LMFDB - Embedded newform 1320.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1320,2,Mod(1,1320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1320.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:	$10.5402530668$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +3.12311 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +3.12311 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.12311 q^{13} +1.00000 q^{15} -5.12311 q^{17} -3.12311 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -1.00000 q^{33} -3.12311 q^{35} +6.00000 q^{37} -5.12311 q^{39} +6.00000 q^{41} -3.12311 q^{43} -1.00000 q^{45} +4.00000 q^{47} +2.75379 q^{49} +5.12311 q^{51} +8.24621 q^{53} -1.00000 q^{55} +4.00000 q^{59} +10.0000 q^{61} +3.12311 q^{63} -5.12311 q^{65} +10.2462 q^{67} +4.00000 q^{69} +10.2462 q^{71} +13.1231 q^{73} -1.00000 q^{75} +3.12311 q^{77} -14.2462 q^{79} +1.00000 q^{81} -13.3693 q^{83} +5.12311 q^{85} -2.00000 q^{87} +0.246211 q^{89} +16.0000 q^{91} -4.24621 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} - 2 q^{17} + 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 2 q^{33} + 2 q^{35} + 12 q^{37} - 2 q^{39} + 12 q^{41} + 2 q^{43} - 2 q^{45} + 8 q^{47} + 22 q^{49} + 2 q^{51} - 2 q^{55} + 8 q^{59} + 20 q^{61} - 2 q^{63} - 2 q^{65} + 4 q^{67} + 8 q^{69} + 4 q^{71} + 18 q^{73} - 2 q^{75} - 2 q^{77} - 12 q^{79} + 2 q^{81} - 2 q^{83} + 2 q^{85} - 4 q^{87} - 16 q^{89} + 32 q^{91} + 8 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 - 2 * q^17 + 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 - 2 * q^33 + 2 * q^35 + 12 * q^37 - 2 * q^39 + 12 * q^41 + 2 * q^43 - 2 * q^45 + 8 * q^47 + 22 * q^49 + 2 * q^51 - 2 * q^55 + 8 * q^59 + 20 * q^61 - 2 * q^63 - 2 * q^65 + 4 * q^67 + 8 * q^69 + 4 * q^71 + 18 * q^73 - 2 * q^75 - 2 * q^77 - 12 * q^79 + 2 * q^81 - 2 * q^83 + 2 * q^85 - 4 * q^87 - 16 * q^89 + 32 * q^91 + 8 * q^97 + 2 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.12311	1.18042	0.590211	−	0.807249i	$-0.299044\pi$
$7$	3.12311	1.18042	0.590211	+	0.807249i	$0.299044\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.12311	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$13$	5.12311	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−3.12311	−0.681518
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−3.12311	−0.527901
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	−5.12311	−0.820353
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−3.12311	−0.476269	−0.238135	−	0.971232i	$-0.576536\pi$
$43$	−3.12311	−0.476269	−0.238135	+	0.971232i	$0.576536\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	5.12311	0.717378
$52$	0	0
$53$	8.24621	1.13270	0.566352	−	0.824163i	$-0.308354\pi$
$53$	8.24621	1.13270	0.566352	+	0.824163i	$0.308354\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	3.12311	0.393474
$64$	0	0
$65$	−5.12311	−0.635443
$66$	0	0
$67$	10.2462	1.25177	0.625887	−	0.779914i	$-0.284737\pi$
$67$	10.2462	1.25177	0.625887	+	0.779914i	$0.284737\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	10.2462	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$71$	10.2462	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$72$	0	0
$73$	13.1231	1.53594	0.767972	−	0.640484i	$-0.221266\pi$
$73$	13.1231	1.53594	0.767972	+	0.640484i	$0.221266\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	3.12311	0.355911
$78$	0	0
$79$	−14.2462	−1.60282	−0.801412	−	0.598113i	$-0.795918\pi$
$79$	−14.2462	−1.60282	−0.801412	+	0.598113i	$0.795918\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.3693	−1.46747	−0.733737	−	0.679434i	$-0.762225\pi$
$83$	−13.3693	−1.46747	−0.733737	+	0.679434i	$0.762225\pi$
$84$	0	0
$85$	5.12311	0.555679
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	0.246211	0.0260983	0.0130492	−	0.999915i	$-0.495846\pi$
$89$	0.246211	0.0260983	0.0130492	+	0.999915i	$0.495846\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.24621	−0.431137	−0.215569	−	0.976489i	$-0.569161\pi$
$97$	−4.24621	−0.431137	−0.215569	+	0.976489i	$0.569161\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	3.12311	0.304784
