LMFDB - Embedded newform 6936.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6936,2,Mod(1,6936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6936, 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("6936.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6936 = 2^{3} \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6936.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:	$55.3842388420$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 408)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	6936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.27492 q^{5} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.27492 q^{5} -4.00000 q^{7} +1.00000 q^{9} +5.27492 q^{11} -3.27492 q^{13} +3.27492 q^{15} -5.27492 q^{19} +4.00000 q^{21} +5.27492 q^{23} +5.72508 q^{25} -1.00000 q^{27} -2.00000 q^{29} -6.54983 q^{31} -5.27492 q^{33} +13.0997 q^{35} -4.54983 q^{37} +3.27492 q^{39} -0.725083 q^{41} -2.72508 q^{43} -3.27492 q^{45} -10.5498 q^{47} +9.00000 q^{49} -10.0000 q^{53} -17.2749 q^{55} +5.27492 q^{57} -6.54983 q^{59} -12.5498 q^{61} -4.00000 q^{63} +10.7251 q^{65} +4.00000 q^{67} -5.27492 q^{69} +4.00000 q^{71} -10.0000 q^{73} -5.72508 q^{75} -21.0997 q^{77} -6.54983 q^{79} +1.00000 q^{81} +6.54983 q^{83} +2.00000 q^{87} +12.5498 q^{89} +13.0997 q^{91} +6.54983 q^{93} +17.2749 q^{95} +8.54983 q^{97} +5.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + q^{5} - 8 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + q^5 - 8 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + q^{5} - 8 q^{7} + 2 q^{9} + 3 q^{11} + q^{13} - q^{15} - 3 q^{19} + 8 q^{21} + 3 q^{23} + 19 q^{25} - 2 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} - 4 q^{35} + 6 q^{37} - q^{39} - 9 q^{41} - 13 q^{43} + q^{45} - 6 q^{47} + 18 q^{49} - 20 q^{53} - 27 q^{55} + 3 q^{57} + 2 q^{59} - 10 q^{61} - 8 q^{63} + 29 q^{65} + 8 q^{67} - 3 q^{69} + 8 q^{71} - 20 q^{73} - 19 q^{75} - 12 q^{77} + 2 q^{79} + 2 q^{81} - 2 q^{83} + 4 q^{87} + 10 q^{89} - 4 q^{91} - 2 q^{93} + 27 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + q^5 - 8 * q^7 + 2 * q^9 + 3 * q^11 + q^13 - q^15 - 3 * q^19 + 8 * q^21 + 3 * q^23 + 19 * q^25 - 2 * q^27 - 4 * q^29 + 2 * q^31 - 3 * q^33 - 4 * q^35 + 6 * q^37 - q^39 - 9 * q^41 - 13 * q^43 + q^45 - 6 * q^47 + 18 * q^49 - 20 * q^53 - 27 * q^55 + 3 * q^57 + 2 * q^59 - 10 * q^61 - 8 * q^63 + 29 * q^65 + 8 * q^67 - 3 * q^69 + 8 * q^71 - 20 * q^73 - 19 * q^75 - 12 * q^77 + 2 * q^79 + 2 * q^81 - 2 * q^83 + 4 * q^87 + 10 * q^89 - 4 * q^91 - 2 * q^93 + 27 * q^95 + 2 * q^97 + 3 * 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
$5$	−3.27492	−1.46459	−0.732294	−	0.680989i	$-0.761550\pi$
$5$	−3.27492	−1.46459	−0.732294	+	0.680989i	$0.761550\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.27492	1.59045	0.795224	−	0.606316i	$-0.207353\pi$
$11$	5.27492	1.59045	0.795224	+	0.606316i	$0.207353\pi$
$12$	0	0
$13$	−3.27492	−0.908299	−0.454149	−	0.890926i	$-0.650057\pi$
$13$	−3.27492	−0.908299	−0.454149	+	0.890926i	$0.650057\pi$
$14$	0	0
$15$	3.27492	0.845580
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−5.27492	−1.21015	−0.605075	−	0.796169i	$-0.706857\pi$
$19$	−5.27492	−1.21015	−0.605075	+	0.796169i	$0.706857\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	5.27492	1.09990	0.549948	−	0.835199i	$-0.314647\pi$
$23$	5.27492	1.09990	0.549948	+	0.835199i	$0.314647\pi$
$24$	0	0
$25$	5.72508	1.14502
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−6.54983	−1.17638	−0.588192	−	0.808721i	$-0.700160\pi$
$31$	−6.54983	−1.17638	−0.588192	+	0.808721i	$0.700160\pi$
$32$	0	0
$33$	−5.27492	−0.918245
$34$	0	0
$35$	13.0997	2.21425
$36$	0	0
$37$	−4.54983	−0.747988	−0.373994	−	0.927431i	$-0.622012\pi$
$37$	−4.54983	−0.747988	−0.373994	+	0.927431i	$0.622012\pi$
$38$	0	0
$39$	3.27492	0.524406
$40$	0	0
$41$	−0.725083	−0.113239	−0.0566195	−	0.998396i	$-0.518032\pi$
$41$	−0.725083	−0.113239	−0.0566195	+	0.998396i	$0.518032\pi$
$42$	0	0
$43$	−2.72508	−0.415571	−0.207786	−	0.978174i	$-0.566626\pi$
$43$	−2.72508	−0.415571	−0.207786	+	0.978174i	$0.566626\pi$
$44$	0	0
$45$	−3.27492	−0.488196
$46$	0	0
$47$	−10.5498	−1.53885	−0.769426	−	0.638736i	$-0.779458\pi$
$47$	−10.5498	−1.53885	−0.769426	+	0.638736i	$0.779458\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	−17.2749	−2.32935
$56$	0	0
$57$	5.27492	0.698680
$58$	0	0
