45.2054	−1.67657	−0.838287	+	0.545229i	$0.816443\pi$
$728$	0	0
$729$	−43.6980	−1.61844
$730$	0	0
$731$	15.9987	0.591732
$732$	0	0
$733$	51.4631	1.90083	0.950416	−	0.310980i	$-0.100657\pi$
$733$	51.4631	1.90083	0.950416	+	0.310980i	$0.100657\pi$
$734$	0	0
$735$	−29.4644	−1.08681
$736$	0	0
$737$	13.9065	0.512252
$738$	0	0
$739$	12.2150	0.449338	0.224669	−	0.974435i	$-0.427870\pi$
$739$	12.2150	0.449338	0.224669	+	0.974435i	$0.427870\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	51.5182	1.89002	0.945010	−	0.327040i	$-0.106051\pi$
$743$	51.5182	1.89002	0.945010	+	0.327040i	$0.106051\pi$
$744$	0	0
$745$	13.2838	0.486681
$746$	0	0
$747$	0.190293	0.00696244
$748$	0	0
$749$	8.04892	0.294101
$750$	0	0
$751$	21.3250	0.778159	0.389079	−	0.921204i	$-0.372793\pi$
$751$	21.3250	0.778159	0.389079	+	0.921204i	$0.372793\pi$
$752$	0	0
$753$	−14.8116	−0.539766
$754$	0	0
$755$	−15.0519	−0.547795
$756$	0	0
$757$	−28.5254	−1.03677	−0.518387	−	0.855146i	$-0.673467\pi$
$757$	−28.5254	−1.03677	−0.518387	+	0.855146i	$0.673467\pi$
$758$	0	0
$759$	−55.6021	−2.01823
$760$	0	0
$761$	31.1661	1.12977	0.564886	−	0.825169i	$-0.308920\pi$
$761$	31.1661	1.12977	0.564886	+	0.825169i	$0.308920\pi$
$762$	0	0
$763$	2.09783	0.0759467
$764$	0	0
$765$	76.5357	2.76715
$766$	0	0
$767$	0	0
$768$	0	0
$769$	26.3811	0.951325	0.475663	−	0.879628i	$-0.342208\pi$
$769$	26.3811	0.951325	0.475663	+	0.879628i	$0.342208\pi$
$770$	0	0
$771$	−54.5260	−1.96371
$772$	0	0
$773$	−1.67696	−0.0603159	−0.0301580	−	0.999545i	$-0.509601\pi$
$773$	−1.67696	−0.0603159	−0.0301580	+	0.999545i	$0.509601\pi$
$774$	0	0
$775$	10.2121	0.366828
$776$	0	0
$777$	25.9855	0.932226
$778$	0	0
$779$	−11.8170	−0.423388
$780$	0	0
$781$	−69.8563	−2.49966
$782$	0	0
$783$	−31.5104	−1.12609
$784$	0	0
$785$	31.4644	1.12301
$786$	0	0
$787$	4.58881	0.163573	0.0817867	−	0.996650i	$-0.473937\pi$
$787$	4.58881	0.163573	0.0817867	+	0.996650i	$0.473937\pi$
$788$	0	0
$789$	54.3484	1.93485
$790$	0	0
$791$	9.27844	0.329903
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−84.4965	−2.99678
$796$	0	0
$797$	38.0605	1.34817	0.674086	−	0.738652i	$-0.264537\pi$
$797$	38.0605	1.34817	0.674086	+	0.738652i	$0.264537\pi$
$798$	0	0
$799$	8.53511	0.301950
$800$	0	0
$801$	36.1032	1.27564
$802$	0	0
$803$	−12.4983	−0.441054
$804$	0	0
$805$	15.6407	0.551263
$806$	0	0
$807$	10.1250	0.356416
$808$	0	0
$809$	47.0616	1.65460	0.827299	−	0.561762i	$-0.189876\pi$
$809$	47.0616	1.65460	0.827299	+	0.561762i	$0.189876\pi$
$810$	0	0
$811$	−6.21744	−0.218324	−0.109162	−	0.994024i	$-0.534817\pi$
$811$	−6.21744	−0.218324	−0.109162	+	0.994024i	$0.534817\pi$
$812$	0	0
$813$	8.82072	0.309356
$814$	0	0
$815$	−2.91185	−0.101998
$816$	0	0
$817$	−4.73019	−0.165488
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.6843	0.547384	0.273692	−	0.961817i	$-0.411755\pi$
$821$	15.6843	0.547384	0.273692	+	0.961817i	$0.411755\pi$
$822$	0	0
$823$	−25.2306	−0.879483	−0.439741	−	0.898124i	$-0.644930\pi$