$106$	0	0
$107$	−13.3693	−1.29246	−0.646230	−	0.763142i	$-0.723655\pi$
$107$	−13.3693	−1.29246	−0.646230	+	0.763142i	$0.723655\pi$
$108$	0	0
$109$	3.75379	0.359548	0.179774	−	0.983708i	$-0.442463\pi$
$109$	3.75379	0.359548	0.179774	+	0.983708i	$0.442463\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	0.246211	0.0231616	0.0115808	−	0.999933i	$-0.496314\pi$
$113$	0.246211	0.0231616	0.0115808	+	0.999933i	$0.496314\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	5.12311	0.473631
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.12311	−0.277131	−0.138565	−	0.990353i	$-0.544249\pi$
$127$	−3.12311	−0.277131	−0.138565	+	0.990353i	$0.544249\pi$
$128$	0	0
$129$	3.12311	0.274974
$130$	0	0
$131$	18.2462	1.59418	0.797089	−	0.603861i	$-0.206372\pi$
$131$	18.2462	1.59418	0.797089	+	0.603861i	$0.206372\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	3.75379	0.320708	0.160354	−	0.987060i	$-0.448736\pi$
$137$	3.75379	0.320708	0.160354	+	0.987060i	$0.448736\pi$
$138$	0	0
$139$	14.2462	1.20835	0.604174	−	0.796852i	$-0.293503\pi$
$139$	14.2462	1.20835	0.604174	+	0.796852i	$0.293503\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	5.12311	0.428416
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	−2.75379	−0.227129
$148$	0	0
$149$	0.246211	0.0201704	0.0100852	−	0.999949i	$-0.496790\pi$
$149$	0.246211	0.0201704	0.0100852	+	0.999949i	$0.496790\pi$
$150$	0	0
$151$	−6.24621	−0.508309	−0.254155	−	0.967164i	$-0.581797\pi$
$151$	−6.24621	−0.508309	−0.254155	+	0.967164i	$0.581797\pi$
$152$	0	0
$153$	−5.12311	−0.414179
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.2462	0.977354	0.488677	−	0.872465i	$-0.337480\pi$
$157$	12.2462	0.977354	0.488677	+	0.872465i	$0.337480\pi$
$158$	0	0
$159$	−8.24621	−0.653967
$160$	0	0
$161$	−12.4924	−0.984541
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	−19.1231	−1.47979	−0.739895	−	0.672722i	$-0.765125\pi$
$167$	−19.1231	−1.47979	−0.739895	+	0.672722i	$0.765125\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−21.6155	−1.64340	−0.821699	−	0.569922i	$-0.806974\pi$
$173$	−21.6155	−1.64340	−0.821699	+	0.569922i	$0.806974\pi$
$174$	0	0
$175$	3.12311	0.236085
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	16.4924	1.23270	0.616351	−	0.787472i	$-0.288610\pi$
$179$	16.4924	1.23270	0.616351	+	0.787472i	$0.288610\pi$
$180$	0	0
$181$	7.75379	0.576335	0.288167	−	0.957580i	$-0.406954\pi$
$181$	7.75379	0.576335	0.288167	+	0.957580i	$0.406954\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−5.12311	−0.374639
$188$	0	0
$189$	−3.12311	−0.227173
$190$	0	0
$191$	−2.24621	−0.162530	−0.0812651	−	0.996693i	$-0.525896\pi$
$191$	−2.24621	−0.162530	−0.0812651	+	0.996693i	$0.525896\pi$
$192$	0	0
$193$	−23.3693	−1.68216	−0.841080	−	0.540911i	$-0.818080\pi$
$193$	−23.3693	−1.68216	−0.841080	+	0.540911i	$0.818080\pi$
$194$	0	0
$195$	5.12311	0.366873
$196$	0	0
$197$	−1.12311	−0.0800180	−0.0400090	−	0.999199i	$-0.512739\pi$
$197$	−1.12311	−0.0800180	−0.0400090	+	0.999199i	$0.512739\pi$
$198$	0	0
$199$	12.4924	0.885564	0.442782	−	0.896629i	$-0.353991\pi$
$199$	12.4924	0.885564	0.442782	+	0.896629i	$0.353991\pi$
$200$	0	0
$201$	−10.2462	−0.722712
$202$	0	0
$203$	6.24621	0.438398
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	−10.2462	−0.702059
$214$	0	0
$215$	3.12311	0.212994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.1231	−0.886777
$220$	0	0
$221$	−26.2462	−1.76551
$222$	0	0
$223$	−26.7386	−1.79055	−0.895276	−	0.445513i	$-0.853021\pi$
$223$	−26.7386	−1.79055	−0.895276	+	0.445513i	$0.853021\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	16.8769	1.12016	0.560079	−	0.828439i	$-0.310771\pi$
$227$	16.8769	1.12016	0.560079	+	0.828439i	$0.310771\pi$
$228$	0	0
$229$	−14.4924	−0.957686	−0.478843	−	0.877900i	$-0.658944\pi$
$229$	−14.4924	−0.957686	−0.478843	+	0.877900i	$0.658944\pi$
$230$	0	0
$231$	−3.12311	−0.205485
$232$	0	0
$233$	−14.8769	−0.974618	−0.487309	−	0.873230i	$-0.662021\pi$
