LMFDB - Embedded newform 6384.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 798)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +0.763932 q^{13} -3.23607 q^{15} -6.47214 q^{17} +1.00000 q^{19} -1.00000 q^{21} -7.70820 q^{23} +5.47214 q^{25} +1.00000 q^{27} -0.472136 q^{29} +5.70820 q^{31} +2.00000 q^{33} +3.23607 q^{35} +5.23607 q^{37} +0.763932 q^{39} +10.9443 q^{41} -10.4721 q^{43} -3.23607 q^{45} -7.23607 q^{47} +1.00000 q^{49} -6.47214 q^{51} -0.472136 q^{53} -6.47214 q^{55} +1.00000 q^{57} -4.94427 q^{59} +3.52786 q^{61} -1.00000 q^{63} -2.47214 q^{65} +6.94427 q^{67} -7.70820 q^{69} -5.52786 q^{71} +6.00000 q^{73} +5.47214 q^{75} -2.00000 q^{77} +10.1803 q^{79} +1.00000 q^{81} +8.94427 q^{83} +20.9443 q^{85} -0.472136 q^{87} +14.9443 q^{89} -0.763932 q^{91} +5.70820 q^{93} -3.23607 q^{95} +1.52786 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{11} + 6 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{31} + 4 q^{33} + 2 q^{35} + 6 q^{37} + 6 q^{39} + 4 q^{41} - 12 q^{43} - 2 q^{45} - 10 q^{47} + 2 q^{49} - 4 q^{51} + 8 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} + 16 q^{61} - 2 q^{63} + 4 q^{65} - 4 q^{67} - 2 q^{69} - 20 q^{71} + 12 q^{73} + 2 q^{75} - 4 q^{77} - 2 q^{79} + 2 q^{81} + 24 q^{85} + 8 q^{87} + 12 q^{89} - 6 q^{91} - 2 q^{93} - 2 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 + 6 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^31 + 4 * q^33 + 2 * q^35 + 6 * q^37 + 6 * q^39 + 4 * q^41 - 12 * q^43 - 2 * q^45 - 10 * q^47 + 2 * q^49 - 4 * q^51 + 8 * q^53 - 4 * q^55 + 2 * q^57 + 8 * q^59 + 16 * q^61 - 2 * q^63 + 4 * q^65 - 4 * q^67 - 2 * q^69 - 20 * q^71 + 12 * q^73 + 2 * q^75 - 4 * q^77 - 2 * q^79 + 2 * q^81 + 24 * q^85 + 8 * q^87 + 12 * q^89 - 6 * q^91 - 2 * q^93 - 2 * q^95 + 12 * q^97 + 4 * 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
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	0.763932	0.211877	0.105938	−	0.994373i	$-0.466215\pi$
$13$	0.763932	0.211877	0.105938	+	0.994373i	$0.466215\pi$
$14$	0	0
$15$	−3.23607	−0.835549
$16$	0	0
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−7.70820	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−7.70820	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	0	0
$31$	5.70820	1.02522	0.512612	−	0.858620i	$-0.328678\pi$
$31$	5.70820	1.02522	0.512612	+	0.858620i	$0.328678\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	3.23607	0.546995
$36$	0	0
$37$	5.23607	0.860804	0.430402	−	0.902637i	$-0.358372\pi$
$37$	5.23607	0.860804	0.430402	+	0.902637i	$0.358372\pi$
$38$	0	0
$39$	0.763932	0.122327
$40$	0	0
$41$	10.9443	1.70921	0.854604	−	0.519280i	$-0.173800\pi$
$41$	10.9443	1.70921	0.854604	+	0.519280i	$0.173800\pi$
$42$	0	0
$43$	−10.4721	−1.59699	−0.798493	−	0.602004i	$-0.794369\pi$
$43$	−10.4721	−1.59699	−0.798493	+	0.602004i	$0.794369\pi$
$44$	0	0
$45$	−3.23607	−0.482405
$46$	0	0
$47$	−7.23607	−1.05549	−0.527744	−	0.849403i	$-0.676962\pi$
$47$	−7.23607	−1.05549	−0.527744	+	0.849403i	$0.676962\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.47214	−0.906280
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	−6.47214	−0.872703
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−4.94427	−0.643689	−0.321845	−	0.946792i	$-0.604303\pi$
$59$	−4.94427	−0.643689	−0.321845	+	0.946792i	$0.604303\pi$
$60$	0	0
$61$	3.52786	0.451697	0.225848	−	0.974162i	$-0.427485\pi$
$61$	3.52786	0.451697	0.225848	+	0.974162i	$0.427485\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.47214	−0.306631
$66$	0	0
$67$	6.94427	0.848378	0.424189	−	0.905574i	$-0.360559\pi$
$67$	6.94427	0.848378	0.424189	+	0.905574i	$0.360559\pi$
$68$	0	0
$69$	−7.70820	−0.927959
$70$	0	0
$71$	−5.52786	−0.656037	−0.328018	−	0.944671i	$-0.606381\pi$
$71$	−5.52786	−0.656037	−0.328018	+	0.944671i	$0.606381\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	5.47214	0.631868
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	10.1803	1.14538	0.572689	−	0.819773i	$-0.305900\pi$
$79$	10.1803	1.14538	0.572689	+	0.819773i	$0.305900\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.94427	0.981761	0.490881	−	0.871227i	$-0.336675\pi$
$83$	8.94427	0.981761	0.490881	+	0.871227i	$0.336675\pi$
$84$	0	0
$85$	20.9443	2.27173
$86$	0	0
$87$	−0.472136	−0.0506183
$88$	0	0