$59$	−6.54983	−0.852716	−0.426358	−	0.904555i	$-0.640204\pi$
$59$	−6.54983	−0.852716	−0.426358	+	0.904555i	$0.640204\pi$
$60$	0	0
$61$	−12.5498	−1.60684	−0.803421	−	0.595412i	$-0.796989\pi$
$61$	−12.5498	−1.60684	−0.803421	+	0.595412i	$0.796989\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	10.7251	1.33028
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−5.27492	−0.635025
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	−5.72508	−0.661076
$76$	0	0
$77$	−21.0997	−2.40453
$78$	0	0
$79$	−6.54983	−0.736914	−0.368457	−	0.929645i	$-0.620114\pi$
$79$	−6.54983	−0.736914	−0.368457	+	0.929645i	$0.620114\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.54983	0.718938	0.359469	−	0.933157i	$-0.382958\pi$
$83$	6.54983	0.718938	0.359469	+	0.933157i	$0.382958\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	12.5498	1.33028	0.665140	−	0.746719i	$-0.268372\pi$
$89$	12.5498	1.33028	0.665140	+	0.746719i	$0.268372\pi$
$90$	0	0
$91$	13.0997	1.37322
$92$	0	0
$93$	6.54983	0.679186
$94$	0	0
$95$	17.2749	1.77237
$96$	0	0
$97$	8.54983	0.868104	0.434052	−	0.900888i	$-0.357083\pi$
$97$	8.54983	0.868104	0.434052	+	0.900888i	$0.357083\pi$
$98$	0	0
$99$	5.27492	0.530149
$100$	0	0
$101$	−12.5498	−1.24876	−0.624378	−	0.781123i	$-0.714647\pi$
$101$	−12.5498	−1.24876	−0.624378	+	0.781123i	$0.714647\pi$
$102$	0	0
$103$	−19.8248	−1.95339	−0.976695	−	0.214630i	$-0.931145\pi$
$103$	−19.8248	−1.95339	−0.976695	+	0.214630i	$0.931145\pi$
$104$	0	0
$105$	−13.0997	−1.27840
$106$	0	0
$107$	−15.8248	−1.52984	−0.764918	−	0.644127i	$-0.777221\pi$
$107$	−15.8248	−1.52984	−0.764918	+	0.644127i	$0.777221\pi$
$108$	0	0
$109$	16.5498	1.58519	0.792593	−	0.609751i	$-0.208730\pi$
$109$	16.5498	1.58519	0.792593	+	0.609751i	$0.208730\pi$
$110$	0	0
$111$	4.54983	0.431851
$112$	0	0
$113$	−3.27492	−0.308078	−0.154039	−	0.988065i	$-0.549228\pi$
$113$	−3.27492	−0.308078	−0.154039	+	0.988065i	$0.549228\pi$
$114$	0	0
$115$	−17.2749	−1.61089
$116$	0	0
$117$	−3.27492	−0.302766
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.8248	1.52952
$122$	0	0
$123$	0.725083	0.0653785
$124$	0	0
$125$	−2.37459	−0.212389
$126$	0	0
$127$	19.8248	1.75916	0.879581	−	0.475749i	$-0.157823\pi$
$127$	19.8248	1.75916	0.879581	+	0.475749i	$0.157823\pi$
$128$	0	0
$129$	2.72508	0.239930
$130$	0	0
$131$	10.7251	0.937055	0.468527	−	0.883449i	$-0.344785\pi$
$131$	10.7251	0.937055	0.468527	+	0.883449i	$0.344785\pi$
$132$	0	0
$133$	21.0997	1.82957
$134$	0	0
$135$	3.27492	0.281860
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−6.54983	−0.555550	−0.277775	−	0.960646i	$-0.589597\pi$
$139$	−6.54983	−0.555550	−0.277775	+	0.960646i	$0.589597\pi$
$140$	0	0
$141$	10.5498	0.888456
$142$	0	0
$143$	−17.2749	−1.44460
$144$	0	0
$145$	6.54983	0.543934
$146$	0	0
$147$	−9.00000	−0.742307
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	21.4502	1.72292
$156$	0	0
$157$	−16.3746	−1.30683	−0.653417	−	0.756998i	$-0.726665\pi$
$157$	−16.3746	−1.30683	−0.653417	+	0.756998i	$0.726665\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	−21.0997	−1.66289
$162$	0	0
$163$	−6.54983	−0.513023	−0.256511	−	0.966541i	$-0.582573\pi$
$163$	−6.54983	−0.513023	−0.256511	+	0.966541i	$0.582573\pi$
$164$	0	0
$165$	17.2749	1.34485
$166$	0	0
$167$	−15.8248	−1.22456	−0.612278	−	0.790643i	$-0.709747\pi$
$167$	−15.8248	−1.22456	−0.612278	+	0.790643i	$0.709747\pi$
$168$	0	0
$169$	−2.27492	−0.174994
$170$	0	0
$171$	−5.27492	−0.403383
$172$	0	0
$173$	9.82475	0.746962	0.373481	−	0.927638i	$-0.378164\pi$
$173$	9.82475	0.746962	0.373481	+	0.927638i	$0.378164\pi$
$174$	0	0
$175$	−22.9003	−1.73110
$176$	0	0
$177$	6.54983	0.492316
$178$	0	0
$179$	9.45017	0.706339	0.353169	−	0.935559i	$-0.385104\pi$
$179$	9.45017	0.706339	0.353169	+	0.935559i	$0.385104\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	12.5498	0.927710
$184$	0	0
$185$	14.9003	1.09549
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	10.5498	0.763359	0.381680	−	0.924295i	$-0.375346\pi$
$191$	10.5498	0.763359	0.381680	+	0.924295i	$0.375346\pi$
$192$	0	0
$193$	−15.0997	−1.08690	−0.543449	−	0.839442i	$-0.682882\pi$
$193$	−15.0997	−1.08690	−0.543449	+	0.839442i	$0.682882\pi$
$194$	0	0
$195$	−10.7251	−0.768039
$196$	0	0
$197$	4.72508	0.336648	0.168324	−	0.985732i	$-0.446164\pi$
$197$	4.72508	0.336648	0.168324	+	0.985732i	$0.446164\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	2.37459	0.165848