$233$	−14.8769	−0.974618	−0.487309	+	0.873230i	$0.662021\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	14.2462	0.925391
$238$	0	0
$239$	−9.75379	−0.630920	−0.315460	−	0.948939i	$-0.602159\pi$
$239$	−9.75379	−0.630920	−0.315460	+	0.948939i	$0.602159\pi$
$240$	0	0
$241$	20.7386	1.33589	0.667946	−	0.744209i	$-0.267174\pi$
$241$	20.7386	1.33589	0.667946	+	0.744209i	$0.267174\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.75379	−0.175933
$246$	0	0
$247$	0	0
$248$	0	0
$249$	13.3693	0.847246
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	−5.12311	−0.320821
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	18.7386	1.16436
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	23.6155	1.45620	0.728098	−	0.685473i	$-0.240405\pi$
$263$	23.6155	1.45620	0.728098	+	0.685473i	$0.240405\pi$
$264$	0	0
$265$	−8.24621	−0.506561
$266$	0	0
$267$	−0.246211	−0.0150679
$268$	0	0
$269$	−12.2462	−0.746665	−0.373332	−	0.927698i	$-0.621785\pi$
$269$	−12.2462	−0.746665	−0.373332	+	0.927698i	$0.621785\pi$
$270$	0	0
$271$	1.75379	0.106535	0.0532675	−	0.998580i	$-0.483036\pi$
$271$	1.75379	0.106535	0.0532675	+	0.998580i	$0.483036\pi$
$272$	0	0
$273$	−16.0000	−0.968364
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	17.6155	1.05841	0.529207	−	0.848493i	$-0.322489\pi$
$277$	17.6155	1.05841	0.529207	+	0.848493i	$0.322489\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−19.1231	−1.13675	−0.568375	−	0.822769i	$-0.692428\pi$
$283$	−19.1231	−1.13675	−0.568375	+	0.822769i	$0.692428\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.7386	1.10611
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	4.24621	0.248917
$292$	0	0
$293$	−7.36932	−0.430520	−0.215260	−	0.976557i	$-0.569060\pi$
$293$	−7.36932	−0.430520	−0.215260	+	0.976557i	$0.569060\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−20.4924	−1.18511
$300$	0	0
$301$	−9.75379	−0.562199
$302$	0	0
$303$	−16.2462	−0.933320
$304$	0	0
$305$	−10.0000	−0.572598
$306$	0	0
$307$	4.87689	0.278339	0.139170	−	0.990269i	$-0.455557\pi$
$307$	4.87689	0.278339	0.139170	+	0.990269i	$0.455557\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	5.75379	0.326267	0.163134	−	0.986604i	$-0.447840\pi$
$311$	5.75379	0.326267	0.163134	+	0.986604i	$0.447840\pi$
$312$	0	0
$313$	34.9848	1.97746	0.988730	−	0.149709i	$-0.0478335\pi$
$313$	34.9848	1.97746	0.988730	+	0.149709i	$0.0478335\pi$
$314$	0	0
$315$	−3.12311	−0.175967
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	13.3693	0.746203
$322$	0	0
$323$	0	0
$324$	0	0
$325$	5.12311	0.284179
$326$	0	0
$327$	−3.75379	−0.207585
$328$	0	0
$329$	12.4924	0.688730
$330$	0	0
$331$	−0.492423	−0.0270660	−0.0135330	−	0.999908i	$-0.504308\pi$
$331$	−0.492423	−0.0270660	−0.0135330	+	0.999908i	$0.504308\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−10.2462	−0.559810
$336$	0	0
$337$	21.1231	1.15065	0.575324	−	0.817925i	$-0.304876\pi$
$337$	21.1231	1.15065	0.575324	+	0.817925i	$0.304876\pi$
$338$	0	0
$339$	−0.246211	−0.0133724
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.2614	−0.716046
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−21.3693	−1.14717	−0.573583	−	0.819148i	$-0.694447\pi$
$347$	−21.3693	−1.14717	−0.573583	+	0.819148i	$0.694447\pi$
$348$	0	0
$349$	30.4924	1.63222	0.816111	−	0.577895i	$-0.196126\pi$
$349$	30.4924	1.63222	0.816111	+	0.577895i	$0.196126\pi$
$350$	0	0
$351$	−5.12311	−0.273451
$352$	0	0
$353$	−26.4924	−1.41005	−0.705025	−	0.709183i	$-0.749064\pi$
$353$	−26.4924	−1.41005	−0.705025	+	0.709183i	$0.749064\pi$
$354$	0	0
$355$	−10.2462	−0.543812
$356$	0	0
$357$	16.0000	0.846810
$358$	0	0
$359$	−22.2462	−1.17411	−0.587055	−	0.809547i	$-0.699713\pi$
$359$	−22.2462	−1.17411	−0.587055	+	0.809547i	$0.699713\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−13.1231	−0.686895
$366$	0	0
$367$	−26.7386	−1.39575	−0.697873	−	0.716222i	$-0.745870\pi$
$367$	−26.7386	−1.39575	−0.697873	+	0.716222i	$0.745870\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	25.7538	1.33707
$372$	0	0
$373$	−13.6155	−0.704985	−0.352493	−	0.935815i	$-0.614666\pi$