$89$	14.9443	1.58409	0.792045	−	0.610463i	$-0.209017\pi$
$89$	14.9443	1.58409	0.792045	+	0.610463i	$0.209017\pi$
$90$	0	0
$91$	−0.763932	−0.0800818
$92$	0	0
$93$	5.70820	0.591913
$94$	0	0
$95$	−3.23607	−0.332014
$96$	0	0
$97$	1.52786	0.155131	0.0775655	−	0.996987i	$-0.475285\pi$
$97$	1.52786	0.155131	0.0775655	+	0.996987i	$0.475285\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	−6.29180	−0.626057	−0.313029	−	0.949744i	$-0.601344\pi$
$101$	−6.29180	−0.626057	−0.313029	+	0.949744i	$0.601344\pi$
$102$	0	0
$103$	−18.6525	−1.83788	−0.918942	−	0.394394i	$-0.870955\pi$
$103$	−18.6525	−1.83788	−0.918942	+	0.394394i	$0.870955\pi$
$104$	0	0
$105$	3.23607	0.315808
$106$	0	0
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	0	0
$109$	11.7082	1.12144	0.560721	−	0.828005i	$-0.310524\pi$
$109$	11.7082	1.12144	0.560721	+	0.828005i	$0.310524\pi$
$110$	0	0
$111$	5.23607	0.496986
$112$	0	0
$113$	3.52786	0.331874	0.165937	−	0.986136i	$-0.446935\pi$
$113$	3.52786	0.331874	0.165937	+	0.986136i	$0.446935\pi$
$114$	0	0
$115$	24.9443	2.32607
$116$	0	0
$117$	0.763932	0.0706255
$118$	0	0
$119$	6.47214	0.593300
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	10.9443	0.986812
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−8.29180	−0.735778	−0.367889	−	0.929870i	$-0.619919\pi$
$127$	−8.29180	−0.735778	−0.367889	+	0.929870i	$0.619919\pi$
$128$	0	0
$129$	−10.4721	−0.922020
$130$	0	0
$131$	20.9443	1.82991	0.914955	−	0.403556i	$-0.132226\pi$
$131$	20.9443	1.82991	0.914955	+	0.403556i	$0.132226\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−3.23607	−0.278516
$136$	0	0
$137$	17.4164	1.48798	0.743992	−	0.668188i	$-0.232930\pi$
$137$	17.4164	1.48798	0.743992	+	0.668188i	$0.232930\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−7.23607	−0.609387
$142$	0	0
$143$	1.52786	0.127766
$144$	0	0
$145$	1.52786	0.126882
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−10.1803	−0.834006	−0.417003	−	0.908905i	$-0.636920\pi$
$149$	−10.1803	−0.834006	−0.417003	+	0.908905i	$0.636920\pi$
$150$	0	0
$151$	0.291796	0.0237460	0.0118730	−	0.999930i	$-0.496221\pi$
$151$	0.291796	0.0237460	0.0118730	+	0.999930i	$0.496221\pi$
$152$	0	0
$153$	−6.47214	−0.523241
$154$	0	0
$155$	−18.4721	−1.48372
$156$	0	0
$157$	4.47214	0.356915	0.178458	−	0.983948i	$-0.442889\pi$
$157$	4.47214	0.356915	0.178458	+	0.983948i	$0.442889\pi$
$158$	0	0
$159$	−0.472136	−0.0374428
$160$	0	0
$161$	7.70820	0.607492
$162$	0	0
$163$	3.05573	0.239343	0.119672	−	0.992814i	$-0.461816\pi$
$163$	3.05573	0.239343	0.119672	+	0.992814i	$0.461816\pi$
$164$	0	0
$165$	−6.47214	−0.503855
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−23.8885	−1.81621	−0.908106	−	0.418740i	$-0.862472\pi$
$173$	−23.8885	−1.81621	−0.908106	+	0.418740i	$0.862472\pi$
$174$	0	0
$175$	−5.47214	−0.413655
$176$	0	0
$177$	−4.94427	−0.371634
$178$	0	0
$179$	−11.4164	−0.853302	−0.426651	−	0.904416i	$-0.640307\pi$
$179$	−11.4164	−0.853302	−0.426651	+	0.904416i	$0.640307\pi$
$180$	0	0
$181$	−4.76393	−0.354100	−0.177050	−	0.984202i	$-0.556655\pi$
$181$	−4.76393	−0.354100	−0.177050	+	0.984202i	$0.556655\pi$
$182$	0	0
$183$	3.52786	0.260787
$184$	0	0
$185$	−16.9443	−1.24577
$186$	0	0
$187$	−12.9443	−0.946579
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	26.1803	1.89434	0.947171	−	0.320728i	$-0.103927\pi$
$191$	26.1803	1.89434	0.947171	+	0.320728i	$0.103927\pi$
$192$	0	0
$193$	8.47214	0.609838	0.304919	−	0.952378i	$-0.401371\pi$
$193$	8.47214	0.609838	0.304919	+	0.952378i	$0.401371\pi$
$194$	0	0
$195$	−2.47214	−0.177033
$196$	0	0
$197$	15.7082	1.11916	0.559582	−	0.828775i	$-0.310962\pi$
$197$	15.7082	1.11916	0.559582	+	0.828775i	$0.310962\pi$
$198$	0	0
$199$	14.4721	1.02590	0.512951	−	0.858418i	$-0.328552\pi$
$199$	14.4721	1.02590	0.512951	+	0.858418i	$0.328552\pi$
$200$	0	0
$201$	6.94427	0.489811
$202$	0	0
$203$	0.472136	0.0331374
$204$	0	0
$205$	−35.4164	−2.47359
$206$	0	0
$207$	−7.70820	−0.535757
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−1.41641	−0.0975095	−0.0487548	−	0.998811i	$-0.515525\pi$
$211$	−1.41641	−0.0975095	−0.0487548	+	0.998811i	$0.515525\pi$
$212$	0	0
$213$	−5.52786	−0.378763
$214$	0	0
$215$	33.8885	2.31118
$216$	0	0
$217$	−5.70820	−0.387498
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−4.94427	−0.332588
