LMFDB - Embedded newform 6936.2.a.u.1.2

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			1.2
Root			$4.27492$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.27492 q^{5} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.27492 q^{5} -4.00000 q^{7} +1.00000 q^{9} -2.27492 q^{11} +4.27492 q^{13} -4.27492 q^{15} +2.27492 q^{19} +4.00000 q^{21} -2.27492 q^{23} +13.2749 q^{25} -1.00000 q^{27} -2.00000 q^{29} +8.54983 q^{31} +2.27492 q^{33} -17.0997 q^{35} +10.5498 q^{37} -4.27492 q^{39} -8.27492 q^{41} -10.2749 q^{43} +4.27492 q^{45} +4.54983 q^{47} +9.00000 q^{49} -10.0000 q^{53} -9.72508 q^{55} -2.27492 q^{57} +8.54983 q^{59} +2.54983 q^{61} -4.00000 q^{63} +18.2749 q^{65} +4.00000 q^{67} +2.27492 q^{69} +4.00000 q^{71} -10.0000 q^{73} -13.2749 q^{75} +9.09967 q^{77} +8.54983 q^{79} +1.00000 q^{81} -8.54983 q^{83} +2.00000 q^{87} -2.54983 q^{89} -17.0997 q^{91} -8.54983 q^{93} +9.72508 q^{95} -6.54983 q^{97} -2.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.27492	1.91180	0.955901	−	0.293691i	$-0.0948835\pi$
$5$	4.27492	1.91180	0.955901	+	0.293691i	$0.0948835\pi$
$6$	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
$11$	−2.27492	−0.685913	−0.342957	−	0.939351i	$-0.611428\pi$
$11$	−2.27492	−0.685913	−0.342957	+	0.939351i	$0.611428\pi$
$12$	0	0
$13$	4.27492	1.18565	0.592824	−	0.805332i	$-0.298013\pi$
$13$	4.27492	1.18565	0.592824	+	0.805332i	$0.298013\pi$
$14$	0	0
$15$	−4.27492	−1.10378
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	2.27492	0.521902	0.260951	−	0.965352i	$-0.415964\pi$
$19$	2.27492	0.521902	0.260951	+	0.965352i	$0.415964\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	−2.27492	−0.474353	−0.237177	−	0.971467i	$-0.576222\pi$
$23$	−2.27492	−0.474353	−0.237177	+	0.971467i	$0.576222\pi$
$24$	0	0
$25$	13.2749	2.65498
$26$	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
$31$	8.54983	1.53560	0.767798	−	0.640692i	$-0.221353\pi$
$31$	8.54983	1.53560	0.767798	+	0.640692i	$0.221353\pi$
$32$	0	0
$33$	2.27492	0.396012
$34$	0	0
$35$	−17.0997	−2.89037
$36$	0	0
$37$	10.5498	1.73438	0.867191	−	0.497976i	$-0.165923\pi$
$37$	10.5498	1.73438	0.867191	+	0.497976i	$0.165923\pi$
$38$	0	0
$39$	−4.27492	−0.684535
$40$	0	0
$41$	−8.27492	−1.29232	−0.646162	−	0.763200i	$-0.723627\pi$
$41$	−8.27492	−1.29232	−0.646162	+	0.763200i	$0.723627\pi$
$42$	0	0
$43$	−10.2749	−1.56691	−0.783455	−	0.621448i	$-0.786545\pi$
$43$	−10.2749	−1.56691	−0.783455	+	0.621448i	$0.786545\pi$
$44$	0	0
$45$	4.27492	0.637267
$46$	0	0
$47$	4.54983	0.663662	0.331831	−	0.943339i	$-0.392334\pi$
$47$	4.54983	0.663662	0.331831	+	0.943339i	$0.392334\pi$
$48$	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
$55$	−9.72508	−1.31133
$56$	0	0
$57$	−2.27492	−0.301320
$58$	0	0
$59$	8.54983	1.11309	0.556547	−	0.830816i	$-0.312126\pi$
$59$	8.54983	1.11309	0.556547	+	0.830816i	$0.312126\pi$
$60$	0	0
$61$	2.54983	0.326473	0.163236	−	0.986587i	$-0.447807\pi$
$61$	2.54983	0.326473	0.163236	+	0.986587i	$0.447807\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	18.2749	2.26672
$66$	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
$69$	2.27492	0.273868
$70$	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
$75$	−13.2749	−1.53286
$76$	0	0
$77$	9.09967	1.03700
$78$	0	0
$79$	8.54983	0.961932	0.480966	−	0.876739i	$-0.340286\pi$
$79$	8.54983	0.961932	0.480966	+	0.876739i	$0.340286\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.54983	−0.938466	−0.469233	−	0.883074i	$-0.655470\pi$
$83$	−8.54983	−0.938466	−0.469233	+	0.883074i	$0.655470\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−2.54983	−0.270282	−0.135141	−	0.990826i	$-0.543149\pi$
$89$	−2.54983	−0.270282	−0.135141	+	0.990826i	$0.543149\pi$
$90$	0	0
$91$	−17.0997	−1.79253
$92$	0	0
$93$	−8.54983	−0.886577
$94$	0	0
$95$	9.72508	0.997772
$96$	0	0
$97$	−6.54983	−0.665035	−0.332517	−	0.943097i	$-0.607898\pi$
$97$	−6.54983	−0.665035	−0.332517	+	0.943097i	$0.607898\pi$
$98$	0	0