$823$	−25.2306	−0.879483	−0.439741	+	0.898124i	$0.644930\pi$
$824$	0	0
$825$	−51.1691	−1.78148
$826$	0	0
$827$	51.8582	1.80328	0.901642	−	0.432483i	$-0.142362\pi$
$827$	51.8582	1.80328	0.901642	+	0.432483i	$0.142362\pi$
$828$	0	0
$829$	−23.7192	−0.823801	−0.411900	−	0.911229i	$-0.635135\pi$
$829$	−23.7192	−0.823801	−0.411900	+	0.911229i	$0.635135\pi$
$830$	0	0
$831$	−27.1987	−0.943511
$832$	0	0
$833$	21.1333	0.732227
$834$	0	0
$835$	7.87800	0.272630
$836$	0	0
$837$	−18.5767	−0.642106
$838$	0	0
$839$	−13.0315	−0.449896	−0.224948	−	0.974371i	$-0.572221\pi$
$839$	−13.0315	−0.449896	−0.224948	+	0.974371i	$0.572221\pi$
$840$	0	0
$841$	7.91856	0.273054
$842$	0	0
$843$	34.6233	1.19249
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−54.1202	−1.85959
$848$	0	0
$849$	40.6655	1.39564
$850$	0	0
$851$	15.9438	0.546546
$852$	0	0
$853$	−18.4155	−0.630535	−0.315267	−	0.949003i	$-0.602094\pi$
$853$	−18.4155	−0.630535	−0.315267	+	0.949003i	$0.602094\pi$
$854$	0	0
$855$	−22.6286	−0.773883
$856$	0	0
$857$	−6.54288	−0.223500	−0.111750	−	0.993736i	$-0.535646\pi$
$857$	−6.54288	−0.223500	−0.111750	+	0.993736i	$0.535646\pi$
$858$	0	0
$859$	−38.9396	−1.32860	−0.664301	−	0.747465i	$-0.731271\pi$
$859$	−38.9396	−1.32860	−0.664301	+	0.747465i	$0.731271\pi$
$860$	0	0
$861$	−35.8364	−1.22130
$862$	0	0
$863$	40.8732	1.39134	0.695670	−	0.718361i	$-0.255107\pi$
$863$	40.8732	1.39134	0.695670	+	0.718361i	$0.255107\pi$
$864$	0	0
$865$	59.3551	2.01813
$866$	0	0
$867$	−41.2121	−1.39964
$868$	0	0
$869$	−74.0810	−2.51302
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.29052	0.0775224
$874$	0	0
$875$	−10.8509	−0.366826
$876$	0	0
$877$	−25.9433	−0.876043	−0.438022	−	0.898964i	$-0.644321\pi$
$877$	−25.9433	−0.876043	−0.438022	+	0.898964i	$0.644321\pi$
$878$	0	0
$879$	39.4553	1.33079
$880$	0	0
$881$	−56.5575	−1.90547	−0.952735	−	0.303803i	$-0.901743\pi$
$881$	−56.5575	−1.90547	−0.952735	+	0.303803i	$0.901743\pi$
$882$	0	0
$883$	−11.4910	−0.386702	−0.193351	−	0.981130i	$-0.561936\pi$
$883$	−11.4910	−0.386702	−0.193351	+	0.981130i	$0.561936\pi$
$884$	0	0
$885$	72.8098	2.44748
$886$	0	0
$887$	−4.51035	−0.151443	−0.0757214	−	0.997129i	$-0.524126\pi$
$887$	−4.51035	−0.151443	−0.0757214	+	0.997129i	$0.524126\pi$
$888$	0	0
$889$	18.6722	0.626244
$890$	0	0
$891$	−0.139456	−0.00467196
$892$	0	0
$893$	−2.52350	−0.0844457
$894$	0	0
$895$	−51.4349	−1.71928
$896$	0	0
$897$	0	0
$898$	0	0
$899$	21.7651	0.725907
$900$	0	0
$901$	60.6051	2.01905
$902$	0	0
$903$	−14.3448	−0.477366
$904$	0	0
$905$	57.4312	1.90908
$906$	0	0
$907$	19.9675	0.663009	0.331505	−	0.943454i	$-0.392444\pi$
$907$	19.9675	0.663009	0.331505	+	0.943454i	$0.392444\pi$
$908$	0	0
$909$	32.1661	1.06688
$910$	0	0
$911$	4.73855	0.156995	0.0784975	−	0.996914i	$-0.474988\pi$
$911$	4.73855	0.156995	0.0784975	+	0.996914i	$0.474988\pi$
$912$	0	0
$913$	0.251291	0.00831653
$914$	0	0
$915$	−100.719	−3.32968
$916$	0	0
$917$	−23.1250	−0.763654
$918$	0	0