$222$	0	0
$223$	9.70820	0.650109	0.325055	−	0.945695i	$-0.394617\pi$
$223$	9.70820	0.650109	0.325055	+	0.945695i	$0.394617\pi$
$224$	0	0
$225$	5.47214	0.364809
$226$	0	0
$227$	24.9443	1.65561	0.827805	−	0.561016i	$-0.189590\pi$
$227$	24.9443	1.65561	0.827805	+	0.561016i	$0.189590\pi$
$228$	0	0
$229$	11.8885	0.785617	0.392809	−	0.919620i	$-0.371504\pi$
$229$	11.8885	0.785617	0.392809	+	0.919620i	$0.371504\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	23.4164	1.52752
$236$	0	0
$237$	10.1803	0.661284
$238$	0	0
$239$	8.65248	0.559682	0.279841	−	0.960046i	$-0.409718\pi$
$239$	8.65248	0.559682	0.279841	+	0.960046i	$0.409718\pi$
$240$	0	0
$241$	28.3607	1.82687	0.913436	−	0.406982i	$-0.133419\pi$
$241$	28.3607	1.82687	0.913436	+	0.406982i	$0.133419\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	0.763932	0.0486078
$248$	0	0
$249$	8.94427	0.566820
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−15.4164	−0.969221
$254$	0	0
$255$	20.9443	1.31158
$256$	0	0
$257$	15.8885	0.991100	0.495550	−	0.868579i	$-0.334967\pi$
$257$	15.8885	0.991100	0.495550	+	0.868579i	$0.334967\pi$
$258$	0	0
$259$	−5.23607	−0.325353
$260$	0	0
$261$	−0.472136	−0.0292245
$262$	0	0
$263$	−18.1803	−1.12105	−0.560524	−	0.828138i	$-0.689400\pi$
$263$	−18.1803	−1.12105	−0.560524	+	0.828138i	$0.689400\pi$
$264$	0	0
$265$	1.52786	0.0938559
$266$	0	0
$267$	14.9443	0.914575
$268$	0	0
$269$	19.5279	1.19063	0.595317	−	0.803491i	$-0.297026\pi$
$269$	19.5279	1.19063	0.595317	+	0.803491i	$0.297026\pi$
$270$	0	0
$271$	4.94427	0.300343	0.150172	−	0.988660i	$-0.452017\pi$
$271$	4.94427	0.300343	0.150172	+	0.988660i	$0.452017\pi$
$272$	0	0
$273$	−0.763932	−0.0462353
$274$	0	0
$275$	10.9443	0.659964
$276$	0	0
$277$	−11.8885	−0.714313	−0.357157	−	0.934044i	$-0.616254\pi$
$277$	−11.8885	−0.714313	−0.357157	+	0.934044i	$0.616254\pi$
$278$	0	0
$279$	5.70820	0.341741
$280$	0	0
$281$	21.4164	1.27760	0.638798	−	0.769375i	$-0.279432\pi$
$281$	21.4164	1.27760	0.638798	+	0.769375i	$0.279432\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	−3.23607	−0.191688
$286$	0	0
$287$	−10.9443	−0.646020
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	1.52786	0.0895650
$292$	0	0
$293$	0.472136	0.0275825	0.0137912	−	0.999905i	$-0.495610\pi$
$293$	0.472136	0.0275825	0.0137912	+	0.999905i	$0.495610\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−5.88854	−0.340543
$300$	0	0
$301$	10.4721	0.603604
$302$	0	0
$303$	−6.29180	−0.361454
$304$	0	0
$305$	−11.4164	−0.653702
$306$	0	0
$307$	−14.4721	−0.825968	−0.412984	−	0.910738i	$-0.635514\pi$
$307$	−14.4721	−0.825968	−0.412984	+	0.910738i	$0.635514\pi$
$308$	0	0
$309$	−18.6525	−1.06110
$310$	0	0
$311$	12.7639	0.723776	0.361888	−	0.932222i	$-0.382132\pi$
$311$	12.7639	0.723776	0.361888	+	0.932222i	$0.382132\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	0	0
$315$	3.23607	0.182332
$316$	0	0
$317$	3.52786	0.198145	0.0990723	−	0.995080i	$-0.468412\pi$
$317$	3.52786	0.198145	0.0990723	+	0.995080i	$0.468412\pi$
$318$	0	0
$319$	−0.944272	−0.0528691
$320$	0	0
$321$	−8.94427	−0.499221
$322$	0	0
$323$	−6.47214	−0.360119
$324$	0	0
$325$	4.18034	0.231884
$326$	0	0
$327$	11.7082	0.647465
$328$	0	0
$329$	7.23607	0.398937
$330$	0	0
$331$	1.05573	0.0580281	0.0290140	−	0.999579i	$-0.490763\pi$
$331$	1.05573	0.0580281	0.0290140	+	0.999579i	$0.490763\pi$
$332$	0	0
$333$	5.23607	0.286935
$334$	0	0
$335$	−22.4721	−1.22778
$336$	0	0
$337$	18.9443	1.03196	0.515980	−	0.856601i	$-0.327428\pi$
$337$	18.9443	1.03196	0.515980	+	0.856601i	$0.327428\pi$
$338$	0	0
$339$	3.52786	0.191607
$340$	0	0
$341$	11.4164	0.618233
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	24.9443	1.34295
$346$	0	0
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	22.3607	1.19694	0.598470	−	0.801145i	$-0.295776\pi$
$349$	22.3607	1.19694	0.598470	+	0.801145i	$0.295776\pi$
$350$	0	0
$351$	0.763932	0.0407757
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	17.8885	0.949425
$356$	0	0
$357$	6.47214	0.342542
$358$	0	0
$359$	−27.7082	−1.46238	−0.731192	−	0.682172i	$-0.761035\pi$
$359$	−27.7082	−1.46238	−0.731192	+	0.682172i	$0.761035\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−19.4164	−1.01630
$366$	0	0
$367$	36.9443	1.92848	0.964238	−	0.265039i	$-0.0853849\pi$