$99$	−2.27492	−0.228638
$100$	0	0
$101$	2.54983	0.253718	0.126859	−	0.991921i	$-0.459510\pi$
$101$	2.54983	0.253718	0.126859	+	0.991921i	$0.459510\pi$
$102$	0	0
$103$	2.82475	0.278331	0.139166	−	0.990269i	$-0.455558\pi$
$103$	2.82475	0.278331	0.139166	+	0.990269i	$0.455558\pi$
$104$	0	0
$105$	17.0997	1.66876
$106$	0	0
$107$	6.82475	0.659774	0.329887	−	0.944020i	$-0.392989\pi$
$107$	6.82475	0.659774	0.329887	+	0.944020i	$0.392989\pi$
$108$	0	0
$109$	1.45017	0.138901	0.0694503	−	0.997585i	$-0.477875\pi$
$109$	1.45017	0.138901	0.0694503	+	0.997585i	$0.477875\pi$
$110$	0	0
$111$	−10.5498	−1.00135
$112$	0	0
$113$	4.27492	0.402150	0.201075	−	0.979576i	$-0.435556\pi$
$113$	4.27492	0.402150	0.201075	+	0.979576i	$0.435556\pi$
$114$	0	0
$115$	−9.72508	−0.906869
$116$	0	0
$117$	4.27492	0.395216
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.82475	−0.529523
$122$	0	0
$123$	8.27492	0.746124
$124$	0	0
$125$	35.3746	3.16400
$126$	0	0
$127$	−2.82475	−0.250656	−0.125328	−	0.992115i	$-0.539998\pi$
$127$	−2.82475	−0.250656	−0.125328	+	0.992115i	$0.539998\pi$
$128$	0	0
$129$	10.2749	0.904656
$130$	0	0
$131$	18.2749	1.59669	0.798343	−	0.602202i	$-0.205710\pi$
$131$	18.2749	1.59669	0.798343	+	0.602202i	$0.205710\pi$
$132$	0	0
$133$	−9.09967	−0.789041
$134$	0	0
$135$	−4.27492	−0.367926
$136$	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
$139$	8.54983	0.725187	0.362594	−	0.931947i	$-0.381891\pi$
$139$	8.54983	0.725187	0.362594	+	0.931947i	$0.381891\pi$
$140$	0	0
$141$	−4.54983	−0.383165
$142$	0	0
$143$	−9.72508	−0.813252
$144$	0	0
$145$	−8.54983	−0.710025
$146$	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
$155$	36.5498	2.93575
$156$	0	0
$157$	21.3746	1.70588	0.852939	−	0.522011i	$-0.174818\pi$
$157$	21.3746	1.70588	0.852939	+	0.522011i	$0.174818\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	9.09967	0.717154
$162$	0	0
$163$	8.54983	0.669675	0.334837	−	0.942276i	$-0.391319\pi$
$163$	8.54983	0.669675	0.334837	+	0.942276i	$0.391319\pi$
$164$	0	0
$165$	9.72508	0.757097
$166$	0	0
$167$	6.82475	0.528115	0.264058	−	0.964507i	$-0.414939\pi$
$167$	6.82475	0.528115	0.264058	+	0.964507i	$0.414939\pi$
$168$	0	0
$169$	5.27492	0.405763
$170$	0	0
$171$	2.27492	0.173967
$172$	0	0
$173$	−12.8248	−0.975048	−0.487524	−	0.873110i	$-0.662100\pi$
$173$	−12.8248	−0.975048	−0.487524	+	0.873110i	$0.662100\pi$
$174$	0	0
$175$	−53.0997	−4.01396
$176$	0	0
$177$	−8.54983	−0.642645
$178$	0	0
$179$	24.5498	1.83494	0.917470	−	0.397804i	$-0.130228\pi$
$179$	24.5498	1.83494	0.917470	+	0.397804i	$0.130228\pi$
$180$	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
$183$	−2.54983	−0.188489
$184$	0	0
$185$	45.0997	3.31579
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	−4.54983	−0.329214	−0.164607	−	0.986359i	$-0.552636\pi$
$191$	−4.54983	−0.329214	−0.164607	+	0.986359i	$0.552636\pi$
$192$	0	0
$193$	15.0997	1.08690	0.543449	−	0.839442i	$-0.317118\pi$
$193$	15.0997	1.08690	0.543449	+	0.839442i	$0.317118\pi$
$194$	0	0
$195$	−18.2749	−1.30869
$196$	0	0
$197$	12.2749	0.874552	0.437276	−	0.899327i	$-0.355943\pi$
$197$	12.2749	0.874552	0.437276	+	0.899327i	$0.355943\pi$
$198$	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
$205$	−35.3746	−2.47067
$206$	0	0
$207$	−2.27492	−0.158118
$208$	0	0
$209$	−5.17525	−0.357979
$210$	0	0
$211$	16.5498	1.13934	0.569669	−	0.821874i	$-0.307071\pi$
$211$	16.5498	1.13934	0.569669	+	0.821874i	$0.307071\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	−43.9244	−2.99562
$216$	0	0
$217$	−34.1993	−2.32160
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.2749	0.955919	0.477960	−	0.878382i	$-0.341377\pi$
$223$	14.2749	0.955919	0.477960	+	0.878382i	$0.341377\pi$
$224$	0	0
$225$	13.2749	0.884994
$226$	0	0
$227$	−5.72508	−0.379987	−0.189994	−	0.981785i	$-0.560847\pi$
$227$	−5.72508	−0.379987	−0.189994	+	0.981785i	$0.560847\pi$
$228$	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
$231$	−9.09967	−0.598714
$232$	0	0