$206$	0	0
$207$	5.27492	0.366632
$208$	0	0
$209$	−27.8248	−1.92468
$210$	0	0
$211$	1.45017	0.0998335	0.0499168	−	0.998753i	$-0.484104\pi$
$211$	1.45017	0.0998335	0.0499168	+	0.998753i	$0.484104\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	8.92442	0.608640
$216$	0	0
$217$	26.1993	1.77853
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6.72508	0.450345	0.225172	−	0.974319i	$-0.427705\pi$
$223$	6.72508	0.450345	0.225172	+	0.974319i	$0.427705\pi$
$224$	0	0
$225$	5.72508	0.381672
$226$	0	0
$227$	−13.2749	−0.881087	−0.440544	−	0.897731i	$-0.645214\pi$
$227$	−13.2749	−0.881087	−0.440544	+	0.897731i	$0.645214\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	21.0997	1.38826
$232$	0	0
$233$	7.27492	0.476596	0.238298	−	0.971192i	$-0.423411\pi$
$233$	7.27492	0.476596	0.238298	+	0.971192i	$0.423411\pi$
$234$	0	0
$235$	34.5498	2.25378
$236$	0	0
$237$	6.54983	0.425457
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	29.6495	1.90989	0.954946	−	0.296779i	$-0.0959125\pi$
$241$	29.6495	1.90989	0.954946	+	0.296779i	$0.0959125\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−29.4743	−1.88304
$246$	0	0
$247$	17.2749	1.09918
$248$	0	0
$249$	−6.54983	−0.415079
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	27.8248	1.74933
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.6495	1.59997	0.799986	−	0.600019i	$-0.204840\pi$
$257$	25.6495	1.59997	0.799986	+	0.600019i	$0.204840\pi$
$258$	0	0
$259$	18.1993	1.13085
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	32.7492	2.01177
$266$	0	0
$267$	−12.5498	−0.768037
$268$	0	0
$269$	−8.72508	−0.531978	−0.265989	−	0.963976i	$-0.585698\pi$
$269$	−8.72508	−0.531978	−0.265989	+	0.963976i	$0.585698\pi$
$270$	0	0
$271$	19.8248	1.20427	0.602134	−	0.798395i	$-0.294317\pi$
$271$	19.8248	1.20427	0.602134	+	0.798395i	$0.294317\pi$
$272$	0	0
$273$	−13.0997	−0.792828
$274$	0	0
$275$	30.1993	1.82109
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−6.54983	−0.392128
$280$	0	0
$281$	−24.5498	−1.46452	−0.732260	−	0.681025i	$-0.761534\pi$
$281$	−24.5498	−1.46452	−0.732260	+	0.681025i	$0.761534\pi$
$282$	0	0
$283$	−6.54983	−0.389347	−0.194674	−	0.980868i	$-0.562365\pi$
$283$	−6.54983	−0.389347	−0.194674	+	0.980868i	$0.562365\pi$
$284$	0	0
$285$	−17.2749	−1.02328
$286$	0	0
$287$	2.90033	0.171201
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−8.54983	−0.501200
$292$	0	0
$293$	16.5498	0.966852	0.483426	−	0.875385i	$-0.339392\pi$
$293$	16.5498	0.966852	0.483426	+	0.875385i	$0.339392\pi$
$294$	0	0
$295$	21.4502	1.24888
$296$	0	0
$297$	−5.27492	−0.306082
$298$	0	0
$299$	−17.2749	−0.999034
$300$	0	0
$301$	10.9003	0.628285
$302$	0	0
$303$	12.5498	0.720969
$304$	0	0
$305$	41.0997	2.35336
$306$	0	0
$307$	−30.1993	−1.72357	−0.861784	−	0.507276i	$-0.830652\pi$
$307$	−30.1993	−1.72357	−0.861784	+	0.507276i	$0.830652\pi$
$308$	0	0
$309$	19.8248	1.12779
$310$	0	0
$311$	−9.09967	−0.515995	−0.257997	−	0.966146i	$-0.583063\pi$
$311$	−9.09967	−0.515995	−0.257997	+	0.966146i	$0.583063\pi$
$312$	0	0
$313$	16.5498	0.935452	0.467726	−	0.883874i	$-0.345073\pi$
$313$	16.5498	0.935452	0.467726	+	0.883874i	$0.345073\pi$
$314$	0	0
$315$	13.0997	0.738083
$316$	0	0
$317$	11.0997	0.623420	0.311710	−	0.950177i	$-0.399098\pi$
$317$	11.0997	0.623420	0.311710	+	0.950177i	$0.399098\pi$
$318$	0	0
$319$	−10.5498	−0.590677
$320$	0	0
$321$	15.8248	0.883252
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−18.7492	−1.04002
$326$	0	0
$327$	−16.5498	−0.915208
$328$	0	0
$329$	42.1993	2.32652
$330$	0	0
$331$	−0.175248	−0.00963252	−0.00481626	−	0.999988i	$-0.501533\pi$
$331$	−0.175248	−0.00963252	−0.00481626	+	0.999988i	$0.501533\pi$
$332$	0	0
$333$	−4.54983	−0.249329
$334$	0	0
$335$	−13.0997	−0.715711
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	3.27492	0.177869
$340$	0	0
$341$	−34.5498	−1.87098
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	17.2749	0.930050
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−16.7251	−0.895273	−0.447637	−	0.894216i	$-0.647734\pi$
$349$	−16.7251	−0.895273	−0.447637	+	0.894216i	$0.647734\pi$
$350$	0	0
$351$	3.27492	0.174802
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−13.0997	−0.695258
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.90033	−0.153074	−0.0765368	−	0.997067i	$-0.524386\pi$
$359$	−2.90033	−0.153074	−0.0765368	+	0.997067i	$0.524386\pi$