$367$	36.9443	1.92848	0.964238	+	0.265039i	$0.0853849\pi$
$368$	0	0
$369$	10.9443	0.569736
$370$	0	0
$371$	0.472136	0.0245121
$372$	0	0
$373$	−11.7082	−0.606228	−0.303114	−	0.952954i	$-0.598026\pi$
$373$	−11.7082	−0.606228	−0.303114	+	0.952954i	$0.598026\pi$
$374$	0	0
$375$	−1.52786	−0.0788986
$376$	0	0
$377$	−0.360680	−0.0185760
$378$	0	0
$379$	−6.94427	−0.356703	−0.178352	−	0.983967i	$-0.557076\pi$
$379$	−6.94427	−0.356703	−0.178352	+	0.983967i	$0.557076\pi$
$380$	0	0
$381$	−8.29180	−0.424802
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	6.47214	0.329851
$386$	0	0
$387$	−10.4721	−0.532329
$388$	0	0
$389$	−2.76393	−0.140137	−0.0700685	−	0.997542i	$-0.522322\pi$
$389$	−2.76393	−0.140137	−0.0700685	+	0.997542i	$0.522322\pi$
$390$	0	0
$391$	49.8885	2.52297
$392$	0	0
$393$	20.9443	1.05650
$394$	0	0
$395$	−32.9443	−1.65761
$396$	0	0
$397$	10.9443	0.549277	0.274639	−	0.961548i	$-0.411442\pi$
$397$	10.9443	0.549277	0.274639	+	0.961548i	$0.411442\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−11.5279	−0.575674	−0.287837	−	0.957679i	$-0.592936\pi$
$401$	−11.5279	−0.575674	−0.287837	+	0.957679i	$0.592936\pi$
$402$	0	0
$403$	4.36068	0.217221
$404$	0	0
$405$	−3.23607	−0.160802
$406$	0	0
$407$	10.4721	0.519085
$408$	0	0
$409$	14.4721	0.715601	0.357801	−	0.933798i	$-0.383527\pi$
$409$	14.4721	0.715601	0.357801	+	0.933798i	$0.383527\pi$
$410$	0	0
$411$	17.4164	0.859088
$412$	0	0
$413$	4.94427	0.243292
$414$	0	0
$415$	−28.9443	−1.42082
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	−24.3607	−1.19010	−0.595049	−	0.803690i	$-0.702867\pi$
$419$	−24.3607	−1.19010	−0.595049	+	0.803690i	$0.702867\pi$
$420$	0	0
$421$	−6.76393	−0.329654	−0.164827	−	0.986323i	$-0.552707\pi$
$421$	−6.76393	−0.329654	−0.164827	+	0.986323i	$0.552707\pi$
$422$	0	0
$423$	−7.23607	−0.351830
$424$	0	0
$425$	−35.4164	−1.71795
$426$	0	0
$427$	−3.52786	−0.170725
$428$	0	0
$429$	1.52786	0.0737660
$430$	0	0
$431$	18.4721	0.889771	0.444886	−	0.895587i	$-0.353244\pi$
$431$	18.4721	0.889771	0.444886	+	0.895587i	$0.353244\pi$
$432$	0	0
$433$	−22.4721	−1.07994	−0.539971	−	0.841684i	$-0.681565\pi$
$433$	−22.4721	−1.07994	−0.539971	+	0.841684i	$0.681565\pi$
$434$	0	0
$435$	1.52786	0.0732555
$436$	0	0
$437$	−7.70820	−0.368733
$438$	0	0
$439$	18.6525	0.890234	0.445117	−	0.895472i	$-0.353162\pi$
$439$	18.6525	0.890234	0.445117	+	0.895472i	$0.353162\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−23.8885	−1.13498	−0.567489	−	0.823381i	$-0.692085\pi$
$443$	−23.8885	−1.13498	−0.567489	+	0.823381i	$0.692085\pi$
$444$	0	0
$445$	−48.3607	−2.29252
$446$	0	0
$447$	−10.1803	−0.481514
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	21.8885	1.03069
$452$	0	0
$453$	0.291796	0.0137098
$454$	0	0
$455$	2.47214	0.115896
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	−6.47214	−0.302093
$460$	0	0
$461$	−20.7639	−0.967073	−0.483536	−	0.875324i	$-0.660648\pi$
$461$	−20.7639	−0.967073	−0.483536	+	0.875324i	$0.660648\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	−18.4721	−0.856625
$466$	0	0
$467$	5.88854	0.272489	0.136245	−	0.990675i	$-0.456497\pi$
$467$	5.88854	0.272489	0.136245	+	0.990675i	$0.456497\pi$
$468$	0	0
$469$	−6.94427	−0.320657
$470$	0	0
$471$	4.47214	0.206065
$472$	0	0
$473$	−20.9443	−0.963019
$474$	0	0
$475$	5.47214	0.251079
$476$	0	0
$477$	−0.472136	−0.0216176
$478$	0	0
$479$	3.23607	0.147860	0.0739299	−	0.997263i	$-0.476446\pi$
$479$	3.23607	0.147860	0.0739299	+	0.997263i	$0.476446\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	7.70820	0.350735
$484$	0	0
$485$	−4.94427	−0.224508
$486$	0	0
$487$	−14.7639	−0.669018	−0.334509	−	0.942393i	$-0.608570\pi$
$487$	−14.7639	−0.669018	−0.334509	+	0.942393i	$0.608570\pi$
$488$	0	0
$489$	3.05573	0.138185
$490$	0	0
$491$	−32.4721	−1.46545	−0.732723	−	0.680526i	$-0.761751\pi$
$491$	−32.4721	−1.46545	−0.732723	+	0.680526i	$0.761751\pi$
$492$	0	0
$493$	3.05573	0.137623
$494$	0	0
$495$	−6.47214	−0.290901
$496$	0	0
$497$	5.52786	0.247959
$498$	0	0
$499$	36.3607	1.62773	0.813864	−	0.581056i	$-0.197360\pi$
$499$	36.3607	1.62773	0.813864	+	0.581056i	$0.197360\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−29.1246	−1.29860	−0.649301	−	0.760531i	$-0.724939\pi$
$503$	−29.1246	−1.29860	−0.649301	+	0.760531i	$0.724939\pi$
$504$	0	0
$505$	20.3607	0.906038
$506$	0	0