$233$	−0.274917	−0.0180104	−0.00900521	−	0.999959i	$-0.502866\pi$
$233$	−0.274917	−0.0180104	−0.00900521	+	0.999959i	$0.502866\pi$
$234$	0	0
$235$	19.4502	1.26879
$236$	0	0
$237$	−8.54983	−0.555371
$238$	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
$241$	−15.6495	−1.00807	−0.504037	−	0.863682i	$-0.668152\pi$
$241$	−15.6495	−1.00807	−0.504037	+	0.863682i	$0.668152\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	38.4743	2.45803
$246$	0	0
$247$	9.72508	0.618792
$248$	0	0
$249$	8.54983	0.541824
$250$	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
$253$	5.17525	0.325365
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.6495	−1.22570	−0.612851	−	0.790198i	$-0.709977\pi$
$257$	−19.6495	−1.22570	−0.612851	+	0.790198i	$0.709977\pi$
$258$	0	0
$259$	−42.1993	−2.62214
$260$	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
$265$	−42.7492	−2.62606
$266$	0	0
$267$	2.54983	0.156047
$268$	0	0
$269$	−16.2749	−0.992299	−0.496150	−	0.868237i	$-0.665253\pi$
$269$	−16.2749	−0.992299	−0.496150	+	0.868237i	$0.665253\pi$
$270$	0	0
$271$	−2.82475	−0.171591	−0.0857957	−	0.996313i	$-0.527343\pi$
$271$	−2.82475	−0.171591	−0.0857957	+	0.996313i	$0.527343\pi$
$272$	0	0
$273$	17.0997	1.03492
$274$	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
$279$	8.54983	0.511865
$280$	0	0
$281$	−9.45017	−0.563750	−0.281875	−	0.959451i	$-0.590956\pi$
$281$	−9.45017	−0.563750	−0.281875	+	0.959451i	$0.590956\pi$
$282$	0	0
$283$	8.54983	0.508235	0.254117	−	0.967173i	$-0.418215\pi$
$283$	8.54983	0.508235	0.254117	+	0.967173i	$0.418215\pi$
$284$	0	0
$285$	−9.72508	−0.576064
$286$	0	0
$287$	33.0997	1.95381
$288$	0	0
$289$	0	0
$290$	0	0
$291$	6.54983	0.383958
$292$	0	0
$293$	1.45017	0.0847196	0.0423598	−	0.999102i	$-0.486512\pi$
$293$	1.45017	0.0847196	0.0423598	+	0.999102i	$0.486512\pi$
$294$	0	0
$295$	36.5498	2.12801
$296$	0	0
$297$	2.27492	0.132004
$298$	0	0
$299$	−9.72508	−0.562416
$300$	0	0
$301$	41.0997	2.36895
$302$	0	0
$303$	−2.54983	−0.146484
$304$	0	0
$305$	10.9003	0.624151
$306$	0	0
$307$	30.1993	1.72357	0.861784	−	0.507276i	$-0.169348\pi$
$307$	30.1993	1.72357	0.861784	+	0.507276i	$0.169348\pi$
$308$	0	0
$309$	−2.82475	−0.160695
$310$	0	0
$311$	21.0997	1.19645	0.598226	−	0.801327i	$-0.295872\pi$
$311$	21.0997	1.19645	0.598226	+	0.801327i	$0.295872\pi$
$312$	0	0
$313$	1.45017	0.0819682	0.0409841	−	0.999160i	$-0.486951\pi$
$313$	1.45017	0.0819682	0.0409841	+	0.999160i	$0.486951\pi$
$314$	0	0
$315$	−17.0997	−0.963457
$316$	0	0
$317$	−19.0997	−1.07274	−0.536372	−	0.843982i	$-0.680206\pi$
$317$	−19.0997	−1.07274	−0.536372	+	0.843982i	$0.680206\pi$
$318$	0	0
$319$	4.54983	0.254742
$320$	0	0
$321$	−6.82475	−0.380920
$322$	0	0
$323$	0	0
$324$	0	0
$325$	56.7492	3.14788
$326$	0	0
$327$	−1.45017	−0.0801943
$328$	0	0
$329$	−18.1993	−1.00336
$330$	0	0
$331$	−22.8248	−1.25456	−0.627281	−	0.778793i	$-0.715832\pi$
$331$	−22.8248	−1.25456	−0.627281	+	0.778793i	$0.715832\pi$
$332$	0	0
$333$	10.5498	0.578127
$334$	0	0
$335$	17.0997	0.934255
$336$	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
$339$	−4.27492	−0.232182
$340$	0	0
$341$	−19.4502	−1.05329
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	9.72508	0.523581
$346$	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
$349$	−24.2749	−1.29941	−0.649703	−	0.760188i	$-0.725107\pi$
$349$	−24.2749	−1.29941	−0.649703	+	0.760188i	$0.725107\pi$
$350$	0	0
$351$	−4.27492	−0.228178
$352$	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
$355$	17.0997	0.907556
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.0997	−1.74693	−0.873467	−	0.486884i	$-0.838134\pi$
$359$	−33.0997	−1.74693	−0.873467	+	0.486884i	$0.838134\pi$
$360$	0	0
$361$	−13.8248	−0.727619
$362$	0	0
$363$	5.82475	0.305720
$364$	0	0
$365$	−42.7492	−2.23759
$366$	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
$369$	−8.27492	−0.430775
$370$	0	0
$371$	40.0000	2.07670
$372$	0	0
$373$	4.90033	0.253730	0.126865	−	0.991920i	$-0.459509\pi$