$919$	−12.0140	−0.396305	−0.198153	−	0.980171i	$-0.563494\pi$
$919$	−12.0140	−0.396305	−0.198153	+	0.980171i	$0.563494\pi$
$920$	0	0
$921$	−56.0253	−1.84610
$922$	0	0
$923$	0	0
$924$	0	0
$925$	14.6726	0.482434
$926$	0	0
$927$	5.88471	0.193279
$928$	0	0
$929$	25.2194	0.827420	0.413710	−	0.910409i	$-0.364233\pi$
$929$	25.2194	0.827420	0.413710	+	0.910409i	$0.364233\pi$
$930$	0	0
$931$	−6.24831	−0.204780
$932$	0	0
$933$	−60.5937	−1.98375
$934$	0	0
$935$	101.069	3.30533
$936$	0	0
$937$	−9.69143	−0.316605	−0.158303	−	0.987391i	$-0.550602\pi$
$937$	−9.69143	−0.316605	−0.158303	+	0.987391i	$0.550602\pi$
$938$	0	0
$939$	−3.98121	−0.129922
$940$	0	0
$941$	−5.46117	−0.178029	−0.0890146	−	0.996030i	$-0.528372\pi$
$941$	−5.46117	−0.178029	−0.0890146	+	0.996030i	$0.528372\pi$
$942$	0	0
$943$	−21.9879	−0.716025
$944$	0	0
$945$	−26.1836	−0.851752
$946$	0	0
$947$	−33.8611	−1.10034	−0.550170	−	0.835053i	$-0.685437\pi$
$947$	−33.8611	−1.10034	−0.550170	+	0.835053i	$0.685437\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	27.0616	0.877532
$952$	0	0
$953$	−47.4446	−1.53688	−0.768440	−	0.639922i	$-0.778967\pi$
$953$	−47.4446	−1.53688	−0.768440	+	0.639922i	$0.778967\pi$
$954$	0	0
$955$	9.61894	0.311262
$956$	0	0
$957$	−109.057	−3.52532
$958$	0	0
$959$	1.42758	0.0460991
$960$	0	0
$961$	−18.1685	−0.586081
$962$	0	0
$963$	−21.6679	−0.698237
$964$	0	0
$965$	−4.71379	−0.151742
$966$	0	0
$967$	52.5163	1.68881	0.844406	−	0.535705i	$-0.179954\pi$
$967$	52.5163	1.68881	0.844406	+	0.535705i	$0.179954\pi$
$968$	0	0
$969$	26.2680	0.843850
$970$	0	0
$971$	−1.18705	−0.0380941	−0.0190471	−	0.999819i	$-0.506063\pi$
$971$	−1.18705	−0.0380941	−0.0190471	+	0.999819i	$0.506063\pi$
$972$	0	0
$973$	10.6213	0.340504
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.0455	−1.08921	−0.544605	−	0.838692i	$-0.683321\pi$
$977$	−34.0455	−1.08921	−0.544605	+	0.838692i	$0.683321\pi$
$978$	0	0
$979$	47.6762	1.52374
$980$	0	0
$981$	−5.64742	−0.180308
$982$	0	0
$983$	38.5338	1.22904	0.614518	−	0.788903i	$-0.289350\pi$
$983$	38.5338	1.22904	0.614518	+	0.788903i	$0.289350\pi$
$984$	0	0
$985$	−75.5169	−2.40617
$986$	0	0
$987$	−7.65279	−0.243591
$988$	0	0
$989$	−8.80146	−0.279870
$990$	0	0
$991$	44.3752	1.40962	0.704812	−	0.709394i	$-0.251031\pi$
$991$	44.3752	1.40962	0.704812	+	0.709394i	$0.251031\pi$
$992$	0	0
$993$	89.4814	2.83961
$994$	0	0
$995$	56.2640	1.78369
$996$	0	0
$997$	−29.2355	−0.925897	−0.462949	−	0.886385i	$-0.653209\pi$
$997$	−29.2355	−0.925897	−0.462949	+	0.886385i	$0.653209\pi$
$998$	0	0
$999$	−26.6910	−0.844464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.a.i.1.1	✓	3
4.3	odd	2		2704.2.a.bb.1.3		3
13.2	odd	12		1352.2.o.g.1161.6		12
13.3	even	3		1352.2.i.i.529.3		6
13.4	even	6		1352.2.i.j.1329.3		6
13.5	odd	4		1352.2.f.e.337.2		6
13.6	odd	12		1352.2.o.g.361.6		12
13.7	odd	12		1352.2.o.g.361.5		12
13.8	odd	4		1352.2.f.e.337.1		6
13.9	even	3		1352.2.i.i.1329.3		6
13.10	even	6		1352.2.i.j.529.3		6