$507$	−12.4164	−0.551432
$508$	0	0
$509$	−15.5279	−0.688260	−0.344130	−	0.938922i	$-0.611826\pi$
$509$	−15.5279	−0.688260	−0.344130	+	0.938922i	$0.611826\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	60.3607	2.65981
$516$	0	0
$517$	−14.4721	−0.636484
$518$	0	0
$519$	−23.8885	−1.04859
$520$	0	0
$521$	24.8328	1.08795	0.543973	−	0.839103i	$-0.316920\pi$
$521$	24.8328	1.08795	0.543973	+	0.839103i	$0.316920\pi$
$522$	0	0
$523$	−6.47214	−0.283007	−0.141503	−	0.989938i	$-0.545194\pi$
$523$	−6.47214	−0.283007	−0.141503	+	0.989938i	$0.545194\pi$
$524$	0	0
$525$	−5.47214	−0.238824
$526$	0	0
$527$	−36.9443	−1.60932
$528$	0	0
$529$	36.4164	1.58332
$530$	0	0
$531$	−4.94427	−0.214563
$532$	0	0
$533$	8.36068	0.362141
$534$	0	0
$535$	28.9443	1.25137
$536$	0	0
$537$	−11.4164	−0.492654
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−28.8328	−1.23962	−0.619810	−	0.784752i	$-0.712790\pi$
$541$	−28.8328	−1.23962	−0.619810	+	0.784752i	$0.712790\pi$
$542$	0	0
$543$	−4.76393	−0.204440
$544$	0	0
$545$	−37.8885	−1.62297
$546$	0	0
$547$	18.0000	0.769624	0.384812	−	0.922995i	$-0.374266\pi$
$547$	18.0000	0.769624	0.384812	+	0.922995i	$0.374266\pi$
$548$	0	0
$549$	3.52786	0.150566
$550$	0	0
$551$	−0.472136	−0.0201137
$552$	0	0
$553$	−10.1803	−0.432912
$554$	0	0
$555$	−16.9443	−0.719244
$556$	0	0
$557$	30.5410	1.29406	0.647032	−	0.762463i	$-0.276010\pi$
$557$	30.5410	1.29406	0.647032	+	0.762463i	$0.276010\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−12.9443	−0.546508
$562$	0	0
$563$	−25.8885	−1.09107	−0.545536	−	0.838087i	$-0.683674\pi$
$563$	−25.8885	−1.09107	−0.545536	+	0.838087i	$0.683674\pi$
$564$	0	0
$565$	−11.4164	−0.480292
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	30.9443	1.29725	0.648626	−	0.761108i	$-0.275344\pi$
$569$	30.9443	1.29725	0.648626	+	0.761108i	$0.275344\pi$
$570$	0	0
$571$	17.5279	0.733518	0.366759	−	0.930316i	$-0.380467\pi$
$571$	17.5279	0.733518	0.366759	+	0.930316i	$0.380467\pi$
$572$	0	0
$573$	26.1803	1.09370
$574$	0	0
$575$	−42.1803	−1.75904
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	8.47214	0.352090
$580$	0	0
$581$	−8.94427	−0.371071
$582$	0	0
$583$	−0.944272	−0.0391077
$584$	0	0
$585$	−2.47214	−0.102210
$586$	0	0
$587$	7.41641	0.306108	0.153054	−	0.988218i	$-0.451089\pi$
$587$	7.41641	0.306108	0.153054	+	0.988218i	$0.451089\pi$
$588$	0	0
$589$	5.70820	0.235202
$590$	0	0
$591$	15.7082	0.646149
$592$	0	0
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	0	0
$595$	−20.9443	−0.858631
$596$	0	0
$597$	14.4721	0.592305
$598$	0	0
$599$	12.3607	0.505044	0.252522	−	0.967591i	$-0.418740\pi$
$599$	12.3607	0.505044	0.252522	+	0.967591i	$0.418740\pi$
$600$	0	0
$601$	4.94427	0.201681	0.100841	−	0.994903i	$-0.467847\pi$
$601$	4.94427	0.201681	0.100841	+	0.994903i	$0.467847\pi$
$602$	0	0
$603$	6.94427	0.282793
$604$	0	0
$605$	22.6525	0.920954
$606$	0	0
$607$	−24.5410	−0.996089	−0.498045	−	0.867151i	$-0.665948\pi$
$607$	−24.5410	−0.996089	−0.498045	+	0.867151i	$0.665948\pi$
$608$	0	0
$609$	0.472136	0.0191319
$610$	0	0
$611$	−5.52786	−0.223633
$612$	0	0
$613$	30.3607	1.22626	0.613128	−	0.789983i	$-0.289911\pi$
$613$	30.3607	1.22626	0.613128	+	0.789983i	$0.289911\pi$
$614$	0	0
$615$	−35.4164	−1.42813
$616$	0	0
$617$	9.05573	0.364570	0.182285	−	0.983246i	$-0.441651\pi$
$617$	9.05573	0.364570	0.182285	+	0.983246i	$0.441651\pi$
$618$	0	0
$619$	−26.8328	−1.07850	−0.539251	−	0.842145i	$-0.681293\pi$
$619$	−26.8328	−1.07850	−0.539251	+	0.842145i	$0.681293\pi$
$620$	0	0
$621$	−7.70820	−0.309320
$622$	0	0
$623$	−14.9443	−0.598730
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−33.8885	−1.35122
$630$	0	0
$631$	12.3607	0.492071	0.246035	−	0.969261i	$-0.420872\pi$
$631$	12.3607	0.492071	0.246035	+	0.969261i	$0.420872\pi$
$632$	0	0
$633$	−1.41641	−0.0562972
$634$	0	0
$635$	26.8328	1.06483
$636$	0	0
$637$	0.763932	0.0302681
$638$	0	0
$639$	−5.52786	−0.218679
$640$	0	0
$641$	−46.9443	−1.85419	−0.927094	−	0.374830i	$-0.877701\pi$
$641$	−46.9443	−1.85419	−0.927094	+	0.374830i	$0.877701\pi$
$642$	0	0
$643$	−40.7214	−1.60589	−0.802947	−	0.596051i	$-0.796736\pi$
$643$	−40.7214	−1.60589	−0.802947	+	0.596051i	$0.796736\pi$
$644$	0	0
$645$	33.8885	1.33436
$646$	0	0
$647$	45.1246	1.77403	0.887016	−	0.461739i	$-0.152774\pi$