$373$	4.90033	0.253730	0.126865	+	0.991920i	$0.459509\pi$
$374$	0	0
$375$	−35.3746	−1.82674
$376$	0	0
$377$	−8.54983	−0.440339
$378$	0	0
$379$	−21.0997	−1.08382	−0.541909	−	0.840437i	$-0.682298\pi$
$379$	−21.0997	−1.08382	−0.541909	+	0.840437i	$0.682298\pi$
$380$	0	0
$381$	2.82475	0.144716
$382$	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
$385$	38.9003	1.98254
$386$	0	0
$387$	−10.2749	−0.522303
$388$	0	0
$389$	26.5498	1.34613	0.673065	−	0.739583i	$-0.264977\pi$
$389$	26.5498	1.34613	0.673065	+	0.739583i	$0.264977\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−18.2749	−0.921848
$394$	0	0
$395$	36.5498	1.83902
$396$	0	0
$397$	34.5498	1.73401	0.867003	−	0.498302i	$-0.166043\pi$
$397$	34.5498	1.73401	0.867003	+	0.498302i	$0.166043\pi$
$398$	0	0
$399$	9.09967	0.455553
$400$	0	0
$401$	8.82475	0.440687	0.220344	−	0.975422i	$-0.429282\pi$
$401$	8.82475	0.440687	0.220344	+	0.975422i	$0.429282\pi$
$402$	0	0
$403$	36.5498	1.82068
$404$	0	0
$405$	4.27492	0.212422
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	27.7251	1.37092	0.685459	−	0.728112i	$-0.259602\pi$
$409$	27.7251	1.37092	0.685459	+	0.728112i	$0.259602\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	−34.1993	−1.68284
$414$	0	0
$415$	−36.5498	−1.79416
$416$	0	0
$417$	−8.54983	−0.418687
$418$	0	0
$419$	30.1993	1.47533	0.737667	−	0.675165i	$-0.235927\pi$
$419$	30.1993	1.47533	0.737667	+	0.675165i	$0.235927\pi$
$420$	0	0
$421$	−3.72508	−0.181549	−0.0907747	−	0.995871i	$-0.528934\pi$
$421$	−3.72508	−0.181549	−0.0907747	+	0.995871i	$0.528934\pi$
$422$	0	0
$423$	4.54983	0.221221
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.1993	−0.493581
$428$	0	0
$429$	9.72508	0.469531
$430$	0	0
$431$	13.0997	0.630989	0.315494	−	0.948927i	$-0.397830\pi$
$431$	13.0997	0.630989	0.315494	+	0.948927i	$0.397830\pi$
$432$	0	0
$433$	16.2749	0.782123	0.391061	−	0.920365i	$-0.372108\pi$
$433$	16.2749	0.782123	0.391061	+	0.920365i	$0.372108\pi$
$434$	0	0
$435$	8.54983	0.409933
$436$	0	0
$437$	−5.17525	−0.247566
$438$	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
$443$	16.5498	0.786306	0.393153	−	0.919473i	$-0.371384\pi$
$443$	16.5498	0.786306	0.393153	+	0.919473i	$0.371384\pi$
$444$	0	0
$445$	−10.9003	−0.516725
$446$	0	0
$447$	−14.0000	−0.662177
$448$	0	0
$449$	−27.0997	−1.27891	−0.639456	−	0.768828i	$-0.720840\pi$
$449$	−27.0997	−1.27891	−0.639456	+	0.768828i	$0.720840\pi$
$450$	0	0
$451$	18.8248	0.886423
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−73.0997	−3.42697
$456$	0	0
$457$	−20.2749	−0.948421	−0.474210	−	0.880412i	$-0.657266\pi$
$457$	−20.2749	−0.948421	−0.474210	+	0.880412i	$0.657266\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.0997	1.07586	0.537929	−	0.842990i	$-0.319207\pi$
$461$	23.0997	1.07586	0.537929	+	0.842990i	$0.319207\pi$
$462$	0	0
$463$	34.1993	1.58938	0.794689	−	0.607017i	$-0.207634\pi$
$463$	34.1993	1.58938	0.794689	+	0.607017i	$0.207634\pi$
$464$	0	0
$465$	−36.5498	−1.69496
$466$	0	0
$467$	24.5498	1.13603	0.568015	−	0.823018i	$-0.307711\pi$
$467$	24.5498	1.13603	0.568015	+	0.823018i	$0.307711\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−21.3746	−0.984889
$472$	0	0
$473$	23.3746	1.07476
$474$	0	0
$475$	30.1993	1.38564
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	38.8248	1.77395	0.886974	−	0.461819i	$-0.152803\pi$
$479$	38.8248	1.77395	0.886974	+	0.461819i	$0.152803\pi$
$480$	0	0
$481$	45.0997	2.05637
$482$	0	0
$483$	−9.09967	−0.414049
$484$	0	0
$485$	−28.0000	−1.27141
$486$	0	0
$487$	−8.54983	−0.387430	−0.193715	−	0.981058i	$-0.562054\pi$
$487$	−8.54983	−0.387430	−0.193715	+	0.981058i	$0.562054\pi$
$488$	0	0
$489$	−8.54983	−0.386637
$490$	0	0
$491$	−8.54983	−0.385849	−0.192924	−	0.981214i	$-0.561797\pi$
$491$	−8.54983	−0.385849	−0.192924	+	0.981214i	$0.561797\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−9.72508	−0.437110
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−8.54983	−0.382743	−0.191372	−	0.981518i	$-0.561294\pi$