13.11	odd	12		1352.2.o.g.1161.5		12
13.12	even	2		1352.2.a.j.1.1	yes	3
52.31	even	4		2704.2.f.p.337.6		6
52.47	even	4		2704.2.f.p.337.5		6
52.51	odd	2		2704.2.a.bc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1352.2.a.i.1.1	✓	3	1.1	even	1	trivial
1352.2.a.j.1.1	yes	3	13.12	even	2
1352.2.f.e.337.1		6	13.8	odd	4
1352.2.f.e.337.2		6	13.5	odd	4
1352.2.i.i.529.3		6	13.3	even	3
1352.2.i.i.1329.3		6	13.9	even	3
1352.2.i.j.529.3		6	13.10	even	6
1352.2.i.j.1329.3		6	13.4	even	6
1352.2.o.g.361.5		12	13.7	odd	12
1352.2.o.g.361.6		12	13.6	odd	12
1352.2.o.g.1161.5		12	13.11	odd	12
1352.2.o.g.1161.6		12	13.2	odd	12
2704.2.a.bb.1.3		3	4.3	odd	2
2704.2.a.bc.1.3		3	52.51	odd	2
2704.2.f.p.337.5		6	52.47	even	4
2704.2.f.p.337.6		6	52.31	even	4

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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

\(f(q)\)	\(=\)	\(q-2.80194 q^{3} -2.80194 q^{5} -1.80194 q^{7} +4.85086 q^{9} +O(q^{10})\)	\(q-2.80194 q^{3} -2.80194 q^{5} -1.80194 q^{7} +4.85086 q^{9} +6.40581 q^{11} +7.85086 q^{15} -5.63102 q^{17} +1.66487 q^{19} +5.04892 q^{21} +3.09783 q^{23} +2.85086 q^{25} -5.18598 q^{27} +6.07606 q^{29} +3.58211 q^{31} -17.9487 q^{33} +5.04892 q^{35} +5.14675 q^{37} -7.09783 q^{41} -2.84117 q^{43} -13.5918 q^{45} -1.51573 q^{47} -3.75302 q^{49} +15.7778 q^{51} -10.7627 q^{53} -17.9487 q^{55} -4.66487 q^{57} +9.27413 q^{59} -12.8291 q^{61} -8.74094 q^{63} +2.17092 q^{67} -8.67994 q^{69} -10.9051 q^{71} -1.95108 q^{73} -7.98792 q^{75} -11.5429 q^{77} -11.5646 q^{79} -0.0217703 q^{81} +0.0392287 q^{83} +15.7778 q^{85} -17.0248 q^{87} +7.44265 q^{89} -10.0368 q^{93} -4.66487 q^{95} +0.472189 q^{97} +31.0737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 4 q^{5} - q^{7} + q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 - 4 * q^5 - q^7 + q^9	\( 3 q - 4 q^{3} - 4 q^{5} - q^{7} + q^{9} + 6 q^{11} + 10 q^{15} - 2 q^{17} + 6 q^{19} + 6 q^{21} - 9 q^{23} - 5 q^{25} - q^{27} + 3 q^{29} + 5 q^{31} - 22 q^{33} + 6 q^{35} - 12 q^{37} - 3 q^{41} - 17 q^{43} - 13 q^{45} + 8 q^{47} - 16 q^{49} + 5 q^{51} - 15 q^{53} - 22 q^{55} - 15 q^{57} + 17 q^{59} - 28 q^{61} - 12 q^{63} + 17 q^{67} - 2 q^{69} - 7 q^{71} - 15 q^{73} - 5 q^{75} - 16 q^{77} - 13 q^{79} + 3 q^{81} + 13 q^{83} + 5 q^{85} - 4 q^{87} - 19 q^{89} - 2 q^{93} - 15 q^{95} - 5 q^{97} + 37 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 4 * q^5 - q^7 + q^9 + 6 * q^11 + 10 * q^15 - 2 * q^17 + 6 * q^19 + 6 * q^21 - 9 * q^23 - 5 * q^25 - q^27 + 3 * q^29 + 5 * q^31 - 22 * q^33 + 6 * q^35 - 12 * q^37 - 3 * q^41 - 17 * q^43 - 13 * q^45 + 8 * q^47 - 16 * q^49 + 5 * q^51 - 15 * q^53 - 22 * q^55 - 15 * q^57 + 17 * q^59 - 28 * q^61 - 12 * q^63 + 17 * q^67 - 2 * q^69 - 7 * q^71 - 15 * q^73 - 5 * q^75 - 16 * q^77 - 13 * q^79 + 3 * q^81 + 13 * q^83 + 5 * q^85 - 4 * q^87 - 19 * q^89 - 2 * q^93 - 15 * q^95 - 5 * q^97 + 37 * q^99

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