$647$	45.1246	1.77403	0.887016	+	0.461739i	$0.152774\pi$
$648$	0	0
$649$	−9.88854	−0.388159
$650$	0	0
$651$	−5.70820	−0.223722
$652$	0	0
$653$	−43.1246	−1.68760	−0.843798	−	0.536661i	$-0.819686\pi$
$653$	−43.1246	−1.68760	−0.843798	+	0.536661i	$0.819686\pi$
$654$	0	0
$655$	−67.7771	−2.64827
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	−33.5279	−1.30606	−0.653030	−	0.757332i	$-0.726502\pi$
$659$	−33.5279	−1.30606	−0.653030	+	0.757332i	$0.726502\pi$
$660$	0	0
$661$	−39.0132	−1.51744	−0.758718	−	0.651419i	$-0.774174\pi$
$661$	−39.0132	−1.51744	−0.758718	+	0.651419i	$0.774174\pi$
$662$	0	0
$663$	−4.94427	−0.192020
$664$	0	0
$665$	3.23607	0.125489
$666$	0	0
$667$	3.63932	0.140915
$668$	0	0
$669$	9.70820	0.375341
$670$	0	0
$671$	7.05573	0.272383
$672$	0	0
$673$	−27.5279	−1.06112	−0.530561	−	0.847647i	$-0.678019\pi$
$673$	−27.5279	−1.06112	−0.530561	+	0.847647i	$0.678019\pi$
$674$	0	0
$675$	5.47214	0.210623
$676$	0	0
$677$	−1.63932	−0.0630042	−0.0315021	−	0.999504i	$-0.510029\pi$
$677$	−1.63932	−0.0630042	−0.0315021	+	0.999504i	$0.510029\pi$
$678$	0	0
$679$	−1.52786	−0.0586340
$680$	0	0
$681$	24.9443	0.955867
$682$	0	0
$683$	4.36068	0.166857	0.0834284	−	0.996514i	$-0.473413\pi$
$683$	4.36068	0.166857	0.0834284	+	0.996514i	$0.473413\pi$
$684$	0	0
$685$	−56.3607	−2.15343
$686$	0	0
$687$	11.8885	0.453576
$688$	0	0
$689$	−0.360680	−0.0137408
$690$	0	0
$691$	−21.8885	−0.832679	−0.416340	−	0.909209i	$-0.636687\pi$
$691$	−21.8885	−0.832679	−0.416340	+	0.909209i	$0.636687\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	−51.7771	−1.96402
$696$	0	0
$697$	−70.8328	−2.68298
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	−34.7639	−1.31302	−0.656508	−	0.754319i	$-0.727967\pi$
$701$	−34.7639	−1.31302	−0.656508	+	0.754319i	$0.727967\pi$
$702$	0	0
$703$	5.23607	0.197482
$704$	0	0
$705$	23.4164	0.881913
$706$	0	0
$707$	6.29180	0.236627
$708$	0	0
$709$	37.4164	1.40520	0.702601	−	0.711584i	$-0.252022\pi$
$709$	37.4164	1.40520	0.702601	+	0.711584i	$0.252022\pi$
$710$	0	0
$711$	10.1803	0.381793
$712$	0	0
$713$	−44.0000	−1.64781
$714$	0	0
$715$	−4.94427	−0.184905
$716$	0	0
$717$	8.65248	0.323133
$718$	0	0
$719$	48.5410	1.81027	0.905137	−	0.425119i	$-0.139768\pi$
$719$	48.5410	1.81027	0.905137	+	0.425119i	$0.139768\pi$
$720$	0	0
$721$	18.6525	0.694655
$722$	0	0
$723$	28.3607	1.05475
$724$	0	0
$725$	−2.58359	−0.0959522
$726$	0	0
$727$	1.52786	0.0566653	0.0283327	−	0.999599i	$-0.490980\pi$
$727$	1.52786	0.0566653	0.0283327	+	0.999599i	$0.490980\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	67.7771	2.50683
$732$	0	0
$733$	43.3050	1.59950	0.799752	−	0.600330i	$-0.204964\pi$
$733$	43.3050	1.59950	0.799752	+	0.600330i	$0.204964\pi$
$734$	0	0
$735$	−3.23607	−0.119364
$736$	0	0
$737$	13.8885	0.511591
$738$	0	0
$739$	−15.0557	−0.553834	−0.276917	−	0.960894i	$-0.589313\pi$
$739$	−15.0557	−0.553834	−0.276917	+	0.960894i	$0.589313\pi$
$740$	0	0
$741$	0.763932	0.0280637
$742$	0	0
$743$	52.9443	1.94234	0.971168	−	0.238394i	$-0.0766210\pi$
$743$	52.9443	1.94234	0.971168	+	0.238394i	$0.0766210\pi$
$744$	0	0
$745$	32.9443	1.20698
$746$	0	0
$747$	8.94427	0.327254
$748$	0	0
$749$	8.94427	0.326817
$750$	0	0
$751$	−31.4853	−1.14891	−0.574457	−	0.818535i	$-0.694787\pi$
$751$	−31.4853	−1.14891	−0.574457	+	0.818535i	$0.694787\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.944272	−0.0343656
$756$	0	0
$757$	−23.3050	−0.847033	−0.423516	−	0.905888i	$-0.639204\pi$
$757$	−23.3050	−0.847033	−0.423516	+	0.905888i	$0.639204\pi$
$758$	0	0
$759$	−15.4164	−0.559580
$760$	0	0
$761$	−28.3607	−1.02807	−0.514037	−	0.857768i	$-0.671851\pi$
$761$	−28.3607	−1.02807	−0.514037	+	0.857768i	$0.671851\pi$
$762$	0	0
$763$	−11.7082	−0.423865
$764$	0	0
$765$	20.9443	0.757242
$766$	0	0
$767$	−3.77709	−0.136383
$768$	0	0
$769$	−11.8885	−0.428712	−0.214356	−	0.976756i	$-0.568765\pi$
$769$	−11.8885	−0.428712	−0.214356	+	0.976756i	$0.568765\pi$
$770$	0	0
$771$	15.8885	0.572212
$772$	0	0
$773$	40.4721	1.45568	0.727841	−	0.685746i	$-0.240524\pi$
$773$	40.4721	1.45568	0.727841	+	0.685746i	$0.240524\pi$
$774$	0	0
$775$	31.2361	1.12203
$776$	0	0
$777$	−5.23607	−0.187843
$778$	0	0
$779$	10.9443	0.392119
$780$	0	0
$781$	−11.0557	−0.395605
$782$	0	0
$783$	−0.472136	−0.0168728
$784$	0	0
$785$	−14.4721	−0.516533
$786$	0	0