$499$	−8.54983	−0.382743	−0.191372	+	0.981518i	$0.561294\pi$
$500$	0	0
$501$	−6.82475	−0.304907
$502$	0	0
$503$	10.2749	0.458136	0.229068	−	0.973410i	$-0.426432\pi$
$503$	10.2749	0.458136	0.229068	+	0.973410i	$0.426432\pi$
$504$	0	0
$505$	10.9003	0.485058
$506$	0	0
$507$	−5.27492	−0.234267
$508$	0	0
$509$	27.6495	1.22554	0.612771	−	0.790260i	$-0.290055\pi$
$509$	27.6495	1.22554	0.612771	+	0.790260i	$0.290055\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	0	0
$513$	−2.27492	−0.100440
$514$	0	0
$515$	12.0756	0.532114
$516$	0	0
$517$	−10.3505	−0.455214
$518$	0	0
$519$	12.8248	0.562944
$520$	0	0
$521$	41.9244	1.83674	0.918371	−	0.395720i	$-0.129505\pi$
$521$	41.9244	1.83674	0.918371	+	0.395720i	$0.129505\pi$
$522$	0	0
$523$	−38.1993	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$523$	−38.1993	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$524$	0	0
$525$	53.0997	2.31746
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−17.8248	−0.774989
$530$	0	0
$531$	8.54983	0.371031
$532$	0	0
$533$	−35.3746	−1.53224
$534$	0	0
$535$	29.1752	1.26136
$536$	0	0
$537$	−24.5498	−1.05940
$538$	0	0
$539$	−20.4743	−0.881889
$540$	0	0
$541$	9.45017	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$541$	9.45017	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	6.19934	0.265550
$546$	0	0
$547$	−2.90033	−0.124009	−0.0620046	−	0.998076i	$-0.519749\pi$
$547$	−2.90033	−0.124009	−0.0620046	+	0.998076i	$0.519749\pi$
$548$	0	0
$549$	2.54983	0.108824
$550$	0	0
$551$	−4.54983	−0.193829
$552$	0	0
$553$	−34.1993	−1.45430
$554$	0	0
$555$	−45.0997	−1.91437
$556$	0	0
$557$	32.1993	1.36433	0.682165	−	0.731198i	$-0.261039\pi$
$557$	32.1993	1.36433	0.682165	+	0.731198i	$0.261039\pi$
$558$	0	0
$559$	−43.9244	−1.85781
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.5498	0.697492	0.348746	−	0.937217i	$-0.386608\pi$
$563$	16.5498	0.697492	0.348746	+	0.937217i	$0.386608\pi$
$564$	0	0
$565$	18.2749	0.768832
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	6.54983	0.274583	0.137292	−	0.990531i	$-0.456160\pi$
$569$	6.54983	0.274583	0.137292	+	0.990531i	$0.456160\pi$
$570$	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
$573$	4.54983	0.190072
$574$	0	0
$575$	−30.1993	−1.25940
$576$	0	0
$577$	−5.37459	−0.223747	−0.111873	−	0.993722i	$-0.535685\pi$
$577$	−5.37459	−0.223747	−0.111873	+	0.993722i	$0.535685\pi$
$578$	0	0
$579$	−15.0997	−0.627521
$580$	0	0
$581$	34.1993	1.41883
$582$	0	0
$583$	22.7492	0.942174
$584$	0	0
$585$	18.2749	0.755575
$586$	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
$589$	19.4502	0.801430
$590$	0	0
$591$	−12.2749	−0.504923
$592$	0	0
$593$	−16.1993	−0.665227	−0.332614	−	0.943063i	$-0.607930\pi$
$593$	−16.1993	−0.665227	−0.332614	+	0.943063i	$0.607930\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.00000	0.163709
$598$	0	0
$599$	−20.5498	−0.839643	−0.419822	−	0.907607i	$-0.637907\pi$
$599$	−20.5498	−0.839643	−0.419822	+	0.907607i	$0.637907\pi$
$600$	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
$605$	−24.9003	−1.01234
$606$	0	0
$607$	−48.5498	−1.97058	−0.985288	−	0.170899i	$-0.945333\pi$
$607$	−48.5498	−1.97058	−0.985288	+	0.170899i	$0.945333\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	19.4502	0.786869
$612$	0	0
$613$	−36.8248	−1.48734	−0.743669	−	0.668548i	$-0.766916\pi$
$613$	−36.8248	−1.48734	−0.743669	+	0.668548i	$0.766916\pi$
$614$	0	0
$615$	35.3746	1.42644
$616$	0	0
$617$	−8.90033	−0.358314	−0.179157	−	0.983821i	$-0.557337\pi$
$617$	−8.90033	−0.358314	−0.179157	+	0.983821i	$0.557337\pi$
$618$	0	0
$619$	−15.4502	−0.620995	−0.310497	−	0.950574i	$-0.600496\pi$
$619$	−15.4502	−0.620995	−0.310497	+	0.950574i	$0.600496\pi$
$620$	0	0
$621$	2.27492	0.0912893
$622$	0	0
$623$	10.1993	0.408628
$624$	0	0
$625$	84.8488	3.39395
$626$	0	0
$627$	5.17525	0.206680
$628$	0	0
$629$	0	0
$630$	0	0
$631$	13.1752	0.524498	0.262249	−	0.965000i	$-0.415536\pi$