$373$	−13.6155	−0.704985	−0.352493	+	0.935815i	$0.614666\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	10.2462	0.527707
$378$	0	0
$379$	−0.492423	−0.0252940	−0.0126470	−	0.999920i	$-0.504026\pi$
$379$	−0.492423	−0.0252940	−0.0126470	+	0.999920i	$0.504026\pi$
$380$	0	0
$381$	3.12311	0.160002
$382$	0	0
$383$	10.2462	0.523557	0.261778	−	0.965128i	$-0.415691\pi$
$383$	10.2462	0.523557	0.261778	+	0.965128i	$0.415691\pi$
$384$	0	0
$385$	−3.12311	−0.159168
$386$	0	0
$387$	−3.12311	−0.158756
$388$	0	0
$389$	−24.7386	−1.25430	−0.627149	−	0.778899i	$-0.715778\pi$
$389$	−24.7386	−1.25430	−0.627149	+	0.778899i	$0.715778\pi$
$390$	0	0
$391$	20.4924	1.03635
$392$	0	0
$393$	−18.2462	−0.920400
$394$	0	0
$395$	14.2462	0.716805
$396$	0	0
$397$	−28.7386	−1.44235	−0.721175	−	0.692753i	$-0.756398\pi$
$397$	−28.7386	−1.44235	−0.721175	+	0.692753i	$0.756398\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.24621	0.411796	0.205898	−	0.978573i	$-0.433988\pi$
$401$	8.24621	0.411796	0.205898	+	0.978573i	$0.433988\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−30.9848	−1.53210	−0.766051	−	0.642780i	$-0.777781\pi$
$409$	−30.9848	−1.53210	−0.766051	+	0.642780i	$0.777781\pi$
$410$	0	0
$411$	−3.75379	−0.185161
$412$	0	0
$413$	12.4924	0.614712
$414$	0	0
$415$	13.3693	0.656274
$416$	0	0
$417$	−14.2462	−0.697640
$418$	0	0
$419$	−36.9848	−1.80683	−0.903414	−	0.428769i	$-0.858947\pi$
$419$	−36.9848	−1.80683	−0.903414	+	0.428769i	$0.858947\pi$
$420$	0	0
$421$	−32.2462	−1.57158	−0.785792	−	0.618491i	$-0.787744\pi$
$421$	−32.2462	−1.57158	−0.785792	+	0.618491i	$0.787744\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	−5.12311	−0.248507
$426$	0	0
$427$	31.2311	1.51138
$428$	0	0
$429$	−5.12311	−0.247346
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	20.7386	0.996635	0.498318	−	0.866995i	$-0.333951\pi$
$433$	20.7386	0.996635	0.498318	+	0.866995i	$0.333951\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−4.49242	−0.214412	−0.107206	−	0.994237i	$-0.534190\pi$
$439$	−4.49242	−0.214412	−0.107206	+	0.994237i	$0.534190\pi$
$440$	0	0
$441$	2.75379	0.131133
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	−0.246211	−0.0116715
$446$	0	0
$447$	−0.246211	−0.0116454
$448$	0	0
$449$	0.246211	0.0116194	0.00580971	−	0.999983i	$-0.498151\pi$
$449$	0.246211	0.0116194	0.00580971	+	0.999983i	$0.498151\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	6.24621	0.293473
$454$	0	0
$455$	−16.0000	−0.750092
$456$	0	0
$457$	−13.6155	−0.636908	−0.318454	−	0.947938i	$-0.603164\pi$
$457$	−13.6155	−0.636908	−0.318454	+	0.947938i	$0.603164\pi$
$458$	0	0
$459$	5.12311	0.239126
$460$	0	0
$461$	8.24621	0.384064	0.192032	−	0.981389i	$-0.438492\pi$
$461$	8.24621	0.384064	0.192032	+	0.981389i	$0.438492\pi$
$462$	0	0
$463$	−6.24621	−0.290286	−0.145143	−	0.989411i	$-0.546364\pi$
$463$	−6.24621	−0.290286	−0.145143	+	0.989411i	$0.546364\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.2462	0.474138	0.237069	−	0.971493i	$-0.423813\pi$
$467$	10.2462	0.474138	0.237069	+	0.971493i	$0.423813\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	−12.2462	−0.564276
$472$	0	0
$473$	−3.12311	−0.143601
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.24621	0.377568
$478$	0	0
$479$	14.2462	0.650926	0.325463	−	0.945555i	$-0.394480\pi$
$479$	14.2462	0.650926	0.325463	+	0.945555i	$0.394480\pi$
$480$	0	0
$481$	30.7386	1.40156
$482$	0	0
$483$	12.4924	0.568425
$484$	0	0
$485$	4.24621	0.192811
$486$	0	0
$487$	9.75379	0.441986	0.220993	−	0.975275i	$-0.429070\pi$
$487$	9.75379	0.441986	0.220993	+	0.975275i	$0.429070\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−10.2462	−0.461466
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	19.1231	0.854357
$502$	0	0
$503$	25.3693	1.13116	0.565581	−	0.824693i	$-0.308652\pi$
$503$	25.3693	1.13116	0.565581	+	0.824693i	$0.308652\pi$
$504$	0	0
$505$	−16.2462	−0.722947
$506$	0	0
$507$	−13.2462	−0.588285
$508$	0	0
$509$	−44.2462	−1.96118	−0.980589	−	0.196072i	$-0.937181\pi$
$509$	−44.2462	−1.96118	−0.980589	+	0.196072i	$0.937181\pi$
$510$	0	0
$511$	40.9848	1.81306