$787$	−14.8328	−0.528733	−0.264366	−	0.964422i	$-0.585163\pi$
$787$	−14.8328	−0.528733	−0.264366	+	0.964422i	$0.585163\pi$
$788$	0	0
$789$	−18.1803	−0.647237
$790$	0	0
$791$	−3.52786	−0.125436
$792$	0	0
$793$	2.69505	0.0957040
$794$	0	0
$795$	1.52786	0.0541878
$796$	0	0
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	46.8328	1.65683
$800$	0	0
$801$	14.9443	0.528030
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	−24.9443	−0.879170
$806$	0	0
$807$	19.5279	0.687413
$808$	0	0
$809$	6.58359	0.231467	0.115733	−	0.993280i	$-0.463078\pi$
$809$	6.58359	0.231467	0.115733	+	0.993280i	$0.463078\pi$
$810$	0	0
$811$	4.58359	0.160952	0.0804758	−	0.996757i	$-0.474356\pi$
$811$	4.58359	0.160952	0.0804758	+	0.996757i	$0.474356\pi$
$812$	0	0
$813$	4.94427	0.173403
$814$	0	0
$815$	−9.88854	−0.346381
$816$	0	0
$817$	−10.4721	−0.366374
$818$	0	0
$819$	−0.763932	−0.0266939
$820$	0	0
$821$	−35.7082	−1.24622	−0.623112	−	0.782132i	$-0.714132\pi$
$821$	−35.7082	−1.24622	−0.623112	+	0.782132i	$0.714132\pi$
$822$	0	0
$823$	19.7771	0.689386	0.344693	−	0.938715i	$-0.387983\pi$
$823$	19.7771	0.689386	0.344693	+	0.938715i	$0.387983\pi$
$824$	0	0
$825$	10.9443	0.381031
$826$	0	0
$827$	−6.47214	−0.225058	−0.112529	−	0.993648i	$-0.535895\pi$
$827$	−6.47214	−0.225058	−0.112529	+	0.993648i	$0.535895\pi$
$828$	0	0
$829$	36.1803	1.25660	0.628298	−	0.777973i	$-0.283752\pi$
$829$	36.1803	1.25660	0.628298	+	0.777973i	$0.283752\pi$
$830$	0	0
$831$	−11.8885	−0.412409
$832$	0	0
$833$	−6.47214	−0.224246
$834$	0	0
$835$	25.8885	0.895910
$836$	0	0
$837$	5.70820	0.197304
$838$	0	0
$839$	−6.11146	−0.210991	−0.105495	−	0.994420i	$-0.533643\pi$
$839$	−6.11146	−0.210991	−0.105495	+	0.994420i	$0.533643\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	21.4164	0.737620
$844$	0	0
$845$	40.1803	1.38225
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	−40.3607	−1.38355
$852$	0	0
$853$	−10.5836	−0.362375	−0.181188	−	0.983449i	$-0.557994\pi$
$853$	−10.5836	−0.362375	−0.181188	+	0.983449i	$0.557994\pi$
$854$	0	0
$855$	−3.23607	−0.110671
$856$	0	0
$857$	−23.8885	−0.816017	−0.408009	−	0.912978i	$-0.633777\pi$
$857$	−23.8885	−0.816017	−0.408009	+	0.912978i	$0.633777\pi$
$858$	0	0
$859$	−18.1115	−0.617955	−0.308977	−	0.951069i	$-0.599987\pi$
$859$	−18.1115	−0.617955	−0.308977	+	0.951069i	$0.599987\pi$
$860$	0	0
$861$	−10.9443	−0.372980
$862$	0	0
$863$	−27.0557	−0.920988	−0.460494	−	0.887663i	$-0.652328\pi$
$863$	−27.0557	−0.920988	−0.460494	+	0.887663i	$0.652328\pi$
$864$	0	0
$865$	77.3050	2.62845
$866$	0	0
$867$	24.8885	0.845259
$868$	0	0
$869$	20.3607	0.690689
$870$	0	0
$871$	5.30495	0.179751
$872$	0	0
$873$	1.52786	0.0517104
$874$	0	0
$875$	1.52786	0.0516512
$876$	0	0
$877$	−15.7082	−0.530428	−0.265214	−	0.964190i	$-0.585443\pi$
$877$	−15.7082	−0.530428	−0.265214	+	0.964190i	$0.585443\pi$
$878$	0	0
$879$	0.472136	0.0159248
$880$	0	0
$881$	−19.0557	−0.642004	−0.321002	−	0.947079i	$-0.604020\pi$
$881$	−19.0557	−0.642004	−0.321002	+	0.947079i	$0.604020\pi$
$882$	0	0
$883$	−44.3607	−1.49286	−0.746428	−	0.665466i	$-0.768233\pi$
$883$	−44.3607	−1.49286	−0.746428	+	0.665466i	$0.768233\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	−55.1935	−1.85322	−0.926608	−	0.376028i	$-0.877289\pi$
$887$	−55.1935	−1.85322	−0.926608	+	0.376028i	$0.877289\pi$
$888$	0	0
$889$	8.29180	0.278098
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	−7.23607	−0.242146
$894$	0	0
$895$	36.9443	1.23491
$896$	0	0
$897$	−5.88854	−0.196613
$898$	0	0
$899$	−2.69505	−0.0898849
$900$	0	0
$901$	3.05573	0.101801
$902$	0	0
$903$	10.4721	0.348491
$904$	0	0
$905$	15.4164	0.512459
$906$	0	0
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	0	0
$909$	−6.29180	−0.208686
$910$	0	0
$911$	13.5279	0.448198	0.224099	−	0.974566i	$-0.428056\pi$
$911$	13.5279	0.448198	0.224099	+	0.974566i	$0.428056\pi$
$912$	0	0
$913$	17.8885	0.592024
$914$	0	0
$915$	−11.4164	−0.377415
$916$	0	0
$917$	−20.9443	−0.691641
$918$	0	0
$919$	30.8328	1.01708	0.508540	−	0.861038i	$-0.330185\pi$
$919$	30.8328	1.01708	0.508540	+	0.861038i	$0.330185\pi$
$920$	0	0
$921$	−14.4721	−0.476873
$922$	0	0
$923$	−4.22291	−0.138999
$924$	0	0
$925$	28.6525	0.942088
$926$	0	0
$927$	−18.6525	−0.612628
$928$	0	0
$929$	−12.5836	−0.412854	−0.206427	−	0.978462i	$-0.566184\pi$