$631$	13.1752	0.524498	0.262249	+	0.965000i	$0.415536\pi$
$632$	0	0
$633$	−16.5498	−0.657797
$634$	0	0
$635$	−12.0756	−0.479205
$636$	0	0
$637$	38.4743	1.52441
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	12.2749	0.484830	0.242415	−	0.970173i	$-0.422060\pi$
$641$	12.2749	0.484830	0.242415	+	0.970173i	$0.422060\pi$
$642$	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
$645$	43.9244	1.72952
$646$	0	0
$647$	−12.5498	−0.493385	−0.246692	−	0.969094i	$-0.579344\pi$
$647$	−12.5498	−0.493385	−0.246692	+	0.969094i	$0.579344\pi$
$648$	0	0
$649$	−19.4502	−0.763486
$650$	0	0
$651$	34.1993	1.34038
$652$	0	0
$653$	−21.9244	−0.857969	−0.428984	−	0.903312i	$-0.641128\pi$
$653$	−21.9244	−0.857969	−0.428984	+	0.903312i	$0.641128\pi$
$654$	0	0
$655$	78.1238	3.05255
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−24.5498	−0.956326	−0.478163	−	0.878271i	$-0.658697\pi$
$659$	−24.5498	−0.956326	−0.478163	+	0.878271i	$0.658697\pi$
$660$	0	0
$661$	−7.17525	−0.279085	−0.139542	−	0.990216i	$-0.544563\pi$
$661$	−7.17525	−0.279085	−0.139542	+	0.990216i	$0.544563\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−38.9003	−1.50849
$666$	0	0
$667$	4.54983	0.176170
$668$	0	0
$669$	−14.2749	−0.551900
$670$	0	0
$671$	−5.80066	−0.223932
$672$	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
$675$	−13.2749	−0.510952
$676$	0	0
$677$	−7.17525	−0.275767	−0.137884	−	0.990448i	$-0.544030\pi$
$677$	−7.17525	−0.275767	−0.137884	+	0.990448i	$0.544030\pi$
$678$	0	0
$679$	26.1993	1.00544
$680$	0	0
$681$	5.72508	0.219386
$682$	0	0
$683$	20.4743	0.783426	0.391713	−	0.920088i	$-0.371883\pi$
$683$	20.4743	0.783426	0.391713	+	0.920088i	$0.371883\pi$
$684$	0	0
$685$	−25.6495	−0.980017
$686$	0	0
$687$	−22.0000	−0.839352
$688$	0	0
$689$	−42.7492	−1.62861
$690$	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
$693$	9.09967	0.345668
$694$	0	0
$695$	36.5498	1.38641
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0.274917	0.0103983
$700$	0	0
$701$	10.5498	0.398462	0.199231	−	0.979953i	$-0.436156\pi$
$701$	10.5498	0.398462	0.199231	+	0.979953i	$0.436156\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	−19.4502	−0.732536
$706$	0	0
$707$	−10.1993	−0.383586
$708$	0	0
$709$	15.0997	0.567080	0.283540	−	0.958960i	$-0.408491\pi$
$709$	15.0997	0.567080	0.283540	+	0.958960i	$0.408491\pi$
$710$	0	0
$711$	8.54983	0.320644
$712$	0	0
$713$	−19.4502	−0.728414
$714$	0	0
$715$	−41.5739	−1.55478
$716$	0	0
$717$	8.00000	0.298765
$718$	0	0
$719$	−14.8248	−0.552870	−0.276435	−	0.961033i	$-0.589153\pi$
$719$	−14.8248	−0.552870	−0.276435	+	0.961033i	$0.589153\pi$
$720$	0	0
$721$	−11.2990	−0.420797
$722$	0	0
$723$	15.6495	0.582011
$724$	0	0
$725$	−26.5498	−0.986036
$726$	0	0
$727$	50.1993	1.86179	0.930895	−	0.365286i	$-0.119029\pi$
$727$	50.1993	1.86179	0.930895	+	0.365286i	$0.119029\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	47.0997	1.73967	0.869833	−	0.493346i	$-0.164226\pi$
$733$	47.0997	1.73967	0.869833	+	0.493346i	$0.164226\pi$
$734$	0	0
$735$	−38.4743	−1.41914
$736$	0	0
$737$	−9.09967	−0.335191
$738$	0	0
$739$	36.4743	1.34173	0.670864	−	0.741581i	$-0.265924\pi$
$739$	36.4743	1.34173	0.670864	+	0.741581i	$0.265924\pi$
$740$	0	0
$741$	−9.72508	−0.357260
$742$	0	0
$743$	−13.0997	−0.480580	−0.240290	−	0.970701i	$-0.577243\pi$
$743$	−13.0997	−0.480580	−0.240290	+	0.970701i	$0.577243\pi$
$744$	0	0
$745$	59.8488	2.19269
$746$	0	0
$747$	−8.54983	−0.312822
$748$	0	0
$749$	−27.2990	−0.997484
$750$	0	0
$751$	−49.6495	−1.81174	−0.905868	−	0.423560i	$-0.860780\pi$
$751$	−49.6495	−1.81174	−0.905868	+	0.423560i	$0.860780\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.9244	1.52377	0.761884	−	0.647713i	$-0.224274\pi$
$757$	41.9244	1.52377	0.761884	+	0.647713i	$0.224274\pi$
$758$	0	0
$759$	−5.17525	−0.187850
$760$	0	0
$761$	−15.0997	−0.547363	−0.273681	−	0.961820i	$-0.588241\pi$
$761$	−15.0997	−0.547363	−0.273681	+	0.961820i	$0.588241\pi$
$762$	0	0
$763$	−5.80066	−0.209998