$360$	0	0
$361$	8.82475	0.464461
$362$	0	0
$363$	−16.8248	−0.883070
$364$	0	0
$365$	32.7492	1.71417
$366$	0	0
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	0	0
$369$	−0.725083	−0.0377463
$370$	0	0
$371$	40.0000	2.07670
$372$	0	0
$373$	35.0997	1.81739	0.908696	−	0.417459i	$-0.137079\pi$
$373$	35.0997	1.81739	0.908696	+	0.417459i	$0.137079\pi$
$374$	0	0
$375$	2.37459	0.122623
$376$	0	0
$377$	6.54983	0.337334
$378$	0	0
$379$	9.09967	0.467419	0.233709	−	0.972307i	$-0.424914\pi$
$379$	9.09967	0.467419	0.233709	+	0.972307i	$0.424914\pi$
$380$	0	0
$381$	−19.8248	−1.01565
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	69.0997	3.52165
$386$	0	0
$387$	−2.72508	−0.138524
$388$	0	0
$389$	11.4502	0.580546	0.290273	−	0.956944i	$-0.406254\pi$
$389$	11.4502	0.580546	0.290273	+	0.956944i	$0.406254\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−10.7251	−0.541009
$394$	0	0
$395$	21.4502	1.07927
$396$	0	0
$397$	19.4502	0.976176	0.488088	−	0.872794i	$-0.337695\pi$
$397$	19.4502	0.976176	0.488088	+	0.872794i	$0.337695\pi$
$398$	0	0
$399$	−21.0997	−1.05630
$400$	0	0
$401$	−13.8248	−0.690375	−0.345188	−	0.938534i	$-0.612185\pi$
$401$	−13.8248	−0.690375	−0.345188	+	0.938534i	$0.612185\pi$
$402$	0	0
$403$	21.4502	1.06851
$404$	0	0
$405$	−3.27492	−0.162732
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	35.2749	1.74423	0.872116	−	0.489299i	$-0.162747\pi$
$409$	35.2749	1.74423	0.872116	+	0.489299i	$0.162747\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	26.1993	1.28919
$414$	0	0
$415$	−21.4502	−1.05295
$416$	0	0
$417$	6.54983	0.320747
$418$	0	0
$419$	−30.1993	−1.47533	−0.737667	−	0.675165i	$-0.764073\pi$
$419$	−30.1993	−1.47533	−0.737667	+	0.675165i	$0.764073\pi$
$420$	0	0
$421$	−11.2749	−0.549506	−0.274753	−	0.961515i	$-0.588596\pi$
$421$	−11.2749	−0.549506	−0.274753	+	0.961515i	$0.588596\pi$
$422$	0	0
$423$	−10.5498	−0.512951
$424$	0	0
$425$	0	0
$426$	0	0
$427$	50.1993	2.42932
$428$	0	0
$429$	17.2749	0.834041
$430$	0	0
$431$	−17.0997	−0.823662	−0.411831	−	0.911260i	$-0.635111\pi$
$431$	−17.0997	−0.823662	−0.411831	+	0.911260i	$0.635111\pi$
$432$	0	0
$433$	8.72508	0.419301	0.209650	−	0.977776i	$-0.432767\pi$
$433$	8.72508	0.419301	0.209650	+	0.977776i	$0.432767\pi$
$434$	0	0
$435$	−6.54983	−0.314041
$436$	0	0
$437$	−27.8248	−1.33104
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	1.45017	0.0688994	0.0344497	−	0.999406i	$-0.489032\pi$
$443$	1.45017	0.0688994	0.0344497	+	0.999406i	$0.489032\pi$
$444$	0	0
$445$	−41.0997	−1.94831
$446$	0	0
$447$	−14.0000	−0.662177
$448$	0	0
$449$	3.09967	0.146282	0.0731412	−	0.997322i	$-0.476698\pi$
$449$	3.09967	0.146282	0.0731412	+	0.997322i	$0.476698\pi$
$450$	0	0
$451$	−3.82475	−0.180101
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−42.9003	−2.01120
$456$	0	0
$457$	−12.7251	−0.595254	−0.297627	−	0.954682i	$-0.596195\pi$
$457$	−12.7251	−0.595254	−0.297627	+	0.954682i	$0.596195\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.09967	−0.330665	−0.165332	−	0.986238i	$-0.552870\pi$
$461$	−7.09967	−0.330665	−0.165332	+	0.986238i	$0.552870\pi$
$462$	0	0
$463$	−26.1993	−1.21759	−0.608793	−	0.793329i	$-0.708346\pi$
$463$	−26.1993	−1.21759	−0.608793	+	0.793329i	$0.708346\pi$
$464$	0	0
$465$	−21.4502	−0.994728
$466$	0	0
$467$	9.45017	0.437302	0.218651	−	0.975803i	$-0.429834\pi$
$467$	9.45017	0.437302	0.218651	+	0.975803i	$0.429834\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	16.3746	0.754501
$472$	0	0
$473$	−14.3746	−0.660944
$474$	0	0
$475$	−30.1993	−1.38564
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	16.1752	0.739066	0.369533	−	0.929218i	$-0.379518\pi$
$479$	16.1752	0.739066	0.369533	+	0.929218i	$0.379518\pi$
$480$	0	0
$481$	14.9003	0.679397
$482$	0	0
$483$	21.0997	0.960068
$484$	0	0
$485$	−28.0000	−1.27141
$486$	0	0
$487$	6.54983	0.296801	0.148401	−	0.988927i	$-0.452587\pi$
$487$	6.54983	0.296801	0.148401	+	0.988927i	$0.452587\pi$
$488$	0	0
$489$	6.54983	0.296194
$490$	0	0
$491$	6.54983	0.295590	0.147795	−	0.989018i	$-0.452782\pi$
$491$	6.54983	0.295590	0.147795	+	0.989018i	$0.452782\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−17.2749	−0.776450
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	6.54983	0.293211	0.146605	−	0.989195i	$-0.453165\pi$
$499$	6.54983	0.293211	0.146605	+	0.989195i	$0.453165\pi$
$500$	0	0