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	4.00000	0.175920
$518$	0	0
$519$	21.6155	0.948816
$520$	0	0
$521$	16.2462	0.711759	0.355880	−	0.934532i	$-0.384181\pi$
$521$	16.2462	0.711759	0.355880	+	0.934532i	$0.384181\pi$
$522$	0	0
$523$	−12.8769	−0.563067	−0.281534	−	0.959551i	$-0.590843\pi$
$523$	−12.8769	−0.563067	−0.281534	+	0.959551i	$0.590843\pi$
$524$	0	0
$525$	−3.12311	−0.136304
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	30.7386	1.33144
$534$	0	0
$535$	13.3693	0.578006
$536$	0	0
$537$	−16.4924	−0.711701
$538$	0	0
$539$	2.75379	0.118614
$540$	0	0
$541$	−28.2462	−1.21440	−0.607200	−	0.794549i	$-0.707707\pi$
$541$	−28.2462	−1.21440	−0.607200	+	0.794549i	$0.707707\pi$
$542$	0	0
$543$	−7.75379	−0.332747
$544$	0	0
$545$	−3.75379	−0.160795
$546$	0	0
$547$	−25.3693	−1.08471	−0.542357	−	0.840148i	$-0.682468\pi$
$547$	−25.3693	−1.08471	−0.542357	+	0.840148i	$0.682468\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−44.4924	−1.89201
$554$	0	0
$555$	6.00000	0.254686
$556$	0	0
$557$	35.3693	1.49865	0.749323	−	0.662205i	$-0.230379\pi$
$557$	35.3693	1.49865	0.749323	+	0.662205i	$0.230379\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	5.12311	0.216298
$562$	0	0
$563$	5.36932	0.226290	0.113145	−	0.993579i	$-0.463908\pi$
$563$	5.36932	0.226290	0.113145	+	0.993579i	$0.463908\pi$
$564$	0	0
$565$	−0.246211	−0.0103582
$566$	0	0
$567$	3.12311	0.131158
$568$	0	0
$569$	32.7386	1.37247	0.686237	−	0.727378i	$-0.259261\pi$
$569$	32.7386	1.37247	0.686237	+	0.727378i	$0.259261\pi$
$570$	0	0
$571$	−2.73863	−0.114608	−0.0573041	−	0.998357i	$-0.518250\pi$
$571$	−2.73863	−0.114608	−0.0573041	+	0.998357i	$0.518250\pi$
$572$	0	0
$573$	2.24621	0.0938368
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	40.2462	1.67547	0.837736	−	0.546076i	$-0.183879\pi$
$577$	40.2462	1.67547	0.837736	+	0.546076i	$0.183879\pi$
$578$	0	0
$579$	23.3693	0.971196
$580$	0	0
$581$	−41.7538	−1.73224
$582$	0	0
$583$	8.24621	0.341523
$584$	0	0
$585$	−5.12311	−0.211814
$586$	0	0
$587$	38.7386	1.59891	0.799457	−	0.600723i	$-0.205121\pi$
$587$	38.7386	1.59891	0.799457	+	0.600723i	$0.205121\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	1.12311	0.0461984
$592$	0	0
$593$	34.8769	1.43222	0.716111	−	0.697986i	$-0.245920\pi$
$593$	34.8769	1.43222	0.716111	+	0.697986i	$0.245920\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	−12.4924	−0.511281
$598$	0	0
$599$	−38.7386	−1.58282	−0.791409	−	0.611287i	$-0.790652\pi$
$599$	−38.7386	−1.58282	−0.791409	+	0.611287i	$0.790652\pi$
$600$	0	0
$601$	−24.7386	−1.00911	−0.504555	−	0.863380i	$-0.668343\pi$
$601$	−24.7386	−1.00911	−0.504555	+	0.863380i	$0.668343\pi$
$602$	0	0
$603$	10.2462	0.417258
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−19.1231	−0.776183	−0.388091	−	0.921621i	$-0.626866\pi$
$607$	−19.1231	−0.776183	−0.388091	+	0.921621i	$0.626866\pi$
$608$	0	0
$609$	−6.24621	−0.253109
$610$	0	0
$611$	20.4924	0.829035
$612$	0	0
$613$	6.87689	0.277755	0.138878	−	0.990310i	$-0.455651\pi$
$613$	6.87689	0.277755	0.138878	+	0.990310i	$0.455651\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−7.75379	−0.312156	−0.156078	−	0.987745i	$-0.549885\pi$
$617$	−7.75379	−0.312156	−0.156078	+	0.987745i	$0.549885\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	0.768944	0.0308071
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−30.7386	−1.22563
$630$	0	0
$631$	−36.4924	−1.45274	−0.726370	−	0.687304i	$-0.758794\pi$
$631$	−36.4924	−1.45274	−0.726370	+	0.687304i	$0.758794\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	3.12311	0.123937
$636$	0	0
$637$	14.1080	0.558977
$638$	0	0
$639$	10.2462	0.405334
$640$	0	0
$641$	3.75379	0.148266	0.0741329	−	0.997248i	$-0.476381\pi$
$641$	3.75379	0.148266	0.0741329	+	0.997248i	$0.476381\pi$
$642$	0	0
$643$	−0.492423	−0.0194192	−0.00970962	−	0.999953i	$-0.503091\pi$
$643$	−0.492423	−0.0194192	−0.00970962	+	0.999953i	$0.503091\pi$
$644$	0	0
$645$	−3.12311	−0.122972
$646$	0	0
$647$	−22.7386	−0.893948	−0.446974	−	0.894547i	$-0.647498\pi$