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.5498	1.31974
$768$	0	0
$769$	10.6254	0.383162	0.191581	−	0.981477i	$-0.438638\pi$
$769$	10.6254	0.383162	0.191581	+	0.981477i	$0.438638\pi$
$770$	0	0
$771$	19.6495	0.707660
$772$	0	0
$773$	17.4502	0.627639	0.313819	−	0.949483i	$-0.398391\pi$
$773$	17.4502	0.627639	0.313819	+	0.949483i	$0.398391\pi$
$774$	0	0
$775$	113.498	4.07698
$776$	0	0
$777$	42.1993	1.51389
$778$	0	0
$779$	−18.8248	−0.674467
$780$	0	0
$781$	−9.09967	−0.325612
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	91.3746	3.26130
$786$	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
$791$	−17.0997	−0.607994
$792$	0	0
$793$	10.9003	0.387082
$794$	0	0
$795$	42.7492	1.51616
$796$	0	0
$797$	−55.6495	−1.97121	−0.985603	−	0.169075i	$-0.945922\pi$
$797$	−55.6495	−1.97121	−0.985603	+	0.169075i	$0.945922\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2.54983	−0.0900940
$802$	0	0
$803$	22.7492	0.802801
$804$	0	0
$805$	38.9003	1.37106
$806$	0	0
$807$	16.2749	0.572904
$808$	0	0
$809$	39.7251	1.39666	0.698330	−	0.715776i	$-0.253927\pi$
$809$	39.7251	1.39666	0.698330	+	0.715776i	$0.253927\pi$
$810$	0	0
$811$	−33.6495	−1.18159	−0.590797	−	0.806820i	$-0.701187\pi$
$811$	−33.6495	−1.18159	−0.590797	+	0.806820i	$0.701187\pi$
$812$	0	0
$813$	2.82475	0.0990684
$814$	0	0
$815$	36.5498	1.28028
$816$	0	0
$817$	−23.3746	−0.817773
$818$	0	0
$819$	−17.0997	−0.597511
$820$	0	0
$821$	44.2749	1.54521	0.772603	−	0.634890i	$-0.218955\pi$
$821$	44.2749	1.54521	0.772603	+	0.634890i	$0.218955\pi$
$822$	0	0
$823$	−5.09967	−0.177763	−0.0888816	−	0.996042i	$-0.528329\pi$
$823$	−5.09967	−0.177763	−0.0888816	+	0.996042i	$0.528329\pi$
$824$	0	0
$825$	30.1993	1.05141
$826$	0	0
$827$	−55.9244	−1.94468	−0.972341	−	0.233564i	$-0.924961\pi$
$827$	−55.9244	−1.94468	−0.972341	+	0.233564i	$0.924961\pi$
$828$	0	0
$829$	−20.1993	−0.701552	−0.350776	−	0.936459i	$-0.614082\pi$
$829$	−20.1993	−0.701552	−0.350776	+	0.936459i	$0.614082\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	0	0
$834$	0	0
$835$	29.1752	1.00965
$836$	0	0
$837$	−8.54983	−0.295526
$838$	0	0
$839$	−15.9244	−0.549772	−0.274886	−	0.961477i	$-0.588640\pi$
$839$	−15.9244	−0.549772	−0.274886	+	0.961477i	$0.588640\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	9.45017	0.325481
$844$	0	0
$845$	22.5498	0.775738
$846$	0	0
$847$	23.2990	0.800563
$848$	0	0
$849$	−8.54983	−0.293430
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	−28.1993	−0.965527	−0.482763	−	0.875751i	$-0.660367\pi$
$853$	−28.1993	−0.965527	−0.482763	+	0.875751i	$0.660367\pi$
$854$	0	0
$855$	9.72508	0.332591
$856$	0	0
$857$	−28.1993	−0.963271	−0.481636	−	0.876372i	$-0.659957\pi$
$857$	−28.1993	−0.963271	−0.481636	+	0.876372i	$0.659957\pi$
$858$	0	0
$859$	−14.1993	−0.484475	−0.242238	−	0.970217i	$-0.577881\pi$
$859$	−14.1993	−0.484475	−0.242238	+	0.970217i	$0.577881\pi$
$860$	0	0
$861$	−33.0997	−1.12803
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−54.8248	−1.86410
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−19.4502	−0.659802
$870$	0	0
$871$	17.0997	0.579400
$872$	0	0
$873$	−6.54983	−0.221678
$874$	0	0
$875$	−141.498	−4.78352
$876$	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
$879$	−1.45017	−0.0489129
$880$	0	0
$881$	−3.09967	−0.104430	−0.0522152	−	0.998636i	$-0.516628\pi$
$881$	−3.09967	−0.104430	−0.0522152	+	0.998636i	$0.516628\pi$
$882$	0	0
$883$	−38.8248	−1.30656	−0.653278	−	0.757118i	$-0.726607\pi$
$883$	−38.8248	−1.30656	−0.653278	+	0.757118i	$0.726607\pi$
$884$	0	0
$885$	−36.5498	−1.22861
$886$	0	0
$887$	20.4743	0.687458	0.343729	−	0.939069i	$-0.388310\pi$
$887$	20.4743	0.687458	0.343729	+	0.939069i	$0.388310\pi$
$888$	0	0
$889$	11.2990	0.378957
$890$	0	0
$891$	−2.27492	−0.0762126
$892$	0	0
$893$	10.3505	0.346366
$894$	0	0
$895$	104.949	3.50804
$896$	0	0
$897$	9.72508	0.324711
$898$	0	0
$899$	−17.0997	−0.570306
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−41.0997	−1.36771