$501$	15.8248	0.706998
$502$	0	0
$503$	2.72508	0.121505	0.0607527	−	0.998153i	$-0.480650\pi$
$503$	2.72508	0.121505	0.0607527	+	0.998153i	$0.480650\pi$
$504$	0	0
$505$	41.0997	1.82891
$506$	0	0
$507$	2.27492	0.101033
$508$	0	0
$509$	−17.6495	−0.782300	−0.391150	−	0.920327i	$-0.627923\pi$
$509$	−17.6495	−0.782300	−0.391150	+	0.920327i	$0.627923\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	0	0
$513$	5.27492	0.232893
$514$	0	0
$515$	64.9244	2.86091
$516$	0	0
$517$	−55.6495	−2.44746
$518$	0	0
$519$	−9.82475	−0.431259
$520$	0	0
$521$	−10.9244	−0.478608	−0.239304	−	0.970945i	$-0.576919\pi$
$521$	−10.9244	−0.478608	−0.239304	+	0.970945i	$0.576919\pi$
$522$	0	0
$523$	22.1993	0.970709	0.485355	−	0.874317i	$-0.338691\pi$
$523$	22.1993	0.970709	0.485355	+	0.874317i	$0.338691\pi$
$524$	0	0
$525$	22.9003	0.999452
$526$	0	0
$527$	0	0
$528$	0	0
$529$	4.82475	0.209772
$530$	0	0
$531$	−6.54983	−0.284239
$532$	0	0
$533$	2.37459	0.102855
$534$	0	0
$535$	51.8248	2.24058
$536$	0	0
$537$	−9.45017	−0.407805
$538$	0	0
$539$	47.4743	2.04486
$540$	0	0
$541$	24.5498	1.05548	0.527740	−	0.849406i	$-0.323040\pi$
$541$	24.5498	1.05548	0.527740	+	0.849406i	$0.323040\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	−54.1993	−2.32164
$546$	0	0
$547$	−33.0997	−1.41524	−0.707620	−	0.706593i	$-0.750231\pi$
$547$	−33.0997	−1.41524	−0.707620	+	0.706593i	$0.750231\pi$
$548$	0	0
$549$	−12.5498	−0.535614
$550$	0	0
$551$	10.5498	0.449438
$552$	0	0
$553$	26.1993	1.11411
$554$	0	0
$555$	−14.9003	−0.632484
$556$	0	0
$557$	−28.1993	−1.19484	−0.597422	−	0.801927i	$-0.703808\pi$
$557$	−28.1993	−1.19484	−0.597422	+	0.801927i	$0.703808\pi$
$558$	0	0
$559$	8.92442	0.377463
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.45017	0.0611172	0.0305586	−	0.999533i	$-0.490271\pi$
$563$	1.45017	0.0611172	0.0305586	+	0.999533i	$0.490271\pi$
$564$	0	0
$565$	10.7251	0.451208
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−8.54983	−0.358428	−0.179214	−	0.983810i	$-0.557355\pi$
$569$	−8.54983	−0.358428	−0.179214	+	0.983810i	$0.557355\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	−10.5498	−0.440726
$574$	0	0
$575$	30.1993	1.25940
$576$	0	0
$577$	32.3746	1.34777	0.673886	−	0.738835i	$-0.264624\pi$
$577$	32.3746	1.34777	0.673886	+	0.738835i	$0.264624\pi$
$578$	0	0
$579$	15.0997	0.627521
$580$	0	0
$581$	−26.1993	−1.08693
$582$	0	0
$583$	−52.7492	−2.18465
$584$	0	0
$585$	10.7251	0.443428
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	34.5498	1.42360
$590$	0	0
$591$	−4.72508	−0.194364
$592$	0	0
$593$	44.1993	1.81505	0.907525	−	0.419999i	$-0.137970\pi$
$593$	44.1993	1.81505	0.907525	+	0.419999i	$0.137970\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.00000	0.163709
$598$	0	0
$599$	−5.45017	−0.222688	−0.111344	−	0.993782i	$-0.535515\pi$
$599$	−5.45017	−0.222688	−0.111344	+	0.993782i	$0.535515\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	−55.0997	−2.24012
$606$	0	0
$607$	−33.4502	−1.35770	−0.678850	−	0.734277i	$-0.737521\pi$
$607$	−33.4502	−1.35770	−0.678850	+	0.734277i	$0.737521\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	34.5498	1.39774
$612$	0	0
$613$	−14.1752	−0.572533	−0.286266	−	0.958150i	$-0.592414\pi$
$613$	−14.1752	−0.572533	−0.286266	+	0.958150i	$0.592414\pi$
$614$	0	0
$615$	−2.37459	−0.0957526
$616$	0	0
$617$	−39.0997	−1.57409	−0.787047	−	0.616893i	$-0.788391\pi$
$617$	−39.0997	−1.57409	−0.787047	+	0.616893i	$0.788391\pi$
$618$	0	0
$619$	−30.5498	−1.22790	−0.613951	−	0.789344i	$-0.710421\pi$
$619$	−30.5498	−1.22790	−0.613951	+	0.789344i	$0.710421\pi$
$620$	0	0
$621$	−5.27492	−0.211675
$622$	0	0
$623$	−50.1993	−2.01119
$624$	0	0
$625$	−20.8488	−0.833954
$626$	0	0
$627$	27.8248	1.11121
$628$	0	0
$629$	0	0
$630$	0	0
$631$	35.8248	1.42616	0.713080	−	0.701082i	$-0.247299\pi$
$631$	35.8248	1.42616	0.713080	+	0.701082i	$0.247299\pi$
$632$	0	0
$633$	−1.45017	−0.0576389
$634$	0	0
$635$	−64.9244	−2.57645
$636$	0	0
$637$	−29.4743	−1.16781
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	4.72508	0.186630	0.0933148	−	0.995637i	$-0.470254\pi$
$641$	4.72508	0.186630	0.0933148	+	0.995637i	$0.470254\pi$
$642$	0	0
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	−8.92442	−0.351399
$646$	0	0
$647$	2.54983	0.100244	0.0501222	−	0.998743i	$-0.484039\pi$
$647$	2.54983	0.100244	0.0501222	+	0.998743i	$0.484039\pi$
$648$	0	0
$649$	−34.5498	−1.35620