$929$	−12.5836	−0.412854	−0.206427	+	0.978462i	$0.566184\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	12.7639	0.417872
$934$	0	0
$935$	41.8885	1.36990
$936$	0	0
$937$	37.7771	1.23412	0.617062	−	0.786915i	$-0.288323\pi$
$937$	37.7771	1.23412	0.617062	+	0.786915i	$0.288323\pi$
$938$	0	0
$939$	30.0000	0.979013
$940$	0	0
$941$	−7.52786	−0.245401	−0.122701	−	0.992444i	$-0.539156\pi$
$941$	−7.52786	−0.245401	−0.122701	+	0.992444i	$0.539156\pi$
$942$	0	0
$943$	−84.3607	−2.74716
$944$	0	0
$945$	3.23607	0.105269
$946$	0	0
$947$	48.2492	1.56789	0.783945	−	0.620831i	$-0.213205\pi$
$947$	48.2492	1.56789	0.783945	+	0.620831i	$0.213205\pi$
$948$	0	0
$949$	4.58359	0.148790
$950$	0	0
$951$	3.52786	0.114399
$952$	0	0
$953$	45.4164	1.47118	0.735591	−	0.677426i	$-0.236905\pi$
$953$	45.4164	1.47118	0.735591	+	0.677426i	$0.236905\pi$
$954$	0	0
$955$	−84.7214	−2.74152
$956$	0	0
$957$	−0.944272	−0.0305240
$958$	0	0
$959$	−17.4164	−0.562405
$960$	0	0
$961$	1.58359	0.0510836
$962$	0	0
$963$	−8.94427	−0.288225
$964$	0	0
$965$	−27.4164	−0.882565
$966$	0	0
$967$	−7.41641	−0.238496	−0.119248	−	0.992865i	$-0.538048\pi$
$967$	−7.41641	−0.238496	−0.119248	+	0.992865i	$0.538048\pi$
$968$	0	0
$969$	−6.47214	−0.207915
$970$	0	0
$971$	−44.7214	−1.43518	−0.717588	−	0.696467i	$-0.754754\pi$
$971$	−44.7214	−1.43518	−0.717588	+	0.696467i	$0.754754\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	4.18034	0.133878
$976$	0	0
$977$	−21.4164	−0.685172	−0.342586	−	0.939487i	$-0.611303\pi$
$977$	−21.4164	−0.685172	−0.342586	+	0.939487i	$0.611303\pi$
$978$	0	0
$979$	29.8885	0.955242
$980$	0	0
$981$	11.7082	0.373814
$982$	0	0
$983$	45.3050	1.44500	0.722502	−	0.691369i	$-0.242992\pi$
$983$	45.3050	1.44500	0.722502	+	0.691369i	$0.242992\pi$
$984$	0	0
$985$	−50.8328	−1.61967
$986$	0	0
$987$	7.23607	0.230327
$988$	0	0
$989$	80.7214	2.56679
$990$	0	0
$991$	−36.2918	−1.15285	−0.576423	−	0.817151i	$-0.695552\pi$
$991$	−36.2918	−1.15285	−0.576423	+	0.817151i	$0.695552\pi$
$992$	0	0
$993$	1.05573	0.0335025
$994$	0	0
$995$	−46.8328	−1.48470
$996$	0	0
$997$	39.5279	1.25186	0.625930	−	0.779879i	$-0.284720\pi$
$997$	39.5279	1.25186	0.625930	+	0.779879i	$0.284720\pi$
$998$	0	0
$999$	5.23607	0.165662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bn.1.1		2
4.3	odd	2		798.2.a.j.1.1	✓	2
12.11	even	2		2394.2.a.z.1.2		2
28.27	even	2		5586.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.a.j.1.1	✓	2	4.3	odd	2
2394.2.a.z.1.2		2	12.11	even	2
5586.2.a.be.1.2		2	28.27	even	2
6384.2.a.bn.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +0.763932 q^{13} -3.23607 q^{15} -6.47214 q^{17} +1.00000 q^{19} -1.00000 q^{21} -7.70820 q^{23} +5.47214 q^{25} +1.00000 q^{27} -0.472136 q^{29} +5.70820 q^{31} +2.00000 q^{33} +3.23607 q^{35} +5.23607 q^{37} +0.763932 q^{39} +10.9443 q^{41} -10.4721 q^{43} -3.23607 q^{45} -7.23607 q^{47} +1.00000 q^{49} -6.47214 q^{51} -0.472136 q^{53} -6.47214 q^{55} +1.00000 q^{57} -4.94427 q^{59} +3.52786 q^{61} -1.00000 q^{63} -2.47214 q^{65} +6.94427 q^{67} -7.70820 q^{69} -5.52786 q^{71} +6.00000 q^{73} +5.47214 q^{75} -2.00000 q^{77} +10.1803 q^{79} +1.00000 q^{81} +8.94427 q^{83} +20.9443 q^{85} -0.472136 q^{87} +14.9443 q^{89} -0.763932 q^{91} +5.70820 q^{93} -3.23607 q^{95} +1.52786 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} + 6 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{31} + 4 q^{33} + 2 q^{35} + 6 q^{37} + 6 q^{39} + 4 q^{41} - 12 q^{43} - 2 q^{45} - 10 q^{47} + 2 q^{49} - 4 q^{51} + 8 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} + 16 q^{61} - 2 q^{63} + 4 q^{65} - 4 q^{67} - 2 q^{69} - 20 q^{71} + 12 q^{73} + 2 q^{75} - 4 q^{77} - 2 q^{79} + 2 q^{81} + 24 q^{85} + 8 q^{87} + 12 q^{89} - 6 q^{91} - 2 q^{93} - 2 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 + 6 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^31 + 4 * q^33 + 2 * q^35 + 6 * q^37 + 6 * q^39 + 4 * q^41 - 12 * q^43 - 2 * q^45 - 10 * q^47 + 2 * q^49 - 4 * q^51 + 8 * q^53 - 4 * q^55 + 2 * q^57 + 8 * q^59 + 16 * q^61 - 2 * q^63 + 4 * q^65 - 4 * q^67 - 2 * q^69 - 20 * q^71 + 12 * q^73 + 2 * q^75 - 4 * q^77 - 2 * q^79 + 2 * q^81 + 24 * q^85 + 8 * q^87 + 12 * q^89 - 6 * q^91 - 2 * q^93 - 2 * q^95 + 12 * q^97 + 4 * q^99

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