$904$	0	0
$905$	59.8488	1.98944
$906$	0	0
$907$	6.35050	0.210865	0.105432	−	0.994426i	$-0.466377\pi$
$907$	6.35050	0.210865	0.105432	+	0.994426i	$0.466377\pi$
$908$	0	0
$909$	2.54983	0.0845727
$910$	0	0
$911$	−27.3746	−0.906960	−0.453480	−	0.891266i	$-0.649818\pi$
$911$	−27.3746	−0.906960	−0.453480	+	0.891266i	$0.649818\pi$
$912$	0	0
$913$	19.4502	0.643707
$914$	0	0
$915$	−10.9003	−0.360354
$916$	0	0
$917$	−73.0997	−2.41396
$918$	0	0
$919$	23.3746	0.771056	0.385528	−	0.922696i	$-0.374019\pi$
$919$	23.3746	0.771056	0.385528	+	0.922696i	$0.374019\pi$
$920$	0	0
$921$	−30.1993	−0.995102
$922$	0	0
$923$	17.0997	0.562842
$924$	0	0
$925$	140.048	4.60476
$926$	0	0
$927$	2.82475	0.0927770
$928$	0	0
$929$	−21.9244	−0.719317	−0.359658	−	0.933084i	$-0.617107\pi$
$929$	−21.9244	−0.719317	−0.359658	+	0.933084i	$0.617107\pi$
$930$	0	0
$931$	20.4743	0.671017
$932$	0	0
$933$	−21.0997	−0.690772
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.1993	0.398535	0.199267	−	0.979945i	$-0.436144\pi$
$937$	12.1993	0.398535	0.199267	+	0.979945i	$0.436144\pi$
$938$	0	0
$939$	−1.45017	−0.0473244
$940$	0	0
$941$	−19.0997	−0.622631	−0.311316	−	0.950307i	$-0.600770\pi$
$941$	−19.0997	−0.622631	−0.311316	+	0.950307i	$0.600770\pi$
$942$	0	0
$943$	18.8248	0.613018
$944$	0	0
$945$	17.0997	0.556252
$946$	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
$949$	−42.7492	−1.38770
$950$	0	0
$951$	19.0997	0.619349
$952$	0	0
$953$	22.5498	0.730461	0.365230	−	0.930917i	$-0.380990\pi$
$953$	22.5498	0.730461	0.365230	+	0.930917i	$0.380990\pi$
$954$	0	0
$955$	−19.4502	−0.629393
$956$	0	0
$957$	−4.54983	−0.147075
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	42.0997	1.35805
$962$	0	0
$963$	6.82475	0.219925
$964$	0	0
$965$	64.5498	2.07793
$966$	0	0
$967$	−22.2749	−0.716313	−0.358157	−	0.933662i	$-0.616595\pi$
$967$	−22.2749	−0.716313	−0.358157	+	0.933662i	$0.616595\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−0.549834	−0.0176450	−0.00882251	−	0.999961i	$-0.502808\pi$
$971$	−0.549834	−0.0176450	−0.00882251	+	0.999961i	$0.502808\pi$
$972$	0	0
$973$	−34.1993	−1.09638
$974$	0	0
$975$	−56.7492	−1.81743
$976$	0	0
$977$	27.0997	0.866995	0.433498	−	0.901155i	$-0.357279\pi$
$977$	27.0997	0.866995	0.433498	+	0.901155i	$0.357279\pi$
$978$	0	0
$979$	5.80066	0.185390
$980$	0	0
$981$	1.45017	0.0463002
$982$	0	0
$983$	−47.9244	−1.52855	−0.764276	−	0.644890i	$-0.776903\pi$
$983$	−47.9244	−1.52855	−0.764276	+	0.644890i	$0.776903\pi$
$984$	0	0
$985$	52.4743	1.67197
$986$	0	0
$987$	18.1993	0.579291
$988$	0	0
$989$	23.3746	0.743269
$990$	0	0
$991$	10.9003	0.346260	0.173130	−	0.984899i	$-0.444612\pi$
$991$	10.9003	0.346260	0.173130	+	0.984899i	$0.444612\pi$
$992$	0	0
$993$	22.8248	0.724322
$994$	0	0
$995$	−17.0997	−0.542096
$996$	0	0
$997$	7.09967	0.224849	0.112424	−	0.993660i	$-0.464138\pi$
$997$	7.09967	0.224849	0.112424	+	0.993660i	$0.464138\pi$
$998$	0	0
$999$	−10.5498	−0.333782

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.27492 q^{5} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.27492 q^{5} -4.00000 q^{7} +1.00000 q^{9} -2.27492 q^{11} +4.27492 q^{13} -4.27492 q^{15} +2.27492 q^{19} +4.00000 q^{21} -2.27492 q^{23} +13.2749 q^{25} -1.00000 q^{27} -2.00000 q^{29} +8.54983 q^{31} +2.27492 q^{33} -17.0997 q^{35} +10.5498 q^{37} -4.27492 q^{39} -8.27492 q^{41} -10.2749 q^{43} +4.27492 q^{45} +4.54983 q^{47} +9.00000 q^{49} -10.0000 q^{53} -9.72508 q^{55} -2.27492 q^{57} +8.54983 q^{59} +2.54983 q^{61} -4.00000 q^{63} +18.2749 q^{65} +4.00000 q^{67} +2.27492 q^{69} +4.00000 q^{71} -10.0000 q^{73} -13.2749 q^{75} +9.09967 q^{77} +8.54983 q^{79} +1.00000 q^{81} -8.54983 q^{83} +2.00000 q^{87} -2.54983 q^{89} -17.0997 q^{91} -8.54983 q^{93} +9.72508 q^{95} -6.54983 q^{97} -2.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} - 8 q^{7} + 2 q^{9}+O(q^{10}) \) 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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