$650$	0	0
$651$	−26.1993	−1.02683
$652$	0	0
$653$	30.9244	1.21017	0.605083	−	0.796162i	$-0.293140\pi$
$653$	30.9244	1.21017	0.605083	+	0.796162i	$0.293140\pi$
$654$	0	0
$655$	−35.1238	−1.37240
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−9.45017	−0.368126	−0.184063	−	0.982914i	$-0.558925\pi$
$659$	−9.45017	−0.368126	−0.184063	+	0.982914i	$0.558925\pi$
$660$	0	0
$661$	−29.8248	−1.16005	−0.580024	−	0.814599i	$-0.696957\pi$
$661$	−29.8248	−1.16005	−0.580024	+	0.814599i	$0.696957\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−69.0997	−2.67957
$666$	0	0
$667$	−10.5498	−0.408491
$668$	0	0
$669$	−6.72508	−0.260007
$670$	0	0
$671$	−66.1993	−2.55560
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	0	0
$675$	−5.72508	−0.220359
$676$	0	0
$677$	−29.8248	−1.14626	−0.573129	−	0.819465i	$-0.694271\pi$
$677$	−29.8248	−1.14626	−0.573129	+	0.819465i	$0.694271\pi$
$678$	0	0
$679$	−34.1993	−1.31245
$680$	0	0
$681$	13.2749	0.508696
$682$	0	0
$683$	−47.4743	−1.81655	−0.908276	−	0.418372i	$-0.862601\pi$
$683$	−47.4743	−1.81655	−0.908276	+	0.418372i	$0.862601\pi$
$684$	0	0
$685$	19.6495	0.750769
$686$	0	0
$687$	−22.0000	−0.839352
$688$	0	0
$689$	32.7492	1.24764
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	−21.0997	−0.801510
$694$	0	0
$695$	21.4502	0.813651
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−7.27492	−0.275163
$700$	0	0
$701$	−4.54983	−0.171845	−0.0859224	−	0.996302i	$-0.527384\pi$
$701$	−4.54983	−0.171845	−0.0859224	+	0.996302i	$0.527384\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	−34.5498	−1.30122
$706$	0	0
$707$	50.1993	1.88794
$708$	0	0
$709$	−15.0997	−0.567080	−0.283540	−	0.958960i	$-0.591509\pi$
$709$	−15.0997	−0.567080	−0.283540	+	0.958960i	$0.591509\pi$
$710$	0	0
$711$	−6.54983	−0.245638
$712$	0	0
$713$	−34.5498	−1.29390
$714$	0	0
$715$	56.5739	2.11574
$716$	0	0
$717$	8.00000	0.298765
$718$	0	0
$719$	7.82475	0.291814	0.145907	−	0.989298i	$-0.453390\pi$
$719$	7.82475	0.291814	0.145907	+	0.989298i	$0.453390\pi$
$720$	0	0
$721$	79.2990	2.95325
$722$	0	0
$723$	−29.6495	−1.10268
$724$	0	0
$725$	−11.4502	−0.425248
$726$	0	0
$727$	−10.1993	−0.378272	−0.189136	−	0.981951i	$-0.560569\pi$
$727$	−10.1993	−0.378272	−0.189136	+	0.981951i	$0.560569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	16.9003	0.624228	0.312114	−	0.950045i	$-0.398963\pi$
$733$	16.9003	0.624228	0.312114	+	0.950045i	$0.398963\pi$
$734$	0	0
$735$	29.4743	1.08717
$736$	0	0
$737$	21.0997	0.777216
$738$	0	0
$739$	−31.4743	−1.15780	−0.578900	−	0.815399i	$-0.696518\pi$
$739$	−31.4743	−1.15780	−0.578900	+	0.815399i	$0.696518\pi$
$740$	0	0
$741$	−17.2749	−0.634610
$742$	0	0
$743$	17.0997	0.627326	0.313663	−	0.949534i	$-0.398444\pi$
$743$	17.0997	0.627326	0.313663	+	0.949534i	$0.398444\pi$
$744$	0	0
$745$	−45.8488	−1.67977
$746$	0	0
$747$	6.54983	0.239646
$748$	0	0
$749$	63.2990	2.31290
$750$	0	0
$751$	−4.35050	−0.158752	−0.0793759	−	0.996845i	$-0.525293\pi$
$751$	−4.35050	−0.158752	−0.0793759	+	0.996845i	$0.525293\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10.9244	−0.397055	−0.198527	−	0.980095i	$-0.563616\pi$
$757$	−10.9244	−0.397055	−0.198527	+	0.980095i	$0.563616\pi$
$758$	0	0
$759$	−27.8248	−1.00997
$760$	0	0
$761$	15.0997	0.547363	0.273681	−	0.961820i	$-0.411759\pi$
$761$	15.0997	0.547363	0.273681	+	0.961820i	$0.411759\pi$
$762$	0	0
$763$	−66.1993	−2.39658
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.4502	0.774521
$768$	0	0
$769$	48.3746	1.74443	0.872216	−	0.489121i	$-0.162682\pi$
$769$	48.3746	1.74443	0.872216	+	0.489121i	$0.162682\pi$
$770$	0	0
$771$	−25.6495	−0.923744
$772$	0	0
$773$	32.5498	1.17074	0.585368	−	0.810768i	$-0.300950\pi$
$773$	32.5498	1.17074	0.585368	+	0.810768i	$0.300950\pi$
$774$	0	0
$775$	−37.4983	−1.34698
$776$	0	0
$777$	−18.1993	−0.652898
$778$	0	0
$779$	3.82475	0.137036
$780$	0	0
$781$	21.0997	0.755006
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	53.6254	1.91397
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.0997	0.465771
$792$	0	0
$793$	41.0997	1.45949
$794$	0	0
$795$	−32.7492	−1.16149
$796$	0	0
$797$	−10.3505	−0.366633	−0.183317	−	0.983054i	$-0.558683\pi$
$797$	−10.3505	−0.366633	−0.183317	+	0.983054i	$0.558683\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	12.5498	0.443427
$802$	0	0
$803$	−52.7492	−1.86148
$804$	0	0
$805$	69.0997	2.43544
$806$	0	0
$807$	8.72508	0.307137