$647$	−22.7386	−0.893948	−0.446974	+	0.894547i	$0.647498\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.7386	−1.28116	−0.640581	−	0.767891i	$-0.721306\pi$
$653$	−32.7386	−1.28116	−0.640581	+	0.767891i	$0.721306\pi$
$654$	0	0
$655$	−18.2462	−0.712938
$656$	0	0
$657$	13.1231	0.511981
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	0	0
$663$	26.2462	1.01932
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	26.7386	1.03378
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	−39.3693	−1.51758	−0.758788	−	0.651338i	$-0.774208\pi$
$673$	−39.3693	−1.51758	−0.758788	+	0.651338i	$0.774208\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	33.6155	1.29195	0.645975	−	0.763359i	$-0.276451\pi$
$677$	33.6155	1.29195	0.645975	+	0.763359i	$0.276451\pi$
$678$	0	0
$679$	−13.2614	−0.508925
$680$	0	0
$681$	−16.8769	−0.646724
$682$	0	0
$683$	21.7538	0.832386	0.416193	−	0.909276i	$-0.363364\pi$
$683$	21.7538	0.832386	0.416193	+	0.909276i	$0.363364\pi$
$684$	0	0
$685$	−3.75379	−0.143425
$686$	0	0
$687$	14.4924	0.552920
$688$	0	0
$689$	42.2462	1.60945
$690$	0	0
$691$	24.4924	0.931736	0.465868	−	0.884854i	$-0.345742\pi$
$691$	24.4924	0.931736	0.465868	+	0.884854i	$0.345742\pi$
$692$	0	0
$693$	3.12311	0.118637
$694$	0	0
$695$	−14.2462	−0.540390
$696$	0	0
$697$	−30.7386	−1.16431
$698$	0	0
$699$	14.8769	0.562696
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	50.7386	1.90822
$708$	0	0
$709$	20.2462	0.760362	0.380181	−	0.924912i	$-0.375862\pi$
$709$	20.2462	0.760362	0.380181	+	0.924912i	$0.375862\pi$
$710$	0	0
$711$	−14.2462	−0.534275
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−5.12311	−0.191593
$716$	0	0
$717$	9.75379	0.364262
$718$	0	0
$719$	−10.2462	−0.382119	−0.191060	−	0.981578i	$-0.561192\pi$
$719$	−10.2462	−0.382119	−0.191060	+	0.981578i	$0.561192\pi$
$720$	0	0
$721$	−24.9848	−0.930484
$722$	0	0
$723$	−20.7386	−0.771278
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−40.9848	−1.52004	−0.760022	−	0.649897i	$-0.774812\pi$
$727$	−40.9848	−1.52004	−0.760022	+	0.649897i	$0.774812\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	14.8769	0.549491	0.274745	−	0.961517i	$-0.411406\pi$
$733$	14.8769	0.549491	0.274745	+	0.961517i	$0.411406\pi$
$734$	0	0
$735$	2.75379	0.101575
$736$	0	0
$737$	10.2462	0.377424
$738$	0	0
$739$	9.75379	0.358799	0.179399	−	0.983776i	$-0.442585\pi$
$739$	9.75379	0.358799	0.179399	+	0.983776i	$0.442585\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.36932	−0.0502354	−0.0251177	−	0.999685i	$-0.507996\pi$
$743$	−1.36932	−0.0502354	−0.0251177	+	0.999685i	$0.507996\pi$
$744$	0	0
$745$	−0.246211	−0.00902048
$746$	0	0
$747$	−13.3693	−0.489158
$748$	0	0
$749$	−41.7538	−1.52565
$750$	0	0
$751$	40.9848	1.49556	0.747779	−	0.663948i	$-0.231120\pi$
$751$	40.9848	1.49556	0.747779	+	0.663948i	$0.231120\pi$
$752$	0	0
$753$	4.00000	0.145768
$754$	0	0
$755$	6.24621	0.227323
$756$	0	0
$757$	12.2462	0.445096	0.222548	−	0.974922i	$-0.428563\pi$
$757$	12.2462	0.445096	0.222548	+	0.974922i	$0.428563\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	−8.24621	−0.298925	−0.149462	−	0.988767i	$-0.547754\pi$
$761$	−8.24621	−0.298925	−0.149462	+	0.988767i	$0.547754\pi$
$762$	0	0
$763$	11.7235	0.424418
$764$	0	0
$765$	5.12311	0.185226
$766$	0	0
$767$	20.4924	0.739938
$768$	0	0
$769$	−31.7538	−1.14507	−0.572535	−	0.819880i	$-0.694040\pi$
$769$	−31.7538	−1.14507	−0.572535	+	0.819880i	$0.694040\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	−36.2462	−1.30369	−0.651843	−	0.758354i	$-0.726004\pi$
$773$	−36.2462	−1.30369	−0.651843	+	0.758354i	$0.726004\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−18.7386	−0.672245
$778$	0	0
$779$	0	0
$780$	0	0
$781$	10.2462	0.366638
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−12.2462	−0.437086
$786$	0	0
$787$	49.3693	1.75983	0.879913	−	0.475135i	$-0.157601\pi$
$787$	49.3693	1.75983	0.879913	+	0.475135i	$0.157601\pi$
$788$	0	0
$789$	−23.6155	−0.840735
$790$	0	0
$791$	0.768944	0.0273405
$792$	0	0
$793$	51.2311	1.81927
$794$	0	0
$795$	8.24621	0.292463
$796$	0	0