$808$	0	0
$809$	47.2749	1.66210	0.831049	−	0.556200i	$-0.187741\pi$
$809$	47.2749	1.66210	0.831049	+	0.556200i	$0.187741\pi$
$810$	0	0
$811$	11.6495	0.409069	0.204535	−	0.978859i	$-0.434432\pi$
$811$	11.6495	0.409069	0.204535	+	0.978859i	$0.434432\pi$
$812$	0	0
$813$	−19.8248	−0.695284
$814$	0	0
$815$	21.4502	0.751367
$816$	0	0
$817$	14.3746	0.502903
$818$	0	0
$819$	13.0997	0.457739
$820$	0	0
$821$	36.7251	1.28171	0.640857	−	0.767660i	$-0.278579\pi$
$821$	36.7251	1.28171	0.640857	+	0.767660i	$0.278579\pi$
$822$	0	0
$823$	25.0997	0.874919	0.437460	−	0.899238i	$-0.355878\pi$
$823$	25.0997	0.874919	0.437460	+	0.899238i	$0.355878\pi$
$824$	0	0
$825$	−30.1993	−1.05141
$826$	0	0
$827$	−3.07558	−0.106948	−0.0534742	−	0.998569i	$-0.517029\pi$
$827$	−3.07558	−0.106948	−0.0534742	+	0.998569i	$0.517029\pi$
$828$	0	0
$829$	40.1993	1.39618	0.698090	−	0.716010i	$-0.254033\pi$
$829$	40.1993	1.39618	0.698090	+	0.716010i	$0.254033\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	0	0
$834$	0	0
$835$	51.8248	1.79347
$836$	0	0
$837$	6.54983	0.226395
$838$	0	0
$839$	36.9244	1.27477	0.637386	−	0.770544i	$-0.280016\pi$
$839$	36.9244	1.27477	0.637386	+	0.770544i	$0.280016\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	24.5498	0.845541
$844$	0	0
$845$	7.45017	0.256293
$846$	0	0
$847$	−67.2990	−2.31242
$848$	0	0
$849$	6.54983	0.224790
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	32.1993	1.10248	0.551242	−	0.834345i	$-0.314154\pi$
$853$	32.1993	1.10248	0.551242	+	0.834345i	$0.314154\pi$
$854$	0	0
$855$	17.2749	0.590790
$856$	0	0
$857$	32.1993	1.09991	0.549954	−	0.835195i	$-0.314645\pi$
$857$	32.1993	1.09991	0.549954	+	0.835195i	$0.314645\pi$
$858$	0	0
$859$	46.1993	1.57630	0.788151	−	0.615483i	$-0.211039\pi$
$859$	46.1993	1.57630	0.788151	+	0.615483i	$0.211039\pi$
$860$	0	0
$861$	−2.90033	−0.0988430
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−32.1752	−1.09399
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−34.5498	−1.17202
$870$	0	0
$871$	−13.0997	−0.443865
$872$	0	0
$873$	8.54983	0.289368
$874$	0	0
$875$	9.49834	0.321103
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	0	0
$879$	−16.5498	−0.558212
$880$	0	0
$881$	27.0997	0.913011	0.456506	−	0.889721i	$-0.349101\pi$
$881$	27.0997	0.913011	0.456506	+	0.889721i	$0.349101\pi$
$882$	0	0
$883$	−16.1752	−0.544340	−0.272170	−	0.962249i	$-0.587741\pi$
$883$	−16.1752	−0.544340	−0.272170	+	0.962249i	$0.587741\pi$
$884$	0	0
$885$	−21.4502	−0.721039
$886$	0	0
$887$	−47.4743	−1.59403	−0.797015	−	0.603960i	$-0.793589\pi$
$887$	−47.4743	−1.59403	−0.797015	+	0.603960i	$0.793589\pi$
$888$	0	0
$889$	−79.2990	−2.65960
$890$	0	0
$891$	5.27492	0.176716
$892$	0	0
$893$	55.6495	1.86224
$894$	0	0
$895$	−30.9485	−1.03449
$896$	0	0
$897$	17.2749	0.576793
$898$	0	0
$899$	13.0997	0.436898
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−10.9003	−0.362740
$904$	0	0
$905$	−45.8488	−1.52407
$906$	0	0
$907$	51.6495	1.71499	0.857497	−	0.514489i	$-0.172018\pi$
$907$	51.6495	1.71499	0.857497	+	0.514489i	$0.172018\pi$
$908$	0	0
$909$	−12.5498	−0.416252
$910$	0	0
$911$	10.3746	0.343725	0.171863	−	0.985121i	$-0.445021\pi$
$911$	10.3746	0.343725	0.171863	+	0.985121i	$0.445021\pi$
$912$	0	0
$913$	34.5498	1.14343
$914$	0	0
$915$	−41.0997	−1.35871
$916$	0	0
$917$	−42.9003	−1.41669
$918$	0	0
$919$	−14.3746	−0.474174	−0.237087	−	0.971488i	$-0.576193\pi$
$919$	−14.3746	−0.474174	−0.237087	+	0.971488i	$0.576193\pi$
$920$	0	0
$921$	30.1993	0.995102
$922$	0	0
$923$	−13.0997	−0.431181
$924$	0	0
$925$	−26.0482	−0.856459
$926$	0	0
$927$	−19.8248	−0.651130
$928$	0	0
$929$	30.9244	1.01460	0.507299	−	0.861770i	$-0.330644\pi$
$929$	30.9244	1.01460	0.507299	+	0.861770i	$0.330644\pi$
$930$	0	0
$931$	−47.4743	−1.55591
$932$	0	0
$933$	9.09967	0.297910
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−48.1993	−1.57460	−0.787302	−	0.616568i	$-0.788523\pi$
$937$	−48.1993	−1.57460	−0.787302	+	0.616568i	$0.788523\pi$
$938$	0	0
$939$	−16.5498	−0.540083
$940$	0	0
$941$	11.0997	0.361839	0.180919	−	0.983498i	$-0.442093\pi$
$941$	11.0997	0.361839	0.180919	+	0.983498i	$0.442093\pi$
$942$	0	0
$943$	−3.82475	−0.124551
$944$	0	0
$945$	−13.0997	−0.426132
$946$	0	0
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	0	0
$949$	32.7492	1.06308
$950$	0	0
$951$	−11.0997	−0.359931
$952$	0	0
$953$	7.45017	0.241335	0.120667	−	0.992693i	$-0.461497\pi$