$797$	−42.4924	−1.50516	−0.752579	−	0.658501i	$-0.771191\pi$
$797$	−42.4924	−1.50516	−0.752579	+	0.658501i	$0.771191\pi$
$798$	0	0
$799$	−20.4924	−0.724970
$800$	0	0
$801$	0.246211	0.00869945
$802$	0	0
$803$	13.1231	0.463104
$804$	0	0
$805$	12.4924	0.440300
$806$	0	0
$807$	12.2462	0.431087
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−47.2311	−1.65851	−0.829253	−	0.558873i	$-0.811234\pi$
$811$	−47.2311	−1.65851	−0.829253	+	0.558873i	$0.811234\pi$
$812$	0	0
$813$	−1.75379	−0.0615081
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	16.0000	0.559085
$820$	0	0
$821$	34.9848	1.22098	0.610490	−	0.792024i	$-0.290973\pi$
$821$	34.9848	1.22098	0.610490	+	0.792024i	$0.290973\pi$
$822$	0	0
$823$	−40.9848	−1.42864	−0.714321	−	0.699818i	$-0.753264\pi$
$823$	−40.9848	−1.42864	−0.714321	+	0.699818i	$0.753264\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−37.3693	−1.29946	−0.649729	−	0.760166i	$-0.725118\pi$
$827$	−37.3693	−1.29946	−0.649729	+	0.760166i	$0.725118\pi$
$828$	0	0
$829$	0.738634	0.0256538	0.0128269	−	0.999918i	$-0.495917\pi$
$829$	0.738634	0.0256538	0.0128269	+	0.999918i	$0.495917\pi$
$830$	0	0
$831$	−17.6155	−0.611076
$832$	0	0
$833$	−14.1080	−0.488812
$834$	0	0
$835$	19.1231	0.661782
$836$	0	0
$837$	0	0
$838$	0	0
$839$	22.7386	0.785025	0.392512	−	0.919747i	$-0.371606\pi$
$839$	22.7386	0.785025	0.392512	+	0.919747i	$0.371606\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	10.0000	0.344418
$844$	0	0
$845$	−13.2462	−0.455684
$846$	0	0
$847$	3.12311	0.107311
$848$	0	0
$849$	19.1231	0.656303
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	51.3693	1.75885	0.879426	−	0.476036i	$-0.157927\pi$
$853$	51.3693	1.75885	0.879426	+	0.476036i	$0.157927\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.38447	0.218089	0.109045	−	0.994037i	$-0.465221\pi$
$857$	6.38447	0.218089	0.109045	+	0.994037i	$0.465221\pi$
$858$	0	0
$859$	24.4924	0.835671	0.417835	−	0.908523i	$-0.362789\pi$
$859$	24.4924	0.835671	0.417835	+	0.908523i	$0.362789\pi$
$860$	0	0
$861$	−18.7386	−0.638611
$862$	0	0
$863$	−6.73863	−0.229386	−0.114693	−	0.993401i	$-0.536588\pi$
$863$	−6.73863	−0.229386	−0.114693	+	0.993401i	$0.536588\pi$
$864$	0	0
$865$	21.6155	0.734950
$866$	0	0
$867$	−9.24621	−0.314018
$868$	0	0
$869$	−14.2462	−0.483270
$870$	0	0
$871$	52.4924	1.77864
$872$	0	0
$873$	−4.24621	−0.143712
$874$	0	0
$875$	−3.12311	−0.105580
$876$	0	0
$877$	−12.6307	−0.426508	−0.213254	−	0.976997i	$-0.568406\pi$
$877$	−12.6307	−0.426508	−0.213254	+	0.976997i	$0.568406\pi$
$878$	0	0
$879$	7.36932	0.248561
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	−14.6307	−0.491250	−0.245625	−	0.969365i	$-0.578993\pi$
$887$	−14.6307	−0.491250	−0.245625	+	0.969365i	$0.578993\pi$
$888$	0	0
$889$	−9.75379	−0.327132
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−16.4924	−0.551281
$896$	0	0
$897$	20.4924	0.684222
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−42.2462	−1.40743
$902$	0	0
$903$	9.75379	0.324586
$904$	0	0
$905$	−7.75379	−0.257745
$906$	0	0
$907$	−18.2462	−0.605856	−0.302928	−	0.953014i	$-0.597964\pi$
$907$	−18.2462	−0.605856	−0.302928	+	0.953014i	$0.597964\pi$
$908$	0	0
$909$	16.2462	0.538853
$910$	0	0
$911$	18.2462	0.604524	0.302262	−	0.953225i	$-0.402258\pi$
$911$	18.2462	0.604524	0.302262	+	0.953225i	$0.402258\pi$
$912$	0	0
$913$	−13.3693	−0.442460
$914$	0	0
$915$	10.0000	0.330590
$916$	0	0
$917$	56.9848	1.88181
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−4.87689	−0.160699
$922$	0	0
$923$	52.4924	1.72781
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−38.9848	−1.27905	−0.639526	−	0.768770i	$-0.720869\pi$
$929$	−38.9848	−1.27905	−0.639526	+	0.768770i	$0.720869\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5.75379	−0.188371
$934$	0	0
$935$	5.12311	0.167543
$936$	0	0
$937$	27.3693	0.894117	0.447058	−	0.894505i	$-0.352472\pi$
$937$	27.3693	0.894117	0.447058	+	0.894505i	$0.352472\pi$
$938$	0	0
$939$	−34.9848	−1.14169
$940$	0	0
$941$	−29.2311	−0.952905	−0.476453	−	0.879200i	$-0.658078\pi$
$941$	−29.2311	−0.952905	−0.476453	+	0.879200i	$0.658078\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	3.12311	0.101595
$946$	0	0
$947$	−29.7538	−0.966868	−0.483434	−	0.875381i	$-0.660611\pi$
$947$	−29.7538	−0.966868	−0.483434	+	0.875381i	$0.660611\pi$
$948$	0	0
$949$	67.2311	2.18241
$950$	0	0
$951$	22.0000	0.713399
$952$	0	0
$953$	53.6155	1.73678	0.868389	−	0.495884i	$-0.165156\pi$
$953$	53.6155	1.73678	0.868389	+	0.495884i	$0.165156\pi$
$954$	0	0
$955$	2.24621	0.0726857
$956$	0	0
$957$	−2.00000	−0.0646508
$958$	0	0
$959$	11.7235	0.378571
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−13.3693	−0.430820
$964$	0	0
$965$	23.3693	0.752285
$966$	0	0
$967$	13.8617	0.445763	0.222882	−	0.974845i	$-0.428454\pi$
$967$	13.8617	0.445763	0.222882	+	0.974845i	$0.428454\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.492423	0.0158026	0.00790130	−	0.999969i	$-0.497485\pi$
$971$	0.492423	0.0158026	0.00790130	+	0.999969i	$0.497485\pi$
$972$	0	0
$973$	44.4924	1.42636
$974$	0	0
$975$	−5.12311	−0.164071
$976$	0	0
$977$	24.2462	0.775705	0.387853	−	0.921721i	$-0.373217\pi$
$977$	24.2462	0.775705	0.387853	+	0.921721i	$0.373217\pi$
$978$	0	0
$979$	0.246211	0.00786895
$980$	0	0
$981$	3.75379	0.119849
$982$	0	0
$983$	37.7538	1.20416	0.602079	−	0.798436i	$-0.294339\pi$
$983$	37.7538	1.20416	0.602079	+	0.798436i	$0.294339\pi$
$984$	0	0
$985$	1.12311	0.0357851
$986$	0	0
$987$	−12.4924	−0.397638
$988$	0	0
$989$	12.4924	0.397236
$990$	0	0
$991$	53.4773	1.69876	0.849381	−	0.527781i	$-0.176976\pi$
$991$	53.4773	1.69876	0.849381	+	0.527781i	$0.176976\pi$
$992$	0	0
$993$	0.492423	0.0156266
$994$	0	0
$995$	−12.4924	−0.396036
$996$	0	0
$997$	−13.6155	−0.431208	−0.215604	−	0.976481i	$-0.569172\pi$
$997$	−13.6155	−0.431208	−0.215604	+	0.976481i	$0.569172\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1320.2.a.o.1.2	✓	2
3.2	odd	2		3960.2.a.bc.1.2		2
4.3	odd	2		2640.2.a.ba.1.1		2
5.2	odd	4		6600.2.d.bc.1849.4		4
5.3	odd	4		6600.2.d.bc.1849.1		4
5.4	even	2		6600.2.a.bl.1.1		2
12.11	even	2		7920.2.a.ce.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.a.o.1.2	✓	2	1.1	even	1	trivial
2640.2.a.ba.1.1		2	4.3	odd	2
3960.2.a.bc.1.2		2	3.2	odd	2
6600.2.a.bl.1.1		2	5.4	even	2
6600.2.d.bc.1849.1		4	5.3	odd	4
6600.2.d.bc.1849.4		4	5.2	odd	4
7920.2.a.ce.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1320.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.12311 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.12311 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.12311 q^{13} +1.00000 q^{15} -5.12311 q^{17} -3.12311 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -1.00000 q^{33} -3.12311 q^{35} +6.00000 q^{37} -5.12311 q^{39} +6.00000 q^{41} -3.12311 q^{43} -1.00000 q^{45} +4.00000 q^{47} +2.75379 q^{49} +5.12311 q^{51} +8.24621 q^{53} -1.00000 q^{55} +4.00000 q^{59} +10.0000 q^{61} +3.12311 q^{63} -5.12311 q^{65} +10.2462 q^{67} +4.00000 q^{69} +10.2462 q^{71} +13.1231 q^{73} -1.00000 q^{75} +3.12311 q^{77} -14.2462 q^{79} +1.00000 q^{81} -13.3693 q^{83} +5.12311 q^{85} -2.00000 q^{87} +0.246211 q^{89} +16.0000 q^{91} -4.24621 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} - 2 q^{17} + 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 2 q^{33} + 2 q^{35} + 12 q^{37} - 2 q^{39} + 12 q^{41} + 2 q^{43} - 2 q^{45} + 8 q^{47} + 22 q^{49} + 2 q^{51} - 2 q^{55} + 8 q^{59} + 20 q^{61} - 2 q^{63} - 2 q^{65} + 4 q^{67} + 8 q^{69} + 4 q^{71} + 18 q^{73} - 2 q^{75} - 2 q^{77} - 12 q^{79} + 2 q^{81} - 2 q^{83} + 2 q^{85} - 4 q^{87} - 16 q^{89} + 32 q^{91} + 8 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 - 2 * q^17 + 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 - 2 * q^33 + 2 * q^35 + 12 * q^37 - 2 * q^39 + 12 * q^41 + 2 * q^43 - 2 * q^45 + 8 * q^47 + 22 * q^49 + 2 * q^51 - 2 * q^55 + 8 * q^59 + 20 * q^61 - 2 * q^63 - 2 * q^65 + 4 * q^67 + 8 * q^69 + 4 * q^71 + 18 * q^73 - 2 * q^75 - 2 * q^77 - 12 * q^79 + 2 * q^81 - 2 * q^83 + 2 * q^85 - 4 * q^87 - 16 * q^89 + 32 * q^91 + 8 * q^97 + 2 * q^99

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