$953$	7.45017	0.241335	0.120667	+	0.992693i	$0.461497\pi$
$954$	0	0
$955$	−34.5498	−1.11801
$956$	0	0
$957$	10.5498	0.341028
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	11.9003	0.383882
$962$	0	0
$963$	−15.8248	−0.509945
$964$	0	0
$965$	49.4502	1.59186
$966$	0	0
$967$	−14.7251	−0.473527	−0.236763	−	0.971567i	$-0.576087\pi$
$967$	−14.7251	−0.473527	−0.236763	+	0.971567i	$0.576087\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.5498	0.466926	0.233463	−	0.972366i	$-0.424994\pi$
$971$	14.5498	0.466926	0.233463	+	0.972366i	$0.424994\pi$
$972$	0	0
$973$	26.1993	0.839912
$974$	0	0
$975$	18.7492	0.600454
$976$	0	0
$977$	−3.09967	−0.0991672	−0.0495836	−	0.998770i	$-0.515789\pi$
$977$	−3.09967	−0.0991672	−0.0495836	+	0.998770i	$0.515789\pi$
$978$	0	0
$979$	66.1993	2.11574
$980$	0	0
$981$	16.5498	0.528396
$982$	0	0
$983$	4.92442	0.157065	0.0785323	−	0.996912i	$-0.474977\pi$
$983$	4.92442	0.157065	0.0785323	+	0.996912i	$0.474977\pi$
$984$	0	0
$985$	−15.4743	−0.493051
$986$	0	0
$987$	−42.1993	−1.34322
$988$	0	0
$989$	−14.3746	−0.457085
$990$	0	0
$991$	41.0997	1.30557	0.652787	−	0.757542i	$-0.273600\pi$
$991$	41.0997	1.30557	0.652787	+	0.757542i	$0.273600\pi$
$992$	0	0
$993$	0.175248	0.00556134
$994$	0	0
$995$	13.0997	0.415287
$996$	0	0
$997$	−23.0997	−0.731574	−0.365787	−	0.930699i	$-0.619200\pi$
$997$	−23.0997	−0.731574	−0.365787	+	0.930699i	$0.619200\pi$
$998$	0	0
$999$	4.54983	0.143950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6936.2.a.u.1.1		2
17.16	even	2		408.2.a.f.1.2	✓	2
51.50	odd	2		1224.2.a.k.1.1		2
68.67	odd	2		816.2.a.k.1.2		2
136.67	odd	2		3264.2.a.bn.1.1		2
136.101	even	2		3264.2.a.bj.1.1		2
204.203	even	2		2448.2.a.z.1.1		2
408.101	odd	2		9792.2.a.co.1.2		2
408.203	even	2		9792.2.a.cl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.a.f.1.2	✓	2	17.16	even	2
816.2.a.k.1.2		2	68.67	odd	2
1224.2.a.k.1.1		2	51.50	odd	2
2448.2.a.z.1.1		2	204.203	even	2
3264.2.a.bj.1.1		2	136.101	even	2
3264.2.a.bn.1.1		2	136.67	odd	2
6936.2.a.u.1.1		2	1.1	even	1	trivial
9792.2.a.cl.1.2		2	408.203	even	2
9792.2.a.co.1.2		2	408.101	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:	\( N \)	\(=\)	\( 6936 = 2^{3} \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6936.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.27492 q^{5} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.27492 q^{5} -4.00000 q^{7} +1.00000 q^{9} +5.27492 q^{11} -3.27492 q^{13} +3.27492 q^{15} -5.27492 q^{19} +4.00000 q^{21} +5.27492 q^{23} +5.72508 q^{25} -1.00000 q^{27} -2.00000 q^{29} -6.54983 q^{31} -5.27492 q^{33} +13.0997 q^{35} -4.54983 q^{37} +3.27492 q^{39} -0.725083 q^{41} -2.72508 q^{43} -3.27492 q^{45} -10.5498 q^{47} +9.00000 q^{49} -10.0000 q^{53} -17.2749 q^{55} +5.27492 q^{57} -6.54983 q^{59} -12.5498 q^{61} -4.00000 q^{63} +10.7251 q^{65} +4.00000 q^{67} -5.27492 q^{69} +4.00000 q^{71} -10.0000 q^{73} -5.72508 q^{75} -21.0997 q^{77} -6.54983 q^{79} +1.00000 q^{81} +6.54983 q^{83} +2.00000 q^{87} +12.5498 q^{89} +13.0997 q^{91} +6.54983 q^{93} +17.2749 q^{95} +8.54983 q^{97} +5.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} - 8 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + q^5 - 8 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + q^{5} - 8 q^{7} + 2 q^{9} + 3 q^{11} + q^{13} - q^{15} - 3 q^{19} + 8 q^{21} + 3 q^{23} + 19 q^{25} - 2 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} - 4 q^{35} + 6 q^{37} - q^{39} - 9 q^{41} - 13 q^{43} + q^{45} - 6 q^{47} + 18 q^{49} - 20 q^{53} - 27 q^{55} + 3 q^{57} + 2 q^{59} - 10 q^{61} - 8 q^{63} + 29 q^{65} + 8 q^{67} - 3 q^{69} + 8 q^{71} - 20 q^{73} - 19 q^{75} - 12 q^{77} + 2 q^{79} + 2 q^{81} - 2 q^{83} + 4 q^{87} + 10 q^{89} - 4 q^{91} - 2 q^{93} + 27 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + q^5 - 8 * q^7 + 2 * q^9 + 3 * q^11 + q^13 - q^15 - 3 * q^19 + 8 * q^21 + 3 * q^23 + 19 * q^25 - 2 * q^27 - 4 * q^29 + 2 * q^31 - 3 * q^33 - 4 * q^35 + 6 * q^37 - q^39 - 9 * q^41 - 13 * q^43 + q^45 - 6 * q^47 + 18 * q^49 - 20 * q^53 - 27 * q^55 + 3 * q^57 + 2 * q^59 - 10 * q^61 - 8 * q^63 + 29 * q^65 + 8 * q^67 - 3 * q^69 + 8 * q^71 - 20 * q^73 - 19 * q^75 - 12 * q^77 + 2 * q^79 + 2 * q^81 - 2 * q^83 + 4 * q^87 + 10 * q^89 - 4 * q^91 - 2 * q^93 + 27 * q^95 + 2 * q^97 + 3 * q^99

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